diff --git a/LeanPool.lean b/LeanPool.lean
index 56b67b59e0..c2ee331250 100644
--- a/LeanPool.lean
+++ b/LeanPool.lean
@@ -6031,6 +6031,60 @@ public import LeanPool.MooreBound.PrimeNumberTheoremAnd.Wiener
 public import LeanPool.MoserLatticeColorings
 public import LeanPool.MoserLatticeColorings.Basic
 public import LeanPool.MoserLatticeColorings.Ring
+public import LeanPool.MovingSofa
+public import LeanPool.MovingSofa.Development.Geometry.Applications.Development001
+public import LeanPool.MovingSofa.Development.Geometry.Applications.Development002
+public import LeanPool.MovingSofa.Development.Geometry.Applications.Development003
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development001
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development002
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development003
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development004
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development005
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development006
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development007
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development008
+public import LeanPool.MovingSofa.Development.IntervalArithmetic.Foundations.Development001
+public import LeanPool.MovingSofa.Development.IntervalArithmetic.Foundations.Development002
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle001
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle002
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle003
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle004
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle006
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle007
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle008
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle001
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle002
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle003
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle004
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle005
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle006
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle007
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle008
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle009
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle010
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle011
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle012
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle013
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle014
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle015
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle016
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle017
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle018
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle019
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle020
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle021
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle022
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle023
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle024
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle025
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle026
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle027
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle028
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle029
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle030
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle031
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle032
 public import LeanPool.MulticolorTriangleRamsey
 public import LeanPool.NagataFactoriality
 public import LeanPool.NagataFactoriality.NagataFactoriality
diff --git a/LeanPool/MovingSofa.lean b/LeanPool/MovingSofa.lean
new file mode 100644
index 0000000000..e084957c6b
--- /dev/null
+++ b/LeanPool/MovingSofa.lean
@@ -0,0 +1,71 @@
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+
+module
+
+public import LeanPool.MovingSofa.Development.Geometry.Applications.Development003
+
+/-!
+# Optimality of Gerver's sofa
+
+Source: arxiv:2411.19826, url:https://github.com/deancureton/MovingSofa/tree/4d5569131940815f47a9ccf3e90a4c5043c56127
+Authors: Dean Cureton, Dawid Trela, Rado Kirov, Kirill Alkhimov
+Status: verified
+Main declarations: `MovingSofa.sofaConstant_eq_volume_gerversSofa`
+Tags: convex-geometry, moving-sofa-problem, geometric-optimization, interval-arithmetic
+MSC: 52A40, 52A10, 49Q10
+-/
+
+/-!
+## Imported sources and provenance
+
+The main formalization is from Dean Cureton's `deancureton/MovingSofa` at
+`4d5569131940815f47a9ccf3e90a4c5043c56127` (Apache-2.0). The proofs were produced using
+Codex and Claude Code under Cureton's direction. Completion was announced on 20 September 2026.
+
+The coherent dependency closure also contains:
+
+* Dawid Trela's `dawidmtrela-dotcom/GerverSofaLean`, release v1.1.0
+  (MIT; notice below), including its Part F bridge.
+* `alerad/leancert` at `571a228555ae38742448854be81d7b59d994a8e3` (Apache-2.0):
+  pure interval arithmetic and its soundness proofs, without the metaprogramming front end.
+* Rado Kirov's `rkirov/jordan_pick` at
+  `b3c9b7cf7358bf81a077d78ad67e6e8247869ddd` (Apache-2.0): Jordan separation and Brouwer.
+* The Tau Ceti contributors: `TauCetiProject/TauCeti` at
+  `c52a81811e4626d2e9769fe9b0c701168c211332` (Apache-2.0):
+  variation, filled hulls, and supporting topology.
+* The Formal Conjectures Authors: `google-deepmind/formal-conjectures` at
+  `ddfbaf90f4482030d88aae5233fe933874296a23`
+  (Apache-2.0): the moving-sofa definitions, with its open parameter theorem proved here.
+
+Jonathan Ho's isoperimetric infrastructure is reused from `LeanPool.Isoperimetric`.
+-/
+
+/-
+The GerverSofa dependency retains its original MIT licence:
+
+MIT License
+
+Copyright (c) 2026 Dawid Trela
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.
+-/
diff --git a/LeanPool/MovingSofa/Development/Geometry/Applications/Development001.lean b/LeanPool/MovingSofa/Development/Geometry/Applications/Development001.lean
new file mode 100644
index 0000000000..6d691edd82
--- /dev/null
+++ b/LeanPool/MovingSofa/Development/Geometry/Applications/Development001.lean
@@ -0,0 +1,10227 @@
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+module
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development006
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development007
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development005
+
+
+
+
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development004
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development002
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development001
+
+
+
+
+/-!
+# Moving sofa: related mathematical developments
+
+* `Area.Applications.Development001`.
+* `Area.Applications.Development002`.
+* `Area.Applications.Development003`.
+* `Cap.Applications.Development001`.
+* `Gerver.Applications.Development001`.
+* `Gerver.Applications.Development002`.
+* `Polygon.Applications.Development001`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Area.Middle`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-! # The middle part of the niche
+
+`capMiddle_area_lower_bound` bounds the area of the part of a special cap's niche outside both
+distinguished tail half-planes from below by the three signed areas of the middle fan: the two
+wedge triangles and the inner-corner arc.
+
+The geometry is that of the paper's proof.  Writing `φ` for the right distinguished angle,
+`l = π/2 - φ` for the left one and `k = tan φ`, the two distinguished tail support lines are
+`X = w - k y` and `X = z + k y`, where `w` and `z` are the horizontal coordinates of the two fan
+points `W` and `Z`; the inner corner runs from the first line to the second through the open cone
+between them, with strictly decreasing horizontal coordinate.  Adding a base line `y = -h` strictly
+below the arc turns the arc together with the two side segments into a strictly monotone
+three-piece roof over that base line, and the region under the roof exceeds the base trapezoid by
+the asserted three signed areas.  Every point of the open region above the trapezoid lies in the
+niche, because the roof point vertically above it exhibits a time whose open inward quadrant
+contains it.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The middle area bound obtained from cap area, endpoint segments and the corner-path
+integral. -/
+def upperBoundMiddle (K : SpecialCapSpace) : ℝ :=
+  ClassicalResults.area (K.val.val : Set Point) +
+    segmentArea (distinguishedCapSides K.val).2.fanPoint
+      (distinguishedCapSides K.val).2.corner -
+    curveAreaFunctional (capMiddleBV K) +
+    segmentArea (distinguishedCapSides K.val).1.corner
+      (distinguishedCapSides K.val).1.fanPoint
+
+/-! ### Arithmetic of a side segment of the middle cone
+
+The two side segments of the four-piece loop run along the two tail support lines, so their
+displacements `(Δ₀, Δ₁)` satisfy `Δ₀ cos = Δ₁ sin` for the relevant frame angle.  The two lemmas
+below are the resulting sign computations for a point `δ` below such a segment. -/
+
+/-- On a segment parallel to the frame tangent, the frame's normal coordinate of a point below the
+segment is the vertical drop only. -/
+private theorem segment_cross_eq {cs sn Δ0 Δ1 u δ : ℝ} (hΔ : Δ0 * cs = Δ1 * sn) :
+    u * Δ0 * cs + (-(u * Δ1) - δ) * sn = -(δ * sn) := by linear_combination u * hΔ
+
+/-- On a segment parallel to the frame tangent, the frame's tangent coordinate of a point below the
+segment is negative. -/
+private theorem segment_normal_neg {cs sn Δ0 Δ1 u δ : ℝ} (hcs : 0 < cs)
+    (hpy : sn ^ 2 + cs ^ 2 = 1) (hΔ : Δ0 * cs = Δ1 * sn) (hΔ1 : 0 < Δ1)
+    (hu0 : 0 ≤ u) (hδ : 0 < δ) :
+    -(u * Δ0) * sn + (-(u * Δ1) - δ) * cs < 0 := by
+  have key : (-(u * Δ0) * sn + (-(u * Δ1) - δ) * cs) * cs = -(u * Δ1) - δ * cs ^ 2 := by
+    linear_combination (-(u * sn)) * hΔ + (-(u * Δ1)) * hpy
+  have hneg : (-(u * Δ0) * sn + (-(u * Δ1) - δ) * cs) * cs < 0 * cs := by
+    rw [key, zero_mul]
+    have h1 : 0 ≤ u * Δ1 := mul_nonneg hu0 hΔ1.le
+    have h2 : 0 < δ * cs ^ 2 := mul_pos hδ (pow_pos hcs 2)
+    linarith
+  exact lt_of_mul_lt_mul_right hneg hcs.le
+
+/-! ### The cone between the two distinguished tail support lines -/
+
+/-- Both distinguished fan points lie on the horizontal axis, at the horizontal intercepts of the
+two distinguished tail support lines. -/
+private theorem distinguishedCapSides_fanPoint_coordinates (K : SpecialCapSpace) :
+    (distinguishedCapSides K.val).1.fanPoint 0 =
+        (supportValue (K.val.val : Set Point) (paperGerverConstants.2.1 : Real.Angle) - 1) /
+          Real.cos paperGerverConstants.2.1 ∧
+      (distinguishedCapSides K.val).1.fanPoint 1 = 0 ∧
+      (distinguishedCapSides K.val).2.fanPoint 0 =
+        -((supportValue (K.val.val : Set Point)
+              ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+          Real.cos paperGerverConstants.2.1) ∧
+      (distinguishedCapSides K.val).2.fanPoint 1 = 0 := by
+  obtain ⟨-, -, hsum⟩ := paperGerverConstants_snd_mem_Ioo
+  have hsinl : Real.sin paperGerverConstants.2.2 = Real.cos paperGerverConstants.2.1 := by
+    rw [show paperGerverConstants.2.2 = Real.pi / 2 - paperGerverConstants.2.1 by linarith,
+      Real.sin_pi_div_two_sub]
+  obtain ⟨hW0, hW1⟩ := wedgeEndpoints_fst_coords K.val paperGerverConstants.2.1
+  obtain ⟨hZ0, hZ1⟩ := wedgeEndpoints_snd_coords K.val paperGerverConstants.2.2
+  rw [hsinl] at hZ0
+  rw [distinguishedCapSides_fst_fanPoint, distinguishedCapSides_snd_fanPoint]
+  exact ⟨hW0, hW1, hZ0, hZ1⟩
+
+/-- The left fan point lies strictly to the left of the right one: otherwise the right fan point
+would lie in the cap and in both tail half-planes, which the separation lemma forbids. -/
+private theorem distinguishedCapSides_fanPoint_fst_lt (K : SpecialCapSpace) :
+    (distinguishedCapSides K.val).2.fanPoint 0 < (distinguishedCapSides K.val).1.fanPoint 0 := by
+  obtain ⟨hrIoo, -, hsum⟩ := paperGerverConstants_snd_mem_Ioo
+  obtain ⟨hW0, hW1, hZ0, -⟩ := distinguishedCapSides_fanPoint_coordinates K
+  have hcosr : 0 < Real.cos paperGerverConstants.2.1 :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos, hrIoo.1], hrIoo.2⟩
+  have hsinl : Real.sin paperGerverConstants.2.2 = Real.cos paperGerverConstants.2.1 := by
+    rw [show paperGerverConstants.2.2 = Real.pi / 2 - paperGerverConstants.2.1 by linarith,
+      Real.sin_pi_div_two_sub]
+  have hcosladd : Real.cos (paperGerverConstants.2.2 + Real.pi / 2) =
+      -Real.cos paperGerverConstants.2.1 := by
+    rw [Real.cos_add, Real.cos_pi_div_two, Real.sin_pi_div_two, hsinl]
+    ring
+  have hsr1 : supportValue (K.val.val : Set Point)
+      (paperGerverConstants.2.1 : Real.Angle) - 1 =
+      (distinguishedCapSides K.val).1.fanPoint 0 * Real.cos paperGerverConstants.2.1 := by
+    rw [hW0]
+    field_simp
+  have hsl1 : supportValue (K.val.val : Set Point)
+      ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) - 1 =
+      -((distinguishedCapSides K.val).2.fanPoint 0 * Real.cos paperGerverConstants.2.1) := by
+    rw [hZ0]
+    field_simp
+  by_contra hcon
+  rw [not_lt] at hcon
+  have hmul := mul_le_mul_of_nonneg_right hcon hcosr.le
+  have h1 : (distinguishedCapSides K.val).1.fanPoint ∈
+      (distinguishedCapSides K.val).1.upperHalfPlane := by
+    change _ ≤ inner ℝ _ (normalVector (paperGerverConstants.2.1 : Real.Angle))
+    rw [inner_normalVector_real, hW1, hsr1]
+    simp
+  have h2 : (distinguishedCapSides K.val).1.fanPoint ∈
+      (distinguishedCapSides K.val).2.upperHalfPlane := by
+    change _ ≤ inner ℝ _
+      (normalVector ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle))
+    rw [inner_normalVector_real, hW1, hcosladd, hsl1]
+    linarith only [hmul]
+  exact Set.eq_empty_iff_forall_notMem.mp (cap_and_niche_tail_separation K).1 _
+    ⟨⟨(specialCap_wedgeEndpoints_in_bottomEdge K).1.1.1, h1⟩, h2⟩
+
+/-- The inner corner traverses the closed cone between the two distinguished tail support lines
+`X = w - k y` and `X = z + k y` on the middle interval, with its two endpoints on the two lines.
+Here `k = tan φ` is encoded by `hk`, and `w`, `z` by the two intercept equations. -/
+private theorem capInnerCorner_mem_cone (K : SpecialCapSpace) {k w z : ℝ}
+    (hk : k * Real.cos paperGerverConstants.2.1 = Real.sin paperGerverConstants.2.1)
+    (hw : supportValue (K.val.val : Set Point) (paperGerverConstants.2.1 : Real.Angle) - 1 =
+      w * Real.cos paperGerverConstants.2.1)
+    (hz : supportValue (K.val.val : Set Point)
+        ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) - 1 =
+      -(z * Real.cos paperGerverConstants.2.1)) :
+    capInnerCorner K.val paperGerverConstants.2.1 0 +
+        k * capInnerCorner K.val paperGerverConstants.2.1 1 = w ∧
+      capInnerCorner K.val paperGerverConstants.2.2 0 -
+        k * capInnerCorner K.val paperGerverConstants.2.2 1 = z ∧
+      (∀ t ∈ Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2,
+        capInnerCorner K.val t 0 + k * capInnerCorner K.val t 1 ≤ w) ∧
+      (∀ t ∈ Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2,
+        z ≤ capInnerCorner K.val t 0 - k * capInnerCorner K.val t 1) := by
+  obtain ⟨hrIoo, hlIoo, hsum⟩ := paperGerverConstants_snd_mem_Ioo
+  set r : ℝ := paperGerverConstants.2.1 with hrdef
+  set l : ℝ := paperGerverConstants.2.2 with hldef
+  set xc : ℝ → Point := capInnerCorner K.val with hxcdef
+  have hcosr : 0 < Real.cos r :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos, hrIoo.1], hrIoo.2⟩
+  have hsinl : Real.sin l = Real.cos r := by
+    rw [show l = Real.pi / 2 - r by linarith, Real.sin_pi_div_two_sub]
+  have hcosl : Real.cos l = Real.sin r := by
+    rw [show l = Real.pi / 2 - r by linarith, Real.cos_pi_div_two_sub]
+  have hcosladd : Real.cos (l + Real.pi / 2) = -Real.cos r := by
+    rw [Real.cos_add, Real.cos_pi_div_two, Real.sin_pi_div_two, hsinl]
+    ring
+  have hsinladd : Real.sin (l + Real.pi / 2) = Real.sin r := by
+    rw [Real.sin_add, Real.cos_pi_div_two, Real.sin_pi_div_two, hcosl]
+    ring
+  -- the right endpoint is on the right line, the left endpoint on the left line
+  have hXR : xc r 0 + k * xc r 1 = w := by
+    have h := inner_capInnerCorner_normalVector K.val r
+    rw [inner_normalVector_real, hw] at h
+    refine mul_right_cancel₀ (ne_of_gt hcosr) ?_
+    linear_combination h + xc r 1 * hk
+  have hXL : xc l 0 - k * xc l 1 = z := by
+    have h := inner_capInnerCorner_tangentVector K.val l
+    rw [inner_tangentVector_real, hsinl, hcosl, hz] at h
+    refine mul_right_cancel₀ (ne_of_gt hcosr) ?_
+    linear_combination -h - xc l 1 * hk
+  -- strictly inside the cone away from the two endpoints, by the monotonicity intervals
+  obtain ⟨hmonoR, hmonoL, -, -⟩ := cap_tail_monotonicity_intervals K
+  have hnotR : ∀ t ∈ Set.Ioc r (Real.pi / 2), xc t 0 + k * xc t 1 < w := by
+    intro t ht
+    have h : ¬ (supportValue (K.val.val : Set Point) (r : Real.Angle) - 1 ≤
+        inner ℝ (xc t) (normalVector (r : Real.Angle))) :=
+      fun hc ↦ (hmonoR t ht).1 hc
+    rw [not_le, inner_normalVector_real, hw] at h
+    refine lt_of_mul_lt_mul_right ?_ hcosr.le
+    have he : (xc t 0 + k * xc t 1) * Real.cos r =
+        xc t 0 * Real.cos r + xc t 1 * Real.sin r := by
+      linear_combination xc t 1 * hk
+    rw [he]
+    exact h
+  have hnotL : ∀ t ∈ Set.Ico (0 : ℝ) l, z < xc t 0 - k * xc t 1 := by
+    intro t ht
+    have h : ¬ (supportValue (K.val.val : Set Point) ((l + Real.pi / 2 : ℝ) : Real.Angle) - 1 ≤
+        inner ℝ (xc t) (normalVector ((l + Real.pi / 2 : ℝ) : Real.Angle))) :=
+      fun hc ↦ (hmonoL t ht).1 hc
+    rw [not_le, inner_normalVector_real, hcosladd, hsinladd, hz] at h
+    refine lt_of_mul_lt_mul_right ?_ hcosr.le
+    have he2 : (xc t 0 - k * xc t 1) * Real.cos r =
+        xc t 0 * Real.cos r - xc t 1 * Real.sin r := by
+      linear_combination -(xc t 1) * hk
+    rw [he2]
+    linarith only [h]
+  refine ⟨hXR, hXL, fun t ht ↦ ?_, fun t ht ↦ ?_⟩
+  · rcases eq_or_lt_of_le ht.1 with heq | hlt
+    · rw [← heq, hXR]
+    · exact (hnotR t ⟨hlt, le_trans ht.2 hlIoo.2.le⟩).le
+  · rcases eq_or_lt_of_le ht.2 with heq | hlt
+    · rw [heq, hXL]
+    · exact (hnotL t ⟨le_trans hrIoo.1.le ht.1, hlt⟩).le
+
+/-- A horizontal line strictly below the inner corner on a compact subinterval of `[0, π/2]`. -/
+private theorem exists_lt_capInnerCorner_snd (K : SpecialCapSpace) {a b : ℝ} (hab : a ≤ b)
+    (ha : 0 ≤ a) (hb : b ≤ Real.pi / 2) :
+    ∃ h : ℝ, 0 < h ∧ ∀ t ∈ Set.Icc a b, -h < capInnerCorner K.val t 1 := by
+  have hsub : Set.Icc a b ⊆ Set.Icc (0 : ℝ) (Real.pi / 2) := fun t ht ↦
+    ⟨le_trans ha ht.1, le_trans ht.2 hb⟩
+  have hcont : ContinuousOn (fun t ↦ capInnerCorner K.val t 1) (Set.Icc a b) :=
+    (PiLp.continuous_apply 2 _ 1).comp_continuousOn (K.property.1.2.1.continuousOn.mono hsub)
+  obtain ⟨t0, -, hmin⟩ := isCompact_Icc.exists_isMinOn (⟨a, le_rfl, hab⟩ : (Set.Icc a b).Nonempty)
+    hcont
+  refine ⟨|capInnerCorner K.val t0 1| + 1, by positivity, fun t ht ↦ ?_⟩
+  have hle : capInnerCorner K.val t0 1 ≤ capInnerCorner K.val t 1 := hmin ht
+  linarith only [hle, neg_abs_le (capInnerCorner K.val t0 1)]
+
+/-- The area of the trapezoid cut from the strip `-h ≤ y ≤ 0` by two lines of opposite slope
+meeting the horizontal axis at `z` and `w`. -/
+private theorem volume_slantedTrapezoid {z w k h : ℝ} (hk : 0 ≤ k) (hzw : z ≤ w) (hh : 0 ≤ h) :
+    MeasureTheory.volume {p : Point | p 1 ∈ Set.Icc (-h) 0 ∧
+        p 0 ∈ Set.Icc (z + k * p 1) (w - k * p 1)} =
+      ENNReal.ofReal (h * (w - z) + k * h ^ 2) := by
+  have hkh : 0 ≤ k * h := mul_nonneg hk hh
+  rw [show {p : Point | p 1 ∈ Set.Icc (-h) 0 ∧
+        p 0 ∈ Set.Icc (z + k * p 1) (w - k * p 1)} =
+      {p : Point | p 1 ∈ Set.Icc (-h) 0 ∧
+        p 0 ∈ Set.Icc (z + k * p 1) (w + -k * p 1)} from by
+      refine Set.ext fun p ↦ ?_
+      simp only [Set.mem_ofPred_eq, show w + -k * p 1 = w - k * p 1 from by ring],
+    EuclideanSpace.volume_horizontalTrapezoid_band (by linarith) (by linarith) (by linarith)]
+  congr 1
+  ring
+
+/-! ### The niche contains the open region above the base trapezoid -/
+
+/-- Every point strictly below the three-piece roof made of the inner-corner arc and the two side
+segments, and outside the base trapezoid, lies in the niche outside both distinguished tail
+half-planes.  The roof point vertically above such a point exhibits a time of the closed middle
+interval whose open inward quadrant contains it, and the strict cone bounds plus the point's
+positive height put it in the fan and outside the two half-planes. -/
+private theorem region_under_roof_diff_subset (K : SpecialCapSpace) {k w z h : ℝ} (hk0 : 0 < k)
+    (hk : k * Real.cos paperGerverConstants.2.1 = Real.sin paperGerverConstants.2.1)
+    (hw : supportValue (K.val.val : Set Point) (paperGerverConstants.2.1 : Real.Angle) - 1 =
+      w * Real.cos paperGerverConstants.2.1)
+    (hz : supportValue (K.val.val : Set Point)
+        ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) - 1 =
+      -(z * Real.cos paperGerverConstants.2.1))
+    (P Q : Point) (γ : ContinuousBVPaths 0 3) (hP1 : P 1 = -h) (hQ1 : Q 1 = -h)
+    (hγ0 : ∀ s : Set.Icc (0 : ℝ) 3, (s : ℝ) ≤ 1 →
+      γ.val s = P + (s : ℝ) • (capInnerCorner K.val paperGerverConstants.2.2 - P))
+    (hγ1 : ∀ s : Set.Icc (0 : ℝ) 3, 1 ≤ (s : ℝ) → (s : ℝ) ≤ 2 →
+      γ.val s = capInnerCorner K.val (paperGerverConstants.2.2 +
+        ((s : ℝ) - 1) * (paperGerverConstants.2.1 - paperGerverConstants.2.2)))
+    (hγ2 : ∀ s : Set.Icc (0 : ℝ) 3, 2 ≤ (s : ℝ) →
+      γ.val s = capInnerCorner K.val paperGerverConstants.2.1 +
+        ((s : ℝ) - 2) • (Q - capInnerCorner K.val paperGerverConstants.2.1))
+    (hconeR : ∀ s : Set.Icc (0 : ℝ) 3, γ.val s 0 + k * γ.val s 1 ≤ w)
+    (hconeL : ∀ s : Set.Icc (0 : ℝ) 3, z ≤ γ.val s 0 - k * γ.val s 1)
+    (hPpar : (capInnerCorner K.val paperGerverConstants.2.2 0 - P 0) *
+        Real.cos paperGerverConstants.2.1 =
+      (capInnerCorner K.val paperGerverConstants.2.2 1 - P 1) *
+        Real.sin paperGerverConstants.2.1)
+    (hQpar : (Q 0 - capInnerCorner K.val paperGerverConstants.2.1 0) *
+        Real.cos paperGerverConstants.2.1 =
+      (capInnerCorner K.val paperGerverConstants.2.1 1 - Q 1) *
+        Real.sin paperGerverConstants.2.1) :
+    {p : Point | ∃ s : Set.Icc (0 : ℝ) 3, p 0 = γ.val s 0 ∧ -h < p 1 ∧ p 1 < γ.val s 1} \
+        {p : Point | p 1 ∈ Set.Icc (-h) 0 ∧ p 0 ∈ Set.Icc (z + k * p 1) (w - k * p 1)} ⊆
+      (capNiche K.val \ (distinguishedCapSides K.val).1.upperHalfPlane) \
+        (distinguishedCapSides K.val).2.upperHalfPlane := by
+  obtain ⟨hrIoo, hlIoo, hsum⟩ := paperGerverConstants_snd_mem_Ioo
+  set r : ℝ := paperGerverConstants.2.1 with hrdef
+  set l : ℝ := paperGerverConstants.2.2 with hldef
+  set xc : ℝ → Point := capInnerCorner K.val with hxcdef
+  have hrl : r < l := paperGerverConstants_snd_fst_lt_snd_snd
+  have hcosr : 0 < Real.cos r :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos, hrIoo.1], hrIoo.2⟩
+  have hsinr : 0 < Real.sin r :=
+    Real.sin_pos_of_pos_of_lt_pi hrIoo.1 (by linarith [Real.pi_pos, hrIoo.2])
+  have hpy : Real.sin r ^ 2 + Real.cos r ^ 2 = 1 := Real.sin_sq_add_cos_sq r
+  have hsinl : Real.sin l = Real.cos r := by
+    rw [show l = Real.pi / 2 - r by linarith, Real.sin_pi_div_two_sub]
+  have hcosl : Real.cos l = Real.sin r := by
+    rw [show l = Real.pi / 2 - r by linarith, Real.cos_pi_div_two_sub]
+  have hcosladd : Real.cos (l + Real.pi / 2) = -Real.cos r := by
+    rw [Real.cos_add, Real.cos_pi_div_two, Real.sin_pi_div_two, hsinl]
+    ring
+  have hsinladd : Real.sin (l + Real.pi / 2) = Real.sin r := by
+    rw [Real.sin_add, Real.cos_pi_div_two, Real.sin_pi_div_two, hcosl]
+    ring
+  have hcoord : ∀ (p q : Point) (c : ℝ) (i : Fin 2),
+      (p + c • (q - p)) i = p i + c * (q i - p i) := fun _ _ _ _ ↦ by simp
+  rintro p ⟨hpP, hpR⟩
+  obtain ⟨s, hs0, hs1, hs2⟩ := hpP
+  obtain ⟨δ, hδ0, hδeq⟩ : ∃ δ : ℝ, 0 < δ ∧ p 1 = γ.val s 1 - δ :=
+    ⟨γ.val s 1 - p 1, by linarith only [hs2], by ring⟩
+  have hkp : k * p 1 < k * γ.val s 1 := mul_lt_mul_of_pos_left hs2 hk0
+  -- the point is strictly inside the open cone
+  have hstrictR : p 0 + k * p 1 < w := by
+    have h := hconeR s
+    rw [hs0]
+    linarith only [h, hkp]
+  have hstrictL : z < p 0 - k * p 1 := by
+    have h := hconeL s
+    rw [hs0]
+    linarith only [h, hkp]
+  -- being outside the trapezoid, its height is positive
+  have hp1pos : 0 < p 1 := by
+    by_contra hcon
+    rw [not_lt] at hcon
+    exact hpR ⟨⟨hs1.le, hcon⟩, ⟨by linarith only [hstrictL], by linarith only [hstrictR]⟩⟩
+  have hnotHR : p ∉ (distinguishedCapSides K.val).1.upperHalfPlane := by
+    intro hc
+    replace hc : supportValue (K.val.val : Set Point) (r : Real.Angle) - 1 ≤
+      inner ℝ p (normalVector (r : Real.Angle)) := hc
+    rw [inner_normalVector_real, hw] at hc
+    have hm : (p 0 + k * p 1) * Real.cos r < w * Real.cos r :=
+      mul_lt_mul_of_pos_right hstrictR hcosr
+    have he : (p 0 + k * p 1) * Real.cos r = p 0 * Real.cos r + p 1 * Real.sin r := by
+      linear_combination p 1 * hk
+    rw [he] at hm
+    linarith only [hc, hm]
+  have hnotHL : p ∉ (distinguishedCapSides K.val).2.upperHalfPlane := by
+    intro hc
+    replace hc : supportValue (K.val.val : Set Point) ((l + Real.pi / 2 : ℝ) : Real.Angle) - 1 ≤
+      inner ℝ p (normalVector ((l + Real.pi / 2 : ℝ) : Real.Angle)) := hc
+    rw [inner_normalVector_real, hcosladd, hsinladd, hz] at hc
+    have hm : z * Real.cos r < (p 0 - k * p 1) * Real.cos r :=
+      mul_lt_mul_of_pos_right hstrictL hcosr
+    have he : (p 0 - k * p 1) * Real.cos r = p 0 * Real.cos r - p 1 * Real.sin r := by
+      linear_combination -(p 1) * hk
+    rw [he] at hm
+    linarith only [hc, hm]
+  have hfan : p ∈ capFan (Real.pi / 2) := by
+    refine ⟨?_, ?_⟩ <;>
+      · change (0 : ℝ) ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+        rw [inner_normalVector_pi_div_two]
+        exact hp1pos.le
+  -- the roof point above `p` provides the time whose inward quadrant contains `p`
+  have hquad : ∃ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+      p ∈ innerQuadrant (K.val.val : Set Point) t := by
+    rcases le_or_gt (s : ℝ) 1 with hc | hc
+    · -- the left side segment, with witness time `l`
+      have hγs1 : γ.val s 1 = P 1 + (s : ℝ) * (xc l 1 - P 1) := by
+        rw [hγ0 s hc, hcoord]
+      have hΔ1 : 0 < xc l 1 - P 1 := by
+        nlinarith only [hγs1, hs1, hs2, hP1, s.property.1]
+      have hΔ : (xc l 0 - P 0) * Real.sin l = (xc l 1 - P 1) * Real.cos l := by
+        rw [hsinl, hcosl]
+        exact hPpar
+      have e0 : p 0 - xc l 0 = -((1 - (s : ℝ)) * (xc l 0 - P 0)) := by
+        rw [hs0, hγ0 s hc, hcoord]
+        ring
+      have e1 : p 1 - xc l 1 = -((1 - (s : ℝ)) * (xc l 1 - P 1)) - δ := by
+        rw [hδeq, hγ0 s hc, hcoord]
+        ring
+      refine ⟨l, ⟨hlIoo.1, hlIoo.2⟩, mem_innerQuadrant_of_frame_coordinates_neg K.val l p ?_ ?_⟩
+      · rw [e0, e1]
+        have hnn := segment_normal_neg (cs := Real.sin l) (sn := Real.cos l)
+          (u := 1 - (s : ℝ)) (δ := δ) (by rw [hsinl]; exact hcosr)
+          (by rw [hsinl, hcosl]; linarith only [hpy]) hΔ hΔ1 (by linarith only [hc]) hδ0
+        linarith only [hnn]
+      · rw [e0, e1]
+        have hcr := segment_cross_eq (cs := Real.sin l) (sn := Real.cos l)
+          (u := 1 - (s : ℝ)) (δ := δ) hΔ
+        have hps : 0 < δ * Real.cos l := by
+          rw [hcosl]
+          exact mul_pos hδ0 hsinr
+        linarith only [hcr, hps]
+    · rcases le_or_gt (s : ℝ) 2 with hc2 | hc2
+      · -- the middle arc, with witness time in `[r, l]`
+        have htI := mem_Icc_roofArcTime hrl hc.le hc2
+        set t : ℝ := l + ((s : ℝ) - 1) * (r - l) with htdef
+        have hgs : γ.val s = xc t := hγ1 s hc.le hc2
+        have e0 : p 0 - xc t 0 = 0 := by rw [hs0, hgs]; ring
+        have e1 : p 1 - xc t 1 = -δ := by rw [hδeq, hgs]; ring
+        have ht0 : 0 < t := lt_of_lt_of_le hrIoo.1 htI.1
+        have htpi : t < Real.pi / 2 := lt_of_le_of_lt htI.2 hlIoo.2
+        have hsint : 0 < Real.sin t :=
+          Real.sin_pos_of_pos_of_lt_pi ht0 (by linarith only [Real.pi_pos, htpi])
+        have hcost : 0 < Real.cos t :=
+          Real.cos_pos_of_mem_Ioo ⟨by linarith only [Real.pi_pos, ht0], htpi⟩
+        refine ⟨t, ⟨ht0, htpi⟩, mem_innerQuadrant_of_frame_coordinates_neg K.val t p ?_ ?_⟩
+        · rw [e0, e1]
+          have hps := mul_pos hδ0 hsint
+          linarith only [hps]
+        · rw [e0, e1]
+          have hps := mul_pos hδ0 hcost
+          linarith only [hps]
+      · -- the right side segment, with witness time `r`
+        have hγs1 : γ.val s 1 = xc r 1 + ((s : ℝ) - 2) * (Q 1 - xc r 1) := by
+          rw [hγ2 s hc2.le, hcoord]
+        have hΔ1 : 0 < xc r 1 - Q 1 := by
+          nlinarith only [hγs1, hs1, hs2, hQ1, hc2, s.property.2]
+        have e0 : p 0 - xc r 0 = ((s : ℝ) - 2) * (Q 0 - xc r 0) := by
+          rw [hs0, hγ2 s hc2.le, hcoord]
+          ring
+        have e1 : p 1 - xc r 1 = -(((s : ℝ) - 2) * (xc r 1 - Q 1)) - δ := by
+          rw [hδeq, hγ2 s hc2.le, hcoord]
+          ring
+        refine ⟨r, ⟨hrIoo.1, hrIoo.2⟩,
+          mem_innerQuadrant_of_frame_coordinates_neg K.val r p ?_ ?_⟩
+        · rw [e0, e1]
+          have hcr := segment_cross_eq (cs := Real.cos r) (sn := Real.sin r)
+            (u := (s : ℝ) - 2) (δ := δ) hQpar
+          have hps := mul_pos hδ0 hsinr
+          linarith only [hcr, hps]
+        · rw [e0, e1]
+          have hnn := segment_normal_neg (cs := Real.cos r) (sn := Real.sin r)
+            (u := (s : ℝ) - 2) (δ := δ) hcosr hpy hQpar hΔ1 (by linarith only [hc2]) hδ0
+          linarith only [hnn]
+  obtain ⟨t, htmem, hq⟩ := hquad
+  exact ⟨⟨⟨hfan, Set.mem_biUnion htmem hq⟩, hnotHR⟩, hnotHL⟩
+
+/-! ### The lower estimate -/
+
+theorem capMiddle_area_lower_bound (K : SpecialCapSpace) :
+    segmentArea (distinguishedCapSides K.val).1.fanPoint
+        (distinguishedCapSides K.val).1.corner +
+      curveAreaFunctional (capMiddleBV K) +
+      segmentArea (distinguishedCapSides K.val).2.corner
+        (distinguishedCapSides K.val).2.fanPoint ≤
+    ClassicalResults.area
+      ((capNiche K.val \ (distinguishedCapSides K.val).1.upperHalfPlane) \
+        (distinguishedCapSides K.val).2.upperHalfPlane) := by
+  obtain ⟨hW0, hW1, hZ0, hZ1⟩ := distinguishedCapSides_fanPoint_coordinates K
+  obtain ⟨hrIoo, hlIoo, hsum⟩ := paperGerverConstants_snd_mem_Ioo
+  set r : ℝ := paperGerverConstants.2.1 with hrdef
+  set l : ℝ := paperGerverConstants.2.2 with hldef
+  have hrl : r < l := paperGerverConstants_snd_fst_lt_snd_snd
+  have hcosr : 0 < Real.cos r :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos, hrIoo.1], hrIoo.2⟩
+  have hsinr : 0 < Real.sin r :=
+    Real.sin_pos_of_pos_of_lt_pi hrIoo.1 (by linarith [Real.pi_pos, hrIoo.2])
+  -- the cone's slope `k = tan φ` and the horizontal intercepts `w`, `z` of its two lines
+  set kk : ℝ := Real.sin r / Real.cos r with hkkdef
+  have hkk0 : 0 < kk := div_pos hsinr hcosr
+  have hkcos : kk * Real.cos r = Real.sin r := by
+    rw [hkkdef]
+    field_simp
+  set xc : ℝ → Point := capInnerCorner K.val with hxcdef
+  set W : Point := (distinguishedCapSides K.val).1.fanPoint with hWdef
+  set Z : Point := (distinguishedCapSides K.val).2.fanPoint with hZdef
+  have hsr1 : supportValue (K.val.val : Set Point) (r : Real.Angle) - 1 = W 0 * Real.cos r := by
+    rw [hW0, div_mul_cancel₀ _ (ne_of_gt hcosr)]
+  have hsl1 : supportValue (K.val.val : Set Point) ((l + Real.pi / 2 : ℝ) : Real.Angle) - 1 =
+      -(Z 0 * Real.cos r) := by
+    rw [hZ0, neg_mul, div_mul_cancel₀ _ (ne_of_gt hcosr), neg_neg]
+  have hzw : Z 0 < W 0 := distinguishedCapSides_fanPoint_fst_lt K
+  -- the arc runs through the closed cone from one boundary line to the other
+  obtain ⟨hXR, hXL, harcR, harcL⟩ := capInnerCorner_mem_cone K hkcos hsr1 hsl1
+  -- a base line `y = -hgt` strictly below the arc, and the two base corners on it
+  obtain ⟨hgt, hgt0, hgtlow⟩ := exists_lt_capInnerCorner_snd K hrl.le hrIoo.1.le hlIoo.2.le
+  have hkh : (0 : ℝ) ≤ kk * hgt := mul_nonneg hkk0.le hgt0.le
+  set Bp : Point := !₂[W 0 + kk * hgt, -hgt] with hBpdef
+  set Dp : Point := !₂[Z 0 - kk * hgt, -hgt] with hDpdef
+  have hBp0 : Bp 0 = W 0 + kk * hgt := rfl
+  have hBp1 : Bp 1 = -hgt := rfl
+  have hDp0 : Dp 0 = Z 0 - kk * hgt := rfl
+  have hDp1 : Dp 1 = -hgt := rfl
+  -- the three-piece roof `Dp → x(l) → arc → x(r) → Bp`
+  obtain ⟨γ, hγ0, hγ1, hγ2, hγmono⟩ :=
+    exists_strictMono_roof_of_strictAntiOn hrl (capMiddleBV K) xc Dp Bp (fun _ _ ↦ rfl)
+      (strictAntiOn_capInnerCorner_fst K hrIoo.1.le hlIoo.2.le)
+      (by
+        rw [hDp0]
+        have h : 0 < kk * (xc l 1 + hgt) :=
+          mul_pos hkk0 (by linarith only [hgtlow l ⟨hrl.le, le_rfl⟩])
+        linarith only [hXL, h])
+      (by
+        rw [hBp0]
+        have h : 0 < kk * (xc r 1 + hgt) :=
+          mul_pos hkk0 (by linarith only [hgtlow r ⟨le_rfl, hrl.le⟩])
+        linarith only [hXR, h])
+  -- the roof stays in the closed cone above the base line
+  have hγconeR : ∀ s : Set.Icc (0 : ℝ) 3, γ.val s 0 + kk * γ.val s 1 ≤ W 0 := fun s ↦ by
+    have h := le_of_threePieceRoof (α := 1) (β := kk) (cc := W 0) hrl xc Dp Bp γ hγ0 hγ1 hγ2
+      (by rw [hDp0, hDp1]; linarith only [hzw, hkh])
+      (by rw [hBp0, hBp1]; linarith only []) (fun t ht ↦ by linarith only [harcR t ht]) s
+    linarith only [h]
+  have hγconeL : ∀ s : Set.Icc (0 : ℝ) 3, Z 0 ≤ γ.val s 0 - kk * γ.val s 1 := fun s ↦ by
+    have h := le_of_threePieceRoof (α := -1) (β := kk) (cc := -Z 0) hrl xc Dp Bp γ hγ0 hγ1 hγ2
+      (by rw [hDp0, hDp1]; linarith only [])
+      (by rw [hBp0, hBp1]; linarith only [hzw, hkh])
+      (fun t ht ↦ by linarith only [harcL t ht]) s
+    linarith only [h]
+  have hγlow : ∀ s : Set.Icc (0 : ℝ) 3, -hgt ≤ γ.val s 1 := fun s ↦ by
+    have h := le_of_threePieceRoof (α := 0) (β := -1) (cc := hgt) hrl xc Dp Bp γ hγ0 hγ1 hγ2
+      (by rw [hDp1]; linarith only []) (by rw [hBp1]; linarith only [])
+      (fun t ht ↦ by linarith only [hgtlow t ht]) s
+    linarith only [h]
+  -- the closed region under the roof and the base trapezoid
+  set Pu : Set Point :=
+    {p : Point | ∃ s : Set.Icc (0 : ℝ) 3, p 0 = γ.val s 0 ∧ -hgt ≤ p 1 ∧ p 1 ≤ γ.val s 1}
+    with hPudef
+  set Rg : Set Point :=
+    {p : Point | p 1 ∈ Set.Icc (-hgt) 0 ∧ p 0 ∈ Set.Icc (Z 0 + kk * p 1) (W 0 - kk * p 1)}
+    with hRgdef
+  have hRgarea : MeasureTheory.volume Rg =
+      ENNReal.ofReal (hgt * (W 0 - Z 0) + kk * hgt ^ 2) :=
+    volume_slantedTrapezoid hkk0.le hzw.le hgt0.le
+  have hRgreal : ClassicalResults.area Rg = hgt * (W 0 - Z 0) + kk * hgt ^ 2 := by
+    rw [ClassicalResults.area, hRgarea, ENNReal.toReal_ofReal]
+    positivity
+  -- the four-piece signed area of the loop bounding the region
+  have hPuarea : ClassicalResults.area Pu =
+      segmentArea Bp (xc r) + curveAreaFunctional (capMiddleBV K) +
+        segmentArea (xc l) Dp + segmentArea Dp Bp :=
+    area_region_under_strictMono_roof hrl (capMiddleBV K) xc Dp Bp γ (fun _ _ ↦ rfl) hDp1 hBp1
+      hγ0 hγ1 hγ2 hγmono hγlow
+  -- the strict region above the trapezoid lies in the niche outside both half-planes
+  have hincl := region_under_roof_diff_subset K hkk0 hkcos hsr1 hsl1 Dp Bp γ hDp1 hBp1
+    hγ0 hγ1 hγ2 hγconeR hγconeL
+    (by
+      rw [hDp0, hDp1]
+      linear_combination Real.cos r * hXL + (xc l 1 + hgt) * hkcos)
+    (by
+      rw [hBp0, hBp1]
+      linear_combination (-Real.cos r) * hXR + (xc r 1 + hgt) * hkcos)
+  -- comparing the three areas
+  have hTgtfin : MeasureTheory.volume
+      ((capNiche K.val \ (distinguishedCapSides K.val).1.upperHalfPlane) \
+        (distinguishedCapSides K.val).2.upperHalfPlane) ≠ ⊤ :=
+    ne_top_of_le_ne_top (niche_uniform_bounds.1 _ K.val).2.2.1.ne
+      (MeasureTheory.measure_mono (Set.sdiff_subset.trans Set.sdiff_subset))
+  have hbase : ∀ p : Point, (∃ s : Set.Icc (0 : ℝ) 3, p 0 = γ.val s 0) → p 1 = -hgt → p ∈ Rg := by
+    rintro p ⟨s, hs0⟩ hp1
+    have h1 : γ.val ⟨0, le_rfl, by norm_num⟩ 0 ≤ γ.val s 0 :=
+      hγmono.monotone (show (⟨0, le_rfl, by norm_num⟩ : Set.Icc (0 : ℝ) 3) ≤ s from s.property.1)
+    have h2 : γ.val s 0 ≤ γ.val ⟨3, by norm_num, le_rfl⟩ 0 :=
+      hγmono.monotone (show s ≤ (⟨3, by norm_num, le_rfl⟩ : Set.Icc (0 : ℝ) 3) from s.property.2)
+    rw [show γ.val ⟨0, le_rfl, by norm_num⟩ = Dp from by rw [hγ0 _ (by norm_num)]; simp] at h1
+    rw [show γ.val ⟨3, by norm_num, le_rfl⟩ = Bp from by rw [hγ2 _ (by norm_num)]; norm_num] at h2
+    rw [hDp0] at h1
+    rw [hBp0] at h2
+    refine ⟨⟨le_of_eq hp1.symm, by rw [hp1]; linarith only [hgt0]⟩, ?_⟩
+    rw [hp1, hs0]
+    exact ⟨by linarith only [h1], by linarith only [h2]⟩
+  have hreal := area_region_under_roof_le γ hγmono Rg _
+    (by rw [hRgarea]; exact ENNReal.ofReal_ne_top) hTgtfin hbase hincl
+  -- the signed-area cancellation along the base line
+  have harith : segmentArea W (xc r) + curveAreaFunctional (capMiddleBV K) +
+      segmentArea (xc l) Z = ClassicalResults.area Pu - ClassicalResults.area Rg := by
+    rw [hPuarea, hRgreal]
+    simp only [segmentArea, planeCrossProduct, hBp0, hBp1, hDp0, hDp1, hW1, hZ1]
+    linear_combination (-(hgt / 2)) * hXR + (hgt / 2) * hXL
+  rw [show (distinguishedCapSides K.val).1.corner = xc r from rfl,
+    show (distinguishedCapSides K.val).2.corner = xc l from rfl]
+  linarith [harith, hreal]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Area.Mamikon.Properties`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Area / Mamikon / Properties
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- The tangential displacement from an exposed-edge endpoint to the path, extended by zero. -/
+def mamikonOffset (K : ConvexBody Point) {a b : ℝ} (z : ContinuousBVPaths a b)
+    (t : ℝ) : ℝ :=
+  if ht : t ∈ Set.Icc a b then
+    inner ℝ (z.val ⟨t, ht⟩ - (edgeVertices K (t : Real.Angle)).1)
+      (tangentVector (t : Real.Angle))
+  else 0
+
+/-- Half the square integral of a bounded measurable family of integrands that is pointwise
+convex-linear in its parameter is a quadratic and convex functional of that parameter. -/
+theorem sqIntegral_quadratic_convex {α : Type*} {Ω : Type*} [MeasurableSpace Ω]
+    (c : unitInterval → α → α → α) (μ : Measure Ω) [IsFiniteMeasure μ] (g : α → Ω → ℝ)
+    (hmeas : ∀ x, Measurable (g x)) (hbdd : ∀ x, ∃ C, ∀ ω, |g x ω| ≤ C)
+    (hlin : ∀ (s : unitInterval) (x y : α) (ω : Ω),
+      g (c s x y) ω = (1 - (s : ℝ)) * g x ω + (s : ℝ) * g y ω)
+    (f : α → ℝ) (hf : ∀ x, f x = (∫ ω, g x ω ^ 2 ∂μ) / 2) :
+    IsQuadraticFunctional c f ∧ IsConvexFunctional c f false := by
+  have hint : ∀ x y, Integrable (fun ω ↦ g x ω * g y ω) μ := by
+    intro x y
+    obtain ⟨Cx, hCx⟩ := hbdd x
+    obtain ⟨Cy, hCy⟩ := hbdd y
+    refine Integrable.mono' (integrable_const (|Cx| * |Cy|))
+      ((hmeas x).mul (hmeas y)).aestronglyMeasurable (Filter.Eventually.of_forall fun ω ↦ ?_)
+    rw [Real.norm_eq_abs, abs_mul]
+    exact mul_le_mul ((hCx ω).trans (le_abs_self _)) ((hCy ω).trans (le_abs_self _))
+      (abs_nonneg _) (abs_nonneg _)
+  set B : α → α → ℝ := fun x y ↦ (∫ ω, g x ω * g y ω ∂μ) / 2
+  have hBval : ∀ x y, B x y = (∫ ω, g x ω * g y ω ∂μ) / 2 := fun _ _ ↦ rfl
+  have hBsymm : ∀ x y, B x y = B y x := fun x y ↦ by
+    rw [hBval, hBval, integral_congr_ae (Filter.Eventually.of_forall fun ω ↦ mul_comm _ _)]
+  have hBleft : ∀ (s : unitInterval) (x y z : α),
+      B (c s x y) z = realCombination s (B x z) (B y z) := by
+    intro s x y z
+    rw [hBval, hBval, hBval, realCombination,
+      integral_congr_ae (Filter.Eventually.of_forall fun ω ↦
+        show g (c s x y) ω * g z ω =
+            (1 - (s : ℝ)) * (g x ω * g z ω) + (s : ℝ) * (g y ω * g z ω) by
+          rw [hlin]; ring),
+      integral_add ((hint x z).const_mul _) ((hint y z).const_mul _),
+      integral_const_mul, integral_const_mul]
+    ring
+  have hB : IsConvexBilinear c c realCombination B := by
+    refine ⟨fun x s y z ↦ ?_, fun z s x y ↦ hBleft s x y z⟩
+    rw [hBsymm x (c s y z), hBleft s y z x, hBsymm y x, hBsymm z x]
+  have hfB : ∀ x, f x = B x x := fun x ↦ by
+    rw [hf x, hBval, integral_congr_ae (Filter.Eventually.of_forall fun ω ↦ pow_two (g x ω))]
+  refine ⟨⟨B, hB, hfB⟩, fun s x y ↦ ?_⟩
+  change f (c s x y) ≤ realCombination s (f x) (f y)
+  have hnn : 0 ≤ B x x - B x y - B y x + B y y := by
+    have i1 : Integrable (fun ω ↦ g x ω * g x ω - g x ω * g y ω) μ := (hint x x).sub (hint x y)
+    have i2 : Integrable (fun ω ↦ g x ω * g x ω - g x ω * g y ω - g y ω * g x ω) μ :=
+      i1.sub (hint y x)
+    have hsplit : (∫ ω, (g x ω * g x ω - g x ω * g y ω - g y ω * g x ω + g y ω * g y ω) ∂μ) =
+        (∫ ω, g x ω * g x ω ∂μ) - (∫ ω, g x ω * g y ω ∂μ) - (∫ ω, g y ω * g x ω ∂μ) +
+          ∫ ω, g y ω * g y ω ∂μ := by
+      rw [integral_add i2 (hint y y), integral_sub i1 (hint y x),
+        integral_sub (hint x x) (hint x y)]
+    have hpos : 0 ≤ ∫ ω, (g x ω * g x ω - g x ω * g y ω - g y ω * g x ω + g y ω * g y ω) ∂μ :=
+      integral_nonneg fun ω ↦ by
+        change (0 : ℝ) ≤ g x ω * g x ω - g x ω * g y ω - g y ω * g x ω + g y ω * g y ω
+        nlinarith [sq_nonneg (g x ω - g y ω)]
+    rw [hsplit] at hpos
+    rw [hBval, hBval, hBval, hBval]
+    linarith
+  have hexp : B (c s x y) (c s x y) = (1 - (s : ℝ)) ^ 2 * B x x +
+      (s : ℝ) * (1 - (s : ℝ)) * (B x y + B y x) + (s : ℝ) ^ 2 * B y y := by
+    rw [hBleft s x y (c s x y), realCombination, hB.1 x s x y, hB.1 y s x y, realCombination,
+      realCombination]
+    ring
+  rw [hfB, hfB, hfB, realCombination, hexp]
+  nlinarith [mul_nonneg (mul_nonneg s.2.1 (sub_nonneg.mpr s.2.2)) hnn]
+
+/-- The Mamikon offset is convex-linear in the body along a convex-linear family of paths. -/
+theorem mamikonOffset_convexBodyCombination (s : unitInterval) (K L : ConvexBody Point)
+    {a b : ℝ} (zK zL : ContinuousBVPaths a b) (t : ℝ) :
+    mamikonOffset (convexBodyCombination s K L) (bvPathCombination s zK zL) t =
+      (1 - (s : ℝ)) * mamikonOffset K zK t + (s : ℝ) * mamikonOffset L zL t := by
+  by_cases ht : t ∈ Set.Icc a b
+  · have hz : (bvPathCombination s zK zL).val ⟨t, ht⟩ =
+        (1 - (s : ℝ)) • zK.val ⟨t, ht⟩ + (s : ℝ) • zL.val ⟨t, ht⟩ := rfl
+    have hv := ((convexBody_maps_linear s K L).2.1 (t : Real.Angle)).1
+    simp only [mamikonOffset, dite_eq_left ht, hz, hv, inner_sub_left, inner_add_left,
+      real_inner_smul_left]
+    ring
+  · simp only [mamikonOffset, dite_eq_right ht]
+    ring
+
+private theorem mamikon_frame_identity (K : ConvexBody Point) {a b : ℝ}
+    (hab : a < b) (hturn : b ≤ a + 2 * Real.pi) (z : ContinuousBVPaths a b)
+    (S : Set (Set.Icc a b)) (hSmeas : MeasurableSet S)
+    (hSa : ∀ t ∈ S, a < (t : ℝ))
+    (W Z U T D : Fin 2 → RightContinuousIntervalBV a b)
+    (hW : ∀ i t, (W i).toFun t = (edgeVertices K ((t : ℝ) : Real.Angle)).1 i)
+    (hWm : ∀ i (E : Set (Set.Icc a b)), MeasurableSet E →
+      (∀ t ∈ E, a < (t : ℝ)) →
+      intervalStieltjesMeasure (W i) E =
+        ∫ u in (fun t : Set.Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+          tangentVector u i ∂surfaceAreaMeasure K)
+    (hZ : ∀ i t, (Z i).toFun t = z.val t i)
+    (hU : ∀ i t, (U i).toFun t = normalVector ((t : ℝ) : Real.Angle) i)
+    (hUd : ∀ i, HasIntervalStieltjesDensity (U i)
+      (fun t ↦ tangentVector (t : Real.Angle) i))
+    (hT : ∀ i t, (T i).toFun t = tangentVector ((t : ℝ) : Real.Angle) i)
+    (hUc : ∀ i, Continuous (U i).toFun)
+    (hDval : ∀ i t, (D i).toFun t =
+      z.val t i - (edgeVertices K ((t : ℝ) : Real.Angle)).1 i)
+    (A : RightContinuousIntervalBV a b)
+    (hA : ∀ t, A.toFun t = ∑ i, (D i).toFun t * (T i).toFun t)
+    (hAmeas : Measurable A.toFun)
+    (hzt : ∀ t : Set.Icc a b, inner ℝ (z.val t) (normalVector ((t : ℝ) : Real.Angle)) =
+      supportValue K ((t : ℝ) : Real.Angle))
+    (CA : ℝ) (hCA : ∀ t, ‖A.toFun t‖ ≤ CA) (hCA0 : 0 ≤ CA)
+    (hAUb : ∀ i t, ‖A.toFun t * (U i).toFun t‖ ≤ CA)
+    (hperpframe : ∀ u : Real.Angle,
+      normalVector u 0 * tangentVector u 0 + normalVector u 1 * tangentVector u 1 = 0) :
+    intervalStieltjesIntegral (Z 0) (fun t ↦ A.toFun t * (U 0).toFun t) S +
+      intervalStieltjesIntegral (Z 1) (fun t ↦ A.toFun t * (U 1).toFun t) S =
+    -(∫ t in S, (A.toFun t) ^ 2 ∂volume.comap (Subtype.val : Set.Icc a b → ℝ)) := by
+  -- the integrated product rule against the rotating normal frame
+  choose zu hzu hzum using fun i : Fin 2 ↦
+    intervalStieltjes_product a b (Z i) (U i) (Or.inr (hUc i))
+  choose wu hwu hwum using fun i : Fin 2 ↦
+    intervalStieltjes_product a b (W i) (U i) (Or.inr (hUc i))
+  obtain ⟨ZU, hZU, hZUm⟩ := intervalStieltjes_linear_combination a b (zu 0) (zu 1) 1 1
+  obtain ⟨WU, hWU, hWUm⟩ := intervalStieltjes_linear_combination a b (wu 0) (wu 1) 1 1
+  have hZUWU : ZU = WU := by
+    refine RightContinuousIntervalBV.toFun_injective (funext fun t ↦ ?_)
+    rw [hZU t, hWU t, hzu 0 t, hzu 1 t, hwu 0 t, hwu 1 t, hZ, hZ, hW, hW, hU, hU]
+    have h1 := hzt t
+    have h2 : inner ℝ (edgeVertices K ((t : ℝ) : Real.Angle)).1
+        (normalVector ((t : ℝ) : Real.Angle)) = supportValue K ((t : ℝ) : Real.Angle) :=
+      (edgeVertices_fst_mem K ((t : ℝ) : Real.Angle)).2
+    simp only [PiLp.inner_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two] at h1 h2
+    linear_combination h1 - h2
+  have hAint : ∀ F : RightContinuousIntervalBV a b,
+      (intervalStieltjesMeasure F).Integrable A.toFun := by
+    intro F
+    have _ : IsFiniteMeasure (intervalStieltjesMeasure F).variation :=
+      BoundedVariationOn.instIsFiniteMeasureVariationVectorMeasure F.boundedVariation
+    exact A.boundedVariation.integrable
+  have hsplitsum : ∀ F G H : RightContinuousIntervalBV a b,
+      intervalStieltjesMeasure H =
+          (1 : ℝ) • intervalStieltjesMeasure F + (1 : ℝ) • intervalStieltjesMeasure G →
+      intervalStieltjesIntegral H A.toFun S =
+        intervalStieltjesIntegral F A.toFun S + intervalStieltjesIntegral G A.toFun S := by
+    intro F G H hH
+    unfold intervalStieltjesIntegral
+    rw [hH]
+    simp only [one_smul]
+    rw [VectorMeasure.restrict_add,
+      VectorMeasure.integral_add_vectorMeasure (hAint F).integrableOn (hAint G).integrableOn]
+  have hH2 : ∀ (i : Fin 2) (F P : RightContinuousIntervalBV a b),
+      (∀ t, P.toFun t = F.toFun t * (U i).toFun t) →
+      intervalStieltjesIntegral P A.toFun S =
+        intervalStieltjesIntegral F (fun t ↦ A.toFun t * (U i).toFun t) S +
+          intervalStieltjesIntegral (U i) (fun t ↦ A.toFun t * F.toFun t) S := fun i F P hP ↦
+    intervalStieltjesIntegral_product_of_bounded hab.le F (U i) P (hUc i) hP A.toFun
+      hAmeas CA hCA S hSmeas
+  have hkey := hsplitsum (zu 0) (zu 1) ZU hZUm
+  rw [hZUWU, hsplitsum (wu 0) (wu 1) WU hWUm, hH2 0 (Z 0) (zu 0) (hzu 0),
+    hH2 1 (Z 1) (zu 1) (hzu 1), hH2 0 (W 0) (wu 0) (hwu 0), hH2 1 (W 1) (wu 1) (hwu 1)] at hkey
+  -- the vertex measure pairs to zero against the rotating normal frame
+  have hWzero : intervalStieltjesIntegral (W 0) (fun t ↦ A.toFun t * (U 0).toFun t) S +
+      intervalStieltjesIntegral (W 1) (fun t ↦ A.toFun t * (U 1).toFun t) S = 0 := by
+    have hdot := sum_intervalStieltjesIntegral_positiveVertex_dot K hab hturn W hWm
+      (fun i t ↦ A.toFun t * (U i).toFun t) CA hCA0
+      (fun i ↦ hAmeas.mul (U i).boundedVariation.measurable) hAUb (fun _ ↦ 0)
+      (by
+        intro t _
+        rw [Fin.sum_univ_two]
+        simp only [hU]
+        linear_combination (A.toFun t) * hperpframe ((t : ℝ) : Real.Angle))
+      S hSmeas hSa
+    simpa [Fin.sum_univ_two] using hdot
+  -- the Lebesgue density of the rotating frame
+  have hfinite : IsFiniteMeasure (volume.comap (Subtype.val : Set.Icc a b → ℝ)) := ⟨by
+    rw [comap_subtype_coe_apply measurableSet_Icc]
+    simp only [Set.image_univ, Subtype.range_val]
+    exact measure_Icc_lt_top⟩
+  let _ := hfinite
+  have hdens : ∀ (i : Fin 2) (F : RightContinuousIntervalBV a b),
+      intervalStieltjesIntegral (U i) (fun t ↦ A.toFun t * F.toFun t) S =
+        ∫ t in S, A.toFun t * F.toFun t * tangentVector ((t : ℝ) : Real.Angle) i
+          ∂volume.comap (Subtype.val : Set.Icc a b → ℝ) := fun i F ↦
+    intervalStieltjesIntegral_eq_integral_mul_of_density_bv hab.le (U i) (hUd i)
+      (A.boundedVariation.bilinear_comp F.boundedVariation (ContinuousLinearMap.mul ℝ ℝ)) S hSmeas
+  have hintd : ∀ (i : Fin 2) (F : RightContinuousIntervalBV a b),
+      Integrable (fun t : Set.Icc a b ↦
+        A.toFun t * F.toFun t * tangentVector ((t : ℝ) : Real.Angle) i)
+        (volume.comap (Subtype.val : Set.Icc a b → ℝ)) := by
+    intro i F
+    refine (A.boundedVariation.bilinear_comp F.boundedVariation
+      (ContinuousLinearMap.mul ℝ ℝ)).integrable.mul_bdd (c := 1) ?_ ?_
+    · have h : Continuous fun u : Real.Angle ↦ tangentVector u i := by
+        fin_cases i
+        · exact Real.Angle.continuous_sin.neg
+        · exact Real.Angle.continuous_cos
+      exact (h.comp (Real.Angle.continuous_coe.comp continuous_subtype_val)).aestronglyMeasurable
+    · filter_upwards with t
+      fin_cases i
+      · simpa [tangentVector, frame] using Real.abs_sin_le_one (t : ℝ)
+      · simpa [tangentVector, frame] using Real.abs_cos_le_one (t : ℝ)
+  have haddZ : Integrable (fun t : Set.Icc a b ↦
+      A.toFun t * (Z 0).toFun t * tangentVector ((t : ℝ) : Real.Angle) 0 +
+        A.toFun t * (Z 1).toFun t * tangentVector ((t : ℝ) : Real.Angle) 1)
+      (volume.comap (Subtype.val : Set.Icc a b → ℝ)) := (hintd 0 (Z 0)).add (hintd 1 (Z 1))
+  have haddW : Integrable (fun t : Set.Icc a b ↦
+      A.toFun t * (W 0).toFun t * tangentVector ((t : ℝ) : Real.Angle) 0 +
+        A.toFun t * (W 1).toFun t * tangentVector ((t : ℝ) : Real.Angle) 1)
+      (volume.comap (Subtype.val : Set.Icc a b → ℝ)) := (hintd 0 (W 0)).add (hintd 1 (W 1))
+  have hsquare : (intervalStieltjesIntegral (U 0) (fun t ↦ A.toFun t * (Z 0).toFun t) S +
+        intervalStieltjesIntegral (U 1) (fun t ↦ A.toFun t * (Z 1).toFun t) S) -
+      (intervalStieltjesIntegral (U 0) (fun t ↦ A.toFun t * (W 0).toFun t) S +
+        intervalStieltjesIntegral (U 1) (fun t ↦ A.toFun t * (W 1).toFun t) S) =
+      ∫ t in S, (A.toFun t) ^ 2 ∂volume.comap (Subtype.val : Set.Icc a b → ℝ) := by
+    rw [hdens 0 (Z 0), hdens 1 (Z 1), hdens 0 (W 0), hdens 1 (W 1),
+      ← integral_add (hintd 0 (Z 0)).integrableOn (hintd 1 (Z 1)).integrableOn,
+      ← integral_add (hintd 0 (W 0)).integrableOn (hintd 1 (W 1)).integrableOn,
+      ← integral_sub haddZ.integrableOn haddW.integrableOn]
+    refine setIntegral_congr_fun hSmeas fun t _ ↦ ?_
+    have h := hA t
+    rw [Fin.sum_univ_two, hDval, hDval, hT, hT] at h
+    simp only [hZ, hW]
+    linear_combination (-A.toFun t) * h
+  linarith [hkey, hWzero, hsquare]
+
+private theorem mamikon_arc_identity (K : ConvexBody Point) {a b : ℝ}
+    (hab : a < b) (hba : b < a + Real.pi) (hturn : b ≤ a + 2 * Real.pi)
+    (z : ContinuousBVPaths a b) (S : Set (Set.Icc a b))
+    (hSmeas : MeasurableSet S) (hSa : ∀ t ∈ S, a < (t : ℝ))
+    (W Z : Fin 2 → RightContinuousIntervalBV a b)
+    (hWm : ∀ i (E : Set (Set.Icc a b)), MeasurableSet E →
+      (∀ t ∈ E, a < (t : ℝ)) →
+      intervalStieltjesMeasure (W i) E =
+        ∫ u in (fun t : Set.Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+          tangentVector u i ∂surfaceAreaMeasure K)
+    (hZ : ∀ i t, (Z i).toFun t = z.val t i)
+    (hzt : ∀ t : Set.Icc a b, inner ℝ (z.val t) (normalVector ((t : ℝ) : Real.Angle)) =
+      supportValue K ((t : ℝ) : Real.Angle))
+    (himg : (fun t : Set.Icc a b ↦ ((t : ℝ) : Real.Angle)) '' S =
+      (fun t : ℝ ↦ (t : Real.Angle)) '' Set.Ioo a b) :
+    intervalStieltjesIntegral (W 1) (Z 0).toFun S -
+      intervalStieltjesIntegral (W 0) (Z 1).toFun S = 2 * convexArcArea K a b := by
+  obtain ⟨C0, hC0⟩ := (Z 0).exists_norm_bound hab.le
+  obtain ⟨C1, hC1⟩ := (Z 1).exists_norm_bound hab.le
+  have hCz0 : 0 ≤ max C0 C1 := le_trans (norm_nonneg _) ((hC0 ⟨a, le_rfl, hab.le⟩).trans
+    (le_max_left _ _))
+  have hdot := sum_intervalStieltjesIntegral_positiveVertex_dot K hab hturn W hWm
+    ![fun t ↦ -(Z 1).toFun t, (Z 0).toFun] (max C0 C1) hCz0
+    (by
+      intro i
+      fin_cases i
+      · exact ((Z 1).boundedVariation.measurable).neg
+      · exact (Z 0).boundedVariation.measurable)
+    (by
+      intro i t
+      fin_cases i
+      · simpa using (hC1 t).trans (le_max_right _ _)
+      · simpa using (hC0 t).trans (le_max_left _ _))
+    (fun u ↦ supportValue K u)
+    (by
+      intro t _
+      have h := planeCrossProduct_tangentVector (z.val t) ((t : ℝ) : Real.Angle)
+      rw [hzt t] at h
+      simp only [planeCrossProduct] at h
+      simp only [Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one, hZ]
+      linarith)
+    S hSmeas hSa
+  rw [Fin.sum_univ_two] at hdot
+  simp only [Matrix.cons_val_zero, Matrix.cons_val_one] at hdot
+  rw [show intervalStieltjesIntegral (W 0) (fun t ↦ -(Z 1).toFun t) S =
+      -intervalStieltjesIntegral (W 0) (Z 1).toFun S from
+    VectorMeasure.integral_fun_neg _ _ _] at hdot
+  rw [himg] at hdot
+  have harea := (convexArc_area a b hab hba).2.1 K
+  rw [← hdot] at harea
+  linarith
+
+private theorem mamikon_endpoint_identity (K : ConvexBody Point) {a b : ℝ}
+    (hab : a < b) (z : ContinuousBVPaths a b)
+    (W Z : Fin 2 → RightContinuousIntervalBV a b)
+    (hW : ∀ i t, (W i).toFun t = (edgeVertices K ((t : ℝ) : Real.Angle)).1 i)
+    (hZ : ∀ i t, (Z i).toFun t = z.val t i)
+    (hZc : ∀ i, Continuous (Z i).toFun) :
+    let a' : Set.Icc a b := ⟨a, le_rfl, hab.le⟩
+    let b' : Set.Icc a b := ⟨b, hab.le, le_rfl⟩
+    let S : Set (Set.Icc a b) := Set.Ioo a' b'
+    (intervalStieltjesIntegral (W 1) (Z 0).toFun S +
+           intervalStieltjesIntegral (Z 0) (W 1).toFun S) -
+         (intervalStieltjesIntegral (W 0) (Z 1).toFun S +
+           intervalStieltjesIntegral (Z 1) (W 0).toFun S) =
+         2 * segmentArea (edgeVertices K ((a : ℝ) : Real.Angle)).1 (z.val a') +
+           2 * segmentArea (z.val b') (edgeVertices K ((b : ℝ) : Real.Angle)).2 := by
+  let a' : Set.Icc a b := ⟨a, le_rfl, hab.le⟩
+  let b' : Set.Icc a b := ⟨b, hab.le, le_rfl⟩
+  let S : Set (Set.Icc a b) := Set.Ioo a' b'
+  have hS : S = Set.Ioo a' b' := rfl
+  have hSmeas : MeasurableSet S := measurableSet_Ioo
+  -- the product-rule endpoint identity
+  obtain ⟨p1, hp1, hp1m⟩ := intervalStieltjes_product a b (W 1) (Z 0) (Or.inr (hZc 0))
+  obtain ⟨p0, hp0, hp0m⟩ := intervalStieltjes_product a b (W 0) (Z 1) (Or.inr (hZc 1))
+  have hab' : a' < b' := hab
+  have hIoo : ∀ P : RightContinuousIntervalBV a b,
+      intervalStieltjesMeasure P S = Function.leftLim P.toFun b' - P.toFun a' := by
+    intro P
+    rw [hS, intervalStieltjesMeasure, P.boundedVariation.vectorMeasure_Ioo hab',
+      (P.right_continuous a').rightLim_eq]
+  have hleftLimP : ∀ (i j : Fin 2) (P : RightContinuousIntervalBV a b),
+      (∀ t, P.toFun t = (W i).toFun t * (Z j).toFun t) →
+      Function.leftLim P.toFun b' =
+        (edgeVertices K ((b : ℝ) : Real.Angle)).2 i * z.val b' j := by
+    intro i j P hP
+    have hne : (nhdsWithin b' (Set.Iio b')).NeBot := nhdsLT_neBot_of_exists_lt ⟨a', hab'⟩
+    let _ := hne
+    refine tendsto_nhds_unique (P.boundedVariation.tendsto_leftLim b') ?_
+    have h1 := (W i).boundedVariation.tendsto_leftLim b'
+    rw [leftLim_positiveVertex_coordinate K W hW i b' hab] at h1
+    have h2 : Filter.Tendsto (Z j).toFun (nhdsWithin b' (Set.Iio b')) (nhds ((Z j).toFun b')) :=
+      (hZc j).continuousAt.continuousWithinAt
+    rw [hZ j b'] at h2
+    exact (h1.mul h2).congr fun t ↦ (hP t).symm
+  have hEndpoint : (intervalStieltjesIntegral (W 1) (Z 0).toFun S +
+        intervalStieltjesIntegral (Z 0) (W 1).toFun S) -
+      (intervalStieltjesIntegral (W 0) (Z 1).toFun S +
+        intervalStieltjesIntegral (Z 1) (W 0).toFun S) =
+      2 * segmentArea (edgeVertices K ((a : ℝ) : Real.Angle)).1 (z.val a') +
+        2 * segmentArea (z.val b') (edgeVertices K ((b : ℝ) : Real.Angle)).2 := by
+    rw [← hp1m S hSmeas, ← hp0m S hSmeas, hIoo p1, hIoo p0,
+      hleftLimP 1 0 p1 hp1, hleftLimP 0 1 p0 hp0, hp1 a', hp0 a', hW, hW, hZ, hZ]
+    simp only [segmentArea, planeCrossProduct]
+    ring
+  exact hEndpoint
+
+theorem mamikon_integral (K : ConvexBody Point) (a b : ℝ)
+    (hab : a < b) (hba : b < a + Real.pi) (z : ContinuousBVPaths a b)
+    (hz : ∀ t : Set.Icc a b,
+      z.val t ∈ (supportingLineHalfPlane K (t.val : Real.Angle)).1) :
+    Measurable (mamikonOffset K z) ∧
+    (∃ C : ℝ, ∀ t, |mamikonOffset K z t| ≤ C) ∧
+    (∀ t : Set.Icc a b, z.val t = (edgeVertices K (t.val : Real.Angle)).1 +
+      mamikonOffset K z t.val • tangentVector (t.val : Real.Angle)) ∧
+    mamikonFunctional K a b hab hba z hz =
+      (∫ t in Set.Icc a b, (mamikonOffset K z t) ^ 2) / 2 := by
+  have hturn : b ≤ a + 2 * Real.pi := by linarith [Real.pi_pos]
+  set a' : Set.Icc a b := ⟨a, le_rfl, hab.le⟩ with ha'
+  set b' : Set.Icc a b := ⟨b, hab.le, le_rfl⟩ with hb'
+  set S : Set (Set.Icc a b) := Set.Ioo a' b' with hS
+  have hSmeas : MeasurableSet S := measurableSet_Ioo
+  have hSa : ∀ t ∈ S, a < (t : ℝ) := fun t ht ↦ ht.1
+  -- the frame data
+  obtain ⟨W, hW, hWm⟩ := positiveVertex_stieltjes_surface K a b hab hturn
+  set Z : Fin 2 → RightContinuousIntervalBV a b := fun i ↦ continuousBVCoordinate z i with hZdef
+  have hZ : ∀ i (t : Set.Icc a b), (Z i).toFun t = z.val t i := fun _ _ ↦ rfl
+  have hZc : ∀ i, Continuous (Z i).toFun := fun i ↦
+    (PiLp.continuous_apply 2 _ i).comp z.property.1
+  choose U hU hUd using fun i : Fin 2 ↦ exists_normalVector_coordinate_intervalBV hab.le i
+  choose T hT hTd using fun i : Fin 2 ↦ exists_tangentVector_coordinate_intervalBV hab.le i
+  have hUc : ∀ i, Continuous (U i).toFun := by
+    intro i
+    have h : Continuous fun u : Real.Angle ↦ normalVector u i := by
+      fin_cases i
+      · exact Real.Angle.continuous_cos
+      · exact Real.Angle.continuous_sin
+    exact (h.comp (Real.Angle.continuous_coe.comp continuous_subtype_val)).congr
+      fun t ↦ (hU i t).symm
+  have hTc : ∀ i, Continuous (T i).toFun := by
+    intro i
+    have h : Continuous fun u : Real.Angle ↦ tangentVector u i := by
+      fin_cases i
+      · exact Real.Angle.continuous_sin.neg
+      · exact Real.Angle.continuous_cos
+    exact (h.comp (Real.Angle.continuous_coe.comp continuous_subtype_val)).congr
+      fun t ↦ (hT i t).symm
+  choose D hD hDm using fun i : Fin 2 ↦
+    intervalStieltjes_linear_combination a b (Z i) (W i) 1 (-1)
+  have hDval : ∀ i (t : Set.Icc a b),
+      (D i).toFun t = z.val t i - (edgeVertices K ((t : ℝ) : Real.Angle)).1 i := by
+    intro i t
+    rw [hD i t, hZ i t, hW i t]
+    ring
+  obtain ⟨A, hA, hAm⟩ := intervalStieltjes_inner_fin_two D T hTc
+  have hAmeas : Measurable A.toFun := A.boundedVariation.measurable
+  -- the offset is the tangent projection of the difference
+  have hoffset : ∀ t : Set.Icc a b, mamikonOffset K z (t : ℝ) = A.toFun t := by
+    intro t
+    rw [hA t]
+    simp only [mamikonOffset, dite_eq_left t.property, Subtype.coe_eta, hDval, hT,
+      PiLp.inner_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, PiLp.sub_apply]
+    ring
+  have hoffsetval : ∀ t : Set.Icc a b, mamikonOffset K z (t : ℝ) =
+      inner ℝ (z.val t - (edgeVertices K ((t : ℝ) : Real.Angle)).1)
+        (tangentVector ((t : ℝ) : Real.Angle)) := fun t ↦ by
+    simp only [mamikonOffset, dite_eq_left t.property, Subtype.coe_eta]
+  have hzt : ∀ t : Set.Icc a b,
+      inner ℝ (z.val t) (normalVector ((t : ℝ) : Real.Angle)) =
+        supportValue K ((t : ℝ) : Real.Angle) := fun t ↦ hz t
+  -- measurability and boundedness of the offset
+  have hmeas : Measurable (mamikonOffset K z) := by
+    refine measurable_of_restrict_of_restrict_compl (s := Set.Icc a b) measurableSet_Icc ?_ ?_
+    · change Measurable fun t : Set.Icc a b ↦ mamikonOffset K z (t : ℝ)
+      rw [funext hoffset]
+      exact hAmeas
+    · change Measurable fun t : ((Set.Icc a b)ᶜ : Set ℝ) ↦ mamikonOffset K z (t : ℝ)
+      rw [show (fun t : ((Set.Icc a b)ᶜ : Set ℝ) ↦ mamikonOffset K z (t : ℝ)) = fun _ ↦ 0 from
+        funext fun t ↦ dite_eq_right t.property]
+      exact measurable_const
+  obtain ⟨CA, hCA⟩ := A.exists_norm_bound hab.le
+  have hbound : ∀ t : ℝ, |mamikonOffset K z t| ≤ max CA 0 := by
+    intro t
+    by_cases ht : t ∈ Set.Icc a b
+    · have h := hCA ⟨t, ht⟩
+      rw [Real.norm_eq_abs] at h
+      rw [hoffset ⟨t, ht⟩]
+      exact h.trans (le_max_left _ _)
+    · simp only [mamikonOffset, dite_eq_right ht, abs_zero]
+      exact le_max_right _ _
+  have hrecon : ∀ t : Set.Icc a b, z.val t = (edgeVertices K ((t : ℝ) : Real.Angle)).1 +
+      mamikonOffset K z (t : ℝ) • tangentVector ((t : ℝ) : Real.Angle) := by
+    intro t
+    have hperp : inner ℝ (z.val t - (edgeVertices K ((t : ℝ) : Real.Angle)).1)
+        (normalVector ((t : ℝ) : Real.Angle)) = 0 := by
+      rw [inner_sub_left, hzt t, (edgeVertices_fst_mem K ((t : ℝ) : Real.Angle)).2, sub_self]
+    have hdec := inner_normalVector_smul_add_inner_tangentVector_smul
+      (z.val t - (edgeVertices K ((t : ℝ) : Real.Angle)).1) ((t : ℝ) : Real.Angle)
+    rw [hperp, zero_smul, zero_add] at hdec
+    rw [hoffsetval t, hdec, add_sub_cancel]
+  refine ⟨hmeas, ⟨max CA 0, hbound⟩, hrecon, ?_⟩
+  have hCA0 : 0 ≤ CA := le_trans (norm_nonneg _) (hCA a')
+  have himg : (fun t : Set.Icc a b ↦ ((t : ℝ) : Real.Angle)) '' S =
+      (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b := by
+    ext u
+    constructor
+    · rintro ⟨t, ht, rfl⟩
+      exact ⟨(t : ℝ), ⟨ht.1, ht.2⟩, rfl⟩
+    · rintro ⟨s, hs, rfl⟩
+      exact ⟨⟨s, hs.1.le, hs.2.le⟩, ⟨hs.1, hs.2⟩, rfl⟩
+  have hUb : ∀ (i : Fin 2) (t : Set.Icc a b), ‖(U i).toFun t‖ ≤ 1 := by
+    intro i t
+    simp only [hU]
+    fin_cases i
+    · simpa [normalVector, frame] using Real.abs_cos_le_one (t : ℝ)
+    · simpa [normalVector, frame] using Real.abs_sin_le_one (t : ℝ)
+  have hAUb : ∀ (i : Fin 2) (t : Set.Icc a b), ‖A.toFun t * (U i).toFun t‖ ≤ CA := by
+    intro i t
+    rw [norm_mul]
+    calc ‖A.toFun t‖ * ‖(U i).toFun t‖ ≤ CA * 1 :=
+          mul_le_mul (hCA t) (hUb i t) (norm_nonneg _) hCA0
+      _ = CA := mul_one CA
+  have hperpframe : ∀ u : Real.Angle,
+      normalVector u 0 * tangentVector u 0 + normalVector u 1 * tangentVector u 1 = 0 := by
+    intro u
+    simp only [normalVector, tangentVector, frame, Matrix.cons_val_zero, Matrix.cons_val_one]
+    ring
+  have hTU0 : ∀ t : Set.Icc a b, (T 0).toFun t = -(U 1).toFun t := by
+    intro t
+    rw [hT, hU]
+    simp [normalVector, tangentVector, frame]
+  have hTU1 : ∀ t : Set.Icc a b, (T 1).toFun t = (U 0).toFun t := by
+    intro t
+    rw [hT, hU]
+    simp [normalVector, tangentVector, frame]
+  have hDA : ∀ (i : Fin 2) (t : Set.Icc a b), (D i).toFun t = A.toFun t * (T i).toFun t := by
+    intro i t
+    have h := congrArg (fun p : Point ↦ p i) (hrecon t)
+    simp only [PiLp.add_apply, PiLp.smul_apply, smul_eq_mul] at h
+    rw [hDval i t, hT i t, ← hoffset t]
+    linarith [h]
+  have hDU0 : ∀ t : Set.Icc a b,
+      (Z 0).toFun t - (W 0).toFun t = -(A.toFun t * (U 1).toFun t) := by
+    intro t
+    have h := hDA 0 t
+    rw [hD 0 t, hTU0 t] at h
+    rw [show (Z 0).toFun t - (W 0).toFun t = 1 * (Z 0).toFun t + -1 * (W 0).toFun t by ring, h]
+    ring
+  have hDU1 : ∀ t : Set.Icc a b,
+      (Z 1).toFun t - (W 1).toFun t = A.toFun t * (U 0).toFun t := by
+    intro t
+    have h := hDA 1 t
+    rw [hD 1 t, hTU1 t] at h
+    rw [show (Z 1).toFun t - (W 1).toFun t = 1 * (Z 1).toFun t + -1 * (W 1).toFun t by ring, h]
+  -- integrability of interval-BV integrands
+  have hFint : ∀ F G : RightContinuousIntervalBV a b,
+      ((intervalStieltjesMeasure G).restrict S).Integrable F.toFun := by
+    intro F G
+    have _ : IsFiniteMeasure (intervalStieltjesMeasure G).variation :=
+      BoundedVariationOn.instIsFiniteMeasureVariationVectorMeasure G.boundedVariation
+    exact (show (intervalStieltjesMeasure G).Integrable F.toFun from
+      F.boundedVariation.integrable).integrableOn
+  have hsubint : ∀ F G H : RightContinuousIntervalBV a b,
+      intervalStieltjesIntegral H (fun t ↦ F.toFun t - G.toFun t) S =
+        intervalStieltjesIntegral H F.toFun S - intervalStieltjesIntegral H G.toFun S :=
+    fun F G H ↦ VectorMeasure.integral_fun_sub (hFint F H) (hFint G H)
+  -- the curve area functional over the open interval
+  have hJ : 2 * curveAreaFunctional z =
+      intervalStieltjesIntegral (Z 1) (Z 0).toFun S -
+        intervalStieltjesIntegral (Z 0) (Z 1).toFun S := by
+    rw [curveAreaFunctional,
+      show (fun t : Set.Icc a b ↦ z.val t 0) = (Z 0).toFun from rfl,
+      show (fun t : Set.Icc a b ↦ z.val t 1) = (Z 1).toFun from rfl,
+      show continuousBVCoordinate z 1 = Z 1 from rfl,
+      show continuousBVCoordinate z 0 = Z 0 from rfl,
+      intervalStieltjesIntegral_univ_eq_Ioo_of_continuous hab.le (Z 1) (hZc 1) _ (hZc 0),
+      intervalStieltjesIntegral_univ_eq_Ioo_of_continuous hab.le (Z 0) (hZc 0) _ (hZc 1)]
+    rw [← hS]
+    ring
+  have hArc := mamikon_arc_identity K hab hba hturn z S hSmeas hSa W Z hWm hZ hzt himg
+  have hEndpoint := mamikon_endpoint_identity K hab z W Z hW hZ hZc
+  have hframe := mamikon_frame_identity K hab hturn z S hSmeas hSa W Z U T D
+    hW hWm hZ hU hUd hT hUc hDval A hA hAmeas hzt CA hCA hCA0 hAUb hperpframe
+  -- the offset difference of the two coordinate integrals
+  have hGval : (intervalStieltjesIntegral (Z 1) (Z 0).toFun S -
+        intervalStieltjesIntegral (Z 1) (W 0).toFun S) -
+      (intervalStieltjesIntegral (Z 0) (Z 1).toFun S -
+        intervalStieltjesIntegral (Z 0) (W 1).toFun S) =
+      -(intervalStieltjesIntegral (Z 0) (fun t ↦ A.toFun t * (U 0).toFun t) S +
+        intervalStieltjesIntegral (Z 1) (fun t ↦ A.toFun t * (U 1).toFun t) S) := by
+    rw [← hsubint (Z 0) (W 0) (Z 1), ← hsubint (Z 1) (W 1) (Z 0),
+      show intervalStieltjesIntegral (Z 1) (fun t ↦ (Z 0).toFun t - (W 0).toFun t) S =
+        intervalStieltjesIntegral (Z 1) (fun t ↦ -(A.toFun t * (U 1).toFun t)) S from
+      VectorMeasure.setIntegral_congr_fun fun t _ ↦ hDU0 t,
+      show intervalStieltjesIntegral (Z 0) (fun t ↦ (Z 1).toFun t - (W 1).toFun t) S =
+        intervalStieltjesIntegral (Z 0) (fun t ↦ A.toFun t * (U 0).toFun t) S from
+      VectorMeasure.setIntegral_congr_fun fun t _ ↦ hDU1 t,
+      show intervalStieltjesIntegral (Z 1) (fun t ↦ -(A.toFun t * (U 1).toFun t)) S =
+        -intervalStieltjesIntegral (Z 1) (fun t ↦ A.toFun t * (U 1).toFun t) S from
+      VectorMeasure.integral_fun_neg _ _ _]
+    ring
+  -- the open-interval subtype integral is the closed-interval Lebesgue integral
+  have hLeb : ∫ t in S, (A.toFun t) ^ 2 ∂volume.comap (Subtype.val : Set.Icc a b → ℝ) =
+      ∫ t in Set.Icc a b, (mamikonOffset K z t) ^ 2 := by
+    have hpre : {x : Set.Icc a b | (x : ℝ) ∈ Set.Ioo a b} = S := by
+      ext t
+      exact ⟨fun h ↦ ⟨h.1, h.2⟩, fun h ↦ ⟨h.1, h.2⟩⟩
+    have hsubtype := integral_subtype_preimage (μ := volume) (s := Set.Icc a b)
+      (t := Set.Ioo a b) measurableSet_Icc measurableSet_Ioo
+      (fun s : ℝ ↦ (mamikonOffset K z s) ^ 2)
+    rw [hpre, Measure.restrict_restrict_of_subset Set.Ioo_subset_Icc_self] at hsubtype
+    rw [show ∫ t in S, (A.toFun t) ^ 2 ∂volume.comap (Subtype.val : Set.Icc a b → ℝ) =
+        ∫ t in S, (mamikonOffset K z (t : ℝ)) ^ 2
+          ∂volume.comap (Subtype.val : Set.Icc a b → ℝ) from
+      setIntegral_congr_fun hSmeas fun t _ ↦ by rw [hoffset t], hsubtype]
+    exact setIntegral_congr_set Ioo_ae_eq_Icc
+  -- assemble
+  unfold mamikonFunctional
+  rw [← ha', ← hb']
+  linarith [hJ, hArc, hEndpoint, hframe, hGval, hLeb]
+
+theorem mamikon_quadratic_convex (a b : ℝ) (hab : a < b) (hba : b < a + Real.pi)
+    (F : ConvexBody Point → ContinuousBVPaths a b)
+    (hF : ∀ K : ConvexBody Point, ∀ t : Set.Icc a b,
+      (F K).val t ∈ (supportingLineHalfPlane K (t.val : Real.Angle)).1)
+    (hlinear : IsConvexLinear convexBodyCombination bvPathCombination F) :
+    IsQuadraticFunctional convexBodyCombination
+        (fun K ↦ mamikonFunctional K a b hab hba (F K) (hF K)) ∧
+      IsConvexFunctional convexBodyCombination
+        (fun K ↦ mamikonFunctional K a b hab hba (F K) (hF K)) false := by
+  have hdata := fun K : ConvexBody Point ↦ mamikon_integral K a b hab hba (F K) (hF K)
+  refine sqIntegral_quadratic_convex convexBodyCombination (volume.restrict (Set.Icc a b))
+    (fun K ↦ mamikonOffset K (F K)) (fun K ↦ (hdata K).1) (fun K ↦ (hdata K).2.1) ?_ _
+    (fun K ↦ (hdata K).2.2.2)
+  intro s K L t
+  rw [show F (convexBodyCombination s K L) = bvPathCombination s (F K) (F L) from hlinear s K L]
+  exact mamikonOffset_convexBodyCombination s K L (F K) (F L) t
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Area.Mamikon.Tails`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Area / Mamikon / Tails
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The signed area between a convex boundary arc and its endpoint tangent segments. -/
+def tangentMamikonValue (K : ConvexBody Point) (a b : ℝ) : ℝ :=
+  segmentArea (edgeVertices K (a : Real.Angle)).1
+      (supportingIntersection K (a : Real.Angle) (b : Real.Angle)) +
+    segmentArea (supportingIntersection K (a : Real.Angle) (b : Real.Angle))
+      (edgeVertices K (b : Real.Angle)).2 - convexArcArea K a b
+
+/-- The Mamikon value of the right tail on its Gerver angle interval. -/
+def rightTailMamikon (B : ConvexBody Point) : ℝ :=
+  tangentMamikonValue B (Real.pi + paperGerverConstants.2.1) (3 * Real.pi / 2)
+
+/-- The Mamikon value of the left tail on its Gerver angle interval. -/
+def leftTailMamikon (D : ConvexBody Point) : ℝ :=
+  tangentMamikonValue D (3 * Real.pi / 2) (3 * Real.pi / 2 + paperGerverConstants.2.2)
+
+/-- Bundle the right and left tail Mamikon functionals. -/
+def tailMamikonFunctionals : (ConvexBody Point → ℝ) × (ConvexBody Point → ℝ) :=
+  (rightTailMamikon, leftTailMamikon)
+
+/-- The support and endpoint identities of a cap-tail triple, in segment-area form. -/
+theorem tailFanPoint_identities
+    (K : RightAngleCapSpace) (B D : ConvexBody Point) {r l : ℝ}
+    (hr : r ∈ Set.Ioo 0 (Real.pi / 2)) (hl : l ∈ Set.Ioo 0 (Real.pi / 2))
+    (hB1 : supportValue (B : Set Point) ((Real.pi + r : ℝ) : Real.Angle) =
+      1 - supportValue (K.1 : Set Point) ((r : ℝ) : Real.Angle))
+    (hB2 : supportValue (B : Set Point) ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0)
+    (hD1 : supportValue (D : Set Point) ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0)
+    (hD2 : supportValue (D : Set Point) ((3 * Real.pi / 2 + l : ℝ) : Real.Angle) =
+      1 - supportValue (K.1 : Set Point) ((Real.pi / 2 + l : ℝ) : Real.Angle)) :
+    supportingIntersection B ((Real.pi + r : ℝ) : Real.Angle)
+        ((3 * Real.pi / 2 : ℝ) : Real.Angle) = (wedgeEndpoints K r).1 ∧
+    supportingIntersection D ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+        ((3 * Real.pi / 2 + l : ℝ) : Real.Angle) = (wedgeEndpoints K l).2 ∧
+    segmentArea (wedgeEndpoints K r).1
+        (edgeVertices B ((3 * Real.pi / 2 : ℝ) : Real.Angle)).2 = 0 ∧
+    segmentArea (edgeVertices D ((3 * Real.pi / 2 : ℝ) : Real.Angle)).1
+        (wedgeEndpoints K l).2 = 0 ∧
+    segmentArea (capInnerCorner K r) (wedgeEndpoints K r).1 -
+        segmentArea (edgeVertices B ((Real.pi + r : ℝ) : Real.Angle)).1
+          (wedgeEndpoints K r).1 =
+      segmentArea (capInnerCorner K r)
+        (edgeVertices B ((Real.pi + r : ℝ) : Real.Angle)).1 ∧
+    segmentArea (wedgeEndpoints K l).2 (capInnerCorner K l) -
+        segmentArea (wedgeEndpoints K l).2
+          (edgeVertices D ((3 * Real.pi / 2 + l : ℝ) : Real.Angle)).2 =
+      segmentArea (edgeVertices D ((3 * Real.pi / 2 + l : ℝ) : Real.Angle)).2
+        (capInnerCorner K l) := by
+  obtain ⟨hr0, hr2⟩ := hr
+  obtain ⟨hl0, hl2⟩ := hl
+  have hpi := Real.pi_pos
+  have hcosr : 0 < Real.cos r := Real.cos_pos_of_mem_Ioo ⟨by linarith, hr2⟩
+  have hsinl : 0 < Real.sin l := Real.sin_pos_of_pos_of_lt_pi hl0 (by linarith)
+  have hcast : ∀ a b : ℝ, a = b → ((a : ℝ) : Real.Angle) = ((b : ℝ) : Real.Angle) :=
+    fun a b h => by rw [h]
+  -- opposite normals
+  have hnr : normalVector ((Real.pi + r : ℝ) : Real.Angle) =
+      -normalVector ((r : ℝ) : Real.Angle) := by
+    rw [hcast _ _ (show Real.pi + r = r + Real.pi by ring), normalVector_add_pi]
+  have hn32 : normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle) =
+      -normalVector ((Real.pi / 2 : ℝ) : Real.Angle) := by
+    rw [hcast _ _ (show 3 * Real.pi / 2 = Real.pi / 2 + Real.pi by ring), normalVector_add_pi]
+  have hn32l : normalVector ((3 * Real.pi / 2 + l : ℝ) : Real.Angle) =
+      -normalVector ((Real.pi / 2 + l : ℝ) : Real.Angle) := by
+    rw [hcast _ _ (show 3 * Real.pi / 2 + l = (Real.pi / 2 + l) + Real.pi by ring),
+      normalVector_add_pi]
+  have hnl : normalVector ((Real.pi / 2 + l : ℝ) : Real.Angle) =
+      tangentVector ((l : ℝ) : Real.Angle) := by
+    rw [hcast _ _ (show Real.pi / 2 + l = l + Real.pi / 2 by ring),
+      normalVector_add_pi_div_two_real]
+  have hKl : supportValue (K.1 : Set Point) ((l + Real.pi / 2 : ℝ) : Real.Angle) =
+      supportValue (K.1 : Set Point) ((Real.pi / 2 + l : ℝ) : Real.Angle) := by
+    rw [hcast _ _ (show l + Real.pi / 2 = Real.pi / 2 + l by ring)]
+  -- the two fan points
+  have h1 : inner ℝ (wedgeEndpoints K r).1 (normalVector ((r : ℝ) : Real.Angle)) =
+      supportValue (K.1 : Set Point) ((r : ℝ) : Real.Angle) - 1 := by
+    change inner ℝ (((supportValue (K.1 : Set Point) ((r : ℝ) : Real.Angle) - 1) /
+      Real.cos r) • normalVector 0) _ = _
+    rw [real_inner_smul_left, ← Real.Angle.coe_zero, inner_normalVector_normalVector,
+      zero_sub, Real.cos_neg]
+    field_simp
+  have h2 : inner ℝ (wedgeEndpoints K r).1
+      (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := by
+    change inner ℝ (((supportValue (K.1 : Set Point) ((r : ℝ) : Real.Angle) - 1) /
+      Real.cos r) • normalVector 0) _ = _
+    rw [real_inner_smul_left, ← Real.Angle.coe_zero, inner_normalVector_normalVector,
+      zero_sub, Real.cos_neg, Real.cos_pi_div_two, mul_zero]
+  have h5 : inner ℝ (wedgeEndpoints K l).2
+      (normalVector ((Real.pi / 2 + l : ℝ) : Real.Angle)) =
+      supportValue (K.1 : Set Point) ((Real.pi / 2 + l : ℝ) : Real.Angle) - 1 := by
+    change inner ℝ (((supportValue (K.1 : Set Point) ((l + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+      Real.cos (Real.pi / 2 - l)) • tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) _ = _
+    rw [real_inner_smul_left, inner_tangentVector_normalVector_real,
+      Real.cos_pi_div_two_sub, hKl, show Real.pi / 2 + l - Real.pi / 2 = l by ring]
+    field_simp
+  have h6 : inner ℝ (wedgeEndpoints K l).2
+      (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := by
+    change inner ℝ (((supportValue (K.1 : Set Point) ((l + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+      Real.cos (Real.pi / 2 - l)) • tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) _ = _
+    rw [real_inner_smul_left, inner_tangentVector_normalVector_real, sub_self,
+      Real.sin_zero, mul_zero]
+  -- the two supporting intersections
+  have hs1 : Real.sin (3 * Real.pi / 2 - (Real.pi + r)) ≠ 0 := by
+    rw [show 3 * Real.pi / 2 - (Real.pi + r) = Real.pi / 2 - r by ring, Real.sin_pi_div_two_sub]
+    exact ne_of_gt hcosr
+  have hs2 : Real.sin (3 * Real.pi / 2 + l - 3 * Real.pi / 2) ≠ 0 := by
+    rw [show 3 * Real.pi / 2 + l - 3 * Real.pi / 2 = l by ring]
+    exact ne_of_gt hsinl
+  have h3 : inner ℝ (supportingIntersection B ((Real.pi + r : ℝ) : Real.Angle)
+      ((3 * Real.pi / 2 : ℝ) : Real.Angle)) (normalVector ((r : ℝ) : Real.Angle)) =
+      supportValue (K.1 : Set Point) ((r : ℝ) : Real.Angle) - 1 := by
+    have h := supportingIntersection_inner_left B (Real.pi + r) (3 * Real.pi / 2)
+    rw [hnr, inner_neg_right, hB1] at h
+    linarith
+  have h4 : inner ℝ (supportingIntersection B ((Real.pi + r : ℝ) : Real.Angle)
+      ((3 * Real.pi / 2 : ℝ) : Real.Angle))
+      (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := by
+    have h := supportingIntersection_inner_right B (Real.pi + r) (3 * Real.pi / 2) hs1
+    rw [hn32, inner_neg_right, hB2] at h
+    linarith
+  have h7 : inner ℝ (supportingIntersection D ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+      ((3 * Real.pi / 2 + l : ℝ) : Real.Angle))
+      (normalVector ((Real.pi / 2 + l : ℝ) : Real.Angle)) =
+      supportValue (K.1 : Set Point) ((Real.pi / 2 + l : ℝ) : Real.Angle) - 1 := by
+    have h := supportingIntersection_inner_right D (3 * Real.pi / 2)
+      (3 * Real.pi / 2 + l) hs2
+    rw [hn32l, inner_neg_right, hD2] at h
+    linarith
+  have h8 : inner ℝ (supportingIntersection D ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+      ((3 * Real.pi / 2 + l : ℝ) : Real.Angle))
+      (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := by
+    have h := supportingIntersection_inner_left D (3 * Real.pi / 2) (3 * Real.pi / 2 + l)
+    rw [hn32, inner_neg_right, hD1] at h
+    linarith
+  -- the cap corners and the tail vertices
+  have h9 : inner ℝ (capInnerCorner K r) (normalVector ((r : ℝ) : Real.Angle)) =
+      supportValue (K.1 : Set Point) ((r : ℝ) : Real.Angle) - 1 := (inner_capInnerCorner K r).1
+  have h13 : inner ℝ (capInnerCorner K l)
+      (normalVector ((Real.pi / 2 + l : ℝ) : Real.Angle)) =
+      supportValue (K.1 : Set Point) ((Real.pi / 2 + l : ℝ) : Real.Angle) - 1 := by
+    rw [hnl, (inner_capInnerCorner K l).2, hKl]
+  have h10 : inner ℝ (edgeVertices B ((Real.pi + r : ℝ) : Real.Angle)).1
+      (normalVector ((r : ℝ) : Real.Angle)) =
+      supportValue (K.1 : Set Point) ((r : ℝ) : Real.Angle) - 1 := by
+    have h : inner ℝ (edgeVertices B ((Real.pi + r : ℝ) : Real.Angle)).1
+        (normalVector ((Real.pi + r : ℝ) : Real.Angle)) =
+        supportValue (B : Set Point) ((Real.pi + r : ℝ) : Real.Angle) :=
+      (edgeVertices_fst_mem B ((Real.pi + r : ℝ) : Real.Angle)).2
+    rw [hnr, inner_neg_right, hB1] at h
+    linarith
+  have h11 : inner ℝ (edgeVertices B ((3 * Real.pi / 2 : ℝ) : Real.Angle)).2
+      (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := by
+    have h : inner ℝ (edgeVertices B ((3 * Real.pi / 2 : ℝ) : Real.Angle)).2
+        (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) =
+        supportValue (B : Set Point) ((3 * Real.pi / 2 : ℝ) : Real.Angle) :=
+      (edgeVertices_snd_mem B ((3 * Real.pi / 2 : ℝ) : Real.Angle)).2
+    rw [hn32, inner_neg_right, hB2] at h
+    linarith
+  have h12 : inner ℝ (edgeVertices D ((3 * Real.pi / 2 : ℝ) : Real.Angle)).1
+      (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := by
+    have h : inner ℝ (edgeVertices D ((3 * Real.pi / 2 : ℝ) : Real.Angle)).1
+        (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) =
+        supportValue (D : Set Point) ((3 * Real.pi / 2 : ℝ) : Real.Angle) :=
+      (edgeVertices_fst_mem D ((3 * Real.pi / 2 : ℝ) : Real.Angle)).2
+    rw [hn32, inner_neg_right, hD1] at h
+    linarith
+  have h14 : inner ℝ (edgeVertices D ((3 * Real.pi / 2 + l : ℝ) : Real.Angle)).2
+      (normalVector ((Real.pi / 2 + l : ℝ) : Real.Angle)) =
+      supportValue (K.1 : Set Point) ((Real.pi / 2 + l : ℝ) : Real.Angle) - 1 := by
+    have h : inner ℝ (edgeVertices D ((3 * Real.pi / 2 + l : ℝ) : Real.Angle)).2
+        (normalVector ((3 * Real.pi / 2 + l : ℝ) : Real.Angle)) =
+        supportValue (D : Set Point) ((3 * Real.pi / 2 + l : ℝ) : Real.Angle) :=
+      (edgeVertices_snd_mem D ((3 * Real.pi / 2 + l : ℝ) : Real.Angle)).2
+    rw [hn32l, inner_neg_right, hD2] at h
+    linarith
+  refine ⟨?_, ?_, segmentArea_eq_zero_of_inner_normalVector_eq_zero h2 h11,
+    segmentArea_eq_zero_of_inner_normalVector_eq_zero h12 h6,
+    segmentArea_sub_segmentArea_of_inner_normalVector_eq h9 h10 h1, ?_⟩
+  · refine eq_of_inner_normalVector_eq (s := r) (t := Real.pi / 2) ?_
+      (h3.trans h1.symm) (h4.trans h2.symm)
+    rw [show r - Real.pi / 2 = -(Real.pi / 2 - r) by ring, Real.sin_neg,
+      Real.sin_pi_div_two_sub]
+    exact neg_ne_zero.mpr (ne_of_gt hcosr)
+  · refine eq_of_inner_normalVector_eq (s := Real.pi / 2 + l) (t := Real.pi / 2) ?_
+      (h7.trans h5.symm) (h8.trans h6.symm)
+    rw [show Real.pi / 2 + l - Real.pi / 2 = l by ring]
+    exact ne_of_gt hsinl
+  · have h := segmentArea_sub_segmentArea_of_inner_normalVector_eq h13 h14 h5
+    linarith [segmentArea_swap (wedgeEndpoints K l).2 (capInnerCorner K l),
+      segmentArea_swap (wedgeEndpoints K l).2
+        (edgeVertices D ((3 * Real.pi / 2 + l : ℝ) : Real.Angle)).2,
+      segmentArea_swap (capInnerCorner K l)
+        (edgeVertices D ((3 * Real.pi / 2 + l : ℝ) : Real.Angle)).2]
+
+theorem upperBoundQ_decomposition (X : CapTailSpace) :
+    upperBoundQ X = upperBoundMiddle X.cap - rightTailMamikon X.rightBody -
+      leftTailMamikon X.leftBody := by
+  obtain ⟨hr, hl, -⟩ := paperGerverConstants_snd_mem_Ioo
+  have hcast : ∀ a b : ℝ, a = b → ((a : ℝ) : Real.Angle) = ((b : ℝ) : Real.Angle) :=
+    fun a b h => by rw [h]
+  have htop : supportValue (X.cap.1.1 : Set Point) ((Real.pi / 2 : ℝ) : Real.Angle) = 1 :=
+    X.cap.1.property.2.2.2.1
+  have hB1 : supportValue (X.rightBody : Set Point)
+      ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle) =
+      1 - supportValue (X.cap.1.1 : Set Point)
+        ((paperGerverConstants.2.1 : ℝ) : Real.Angle) := by
+    have h := X.right_eq paperGerverConstants.2.1 (by simp)
+    linarith
+  have hB2 : supportValue (X.rightBody : Set Point)
+      ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := by
+    have h := X.right_eq (Real.pi / 2) (by simp)
+    rw [hcast _ _ (show Real.pi + Real.pi / 2 = 3 * Real.pi / 2 by ring), htop] at h
+    linarith
+  have hD1 : supportValue (X.leftBody : Set Point)
+      ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := by
+    have h := X.left_eq 0 (by simp)
+    rw [hcast _ _ (show Real.pi / 2 + (0 : ℝ) = Real.pi / 2 by ring),
+      hcast _ _ (show 3 * Real.pi / 2 + (0 : ℝ) = 3 * Real.pi / 2 by ring), htop] at h
+    linarith
+  have hD2 : supportValue (X.leftBody : Set Point)
+      ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle) =
+      1 - supportValue (X.cap.1.1 : Set Point)
+        ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle) := by
+    have h := X.left_eq paperGerverConstants.2.2 (by simp)
+    linarith
+  obtain ⟨e1, e2, e3, e4, e5, e6⟩ :=
+    tailFanPoint_identities X.cap.val X.rightBody X.leftBody hr hl hB1 hB2 hD1 hD2
+  simp only [upperBoundQ, upperBoundMiddle, rightTailMamikon, leftTailMamikon,
+    tangentMamikonValue, rightLeftTailArcs, distinguishedCapSides]
+  rw [e1, e2]
+  linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Cap.PolylineBoundary`.
+* `Cap.PolylineDefinition`.
+* `Cap.Polyline`.
+* `Cap.UpperBoundary.Polygon`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Polyline Boundary
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The slope and intercept of the inner B-wall line at angle `t`. -/
+def capBAffine {Θ : AngleSet} (K : PolygonCapSpace Θ) (t : ℝ) : ℝ × ℝ :=
+  (-Real.cos t / Real.sin t,
+    (supportValue K.val.val (t : Real.Angle) - 1) / Real.sin t)
+
+/-- The slope and intercept of the inner D-wall line at angle `t`. -/
+def capDAffine {Θ : AngleSet} (K : PolygonCapSpace Θ) (t : ℝ) : ℝ × ℝ :=
+  (Real.sin t / Real.cos t,
+    (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) / Real.cos t)
+
+/-- The slope and zero intercept of the slanted fan boundary. -/
+def fanAffine (ω : ℝ) : ℝ × ℝ :=
+  (-Real.cos ω / Real.sin ω, 0)
+
+private def capAffinePieces {Θ : AngleSet} (K : PolygonCapSpace Θ) : Finset (ℝ × ℝ) :=
+  {(0, 0), fanAffine Θ.angle} ∪
+    Θ.directions.biUnion fun t ↦ {capBAffine K t, capDAffine K t}
+
+/-- The lower boundary height of a polygon cap after removing its niche. -/
+def capBoundaryHeight {Θ : AngleSet} (K : PolygonCapSpace Θ) (x : ℝ) : ℝ :=
+  max 0 (max (affineValue (fanAffine Θ.angle) x)
+    (Θ.directions.sup' Θ.nonempty fun t ↦
+      min (affineValue (capBAffine K t) x) (affineValue (capDAffine K t) x)))
+
+private theorem continuous_capBoundaryHeight {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    Continuous (capBoundaryHeight K) := by
+  unfold capBoundaryHeight
+  apply Continuous.max continuous_const
+  apply Continuous.max (continuous_affineValue _)
+  apply Continuous.finset_sup'_apply Θ.nonempty
+  intro t ht
+  exact (continuous_affineValue _).min (continuous_affineValue _)
+
+private theorem capBoundaryHeight_eq_affineValue {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (x : ℝ) :
+    ∃ c ∈ capAffinePieces K, capBoundaryHeight K x = affineValue c x := by
+  obtain hzero | hrest := max_choice 0
+    (max (affineValue (fanAffine Θ.angle) x)
+      (Θ.directions.sup' Θ.nonempty fun t ↦
+        min (affineValue (capBAffine K t) x) (affineValue (capDAffine K t) x)))
+  · refine ⟨(0, 0), by simp [capAffinePieces], ?_⟩
+    simpa [capBoundaryHeight, affineValue] using hzero
+  · obtain hfan | hsup := max_choice (affineValue (fanAffine Θ.angle) x)
+      (Θ.directions.sup' Θ.nonempty fun t ↦
+        min (affineValue (capBAffine K t) x) (affineValue (capDAffine K t) x))
+    · refine ⟨fanAffine Θ.angle, by simp [capAffinePieces], ?_⟩
+      rw [capBoundaryHeight, hrest, hfan]
+    · obtain ⟨t, ht, htEq⟩ := Finset.exists_mem_eq_sup' Θ.nonempty
+        (fun t ↦ min (affineValue (capBAffine K t) x)
+          (affineValue (capDAffine K t) x))
+      obtain hB | hD := min_choice (affineValue (capBAffine K t) x)
+        (affineValue (capDAffine K t) x)
+      · refine ⟨capBAffine K t, ?_, ?_⟩
+        · apply Finset.mem_union.mpr
+          right
+          exact Finset.mem_biUnion.mpr ⟨t, ht, by simp⟩
+        · rw [capBoundaryHeight, hrest, hsup, htEq, hB]
+      · refine ⟨capDAffine K t, ?_, ?_⟩
+        · apply Finset.mem_union.mpr
+          right
+          exact Finset.mem_biUnion.mpr ⟨t, ht, by simp⟩
+        · rw [capBoundaryHeight, hrest, hsup, htEq, hD]
+
+private theorem capAffinePieces_has_normal {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    {c : ℝ × ℝ} (hc : c ∈ capAffinePieces K) :
+    ∃ t ∈ angleDomain Θ, Real.cos t + c.1 * Real.sin t = 0 := by
+  rw [capAffinePieces, Finset.mem_union] at hc
+  rcases hc with hc | hc
+  · simp only [Finset.mem_insert, Finset.mem_singleton] at hc
+    rcases hc with rfl | rfl
+    · refine ⟨Real.pi / 2, ?_, ?_⟩
+      · simp [angleDomain]
+      · simp
+    · refine ⟨Θ.angle, ?_, ?_⟩
+      · simp [angleDomain]
+      · simp only [fanAffine]
+        have hs : Real.sin Θ.angle ≠ 0 := (Real.sin_pos_of_pos_of_lt_pi Θ.angle_pos
+          (Θ.angle_le.trans_lt (by linarith [Real.pi_pos]))).ne'
+        field_simp
+        ring
+  · obtain ⟨t, ht, hc⟩ := Finset.mem_biUnion.mp hc
+    simp only [Finset.mem_insert, Finset.mem_singleton] at hc
+    rcases hc with rfl | rfl
+    · refine ⟨t, by simp [angleDomain, ht], ?_⟩
+      simp only [capBAffine]
+      have hs : Real.sin t ≠ 0 := (Real.sin_pos_of_pos_of_lt_pi
+        (Θ.interior t ht).1
+        ((Θ.interior t ht).2.trans_le Θ.angle_le |>.trans
+          (by linarith [Real.pi_pos]))).ne'
+      field_simp
+      ring
+    · refine ⟨t + Real.pi / 2, ?_, ?_⟩
+      · simp [angleDomain, ht]
+      · simp only [capDAffine, Real.cos_add_pi_div_two, Real.sin_add_pi_div_two]
+        have hc : Real.cos t ≠ 0 := (Real.cos_pos_of_mem_Ioo
+          ⟨by linarith [Real.pi_pos, (Θ.interior t ht).1],
+            (Θ.interior t ht).2.trans_le Θ.angle_le⟩).ne'
+        field_simp
+        ring
+
+private theorem mem_innerQuadrant_iff_lt_min_affine {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {t : ℝ} (ht : t ∈ Θ.directions) (p : Point) :
+    p ∈ innerQuadrant K.val.val t ↔
+      p 1 < min (affineValue (capBAffine K t) (p 0))
+        (affineValue (capDAffine K t) (p 0)) := by
+  have htt := Θ.interior t ht
+  have hs : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi htt.1
+    ((htt.2.trans_le Θ.angle_le).trans (by linarith [Real.pi_pos]))
+  have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, htt.1], htt.2.trans_le Θ.angle_le⟩
+  simp only [innerQuadrant, normalHalfPlane, Set.mem_inter_iff,
+    Bool.false_eq_true, ite_false, ite_true]
+  simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply,
+    RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue, Matrix.cons_val_zero,
+    Matrix.cons_val_one, Matrix.cons_val_fin_one, Set.mem_ofPred_eq, Real.cos_add,
+    Real.cos_pi_div_two, mul_zero, Real.sin_pi_div_two, mul_one, zero_sub, Real.sin_add,
+    zero_add, neg_mul, neg_add_lt_iff_lt_add, lt_min_iff]
+  change
+    (Real.cos t * p 0 + Real.sin t * p 1 <
+        supportValue K.val.val (t : Real.Angle) - 1 ∧
+      Real.cos t * p 1 < Real.sin t * p 0 +
+        (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1)) ↔ _
+  simp only [affineValue, capBAffine, capDAffine]
+  have heqB : -Real.cos t / Real.sin t * p 0 +
+      (supportValue K.val.val (t : Real.Angle) - 1) / Real.sin t =
+      (supportValue K.val.val (t : Real.Angle) - 1 - Real.cos t * p 0) /
+        Real.sin t := by
+    field_simp
+    ring
+  have heqD : Real.sin t / Real.cos t * p 0 +
+      (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+        Real.cos t =
+      (Real.sin t * p 0 +
+        (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1)) /
+        Real.cos t := by
+    field_simp
+  rw [heqB, heqD]
+  constructor
+  · rintro ⟨hB, hD⟩
+    constructor
+    · apply (lt_div_iff₀ hs).2
+      nlinarith
+    · apply (lt_div_iff₀ hc).2
+      nlinarith
+  · rintro ⟨hB, hD⟩
+    constructor
+    · apply (lt_div_iff₀ hs).mp at hB
+      nlinarith
+    · apply (lt_div_iff₀ hc).mp at hD
+      nlinarith
+
+private theorem mem_capFan_iff_max_affine_le {Θ : AngleSet}
+    (p : Point) :
+    p ∈ capFan Θ.angle ↔
+      max 0 (affineValue (fanAffine Θ.angle) (p 0)) ≤ p 1 := by
+  have hs : 0 < Real.sin Θ.angle := Real.sin_pos_of_pos_of_lt_pi Θ.angle_pos
+    (Θ.angle_le.trans_lt (by linarith [Real.pi_pos]))
+  simp only [capFan, normalHalfPlane, ↓reduceIte, Bool.false_eq_true, normalVector, frame,
+    Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply, RCLike.inner_apply, conj_trivial,
+    Fin.sum_univ_two, Fin.isValue, Matrix.cons_val_zero, Matrix.cons_val_one,
+    Matrix.cons_val_fin_one, Real.cos_pi_div_two, Real.sin_pi_div_two, zero_mul, one_mul,
+    zero_add, Set.mem_inter_iff, Set.mem_ofPred_eq, affineValue, fanAffine, add_zero,
+    sup_le_iff]
+  have heq : -Real.cos Θ.angle / Real.sin Θ.angle * p 0 =
+      (-Real.cos Θ.angle * p 0) / Real.sin Θ.angle := by ring
+  rw [heq]
+  constructor
+  · rintro ⟨hfan, hy⟩
+    exact ⟨hy, (div_le_iff₀ hs).2 (by nlinarith)⟩
+  · rintro ⟨hy, hfan⟩
+    apply (div_le_iff₀ hs).mp at hfan
+    exact ⟨by nlinarith, hy⟩
+
+/-- The fan outside the polygon niche is the epigraph of its boundary height. -/
+theorem capFan_sdiff_polygonNiche_eq_epigraph {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) :
+    capFan Θ.angle \ polygonNiche Θ K.val =
+      {p | capBoundaryHeight K (p 0) ≤ p 1} := by
+  ext p
+  constructor
+  · rintro ⟨hfan, hniche⟩
+    have hnotUnion : p ∉ ⋃ t ∈ Θ.directions, innerQuadrant K.val.val t := by
+      intro hp
+      exact hniche ⟨hfan, hp⟩
+    have hfan' := (mem_capFan_iff_max_affine_le (Θ := Θ) p).mp hfan
+    change capBoundaryHeight K (p 0) ≤ p 1
+    unfold capBoundaryHeight
+    apply max_le
+    · exact (le_max_left _ _).trans hfan'
+    · apply max_le
+      · exact (le_max_right _ _).trans hfan'
+      · apply (Finset.sup'_le_iff Θ.nonempty _).2
+        intro t ht
+        apply le_of_not_gt
+        intro hlt
+        apply hnotUnion
+        exact Set.mem_iUnion₂.mpr
+          ⟨t, ht, (mem_innerQuadrant_iff_lt_min_affine K ht p).mpr hlt⟩
+  · intro hp
+    change capBoundaryHeight K (p 0) ≤ p 1 at hp
+    unfold capBoundaryHeight at hp
+    have hzero : 0 ≤ p 1 := by
+      exact (le_max_left 0 _).trans hp
+    have hfanLine : affineValue (fanAffine Θ.angle) (p 0) ≤ p 1 := by
+      exact (le_max_left _ _).trans <|
+        (le_max_right 0 _).trans hp
+    have hsup : (Θ.directions.sup' Θ.nonempty fun t ↦
+        min (affineValue (capBAffine K t) (p 0))
+          (affineValue (capDAffine K t) (p 0))) ≤ p 1 := by
+      exact (le_max_right _ _).trans <|
+        (le_max_right 0 _).trans hp
+    have hfan : p ∈ capFan Θ.angle :=
+      (mem_capFan_iff_max_affine_le (Θ := Θ) p).mpr (max_le hzero hfanLine)
+    refine ⟨hfan, ?_⟩
+    rintro ⟨-, hunion⟩
+    obtain ⟨t, ht, hquad⟩ := Set.mem_iUnion₂.mp hunion
+    have hlt := (mem_innerQuadrant_iff_lt_min_affine K ht p).mp hquad
+    have hle := ((Finset.sup'_le_iff Θ.nonempty _).1 hsup) t ht
+    exact (not_lt_of_ge hle) hlt
+
+/-- The cap outside its polygon niche is closed, with frontier the boundary graph. -/
+theorem capFan_sdiff_polygonNiche_closed_frontier {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) :
+    IsClosed (capFan Θ.angle \ polygonNiche Θ K.val) ∧
+    frontier (capFan Θ.angle \ polygonNiche Θ K.val) =
+      Set.range (pointOnGraph (capBoundaryHeight K)) := by
+  rw [capFan_sdiff_polygonNiche_eq_epigraph K]
+  exact ⟨isClosed_verticalEpigraph (continuous_capBoundaryHeight K),
+    frontier_verticalEpigraph (continuous_capBoundaryHeight K)⟩
+
+/-- Every compact interval of the polygon-cap boundary graph is a finite polyline. -/
+theorem exists_capBoundary_graphPolyline {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {a b : ℝ} (hab : a < b) :
+    ∃ p : XMonotonePolylineData,
+      0 < p.edges ∧
+      p.vertices 0 = pointOnGraph (capBoundaryHeight K) a ∧
+      p.vertices (Fin.last p.edges) = pointOnGraph (capBoundaryHeight K) b ∧
+      p.carrier = pointOnGraph (capBoundaryHeight K) '' Set.Icc a b ∧
+      (∀ i : Fin p.edges, ∃ t ∈ angleDomain Θ,
+        inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+          (normalVector (t : Real.Angle)) = 0) := by
+  obtain ⟨p, hp, hp0, hp1, hpcarrier, hgraph, hpieces⟩ :=
+    exists_graphPolyline_of_finite_affine_selector hab (capAffinePieces K)
+      (continuous_capBoundaryHeight K).continuousOn
+      (fun x _ ↦ capBoundaryHeight_eq_affineValue K x)
+  refine ⟨p, hp, hp0, hp1, hpcarrier, ?_⟩
+  intro i
+  obtain ⟨c, hcL, hc⟩ := hpieces i
+  obtain ⟨t, ht, horth⟩ := capAffinePieces_has_normal K hcL
+  refine ⟨t, ht, ?_⟩
+  have hab' : p.vertices i.castSucc 0 ≤ p.vertices i.succ 0 :=
+    (p.increasing i.castSucc_lt_succ).le
+  rw [hgraph i.succ, hgraph i.castSucc]
+  change Set.EqOn (capBoundaryHeight K) (fun x ↦ c.1 * x + c.2)
+    (Set.Icc (p.vertices i.castSucc 0) (p.vertices i.succ 0)) at hc
+  exact inner_pointOnGraph_sub_normalVector_eq_zero
+    (m := c.1) (c := c.2) (t := t)
+    hc hab' horth
+
+/-- The lower endpoint of the zero-angle cap face lies on the horizontal axis. -/
+theorem capVertices_zero_snd_eq {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    (capVertices K.val 0).1.2 =
+      supportValue K.val.val (0 : Real.Angle) • normalVector (0 : Real.Angle) := by
+  let A := supportValue K.val.val (0 : Real.Angle) • normalVector (0 : Real.Angle)
+  have hAK : A ∈ (K.val.val : Set Point) :=
+    supportValue_zero_smul_normalVector_mem K.val
+  have hAedge : A ∈ exposedEdge K.val.val (0 : Real.Angle) := by
+    refine ⟨hAK, ?_⟩
+    change inner ℝ A (normalVector (0 : Real.Angle)) =
+      supportValue K.val.val (0 : Real.Angle)
+    dsimp [A]
+    simp [normalVector, frame, PiLp.inner_apply]
+  change (edgeVertices K.val.val (0 : Real.Angle)).2 = A
+  apply edgeVertices_snd_eq_of_tangent_isLeast K.val.val _ hAedge
+  intro q hq
+  have hqFan := K.val.subset_capFan hq.1
+  have hqy : 0 ≤ q 1 := by
+    have h := hqFan.2
+    change 0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at h
+    simpa [normalVector, frame, PiLp.inner_apply] using h
+  simp only [normalVector, frame, Real.Angle.cos_zero, Real.Angle.sin_zero, neg_zero,
+    tangentVector, PiLp.inner_apply, PiLp.smul_apply, smul_eq_mul, RCLike.inner_apply,
+    conj_trivial, Fin.sum_univ_two, Fin.isValue, Matrix.cons_val_zero, mul_one, zero_mul,
+    Matrix.cons_val_one, Matrix.cons_val_fin_one, mul_zero, add_zero, one_mul, zero_add,
+    ge_iff_le, A]
+  exact hqy
+
+/-- The terminal positive cap contact lies on the lower fan ray. -/
+theorem capVertices_angle_fst_eq {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    (capVertices K.val Θ.angle).2.1 =
+      supportValue K.val.val ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) •
+        tangentVector (Θ.angle : Real.Angle) := by
+  let L := supportValue K.val.val
+    ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)
+  let C := L • tangentVector (Θ.angle : Real.Angle)
+  have hCK : C ∈ (K.val.val : Set Point) := by
+    by_cases hω : Θ.angle = Real.pi / 2
+    · have hmem := supportValue_pi_smul_normalVector_mem_of_eq K.val hω
+      simpa [C, L, hω, tangentVector, normalVector, frame, PiLp.inner_apply] using hmem
+    · exact supportValue_add_pi_div_two_smul_tangentVector_mem_of_lt K.val
+        (Θ.angle_le.lt_of_ne hω)
+  have hCedge : C ∈ exposedEdge K.val.val
+      ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) := by
+    refine ⟨hCK, ?_⟩
+    change inner ℝ C
+      (normalVector ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)) = L
+    rw [show ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) =
+        (Θ.angle : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) by rfl,
+      normalVector_add_pi_div_two, real_inner_smul_left,
+      inner_tangentVector_self]
+    ring
+  change (edgeVertices K.val.val
+    ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)).1 = C
+  apply edgeVertices_fst_eq_of_tangent_isGreatest K.val.val _ hCedge
+  intro q hq
+  have hqFan := K.val.subset_capFan hq.1
+  have hqu : 0 ≤ inner ℝ q (normalVector (Θ.angle : Real.Angle)) := hqFan.1
+  rw [tangentVector_add_pi_div_two]
+  simp only [inner_neg_right]
+  have hCzero : inner ℝ C (normalVector (Θ.angle : Real.Angle)) = 0 := by
+    dsimp [C]
+    rw [real_inner_smul_left, real_inner_comm,
+      inner_normalVector_tangentVector]
+    simp
+  rw [hCzero]
+  simpa using neg_nonpos.mpr hqu
+
+/-- The terminal cap contacts occur in strictly increasing horizontal order. -/
+theorem polygonCap_left_x_lt_right_x {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    ((capVertices K.val Θ.angle).2.1) 0 < ((capVertices K.val 0).1.2) 0 := by
+  let C := (capVertices K.val Θ.angle).2.1
+  let A := (capVertices K.val 0).1.2
+  have hq_le_A : ∀ q ∈ (K.val.val : Set Point), q 0 ≤ A 0 := by
+    intro q hq
+    have hq' := inner_le_supportValue K.val.val hq (0 : Real.Angle)
+    have hA := (edgeVertices_snd_mem K.val.val (0 : Real.Angle)).2
+    change inner ℝ A (normalVector (0 : Real.Angle)) =
+      supportValue K.val.val (0 : Real.Angle) at hA
+    calc
+      q 0 = inner ℝ q (normalVector (0 : Real.Angle)) := by
+        simp [normalVector, frame, PiLp.inner_apply]
+      _ ≤ supportValue K.val.val (0 : Real.Angle) := hq'
+      _ = inner ℝ A (normalVector (0 : Real.Angle)) := hA.symm
+      _ = A 0 := by simp [normalVector, frame, PiLp.inner_apply]
+  by_cases hω : Θ.angle = Real.pi / 2
+  · have hC_le_q : ∀ q ∈ (K.val.val : Set Point), C 0 ≤ q 0 := by
+      intro q hq
+      have hq' := inner_le_supportValue K.val.val hq (Real.pi : Real.Angle)
+      have hC := (edgeVertices_fst_mem K.val.val
+        ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)).2
+      change inner ℝ C
+        (normalVector ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)) =
+          supportValue K.val.val ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) at hC
+      have hang : ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) =
+          (Real.pi : Real.Angle) := by
+        congr 1
+        rw [hω]
+        ring
+      rw [hang] at hC
+      have hqcoord : -q 0 ≤ supportValue K.val.val (Real.pi : Real.Angle) := by
+        simpa [normalVector, frame, PiLp.inner_apply] using hq'
+      have hCcoord : -C 0 = supportValue K.val.val (Real.pi : Real.Angle) := by
+        simpa [normalVector, frame, PiLp.inner_apply] using hC
+      linarith
+    by_contra hnot
+    have hAC : A 0 = C 0 := by
+      apply le_antisymm (le_of_not_gt hnot)
+      exact hC_le_q A (edgeVertices_snd_mem K.val.val (0 : Real.Angle)).1
+    have hx : ∀ q ∈ (K.val.val : Set Point), q 0 = A 0 := by
+      intro q hq
+      exact le_antisymm (hq_le_A q hq) (hAC.trans_le (hC_le_q q hq))
+    obtain ⟨u, huK, hu⟩ := exists_mem_inner_eq_supportValue K.val.val
+      ((Real.pi / 2 : ℝ) : Real.Angle)
+    obtain ⟨l, hlK, hl⟩ := exists_mem_inner_eq_supportValue K.val.val
+      ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+    have huy : u 1 = 1 := by
+      rw [K.val.property.2.2.2.1] at hu
+      simpa [normalVector, frame, PiLp.inner_apply] using hu
+    have hly : l 1 = 0 := by
+      rw [K.val.property.2.2.2.2.2.1, inner_normalVector_three_pi_div_two] at hl
+      linarith
+    let m : Point := (2 : ℝ)⁻¹ • (u + l)
+    let N : Set Real.Angle :=
+      ((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain Θ) ∪ capLowerNormals Θ.angle
+    let U : Set Point := ⋂ t ∈ N,
+      {p | inner ℝ p (normalVector t) < supportValue K.val.val t}
+    have hdomain_finite : (angleDomain Θ).Finite := by
+      unfold angleDomain
+      exact (Θ.directions.finite_toSet.union
+        (Θ.directions.finite_toSet.image (fun t ↦ t + Real.pi / 2))).union
+          (Set.toFinite _)
+    have hlower_finite : (capLowerNormals Θ.angle).Finite := by
+      simp [capLowerNormals]
+    have hNfinite : N.Finite := by
+      dsimp [N]
+      exact (hdomain_finite.image fun t : ℝ ↦ (t : Real.Angle)).union hlower_finite
+    have hUopen : IsOpen U := by
+      dsimp [U]
+      apply hNfinite.isOpen_biInter
+      intro t ht
+      exact isOpen_lt (by fun_prop) continuous_const
+    have hmU : m ∈ U := by
+      dsimp [U]
+      simp only [Set.mem_iInter]
+      intro t htN
+      rcases htN with ⟨r, hr, rfl⟩ | htLower
+      · have hrI : r ∈ Set.Ioo 0 Real.pi := by
+          rcases hr with (hr | ⟨s, hs, rfl⟩) | hr
+          · exact ⟨(Θ.interior r hr).1,
+              by linarith [(Θ.interior r hr).2, Θ.angle_le, Real.pi_pos]⟩
+          · exact ⟨by linarith [(Θ.interior s hs).1, Real.pi_pos],
+              by linarith [(Θ.interior s hs).2, hω, Real.pi_pos]⟩
+          · rcases hr with rfl | hr
+            · rw [hω]
+              constructor <;> linarith [Real.pi_pos]
+            · have hr : r = Real.pi / 2 := hr
+              rw [hr]
+              constructor <;> linarith [Real.pi_pos]
+        have hsin : 0 < Real.sin r := Real.sin_pos_of_pos_of_lt_pi hrI.1 hrI.2
+        have htop : inner ℝ m (normalVector (r : Real.Angle)) <
+            inner ℝ u (normalVector (r : Real.Angle)) := by
+          have hux : u 0 = l 0 := (hx u huK).trans (hx l hlK).symm
+          simp only [smul_add, normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe,
+            PiLp.inner_apply, PiLp.add_apply, PiLp.smul_apply, smul_eq_mul, RCLike.inner_apply,
+            conj_trivial, Fin.sum_univ_two, Fin.isValue, Matrix.cons_val_zero,
+            Matrix.cons_val_one, Matrix.cons_val_fin_one, huy, mul_one, hly, mul_zero, add_zero,
+            gt_iff_lt, m]
+          rw [hux]
+          norm_num
+          nlinarith
+        exact htop.trans_le (inner_le_supportValue K.val.val huK _)
+      · simp only [capLowerNormals, Set.mem_insert_iff, Set.mem_singleton_iff] at htLower
+        rcases htLower with rfl | rfl
+        · rw [K.val.property.2.2.2.2.1]
+          norm_num [m, hω, normalVector, frame, PiLp.inner_apply, huy, hly]
+        · rw [K.val.property.2.2.2.2.2.1]
+          change inner ℝ m
+            (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) < 0
+          rw [inner_normalVector_three_pi_div_two]
+          simp [m, huy, hly]
+    obtain ⟨ε, hε, hball⟩ := (Metric.isOpen_iff.mp hUopen) m hmU
+    let q : Point := m + (ε / 2) • normalVector (0 : Real.Angle)
+    have hqm : q ∈ Metric.ball m ε := by
+      rw [Metric.mem_ball, dist_eq_norm]
+      simp only [q, add_sub_cancel_left, norm_smul]
+      rw [Real.norm_eq_abs, abs_of_pos (half_pos hε)]
+      have hn : ‖normalVector (0 : Real.Angle)‖ = 1 := by
+        simpa only [Real.Angle.coe_zero] using norm_normalVector_real 0
+      rw [hn, mul_one]
+      linarith
+    have hqU : q ∈ U := hball hqm
+    have hqK : q ∈ (K.val.val : Set Point) := by
+      rw [K.property.eq_iInter_supportValue]
+      simp only [Set.mem_iInter]
+      intro t ht
+      have hqt := Set.mem_iInter.mp (Set.mem_iInter.mp hqU t) ht
+      change inner ℝ q (normalVector t) < supportValue K.val.val t at hqt
+      exact hqt.le
+    have hq0 : q 0 = A 0 + ε / 2 := by
+      simp [q, m, normalVector, frame, hx u huK, hx l hlK]
+      ring
+    have := hq_le_A q hqK
+    rw [hq0] at this
+    linarith
+  · have hωlt : Θ.angle < Real.pi / 2 := Θ.angle_le.lt_of_ne hω
+    rw [capVertices_zero_snd_eq K, capVertices_angle_fst_eq K]
+    have hA0 : 0 < supportValue K.val.val (0 : Real.Angle) := by
+      obtain ⟨p, hpK, hp⟩ := exists_mem_inner_eq_supportValue K.val.val
+        (Θ.angle : Real.Angle)
+      have hpy : p 1 ≤ 1 := by
+        have hpy' := inner_le_supportValue K.val.val hpK
+          ((Real.pi / 2 : ℝ) : Real.Angle)
+        rw [K.val.property.2.2.2.1] at hpy'
+        simpa [normalVector, frame, PiLp.inner_apply] using hpy'
+      rw [K.val.property.2.2.1] at hp
+      have hcos : 0 < Real.cos Θ.angle := Real.cos_pos_of_mem_Ioo
+        ⟨by linarith [Θ.angle_pos, Real.pi_pos], hωlt⟩
+      have hsin_lt : Real.sin Θ.angle < 1 := by
+        nlinarith only [Real.sin_sq_add_cos_sq Θ.angle, sq_pos_of_pos hcos]
+      have hpx : 0 < p 0 := by
+        simp [normalVector, frame, PiLp.inner_apply] at hp
+        have hsin : 0 ≤ Real.sin Θ.angle :=
+          (Real.sin_pos_of_pos_of_lt_pi Θ.angle_pos
+            (Θ.angle_le.trans_lt (by linarith [Real.pi_pos]))).le
+        have hmul := mul_le_mul_of_nonneg_left hpy hsin
+        nlinarith
+      exact hpx.trans_le (by
+        have := inner_le_supportValue K.val.val hpK (0 : Real.Angle)
+        simpa [normalVector, frame, PiLp.inner_apply] using this)
+    have hL0 : 0 ≤ supportValue K.val.val
+        ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) :=
+      supportValue_nonneg_of_mem_capUpperAngles K.val hωlt
+        (Or.inr ⟨by linarith [Θ.angle_pos, Real.pi_pos], le_rfl⟩)
+    have hs : 0 < Real.sin Θ.angle := Real.sin_pos_of_pos_of_lt_pi Θ.angle_pos
+      (Θ.angle_le.trans_lt (by linarith [Real.pi_pos]))
+    have hprod : 0 ≤ supportValue K.val.val
+        ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) * Real.sin Θ.angle :=
+      mul_nonneg hL0 hs.le
+    have hprod' : 0 ≤ supportValue K.val.val
+        ((Θ.angle : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle)) *
+          Real.sin Θ.angle := by
+      simpa only [Real.Angle.coe_add] using hprod
+    simp [normalVector, tangentVector, frame]
+    nlinarith
+
+private theorem capBoundaryHeight_eq_zero_of_right {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {x : ℝ}
+    (hx : ((capVertices K.val 0).1.2) 0 ≤ x) :
+    capBoundaryHeight K x = 0 := by
+  let A := (capVertices K.val 0).1.2
+  have hAeq := capVertices_zero_snd_eq K
+  have hsinω : 0 < Real.sin Θ.angle := Real.sin_pos_of_pos_of_lt_pi Θ.angle_pos
+    (Θ.angle_le.trans_lt (by linarith [Real.pi_pos]))
+  have hcosω : 0 ≤ Real.cos Θ.angle := Real.cos_nonneg_of_mem_Icc
+    ⟨by linarith [Θ.angle_pos, Real.pi_pos], Θ.angle_le⟩
+  have hAfan := K.val.subset_capFan (edgeVertices_snd_mem K.val.val
+    (0 : Real.Angle)).1
+  have hfanA : affineValue (fanAffine Θ.angle) (A 0) ≤ 0 := by
+    change A = supportValue K.val.val (0 : Real.Angle) •
+      normalVector (0 : Real.Angle) at hAeq
+    change A ∈ capFan Θ.angle at hAfan
+    have h := hAfan.1
+    change 0 ≤ inner ℝ A (normalVector (Θ.angle : Real.Angle)) at h
+    rw [hAeq] at h
+    simp only [normalVector, frame, Real.Angle.cos_zero, Real.Angle.sin_zero, neg_zero,
+      Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply, PiLp.smul_apply,
+      smul_eq_mul, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue,
+      Matrix.cons_val_zero, mul_one, Matrix.cons_val_one, Matrix.cons_val_fin_one,
+      mul_zero, add_zero] at h
+    have hAcoord : A 0 = supportValue K.val.val (0 : Real.Angle) := by
+      rw [hAeq]
+      simp [normalVector, frame]
+    rw [hAcoord]
+    simp only [affineValue, fanAffine, add_zero]
+    have heq : -Real.cos Θ.angle / Real.sin Θ.angle *
+        supportValue K.val.val (0 : Real.Angle) =
+        (-Real.cos Θ.angle * supportValue K.val.val (0 : Real.Angle)) /
+          Real.sin Θ.angle := by ring
+    rw [heq]
+    apply div_nonpos_of_nonpos_of_nonneg
+    · nlinarith [h]
+    · exact hsinω.le
+  have hfan : affineValue (fanAffine Θ.angle) x ≤ 0 := by
+    simp only [affineValue, fanAffine, add_zero] at hfanA ⊢
+    have hslope : -Real.cos Θ.angle / Real.sin Θ.angle ≤ 0 :=
+      div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hcosω) hsinω.le
+    exact (mul_le_mul_of_nonpos_left hx hslope).trans hfanA
+  have hpieces : (Θ.directions.sup' Θ.nonempty fun t ↦
+      min (affineValue (capBAffine K t) x)
+        (affineValue (capDAffine K t) x)) ≤ 0 := by
+    apply (Finset.sup'_le_iff Θ.nonempty _).2
+    intro t ht
+    apply (min_le_left _ _).trans
+    have htt := Θ.interior t ht
+    have hsint : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi htt.1
+      ((htt.2.trans_le Θ.angle_le).trans (by linarith [Real.pi_pos]))
+    have hcost : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [htt.1, Real.pi_pos], htt.2.trans_le Θ.angle_le⟩
+    have hgapBounds := wedgeGaps_positive_lower_bound K.val t htt
+    have hgap : 0 < (wedgeGaps K.val t).1 :=
+      hgapBounds.2.1.trans_le hgapBounds.1
+    rw [wedgeGaps_fst_eq_supportValue] at hgap
+    have hAcoord : A 0 = supportValue K.val.val (0 : Real.Angle) := by
+      change A = supportValue K.val.val (0 : Real.Angle) •
+        normalVector (0 : Real.Angle) at hAeq
+      rw [hAeq]
+      simp [normalVector, frame]
+    have hnum : supportValue K.val.val (t : Real.Angle) - 1 - Real.cos t * x < 0 := by
+      have hdiv := (sub_pos.mp hgap)
+      have hmul := (div_lt_iff₀ hcost).mp hdiv
+      rw [hAcoord] at hx
+      nlinarith
+    simp only [affineValue, capBAffine]
+    have heq : -Real.cos t / Real.sin t * x +
+        (supportValue K.val.val (t : Real.Angle) - 1) / Real.sin t =
+        (supportValue K.val.val (t : Real.Angle) - 1 - Real.cos t * x) /
+          Real.sin t := by
+      field_simp
+      ring
+    rw [heq]
+    exact div_nonpos_of_nonpos_of_nonneg hnum.le hsint.le
+  unfold capBoundaryHeight
+  rw [max_eq_left (max_le hfan hpieces)]
+
+private theorem capBoundaryHeight_eq_fanAffine_of_left {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {x : ℝ}
+    (hx : x ≤ ((capVertices K.val Θ.angle).2.1) 0) :
+    capBoundaryHeight K x = affineValue (fanAffine Θ.angle) x := by
+  let C := (capVertices K.val Θ.angle).2.1
+  let L := supportValue K.val.val
+    ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)
+  have hCeq := capVertices_angle_fst_eq K
+  have hsinω : 0 < Real.sin Θ.angle := Real.sin_pos_of_pos_of_lt_pi Θ.angle_pos
+    (Θ.angle_le.trans_lt (by linarith [Real.pi_pos]))
+  have hcosω : 0 ≤ Real.cos Θ.angle := Real.cos_nonneg_of_mem_Icc
+    ⟨by linarith [Θ.angle_pos, Real.pi_pos], Θ.angle_le⟩
+  have hfan_nonneg : 0 ≤ affineValue (fanAffine Θ.angle) x := by
+    by_cases hω : Θ.angle = Real.pi / 2
+    · simp [affineValue, fanAffine, hω]
+    · have hωlt : Θ.angle < Real.pi / 2 := Θ.angle_le.lt_of_ne hω
+      have hL : 0 ≤ L := supportValue_nonneg_of_mem_capUpperAngles K.val hωlt
+        (Or.inr ⟨by linarith [Θ.angle_pos, Real.pi_pos], le_rfl⟩)
+      have hCx : C 0 = -L * Real.sin Θ.angle := by
+        change C = L • tangentVector (Θ.angle : Real.Angle) at hCeq
+        rw [hCeq]
+        simp [tangentVector, frame]
+      have hx0 : x ≤ 0 := by
+        rw [hCx] at hx
+        nlinarith [mul_nonneg hL hsinω.le]
+      simp only [affineValue, fanAffine, add_zero]
+      exact mul_nonneg_of_nonpos_of_nonpos
+        (div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hcosω) hsinω.le) hx0
+  have hpieces : (Θ.directions.sup' Θ.nonempty fun t ↦
+      min (affineValue (capBAffine K t) x)
+        (affineValue (capDAffine K t) x)) ≤
+      affineValue (fanAffine Θ.angle) x := by
+    apply (Finset.sup'_le_iff Θ.nonempty _).2
+    intro t ht
+    apply (min_le_right _ _).trans
+    have htt := Θ.interior t ht
+    have hcost : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [htt.1, Real.pi_pos], htt.2.trans_le Θ.angle_le⟩
+    have hcosδ : 0 < Real.cos (Θ.angle - t) := Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [htt.2, Real.pi_pos], by linarith [htt.1, Θ.angle_le]⟩
+    have hgapBounds := wedgeGaps_positive_lower_bound K.val t htt
+    have hgap : 0 < (wedgeGaps K.val t).2 :=
+      hgapBounds.2.2.2.trans_le hgapBounds.2.2.1
+    rw [wedgeGaps_snd_eq_supportValue] at hgap
+    have hCx : C 0 = -L * Real.sin Θ.angle := by
+      change C = L • tangentVector (Θ.angle : Real.Angle) at hCeq
+      rw [hCeq]
+      simp [tangentVector, frame]
+    have hsupport :
+        supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 <
+          L * Real.cos (Θ.angle - t) := by
+      have hdiv := sub_pos.mp hgap
+      exact (div_lt_iff₀ hcosδ).mp hdiv
+    have hnum : Real.cos (Θ.angle - t) / Real.sin Θ.angle * x +
+        supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 < 0 := by
+      have hcoef : 0 < Real.cos (Θ.angle - t) / Real.sin Θ.angle :=
+        div_pos hcosδ hsinω
+      have hxC : x ≤ C 0 := by simpa [C] using hx
+      have hx' : x ≤ -L * Real.sin Θ.angle := hCx ▸ hxC
+      have hmul := mul_le_mul_of_nonneg_left hx' hcoef.le
+      have hrhs : Real.cos (Θ.angle - t) / Real.sin Θ.angle *
+          (-L * Real.sin Θ.angle) = -L * Real.cos (Θ.angle - t) := by
+        field_simp [hsinω.ne']
+      rw [hrhs] at hmul
+      nlinarith
+    have hdiff : affineValue (capDAffine K t) x -
+        affineValue (fanAffine Θ.angle) x =
+        (Real.cos (Θ.angle - t) / Real.sin Θ.angle * x +
+          supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+            Real.cos t := by
+      simp only [affineValue, capDAffine, fanAffine]
+      rw [Real.cos_sub]
+      field_simp [hcost.ne', hsinω.ne']
+      ring
+    rw [sub_eq_iff_eq_add] at hdiff
+    rw [hdiff]
+    have hdiv := div_nonpos_of_nonpos_of_nonneg hnum.le hcost.le
+    linarith
+  unfold capBoundaryHeight
+  rw [max_eq_left hpieces, max_eq_right hfan_nonneg]
+
+/-- The right cap contact lies on the boundary graph. -/
+theorem pointOnGraph_capBoundaryHeight_rightEndpoint {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) :
+    pointOnGraph (capBoundaryHeight K) (((capVertices K.val 0).1.2) 0) =
+      (capVertices K.val 0).1.2 := by
+  have hAeq := capVertices_zero_snd_eq K
+  have hh := capBoundaryHeight_eq_zero_of_right K
+    (x := ((capVertices K.val 0).1.2) 0) le_rfl
+  ext i
+  fin_cases i
+  · rfl
+  · change capBoundaryHeight K (((capVertices K.val 0).1.2) 0) =
+      ((capVertices K.val 0).1.2) 1
+    rw [hh, hAeq]
+    simp [normalVector, frame]
+
+/-- The left cap contact lies on the boundary graph. -/
+theorem pointOnGraph_capBoundaryHeight_leftEndpoint {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) :
+    pointOnGraph (capBoundaryHeight K) (((capVertices K.val Θ.angle).2.1) 0) =
+      (capVertices K.val Θ.angle).2.1 := by
+  let C := (capVertices K.val Θ.angle).2.1
+  let L := supportValue K.val.val
+    ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)
+  have hCeq := capVertices_angle_fst_eq K
+  have hh := capBoundaryHeight_eq_fanAffine_of_left K
+    (x := ((capVertices K.val Θ.angle).2.1) 0) le_rfl
+  change C = L • tangentVector (Θ.angle : Real.Angle) at hCeq
+  ext i
+  fin_cases i
+  · rfl
+  · change capBoundaryHeight K (C 0) = C 1
+    change capBoundaryHeight K (((capVertices K.val Θ.angle).2.1) 0) = C 1
+    rw [hh]
+    have hCxy : C 0 = -L * Real.sin Θ.angle ∧
+        C 1 = L * Real.cos Θ.angle := by
+      rw [hCeq]
+      simp [tangentVector, frame]
+    rw [hCxy.1, hCxy.2]
+    simp [affineValue, fanAffine]
+    field_simp [(Real.sin_pos_of_pos_of_lt_pi Θ.angle_pos
+      (Θ.angle_le.trans_lt (by linarith [Real.pi_pos]))).ne']
+
+/-- The open ray to the right of a polygon-cap boundary is the right graph tail. -/
+theorem capRightOpenRay_eq_graph_image {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) :
+    {q | ∃ s : ℝ, 0 < s ∧
+      q = (capVertices K.val 0).1.2 + s • normalVector 0} =
+      pointOnGraph (capBoundaryHeight K) ''
+        Set.Ioi (((capVertices K.val 0).1.2) 0) := by
+  let A := (capVertices K.val 0).1.2
+  have hAeq := capVertices_zero_snd_eq K
+  have hAy : A 1 = 0 := by
+    change A = supportValue K.val.val (0 : Real.Angle) •
+      normalVector (0 : Real.Angle) at hAeq
+    rw [hAeq]
+    simp [normalVector, frame]
+  ext q
+  constructor
+  · rintro ⟨s, hs, rfl⟩
+    refine ⟨A 0 + s, ?_, ?_⟩
+    · change ((capVertices K.val 0).1.2) 0 < A 0 + s
+      simpa [A] using (show A 0 < A 0 + s by linarith)
+    · have hh := capBoundaryHeight_eq_zero_of_right K
+        (x := A 0 + s) (by simpa [A] using (show A 0 ≤ A 0 + s by linarith))
+      ext i
+      fin_cases i <;> simp [pointOnGraph, hh, A, hAy, normalVector, frame]
+  · rintro ⟨x, hx, rfl⟩
+    refine ⟨x - A 0, ?_, ?_⟩
+    · exact sub_pos.mpr (by simpa [A] using hx)
+    · have hh := capBoundaryHeight_eq_zero_of_right K
+        (x := x) (by simpa [A] using hx.le)
+      ext i
+      fin_cases i <;> simp [pointOnGraph, hh, A, hAy, normalVector, frame]
+
+/-- The open ray to the left of a polygon-cap boundary is the left graph tail. -/
+theorem capLeftOpenRay_eq_graph_image {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) :
+    {q | ∃ s : ℝ, 0 < s ∧ q = (capVertices K.val Θ.angle).2.1 +
+      s • tangentVector (Θ.angle : Real.Angle)} =
+      pointOnGraph (capBoundaryHeight K) ''
+        Set.Iio (((capVertices K.val Θ.angle).2.1) 0) := by
+  let C := (capVertices K.val Θ.angle).2.1
+  let L := supportValue K.val.val
+    ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)
+  have hCeq := capVertices_angle_fst_eq K
+  have hs : 0 < Real.sin Θ.angle := Real.sin_pos_of_pos_of_lt_pi Θ.angle_pos
+    (Θ.angle_le.trans_lt (by linarith [Real.pi_pos]))
+  have hCxy : C 0 = -L * Real.sin Θ.angle ∧ C 1 = L * Real.cos Θ.angle := by
+    change C = L • tangentVector (Θ.angle : Real.Angle) at hCeq
+    rw [hCeq]
+    simp [tangentVector, frame]
+  ext q
+  constructor
+  · rintro ⟨s, hspos, rfl⟩
+    let x := C 0 - s * Real.sin Θ.angle
+    have hx : x < C 0 := by dsimp [x]; nlinarith
+    refine ⟨x, by simpa [C] using hx, ?_⟩
+    have hh := capBoundaryHeight_eq_fanAffine_of_left K
+      (x := x) (by simpa [C] using hx.le)
+    ext i
+    fin_cases i
+    · change x = C 0 + s * (-Real.sin Θ.angle)
+      dsimp [x]
+      ring
+    · change capBoundaryHeight K x = C 1 + s * Real.cos Θ.angle
+      rw [hh, hCxy.2]
+      simp only [affineValue, fanAffine, add_zero]
+      dsimp [x]
+      rw [hCxy.1]
+      field_simp [hs.ne']
+      ring
+  · rintro ⟨x, hx, rfl⟩
+    let s := (C 0 - x) / Real.sin Θ.angle
+    have hspos : 0 < s := div_pos (sub_pos.mpr (by simpa [C] using hx)) hs
+    refine ⟨s, hspos, ?_⟩
+    have hh := capBoundaryHeight_eq_fanAffine_of_left K
+      (x := x) (by simpa [C] using hx.le)
+    ext i
+    fin_cases i
+    · change x = C 0 + s * (-Real.sin Θ.angle)
+      dsimp [s]
+      field_simp [hs.ne']
+      ring
+    · change capBoundaryHeight K x = C 1 + s * Real.cos Θ.angle
+      rw [hh, hCxy.2]
+      simp only [affineValue, fanAffine, add_zero]
+      dsimp [s]
+      rw [hCxy.1]
+      field_simp [hs.ne']
+      ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Polyline Definition
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The open ray starting at `p` in direction `v`, excluding its initial point. -/
+def openRay (p v : Point) : Set Point :=
+  {q | ∃ s : ℝ, 0 < s ∧ q = p + s • v}
+
+/-- The polyline and two disjoint endpoint rays describe the fan-minus-niche frontier. -/
+def IsCapPolyline {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (p : XMonotonePolylineData) : Prop :=
+  0 < p.edges ∧
+  p.vertices 0 = (capVertices K.val Θ.angle).2.1 ∧
+  p.vertices (Fin.last p.edges) = (capVertices K.val 0).1.2 ∧
+  (∀ i : Fin p.edges, ∃ t : angleDomain Θ,
+    inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+      (normalVector (t.val : Real.Angle)) = 0) ∧
+  IsClosed (capFan Θ.angle \ polygonNiche Θ K.val) ∧
+  frontier (capFan Θ.angle \ polygonNiche Θ K.val) =
+    openRay ((capVertices K.val Θ.angle).2.1) (tangentVector (Θ.angle : Real.Angle)) ∪
+      p.carrier ∪ openRay ((capVertices K.val 0).1.2) (normalVector 0) ∧
+  Disjoint (openRay ((capVertices K.val Θ.angle).2.1)
+    (tangentVector (Θ.angle : Real.Angle))) p.carrier ∧
+  Disjoint p.carrier (openRay ((capVertices K.val 0).1.2) (normalVector 0)) ∧
+  Disjoint (openRay ((capVertices K.val Θ.angle).2.1)
+    (tangentVector (Θ.angle : Real.Angle)))
+    (openRay ((capVertices K.val 0).1.2) (normalVector 0))
+
+theorem polygonCap_polyline {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    ∃ p : XMonotonePolylineData, IsCapPolyline K p := by
+  let C := (capVertices K.val Θ.angle).2.1
+  let A := (capVertices K.val 0).1.2
+  have hCA : C 0 < A 0 := polygonCap_left_x_lt_right_x K
+  obtain ⟨p, hpedge, hpC, hpA, hcarrier, hedge⟩ :=
+    exists_capBoundary_graphPolyline K hCA
+  refine ⟨p, hpedge, ?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
+  · rw [hpC]
+    exact pointOnGraph_capBoundaryHeight_leftEndpoint K
+  · rw [hpA]
+    exact pointOnGraph_capBoundaryHeight_rightEndpoint K
+  · intro i
+    obtain ⟨t, ht, hi⟩ := hedge i
+    exact ⟨⟨t, ht⟩, hi⟩
+  · exact (capFan_sdiff_polygonNiche_closed_frontier K).1
+  · rw [(capFan_sdiff_polygonNiche_closed_frontier K).2,
+      show openRay ((capVertices K.val Θ.angle).2.1)
+        (tangentVector (Θ.angle : Real.Angle)) = _ by
+          simpa [openRay] using capLeftOpenRay_eq_graph_image K,
+      show openRay ((capVertices K.val 0).1.2) (normalVector 0) = _ by
+          simpa [openRay] using capRightOpenRay_eq_graph_image K, hcarrier]
+    ext q
+    constructor
+    · rintro ⟨x, rfl⟩
+      rcases lt_or_ge x (C 0) with hx | hx
+      · apply Or.inl
+        apply Or.inl
+        exact ⟨x, by simpa [C] using hx, rfl⟩
+      · rcases le_or_gt x (A 0) with hxA | hxA
+        · apply Or.inl
+          apply Or.inr
+          exact ⟨x, ⟨hx, hxA⟩, rfl⟩
+        · apply Or.inr
+          exact ⟨x, by simpa [A] using hxA, rfl⟩
+    · rintro ((⟨x, hx, rfl⟩ | ⟨x, hx, rfl⟩) | ⟨x, hx, rfl⟩) <;>
+        exact ⟨x, rfl⟩
+  · rw [show openRay ((capVertices K.val Θ.angle).2.1)
+      (tangentVector (Θ.angle : Real.Angle)) = _ by
+        simpa [openRay] using capLeftOpenRay_eq_graph_image K, hcarrier]
+    apply pointOnGraph_image_disjoint
+    rw [Set.disjoint_left]
+    intro x hx hy
+    have hx' : x < C 0 := by simpa [C] using hx
+    exact (not_lt_of_ge hy.1) hx'
+  · rw [hcarrier, show openRay ((capVertices K.val 0).1.2) (normalVector 0) = _ by
+      simpa [openRay] using capRightOpenRay_eq_graph_image K]
+    apply pointOnGraph_image_disjoint
+    rw [Set.disjoint_left]
+    intro x hx hy
+    have hy' : A 0 < x := by simpa [A] using hy
+    exact (not_lt_of_ge hx.2) hy'
+  · rw [show openRay ((capVertices K.val Θ.angle).2.1)
+      (tangentVector (Θ.angle : Real.Angle)) = _ by
+        simpa [openRay] using capLeftOpenRay_eq_graph_image K,
+    show openRay ((capVertices K.val 0).1.2) (normalVector 0) = _ by
+      simpa [openRay] using capRightOpenRay_eq_graph_image K]
+    apply pointOnGraph_image_disjoint
+    rw [Set.disjoint_left]
+    intro x hx hy
+    have hx' : x < C 0 := by simpa [C] using hx
+    have hy' : A 0 < x := by simpa [A] using hy
+    linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Polyline
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A chosen monotone polyline describing the polygonal cap’s niche boundary. -/
+def polygonCapPolyline {Θ : AngleSet} (K : PolygonCapSpace Θ) : XMonotonePolylineData :=
+  (polygonCap_polyline K).choose
+
+/-- Sum the lengths of polyline edges perpendicular to a prescribed wall direction. -/
+def polygonCapPolylineLength {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (t : angleDomain Θ) : ℝ := by
+  classical
+  let p := polygonCapPolyline K
+  exact ∑ i : Fin p.edges,
+    if inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+      (normalVector (t.val : Real.Angle)) = 0 then
+      dist (p.vertices i.castSucc) (p.vertices i.succ) else 0
+
+/-- The cap’s surface measure at each wall direction equals the corresponding polyline length. -/
+def IsBalancedPolygonCap {Θ : AngleSet} (K : PolygonCapSpace Θ) : Prop :=
+  ∀ t : angleDomain Θ, surfaceAreaMeasure K.val.val {(t.val : Real.Angle)} =
+    ENNReal.ofReal (polygonCapPolylineLength K t)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Upper Boundary / Polygon
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open MeasureTheory
+
+private lemma PolygonCapSpace.eq_of_mem_of_fst_eq_of_fst_extremal {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {a b : Point}
+    (ha : a ∈ (K.val.val : Set Point)) (hb : b ∈ (K.val.val : Set Point))
+    (hab : a 0 = b 0)
+    (hext : (∀ q ∈ (K.val.val : Set Point), q 0 ≤ a 0) ∨
+      (∀ q ∈ (K.val.val : Set Point), a 0 ≤ q 0)) : a = b := by
+  apply ConvexBody.eq_of_mem_of_fst_eq_of_fst_extremal K.val.val
+    (((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain Θ) ∪ capLowerNormals Θ.angle)
+    _ K.property _ ha hb hab hext
+  · have hfinite : (angleDomain Θ).Finite := by
+      unfold angleDomain
+      exact (Θ.directions.finite_toSet.union
+        (Θ.directions.finite_toSet.image (fun t ↦ t + Real.pi / 2))).union
+          (Set.toFinite _)
+    exact (hfinite.image _).union (by simp [capLowerNormals])
+  · intro t ht
+    rcases ht with ⟨r, hr, rfl⟩ | ht
+    · exact (Real.sin_pos_of_pos_of_lt_pi
+        (angleDomain_subset_Ioo Θ hr).1 (angleDomain_subset_Ioo Θ hr).2).ne'
+    · simp only [capLowerNormals, Set.mem_insert_iff, Set.mem_singleton_iff] at ht
+      rcases ht with rfl | rfl
+      · change Real.sin (Θ.angle + Real.pi) ≠ 0
+        rw [Real.sin_add_pi]
+        exact neg_ne_zero.mpr (Real.sin_pos_of_pos_of_lt_pi Θ.angle_pos
+          (by linarith [Θ.angle_le, Real.pi_pos])).ne'
+      · change Real.sin (3 * Real.pi / 2) ≠ 0
+        norm_num [show 3 * Real.pi / 2 = Real.pi + Real.pi / 2 by ring,
+          Real.sin_add]
+
+private def upperSupportAffine (K : ConvexBody Point) (t : ℝ) : ℝ × ℝ :=
+  (-Real.cos t / Real.sin t, supportValue K (t : Real.Angle) / Real.sin t)
+
+private def upperSupportHeight (K : ConvexBody Point) (D : Finset ℝ)
+    (hne : D.Nonempty) (x : ℝ) : ℝ :=
+  D.inf' hne (fun t ↦ affineValue (upperSupportAffine K t) x)
+
+private lemma continuous_upperSupportHeight (K : ConvexBody Point) (D : Finset ℝ)
+    (hne : D.Nonempty) : Continuous (upperSupportHeight K D hne) := by
+  apply Continuous.finset_inf'_apply hne
+  intro t ht
+  exact continuous_affineValue _
+
+private lemma upperSupportHeight_eq_affineValue (K : ConvexBody Point) (D : Finset ℝ)
+    (hne : D.Nonempty) (x : ℝ) :
+    ∃ c ∈ D.image (upperSupportAffine K),
+      upperSupportHeight K D hne x = affineValue c x := by
+  obtain ⟨t, ht, heq⟩ := Finset.exists_mem_eq_inf' hne
+    (fun t ↦ affineValue (upperSupportAffine K t) x)
+  exact ⟨upperSupportAffine K t, Finset.mem_image.mpr ⟨t, ht, rfl⟩, heq⟩
+
+private lemma inner_le_support_iff_snd_le_affine (K : ConvexBody Point)
+    {t : ℝ} (ht : 0 < Real.sin t) (q : Point) :
+    inner ℝ q (normalVector (t : Real.Angle)) ≤ supportValue K (t : Real.Angle) ↔
+      q 1 ≤ affineValue (upperSupportAffine K t) (q 0) := by
+  have hformula : affineValue (upperSupportAffine K t) (q 0) =
+      (supportValue K (t : Real.Angle) - q 0 * Real.cos t) / Real.sin t := by
+    simp only [affineValue, upperSupportAffine]
+    ring
+  rw [hformula, le_div_iff₀ ht]
+  simp only [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+  simp only [Fin.isValue, Real.Angle.cos_coe, Real.Angle.sin_coe, Matrix.cons_val_zero,
+    RCLike.inner_apply, conj_trivial, Matrix.cons_val_one, Matrix.cons_val_fin_one]
+  constructor <;> intro h <;> nlinarith
+
+private lemma snd_le_upperSupportHeight_iff (K : ConvexBody Point)
+    (D : Finset ℝ) (hne : D.Nonempty) (hD : ∀ t ∈ D, 0 < Real.sin t) (q : Point) :
+    q 1 ≤ upperSupportHeight K D hne (q 0) ↔
+      ∀ t ∈ D, inner ℝ q (normalVector (t : Real.Angle)) ≤
+        supportValue K (t : Real.Angle) := by
+  rw [upperSupportHeight, Finset.le_inf'_iff]
+  exact forall_congr' fun t ↦ forall_congr' fun ht ↦
+    (inner_le_support_iff_snd_le_affine K (hD t ht) q).symm
+
+private lemma PolygonCapSpace.mem_iff_mem_capFan_and_support_le {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (p : Point) :
+    p ∈ (K.val.val : Set Point) ↔ p ∈ capFan Θ.angle ∧
+      ∀ t ∈ angleDomain Θ,
+        inner ℝ p (normalVector (t : Real.Angle)) ≤ supportValue K.val.val (t : Real.Angle) := by
+  constructor
+  · intro hp
+    exact ⟨K.val.subset_capFan hp, fun t _ ↦ inner_le_supportValue K.val.val hp _⟩
+  · rintro ⟨hp, hupper⟩
+    rw [K.property.eq_iInter_supportValue]
+    simp only [Set.mem_iInter]
+    intro a ha
+    rcases ha with ⟨t, ht, rfl⟩ | ha
+    · exact hupper t ht
+    rcases ha with rfl | rfl
+    · change inner ℝ p (normalVector ((Θ.angle + Real.pi : ℝ) : Real.Angle)) ≤ _
+      rw [K.val.property.2.2.2.2.1, normalVector_add_pi, inner_neg_right]
+      exact neg_nonpos.mpr hp.1
+    · have hang : ((3 * Real.pi / 2 : ℝ) : Real.Angle) =
+          (((Real.pi / 2 + Real.pi : ℝ) : Real.Angle)) := by congr 1; ring
+      change inner ℝ p (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) ≤ _
+      rw [K.val.property.2.2.2.2.2.1, hang, normalVector_add_pi, inner_neg_right]
+      exact neg_nonpos.mpr hp.2
+
+private lemma mem_capFan_of_fst_eq_of_snd_le {ω : ℝ} (hω : 0 ≤ Real.sin ω)
+    {p q : Point} (hp : p ∈ capFan ω) (hx : p 0 = q 0) (hy : p 1 ≤ q 1) :
+    q ∈ capFan ω := by
+  change 0 ≤ inner ℝ q (normalVector (ω : Real.Angle)) ∧
+    0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+  change 0 ≤ inner ℝ p (normalVector (ω : Real.Angle)) ∧
+    0 ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at hp
+  simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe,
+    PiLp.inner_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue,
+    Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one,
+    Real.cos_pi_div_two, Real.sin_pi_div_two, zero_mul, one_mul, zero_add] at hp ⊢
+  rw [hx] at hp
+  constructor
+  · nlinarith [hp.1, mul_nonneg hω (sub_nonneg.mpr hy)]
+  · exact hp.2.trans hy
+
+private lemma pointOnGraph_upperSupportHeight_mem {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (D : Finset ℝ) (hne : D.Nonempty) (hD : (D : Set ℝ) = angleDomain Θ)
+    (hsin : ∀ t ∈ D, 0 < Real.sin t) {x : ℝ}
+    (hx : ∃ q ∈ (K.val.val : Set Point), q 0 = x) :
+    pointOnGraph (upperSupportHeight K.val.val D hne) x ∈ (K.val.val : Set Point) := by
+  obtain ⟨q, hq, hqx⟩ := hx
+  have hqle : q 1 ≤ upperSupportHeight K.val.val D hne x := by
+    rw [← hqx, snd_le_upperSupportHeight_iff K.val.val D hne hsin]
+    exact fun t _ ↦ inner_le_supportValue K.val.val hq _
+  apply (K.mem_iff_mem_capFan_and_support_le _).mpr
+  constructor
+  · apply mem_capFan_of_fst_eq_of_snd_le
+      (Real.sin_nonneg_of_nonneg_of_le_pi Θ.angle_pos.le
+        (Θ.angle_le.trans (by linarith [Real.pi_pos]))) (K.val.subset_capFan hq)
+    · exact hqx
+    · exact hqle
+  · intro t ht
+    have htD : t ∈ D := by change t ∈ (D : Set ℝ); rwa [hD]
+    apply (inner_le_support_iff_snd_le_affine K.val.val (hsin t htD) _).mpr
+    change upperSupportHeight K.val.val D hne x ≤
+      affineValue (upperSupportAffine K.val.val t) x
+    exact Finset.inf'_le _ htD
+
+private lemma exists_upperSupportHeight_polyline {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (D : Finset ℝ) (hne : D.Nonempty) (hD : (D : Set ℝ) = angleDomain Θ)
+    (hsin : ∀ t ∈ D, 0 < Real.sin t) {a b : ℝ} (hab : a < b)
+    (hfeasible : ∀ x ∈ Set.Icc a b, ∃ q ∈ (K.val.val : Set Point), q 0 = x) :
+    ∃ p : XMonotonePolylineData,
+      0 < p.edges ∧
+      p.vertices 0 = pointOnGraph (upperSupportHeight K.val.val D hne) a ∧
+      p.vertices (Fin.last p.edges) = pointOnGraph (upperSupportHeight K.val.val D hne) b ∧
+      p.carrier = pointOnGraph (upperSupportHeight K.val.val D hne) '' Set.Icc a b ∧
+      p.carrier ⊆ (K.val.val : Set Point) ∧
+      ∀ i : Fin p.edges, ∃ t ∈ D,
+        inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+          (normalVector (t : Real.Angle)) = 0 ∧
+        inner ℝ (p.vertices i.castSucc) (normalVector (t : Real.Angle)) =
+          supportValue K.val.val (t : Real.Angle) := by
+  obtain ⟨p, hp, hpa, hpb, hcarrier, hvertices, hpieces⟩ :=
+    exists_graphPolyline_of_finite_affine_selector hab (D.image (upperSupportAffine K.val.val))
+      (continuous_upperSupportHeight K.val.val D hne).continuousOn
+      (fun x _ ↦ upperSupportHeight_eq_affineValue K.val.val D hne x)
+  refine ⟨p, hp, hpa, hpb, hcarrier, ?_, ?_⟩
+  · rw [hcarrier]
+    rintro q ⟨x, hx, rfl⟩
+    exact pointOnGraph_upperSupportHeight_mem K D hne hD hsin (hfeasible x hx)
+  · intro i
+    obtain ⟨c, hc, heq⟩ := hpieces i
+    obtain ⟨t, ht, rfl⟩ := Finset.mem_image.mp hc
+    refine ⟨t, ht, ?_, ?_⟩
+    · rw [hvertices i.succ, hvertices i.castSucc]
+      apply inner_pointOnGraph_sub_normalVector_eq_zero heq
+        (p.increasing i.castSucc_lt_succ).le
+      dsimp [upperSupportAffine]
+      field_simp [(hsin t ht).ne']
+      ring
+    · have hstart := heq ⟨le_rfl, (p.increasing i.castSucc_lt_succ).le⟩
+      rw [hvertices i.castSucc]
+      simp only [pointOnGraph, normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+      simp only [Real.Angle.cos_coe, Real.Angle.sin_coe]
+      change Real.cos t * (p.vertices i.castSucc) 0 +
+        Real.sin t * upperSupportHeight K.val.val D hne ((p.vertices i.castSucc) 0) = _
+      rw [hstart]
+      dsimp [upperSupportAffine, affineValue]
+      field_simp [(hsin t ht).ne']
+      ring
+
+private lemma exists_upperSupportHeight_polyline_with_endpoints {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (D : Finset ℝ) (hne : D.Nonempty)
+    (hD : (D : Set ℝ) = angleDomain Θ) {a b : Point}
+    (ha : a ∈ (K.val.val : Set Point)) (hb : b ∈ (K.val.val : Set Point))
+    (hab : a 0 < b 0)
+    (hmin : ∀ q ∈ (K.val.val : Set Point), a 0 ≤ q 0)
+    (hmax : ∀ q ∈ (K.val.val : Set Point), q 0 ≤ b 0) :
+    ∃ p : XMonotonePolylineData, 0 < p.edges ∧ p.vertices 0 = a ∧
+      p.vertices (Fin.last p.edges) = b ∧
+      p.carrier = pointOnGraph (upperSupportHeight K.val.val D hne) '' Set.Icc (a 0) (b 0) ∧
+      p.carrier ⊆ (K.val.val : Set Point) ∧
+      ∀ i : Fin p.edges, ∃ t ∈ D,
+        inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+          (normalVector (t : Real.Angle)) = 0 ∧
+        inner ℝ (p.vertices i.castSucc) (normalVector (t : Real.Angle)) =
+          supportValue K.val.val (t : Real.Angle) := by
+  have hsin : ∀ t ∈ D, 0 < Real.sin t := by
+    intro t ht
+    have ht' : t ∈ angleDomain Θ := hD ▸ ht
+    exact Real.sin_pos_of_pos_of_lt_pi (angleDomain_subset_Ioo Θ ht').1
+      (angleDomain_subset_Ioo Θ ht').2
+  obtain ⟨p, hp, hpa, hpb, hc, hK, hn⟩ :=
+    exists_upperSupportHeight_polyline K D hne hD hsin hab
+      (fun _ hx ↦ ConvexBody.exists_mem_fst_eq_of_mem_Icc K.val.val ha hb hab hx)
+  refine ⟨p, hp, ?_, ?_, hc, hK, hn⟩
+  · rw [hpa]
+    apply K.eq_of_mem_of_fst_eq_of_fst_extremal
+      (pointOnGraph_upperSupportHeight_mem K D hne hD hsin ⟨a, ha, rfl⟩) ha rfl
+    exact Or.inr hmin
+  · rw [hpb]
+    apply K.eq_of_mem_of_fst_eq_of_fst_extremal
+      (pointOnGraph_upperSupportHeight_mem K D hne hD hsin ⟨b, hb, rfl⟩) hb rfl
+    exact Or.inl hmax
+
+/-- The terminal cap contacts bound every horizontal coordinate of a polygon cap. -/
+lemma PolygonCapSpace.capVertices_fst_bounds {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    {q : Point} (hq : q ∈ (K.val.val : Set Point)) :
+    ((capVertices K.val Θ.angle).2.1) 0 ≤ q 0 ∧
+      q 0 ≤ ((capVertices K.val 0).1.2) 0 := by
+  constructor
+  · rw [capVertices_angle_fst_eq]
+    have hu := (K.val.subset_capFan hq).1
+    have hv := inner_le_supportValue K.val.val hq
+      ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)
+    have hs : 0 ≤ Real.sin Θ.angle := (Real.sin_pos_of_pos_of_lt_pi Θ.angle_pos
+      (by linarith [Θ.angle_le, Real.pi_pos])).le
+    have hc : 0 ≤ Real.cos Θ.angle := Real.cos_nonneg_of_mem_Icc
+      ⟨by linarith [Θ.angle_pos, Real.pi_pos], Θ.angle_le⟩
+    change 0 ≤ inner ℝ q (normalVector (Θ.angle : Real.Angle)) at hu
+    simp [normalVector, tangentVector, frame, PiLp.inner_apply,
+      Real.Angle.cos_add_pi_div_two, Real.Angle.sin_add_pi_div_two] at hu hv ⊢
+    nlinarith [congrArg (fun z : ℝ ↦ z * q 0) (Real.sin_sq_add_cos_sq Θ.angle),
+      mul_nonneg hu hc, mul_le_mul_of_nonneg_right hv hs,
+      sq_nonneg (Real.sin Θ.angle), sq_nonneg (Real.cos Θ.angle)]
+  · have hq' := inner_le_supportValue K.val.val hq (0 : Real.Angle)
+    have hA := (edgeVertices_snd_mem K.val.val (0 : Real.Angle)).2
+    change inner ℝ ((capVertices K.val 0).1.2) (normalVector (0 : Real.Angle)) =
+      supportValue K.val.val (0 : Real.Angle) at hA
+    simp [normalVector, frame, PiLp.inner_apply] at hq' hA
+    linarith
+
+private lemma PolygonCapSpace.exists_upperBoundary_polyline {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (D : Finset ℝ) (hne : D.Nonempty)
+    (hD : (D : Set ℝ) = angleDomain Θ) :
+    ∃ p : XMonotonePolylineData, 0 < p.edges ∧
+      p.vertices 0 = (capVertices K.val Θ.angle).2.1 ∧
+      p.vertices (Fin.last p.edges) = (capVertices K.val 0).1.2 ∧
+      p.carrier = pointOnGraph (upperSupportHeight K.val.val D hne) ''
+        Set.Icc (((capVertices K.val Θ.angle).2.1) 0)
+          (((capVertices K.val 0).1.2) 0) ∧
+      p.carrier ⊆ (K.val.val : Set Point) ∧
+      ∀ i : Fin p.edges, ∃ t ∈ D,
+        inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+          (normalVector (t : Real.Angle)) = 0 ∧
+        inner ℝ (p.vertices i.castSucc) (normalVector (t : Real.Angle)) =
+          supportValue K.val.val (t : Real.Angle) := by
+  apply exists_upperSupportHeight_polyline_with_endpoints K D hne hD
+    (edgeVertices_fst_mem K.val.val
+      ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)).1
+    (edgeVertices_snd_mem K.val.val (0 : Real.Angle)).1
+    (polygonCap_left_x_lt_right_x K)
+  · exact fun q hq ↦ (K.capVertices_fst_bounds hq).1
+  · exact fun q hq ↦ (K.capVertices_fst_bounds hq).2
+
+private lemma PolygonCapSpace.exposedEdge_eq_upperSupportGraph_inter {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (D : Finset ℝ) (hne : D.Nonempty)
+    (hD : (D : Set ℝ) = angleDomain Θ) {t : ℝ} (ht : t ∈ D) :
+    exposedEdge K.val.val (t : Real.Angle) =
+      (pointOnGraph (upperSupportHeight K.val.val D hne) ''
+        Set.Icc (((capVertices K.val Θ.angle).2.1) 0)
+          (((capVertices K.val 0).1.2) 0)) ∩
+      {q | inner ℝ q (normalVector (t : Real.Angle)) =
+        supportValue K.val.val (t : Real.Angle)} := by
+  have hsin : ∀ r ∈ D, 0 < Real.sin r := by
+    intro r hr
+    have hr' : r ∈ angleDomain Θ := hD ▸ hr
+    exact Real.sin_pos_of_pos_of_lt_pi (angleDomain_subset_Ioo Θ hr').1
+      (angleDomain_subset_Ioo Θ hr').2
+  ext q
+  constructor
+  · rintro ⟨hq, heq⟩
+    refine ⟨⟨q 0, K.capVertices_fst_bounds hq, ?_⟩, heq⟩
+    have hle : q 1 ≤ upperSupportHeight K.val.val D hne (q 0) :=
+      (snd_le_upperSupportHeight_iff K.val.val D hne hsin q).mpr
+        (fun r _ ↦ inner_le_supportValue K.val.val hq _)
+    have hge := Finset.inf'_le (fun r ↦ affineValue (upperSupportAffine K.val.val r) (q 0)) ht
+    have haff : affineValue (upperSupportAffine K.val.val t) (q 0) = q 1 := by
+      change inner ℝ q (normalVector (t : Real.Angle)) =
+        supportValue K.val.val (t : Real.Angle) at heq
+      simp [normalVector, frame, PiLp.inner_apply] at heq
+      dsimp [affineValue, upperSupportAffine]
+      field_simp [(hsin t ht).ne']
+      nlinarith [heq]
+    have hy : upperSupportHeight K.val.val D hne (q 0) = q 1 :=
+      le_antisymm (haff ▸ hge) hle
+    ext i
+    fin_cases i
+    · rfl
+    · exact hy
+  · rintro ⟨⟨x, hx, rfl⟩, heq⟩
+    refine ⟨pointOnGraph_upperSupportHeight_mem K D hne hD hsin ?_, heq⟩
+    exact ConvexBody.exists_mem_fst_eq_of_mem_Icc K.val.val
+      (edgeVertices_fst_mem K.val.val
+        ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)).1
+      (edgeVertices_snd_mem K.val.val (0 : Real.Angle)).1
+      (polygonCap_left_x_lt_right_x K) hx
+
+/-- The sine-weighted exposed-face lengths equal the horizontal separation of cap contacts. -/
+lemma PolygonCapSpace.sum_hausdorffMeasure_exposedEdge_mul_sin {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (D : Finset ℝ) (hne : D.Nonempty)
+    (hD : (D : Set ℝ) = angleDomain Θ) :
+    ∑ t ∈ D, (Measure.hausdorffMeasure 1 (exposedEdge K.val.val (t : Real.Angle))).toReal *
+      Real.sin t = ((capVertices K.val 0).1.2) 0 - ((capVertices K.val Θ.angle).2.1) 0 := by
+  classical
+  obtain ⟨p, hp, hpa, hpb, hc, hK, hn⟩ := K.exists_upperBoundary_polyline D hne hD
+  have hangle : ∀ t ∈ D, t ∈ Set.Ioo 0 Real.pi := by
+    intro t ht
+    exact angleDomain_subset_Ioo Θ (hD ▸ ht)
+  have hface (t : ℝ) (ht : t ∈ D) :
+      (Measure.hausdorffMeasure 1 (exposedEdge K.val.val (t : Real.Angle))).toReal =
+        ∑ i : Fin p.edges,
+          if inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+            (normalVector (t : Real.Angle)) = 0 then
+            dist (p.vertices i.castSucc) (p.vertices i.succ) else 0 := by
+    rw [K.exposedEdge_eq_upperSupportGraph_inter D hne hD ht, ← hc]
+    apply p.toReal_hausdorffMeasure_carrier_inter_hyperplane
+    intro i hi
+    obtain ⟨r, hr, horth, hstart⟩ := hn i
+    have hrt := eq_of_inner_sub_normalVector_eq_zero (hangle r hr) (hangle t ht)
+      (p.increasing i.castSucc_lt_succ) horth hi
+    subst r
+    intro q hq
+    rw [segment_eq_image'] at hq
+    obtain ⟨c, hcoef, rfl⟩ := hq
+    change inner ℝ (p.vertices i.castSucc + c •
+      (p.vertices i.succ - p.vertices i.castSucc)) (normalVector (t : Real.Angle)) = _
+    rw [inner_add_left, real_inner_smul_left, hi, mul_zero, add_zero, hstart]
+  calc
+    _ = ∑ t ∈ D, (∑ i : Fin p.edges,
+        if inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+          (normalVector (t : Real.Angle)) = 0 then
+          dist (p.vertices i.castSucc) (p.vertices i.succ) else 0) * Real.sin t := by
+      apply Finset.sum_congr rfl
+      intro t ht
+      rw [hface t ht]
+    _ = _ := by
+      rw [p.sum_normal_lengths_mul_sin D hangle
+        (fun i ↦ by obtain ⟨t, ht, ho, _⟩ := hn i; exact ⟨t, ht, ho⟩), hpa, hpb]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Gerver.Area.Evaluator`.
+* `Gerver.Area.CapFan`.
+* `Gerver.Area.NicheCover`.
+* `Gerver.Area`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Soundness of the contact evaluator
+
+`GerverAreaCert.evalZ s z kind` evaluates one of the five phase formulas of Gerver's sofa,
+and one of its four contact curves, on an interval of rotation angles.  This module proves
+that it encloses the analytic value: `gerverBranch_eq` identifies the branch that the
+piecewise vendor definitions `GerverSofa.Romik.path` and `GerverSofa.PartC.alphaBetaAt`
+select, the five per-stage lemmas verify the phase formulas and the two velocity
+coefficients against `GerverSofa.Romik.path1 … path5` and
+`GerverSofa.Romik.alphaBeta1 … alphaBeta5`, and `evalZ_sound` assembles them through the
+coordinate dictionary `fromPlane_paperGerverContacts`.  `contactZ_sound` and
+`evalZ_interval_sound` specialise this to a grid angle and to a whole grid subinterval.
+-/
+
+/-! ### The contact evaluator -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open GerverAreaCert
+
+/-- The five certified phase maps, selected by stage index. -/
+def gerverBranchPath : ℕ → ℝ → GerverSofa.Point
+  | 1 => GerverSofa.Romik.path1 GerverSofa.PartB.params
+  | 2 => GerverSofa.Romik.path2 GerverSofa.PartB.params
+  | 3 => GerverSofa.Romik.path3 GerverSofa.PartB.params
+  | 4 => GerverSofa.Romik.path4 GerverSofa.PartB.params
+  | _ => GerverSofa.Romik.path5 GerverSofa.PartB.params
+
+/-- The five certified velocity-coefficient pairs, selected by stage index. -/
+def gerverBranchAlphaBeta : ℕ → ℝ → GerverSofa.Point
+  | 1 => GerverSofa.Romik.alphaBeta1 GerverSofa.PartB.params
+  | 2 => GerverSofa.Romik.alphaBeta2 GerverSofa.PartB.params
+  | 3 => GerverSofa.Romik.alphaBeta3 GerverSofa.PartB.params
+  | 4 => GerverSofa.Romik.alphaBeta4 GerverSofa.PartB.params
+  | _ => GerverSofa.Romik.alphaBeta5 GerverSofa.PartB.params
+
+/-- On a grid angle of stage `s` both piecewise definitions select branch `s`. -/
+theorem gerverBranch_eq (s : ℕ) (hs1 : 1 ≤ s) (hs5 : s ≤ 5) {x : ℝ}
+    (hxhi : x ≤ gerverStageTime s) (hxlo : 2 ≤ s → gerverStageTime (s - 1) < x) :
+    GerverSofa.Romik.path GerverSofa.PartB.params x = gerverBranchPath s x ∧
+      GerverSofa.PartC.alphaBetaAt x = gerverBranchAlphaBeta s x := by
+  have o12 : GerverSofa.PartB.params.phi < GerverSofa.PartB.params.theta := by
+    have := gerverStageTime_lt_succ 1 (by norm_num)
+    rwa [gerverStageTime_one, gerverStageTime_two] at this
+  have o23 : GerverSofa.PartB.params.theta < GerverSofa.PartC.eta := by
+    have := gerverStageTime_lt_succ 2 (by norm_num)
+    rwa [gerverStageTime_two, gerverStageTime_three] at this
+  have o34 : GerverSofa.PartC.eta < GerverSofa.PartC.tau := by
+    have := gerverStageTime_lt_succ 3 (by norm_num)
+    rwa [gerverStageTime_three, gerverStageTime_four] at this
+  have hpath : GerverSofa.Romik.path GerverSofa.PartB.params x =
+      if x ≤ GerverSofa.PartB.params.phi then
+        GerverSofa.Romik.path1 GerverSofa.PartB.params x
+      else if x ≤ GerverSofa.PartB.params.theta then
+        GerverSofa.Romik.path2 GerverSofa.PartB.params x
+      else if x ≤ GerverSofa.PartC.eta then
+        GerverSofa.Romik.path3 GerverSofa.PartB.params x
+      else if x ≤ GerverSofa.PartC.tau then
+        GerverSofa.Romik.path4 GerverSofa.PartB.params x
+      else GerverSofa.Romik.path5 GerverSofa.PartB.params x := rfl
+  have hab : GerverSofa.PartC.alphaBetaAt x =
+      if x ≤ GerverSofa.PartB.params.phi then
+        GerverSofa.Romik.alphaBeta1 GerverSofa.PartB.params x
+      else if x ≤ GerverSofa.PartB.params.theta then
+        GerverSofa.Romik.alphaBeta2 GerverSofa.PartB.params x
+      else if x ≤ GerverSofa.PartC.eta then
+        GerverSofa.Romik.alphaBeta3 GerverSofa.PartB.params x
+      else if x ≤ GerverSofa.PartC.tau then
+        GerverSofa.Romik.alphaBeta4 GerverSofa.PartB.params x
+      else GerverSofa.Romik.alphaBeta5 GerverSofa.PartB.params x := rfl
+  interval_cases s
+  · rw [gerverStageTime_one] at hxhi
+    refine ⟨?_, ?_⟩
+    · rw [hpath, ite_eq_left hxhi]; rfl
+    · rw [hab, ite_eq_left hxhi]; rfl
+  · rw [gerverStageTime_two] at hxhi
+    have h1 : ¬ x ≤ GerverSofa.PartB.params.phi := by
+      have := hxlo (by norm_num)
+      rw [gerverStageTime_one] at this
+      exact not_le.mpr this
+    refine ⟨?_, ?_⟩
+    · rw [hpath, ite_eq_right h1, ite_eq_left hxhi]; rfl
+    · rw [hab, ite_eq_right h1, ite_eq_left hxhi]; rfl
+  · rw [gerverStageTime_three] at hxhi
+    have h2 : GerverSofa.PartB.params.theta < x := by
+      have := hxlo (by norm_num)
+      rwa [gerverStageTime_two] at this
+    have h1 : ¬ x ≤ GerverSofa.PartB.params.phi := not_le.mpr (o12.trans h2)
+    refine ⟨?_, ?_⟩
+    · rw [hpath, ite_eq_right h1, ite_eq_right (not_le.mpr h2), ite_eq_left hxhi]; rfl
+    · rw [hab, ite_eq_right h1, ite_eq_right (not_le.mpr h2), ite_eq_left hxhi]; rfl
+  · rw [gerverStageTime_four] at hxhi
+    have h3 : GerverSofa.PartC.eta < x := by
+      have := hxlo (by norm_num)
+      rwa [gerverStageTime_three] at this
+    have h2 : GerverSofa.PartB.params.theta < x := o23.trans h3
+    have h1 : ¬ x ≤ GerverSofa.PartB.params.phi := not_le.mpr (o12.trans h2)
+    refine ⟨?_, ?_⟩
+    · rw [hpath, ite_eq_right h1, ite_eq_right (not_le.mpr h2), ite_eq_right (not_le.mpr h3),
+        ite_eq_left hxhi]
+      rfl
+    · rw [hab, ite_eq_right h1, ite_eq_right (not_le.mpr h2), ite_eq_right (not_le.mpr h3),
+        ite_eq_left hxhi]
+      rfl
+  · have h4 : GerverSofa.PartC.tau < x := by
+      have := hxlo (by norm_num)
+      rwa [gerverStageTime_four] at this
+    have h3 : GerverSofa.PartC.eta < x := o34.trans h4
+    have h2 : GerverSofa.PartB.params.theta < x := o23.trans h3
+    have h1 : ¬ x ≤ GerverSofa.PartB.params.phi := not_le.mpr (o12.trans h2)
+    refine ⟨?_, ?_⟩
+    · rw [hpath, ite_eq_right h1, ite_eq_right (not_le.mpr h2), ite_eq_right (not_le.mpr h3),
+        ite_eq_right (not_le.mpr h4)]
+      rfl
+    · rw [hab, ite_eq_right h1, ite_eq_right (not_le.mpr h2), ite_eq_right (not_le.mpr h3),
+        ite_eq_right (not_le.mpr h4)]
+      rfl
+
+/-- Transport an enclosure along an equality of the enclosed real number. -/
+private theorem contains_of_eq_real {z : SI} {a b : ℝ} (hab : a = b) (h : SI.Contains z a) :
+    SI.Contains z b := hab ▸ h
+
+open SI in
+/-- Stage-1 interval soundness of the phase map and of the two velocity offsets. -/
+private theorem evalZ_stage1_sound {z : SI} {x : ℝ}
+    (hsin : SI.Contains (trigZ z).1 (Real.sin x))
+    (hcos : SI.Contains (trigZ z).2 (Real.cos x)) :
+    SI.Contains (evalZ 1 z 0).1 (gerverBranchPath 1 x).1 ∧
+      SI.Contains (evalZ 1 z 0).2 (gerverBranchPath 1 x).2 ∧
+      SI.Contains (evalZ 1 z 2).1
+        ((gerverBranchPath 1 x).1 - (gerverBranchAlphaBeta 1 x).1 * Real.sin x) ∧
+      SI.Contains (evalZ 1 z 2).2
+        ((gerverBranchPath 1 x).2 + (gerverBranchAlphaBeta 1 x).1 * Real.cos x) ∧
+      SI.Contains (evalZ 1 z 4).1
+        ((gerverBranchPath 1 x).1 - (gerverBranchAlphaBeta 1 x).2 * Real.cos x) ∧
+      SI.Contains (evalZ 1 z 4).2
+        ((gerverBranchPath 1 x).2 - (gerverBranchAlphaBeta 1 x).2 * Real.sin x) := by
+  obtain ⟨hk11, hk12, -, -, -, -, -, -, -, -, ha1, ha2, -⟩ := contains_params
+  have hf := contains_add (contains_add (contains_mul ha1 hcos) (contains_mul ha2 hsin))
+    (contains_ratI (a := -1) (b := 1) (by norm_num))
+  have hg := contains_add (contains_add (contains_mul (contains_neg ha2) hcos)
+    (contains_mul ha1 hsin)) (contains_ratI (a := -1) (b := 2) (by norm_num))
+  have hal := contains_add (contains_add (contains_mul (contains_imul (-2) ha1) hsin)
+    (contains_mul (contains_imul 2 ha2) hcos)) (contains_ratI (a := 1) (b := 2) (by norm_num))
+  have hbe := contains_add (contains_add (contains_mul (contains_imul 2 ha1) hcos)
+    (contains_mul (contains_imul 2 ha2) hsin)) (contains_ratI (a := -1) (b := 1) (by norm_num))
+  have hx := contains_add (contains_sub (contains_mul hcos hf) (contains_mul hsin hg)) hk11
+  have hy := contains_add (contains_add (contains_mul hsin hf) (contains_mul hcos hg)) hk12
+  refine ⟨contains_of_eq_real ?_ hx, contains_of_eq_real ?_ hy,
+    contains_of_eq_real ?_ (contains_sub hx (contains_mul hal hsin)),
+    contains_of_eq_real ?_ (contains_add hy (contains_mul hal hcos)),
+    contains_of_eq_real ?_ (contains_sub hx (contains_mul hbe hcos)),
+    contains_of_eq_real ?_ (contains_sub hy (contains_mul hbe hsin))⟩ <;>
+  · simp only [gerverBranchPath, gerverBranchAlphaBeta, GerverSofa.Romik.path1,
+      GerverSofa.Romik.alphaBeta1, GerverSofa.Romik.addK, GerverSofa.Romik.rot]
+    push_cast
+    ring
+
+open SI in
+/-- Stage-2 interval soundness of the phase map and of the two velocity offsets. -/
+private theorem evalZ_stage2_sound {z : SI} {x : ℝ} (hz : SI.Contains z x)
+    (hsin : SI.Contains (trigZ z).1 (Real.sin x))
+    (hcos : SI.Contains (trigZ z).2 (Real.cos x)) :
+    SI.Contains (evalZ 2 z 0).1 (gerverBranchPath 2 x).1 ∧
+      SI.Contains (evalZ 2 z 0).2 (gerverBranchPath 2 x).2 ∧
+      SI.Contains (evalZ 2 z 2).1
+        ((gerverBranchPath 2 x).1 - (gerverBranchAlphaBeta 2 x).1 * Real.sin x) ∧
+      SI.Contains (evalZ 2 z 2).2
+        ((gerverBranchPath 2 x).2 + (gerverBranchAlphaBeta 2 x).1 * Real.cos x) ∧
+      SI.Contains (evalZ 2 z 4).1
+        ((gerverBranchPath 2 x).1 - (gerverBranchAlphaBeta 2 x).2 * Real.cos x) ∧
+      SI.Contains (evalZ 2 z 4).2
+        ((gerverBranchPath 2 x).2 - (gerverBranchAlphaBeta 2 x).2 * Real.sin x) := by
+  obtain ⟨-, -, hk21, hk22, -, -, -, -, -, -, -, -, hb1, hb2, -⟩ := contains_params
+  have hf := contains_add (contains_add
+    (contains_divn (b := 4) (by norm_num) (contains_neg (contains_mul hz hz)))
+    (contains_mul hb1 hz)) hb2
+  have hg := contains_add (contains_add (contains_divn (b := 2) (by norm_num) hz)
+    (contains_neg hb1)) (contains_ratI (a := -1) (b := 1) (by norm_num))
+  have hal := contains_add (contains_add (contains_ratI (a := 1) (b := 1) (by norm_num))
+    (contains_imul 2 hb1)) (contains_neg hz)
+  have hbe := contains_add hf (contains_ratI (a := 1) (b := 2) (by norm_num))
+  have hx := contains_add (contains_sub (contains_mul hcos hf) (contains_mul hsin hg)) hk21
+  have hy := contains_add (contains_add (contains_mul hsin hf) (contains_mul hcos hg)) hk22
+  refine ⟨contains_of_eq_real ?_ hx, contains_of_eq_real ?_ hy,
+    contains_of_eq_real ?_ (contains_sub hx (contains_mul hal hsin)),
+    contains_of_eq_real ?_ (contains_add hy (contains_mul hal hcos)),
+    contains_of_eq_real ?_ (contains_sub hx (contains_mul hbe hcos)),
+    contains_of_eq_real ?_ (contains_sub hy (contains_mul hbe hsin))⟩ <;>
+  · simp only [gerverBranchPath, gerverBranchAlphaBeta, GerverSofa.Romik.path2,
+      GerverSofa.Romik.alphaBeta2, GerverSofa.Romik.addK, GerverSofa.Romik.rot]
+    push_cast
+    ring
+
+open SI in
+/-- Stage-3 interval soundness of the phase map and of the two velocity offsets. -/
+private theorem evalZ_stage3_sound {z : SI} {x : ℝ} (hz : SI.Contains z x)
+    (hsin : SI.Contains (trigZ z).1 (Real.sin x))
+    (hcos : SI.Contains (trigZ z).2 (Real.cos x)) :
+    SI.Contains (evalZ 3 z 0).1 (gerverBranchPath 3 x).1 ∧
+      SI.Contains (evalZ 3 z 0).2 (gerverBranchPath 3 x).2 ∧
+      SI.Contains (evalZ 3 z 2).1
+        ((gerverBranchPath 3 x).1 - (gerverBranchAlphaBeta 3 x).1 * Real.sin x) ∧
+      SI.Contains (evalZ 3 z 2).2
+        ((gerverBranchPath 3 x).2 + (gerverBranchAlphaBeta 3 x).1 * Real.cos x) ∧
+      SI.Contains (evalZ 3 z 4).1
+        ((gerverBranchPath 3 x).1 - (gerverBranchAlphaBeta 3 x).2 * Real.cos x) ∧
+      SI.Contains (evalZ 3 z 4).2
+        ((gerverBranchPath 3 x).2 - (gerverBranchAlphaBeta 3 x).2 * Real.sin x) := by
+  obtain ⟨-, -, -, -, hk31, hk32, -, -, -, -, -, -, -, -, hc1, hc2, -⟩ := contains_params
+  have hf := contains_sub hc1 hz
+  have hg := contains_add hc2 hz
+  have hal := contains_add (contains_ratI (a := -1) (b := 1) (by norm_num)) (contains_neg hg)
+  have hbe := contains_add (contains_ratI (a := 1) (b := 1) (by norm_num)) hf
+  have hx := contains_add (contains_sub (contains_mul hcos hf) (contains_mul hsin hg)) hk31
+  have hy := contains_add (contains_add (contains_mul hsin hf) (contains_mul hcos hg)) hk32
+  refine ⟨contains_of_eq_real ?_ hx, contains_of_eq_real ?_ hy,
+    contains_of_eq_real ?_ (contains_sub hx (contains_mul hal hsin)),
+    contains_of_eq_real ?_ (contains_add hy (contains_mul hal hcos)),
+    contains_of_eq_real ?_ (contains_sub hx (contains_mul hbe hcos)),
+    contains_of_eq_real ?_ (contains_sub hy (contains_mul hbe hsin))⟩ <;>
+    simp only [gerverBranchPath, gerverBranchAlphaBeta, GerverSofa.Romik.path3,
+      GerverSofa.Romik.alphaBeta3, GerverSofa.Romik.addK, GerverSofa.Romik.rot] <;>
+    push_cast <;> ring
+
+open SI in
+/-- Stage-4 interval soundness of the phase map and of the two velocity offsets. -/
+private theorem evalZ_stage4_sound {z : SI} {x : ℝ} (hz : SI.Contains z x)
+    (hsin : SI.Contains (trigZ z).1 (Real.sin x))
+    (hcos : SI.Contains (trigZ z).2 (Real.cos x)) :
+    SI.Contains (evalZ 4 z 0).1 (gerverBranchPath 4 x).1 ∧
+      SI.Contains (evalZ 4 z 0).2 (gerverBranchPath 4 x).2 ∧
+      SI.Contains (evalZ 4 z 2).1
+        ((gerverBranchPath 4 x).1 - (gerverBranchAlphaBeta 4 x).1 * Real.sin x) ∧
+      SI.Contains (evalZ 4 z 2).2
+        ((gerverBranchPath 4 x).2 + (gerverBranchAlphaBeta 4 x).1 * Real.cos x) ∧
+      SI.Contains (evalZ 4 z 4).1
+        ((gerverBranchPath 4 x).1 - (gerverBranchAlphaBeta 4 x).2 * Real.cos x) ∧
+      SI.Contains (evalZ 4 z 4).2
+        ((gerverBranchPath 4 x).2 - (gerverBranchAlphaBeta 4 x).2 * Real.sin x) := by
+  obtain ⟨-, -, -, -, -, -, hk41, hk42, -, -, -, -, -, -, -, -, hd1, hd2, -⟩ := contains_params
+  have hf := contains_add (contains_add
+    (contains_neg (contains_divn (b := 2) (by norm_num) hz)) hd1)
+    (contains_ratI (a := -1) (b := 1) (by norm_num))
+  have hg := contains_add (contains_add
+    (contains_divn (b := 4) (by norm_num) (contains_neg (contains_mul hz hz)))
+    (contains_mul hd1 hz)) hd2
+  have hal := contains_add (contains_neg hg) (contains_ratI (a := -1) (b := 2) (by norm_num))
+  have hbe := contains_add (contains_add (contains_imul 2 hd1)
+    (contains_ratI (a := -1) (b := 1) (by norm_num))) (contains_neg hz)
+  have hx := contains_add (contains_sub (contains_mul hcos hf) (contains_mul hsin hg)) hk41
+  have hy := contains_add (contains_add (contains_mul hsin hf) (contains_mul hcos hg)) hk42
+  refine ⟨contains_of_eq_real ?_ hx, contains_of_eq_real ?_ hy,
+    contains_of_eq_real ?_ (contains_sub hx (contains_mul hal hsin)),
+    contains_of_eq_real ?_ (contains_add hy (contains_mul hal hcos)),
+    contains_of_eq_real ?_ (contains_sub hx (contains_mul hbe hcos)),
+    contains_of_eq_real ?_ (contains_sub hy (contains_mul hbe hsin))⟩ <;>
+  · simp only [gerverBranchPath, gerverBranchAlphaBeta, GerverSofa.Romik.path4,
+      GerverSofa.Romik.alphaBeta4, GerverSofa.Romik.addK, GerverSofa.Romik.rot]
+    push_cast
+    ring
+
+open SI in
+/-- Stage-5 interval soundness of the phase map and of the two velocity offsets. -/
+private theorem evalZ_stage5_sound {z : SI} {x : ℝ}
+    (hsin : SI.Contains (trigZ z).1 (Real.sin x))
+    (hcos : SI.Contains (trigZ z).2 (Real.cos x)) :
+    SI.Contains (evalZ 5 z 0).1 (gerverBranchPath 5 x).1 ∧
+      SI.Contains (evalZ 5 z 0).2 (gerverBranchPath 5 x).2 ∧
+      SI.Contains (evalZ 5 z 2).1
+        ((gerverBranchPath 5 x).1 - (gerverBranchAlphaBeta 5 x).1 * Real.sin x) ∧
+      SI.Contains (evalZ 5 z 2).2
+        ((gerverBranchPath 5 x).2 + (gerverBranchAlphaBeta 5 x).1 * Real.cos x) ∧
+      SI.Contains (evalZ 5 z 4).1
+        ((gerverBranchPath 5 x).1 - (gerverBranchAlphaBeta 5 x).2 * Real.cos x) ∧
+      SI.Contains (evalZ 5 z 4).2
+        ((gerverBranchPath 5 x).2 - (gerverBranchAlphaBeta 5 x).2 * Real.sin x) := by
+  obtain ⟨-, -, -, -, -, -, -, -, hk51, hk52, -, -, -, -, -, -, -, -, he1, he2, -⟩ :=
+    contains_params
+  have hf := contains_add (contains_add (contains_mul he1 hcos) (contains_mul he2 hsin))
+    (contains_ratI (a := -1) (b := 2) (by norm_num))
+  have hg := contains_add (contains_add (contains_mul (contains_neg he2) hcos)
+    (contains_mul he1 hsin)) (contains_ratI (a := -1) (b := 1) (by norm_num))
+  have hal := contains_add (contains_add (contains_ratI (a := 1) (b := 1) (by norm_num))
+    (contains_mul (contains_imul (-2) he1) hsin)) (contains_mul (contains_imul 2 he2) hcos)
+  have hbe := contains_add (contains_add (contains_mul (contains_imul 2 he1) hcos)
+    (contains_mul (contains_imul 2 he2) hsin)) (contains_ratI (a := -1) (b := 2) (by norm_num))
+  have hx := contains_add (contains_sub (contains_mul hcos hf) (contains_mul hsin hg)) hk51
+  have hy := contains_add (contains_add (contains_mul hsin hf) (contains_mul hcos hg)) hk52
+  refine ⟨contains_of_eq_real ?_ hx, contains_of_eq_real ?_ hy,
+    contains_of_eq_real ?_ (contains_sub hx (contains_mul hal hsin)),
+    contains_of_eq_real ?_ (contains_add hy (contains_mul hal hcos)),
+    contains_of_eq_real ?_ (contains_sub hx (contains_mul hbe hcos)),
+    contains_of_eq_real ?_ (contains_sub hy (contains_mul hbe hsin))⟩ <;>
+  · simp only [gerverBranchPath, gerverBranchAlphaBeta, GerverSofa.Romik.path5,
+      GerverSofa.Romik.alphaBeta5, GerverSofa.Romik.addK, GerverSofa.Romik.rot]
+    push_cast
+    ring
+
+/-- Per-stage soundness of the phase-map evaluation, i.e. of the `kind = 0` output.
+Pure interval arithmetic against `GerverSofa.Romik.path1 … path5`: five cases, each a chain
+of `SI.contains_*` applications on top of `contains_params`, `hsin` and `hcos`. -/
+theorem evalZ_zero_sound (s : ℕ) (hs1 : 1 ≤ s) (hs5 : s ≤ 5) {z : SI} {x : ℝ}
+    (hz : SI.Contains z x) (hsin : SI.Contains (trigZ z).1 (Real.sin x))
+    (hcos : SI.Contains (trigZ z).2 (Real.cos x)) :
+    SI.Contains (evalZ s z 0).1 (gerverBranchPath s x).1 ∧
+      SI.Contains (evalZ s z 0).2 (gerverBranchPath s x).2 := by
+  interval_cases s
+  · exact ⟨(evalZ_stage1_sound hsin hcos).1, (evalZ_stage1_sound hsin hcos).2.1⟩
+  · exact ⟨(evalZ_stage2_sound hz hsin hcos).1, (evalZ_stage2_sound hz hsin hcos).2.1⟩
+  · exact ⟨(evalZ_stage3_sound hz hsin hcos).1, (evalZ_stage3_sound hz hsin hcos).2.1⟩
+  · exact ⟨(evalZ_stage4_sound hz hsin hcos).1, (evalZ_stage4_sound hz hsin hcos).2.1⟩
+  · exact ⟨(evalZ_stage5_sound hsin hcos).1, (evalZ_stage5_sound hsin hcos).2.1⟩
+
+/-- Per-stage soundness of the `B` and `D` offsets, i.e. of the two velocity
+coefficients `α` and `β` against `GerverSofa.Romik.alphaBeta1 … alphaBeta5`.  Note
+`evalZ s z 1 = evalZ s z 2` shifted by `(cos x, sin x)` and
+`evalZ s z 3 = evalZ s z 4` shifted by `(-sin x, cos x)`, so the remaining two kinds need no
+separate stage analysis. -/
+theorem evalZ_offset_sound (s : ℕ) (hs1 : 1 ≤ s) (hs5 : s ≤ 5) {z : SI} {x : ℝ}
+    (hz : SI.Contains z x) (hsin : SI.Contains (trigZ z).1 (Real.sin x))
+    (hcos : SI.Contains (trigZ z).2 (Real.cos x)) :
+    SI.Contains (evalZ s z 2).1
+        ((gerverBranchPath s x).1 - (gerverBranchAlphaBeta s x).1 * Real.sin x) ∧
+      SI.Contains (evalZ s z 2).2
+        ((gerverBranchPath s x).2 + (gerverBranchAlphaBeta s x).1 * Real.cos x) ∧
+      SI.Contains (evalZ s z 4).1
+        ((gerverBranchPath s x).1 - (gerverBranchAlphaBeta s x).2 * Real.cos x) ∧
+      SI.Contains (evalZ s z 4).2
+        ((gerverBranchPath s x).2 - (gerverBranchAlphaBeta s x).2 * Real.sin x) := by
+  interval_cases s
+  · exact (evalZ_stage1_sound hsin hcos).2.2
+  · exact (evalZ_stage2_sound hz hsin hcos).2.2
+  · exact (evalZ_stage3_sound hz hsin hcos).2.2
+  · exact (evalZ_stage4_sound hz hsin hcos).2.2
+  · exact (evalZ_stage5_sound hsin hcos).2.2
+
+/-- Soundness of the executable phase/contact evaluator: on the branch that the
+piecewise definitions select, the interval evaluation encloses both coordinates of the
+selected curve.  Reduces to `gerverBranch_eq`, `evalZ_zero_sound`, `evalZ_offset_sound`,
+`trigZ_sound` and the coordinate dictionary
+`fromPlane_paperGerverContacts` (`GerverSofa.u t = (cos t, sin t)`,
+`GerverSofa.v t = (-sin t, cos t)`). -/
+theorem evalZ_sound (s : ℕ) (hs1 : 1 ≤ s) (hs5 : s ≤ 5) {z : SI} {x : ℝ}
+    (hz : SI.Contains z x) (hx0 : 0 ≤ x) (hxT : x ≤ Real.pi / 2)
+    (hxhi : x ≤ gerverStageTime s) (hxlo : 2 ≤ s → gerverStageTime (s - 1) < x)
+    (kind : ℕ) (hk : kind ≤ 4) :
+    SI.Contains (evalZ s z kind).1 (gerverContactPoint kind x 0) ∧
+      SI.Contains (evalZ s z kind).2 (gerverContactPoint kind x 1) := by
+  have hx2 : x ≤ 2 := by linarith [Real.pi_lt_d2]
+  obtain ⟨hsin, hcos⟩ := trigZ_sound hz hx0 hx2
+  obtain ⟨hp, hab⟩ := gerverBranch_eq s hs1 hs5 hxhi hxlo
+  obtain ⟨h0x, h0y⟩ := evalZ_zero_sound s hs1 hs5 hz hsin hcos
+  obtain ⟨h2x, h2y, h4x, h4y⟩ := evalZ_offset_sound s hs1 hs5 hz hsin hcos
+  have ht : x ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨hx0, hxT⟩
+  -- The four contact curves in terms of the branch phase map and coefficients.
+  set P : ℝ × ℝ := gerverBranchPath s x with hPdef
+  set AB : ℝ × ℝ := gerverBranchAlphaBeta s x with hABdef
+  have hpath1 : (GerverSofa.Romik.path GerverSofa.PartB.params x).1 = P.1 := by rw [hp]
+  have hpath2 : (GerverSofa.Romik.path GerverSofa.PartB.params x).2 = P.2 := by rw [hp]
+  have hal : (GerverSofa.PartC.alphaBetaAt x).1 = AB.1 := by rw [hab]
+  have hbe : (GerverSofa.PartC.alphaBetaAt x).2 = AB.2 := by rw [hab]
+  have hdA1 : gerverContactPoint 1 x 0 = P.1 - AB.1 * Real.sin x + Real.cos x := by
+    have h' : gerverContactPoint 1 x 0 = (GerverSofa.PartC.A x).1 :=
+      congrArg Prod.fst (fromPlane_paperGerverContacts x ht 0)
+    rw [h']
+    change (GerverSofa.Romik.path GerverSofa.PartB.params x).1 +
+      (GerverSofa.PartC.alphaBetaAt x).1 * (GerverSofa.v x).1 + (GerverSofa.u x).1 = _
+    rw [hpath1, hal]
+    change P.1 + AB.1 * (-Real.sin x) + Real.cos x = _
+    ring
+  have hdA2 : gerverContactPoint 1 x 1 = P.2 + AB.1 * Real.cos x + Real.sin x := by
+    have h' : gerverContactPoint 1 x 1 = (GerverSofa.PartC.A x).2 :=
+      congrArg Prod.snd (fromPlane_paperGerverContacts x ht 0)
+    rw [h']
+    change (GerverSofa.Romik.path GerverSofa.PartB.params x).2 +
+      (GerverSofa.PartC.alphaBetaAt x).1 * (GerverSofa.v x).2 + (GerverSofa.u x).2 = _
+    rw [hpath2, hal]
+    change P.2 + AB.1 * Real.cos x + Real.sin x = _
+    ring
+  have hdB1 : gerverContactPoint 2 x 0 = P.1 - AB.1 * Real.sin x := by
+    have h' : gerverContactPoint 2 x 0 = (GerverSofa.PartC.B x).1 :=
+      congrArg Prod.fst (fromPlane_paperGerverContacts x ht 1)
+    rw [h']
+    change (GerverSofa.Romik.path GerverSofa.PartB.params x).1 +
+      (GerverSofa.PartC.alphaBetaAt x).1 * (GerverSofa.v x).1 = _
+    rw [hpath1, hal]
+    change P.1 + AB.1 * (-Real.sin x) = _
+    ring
+  have hdB2 : gerverContactPoint 2 x 1 = P.2 + AB.1 * Real.cos x := by
+    have h' : gerverContactPoint 2 x 1 = (GerverSofa.PartC.B x).2 :=
+      congrArg Prod.snd (fromPlane_paperGerverContacts x ht 1)
+    rw [h']
+    change (GerverSofa.Romik.path GerverSofa.PartB.params x).2 +
+      (GerverSofa.PartC.alphaBetaAt x).1 * (GerverSofa.v x).2 = _
+    rw [hpath2, hal]
+    rfl
+  have hdC1 : gerverContactPoint 3 x 0 = P.1 - AB.2 * Real.cos x - Real.sin x := by
+    have h' : gerverContactPoint 3 x 0 = (GerverSofa.PartC.C x).1 :=
+      congrArg Prod.fst (fromPlane_paperGerverContacts x ht 2)
+    rw [h']
+    change (GerverSofa.Romik.path GerverSofa.PartB.params x).1 -
+      (GerverSofa.PartC.alphaBetaAt x).2 * (GerverSofa.u x).1 + (GerverSofa.v x).1 = _
+    rw [hpath1, hbe]
+    change P.1 - AB.2 * Real.cos x + -Real.sin x = _
+    ring
+  have hdC2 : gerverContactPoint 3 x 1 = P.2 - AB.2 * Real.sin x + Real.cos x := by
+    have h' : gerverContactPoint 3 x 1 = (GerverSofa.PartC.C x).2 :=
+      congrArg Prod.snd (fromPlane_paperGerverContacts x ht 2)
+    rw [h']
+    change (GerverSofa.Romik.path GerverSofa.PartB.params x).2 -
+      (GerverSofa.PartC.alphaBetaAt x).2 * (GerverSofa.u x).2 + (GerverSofa.v x).2 = _
+    rw [hpath2, hbe]
+    rfl
+  have hdD1 : gerverContactPoint 4 x 0 = P.1 - AB.2 * Real.cos x := by
+    have h' : gerverContactPoint 4 x 0 = (GerverSofa.PartC.D x).1 :=
+      congrArg Prod.fst (fromPlane_paperGerverContacts x ht 3)
+    rw [h']
+    change (GerverSofa.Romik.path GerverSofa.PartB.params x).1 -
+      (GerverSofa.PartC.alphaBetaAt x).2 * (GerverSofa.u x).1 = _
+    rw [hpath1, hbe]
+    rfl
+  have hdD2 : gerverContactPoint 4 x 1 = P.2 - AB.2 * Real.sin x := by
+    have h' : gerverContactPoint 4 x 1 = (GerverSofa.PartC.D x).2 :=
+      congrArg Prod.snd (fromPlane_paperGerverContacts x ht 3)
+    rw [h']
+    change (GerverSofa.Romik.path GerverSofa.PartB.params x).2 -
+      (GerverSofa.PartC.alphaBetaAt x).2 * (GerverSofa.u x).2 = _
+    rw [hpath2, hbe]
+    rfl
+  have hd01 : gerverContactPoint 0 x 0 = P.1 := hpath1
+  have hd02 : gerverContactPoint 0 x 1 = P.2 := hpath2
+  -- The `A` and `C` outputs are the `B` and `D` outputs shifted by the frame vectors.
+  have eA1 : (evalZ s z 1).1 = SI.add (evalZ s z 2).1 (trigZ z).2 := rfl
+  have eA2 : (evalZ s z 1).2 = SI.add (evalZ s z 2).2 (trigZ z).1 := rfl
+  have eC1 : (evalZ s z 3).1 = SI.sub (evalZ s z 4).1 (trigZ z).1 := rfl
+  have eC2 : (evalZ s z 3).2 = SI.add (evalZ s z 4).2 (trigZ z).2 := rfl
+  interval_cases kind
+  · exact ⟨hd01 ▸ h0x, hd02 ▸ h0y⟩
+  · refine ⟨?_, ?_⟩
+    · rw [hdA1, eA1]
+      exact SI.contains_add h2x hcos
+    · rw [hdA2, eA2]
+      exact SI.contains_add h2y hsin
+  · exact ⟨hdB1 ▸ h2x, hdB2 ▸ h2y⟩
+  · refine ⟨?_, ?_⟩
+    · rw [hdC1, eC1]
+      exact SI.contains_sub h4x hsin
+    · rw [hdC2, eC2]
+      exact SI.contains_add h4y hcos
+  · exact ⟨hdD1 ▸ h4x, hdD2 ▸ h4y⟩
+
+/-- The left endpoint of the branch selected at the right end of a grid subinterval
+does not exceed the left end of that subinterval. -/
+theorem stageTime_pred_le_gerverGridTime (m : ℕ) (hm : m + 1 ≤ 5 * NN)
+    (h2 : 2 ≤ stageOf (m + 1)) :
+    gerverStageTime (stageOf (m + 1) - 1) ≤ gerverGridTime m := by
+  have hNN : 0 < NN := by norm_num [NN]
+  rw [stageOf] at h2 ⊢
+  split_ifs at h2 ⊢ with h
+  · have hd : 2 ≤ (m + 1) / NN := by
+      rcases le_max_iff.mp h2 with h' | h'
+      · exact absurd h' (by norm_num)
+      · exact h'
+    have hk : (m + 1) / NN * NN = m + 1 := Nat.div_mul_cancel (Nat.dvd_of_mod_eq_zero h)
+    have hstep : ((m + 1) / NN - 1) * NN = m + 1 - NN := by rw [Nat.sub_one_mul, hk]
+    have hle : ((m + 1) / NN - 1) * NN ≤ m := by rw [hstep]; omega
+    rw [max_eq_right (Nat.le_of_succ_le hd), ← gerverGridTime_mul_NN ((m + 1) / NN - 1)]
+    exact gerverGridTime_le_of_le hle (by omega)
+  · rw [Nat.add_sub_cancel]
+    have hle1 : (m + 1) / NN * NN ≤ m + 1 := Nat.div_mul_le_self (m + 1) NN
+    have hne : (m + 1) / NN * NN ≠ m + 1 := by
+      intro he
+      exact h (by rw [← he]; exact Nat.mul_mod_left _ _)
+    have hle : (m + 1) / NN * NN ≤ m := Nat.lt_succ_iff.mp (lt_of_le_of_ne hle1 hne)
+    rw [← gerverGridTime_mul_NN ((m + 1) / NN)]
+    exact gerverGridTime_le_of_le hle (by omega)
+
+/-- The evaluator at a grid angle. -/
+theorem contactZ_sound (m : ℕ) (hm : m ≤ 5 * NN) (kind : ℕ) (hk : kind ≤ 4) :
+    SI.Contains (contactZ m kind).1 (gerverContactPoint kind (gerverGridTime m) 0) ∧
+      SI.Contains (contactZ m kind).2 (gerverContactPoint kind (gerverGridTime m) 1) := by
+  obtain ⟨h1, h5⟩ := stageOf_mem m hm
+  obtain ⟨hg0, hgT⟩ := gerverGridTime_mem_Icc m hm
+  exact evalZ_sound (stageOf m) h1 h5 (ttZ_sound m hm) hg0 hgT
+    (gerverGridTime_le_stageTime m hm) (fun h ↦ stageTime_lt_gerverGridTime m hm h) kind hk
+
+/-- The evaluator over a whole grid subinterval, on the branch selected at its right
+endpoint (which is the branch of every angle in the half-open subinterval). -/
+theorem evalZ_interval_sound (m : ℕ) (hm : m + 1 ≤ 5 * NN) (kind : ℕ) (hk : kind ≤ 4)
+    {x : ℝ} (hx : x ∈ Set.Ioc (gerverGridTime m) (gerverGridTime (m + 1))) :
+    SI.Contains (evalZ (stageOf (m + 1))
+        ⟨(ttZ m).lo, (ttZ (m + 1)).hi⟩ kind).1 (gerverContactPoint kind x 0) ∧
+      SI.Contains (evalZ (stageOf (m + 1))
+        ⟨(ttZ m).lo, (ttZ (m + 1)).hi⟩ kind).2 (gerverContactPoint kind x 1) := by
+  have hmm : m ≤ 5 * NN := by omega
+  obtain ⟨hxl, hxr⟩ := hx
+  obtain ⟨h1, h5⟩ := stageOf_mem (m + 1) hm
+  obtain ⟨hg0, -⟩ := gerverGridTime_mem_Icc m hmm
+  obtain ⟨-, hgT⟩ := gerverGridTime_mem_Icc (m + 1) hm
+  have hM := SI.Mpos
+  have hz : SI.Contains (⟨(ttZ m).lo, (ttZ (m + 1)).hi⟩ : SI) x := by
+    obtain ⟨ha, -⟩ := ttZ_sound m hmm
+    obtain ⟨-, hb⟩ := ttZ_sound (m + 1) hm
+    exact ⟨ha.trans (mul_le_mul_of_nonneg_left hxl.le hM.le),
+      (mul_le_mul_of_nonneg_left hxr hM.le).trans hb⟩
+  refine evalZ_sound (stageOf (m + 1)) h1 h5 hz (hg0.trans hxl.le) (hxr.trans hgT)
+    (hxr.trans (gerverGridTime_le_stageTime (m + 1) hm)) (fun h2 ↦ ?_) kind hk
+  exact lt_of_le_of_lt (stageTime_pred_le_gerverGridTime m hm h2) hxl
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The support-contact fan of the Gerver outer cap
+
+The literal Gerver outer cap is a compact convex subset of the plane lying in the closed
+quadrant above the fan anchor `L = C (π / 2)`, which sits on the wall.  The `640` listed
+contacts `A (t)` and `C (t)` at the grid angles attain the cap support at the strictly
+increasing normals `t` and `t + π / 2` of `[0, π)`, so `supportContact_fan_area` bounds the
+cap area below by half the shoelace sum of the fan over the anchor.  The certificate
+encloses that sum (`capDoubledZ_sound`), and its kernel-checked numeric conclusion
+`GerverAreaCert.capOK_true` turns the enclosure into `28609 / 10000 ≤ |K₀|`.
+-/
+
+/-! ### Elementary geometry of the outer cap -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open MeasureTheory
+open GerverAreaCert
+
+/-- The paper path starts at the origin. -/
+theorem paperGerverPath_zero : paperGerverPath 0 = 0 := by
+  have hreg := gerver_direct_path_regularity GerverSofa.PartB.params
+    GerverSofa.PartB.params_mem GerverSofa.PartB.params_equations
+  change GerverSofa.PartF.Coordinates.toPlane
+    (GerverSofa.Romik.path GerverSofa.PartB.params 0) = 0
+  rw [hreg.2.1]
+  ext i
+  fin_cases i <;> rfl
+
+/-- The outer cap written as an intersection of closed half-planes. -/
+theorem gerverOuterCap_eq_iInter :
+    gerverOuterCap =
+      {q : Point | 0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))} ∩
+        ⋂ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+          ({q : Point | inner ℝ q (normalVector (t : Real.Angle)) ≤
+              inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) + 1} ∩
+            {q : Point | inner ℝ q (tangentVector (t : Real.Angle)) ≤
+              inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) + 1}) := by
+  ext q
+  simp only [gerverOuterCap, Set.mem_ofPred_eq, Set.mem_inter_iff, Set.mem_iInter₂,
+    inner_normalVector_pi_div_two]
+
+/-- The outer cap is closed, being an intersection of closed half-spaces. -/
+theorem isClosed_gerverOuterCap : IsClosed gerverOuterCap := by
+  rw [gerverOuterCap_eq_iInter]
+  refine IsClosed.inter (isClosed_le continuous_const
+    (continuous_id.inner continuous_const)) ?_
+  exact isClosed_biInter fun t _ ↦ IsClosed.inter
+    (isClosed_le (continuous_id.inner continuous_const) continuous_const)
+    (isClosed_le (continuous_id.inner continuous_const) continuous_const)
+
+/-- The outer cap is convex, being an intersection of half-spaces. -/
+theorem convex_gerverOuterCap : Convex ℝ gerverOuterCap := by
+  rw [gerverOuterCap_eq_iInter]
+  refine Convex.inter (convex_halfSpace_ge (isLinearMap_inner_left _) 0) ?_
+  exact convex_iInter₂ fun t _ ↦ Convex.inter
+    (convex_halfSpace_le (isLinearMap_inner_left _) _)
+    (convex_halfSpace_le (isLinearMap_inner_left _) _)
+
+/-- The outer cap is contained in an explicit coordinate rectangle. -/
+theorem gerverOuterCap_subset_box :
+    gerverOuterCap ⊆ {p : Point |
+      p 0 ∈ Set.Icc (-(inner ℝ (paperGerverPath (Real.pi / 2))
+        (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) + 1)) 1 ∧ p 1 ∈ Set.Icc 0 1} := by
+  have hTpos : (0 : ℝ) < Real.pi / 2 := by positivity
+  intro q hq
+  obtain ⟨hy, hcap⟩ := hq
+  have h0 := hcap 0 ⟨le_rfl, hTpos.le⟩
+  have hT := hcap (Real.pi / 2) ⟨hTpos.le, le_rfl⟩
+  rw [paperGerverPath_zero, inner_normalVector_zero, inner_normalVector_zero,
+    inner_tangentVector_zero, inner_tangentVector_zero] at h0
+  refine ⟨⟨?_, ?_⟩, hy, ?_⟩
+  · have h := hT.2
+    rw [inner_tangentVector_pi_div_two] at h
+    linarith
+  · have h := h0.1
+    simpa using h
+  · have h := h0.2
+    simpa using h
+
+/-- The outer cap is compact: it is closed and contained in a coordinate rectangle. -/
+theorem isCompact_gerverOuterCap : IsCompact gerverOuterCap := by
+  refine Metric.isCompact_of_isClosed_isBounded isClosed_gerverOuterCap ?_
+  set c : ℝ := -(inner ℝ (paperGerverPath (Real.pi / 2))
+    (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) + 1) with hc
+  set K : ℝ := |c| + 1 with hK
+  rw [isBounded_iff_forall_norm_le]
+  refine ⟨2 * K, fun q hq ↦ ?_⟩
+  obtain ⟨⟨hx0, hx1⟩, hy0, hy1⟩ := gerverOuterCap_subset_box hq
+  have hK1 : (1 : ℝ) ≤ K := by rw [hK]; linarith [abs_nonneg c]
+  have hsq : ‖q‖ ^ 2 = q 0 ^ 2 + q 1 ^ 2 := by
+    rw [EuclideanSpace.norm_eq, Real.sq_sqrt (by positivity)]
+    simp [Fin.sum_univ_two, sq_abs]
+  have hx : |q 0| ≤ K := by
+    rw [abs_le]
+    refine ⟨?_, by rw [hK]; linarith [abs_nonneg c]⟩
+    have hcle : -|c| ≤ c := neg_abs_le c
+    rw [hK]
+    linarith [hx0]
+  have hy : |q 1| ≤ K := by
+    rw [abs_le]
+    exact ⟨by linarith, by linarith⟩
+  have hx2 : q 0 ^ 2 ≤ K ^ 2 := by nlinarith [sq_abs (q 0), abs_nonneg (q 0)]
+  have hy2 : q 1 ^ 2 ≤ K ^ 2 := by nlinarith [sq_abs (q 1), abs_nonneg (q 1)]
+  nlinarith [norm_nonneg q, hsq, hx2, hy2, hK1]
+
+/-- The outer cap is Borel measurable. -/
+theorem measurableSet_gerverOuterCap : MeasurableSet gerverOuterCap :=
+  isClosed_gerverOuterCap.measurableSet
+
+/-- The outer cap has finite planar volume. -/
+theorem volume_gerverOuterCap_lt_top : volume gerverOuterCap < ⊤ :=
+  isCompact_gerverOuterCap.measure_lt_top
+
+/-! ### The support-contact fan and the cap lower bound -/
+
+/-- The fan anchor `L = C (π / 2)`. -/
+def gerverFanAnchor : Point := paperGerverContacts (Real.pi / 2) 2
+
+/-- The `i`-th listed support contact, `i < fanCount`. -/
+def gerverFanPoint (i : ℕ) : Point :=
+  gerverContactPoint (fanKind i) (gerverGridTime (fanIdx i))
+
+/-- The support normal of the `i`-th listed contact. -/
+def gerverFanNormal (i : ℕ) : ℝ :=
+  if i ≤ 5 * NN then gerverGridTime i else gerverGridTime (i - 5 * NN) + Real.pi / 2
+
+/-- The support value of the first outer contact in its own normal direction. -/
+private theorem inner_paperGerverContacts_zero_normalVector (t : ℝ) :
+    inner ℝ (paperGerverContacts t 0) (normalVector (t : Real.Angle)) =
+      inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) + 1 := by
+  change inner ℝ (paperGerverPath t +
+    (paperGerverVelocityComponents t).1 • tangentVector (t : Real.Angle) +
+      normalVector (t : Real.Angle)) (normalVector (t : Real.Angle)) = _
+  rw [inner_add_left, inner_add_left, real_inner_smul_left,
+    inner_tangentVector_normalVector_real, inner_normalVector_self]
+  simp
+
+/-- The support value of the third outer contact in the rotated normal direction. -/
+private theorem inner_paperGerverContacts_two_tangentVector (t : ℝ) :
+    inner ℝ (paperGerverContacts t 2) (tangentVector (t : Real.Angle)) =
+      inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) + 1 := by
+  change inner ℝ (paperGerverPath t -
+    (paperGerverVelocityComponents t).2 • normalVector (t : Real.Angle) +
+      tangentVector (t : Real.Angle)) (tangentVector (t : Real.Angle)) = _
+  rw [inner_add_left, inner_sub_left, real_inner_smul_left,
+    inner_normalVector_tangentVector, inner_tangentVector_self]
+  ring
+
+/-- `A t` attains the cap support at normal `t`. -/
+theorem gerverOuterCap_support_A (t : ℝ) (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2))
+    {q : Point} (hq : q ∈ gerverOuterCap) :
+    inner ℝ q (normalVector (t : Real.Angle)) ≤
+      inner ℝ (paperGerverContacts t 0) (normalVector (t : Real.Angle)) := by
+  rw [inner_paperGerverContacts_zero_normalVector]
+  exact (hq.2 t ht).1
+
+/-- `C t` attains the cap support at normal `t + π / 2`. -/
+theorem gerverOuterCap_support_C (t : ℝ) (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2))
+    {q : Point} (hq : q ∈ gerverOuterCap) :
+    inner ℝ q (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) ≤
+      inner ℝ (paperGerverContacts t 2) (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) := by
+  rw [normalVector_add_pi_div_two_real, inner_paperGerverContacts_two_tangentVector]
+  exact (hq.2 t ht).2
+
+/-- The anchor sits on the wall. -/
+theorem gerverFanAnchor_snd : gerverFanAnchor 1 = 0 := by
+  have hTpos : (0 : ℝ) < Real.pi / 2 := by positivity
+  have hT : (Real.pi / 2) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨hTpos.le, le_rfl⟩
+  have h := congrArg Prod.snd (fromPlane_paperGerverContacts (Real.pi / 2) hT 2)
+  have hC : (GerverSofa.PartC.C (Real.pi / 2)).2 = 0 :=
+    GerverSofa.PartC.Stage2.C_T_snd_zero
+  change (GerverSofa.PartF.Coordinates.fromPlane (paperGerverContacts (Real.pi / 2) 2)).2 = 0
+  rw [h]
+  simpa using hC
+
+/-- The cap lies in the closed quadrant above the anchor. -/
+theorem gerverOuterCap_quadrant {q : Point} (hq : q ∈ gerverOuterCap) :
+    gerverFanAnchor 0 ≤ q 0 ∧ 0 ≤ q 1 := by
+  have hTpos : (0 : ℝ) < Real.pi / 2 := by positivity
+  have hT : (Real.pi / 2) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨hTpos.le, le_rfl⟩
+  refine ⟨?_, hq.1⟩
+  have h := gerverOuterCap_support_C (Real.pi / 2) hT hq
+  have hsplit : (Real.pi / 2 + Real.pi / 2 : ℝ) = 0 + Real.pi := by ring
+  rw [hsplit, normalVector_add_pi, inner_neg_right, inner_neg_right,
+    inner_normalVector_zero, inner_normalVector_zero] at h
+  change gerverFanAnchor 0 ≤ q 0
+  simp only [gerverFanAnchor]
+  linarith
+
+/-- The listed support normals are strictly increasing. -/
+theorem gerverFanNormal_lt_succ (i : ℕ) (hi : i + 1 < fanCount) :
+    gerverFanNormal i < gerverFanNormal (i + 1) := by
+  have hNN : NN = 64 := rfl
+  simp only [fanCount] at hi
+  simp only [gerverFanNormal]
+  split_ifs with h1 h2
+  · exact gerverGridTime_lt_succ i (by omega)
+  · have hi5 : i = 5 * NN := by omega
+    subst hi5
+    rw [Nat.add_sub_cancel_left, gerverGridTime_top]
+    have h0 : gerverGridTime 0 < gerverGridTime 1 := gerverGridTime_lt_succ 0 (by omega)
+    rw [gerverGridTime_zero] at h0
+    linarith
+  · omega
+  · have hrw : i + 1 - 5 * NN = (i - 5 * NN) + 1 := by omega
+    rw [hrw]
+    have h := gerverGridTime_lt_succ (i - 5 * NN) (by omega)
+    linarith
+
+/-- The listed support normals lie in `[0, π)`. -/
+theorem gerverFanNormal_mem_Ico (i : ℕ) (hi : i < fanCount) :
+    0 ≤ gerverFanNormal i ∧ gerverFanNormal i < Real.pi := by
+  have hNN : NN = 64 := rfl
+  simp only [fanCount] at hi
+  have hTpos : (0 : ℝ) < Real.pi / 2 := by positivity
+  simp only [gerverFanNormal]
+  split_ifs with h1
+  · obtain ⟨hlo, hhi⟩ := gerverGridTime_mem_Icc i h1
+    exact ⟨hlo, by linarith⟩
+  · have hlt : i - 5 * NN < 5 * NN := by omega
+    have hstrict := gerverGridTime_lt_of_lt hlt le_rfl
+    rw [gerverGridTime_top] at hstrict
+    obtain ⟨hlo, -⟩ := gerverGridTime_mem_Icc (i - 5 * NN) hlt.le
+    exact ⟨by linarith, by linarith⟩
+
+/-- The `kind = 1` slot of the contact dictionary is the first outer contact. -/
+private theorem gerverContactPoint_one (x : ℝ) :
+    gerverContactPoint 1 x = paperGerverContacts x 0 := rfl
+
+/-- The `kind = 3` slot of the contact dictionary is the third outer contact. -/
+private theorem gerverContactPoint_three (x : ℝ) :
+    gerverContactPoint 3 x = paperGerverContacts x 2 := rfl
+
+/-- Every listed contact lies in the cap. -/
+theorem gerverFanPoint_mem (i : ℕ) (hi : i < fanCount) :
+    gerverFanPoint i ∈ gerverOuterCap := by
+  have hNN : NN = 64 := rfl
+  simp only [fanCount] at hi
+  simp only [gerverFanPoint, fanKind, fanIdx]
+  split_ifs with h1
+  · rw [gerverContactPoint_one]
+    exact gerver_outer_contact_A _ (gerverGridTime_mem_Icc i h1)
+  · rw [gerverContactPoint_three]
+    exact gerver_outer_contact_C _ (gerverGridTime_mem_Icc (i - 5 * NN) (by omega))
+
+/-- Every listed contact attains the cap support at its listed normal. -/
+theorem gerverFanPoint_support (i : ℕ) (hi : i < fanCount) {q : Point}
+    (hq : q ∈ gerverOuterCap) :
+    inner ℝ q (normalVector (gerverFanNormal i : Real.Angle)) ≤
+      inner ℝ (gerverFanPoint i) (normalVector (gerverFanNormal i : Real.Angle)) := by
+  have hNN : NN = 64 := rfl
+  simp only [fanCount] at hi
+  simp only [gerverFanPoint, gerverFanNormal, fanKind, fanIdx]
+  split_ifs with h1
+  · rw [gerverContactPoint_one]
+    exact gerverOuterCap_support_A _ (gerverGridTime_mem_Icc i h1) hq
+  · rw [gerverContactPoint_three]
+    exact gerverOuterCap_support_C _ (gerverGridTime_mem_Icc (i - 5 * NN) (by omega)) hq
+
+/-- The certificate encloses both coordinates of every listed fan contact. -/
+private theorem fanZ_sound (i : ℕ) (hi : i < fanCount) :
+    SI.Contains (fanZ i).1 (gerverFanPoint i 0) ∧
+      SI.Contains (fanZ i).2 (gerverFanPoint i 1) := by
+  have hNN : NN = 64 := rfl
+  simp only [fanCount] at hi
+  have hidx : fanIdx i ≤ 5 * NN := by
+    simp only [fanIdx]
+    split_ifs with h <;> omega
+  have hkind : fanKind i ≤ 4 := by
+    simp only [fanKind]
+    split_ifs <;> norm_num
+  exact contactZ_sound (fanIdx i) hidx (fanKind i) hkind
+
+/-- The certificate encloses both coordinates of the fan anchor. -/
+private theorem anchorZ_sound :
+    SI.Contains anchorZ.1 (gerverFanAnchor 0) ∧ SI.Contains anchorZ.2 (gerverFanAnchor 1) := by
+  have h := contactZ_sound (5 * NN) le_rfl 3 (by norm_num)
+  rwa [gerverGridTime_top] at h
+
+/-- The certificate encloses the fan shoelace sum. -/
+theorem capDoubledZ_sound :
+    SI.Contains capDoubledZ (∑ i ∈ Finset.range (fanCount - 1),
+      planeCrossProduct (gerverFanPoint i - gerverFanAnchor)
+        (gerverFanPoint (i + 1) - gerverFanAnchor)) := by
+  have hNN : NN = 64 := rfl
+  refine SI.contains_foldl_range (fanCount - 1) fun i hi => ?_
+  simp only [fanCount] at hi
+  obtain ⟨ha1, ha2⟩ := fanZ_sound i (by simp only [fanCount]; omega)
+  obtain ⟨hb1, hb2⟩ := fanZ_sound (i + 1) (by simp only [fanCount]; omega)
+  obtain ⟨hL1, hL2⟩ := anchorZ_sound
+  have hcross : planeCrossProduct (gerverFanPoint i - gerverFanAnchor)
+      (gerverFanPoint (i + 1) - gerverFanAnchor) =
+      (gerverFanPoint i 0 - gerverFanAnchor 0) *
+          (gerverFanPoint (i + 1) 1 - gerverFanAnchor 1) -
+        (gerverFanPoint i 1 - gerverFanAnchor 1) *
+          (gerverFanPoint (i + 1) 0 - gerverFanAnchor 0) := by
+    simp [planeCrossProduct]
+  rw [hcross]
+  exact SI.contains_sub
+    (SI.contains_mul (SI.contains_sub ha1 hL1) (SI.contains_sub hb2 hL2))
+    (SI.contains_mul (SI.contains_sub ha2 hL2) (SI.contains_sub hb1 hL1))
+
+/-- The cap area lower bound. -/
+theorem gerverOuterCap_area_certified_lower_bound :
+    (28609 : ℝ) / 10000 ≤ ClassicalResults.area gerverOuterCap := by
+  have hNN : NN = 64 := rfl
+  have hTpos : (0 : ℝ) < Real.pi / 2 := by positivity
+  have hcount : fanCount = 640 := rfl
+  -- The fan data on `Fin 640`.
+  have hlt : ∀ i : Fin 640, (i : ℕ) < fanCount := fun i => by
+    rw [hcount]; exact i.isLt
+  have hmono : StrictMono fun i : Fin 640 => gerverFanNormal (i : ℕ) := by
+    refine Fin.strictMono_iff_lt_succ.2 fun i => ?_
+    simpa using gerverFanNormal_lt_succ (i : ℕ) (by rw [hcount]; omega)
+  obtain ⟨-, hfan⟩ := supportContact_fan_area gerverOuterCap
+    isCompact_gerverOuterCap convex_gerverOuterCap gerverFanAnchor
+    (gerver_outer_contact_C (Real.pi / 2) ⟨hTpos.le, le_rfl⟩)
+    gerverFanAnchor_snd (fun _ hq => gerverOuterCap_quadrant hq) 639
+    (fun i : Fin 640 => gerverFanNormal (i : ℕ)) hmono
+    (fun i => gerverFanNormal_mem_Ico (i : ℕ) (hlt i))
+    (fun i : Fin 640 => gerverFanPoint (i : ℕ)) (fun i => gerverFanPoint_mem (i : ℕ) (hlt i))
+    (fun i q hq => gerverFanPoint_support (i : ℕ) (hlt i) hq)
+  -- Identify the `Fin`-indexed fan sum with the certificate's range sum.
+  rw [show (∑ i : Fin 639, planeCrossProduct
+        (gerverFanPoint (i.castSucc : Fin 640) - gerverFanAnchor)
+        (gerverFanPoint (i.succ : Fin 640) - gerverFanAnchor)) =
+      ∑ i ∈ Finset.range (fanCount - 1),
+        planeCrossProduct (gerverFanPoint i - gerverFanAnchor)
+          (gerverFanPoint (i + 1) - gerverFanAnchor) by
+    rw [hcount]
+    exact Fin.sum_univ_eq_sum_range (fun i => planeCrossProduct
+      (gerverFanPoint i - gerverFanAnchor) (gerverFanPoint (i + 1) - gerverFanAnchor)) 639] at hfan
+  -- The kernel-checked numeric inequality.  The fan sum and the certificate endpoint are
+  -- abstracted into local variables first, and the arithmetic tactics are used in their
+  -- `only` form, so that no tactic ever tries to evaluate the 639-term fold.
+  obtain ⟨S, hS⟩ : ∃ S : ℝ, (∑ i ∈ Finset.range (fanCount - 1),
+      planeCrossProduct (gerverFanPoint i - gerverFanAnchor)
+        (gerverFanPoint (i + 1) - gerverFanAnchor)) = S := ⟨_, rfl⟩
+  obtain ⟨z, hz⟩ : ∃ z : ℤ, capDoubledZ.lo = z := ⟨_, rfl⟩
+  obtain ⟨hlo, -⟩ := capDoubledZ_sound
+  rw [hS] at hfan
+  rw [hS, hz] at hlo
+  have hOK : 2 * 28609 * M ≤ 10000 * capDoubledZ.lo := by
+    have h := capOK_true
+    unfold capOK at h
+    exact of_decide_eq_true h
+  rw [hz] at hOK
+  have hOK' : (2 * 28609 * (M : ℝ)) ≤ 10000 * (z : ℝ) := by exact_mod_cast hOK
+  have hM : (0 : ℝ) < (M : ℝ) := SI.Mpos
+  have hkey : 2 * 28609 ≤ 10000 * S := by nlinarith only [hlo, hOK', hM]
+  refine le_trans ?_ hfan
+  linarith only [hkey]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The rectangle cover of the literal Gerver niche
+
+The literal niche is the union of the strict vertical fills under the three pieces of its
+roof (`gerver_niche_vertical_fills`).  Each piece is traversed with monotone horizontal
+coordinate, so subdividing the seven monotone roof stretches at the grid angles covers the
+niche by `7 * NN` coordinate rectangles whose widths and heights the certificate encloses
+(`gerverNicheRect_covers`).  The kernel-checked numeric conclusion
+`GerverAreaCert.nicheOK_true` bounds the total rectangle area, hence `|N₀| ≤ 3301 / 5000`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open MeasureTheory
+open GerverAreaCert
+
+/-- The literal niche is Borel measurable: it is a closed half-plane intersected with a
+countable union of open sets. -/
+theorem measurableSet_gerverLiteralNiche : MeasurableSet gerverLiteralNiche := by
+  have h : gerverLiteralNiche =
+      {q : Point | 0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))} ∩
+        ⋃ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+          ({q : Point | inner ℝ (q - paperGerverPath t) (normalVector (t : Real.Angle)) < 0} ∩
+            {q : Point |
+              inner ℝ (q - paperGerverPath t) (tangentVector (t : Real.Angle)) < 0}) := by
+    ext q
+    simp only [gerverLiteralNiche, Set.mem_ofPred_eq, Set.mem_inter_iff, Set.mem_iUnion₂,
+      inner_normalVector_pi_div_two]
+    tauto
+  rw [h]
+  refine MeasurableSet.inter
+    ((isClosed_le continuous_const (continuous_id.inner continuous_const)).measurableSet) ?_
+  refine IsOpen.measurableSet (isOpen_biUnion fun t _ ↦ IsOpen.inter ?_ ?_)
+  · exact isOpen_lt ((continuous_id.sub continuous_const).inner continuous_const) continuous_const
+  · exact isOpen_lt ((continuous_id.sub continuous_const).inner continuous_const) continuous_const
+
+/-! ### The rectangle cover and the niche upper bound -/
+
+/-- The covering rectangle of the `j`-th subinterval of roof piece `r`. -/
+def gerverNicheRect (r j : ℕ) : Set Point :=
+  {p : Point | p 0 ∈ Set.Icc (((rectLoZ r j : ℤ) : ℝ) / (M : ℝ))
+      (((rectHiZ r j : ℤ) : ℝ) / (M : ℝ)) ∧
+    p 1 ∈ Set.Icc 0 (((rectHZ r j : ℤ) : ℝ) / (M : ℝ))}
+
+/-- A strict vertical fill splits along a subdivision of its parameter interval. -/
+theorem strictVerticalFill_Icc_union {f : ℝ → Point} {a b c : ℝ} (hab : a ≤ b) (hbc : b ≤ c) :
+    strictVerticalFill f (Set.Icc a c) =
+      strictVerticalFill f (Set.Icc a b) ∪ strictVerticalFill f (Set.Icc b c) := by
+  ext q
+  simp only [strictVerticalFill, Set.mem_ofPred_eq, Set.mem_union, Set.mem_Icc]
+  constructor
+  · rintro ⟨t, ⟨h1, h2⟩, h3⟩
+    rcases le_total t b with h | h
+    · exact Or.inl ⟨t, ⟨h1, h⟩, h3⟩
+    · exact Or.inr ⟨t, ⟨h, h2⟩, h3⟩
+  · rintro (⟨t, ⟨h1, h2⟩, h3⟩ | ⟨t, ⟨h1, h2⟩, h3⟩)
+    · exact ⟨t, ⟨h1, h2.trans hbc⟩, h3⟩
+    · exact ⟨t, ⟨hab.trans h1, h2⟩, h3⟩
+
+/-- The strictly increasing chain of the six `ℕ`-indexed stage endpoints. -/
+private theorem gerverStageTime_chain :
+    gerverStageTime 0 = 0 ∧ gerverStageTime 0 < gerverStageTime 1 ∧
+      gerverStageTime 1 < gerverStageTime 2 ∧ gerverStageTime 2 < gerverStageTime 3 ∧
+      gerverStageTime 3 < gerverStageTime 4 ∧ gerverStageTime 4 < gerverStageTime 5 ∧
+      gerverStageTime 5 = Real.pi / 2 :=
+  ⟨gerverStageTime_zero, gerverStageTime_lt_succ 0 (by norm_num),
+    gerverStageTime_lt_succ 1 (by norm_num), gerverStageTime_lt_succ 2 (by norm_num),
+    gerverStageTime_lt_succ 3 (by norm_num), gerverStageTime_lt_succ 4 (by norm_num),
+    gerverStageTime_five⟩
+
+/-- `gerverRoofReverseTime` written in the `ℕ`-indexed stage endpoints. -/
+private theorem gerverRoofReverseTime_eq_stageTime (s : ℝ) :
+    gerverRoofReverseTime s = gerverStageTime 4 -
+      (gerverStageTime 4 - gerverStageTime 1) / (gerverStageTime 3 - gerverStageTime 2) *
+        (s - gerverStageTime 2) := rfl
+
+/-- The first roof piece, in the `ℕ`-indexed stage endpoints. -/
+private theorem gerverNicheRoof_eq_D {s : ℝ} (hs : s ∈ Set.Icc (0 : ℝ) (Real.pi / 2))
+    (h : s ≤ gerverStageTime 2) :
+    gerverNicheRoof ⟨s, hs⟩ = paperGerverContacts s 3 :=
+  gerverNicheRoof_of_le_two hs h
+
+/-- The middle roof piece, in the `ℕ`-indexed stage endpoints. -/
+private theorem gerverNicheRoof_eq_path {s : ℝ} (hs : s ∈ Set.Icc (0 : ℝ) (Real.pi / 2))
+    (h2 : gerverStageTime 2 ≤ s) (h3 : s ≤ gerverStageTime 3) :
+    gerverNicheRoof ⟨s, hs⟩ = paperGerverPath (gerverRoofReverseTime s) :=
+  gerverNicheRoof_mid hs h2 h3
+
+/-- The last roof piece, in the `ℕ`-indexed stage endpoints. -/
+private theorem gerverNicheRoof_eq_B {s : ℝ} (hs : s ∈ Set.Icc (0 : ℝ) (Real.pi / 2))
+    (h3 : gerverStageTime 3 ≤ s) :
+    gerverNicheRoof ⟨s, hs⟩ = paperGerverContacts s 1 :=
+  gerverNicheRoof_of_ge_three hs h3
+
+/-- Two roof arguments compare as their underlying reals. -/
+private theorem gerverNicheRoof_fst_le_of_le {a b : ℝ} (ha : a ∈ Set.Icc (0 : ℝ) (Real.pi / 2))
+    (hb : b ∈ Set.Icc (0 : ℝ) (Real.pi / 2)) (hab : a ≤ b) :
+    gerverNicheRoof ⟨a, ha⟩ 0 ≤ gerverNicheRoof ⟨b, hb⟩ 0 :=
+  gerver_niche_roof_strictMono.monotone (Subtype.mk_le_mk.mpr hab)
+
+/-- The `D` piece has monotone horizontal coordinate. -/
+private theorem monotoneOn_paperGerverContacts_three_fst :
+    MonotoneOn (fun t ↦ paperGerverContacts t 3 0)
+      (Set.Icc (gerverStageTime 0) (gerverStageTime 2)) := by
+  obtain ⟨h0, c01, c12, c23, c34, c45, h5⟩ := gerverStageTime_chain
+  intro a ha b hb hab
+  have hA : a ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+    ⟨by linarith [ha.1], by linarith [ha.2]⟩
+  have hB : b ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+    ⟨by linarith [hb.1], by linarith [hb.2]⟩
+  have h := gerverNicheRoof_fst_le_of_le hA hB hab
+  rwa [gerverNicheRoof_eq_D hA ha.2, gerverNicheRoof_eq_D hB hb.2] at h
+
+/-- The `B` piece has monotone horizontal coordinate. -/
+private theorem monotoneOn_paperGerverContacts_one_fst :
+    MonotoneOn (fun t ↦ paperGerverContacts t 1 0)
+      (Set.Icc (gerverStageTime 3) (gerverStageTime 5)) := by
+  obtain ⟨h0, c01, c12, c23, c34, c45, h5⟩ := gerverStageTime_chain
+  intro a ha b hb hab
+  have hA : a ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+    ⟨by linarith [ha.1], by linarith [ha.2]⟩
+  have hB : b ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+    ⟨by linarith [hb.1], by linarith [hb.2]⟩
+  have h := gerverNicheRoof_fst_le_of_le hA hB hab
+  rwa [gerverNicheRoof_eq_B hA ha.1, gerverNicheRoof_eq_B hB hb.1] at h
+
+/-- The ambient path has antitone horizontal coordinate over the middle roof piece. -/
+private theorem antitoneOn_paperGerverPath_fst :
+    AntitoneOn (fun t ↦ paperGerverPath t 0)
+      (Set.Icc (gerverStageTime 1) (gerverStageTime 4)) := by
+  obtain ⟨h0, c01, c12, c23, c34, c45, h5⟩ := gerverStageTime_chain
+  have hne32 : gerverStageTime 3 - gerverStageTime 2 ≠ 0 :=
+    sub_ne_zero_of_ne (by intro hz; linarith)
+  have hne41 : gerverStageTime 4 - gerverStageTime 1 ≠ 0 :=
+    sub_ne_zero_of_ne (by intro hz; linarith)
+  have hc : 0 < (gerverStageTime 4 - gerverStageTime 1) /
+      (gerverStageTime 3 - gerverStageTime 2) := div_pos (by linarith) (by linarith)
+  have hcne : (gerverStageTime 4 - gerverStageTime 1) /
+      (gerverStageTime 3 - gerverStageTime 2) ≠ 0 := ne_of_gt hc
+  set c := (gerverStageTime 4 - gerverStageTime 1) /
+    (gerverStageTime 3 - gerverStageTime 2) with hcdef
+  -- The reverse-time map is inverted by `t ↦ e₂ + (e₄ - t) / c`.
+  have hinv : ∀ t : ℝ, gerverRoofReverseTime (gerverStageTime 2 +
+      (gerverStageTime 4 - t) / c) = t := by
+    intro t
+    rw [gerverRoofReverseTime_eq_stageTime, ← hcdef]
+    field_simp
+    ring
+  have heq : (gerverStageTime 4 - gerverStageTime 1) / c =
+      gerverStageTime 3 - gerverStageTime 2 := by
+    rw [hcdef]
+    field_simp
+  have hmem : ∀ t : ℝ, gerverStageTime 1 ≤ t → t ≤ gerverStageTime 4 →
+      gerverStageTime 2 ≤ gerverStageTime 2 + (gerverStageTime 4 - t) / c ∧
+        gerverStageTime 2 + (gerverStageTime 4 - t) / c ≤ gerverStageTime 3 := by
+    intro t ht1 ht4
+    have hnn : 0 ≤ (gerverStageTime 4 - t) / c := div_nonneg (by linarith) hc.le
+    have hstep : (gerverStageTime 4 - gerverStageTime 1) / c -
+        (gerverStageTime 4 - t) / c = (t - gerverStageTime 1) / c := by
+      field_simp
+      ring
+    have hnn2 : 0 ≤ (t - gerverStageTime 1) / c := div_nonneg (by linarith) hc.le
+    exact ⟨by linarith, by linarith⟩
+  intro a ha b hb hab
+  obtain ⟨ha2, ha3⟩ := hmem a ha.1 ha.2
+  obtain ⟨hb2, hb3⟩ := hmem b hb.1 hb.2
+  have hAmem : gerverStageTime 2 + (gerverStageTime 4 - a) / c ∈
+      Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨by linarith, by linarith⟩
+  have hBmem : gerverStageTime 2 + (gerverStageTime 4 - b) / c ∈
+      Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨by linarith, by linarith⟩
+  have hba : gerverStageTime 2 + (gerverStageTime 4 - b) / c ≤
+      gerverStageTime 2 + (gerverStageTime 4 - a) / c := by
+    have hsub : (gerverStageTime 4 - a) / c - (gerverStageTime 4 - b) / c = (b - a) / c := by
+      field_simp
+      ring
+    have hnn : 0 ≤ (b - a) / c := div_nonneg (by linarith) hc.le
+    linarith
+  have h := gerverNicheRoof_fst_le_of_le hBmem hAmem hba
+  rwa [gerverNicheRoof_eq_path hBmem hb2 hb3, gerverNicheRoof_eq_path hAmem ha2 ha3,
+    hinv a, hinv b] at h
+
+/-- Each roof piece has monotone horizontal coordinate in the direction recorded by
+`rowFwd`. -/
+theorem gerverNicheRow_monotone (r : ℕ) (hr : r < 7) :
+    (rowFwd r = true → MonotoneOn (fun t ↦ gerverContactPoint (rowKind r) t 0)
+        (Set.Icc (gerverStageTime (rowStage r - 1)) (gerverStageTime (rowStage r)))) ∧
+      (rowFwd r = false → AntitoneOn (fun t ↦ gerverContactPoint (rowKind r) t 0)
+        (Set.Icc (gerverStageTime (rowStage r - 1)) (gerverStageTime (rowStage r)))) := by
+  obtain ⟨-, c01, c12, c23, c34, c45, -⟩ := gerverStageTime_chain
+  have sD0 : Set.Icc (gerverStageTime 0) (gerverStageTime 1) ⊆
+      Set.Icc (gerverStageTime 0) (gerverStageTime 2) := Set.Icc_subset_Icc le_rfl c12.le
+  have sD1 : Set.Icc (gerverStageTime 1) (gerverStageTime 2) ⊆
+      Set.Icc (gerverStageTime 0) (gerverStageTime 2) := Set.Icc_subset_Icc c01.le le_rfl
+  have sP2 : Set.Icc (gerverStageTime 3) (gerverStageTime 4) ⊆
+      Set.Icc (gerverStageTime 1) (gerverStageTime 4) :=
+    Set.Icc_subset_Icc (c12.trans c23).le le_rfl
+  have sP3 : Set.Icc (gerverStageTime 2) (gerverStageTime 3) ⊆
+      Set.Icc (gerverStageTime 1) (gerverStageTime 4) := Set.Icc_subset_Icc c12.le c34.le
+  have sP4 : Set.Icc (gerverStageTime 1) (gerverStageTime 2) ⊆
+      Set.Icc (gerverStageTime 1) (gerverStageTime 4) :=
+    Set.Icc_subset_Icc le_rfl (c23.trans c34).le
+  have sB5 : Set.Icc (gerverStageTime 3) (gerverStageTime 4) ⊆
+      Set.Icc (gerverStageTime 3) (gerverStageTime 5) := Set.Icc_subset_Icc le_rfl c45.le
+  have sB6 : Set.Icc (gerverStageTime 4) (gerverStageTime 5) ⊆
+      Set.Icc (gerverStageTime 3) (gerverStageTime 5) := Set.Icc_subset_Icc c34.le le_rfl
+  interval_cases r
+  · exact ⟨fun _ ↦ monotoneOn_paperGerverContacts_three_fst.mono sD0, fun h ↦ absurd h (by decide)⟩
+  · exact ⟨fun _ ↦ monotoneOn_paperGerverContacts_three_fst.mono sD1, fun h ↦ absurd h (by decide)⟩
+  · exact ⟨fun h ↦ absurd h (by decide), fun _ ↦ antitoneOn_paperGerverPath_fst.mono sP2⟩
+  · exact ⟨fun h ↦ absurd h (by decide), fun _ ↦ antitoneOn_paperGerverPath_fst.mono sP3⟩
+  · exact ⟨fun h ↦ absurd h (by decide), fun _ ↦ antitoneOn_paperGerverPath_fst.mono sP4⟩
+  · exact ⟨fun _ ↦ monotoneOn_paperGerverContacts_one_fst.mono sB5, fun h ↦ absurd h (by decide)⟩
+  · exact ⟨fun _ ↦ monotoneOn_paperGerverContacts_one_fst.mono sB6, fun h ↦ absurd h (by decide)⟩
+
+/-- An integer lower bound at scale `M` read as a bound on the real quotient. -/
+private theorem div_M_le_of_le {p : ℤ} {x : ℝ} (h : (p : ℝ) ≤ (M : ℝ) * x) :
+    (p : ℝ) / (M : ℝ) ≤ x :=
+  (div_le_iff₀ SI.Mpos).2 (by linarith)
+
+/-- An integer upper bound at scale `M` read as a bound on the real quotient. -/
+private theorem le_div_M_of_le {p : ℤ} {x : ℝ} (h : (M : ℝ) * x ≤ (p : ℝ)) :
+    x ≤ (p : ℝ) / (M : ℝ) :=
+  (le_div_iff₀ SI.Mpos).2 (by linarith)
+
+/-- A strict vertical fill over an increasing subdivision of its parameter interval is the
+union of the fills of the pieces.  Iterated form of `strictVerticalFill_Icc_union`. -/
+private theorem strictVerticalFill_Icc_biUnion {f : ℝ → Point} (a : ℕ → ℝ) :
+    ∀ n : ℕ, 0 < n → (∀ i j, i ≤ j → j ≤ n → a i ≤ a j) →
+      strictVerticalFill f (Set.Icc (a 0) (a n)) =
+        ⋃ i ∈ Finset.range n, strictVerticalFill f (Set.Icc (a i) (a (i + 1))) := by
+  intro n
+  induction n with
+  | zero => intro h; exact absurd h (by omega)
+  | succ n ih =>
+    intro _ ha
+    rcases Nat.eq_zero_or_pos n with rfl | hpos
+    · simp
+    · rw [strictVerticalFill_Icc_union (ha 0 n (Nat.zero_le _) (by omega))
+        (ha n (n + 1) (by omega) le_rfl),
+        ih hpos fun i j hij hj => ha i j hij (by omega),
+        Finset.range_add_one, Finset.set_biUnion_insert, Set.union_comm]
+
+/-- Each covering rectangle contains the strict vertical fill of its subinterval. -/
+theorem gerverNicheRect_covers (r j : ℕ) (hr : r < 7) (hj : j < NN) :
+    strictVerticalFill (gerverContactPoint (rowKind r))
+        (Set.Icc (gerverGridTime (rowBase r j)) (gerverGridTime (rowBase r j + 1))) ⊆
+      gerverNicheRect r j := by
+  have hNN : NN = 64 := rfl
+  have hM := SI.Mpos
+  have hk : rowKind r ≤ 4 := by interval_cases r <;> decide
+  have hS1 : 1 ≤ rowStage r := by interval_cases r <;> decide
+  have hS5 : rowStage r ≤ 5 := by interval_cases r <;> decide
+  have hj64 : j < 64 := by rwa [hNN] at hj
+  have hm1 : rowBase r j + 1 ≤ 5 * NN := by simp only [rowBase, hNN]; omega
+  have hm : rowBase r j ≤ 5 * NN := by omega
+  have hlo : gerverStageTime (rowStage r - 1) ≤ gerverGridTime (rowBase r j) := by
+    have h := gerverGridTime_le_of_le
+      (show (rowStage r - 1) * NN ≤ rowBase r j by simp only [rowBase]; omega) hm
+    rwa [gerverGridTime_mul_NN] at h
+  have hhi : gerverGridTime (rowBase r j + 1) ≤ gerverStageTime (rowStage r) := by
+    have h := gerverGridTime_le_of_le
+      (show rowBase r j + 1 ≤ rowStage r * NN by simp only [rowBase, hNN]; omega)
+      (Nat.mul_le_mul_right NN hS5)
+    rwa [gerverGridTime_mul_NN] at h
+  have hstep : gerverGridTime (rowBase r j) ≤ gerverGridTime (rowBase r j + 1) :=
+    gerverGridTime_le_of_le (Nat.le_succ _) hm1
+  have hmemL : gerverGridTime (rowBase r j) ∈
+      Set.Icc (gerverStageTime (rowStage r - 1)) (gerverStageTime (rowStage r)) :=
+    ⟨hlo, hstep.trans hhi⟩
+  have hmemR : gerverGridTime (rowBase r j + 1) ∈
+      Set.Icc (gerverStageTime (rowStage r - 1)) (gerverStageTime (rowStage r)) :=
+    ⟨hlo.trans hstep, hhi⟩
+  obtain ⟨hmono, hanti⟩ := gerverNicheRow_monotone r hr
+  rintro q ⟨t, ⟨htl, htr⟩, hq0, hq1a, hq1b⟩
+  have hmemt : t ∈ Set.Icc (gerverStageTime (rowStage r - 1)) (gerverStageTime (rowStage r)) :=
+    ⟨hlo.trans htl, htr.trans hhi⟩
+  have hHmax : rectHZ r j = max
+      (evalZ (stageOf (rowBase r j + 1))
+        ⟨(ttZ (rowBase r j)).lo, (ttZ (rowBase r j + 1)).hi⟩ (rowKind r)).2.hi
+      (contactZ (rowBase r j) (rowKind r)).2.hi := by
+    simp only [rectHZ, SI.imax_eq_max]
+  have hH : (M : ℝ) * gerverContactPoint (rowKind r) t 1 ≤ (rectHZ r j : ℝ) := by
+    rcases eq_or_lt_of_le htl with heq | hlt
+    · have h := (contactZ_sound (rowBase r j) hm (rowKind r) hk).2.2
+      rw [← heq]
+      refine h.trans ?_
+      rw [hHmax]
+      exact_mod_cast le_max_right _ _
+    · have h := (evalZ_interval_sound (rowBase r j) hm1 (rowKind r) hk ⟨hlt, htr⟩).2.2
+      refine h.trans ?_
+      rw [hHmax]
+      exact_mod_cast le_max_left _ _
+  refine ⟨⟨?_, ?_⟩, hq1a, ?_⟩
+  · rcases Bool.eq_false_or_eq_true (rowFwd r) with hf | hf
+    · have hlo' : rectLoZ r j = (contactZ (rowBase r j) (rowKind r)).1.lo := by
+        simp [rectLoZ, hf]
+      have hx : gerverContactPoint (rowKind r) (gerverGridTime (rowBase r j)) 0 ≤
+          gerverContactPoint (rowKind r) t 0 := hmono hf hmemL hmemt htl
+      have h := (contactZ_sound (rowBase r j) hm (rowKind r) hk).1.1
+      refine div_M_le_of_le ?_
+      rw [hq0, hlo']
+      exact h.trans (mul_le_mul_of_nonneg_left hx hM.le)
+    · have hlo' : rectLoZ r j = (contactZ (rowBase r j + 1) (rowKind r)).1.lo := by
+        simp [rectLoZ, hf]
+      have hx : gerverContactPoint (rowKind r) (gerverGridTime (rowBase r j + 1)) 0 ≤
+          gerverContactPoint (rowKind r) t 0 := hanti hf hmemt hmemR htr
+      have h := (contactZ_sound (rowBase r j + 1) hm1 (rowKind r) hk).1.1
+      refine div_M_le_of_le ?_
+      rw [hq0, hlo']
+      exact h.trans (mul_le_mul_of_nonneg_left hx hM.le)
+  · rcases Bool.eq_false_or_eq_true (rowFwd r) with hf | hf
+    · have hhi' : rectHiZ r j = (contactZ (rowBase r j + 1) (rowKind r)).1.hi := by
+        simp [rectHiZ, hf]
+      have hx : gerverContactPoint (rowKind r) t 0 ≤
+          gerverContactPoint (rowKind r) (gerverGridTime (rowBase r j + 1)) 0 :=
+        hmono hf hmemt hmemR htr
+      have h := (contactZ_sound (rowBase r j + 1) hm1 (rowKind r) hk).1.2
+      refine le_div_M_of_le ?_
+      rw [hq0, hhi']
+      exact (mul_le_mul_of_nonneg_left hx hM.le).trans h
+    · have hhi' : rectHiZ r j = (contactZ (rowBase r j) (rowKind r)).1.hi := by
+        simp [rectHiZ, hf]
+      have hx : gerverContactPoint (rowKind r) t 0 ≤
+          gerverContactPoint (rowKind r) (gerverGridTime (rowBase r j)) 0 :=
+        hanti hf hmemL hmemt htl
+      have h := (contactZ_sound (rowBase r j) hm (rowKind r) hk).1.2
+      refine le_div_M_of_le ?_
+      rw [hq0, hhi']
+      exact (mul_le_mul_of_nonneg_left hx hM.le).trans h
+  · have h2 : (M : ℝ) * q 1 ≤ (M : ℝ) * gerverContactPoint (rowKind r) t 1 :=
+      mul_le_mul_of_nonneg_left hq1b.le hM.le
+    exact le_div_M_of_le (h2.trans hH)
+
+/-- The `NN` rectangles of a row cover the strict vertical fill of its whole stage. -/
+private theorem strictVerticalFill_row_subset (r : ℕ) (hr : r < 7) :
+    strictVerticalFill (gerverContactPoint (rowKind r))
+        (Set.Icc (gerverStageTime (rowStage r - 1)) (gerverStageTime (rowStage r))) ⊆
+      ⋃ j ∈ Finset.range NN, gerverNicheRect r j := by
+  have hNN : NN = 64 := rfl
+  have hS1 : 1 ≤ rowStage r := by interval_cases r <;> decide
+  have hS5 : rowStage r ≤ 5 := by interval_cases r <;> decide
+  have ha0 : gerverGridTime (rowBase r 0) = gerverStageTime (rowStage r - 1) := by
+    simpa [rowBase] using gerverGridTime_mul_NN (rowStage r - 1)
+  have haN : gerverGridTime (rowBase r NN) = gerverStageTime (rowStage r) := by
+    have hb : rowBase r NN = rowStage r * NN := by simp only [rowBase, hNN]; omega
+    rw [hb, gerverGridTime_mul_NN]
+  have hmono : ∀ i j, i ≤ j → j ≤ NN →
+      gerverGridTime (rowBase r i) ≤ gerverGridTime (rowBase r j) := by
+    intro i j hij hj
+    refine gerverGridTime_le_of_le (by simp only [rowBase]; omega) ?_
+    simp only [rowBase, hNN]; omega
+  have hsplit : strictVerticalFill (gerverContactPoint (rowKind r))
+        (Set.Icc (gerverGridTime (rowBase r 0)) (gerverGridTime (rowBase r NN))) =
+      ⋃ i ∈ Finset.range NN, strictVerticalFill (gerverContactPoint (rowKind r))
+        (Set.Icc (gerverGridTime (rowBase r i)) (gerverGridTime (rowBase r i + 1))) :=
+    strictVerticalFill_Icc_biUnion (fun i => gerverGridTime (rowBase r i)) NN
+      (by simp [hNN]) hmono
+  rw [← ha0, ← haN, hsplit]
+  refine Set.iUnion₂_subset fun i hi q hq => ?_
+  simp only [Set.mem_iUnion, exists_prop]
+  exact ⟨i, hi, gerverNicheRect_covers r i hr (Finset.mem_range.1 hi) hq⟩
+
+/-- The `7 * NN` rectangles cover the literal niche. -/
+theorem gerverLiteralNiche_subset_rects :
+    gerverLiteralNiche ⊆
+      ⋃ r ∈ Finset.range 7, ⋃ j ∈ Finset.range NN, gerverNicheRect r j := by
+  have e0 : gerverStageTimes 0 = gerverStageTime 0 := rfl
+  have e1 : gerverStageTimes 1 = gerverStageTime 1 := rfl
+  have e2 : gerverStageTimes 2 = gerverStageTime 2 := rfl
+  have e3 : gerverStageTimes 3 = gerverStageTime 3 := rfl
+  have e4 : gerverStageTimes 4 = gerverStageTime 4 := rfl
+  have e5 : gerverStageTimes 5 = gerverStageTime 5 := rfl
+  have h01 : gerverStageTime 0 ≤ gerverStageTime 1 := (gerverStageTime_lt_succ 0 (by norm_num)).le
+  have h12 : gerverStageTime 1 ≤ gerverStageTime 2 := (gerverStageTime_lt_succ 1 (by norm_num)).le
+  have h23 : gerverStageTime 2 ≤ gerverStageTime 3 := (gerverStageTime_lt_succ 2 (by norm_num)).le
+  have h34 : gerverStageTime 3 ≤ gerverStageTime 4 := (gerverStageTime_lt_succ 3 (by norm_num)).le
+  have h45 : gerverStageTime 4 ≤ gerverStageTime 5 := (gerverStageTime_lt_succ 4 (by norm_num)).le
+  have hrow : ∀ r, r < 7 → strictVerticalFill (gerverContactPoint (rowKind r))
+      (Set.Icc (gerverStageTime (rowStage r - 1)) (gerverStageTime (rowStage r))) ⊆
+      ⋃ r ∈ Finset.range 7, ⋃ j ∈ Finset.range NN, gerverNicheRect r j := by
+    intro r hr q hq
+    have h := strictVerticalFill_row_subset r hr hq
+    simp only [Set.mem_iUnion, exists_prop] at h ⊢
+    obtain ⟨j, hj, hjq⟩ := h
+    exact ⟨r, Finset.mem_range.2 hr, j, hj, hjq⟩
+  rw [gerver_niche_vertical_fills, e0, e1, e2, e3, e4, e5]
+  refine Set.union_subset (Set.union_subset ?_ ?_) ?_
+  · rw [strictVerticalFill_Icc_union h01 h12]
+    exact Set.union_subset (hrow 0 (by norm_num)) (hrow 1 (by norm_num))
+  · rw [strictVerticalFill_Icc_union h12 (h23.trans h34), strictVerticalFill_Icc_union h23 h34]
+    exact Set.union_subset (hrow 4 (by norm_num))
+      (Set.union_subset (hrow 3 (by norm_num)) (hrow 2 (by norm_num)))
+  · rw [strictVerticalFill_Icc_union h34 h45]
+    exact Set.union_subset (hrow 5 (by norm_num)) (hrow 6 (by norm_num))
+
+/-- The planar volume of one covering rectangle. -/
+theorem volume_gerverNicheRect (r j : ℕ) :
+    volume (gerverNicheRect r j) =
+      ENNReal.ofReal ((((rectHiZ r j - rectLoZ r j : ℤ)) : ℝ) / (M : ℝ)) *
+        ENNReal.ofReal (((rectHZ r j : ℤ) : ℝ) / (M : ℝ)) := by
+  rw [gerverNicheRect, EuclideanSpace.volume_setOf_apply_mem_Icc]
+  congr 2
+  · push_cast
+    ring
+  · ring
+
+/-- The certificate bounds the total rectangle volume. -/
+theorem gerverNicheRect_volume_sum_le :
+    ∑ r ∈ Finset.range 7, ∑ j ∈ Finset.range NN, volume (gerverNicheRect r j) ≤
+      ENNReal.ofReal ((3301 : ℝ) / 5000) := by
+  have hM := SI.Mpos
+  have hQ : (0 : ℝ) < (M : ℝ) * (M : ℝ) := mul_pos hM hM
+  -- `unfold` rather than a type ascription: matching `nicheOK` against `decide _` by
+  -- unification would force the elaborator to evaluate the whole certificate.
+  have hcert : 5000 * nicheSumZ ≤ 3301 * M * M := by
+    have h := nicheOK_true
+    unfold nicheOK at h
+    exact of_decide_eq_true h
+  have harea : ∀ r j : ℕ,
+      rectAreaZ r j = max 0 (rectHiZ r j - rectLoZ r j) * max 0 (rectHZ r j) := by
+    intro r j
+    simp only [rectAreaZ, SI.imax_eq_max]
+  have hnn : ∀ r j : ℕ, (0 : ℝ) ≤ (rectAreaZ r j : ℝ) / ((M : ℝ) * (M : ℝ)) := by
+    intro r j
+    refine div_nonneg ?_ hQ.le
+    have : (0 : ℤ) ≤ rectAreaZ r j := by
+      rw [harea]
+      exact mul_nonneg (le_max_left _ _) (le_max_left _ _)
+    exact_mod_cast this
+  have hcell : ∀ r j : ℕ, volume (gerverNicheRect r j) ≤
+      ENNReal.ofReal ((rectAreaZ r j : ℝ) / ((M : ℝ) * (M : ℝ))) := by
+    intro r j
+    have hA : ((rectHiZ r j - rectLoZ r j : ℤ) : ℝ) / (M : ℝ) ≤
+        ((max 0 (rectHiZ r j - rectLoZ r j) : ℤ) : ℝ) / (M : ℝ) := by
+      gcongr
+      exact_mod_cast le_max_right (0 : ℤ) (rectHiZ r j - rectLoZ r j)
+    have hB : ((rectHZ r j : ℤ) : ℝ) / (M : ℝ) ≤
+        ((max 0 (rectHZ r j) : ℤ) : ℝ) / (M : ℝ) := by
+      gcongr
+      exact_mod_cast le_max_right (0 : ℤ) (rectHZ r j)
+    have hA0 : (0 : ℝ) ≤ ((max 0 (rectHiZ r j - rectLoZ r j) : ℤ) : ℝ) / (M : ℝ) := by
+      refine div_nonneg ?_ hM.le
+      exact_mod_cast le_max_left (0 : ℤ) (rectHiZ r j - rectLoZ r j)
+    rw [volume_gerverNicheRect]
+    refine le_trans (mul_le_mul' (ENNReal.ofReal_le_ofReal hA) (ENNReal.ofReal_le_ofReal hB)) ?_
+    rw [← ENNReal.ofReal_mul hA0]
+    refine ENNReal.ofReal_le_ofReal (le_of_eq ?_)
+    rw [harea]
+    push_cast
+    field_simp
+  have hns : nicheSumZ = ∑ r ∈ Finset.range 7, ∑ j ∈ Finset.range NN, rectAreaZ r j := by
+    rw [nicheSumZ, SI.foldl_range_int]
+    exact Finset.sum_congr rfl fun r _ => by rw [rowSumZ, SI.foldl_range_int]
+  have hsum : ∑ r ∈ Finset.range 7, ∑ j ∈ Finset.range NN,
+      (rectAreaZ r j : ℝ) / ((M : ℝ) * (M : ℝ)) =
+        (nicheSumZ : ℝ) / ((M : ℝ) * (M : ℝ)) := by
+    rw [hns]
+    push_cast
+    simp only [← Finset.sum_div]
+  calc ∑ r ∈ Finset.range 7, ∑ j ∈ Finset.range NN, volume (gerverNicheRect r j)
+      ≤ ∑ r ∈ Finset.range 7, ∑ j ∈ Finset.range NN,
+          ENNReal.ofReal ((rectAreaZ r j : ℝ) / ((M : ℝ) * (M : ℝ))) :=
+        Finset.sum_le_sum fun r _ => Finset.sum_le_sum fun j _ => hcell r j
+    _ = ∑ r ∈ Finset.range 7, ENNReal.ofReal
+          (∑ j ∈ Finset.range NN, (rectAreaZ r j : ℝ) / ((M : ℝ) * (M : ℝ))) :=
+        Finset.sum_congr rfl fun r _ =>
+          (ENNReal.ofReal_sum_of_nonneg fun j _ => hnn r j).symm
+    _ = ENNReal.ofReal (∑ r ∈ Finset.range 7, ∑ j ∈ Finset.range NN,
+          (rectAreaZ r j : ℝ) / ((M : ℝ) * (M : ℝ))) :=
+        (ENNReal.ofReal_sum_of_nonneg fun r _ => Finset.sum_nonneg fun j _ => hnn r j).symm
+    _ ≤ ENNReal.ofReal ((3301 : ℝ) / 5000) := by
+        refine ENNReal.ofReal_le_ofReal ?_
+        rw [hsum, div_le_div_iff₀ hQ (by norm_num)]
+        have hR : (5000 : ℝ) * (nicheSumZ : ℝ) ≤ 3301 * (M : ℝ) * (M : ℝ) := by
+          exact_mod_cast hcert
+        linarith
+
+/-- The niche volume bound. -/
+theorem gerver_niche_volume_le :
+    volume gerverLiteralNiche ≤ ENNReal.ofReal ((3301 : ℝ) / 5000) := by
+  calc volume gerverLiteralNiche
+      ≤ volume (⋃ r ∈ Finset.range 7, ⋃ j ∈ Finset.range NN, gerverNicheRect r j) :=
+        measure_mono gerverLiteralNiche_subset_rects
+    _ ≤ ∑ r ∈ Finset.range 7, volume (⋃ j ∈ Finset.range NN, gerverNicheRect r j) :=
+        measure_biUnion_finset_le _ _
+    _ ≤ ∑ r ∈ Finset.range 7, ∑ j ∈ Finset.range NN, volume (gerverNicheRect r j) :=
+        Finset.sum_le_sum fun _ _ => measure_biUnion_finset_le _ _
+    _ ≤ ENNReal.ofReal ((3301 : ℝ) / 5000) := gerverNicheRect_volume_sum_le
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Rational area bounds for the canonical Gerver sofa
+
+The canonical Gerver sofa is the paper's literal set (`gerver_canonical_paper_literal`), the
+difference of the literal outer cap and the literal niche.  The cap and the niche are Borel
+of finite area with `28609 / 10000 ≤ |K₀|` and `|N₀| ≤ 3301 / 5000`
+(`gerver_geometric_area_bounds`), both numeric bounds coming from the kernel-checked
+certificate `MovingSofa.Gerver.AreaCertificate` through the fan bound of
+`MovingSofa.Gerver.Area.CapFan` and the rectangle cover of
+`MovingSofa.Gerver.Area.NicheCover`.  Subadditivity of area then gives
+`11 / 5 ≤ |G|` (`gerver_area_lower_bound`).
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open MeasureTheory
+
+theorem gerver_canonical_paper_literal :
+    gerversSofa = paperGerverSofa ∧ paperGerverSofa = gerverLiteralSofa := by
+  refine ⟨?_, paperGerverSofa_eq_literal⟩
+  have hTpos : (0 : ℝ) < Real.pi / 2 := by positivity
+  have hT0 : (0 : ℝ) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨le_rfl, hTpos.le⟩
+  have hTT : (Real.pi / 2) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨hTpos.le, le_rfl⟩
+  -- Translate-then-rotate splits into rotation of the point plus rotation of the shift.
+  have hrt : ∀ (α : Real.Angle) (v s : Point),
+      rotateTranslate α v s = rotationMap α s + rotationMap α v := by
+    intro α v s
+    change (EuclideanGeometry.o.rotation α) (s + v) = _
+    exact map_add _ _ _
+  -- The canonical placements agree with the paper ones on `[0, π/2]`.
+  have himg : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2), ∀ S : Set Point,
+      rotateTranslate (t : Real.Angle) (GerversSofa.p t) '' S =
+        (fun s ↦ rotationMap (t : Real.Angle) s + paperGerverPath t) '' S := by
+    intro t ht S
+    apply Set.image_congr'
+    intro s
+    rw [hrt, canonical_path_rotation_eq_paper t ht]
+  -- Membership in a rotated-translated coordinate region is read off in the moving frame.
+  have himage : ∀ (P : ℝ → ℝ → Prop) (S : Set Point),
+      (∀ s : Point, s ∈ S ↔ P (s 0) (s 1)) →
+      ∀ (t : Real.Angle) (v z : Point),
+        z ∈ (fun s ↦ rotationMap t s + v) '' S ↔
+          P (inner ℝ (z - v) (normalVector t)) (inner ℝ (z - v) (tangentVector t)) := by
+    intro P S hS t v z
+    constructor
+    · rintro ⟨s, hs, rfl⟩
+      have h0 : inner ℝ (rotationMap t s + v - v) (normalVector t) = s 0 := by
+        rw [add_sub_cancel_right, inner_rotationMap_normalVector]
+      have h1 : inner ℝ (rotationMap t s + v - v) (tangentVector t) = s 1 := by
+        rw [add_sub_cancel_right, inner_rotationMap_tangentVector]
+      rw [h0, h1]
+      exact (hS s).1 hs
+    · intro h
+      obtain ⟨s, hs⟩ := (EuclideanGeometry.o.rotation t).surjective (z - v)
+      have hsz : rotationMap t s = z - v := hs
+      have h0 : s 0 = inner ℝ (z - v) (normalVector t) := by
+        rw [← inner_rotationMap_normalVector s t, hsz]
+      have h1 : s 1 = inner ℝ (z - v) (tangentVector t) := by
+        rw [← inner_rotationMap_tangentVector s t, hsz]
+      refine ⟨s, (hS s).2 ?_, ?_⟩
+      · rw [h0, h1]; exact h
+      · change rotationMap t s + v = z
+        rw [hsz]; abel
+  have hhoriz : ∀ s : Point, s ∈ horizontalHallway ↔ s 0 ≤ 1 ∧ 0 ≤ s 1 ∧ s 1 ≤ 1 :=
+    fun s => ⟨mem_horizontalHallway_coordinates,
+      fun h => mem_horizontalHallway_of_coordinates s h.1 ⟨h.2.1, h.2.2⟩⟩
+  have hvert : ∀ s : Point, s ∈ verticalHallway ↔ 0 ≤ s 0 ∧ s 0 ≤ 1 ∧ s 1 ≤ 1 :=
+    fun s => ⟨mem_verticalHallway_coordinates,
+      fun h => mem_verticalHallway_of_coordinates s ⟨h.1, h.2.1⟩ h.2.2⟩
+  have hhall : ∀ s : Point, s ∈ hallway ↔ (s 0 ≤ 1 ∧ s 1 ≤ 1) ∧ (0 ≤ s 0 ∨ 0 ≤ s 1) :=
+    mem_hallway_iff
+  -- Endpoint normalisations of the paper path.
+  have hreg := gerver_direct_path_regularity GerverSofa.PartB.params
+    GerverSofa.PartB.params_mem GerverSofa.PartB.params_equations
+  have hpath0 : paperGerverPath 0 = 0 := by
+    change GerverSofa.PartF.Coordinates.toPlane
+      (GerverSofa.Romik.path GerverSofa.PartB.params 0) = 0
+    rw [hreg.2.1]
+    ext i
+    fin_cases i <;> rfl
+  have hpathT : paperGerverPath (Real.pi / 2) 1 = 0 :=
+    GerverSofa.Romik.path_end_y_zero_of_mem_box_and_equations
+      GerverSofa.PartB.params_mem GerverSofa.PartB.params_equations
+  have hstrip : ∀ z : Point,
+      z ∈ (strips (Real.pi / 2)).1 ∩ (strips (Real.pi / 2)).2.2 ↔ 0 ≤ z 1 ∧ z 1 ≤ 1 := by
+    intro z
+    have h := mem_stripParallelogram_iff (Real.pi / 2) z
+    rw [inner_normalVector_pi_div_two] at h
+    have hset : (strips (Real.pi / 2)).1 ∩ (strips (Real.pi / 2)).2.2 =
+        (stripParallelogram (Real.pi / 2)).1 := rfl
+    rw [hset, h]
+    tauto
+  -- Rewrite both endpoint arms and the family of hallways in the paper frame.
+  have hzero : ((0 : ℝ) : Real.Angle) = (0 : Real.Angle) := Real.Angle.coe_zero
+  have hgs : gerversSofa =
+      ((fun s ↦ rotationMap ((0 : ℝ) : Real.Angle) s + paperGerverPath 0) ''
+          horizontalHallway ∩
+        (fun s ↦ rotationMap ((Real.pi / 2 : ℝ) : Real.Angle) s +
+          paperGerverPath (Real.pi / 2)) '' verticalHallway) ∩
+      ⋂ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+        (fun s ↦ rotationMap (t : Real.Angle) s + paperGerverPath t) '' hallway := by
+    change rotateTranslate 0 (GerversSofa.p 0) '' horizontalHallway ∩
+        rotateTranslate ((Real.pi / 2 : ℝ) : Real.Angle)
+          (GerversSofa.p (Real.pi / 2)) '' verticalHallway ∩
+        (⋂ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+          rotateTranslate (t : Real.Angle) (GerversSofa.p t) '' hallway) = _
+    rw [← hzero, himg 0 hT0 horizontalHallway,
+      himg (Real.pi / 2) hTT verticalHallway,
+      Set.iInter₂_congr (fun t (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)) => himg t ht hallway)]
+  have hpaper : paperGerverSofa =
+      {z : Point | 0 ≤ z 1 ∧ z 1 ≤ 1} ∩
+        ⋂ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+          (fun s ↦ rotationMap (t : Real.Angle) s + paperGerverPath t) '' hallway := by
+    change (strips (Real.pi / 2)).1 ∩ (strips (Real.pi / 2)).2.2 ∩ _ = _
+    rw [Set.ext hstrip]
+    rfl
+  rw [hgs, hpaper]
+  ext q
+  have hn0 : inner ℝ (q - paperGerverPath 0) (normalVector ((0 : ℝ) : Real.Angle)) = q 0 := by
+    rw [hpath0, sub_zero, inner_normalVector_zero]
+  have hg0 : inner ℝ (q - paperGerverPath 0) (tangentVector ((0 : ℝ) : Real.Angle)) = q 1 := by
+    rw [hpath0, sub_zero, inner_tangentVector_zero]
+  have hnT : inner ℝ (q - paperGerverPath (Real.pi / 2))
+      (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = q 1 := by
+    rw [inner_normalVector_pi_div_two]
+    change q 1 - paperGerverPath (Real.pi / 2) 1 = q 1
+    rw [hpathT, sub_zero]
+  have hAhoriz := himage (fun a b => a ≤ 1 ∧ 0 ≤ b ∧ b ≤ 1) horizontalHallway hhoriz
+  have hAvert := himage (fun a b => 0 ≤ a ∧ a ≤ 1 ∧ b ≤ 1) verticalHallway hvert
+  have hAhall := himage (fun a b => (a ≤ 1 ∧ b ≤ 1) ∧ (0 ≤ a ∨ 0 ≤ b)) hallway hhall
+  simp only [Set.mem_inter_iff, Set.mem_iInter₂, Set.mem_ofPred_eq, hAhoriz, hAvert, hAhall,
+    hn0, hg0, hnT]
+  constructor
+  · rintro ⟨⟨⟨-, hb0, hb1⟩, -⟩, hC⟩
+    exact ⟨⟨hb0, hb1⟩, hC⟩
+  · rintro ⟨⟨hb0, hb1⟩, hC⟩
+    refine ⟨⟨⟨?_, hb0, hb1⟩, hb0, hb1, ?_⟩, hC⟩
+    · have h := (hC 0 hT0).1.1
+      rwa [hn0] at h
+    · exact (hC (Real.pi / 2) hTT).1.2
+
+theorem gerver_geometric_area_bounds :
+    MeasurableSet gerverOuterCap ∧ volume gerverOuterCap < ⊤ ∧
+    MeasurableSet gerverLiteralNiche ∧ volume gerverLiteralNiche < ⊤ ∧
+    (28609 : ℝ) / 10000 ≤ ClassicalResults.area gerverOuterCap ∧
+    ClassicalResults.area gerverLiteralNiche ≤ (3301 : ℝ) / 5000 := by
+  have hfin : volume gerverLiteralNiche ≤ ENNReal.ofReal ((3301 : ℝ) / 5000) :=
+    gerver_niche_volume_le
+  refine ⟨measurableSet_gerverOuterCap, volume_gerverOuterCap_lt_top,
+    measurableSet_gerverLiteralNiche, lt_of_le_of_lt hfin ENNReal.ofReal_lt_top,
+    gerverOuterCap_area_certified_lower_bound, ?_⟩
+  calc ClassicalResults.area gerverLiteralNiche
+      = (volume gerverLiteralNiche).toReal := rfl
+    _ ≤ (ENNReal.ofReal ((3301 : ℝ) / 5000)).toReal :=
+        ENNReal.toReal_mono ENNReal.ofReal_ne_top hfin
+    _ = (3301 : ℝ) / 5000 := ENNReal.toReal_ofReal (by norm_num)
+
+theorem gerver_area_lower_bound :
+    volume gerversSofa < ⊤ ∧ (11 : ℝ) / 5 ≤ ClassicalResults.area gerversSofa := by
+  obtain ⟨-, hKtop, -, hNtop, hKarea, hNarea⟩ := gerver_geometric_area_bounds
+  have hG : gerversSofa = gerverOuterCap \ gerverLiteralNiche :=
+    gerver_canonical_paper_literal.1.trans gerver_canonical_paper_literal.2
+  have hGtop : volume gerversSofa < ⊤ :=
+    lt_of_le_of_lt (measure_mono (hG ▸ Set.sdiff_subset)) hKtop
+  refine ⟨hGtop, ?_⟩
+  -- The cap is covered by the sofa together with the niche, so areas are subadditive.
+  have hcover : volume gerverOuterCap ≤ volume gerversSofa + volume gerverLiteralNiche := by
+    refine le_trans (measure_mono ?_) (measure_union_le _ _)
+    rw [hG]
+    exact Set.subset_sdiff_union _ _
+  have hreal : ClassicalResults.area gerverOuterCap ≤
+      ClassicalResults.area gerversSofa + ClassicalResults.area gerverLiteralNiche := by
+    have h := ENNReal.toReal_mono (by finiteness) hcover
+    rwa [ENNReal.toReal_add hGtop.ne hNtop.ne] at h
+  linarith only [hreal, hKarea, hNarea]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Gerver.CapIdentification`.
+* `Gerver.Niche.Identification`.
+* `Gerver.SurfaceDensity`.
+* `Gerver.VelocityAndCapArea`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver / Cap Identification
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private theorem gerver_cap_fiber_has_sofa_point
+    (hKcomp : IsCompact gerverOuterCap) :
+    ∀ q ∈ gerverOuterCap,
+     ∃ p ∈ gerverLiteralSofa, p 0 = q 0 ∧ q 1 ≤ p 1 := by
+  intro q hq
+  have hFcomp : IsCompact (gerverOuterCap ∩ {z : Point |
+      inner ℝ z (normalVector ((0 : ℝ) : Real.Angle)) =
+        inner ℝ q (normalVector ((0 : ℝ) : Real.Angle))}) :=
+    hKcomp.inter_right (isClosed_eq (continuous_id.inner continuous_const) continuous_const)
+  obtain ⟨p, hpF, hpmax⟩ := hFcomp.exists_isMaxOn ⟨q, hq, rfl⟩
+    (f := fun z : Point => inner ℝ z (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)))
+    (continuous_id.inner continuous_const).continuousOn
+  have hpx : p 0 = q 0 := by
+    have h : inner ℝ p (normalVector ((0 : ℝ) : Real.Angle)) =
+      inner ℝ q (normalVector ((0 : ℝ) : Real.Angle)) := hpF.2
+    rwa [inner_normalVector_zero, inner_normalVector_zero] at h
+  have hpy : q 1 ≤ p 1 := by
+    have h : inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) ≤
+      inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) := hpmax ⟨hq, rfl⟩
+    rwa [inner_normalVector_pi_div_two, inner_normalVector_pi_div_two] at h
+  have hnofill : ∀ (f : ℝ → Point) (I : Set ℝ), (∀ t ∈ I, f t ∈ gerverOuterCap) →
+      p ∉ strictVerticalFill f I := by
+    rintro f I hf ⟨t, ht, h0, -, h2⟩
+    have hmem : f t ∈ gerverOuterCap ∩ {z : Point |
+        inner ℝ z (normalVector ((0 : ℝ) : Real.Angle)) =
+          inner ℝ q (normalVector ((0 : ℝ) : Real.Angle))} := by
+      refine ⟨hf t ht, ?_⟩
+      change inner ℝ (f t) (normalVector ((0 : ℝ) : Real.Angle)) =
+        inner ℝ q (normalVector ((0 : ℝ) : Real.Angle))
+      rw [inner_normalVector_zero, inner_normalVector_zero, ← h0, hpx]
+    have hle : inner ℝ (f t) (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) ≤
+      inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) := hpmax hmem
+    rw [inner_normalVector_pi_div_two, inner_normalVector_pi_div_two] at hle
+    linarith
+  obtain ⟨hroofD, hroofx, hroofB⟩ := gerver_niche_roof_membership
+  refine ⟨p, ⟨hpF.1, ?_⟩, hpx, hpy⟩
+  rw [gerver_niche_vertical_fills]
+  rintro ((h | h) | h)
+  · exact hnofill _ _ hroofD h
+  · exact hnofill _ _ hroofx h
+  · exact hnofill _ _ hroofB h
+
+private theorem gerver_cap_contact_faces (Kb : ConvexBody Point)
+    (hKset : (Kb : Set Point) = gerverOuterCap)
+    (hKcap : IsCap (Real.pi / 2) Kb)
+    (hC1 : ContDiff ℝ 1 (GerverSofa.Romik.path GerverSofa.PartB.params))
+    (hAn : ∀ t : ℝ, inner ℝ (paperGerverContacts t 0) (normalVector (t : Real.Angle)) =
+     inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) + 1)
+    (hAt : ∀ t : ℝ, inner ℝ (paperGerverContacts t 0) (tangentVector (t : Real.Angle)) =
+     inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) +
+       (paperGerverVelocityComponents t).1)
+    (hCn : ∀ t : ℝ, inner ℝ (paperGerverContacts t 2) (normalVector (t : Real.Angle)) =
+     inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) -
+       (paperGerverVelocityComponents t).2)
+    (hCt : ∀ t : ℝ, inner ℝ (paperGerverContacts t 2) (tangentVector (t : Real.Angle)) =
+     inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) + 1)
+    (hsupK : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+     supportValue gerverOuterCap (t : Real.Angle) =
+       inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) + 1 ∧
+     supportValue gerverOuterCap ((Real.pi / 2 + t : ℝ) : Real.Angle) =
+       inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) + 1)
+    (hTpos : (0 : ℝ) < Real.pi / 2) :
+    ∃ K : CapSpace (Real.pi / 2), (K.val : Set Point) = gerverOuterCap ∧
+     (∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+       exposedEdge K.val (t : Real.Angle) = {paperGerverContacts t 0} ∧
+       exposedEdge K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) = {paperGerverContacts t 2}) ∧
+     paperGerverContacts 0 0 = (edgeVertices K.val (0 : Real.Angle)).1 ∧
+     paperGerverContacts (Real.pi / 2) 0 =
+       (edgeVertices K.val ((Real.pi / 2 : ℝ) : Real.Angle)).2 ∧
+     paperGerverContacts 0 2 = (edgeVertices K.val ((Real.pi / 2 : ℝ) : Real.Angle)).1 ∧
+     paperGerverContacts (Real.pi / 2) 2 = (edgeVertices K.val (Real.pi : Real.Angle)).2 := by
+  -- a plane point is determined by its two frame coordinates
+  have hpteq : ∀ (a : Real.Angle) (p q : Point),
+      inner ℝ p (normalVector a) = inner ℝ q (normalVector a) →
+      inner ℝ p (tangentVector a) = inner ℝ q (tangentVector a) → p = q := by
+    intro a p q h1 h2
+    rw [← inner_normalVector_smul_add_inner_tangentVector_smul p a,
+      ← inner_normalVector_smul_add_inner_tangentVector_smul q a, h1, h2]
+  have hdx : ∀ t : ℝ, HasDerivAt paperGerverPath (deriv paperGerverPath t) t := fun t => by
+    rw [deriv_paperGerverPath hC1 t]
+    exact hasDerivAt_paperGerverPath hC1 t
+  -- exposed faces of the certified body, read in the cap's own description
+  have hexp : ∀ (a : Real.Angle) (z : Point), z ∈ exposedEdge Kb a ↔
+      (z ∈ gerverOuterCap ∧
+        inner ℝ z (normalVector a) = supportValue gerverOuterCap a) := by
+    intro a z
+    constructor
+    · intro h
+      refine ⟨hKset ▸ h.1, ?_⟩
+      have h2 : inner ℝ z (normalVector a) = supportValue (Kb : Set Point) a := h.2
+      rwa [hKset] at h2
+    · intro h
+      refine ⟨hKset ▸ h.1, ?_⟩
+      change inner ℝ z (normalVector a) = supportValue (Kb : Set Point) a
+      rw [hKset]
+      exact h.2
+  have hsingleV : ∀ (a : Real.Angle) (p : Point), exposedEdge Kb a = {p} →
+      (edgeVertices Kb a).1 = p ∧ (edgeVertices Kb a).2 = p := fun a p h =>
+    ⟨Set.mem_singleton_iff.1 (h ▸ edgeVertices_fst_mem Kb a),
+      Set.mem_singleton_iff.1 (h ▸ edgeVertices_snd_mem Kb a)⟩
+  -- interior uniqueness in the normal family
+  have hnormaledge : ∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2), ∀ z ∈ gerverOuterCap,
+      inner ℝ z (normalVector (t : Real.Angle)) =
+        inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) + 1 →
+      z = paperGerverContacts t 0 := by
+    intro t ht z hz heq
+    have hgd : HasDerivAt (fun s : ℝ =>
+        (inner ℝ (paperGerverPath s) (normalVector (s : Real.Angle)) + 1) -
+          inner ℝ z (normalVector (s : Real.Angle)))
+        ((inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) +
+          (paperGerverVelocityComponents t).1) -
+          inner ℝ z (tangentVector (t : Real.Angle))) t :=
+      (((hdx t).inner ℝ (hasDerivAt_normalVector t)).add_const 1).sub
+        (hasDerivAt_inner_normalVector z t)
+    have hgmin : IsLocalMin (fun s : ℝ =>
+        (inner ℝ (paperGerverPath s) (normalVector (s : Real.Angle)) + 1) -
+          inner ℝ z (normalVector (s : Real.Angle))) t := by
+      refine IsMinOn.isLocalMin ?_ (Icc_mem_nhds ht.1 ht.2)
+      intro s hs
+      have h := (hz.2 s hs).1
+      change (inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) + 1) -
+          inner ℝ z (normalVector (t : Real.Angle)) ≤
+        (inner ℝ (paperGerverPath s) (normalVector (s : Real.Angle)) + 1) -
+          inner ℝ z (normalVector (s : Real.Angle))
+      linarith
+    have hzero := hgmin.hasDerivAt_eq_zero hgd
+    refine hpteq (t : Real.Angle) z (paperGerverContacts t 0) ?_ ?_
+    · rw [hAn t]; exact heq
+    · rw [hAt t]; linarith
+  -- interior uniqueness in the tangent family
+  have htangentedge : ∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2), ∀ z ∈ gerverOuterCap,
+      inner ℝ z (tangentVector (t : Real.Angle)) =
+        inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) + 1 →
+      z = paperGerverContacts t 2 := by
+    intro t ht z hz heq
+    have hzt : HasDerivAt (fun s : ℝ => inner ℝ z (tangentVector (s : Real.Angle)))
+        (-inner ℝ z (normalVector (t : Real.Angle))) t := by
+      simpa using (hasDerivAt_const t z).inner ℝ (hasDerivAt_tangentVector t)
+    have hgd : HasDerivAt (fun s : ℝ =>
+        (inner ℝ (paperGerverPath s) (tangentVector (s : Real.Angle)) + 1) -
+          inner ℝ z (tangentVector (s : Real.Angle)))
+        ((inner ℝ (paperGerverPath t) (-normalVector (t : Real.Angle)) +
+          (paperGerverVelocityComponents t).2) -
+          -inner ℝ z (normalVector (t : Real.Angle))) t :=
+      (((hdx t).inner ℝ (hasDerivAt_tangentVector t)).add_const 1).sub hzt
+    have hgmin : IsLocalMin (fun s : ℝ =>
+        (inner ℝ (paperGerverPath s) (tangentVector (s : Real.Angle)) + 1) -
+          inner ℝ z (tangentVector (s : Real.Angle))) t := by
+      refine IsMinOn.isLocalMin ?_ (Icc_mem_nhds ht.1 ht.2)
+      intro s hs
+      have h := (hz.2 s hs).2
+      change (inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) + 1) -
+          inner ℝ z (tangentVector (t : Real.Angle)) ≤
+        (inner ℝ (paperGerverPath s) (tangentVector (s : Real.Angle)) + 1) -
+          inner ℝ z (tangentVector (s : Real.Angle))
+      linarith
+    have hzero := hgmin.hasDerivAt_eq_zero hgd
+    rw [inner_neg_right] at hzero
+    refine hpteq (t : Real.Angle) z (paperGerverContacts t 2) ?_ ?_
+    · rw [hCn t]; linarith
+    · rw [hCt t]; exact heq
+  -- the interior exposed faces are the two contact singletons
+  have hedgeA : ∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+      exposedEdge Kb (t : Real.Angle) = {paperGerverContacts t 0} := by
+    intro t ht
+    have htI : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := Set.Ioo_subset_Icc_self ht
+    refine Set.eq_singleton_iff_unique_mem.2
+      ⟨(hexp _ _).2 ⟨gerver_outer_contact_A t htI, ?_⟩, fun z hz => ?_⟩
+    · rw [hAn t, (hsupK t htI).1]
+    · rw [hexp] at hz
+      exact hnormaledge t ht z hz.1 (by rw [hz.2, (hsupK t htI).1])
+  have hedgeC : ∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+      exposedEdge Kb ((t + Real.pi / 2 : ℝ) : Real.Angle) = {paperGerverContacts t 2} := by
+    intro t ht
+    have htI : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := Set.Ioo_subset_Icc_self ht
+    have hang : ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+        ((Real.pi / 2 + t : ℝ) : Real.Angle) := by rw [add_comm]
+    have hsv : supportValue gerverOuterCap ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+        inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) + 1 := by
+      rw [hang]; exact (hsupK t htI).2
+    refine Set.eq_singleton_iff_unique_mem.2
+      ⟨(hexp _ _).2 ⟨gerver_outer_contact_C t htI, ?_⟩, fun z hz => ?_⟩
+    · rw [normalVector_add_pi_div_two_real, hCt t, hsv]
+    · rw [hexp] at hz
+      refine htangentedge t ht z hz.1 ?_
+      have h := hz.2
+      rwa [normalVector_add_pi_div_two_real, hsv] at h
+  -- the four contact curves are continuous
+  have hcontacts : Continuous paperGerverContacts := paperGerverContactData_properties.2.1
+  have hcontA : Continuous fun s : ℝ => paperGerverContacts s 0 :=
+    (continuous_apply 0).comp hcontacts
+  have hcontC : Continuous fun s : ℝ => paperGerverContacts s 2 :=
+    (continuous_apply 2).comp hcontacts
+  -- vertex limits at the four endpoints
+  have hpi : (0 : ℝ) < Real.pi := Real.pi_pos
+  have heventA : ∀ᶠ s : ℝ in nhdsWithin 0 (Set.Ioi 0),
+      paperGerverContacts s 0 = (edgeVertices Kb (s : Real.Angle)).1 := by
+    filter_upwards [Ioo_mem_nhdsGT hTpos] with s hs
+    exact ((hsingleV _ _ (hedgeA s hs)).1).symm
+  have heventA' : ∀ᶠ s : ℝ in nhdsWithin (Real.pi / 2) (Set.Iio (Real.pi / 2)),
+      paperGerverContacts s 0 = (edgeVertices Kb (s : Real.Angle)).1 := by
+    filter_upwards [Ioo_mem_nhdsLT hTpos] with s hs
+    exact ((hsingleV _ _ (hedgeA s hs)).1).symm
+  have hCshift : ∀ s : ℝ, s ∈ Set.Ioo (Real.pi / 2) Real.pi →
+      paperGerverContacts (s - Real.pi / 2) 2 = (edgeVertices Kb (s : Real.Angle)).1 := by
+    intro s hs
+    have hs' : s - Real.pi / 2 ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) :=
+      ⟨by linarith [hs.1], by linarith [hs.2]⟩
+    have hang : ((s : ℝ) : Real.Angle) =
+        ((s - Real.pi / 2 + Real.pi / 2 : ℝ) : Real.Angle) := by rw [sub_add_cancel]
+    rw [hang]
+    exact ((hsingleV _ _ (hedgeC (s - Real.pi / 2) hs')).1).symm
+  have heventC : ∀ᶠ s : ℝ in nhdsWithin (Real.pi / 2) (Set.Ioi (Real.pi / 2)),
+      paperGerverContacts (s - Real.pi / 2) 2 = (edgeVertices Kb (s : Real.Angle)).1 := by
+    filter_upwards [Ioo_mem_nhdsGT (show Real.pi / 2 < Real.pi by linarith)] with s hs
+    exact hCshift s hs
+  have heventC' : ∀ᶠ s : ℝ in nhdsWithin Real.pi (Set.Iio Real.pi),
+      paperGerverContacts (s - Real.pi / 2) 2 = (edgeVertices Kb (s : Real.Angle)).1 := by
+    filter_upwards [Ioo_mem_nhdsLT (show Real.pi / 2 < Real.pi by linarith)] with s hs
+    exact hCshift s hs
+  have hcontCshift : Continuous fun s : ℝ => paperGerverContacts (s - Real.pi / 2) 2 :=
+    hcontC.comp (continuous_id.sub continuous_const)
+  refine ⟨⟨Kb, hKcap⟩, hKset, fun t ht => ⟨hedgeA t ht, hedgeC t ht⟩, ?_, ?_, ?_, ?_⟩
+  · rw [← Real.Angle.coe_zero]
+    refine tendsto_nhds_unique ?_ (contact_oneSided_limits Kb 0).1
+    exact Filter.Tendsto.congr' heventA ((hcontA.tendsto 0).mono_left nhdsWithin_le_nhds)
+  · refine tendsto_nhds_unique ?_ (contact_oneSided_limits Kb (Real.pi / 2)).2.2.2.1
+    exact Filter.Tendsto.congr' heventA'
+      ((hcontA.tendsto (Real.pi / 2)).mono_left nhdsWithin_le_nhds)
+  · refine tendsto_nhds_unique ?_ (contact_oneSided_limits Kb (Real.pi / 2)).1
+    have h : Filter.Tendsto (fun s : ℝ => paperGerverContacts (s - Real.pi / 2) 2)
+        (nhdsWithin (Real.pi / 2) (Set.Ioi (Real.pi / 2)))
+        (nhds (paperGerverContacts 0 2)) := by
+      have := (hcontCshift.tendsto (Real.pi / 2)).mono_left
+        (nhdsWithin_le_nhds (s := Set.Ioi (Real.pi / 2)))
+      rwa [sub_self] at this
+    exact Filter.Tendsto.congr' heventC h
+  · refine tendsto_nhds_unique ?_ (contact_oneSided_limits Kb Real.pi).2.2.2.1
+    have h : Filter.Tendsto (fun s : ℝ => paperGerverContacts (s - Real.pi / 2) 2)
+        (nhdsWithin Real.pi (Set.Iio Real.pi))
+        (nhds (paperGerverContacts (Real.pi / 2) 2)) := by
+      have := (hcontCshift.tendsto Real.pi).mono_left
+        (nhdsWithin_le_nhds (s := Set.Iio Real.pi))
+      rwa [show Real.pi - Real.pi / 2 = Real.pi / 2 by ring] at this
+    exact Filter.Tendsto.congr' heventC' h
+
+private theorem gerver_literal_standard_position
+    (hTpos : (0 : ℝ) < Real.pi / 2)
+    (hTmem : (Real.pi / 2) ∈ Set.Icc (0 : ℝ) (Real.pi / 2))
+    (hGcomp : IsCompact gerverLiteralSofa)
+    (hmotioncont : Continuous fun r : unitInterval =>
+     rotateTranslate ((-(r.val * (Real.pi / 2)) : ℝ) : Real.Angle)
+       (-paperGerverPath (r.val * (Real.pi / 2))))
+    (hpath0 : paperGerverPath 0 = 0)
+    (hpathTy : paperGerverPath (Real.pi / 2) 1 = 0)
+    (hcapx1 : ∀ q ∈ gerverOuterCap, q 0 ≤ 1)
+    (hcapy1 : ∀ q ∈ gerverOuterCap, q 1 ≤ 1)
+    (hsv1 : supportValue gerverLiteralSofa ((Real.pi / 2 : ℝ) : Real.Angle) = 1) :
+    IsStandardPosition gerverLiteralSofa (Real.pi / 2) := by
+  -- ### Frame readers for the clockwise motion
+  have hinnertan : ∀ (w : Point) (t : ℝ), inner ℝ w (tangentVector (t : Real.Angle)) =
+      -(w 0 * Real.sin t) + w 1 * Real.cos t := by
+    intro w t
+    simp [tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+    ring
+  have hmapply : ∀ (t : ℝ) (v p : Point),
+      rotateTranslate ((-t : ℝ) : Real.Angle) (-v) p =
+        rotationMap ((-t : ℝ) : Real.Angle) (p - v) := by
+    intro t v p
+    change (EuclideanGeometry.o.rotation ((-t : ℝ) : Real.Angle)) (p + -v) =
+      (EuclideanGeometry.o.rotation ((-t : ℝ) : Real.Angle)) (p - v)
+    rw [← sub_eq_add_neg]
+  have hrotinv : ∀ (t : ℝ) (w : Point),
+      rotationMap ((-t : ℝ) : Real.Angle) (rotationMap ((t : ℝ) : Real.Angle) w) = w := by
+    intro t w
+    change (EuclideanGeometry.o.rotation ((-t : ℝ) : Real.Angle))
+      ((EuclideanGeometry.o.rotation ((t : ℝ) : Real.Angle)) w) = w
+    rw [Real.Angle.coe_neg, ← Orientation.rotation_symm]
+    exact (EuclideanGeometry.o.rotation ((t : ℝ) : Real.Angle)).symm_apply_apply w
+  have hcoord0 : ∀ (t : ℝ) (w : Point), rotationMap ((-t : ℝ) : Real.Angle) w 0 =
+      inner ℝ w (normalVector (t : Real.Angle)) := by
+    intro t w
+    rw [rotationMap_apply_zero, inner_normalVector_real, Real.Angle.cos_coe, Real.Angle.sin_coe,
+      Real.cos_neg, Real.sin_neg]
+    ring
+  have hcoord1 : ∀ (t : ℝ) (w : Point), rotationMap ((-t : ℝ) : Real.Angle) w 1 =
+      inner ℝ w (tangentVector (t : Real.Angle)) := by
+    intro t w
+    rw [rotationMap_apply_one, hinnertan, Real.Angle.cos_coe, Real.Angle.sin_coe,
+      Real.cos_neg, Real.sin_neg]
+    ring
+  refine ⟨hGcomp, ⟨fun r => rotateTranslate ((-(r.val * (Real.pi / 2)) : ℝ) : Real.Angle)
+      (-paperGerverPath (r.val * (Real.pi / 2))),
+    ⟨gerver_literal_connected, hGcomp.isClosed, ?_, ⟨0, ?_⟩, ?_, ?_, ?_, ?_⟩,
+    fun r => -(r.val * (Real.pi / 2)),
+    (continuous_subtype_val.mul continuous_const).neg,
+    by simp, by simp, ?_⟩,
+    hTpos, le_rfl, hsv1, hsv1⟩
+  · exact hmotioncont
+  · -- the motion starts at the identity
+    intro p
+    rw [hmapply]
+    simp only [Set.Icc.coe_zero, zero_mul, neg_zero, Real.Angle.coe_zero, hpath0,
+      sub_zero, add_zero]
+    change (EuclideanGeometry.o.rotation 0) p = p
+    simp
+  · -- each placement is a rotation followed by a translation
+    intro r
+    refine ⟨(((-(r.val * (Real.pi / 2))) : ℝ) : Real.Angle), fun p => ?_⟩
+    rw [hmapply, hmapply]
+    simp only [rotationMap, zero_sub, map_sub, map_neg]
+    abel
+  · -- the initial placement lands in the horizontal arm
+    rintro _ ⟨z, hz, rfl⟩
+    have heq : rotateTranslate
+        ((-((0 : unitInterval).val * (Real.pi / 2)) : ℝ) : Real.Angle)
+        (-paperGerverPath ((0 : unitInterval).val * (Real.pi / 2))) z = z := by
+      rw [hmapply]
+      simp only [Set.Icc.coe_zero, zero_mul, neg_zero, Real.Angle.coe_zero, hpath0,
+        sub_zero]
+      change (EuclideanGeometry.o.rotation 0) z = z
+      simp
+    rw [heq]
+    exact mem_horizontalHallway_of_coordinates z (hcapx1 z hz.1)
+      ⟨hz.1.1, hcapy1 z hz.1⟩
+  · -- every intermediate placement lands in the hallway
+    intro r
+    rintro _ ⟨z, hz, rfl⟩
+    have ht : r.val * (Real.pi / 2) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+      ⟨mul_nonneg r.2.1 hTpos.le, by
+        nlinarith [r.2.2, hTpos]⟩
+    have hzP : z ∈ paperGerverSofa := by rw [paperGerverSofa_eq_literal]; exact hz
+    obtain ⟨w, hw, hwz⟩ :=
+      Set.mem_iInter₂.1 hzP.2 (r.val * (Real.pi / 2)) ht
+    rw [hmapply]
+    have hzw : z - paperGerverPath (r.val * (Real.pi / 2)) =
+        rotationMap ((r.val * (Real.pi / 2) : ℝ) : Real.Angle) w := by
+      rw [← hwz]
+      change rotationMap ((r.val * (Real.pi / 2) : ℝ) : Real.Angle) w +
+          paperGerverPath (r.val * (Real.pi / 2)) -
+          paperGerverPath (r.val * (Real.pi / 2)) =
+        rotationMap ((r.val * (Real.pi / 2) : ℝ) : Real.Angle) w
+      abel
+    rw [hzw, hrotinv]
+    exact hw
+  · -- the final placement lands in the vertical arm
+    rintro _ ⟨z, hz, rfl⟩
+    have hTm' : (1 : unitInterval).val * (Real.pi / 2) = Real.pi / 2 := by
+      rw [Set.Icc.coe_one, one_mul]
+    rw [hmapply]
+    refine mem_verticalHallway_of_coordinates _ ⟨?_, ?_⟩ ?_
+    · rw [hcoord0, hTm', inner_sub_left, inner_normalVector_pi_div_two,
+        inner_normalVector_pi_div_two, hpathTy, sub_zero]
+      exact hz.1.1
+    · rw [hcoord0, hTm', inner_sub_left, inner_normalVector_pi_div_two,
+        inner_normalVector_pi_div_two, hpathTy, sub_zero]
+      exact hcapy1 z hz.1
+    · rw [hcoord1, hTm', inner_sub_left]
+      have h := (hz.1.2 (Real.pi / 2) hTmem).2
+      linarith
+  · -- the motion realizes the lifted clockwise angle
+    intro r p
+    rw [hmapply, hmapply]
+    simp only [rotationMap, zero_sub, map_sub, map_neg]
+    abel
+
+private theorem gerver_literal_cap_eq
+    (hsupG : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      supportValue gerverLiteralSofa (t : Real.Angle) =
+        inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) + 1 ∧
+      supportValue gerverLiteralSofa ((Real.pi / 2 + t : ℝ) : Real.Angle) =
+        inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) + 1)
+    (hcapy1 : ∀ q ∈ gerverOuterCap, q 1 ≤ 1) :
+    capOfSofa gerverLiteralSofa (Real.pi / 2) = gerverOuterCap := by
+  have hangsum : ∀ t : ℝ, (t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) =
+      ((Real.pi / 2 + t : ℝ) : Real.Angle) := fun t => by
+    rw [← Real.Angle.coe_add, add_comm]
+  have houter : ∀ t : ℝ,
+      (rotatingHallwayParts gerverLiteralSofa (t : Real.Angle)).outerQuadrant =
+        normalHalfPlane (t : Real.Angle)
+            (supportValue gerverLiteralSofa (t : Real.Angle)) false false ∩
+          normalHalfPlane ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle))
+            (supportValue gerverLiteralSofa
+              ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle))) false false :=
+    fun t => (rotatingHallwayParts_formulas gerverLiteralSofa
+      (t : Real.Angle)).2.2.2.2.2.2.2.1
+  ext z
+  constructor
+  · rintro ⟨hstrip, hint⟩
+    rw [mem_stripParallelogram_iff] at hstrip
+    refine ⟨hstrip.1.1, fun t ht => ?_⟩
+    have hq : z ∈ (rotatingHallwayParts gerverLiteralSofa (t : Real.Angle)).outerQuadrant :=
+      Set.mem_iInter₂.1 hint t ht
+    rw [houter t] at hq
+    have h1 : inner ℝ z (normalVector (t : Real.Angle)) ≤
+      supportValue gerverLiteralSofa (t : Real.Angle) := hq.1
+    have h2 : inner ℝ z
+        (normalVector ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle))) ≤
+      supportValue gerverLiteralSofa
+        ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle)) := hq.2
+    rw [normalVector_add_pi_div_two, hangsum t, (hsupG t ht).2] at h2
+    rw [(hsupG t ht).1] at h1
+    exact ⟨h1, h2⟩
+  · intro hz
+    refine ⟨?_, Set.mem_iInter₂.2 fun t ht => ?_⟩
+    · rw [mem_stripParallelogram_iff, inner_normalVector_pi_div_two]
+      exact ⟨⟨hz.1, hcapy1 z hz⟩, hz.1, hcapy1 z hz⟩
+    · rw [houter t]
+      refine ⟨?_, ?_⟩
+      · change inner ℝ z (normalVector (t : Real.Angle)) ≤
+          supportValue gerverLiteralSofa (t : Real.Angle)
+        rw [(hsupG t ht).1]
+        exact (hz.2 t ht).1
+      · change inner ℝ z
+            (normalVector ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle))) ≤
+          supportValue gerverLiteralSofa
+            ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle))
+        rw [normalVector_add_pi_div_two, hangsum t, (hsupG t ht).2]
+        exact (hz.2 t ht).2
+
+theorem gerver_capSupport_identification :
+    (∀ q ∈ gerverOuterCap, ∃ p ∈ gerverLiteralSofa, p 0 = q 0 ∧ q 1 ≤ p 1) ∧
+    paperGerverSofa = gerverLiteralSofa ∧
+    IsStandardPosition gerverLiteralSofa (Real.pi / 2) ∧
+    (∀ s ∈ Set.Icc (0 : ℝ) Real.pi,
+      supportValue gerverLiteralSofa (s : Real.Angle) = supportValue gerverOuterCap (s :
+        Real.Angle)) ∧
+    capOfSofa gerverLiteralSofa (Real.pi / 2) = gerverOuterCap ∧
+    (∃ K : CapSpace (Real.pi / 2), (K.val : Set Point) = gerverOuterCap ∧
+      (∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+        exposedEdge K.val (t : Real.Angle) = {paperGerverContacts t 0} ∧
+        exposedEdge K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) = {paperGerverContacts t 2}) ∧
+      paperGerverContacts 0 0 = (edgeVertices K.val (0 : Real.Angle)).1 ∧
+      paperGerverContacts (Real.pi / 2) 0 =
+        (edgeVertices K.val ((Real.pi / 2 : ℝ) : Real.Angle)).2 ∧
+      paperGerverContacts 0 2 = (edgeVertices K.val ((Real.pi / 2 : ℝ) : Real.Angle)).1 ∧
+      paperGerverContacts (Real.pi / 2) 2 = (edgeVertices K.val (Real.pi : Real.Angle)).2) ∧
+    (∀ S ∈ ({gerverOuterCap, gerverLiteralSofa} : Set (Set Point)),
+      ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+        supportValue S (t : Real.Angle) =
+          inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) + 1 ∧
+        supportValue S ((Real.pi / 2 + t : ℝ) : Real.Angle) =
+          inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) + 1) := by
+  have hTpos : (0 : ℝ) < Real.pi / 2 := by positivity
+  have h0mem : (0 : ℝ) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨le_rfl, hTpos.le⟩
+  have hTmem : (Real.pi / 2) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨hTpos.le, le_rfl⟩
+  -- ### Path regularity and endpoints
+  have hreg := gerver_direct_path_regularity GerverSofa.PartB.params
+    GerverSofa.PartB.params_mem GerverSofa.PartB.params_equations
+  have hC1 : ContDiff ℝ 1 (GerverSofa.Romik.path GerverSofa.PartB.params) := hreg.1
+  have hpathcont : Continuous paperGerverPath := continuous_paperGerverPath hC1
+  -- The clockwise motion used for the moving-sofa property, and its continuity.
+  have hmotioncont : Continuous fun r : unitInterval =>
+      rotateTranslate ((-(r.val * (Real.pi / 2)) : ℝ) : Real.Angle)
+        (-paperGerverPath (r.val * (Real.pi / 2))) := by
+    have hpair : Continuous fun r : unitInterval =>
+        ((((-(r.val * (Real.pi / 2))) : ℝ) : Real.Angle),
+          -paperGerverPath (r.val * (Real.pi / 2))) :=
+      (Real.Angle.continuous_coe.comp (continuous_subtype_val.mul continuous_const).neg).prodMk
+        ((hpathcont.comp (continuous_subtype_val.mul continuous_const)).neg)
+    have heq : (fun r : unitInterval =>
+          rotateTranslate ((-(r.val * (Real.pi / 2)) : ℝ) : Real.Angle)
+            (-paperGerverPath (r.val * (Real.pi / 2)))) =
+        (fun q : Real.Angle × Point => (AffineIsometryEquiv.vaddConst ℝ q.2).trans
+          (EuclideanGeometry.o.rotation q.1).toAffineIsometryEquiv) ∘
+        (fun r : unitInterval => ((((-(r.val * (Real.pi / 2))) : ℝ) : Real.Angle),
+          -paperGerverPath (r.val * (Real.pi / 2)))) := rfl
+    rw [heq]
+    exact continuous_vaddConst_trans_rotation.comp hpair
+  have hpath0 : paperGerverPath 0 = 0 := by
+    change GerverSofa.PartF.Coordinates.toPlane
+      (GerverSofa.Romik.path GerverSofa.PartB.params 0) = 0
+    rw [hreg.2.1]
+    ext i
+    fin_cases i <;> rfl
+  have hpathTy : paperGerverPath (Real.pi / 2) 1 = 0 :=
+    GerverSofa.Romik.path_end_y_zero_of_mem_box_and_equations
+      GerverSofa.PartB.params_mem GerverSofa.PartB.params_equations
+  -- ### Coordinate images and compactness
+  have hKimg : gerverOuterCap =
+      GerverSofa.PartF.Coordinates.toPlane '' GerverSofa.Romik.K0 GerverSofa.PartB.params := by
+    rw [GerverSofa.PartF.Coordinates.image_eq_preimage]
+    exact Set.ext mem_gerverOuterCap_iff
+  have hKcomp : IsCompact gerverOuterCap := by
+    rw [hKimg]
+    exact GerverSofa.PartC.Stage4.K_compact_direct.image
+      GerverSofa.PartF.Coordinates.continuous_toPlane
+  have hGimg : gerverLiteralSofa =
+      GerverSofa.PartF.Coordinates.toPlane '' GerverSofa.PartC.G := by
+    rw [GerverSofa.PartF.Coordinates.image_eq_preimage]
+    exact Set.ext fun q =>
+      and_congr (mem_gerverOuterCap_iff q) (not_congr (mem_gerverLiteralNiche_iff q))
+  have hGcomp : IsCompact gerverLiteralSofa := by
+    rw [hGimg]
+    exact GerverSofa.PartC.Stage4.G_compact_direct.image
+      GerverSofa.PartF.Coordinates.continuous_toPlane
+  -- ### Contact coordinates in the moving frame
+  have hvn : ∀ t : ℝ, inner ℝ (tangentVector (t : Real.Angle))
+      (normalVector (t : Real.Angle)) = 0 := fun t => by
+    rw [real_inner_comm, inner_normalVector_tangentVector]
+  have hAn : ∀ t : ℝ, inner ℝ (paperGerverContacts t 0) (normalVector (t : Real.Angle)) =
+      inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) + 1 := fun t => by
+    simp only [paperGerverContacts, Matrix.cons_val_zero, inner_add_left, real_inner_smul_left,
+      inner_normalVector_self, hvn t]
+    ring
+  have hAt : ∀ t : ℝ, inner ℝ (paperGerverContacts t 0) (tangentVector (t : Real.Angle)) =
+      inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) +
+        (paperGerverVelocityComponents t).1 := fun t => by
+    simp only [paperGerverContacts, Matrix.cons_val_zero, inner_add_left, real_inner_smul_left,
+      inner_tangentVector_self, inner_normalVector_tangentVector]
+    ring
+  have hCn : ∀ t : ℝ, inner ℝ (paperGerverContacts t 2) (normalVector (t : Real.Angle)) =
+      inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) -
+        (paperGerverVelocityComponents t).2 := fun t => by
+    simp only [paperGerverContacts, Matrix.cons_val_two, Matrix.tail_cons, Matrix.head_cons,
+      inner_add_left, inner_sub_left, real_inner_smul_left, inner_normalVector_self, hvn t]
+    ring
+  have hCt : ∀ t : ℝ, inner ℝ (paperGerverContacts t 2) (tangentVector (t : Real.Angle)) =
+      inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) + 1 := fun t => by
+    simp only [paperGerverContacts, Matrix.cons_val_two, Matrix.tail_cons, Matrix.head_cons,
+      inner_add_left, inner_sub_left, real_inner_smul_left, inner_tangentVector_self,
+      inner_normalVector_tangentVector]
+    ring
+  -- ### The cap is the outer path constraint set
+  have hOPCS : outerPathConstraintSet
+      (fun t : Set.Icc (0 : ℝ) (Real.pi / 2) => paperGerverPath t.val) = gerverOuterCap := by
+    ext q
+    constructor
+    · rintro ⟨hy, h⟩
+      exact ⟨hy, fun t ht => h ⟨t, ht⟩⟩
+    · rintro ⟨hy, h⟩
+      exact ⟨hy, fun t => h t.val t.2⟩
+  -- ### The two normalizing contacts
+  have halpha0 : (paperGerverVelocityComponents 0).1 = 0 := by
+    rw [paperGerverVelocityComponents_eq_alphaBetaAt 0 h0mem]
+    have hab : GerverSofa.PartC.alphaBetaAt 0 =
+        GerverSofa.Romik.alphaBeta1 GerverSofa.PartB.params 0 := by
+      rw [GerverSofa.PartC.alphaBetaAt]
+      simp [GerverSofa.PartC.Stage4.phi_pos.le]
+    have ha2 := GerverSofa.Romik.a2_eq_neg_quarter_of_equations GerverSofa.PartB.params_equations
+    rw [hab]
+    simp only [GerverSofa.Romik.alphaBeta1, Real.sin_zero, Real.cos_zero, ha2]
+    ring
+  have hA0y : paperGerverContacts 0 0 1 = 0 := by
+    have h := inner_tangentVector_zero (paperGerverContacts 0 0)
+    rw [hAt 0, hpath0, inner_zero_left, halpha0] at h
+    linarith
+  have hC0y : paperGerverContacts 0 2 1 = 1 := by
+    have h := inner_tangentVector_zero (paperGerverContacts 0 2)
+    rw [hCt 0, hpath0, inner_zero_left] at h
+    linarith
+  obtain ⟨Kb, hKset, hKcap⟩ := outerPathConstraintSet_isCap
+    (fun t : Set.Icc (0 : ℝ) (Real.pi / 2) => paperGerverPath t.val)
+    (by simpa using hpath0)
+    ⟨paperGerverContacts 0 0, by rw [hOPCS]; exact gerver_outer_contact_A 0 h0mem, hA0y⟩
+    ⟨paperGerverContacts 0 2, by rw [hOPCS]; exact gerver_outer_contact_C 0 h0mem, hC0y⟩
+  rw [hOPCS] at hKset
+  -- ### Conjunct 1: every cap fibre has a sofa point at least as high
+  have hfiber := gerver_cap_fiber_has_sofa_point hKcomp
+  -- ### The support values of the cap
+  have hcapy1 : ∀ q ∈ gerverOuterCap, q 1 ≤ 1 := by
+    rintro q ⟨-, hc⟩
+    have h := (hc 0 h0mem).2
+    rw [hpath0, inner_zero_left, inner_tangentVector_zero] at h
+    linarith
+  have hcapx1 : ∀ q ∈ gerverOuterCap, q 0 ≤ 1 := by
+    rintro q ⟨-, hc⟩
+    have h := (hc 0 h0mem).1
+    rw [hpath0, inner_zero_left, inner_normalVector_zero] at h
+    linarith
+  have hsupK : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      supportValue gerverOuterCap (t : Real.Angle) =
+        inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) + 1 ∧
+      supportValue gerverOuterCap ((Real.pi / 2 + t : ℝ) : Real.Angle) =
+        inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) + 1 := by
+    intro t ht
+    have hAmem := gerver_outer_contact_A t ht
+    have hCmem := gerver_outer_contact_C t ht
+    have hang : ((Real.pi / 2 + t : ℝ) : Real.Angle) =
+        ((t + Real.pi / 2 : ℝ) : Real.Angle) := by rw [add_comm]
+    refine ⟨?_, ?_⟩
+    · simp only [supportValue]
+      refine IsGreatest.csSup_eq ⟨⟨paperGerverContacts t 0, hAmem, hAn t⟩, ?_⟩
+      rintro _ ⟨z, hz, rfl⟩
+      exact (hz.2 t ht).1
+    · rw [supportValue, hang, normalVector_add_pi_div_two_real]
+      refine IsGreatest.csSup_eq ⟨⟨paperGerverContacts t 2, hCmem, hCt t⟩, ?_⟩
+      rintro _ ⟨z, hz, rfl⟩
+      exact (hz.2 t ht).2
+  -- ### Conjunct 4: the sofa and the cap have the same upper support
+  have hGne : gerverLiteralSofa.Nonempty := gerver_literal_connected.1
+  have hsupeq : ∀ s ∈ Set.Icc (0 : ℝ) Real.pi,
+      supportValue gerverLiteralSofa (s : Real.Angle) =
+        supportValue gerverOuterCap (s : Real.Angle) := by
+    intro s hs
+    have hsin : 0 ≤ Real.sin s := Real.sin_nonneg_of_nonneg_of_le_pi hs.1 hs.2
+    refine (supportValue_eq_of_subset_of_inner_le hGne (fun z hz => hz.1)
+      (s : Real.Angle) ?_).symm
+    intro z hz
+    obtain ⟨w, hw, hwx, hwy⟩ := hfiber z hz
+    have h1 : inner ℝ z (normalVector (s : Real.Angle)) ≤
+        inner ℝ w (normalVector (s : Real.Angle)) := by
+      rw [inner_normalVector_real, inner_normalVector_real, hwx]
+      have := mul_le_mul_of_nonneg_right hwy hsin
+      linarith
+    exact h1.trans (inner_le_supportValue_of_isCompact hGcomp hw _)
+  have hsupG : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      supportValue gerverLiteralSofa (t : Real.Angle) =
+        inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) + 1 ∧
+      supportValue gerverLiteralSofa ((Real.pi / 2 + t : ℝ) : Real.Angle) =
+        inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) + 1 := by
+    intro t ht
+    refine ⟨?_, ?_⟩
+    · rw [hsupeq t ⟨ht.1, by linarith [ht.2, Real.pi_pos]⟩]
+      exact (hsupK t ht).1
+    · rw [hsupeq (Real.pi / 2 + t) ⟨by linarith [ht.1], by linarith [ht.2]⟩]
+      exact (hsupK t ht).2
+  -- ### Conjunct 3: standard position
+  have hsv1 : supportValue gerverLiteralSofa ((Real.pi / 2 : ℝ) : Real.Angle) = 1 := by
+    have h := (hsupG 0 h0mem).2
+    rw [add_zero, hpath0, inner_zero_left] at h
+    linarith
+  have hstd := gerver_literal_standard_position hTpos hTmem hGcomp hmotioncont hpath0 hpathTy
+    hcapx1 hcapy1 hsv1
+  -- ### Conjunct 5: the paper cap of the sofa is the certified cap
+  have hcapOf := gerver_literal_cap_eq hsupG hcapy1
+  -- ### Conjunct 6: the cap representative and its contact faces
+  have hfaces := gerver_cap_contact_faces Kb hKset hKcap hC1 hAn hAt hCn hCt hsupK hTpos
+  refine ⟨hfiber, paperGerverSofa_eq_literal, hstd, hsupeq, hcapOf, hfaces, ?_⟩
+  rintro S (rfl | rfl)
+  · exact hsupK
+  · exact hsupG
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver / Niche / Identification
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem gerver_paperNiche_identification :
+    (∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      supportingHallway gerverOuterCap (t : Real.Angle) =
+        supportingHallway paperGerverSofa (t : Real.Angle) ∧
+      supportingHallway paperGerverSofa (t : Real.Angle) =
+        (fun p ↦ rotationMap (t : Real.Angle) p + paperGerverPath t) '' hallway ∧
+      (rotatingHallwayParts gerverOuterCap (t : Real.Angle)).innerCorner = paperGerverPath t) ∧
+    (∃ K : CapSpace (Real.pi / 2), (K.val : Set Point) = gerverOuterCap ∧
+      capNiche K = gerverLiteralNiche) ∧
+    monotonization paperGerverSofa (Real.pi / 2) = paperGerverSofa ∧
+    IsMonotoneSofa paperGerverSofa := by
+  obtain ⟨-, hGeq, hstd, -, -, ⟨K, hKset, -⟩, hsup⟩ := gerver_capSupport_identification
+  have hangsum : ∀ t : ℝ, (t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) =
+      ((Real.pi / 2 + t : ℝ) : Real.Angle) := fun t => by
+    rw [← Real.Angle.coe_add, add_comm]
+  -- ### The supporting placement is the paper motion
+  have hplace : ∀ S ∈ ({gerverOuterCap, gerverLiteralSofa} : Set (Set Point)),
+      ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2), ∀ p : Point,
+      supportingPlacement S (t : Real.Angle) p =
+        rotationMap (t : Real.Angle) p + paperGerverPath t := by
+    intro S hS t ht p
+    rw [supportingPlacement, hangsum t, (hsup S hS t ht).1, (hsup S hS t ht).2,
+      add_sub_cancel_right, add_sub_cancel_right, add_assoc,
+      inner_normalVector_smul_add_inner_tangentVector_smul]
+  have hplaceK : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2), ∀ p : Point,
+      supportingPlacement gerverOuterCap (t : Real.Angle) p =
+        rotationMap (t : Real.Angle) p + paperGerverPath t :=
+    hplace gerverOuterCap (Or.inl rfl)
+  have hplaceG : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2), ∀ p : Point,
+      supportingPlacement paperGerverSofa (t : Real.Angle) p =
+        rotationMap (t : Real.Angle) p + paperGerverPath t := by
+    rw [hGeq]
+    exact hplace gerverLiteralSofa (Or.inr rfl)
+  have hhallG : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      supportingHallway paperGerverSofa (t : Real.Angle) =
+        (fun p ↦ rotationMap (t : Real.Angle) p + paperGerverPath t) '' hallway :=
+    fun t ht => Set.image_congr' (hplaceG t ht)
+  have hhallK : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      supportingHallway gerverOuterCap (t : Real.Angle) =
+        (fun p ↦ rotationMap (t : Real.Angle) p + paperGerverPath t) '' hallway :=
+    fun t ht => Set.image_congr' (hplaceK t ht)
+  -- ### The monotonization fixed point
+  have hmono : monotonization paperGerverSofa (Real.pi / 2) = paperGerverSofa := by
+    rw [monotonization, Set.iInter₂_congr hhallG]
+    rfl
+  refine ⟨fun t ht => ⟨(hhallK t ht).trans (hhallG t ht).symm, hhallG t ht, ?_⟩, ⟨K, hKset, ?_⟩,
+    hmono, paperGerverSofa, Real.pi / 2, by rw [hGeq]; exact hstd, hmono.symm⟩
+  · change supportingPlacement gerverOuterCap (t : Real.Angle) hallwayParts.innerCorner =
+      paperGerverPath t
+    rw [hplaceK t ht]
+    change rotationMap (t : Real.Angle) 0 + paperGerverPath t = paperGerverPath t
+    rw [rotationMap, map_zero, zero_add]
+  -- ### The cap niche is the literal niche
+  · have hfan : ∀ q : Point, q ∈ capFan (Real.pi / 2) ↔ 0 ≤ q 1 := by
+      intro q
+      have h : (q ∈ capFan (Real.pi / 2)) ↔
+          ((0 : ℝ) ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) ∧
+            (0 : ℝ) ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))) := Iff.rfl
+      rw [h, inner_normalVector_pi_div_two, and_self]
+    have hquad : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2), ∀ q : Point,
+        q ∈ innerQuadrant gerverOuterCap t ↔
+          inner ℝ (q - paperGerverPath t) (normalVector (t : Real.Angle)) < 0 ∧
+            inner ℝ (q - paperGerverPath t) (tangentVector (t : Real.Angle)) < 0 := by
+      intro t ht q
+      have hang : ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+          ((Real.pi / 2 + t : ℝ) : Real.Angle) := by rw [add_comm]
+      have h : (q ∈ innerQuadrant gerverOuterCap t) ↔
+          (inner ℝ q (normalVector (t : Real.Angle)) <
+              supportValue gerverOuterCap (t : Real.Angle) - 1 ∧
+            inner ℝ q (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) <
+              supportValue gerverOuterCap ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) := Iff.rfl
+      rw [h, normalVector_add_pi_div_two_real, hang,
+        (hsup gerverOuterCap (Or.inl rfl) t ht).1, (hsup gerverOuterCap (Or.inl rfl) t ht).2,
+        add_sub_cancel_right, add_sub_cancel_right, inner_sub_left, inner_sub_left,
+        sub_neg, sub_neg]
+    rw [capNiche, hKset]
+    ext q
+    constructor
+    · rintro ⟨hq, hmem⟩
+      obtain ⟨t, ht, hqt⟩ := Set.mem_iUnion₂.1 hmem
+      exact ⟨(hfan q).1 hq, t, ht, (hquad t (Set.Ioo_subset_Icc_self ht) q).1 hqt⟩
+    · rintro ⟨hy, t, ht, hqt⟩
+      exact ⟨(hfan q).2 hy,
+        Set.mem_iUnion₂.2 ⟨t, ht, (hquad t (Set.Ioo_subset_Icc_self ht) q).2 hqt⟩⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Surface densities of the certified Gerver cap
+
+The certified Gerver cap carries the two envelope densities `GerverSofa.PartC.Stage2.rhoA` and
+`rhoC` of the vendor development: its surface-area measure is `rhoA (t) dt` on the angular arc
+`[0, π/2)` and `rhoC (t - π/2) dt` on `(π/2, π]`.
+
+The argument is stage by stage.  On each of the five closed stages the selected (positive) vertex
+of the cap is a globally analytic branch of the certified phase curves, with derivative
+`rhoA t • v t` for the first contact and `-rhoC t • u t` for the third, so
+`surfaceAreaMeasure_angleImage_eq_withDensity_of_hasDerivAt` identifies the surface measure with
+the exact Lebesgue density on that stage.  The stages are then glued with
+`measure_angleImage_eq_of_union`; only the last `rhoA` stage needs an interior exhaustion,
+because the positive vertex jumps at `π/2`.
+
+The same stagewise phase curves describe the two *inner* contacts, because `B = A - u` and
+`D = C - v` differ from the outer contacts by a frame vector
+(`paperGerverContacts_one_eq_sub`, `paperGerverContacts_three_eq_sub`).  Subtracting the frame
+vector from a phase curve therefore produces a globally differentiable branch curve for `B` on
+the last two stages and for `D` on the first two, with the nonnegative speeds `1 - rhoA` and
+`1 - rhoC` (`exists_branch_paperGerverContacts_one`,
+`exists_branch_paperGerverContacts_three`); the two speed bounds are again coarse box bounds on
+the certified parameters.
+-/
+
+/-! ### Coordinate transport of derivatives -/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped NNReal
+
+namespace MovingSofa
+
+open MeasureTheory Set
+open scoped ENNReal
+
+/-- The vendor coordinate identification is homogeneous. -/
+private theorem toPlane_smul (c : ℝ) (q : GerverSofa.Point) :
+    GerverSofa.PartF.Coordinates.toPlane (c • q) =
+      c • GerverSofa.PartF.Coordinates.toPlane q := by
+  ext i
+  fin_cases i <;> rfl
+
+/-! ### The certified switching angles -/
+
+/-- The paper's two switching angles are the vendor parameter angles, strictly ordered. -/
+private theorem paperGerver_switchAngles :
+    GerverSofa.PartB.params.phi = GerversSofa.φ ∧
+      GerverSofa.PartB.params.theta = GerversSofa.θ ∧
+      GerverSofa.PartB.params.phi < GerverSofa.PartB.params.theta ∧
+      GerverSofa.PartB.params.theta < Real.pi / 4 := by
+  obtain ⟨-, hphi, htheta, -, hphitheta, hthetalt, -⟩ :=
+    gerver_parameter_identification.1 GerverSofa.PartB.params GerverSofa.PartB.params_mem
+      GerverSofa.PartB.params_equations
+  have hphi' : GerverSofa.PartB.params.phi = GerversSofa.φ := hphi.trans selected_phi
+  have htheta' : GerverSofa.PartB.params.theta = GerversSofa.θ := htheta.trans selected_theta
+  refine ⟨hphi', htheta', ?_, ?_⟩
+  · rw [hphi', htheta']; exact hphitheta
+  · rw [htheta']; exact hthetalt
+
+/-! ### The contact curves as transported certified phase curves -/
+
+/-- The first contact of the paper Gerver cap is the transported point `x + α v + u`. -/
+private theorem paperGerverContacts_zero_eq_toPlane {X W : ℝ → GerverSofa.Point} {t : ℝ}
+    (hx : GerverSofa.Romik.path GerverSofa.PartB.params t = X t)
+    (hw : paperGerverVelocityComponents t = W t) :
+    paperGerverContacts t 0 =
+      GerverSofa.PartF.Coordinates.toPlane (X t + (W t).1 • GerverSofa.v t + GerverSofa.u t) := by
+  have hp : paperGerverPath t = GerverSofa.PartF.Coordinates.toPlane (X t) := by
+    change GerverSofa.PartF.Coordinates.toPlane
+      (GerverSofa.Romik.path GerverSofa.PartB.params t) = _
+    rw [hx]
+  simp only [paperGerverContacts, Matrix.cons_val_zero, hw, hp]
+  ext i
+  fin_cases i <;> rfl
+
+/-- The third contact of the paper Gerver cap is the transported point `x - β u + v`. -/
+private theorem paperGerverContacts_two_eq_toPlane {X W : ℝ → GerverSofa.Point} {t : ℝ}
+    (hx : GerverSofa.Romik.path GerverSofa.PartB.params t = X t)
+    (hw : paperGerverVelocityComponents t = W t) :
+    paperGerverContacts t 2 =
+      GerverSofa.PartF.Coordinates.toPlane (X t - (W t).2 • GerverSofa.u t + GerverSofa.v t) := by
+  have hp : paperGerverPath t = GerverSofa.PartF.Coordinates.toPlane (X t) := by
+    change GerverSofa.PartF.Coordinates.toPlane
+      (GerverSofa.Romik.path GerverSofa.PartB.params t) = _
+    rw [hx]
+  simp only [paperGerverContacts, hw, hp]
+  ext i
+  fin_cases i <;> rfl
+
+open GerverSofa.PartC.Stage2 GerverSofa.Romik in
+/-- On each of the five stages the two contact curves are transported analytic phase curves whose
+derivatives are `rA • v` and `-rC • u` for continuous factors agreeing with `rhoA`, `rhoC`. -/
+private theorem exists_phaseCurves_of_stage (j : Fin 5) :
+    ∃ (PA PC : ℝ → GerverSofa.Point) (rA rC : ℝ → ℝ),
+      (∀ t, HasDerivAt PA (rA t • GerverSofa.v t) t) ∧
+      (∀ t, HasDerivAt PC (-(rC t) • GerverSofa.u t) t) ∧
+      Continuous rA ∧ Continuous rC ∧
+      (∀ t ∈ gerverStageIntervals j,
+        paperGerverContacts t 0 = GerverSofa.PartF.Coordinates.toPlane (PA t) ∧
+        paperGerverContacts t 2 = GerverSofa.PartF.Coordinates.toPlane (PC t)) ∧
+      (∀ t ∈ Set.Ioc (gerverStageTimes j.castSucc) (gerverStageTimes j.succ),
+        rhoA t = rA t ∧ rhoC t = rC t) := by
+  classical
+  obtain ⟨hphi', htheta', hφθ', hθq'⟩ := paperGerver_switchAngles
+  have heqs : Equations GerverSofa.PartB.params := GerverSofa.PartB.params_equations
+  have hφθ : GerversSofa.φ < GerversSofa.θ := by rw [← hphi', ← htheta']; exact hφθ'
+  have hθq : GerversSofa.θ < Real.pi / 4 := by rw [← htheta']; exact hθq'
+  have hθη : GerversSofa.θ ≤ Real.pi / 2 - GerversSofa.θ := by
+    have := GerverSofa.PartC.switchOrder.theta_le_eta
+    rw [htheta'] at this; exact this
+  have hητ : Real.pi / 2 - GerversSofa.θ ≤ Real.pi / 2 - GerversSofa.φ := by linarith
+  have heta : GerverSofa.PartC.eta = Real.pi / 2 - GerversSofa.θ := by
+    simp only [GerverSofa.PartC.eta, GerverSofa.PartC.T, htheta']
+  have htau : GerverSofa.PartC.tau = Real.pi / 2 - GerversSofa.φ := by
+    simp only [GerverSofa.PartC.tau, GerverSofa.PartC.T, hphi']
+  have hvel : ∀ (i : Fin 5), ∀ t ∈ gerverStageIntervals i,
+      paperGerverVelocityComponents t = gerverBranchVelocityComponents i t :=
+    paperGerverContactData_properties.2.2.2
+  fin_cases j
+  · refine ⟨phaseA1, phaseC1, fun _ ↦ 0, fun _ ↦ 1 / 2, fun t ↦ A1_hasDerivAt_public t,
+      fun t ↦ C1_hasDerivAt_public t, continuous_const, continuous_const, ?_, ?_⟩
+    · change ∀ t ∈ Set.Icc (0 : ℝ) GerversSofa.φ, _
+      intro t ht
+      have hx : path GerverSofa.PartB.params t = path1 GerverSofa.PartB.params t :=
+        path_eq_path1_of_mem_Icc _ (by rw [hphi']; exact ht)
+      have hw : paperGerverVelocityComponents t = alphaBeta1 GerverSofa.PartB.params t :=
+        hvel 0 t ht
+      exact ⟨paperGerverContacts_zero_eq_toPlane hx hw, paperGerverContacts_two_eq_toPlane hx hw⟩
+    · change ∀ t ∈ Set.Ioc (0 : ℝ) GerversSofa.φ, _
+      intro t ht
+      refine ⟨?_, ?_⟩ <;> simp only [rhoA, rhoC, hphi', htheta', heta, htau] <;>
+        split_ifs with h1 h2 h3 h4 <;>
+        first
+          | rfl
+          | (exfalso; linarith [ht.1, ht.2, hφθ, hθη, hητ])
+  · refine ⟨phaseA2, phaseC2,
+      fun t ↦ -(1 / 4 : ℝ) * t * t + GerverSofa.PartC.params.b1 * t +
+        GerverSofa.PartC.params.b2 + 1 / 2,
+      fun t ↦ t / 2 - GerverSofa.PartC.params.b1,
+      fun t ↦ A2_hasDerivAt_public t, fun t ↦ C2_hasDerivAt_public t, by fun_prop, by fun_prop,
+      ?_, ?_⟩
+    · change ∀ t ∈ Set.Icc GerversSofa.φ GerversSofa.θ, _
+      intro t ht
+      have hx : path GerverSofa.PartB.params t = path2 GerverSofa.PartB.params t :=
+        path_eq_path2_of_mem_Icc heqs (by rw [hphi', htheta']; exact ht)
+      have hw : paperGerverVelocityComponents t = alphaBeta2 GerverSofa.PartB.params t :=
+        hvel 1 t ht
+      exact ⟨paperGerverContacts_zero_eq_toPlane hx hw, paperGerverContacts_two_eq_toPlane hx hw⟩
+    · change ∀ t ∈ Set.Ioc GerversSofa.φ GerversSofa.θ, _
+      intro t ht
+      refine ⟨?_, ?_⟩ <;> simp only [rhoA, rhoC, hphi', htheta', heta, htau] <;>
+        split_ifs with h1 h2 h3 h4 <;>
+        first
+          | rfl
+          | (exfalso; linarith [ht.1, ht.2, hφθ, hθη, hητ])
+  · refine ⟨phaseA3, phaseC3, fun t ↦ 1 + GerverSofa.PartC.params.c1 - t,
+      fun t ↦ 1 + GerverSofa.PartC.params.c2 + t,
+      fun t ↦ A3_hasDerivAt_public t, fun t ↦ C3_hasDerivAt_public t, by fun_prop, by fun_prop,
+      ?_, ?_⟩
+    · change ∀ t ∈ Set.Icc GerversSofa.θ (Real.pi / 2 - GerversSofa.θ), _
+      intro t ht
+      have hx : path GerverSofa.PartB.params t = path3 GerverSofa.PartB.params t :=
+        path_eq_path3_of_mem_Icc heqs hφθ' (by rw [htheta']; exact ht)
+      have hw : paperGerverVelocityComponents t = alphaBeta3 GerverSofa.PartB.params t :=
+        hvel 2 t ht
+      exact ⟨paperGerverContacts_zero_eq_toPlane hx hw, paperGerverContacts_two_eq_toPlane hx hw⟩
+    · change ∀ t ∈ Set.Ioc GerversSofa.θ (Real.pi / 2 - GerversSofa.θ), _
+      intro t ht
+      refine ⟨?_, ?_⟩ <;> simp only [rhoA, rhoC, hphi', htheta', heta, htau] <;>
+        split_ifs with h1 h2 h3 h4 <;>
+        first
+          | rfl
+          | (exfalso; linarith [ht.1, ht.2, hφθ, hθη, hητ])
+  · refine ⟨phaseA4, phaseC4, fun t ↦ GerverSofa.PartC.params.d1 - t / 2,
+      fun t ↦ -(1 / 4 : ℝ) * t * t + GerverSofa.PartC.params.d1 * t +
+        GerverSofa.PartC.params.d2 + 1 / 2,
+      fun t ↦ A4_hasDerivAt_public t, fun t ↦ C4_hasDerivAt_public t, by fun_prop, by fun_prop,
+      ?_, ?_⟩
+    · change ∀ t ∈ Set.Icc (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ), _
+      intro t ht
+      have hx : path GerverSofa.PartB.params t = path4 GerverSofa.PartB.params t :=
+        path_eq_path4_of_mem_Icc heqs hφθ' hθq' (by rw [hphi', htheta']; exact ht)
+      have hw : paperGerverVelocityComponents t = alphaBeta4 GerverSofa.PartB.params t :=
+        hvel 3 t ht
+      exact ⟨paperGerverContacts_zero_eq_toPlane hx hw, paperGerverContacts_two_eq_toPlane hx hw⟩
+    · change ∀ t ∈ Set.Ioc (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ), _
+      intro t ht
+      refine ⟨?_, ?_⟩ <;> simp only [rhoA, rhoC, hphi', htheta', heta, htau] <;>
+        split_ifs with h1 h2 h3 h4 <;>
+        first
+          | rfl
+          | (exfalso; linarith [ht.1, ht.2, hφθ, hθη, hητ])
+  · refine ⟨phaseA5, phaseC5, fun _ ↦ 1 / 2, fun _ ↦ 0, fun t ↦ A5_hasDerivAt_public t,
+      fun t ↦ by simpa using C5_hasDerivAt_public t, continuous_const, continuous_const, ?_, ?_⟩
+    · change ∀ t ∈ Set.Icc (Real.pi / 2 - GerversSofa.φ) (Real.pi / 2), _
+      intro t ht
+      have hx : path GerverSofa.PartB.params t = path5 GerverSofa.PartB.params t :=
+        path_eq_path5_of_mem_Icc heqs hφθ' hθq' (by rw [hphi']; exact ht)
+      have hw : paperGerverVelocityComponents t = alphaBeta5 GerverSofa.PartB.params t :=
+        hvel 4 t ht
+      exact ⟨paperGerverContacts_zero_eq_toPlane hx hw, paperGerverContacts_two_eq_toPlane hx hw⟩
+    · change ∀ t ∈ Set.Ioc (Real.pi / 2 - GerversSofa.φ) (Real.pi / 2), _
+      intro t ht
+      refine ⟨?_, ?_⟩ <;> simp only [rhoA, rhoC, hphi', htheta', heta, htau] <;>
+        split_ifs with h1 h2 h3 h4 <;>
+        first
+          | rfl
+          | (exfalso; linarith [ht.1, ht.2, hφθ, hθη, hητ])
+
+/-! ### Contact derivatives on the open stages -/
+
+/-- A branch curve agreeing with a global curve on a closed stage computes the global derivative
+at every interior parameter of that stage. -/
+theorem hasDerivAt_of_eqOn_stage {G F : ℝ → Point} {w : ℝ → Point} {i : Fin 5}
+    (hF : ∀ s, HasDerivAt F (w s) s) (hGF : ∀ t ∈ gerverStageIntervals i, G t = F t)
+    {t : ℝ} (ht : t ∈ Set.Ioo (gerverStageTimes i.castSucc) (gerverStageTimes i.succ)) :
+    HasDerivAt G (w t) t :=
+  (hF t).congr_of_eventuallyEq (Filter.eventuallyEq_of_mem (Ioo_mem_nhds ht.1 ht.2)
+    fun u hu ↦ hGF u (Set.Ioo_subset_Icc_self hu))
+
+/-- The transported phase curves of a stage are differentiable everywhere, with the two branch
+speeds against the moving frame. -/
+private theorem hasDerivAt_toPlane_phaseCurves {PA PC : ℝ → GerverSofa.Point}
+    {rA rC : ℝ → ℝ} (hPA : ∀ t, HasDerivAt PA (rA t • GerverSofa.v t) t)
+    (hPC : ∀ t, HasDerivAt PC (-(rC t) • GerverSofa.u t) t) :
+    (∀ s, HasDerivAt (fun u ↦ GerverSofa.PartF.Coordinates.toPlane (PA u))
+        (rA s • tangentVector (s : Real.Angle)) s) ∧
+      ∀ s, HasDerivAt (fun u ↦ GerverSofa.PartF.Coordinates.toPlane (PC u))
+        (-(rC s) • normalVector (s : Real.Angle)) s := by
+  refine ⟨fun s ↦ ?_, fun s ↦ ?_⟩
+  · have h := hasDerivAt_toPlane (hPA s)
+    rwa [toPlane_smul] at h
+  · have h := hasDerivAt_toPlane (hPC s)
+    rwa [toPlane_smul] at h
+
+/-- Inside each open stage the two contact curves are differentiable with the envelope
+densities as their speed factors. -/
+private theorem hasDerivAt_paperGerverContacts_of_mem_stage (j : Fin 5) {t : ℝ}
+    (ht : t ∈ Set.Ioo (gerverStageTimes j.castSucc) (gerverStageTimes j.succ)) :
+    HasDerivAt (fun u ↦ paperGerverContacts u 0)
+        (GerverSofa.PartC.Stage2.rhoA t • tangentVector (t : Real.Angle)) t ∧
+      HasDerivAt (fun u ↦ paperGerverContacts u 2)
+        (-(GerverSofa.PartC.Stage2.rhoC t) • normalVector (t : Real.Angle)) t := by
+  obtain ⟨PA, PC, rA, rC, hPA, hPC, -, -, hmem, hrho⟩ := exists_phaseCurves_of_stage j
+  obtain ⟨hrA, hrC⟩ := hrho t (Set.Ioo_subset_Ioc_self ht)
+  obtain ⟨hA, hC⟩ := hasDerivAt_toPlane_phaseCurves hPA hPC
+  refine ⟨?_, ?_⟩
+  · rw [hrA]
+    exact hasDerivAt_of_eqOn_stage hA (fun u hu ↦ (hmem u hu).1) ht
+  · rw [hrC]
+    exact hasDerivAt_of_eqOn_stage hC (fun u hu ↦ (hmem u hu).2) ht
+
+/-! ### Singleton faces have no surface atom -/
+
+/-- A face whose positive tangent endpoint does not exceed its negative one carries no surface
+atom: the atom is exactly the tangential gap between the two endpoints. -/
+private theorem surfaceAreaMeasure_singleton_eq_zero_of_inner_edgeVertices_le
+    (K : ConvexBody Point) (t : Real.Angle)
+    (h : inner ℝ (edgeVertices K t).1 (tangentVector t) ≤
+      inner ℝ (edgeVertices K t).2 (tangentVector t)) :
+    surfaceAreaMeasure K {t} = 0 := by
+  have hfin : IsFiniteMeasure (surfaceAreaMeasure K) := (surfaceAreaMeasure_face_union K).1
+  have hsplit := (surfaceAreaMeasure_atom_length K t).2.2
+  have hone : inner ℝ (tangentVector t) (tangentVector t) = 1 := by
+    rw [real_inner_self_eq_norm_sq, norm_tangentVector]
+    norm_num
+  have hinner : inner ℝ (edgeVertices K t).1 (tangentVector t) =
+      inner ℝ (edgeVertices K t).2 (tangentVector t) +
+        (surfaceAreaMeasure K {t}).toReal := by
+    rw [hsplit, inner_add_left, real_inner_smul_left, hone, mul_one]
+  have hle : (surfaceAreaMeasure K {t}).toReal ≤ 0 := by linarith
+  have hzero : (surfaceAreaMeasure K {t}).toReal = 0 :=
+    le_antisymm hle ENNReal.toReal_nonneg
+  rcases (ENNReal.toReal_eq_zero_iff _).1 hzero with h0 | htop
+  · exact h0
+  · exact absurd htop (measure_ne_top _ _)
+
+/-! ### Measurability and boundedness of the two envelope densities -/
+
+private theorem measurable_rhoA : Measurable GerverSofa.PartC.Stage2.rhoA := by
+  unfold GerverSofa.PartC.Stage2.rhoA
+  refine Measurable.ite measurableSet_Iic measurable_const ?_
+  refine Measurable.ite measurableSet_Iic (by fun_prop) ?_
+  refine Measurable.ite measurableSet_Iic (by fun_prop) ?_
+  exact Measurable.ite measurableSet_Iic (by fun_prop) measurable_const
+
+private theorem measurable_rhoC : Measurable GerverSofa.PartC.Stage2.rhoC := by
+  unfold GerverSofa.PartC.Stage2.rhoC
+  refine Measurable.ite measurableSet_Iic measurable_const ?_
+  refine Measurable.ite measurableSet_Iic (by fun_prop) ?_
+  refine Measurable.ite measurableSet_Iic (by fun_prop) ?_
+  exact Measurable.ite measurableSet_Iic (by fun_prop) measurable_const
+
+open GerverSofa.PartC.Stage2 in
+/-- Each of the finitely many branch polynomials of the two envelope densities is continuous,
+hence bounded on the compact physical interval. -/
+private theorem exists_bound_rhoA_rhoC : ∃ M : ℝ, ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+    rhoA t ≤ M ∧ rhoC t ≤ M := by
+  have key : ∀ f : ℝ → ℝ, Continuous f →
+      ∃ M : ℝ, ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2), f t ≤ M := by
+    intro f hf
+    obtain ⟨x, hx, hmax⟩ := isCompact_Icc.exists_isMaxOn
+      (Set.nonempty_Icc.2 (show (0 : ℝ) ≤ Real.pi / 2 by linarith [Real.pi_pos]))
+      hf.continuousOn
+    exact ⟨f x, fun t ht ↦ isMaxOn_iff.1 hmax t ht⟩
+  obtain ⟨M1, h1⟩ := key (fun t ↦ -(1 / 4 : ℝ) * t * t + GerverSofa.PartC.params.b1 * t +
+    GerverSofa.PartC.params.b2 + 1 / 2) (by fun_prop)
+  obtain ⟨M2, h2⟩ := key (fun t ↦ 1 + GerverSofa.PartC.params.c1 - t) (by fun_prop)
+  obtain ⟨M3, h3⟩ := key (fun t ↦ GerverSofa.PartC.params.d1 - t / 2) (by fun_prop)
+  obtain ⟨M4, h4⟩ := key (fun t ↦ t / 2 - GerverSofa.PartC.params.b1) (by fun_prop)
+  obtain ⟨M5, h5⟩ := key (fun t ↦ 1 + GerverSofa.PartC.params.c2 + t) (by fun_prop)
+  obtain ⟨M6, h6⟩ := key (fun t ↦ -(1 / 4 : ℝ) * t * t + GerverSofa.PartC.params.d1 * t +
+    GerverSofa.PartC.params.d2 + 1 / 2) (by fun_prop)
+  refine ⟨max (max (max M1 M2) (max M3 M4)) (max (max M5 M6) 1), fun t ht ↦ ⟨?_, ?_⟩⟩
+  · rw [rhoA]
+    split_ifs
+    · exact le_trans (by norm_num) (le_max_right _ _ |>.trans' (le_max_right _ _))
+    · exact (h1 t ht).trans (le_max_of_le_left (le_max_of_le_left (le_max_left _ _)))
+    · exact (h2 t ht).trans (le_max_of_le_left (le_max_of_le_left (le_max_right _ _)))
+    · exact (h3 t ht).trans (le_max_of_le_left (le_max_of_le_right (le_max_left _ _)))
+    · exact le_trans (by norm_num) (le_max_of_le_right (le_max_right _ _))
+  · rw [rhoC]
+    split_ifs
+    · exact le_trans (by norm_num) (le_max_of_le_right (le_max_right _ _))
+    · exact (h4 t ht).trans (le_max_of_le_left (le_max_of_le_right (le_max_right _ _)))
+    · exact (h5 t ht).trans (le_max_of_le_right (le_max_of_le_left (le_max_left _ _)))
+    · exact (h6 t ht).trans (le_max_of_le_right (le_max_of_le_left (le_max_right _ _)))
+    · exact le_trans (by norm_num) (le_max_of_le_right (le_max_right _ _))
+
+/-! ### Elementary facts about the stage endpoints and the frame -/
+
+/-- The two Gerver switch angles satisfy `0 < φ < θ < π/4`. -/
+private theorem gerver_switch_angle_bounds :
+    (0 : ℝ) < GerversSofa.φ ∧ GerversSofa.φ < GerversSofa.θ ∧ GerversSofa.θ < Real.pi / 4 := by
+  obtain ⟨-, -, -, hpos, hlt, hq, -⟩ :=
+    gerver_parameter_identification.1 GerverSofa.PartB.params GerverSofa.PartB.params_mem
+      GerverSofa.PartB.params_equations
+  refine ⟨selected_phi ▸ hpos, ?_, selected_theta ▸ hq⟩
+  rw [← selected_phi, ← selected_theta]
+  exact hlt
+
+private theorem gerverStageTimes_nonneg (k : Fin 6) : 0 ≤ gerverStageTimes k := by
+  obtain ⟨h0, h1, h2⟩ := gerver_switch_angle_bounds
+  have hpi := Real.pi_gt_three
+  fin_cases k
+  · change (0 : ℝ) ≤ 0
+    exact le_refl 0
+  · change (0 : ℝ) ≤ GerversSofa.φ
+    linarith
+  · change (0 : ℝ) ≤ GerversSofa.θ
+    linarith
+  · change (0 : ℝ) ≤ Real.pi / 2 - GerversSofa.θ
+    linarith
+  · change (0 : ℝ) ≤ Real.pi / 2 - GerversSofa.φ
+    linarith
+  · change (0 : ℝ) ≤ Real.pi / 2
+    linarith
+
+private theorem gerverStageTimes_le_pi_div_two (k : Fin 6) : gerverStageTimes k ≤ Real.pi / 2 := by
+  obtain ⟨h0, h1, h2⟩ := gerver_switch_angle_bounds
+  have hpi := Real.pi_gt_three
+  fin_cases k
+  · change (0 : ℝ) ≤ Real.pi / 2
+    linarith
+  · change GerversSofa.φ ≤ Real.pi / 2
+    linarith
+  · change GerversSofa.θ ≤ Real.pi / 2
+    linarith
+  · change Real.pi / 2 - GerversSofa.θ ≤ Real.pi / 2
+    linarith
+  · change Real.pi / 2 - GerversSofa.φ ≤ Real.pi / 2
+    linarith
+  · change Real.pi / 2 ≤ Real.pi / 2
+    exact le_refl _
+
+private theorem gerverStageTimes_lt_succ (j : Fin 5) :
+    gerverStageTimes j.castSucc < gerverStageTimes j.succ := by
+  obtain ⟨h0, h1, h2⟩ := gerver_switch_angle_bounds
+  have hpi := Real.pi_gt_three
+  fin_cases j
+  · change (0 : ℝ) < GerversSofa.φ
+    linarith
+  · change GerversSofa.φ < GerversSofa.θ
+    linarith
+  · change GerversSofa.θ < Real.pi / 2 - GerversSofa.θ
+    linarith
+  · change Real.pi / 2 - GerversSofa.θ < Real.pi / 2 - GerversSofa.φ
+    linarith
+  · change Real.pi / 2 - GerversSofa.φ < Real.pi / 2
+    linarith
+
+private theorem gerverStageTimes_four_lt_pi_div_two : gerverStageTimes 4 < Real.pi / 2 := by
+  have h : gerverStageTimes 4 < gerverStageTimes 5 := gerverStageTimes_lt_succ 4
+  have h5 : gerverStageTimes 5 = Real.pi / 2 := rfl
+  linarith
+
+/-! ### Almost every parameter lies in an open stage -/
+
+/-- A parameter of the rotation interval that is none of the six stage times lies in an open
+stage. -/
+theorem mem_openStage_of_mem_Ico {t : ℝ} (ht : t ∈ Set.Ico (0 : ℝ) (Real.pi / 2))
+    (hne : ∀ k : Fin 6, t ≠ gerverStageTimes k) :
+    ∃ j : Fin 5, t ∈ Set.Ioo (gerverStageTimes j.castSucc) (gerverStageTimes j.succ) := by
+  have hlow : gerverStageTimes 0 < t :=
+    lt_of_le_of_ne (gerverStageTimes_zero ▸ ht.1) fun h ↦ hne 0 h.symm
+  have hhigh : t < gerverStageTimes 5 := ht.2
+  rcases lt_or_gt_of_ne (hne 1) with h1 | h1
+  · exact ⟨0, hlow, h1⟩
+  · rcases lt_or_gt_of_ne (hne 2) with h2 | h2
+    · exact ⟨1, h1, h2⟩
+    · rcases lt_or_gt_of_ne (hne 3) with h3 | h3
+      · exact ⟨2, h2, h3⟩
+      · rcases lt_or_gt_of_ne (hne 4) with h4 | h4
+        · exact ⟨3, h3, h4⟩
+        · exact ⟨4, h4, hhigh⟩
+
+/-- Almost every parameter of a measurable subset of the shifted rotation interval lies in a
+shifted open stage. -/
+theorem ae_mem_openStage (c : ℝ) {S : Set ℝ} (hS : MeasurableSet S)
+    (hSsub : S ⊆ Set.Icc c (c + Real.pi / 2)) :
+    ∀ᵐ u ∂volume.restrict S, ∃ j : Fin 5,
+      u - c ∈ Set.Ioo (gerverStageTimes j.castSucc) (gerverStageTimes j.succ) := by
+  refine ae_restrict_mem_of_countable_diff hS
+    (((Set.finite_range gerverStageTimes).countable).image fun x ↦ x + c) ?_
+  rintro u ⟨hu, hu'⟩
+  by_contra hcon
+  have hne : ∀ k : Fin 6, u - c ≠ gerverStageTimes k := by
+    intro k hk
+    exact hcon ⟨gerverStageTimes k, Set.mem_range_self k, by rw [← hk]; ring⟩
+  have hmem : u - c ∈ Set.Ico (0 : ℝ) (Real.pi / 2) := by
+    refine ⟨by linarith [(hSsub hu).1], ?_⟩
+    have h5 : u - c ≠ gerverStageTimes 5 := hne 5
+    rw [show gerverStageTimes 5 = Real.pi / 2 from rfl] at h5
+    exact lt_of_le_of_ne (by linarith [(hSsub hu).2]) h5
+  exact hu' (mem_openStage_of_mem_Ico hmem hne)
+
+/-! ### Branch curves for the two inner contacts
+
+The second contact `B = x + α v` is the first contact `A = x + α v + u` translated by `-u`, and
+the fourth contact `D = x - β u` is the third contact `C = x - β u + v` translated by `-v`.  So
+subtracting a frame vector from the transported phase curve of a stage produces a globally
+differentiable branch curve for `B` and for `D`, whose speed against the frame is `1 - rhoA`
+resp. `1 - rhoC`.  Both are nonnegative exactly where the inner contact is a genuine contact:
+after the third stage time for `B` and before the second for `D`.
+-/
+
+/-- The second contact curve is the first one translated by `-u_t`. -/
+theorem paperGerverContacts_one_eq_sub (t : ℝ) :
+    paperGerverContacts t 1 = paperGerverContacts t 0 - normalVector (t : Real.Angle) := by
+  change _ = paperGerverPath t + (paperGerverVelocityComponents t).1 •
+    tangentVector (t : Real.Angle) + normalVector (t : Real.Angle) - normalVector _
+  rw [add_sub_cancel_right]
+  rfl
+
+/-- The fourth contact curve is the third one translated by `-v_t`. -/
+theorem paperGerverContacts_three_eq_sub (t : ℝ) :
+    paperGerverContacts t 3 = paperGerverContacts t 2 - tangentVector (t : Real.Angle) := by
+  change _ = paperGerverPath t - (paperGerverVelocityComponents t).2 •
+    normalVector (t : Real.Angle) + tangentVector (t : Real.Angle) - tangentVector _
+  rw [add_sub_cancel_right]
+  rfl
+
+open GerverSofa.PartC.Stage2 in
+/-- After the third stage time the first contact never outruns the rotating frame: its tangential
+speed `rhoA` is at most the unit speed of `u`.  On the fourth stage this is the coarse bound
+`d₁ ≤ 33/25` against `t > π/2 - θ > 4/5`; on the fifth the speed is the constant `1/2`. -/
+private theorem rhoA_le_one_of_gerverStageTimes_three_lt {t : ℝ}
+    (ht : gerverStageTimes 3 < t) : rhoA t ≤ 1 := by
+  have h1 : gerverStageTimes 1 = GerverSofa.PartB.params.phi := gerverStageTimes_one
+  have h2 : gerverStageTimes 2 = GerverSofa.PartB.params.theta := gerverStageTimes_two
+  have h3 : gerverStageTimes 3 = GerverSofa.PartC.eta := gerverStageTimes_three
+  have h4 : gerverStageTimes 4 = GerverSofa.PartC.tau := gerverStageTimes_four
+  have h12 : gerverStageTimes 1 < gerverStageTimes 2 := gerverStageTimes_lt_succ 1
+  have h23 : gerverStageTimes 2 < gerverStageTimes 3 := gerverStageTimes_lt_succ 2
+  have h32 : gerverStageTimes 3 = Real.pi / 2 - gerverStageTimes 2 := rfl
+  have hd1 := gerverDirectBox_d1_upper_bound GerverSofa.PartB.params_mem
+  have hθ := gerverStageTimes_two_le_seven_div_ten
+  have hpi := Real.pi_gt_three
+  rw [rhoA]
+  split_ifs with hphi htheta heta
+  · exact absurd (h1 ▸ hphi) (by linarith)
+  · exact absurd (h2 ▸ htheta) (by linarith)
+  · exact absurd (h3 ▸ heta) (by linarith)
+  · linarith
+  · norm_num
+
+open GerverSofa.PartC.Stage2 in
+/-- Before the second stage time the third contact never outruns the rotating frame: its normal
+speed `rhoC` is at most the unit speed of `v`.  On the first stage the speed is the constant
+`1/2`; on the second this is the coarse bound `b₁ ≥ -53/100` against `t ≤ θ ≤ 7/10`. -/
+private theorem rhoC_le_one_of_le_gerverStageTimes_two {t : ℝ}
+    (ht : t ≤ gerverStageTimes 2) : rhoC t ≤ 1 := by
+  have h2 : gerverStageTimes 2 = GerverSofa.PartB.params.theta := gerverStageTimes_two
+  have hb1 := gerverDirectBox_b1_lower_bound GerverSofa.PartB.params_mem
+  have hθ := gerverStageTimes_two_le_seven_div_ten
+  rw [rhoC]
+  split_ifs with hphi htheta
+  · norm_num
+  · linarith
+  · exact absurd (h2 ▸ ht) (by linarith)
+  · exact absurd (h2 ▸ ht) (by linarith)
+  · exact absurd (h2 ▸ ht) (by linarith)
+
+/-- On each of the last two stages the second contact curve agrees with a globally differentiable
+branch curve whose derivative is `-(g t) • v_t` for a continuous factor `g` that is nonnegative
+on the stage. -/
+theorem exists_branch_paperGerverContacts_one {i : Fin 5} (hi : i = 3 ∨ i = 4) :
+    ∃ (F : ℝ → Point) (g : ℝ → ℝ),
+      (∀ s, HasDerivAt F (-(g s) • tangentVector (s : Real.Angle)) s) ∧ Continuous g ∧
+      (∀ t ∈ Set.Ioc (gerverStageTimes i.castSucc) (gerverStageTimes i.succ), 0 ≤ g t) ∧
+      (∀ t ∈ gerverStageIntervals i, paperGerverContacts t 1 = F t) := by
+  obtain ⟨PA, PC, rA, rC, hPA, hPC, hcA, -, hmem, hrho⟩ := exists_phaseCurves_of_stage i
+  obtain ⟨hA, -⟩ := hasDerivAt_toPlane_phaseCurves hPA hPC
+  have h3i : gerverStageTimes 3 ≤ gerverStageTimes i.castSucc :=
+    gerverStageTimes_strictMono.monotone (by rcases hi with rfl | rfl <;> decide)
+  refine ⟨fun s ↦ GerverSofa.PartF.Coordinates.toPlane (PA s) - normalVector (s : Real.Angle),
+    fun s ↦ 1 - rA s, fun s ↦ ?_, continuous_const.sub hcA, fun t ht ↦ ?_, fun t ht ↦ ?_⟩
+  · refine ((hA s).sub (hasDerivAt_normalVector s)).congr_deriv ?_
+    module
+  · have hrA := (hrho t ht).1
+    linarith [rhoA_le_one_of_gerverStageTimes_three_lt (lt_of_le_of_lt h3i ht.1)]
+  · rw [paperGerverContacts_one_eq_sub, (hmem t ht).1]
+
+/-- On each of the first two stages the fourth contact curve agrees with a globally differentiable
+branch curve whose derivative is `g t • u_t` for a continuous factor `g` that is nonnegative on
+the stage. -/
+theorem exists_branch_paperGerverContacts_three {i : Fin 5} (hi : i = 0 ∨ i = 1) :
+    ∃ (F : ℝ → Point) (g : ℝ → ℝ),
+      (∀ s, HasDerivAt F (g s • normalVector (s : Real.Angle)) s) ∧ Continuous g ∧
+      (∀ t ∈ Set.Ioc (gerverStageTimes i.castSucc) (gerverStageTimes i.succ), 0 ≤ g t) ∧
+      (∀ t ∈ gerverStageIntervals i, paperGerverContacts t 3 = F t) := by
+  obtain ⟨PA, PC, rA, rC, hPA, hPC, -, hcC, hmem, hrho⟩ := exists_phaseCurves_of_stage i
+  obtain ⟨-, hC⟩ := hasDerivAt_toPlane_phaseCurves hPA hPC
+  have hi2 : gerverStageTimes i.succ ≤ gerverStageTimes 2 :=
+    gerverStageTimes_strictMono.monotone (by rcases hi with rfl | rfl <;> decide)
+  refine ⟨fun s ↦ GerverSofa.PartF.Coordinates.toPlane (PC s) - tangentVector (s : Real.Angle),
+    fun s ↦ 1 - rC s, fun s ↦ ?_, continuous_const.sub hcC, fun t ht ↦ ?_, fun t ht ↦ ?_⟩
+  · refine ((hC s).sub (hasDerivAt_tangentVector s)).congr_deriv ?_
+    module
+  · have hrC := (hrho t ht).2
+    linarith [rhoC_le_one_of_le_gerverStageTimes_two (le_trans ht.2 hi2)]
+  · rw [paperGerverContacts_three_eq_sub, (hmem t ht).2]
+
+/-! ### The certified cap witness and its positive vertex curves -/
+
+/-- The certified cap witness of support identification, together with the identification of its
+positive vertex by the two contact curves on the whole closed parameter interval and the
+vanishing of the surface atom at the included endpoint `0`.
+
+Both included endpoint faces are singletons: at `0` the first contact is the anchor `(1, 0)`
+and the cap lies in `0 ≤ q 1`, and at `π` the third contact has vanishing second coordinate,
+so in each case the positive tangent endpoint cannot exceed the negative one. -/
+private theorem exists_gerverCap_positiveVertex :
+    ∃ K : CapSpace (Real.pi / 2),
+      (K.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2) ∧
+      surfaceAreaMeasure K.val {((0 : ℝ) : Real.Angle)} = 0 ∧
+      (∀ t ∈ Set.Ico (0 : ℝ) (Real.pi / 2),
+        (edgeVertices K.val ((t : ℝ) : Real.Angle)).1 = paperGerverContacts t 0) ∧
+      (∀ s ∈ Set.Icc (Real.pi / 2) Real.pi,
+        (edgeVertices K.val ((s : ℝ) : Real.Angle)).1 =
+          paperGerverContacts (s - Real.pi / 2) 2) := by
+  have hpi := Real.pi_gt_three
+  have hTpos : (0 : ℝ) < Real.pi / 2 := by linarith
+  have h0mem : (0 : ℝ) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨le_rfl, hTpos.le⟩
+  have hTmem : Real.pi / 2 ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨hTpos.le, le_rfl⟩
+  obtain ⟨-, hpaper, -, -, hcapEq, ⟨K, hKset, hedges, hA0, -, hC0, hCT⟩, -⟩ :=
+    gerver_capSupport_identification
+  have hKupper : ∀ q ∈ (K.val : Set Point), 0 ≤ q 1 := by
+    intro q hq
+    rw [hKset] at hq
+    exact hq.1
+  -- the two included endpoint contacts lie on the lower fan boundary
+  have hcontactA0y : paperGerverContacts 0 0 1 = 0 := by
+    have h := fromPlane_paperGerverContacts 0 h0mem 0
+    have h2 : paperGerverContacts 0 0 =
+        GerverSofa.PartF.Coordinates.toPlane (GerverSofa.PartC.A 0) := by
+      rw [← GerverSofa.PartF.Coordinates.toPlane_fromPlane (paperGerverContacts 0 0), h]
+      rfl
+    rw [h2, GerverSofa.PartC.Stage2.A_zero_eq_anchor]
+    rfl
+  have hcontactCTy : paperGerverContacts (Real.pi / 2) 2 1 = 0 := by
+    have h := fromPlane_paperGerverContacts (Real.pi / 2) hTmem 2
+    have h2 : paperGerverContacts (Real.pi / 2) 2 =
+        GerverSofa.PartF.Coordinates.toPlane (GerverSofa.PartC.C (Real.pi / 2)) := by
+      rw [← GerverSofa.PartF.Coordinates.toPlane_fromPlane
+        (paperGerverContacts (Real.pi / 2) 2), h]
+      rfl
+    rw [h2]
+    exact GerverSofa.PartC.Stage2.C_T_snd_zero
+  -- hence both included endpoint faces are singletons and carry no atom
+  have hatom0 : surfaceAreaMeasure K.val {((0 : ℝ) : Real.Angle)} = 0 := by
+    refine surfaceAreaMeasure_singleton_eq_zero_of_inner_edgeVertices_le K.val _ ?_
+    have hfst : (edgeVertices K.val ((0 : ℝ) : Real.Angle)).1 = paperGerverContacts 0 0 := by
+      rw [Real.Angle.coe_zero]
+      exact hA0.symm
+    rw [inner_tangentVector_zero, inner_tangentVector_zero, hfst, hcontactA0y]
+    exact hKupper _ (edgeVertices_snd_mem K.val _).1
+  have hatompi : surfaceAreaMeasure K.val {((Real.pi : ℝ) : Real.Angle)} = 0 := by
+    refine surfaceAreaMeasure_singleton_eq_zero_of_inner_edgeVertices_le K.val _ ?_
+    rw [inner_tangentVector_pi, inner_tangentVector_pi, ← hCT, hcontactCTy]
+    have h := hKupper _ (edgeVertices_fst_mem K.val ((Real.pi : ℝ) : Real.Angle)).1
+    linarith
+  refine ⟨K, by rw [hpaper, hcapEq]; exact hKset, hatom0, ?_, ?_⟩
+  · intro t ht
+    rcases eq_or_lt_of_le ht.1 with h | h
+    · rw [← h, Real.Angle.coe_zero]
+      exact hA0.symm
+    · rw [edgeVertices_eq_of_exposedEdge_singleton (hedges t ⟨h, ht.2⟩).1]
+  · intro s hs
+    rcases eq_or_lt_of_le hs.1 with h1 | h1
+    · rw [← h1, sub_self]
+      exact hC0.symm
+    rcases eq_or_lt_of_le hs.2 with h2 | h2
+    · have hatom := (surfaceAreaMeasure_atom_length K.val ((Real.pi : ℝ) : Real.Angle)).2.2
+      rw [hatompi] at hatom
+      simp only [ENNReal.toReal_zero, zero_smul, add_zero] at hatom
+      rw [h2, show Real.pi - Real.pi / 2 = Real.pi / 2 from by ring, hatom, ← hCT]
+    · have htmem : s - Real.pi / 2 ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) :=
+        ⟨by linarith, by linarith⟩
+      have h := (hedges (s - Real.pi / 2) htmem).2
+      rw [show s - Real.pi / 2 + Real.pi / 2 = s from by ring] at h
+      rw [edgeVertices_eq_of_exposedEdge_singleton h]
+
+/-! ### The stagewise density identities -/
+
+/-- On a subinterval of a single stage, strictly below the switch `π/2`, the surface measure of
+an angular image is the `rhoA`-weighted Lebesgue measure. -/
+private theorem surfaceAreaMeasure_angleImage_eq_withDensity_rhoA_of_stage (K : ConvexBody Point)
+    (hvertexA : ∀ t ∈ Set.Ico (0 : ℝ) (Real.pi / 2),
+      (edgeVertices K ((t : ℝ) : Real.Angle)).1 = paperGerverContacts t 0)
+    (j : Fin 5) (a b : ℝ) (hja : gerverStageTimes j.castSucc ≤ a)
+    (hjb : b ≤ gerverStageTimes j.succ) (hbT : b < Real.pi / 2)
+    (S : Set ℝ) (hS : MeasurableSet S) (hSsub : S ⊆ Set.Ioc a b) :
+    surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+      volume.withDensity (fun s ↦ ENNReal.ofReal (GerverSofa.PartC.Stage2.rhoA s)) S := by
+  have hpi := Real.pi_gt_three
+  rcases le_or_gt b a with hle | hab
+  · have hempty : S = ∅ := by
+      refine Set.eq_empty_iff_forall_notMem.2 fun x hx ↦ ?_
+      have hx' := hSsub hx
+      rw [Set.mem_Ioc] at hx'
+      linarith [hx'.1, hx'.2]
+    rw [hempty]
+    simp
+  obtain ⟨PA, PC, rA, rC, hPA, hPC, hcA, hcC, hmem, hrho⟩ := exists_phaseCurves_of_stage j
+  have ha0 : 0 ≤ a := le_trans (gerverStageTimes_nonneg _) hja
+  refine surfaceAreaMeasure_angleImage_eq_withDensity_of_hasDerivAt K hab (by linarith)
+    (fun s ↦ GerverSofa.PartF.Coordinates.toPlane (PA s)) rA
+    GerverSofa.PartC.Stage2.rhoA (fun s ↦ ?_) hcA ?_ ?_ ?_ S hS hSsub
+  · have h := hasDerivAt_toPlane (hPA s)
+    rw [toPlane_smul] at h
+    exact h
+  · intro s hs
+    rw [hvertexA s ⟨le_trans ha0 hs.1, lt_of_le_of_lt hs.2 hbT⟩]
+    exact (hmem s ⟨le_trans hja hs.1, le_trans hs.2 hjb⟩).1
+  · intro s hs
+    exact (hrho s ⟨lt_of_le_of_lt hja hs.1, le_trans hs.2 hjb⟩).1
+  · intro s hs
+    have hs0 : (0 : ℝ) ≤ s := by linarith [hs.1]
+    have hsT : s ≤ Real.pi / 2 := by linarith [hs.2]
+    exact GerverSofa.PartC.Stage2.rhoA_nonneg ⟨hs0, hsT⟩
+
+/-- On a subinterval of a single shifted stage the surface measure of an angular image is the
+translated `rhoC`-weighted Lebesgue measure. -/
+private theorem surfaceAreaMeasure_angleImage_eq_withDensity_rhoC_of_stage (K : ConvexBody Point)
+    (hvertexC : ∀ s ∈ Set.Icc (Real.pi / 2) Real.pi,
+      (edgeVertices K ((s : ℝ) : Real.Angle)).1 = paperGerverContacts (s - Real.pi / 2) 2)
+    (j : Fin 5) (a b : ℝ) (hja : Real.pi / 2 + gerverStageTimes j.castSucc ≤ a)
+    (hjb : b ≤ Real.pi / 2 + gerverStageTimes j.succ)
+    (S : Set ℝ) (hS : MeasurableSet S) (hSsub : S ⊆ Set.Ioc a b) :
+    surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+      volume.withDensity
+        (fun u ↦ ENNReal.ofReal (GerverSofa.PartC.Stage2.rhoC (u - Real.pi / 2))) S := by
+  have hpi := Real.pi_gt_three
+  rcases le_or_gt b a with hle | hab
+  · have hempty : S = ∅ := by
+      refine Set.eq_empty_iff_forall_notMem.2 fun x hx ↦ ?_
+      have hx' := hSsub hx
+      rw [Set.mem_Ioc] at hx'
+      linarith [hx'.1, hx'.2]
+    rw [hempty]
+    simp
+  obtain ⟨PA, PC, rA, rC, hPA, hPC, hcA, hcC, hmem, hrho⟩ := exists_phaseCurves_of_stage j
+  have ha0 : Real.pi / 2 ≤ a := le_trans (by linarith [gerverStageTimes_nonneg j.castSucc]) hja
+  have hbpi : b ≤ Real.pi := by
+    have := gerverStageTimes_le_pi_div_two j.succ
+    linarith
+  refine surfaceAreaMeasure_angleImage_eq_withDensity_of_hasDerivAt K hab (by linarith)
+    (fun s ↦ GerverSofa.PartF.Coordinates.toPlane (PC (s - Real.pi / 2)))
+    (fun s ↦ rC (s - Real.pi / 2))
+    (fun u ↦ GerverSofa.PartC.Stage2.rhoC (u - Real.pi / 2)) (fun s ↦ ?_)
+    (hcC.comp (continuous_id.sub continuous_const)) ?_ ?_ ?_ S hS hSsub
+  · have hinner : HasDerivAt (fun x : ℝ ↦ x - Real.pi / 2) 1 s :=
+      (hasDerivAt_id s).sub_const _
+    have h1 : HasDerivAt (fun x : ℝ ↦ PC (x - Real.pi / 2))
+        (-(rC (s - Real.pi / 2)) • GerverSofa.u (s - Real.pi / 2)) s := by
+      simpa [Function.comp_def] using (hPC (s - Real.pi / 2)).scomp s hinner
+    have h2 := hasDerivAt_toPlane h1
+    rw [toPlane_smul] at h2
+    have hangle : tangentVector ((s : ℝ) : Real.Angle) =
+        -normalVector (((s - Real.pi / 2 : ℝ)) : Real.Angle) := by
+      have h := tangentVector_add_pi_div_two (s - Real.pi / 2)
+      rw [show s - Real.pi / 2 + Real.pi / 2 = s from by ring] at h
+      exact h
+    rw [hangle, smul_neg, ← neg_smul]
+    exact h2
+  · intro s hs
+    rw [hvertexC s ⟨le_trans ha0 hs.1, le_trans hs.2 hbpi⟩]
+    exact (hmem (s - Real.pi / 2) ⟨by linarith [hs.1], by linarith [hs.2]⟩).2
+  · intro s hs
+    exact (hrho (s - Real.pi / 2) ⟨by linarith [hs.1], by linarith [hs.2]⟩).2
+  · intro s hs
+    have hs0 : (0 : ℝ) ≤ s - Real.pi / 2 := by linarith [hs.1]
+    have hsT : s - Real.pi / 2 ≤ Real.pi / 2 := by linarith [hs.2, hbpi]
+    exact GerverSofa.PartC.Stage2.rhoC_nonneg ⟨hs0, hsT⟩
+
+/-! ### Exhausting the last stage of the first arc from inside -/
+
+/-- The positive vertex jumps at `π/2`, so the last `rhoA` stage is reached by exhausting its
+open interval by half-open subintervals. -/
+private theorem surfaceAreaMeasure_angleImage_eq_withDensity_rhoA_of_final_stage
+    (K : ConvexBody Point)
+    (hvertexA : ∀ t ∈ Set.Ico (0 : ℝ) (Real.pi / 2),
+      (edgeVertices K ((t : ℝ) : Real.Angle)).1 = paperGerverContacts t 0)
+    (S : Set ℝ) (hS : MeasurableSet S) (hSsub : S ⊆ Set.Ioo (gerverStageTimes 4) (Real.pi / 2)) :
+    surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+      volume.withDensity (fun s ↦ ENNReal.ofReal (GerverSofa.PartC.Stage2.rhoA s)) S := by
+  have h4T := gerverStageTimes_four_lt_pi_div_two
+  have hpos : 0 < Real.pi / 2 - gerverStageTimes 4 := by linarith
+  have hbnlt : ∀ n : ℕ, Real.pi / 2 -
+      (Real.pi / 2 - gerverStageTimes 4) / ((n : ℝ) + 2) < Real.pi / 2 := by
+    intro n
+    have h : 0 < (Real.pi / 2 - gerverStageTimes 4) / ((n : ℝ) + 2) := by positivity
+    linarith
+  refine measure_angleImage_eq_of_iUnion
+    (J := fun n : ℕ ↦ Set.Ioc (gerverStageTimes 4)
+      (Real.pi / 2 - (Real.pi / 2 - gerverStageTimes 4) / ((n : ℝ) + 2)))
+    ?_ (fun n ↦ measurableSet_Ioc) ?_ S hS ?_
+  · intro m n hmn
+    refine Set.Ioc_subset_Ioc_right ?_
+    have hmn' : ((m : ℝ) + 2) ≤ ((n : ℝ) + 2) := by
+      have : (m : ℝ) ≤ (n : ℝ) := Nat.cast_le.2 hmn
+      linarith
+    have hd : (Real.pi / 2 - gerverStageTimes 4) / ((n : ℝ) + 2) ≤
+        (Real.pi / 2 - gerverStageTimes 4) / ((m : ℝ) + 2) := by
+      gcongr
+    linarith
+  · intro n S' hS' hS'sub
+    exact surfaceAreaMeasure_angleImage_eq_withDensity_rhoA_of_stage K hvertexA 4 _ _
+      (le_refl _) (hbnlt n).le (hbnlt n) S' hS' hS'sub
+  · intro x hx
+    have hx' := hSsub hx
+    obtain ⟨n, hn⟩ := exists_nat_gt ((Real.pi / 2 - gerverStageTimes 4) / (Real.pi / 2 - x))
+    refine Set.mem_iUnion.2 ⟨n, ⟨hx'.1, ?_⟩⟩
+    have h1 : 0 < Real.pi / 2 - x := by linarith [hx'.2]
+    rw [div_lt_iff₀ h1] at hn
+    have h4 : (Real.pi / 2 - gerverStageTimes 4) / ((n : ℝ) + 2) < Real.pi / 2 - x := by
+      rw [div_lt_iff₀ (by positivity)]
+      nlinarith [h1]
+    linarith
+
+/-! ### Gluing two adjacent arcs -/
+
+/-- Two adjacent windows inside `[0, π/2]` on which the `rhoA` identity holds may be merged. -/
+private theorem surfaceAreaMeasure_angleImage_eq_withDensity_rhoA_of_union (K : ConvexBody Point)
+    (x y z : ℝ) (hx : 0 ≤ x) (hxy : x ≤ y) (hyz : y ≤ z) (hz : z ≤ Real.pi / 2)
+    (hI : ∀ S, MeasurableSet S → S ⊆ Set.Icc x y →
+      surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+        volume.withDensity (fun s ↦ ENNReal.ofReal (GerverSofa.PartC.Stage2.rhoA s)) S)
+    (hJ : ∀ S, MeasurableSet S → S ⊆ Set.Ioc y z →
+      surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+        volume.withDensity (fun s ↦ ENNReal.ofReal (GerverSofa.PartC.Stage2.rhoA s)) S)
+    (S : Set ℝ) (hS : MeasurableSet S) (hSsub : S ⊆ Set.Icc x z) :
+    surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+      volume.withDensity (fun s ↦ ENNReal.ofReal (GerverSofa.PartC.Stage2.rhoA s)) S := by
+  have hpi := Real.pi_gt_three
+  refine measure_angleImage_eq_of_union (c := -1) (d := Real.pi / 2) (by linarith)
+    (I := Set.Icc x y) (J := Set.Ioc y z)
+    (fun u hu ↦ ⟨by linarith [hu.1], by linarith [hu.2]⟩)
+    (fun u hu ↦ ⟨by linarith [hu.1], by linarith [hu.2]⟩)
+    measurableSet_Icc measurableSet_Ioc ?_ hI hJ S hS ?_
+  · rw [Set.disjoint_left]
+    intro u hu hu'
+    linarith [hu.2, hu'.1]
+  · rw [Set.Icc_union_Ioc_eq_Icc hxy hyz]
+    exact hSsub
+
+/-- Two adjacent windows inside `[π/2, π]` on which the `rhoC` identity holds may be merged. -/
+private theorem surfaceAreaMeasure_angleImage_eq_withDensity_rhoC_of_union (K : ConvexBody Point)
+    (x y z : ℝ) (hx : Real.pi / 2 ≤ x) (hxy : x ≤ y) (hyz : y ≤ z) (hz : z ≤ Real.pi)
+    (hI : ∀ S, MeasurableSet S → S ⊆ Set.Ioc x y →
+      surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+        volume.withDensity
+          (fun u ↦ ENNReal.ofReal (GerverSofa.PartC.Stage2.rhoC (u - Real.pi / 2))) S)
+    (hJ : ∀ S, MeasurableSet S → S ⊆ Set.Ioc y z →
+      surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+        volume.withDensity
+          (fun u ↦ ENNReal.ofReal (GerverSofa.PartC.Stage2.rhoC (u - Real.pi / 2))) S)
+    (S : Set ℝ) (hS : MeasurableSet S) (hSsub : S ⊆ Set.Ioc x z) :
+    surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+      volume.withDensity
+        (fun u ↦ ENNReal.ofReal (GerverSofa.PartC.Stage2.rhoC (u - Real.pi / 2))) S := by
+  have hpi := Real.pi_gt_three
+  refine measure_angleImage_eq_of_union (c := Real.pi / 2) (d := Real.pi) (by linarith)
+    (I := Set.Ioc x y) (J := Set.Ioc y z)
+    (fun u hu ↦ ⟨by linarith [hu.1], by linarith [hu.2]⟩)
+    (fun u hu ↦ ⟨by linarith [hu.1], by linarith [hu.2]⟩)
+    measurableSet_Ioc measurableSet_Ioc ?_ hI hJ S hS ?_
+  · rw [Set.disjoint_left]
+    intro u hu hu'
+    linarith [hu.2, hu'.1]
+  · rw [Set.Ioc_union_Ioc_eq_Ioc hxy hyz]
+    exact hSsub
+
+/-! ### The density identity on each full arc -/
+
+/-- The surface measure of the angular image of any measurable subset of `[0, π/2)` is the
+`rhoA`-weighted Lebesgue measure. -/
+private theorem surfaceAreaMeasure_angleImage_eq_withDensity_rhoA (K : ConvexBody Point)
+    (hatom0 : surfaceAreaMeasure K {((0 : ℝ) : Real.Angle)} = 0)
+    (hvertexA : ∀ t ∈ Set.Ico (0 : ℝ) (Real.pi / 2),
+      (edgeVertices K ((t : ℝ) : Real.Angle)).1 = paperGerverContacts t 0)
+    (S : Set ℝ) (hS : MeasurableSet S) (hSsub : S ⊆ Set.Ico 0 (Real.pi / 2)) :
+    surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+      volume.withDensity (fun s ↦ ENNReal.ofReal (GerverSofa.PartC.Stage2.rhoA s)) S := by
+  have hpi := Real.pi_gt_three
+  have hT0 : gerverStageTimes 0 = 0 := rfl
+  have hlt01 : gerverStageTimes 0 < gerverStageTimes 1 := gerverStageTimes_lt_succ 0
+  have hlt12 : gerverStageTimes 1 < gerverStageTimes 2 := gerverStageTimes_lt_succ 1
+  have hlt23 : gerverStageTimes 2 < gerverStageTimes 3 := gerverStageTimes_lt_succ 2
+  have hlt34 : gerverStageTimes 3 < gerverStageTimes 4 := gerverStageTimes_lt_succ 3
+  have h4T := gerverStageTimes_four_lt_pi_div_two
+  have h1T : gerverStageTimes 1 < Real.pi / 2 := by linarith
+  have h2T : gerverStageTimes 2 < Real.pi / 2 := by linarith
+  have h3T : gerverStageTimes 3 < Real.pi / 2 := by linarith
+  have h04 : gerverStageTimes 0 ≤ gerverStageTimes 4 := by linarith
+  -- the left endpoint carries no atom
+  have hzeroA : ∀ S, MeasurableSet S →
+      S ⊆ Set.Icc (gerverStageTimes 0) (gerverStageTimes 0) →
+      surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+        volume.withDensity (fun s ↦ ENNReal.ofReal (GerverSofa.PartC.Stage2.rhoA s)) S := by
+    intro S hS hSsub
+    have h1 : surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) = 0 := by
+      refine measure_mono_null ?_ hatom0
+      rintro u ⟨x, hx, rfl⟩
+      have hx' := hSsub hx
+      rw [Set.Icc_self, Set.mem_singleton_iff, hT0] at hx'
+      rw [hx']
+      exact rfl
+    have h2 : volume.withDensity
+        (fun s ↦ ENNReal.ofReal (GerverSofa.PartC.Stage2.rhoA s)) S = 0 := by
+      refine measure_mono_null hSsub ?_
+      rw [Set.Icc_self]
+      exact (withDensity_absolutelyContinuous volume _) (measure_singleton _)
+    rw [h1, h2]
+  -- glue the left endpoint and the first four stages
+  have hstep1 := surfaceAreaMeasure_angleImage_eq_withDensity_rhoA_of_union K
+    (gerverStageTimes 0) (gerverStageTimes 0) (gerverStageTimes 1)
+    hT0.ge le_rfl hlt01.le h1T.le hzeroA
+    (surfaceAreaMeasure_angleImage_eq_withDensity_rhoA_of_stage K hvertexA 0 _ _
+      (le_refl _) (le_refl _) h1T)
+  have hstep2 := surfaceAreaMeasure_angleImage_eq_withDensity_rhoA_of_union K
+    (gerverStageTimes 0) (gerverStageTimes 1) (gerverStageTimes 2)
+    hT0.ge hlt01.le hlt12.le h2T.le hstep1
+    (surfaceAreaMeasure_angleImage_eq_withDensity_rhoA_of_stage K hvertexA 1 _ _
+      (le_refl _) (le_refl _) h2T)
+  have hstep3 := surfaceAreaMeasure_angleImage_eq_withDensity_rhoA_of_union K
+    (gerverStageTimes 0) (gerverStageTimes 2) (gerverStageTimes 3)
+    hT0.ge (by linarith) hlt23.le h3T.le hstep2
+    (surfaceAreaMeasure_angleImage_eq_withDensity_rhoA_of_stage K hvertexA 2 _ _
+      (le_refl _) (le_refl _) h3T)
+  have hstep4 := surfaceAreaMeasure_angleImage_eq_withDensity_rhoA_of_union K
+    (gerverStageTimes 0) (gerverStageTimes 3) (gerverStageTimes 4)
+    hT0.ge (by linarith) hlt34.le h4T.le hstep3
+    (surfaceAreaMeasure_angleImage_eq_withDensity_rhoA_of_stage K hvertexA 3 _ _
+      (le_refl _) (le_refl _) h4T)
+  -- glue the exhausted last stage
+  refine measure_angleImage_eq_of_union (c := -1) (d := Real.pi / 2) (by linarith)
+    (I := Set.Icc (gerverStageTimes 0) (gerverStageTimes 4))
+    (J := Set.Ioo (gerverStageTimes 4) (Real.pi / 2))
+    (fun u hu ↦ ⟨by linarith [hu.1, hT0], by linarith [hu.2, h4T]⟩)
+    (fun u hu ↦ ⟨by linarith [hu.1, hT0, h04], by linarith [hu.2]⟩)
+    measurableSet_Icc measurableSet_Ioo ?_ hstep4
+    (surfaceAreaMeasure_angleImage_eq_withDensity_rhoA_of_final_stage K hvertexA) S hS ?_
+  · rw [Set.disjoint_left]
+    intro u hu hu'
+    linarith [hu.2, hu'.1]
+  · rw [Set.Icc_union_Ioo_eq_Ico h04 h4T, hT0]
+    exact hSsub
+
+/-- The surface measure of the angular image of any measurable subset of `(π/2, π]` is the
+translated `rhoC`-weighted Lebesgue measure. -/
+private theorem surfaceAreaMeasure_angleImage_eq_withDensity_rhoC (K : ConvexBody Point)
+    (hvertexC : ∀ s ∈ Set.Icc (Real.pi / 2) Real.pi,
+      (edgeVertices K ((s : ℝ) : Real.Angle)).1 = paperGerverContacts (s - Real.pi / 2) 2)
+    (S : Set ℝ) (hS : MeasurableSet S)
+    (hSsub : S ⊆ Set.Ioc (Real.pi / 2) (Real.pi / 2 + Real.pi / 2)) :
+    surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+      volume.withDensity
+        (fun u ↦ ENNReal.ofReal (GerverSofa.PartC.Stage2.rhoC (u - Real.pi / 2))) S := by
+  have hpi := Real.pi_gt_three
+  have hT0 : gerverStageTimes 0 = 0 := rfl
+  have hT5 : gerverStageTimes 5 = Real.pi / 2 := rfl
+  have hlt01 : gerverStageTimes 0 < gerverStageTimes 1 := gerverStageTimes_lt_succ 0
+  have hlt12 : gerverStageTimes 1 < gerverStageTimes 2 := gerverStageTimes_lt_succ 1
+  have hlt23 : gerverStageTimes 2 < gerverStageTimes 3 := gerverStageTimes_lt_succ 2
+  have hlt34 : gerverStageTimes 3 < gerverStageTimes 4 := gerverStageTimes_lt_succ 3
+  have hlt45 : gerverStageTimes 4 < gerverStageTimes 5 := gerverStageTimes_lt_succ 4
+  have hc0 := surfaceAreaMeasure_angleImage_eq_withDensity_rhoC_of_stage K hvertexC 0
+    (Real.pi / 2 + gerverStageTimes 0) (Real.pi / 2 + gerverStageTimes 1) (le_refl _) (le_refl _)
+  have hc1 := surfaceAreaMeasure_angleImage_eq_withDensity_rhoC_of_stage K hvertexC 1
+    (Real.pi / 2 + gerverStageTimes 1) (Real.pi / 2 + gerverStageTimes 2) (le_refl _) (le_refl _)
+  have hc2 := surfaceAreaMeasure_angleImage_eq_withDensity_rhoC_of_stage K hvertexC 2
+    (Real.pi / 2 + gerverStageTimes 2) (Real.pi / 2 + gerverStageTimes 3) (le_refl _) (le_refl _)
+  have hc3 := surfaceAreaMeasure_angleImage_eq_withDensity_rhoC_of_stage K hvertexC 3
+    (Real.pi / 2 + gerverStageTimes 3) (Real.pi / 2 + gerverStageTimes 4) (le_refl _) (le_refl _)
+  have hc4 := surfaceAreaMeasure_angleImage_eq_withDensity_rhoC_of_stage K hvertexC 4
+    (Real.pi / 2 + gerverStageTimes 4) (Real.pi / 2 + gerverStageTimes 5) (le_refl _) (le_refl _)
+  have hstep1 := surfaceAreaMeasure_angleImage_eq_withDensity_rhoC_of_union K
+    (Real.pi / 2 + gerverStageTimes 0)
+    (Real.pi / 2 + gerverStageTimes 1) (Real.pi / 2 + gerverStageTimes 2)
+    (by linarith [gerverStageTimes_nonneg 0]) (by linarith) (by linarith)
+    (by linarith [gerverStageTimes_le_pi_div_two 2]) hc0 hc1
+  have hstep2 := surfaceAreaMeasure_angleImage_eq_withDensity_rhoC_of_union K
+    (Real.pi / 2 + gerverStageTimes 0)
+    (Real.pi / 2 + gerverStageTimes 2) (Real.pi / 2 + gerverStageTimes 3)
+    (by linarith [gerverStageTimes_nonneg 0]) (by linarith) (by linarith)
+    (by linarith [gerverStageTimes_le_pi_div_two 3]) hstep1 hc2
+  have hstep3 := surfaceAreaMeasure_angleImage_eq_withDensity_rhoC_of_union K
+    (Real.pi / 2 + gerverStageTimes 0)
+    (Real.pi / 2 + gerverStageTimes 3) (Real.pi / 2 + gerverStageTimes 4)
+    (by linarith [gerverStageTimes_nonneg 0]) (by linarith) (by linarith)
+    (by linarith [gerverStageTimes_le_pi_div_two 4]) hstep2 hc3
+  have hstep4 := surfaceAreaMeasure_angleImage_eq_withDensity_rhoC_of_union K
+    (Real.pi / 2 + gerverStageTimes 0)
+    (Real.pi / 2 + gerverStageTimes 4) (Real.pi / 2 + gerverStageTimes 5)
+    (by linarith [gerverStageTimes_nonneg 0]) (by linarith) (by linarith)
+    (by linarith [gerverStageTimes_le_pi_div_two 5]) hstep3 hc4
+  refine hstep4 S hS ?_
+  rw [hT0, hT5, add_zero]
+  exact hSsub
+
+theorem gerver_surface_densities :
+    ∃ K : RightAngleCapSpace,
+      (K.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2) ∧
+      ∃ r s : ℝ → ℝ≥0, HasCapDensities K r s ∧
+        (∃ M : ℝ, ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+          (r t : ℝ) ≤ M ∧ (s t : ℝ) ≤ M) ∧
+        (∀ i : Fin 5, ∀ t ∈ Set.Ioo (gerverStageTimes i.castSucc) (gerverStageTimes i.succ),
+          HasDerivAt (fun u ↦ paperGerverContacts u 0)
+            ((r t : ℝ) • tangentVector (t : Real.Angle)) t ∧
+          HasDerivAt (fun u ↦ paperGerverContacts u 2)
+            (-(s t : ℝ) • normalVector (t : Real.Angle)) t) := by
+  obtain ⟨K, hKcap, hatom0, hvertexA, hvertexC⟩ := exists_gerverCap_positiveVertex
+  obtain ⟨M, hM⟩ := exists_bound_rhoA_rhoC
+  refine ⟨K, hKcap, fun t ↦ Real.toNNReal (GerverSofa.PartC.Stage2.rhoA t),
+    fun t ↦ Real.toNNReal (GerverSofa.PartC.Stage2.rhoC t),
+    ⟨measurable_real_toNNReal.comp measurable_rhoA,
+      measurable_real_toNNReal.comp measurable_rhoC, ?_, ?_⟩,
+    ⟨max M 0, fun t ht ↦ ⟨?_, ?_⟩⟩, ?_⟩
+  · exact surfaceAreaMeasure_restrict_eq_map_withDensity measurableSet_Ico
+      (surfaceAreaMeasure_angleImage_eq_withDensity_rhoA K.val hatom0 hvertexA)
+  · rw [show Set.Ioc (Real.pi / 2) Real.pi =
+      Set.Ioc (Real.pi / 2) (Real.pi / 2 + Real.pi / 2) from by
+        rw [show Real.pi / 2 + Real.pi / 2 = Real.pi from by ring]]
+    exact surfaceAreaMeasure_restrict_eq_map_add_withDensity
+      (surfaceAreaMeasure_angleImage_eq_withDensity_rhoC K.val hvertexC)
+  · rw [Real.coe_toNNReal']
+    exact max_le_max (hM t ht).1 le_rfl
+  · rw [Real.coe_toNNReal']
+    exact max_le_max (hM t ht).2 le_rfl
+  · intro i t ht
+    have htIcc : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+      ⟨le_trans (gerverStageTimes_nonneg _) ht.1.le,
+        le_trans ht.2.le (gerverStageTimes_le_pi_div_two _)⟩
+    obtain ⟨hA, hC⟩ := hasDerivAt_paperGerverContacts_of_mem_stage i ht
+    rw [Real.coe_toNNReal _ (GerverSofa.PartC.Stage2.rhoA_nonneg htIcc),
+      Real.coe_toNNReal _ (GerverSofa.PartC.Stage2.rhoC_nonneg htIcc)]
+    exact ⟨hA, hC⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver / Velocity And Cap Area
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem gerver_strict_velocity (t : ℝ) (ht : t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2)) :
+    (paperGerverVelocityComponents t).1 < 0 ∧ 0 < (paperGerverVelocityComponents t).2 ∧
+    inner ℝ (deriv paperGerverPath t) (normalVector (t : Real.Angle)) < 0 ∧
+    0 < inner ℝ (deriv paperGerverPath t) (tangentVector (t : Real.Angle)) := by
+  open GerverSofa.Romik in
+  -- The certified direct parameter vector, its box enclosures and its equations.
+  obtain ⟨p, hp⟩ : ∃ p : Params, GerverSofa.PartB.params = p := ⟨_, rfl⟩
+  have hmem : p ∈ gerverDirectBox := hp ▸ GerverSofa.PartB.params_mem
+  have heqs : gerverDirectEquations p := hp ▸ GerverSofa.PartB.params_equations
+  obtain ⟨-, hphieq, hthetaeq, hphipos, -, -, -⟩ :=
+    gerver_parameter_identification.1 p hmem heqs
+  have hphi' : p.phi = GerversSofa.φ := hphieq.trans selected_phi
+  have htheta' : p.theta = GerversSofa.θ := hthetaeq.trans selected_theta
+  have hphi0 : (0 : ℝ) < p.phi := by rw [hphieq]; exact hphipos
+  have ha2 : p.a2 = -(1 / 4 : ℝ) := a2_eq_neg_quarter_of_equations heqs
+  have he1 : p.e1 = p.a1 := e1_eq_a1_of_equations heqs
+  have he2 : p.e2 = -p.a2 := e2_eq_neg_a2_of_equations heqs
+  have hbox : p ∈ GerverSofa.Romik.box := hmem
+  dsimp only [GerverSofa.Romik.box, GerverSofa.qR, Set.mem_ofPred_eq] at hbox
+  obtain ⟨-, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -,
+      ha1lo, -, -, -, hb1lo, hb1hi, hb2lo, -, hc1lo, -, hc2lo, -, hd1lo, -, hd2lo, -,
+      -, -, -, -, -, hphihi, -, hthetahi⟩ := hbox
+  have ha1 : (6 : ℝ) / 5 ≤ p.a1 := le_trans (by norm_num) ha1lo
+  have hb1u : p.b1 ≤ -(1 / 2 : ℝ) := le_trans hb1hi (by norm_num)
+  have hb1l : -(53 / 100 : ℝ) ≤ p.b1 := le_trans (by norm_num) hb1lo
+  have hb2 : (9 : ℝ) / 10 ≤ p.b2 := le_trans (by norm_num) hb2lo
+  have hc1 : (3 : ℝ) / 5 ≤ p.c1 := le_trans (by norm_num) hc1lo
+  have hc2 : (-1 : ℝ) ≤ p.c2 := le_trans (by norm_num) hc2lo
+  have hd1 : (13 : ℝ) / 10 ≤ p.d1 := le_trans (by norm_num) hd1lo
+  have hd2 : -(53 / 100 : ℝ) ≤ p.d2 := le_trans (by norm_num) hd2lo
+  have hphiu : p.phi ≤ (1 : ℝ) / 20 := le_trans hphihi (by norm_num)
+  have hthetau : p.theta ≤ (7 : ℝ) / 10 := le_trans hthetahi (by norm_num)
+  have hpilo : (3 : ℝ) < Real.pi := Real.pi_gt_three
+  have hpihi : Real.pi < (63 : ℝ) / 20 := by linarith only [Real.pi_lt_d2]
+  -- The two strict first-phase estimates, also used on the reflected fifth phase.
+  have hkeyA : ∀ s : ℝ, 0 ≤ s → s ≤ (1 : ℝ) / 20 →
+      (alphaBeta1 p s).1 ≤ -(2 * s) := by
+    intro s hs0 hs20
+    have hspi : s ≤ Real.pi := by linarith only [hs20, hpilo]
+    have hsin0 : 0 ≤ Real.sin s := Real.sin_nonneg_of_nonneg_of_le_pi hs0 hspi
+    have hsinlo : s - s ^ 3 / 6 ≤ Real.sin s := Real.sin_ge_sub_cube hs0
+    have hcoslo : 1 - s ^ 2 / 2 ≤ Real.cos s := Real.one_sub_sq_div_two_le_cos
+    have hmul : (12 / 5 : ℝ) * Real.sin s ≤ 2 * p.a1 * Real.sin s := by
+      nlinarith only [ha1, hsin0]
+    have hquad : 0 ≤ s * ((1 / 20 : ℝ) - s) := mul_nonneg hs0 (by linarith only [hs20])
+    have hcube : 0 ≤ s ^ 2 * ((1 / 20 : ℝ) - s) :=
+      mul_nonneg (sq_nonneg s) (by linarith only [hs20])
+    dsimp [alphaBeta1]
+    rw [ha2]
+    linarith only [hmul, hsinlo, hcoslo, hquad, hcube, hs0]
+  have hkeyB : ∀ s : ℝ, 0 ≤ s → s ≤ (1 : ℝ) / 20 →
+      (343 : ℝ) / 250 ≤ (alphaBeta1 p s).2 := by
+    intro s hs0 hs20
+    have hspi : s ≤ Real.pi := by linarith only [hs20, hpilo]
+    have hsinhi : Real.sin s ≤ s := Real.sin_le hs0
+    have hcoslo : 1 - s ^ 2 / 2 ≤ Real.cos s := Real.one_sub_sq_div_two_le_cos
+    have hcos0 : 0 ≤ Real.cos s :=
+      Real.cos_nonneg_of_mem_Icc ⟨by linarith only [hs0, hpilo], by linarith only [hs20, hpilo]⟩
+    have hmul : (12 / 5 : ℝ) * Real.cos s ≤ 2 * p.a1 * Real.cos s := by
+      nlinarith only [ha1, hcos0]
+    have hquad : 0 ≤ s * ((1 / 20 : ℝ) - s) := mul_nonneg hs0 (by linarith only [hs20])
+    dsimp [alphaBeta1]
+    rw [ha2]
+    linarith only [hmul, hsinhi, hcoslo, hquad, hs0, hs20]
+  -- The fifth phase is the reflection of the first one through the angle `π/4`.
+  have hrefl : ∀ s : ℝ, alphaBeta5 p (Real.pi / 2 - s) =
+      (-(alphaBeta1 p s).2, -(alphaBeta1 p s).1) := by
+    intro s
+    dsimp [alphaBeta5, alphaBeta1]
+    rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub, he1, he2]
+    simp only [Prod.mk.injEq]
+    constructor <;> ring
+  -- The five closed stage intervals in terms of the direct switching angles.
+  have hI0 : gerverStageIntervals 0 = Set.Icc 0 p.phi := by
+    simp [gerverStageIntervals, gerverStageTimes, hphi']
+  have hI1 : gerverStageIntervals 1 = Set.Icc p.phi p.theta := by
+    simp [gerverStageIntervals, gerverStageTimes, hphi', htheta']
+  have hI2 : gerverStageIntervals 2 = Set.Icc p.theta (Real.pi / 2 - p.theta) := by
+    simp [gerverStageIntervals, gerverStageTimes, htheta']
+  have hI3 : gerverStageIntervals 3 =
+      Set.Icc (Real.pi / 2 - p.theta) (Real.pi / 2 - p.phi) := by
+    simp [gerverStageIntervals, gerverStageTimes, hphi', htheta']
+  have hI4 : gerverStageIntervals 4 = Set.Icc (Real.pi / 2 - p.phi) (Real.pi / 2) := by
+    simp [gerverStageIntervals, gerverStageTimes, hphi']
+  have hval := paperGerverContactData_properties.2.2.2
+  have hmain : (paperGerverVelocityComponents t).1 < 0 ∧
+      0 < (paperGerverVelocityComponents t).2 := by
+    rcases le_or_gt t p.phi with hs1 | hs1
+    · -- Phase 1: `α ≤ -2t < 0` and `β ≥ 343/250 > 0`.
+      have hv : paperGerverVelocityComponents t = alphaBeta1 p t := by
+        rw [hval 0 t (by rw [hI0]; exact ⟨ht.1.le, hs1⟩)]
+        simp [gerverBranchVelocityComponents, hp]
+      have htu : t ≤ (1 : ℝ) / 20 := le_trans hs1 hphiu
+      rw [hv]
+      exact ⟨by linarith only [hkeyA t ht.1.le htu, ht.1],
+        by linarith only [hkeyB t ht.1.le htu]⟩
+    rcases le_or_gt t p.theta with hs2 | hs2
+    · -- Phase 2: `α ≤ -t < 0` and `β ≥ 1813/2000 > 0`.
+      have hv : paperGerverVelocityComponents t = alphaBeta2 p t := by
+        rw [hval 1 t (by rw [hI1]; exact ⟨hs1.le, hs2⟩)]
+        simp [gerverBranchVelocityComponents, hp]
+      have htu : t ≤ (7 : ℝ) / 10 := le_trans hs2 hthetau
+      have hbt : 0 ≤ (p.b1 + 53 / 100) * t := mul_nonneg (by linarith only [hb1l]) ht.1.le
+      have hst : 0 ≤ ((7 : ℝ) / 10 - t) * t := mul_nonneg (by linarith only [htu]) ht.1.le
+      rw [hv]
+      dsimp [alphaBeta2]
+      exact ⟨by linarith only [hb1u, ht.1], by linarith only [hb2, hbt, hst, htu]⟩
+    rcases le_or_gt t (Real.pi / 2 - p.theta) with hs3 | hs3
+    · -- Phase 3: `α ≤ -t < 0` and `β ≥ 8/5 - t > 0`.
+      have hv : paperGerverVelocityComponents t = alphaBeta3 p t := by
+        rw [hval 2 t (by rw [hI2]; exact ⟨hs2.le, hs3⟩)]
+        simp [gerverBranchVelocityComponents, hp]
+      rw [hv]
+      dsimp [alphaBeta3]
+      exact ⟨by linarith only [hc2, ht.1], by linarith only [hc1, ht.2, hpihi]⟩
+    rcases le_or_gt t (Real.pi / 2 - p.phi) with hs4 | hs4
+    · -- Phase 4: `-α > 69/100 > 0` and `β ≥ 8/5 - t > 0`.
+      have hv : paperGerverVelocityComponents t = alphaBeta4 p t := by
+        rw [hval 3 t (by rw [hI3]; exact ⟨hs3.le, hs4⟩)]
+        simp [gerverBranchVelocityComponents, hp]
+      have htlo : (4 : ℝ) / 5 < t := by linarith only [hs3, hthetau, hpilo]
+      have hthi : t < (8 : ℝ) / 5 := by linarith only [ht.2, hpihi]
+      have hcoef : (9 : ℝ) / 10 ≤ p.d1 - t / 4 := by linarith only [hd1, hthi]
+      have hprod : (18 : ℝ) / 25 < t * (p.d1 - t / 4) := by nlinarith only [htlo, hcoef]
+      rw [hv]
+      dsimp [alphaBeta4]
+      exact ⟨by linarith only [hprod, hd2], by linarith only [hd1, hthi]⟩
+    · -- Phase 5: the reflected first-phase estimates at `s = π/2 - t ∈ (0, φ]`.
+      have hv : paperGerverVelocityComponents t = alphaBeta5 p t := by
+        rw [hval 4 t (by rw [hI4]; exact ⟨hs4.le, ht.2.le⟩)]
+        simp [gerverBranchVelocityComponents, hp]
+      have hs0 : 0 < Real.pi / 2 - t := by linarith only [ht.2]
+      have hsu : Real.pi / 2 - t ≤ (1 : ℝ) / 20 := by linarith only [hs4, hphiu]
+      have hA := hkeyA (Real.pi / 2 - t) hs0.le hsu
+      have hB := hkeyB (Real.pi / 2 - t) hs0.le hsu
+      have h5 := hrefl (Real.pi / 2 - t)
+      rw [show Real.pi / 2 - (Real.pi / 2 - t) = t by ring] at h5
+      have hfst : (alphaBeta5 p t).1 = -(alphaBeta1 p (Real.pi / 2 - t)).2 := by rw [h5]
+      have hsnd : (alphaBeta5 p t).2 = -(alphaBeta1 p (Real.pi / 2 - t)).1 := by rw [h5]
+      rw [hv, hfst, hsnd]
+      exact ⟨by linarith only [hB], by linarith only [hA, hs0]⟩
+  exact ⟨hmain.1, hmain.2, hmain.1, hmain.2⟩
+
+/-- The tangential velocity component is nonpositive on the whole closed rotation interval:
+the strict inequality of `gerver_strict_velocity` holds on the open interval, and the
+component is continuous, so the closed condition propagates to the two endpoints. -/
+theorem paperGerverVelocityComponents_fst_nonpos {t : ℝ}
+    (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)) : (paperGerverVelocityComponents t).1 ≤ 0 := by
+  have hpi : (0 : ℝ) < Real.pi / 2 := by positivity
+  have hcl : IsClosed {s : ℝ | (paperGerverVelocityComponents s).1 ≤ 0} :=
+    isClosed_le (continuous_fst.comp paperGerverContactData_properties.1) continuous_const
+  have h := closure_minimal
+    (fun s hs => (gerver_strict_velocity s hs).1.le : Set.Ioo (0 : ℝ) (Real.pi / 2) ⊆ _) hcl
+  rw [closure_Ioo hpi.ne] at h
+  exact h ht
+
+/-- The normal velocity component is nonnegative on the whole closed rotation interval; see
+`paperGerverVelocityComponents_fst_nonpos` for the argument. -/
+theorem paperGerverVelocityComponents_snd_nonneg {t : ℝ}
+    (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)) : 0 ≤ (paperGerverVelocityComponents t).2 := by
+  have hpi : (0 : ℝ) < Real.pi / 2 := by positivity
+  have hcl : IsClosed {s : ℝ | 0 ≤ (paperGerverVelocityComponents s).2} :=
+    isClosed_le continuous_const (continuous_snd.comp paperGerverContactData_properties.1)
+  have h := closure_minimal
+    (fun s hs => (gerver_strict_velocity s hs).2.1.le : Set.Ioo (0 : ℝ) (Real.pi / 2) ⊆ _) hcl
+  rw [closure_Ioo hpi.ne] at h
+  exact h ht
+
+theorem gerver_cap_area_lower_bound :
+    2 * GerverSofa.PartB.params.a1 ≤ ClassicalResults.area gerverOuterCap ∧
+    (12 : ℝ) / 5 ≤ 2 * GerverSofa.PartB.params.a1 ∧ (11 : ℝ) / 5 < 12 / 5 := by
+  open GerverSofa.Romik in
+  -- The certified direct parameter vector, its box enclosures and its equations.
+  obtain ⟨p, hp⟩ : ∃ p : Params, GerverSofa.PartB.params = p := ⟨_, rfl⟩
+  rw [hp]
+  have hmem : p ∈ gerverDirectBox := hp ▸ GerverSofa.PartB.params_mem
+  have heqs : gerverDirectEquations p := hp ▸ GerverSofa.PartB.params_equations
+  obtain ⟨-, hphieq, hthetaeq, hphipos, hphitheta, hthetalt, -⟩ :=
+    gerver_parameter_identification.1 p hmem heqs
+  have hphi' : p.phi = GerversSofa.φ := hphieq.trans selected_phi
+  have h0 : (0 : ℝ) < p.phi := by rw [hphieq]; exact hphipos
+  have h1 : p.phi < p.theta := by rw [hphieq, hthetaeq]; exact hphitheta
+  have h2 : p.theta < Real.pi / 4 := by rw [hthetaeq]; exact hthetalt
+  have hpi : (3 : ℝ) < Real.pi := Real.pi_gt_three
+  have hbox : p ∈ GerverSofa.Romik.box := hmem
+  dsimp only [GerverSofa.Romik.box, GerverSofa.qR, Set.mem_ofPred_eq] at hbox
+  obtain ⟨-, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, hk51lo, hk51hi, -, -,
+      ha1lo, ha1hi, -, -, -, -, -, -, -, -, -, -, -, -, -, -,
+      -, -, -, -, -, -, -, -⟩ := hbox
+  have ha1 : (6 : ℝ) / 5 ≤ p.a1 := le_trans (by norm_num) ha1lo
+  have ha1' : p.a1 ≤ (61 : ℝ) / 50 := le_trans ha1hi (by norm_num)
+  have hk51 : -(51 : ℝ) / 50 ≤ p.k51 := le_trans (by norm_num) hk51lo
+  have hk51' : p.k51 ≤ -1 := le_trans hk51hi (by norm_num)
+  have ha2 : p.a2 = -(1 / 4 : ℝ) := a2_eq_neg_quarter_of_equations heqs
+  have he1 : p.e1 = p.a1 := e1_eq_a1_of_equations heqs
+  have he2 : p.e2 = -p.a2 := e2_eq_neg_a2_of_equations heqs
+  have hk52 : p.k52 = (1 / 4 : ℝ) := k52_eq_quarter_of_equations heqs
+  -- The two path endpoints: `x(0) = 0` and `x(π/2) = (1 - a₁ + k₅₁, 0)`.
+  obtain ⟨-, hzero, -⟩ := gerver_direct_path_regularity p hmem heqs
+  have hpath0 : paperGerverPath 0 = 0 := by
+    change GerverSofa.PartF.Coordinates.toPlane (path GerverSofa.PartB.params 0) = 0
+    rw [hp, hzero]
+    ext i
+    fin_cases i <;> rfl
+  have hpathT : path p (Real.pi / 2) = (1 - p.a1 + p.k51, 0) := by
+    rw [path_eq_path5_of_mem_Icc heqs h1 h2 ⟨by linarith, le_rfl⟩]
+    simp only [path5, addK, rot, Real.cos_pi_div_two, Real.sin_pi_div_two, he1, he2, ha2, hk52,
+      Prod.mk.injEq]
+    constructor <;> ring
+  have hxT0 : paperGerverPath (Real.pi / 2) 0 = 1 - p.a1 + p.k51 := by
+    change (path GerverSofa.PartB.params (Real.pi / 2)).1 = _
+    rw [hp, hpathT]
+  have hxT1 : paperGerverPath (Real.pi / 2) 1 = 0 := by
+    change (path GerverSofa.PartB.params (Real.pi / 2)).2 = _
+    rw [hp, hpathT]
+  -- The velocity components at the two endpoints.
+  have hvel := paperGerverContactData_properties.2.2.2
+  have hI0 : gerverStageIntervals 0 = Set.Icc 0 p.phi := by
+    simp [gerverStageIntervals, gerverStageTimes, hphi']
+  have hI4 : gerverStageIntervals 4 = Set.Icc (Real.pi / 2 - p.phi) (Real.pi / 2) := by
+    simp [gerverStageIntervals, gerverStageTimes, hphi']
+  have halpha0 : (paperGerverVelocityComponents 0).1 = 0 := by
+    rw [hvel 0 0 (by rw [hI0]; exact ⟨le_rfl, h0.le⟩)]
+    simp [gerverBranchVelocityComponents, hp, alphaBeta1, ha2]
+    norm_num
+  have hbeta0 : (paperGerverVelocityComponents 0).2 = 2 * p.a1 - 1 := by
+    rw [hvel 0 0 (by rw [hI0]; exact ⟨le_rfl, h0.le⟩)]
+    simp [gerverBranchVelocityComponents, hp, alphaBeta1, ha2]
+  have halphaT : (paperGerverVelocityComponents (Real.pi / 2)).1 = 1 - 2 * p.a1 := by
+    rw [hvel 4 (Real.pi / 2) (by rw [hI4]; exact ⟨by linarith, le_rfl⟩)]
+    simp [gerverBranchVelocityComponents, hp, alphaBeta5, he1]
+  have hbetaT : (paperGerverVelocityComponents (Real.pi / 2)).2 = 0 := by
+    rw [hvel 4 (Real.pi / 2) (by rw [hI4]; exact ⟨by linarith, le_rfl⟩)]
+    simp [gerverBranchVelocityComponents, hp, alphaBeta5, he2, ha2]
+    norm_num
+  -- The four displayed cap contacts, in coordinates.
+  have hpteq : ∀ z w : Point, z 0 = w 0 → z 1 = w 1 → z = w := by
+    intro z w hz hw
+    ext i
+    fin_cases i
+    · exact hz
+    · exact hw
+  have hA0 : paperGerverContacts 0 0 = (!₂[1, 0] : Point) := by
+    refine hpteq _ _ ?_ ?_ <;>
+      simp [paperGerverContacts, normalVector, tangentVector, frame, halpha0, hpath0]
+  have hC0 : paperGerverContacts 0 2 = (!₂[1 - 2 * p.a1, 1] : Point) := by
+    refine hpteq _ _ ?_ ?_ <;>
+      simp [paperGerverContacts, normalVector, tangentVector, frame, hbeta0, hpath0]
+  have hAT : paperGerverContacts (Real.pi / 2) 0 = (!₂[p.a1 + p.k51, 1] : Point) := by
+    refine hpteq _ _ ?_ ?_ <;>
+      simp [paperGerverContacts, normalVector, tangentVector, frame, halphaT, hxT0, hxT1]
+    ring
+  have hCT : paperGerverContacts (Real.pi / 2) 2 = (!₂[p.k51 - p.a1, 0] : Point) := by
+    refine hpteq _ _ ?_ ?_ <;>
+      simp [paperGerverContacts, normalVector, tangentVector, frame, hbetaT, hxT0, hxT1]
+    ring
+  -- The cap is a convex body, so it is convex and of finite area.
+  obtain ⟨-, -, -, -, -, ⟨K, hKset, -⟩, -⟩ := gerver_capSupport_identification
+  have hconv : Convex ℝ gerverOuterCap := hKset ▸ K.val.convex'
+  have hcomp : IsCompact gerverOuterCap := hKset ▸ K.val.isCompact'
+  -- The inscribed trapezoid and its area.
+  have hsub := EuclideanSpace.horizontalTrapezoid_subset_of_convex hconv
+    (l₀ := p.k51 - p.a1) (r₀ := 1) (l₁ := 1 - 2 * p.a1) (r₁ := p.a1 + p.k51)
+    (hCT ▸ gerver_outer_contact_C (Real.pi / 2) ⟨by positivity, le_rfl⟩)
+    (hA0 ▸ gerver_outer_contact_A 0 ⟨le_rfl, by positivity⟩)
+    (hC0 ▸ gerver_outer_contact_C 0 ⟨le_rfl, by positivity⟩)
+    (hAT ▸ gerver_outer_contact_A (Real.pi / 2) ⟨by positivity, le_rfl⟩)
+  have hvol := EuclideanSpace.volume_horizontalTrapezoid (l₀ := p.k51 - p.a1) (r₀ := 1)
+    (l₁ := 1 - 2 * p.a1) (r₁ := p.a1 + p.k51) (by linarith) (by linarith)
+  have hkey : ENNReal.ofReal (2 * p.a1) ≤ MeasureTheory.volume gerverOuterCap := by
+    rw [show (2 * p.a1 : ℝ) =
+      (1 - (p.k51 - p.a1) + (p.a1 + p.k51 - (1 - 2 * p.a1))) / 2 by ring, ← hvol]
+    exact MeasureTheory.measure_mono hsub
+  refine ⟨?_, by linarith, by norm_num⟩
+  rw [ClassicalResults.area]
+  exact (ENNReal.ofReal_le_iff_le_toReal hcomp.measure_lt_top.ne).1 hkey
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Polygon.BalancedContainment`.
+* `Polygon.Polyline.Length.Basic`.
+* `Polygon.Polyline.Length.Boundary`.
+* `Polygon.Polyline.Length`.
+* `Polygon.Balancing.Coefficients`.
+* `Polygon.Balancing.Estimate`.
+* `Polygon.Balancing`.
+* `Polygon.EdgeNormals`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Balanced Containment
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Balance bounds the polyline projections by the surface-area atoms. -/
+theorem polygonCapPolyline_inner_le_of_balanced {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hK : IsBalancedPolygonCap K)
+    (D : Finset ℝ) (hD : (D : Set ℝ) = angleDomain Θ) (s : ℝ)
+    {q : Point} (hq : q ∈ (polygonCapPolyline K).carrier) :
+    inner ℝ q (normalVector (s : Real.Angle)) ≤
+      inner ℝ ((capVertices K.val 0).1.2) (normalVector (s : Real.Angle)) +
+      ∑ u ∈ D, (surfaceAreaMeasure K.val.val {(u : Real.Angle)}).toReal *
+        max (Real.sin (s - u)) 0 := by
+  classical
+  have hp : IsCapPolyline K (polygonCapPolyline K) := (polygonCap_polyline K).choose_spec
+  have hmem (u : ℝ) (hu : u ∈ D) : u ∈ angleDomain Θ := by
+    rw [← hD]
+    exact hu
+  have hb := (polygonCapPolyline K).inner_le_endpoint_add_sum_normal_lengths D
+    (fun u hu ↦ angleDomain_subset_Ioo Θ (hmem u hu))
+    (fun i ↦ by
+      obtain ⟨t, ht⟩ := hp.2.2.2.1 i
+      refine ⟨t.val, ?_, ht⟩
+      have := t.property
+      simpa only [← hD, Finset.mem_coe] using this) s hq
+  rw [hp.2.2.1] at hb
+  convert hb using 2
+  apply Finset.sum_congr rfl
+  intro u hu
+  have heq := hK ⟨u, hmem u hu⟩
+  rw [heq, ENNReal.toReal_ofReal]
+  · rfl
+  · unfold polygonCapPolylineLength
+    positivity
+
+/-- A balanced polygon cap bounds every upper-normal projection of its polyline. -/
+theorem polygonCapPolyline_inner_le_support_of_balanced {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hK : IsBalancedPolygonCap K)
+    (D : Finset ℝ) (hD : (D : Set ℝ) = angleDomain Θ)
+    {s : ℝ} (hs : s ∈ Set.Ioo 0 Real.pi)
+    {q : Point} (hq : q ∈ (polygonCapPolyline K).carrier) :
+    inner ℝ q (normalVector (s : Real.Angle)) ≤ supportValue K.val.val (s : Real.Angle) := by
+  have hpoly := polygonCapPolyline_inner_le_of_balanced K hK D hD s hq
+  have hsum := sum_surfaceAreaMeasure_mul_pos_sin_le K.val.val D
+    (fun t ht ↦ angleDomain_subset_Ioo Θ (by rw [← hD]; exact ht)) hs
+  have hstart := inner_negativeVertex_zero_le_positiveVertex K.val.val ⟨hs.1.le, hs.2.le⟩
+  change inner ℝ q (normalVector (s : Real.Angle)) ≤
+    inner ℝ (edgeVertices K.val.val 0).2 (normalVector (s : Real.Angle)) + _ at hpoly
+  rw [real_inner_comm (normalVector (s : Real.Angle)) (edgeVertices K.val.val 0).2] at hpoly
+  linarith
+
+/-- The polyline of a balanced polygon cap lies inside the cap. -/
+theorem polygonCapPolyline_subset_of_balanced {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hK : IsBalancedPolygonCap K) :
+    (polygonCapPolyline K).carrier ⊆ (K.val.val : Set Point) := by
+  classical
+  let D := Θ.directions ∪ Θ.directions.image (fun t ↦ t + Real.pi / 2) ∪
+    {Θ.angle, Real.pi / 2}
+  have hD : (D : Set ℝ) = angleDomain Θ := by
+    simp [D, angleDomain]
+  intro q hq
+  have hp : IsCapPolyline K (polygonCapPolyline K) := (polygonCap_polyline K).choose_spec
+  have hfront : q ∈ frontier (capFan Θ.angle \ polygonNiche Θ K.val) := by
+    rw [hp.2.2.2.2.2.1]
+    exact Or.inl (Or.inr hq)
+  have hfan : q ∈ capFan Θ.angle := by
+    have hmem := frontier_subset_closure hfront
+    rw [hp.2.2.2.2.1.closure_eq] at hmem
+    exact hmem.1
+  rw [K.property.eq_iInter_supportValue]
+  simp only [Set.mem_iInter]
+  intro a ha
+  rcases ha with ⟨t, ht, rfl⟩ | ha
+  · exact polygonCapPolyline_inner_le_support_of_balanced K hK D hD
+      (angleDomain_subset_Ioo Θ ht) hq
+  rcases ha with rfl | rfl
+  · change inner ℝ q (normalVector ((Θ.angle + Real.pi : ℝ) : Real.Angle)) ≤
+      supportValue K.val.val ((Θ.angle + Real.pi : ℝ) : Real.Angle)
+    rw [K.val.property.2.2.2.2.1, normalVector_add_pi, inner_neg_right]
+    exact neg_nonpos.mpr hfan.1
+  · have hang : ((3 * Real.pi / 2 : ℝ) : Real.Angle) =
+        (((Real.pi / 2 + Real.pi : ℝ) : Real.Angle)) := by
+      congr 1
+      ring
+    change inner ℝ q (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) ≤
+      supportValue K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+    rw [K.val.property.2.2.2.2.2.1, hang, normalVector_add_pi, inner_neg_right]
+    exact neg_nonpos.mpr hfan.2
+
+/-- A polygon niche lies in its cap whenever its upper boundary polyline does. -/
+theorem polygonNiche_subset_of_polyline_subset {Θ : AngleSet}
+    (K : PolygonCapSpace Θ)
+    (hpoly : (polygonCapPolyline K).carrier ⊆ (K.val.val : Set Point)) :
+    polygonNiche Θ K.val ⊆ (K.val.val : Set Point) := by
+  intro q hq
+  have hfan : q ∈ capFan Θ.angle := hq.1
+  have hheight : q 1 < capBoundaryHeight K (q 0) := by
+    by_contra hn
+    have hmem : q ∈ capFan Θ.angle \ polygonNiche Θ K.val := by
+      rw [capFan_sdiff_polygonNiche_eq_epigraph K]
+      exact le_of_not_gt hn
+    exact hmem.2 hq
+  let r := pointOnGraph (capBoundaryHeight K) (q 0)
+  have hrx : r 0 = q 0 := by simp [r, pointOnGraph]
+  have hry : q 1 < r 1 := by simpa [r, pointOnGraph] using hheight
+  have hrfront : r ∈ frontier (capFan Θ.angle \ polygonNiche Θ K.val) := by
+    rw [(capFan_sdiff_polygonNiche_closed_frontier K).2]
+    exact ⟨q 0, rfl⟩
+  have hp : IsCapPolyline K (polygonCapPolyline K) := (polygonCap_polyline K).choose_spec
+  rw [hp.2.2.2.2.2.1] at hrfront
+  have hrpoly : r ∈ (polygonCapPolyline K).carrier := by
+    rcases hrfront with (hrleft | hrpoly) | hrright
+    · obtain ⟨c, hc, hr⟩ := hrleft
+      have hrzero : inner ℝ r (normalVector (Θ.angle : Real.Angle)) = 0 := by
+        rw [hr, capVertices_angle_fst_eq K, inner_add_left,
+          real_inner_smul_left, real_inner_smul_left]
+        have hz : inner ℝ (tangentVector (Θ.angle : Real.Angle))
+            (normalVector (Θ.angle : Real.Angle)) = 0 := by
+          rw [real_inner_comm (normalVector (Θ.angle : Real.Angle))]
+          exact inner_normalVector_tangentVector Θ.angle
+        rw [hz]
+        ring
+      have hqnonneg := hfan.1
+      change 0 ≤ inner ℝ q (normalVector (Θ.angle : Real.Angle)) at hqnonneg
+      have hs : 0 < Real.sin Θ.angle := Real.sin_pos_of_pos_of_lt_pi Θ.angle_pos
+        (by linarith [Θ.angle_le, Real.pi_pos])
+      simp [normalVector, frame, PiLp.inner_apply, hrx] at hrzero hqnonneg
+      exfalso
+      nlinarith
+    · exact hrpoly
+    · obtain ⟨c, hc, hr⟩ := hrright
+      have hrzero : r 1 = 0 := by
+        rw [hr, capVertices_zero_snd_eq K]
+        simp [normalVector, frame]
+      have hqnonneg := hfan.2
+      change 0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at hqnonneg
+      simp [normalVector, frame, PiLp.inner_apply] at hqnonneg
+      exfalso
+      linarith
+  exact K.val.mem_of_fst_eq_of_snd_le (hpoly hrpoly) hfan hrx.symm hry.le
+
+/-- Every balanced polygon cap contains its polygon niche. -/
+theorem polygonNiche_subset_of_balanced {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hK : IsBalancedPolygonCap K) :
+    polygonNiche Θ K.val ⊆ (K.val.val : Set Point) :=
+  polygonNiche_subset_of_polyline_subset K (polygonCapPolyline_subset_of_balanced K hK)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Polyline / Length / Basic
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Extend the directional polyline-length function by zero outside the angle domain. -/
+def polygonPolylineLengthAt {Θ : AngleSet} (K : PolygonCapSpace Θ) (t : ℝ) : ℝ := by
+  classical
+  exact if ht : t ∈ angleDomain Θ then polygonCapPolylineLength K ⟨t, ht⟩ else 0
+
+/-- The one-dimensional Hausdorff measure of the niche frontier inside a specified set. -/
+def nicheBoundaryLength {Θ : AngleSet} (K : PolygonCapSpace Θ) (S : Set Point) : ℝ :=
+  (MeasureTheory.Measure.hausdorffMeasure 1 (frontier (polygonNiche Θ K.val) ∩ S)).toReal
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Polyline / Length / Boundary
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+private lemma fst_mem_Icc_of_mem_segment_mc4d199c {a b q : Point}
+    (hab : a 0 ≤ b 0) (hq : q ∈ segment ℝ a b) : q 0 ∈ Set.Icc (a 0) (b 0) := by
+  rw [segment_eq_image'] at hq
+  obtain ⟨r, hr, rfl⟩ := hq
+  change a 0 + r * (b 0 - a 0) ∈ Set.Icc (a 0) (b 0)
+  constructor <;> nlinarith [hr.1, hr.2]
+
+private theorem frontier_inter_open_eq_frontier_sdiff_open_on_interior
+    {F X : Set Point} :
+    frontier (F ∩ X) ∩ interior F = frontier (F \ X) ∩ interior F := by
+  have hInt : IsOpen (interior F) := isOpen_interior
+  calc
+    frontier (F ∩ X) ∩ interior F =
+        frontier ((F ∩ X) ∩ interior F) ∩ interior F :=
+      (frontier_inter_open_inter hInt).symm
+    _ = frontier (X ∩ interior F) ∩ interior F := by
+      congr 2
+      ext p
+      simp only [Set.mem_inter_iff]
+      constructor
+      · rintro ⟨⟨-, hpX⟩, hpF⟩
+        exact ⟨hpX, hpF⟩
+      · rintro ⟨hpX, hpF⟩
+        exact ⟨⟨interior_subset hpF, hpX⟩, hpF⟩
+    _ = frontier X ∩ interior F := frontier_inter_open_inter hInt
+    _ = frontier Xᶜ ∩ interior F := by rw [frontier_compl]
+    _ = frontier (Xᶜ ∩ interior F) ∩ interior F :=
+      (frontier_inter_open_inter hInt).symm
+    _ = frontier ((F \ X) ∩ interior F) ∩ interior F := by
+      congr 2
+      ext p
+      simp only [Set.mem_inter_iff, Set.mem_compl_iff, Set.mem_sdiff]
+      constructor
+      · rintro ⟨hpX, hpF⟩
+        exact ⟨⟨interior_subset hpF, hpX⟩, hpF⟩
+      · rintro ⟨⟨-, hpX⟩, hpF⟩
+        exact ⟨hpX, hpF⟩
+    _ = frontier (F \ X) ∩ interior F := frontier_inter_open_inter hInt
+
+private theorem frontier_inter_open_symmDiff_subset_frontier
+    {F X : Set Point} (hF : IsClosed F) :
+    (frontier (F ∩ X) \ frontier (F \ X)) ∪
+        (frontier (F \ X) \ frontier (F ∩ X)) ⊆ frontier F := by
+  have heq := frontier_inter_open_eq_frontier_sdiff_open_on_interior (F := F) (X := X)
+  intro p hp
+  by_contra hpF
+  have hmemF_of_left (hpN : p ∈ frontier (F ∩ X)) : p ∈ F := by
+    have hpcl : p ∈ closure (F ∩ X) := frontier_subset_closure hpN
+    exact hF.closure_eq ▸ closure_mono Set.inter_subset_left hpcl
+  have hmemF_of_right (hpC : p ∈ frontier (F \ X)) : p ∈ F := by
+    have hpcl : p ∈ closure (F \ X) := frontier_subset_closure hpC
+    exact hF.closure_eq ▸ closure_mono Set.sdiff_subset hpcl
+  rcases hp with hp | hp
+  · have hpInt : p ∈ interior F := by
+      simpa using (mem_frontier_iff_notMem_interior (hmemF_of_left hp.1)).not.mp hpF
+    have : p ∈ frontier (F \ X) ∩ interior F := heq ▸ ⟨hp.1, hpInt⟩
+    exact hp.2 this.1
+  · have hpInt : p ∈ interior F := by
+      simpa using (mem_frontier_iff_notMem_interior (hmemF_of_right hp.1)).not.mp hpF
+    have : p ∈ frontier (F ∩ X) ∩ interior F := heq.symm ▸ ⟨hp.1, hpInt⟩
+    exact hp.2 this.1
+
+private theorem measure_eq_of_symmDiff_subset_null
+    (μ : Measure Point) {A B E : Set Point}
+    (hsub : (A \ B) ∪ (B \ A) ⊆ E) (hE : μ E = 0) : μ A = μ B := by
+  apply measure_congr
+  rw [ae_eq_set]
+  constructor
+  · exact measure_mono_null (fun _ hp ↦ hsub (Or.inl hp)) hE
+  · exact measure_mono_null (fun _ hp ↦ hsub (Or.inr hp)) hE
+
+/-- The fan clipping the niche is closed. -/
+theorem isClosed_capFan (ω : ℝ) : IsClosed (capFan ω) := by
+  unfold capFan normalHalfPlane
+  exact (isClosed_le continuous_const (by fun_prop)).inter
+    (isClosed_le continuous_const (by fun_prop))
+
+/-- The inward quadrant of a supporting hallway is open. -/
+theorem isOpen_innerQuadrant (S : Set Point) (t : ℝ) :
+    IsOpen (innerQuadrant S t) := by
+  unfold innerQuadrant normalHalfPlane
+  exact (isOpen_lt (by fun_prop) continuous_const).inter
+    (isOpen_lt (by fun_prop) continuous_const)
+
+private theorem frontier_normalHalfPlane_lower_strict_subset_normalLine
+    (t : Real.Angle) (h : ℝ) :
+    frontier (normalHalfPlane t h false true) ⊆ normalLine t h := by
+  change frontier {p : Point | inner ℝ p (normalVector t) < h} ⊆
+    {p | inner ℝ p (normalVector t) = h}
+  exact frontier_lt_subset_eq (by fun_prop) continuous_const
+
+private theorem frontier_normalHalfPlane_upper_closed_subset_normalLine
+    (t : Real.Angle) (h : ℝ) :
+    frontier (normalHalfPlane t h true false) ⊆ normalLine t h := by
+  change frontier {p : Point | h ≤ inner ℝ p (normalVector t)} ⊆
+    {p | inner ℝ p (normalVector t) = h}
+  exact frontier_ge_subset_eq continuous_const (by fun_prop)
+
+private def polygonCapBoundaryLines {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    Finset (Real.Angle × ℝ) := by
+  classical
+  exact {(((Θ.angle : ℝ) : Real.Angle), 0),
+      (((Real.pi / 2 : ℝ) : Real.Angle), 0)} ∪
+    Θ.directions.biUnion fun t ↦
+      {(((t : ℝ) : Real.Angle), supportValue K.val.val (t : Real.Angle) - 1),
+        (((t + Real.pi / 2 : ℝ) : Real.Angle),
+          supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1)}
+
+private theorem frontier_capFan_subset_boundaryLines {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) :
+    frontier (capFan Θ.angle) ⊆
+      ⋃ l ∈ polygonCapBoundaryLines K, normalLine l.1 l.2 := by
+  intro p hp
+  have hp' := frontier_inter_subset
+    (normalHalfPlane (Θ.angle : Real.Angle) 0 true false)
+    (normalHalfPlane ((Real.pi / 2 : ℝ) : Real.Angle) 0 true false) hp
+  rcases hp' with hp' | hp'
+  · have hline := frontier_normalHalfPlane_upper_closed_subset_normalLine
+      (Θ.angle : Real.Angle) 0 hp'.1
+    apply Set.mem_iUnion₂.mpr
+    exact ⟨(((Θ.angle : ℝ) : Real.Angle), 0), by simp [polygonCapBoundaryLines], hline⟩
+  · have hline := frontier_normalHalfPlane_upper_closed_subset_normalLine
+      ((Real.pi / 2 : ℝ) : Real.Angle) 0 hp'.2
+    apply Set.mem_iUnion₂.mpr
+    exact ⟨(((Real.pi / 2 : ℝ) : Real.Angle), 0),
+      by simp [polygonCapBoundaryLines], hline⟩
+
+private theorem frontier_innerQuadrant_subset_boundaryLines {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {t : ℝ} (ht : t ∈ Θ.directions) :
+    frontier (innerQuadrant K.val.val t) ⊆
+      ⋃ l ∈ polygonCapBoundaryLines K, normalLine l.1 l.2 := by
+  classical
+  intro p hp
+  have hp' := frontier_inter_subset
+    (normalHalfPlane (t : Real.Angle) (supportValue K.val.val (t : Real.Angle) - 1)
+      false true)
+    (normalHalfPlane ((t + Real.pi / 2 : ℝ) : Real.Angle)
+      (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) false true) hp
+  rcases hp' with hp' | hp'
+  · have hline := frontier_normalHalfPlane_lower_strict_subset_normalLine
+      (t : Real.Angle) (supportValue K.val.val (t : Real.Angle) - 1) hp'.1
+    apply Set.mem_iUnion₂.mpr
+    refine ⟨(((t : ℝ) : Real.Angle), supportValue K.val.val (t : Real.Angle) - 1),
+      ?_, hline⟩
+    simp only [polygonCapBoundaryLines, Finset.mem_union]
+    right
+    exact Finset.mem_biUnion.mpr ⟨t, ht, by simp⟩
+  · have hline := frontier_normalHalfPlane_lower_strict_subset_normalLine
+      ((t + Real.pi / 2 : ℝ) : Real.Angle)
+      (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) hp'.2
+    apply Set.mem_iUnion₂.mpr
+    refine ⟨(((t + Real.pi / 2 : ℝ) : Real.Angle),
+      supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1), ?_, hline⟩
+    simp only [polygonCapBoundaryLines, Finset.mem_union]
+    right
+    exact Finset.mem_biUnion.mpr ⟨t, ht, by simp⟩
+
+private theorem frontier_polygonComplement_subset_boundaryLines {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) :
+    frontier (capFan Θ.angle \ polygonNiche Θ K.val) ⊆
+      ⋃ l ∈ polygonCapBoundaryLines K, normalLine l.1 l.2 := by
+  let X : Set Point := ⋃ t ∈ Θ.directions, innerQuadrant K.val.val t
+  have hset : capFan Θ.angle \ polygonNiche Θ K.val = capFan Θ.angle \ X := by
+    simp [polygonNiche, X]
+  rw [hset, Set.sdiff_eq]
+  intro p hp
+  have hp' := frontier_inter_subset (capFan Θ.angle) Xᶜ hp
+  rcases hp' with hp | hp
+  · exact frontier_capFan_subset_boundaryLines K hp.1
+  · have hpX : p ∈ frontier X := by simpa [frontier_compl] using hp.2
+    have hmem := Finset.frontier_biUnion_subset Θ.directions
+      (fun t ↦ innerQuadrant K.val.val t) hpX
+    obtain ⟨t, ht, hpt⟩ := Set.mem_iUnion₂.mp hmem
+    exact frontier_innerQuadrant_subset_boundaryLines K ht hpt
+
+private theorem real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi
+    {v : Point} (hv : v ≠ 0) {s t : ℝ}
+    (hs : s ∈ Set.Ioo 0 Real.pi) (ht : t ∈ Set.Ioo 0 Real.pi)
+    (hvs : inner ℝ v (normalVector (s : Real.Angle)) = 0)
+    (hvt : inner ℝ v (normalVector (t : Real.Angle)) = 0) : s = t := by
+  apply eq_of_inner_sub_normalVector_eq_zero_of_ne (a := 0) (b := v) hs ht hv.symm
+  · simpa using hvs
+  · simpa using hvt
+
+private theorem normalLine_inter_normalLine_subsingleton {s t c d : ℝ}
+    (hs : s ∈ Set.Ioo 0 Real.pi) (ht : t ∈ Set.Ioo 0 Real.pi) (hst : s ≠ t) :
+    (normalLine (s : Real.Angle) c ∩ normalLine (t : Real.Angle) d).Subsingleton := by
+  intro p hp q hq
+  by_contra hpq
+  have hv : p - q ≠ 0 := sub_ne_zero.mpr hpq
+  have hsorth : inner ℝ (p - q) (normalVector (s : Real.Angle)) = 0 := by
+    have hps : inner ℝ p (normalVector (s : Real.Angle)) = c := hp.1
+    have hqs : inner ℝ q (normalVector (s : Real.Angle)) = c := hq.1
+    rw [inner_sub_left, hps, hqs, sub_self]
+  have htorth : inner ℝ (p - q) (normalVector (t : Real.Angle)) = 0 := by
+    have hpt : inner ℝ p (normalVector (t : Real.Angle)) = d := hp.2
+    have hqt : inner ℝ q (normalVector (t : Real.Angle)) = d := hq.2
+    rw [inner_sub_left, hpt, hqt, sub_self]
+  exact hst (real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi hv hs ht hsorth htorth)
+
+private theorem hausdorffMeasure_normalLine_inter_normalLine_eq_zero {s t c d : ℝ}
+    (hs : s ∈ Set.Ioo 0 Real.pi) (ht : t ∈ Set.Ioo 0 Real.pi) (hst : s ≠ t) :
+    Measure.hausdorffMeasure 1
+      (normalLine (s : Real.Angle) c ∩ normalLine (t : Real.Angle) d) = 0 := by
+  have := Measure.nullSingletonClass_hausdorff Point (by norm_num : (0 : ℝ) < 1)
+  exact (normalLine_inter_normalLine_subsingleton hs ht hst).countable.measure_zero _
+
+private theorem openRay_inter_hyperplane_subsingleton {p v n : Point} {c : ℝ}
+    (hvn : inner ℝ v n ≠ 0) :
+    (openRay p v ∩ {q | inner ℝ q n = c}).Subsingleton := by
+  rintro x ⟨⟨r, hr, rfl⟩, hxr⟩ y ⟨⟨s, hs, rfl⟩, hyr⟩
+  have hrs : r * inner ℝ v n = s * inner ℝ v n := by
+    simp only [Set.mem_ofPred_eq, inner_add_left, real_inner_smul_left] at hxr hyr
+    linarith
+  rw [mul_right_cancel₀ hvn hrs]
+
+private theorem hausdorffMeasure_openRay_inter_hyperplane_eq_zero {p v n : Point} {c : ℝ}
+    (hvn : inner ℝ v n ≠ 0) :
+    Measure.hausdorffMeasure 1 (openRay p v ∩ {q | inner ℝ q n = c}) = 0 := by
+  have := Measure.nullSingletonClass_hausdorff Point (by norm_num : (0 : ℝ) < 1)
+  exact (openRay_inter_hyperplane_subsingleton hvn).countable.measure_zero _
+
+private theorem hausdorffMeasure_frontier_capFan_inter_normalLine_eq_zero
+    {ω t c : ℝ} (hω : ω ∈ Set.Ioo 0 Real.pi) (ht : t ∈ Set.Ioo 0 Real.pi)
+    (htω : t ≠ ω) (htT : t ≠ Real.pi / 2) :
+    Measure.hausdorffMeasure 1
+      (frontier (capFan ω) ∩ normalLine (t : Real.Angle) c) = 0 := by
+  apply measure_mono_null (t :=
+    (normalLine (ω : Real.Angle) 0 ∩ normalLine (t : Real.Angle) c) ∪
+      (normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0 ∩
+        normalLine (t : Real.Angle) c))
+  · rintro p ⟨hp, hpt⟩
+    have hp' := frontier_inter_subset
+      (normalHalfPlane (ω : Real.Angle) 0 true false)
+      (normalHalfPlane ((Real.pi / 2 : ℝ) : Real.Angle) 0 true false) hp
+    rcases hp' with hp' | hp'
+    · exact Or.inl ⟨frontier_normalHalfPlane_upper_closed_subset_normalLine _ _ hp'.1, hpt⟩
+    · exact Or.inr ⟨frontier_normalHalfPlane_upper_closed_subset_normalLine _ _ hp'.2, hpt⟩
+  · rw [measure_union_null]
+    · exact hausdorffMeasure_normalLine_inter_normalLine_eq_zero hω ht htω.symm
+    · exact hausdorffMeasure_normalLine_inter_normalLine_eq_zero
+        ⟨by positivity, by linarith [Real.pi_pos]⟩ ht htT.symm
+
+private theorem frontier_capFan_inter_normalLine_countable
+    {ω t c : ℝ} (hω : ω ∈ Set.Ioo 0 Real.pi) (ht : t ∈ Set.Ioo 0 Real.pi)
+    (htω : t ≠ ω) (htT : t ≠ Real.pi / 2) :
+    (frontier (capFan ω) ∩ normalLine (t : Real.Angle) c).Countable := by
+  refine ((normalLine_inter_normalLine_subsingleton hω ht htω.symm
+      (c := 0) (d := c)).countable.union
+    (normalLine_inter_normalLine_subsingleton
+      (s := Real.pi / 2) (t := t) (c := 0) (d := c)
+      ⟨by positivity, by linarith [Real.pi_pos]⟩ ht htT.symm).countable).mono ?_
+  rintro p ⟨hp, hpt⟩
+  have hp' := frontier_inter_subset
+    (normalHalfPlane (ω : Real.Angle) 0 true false)
+    (normalHalfPlane ((Real.pi / 2 : ℝ) : Real.Angle) 0 true false) hp
+  rcases hp' with hp' | hp'
+  · exact Or.inl ⟨frontier_normalHalfPlane_upper_closed_subset_normalLine _ _ hp'.1, hpt⟩
+  · exact Or.inr ⟨frontier_normalHalfPlane_upper_closed_subset_normalLine _ _ hp'.2, hpt⟩
+
+private theorem hausdorffMeasure_frontier_polygonNiche_inter_normalLine_eq_complement
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) {t c : ℝ}
+    (ht : t ∈ Set.Ioo 0 Real.pi) (htω : t ≠ Θ.angle)
+    (htT : t ≠ Real.pi / 2) :
+    Measure.hausdorffMeasure 1
+        (frontier (polygonNiche Θ K.val) ∩ normalLine (t : Real.Angle) c) =
+      Measure.hausdorffMeasure 1
+        (frontier (capFan Θ.angle \ polygonNiche Θ K.val) ∩
+          normalLine (t : Real.Angle) c) := by
+  let X : Set Point := ⋃ s ∈ Θ.directions, innerQuadrant K.val.val s
+  have hN : polygonNiche Θ K.val = capFan Θ.angle ∩ X := by simp [polygonNiche, X]
+  have hC : capFan Θ.angle \ polygonNiche Θ K.val = capFan Θ.angle \ X := by
+    simp [polygonNiche, X]
+  rw [hC, hN]
+  apply measure_eq_of_symmDiff_subset_null (E :=
+    frontier (capFan Θ.angle) ∩ normalLine (t : Real.Angle) c)
+  · intro p hp
+    rcases hp with hp | hp
+    · refine ⟨frontier_inter_open_symmDiff_subset_frontier
+          (isClosed_capFan Θ.angle) (Or.inl ⟨hp.1.1, ?_⟩), hp.1.2⟩
+      intro hpC
+      exact hp.2 ⟨hpC, hp.1.2⟩
+    · refine ⟨frontier_inter_open_symmDiff_subset_frontier
+          (isClosed_capFan Θ.angle) (Or.inr ⟨hp.1.1, ?_⟩), hp.1.2⟩
+      intro hpN
+      exact hp.2 ⟨hpN, hp.1.2⟩
+  · exact hausdorffMeasure_frontier_capFan_inter_normalLine_eq_zero
+      ⟨Θ.angle_pos, Θ.angle_le.trans_lt (by linarith [Real.pi_pos])⟩ ht htω htT
+
+private theorem hausdorffMeasure_frontier_polygonComplement_inter_normalLine_eq_carrier
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) {t c : ℝ}
+    (hleft : inner ℝ (tangentVector (Θ.angle : Real.Angle))
+      (normalVector (t : Real.Angle)) ≠ 0)
+    (hright : inner ℝ (normalVector 0) (normalVector (t : Real.Angle)) ≠ 0) :
+    Measure.hausdorffMeasure 1
+        (frontier (capFan Θ.angle \ polygonNiche Θ K.val) ∩
+          normalLine (t : Real.Angle) c) =
+      Measure.hausdorffMeasure 1
+        ((polygonCapPolyline K).carrier ∩ normalLine (t : Real.Angle) c) := by
+  let p := polygonCapPolyline K
+  have hp : IsCapPolyline K p := (polygonCap_polyline K).choose_spec
+  let L := openRay ((capVertices K.val Θ.angle).2.1)
+    (tangentVector (Θ.angle : Real.Angle))
+  let R := openRay ((capVertices K.val 0).1.2) (normalVector 0)
+  have hnullL : Measure.hausdorffMeasure 1
+      (L ∩ normalLine (t : Real.Angle) c) = 0 := by
+    exact hausdorffMeasure_openRay_inter_hyperplane_eq_zero hleft
+  have hnullR : Measure.hausdorffMeasure 1
+      (R ∩ normalLine (t : Real.Angle) c) = 0 := by
+    exact hausdorffMeasure_openRay_inter_hyperplane_eq_zero hright
+  apply measure_eq_of_symmDiff_subset_null
+    (E := (L ∩ normalLine (t : Real.Angle) c) ∪
+      (R ∩ normalLine (t : Real.Angle) c))
+  · intro q hq
+    rcases hq with hq | hq
+    · rw [hp.2.2.2.2.2.1] at hq
+      rcases hq.1.1 with hL | hR
+      · rcases hL with hL | hcarrier
+        · exact Or.inl ⟨hL, hq.1.2⟩
+        · exact False.elim (hq.2 ⟨hcarrier, hq.1.2⟩)
+      · exact Or.inr ⟨hR, hq.1.2⟩
+    · exact False.elim (hq.2 ⟨by
+        rw [hp.2.2.2.2.2.1]
+        exact Or.inl (Or.inr hq.1.1), hq.1.2⟩)
+  · rw [measure_union_null hnullL hnullR]
+
+private theorem edge_direction_ne_zero (p : XMonotonePolylineData)
+    (i : Fin p.edges) : p.vertices i.succ - p.vertices i.castSucc ≠ 0 := by
+  rw [sub_ne_zero]
+  intro h
+  have hcoord := congrArg (fun q : Point ↦ q 0) h
+  exact (ne_of_gt (p.increasing i.castSucc_lt_succ)) hcoord
+
+private theorem IsCapPolyline.edge_subset_boundaryLine {Θ : AngleSet}
+    {K : PolygonCapSpace Θ} {p : XMonotonePolylineData}
+    (hp : IsCapPolyline K p) (i : Fin p.edges) :
+    ∃ l ∈ polygonCapBoundaryLines K,
+      segment ℝ (p.vertices i.castSucc) (p.vertices i.succ) ⊆
+        normalLine l.1 l.2 := by
+  classical
+  simpa only [normalLine] using EuclideanGeometry.exists_segment_subset_hyperplane_of_finite_cover
+    (I := polygonCapBoundaryLines K) (n := fun l ↦ normalVector l.1) (c := Prod.snd)
+    (a := p.vertices i.castSucc) (b := p.vertices i.succ)
+    (sub_ne_zero.mp (edge_direction_ne_zero p i)).symm (by
+      intro q hq
+      have hcarrier : q ∈ p.carrier := by
+        rw [XMonotonePolylineData.carrier]
+        exact Set.mem_iUnion.mpr ⟨i, hq⟩
+      have hfrontier : q ∈ frontier (capFan Θ.angle \ polygonNiche Θ K.val) := by
+        rw [hp.2.2.2.2.2.1]
+        exact Or.inl (Or.inr hcarrier)
+      exact frontier_polygonComplement_subset_boundaryLines K hfrontier)
+
+private theorem IsCapPolyline.edge_subset_bLine_of_orthogonal {Θ : AngleSet}
+    {K : PolygonCapSpace Θ} {p : XMonotonePolylineData}
+    (hp : IsCapPolyline K p) {t : ℝ} (ht : t ∈ Θ.directions)
+    (i : Fin p.edges)
+    (hi : inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+      (normalVector (t : Real.Angle)) = 0) :
+    segment ℝ (p.vertices i.castSucc) (p.vertices i.succ) ⊆
+      normalLine (t : Real.Angle) (supportValue K.val.val (t : Real.Angle) - 1) := by
+  classical
+  obtain ⟨l, hl, hline⟩ := hp.edge_subset_boundaryLine i
+  have hleft := hline (left_mem_segment ℝ _ _)
+  have hright := hline (right_mem_segment ℝ _ _)
+  have hlorth : inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+      (normalVector l.1) = 0 := by
+    change inner ℝ (p.vertices i.castSucc) (normalVector l.1) = l.2 at hleft
+    change inner ℝ (p.vertices i.succ) (normalVector l.1) = l.2 at hright
+    rw [inner_sub_left, hright, hleft, sub_self]
+  have htI : t ∈ Set.Ioo 0 Real.pi :=
+    ⟨(Θ.interior t ht).1,
+      ((Θ.interior t ht).2.trans_le Θ.angle_le).trans (by linarith [Real.pi_pos])⟩
+  simp only [polygonCapBoundaryLines, Finset.mem_union] at hl
+  rcases hl with hl | hl
+  · simp only [Finset.mem_insert, Finset.mem_singleton] at hl
+    rcases hl with rfl | rfl
+    · have heq := real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi
+        (edge_direction_ne_zero p i)
+        ⟨Θ.angle_pos, Θ.angle_le.trans_lt (by linarith [Real.pi_pos])⟩ htI hlorth hi
+      linarith [(Θ.interior t ht).2]
+    · have heq := real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi
+        (edge_direction_ne_zero p i)
+        ⟨by positivity, by linarith [Real.pi_pos]⟩ htI hlorth hi
+      linarith [(Θ.interior t ht).2, Θ.angle_le]
+  · obtain ⟨s, hs, hsl⟩ := Finset.mem_biUnion.mp hl
+    simp only [Finset.mem_insert, Finset.mem_singleton] at hsl
+    rcases hsl with rfl | rfl
+    · have heq := real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi
+        (edge_direction_ne_zero p i)
+        ⟨(Θ.interior s hs).1,
+          ((Θ.interior s hs).2.trans_le Θ.angle_le).trans
+            (by linarith [Real.pi_pos])⟩ htI hlorth hi
+      simpa [heq] using hline
+    · have hsI : s + Real.pi / 2 ∈ Set.Ioo 0 Real.pi := by
+        constructor
+        · linarith [(Θ.interior s hs).1, Real.pi_pos]
+        · linarith [(Θ.interior s hs).2, Θ.angle_le]
+      have heq := real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi
+        (edge_direction_ne_zero p i) hsI htI hlorth hi
+      linarith [(Θ.interior t ht).2, Θ.angle_le, (Θ.interior s hs).1, Real.pi_pos]
+
+private theorem IsCapPolyline.edge_subset_dLine_of_orthogonal {Θ : AngleSet}
+    {K : PolygonCapSpace Θ} {p : XMonotonePolylineData}
+    (hp : IsCapPolyline K p) {t : ℝ} (ht : t ∈ Θ.directions)
+    (i : Fin p.edges)
+    (hi : inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+      (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) = 0) :
+    segment ℝ (p.vertices i.castSucc) (p.vertices i.succ) ⊆
+      normalLine ((t + Real.pi / 2 : ℝ) : Real.Angle)
+        (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) := by
+  classical
+  obtain ⟨l, hl, hline⟩ := hp.edge_subset_boundaryLine i
+  have hleft := hline (left_mem_segment ℝ _ _)
+  have hright := hline (right_mem_segment ℝ _ _)
+  have hlorth : inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+      (normalVector l.1) = 0 := by
+    change inner ℝ (p.vertices i.castSucc) (normalVector l.1) = l.2 at hleft
+    change inner ℝ (p.vertices i.succ) (normalVector l.1) = l.2 at hright
+    rw [inner_sub_left, hright, hleft, sub_self]
+  have htI : t + Real.pi / 2 ∈ Set.Ioo 0 Real.pi := by
+    constructor
+    · linarith [(Θ.interior t ht).1, Real.pi_pos]
+    · linarith [(Θ.interior t ht).2, Θ.angle_le]
+  simp only [polygonCapBoundaryLines, Finset.mem_union] at hl
+  rcases hl with hl | hl
+  · simp only [Finset.mem_insert, Finset.mem_singleton] at hl
+    rcases hl with rfl | rfl
+    · have heq := real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi
+        (edge_direction_ne_zero p i)
+        ⟨Θ.angle_pos, Θ.angle_le.trans_lt (by linarith [Real.pi_pos])⟩ htI hlorth hi
+      linarith [(Θ.interior t ht).1, Θ.angle_le, Real.pi_pos]
+    · have heq := real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi
+        (edge_direction_ne_zero p i)
+        ⟨by positivity, by linarith [Real.pi_pos]⟩ htI hlorth hi
+      linarith [(Θ.interior t ht).1]
+  · obtain ⟨s, hs, hsl⟩ := Finset.mem_biUnion.mp hl
+    simp only [Finset.mem_insert, Finset.mem_singleton] at hsl
+    rcases hsl with rfl | rfl
+    · have hsI : s ∈ Set.Ioo 0 Real.pi :=
+        ⟨(Θ.interior s hs).1,
+          ((Θ.interior s hs).2.trans_le Θ.angle_le).trans
+            (by linarith [Real.pi_pos])⟩
+      have heq := real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi
+        (edge_direction_ne_zero p i) hsI htI hlorth hi
+      linarith [(Θ.interior s hs).2, (Θ.interior t ht).1, Θ.angle_le]
+    · have hsI : s + Real.pi / 2 ∈ Set.Ioo 0 Real.pi := by
+        constructor
+        · linarith [(Θ.interior s hs).1, Real.pi_pos]
+        · linarith [(Θ.interior s hs).2, Θ.angle_le]
+      have heq := real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi
+        (edge_direction_ne_zero p i) hsI htI hlorth hi
+      have hst : s = t := by linarith
+      simpa [hst] using hline
+
+private theorem polygonCapPolyline_spec {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    IsCapPolyline K (polygonCapPolyline K) := by
+  exact (polygonCap_polyline K).choose_spec
+
+private theorem polygonCapPolyline_bLine_length {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {t : ℝ} (ht : t ∈ Θ.directions) :
+    (Measure.hausdorffMeasure 1
+      ((polygonCapPolyline K).carrier ∩
+        normalLine (t : Real.Angle) (supportValue K.val.val (t : Real.Angle) - 1))).toReal =
+      polygonPolylineLengthAt K t := by
+  let p := polygonCapPolyline K
+  have hp : IsCapPolyline K p := polygonCapPolyline_spec K
+  have hmeasure := p.toReal_hausdorffMeasure_carrier_inter_hyperplane
+    (normalVector (t : Real.Angle)) (supportValue K.val.val (t : Real.Angle) - 1)
+    (fun i hi ↦ hp.edge_subset_bLine_of_orthogonal ht i hi)
+  have htDomain : t ∈ angleDomain Θ := by
+    exact Or.inl (Or.inl ht)
+  rw [show normalLine (t : Real.Angle)
+      (supportValue K.val.val (t : Real.Angle) - 1) =
+        {q | inner ℝ q (normalVector (t : Real.Angle)) =
+          supportValue K.val.val (t : Real.Angle) - 1} by rfl,
+    hmeasure]
+  simp [polygonPolylineLengthAt, htDomain, polygonCapPolylineLength, p]
+
+private theorem polygonCapPolyline_dLine_length {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {t : ℝ} (ht : t ∈ Θ.directions) :
+    (Measure.hausdorffMeasure 1
+      ((polygonCapPolyline K).carrier ∩
+        normalLine ((t + Real.pi / 2 : ℝ) : Real.Angle)
+          (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1))).toReal =
+      polygonPolylineLengthAt K (t + Real.pi / 2) := by
+  let p := polygonCapPolyline K
+  have hp : IsCapPolyline K p := polygonCapPolyline_spec K
+  have hmeasure := p.toReal_hausdorffMeasure_carrier_inter_hyperplane
+    (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle))
+    (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1)
+    (fun i hi ↦ hp.edge_subset_dLine_of_orthogonal ht i hi)
+  have htDomain : t + Real.pi / 2 ∈ angleDomain Θ := by
+    exact Or.inl (Or.inr ⟨t, ht, rfl⟩)
+  rw [show normalLine ((t + Real.pi / 2 : ℝ) : Real.Angle)
+      (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) =
+        {q | inner ℝ q (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) =
+          supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1} by rfl,
+    hmeasure]
+  simp [polygonPolylineLengthAt, htDomain, polygonCapPolylineLength, p]
+
+private theorem tangentVector_angle_inner_normalVector_direction_ne_zero
+    {Θ : AngleSet} {t : ℝ} (ht : t ∈ Θ.directions) :
+    inner ℝ (tangentVector (Θ.angle : Real.Angle))
+      (normalVector (t : Real.Angle)) ≠ 0 := by
+  have hsin : 0 < Real.sin (Θ.angle - t) := Real.sin_pos_of_pos_of_lt_pi
+    (by linarith [(Θ.interior t ht).2])
+    (by linarith [(Θ.interior t ht).1, Θ.angle_le, Real.pi_pos])
+  have h := sin_sub_eq_neg_inner_normalVector_tangentVector
+    (t : Real.Angle) (Θ.angle : Real.Angle)
+  change Real.sin (Θ.angle - t) =
+    -inner ℝ (normalVector (t : Real.Angle))
+      (tangentVector (Θ.angle : Real.Angle)) at h
+  rw [real_inner_comm]
+  linarith
+
+private theorem tangentVector_angle_inner_normalVector_direction_add_ne_zero
+    {Θ : AngleSet} {t : ℝ} (ht : t ∈ Θ.directions) :
+    inner ℝ (tangentVector (Θ.angle : Real.Angle))
+      (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) ≠ 0 := by
+  have hneg : -Real.pi < Θ.angle - (t + Real.pi / 2) := by
+    linarith [(Θ.interior t ht).2, Real.pi_pos]
+  have hzero : Θ.angle - (t + Real.pi / 2) < 0 := by
+    linarith [(Θ.interior t ht).1, Θ.angle_le]
+  have hsin : Real.sin (Θ.angle - (t + Real.pi / 2)) < 0 :=
+    Real.sin_neg_of_neg_of_neg_pi_lt hzero hneg
+  have h := sin_sub_eq_neg_inner_normalVector_tangentVector
+    ((t + Real.pi / 2 : ℝ) : Real.Angle) (Θ.angle : Real.Angle)
+  change Real.sin (Θ.angle - (t + Real.pi / 2)) =
+    -inner ℝ (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle))
+      (tangentVector (Θ.angle : Real.Angle)) at h
+  rw [real_inner_comm]
+  linarith
+
+private theorem normalVector_zero_inner_normalVector_direction_ne_zero
+    {Θ : AngleSet} {t : ℝ} (ht : t ∈ Θ.directions) :
+    inner ℝ (normalVector 0) (normalVector (t : Real.Angle)) ≠ 0 := by
+  rw [show (0 : Real.Angle) = ((0 : ℝ) : Real.Angle) by rfl,
+    inner_normalVector_normalVector]
+  have hcos : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, (Θ.interior t ht).1],
+      (Θ.interior t ht).2.trans_le Θ.angle_le⟩
+  simpa [Real.cos_neg] using hcos.ne'
+
+private theorem normalVector_zero_inner_normalVector_direction_add_ne_zero
+    {Θ : AngleSet} {t : ℝ} (ht : t ∈ Θ.directions) :
+    inner ℝ (normalVector 0)
+      (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) ≠ 0 := by
+  rw [show (0 : Real.Angle) = ((0 : ℝ) : Real.Angle) by rfl,
+    inner_normalVector_normalVector]
+  have hsin : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi
+    (Θ.interior t ht).1
+    (by linarith [(Θ.interior t ht).2, Θ.angle_le, Real.pi_pos])
+  rw [show 0 - (t + Real.pi / 2) = -(t + Real.pi / 2) by ring,
+    Real.cos_neg, Real.cos_add_pi_div_two]
+  exact neg_ne_zero.mpr hsin.ne'
+
+private theorem polygonNiche_bLine_length {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {t : ℝ} (ht : t ∈ Θ.directions) :
+    (Measure.hausdorffMeasure 1
+      (frontier (polygonNiche Θ K.val) ∩
+        normalLine (t : Real.Angle) (supportValue K.val.val (t : Real.Angle) - 1))).toReal =
+      polygonPolylineLengthAt K t := by
+  have htI : t ∈ Set.Ioo 0 Real.pi :=
+    ⟨(Θ.interior t ht).1,
+      ((Θ.interior t ht).2.trans_le Θ.angle_le).trans (by linarith [Real.pi_pos])⟩
+  rw [hausdorffMeasure_frontier_polygonNiche_inter_normalLine_eq_complement K htI
+      (ne_of_lt (Θ.interior t ht).2) (ne_of_lt
+        ((Θ.interior t ht).2.trans_le Θ.angle_le)),
+    hausdorffMeasure_frontier_polygonComplement_inter_normalLine_eq_carrier K
+      (tangentVector_angle_inner_normalVector_direction_ne_zero ht)
+      (normalVector_zero_inner_normalVector_direction_ne_zero ht)]
+  exact polygonCapPolyline_bLine_length K ht
+
+private theorem polygonNiche_dLine_length {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {t : ℝ} (ht : t ∈ Θ.directions) :
+    (Measure.hausdorffMeasure 1
+      (frontier (polygonNiche Θ K.val) ∩
+        normalLine ((t + Real.pi / 2 : ℝ) : Real.Angle)
+          (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1))).toReal =
+      polygonPolylineLengthAt K (t + Real.pi / 2) := by
+  have htI : t + Real.pi / 2 ∈ Set.Ioo 0 Real.pi := by
+    constructor
+    · linarith [(Θ.interior t ht).1, Real.pi_pos]
+    · linarith [(Θ.interior t ht).2, Θ.angle_le]
+  rw [hausdorffMeasure_frontier_polygonNiche_inter_normalLine_eq_complement K htI
+      (by linarith [(Θ.interior t ht).1, Θ.angle_le, Real.pi_pos])
+      (by linarith [(Θ.interior t ht).1]),
+    hausdorffMeasure_frontier_polygonComplement_inter_normalLine_eq_carrier K
+      (tangentVector_angle_inner_normalVector_direction_add_ne_zero ht)
+      (normalVector_zero_inner_normalVector_direction_add_ne_zero ht)]
+  exact polygonCapPolyline_dLine_length K ht
+
+private theorem frontier_innerQuadrant_inter_bLine_subset_bRay
+    (s : Set Point) (t : ℝ) :
+    frontier (innerQuadrant s t) ∩
+        normalLine (t : Real.Angle) (supportValue s (t : Real.Angle) - 1) ⊆
+      (rotatingHallwayParts s (t : Real.Angle)).bRay := by
+  intro p hp
+  rw [mem_rotatingHallwayParts_bRay_iff]
+  refine ⟨hp.2, ?_⟩
+  apply closure_minimal (t := {q | inner ℝ q (tangentVector (t : Real.Angle)) ≤
+      supportValue s ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1})
+      (fun q hq ↦ ?_) (isClosed_le (by fun_prop) continuous_const)
+      (frontier_subset_closure hp.1)
+  change q ∈ innerQuadrant s t at hq
+  have h := hq.2
+  change inner ℝ q (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) <
+    supportValue s ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 at h
+  rw [show ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+    (t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) by rfl,
+    normalVector_add_pi_div_two] at h
+  exact h.le
+
+private theorem frontier_innerQuadrant_inter_dLine_subset_dRay
+    (s : Set Point) (t : ℝ) :
+    frontier (innerQuadrant s t) ∩
+        normalLine ((t + Real.pi / 2 : ℝ) : Real.Angle)
+          (supportValue s ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) ⊆
+      (rotatingHallwayParts s (t : Real.Angle)).dRay := by
+  intro p hp
+  rw [mem_rotatingHallwayParts_dRay_iff]
+  refine ⟨?_, ?_⟩
+  · apply closure_minimal (t := {q | inner ℝ q (normalVector (t : Real.Angle)) ≤
+        supportValue s (t : Real.Angle) - 1})
+        (fun q hq ↦ ?_) (isClosed_le (by fun_prop) continuous_const)
+        (frontier_subset_closure hp.1)
+    change q ∈ innerQuadrant s t at hq
+    have h := hq.1
+    change inner ℝ q (normalVector (t : Real.Angle)) <
+      supportValue s (t : Real.Angle) - 1 at h
+    exact h.le
+  · have hline := hp.2
+    change inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) =
+      supportValue s ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 at hline
+    rw [show ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+      (t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) by rfl,
+      normalVector_add_pi_div_two] at hline
+    exact hline
+
+private theorem frontier_innerQuadrant_inter_bLine_countable_of_ne
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) {s t : ℝ}
+    (hs : s ∈ Θ.directions) (ht : t ∈ Θ.directions) (hst : s ≠ t) :
+    (frontier (innerQuadrant K.val.val s) ∩
+      normalLine (t : Real.Angle) (supportValue K.val.val (t : Real.Angle) - 1)).Countable := by
+  have hsI : s ∈ Set.Ioo 0 Real.pi :=
+    ⟨(Θ.interior s hs).1,
+      ((Θ.interior s hs).2.trans_le Θ.angle_le).trans (by linarith [Real.pi_pos])⟩
+  have htI : t ∈ Set.Ioo 0 Real.pi :=
+    ⟨(Θ.interior t ht).1,
+      ((Θ.interior t ht).2.trans_le Θ.angle_le).trans (by linarith [Real.pi_pos])⟩
+  have hsTI : s + Real.pi / 2 ∈ Set.Ioo 0 Real.pi := by
+    constructor
+    · linarith [(Θ.interior s hs).1, Real.pi_pos]
+    · linarith [(Θ.interior s hs).2, Θ.angle_le]
+  have hfirst := normalLine_inter_normalLine_subsingleton hsI htI hst
+    (c := supportValue K.val.val (s : Real.Angle) - 1)
+    (d := supportValue K.val.val (t : Real.Angle) - 1)
+  have hsecond := normalLine_inter_normalLine_subsingleton hsTI htI
+    (by linarith [(Θ.interior s hs).1, (Θ.interior t ht).2, Θ.angle_le])
+    (c := supportValue K.val.val ((s + Real.pi / 2 : ℝ) : Real.Angle) - 1)
+    (d := supportValue K.val.val (t : Real.Angle) - 1)
+  refine (hfirst.countable.union hsecond.countable).mono ?_
+  rintro p ⟨hp, hpt⟩
+  have hp' := frontier_inter_subset
+    (normalHalfPlane (s : Real.Angle) (supportValue K.val.val (s : Real.Angle) - 1)
+      false true)
+    (normalHalfPlane ((s + Real.pi / 2 : ℝ) : Real.Angle)
+      (supportValue K.val.val ((s + Real.pi / 2 : ℝ) : Real.Angle) - 1) false true) hp
+  rcases hp' with hp' | hp'
+  · exact Or.inl
+      ⟨frontier_normalHalfPlane_lower_strict_subset_normalLine _ _ hp'.1, hpt⟩
+  · exact Or.inr
+      ⟨frontier_normalHalfPlane_lower_strict_subset_normalLine _ _ hp'.2, hpt⟩
+
+private theorem frontier_innerQuadrant_inter_dLine_countable_of_ne
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) {s t : ℝ}
+    (hs : s ∈ Θ.directions) (ht : t ∈ Θ.directions) (hst : s ≠ t) :
+    (frontier (innerQuadrant K.val.val s) ∩
+      normalLine ((t + Real.pi / 2 : ℝ) : Real.Angle)
+        (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1)).Countable := by
+  have hsI : s ∈ Set.Ioo 0 Real.pi :=
+    ⟨(Θ.interior s hs).1,
+      ((Θ.interior s hs).2.trans_le Θ.angle_le).trans (by linarith [Real.pi_pos])⟩
+  have hsTI : s + Real.pi / 2 ∈ Set.Ioo 0 Real.pi := by
+    constructor
+    · linarith [(Θ.interior s hs).1, Real.pi_pos]
+    · linarith [(Θ.interior s hs).2, Θ.angle_le]
+  have htTI : t + Real.pi / 2 ∈ Set.Ioo 0 Real.pi := by
+    constructor
+    · linarith [(Θ.interior t ht).1, Real.pi_pos]
+    · linarith [(Θ.interior t ht).2, Θ.angle_le]
+  have hfirst := normalLine_inter_normalLine_subsingleton hsI htTI
+    (by linarith [(Θ.interior s hs).2, (Θ.interior t ht).1, Θ.angle_le])
+    (c := supportValue K.val.val (s : Real.Angle) - 1)
+    (d := supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1)
+  have hsecond := normalLine_inter_normalLine_subsingleton hsTI htTI
+    (by
+      intro h
+      apply hst
+      linarith)
+    (c := supportValue K.val.val ((s + Real.pi / 2 : ℝ) : Real.Angle) - 1)
+    (d := supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1)
+  refine (hfirst.countable.union hsecond.countable).mono ?_
+  rintro p ⟨hp, hpt⟩
+  have hp' := frontier_inter_subset
+    (normalHalfPlane (s : Real.Angle) (supportValue K.val.val (s : Real.Angle) - 1)
+      false true)
+    (normalHalfPlane ((s + Real.pi / 2 : ℝ) : Real.Angle)
+      (supportValue K.val.val ((s + Real.pi / 2 : ℝ) : Real.Angle) - 1) false true) hp
+  rcases hp' with hp' | hp'
+  · exact Or.inl
+      ⟨frontier_normalHalfPlane_lower_strict_subset_normalLine _ _ hp'.1, hpt⟩
+  · exact Or.inr
+      ⟨frontier_normalHalfPlane_lower_strict_subset_normalLine _ _ hp'.2, hpt⟩
+
+private theorem frontier_polygonNiche_bLine_sdiff_bRay_countable
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) {t : ℝ} (ht : t ∈ Θ.directions) :
+    ((frontier (polygonNiche Θ K.val) ∩
+        normalLine (t : Real.Angle) (supportValue K.val.val (t : Real.Angle) - 1)) \
+      (rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).bRay).Countable := by
+  let L := normalLine (t : Real.Angle) (supportValue K.val.val (t : Real.Angle) - 1)
+  let R := (rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).bRay
+  let X : Set Point := ⋃ s ∈ Θ.directions, innerQuadrant K.val.val s
+  let E : Θ.directions → Set Point := fun s ↦
+    (frontier (innerQuadrant K.val.val s) ∩ L) \ R
+  have htI : t ∈ Set.Ioo 0 Real.pi :=
+    ⟨(Θ.interior t ht).1,
+      ((Θ.interior t ht).2.trans_le Θ.angle_le).trans (by linarith [Real.pi_pos])⟩
+  have hfan : (frontier (capFan Θ.angle) ∩ L).Countable :=
+    frontier_capFan_inter_normalLine_countable
+      ⟨Θ.angle_pos, Θ.angle_le.trans_lt (by linarith [Real.pi_pos])⟩ htI
+      (ne_of_lt (Θ.interior t ht).2)
+      (ne_of_lt ((Θ.interior t ht).2.trans_le Θ.angle_le))
+  have hE (s : Θ.directions) : (E s).Countable := by
+    by_cases hst : (s : ℝ) = t
+    · apply Set.countable_empty.mono
+      intro p hp
+      exfalso
+      apply hp.2
+      apply frontier_innerQuadrant_inter_bLine_subset_bRay
+        (s := (K.val.val : Set Point)) (t := t)
+      simpa [E, L, R, hst] using hp.1
+    · exact (frontier_innerQuadrant_inter_bLine_countable_of_ne K s.property ht hst).mono
+        (by intro p hp; exact hp.1)
+  refine (hfan.union (Set.countable_iUnion hE)).mono ?_
+  intro p hp
+  have hp' : p ∈ frontier (capFan Θ.angle ∩ X) := by
+    simpa [polygonNiche, X] using hp.1.1
+  rcases frontier_inter_subset (capFan Θ.angle) X hp' with hpF | hpX
+  · exact Or.inl ⟨hpF.1, hp.1.2⟩
+  · right
+    have hmem := Finset.frontier_biUnion_subset Θ.directions
+      (fun s ↦ innerQuadrant K.val.val s) hpX.2
+    obtain ⟨s, hs, hps⟩ := Set.mem_iUnion₂.mp hmem
+    exact Set.mem_iUnion.mpr ⟨⟨s, hs⟩, ⟨⟨hps, hp.1.2⟩, hp.2⟩⟩
+
+private theorem frontier_polygonNiche_dLine_sdiff_dRay_countable
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) {t : ℝ} (ht : t ∈ Θ.directions) :
+    ((frontier (polygonNiche Θ K.val) ∩
+        normalLine ((t + Real.pi / 2 : ℝ) : Real.Angle)
+          (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1)) \
+      (rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).dRay).Countable := by
+  let L := normalLine ((t + Real.pi / 2 : ℝ) : Real.Angle)
+    (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1)
+  let R := (rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).dRay
+  let X : Set Point := ⋃ s ∈ Θ.directions, innerQuadrant K.val.val s
+  let E : Θ.directions → Set Point := fun s ↦
+    (frontier (innerQuadrant K.val.val s) ∩ L) \ R
+  have htI : t + Real.pi / 2 ∈ Set.Ioo 0 Real.pi := by
+    constructor
+    · linarith [(Θ.interior t ht).1, Real.pi_pos]
+    · linarith [(Θ.interior t ht).2, Θ.angle_le]
+  have hfan : (frontier (capFan Θ.angle) ∩ L).Countable :=
+    frontier_capFan_inter_normalLine_countable
+      ⟨Θ.angle_pos, Θ.angle_le.trans_lt (by linarith [Real.pi_pos])⟩ htI
+      (by linarith [(Θ.interior t ht).1, Θ.angle_le, Real.pi_pos])
+      (by linarith [(Θ.interior t ht).1])
+  have hE (s : Θ.directions) : (E s).Countable := by
+    by_cases hst : (s : ℝ) = t
+    · apply Set.countable_empty.mono
+      intro p hp
+      exfalso
+      apply hp.2
+      apply frontier_innerQuadrant_inter_dLine_subset_dRay
+        (s := (K.val.val : Set Point)) (t := t)
+      simpa [E, L, R, hst] using hp.1
+    · exact (frontier_innerQuadrant_inter_dLine_countable_of_ne K s.property ht hst).mono
+        (by intro p hp; exact hp.1)
+  refine (hfan.union (Set.countable_iUnion hE)).mono ?_
+  intro p hp
+  have hp' : p ∈ frontier (capFan Θ.angle ∩ X) := by
+    simpa [polygonNiche, X] using hp.1.1
+  rcases frontier_inter_subset (capFan Θ.angle) X hp' with hpF | hpX
+  · exact Or.inl ⟨hpF.1, hp.1.2⟩
+  · right
+    have hmem := Finset.frontier_biUnion_subset Θ.directions
+      (fun s ↦ innerQuadrant K.val.val s) hpX.2
+    obtain ⟨s, hs, hps⟩ := Set.mem_iUnion₂.mp hmem
+    exact Set.mem_iUnion.mpr ⟨⟨s, hs⟩, ⟨⟨hps, hp.1.2⟩, hp.2⟩⟩
+
+private theorem rotatingHallwayParts_bRay_subset_bLine
+    (s : Set Point) (t : ℝ) :
+    (rotatingHallwayParts s (t : Real.Angle)).bRay ⊆
+      normalLine (t : Real.Angle) (supportValue s (t : Real.Angle) - 1) := by
+  intro p hp
+  exact (mem_rotatingHallwayParts_bRay_iff s (t : Real.Angle) p).mp hp |>.1
+
+private theorem rotatingHallwayParts_dRay_subset_dLine
+    (s : Set Point) (t : ℝ) :
+    (rotatingHallwayParts s (t : Real.Angle)).dRay ⊆
+      normalLine ((t + Real.pi / 2 : ℝ) : Real.Angle)
+        (supportValue s ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) := by
+  intro p hp
+  have h := (mem_rotatingHallwayParts_dRay_iff s (t : Real.Angle) p).mp hp |>.2
+  change inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) =
+    supportValue s ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1
+  rw [show ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+    (t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) by rfl,
+    normalVector_add_pi_div_two]
+  exact h
+
+private theorem polygonNiche_bRay_length {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {t : ℝ} (ht : t ∈ Θ.directions) :
+    (Measure.hausdorffMeasure 1
+      (frontier (polygonNiche Θ K.val) ∩
+        (rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).bRay)).toReal =
+      polygonPolylineLengthAt K t := by
+  let L := normalLine (t : Real.Angle) (supportValue K.val.val (t : Real.Angle) - 1)
+  let R := (rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).bRay
+  have hRL : R ⊆ L := rotatingHallwayParts_bRay_subset_bLine K.val.val t
+  have hmeasure : Measure.hausdorffMeasure 1
+      (frontier (polygonNiche Θ K.val) ∩ L) =
+      Measure.hausdorffMeasure 1 (frontier (polygonNiche Θ K.val) ∩ R) := by
+    apply measure_eq_of_symmDiff_subset_null (E :=
+      (frontier (polygonNiche Θ K.val) ∩ L) \ R)
+    · intro p hp
+      rcases hp with hp | hp
+      · exact ⟨hp.1, fun hpR ↦ hp.2 ⟨hp.1.1, hpR⟩⟩
+      · exfalso
+        exact hp.2 ⟨hp.1.1, hRL hp.1.2⟩
+    · have := Measure.nullSingletonClass_hausdorff Point (by norm_num : (0 : ℝ) < 1)
+      exact (frontier_polygonNiche_bLine_sdiff_bRay_countable K ht).measure_zero _
+  rw [← hmeasure]
+  exact polygonNiche_bLine_length K ht
+
+private theorem polygonNiche_dRay_length {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {t : ℝ} (ht : t ∈ Θ.directions) :
+    (Measure.hausdorffMeasure 1
+      (frontier (polygonNiche Θ K.val) ∩
+        (rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).dRay)).toReal =
+      polygonPolylineLengthAt K (t + Real.pi / 2) := by
+  let L := normalLine ((t + Real.pi / 2 : ℝ) : Real.Angle)
+    (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1)
+  let R := (rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).dRay
+  have hRL : R ⊆ L := rotatingHallwayParts_dRay_subset_dLine K.val.val t
+  have hmeasure : Measure.hausdorffMeasure 1
+      (frontier (polygonNiche Θ K.val) ∩ L) =
+      Measure.hausdorffMeasure 1 (frontier (polygonNiche Θ K.val) ∩ R) := by
+    apply measure_eq_of_symmDiff_subset_null (E :=
+      (frontier (polygonNiche Θ K.val) ∩ L) \ R)
+    · intro p hp
+      rcases hp with hp | hp
+      · exact ⟨hp.1, fun hpR ↦ hp.2 ⟨hp.1.1, hpR⟩⟩
+      · exfalso
+        exact hp.2 ⟨hp.1.1, hRL hp.1.2⟩
+    · have := Measure.nullSingletonClass_hausdorff Point (by norm_num : (0 : ℝ) < 1)
+      exact (frontier_polygonNiche_dLine_sdiff_dRay_countable K ht).measure_zero _
+  rw [← hmeasure]
+  exact polygonNiche_dLine_length K ht
+
+private theorem exposedEdge_bottom_eq_segment_zero_right {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hω : Θ.angle < Real.pi / 2) :
+    exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) =
+      segment ℝ 0 (capVertices K.val 0).1.2 := by
+  let A := (capVertices K.val 0).1.2
+  have hAeq := capVertices_zero_snd_eq K
+  change A = supportValue K.val.val (0 : Real.Angle) •
+    normalVector (0 : Real.Angle) at hAeq
+  have htangent : tangentVector ((3 * Real.pi / 2 : ℝ) : Real.Angle) =
+      normalVector (0 : Real.Angle) := by
+    ext i
+    fin_cases i <;>
+      simp [tangentVector, normalVector, frame,
+        show 3 * Real.pi / 2 = Real.pi + Real.pi / 2 by ring,
+        Real.sin_add, Real.cos_add, -Real.Angle.coe_add]
+  have hAK : A ∈ (K.val.val : Set Point) := by
+    rw [hAeq]
+    exact supportValue_zero_smul_normalVector_mem K.val
+  have hAedge : A ∈ exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+    refine ⟨hAK, ?_⟩
+    change inner ℝ A (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) =
+      supportValue K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+    rw [K.val.property.2.2.2.2.2.1, hAeq, inner_normalVector_three_pi_div_two]
+    simp [normalVector, frame]
+  have h0edge : (0 : Point) ∈
+      exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+    refine ⟨zero_mem_cap_of_lt K.val hω, ?_⟩
+    change inner ℝ (0 : Point) (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) =
+      supportValue K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+    rw [K.val.property.2.2.2.2.2.1]
+    simp
+  have hfst : (edgeVertices K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle)).1 = A := by
+    apply edgeVertices_fst_eq_of_tangent_isGreatest K.val.val _ hAedge
+    intro q hq
+    have hqx := inner_le_supportValue K.val.val hq.1 (0 : Real.Angle)
+    have hA := (edgeVertices_snd_mem K.val.val (0 : Real.Angle)).2
+    change inner ℝ A (normalVector (0 : Real.Angle)) =
+      supportValue K.val.val (0 : Real.Angle) at hA
+    rw [htangent]
+    exact hqx.trans_eq hA.symm
+  have hsnd : (edgeVertices K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle)).2 = 0 := by
+    apply edgeVertices_snd_eq_of_tangent_isLeast K.val.val _ h0edge
+    intro q hq
+    have hfan := K.val.subset_capFan hq.1
+    have hqy : q 1 = 0 := by
+      have h := hq.2
+      change inner ℝ q (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) =
+        supportValue K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) at h
+      rw [K.val.property.2.2.2.2.2.1] at h
+      simpa only [inner_normalVector_three_pi_div_two, neg_eq_zero] using h
+    have hcos : 0 < Real.cos Θ.angle := Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [Θ.angle_pos, Real.pi_pos], hω⟩
+    have hqx : 0 ≤ q 0 := by
+      have h := hfan.1
+      change 0 ≤ inner ℝ q (normalVector (Θ.angle : Real.Angle)) at h
+      simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply,
+        RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue,
+        Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one, hqy, mul_zero,
+        add_zero] at h
+      exact nonneg_of_mul_nonneg_left (by simpa [mul_comm] using h) hcos
+    rw [htangent]
+    simpa [normalVector, frame, PiLp.inner_apply] using hqx
+  rw [exposedEdge_eq_segment_edgeVertices, hfst, hsnd]
+
+private theorem exposedEdge_left_eq_segment_zero_left {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hω : Θ.angle < Real.pi / 2) :
+    exposedEdge K.val.val ((Θ.angle + Real.pi : ℝ) : Real.Angle) =
+      segment ℝ 0 (capVertices K.val Θ.angle).2.1 := by
+  let C := (capVertices K.val Θ.angle).2.1
+  let L := supportValue K.val.val ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)
+  have hCeq := capVertices_angle_fst_eq K
+  change C = L • tangentVector (Θ.angle : Real.Angle) at hCeq
+  have hnormal : normalVector ((Θ.angle + Real.pi : ℝ) : Real.Angle) =
+      -normalVector (Θ.angle : Real.Angle) := normalVector_add_pi Θ.angle
+  have htangent : tangentVector ((Θ.angle + Real.pi : ℝ) : Real.Angle) =
+      -tangentVector (Θ.angle : Real.Angle) := by
+    ext i
+    fin_cases i <;>
+      simp [tangentVector, frame, Real.sin_add, Real.cos_add, -Real.Angle.coe_add]
+  have hCK : C ∈ (K.val.val : Set Point) := by
+    rw [hCeq]
+    exact supportValue_add_pi_div_two_smul_tangentVector_mem_of_lt K.val hω
+  have hCedge : C ∈ exposedEdge K.val.val
+      ((Θ.angle + Real.pi : ℝ) : Real.Angle) := by
+    refine ⟨hCK, ?_⟩
+    change inner ℝ C (normalVector ((Θ.angle + Real.pi : ℝ) : Real.Angle)) =
+      supportValue K.val.val ((Θ.angle + Real.pi : ℝ) : Real.Angle)
+    rw [K.val.property.2.2.2.2.1, hnormal, hCeq, inner_neg_right,
+      real_inner_smul_left, real_inner_comm, inner_normalVector_tangentVector]
+    simp
+  have h0edge : (0 : Point) ∈
+      exposedEdge K.val.val ((Θ.angle + Real.pi : ℝ) : Real.Angle) := by
+    refine ⟨zero_mem_cap_of_lt K.val hω, ?_⟩
+    change inner ℝ (0 : Point) (normalVector ((Θ.angle + Real.pi : ℝ) : Real.Angle)) =
+      supportValue K.val.val ((Θ.angle + Real.pi : ℝ) : Real.Angle)
+    rw [K.val.property.2.2.2.2.1]
+    simp
+  have hfst : (edgeVertices K.val.val
+      ((Θ.angle + Real.pi : ℝ) : Real.Angle)).1 = 0 := by
+    apply edgeVertices_fst_eq_of_tangent_isGreatest K.val.val _ h0edge
+    intro q hq
+    have hfan := K.val.subset_capFan hq.1
+    have hqn : inner ℝ q (normalVector (Θ.angle : Real.Angle)) = 0 := by
+      have h := hq.2
+      change inner ℝ q (normalVector ((Θ.angle + Real.pi : ℝ) : Real.Angle)) =
+        supportValue K.val.val ((Θ.angle + Real.pi : ℝ) : Real.Angle) at h
+      rw [K.val.property.2.2.2.2.1, hnormal, inner_neg_right] at h
+      linarith
+    let μ := inner ℝ q (tangentVector (Θ.angle : Real.Angle))
+    have hqeq : μ • tangentVector (Θ.angle : Real.Angle) = q := by
+      have hframe := inner_normalVector_smul_add_inner_tangentVector_smul
+        q (Θ.angle : Real.Angle)
+      rw [hqn, zero_smul, zero_add] at hframe
+      exact hframe
+    have hμ : 0 ≤ μ := by
+      have hqy := hfan.2
+      change 0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at hqy
+      have hcoord := congrArg (fun p : Point ↦ p 1) hqeq
+      change μ * Real.cos Θ.angle = q 1 at hcoord
+      have hqy' : 0 ≤ q 1 := by
+        simpa [normalVector, frame, PiLp.inner_apply] using hqy
+      rw [← hcoord] at hqy'
+      exact nonneg_of_mul_nonneg_left hqy' (Real.cos_pos_of_mem_Ioo
+        ⟨by linarith [Θ.angle_pos, Real.pi_pos], hω⟩)
+    rw [htangent, inner_neg_right]
+    simpa using neg_nonpos.mpr hμ
+  have hsnd : (edgeVertices K.val.val
+      ((Θ.angle + Real.pi : ℝ) : Real.Angle)).2 = C := by
+    apply edgeVertices_snd_eq_of_tangent_isLeast K.val.val _ hCedge
+    intro q hq
+    have hqL := inner_le_supportValue K.val.val hq.1
+      ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)
+    rw [show ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) =
+      (Θ.angle : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) by rfl,
+      normalVector_add_pi_div_two] at hqL
+    change inner ℝ q (tangentVector (Θ.angle : Real.Angle)) ≤ L at hqL
+    rw [htangent, hCeq, inner_neg_right, inner_neg_right, real_inner_smul_left,
+      inner_tangentVector_self]
+    linarith
+  rw [exposedEdge_eq_segment_edgeVertices, hfst, hsnd, segment_symm]
+
+private theorem exposedEdge_bottom_eq_segment_left_right {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hω : Θ.angle = Real.pi / 2) :
+    exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) =
+      segment ℝ (capVertices K.val Θ.angle).2.1 (capVertices K.val 0).1.2 := by
+  let A := (capVertices K.val 0).1.2
+  let C := (capVertices K.val Θ.angle).2.1
+  have hAeq := capVertices_zero_snd_eq K
+  change A = supportValue K.val.val (0 : Real.Angle) •
+    normalVector (0 : Real.Angle) at hAeq
+  have hCeq := capVertices_angle_fst_eq K
+  change C = supportValue K.val.val ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) •
+    tangentVector (Θ.angle : Real.Angle) at hCeq
+  have htangent : tangentVector ((3 * Real.pi / 2 : ℝ) : Real.Angle) =
+      normalVector (0 : Real.Angle) := by
+    ext i
+    fin_cases i <;>
+      simp [tangentVector, normalVector, frame,
+        show 3 * Real.pi / 2 = Real.pi + Real.pi / 2 by ring,
+        Real.sin_add, Real.cos_add, -Real.Angle.coe_add]
+  have hAK : A ∈ (K.val.val : Set Point) := by
+    rw [hAeq]
+    exact supportValue_zero_smul_normalVector_mem K.val
+  have hCK : C ∈ (K.val.val : Set Point) := by
+    exact (edgeVertices_fst_mem K.val.val
+      ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)).1
+  have hAedge : A ∈ exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+    refine ⟨hAK, ?_⟩
+    change inner ℝ A (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) =
+      supportValue K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+    rw [K.val.property.2.2.2.2.2.1, hAeq, inner_normalVector_three_pi_div_two]
+    simp [normalVector, frame]
+  have hCedge : C ∈ exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+    refine ⟨hCK, ?_⟩
+    change inner ℝ C (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) =
+      supportValue K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+    rw [K.val.property.2.2.2.2.2.1, hCeq, inner_normalVector_three_pi_div_two]
+    simp [hω, tangentVector, frame]
+  have hfst : (edgeVertices K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle)).1 = A := by
+    apply edgeVertices_fst_eq_of_tangent_isGreatest K.val.val _ hAedge
+    intro q hq
+    have hqx := inner_le_supportValue K.val.val hq.1 (0 : Real.Angle)
+    have hA := (edgeVertices_snd_mem K.val.val (0 : Real.Angle)).2
+    change inner ℝ A (normalVector (0 : Real.Angle)) =
+      supportValue K.val.val (0 : Real.Angle) at hA
+    rw [htangent]
+    exact hqx.trans_eq hA.symm
+  have hsnd : (edgeVertices K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle)).2 = C := by
+    apply edgeVertices_snd_eq_of_tangent_isLeast K.val.val _ hCedge
+    intro q hq
+    have hqx := inner_le_supportValue K.val.val hq.1 (Real.pi : Real.Angle)
+    have hC := (edgeVertices_fst_mem K.val.val
+      ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)).2
+    change inner ℝ C
+      (normalVector ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle)) =
+        supportValue K.val.val ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) at hC
+    have hang : ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) =
+        (Real.pi : Real.Angle) := by
+      congr 1
+      rw [hω]
+      ring
+    rw [hang] at hC
+    rw [htangent]
+    simpa [normalVector, frame, PiLp.inner_apply] using hqx.trans_eq hC.symm
+  rw [exposedEdge_eq_segment_edgeVertices, hfst, hsnd]
+
+private theorem XMonotonePolylineData.carrier_fst_mem_Icc
+    (p : XMonotonePolylineData) {q : Point} (hq : q ∈ p.carrier) :
+    q 0 ∈ Set.Icc ((p.vertices 0) 0) ((p.vertices (Fin.last p.edges)) 0) := by
+  rw [XMonotonePolylineData.carrier] at hq
+  obtain ⟨i, hqi⟩ := Set.mem_iUnion.mp hq
+  have hi := fst_mem_Icc_of_mem_segment_mc4d199c (p.increasing i.castSucc_lt_succ).le hqi
+  exact ⟨(p.increasing.monotone (Fin.zero_le i.castSucc)).trans hi.1,
+    hi.2.trans (p.increasing.monotone (Fin.le_last i.succ))⟩
+
+private theorem mem_segment_of_fst_mem_Icc_of_mem_normalLine
+    {a b q : Point} {t c : ℝ} (hab : a 0 < b 0)
+    (ht : t ∈ Set.Ioo 0 Real.pi)
+    (ha : a ∈ normalLine (t : Real.Angle) c)
+    (hb : b ∈ normalLine (t : Real.Angle) c)
+    (hq : q ∈ normalLine (t : Real.Angle) c)
+    (hx : q 0 ∈ Set.Icc (a 0) (b 0)) : q ∈ segment ℝ a b := by
+  let r := (q 0 - a 0) / (b 0 - a 0)
+  have hden : 0 < b 0 - a 0 := sub_pos.mpr hab
+  have hr : r ∈ Set.Icc (0 : ℝ) 1 := by
+    constructor
+    · exact div_nonneg (sub_nonneg.mpr hx.1) hden.le
+    · exact (div_le_one hden).2 (by linarith [hx.2])
+  rw [segment_eq_image']
+  refine ⟨r, hr, ?_⟩
+  change a + r • (b - a) = q
+  have hxcoord : (a + r • (b - a)) 0 = q 0 := by
+    dsimp [r]
+    field_simp
+    ring
+  have hline : a + r • (b - a) ∈ normalLine (t : Real.Angle) c := by
+    change inner ℝ (a + r • (b - a)) (normalVector (t : Real.Angle)) = c
+    change inner ℝ a (normalVector (t : Real.Angle)) = c at ha
+    change inner ℝ b (normalVector (t : Real.Angle)) = c at hb
+    rw [inner_add_left, real_inner_smul_left, inner_sub_left, ha, hb, sub_self,
+      mul_zero, add_zero]
+  ext i
+  fin_cases i
+  · exact hxcoord
+  · change (a + r • (b - a)) 1 = q 1
+    change inner ℝ (a + r • (b - a)) (normalVector (t : Real.Angle)) = c at hline
+    change inner ℝ q (normalVector (t : Real.Angle)) = c at hq
+    simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two] at hline hq
+    simp only [PiLp.add_apply, PiLp.smul_apply, PiLp.sub_apply, smul_eq_mul] at hxcoord ⊢
+    apply mul_left_cancel₀ (Real.sin_pos_of_pos_of_lt_pi ht.1 ht.2).ne'
+    linear_combination hline - hq - Real.cos t * hxcoord
+
+private theorem eq_of_fst_eq_of_mem_normalLine {p q : Point} {t c : ℝ}
+    (ht : t ∈ Set.Ioo 0 Real.pi)
+    (hp : p ∈ normalLine (t : Real.Angle) c)
+    (hq : q ∈ normalLine (t : Real.Angle) c) (hx : p 0 = q 0) : p = q := by
+  ext i
+  fin_cases i
+  · exact hx
+  · change p 1 = q 1
+    change inner ℝ p (normalVector (t : Real.Angle)) = c at hp
+    change inner ℝ q (normalVector (t : Real.Angle)) = c at hq
+    simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two] at hp hq
+    apply mul_left_cancel₀ (Real.sin_pos_of_pos_of_lt_pi ht.1 ht.2).ne'
+    linear_combination hp - hq - Real.cos t * hx
+
+private theorem polygonCapPolyline_carrier_inter_bottom_subset_exposedEdge_of_lt
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) (hω : Θ.angle < Real.pi / 2) :
+    (polygonCapPolyline K).carrier ∩
+        normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0 ⊆
+      exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+  let p := polygonCapPolyline K
+  let A := (capVertices K.val 0).1.2
+  let C := (capVertices K.val Θ.angle).2.1
+  have hp : IsCapPolyline K p := polygonCapPolyline_spec K
+  have hAeq := capVertices_zero_snd_eq K
+  change A = supportValue K.val.val (0 : Real.Angle) •
+    normalVector (0 : Real.Angle) at hAeq
+  have hCeq := capVertices_angle_fst_eq K
+  change C = supportValue K.val.val ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) •
+    tangentVector (Θ.angle : Real.Angle) at hCeq
+  have hL : 0 ≤ supportValue K.val.val
+      ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) :=
+    supportValue_nonneg_of_mem_capUpperAngles K.val hω
+      (Or.inr ⟨by linarith [Θ.angle_pos, Real.pi_pos], le_rfl⟩)
+  have hC0 : C 0 ≤ 0 := by
+    rw [hCeq]
+    simp [tangentVector, frame]
+    simpa only [Real.Angle.coe_add] using mul_nonneg hL
+      (Real.sin_nonneg_of_nonneg_of_le_pi Θ.angle_pos.le
+        (Θ.angle_le.trans (by linarith [Real.pi_pos])))
+  have hA0support : 0 < supportValue K.val.val (0 : Real.Angle) := by
+    obtain ⟨u, huK, hu⟩ := exists_mem_inner_eq_supportValue K.val.val
+      (Θ.angle : Real.Angle)
+    have huy : u 1 ≤ 1 := by
+      have huy' := inner_le_supportValue K.val.val huK
+        ((Real.pi / 2 : ℝ) : Real.Angle)
+      rw [K.val.property.2.2.2.1] at huy'
+      simpa [normalVector, frame, PiLp.inner_apply] using huy'
+    rw [K.val.property.2.2.1] at hu
+    have hcos : 0 < Real.cos Θ.angle := Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [Θ.angle_pos, Real.pi_pos], hω⟩
+    have hsin_lt : Real.sin Θ.angle < 1 := by
+      nlinarith only [Real.sin_sq_add_cos_sq Θ.angle, sq_pos_of_pos hcos]
+    have hux : 0 < u 0 := by
+      simp [normalVector, frame, PiLp.inner_apply] at hu
+      have hsin : 0 ≤ Real.sin Θ.angle :=
+        (Real.sin_pos_of_pos_of_lt_pi Θ.angle_pos
+          (Θ.angle_le.trans_lt (by linarith [Real.pi_pos]))).le
+      have hmul := mul_le_mul_of_nonneg_left huy hsin
+      nlinarith
+    exact hux.trans_le (by
+      have := inner_le_supportValue K.val.val huK (0 : Real.Angle)
+      simpa [normalVector, frame, PiLp.inner_apply] using this)
+  have hA0 : 0 < A 0 := by
+    rw [hAeq]
+    simpa [normalVector, frame] using hA0support
+  intro q hq
+  rw [exposedEdge_bottom_eq_segment_zero_right K hω]
+  apply mem_segment_of_fst_mem_Icc_of_mem_normalLine hA0
+    (t := Real.pi / 2) (c := 0)
+    ⟨by positivity, by linarith [Real.pi_pos]⟩
+  · simp [normalLine, normalVector, frame, PiLp.inner_apply]
+  · change inner ℝ (capVertices K.val 0).1.2
+      (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0
+    rw [capVertices_zero_snd_eq K]
+    simp [normalVector, frame, PiLp.inner_apply]
+  · exact hq.2
+  · have hxbounds := p.carrier_fst_mem_Icc hq.1
+    rw [hp.2.1, hp.2.2.1] at hxbounds
+    refine ⟨?_, hxbounds.2⟩
+    have hfront : q ∈ frontier (capFan Θ.angle \ polygonNiche Θ K.val) := by
+      rw [hp.2.2.2.2.2.1]
+      exact Or.inl (Or.inr hq.1)
+    have hclosed : IsClosed (capFan Θ.angle \ polygonNiche Θ K.val) := hp.2.2.2.2.1
+    have hfan : q ∈ capFan Θ.angle :=
+      (hclosed.closure_eq ▸ frontier_subset_closure hfront).1
+    have hqy : q 1 = 0 := by
+      have h := hq.2
+      change inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 at h
+      simpa [normalVector, frame, PiLp.inner_apply] using h
+    have hcos : 0 < Real.cos Θ.angle := Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [Θ.angle_pos, Real.pi_pos], hω⟩
+    have h := hfan.1
+    change 0 ≤ inner ℝ q (normalVector (Θ.angle : Real.Angle)) at h
+    simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply,
+      RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue,
+      Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one, hqy, mul_zero,
+      add_zero] at h
+    exact nonneg_of_mul_nonneg_left (by simpa [mul_comm] using h) hcos
+
+private theorem polygonCapPolyline_carrier_inter_left_subset_exposedEdge_of_lt
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) (hω : Θ.angle < Real.pi / 2) :
+    (polygonCapPolyline K).carrier ∩ normalLine (Θ.angle : Real.Angle) 0 ⊆
+      exposedEdge K.val.val ((Θ.angle + Real.pi : ℝ) : Real.Angle) := by
+  let p := polygonCapPolyline K
+  let C := (capVertices K.val Θ.angle).2.1
+  have hp : IsCapPolyline K p := polygonCapPolyline_spec K
+  have hCeq := capVertices_angle_fst_eq K
+  change C = supportValue K.val.val ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) •
+    tangentVector (Θ.angle : Real.Angle) at hCeq
+  have hL : 0 ≤ supportValue K.val.val
+      ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) :=
+    supportValue_nonneg_of_mem_capUpperAngles K.val hω
+      (Or.inr ⟨by linarith [Θ.angle_pos, Real.pi_pos], le_rfl⟩)
+  have hC0 : C 0 ≤ 0 := by
+    rw [hCeq]
+    simp [tangentVector, frame]
+    simpa only [Real.Angle.coe_add] using mul_nonneg hL
+      (Real.sin_nonneg_of_nonneg_of_le_pi Θ.angle_pos.le
+        (Θ.angle_le.trans (by linarith [Real.pi_pos])))
+  intro q hq
+  rw [exposedEdge_left_eq_segment_zero_left K hω, segment_symm]
+  have hfront : q ∈ frontier (capFan Θ.angle \ polygonNiche Θ K.val) := by
+    rw [hp.2.2.2.2.2.1]
+    exact Or.inl (Or.inr hq.1)
+  have hclosed : IsClosed (capFan Θ.angle \ polygonNiche Θ K.val) := hp.2.2.2.2.1
+  have hfan : q ∈ capFan Θ.angle :=
+    (hclosed.closure_eq ▸ frontier_subset_closure hfront).1
+  have hxbounds := p.carrier_fst_mem_Icc hq.1
+  rw [hp.2.1, hp.2.2.1] at hxbounds
+  have hq0 : q 0 ≤ 0 := by
+    have hline := hq.2
+    change inner ℝ q (normalVector (Θ.angle : Real.Angle)) = 0 at hline
+    have hqy := hfan.2
+    change 0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at hqy
+    have hqy' : 0 ≤ q 1 := by
+      simpa [normalVector, frame, PiLp.inner_apply] using hqy
+    simp [normalVector, frame, PiLp.inner_apply] at hline
+    have hcos : 0 < Real.cos Θ.angle := Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [Θ.angle_pos, Real.pi_pos], hω⟩
+    have hsin : 0 < Real.sin Θ.angle := Real.sin_pos_of_pos_of_lt_pi Θ.angle_pos
+      (by linarith [Θ.angle_le, Real.pi_pos])
+    nlinarith
+  have hCline : C ∈ normalLine (Θ.angle : Real.Angle) 0 := by
+    change inner ℝ C (normalVector (Θ.angle : Real.Angle)) = 0
+    rw [hCeq, real_inner_smul_left, real_inner_comm, inner_normalVector_tangentVector]
+    simp
+  have h0line : (0 : Point) ∈ normalLine (Θ.angle : Real.Angle) 0 := by
+    simp [normalLine]
+  by_cases hCstrict : C 0 < 0
+  · exact mem_segment_of_fst_mem_Icc_of_mem_normalLine hCstrict
+      ⟨Θ.angle_pos, Θ.angle_le.trans_lt (by linarith [Real.pi_pos])⟩
+      hCline h0line hq.2 ⟨hxbounds.1, hq0⟩
+  · have hCzero : C 0 = 0 := le_antisymm hC0 (le_of_not_gt hCstrict)
+    have hqzero : q 0 = C 0 := by linarith [hxbounds.1, hq0]
+    have hqC := eq_of_fst_eq_of_mem_normalLine
+      ⟨Θ.angle_pos, Θ.angle_le.trans_lt (by linarith [Real.pi_pos])⟩
+      hq.2 hCline hqzero
+    rw [hqC]
+    exact left_mem_segment ℝ C 0
+
+private theorem polygonCapPolyline_carrier_inter_bottom_subset_exposedEdge_of_eq
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) (hω : Θ.angle = Real.pi / 2) :
+    (polygonCapPolyline K).carrier ∩
+        normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0 ⊆
+      exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+  let p := polygonCapPolyline K
+  let A := (capVertices K.val 0).1.2
+  let C := (capVertices K.val Θ.angle).2.1
+  have hp : IsCapPolyline K p := polygonCapPolyline_spec K
+  have hAeq := capVertices_zero_snd_eq K
+  change A = supportValue K.val.val (0 : Real.Angle) •
+    normalVector (0 : Real.Angle) at hAeq
+  have hCeq := capVertices_angle_fst_eq K
+  change C = supportValue K.val.val ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) •
+    tangentVector (Θ.angle : Real.Angle) at hCeq
+  intro q hq
+  rw [exposedEdge_bottom_eq_segment_left_right K hω]
+  apply mem_segment_of_fst_mem_Icc_of_mem_normalLine (polygonCap_left_x_lt_right_x K)
+    (t := Real.pi / 2) (c := 0)
+    ⟨by positivity, by linarith [Real.pi_pos]⟩
+  · change inner ℝ C (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0
+    rw [hCeq]
+    simp [hω, tangentVector, normalVector, frame, PiLp.inner_apply]
+  · change inner ℝ A (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0
+    rw [hAeq]
+    simp [normalVector, frame, PiLp.inner_apply]
+  · exact hq.2
+  · have hxbounds := p.carrier_fst_mem_Icc hq.1
+    simpa [hp.2.1, hp.2.2.1] using hxbounds
+
+private theorem IsCapPolyline.edge_subset_fanLine_of_orthogonal
+    {Θ : AngleSet} {K : PolygonCapSpace Θ} {p : XMonotonePolylineData}
+    (hp : IsCapPolyline K p) {t : ℝ} (ht : t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ))
+    (i : Fin p.edges)
+    (hi : inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+      (normalVector (t : Real.Angle)) = 0) :
+    segment ℝ (p.vertices i.castSucc) (p.vertices i.succ) ⊆
+      normalLine (t : Real.Angle) 0 := by
+  classical
+  obtain ⟨l, hl, hline⟩ := hp.edge_subset_boundaryLine i
+  have hleft := hline (left_mem_segment ℝ _ _)
+  have hright := hline (right_mem_segment ℝ _ _)
+  have hlorth : inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+      (normalVector l.1) = 0 := by
+    change inner ℝ (p.vertices i.castSucc) (normalVector l.1) = l.2 at hleft
+    change inner ℝ (p.vertices i.succ) (normalVector l.1) = l.2 at hright
+    rw [inner_sub_left, hright, hleft, sub_self]
+  have hωI : Θ.angle ∈ Set.Ioo 0 Real.pi :=
+    ⟨Θ.angle_pos, Θ.angle_le.trans_lt (by linarith [Real.pi_pos])⟩
+  have hTI : Real.pi / 2 ∈ Set.Ioo 0 Real.pi :=
+    ⟨by positivity, by linarith [Real.pi_pos]⟩
+  rcases Set.mem_insert_iff.mp ht with rfl | ht
+  · simp only [polygonCapBoundaryLines, Finset.mem_union] at hl
+    rcases hl with hl | hl
+    · simp only [Finset.mem_insert, Finset.mem_singleton] at hl
+      rcases hl with rfl | rfl
+      · exact hline
+      · have heq := real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi
+          (edge_direction_ne_zero p i) hTI hωI hlorth hi
+        simpa [heq] using hline
+    · obtain ⟨s, hs, hsl⟩ := Finset.mem_biUnion.mp hl
+      simp only [Finset.mem_insert, Finset.mem_singleton] at hsl
+      rcases hsl with rfl | rfl
+      · have hsI : s ∈ Set.Ioo 0 Real.pi :=
+          ⟨(Θ.interior s hs).1,
+            ((Θ.interior s hs).2.trans_le Θ.angle_le).trans
+              (by linarith [Real.pi_pos])⟩
+        have heq := real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi
+          (edge_direction_ne_zero p i) hsI hωI hlorth hi
+        linarith [(Θ.interior s hs).2]
+      · have hsI : s + Real.pi / 2 ∈ Set.Ioo 0 Real.pi := by
+          constructor
+          · linarith [(Θ.interior s hs).1, Real.pi_pos]
+          · linarith [(Θ.interior s hs).2, Θ.angle_le]
+        have heq := real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi
+          (edge_direction_ne_zero p i) hsI hωI hlorth hi
+        linarith [(Θ.interior s hs).1, Θ.angle_le, Real.pi_pos]
+  · have ht : t = Real.pi / 2 := Set.mem_singleton_iff.mp ht
+    subst t
+    simp only [polygonCapBoundaryLines, Finset.mem_union] at hl
+    rcases hl with hl | hl
+    · simp only [Finset.mem_insert, Finset.mem_singleton] at hl
+      rcases hl with rfl | rfl
+      · have heq := real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi
+          (edge_direction_ne_zero p i) hωI hTI hlorth hi
+        simpa [heq] using hline
+      · exact hline
+    · obtain ⟨s, hs, hsl⟩ := Finset.mem_biUnion.mp hl
+      simp only [Finset.mem_insert, Finset.mem_singleton] at hsl
+      rcases hsl with rfl | rfl
+      · have hsI : s ∈ Set.Ioo 0 Real.pi :=
+          ⟨(Θ.interior s hs).1,
+            ((Θ.interior s hs).2.trans_le Θ.angle_le).trans
+              (by linarith [Real.pi_pos])⟩
+        have heq := real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi
+          (edge_direction_ne_zero p i) hsI hTI hlorth hi
+        linarith [(Θ.interior s hs).2, Θ.angle_le]
+      · have hsI : s + Real.pi / 2 ∈ Set.Ioo 0 Real.pi := by
+          constructor
+          · linarith [(Θ.interior s hs).1, Real.pi_pos]
+          · linarith [(Θ.interior s hs).2, Θ.angle_le]
+        have heq := real_eq_of_normal_orthogonal_of_mem_Ioo_zero_pi
+          (edge_direction_ne_zero p i) hsI hTI hlorth hi
+        linarith [(Θ.interior s hs).1]
+
+private theorem polygonCapPolyline_fanLine_length {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {t : ℝ} (ht : t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    (Measure.hausdorffMeasure 1
+      ((polygonCapPolyline K).carrier ∩ normalLine (t : Real.Angle) 0)).toReal =
+      polygonPolylineLengthAt K t := by
+  let p := polygonCapPolyline K
+  have hp : IsCapPolyline K p := polygonCapPolyline_spec K
+  have hmeasure := p.toReal_hausdorffMeasure_carrier_inter_hyperplane
+    (normalVector (t : Real.Angle)) 0
+    (fun i hi ↦ hp.edge_subset_fanLine_of_orthogonal ht i hi)
+  have htDomain : t ∈ angleDomain Θ := Or.inr ht
+  rw [show normalLine (t : Real.Angle) 0 =
+      {q | inner ℝ q (normalVector (t : Real.Angle)) = 0} by rfl,
+    hmeasure]
+  simp [polygonPolylineLengthAt, htDomain, polygonCapPolylineLength, p]
+
+private theorem polygonNiche_inter_bottom_subset_exposedEdge_of_lt
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) (hω : Θ.angle < Real.pi / 2) :
+    polygonNiche Θ K.val ∩ normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0 ⊆
+      exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+  intro q hq
+  rw [exposedEdge_bottom_eq_segment_zero_right K hω]
+  rcases Set.mem_iUnion.mp hq.1.2 with ⟨t, ht⟩
+  rcases Set.mem_iUnion.mp ht with ⟨htΘ, hqt⟩
+  have htt := Θ.interior t htΘ
+  have hcost : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [htt.1, Real.pi_pos], htt.2.trans_le Θ.angle_le⟩
+  have hgapBounds := wedgeGaps_positive_lower_bound K.val t htt
+  have hgap : 0 < (wedgeGaps K.val t).1 := hgapBounds.2.1.trans_le hgapBounds.1
+  have hqy : q 1 = 0 := by
+    have hline := hq.2
+    change inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 at hline
+    simpa [normalVector, frame, PiLp.inner_apply] using hline
+  have hqx_endpoint : q 0 < (wedgeEndpoints K.val t).1 0 := by
+    have hb := hqt.1
+    change inner ℝ q (normalVector (t : Real.Angle)) <
+      supportValue K.val.val (t : Real.Angle) - 1 at hb
+    simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply,
+      RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue,
+      Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one, hqy, mul_zero,
+      add_zero] at hb
+    simp only [wedgeEndpoints]
+    simp only [Fin.isValue, normalVector, frame, Real.Angle.cos_zero, Real.Angle.sin_zero, neg_zero,
+      PiLp.smul_apply, Matrix.cons_val_zero, smul_eq_mul, mul_one, gt_iff_lt]
+    exact (lt_div_iff₀ hcost).2 (by simpa [mul_comm] using hb)
+  have hendpoint_A : (wedgeEndpoints K.val t).1 0 < (capVertices K.val 0).1.2 0 := by
+    simpa [wedgeGaps, inner_sub_left, normalVector, frame, PiLp.inner_apply] using hgap
+  have hqx_nonneg : 0 ≤ q 0 := by
+    have hfan := hq.1.1.1
+    change 0 ≤ inner ℝ q (normalVector (Θ.angle : Real.Angle)) at hfan
+    simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe,
+      PiLp.inner_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue,
+      Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one, hqy, mul_zero,
+      add_zero] at hfan
+    have hcosω : 0 < Real.cos Θ.angle := Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [Θ.angle_pos, Real.pi_pos], hω⟩
+    exact nonneg_of_mul_nonneg_left (by simpa [mul_comm] using hfan) hcosω
+  have hApos : 0 < (capVertices K.val 0).1.2 0 :=
+    hqx_nonneg.trans_lt (hqx_endpoint.trans hendpoint_A)
+  apply mem_segment_of_fst_mem_Icc_of_mem_normalLine hApos
+    (t := Real.pi / 2) (c := 0)
+    ⟨by positivity, by linarith [Real.pi_pos]⟩
+  · simp [normalLine, normalVector, frame, PiLp.inner_apply]
+  · rw [capVertices_zero_snd_eq K]
+    simp [normalLine, normalVector, frame, PiLp.inner_apply]
+  · exact hq.2
+  · exact ⟨hqx_nonneg, (hqx_endpoint.trans hendpoint_A).le⟩
+
+private theorem mem_segment_of_inner_tangent_mem_Icc_of_mem_normalLine
+    {a b q : Point} {t : Real.Angle} {c : ℝ}
+    (hab : inner ℝ a (tangentVector t) < inner ℝ b (tangentVector t))
+    (ha : a ∈ normalLine t c) (hb : b ∈ normalLine t c)
+    (hq : q ∈ normalLine t c)
+    (hx : inner ℝ q (tangentVector t) ∈
+      Set.Icc (inner ℝ a (tangentVector t)) (inner ℝ b (tangentVector t))) :
+    q ∈ segment ℝ a b := by
+  let r := (inner ℝ q (tangentVector t) - inner ℝ a (tangentVector t)) /
+    (inner ℝ b (tangentVector t) - inner ℝ a (tangentVector t))
+  have hden : 0 < inner ℝ b (tangentVector t) - inner ℝ a (tangentVector t) :=
+    sub_pos.mpr hab
+  have hr : r ∈ Set.Icc (0 : ℝ) 1 := by
+    constructor
+    · exact div_nonneg (sub_nonneg.mpr hx.1) hden.le
+    · exact (div_le_one hden).2 (by linarith [hx.2])
+  rw [segment_eq_image']
+  refine ⟨r, hr, ?_⟩
+  change a + r • (b - a) = q
+  have hn : inner ℝ (a + r • (b - a)) (normalVector t) =
+      inner ℝ q (normalVector t)
+      := by
+    change inner ℝ a (normalVector t) = c at ha
+    change inner ℝ b (normalVector t) = c at hb
+    change inner ℝ q (normalVector t) = c at hq
+    rw [inner_add_left, real_inner_smul_left, inner_sub_left, ha, hb, sub_self,
+      mul_zero, add_zero, hq]
+  have htangent : inner ℝ (a + r • (b - a)) (tangentVector t) =
+      inner ℝ q (tangentVector t)
+      := by
+    rw [inner_add_left, real_inner_smul_left, inner_sub_left]
+    dsimp [r]
+    field_simp
+    ring
+  rw [← inner_normalVector_smul_add_inner_tangentVector_smul
+      (a + r • (b - a)) t,
+    ← inner_normalVector_smul_add_inner_tangentVector_smul q t, hn, htangent]
+
+private theorem polygonNiche_inter_left_subset_exposedEdge_of_lt
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) (hω : Θ.angle < Real.pi / 2) :
+    polygonNiche Θ K.val ∩ normalLine (Θ.angle : Real.Angle) 0 ⊆
+      exposedEdge K.val.val ((Θ.angle + Real.pi : ℝ) : Real.Angle) := by
+  intro q hq
+  rw [exposedEdge_left_eq_segment_zero_left K hω]
+  rcases Set.mem_iUnion.mp hq.1.2 with ⟨t, ht⟩
+  rcases Set.mem_iUnion.mp ht with ⟨htΘ, hqt⟩
+  have htt := Θ.interior t htΘ
+  have hcosδ : 0 < Real.cos (Θ.angle - t) := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [htt.2, Real.pi_pos], by linarith [htt.1, Θ.angle_le]⟩
+  have hgapBounds := wedgeGaps_positive_lower_bound K.val t htt
+  have hgap : 0 < (wedgeGaps K.val t).2 :=
+    hgapBounds.2.2.2.trans_le hgapBounds.2.2.1
+  have hqcoord_endpoint : inner ℝ q (tangentVector (Θ.angle : Real.Angle)) <
+      inner ℝ (wedgeEndpoints K.val t).2
+        (tangentVector (Θ.angle : Real.Angle)) := by
+    have hd := hqt.2
+    change inner ℝ q (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) <
+      supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 at hd
+    have hqdecomp := inner_normalVector_smul_add_inner_tangentVector_smul
+      q (Θ.angle : Real.Angle)
+    have hqnormal := hq.2
+    change inner ℝ q (normalVector (Θ.angle : Real.Angle)) = 0 at hqnormal
+    rw [hqnormal, zero_smul, zero_add] at hqdecomp
+    rw [← hqdecomp, real_inner_smul_left] at hd
+    have hinner : inner ℝ (tangentVector (Θ.angle : Real.Angle))
+        (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) =
+        Real.cos (Θ.angle - t) := by
+      rw [Real.Angle.coe_add, normalVector_add_pi_div_two]
+      simp [tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two,
+        Real.cos_sub]
+      ring_nf
+    rw [hinner] at hd
+    have hendpoint : inner ℝ (wedgeEndpoints K.val t).2
+        (tangentVector (Θ.angle : Real.Angle)) =
+        (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+          Real.cos (Θ.angle - t) := by
+      simp only [wedgeEndpoints, real_inner_smul_left]
+      rw [show inner ℝ (tangentVector (Θ.angle : Real.Angle))
+        (tangentVector (Θ.angle : Real.Angle)) = 1 by
+          exact inner_tangentVector_self Θ.angle, mul_one]
+    rw [hendpoint]
+    exact (lt_div_iff₀ hcosδ).2 (by linarith [hd])
+  have hendpoint_C : inner ℝ (wedgeEndpoints K.val t).2
+      (tangentVector (Θ.angle : Real.Angle)) <
+      inner ℝ (capVertices K.val Θ.angle).2.1
+        (tangentVector (Θ.angle : Real.Angle)) := by
+    simpa [wedgeGaps, inner_sub_left] using hgap
+  have hqcoord_nonneg : 0 ≤ inner ℝ q (tangentVector (Θ.angle : Real.Angle)) := by
+    have hfan := hq.1.1.2
+    change 0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at hfan
+    have hqdecomp := inner_normalVector_smul_add_inner_tangentVector_smul
+      q (Θ.angle : Real.Angle)
+    have hqnormal := hq.2
+    change inner ℝ q (normalVector (Θ.angle : Real.Angle)) = 0 at hqnormal
+    rw [hqnormal, zero_smul, zero_add] at hqdecomp
+    rw [← hqdecomp, real_inner_smul_left] at hfan
+    have hcosω : 0 < Real.cos Θ.angle := Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [Θ.angle_pos, Real.pi_pos], hω⟩
+    have hinner : inner ℝ (tangentVector (Θ.angle : Real.Angle))
+        (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = Real.cos Θ.angle := by
+      simp [tangentVector, normalVector, frame, PiLp.inner_apply,
+        Fin.sum_univ_two]
+    rw [hinner] at hfan
+    exact nonneg_of_mul_nonneg_left hfan hcosω
+  have hCpos : 0 < inner ℝ (capVertices K.val Θ.angle).2.1
+      (tangentVector (Θ.angle : Real.Angle)) :=
+    hqcoord_nonneg.trans_lt (hqcoord_endpoint.trans hendpoint_C)
+  apply mem_segment_of_inner_tangent_mem_Icc_of_mem_normalLine
+    (t := (Θ.angle : Real.Angle)) (c := 0) (by simpa using hCpos)
+  · simp [normalLine]
+  · have hCeq := capVertices_angle_fst_eq K
+    change inner ℝ (capVertices K.val Θ.angle).2.1
+      (normalVector (Θ.angle : Real.Angle)) = 0
+    rw [hCeq, real_inner_smul_left, real_inner_comm,
+      inner_normalVector_tangentVector, mul_zero]
+  · exact hq.2
+  · exact ⟨by simpa using hqcoord_nonneg, (hqcoord_endpoint.trans hendpoint_C).le⟩
+
+private theorem polygonNiche_inter_bottom_subset_exposedEdge_of_eq
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) (hω : Θ.angle = Real.pi / 2) :
+    polygonNiche Θ K.val ∩ normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0 ⊆
+      exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+  intro q hq
+  rw [exposedEdge_bottom_eq_segment_left_right K hω]
+  rcases Set.mem_iUnion.mp hq.1.2 with ⟨t, ht⟩
+  rcases Set.mem_iUnion.mp ht with ⟨htΘ, hqt⟩
+  have htt := Θ.interior t htΘ
+  have hcost : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [htt.1, Real.pi_pos], htt.2.trans_le Θ.angle_le⟩
+  have hcosδ : 0 < Real.cos (Θ.angle - t) := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [htt.2, Real.pi_pos], by linarith [htt.1, Θ.angle_le]⟩
+  have hgapBounds := wedgeGaps_positive_lower_bound K.val t htt
+  have hgapA : 0 < (wedgeGaps K.val t).1 := hgapBounds.2.1.trans_le hgapBounds.1
+  have hgapC : 0 < (wedgeGaps K.val t).2 :=
+    hgapBounds.2.2.2.trans_le hgapBounds.2.2.1
+  have hqy : q 1 = 0 := by
+    have hline := hq.2
+    change inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 at hline
+    simpa [normalVector, frame, PiLp.inner_apply] using hline
+  have hqx_endpoint : q 0 < (wedgeEndpoints K.val t).1 0 := by
+    have hb := hqt.1
+    change inner ℝ q (normalVector (t : Real.Angle)) <
+      supportValue K.val.val (t : Real.Angle) - 1 at hb
+    simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply,
+      RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue,
+      Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one, hqy, mul_zero,
+      add_zero] at hb
+    simp only [wedgeEndpoints]
+    simp only [Fin.isValue, normalVector, frame, Real.Angle.cos_zero, Real.Angle.sin_zero, neg_zero,
+      PiLp.smul_apply, Matrix.cons_val_zero, smul_eq_mul, mul_one, gt_iff_lt]
+    exact (lt_div_iff₀ hcost).2 (by simpa [mul_comm] using hb)
+  have hendpoint_A : (wedgeEndpoints K.val t).1 0 < (capVertices K.val 0).1.2 0 := by
+    simpa [wedgeGaps, inner_sub_left, normalVector, frame, PiLp.inner_apply] using hgapA
+  have hqcoord_endpoint : inner ℝ q
+      (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) <
+      inner ℝ (wedgeEndpoints K.val t).2
+        (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) := by
+    have hd := hqt.2
+    change inner ℝ q (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) <
+      supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 at hd
+    have hqdecomp := inner_normalVector_smul_add_inner_tangentVector_smul
+      q ((Real.pi / 2 : ℝ) : Real.Angle)
+    have hqnormal := hq.2
+    change inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 at hqnormal
+    rw [hqnormal, zero_smul, zero_add] at hqdecomp
+    rw [← hqdecomp, real_inner_smul_left] at hd
+    have hinner : inner ℝ (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle))
+        (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) =
+        Real.cos (Real.pi / 2 - t) := by
+      rw [Real.Angle.coe_add, normalVector_add_pi_div_two]
+      simp [tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two,
+        Real.cos_sub]
+    rw [hinner] at hd
+    have hcosEq : Real.cos (Θ.angle - t) = Real.cos (Real.pi / 2 - t) := by
+      rw [hω]
+    have hendpoint : inner ℝ (wedgeEndpoints K.val t).2
+        (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) =
+        (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+          Real.cos (Real.pi / 2 - t) := by
+      calc
+        _ = (supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+            Real.cos (Θ.angle - t) := by
+          rw [show tangentVector ((Real.pi / 2 : ℝ) : Real.Angle) =
+              tangentVector (Θ.angle : Real.Angle) by rw [hω]]
+          simp only [wedgeEndpoints, real_inner_smul_left]
+          rw [show inner ℝ (tangentVector (Θ.angle : Real.Angle))
+            (tangentVector (Θ.angle : Real.Angle)) = 1 by
+              exact inner_tangentVector_self Θ.angle, mul_one]
+        _ = _ := by rw [hcosEq]
+    rw [hendpoint]
+    have hcosT : 0 < Real.cos (Real.pi / 2 - t) := by simpa [hω] using hcosδ
+    exact (lt_div_iff₀ hcosT).2 (by linarith [hd])
+  have hendpoint_C : inner ℝ (wedgeEndpoints K.val t).2
+      (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) <
+      inner ℝ (capVertices K.val Θ.angle).2.1
+        (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) := by
+    rw [← hω]
+    simpa [wedgeGaps, inner_sub_left] using hgapC
+  have hCx_lt_qx : (capVertices K.val Θ.angle).2.1 0 < q 0 := by
+    have h := hqcoord_endpoint.trans hendpoint_C
+    simp [tangentVector, frame, PiLp.inner_apply, hω] at h
+    simpa [hω] using h
+  have hqx_lt_Ax : q 0 < (capVertices K.val 0).1.2 0 :=
+    hqx_endpoint.trans hendpoint_A
+  apply mem_segment_of_fst_mem_Icc_of_mem_normalLine (polygonCap_left_x_lt_right_x K)
+    (t := Real.pi / 2) (c := 0)
+    ⟨by positivity, by linarith [Real.pi_pos]⟩
+  · rw [capVertices_angle_fst_eq K]
+    simp [normalLine, normalVector, tangentVector, frame, PiLp.inner_apply, hω]
+  · rw [capVertices_zero_snd_eq K]
+    simp [normalLine, normalVector, frame, PiLp.inner_apply]
+  · exact hq.2
+  · exact ⟨hCx_lt_qx.le, hqx_lt_Ax.le⟩
+
+private theorem mem_frontier_normalHalfPlane_upper_closed_of_mem_normalLine
+    (t : ℝ) (c : ℝ) {q : Point} (hq : q ∈ normalLine (t : Real.Angle) c) :
+    q ∈ frontier (normalHalfPlane (t : Real.Angle) c true false) := by
+  have hqH : q ∈ normalHalfPlane (t : Real.Angle) c true false := by
+    change c ≤ inner ℝ q (normalVector (t : Real.Angle))
+    exact hq.ge
+  apply (mem_frontier_iff_notMem_interior hqH).mpr
+  intro hqInt
+  obtain ⟨ε, hε, hball⟩ := Metric.isOpen_iff.mp isOpen_interior q hqInt
+  let e := ε / 2
+  let z := q - e • normalVector (t : Real.Angle)
+  have he : 0 < e := half_pos hε
+  have hzball : z ∈ Metric.ball q ε := by
+    change dist z q < ε
+    simp [z, norm_smul, norm_normalVector_real, abs_of_pos he]
+    dsimp [e]
+    linarith
+  have hz := interior_subset (hball hzball)
+  change c ≤ inner ℝ z (normalVector (t : Real.Angle)) at hz
+  change inner ℝ q (normalVector (t : Real.Angle)) = c at hq
+  dsimp [z] at hz
+  rw [inner_sub_left, real_inner_smul_left, inner_normalVector_self, mul_one, hq] at hz
+  linarith
+
+private theorem capFan_inter_fanLine_subset_frontier {Θ : AngleSet} {t : ℝ}
+    (ht : t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    capFan Θ.angle ∩ normalLine (t : Real.Angle) 0 ⊆ frontier (capFan Θ.angle) := by
+  intro q hq
+  apply (mem_frontier_iff_notMem_interior hq.1).mpr
+  intro hqInt
+  rcases Set.mem_insert_iff.mp ht with rfl | ht
+  · have hInt := interior_mono Set.inter_subset_left hqInt
+    have hfront := mem_frontier_normalHalfPlane_upper_closed_of_mem_normalLine
+      Θ.angle 0 hq.2
+    exact (mem_frontier_iff_notMem_interior hq.1.1).mp hfront hInt
+  · have ht : t = Real.pi / 2 := Set.mem_singleton_iff.mp ht
+    subst t
+    have hInt := interior_mono Set.inter_subset_right hqInt
+    have hfront := mem_frontier_normalHalfPlane_upper_closed_of_mem_normalLine
+      (Real.pi / 2) 0 hq.2
+    exact (mem_frontier_iff_notMem_interior hq.1.2).mp hfront hInt
+
+private theorem frontier_innerQuadrant_inter_fanLine_countable
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) {s t : ℝ}
+    (hs : s ∈ Θ.directions) (ht : t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    (frontier (innerQuadrant K.val.val s) ∩ normalLine (t : Real.Angle) 0).Countable := by
+  have hsI : s ∈ Set.Ioo 0 Real.pi :=
+    ⟨(Θ.interior s hs).1,
+      ((Θ.interior s hs).2.trans_le Θ.angle_le).trans (by linarith [Real.pi_pos])⟩
+  have hsTI : s + Real.pi / 2 ∈ Set.Ioo 0 Real.pi := by
+    constructor
+    · linarith [(Θ.interior s hs).1, Real.pi_pos]
+    · linarith [(Θ.interior s hs).2, Θ.angle_le]
+  have htI : t ∈ Set.Ioo 0 Real.pi := by
+    rcases Set.mem_insert_iff.mp ht with rfl | ht
+    · exact ⟨Θ.angle_pos, Θ.angle_le.trans_lt (by linarith [Real.pi_pos])⟩
+    · rw [Set.mem_singleton_iff.mp ht]
+      exact ⟨by positivity, by linarith [Real.pi_pos]⟩
+  have hst : s ≠ t := by
+    rcases Set.mem_insert_iff.mp ht with rfl | ht
+    · exact ne_of_lt (Θ.interior s hs).2
+    · rw [Set.mem_singleton_iff.mp ht]
+      exact ne_of_lt ((Θ.interior s hs).2.trans_le Θ.angle_le)
+  have hsTt : s + Real.pi / 2 ≠ t := by
+    rcases Set.mem_insert_iff.mp ht with rfl | ht
+    · intro heq
+      linarith [(Θ.interior s hs).1, Θ.angle_le]
+    · rw [Set.mem_singleton_iff.mp ht]
+      exact ne_of_gt (by linarith [(Θ.interior s hs).1])
+  refine ((normalLine_inter_normalLine_subsingleton hsI htI hst
+      (c := supportValue K.val.val (s : Real.Angle) - 1) (d := 0)).countable.union
+    (normalLine_inter_normalLine_subsingleton hsTI htI hsTt
+      (c := supportValue K.val.val ((s + Real.pi / 2 : ℝ) : Real.Angle) - 1)
+      (d := 0)).countable).mono ?_
+  rintro q ⟨hqfront, hqline⟩
+  have hfront := frontier_inter_subset
+    (normalHalfPlane (s : Real.Angle) (supportValue K.val.val (s : Real.Angle) - 1)
+      false true)
+    (normalHalfPlane ((s + Real.pi / 2 : ℝ) : Real.Angle)
+      (supportValue K.val.val ((s + Real.pi / 2 : ℝ) : Real.Angle) - 1)
+      false true) hqfront
+  rcases hfront with hfront | hfront
+  · exact Or.inl ⟨frontier_normalHalfPlane_lower_strict_subset_normalLine _ _ hfront.1,
+      hqline⟩
+  · exact Or.inr ⟨frontier_normalHalfPlane_lower_strict_subset_normalLine _ _ hfront.2,
+      hqline⟩
+
+private theorem frontier_polygonNiche_inter_fanLine_eq_niche_inter
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) {t : ℝ}
+    (ht : t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    Measure.hausdorffMeasure 1
+        (frontier (polygonNiche Θ K.val) ∩ normalLine (t : Real.Angle) 0) =
+      Measure.hausdorffMeasure 1
+        (polygonNiche Θ K.val ∩ normalLine (t : Real.Angle) 0) := by
+  let X : Set Point := ⋃ s ∈ Θ.directions, innerQuadrant K.val.val s
+  have hN : polygonNiche Θ K.val = capFan Θ.angle ∩ X := by simp [polygonNiche, X]
+  rw [hN]
+  apply measure_eq_of_symmDiff_subset_null (E := frontier X ∩ normalLine (t : Real.Angle) 0)
+  · intro q hq
+    rcases hq with hq | hq
+    · refine ⟨?_, hq.1.2⟩
+      rw [← closure_sdiff_interior]
+      refine ⟨closure_mono Set.inter_subset_right (frontier_subset_closure hq.1.1), ?_⟩
+      intro hqInt
+      have hqX : q ∈ X := interior_subset hqInt
+      have hqF : q ∈ capFan Θ.angle := by
+        have hqcl := frontier_subset_closure hq.1.1
+        exact (isClosed_capFan Θ.angle).closure_eq ▸
+          closure_mono Set.inter_subset_left hqcl
+      exact hq.2 ⟨⟨hqF, hqX⟩, hq.1.2⟩
+    · have hqFront : q ∈ frontier (capFan Θ.angle ∩ X) := by
+        apply (mem_frontier_iff_notMem_interior hq.1.1).mpr
+        intro hqInt
+        have hqFInt := interior_mono Set.inter_subset_left hqInt
+        exact (mem_frontier_iff_notMem_interior hq.1.1.1).mp
+          (capFan_inter_fanLine_subset_frontier ht ⟨hq.1.1.1, hq.1.2⟩) hqFInt
+      exact (hq.2 ⟨hqFront, hq.1.2⟩).elim
+  · have hcount : (frontier X ∩ normalLine (t : Real.Angle) 0).Countable := by
+      refine (Set.Countable.biUnion Θ.directions.countable_toSet fun s hs ↦
+        frontier_innerQuadrant_inter_fanLine_countable K hs ht).mono ?_
+      rintro q ⟨hqX, hqline⟩
+      have hqUnion := Finset.frontier_biUnion_subset Θ.directions
+        (fun s ↦ innerQuadrant K.val.val s) hqX
+      obtain ⟨s, hs, hqs⟩ := Set.mem_iUnion₂.mp hqUnion
+      exact Set.mem_iUnion₂.mpr ⟨s, hs, ⟨hqs, hqline⟩⟩
+    have := Measure.nullSingletonClass_hausdorff Point (by norm_num : (0 : ℝ) < 1)
+    exact hcount.measure_zero (Measure.hausdorffMeasure 1)
+
+private theorem capVertices_zero_fst_pos_of_lt {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hω : Θ.angle < Real.pi / 2) :
+    0 < (capVertices K.val 0).1.2 0 := by
+  obtain ⟨u, huK, hu⟩ := exists_mem_inner_eq_supportValue K.val.val
+    (Θ.angle : Real.Angle)
+  have huy : u 1 ≤ 1 := by
+    have huy' := inner_le_supportValue K.val.val huK
+      ((Real.pi / 2 : ℝ) : Real.Angle)
+    rw [K.val.property.2.2.2.1] at huy'
+    simpa [normalVector, frame, PiLp.inner_apply] using huy'
+  have hcos : 0 < Real.cos Θ.angle := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Θ.angle_pos, Real.pi_pos], hω⟩
+  have hsin_lt : Real.sin Θ.angle < 1 := by
+    nlinarith only [Real.sin_sq_add_cos_sq Θ.angle, sq_pos_of_pos hcos]
+  have hux : 0 < u 0 := by
+    rw [K.val.property.2.2.1] at hu
+    simp [normalVector, frame, PiLp.inner_apply] at hu
+    have hsin : 0 ≤ Real.sin Θ.angle :=
+      (Real.sin_pos_of_pos_of_lt_pi Θ.angle_pos
+        (Θ.angle_le.trans_lt (by linarith [Real.pi_pos]))).le
+    have hmul := mul_le_mul_of_nonneg_left huy hsin
+    nlinarith
+  have hsupport : 0 < supportValue K.val.val (0 : Real.Angle) :=
+    hux.trans_le (by
+      have := inner_le_supportValue K.val.val huK (0 : Real.Angle)
+      simpa [normalVector, frame, PiLp.inner_apply] using this)
+  rw [capVertices_zero_snd_eq K]
+  simpa [normalVector, frame] using hsupport
+
+private theorem hausdorffMeasure_bottom_exposedEdge_sdiff_niche_eq_carrier_of_lt
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) (hω : Θ.angle < Real.pi / 2) :
+    Measure.hausdorffMeasure 1
+        ((exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) \
+          polygonNiche Θ K.val) ∩ normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0) =
+      Measure.hausdorffMeasure 1
+        ((polygonCapPolyline K).carrier ∩
+          normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0) := by
+  let p := polygonCapPolyline K
+  let A := (capVertices K.val 0).1.2
+  let C := (capVertices K.val Θ.angle).2.1
+  let L := openRay C (tangentVector (Θ.angle : Real.Angle))
+  have hp : IsCapPolyline K p := polygonCapPolyline_spec K
+  have hclosed : IsClosed (capFan Θ.angle \ polygonNiche Θ K.val) := hp.2.2.2.2.1
+  apply measure_eq_of_symmDiff_subset_null (E :=
+    L ∩ normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0)
+  · intro q hq
+    rcases hq with hq | hq
+    · have hqF : q ∈ capFan Θ.angle := K.val.subset_capFan hq.1.1.1.1
+      have hqFrontF := capFan_inter_fanLine_subset_frontier
+        (Θ := Θ) (t := Real.pi / 2) (by simp) ⟨hqF, hq.1.2⟩
+      have hqC : q ∈ capFan Θ.angle \ polygonNiche Θ K.val := ⟨hqF, hq.1.1.2⟩
+      have hqFrontC : q ∈ frontier (capFan Θ.angle \ polygonNiche Θ K.val) := by
+        apply (mem_frontier_iff_notMem_interior hqC).mpr
+        intro hqInt
+        exact (mem_frontier_iff_notMem_interior hqF).mp hqFrontF
+          (interior_mono Set.sdiff_subset hqInt)
+      rw [hp.2.2.2.2.2.1] at hqFrontC
+      rcases hqFrontC with hqL | hqR
+      · rcases hqL with hqL | hqcarrier
+        · exact ⟨hqL, hq.1.2⟩
+        · exact (hq.2 ⟨hqcarrier, hq.1.2⟩).elim
+      · rcases hqR with ⟨r, hr, hqr⟩
+        have hD := hq.1.1.1
+        rw [exposedEdge_bottom_eq_segment_zero_right K hω] at hD
+        have hApos : 0 < A 0 := capVertices_zero_fst_pos_of_lt K hω
+        have hqx := (fst_mem_Icc_of_mem_segment_mc4d199c hApos.le hD).2
+        have hqx' : A 0 < q 0 := by
+          rw [hqr]
+          simp [A, normalVector, frame]
+          linarith
+        linarith
+    · have hD := polygonCapPolyline_carrier_inter_bottom_subset_exposedEdge_of_lt
+        K hω hq.1
+      have hqFrontC : q ∈ frontier (capFan Θ.angle \ polygonNiche Θ K.val) := by
+        rw [hp.2.2.2.2.2.1]
+        exact Or.inl (Or.inr hq.1.1)
+      have hqC : q ∈ capFan Θ.angle \ polygonNiche Θ K.val :=
+        hclosed.closure_eq ▸ frontier_subset_closure hqFrontC
+      exact (hq.2 ⟨⟨hD, hqC.2⟩, hq.1.2⟩).elim
+  · have hinner : inner ℝ (tangentVector (Θ.angle : Real.Angle))
+        (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) ≠ 0 := by
+      have hcos : 0 < Real.cos Θ.angle := Real.cos_pos_of_mem_Ioo
+        ⟨by linarith [Θ.angle_pos, Real.pi_pos], hω⟩
+      simpa [tangentVector, normalVector, frame, PiLp.inner_apply,
+        Fin.sum_univ_two] using hcos.ne'
+    exact hausdorffMeasure_openRay_inter_hyperplane_eq_zero hinner
+
+private theorem hausdorffMeasure_left_exposedEdge_sdiff_niche_eq_carrier_of_lt
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) (hω : Θ.angle < Real.pi / 2) :
+    Measure.hausdorffMeasure 1
+        ((exposedEdge K.val.val ((Θ.angle + Real.pi : ℝ) : Real.Angle) \
+          polygonNiche Θ K.val) ∩ normalLine (Θ.angle : Real.Angle) 0) =
+      Measure.hausdorffMeasure 1
+        ((polygonCapPolyline K).carrier ∩ normalLine (Θ.angle : Real.Angle) 0) := by
+  let p := polygonCapPolyline K
+  let A := (capVertices K.val 0).1.2
+  let C := (capVertices K.val Θ.angle).2.1
+  let R := openRay A (normalVector 0)
+  have hp : IsCapPolyline K p := polygonCapPolyline_spec K
+  have hclosed : IsClosed (capFan Θ.angle \ polygonNiche Θ K.val) := hp.2.2.2.2.1
+  have hCeq := capVertices_angle_fst_eq K
+  change C = supportValue K.val.val
+    ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) •
+      tangentVector (Θ.angle : Real.Angle) at hCeq
+  have hL : 0 ≤ supportValue K.val.val
+      ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) :=
+    supportValue_nonneg_of_mem_capUpperAngles K.val hω
+      (Or.inr ⟨by linarith [Θ.angle_pos, Real.pi_pos], le_rfl⟩)
+  have hCcoord : 0 ≤ inner ℝ C (tangentVector (Θ.angle : Real.Angle)) := by
+    rw [hCeq, real_inner_smul_left]
+    rw [show inner ℝ (tangentVector (Θ.angle : Real.Angle))
+      (tangentVector (Θ.angle : Real.Angle)) = 1 by
+        exact inner_tangentVector_self Θ.angle, mul_one]
+    exact hL
+  apply measure_eq_of_symmDiff_subset_null (E :=
+    R ∩ normalLine (Θ.angle : Real.Angle) 0)
+  · intro q hq
+    rcases hq with hq | hq
+    · have hqF : q ∈ capFan Θ.angle := K.val.subset_capFan hq.1.1.1.1
+      have hqFrontF := capFan_inter_fanLine_subset_frontier
+        (Θ := Θ) (t := Θ.angle) (by simp) ⟨hqF, hq.1.2⟩
+      have hqC : q ∈ capFan Θ.angle \ polygonNiche Θ K.val := ⟨hqF, hq.1.1.2⟩
+      have hqFrontC : q ∈ frontier (capFan Θ.angle \ polygonNiche Θ K.val) := by
+        apply (mem_frontier_iff_notMem_interior hqC).mpr
+        intro hqInt
+        exact (mem_frontier_iff_notMem_interior hqF).mp hqFrontF
+          (interior_mono Set.sdiff_subset hqInt)
+      rw [hp.2.2.2.2.2.1] at hqFrontC
+      rcases hqFrontC with hqL | hqR
+      · rcases hqL with hqL | hqcarrier
+        · rcases hqL with ⟨r, hr, hqr⟩
+          have hD := hq.1.1.1
+          rw [exposedEdge_left_eq_segment_zero_left K hω] at hD
+          rcases hD with ⟨u, v, hu, hv, huv, hqseg⟩
+          have hvle : v ≤ 1 := by linarith
+          have hqcoord_le : inner ℝ q (tangentVector (Θ.angle : Real.Angle)) ≤
+              inner ℝ C (tangentVector (Θ.angle : Real.Angle)) := by
+            rw [← hqseg, inner_add_left, real_inner_smul_left,
+              real_inner_smul_left, inner_zero_left, mul_zero, zero_add]
+            exact mul_le_of_le_one_left hCcoord hvle
+          have hqcoord_gt : inner ℝ C (tangentVector (Θ.angle : Real.Angle)) <
+              inner ℝ q (tangentVector (Θ.angle : Real.Angle)) := by
+            change inner ℝ (capVertices K.val Θ.angle).2.1
+                (tangentVector (Θ.angle : Real.Angle)) <
+              inner ℝ q (tangentVector (Θ.angle : Real.Angle))
+            rw [hqr, inner_add_left, real_inner_smul_left]
+            rw [show inner ℝ (tangentVector (Θ.angle : Real.Angle))
+              (tangentVector (Θ.angle : Real.Angle)) = 1 by
+                exact inner_tangentVector_self Θ.angle]
+            simp only [mul_one, lt_add_iff_pos_right]
+            exact hr
+          exact (not_lt_of_ge hqcoord_le hqcoord_gt).elim
+        · exact (hq.2 ⟨hqcarrier, hq.1.2⟩).elim
+      · exact ⟨hqR, hq.1.2⟩
+    · have hD := polygonCapPolyline_carrier_inter_left_subset_exposedEdge_of_lt
+        K hω hq.1
+      have hqFrontC : q ∈ frontier (capFan Θ.angle \ polygonNiche Θ K.val) := by
+        rw [hp.2.2.2.2.2.1]
+        exact Or.inl (Or.inr hq.1.1)
+      have hqC : q ∈ capFan Θ.angle \ polygonNiche Θ K.val :=
+        hclosed.closure_eq ▸ frontier_subset_closure hqFrontC
+      exact (hq.2 ⟨⟨hD, hqC.2⟩, hq.1.2⟩).elim
+  · have hinner : inner ℝ (normalVector 0)
+        (normalVector (Θ.angle : Real.Angle)) ≠ 0 := by
+      have hcos : 0 < Real.cos Θ.angle := Real.cos_pos_of_mem_Ioo
+        ⟨by linarith [Θ.angle_pos, Real.pi_pos], hω⟩
+      rw [show (0 : Real.Angle) = ((0 : ℝ) : Real.Angle) by rfl,
+        inner_normalVector_normalVector, zero_sub, Real.cos_neg]
+      exact hcos.ne'
+    exact hausdorffMeasure_openRay_inter_hyperplane_eq_zero hinner
+
+private theorem bottom_exposedEdge_sdiff_niche_inter_eq_carrier_of_eq
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) (hω : Θ.angle = Real.pi / 2) :
+    (exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) \
+        polygonNiche Θ K.val) ∩ normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0 =
+      (polygonCapPolyline K).carrier ∩
+        normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0 := by
+  let p := polygonCapPolyline K
+  let A := (capVertices K.val 0).1.2
+  let C := (capVertices K.val Θ.angle).2.1
+  have hp : IsCapPolyline K p := polygonCapPolyline_spec K
+  have hclosed : IsClosed (capFan Θ.angle \ polygonNiche Θ K.val) := hp.2.2.2.2.1
+  ext q
+  constructor
+  · intro hq
+    have hqF : q ∈ capFan Θ.angle := K.val.subset_capFan hq.1.1.1
+    have hqFrontF := capFan_inter_fanLine_subset_frontier
+      (Θ := Θ) (t := Real.pi / 2) (by simp) ⟨hqF, hq.2⟩
+    have hqC : q ∈ capFan Θ.angle \ polygonNiche Θ K.val := ⟨hqF, hq.1.2⟩
+    have hqFrontC : q ∈ frontier (capFan Θ.angle \ polygonNiche Θ K.val) := by
+      apply (mem_frontier_iff_notMem_interior hqC).mpr
+      intro hqInt
+      exact (mem_frontier_iff_notMem_interior hqF).mp hqFrontF
+        (interior_mono Set.sdiff_subset hqInt)
+    rw [hp.2.2.2.2.2.1] at hqFrontC
+    rcases hqFrontC with hqL | hqR
+    · rcases hqL with hqL | hqcarrier
+      · rcases hqL with ⟨r, hr, hqr⟩
+        have hD := hq.1.1
+        rw [exposedEdge_bottom_eq_segment_left_right K hω] at hD
+        have hqx := fst_mem_Icc_of_mem_segment_mc4d199c
+          (polygonCap_left_x_lt_right_x K).le hD
+        have hqx' : q 0 < C 0 := by
+          rw [hqr]
+          simp only [Fin.isValue, hω, tangentVector, frame, Real.Angle.cos_coe, Real.cos_pi_div_two,
+            Real.Angle.sin_coe, Real.sin_pi_div_two, PiLp.add_apply, PiLp.smul_apply,
+            Matrix.cons_val_zero, smul_eq_mul, mul_neg, mul_one, add_lt_iff_neg_left,
+            Left.neg_neg_iff, C]
+          exact hr
+        exact (not_lt_of_ge (by simpa [C] using hqx.1) hqx').elim
+      · exact ⟨hqcarrier, hq.2⟩
+    · rcases hqR with ⟨r, hr, hqr⟩
+      have hD := hq.1.1
+      rw [exposedEdge_bottom_eq_segment_left_right K hω] at hD
+      have hqx := fst_mem_Icc_of_mem_segment_mc4d199c
+        (polygonCap_left_x_lt_right_x K).le hD
+      have hqx' : A 0 < q 0 := by
+        rw [hqr]
+        simp only [Fin.isValue, normalVector, frame, Real.Angle.cos_zero,
+          Real.Angle.sin_zero, neg_zero,
+          PiLp.add_apply, PiLp.smul_apply, Matrix.cons_val_zero, smul_eq_mul, mul_one,
+          lt_add_iff_pos_right, A]
+        exact hr
+      exact (not_lt_of_ge (by simpa [A] using hqx.2) hqx').elim
+  · intro hq
+    have hD := polygonCapPolyline_carrier_inter_bottom_subset_exposedEdge_of_eq
+      K hω hq
+    have hqFrontC : q ∈ frontier (capFan Θ.angle \ polygonNiche Θ K.val) := by
+      rw [hp.2.2.2.2.2.1]
+      exact Or.inl (Or.inr hq.1)
+    have hqC : q ∈ capFan Θ.angle \ polygonNiche Θ K.val :=
+      hclosed.closure_eq ▸ frontier_subset_closure hqFrontC
+    exact ⟨⟨hD, hqC.2⟩, hq.2⟩
+
+private theorem segment_subset_normalLine {a b : Point} {t : Real.Angle} {c : ℝ}
+    (ha : a ∈ normalLine t c) (hb : b ∈ normalLine t c) :
+    segment ℝ a b ⊆ normalLine t c := by
+  rintro q ⟨u, v, hu, hv, huv, rfl⟩
+  change inner ℝ (u • a + v • b) (normalVector t) = c
+  change inner ℝ a (normalVector t) = c at ha
+  change inner ℝ b (normalVector t) = c at hb
+  rw [inner_add_left, real_inner_smul_left, real_inner_smul_left, ha, hb]
+  linear_combination c * huv
+
+private theorem hausdorffMeasure_niche_inter_eq_exposedEdge_sub_carrier
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) {t : ℝ}
+    (hDline : exposedEdge K.val.val ((t + Real.pi : ℝ) : Real.Angle) ⊆
+      normalLine (t : Real.Angle) 0)
+    (hNsub : polygonNiche Θ K.val ∩ normalLine (t : Real.Angle) 0 ⊆
+      exposedEdge K.val.val ((t + Real.pi : ℝ) : Real.Angle))
+    (hcomp : Measure.hausdorffMeasure 1
+        ((exposedEdge K.val.val ((t + Real.pi : ℝ) : Real.Angle) \
+          polygonNiche Θ K.val) ∩ normalLine (t : Real.Angle) 0) =
+      Measure.hausdorffMeasure 1
+        ((polygonCapPolyline K).carrier ∩ normalLine (t : Real.Angle) 0)) :
+    (Measure.hausdorffMeasure 1
+      (polygonNiche Θ K.val ∩ normalLine (t : Real.Angle) 0)).toReal =
+      (Measure.hausdorffMeasure 1
+        (exposedEdge K.val.val ((t + Real.pi : ℝ) : Real.Angle))).toReal -
+      (Measure.hausdorffMeasure 1
+        ((polygonCapPolyline K).carrier ∩ normalLine (t : Real.Angle) 0)).toReal := by
+  let μ : Measure Point := Measure.hausdorffMeasure 1
+  let D := exposedEdge K.val.val ((t + Real.pi : ℝ) : Real.Angle)
+  let M := polygonNiche Θ K.val ∩ normalLine (t : Real.Angle) 0
+  have hdiff : D \ M =
+      (D \ polygonNiche Θ K.val) ∩ normalLine (t : Real.Angle) 0 := by
+    ext q
+    constructor
+    · intro hq
+      exact ⟨⟨hq.1, fun hqN ↦ hq.2 ⟨hqN, hDline hq.1⟩⟩, hDline hq.1⟩
+    · intro hq
+      exact ⟨hq.1.1, fun hqM ↦ hq.1.2 hqM.1⟩
+  have hNMeas : MeasurableSet (polygonNiche Θ K.val) := by
+    exact (isClosed_capFan Θ.angle).measurableSet.inter
+      (isOpen_iUnion fun s ↦ isOpen_iUnion fun _ ↦
+        isOpen_innerQuadrant K.val.val s).measurableSet
+  have hMMeas : MeasurableSet M :=
+    hNMeas.inter (isClosed_eq (by fun_prop) continuous_const).measurableSet
+  have hDfinite : μ D ≠ ⊤ := by
+    dsimp [μ, D]
+    rw [exposedEdge_eq_segment_edgeVertices, MeasureTheory.hausdorffMeasure_segment,
+      edist_dist]
+    simp
+  have hreal := MeasureTheory.measureReal_sdiff (μ := μ) hNsub hMMeas hDfinite
+  change (μ (D \ M)).toReal = (μ D).toReal - (μ M).toReal at hreal
+  rw [hdiff] at hreal
+  have hcompReal := congrArg ENNReal.toReal hcomp
+  dsimp [μ, D, M] at hreal hcompReal ⊢
+  rw [hcompReal] at hreal
+  linarith
+
+private theorem bottom_exposedEdge_subset_fanLine_of_lt {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hω : Θ.angle < Real.pi / 2) :
+    exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) ⊆
+      normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0 := by
+  rw [exposedEdge_bottom_eq_segment_zero_right K hω]
+  apply segment_subset_normalLine
+  · simp [normalLine]
+  · rw [capVertices_zero_snd_eq K]
+    simp [normalLine, normalVector, frame, PiLp.inner_apply]
+
+private theorem left_exposedEdge_subset_fanLine_of_lt {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hω : Θ.angle < Real.pi / 2) :
+    exposedEdge K.val.val ((Θ.angle + Real.pi : ℝ) : Real.Angle) ⊆
+      normalLine (Θ.angle : Real.Angle) 0 := by
+  rw [exposedEdge_left_eq_segment_zero_left K hω]
+  apply segment_subset_normalLine
+  · simp [normalLine]
+  · change inner ℝ (capVertices K.val Θ.angle).2.1
+      (normalVector (Θ.angle : Real.Angle)) = 0
+    rw [capVertices_angle_fst_eq K, real_inner_smul_left,
+      show inner ℝ (tangentVector (Θ.angle : Real.Angle))
+        (normalVector (Θ.angle : Real.Angle)) = 0 by
+          rw [real_inner_comm, inner_normalVector_tangentVector], mul_zero]
+
+private theorem bottom_exposedEdge_subset_fanLine_of_eq {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hω : Θ.angle = Real.pi / 2) :
+    exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) ⊆
+      normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0 := by
+  rw [exposedEdge_bottom_eq_segment_left_right K hω]
+  apply segment_subset_normalLine
+  · rw [capVertices_angle_fst_eq K]
+    simp [normalLine, normalVector, tangentVector, frame, PiLp.inner_apply, hω]
+  · rw [capVertices_zero_snd_eq K]
+    simp [normalLine, normalVector, frame, PiLp.inner_apply]
+
+private theorem polygonNiche_fanLine_length {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {t : ℝ} (ht : t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    (Measure.hausdorffMeasure 1
+      (polygonNiche Θ K.val ∩ normalLine (t : Real.Angle) 0)).toReal =
+      (Measure.hausdorffMeasure 1
+        (exposedEdge K.val.val ((t + Real.pi : ℝ) : Real.Angle))).toReal -
+      polygonPolylineLengthAt K t := by
+  by_cases hω : Θ.angle < Real.pi / 2
+  · rcases Set.mem_insert_iff.mp ht with rfl | ht
+    · rw [← polygonCapPolyline_fanLine_length K (by simp)]
+      exact hausdorffMeasure_niche_inter_eq_exposedEdge_sub_carrier K
+        (left_exposedEdge_subset_fanLine_of_lt K hω)
+        (polygonNiche_inter_left_subset_exposedEdge_of_lt K hω)
+        (hausdorffMeasure_left_exposedEdge_sdiff_niche_eq_carrier_of_lt K hω)
+    · have ht : t = Real.pi / 2 := Set.mem_singleton_iff.mp ht
+      subst t
+      rw [← polygonCapPolyline_fanLine_length K (by simp)]
+      apply hausdorffMeasure_niche_inter_eq_exposedEdge_sub_carrier K
+      · convert bottom_exposedEdge_subset_fanLine_of_lt K hω using 1; ring_nf
+      · convert polygonNiche_inter_bottom_subset_exposedEdge_of_lt K hω using 1; ring_nf
+      · convert hausdorffMeasure_bottom_exposedEdge_sdiff_niche_eq_carrier_of_lt
+          K hω using 1; ring_nf
+  · have hωeq : Θ.angle = Real.pi / 2 := le_antisymm Θ.angle_le (le_of_not_gt hω)
+    have htT : t = Real.pi / 2 := by
+      rcases Set.mem_insert_iff.mp ht with ht | ht
+      · exact ht.trans hωeq
+      · exact Set.mem_singleton_iff.mp ht
+    subst t
+    rw [← polygonCapPolyline_fanLine_length K (by simp)]
+    apply hausdorffMeasure_niche_inter_eq_exposedEdge_sub_carrier K
+    · convert bottom_exposedEdge_subset_fanLine_of_eq K hωeq using 1; ring_nf
+    · convert polygonNiche_inter_bottom_subset_exposedEdge_of_eq K hωeq using 1; ring_nf
+    · convert congrArg (Measure.hausdorffMeasure 1)
+        (bottom_exposedEdge_sdiff_niche_inter_eq_carrier_of_eq K hωeq) using 1;
+        ring_nf
+
+/-- The niche trace on a fan line is the lower face length minus polyline length. -/
+theorem polygonNiche_fanLine_lengths {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {t : ℝ} (ht : t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    nicheBoundaryLength K (normalLine t 0) =
+        (surfaceAreaMeasure K.val.val {((t + Real.pi : ℝ) : Real.Angle)}).toReal -
+          polygonPolylineLengthAt K t ∧
+      (Measure.hausdorffMeasure 1
+        (polygonNiche Θ K.val ∩ normalLine t 0)).toReal =
+        (surfaceAreaMeasure K.val.val {((t + Real.pi : ℝ) : Real.Angle)}).toReal -
+          polygonPolylineLengthAt K t := by
+  have hN := polygonNiche_fanLine_length K ht
+  have hatom := congrArg ENNReal.toReal
+    (surfaceAreaMeasure_atom_length K.val.val
+      ((t + Real.pi : ℝ) : Real.Angle)).1
+  rw [← hatom] at hN
+  refine ⟨?_, hN⟩
+  unfold nicheBoundaryLength
+  rw [frontier_polygonNiche_inter_fanLine_eq_niche_inter K ht]
+  exact hN
+
+/-- Inner-wall and inner-ray niche lengths agree with the corresponding polyline lengths. -/
+theorem polygonNiche_wall_lengths {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    {t : ℝ} (ht : t ∈ Θ.directions) :
+    nicheBoundaryLength K (rotatingHallwayParts (K.val.val : Set Point) t).b =
+      polygonPolylineLengthAt K t ∧
+    nicheBoundaryLength K (rotatingHallwayParts (K.val.val : Set Point) t).bRay =
+      polygonPolylineLengthAt K t ∧
+    nicheBoundaryLength K (rotatingHallwayParts (K.val.val : Set Point) t).d =
+      polygonPolylineLengthAt K (t + Real.pi / 2) ∧
+    nicheBoundaryLength K (rotatingHallwayParts (K.val.val : Set Point) t).dRay =
+      polygonPolylineLengthAt K (t + Real.pi / 2) := by
+  have hf := rotatingHallwayParts_formulas
+    (K.val.val : Set Point) (t : Real.Angle)
+  refine ⟨?_, ?_, ?_, ?_⟩
+  · unfold nicheBoundaryLength
+    rw [hf.2.2.2.2.1]
+    exact polygonNiche_bLine_length K ht
+  · unfold nicheBoundaryLength
+    exact polygonNiche_bRay_length K ht
+  · unfold nicheBoundaryLength
+    rw [hf.2.2.2.2.2.2.1]
+    exact polygonNiche_dLine_length K ht
+  · unfold nicheBoundaryLength
+    exact polygonNiche_dRay_length K ht
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Polyline / Length
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem polygonCap_polyline_lengths {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    (∀ t ∈ Θ.directions,
+      nicheBoundaryLength K (rotatingHallwayParts (K.val.val : Set Point) t).b =
+        polygonPolylineLengthAt K t ∧
+      nicheBoundaryLength K (rotatingHallwayParts (K.val.val : Set Point) t).bRay =
+        polygonPolylineLengthAt K t ∧
+      nicheBoundaryLength K (rotatingHallwayParts (K.val.val : Set Point) t).d =
+        polygonPolylineLengthAt K (t + Real.pi / 2) ∧
+      nicheBoundaryLength K (rotatingHallwayParts (K.val.val : Set Point) t).dRay =
+        polygonPolylineLengthAt K (t + Real.pi / 2)) ∧
+    (∀ t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+      nicheBoundaryLength K (normalLine t 0) =
+        (surfaceAreaMeasure K.val.val {((t + Real.pi : ℝ) : Real.Angle)}).toReal -
+          polygonPolylineLengthAt K t ∧
+      (MeasureTheory.Measure.hausdorffMeasure 1
+        (polygonNiche Θ K.val ∩ normalLine t 0)).toReal =
+        (surfaceAreaMeasure K.val.val {((t + Real.pi : ℝ) : Real.Angle)}).toReal -
+          polygonPolylineLengthAt K t) := by
+  exact ⟨fun _ ht ↦ polygonNiche_wall_lengths K ht,
+    fun _ ht ↦ polygonNiche_fanLine_lengths K ht⟩
+
+private lemma polygonCapPolyline_sum_length_mul_sin {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (D : Finset ℝ) (hD : (D : Set ℝ) = angleDomain Θ) :
+    ∑ t ∈ D, polygonPolylineLengthAt K t * Real.sin t =
+      ((capVertices K.val 0).1.2) 0 - ((capVertices K.val Θ.angle).2.1) 0 := by
+  classical
+  let p := polygonCapPolyline K
+  have hp : IsCapPolyline K p := (polygonCap_polyline K).choose_spec
+  have hmem (t : ℝ) : t ∈ D ↔ t ∈ angleDomain Θ := by
+    change t ∈ (D : Set ℝ) ↔ _
+    rw [hD]
+  have h := p.sum_normal_lengths_mul_sin D
+    (fun t ht ↦ angleDomain_subset_Ioo Θ ((hmem t).mp ht)) (by
+      intro i
+      obtain ⟨t, ht⟩ := hp.2.2.2.1 i
+      exact ⟨t.val, (hmem t.val).mpr t.property, ht⟩)
+  have heq : ∑ t ∈ D, polygonPolylineLengthAt K t * Real.sin t =
+      ∑ t ∈ D, (∑ i : Fin p.edges,
+        if inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+          (normalVector (t : Real.Angle)) = 0 then
+          dist (p.vertices i.castSucc) (p.vertices i.succ) else 0) * Real.sin t := by
+    apply Finset.sum_congr rfl
+    intro t ht
+    simp only [polygonPolylineLengthAt, dite_eq_left ((hmem t).mp ht), polygonCapPolylineLength]
+    rfl
+  rw [heq, h, hp.2.1, hp.2.2.1]
+
+theorem polygonCap_not_balanced_positive {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (h : ¬ IsBalancedPolygonCap K) :
+    ∃ t : angleDomain Θ,
+      ENNReal.ofReal (polygonCapPolylineLength K t) <
+        surfaceAreaMeasure K.val.val {(t.val : Real.Angle)} := by
+  classical
+  by_contra hnot
+  push Not at hnot
+  have hfinite : (angleDomain Θ).Finite := by
+    unfold angleDomain
+    exact (Θ.directions.finite_toSet.union
+      (Θ.directions.finite_toSet.image (fun t ↦ t + Real.pi / 2))).union (Set.toFinite _)
+  let D := hfinite.toFinset
+  have hD : (D : Set ℝ) = angleDomain Θ := hfinite.coe_toFinset
+  have hne : D.Nonempty := by
+    refine ⟨Θ.angle, ?_⟩
+    change Θ.angle ∈ hfinite.toFinset
+    simp [angleDomain]
+  have hmem (t : ℝ) : t ∈ D ↔ t ∈ angleDomain Θ := by
+    change t ∈ (D : Set ℝ) ↔ _
+    rw [hD]
+  have hpoly (t : angleDomain Θ) : 0 ≤ polygonCapPolylineLength K t := by
+    unfold polygonCapPolylineLength
+    exact Finset.sum_nonneg (fun _ _ ↦ by split_ifs <;> positivity)
+  have htop (t : ℝ) : surfaceAreaMeasure K.val.val {(t : Real.Angle)} ≠ ⊤ := by
+    rw [(surfaceAreaMeasure_atom_length K.val.val (t : Real.Angle)).2.1]
+    exact ENNReal.ofReal_ne_top
+  have hle (t : ℝ) (ht : t ∈ D) :
+      (surfaceAreaMeasure K.val.val {(t : Real.Angle)}).toReal ≤ polygonPolylineLengthAt K t := by
+    have hh := ENNReal.toReal_mono ENNReal.ofReal_ne_top (hnot ⟨t, (hmem t).mp ht⟩)
+    simpa [polygonPolylineLengthAt, (hmem t).mp ht, ENNReal.toReal_ofReal
+      (hpoly ⟨t, (hmem t).mp ht⟩)] using hh
+  have hsum : ∑ t ∈ D,
+      (polygonPolylineLengthAt K t -
+        (surfaceAreaMeasure K.val.val {(t : Real.Angle)}).toReal) * Real.sin t = 0 := by
+    simp_rw [sub_mul]
+    rw [Finset.sum_sub_distrib, polygonCapPolyline_sum_length_mul_sin K D hD]
+    have hh := K.sum_hausdorffMeasure_exposedEdge_mul_sin D hne hD
+    have heq : ∑ t ∈ D,
+        (surfaceAreaMeasure K.val.val {(t : Real.Angle)}).toReal * Real.sin t =
+        ((capVertices K.val 0).1.2) 0 - ((capVertices K.val Θ.angle).2.1) 0 := by
+      simpa only [(surfaceAreaMeasure_atom_length K.val.val _).1] using hh
+    rw [heq, sub_self]
+  have hzero := (Finset.sum_eq_zero_iff_of_nonneg (fun t ht ↦
+    mul_nonneg (sub_nonneg.mpr (hle t ht))
+      (Real.sin_pos_of_pos_of_lt_pi (angleDomain_subset_Ioo Θ ((hmem t).mp ht)).1
+        (angleDomain_subset_Ioo Θ ((hmem t).mp ht)).2).le)).mp hsum
+  apply h
+  intro t
+  have ht := (hmem t.val).mpr t.property
+  have hz := hzero t.val ht
+  have hs := Real.sin_pos_of_pos_of_lt_pi (angleDomain_subset_Ioo Θ t.property).1
+    (angleDomain_subset_Ioo Θ t.property).2
+  have heq := sub_eq_zero.mp ((mul_eq_zero.mp hz).resolve_right hs.ne')
+  apply (ENNReal.toReal_eq_toReal_iff' (htop t.val) ENNReal.ofReal_ne_top).mp
+  simpa [polygonPolylineLengthAt, t.property, ENNReal.toReal_ofReal (hpoly t)] using heq.symm
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Balancing / Coefficients
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace MovingSofa
+
+/-- A convex body's frontier on a supporting line is its exposed edge. -/
+theorem frontier_inter_supportingLine_eq_exposedEdge (K : ConvexBody Point)
+    (t : Real.Angle) :
+    frontier (K : Set Point) ∩ normalLine t (supportValue K t) = exposedEdge K t := by
+  ext p
+  constructor
+  · rintro ⟨hp, hline⟩
+    exact ⟨K.isCompact.isClosed.closure_subset (frontier_subset_closure hp), hline⟩
+  · intro hp
+    refine ⟨mem_frontier_of_mem_of_isExteriorNormal K (a := t) hp.1 ?_, hp.2⟩
+    intro q hq
+    change inner ℝ (q - p) (normalVector t) ≤ 0
+    rw [inner_sub_left, show inner ℝ p (normalVector t) = supportValue K t from hp.2]
+    exact sub_nonpos.mpr (inner_le_supportValue K hq t)
+
+/-- The surface-area atom is the length of the frontier on its supporting line. -/
+theorem hausdorffMeasure_frontier_inter_supportingLine (K : ConvexBody Point)
+    (t : Real.Angle) :
+    MeasureTheory.Measure.hausdorffMeasure 1
+      (frontier (K : Set Point) ∩ normalLine t (supportValue K t)) =
+      surfaceAreaMeasure K {t} := by
+  rw [frontier_inter_supportingLine_eq_exposedEdge]
+  exact (surfaceAreaMeasure_atom_length K t).1.symm
+
+/-- An endpoint cap's lower wall has the surface-area atom of the opposite normal. -/
+theorem hausdorffMeasure_frontier_inter_lowerLine_of_endpoint {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {t : ℝ}
+    (ht : t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    MeasureTheory.Measure.hausdorffMeasure 1
+      (frontier (K.val.val : Set Point) ∩
+        normalLine (t : Real.Angle) (supportValue K.val.val (t : Real.Angle) - 1)) =
+      surfaceAreaMeasure K.val.val {((t + Real.pi : ℝ) : Real.Angle)} := by
+  have htSupport : supportValue K.val.val (t : Real.Angle) = 1 := by
+    rcases Set.mem_insert_iff.mp ht with ht | ht
+    · subst t
+      exact K.val.property.2.2.1
+    · have ht' := Set.mem_singleton_iff.mp ht
+      subst t
+      exact K.val.property.2.2.2.1
+  have htOpposite :
+      supportValue K.val.val ((t + Real.pi : ℝ) : Real.Angle) = 0 := by
+    rcases Set.mem_insert_iff.mp ht with ht | ht
+    · subst t
+      exact K.val.property.2.2.2.2.1
+    · have ht' := Set.mem_singleton_iff.mp ht
+      subst t
+      convert K.val.property.2.2.2.2.2.1 using 1
+      ring_nf
+  rw [htSupport, sub_self, show normalLine (t : Real.Angle) 0 =
+      normalLine ((t + Real.pi : ℝ) : Real.Angle) 0 by
+    ext p
+    change inner ℝ p (normalVector (t : Real.Angle)) = 0 ↔
+      inner ℝ p (normalVector ((t : Real.Angle) + Real.pi)) = 0
+    rw [normalVector_add_pi_angle]
+    simp]
+  rw [← htOpposite]
+  exact hausdorffMeasure_frontier_inter_supportingLine K.val.val _
+
+/-- Away from endpoint normals, the niche boundary on the lower wall has polyline length. -/
+theorem nicheBoundaryLength_lowerLine_of_not_endpoint {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (t : angleDomain Θ)
+    (ht : t.val ∉ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    nicheBoundaryLength K
+        (normalLine (t.val : Real.Angle)
+          (supportValue K.val.val (t.val : Real.Angle) - 1)) =
+      polygonCapPolylineLength K t := by
+  rw [show polygonCapPolylineLength K t = polygonPolylineLengthAt K t.val by
+    simp [polygonPolylineLengthAt, t.property]]
+  rcases t.property with htInner | htEndpoint
+  · rcases htInner with htDirection | ⟨s, hs, hst⟩
+    · have hline := (rotatingHallwayParts_formulas
+        (K.val.val : Set Point) (t.val : Real.Angle)).2.2.2.2.1
+      rw [← hline]
+      exact ((polygonCap_polyline_lengths K).1 t.val htDirection).1
+    · have hline := (rotatingHallwayParts_formulas
+        (K.val.val : Set Point) (s : Real.Angle)).2.2.2.2.2.2.1
+      rw [← hst]
+      change nicheBoundaryLength K
+          (normalLine ((s + Real.pi / 2 : ℝ) : Real.Angle)
+            (supportValue K.val.val ((s + Real.pi / 2 : ℝ) : Real.Angle) - 1)) =
+        polygonPolylineLengthAt K (s + Real.pi / 2)
+      have hline' :
+          (rotatingHallwayParts (K.val.val : Set Point) (s : Real.Angle)).d =
+            normalLine ((s + Real.pi / 2 : ℝ) : Real.Angle)
+              (supportValue K.val.val ((s + Real.pi / 2 : ℝ) : Real.Angle) - 1) := by
+        simpa only [Real.Angle.coe_add] using hline
+      rw [← hline']
+      exact ((polygonCap_polyline_lengths K).1 s hs).2.2.1
+  · exact False.elim (ht htEndpoint)
+
+/-- At an endpoint normal, the lower-wall niche length is the opposite-face length
+minus the polyline length. -/
+theorem nicheBoundaryLength_lowerLine_of_endpoint {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (t : angleDomain Θ)
+    (ht : t.val ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    nicheBoundaryLength K
+        (normalLine (t.val : Real.Angle)
+          (supportValue K.val.val (t.val : Real.Angle) - 1)) =
+      (surfaceAreaMeasure K.val.val
+        {((t.val + Real.pi : ℝ) : Real.Angle)}).toReal -
+        polygonCapPolylineLength K t := by
+  have htSupport : supportValue K.val.val (t.val : Real.Angle) = 1 := by
+    rcases Set.mem_insert_iff.mp ht with ht | ht
+    · rw [ht]
+      exact K.val.property.2.2.1
+    · rw [Set.mem_singleton_iff.mp ht]
+      exact K.val.property.2.2.2.1
+  rw [htSupport, sub_self]
+  simpa [polygonPolylineLengthAt, t.property] using
+    (polygonCap_polyline_lengths K).2 t.val ht |>.1
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Balancing / Estimate
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private def polygonCapSupportHeight {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    PolygonHeightSpace Θ :=
+  fun s ↦ supportValue K.val.val (s.val : Real.Angle)
+
+private theorem raisedPolygonSupport_eq_update {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (t : angleDomain Θ) (varepsilon : ℝ) :
+    raisedPolygonSupport K t varepsilon =
+      Function.update (polygonCapSupportHeight K) t
+        (polygonCapSupportHeight K t + varepsilon) := by
+  classical
+  unfold raisedPolygonSupport polygonCapSupportHeight PolygonHeightSpace
+  funext s
+  by_cases hst : s = t
+  · subst s
+    simp
+  · simp [Function.update, hst]
+
+private theorem raisedPolygonSupport_zero {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (t : angleDomain Θ) : raisedPolygonSupport K t 0 = polygonCapSupportHeight K := by
+  unfold raisedPolygonSupport polygonCapSupportHeight PolygonHeightSpace
+  funext s
+  simp
+
+private theorem update_sub_one {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    (t : angleDomain Θ) (varepsilon : ℝ) :
+    Function.update (fun s ↦ h s - 1) t (h t - 1 + varepsilon) =
+      fun s ↦ Function.update h t (h t + varepsilon) s - 1 := by
+  classical
+  change angleDomain Θ → ℝ at h
+  funext s
+  by_cases hst : s = t
+  · subst s
+    simp
+    ring
+  · simp [Function.update, hst]
+
+private theorem polygonHeightCap_supportHeight {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    polygonHeightCap (polygonCapSupportHeight K) = K.val.val := by
+  let K' : PolygonCapTranslateSpace Θ :=
+    ⟨K.val.val, ⟨K, 0, by simp⟩⟩
+  change polygonHeightCap
+    (fun s ↦ supportValue K.val.val (s.val : Real.Angle)) = K.val.val
+  exact polygonHeightCap_of_translate K'
+
+private theorem polygonHeightNiche_supportHeight {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    polygonHeightNiche (polygonCapSupportHeight K) = polygonNiche Θ K.val := by
+  change polygonHeightNiche
+    (fun s ↦ supportValue K.val.val (s.val : Real.Angle)) = polygonNiche Θ K.val
+  exact (polygonHeightNiche_of_cap K).1
+
+private theorem independentWallCap_update_eq_of_not_endpoint {Θ : AngleSet}
+    (h : PolygonHeightSpace Θ) (t : angleDomain Θ) (varepsilon : ℝ)
+    (ht : t.val ∉ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    independentWallCap (Function.update h t (h t + varepsilon)) (fun s ↦ h s - 1) =
+      polygonHeightCap (Function.update h t (h t + varepsilon)) := by
+  classical
+  calc
+    independentWallCap (Function.update h t (h t + varepsilon)) (fun s ↦ h s - 1) =
+        independentWallCap (Function.update h t (h t + varepsilon))
+          (fun s ↦ Function.update h t (h t + varepsilon) s - 1) := by
+      ext p
+      simp only [independentWallCap, Set.mem_inter_iff, Set.mem_iInter]
+      apply and_congr
+      · apply forall_congr'
+        intro s
+        apply forall_congr'
+        intro hs
+        have hst : (⟨s, Or.inr hs⟩ : angleDomain Θ) ≠ t := by
+          intro hst
+          apply ht
+          have hval : s = t.val := congrArg Subtype.val hst
+          rwa [hval] at hs
+        simp [polygonHeightValue, Function.update, hst]
+      · rfl
+    _ = polygonHeightCap (Function.update h t (h t + varepsilon)) :=
+      independentWallCap_sub_one (Θ := Θ) _
+
+private theorem independentWallCap_update_both_eq {Θ : AngleSet}
+    (h : PolygonHeightSpace Θ) (t : angleDomain Θ) (varepsilon : ℝ) :
+    independentWallCap (Function.update h t (h t + varepsilon))
+        (Function.update (fun s ↦ h s - 1) t (h t - 1 + varepsilon)) =
+      polygonHeightCap (Function.update h t (h t + varepsilon)) := by
+  rw [update_sub_one]
+  exact independentWallCap_sub_one (Θ := Θ) _
+
+private theorem independentWallNiche_update_eq {Θ : AngleSet}
+    (h : PolygonHeightSpace Θ) (t : angleDomain Θ) (varepsilon : ℝ) :
+    independentWallNiche
+        (Function.update (fun s ↦ h s - 1) t (h t - 1 + varepsilon)) =
+      polygonHeightNiche (Function.update h t (h t + varepsilon)) := by
+  rw [update_sub_one]
+  exact independentWallNiche_sub_one (Θ := Θ) _
+
+/-- The balancing area estimate when the perturbed normal is not an endpoint. -/
+theorem polygonCap_balancing_estimate_of_not_endpoint {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (t : angleDomain Θ)
+    (ht : t.val ∉ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    ∃ C eta : ℝ, 0 ≤ C ∧ 0 < eta ∧ ∀ varepsilon : ℝ,
+      0 ≤ varepsilon → varepsilon ≤ eta →
+      |polygonHeightArea (raisedPolygonSupport K t varepsilon) -
+        polygonHeightArea (raisedPolygonSupport K t 0) -
+        ((surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}).toReal -
+          polygonCapPolylineLength K t) * varepsilon| ≤ C * varepsilon ^ 2 := by
+  classical
+  let h := polygonCapSupportHeight K
+  obtain ⟨⟨nc, Hc, hHc, hSc⟩, ⟨nn, Hn, hHn, hSn⟩⟩ := polygonCap_niche_simpleNef Θ h
+  have htInner : t.val ∈ (Θ.directions : Set ℝ) ∪
+      ((fun s : ℝ ↦ s + Real.pi / 2) '' Θ.directions) := by
+    rcases t.property with htInner | htEndpoint
+    · exact htInner
+    · exact False.elim (ht htEndpoint)
+  have hcMem :
+      (⟨(t.val : Real.Angle), h t, false, false⟩ : PlanarHalfPlaneData) ∈
+        Set.range Hc := by
+    rw [hHc]
+    exact Or.inl ⟨t.val, t.property, by simp [polygonHeightValue]⟩
+  obtain ⟨ic, hic⟩ := hcMem
+  have hnMem :
+      (⟨(t.val : Real.Angle), h t - 1, false, true⟩ : PlanarHalfPlaneData) ∈
+        Set.range Hn := by
+    rw [hHn]
+    exact Or.inl ⟨t.val, htInner, by simp [polygonHeightValue, t.property]⟩
+  obtain ⟨in_, hin⟩ := hnMem
+  obtain ⟨Cc, etac, hCc, hetac, hcap⟩ :=
+    polygonCap_upper_wall_area_variation h Hc hHc hSc.1 ic t hic
+  obtain ⟨tn, Cn, etan, htnAngle, hCn, hetan, hniche⟩ :=
+    polygonNiche_wall_area_variation h Hn hHn hSn.1 in_
+  have hinAngle : (Hn in_).angle = (t.val : Real.Angle) := by rw [hin]
+  have htn : tn = t := angleDomain_coe_injective Θ (htnAngle.symm.trans hinAngle)
+  subst tn
+  have hcapCoeff :
+      (MeasureTheory.Measure.hausdorffMeasure 1
+        (frontier (polygonHeightCap h) ∩ (Hc ic).boundaryLine)).toReal =
+        (surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}).toReal := by
+    rw [polygonHeightCap_supportHeight K, hic]
+    simpa [PlanarHalfPlaneData.boundaryLine, h, polygonCapSupportHeight] using
+      congrArg ENNReal.toReal
+        (hausdorffMeasure_frontier_inter_supportingLine K.val.val
+          (t.val : Real.Angle))
+  have hnicheCoeff :
+      (MeasureTheory.Measure.hausdorffMeasure 1
+        (frontier (polygonHeightNiche h) ∩ (Hn in_).boundaryLine)).toReal =
+        polygonCapPolylineLength K t := by
+    rw [polygonHeightNiche_supportHeight K, hin]
+    simpa [nicheBoundaryLength, PlanarHalfPlaneData.boundaryLine, h,
+      polygonCapSupportHeight] using nicheBoundaryLength_lowerLine_of_not_endpoint K t ht
+  refine ⟨Cc + Cn, min etac etan, add_nonneg hCc hCn, lt_min hetac hetan, ?_⟩
+  intro varepsilon hvarepsilon hvarepsilonEta
+  have hvarepsilonC : |varepsilon| ≤ etac := by
+    rw [abs_of_nonneg hvarepsilon]
+    exact hvarepsilonEta.trans (min_le_left _ _)
+  have hvarepsilonN : |varepsilon| ≤ etan := by
+    rw [abs_of_nonneg hvarepsilon]
+    exact hvarepsilonEta.trans (min_le_right _ _)
+  have hc := hcap varepsilon hvarepsilonC
+  have hn := hniche varepsilon hvarepsilonN
+  rw [hcapCoeff, independentWallCap_update_eq_of_not_endpoint h t varepsilon ht] at hc
+  rw [hnicheCoeff, independentWallNiche_update_eq h t varepsilon] at hn
+  simp only [hin, Bool.false_eq_true, ite_false, one_mul] at hn
+  rw [raisedPolygonSupport_eq_update K t varepsilon, raisedPolygonSupport_zero K t,
+    polygonHeightArea]
+  change
+    |(ClassicalResults.area (polygonHeightCap (Function.update h t (h t + varepsilon))) -
+        ClassicalResults.area (polygonHeightNiche (Function.update h t (h t + varepsilon)))) -
+      (ClassicalResults.area (polygonHeightCap h) -
+        ClassicalResults.area (polygonHeightNiche h)) -
+      ((surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}).toReal -
+        polygonCapPolylineLength K t) * varepsilon| ≤
+      (Cc + Cn) * varepsilon ^ 2
+  calc
+    _ = |(ClassicalResults.area
+          (polygonHeightCap (Function.update h t (h t + varepsilon))) -
+        ClassicalResults.area (polygonHeightCap h) -
+        (surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}).toReal * varepsilon) -
+      (ClassicalResults.area
+          (polygonHeightNiche (Function.update h t (h t + varepsilon))) -
+        ClassicalResults.area (polygonHeightNiche h) -
+        polygonCapPolylineLength K t * varepsilon)| := by ring_nf
+    _ ≤ |ClassicalResults.area
+          (polygonHeightCap (Function.update h t (h t + varepsilon))) -
+        ClassicalResults.area (polygonHeightCap h) -
+        (surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}).toReal * varepsilon| +
+      |ClassicalResults.area
+          (polygonHeightNiche (Function.update h t (h t + varepsilon))) -
+        ClassicalResults.area (polygonHeightNiche h) -
+        polygonCapPolylineLength K t * varepsilon| := abs_sub _ _
+    _ ≤ Cc * varepsilon ^ 2 + Cn * varepsilon ^ 2 := add_le_add hc hn
+    _ = (Cc + Cn) * varepsilon ^ 2 := by ring
+
+/-- The balancing area estimate for a simultaneous endpoint-wall displacement. -/
+theorem polygonCap_balancing_estimate_of_endpoint {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (t : angleDomain Θ)
+    (ht : t.val ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    ∃ C eta : ℝ, 0 ≤ C ∧ 0 < eta ∧ ∀ varepsilon : ℝ,
+      0 ≤ varepsilon → varepsilon ≤ eta →
+      |polygonHeightArea (raisedPolygonSupport K t varepsilon) -
+        polygonHeightArea (raisedPolygonSupport K t 0) -
+        ((surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}).toReal -
+          polygonCapPolylineLength K t) * varepsilon| ≤ C * varepsilon ^ 2 := by
+  classical
+  let h := polygonCapSupportHeight K
+  obtain ⟨⟨nc, Hc, hHc, hSc⟩, ⟨nn, Hn, hHn, hSn⟩⟩ := polygonCap_niche_simpleNef Θ h
+  have hcuMem :
+      (⟨(t.val : Real.Angle), h t, false, false⟩ : PlanarHalfPlaneData) ∈
+        Set.range Hc := by
+    rw [hHc]
+    exact Or.inl ⟨t.val, t.property, by simp [polygonHeightValue]⟩
+  obtain ⟨icu, hicu⟩ := hcuMem
+  have hclMem :
+      (⟨(t.val : Real.Angle), h t - 1, true, false⟩ : PlanarHalfPlaneData) ∈
+        Set.range Hc := by
+    rw [hHc]
+    exact Or.inr ⟨t.val, ht, by simp [polygonHeightValue, t.property]⟩
+  obtain ⟨icl, hicl⟩ := hclMem
+  have hnMem :
+      (⟨(t.val : Real.Angle), h t - 1, true, false⟩ : PlanarHalfPlaneData) ∈
+        Set.range Hn := by
+    rw [hHn]
+    exact Or.inr ⟨t.val, ht, by simp [polygonHeightValue, t.property]⟩
+  obtain ⟨in_, hin⟩ := hnMem
+  obtain ⟨Cu, etau, hCu, hetau, hcapUpper⟩ :=
+    polygonCap_upper_wall_area_variation h Hc hHc hSc.1 icu t hicu
+  obtain ⟨Cl, etal, hCl, hetal, hcapLower⟩ :=
+    polygonCap_lower_wall_area_variation h Hc hHc hSc.1 icl t hicl
+  obtain ⟨tn, Cn, etan, htnAngle, hCn, hetan, hniche⟩ :=
+    polygonNiche_wall_area_variation h Hn hHn hSn.1 in_
+  have hinAngle : (Hn in_).angle = (t.val : Real.Angle) := by rw [hin]
+  have htn : tn = t := angleDomain_coe_injective Θ (htnAngle.symm.trans hinAngle)
+  subst tn
+  obtain ⟨R, epsilonZero, hR, hepsilonZero, huniform⟩ :=
+    polygonPerturbation_uniform_bounds Θ h
+  have hupperCoeff :
+      (MeasureTheory.Measure.hausdorffMeasure 1
+        (frontier (polygonHeightCap h) ∩ (Hc icu).boundaryLine)).toReal =
+        (surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}).toReal := by
+    rw [polygonHeightCap_supportHeight K, hicu]
+    simpa [PlanarHalfPlaneData.boundaryLine, h, polygonCapSupportHeight] using
+      congrArg ENNReal.toReal
+        (hausdorffMeasure_frontier_inter_supportingLine K.val.val
+          (t.val : Real.Angle))
+  have hlowerCoeff :
+      (MeasureTheory.Measure.hausdorffMeasure 1
+        (frontier (polygonHeightCap h) ∩ (Hc icl).boundaryLine)).toReal =
+        (surfaceAreaMeasure K.val.val
+          {((t.val + Real.pi : ℝ) : Real.Angle)}).toReal := by
+    rw [polygonHeightCap_supportHeight K, hicl]
+    simpa [PlanarHalfPlaneData.boundaryLine, h, polygonCapSupportHeight] using
+      congrArg ENNReal.toReal
+        (hausdorffMeasure_frontier_inter_lowerLine_of_endpoint K ht)
+  have hnicheCoeff :
+      (MeasureTheory.Measure.hausdorffMeasure 1
+        (frontier (polygonHeightNiche h) ∩ (Hn in_).boundaryLine)).toReal =
+        (surfaceAreaMeasure K.val.val
+          {((t.val + Real.pi : ℝ) : Real.Angle)}).toReal -
+          polygonCapPolylineLength K t := by
+    rw [polygonHeightNiche_supportHeight K, hin]
+    simpa [nicheBoundaryLength, PlanarHalfPlaneData.boundaryLine, h,
+      polygonCapSupportHeight] using nicheBoundaryLength_lowerLine_of_endpoint K t ht
+  let eta := min etau (min etal (min etan (min epsilonZero 1)))
+  refine ⟨Cu + Cl + Cn, eta, add_nonneg (add_nonneg hCu hCl) hCn,
+    lt_min hetau (lt_min hetal (lt_min hetan (lt_min hepsilonZero zero_lt_one))), ?_⟩
+  intro varepsilon hvarepsilon hvarepsilonEta
+  have hvarepsilonU : |varepsilon| ≤ etau := by
+    rw [abs_of_nonneg hvarepsilon]
+    exact hvarepsilonEta.trans (by simp [eta])
+  have hvarepsilonL : |varepsilon| ≤ etal := by
+    rw [abs_of_nonneg hvarepsilon]
+    exact hvarepsilonEta.trans (by simp [eta])
+  have hvarepsilonN : |varepsilon| ≤ etan := by
+    rw [abs_of_nonneg hvarepsilon]
+    exact hvarepsilonEta.trans (by simp [eta])
+  have hvarepsilonZero : varepsilon ≤ epsilonZero :=
+    hvarepsilonEta.trans (by simp [eta])
+  have hvarepsilonOne : varepsilon ≤ 1 :=
+    hvarepsilonEta.trans (by simp [eta])
+  have hu := hcapUpper varepsilon hvarepsilonU
+  have hl := hcapLower varepsilon hvarepsilonL
+  have hn := hniche varepsilon hvarepsilonN
+  rw [hupperCoeff] at hu
+  rw [hlowerCoeff] at hl
+  rw [hnicheCoeff, independentWallNiche_update_eq h t varepsilon] at hn
+  simp only [hin, ↓reduceIte, Real.Angle.coe_add, neg_mul, one_mul, neg_sub] at hn
+  have hchangeUpper (s : angleDomain Θ) :
+      |Function.update h t (h t + varepsilon) s - h s| ≤ epsilonZero := by
+    by_cases hst : s = t
+    · subst s
+      have hself : Function.update h t (h t + varepsilon) t = h t + varepsilon :=
+        Function.update_self t (h t + varepsilon) h
+      rw [hself]
+      simpa [abs_of_nonneg hvarepsilon] using hvarepsilonZero
+    · simp [Function.update, hst, le_of_lt hepsilonZero]
+  have hchangeLower (s : angleDomain Θ) :
+      |Function.update (fun r ↦ h r - 1) t (h t - 1 + varepsilon) s -
+        (h s - 1)| ≤ epsilonZero := by
+    by_cases hst : s = t
+    · subst s
+      have hself :
+          Function.update (fun r ↦ h r - 1) t (h t - 1 + varepsilon) t =
+            h t - 1 + varepsilon :=
+        Function.update_self t (h t - 1 + varepsilon) (fun r ↦ h r - 1)
+      rw [hself]
+      simpa [abs_of_nonneg hvarepsilon] using hvarepsilonZero
+    · simp [Function.update, hst, le_of_lt hepsilonZero]
+  have hbound (upper lower : PolygonHeightSpace Θ)
+      (hupper : upper = h ∨ upper = Function.update h t (h t + varepsilon))
+      (hlower : lower = (fun s ↦ h s - 1) ∨
+        lower = Function.update (fun s ↦ h s - 1) t (h t - 1 + varepsilon)) :
+      independentWallCap upper lower ⊆ Metric.closedBall 0 R := by
+    apply (huniform upper lower ?_ ?_).1
+    · rcases hupper with rfl | rfl
+      · intro s
+        simp [le_of_lt hepsilonZero]
+      · exact hchangeUpper
+    · rcases hlower with rfl | rfl
+      · intro s
+        simp [le_of_lt hepsilonZero]
+      · exact hchangeLower
+  have hadd := independentWallCap_area_update_add h t varepsilon R
+    hvarepsilon hvarepsilonOne hbound
+  rw [independentWallCap_update_both_eq h t varepsilon,
+    independentWallCap_sub_one (Θ := Θ)] at hadd
+  rw [raisedPolygonSupport_eq_update K t varepsilon, raisedPolygonSupport_zero K t,
+    polygonHeightArea]
+  change
+    |(ClassicalResults.area (polygonHeightCap (Function.update h t (h t + varepsilon))) -
+        ClassicalResults.area (polygonHeightNiche (Function.update h t (h t + varepsilon)))) -
+      (ClassicalResults.area (polygonHeightCap h) -
+        ClassicalResults.area (polygonHeightNiche h)) -
+      ((surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}).toReal -
+        polygonCapPolylineLength K t) * varepsilon| ≤
+      (Cu + Cl + Cn) * varepsilon ^ 2
+  let upperResidual :=
+    ClassicalResults.area
+        (independentWallCap (Function.update h t (h t + varepsilon)) (fun s ↦ h s - 1)) -
+      ClassicalResults.area (polygonHeightCap h) -
+      (surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}).toReal * varepsilon
+  let lowerResidual :=
+    ClassicalResults.area
+        (independentWallCap h
+          (Function.update (fun s ↦ h s - 1) t (h t - 1 + varepsilon))) -
+      ClassicalResults.area (polygonHeightCap h) +
+      (surfaceAreaMeasure K.val.val
+        {((t.val + Real.pi : ℝ) : Real.Angle)}).toReal * varepsilon
+  let nicheResidual :=
+    ClassicalResults.area (polygonHeightNiche (Function.update h t (h t + varepsilon))) -
+      ClassicalResults.area (polygonHeightNiche h) +
+      ((surfaceAreaMeasure K.val.val
+        {((t.val + Real.pi : ℝ) : Real.Angle)}).toReal -
+        polygonCapPolylineLength K t) * varepsilon
+  have hn' : |nicheResidual| ≤ Cn * varepsilon ^ 2 := by
+    dsimp [nicheResidual]
+    convert hn using 1
+    ring_nf
+  have hresidual :
+      (ClassicalResults.area (polygonHeightCap (Function.update h t (h t + varepsilon))) -
+          ClassicalResults.area
+            (polygonHeightNiche (Function.update h t (h t + varepsilon)))) -
+        (ClassicalResults.area (polygonHeightCap h) -
+          ClassicalResults.area (polygonHeightNiche h)) -
+        ((surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}).toReal -
+          polygonCapPolylineLength K t) * varepsilon =
+      upperResidual + lowerResidual - nicheResidual := by
+    dsimp [upperResidual, lowerResidual, nicheResidual]
+    linarith [hadd]
+  rw [hresidual]
+  calc
+    |upperResidual + lowerResidual - nicheResidual| ≤
+        |upperResidual + lowerResidual| + |nicheResidual| := abs_sub _ _
+    _ ≤ (|upperResidual| + |lowerResidual|) + |nicheResidual| :=
+      add_le_add (abs_add_le _ _) (le_refl _)
+    _ ≤ (Cu * varepsilon ^ 2 + Cl * varepsilon ^ 2) + Cn * varepsilon ^ 2 := by
+      exact add_le_add (add_le_add hu hl) hn'
+    _ = (Cu + Cl + Cn) * varepsilon ^ 2 := by ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Balancing
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private lemma exists_pos_lt_of_quadratic_error {f : ℝ → ℝ} {a C η δ : ℝ}
+    (ha : 0 < a) (hC : 0 ≤ C) (hη : 0 < η) (hδ : 0 < δ)
+    (herror : ∀ ε, 0 ≤ ε → ε ≤ η → |f ε - f 0 - a * ε| ≤ C * ε ^ 2) :
+    ∃ ε, 0 < ε ∧ ε < δ ∧ f 0 < f ε := by
+  let ε := min (η / 2) (min (δ / 2) (a / (2 * (C + 1))))
+  have hε : 0 < ε := lt_min (half_pos hη)
+    (lt_min (half_pos hδ) (div_pos ha (by positivity)))
+  have hεη : ε ≤ η := (min_le_left _ _).trans (by linarith)
+  have hεδ : ε < δ := (min_le_right _ _).trans_lt
+    ((min_le_left _ _).trans_lt (by linarith))
+  have hεa : ε ≤ a / (2 * (C + 1)) := (min_le_right _ _).trans (min_le_right _ _)
+  have hprod : ε * (2 * (C + 1)) ≤ a :=
+    (le_div_iff₀ (by positivity : 0 < 2 * (C + 1))).mp hεa
+  have hsmall : C * ε < a := by nlinarith
+  have hquad : C * ε ^ 2 < a * ε := by nlinarith [mul_pos (sub_pos.mpr hsmall) hε]
+  have hlower := (abs_le.mp (herror ε hε.le hεη)).1
+  exact ⟨ε, hε, hεδ, by linarith⟩
+
+/-- A support-height increment has the balancing first-order area term. -/
+theorem polygonCap_balancing_estimate {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (t : angleDomain Θ) :
+    ∃ C η : ℝ, 0 ≤ C ∧ 0 < η ∧ ∀ ε : ℝ, 0 ≤ ε → ε ≤ η →
+      |polygonHeightArea (raisedPolygonSupport K t ε) -
+        polygonHeightArea (raisedPolygonSupport K t 0) -
+        ((surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}).toReal -
+          polygonCapPolylineLength K t) * ε| ≤ C * ε ^ 2 := by
+  by_cases ht : t.val ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)
+  · exact polygonCap_balancing_estimate_of_endpoint K t ht
+  · exact polygonCap_balancing_estimate_of_not_endpoint K t ht
+
+/-- A maximum polygon cap has balanced boundary coefficients. -/
+theorem maximumPolygonCap_balanced {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (hK : IsMaximumPolygonCap Θ K) : IsBalancedPolygonCap K := by
+  classical
+  by_contra hnot
+  obtain ⟨t, ht⟩ := polygonCap_not_balanced_positive K hnot
+  have hnonneg : 0 ≤ polygonCapPolylineLength K t := by
+    unfold polygonCapPolylineLength
+    exact Finset.sum_nonneg (fun _ _ ↦ by split_ifs <;> positivity)
+  have hfinite : surfaceAreaMeasure K.val.val {(t.val : Real.Angle)} ≠ ⊤ := by
+    let := (surfaceAreaMeasure_face_union K.val.val).1
+    exact MeasureTheory.measure_ne_top _ _
+  have hpos : 0 < surfaceAreaMeasure K.val.val {(t.val : Real.Angle)} :=
+    lt_of_le_of_lt zero_le ht
+  have hgain : 0 < (surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}).toReal -
+      polygonCapPolylineLength K t := by
+    have hh := (ENNReal.toReal_lt_toReal ENNReal.ofReal_ne_top hfinite).mpr ht
+    rw [ENNReal.toReal_ofReal hnonneg] at hh
+    exact sub_pos.mpr hh
+  obtain ⟨C, η, hC, hη, herror⟩ := polygonCap_balancing_estimate K t
+  obtain ⟨δ, hδ, hfeasible⟩ := polygonCap_positive_height_increment K t hpos
+  obtain ⟨ε, hε, hεδ, harea⟩ := exists_pos_lt_of_quadratic_error hgain hC hη hδ herror
+  obtain ⟨K', hK'⟩ := hfeasible ε hε hεδ
+  obtain ⟨L, v, htranslate⟩ := K'.property
+  have hreduce := polygonHeightArea_le_translateArea (raisedPolygonSupport K t ε) K' hK'.symm
+  rw [(polygonTranslateExtensions_eq L v K' htranslate).2] at hreduce
+  have hzero : raisedPolygonSupport K t 0 =
+      (fun s ↦ supportValue (K.val.val : Set Point) (s.val : Real.Angle)) := by
+    funext s
+    simp [raisedPolygonSupport]
+  rw [hzero, (polygonHeightNiche_of_cap K).2] at harea
+  exact (not_lt_of_ge (hK.2 L)) (harea.trans_le hreduce)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Edge normals and contact vertices of a finite half-plane intersection
+
+A convex body presented as a finite intersection of closed half-planes has only finitely many
+possible contact points and only finitely many possible proper edge normals. This file records
+both facts in the form used by the discrete estimates on polygon caps:
+
+* `exists_active_constraint_of_mem_notMem_interior`: a boundary point activates a constraint;
+* `exists_active_constraint_of_forall_notMem_add_smul`: a direction that immediately leaves the
+  body activates a constraint increasing along it;
+* `properEdgeNormal_eq_constraint_or_add_pi`: a nondegenerate exposed edge has a constraint
+  normal, up to half a turn;
+* `properEdgeNormal_eq_constraint`: the same, without the antipodal alternative;
+* `PolygonCapSpace.properEdgeNormal_mem_allowed_or_antipodal` and
+  `PolygonCapSpace.properEdgeNormal_mem_allowed`: the polygon-cap versions;
+* `finiteConstraintVertices`: the finite set of transversal constraint-line intersections, which
+  contains every singleton exposed edge;
+* `PolygonCapSpace.surfaceAreaMeasure_compl_properEdgeNormals_eq_zero`: the surface measure of a
+  polygon cap is carried by its proper edge normals.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A point of a finite half-plane intersection off its interior lies on an active constraint. -/
+theorem exists_active_constraint_of_mem_notMem_interior
+    (C : Set (Real.Angle × ℝ)) (hC : C.Finite) (S : Set Point)
+    (hS : S = ⋂ c ∈ C, normalHalfPlane c.1 c.2 false false)
+    {x : Point} (hxS : x ∈ S) (hxint : x ∉ interior S) :
+    ∃ c ∈ C, inner ℝ x (normalVector c.1) = c.2 := by
+  by_contra h
+  push Not at h
+  apply hxint
+  rw [hS, hC.interior_biInter]
+  simp only [Set.mem_iInter]
+  intro c hc
+  have hxc : inner ℝ x (normalVector c.1) < c.2 := by
+    rw [hS] at hxS
+    have hle := Set.mem_iInter.mp (Set.mem_iInter.mp hxS c) hc
+    change inner ℝ x (normalVector c.1) ≤ c.2 at hle
+    exact lt_of_le_of_ne hle (h c hc)
+  have hUopen : IsOpen {p : Point | inner ℝ p (normalVector c.1) < c.2} :=
+    isOpen_lt (by fun_prop) continuous_const
+  apply mem_interior_iff_mem_nhds.mpr
+  apply Filter.mem_of_superset (hUopen.mem_nhds hxc)
+  intro p hp
+  change inner ℝ p (normalVector c.1) < c.2 at hp
+  change inner ℝ p (normalVector c.1) ≤ c.2
+  exact hp.le
+
+/-- A nondegenerate exposed edge has the normal angle of one of the constraints, up to half a
+turn. -/
+theorem properEdgeNormal_eq_constraint_or_add_pi
+    (K : ConvexBody Point) (C : Set (Real.Angle × ℝ)) (hC : C.Finite)
+    (hK : (K : Set Point) = ⋂ c ∈ C, normalHalfPlane c.1 c.2 false false)
+    (t : Real.Angle) (ht : (edgeVertices K t).1 ≠ (edgeVertices K t).2) :
+    ∃ c ∈ C, t = c.1 ∨ t = c.1 + (Real.pi : Real.Angle) := by
+  set p := (edgeVertices K t).1 with hpdef
+  set q := (edgeVertices K t).2 with hqdef
+  have hp : p ∈ exposedEdge K t := edgeVertices_fst_mem K t
+  have hq : q ∈ exposedEdge K t := edgeVertices_snd_mem K t
+  have hxedge : midpoint ℝ p q ∈ exposedEdge K t :=
+    (convex_exposedEdge K t).segment_subset hp hq (midpoint_mem_segment p q)
+  have hxfront : midpoint ℝ p q ∈ frontier (K : Set Point) := by
+    rw [← frontier_inter_supportingLine_eq_exposedEdge K t] at hxedge
+    exact hxedge.1
+  obtain ⟨c, hcC, hcx⟩ := exists_active_constraint_of_mem_notMem_interior
+    C hC (K : Set Point) hK hxedge.1 hxfront.2
+  refine ⟨c, hcC, ?_⟩
+  have hpc : inner ℝ p (normalVector c.1) ≤ c.2 := by
+    have hpK : p ∈ (K : Set Point) := hp.1
+    rw [hK] at hpK
+    exact Set.mem_iInter.mp (Set.mem_iInter.mp hpK c) hcC
+  have hqc : inner ℝ q (normalVector c.1) ≤ c.2 := by
+    have hqK : q ∈ (K : Set Point) := hq.1
+    rw [hK] at hqK
+    exact Set.mem_iInter.mp (Set.mem_iInter.mp hqK c) hcC
+  rw [midpoint_eq_smul_add, inner_smul_left, inner_add_left] at hcx
+  norm_num at hcx
+  have hpcEq : inner ℝ p (normalVector c.1) = c.2 := by linarith
+  have hqcEq : inner ℝ q (normalVector c.1) = c.2 := by linarith
+  have hvt : inner ℝ (p - q) (normalVector t) = 0 := by
+    rw [inner_sub_left, hp.2, hq.2, sub_self]
+  have hvc : inner ℝ (p - q) (normalVector c.1) = 0 := by
+    rw [inner_sub_left, hpcEq, hqcEq, sub_self]
+  exact normalVector_eq_or_eq_add_pi_of_orthogonal (sub_ne_zero.mpr ht) hvt hvc
+
+/-- If a direction immediately leaves a finite intersection of closed half-planes at a point of
+it, some constraint is active there and increases along that direction. -/
+theorem exists_active_constraint_of_forall_notMem_add_smul
+    (C : Set (Real.Angle × ℝ)) (hC : C.Finite) (S : Set Point)
+    (hS : S = ⋂ c ∈ C, normalHalfPlane c.1 c.2 false false)
+    {x w : Point} (hxS : x ∈ S) (hw : ∀ r : ℝ, 0 < r → x + r • w ∉ S) :
+    ∃ c ∈ C, inner ℝ x (normalVector c.1) = c.2 ∧ 0 < inner ℝ w (normalVector c.1) := by
+  by_contra hcon
+  push Not at hcon
+  have hmem : ∀ c ∈ C, inner ℝ x (normalVector c.1) ≤ c.2 := by
+    intro c hc
+    rw [hS] at hxS
+    have hle := Set.mem_iInter.mp (Set.mem_iInter.mp hxS c) hc
+    change inner ℝ x (normalVector c.1) ≤ c.2 at hle
+    exact hle
+  have hall : ∀ᶠ r : ℝ in nhdsWithin 0 (Set.Ioi 0), ∀ c ∈ C,
+      inner ℝ (x + r • w) (normalVector c.1) ≤ c.2 := by
+    refine hC.eventually_all.mpr ?_
+    intro c hc
+    by_cases hact : inner ℝ x (normalVector c.1) = c.2
+    · have hwc : inner ℝ w (normalVector c.1) ≤ 0 := hcon c hc hact
+      filter_upwards [self_mem_nhdsWithin] with r hr
+      rw [inner_add_left, real_inner_smul_left, hact]
+      have hrw : r * inner ℝ w (normalVector c.1) ≤ 0 :=
+        mul_nonpos_of_nonneg_of_nonpos (le_of_lt hr) hwc
+      linarith
+    · have hlt : inner ℝ x (normalVector c.1) < c.2 := lt_of_le_of_ne (hmem c hc) hact
+      have hcont : Continuous fun r : ℝ ↦ inner ℝ (x + r • w) (normalVector c.1) := by
+        fun_prop
+      have h0 : inner ℝ (x + (0 : ℝ) • w) (normalVector c.1) < c.2 := by simpa using hlt
+      have hev := (hcont.tendsto 0).eventually (gt_mem_nhds h0)
+      exact (hev.filter_mono nhdsWithin_le_nhds).mono fun r hr ↦ hr.le
+  obtain ⟨r, hrall, hrpos⟩ := (hall.and self_mem_nhdsWithin).exists
+  refine hw r hrpos ?_
+  rw [hS]
+  refine Set.mem_iInter.2 fun c ↦ Set.mem_iInter.2 fun hc ↦ ?_
+  change inner ℝ (x + r • w) (normalVector c.1) ≤ c.2
+  exact hrall c hc
+
+/-- The normal of a nondegenerate exposed edge of a finite intersection of closed half-planes is
+itself a constraint normal. -/
+theorem properEdgeNormal_eq_constraint (K : ConvexBody Point)
+    (C : Set (Real.Angle × ℝ)) (hC : C.Finite)
+    (hK : (K : Set Point) = ⋂ c ∈ C, normalHalfPlane c.1 c.2 false false)
+    (t : Real.Angle) (ht : (edgeVertices K t).1 ≠ (edgeVertices K t).2) :
+    ∃ c ∈ C, t = c.1 := by
+  set p := (edgeVertices K t).1 with hpdef
+  set q := (edgeVertices K t).2 with hqdef
+  have hp : p ∈ exposedEdge K t := edgeVertices_fst_mem K t
+  have hq : q ∈ exposedEdge K t := edgeVertices_snd_mem K t
+  have hx : (1 / 2 : ℝ) • p + (1 / 2 : ℝ) • q ∈ exposedEdge K t :=
+    (convex_exposedEdge K t) hp hq (by norm_num) (by norm_num) (by norm_num)
+  have hout : ∀ r : ℝ, 0 < r →
+      (1 / 2 : ℝ) • p + (1 / 2 : ℝ) • q + r • normalVector t ∉ (K : Set Point) := by
+    intro r hr hmemK
+    have hle := inner_le_supportValue K hmemK t
+    rw [inner_add_left, real_inner_smul_left, inner_normalVector_self_angle, mul_one,
+      hx.2] at hle
+    linarith
+  obtain ⟨c, hcC, hact, hpos⟩ :=
+    exists_active_constraint_of_forall_notMem_add_smul C hC _ hK hx.1 hout
+  have hpc : inner ℝ p (normalVector c.1) ≤ c.2 := by
+    have hpK : p ∈ (K : Set Point) := hp.1
+    rw [hK] at hpK
+    have h := Set.mem_iInter.mp (Set.mem_iInter.mp hpK c) hcC
+    change inner ℝ p (normalVector c.1) ≤ c.2 at h
+    exact h
+  have hqc : inner ℝ q (normalVector c.1) ≤ c.2 := by
+    have hqK : q ∈ (K : Set Point) := hq.1
+    rw [hK] at hqK
+    have h := Set.mem_iInter.mp (Set.mem_iInter.mp hqK c) hcC
+    change inner ℝ q (normalVector c.1) ≤ c.2 at h
+    exact h
+  have hmid : inner ℝ ((1 / 2 : ℝ) • p + (1 / 2 : ℝ) • q) (normalVector c.1) =
+      (inner ℝ p (normalVector c.1) + inner ℝ q (normalVector c.1)) / 2 := by
+    rw [inner_add_left, real_inner_smul_left, real_inner_smul_left]
+    ring
+  rw [hmid] at hact
+  have hpceq : inner ℝ p (normalVector c.1) = c.2 := by linarith
+  have hqceq : inner ℝ q (normalVector c.1) = c.2 := by linarith
+  have hvt : inner ℝ (p - q) (normalVector t) = 0 := by
+    rw [inner_sub_left, hp.2, hq.2, sub_self]
+  have hvc : inner ℝ (p - q) (normalVector c.1) = 0 := by
+    rw [inner_sub_left, hpceq, hqceq, sub_self]
+  rcases normalVector_eq_or_eq_add_pi_of_orthogonal (sub_ne_zero.mpr ht) hvt hvc with h | h
+  · exact ⟨c, hcC, h⟩
+  · exfalso
+    rw [h, normalVector_add_pi_angle, inner_neg_left, inner_normalVector_self_angle] at hpos
+    linarith
+
+/-- The allowed normal set of a polygon cap is finite. -/
+theorem finite_polygonCapNormals (Θ : AngleSet) :
+    (((fun r : ℝ ↦ (r : Real.Angle)) '' angleDomain Θ) ∪
+      capLowerNormals Θ.angle).Finite := by
+  have hdomain : (angleDomain Θ).Finite := by
+    unfold angleDomain
+    exact ((Θ.directions.finite_toSet.union
+      (Θ.directions.finite_toSet.image (fun t ↦ t + Real.pi / 2))).union
+        ((Set.finite_singleton (Real.pi / 2)).insert Θ.angle))
+  exact (hdomain.image _).union ((Set.finite_singleton _).insert _)
+
+/-- Every nondegenerate exposed edge of a polygon cap has an allowed or antipodal normal. -/
+theorem PolygonCapSpace.properEdgeNormal_mem_allowed_or_antipodal
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) (t : Real.Angle)
+    (ht : (edgeVertices K.val.val t).1 ≠ (edgeVertices K.val.val t).2) :
+    t ∈ ((fun r : ℝ ↦ (r : Real.Angle)) '' angleDomain Θ) ∪
+        capLowerNormals Θ.angle ∪
+      ((fun u : Real.Angle ↦ u + (Real.pi : Real.Angle)) ''
+        (((fun r : ℝ ↦ (r : Real.Angle)) '' angleDomain Θ) ∪
+          capLowerNormals Θ.angle)) := by
+  obtain ⟨C, hC, hCN, hKC⟩ := K.property.finite_constraints (finite_polygonCapNormals Θ)
+  obtain ⟨c, hcC, htc | htc⟩ :=
+    properEdgeNormal_eq_constraint_or_add_pi K.val.val C hC hKC t ht
+  · exact Or.inl (htc ▸ hCN c hcC)
+  · exact Or.inr ⟨c.1, hCN c hcC, htc.symm⟩
+
+/-- Every proper edge normal of a polygon cap is an allowed normal. -/
+theorem PolygonCapSpace.properEdgeNormal_mem_allowed {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (t : Real.Angle)
+    (ht : (edgeVertices K.val.val t).1 ≠ (edgeVertices K.val.val t).2) :
+    t ∈ ((fun r : ℝ ↦ (r : Real.Angle)) '' angleDomain Θ) ∪ capLowerNormals Θ.angle := by
+  obtain ⟨C, hC, hCN, hKC⟩ := K.property.finite_constraints (finite_polygonCapNormals Θ)
+  obtain ⟨c, hcC, htc⟩ := properEdgeNormal_eq_constraint K.val.val C hC hKC t ht
+  exact htc ▸ hCN c hcC
+
+/-- The finitely many transversal intersection points of a finite constraint family. -/
+def finiteConstraintVertices (C : Set (Real.Angle × ℝ)) : Set Point := by
+  classical
+  exact ⋃ c ∈ C, ⋃ d ∈ C,
+    if c.1 = d.1 ∨ c.1 = d.1 + (Real.pi : Real.Angle) then ∅
+    else normalLine c.1 c.2 ∩ normalLine d.1 d.2
+
+/-- A finite constraint family has finitely many transversal intersection points. -/
+theorem finite_finiteConstraintVertices (C : Set (Real.Angle × ℝ))
+    (hC : C.Finite) : (finiteConstraintVertices C).Finite := by
+  classical
+  unfold finiteConstraintVertices
+  refine hC.biUnion fun c _ ↦ hC.biUnion fun d _ ↦ ?_
+  split_ifs with hparallel
+  · exact Set.finite_empty
+  · apply Set.Subsingleton.finite
+    intro p hp q hq
+    simp only [Set.mem_inter_iff, normalLine, Set.mem_ofPred_eq] at hp hq
+    by_contra hpq
+    have hcorth : inner ℝ (p - q) (normalVector c.1) = 0 := by
+      rw [inner_sub_left, hp.1, hq.1, sub_self]
+    have hdorth : inner ℝ (p - q) (normalVector d.1) = 0 := by
+      rw [inner_sub_left, hp.2, hq.2, sub_self]
+    exact hparallel (normalVector_eq_or_eq_add_pi_of_orthogonal
+      (sub_ne_zero.mpr hpq) hcorth hdorth)
+
+/-- A singleton exposed edge of a finite half-plane intersection is a constraint vertex. -/
+theorem singleton_exposedEdge_mem_finiteConstraintVertices
+    (C : Set (Real.Angle × ℝ)) (hC : C.Finite) (K : ConvexBody Point)
+    (hK : (K : Set Point) = ⋂ c ∈ C, normalHalfPlane c.1 c.2 false false)
+    {t : Real.Angle} {p : Point} (hp : exposedEdge K t = {p}) :
+    p ∈ finiteConstraintVertices C := by
+  classical
+  have hpedge : p ∈ exposedEdge K t := by rw [hp]; exact Set.mem_singleton p
+  have hpfront : p ∈ frontier (K : Set Point) := by
+    rw [← frontier_inter_supportingLine_eq_exposedEdge K t] at hpedge
+    exact hpedge.1
+  obtain ⟨c, hc, hpc⟩ := exists_active_constraint_of_mem_notMem_interior
+    C hC (K : Set Point) hK hpedge.1 hpfront.2
+  by_contra hnot
+  have hparallel (d : Real.Angle × ℝ) (hd : d ∈ C)
+      (hpd : inner ℝ p (normalVector d.1) = d.2) :
+      c.1 = d.1 ∨ c.1 = d.1 + (Real.pi : Real.Angle) := by
+    by_contra hnon
+    apply hnot
+    unfold finiteConstraintVertices
+    exact Set.mem_iUnion.2 ⟨c, Set.mem_iUnion.2 ⟨hc, Set.mem_iUnion.2 ⟨d,
+      Set.mem_iUnion.2 ⟨hd, by simp only [hnon, ↓reduceIte]; exact ⟨hpc, hpd⟩⟩⟩⟩⟩
+  have horth : inner ℝ (tangentVector c.1) (normalVector c.1) = 0 := by
+    induction c.1 using Real.Angle.induction_on with
+    | _ r => rw [real_inner_comm, inner_normalVector_tangentVector]
+  have hvne : tangentVector c.1 ≠ 0 := by
+    have hvself : inner ℝ (tangentVector c.1) (tangentVector c.1) = 1 := by
+      induction c.1 using Real.Angle.induction_on with
+      | _ r => exact inner_tangentVector_self r
+    intro hv
+    simp [hv] at hvself
+  have hevent : ∀ᶠ r : ℝ in nhds 0, p + r • tangentVector c.1 ∈ (K : Set Point) := by
+    have hall : ∀ᶠ r : ℝ in nhds 0, ∀ d ∈ C,
+        inner ℝ (p + r • tangentVector c.1) (normalVector d.1) ≤ d.2 := by
+      apply hC.eventually_all.mpr
+      intro d hd
+      by_cases hpd : inner ℝ p (normalVector d.1) = d.2
+      · have hvd : inner ℝ (tangentVector c.1) (normalVector d.1) = 0 := by
+          rcases hparallel d hd hpd with heq | heq
+          · rwa [← heq]
+          · have hnn : normalVector d.1 = -normalVector c.1 := by
+              rw [heq, normalVector_add_pi_angle, neg_neg]
+            rw [hnn, inner_neg_right, horth, neg_zero]
+        exact Filter.Eventually.of_forall fun r ↦ by
+          rw [inner_add_left, real_inner_smul_left, hvd, mul_zero, add_zero, hpd]
+      · have hple : inner ℝ p (normalVector d.1) ≤ d.2 := by
+          have hm := hpedge.1
+          rw [hK] at hm
+          exact Set.mem_iInter.mp (Set.mem_iInter.mp hm d) hd
+        have hlt : inner ℝ p (normalVector d.1) < d.2 := lt_of_le_of_ne hple hpd
+        have hcont : Continuous
+            (fun r : ℝ ↦ inner ℝ (p + r • tangentVector c.1) (normalVector d.1)) := by
+          fun_prop
+        have hzero : inner ℝ (p + (0 : ℝ) • tangentVector c.1) (normalVector d.1) < d.2 := by
+          simpa using hlt
+        filter_upwards [(hcont.tendsto 0).eventually (gt_mem_nhds hzero)] with r hr
+        exact hr.le
+    filter_upwards [hall] with r hr
+    rw [hK]
+    exact Set.mem_iInter.2 fun d ↦ Set.mem_iInter.2 fun hd ↦ hr d hd
+  obtain ⟨ε, hε, hball⟩ := Metric.mem_nhds_iff.mp hevent
+  have hplus : p + (ε / 2) • tangentVector c.1 ∈ (K : Set Point) := hball (by
+    simp only [Metric.mem_ball, dist_zero_right, Real.norm_eq_abs]
+    rw [abs_of_pos (by positivity : 0 < ε / 2)]
+    linarith)
+  have hminus : p + (-(ε / 2)) • tangentVector c.1 ∈ (K : Set Point) := hball (by
+    simp only [Metric.mem_ball, dist_zero_right, Real.norm_eq_abs, abs_neg]
+    rw [abs_of_pos (by positivity : 0 < ε / 2)]
+    linarith)
+  have hplusle := inner_le_supportValue K hplus t
+  have hminusle := inner_le_supportValue K hminus t
+  rw [inner_add_left, real_inner_smul_left, hpedge.2] at hplusle hminusle
+  have hvt : inner ℝ (tangentVector c.1) (normalVector t) = 0 := by nlinarith
+  have hplusface : p + (ε / 2) • tangentVector c.1 ∈ exposedEdge K t := by
+    refine ⟨hplus, ?_⟩
+    change inner ℝ (p + (ε / 2) • tangentVector c.1) (normalVector t) = supportValue K t
+    rw [inner_add_left, real_inner_smul_left, hvt, mul_zero, add_zero, hpedge.2]
+  rw [hp, Set.mem_singleton_iff] at hplusface
+  have hzero : (ε / 2) • tangentVector c.1 = 0 :=
+    add_left_cancel (show p + (ε / 2) • tangentVector c.1 = p + 0 by simpa using hplusface)
+  exact hvne ((smul_eq_zero.mp hzero).resolve_left (ne_of_gt (by positivity)))
+
+/-- The transversal constraint vertices form a one-dimensional null set. -/
+theorem finiteConstraintVertices_measure_zero (C : Set (Real.Angle × ℝ))
+    (hC : C.Finite) :
+    MeasureTheory.Measure.hausdorffMeasure 1 (finiteConstraintVertices C) = 0 := by
+  let _ := MeasureTheory.Measure.nullSingletonClass_hausdorff Point (by norm_num : (0 : ℝ) < 1)
+  exact (finite_finiteConstraintVertices C hC).measure_zero _
+
+/-- Surface measure of a finite intersection of closed half-planes is carried by its proper
+edge normals. -/
+theorem surfaceAreaMeasure_compl_properEdgeNormals_eq_zero_of_finite_constraints
+    (C : Set (Real.Angle × ℝ)) (hC : C.Finite) (K : ConvexBody Point)
+    (hK : (K : Set Point) = ⋂ c ∈ C, normalHalfPlane c.1 c.2 false false) :
+    surfaceAreaMeasure K {t | (edgeVertices K t).1 = (edgeVertices K t).2} = 0 := by
+  have hN : {t | (edgeVertices K t).1 ≠ (edgeVertices K t).2}.Finite := by
+    apply (hC.image Prod.fst).subset
+    intro t ht
+    obtain ⟨c, hc, heq⟩ := properEdgeNormal_eq_constraint K C hC hK t ht
+    exact ⟨c, hc, heq.symm⟩
+  apply surfaceAreaMeasure_compl_properEdgeNormals_eq_zero_of_finite_carrier
+    K hN (finiteConstraintVertices C) (finiteConstraintVertices_measure_zero C hC)
+  intro p hp
+  obtain ⟨t, ht, hp⟩ := Set.mem_iUnion₂.mp hp
+  have hsingle : exposedEdge K t = {(edgeVertices K t).1} := by
+    rw [exposedEdge_eq_segment_edgeVertices, ht, segment_same]
+  have hpeq : p = (edgeVertices K t).1 := by simpa [hsingle] using hp
+  exact hpeq ▸ singleton_exposedEdge_mem_finiteConstraintVertices C hC K hK hsingle
+
+/-- Surface measure of a polygon cap is carried by its proper edge normals. -/
+theorem PolygonCapSpace.surfaceAreaMeasure_compl_properEdgeNormals_eq_zero
+    {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    surfaceAreaMeasure K.val.val
+      {t | (edgeVertices K.val.val t).1 = (edgeVertices K.val.val t).2} = 0 := by
+  obtain ⟨C, hC, _, hKC⟩ := K.property.finite_constraints (finite_polygonCapNormals Θ)
+  exact surfaceAreaMeasure_compl_properEdgeNormals_eq_zero_of_finite_constraints
+    C hC K.val.val hKC
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/Development/Geometry/Applications/Development002.lean b/LeanPool/MovingSofa/Development/Geometry/Applications/Development002.lean
new file mode 100644
index 0000000000..a4c274c9cd
--- /dev/null
+++ b/LeanPool/MovingSofa/Development/Geometry/Applications/Development002.lean
@@ -0,0 +1,7720 @@
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+module
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development005
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development006
+public import LeanPool.MovingSofa.Development.Geometry.Applications.Development001
+
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development007
+
+
+
+
+
+
+
+
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development001
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development002
+
+
+
+
+
+/-!
+# Moving sofa: related mathematical developments
+
+* `Bounds.Applications.Development001`.
+* `Cap.Applications.Development002`.
+* `Polygon.Applications.Development002`.
+* `Cap.Applications.Development003`.
+* `Bounds.Applications.Development002`.
+* `Cap.Applications.Development004`.
+* `Gerver.Applications.Development003`.
+* `Cap.Applications.Development005`.
+* `Cap.Applications.Development006`.
+* `Area.Applications.Development004`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Bounds.Arm.Estimates`.
+* `Bounds.WedgeGap.Infimum`.
+* `Bounds.WedgeGap.Limit`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Bounds / Arm / Estimates
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter MeasureTheory Set
+open scoped Topology BoundedContinuousFunction
+
+namespace MovingSofa
+
+private def positiveArmKernel (t : ℝ) (u : Real.Angle) : ℝ :=
+  if 0 < (u - (t : Real.Angle)).sin ∧ 0 ≤ (u - (t : Real.Angle)).cos then
+    (u - (t : Real.Angle)).sin
+  else 0
+
+private theorem measurable_positiveArmKernel :
+    Measurable (Function.uncurry positiveArmKernel) := by
+  have hs : Continuous (fun z : ℝ × Real.Angle ↦
+      (z.2 - (z.1 : Real.Angle)).sin) :=
+    Real.Angle.continuous_sin.comp
+      (continuous_snd.sub (Real.Angle.continuous_coe.comp continuous_fst))
+  have hc : Continuous (fun z : ℝ × Real.Angle ↦
+      (z.2 - (z.1 : Real.Angle)).cos) :=
+    Real.Angle.continuous_cos.comp
+      (continuous_snd.sub (Real.Angle.continuous_coe.comp continuous_fst))
+  exact hs.measurable.piecewise
+    ((measurableSet_lt measurable_const hs.measurable).inter
+      (measurableSet_le measurable_const hc.measurable)) measurable_const
+
+private theorem mem_positiveArmArc_iff {t : ℝ} (u : Real.Angle) :
+    u ∈ (fun s : ℝ ↦ (s : Real.Angle)) '' Ioc t (t + Real.pi / 2) ↔
+      0 < (u - (t : Real.Angle)).sin ∧ 0 ≤ (u - (t : Real.Angle)).cos := by
+  constructor
+  · rintro ⟨x, hx, rfl⟩
+    simp only [← Real.Angle.coe_sub, Real.Angle.sin_coe, Real.Angle.cos_coe]
+    exact ⟨Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hx.1)
+      (by linarith [hx.2, Real.pi_pos]),
+      Real.cos_nonneg_of_mem_Icc ⟨by linarith [Real.pi_pos, hx.1], by linarith [hx.2]⟩⟩
+  · intro hu
+    let d := (u - (t : Real.Angle)).toReal
+    have hdpos : 0 < d := by
+      have hsign : (u - (t : Real.Angle)).sign = 1 := by
+        simpa only [Real.Angle.sign, sign_eq_one_iff] using hu.1
+      exact (Real.Angle.toReal_mem_Ioo_iff_sign_pos.mpr hsign).1
+    have hdle : d ≤ Real.pi / 2 := by
+      have habs := Real.Angle.cos_nonneg_iff_abs_toReal_le_pi_div_two.1 hu.2
+      exact (le_abs_self d).trans habs
+    refine ⟨t + d, ⟨by linarith, by linarith⟩, ?_⟩
+    change (((t + d : ℝ) : Real.Angle)) = u
+    calc
+      (((t + d : ℝ) : Real.Angle)) = (t : Real.Angle) + (d : Real.Angle) :=
+        Real.Angle.coe_add t d
+      _ = (t : Real.Angle) + (u - (t : Real.Angle)) := by
+        congr 1
+        exact Real.Angle.coe_toReal (u - (t : Real.Angle))
+      _ = u := by abel
+
+private theorem positiveArmKernel_eq_indicator (t : ℝ) :
+    positiveArmKernel t =
+      ((fun s : ℝ ↦ (s : Real.Angle)) '' Ioc t (t + Real.pi / 2)).indicator
+        (fun u ↦ (u - (t : Real.Angle)).sin) := by
+  classical
+  funext u
+  rw [Set.indicator_apply]
+  simp only [positiveArmKernel, mem_positiveArmArc_iff]
+
+private theorem stronglyMeasurable_integral_positiveArmKernel (μ : Measure Real.Angle)
+    [SFinite μ] : StronglyMeasurable (fun t ↦ ∫ u, positiveArmKernel t u ∂μ) := by
+  exact measurable_positiveArmKernel.stronglyMeasurable.integral_prod_right
+
+/-- The positive tangent arm length is almost everywhere strongly measurable. -/
+theorem aestronglyMeasurable_tangentArm_fst (C : RightAngleCapSpace) :
+    AEStronglyMeasurable (fun t ↦ (tangentArmLengths C t).2.1)
+      (volume.restrict (Ioc 0 (Real.pi / 2))) := by
+  let _ : IsFiniteMeasure (surfaceAreaMeasure C.val) :=
+    (surfaceAreaMeasure_face_union C.val).1
+  apply (stronglyMeasurable_integral_positiveArmKernel
+    (surfaceAreaMeasure C.val)).aestronglyMeasurable.congr
+  filter_upwards [] with t
+  rw [tangentArm_convolution C t, positiveArmKernel_eq_indicator]
+  exact integral_indicator (μ := surfaceAreaMeasure C.val)
+    (f := fun u : Real.Angle ↦ (u - (t : Real.Angle)).sin)
+    (Real.Angle.measurableSet_image_Ioc t (t + Real.pi / 2))
+
+/-- The positive tangent arm length is nonnegative: it is the integral of the sine of a
+quarter-turn of normal directions against the surface area measure. -/
+theorem tangentArm_fst_nonneg (C : RightAngleCapSpace) {t : ℝ} :
+    0 ≤ (tangentArmLengths C t).2.1 := by
+  rw [tangentArm_convolution C t]
+  refine setIntegral_nonneg (Real.Angle.measurableSet_image_Ioc t (t + Real.pi / 2)) ?_
+  rintro u ⟨v, hv, rfl⟩
+  rw [← Real.Angle.coe_sub, Real.Angle.sin_coe]
+  exact Real.sin_nonneg_of_nonneg_of_le_pi (by linarith [hv.1])
+    (by linarith [hv.2, Real.pi_pos])
+
+/-- Both right tangent arm lengths of a right-angle cap are nonnegative: the outer corner
+realizes the support value in the tangent direction, while the two contacts lie in the cap. -/
+theorem tangentArmLengths_right_nonneg (K : RightAngleCapSpace) (t : ℝ) :
+    0 ≤ (tangentArmLengths K t).1.1 ∧ 0 ≤ (tangentArmLengths K t).1.2 := by
+  have hy : inner ℝ (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).outerCorner
+      (tangentVector (t : Real.Angle)) =
+      supportValue K.val ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle)) := by
+    change inner ℝ (supportingPlacement (K.val : Set Point) (t : Real.Angle)
+      hallwayParts.outerCorner) _ = _
+    rw [inner_supportingPlacement_tangentVector]
+    simp [hallwayParts]
+  have hmem (p : Point) (hp : p ∈ (K.val : Set Point)) :
+      inner ℝ p (tangentVector (t : Real.Angle)) ≤
+        supportValue K.val ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle)) := by
+    rw [← normalVector_add_pi_div_two (t : Real.Angle)]
+    exact inner_le_supportValue K.val hp _
+  simp only [tangentArmLengths, capVertices, inner_sub_left, hy]
+  exact ⟨sub_nonneg.mpr (hmem _ (edgeVertices_fst_mem K.val _).1),
+    sub_nonneg.mpr (hmem _ (edgeVertices_snd_mem K.val _).1)⟩
+
+/-- The positive tangent arm length is bounded by the total mass of the surface area measure. -/
+theorem abs_tangentArm_fst_le (C : RightAngleCapSpace) {t : ℝ} :
+    |(tangentArmLengths C t).2.1| ≤ (surfaceAreaMeasure C.val).real Set.univ := by
+  let _ : IsFiniteMeasure (surfaceAreaMeasure C.val) :=
+    (surfaceAreaMeasure_face_union C.val).1
+  rw [tangentArm_convolution C t]
+  change ‖∫ u in (fun s : ℝ ↦ (s : Real.Angle)) '' Ioc t (t + Real.pi / 2),
+    (u - (t : Real.Angle)).sin ∂surfaceAreaMeasure C.val‖ ≤ _
+  calc
+    _ ≤ 1 * (surfaceAreaMeasure C.val).real
+        ((fun s : ℝ ↦ (s : Real.Angle)) '' Ioc t (t + Real.pi / 2)) := by
+      apply norm_setIntegral_le_of_norm_le_const
+      · finiteness
+      · intro u hu
+        simp only [Real.norm_eq_abs]
+        rw [← Real.Angle.sin_toReal]
+        exact Real.abs_sin_le_one _
+    _ ≤ (surfaceAreaMeasure C.val).real Set.univ := by
+      simpa only [one_mul] using
+        measureReal_mono (μ := surfaceAreaMeasure C.val) (Set.subset_univ _)
+          (measure_ne_top _ _)
+
+/-- The positive tangent arm length is bounded by the total surface mass, hence integrable. -/
+theorem intervalIntegrable_tangentArm_fst (C : RightAngleCapSpace) :
+    IntervalIntegrable (fun t ↦ (tangentArmLengths C t).2.1)
+      volume 0 (Real.pi / 2) := by
+  apply (intervalIntegrable_iff_integrableOn_Ioc_of_le (by positivity)).2
+  apply (integrableOn_const
+    (μ := volume) (s := Ioc 0 (Real.pi / 2)) (measure_Ioc_lt_top.ne)
+    (C := (surfaceAreaMeasure C.val).real Set.univ)).mono'
+      (aestronglyMeasurable_tangentArm_fst C)
+  filter_upwards [] with t
+  exact abs_tangentArm_fst_le C
+
+private theorem maximumPolygonCap_diam_le (n : ℕ) (K : RightAngleCapSpace)
+    (hK : IsMaximumPolygonCapSteps n K) : Metric.diam (K.val : Set Point) ≤ 5 := by
+  classical
+  obtain ⟨hn, ⟨k, hk⟩, P, hPK, hmax⟩ := hK
+  have hs2 : Real.sqrt 2 * Real.sqrt 2 = 2 := Real.mul_self_sqrt (by norm_num)
+  have hs2pos : (0 : ℝ) < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num)
+  have hs2lt : Real.sqrt 2 < 3 / 2 := by nlinarith
+  have hinner : ∀ (p : Point) (s : ℝ),
+      inner ℝ p (normalVector (s : Real.Angle)) = p 0 * Real.cos s + p 1 * Real.sin s := by
+    intro p s
+    simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two, Real.Angle.cos_coe,
+      Real.Angle.sin_coe]
+    ring
+  have hc1 : Real.cos (Real.pi / 4) = Real.sqrt 2 / 2 := Real.cos_pi_div_four
+  have hsn1 : Real.sin (Real.pi / 4) = Real.sqrt 2 / 2 := Real.sin_pi_div_four
+  have hc2 : Real.cos (Real.pi / 4 + Real.pi / 2) = -(Real.sqrt 2 / 2) := by
+    rw [Real.cos_add, Real.cos_pi_div_four, Real.sin_pi_div_four, Real.cos_pi_div_two,
+      Real.sin_pi_div_two]
+    ring
+  have hsn2 : Real.sin (Real.pi / 4 + Real.pi / 2) = Real.sqrt 2 / 2 := by
+    rw [Real.sin_add, Real.cos_pi_div_four, Real.sin_pi_div_four, Real.cos_pi_div_two,
+      Real.sin_pi_div_two]
+    ring
+  set h1 : ℝ := supportValue (K.val : Set Point) ((Real.pi / 4 : ℝ) : Real.Angle)
+  set h2 : ℝ := supportValue (K.val : Set Point)
+    ((Real.pi / 4 + Real.pi / 2 : ℝ) : Real.Angle)
+  have htop : ∀ p ∈ (K.val : Set Point), p 1 ≤ 1 := by
+    intro p hp
+    have hle := inner_le_supportValue K.val hp ((Real.pi / 2 : ℝ) : Real.Angle)
+    rw [hinner, Real.cos_pi_div_two, Real.sin_pi_div_two, K.property.2.2.2.1] at hle
+    linarith
+  have hbot : ∀ p ∈ (K.val : Set Point), 0 ≤ p 1 := by
+    intro p hp
+    have hle := inner_le_supportValue K.val hp ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+    have hcc : Real.cos (3 * Real.pi / 2) = 0 := by
+      rw [show (3 : ℝ) * Real.pi / 2 = Real.pi + Real.pi / 2 by ring, Real.cos_add,
+        Real.cos_pi, Real.sin_pi, Real.cos_pi_div_two, Real.sin_pi_div_two]
+      ring
+    have hss : Real.sin (3 * Real.pi / 2) = -1 := by
+      rw [show (3 : ℝ) * Real.pi / 2 = Real.pi + Real.pi / 2 by ring, Real.sin_add,
+        Real.cos_pi, Real.sin_pi, Real.cos_pi_div_two, Real.sin_pi_div_two]
+      ring
+    rw [hinner, hcc, hss, K.property.2.2.2.2.2.1] at hle
+    linarith
+  have hsupp1 : ∀ p ∈ (K.val : Set Point),
+      p 0 * (Real.sqrt 2 / 2) + p 1 * (Real.sqrt 2 / 2) ≤ h1 := by
+    intro p hp
+    have hle := inner_le_supportValue K.val hp ((Real.pi / 4 : ℝ) : Real.Angle)
+    rwa [hinner, hc1, hsn1] at hle
+  have hsupp2 : ∀ p ∈ (K.val : Set Point),
+      p 0 * -(Real.sqrt 2 / 2) + p 1 * (Real.sqrt 2 / 2) ≤ h2 := by
+    intro p hp
+    have hle := inner_le_supportValue K.val hp ((Real.pi / 4 + Real.pi / 2 : ℝ) : Real.Angle)
+    rwa [hinner, hc2, hsn2] at hle
+  -- the diagonal angle belongs to the dyadic grid
+  have hdir : (rightAngleSet n hn).directions =
+      (Finset.Ioo 0 n).image (fun i : ℕ ↦ (i : ℝ) / n * (Real.pi / 2)) := rfl
+  have hquarter : Real.pi / 4 ∈ (rightAngleSet n hn).directions := by
+    obtain ⟨m, rfl⟩ : ∃ m, k = m + 1 := by
+      cases k with
+      | zero => rw [pow_zero] at hk; omega
+      | succ m => exact ⟨m, rfl⟩
+    rw [hdir]
+    refine Finset.mem_image.mpr ⟨2 ^ m, Finset.mem_Ioo.mpr ⟨?_, ?_⟩, ?_⟩
+    · positivity
+    · rw [hk]
+      exact Nat.pow_lt_pow_right (by norm_num) (Nat.lt_succ_self m)
+    · rw [hk]
+      have hpos : (0 : ℝ) < 2 ^ m := by positivity
+      push_cast
+      rw [pow_succ]
+      field_simp
+      ring
+  -- the inner corner at the diagonal angle has height at most one
+  have hsum : h1 + h2 ≤ 2 + Real.sqrt 2 := by
+    by_contra hcon
+    rw [not_le] at hcon
+    set e : ℝ := (h1 + h2 - 2 - Real.sqrt 2) / 4 with he
+    have hepos : 0 < e := by rw [he]; linarith
+    set q : Point := !₂[Real.sqrt 2 / 2 * (h1 - h2),
+      Real.sqrt 2 / 2 * (h1 + h2 - 2 - 2 * e)]
+    have hq0 : q 0 = Real.sqrt 2 / 2 * (h1 - h2) := rfl
+    have hq1 : q 1 = Real.sqrt 2 / 2 * (h1 + h2 - 2 - 2 * e) := rfl
+    have hqheight : 1 < q 1 := by
+      rw [hq1, he]
+      nlinarith
+    have hqu : inner ℝ q (normalVector ((Real.pi / 4 : ℝ) : Real.Angle)) = h1 - 1 - e := by
+      rw [hinner, hq0, hq1, hc1, hsn1]
+      linear_combination ((h1 - 1 - e) / 2) * hs2
+    have hqv : inner ℝ q (normalVector ((Real.pi / 4 + Real.pi / 2 : ℝ) : Real.Angle))
+        = h2 - 1 - e := by
+      rw [hinner, hq0, hq1, hc2, hsn2]
+      linear_combination ((h2 - 1 - e) / 2) * hs2
+    have hqe : inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = q 1 := by
+      rw [hinner, Real.cos_pi_div_two, Real.sin_pi_div_two]
+      ring
+    have hangle : (rightAngleSet n hn).angle = Real.pi / 2 := rfl
+    have hfan : q ∈ capFan (rightAngleSet n hn).angle := by
+      rw [hangle]
+      refine ⟨?_, ?_⟩ <;>
+        · change (0 : ℝ) ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+          rw [hqe]
+          linarith
+    have hquad : q ∈ innerQuadrant (P.val.val : Set Point) (Real.pi / 4) := by
+      rw [hPK]
+      refine ⟨?_, ?_⟩
+      · change inner ℝ q (normalVector ((Real.pi / 4 : ℝ) : Real.Angle)) < h1 - 1
+        rw [hqu]
+        linarith
+      · change inner ℝ q (normalVector ((Real.pi / 4 + Real.pi / 2 : ℝ) : Real.Angle)) < h2 - 1
+        rw [hqv]
+        linarith
+    have hmemniche : q ∈ polygonNiche (rightAngleSet n hn) P.val :=
+      ⟨hfan, Set.mem_biUnion hquarter hquad⟩
+    have hsub := polygonNiche_subset_of_balanced P (maximumPolygonCap_balanced P hmax)
+    have hqK : q ∈ (P.val.val : Set Point) := hsub hmemniche
+    rw [hPK] at hqK
+    exact absurd (htop q hqK) (not_le.mpr hqheight)
+  -- the outer corner height and the trapezoid containing the cap
+  set hgt : ℝ := Real.sqrt 2 / 2 * (h1 + h2) with hgtdef
+  set ctr : ℝ := Real.sqrt 2 / 2 * (h1 - h2) with hctrdef
+  have hgtle : hgt ≤ 1 + Real.sqrt 2 := by
+    rw [hgtdef]
+    nlinarith
+  have hsum1 : hgt + ctr = Real.sqrt 2 * h1 := by rw [hgtdef, hctrdef]; ring
+  have hsum2 : hgt - ctr = Real.sqrt 2 * h2 := by rw [hgtdef, hctrdef]; ring
+  have htrap1 : ∀ p ∈ (K.val : Set Point), p 0 + p 1 ≤ hgt + ctr := by
+    intro p hp
+    have hm := mul_le_mul_of_nonneg_left (hsupp1 p hp) hs2pos.le
+    have hkey : Real.sqrt 2 * (p 0 * (Real.sqrt 2 / 2) + p 1 * (Real.sqrt 2 / 2))
+        = p 0 + p 1 := by linear_combination (p 0 / 2 + p 1 / 2) * hs2
+    rw [hkey] at hm
+    linarith
+  have htrap2 : ∀ p ∈ (K.val : Set Point), p 1 - p 0 ≤ hgt - ctr := by
+    intro p hp
+    have hm := mul_le_mul_of_nonneg_left (hsupp2 p hp) hs2pos.le
+    have hkey : Real.sqrt 2 * (p 0 * -(Real.sqrt 2 / 2) + p 1 * (Real.sqrt 2 / 2))
+        = p 1 - p 0 := by linear_combination (p 1 / 2 - p 0 / 2) * hs2
+    rw [hkey] at hm
+    linarith
+  have hgtge : 1 ≤ hgt := by
+    have hle : supportValue (K.val : Set Point) ((Real.pi / 2 : ℝ) : Real.Angle) ≤ hgt := by
+      apply csSup_le (K.val.nonempty.image _)
+      rintro _ ⟨p, hp, rfl⟩
+      change inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) ≤ hgt
+      rw [hinner, Real.cos_pi_div_two, Real.sin_pi_div_two]
+      have h1p := htrap1 p hp
+      have h2p := htrap2 p hp
+      linarith
+    rw [K.property.2.2.2.1] at hle
+    exact hle
+  -- every distance inside the trapezoid is at most the length of its bottom side
+  have hdiam : Metric.diam (K.val : Set Point) ≤ 2 * hgt := by
+    refine Metric.diam_le_of_forall_dist_le (by linarith) ?_
+    intro p hp q hq
+    have hp1 := htrap1 p hp
+    have hp2 := htrap2 p hp
+    have hq1 := htrap1 q hq
+    have hq2 := htrap2 q hq
+    have hpt := htop p hp
+    have hpb := hbot p hp
+    have hqt := htop q hq
+    have hqb := hbot q hq
+    have hd : dist p q = Real.sqrt ((p 0 - q 0) ^ 2 + (p 1 - q 1) ^ 2) := by
+      rw [EuclideanSpace.dist_eq, Fin.sum_univ_two]
+      simp [Real.dist_eq, sq_abs]
+    rw [hd, show (2 : ℝ) * hgt = Real.sqrt ((2 * hgt) ^ 2) from
+      (Real.sqrt_sq (by linarith)).symm]
+    apply Real.sqrt_le_sqrt
+    have kx : (p 0 - q 0) ^ 2 ≤ (2 * hgt - p 1 - q 1) ^ 2 := by
+      nlinarith [mul_nonneg (by linarith : (0 : ℝ) ≤ 2 * hgt - p 1 - q 1 - (p 0 - q 0))
+        (by linarith : (0 : ℝ) ≤ 2 * hgt - p 1 - q 1 + (p 0 - q 0))]
+    nlinarith [kx, mul_nonneg (by linarith : (0 : ℝ) ≤ 4 * hgt - 2)
+      (by linarith : (0 : ℝ) ≤ p 1 + q 1),
+      mul_nonneg hpb (by linarith : (0 : ℝ) ≤ 1 - p 1),
+      mul_nonneg hqb (by linarith : (0 : ℝ) ≤ 1 - q 1)]
+  have hdiam5 : Metric.diam (K.val : Set Point) ≤ 5 := by linarith
+  exact hdiam5
+
+private theorem tangentArmLengths_le_of_diam (K : RightAngleCapSpace)
+    (hdiam5 : Metric.diam (K.val : Set Point) ≤ 5) :
+    ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      (tangentArmLengths K t).1.1 ∈ Set.Icc (0 : ℝ) 5 ∧
+      (tangentArmLengths K t).1.2 ∈ Set.Icc (0 : ℝ) 5 ∧
+      (tangentArmLengths K t).2.1 ∈ Set.Icc (0 : ℝ) 5 ∧
+      (tangentArmLengths K t).2.2 ∈ Set.Icc (0 : ℝ) 5 := by
+  intro t ht
+  obtain ⟨hAp, hAm, hCp, hCm⟩ := capTangentArm_identities K t
+  have huu : inner ℝ (normalVector (t : Real.Angle)) (normalVector (t : Real.Angle)) = 1 :=
+    inner_normalVector_self t
+  have hvv : inner ℝ (tangentVector (t : Real.Angle)) (tangentVector (t : Real.Angle)) = 1 :=
+    inner_tangentVector_self t
+  have huv : inner ℝ (normalVector (t : Real.Angle)) (tangentVector (t : Real.Angle)) = 0 :=
+    inner_normalVector_tangentVector t
+  have hvu : inner ℝ (tangentVector (t : Real.Angle)) (normalVector (t : Real.Angle)) = 0 := by
+    rw [real_inner_comm, huv]
+  have hnu : ‖normalVector (t : Real.Angle)‖ = 1 := norm_normalVector_real t
+  have hvn : normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+      tangentVector (t : Real.Angle) := by
+    rw [Real.Angle.coe_add, normalVector_add_pi_div_two]
+  have hnv : ‖tangentVector (t : Real.Angle)‖ = 1 := by
+    rw [← hvn]
+    exact norm_normalVector_real _
+  have hyu : inner ℝ (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).outerCorner
+      (normalVector (t : Real.Angle)) = supportValue (K.val : Set Point) (t : Real.Angle) := by
+    rw [outerCorner_eq_support_sum, inner_add_left, real_inner_smul_left, real_inner_smul_left,
+      huu, hvu]
+    ring
+  have hyv : inner ℝ (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).outerCorner
+      (tangentVector (t : Real.Angle)) =
+      supportValue (K.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) := by
+    rw [outerCorner_eq_support_sum, inner_add_left, real_inner_smul_left, real_inner_smul_left,
+      huv, hvv]
+    ring
+  have hApmem : (capVertices K t).1.1 ∈ (K.val : Set Point) :=
+    (edgeVertices_fst_mem K.val (t : Real.Angle)).1
+  have hAmmem : (capVertices K t).1.2 ∈ (K.val : Set Point) :=
+    (edgeVertices_snd_mem K.val (t : Real.Angle)).1
+  have hCpmem : (capVertices K t).2.1 ∈ (K.val : Set Point) :=
+    (edgeVertices_fst_mem K.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).1
+  have hCmmem : (capVertices K t).2.2 ∈ (K.val : Set Point) :=
+    (edgeVertices_snd_mem K.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).1
+  have hsupv : ∀ p ∈ (K.val : Set Point),
+      inner ℝ p (tangentVector (t : Real.Angle)) ≤
+        supportValue (K.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) := by
+    intro p hp
+    have hle := inner_le_supportValue K.val hp ((t + Real.pi / 2 : ℝ) : Real.Angle)
+    rwa [hvn] at hle
+  have hsupu : ∀ p ∈ (K.val : Set Point),
+      inner ℝ p (normalVector (t : Real.Angle)) ≤
+        supportValue (K.val : Set Point) (t : Real.Angle) :=
+    fun p hp ↦ inner_le_supportValue K.val hp (t : Real.Angle)
+  -- the arm identities in projected form
+  have harm : ∀ (A C : Point) (f g : ℝ),
+      (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).outerCorner =
+        A + f • tangentVector (t : Real.Angle) →
+      (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).outerCorner =
+        C + g • normalVector (t : Real.Angle) →
+      A ∈ (K.val : Set Point) → C ∈ (K.val : Set Point) →
+      (f ∈ Set.Icc (0 : ℝ) 5 ∧ g ∈ Set.Icc (0 : ℝ) 5) := by
+    intro A C f g hfA hgC hAmem hCmem
+    have hfv : inner ℝ A (tangentVector (t : Real.Angle)) + f =
+        supportValue (K.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) := by
+      rw [← hyv, hfA, inner_add_left, real_inner_smul_left, hvv]
+      ring
+    have hgu : inner ℝ C (normalVector (t : Real.Angle)) + g =
+        supportValue (K.val : Set Point) (t : Real.Angle) := by
+      rw [← hyu, hgC, inner_add_left, real_inner_smul_left, huu]
+      ring
+    have hAC : A + f • tangentVector (t : Real.Angle) = C + g • normalVector (t : Real.Angle) :=
+      hfA ▸ hgC
+    have hprojv := congrArg (fun z : Point ↦ inner ℝ z (tangentVector (t : Real.Angle))) hAC
+    have hproju := congrArg (fun z : Point ↦ inner ℝ z (normalVector (t : Real.Angle))) hAC
+    simp only [inner_add_left, real_inner_smul_left, hvv, hvu, huu, huv, mul_one,
+      mul_zero, add_zero] at hprojv hproju
+    have hdist : dist A C ≤ Metric.diam (K.val : Set Point) :=
+      Metric.dist_le_diam_of_mem K.val.isCompact.isBounded hAmem hCmem
+    have hfeq : f = inner ℝ (C - A) (tangentVector (t : Real.Angle)) := by
+      rw [inner_sub_left]
+      linarith
+    have hgeq : g = inner ℝ (A - C) (normalVector (t : Real.Angle)) := by
+      rw [inner_sub_left]
+      linarith
+    constructor
+    · refine ⟨by linarith [hsupv A hAmem], ?_⟩
+      have hb : inner ℝ (C - A) (tangentVector (t : Real.Angle)) ≤
+          ‖C - A‖ * ‖tangentVector (t : Real.Angle)‖ := real_inner_le_norm _ _
+      rw [hnv, mul_one, norm_sub_rev, ← dist_eq_norm] at hb
+      rw [hfeq]
+      linarith
+    · refine ⟨by linarith [hsupu C hCmem], ?_⟩
+      have hb : inner ℝ (A - C) (normalVector (t : Real.Angle)) ≤
+          ‖A - C‖ * ‖normalVector (t : Real.Angle)‖ := real_inner_le_norm _ _
+      rw [hnu, mul_one, ← dist_eq_norm] at hb
+      rw [hgeq]
+      linarith
+  obtain ⟨hf1, hg1⟩ := harm (capVertices K t).1.1 (capVertices K t).2.1
+    (tangentArmLengths K t).1.1 (tangentArmLengths K t).2.1 hAp hCp hApmem hCpmem
+  obtain ⟨hf2, hg2⟩ := harm (capVertices K t).1.2 (capVertices K t).2.2
+    (tangentArmLengths K t).1.2 (tangentArmLengths K t).2.2 hAm hCm hAmmem hCmmem
+  exact ⟨hf1, hf2, hg1, hg2⟩
+
+
+theorem maximumPolygonCap_arm_bound (n : ℕ) (K : RightAngleCapSpace)
+    (hK : IsMaximumPolygonCapSteps n K) :
+    Metric.diam (K.val : Set Point) ≤ 5 ∧
+    ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      (tangentArmLengths K t).1.1 ∈ Set.Icc (0 : ℝ) 5 ∧
+      (tangentArmLengths K t).1.2 ∈ Set.Icc (0 : ℝ) 5 ∧
+      (tangentArmLengths K t).2.1 ∈ Set.Icc (0 : ℝ) 5 ∧
+      (tangentArmLengths K t).2.2 ∈ Set.Icc (0 : ℝ) 5 := by
+  have hdiam := maximumPolygonCap_diam_le n K hK
+  exact ⟨hdiam, tangentArmLengths_le_of_diam K hdiam⟩
+
+/-! ### Helpers for the discrete arm bound -/
+
+/-- Two representatives of the same angle in one turn agree. -/
+private theorem coe_angle_inj_Ioc {x y : ℝ} (hx : x ∈ Set.Ioc (0 : ℝ) (2 * Real.pi))
+    (hy : y ∈ Set.Ioc (0 : ℝ) (2 * Real.pi)) (h : (x : Real.Angle) = (y : Real.Angle)) :
+    x = y :=
+  Real.Angle.injOn_coe_Ioc (a := 0) (b := 2 * Real.pi) (by linarith) hx hy h
+
+/-- Between consecutive grid normals a right-angle polygon cap has no exposed edge, both at
+the direction itself and at the direction shifted by a right angle. -/
+private theorem rightAnglePolygonCap_edgeVertices_eq (n : ℕ) (hn : 2 ≤ n)
+    (K : PolygonCapSpace (rightAngleSet n hn)) {s : ℝ}
+    (hs : s ∈ Set.Ioo (0 : ℝ) (Real.pi / 2))
+    (hgrid : s ∉ (rightAngleSet n hn).directions) :
+    (edgeVertices K.val.val (s : Real.Angle)).1 =
+        (edgeVertices K.val.val (s : Real.Angle)).2 ∧
+      (edgeVertices K.val.val ((s + Real.pi / 2 : ℝ) : Real.Angle)).1 =
+        (edgeVertices K.val.val ((s + Real.pi / 2 : ℝ) : Real.Angle)).2 := by
+  have hpi := Real.pi_pos
+  have hdirI : ∀ r ∈ (rightAngleSet n hn).directions, r ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) :=
+    fun r hr ↦ (rightAngleSet n hn).interior r hr
+  -- the possible normals of a nondegenerate exposed edge in the upper half turn
+  have hkey : ∀ x : ℝ, x ∈ Set.Ioo (0 : ℝ) Real.pi →
+      (edgeVertices K.val.val (x : Real.Angle)).1 ≠
+        (edgeVertices K.val.val (x : Real.Angle)).2 →
+      x ∈ (rightAngleSet n hn).directions ∨
+        (∃ r ∈ (rightAngleSet n hn).directions, x = r + Real.pi / 2) ∨
+        x = Real.pi / 2 := by
+    intro x hx hne
+    have hxIoc : x ∈ Set.Ioc (0 : ℝ) (2 * Real.pi) := ⟨hx.1, by linarith [hx.2]⟩
+    have hlow : ∀ z : Real.Angle, z ∈ capLowerNormals (rightAngleSet n hn).angle →
+        z = ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+      intro z hz
+      rcases hz with hz | hz
+      · rw [hz]
+        congr 1
+        show (rightAngleSet n hn).angle + Real.pi = 3 * Real.pi / 2
+        change Real.pi / 2 + Real.pi = 3 * Real.pi / 2
+        ring
+      · exact hz
+    rcases K.properEdgeNormal_mem_allowed_or_antipodal (x : Real.Angle) hne with
+      (⟨r, hr, hrx⟩ | hlowx) | ⟨z, hz, hzx⟩
+    · have hrI := angleDomain_subset_Ioo (rightAngleSet n hn) hr
+      have hxr : x = r := coe_angle_inj_Ioc hxIoc ⟨hrI.1, by linarith [hrI.2]⟩ hrx.symm
+      subst hxr
+      rcases hr with (hr | ⟨r, hr, hrx⟩) | hr
+      · exact Or.inl hr
+      · exact Or.inr (Or.inl ⟨r, hr, hrx.symm⟩)
+      · rcases hr with hr | hr
+        · exact Or.inr (Or.inr hr)
+        · exact Or.inr (Or.inr hr)
+    · exfalso
+      have hx3 : x = 3 * Real.pi / 2 :=
+        coe_angle_inj_Ioc hxIoc ⟨by linarith, by linarith⟩ (hlow _ hlowx)
+      linarith [hx.2]
+    · rcases hz with ⟨r, hr, rfl⟩ | hz
+      · exfalso
+        have hrI := angleDomain_subset_Ioo (rightAngleSet n hn) hr
+        have hxr : x = r + Real.pi := by
+          refine coe_angle_inj_Ioc hxIoc ⟨by linarith [hrI.1], by linarith [hrI.2]⟩ ?_
+          rw [Real.Angle.coe_add]
+          exact hzx.symm
+        linarith [hx.2, hrI.1]
+      · right; right
+        refine coe_angle_inj_Ioc hxIoc ⟨by linarith, by linarith⟩ ?_
+        have hxz : (x : Real.Angle) =
+            ((3 * Real.pi / 2 : ℝ) : Real.Angle) + (Real.pi : Real.Angle) := by
+          rw [← hzx, hlow _ hz]
+        rw [hxz, ← Real.Angle.coe_add, Real.Angle.angle_eq_iff_two_pi_dvd_sub]
+        exact ⟨1, by push_cast; ring⟩
+  constructor
+  · by_contra hne
+    rcases hkey s ⟨hs.1, by linarith [hs.2]⟩ hne with h | ⟨r, hr, hrs⟩ | h
+    · exact hgrid h
+    · linarith [(hdirI r hr).1, hs.2]
+    · linarith [hs.2]
+  · by_contra hne
+    rcases hkey (s + Real.pi / 2) ⟨by linarith [hs.1], by linarith [hs.2]⟩ hne with
+      h | ⟨r, hr, hrs⟩ | h
+    · linarith [(hdirI _ h).2, hs.1]
+    · exact hgrid (by rwa [show s = r by linarith])
+    · linarith [hs.1]
+
+/-- On a cell of degenerate exposed edges whose contacts lie in a finite set, the contact point
+is constant and agrees with both endpoint conventions. -/
+private theorem exists_edgeVertices_const_of_degenerate (K : ConvexBody Point)
+    {S : Set Point} (hS : S.Finite) {a b : ℝ} (hab : a < b)
+    (hdeg : ∀ s ∈ Set.Ioo a b,
+      (edgeVertices K (s : Real.Angle)).1 = (edgeVertices K (s : Real.Angle)).2)
+    (hmem : ∀ s ∈ Set.Ioo a b, (edgeVertices K (s : Real.Angle)).1 ∈ S) :
+    ∃ A : Point, (∀ s ∈ Set.Ioo a b, (edgeVertices K (s : Real.Angle)).1 = A) ∧
+      (edgeVertices K (a : Real.Angle)).1 = A ∧
+      (edgeVertices K (b : Real.Angle)).2 = A := by
+  classical
+  have hloc : ∀ u ∈ Set.Ioo a b, ∀ᶠ r : ℝ in 𝓝 u,
+      (edgeVertices K (r : Real.Angle)).1 = (edgeVertices K (u : Real.Angle)).1 := by
+    intro u hu
+    have hIoo : Set.Ioo a b ∈ 𝓝 u := isOpen_Ioo.mem_nhds hu
+    have hnbhd : (S \ {(edgeVertices K (u : Real.Angle)).1})ᶜ ∈
+        𝓝 (edgeVertices K (u : Real.Angle)).1 :=
+      (hS.sdiff (t := {(edgeVertices K (u : Real.Angle)).1})).isClosed.isOpen_compl.mem_nhds
+        (by simp)
+    have hright := (contact_oneSided_limits K u).1 hnbhd
+    have hleft := (contact_oneSided_limits K u).2.2.2.1
+    rw [← hdeg u hu] at hleft
+    have hleft' := hleft hnbhd
+    have hfin : ∀ᶠ r : ℝ in 𝓝[≠] u,
+        (edgeVertices K (r : Real.Angle)).1 = (edgeVertices K (u : Real.Angle)).1 := by
+      rw [← nhdsLT_sup_nhdsGT]
+      refine Filter.eventually_sup.mpr ⟨?_, ?_⟩
+      · filter_upwards [hleft', nhdsWithin_le_nhds hIoo] with r hr hrI
+        by_contra hcon
+        exact hr ⟨hmem r hrI, hcon⟩
+      · filter_upwards [hright, nhdsWithin_le_nhds hIoo] with r hr hrI
+        by_contra hcon
+        exact hr ⟨hmem r hrI, hcon⟩
+    rw [← nhdsNE_sup_pure u]
+    exact Filter.eventually_sup.mpr ⟨hfin, Filter.eventually_pure.mpr rfl⟩
+  have hmI : (a + b) / 2 ∈ Set.Ioo a b := ⟨by linarith, by linarith⟩
+  have hconst : ∀ s ∈ Set.Ioo a b, (edgeVertices K (s : Real.Angle)).1 =
+      (edgeVertices K (((a + b) / 2 : ℝ) : Real.Angle)).1 := by
+    intro s hs
+    by_contra hne
+    have hUopen : IsOpen {r : ℝ | ∀ᶠ x : ℝ in 𝓝 r, (edgeVertices K (x : Real.Angle)).1 =
+        (edgeVertices K (((a + b) / 2 : ℝ) : Real.Angle)).1} :=
+      isOpen_iff_mem_nhds.2 fun r hr ↦ eventually_eventually_nhds.2 hr
+    have hVopen : IsOpen {r : ℝ | ∀ᶠ x : ℝ in 𝓝 r, (edgeVertices K (x : Real.Angle)).1 ≠
+        (edgeVertices K (((a + b) / 2 : ℝ) : Real.Angle)).1} :=
+      isOpen_iff_mem_nhds.2 fun r hr ↦ eventually_eventually_nhds.2 hr
+    have hcover : Set.Ioo a b ⊆
+        {r : ℝ | ∀ᶠ x : ℝ in 𝓝 r, (edgeVertices K (x : Real.Angle)).1 =
+          (edgeVertices K (((a + b) / 2 : ℝ) : Real.Angle)).1} ∪
+        {r : ℝ | ∀ᶠ x : ℝ in 𝓝 r, (edgeVertices K (x : Real.Angle)).1 ≠
+          (edgeVertices K (((a + b) / 2 : ℝ) : Real.Angle)).1} := by
+      intro r hr
+      by_cases hcase : (edgeVertices K (r : Real.Angle)).1 =
+          (edgeVertices K (((a + b) / 2 : ℝ) : Real.Angle)).1
+      · exact Or.inl (by filter_upwards [hloc r hr] with x hx; rw [hx, hcase])
+      · exact Or.inr (by filter_upwards [hloc r hr] with x hx; rw [hx]; exact hcase)
+    obtain ⟨x, _, hxU, hxV⟩ := isPreconnected_Ioo _ _ hUopen hVopen hcover
+      ⟨(a + b) / 2, hmI, hloc _ hmI⟩
+      ⟨s, hs, by filter_upwards [hloc s hs] with x hx; rw [hx]; exact hne⟩
+    obtain ⟨y, hy1, hy2⟩ := (hxU.and hxV).exists
+    exact hy2 hy1
+  refine ⟨(edgeVertices K (((a + b) / 2 : ℝ) : Real.Angle)).1, hconst, ?_, ?_⟩
+  · refine tendsto_nhds_unique (contact_oneSided_limits K a).1
+      (Filter.Tendsto.congr' ?_ tendsto_const_nhds)
+    filter_upwards [Ioo_mem_nhdsGT hab] with r hr
+    exact (hconst r hr).symm
+  · refine tendsto_nhds_unique (contact_oneSided_limits K b).2.2.2.1
+      (Filter.Tendsto.congr' ?_ tendsto_const_nhds)
+    filter_upwards [Ioo_mem_nhdsLT hab] with r hr
+    exact (hconst r hr).symm
+
+private theorem normalVector_chord_norm_le (t δ : ℝ) (hδnonneg : 0 ≤ δ) :
+    ‖(1 - Real.cos δ) • normalVector (t : Real.Angle) -
+      Real.sin δ • tangentVector (t : Real.Angle)‖ ≤ δ := by
+  have huu : inner ℝ (normalVector (t : Real.Angle)) (normalVector (t : Real.Angle)) = 1 :=
+    inner_normalVector_self t
+  have hvv : inner ℝ (tangentVector (t : Real.Angle)) (tangentVector (t : Real.Angle)) = 1 :=
+    inner_tangentVector_self t
+  have huv : inner ℝ (normalVector (t : Real.Angle)) (tangentVector (t : Real.Angle)) = 0 :=
+    inner_normalVector_tangentVector t
+  have hvu : inner ℝ (tangentVector (t : Real.Angle)) (normalVector (t : Real.Angle)) = 0 := by
+    rw [real_inner_comm]
+    exact huv
+  have hsqn : ‖(1 - Real.cos (δ)) • normalVector (t : Real.Angle) -
+      Real.sin (δ) • tangentVector (t : Real.Angle)‖ ^ 2 =
+      2 - 2 * Real.cos (δ) := by
+    rw [← real_inner_self_eq_norm_sq]
+    simp only [inner_sub_left, inner_sub_right, real_inner_smul_left, real_inner_smul_right,
+      huu, hvv, huv, hvu]
+    nlinarith [Real.sin_sq_add_cos_sq (δ)]
+  have hznorm : ‖(1 - Real.cos (δ)) • normalVector (t : Real.Angle) -
+      Real.sin (δ) • tangentVector (t : Real.Angle)‖ ≤
+      δ := by
+    have hcosb : 1 - δ ^ 2 / 2 ≤ Real.cos (δ) :=
+      Real.one_sub_sq_div_two_le_cos
+    nlinarith [norm_nonneg ((1 - Real.cos (δ)) • normalVector (t : Real.Angle) -
+      Real.sin (δ) • tangentVector (t : Real.Angle)), hsqn, hδnonneg]
+  exact hznorm
+
+private theorem tangentArmLengths_snd_support (K : RightAngleCapSpace) :
+    ∀ s : ℝ,
+     (tangentArmLengths K s).2.1 = supportValue (K.val : Set Point) (s : Real.Angle) -
+       inner ℝ (capVertices K s).2.1 (normalVector (s : Real.Angle)) ∧
+     (tangentArmLengths K s).2.2 = supportValue (K.val : Set Point) (s : Real.Angle) -
+       inner ℝ (capVertices K s).2.2 (normalVector (s : Real.Angle)) := by
+  have hy : ∀ s : ℝ, inner ℝ
+      (rotatingHallwayParts (K.val : Set Point) (s : Real.Angle)).outerCorner
+      (normalVector (s : Real.Angle)) = supportValue (K.val : Set Point) (s : Real.Angle) := by
+    intro s
+    have hvu : inner ℝ (tangentVector (s : Real.Angle)) (normalVector (s : Real.Angle)) = 0 := by
+      rw [real_inner_comm]
+      exact inner_normalVector_tangentVector s
+    rw [outerCorner_eq_support_sum, inner_add_left, real_inner_smul_left, real_inner_smul_left,
+      inner_normalVector_self s, hvu]
+    ring
+  intro s
+  constructor <;>
+    · change inner ℝ (_ - _) (normalVector (s : Real.Angle)) = _
+      rw [inner_sub_left, hy s]
+
+theorem maximumPolygonCap_arm_cell (n : ℕ) (hn : 2 ≤ n) (K : RightAngleCapSpace)
+    (hK : IsMaximumPolygonCapSteps n K) (t : ℝ)
+    (ht : t = 0 ∨ t ∈ (rightAngleSet n hn).directions) :
+    (∀ u ∈ Set.Ioo t (t + polygonStepSize n),
+      (tangentArmLengths K u).2.1 ≤ (tangentArmLengths K t).2.1 ∧
+      (tangentArmLengths K u).2.1 = (tangentArmLengths K u).2.2 ∧
+      (tangentArmLengths K (t + polygonStepSize n)).2.2 ≤
+        (tangentArmLengths K u).2.2) ∧
+    (tangentArmLengths K t).2.1 -
+      (tangentArmLengths K (t + polygonStepSize n)).2.2 ≤ 5 * polygonStepSize n := by
+  classical
+  obtain ⟨hdiam, harmbd⟩ := maximumPolygonCap_arm_bound n K hK
+  obtain ⟨hn2, -, P, hPK, -⟩ := hK
+  have hPKval : (P.val.val : ConvexBody Point) = K.val := congrArg Subtype.val hPK
+  have hpi := Real.pi_pos
+  have hnR : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
+  have hnR2 : (2 : ℝ) ≤ n := by exact_mod_cast hn
+  have hδpos : 0 < polygonStepSize n := div_pos (by linarith) hnR
+  have hδle : polygonStepSize n ≤ Real.pi / 4 := by
+    change Real.pi / 2 / n ≤ Real.pi / 4
+    rw [div_le_iff₀ hnR]
+    nlinarith
+  -- the cell lies in the parameter range and contains no grid direction
+  have hcell0 : 0 ≤ t ∧ t + polygonStepSize n ≤ Real.pi / 2 := by
+    rcases ht with rfl | ht'
+    · exact ⟨le_rfl, by linarith⟩
+    · obtain ⟨h1, h2⟩ := rightAngleSet_direction_bounds n hn ht'
+      exact ⟨by linarith, by linarith⟩
+  obtain ⟨ht0, htδ⟩ := hcell0
+  have hnogrid : ∀ s ∈ Set.Ioo t (t + polygonStepSize n),
+      s ∉ (rightAngleSet n hn).directions := by
+    intro s hs hcon
+    rcases ht with rfl | ht'
+    · linarith [(rightAngleSet_direction_bounds n hn hcon).1, hs.2]
+    · rcases rightAngleSet_direction_le_or_succ_le n hn hcon ht' with h | h
+      · linarith [hs.1]
+      · linarith [hs.2]
+  -- no exposed edge has its normal inside the open cell, shifted or not
+  have hdegen : ∀ s ∈ Set.Ioo t (t + polygonStepSize n),
+      (edgeVertices K.val (s : Real.Angle)).1 = (edgeVertices K.val (s : Real.Angle)).2 ∧
+      (edgeVertices K.val ((s + Real.pi / 2 : ℝ) : Real.Angle)).1 =
+        (edgeVertices K.val ((s + Real.pi / 2 : ℝ) : Real.Angle)).2 := by
+    intro s hs
+    have hsI : s ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) :=
+      ⟨by linarith [hs.1], by linarith [hs.2]⟩
+    have h := rightAnglePolygonCap_edgeVertices_eq n hn2 P hsI (hnogrid s hs)
+    rwa [hPKval] at h
+  have hdegen2 : ∀ s ∈ Set.Ioo (t + Real.pi / 2) (t + polygonStepSize n + Real.pi / 2),
+      (edgeVertices K.val (s : Real.Angle)).1 = (edgeVertices K.val (s : Real.Angle)).2 := by
+    intro s hs
+    have hr : s - Real.pi / 2 ∈ Set.Ioo t (t + polygonStepSize n) :=
+      ⟨by linarith [hs.1], by linarith [hs.2]⟩
+    have h := (hdegen _ hr).2
+    rwa [show s - Real.pi / 2 + Real.pi / 2 = s by ring] at h
+  -- singleton contacts range over a finite set of constraint vertices
+  have hdomain : (angleDomain (rightAngleSet n hn2)).Finite :=
+    (((rightAngleSet n hn2).directions.finite_toSet.union
+      ((rightAngleSet n hn2).directions.finite_toSet.image
+        (fun r ↦ r + Real.pi / 2))).union
+      ((Set.finite_singleton (Real.pi / 2)).insert (rightAngleSet n hn2).angle))
+  have hNfin : (((fun r : ℝ ↦ (r : Real.Angle)) '' angleDomain (rightAngleSet n hn2)) ∪
+      capLowerNormals (rightAngleSet n hn2).angle).Finite :=
+    (hdomain.image (fun r : ℝ ↦ (r : Real.Angle))).union
+      ((Set.finite_singleton _).insert _)
+  obtain ⟨Cs, hCsfin, -, hKCs⟩ := P.property.finite_constraints hNfin
+  rw [hPKval] at hKCs
+  have hSfin : (finiteConstraintVertices Cs).Finite :=
+    finite_finiteConstraintVertices Cs hCsfin
+  have hSmem : ∀ θ : Real.Angle, (edgeVertices K.val θ).1 = (edgeVertices K.val θ).2 →
+      (edgeVertices K.val θ).1 ∈ finiteConstraintVertices Cs := by
+    intro θ hθ
+    refine singleton_exposedEdge_mem_finiteConstraintVertices Cs hCsfin K.val hKCs (t := θ) ?_
+    rw [exposedEdge_eq_segment_edgeVertices, ← hθ, segment_same]
+  -- the two contact points that are constant across the cell
+  obtain ⟨A, hAcell, hAleft, hAright⟩ := exists_edgeVertices_const_of_degenerate K.val hSfin
+    (a := t) (b := t + polygonStepSize n) (by linarith)
+    (fun s hs ↦ (hdegen s hs).1) (fun s hs ↦ hSmem _ (hdegen s hs).1)
+  obtain ⟨Cp, hCcell, hCleft, hCright⟩ := exists_edgeVertices_const_of_degenerate K.val hSfin
+    (a := t + Real.pi / 2) (b := t + polygonStepSize n + Real.pi / 2) (by linarith)
+    hdegen2 (fun s hs ↦ hSmem _ (hdegen2 s hs))
+  -- the arm lengths in support-function coordinates
+  have harm := tangentArmLengths_snd_support K
+  have hAK : A ∈ (K.val : Set Point) := by
+    rw [← hAleft]
+    exact (edgeVertices_fst_mem K.val (t : Real.Angle)).1
+  have hCK : Cp ∈ (K.val : Set Point) := by
+    rw [← hCleft]
+    exact (edgeVertices_fst_mem K.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).1
+  have hAsup : ∀ s : ℝ, (edgeVertices K.val (s : Real.Angle)).1 = A →
+      inner ℝ A (normalVector (s : Real.Angle)) =
+        supportValue (K.val : Set Point) (s : Real.Angle) := by
+    intro s hsA
+    rw [← hsA]
+    exact (edgeVertices_fst_mem K.val (s : Real.Angle)).2
+  have hAsup' : ∀ s : ℝ, (edgeVertices K.val (s : Real.Angle)).2 = A →
+      inner ℝ A (normalVector (s : Real.Angle)) =
+        supportValue (K.val : Set Point) (s : Real.Angle) := by
+    intro s hsA
+    rw [← hsA]
+    exact (edgeVertices_snd_mem K.val (s : Real.Angle)).2
+  -- the smooth comparison function and its sign conditions
+  have hadd : ∀ s ε : ℝ, inner ℝ (A - Cp) (normalVector ((s + ε : ℝ) : Real.Angle)) =
+      Real.cos ε * inner ℝ (A - Cp) (normalVector (s : Real.Angle)) +
+      Real.sin ε * inner ℝ (A - Cp) (tangentVector (s : Real.Angle)) := by
+    intro s ε
+    rw [normalVector_add_real, inner_add_right, real_inner_smul_right, real_inner_smul_right]
+  have hDsign : ∀ s : ℝ, (edgeVertices K.val ((s + Real.pi / 2 : ℝ) : Real.Angle)).1 = Cp →
+      inner ℝ (A - Cp) (tangentVector (s : Real.Angle)) ≤ 0 := by
+    intro s hsC
+    rw [← normalVector_add_pi_div_two_real s, inner_sub_left]
+    have hA' := inner_le_supportValue K.val hAK ((s + Real.pi / 2 : ℝ) : Real.Angle)
+    have hC' : inner ℝ Cp (normalVector ((s + Real.pi / 2 : ℝ) : Real.Angle)) =
+        supportValue (K.val : Set Point) ((s + Real.pi / 2 : ℝ) : Real.Angle) := by
+      rw [← hsC]
+      exact (edgeVertices_fst_mem K.val ((s + Real.pi / 2 : ℝ) : Real.Angle)).2
+    linarith
+  have hgt : (tangentArmLengths K t).2.1 =
+      inner ℝ (A - Cp) (normalVector (t : Real.Angle)) := by
+    rw [(harm t).1, inner_sub_left, hAsup t hAleft,
+      show (capVertices K t).2.1 = Cp from hCleft]
+  have hgb : (tangentArmLengths K (t + polygonStepSize n)).2.2 =
+      inner ℝ (A - Cp) (normalVector ((t + polygonStepSize n : ℝ) : Real.Angle)) := by
+    rw [(harm (t + polygonStepSize n)).2, inner_sub_left,
+      hAsup' (t + polygonStepSize n) hAright,
+      show (capVertices K (t + polygonStepSize n)).2.2 = Cp from hCright]
+  have hgu : ∀ u ∈ Set.Ioo t (t + polygonStepSize n),
+      (tangentArmLengths K u).2.1 = inner ℝ (A - Cp) (normalVector (u : Real.Angle)) ∧
+      (tangentArmLengths K u).2.2 = inner ℝ (A - Cp) (normalVector (u : Real.Angle)) := by
+    intro u hu
+    have hCu : (edgeVertices K.val ((u + Real.pi / 2 : ℝ) : Real.Angle)).1 = Cp :=
+      hCcell (u + Real.pi / 2) ⟨by linarith [hu.1], by linarith [hu.2]⟩
+    have hCu2 : (capVertices K u).2.2 = Cp := by
+      rw [show (capVertices K u).2.2 =
+        (edgeVertices K.val ((u + Real.pi / 2 : ℝ) : Real.Angle)).2 from rfl, ← (hdegen u hu).2]
+      exact hCu
+    refine ⟨?_, ?_⟩
+    · rw [(harm u).1, inner_sub_left, hAsup u (hAcell u hu),
+        show (capVertices K u).2.1 = Cp from hCu]
+    · rw [(harm u).2, inner_sub_left, hAsup u (hAcell u hu), hCu2]
+  have hGt0 : 0 ≤ inner ℝ (A - Cp) (normalVector (t : Real.Angle)) := by
+    rw [← hgt]
+    exact ((harmbd t ⟨ht0, by linarith⟩).2.2.1).1
+  have hGt5 : inner ℝ (A - Cp) (normalVector (t : Real.Angle)) ≤ 5 := by
+    rw [← hgt]
+    exact ((harmbd t ⟨ht0, by linarith⟩).2.2.1).2
+  constructor
+  · intro u hu
+    have hCu : (edgeVertices K.val ((u + Real.pi / 2 : ℝ) : Real.Angle)).1 = Cp :=
+      hCcell (u + Real.pi / 2) ⟨by linarith [hu.1], by linarith [hu.2]⟩
+    have hGu0 : 0 ≤ inner ℝ (A - Cp) (normalVector (u : Real.Angle)) := by
+      rw [← (hgu u hu).2]
+      exact ((harmbd u ⟨by linarith [hu.1], by linarith [hu.2]⟩).2.2.2).1
+    have hkey : inner ℝ (A - Cp) (normalVector (u : Real.Angle)) =
+        Real.cos (u - t) * inner ℝ (A - Cp) (normalVector (t : Real.Angle)) +
+        Real.sin (u - t) * inner ℝ (A - Cp) (tangentVector (t : Real.Angle)) := by
+      rw [show ((u : ℝ) : Real.Angle) = ((t + (u - t) : ℝ) : Real.Angle) by
+        rw [show t + (u - t) = u by ring], hadd]
+    have hkey2 : inner ℝ (A - Cp) (normalVector ((t + polygonStepSize n : ℝ) : Real.Angle)) =
+        Real.cos (t + polygonStepSize n - u) *
+            inner ℝ (A - Cp) (normalVector (u : Real.Angle)) +
+          Real.sin (t + polygonStepSize n - u) *
+            inner ℝ (A - Cp) (tangentVector (u : Real.Angle)) := by
+      rw [show ((t + polygonStepSize n : ℝ) : Real.Angle) =
+        ((u + (t + polygonStepSize n - u) : ℝ) : Real.Angle) by
+          rw [show u + (t + polygonStepSize n - u) = t + polygonStepSize n by ring], hadd]
+    have hD := hDsign t hCleft
+    have hDu := hDsign u hCu
+    have hc1 : Real.cos (u - t) ≤ 1 := Real.cos_le_one _
+    have hs1 : 0 ≤ Real.sin (u - t) :=
+      Real.sin_nonneg_of_nonneg_of_le_pi (by linarith [hu.1]) (by linarith [hu.2])
+    have hc2 : Real.cos (t + polygonStepSize n - u) ≤ 1 := Real.cos_le_one _
+    have hs2 : 0 ≤ Real.sin (t + polygonStepSize n - u) :=
+      Real.sin_nonneg_of_nonneg_of_le_pi (by linarith [hu.2]) (by linarith [hu.1])
+    refine ⟨?_, ?_, ?_⟩
+    · rw [(hgu u hu).1, hgt, hkey]
+      linarith [mul_nonneg (sub_nonneg.mpr hc1) hGt0,
+        mul_nonneg hs1 (neg_nonneg.mpr hD)]
+    · rw [(hgu u hu).1, (hgu u hu).2]
+    · rw [hgb, (hgu u hu).2, hkey2]
+      linarith [mul_nonneg (sub_nonneg.mpr hc2) hGu0,
+        mul_nonneg hs2 (neg_nonneg.mpr hDu)]
+  · have hD := hDsign t hCleft
+    have hz : (tangentArmLengths K t).2.1 -
+        (tangentArmLengths K (t + polygonStepSize n)).2.2 =
+        inner ℝ (A - Cp)
+          ((1 - Real.cos (polygonStepSize n)) • normalVector (t : Real.Angle) -
+            Real.sin (polygonStepSize n) • tangentVector (t : Real.Angle)) := by
+      rw [hgt, hgb, hadd t (polygonStepSize n), inner_sub_right, real_inner_smul_right,
+        real_inner_smul_right]
+      ring
+    have hznorm := normalVector_chord_norm_le t (polygonStepSize n) hδpos.le
+    have hwnorm : ‖A - Cp‖ ≤ 5 := by
+      rw [← dist_eq_norm]
+      exact le_trans (Metric.dist_le_diam_of_mem K.val.isCompact.isBounded hAK hCK) hdiam
+    rw [hz]
+    calc inner ℝ (A - Cp)
+          ((1 - Real.cos (polygonStepSize n)) • normalVector (t : Real.Angle) -
+            Real.sin (polygonStepSize n) • tangentVector (t : Real.Angle))
+        ≤ ‖A - Cp‖ * ‖(1 - Real.cos (polygonStepSize n)) • normalVector (t : Real.Angle) -
+            Real.sin (polygonStepSize n) • tangentVector (t : Real.Angle)‖ :=
+          real_inner_le_norm _ _
+      _ ≤ 5 * polygonStepSize n :=
+          mul_le_mul hwnorm hznorm (norm_nonneg _) (by norm_num)
+
+theorem polygonCap_arm_integral_limit (K : ℕ → RightAngleCapSpace)
+    (L : RightAngleCapSpace)
+    (hlim : Tendsto (fun i ↦ Metric.hausdorffDist ((K i).val : Set Point)
+      (L.val : Set Point)) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ ∫ t in (0 : ℝ)..(Real.pi / 2),
+      |(tangentArmLengths (K i) t).2.1 - (tangentArmLengths L t).2.1|)
+      atTop (𝓝 0) := by
+  let μs : ℕ → FiniteMeasure Real.Angle := fun n ↦
+    ⟨surfaceAreaMeasure (K n).val, (surfaceAreaMeasure_face_union (K n).val).1⟩
+  let μ : FiniteMeasure Real.Angle :=
+    ⟨surfaceAreaMeasure L.val, (surfaceAreaMeasure_face_union L.val).1⟩
+  have hμ : Tendsto μs atTop (𝓝 μ) := by
+    apply FiniteMeasure.tendsto_iff_forall_integral_tendsto.mpr
+    intro f
+    exact surfaceAreaMeasure_weak_continuity (fun n ↦ (K n).val) L.val hlim f f.continuous
+  have hmass : Tendsto (fun n ↦ (μs n).mass) atTop (𝓝 μ.mass) := hμ.mass
+  obtain ⟨M, hM⟩ := hmass.bddAbove_range
+  have hmass_le (n : ℕ) : (μs n).mass ≤ M := hM ⟨n, rfl⟩
+  have hcoe_inj : Set.InjOn (fun t : ℝ ↦ (t : Real.Angle))
+      (Ioc 0 (Real.pi / 2)) :=
+    Real.Angle.injOn_coe_Ioc (by linarith [Real.pi_pos])
+  have hshift_inj : Set.InjOn
+      (fun t : ℝ ↦ ((t + Real.pi / 2 : ℝ) : Real.Angle))
+      (Ioc 0 (Real.pi / 2)) := by
+    intro x hx y hy hxy
+    apply hcoe_inj hx hy
+    simp only [Real.Angle.coe_add] at hxy
+    exact add_right_cancel hxy
+  have hae_left : ∀ᵐ t : ℝ ∂volume.restrict (Ioc 0 (Real.pi / 2)),
+      (μ : Measure Real.Angle) {(t : Real.Angle)} = 0 :=
+    ae_measure_singleton_comp_eq_zero_of_injOn (μ : Measure Real.Angle)
+      measurableSet_Ioc hcoe_inj
+  have hae_right : ∀ᵐ t : ℝ ∂volume.restrict (Ioc 0 (Real.pi / 2)),
+      (μ : Measure Real.Angle) {((t + Real.pi / 2 : ℝ) : Real.Angle)} = 0 :=
+    ae_measure_singleton_comp_eq_zero_of_injOn (μ : Measure Real.Angle)
+      measurableSet_Ioc hshift_inj
+  have hpoint : ∀ᵐ t ∂volume.restrict (Ioc 0 (Real.pi / 2)),
+      Tendsto (fun n ↦ (tangentArmLengths (K n) t).2.1) atTop
+        (𝓝 (tangentArmLengths L t).2.1) := by
+    filter_upwards [ae_restrict_mem measurableSet_Ioc, hae_left, hae_right] with t ht hleft hright
+    let A := (fun s : ℝ ↦ (s : Real.Angle)) '' Ioc t (t + Real.pi / 2)
+    have hfrontier : (μ : Measure Real.Angle) (frontier A) = 0 := by
+      apply measure_mono_null (Real.Angle.frontier_image_Ioc_subset
+        (by linarith [Real.pi_pos]))
+      simp only [insert_eq, measure_union_null hleft hright]
+    have hAmeas : MeasurableSet A :=
+      Real.Angle.measurableSet_image_Ioc t (t + Real.pi / 2)
+    have hfrontier' : μ (frontier A) = 0 := by
+      rw [← ENNReal.coe_eq_zero, FiniteMeasure.ennreal_coeFn_eq_coeFn_toMeasure]
+      exact hfrontier
+    have hres := FiniteMeasure.tendsto_restrict_of_null_frontier
+      hAmeas hμ hfrontier'
+    let f : Real.Angle →ᵇ ℝ := BoundedContinuousFunction.mkOfCompact
+      ⟨fun u ↦ (u - (t : Real.Angle)).sin,
+        Real.Angle.continuous_sin.comp (continuous_id.sub continuous_const)⟩
+    have hint := FiniteMeasure.tendsto_iff_forall_integral_tendsto.mp hres f
+    change Tendsto (fun n ↦ ∫ u in A,
+      (u - (t : Real.Angle)).sin ∂surfaceAreaMeasure (K n).val) atTop
+      (𝓝 (∫ u in A, (u - (t : Real.Angle)).sin ∂surfaceAreaMeasure L.val)) at hint
+    have hKt (n : ℕ) := tangentArm_convolution (K n) t
+    have hLt := tangentArm_convolution L t
+    simpa only [A, hKt, hLt] using hint
+  have hmeas_arm := aestronglyMeasurable_tangentArm_fst
+  have hmeas : ∀ n, AEStronglyMeasurable
+      (fun t ↦ |(tangentArmLengths (K n) t).2.1 - (tangentArmLengths L t).2.1|)
+      (volume.restrict (Ioc 0 (Real.pi / 2))) := by
+    intro n
+    have hm := ((hmeas_arm (K n)).sub (hmeas_arm L)).norm
+    convert hm using 1
+  have hbound : ∀ n, ∀ᵐ t ∂volume.restrict (Ioc 0 (Real.pi / 2)),
+      ‖|(tangentArmLengths (K n) t).2.1 - (tangentArmLengths L t).2.1|‖ ≤
+        (M : ℝ) + (μ.mass : ℝ) := by
+    intro n
+    filter_upwards [] with t
+    rw [Real.norm_eq_abs, abs_abs]
+    calc
+      |(tangentArmLengths (K n) t).2.1 - (tangentArmLengths L t).2.1| ≤
+          |(tangentArmLengths (K n) t).2.1| + |(tangentArmLengths L t).2.1| :=
+        abs_sub _ _
+      _ ≤ (surfaceAreaMeasure (K n).val).real Set.univ +
+          (surfaceAreaMeasure L.val).real Set.univ :=
+        add_le_add (abs_tangentArm_fst_le (K n) (t := t))
+          (abs_tangentArm_fst_le L (t := t))
+      _ = ((μs n).mass : ℝ) + (μ.mass : ℝ) := by
+        change (μs n : Measure Real.Angle).real univ +
+          (μ : Measure Real.Angle).real univ = _
+        simp [Measure.real, ← FiniteMeasure.ennreal_mass]
+      _ ≤ (M : ℝ) + (μ.mass : ℝ) := by
+        have hn : ((μs n).mass : ℝ) ≤ (M : ℝ) := by exact_mod_cast hmass_le n
+        exact add_le_add hn le_rfl
+  have hlim_zero : ∀ᵐ t ∂volume.restrict (Ioc 0 (Real.pi / 2)),
+      Tendsto (fun n ↦ |(tangentArmLengths (K n) t).2.1 -
+        (tangentArmLengths L t).2.1|) atTop (𝓝 0) := by
+    filter_upwards [hpoint] with t ht
+    have hc : Tendsto (fun _ : ℕ ↦ (tangentArmLengths L t).2.1) atTop
+        (𝓝 (tangentArmLengths L t).2.1) := tendsto_const_nhds
+    simpa only [sub_self, abs_zero] using (ht.sub hc).abs
+  have hdom := tendsto_integral_of_dominated_convergence
+    (μ := volume.restrict (Ioc 0 (Real.pi / 2)))
+    (fun _ ↦ (M : ℝ) + (μ.mass : ℝ)) hmeas
+    (integrableOn_const (μ := volume) (s := Ioc 0 (Real.pi / 2))
+      (measure_Ioc_lt_top.ne)) hbound hlim_zero
+  simpa only [intervalIntegral.integral_of_le (by positivity : (0 : ℝ) ≤ Real.pi / 2),
+    integral_zero] using hdom
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Bounds / Wedge Gap / Infimum
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter MeasureTheory Set
+open scoped Topology
+
+namespace MovingSofa
+
+/-- The two infima of the wedge-gap components over interior rotation angles. -/
+def wedgeGapInfimum {ω : ℝ} (K : CapSpace ω) : ℝ × ℝ :=
+  (sInf ((fun t ↦ (wedgeGaps K t).1) '' Set.Ioo 0 ω),
+    sInf ((fun t ↦ (wedgeGaps K t).2) '' Set.Ioo 0 ω))
+
+private theorem bddBelow_wedgeGaps_fst {ω : ℝ} (K : CapSpace ω) :
+    BddBelow ((fun t ↦ (wedgeGaps K t).1) '' Ioo 0 ω) := by
+  refine ⟨0, ?_⟩
+  rintro _ ⟨t, ht, rfl⟩
+  have h := wedgeGaps_positive_lower_bound K t ht
+  exact h.2.1.le.trans h.1
+
+private theorem wedgeGapInfimum_fst_nonneg {ω : ℝ} (K : CapSpace ω) :
+    0 ≤ (wedgeGapInfimum K).1 := by
+  rw [wedgeGapInfimum]
+  apply le_csInf
+  · exact (nonempty_Ioo.mpr K.property.1).image _
+  · rintro _ ⟨t, ht, rfl⟩
+    have h := wedgeGaps_positive_lower_bound K t ht
+    exact h.2.1.le.trans h.1
+
+private theorem wedgeGapInfimum_fst_le_supportValue_zero {ω : ℝ}
+    (K : CapSpace ω) (hω : ω < Real.pi / 2) :
+    (wedgeGapInfimum K).1 ≤ supportValue K.val (0 : Real.Angle) := by
+  let f := fun t : ℝ ↦ supportValue K.val (0 : Real.Angle) -
+    (supportValue K.val (t : Real.Angle) - 1) / Real.cos t
+  have hcos : Real.cos ω ≠ 0 := (Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [K.property.1, Real.pi_pos], hω⟩).ne'
+  have hf : ContinuousAt f ω := continuousAt_const.sub
+    (((continuous_supportValue_real K.val).continuousAt.sub continuousAt_const).div
+      Real.continuous_cos.continuousAt hcos)
+  have hf' : Tendsto f (𝓝[<] ω) (𝓝 (f ω)) :=
+    hf.tendsto.mono_left inf_le_left
+  have hlim : Tendsto f (𝓝[<] ω)
+      (𝓝 (supportValue K.val (0 : Real.Angle))) := by
+    convert hf' using 1
+    simp [f, K.property.2.2.1]
+  apply ge_of_tendsto hlim
+  filter_upwards [Ioo_mem_nhdsLT K.property.1] with t ht
+  rw [wedgeGapInfimum]
+  have hle := csInf_le (bddBelow_wedgeGaps_fst K) ⟨t, ht, rfl⟩
+  simpa only [f, wedgeGaps_fst_eq_supportValue] using hle
+
+private def bottomGapSegment {ω : ℝ} (K : CapSpace ω) : Set Point :=
+  let a := supportValue K.val (0 : Real.Angle)
+  let w := (wedgeGapInfimum K).1
+  segment ℝ ((a - w) • normalVector (0 : Real.Angle))
+    (a • normalVector (0 : Real.Angle))
+
+private theorem point_eq_fst_smul_normalVector_zero_of_mem_normalLine
+    {p : Point} (hp : p ∈ normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0) :
+    p = p 0 • normalVector (0 : Real.Angle) := by
+  have hpy : p 1 = 0 := by
+    change inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 at hp
+    simpa [normalVector, frame, PiLp.inner_apply] using hp
+  ext i
+  fin_cases i <;> simp [normalVector, frame, hpy]
+
+private theorem bottomGapSegment_subset_exposedEdge {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hω : Θ.angle < Real.pi / 2) :
+    bottomGapSegment K.val ⊆
+      exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+  let a := supportValue K.val.val (0 : Real.Angle)
+  let w := (wedgeGapInfimum K.val).1
+  let A := a • normalVector (0 : Real.Angle)
+  let B := (a - w) • normalVector (0 : Real.Angle)
+  have hzeroK : (0 : Point) ∈ (K.val.val : Set Point) := zero_mem_cap_of_lt K.val hω
+  have hAK : A ∈ (K.val.val : Set Point) := by
+    exact supportValue_zero_smul_normalVector_mem K.val
+  have hw0 : 0 ≤ w := wedgeGapInfimum_fst_nonneg K.val
+  have hwa : w ≤ a := wedgeGapInfimum_fst_le_supportValue_zero K.val hω
+  have hBK : B ∈ (K.val.val : Set Point) := by
+    exact K.val.val.convex.smul_mem_of_nonneg_of_le hzeroK hAK
+      (sub_nonneg.mpr hwa) (by linarith)
+  have hbottom : supportValue K.val.val
+      ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := K.val.property.2.2.2.2.2.1
+  have hAedge : A ∈ exposedEdge K.val.val
+      ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+    refine ⟨hAK, ?_⟩
+    change inner ℝ A (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) = _
+    rw [hbottom]
+    rw [inner_normalVector_three_pi_div_two]
+    simp [A, normalVector, frame]
+  have hBedge : B ∈ exposedEdge K.val.val
+      ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+    refine ⟨hBK, ?_⟩
+    change inner ℝ B (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) = _
+    rw [hbottom]
+    rw [inner_normalVector_three_pi_div_two]
+    simp [B, normalVector, frame]
+  exact (convex_exposedEdge K.val.val _).segment_subset hBedge hAedge
+
+private theorem polygonNiche_fanLine_fst_lt_gapStart {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {q : Point}
+    (hq : q ∈ polygonNiche Θ K.val ∩
+      normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0) :
+    q 0 < supportValue K.val.val (0 : Real.Angle) -
+      (wedgeGapInfimum K.val).1 := by
+  obtain ⟨t, ht, hqt⟩ := Set.mem_iUnion₂.mp hq.1.2
+  have htt := Θ.interior t ht
+  have hcost : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [htt.1, Real.pi_pos], htt.2.trans_le Θ.angle_le⟩
+  have hqy : q 1 = 0 := by
+    have hline := hq.2
+    change inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 at hline
+    simpa [normalVector, frame, PiLp.inner_apply] using hline
+  have hqx_endpoint : q 0 < (wedgeEndpoints K.val t).1 0 := by
+    have hb := hqt.1
+    change inner ℝ q (normalVector (t : Real.Angle)) <
+      supportValue K.val.val (t : Real.Angle) - 1 at hb
+    simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply,
+      RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue,
+      Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one, hqy, mul_zero,
+      add_zero] at hb
+    simp only [wedgeEndpoints]
+    simp only [Fin.isValue, normalVector, frame, Real.Angle.cos_zero, Real.Angle.sin_zero, neg_zero,
+      PiLp.smul_apply, Matrix.cons_val_zero, smul_eq_mul, mul_one, gt_iff_lt]
+    exact (lt_div_iff₀ hcost).2 (by simpa [mul_comm] using hb)
+  have hinf : (wedgeGapInfimum K.val).1 ≤ (wedgeGaps K.val t).1 := by
+    rw [wedgeGapInfimum]
+    exact csInf_le (bddBelow_wedgeGaps_fst K.val) ⟨t, htt, rfl⟩
+  have hgap : (wedgeGaps K.val t).1 =
+      supportValue K.val.val (0 : Real.Angle) - (wedgeEndpoints K.val t).1 0 := by
+    rw [wedgeGaps_fst_eq_supportValue]
+    simp [wedgeEndpoints, normalVector, frame]
+  rw [hgap] at hinf
+  linarith
+
+private theorem polygonNiche_fanLine_subset_bottom_exposedEdge {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hω : Θ.angle < Real.pi / 2) :
+    polygonNiche Θ K.val ∩ normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0 ⊆
+      exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+  intro q hq
+  let a := supportValue K.val.val (0 : Real.Angle)
+  let A := a • normalVector (0 : Real.Angle)
+  have hzeroK : (0 : Point) ∈ (K.val.val : Set Point) := zero_mem_cap_of_lt K.val hω
+  have hAK : A ∈ (K.val.val : Set Point) := supportValue_zero_smul_normalVector_mem K.val
+  have hqx0 : 0 ≤ q 0 := by
+    have hfan := hq.1.1.1
+    have hqy : q 1 = 0 := by
+      have hline := hq.2
+      change inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 at hline
+      simpa [normalVector, frame, PiLp.inner_apply] using hline
+    change 0 ≤ inner ℝ q (normalVector (Θ.angle : Real.Angle)) at hfan
+    simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe,
+      PiLp.inner_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue,
+      Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one, hqy, mul_zero,
+      add_zero] at hfan
+    exact nonneg_of_mul_nonneg_left (by simpa [mul_comm] using hfan)
+      (Real.cos_pos_of_mem_Ioo
+        ⟨by linarith [Θ.angle_pos, Real.pi_pos], hω⟩)
+  have hqxa : q 0 ≤ a := by
+    have hlt := polygonNiche_fanLine_fst_lt_gapStart K hq
+    have hw0 := wedgeGapInfimum_fst_nonneg K.val
+    dsimp only [a]
+    linarith
+  have hqeq := point_eq_fst_smul_normalVector_zero_of_mem_normalLine hq.2
+  have hqK : q ∈ (K.val.val : Set Point) := by
+    rw [hqeq]
+    exact K.val.val.convex.smul_mem_of_nonneg_of_le hzeroK hAK hqx0 hqxa
+  refine ⟨hqK, ?_⟩
+  change inner ℝ q (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) =
+    supportValue K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+  rw [K.val.property.2.2.2.2.2.1]
+  have hqy : q 1 = 0 := by
+    have hline := hq.2
+    change inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 at hline
+    simpa [normalVector, frame, PiLp.inner_apply] using hline
+  rw [inner_normalVector_three_pi_div_two]
+  simp [hqy]
+
+private theorem bottomGapSegment_disjoint_polygonNiche_fanLine {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) :
+    Disjoint (bottomGapSegment K.val)
+      (polygonNiche Θ K.val ∩ normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0) := by
+  rw [Set.disjoint_left]
+  intro q hqS hqN
+  let a := supportValue K.val.val (0 : Real.Angle)
+  let w := (wedgeGapInfimum K.val).1
+  let B := (a - w) • normalVector (0 : Real.Angle)
+  let A := a • normalVector (0 : Real.Angle)
+  have hw0 : 0 ≤ w := wedgeGapInfimum_fst_nonneg K.val
+  have hqx : a - w ≤ q 0 := by
+    change q ∈ segment ℝ B A at hqS
+    rw [segment_eq_image'] at hqS
+    obtain ⟨r, hr, rfl⟩ := hqS
+    have heq : (B + r • (A - B)) 0 = a - w + r * w := by
+      simp [B, A, normalVector, frame]
+    rw [heq]
+    nlinarith [hr.1]
+  have hlt := polygonNiche_fanLine_fst_lt_gapStart K hqN
+  exact (not_lt_of_ge hqx) (by simpa only [a, w] using hlt)
+
+private theorem hausdorffMeasure_bottomGapSegment {ω : ℝ} (K : CapSpace ω) :
+    (Measure.hausdorffMeasure 1 (bottomGapSegment K)).toReal =
+      (wedgeGapInfimum K).1 := by
+  have hw0 := wedgeGapInfimum_fst_nonneg K
+  rw [show bottomGapSegment K =
+      segment ℝ
+        ((supportValue K.val (0 : Real.Angle) - (wedgeGapInfimum K).1) •
+          normalVector (0 : Real.Angle))
+        (supportValue K.val (0 : Real.Angle) • normalVector (0 : Real.Angle)) by rfl,
+    MeasureTheory.hausdorffMeasure_segment, edist_dist, ENNReal.toReal_ofReal dist_nonneg,
+    dist_eq_norm]
+  have hdiff :
+      (supportValue K.val (0 : Real.Angle) - (wedgeGapInfimum K).1) •
+          normalVector (0 : Real.Angle) -
+        supportValue K.val (0 : Real.Angle) • normalVector (0 : Real.Angle) =
+      (-(wedgeGapInfimum K).1) • normalVector (0 : Real.Angle) := by module
+  rw [hdiff, norm_smul, Real.norm_eq_abs, abs_neg, abs_of_nonneg hw0,
+    norm_normalVector, mul_one]
+
+private theorem wedgeGapInfimum_fst_le_surface_of_balanced {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hω : Θ.angle < Real.pi / 2)
+    (hbal : IsBalancedPolygonCap K) :
+    (wedgeGapInfimum K.val).1 ≤
+      (surfaceAreaMeasure K.val.val
+        {((Real.pi / 2 : ℝ) : Real.Angle)}).toReal := by
+  let μ : Measure Point := Measure.hausdorffMeasure 1
+  let S := bottomGapSegment K.val
+  let M := polygonNiche Θ K.val ∩
+    normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0
+  let E := exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+  have hSsub : S ⊆ E := bottomGapSegment_subset_exposedEdge K hω
+  have hMsub : M ⊆ E := polygonNiche_fanLine_subset_bottom_exposedEdge K hω
+  have hSMsub : S ∪ M ⊆ E := Set.union_subset hSsub hMsub
+  have hdisj : Disjoint S M := bottomGapSegment_disjoint_polygonNiche_fanLine K
+  have hMMeas : MeasurableSet M := by
+    apply (measurableSet_polygonNiche Θ K.val).inter
+    exact (isClosed_eq (by fun_prop) continuous_const).measurableSet
+  have hSne : μ S ≠ ⊤ := by
+    dsimp only [μ, S]
+    rw [show bottomGapSegment K.val =
+        segment ℝ
+          ((supportValue K.val.val (0 : Real.Angle) - (wedgeGapInfimum K.val).1) •
+            normalVector (0 : Real.Angle))
+          (supportValue K.val.val (0 : Real.Angle) • normalVector (0 : Real.Angle)) by rfl,
+      MeasureTheory.hausdorffMeasure_segment, edist_dist]
+    exact ENNReal.ofReal_ne_top
+  have hatom := surfaceAreaMeasure_atom_length K.val.val
+    ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+  have hEne : μ E ≠ ⊤ := by
+    dsimp only [μ, E]
+    rw [← hatom.1, hatom.2.1]
+    exact ENNReal.ofReal_ne_top
+  have hMne : μ M ≠ ⊤ := measure_ne_top_of_subset hMsub hEne
+  have hunion := measureReal_union hdisj hMMeas hSne hMne
+  have hmono : μ.real (S ∪ M) ≤ μ.real E :=
+    ENNReal.toReal_mono hEne (measure_mono hSMsub)
+  have hSreal : μ.real S = (wedgeGapInfimum K.val).1 := by
+    exact hausdorffMeasure_bottomGapSegment K.val
+  have hMreal : μ.real M =
+      (surfaceAreaMeasure K.val.val
+        {((3 * Real.pi / 2 : ℝ) : Real.Angle)}).toReal -
+        polygonPolylineLengthAt K (Real.pi / 2) := by
+    have h := (polygonCap_polyline_lengths K).2 (Real.pi / 2) (by simp)
+    dsimp only [μ, M]
+    change (Measure.hausdorffMeasure 1
+      (polygonNiche Θ K.val ∩
+        normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0)).toReal = _
+    have hang : (((Real.pi / 2 + Real.pi : ℝ) : Real.Angle)) =
+        ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+      congr 1
+      ring
+    simpa only [hang] using h.2
+  have hEreal : μ.real E =
+      (surfaceAreaMeasure K.val.val
+        {((3 * Real.pi / 2 : ℝ) : Real.Angle)}).toReal := by
+    exact (congrArg ENNReal.toReal hatom.1).symm
+  have htT : Real.pi / 2 ∈ angleDomain Θ := by simp [angleDomain]
+  let tT : angleDomain Θ := ⟨Real.pi / 2, htT⟩
+  have hpoly0 : 0 ≤ polygonCapPolylineLength K tT := by
+    unfold polygonCapPolylineLength
+    exact Finset.sum_nonneg fun _ _ ↦ by split_ifs <;> positivity
+  have hbalReal :
+      (surfaceAreaMeasure K.val.val
+        {((Real.pi / 2 : ℝ) : Real.Angle)}).toReal =
+        polygonCapPolylineLength K tT := by
+    simpa only [tT, ENNReal.toReal_ofReal hpoly0] using
+      congrArg ENNReal.toReal (hbal tT)
+  have hlength : polygonPolylineLengthAt K (Real.pi / 2) =
+      (surfaceAreaMeasure K.val.val
+        {((Real.pi / 2 : ℝ) : Real.Angle)}).toReal := by
+    simp only [polygonPolylineLengthAt, dite_eq_left htT]
+    exact hbalReal.symm
+  rw [hunion, hSreal, hMreal, hEreal, hlength] at hmono
+  linarith
+
+theorem maximumPolygonCap_gap_le_surface {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (hK : IsMaximumPolygonCap Θ K) (hω : Θ.angle < Real.pi / 2) :
+    (wedgeGapInfimum K.val).1 ≤
+      (surfaceAreaMeasure K.val.val {((Real.pi / 2 : ℝ) : Real.Angle)}).toReal ∧
+    (wedgeGapInfimum K.val).2 ≤
+      (surfaceAreaMeasure K.val.val {(Θ.angle : Real.Angle)}).toReal := by
+  have hbal := maximumPolygonCap_balanced K hK
+  refine ⟨wedgeGapInfimum_fst_le_surface_of_balanced K hω hbal, ?_⟩
+  obtain ⟨Q, hQcarrier, hQmax⟩ := maximumPolygonCap_mirror Θ K hK
+  obtain ⟨P, hPcarrier, _hHall, hgap, _hupper, _hniche, hsurface⟩ :=
+    cap_mirror_features K.val
+  have hPQ : P = Q.val := by
+    apply Subtype.ext
+    apply ConvexBody.ext
+    exact hPcarrier.trans hQcarrier.symm
+  have hQbal := maximumPolygonCap_balanced Q hQmax
+  have hQright := wedgeGapInfimum_fst_le_surface_of_balanced Q hω hQbal
+  have himage :
+      (fun t ↦ (wedgeGaps Q.val t).1) '' Ioo 0 Θ.angle =
+        (fun t ↦ (wedgeGaps K.val t).2) '' Ioo 0 Θ.angle := by
+    ext x
+    constructor
+    · rintro ⟨t, ht, rfl⟩
+      rcases ht with ⟨ht0, htω⟩
+      refine ⟨Θ.angle - t, ⟨by linarith, by linarith⟩, ?_⟩
+      have hg := (hgap t ⟨ht0, htω⟩).2.2.1
+      rw [hPQ] at hg
+      change (wedgeGaps Q.val t).1 = (wedgeGaps K.val (Θ.angle - t)).2 at hg
+      exact hg.symm
+    · rintro ⟨t, ht, rfl⟩
+      rcases ht with ⟨ht0, htω⟩
+      have hcomp : Θ.angle - t ∈ Ioo 0 Θ.angle := ⟨by linarith, by linarith⟩
+      refine ⟨Θ.angle - t, hcomp, ?_⟩
+      have hg := (hgap (Θ.angle - t) hcomp).2.2.1
+      rw [hPQ] at hg
+      change (wedgeGaps Q.val (Θ.angle - t)).1 =
+        (wedgeGaps K.val (Θ.angle - (Θ.angle - t))).2 at hg
+      simpa only [sub_sub_cancel] using hg
+  have hinf : (wedgeGapInfimum Q.val).1 = (wedgeGapInfimum K.val).2 := by
+    change sInf ((fun t ↦ (wedgeGaps Q.val t).1) '' Ioo 0 Θ.angle) =
+      sInf ((fun t ↦ (wedgeGaps K.val t).2) '' Ioo 0 Θ.angle)
+    exact congrArg sInf himage
+  have hsurface' :
+      surfaceAreaMeasure Q.val.val {((Real.pi / 2 : ℝ) : Real.Angle)} =
+        surfaceAreaMeasure K.val.val {(Θ.angle : Real.Angle)} := by
+    have hs := hsurface {((Real.pi / 2 : ℝ) : Real.Angle)}
+      (measurableSet_singleton _)
+    rw [hPQ] at hs
+    rw [Set.image_singleton] at hs
+    have hang :
+        ((Θ.angle + Real.pi / 2 : ℝ) : Real.Angle) -
+            ((Real.pi / 2 : ℝ) : Real.Angle) =
+          (Θ.angle : Real.Angle) := by
+      rw [← Real.Angle.coe_sub]
+      congr 1
+      ring
+    simpa only [hang] using hs
+  rw [hinf, hsurface'] at hQright
+  exact hQright
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Bounds / Wedge Gap / Limit
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter MeasureTheory Set
+open scoped Topology
+
+namespace MovingSofa
+
+theorem wedgeGapInfimum_hausdorff_bound {ω : ℝ} (K L : CapSpace ω)
+    (hω : ω < Real.pi / 2) :
+    |(wedgeGapInfimum K).1 - (wedgeGapInfimum L).1| ≤
+      (1 + (Real.cos ω)⁻¹) * Metric.hausdorffDist (K.val : Set Point) L.val := by
+  let d := Metric.hausdorffDist (K.val : Set Point) L.val
+  let C := (1 + (Real.cos ω)⁻¹) * d
+  let f := fun t : ℝ ↦ (wedgeGaps K t).1
+  let g := fun t : ℝ ↦ (wedgeGaps L t).1
+  have hω0 : 0 < ω := K.property.1
+  have hcosω : 0 < Real.cos ω := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, hω0], hω⟩
+  have hcost {t : ℝ} (ht : t ∈ Icc 0 ω) : 0 < Real.cos t :=
+    Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [Real.pi_pos, ht.1], ht.2.trans_lt hω⟩
+  have hsupport := compactSet_support_continuity (K.val : Set Point) (L.val : Set Point)
+    K.val.nonempty K.val.isCompact L.val.nonempty L.val.isCompact
+  have hsupport' (t : ℝ) :
+      |supportValue K.val (t : Real.Angle) - supportValue L.val (t : Real.Angle)| ≤ d := by
+    change |vectorSupport (K.val : Set Point) (normalVector (t : Real.Angle)) -
+      vectorSupport (L.val : Set Point) (normalVector (t : Real.Angle))| ≤ d
+    exact hsupport.2.1 _ (norm_normalVector_real t)
+  have hpoint (t : ℝ) (ht : t ∈ Ioo 0 ω) : |f t - g t| ≤ C := by
+    have hct : 0 < Real.cos t := hcost ⟨ht.1.le, ht.2.le⟩
+    have hcos_le : Real.cos ω ≤ Real.cos t :=
+      Real.cos_le_cos_of_nonneg_of_le_pi ht.1.le
+        (by linarith [Real.pi_pos, hω]) ht.2.le
+    dsimp only [f, g]
+    rw [wedgeGaps_fst_eq_supportValue, wedgeGaps_fst_eq_supportValue]
+    calc
+      |(supportValue K.val (0 : Real.Angle) -
+          (supportValue K.val (t : Real.Angle) - 1) / Real.cos t) -
+        (supportValue L.val (0 : Real.Angle) -
+          (supportValue L.val (t : Real.Angle) - 1) / Real.cos t)| =
+          |(supportValue K.val (0 : Real.Angle) -
+              supportValue L.val (0 : Real.Angle)) -
+            (supportValue K.val (t : Real.Angle) -
+              supportValue L.val (t : Real.Angle)) / Real.cos t| := by
+            congr 1
+            field_simp
+            ring
+      _ ≤ |supportValue K.val (0 : Real.Angle) -
+            supportValue L.val (0 : Real.Angle)| +
+          |(supportValue K.val (t : Real.Angle) -
+            supportValue L.val (t : Real.Angle)) / Real.cos t| := abs_sub _ _
+      _ = |supportValue K.val (0 : Real.Angle) -
+            supportValue L.val (0 : Real.Angle)| +
+          |supportValue K.val (t : Real.Angle) -
+            supportValue L.val (t : Real.Angle)| / Real.cos t := by
+            rw [abs_div, abs_of_pos hct]
+      _ ≤ d + d / Real.cos t := add_le_add (hsupport' 0)
+        (div_le_div_of_nonneg_right (hsupport' t) hct.le)
+      _ ≤ d + d / Real.cos ω := add_le_add le_rfl
+        (div_le_div_of_nonneg_left Metric.hausdorffDist_nonneg hcosω hcos_le)
+      _ = C := by simp only [C, div_eq_mul_inv]; ring
+  have hf_cont : ContinuousOn f (Icc 0 ω) := by
+    have hnum : Continuous fun t : ℝ ↦
+        supportValue K.val (t : Real.Angle) - 1 :=
+      (hsupport.2.2.1.comp Real.Angle.continuous_coe).sub continuous_const
+    have hquot : ContinuousOn (fun t : ℝ ↦
+        (supportValue K.val (t : Real.Angle) - 1) / Real.cos t) (Icc 0 ω) :=
+      hnum.continuousOn.div Real.continuous_cos.continuousOn
+        (fun t ht ↦ (hcost ht).ne')
+    have hf_eq : f = fun t : ℝ ↦ supportValue K.val (0 : Real.Angle) -
+        (supportValue K.val (t : Real.Angle) - 1) / Real.cos t := by
+      funext t
+      exact wedgeGaps_fst_eq_supportValue K t
+    rw [hf_eq]
+    convert continuousOn_const.sub hquot using 1
+  have hsupportL := compactSet_support_continuity (L.val : Set Point) (K.val : Set Point)
+    L.val.nonempty L.val.isCompact K.val.nonempty K.val.isCompact
+  have hg_cont : ContinuousOn g (Icc 0 ω) := by
+    have hnum : Continuous fun t : ℝ ↦
+        supportValue L.val (t : Real.Angle) - 1 :=
+      (hsupportL.2.2.1.comp Real.Angle.continuous_coe).sub continuous_const
+    have hquot : ContinuousOn (fun t : ℝ ↦
+        (supportValue L.val (t : Real.Angle) - 1) / Real.cos t) (Icc 0 ω) :=
+      hnum.continuousOn.div Real.continuous_cos.continuousOn
+        (fun t ht ↦ (hcost ht).ne')
+    have hg_eq : g = fun t : ℝ ↦ supportValue L.val (0 : Real.Angle) -
+        (supportValue L.val (t : Real.Angle) - 1) / Real.cos t := by
+      funext t
+      exact wedgeGaps_fst_eq_supportValue L t
+    rw [hg_eq]
+    convert continuousOn_const.sub hquot using 1
+  have hf : BddBelow (f '' Ioo 0 ω) :=
+    (IsCompact.bddBelow_image isCompact_Icc hf_cont).mono (image_mono Ioo_subset_Icc_self)
+  have hg : BddBelow (g '' Ioo 0 ω) :=
+    (IsCompact.bddBelow_image isCompact_Icc hg_cont).mono (image_mono Ioo_subset_Icc_self)
+  simpa only [wedgeGapInfimum, f, g, C, d] using
+    abs_sInf_image_sub_sInf_image_le (nonempty_Ioo.mpr hω0) f g hf hg hpoint
+
+/-- The left wedge-gap infimum obeys the same Hausdorff estimate as the right one. -/
+theorem wedgeGapInfimum_snd_hausdorff_bound {ω : ℝ} (K L : CapSpace ω)
+    (hω : ω < Real.pi / 2) :
+    |(wedgeGapInfimum K).2 - (wedgeGapInfimum L).2| ≤
+      (1 + (Real.cos ω)⁻¹) * Metric.hausdorffDist (K.val : Set Point) L.val := by
+  let d := Metric.hausdorffDist (K.val : Set Point) L.val
+  let C := (1 + (Real.cos ω)⁻¹) * d
+  let f := fun t : ℝ ↦ (wedgeGaps K t).2
+  let g := fun t : ℝ ↦ (wedgeGaps L t).2
+  have hω0 : 0 < ω := K.property.1
+  have hcosω : 0 < Real.cos ω := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, hω0], hω⟩
+  have hcosδ {t : ℝ} (ht : t ∈ Icc 0 ω) : 0 < Real.cos (ω - t) :=
+    Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [ht.2], by linarith [Real.pi_pos, ht.1, hω]⟩
+  have hsupport := compactSet_support_continuity (K.val : Set Point) (L.val : Set Point)
+    K.val.nonempty K.val.isCompact L.val.nonempty L.val.isCompact
+  have hsupport' (t : ℝ) :
+      |supportValue K.val (t : Real.Angle) - supportValue L.val (t : Real.Angle)| ≤ d := by
+    change |vectorSupport (K.val : Set Point) (normalVector (t : Real.Angle)) -
+      vectorSupport (L.val : Set Point) (normalVector (t : Real.Angle))| ≤ d
+    exact hsupport.2.1 _ (norm_normalVector_real t)
+  have hpoint (t : ℝ) (ht : t ∈ Ioo 0 ω) : |f t - g t| ≤ C := by
+    have hct : 0 < Real.cos (ω - t) := hcosδ ⟨ht.1.le, ht.2.le⟩
+    have hcos_le : Real.cos ω ≤ Real.cos (ω - t) :=
+      Real.cos_le_cos_of_nonneg_of_le_pi (sub_nonneg.mpr ht.2.le)
+        (by linarith [Real.pi_pos, hω]) (by linarith [ht.1])
+    dsimp only [f, g]
+    rw [wedgeGaps_snd_eq_supportValue, wedgeGaps_snd_eq_supportValue]
+    calc
+      |(supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) -
+          (supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+            Real.cos (ω - t)) -
+        (supportValue L.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) -
+          (supportValue L.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+            Real.cos (ω - t))| =
+          |(supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) -
+              supportValue L.val ((ω + Real.pi / 2 : ℝ) : Real.Angle)) -
+            (supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) -
+              supportValue L.val ((t + Real.pi / 2 : ℝ) : Real.Angle)) /
+                Real.cos (ω - t)| := by
+            congr 1
+            field_simp
+            ring
+      _ ≤ |supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) -
+            supportValue L.val ((ω + Real.pi / 2 : ℝ) : Real.Angle)| +
+          |(supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) -
+            supportValue L.val ((t + Real.pi / 2 : ℝ) : Real.Angle)) /
+              Real.cos (ω - t)| := abs_sub _ _
+      _ = |supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) -
+            supportValue L.val ((ω + Real.pi / 2 : ℝ) : Real.Angle)| +
+          |supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) -
+            supportValue L.val ((t + Real.pi / 2 : ℝ) : Real.Angle)| /
+              Real.cos (ω - t) := by
+            rw [abs_div, abs_of_pos hct]
+      _ ≤ d + d / Real.cos (ω - t) := add_le_add (hsupport' (ω + Real.pi / 2))
+        (div_le_div_of_nonneg_right (hsupport' (t + Real.pi / 2)) hct.le)
+      _ ≤ d + d / Real.cos ω := add_le_add le_rfl
+        (div_le_div_of_nonneg_left Metric.hausdorffDist_nonneg hcosω hcos_le)
+      _ = C := by simp only [C, div_eq_mul_inv]; ring
+  have hf_cont : ContinuousOn f (Icc 0 ω) := by
+    have hnum : Continuous fun t : ℝ ↦
+        supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 :=
+      (hsupport.2.2.1.comp
+        (Real.Angle.continuous_coe.comp (continuous_id.add continuous_const))).sub
+          continuous_const
+    have hquot : ContinuousOn (fun t : ℝ ↦
+        (supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+          Real.cos (ω - t)) (Icc 0 ω) :=
+      hnum.continuousOn.div
+        (Real.continuous_cos.comp (continuous_const.sub continuous_id)).continuousOn
+        (fun t ht ↦ (hcosδ ht).ne')
+    have hf_eq : f = fun t : ℝ ↦
+        supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) -
+          (supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+            Real.cos (ω - t) := by
+      funext t
+      exact wedgeGaps_snd_eq_supportValue K t
+    rw [hf_eq]
+    exact continuousOn_const.sub hquot
+  have hsupportL := compactSet_support_continuity (L.val : Set Point) (K.val : Set Point)
+    L.val.nonempty L.val.isCompact K.val.nonempty K.val.isCompact
+  have hg_cont : ContinuousOn g (Icc 0 ω) := by
+    have hnum : Continuous fun t : ℝ ↦
+        supportValue L.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 :=
+      (hsupportL.2.2.1.comp
+        (Real.Angle.continuous_coe.comp (continuous_id.add continuous_const))).sub
+          continuous_const
+    have hquot : ContinuousOn (fun t : ℝ ↦
+        (supportValue L.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+          Real.cos (ω - t)) (Icc 0 ω) :=
+      hnum.continuousOn.div
+        (Real.continuous_cos.comp (continuous_const.sub continuous_id)).continuousOn
+        (fun t ht ↦ (hcosδ ht).ne')
+    have hg_eq : g = fun t : ℝ ↦
+        supportValue L.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) -
+          (supportValue L.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+            Real.cos (ω - t) := by
+      funext t
+      exact wedgeGaps_snd_eq_supportValue L t
+    rw [hg_eq]
+    exact continuousOn_const.sub hquot
+  have hf : BddBelow (f '' Ioo 0 ω) :=
+    (IsCompact.bddBelow_image isCompact_Icc hf_cont).mono (image_mono Ioo_subset_Icc_self)
+  have hg : BddBelow (g '' Ioo 0 ω) :=
+    (IsCompact.bddBelow_image isCompact_Icc hg_cont).mono (image_mono Ioo_subset_Icc_self)
+  simpa only [wedgeGapInfimum, f, g, C, d] using
+    abs_sInf_image_sub_sInf_image_le (nonempty_Ioo.mpr hω0) f g hf hg hpoint
+
+theorem balancedMaximumCap_gap_le_surface {ω : ℝ} (K : CapSpace ω)
+    (hK : IsBalancedMaximumCap K) (hω : ω < Real.pi / 2) :
+    (wedgeGapInfimum K).1 ≤
+      (surfaceAreaMeasure K.val {((Real.pi / 2 : ℝ) : Real.Angle)}).toReal ∧
+    (wedgeGapInfimum K).2 ≤
+      (surfaceAreaMeasure K.val {(ω : Real.Angle)}).toReal := by
+  obtain ⟨n, hn, _hmono, _hdyadic, P, hmax, hlim⟩ := hK
+  let Θ (i : ℕ) := uniformAngleSet ω K.property.1 K.property.2.1 (n i) (hn i)
+  have hpolygon (i : ℕ) := maximumPolygonCap_gap_le_surface (P i) (hmax i) hω
+  have hfst : Tendsto (fun i ↦ (wedgeGapInfimum (P i).val).1) atTop
+      (𝓝 (wedgeGapInfimum K).1) := by
+    apply tendsto_iff_dist_tendsto_zero.mpr
+    apply squeeze_zero (fun _ ↦ dist_nonneg) _ (by
+      simpa only [mul_zero] using
+        Tendsto.const_mul (1 + (Real.cos ω)⁻¹) hlim)
+    intro i
+    simpa only [Real.dist_eq, uniformAngleSet] using
+      wedgeGapInfimum_hausdorff_bound (P i).val K hω
+  have hsnd : Tendsto (fun i ↦ (wedgeGapInfimum (P i).val).2) atTop
+      (𝓝 (wedgeGapInfimum K).2) := by
+    apply tendsto_iff_dist_tendsto_zero.mpr
+    apply squeeze_zero (fun _ ↦ dist_nonneg) _ (by
+      simpa only [mul_zero] using
+        Tendsto.const_mul (1 + (Real.cos ω)⁻¹) hlim)
+    intro i
+    simpa only [Real.dist_eq, uniformAngleSet] using
+      wedgeGapInfimum_snd_hausdorff_bound (P i).val K hω
+  constructor
+  · exact le_surfaceAreaMeasure_atom_of_tendsto hlim hfst (fun i ↦ (hpolygon i).1)
+  · apply le_surfaceAreaMeasure_atom_of_tendsto hlim hsnd
+    intro i
+    simpa only [Θ, uniformAngleSet] using (hpolygon i).2
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Cap.BalancedExistence`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Balanced Existence
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter
+open scoped Topology
+
+namespace MovingSofa
+
+private def polygonSeedSet (Θ : AngleSet) : Set Point :=
+  capFan Θ.angle ∩ ⋂ t ∈ angleDomain Θ,
+    normalHalfPlane (t : Real.Angle) 1 false false
+
+private theorem mem_polygonSeedSet_iff (Θ : AngleSet) (p : Point) :
+    p ∈ polygonSeedSet Θ ↔
+      p ∈ capFan Θ.angle ∧
+        ∀ t ∈ angleDomain Θ, inner ℝ p (normalVector (t : Real.Angle)) ≤ 1 := by
+  simp [polygonSeedSet, normalHalfPlane]
+
+private theorem normalVector_mem_polygonSeedSet (Θ : AngleSet) {t : ℝ}
+    (ht : t ∈ angleDomain Θ) : normalVector (t : Real.Angle) ∈ polygonSeedSet Θ := by
+  apply (mem_polygonSeedSet_iff Θ _).mpr
+  have htupper : t ∈ capUpperAngles Θ.angle := by
+    rcases ht with (ht | ⟨s, hs, rfl⟩) | ht
+    · exact Or.inl ⟨(Θ.interior t ht).1.le, (Θ.interior t ht).2.le⟩
+    · exact Or.inr ⟨by dsimp; linarith [(Θ.interior s hs).1],
+        by dsimp; linarith [(Θ.interior s hs).2]⟩
+    · rcases ht with rfl | rfl
+      · exact Or.inl ⟨Θ.angle_pos.le, le_rfl⟩
+      · exact Or.inr ⟨le_rfl, by linarith [Θ.angle_pos]⟩
+  have ht0 : 0 ≤ t := by
+    rcases htupper with ⟨h0, _⟩ | ⟨hT, _⟩
+    · exact h0
+    · linarith [Real.pi_pos]
+  have htpi : t ≤ Real.pi := by
+    rcases htupper with ⟨_, hω⟩ | ⟨_, hω⟩
+    · linarith [Θ.angle_le, Real.pi_pos]
+    · linarith [Θ.angle_le]
+  have htω : -(Real.pi / 2) ≤ t - Θ.angle ∧ t - Θ.angle ≤ Real.pi / 2 := by
+    rcases htupper with ⟨h0, hω⟩ | ⟨hT, hω⟩ <;>
+      constructor <;> linarith [Θ.angle_le]
+  constructor
+  · constructor
+    · change 0 ≤ inner ℝ (normalVector (t : Real.Angle))
+        (normalVector (Θ.angle : Real.Angle))
+      rw [inner_normalVector_normalVector]
+      exact Real.cos_nonneg_of_mem_Icc htω
+    · change 0 ≤ inner ℝ (normalVector (t : Real.Angle))
+        (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+      simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, Real.cos_pi_div_two,
+        Real.sin_pi_div_two, PiLp.inner_apply, RCLike.inner_apply, conj_trivial,
+        Fin.sum_univ_two, Fin.isValue, Matrix.cons_val_zero, zero_mul, Matrix.cons_val_one,
+        Matrix.cons_val_fin_one, one_mul, zero_add]
+      exact Real.sin_nonneg_of_nonneg_of_le_pi ht0 htpi
+  · intro s hs
+    calc
+      inner ℝ (normalVector (t : Real.Angle)) (normalVector (s : Real.Angle)) ≤
+          ‖normalVector (t : Real.Angle)‖ * ‖normalVector (s : Real.Angle)‖ :=
+        real_inner_le_norm _ _
+      _ = 1 := by rw [norm_normalVector_real, norm_normalVector_real, mul_one]
+
+private theorem zero_mem_polygonSeedSet (Θ : AngleSet) :
+    (0 : Point) ∈ polygonSeedSet Θ := by
+  rw [mem_polygonSeedSet_iff]
+  simp [capFan, normalHalfPlane]
+
+private theorem convex_polygonSeedSet (Θ : AngleSet) : Convex ℝ (polygonSeedSet Θ) := by
+  rw [polygonSeedSet]
+  apply Convex.inter
+  · apply Convex.inter
+    · exact convex_halfSpace_ge
+        ⟨fun x y ↦ inner_add_left x y _, fun a x ↦ by simp [real_inner_smul_left]⟩ 0
+    · exact convex_halfSpace_ge
+        ⟨fun x y ↦ inner_add_left x y _, fun a x ↦ by simp [real_inner_smul_left]⟩ 0
+  · apply convex_iInter
+    intro t
+    apply convex_iInter
+    intro _
+    exact convex_halfSpace_le
+      ⟨fun x y ↦ inner_add_left x y _, fun a x ↦ by simp [real_inner_smul_left]⟩ 1
+
+private theorem isClosed_polygonSeedSet (Θ : AngleSet) : IsClosed (polygonSeedSet Θ) := by
+  rw [polygonSeedSet]
+  apply IsClosed.inter
+  · apply IsClosed.inter <;> exact isClosed_le continuous_const (by fun_prop)
+  · apply isClosed_iInter
+    intro t
+    apply isClosed_iInter
+    intro _
+    exact isClosed_le (by fun_prop) continuous_const
+
+private theorem isBounded_polygonSeedSet (Θ : AngleSet) :
+    Bornology.IsBounded (polygonSeedSet Θ) := by
+  obtain ⟨t, ht⟩ := Θ.nonempty
+  have hti := Θ.interior t ht
+  have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, hti.1], hti.2.trans_le Θ.angle_le⟩
+  have hs : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi hti.1
+    (by linarith [hti.2, Θ.angle_le, Real.pi_pos])
+  let l := -1 / Real.sin t
+  let r := 1 / Real.cos t
+  let M := |l| + |r|
+  have hM : 0 ≤ M := by dsimp [M]; positivity
+  apply isBounded_iff_forall_norm_le.mpr
+  refine ⟨M + 1, ?_⟩
+  intro p hp
+  obtain ⟨hfan, hupp⟩ := (mem_polygonSeedSet_iff Θ p).mp hp
+  have htmem : t ∈ angleDomain Θ := Or.inl (Or.inl ht)
+  have htmem' : t + Real.pi / 2 ∈ angleDomain Θ :=
+    Or.inl (Or.inr ⟨t, ht, rfl⟩)
+  have ha := hupp t htmem
+  have hb := hupp (t + Real.pi / 2) htmem'
+  have hy0 : 0 ≤ p 1 := by
+    have := hfan.2
+    simpa [normalHalfPlane, normalVector, frame, PiLp.inner_apply] using this
+  have hy1 : p 1 ≤ 1 := by
+    have htop := hupp (Real.pi / 2) (Or.inr (Or.inr rfl))
+    simpa [normalVector, frame, PiLp.inner_apply] using htop
+  simp [normalVector, frame, PiLp.inner_apply, Real.cos_add, Real.sin_add,
+    -Real.Angle.coe_add] at ha hb
+  have hl : l ≤ p 0 := by
+    apply (div_le_iff₀ hs).mpr
+    dsimp [l]
+    nlinarith [mul_nonneg hc.le hy0]
+  have hr : p 0 ≤ r := by
+    apply (le_div_iff₀ hc).mpr
+    dsimp [r]
+    nlinarith [mul_nonneg hs.le hy0]
+  have hx : |p 0| ≤ M := by
+    apply abs_le.mpr
+    dsimp [M]
+    constructor <;> linarith [neg_abs_le l, le_abs_self r, abs_nonneg l, abs_nonneg r]
+  have hx2 := (sq_le_sq₀ (abs_nonneg (p 0)) hM).mpr hx
+  have hy2 : (p 1) ^ 2 ≤ 1 := by nlinarith
+  have hn := EuclideanSpace.norm_sq_eq p
+  simp only [Fin.sum_univ_two, Real.norm_eq_abs, sq_abs] at hn hx2
+  nlinarith [norm_nonneg p]
+
+private def polygonSeedBody (Θ : AngleSet) : ConvexBody Point where
+  carrier := polygonSeedSet Θ
+  convex' := convex_polygonSeedSet Θ
+  isCompact' := Metric.isCompact_iff_isClosed_bounded.mpr
+    ⟨isClosed_polygonSeedSet Θ, isBounded_polygonSeedSet Θ⟩
+  nonempty' := ⟨0, zero_mem_polygonSeedSet Θ⟩
+
+private theorem supportValue_polygonSeedBody_upper (Θ : AngleSet) {t : ℝ}
+    (ht : t ∈ angleDomain Θ) :
+    supportValue (polygonSeedBody Θ) (t : Real.Angle) = 1 := by
+  apply le_antisymm
+  · apply supportValue_le_of_subset_normalHalfPlane
+    intro p hp
+    exact ((mem_polygonSeedSet_iff Θ p).mp hp).2 t ht
+  · have hmem := normalVector_mem_polygonSeedSet Θ ht
+    have hle := inner_le_supportValue (polygonSeedBody Θ) hmem (t : Real.Angle)
+    rwa [inner_normalVector_self] at hle
+
+private theorem supportValue_polygonSeedBody_lower_angle (Θ : AngleSet) :
+    supportValue (polygonSeedBody Θ) ((Θ.angle + Real.pi : ℝ) : Real.Angle) = 0 := by
+  apply le_antisymm
+  · apply csSup_le ((polygonSeedBody Θ).nonempty.image _)
+    rintro _ ⟨p, hp, rfl⟩
+    have hfan := ((mem_polygonSeedSet_iff Θ p).mp hp).1.1
+    change 0 ≤ inner ℝ p (normalVector (Θ.angle : Real.Angle)) at hfan
+    change inner ℝ p (normalVector ((Θ.angle + Real.pi : ℝ) : Real.Angle)) ≤ 0
+    rw [normalVector_add_pi, inner_neg_right]
+    exact neg_nonpos.mpr hfan
+  · apply le_csSup ((polygonSeedBody Θ).isCompact.image
+      (continuous_inner.comp (continuous_id.prodMk continuous_const))).bddAbove
+    exact ⟨0, zero_mem_polygonSeedSet Θ, by simp⟩
+
+private theorem supportValue_polygonSeedBody_lower_vertical (Θ : AngleSet) :
+    supportValue (polygonSeedBody Θ) ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := by
+  apply le_antisymm
+  · apply csSup_le ((polygonSeedBody Θ).nonempty.image _)
+    rintro _ ⟨p, hp, rfl⟩
+    have hfan := ((mem_polygonSeedSet_iff Θ p).mp hp).1.2
+    change 0 ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at hfan
+    change inner ℝ p (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) ≤ 0
+    rw [show (3 * Real.pi / 2 : ℝ) = Real.pi / 2 + Real.pi by ring,
+      normalVector_add_pi, inner_neg_right]
+    exact neg_nonpos.mpr hfan
+  · apply le_csSup ((polygonSeedBody Θ).isCompact.image
+      (continuous_inner.comp (continuous_id.prodMk continuous_const))).bddAbove
+    exact ⟨0, zero_mem_polygonSeedSet Θ, by simp⟩
+
+private theorem polygonSeedBody_representation (Θ : AngleSet) :
+    HasHalfPlaneRepresentation (polygonSeedBody Θ)
+      (((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain Θ) ∪
+        capLowerNormals Θ.angle) := by
+  let C : Set (Real.Angle × ℝ) :=
+    (fun t : ℝ ↦ ((t : Real.Angle), 1)) '' angleDomain Θ ∪
+      {(((Θ.angle + Real.pi : ℝ) : Real.Angle), 0),
+        (((3 * Real.pi / 2 : ℝ) : Real.Angle), 0)}
+  refine ⟨C, ?_, ?_⟩
+  · rintro c (⟨t, ht, rfl⟩ | hc)
+    · exact Or.inl ⟨t, ht, rfl⟩
+    · rcases hc with rfl | hc
+      · exact Or.inr (Or.inl rfl)
+      · have hc' : c = ((((3 * Real.pi / 2 : ℝ) : Real.Angle), 0)) := hc
+        subst c
+        exact Or.inr (Or.inr rfl)
+  · ext p
+    change p ∈ polygonSeedSet Θ ↔ _
+    rw [mem_polygonSeedSet_iff]
+    simp only [C, Set.mem_iInter, Set.mem_union, Set.mem_image, Set.mem_insert_iff,
+      Set.mem_singleton_iff, normalHalfPlane, Bool.false_eq_true, ↓reduceIte]
+    constructor
+    · rintro ⟨hfan, hupp⟩ c (⟨t, ht, rfl⟩ | hc)
+      · exact hupp t ht
+      · rcases hc with rfl | rfl
+        · change inner ℝ p (normalVector ((Θ.angle + Real.pi : ℝ) : Real.Angle)) ≤ 0
+          rw [normalVector_add_pi, inner_neg_right]
+          exact neg_nonpos.mpr hfan.1
+        · change inner ℝ p (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) ≤ 0
+          rw [inner_normalVector_three_pi_div_two]
+          have hy : 0 ≤ p 1 := by
+            simpa [capFan, normalHalfPlane, normalVector, frame, PiLp.inner_apply] using hfan.2
+          exact neg_nonpos.mpr hy
+    · intro hp
+      constructor
+      · constructor
+        · change 0 ≤ inner ℝ p (normalVector (Θ.angle : Real.Angle))
+          have h := hp (((Θ.angle + Real.pi : ℝ) : Real.Angle), 0)
+            (Or.inr (Or.inl rfl))
+          change inner ℝ p (normalVector ((Θ.angle + Real.pi : ℝ) : Real.Angle)) ≤ 0 at h
+          rw [normalVector_add_pi, inner_neg_right] at h
+          exact neg_nonpos.mp h
+        · have h := hp (((3 * Real.pi / 2 : ℝ) : Real.Angle), 0)
+            (Or.inr (Or.inr rfl))
+          change inner ℝ p (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) ≤ 0 at h
+          rw [inner_normalVector_three_pi_div_two] at h
+          change p ∈ normalHalfPlane ((Real.pi / 2 : ℝ) : Real.Angle) 0 true false
+          simpa [normalHalfPlane, normalVector, frame, PiLp.inner_apply] using h
+      · intro t ht
+        exact hp ((t : Real.Angle), 1) (Or.inl ⟨t, ht, rfl⟩)
+
+private def polygonSeedCap (Θ : AngleSet) : PolygonCapSpace Θ := by
+  let K := polygonSeedBody Θ
+  have hrepr := polygonSeedBody_representation Θ
+  have hdomain : angleDomain Θ ⊆ capUpperAngles Θ.angle := by
+    rintro t ((ht | ⟨s, hs, rfl⟩) | ht)
+    · exact Or.inl ⟨(Θ.interior t ht).1.le, (Θ.interior t ht).2.le⟩
+    · exact Or.inr ⟨by dsimp; linarith [(Θ.interior s hs).1],
+        by dsimp; linarith [(Θ.interior s hs).2]⟩
+    · rcases ht with rfl | rfl
+      · exact Or.inl ⟨Θ.angle_pos.le, le_rfl⟩
+      · exact Or.inr ⟨le_rfl, by linarith [Θ.angle_pos]⟩
+  have hcaprepr : HasHalfPlaneRepresentation K
+      (((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles Θ.angle) ∪
+        capLowerNormals Θ.angle) := by
+    obtain ⟨C, hC, hKC⟩ := hrepr
+    refine ⟨C, ?_, hKC⟩
+    intro c hc
+    rcases hC c hc with ⟨t, ht, heq⟩ | ht
+    · exact Or.inl ⟨t, hdomain ht, heq⟩
+    · exact Or.inr ht
+  have hcap : IsCap Θ.angle K := by
+    refine ⟨Θ.angle_pos, Θ.angle_le, ?_, ?_, ?_, ?_, hcaprepr⟩
+    · exact supportValue_polygonSeedBody_upper Θ (Or.inr (Or.inl rfl))
+    · exact supportValue_polygonSeedBody_upper Θ (Or.inr (Or.inr rfl))
+    · exact supportValue_polygonSeedBody_lower_angle Θ
+    · exact supportValue_polygonSeedBody_lower_vertical Θ
+  exact ⟨⟨K, hcap⟩, hrepr⟩
+
+private theorem supportValue_polygonSeedCap_upper (Θ : AngleSet) {t : ℝ}
+    (ht : t ∈ angleDomain Θ) :
+    supportValue (polygonSeedCap Θ).val.val (t : Real.Angle) = 1 := by
+  change supportValue (polygonSeedBody Θ) (t : Real.Angle) = 1
+  exact supportValue_polygonSeedBody_upper Θ ht
+
+private theorem polygonNiche_polygonSeedCap (Θ : AngleSet) :
+    polygonNiche Θ (polygonSeedCap Θ).val = ∅ := by
+  apply Set.eq_empty_iff_forall_notMem.mpr
+  intro p hp
+  obtain ⟨hfan, hq⟩ := hp
+  obtain ⟨t, ht, hq⟩ := Set.mem_iUnion₂.mp hq
+  have hti := Θ.interior t ht
+  have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, hti.1], hti.2.trans_le Θ.angle_le⟩
+  have hs : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi hti.1
+    (by linarith [hti.2, Θ.angle_le, Real.pi_pos])
+  have hy : 0 ≤ p 1 := by
+    have := hfan.2
+    simpa [capFan, normalHalfPlane, normalVector, frame, PiLp.inner_apply] using this
+  have h₁ := hq.1
+  have h₂ := hq.2
+  change inner ℝ p (normalVector (t : Real.Angle)) <
+    supportValue (polygonSeedCap Θ).val.val (t : Real.Angle) - 1 at h₁
+  change inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) <
+    supportValue (polygonSeedCap Θ).val.val
+      ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 at h₂
+  rw [supportValue_polygonSeedCap_upper Θ (Or.inl (Or.inl ht))] at h₁
+  rw [supportValue_polygonSeedCap_upper Θ (Or.inl (Or.inr ⟨t, ht, rfl⟩))] at h₂
+  simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe,
+    PiLp.inner_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue,
+    Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one, sub_self,
+    Real.cos_add, Real.cos_pi_div_two, mul_zero, Real.sin_pi_div_two, mul_one,
+    zero_sub, Real.sin_add, zero_add, neg_mul, neg_add_lt_iff_lt_add, add_zero] at h₁ h₂
+  have h₁' := mul_lt_mul_of_pos_left h₁ hs
+  have h₂' := mul_lt_mul_of_pos_left h₂ hc
+  nlinarith [Real.sin_sq_add_cos_sq t]
+
+private theorem polygonSeedCap_area_nonneg (Θ : AngleSet) :
+    0 ≤ polygonAreaFunctional Θ (polygonSeedCap Θ).val := by
+  rw [(polygonArea_upperBound Θ).1, polygonNiche_polygonSeedCap]
+  simp [ClassicalResults.area]
+
+private theorem angleDomain_distance_midpoint (Θ : AngleSet) {t : ℝ}
+    (ht : t ∈ angleDomain Θ) :
+    Real.pi / 4 - Θ.angle / 2 ≤
+        |t - (Real.pi / 4 + Θ.angle / 2)| ∧
+      |t - (Real.pi / 4 + Θ.angle / 2)| ≤ Real.pi / 2 := by
+  have htupper : t ∈ capUpperAngles Θ.angle := by
+    rcases ht with (ht | ⟨s, hs, rfl⟩) | ht
+    · exact Or.inl ⟨(Θ.interior t ht).1.le, (Θ.interior t ht).2.le⟩
+    · exact Or.inr ⟨by dsimp; linarith [(Θ.interior s hs).1],
+        by dsimp; linarith [(Θ.interior s hs).2]⟩
+    · rcases ht with rfl | rfl
+      · exact Or.inl ⟨Θ.angle_pos.le, le_rfl⟩
+      · exact Or.inr ⟨le_rfl, by linarith [Θ.angle_pos]⟩
+  rcases htupper with ⟨ht0, htω⟩ | ⟨htT, htωT⟩
+  · rw [abs_of_nonpos (by linarith [Θ.angle_le])]
+    constructor <;> linarith [Θ.angle_le, Real.pi_pos]
+  · rw [abs_of_nonneg (by linarith [Θ.angle_le])]
+    constructor <;> linarith [Θ.angle_le]
+
+private theorem stripParallelogram_top_mem_polygonSeedCap (Θ : AngleSet) :
+    (stripParallelogram Θ.angle).2.2 ∈ ((polygonSeedCap Θ).val.val : Set Point) := by
+  change (stripParallelogram Θ.angle).2.2 ∈ polygonSeedSet Θ
+  apply (mem_polygonSeedSet_iff Θ _).mpr
+  let β := Real.pi / 4 - Θ.angle / 2
+  let m := Real.pi / 4 + Θ.angle / 2
+  have hβ0 : 0 ≤ β := by dsimp [β]; linarith [Θ.angle_le]
+  have hβlt : β < Real.pi / 2 := by
+    dsimp [β]
+    linarith [Θ.angle_pos, Real.pi_pos]
+  have hcosβ : 0 < Real.cos β := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos], hβlt⟩
+  have htanβ : 0 ≤ Real.tan β :=
+    Real.tan_nonneg_of_nonneg_of_le_pi_div_two hβ0 hβlt.le
+  have hcosω : 0 ≤ Real.cos Θ.angle := Real.cos_nonneg_of_mem_Icc
+    ⟨by linarith [Θ.angle_pos, Real.pi_pos], Θ.angle_le⟩
+  have hsinω : 0 ≤ Real.sin Θ.angle := Real.sin_nonneg_of_nonneg_of_le_pi
+    Θ.angle_pos.le (by linarith [Θ.angle_le])
+  constructor
+  · constructor
+    · change 0 ≤ inner ℝ (!₂[Real.tan β, 1] : Point)
+        (normalVector (Θ.angle : Real.Angle))
+      simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply,
+        RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue, Matrix.cons_val_zero,
+        Matrix.cons_val_one, Matrix.cons_val_fin_one, mul_one]
+      positivity
+    · change 0 ≤ inner ℝ (!₂[Real.tan β, 1] : Point)
+        (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+      simp [normalVector, frame, PiLp.inner_apply]
+  · intro t ht
+    have hd := angleDomain_distance_midpoint Θ ht
+    have hdpi : |t - m| ≤ Real.pi := by dsimp [m]; linarith [hd.2, Real.pi_pos]
+    have hcosabs : Real.cos |t - m| ≤ Real.cos β :=
+      Real.cos_le_cos_of_nonneg_of_le_pi hβ0 hdpi (by simpa [β, m] using hd.1)
+    have hcos : Real.cos (t - m) ≤ Real.cos β := by
+      simpa only [Real.cos_abs] using hcosabs
+    change inner ℝ (!₂[Real.tan β, 1] : Point) (normalVector (t : Real.Angle)) ≤ 1
+    simp only [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two,
+      Matrix.cons_val_zero, Matrix.cons_val_one, Real.inner_apply]
+    have htrig : Real.sin β * Real.cos t + Real.cos β * Real.sin t =
+        Real.cos (t - m) := by
+      calc
+        Real.sin β * Real.cos t + Real.cos β * Real.sin t =
+            Real.sin (β + t) := by rw [Real.sin_add]
+        _ = Real.cos (Real.pi / 2 - (β + t)) := by rw [Real.cos_pi_div_two_sub]
+        _ = Real.cos (m - t) := by congr 1; dsimp [β, m]; ring
+        _ = Real.cos (t - m) := by rw [← Real.cos_neg]; congr 1; ring
+    apply le_of_mul_le_mul_left ?_ hcosβ
+    rw [Real.tan_eq_sin_div_cos]
+    field_simp [hcosβ.ne']
+    change Real.sin β * Real.cos t + Real.cos β * Real.sin t ≤ Real.cos β
+    rw [htrig]
+    exact hcos
+
+private def IsPolygonMaximizationCandidate (Θ : AngleSet) (K : PolygonCapSpace Θ) : Prop :=
+  (stripParallelogram Θ.angle).2.2 ∈ (K.val.val : Set Point) ∧
+    0 ≤ polygonAreaFunctional Θ K.val
+
+private theorem exists_compact_polygonCap_bound (ω : ℝ) (hω : 0 < ω)
+    (hω' : ω ≤ Real.pi / 2) (t : ℝ) (ht : t ∈ Set.Ioo 0 ω) :
+    ∃ A : Set Point, IsCompact A ∧ ∀ (Θ : AngleSet), Θ.angle = ω →
+      t ∈ Θ.directions → ∀ K : PolygonCapSpace Θ,
+        IsPolygonMaximizationCandidate Θ K → (K.val.val : Set Point) ⊆ A := by
+  obtain ⟨c, hc, hwidth⟩ := polygonCap_width_bound ω t hω hω' ht
+  let o := (stripParallelogram ω).2.2
+  let R := c + |o 0| + 2
+  refine ⟨Metric.closedBall 0 R, isCompact_closedBall 0 R, ?_⟩
+  intro Θ hΘ hdir K hK p hp
+  subst ω
+  have hw := hwidth Θ rfl hdir K hK.2
+  have hp0 := inner_le_supportValue K.val.val hp (0 : Real.Angle)
+  have hpπ := inner_le_supportValue K.val.val hp (Real.pi : Real.Angle)
+  have ho0 := inner_le_supportValue K.val.val hK.1 (0 : Real.Angle)
+  have hoπ := inner_le_supportValue K.val.val hK.1 (Real.pi : Real.Angle)
+  simp [directionalWidth, supportValue, normalVector, frame, PiLp.inner_apply]
+    at hw hp0 hpπ ho0 hoπ
+  have hpx : |p 0| ≤ c + |o 0| := by
+    rw [abs_le]
+    constructor
+    · linarith [neg_abs_le (o 0)]
+    · linarith [le_abs_self (o 0)]
+  have hpy := K.val.mem_horizontalStrip hp
+  have hR0 : 0 ≤ R := by dsimp [R]; positivity
+  rw [Metric.mem_closedBall]
+  have hpx2 := (sq_le_sq₀ (abs_nonneg (p 0)) (by positivity : 0 ≤ c + |o 0|)).mpr hpx
+  have hpy2 : (p 1) ^ 2 ≤ 1 := by nlinarith [hpy.1, hpy.2]
+  have ha0 : 0 ≤ c + |o 0| := by positivity
+  have hn := EuclideanSpace.norm_sq_eq p
+  simp only [Fin.sum_univ_two, Real.norm_eq_abs, sq_abs] at hn hpx2
+  have hsquare : ‖p‖ ^ 2 ≤ R ^ 2 := by
+    dsimp [R]
+    nlinarith
+  simpa [dist_eq_norm] using
+    ((sq_le_sq₀ (norm_nonneg p) hR0).mp hsquare)
+
+private theorem exists_compact_polygonCandidate_bound (Θ : AngleSet) :
+    ∃ A : Set Point, IsCompact A ∧ ∀ K : PolygonCapSpace Θ,
+      IsPolygonMaximizationCandidate Θ K → (K.val.val : Set Point) ⊆ A := by
+  obtain ⟨t, ht⟩ := Θ.nonempty
+  obtain ⟨A, hA, hbound⟩ := exists_compact_polygonCap_bound Θ.angle
+    Θ.angle_pos Θ.angle_le t (Θ.interior t ht)
+  exact ⟨A, hA, fun K hK ↦ hbound Θ rfl ht K hK⟩
+
+private theorem tendsto_supportValue_of_hausdorffBodies
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hlim : Tendsto (fun i ↦ Metric.hausdorffDist (K i : Set Point)
+      (L : Set Point)) atTop (𝓝 0)) (t : ℝ) :
+    Tendsto (fun i ↦ supportValue (K i) (t : Real.Angle)) atTop
+      (𝓝 (supportValue L (t : Real.Angle))) := by
+  apply tendsto_iff_dist_tendsto_zero.mpr
+  apply squeeze_zero (fun _ ↦ dist_nonneg) _ hlim
+  intro i
+  simpa only [Real.dist_eq, vectorSupport, supportValue] using
+    (compactSet_support_continuity (K i) L (K i).nonempty
+      (K i).isCompact L.nonempty L.isCompact).2.1
+        (normalVector (t : Real.Angle)) (norm_normalVector_real t)
+
+private theorem polygonCandidate_values_bddAbove (Θ : AngleSet) :
+    BddAbove {r : ℝ | ∃ K : PolygonCapSpace Θ,
+      IsPolygonMaximizationCandidate Θ K ∧ polygonAreaFunctional Θ K.val = r} := by
+  obtain ⟨A, hA, hbound⟩ := exists_compact_polygonCandidate_bound Θ
+  refine ⟨ClassicalResults.area A, ?_⟩
+  rintro r ⟨K, hK, rfl⟩
+  rw [(polygonArea_upperBound Θ).1 K]
+  calc
+    ClassicalResults.area (K.val.val : Set Point) -
+        ClassicalResults.area (polygonNiche Θ K.val) ≤
+        ClassicalResults.area (K.val.val : Set Point) := by
+      linarith [show 0 ≤ ClassicalResults.area (polygonNiche Θ K.val) from
+        ENNReal.toReal_nonneg]
+    _ ≤ ClassicalResults.area A := by
+      exact ENNReal.toReal_mono hA.measure_ne_top (MeasureTheory.measure_mono (hbound K hK))
+
+private theorem supportValue_limit_eq {K : ℕ → ConvexBody Point} {L : ConvexBody Point}
+    (hlim : Tendsto (fun i ↦ Metric.hausdorffDist (K i : Set Point)
+      (L : Set Point)) atTop (𝓝 0)) (t c : ℝ)
+    (hvalue : ∀ i, supportValue (K i) (t : Real.Angle) = c) :
+    supportValue L (t : Real.Angle) = c := by
+  apply tendsto_nhds_unique (tendsto_supportValue_of_hausdorffBodies K L hlim t)
+  simpa only [hvalue] using (tendsto_const_nhds : Tendsto (fun _ : ℕ ↦ c) atTop (𝓝 c))
+
+private theorem mem_hausdorffLimit {K : ℕ → ConvexBody Point} {L : ConvexBody Point}
+    (hlim : Tendsto (fun i ↦ Metric.hausdorffDist (K i : Set Point)
+    (L : Set Point)) atTop (𝓝 0)) {p : Point} (hp : ∀ i, p ∈ (K i : Set Point)) :
+    p ∈ (L : Set Point) := by
+  have hle : (fun _ : ℕ ↦ Metric.infDist p (L : Set Point)) ≤ᶠ[atTop]
+      (fun i ↦ Metric.hausdorffDist (K i : Set Point) (L : Set Point)) := by
+    filter_upwards [] with i
+    exact Metric.infDist_le_hausdorffDist_of_mem (hp i)
+      (Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded
+        (K i).nonempty L.nonempty (K i).isCompact.isBounded L.isCompact.isBounded)
+  have hz : Metric.infDist p (L : Set Point) ≤ 0 :=
+    le_of_tendsto_of_tendsto tendsto_const_nhds hlim hle
+  exact (L.isClosed.mem_iff_infDist_zero L.nonempty).mpr
+    (le_antisymm hz Metric.infDist_nonneg)
+
+private theorem exists_polygonCap_of_hausdorffLimit (Θ : AngleSet)
+    (K : ℕ → PolygonCapSpace Θ) (L : ConvexBody Point)
+    (hlim : Tendsto (fun i ↦ Metric.hausdorffDist (K i).val.val (L : Set Point))
+      atTop (𝓝 0)) :
+    ∃ P : PolygonCapSpace Θ, (P.val.val : Set Point) = (L : Set Point) := by
+  let N : Set Real.Angle :=
+    ((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain Θ) ∪ capLowerNormals Θ.angle
+  let U : Set Point := normalVector '' N
+  have hU : ∀ u ∈ U, ‖u‖ = 1 := by
+    rintro u ⟨a, ha, rfl⟩
+    rcases ha with ⟨t, _, rfl⟩ | ha
+    · exact norm_normalVector_real t
+    · rcases ha with rfl | rfl
+      · exact norm_normalVector_real (Θ.angle + Real.pi)
+      · exact norm_normalVector_real (3 * Real.pi / 2)
+  have hK (i : ℕ) : ((K i).val.val : Set Point) = ⋂ u ∈ U,
+      {x | inner ℝ x u ≤ vectorSupport (K i).val.val u} := by
+    calc
+      ((K i).val.val : Set Point) = ⋂ a ∈ N,
+          normalHalfPlane a (supportValue (K i).val.val a) false false :=
+        (K i).property.eq_iInter_supportValue
+      _ = ⋂ u ∈ U, {x | inner ℝ x u ≤ vectorSupport (K i).val.val u} := by
+        ext p
+        simp [U, normalHalfPlane, supportValue, vectorSupport]
+  have hclosed := fixedNormalBody_closed U hU (fun i ↦ (K i).val.val) L hK hlim
+  have hrepr : HasHalfPlaneRepresentation L N := by
+    refine ⟨(fun t ↦ (t, supportValue L t)) '' N, ?_, ?_⟩
+    · rintro _ ⟨t, ht, rfl⟩
+      exact ht
+    · calc
+        (L : Set Point) = ⋂ u ∈ U,
+            {x | inner ℝ x u ≤ vectorSupport L u} := hclosed
+        _ = ⋂ t ∈ N, normalHalfPlane t (supportValue L t) false false := by
+          ext p
+          simp [U, normalHalfPlane, supportValue, vectorSupport]
+        _ = ⋂ c ∈ (fun t ↦ (t, supportValue L t)) '' N,
+            normalHalfPlane c.1 c.2 false false := by
+          ext p
+          simp
+  have hω := supportValue_limit_eq hlim Θ.angle 1
+    (fun i ↦ (K i).val.property.2.2.1)
+  have hT := supportValue_limit_eq hlim (Real.pi / 2) 1
+    (fun i ↦ (K i).val.property.2.2.2.1)
+  have hωπ := supportValue_limit_eq hlim (Θ.angle + Real.pi) 0
+    (fun i ↦ (K i).val.property.2.2.2.2.1)
+  have h3T := supportValue_limit_eq hlim (3 * Real.pi / 2) 0
+    (fun i ↦ (K i).val.property.2.2.2.2.2.1)
+  have hdomain : angleDomain Θ ⊆ capUpperAngles Θ.angle := by
+    rintro t ((ht | ⟨s, hs, rfl⟩) | ht)
+    · exact Or.inl ⟨(Θ.interior t ht).1.le, (Θ.interior t ht).2.le⟩
+    · exact Or.inr ⟨by dsimp; linarith [(Θ.interior s hs).1],
+        by dsimp; linarith [(Θ.interior s hs).2]⟩
+    · rcases ht with rfl | rfl
+      · exact Or.inl ⟨Θ.angle_pos.le, le_rfl⟩
+      · exact Or.inr ⟨le_rfl, by linarith [Θ.angle_pos]⟩
+  have hcaprepr : HasHalfPlaneRepresentation L
+      (((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles Θ.angle) ∪
+        capLowerNormals Θ.angle) := by
+    obtain ⟨C, hCN, hLC⟩ := hrepr
+    refine ⟨C, ?_, hLC⟩
+    intro c hc
+    rcases hCN c hc with ⟨t, ht, heq⟩ | ht
+    · exact Or.inl ⟨t, hdomain ht, heq⟩
+    · exact Or.inr ht
+  have hcap : IsCap Θ.angle L :=
+    ⟨Θ.angle_pos, Θ.angle_le, hω, hT, hωπ, h3T, hcaprepr⟩
+  exact ⟨⟨⟨L, hcap⟩, hrepr⟩, rfl⟩
+
+private theorem exists_maximum_polygonCandidate (Θ : AngleSet) :
+    ∃ K : PolygonCapSpace Θ, IsPolygonMaximizationCandidate Θ K ∧
+      ∀ L : PolygonCapSpace Θ, IsPolygonMaximizationCandidate Θ L →
+        polygonAreaFunctional Θ L.val ≤ polygonAreaFunctional Θ K.val := by
+  let S : Set ℝ := {r | ∃ K : PolygonCapSpace Θ,
+    IsPolygonMaximizationCandidate Θ K ∧ polygonAreaFunctional Θ K.val = r}
+  have hS : S.Nonempty := by
+    refine ⟨polygonAreaFunctional Θ (polygonSeedCap Θ).val, polygonSeedCap Θ, ?_, rfl⟩
+    exact ⟨stripParallelogram_top_mem_polygonSeedCap Θ, polygonSeedCap_area_nonneg Θ⟩
+  have hSbdd : BddAbove S := polygonCandidate_values_bddAbove Θ
+  obtain ⟨u, _, hu, huS⟩ := exists_seq_tendsto_sSup hS hSbdd
+  choose K hK hKu using huS
+  obtain ⟨A, hA, hbound⟩ := exists_compact_polygonCandidate_bound Θ
+  obtain ⟨φ, L, hφ, hlim⟩ := convexBody_selection A hA (fun i ↦ (K i).val.val)
+    (fun i ↦ hbound (K i) (hK i))
+  obtain ⟨P, hPL⟩ := exists_polygonCap_of_hausdorffLimit Θ (fun i ↦ K (φ i)) L hlim
+  have hlimP : Tendsto (fun i ↦ Metric.hausdorffDist ((K (φ i)).val.val : Set Point)
+      (P.val.val : Set Point)) atTop (𝓝 0) := by
+    simpa only [hPL] using hlim
+  have hcont := (polygonArea_continuity Θ (fun i ↦ (K (φ i)).val) P.val hlimP).2
+  have huφ : Tendsto (fun i ↦ u (φ i)) atTop (𝓝 (sSup S)) :=
+    hu.comp hφ.tendsto_atTop
+  have hfuncSup : polygonAreaFunctional Θ P.val = sSup S := by
+    apply tendsto_nhds_unique hcont
+    simpa only [hKu] using huφ
+  have hPtop : (stripParallelogram Θ.angle).2.2 ∈ (P.val.val : Set Point) := by
+    rw [hPL]
+    exact mem_hausdorffLimit hlim (fun i ↦ (hK (φ i)).1)
+  have hPnonneg : 0 ≤ polygonAreaFunctional Θ P.val := by
+    apply ge_of_tendsto hcont
+    exact Eventually.of_forall fun i ↦ (hK (φ i)).2
+  refine ⟨P, ⟨hPtop, hPnonneg⟩, ?_⟩
+  intro Q hQ
+  rw [hfuncSup]
+  exact le_csSup hSbdd ⟨Q, hQ, rfl⟩
+
+private theorem innerQuadrant_translate (K : ConvexBody Point) (v : Point) (t : ℝ) :
+    innerQuadrant (ConvexBody.translate K v : Set Point) t =
+      (fun p ↦ p + v) '' innerQuadrant (K : Set Point) t := by
+  have hsupp (a : Real.Angle) : supportValue (ConvexBody.translate K v) a =
+      supportValue K a + inner ℝ v (normalVector a) := supportValue_image_add K v a
+  ext q
+  simp only [innerQuadrant, normalHalfPlane, Set.mem_inter_iff, Bool.false_eq_true,
+    ↓reduceIte, Set.mem_image]
+  rw [hsupp, hsupp]
+  constructor
+  · intro h
+    change inner ℝ q (normalVector (t : Real.Angle)) <
+        supportValue K (t : Real.Angle) + inner ℝ v (normalVector (t : Real.Angle)) - 1 ∧
+      inner ℝ q (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) <
+        supportValue K ((t + Real.pi / 2 : ℝ) : Real.Angle) +
+          inner ℝ v (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) - 1 at h
+    refine ⟨q - v, ?_, by simp⟩
+    constructor
+    · change inner ℝ (q - v) (normalVector (t : Real.Angle)) < _
+      rw [inner_sub_left]
+      linarith [h.1]
+    · change inner ℝ (q - v) (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) < _
+      rw [inner_sub_left]
+      linarith [h.2]
+  · rintro ⟨p, hp, rfl⟩
+    change inner ℝ p (normalVector (t : Real.Angle)) < supportValue K (t : Real.Angle) - 1 ∧
+      inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) <
+        supportValue K ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 at hp
+    constructor
+    · change inner ℝ (p + v) (normalVector (t : Real.Angle)) < _
+      rw [inner_add_left]
+      linarith [hp.1]
+    · change inner ℝ (p + v) (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) < _
+      rw [inner_add_left]
+      linarith [hp.2]
+
+private theorem exists_candidate_translate_rightAngle (Θ : AngleSet)
+    (hΘ : Θ.angle = Real.pi / 2) (K : PolygonCapSpace Θ)
+    (hK : 0 ≤ polygonAreaFunctional Θ K.val) :
+    ∃ P : PolygonCapSpace Θ, IsPolygonMaximizationCandidate Θ P ∧
+      polygonAreaFunctional Θ P.val = polygonAreaFunctional Θ K.val := by
+  obtain ⟨p, hp, hpeq⟩ := (K.val.val.isCompact.image
+    (continuous_inner.comp (continuous_id.prodMk continuous_const))).sSup_mem
+      (K.val.val.nonempty.image
+        (fun q ↦ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))))
+  have hp1 : p 1 = 1 := by
+    change inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) =
+      supportValue K.val.val ((Real.pi / 2 : ℝ) : Real.Angle) at hpeq
+    rw [K.val.property.2.2.2.1] at hpeq
+    simpa [normalVector, frame, PiLp.inner_apply] using hpeq
+  let v : Point := !₂[-p 0, 0]
+  let L := ConvexBody.translate K.val.val v
+  have hv (a : Real.Angle) (ha : a = ((Real.pi / 2 : ℝ) : Real.Angle) ∨
+      a = ((3 * Real.pi / 2 : ℝ) : Real.Angle)) :
+      inner ℝ v (normalVector a) = 0 := by
+    rcases ha with rfl | rfl <;>
+      simp [v, normalVector, frame, PiLp.inner_apply,
+        show 3 * Real.pi / 2 = Real.pi + Real.pi / 2 by ring,
+        Real.cos_add, Real.sin_add, -Real.Angle.coe_add]
+  have hsupp (a : Real.Angle) : supportValue L a = supportValue K.val.val a +
+      inner ℝ v (normalVector a) := supportValue_image_add K.val.val v a
+  have hrepr := K.property.translate v
+  have hcaprepr := K.val.property.2.2.2.2.2.2.translate v
+  have htopω : supportValue K.val.val ((Real.pi / 2 : ℝ) : Real.Angle) = 1 := by
+    simpa only [hΘ] using K.val.property.2.2.1
+  have hbotω : supportValue K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := by
+    have ha : (((Θ.angle + Real.pi : ℝ) : Real.Angle)) =
+        ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by rw [hΘ]; congr 1; ring
+    simpa only [ha] using K.val.property.2.2.2.2.1
+  have hcap : IsCap Θ.angle L := by
+    refine ⟨Θ.angle_pos, Θ.angle_le, ?_, ?_, ?_, ?_, hcaprepr⟩
+    · rw [hΘ, hsupp, htopω, hv _ (Or.inl rfl), add_zero]
+    · rw [hsupp, K.val.property.2.2.2.1, hv _ (Or.inl rfl), add_zero]
+    · have ha : (((Θ.angle + Real.pi : ℝ) : Real.Angle)) =
+          ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by rw [hΘ]; congr 1; ring
+      rw [ha, hsupp, hbotω, hv _ (Or.inr rfl), add_zero]
+    · rw [hsupp, K.val.property.2.2.2.2.2.1, hv _ (Or.inr rfl), add_zero]
+  let P : PolygonCapSpace Θ := ⟨⟨L, hcap⟩, hrepr⟩
+  have htop : (stripParallelogram Θ.angle).2.2 ∈ (P.val.val : Set Point) := by
+    refine ⟨p, hp, ?_⟩
+    ext i
+    fin_cases i
+    · simp only [Fin.isValue, Fin.zero_eta, PiLp.add_apply, Matrix.cons_val_zero, add_neg_cancel,
+      stripParallelogram, hΘ, v]
+      rw [show Real.pi / 4 - Real.pi / 2 / 2 = 0 by ring, Real.tan_zero]
+    · simp [v, hΘ, stripParallelogram, hp1]
+  have hfan : capFan Θ.angle = (fun q ↦ q + v) '' capFan Θ.angle := by
+    ext q
+    rw [hΘ]
+    simp only [capFan, normalHalfPlane, Set.mem_inter_iff, Bool.false_eq_true,
+      ↓reduceIte, Set.mem_image]
+    constructor
+    · intro hq
+      refine ⟨q - v, ?_, by simp⟩
+      simpa [v, normalVector, frame, PiLp.inner_apply, inner_sub_left] using hq
+    · rintro ⟨r, hr, rfl⟩
+      simpa [v, normalVector, frame, PiLp.inner_apply, inner_add_left] using hr
+  have hniche : polygonNiche Θ P.val =
+      (fun q ↦ q + v) '' polygonNiche Θ K.val := by
+    have hquad (t : ℝ) : innerQuadrant (P.val.val : Set Point) t =
+        (fun q ↦ q + v) '' innerQuadrant (K.val.val : Set Point) t :=
+      innerQuadrant_translate K.val.val v t
+    have hunion : (⋃ t ∈ Θ.directions, innerQuadrant (P.val.val : Set Point) t) =
+        (fun q ↦ q + v) '' ⋃ t ∈ Θ.directions,
+          innerQuadrant (K.val.val : Set Point) t := by
+      rw [Set.image_iUnion]
+      congr 1
+      funext t
+      rw [Set.image_iUnion]
+      congr 1
+      funext ht
+      exact hquad t
+    unfold polygonNiche
+    calc
+      capFan Θ.angle ∩ ⋃ t ∈ Θ.directions, innerQuadrant (P.val.val : Set Point) t =
+          (fun q ↦ q + v) '' capFan Θ.angle ∩
+            (fun q ↦ q + v) '' ⋃ t ∈ Θ.directions,
+              innerQuadrant (K.val.val : Set Point) t :=
+        congrArg₂ (· ∩ ·) hfan hunion
+      _ = (fun q ↦ q + v) '' (capFan Θ.angle ∩ ⋃ t ∈ Θ.directions,
+          innerQuadrant (K.val.val : Set Point) t) :=
+        (Set.image_inter (Equiv.addRight v).injective).symm
+  have harea : polygonAreaFunctional Θ P.val = polygonAreaFunctional Θ K.val := by
+    rw [(polygonArea_upperBound Θ).1 P, (polygonArea_upperBound Θ).1 K]
+    change ClassicalResults.area ((fun q ↦ q + v) '' (K.val.val : Set Point)) -
+        ClassicalResults.area (polygonNiche Θ P.val) = _
+    rw [ClassicalResults.area_image_add, hniche, ClassicalResults.area_image_add]
+  exact ⟨P, ⟨htop, harea.symm ▸ hK⟩, harea⟩
+
+theorem exists_maximumPolygonCap (Θ : AngleSet) :
+    ∃ K : PolygonCapSpace Θ, IsMaximumPolygonCap Θ K ∧ 0 ≤ polygonAreaFunctional Θ K.val := by
+  obtain ⟨K, hK, hmax⟩ := exists_maximum_polygonCandidate Θ
+  refine ⟨K, ⟨hK.1, ?_⟩, hK.2⟩
+  intro L
+  by_cases hL : 0 ≤ polygonAreaFunctional Θ L.val
+  · rcases lt_or_eq_of_le Θ.angle_le with hΘ | hΘ
+    · exact hmax L ⟨stripParallelogram_top_mem_of_angle_lt Θ hΘ L, hL⟩
+    · obtain ⟨P, hP, heq⟩ := exists_candidate_translate_rightAngle Θ hΘ L hL
+      rw [← heq]
+      exact hmax P hP
+  · exact (le_of_not_ge hL).trans hK.2
+
+theorem maximumPolygonCap_niche_subset {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (hK : IsMaximumPolygonCap Θ K) : polygonNiche Θ K.val ⊆ (K.val.val : Set Point) := by
+  exact polygonNiche_subset_of_balanced K (maximumPolygonCap_balanced K hK)
+
+private theorem isCap_of_hausdorffLimit {ω : ℝ} (K : ℕ → CapSpace ω)
+    (L : ConvexBody Point)
+    (hlim : Tendsto (fun i ↦ Metric.hausdorffDist ((K i).val : Set Point)
+      (L : Set Point)) atTop (𝓝 0)) : IsCap ω L := by
+  let N : Set Real.Angle :=
+    ((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles ω) ∪ capLowerNormals ω
+  let U : Set Point := normalVector '' N
+  have hU : ∀ u ∈ U, ‖u‖ = 1 := by
+    rintro u ⟨a, _, rfl⟩
+    induction a using Real.Angle.induction_on with
+    | _ a => exact norm_normalVector_real a
+  have hrepr (i : ℕ) : ((K i).val : Set Point) = ⋂ u ∈ U,
+      {x | inner ℝ x u ≤ vectorSupport (K i).val u} := by
+    calc
+      ((K i).val : Set Point) = ⋂ a ∈ N,
+          normalHalfPlane a (supportValue (K i).val a) false false :=
+        (K i).property.2.2.2.2.2.2.eq_iInter_supportValue
+      _ = ⋂ u ∈ U, {x | inner ℝ x u ≤ vectorSupport (K i).val u} := by
+        ext p
+        simp [U, normalHalfPlane, supportValue, vectorSupport]
+  have hclosed := fixedNormalBody_closed U hU (fun i ↦ (K i).val) L hrepr hlim
+  refine ⟨(K 0).property.1, (K 0).property.2.1,
+    supportValue_limit_eq hlim ω 1 (fun i ↦ (K i).property.2.2.1),
+    supportValue_limit_eq hlim (Real.pi / 2) 1 (fun i ↦ (K i).property.2.2.2.1),
+    supportValue_limit_eq hlim (ω + Real.pi) 0 (fun i ↦ (K i).property.2.2.2.2.1),
+    supportValue_limit_eq hlim (3 * Real.pi / 2) 0
+      (fun i ↦ (K i).property.2.2.2.2.2.1), ?_⟩
+  refine ⟨(fun t ↦ (t, supportValue L t)) '' N, ?_, ?_⟩
+  · rintro _ ⟨t, ht, rfl⟩
+    exact ht
+  · calc
+      (L : Set Point) = ⋂ u ∈ U,
+          {x | inner ℝ x u ≤ vectorSupport L u} := hclosed
+      _ = ⋂ c ∈ (fun t ↦ (t, supportValue L t)) '' N,
+          normalHalfPlane c.1 c.2 false false := by
+        ext p
+        simp [U, normalHalfPlane, supportValue, vectorSupport]
+
+theorem exists_balancedMaximumCap (ω : ℝ) (hω : 0 < ω) (hω' : ω ≤ Real.pi / 2) :
+    ∃ K : CapSpace ω, IsBalancedMaximumCap K := by
+  let n : ℕ → ℕ := fun i ↦ 2 ^ (i + 1)
+  have hn (i : ℕ) : 2 ≤ n i := by
+    dsimp [n]
+    have h : 1 ≤ 2 ^ i := Nat.one_le_pow i 2 (by omega)
+    rw [pow_succ]
+    omega
+  let Θ (i : ℕ) := uniformAngleSet ω hω hω' (n i) (hn i)
+  have hmid (i : ℕ) : ω / 2 ∈ (Θ i).directions := by
+    apply Finset.mem_image.mpr
+    refine ⟨2 ^ i, Finset.mem_Ioo.mpr ⟨by positivity, ?_⟩, ?_⟩
+    · change 2 ^ i < 2 ^ (i + 1)
+      exact pow_lt_pow_right₀ (by norm_num) (Nat.lt_succ_self i)
+    · simp only [n, Nat.cast_pow, Nat.cast_ofNat, pow_succ]
+      field_simp
+      push_cast
+      ring
+  choose Q hQmax hQnonneg using fun i ↦ exists_maximumPolygonCap (Θ i)
+  obtain ⟨A, hA, hbound⟩ := exists_compact_polygonCap_bound ω hω hω' (ω / 2)
+    ⟨by linarith, by linarith⟩
+  obtain ⟨φ, L, hφ, hlim⟩ := convexBody_selection A hA (fun i ↦ (Q i).val.val)
+    (fun i ↦ hbound (Θ i) rfl (hmid i) (Q i) ⟨(hQmax i).1, hQnonneg i⟩)
+  let K : CapSpace ω := ⟨L, isCap_of_hausdorffLimit (fun i ↦ (Q (φ i)).val) L hlim⟩
+  refine ⟨K, (fun i ↦ n (φ i)), (fun i ↦ hn (φ i)), ?_, ?_, ?_⟩
+  · intro i j hij
+    apply pow_lt_pow_right₀ (by norm_num)
+    exact Nat.add_lt_add_right (hφ hij) 1
+  · exact fun i ↦ ⟨φ i + 1, rfl⟩
+  · exact ⟨(fun i ↦ Q (φ i)), (fun i ↦ hQmax (φ i)), hlim⟩
+
+theorem balancedMaximumCap_niche_subset {ω : ℝ} (K : CapSpace ω)
+    (hK : IsBalancedMaximumCap K) : capNiche K ⊆ (K.val : Set Point) := by
+  obtain ⟨n, hn, hmono, hdyadic, P, hmax, hlim⟩ := hK
+  exact capNiche_subset_of_uniform_polygonNiche_subset K n hn hmono hdyadic P
+    (fun i ↦ maximumPolygonCap_niche_subset (P i) (hmax i)) hlim
+
+theorem balancedMaximumCap_maximizes_area {ω : ℝ} (K : CapSpace ω)
+    (hK : IsBalancedMaximumCap K) :
+    ∀ L : CapSpace ω, capAreaFunctional L ≤ capAreaFunctional K := by
+  obtain ⟨n, hn, hmono, hdyadic, P, hmax, hlim⟩ := hK
+  let Θ (i : ℕ) := uniformAngleSet ω K.property.1 K.property.2.1 (n i) (hn i)
+  have harealim := convexArea_hausdorff_continuity (fun i ↦ (P i).val.val) K.val hlim
+  have hniche := maximizingPolygon_nicheArea_limit ω K.property.1 K.property.2.1
+    n hn hmono hdyadic P hmax K hlim
+  have hfunc : Tendsto (fun i ↦ polygonAreaFunctional (Θ i) (P i).val) atTop
+      (𝓝 (capAreaFunctional K)) := by
+    have hsub := harealim.sub hniche
+    convert hsub using 1
+    · funext i
+      exact (polygonArea_upperBound (Θ i)).1 (P i)
+    · rfl
+  intro L
+  apply ge_of_tendsto hfunc
+  filter_upwards [] with i
+  exact ((polygonArea_upperBound (Θ i)).2 L).trans
+    (polygonAreaFunctional_le_maximum (Θ i) (P i) (hmax i) L)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Polygon.BalancedInequalities`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Balanced Inequalities
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- A real angle strictly inside one full turn is not the zero angle. -/
+private theorem coe_ne_zero_of_pos_of_lt_two_pi {x : ℝ} (h0 : 0 < x)
+    (h2 : x < 2 * Real.pi) : ((x : ℝ) : Real.Angle) ≠ (0 : Real.Angle) := by
+  rw [← Real.Angle.coe_zero]
+  refine Real.Angle.coe_ne_coe_of_abs_sub_lt (by linarith) ?_
+  rw [abs_lt]
+  constructor <;> linarith [Real.pi_pos]
+
+/-- Adding half a turn keeps a small positive angle away from zero. -/
+private theorem coe_add_pi_ne_zero_of_pos_of_lt_pi {x : ℝ} (h0 : 0 < x)
+    (h1 : x < Real.pi) :
+    ((x : ℝ) : Real.Angle) + ((Real.pi : ℝ) : Real.Angle) ≠ (0 : Real.Angle) := by
+  rw [← Real.Angle.coe_add]
+  exact coe_ne_zero_of_pos_of_lt_two_pi (by linarith) (by linarith)
+
+/-- A right-angle polygon cap has no face with horizontal outward normal. -/
+private theorem rightAngle_edgeVertices_zero_eq (n : ℕ) (hn : 2 ≤ n)
+    (K : PolygonCapSpace (rightAngleSet n hn)) :
+    (edgeVertices K.val.val (0 : Real.Angle)).1 =
+      (edgeVertices K.val.val (0 : Real.Angle)).2 := by
+  by_contra hne
+  have hpi := Real.pi_pos
+  have hangle : (rightAngleSet n hn).angle = Real.pi / 2 := rfl
+  have hlow : ∀ u ∈ capLowerNormals (rightAngleSet n hn).angle,
+      u = ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+    intro u hu
+    simp only [capLowerNormals, hangle, Set.mem_insert_iff, Set.mem_singleton_iff] at hu
+    rcases hu with rfl | rfl
+    · congr 1
+      ring
+    · rfl
+  have h3zero : ((3 * Real.pi / 2 : ℝ) : Real.Angle) ≠ (0 : Real.Angle) :=
+    coe_ne_zero_of_pos_of_lt_two_pi (by linarith) (by linarith)
+  have h3pi : ((3 * Real.pi / 2 : ℝ) : Real.Angle) + ((Real.pi : ℝ) : Real.Angle) ≠
+      (0 : Real.Angle) := by
+    have hshift : ((3 * Real.pi / 2 : ℝ) : Real.Angle) + ((Real.pi : ℝ) : Real.Angle) =
+        ((Real.pi / 2 : ℝ) : Real.Angle) := by
+      rw [← Real.Angle.coe_add, Real.Angle.angle_eq_iff_two_pi_dvd_sub]
+      exact ⟨1, by push_cast; ring⟩
+    rw [hshift]
+    exact coe_ne_zero_of_pos_of_lt_two_pi (by linarith) (by linarith)
+  have hmain := K.properEdgeNormal_mem_allowed_or_antipodal (0 : Real.Angle) hne
+  rcases hmain with (⟨r, hr, hr0⟩ | hlowmem) | ⟨u, huN, hu0⟩
+  · exact coe_ne_zero_of_pos_of_lt_two_pi
+      (angleDomain_subset_Ioo _ hr).1
+      (by linarith [(angleDomain_subset_Ioo _ hr).2]) hr0
+  · exact h3zero (hlow _ hlowmem).symm
+  · rcases huN with ⟨r, hr, rfl⟩ | hlowmem
+    · exact coe_add_pi_ne_zero_of_pos_of_lt_pi
+        (angleDomain_subset_Ioo _ hr).1 (angleDomain_subset_Ioo _ hr).2 hu0
+    · rw [hlow _ hlowmem] at hu0
+      exact h3pi hu0
+
+/-- The surface measure of a right-angle polygon cap has no horizontal-normal atom. -/
+private theorem rightAngle_surfaceAreaMeasure_zero (n : ℕ) (hn : 2 ≤ n)
+    (K : PolygonCapSpace (rightAngleSet n hn)) :
+    surfaceAreaMeasure K.val.val {((0 : ℝ) : Real.Angle)} = 0 := by
+  rw [Real.Angle.coe_zero, (surfaceAreaMeasure_atom_length K.val.val (0 : Real.Angle)).2.1,
+    rightAngle_edgeVertices_zero_eq n hn K]
+  simp
+
+/-- The real-arithmetic core of the discrete balancing inequality. -/
+private theorem magicBound_of_le {δ T c s gp gm : ℝ} (hδpos : 0 < δ) (hδ45 : δ ≤ 4 / 5)
+    (hTub : T ≤ δ + 4 / 3 * δ ^ 3) (hTpos : 0 < T)
+    (hcub : c ≤ δ / 2 + 4 / 3 * (δ / 2) ^ 3) (hcpos : 0 < c)
+    (hgp5 : gp ≤ 5) (hgm0 : 0 ≤ gm) (hgle : gm ≤ gp)
+    (hsum : s ≤ T * max 0 (gm - 1 + c) + T * max 0 (1 - gp + c) + max 0 (2 * c - s)) :
+    s ≤ max |gp - 1| ((|gp - 1| + 1) / 2) * δ + 8 * δ ^ 2 := by
+  have hW0 : 0 ≤ max 0 (gm - 1) + max 0 (1 - gp) := by positivity
+  have hMsum : max 0 (gm - 1) + max 0 (1 - gp) ≤ |gp - 1| := by
+    rcases le_total gp 1 with hg | hg
+    · rw [abs_of_nonpos (by linarith), max_eq_left (show gm - 1 ≤ 0 by linarith),
+        max_eq_right (show (0 : ℝ) ≤ 1 - gp by linarith)]
+      linarith
+    · rw [abs_of_nonneg (by linarith), max_eq_left (show (1 : ℝ) - gp ≤ 0 by linarith)]
+      rcases le_total gm 1 with h2 | h2
+      · rw [max_eq_left (show gm - 1 ≤ 0 by linarith)]
+        linarith
+      · rw [max_eq_right (show (0 : ℝ) ≤ gm - 1 by linarith)]
+        linarith
+  have hMsum4 : max 0 (gm - 1) + max 0 (1 - gp) ≤ 4 := by
+    rcases le_total gm 1 with h2 | h2
+    · rw [max_eq_left (show gm - 1 ≤ 0 by linarith)]
+      have h3 : max 0 (1 - gp) ≤ 1 := max_le (by norm_num) (by linarith)
+      linarith
+    · rw [max_eq_left (show (1 : ℝ) - gp ≤ 0 by linarith),
+        max_eq_right (show (0 : ℝ) ≤ gm - 1 by linarith)]
+      linarith
+  have hstep1 : max 0 (gm - 1 + c) ≤ max 0 (gm - 1) + c := by
+    rcases le_total (gm - 1 + c) 0 with h | h
+    · rw [max_eq_left h]
+      linarith [le_max_left (0 : ℝ) (gm - 1)]
+    · rw [max_eq_right h]
+      linarith [le_max_right (0 : ℝ) (gm - 1)]
+  have hstep2 : max 0 (1 - gp + c) ≤ max 0 (1 - gp) + c := by
+    rcases le_total (1 - gp + c) 0 with h | h
+    · rw [max_eq_left h]
+      linarith [le_max_left (0 : ℝ) (1 - gp)]
+    · rw [max_eq_right h]
+      linarith [le_max_right (0 : ℝ) (1 - gp)]
+  have e1 : T * max 0 (gm - 1 + c) ≤ T * (max 0 (gm - 1) + c) :=
+    mul_le_mul_of_nonneg_left hstep1 hTpos.le
+  have e2 : T * max 0 (1 - gp + c) ≤ T * (max 0 (1 - gp) + c) :=
+    mul_le_mul_of_nonneg_left hstep2 hTpos.le
+  have hcombine : s ≤ T * (max 0 (gm - 1) + max 0 (1 - gp)) + 2 * (T * c) +
+      max 0 (2 * c - s) := by nlinarith only [e1, e2, hsum]
+  have p3 : δ ^ 3 ≤ 4 / 5 * δ ^ 2 := by
+    nlinarith only [mul_nonneg (sq_nonneg δ) (sub_nonneg.mpr hδ45)]
+  have hδsq : δ ^ 2 ≤ 16 / 25 := by nlinarith only [hδpos, hδ45]
+  have p4 : δ ^ 4 ≤ 16 / 25 * δ ^ 2 := by
+    nlinarith only [mul_nonneg (sq_nonneg δ) (sub_nonneg.mpr hδsq)]
+  have p6 : δ ^ 6 ≤ 1 / 2 * δ ^ 2 := by
+    nlinarith only [mul_nonneg (sq_nonneg δ)
+      (show (0 : ℝ) ≤ 1 / 2 - δ ^ 4 by nlinarith only [p4, hδsq])]
+  have f1 : T * (max 0 (gm - 1) + max 0 (1 - gp)) ≤
+      (δ + 4 / 3 * δ ^ 3) * (max 0 (gm - 1) + max 0 (1 - gp)) :=
+    mul_le_mul_of_nonneg_right hTub hW0
+  have f2 : δ * (max 0 (gm - 1) + max 0 (1 - gp)) ≤ δ * |gp - 1| :=
+    mul_le_mul_of_nonneg_left hMsum hδpos.le
+  have f3 : 4 / 3 * δ ^ 3 * (max 0 (gm - 1) + max 0 (1 - gp)) ≤ 4 / 3 * δ ^ 3 * 4 :=
+    mul_le_mul_of_nonneg_left hMsum4 (by positivity)
+  have f4 : T * c ≤ (δ + 4 / 3 * δ ^ 3) * (δ / 2 + 4 / 3 * (δ / 2) ^ 3) :=
+    mul_le_mul hTub hcub hcpos.le (by positivity)
+  have hE : T * (max 0 (gm - 1) + max 0 (1 - gp)) + 2 * (T * c) ≤
+      |gp - 1| * δ + 7 * δ ^ 2 := by
+    nlinarith only [f1, f2, f3, f4, p3, p4, p6]
+  have hak1 : |gp - 1| * δ ≤ max |gp - 1| ((|gp - 1| + 1) / 2) * δ :=
+    mul_le_mul_of_nonneg_right (le_max_left _ _) hδpos.le
+  have hak2 : (|gp - 1| + 1) / 2 * δ ≤ max |gp - 1| ((|gp - 1| + 1) / 2) * δ :=
+    mul_le_mul_of_nonneg_right (le_max_right _ _) hδpos.le
+  have h2c : 2 * c ≤ δ + 1 / 3 * δ ^ 3 := by nlinarith only [hcub]
+  rcases le_total (2 * c - s) 0 with hcase | hcase
+  · rw [max_eq_left hcase] at hcombine
+    nlinarith only [hcombine, hE, hak1, sq_nonneg δ]
+  · rw [max_eq_right hcase] at hcombine
+    nlinarith only [hcombine, hE, hak2, h2c, p3, sq_nonneg δ]
+
+private theorem bRay_horizontal_boundary_measure_zero (K : ConvexBody Point)
+    (t : ℝ) (htIoo : t ∈ Set.Ioo 0 (Real.pi / 2)) :
+    Measure.hausdorffMeasure 1
+       ((rotatingHallwayParts (K : Set Point) (t : Real.Angle)).bRay ∩
+         {p : Point | inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0}) = 0 := by
+  have hpi := Real.pi_pos
+  have hns : NullSingletonClass (Measure.hausdorffMeasure 1 : Measure Point) :=
+    Measure.nullSingletonClass_hausdorff Point one_pos
+  refine Set.Subsingleton.measure_zero ?_ _
+  intro x hx y hy
+  by_contra hxy
+  have h1 : inner ℝ (x - y) (normalVector (t : Real.Angle)) = 0 := by
+    rw [inner_sub_left, ((mem_rotatingHallwayParts_bRay_iff _ _ x).mp hx.1).1,
+      ((mem_rotatingHallwayParts_bRay_iff _ _ y).mp hy.1).1, sub_self]
+  have hx2 : inner ℝ x (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := hx.2
+  have hy2 : inner ℝ y (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := hy.2
+  have h2 : inner ℝ (x - y) (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := by
+    rw [inner_sub_left, hx2, hy2, sub_self]
+  rcases normalVector_eq_or_eq_add_pi_of_orthogonal (sub_ne_zero.mpr hxy) h1 h2 with hA | hA
+  · exact Real.Angle.coe_ne_coe_of_abs_sub_lt (by linarith [htIoo.2])
+      (by rw [abs_lt]; constructor <;> linarith [htIoo.1, htIoo.2]) hA
+  · rw [← Real.Angle.coe_add] at hA
+    exact Real.Angle.coe_ne_coe_of_abs_sub_lt (by linarith [htIoo.2])
+      (by rw [abs_lt]; constructor <;> linarith [htIoo.1, htIoo.2]) hA
+
+/-- The magic function `k₀` read as a function on all reals through truncation. -/
+def magicDensity (x : ℝ) : ℝ := magicFunctions.1 x.toNNReal
+
+theorem maximumPolygonCap_surfaceAtom_bound (n : ℕ) (hn : 2 ≤ n)
+    (K : RightAngleCapSpace) (hK : IsMaximumPolygonCapSteps n K) (t : ℝ)
+    (ht : t = 0 ∨ t ∈ (rightAngleSet n hn).directions) :
+    surfaceAreaMeasure K.val {(t : Real.Angle)} ≤
+      ENNReal.ofReal (magicFunctions.1 (Real.toNNReal (tangentArmLengths K t).2.1) *
+        polygonStepSize n + 8 * polygonStepSize n ^ 2) := by
+  classical
+  obtain ⟨hn2, hdy, P, hPK, hmax⟩ := hK
+  subst hPK
+  have hpi := Real.pi_pos
+  have hnR : (2 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn
+  have hδpos : 0 < polygonStepSize n := by
+    have hn0 : (0 : ℝ) < n := by linarith
+    simp only [polygonStepSize]
+    positivity
+  have hδle : polygonStepSize n ≤ Real.pi / 4 := by
+    have hn0 : (0 : ℝ) < n := by linarith
+    rw [polygonStepSize, div_le_iff₀ hn0]
+    nlinarith
+  rcases ht with rfl | ht
+  · rw [rightAngle_surfaceAreaMeasure_zero n hn2 P]
+    exact zero_le
+  · have htIoo : t ∈ Set.Ioo 0 (Real.pi / 2) := (rightAngleSet n hn2).interior t ht
+    have hcap : supportValue (P.val.val : Set Point) ((Real.pi / 2 : ℝ) : Real.Angle) = 1 :=
+      P.val.property.2.2.2.1
+    have htdom : t ∈ angleDomain (rightAngleSet n hn2) :=
+      Set.mem_union_left _ (Set.mem_union_left _ (Finset.mem_coe.mpr ht))
+    have hatom : surfaceAreaMeasure P.val.val {(t : Real.Angle)} =
+        ENNReal.ofReal (polygonCapPolylineLength P ⟨t, htdom⟩) :=
+      maximumPolygonCap_balanced P hmax ⟨t, htdom⟩
+    have hlenat : polygonPolylineLengthAt P t = polygonCapPolylineLength P ⟨t, htdom⟩ := by
+      simp only [polygonPolylineLengthAt, htdom, ↓reduceDIte]
+    have hwall : (Measure.hausdorffMeasure 1
+        (frontier (polygonNiche (rightAngleSet n hn2) P.val) ∩
+          (rotatingHallwayParts (P.val.val : Set Point) (t : Real.Angle)).bRay)).toReal =
+        polygonPolylineLengthAt P t := (polygonNiche_wall_lengths P ht).2.1
+    have hτnonneg : 0 ≤ polygonCapPolylineLength P ⟨t, htdom⟩ := by
+      rw [← hlenat, ← hwall]
+      exact ENNReal.toReal_nonneg
+    -- the three pieces
+    have hlegs := maximumPolygonCap_leg_lengths n hn2 P t ht
+    have hsubset : frontier (polygonNiche (rightAngleSet n hn2) P.val) ∩
+          (rotatingHallwayParts (P.val.val : Set Point) (t : Real.Angle)).bRay ⊆
+        (((rotatingHallwayParts (P.val.val : Set Point) (t : Real.Angle)).bRay ∩
+            {p : Point | inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0}) ∪
+          ((rotatingHallwayParts (P.val.val : Set Point) (t : Real.Angle)).bRay ∩
+            (innerWallUpperHalfPlanes P.val (t - polygonStepSize n)).2)) ∪
+          ((rotatingHallwayParts (P.val.val : Set Point) (t : Real.Angle)).bRay ∩
+            (innerWallUpperHalfPlanes P.val (t + polygonStepSize n)).2) ∪
+          ({p : Point | inner ℝ p (normalVector (t : Real.Angle)) =
+              supportValue (P.val.val : Set Point) (t : Real.Angle) - 1} ∩
+            ((innerWallUpperHalfPlanes P.val (t - polygonStepSize n)).1 ∩
+              (innerWallUpperHalfPlanes P.val (t + polygonStepSize n)).1)) := by
+      rintro p ⟨hpfront, hpray⟩
+      have hpline : inner ℝ p (normalVector (t : Real.Angle)) =
+          supportValue (P.val.val : Set Point) (t : Real.Angle) - 1 :=
+        ((mem_rotatingHallwayParts_bRay_iff _ _ p).mp hpray).1
+      by_cases hzero : inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0
+      · exact Or.inl (Or.inl (Or.inl ⟨hpray, hzero⟩))
+      · have hclos : p ∈ capFan (rightAngleSet n hn2).angle := by
+          have hsub : closure (polygonNiche (rightAngleSet n hn2) P.val) ⊆
+              capFan (rightAngleSet n hn2).angle :=
+            (isClosed_capFan _).closure_subset_iff.mpr (fun q hq ↦ hq.1)
+          exact hsub hpfront.1
+        have hppos : 0 < inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) :=
+          lt_of_le_of_ne hclos.2 (Ne.symm hzero)
+        have hnotN : p ∉ polygonNiche (rightAngleSet n hn2) P.val := by
+          intro hpN
+          refine hpfront.2 ?_
+          have hUopen : IsOpen {q : Point |
+              0 < inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))} :=
+            isOpen_lt continuous_const (by fun_prop)
+          have hVopen : IsOpen (⋃ u ∈ (rightAngleSet n hn2).directions,
+              innerQuadrant (P.val.val : Set Point) u) :=
+            isOpen_biUnion (fun u _ ↦ isOpen_innerQuadrant _ u)
+          refine interior_maximal ?_ (hUopen.inter hVopen) ⟨hppos, hpN.2⟩
+          rintro q ⟨hq1, hq2⟩
+          have hq1' : (0 : ℝ) < inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) := hq1
+          exact ⟨⟨hq1'.le, hq1'.le⟩, hq2⟩
+        have hquad : ∀ u : ℝ,
+            (u = 0 ∨ u = Real.pi / 2 ∨ u ∈ (rightAngleSet n hn2).directions) →
+            p ∉ innerQuadrant (P.val.val : Set Point) u := by
+          rintro u (rfl | rfl | hu)
+          · intro hq
+            have h2 : inner ℝ p (normalVector ((0 + Real.pi / 2 : ℝ) : Real.Angle)) <
+                supportValue (P.val.val : Set Point)
+                  ((0 + Real.pi / 2 : ℝ) : Real.Angle) - 1 := hq.2
+            rw [show (0 + Real.pi / 2 : ℝ) = Real.pi / 2 by ring, hcap] at h2
+            linarith
+          · intro hq
+            have h1 : inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) <
+                supportValue (P.val.val : Set Point)
+                  ((Real.pi / 2 : ℝ) : Real.Angle) - 1 := hq.1
+            rw [hcap] at h1
+            linarith
+          · intro hq
+            exact hnotN ⟨hclos, Set.mem_biUnion hu hq⟩
+        have hq1 : p ∉ innerQuadrant (P.val.val : Set Point) (t - polygonStepSize n) :=
+          hquad _ ((rightAngleSet_sub_step n hn2 ht).imp id Or.inr)
+        have hq2 : p ∉ innerQuadrant (P.val.val : Set Point) (t + polygonStepSize n) :=
+          hquad _ (Or.inr ((rightAngleSet_add_step n hn2 ht).imp id id))
+        rcases (notMem_innerQuadrant_iff _ _ _).mp hq1 with hb1 | hd1
+        · rcases (notMem_innerQuadrant_iff _ _ _).mp hq2 with hb2 | hd2
+          · exact Or.inr ⟨hpline, hb1, hb2⟩
+          · exact Or.inl (Or.inr ⟨hpray, hd2⟩)
+        · exact Or.inl (Or.inl (Or.inr ⟨hpray, hd1⟩))
+    have hz := bRay_horizontal_boundary_measure_zero P.val.val t htIoo
+    have hslice := hausdorffMeasure_faceLine_inter_innerWalls_le P.val.val t
+      (polygonStepSize n) hδpos (by linarith)
+    have hbound : Measure.hausdorffMeasure 1
+        (frontier (polygonNiche (rightAngleSet n hn2) P.val) ∩
+          (rotatingHallwayParts (P.val.val : Set Point) (t : Real.Angle)).bRay) ≤
+        ENNReal.ofReal (Real.tan (polygonStepSize n) *
+            max 0 ((tangentArmLengths P.val t).2.2 - 1 +
+              Real.tan (polygonStepSize n / 2))) +
+          ENNReal.ofReal (Real.tan (polygonStepSize n) *
+            max 0 (1 - (tangentArmLengths P.val t).2.1 +
+              Real.tan (polygonStepSize n / 2))) +
+          ENNReal.ofReal (max 0 (2 * Real.tan (polygonStepSize n / 2) -
+            (surfaceAreaMeasure P.val.val {(t : Real.Angle)}).toReal)) := by
+      have hAB : Measure.hausdorffMeasure 1
+          (((rotatingHallwayParts (P.val.val : Set Point) (t : Real.Angle)).bRay ∩
+              {p : Point | inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0}) ∪
+            ((rotatingHallwayParts (P.val.val : Set Point) (t : Real.Angle)).bRay ∩
+              (innerWallUpperHalfPlanes P.val (t - polygonStepSize n)).2)) ≤
+          ENNReal.ofReal (Real.tan (polygonStepSize n) *
+            max 0 ((tangentArmLengths P.val t).2.2 - 1 +
+              Real.tan (polygonStepSize n / 2))) := by
+        refine (measure_union_le _ _).trans ?_
+        rw [hz, hlegs.1, zero_add]
+      exact (measure_mono hsubset).trans ((measure_union_le _ _).trans
+        (add_le_add ((measure_union_le _ _).trans
+          (add_le_add hAB (le_of_eq hlegs.2))) hslice))
+    have harm := (maximumPolygonCap_arm_bound n P.val ⟨hn2, hdy, P, rfl, hmax⟩).2 t
+      ⟨htIoo.1.le, htIoo.2.le⟩
+    have hgp0 : 0 ≤ (tangentArmLengths P.val t).2.1 := harm.2.2.1.1
+    have hgp5 : (tangentArmLengths P.val t).2.1 ≤ 5 := harm.2.2.1.2
+    have hgm0 : 0 ≤ (tangentArmLengths P.val t).2.2 := harm.2.2.2.1
+    have harm1 : (tangentArmLengths P.val t).2.1 =
+        inner ℝ ((rotatingHallwayParts (P.val.val : Set Point) (t : Real.Angle)).outerCorner -
+          (edgeVertices P.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).1)
+          (normalVector (t : Real.Angle)) := rfl
+    have harm2 : (tangentArmLengths P.val t).2.2 =
+        inner ℝ ((rotatingHallwayParts (P.val.val : Set Point) (t : Real.Angle)).outerCorner -
+          (edgeVertices P.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).2)
+          (normalVector (t : Real.Angle)) := rfl
+    have hgle : (tangentArmLengths P.val t).2.2 ≤ (tangentArmLengths P.val t).2.1 := by
+      have hC : (edgeVertices P.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).1 =
+          (edgeVertices P.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).2 -
+            (surfaceAreaMeasure P.val.val {((t + Real.pi / 2 : ℝ) : Real.Angle)}).toReal •
+              normalVector (t : Real.Angle) := by
+        rw [(surfaceAreaMeasure_atom_length P.val.val
+          ((t + Real.pi / 2 : ℝ) : Real.Angle)).2.2, tangentVector_add_pi_div_two]
+        module
+      rw [harm1, harm2, hC, show (rotatingHallwayParts (P.val.val : Set Point)
+            (t : Real.Angle)).outerCorner -
+          ((edgeVertices P.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).2 -
+            (surfaceAreaMeasure P.val.val {((t + Real.pi / 2 : ℝ) : Real.Angle)}).toReal •
+              normalVector (t : Real.Angle)) =
+          ((rotatingHallwayParts (P.val.val : Set Point) (t : Real.Angle)).outerCorner -
+            (edgeVertices P.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).2) +
+            (surfaceAreaMeasure P.val.val {((t + Real.pi / 2 : ℝ) : Real.Angle)}).toReal •
+              normalVector (t : Real.Angle) from by module,
+        inner_add_left, real_inner_smul_left, inner_normalVector_self, mul_one]
+      have := ENNReal.toReal_nonneg (a := surfaceAreaMeasure P.val.val
+        {((t + Real.pi / 2 : ℝ) : Real.Angle)})
+      linarith
+    have hstoReal : (surfaceAreaMeasure P.val.val {(t : Real.Angle)}).toReal =
+        polygonCapPolylineLength P ⟨t, htdom⟩ := by
+      rw [hatom, ENNReal.toReal_ofReal hτnonneg]
+    have hXeq : (surfaceAreaMeasure P.val.val {(t : Real.Angle)}).toReal =
+        (Measure.hausdorffMeasure 1
+          (frontier (polygonNiche (rightAngleSet n hn2) P.val) ∩
+            (rotatingHallwayParts (P.val.val : Set Point) (t : Real.Angle)).bRay)).toReal := by
+      rw [hstoReal, ← hlenat]
+      exact hwall.symm
+    have hTpos : 0 < Real.tan (polygonStepSize n) :=
+      Real.tan_pos_of_pos_of_lt_pi_div_two hδpos (by linarith)
+    have hcpos : 0 < Real.tan (polygonStepSize n / 2) :=
+      Real.tan_pos_of_pos_of_lt_pi_div_two (by linarith) (by linarith)
+    rw [← ENNReal.ofReal_add (by positivity) (by positivity),
+      ← ENNReal.ofReal_add (by positivity) (le_max_left _ _)] at hbound
+    have hsum : (surfaceAreaMeasure P.val.val {(t : Real.Angle)}).toReal ≤
+        Real.tan (polygonStepSize n) *
+            max 0 ((tangentArmLengths P.val t).2.2 - 1 + Real.tan (polygonStepSize n / 2)) +
+          Real.tan (polygonStepSize n) *
+            max 0 (1 - (tangentArmLengths P.val t).2.1 + Real.tan (polygonStepSize n / 2)) +
+          max 0 (2 * Real.tan (polygonStepSize n / 2) -
+            (surfaceAreaMeasure P.val.val {(t : Real.Angle)}).toReal) := by
+      conv_lhs => rw [hXeq]
+      refine (ENNReal.toReal_mono ENNReal.ofReal_ne_top hbound).trans_eq ?_
+      exact ENNReal.toReal_ofReal (by positivity)
+    have hmagic : magicFunctions.1 (Real.toNNReal (tangentArmLengths P.val t).2.1) =
+        max |(tangentArmLengths P.val t).2.1 - 1|
+          ((|(tangentArmLengths P.val t).2.1 - 1| + 1) / 2) := by
+      simp only [magicFunctions, Real.coe_toNNReal _ hgp0]
+    rw [hatom]
+    refine ENNReal.ofReal_le_ofReal ?_
+    rw [← hstoReal, hmagic]
+    refine magicBound_of_le hδpos ?_ ?_ hTpos ?_ hcpos hgp5 hgm0 hgle hsum
+    · linarith [Real.pi_lt_d2]
+    · exact Real.tan_le_self_add_cube hδpos.le (by linarith [Real.pi_le_four])
+    · exact Real.tan_le_self_add_cube (by linarith) (by linarith [Real.pi_le_four])
+
+section Domination
+
+open Filter Set
+open scoped Topology
+
+/-! ### The magic density -/
+
+private theorem magicDensity_eq (x : ℝ) :
+    magicDensity x = max |max x 0 - 1| ((|max x 0 - 1| + 1) / 2) := by
+  simp [magicDensity, magicFunctions]
+
+/-- The magic function `k₀` is nonnegative. -/
+theorem magicDensity_nonneg (x : ℝ) : 0 ≤ magicDensity x := by
+  rw [magicDensity_eq]
+  exact (abs_nonneg _).trans (le_max_left _ _)
+
+private theorem continuous_magicDensity : Continuous magicDensity := by
+  unfold magicDensity magicFunctions
+  fun_prop
+
+/-- The magic function `k₀` is `1`-Lipschitz. -/
+private theorem abs_magicDensity_sub_le (x y : ℝ) :
+    |magicDensity x - magicDensity y| ≤ |x - y| := by
+  rw [magicDensity_eq, magicDensity_eq]
+  have hxy : |max x 0 - max y 0| ≤ |x - y| := by
+    calc
+      |max x 0 - max y 0| ≤ max |x - y| |(0 : ℝ) - 0| :=
+        abs_max_sub_max_le_max x 0 y 0
+      _ = |x - y| := by simp
+  have habs : abs (|max x 0 - 1| - |max y 0 - 1|) ≤ |x - y| := by
+    calc
+      abs (|max x 0 - 1| - |max y 0 - 1|) ≤
+          |(max x 0 - 1) - (max y 0 - 1)| := abs_abs_sub_abs_le_abs_sub _ _
+      _ = |max x 0 - max y 0| := by ring_nf
+      _ ≤ |x - y| := hxy
+  have hhalf :
+      |((|max x 0 - 1| + 1) / 2) - ((|max y 0 - 1| + 1) / 2)| ≤ |x - y| := by
+    rw [show ((|max x 0 - 1| + 1) / 2) - ((|max y 0 - 1| + 1) / 2) =
+      (|max x 0 - 1| - |max y 0 - 1|) / 2 by ring, abs_div]
+    norm_num
+    linarith [abs_nonneg (|max x 0 - 1| - |max y 0 - 1|)]
+  calc
+    |max |max x 0 - 1| ((|max x 0 - 1| + 1) / 2) -
+        max |max y 0 - 1| ((|max y 0 - 1| + 1) / 2)| ≤
+        max (abs (|max x 0 - 1| - |max y 0 - 1|))
+          |((|max x 0 - 1| + 1) / 2) - ((|max y 0 - 1| + 1) / 2)| :=
+      abs_max_sub_max_le_max _ _ _ _
+    _ ≤ |x - y| := max_le habs hhalf
+
+/-- The magic function `k₀` grows at most linearly: `k₀ x ≤ |x| + 1`. -/
+theorem magicDensity_le_abs_add_one (x : ℝ) : magicDensity x ≤ |x| + 1 := by
+  have hzero : magicDensity 0 = 1 := by norm_num [magicDensity, magicFunctions]
+  have h := abs_magicDensity_sub_le x 0
+  rw [hzero, sub_zero] at h
+  linarith [le_abs_self (magicDensity x - 1)]
+
+/-! ### Integrability of the magic density -/
+
+/-- The magic density of the positive tangent arm length is almost everywhere strongly
+measurable on the rotation interval. -/
+theorem aestronglyMeasurable_magicDensity_tangentArm_fst (C : RightAngleCapSpace) :
+    AEStronglyMeasurable (fun t ↦ magicDensity (tangentArmLengths C t).2.1)
+      (volume.restrict (Ioc 0 (Real.pi / 2))) :=
+  continuous_magicDensity.comp_aestronglyMeasurable (aestronglyMeasurable_tangentArm_fst C)
+
+/-- The magic density of the positive tangent arm length is integrable on the rotation
+interval. -/
+private theorem intervalIntegrable_magicDensity_tangentArm_fst (C : RightAngleCapSpace) :
+    IntervalIntegrable (fun t ↦ magicDensity (tangentArmLengths C t).2.1)
+      volume 0 (Real.pi / 2) := by
+  have hdensity' : AEStronglyMeasurable
+      (fun t ↦ magicDensity (tangentArmLengths C t).2.1)
+      (volume.restrict (uIoc 0 (Real.pi / 2))) := by
+    simpa only [uIoc_of_le (by positivity : (0 : ℝ) ≤ Real.pi / 2)] using
+      aestronglyMeasurable_magicDensity_tangentArm_fst C
+  have harmInt := intervalIntegrable_tangentArm_fst C
+  apply (harmInt.norm.add (intervalIntegrable_const (c := (1 : ℝ)))).mono_fun hdensity'
+  filter_upwards [] with t
+  change |magicDensity (tangentArmLengths C t).2.1| ≤
+    abs (|(tangentArmLengths C t).2.1| + 1)
+  rw [abs_of_nonneg (magicDensity_nonneg _),
+    abs_of_nonneg (add_nonneg (abs_nonneg (tangentArmLengths C t).2.1) zero_le_one)]
+  exact magicDensity_le_abs_add_one _
+
+private theorem integrableOn_magicDensity_tangentArm_fst (C : RightAngleCapSpace)
+    {S : Set ℝ} (hS : S ⊆ Icc (0 : ℝ) (Real.pi / 2)) :
+    IntegrableOn (fun t ↦ magicDensity (tangentArmLengths C t).2.1) S volume := by
+  have hIoc := (intervalIntegrable_iff_integrableOn_Ioc_of_le
+    (by positivity : (0 : ℝ) ≤ Real.pi / 2)).1
+      (intervalIntegrable_magicDensity_tangentArm_fst C)
+  have hIcc : IntegrableOn (fun t ↦ magicDensity (tangentArmLengths C t).2.1)
+      (Icc (0 : ℝ) (Real.pi / 2)) volume := by
+    rwa [IntegrableOn, Measure.restrict_congr_set Ioc_ae_eq_Icc] at hIoc
+  exact hIcc.mono_set hS
+
+/-! ### The density measure of a cap -/
+
+/-- The measure `k₀(g⁺_K(t)) dt` on the rotation interval. -/
+private def armDensityMeasure (C : RightAngleCapSpace) : Measure ℝ :=
+  (volume.restrict (Ioc 0 (Real.pi / 2))).withDensity
+    (fun t ↦ ENNReal.ofReal (magicDensity (tangentArmLengths C t).2.1))
+
+private theorem armDensityMeasure_apply (C : RightAngleCapSpace) {S : Set ℝ}
+    (hS : MeasurableSet S) :
+    armDensityMeasure C S = ∫⁻ t in S ∩ Ioc 0 (Real.pi / 2),
+      ENNReal.ofReal (magicDensity (tangentArmLengths C t).2.1) := by
+  rw [armDensityMeasure, withDensity_apply _ hS, Measure.restrict_restrict hS]
+
+private theorem lintegral_magicDensity_eq_ofReal (C : RightAngleCapSpace) {S : Set ℝ}
+    (hS : S ⊆ Icc (0 : ℝ) (Real.pi / 2)) :
+    ∫⁻ t in S, ENNReal.ofReal (magicDensity (tangentArmLengths C t).2.1) =
+      ENNReal.ofReal (∫ t in S, magicDensity (tangentArmLengths C t).2.1) := by
+  rw [← ofReal_integral_eq_lintegral_ofReal
+    (integrableOn_magicDensity_tangentArm_fst C hS)
+    (Filter.Eventually.of_forall fun t ↦ magicDensity_nonneg _)]
+
+private theorem armDensityMeasure_ne_top (C : RightAngleCapSpace) :
+    armDensityMeasure C Set.univ ≠ ⊤ := by
+  rw [armDensityMeasure_apply C MeasurableSet.univ, Set.univ_inter,
+    lintegral_magicDensity_eq_ofReal C Ioc_subset_Icc_self]
+  exact ENNReal.ofReal_ne_top
+
+private theorem armDensityMeasure_eq_ofReal_integral (C : RightAngleCapSpace) {S : Set ℝ}
+    (hSm : MeasurableSet S) (hS : S ⊆ Ico (0 : ℝ) (Real.pi / 2)) :
+    armDensityMeasure C S =
+      ENNReal.ofReal (∫ t in S, magicDensity (tangentArmLengths C t).2.1) := by
+  have hnull : S ∩ Ioc 0 (Real.pi / 2) =ᵐ[volume] S := by
+    have hsub : S \ (S ∩ Ioc 0 (Real.pi / 2)) ⊆ {0} := by
+      intro x hx
+      have hxS := hx.1
+      have hx2 : x ∉ Ioc 0 (Real.pi / 2) := fun h ↦ hx.2 ⟨hxS, h⟩
+      have hxIco := hS hxS
+      have : x = 0 := by
+        by_contra hne
+        exact hx2 ⟨lt_of_le_of_ne hxIco.1 (Ne.symm hne), hxIco.2.le⟩
+      simp [this]
+    refine (Filter.EventuallyEq.symm ?_)
+    refine (ae_eq_set.mpr ⟨?_, ?_⟩)
+    · exact measure_mono_null hsub (measure_singleton 0)
+    · simp [Set.sdiff_eq_empty.2 Set.inter_subset_left]
+  rw [armDensityMeasure_apply C hSm, setLIntegral_congr hnull,
+    lintegral_magicDensity_eq_ofReal C (hS.trans Ico_subset_Icc_self)]
+
+/-- Comparing two cap densities costs at most the `L¹` distance of their arm lengths. -/
+private theorem armDensityMeasure_le_add_l1 (C D : RightAngleCapSpace) {S : Set ℝ}
+    (hS : MeasurableSet S) :
+    armDensityMeasure C S ≤ armDensityMeasure D S +
+      ENNReal.ofReal (∫ t in (0 : ℝ)..(Real.pi / 2),
+        |magicDensity (tangentArmLengths C t).2.1 -
+          magicDensity (tangentArmLengths D t).2.1|) := by
+  have hT : (0 : ℝ) ≤ Real.pi / 2 := by positivity
+  have hdiff : IntervalIntegrable (fun t ↦ |magicDensity (tangentArmLengths C t).2.1 -
+      magicDensity (tangentArmLengths D t).2.1|) volume 0 (Real.pi / 2) := by
+    simpa only [Real.norm_eq_abs] using
+      ((intervalIntegrable_magicDensity_tangentArm_fst C).sub
+        (intervalIntegrable_magicDensity_tangentArm_fst D)).norm
+  have hdiffIoc := (intervalIntegrable_iff_integrableOn_Ioc_of_le hT).1 hdiff
+  have hmeasD : AEMeasurable
+      (fun t ↦ ENNReal.ofReal (magicDensity (tangentArmLengths D t).2.1))
+      (volume.restrict (S ∩ Ioc 0 (Real.pi / 2))) := by
+    refine (ENNReal.measurable_ofReal.comp_aemeasurable ?_)
+    exact ((continuous_magicDensity.comp_aestronglyMeasurable
+      (aestronglyMeasurable_tangentArm_fst D)).aemeasurable).mono_measure
+        (Measure.restrict_mono Set.inter_subset_right le_rfl)
+  rw [armDensityMeasure_apply C hS, armDensityMeasure_apply D hS]
+  calc
+    ∫⁻ t in S ∩ Ioc 0 (Real.pi / 2),
+        ENNReal.ofReal (magicDensity (tangentArmLengths C t).2.1) ≤
+        ∫⁻ t in S ∩ Ioc 0 (Real.pi / 2),
+          (ENNReal.ofReal (magicDensity (tangentArmLengths D t).2.1) +
+            ENNReal.ofReal |magicDensity (tangentArmLengths C t).2.1 -
+              magicDensity (tangentArmLengths D t).2.1|) := by
+      refine lintegral_mono fun t ↦ ?_
+      rw [← ENNReal.ofReal_add (magicDensity_nonneg _) (abs_nonneg _)]
+      refine ENNReal.ofReal_le_ofReal ?_
+      linarith [le_abs_self (magicDensity (tangentArmLengths C t).2.1 -
+        magicDensity (tangentArmLengths D t).2.1)]
+    _ = (∫⁻ t in S ∩ Ioc 0 (Real.pi / 2),
+          ENNReal.ofReal (magicDensity (tangentArmLengths D t).2.1)) +
+        ∫⁻ t in S ∩ Ioc 0 (Real.pi / 2),
+          ENNReal.ofReal |magicDensity (tangentArmLengths C t).2.1 -
+            magicDensity (tangentArmLengths D t).2.1| := lintegral_add_left' hmeasD _
+    _ ≤ (∫⁻ t in S ∩ Ioc 0 (Real.pi / 2),
+          ENNReal.ofReal (magicDensity (tangentArmLengths D t).2.1)) +
+        ∫⁻ t in Ioc 0 (Real.pi / 2),
+          ENNReal.ofReal |magicDensity (tangentArmLengths C t).2.1 -
+            magicDensity (tangentArmLengths D t).2.1| := by
+      gcongr
+      exact Set.inter_subset_right
+    _ = _ := by
+      rw [← ofReal_integral_eq_lintegral_ofReal hdiffIoc
+        (Filter.Eventually.of_forall fun t ↦ abs_nonneg _),
+        intervalIntegral.integral_of_le hT]
+
+/-! ### Surface measure support of a right-angle polygon cap -/
+
+/-- Angular images of Borel subsets of the half-open rotation interval are Borel. -/
+private theorem measurableSet_coe_image_of_subset_Ico {E : Set ℝ} (hE : MeasurableSet E)
+    (hET : E ⊆ Ico (0 : ℝ) (Real.pi / 2)) :
+    MeasurableSet ((fun t : ℝ ↦ (t : Real.Angle)) '' E) :=
+  Real.Angle.measurableSet_image_of_subset_Ioc (a := -Real.pi) (b := Real.pi)
+    (by linarith [Real.pi_pos]) hE fun t ht ↦
+      ⟨by linarith [Real.pi_pos, (hET ht).1],
+        (hET ht).2.le.trans (half_le_self Real.pi_pos.le)⟩
+
+private theorem rightAngle_properEdgeNormal_real (n : ℕ) (hn : 2 ≤ n)
+    (K : PolygonCapSpace (rightAngleSet n hn)) {s : ℝ} (hs : s ∈ Ioc (-Real.pi) Real.pi)
+    (hedge : (edgeVertices K.val.val (s : Real.Angle)).1 ≠
+      (edgeVertices K.val.val (s : Real.Angle)).2) :
+    s ∈ angleDomain (rightAngleSet n hn) ∨ s = -(Real.pi / 2) := by
+  have hpi := Real.pi_pos
+  have h := K.properEdgeNormal_mem_allowed (s : Real.Angle) hedge
+  rw [rightAngle_polygon_normals_eq n hn] at h
+  obtain ⟨r, hr, hrs⟩ := h
+  rcases hr with hr | rfl
+  · left
+    have hrb := angleDomain_subset_Ioo (rightAngleSet n hn) hr
+    have hsr : s = r := Real.Angle.injOn_coe_Ioc (a := -Real.pi) (b := Real.pi)
+      (by linarith) hs ⟨by linarith [hrb.1], hrb.2.le⟩ hrs.symm
+    exact hsr ▸ hr
+  · right
+    have hrs' : ((3 * Real.pi / 2 : ℝ) : Real.Angle) = ((s : ℝ) : Real.Angle) := hrs
+    refine Real.Angle.injOn_coe_Ioc (a := -Real.pi) (b := Real.pi) (by linarith) hs
+      ⟨by linarith, by linarith⟩ ?_
+    change ((s : ℝ) : Real.Angle) = ((-(Real.pi / 2) : ℝ) : Real.Angle)
+    rw [← hrs', Real.Angle.angle_eq_iff_two_pi_dvd_sub]
+    exact ⟨1, by push_cast; ring⟩
+
+private theorem rightAngle_properEdgeNormal_mem_directions (n : ℕ) (hn : 2 ≤ n)
+    (K : PolygonCapSpace (rightAngleSet n hn)) {s : ℝ}
+    (hs : s ∈ Ico (0 : ℝ) (Real.pi / 2))
+    (hedge : (edgeVertices K.val.val (s : Real.Angle)).1 ≠
+      (edgeVertices K.val.val (s : Real.Angle)).2) :
+    s ∈ (rightAngleSet n hn).directions := by
+  have hpi := Real.pi_pos
+  have hsIoc : s ∈ Ioc (-Real.pi) Real.pi :=
+    ⟨by linarith [hs.1], by linarith [hs.2]⟩
+  rcases rightAngle_properEdgeNormal_real n hn K hsIoc hedge with hdom | hneg
+  · rcases hdom with (hd | ⟨r, hr, hrs⟩) | hd
+    · exact hd
+    · exact absurd hs.2 (by
+        have hr0 := ((rightAngleSet n hn).interior r hr).1
+        rw [← hrs]
+        push Not
+        linarith)
+    · exfalso
+      have hangle : (rightAngleSet n hn).angle = Real.pi / 2 := rfl
+      simp only [hangle, Set.mem_insert_iff, Set.mem_singleton_iff] at hd
+      rcases hd with hd | hd <;> (rw [hd] at hs; linarith [hs.2])
+  · exfalso
+    rw [hneg] at hs
+    linarith [hs.1]
+
+private theorem rightAngle_edgeVertices_eq_of_mem_Ioo_neg (n : ℕ) (hn : 2 ≤ n)
+    (K : PolygonCapSpace (rightAngleSet n hn)) {s : ℝ}
+    (hs : s ∈ Ioo (-(Real.pi / 2)) 0) :
+    (edgeVertices K.val.val (s : Real.Angle)).1 =
+      (edgeVertices K.val.val (s : Real.Angle)).2 := by
+  have hpi := Real.pi_pos
+  by_contra hne
+  have hsIoc : s ∈ Ioc (-Real.pi) Real.pi :=
+    ⟨by linarith [hs.1], by linarith [hs.2]⟩
+  rcases rightAngle_properEdgeNormal_real n hn K hsIoc hne with hdom | hneg
+  · have hrb := angleDomain_subset_Ioo (rightAngleSet n hn) hdom
+    linarith [hrb.1, hs.2]
+  · rw [hneg] at hs
+    linarith [hs.1]
+
+private theorem surfaceAreaMeasure_rightAngle_compl_grid_eq_zero (n : ℕ) (hn : 2 ≤ n)
+    (K : PolygonCapSpace (rightAngleSet n hn)) :
+    surfaceAreaMeasure K.val.val
+      (((fun s : ℝ ↦ (s : Real.Angle)) '' Ico (0 : ℝ) (Real.pi / 2)) \
+        ((fun s : ℝ ↦ (s : Real.Angle)) ''
+          ({0} ∪ ((rightAngleSet n hn).directions : Set ℝ)))) = 0 := by
+  apply measure_mono_null _ K.surfaceAreaMeasure_compl_properEdgeNormals_eq_zero
+  rintro u ⟨⟨s, hs, rfl⟩, hnot⟩
+  by_contra hne
+  exact hnot ⟨s, Or.inr (rightAngle_properEdgeNormal_mem_directions n hn K hs hne), rfl⟩
+
+private theorem surfaceAreaMeasure_rightAngle_Ioo_neg_eq_zero (n : ℕ) (hn : 2 ≤ n)
+    (K : PolygonCapSpace (rightAngleSet n hn)) :
+    surfaceAreaMeasure K.val.val
+      ((fun s : ℝ ↦ (s : Real.Angle)) '' Ioo (-(Real.pi / 2)) 0) = 0 := by
+  apply measure_mono_null _ K.surfaceAreaMeasure_compl_properEdgeNormals_eq_zero
+  rintro u ⟨s, hs, rfl⟩
+  exact rightAngle_edgeVertices_eq_of_mem_Ioo_neg n hn K hs
+
+/-! ### Discrete cell bounds -/
+
+private theorem maximumPolygonCap_arm_cell_abs_le (n : ℕ) (hn : 2 ≤ n)
+    (K : RightAngleCapSpace) (hK : IsMaximumPolygonCapSteps n K) (t u : ℝ)
+    (ht : t = 0 ∨ t ∈ (rightAngleSet n hn).directions)
+    (hu : u ∈ Ioo t (t + polygonStepSize n)) :
+    |(tangentArmLengths K u).2.1 - (tangentArmLengths K t).2.1| ≤
+      5 * polygonStepSize n := by
+  obtain ⟨hcell, hdrop⟩ := maximumPolygonCap_arm_cell n hn K hK t ht
+  have hu' := hcell u hu
+  have hlower :
+      (tangentArmLengths K t).2.1 - 5 * polygonStepSize n ≤
+        (tangentArmLengths K u).2.1 := by
+    rw [hu'.2.1]
+    linarith [hu'.2.2]
+  rw [abs_le]
+  constructor <;> linarith [hu'.1]
+
+private theorem magicDensity_mul_le_integral_add_of_cell_bound
+    (f : ℝ → ℝ) (t δ c : ℝ) (hδ : 0 ≤ δ)
+    (hint : IntervalIntegrable (fun u ↦ magicDensity (f u)) volume t (t + δ))
+    (hvar : ∀ᵐ u ∂volume.restrict (Icc t (t + δ)), |f u - f t| ≤ c) :
+    magicDensity (f t) * δ ≤
+      (∫ u in t..(t + δ), magicDensity (f u)) + c * δ := by
+  have hp : ∀ᵐ u ∂volume.restrict (Icc t (t + δ)),
+      magicDensity (f t) ≤ magicDensity (f u) + c := by
+    filter_upwards [hvar] with u hu
+    have hle : magicDensity (f t) - magicDensity (f u) ≤ c := by
+      calc
+        magicDensity (f t) - magicDensity (f u) ≤
+            |magicDensity (f t) - magicDensity (f u)| := le_abs_self _
+        _ ≤ |f t - f u| := abs_magicDensity_sub_le _ _
+        _ = |f u - f t| := abs_sub_comm _ _
+        _ ≤ c := hu
+    linarith
+  have hc : IntervalIntegrable (fun _ : ℝ ↦ c) volume t (t + δ) :=
+    intervalIntegrable_const
+  have hi := intervalIntegral.integral_mono_ae_restrict
+    (show t ≤ t + δ by linarith)
+    (intervalIntegrable_const :
+      IntervalIntegrable (fun _ : ℝ ↦ magicDensity (f t)) volume t (t + δ))
+    (hint.add hc) hp
+  rw [intervalIntegral.integral_add hint hc, intervalIntegral.integral_const,
+    intervalIntegral.integral_const] at hi
+  simpa only [add_sub_cancel_left, smul_eq_mul, mul_comm] using hi
+
+private theorem maximumPolygonCap_magicDensity_cell_bound
+    (n : ℕ) (hn : 2 ≤ n) (K : RightAngleCapSpace)
+    (hK : IsMaximumPolygonCapSteps n K) (t : ℝ)
+    (ht : t = 0 ∨ t ∈ (rightAngleSet n hn).directions)
+    (hint : IntervalIntegrable
+      (fun u ↦ magicDensity (tangentArmLengths K u).2.1)
+      volume t (t + polygonStepSize n)) :
+    magicDensity (tangentArmLengths K t).2.1 * polygonStepSize n ≤
+      (∫ u in t..(t + polygonStepSize n),
+        magicDensity (tangentArmLengths K u).2.1) +
+        5 * polygonStepSize n ^ 2 := by
+  have haeIoo : ∀ᵐ u ∂volume.restrict
+      (Icc t (t + polygonStepSize n)), u ∈ Ioo t (t + polygonStepSize n) := by
+    rw [ae_iff]
+    change (volume.restrict (Icc t (t + polygonStepSize n)))
+      (Ioo t (t + polygonStepSize n))ᶜ = 0
+    rw [Measure.restrict_apply measurableSet_Ioo.compl]
+    have hsub : (Ioo t (t + polygonStepSize n))ᶜ ∩
+        Icc t (t + polygonStepSize n) ⊆ {t, t + polygonStepSize n} := by
+      intro x hx
+      simp only [mem_inter_iff, mem_Icc, mem_compl_iff, mem_Ioo, not_and_or,
+        not_lt, Set.mem_insert_iff, Set.mem_singleton_iff] at hx ⊢
+      rcases hx with ⟨hxt | hux, htx, hxu⟩
+      · exact Or.inl (le_antisymm hxt htx)
+      · exact Or.inr (le_antisymm hxu hux)
+    exact measure_mono_null hsub
+      ((Set.finite_singleton (t + polygonStepSize n)).insert t |>.measure_zero volume)
+  have hvar : ∀ᵐ u ∂volume.restrict (Icc t (t + polygonStepSize n)),
+      |(tangentArmLengths K u).2.1 - (tangentArmLengths K t).2.1| ≤
+        5 * polygonStepSize n := by
+    filter_upwards [haeIoo] with u hu
+    exact maximumPolygonCap_arm_cell_abs_le n hn K hK t u ht hu
+  have hδ : 0 ≤ polygonStepSize n := by
+    simp only [polygonStepSize]
+    positivity
+  have h := magicDensity_mul_le_integral_add_of_cell_bound
+    (fun u ↦ (tangentArmLengths K u).2.1) t (polygonStepSize n)
+      (5 * polygonStepSize n) hδ hint hvar
+  nlinarith [sq_nonneg (polygonStepSize n)]
+
+private theorem maximumPolygonCap_surfaceAtom_le_lintegral_cell
+    (n : ℕ) (hn : 2 ≤ n) (K : RightAngleCapSpace)
+    (hK : IsMaximumPolygonCapSteps n K) (t : ℝ)
+    (ht : t = 0 ∨ t ∈ (rightAngleSet n hn).directions)
+    (hmem : Icc t (t + polygonStepSize n) ⊆ Icc (0 : ℝ) (Real.pi / 2)) :
+    surfaceAreaMeasure K.val {(t : Real.Angle)} ≤
+      (∫⁻ u in Ioc t (t + polygonStepSize n),
+        ENNReal.ofReal (magicDensity (tangentArmLengths K u).2.1)) +
+      ENNReal.ofReal (13 * polygonStepSize n ^ 2) := by
+  have hnR : (0 : ℝ) < n := by
+    have : 0 < n := lt_of_lt_of_le zero_lt_two hn
+    exact_mod_cast this
+  have hδ : 0 < polygonStepSize n :=
+    div_pos (by linarith [Real.pi_pos]) hnR
+  have hle : t ≤ t + polygonStepSize n := by linarith
+  have hint : IntervalIntegrable
+      (fun u ↦ magicDensity (tangentArmLengths K u).2.1) volume t (t + polygonStepSize n) := by
+    apply (intervalIntegrable_magicDensity_tangentArm_fst K).mono_set
+    rw [uIcc_of_le hle, uIcc_of_le (by positivity : (0 : ℝ) ≤ Real.pi / 2)]
+    exact hmem
+  have hcell := maximumPolygonCap_magicDensity_cell_bound n hn K hK t ht hint
+  have hnonneg : 0 ≤ ∫ u in t..(t + polygonStepSize n),
+      magicDensity (tangentArmLengths K u).2.1 :=
+    intervalIntegral.integral_nonneg hle fun u _ ↦ magicDensity_nonneg _
+  have hlint : ENNReal.ofReal (∫ u in t..(t + polygonStepSize n),
+        magicDensity (tangentArmLengths K u).2.1) =
+      ∫⁻ u in Ioc t (t + polygonStepSize n),
+        ENNReal.ofReal (magicDensity (tangentArmLengths K u).2.1) := by
+    rw [intervalIntegral.integral_of_le hle]
+    exact ofReal_integral_eq_lintegral_ofReal
+      ((intervalIntegrable_iff_integrableOn_Ioc_of_le hle).1 hint)
+      (Filter.Eventually.of_forall fun u ↦ magicDensity_nonneg _)
+  calc
+    surfaceAreaMeasure K.val {(t : Real.Angle)} ≤
+        ENNReal.ofReal (magicFunctions.1 (Real.toNNReal (tangentArmLengths K t).2.1) *
+          polygonStepSize n + 8 * polygonStepSize n ^ 2) :=
+      maximumPolygonCap_surfaceAtom_bound n hn K hK t ht
+    _ ≤ ENNReal.ofReal ((∫ u in t..(t + polygonStepSize n),
+          magicDensity (tangentArmLengths K u).2.1) + 13 * polygonStepSize n ^ 2) := by
+      refine ENNReal.ofReal_le_ofReal ?_
+      change magicDensity (tangentArmLengths K t).2.1 * polygonStepSize n +
+        8 * polygonStepSize n ^ 2 ≤ _
+      linarith
+    _ = _ := by rw [ENNReal.ofReal_add hnonneg (by positivity), hlint]
+
+/-! ### The per-level grid bound -/
+
+private theorem maximumPolygonCap_surface_image_le_thickening
+    (n : ℕ) (hn : 2 ≤ n) (K : RightAngleCapSpace) (hK : IsMaximumPolygonCapSteps n K)
+    (S : Set ℝ) (hSm : MeasurableSet S) (hST : S ⊆ Ico (0 : ℝ) (Real.pi / 2))
+    {ε : ℝ} (hε : polygonStepSize n < ε) :
+    surfaceAreaMeasure K.val ((fun t : ℝ ↦ (t : Real.Angle)) '' S) ≤
+      armDensityMeasure K (Metric.thickening ε S) +
+        ENNReal.ofReal (13 * (Real.pi / 2) * polygonStepSize n) := by
+  classical
+  have hpi := Real.pi_pos
+  have hK0 := hK
+  obtain ⟨hn', hdy, P, hPK, hmax⟩ := hK
+  have hnR : (0 : ℝ) < n := by
+    have : 0 < n := lt_of_lt_of_le zero_lt_two hn
+    exact_mod_cast this
+  have hδpos : 0 < polygonStepSize n := div_pos (by linarith) hnR
+  have hnδ : (n : ℝ) * polygonStepSize n = Real.pi / 2 := by
+    rw [polygonStepSize]
+    field_simp
+  have hinj : Set.InjOn (fun t : ℝ ↦ (t : Real.Angle)) (Ico (0 : ℝ) (Real.pi / 2)) := by
+    apply (Real.Angle.injOn_coe_Ioc (a := -Real.pi) (b := Real.pi) (by linarith)).mono
+    intro t ht
+    exact ⟨by linarith [ht.1], by linarith [ht.2]⟩
+  have hgridmem : ∀ i : ℕ, i < n →
+      (i : ℝ) * polygonStepSize n ∈ Ico (0 : ℝ) (Real.pi / 2) := by
+    intro i hi
+    have hin : (i : ℝ) < n := by exact_mod_cast hi
+    refine ⟨mul_nonneg (Nat.cast_nonneg i) hδpos.le, ?_⟩
+    calc (i : ℝ) * polygonStepSize n < (n : ℝ) * polygonStepSize n := by
+          exact mul_lt_mul_of_pos_right hin hδpos
+      _ = Real.pi / 2 := hnδ
+  have hcellIcc : ∀ i : ℕ, i < n →
+      Icc ((i : ℝ) * polygonStepSize n) ((i : ℝ) * polygonStepSize n + polygonStepSize n) ⊆
+        Icc (0 : ℝ) (Real.pi / 2) := by
+    intro i hi x hx
+    have hsucc : ((i : ℝ) + 1) ≤ (n : ℝ) := by exact_mod_cast Nat.succ_le_of_lt hi
+    refine ⟨le_trans (mul_nonneg (Nat.cast_nonneg i) hδpos.le) hx.1, hx.2.trans ?_⟩
+    calc (i : ℝ) * polygonStepSize n + polygonStepSize n =
+          ((i : ℝ) + 1) * polygonStepSize n := by ring
+      _ ≤ (n : ℝ) * polygonStepSize n := mul_le_mul_of_nonneg_right hsucc hδpos.le
+      _ = Real.pi / 2 := hnδ
+  have hgriddir : ∀ i : ℕ, i < n →
+      (i : ℝ) * polygonStepSize n = 0 ∨
+        (i : ℝ) * polygonStepSize n ∈ (rightAngleSet n hn').directions := by
+    intro i hi
+    by_cases hi0 : i = 0
+    · exact Or.inl (by simp [hi0])
+    · exact Or.inr ((mem_rightAngleSet_directions_iff n hn' _).2
+        ⟨i, Finset.mem_Ioo.2 ⟨Nat.pos_of_ne_zero hi0, hi⟩, rfl⟩)
+  -- the surface measure restricted to the arc of interest
+  have hSimage : MeasurableSet ((fun t : ℝ ↦ (t : Real.Angle)) '' S) :=
+    measurableSet_coe_image_of_subset_Ico hSm hST
+  have hμS : ((surfaceAreaMeasure K.val).restrict
+        ((fun t : ℝ ↦ (t : Real.Angle)) '' Ico (0 : ℝ) (Real.pi / 2)))
+        ((fun t : ℝ ↦ (t : Real.Angle)) '' S) =
+      surfaceAreaMeasure K.val ((fun t : ℝ ↦ (t : Real.Angle)) '' S) := by
+    rw [Measure.restrict_apply hSimage,
+      Set.inter_eq_left.2 (Set.image_mono hST)]
+  -- the atom decomposition over the grid
+  have hbound := measure_le_sum_of_measure_compl_image_eq_zero
+    ((surfaceAreaMeasure K.val).restrict
+      ((fun t : ℝ ↦ (t : Real.Angle)) '' Ico (0 : ℝ) (Real.pi / 2)))
+    (Finset.range n)
+    ((Finset.range n).filter (fun i : ℕ ↦ ((i : ℝ) * polygonStepSize n) ∈ S))
+    (fun i : ℕ ↦ (((i : ℝ) * polygonStepSize n : ℝ) : Real.Angle))
+    (fun i : ℕ ↦ (∫⁻ u in Ioc ((i : ℝ) * polygonStepSize n)
+        ((i : ℝ) * polygonStepSize n + polygonStepSize n),
+        ENNReal.ofReal (magicDensity (tangentArmLengths K u).2.1)) +
+      ENNReal.ofReal (13 * polygonStepSize n ^ 2))
+    ?_ ?_ ?_ ((fun t : ℝ ↦ (t : Real.Angle)) '' S) hSimage ?_
+  · rw [hμS] at hbound
+    refine hbound.trans ?_
+    rw [Finset.sum_add_distrib]
+    have hdisj : ((Finset.range n).filter
+          (fun i : ℕ ↦ ((i : ℝ) * polygonStepSize n) ∈ S) : Set ℕ).PairwiseDisjoint
+        (fun i : ℕ ↦ Ioc ((i : ℝ) * polygonStepSize n)
+          ((i : ℝ) * polygonStepSize n + polygonStepSize n)) := by
+      intro i _ j _ hij
+      simp only [Function.onFun, Set.disjoint_left]
+      intro x hx hx'
+      rcases lt_or_gt_of_ne hij with h | h
+      · have hle : (i : ℝ) + 1 ≤ (j : ℝ) := by exact_mod_cast Nat.succ_le_of_lt h
+        have : (i : ℝ) * polygonStepSize n + polygonStepSize n ≤
+            (j : ℝ) * polygonStepSize n := by
+          nlinarith [hδpos]
+        linarith [hx.2, hx'.1]
+      · have hle : (j : ℝ) + 1 ≤ (i : ℝ) := by exact_mod_cast Nat.succ_le_of_lt h
+        have : (j : ℝ) * polygonStepSize n + polygonStepSize n ≤
+            (i : ℝ) * polygonStepSize n := by
+          nlinarith [hδpos]
+        linarith [hx.1, hx'.2]
+    have hsumlint : ∑ i ∈ (Finset.range n).filter
+          (fun i : ℕ ↦ ((i : ℝ) * polygonStepSize n) ∈ S),
+          (∫⁻ u in Ioc ((i : ℝ) * polygonStepSize n)
+            ((i : ℝ) * polygonStepSize n + polygonStepSize n),
+            ENNReal.ofReal (magicDensity (tangentArmLengths K u).2.1)) =
+        ∫⁻ u in ⋃ i ∈ (Finset.range n).filter
+          (fun i : ℕ ↦ ((i : ℝ) * polygonStepSize n) ∈ S),
+            Ioc ((i : ℝ) * polygonStepSize n)
+              ((i : ℝ) * polygonStepSize n + polygonStepSize n),
+          ENNReal.ofReal (magicDensity (tangentArmLengths K u).2.1) :=
+      (lintegral_biUnion_finset hdisj (fun i _ ↦ measurableSet_Ioc) _).symm
+    have hsubset : (⋃ i ∈ (Finset.range n).filter
+          (fun i : ℕ ↦ ((i : ℝ) * polygonStepSize n) ∈ S),
+            Ioc ((i : ℝ) * polygonStepSize n)
+              ((i : ℝ) * polygonStepSize n + polygonStepSize n)) ⊆
+        Metric.thickening ε S ∩ Ioc 0 (Real.pi / 2) := by
+      intro x hx
+      obtain ⟨i, hi, hxi⟩ := Set.mem_iUnion₂.mp hx
+      obtain ⟨hiR, hiS⟩ := Finset.mem_filter.1 hi
+      have hi' : i < n := Finset.mem_range.1 hiR
+      refine ⟨?_, ?_⟩
+      · refine Metric.mem_thickening_iff.2 ⟨(i : ℝ) * polygonStepSize n, hiS, ?_⟩
+        rw [Real.dist_eq, abs_of_nonneg (by linarith [hxi.1])]
+        linarith [hxi.2]
+      · exact ⟨lt_of_le_of_lt (hgridmem i hi').1 hxi.1,
+          (hcellIcc i hi' ⟨hxi.1.le, hxi.2⟩).2⟩
+    have hcard : (((Finset.range n).filter
+          (fun i : ℕ ↦ ((i : ℝ) * polygonStepSize n) ∈ S)).card : ENNReal) ≤
+        (n : ENNReal) := by
+      have := Finset.card_filter_le (Finset.range n)
+        (fun i : ℕ ↦ ((i : ℝ) * polygonStepSize n) ∈ S)
+      rw [Finset.card_range] at this
+      exact_mod_cast this
+    have hconst : ∑ _i ∈ (Finset.range n).filter
+          (fun i : ℕ ↦ ((i : ℝ) * polygonStepSize n) ∈ S),
+          ENNReal.ofReal (13 * polygonStepSize n ^ 2) ≤
+        ENNReal.ofReal (13 * (Real.pi / 2) * polygonStepSize n) := by
+      rw [Finset.sum_const, nsmul_eq_mul]
+      calc
+        (((Finset.range n).filter
+            (fun i : ℕ ↦ ((i : ℝ) * polygonStepSize n) ∈ S)).card : ENNReal) *
+              ENNReal.ofReal (13 * polygonStepSize n ^ 2) ≤
+            (n : ENNReal) * ENNReal.ofReal (13 * polygonStepSize n ^ 2) := by
+          gcongr
+        _ = ENNReal.ofReal (13 * (Real.pi / 2) * polygonStepSize n) := by
+          rw [show ((n : ENNReal)) = ENNReal.ofReal ((n : ℝ)) by
+            simp [ENNReal.ofReal_natCast],
+            ← ENNReal.ofReal_mul (Nat.cast_nonneg n)]
+          congr 1
+          nlinarith [hnδ]
+    refine add_le_add ?_ hconst
+    rw [hsumlint, armDensityMeasure_apply K Metric.isOpen_thickening.measurableSet]
+    exact lintegral_mono_set hsubset
+  · -- injectivity of the grid map
+    intro i hi j hj hij
+    have hi' : i < n := Finset.mem_range.1 (Finset.mem_coe.1 hi)
+    have hj' : j < n := Finset.mem_range.1 (Finset.mem_coe.1 hj)
+    have heq : (i : ℝ) * polygonStepSize n = (j : ℝ) * polygonStepSize n :=
+      hinj (hgridmem i hi') (hgridmem j hj') hij
+    have hij' : (i : ℝ) = (j : ℝ) := mul_right_cancel₀ hδpos.ne' heq
+    exact_mod_cast hij'
+  · -- the surface measure is carried by the grid
+    have himagegrid :
+        (fun i : ℕ ↦ (((i : ℝ) * polygonStepSize n : ℝ) : Real.Angle)) ''
+            (Finset.range n : Set ℕ) =
+          (fun s : ℝ ↦ (s : Real.Angle)) ''
+            ({0} ∪ ((rightAngleSet n hn').directions : Set ℝ)) := by
+      ext u
+      constructor
+      · rintro ⟨i, hi, rfl⟩
+        have hi' : i < n := Finset.mem_range.1 (Finset.mem_coe.1 hi)
+        rcases hgriddir i hi' with h0 | hdir
+        · exact ⟨(i : ℝ) * polygonStepSize n, Or.inl h0, rfl⟩
+        · exact ⟨(i : ℝ) * polygonStepSize n, Or.inr hdir, rfl⟩
+      · rintro ⟨s, hs, rfl⟩
+        rcases hs with hs | hs
+        · refine ⟨0, Finset.mem_coe.2 (Finset.mem_range.2 (by omega)), ?_⟩
+          simp only [Set.mem_singleton_iff] at hs
+          simp [hs]
+        · obtain ⟨i, hi, rfl⟩ := (mem_rightAngleSet_directions_iff n hn' s).1 hs
+          exact ⟨i, Finset.mem_coe.2 (Finset.mem_range.2 (Finset.mem_Ioo.1 hi).2), rfl⟩
+    rw [himagegrid, Measure.restrict_apply
+      (((Set.finite_singleton (0 : ℝ)).union
+        (rightAngleSet n hn').directions.finite_toSet).image
+          (fun s : ℝ ↦ (s : Real.Angle))).measurableSet.compl,
+      Set.inter_comm]
+    have hsupp := surfaceAreaMeasure_rightAngle_compl_grid_eq_zero n hn' P
+    rw [hPK] at hsupp
+    exact hsupp
+  · -- the atom bound at each grid point
+    intro i hi
+    have hi' : i < n := Finset.mem_range.1 hi
+    refine le_trans (Measure.restrict_apply_le _ _) ?_
+    exact maximumPolygonCap_surfaceAtom_le_lintegral_cell n hn' K hK0
+      ((i : ℝ) * polygonStepSize n) (hgriddir i hi') (hcellIcc i hi')
+  · -- only grid points inside `S` contribute
+    intro i hi hmem
+    have hi' : i < n := Finset.mem_range.1 hi
+    obtain ⟨s, hsS, hseq⟩ := hmem
+    have hs : s = (i : ℝ) * polygonStepSize n :=
+      hinj (hST hsS) (hgridmem i hi') hseq
+    exact Finset.mem_filter.2 ⟨hi, hs ▸ hsS⟩
+
+/-! ### Passing to the limit -/
+
+private theorem tendsto_integral_abs_magicDensity_sub_of_dominated
+    (f g : ℕ → ℝ → ℝ) (hfi : ∀ i, IntervalIntegrable (f i) volume 0 (Real.pi / 2))
+    (hgi : ∀ i, IntervalIntegrable (g i) volume 0 (Real.pi / 2))
+    (hlim : Tendsto (fun i ↦ ∫ t in (0 : ℝ)..(Real.pi / 2), |f i t - g i t|)
+      atTop (𝓝 0)) :
+    Tendsto (fun i ↦ ∫ t in (0 : ℝ)..(Real.pi / 2),
+      |magicDensity (f i t) - magicDensity (g i t)|) atTop (𝓝 0) := by
+  have hsource (i : ℕ) :
+      IntervalIntegrable (fun t ↦ |f i t - g i t|) volume 0 (Real.pi / 2) :=
+    ((hfi i).sub (hgi i)).norm
+  have htarget (i : ℕ) : IntervalIntegrable
+      (fun t ↦ |magicDensity (f i t) - magicDensity (g i t)|)
+      volume 0 (Real.pi / 2) := by
+    apply (hsource i).mono_fun
+    · rw [uIoc_of_le (by positivity : (0 : ℝ) ≤ Real.pi / 2)]
+      convert ((continuous_magicDensity.comp_aestronglyMeasurable
+        (hfi i).aestronglyMeasurable).sub
+        (continuous_magicDensity.comp_aestronglyMeasurable
+          (hgi i).aestronglyMeasurable)).norm using 1
+    · filter_upwards [] with t
+      simpa only [Real.norm_eq_abs, abs_abs] using abs_magicDensity_sub_le (f i t) (g i t)
+  apply squeeze_zero
+  · intro i
+    exact intervalIntegral.integral_nonneg (by positivity) (fun _ _ ↦ abs_nonneg _)
+  · intro i
+    apply intervalIntegral.integral_mono_on (by positivity) (htarget i) (hsource i)
+    intro t _
+    exact abs_magicDensity_sub_le (f i t) (g i t)
+  · exact hlim
+
+private theorem IsBalancedMaximumCap.approximatingPolygonCaps
+    (K : RightAngleCapSpace) (hK : IsBalancedMaximumCap K) :
+    ∃ (n : ℕ → ℕ) (P : ℕ → RightAngleCapSpace), StrictMono n ∧
+      (∀ i, IsMaximumPolygonCapSteps (n i) (P i)) ∧
+      Tendsto (fun i ↦ Metric.hausdorffDist ((P i).val : Set Point) (K.val : Set Point))
+        atTop (𝓝 0) ∧
+      Tendsto (fun i ↦ ∫ t in (0 : ℝ)..(Real.pi / 2),
+        |magicDensity (tangentArmLengths (P i) t).2.1 -
+          magicDensity (tangentArmLengths K t).2.1|) atTop (𝓝 0) := by
+  obtain ⟨n, hn, hmono, hdyadic, Q, hQmax, hQlim⟩ := hK
+  refine ⟨n, fun i ↦ (Q i).val, hmono, fun i ↦ ⟨hn i, hdyadic i, Q i, rfl, hQmax i⟩,
+    hQlim, ?_⟩
+  apply tendsto_integral_abs_magicDensity_sub_of_dominated
+    (fun i t ↦ (tangentArmLengths ((Q i).val) t).2.1)
+    (fun _ t ↦ (tangentArmLengths K t).2.1)
+    (fun i ↦ intervalIntegrable_tangentArm_fst _)
+    (fun i ↦ intervalIntegrable_tangentArm_fst K)
+  exact polygonCap_arm_integral_limit (fun i ↦ (Q i).val) K hQlim
+
+/-- The limit of the discrete inequalities on a compact subset of the rotation interval. -/
+private theorem balancedMaximumCap_surface_isCompact_le (K : RightAngleCapSpace)
+    (hK : IsBalancedMaximumCap K) {C : Set ℝ} (hC : IsCompact C)
+    (hCT : C ⊆ Ico (0 : ℝ) (Real.pi / 2)) :
+    surfaceAreaMeasure K.val ((fun t : ℝ ↦ (t : Real.Angle)) '' C) ≤
+      armDensityMeasure K C := by
+  have hpi := Real.pi_pos
+  obtain ⟨n, P, hmono, hPmax, hPlim, hL1⟩ :=
+    IsBalancedMaximumCap.approximatingPolygonCaps K hK
+  set err : ℕ → ℝ := fun i ↦ ∫ t in (0 : ℝ)..(Real.pi / 2),
+    |magicDensity (tangentArmLengths (P i) t).2.1 -
+      magicDensity (tangentArmLengths K t).2.1| with herrdef
+  have herr0 : ∀ i, 0 ≤ err i := fun i ↦
+    intervalIntegral.integral_nonneg (by positivity) fun t _ ↦ abs_nonneg _
+  have hδpos : ∀ i, 0 < polygonStepSize (n i) := by
+    intro i
+    obtain ⟨hn', _⟩ := hPmax i
+    have hnR : (0 : ℝ) < n i := by
+      have : 0 < n i := lt_of_lt_of_le zero_lt_two hn'
+      exact_mod_cast this
+    exact div_pos (by linarith) hnR
+  have hδtend : Tendsto (fun i ↦ polygonStepSize (n i)) atTop (𝓝 0) := by
+    have h := (tendsto_const_div_atTop_nhds_zero_nat (Real.pi / 2)).comp
+      hmono.tendsto_atTop
+    simpa only [polygonStepSize, Function.comp_def] using h
+  -- weak convergence of the surface measures
+  let μs : ℕ → FiniteMeasure Real.Angle := fun i ↦
+    ⟨surfaceAreaMeasure (P i).val, (surfaceAreaMeasure_face_union (P i).val).1⟩
+  let μ : FiniteMeasure Real.Angle :=
+    ⟨surfaceAreaMeasure K.val, (surfaceAreaMeasure_face_union K.val).1⟩
+  have hweak : Tendsto μs atTop (𝓝 μ) := by
+    apply MeasureTheory.FiniteMeasure.tendsto_iff_forall_integral_tendsto.mpr
+    intro f
+    exact surfaceAreaMeasure_weak_continuity (fun i ↦ (P i).val) K.val hPlim f f.continuous
+  -- the estimate at every positive thickening radius
+  have hkey : ∀ η : ℝ, 0 < η →
+      surfaceAreaMeasure K.val ((fun t : ℝ ↦ (t : Real.Angle)) '' C) ≤
+        armDensityMeasure K (Metric.thickening (2 * η) C) := by
+    intro η hη
+    set V := (Metric.thickening η C ∩ Ioo (-(Real.pi / 2)) (Real.pi / 2)) ∪
+      Ioo (-(Real.pi / 2)) 0 with hVdef
+    have hVopen : IsOpen V :=
+      (Metric.isOpen_thickening.inter isOpen_Ioo).union isOpen_Ioo
+    have hCV : C ⊆ V := by
+      intro x hx
+      exact Or.inl ⟨Metric.self_subset_thickening hη C hx,
+        ⟨by linarith [(hCT hx).1], (hCT hx).2⟩⟩
+    have hVsubIoo : V ⊆ Ioo (-(Real.pi / 2)) (Real.pi / 2) := by
+      rintro x (hx | hx)
+      · exact hx.2
+      · exact ⟨hx.1, by linarith [hx.2]⟩
+    have hVsubThick : V ∩ Ico (0 : ℝ) (Real.pi / 2) ⊆ Metric.thickening η C := by
+      rintro x ⟨hxV, hxI⟩
+      rcases hxV with hx | hx
+      · exact hx.1
+      · exact absurd hxI.1 (by simpa using hx.2)
+    have hVmeas : MeasurableSet (V ∩ Ico (0 : ℝ) (Real.pi / 2)) :=
+      hVopen.measurableSet.inter measurableSet_Ico
+    have hφVopen : IsOpen ((fun t : ℝ ↦ (t : Real.Angle)) '' V) :=
+      QuotientAddGroup.isOpenMap_coe _ hVopen
+    have hstep1 : surfaceAreaMeasure K.val ((fun t : ℝ ↦ (t : Real.Angle)) '' C) ≤
+        Filter.liminf (fun i ↦ surfaceAreaMeasure (P i).val
+          ((fun t : ℝ ↦ (t : Real.Angle)) '' V)) atTop := by
+      refine le_trans (measure_mono (Set.image_mono hCV)) ?_
+      exact FiniteMeasure.le_liminf_measure_open_of_tendsto (μ := μ) (μs := μs)
+        hweak hφVopen
+    have hev : ∀ᶠ i in atTop, surfaceAreaMeasure (P i).val
+        ((fun t : ℝ ↦ (t : Real.Angle)) '' V) ≤
+        armDensityMeasure K (Metric.thickening (2 * η) C) +
+          ENNReal.ofReal (err i + 13 * (Real.pi / 2) * polygonStepSize (n i)) := by
+      filter_upwards [hδtend.eventually (gt_mem_nhds (show (0 : ℝ) < η / 2 by positivity))]
+        with i hi
+      obtain ⟨hn', hdy, Q, hQK, hQmax⟩ := hPmax i
+      have hQval : Q.val.val = (P i).val := by rw [hQK]
+      have hneg : surfaceAreaMeasure (P i).val
+          ((fun s : ℝ ↦ (s : Real.Angle)) '' Ioo (-(Real.pi / 2)) 0) = 0 := by
+        rw [← hQval]
+        exact surfaceAreaMeasure_rightAngle_Ioo_neg_eq_zero (n i) hn' Q
+      have hsplit : ((fun t : ℝ ↦ (t : Real.Angle)) '' V) ⊆
+          ((fun t : ℝ ↦ (t : Real.Angle)) '' (V ∩ Ico (0 : ℝ) (Real.pi / 2))) ∪
+            ((fun s : ℝ ↦ (s : Real.Angle)) '' Ioo (-(Real.pi / 2)) 0) := by
+        rintro u ⟨x, hxV, rfl⟩
+        rcases lt_or_ge x 0 with hx | hx
+        · exact Or.inr ⟨x, ⟨(hVsubIoo hxV).1, hx⟩, rfl⟩
+        · exact Or.inl ⟨x, ⟨hxV, ⟨hx, (hVsubIoo hxV).2⟩⟩, rfl⟩
+      have hthick : Metric.thickening (2 * polygonStepSize (n i))
+          (V ∩ Ico (0 : ℝ) (Real.pi / 2)) ⊆ Metric.thickening (2 * η) C := by
+        calc
+          Metric.thickening (2 * polygonStepSize (n i))
+              (V ∩ Ico (0 : ℝ) (Real.pi / 2)) ⊆
+              Metric.thickening (2 * polygonStepSize (n i))
+                (Metric.thickening η C) :=
+            Metric.thickening_subset_of_subset _ hVsubThick
+          _ ⊆ Metric.thickening (2 * polygonStepSize (n i) + η) C :=
+            Metric.thickening_thickening_subset _ _ _
+          _ ⊆ Metric.thickening (2 * η) C :=
+            Metric.thickening_mono (by linarith) C
+      calc
+        surfaceAreaMeasure (P i).val ((fun t : ℝ ↦ (t : Real.Angle)) '' V) ≤
+            surfaceAreaMeasure (P i).val
+                ((fun t : ℝ ↦ (t : Real.Angle)) '' (V ∩ Ico (0 : ℝ) (Real.pi / 2))) +
+              surfaceAreaMeasure (P i).val
+                ((fun s : ℝ ↦ (s : Real.Angle)) '' Ioo (-(Real.pi / 2)) 0) :=
+          (measure_mono hsplit).trans (measure_union_le _ _)
+        _ = surfaceAreaMeasure (P i).val
+            ((fun t : ℝ ↦ (t : Real.Angle)) '' (V ∩ Ico (0 : ℝ) (Real.pi / 2))) := by
+          rw [hneg, add_zero]
+        _ ≤ armDensityMeasure (P i) (Metric.thickening (2 * polygonStepSize (n i))
+              (V ∩ Ico (0 : ℝ) (Real.pi / 2))) +
+            ENNReal.ofReal (13 * (Real.pi / 2) * polygonStepSize (n i)) :=
+          maximumPolygonCap_surface_image_le_thickening (n i) hn' (P i) (hPmax i)
+            (V ∩ Ico (0 : ℝ) (Real.pi / 2)) hVmeas Set.inter_subset_right
+            (by linarith [hδpos i])
+        _ ≤ armDensityMeasure (P i) (Metric.thickening (2 * η) C) +
+            ENNReal.ofReal (13 * (Real.pi / 2) * polygonStepSize (n i)) := by
+          gcongr
+        _ ≤ (armDensityMeasure K (Metric.thickening (2 * η) C) +
+              ENNReal.ofReal (err i)) +
+            ENNReal.ofReal (13 * (Real.pi / 2) * polygonStepSize (n i)) := by
+          gcongr
+          exact armDensityMeasure_le_add_l1 (P i) K Metric.isOpen_thickening.measurableSet
+        _ = armDensityMeasure K (Metric.thickening (2 * η) C) +
+            ENNReal.ofReal (err i + 13 * (Real.pi / 2) * polygonStepSize (n i)) := by
+          rw [ENNReal.ofReal_add (herr0 i)
+            (mul_nonneg (by positivity) (hδpos i).le), add_assoc]
+    have htend : Tendsto (fun i ↦ armDensityMeasure K (Metric.thickening (2 * η) C) +
+        ENNReal.ofReal (err i + 13 * (Real.pi / 2) * polygonStepSize (n i))) atTop
+        (𝓝 (armDensityMeasure K (Metric.thickening (2 * η) C))) := by
+      have hsum : Tendsto (fun i ↦ err i + 13 * (Real.pi / 2) * polygonStepSize (n i))
+          atTop (𝓝 0) := by
+        have h13 : Tendsto (fun i ↦ 13 * (Real.pi / 2) * polygonStepSize (n i))
+            atTop (𝓝 0) := by
+          simpa using (tendsto_const_nhds (x := 13 * (Real.pi / 2))
+            (f := atTop (α := ℕ))).mul hδtend
+        simpa using hL1.add h13
+      have h0 : Tendsto (fun i ↦ ENNReal.ofReal
+          (err i + 13 * (Real.pi / 2) * polygonStepSize (n i))) atTop (𝓝 0) := by
+        simpa using ENNReal.tendsto_ofReal hsum
+      simpa using (tendsto_const_nhds
+        (x := armDensityMeasure K (Metric.thickening (2 * η) C))
+        (f := atTop (α := ℕ))).add h0
+    exact hstep1.trans ((Filter.liminf_le_liminf hev).trans (le_of_eq htend.liminf_eq))
+  -- shrink the radius to zero
+  have hinter : (⋂ m : ℕ, Metric.thickening (2 * (1 / (m + 1) : ℝ)) C) = C := by
+    refine Set.Subset.antisymm ?_ ?_
+    · intro x hx
+      have hclos : x ∈ closure C := by
+        rw [Metric.mem_closure_iff]
+        intro r hr
+        obtain ⟨m, hm⟩ := exists_nat_gt (2 / r)
+        obtain ⟨y, hyC, hy⟩ := Metric.mem_thickening_iff.1 (Set.mem_iInter.1 hx m)
+        refine ⟨y, hyC, lt_of_lt_of_le hy ?_⟩
+        have hmpos : (0 : ℝ) < (m : ℝ) + 1 := by positivity
+        rw [mul_one_div, div_le_iff₀ hmpos]
+        have h2r : 2 / r < (m : ℝ) := hm
+        have : 2 < r * (m : ℝ) := by
+          rw [div_lt_iff₀ hr] at h2r
+          linarith
+        nlinarith
+      rwa [hC.isClosed.closure_eq] at hclos
+    · exact Set.subset_iInter fun m ↦
+        Metric.self_subset_thickening (by positivity) C
+  have hanti : Antitone (fun m : ℕ ↦ Metric.thickening (2 * (1 / (m + 1) : ℝ)) C) := by
+    intro a b hab
+    refine Metric.thickening_mono ?_ C
+    have ha : (0 : ℝ) < (a : ℝ) + 1 := by positivity
+    have hb : (0 : ℝ) < (b : ℝ) + 1 := by positivity
+    have hab' : (a : ℝ) ≤ (b : ℝ) := by exact_mod_cast hab
+    rw [mul_one_div, mul_one_div]
+    gcongr
+  have hfin : ∃ m : ℕ, armDensityMeasure K
+      (Metric.thickening (2 * (1 / (m + 1) : ℝ)) C) ≠ ⊤ := by
+    refine ⟨0, ?_⟩
+    exact ne_top_of_le_ne_top (armDensityMeasure_ne_top K)
+      (measure_mono (Set.subset_univ _))
+  have hseq := tendsto_measure_iInter_atTop
+    (μ := armDensityMeasure K)
+    (s := fun m : ℕ ↦ Metric.thickening (2 * (1 / (m + 1) : ℝ)) C)
+    (fun m ↦ Metric.isOpen_thickening.measurableSet.nullMeasurableSet) hanti hfin
+  rw [hinter] at hseq
+  refine ge_of_tendsto hseq (Filter.Eventually.of_forall fun m ↦ ?_)
+  exact hkey _ (by positivity)
+
+private theorem exists_isCompact_preimage {E : Set ℝ}
+    (hET : E ⊆ Ico (0 : ℝ) (Real.pi / 2)) {F : Set Real.Angle} (hF : IsCompact F)
+    (hFE : F ⊆ (fun t : ℝ ↦ (t : Real.Angle)) '' E) :
+    ∃ C : Set ℝ, IsCompact C ∧ C ⊆ E ∧ F ⊆ (fun t : ℝ ↦ (t : Real.Angle)) '' C := by
+  have hpi := Real.pi_pos
+  have hinj : Set.InjOn (fun t : ℝ ↦ (t : Real.Angle)) (Icc (0 : ℝ) (Real.pi / 2)) := by
+    apply (Real.Angle.injOn_coe_Ioc (a := -Real.pi) (b := Real.pi)
+      (by linarith)).mono
+    intro t ht
+    exact ⟨by linarith [ht.1], by linarith [ht.2]⟩
+  have hclosed : IsClosed ((fun t : ℝ ↦ (t : Real.Angle)) ⁻¹' F) :=
+    hF.isClosed.preimage Real.Angle.continuous_coe
+  refine ⟨(fun t : ℝ ↦ (t : Real.Angle)) ⁻¹' F ∩ Icc 0 (Real.pi / 2),
+    isCompact_Icc.inter_left hclosed, ?_, ?_⟩
+  · rintro t ⟨htF, htI⟩
+    obtain ⟨e, heE, het⟩ := hFE htF
+    have : e = t := hinj (Ico_subset_Icc_self (hET heE)) htI het
+    exact this ▸ heE
+  · intro u hu
+    obtain ⟨e, heE, rfl⟩ := hFE hu
+    exact ⟨e, ⟨hu, Ico_subset_Icc_self (hET heE)⟩, rfl⟩
+
+end Domination
+
+theorem balancedMaximumCap_surface_domination (K : RightAngleCapSpace)
+    (hK : IsBalancedMaximumCap K) (E : Set ℝ) (hE : MeasurableSet E)
+    (hET : E ⊆ Set.Ico (0 : ℝ) (Real.pi / 2)) :
+    surfaceAreaMeasure K.val ((fun t : ℝ ↦ (t : Real.Angle)) '' E) ≤
+      ENNReal.ofReal (∫ t in E,
+        magicFunctions.1 (Real.toNNReal (tangentArmLengths K t).2.1)) := by
+  classical
+  let _ : IsFiniteMeasure (surfaceAreaMeasure K.val) :=
+    (surfaceAreaMeasure_face_union K.val).1
+  have hEimage : MeasurableSet ((fun t : ℝ ↦ (t : Real.Angle)) '' E) :=
+    measurableSet_coe_image_of_subset_Ico hE hET
+  rw [show ENNReal.ofReal (∫ t in E, magicFunctions.1
+      (Real.toNNReal (tangentArmLengths K t).2.1)) = armDensityMeasure K E from
+    (armDensityMeasure_eq_ofReal_integral K hE hET).symm,
+    MeasurableSet.measure_eq_iSup_isCompact hEimage (surfaceAreaMeasure K.val)]
+  refine iSup_le fun F ↦ iSup_le fun hFE ↦ iSup_le fun hFcomp ↦ ?_
+  obtain ⟨C, hCcomp, hCE, hFC⟩ := exists_isCompact_preimage hET hFcomp hFE
+  exact (measure_mono hFC).trans
+    ((balancedMaximumCap_surface_isCompact_le K hK hCcomp (hCE.trans hET)).trans
+      (measure_mono hCE))
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Cap.DensityExistence`.
+* `Cap.Interpolation`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Existence and uniqueness of the surface densities of a balanced maximum cap
+
+A balanced maximum cap of rotation angle `π / 2` satisfies the density clause of the
+injectivity condition: its surface area measure has nonnegative measurable densities on the
+quarter arcs `[0, π / 2)` and `(π / 2, π]`, unique up to Lebesgue-null sets.
+
+On the first arc the density comes from the limiting inequality
+`balancedMaximumCap_surface_domination` together with the Radon–Nikodym construction
+`exists_nnreal_density_of_domination`. On the second arc the mirror image of the cap is again a
+balanced maximum cap, and the surface-measure identity of `cap_mirror_features` transports its
+first-arc density back along the reflection `a ↦ π - a` of normal angles.
+
+On the first arc the domination inequality also bounds the density: the real density produced by
+`exists_capDensity_right_le_magicDensity` is at most `k₀` of the positive tangent arm length.
+-/
+
+/-! ### A continuous section of the angular projection -/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped NNReal ENNReal
+
+namespace MovingSofa
+
+/-- A continuous left inverse of the angle coercion on `[-(π / 2), π / 2]`. -/
+private def angleSection (a : Real.Angle) : ℝ := Real.arcsin a.sin
+
+private theorem continuous_angleSection : Continuous angleSection :=
+  Real.continuous_arcsin.comp Real.Angle.continuous_sin
+
+private theorem angleSection_coe {t : ℝ} (ht : t ∈ Icc (-(Real.pi / 2)) (Real.pi / 2)) :
+    angleSection (t : Real.Angle) = t := by
+  rw [angleSection, Real.Angle.sin_coe, Real.arcsin_sin ht.1 ht.2]
+
+/-- After a quarter-turn shift, `angleSection` inverts the angle coercion on `[0, π]`. -/
+private theorem angleSection_sub_coe {t : ℝ} (ht : t ∈ Icc (-(Real.pi / 2)) (Real.pi / 2)) :
+    angleSection (((t + Real.pi / 2 : ℝ) : Real.Angle) -
+      ((Real.pi / 2 : ℝ) : Real.Angle)) = t := by
+  rw [← Real.Angle.coe_sub, add_sub_cancel_right, angleSection_coe ht]
+
+/-- The second quarter arc, parametrized from `0` by a quarter-turn shift. -/
+private theorem image_coe_add_pi_div_two_Ioc :
+    (fun t : ℝ ↦ ((t + Real.pi / 2 : ℝ) : Real.Angle)) '' Ioc 0 (Real.pi / 2) =
+      (fun t : ℝ ↦ (t : Real.Angle)) '' Ioc (Real.pi / 2) Real.pi := by
+  rw [show (fun t : ℝ ↦ ((t + Real.pi / 2 : ℝ) : Real.Angle)) =
+      (fun u : ℝ ↦ (u : Real.Angle)) ∘ (fun t : ℝ ↦ t + Real.pi / 2) from rfl,
+    Set.image_comp, Set.image_add_const_Ioc, zero_add, add_halves]
+
+/-- If `ψ` inverts `φ` on `S`, then cutting a `ψ`-preimage down to the `φ`-image of `S` is the
+same as taking the `φ`-image of the corresponding part of `S`. -/
+private theorem preimage_inter_image_eq_image_inter {α β : Type*} {φ : α → β} {ψ : β → α}
+    {S : Set α} (hsec : ∀ t ∈ S, ψ (φ t) = t) (E : Set α) :
+    ψ ⁻¹' E ∩ φ '' S = φ '' (E ∩ S) := by
+  ext a
+  constructor
+  · rintro ⟨haE, t, ht, rfl⟩
+    rw [mem_preimage, hsec t ht] at haE
+    exact ⟨t, ⟨haE, ht⟩, rfl⟩
+  · rintro ⟨t, ⟨htE, ht⟩, rfl⟩
+    exact ⟨by rw [mem_preimage, hsec t ht]; exact htE, t, ht, rfl⟩
+
+/-! ### Densities on an arc from a domination inequality -/
+
+/-- Transport of the bounded-density construction to an arc of normal angles. If the angular
+parametrization `φ` has a measurable left inverse on a measurable set `S` that fills
+`[0, π / 2]` up to a null set, and the mass that `μ₀` gives to `φ '' E` is dominated by the
+integral of a bounded nonnegative function `f` over `E ⊆ S`, then `μ₀` restricted to the arc
+`φ '' S` is the pushforward along `φ` of a weighted Lebesgue measure on `S`. -/
+private theorem exists_arcDensity_of_domination (μ₀ : Measure Real.Angle) [IsFiniteMeasure μ₀]
+    {φ : ℝ → Real.Angle} (hφ : Measurable φ) {ψ : Real.Angle → ℝ} (hψ : Measurable ψ)
+    {S : Set ℝ} (hS : MeasurableSet S) (hSae : S =ᵐ[volume] Icc (0 : ℝ) (Real.pi / 2))
+    (hSsub : S ⊆ Icc (0 : ℝ) (Real.pi / 2)) (hA : MeasurableSet (φ '' S))
+    (hsec : ∀ t ∈ S, ψ (φ t) = t)
+    (f : ℝ → ℝ) (hf : AEStronglyMeasurable f (volume.restrict (Icc (0 : ℝ) (Real.pi / 2))))
+    (hf_nonneg : ∀ t ∈ Icc (0 : ℝ) (Real.pi / 2), 0 ≤ f t)
+    {M : ℝ} (hf_bound : ∀ t ∈ Icc (0 : ℝ) (Real.pi / 2), f t ≤ M)
+    (hdom : ∀ E : Set ℝ, MeasurableSet E → E ⊆ S →
+      μ₀ (φ '' E) ≤ ENNReal.ofReal (∫ t in E, f t)) :
+    ∃ r : ℝ → ℝ≥0, Measurable r ∧
+      μ₀.restrict (φ '' S) =
+        Measure.map φ ((volume.restrict S).withDensity fun t ↦ (r t : ℝ≥0∞)) := by
+  classical
+  set μ : Measure ℝ := Measure.map ψ (μ₀.restrict (φ '' S)) with hμ
+  have hμ_apply : ∀ E : Set ℝ, MeasurableSet E → μ E = μ₀ (φ '' (E ∩ S)) := fun E hE ↦ by
+    rw [hμ, Measure.map_apply hψ hE, Measure.restrict_apply (hψ hE),
+      preimage_inter_image_eq_image_inter hsec E]
+  have : IsFiniteMeasure μ := by
+    refine ⟨?_⟩
+    rw [hμ_apply univ MeasurableSet.univ]
+    exact measure_lt_top _ _
+  have hμ_compl : μ (Icc (0 : ℝ) (Real.pi / 2))ᶜ = 0 := by
+    rw [hμ_apply _ measurableSet_Icc.compl,
+      show (Icc (0 : ℝ) (Real.pi / 2))ᶜ ∩ S = ∅ from
+        eq_empty_of_subset_empty fun x hx ↦ hx.1 (hSsub hx.2),
+      Set.image_empty, measure_empty]
+  have hdom' : ∀ E : Set ℝ, MeasurableSet E → E ⊆ Icc (0 : ℝ) (Real.pi / 2) →
+      μ E ≤ ENNReal.ofReal (∫ t in E, f t) := by
+    intro E hE hEsub
+    have hES : E ∩ S =ᵐ[volume] E := by
+      have h := Filter.EventuallyEqSet.inter (Filter.EventuallyEq.refl _ E) hSae
+      rwa [inter_eq_left.mpr hEsub] at h
+    rw [hμ_apply E hE]
+    exact (hdom _ (hE.inter hS) inter_subset_right).trans
+      (le_of_eq (congrArg ENNReal.ofReal (setIntegral_congr_set hES)))
+  obtain ⟨r, hr, hreq⟩ :=
+    exists_nnreal_density_of_domination μ hμ_compl f hf hf_nonneg hf_bound hdom'
+  refine ⟨r, hr, ?_⟩
+  have hmap : Measure.map φ μ = μ₀.restrict (φ '' S) := by
+    rw [hμ]
+    refine Measure.map_map_of_ae_leftInverse hψ hφ ?_
+    filter_upwards [ae_restrict_mem hA] with a ha
+    obtain ⟨t, ht, rfl⟩ := ha
+    rw [hsec t ht]
+  rw [← hmap, hreq, ← Measure.restrict_congr_set hSae]
+
+/-! ### The density on the first arc -/
+
+private theorem exists_capDensity_right (K : RightAngleCapSpace)
+    (hK : IsBalancedMaximumCap K) :
+    ∃ r : ℝ → ℝ≥0, Measurable r ∧
+      (surfaceAreaMeasure K.1).restrict
+          ((fun t : ℝ ↦ (t : Real.Angle)) '' Ico 0 (Real.pi / 2)) =
+        Measure.map (fun t : ℝ ↦ (t : Real.Angle))
+          ((volume.restrict (Ico 0 (Real.pi / 2))).withDensity
+            (fun t ↦ (r t : ℝ≥0∞))) := by
+  have hpi := Real.pi_pos
+  have _ : IsFiniteMeasure (surfaceAreaMeasure K.val) :=
+    (surfaceAreaMeasure_face_union K.val).1
+  have hmeas : AEStronglyMeasurable (fun t ↦ magicDensity (tangentArmLengths K t).2.1)
+      (volume.restrict (Icc (0 : ℝ) (Real.pi / 2))) := by
+    rw [← Measure.restrict_congr_set Ioc_ae_eq_Icc]
+    exact aestronglyMeasurable_magicDensity_tangentArm_fst K
+  refine exists_arcDensity_of_domination (surfaceAreaMeasure K.val)
+    Real.Angle.continuous_coe.measurable continuous_angleSection.measurable
+    measurableSet_Ico Ico_ae_eq_Icc Ico_subset_Icc_self ?_
+    (fun t ht ↦ angleSection_coe ⟨by linarith [ht.1], ht.2.le⟩)
+    (fun t ↦ magicDensity (tangentArmLengths K t).2.1) hmeas
+    (fun t _ ↦ magicDensity_nonneg _)
+    (M := (surfaceAreaMeasure K.val).real univ + 1)
+    (fun t _ ↦ (magicDensity_le_abs_add_one _).trans
+      (by linarith [abs_tangentArm_fst_le K (t := t)]))
+    fun E hE hEsub ↦ balancedMaximumCap_surface_domination K hK E hE hEsub
+  exact Real.Angle.measurableSet_image_of_subset_Ioc (a := -Real.pi) (b := Real.pi)
+    (by linarith) measurableSet_Ico fun t ht ↦ ⟨by linarith [ht.1], by linarith [ht.2]⟩
+
+/-- On the first quarter arc the surface area measure of a balanced maximum cap is the integral
+of a real density on the rotation interval which is bounded by the magic density of the positive
+tangent arm length. This refines `exists_capDensity_right`, which only records the existence of a
+density, by the pointwise bound carried by the limiting inequality
+`balancedMaximumCap_surface_domination`. -/
+theorem exists_capDensity_right_le_magicDensity (K : RightAngleCapSpace)
+    (hK : IsBalancedMaximumCap K) :
+    ∃ ρ : ℝ → ℝ, IntegrableOn ρ (Icc 0 (Real.pi / 2)) volume ∧
+      (∀ᵐ t ∂volume.restrict (Icc 0 (Real.pi / 2)),
+        0 ≤ ρ t ∧ ρ t ≤ magicDensity (tangentArmLengths K t).2.1) ∧
+      ∀ E : Set ℝ, MeasurableSet E → E ⊆ Ico 0 (Real.pi / 2) →
+        (surfaceAreaMeasure K.val ((fun t : ℝ ↦ (t : Real.Angle)) '' E)).toReal =
+          ∫ t in E, ρ t := by
+  have hpi := Real.pi_pos
+  have _ : IsFiniteMeasure (surfaceAreaMeasure K.val) :=
+    (surfaceAreaMeasure_face_union K.val).1
+  have hsec : ∀ t ∈ Ico (0 : ℝ) (Real.pi / 2), angleSection (t : Real.Angle) = t :=
+    fun t ht ↦ angleSection_coe ⟨by linarith [ht.1], ht.2.le⟩
+  set μ : Measure ℝ := Measure.map angleSection ((surfaceAreaMeasure K.val).restrict
+    ((fun t : ℝ ↦ (t : Real.Angle)) '' Ico 0 (Real.pi / 2))) with hμdef
+  have hμ_apply : ∀ E : Set ℝ, MeasurableSet E →
+      μ E = surfaceAreaMeasure K.val
+        ((fun t : ℝ ↦ (t : Real.Angle)) '' (E ∩ Ico 0 (Real.pi / 2))) := fun E hE ↦ by
+    rw [hμdef, Measure.map_apply continuous_angleSection.measurable hE,
+      Measure.restrict_apply (continuous_angleSection.measurable hE),
+      preimage_inter_image_eq_image_inter hsec E]
+  have _ : IsFiniteMeasure μ := by
+    refine ⟨?_⟩
+    rw [hμ_apply univ MeasurableSet.univ]
+    exact measure_lt_top _ _
+  have hμ_compl : μ (Icc (0 : ℝ) (Real.pi / 2))ᶜ = 0 := by
+    rw [hμ_apply _ measurableSet_Icc.compl,
+      show (Icc (0 : ℝ) (Real.pi / 2))ᶜ ∩ Ico 0 (Real.pi / 2) = ∅ from
+        eq_empty_of_subset_empty fun x hx ↦ hx.1 (Ico_subset_Icc_self hx.2),
+      Set.image_empty, measure_empty]
+  have hmeas : AEStronglyMeasurable (fun t ↦ magicDensity (tangentArmLengths K t).2.1)
+      (volume.restrict (Icc (0 : ℝ) (Real.pi / 2))) := by
+    rw [← Measure.restrict_congr_set Ioc_ae_eq_Icc]
+    exact aestronglyMeasurable_magicDensity_tangentArm_fst K
+  have hdom : ∀ E : Set ℝ, MeasurableSet E → E ⊆ Icc 0 (Real.pi / 2) →
+      μ E ≤ ENNReal.ofReal (∫ t in E, magicDensity (tangentArmLengths K t).2.1) := by
+    intro E hE hEsub
+    have hES : E ∩ Ico (0 : ℝ) (Real.pi / 2) =ᵐ[volume] E := by
+      have h := Filter.EventuallyEqSet.inter (Filter.EventuallyEq.refl _ E)
+        (Ico_ae_eq_Icc (μ := volume) (a := (0 : ℝ)) (b := Real.pi / 2))
+      rwa [inter_eq_left.mpr hEsub] at h
+    rw [hμ_apply E hE]
+    exact (balancedMaximumCap_surface_domination K hK _ (hE.inter measurableSet_Ico)
+      inter_subset_right).trans
+      (le_of_eq (congrArg ENNReal.ofReal (setIntegral_congr_set hES)))
+  obtain ⟨ρ, -, hρint, hρle, hρeq⟩ := exists_density_le_of_domination μ hμ_compl
+    (fun t ↦ magicDensity (tangentArmLengths K t).2.1) hmeas (fun t _ ↦ magicDensity_nonneg _)
+    (M := (surfaceAreaMeasure K.val).real univ + 1)
+    (fun t _ ↦ (magicDensity_le_abs_add_one _).trans
+      (by linarith [abs_tangentArm_fst_le K (t := t)])) hdom
+  refine ⟨ρ, hρint, hρle, fun E hE hEsub ↦ ?_⟩
+  have hEIcc : E ⊆ Icc (0 : ℝ) (Real.pi / 2) := hEsub.trans Ico_subset_Icc_self
+  have hEint : Integrable ρ (volume.restrict E) := hρint.mono_set hEIcc
+  have hEnn : 0 ≤ᵐ[volume.restrict E] ρ :=
+    (ae_restrict_of_ae_restrict_of_subset hEIcc hρle).mono fun t h ↦ h.1
+  have hEμ : μ E = surfaceAreaMeasure K.val ((fun t : ℝ ↦ (t : Real.Angle)) '' E) := by
+    rw [hμ_apply E hE, inter_eq_left.mpr hEsub]
+  rw [← hEμ, hρeq, withDensity_apply _ hE, Measure.restrict_restrict_of_subset hEIcc,
+    ← ofReal_integral_eq_lintegral_ofReal hEint hEnn]
+  exact ENNReal.toReal_ofReal (integral_nonneg_of_ae hEnn)
+
+/-! ### The mirrored cap and the density on the second arc -/
+
+/-- The mirror image of a balanced maximum right-angle cap is again a balanced maximum cap, and
+the surface area measure of the cap is obtained from that of its mirror image by reflecting
+normal angles through `a ↦ π - a`. -/
+private theorem exists_mirror_balancedMaximumCap (K : RightAngleCapSpace)
+    (hK : IsBalancedMaximumCap K) :
+    ∃ P : RightAngleCapSpace, IsBalancedMaximumCap P ∧
+      ∀ E : Set Real.Angle, MeasurableSet E →
+        surfaceAreaMeasure K.val E =
+          surfaceAreaMeasure P.val
+            ((fun a : Real.Angle ↦ ((Real.pi : ℝ) : Real.Angle) - a) '' E) := by
+  obtain ⟨P, hPset, hPbal⟩ := balancedMaximumCap_mirror K hK
+  obtain ⟨Q, hQset, -, -, -, -, hQmeas⟩ := cap_mirror_features K
+  have hQP : Q = P := Subtype.ext (ConvexBody.ext (by rw [hQset, hPset]))
+  have hF : Function.Involutive (fun a : Real.Angle ↦ ((Real.pi : ℝ) : Real.Angle) - a) :=
+    fun a ↦ sub_sub_cancel _ _
+  refine ⟨P, hPbal, fun E hE ↦ ?_⟩
+  have hFEmeas : MeasurableSet
+      ((fun a : Real.Angle ↦ ((Real.pi : ℝ) : Real.Angle) - a) '' E) := by
+    rw [Set.image_eq_preimage_of_inverse hF hF]
+    exact (continuous_const.sub continuous_id).measurable hE
+  have himg : (fun a : Real.Angle ↦ ((Real.pi / 2 + Real.pi / 2 : ℝ) : Real.Angle) - a) ''
+      ((fun a : Real.Angle ↦ ((Real.pi : ℝ) : Real.Angle) - a) '' E) = E := by
+    rw [show ((Real.pi / 2 + Real.pi / 2 : ℝ) : Real.Angle) = ((Real.pi : ℝ) : Real.Angle) from
+      by rw [add_halves], Set.image_image, show (fun a : Real.Angle ↦
+        ((Real.pi : ℝ) : Real.Angle) - (((Real.pi : ℝ) : Real.Angle) - a)) = fun a ↦ a from
+      funext hF, Set.image_id']
+  have hkey := hQmeas _ hFEmeas
+  rw [hQP, himg] at hkey
+  exact hkey.symm
+
+private theorem exists_capDensity_left (K : RightAngleCapSpace)
+    (hK : IsBalancedMaximumCap K) :
+    ∃ s : ℝ → ℝ≥0, Measurable s ∧
+      (surfaceAreaMeasure K.1).restrict
+          ((fun t : ℝ ↦ (t : Real.Angle)) '' Ioc (Real.pi / 2) Real.pi) =
+        Measure.map (fun t : ℝ ↦ ((t + Real.pi / 2 : ℝ) : Real.Angle))
+          ((volume.restrict (Ioc 0 (Real.pi / 2))).withDensity
+            (fun t ↦ (s t : ℝ≥0∞))) := by
+  obtain ⟨P, hPbal, hPmeas⟩ := exists_mirror_balancedMaximumCap K hK
+  have hpi := Real.pi_pos
+  have _ : IsFiniteMeasure (surfaceAreaMeasure K.val) :=
+    (surfaceAreaMeasure_face_union K.val).1
+  have hrefl : Function.Involutive (fun t : ℝ ↦ Real.pi / 2 - t) := fun t ↦ sub_sub_cancel _ _
+  have hmp : MeasurePreserving (fun t : ℝ ↦ Real.pi / 2 - t)
+      (volume.restrict (Icc (0 : ℝ) (Real.pi / 2)))
+      (volume.restrict (Icc (0 : ℝ) (Real.pi / 2))) := by
+    have h := (Measure.measurePreserving_sub_left volume (Real.pi / 2)).restrict_preimage
+      (measurableSet_Icc (a := (0 : ℝ)) (b := Real.pi / 2))
+    have hpre : (fun t : ℝ ↦ Real.pi / 2 - t) ⁻¹' Icc (0 : ℝ) (Real.pi / 2) =
+        Icc (0 : ℝ) (Real.pi / 2) := by
+      ext x
+      simp only [mem_preimage, mem_Icc]
+      constructor <;> intro hx <;> exact ⟨by linarith [hx.2], by linarith [hx.1]⟩
+    rwa [hpre] at h
+  have hmeas : AEStronglyMeasurable (fun u ↦ magicDensity (tangentArmLengths P u).2.1)
+      (volume.restrict (Icc (0 : ℝ) (Real.pi / 2))) := by
+    rw [← Measure.restrict_congr_set Ioc_ae_eq_Icc]
+    exact aestronglyMeasurable_magicDensity_tangentArm_fst P
+  have hbound : ∀ t ∈ Icc (0 : ℝ) (Real.pi / 2),
+      magicDensity (tangentArmLengths P (Real.pi / 2 - t)).2.1 ≤
+        (surfaceAreaMeasure P.val).real univ + 1 := by
+    intro t _
+    refine (magicDensity_le_abs_add_one _).trans ?_
+    have harm := abs_tangentArm_fst_le P (t := Real.pi / 2 - t)
+    linarith
+  rw [← image_coe_add_pi_div_two_Ioc]
+  refine exists_arcDensity_of_domination (surfaceAreaMeasure K.val)
+    (φ := fun t : ℝ ↦ ((t + Real.pi / 2 : ℝ) : Real.Angle))
+    (ψ := fun a : Real.Angle ↦ angleSection (a - ((Real.pi / 2 : ℝ) : Real.Angle)))
+    (Real.Angle.continuous_coe.comp (continuous_id.add continuous_const)).measurable
+    (continuous_angleSection.comp (continuous_id.sub continuous_const)).measurable
+    measurableSet_Ioc Ioc_ae_eq_Icc Ioc_subset_Icc_self ?_
+    (fun t ht ↦ angleSection_sub_coe ⟨by linarith [ht.1], ht.2⟩)
+    (fun t ↦ magicDensity (tangentArmLengths P (Real.pi / 2 - t)).2.1)
+    (hmeas.comp_measurePreserving hmp) (fun t _ ↦ magicDensity_nonneg _) hbound ?_
+  · rw [image_coe_add_pi_div_two_Ioc]
+    exact Real.Angle.measurableSet_image_of_subset_Ioc (a := -Real.pi) (b := Real.pi)
+      (by linarith) measurableSet_Ioc fun t ht ↦ ⟨by linarith [ht.1], ht.2⟩
+  · intro E hE hEsub
+    have hgmeas : MeasurableSet ((fun t : ℝ ↦ Real.pi / 2 - t) '' E) := by
+      rw [Set.image_eq_preimage_of_inverse hrefl hrefl]
+      exact (measurable_const.sub measurable_id) hE
+    have hgsub : (fun t : ℝ ↦ Real.pi / 2 - t) '' E ⊆ Ico 0 (Real.pi / 2) := by
+      rintro u ⟨t, ht, rfl⟩
+      exact ⟨by linarith [(hEsub ht).2], by linarith [(hEsub ht).1]⟩
+    have hEarc : MeasurableSet
+        ((fun t : ℝ ↦ ((t + Real.pi / 2 : ℝ) : Real.Angle)) '' E) := by
+      rw [show (fun t : ℝ ↦ ((t + Real.pi / 2 : ℝ) : Real.Angle)) =
+          (fun u : ℝ ↦ (u : Real.Angle)) ∘ (fun t : ℝ ↦ t + Real.pi / 2) from rfl,
+        Set.image_comp]
+      refine Real.Angle.measurableSet_image_of_subset_Ioc (a := -Real.pi) (b := Real.pi)
+        (by linarith) ?_ ?_
+      · rw [Set.image_add_right]
+        exact (measurable_id.add_const _) hE
+      · rintro u ⟨t, ht, rfl⟩
+        exact ⟨by linarith [(hEsub ht).1], by linarith [(hEsub ht).2]⟩
+    have hmirror : (fun a : Real.Angle ↦ ((Real.pi : ℝ) : Real.Angle) - a) ''
+        ((fun t : ℝ ↦ ((t + Real.pi / 2 : ℝ) : Real.Angle)) '' E) =
+        (fun u : ℝ ↦ (u : Real.Angle)) '' ((fun t : ℝ ↦ Real.pi / 2 - t) '' E) := by
+      rw [Set.image_image, Set.image_image]
+      refine Set.image_congr' fun t ↦ ?_
+      show ((Real.pi : ℝ) : Real.Angle) - ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+        ((Real.pi / 2 - t : ℝ) : Real.Angle)
+      rw [← Real.Angle.coe_sub]
+      congr 1
+      ring
+    calc surfaceAreaMeasure K.val
+          ((fun t : ℝ ↦ ((t + Real.pi / 2 : ℝ) : Real.Angle)) '' E)
+        = surfaceAreaMeasure P.val ((fun u : ℝ ↦ (u : Real.Angle)) ''
+            ((fun t : ℝ ↦ Real.pi / 2 - t) '' E)) := by rw [hPmeas _ hEarc, hmirror]
+      _ ≤ ENNReal.ofReal (∫ t in (fun t : ℝ ↦ Real.pi / 2 - t) '' E,
+            magicDensity (tangentArmLengths P t).2.1) :=
+          balancedMaximumCap_surface_domination P hPbal _ hgmeas hgsub
+      _ = ENNReal.ofReal
+            (∫ t in E, magicDensity (tangentArmLengths P (Real.pi / 2 - t)).2.1) :=
+          congrArg ENNReal.ofReal
+            ((Measure.measurePreserving_sub_left volume (Real.pi / 2)).setIntegral_image_emb
+              (measurableEmbedding_subLeft _) _ E)
+
+/-! ### Uniqueness of the densities -/
+
+/-- Two measurable weights on `S` whose pushforwards along `φ` agree are almost everywhere
+equal, provided `φ` has a measurable left inverse on `S`. -/
+private theorem ae_eq_of_map_withDensity_eq {S : Set ℝ} (hS : MeasurableSet S)
+    {φ : ℝ → Real.Angle} (hφ : Measurable φ) {ψ : Real.Angle → ℝ} (hψ : Measurable ψ)
+    (hsec : ∀ t ∈ S, ψ (φ t) = t) {w₁ w₂ : ℝ → ℝ≥0}
+    (h₁ : Measurable w₁) (h₂ : Measurable w₂)
+    (h : Measure.map φ ((volume.restrict S).withDensity (fun t ↦ (w₁ t : ℝ≥0∞))) =
+      Measure.map φ ((volume.restrict S).withDensity (fun t ↦ (w₂ t : ℝ≥0∞)))) :
+    w₁ =ᵐ[volume.restrict S] w₂ := by
+  have key : ∀ w : ℝ → ℝ≥0,
+      Measure.map ψ (Measure.map φ ((volume.restrict S).withDensity
+        (fun t ↦ (w t : ℝ≥0∞)))) =
+      (volume.restrict S).withDensity (fun t ↦ (w t : ℝ≥0∞)) := by
+    intro w
+    refine Measure.map_map_of_ae_leftInverse hφ hψ ?_
+    filter_upwards [(withDensity_absolutelyContinuous (volume.restrict S)
+      (fun t ↦ (w t : ℝ≥0∞))).ae_le (ae_restrict_mem hS)] with t ht
+    exact hsec t ht
+  have hν : (volume.restrict S).withDensity (fun t ↦ (w₁ t : ℝ≥0∞)) =
+      (volume.restrict S).withDensity (fun t ↦ (w₂ t : ℝ≥0∞)) := by
+    rw [← key w₁, ← key w₂, h]
+  filter_upwards [(withDensity_eq_iff_of_sigmaFinite
+    h₁.coe_nnreal_ennreal.aemeasurable h₂.coe_nnreal_ennreal.aemeasurable).mp hν] with t ht
+  exact ENNReal.coe_inj.mp ht
+
+/-- A right-angle cap determines its two surface densities up to Lebesgue-null sets: any two
+pairs of densities of the same cap agree almost everywhere on `[0, π / 2]`. -/
+theorem HasCapDensities.ae_eq {K : RightAngleCapSpace} {r s r' s' : ℝ → ℝ≥0}
+    (h : HasCapDensities K r s) (h' : HasCapDensities K r' s') :
+    (r =ᵐ[volume.restrict (Set.Icc 0 (Real.pi / 2))] r') ∧
+    (s =ᵐ[volume.restrict (Set.Icc 0 (Real.pi / 2))] s') := by
+  have hpi := Real.pi_pos
+  refine ⟨?_, ?_⟩
+  · rw [← Measure.restrict_congr_set (Ico_ae_eq_Icc (a := (0 : ℝ)) (b := Real.pi / 2))]
+    exact ae_eq_of_map_withDensity_eq measurableSet_Ico
+      Real.Angle.continuous_coe.measurable continuous_angleSection.measurable
+      (fun t ht ↦ angleSection_coe ⟨by linarith [ht.1], ht.2.le⟩)
+      h.1 h'.1 (h.2.2.1.symm.trans h'.2.2.1)
+  · rw [← Measure.restrict_congr_set (Ioc_ae_eq_Icc (a := (0 : ℝ)) (b := Real.pi / 2))]
+    exact ae_eq_of_map_withDensity_eq measurableSet_Ioc
+      (φ := fun t : ℝ ↦ ((t + Real.pi / 2 : ℝ) : Real.Angle))
+      (ψ := fun a : Real.Angle ↦ angleSection (a - ((Real.pi / 2 : ℝ) : Real.Angle)))
+      (Real.Angle.continuous_coe.comp (continuous_id.add continuous_const)).measurable
+      (continuous_angleSection.comp (continuous_id.sub continuous_const)).measurable
+      (fun t ht ↦ angleSection_sub_coe ⟨by linarith [ht.1], ht.2⟩)
+      h.2.1 h'.2.1 (h.2.2.2.symm.trans h'.2.2.2)
+
+theorem balancedMaximumCap_hasDensities (K : RightAngleCapSpace)
+    (hK : IsBalancedMaximumCap K) :
+    ∃ r s : ℝ → NNReal, HasCapDensities K r s ∧
+      ∀ r' s', HasCapDensities K r' s' →
+        (r =ᵐ[volume.restrict (Set.Icc 0 (Real.pi / 2))] r') ∧
+        (s =ᵐ[volume.restrict (Set.Icc 0 (Real.pi / 2))] s') := by
+  obtain ⟨r, hrmeas, hr⟩ := exists_capDensity_right K hK
+  obtain ⟨s, hsmeas, hs⟩ := exists_capDensity_left K hK
+  have h : HasCapDensities K r s := ⟨hrmeas, hsmeas, hr, hs⟩
+  exact ⟨r, s, h, fun _ _ h' ↦ h.ae_eq h'⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Minkowski interpolation of right-angle caps
+
+The cap conditions of `IsCap (π / 2)` and the injectivity condition of
+`SatisfiesInjectivityCondition` are preserved by the Minkowski interpolation
+`convexBodyCombination t K L = (1 - t) • K + t • L` of two right-angle caps.  Each statement
+about the interpolation is phrased for an arbitrary cap whose underlying convex body is that
+interpolation, so that it applies both to the cap produced by `isCap_convexBodyCombination`
+and to an interpolated cap obtained by choice.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+open scoped NNReal ENNReal Pointwise unitInterval
+
+namespace MovingSofa
+
+/-- Lowering a point of an interpolated right-angle cap onto the base line keeps it inside,
+because the same projection can be applied to both summands. -/
+theorem base_projection_mem_convexBodyCombination (t : I) (K L : CapSpace (Real.pi / 2))
+    {q : Point} (hq : q ∈ (convexBodyCombination t K.val L.val : Set Point)) :
+    q - q 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle) ∈
+      (convexBodyCombination t K.val L.val : Set Point) := by
+  obtain ⟨a, ha, b, hb, rfl⟩ := (mem_convexBodyCombination_iff t K.val L.val q).1 hq
+  refine (mem_convexBodyCombination_iff t K.val L.val _).2
+    ⟨_, K.base_projection_mem ha, _, L.base_projection_mem hb, ?_⟩
+  have hcoord : ((1 - (t : ℝ)) • a + (t : ℝ) • b) 1 =
+      (1 - (t : ℝ)) * a 1 + (t : ℝ) * b 1 := by
+    simp only [PiLp.add_apply, PiLp.smul_apply, smul_eq_mul]
+  rw [hcoord]
+  module
+
+/-- Right-angle caps are closed under Minkowski interpolation. -/
+theorem isCap_convexBodyCombination (t : I) (K L : RightAngleCapSpace) :
+    IsCap (Real.pi / 2) (convexBodyCombination t K.val L.val) := by
+  have hsup := supportValue_convexBodyCombination t K.val L.val
+  have hbase : supportValue (convexBodyCombination t K.val L.val)
+      ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := by
+    rw [hsup, K.property.2.2.2.2.2.1, L.property.2.2.2.2.2.1]; ring
+  refine ⟨by positivity, le_rfl, ?_, ?_, ?_, hbase,
+    hasHalfPlaneRepresentation_of_base_projection _ hbase
+      (fun q hq ↦ base_projection_mem_convexBodyCombination t K L hq)⟩
+  · rw [hsup, K.property.2.2.1, L.property.2.2.1]; ring
+  · rw [hsup, K.property.2.2.2.1, L.property.2.2.2.1]; ring
+  · rw [hsup, K.property.2.2.2.2.1, L.property.2.2.2.2.1]; ring
+
+/-- An interpolated right-angle cap carries the interpolated surface densities: its surface
+measure is the interpolation of the two surface measures, and pushing a weighted measure
+forward is linear in the weight. -/
+theorem hasCapDensities_of_eq_convexBodyCombination {t : I} {K L M : RightAngleCapSpace}
+    (hM : (M.val : ConvexBody Point) = convexBodyCombination t K.val L.val)
+    {r₁ s₁ r₂ s₂ : ℝ → ℝ≥0} (hK : HasCapDensities K r₁ s₁) (hL : HasCapDensities L r₂ s₂) :
+    HasCapDensities M (fun x ↦ (1 - (t : ℝ)).toNNReal * r₁ x + (t : ℝ).toNNReal * r₂ x)
+      (fun x ↦ (1 - (t : ℝ)).toNNReal * s₁ x + (t : ℝ).toNNReal * s₂ x) := by
+  have hmeasure : surfaceAreaMeasure M.val =
+      ENNReal.ofReal (1 - (t : ℝ)) • surfaceAreaMeasure K.val +
+        ENNReal.ofReal (t : ℝ) • surfaceAreaMeasure L.val := by
+    rw [hM]; exact surfaceAreaMeasure_convexBodyCombination t K.val L.val
+  refine ⟨(measurable_const.mul hK.1).add (measurable_const.mul hL.1),
+    (measurable_const.mul hK.2.1).add (measurable_const.mul hL.2.1), ?_, ?_⟩
+  · rw [hmeasure, Measure.restrict_add, Measure.restrict_smul, Measure.restrict_smul,
+      hK.2.2.1, hL.2.2.1]
+    exact (Measure.map_withDensity_eq_smul_add_smul _ Real.Angle.continuous_coe.measurable _ _
+      hK.1.coe_nnreal_ennreal hL.1.coe_nnreal_ennreal (fun x ↦ by simp [ENNReal.ofReal])).symm
+  · rw [hmeasure, Measure.restrict_add, Measure.restrict_smul, Measure.restrict_smul,
+      hK.2.2.2, hL.2.2.2]
+    exact (Measure.map_withDensity_eq_smul_add_smul _
+      (Real.Angle.continuous_coe.comp (continuous_id.add continuous_const)).measurable _ _
+      hK.2.1.coe_nnreal_ennreal hL.2.1.coe_nnreal_ennreal
+      (fun x ↦ by simp [ENNReal.ofReal])).symm
+
+/-- The inner corner of an interpolated cap is the interpolation of the two inner corners. -/
+theorem capInnerCorner_of_eq_convexBodyCombination {t : I} {K L M : RightAngleCapSpace}
+    (hM : (M.val : ConvexBody Point) = convexBodyCombination t K.val L.val) (x : ℝ) :
+    capInnerCorner M x =
+      (1 - (t : ℝ)) • capInnerCorner K x + (t : ℝ) • capInnerCorner L x := by
+  simp only [capInnerCorner, (rotatingHallwayParts_formulas _ _).2.1, hM,
+    supportValue_convexBodyCombination]
+  module
+
+/-- The injectivity condition is inherited by Minkowski interpolations of right-angle caps: the
+inner corner interpolates, so its two frame velocity components are the same combinations of
+the original ones, and a combination with nonnegative weights summing to one preserves their
+strict signs even at the two degenerate weights. -/
+theorem satisfiesInjectivityCondition_of_eq_convexBodyCombination {t : I}
+    {K L M : RightAngleCapSpace}
+    (hM : (M.val : ConvexBody Point) = convexBodyCombination t K.val L.val)
+    (hK : SatisfiesInjectivityCondition K) (hL : SatisfiesInjectivityCondition L) :
+    SatisfiesInjectivityCondition M := by
+  obtain ⟨⟨r₁, s₁, hd₁, -⟩, hC₁, hsign₁⟩ := hK
+  obtain ⟨⟨r₂, s₂, hd₂, -⟩, hC₂, hsign₂⟩ := hL
+  have hcornerfun : capInnerCorner M =
+      fun x ↦ (1 - (t : ℝ)) • capInnerCorner K x + (t : ℝ) • capInnerCorner L x :=
+    funext (capInnerCorner_of_eq_convexBodyCombination hM)
+  have hpi : (0 : ℝ) < Real.pi / 2 := by positivity
+  have ht0 : (0 : ℝ) ≤ (t : ℝ) := t.2.1
+  have ht1 : (0 : ℝ) ≤ 1 - (t : ℝ) := sub_nonneg.mpr t.2.2
+  -- A combination of two negative quantities with weights summing to one is negative.
+  have hneg : ∀ a b : ℝ, a < 0 → b < 0 → (1 - (t : ℝ)) * a + (t : ℝ) * b < 0 := by
+    intro a b ha hb
+    rcases eq_or_lt_of_le ht0 with h | h
+    · rw [← h]; simpa using ha
+    · nlinarith [mul_nonpos_of_nonneg_of_nonpos ht1 ha.le, mul_neg_of_pos_of_neg h hb]
+  have hpos : ∀ a b : ℝ, 0 < a → 0 < b → 0 < (1 - (t : ℝ)) * a + (t : ℝ) * b := by
+    intro a b ha hb
+    have := hneg (-a) (-b) (by linarith) (by linarith)
+    linarith
+  -- Density uniqueness needs no proof here: it holds for every right-angle cap.
+  have hdens := hasCapDensities_of_eq_convexBodyCombination hM hd₁ hd₂
+  refine ⟨⟨_, _, hdens, fun _ _ h' ↦ hdens.ae_eq h'⟩, ?_, ?_⟩
+  · rw [hcornerfun]
+    exact (hC₁.const_smul _).add (hC₂.const_smul _)
+  · intro x hx
+    have hxmem : x ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨hx.1.le, hx.2.le⟩
+    have hdK := ((hC₁.differentiableOn one_ne_zero) x hxmem).hasDerivWithinAt
+    have hdL := ((hC₂.differentiableOn one_ne_zero) x hxmem).hasDerivWithinAt
+    have hdM : HasDerivWithinAt (capInnerCorner M)
+        ((1 - (t : ℝ)) • derivWithin (capInnerCorner K) (Set.Icc 0 (Real.pi / 2)) x +
+          (t : ℝ) • derivWithin (capInnerCorner L) (Set.Icc 0 (Real.pi / 2)) x)
+        (Set.Icc 0 (Real.pi / 2)) x := by
+      rw [hcornerfun]
+      exact (hdK.const_smul (1 - (t : ℝ))).add (hdL.const_smul (t : ℝ))
+    rw [hdM.derivWithin (uniqueDiffOn_Icc hpi x hxmem), inner_add_left, inner_add_left,
+      real_inner_smul_left, real_inner_smul_left, real_inner_smul_left, real_inner_smul_left]
+    exact ⟨hneg _ _ (hsign₁ x hx).1 (hsign₂ x hx).1,
+      hpos _ _ (hsign₁ x hx).2 (hsign₂ x hx).2⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Bounds.Arm.Regularity`.
+* `Bounds.Lower.Sequence`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Regularity of the arm length function of a balanced maximum cap
+
+For a balanced maximum cap `K` of rotation angle `π / 2` the arm length function `f_K` is
+absolutely continuous on `[0, π / 2]` and satisfies `f_K' ≥ m₀ ∘ g_K` almost everywhere on
+`(0, π / 2)`.
+
+The argument combines three inputs. The differentiation identity
+`positiveArm_stieltjes_surface` for the positive arm length, read on an interval `(0, t]` with
+`t < π / 2`, expresses `f_K(t) - f_K(0)` as the integral of `g_K` minus the surface measure of the
+arc traversed. The surface measure on that arc has a real density bounded by `k₀ ∘ g_K`, by
+`exists_capDensity_right_le_magicDensity`. Together they present `f_K` on `[0, π / 2)` as the
+primitive of `w = g_K - ρ`; continuity of `f_K` on the closed interval
+(`nondegenerateCap_continuity`) upgrades the representation to `[0, π / 2]`, whence absolute
+continuity and `f_K' = w ≥ g_K - k₀ ∘ g_K = m₀ ∘ g_K` almost everywhere.
+
+The module also records the pointwise data accompanying that differential inequality: both arm
+length functions of a nondegenerate cap are nonnegative, and the right one has initial value
+`f_K(0) = 1`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- A function on the interval subtype `Set.Icc a b`, extended to `ℝ` by zero. -/
+def extendIntervalScalar {a b : ℝ} (f : Set.Icc a b → ℝ) (t : ℝ) : ℝ := by
+  classical
+  exact if h : t ∈ Set.Icc a b then f ⟨t, h⟩ else 0
+
+/-- Inside its interval the zero extension agrees with the original subtype function. -/
+theorem extendIntervalScalar_of_mem {a b : ℝ} (f : Set.Icc a b → ℝ) {t : ℝ}
+    (ht : t ∈ Set.Icc a b) : extendIntervalScalar f t = f ⟨t, ht⟩ := by
+  classical
+  simp only [extendIntervalScalar, dite_eq_left ht]
+
+/-- The right arm length of a nondegenerate right-angle cap at the horizontal normal equals one.
+Lowering the contact there onto the base line stays inside the cap and meets the same supporting
+line, so the infimum of the tangent heights of that edge is at most zero; the contacts coincide,
+so the contact itself has height zero, and the outer corner at that angle has height
+`h_K(π / 2) = 1`. -/
+theorem tangentArmLengths_right_zero_eq_one (K : RightAngleCapSpace)
+    (hD : ∃ r s, HasCapDensities K r s) : (tangentArmLengths K 0).1.1 = 1 := by
+  have hAmem : (edgeVertices K.val ((0 : ℝ) : Real.Angle)).1 ∈
+      exposedEdge K.val ((0 : ℝ) : Real.Angle) := edgeVertices_fst_mem _ _
+  set A : Point := (edgeVertices K.val ((0 : ℝ) : Real.Angle)).1 with hAdef
+  set B : Point := A - A 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle) with hBdef
+  -- the lowered contact lies on the same supporting line and on the base line
+  have hBline : B ∈ exposedEdge K.val ((0 : ℝ) : Real.Angle) := by
+    refine ⟨K.base_projection_mem hAmem.1, ?_⟩
+    change inner ℝ B (normalVector ((0 : ℝ) : Real.Angle)) = supportValue K.val _
+    rw [hBdef, inner_sub_left, real_inner_smul_left, inner_normalVector_normalVector]
+    simp only [sub_zero, Real.cos_pi_div_two, mul_zero, sub_zero]
+    exact hAmem.2
+  have hBtan : inner ℝ B (tangentVector ((0 : ℝ) : Real.Angle)) = 0 := by
+    rw [hBdef, inner_sub_left, real_inner_smul_left, inner_tangentVector_zero,
+      ← normalVector_add_pi_div_two_real 0, inner_normalVector_normalVector]
+    norm_num
+  -- the two contacts coincide, so the edge is a single point of height zero
+  have hcontact := (capDensities_contact_eq K hD).1 0 ⟨le_rfl, by positivity⟩
+  have hAeq : A = (edgeVertices K.val ((0 : ℝ) : Real.Angle)).2 := hcontact.1
+  have hA1le : A 1 ≤ 0 := by
+    have h1 : A 1 = sInf ((fun p ↦ inner ℝ p (tangentVector ((0 : ℝ) : Real.Angle))) ''
+        exposedEdge K.val ((0 : ℝ) : Real.Angle)) := by
+      rw [← inner_tangentVector_zero A, hAeq, inner_edgeVertices_snd_tangent]
+    rw [h1]
+    refine csInf_le ?_ ⟨B, hBline, hBtan⟩
+    exact ((isCompact_exposedEdge K.val _).image
+      (continuous_id.inner continuous_const)).bddBelow
+  have hA1ge : 0 ≤ A 1 := by
+    simpa only [inner_normalVector_pi_div_two] using
+      K.inner_normalVector_pi_div_two_nonneg hAmem.1
+  have hA1 : A 1 = 0 := le_antisymm hA1le hA1ge
+  -- the outer corner at the horizontal normal has unit height
+  have hy : inner ℝ
+      (rotatingHallwayParts (K.val : Set Point) ((0 : ℝ) : Real.Angle)).outerCorner
+      (tangentVector ((0 : ℝ) : Real.Angle)) = 1 := by
+    change inner ℝ (supportingPlacement (K.val : Set Point) ((0 : ℝ) : Real.Angle)
+      hallwayParts.outerCorner) _ = 1
+    rw [inner_supportingPlacement_tangentVector, Real.Angle.coe_zero, zero_add,
+      K.property.2.2.2.1]
+    simp [hallwayParts]
+  simp only [tangentArmLengths, capVertices, inner_sub_left, hy, ← hAdef,
+    inner_tangentVector_zero, hA1, sub_zero]
+
+/-- Both arm length functions of a nondegenerate right-angle cap are nonnegative. -/
+theorem nondegenerateCapData_arm_nonneg (K : RightAngleCapSpace)
+    (hD : ∃ r s, HasCapDensities K r s) (t : Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    0 ≤ (nondegenerateCapData K hD).2.1 t ∧ 0 ≤ (nondegenerateCapData K hD).2.2 t := by
+  refine ⟨?_, tangentArm_fst_nonneg K⟩
+  simp only [nondegenerateCapData]
+  split_ifs
+  · exact (tangentArmLengths_right_nonneg K t).2
+  · exact (tangentArmLengths_right_nonneg K t).1
+
+/-- Absolute continuity and the almost everywhere derivative of a function `G` that agrees on
+`[a, b]` with a function `F` which is a primitive of `w` on the half-open interval `[a, b)`. Only
+continuity of `F` at `b` is needed there, so no limit of the primitive has to be computed. -/
+private theorem absolutelyContinuousOnInterval_and_ae_hasDerivAt_of_eqOn_Ico
+    {a b : ℝ} (hab : a < b) {F G w : ℝ → ℝ} (hF : ContinuousOn F (Set.Icc a b))
+    (hw : IntervalIntegrable w volume a b)
+    (hFw : ∀ t ∈ Set.Ico a b, F t = F a + ∫ y in a..t, w y)
+    (hGF : Set.EqOn G F (Set.Icc a b)) :
+    AbsolutelyContinuousOnInterval G a b ∧
+      ∀ᵐ t ∂volume.restrict (Set.Ioo a b), HasDerivAt G (w t) t := by
+  set Φ : ℝ → ℝ := fun x ↦ F a + ∫ y in a..x, w y
+  have hΦAC : AbsolutelyContinuousOnInterval Φ a b :=
+    ((LipschitzWith.const (F a)).lipschitzOnWith.absolutelyContinuousOnInterval).add
+      (hw.absolutelyContinuousOnInterval_intervalIntegral (c := a) (by simp [hab.le]))
+  have hΦcont : ContinuousOn Φ (Set.Icc a b) := by
+    simpa only [Set.uIcc_of_le hab.le] using hΦAC.continuousOn
+  have heq : Set.EqOn G Φ (Set.Icc a b) := by
+    refine hGF.trans (Set.EqOn.of_subset_closure (fun x hx ↦ hFw x hx) hF hΦcont
+      Set.Ico_subset_Icc_self ?_)
+    rw [closure_Ico hab.ne]
+  refine ⟨hΦAC.congr fun x hx ↦
+    (heq (by simpa only [Set.uIcc_of_le hab.le] using hx)).symm, ?_⟩
+  refine (ae_restrict_iff' measurableSet_Ioo).2 ?_
+  filter_upwards [hw.ae_hasDerivAt_integral] with x hx hxo
+  have hxu : x ∈ Set.uIcc a b := by
+    simpa only [Set.uIcc_of_le hab.le] using ⟨hxo.1.le, hxo.2.le⟩
+  refine ((hx hxu a (by simp [hab.le])).const_add (F a)).congr_of_eventuallyEq ?_
+  filter_upwards [isOpen_Ioo.mem_nhds hxo] with y hyo
+  exact heq ⟨hyo.1.le, hyo.2.le⟩
+
+/-- The differentiation identity for the positive arm length, restricted to an interval `(0, t]`
+with `t < π / 2`: the positive arm length increases by the integral of the positive tangent arm
+length minus the surface measure of the arc of normal directions traversed. Restricting to
+`t < π / 2` keeps the possibly nonzero top-face atom of the surface measure out of the
+identity. -/
+private theorem positiveArm_sub_eq_integral_sub_surface (K : RightAngleCapSpace) {t : ℝ}
+    (ht : t ∈ Set.Ico (0 : ℝ) (Real.pi / 2)) :
+    (tangentArmLengths K t).1.1 - (tangentArmLengths K 0).1.1 =
+      (∫ u in Set.Ioc (0 : ℝ) t, (tangentArmLengths K u).2.1) -
+        (surfaceAreaMeasure K.val
+          ((fun u : ℝ ↦ (u : Real.Angle)) '' Set.Ioc (0 : ℝ) t)).toReal := by
+  have hle : (0 : ℝ) ≤ Real.pi / 2 := by positivity
+  have hmem0 : (0 : ℝ) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨le_rfl, hle⟩
+  have htmem : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨ht.1, ht.2.le⟩
+  obtain ⟨F, hF, hinc⟩ := positiveArm_stieltjes_surface K
+  have hzx : (⟨0, hmem0⟩ : Set.Icc (0 : ℝ) (Real.pi / 2)) ≤ ⟨t, htmem⟩ :=
+    Subtype.mk_le_mk.mpr ht.1
+  have himg : (fun u : Set.Icc (0 : ℝ) (Real.pi / 2) ↦ (u : ℝ)) ''
+      Set.Ioc (⟨0, hmem0⟩ : Set.Icc (0 : ℝ) (Real.pi / 2)) ⟨t, htmem⟩ =
+      Set.Ioc (0 : ℝ) t := by
+    apply Set.eq_of_subset_of_subset
+    · rintro y ⟨u, hu, rfl⟩
+      exact ⟨hu.1, hu.2⟩
+    · intro y hy
+      exact ⟨⟨y, ⟨hy.1.le, hy.2.trans ht.2.le⟩⟩, ⟨hy.1, hy.2⟩, rfl⟩
+  have himg2 : (fun u : Set.Icc (0 : ℝ) (Real.pi / 2) ↦ ((u : ℝ) : Real.Angle)) ''
+      Set.Ioc (⟨0, hmem0⟩ : Set.Icc (0 : ℝ) (Real.pi / 2)) ⟨t, htmem⟩ =
+      (fun y : ℝ ↦ (y : Real.Angle)) '' Set.Ioc (0 : ℝ) t := by
+    rw [← himg, Set.image_image]
+  have hEpos : ∀ y ∈ Set.Ioc (⟨0, hmem0⟩ : Set.Icc (0 : ℝ) (Real.pi / 2)) ⟨t, htmem⟩,
+      0 < (y : ℝ) := fun y hy ↦ hy.1
+  have h := hinc _ measurableSet_Ioc hEpos
+  rw [intervalStieltjesMeasure_Ioc F _ _ hzx, himg, himg2, hF, hF] at h
+  push_cast at h ⊢
+  linarith [h]
+
+theorem balancedMaximumCap_arm_regularity (K : RightAngleCapSpace)
+    (hK : IsBalancedMaximumCap K) :
+    ∃ hD : ∃ r s, HasCapDensities K r s,
+      AbsolutelyContinuousOnInterval
+        (extendIntervalScalar (nondegenerateCapData K hD).2.1) 0 (Real.pi / 2) ∧
+      ∀ᵐ t ∂volume.restrict (Set.Ioo (0 : ℝ) (Real.pi / 2)),
+        magicFunctions.2
+          (Real.toNNReal (extendIntervalScalar (nondegenerateCapData K hD).2.2 t)) ≤
+        deriv (extendIntervalScalar (nondegenerateCapData K hD).2.1) t := by
+  have hTpos : (0 : ℝ) < Real.pi / 2 := by positivity
+  have hle : (0 : ℝ) ≤ Real.pi / 2 := hTpos.le
+  obtain ⟨rd, sd, hrs, -⟩ := balancedMaximumCap_hasDensities K hK
+  have hD : ∃ r s, HasCapDensities K r s := ⟨rd, sd, hrs⟩
+  refine ⟨hD, ?_⟩
+  obtain ⟨-, -, hFcont, hGcont, -, -, -⟩ := nondegenerateCap_continuity K hD
+  -- the continuous representatives of `f_K` and `g_K`, obtained by clamping the parameter
+  set fc : ℝ → ℝ :=
+    fun t ↦ (nondegenerateCapData K hD).2.1 (Set.projIcc 0 (Real.pi / 2) hle t) with hfcdef
+  set gc : ℝ → ℝ :=
+    fun t ↦ (nondegenerateCapData K hD).2.2 (Set.projIcc 0 (Real.pi / 2) hle t) with hgcdef
+  have hfccont : Continuous fc := hFcont.comp continuous_projIcc
+  have hgccont : Continuous gc := hGcont.comp continuous_projIcc
+  have hext_f : Set.EqOn (extendIntervalScalar (nondegenerateCapData K hD).2.1) fc
+      (Set.Icc 0 (Real.pi / 2)) := by
+    intro x hx
+    rw [extendIntervalScalar_of_mem _ hx, hfcdef]
+    simp only [Set.projIcc_of_mem hle hx]
+  have hext_g : ∀ x ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      extendIntervalScalar (nondegenerateCapData K hD).2.2 x = gc x := by
+    intro x hx
+    rw [extendIntervalScalar_of_mem _ hx, hgcdef]
+    simp only [Set.projIcc_of_mem hle hx]
+  -- on `[0, π / 2)` the arm length function is the positive arm length `f⁺_K`
+  have hfcval : ∀ x ∈ Set.Ico (0 : ℝ) (Real.pi / 2), fc x = (tangentArmLengths K x).1.1 := by
+    intro x hx
+    rw [hfcdef]
+    simp only [Set.projIcc_of_mem hle (Set.Ico_subset_Icc_self hx), nondegenerateCapData,
+      ite_eq_right (ne_of_lt hx.2)]
+  have hgcval : ∀ x ∈ Set.Icc (0 : ℝ) (Real.pi / 2), gc x = (tangentArmLengths K x).2.1 := by
+    intro x hx
+    rw [hgcdef]
+    simp only [Set.projIcc_of_mem hle hx]
+    rfl
+  have hgcnonneg : ∀ x, 0 ≤ gc x := fun x ↦ by
+    rw [hgcdef]
+    exact tangentArm_fst_nonneg K
+  -- the density of the surface measure on the first arc, bounded by `k₀ ∘ g_K`
+  obtain ⟨ρ, hρint, hρle, hρeq⟩ := exists_capDensity_right_le_magicDensity K hK
+  set w : ℝ → ℝ := fun t ↦ gc t - ρ t
+  have hwint : IntervalIntegrable w volume 0 (Real.pi / 2) :=
+    (intervalIntegrable_iff_integrableOn_Icc_of_le hle).2
+      ((hgccont.continuousOn.integrableOn_Icc).sub hρint)
+  -- the integral representation of the arm length function on `[0, π / 2)`
+  have hrepr : ∀ t ∈ Set.Ico (0 : ℝ) (Real.pi / 2),
+      fc t = fc 0 + ∫ y in (0 : ℝ)..t, w y := by
+    intro t ht
+    have hIocsub : Set.Ioc (0 : ℝ) t ⊆ Set.Ico 0 (Real.pi / 2) :=
+      fun y hy ↦ ⟨hy.1.le, lt_of_le_of_lt hy.2 ht.2⟩
+    have hIccsub : Set.Ioc (0 : ℝ) t ⊆ Set.Icc 0 (Real.pi / 2) :=
+      hIocsub.trans Set.Ico_subset_Icc_self
+    have hsplit : (∫ y in Set.Ioc (0 : ℝ) t, w y) =
+        (∫ u in Set.Ioc (0 : ℝ) t, gc u) - ∫ y in Set.Ioc (0 : ℝ) t, ρ y :=
+      integral_sub (hgccont.continuousOn.integrableOn_Icc.mono_set hIccsub)
+        (hρint.mono_set hIccsub)
+    have harm : (∫ u in Set.Ioc (0 : ℝ) t, gc u) =
+        ∫ u in Set.Ioc (0 : ℝ) t, (tangentArmLengths K u).2.1 :=
+      setIntegral_congr_fun measurableSet_Ioc fun u hu ↦ hgcval u (hIccsub hu)
+    rw [intervalIntegral.integral_of_le ht.1, hsplit, harm,
+      ← hρeq _ measurableSet_Ioc hIocsub, hfcval t ht, hfcval 0 ⟨le_rfl, hTpos⟩]
+    linarith [positiveArm_sub_eq_integral_sub_surface K ht]
+  -- absolute continuity on `[0, π / 2]` and the almost everywhere derivative on `(0, π / 2)`
+  obtain ⟨hAC, hderiv⟩ := absolutelyContinuousOnInterval_and_ae_hasDerivAt_of_eqOn_Ico
+    hTpos hfccont.continuousOn hwint hrepr hext_f
+  refine ⟨hAC, ?_⟩
+  have hρIoo : ∀ᵐ t ∂volume.restrict (Set.Ioo (0 : ℝ) (Real.pi / 2)),
+      0 ≤ ρ t ∧ ρ t ≤ magicDensity (tangentArmLengths K t).2.1 :=
+    ae_restrict_of_ae_restrict_of_subset Set.Ioo_subset_Icc_self hρle
+  filter_upwards [hderiv, hρIoo, ae_restrict_mem measurableSet_Ioo] with t h1 h2 h3
+  have h3' : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := Set.Ioo_subset_Icc_self h3
+  rw [h1.deriv, hext_g t h3']
+  have hval : magicFunctions.2 (Real.toNNReal (gc t)) = gc t - magicDensity (gc t) := by
+    simp only [magicDensity, magicFunctions, Real.coe_toNNReal _ (hgcnonneg t)]
+  have hw : w t = gc t - ρ t := rfl
+  rw [hval, hw, hgcval t h3']
+  linarith [h2.2]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Bounds / Lower / Sequence
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The nonnegative-real version of the continuous lower-bound profile. -/
+def nonnegativeLowerBoundProfile (c : Set.Icc (0 : ℝ) 1) :
+    C(Set.Icc (0 : ℝ) (Real.pi / 2), NNReal) where
+  toFun x := Real.toNNReal (lowerBoundProfile c x)
+  continuous_toFun := continuous_real_toNNReal.comp (lowerBoundProfile c).continuous
+
+theorem armIntegralOperator_monotone
+    (f g : C(Set.Icc (0 : ℝ) (Real.pi / 2), NNReal))
+    (hfg : ∀ x, f x ≤ g x) :
+    ∀ x, armIntegralOperator f x ≤ armIntegralOperator g x := by
+  have hc (k : C(Set.Icc (0 : ℝ) (Real.pi / 2), NNReal)) :
+      Continuous (fun u : ℝ ↦ magicFunctions.2
+        (k (Set.projIcc 0 (Real.pi / 2) (by positivity) (Real.pi / 2 - u)))) := by
+    unfold magicFunctions
+    fun_prop
+  intro x
+  change (1 : ℝ) + _ ≤ 1 + _
+  apply add_le_add le_rfl
+  exact intervalIntegral.integral_mono x.property.1
+    ((hc f).intervalIntegrable _ _) ((hc g).intervalIntegrable _ _)
+    (fun u ↦ magicFunctions_snd_monotone (hfg _))
+
+private theorem magicFunction_eq_affine_of_mem_Icc (y : ℝ) (hy0 : 0 ≤ y) (hy1 : y ≤ 1) :
+    magicFunctions.2 (Real.toNNReal y) = 3 / 2 * y - 1 := by
+  simp only [magicFunctions, Real.toNNReal_of_nonneg hy0, NNReal.coe_mk, abs_of_nonpos (by
+    linarith : y - 1 ≤ 0), neg_sub]
+  rw [max_eq_right (by linarith)]
+  ring
+
+private theorem armIntegralOperator_nonnegativeLowerBoundProfile_eq
+    (c : ℝ) (hc : 0 ≤ c) (hc' : c ≤ 2 / 3)
+    (x : Set.Icc (0 : ℝ) (Real.pi / 2)) :
+  armIntegralOperator (nonnegativeLowerBoundProfile ⟨c, ⟨hc, by linarith⟩⟩) x =
+    1 - (x : ℝ) + (3 / 2) *
+      (∫ u in (0 : ℝ)..(x : ℝ), max (1 - (Real.pi / 2 - u)) c) := by
+  have hintegrand (u : ℝ) (hu : u ∈ Set.uIcc (0 : ℝ) (x : ℝ)) :
+      magicFunctions.2
+          ((nonnegativeLowerBoundProfile ⟨c, hc, by linarith⟩)
+            (Set.projIcc 0 (Real.pi / 2) (by positivity) (Real.pi / 2 - u))) =
+        3 / 2 * max (1 - (Real.pi / 2 - u)) c - 1 := by
+    rw [Set.uIcc_of_le x.property.1] at hu
+    have hu0 : 0 ≤ u := hu.1
+    have huT : Real.pi / 2 - u ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := by
+      constructor <;> linarith [hu.2, x.property.2, Real.pi_pos]
+    rw [Set.projIcc_of_mem (by positivity) huT]
+    apply magicFunction_eq_affine_of_mem_Icc
+    · exact le_max_of_le_right hc
+    · apply max_le
+      · linarith [huT.1]
+      · linarith
+  change
+    (1 + ∫ u in (0 : ℝ)..(x : ℝ),
+        magicFunctions.2
+          ((nonnegativeLowerBoundProfile ⟨c, ⟨hc, by linarith⟩⟩)
+            (Set.projIcc 0 (Real.pi / 2) (by positivity) (Real.pi / 2 - u)))) = _
+  have hcont : Continuous (fun u : ℝ ↦ max (1 - (Real.pi / 2 - u)) c) := by
+    fun_prop
+  rw [intervalIntegral.integral_congr (fun u hu ↦ hintegrand u hu)]
+  have hsplit :
+      (∫ u in (0 : ℝ)..(x : ℝ), 3 / 2 * max (1 - (Real.pi / 2 - u)) c - 1) =
+        (3 / 2) * (∫ u in (0 : ℝ)..(x : ℝ), max (1 - (Real.pi / 2 - u)) c) - (x : ℝ) := by
+    calc
+      _ = (∫ u in (0 : ℝ)..(x : ℝ), 3 / 2 * max (1 - (Real.pi / 2 - u)) c) -
+            ∫ u in (0 : ℝ)..(x : ℝ), (1 : ℝ) := by
+        rw [intervalIntegral.integral_sub ((hcont.const_mul (3 / 2)).intervalIntegrable _ _)
+          intervalIntegrable_const]
+      _ = _ := by
+        nth_rewrite 1 [intervalIntegral.integral_const_mul]
+        nth_rewrite 1 [intervalIntegral.integral_const]
+        simp only [smul_eq_mul, sub_zero]
+        ring
+  rw [hsplit]
+  ring
+
+private theorem split_le_integral_lowerBoundProfile (c : ℝ) (hc : 0 ≤ c)
+    (x : Set.Icc (0 : ℝ) (Real.pi / 2)) (hsx : Real.pi / 2 - 1 + c ≤ (x : ℝ)) :
+    c * (Real.pi / 2 - 1 + c) + (((x : ℝ) + 1 - Real.pi / 2) ^ 2 - c ^ 2) / 2 ≤
+      ∫ u in (0 : ℝ)..(x : ℝ), max (1 - (Real.pi / 2 - u)) c := by
+  have hcont : Continuous (fun u : ℝ ↦ max (1 - (Real.pi / 2 - u)) c) := by
+    fun_prop
+  have hlin (s t : ℝ) :
+      (∫ u in s..t, 1 - Real.pi / 2 + u) =
+        ((t + 1 - Real.pi / 2) ^ 2 - (s + 1 - Real.pi / 2) ^ 2) / 2 := by
+    rw [intervalIntegral.integral_add (f := fun _ : ℝ ↦ 1 - Real.pi / 2)
+      (g := fun u : ℝ ↦ u)
+      (a := s) (b := t) (intervalIntegrable_const)
+      (continuous_id.intervalIntegrable _ _)]
+    rw [intervalIntegral.integral_const, integral_id]
+    ring
+  let s : ℝ := Real.pi / 2 - 1 + c
+  have hq : Continuous (fun u : ℝ ↦ max (1 - (Real.pi / 2 - u)) c) := hcont
+  have hfirst := intervalIntegral.integral_mono (μ := MeasureTheory.volume)
+    (f := fun _ : ℝ ↦ c) (g := fun u : ℝ ↦ max (1 - (Real.pi / 2 - u)) c)
+    (a := (0 : ℝ)) (b := s) (by dsimp [s]; linarith [Real.pi_gt_three])
+    (intervalIntegrable_const (μ := MeasureTheory.volume)) (hq.intervalIntegrable _ _)
+    (fun u ↦ (le_max_right _ _ : c ≤ max (1 - (Real.pi / 2 - u)) c))
+  have hcontlin : Continuous (fun u : ℝ ↦ 1 - Real.pi / 2 + u) := by fun_prop
+  have hsecond := intervalIntegral.integral_mono (μ := MeasureTheory.volume)
+    (f := fun u : ℝ ↦ 1 - Real.pi / 2 + u)
+    (g := fun u : ℝ ↦ max (1 - (Real.pi / 2 - u)) c) (a := s) (b := (x : ℝ)) hsx
+    (hcontlin.intervalIntegrable _ _) (hq.intervalIntegrable _ _)
+    (fun u ↦ by
+      change 1 - Real.pi / 2 + u ≤ max (1 - (Real.pi / 2 - u)) c
+      have hu : 1 - Real.pi / 2 + u = 1 - (Real.pi / 2 - u) := by ring
+      exact hu.le.trans (le_max_left _ _))
+  have hadd := intervalIntegral.integral_add_adjacent_intervals (μ := MeasureTheory.volume)
+    (hq.intervalIntegrable (0 : ℝ) s) (hq.intervalIntegrable s (x : ℝ))
+  rw [hlin s (x : ℝ)] at hsecond
+  have hfirst' : c * s ≤ ∫ u in (0 : ℝ)..s,
+      max (1 - (Real.pi / 2 - u)) c := by
+    convert hfirst using 1
+    simp [intervalIntegral.integral_const, smul_eq_mul, mul_comm]
+  have hsum : c * s + ∫ u in s..(x : ℝ), max (1 - (Real.pi / 2 - u)) c ≤
+      ∫ u in (0 : ℝ)..(x : ℝ), max (1 - (Real.pi / 2 - u)) c := by
+    calc
+      _ ≤ (∫ u in (0 : ℝ)..s, max (1 - (Real.pi / 2 - u)) c) +
+          ∫ u in s..(x : ℝ), max (1 - (Real.pi / 2 - u)) c :=
+        by
+          simpa [add_comm, add_left_comm, add_assoc] using
+            add_le_add_right hfirst' (∫ u in s..(x : ℝ),
+              max (1 - (Real.pi / 2 - u)) c)
+      _ = _ := hadd
+  calc
+    c * s + (((x : ℝ) + 1 - Real.pi / 2) ^ 2 - c ^ 2) / 2 ≤
+        c * s + ∫ u in s..(x : ℝ), max (1 - (Real.pi / 2 - u)) c := by
+      dsimp [s] at *
+      nlinarith [hsecond]
+    _ ≤ _ := hsum
+
+theorem armIntegralOperator_profile_step (c : ℝ) (hc : 0 ≤ c) (hc' : c ≤ 2 / 3) :
+    ∀ x, lowerBoundProfile ⟨c + 1 / 12, by constructor <;> linarith⟩ x ≤
+      armIntegralOperator
+        (nonnegativeLowerBoundProfile ⟨c, hc, by linarith⟩) x := by
+  intro x
+  rw [armIntegralOperator_nonnegativeLowerBoundProfile_eq c hc hc' x]
+  simp only [lowerBoundProfile, ContinuousMap.coe_mk]
+  have hcont : Continuous (fun u : ℝ ↦ max (1 - (Real.pi / 2 - u)) c) := by
+    fun_prop
+  have hI : c * (x : ℝ) ≤
+      ∫ u in (0 : ℝ)..(x : ℝ), max (1 - (Real.pi / 2 - u)) c := by
+    have h := intervalIntegral.integral_mono (μ := MeasureTheory.volume)
+      (f := fun _ : ℝ ↦ c) (g := fun u : ℝ ↦ max (1 - (Real.pi / 2 - u)) c) x.property.1
+      (intervalIntegrable_const (μ := MeasureTheory.volume)) (hcont.intervalIntegrable _ _)
+      (fun u ↦ (le_max_right _ _ : c ≤ max (1 - (Real.pi / 2 - u)) c))
+    convert h using 1
+    simp [intervalIntegral.integral_const, smul_eq_mul, mul_comm]
+  let s : ℝ := Real.pi / 2 - 1 + c
+  apply max_le
+  · have hnonneg : 0 ≤ ∫ u in (0 : ℝ)..(x : ℝ), max (1 - (Real.pi / 2 - u)) c := by
+      exact intervalIntegral.integral_nonneg x.property.1
+        (fun _ _ ↦ le_max_of_le_right hc)
+    linarith
+  · have hpi : Real.pi ≤ 22 / 7 := by linarith [Real.pi_lt_d20]
+    have hmul : 0 ≤ (22 / 7 - Real.pi) * (2 / 3 - c) :=
+      mul_nonneg (sub_nonneg.mpr hpi) (sub_nonneg.mpr hc')
+    by_cases hsx : s ≤ (x : ℝ)
+    · have hs := split_le_integral_lowerBoundProfile c hc x hsx
+      nlinarith [sq_nonneg ((x : ℝ) - (Real.pi / 2 - 1 / 3)),
+        sq_nonneg (c - 2 / 21), hmul]
+    · have hxs : (x : ℝ) ≤ s := le_of_not_ge hsx
+      have hprod : 0 ≤ (s - (x : ℝ)) * (1 - 3 / 2 * c) :=
+        mul_nonneg (sub_nonneg.mpr hxs) (by linarith)
+      dsimp [s] at hxs hprod ⊢
+      nlinarith [hI, sq_nonneg (c - 7 / 6 + Real.pi / 4), hprod, hmul]
+
+private theorem armIntegralOperator_le_sequence_succ (n : ℕ)
+    (x : Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    armIntegralOperator (armLowerBoundSequence n) x ≤
+      (armLowerBoundSequence (n + 1) x : ℝ) := by
+  exact (Real.le_coe_toNNReal _).trans (NNReal.coe_le_coe.mpr (le_max_right _ _))
+
+private theorem lowerBoundProfile_zero_le_one
+    (x : Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    lowerBoundProfile ⟨0, by constructor <;> norm_num⟩ x ≤
+      (armLowerBoundSequence 1 x : ℝ) := by
+  change max (1 - (x : ℝ)) 0 ≤ _
+  simp only [armLowerBoundSequence, ContinuousMap.coe_mk, NNReal.coe_max,
+    NNReal.coe_zero]
+  simp only [armIntegralOperator, magicFunctions, ContinuousMap.coe_mk, NNReal.coe_zero, zero_sub,
+    abs_neg, abs_one, add_self_div_two, max_self, intervalIntegral.integral_neg,
+    intervalIntegral.integral_const, sub_zero, smul_eq_mul, mul_one,
+    Real.coe_toNNReal', le_max_iff, le_add_neg_iff_add_le, zero_add, Std.le_refl, or_true,
+    sup_of_le_right, sup_le_iff, tsub_le_iff_right, and_true]
+  by_cases hx : 1 ≤ (x : ℝ)
+  · exact Or.inr hx
+  · left
+    rw [max_eq_left (by linarith)]
+    linarith
+
+private theorem lowerBoundProfile_le_sequence_succ (n : ℕ) (hn : n ≤ 9)
+    (x : Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    lowerBoundProfile ⟨(n : ℝ) / 12, by
+      have hnR : (n : ℝ) ≤ 9 := by exact_mod_cast hn
+      constructor
+      · positivity
+      · linarith⟩ x ≤
+      (armLowerBoundSequence (n + 1) x : ℝ) := by
+  induction n generalizing x with
+  | zero => simpa [lowerBoundProfile] using lowerBoundProfile_zero_le_one x
+  | succ n ih =>
+      have hn' : n ≤ 9 := by omega
+      have hn8 : n ≤ 8 := by omega
+      have hn8r : (n : ℝ) ≤ 8 := by exact_mod_cast hn8
+      have hprof := armIntegralOperator_profile_step ((n : ℝ) / 12) (by positivity)
+        (by nlinarith)
+      have hmon := armIntegralOperator_monotone
+        (nonnegativeLowerBoundProfile ⟨(n : ℝ) / 12, by
+          have hnR : (n : ℝ) ≤ 9 := by exact_mod_cast hn'
+          constructor <;> nlinarith [Nat.cast_nonneg (α := ℝ) n]⟩)
+        (armLowerBoundSequence (n + 1)) (by
+          intro y
+          exact Real.toNNReal_le_iff_le_coe.mpr (ih hn' y))
+      have hop : lowerBoundProfile ⟨(n + 1) / 12, by
+          have hnR : ((n + 1 : ℕ) : ℝ) ≤ 9 := by exact_mod_cast hn
+          constructor <;> nlinarith [Nat.cast_nonneg (α := ℝ) n]⟩ x ≤
+          armIntegralOperator (armLowerBoundSequence (n + 1)) x := by
+        have heq : (n : ℝ) / 12 + 1 / 12 = (n + 1) / 12 := by ring
+        simpa only [heq] using (hprof x).trans (hmon x)
+      simpa only [Nat.cast_add, Nat.cast_one] using
+        hop.trans (armIntegralOperator_le_sequence_succ (n + 1) x)
+
+theorem armLowerBoundSequence_threshold :
+    ∀ x : Set.Icc (0 : ℝ) (Real.pi / 2), 0 < (x : ℝ) →
+      1 < (armLowerBoundSequence 11 x : ℝ) := by
+  intro x hx
+  let f : C(Set.Icc (0 : ℝ) (Real.pi / 2), NNReal) := ⟨fun _ ↦ 3 / 4, continuous_const⟩
+  have hbound (y : Set.Icc (0 : ℝ) (Real.pi / 2)) : f y ≤ armLowerBoundSequence 10 y := by
+    have h := lowerBoundProfile_le_sequence_succ 9 (by omega) y
+    have hreal : (3 / 4 : ℝ) ≤ (armLowerBoundSequence 10 y : ℝ) := by
+      norm_num [lowerBoundProfile] at h
+      exact h.2
+    exact_mod_cast hreal
+  have hmon := armIntegralOperator_monotone f (armLowerBoundSequence 10) hbound x
+  have hconst : armIntegralOperator f x = 1 + (x : ℝ) / 8 := by
+    norm_num [armIntegralOperator, f, magicFunctions, intervalIntegral.integral_const]
+  rw [hconst] at hmon
+  have hstep := armIntegralOperator_le_sequence_succ 10 x
+  linarith
+
+/-- The mirror reflection turns the right arm length function of a nondegenerate right-angle cap
+into the reflected left arm length function of the original cap. At the two endpoints the
+prescribed one-sided conventions match: `f⁺` of the mirror at `π / 2 - t = 0` is the terminal
+`g⁻`, and `f⁻` of the mirror at `π / 2` is the prescribed `g⁺` at `0`. -/
+private theorem nondegenerateCapData_arm_mirror (K P : RightAngleCapSpace)
+    (hP : (P.val : Set Point) = mirrorReflection (Real.pi / 2) '' (K.val : Set Point))
+    (hDK : ∃ r s, HasCapDensities K r s) (hDP : ∃ r s, HasCapDensities P r s)
+    (t : ℝ) (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    (nondegenerateCapData P hDP).2.1
+        ⟨Real.pi / 2 - t, ⟨by linarith [ht.2], by linarith [ht.1]⟩⟩ =
+      (nondegenerateCapData K hDK).2.2 ⟨t, ht⟩ := by
+  have hT : Real.pi / 2 - t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+    ⟨by linarith [ht.2], by linarith [ht.1]⟩
+  obtain ⟨h1, h2, -, -⟩ := tangentArms_mirror K P hP (Real.pi / 2 - t) hT
+  rw [show Real.pi / 2 - (Real.pi / 2 - t) = t by ring] at h1 h2
+  simp only [nondegenerateCapData]
+  split_ifs with hcase
+  · exact h2
+  · have htpos : 0 < t := lt_of_le_of_ne ht.1 fun h ↦ hcase (by rw [← h]; ring)
+    rw [h1]
+    exact ((capDensities_contact_eq K hDK).2 t ⟨htpos, ht.2⟩).2.symm
+
+/-- The integral operator is a lower bound for the right arm length function of a balanced
+maximum cap, for every continuous profile `f` with `f(π / 2 - u) ≤ g_K(u)`. Absolute continuity
+of `f_K` on `[0, t]` turns it into the integral of its derivative, the differential inequality
+`f_K' ≥ m₀ ∘ g_K` bounds that derivative from below almost everywhere, and `m₀` is nondecreasing;
+the initial value `f_K(0) = 1` supplies the constant term of the operator. -/
+private theorem armIntegralOperator_le_nondegenerateCapData_right (K : RightAngleCapSpace)
+    (hK : IsBalancedMaximumCap K) (hD : ∃ r s, HasCapDensities K r s)
+    (f : C(Set.Icc (0 : ℝ) (Real.pi / 2), NNReal))
+    (hf : ∀ (u : ℝ) (hu : u ∈ Set.Icc (0 : ℝ) (Real.pi / 2)),
+      (f ⟨Real.pi / 2 - u, ⟨by linarith [hu.2], by linarith [hu.1]⟩⟩ : ℝ) ≤
+        (nondegenerateCapData K hD).2.2 ⟨u, hu⟩)
+    (t : ℝ) (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    armIntegralOperator f ⟨t, ht⟩ ≤ (nondegenerateCapData K hD).2.1 ⟨t, ht⟩ := by
+  have hTle : (0 : ℝ) ≤ Real.pi / 2 := by positivity
+  obtain ⟨hD', hAC, hderiv⟩ := balancedMaximumCap_arm_regularity K hK
+  -- proof irrelevance retypes the regularity statement for the witness `hD`
+  have hACf : AbsolutelyContinuousOnInterval
+      (extendIntervalScalar (nondegenerateCapData K hD).2.1) 0 (Real.pi / 2) := hAC
+  have hderivf : ∀ᵐ u ∂MeasureTheory.volume.restrict (Set.Ioo (0 : ℝ) (Real.pi / 2)),
+      magicFunctions.2
+          (Real.toNNReal (extendIntervalScalar (nondegenerateCapData K hD).2.2 u)) ≤
+        deriv (extendIntervalScalar (nondegenerateCapData K hD).2.1) u := hderiv
+  have hACt := hACf.mono (by
+    rw [Set.uIcc_of_le ht.1, Set.uIcc_of_le hTle]
+    exact Set.Icc_subset_Icc le_rfl ht.2)
+  have hFTC := hACt.integral_deriv_eq_sub
+  have hF0 : extendIntervalScalar (nondegenerateCapData K hD).2.1 0 = 1 := by
+    rw [extendIntervalScalar_of_mem _ ⟨le_rfl, hTle⟩]
+    simp only [nondegenerateCapData,
+      ite_eq_right (ne_of_lt (show (0 : ℝ) < Real.pi / 2 by positivity))]
+    exact tangentArmLengths_right_zero_eq_one K hD
+  have hFt : extendIntervalScalar (nondegenerateCapData K hD).2.1 t =
+      (nondegenerateCapData K hD).2.1 ⟨t, ht⟩ := extendIntervalScalar_of_mem _ ht
+  have hcont : Continuous (fun u : ℝ ↦ magicFunctions.2
+      (f (Set.projIcc 0 (Real.pi / 2) hTle (Real.pi / 2 - u)))) := by
+    unfold magicFunctions
+    fun_prop
+  -- the integrand of the operator is dominated by the derivative almost everywhere
+  have hae : (fun u : ℝ ↦ magicFunctions.2
+        (f (Set.projIcc 0 (Real.pi / 2) hTle (Real.pi / 2 - u)))) ≤ᵐ[
+      MeasureTheory.volume.restrict (Set.Icc (0 : ℝ) t)]
+      deriv (extendIntervalScalar (nondegenerateCapData K hD).2.1) := by
+    rw [← MeasureTheory.restrict_Ioo_eq_restrict_Icc]
+    filter_upwards [MeasureTheory.ae_restrict_of_ae_restrict_of_subset
+        (Set.Ioo_subset_Ioo le_rfl ht.2) hderivf,
+      MeasureTheory.ae_restrict_mem measurableSet_Ioo] with u h1 h2
+    refine le_trans ?_ h1
+    have humem : u ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨h2.1.le, h2.2.le.trans ht.2⟩
+    have hTu : Real.pi / 2 - u ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+      ⟨by linarith [humem.2], by linarith [humem.1]⟩
+    apply magicFunctions_snd_monotone
+    rw [Set.projIcc_of_mem hTle hTu, extendIntervalScalar_of_mem _ humem,
+      Real.le_toNNReal_iff_coe_le (nondegenerateCapData_arm_nonneg K hD ⟨u, humem⟩).2]
+    exact hf u humem
+  have hmono := intervalIntegral.integral_mono_ae_restrict ht.1
+    (hcont.intervalIntegrable _ _) hACt.intervalIntegrable_deriv hae
+  have hval : armIntegralOperator f ⟨t, ht⟩ =
+      1 + ∫ u in (0 : ℝ)..t, magicFunctions.2
+        (f (Set.projIcc 0 (Real.pi / 2) hTle (Real.pi / 2 - u))) := rfl
+  rw [hval, ← hFt]
+  linarith [hFTC, hmono, hF0]
+
+/-- The simultaneous lower bound for the two arm length functions of a balanced maximum
+right-angle cap. The induction on `n` quantifies over every balanced maximum cap, because the
+bound for the left arm is obtained by applying the bound for the right arm to the mirror cap. -/
+private theorem armLowerBoundSequence_le_nondegenerateCapData (n : ℕ) :
+    ∀ (K : RightAngleCapSpace), IsBalancedMaximumCap K →
+      ∀ (hD : ∃ r s, HasCapDensities K r s) (t : ℝ)
+        (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)),
+        (armLowerBoundSequence n ⟨t, ht⟩ : ℝ) ≤
+          (nondegenerateCapData K hD).2.1 ⟨t, ht⟩ ∧
+        (armLowerBoundSequence n
+            ⟨Real.pi / 2 - t, ⟨by linarith [ht.2], by linarith [ht.1]⟩⟩ : ℝ) ≤
+          (nondegenerateCapData K hD).2.2 ⟨t, ht⟩ := by
+  induction n with
+  | zero =>
+      intro K hK hD t ht
+      simpa only [armLowerBoundSequence, ContinuousMap.coe_mk, NNReal.coe_zero] using
+        nondegenerateCapData_arm_nonneg K hD ⟨t, ht⟩
+  | succ n ih =>
+      -- the bound for the right arm, proved for every balanced maximum cap at once
+      have hfirst : ∀ (K : RightAngleCapSpace), IsBalancedMaximumCap K →
+          ∀ (hD : ∃ r s, HasCapDensities K r s) (t : ℝ)
+            (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)),
+            (armLowerBoundSequence (n + 1) ⟨t, ht⟩ : ℝ) ≤
+              (nondegenerateCapData K hD).2.1 ⟨t, ht⟩ := by
+        intro K hK hD t ht
+        have hop := armIntegralOperator_le_nondegenerateCapData_right K hK hD
+          (armLowerBoundSequence n) (fun u hu ↦ (ih K hK hD u hu).2) t ht
+        simp only [armLowerBoundSequence, ContinuousMap.coe_mk, NNReal.coe_max,
+          Real.coe_toNNReal']
+        exact max_le (ih K hK hD t ht).1
+          (max_le hop (nondegenerateCapData_arm_nonneg K hD ⟨t, ht⟩).1)
+      intro K hK hD t ht
+      refine ⟨hfirst K hK hD t ht, ?_⟩
+      obtain ⟨P, hP, hPbal⟩ := balancedMaximumCap_mirror K hK
+      obtain ⟨r, s, hrs, -⟩ := balancedMaximumCap_hasDensities P hPbal
+      have hT : Real.pi / 2 - t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+        ⟨by linarith [ht.2], by linarith [ht.1]⟩
+      rw [← nondegenerateCapData_arm_mirror K P hP hD ⟨r, s, hrs⟩ t ht]
+      exact hfirst P hPbal ⟨r, s, hrs⟩ (Real.pi / 2 - t) hT
+
+theorem balancedMaximumCap_sequence_bound (K : RightAngleCapSpace)
+    (hK : IsBalancedMaximumCap K) :
+    ∃ hD : ∃ r s, HasCapDensities K r s,
+      ∀ (n : ℕ) (t : Set.Icc (0 : ℝ) (Real.pi / 2)),
+        (armLowerBoundSequence n t : ℝ) ≤ (nondegenerateCapData K hD).2.1 t ∧
+        (armLowerBoundSequence n
+          ⟨Real.pi / 2 - t, by constructor <;> linarith [t.property.1, t.property.2]⟩ : ℝ) ≤
+          (nondegenerateCapData K hD).2.2 t := by
+  obtain ⟨r, s, hrs, -⟩ := balancedMaximumCap_hasDensities K hK
+  refine ⟨⟨r, s, hrs⟩, fun n t ↦ ?_⟩
+  exact armLowerBoundSequence_le_nondegenerateCapData n K hK ⟨r, s, hrs⟩ t t.property
+
+theorem balancedMaximumCap_arm_gt_one (K : RightAngleCapSpace)
+    (hK : IsBalancedMaximumCap K) :
+    ∃ hD : ∃ r s, HasCapDensities K r s,
+      (∀ t : Set.Icc (0 : ℝ) (Real.pi / 2), 0 < (t : ℝ) →
+        1 < (nondegenerateCapData K hD).2.1 t) ∧
+      (∀ t : Set.Icc (0 : ℝ) (Real.pi / 2), (t : ℝ) < Real.pi / 2 →
+        1 < (nondegenerateCapData K hD).2.2 t) := by
+  obtain ⟨hD, hbound⟩ := balancedMaximumCap_sequence_bound K hK
+  refine ⟨hD, fun t ht ↦ ?_, fun t ht ↦ ?_⟩
+  · exact (armLowerBoundSequence_threshold t ht).trans_le (hbound 11 t).1
+  · have hT : Real.pi / 2 - (t : ℝ) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := by
+      constructor <;> linarith [t.property.1, t.property.2]
+    exact (armLowerBoundSequence_threshold ⟨Real.pi / 2 - (t : ℝ), hT⟩
+      (by linarith)).trans_le (hbound 11 t).2
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Cap.Injectivity`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The injectivity condition for balanced maximum caps
+
+Every balanced maximum cap of rotation angle `π / 2` satisfies the injectivity condition: its
+surface measure has densities that are unique up to null sets, its inner corner is continuously
+differentiable, and the two frame components of the corner velocity have strict signs on the open
+rotation interval.
+
+The theorem lives downstream of `MovingSofa/Cap/Regularity.lean` because its inputs
+`balancedMaximumCap_hasDensities` and `balancedMaximumCap_arm_gt_one` depend on that module.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem balancedMaximumCap_injectivity (K : RightAngleCapSpace)
+    (hK : IsBalancedMaximumCap K) : SatisfiesInjectivityCondition K := by
+  obtain ⟨hD, hf, hg⟩ := balancedMaximumCap_arm_gt_one K hK
+  obtain ⟨-, -, -, -, hC1, -, hderiv⟩ := nondegenerateCap_continuity K hD
+  refine ⟨balancedMaximumCap_hasDensities K hK, hC1, fun t ht ↦ ?_⟩
+  have hmem : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨ht.1.le, ht.2.le⟩
+  have hvn : inner ℝ (tangentVector (t : Real.Angle)) (normalVector (t : Real.Angle)) = 0 := by
+    rw [real_inner_comm]; exact inner_normalVector_tangentVector t
+  rw [(hderiv ⟨t, hmem⟩).1.derivWithin (uniqueDiffOn_Icc (by positivity) t hmem)]
+  refine ⟨?_, ?_⟩
+  · have h := hf ⟨t, hmem⟩ ht.1
+    simp only [inner_add_left, real_inner_smul_left, inner_normalVector_self, hvn, mul_one,
+      mul_zero, add_zero]
+    linarith
+  · have h := hg ⟨t, hmem⟩ ht.2
+    simp only [inner_add_left, real_inner_smul_left, inner_normalVector_tangentVector,
+      inner_tangentVector_self, mul_one, mul_zero, zero_add]
+    linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Gerver.Injectivity`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver / Injectivity
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- On the rotation interval the inner corner of the cap of Gerver's sofa is the certified
+direct Gerver path.  The rotating-hallway coordinates of the inner corner are the cap's two
+support values at `t` and `t + π / 2`, and `gerver_capSupport_identification` evaluates those
+at the frame coordinates of the paper path. -/
+theorem capInnerCorner_eq_paperGerverPath (K : RightAngleCapSpace)
+    (hK : (K.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2))
+    {t : ℝ} (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    capInnerCorner K t = paperGerverPath t := by
+  obtain ⟨-, hGeq, -, -, hcapeq, -, hsup⟩ := gerver_capSupport_identification
+  -- The certified cap is the outer half-plane cap, so its support values are the paper ones.
+  have hKmem : (K.val : Set Point) ∈
+      ({gerverOuterCap, gerverLiteralSofa} : Set (Set Point)) := by
+    refine Or.inl ?_
+    rw [hK, hGeq, hcapeq]
+  have hangsum : (t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) =
+      ((Real.pi / 2 + t : ℝ) : Real.Angle) := by
+    rw [← Real.Angle.coe_add, add_comm]
+  rw [capInnerCorner,
+    (rotatingHallwayParts_formulas (K.val : Set Point) (t : Real.Angle)).2.1, hangsum,
+    (hsup _ hKmem t ht).1, (hsup _ hKmem t ht).2, add_sub_cancel_right,
+    add_sub_cancel_right, inner_normalVector_smul_add_inner_tangentVector_smul]
+
+/-- On the open rotation interval the inner-corner velocity of the cap of Gerver's sofa is the
+velocity of the certified direct Gerver path. -/
+theorem derivWithin_capInnerCorner_eq_deriv_paperGerverPath (K : RightAngleCapSpace)
+    (hK : (K.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2))
+    {t : ℝ} (ht : t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2)) :
+    derivWithin (capInnerCorner K) (Set.Icc 0 (Real.pi / 2)) t = deriv paperGerverPath t := by
+  have hcorner : ∀ s ∈ Set.Icc (0 : ℝ) (Real.pi / 2), capInnerCorner K s = paperGerverPath s :=
+    fun _ hs ↦ capInnerCorner_eq_paperGerverPath K hK hs
+  rw [derivWithin_congr hcorner (hcorner t ⟨ht.1.le, ht.2.le⟩),
+    derivWithin_of_mem_nhds (Icc_mem_nhds ht.1 ht.2)]
+
+theorem paperGerverCap_injectivity :
+    ∃ K : RightAngleCapSpace,
+      (K.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2) ∧
+      SatisfiesInjectivityCondition K := by
+  obtain ⟨K, hKset, r, s, hdens, -, -⟩ := gerver_surface_densities
+  have hcorner : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      capInnerCorner K t = paperGerverPath t :=
+    fun _ ht ↦ capInnerCorner_eq_paperGerverPath K hKset ht
+  refine ⟨K, hKset, ⟨r, s, hdens, fun r' s' h' ↦ hdens.ae_eq h'⟩,
+    contDiff_paperGerverPath.contDiffOn.congr hcorner, fun t ht ↦ ?_⟩
+  rw [derivWithin_capInnerCorner_eq_deriv_paperGerverPath K hKset ht]
+  exact ⟨(gerver_strict_velocity t ht).2.2.1, (gerver_strict_velocity t ht).2.2.2⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Cap.Special.Domain`.
+* `Cap.Special.AreaVariation`.
+* `Cap.Tail.Interpolation`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Special / Domain
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval
+
+namespace MovingSofa
+
+/-- Choose the special cap representing a convex body combination, with a fallback to the
+first cap. -/
+def specialCapCombination (t : I) (K L : SpecialCapSpace) : SpecialCapSpace := by
+  classical
+  exact if h : ∃ M : SpecialCapSpace,
+    M.val.val = convexBodyCombination t K.val.val L.val.val then h.choose else K
+
+theorem specialCap_isConvexDomain :
+    (∀ t K L, (specialCapCombination t K L).val.val =
+      convexBodyCombination t K.val.val L.val.val) ∧
+    IsConvexDomain.{0, 0} specialCapCombination ∧
+    (∀ K : RightAngleCapSpace, IsBalancedMaximumCap K →
+      ∃ L : SpecialCapSpace, L.val = K) ∧
+    (∃ K : SpecialCapSpace,
+      (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2)) := by
+  -- ### The special class is closed under Minkowski interpolation
+  have hclosed : ∀ (t : I) (K L : SpecialCapSpace), ∃ M : SpecialCapSpace,
+      M.val.val = convexBodyCombination t K.val.val L.val.val := fun t K L ↦
+    ⟨⟨⟨convexBodyCombination t K.val.val L.val.val,
+        isCap_convexBodyCombination t K.val L.val⟩,
+      satisfiesInjectivityCondition_of_eq_convexBodyCombination rfl K.property.1 L.property.1,
+      convexBody_area_superlevel _ _ K.property.2 L.property.2 t⟩, rfl⟩
+  have h1 : ∀ (t : I) (K L : SpecialCapSpace), (specialCapCombination t K L).val.val =
+      convexBodyCombination t K.val.val L.val.val := by
+    intro t K L
+    have h := hclosed t K L
+    rw [specialCapCombination, dite_eq_left h]
+    exact h.choose_spec
+  -- ### The sofa area functional is bounded by the cap area
+  have hfunc : ∀ C : RightAngleCapSpace,
+      capAreaFunctional C ≤ ClassicalResults.area (C.val : Set Point) := by
+    intro C
+    have h : (0 : ℝ) ≤ ClassicalResults.area (capNiche C) := ENNReal.toReal_nonneg
+    simp only [capAreaFunctional]
+    linarith
+  -- ### Gerver's cap is a special cap
+  obtain ⟨KG, hKGset, hKGinj⟩ := paperGerverCap_injectivity
+  have hstd : IsStandardPosition paperGerverSofa (Real.pi / 2) :=
+    gerver_capSupport_identification.2.1 ▸ gerver_capSupport_identification.2.2.1
+  have hGfunc : capAreaFunctional KG = ClassicalResults.area paperGerverSofa :=
+    capAreaFunctional_eq_sofaArea paperGerverSofa (Real.pi / 2)
+      ⟨paperGerverSofa, hstd, gerver_paperNiche_identification.2.2.1.symm⟩ KG hKGset
+  have hGbound : (11 : ℝ) / 5 ≤ capAreaFunctional KG := by
+    rw [hGfunc, ← gerver_canonical_paper_literal.1]
+    exact gerver_area_lower_bound.2
+  have hGarea : (11 : ℝ) / 5 ≤ ClassicalResults.area (KG.val : Set Point) :=
+    hGbound.trans (hfunc KG)
+  refine ⟨h1, ?_, ?_, ⟨⟨KG, hKGinj, hGarea⟩, hKGset⟩⟩
+  -- ### The convex-domain structure restricts from the ambient body domain
+  · obtain ⟨V, e, hinj, -, hcomb⟩ := convexBody_isConvexDomain
+    refine ⟨V, fun K ↦ e K.val.val, ?_, ?_, fun t K L ↦
+      (congrArg e (h1 t K L)).trans (hcomb t K.val.val L.val.val)⟩
+    · intro K L h
+      exact Subtype.ext (Subtype.ext (hinj h))
+    · rintro _ ⟨K, rfl⟩ _ ⟨L, rfl⟩ a b ha hb hab
+      have hb1 : b ≤ 1 := by linarith
+      refine ⟨specialCapCombination ⟨b, hb, hb1⟩ K L, ?_⟩
+      have hba : 1 - b = a := by linarith
+      simpa only [hba] using (congrArg e (h1 ⟨b, hb, hb1⟩ K L)).trans
+        (hcomb ⟨b, hb, hb1⟩ K.val.val L.val.val)
+  -- ### Every balanced maximum cap is special
+  · intro K hK
+    refine ⟨⟨K, balancedMaximumCap_injectivity K hK, ?_⟩, rfl⟩
+    exact hGbound.trans ((balancedMaximumCap_maximizes_area K hK KG).trans (hfunc K))
+
+/-- The extreme face vertices and the intersections of supporting lines of a special cap are
+convex-linear along `specialCapCombination`: the underlying bodies interpolate, and both quantities
+are convex-linear in the body. -/
+theorem specialCap_maps_linear (t : I) (K L : SpecialCapSpace) :
+    (∀ a : Real.Angle,
+      (edgeVertices (specialCapCombination t K L).val.val a).1 =
+          (1 - (t : ℝ)) • (edgeVertices K.val.val a).1 +
+            (t : ℝ) • (edgeVertices L.val.val a).1 ∧
+        (edgeVertices (specialCapCombination t K L).val.val a).2 =
+          (1 - (t : ℝ)) • (edgeVertices K.val.val a).2 +
+            (t : ℝ) • (edgeVertices L.val.val a).2) ∧
+    ∀ a b : ℝ, a < b → b < a + Real.pi →
+      supportingIntersection (specialCapCombination t K L).val.val (a : Real.Angle)
+          (b : Real.Angle) =
+        (1 - (t : ℝ)) • supportingIntersection K.val.val (a : Real.Angle) (b : Real.Angle) +
+          (t : ℝ) • supportingIntersection L.val.val (a : Real.Angle) (b : Real.Angle) := by
+  obtain ⟨-, hev, hsi, -⟩ := convexBody_maps_linear t K.val.val L.val.val
+  rw [specialCap_isConvexDomain.1 t K L]
+  exact ⟨hev, hsi⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Special / Area Variation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+theorem specialCapArea_variation :
+    IsQuadraticFunctional specialCapCombination
+      (fun K ↦ ClassicalResults.area (K.val.val : Set Point)) ∧
+    ∀ K L : SpecialCapSpace,
+      convexDirectionalDerivative specialCapCombination
+        (fun M ↦ ClassicalResults.area (M.val.val : Set Point)) K L =
+      ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Icc 0 Real.pi,
+        (supportValue L.val.val t - supportValue K.val.val t) ∂surfaceAreaMeasure K.val.val := by
+  obtain ⟨harea, hbil, -⟩ := convexBody_area_support_integral
+  have hcomb := specialCap_isConvexDomain.1
+  -- ### Quadraticity is the ambient bilinear form restricted to the special caps
+  refine ⟨⟨fun K L ↦ (1 / 2 : ℝ) * ∫ a : Real.Angle,
+      supportValue K.val.val a ∂surfaceAreaMeasure L.val.val, ⟨?_, ?_⟩,
+    fun K ↦ harea K.val.val⟩, ?_⟩
+  · intro K t L M
+    change (1 / 2 : ℝ) * ∫ a : Real.Angle, supportValue K.val.val a
+        ∂surfaceAreaMeasure (specialCapCombination t L M).val.val = _
+    rw [hcomb t L M]
+    exact hbil.1 K.val.val t L.val.val M.val.val
+  · intro M t K L
+    change (1 / 2 : ℝ) * ∫ a : Real.Angle,
+        supportValue (specialCapCombination t K L).val.val a
+        ∂surfaceAreaMeasure M.val.val = _
+    rw [hcomb t K L]
+    exact hbil.2 M.val.val t K.val.val L.val.val
+  intro K L
+  -- ### The support difference is integrable against the finite surface area measure
+  let _ : IsFiniteMeasure (surfaceAreaMeasure K.val.val) :=
+    (surfaceAreaMeasure_face_union K.val.val).1
+  have hint : ∀ M : ConvexBody Point,
+      Integrable (supportValue M) (surfaceAreaMeasure K.val.val) := by
+    intro M
+    have hcont : Continuous (supportValue M) :=
+      (compactSet_support_continuity M M M.nonempty M.isCompact M.nonempty M.isCompact).2.2.1
+    exact hcont.integrable_of_hasCompactSupport
+      (isCompact_univ.of_isClosed_subset isClosed_closure (Set.subset_univ _))
+  have hdiff : Integrable (fun t ↦ supportValue L.val.val t - supportValue K.val.val t)
+      (surfaceAreaMeasure K.val.val) := (hint L.val.val).sub (hint K.val.val)
+  -- ### Interpolating special caps is interpolating convex bodies, so the two segment
+  -- functionals are equal as functions and the mixed-area derivative transports
+  have hseg : segmentFunctional specialCapCombination
+        (fun M : SpecialCapSpace ↦ ClassicalResults.area (M.val.val : Set Point)) K L =
+      segmentFunctional convexBodyCombination
+        (fun M : ConvexBody Point ↦ ClassicalResults.area (M : Set Point))
+        K.val.val L.val.val := by
+    funext t
+    by_cases ht : t ∈ Set.Icc (0 : ℝ) 1
+    · simp only [segmentFunctional, ht, ↓reduceDIte]
+      exact congrArg (fun s : Set Point ↦ ClassicalResults.area s)
+        (congrArg (fun M : ConvexBody Point ↦ (M : Set Point)) (hcomb ⟨t, ht⟩ K L))
+    · simp only [segmentFunctional, ht, ↓reduceDIte]
+  -- ### Both caps have vanishing base support value, so the lower normals contribute nothing
+  have hzero := K.val.setIntegral_compl_image_Icc_zero_pi_eq_zero
+    (fun t ↦ supportValue L.val.val t - supportValue K.val.val t)
+    (by rw [K.val.property.2.2.2.2.2.1, L.val.property.2.2.2.2.2.1, sub_zero])
+  have hS : MeasurableSet ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Icc 0 Real.pi) :=
+    (isCompact_Icc.image Real.Angle.continuous_coe).isClosed.measurableSet
+  rw [convexDirectionalDerivative, hseg,
+    ((supportMeasure_mixedArea_symmetry K.val.val L.val.val).2).derivWithin
+      (uniqueDiffOn_Icc_zero_one.uniqueDiffWithinAt (by norm_num)),
+    ← integral_add_compl hS hdiff, hzero, add_zero]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Tail / Interpolation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval
+
+namespace MovingSofa
+
+/-- The cap and both tail bodies are the corresponding convex body combinations. -/
+def IsCapTailCombination (t : I) (X Y Z : CapTailSpace) : Prop :=
+  Z.cap.val.val = convexBodyCombination t X.cap.val.val Y.cap.val.val ∧
+    Z.rightBody = convexBodyCombination t X.rightBody Y.rightBody ∧
+    Z.leftBody = convexBodyCombination t X.leftBody Y.leftBody
+
+/-- Choose a cap-tail triple representing componentwise convex combination, with a fallback to
+`X`. -/
+def capTailCombination (t : I) (X Y : CapTailSpace) : CapTailSpace := by
+  classical
+  exact if h : ∃ Z, IsCapTailCombination t X Y Z then h.choose else X
+
+theorem capTail_isConvexDomain :
+    (∀ t X Y, IsCapTailCombination t X Y (capTailCombination t X Y)) ∧
+    IsConvexDomain.{0, 0} capTailCombination := by
+  -- A cap-tail triple is determined by its three convex bodies.
+  have hext : ∀ X Y : CapTailSpace, X.cap = Y.cap → X.rightBody = Y.rightBody →
+      X.leftBody = Y.leftBody → X = Y := by
+    intro X Y hcap hright hleft
+    revert hcap hright hleft
+    obtain ⟨c₁, r₁, l₁, -, -, -, -, -, -⟩ := X
+    obtain ⟨c₂, r₂, l₂, -, -, -, -, -, -⟩ := Y
+    intro hcap hright hleft
+    subst hcap; subst hright; subst hleft
+    rfl
+  -- Minkowski interpolation is monotone in both of its convex-body arguments.
+  have hmono : ∀ (t : I) (A B C D : ConvexBody Point), (A : Set Point) ⊆ (C : Set Point) →
+      (B : Set Point) ⊆ (D : Set Point) →
+      (convexBodyCombination t A B : Set Point) ⊆
+        (convexBodyCombination t C D : Set Point) := by
+    intro t A B C D hAC hBD z hz
+    obtain ⟨x, hx, y, hy, rfl⟩ := (mem_convexBodyCombination_iff t A B z).1 hz
+    exact (mem_convexBodyCombination_iff t C D _).2 ⟨x, hAC hx, y, hBD hy, rfl⟩
+  -- ### The cap-tail conditions are closed under componentwise Minkowski interpolation
+  have hclosed : ∀ (t : I) (X Y : CapTailSpace), ∃ Z, IsCapTailCombination t X Y Z := by
+    intro t X Y
+    have hKval : (specialCapCombination t X.cap Y.cap).val.val =
+        convexBodyCombination t X.cap.val.val Y.cap.val.val :=
+      specialCap_isConvexDomain.1 t X.cap Y.cap
+    have hlin : ∀ (K L : ConvexBody Point) (a : Real.Angle),
+        supportValue (convexBodyCombination t K L) a =
+          (1 - (t : ℝ)) * supportValue K a + (t : ℝ) * supportValue L a :=
+      fun K L ↦ (convexBody_maps_linear t K L).1
+    have ht0 : (0 : ℝ) ≤ (t : ℝ) := t.2.1
+    have ht1 : (0 : ℝ) ≤ 1 - (t : ℝ) := sub_nonneg.mpr t.2.2
+    have hb : ∀ a b c d : ℝ, a + b ≤ 1 → c + d ≤ 1 →
+        ((1 - (t : ℝ)) * a + (t : ℝ) * c) + ((1 - (t : ℝ)) * b + (t : ℝ) * d) ≤ 1 := by
+      intro a b c d h₁ h₂
+      nlinarith [mul_le_mul_of_nonneg_left h₁ ht1, mul_le_mul_of_nonneg_left h₂ ht0]
+    have he : ∀ a b c d : ℝ, a + b = 1 → c + d = 1 →
+        ((1 - (t : ℝ)) * a + (t : ℝ) * c) + ((1 - (t : ℝ)) * b + (t : ℝ) * d) = 1 := by
+      intro a b c d h₁ h₂
+      linear_combination (1 - (t : ℝ)) * h₁ + (t : ℝ) * h₂
+    refine ⟨⟨specialCapCombination t X.cap Y.cap,
+      convexBodyCombination t X.rightBody Y.rightBody,
+      convexBodyCombination t X.leftBody Y.leftBody,
+      ?_, ?_, ?_, ?_, ?_, ?_⟩, hKval, rfl, rfl⟩
+    · rw [hKval]
+      exact hmono t _ _ _ _ X.right_subset Y.right_subset
+    · rw [hKval]
+      exact hmono t _ _ _ _ X.left_subset Y.left_subset
+    · intro s hs
+      rw [hKval, hlin, hlin]
+      exact hb _ _ _ _ (X.right_bound s hs) (Y.right_bound s hs)
+    · intro s hs
+      rw [hKval, hlin, hlin]
+      exact he _ _ _ _ (X.right_eq s hs) (Y.right_eq s hs)
+    · intro s hs
+      rw [hKval, hlin, hlin]
+      exact hb _ _ _ _ (X.left_bound s hs) (Y.left_bound s hs)
+    · intro s hs
+      rw [hKval, hlin, hlin]
+      exact he _ _ _ _ (X.left_eq s hs) (Y.left_eq s hs)
+  -- ### The totalized selection therefore always matches
+  have h1 : ∀ (t : I) (X Y : CapTailSpace),
+      IsCapTailCombination t X Y (capTailCombination t X Y) := by
+    intro t X Y
+    have h := hclosed t X Y
+    rw [capTailCombination, dite_eq_left h]
+    exact h.choose_spec
+  refine ⟨h1, ?_⟩
+  -- ### The convex-domain structure is inherited from three copies of the body domain
+  obtain ⟨V, e, hinj, -, hcomb⟩ := convexBody_isConvexDomain
+  refine ⟨ModuleCat.of ℝ (V × V × V),
+    fun X ↦ (e X.cap.val.val, e X.rightBody, e X.leftBody), ?_, ?_, ?_⟩
+  · intro X Y h
+    simp only [Prod.mk.injEq] at h
+    exact hext X Y (Subtype.ext (Subtype.ext (hinj h.1))) (hinj h.2.1) (hinj h.2.2)
+  · rintro _ ⟨X, rfl⟩ _ ⟨Y, rfl⟩ a b ha hb hab
+    have hb1 : b ≤ 1 := by linarith
+    have hba : 1 - b = a := by linarith
+    obtain ⟨hc, hr, hl⟩ := h1 ⟨b, hb, hb1⟩ X Y
+    refine ⟨capTailCombination ⟨b, hb, hb1⟩ X Y, ?_⟩
+    simp only [Prod.smul_mk, Prod.mk_add_mk, Prod.mk.injEq]
+    refine ⟨?_, ?_, ?_⟩
+    · simpa only [hba] using (congrArg e hc).trans (hcomb ⟨b, hb, hb1⟩ _ _)
+    · simpa only [hba] using (congrArg e hr).trans (hcomb ⟨b, hb, hb1⟩ _ _)
+    · simpa only [hba] using (congrArg e hl).trans (hcomb ⟨b, hb, hb1⟩ _ _)
+  · intro t X Y
+    obtain ⟨hc, hr, hl⟩ := h1 t X Y
+    simp only [Prod.smul_mk, Prod.mk_add_mk, Prod.mk.injEq]
+    exact ⟨(congrArg e hc).trans (hcomb t _ _), (congrArg e hr).trans (hcomb t _ _),
+      (congrArg e hl).trans (hcomb t _ _)⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Cap.InnerCornerVariation`.
+* `Cap.CornerModuloLinear`.
+* `Cap.Tail.AreaBounds`.
+* `Cap.UpperBoundaryTracing`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The inner-corner variation on the middle window
+
+On the middle window `I = [φᴿ, φᴸ]` the inner corner of a special cap is
+`(h_K(t) - 1) • u_t + (h_K(t + π/2) - 1) • v_t`, an affine expression in two support values, so it
+depends convex-linearly on the cap. `capInnerCorner_variation` pulls the quadratic curve-area
+functional back along that convex-linear map, which gives quadraticity of
+`K ↦ 𝒥(x_K|_I)` and reduces its directional derivative to the curve-variation formula.
+
+The remaining work is to recognize the two mixed Stieltjes integrals of that formula as the
+pairing of the support increment against the corner measure. The inner corner is `C¹` on the cap
+domain by the injectivity condition, so its Stieltjes measure is its classical velocity times
+Lebesgue measure; the frame identity `planeCrossProduct_eq_inner_frame` then turns the pointwise
+cross product into the corner density against the support increment, on `I` and on `I + π/2`
+separately.
+-/
+
+/-! ### The middle window -/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- The two distinguished Gerver angles are positive, ordered, and below `π / 2`, so the middle
+window `[φᴿ, φᴸ]` and its quarter turn are disjoint subsets of `[0, π]`. -/
+private theorem middleWindow_bounds :
+    0 < paperGerverConstants.2.1 ∧ paperGerverConstants.2.1 ≤ paperGerverConstants.2.2 ∧
+      paperGerverConstants.2.2 < Real.pi / 2 := by
+  obtain ⟨hr, hl, hsum⟩ := paperGerverConstants_snd_mem_Ioo
+  have hle : paperGerverConstants.2.1 ≤ Real.pi / 4 := by
+    have h := GerversSofa.ABφθSpec.existsUnique.choose_spec.1
+    exact le_trans h.2.1 h.2.2.1
+  exact ⟨hr.1, by linarith only [hle, hsum], hl.2⟩
+
+/-- The middle window lies inside the cap's angular domain. -/
+private theorem middleWindow_subset :
+    Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2 ⊆ Set.Icc 0 (Real.pi / 2) := by
+  obtain ⟨hrpos, -, hlpi⟩ := middleWindow_bounds
+  exact Set.Icc_subset_Icc hrpos.le hlpi.le
+
+/-! ### Regularity of the inner corner of a special cap -/
+
+/-- The inner corner of a special cap is continuously differentiable on the cap domain. -/
+private theorem specialCap_contDiffOn (K : SpecialCapSpace) :
+    ContDiffOn ℝ 1 (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2)) :=
+  K.property.1.2.1
+
+/-- At an interior time the inner corner of a special cap has an honest derivative. -/
+private theorem specialCap_hasDerivAt (K : SpecialCapSpace) {t : ℝ}
+    (ht : t ∈ Set.Ioo 0 (Real.pi / 2)) :
+    HasDerivAt (capInnerCorner K.val)
+      (derivWithin (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2)) t) t :=
+  ((specialCap_contDiffOn K).differentiableOn one_ne_zero t
+    (Set.Ioo_subset_Icc_self ht)).hasDerivWithinAt.hasDerivAt (Icc_mem_nhds ht.1 ht.2)
+
+/-! ### The corner density on the two middle windows -/
+
+/-- On the right middle window the corner density is the tangent velocity component. -/
+private theorem capCornerDensity_eq_right (K : SpecialCapSpace) {s : ℝ}
+    (hs : s ∈ Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2) :
+    capCornerDensity K s = (capVelocityCoefficients K s).2 := by
+  obtain ⟨hrpos, -, hlpi⟩ := middleWindow_bounds
+  rw [capCornerDensity, ite_eq_left ⟨lt_of_lt_of_le hrpos hs.1, hs.2.trans hlpi.le⟩]
+
+/-- On the left middle window the corner density is the shifted normal velocity component. -/
+private theorem capCornerDensity_eq_left (K : SpecialCapSpace) {s : ℝ}
+    (hs : s ∈ Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+      (Real.pi / 2 + paperGerverConstants.2.2)) :
+    capCornerDensity K s = -(capVelocityCoefficients K (s - Real.pi / 2)).1 := by
+  obtain ⟨hrpos, -, hlpi⟩ := middleWindow_bounds
+  have h1 : Real.pi / 2 < s := by linarith [hs.1]
+  have h2 : s ≤ Real.pi := by linarith [hs.2]
+  rw [capCornerDensity, ite_eq_right (by rintro ⟨-, h⟩; linarith), ite_eq_left ⟨h1, h2⟩]
+
+/-- The corner density is nonnegative on the middle windows, by the injectivity signs. -/
+private theorem capCornerDensity_nonneg_middle (K : SpecialCapSpace) {s : ℝ}
+    (hs : s ∈ Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2 ∪
+      Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+        (Real.pi / 2 + paperGerverConstants.2.2)) :
+    0 ≤ capCornerDensity K s := by
+  obtain ⟨hrpos, -, hlpi⟩ := middleWindow_bounds
+  have hsign := K.property.1.2.2
+  rcases hs with hs | hs
+  · rw [capCornerDensity_eq_right K hs]
+    exact (hsign s ⟨lt_of_lt_of_le hrpos hs.1, lt_of_le_of_lt hs.2 hlpi⟩).2.le
+  · rw [capCornerDensity_eq_left K hs]
+    have hmem : s - Real.pi / 2 ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) :=
+      ⟨by linarith [hs.1], by linarith [hs.2]⟩
+    exact neg_nonneg.mpr (hsign _ hmem).1.le
+
+/-- The corner density is continuous on each of the two middle windows. -/
+private theorem continuousOn_capCornerDensity_middle (K : SpecialCapSpace) :
+    ContinuousOn (capCornerDensity K)
+        (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2) ∧
+      ContinuousOn (capCornerDensity K)
+        (Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+          (Real.pi / 2 + paperGerverConstants.2.2)) := by
+  obtain ⟨hα, hβ⟩ := continuousOn_capVelocityCoefficients K
+  refine ⟨(hβ.mono middleWindow_subset).congr fun s hs ↦ capCornerDensity_eq_right K hs, ?_⟩
+  have hmaps : Set.MapsTo (fun s : ℝ ↦ s - Real.pi / 2)
+      (Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+        (Real.pi / 2 + paperGerverConstants.2.2))
+      (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2) := by
+    intro s hs
+    exact ⟨by linarith [hs.1], by linarith [hs.2]⟩
+  refine ContinuousOn.congr ?_ fun s hs ↦ capCornerDensity_eq_left K hs
+  exact (((hα.mono middleWindow_subset).comp
+    (continuous_id.sub continuous_const).continuousOn hmaps)).neg
+
+/-! ### Integrating against the corner angle measure -/
+
+/-- On a parameter window inside `[0, π]` the corner angle measure integrates a continuous
+function against the real corner density. -/
+private theorem setIntegral_capCornerAngleMeasure (K : SpecialCapSpace) {S : Set ℝ}
+    (hSmeas : MeasurableSet S) (hS : S ⊆ Set.Icc 0 Real.pi)
+    (hd : AEMeasurable (capCornerDensity K) (volume.restrict S))
+    (hdpos : ∀ s ∈ S, 0 ≤ capCornerDensity K s) {f : Real.Angle → ℝ} (hf : Continuous f) :
+    ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' S, f t ∂capCornerAngleMeasure K =
+      ∫ s in S, capCornerDensity K s * f (s : Real.Angle) := by
+  have hcoe : Measurable fun s : ℝ ↦ (s : Real.Angle) := Real.Angle.continuous_coe.measurable
+  have hturn : Real.pi ≤ -1 + 2 * Real.pi := by linarith [Real.pi_gt_three]
+  have hSIoc : S ⊆ Set.Ioc (-1) Real.pi := fun s hs ↦ ⟨by linarith [(hS hs).1], (hS hs).2⟩
+  have himg : MeasurableSet ((fun s : ℝ ↦ (s : Real.Angle)) '' S) :=
+    Real.Angle.measurableSet_image_of_subset_Ioc hturn hSmeas hSIoc
+  have hApre : MeasurableSet ((fun s : ℝ ↦ (s : Real.Angle)) ⁻¹'
+      ((fun s : ℝ ↦ (s : Real.Angle)) '' S)) := himg.preimage hcoe
+  have hAS : ((fun s : ℝ ↦ (s : Real.Angle)) ⁻¹'
+      ((fun s : ℝ ↦ (s : Real.Angle)) '' S)) ∩ Set.Icc 0 Real.pi = S := by
+    refine Set.Subset.antisymm ?_ fun s hs ↦ ⟨⟨s, hs, rfl⟩, hS hs⟩
+    rintro x ⟨⟨s, hs, hxs⟩, hx⟩
+    have hxIoc : x ∈ Set.Ioc (-1) Real.pi := ⟨by linarith [hx.1], hx.2⟩
+    exact (Real.Angle.injOn_coe_Ioc hturn (hSIoc hs) hxIoc hxs) ▸ hs
+  have h1 : ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' S, f t ∂capCornerAngleMeasure K =
+      ∫ s in (fun s : ℝ ↦ (s : Real.Angle)) ⁻¹' ((fun s : ℝ ↦ (s : Real.Angle)) '' S),
+        f (s : Real.Angle) ∂capCornerMeasure K := by
+    rw [capCornerAngleMeasure]
+    exact setIntegral_map himg hf.aestronglyMeasurable hcoe.aemeasurable
+  have hrestrict : (volume.restrict (Set.Icc 0 Real.pi)).restrict
+      ((fun s : ℝ ↦ (s : Real.Angle)) ⁻¹' ((fun s : ℝ ↦ (s : Real.Angle)) '' S)) =
+      volume.restrict S := by
+    rw [Measure.restrict_restrict hApre, hAS]
+  have haem : AEMeasurable (fun s ↦ ENNReal.ofReal (capCornerDensity K s))
+      ((volume.restrict (Set.Icc 0 Real.pi)).restrict
+        ((fun s : ℝ ↦ (s : Real.Angle)) ⁻¹' ((fun s : ℝ ↦ (s : Real.Angle)) '' S))) := by
+    rw [hrestrict]
+    exact ENNReal.measurable_ofReal.comp_aemeasurable hd
+  have h2 : ∫ s in (fun s : ℝ ↦ (s : Real.Angle)) ⁻¹' ((fun s : ℝ ↦ (s : Real.Angle)) '' S),
+        f (s : Real.Angle) ∂capCornerMeasure K =
+      ∫ s in (fun s : ℝ ↦ (s : Real.Angle)) ⁻¹' ((fun s : ℝ ↦ (s : Real.Angle)) '' S),
+        (ENNReal.ofReal (capCornerDensity K s)).toReal • f (s : Real.Angle)
+          ∂volume.restrict (Set.Icc 0 Real.pi) := by
+    rw [capCornerMeasure]
+    exact setIntegral_withDensity_eq_setIntegral_toReal_smul₀ haem
+      (Filter.Eventually.of_forall fun _ ↦ ENNReal.ofReal_lt_top) _ hApre
+  rw [h1, h2]
+  change ∫ s, _ ∂((volume.restrict (Set.Icc 0 Real.pi)).restrict _) = _
+  rw [hrestrict]
+  refine setIntegral_congr_fun hSmeas fun s hs ↦ ?_
+  rw [ENNReal.toReal_ofReal (hdpos s hs), smul_eq_mul]
+
+/-! ### The middle Stieltjes integrals as weighted Lebesgue integrals -/
+
+/-- The middle corner path's Stieltjes measure has the inner-corner velocity as density. -/
+private theorem capMiddle_stieltjesDensity (K : SpecialCapSpace) (i : Fin 2) :
+    HasIntervalStieltjesDensity (continuousBVCoordinate (capMiddleBV K) i)
+      (fun t ↦ derivWithin (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2)) t i) := by
+  obtain ⟨hrpos, hrl, hlpi⟩ := middleWindow_bounds
+  have hproj : ContDiff ℝ 1 fun p : Point ↦ p i :=
+    (EuclideanSpace.proj (𝕜 := ℝ) i).contDiff
+  refine hasIntervalStieltjesDensity_of_hasDerivAt hrl _
+    (fun s ↦ capInnerCorner K.val s i) _ (fun _ ↦ rfl) ?_ ?_ ?_
+  · refine ContDiffOn.absolutelyContinuousOnInterval ?_
+    rw [Set.uIcc_of_le hrl]
+    exact hproj.comp_contDiffOn ((specialCap_contDiffOn K).mono middleWindow_subset)
+  · intro t ht
+    have ht' : t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) := ⟨hrpos.trans ht.1, ht.2.trans hlpi⟩
+    exact (EuclideanSpace.proj (𝕜 := ℝ) i).hasFDerivAt.comp_hasDerivAt t
+      (specialCap_hasDerivAt K ht')
+  · exact (((PiLp.continuous_apply 2 _ i).comp_continuousOn
+      ((continuousOn_derivWithin_capInnerCorner K).mono middleWindow_subset))).integrableOn_compact
+        isCompact_Icc
+
+/-- A mixed Stieltjes integral over the middle window is the corresponding weighted Lebesgue
+integral of the inner-corner velocity. -/
+private theorem capMiddle_stieltjesIntegral_eq (K L : SpecialCapSpace) (i j : Fin 2) :
+    intervalStieltjesIntegral (continuousBVCoordinate (capMiddleBV K) i)
+        (fun t ↦ (capMiddleBV L).val t j - (capMiddleBV K).val t j) Set.univ =
+      ∫ t in Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2,
+        (capInnerCorner L.val t j - capInnerCorner K.val t j) *
+          derivWithin (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2)) t i := by
+  have hq : Continuous fun t : Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2 ↦
+      (capMiddleBV L).val t j - (capMiddleBV K).val t j :=
+    ((PiLp.continuous_apply 2 _ j).comp (capMiddleBV L).property.1).sub
+      ((PiLp.continuous_apply 2 _ j).comp (capMiddleBV K).property.1)
+  have hval : ∀ (M : SpecialCapSpace)
+      (t : Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2),
+      (capMiddleBV M).val t = capInnerCorner M.val t.val := fun _ _ ↦ rfl
+  rw [intervalStieltjesIntegral_eq_integral_mul_of_density _ (capMiddle_stieltjesDensity K i)
+    hq Set.univ MeasurableSet.univ, Measure.restrict_univ]
+  simp only [hval]
+  have key := MeasureTheory.integral_subtype_preimage (μ := volume)
+    (s := Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2) measurableSet_Icc
+    (MeasurableSet.univ (α := ℝ))
+    (fun t : ℝ ↦ (capInnerCorner L.val t j - capInnerCorner K.val t j) *
+      derivWithin (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2)) t i)
+  simp only [Set.mem_univ, Set.ofPred_true, Measure.restrict_univ] at key
+  exact key
+
+/-! ### The variation of the inner corner -/
+
+theorem capInnerCorner_variation :
+    IsConvexLinear specialCapCombination bvPathCombination capMiddleBV ∧
+    IsQuadraticFunctional specialCapCombination (fun K ↦ curveAreaFunctional (capMiddleBV K)) ∧
+    ∀ K L : SpecialCapSpace,
+      convexDirectionalDerivative specialCapCombination
+        (fun M ↦ curveAreaFunctional (capMiddleBV M)) K L =
+      (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) ''
+          (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2 ∪
+            Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+              (Real.pi / 2 + paperGerverConstants.2.2)),
+        (supportValue L.val.val t - supportValue K.val.val t) ∂capCornerAngleMeasure K) +
+        (segmentArea (distinguishedCapSides K.val).2.corner
+            (distinguishedCapSides L.val).2.corner -
+          segmentArea (distinguishedCapSides K.val).1.corner
+            (distinguishedCapSides L.val).1.corner) := by
+  obtain ⟨hrpos, hrl, hlpi⟩ := middleWindow_bounds
+  -- ### The middle corner path is convex-linear
+  have hlinear : IsConvexLinear specialCapCombination bvPathCombination capMiddleBV := by
+    intro t K L
+    refine Subtype.ext (funext fun s ↦ ?_)
+    change capInnerCorner (specialCapCombination t K L).val s.val = _
+    rw [capInnerCorner_of_eq_convexBodyCombination (specialCap_isConvexDomain.1 t K L)]
+    rfl
+  refine ⟨hlinear, (curveArea_variation _ _ hrl).1.comp_isConvexLinear hlinear, ?_⟩
+  intro K L
+  rw [convexDirectionalDerivative_comp_isConvexLinear hlinear _ K L,
+    (curveArea_variation _ _ hrl).2]
+  refine congrArg₂ (· + ·) ?_ rfl
+  -- ### The middle window and the corner density on it
+  have hSmeas : MeasurableSet (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2 ∪
+      Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+        (Real.pi / 2 + paperGerverConstants.2.2)) := measurableSet_Icc.union measurableSet_Icc
+  have hSsub : Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2 ∪
+      Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+        (Real.pi / 2 + paperGerverConstants.2.2) ⊆ Set.Icc 0 Real.pi := by
+    have hpi := Real.pi_pos
+    rintro s (hs | hs)
+    · exact ⟨by linarith [hs.1], by linarith [hs.2]⟩
+    · exact ⟨by linarith [hs.1], by linarith [hs.2]⟩
+  obtain ⟨hdc1, hdc2⟩ := continuousOn_capCornerDensity_middle K
+  have hd : AEMeasurable (capCornerDensity K)
+      (volume.restrict (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2 ∪
+        Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+          (Real.pi / 2 + paperGerverConstants.2.2))) :=
+    (hdc1.union_of_isClosed hdc2 isClosed_Icc isClosed_Icc).aemeasurable hSmeas
+  have hcont (M : ConvexBody Point) : Continuous fun u : Real.Angle ↦ supportValue M u :=
+    (compactSet_support_continuity M M M.nonempty M.isCompact M.nonempty M.isCompact).2.2.1
+  have hfcont : Continuous fun u : Real.Angle ↦ supportValue (L.val.val : Set Point) u -
+      supportValue (K.val.val : Set Point) u := (hcont L.val.val).sub (hcont K.val.val)
+  have hfreal : Continuous fun s : ℝ ↦ supportValue (L.val.val : Set Point) (s : Real.Angle) -
+      supportValue (K.val.val : Set Point) (s : Real.Angle) :=
+    hfcont.comp Real.Angle.continuous_coe
+  -- ### Continuity of the four Lebesgue integrands on the middle window
+  have hXc : ContinuousOn (capInnerCorner K.val)
+      (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2) :=
+    ((specialCap_contDiffOn K).mono middleWindow_subset).continuousOn
+  have hYc : ContinuousOn (capInnerCorner L.val)
+      (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2) :=
+    ((specialCap_contDiffOn L).mono middleWindow_subset).continuousOn
+  have hDc : ContinuousOn (derivWithin (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2)))
+      (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2) :=
+    (continuousOn_derivWithin_capInnerCorner K).mono middleWindow_subset
+  have hmixed (i j : Fin 2) : IntegrableOn
+      (fun t ↦ (capInnerCorner L.val t j - capInnerCorner K.val t j) *
+        derivWithin (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2)) t i)
+      (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2) :=
+    ((((PiLp.continuous_apply 2 _ j).comp_continuousOn hYc).sub
+      ((PiLp.continuous_apply 2 _ j).comp_continuousOn hXc)).mul
+      ((PiLp.continuous_apply 2 _ i).comp_continuousOn hDc)).integrableOn_compact isCompact_Icc
+  have hG1 : IntegrableOn (fun s ↦ capCornerDensity K s *
+      (supportValue (L.val.val : Set Point) (s : Real.Angle) -
+        supportValue (K.val.val : Set Point) (s : Real.Angle)))
+      (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2) :=
+    (hdc1.mul hfreal.continuousOn).integrableOn_compact isCompact_Icc
+  have hG2 : IntegrableOn (fun s ↦ capCornerDensity K s *
+      (supportValue (L.val.val : Set Point) (s : Real.Angle) -
+        supportValue (K.val.val : Set Point) (s : Real.Angle)))
+      (Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+        (Real.pi / 2 + paperGerverConstants.2.2)) :=
+    (hdc2.mul hfreal.continuousOn).integrableOn_compact isCompact_Icc
+  have hmaps : Set.MapsTo (fun t : ℝ ↦ t + Real.pi / 2)
+      (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2)
+      (Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+        (Real.pi / 2 + paperGerverConstants.2.2)) :=
+    fun s hs ↦ ⟨by linarith [hs.1], by linarith [hs.2]⟩
+  have hG2' : IntegrableOn (fun t ↦ capCornerDensity K (t + Real.pi / 2) *
+      (supportValue (L.val.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) -
+        supportValue (K.val.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle)))
+      (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2) :=
+    (((hdc2.mul hfreal.continuousOn).comp
+      (continuous_id.add continuous_const).continuousOn hmaps)).integrableOn_compact isCompact_Icc
+  -- ### Both sides are Lebesgue integrals over the middle window
+  rw [capMiddle_stieltjesIntegral_eq K L 1 0, capMiddle_stieltjesIntegral_eq K L 0 1,
+    setIntegral_capCornerAngleMeasure K hSmeas hSsub hd
+      (fun s hs ↦ capCornerDensity_nonneg_middle K hs) hfcont,
+    setIntegral_union (Set.disjoint_left.mpr fun s hs hs' ↦ by linarith [hs.2, hs'.1])
+      measurableSet_Icc hG1 hG2,
+    integral_Icc_const_add_eq (Real.pi / 2),
+    ← integral_sub (hmixed 1 0) (hmixed 0 1), ← integral_add hG1 hG2']
+  -- ### The pointwise cross-product identity
+  refine setIntegral_congr_fun measurableSet_Icc fun t ht ↦ ?_
+  have hts : t + Real.pi / 2 ∈ Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+      (Real.pi / 2 + paperGerverConstants.2.2) := hmaps ht
+  have hdens1 : capCornerDensity K t =
+      inner ℝ (derivWithin (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2)) t)
+        (tangentVector (t : Real.Angle)) := capCornerDensity_eq_right K ht
+  have hdens2 : capCornerDensity K (t + Real.pi / 2) =
+      -inner ℝ (derivWithin (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2)) t)
+        (normalVector (t : Real.Angle)) := by
+    rw [capCornerDensity_eq_left K hts]
+    simp only [add_sub_cancel_right, capVelocityCoefficients]
+  have hu : inner ℝ (capInnerCorner L.val t - capInnerCorner K.val t)
+      (normalVector (t : Real.Angle)) =
+      supportValue (L.val.val : Set Point) (t : Real.Angle) -
+        supportValue (K.val.val : Set Point) (t : Real.Angle) := by
+    rw [inner_sub_left, inner_capInnerCorner_normalVector, inner_capInnerCorner_normalVector]
+    ring
+  have hv : inner ℝ (capInnerCorner L.val t - capInnerCorner K.val t)
+      (tangentVector (t : Real.Angle)) =
+      supportValue (L.val.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) -
+        supportValue (K.val.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) := by
+    rw [inner_sub_left, inner_capInnerCorner_tangentVector, inner_capInnerCorner_tangentVector]
+    ring
+  have hframe := planeCrossProduct_eq_inner_frame
+    (capInnerCorner L.val t - capInnerCorner K.val t)
+    (derivWithin (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2)) t) t
+  have hs0 : (capInnerCorner L.val t - capInnerCorner K.val t) 0 =
+      capInnerCorner L.val t 0 - capInnerCorner K.val t 0 := by simp
+  have hs1 : (capInnerCorner L.val t - capInnerCorner K.val t) 1 =
+      capInnerCorner L.val t 1 - capInnerCorner K.val t 1 := by simp
+  rw [planeCrossProduct, hs0, hs1] at hframe
+  rw [hdens1, hdens2, ← hu, ← hv]
+  linear_combination hframe
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The outer corner and the two wedge segments, modulo convex-linear functionals
+
+On the middle window `I = [φᴿ, φᴸ]` the outer corner of a cap is its inner corner translated by the
+`K`-independent frame sum `c t = u_t + v_t`, so the two curve-area functionals differ by the two
+mixed Stieltjes cross integrals, each convex-linear in `K`, plus the constant area of `c`.
+
+The same happens at the two ends of the window: the tangent-line intersection point and the wedge
+endpoint differ by a fixed vector, as do the outer and the inner corner, so each of the two
+segment-area comparisons differs by a determinant that is affine in the support values and in the
+inner corner, hence convex-linear in `K` as well.
+-/
+
+/-! ### The outer corner path on the middle window -/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval
+
+namespace MovingSofa
+
+/-- The frame sum `t ↦ u_t + v_t`, as a continuous path of bounded variation. -/
+def frameSumBV (a b : ℝ) : ContinuousBVPaths a b :=
+  continuousBVOfContDiffOn
+    (fun t ↦ normalVector (t : Real.Angle) + tangentVector (t : Real.Angle))
+    ((contDiff_normalVector.add contDiff_tangentVector).of_le
+      (by exact_mod_cast le_top)).contDiffOn
+
+/-- The outer corner of a special cap on the middle window, as a continuous path of bounded
+variation: the inner-corner path translated by the frame sum. -/
+def capOuterMiddleBV (K : SpecialCapSpace) :
+    ContinuousBVPaths paperGerverConstants.2.1 paperGerverConstants.2.2 :=
+  capMiddleBV K + frameSumBV paperGerverConstants.2.1 paperGerverConstants.2.2
+
+/-- The outer middle path of a special cap traces the outer corner of its body. -/
+theorem capOuterMiddleBV_val (K : SpecialCapSpace)
+    (s : Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2) :
+    (capOuterMiddleBV K).val s =
+      (rotatingHallwayParts (K.val.val : Set Point) ((s : ℝ) : Real.Angle)).outerCorner :=
+  (outerCorner_eq_innerCorner_add K.val.val (s : ℝ)).symm
+
+/-! ### The two end segments -/
+
+/-- At the right Gerver angle the tangent-line intersection point is the right wedge endpoint
+translated by a `K`-independent vector. -/
+private theorem supportingIntersection_eq_wedgeEndpoints_add_fst (K : RightAngleCapSpace)
+    (hcos : Real.cos paperGerverConstants.2.1 ≠ 0) :
+    supportingIntersection K.val (paperGerverConstants.2.1 : Real.Angle)
+        ((Real.pi / 2 : ℝ) : Real.Angle) =
+      (wedgeEndpoints K paperGerverConstants.2.1).1 +
+        !₂[(1 - Real.sin paperGerverConstants.2.1) /
+          Real.cos paperGerverConstants.2.1, 1] := by
+  have hangle : ((Real.pi / 2 : ℝ) : Real.Angle) - (paperGerverConstants.2.1 : Real.Angle) =
+      ((Real.pi / 2 - paperGerverConstants.2.1 : ℝ) : Real.Angle) := by rw [Real.Angle.coe_sub]
+  have hpyth := Real.sin_sq_add_cos_sq paperGerverConstants.2.1
+  obtain ⟨hW0, hW1⟩ := wedgeEndpoints_fst_coords K paperGerverConstants.2.1
+  rw [supportingIntersection, hangle, Real.Angle.cos_coe, Real.Angle.sin_coe,
+    Real.cos_pi_div_two_sub, Real.sin_pi_div_two_sub, K.property.2.2.1]
+  ext i
+  fin_cases i <;>
+    simp only [Fin.zero_eta, Fin.mk_one, PiLp.add_apply, PiLp.smul_apply, smul_eq_mul,
+      normalVector, tangentVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe,
+      Matrix.cons_val_zero, Matrix.cons_val_one, hW0, hW1]
+  · field_simp
+    linear_combination (supportValue (K.val : Set Point)
+      (paperGerverConstants.2.1 : Real.Angle)) * hpyth
+  · field_simp
+    ring
+
+/-- At the left Gerver angle the tangent-line intersection point is the left wedge endpoint
+translated by a `K`-independent vector. -/
+private theorem supportingIntersection_eq_wedgeEndpoints_add_snd (K : RightAngleCapSpace)
+    (hsin : Real.sin paperGerverConstants.2.2 ≠ 0) :
+    supportingIntersection K.val ((Real.pi / 2 : ℝ) : Real.Angle)
+        ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle) =
+      (wedgeEndpoints K paperGerverConstants.2.2).2 +
+        !₂[(Real.cos paperGerverConstants.2.2 - 1) /
+          Real.sin paperGerverConstants.2.2, 1] := by
+  have hangle : ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle) -
+      ((Real.pi / 2 : ℝ) : Real.Angle) = (paperGerverConstants.2.2 : Real.Angle) := by
+    rw [← Real.Angle.coe_sub, show Real.pi / 2 + paperGerverConstants.2.2 - Real.pi / 2 =
+      paperGerverConstants.2.2 from by ring]
+  obtain ⟨hZ0, hZ1⟩ := wedgeEndpoints_snd_coords K paperGerverConstants.2.2
+  rw [supportingIntersection, hangle, Real.Angle.cos_coe, Real.Angle.sin_coe,
+    K.property.2.2.1]
+  ext i
+  fin_cases i <;>
+    simp only [Fin.zero_eta, Fin.mk_one, PiLp.add_apply, PiLp.smul_apply, smul_eq_mul,
+      normalVector, tangentVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe,
+      Real.cos_pi_div_two, Real.sin_pi_div_two, Matrix.cons_val_zero, Matrix.cons_val_one,
+      hZ0, hZ1] <;>
+    field_simp <;> ring_nf
+
+/-- A pair of segment areas whose endpoints are fixed translates of a second pair differs from it
+by a determinant, which is convex-linear as soon as the two base points are. -/
+private theorem segmentArea_equivalent_of_translations {α : Type*} (cα : I → α → α → α)
+    {W X P Q : α → Point} {v w : Point} (hP : ∀ K, P K = W K + w) (hQ : ∀ K, Q K = X K + v)
+    (hW : ∀ (t : I) (K L : α), W (cα t K L) = (1 - (t : ℝ)) • W K + (t : ℝ) • W L)
+    (hX : ∀ (t : I) (K L : α), X (cα t K L) = (1 - (t : ℝ)) • X K + (t : ℝ) • X L) :
+    EquivalentModuloConvexLinear cα (fun K ↦ segmentArea (P K) (Q K))
+      (fun K ↦ segmentArea (W K) (X K)) := by
+  intro t K L
+  simp only [hP, hQ, hW, hX, realCombination, segmentArea, planeCrossProduct, PiLp.add_apply,
+    PiLp.smul_apply, smul_eq_mul]
+  ring
+
+/-! ### The main equivalence -/
+
+theorem cornerArea_equivalent_modulo_linear :
+    ∃ F : SpecialCapSpace →
+        ContinuousBVPaths paperGerverConstants.2.1 paperGerverConstants.2.2,
+      (∀ K, (F K).val = fun t ↦
+        (rotatingHallwayParts (K.val.val : Set Point) (t.val : Real.Angle)).outerCorner) ∧
+      EquivalentModuloConvexLinear specialCapCombination
+        (fun K ↦ curveAreaFunctional (F K))
+        (fun K ↦ curveAreaFunctional (capMiddleBV K)) ∧
+      EquivalentModuloConvexLinear specialCapCombination
+        (fun K ↦ segmentArea
+          (supportingIntersection K.val.val (paperGerverConstants.2.1 : Real.Angle)
+            ((Real.pi / 2 : ℝ) : Real.Angle))
+          (rotatingHallwayParts (K.val.val : Set Point)
+            (paperGerverConstants.2.1 : Real.Angle)).outerCorner)
+        (fun K ↦ segmentArea (distinguishedCapSides K.val).1.fanPoint
+          (distinguishedCapSides K.val).1.corner) ∧
+      EquivalentModuloConvexLinear specialCapCombination
+        (fun K ↦ segmentArea
+          (supportingIntersection K.val.val ((Real.pi / 2 : ℝ) : Real.Angle)
+            ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle))
+          (rotatingHallwayParts (K.val.val : Set Point)
+            (paperGerverConstants.2.2 : Real.Angle)).outerCorner)
+        (fun K ↦ segmentArea (distinguishedCapSides K.val).2.fanPoint
+          (distinguishedCapSides K.val).2.corner) := by
+  obtain ⟨hr, hl, -⟩ := paperGerverConstants_snd_mem_Ioo
+  have hrl : paperGerverConstants.2.1 ≤ paperGerverConstants.2.2 :=
+    paperGerverConstants_snd_fst_lt_snd_snd.le
+  refine ⟨capOuterMiddleBV, fun K ↦ funext (capOuterMiddleBV_val K), ?_, ?_, ?_⟩
+  -- ### The middle window: translation by the frame-sum path
+  · intro t K L
+    change curveAreaFunctional (capMiddleBV (specialCapCombination t K L) +
+        frameSumBV paperGerverConstants.2.1 paperGerverConstants.2.2) -
+      curveAreaFunctional (capMiddleBV (specialCapCombination t K L)) = _
+    rw [capInnerCorner_variation.1 t K L]
+    exact curveArea_translation_convexLinear _ t (capMiddleBV K) (capMiddleBV L)
+  -- ### The right end: both endpoints move by fixed vectors
+  · have hcos : 0 < Real.cos paperGerverConstants.2.1 :=
+      Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos, hr.1], hr.2⟩
+    simp only [distinguishedCapSides_fst_fanPoint, distinguishedCapSides_fst_corner]
+    refine segmentArea_equivalent_of_translations specialCapCombination
+      (w := !₂[(1 - Real.sin paperGerverConstants.2.1) / Real.cos paperGerverConstants.2.1, 1])
+      (v := normalVector (paperGerverConstants.2.1 : Real.Angle) +
+        tangentVector (paperGerverConstants.2.1 : Real.Angle))
+      (fun K ↦ supportingIntersection_eq_wedgeEndpoints_add_fst K.val hcos.ne')
+      (fun K ↦ outerCorner_eq_innerCorner_add K.val.val paperGerverConstants.2.1)
+      (fun t K L ↦ ?_) (fun t K L ↦ ?_)
+    · show (wedgeEndpoints (specialCapCombination t K L).val paperGerverConstants.2.1).1 = _
+      simp only [wedgeEndpoints, specialCap_isConvexDomain.1 t K L,
+        supportValue_convexBodyCombination]
+      match_scalars
+      ring
+    · exact capInnerCorner_of_eq_convexBodyCombination (specialCap_isConvexDomain.1 t K L)
+        paperGerverConstants.2.1
+  -- ### The left end: the same, with the complementary trigonometric normalisation
+  · have hsin : 0 < Real.sin paperGerverConstants.2.2 :=
+      Real.sin_pos_of_pos_of_lt_pi hl.1 (by linarith [Real.pi_pos, hl.2])
+    simp only [distinguishedCapSides_snd_fanPoint, distinguishedCapSides_snd_corner]
+    refine segmentArea_equivalent_of_translations specialCapCombination
+      (w := !₂[(Real.cos paperGerverConstants.2.2 - 1) / Real.sin paperGerverConstants.2.2, 1])
+      (v := normalVector (paperGerverConstants.2.2 : Real.Angle) +
+        tangentVector (paperGerverConstants.2.2 : Real.Angle))
+      (fun K ↦ supportingIntersection_eq_wedgeEndpoints_add_snd K.val hsin.ne')
+      (fun K ↦ outerCorner_eq_innerCorner_add K.val.val paperGerverConstants.2.2)
+      (fun t K L ↦ ?_) (fun t K L ↦ ?_)
+    · show (wedgeEndpoints (specialCapCombination t K L).val paperGerverConstants.2.2).2 = _
+      simp only [wedgeEndpoints, specialCap_isConvexDomain.1 t K L,
+        supportValue_convexBodyCombination]
+      match_scalars
+      ring
+    · exact capInnerCorner_of_eq_convexBodyCombination (specialCap_isConvexDomain.1 t K L)
+        paperGerverConstants.2.2
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Tail / Area Bounds
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A point of a special cap above both the right inner wall at `φᴿ` and the bottom axis that
+misses the right canonical tail lies in the niche, on the right side. -/
+private theorem mem_capNiche_inter_right_of_notMem_canonicalTail (K : SpecialCapSpace)
+    {q : Point} (hwall : q ∈ (innerWallUpperHalfPlanes K.val paperGerverConstants.2.1).1)
+    (hup : q ∈ normalHalfPlane ((Real.pi / 2 : ℝ) : Real.Angle) 0 true false)
+    (hqK : q ∈ (K.1.1 : Set Point)) (hq : q ∉ (canonicalTailSets K).1) :
+    q ∈ capNiche K.val ∩ (distinguishedCapSides K.val).1.upperHalfPlane := by
+  obtain ⟨hrIoo, -, -⟩ := paperGerverConstants_snd_mem_Ioo
+  obtain ⟨hmonR, -, -, -⟩ := cap_tail_monotonicity_intervals K
+  have hUpEqR : (innerWallUpperHalfPlanes K.val (Real.pi / 2)).1 =
+      normalHalfPlane ((Real.pi / 2 : ℝ) : Real.Angle) 0 true false := by
+    change normalHalfPlane _ (supportValue (K.1.1 : Set Point) _ - 1) true false = _
+    rw [K.1.property.2.2.2.1, sub_self]
+  obtain ⟨t, ht, hqt⟩ : ∃ t ∈ Set.Icc paperGerverConstants.2.1 (Real.pi / 2),
+      q ∉ (innerWallUpperHalfPlanes K.val t).1 := by
+    by_contra hc
+    exact hq ⟨hqK, Set.mem_iInter₂.mpr fun t ht ↦ not_not.mp fun h ↦ hc ⟨t, ht, h⟩⟩
+  have htlow : paperGerverConstants.2.1 < t :=
+    ht.1.lt_of_ne fun h ↦ hqt (h ▸ hwall)
+  have hthigh : t < Real.pi / 2 :=
+    ht.2.lt_of_ne fun h ↦ hqt (by rw [h, hUpEqR]; exact hup)
+  have hmem : q ∈ (distinguishedCapSides K.val).1.upperHalfPlane ∩
+      innerQuadrant (K.val.val : Set Point) t := by
+    rw [(hmonR t ⟨htlow, ht.2⟩).2]
+    exact ⟨hwall, hqt⟩
+  exact ⟨⟨⟨hup, hup⟩,
+    Set.mem_iUnion₂.mpr ⟨t, ⟨hrIoo.1.trans htlow, hthigh⟩, hmem.2⟩⟩, hwall⟩
+
+/-- A point of a special cap above both the left inner wall at `φᴸ` and the bottom axis that
+misses the left canonical tail lies in the niche, on the left side. -/
+private theorem mem_capNiche_inter_left_of_notMem_canonicalTail (K : SpecialCapSpace)
+    {q : Point} (hwall : q ∈ (innerWallUpperHalfPlanes K.val paperGerverConstants.2.2).2)
+    (hup : q ∈ normalHalfPlane ((Real.pi / 2 : ℝ) : Real.Angle) 0 true false)
+    (hqK : q ∈ (K.1.1 : Set Point)) (hq : q ∉ (canonicalTailSets K).2) :
+    q ∈ capNiche K.val ∩ (distinguishedCapSides K.val).2.upperHalfPlane := by
+  obtain ⟨-, hlIoo, -⟩ := paperGerverConstants_snd_mem_Ioo
+  obtain ⟨-, hmonL, -, -⟩ := cap_tail_monotonicity_intervals K
+  have hUpEqL : (innerWallUpperHalfPlanes K.val 0).2 =
+      normalHalfPlane ((Real.pi / 2 : ℝ) : Real.Angle) 0 true false := by
+    have hzero : ((((0 : ℝ) + Real.pi / 2 : ℝ) : Real.Angle)) =
+        ((Real.pi / 2 : ℝ) : Real.Angle) := by rw [zero_add]
+    change normalHalfPlane (((0 : ℝ) + Real.pi / 2 : ℝ) : Real.Angle)
+      (supportValue (K.1.1 : Set Point) (((0 : ℝ) + Real.pi / 2 : ℝ) : Real.Angle) - 1)
+        true false = _
+    rw [hzero, K.1.property.2.2.2.1, sub_self]
+  obtain ⟨t, ht, hqt⟩ : ∃ t ∈ Set.Icc (0 : ℝ) paperGerverConstants.2.2,
+      q ∉ (innerWallUpperHalfPlanes K.val t).2 := by
+    by_contra hc
+    exact hq ⟨hqK, Set.mem_iInter₂.mpr fun t ht ↦ not_not.mp fun h ↦ hc ⟨t, ht, h⟩⟩
+  have htlow : 0 < t := ht.1.lt_of_ne fun h ↦ hqt (by rw [← h, hUpEqL]; exact hup)
+  have hthigh : t < paperGerverConstants.2.2 := ht.2.lt_of_ne fun h ↦ hqt (h ▸ hwall)
+  have hmem : q ∈ (distinguishedCapSides K.val).2.upperHalfPlane ∩
+      innerQuadrant (K.val.val : Set Point) t := by
+    rw [(hmonL t ⟨ht.1, hthigh⟩).2]
+    exact ⟨hwall, hqt⟩
+  exact ⟨⟨⟨hup, hup⟩,
+    Set.mem_iUnion₂.mpr ⟨t, ⟨htlow, hthigh.trans hlIoo.2⟩, hmem.2⟩⟩, hwall⟩
+
+theorem canonicalTail_niche_area_lower_bounds (K : SpecialCapSpace)
+    (B D : ConvexBody Point)
+    (hB : (B : Set Point) = (canonicalTailSets K).1)
+    (hD : (D : Set Point) = (canonicalTailSets K).2) :
+    segmentArea (rightLeftTailArcs B D).1.startPoint
+        (distinguishedCapSides K.val).1.fanPoint -
+      convexArcArea B (Real.pi + paperGerverConstants.2.1) (3 * Real.pi / 2) ≤
+        ClassicalResults.area
+          (capNiche K.val ∩ (distinguishedCapSides K.val).1.upperHalfPlane) ∧
+    segmentArea (distinguishedCapSides K.val).2.fanPoint
+        (rightLeftTailArcs B D).2.endPoint -
+      convexArcArea D (3 * Real.pi / 2) (3 * Real.pi / 2 + paperGerverConstants.2.2) ≤
+        ClassicalResults.area
+          (capNiche K.val ∩ (distinguishedCapSides K.val).2.upperHalfPlane) := by
+  obtain ⟨hrIoo, hlIoo, -⟩ := paperGerverConstants_snd_mem_Ioo
+  have hpi := Real.pi_pos
+  have hcast : ∀ a b : ℝ, a = b → ((a : ℝ) : Real.Angle) = ((b : ℝ) : Real.Angle) :=
+    fun a b h => by rw [h]
+  have htop : supportValue (K.1.1 : Set Point) ((Real.pi / 2 : ℝ) : Real.Angle) = 1 :=
+    K.1.property.2.2.2.1
+  obtain ⟨-, -, -, hBsub, -, -, -, hDsub, -, hBeq, -, -, -, hDeq, -, -⟩ :=
+    canonicalTailSets_properties K
+  -- endpoint support values of the two tails
+  have hB1 : supportValue (B : Set Point)
+      ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle) =
+      1 - supportValue (K.1.1 : Set Point)
+        ((paperGerverConstants.2.1 : ℝ) : Real.Angle) := by
+    have h := hBeq paperGerverConstants.2.1 (by simp)
+    rw [hB]
+    linarith
+  have hB2 : supportValue (B : Set Point) ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := by
+    have h := hBeq (Real.pi / 2) (by simp)
+    rw [hcast _ _ (show Real.pi + Real.pi / 2 = 3 * Real.pi / 2 by ring), htop] at h
+    rw [hB]
+    linarith
+  have hD1 : supportValue (D : Set Point) ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := by
+    have h := hDeq 0 (by simp)
+    rw [hcast _ _ (show Real.pi / 2 + (0 : ℝ) = Real.pi / 2 by ring),
+      hcast _ _ (show 3 * Real.pi / 2 + (0 : ℝ) = 3 * Real.pi / 2 by ring), htop] at h
+    rw [hD]
+    linarith
+  have hD2 : supportValue (D : Set Point)
+      ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle) =
+      1 - supportValue (K.1.1 : Set Point)
+        ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle) := by
+    have h := hDeq paperGerverConstants.2.2 (by simp)
+    rw [hD]
+    linarith
+  obtain ⟨hOB, hOD, hsegB, hsegD, -, -⟩ :=
+    tailFanPoint_identities K.val B D hrIoo hlIoo hB1 hB2 hD1 hD2
+  obtain ⟨hWmem, hZmem⟩ := specialCap_wedgeEndpoints_in_bottomEdge K
+  have hnichefin : MeasureTheory.volume (capNiche K.val) ≠ ⊤ :=
+    ne_of_lt (niche_uniform_bounds.1 _ K.val).2.2.1
+  have hRfin : ∀ S : Set Point,
+      MeasureTheory.volume (capNiche K.val ∩ S) ≠ ⊤ := fun S ↦
+    ne_top_of_le_ne_top hnichefin (MeasureTheory.measure_mono Set.inter_subset_left)
+  have hKconv : Convex ℝ (K.1.1 : Set Point) := K.1.1.convex
+  have hKclosed : IsClosed (K.1.1 : Set Point) := K.1.1.isCompact.isClosed
+  constructor
+  · -- right tail
+    have hHa : (supportingLineHalfPlane (B : Set Point)
+        ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle)).2 =
+        (distinguishedCapSides K.val).1.upperHalfPlane := by
+      refine supportingLineHalfPlane_snd_eq_normalHalfPlane ?_ ?_
+      · rw [hcast _ _ (show Real.pi + paperGerverConstants.2.1 =
+          paperGerverConstants.2.1 + Real.pi by ring), normalVector_add_pi]
+      · rw [hB1]; ring
+    have hHb : (supportingLineHalfPlane (B : Set Point)
+        ((3 * Real.pi / 2 : ℝ) : Real.Angle)).2 =
+        normalHalfPlane ((Real.pi / 2 : ℝ) : Real.Angle) 0 true false := by
+      refine supportingLineHalfPlane_snd_eq_normalHalfPlane ?_ ?_
+      · rw [hcast _ _ (show 3 * Real.pi / 2 = Real.pi / 2 + Real.pi by ring),
+          normalVector_add_pi]
+      · rw [hB2]; ring
+    have hmain := convexArc_tangentRegion_area_le B
+      (Real.pi + paperGerverConstants.2.1) (3 * Real.pi / 2)
+      (by linarith [hrIoo.2]) (by linarith [hrIoo.1])
+      hKconv hKclosed (by rw [hB]; exact hBsub) (by rw [hOB]; exact hWmem.1.1)
+      (hRfin _) (fun q hq hqK hqB ↦ by
+        have hq' := interior_subset hq
+        rw [hHa, hHb] at hq'
+        exact mem_capNiche_inter_right_of_notMem_canonicalTail K hq'.1 hq'.2 hqK
+          (by rwa [hB] at hqB))
+    rw [hOB] at hmain
+    rw [show (rightLeftTailArcs B D).1.startPoint =
+        (edgeVertices B ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle)).1 from rfl,
+      show (distinguishedCapSides K.val).1.fanPoint =
+        (wedgeEndpoints K.val paperGerverConstants.2.1).1 from rfl]
+    linarith [hmain, hsegB]
+  · -- left tail
+    have hHa : (supportingLineHalfPlane (D : Set Point)
+        ((3 * Real.pi / 2 : ℝ) : Real.Angle)).2 =
+        normalHalfPlane ((Real.pi / 2 : ℝ) : Real.Angle) 0 true false := by
+      refine supportingLineHalfPlane_snd_eq_normalHalfPlane ?_ ?_
+      · rw [hcast _ _ (show 3 * Real.pi / 2 = Real.pi / 2 + Real.pi by ring),
+          normalVector_add_pi]
+      · rw [hD1]; ring
+    have hHb : (supportingLineHalfPlane (D : Set Point)
+        ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle)).2 =
+        (distinguishedCapSides K.val).2.upperHalfPlane := by
+      refine supportingLineHalfPlane_snd_eq_normalHalfPlane ?_ ?_
+      · rw [hcast _ _ (show 3 * Real.pi / 2 + paperGerverConstants.2.2 =
+          (paperGerverConstants.2.2 + Real.pi / 2) + Real.pi by ring), normalVector_add_pi]
+      · rw [hD2, hcast _ _ (show Real.pi / 2 + paperGerverConstants.2.2 =
+          paperGerverConstants.2.2 + Real.pi / 2 by ring)]
+        ring
+    have hmain := convexArc_tangentRegion_area_le D
+      (3 * Real.pi / 2) (3 * Real.pi / 2 + paperGerverConstants.2.2)
+      (by linarith [hlIoo.1]) (by linarith [hlIoo.2])
+      hKconv hKclosed (by rw [hD]; exact hDsub) (by rw [hOD]; exact hZmem.1.1)
+      (hRfin _) (fun q hq hqK hqD ↦ by
+        have hq' := interior_subset hq
+        rw [hHa, hHb] at hq'
+        exact mem_capNiche_inter_left_of_notMem_canonicalTail K hq'.2 hq'.1 hqK
+          (by rwa [hD] at hqD))
+    rw [hOD] at hmain
+    rw [show (rightLeftTailArcs B D).2.endPoint =
+        (edgeVertices D ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle)).2
+        from rfl,
+      show (distinguishedCapSides K.val).2.fanPoint =
+        (wedgeEndpoints K.val paperGerverConstants.2.2).2 from rfl]
+    linarith [hmain, hsegD]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Tracing the area of a special cap along its upper boundary
+
+The area of a convex body is half the integral of its support function against its surface area
+measure.  For a special cap that measure is carried by the closed upper semicircle of normals,
+because the two open quarter arcs of lower normals carry degenerate faces
+(`MovingSofa.CapSpace.surfaceAreaMeasure_image_Ioo_lower_eq_zero`) and the bottom normal carries
+support value zero.  The injectivity condition provides angular densities on the two upper quarter
+circles, so the four distinguished angles `0`, `φᴿ`, `φᴸ` and `π` are not atoms, and the semicircle
+splits — up to a null set — into the four open arcs `(0, φᴿ)`, `(φᴿ, φᴸ)`, `(φᴸ, π / 2)`,
+`(π / 2, π)` and the top normal `π / 2`.  Each open arc is shorter than `π`, so the convex arc area
+formula applies to it, and the top normal contributes half the mass of a single atom
+(`MovingSofa.HasCapDensities.area_eq_upper_arcs_add_top_atom`).  Evaluating the surface area
+measure at that fixed angle is convex-linear on the convex domain of special caps, whence the cap
+area agrees with the sum of the four arc areas modulo convex-linear functionals
+(`MovingSofa.specialCapArea_equivalent_upper_arcs`).
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+open scoped NNReal
+
+namespace MovingSofa
+
+/-- On an open arc shorter than `π` the support-area integral is twice the convex arc area. -/
+private theorem setIntegral_image_Ioo_eq_two_mul_convexArcArea (K : ConvexBody Point)
+    {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi) :
+    ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        supportValue (K : Set Point) t ∂surfaceAreaMeasure K =
+      2 * convexArcArea K a b := by
+  rw [(convexArc_area a b hab hba).2.1 K]
+  ring
+
+/-- Up to half the mass of the atom at its top normal, the area of a right-angle cap carrying
+angular densities is the sum of the areas of the four upper boundary arcs cut out by the two
+distinguished Gerver angles. -/
+theorem HasCapDensities.area_eq_upper_arcs_add_top_atom {C : RightAngleCapSpace}
+    {dr dl : ℝ → ℝ≥0} (hdens : HasCapDensities C dr dl) :
+    ClassicalResults.area (C.val : Set Point) =
+      (convexArcArea C.val 0 paperGerverConstants.2.1 +
+          convexArcArea C.val paperGerverConstants.2.1 paperGerverConstants.2.2 +
+          convexArcArea C.val paperGerverConstants.2.2 (Real.pi / 2) +
+          convexArcArea C.val (Real.pi / 2) Real.pi) +
+        (surfaceAreaMeasure C.val {((Real.pi / 2 : ℝ) : Real.Angle)}).toReal / 2 := by
+  have hpi := Real.pi_pos
+  obtain ⟨⟨hrpos, -⟩, ⟨-, hlT⟩, -⟩ := paperGerverConstants_snd_mem_Ioo
+  have hrl := paperGerverConstants_snd_fst_lt_snd_snd
+  set r := paperGerverConstants.2.1
+  set l := paperGerverConstants.2.2
+  -- ### The surface measure is finite, so the continuous support function is integrable
+  have hfinite : IsFiniteMeasure (surfaceAreaMeasure C.val) :=
+    (surfaceAreaMeasure_face_union C.val).1
+  have hint : Integrable (fun a : Real.Angle ↦ supportValue (C.val : Set Point) a)
+      (surfaceAreaMeasure C.val) :=
+    ((compactSet_support_continuity (C.val : Set Point) (C.val : Set Point) C.val.nonempty'
+      C.val.isCompact' C.val.nonempty' C.val.isCompact').2.2.1).integrable_of_hasCompactSupport
+      (isCompact_univ.of_isClosed_subset isClosed_closure (Set.subset_univ _))
+  -- ### The five pieces of the closed upper semicircle that carry the surface measure
+  set U : Set ℝ := Set.Ioo 0 r ∪ Set.Ioo r l ∪ Set.Ioo l (Real.pi / 2) ∪
+    Set.Ioo (Real.pi / 2) Real.pi ∪ {Real.pi / 2} with hUdef
+  have hs1 : Set.Ioo (0 : ℝ) r ⊆ Set.Ioc 0 (2 * Real.pi) := fun x hx ↦
+    ⟨hx.1, by linarith [hx.2]⟩
+  have hs2 : Set.Ioo r l ⊆ Set.Ioc 0 (2 * Real.pi) := fun x hx ↦
+    ⟨by linarith [hx.1], by linarith [hx.2]⟩
+  have hs3 : Set.Ioo l (Real.pi / 2) ⊆ Set.Ioc 0 (2 * Real.pi) := fun x hx ↦
+    ⟨by linarith [hx.1], by linarith [hx.2]⟩
+  have hs4 : Set.Ioo (Real.pi / 2) Real.pi ⊆ Set.Ioc 0 (2 * Real.pi) := fun x hx ↦
+    ⟨by linarith [hx.1], by linarith [hx.2]⟩
+  have hs5 : ({Real.pi / 2} : Set ℝ) ⊆ Set.Ioc 0 (2 * Real.pi) := by
+    rintro x rfl
+    exact ⟨by linarith, by linarith⟩
+  have hUsub : U ⊆ Set.Ioc 0 (2 * Real.pi) :=
+    Set.union_subset (Set.union_subset (Set.union_subset (Set.union_subset hs1 hs2) hs3) hs4) hs5
+  have hUIcc : U ⊆ Set.Icc 0 Real.pi := by
+    refine Set.union_subset (Set.union_subset (Set.union_subset (Set.union_subset
+      (fun x hx ↦ ⟨hx.1.le, by linarith [hx.2]⟩)
+      (fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩))
+      (fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩))
+      (fun x hx ↦ ⟨by linarith [hx.1], hx.2.le⟩)) ?_
+    rintro x rfl
+    exact ⟨by linarith, by linarith⟩
+  have hUmeas : MeasurableSet U :=
+    (((measurableSet_Ioo.union measurableSet_Ioo).union measurableSet_Ioo).union
+      measurableSet_Ioo).union (measurableSet_singleton _)
+  -- ### Angular images of disjoint pieces of one turn are disjoint, and Borel pieces stay Borel
+  have hdisj : ∀ {X Y : Set ℝ}, X ⊆ Set.Ioc 0 (2 * Real.pi) →
+      Y ⊆ Set.Ioc 0 (2 * Real.pi) → Disjoint X Y →
+      Disjoint ((fun t : ℝ ↦ (t : Real.Angle)) '' X)
+        ((fun t : ℝ ↦ (t : Real.Angle)) '' Y) := by
+    intro X Y hX hY hXY
+    rw [Set.disjoint_left]
+    rintro a ⟨x, hx, rfl⟩ ⟨y, hy, hyx⟩
+    have hyx' : y = x :=
+      Real.Angle.injOn_coe_Ioc (a := 0) (b := 2 * Real.pi) (by linarith) (hY hy) (hX hx) hyx
+    exact Set.disjoint_left.mp hXY hx (hyx' ▸ hy)
+  have hmimg : ∀ {Y : Set ℝ}, MeasurableSet Y → Y ⊆ Set.Ioc 0 (2 * Real.pi) →
+      MeasurableSet ((fun t : ℝ ↦ (t : Real.Angle)) '' Y) := fun hY hYs ↦
+    Real.Angle.measurableSet_image_of_subset_Ioc (by linarith) hY hYs
+  -- ### Splitting the last piece off an angular union
+  have hstep : ∀ {X Y : Set ℝ}, X ⊆ Set.Ioc 0 (2 * Real.pi) →
+      Y ⊆ Set.Ioc 0 (2 * Real.pi) → Disjoint X Y → MeasurableSet Y →
+      ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' (X ∪ Y),
+          supportValue (C.val : Set Point) t ∂surfaceAreaMeasure C.val =
+        (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' X,
+            supportValue (C.val : Set Point) t ∂surfaceAreaMeasure C.val) +
+          ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Y,
+            supportValue (C.val : Set Point) t ∂surfaceAreaMeasure C.val := by
+    intro X Y hX hY hXY hmY
+    rw [Set.image_union, setIntegral_union (hdisj hX hY hXY) (hmimg hmY hY)
+      hint.integrableOn hint.integrableOn]
+  -- ### The four distinguished angles of the upper semicircle are not atoms
+  have hatom0 : surfaceAreaMeasure C.val {((0 : ℝ) : Real.Angle)} = 0 :=
+    hdens.surfaceAreaMeasure_singleton_eq_zero_of_mem_Ico ⟨le_rfl, by positivity⟩
+  have hatomr : surfaceAreaMeasure C.val {(r : Real.Angle)} = 0 :=
+    hdens.surfaceAreaMeasure_singleton_eq_zero_of_mem_Ico ⟨hrpos.le, by linarith⟩
+  have hatoml : surfaceAreaMeasure C.val {(l : Real.Angle)} = 0 :=
+    hdens.surfaceAreaMeasure_singleton_eq_zero_of_mem_Ico ⟨by linarith, hlT⟩
+  have hatompi : surfaceAreaMeasure C.val {((Real.pi : ℝ) : Real.Angle)} = 0 := by
+    have h := hdens.surfaceAreaMeasure_singleton_eq_zero_of_mem_Ioc
+      (t := Real.pi / 2) ⟨by positivity, le_rfl⟩
+    rwa [show (Real.pi / 2 + Real.pi / 2 : ℝ) = Real.pi by ring] at h
+  -- ### The five pieces cover the upper semicircle up to those four angles
+  have hcover : Set.Icc (0 : ℝ) Real.pi ⊆ U ∪ {0, r, l, Real.pi} := by
+    intro x hx
+    rcases eq_or_lt_of_le hx.1 with h0 | h0
+    · exact Or.inr (by simp [← h0])
+    rcases lt_trichotomy x r with h | h | h
+    · exact Or.inl (Or.inl (Or.inl (Or.inl (Or.inl ⟨h0, h⟩))))
+    · exact Or.inr (by simp [h])
+    rcases lt_trichotomy x l with h' | h' | h'
+    · exact Or.inl (Or.inl (Or.inl (Or.inl (Or.inr ⟨h, h'⟩))))
+    · exact Or.inr (by simp [h'])
+    rcases lt_trichotomy x (Real.pi / 2) with h'' | h'' | h''
+    · exact Or.inl (Or.inl (Or.inl (Or.inr ⟨h', h''⟩)))
+    · exact Or.inl (Or.inr h'')
+    rcases eq_or_lt_of_le hx.2 with h₃ | h₃
+    · exact Or.inr (by simp [h₃])
+    · exact Or.inl (Or.inl (Or.inr ⟨h'', h₃⟩))
+  have hnull : surfaceAreaMeasure C.val
+      ((fun t : ℝ ↦ (t : Real.Angle)) '' Set.Icc 0 Real.pi \
+        (fun t : ℝ ↦ (t : Real.Angle)) '' U) = 0 := by
+    refine measure_mono_null (t := {((0 : ℝ) : Real.Angle), (r : Real.Angle), (l : Real.Angle),
+      ((Real.pi : ℝ) : Real.Angle)}) ?_ ?_
+    · rw [Set.sdiff_subset_iff]
+      refine (Set.image_mono hcover).trans ?_
+      rw [Set.image_union]
+      exact Set.union_subset_union_right _ (by simp [Set.image_insert_eq])
+    · simp only [Set.insert_eq]
+      exact measure_union_null hatom0
+        (measure_union_null hatomr (measure_union_null hatoml hatompi))
+  -- ### The support-area integral is carried by the angular image of the five pieces
+  have hSmeas : MeasurableSet ((fun t : ℝ ↦ (t : Real.Angle)) '' Set.Icc 0 Real.pi) :=
+    (isCompact_Icc.image Real.Angle.continuous_coe).isClosed.measurableSet
+  have htotal : ∫ a, supportValue (C.val : Set Point) a ∂surfaceAreaMeasure C.val =
+      ∫ a in (fun t : ℝ ↦ (t : Real.Angle)) '' U,
+        supportValue (C.val : Set Point) a ∂surfaceAreaMeasure C.val := by
+    rw [← integral_add_compl hSmeas hint,
+      C.setIntegral_compl_image_Icc_zero_pi_eq_zero _ C.property.2.2.2.2.2.1, add_zero,
+      ← Set.union_sdiff_cancel (Set.image_mono hUIcc),
+      setIntegral_union Set.disjoint_sdiff_right (hSmeas.diff (hmimg hUmeas hUsub))
+        hint.integrableOn hint.integrableOn,
+      setIntegral_measure_zero _ hnull, add_zero]
+  -- ### The four short arcs contribute their arc areas and the top normal its atom
+  have hI1 : ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo (0 : ℝ) r,
+      supportValue (C.val : Set Point) t ∂surfaceAreaMeasure C.val =
+      2 * convexArcArea C.val 0 r :=
+    setIntegral_image_Ioo_eq_two_mul_convexArcArea C.val hrpos (by linarith)
+  have hI2 : ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo r l,
+      supportValue (C.val : Set Point) t ∂surfaceAreaMeasure C.val =
+      2 * convexArcArea C.val r l :=
+    setIntegral_image_Ioo_eq_two_mul_convexArcArea C.val hrl (by linarith)
+  have hI3 : ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo l (Real.pi / 2),
+      supportValue (C.val : Set Point) t ∂surfaceAreaMeasure C.val =
+      2 * convexArcArea C.val l (Real.pi / 2) :=
+    setIntegral_image_Ioo_eq_two_mul_convexArcArea C.val hlT (by linarith)
+  have hI4 : ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo (Real.pi / 2) Real.pi,
+      supportValue (C.val : Set Point) t ∂surfaceAreaMeasure C.val =
+      2 * convexArcArea C.val (Real.pi / 2) Real.pi :=
+    setIntegral_image_Ioo_eq_two_mul_convexArcArea C.val (by linarith) (by linarith)
+  have hIT : ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' ({Real.pi / 2} : Set ℝ),
+      supportValue (C.val : Set Point) t ∂surfaceAreaMeasure C.val =
+      (surfaceAreaMeasure C.val {((Real.pi / 2 : ℝ) : Real.Angle)}).toReal := by
+    rw [Set.image_singleton, integral_singleton, C.property.2.2.2.1, smul_eq_mul, mul_one,
+      measureReal_def]
+  -- ### Assembling the support-area identity
+  rw [convexBody_area_support_integral.1 C.val, htotal, hUdef,
+    hstep (Set.union_subset (Set.union_subset (Set.union_subset hs1 hs2) hs3) hs4) hs5
+      (by rw [Set.disjoint_singleton_right]
+          rintro (((⟨-, h⟩ | ⟨-, h⟩) | ⟨-, h⟩) | ⟨h, -⟩) <;> linarith)
+      (measurableSet_singleton _),
+    hstep (Set.union_subset (Set.union_subset hs1 hs2) hs3) hs4
+      (by rw [Set.disjoint_left]
+          rintro x ((⟨-, h⟩ | ⟨-, h⟩) | ⟨-, h⟩) ⟨h', -⟩ <;> linarith)
+      measurableSet_Ioo,
+    hstep (Set.union_subset hs1 hs2) hs3
+      (by rw [Set.disjoint_left]
+          rintro x (⟨-, h⟩ | ⟨-, h⟩) ⟨h', -⟩ <;> linarith)
+      measurableSet_Ioo,
+    hstep hs1 hs2
+      (by rw [Set.disjoint_left]
+          rintro x ⟨-, h⟩ ⟨h', -⟩
+          linarith)
+      measurableSet_Ioo,
+    hI1, hI2, hI3, hI4, hIT]
+  ring
+
+theorem specialCapArea_equivalent_upper_arcs :
+    EquivalentModuloConvexLinear specialCapCombination
+      (fun K ↦ ClassicalResults.area (K.val.val : Set Point))
+      (fun K ↦ convexArcArea K.val.val 0 paperGerverConstants.2.1 +
+        convexArcArea K.val.val paperGerverConstants.2.1 paperGerverConstants.2.2 +
+        convexArcArea K.val.val paperGerverConstants.2.2 (Real.pi / 2) +
+        convexArcArea K.val.val (Real.pi / 2) Real.pi) := by
+  -- ### The discrepancy is half the mass of the surface measure at the fixed top normal
+  have key : ∀ M : SpecialCapSpace,
+      ClassicalResults.area (M.val.val : Set Point) -
+        (convexArcArea M.val.val 0 paperGerverConstants.2.1 +
+          convexArcArea M.val.val paperGerverConstants.2.1 paperGerverConstants.2.2 +
+          convexArcArea M.val.val paperGerverConstants.2.2 (Real.pi / 2) +
+          convexArcArea M.val.val (Real.pi / 2) Real.pi) =
+        (surfaceAreaMeasure M.val.val {((Real.pi / 2 : ℝ) : Real.Angle)}).toReal / 2 :=
+    fun M ↦ by
+      obtain ⟨dr, dl, hdens, -⟩ := M.property.1.1
+      rw [hdens.area_eq_upper_arcs_add_top_atom]
+      ring
+  intro t K L
+  have hKfin : IsFiniteMeasure (surfaceAreaMeasure K.val.val) :=
+    (surfaceAreaMeasure_face_union K.val.val).1
+  have hLfin : IsFiniteMeasure (surfaceAreaMeasure L.val.val) :=
+    (surfaceAreaMeasure_face_union L.val.val).1
+  simp only [realCombination, key]
+  rw [specialCap_isConvexDomain.1 t K L, surfaceAreaMeasure_convexBodyCombination,
+    Measure.add_apply, Measure.smul_apply, Measure.smul_apply, smul_eq_mul, smul_eq_mul,
+    ENNReal.toReal_add (ENNReal.mul_ne_top ENNReal.ofReal_ne_top (measure_ne_top _ _))
+      (ENNReal.mul_ne_top ENNReal.ofReal_ne_top (measure_ne_top _ _)),
+    ENNReal.toReal_mul, ENNReal.toReal_mul,
+    ENNReal.toReal_ofReal (by linarith [t.2.2] : (0 : ℝ) ≤ 1 - (t : ℝ)),
+    ENNReal.toReal_ofReal t.2.1]
+  ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Area.Mamikon.Middle`.
+* `Area.Mamikon.TangentValues`.
+* `Area.Mamikon.SofaConvex`.
+* `Area.QVariation`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The middle Mamikon functional of a special cap
+
+`middleMamikon` is the evaluated four-term Mamikon decomposition of the part of a special cap's
+niche cut out by the two straight tangent paths on `[0, φᴿ]` and `[φᴸ, π / 2]`, the outer-corner arc
+on the middle window `[φᴿ, φᴸ]` and the terminal tangent path on `[π / 2, π]`.  This module proves
+that it agrees with `-upperBoundMiddle` modulo convex-linear functionals of the cap.
+
+The four summands already express their straight paths as endpoint segment areas, so the whole
+functional is a sum of eleven signed segment areas, one curve area and four convex arc areas.  The
+four arc areas add up to the cap area modulo a convex-linear functional
+(`specialCapArea_equivalent_upper_arcs`), which supplies the `-|K|` of the upper bound.  Of the
+eleven segments, five are convex-linear and therefore discarded
+(`middleMamikon_segments_isConvexLinear`): all their endpoints move convex-linearly with the cap and
+stay on the two fixed horizontal lines `y = 0` and `y = 1`, the latter being the top supporting
+line, so no determinant of two moving coordinates ever appears.  Two more pairs collapse at the two
+Gerver angles, where the extreme faces are singletons: at `φᴿ` the tangent-line intersection, the
+face and the outer corner are collinear, so the two segments merge
+(`middleMamikon_segments_merge_right`), and at `φᴸ` the tangent-line intersection *is* the outer
+corner, so the two segments cancel (`middleMamikon_segments_cancel_left`).  What remains are exactly
+the three comparisons of `cornerArea_equivalent_modulo_linear`, the last of them reversed.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The signed area between a convex boundary arc and the specified broken straight path. -/
+def straightMamikonValue (K : ConvexBody Point) (a b : ℝ) (p q : Point) : ℝ :=
+  segmentArea (edgeVertices K (a : Real.Angle)).1 p + segmentArea p q +
+    segmentArea q (edgeVertices K (b : Real.Angle)).2 - convexArcArea K a b
+
+/-- The middle Mamikon functional constructed from the special cap’s hallway-corner geometry. -/
+def middleMamikon (K : SpecialCapSpace) : ℝ :=
+  let B := K.val.val
+  let r := paperGerverConstants.2.1
+  let l := paperGerverConstants.2.2
+  let T := Real.pi / 2
+  let y := fun t : ℝ ↦ (rotatingHallwayParts (B : Set Point) (t : Real.Angle)).outerCorner
+  straightMamikonValue B 0 r
+      (supportingIntersection B 0 (T : Real.Angle))
+      (supportingIntersection B (r : Real.Angle) (T : Real.Angle)) +
+    (segmentArea (edgeVertices B (r : Real.Angle)).1 (y r) +
+      curveAreaFunctional (capOuterMiddleBV K) +
+      segmentArea (y l) (edgeVertices B (l : Real.Angle)).2 - convexArcArea B r l) +
+    straightMamikonValue B l T
+      (supportingIntersection B (l : Real.Angle) ((T + l : ℝ) : Real.Angle))
+      (supportingIntersection B (T : Real.Angle) ((T + l : ℝ) : Real.Angle)) +
+    tangentMamikonValue B T Real.pi
+
+/-! ### The singleton faces of a special cap and their positions -/
+
+/-- The extreme faces of a special cap at the upper normals other than the vertical one are
+singletons, because its injectivity condition supplies angular densities. -/
+private theorem specialCap_edgeVertices_eq (K : SpecialCapSpace) {t : ℝ}
+    (ht : t ∈ Set.Icc 0 Real.pi) (htop : t ≠ Real.pi / 2) :
+    (edgeVertices K.val.val (t : Real.Angle)).1 =
+      (edgeVertices K.val.val (t : Real.Angle)).2 := by
+  obtain ⟨r, s, hdens, -⟩ := K.property.1.1
+  exact capDensities_edgeVertices_eq K.val ⟨r, s, hdens⟩ ht htop
+
+/-- Both horizontal extreme faces of a special cap lie on the base line: they are singletons, and a
+singleton horizontal face of a right-angle cap cannot have positive height. -/
+private theorem specialCap_horizontal_edgeVertices_apply_one (K : SpecialCapSpace) :
+    (edgeVertices K.val.val ((0 : ℝ) : Real.Angle)).1 1 = 0 ∧
+      (edgeVertices K.val.val ((Real.pi : ℝ) : Real.Angle)).2 1 = 0 := by
+  have hpi := Real.pi_pos
+  have h0 := specialCap_edgeVertices_eq K (t := 0) ⟨le_rfl, hpi.le⟩
+    (by positivity : (0 : ℝ) < Real.pi / 2).ne
+  have hp := specialCap_edgeVertices_eq K (t := Real.pi) ⟨hpi.le, le_rfl⟩
+    (by linarith : Real.pi / 2 < Real.pi).ne'
+  refine ⟨K.val.edgeVertices_fst_apply_one_eq_zero Real.sin_zero h0, ?_⟩
+  rw [← hp]
+  exact K.val.edgeVertices_fst_apply_one_eq_zero Real.sin_pi hp
+
+/-- At the left Gerver angle the two supporting lines of the middle Mamikon loop are at angular
+difference `π / 2`, so they meet at the outer corner of the supporting hallway. -/
+private theorem specialCap_supportingIntersection_left_eq_outerCorner (K : SpecialCapSpace) :
+    supportingIntersection K.val.val (paperGerverConstants.2.2 : Real.Angle)
+        ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle) =
+      (rotatingHallwayParts (K.val.val : Set Point)
+        (paperGerverConstants.2.2 : Real.Angle)).outerCorner := by
+  rw [show (Real.pi / 2 + paperGerverConstants.2.2 : ℝ) =
+    paperGerverConstants.2.2 + Real.pi / 2 from by ring]
+  exact supportingIntersection_add_pi_div_two_eq_outerCorner _ _
+
+/-! ### The convex-linear segments -/
+
+/-- The signed area of the segment between two convex-linear point functionals of a special cap
+that both keep a constant height is convex-linear. -/
+private theorem segmentArea_isConvexLinear_of_apply_one_eq (P Q : SpecialCapSpace → Point)
+    {a b : ℝ}
+    (hP : ∀ (t : unitInterval) (K L : SpecialCapSpace),
+      P (specialCapCombination t K L) = (1 - (t : ℝ)) • P K + (t : ℝ) • P L)
+    (hQ : ∀ (t : unitInterval) (K L : SpecialCapSpace),
+      Q (specialCapCombination t K L) = (1 - (t : ℝ)) • Q K + (t : ℝ) • Q L)
+    (hPa : ∀ K, P K 1 = a) (hQb : ∀ K, Q K 1 = b) :
+    IsConvexLinear specialCapCombination realCombination fun K ↦ segmentArea (P K) (Q K) :=
+  fun t K L ↦ by
+    change segmentArea (P (specialCapCombination t K L)) (Q (specialCapCombination t K L)) =
+      realCombination t (segmentArea (P K) (Q K)) (segmentArea (P L) (Q L))
+    rw [hP, hQ, realCombination]
+    exact segmentArea_combination_of_apply_one_eq _ ((hPa K).trans (hPa L).symm)
+      ((hQb K).trans (hQb L).symm)
+
+/-- The five straight segments that the reductions leave in place have convex-linear total signed
+area.  In the order of the four-term definition they are the bottom segment at the horizontal normal
+`0`, the chord of the top supporting line from `l_K^T(0)` to `l_K^T(φᴿ)`, the chord from
+`l_K^{T + φᴸ}(T)` to the negative top vertex `v_K^-(T)`, the chord from the positive top vertex
+`v_K^+(T)` to `l_K^π(T)`, and the bottom segment at the normal `π`.  All ten endpoints are
+convex-linear in the cap, and each lies on one of the two fixed horizontal lines `y = 0` — the two
+singleton horizontal contacts — and `y = 1` — the top supporting line. -/
+private theorem middleMamikon_segments_isConvexLinear :
+    IsConvexLinear specialCapCombination realCombination fun K ↦
+      segmentArea (edgeVertices K.val.val ((0 : ℝ) : Real.Angle)).1
+          (supportingIntersection K.val.val 0 ((Real.pi / 2 : ℝ) : Real.Angle)) +
+        segmentArea (supportingIntersection K.val.val 0 ((Real.pi / 2 : ℝ) : Real.Angle))
+          (supportingIntersection K.val.val (paperGerverConstants.2.1 : Real.Angle)
+            ((Real.pi / 2 : ℝ) : Real.Angle)) +
+        segmentArea (supportingIntersection K.val.val ((Real.pi / 2 : ℝ) : Real.Angle)
+            ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle))
+          (edgeVertices K.val.val ((Real.pi / 2 : ℝ) : Real.Angle)).2 +
+        segmentArea (edgeVertices K.val.val ((Real.pi / 2 : ℝ) : Real.Angle)).1
+          (supportingIntersection K.val.val ((Real.pi / 2 : ℝ) : Real.Angle)
+            ((Real.pi : ℝ) : Real.Angle)) +
+        segmentArea (supportingIntersection K.val.val ((Real.pi / 2 : ℝ) : Real.Angle)
+            ((Real.pi : ℝ) : Real.Angle))
+          (edgeVertices K.val.val ((Real.pi : ℝ) : Real.Angle)).2 := by
+  have hpi := Real.pi_pos
+  obtain ⟨hr, hl, -⟩ := paperGerverConstants_snd_mem_Ioo
+  -- ### The endpoints move convex-linearly
+  have hev := fun (a : Real.Angle) (t : unitInterval) (M N : SpecialCapSpace) ↦
+    (specialCap_maps_linear t M N).1 a
+  have hsi := fun (a b : ℝ) (hab : a < b) (hba : b < a + Real.pi) (t : unitInterval)
+    (M N : SpecialCapSpace) ↦ (specialCap_maps_linear t M N).2 a b hab hba
+  have hsi0 : ∀ (t : unitInterval) (M N : SpecialCapSpace),
+      supportingIntersection (specialCapCombination t M N).val.val 0
+          ((Real.pi / 2 : ℝ) : Real.Angle) =
+        (1 - (t : ℝ)) • supportingIntersection M.val.val 0 ((Real.pi / 2 : ℝ) : Real.Angle) +
+          (t : ℝ) • supportingIntersection N.val.val 0 ((Real.pi / 2 : ℝ) : Real.Angle) :=
+    fun t M N ↦ by
+      simpa only [Real.Angle.coe_zero] using
+        hsi 0 (Real.pi / 2) (by positivity) (by linarith) t M N
+  -- ### The endpoints keep their heights
+  have htop : ∀ (M : SpecialCapSpace) (p : Point),
+      inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) =
+        supportValue (M.val.val : Set Point) ((Real.pi / 2 : ℝ) : Real.Angle) → p 1 = 1 :=
+    fun M _ hp ↦ M.val.apply_one_eq_one hp
+  have hsiL : ∀ (M : SpecialCapSpace) (s : ℝ),
+      supportingIntersection M.val.val ((Real.pi / 2 : ℝ) : Real.Angle) (s : Real.Angle) 1 = 1 :=
+    fun M s ↦ htop M _ (supportingIntersection_inner_left _ _ s)
+  have hsiR : ∀ (M : SpecialCapSpace) (s : ℝ), Real.sin (Real.pi / 2 - s) ≠ 0 →
+      supportingIntersection M.val.val (s : Real.Angle) ((Real.pi / 2 : ℝ) : Real.Angle) 1 = 1 :=
+    fun M s hs ↦ htop M _ (supportingIntersection_inner_right _ s _ hs)
+  have hsi0h : ∀ M : SpecialCapSpace,
+      supportingIntersection M.val.val 0 ((Real.pi / 2 : ℝ) : Real.Angle) 1 = 1 := fun M ↦ by
+    simpa only [Real.Angle.coe_zero] using
+      hsiR M 0 (by rw [sub_zero, Real.sin_pi_div_two]; norm_num)
+  have hvtop : ∀ M : SpecialCapSpace,
+      (edgeVertices M.val.val ((Real.pi / 2 : ℝ) : Real.Angle)).1 1 = 1 ∧
+        (edgeVertices M.val.val ((Real.pi / 2 : ℝ) : Real.Angle)).2 1 = 1 :=
+    fun M ↦ ⟨htop M _ (edgeVertices_fst_mem _ _).2, htop M _ (edgeVertices_snd_mem _ _).2⟩
+  have hsinr : Real.sin (Real.pi / 2 - paperGerverConstants.2.1) ≠ 0 := by
+    rw [Real.sin_pi_div_two_sub]
+    exact (Real.cos_pos_of_mem_Ioo ⟨by linarith [hr.1], hr.2⟩).ne'
+  -- ### The five segments
+  have h1 := segmentArea_isConvexLinear_of_apply_one_eq
+    (fun M ↦ (edgeVertices M.val.val ((0 : ℝ) : Real.Angle)).1)
+    (fun M ↦ supportingIntersection M.val.val 0 ((Real.pi / 2 : ℝ) : Real.Angle))
+    (fun t M N ↦ (hev _ t M N).1) hsi0
+    (fun M ↦ (specialCap_horizontal_edgeVertices_apply_one M).1) hsi0h
+  have h2 := segmentArea_isConvexLinear_of_apply_one_eq
+    (fun M ↦ supportingIntersection M.val.val 0 ((Real.pi / 2 : ℝ) : Real.Angle))
+    (fun M ↦ supportingIntersection M.val.val (paperGerverConstants.2.1 : Real.Angle)
+      ((Real.pi / 2 : ℝ) : Real.Angle))
+    hsi0 (hsi paperGerverConstants.2.1 (Real.pi / 2) hr.2 (by linarith [hr.1]))
+    hsi0h (fun M ↦ hsiR M paperGerverConstants.2.1 hsinr)
+  have h3 := segmentArea_isConvexLinear_of_apply_one_eq
+    (fun M ↦ supportingIntersection M.val.val ((Real.pi / 2 : ℝ) : Real.Angle)
+      ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle))
+    (fun M ↦ (edgeVertices M.val.val ((Real.pi / 2 : ℝ) : Real.Angle)).2)
+    (hsi (Real.pi / 2) (Real.pi / 2 + paperGerverConstants.2.2) (by linarith [hl.1])
+      (by linarith [hl.2]))
+    (fun t M N ↦ (hev _ t M N).2)
+    (fun M ↦ hsiL M (Real.pi / 2 + paperGerverConstants.2.2)) (fun M ↦ (hvtop M).2)
+  have h4 := segmentArea_isConvexLinear_of_apply_one_eq
+    (fun M ↦ (edgeVertices M.val.val ((Real.pi / 2 : ℝ) : Real.Angle)).1)
+    (fun M ↦ supportingIntersection M.val.val ((Real.pi / 2 : ℝ) : Real.Angle)
+      ((Real.pi : ℝ) : Real.Angle))
+    (fun t M N ↦ (hev _ t M N).1) (hsi (Real.pi / 2) Real.pi (by linarith) (by linarith))
+    (fun M ↦ (hvtop M).1) (fun M ↦ hsiL M Real.pi)
+  have h5 := segmentArea_isConvexLinear_of_apply_one_eq
+    (fun M ↦ supportingIntersection M.val.val ((Real.pi / 2 : ℝ) : Real.Angle)
+      ((Real.pi : ℝ) : Real.Angle))
+    (fun M ↦ (edgeVertices M.val.val ((Real.pi : ℝ) : Real.Angle)).2)
+    (hsi (Real.pi / 2) Real.pi (by linarith) (by linarith)) (fun t M N ↦ (hev _ t M N).2)
+    (fun M ↦ hsiL M Real.pi)
+    (fun M ↦ (specialCap_horizontal_edgeVertices_apply_one M).2)
+  intro t M N
+  have e1 := h1 t M N
+  have e2 := h2 t M N
+  have e3 := h3 t M N
+  have e4 := h4 t M N
+  have e5 := h5 t M N
+  simp only [realCombination] at e1 e2 e3 e4 e5 ⊢
+  linear_combination e1 + e2 + e3 + e4 + e5
+
+/-! ### The two collapsing pairs at the Gerver angles -/
+
+/-- At the right Gerver angle the tangent-line intersection, the singleton extreme face and the
+outer corner all lie on the same supporting line, so the two segments through the face merge into a
+single chord. -/
+private theorem middleMamikon_segments_merge_right (K : SpecialCapSpace) :
+    segmentArea (supportingIntersection K.val.val (paperGerverConstants.2.1 : Real.Angle)
+          ((Real.pi / 2 : ℝ) : Real.Angle))
+        (edgeVertices K.val.val (paperGerverConstants.2.1 : Real.Angle)).2 +
+      segmentArea (edgeVertices K.val.val (paperGerverConstants.2.1 : Real.Angle)).1
+        (rotatingHallwayParts (K.val.val : Set Point)
+          (paperGerverConstants.2.1 : Real.Angle)).outerCorner =
+    segmentArea (supportingIntersection K.val.val (paperGerverConstants.2.1 : Real.Angle)
+        ((Real.pi / 2 : ℝ) : Real.Angle))
+      (rotatingHallwayParts (K.val.val : Set Point)
+        (paperGerverConstants.2.1 : Real.Angle)).outerCorner := by
+  have hpi := Real.pi_pos
+  obtain ⟨hr, -, -⟩ := paperGerverConstants_snd_mem_Ioo
+  have h1 := supportingIntersection_inner_left K.val.val paperGerverConstants.2.1 (Real.pi / 2)
+  have h2 : inner ℝ (edgeVertices K.val.val (paperGerverConstants.2.1 : Real.Angle)).2
+      (normalVector (paperGerverConstants.2.1 : Real.Angle)) =
+      supportValue (K.val.val : Set Point) (paperGerverConstants.2.1 : Real.Angle) :=
+    (edgeVertices_snd_mem _ _).2
+  have h3 : inner ℝ (rotatingHallwayParts (K.val.val : Set Point)
+      (paperGerverConstants.2.1 : Real.Angle)).outerCorner
+      (normalVector (paperGerverConstants.2.1 : Real.Angle)) =
+      supportValue (K.val.val : Set Point) (paperGerverConstants.2.1 : Real.Angle) := by
+    rw [outerCorner_eq_support_sum, inner_add_left, real_inner_smul_left, real_inner_smul_left,
+      inner_normalVector_self, inner_tangentVector_normalVector_real, sub_self, Real.sin_zero]
+    ring
+  have h := segmentArea_sub_segmentArea_of_inner_normalVector_eq h1 h3 h2
+  rw [specialCap_edgeVertices_eq K ⟨hr.1.le, by linarith [hr.2]⟩ hr.2.ne]
+  linarith [segmentArea_swap (edgeVertices K.val.val
+    (paperGerverConstants.2.1 : Real.Angle)).2
+    (rotatingHallwayParts (K.val.val : Set Point)
+      (paperGerverConstants.2.1 : Real.Angle)).outerCorner]
+
+/-- At the left Gerver angle the extreme face is a single point and the tangent-line intersection is
+the outer corner, so the two segments through the face cancel by antisymmetry. -/
+private theorem middleMamikon_segments_cancel_left (K : SpecialCapSpace) :
+    segmentArea (rotatingHallwayParts (K.val.val : Set Point)
+          (paperGerverConstants.2.2 : Real.Angle)).outerCorner
+        (edgeVertices K.val.val (paperGerverConstants.2.2 : Real.Angle)).2 +
+      segmentArea (edgeVertices K.val.val (paperGerverConstants.2.2 : Real.Angle)).1
+        (supportingIntersection K.val.val (paperGerverConstants.2.2 : Real.Angle)
+          ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle)) = 0 := by
+  have hpi := Real.pi_pos
+  obtain ⟨-, hl, -⟩ := paperGerverConstants_snd_mem_Ioo
+  rw [specialCap_supportingIntersection_left_eq_outerCorner,
+    specialCap_edgeVertices_eq K ⟨hl.1.le, by linarith [hl.2]⟩ hl.2.ne]
+  linarith [segmentArea_swap (rotatingHallwayParts (K.val.val : Set Point)
+    (paperGerverConstants.2.2 : Real.Angle)).outerCorner
+    (edgeVertices K.val.val (paperGerverConstants.2.2 : Real.Angle)).2]
+
+/-! ### The equivalence -/
+
+theorem middleMamikon_equivalent_neg_upperBoundMiddle :
+    EquivalentModuloConvexLinear specialCapCombination middleMamikon
+      (fun K ↦ -upperBoundMiddle K) := by
+  obtain ⟨F, hFval, hmid, hright, hleft⟩ := cornerArea_equivalent_modulo_linear
+  have hF : ∀ M : SpecialCapSpace, F M = capOuterMiddleBV M := fun M ↦
+    Subtype.ext ((hFval M).trans (funext (capOuterMiddleBV_val M)).symm)
+  -- The chord closing the third straight path reverses the left comparison chord.
+  have hrev : ∀ M : SpecialCapSpace,
+      segmentArea (supportingIntersection M.val.val (paperGerverConstants.2.2 : Real.Angle)
+            ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle))
+          (supportingIntersection M.val.val ((Real.pi / 2 : ℝ) : Real.Angle)
+            ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle)) =
+        -segmentArea (supportingIntersection M.val.val ((Real.pi / 2 : ℝ) : Real.Angle)
+            ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle))
+          (rotatingHallwayParts (M.val.val : Set Point)
+            (paperGerverConstants.2.2 : Real.Angle)).outerCorner := fun M ↦ by
+    rw [specialCap_supportingIntersection_left_eq_outerCorner]
+    exact segmentArea_swap _ _
+  -- The right wedge segment of the upper bound runs in the opposite orientation.
+  have hswap : ∀ M : SpecialCapSpace,
+      segmentArea (distinguishedCapSides M.val).1.corner
+          (distinguishedCapSides M.val).1.fanPoint =
+        -segmentArea (distinguishedCapSides M.val).1.fanPoint
+          (distinguishedCapSides M.val).1.corner := fun M ↦ segmentArea_swap _ _
+  intro t K L
+  have e1 := specialCapArea_equivalent_upper_arcs t K L
+  have e2 := hmid t K L
+  have e3 := hright t K L
+  have e4 := hleft t K L
+  have f := middleMamikon_segments_isConvexLinear t K L
+  simp only [hF, realCombination] at e1 e2 e3 e4 f
+  simp only [middleMamikon, straightMamikonValue, tangentMamikonValue, upperBoundMiddle,
+    realCombination, hrev, hswap]
+  linear_combination f + e1 + e2 + e3 - e4 +
+    middleMamikon_segments_merge_right (specialCapCombination t K L) +
+    middleMamikon_segments_cancel_left (specialCapCombination t K L) -
+    (1 - (t : ℝ)) * (middleMamikon_segments_merge_right K +
+      middleMamikon_segments_cancel_left K) -
+    (t : ℝ) * (middleMamikon_segments_merge_right L +
+      middleMamikon_segments_cancel_left L)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Mamikon values of tangent-line chords
+
+For a fixed tangent normal `q`, the tangent-line parametrization of a convex body is a
+continuous bounded-variation path, convex-linear in the body, each of whose values lies on the
+supporting line at its own path parameter.  Mamikon convexity therefore makes the enclosed area
+`straightMamikonValue` a convex quadratic functional of the body, and the terminal case `q = b`
+does the same for `tangentMamikonValue`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The Mamikon value of a straight tangent-line chord is a convex quadratic functional of the
+body. -/
+theorem straightMamikon_quadratic_convex (t a b : ℝ)
+    (ha : a ∈ Set.Ioc (t - Real.pi) t) (hb : b ∈ Set.Ioc (t - Real.pi) t)
+    (hab : a < b) (hba : b < a + Real.pi) :
+    IsQuadraticFunctional convexBodyCombination
+        (fun K ↦ straightMamikonValue K a b (tangentLinePath K t ⟨a, ha⟩)
+          (tangentLinePath K t ⟨b, hb⟩)) ∧
+      IsConvexFunctional convexBodyCombination
+        (fun K ↦ straightMamikonValue K a b (tangentLinePath K t ⟨a, ha⟩)
+          (tangentLinePath K t ⟨b, hb⟩)) false := by
+  obtain ⟨F, hFval, hFlin⟩ := tangentLinePath_convexLinear t a b ha hb hab.le
+  have hF : ∀ (K : ConvexBody Point) (s : Set.Icc a b),
+      (F K).val s ∈ (supportingLineHalfPlane (K : Set Point) ((s : ℝ) : Real.Angle)).1 := by
+    intro K s
+    rw [hFval K]
+    exact tangentLinePath_mem_supportingLine K t
+      ⟨(s : ℝ), lt_of_lt_of_le ha.1 s.property.1, le_trans s.property.2 hb.2⟩
+  have hval : ∀ K : ConvexBody Point,
+      mamikonFunctional K a b hab hba (F K) (hF K) =
+        straightMamikonValue K a b (tangentLinePath K t ⟨a, ha⟩)
+          (tangentLinePath K t ⟨b, hb⟩) := by
+    intro K
+    have hchoose : F K = (tangentLinePath_segment_area K t a b ha hb hab.le).choose :=
+      Subtype.ext ((hFval K).trans
+        (tangentLinePath_segment_area K t a b ha hb hab.le).choose_spec.1.symm)
+    have hstart : (F K).val ⟨a, le_rfl, hab.le⟩ = tangentLinePath K t ⟨a, ha⟩ := by
+      rw [hFval K]; rfl
+    have hend : (F K).val ⟨b, hab.le, le_rfl⟩ = tangentLinePath K t ⟨b, hb⟩ := by
+      rw [hFval K]; rfl
+    have harea : curveAreaFunctional (F K) =
+        segmentArea (tangentLinePath K t ⟨a, ha⟩) (tangentLinePath K t ⟨b, hb⟩) := by
+      rw [hchoose]
+      exact (tangentLinePath_segment_area K t a b ha hb hab.le).choose_spec.2.2.2
+    unfold mamikonFunctional straightMamikonValue
+    rw [hstart, hend, harea]
+  have hmain := mamikon_quadratic_convex a b hab hba F hF hFlin
+  rw [funext hval] at hmain
+  exact hmain
+
+/-- The Mamikon value of a terminal tangent normal is a convex quadratic functional of the body. -/
+theorem tangentMamikon_quadratic_convex (a b : ℝ) (hab : a < b) (hba : b < a + Real.pi) :
+    IsQuadraticFunctional convexBodyCombination (fun K ↦ tangentMamikonValue K a b) ∧
+      IsConvexFunctional convexBodyCombination (fun K ↦ tangentMamikonValue K a b) false := by
+  have ha : a ∈ Set.Ioc (b - Real.pi) b := ⟨by linarith, hab.le⟩
+  have hb : b ∈ Set.Ioc (b - Real.pi) b := ⟨by linarith [Real.pi_pos], le_rfl⟩
+  have hmain := straightMamikon_quadratic_convex b a b ha hb hab hba
+  have hval : ∀ K : ConvexBody Point,
+      straightMamikonValue K a b (tangentLinePath K b ⟨a, ha⟩) (tangentLinePath K b ⟨b, hb⟩) =
+        tangentMamikonValue K a b := by
+    intro K
+    have hstart : tangentLinePath K b ⟨a, ha⟩ =
+        supportingIntersection K (a : Real.Angle) (b : Real.Angle) := by
+      unfold tangentLinePath
+      rw [ite_eq_left hab]
+    have hend : tangentLinePath K b ⟨b, hb⟩ = (edgeVertices K (b : Real.Angle)).2 := by
+      unfold tangentLinePath
+      rw [ite_eq_right (lt_irrefl b)]
+    have hdeg : segmentArea (edgeVertices K (b : Real.Angle)).2
+        (edgeVertices K (b : Real.Angle)).2 = 0 := by
+      have h := segmentArea_swap (edgeVertices K (b : Real.Angle)).2
+        (edgeVertices K (b : Real.Angle)).2
+      linarith
+    unfold straightMamikonValue tangentMamikonValue
+    rw [hstart, hend, hdeg]
+    ring
+  rw [funext hval] at hmain
+  exact hmain
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Area / Mamikon / Sofa Convex
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem sofaMamikon_quadratic_convex :
+    IsConvexFunctional specialCapCombination middleMamikon false ∧
+    IsQuadraticFunctional specialCapCombination middleMamikon ∧
+    IsConvexFunctional convexBodyCombination rightTailMamikon false ∧
+    IsQuadraticFunctional convexBodyCombination rightTailMamikon ∧
+    IsConvexFunctional convexBodyCombination leftTailMamikon false ∧
+    IsQuadraticFunctional convexBodyCombination leftTailMamikon := by
+  obtain ⟨hrIoo, hlIoo, hsum⟩ := paperGerverConstants_snd_mem_Ioo
+  have hpi := Real.pi_pos
+  have hrl := paperGerverConstants_snd_fst_lt_snd_snd
+  have hr0 := hrIoo.1
+  have hr2 := hrIoo.2
+  have hl0 := hlIoo.1
+  have hl2 := hlIoo.2
+  -- ### The two tail functionals are terminal tangent Mamikon values
+  have htailR := tangentMamikon_quadratic_convex (Real.pi + paperGerverConstants.2.1)
+    (3 * Real.pi / 2) (by linarith) (by linarith)
+  have htailL := tangentMamikon_quadratic_convex (3 * Real.pi / 2)
+    (3 * Real.pi / 2 + paperGerverConstants.2.2) (by linarith) (by linarith)
+  -- ### The first middle summand
+  have ha1 : (0 : ℝ) ∈ Set.Ioc (Real.pi / 2 - Real.pi) (Real.pi / 2) :=
+    ⟨by linarith, by linarith⟩
+  have hb1 : paperGerverConstants.2.1 ∈ Set.Ioc (Real.pi / 2 - Real.pi) (Real.pi / 2) :=
+    ⟨by linarith, by linarith⟩
+  have hmid1 := straightMamikon_quadratic_convex (Real.pi / 2) 0 paperGerverConstants.2.1
+    ha1 hb1 (by linarith) (by linarith)
+  -- ### The third middle summand
+  have ha3 : paperGerverConstants.2.2 ∈
+      Set.Ioc (Real.pi / 2 + paperGerverConstants.2.2 - Real.pi)
+        (Real.pi / 2 + paperGerverConstants.2.2) := ⟨by linarith, by linarith⟩
+  have hb3 : (Real.pi / 2 : ℝ) ∈
+      Set.Ioc (Real.pi / 2 + paperGerverConstants.2.2 - Real.pi)
+        (Real.pi / 2 + paperGerverConstants.2.2) := ⟨by linarith, by linarith⟩
+  have hmid3 := straightMamikon_quadratic_convex (Real.pi / 2 + paperGerverConstants.2.2)
+    paperGerverConstants.2.2 (Real.pi / 2) ha3 hb3 (by linarith) (by linarith)
+  -- ### The fourth middle summand
+  have hmid4 := tangentMamikon_quadratic_convex (Real.pi / 2) Real.pi (by linarith) (by linarith)
+  -- ### The second middle summand, the Mamikon value of the outer-corner path
+  obtain ⟨γ, hγ, hγlin⟩ := exists_outerCornerBV_convexLinear paperGerverConstants.2.1
+    paperGerverConstants.2.2
+  have hγmem : ∀ (B : ConvexBody Point)
+      (s : Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2),
+      (γ B).val s ∈ (supportingLineHalfPlane (B : Set Point) ((s : ℝ) : Real.Angle)).1 := by
+    intro B s
+    rw [hγ B s]
+    change inner ℝ (rotatingHallwayParts (B : Set Point) ((s : ℝ) : Real.Angle)).outerCorner
+      (normalVector ((s : ℝ) : Real.Angle)) = supportValue (B : Set Point) ((s : ℝ) : Real.Angle)
+    rw [outerCorner_eq_support_sum B (s : ℝ), inner_add_left, real_inner_smul_left,
+      real_inner_smul_left, inner_normalVector_self, real_inner_comm,
+      inner_normalVector_tangentVector]
+    ring
+  have hmid2 := mamikon_quadratic_convex paperGerverConstants.2.1 paperGerverConstants.2.2
+    hrl (by linarith) γ hγmem hγlin
+  -- ### The four middle summands assemble to the middle functional on all bodies
+  have hquadAll := ((hmid1.1.add hmid2.1).add hmid3.1).add hmid4.1
+  have hconvAll := ((hmid1.2.add hmid2.2).add hmid3.2).add hmid4.2
+  have hbody : IsConvexLinear specialCapCombination convexBodyCombination
+      fun K : SpecialCapSpace ↦ K.val.val := specialCap_isConvexDomain.1
+  have hquad := hquadAll.comp_isConvexLinear hbody
+  have hconv := hconvAll.comp_isConvexLinear hbody
+  have hmideq : (fun K : SpecialCapSpace ↦
+      ((straightMamikonValue K.val.val 0 paperGerverConstants.2.1
+          (tangentLinePath K.val.val (Real.pi / 2) ⟨0, ha1⟩)
+          (tangentLinePath K.val.val (Real.pi / 2) ⟨paperGerverConstants.2.1, hb1⟩) +
+        mamikonFunctional K.val.val paperGerverConstants.2.1 paperGerverConstants.2.2 hrl
+          (by linarith) (γ K.val.val) (hγmem K.val.val)) +
+        straightMamikonValue K.val.val paperGerverConstants.2.2 (Real.pi / 2)
+          (tangentLinePath K.val.val (Real.pi / 2 + paperGerverConstants.2.2)
+            ⟨paperGerverConstants.2.2, ha3⟩)
+          (tangentLinePath K.val.val (Real.pi / 2 + paperGerverConstants.2.2)
+            ⟨Real.pi / 2, hb3⟩)) +
+        tangentMamikonValue K.val.val (Real.pi / 2) Real.pi) = middleMamikon := by
+    funext K
+    have hpath1 : tangentLinePath K.val.val (Real.pi / 2) ⟨0, ha1⟩ =
+        supportingIntersection K.val.val 0 ((Real.pi / 2 : ℝ) : Real.Angle) := by
+      unfold tangentLinePath
+      rw [ite_eq_left (show (0 : ℝ) < Real.pi / 2 by linarith), Real.Angle.coe_zero]
+    have hpath2 : tangentLinePath K.val.val (Real.pi / 2)
+        ⟨paperGerverConstants.2.1, hb1⟩ =
+        supportingIntersection K.val.val ((paperGerverConstants.2.1 : ℝ) : Real.Angle)
+          ((Real.pi / 2 : ℝ) : Real.Angle) := by
+      unfold tangentLinePath
+      rw [ite_eq_left hr2]
+    have hpath3 : tangentLinePath K.val.val (Real.pi / 2 + paperGerverConstants.2.2)
+        ⟨paperGerverConstants.2.2, ha3⟩ =
+        supportingIntersection K.val.val ((paperGerverConstants.2.2 : ℝ) : Real.Angle)
+          ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle) := by
+      unfold tangentLinePath
+      rw [ite_eq_left (by linarith)]
+    have hpath4 : tangentLinePath K.val.val (Real.pi / 2 + paperGerverConstants.2.2)
+        ⟨Real.pi / 2, hb3⟩ =
+        supportingIntersection K.val.val ((Real.pi / 2 : ℝ) : Real.Angle)
+          ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle) := by
+      unfold tangentLinePath
+      rw [ite_eq_left (by linarith)]
+    have hγeq : γ K.val.val = capOuterMiddleBV K :=
+      Subtype.ext (funext fun s ↦ (hγ K.val.val s).trans (capOuterMiddleBV_val K s).symm)
+    have hmam : mamikonFunctional K.val.val paperGerverConstants.2.1
+        paperGerverConstants.2.2 hrl (by linarith) (γ K.val.val) (hγmem K.val.val) =
+        segmentArea (edgeVertices K.val.val ((paperGerverConstants.2.1 : ℝ) : Real.Angle)).1
+            ((rotatingHallwayParts (K.val.val : Set Point)
+              ((paperGerverConstants.2.1 : ℝ) : Real.Angle)).outerCorner) +
+          curveAreaFunctional (capOuterMiddleBV K) +
+          segmentArea ((rotatingHallwayParts (K.val.val : Set Point)
+              ((paperGerverConstants.2.2 : ℝ) : Real.Angle)).outerCorner)
+            (edgeVertices K.val.val ((paperGerverConstants.2.2 : ℝ) : Real.Angle)).2 -
+          convexArcArea K.val.val paperGerverConstants.2.1 paperGerverConstants.2.2 := by
+      unfold mamikonFunctional
+      rw [hγ K.val.val ⟨paperGerverConstants.2.1, le_rfl, hrl.le⟩,
+        hγ K.val.val ⟨paperGerverConstants.2.2, hrl.le, le_rfl⟩, hγeq]
+    change _ = middleMamikon K
+    unfold middleMamikon
+    rw [hpath1, hpath2, hpath3, hpath4, hmam]
+  rw [hmideq] at hquad hconv
+  exact ⟨hconv, hquad, htailR.2, htailR.1, htailL.2, htailL.1⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The directional derivative of the upper bound `𝒬`
+
+`upperBoundQ` is a signed sum of six area functionals of a cap-tail triple: the cap area, the two
+tail arc areas, the inner-corner curve area and the two areas of the segments joining a tail
+endpoint to the corresponding cap corner.  Each summand is quadratic along the barycentric
+interpolation of cap-tail triples, so each segment function is differentiable at the base point
+with the summand's `convexDirectionalDerivative` as its derivative; adding those six derivatives
+computes the derivative of the segment function of `𝒬` itself.
+
+Five of the twelve endpoint contributions cancel in pairs.  The remaining two are the segment
+areas at the far ends of the two tails, and they vanish because the cap-tail constraints force the
+support value of both tails in the direction `3π/2` to be zero, so all four points involved lie on
+the horizontal axis.  What survives is `qVariationIntegral`: the cap surface integral, the
+inner-corner integral, and the two tail integrals rewritten in opposite-angle coordinates.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- The support-function variation integral for the cap and its two tail bodies. -/
+def qVariationIntegral (X Y : CapTailSpace) : ℝ :=
+  (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Icc 0 Real.pi,
+    (supportValue Y.cap.val.val t - supportValue X.cap.val.val t)
+      ∂surfaceAreaMeasure X.cap.val.val) -
+  (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) ''
+      (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2 ∪
+        Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+          (Real.pi / 2 + paperGerverConstants.2.2)),
+    (supportValue Y.cap.val.val t - supportValue X.cap.val.val t)
+      ∂capCornerAngleMeasure X.cap) +
+  (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) ''
+      Set.Ioo paperGerverConstants.2.1 (Real.pi / 2),
+    ((oppositeSurfaceData Y.rightBody).2 t - (oppositeSurfaceData X.rightBody).2 t)
+      ∂(oppositeSurfaceData X.rightBody).1) +
+  (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) ''
+      Set.Ioo (Real.pi / 2) (Real.pi / 2 + paperGerverConstants.2.2),
+    ((oppositeSurfaceData Y.leftBody).2 t - (oppositeSurfaceData X.leftBody).2 t)
+      ∂(oppositeSurfaceData X.leftBody).1)
+
+private theorem capTail_leftPair_isConvexLinear :
+    IsConvexLinear capTailCombination pointPairCombination
+      (fun Z : CapTailSpace ↦ ((rightLeftTailArcs Z.rightBody Z.leftBody).2.endPoint,
+        (distinguishedCapSides Z.cap.val).2.corner)) := by
+  obtain ⟨hcomb, -⟩ := capTail_isConvexDomain
+  intro t Z W
+  have hvertex : (rightLeftTailArcs (capTailCombination t Z W).rightBody
+      (capTailCombination t Z W).leftBody).2.endPoint =
+      (1 - (t : ℝ)) • (rightLeftTailArcs Z.rightBody Z.leftBody).2.endPoint +
+        (t : ℝ) • (rightLeftTailArcs W.rightBody W.leftBody).2.endPoint := by
+    change (edgeVertices (capTailCombination t Z W).leftBody
+      ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle)).2 = _
+    rw [(hcomb t Z W).2.2]
+    exact ((convexBody_maps_linear t Z.leftBody W.leftBody).2.1 _).2
+  have hcorner : (distinguishedCapSides (capTailCombination t Z W).cap.val).2.corner =
+      (1 - (t : ℝ)) • (distinguishedCapSides Z.cap.val).2.corner +
+        (t : ℝ) • (distinguishedCapSides W.cap.val).2.corner :=
+    capInnerCorner_of_eq_convexBodyCombination (hcomb t Z W).1 paperGerverConstants.2.2
+  change ((rightLeftTailArcs (capTailCombination t Z W).rightBody
+    (capTailCombination t Z W).leftBody).2.endPoint,
+    (distinguishedCapSides (capTailCombination t Z W).cap.val).2.corner) = _
+  rw [hvertex, hcorner]
+  rfl
+
+private theorem capTail_rightPair_isConvexLinear :
+    IsConvexLinear capTailCombination pointPairCombination
+      (fun Z : CapTailSpace ↦ ((distinguishedCapSides Z.cap.val).1.corner,
+        (rightLeftTailArcs Z.rightBody Z.leftBody).1.startPoint)) := by
+  obtain ⟨hcomb, -⟩ := capTail_isConvexDomain
+  intro t Z W
+  have hcorner : (distinguishedCapSides (capTailCombination t Z W).cap.val).1.corner =
+      (1 - (t : ℝ)) • (distinguishedCapSides Z.cap.val).1.corner +
+        (t : ℝ) • (distinguishedCapSides W.cap.val).1.corner :=
+    capInnerCorner_of_eq_convexBodyCombination (hcomb t Z W).1 paperGerverConstants.2.1
+  have hvertex : (rightLeftTailArcs (capTailCombination t Z W).rightBody
+      (capTailCombination t Z W).leftBody).1.startPoint =
+      (1 - (t : ℝ)) • (rightLeftTailArcs Z.rightBody Z.leftBody).1.startPoint +
+        (t : ℝ) • (rightLeftTailArcs W.rightBody W.leftBody).1.startPoint := by
+    change (edgeVertices (capTailCombination t Z W).rightBody
+      ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle)).1 = _
+    rw [(hcomb t Z W).2.1]
+    exact ((convexBody_maps_linear t Z.rightBody W.rightBody).2.1 _).1
+  change ((distinguishedCapSides (capTailCombination t Z W).cap.val).1.corner,
+    (rightLeftTailArcs (capTailCombination t Z W).rightBody
+      (capTailCombination t Z W).leftBody).1.startPoint) = _
+  rw [hcorner, hvertex]
+  rfl
+
+private theorem capTail_far_segmentAreas (X Y : CapTailSpace) :
+    (segmentArea (rightLeftTailArcs X.rightBody X.leftBody).1.endPoint
+      (rightLeftTailArcs Y.rightBody Y.leftBody).1.endPoint = 0) ∧
+    (segmentArea (rightLeftTailArcs X.rightBody X.leftBody).2.startPoint
+      (rightLeftTailArcs Y.rightBody Y.leftBody).2.startPoint = 0) := by
+  -- ### The far endpoints of the two tails lie on the horizontal axis
+  have hsupp : ∀ Z : CapTailSpace,
+      supportValue (Z.rightBody : Set Point) ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 ∧
+        supportValue (Z.leftBody : Set Point) ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := by
+    intro Z
+    have hcapval :
+        supportValue (Z.cap.val.val : Set Point) ((Real.pi / 2 : ℝ) : Real.Angle) = 1 :=
+      Z.cap.val.property.2.2.2.1
+    have hr := Z.right_eq (Real.pi / 2) (by simp)
+    have hl := Z.left_eq 0 (by simp)
+    rw [show ((Real.pi + Real.pi / 2 : ℝ) : Real.Angle) =
+      ((3 * Real.pi / 2 : ℝ) : Real.Angle) from by congr 1; ring] at hr
+    rw [show ((Real.pi / 2 + (0 : ℝ) : ℝ) : Real.Angle) =
+        ((Real.pi / 2 : ℝ) : Real.Angle) from by congr 1; ring,
+      show ((3 * Real.pi / 2 + (0 : ℝ) : ℝ) : Real.Angle) =
+        ((3 * Real.pi / 2 : ℝ) : Real.Angle) from by congr 1; ring] at hl
+    exact ⟨by linarith, by linarith⟩
+  have hheight : ∀ K : ConvexBody Point,
+      supportValue (K : Set Point) ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 →
+      (edgeVertices K ((3 * Real.pi / 2 : ℝ) : Real.Angle)).1 1 = 0 ∧
+        (edgeVertices K ((3 * Real.pi / 2 : ℝ) : Real.Angle)).2 1 = 0 := by
+    intro K hK
+    have hn : normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle) =
+        -normalVector ((Real.pi / 2 : ℝ) : Real.Angle) := by
+      rw [show ((3 * Real.pi / 2 : ℝ) : Real.Angle) =
+        ((Real.pi / 2 + Real.pi : ℝ) : Real.Angle) from by congr 1; ring]
+      exact normalVector_add_pi _
+    have key : ∀ p : Point,
+        inner ℝ p (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) = 0 → p 1 = 0 := by
+      intro p hp
+      rw [hn, inner_neg_right, inner_normalVector_real, Real.cos_pi_div_two,
+        Real.sin_pi_div_two] at hp
+      linarith
+    have hfst : inner ℝ (edgeVertices K ((3 * Real.pi / 2 : ℝ) : Real.Angle)).1
+        (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) =
+        supportValue (K : Set Point) ((3 * Real.pi / 2 : ℝ) : Real.Angle) :=
+      (edgeVertices_fst_mem K _).2
+    have hsnd : inner ℝ (edgeVertices K ((3 * Real.pi / 2 : ℝ) : Real.Angle)).2
+        (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) =
+        supportValue (K : Set Point) ((3 * Real.pi / 2 : ℝ) : Real.Angle) :=
+      (edgeVertices_snd_mem K _).2
+    exact ⟨key _ (hfst.trans hK), key _ (hsnd.trans hK)⟩
+  have hzeroSegment : ∀ p q : Point, p 1 = 0 → q 1 = 0 → segmentArea p q = 0 := by
+    intro p q hp hq
+    simp [segmentArea, planeCrossProduct, hp, hq]
+  have hrightFar : segmentArea (rightLeftTailArcs X.rightBody X.leftBody).1.endPoint
+      (rightLeftTailArcs Y.rightBody Y.leftBody).1.endPoint = 0 :=
+    hzeroSegment _ _ (hheight X.rightBody (hsupp X).1).2 (hheight Y.rightBody (hsupp Y).1).2
+  have hleftFar : segmentArea (rightLeftTailArcs X.rightBody X.leftBody).2.startPoint
+      (rightLeftTailArcs Y.rightBody Y.leftBody).2.startPoint = 0 :=
+    hzeroSegment _ _ (hheight X.leftBody (hsupp X).2).1 (hheight Y.leftBody (hsupp Y).2).1
+  exact ⟨hrightFar, hleftFar⟩
+
+private theorem capTail_left_segment_variation (X Y : CapTailSpace)
+    (hL : (rightLeftTailArcs X.rightBody X.leftBody).2.endPoint =
+      (distinguishedCapSides X.cap.val).2.corner) :
+    convexDirectionalDerivative capTailCombination
+     (fun Z : CapTailSpace ↦ segmentArea (rightLeftTailArcs Z.rightBody Z.leftBody).2.endPoint
+       (distinguishedCapSides Z.cap.val).2.corner) X Y =
+     segmentArea (distinguishedCapSides X.cap.val).2.corner
+         (distinguishedCapSides Y.cap.val).2.corner -
+       segmentArea (rightLeftTailArcs X.rightBody X.leftBody).2.endPoint
+         (rightLeftTailArcs Y.rightBody Y.leftBody).2.endPoint := by
+  have hleftPair := capTail_leftPair_isConvexLinear
+  have hvar := segmentArea_variation.2
+    (rightLeftTailArcs X.rightBody X.leftBody).2.endPoint
+    (distinguishedCapSides X.cap.val).2.corner
+    (rightLeftTailArcs Y.rightBody Y.leftBody).2.endPoint
+    (distinguishedCapSides Y.cap.val).2.corner
+  have hbulk : (planeCrossProduct
+      ((rightLeftTailArcs Y.rightBody Y.leftBody).2.endPoint +
+        (distinguishedCapSides Y.cap.val).2.corner)
+      ((distinguishedCapSides X.cap.val).2.corner -
+        (rightLeftTailArcs X.rightBody X.leftBody).2.endPoint) -
+      2 * planeCrossProduct (rightLeftTailArcs X.rightBody X.leftBody).2.endPoint
+        (distinguishedCapSides X.cap.val).2.corner) / 2 = 0 := by
+    rw [← hL]
+    simp only [planeCrossProduct, sub_self, WithLp.ofLp_zero, Pi.zero_apply, mul_zero]
+    ring
+  rw [hbulk, zero_add] at hvar
+  exact (convexDirectionalDerivative_comp_isConvexLinear hleftPair
+    (fun x : Point × Point ↦ segmentArea x.1 x.2) X Y).trans hvar
+
+theorem upperBoundQ_variation (X Y : CapTailSpace)
+    (hR : (rightLeftTailArcs X.rightBody X.leftBody).1.startPoint =
+      (distinguishedCapSides X.cap.val).1.corner)
+    (hL : (rightLeftTailArcs X.rightBody X.leftBody).2.endPoint =
+      (distinguishedCapSides X.cap.val).2.corner) :
+    convexDirectionalDerivative capTailCombination upperBoundQ X Y = qVariationIntegral X Y := by
+  obtain ⟨hcomb, -⟩ := capTail_isConvexDomain
+  obtain ⟨hrIoo, hlIoo, -⟩ := paperGerverConstants_snd_mem_Ioo
+  have hpi := Real.pi_pos
+  -- ### The three components of a cap-tail triple depend convex-linearly on it
+  have hcap : IsConvexLinear capTailCombination specialCapCombination
+      (fun Z : CapTailSpace ↦ Z.cap) := fun t Z W ↦
+    Subtype.ext (Subtype.ext ((hcomb t Z W).1.trans
+      (specialCap_isConvexDomain.1 t Z.cap W.cap).symm))
+  have hright : IsConvexLinear capTailCombination convexBodyCombination
+      (fun Z : CapTailSpace ↦ Z.rightBody) := fun t Z W ↦ (hcomb t Z W).2.1
+  have hleft : IsConvexLinear capTailCombination convexBodyCombination
+      (fun Z : CapTailSpace ↦ Z.leftBody) := fun t Z W ↦ (hcomb t Z W).2.2
+  -- ### The two segment endpoint pairs depend convex-linearly on the triple
+  have hleftPair := capTail_leftPair_isConvexLinear
+  have hrightPair := capTail_rightPair_isConvexLinear
+  -- ### Quadraticity of the six summands of `𝒬`
+  have hq1 : IsQuadraticFunctional capTailCombination
+      (fun Z : CapTailSpace ↦ ClassicalResults.area (Z.cap.val.val : Set Point)) :=
+    specialCapArea_variation.1.comp_isConvexLinear hcap
+  have hq2 : IsQuadraticFunctional capTailCombination
+      (fun Z : CapTailSpace ↦ convexArcArea Z.leftBody (3 * Real.pi / 2)
+        (3 * Real.pi / 2 + paperGerverConstants.2.2)) :=
+    (convexArcArea_variation _ _ (by linarith [hlIoo.1]) (by linarith [hlIoo.2])).1
+      |>.comp_isConvexLinear hleft
+  have hq3 : IsQuadraticFunctional capTailCombination
+      (fun Z : CapTailSpace ↦ segmentArea (rightLeftTailArcs Z.rightBody Z.leftBody).2.endPoint
+        (distinguishedCapSides Z.cap.val).2.corner) :=
+    segmentArea_variation.1.comp_isConvexLinear hleftPair
+  have hq4 : IsQuadraticFunctional capTailCombination
+      (fun Z : CapTailSpace ↦ curveAreaFunctional (capMiddleBV Z.cap)) :=
+    capInnerCorner_variation.2.1.comp_isConvexLinear hcap
+  have hq5 : IsQuadraticFunctional capTailCombination
+      (fun Z : CapTailSpace ↦ segmentArea (distinguishedCapSides Z.cap.val).1.corner
+        (rightLeftTailArcs Z.rightBody Z.leftBody).1.startPoint) :=
+    segmentArea_variation.1.comp_isConvexLinear hrightPair
+  have hq6 : IsQuadraticFunctional capTailCombination
+      (fun Z : CapTailSpace ↦ convexArcArea Z.rightBody (Real.pi + paperGerverConstants.2.1)
+        (3 * Real.pi / 2)) :=
+    (convexArcArea_variation _ _ (by linarith [hrIoo.2]) (by linarith [hrIoo.1])).1
+      |>.comp_isConvexLinear hright
+  -- ### The four summands whose derivatives contribute an integral
+  have hd1 : convexDirectionalDerivative capTailCombination
+      (fun Z : CapTailSpace ↦ ClassicalResults.area (Z.cap.val.val : Set Point)) X Y =
+      ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Icc 0 Real.pi,
+        (supportValue Y.cap.val.val t - supportValue X.cap.val.val t)
+          ∂surfaceAreaMeasure X.cap.val.val :=
+    (convexDirectionalDerivative_comp_isConvexLinear hcap
+      (fun K : SpecialCapSpace ↦ ClassicalResults.area (K.val.val : Set Point)) X Y).trans
+      (specialCapArea_variation.2 X.cap Y.cap)
+  have hd2 : convexDirectionalDerivative capTailCombination
+      (fun Z : CapTailSpace ↦ convexArcArea Z.leftBody (3 * Real.pi / 2)
+        (3 * Real.pi / 2 + paperGerverConstants.2.2)) X Y =
+      (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) ''
+          Set.Ioo (3 * Real.pi / 2) (3 * Real.pi / 2 + paperGerverConstants.2.2),
+        (supportValue Y.leftBody t - supportValue X.leftBody t)
+          ∂surfaceAreaMeasure X.leftBody) +
+        (segmentArea (rightLeftTailArcs X.rightBody X.leftBody).2.endPoint
+            (rightLeftTailArcs Y.rightBody Y.leftBody).2.endPoint -
+          segmentArea (rightLeftTailArcs X.rightBody X.leftBody).2.startPoint
+            (rightLeftTailArcs Y.rightBody Y.leftBody).2.startPoint) :=
+    (convexDirectionalDerivative_comp_isConvexLinear hleft
+      (fun M : ConvexBody Point ↦ convexArcArea M (3 * Real.pi / 2)
+        (3 * Real.pi / 2 + paperGerverConstants.2.2)) X Y).trans
+      ((convexArcArea_variation _ _ (by linarith [hlIoo.1]) (by linarith [hlIoo.2])).2
+        X.leftBody Y.leftBody)
+  have hd4 : convexDirectionalDerivative capTailCombination
+      (fun Z : CapTailSpace ↦ curveAreaFunctional (capMiddleBV Z.cap)) X Y =
+      (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) ''
+          (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2 ∪
+            Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+              (Real.pi / 2 + paperGerverConstants.2.2)),
+        (supportValue Y.cap.val.val t - supportValue X.cap.val.val t)
+          ∂capCornerAngleMeasure X.cap) +
+        (segmentArea (distinguishedCapSides X.cap.val).2.corner
+            (distinguishedCapSides Y.cap.val).2.corner -
+          segmentArea (distinguishedCapSides X.cap.val).1.corner
+            (distinguishedCapSides Y.cap.val).1.corner) :=
+    (convexDirectionalDerivative_comp_isConvexLinear hcap
+      (fun M : SpecialCapSpace ↦ curveAreaFunctional (capMiddleBV M)) X Y).trans
+      (capInnerCorner_variation.2.2 X.cap Y.cap)
+  have hd6 : convexDirectionalDerivative capTailCombination
+      (fun Z : CapTailSpace ↦ convexArcArea Z.rightBody (Real.pi + paperGerverConstants.2.1)
+        (3 * Real.pi / 2)) X Y =
+      (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) ''
+          Set.Ioo (Real.pi + paperGerverConstants.2.1) (3 * Real.pi / 2),
+        (supportValue Y.rightBody t - supportValue X.rightBody t)
+          ∂surfaceAreaMeasure X.rightBody) +
+        (segmentArea (rightLeftTailArcs X.rightBody X.leftBody).1.endPoint
+            (rightLeftTailArcs Y.rightBody Y.leftBody).1.endPoint -
+          segmentArea (rightLeftTailArcs X.rightBody X.leftBody).1.startPoint
+            (rightLeftTailArcs Y.rightBody Y.leftBody).1.startPoint) :=
+    (convexDirectionalDerivative_comp_isConvexLinear hright
+      (fun M : ConvexBody Point ↦ convexArcArea M (Real.pi + paperGerverConstants.2.1)
+        (3 * Real.pi / 2)) X Y).trans
+      ((convexArcArea_variation _ _ (by linarith [hrIoo.2]) (by linarith [hrIoo.1])).2
+        X.rightBody Y.rightBody)
+  -- ### The two segment summands: their bulk terms vanish because the base endpoints coincide
+  have hd3 := capTail_left_segment_variation X Y hL
+  have hd5 : convexDirectionalDerivative capTailCombination
+      (fun Z : CapTailSpace ↦ segmentArea (distinguishedCapSides Z.cap.val).1.corner
+        (rightLeftTailArcs Z.rightBody Z.leftBody).1.startPoint) X Y =
+      segmentArea (rightLeftTailArcs X.rightBody X.leftBody).1.startPoint
+          (rightLeftTailArcs Y.rightBody Y.leftBody).1.startPoint -
+        segmentArea (distinguishedCapSides X.cap.val).1.corner
+          (distinguishedCapSides Y.cap.val).1.corner := by
+    have hvar := segmentArea_variation.2
+      (distinguishedCapSides X.cap.val).1.corner
+      (rightLeftTailArcs X.rightBody X.leftBody).1.startPoint
+      (distinguishedCapSides Y.cap.val).1.corner
+      (rightLeftTailArcs Y.rightBody Y.leftBody).1.startPoint
+    have hbulk : (planeCrossProduct
+        ((distinguishedCapSides Y.cap.val).1.corner +
+          (rightLeftTailArcs Y.rightBody Y.leftBody).1.startPoint)
+        ((rightLeftTailArcs X.rightBody X.leftBody).1.startPoint -
+          (distinguishedCapSides X.cap.val).1.corner) -
+        2 * planeCrossProduct (distinguishedCapSides X.cap.val).1.corner
+          (rightLeftTailArcs X.rightBody X.leftBody).1.startPoint) / 2 = 0 := by
+      rw [hR]
+      simp only [planeCrossProduct, sub_self, WithLp.ofLp_zero, Pi.zero_apply, mul_zero]
+      ring
+    rw [hbulk, zero_add] at hvar
+    exact (convexDirectionalDerivative_comp_isConvexLinear hrightPair
+      (fun x : Point × Point ↦ segmentArea x.1 x.2) X Y).trans hvar
+  -- ### The derivative of `𝒬` is the signed sum of the six derivatives
+  have hsplit : ∀ t : ℝ, segmentFunctional capTailCombination upperBoundQ X Y t =
+      segmentFunctional capTailCombination
+            (fun Z : CapTailSpace ↦ ClassicalResults.area (Z.cap.val.val : Set Point)) X Y t +
+          segmentFunctional capTailCombination
+            (fun Z : CapTailSpace ↦ convexArcArea Z.leftBody (3 * Real.pi / 2)
+              (3 * Real.pi / 2 + paperGerverConstants.2.2)) X Y t +
+          segmentFunctional capTailCombination
+            (fun Z : CapTailSpace ↦ segmentArea
+              (rightLeftTailArcs Z.rightBody Z.leftBody).2.endPoint
+              (distinguishedCapSides Z.cap.val).2.corner) X Y t -
+          segmentFunctional capTailCombination
+            (fun Z : CapTailSpace ↦ curveAreaFunctional (capMiddleBV Z.cap)) X Y t +
+          segmentFunctional capTailCombination
+            (fun Z : CapTailSpace ↦ segmentArea (distinguishedCapSides Z.cap.val).1.corner
+              (rightLeftTailArcs Z.rightBody Z.leftBody).1.startPoint) X Y t +
+          segmentFunctional capTailCombination
+            (fun Z : CapTailSpace ↦ convexArcArea Z.rightBody
+              (Real.pi + paperGerverConstants.2.1) (3 * Real.pi / 2)) X Y t := by
+    intro t
+    simp only [segmentFunctional]
+    split_ifs with ht
+    · rfl
+    · norm_num
+  have hD1 := hq1.hasDerivWithinAt_segmentFunctional X Y
+  have hD2 := hq2.hasDerivWithinAt_segmentFunctional X Y
+  have hD3 := hq3.hasDerivWithinAt_segmentFunctional X Y
+  have hD4 := hq4.hasDerivWithinAt_segmentFunctional X Y
+  have hD5 := hq5.hasDerivWithinAt_segmentFunctional X Y
+  have hD6 := hq6.hasDerivWithinAt_segmentFunctional X Y
+  rw [hd1] at hD1
+  rw [hd2] at hD2
+  rw [hd3] at hD3
+  rw [hd4] at hD4
+  rw [hd5] at hD5
+  rw [hd6] at hD6
+  have hderiv := convexDirectionalDerivative_eq_of_hasDerivWithinAt capTailCombination
+    upperBoundQ X Y ((((((hD1.add hD2).add hD3).sub hD4).add hD5).add hD6).congr
+      (fun t _ ↦ hsplit t) (hsplit 0))
+  obtain ⟨hrightFar, hleftFar⟩ := capTail_far_segmentAreas X Y
+  -- ### The two tail integrals in opposite-angle coordinates
+  have hrightTail : (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) ''
+        Set.Ioo (Real.pi + paperGerverConstants.2.1) (3 * Real.pi / 2),
+      (supportValue Y.rightBody t - supportValue X.rightBody t)
+        ∂surfaceAreaMeasure X.rightBody) =
+      ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) ''
+          Set.Ioo paperGerverConstants.2.1 (Real.pi / 2),
+        ((oppositeSurfaceData Y.rightBody).2 t - (oppositeSurfaceData X.rightBody).2 t)
+          ∂(oppositeSurfaceData X.rightBody).1 :=
+    (setIntegral_oppositeSurfaceData_angleImage_Ioo X.rightBody
+      (a := paperGerverConstants.2.1) (b := Real.pi / 2) (by ring) (by ring)
+      (f := fun u : Real.Angle ↦ supportValue Y.rightBody u - supportValue X.rightBody u)
+      ((continuous_supportValue Y.rightBody).sub
+        (continuous_supportValue X.rightBody)).measurable).symm
+  have hleftTail : (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) ''
+        Set.Ioo (3 * Real.pi / 2) (3 * Real.pi / 2 + paperGerverConstants.2.2),
+      (supportValue Y.leftBody t - supportValue X.leftBody t)
+        ∂surfaceAreaMeasure X.leftBody) =
+      ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) ''
+          Set.Ioo (Real.pi / 2) (Real.pi / 2 + paperGerverConstants.2.2),
+        ((oppositeSurfaceData Y.leftBody).2 t - (oppositeSurfaceData X.leftBody).2 t)
+          ∂(oppositeSurfaceData X.leftBody).1 :=
+    (setIntegral_oppositeSurfaceData_angleImage_Ioo X.leftBody
+      (a := Real.pi / 2) (b := Real.pi / 2 + paperGerverConstants.2.2) (by ring) (by ring)
+      (f := fun u : Real.Angle ↦ supportValue Y.leftBody u - supportValue X.leftBody u)
+      ((continuous_supportValue Y.leftBody).sub
+        (continuous_supportValue X.leftBody)).measurable).symm
+  -- ### Collecting the six contributions
+  rw [hderiv]
+  simp only [qVariationIntegral]
+  linarith [hrightTail, hleftTail, hrightFar, hleftFar]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/Development/Geometry/Applications/Development003.lean b/LeanPool/MovingSofa/Development/Geometry/Applications/Development003.lean
new file mode 100644
index 0000000000..d611810911
--- /dev/null
+++ b/LeanPool/MovingSofa/Development/Geometry/Applications/Development003.lean
@@ -0,0 +1,6022 @@
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+module
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development006
+public import LeanPool.MovingSofa.Development.Geometry.Applications.Development002
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development007
+
+
+
+
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development005
+
+
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development008
+public import LeanPool.MovingSofa.Development.Geometry.Applications.Development001
+
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development003
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development002
+
+
+
+
+/-!
+# Moving sofa: related mathematical developments
+
+* `Gerver.Applications.Development004`.
+* `Bounds.Applications.Development003`.
+* `Sofa.Applications.Development001`.
+* `Motion.Applications.Development001`.
+* `Sofa.Applications.Development002`.
+* `Sofa.Applications.Development003`.
+* `Motion.Applications.Development002`.
+* `Main`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Gerver.Niche.Orientation`.
+* `Gerver.ContactGeometry`.
+* `Gerver.TailGeometry`.
+* `Gerver.MeasureTranslation`.
+* `Gerver.PhaseMeasures`.
+* `Gerver.PhaseIntegration`.
+* `Gerver.QMatch`.
+* `Gerver.QVariation`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Orientation of the certified Gerver niche boundary
+
+The niche of the certified Gerver outer cap is the strict region under the niche roof, and the
+roof is the graph of a continuous height function `f` over the horizontal extent
+`[gerverNicheLeft, gerverNicheRight]` of the roof (`exists_gerverRoofHeight`).  The closure of
+the niche is therefore the closed subgraph of `f`, its interior the open subgraph, and the
+counterclockwise loop `positiveGraphLoop` around that region traces their common frontier.
+
+`gerverNicheBoundary` is the explicit four-piece traversal of that frontier: the second contact
+curve reversed, the ambient path forwards, the fourth contact curve reversed, and the base
+segment.  It runs backwards along the roof through a decreasing piecewise affine parameter
+change (`gerverBoundaryParam`), so it is the positive-graph loop reparametrized by an increasing
+map (`gerverNicheBoundary_orientedJordan`).  The main result `gerver_niche_orientation` collects
+the consequences: the traversal is a continuous path of bounded variation, a counterclockwise
+Jordan parametrization of the frontier of the closed niche, its signed curve area is the area of
+the niche, and no roof point lies in the niche.
+
+The same description of the niche as the strict region under the roof settles the opposite
+inclusion for the fourth boundary piece: every interior point of the base segment does lie in
+the niche (`gerver_bottom_segment_mem_gerverLiteralNiche`).
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The stage-time inequalities of `gerverStageTimes_strictMono`, in the packaged form used
+throughout this file. -/
+private theorem gerverStageTimes_order :
+    gerverStageTimes 0 = 0 ∧ gerverStageTimes 5 = Real.pi / 2 ∧
+      0 < gerverStageTimes 1 ∧ gerverStageTimes 1 < gerverStageTimes 2 ∧
+      gerverStageTimes 2 < gerverStageTimes 3 ∧ gerverStageTimes 3 < gerverStageTimes 4 ∧
+      gerverStageTimes 4 < Real.pi / 2 := by
+  have h := gerverStageTimes_strictMono
+  exact ⟨gerverStageTimes_zero, rfl, h (show (0 : Fin 6) < 1 by decide),
+    h (show (1 : Fin 6) < 2 by decide), h (show (2 : Fin 6) < 3 by decide),
+    h (show (3 : Fin 6) < 4 by decide), h (show (4 : Fin 6) < 5 by decide)⟩
+
+/-! ## The roof as the graph of a continuous height function -/
+
+/-- The left end of the horizontal extent of the niche roof. -/
+private def gerverNicheLeft : ℝ := paperGerverContacts 0 3 0
+
+/-- The right end of the horizontal extent of the niche roof. -/
+private def gerverNicheRight : ℝ := paperGerverContacts (Real.pi / 2) 1 0
+
+private theorem gerverNicheRoof_zero :
+    gerverNicheRoof ⟨0, le_rfl, by positivity⟩ = paperGerverContacts 0 3 := by
+  obtain ⟨-, -, h01, h12, -, -, -⟩ := gerverStageTimes_order
+  exact gerverNicheRoof_of_le_two _ (by linarith)
+
+private theorem gerverNicheRoof_top :
+    gerverNicheRoof ⟨Real.pi / 2, by positivity, le_rfl⟩ =
+      paperGerverContacts (Real.pi / 2) 1 := by
+  obtain ⟨-, -, -, -, -, h34, h45⟩ := gerverStageTimes_order
+  exact gerverNicheRoof_of_ge_three _ (by linarith)
+
+/-- The niche roof is the graph of a continuous function over its horizontal extent, which
+vanishes at the two ends and is positive in between. -/
+private theorem exists_gerverRoofHeight :
+    ∃ f : ℝ → ℝ, Continuous f ∧
+      (∀ s : Set.Icc (0 : ℝ) (Real.pi / 2), f (gerverNicheRoof s 0) = gerverNicheRoof s 1) ∧
+      (∀ c ∈ Set.Icc gerverNicheLeft gerverNicheRight,
+        ∃ s : Set.Icc (0 : ℝ) (Real.pi / 2), gerverNicheRoof s 0 = c) ∧
+      gerverNicheLeft < gerverNicheRight ∧
+      f gerverNicheLeft = 0 ∧ f gerverNicheRight = 0 ∧
+      ∀ c ∈ Set.Ioo gerverNicheLeft gerverNicheRight, 0 < f c := by
+  have hpi : (0 : ℝ) < Real.pi / 2 := by positivity
+  set A : Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨0, le_rfl, hpi.le⟩ with hA
+  set B : Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨Real.pi / 2, hpi.le, le_rfl⟩ with hB
+  have hk : StrictMono fun s : Set.Icc (0 : ℝ) (Real.pi / 2) ↦ gerverNicheRoof s 0 :=
+    gerver_niche_roof_strictMono
+  have hk0 : Continuous fun s : Set.Icc (0 : ℝ) (Real.pi / 2) ↦ gerverNicheRoof s 0 :=
+    (PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 0).comp continuous_gerverNicheRoof
+  have hk1 : Continuous fun s : Set.Icc (0 : ℝ) (Real.pi / 2) ↦ gerverNicheRoof s 1 :=
+    (PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 1).comp continuous_gerverNicheRoof
+  have hAleft : gerverNicheRoof A 0 = gerverNicheLeft := by rw [gerverNicheRoof_zero]; rfl
+  have hBright : gerverNicheRoof B 0 = gerverNicheRight := by rw [gerverNicheRoof_top]; rfl
+  have hab : gerverNicheLeft < gerverNicheRight := by
+    rw [← hAleft, ← hBright]
+    exact hk (show A < B from hpi)
+  have hmem : ∀ s, gerverNicheRoof s 0 ∈ Set.Icc gerverNicheLeft gerverNicheRight := fun s ↦
+    ⟨hAleft ▸ hk.monotone (show A ≤ s from s.2.1), hBright ▸ hk.monotone (show s ≤ B from s.2.2)⟩
+  set kmap : Set.Icc (0 : ℝ) (Real.pi / 2) → Set.Icc gerverNicheLeft gerverNicheRight :=
+    fun s ↦ ⟨gerverNicheRoof s 0, hmem s⟩ with hkmap
+  have hkc : Continuous kmap := hk0.subtype_mk _
+  have hkinj : Function.Injective kmap := fun x y h ↦ hk.injective (congrArg Subtype.val h)
+  have : PreconnectedSpace (Set.Icc (0 : ℝ) (Real.pi / 2)) :=
+    Subtype.preconnectedSpace isPreconnected_Icc
+  have hksurj : Function.Surjective kmap := by
+    intro s
+    obtain ⟨t, ht⟩ := intermediate_value_univ A B hk0 (by rw [hAleft, hBright]; exact s.property)
+    exact ⟨t, Subtype.ext ht⟩
+  set khom : Set.Icc (0 : ℝ) (Real.pi / 2) ≃ₜ Set.Icc gerverNicheLeft gerverNicheRight :=
+    Continuous.homeoOfEquivCompactToT2 (f := Equiv.ofBijective kmap ⟨hkinj, hksurj⟩) hkc
+    with hkhom
+  have hkhom_apply : ∀ s, (khom s : ℝ) = gerverNicheRoof s 0 := fun _ ↦ rfl
+  have hsymm : ∀ x : Set.Icc gerverNicheLeft gerverNicheRight,
+      gerverNicheRoof (khom.symm x) 0 = (x : ℝ) := by
+    intro x
+    rw [← hkhom_apply, khom.apply_symm_apply]
+  refine ⟨fun c ↦ gerverNicheRoof (khom.symm (Set.projIcc _ _ hab.le c)) 1,
+    hk1.comp (khom.symm.continuous.comp continuous_projIcc), ?_, ?_, hab, ?_, ?_, ?_⟩
+  · -- the height function inverts the horizontal coordinate
+    intro s
+    have hval : Set.projIcc gerverNicheLeft gerverNicheRight hab.le (gerverNicheRoof s 0) =
+        khom s := by
+      rw [← hkhom_apply s, Set.projIcc_val]
+    dsimp only
+    rw [hval, khom.symm_apply_apply]
+  · intro c hc
+    exact ⟨khom.symm ⟨c, hc⟩, by rw [hsymm ⟨c, hc⟩]⟩
+  · have hmemL : gerverNicheLeft ∈ Set.Icc gerverNicheLeft gerverNicheRight :=
+      Set.left_mem_Icc.mpr hab.le
+    have hval : Set.projIcc gerverNicheLeft gerverNicheRight hab.le gerverNicheLeft =
+        ⟨gerverNicheLeft, hmemL⟩ := Set.projIcc_of_mem hab.le hmemL
+    have hsA : khom.symm ⟨gerverNicheLeft, hmemL⟩ = A :=
+      hk.injective (by dsimp only; rw [hsymm ⟨gerverNicheLeft, hmemL⟩, hAleft])
+    dsimp only
+    rw [hval, hsA, hA, gerverNicheRoof_zero]
+    exact gerver_niche_piece_endpoints.2.2.2
+  · have hmemR : gerverNicheRight ∈ Set.Icc gerverNicheLeft gerverNicheRight :=
+      Set.right_mem_Icc.mpr hab.le
+    have hval : Set.projIcc gerverNicheLeft gerverNicheRight hab.le gerverNicheRight =
+        ⟨gerverNicheRight, hmemR⟩ := Set.projIcc_of_mem hab.le hmemR
+    have hsB : khom.symm ⟨gerverNicheRight, hmemR⟩ = B :=
+      hk.injective (by dsimp only; rw [hsymm ⟨gerverNicheRight, hmemR⟩, hBright])
+    dsimp only
+    rw [hval, hsB, hB, gerverNicheRoof_top]
+    exact gerver_niche_piece_endpoints.2.2.1
+  · intro c hc
+    have hval : Set.projIcc gerverNicheLeft gerverNicheRight hab.le c =
+        ⟨c, hc.1.le, hc.2.le⟩ := Set.projIcc_of_mem hab.le ⟨hc.1.le, hc.2.le⟩
+    dsimp only
+    rw [hval]
+    set s := khom.symm (⟨c, hc.1.le, hc.2.le⟩ : Set.Icc gerverNicheLeft gerverNicheRight) with hs
+    have hcs : gerverNicheRoof s 0 = c := hsymm _
+    refine gerver_niche_roof_positive s ⟨?_, ?_⟩
+    · rcases eq_or_lt_of_le s.2.1 with h | h
+      · exact absurd (by rw [show s = A from Subtype.ext h.symm, hAleft] at hcs; exact hcs)
+          hc.1.ne
+      · exact h
+    · rcases eq_or_lt_of_le s.2.2 with h | h
+      · exact absurd (by rw [show s = B from Subtype.ext h, hBright] at hcs; exact hcs.symm)
+          hc.2.ne
+      · exact h
+
+/-! ## The literal niche is the strict region under the roof -/
+
+/-- Every central path time is the reverse time of a middle roof parameter. -/
+private theorem exists_gerverRoofReverseTime {t : ℝ}
+    (ht : t ∈ Set.Icc (gerverStageTimes 1) (gerverStageTimes 4)) :
+    ∃ s : ℝ, gerverStageTimes 2 ≤ s ∧ s ≤ gerverStageTimes 3 ∧ gerverRoofReverseTime s = t := by
+  obtain ⟨-, -, -, h12, h23, h34, -⟩ := gerverStageTimes_order
+  have hd : (0 : ℝ) < gerverStageTimes 3 - gerverStageTimes 2 := by linarith
+  have he : (0 : ℝ) < gerverStageTimes 4 - gerverStageTimes 1 := by linarith
+  refine ⟨gerverStageTimes 2 + (gerverStageTimes 4 - t) *
+    (gerverStageTimes 3 - gerverStageTimes 2) / (gerverStageTimes 4 - gerverStageTimes 1),
+    ?_, ?_, ?_⟩
+  · have : 0 ≤ (gerverStageTimes 4 - t) * (gerverStageTimes 3 - gerverStageTimes 2) /
+        (gerverStageTimes 4 - gerverStageTimes 1) :=
+      div_nonneg (mul_nonneg (by linarith [ht.2]) hd.le) he.le
+    linarith
+  · have : (gerverStageTimes 4 - t) * (gerverStageTimes 3 - gerverStageTimes 2) /
+        (gerverStageTimes 4 - gerverStageTimes 1) ≤
+        gerverStageTimes 3 - gerverStageTimes 2 := by
+      rw [div_le_iff₀ he]
+      nlinarith [ht.1, hd]
+    linarith
+  · rw [gerverRoofReverseTime]
+    field_simp
+    ring
+
+/-- The literal niche is the strict vertical region under the niche roof. -/
+private theorem gerverLiteralNiche_eq_roofFill :
+    gerverLiteralNiche =
+      {q : Point | ∃ s : Set.Icc (0 : ℝ) (Real.pi / 2),
+        q 0 = gerverNicheRoof s 0 ∧ 0 ≤ q 1 ∧ q 1 < gerverNicheRoof s 1} := by
+  obtain ⟨h0, h5, h01, h12, h23, h34, h45⟩ := gerverStageTimes_order
+  rw [gerver_niche_vertical_fills]
+  ext q
+  simp only [strictVerticalFill, Set.mem_union, Set.mem_ofPred_eq]
+  constructor
+  · rintro ((⟨t, ht, hq0, hq1, hq2⟩ | ⟨t, ht, hq0, hq1, hq2⟩) | ⟨t, ht, hq0, hq1, hq2⟩)
+    · rw [h0] at ht
+      have hs : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨ht.1, by linarith [ht.2]⟩
+      refine ⟨⟨t, hs⟩, ?_, hq1, ?_⟩
+      · rw [gerverNicheRoof_of_le_two hs ht.2]; exact hq0
+      · rw [gerverNicheRoof_of_le_two hs ht.2]; exact hq2
+    · obtain ⟨s, hs1, hs2, hsrev⟩ := exists_gerverRoofReverseTime ht
+      have hs : s ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨by linarith, by linarith⟩
+      refine ⟨⟨s, hs⟩, ?_, hq1, ?_⟩
+      · rw [gerverNicheRoof_mid hs hs1 hs2, hsrev]; exact hq0
+      · rw [gerverNicheRoof_mid hs hs1 hs2, hsrev]; exact hq2
+    · rw [h5] at ht
+      have hs : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨by linarith [ht.1], ht.2⟩
+      refine ⟨⟨t, hs⟩, ?_, hq1, ?_⟩
+      · rw [gerverNicheRoof_of_ge_three hs ht.1]; exact hq0
+      · rw [gerverNicheRoof_of_ge_three hs ht.1]; exact hq2
+  · rintro ⟨s, hq0, hq1, hq2⟩
+    rcases le_or_gt s.val (gerverStageTimes 2) with hle | hgt
+    · refine Or.inl (Or.inl ⟨s.val, ⟨by rw [h0]; exact s.2.1, hle⟩, ?_, hq1, ?_⟩)
+      · rw [← gerverNicheRoof_of_le_two s.2 hle]; exact hq0
+      · rw [← gerverNicheRoof_of_le_two s.2 hle]; exact hq2
+    rcases le_or_gt s.val (gerverStageTimes 3) with hle3 | hgt3
+    · refine Or.inl (Or.inr ⟨gerverRoofReverseTime s.val,
+        gerverRoofReverseTime_mem_Icc hgt.le hle3, ?_, hq1, ?_⟩)
+      · rw [← gerverNicheRoof_mid s.2 hgt.le hle3]; exact hq0
+      · rw [← gerverNicheRoof_mid s.2 hgt.le hle3]; exact hq2
+    · refine Or.inr ⟨s.val, ⟨hgt3.le, by rw [h5]; exact s.2.2⟩, ?_, hq1, ?_⟩
+      · rw [← gerverNicheRoof_of_ge_three s.2 hgt3.le]; exact hq0
+      · rw [← gerverNicheRoof_of_ge_three s.2 hgt3.le]; exact hq2
+
+/-- The horizontal coordinate of the roof stays in its horizontal extent. -/
+private theorem gerverNicheRoof_fst_mem (s : Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    gerverNicheRoof s 0 ∈ Set.Icc gerverNicheLeft gerverNicheRight := by
+  have hpi : (0 : ℝ) < Real.pi / 2 := by positivity
+  have hk := gerver_niche_roof_strictMono
+  constructor
+  · have h := hk.monotone (show (⟨0, le_rfl, hpi.le⟩ : Set.Icc (0 : ℝ) (Real.pi / 2)) ≤ s
+      from s.2.1)
+    simp only [gerverNicheRoof_zero] at h
+    exact h
+  · have h := hk.monotone (show s ≤ (⟨Real.pi / 2, hpi.le, le_rfl⟩ :
+      Set.Icc (0 : ℝ) (Real.pi / 2)) from s.2.2)
+    simp only [gerverNicheRoof_top] at h
+    exact h
+
+/-- Read through the roof height function, the literal niche is the strict subgraph. -/
+private theorem gerverLiteralNiche_eq_strictSubgraph {f : ℝ → ℝ}
+    (hfgraph : ∀ s : Set.Icc (0 : ℝ) (Real.pi / 2),
+      f (gerverNicheRoof s 0) = gerverNicheRoof s 1)
+    (hfsurj : ∀ c ∈ Set.Icc gerverNicheLeft gerverNicheRight,
+      ∃ s : Set.Icc (0 : ℝ) (Real.pi / 2), gerverNicheRoof s 0 = c) :
+    gerverLiteralNiche = strictSubgraph gerverNicheLeft gerverNicheRight f := by
+  rw [gerverLiteralNiche_eq_roofFill]
+  ext q
+  simp only [strictSubgraph, Set.mem_ofPred_eq]
+  constructor
+  · rintro ⟨s, hq0, hq1, hq2⟩
+    have hmem := gerverNicheRoof_fst_mem s
+    rw [← hq0] at hmem
+    exact ⟨hmem.1, hmem.2, hq1, by rw [hq0, hfgraph s]; exact hq2⟩
+  · rintro ⟨h1, h2, h3, h4⟩
+    obtain ⟨s, hs⟩ := hfsurj (q 0) ⟨h1, h2⟩
+    exact ⟨s, hs.symm, h3, by rw [← hfgraph s, hs]; exact h4⟩
+
+/-- The closed boundary parametrization of the Gerver niche. -/
+def gerverNicheBoundary (s : Set.Icc (0 : ℝ) 4) : Point :=
+  if s.val ≤ 1 then
+    paperGerverContacts
+      (Real.pi / 2 - (Real.pi / 2 - gerverStageTimes 3) * s.val) 1
+  else if s.val ≤ 2 then
+    paperGerverPath
+      (gerverStageTimes 1 + (gerverStageTimes 4 - gerverStageTimes 1) * (s.val - 1))
+  else if s.val ≤ 3 then
+    paperGerverContacts (gerverStageTimes 2 * (3 - s.val)) 3
+  else
+    (4 - s.val) • paperGerverContacts 0 3 +
+      (s.val - 3) • paperGerverContacts (Real.pi / 2) 1
+
+/-! ## The four-piece traversal of the niche boundary -/
+
+/-- On its first piece the niche traversal runs backwards along the second contact curve `B`,
+from `B (π / 2)` at `s = 0` to `B t₃` at `s = 1`. -/
+theorem gerverNicheBoundary_of_le_one {s : ℝ} (hs : s ∈ Set.Icc (0 : ℝ) 4) (h : s ≤ 1) :
+    gerverNicheBoundary ⟨s, hs⟩ =
+      paperGerverContacts (Real.pi / 2 - (Real.pi / 2 - gerverStageTimes 3) * s) 1 :=
+  ite_eq_left h
+
+/-- On its second piece the niche traversal runs forwards along the direct Gerver path, from
+`x t₁` at `s = 1` to `x t₄` at `s = 2`. -/
+theorem gerverNicheBoundary_mid {s : ℝ} (hs : s ∈ Set.Icc (0 : ℝ) 4)
+    (h1 : 1 ≤ s) (h2 : s ≤ 2) :
+    gerverNicheBoundary ⟨s, hs⟩ =
+      paperGerverPath (gerverStageTimes 1 +
+        (gerverStageTimes 4 - gerverStageTimes 1) * (s - 1)) := by
+  rcases eq_or_lt_of_le h1 with heq | hlt
+  · rw [gerverNicheBoundary_of_le_one hs heq.ge, ← heq,
+      show Real.pi / 2 - (Real.pi / 2 - gerverStageTimes 3) * 1 = gerverStageTimes 3 by ring,
+      show gerverStageTimes 1 + (gerverStageTimes 4 - gerverStageTimes 1) * ((1 : ℝ) - 1) =
+        gerverStageTimes 1 by ring]
+    exact gerver_niche_piece_endpoints.1
+  · rw [gerverNicheBoundary, ite_eq_right (not_le.mpr hlt), ite_eq_left h2]
+
+/-- On its third piece the niche traversal runs backwards along the fourth contact curve `D`,
+from `D t₂` at `s = 2` to `D 0` at `s = 3`. -/
+theorem gerverNicheBoundary_third {s : ℝ} (hs : s ∈ Set.Icc (0 : ℝ) 4)
+    (h1 : 2 ≤ s) (h2 : s ≤ 3) :
+    gerverNicheBoundary ⟨s, hs⟩ = paperGerverContacts (gerverStageTimes 2 * (3 - s)) 3 := by
+  rcases eq_or_lt_of_le h1 with heq | hlt
+  · rw [gerverNicheBoundary_mid hs (by linarith) heq.ge, ← heq,
+      show gerverStageTimes 1 + (gerverStageTimes 4 - gerverStageTimes 1) * ((2 : ℝ) - 1) =
+        gerverStageTimes 4 by ring,
+      show gerverStageTimes 2 * ((3 : ℝ) - 2) = gerverStageTimes 2 by ring]
+    exact gerver_niche_piece_endpoints.2.1.symm
+  · rw [gerverNicheBoundary, ite_eq_right (not_le.mpr (by linarith)),
+      ite_eq_right (not_le.mpr hlt), ite_eq_left h2]
+
+/-- On its last piece the niche traversal runs along the base segment of the niche, from `D 0`
+at `s = 3` to `B (π / 2)` at `s = 4`. -/
+theorem gerverNicheBoundary_base {s : ℝ} (hs : s ∈ Set.Icc (0 : ℝ) 4) (h : 3 ≤ s) :
+    gerverNicheBoundary ⟨s, hs⟩ =
+      (4 - s) • paperGerverContacts 0 3 + (s - 3) • paperGerverContacts (Real.pi / 2) 1 := by
+  rcases eq_or_lt_of_le h with heq | hlt
+  · rw [gerverNicheBoundary_third hs (by linarith) heq.ge, ← heq,
+      show gerverStageTimes 2 * ((3 : ℝ) - 3) = 0 by ring]
+    norm_num
+  · rw [gerverNicheBoundary, ite_eq_right (not_le.mpr (by linarith)),
+      ite_eq_right (not_le.mpr (by linarith)), ite_eq_right (not_le.mpr hlt)]
+
+private theorem continuous_gerverNicheBoundary : Continuous gerverNicheBoundary := by
+  have hc1 : Continuous fun s : Set.Icc (0 : ℝ) 4 ↦
+      paperGerverContacts (Real.pi / 2 - (Real.pi / 2 - gerverStageTimes 3) * s.val) 1 :=
+    (continuous_paperGerverContact 1).comp (by fun_prop)
+  have hc2 : Continuous fun s : Set.Icc (0 : ℝ) 4 ↦
+      paperGerverPath (gerverStageTimes 1 +
+        (gerverStageTimes 4 - gerverStageTimes 1) * (s.val - 1)) :=
+    contDiff_paperGerverPath.continuous.comp (by fun_prop)
+  have hc3 : Continuous fun s : Set.Icc (0 : ℝ) 4 ↦
+      paperGerverContacts (gerverStageTimes 2 * (3 - s.val)) 3 :=
+    (continuous_paperGerverContact 3).comp (by fun_prop)
+  have hc4 : Continuous fun s : Set.Icc (0 : ℝ) 4 ↦
+      (4 - s.val) • paperGerverContacts 0 3 +
+        (s.val - 3) • paperGerverContacts (Real.pi / 2) 1 := by fun_prop
+  have hinner2 : Continuous fun s : Set.Icc (0 : ℝ) 4 ↦
+      if s.val ≤ 3 then paperGerverContacts (gerverStageTimes 2 * (3 - s.val)) 3
+      else (4 - s.val) • paperGerverContacts 0 3 +
+        (s.val - 3) • paperGerverContacts (Real.pi / 2) 1 := by
+    refine continuous_if_le continuous_subtype_val continuous_const hc3.continuousOn
+      hc4.continuousOn ?_
+    intro s hsv
+    rw [hsv, show gerverStageTimes 2 * ((3 : ℝ) - 3) = 0 by ring]
+    norm_num
+  have hinner1 : Continuous fun s : Set.Icc (0 : ℝ) 4 ↦
+      if s.val ≤ 2 then
+        paperGerverPath (gerverStageTimes 1 +
+          (gerverStageTimes 4 - gerverStageTimes 1) * (s.val - 1))
+      else if s.val ≤ 3 then paperGerverContacts (gerverStageTimes 2 * (3 - s.val)) 3
+      else (4 - s.val) • paperGerverContacts 0 3 +
+        (s.val - 3) • paperGerverContacts (Real.pi / 2) 1 := by
+    refine continuous_if_le continuous_subtype_val continuous_const hc2.continuousOn
+      hinner2.continuousOn ?_
+    intro s hsv
+    rw [hsv, ite_eq_left (show (2 : ℝ) ≤ 3 by norm_num),
+      show gerverStageTimes 1 + (gerverStageTimes 4 - gerverStageTimes 1) * ((2 : ℝ) - 1) =
+        gerverStageTimes 4 by ring,
+      show gerverStageTimes 2 * ((3 : ℝ) - 2) = gerverStageTimes 2 by ring]
+    exact gerver_niche_piece_endpoints.2.1.symm
+  unfold gerverNicheBoundary
+  refine continuous_if_le continuous_subtype_val continuous_const hc1.continuousOn
+    hinner1.continuousOn ?_
+  intro s hsv
+  rw [hsv, ite_eq_left (show (1 : ℝ) ≤ 2 by norm_num),
+    show Real.pi / 2 - (Real.pi / 2 - gerverStageTimes 3) * (1 : ℝ) = gerverStageTimes 3 by ring,
+    show gerverStageTimes 1 + (gerverStageTimes 4 - gerverStageTimes 1) * ((1 : ℝ) - 1) =
+      gerverStageTimes 1 by ring]
+  exact gerver_niche_piece_endpoints.1
+
+/-- The four-piece niche traversal is a continuous path of bounded variation: each of its six
+analytic pieces is continuously differentiable on a closed parameter interval. -/
+private theorem gerverNicheBoundary_mem_continuousBV :
+    gerverNicheBoundary ∈ continuousBVSubmodule 0 4 := by
+  obtain ⟨h0, h5, h01, h12, h23, h34, h45⟩ := gerverStageTimes_order
+  refine ⟨continuous_gerverNicheBoundary, fun i ↦ ?_⟩
+  have hd3 : (0 : ℝ) < Real.pi / 2 - gerverStageTimes 3 := by linarith
+  set c1 : ℝ := (Real.pi / 2 - gerverStageTimes 4) / (Real.pi / 2 - gerverStageTimes 3) with hc1def
+  set c2 : ℝ := 3 - gerverStageTimes 1 / gerverStageTimes 2 with hc2def
+  have hv : (Real.pi / 2 - gerverStageTimes 3) * c1 = Real.pi / 2 - gerverStageTimes 4 := by
+    rw [hc1def, mul_comm, div_mul_cancel₀ _ hd3.ne']
+  have hu : gerverStageTimes 2 * (gerverStageTimes 1 / gerverStageTimes 2) =
+      gerverStageTimes 1 := by
+    rw [mul_comm, div_mul_cancel₀ _ (show gerverStageTimes 2 ≠ 0 by
+      exact ne_of_gt (by linarith))]
+  have hc10 : (0 : ℝ) ≤ c1 := by rw [hc1def]; exact div_nonneg (by linarith) hd3.le
+  have hc11 : c1 ≤ 1 := by rw [hc1def]; exact (div_le_one hd3).mpr (by linarith)
+  have hq0 : (0 : ℝ) ≤ gerverStageTimes 1 / gerverStageTimes 2 :=
+    div_nonneg (by linarith) (by linarith)
+  have hq1 : gerverStageTimes 1 / gerverStageTimes 2 ≤ 1 :=
+    (div_le_one (by linarith)).mpr (by linarith)
+  have hc22 : (2 : ℝ) ≤ c2 := by rw [hc2def]; linarith
+  have hc23 : c2 ≤ 3 := by rw [hc2def]; linarith
+  -- a contact curve composed with a smooth time change staying in one stage
+  have hcontact : ∀ (j : Fin 4) (k : Fin 5) (arg : ℝ → ℝ) (l r : ℝ), ContDiff ℝ 1 arg →
+      Set.MapsTo arg (Set.Icc l r) (gerverStageIntervals k) →
+      ContDiffOn ℝ 1 (fun s ↦ paperGerverContacts (arg s) j) (Set.Icc l r) :=
+    fun j k arg l r harg hmaps ↦
+      (contDiffOn_paperGerverContact j k).comp harg.contDiffOn hmaps
+  have harg1 : ContDiff ℝ 1 fun s : ℝ ↦
+      Real.pi / 2 - (Real.pi / 2 - gerverStageTimes 3) * s := by fun_prop
+  have harg3 : ContDiff ℝ 1 fun s : ℝ ↦ gerverStageTimes 2 * (3 - s) := by fun_prop
+  -- the six pieces
+  have hp1 : ContDiffOn ℝ 1 (fun s ↦ paperGerverContacts
+      (Real.pi / 2 - (Real.pi / 2 - gerverStageTimes 3) * s) 1) (Set.Icc 0 c1) := by
+    refine hcontact 1 4 _ 0 c1 harg1 ?_
+    intro s hs
+    rw [gerverStageIntervals_four, h5]
+    have hmul := mul_le_mul_of_nonneg_left hs.2 hd3.le
+    rw [hv] at hmul
+    exact ⟨by linarith, by nlinarith [hs.1]⟩
+  have hp2 : ContDiffOn ℝ 1 (fun s ↦ paperGerverContacts
+      (Real.pi / 2 - (Real.pi / 2 - gerverStageTimes 3) * s) 1) (Set.Icc c1 1) := by
+    refine hcontact 1 3 _ c1 1 harg1 ?_
+    intro s hs
+    rw [gerverStageIntervals_three]
+    have hmul := mul_le_mul_of_nonneg_left hs.1 hd3.le
+    rw [hv] at hmul
+    exact ⟨by nlinarith [hs.2], by linarith⟩
+  have hp3 : ContDiffOn ℝ 1 (fun s ↦ paperGerverPath (gerverStageTimes 1 +
+      (gerverStageTimes 4 - gerverStageTimes 1) * (s - 1))) (Set.Icc 1 2) :=
+    (contDiff_paperGerverPath.comp (by fun_prop)).contDiffOn
+  have hp4 : ContDiffOn ℝ 1 (fun s ↦ paperGerverContacts (gerverStageTimes 2 * (3 - s)) 3)
+      (Set.Icc 2 c2) := by
+    refine hcontact 3 1 _ 2 c2 harg3 ?_
+    intro s hs
+    rw [gerverStageIntervals_one]
+    have hmul := mul_le_mul_of_nonneg_left (show 3 - s ≥ 3 - c2 by linarith [hs.2])
+      (show (0:ℝ) ≤ gerverStageTimes 2 by linarith)
+    rw [show (3 : ℝ) - c2 = gerverStageTimes 1 / gerverStageTimes 2 by rw [hc2def]; ring, hu]
+      at hmul
+    exact ⟨hmul, by nlinarith [hs.1]⟩
+  have hp5 : ContDiffOn ℝ 1 (fun s ↦ paperGerverContacts (gerverStageTimes 2 * (3 - s)) 3)
+      (Set.Icc c2 3) := by
+    refine hcontact 3 0 _ c2 3 harg3 ?_
+    intro s hs
+    rw [gerverStageIntervals_zero, h0]
+    have hmul := mul_le_mul_of_nonneg_left (show 3 - s ≤ 3 - c2 by linarith [hs.1])
+      (show (0:ℝ) ≤ gerverStageTimes 2 by linarith)
+    rw [show (3 : ℝ) - c2 = gerverStageTimes 1 / gerverStageTimes 2 by rw [hc2def]; ring, hu]
+      at hmul
+    exact ⟨by nlinarith [hs.2], hmul⟩
+  have hp6 : ContDiffOn ℝ 1 (fun s : ℝ ↦ (4 - s) • paperGerverContacts 0 3 +
+      (s - 3) • paperGerverContacts (Real.pi / 2) 1) (Set.Icc 3 4) := by
+    apply ContDiff.contDiffOn
+    fun_prop
+  -- the variation of each coordinate on each of the six pieces
+  have e1 := boundedVariationOn_coord_Icc_of_contDiffOn gerverNicheBoundary le_rfl hc10
+    (show c1 ≤ 4 by linarith) hp1
+    (fun t ↦ gerverNicheBoundary_of_le_one _ (t.2.2.trans hc11)) i
+  have e2 := boundedVariationOn_coord_Icc_of_contDiffOn gerverNicheBoundary hc10 hc11
+    (show (1 : ℝ) ≤ 4 by norm_num) hp2
+    (fun t ↦ gerverNicheBoundary_of_le_one _ t.2.2) i
+  have e3 := boundedVariationOn_coord_Icc_of_contDiffOn gerverNicheBoundary
+    (show (0 : ℝ) ≤ 1 by norm_num) (show (1 : ℝ) ≤ 2 by norm_num)
+    (show (2 : ℝ) ≤ 4 by norm_num) hp3
+    (fun t ↦ gerverNicheBoundary_mid _ t.2.1 t.2.2) i
+  have e4 := boundedVariationOn_coord_Icc_of_contDiffOn gerverNicheBoundary
+    (show (0 : ℝ) ≤ 2 by norm_num) hc22 (show c2 ≤ 4 by linarith) hp4
+    (fun t ↦ gerverNicheBoundary_third _ t.2.1 (t.2.2.trans hc23)) i
+  have e5 := boundedVariationOn_coord_Icc_of_contDiffOn gerverNicheBoundary
+    (show (0 : ℝ) ≤ c2 by linarith) hc23 (show (3 : ℝ) ≤ 4 by norm_num) hp5
+    (fun t ↦ gerverNicheBoundary_third _ (hc22.trans t.2.1) t.2.2) i
+  have e6 := boundedVariationOn_coord_Icc_of_contDiffOn gerverNicheBoundary
+    (show (0 : ℝ) ≤ 3 by norm_num) (show (3 : ℝ) ≤ 4 by norm_num) le_rfl hp6
+    (fun t ↦ gerverNicheBoundary_base _ t.2.1) i
+  -- glue them along the subdivision `0 ≤ c₁ ≤ 1 ≤ 2 ≤ c₂ ≤ 3 ≤ 4`
+  have hsub : ∀ {x y : ℝ} {hx : x ∈ Set.Icc (0 : ℝ) 4} {hy : y ∈ Set.Icc (0 : ℝ) 4},
+      x ≤ y → (⟨x, hx⟩ : Set.Icc (0 : ℝ) 4) ≤ ⟨y, hy⟩ := fun h ↦ h
+  have q01 : (0 : ℝ) ≤ 1 := by norm_num
+  have q12 : (1 : ℝ) ≤ 2 := by norm_num
+  have q02 : (0 : ℝ) ≤ 2 := by norm_num
+  have q03 : (0 : ℝ) ≤ 3 := by norm_num
+  have q34 : (3 : ℝ) ≤ 4 := by norm_num
+  have q0c2 : (0 : ℝ) ≤ c2 := by linarith
+  have g1 := BoundedVariationOn.Icc_union_Icc (hsub hc10) (hsub hc11) e1 e2
+  have g2 := BoundedVariationOn.Icc_union_Icc (hsub q01) (hsub q12) g1 e3
+  have g3 := BoundedVariationOn.Icc_union_Icc (hsub q02) (hsub hc22) g2 e4
+  have g4 := BoundedVariationOn.Icc_union_Icc (hsub q0c2) (hsub hc23) g3 e5
+  change BoundedVariationOn (fun t : Set.Icc (0 : ℝ) 4 ↦ gerverNicheBoundary t i) Set.univ
+  exact BoundedVariationOn.univ_of_Icc_endpoints (show (0 : ℝ) ≤ 4 by norm_num)
+    (BoundedVariationOn.Icc_union_Icc (hsub q03) (hsub q34) g4 e6)
+
+/-! ## The traversal is the roof, run backwards -/
+
+/-- The decreasing piecewise affine parameter change carrying `[0, 3]` onto the roof
+interval `[0, π/2]`. -/
+private def gerverBoundaryParam (s : ℝ) : ℝ :=
+  if s ≤ 1 then Real.pi / 2 - (Real.pi / 2 - gerverStageTimes 3) * s
+  else if s ≤ 2 then gerverStageTimes 2 + (gerverStageTimes 3 - gerverStageTimes 2) * (2 - s)
+  else gerverStageTimes 2 * (3 - s)
+
+private theorem gerverBoundaryParam_first {s : ℝ} (h : s ≤ 1) :
+    gerverBoundaryParam s = Real.pi / 2 - (Real.pi / 2 - gerverStageTimes 3) * s :=
+  ite_eq_left h
+
+private theorem gerverBoundaryParam_second {s : ℝ} (h1 : 1 ≤ s) (h2 : s ≤ 2) :
+    gerverBoundaryParam s =
+      gerverStageTimes 2 + (gerverStageTimes 3 - gerverStageTimes 2) * (2 - s) := by
+  rcases eq_or_lt_of_le h1 with heq | hlt
+  · rw [gerverBoundaryParam_first heq.ge, ← heq]; ring
+  · rw [gerverBoundaryParam, ite_eq_right (not_le.mpr hlt), ite_eq_left h2]
+
+private theorem gerverBoundaryParam_third {s : ℝ} (h1 : 2 ≤ s) :
+    gerverBoundaryParam s = gerverStageTimes 2 * (3 - s) := by
+  rcases eq_or_lt_of_le h1 with heq | hlt
+  · rw [gerverBoundaryParam_second (by linarith) heq.ge, ← heq]; ring
+  · rw [gerverBoundaryParam, ite_eq_right (not_le.mpr (by linarith)),
+      ite_eq_right (not_le.mpr hlt)]
+
+private theorem gerverBoundaryParam_strictAntiOn :
+    StrictAntiOn gerverBoundaryParam (Set.Icc (0 : ℝ) 3) := by
+  obtain ⟨-, -, h01, h12, h23, h34, h45⟩ := gerverStageTimes_order
+  have hD : (0 : ℝ) < Real.pi / 2 - gerverStageTimes 3 := by linarith
+  have hE : (0 : ℝ) < gerverStageTimes 3 - gerverStageTimes 2 := by linarith
+  have hF : (0 : ℝ) < gerverStageTimes 2 := by linarith
+  have hle1 : ∀ s : ℝ, s ≤ 1 → gerverStageTimes 3 ≤ gerverBoundaryParam s := by
+    intro s hs
+    rw [gerverBoundaryParam_first hs]
+    nlinarith
+  have hlt1 : ∀ s : ℝ, 1 < s → s ≤ 3 → gerverBoundaryParam s < gerverStageTimes 3 := by
+    intro s hs hs3
+    rcases le_or_gt s 2 with h | h
+    · rw [gerverBoundaryParam_second hs.le h]; nlinarith
+    · rw [gerverBoundaryParam_third h.le]; nlinarith
+  have hle2 : ∀ s : ℝ, 1 ≤ s → s ≤ 2 → gerverStageTimes 2 ≤ gerverBoundaryParam s := by
+    intro s hs1 hs2
+    rw [gerverBoundaryParam_second hs1 hs2]
+    nlinarith
+  have hlt2 : ∀ s : ℝ, 2 < s → gerverBoundaryParam s < gerverStageTimes 2 := by
+    intro s hs
+    rw [gerverBoundaryParam_third hs.le]
+    nlinarith
+  intro x hx y hy hxy
+  rcases le_or_gt y 1 with hy1 | hy1
+  · rw [gerverBoundaryParam_first (hxy.le.trans hy1), gerverBoundaryParam_first hy1]
+    nlinarith
+  rcases le_or_gt x 1 with hx1 | hx1
+  · exact lt_of_lt_of_le (hlt1 y hy1 hy.2) (hle1 x hx1)
+  rcases le_or_gt y 2 with hy2 | hy2
+  · rw [gerverBoundaryParam_second hx1.le (hxy.le.trans hy2),
+      gerverBoundaryParam_second (by linarith) hy2]
+    nlinarith
+  rcases le_or_gt x 2 with hx2 | hx2
+  · exact lt_of_lt_of_le (hlt2 y hy2) (hle2 x hx1.le hx2)
+  · rw [gerverBoundaryParam_third hx2.le, gerverBoundaryParam_third (by linarith)]
+    nlinarith
+
+private theorem gerverBoundaryParam_zero : gerverBoundaryParam 0 = Real.pi / 2 := by
+  rw [gerverBoundaryParam_first (by norm_num)]; ring
+
+private theorem gerverBoundaryParam_three : gerverBoundaryParam 3 = 0 := by
+  rw [gerverBoundaryParam_third (by norm_num)]; ring
+
+private theorem gerverBoundaryParam_mem {s : ℝ} (h1 : 0 ≤ s) (h2 : s ≤ 3) :
+    gerverBoundaryParam s ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := by
+  have hanti := gerverBoundaryParam_strictAntiOn.antitoneOn
+  refine ⟨?_, ?_⟩
+  · rw [← gerverBoundaryParam_three]
+    exact hanti ⟨h1, h2⟩ (by norm_num) h2
+  · rw [← gerverBoundaryParam_zero]
+    exact hanti (by norm_num) ⟨h1, h2⟩ h1
+
+/-- On its first three pieces the traversal runs backwards along the niche roof. -/
+private theorem gerverNicheBoundary_eq_roof {s : ℝ} (hs : s ∈ Set.Icc (0 : ℝ) 4) (h : s ≤ 3) :
+    gerverNicheBoundary ⟨s, hs⟩ =
+      gerverNicheRoof ⟨gerverBoundaryParam s, gerverBoundaryParam_mem hs.1 h⟩ := by
+  obtain ⟨-, -, h01, h12, h23, h34, h45⟩ := gerverStageTimes_order
+  have hne : gerverStageTimes 3 - gerverStageTimes 2 ≠ 0 := by
+    exact ne_of_gt (by linarith)
+  rcases le_or_gt s 1 with h1 | h1
+  · rw [gerverNicheBoundary_of_le_one hs h1,
+      gerverNicheRoof_of_ge_three _ (by rw [gerverBoundaryParam_first h1]; nlinarith),
+      gerverBoundaryParam_first h1]
+  rcases le_or_gt s 2 with h2 | h2
+  · rw [gerverNicheBoundary_mid hs h1.le h2,
+      gerverNicheRoof_mid _ (by rw [gerverBoundaryParam_second h1.le h2]; nlinarith)
+        (by rw [gerverBoundaryParam_second h1.le h2]; nlinarith)]
+    congr 1
+    rw [gerverRoofReverseTime, gerverBoundaryParam_second h1.le h2]
+    field_simp
+    ring
+  · rw [gerverNicheBoundary_third hs h2.le h,
+      gerverNicheRoof_of_le_two _ (by rw [gerverBoundaryParam_third h2.le]; nlinarith),
+      gerverBoundaryParam_third h2.le]
+
+/-- The four-piece traversal has the same image as the counterclockwise loop around the
+region under the roof, and is a counterclockwise Jordan parametrization of it. -/
+private theorem gerverNicheBoundary_orientedJordan {f : ℝ → ℝ} (hfc : Continuous f)
+    (hfgraph : ∀ s : Set.Icc (0 : ℝ) (Real.pi / 2),
+      f (gerverNicheRoof s 0) = gerverNicheRoof s 1)
+    (hab : gerverNicheLeft < gerverNicheRight)
+    (hfa : f gerverNicheLeft = 0) (hfb : f gerverNicheRight = 0)
+    (hfpos : ∀ c ∈ Set.Ioo gerverNicheLeft gerverNicheRight, 0 < f c) :
+    Set.range gerverNicheBoundary =
+        Set.range (positiveGraphLoop gerverNicheLeft gerverNicheRight f) ∧
+      IsOrientedJordanParametrization (show (0 : ℝ) ≤ 4 by norm_num)
+        (Set.range (positiveGraphLoop gerverNicheLeft gerverNicheRight f)) true
+        gerverNicheBoundary := by
+  have hba : (0 : ℝ) < gerverNicheRight - gerverNicheLeft := by linarith
+  obtain ⟨hjor, -⟩ := positiveGraphLoop_counterclockwise gerverNicheLeft gerverNicheRight
+    hab f hfc.continuousOn hfa hfb hfpos
+  -- the reparametrization of the loop by the four-piece traversal
+  have hbnd : ∀ s : Set.Icc (0 : ℝ) 4, s.val ≤ 3 →
+      gerverNicheBoundary s 0 ∈ Set.Icc gerverNicheLeft gerverNicheRight := by
+    intro s hs
+    rw [gerverNicheBoundary_eq_roof s.2 hs]
+    exact gerverNicheRoof_fst_mem _
+  have hpsimem : ∀ s : Set.Icc (0 : ℝ) 4,
+      (if s.val ≤ 3 then (gerverNicheRight - gerverNicheBoundary s 0) /
+        (gerverNicheRight - gerverNicheLeft) else s.val - 2) ∈ Set.Icc (0 : ℝ) 2 := by
+    intro s
+    by_cases hs : s.val ≤ 3
+    · obtain ⟨hl, hr⟩ := hbnd s hs
+      rw [ite_eq_left hs]
+      exact ⟨div_nonneg (by linarith) hba.le, by
+        refine le_trans ((div_le_one hba).mpr (by linarith)) (by norm_num)⟩
+    · rw [ite_eq_right hs]
+      exact ⟨by linarith [not_le.mp hs], by linarith [s.2.2]⟩
+  set ψ : Set.Icc (0 : ℝ) 4 → Set.Icc (0 : ℝ) 2 := fun s ↦ ⟨_, hpsimem s⟩ with hψdef
+  have hψval : ∀ s : Set.Icc (0 : ℝ) 4, (ψ s : ℝ) =
+      if s.val ≤ 3 then (gerverNicheRight - gerverNicheBoundary s 0) /
+        (gerverNicheRight - gerverNicheLeft) else s.val - 2 := fun _ ↦ rfl
+  have hψle : ∀ s : Set.Icc (0 : ℝ) 4, s.val ≤ 3 → (ψ s : ℝ) ≤ 1 := by
+    intro s hs
+    rw [hψval, ite_eq_left hs]
+    exact (div_le_one hba).mpr (by linarith [(hbnd s hs).1])
+  have hψeq : ∀ s : Set.Icc (0 : ℝ) 4, s.val ≤ 3 →
+      gerverNicheRight - (gerverNicheRight - gerverNicheLeft) * (ψ s : ℝ) =
+        gerverNicheBoundary s 0 := by
+    intro s hs
+    rw [hψval, ite_eq_left hs, mul_div_cancel₀ _ hba.ne']
+    ring
+  have hbthree : ∀ s : Set.Icc (0 : ℝ) 4, s.val = 3 →
+      gerverNicheBoundary s 0 = gerverNicheLeft := by
+    intro s hs
+    rw [gerverNicheBoundary_eq_roof s.2 hs.le]
+    have h3 : gerverBoundaryParam s.val = 0 := by rw [hs]; exact gerverBoundaryParam_three
+    simp only [h3]
+    rw [gerverNicheRoof_zero]
+    rfl
+  -- the traversal is the loop, reparametrized
+  have hΓψ : ∀ s : Set.Icc (0 : ℝ) 4,
+      gerverNicheBoundary s = positiveGraphLoop gerverNicheLeft gerverNicheRight f (ψ s) := by
+    intro s
+    by_cases hs : s.val ≤ 3
+    · rw [positiveGraphLoop_apply_of_le (hpsimem s) (hψle s hs), hψeq s hs]
+      have hroof := gerverNicheBoundary_eq_roof s.2 hs
+      have hheight : f (gerverNicheBoundary s 0) = gerverNicheBoundary s 1 := by
+        rw [hroof, hfgraph]
+      rw [hheight]
+      exact Point.eq_vecNotation _
+    · have hs' : (3 : ℝ) ≤ s.val := (not_le.mp hs).le
+      rw [positiveGraphLoop_apply_of_not_le (hpsimem s) (by
+        rw [ite_eq_right hs]; exact not_le.mpr (by linarith [not_le.mp hs])),
+        ite_eq_right hs, gerverNicheBoundary_base s.2 hs']
+      have hD : paperGerverContacts 0 3 = !₂[gerverNicheLeft, (0 : ℝ)] := by
+        rw [Point.eq_vecNotation (paperGerverContacts 0 3), gerver_niche_piece_endpoints.2.2.2]
+        rfl
+      have hB : paperGerverContacts (Real.pi / 2) 1 = !₂[gerverNicheRight, (0 : ℝ)] := by
+        rw [Point.eq_vecNotation (paperGerverContacts (Real.pi / 2) 1),
+          gerver_niche_piece_endpoints.2.2.1]
+        rfl
+      rw [hD, hB]
+      ext i
+      fin_cases i
+      · simp
+        ring
+      · simp
+  have hΓfun : gerverNicheBoundary =
+      positiveGraphLoop gerverNicheLeft gerverNicheRight f ∘ ψ := funext hΓψ
+  -- strict monotonicity of the reparametrization
+  have hψmono : StrictMono ψ := by
+    intro s t hst
+    have hstv : s.val < t.val := hst
+    have hlt : (ψ s : ℝ) < (ψ t : ℝ) := by
+      by_cases hs : s.val ≤ 3
+      · by_cases ht : t.val ≤ 3
+        · have hΓ : gerverNicheBoundary t 0 < gerverNicheBoundary s 0 := by
+            rw [gerverNicheBoundary_eq_roof s.2 hs, gerverNicheBoundary_eq_roof t.2 ht]
+            exact gerver_niche_roof_strictMono
+              (show (⟨gerverBoundaryParam t.val, gerverBoundaryParam_mem t.2.1 ht⟩ :
+                  Set.Icc (0 : ℝ) (Real.pi / 2)) <
+                ⟨gerverBoundaryParam s.val, gerverBoundaryParam_mem s.2.1 hs⟩ from
+                gerverBoundaryParam_strictAntiOn ⟨s.2.1, hs⟩ ⟨t.2.1, ht⟩ hstv)
+          rw [hψval s, hψval t, ite_eq_left hs, ite_eq_left ht]
+          have h2 : gerverNicheRight - gerverNicheBoundary s 0 <
+              gerverNicheRight - gerverNicheBoundary t 0 := by linarith
+          gcongr
+        · rw [hψval t, ite_eq_right ht]
+          exact lt_of_le_of_lt (hψle s hs) (by linarith [not_le.mp ht])
+      · have ht : ¬ t.val ≤ 3 := fun h ↦ hs (le_trans hstv.le h)
+        rw [hψval s, hψval t, ite_eq_right hs, ite_eq_right ht]
+        linarith
+    exact hlt
+  -- the reparametrization matches the endpoints and is a continuous surjection
+  have hz0 : (0 : ℝ) ∈ Set.Icc (0 : ℝ) 4 := ⟨le_rfl, by norm_num⟩
+  have hz4 : (4 : ℝ) ∈ Set.Icc (0 : ℝ) 4 := ⟨by norm_num, le_rfl⟩
+  have hu0 : (0 : ℝ) ∈ Set.Icc (0 : ℝ) 2 := ⟨le_rfl, by norm_num⟩
+  have hu2 : (2 : ℝ) ∈ Set.Icc (0 : ℝ) 2 := ⟨by norm_num, le_rfl⟩
+  have hbzero : gerverNicheBoundary ⟨0, hz0⟩ 0 = gerverNicheRight := by
+    rw [gerverNicheBoundary_eq_roof hz0 (by norm_num)]
+    have h3 : gerverBoundaryParam (0 : ℝ) = Real.pi / 2 := gerverBoundaryParam_zero
+    simp only [h3]
+    rw [gerverNicheRoof_top]
+    rfl
+  have hψ0 : ψ ⟨0, hz0⟩ = ⟨0, hu0⟩ := by
+    refine Subtype.ext ?_
+    rw [hψval, ite_eq_left (by norm_num), hbzero, sub_self, zero_div]
+  have hψ4 : ψ ⟨4, hz4⟩ = ⟨2, hu2⟩ := by
+    refine Subtype.ext ?_
+    rw [hψval, ite_eq_right (by norm_num)]
+    norm_num
+  have hψcont : Continuous ψ := by
+    refine Continuous.subtype_mk ?_ _
+    refine continuous_if_le continuous_subtype_val continuous_const
+      ((continuous_const.sub ((PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 0).comp
+        continuous_gerverNicheBoundary)).div_const _).continuousOn
+      (continuous_subtype_val.sub continuous_const).continuousOn ?_
+    intro s hsv
+    rw [hbthree s hsv, hsv, div_self hba.ne']
+    norm_num
+  have : PreconnectedSpace (Set.Icc (0 : ℝ) 4) := Subtype.preconnectedSpace isPreconnected_Icc
+  have hψsurj : Function.Surjective ψ := by
+    intro u
+    have hcont : Continuous fun s : Set.Icc (0 : ℝ) 4 ↦ (ψ s : ℝ) :=
+      continuous_subtype_val.comp hψcont
+    have hmemu : (u : ℝ) ∈ Set.Icc
+        ((fun s : Set.Icc (0 : ℝ) 4 ↦ (ψ s : ℝ)) ⟨0, hz0⟩)
+        ((fun s : Set.Icc (0 : ℝ) 4 ↦ (ψ s : ℝ)) ⟨4, hz4⟩) := by
+      simp only
+      rw [congrArg Subtype.val hψ0, congrArg Subtype.val hψ4]
+      exact u.property
+    obtain ⟨t, ht⟩ := intermediate_value_univ _ _ hcont hmemu
+    exact ⟨t, Subtype.ext ht⟩
+  -- the traversal traces the loop, once
+  have hrange : Set.range gerverNicheBoundary =
+      Set.range (positiveGraphLoop gerverNicheLeft gerverNicheRight f) := by
+    rw [hΓfun, Set.range_comp, hψsurj.range_eq, Set.image_univ]
+  have hinj : Set.InjOn gerverNicheBoundary {t : Set.Icc (0 : ℝ) 4 | (t : ℝ) < 4} := by
+    intro x hx y hy hxy
+    have hlt2 : ∀ z : Set.Icc (0 : ℝ) 4, (z : ℝ) < 4 →
+        (ψ z : Set.Icc (0 : ℝ) 2) ∈ {t : Set.Icc (0 : ℝ) 2 | (t : ℝ) < 2} := by
+      intro z hz
+      have h := hψmono (show z < (⟨4, hz4⟩ : Set.Icc (0 : ℝ) 4) from hz)
+      rw [hψ4] at h
+      exact h
+    refine hψmono.injective (hjor.2.2.2.2.2.1 (hlt2 x hx) (hlt2 y hy) ?_)
+    rw [← hΓψ x, ← hΓψ y]
+    exact hxy
+  have hclosedpath : gerverNicheBoundary ⟨0, hz0⟩ = gerverNicheBoundary ⟨4, hz4⟩ := by
+    rw [hΓψ, hΓψ, hψ0, hψ4]
+    exact hjor.2.2.2.2.1
+  have hwind : ∀ p ∈ jordanInterior
+      (Set.range (positiveGraphLoop gerverNicheLeft gerverNicheRight f)),
+      curveWinding (show (0 : ℝ) ≤ 4 by norm_num) gerverNicheBoundary p = 1 := by
+    intro p hp
+    have hw2 : curveWinding (show (0 : ℝ) ≤ 2 by norm_num)
+        (positiveGraphLoop gerverNicheLeft gerverNicheRight f) p = 1 := by
+      simpa using hjor.2.2.2.2.2.2 p hp
+    have hne : curveWinding (show (0 : ℝ) ≤ 2 by norm_num)
+        (positiveGraphLoop gerverNicheLeft gerverNicheRight f) p ≠ 0 := by
+      rw [hw2]; norm_num
+    have hcomp := curveWinding_comp_of_endpoints (show (0 : ℝ) ≤ 2 by norm_num)
+      (show (0 : ℝ) ≤ 4 by norm_num)
+      (exists_curveAngleLift_of_curveWinding_ne_zero _ hne) hψcont hψ0 hψ4
+    rw [← hΓfun] at hcomp
+    rw [hcomp, hw2]
+  have hOJP : IsOrientedJordanParametrization (show (0 : ℝ) ≤ 4 by norm_num)
+      (Set.range (positiveGraphLoop gerverNicheLeft gerverNicheRight f)) true
+      gerverNicheBoundary :=
+    ⟨by norm_num, hjor.2.1, continuous_gerverNicheBoundary, hrange, hclosedpath, hinj,
+      fun p hp ↦ by simpa using hwind p hp⟩
+  exact ⟨hrange, hOJP⟩
+
+/-- No point of the niche roof lies in the strict region under the roof: the roof is the graph
+of the height function, and that region lies strictly below the graph. -/
+private theorem gerverNicheRoof_notMem_strictSubgraph {f : ℝ → ℝ}
+    (hfgraph : ∀ s : Set.Icc (0 : ℝ) (Real.pi / 2),
+      f (gerverNicheRoof s 0) = gerverNicheRoof s 1)
+    (s : Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    gerverNicheRoof s ∉ strictSubgraph gerverNicheLeft gerverNicheRight f := by
+  intro hmem
+  have h := hmem.2.2.2
+  rw [hfgraph s] at h
+  exact absurd h (lt_irrefl _)
+
+theorem gerver_niche_orientation :
+    ∃ K : RightAngleCapSpace, (K.val : Set Point) = gerverOuterCap ∧
+      ∃ Γ : ContinuousBVPaths 0 4, Γ.val = gerverNicheBoundary ∧
+        IsOrientedJordanParametrization (by norm_num : (0 : ℝ) ≤ 4)
+          (frontier (closure (capNiche K))) true Γ.val ∧
+        closure (capNiche K) = jordanInterior (Set.range Γ.val) ∪ Set.range Γ.val ∧
+        ClassicalResults.area (closure (capNiche K)) = ClassicalResults.area (capNiche K) ∧
+        curveAreaFunctional Γ = ClassicalResults.area (capNiche K) ∧
+        (∀ t ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5),
+          paperGerverContacts t 1 ∉ capNiche K) ∧
+        (∀ t ∈ Set.Icc (gerverStageTimes 1) (gerverStageTimes 4),
+          paperGerverPath t ∉ capNiche K) ∧
+        (∀ t ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2),
+          paperGerverContacts t 3 ∉ capNiche K) := by
+  obtain ⟨-, ⟨K, hKset, hKniche⟩, -, -⟩ := gerver_paperNiche_identification
+  obtain ⟨f, hfc, hfgraph, hfsurj, hab, hfa, hfb, hfpos⟩ := exists_gerverRoofHeight
+  obtain ⟨h0, h5, h01, h12, h23, h34, h45⟩ := gerverStageTimes_order
+  have hfcon : ContinuousOn f (Set.Icc gerverNicheLeft gerverNicheRight) := hfc.continuousOn
+  -- the niche and its closure, read as subgraphs of the roof height
+  have hniche : capNiche K = strictSubgraph gerverNicheLeft gerverNicheRight f := by
+    rw [hKniche, gerverLiteralNiche_eq_strictSubgraph hfgraph hfsurj]
+  have hclos : closure (capNiche K) = closedSubgraph gerverNicheLeft gerverNicheRight f := by
+    rw [hniche, closure_strictSubgraph hab hfcon hfa hfb hfpos]
+  -- the region enclosed by the counterclockwise loop around the same graph
+  have hintU := jordanInterior_range_positiveGraphLoop hab hfcon hfa hfb hfpos
+  have hdiff := closedSubgraph_sdiff_openSubgraph hab hfcon hfa hfb hfpos
+  have hCUR := closedSubgraph_eq_openSubgraph_union_range hab hfcon hfa hfb hfpos
+  have hfront : frontier (closure (capNiche K)) =
+      Set.range (positiveGraphLoop gerverNicheLeft gerverNicheRight f) := by
+    rw [hclos]
+    exact frontier_closedSubgraph hab hfcon hfa hfb hfpos
+  obtain ⟨hrange, hOJP⟩ := gerverNicheBoundary_orientedJordan hfc hfgraph hab hfa hfb hfpos
+  -- the boundary traversal sweeps a null set, so all three regions have the same area
+  have hvolR : MeasureTheory.volume
+      (Set.range (positiveGraphLoop gerverNicheLeft gerverNicheRight f)) = 0 := by
+    rw [← hrange]
+    exact ContinuousBVPaths.volume_range_eq_zero_of_injOn (show (0 : ℝ) ≤ 4 by norm_num)
+      ⟨gerverNicheBoundary, gerverNicheBoundary_mem_continuousBV⟩ hOJP.2.2.2.2.2.1
+  have hCU : MeasureTheory.volume (closedSubgraph gerverNicheLeft gerverNicheRight f) =
+      MeasureTheory.volume (openSubgraph gerverNicheLeft gerverNicheRight f) :=
+    (MeasureTheory.measure_eq_measure_of_null_sdiff openSubgraph_subset_closedSubgraph
+      (by rw [hdiff]; exact hvolR)).symm
+  have hCS : MeasureTheory.volume (closedSubgraph gerverNicheLeft gerverNicheRight f) =
+      MeasureTheory.volume (strictSubgraph gerverNicheLeft gerverNicheRight f) := by
+    refine (MeasureTheory.measure_eq_measure_of_null_sdiff strictSubgraph_subset_closedSubgraph
+      (MeasureTheory.measure_mono_null ?_ hvolR)).symm
+    rw [← hdiff]
+    exact fun q hq ↦ ⟨hq.1, fun hU ↦ hq.2 ⟨hq.1.1, hq.1.2.1, hq.1.2.2.1, hU.2.2.2⟩⟩
+  have hroofnot : ∀ s : Set.Icc (0 : ℝ) (Real.pi / 2), gerverNicheRoof s ∉ capNiche K :=
+    fun s ↦ hniche ▸ gerverNicheRoof_notMem_strictSubgraph hfgraph s
+  refine ⟨K, hKset, ⟨gerverNicheBoundary, gerverNicheBoundary_mem_continuousBV⟩, rfl, ?_, ?_,
+    ?_, ?_, ?_, ?_, ?_⟩
+  · rw [hfront]
+    exact hOJP
+  · rw [hclos, show Set.range (⟨gerverNicheBoundary, gerverNicheBoundary_mem_continuousBV⟩ :
+      ContinuousBVPaths 0 4).val = Set.range gerverNicheBoundary from rfl, hrange, hintU, hCUR]
+  · rw [hclos, hniche]
+    unfold ClassicalResults.area
+    rw [hCS]
+  · rw [curveArea_eq_jordanInterior_area 0 4 (show (0 : ℝ) ≤ 4 by norm_num)
+      (Set.range (positiveGraphLoop gerverNicheLeft gerverNicheRight f))
+      ⟨gerverNicheBoundary, gerverNicheBoundary_mem_continuousBV⟩ hOJP, hintU, hniche]
+    unfold ClassicalResults.area
+    rw [← hCU, hCS]
+  · intro t ht
+    rw [h5] at ht
+    have htI : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨by linarith [ht.1], ht.2⟩
+    rw [← gerverNicheRoof_of_ge_three htI ht.1]
+    exact hroofnot _
+  · intro t ht
+    obtain ⟨σ, hσ1, hσ2, hσ⟩ := exists_gerverRoofReverseTime ht
+    have hσI : σ ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨by linarith, by linarith⟩
+    rw [← hσ, ← gerverNicheRoof_mid hσI hσ1 hσ2]
+    exact hroofnot _
+  · intro t ht
+    rw [h0] at ht
+    have htI : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨ht.1, by linarith [ht.2]⟩
+    rw [← gerverNicheRoof_of_le_two htI ht.2]
+    exact hroofnot _
+
+/-! ## The interior of the base segment -/
+
+/-- The two ends of the base segment of the niche boundary are distinct: they are the images
+of the parameters `3` and `0` of the boundary traversal, which is injective on `[0, 4)`. -/
+theorem gerver_niche_bottom_ends_ne :
+    paperGerverContacts 0 3 ≠ paperGerverContacts (Real.pi / 2) 1 := by
+  obtain ⟨_, _, Γ, hΓ, hOJP, -⟩ := gerver_niche_orientation
+  have hmem0 : (0 : ℝ) ∈ Set.Icc (0 : ℝ) 4 := by norm_num
+  have hmem3 : (3 : ℝ) ∈ Set.Icc (0 : ℝ) 4 := by norm_num
+  have hb0 : gerverNicheBoundary ⟨0, hmem0⟩ = paperGerverContacts (Real.pi / 2) 1 := by
+    norm_num [gerverNicheBoundary]
+  have hb3 : gerverNicheBoundary ⟨3, hmem3⟩ = paperGerverContacts 0 3 := by
+    norm_num [gerverNicheBoundary]
+  intro heq
+  have hinj := hOJP.2.2.2.2.2.1
+  rw [hΓ] at hinj
+  have h := hinj (show ((⟨3, hmem3⟩ : Set.Icc (0 : ℝ) 4) : ℝ) < 4 by norm_num)
+    (show ((⟨0, hmem0⟩ : Set.Icc (0 : ℝ) 4) : ℝ) < 4 by norm_num) (by rw [hb3, hb0, heq])
+  have h' := congrArg Subtype.val h
+  norm_num at h'
+
+/-- Every interior point of the base segment of the niche boundary, which runs from `D(0)` to
+`B(π/2)`, lies in the literal Gerver niche.  Both ends sit on the wall, so the segment is
+horizontal and its interior points have strictly intermediate horizontal coordinate; the
+intermediate value theorem then puts a roof point directly above such a point, where the roof
+has strictly positive height, and the description of the niche as the strict region under the
+roof concludes. -/
+theorem gerver_bottom_segment_mem_gerverLiteralNiche {a : ℝ} (ha : a ∈ Set.Ioo (0 : ℝ) 1) :
+    (1 - a) • paperGerverContacts 0 3 + a • paperGerverContacts (Real.pi / 2) 1 ∈
+      gerverLiteralNiche := by
+  obtain ⟨h0, h5, h01, h12, h23, h34, h45⟩ := gerverStageTimes_order
+  have hpi : (0 : ℝ) < Real.pi / 2 := by positivity
+  have hD0y : paperGerverContacts 0 3 1 = 0 := gerver_niche_piece_endpoints.2.2.2
+  have hBTy : paperGerverContacts (Real.pi / 2) 1 1 = 0 := gerver_niche_piece_endpoints.2.2.1
+  -- The two ends differ, and they agree in their second coordinate, so they differ in the first.
+  have hx : paperGerverContacts 0 3 0 ≠ paperGerverContacts (Real.pi / 2) 1 0 := by
+    intro h
+    refine gerver_niche_bottom_ends_ne ?_
+    ext i
+    fin_cases i
+    · exact h
+    · exact hD0y.trans hBTy.symm
+  set P : Point := (1 - a) • paperGerverContacts 0 3 + a • paperGerverContacts (Real.pi / 2) 1
+    with hPdef
+  have hP0 : P 0 = (1 - a) * paperGerverContacts 0 3 0 +
+      a * paperGerverContacts (Real.pi / 2) 1 0 := by
+    simp [hPdef, PiLp.add_apply, PiLp.smul_apply, smul_eq_mul]
+  have hP1 : P 1 = 0 := by
+    simp [hPdef, PiLp.add_apply, PiLp.smul_apply, smul_eq_mul, hD0y, hBTy]
+  -- The roof, read as a curve of a real parameter, runs from one end of the segment to the other.
+  have hR0 : gerverRoofCurve 0 = paperGerverContacts 0 3 := by
+    rw [gerverRoofCurve, ite_eq_left (show (0 : ℝ) ≤ gerverStageTimes 2 by linarith)]
+  have hRT : gerverRoofCurve (Real.pi / 2) = paperGerverContacts (Real.pi / 2) 1 := by
+    rw [gerverRoofCurve, ite_eq_right (not_le.mpr (by linarith)),
+      ite_eq_right (not_le.mpr (by linarith))]
+  -- The intermediate value theorem produces a roof parameter above `P`.
+  have hcont : ContinuousOn (fun s : ℝ ↦ gerverRoofCurve s 0) (Set.uIcc 0 (Real.pi / 2)) :=
+    ((PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 0).comp
+      continuous_gerverRoofCurve).continuousOn
+  have hPmem : P 0 ∈ Set.uIcc (gerverRoofCurve 0 0) (gerverRoofCurve (Real.pi / 2) 0) := by
+    rw [hR0, hRT, Set.mem_uIcc, hP0]
+    rcases lt_or_gt_of_ne hx with h | h
+    · exact Or.inl ⟨by nlinarith [ha.1, ha.2], by nlinarith [ha.1, ha.2]⟩
+    · exact Or.inr ⟨by nlinarith [ha.1, ha.2], by nlinarith [ha.1, ha.2]⟩
+  obtain ⟨σ, hσmem, hσraw⟩ := intermediate_value_uIcc hcont hPmem
+  have hσ : gerverRoofCurve σ 0 = P 0 := hσraw
+  rw [Set.uIcc_of_le hpi.le] at hσmem
+  -- That parameter is interior, because `P` is not an end of the segment.
+  have hσ0 : σ ≠ 0 := by
+    intro h
+    rw [h, hR0, hP0] at hσ
+    rcases mul_eq_zero.mp (show a * (paperGerverContacts (Real.pi / 2) 1 0 -
+        paperGerverContacts 0 3 0) = 0 by linarith) with h' | h'
+    · exact absurd h' (ne_of_gt ha.1)
+    · exact hx (by linarith)
+  have hσT : σ ≠ Real.pi / 2 := by
+    intro h
+    rw [h, hRT, hP0] at hσ
+    rcases mul_eq_zero.mp (show (1 - a) * (paperGerverContacts (Real.pi / 2) 1 0 -
+        paperGerverContacts 0 3 0) = 0 by linarith) with h' | h'
+    · exact absurd h' (by linarith [ha.2] : (1 : ℝ) - a ≠ 0)
+    · exact hx (by linarith)
+  have hheight : 0 < gerverNicheRoof ⟨σ, hσmem⟩ 1 :=
+    gerver_niche_roof_positive ⟨σ, hσmem⟩
+      ⟨lt_of_le_of_ne hσmem.1 (Ne.symm hσ0), lt_of_le_of_ne hσmem.2 hσT⟩
+  rw [gerverLiteralNiche_eq_roofFill]
+  exact ⟨⟨σ, hσmem⟩, by rw [← gerverRoofCurve_eq ⟨σ, hσmem⟩]; exact hσ.symm,
+    hP1.ge, by rw [hP1]; exact hheight⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The contact geometry of the paper Gerver sofa
+
+Gerver's sofa is a monotone sofa, and its right-angle cap has all the contact geometry the
+upper-bound argument reads off a monotone sofa: selected cap vertices and inner corner given
+by the paper's curves `A`, `C` and `x`, a counterclockwise Jordan traversal of the niche
+boundary, the interior of the base segment inside the niche and the three roof pieces outside
+it, the two inner contact curves on the two inner walls of the supporting hallway, and the
+signs of their stage speeds along the frame directions.
+
+Every clause is about one and the same cap.  The cap is taken from `paperGerverCap_injectivity`
+— which also supplies the density hypothesis that makes the selected contacts well defined —
+and the caps produced by cap-support identification, niche identification and niche
+orientation are identified with it through their common carrier `gerverOuterCap`.  The clauses
+themselves are then assembled from the identification theorems together with
+`gerver_bottom_segment_mem_gerverLiteralNiche`, the frame descriptions
+`mem_rotatingHallwayParts_bRay_iff` and `mem_rotatingHallwayParts_dRay_iff` of the two inner
+walls, the closed-interval velocity signs `paperGerverVelocityComponents_fst_nonpos` and
+`paperGerverVelocityComponents_snd_nonneg`, and the stage speeds
+`hasDerivWithinAt_paperGerverContacts_one_neg_smul` and
+`hasDerivWithinAt_paperGerverContacts_three_pos_smul`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem paperGerver_contact_geometry :
+    IsMonotoneSofa paperGerverSofa ∧
+    ∃ K : RightAngleCapSpace,
+      (K.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2) ∧
+      ∃ hK : ∃ r s, HasCapDensities K r s,
+        (∀ t : Set.Icc (0 : ℝ) (Real.pi / 2),
+          (nondegenerateCapData K hK).1.1 t = paperGerverContacts t 0 ∧
+          (nondegenerateCapData K hK).1.2 t = paperGerverContacts t 2 ∧
+          capInnerCorner K t = paperGerverPath t) ∧
+        (∃ Γ : ContinuousBVPaths 0 4, Γ.val = gerverNicheBoundary ∧
+          IsOrientedJordanParametrization (by norm_num : (0 : ℝ) ≤ 4)
+            (frontier (closure (capNiche K))) true Γ.val ∧
+          closure (capNiche K) = jordanInterior (Set.range Γ.val) ∪ Set.range Γ.val ∧
+          ClassicalResults.area (closure (capNiche K)) = ClassicalResults.area (capNiche K)) ∧
+        (∀ a ∈ Set.Ioo (0 : ℝ) 1,
+          (1 - a) • paperGerverContacts 0 3 +
+            a • paperGerverContacts (Real.pi / 2) 1 ∈ capNiche K) ∧
+        (∀ t ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5),
+          paperGerverContacts t 1 ∉ capNiche K ∧
+          paperGerverContacts t 1 ∈
+            (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).bRay) ∧
+        (∀ t ∈ Set.Icc (gerverStageTimes 1) (gerverStageTimes 4),
+          paperGerverPath t ∉ capNiche K) ∧
+        (∀ t ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2),
+          paperGerverContacts t 3 ∉ capNiche K ∧
+          paperGerverContacts t 3 ∈
+            (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).dRay) ∧
+        (∀ i : Fin 5, i = 3 ∨ i = 4 → ∀ t ∈ gerverStageIntervals i,
+          ∃ c : ℝ, c < 0 ∧
+            HasDerivWithinAt (fun s ↦ paperGerverContacts s 1)
+              (c • tangentVector (t : Real.Angle)) (gerverStageIntervals i) t) ∧
+        (∀ i : Fin 5, i = 0 ∨ i = 1 → ∀ t ∈ gerverStageIntervals i,
+          ∃ c : ℝ, 0 < c ∧
+            HasDerivWithinAt (fun s ↦ paperGerverContacts s 3)
+              (c • normalVector (t : Real.Angle)) (gerverStageIntervals i) t) := by
+  have hpi : (0 : ℝ) < Real.pi / 2 := by positivity
+  have hangsum : ∀ t : ℝ, (t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) =
+      ((Real.pi / 2 + t : ℝ) : Real.Angle) := fun t => by
+    rw [← Real.Angle.coe_add, add_comm]
+  have ht0 : gerverStageTimes 0 = 0 := gerverStageTimes_zero
+  have ht5 : gerverStageTimes 5 = Real.pi / 2 := rfl
+  have h03 : (0 : ℝ) < gerverStageTimes 3 := by
+    rw [← ht0]; exact gerverStageTimes_strictMono (by decide)
+  have h25 : gerverStageTimes 2 < Real.pi / 2 := by
+    rw [← ht5]; exact gerverStageTimes_strictMono (by decide)
+  -- ### One cap serves every clause: the four theorems below all speak about `gerverOuterCap`
+  obtain ⟨K, hKset, ⟨r, s, hdens, -⟩, -, -⟩ := paperGerverCap_injectivity
+  have hK : ∃ r s, HasCapDensities K r s := ⟨r, s, hdens⟩
+  obtain ⟨-, hGeq, -, -, hcapeq, ⟨K₁, hK₁set, hsing, hA0, hAT, hC0, hCT⟩, hsup⟩ :=
+    gerver_capSupport_identification
+  have hKcarrier : (K.val : Set Point) = gerverOuterCap := by rw [hKset, hGeq, hcapeq]
+  have hcapK : ∀ K' : RightAngleCapSpace, (K'.val : Set Point) = gerverOuterCap → K' = K :=
+    fun K' h => Subtype.ext (SetLike.coe_injective (h.trans hKcarrier.symm))
+  rw [hcapK K₁ hK₁set] at hsing hA0 hAT hC0 hCT
+  obtain ⟨-, ⟨K₂, hK₂set, hniche⟩, -, hmono⟩ := gerver_paperNiche_identification
+  rw [hcapK K₂ hK₂set] at hniche
+  obtain ⟨K₃, hK₃set, Γ, hΓ, hOJP, hclos, harea, -, hnotB, hnotx, hnotD⟩ :=
+    gerver_niche_orientation
+  rw [hcapK K₃ hK₃set] at hOJP hclos harea hnotB hnotx hnotD
+  -- ### The two support values of the cap, and the resulting inner corner
+  have hsupK : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      supportValue (K.val : Set Point) (t : Real.Angle) =
+        inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) + 1 ∧
+      supportValue (K.val : Set Point) ((Real.pi / 2 + t : ℝ) : Real.Angle) =
+        inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) + 1 := by
+    rw [hKcarrier]
+    exact hsup gerverOuterCap (Or.inl rfl)
+  have hcorner : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      capInnerCorner K t = paperGerverPath t := by
+    intro t ht
+    rw [capInnerCorner,
+      (rotatingHallwayParts_formulas (K.val : Set Point) (t : Real.Angle)).2.1, hangsum,
+      (hsupK t ht).1, (hsupK t ht).2, add_sub_cancel_right, add_sub_cancel_right,
+      inner_normalVector_smul_add_inner_tangentVector_smul]
+  refine ⟨hmono, K, hKset, hK, ?_, ⟨Γ, hΓ, hOJP, hclos, harea⟩, ?_, ?_, hnotx, ?_,
+    fun i hi t ht => hasDerivWithinAt_paperGerverContacts_one_neg_smul hi ht,
+    fun i hi t ht => hasDerivWithinAt_paperGerverContacts_three_pos_smul hi ht⟩
+  · -- ### The selected cap vertices and the inner corner
+    -- Strictly inside the interval the two exposed faces are singletons; at the two endpoints
+    -- the selected contacts are the frozen one-sided contacts of cap-support identification,
+    -- with the nondegeneracy proposition equating the two vertices at `π / 2`.
+    intro t
+    have htI : (t : ℝ) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := t.2
+    refine ⟨?_, ?_, hcorner t htI⟩
+    · dsimp only [nondegenerateCapData]
+      split_ifs with hT
+      · rw [hT, hAT]
+        rfl
+      · rcases eq_or_lt_of_le htI.1 with h0 | h0
+        · rw [← h0, hA0]
+          exact congrArg (fun a : Real.Angle ↦ (edgeVertices (K.val) a).1) Real.Angle.coe_zero
+        · have hmem : (t : ℝ) ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) :=
+            ⟨h0, lt_of_le_of_ne htI.2 hT⟩
+          change (edgeVertices (K.val) ((t : ℝ) : Real.Angle)).1 = _
+          rw [edgeVertices_eq_of_exposedEdge_singleton (hsing _ hmem).1]
+    · change (edgeVertices (K.val) (((t : ℝ) + Real.pi / 2 : ℝ) : Real.Angle)).1 = _
+      rcases eq_or_lt_of_le htI.1 with h0 | h0
+      · rw [← h0, zero_add, hC0]
+      rcases eq_or_lt_of_le htI.2 with hT | hT
+      · have hnd : (edgeVertices (K.val) ((Real.pi / 2 + Real.pi / 2 : ℝ) : Real.Angle)).1 =
+            (edgeVertices (K.val) ((Real.pi / 2 + Real.pi / 2 : ℝ) : Real.Angle)).2 :=
+          ((capDensities_contact_eq K hK).2 (Real.pi / 2) ⟨hpi, le_rfl⟩).1
+        rw [hT, hnd, hCT]
+        exact congrArg (fun a : Real.Angle ↦ (edgeVertices (K.val) a).2)
+          (congrArg (fun x : ℝ ↦ (x : Real.Angle)) (by ring))
+      · rw [edgeVertices_eq_of_exposedEdge_singleton (hsing _ ⟨h0, hT⟩).2]
+  · -- ### The open base segment lies in the niche
+    intro a ha
+    rw [hniche]
+    exact gerver_bottom_segment_mem_gerverLiteralNiche ha
+  · -- ### The second contact curve lies on the downward inner wall
+    -- With `B = x + α v`, the two support values turn the wall condition into `α t ≤ 0`.
+    intro t ht
+    refine ⟨hnotB t ht, ?_⟩
+    have htI : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+      ⟨by linarith [ht.1], by rw [← ht5]; exact ht.2⟩
+    rw [mem_rotatingHallwayParts_bRay_iff, hangsum, (hsupK t htI).1, (hsupK t htI).2,
+      add_sub_cancel_right, add_sub_cancel_right]
+    have hB : paperGerverContacts t 1 =
+        paperGerverPath t + (paperGerverVelocityComponents t).1 •
+          tangentVector (t : Real.Angle) := rfl
+    rw [hB, inner_add_left, inner_add_left, real_inner_smul_left, real_inner_smul_left,
+      inner_tangentVector_normalVector_real t t, sub_self, Real.sin_zero,
+      inner_tangentVector_self]
+    exact ⟨by ring, by nlinarith [paperGerverVelocityComponents_fst_nonpos htI]⟩
+  · -- ### The fourth contact curve lies on the leftward inner wall
+    -- With `D = x - β u`, the same two support values turn the wall condition into `0 ≤ β t`.
+    intro t ht
+    refine ⟨hnotD t ht, ?_⟩
+    have htI : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+      ⟨by rw [← ht0]; exact ht.1, by linarith [ht.2]⟩
+    rw [mem_rotatingHallwayParts_dRay_iff, hangsum, (hsupK t htI).1, (hsupK t htI).2,
+      add_sub_cancel_right, add_sub_cancel_right]
+    have hD : paperGerverContacts t 3 =
+        paperGerverPath t - (paperGerverVelocityComponents t).2 •
+          normalVector (t : Real.Angle) := rfl
+    rw [hD, inner_sub_left, inner_sub_left, real_inner_smul_left, real_inner_smul_left,
+      inner_normalVector_self, inner_normalVector_tangentVector]
+    exact ⟨by nlinarith [paperGerverVelocityComponents_snd_nonneg htI], by ring⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The left and right tails of the Gerver cap
+
+`gerver_tailGeometry` identifies the two tails cut out of the Gerver cap by the inner hallway
+walls.  Each tail is cut out by a one-parameter family of supporting half-planes whose contact
+points are the two inner Gerver contact curves `B` and `D`, so the envelope-tangency lemmas of
+`MovingSofa/Convex/EnvelopeFace.lean` identify every intervening face of a tail with a single
+point of the corresponding contact curve, and the stagewise monotonicity of the contact curves
+against a fixed frame direction (`MovingSofa/Gerver/StageRegularity.lean`) supplies both the
+containment of the contact curves in the tails and the injectivity of the two
+parametrizations.  The support sums are the cut identity `h_L (s + π) = -m s`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A continuous injective parametrization traces the directed arc with the specified
+endpoints. -/
+def ParametrizesDirectedArc (f : ℝ → Point) (a b : ℝ) (Γ : DirectedArcData) : Prop :=
+  a < b ∧ ContinuousOn f (Set.Icc a b) ∧ Set.InjOn f (Set.Icc a b) ∧
+    f '' Set.Icc a b = Γ.carrier ∧ f a = Γ.startPoint ∧ f b = Γ.endPoint
+
+private def gerverLeftTailSupport (s : ℝ) : ℝ :=
+  inner ℝ (paperGerverPath (s - Real.pi / 2))
+    (tangentVector ((s - Real.pi / 2 : ℝ) : Real.Angle))
+
+private def gerverRightTailSupport (s : ℝ) : ℝ :=
+  inner ℝ (paperGerverPath s) (normalVector (s : Real.Angle))
+
+private theorem gerver_tail_membership (K : SpecialCapSpace) (B D : ConvexBody Point)
+    (hK : (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2))
+    (hB : (B : Set Point) = (canonicalTailSets K).1)
+    (hD : (D : Set Point) = (canonicalTailSets K).2) :
+    (StrictAntiOn (fun s ↦ inner ℝ (paperGerverContacts s 3)
+     (tangentVector ((gerverStageTimes 4 : ℝ) : Real.Angle)))
+     (Set.Icc (gerverStageTimes 0) (gerverStageTimes 2))) ∧
+    (StrictMonoOn (fun s ↦ inner ℝ (paperGerverContacts s 1)
+     (normalVector ((gerverStageTimes 1 : ℝ) : Real.Angle)))
+     (Set.Icc (gerverStageTimes 3) (gerverStageTimes 5))) ∧
+    (∀ t ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2),
+     paperGerverContacts t 3 ∈ (D : Set Point)) ∧
+    (∀ t ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5),
+     paperGerverContacts t 1 ∈ (B : Set Point)) := by
+  -- ### One cap serves every clause, and it carries the literal contact geometry
+  obtain ⟨-, hGeq, -, -, hcapeq, -, -⟩ := gerver_capSupport_identification
+  have hcarrier : (K.val.val : Set Point) = gerverOuterCap := by rw [hK, hGeq, hcapeq]
+  obtain ⟨-, K', hK'set, hdens, hsel, -, -, hBwall, -, hDwall, -, -⟩ :=
+    paperGerver_contact_geometry
+  have hKeq : K' = K.val := Subtype.ext (SetLike.coe_injective (hK'set.trans hK.symm))
+  subst hKeq
+  have ht0 : gerverStageTimes 0 = 0 := gerverStageTimes_zero
+  have ht5 : gerverStageTimes 5 = Real.pi / 2 := rfl
+  have hr : paperGerverConstants.2.1 = gerverStageTimes 1 := rfl
+  have hl : paperGerverConstants.2.2 = gerverStageTimes 4 := rfl
+  have h01' : gerverStageTimes 0 < gerverStageTimes 1 := gerverStageTimes_strictMono (by decide)
+  have h12 : gerverStageTimes 1 < gerverStageTimes 2 := gerverStageTimes_strictMono (by decide)
+  have h23 : gerverStageTimes 2 < gerverStageTimes 3 := gerverStageTimes_strictMono (by decide)
+  have h34 : gerverStageTimes 3 < gerverStageTimes 4 := gerverStageTimes_strictMono (by decide)
+  have h45' : gerverStageTimes 4 < gerverStageTimes 5 := gerverStageTimes_strictMono (by decide)
+  have h01 : (0 : ℝ) < gerverStageTimes 1 := ht0 ▸ h01'
+  have h45 : gerverStageTimes 4 < Real.pi / 2 := ht5 ▸ h45'
+  -- ### The support coordinates of the cap along the path
+  have hcorner : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2), capInnerCorner K.val t = paperGerverPath t :=
+    fun t ht ↦ (hsel ⟨t, ht⟩).2.2
+  have hsupu : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) =
+        supportValue (K.val.val : Set Point) (t : Real.Angle) - 1 := by
+    intro t ht
+    rw [← hcorner t ht]
+    exact inner_capInnerCorner_normalVector K.val t
+  have hsupv : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) =
+        supportValue (K.val.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+    intro t ht
+    rw [← hcorner t ht]
+    exact inner_capInnerCorner_tangentVector K.val t
+  -- ### The two families of supporting lines cutting the tails out of the cap
+  have hmDval : ∀ t : ℝ, gerverLeftTailSupport (t + Real.pi / 2) =
+      inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) := by
+    intro t
+    simp only [gerverLeftTailSupport, add_sub_cancel_right]
+  have hmDsup : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2), gerverLeftTailSupport (t + Real.pi / 2) =
+      supportValue (K.val.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 :=
+    fun t ht ↦ (hmDval t).trans (hsupv t ht)
+  have hcapupper : ∀ q ∈ (K.val.val : Set Point), 0 ≤ q 1 := fun q hq ↦ by
+    simpa only [inner_normalVector_pi_div_two] using
+      K.val.inner_normalVector_pi_div_two_nonneg hq
+  have hcapD : ∀ t ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2),
+      paperGerverContacts t 3 ∈ (K.val.val : Set Point) := fun t ht ↦
+    hcarrier ▸ gerver_niche_roof_membership.1 t ht
+  have hcapB : ∀ t ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5),
+      paperGerverContacts t 1 ∈ (K.val.val : Set Point) := fun t ht ↦
+    hcarrier ▸ gerver_niche_roof_membership.2.2 t ht
+  -- ### Each contact curve stays above its own distinguished wall
+  have hantiD : ∀ i : Fin 5, (i = 0 ∨ i = 1) →
+      StrictAntiOn (fun s ↦ inner ℝ (paperGerverContacts s 3)
+        (tangentVector ((gerverStageTimes 4 : ℝ) : Real.Angle))) (gerverStageIntervals i) := by
+    intro i hi
+    refine strictAntiOn_inner_paperGerverContacts_three hi ?_ ?_ <;> intro s hs <;>
+      rcases hi with rfl | rfl
+    · rw [gerverStageIntervals_zero] at hs; linarith only [hs.2, h12, h23, h34]
+    · rw [gerverStageIntervals_one] at hs; linarith only [hs.2, h23, h34]
+    · rw [gerverStageIntervals_zero] at hs; linarith only [hs.1, ht0, h45, Real.pi_pos]
+    · rw [gerverStageIntervals_one] at hs; linarith only [hs.1, h01, h45, Real.pi_pos]
+  have hmonoB : ∀ i : Fin 5, (i = 3 ∨ i = 4) →
+      StrictMonoOn (fun s ↦ inner ℝ (paperGerverContacts s 1)
+        (normalVector ((gerverStageTimes 1 : ℝ) : Real.Angle))) (gerverStageIntervals i) := by
+    intro i hi
+    refine strictMonoOn_inner_paperGerverContacts_one hi ?_ ?_ <;> intro s hs <;>
+      rcases hi with rfl | rfl
+    · rw [gerverStageIntervals_three] at hs; linarith only [hs.1, h12, h23]
+    · rw [gerverStageIntervals_four] at hs; linarith only [hs.1, h12, h23, h34]
+    · rw [gerverStageIntervals_three] at hs; linarith only [hs.2, h45, h01, Real.pi_pos]
+    · rw [gerverStageIntervals_four] at hs; linarith only [hs.2, ht5, h01, Real.pi_pos]
+  have hstrictD : StrictAntiOn (fun s ↦ inner ℝ (paperGerverContacts s 3)
+      (tangentVector ((gerverStageTimes 4 : ℝ) : Real.Angle)))
+      (Set.Icc (gerverStageTimes 0) (gerverStageTimes 2)) := by
+    rw [← Set.Icc_union_Icc_eq_Icc h01'.le h12.le]
+    exact (gerverStageIntervals_zero ▸ hantiD 0 (Or.inl rfl)).union
+      (gerverStageIntervals_one ▸ hantiD 1 (Or.inr rfl))
+      (isGreatest_Icc h01'.le) (isLeast_Icc h12.le)
+  have hstrictB : StrictMonoOn (fun s ↦ inner ℝ (paperGerverContacts s 1)
+      (normalVector ((gerverStageTimes 1 : ℝ) : Real.Angle)))
+      (Set.Icc (gerverStageTimes 3) (gerverStageTimes 5)) := by
+    rw [← Set.Icc_union_Icc_eq_Icc h34.le h45'.le]
+    exact (gerverStageIntervals_three ▸ hmonoB 3 (Or.inl rfl)).union
+      (gerverStageIntervals_four ▸ hmonoB 4 (Or.inr rfl))
+      (isGreatest_Icc h34.le) (isLeast_Icc h45'.le)
+  have hkeyD : ∀ t ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2),
+      inner ℝ (paperGerverContacts (gerverStageTimes 2) 3)
+          (tangentVector ((gerverStageTimes 4 : ℝ) : Real.Angle)) ≤
+        inner ℝ (paperGerverContacts t 3)
+          (tangentVector ((gerverStageTimes 4 : ℝ) : Real.Angle)) :=
+    fun t ht ↦ hstrictD.antitoneOn ht ⟨ht.1.trans ht.2, le_rfl⟩ ht.2
+  have hkeyB : ∀ t ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5),
+      inner ℝ (paperGerverContacts (gerverStageTimes 3) 1)
+          (normalVector ((gerverStageTimes 1 : ℝ) : Real.Angle)) ≤
+        inner ℝ (paperGerverContacts t 1)
+          (normalVector ((gerverStageTimes 1 : ℝ) : Real.Angle)) :=
+    fun t ht ↦ hstrictB.monotoneOn ⟨le_rfl, ht.1.trans ht.2⟩ ht ht.1
+  have hDfix : ∀ t ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2),
+      paperGerverContacts t 3 ∈ (innerWallUpperHalfPlanes K.val (gerverStageTimes 4)).2 := by
+    intro t ht
+    have hl4 : gerverStageTimes 4 ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+      ⟨by linarith only [h01, h12, h23, h34], h45.le⟩
+    change supportValue (K.val.val : Set Point)
+        ((gerverStageTimes 4 + Real.pi / 2 : ℝ) : Real.Angle) - 1 ≤
+      inner ℝ (paperGerverContacts t 3)
+        (normalVector ((gerverStageTimes 4 + Real.pi / 2 : ℝ) : Real.Angle))
+    rw [← hmDsup _ hl4, hmDval, normalVector_add_pi_div_two_real,
+      ← gerver_niche_piece_endpoints.2.1]
+    exact hkeyD t ht
+  have hBfix : ∀ t ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5),
+      paperGerverContacts t 1 ∈ (innerWallUpperHalfPlanes K.val (gerverStageTimes 1)).1 := by
+    intro t ht
+    have hr1 : gerverStageTimes 1 ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+      ⟨h01.le, by linarith only [h12, h23, h34, h45]⟩
+    change supportValue (K.val.val : Set Point) ((gerverStageTimes 1 : ℝ) : Real.Angle) - 1 ≤
+      inner ℝ (paperGerverContacts t 1) (normalVector ((gerverStageTimes 1 : ℝ) : Real.Angle))
+    rw [← hsupu _ hr1, ← gerver_niche_piece_endpoints.1]
+    exact hkeyB t ht
+  -- ### The remaining walls are excluded by the niche
+  have hiq : ∀ s : ℝ, (rotatingHallwayParts (K.val.val : Set Point) (s : Real.Angle)).innerQuadrant
+      = innerQuadrant (K.val.val : Set Point) s := by
+    intro s
+    rw [(rotatingHallwayParts_formulas (K.val.val : Set Point)
+      (s : Real.Angle)).2.2.2.2.2.2.2.2, innerQuadrant, ← Real.Angle.coe_add]
+  have hDside : (distinguishedCapSides K.val).2.upperHalfPlane =
+      (innerWallUpperHalfPlanes K.val (gerverStageTimes 4)).2 := by rw [← hl]; rfl
+  have hBside : (distinguishedCapSides K.val).1.upperHalfPlane =
+      (innerWallUpperHalfPlanes K.val (gerverStageTimes 1)).1 := by rw [← hr]; rfl
+  have hDwedge : ∀ t ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2),
+      ∀ s ∈ Set.Ioo (0 : ℝ) (gerverStageTimes 4),
+      paperGerverContacts t 3 ∈ (innerWallUpperHalfPlanes K.val s).2 := by
+    intro t ht s hs
+    by_contra hmem
+    have hquad := (cap_tail_monotonicity_intervals K).2.2.2 s
+      ⟨hs.1.le, by rw [hl]; exact hs.2⟩ ⟨hs.1, by linarith only [hs.2, h45]⟩
+    have hfix : paperGerverContacts t 3 ∈ (distinguishedCapSides K.val).2.upperHalfPlane := by
+      rw [hDside]
+      exact hDfix t ht
+    have hin : paperGerverContacts t 3 ∈
+        ((distinguishedCapSides K.val).2.upperHalfPlane ∩ {p : Point | 0 ≤ p 1}) \
+          (innerWallUpperHalfPlanes K.val s).2 :=
+      ⟨⟨hfix, hcapupper _ (hcapD t ht)⟩, hmem⟩
+    rw [← hquad] at hin
+    exact (hDwall t ht).1 ⟨hin.2.1, Set.mem_iUnion₂.2
+      ⟨s, ⟨hs.1, by linarith only [hs.2, h45]⟩, (hiq s) ▸ hin.2.2⟩⟩
+  have hBwedge : ∀ t ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5),
+      ∀ s ∈ Set.Ioo (gerverStageTimes 1) (Real.pi / 2),
+      paperGerverContacts t 1 ∈ (innerWallUpperHalfPlanes K.val s).1 := by
+    intro t ht s hs
+    by_contra hmem
+    have hquad := (cap_tail_monotonicity_intervals K).2.2.1 s
+      ⟨by rw [hr]; exact hs.1, hs.2.le⟩ ⟨by linarith only [hs.1, h01], hs.2⟩
+    have hfix : paperGerverContacts t 1 ∈ (distinguishedCapSides K.val).1.upperHalfPlane := by
+      rw [hBside]
+      exact hBfix t ht
+    have hin : paperGerverContacts t 1 ∈
+        ((distinguishedCapSides K.val).1.upperHalfPlane ∩ {p : Point | 0 ≤ p 1}) \
+          (innerWallUpperHalfPlanes K.val s).1 :=
+      ⟨⟨hfix, hcapupper _ (hcapB t ht)⟩, hmem⟩
+    rw [← hquad] at hin
+    exact (hBwall t ht).1 ⟨hin.2.1, Set.mem_iUnion₂.2
+      ⟨s, ⟨by linarith only [hs.1, h01], hs.2⟩, (hiq s) ▸ hin.2.2⟩⟩
+  -- ### Both contact curves lie in their tails
+  have hDmem : ∀ t ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2),
+      paperGerverContacts t 3 ∈ (D : Set Point) := by
+    intro t ht
+    rw [hD]
+    refine ⟨hcapD t ht, Set.mem_iInter₂.2 fun s hs ↦ ?_⟩
+    rw [hl] at hs
+    rcases eq_or_lt_of_le hs.1 with h0 | h0
+    · change supportValue (K.val.val : Set Point) ((s + Real.pi / 2 : ℝ) : Real.Angle) - 1 ≤
+        inner ℝ (paperGerverContacts t 3) (normalVector ((s + Real.pi / 2 : ℝ) : Real.Angle))
+      rw [← h0, zero_add, K.val.property.2.2.2.1, inner_normalVector_pi_div_two]
+      simpa only [sub_self] using hcapupper _ (hcapD t ht)
+    rcases eq_or_lt_of_le hs.2 with h4 | h4
+    · exact h4 ▸ hDfix t ht
+    · exact hDwedge t ht s ⟨h0, h4⟩
+  have hBmem : ∀ t ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5),
+      paperGerverContacts t 1 ∈ (B : Set Point) := by
+    intro t ht
+    rw [hB]
+    refine ⟨hcapB t ht, Set.mem_iInter₂.2 fun s hs ↦ ?_⟩
+    rw [hr] at hs
+    rcases eq_or_lt_of_le hs.2 with hT | hT
+    · change supportValue (K.val.val : Set Point) (s : Real.Angle) - 1 ≤
+        inner ℝ (paperGerverContacts t 1) (normalVector (s : Real.Angle))
+      rw [hT, K.val.property.2.2.2.1, inner_normalVector_pi_div_two]
+      simpa only [sub_self] using hcapupper _ (hcapB t ht)
+    rcases eq_or_lt_of_le hs.1 with h1 | h1
+    · exact h1 ▸ hBfix t ht
+    · exact hBwedge t ht s ⟨h1, hT⟩
+  -- ### The singleton faces along the two contact curves
+  exact ⟨hstrictD, hstrictB, hDmem, hBmem⟩
+
+private theorem gerver_tail_support_lines (K : SpecialCapSpace) (B D : ConvexBody Point)
+    (hK : (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2))
+    (hB : (B : Set Point) = (canonicalTailSets K).1)
+    (hD : (D : Set Point) = (canonicalTailSets K).2) :
+    (∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+     inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) =
+       supportValue (K.val.val : Set Point) (t : Real.Angle) - 1) ∧
+    (∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+     inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) =
+       supportValue (K.val.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) ∧
+    (∀ t : ℝ, gerverLeftTailSupport (t + Real.pi / 2) =
+     inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle))) ∧
+    (∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2), gerverLeftTailSupport (t + Real.pi / 2) =
+     supportValue (K.val.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) ∧
+    (∀ t : ℝ, HasDerivAt gerverLeftTailSupport
+     (inner ℝ (paperGerverContacts t 3) (tangentVector ((t + Real.pi / 2 : ℝ) : Real.Angle)))
+     (t + Real.pi / 2)) ∧
+    (∀ t : ℝ, HasDerivAt gerverRightTailSupport
+     (inner ℝ (paperGerverContacts t 1) (tangentVector (t : Real.Angle))) t) ∧
+    (∀ q ∈ (D : Set Point), ∀ s ∈ Set.Icc (Real.pi / 2)
+     (gerverStageTimes 4 + Real.pi / 2), gerverLeftTailSupport s ≤ inner ℝ q (normalVector (s :
+       Real.Angle))) ∧
+    (∀ q ∈ (B : Set Point), ∀ s ∈ Set.Icc (gerverStageTimes 1) (Real.pi / 2),
+     gerverRightTailSupport s ≤ inner ℝ q (normalVector (s : Real.Angle))) ∧
+    (∀ t ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2),
+     inner ℝ (paperGerverContacts t 3) (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) =
+       gerverLeftTailSupport (t + Real.pi / 2)) ∧
+    (∀ t ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5),
+     inner ℝ (paperGerverContacts t 1) (normalVector (t : Real.Angle)) = gerverRightTailSupport t)
+       := by
+  -- ### One cap serves every clause, and it carries the literal contact geometry
+  obtain ⟨-, K', hK'set, hdens, hsel, -, -, hBwall, -, hDwall, -, -⟩ :=
+    paperGerver_contact_geometry
+  have hKeq : K' = K.val := Subtype.ext (SetLike.coe_injective (hK'set.trans hK.symm))
+  subst hKeq
+  have ht0 : gerverStageTimes 0 = 0 := gerverStageTimes_zero
+  have ht5 : gerverStageTimes 5 = Real.pi / 2 := rfl
+  have hr : paperGerverConstants.2.1 = gerverStageTimes 1 := rfl
+  have hl : paperGerverConstants.2.2 = gerverStageTimes 4 := rfl
+  have h01' : gerverStageTimes 0 < gerverStageTimes 1 := gerverStageTimes_strictMono (by decide)
+  have h12 : gerverStageTimes 1 < gerverStageTimes 2 := gerverStageTimes_strictMono (by decide)
+  have h23 : gerverStageTimes 2 < gerverStageTimes 3 := gerverStageTimes_strictMono (by decide)
+  have h34 : gerverStageTimes 3 < gerverStageTimes 4 := gerverStageTimes_strictMono (by decide)
+  have h45' : gerverStageTimes 4 < gerverStageTimes 5 := gerverStageTimes_strictMono (by decide)
+  have h01 : (0 : ℝ) < gerverStageTimes 1 := ht0 ▸ h01'
+  have h45 : gerverStageTimes 4 < Real.pi / 2 := ht5 ▸ h45'
+  -- ### The support coordinates of the cap along the path
+  have hcorner : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2), capInnerCorner K.val t = paperGerverPath t :=
+    fun t ht ↦ (hsel ⟨t, ht⟩).2.2
+  have hsupu : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) =
+        supportValue (K.val.val : Set Point) (t : Real.Angle) - 1 := by
+    intro t ht
+    rw [← hcorner t ht]
+    exact inner_capInnerCorner_normalVector K.val t
+  have hsupv : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) =
+        supportValue (K.val.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+    intro t ht
+    rw [← hcorner t ht]
+    exact inner_capInnerCorner_tangentVector K.val t
+  -- ### The two families of supporting lines cutting the tails out of the cap
+  have hmDval : ∀ t : ℝ, gerverLeftTailSupport (t + Real.pi / 2) =
+      inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) := by
+    intro t
+    simp only [gerverLeftTailSupport, add_sub_cancel_right]
+  have hmDsup : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2), gerverLeftTailSupport (t + Real.pi / 2) =
+      supportValue (K.val.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 :=
+    fun t ht ↦ (hmDval t).trans (hsupv t ht)
+  have hmDderiv : ∀ t : ℝ, HasDerivAt gerverLeftTailSupport
+      (inner ℝ (paperGerverContacts t 3) (tangentVector ((t + Real.pi / 2 : ℝ) : Real.Angle)))
+      (t + Real.pi / 2) := by
+    intro t
+    have h := (hasDerivAt_inner_paperGerverPath_tangentVector (t + Real.pi / 2 - Real.pi / 2)).comp
+      (t + Real.pi / 2) ((hasDerivAt_id (t + Real.pi / 2)).sub_const (Real.pi / 2))
+    rw [add_sub_cancel_right] at h
+    rw [tangentVector_add_pi_div_two, inner_neg_right]
+    unfold gerverLeftTailSupport
+    simpa only [Function.comp_def, mul_one, id_eq] using h
+  have hmBderiv : ∀ t : ℝ, HasDerivAt gerverRightTailSupport
+      (inner ℝ (paperGerverContacts t 1) (tangentVector (t : Real.Angle))) t := by
+    intro t
+    unfold gerverRightTailSupport
+    exact hasDerivAt_inner_paperGerverPath_normalVector t
+  have hDle : ∀ q ∈ (D : Set Point), ∀ s ∈ Set.Icc (Real.pi / 2)
+      (gerverStageTimes 4 + Real.pi / 2), gerverLeftTailSupport s ≤ inner ℝ q (normalVector (s :
+        Real.Angle)) := by
+    intro q hq s hs
+    rw [hD] at hq
+    have hs' : s - Real.pi / 2 ∈ Set.Icc (0 : ℝ) paperGerverConstants.2.2 :=
+      ⟨by linarith only [hs.1], by rw [hl]; linarith only [hs.2]⟩
+    have h : supportValue (K.val.val : Set Point)
+        ((s - Real.pi / 2 + Real.pi / 2 : ℝ) : Real.Angle) - 1 ≤
+        inner ℝ q (normalVector ((s - Real.pi / 2 + Real.pi / 2 : ℝ) : Real.Angle)) :=
+      Set.mem_iInter₂.mp hq.2 (s - Real.pi / 2) hs'
+    rw [sub_add_cancel] at h
+    have hmem : s - Real.pi / 2 ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+      ⟨hs'.1, by linarith only [hs'.2, hl ▸ hs'.2, h45, ht5]⟩
+    rw [show gerverLeftTailSupport s = gerverLeftTailSupport (s - Real.pi / 2 + Real.pi / 2) by rw
+      [sub_add_cancel],
+      hmDsup _ hmem, sub_add_cancel]
+    exact h
+  have hBle : ∀ q ∈ (B : Set Point), ∀ s ∈ Set.Icc (gerverStageTimes 1) (Real.pi / 2),
+      gerverRightTailSupport s ≤ inner ℝ q (normalVector (s : Real.Angle)) := by
+    intro q hq s hs
+    rw [hB] at hq
+    have h : supportValue (K.val.val : Set Point) (s : Real.Angle) - 1 ≤
+        inner ℝ q (normalVector (s : Real.Angle)) :=
+      Set.mem_iInter₂.mp hq.2 s ⟨by rw [hr]; exact hs.1, hs.2⟩
+    simp only [gerverRightTailSupport]
+    rw [hsupu s ⟨by linarith only [hs.1, h01], hs.2⟩]
+    exact h
+  -- ### The two contact curves touch their defining lines
+  have hDtouch : ∀ t ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2),
+      inner ℝ (paperGerverContacts t 3) (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) =
+        gerverLeftTailSupport (t + Real.pi / 2) := by
+    intro t ht
+    have htI : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+      ⟨ht0 ▸ ht.1, by linarith only [ht.2, h23, h34, h45]⟩
+    have h := (mem_rotatingHallwayParts_dRay_iff _ _ _).mp (hDwall t ht).2
+    rw [normalVector_add_pi_div_two_real, hmDval t, h.2, hsupv t htI, Real.Angle.coe_add]
+  have hBtouch : ∀ t ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5),
+      inner ℝ (paperGerverContacts t 1) (normalVector (t : Real.Angle)) = gerverRightTailSupport t
+        := by
+    intro t ht
+    have htI : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+      ⟨by linarith only [ht.1, h01, h12, h23], ht5 ▸ ht.2⟩
+    have h := (mem_rotatingHallwayParts_bRay_iff _ _ _).mp (hBwall t ht).2
+    simp only [gerverRightTailSupport]
+    rw [hsupu t htI]
+    exact h.1
+  -- ### The cap lies in the upper half-plane, and both contact curves lie in the cap
+  exact ⟨hsupu, hsupv, hmDval, hmDsup, hmDderiv, hmBderiv, hDle, hBle, hDtouch, hBtouch⟩
+
+private theorem gerver_tail_contact_faces (K : SpecialCapSpace) (B D : ConvexBody Point)
+    (hK : (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2))
+    (hB : (B : Set Point) = (canonicalTailSets K).1)
+    (hD : (D : Set Point) = (canonicalTailSets K).2) :
+    (∀ t ∈ Set.Ioc (gerverStageTimes 0) (gerverStageTimes 2),
+     exposedEdge D ((3 * Real.pi / 2 + t : ℝ) : Real.Angle) = {paperGerverContacts t 3}) ∧
+    (∀ t ∈ Set.Ico (gerverStageTimes 3) (gerverStageTimes 5),
+     exposedEdge B ((Real.pi + t : ℝ) : Real.Angle) = {paperGerverContacts t 1}) ∧
+    ((edgeVertices D ((3 * Real.pi / 2 + gerverStageTimes 4 : ℝ) : Real.Angle)).2 =
+     paperGerverContacts (gerverStageTimes 2) 3) ∧
+    ((edgeVertices B ((Real.pi + gerverStageTimes 1 : ℝ) : Real.Angle)).1 =
+     paperGerverContacts (gerverStageTimes 3) 1) ∧
+    ((edgeVertices D ((3 * Real.pi / 2 : ℝ) : Real.Angle)).1 =
+     paperGerverContacts 0 3) ∧
+    ((edgeVertices B ((3 * Real.pi / 2 : ℝ) : Real.Angle)).2 =
+     paperGerverContacts (Real.pi / 2) 1) ∧
+    (paperGerverContacts (gerverStageTimes 2) 3 ∈
+     exposedEdge D ((3 * Real.pi / 2 + gerverStageTimes 4 : ℝ) : Real.Angle)) ∧
+    (paperGerverContacts (gerverStageTimes 3) 1 ∈
+     exposedEdge B ((Real.pi + gerverStageTimes 1 : ℝ) : Real.Angle)) := by
+  -- ### One cap serves every clause, and it carries the literal contact geometry
+  obtain ⟨-, hGeq, -, -, hcapeq, -, -⟩ := gerver_capSupport_identification
+  have hcarrier : (K.val.val : Set Point) = gerverOuterCap := by rw [hK, hGeq, hcapeq]
+  obtain ⟨-, K', hK'set, hdens, hsel, -, -, hBwall, -, hDwall, -, -⟩ :=
+    paperGerver_contact_geometry
+  have hKeq : K' = K.val := Subtype.ext (SetLike.coe_injective (hK'set.trans hK.symm))
+  subst hKeq
+  obtain ⟨hstrictD, hstrictB, hDmem, hBmem⟩ :=
+    gerver_tail_membership K B D hK hB hD
+  obtain ⟨hsupu, hsupv, hmDval, hmDsup, hmDderiv, hmBderiv, hDle, hBle, hDtouch, hBtouch⟩ :=
+    gerver_tail_support_lines K B D hK hB hD
+  have ht0 : gerverStageTimes 0 = 0 := gerverStageTimes_zero
+  have ht5 : gerverStageTimes 5 = Real.pi / 2 := rfl
+  have h01' : gerverStageTimes 0 < gerverStageTimes 1 := gerverStageTimes_strictMono (by decide)
+  have h12 : gerverStageTimes 1 < gerverStageTimes 2 := gerverStageTimes_strictMono (by decide)
+  have h23 : gerverStageTimes 2 < gerverStageTimes 3 := gerverStageTimes_strictMono (by decide)
+  have h34 : gerverStageTimes 3 < gerverStageTimes 4 := gerverStageTimes_strictMono (by decide)
+  have h45' : gerverStageTimes 4 < gerverStageTimes 5 := gerverStageTimes_strictMono (by decide)
+  have h01 : (0 : ℝ) < gerverStageTimes 1 := ht0 ▸ h01'
+  have h45 : gerverStageTimes 4 < Real.pi / 2 := ht5 ▸ h45'
+  -- ### The support coordinates of the cap along the path
+  have hcast : ∀ x y : ℝ, x = y → ((x : ℝ) : Real.Angle) = ((y : ℝ) : Real.Angle) :=
+    fun x y h ↦ by rw [h]
+  have h02 : (0 : ℝ) < gerverStageTimes 2 := by linarith only [h01, h12]
+  have h24 : gerverStageTimes 2 < gerverStageTimes 4 := by linarith only [h23, h34]
+  have hl4I : gerverStageTimes 4 ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+    ⟨by linarith only [h01, h12, h23, h34], h45.le⟩
+  have hmemD0 : (0 : ℝ) ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2) :=
+    ⟨ht0.le, h02.le⟩
+  have hmemD2 : gerverStageTimes 2 ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2) :=
+    ⟨by linarith only [ht0.le, h02], le_rfl⟩
+  have hmemB3 : gerverStageTimes 3 ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5) :=
+    ⟨le_rfl, by linarith only [ht5.ge, h34, h45]⟩
+  have hmemB5 : Real.pi / 2 ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5) :=
+    ⟨by linarith only [h34, h45], ht5.ge⟩
+  have hDleIoo : ∀ q ∈ (D : Set Point), ∀ s ∈ Set.Ioo (Real.pi / 2)
+      (gerverStageTimes 4 + Real.pi / 2), gerverLeftTailSupport s ≤ inner ℝ q (normalVector (s :
+        Real.Angle)) :=
+    fun q hq s hs ↦ hDle q hq s ⟨hs.1.le, hs.2.le⟩
+  have hBleIoo : ∀ q ∈ (B : Set Point), ∀ s ∈ Set.Ioo (gerverStageTimes 1) (Real.pi / 2),
+      gerverRightTailSupport s ≤ inner ℝ q (normalVector (s : Real.Angle)) :=
+    fun q hq s hs ↦ hBle q hq s ⟨hs.1.le, hs.2.le⟩
+  have hDface : ∀ t ∈ Set.Ioc (gerverStageTimes 0) (gerverStageTimes 2),
+      exposedEdge D ((3 * Real.pi / 2 + t : ℝ) : Real.Angle) = {paperGerverContacts t 3} := by
+    intro t ht
+    have htI : t ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2) := ⟨ht.1.le, ht.2⟩
+    rw [ht0] at ht
+    have h := exposedEdge_add_pi_eq_singleton_of_mem_Ioo (L := D) (m := gerverLeftTailSupport) (t
+      := t + Real.pi / 2)
+      ⟨by linarith only [ht.1], by linarith only [ht.2, h24]⟩ hDleIoo (hDmem t htI)
+      (hDtouch t htI) (hmDderiv t)
+    rwa [hcast _ _ (show t + Real.pi / 2 + Real.pi = 3 * Real.pi / 2 + t by ring)] at h
+  have hBface : ∀ t ∈ Set.Ico (gerverStageTimes 3) (gerverStageTimes 5),
+      exposedEdge B ((Real.pi + t : ℝ) : Real.Angle) = {paperGerverContacts t 1} := by
+    intro t ht
+    have htI : t ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5) := ⟨ht.1, ht.2.le⟩
+    rw [ht5] at ht
+    have h := exposedEdge_add_pi_eq_singleton_of_mem_Ioo (L := B) (m := gerverRightTailSupport) (t
+      := t)
+      ⟨by linarith only [ht.1, h12, h23], ht.2⟩ hBleIoo (hBmem t htI) (hBtouch t htI)
+      (hmBderiv t)
+    rwa [hcast _ _ (show t + Real.pi = Real.pi + t by ring)] at h
+  -- ### The two exact joins are contacts of the outer walls as well
+  have hDjoin : inner ℝ (paperGerverContacts (gerverStageTimes 2) 3)
+      (normalVector ((gerverStageTimes 4 + Real.pi / 2 : ℝ) : Real.Angle)) =
+      gerverLeftTailSupport (gerverStageTimes 4 + Real.pi / 2) := by
+    rw [hmDval, normalVector_add_pi_div_two_real, gerver_niche_piece_endpoints.2.1]
+  have hBjoin : inner ℝ (paperGerverContacts (gerverStageTimes 3) 1)
+      (normalVector ((gerverStageTimes 1 : ℝ) : Real.Angle)) = gerverRightTailSupport
+        (gerverStageTimes 1) := by
+    simp only [gerverRightTailSupport]
+    rw [gerver_niche_piece_endpoints.1]
+  have hDouter : paperGerverContacts (gerverStageTimes 2) 3 ∈
+      exposedEdge D ((3 * Real.pi / 2 + gerverStageTimes 4 : ℝ) : Real.Angle) := by
+    have h := exposedEdge_add_pi_eq_of_forall_le
+      (fun q hq ↦ hDle q hq _ ⟨by linarith only [hl4I.1], le_rfl⟩) (hDmem _ hmemD2) hDjoin
+    rw [hcast _ _ (show gerverStageTimes 4 + Real.pi / 2 + Real.pi =
+      3 * Real.pi / 2 + gerverStageTimes 4 by ring)] at h
+    rw [h]
+    exact ⟨hDmem _ hmemD2, hDjoin⟩
+  have hBouter : paperGerverContacts (gerverStageTimes 3) 1 ∈
+      exposedEdge B ((Real.pi + gerverStageTimes 1 : ℝ) : Real.Angle) := by
+    have h := exposedEdge_add_pi_eq_of_forall_le
+      (fun q hq ↦ hBle q hq _ ⟨le_rfl, by linarith only [h12, h23, h34, h45]⟩)
+      (hBmem _ hmemB3) hBjoin
+    rw [hcast _ _ (show gerverStageTimes 1 + Real.pi = Real.pi + gerverStageTimes 1 by ring)] at h
+    rw [h]
+    exact ⟨hBmem _ hmemB3, hBjoin⟩
+  -- ### The two arc endpoints at the outer normals
+  have hDend : (edgeVertices D ((3 * Real.pi / 2 + gerverStageTimes 4 : ℝ) : Real.Angle)).2 =
+      paperGerverContacts (gerverStageTimes 2) 3 := by
+    have hsin : Real.sin (3 * Real.pi / 2 + gerverStageTimes 2 -
+        (3 * Real.pi / 2 + gerverStageTimes 4)) ≠ 0 := by
+      have : Real.sin (gerverStageTimes 4 - gerverStageTimes 2) > 0 :=
+        Real.sin_pos_of_pos_of_lt_pi (by linarith only [h24])
+          (by linarith only [h45, h02, Real.pi_pos])
+      rw [show 3 * Real.pi / 2 + gerverStageTimes 2 - (3 * Real.pi / 2 + gerverStageTimes 4) =
+        -(gerverStageTimes 4 - gerverStageTimes 2) by ring, Real.sin_neg]
+      linarith only [this]
+    have hp := eq_supportingIntersection_of_mem_exposedEdge hsin
+      ((hDface _ ⟨by linarith only [ht0.le, h02], le_rfl⟩) ▸ rfl) hDouter
+    have hmem : supportingIntersection D ((3 * Real.pi / 2 + gerverStageTimes 2 : ℝ) : Real.Angle)
+        ((3 * Real.pi / 2 + gerverStageTimes 4 : ℝ) : Real.Angle) ∈ D := by
+      rw [← hp]
+      exact hDmem _ hmemD2
+    rw [← supportingIntersection_eq_edgeVertices_snd_of_mem D (by linarith only [h24])
+      (by linarith only [h45, h02, Real.pi_pos]) hmem, ← hp]
+  have hBstart : (edgeVertices B ((Real.pi + gerverStageTimes 1 : ℝ) : Real.Angle)).1 =
+      paperGerverContacts (gerverStageTimes 3) 1 := by
+    have hsin : Real.sin (Real.pi + gerverStageTimes 1 - (Real.pi + gerverStageTimes 3)) ≠ 0 := by
+      have : Real.sin (gerverStageTimes 3 - gerverStageTimes 1) > 0 :=
+        Real.sin_pos_of_pos_of_lt_pi (by linarith only [h12, h23])
+          (by linarith only [h34, h45, h01, Real.pi_pos])
+      rw [show Real.pi + gerverStageTimes 1 - (Real.pi + gerverStageTimes 3) =
+        -(gerverStageTimes 3 - gerverStageTimes 1) by ring, Real.sin_neg]
+      linarith only [this]
+    have hp := eq_supportingIntersection_of_mem_exposedEdge hsin hBouter
+      ((hBface _ ⟨le_rfl, by linarith only [ht5.ge, h34, h45]⟩) ▸ rfl)
+    have hmem : supportingIntersection B ((Real.pi + gerverStageTimes 1 : ℝ) : Real.Angle)
+        ((Real.pi + gerverStageTimes 3 : ℝ) : Real.Angle) ∈ B := by
+      rw [← hp]
+      exact hBmem _ hmemB3
+    rw [← supportingIntersection_eq_edgeVertices_fst_of_mem B (by linarith only [h12, h23])
+      (by linarith only [h34, h45, h01, Real.pi_pos]) hmem, ← hp]
+  have hDstart : (edgeVertices D ((3 * Real.pi / 2 : ℝ) : Real.Angle)).1 =
+      paperGerverContacts 0 3 := by
+    have hpm : inner ℝ (paperGerverContacts 0 3)
+        (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = gerverLeftTailSupport (Real.pi / 2) := by
+      have h := hDtouch 0 hmemD0
+      rwa [zero_add] at h
+    have h := edgeVertices_add_pi_fst_eq_of_lt (L := D) (m := gerverLeftTailSupport)
+      (a := Real.pi / 2) (b := gerverStageTimes 4 + Real.pi / 2)
+      (by linarith only [hl4I.1, h01, h12, h23, h34]) hDleIoo
+      (fun q hq ↦ hDle q hq _ ⟨le_rfl, by linarith only [hl4I.1]⟩) (hDmem 0 hmemD0) hpm
+      (by simpa only [zero_add] using hmDderiv 0)
+    rwa [hcast _ _ (show Real.pi / 2 + Real.pi = 3 * Real.pi / 2 by ring)] at h
+  have hBend : (edgeVertices B ((3 * Real.pi / 2 : ℝ) : Real.Angle)).2 =
+      paperGerverContacts (Real.pi / 2) 1 := by
+    have h := edgeVertices_add_pi_snd_eq_of_lt (L := B) (m := gerverRightTailSupport)
+      (a := gerverStageTimes 1) (b := Real.pi / 2)
+      (by linarith only [h12, h23, h34, h45]) hBleIoo
+      (fun q hq ↦ hBle q hq _ ⟨by linarith only [h12, h23, h34, h45], le_rfl⟩)
+      (hBmem _ hmemB5) (hBtouch _ hmemB5) (hmBderiv _)
+    rwa [hcast _ _ (show Real.pi / 2 + Real.pi = 3 * Real.pi / 2 by ring)] at h
+  exact ⟨hDface, hBface, hDend, hBstart, hDstart, hBend, hDouter, hBouter⟩
+
+theorem gerver_tailGeometry (K : SpecialCapSpace) (B D : ConvexBody Point)
+    (hK : (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2))
+    (hB : (B : Set Point) = (canonicalTailSets K).1)
+    (hD : (D : Set Point) = (canonicalTailSets K).2) :
+    (∀ t ∈ Set.Ioo (gerverStageTimes 0) (gerverStageTimes 2),
+      edgeVertices D ((3 * Real.pi / 2 + t : ℝ) : Real.Angle) =
+        (paperGerverContacts t 3, paperGerverContacts t 3)) ∧
+    (∀ t ∈ Set.Ioo (gerverStageTimes 3) (gerverStageTimes 5),
+      edgeVertices B ((Real.pi + t : ℝ) : Real.Angle) =
+        (paperGerverContacts t 1, paperGerverContacts t 1)) ∧
+    (distinguishedCapSides K.val).2.corner = paperGerverContacts (gerverStageTimes 2) 3 ∧
+    (rightLeftTailArcs B D).2.endPoint = paperGerverContacts (gerverStageTimes 2) 3 ∧
+    edgeVertices D ((3 * Real.pi / 2 + gerverStageTimes 2 : ℝ) : Real.Angle) =
+      (paperGerverContacts (gerverStageTimes 2) 3, paperGerverContacts (gerverStageTimes 2) 3) ∧
+    ParametrizesDirectedArc (fun t ↦ paperGerverContacts t 3)
+      (gerverStageTimes 0) (gerverStageTimes 2) (rightLeftTailArcs B D).2 ∧
+    (distinguishedCapSides K.val).1.corner = paperGerverContacts (gerverStageTimes 3) 1 ∧
+    (rightLeftTailArcs B D).1.startPoint = paperGerverContacts (gerverStageTimes 3) 1 ∧
+    edgeVertices B ((Real.pi + gerverStageTimes 3 : ℝ) : Real.Angle) =
+      (paperGerverContacts (gerverStageTimes 3) 1, paperGerverContacts (gerverStageTimes 3) 1) ∧
+    ParametrizesDirectedArc (fun t ↦ paperGerverContacts t 1)
+      (gerverStageTimes 3) (gerverStageTimes 5) (rightLeftTailArcs B D).1 ∧
+    (∀ t ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2),
+      supportValue K.val.val ((Real.pi / 2 + t : ℝ) : Real.Angle) +
+        supportValue D ((3 * Real.pi / 2 + t : ℝ) : Real.Angle) = 1) ∧
+    (∀ t ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5),
+      supportValue K.val.val (t : Real.Angle) +
+        supportValue B ((Real.pi + t : ℝ) : Real.Angle) = 1) := by
+  -- ### One cap serves every clause, and it carries the literal contact geometry
+  obtain ⟨-, hGeq, -, -, hcapeq, -, -⟩ := gerver_capSupport_identification
+  have hcarrier : (K.val.val : Set Point) = gerverOuterCap := by rw [hK, hGeq, hcapeq]
+  obtain ⟨-, K', hK'set, hdens, hsel, -, -, hBwall, -, hDwall, -, -⟩ :=
+    paperGerver_contact_geometry
+  have hKeq : K' = K.val := Subtype.ext (SetLike.coe_injective (hK'set.trans hK.symm))
+  subst hKeq
+  obtain ⟨hstrictD, hstrictB, hDmem, hBmem⟩ :=
+    gerver_tail_membership K B D hK hB hD
+  obtain ⟨hsupu, hsupv, hmDval, hmDsup, hmDderiv, hmBderiv, hDle, hBle, hDtouch, hBtouch⟩ :=
+    gerver_tail_support_lines K B D hK hB hD
+  obtain ⟨hDface, hBface, hDend, hBstart, hDstart, hBend, hDouter, hBouter⟩ :=
+    gerver_tail_contact_faces K B D hK hB hD
+  have ht0 : gerverStageTimes 0 = 0 := gerverStageTimes_zero
+  have ht5 : gerverStageTimes 5 = Real.pi / 2 := rfl
+  have hr : paperGerverConstants.2.1 = gerverStageTimes 1 := rfl
+  have hl : paperGerverConstants.2.2 = gerverStageTimes 4 := rfl
+  have h01' : gerverStageTimes 0 < gerverStageTimes 1 := gerverStageTimes_strictMono (by decide)
+  have h12 : gerverStageTimes 1 < gerverStageTimes 2 := gerverStageTimes_strictMono (by decide)
+  have h23 : gerverStageTimes 2 < gerverStageTimes 3 := gerverStageTimes_strictMono (by decide)
+  have h34 : gerverStageTimes 3 < gerverStageTimes 4 := gerverStageTimes_strictMono (by decide)
+  have h45' : gerverStageTimes 4 < gerverStageTimes 5 := gerverStageTimes_strictMono (by decide)
+  have h01 : (0 : ℝ) < gerverStageTimes 1 := ht0 ▸ h01'
+  have h45 : gerverStageTimes 4 < Real.pi / 2 := ht5 ▸ h45'
+  -- ### The support coordinates of the cap along the path
+  have hcorner : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2), capInnerCorner K.val t = paperGerverPath t :=
+    fun t ht ↦ (hsel ⟨t, ht⟩).2.2
+  have hcast : ∀ x y : ℝ, x = y → ((x : ℝ) : Real.Angle) = ((y : ℝ) : Real.Angle) :=
+    fun x y h ↦ by rw [h]
+  have h02 : (0 : ℝ) < gerverStageTimes 2 := by linarith only [h01, h12]
+  have h24 : gerverStageTimes 2 < gerverStageTimes 4 := by linarith only [h23, h34]
+  have hl4I : gerverStageTimes 4 ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+    ⟨by linarith only [h01, h12, h23, h34], h45.le⟩
+  have hmemD0 : (0 : ℝ) ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2) :=
+    ⟨ht0.le, h02.le⟩
+  have hmemD2 : gerverStageTimes 2 ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2) :=
+    ⟨by linarith only [ht0.le, h02], le_rfl⟩
+  have hmemB3 : gerverStageTimes 3 ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5) :=
+    ⟨le_rfl, by linarith only [ht5.ge, h34, h45]⟩
+  have hmemB5 : Real.pi / 2 ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5) :=
+    ⟨by linarith only [h34, h45], ht5.ge⟩
+  -- ### Injectivity of the two contact parametrizations
+  have hinjD : Set.InjOn (fun t ↦ paperGerverContacts t 3)
+      (Set.Icc (gerverStageTimes 0) (gerverStageTimes 2)) := fun x hx y hy hxy ↦
+    hstrictD.injOn hx hy (by
+      simpa only using congrArg (fun p : Point ↦ inner ℝ p
+        (tangentVector ((gerverStageTimes 4 : ℝ) : Real.Angle))) hxy)
+  have hinjB : Set.InjOn (fun t ↦ paperGerverContacts t 1)
+      (Set.Icc (gerverStageTimes 3) (gerverStageTimes 5)) := fun x hx y hy hxy ↦
+    hstrictB.injOn hx hy (by
+      simpa only using congrArg (fun p : Point ↦ inner ℝ p
+        (normalVector ((gerverStageTimes 1 : ℝ) : Real.Angle))) hxy)
+  -- ### Assembling the twelve conclusions
+  have hmemD2' : gerverStageTimes 2 ∈ Set.Ioc (gerverStageTimes 0) (gerverStageTimes 2) :=
+    ⟨by linarith only [ht0.le, h02], le_rfl⟩
+  have hmemB3' : gerverStageTimes 3 ∈ Set.Ico (gerverStageTimes 3) (gerverStageTimes 5) :=
+    ⟨le_rfl, by linarith only [ht5.ge, h34, h45]⟩
+  have hr1I : gerverStageTimes 1 ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+    ⟨h01.le, by linarith only [h12, h23, h34, h45]⟩
+  have hDinner : paperGerverContacts (gerverStageTimes 2) 3 ∈
+      exposedEdge D ((3 * Real.pi / 2 + gerverStageTimes 2 : ℝ) : Real.Angle) := by
+    rw [hDface _ hmemD2']
+    rfl
+  have hBinner : paperGerverContacts (gerverStageTimes 3) 1 ∈
+      exposedEdge B ((Real.pi + gerverStageTimes 3 : ℝ) : Real.Angle) := by
+    rw [hBface _ hmemB3']
+    rfl
+  refine ⟨fun t ht ↦ edgeVertices_eq_of_exposedEdge_singleton (hDface t ⟨ht.1, ht.2.le⟩),
+    fun t ht ↦ edgeVertices_eq_of_exposedEdge_singleton (hBface t ⟨ht.1.le, ht.2⟩), ?_, ?_,
+    edgeVertices_eq_of_exposedEdge_singleton (hDface _ hmemD2'), ?_, ?_, ?_,
+    edgeVertices_eq_of_exposedEdge_singleton (hBface _ hmemB3'), ?_, ?_, ?_⟩
+  · change capInnerCorner K.val paperGerverConstants.2.2 = _
+    rw [hl, hcorner _ hl4I]
+    exact gerver_niche_piece_endpoints.2.1.symm
+  · change (edgeVertices D ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle)).2 = _
+    rw [hl]
+    exact hDend
+  · refine ⟨by linarith only [ht0.le, h02], (continuous_paperGerverContact 3).continuousOn,
+      hinjD, ?_, ?_, ?_⟩
+    · change (fun t ↦ paperGerverContacts t 3) ''
+        Set.Icc (gerverStageTimes 0) (gerverStageTimes 2) =
+        convexBoundaryArc D (3 * Real.pi / 2) (3 * Real.pi / 2 + paperGerverConstants.2.2)
+      rw [hl, convexBoundaryArc]
+      refine Set.Subset.antisymm ?_ ?_
+      · rintro p ⟨t, ht, rfl⟩
+        rcases eq_or_lt_of_le ht.1 with h0 | h0
+        · exact Or.inl (Or.inl (by rw [Set.mem_singleton_iff, hDstart, ← h0, ht0]))
+        · refine Or.inl (Or.inr (Set.mem_iUnion₂.2 ⟨3 * Real.pi / 2 + t,
+            ⟨by linarith only [h0, ht0.ge], by linarith only [ht.2, h24]⟩, ?_⟩))
+          rw [hDface t ⟨h0, ht.2⟩]
+          rfl
+      · rintro p ((hp | hp) | hp)
+        · rw [Set.mem_singleton_iff, hDstart] at hp
+          exact ⟨0, hmemD0, hp.symm⟩
+        · obtain ⟨s, hs, hps⟩ := Set.mem_iUnion₂.1 hp
+          rcases le_or_gt (s - 3 * Real.pi / 2) (gerverStageTimes 2) with h | h
+          · have hmem : s - 3 * Real.pi / 2 ∈
+                Set.Ioc (gerverStageTimes 0) (gerverStageTimes 2) :=
+              ⟨by linarith only [hs.1, ht0.le], h⟩
+            have hface := hDface (s - 3 * Real.pi / 2) hmem
+            rw [hcast _ _ (show 3 * Real.pi / 2 + (s - 3 * Real.pi / 2) = s by ring)] at hface
+            rw [hface, Set.mem_singleton_iff] at hps
+            exact ⟨s - 3 * Real.pi / 2, ⟨hmem.1.le, hmem.2⟩, hps.symm⟩
+          · have hface := exposedEdge_eq_singleton_of_mem_exposedEdge_of_mem_Ioo
+              (L := D) (a := 3 * Real.pi / 2 + gerverStageTimes 2)
+              (b := 3 * Real.pi / 2 + gerverStageTimes 4) (s := s)
+              (by linarith only [h45, h02, Real.pi_pos])
+              ⟨by linarith only [h], hs.2⟩ hDinner hDouter
+            rw [hface, Set.mem_singleton_iff] at hps
+            exact ⟨gerverStageTimes 2, hmemD2, hps.symm⟩
+        · rw [Set.mem_singleton_iff, hDend] at hp
+          exact ⟨gerverStageTimes 2, hmemD2, hp.symm⟩
+    · change paperGerverContacts (gerverStageTimes 0) 3 =
+        (edgeVertices D ((3 * Real.pi / 2 : ℝ) : Real.Angle)).1
+      rw [hDstart, ht0]
+    · change paperGerverContacts (gerverStageTimes 2) 3 =
+        (edgeVertices D ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle)).2
+      rw [hl, hDend]
+  · change capInnerCorner K.val paperGerverConstants.2.1 = _
+    rw [hr, hcorner _ hr1I]
+    exact gerver_niche_piece_endpoints.1.symm
+  · change (edgeVertices B ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle)).1 = _
+    rw [hr]
+    exact hBstart
+  · refine ⟨by linarith only [h34, h45, ht5.ge], (continuous_paperGerverContact 1).continuousOn,
+      hinjB, ?_, ?_, ?_⟩
+    · change (fun t ↦ paperGerverContacts t 1) ''
+        Set.Icc (gerverStageTimes 3) (gerverStageTimes 5) =
+        convexBoundaryArc B (Real.pi + paperGerverConstants.2.1) (3 * Real.pi / 2)
+      rw [hr, convexBoundaryArc]
+      refine Set.Subset.antisymm ?_ ?_
+      · rintro p ⟨t, ht, rfl⟩
+        rcases eq_or_lt_of_le ht.2 with hT | hT
+        · exact Or.inr (by rw [Set.mem_singleton_iff, hBend, hT, ht5])
+        · refine Or.inl (Or.inr (Set.mem_iUnion₂.2 ⟨Real.pi + t,
+            ⟨by linarith only [ht.1, h12, h23], by linarith only [hT, ht5.le]⟩, ?_⟩))
+          rw [hBface t ⟨ht.1, hT⟩]
+          rfl
+      · rintro p ((hp | hp) | hp)
+        · rw [Set.mem_singleton_iff, hBstart] at hp
+          exact ⟨gerverStageTimes 3, hmemB3, hp.symm⟩
+        · obtain ⟨s, hs, hps⟩ := Set.mem_iUnion₂.1 hp
+          rcases lt_or_ge (s - Real.pi) (gerverStageTimes 3) with h | h
+          · have hface := exposedEdge_eq_singleton_of_mem_exposedEdge_of_mem_Ioo
+              (L := B) (a := Real.pi + gerverStageTimes 1)
+              (b := Real.pi + gerverStageTimes 3) (s := s)
+              (by linarith only [h34, h45, h01, Real.pi_pos])
+              ⟨hs.1, by linarith only [h]⟩ hBouter hBinner
+            rw [hface, Set.mem_singleton_iff] at hps
+            exact ⟨gerverStageTimes 3, hmemB3, hps.symm⟩
+          · have hmem : s - Real.pi ∈ Set.Ico (gerverStageTimes 3) (gerverStageTimes 5) :=
+              ⟨h, by rw [ht5]; linarith only [hs.2]⟩
+            have hface := hBface (s - Real.pi) hmem
+            rw [hcast _ _ (show Real.pi + (s - Real.pi) = s by ring)] at hface
+            rw [hface, Set.mem_singleton_iff] at hps
+            exact ⟨s - Real.pi, ⟨hmem.1, hmem.2.le⟩, hps.symm⟩
+        · rw [Set.mem_singleton_iff, hBend] at hp
+          exact ⟨Real.pi / 2, hmemB5, hp.symm⟩
+    · change paperGerverContacts (gerverStageTimes 3) 1 =
+        (edgeVertices B ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle)).1
+      rw [hr, hBstart]
+    · change paperGerverContacts (gerverStageTimes 5) 1 =
+        (edgeVertices B ((3 * Real.pi / 2 : ℝ) : Real.Angle)).2
+      rw [hBend, ht5]
+  · intro t ht
+    have htI : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+      ⟨ht0 ▸ ht.1, by linarith only [ht.2, h23, h34, h45]⟩
+    have hsup : supportValue D ((3 * Real.pi / 2 + t : ℝ) : Real.Angle) =
+        -gerverLeftTailSupport (t + Real.pi / 2) := by
+      have h := supportValue_add_pi_eq_neg_of_forall_le (L := D)
+        (fun q hq ↦ hDle q hq (t + Real.pi / 2)
+          ⟨by linarith only [htI.1], by linarith only [ht.2, h23, h34]⟩)
+        (hDmem t ht) (hDtouch t ht)
+      rwa [hcast _ _ (show t + Real.pi / 2 + Real.pi = 3 * Real.pi / 2 + t by ring)] at h
+    rw [hsup, hmDsup t htI, hcast _ _ (show Real.pi / 2 + t = t + Real.pi / 2 by ring)]
+    ring
+  · intro t ht
+    have htI : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+      ⟨by linarith only [ht.1, h01, h12, h23], ht5 ▸ ht.2⟩
+    have hsup : supportValue B ((Real.pi + t : ℝ) : Real.Angle) = -gerverRightTailSupport t := by
+      have h := supportValue_add_pi_eq_neg_of_forall_le (L := B)
+        (fun q hq ↦ hBle q hq t ⟨by linarith only [ht.1, h12, h23], htI.2⟩)
+        (hBmem t ht) (hBtouch t ht)
+      rwa [hcast _ _ (show t + Real.pi = Real.pi + t by ring)] at h
+    rw [hsup]
+    simp only [gerverRightTailSupport]
+    rw [hsupu t htI]
+    ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The four contact densities of Gerver's cap
+
+The rotating frame reads the four contact curves of Gerver's sofa as angular densities of two
+surface measures: the two outer contacts `A`, `C` against the cap `K = C(G)` itself, and the two
+inner contacts `B`, `D` against the two tail bodies through the opposite surface measure.
+
+The two outer clauses are exactly the certified cap densities `gerver_surface_densities`,
+rewritten with the tautological identity `⟨r • v_t, v_t⟩ = r` and translated by the quarter
+turn that separates the `A` arc from the `C` arc.  The two inner clauses use that `B = A - u` and
+`D = C - v` are the outer contacts translated by a frame vector, so each closed stage carries a
+globally differentiable branch curve for them (`exists_branch_paperGerverContacts_one`,
+`exists_branch_paperGerverContacts_three`); the tail geometry identifies those curves with the
+positive vertices of the two tail bodies at the opposite normal, and
+`oppositeSurfaceData_angleImage_eq_withDensity` turns each stage into a Lebesgue density.  The
+stages are glued with `measure_angleImage_eq_of_union`, which puts every interior switch inside a
+window that is closed on the right, so the only atom to compute is the one at the included left
+endpoint `t₃` of the `B` arc.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- Push a real density measure on an angle window to angles modulo a full turn. -/
+def angularDensityMeasure (f : ℝ → ℝ) (S : Set ℝ) : Measure Real.Angle :=
+  Measure.map (fun t : ℝ ↦ (t : Real.Angle))
+    ((volume.restrict S).withDensity (fun t ↦ ENNReal.ofReal (f t)))
+
+/-- The measure has the specified integrable nonnegative density on the angular image of a
+window. -/
+def HasAngularDensity (μ : Measure Real.Angle) (f : ℝ → ℝ) (S : Set ℝ) : Prop :=
+  Integrable f (volume.restrict S) ∧ (∀ᵐ t ∂volume.restrict S, 0 ≤ f t) ∧
+    μ.restrict ((fun t : ℝ ↦ (t : Real.Angle)) '' S) = angularDensityMeasure f S
+
+/-- An angular density identity on a window of at most one turn computes the measure of the
+angular image of every measurable subset of the window. -/
+theorem HasAngularDensity.angleImage_eq_setLIntegral {μ : Measure Real.Angle} {f : ℝ → ℝ}
+    {S : Set ℝ} (h : HasAngularDensity μ f S) {a b : ℝ} (hturn : b ≤ a + 2 * Real.pi)
+    (hS : S ⊆ Set.Ioc a b) {T : Set ℝ} (hT : MeasurableSet T) (hTS : T ⊆ S) :
+    μ ((fun t : ℝ ↦ (t : Real.Angle)) '' T) =
+      ∫⁻ t in T, ENNReal.ofReal (f t) ∂volume := by
+  have himage : MeasurableSet ((fun t : ℝ ↦ (t : Real.Angle)) '' T) :=
+    Real.Angle.measurableSet_image_of_subset_Ioc hturn hT (hTS.trans hS)
+  have hrestrict : μ ((fun t : ℝ ↦ (t : Real.Angle)) '' T) =
+      μ.restrict ((fun t : ℝ ↦ (t : Real.Angle)) '' S)
+        ((fun t : ℝ ↦ (t : Real.Angle)) '' T) := by
+    rw [Measure.restrict_apply himage, Set.inter_eq_left.2 (Set.image_mono hTS)]
+  rw [hrestrict, h.2.2, angularDensityMeasure,
+    Real.Angle.map_coe_withDensity_image_eq_setLIntegral hturn hS hT hTS]
+
+private theorem gerver_angular_density_outer_first (K : SpecialCapSpace)
+    (hK : (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2)) :
+    HasAngularDensity (surfaceAreaMeasure K.val.val)
+      (fun t ↦ inner ℝ (deriv (fun s ↦ paperGerverContacts s 0) t)
+        (tangentVector (t : Real.Angle))) (Set.Ico 0 (Real.pi / 2)) := by
+  -- ### The certified cap carries the two envelope densities
+  obtain ⟨K', hK'set, r, sden, hdens, ⟨M, hM⟩, hstage⟩ := gerver_surface_densities
+  have hKeq : K' = K.val := Subtype.ext (SetLike.coe_injective (hK'set.trans hK.symm))
+  subst hKeq
+  have haeval : ∀ᵐ t ∂volume.restrict (Set.Ico (0 : ℝ) (Real.pi / 2)),
+      inner ℝ (deriv (fun s ↦ paperGerverContacts s 0) t) (tangentVector (t : Real.Angle)) =
+        (r t : ℝ) := by
+    filter_upwards [ae_mem_openStage 0 (S := Set.Ico (0 : ℝ) (Real.pi / 2)) measurableSet_Ico
+      (fun t ht ↦ ⟨ht.1, by linarith [ht.2]⟩)] with t ht
+    obtain ⟨j, hj⟩ := ht
+    rw [sub_zero] at hj
+    rw [(hstage j t hj).1.deriv, real_inner_smul_left, inner_tangentVector_self, mul_one]
+  have hrint : Integrable (fun t ↦ ((r t : ℝ)))
+      (volume.restrict (Set.Ico (0 : ℝ) (Real.pi / 2))) := by
+    have hfin : IsFiniteMeasure (volume.restrict (Set.Ico (0 : ℝ) (Real.pi / 2))) := by
+      refine ⟨?_⟩
+      rw [Measure.restrict_apply_univ, Real.volume_Ico]
+      exact ENNReal.ofReal_lt_top
+    refine Integrable.mono' (integrable_const M)
+      hdens.1.coe_nnreal_real.aestronglyMeasurable ?_
+    filter_upwards [ae_restrict_mem measurableSet_Ico] with t ht
+    rw [Real.norm_eq_abs, abs_of_nonneg (r t).coe_nonneg]
+    exact (hM t ⟨ht.1, ht.2.le⟩).1
+  refine ⟨hrint.congr (Filter.EventuallyEq.symm haeval), ?_, ?_⟩
+  · filter_upwards [haeval] with t ht
+    rw [ht]
+    exact (r t).coe_nonneg
+  · rw [hdens.2.2.1, angularDensityMeasure]
+    congr 1
+    refine withDensity_congr_ae ?_
+    filter_upwards [haeval] with t ht
+    rw [ht, ENNReal.ofReal_coe_nnreal]
+
+private theorem gerver_angular_density_right_tail (K : SpecialCapSpace) (B D : ConvexBody Point)
+    (hK : (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2))
+    (hB : (B : Set Point) = (canonicalTailSets K).1)
+    (hD : (D : Set Point) = (canonicalTailSets K).2) :
+    HasAngularDensity (oppositeSurfaceData B).1
+      (fun t ↦ inner ℝ (-(deriv (fun s ↦ paperGerverContacts s 1) t))
+        (tangentVector (t : Real.Angle)))
+      (Set.Ico (gerverStageTimes 3) (gerverStageTimes 5)) := by
+  -- ### The strictly increasing stage times
+  have h0 : gerverStageTimes 0 = 0 := gerverStageTimes_zero
+  have h5 : gerverStageTimes 5 = Real.pi / 2 := rfl
+  have h01 : gerverStageTimes 0 < gerverStageTimes 1 := gerverStageTimes_strictMono (by decide)
+  have h12 : gerverStageTimes 1 < gerverStageTimes 2 := gerverStageTimes_strictMono (by decide)
+  have h23 : gerverStageTimes 2 < gerverStageTimes 3 := gerverStageTimes_strictMono (by decide)
+  have h34 : gerverStageTimes 3 < gerverStageTimes 4 := gerverStageTimes_strictMono (by decide)
+  have h45 : gerverStageTimes 4 < gerverStageTimes 5 := gerverStageTimes_strictMono (by decide)
+  have hpi : (3 : ℝ) < Real.pi := Real.pi_gt_three
+  -- ### The certified cap carries the two envelope densities
+  obtain ⟨K', hK'set, r, sden, hdens, ⟨M, hM⟩, hstage⟩ := gerver_surface_densities
+  have hKeq : K' = K.val := Subtype.ext (SetLike.coe_injective (hK'set.trans hK.symm))
+  subst hKeq
+  -- ### The two tails are traced by the two inner contact curves
+  obtain ⟨htailD, htailB, -, -, htailD2, -, -, -, htailB3, -, -, -⟩ :=
+    gerver_tailGeometry K B D hK hB hD
+  obtain ⟨fB, hfBdef⟩ : ∃ f : ℝ → ℝ, f = fun t ↦ inner ℝ
+      (-(deriv (fun s ↦ paperGerverContacts s 1) t)) (tangentVector (t : Real.Angle)) :=
+    ⟨_, rfl⟩
+  rw [← hfBdef]
+  obtain ⟨F3, g3, hF3, hg3, hg3nn, hFeq3⟩ :=
+    exists_branch_paperGerverContacts_one (i := 3) (Or.inl rfl)
+  obtain ⟨F4, g4, hF4, hg4, hg4nn, hFeq4⟩ :=
+    exists_branch_paperGerverContacts_one (i := 4) (Or.inr rfl)
+  -- the positive vertex of `B` at normal `π + s` is the inner contact
+  have hvB : ∀ s ∈ Set.Ico (gerverStageTimes 3) (gerverStageTimes 5),
+      (edgeVertices B ((Real.pi + s : ℝ) : Real.Angle)).1 = paperGerverContacts s 1 := by
+    intro s hs
+    rcases eq_or_lt_of_le hs.1 with heq | hlt
+    · rw [← heq, htailB3]
+    · rw [htailB s ⟨hlt, hs.2⟩]
+  -- the target density is the branch speed inside each of the two open stages
+  have hfB3 : ∀ t ∈ Set.Ioo (gerverStageTimes 3) (gerverStageTimes 4), fB t = g3 t := by
+    intro t ht
+    simp only [hfBdef]
+    rw [(hasDerivAt_of_eqOn_stage hF3 hFeq3 ht).deriv, neg_smul, neg_neg, real_inner_smul_left,
+      inner_tangentVector_self, mul_one]
+  have hfB4 : ∀ t ∈ Set.Ioo (gerverStageTimes 4) (gerverStageTimes 5), fB t = g4 t := by
+    intro t ht
+    simp only [hfBdef]
+    rw [(hasDerivAt_of_eqOn_stage hF4 hFeq4 ht).deriv, neg_smul, neg_neg, real_inner_smul_left,
+      inner_tangentVector_self, mul_one]
+  -- the two stage windows
+  have hagree3 : ∀ T, MeasurableSet T →
+      T ⊆ Set.Ioc (gerverStageTimes 3) (gerverStageTimes 4) →
+      (oppositeSurfaceData B).1 ((fun s : ℝ ↦ (s : Real.Angle)) '' T) =
+        volume.withDensity (fun s ↦ ENNReal.ofReal (fB s)) T := by
+    intro T hT hTsub
+    refine oppositeSurfaceData_angleImage_eq_withDensity B h34 (by linarith) F3 fB g3 hF3 hg3
+      hg3nn hfB3 ?_ hT hTsub
+    exact fun s hs ↦ (hvB s ⟨hs.1, by linarith [hs.2]⟩).trans (hFeq3 s hs)
+  have hagree4 : ∀ T, MeasurableSet T →
+      T ⊆ Set.Ioo (gerverStageTimes 4) (gerverStageTimes 5) →
+      (oppositeSurfaceData B).1 ((fun s : ℝ ↦ (s : Real.Angle)) '' T) =
+        volume.withDensity (fun s ↦ ENNReal.ofReal (fB s)) T := by
+    intro T hT hTsub
+    refine oppositeSurfaceData_angleImage_eq_withDensity_of_openRight B h45 (by linarith) F4 fB
+      g4 hF4 hg4 (fun s hs ↦ hg4nn s ⟨hs.1, hs.2.le⟩) hfB4 ?_ hT hTsub
+    exact fun s hs ↦ (hvB s ⟨by linarith [hs.1], hs.2⟩).trans (hFeq4 s ⟨hs.1, hs.2.le⟩)
+  -- the included left endpoint carries no surface atom
+  have hatom : (oppositeSurfaceData B).1
+      ((fun s : ℝ ↦ (s : Real.Angle)) '' {gerverStageTimes 3}) = 0 := by
+    rw [oppositeSurfaceData_angleImage B
+      (by rw [Set.image_singleton]; exact measurableSet_singleton _),
+      Set.image_singleton, Set.image_singleton,
+      (surfaceAreaMeasure_atom_length B
+        ((gerverStageTimes 3 + Real.pi : ℝ) : Real.Angle)).2.1,
+      show ((gerverStageTimes 3 + Real.pi : ℝ) : Real.Angle) =
+        ((Real.pi + gerverStageTimes 3 : ℝ) : Real.Angle) from by rw [add_comm], htailB3]
+    simp
+  have hagreeAtom : ∀ T, MeasurableSet T → T ⊆ {gerverStageTimes 3} →
+      (oppositeSurfaceData B).1 ((fun s : ℝ ↦ (s : Real.Angle)) '' T) =
+        volume.withDensity (fun s ↦ ENNReal.ofReal (fB s)) T := by
+    have hatomν : volume.withDensity (fun s ↦ ENNReal.ofReal (fB s))
+        {gerverStageTimes 3} = 0 := by
+      rw [withDensity_apply _ (measurableSet_singleton _)]
+      exact setLIntegral_measure_zero _ _ (by simp)
+    intro T _ hTsub
+    rw [measure_mono_null (Set.image_mono hTsub) hatom, measure_mono_null hTsub hatomν]
+  -- gluing the closed stage and then the final open stage
+  have hstep1 : ∀ T, MeasurableSet T →
+      T ⊆ Set.Icc (gerverStageTimes 3) (gerverStageTimes 4) →
+      (oppositeSurfaceData B).1 ((fun s : ℝ ↦ (s : Real.Angle)) '' T) =
+        volume.withDensity (fun s ↦ ENNReal.ofReal (fB s)) T := by
+    intro T hT hTsub
+    refine measure_angleImage_eq_of_union (c := gerverStageTimes 3 - 1)
+      (d := gerverStageTimes 5) (by linarith) (I := {gerverStageTimes 3})
+      (J := Set.Ioc (gerverStageTimes 3) (gerverStageTimes 4)) ?_ ?_
+      (measurableSet_singleton _) measurableSet_Ioc ?_ hagreeAtom hagree3 T hT ?_
+    · intro x hx
+      rw [Set.mem_singleton_iff] at hx
+      exact ⟨by linarith, by linarith⟩
+    · exact fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+    · rw [Set.disjoint_singleton_left]
+      simp
+    · intro x hx
+      rcases eq_or_lt_of_le (hTsub hx).1 with heq | hlt
+      · exact Or.inl heq.symm
+      · exact Or.inr ⟨hlt, (hTsub hx).2⟩
+  have hstep2 : ∀ T, MeasurableSet T →
+      T ⊆ Set.Ico (gerverStageTimes 3) (gerverStageTimes 5) →
+      (oppositeSurfaceData B).1 ((fun s : ℝ ↦ (s : Real.Angle)) '' T) =
+        volume.withDensity (fun s ↦ ENNReal.ofReal (fB s)) T := by
+    intro T hT hTsub
+    refine measure_angleImage_eq_of_union (c := gerverStageTimes 3 - 1)
+      (d := gerverStageTimes 5) (by linarith)
+      (I := Set.Icc (gerverStageTimes 3) (gerverStageTimes 4))
+      (J := Set.Ioo (gerverStageTimes 4) (gerverStageTimes 5)) ?_ ?_
+      measurableSet_Icc measurableSet_Ioo ?_ hstep1 hagree4 T hT ?_
+    · exact fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+    · exact fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+    · rw [Set.disjoint_left]
+      exact fun x hx hx' ↦ absurd hx'.1 (not_lt.2 hx.2)
+    · intro x hx
+      rcases le_or_gt x (gerverStageTimes 4) with h | h
+      · exact Or.inl ⟨(hTsub hx).1, h⟩
+      · exact Or.inr ⟨h, (hTsub hx).2⟩
+  -- almost every parameter of the window lies in one of the two open stages
+  have hae : ∀ᵐ t ∂volume.restrict (Set.Ico (gerverStageTimes 3) (gerverStageTimes 5)),
+      t ∈ Set.Ioo (gerverStageTimes 3) (gerverStageTimes 4) ∪
+        Set.Ioo (gerverStageTimes 4) (gerverStageTimes 5) := by
+    refine ae_restrict_mem_of_countable_diff measurableSet_Ico
+      ((Set.countable_singleton (gerverStageTimes 4)).insert (gerverStageTimes 3)) ?_
+    rintro x ⟨hx, hx'⟩
+    rcases eq_or_lt_of_le hx.1 with heq | hlt
+    · exact Or.inl heq.symm
+    · rcases lt_trichotomy x (gerverStageTimes 4) with h | h | h
+      · exact absurd (Or.inl ⟨hlt, h⟩) hx'
+      · exact Or.inr (Set.mem_singleton_iff.2 h)
+      · exact absurd (Or.inr ⟨h, hx.2⟩) hx'
+  have hsplit : Set.Ico (gerverStageTimes 3) (gerverStageTimes 5) =
+      Set.Icc (gerverStageTimes 3) (gerverStageTimes 4) ∪
+        Set.Ioo (gerverStageTimes 4) (gerverStageTimes 5) :=
+    (Set.Icc_union_Ioo_eq_Ico h34.le h45).symm
+  refine ⟨?_, ?_, ?_⟩
+  · have h1 : IntegrableOn fB (Set.Icc (gerverStageTimes 3) (gerverStageTimes 4)) volume := by
+      refine (ContinuousOn.integrableOn_compact isCompact_Icc hg3.continuousOn).congr ?_
+      have hae3 : ∀ᵐ t ∂volume.restrict (Set.Icc (gerverStageTimes 3) (gerverStageTimes 4)),
+          t ∈ Set.Ioo (gerverStageTimes 3) (gerverStageTimes 4) := by
+        refine ae_restrict_mem_of_countable_diff measurableSet_Icc
+          ((Set.countable_singleton (gerverStageTimes 4)).insert (gerverStageTimes 3)) ?_
+        rintro x ⟨hx, hx'⟩
+        rcases eq_or_lt_of_le hx.1 with heq | hlt
+        · exact Or.inl heq.symm
+        · rcases eq_or_lt_of_le hx.2 with heq2 | hlt2
+          · exact Or.inr (Set.mem_singleton_iff.2 heq2)
+          · exact absurd ⟨hlt, hlt2⟩ hx'
+      filter_upwards [hae3] with t ht using (hfB3 t ht).symm
+    have h2 : IntegrableOn fB (Set.Ioo (gerverStageTimes 4) (gerverStageTimes 5)) volume := by
+      refine (((ContinuousOn.integrableOn_compact (isCompact_Icc
+        (a := gerverStageTimes 4) (b := gerverStageTimes 5))
+          hg4.continuousOn)).mono_set Set.Ioo_subset_Icc_self).congr ?_
+      filter_upwards [ae_restrict_mem measurableSet_Ioo] with t ht using (hfB4 t ht).symm
+    rw [show Integrable fB (volume.restrict (Set.Ico (gerverStageTimes 3)
+      (gerverStageTimes 5))) = IntegrableOn fB (Set.Ico (gerverStageTimes 3)
+        (gerverStageTimes 5)) volume from rfl, hsplit]
+    exact h1.union h2
+  · filter_upwards [hae] with t ht
+    rcases ht with ht | ht
+    · rw [hfB3 t ht]
+      exact hg3nn t ⟨ht.1, ht.2.le⟩
+    · rw [hfB4 t ht]
+      exact hg4nn t ⟨ht.1, ht.2.le⟩
+  · rw [angularDensityMeasure]
+    exact measure_restrict_eq_map_withDensity measurableSet_Ico hstep2
+
+private theorem gerver_angular_density_outer_second (K : SpecialCapSpace)
+    (hK : (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2)) :
+    HasAngularDensity (surfaceAreaMeasure K.val.val)
+      (fun t ↦ inner ℝ (-(deriv (fun s ↦ paperGerverContacts s 2) (t - Real.pi / 2)))
+        (normalVector ((t - Real.pi / 2 : ℝ) : Real.Angle)))
+      (Set.Ioc (Real.pi / 2) Real.pi) := by
+  -- ### The certified cap carries the two envelope densities
+  obtain ⟨K', hK'set, r, sden, hdens, ⟨M, hM⟩, hstage⟩ := gerver_surface_densities
+  have hKeq : K' = K.val := Subtype.ext (SetLike.coe_injective (hK'set.trans hK.symm))
+  subst hKeq
+  have haeval : ∀ᵐ u ∂volume.restrict (Set.Ioc (Real.pi / 2) Real.pi),
+      inner ℝ (-(deriv (fun s ↦ paperGerverContacts s 2) (u - Real.pi / 2)))
+          (normalVector ((u - Real.pi / 2 : ℝ) : Real.Angle)) =
+        (sden (u - Real.pi / 2) : ℝ) := by
+    filter_upwards [ae_mem_openStage (Real.pi / 2) (S := Set.Ioc (Real.pi / 2) Real.pi)
+      measurableSet_Ioc (fun u hu ↦ ⟨hu.1.le, by linarith [hu.2]⟩)] with u hu
+    obtain ⟨j, hj⟩ := hu
+    rw [(hstage j _ hj).2.deriv, neg_smul, neg_neg, real_inner_smul_left,
+      inner_normalVector_self, mul_one]
+  have hsint : Integrable (fun u ↦ ((sden (u - Real.pi / 2) : ℝ)))
+      (volume.restrict (Set.Ioc (Real.pi / 2) Real.pi)) := by
+    have hfin : IsFiniteMeasure (volume.restrict (Set.Ioc (Real.pi / 2) Real.pi)) := by
+      refine ⟨?_⟩
+      rw [Measure.restrict_apply_univ, Real.volume_Ioc]
+      exact ENNReal.ofReal_lt_top
+    refine Integrable.mono' (integrable_const M)
+      ((hdens.2.1.coe_nnreal_real.comp (measurable_id.sub_const _)).aestronglyMeasurable) ?_
+    filter_upwards [ae_restrict_mem measurableSet_Ioc] with u hu
+    rw [Real.norm_eq_abs, abs_of_nonneg (sden _).coe_nonneg]
+    exact (hM (u - Real.pi / 2) ⟨by linarith [hu.1], by linarith [hu.2]⟩).2
+  refine ⟨hsint.congr (Filter.EventuallyEq.symm haeval), ?_, ?_⟩
+  · filter_upwards [haeval] with u hu
+    rw [hu]
+    exact (sden _).coe_nonneg
+  · have himg : (fun t : ℝ ↦ t + Real.pi / 2) '' Set.Ioc 0 (Real.pi / 2) =
+        Set.Ioc (Real.pi / 2) Real.pi := by
+      ext u
+      simp only [Set.mem_image, Set.mem_Ioc]
+      constructor
+      · rintro ⟨t, ⟨h1, h2⟩, rfl⟩
+        exact ⟨by linarith, by linarith⟩
+      · rintro ⟨h1, h2⟩
+        exact ⟨u - Real.pi / 2, ⟨by linarith, by linarith⟩, by ring⟩
+    have hcomp : (fun t : ℝ ↦ ((t + Real.pi / 2 : ℝ) : Real.Angle)) =
+        (fun u : ℝ ↦ (u : Real.Angle)) ∘ (fun t : ℝ ↦ t + Real.pi / 2) := rfl
+    rw [hdens.2.2.2, angularDensityMeasure, hcomp,
+      ← Measure.map_map (g := fun u : ℝ ↦ (u : Real.Angle))
+        (f := fun t : ℝ ↦ t + Real.pi / 2) Real.Angle.continuous_coe.measurable (by fun_prop),
+      map_add_right_restrict_withDensity (Real.pi / 2), himg]
+    congr 1
+    refine withDensity_congr_ae ?_
+    filter_upwards [haeval] with u hu
+    rw [hu, ENNReal.ofReal_coe_nnreal]
+
+private theorem gerver_angular_density_left_tail (K : SpecialCapSpace) (B D : ConvexBody Point)
+    (hK : (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2))
+    (hB : (B : Set Point) = (canonicalTailSets K).1)
+    (hD : (D : Set Point) = (canonicalTailSets K).2) :
+    HasAngularDensity (oppositeSurfaceData D).1
+      (fun t ↦ inner ℝ (deriv (fun s ↦ paperGerverContacts s 3) (t - Real.pi / 2))
+        (normalVector ((t - Real.pi / 2 : ℝ) : Real.Angle)))
+      (Set.Ioc (Real.pi / 2 + gerverStageTimes 0)
+        (Real.pi / 2 + gerverStageTimes 2)) := by
+  -- ### The strictly increasing stage times
+  have h0 : gerverStageTimes 0 = 0 := gerverStageTimes_zero
+  have h5 : gerverStageTimes 5 = Real.pi / 2 := rfl
+  have h01 : gerverStageTimes 0 < gerverStageTimes 1 := gerverStageTimes_strictMono (by decide)
+  have h12 : gerverStageTimes 1 < gerverStageTimes 2 := gerverStageTimes_strictMono (by decide)
+  have h23 : gerverStageTimes 2 < gerverStageTimes 3 := gerverStageTimes_strictMono (by decide)
+  have h34 : gerverStageTimes 3 < gerverStageTimes 4 := gerverStageTimes_strictMono (by decide)
+  have h45 : gerverStageTimes 4 < gerverStageTimes 5 := gerverStageTimes_strictMono (by decide)
+  have hpi : (3 : ℝ) < Real.pi := Real.pi_gt_three
+  -- ### The certified cap carries the two envelope densities
+  obtain ⟨K', hK'set, r, sden, hdens, ⟨M, hM⟩, hstage⟩ := gerver_surface_densities
+  have hKeq : K' = K.val := Subtype.ext (SetLike.coe_injective (hK'set.trans hK.symm))
+  subst hKeq
+  -- ### The two tails are traced by the two inner contact curves
+  obtain ⟨htailD, htailB, -, -, htailD2, -, -, -, htailB3, -, -, -⟩ :=
+    gerver_tailGeometry K B D hK hB hD
+  obtain ⟨fD, hfDdef⟩ : ∃ f : ℝ → ℝ, f = fun t ↦ inner ℝ
+      (deriv (fun s ↦ paperGerverContacts s 3) (t - Real.pi / 2))
+      (normalVector ((t - Real.pi / 2 : ℝ) : Real.Angle)) := ⟨_, rfl⟩
+  rw [← hfDdef]
+  obtain ⟨F0, g0, hF0, hg0, hg0nn, hFeq0⟩ :=
+    exists_branch_paperGerverContacts_three (i := 0) (Or.inl rfl)
+  obtain ⟨F1, g1, hF1, hg1, hg1nn, hFeq1⟩ :=
+    exists_branch_paperGerverContacts_three (i := 1) (Or.inr rfl)
+  -- the stage data with the stage times spelled out, so that `linarith` sees them
+  have hg0nn' : ∀ t ∈ Set.Ioc (gerverStageTimes 0) (gerverStageTimes 1), 0 ≤ g0 t := hg0nn
+  have hFeq0' : ∀ t ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 1),
+      paperGerverContacts t 3 = F0 t := hFeq0
+  have hg1nn' : ∀ t ∈ Set.Ioc (gerverStageTimes 1) (gerverStageTimes 2), 0 ≤ g1 t := hg1nn
+  have hFeq1' : ∀ t ∈ Set.Icc (gerverStageTimes 1) (gerverStageTimes 2),
+      paperGerverContacts t 3 = F1 t := hFeq1
+  -- reparametrising the quarter-turn shift turns the normal speed into a tangent speed
+  have hshift : ∀ (F : ℝ → Point) (g : ℝ → ℝ),
+      (∀ s, HasDerivAt F (g s • normalVector (s : Real.Angle)) s) →
+      ∀ u : ℝ, HasDerivAt (fun v : ℝ ↦ F (v - Real.pi / 2))
+        (-(g (u - Real.pi / 2)) • tangentVector (u : Real.Angle)) u := by
+    intro F g hF u
+    have h := (hF (u - Real.pi / 2)).scomp u ((hasDerivAt_id u).sub_const (Real.pi / 2))
+    have hv : tangentVector (u : Real.Angle) =
+        -normalVector ((u - Real.pi / 2 : ℝ) : Real.Angle) := by
+      have h2 := tangentVector_add_pi_div_two (u - Real.pi / 2)
+      rw [show u - Real.pi / 2 + Real.pi / 2 = u from by ring] at h2
+      exact h2
+    rw [hv]
+    refine h.congr_deriv ?_
+    module
+  -- the positive vertex of `D` at normal `π + u` is the inner contact
+  have hvD : ∀ u ∈ Set.Ioc (Real.pi / 2 + gerverStageTimes 0)
+      (Real.pi / 2 + gerverStageTimes 2),
+      (edgeVertices D ((Real.pi + u : ℝ) : Real.Angle)).1 =
+        paperGerverContacts (u - Real.pi / 2) 3 := by
+    intro u hu
+    rw [show ((Real.pi + u : ℝ) : Real.Angle) =
+      ((3 * Real.pi / 2 + (u - Real.pi / 2) : ℝ) : Real.Angle) from by congr 1; ring]
+    rcases eq_or_lt_of_le hu.2 with heq | hlt
+    · rw [show u - Real.pi / 2 = gerverStageTimes 2 from by linarith, htailD2]
+    · rw [htailD (u - Real.pi / 2) ⟨by linarith [hu.1], by linarith⟩]
+  -- the target density is the shifted branch speed inside each of the two open stages
+  have hfD0 : ∀ u ∈ Set.Ioo (Real.pi / 2 + gerverStageTimes 0)
+      (Real.pi / 2 + gerverStageTimes 1), fD u = g0 (u - Real.pi / 2) := by
+    intro u hu
+    simp only [hfDdef]
+    rw [(hasDerivAt_of_eqOn_stage hF0 hFeq0 (show u - Real.pi / 2 ∈
+      Set.Ioo (gerverStageTimes 0) (gerverStageTimes 1) from
+        ⟨by linarith [hu.1], by linarith [hu.2]⟩)).deriv, real_inner_smul_left,
+      inner_normalVector_self, mul_one]
+  have hfD1 : ∀ u ∈ Set.Ioo (Real.pi / 2 + gerverStageTimes 1)
+      (Real.pi / 2 + gerverStageTimes 2), fD u = g1 (u - Real.pi / 2) := by
+    intro u hu
+    simp only [hfDdef]
+    rw [(hasDerivAt_of_eqOn_stage hF1 hFeq1 (show u - Real.pi / 2 ∈
+      Set.Ioo (gerverStageTimes 1) (gerverStageTimes 2) from
+        ⟨by linarith [hu.1], by linarith [hu.2]⟩)).deriv, real_inner_smul_left,
+      inner_normalVector_self, mul_one]
+  -- the two stage windows
+  have hagree0 : ∀ T, MeasurableSet T →
+      T ⊆ Set.Ioc (Real.pi / 2 + gerverStageTimes 0) (Real.pi / 2 + gerverStageTimes 1) →
+      (oppositeSurfaceData D).1 ((fun s : ℝ ↦ (s : Real.Angle)) '' T) =
+        volume.withDensity (fun s ↦ ENNReal.ofReal (fD s)) T := by
+    intro T hT hTsub
+    refine oppositeSurfaceData_angleImage_eq_withDensity_of_openLeft D (by linarith)
+      (by linarith) (fun v ↦ F0 (v - Real.pi / 2)) fD (fun u ↦ g0 (u - Real.pi / 2))
+      (hshift F0 g0 hF0) (hg0.comp (continuous_id.sub continuous_const))
+      (fun s hs ↦ hg0nn' _ ⟨by linarith [hs.1], by linarith [hs.2]⟩) hfD0 ?_ hT hTsub
+    exact fun s hs ↦ (hvD s ⟨hs.1, by linarith [hs.2]⟩).trans
+      (hFeq0' _ ⟨by linarith [hs.1], by linarith [hs.2]⟩)
+  have hagree1 : ∀ T, MeasurableSet T →
+      T ⊆ Set.Ioc (Real.pi / 2 + gerverStageTimes 1) (Real.pi / 2 + gerverStageTimes 2) →
+      (oppositeSurfaceData D).1 ((fun s : ℝ ↦ (s : Real.Angle)) '' T) =
+        volume.withDensity (fun s ↦ ENNReal.ofReal (fD s)) T := by
+    intro T hT hTsub
+    refine oppositeSurfaceData_angleImage_eq_withDensity D (by linarith) (by linarith)
+      (fun v ↦ F1 (v - Real.pi / 2)) fD (fun u ↦ g1 (u - Real.pi / 2)) (hshift F1 g1 hF1)
+      (hg1.comp (continuous_id.sub continuous_const))
+      (fun s hs ↦ hg1nn' _ ⟨by linarith [hs.1], by linarith [hs.2]⟩) hfD1 ?_ hT hTsub
+    exact fun s hs ↦ (hvD s ⟨by linarith [hs.1], by linarith [hs.2]⟩).trans
+      (hFeq1' _ ⟨by linarith [hs.1], by linarith [hs.2]⟩)
+  have hstep : ∀ T, MeasurableSet T →
+      T ⊆ Set.Ioc (Real.pi / 2 + gerverStageTimes 0) (Real.pi / 2 + gerverStageTimes 2) →
+      (oppositeSurfaceData D).1 ((fun s : ℝ ↦ (s : Real.Angle)) '' T) =
+        volume.withDensity (fun s ↦ ENNReal.ofReal (fD s)) T := by
+    intro T hT hTsub
+    refine measure_angleImage_eq_of_union (c := Real.pi / 2 + gerverStageTimes 0)
+      (d := Real.pi / 2 + gerverStageTimes 2) (by linarith)
+      (I := Set.Ioc (Real.pi / 2 + gerverStageTimes 0) (Real.pi / 2 + gerverStageTimes 1))
+      (J := Set.Ioc (Real.pi / 2 + gerverStageTimes 1) (Real.pi / 2 + gerverStageTimes 2)) ?_ ?_
+      measurableSet_Ioc measurableSet_Ioc ?_ hagree0 hagree1 T hT ?_
+    · exact fun x hx ↦ ⟨hx.1, by linarith [hx.2]⟩
+    · exact fun x hx ↦ ⟨by linarith [hx.1], hx.2⟩
+    · rw [Set.disjoint_left]
+      exact fun x hx hx' ↦ absurd hx'.1 (not_lt.2 hx.2)
+    · intro x hx
+      rcases le_or_gt x (Real.pi / 2 + gerverStageTimes 1) with h | h
+      · exact Or.inl ⟨(hTsub hx).1, h⟩
+      · exact Or.inr ⟨h, (hTsub hx).2⟩
+  have hsplit : Set.Ioc (Real.pi / 2 + gerverStageTimes 0) (Real.pi / 2 + gerverStageTimes 2) =
+      Set.Ioc (Real.pi / 2 + gerverStageTimes 0) (Real.pi / 2 + gerverStageTimes 1) ∪
+        Set.Ioc (Real.pi / 2 + gerverStageTimes 1) (Real.pi / 2 + gerverStageTimes 2) :=
+    (Set.Ioc_union_Ioc_eq_Ioc (by linarith) (by linarith)).symm
+  have hae : ∀ᵐ u ∂volume.restrict (Set.Ioc (Real.pi / 2 + gerverStageTimes 0)
+      (Real.pi / 2 + gerverStageTimes 2)),
+      u ∈ Set.Ioo (Real.pi / 2 + gerverStageTimes 0) (Real.pi / 2 + gerverStageTimes 1) ∪
+        Set.Ioo (Real.pi / 2 + gerverStageTimes 1) (Real.pi / 2 + gerverStageTimes 2) := by
+    refine ae_restrict_mem_of_countable_diff measurableSet_Ioc
+      ((Set.countable_singleton (Real.pi / 2 + gerverStageTimes 2)).insert
+        (Real.pi / 2 + gerverStageTimes 1)) ?_
+    rintro x ⟨hx, hx'⟩
+    rcases lt_trichotomy x (Real.pi / 2 + gerverStageTimes 1) with h | h | h
+    · exact absurd (Or.inl ⟨hx.1, h⟩) hx'
+    · exact Or.inl h
+    · rcases eq_or_lt_of_le hx.2 with heq | hlt
+      · exact Or.inr (Set.mem_singleton_iff.2 heq)
+      · exact absurd (Or.inr ⟨h, hlt⟩) hx'
+  refine ⟨?_, ?_, ?_⟩
+  · have hae01 : ∀ a b : ℝ, ∀ᵐ u ∂volume.restrict (Set.Ioc a b),
+        u ∈ Set.Ioo a b := by
+      intro a b
+      refine ae_restrict_mem_of_countable_diff measurableSet_Ioc
+        (Set.countable_singleton b) ?_
+      rintro x ⟨hx, hx'⟩
+      exact le_antisymm hx.2 (not_lt.1 fun h ↦ hx' ⟨hx.1, h⟩)
+    have hint0 : IntegrableOn fD (Set.Ioc (Real.pi / 2 + gerverStageTimes 0)
+        (Real.pi / 2 + gerverStageTimes 1)) volume := by
+      have hc0 : Continuous fun u : ℝ ↦ g0 (u - Real.pi / 2) :=
+        hg0.comp (continuous_id.sub continuous_const)
+      refine (((ContinuousOn.integrableOn_compact (isCompact_Icc
+        (a := Real.pi / 2 + gerverStageTimes 0) (b := Real.pi / 2 + gerverStageTimes 1))
+          hc0.continuousOn)).mono_set Set.Ioc_subset_Icc_self).congr ?_
+      filter_upwards [hae01 _ _] with u hu using (hfD0 u hu).symm
+    have hint1 : IntegrableOn fD (Set.Ioc (Real.pi / 2 + gerverStageTimes 1)
+        (Real.pi / 2 + gerverStageTimes 2)) volume := by
+      have hc1 : Continuous fun u : ℝ ↦ g1 (u - Real.pi / 2) :=
+        hg1.comp (continuous_id.sub continuous_const)
+      refine (((ContinuousOn.integrableOn_compact (isCompact_Icc
+        (a := Real.pi / 2 + gerverStageTimes 1) (b := Real.pi / 2 + gerverStageTimes 2))
+          hc1.continuousOn)).mono_set Set.Ioc_subset_Icc_self).congr ?_
+      filter_upwards [hae01 _ _] with u hu using (hfD1 u hu).symm
+    rw [show Integrable fD (volume.restrict (Set.Ioc (Real.pi / 2 + gerverStageTimes 0)
+      (Real.pi / 2 + gerverStageTimes 2))) = IntegrableOn fD
+        (Set.Ioc (Real.pi / 2 + gerverStageTimes 0)
+          (Real.pi / 2 + gerverStageTimes 2)) volume from rfl, hsplit]
+    exact hint0.union hint1
+  · filter_upwards [hae] with u hu
+    rcases hu with hu | hu
+    · rw [hfD0 u hu]
+      exact hg0nn' _ ⟨by linarith [hu.1], by linarith [hu.2]⟩
+    · rw [hfD1 u hu]
+      exact hg1nn' _ ⟨by linarith [hu.1], by linarith [hu.2]⟩
+  · rw [angularDensityMeasure]
+    exact measure_restrict_eq_map_withDensity measurableSet_Ioc hstep
+
+theorem gerver_measureTranslation (K : SpecialCapSpace) (B D : ConvexBody Point)
+    (hK : (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2))
+    (hB : (B : Set Point) = (canonicalTailSets K).1)
+    (hD : (D : Set Point) = (canonicalTailSets K).2) :
+    HasAngularDensity (surfaceAreaMeasure K.val.val)
+      (fun t ↦ inner ℝ (deriv (fun s ↦ paperGerverContacts s 0) t)
+        (tangentVector (t : Real.Angle))) (Set.Ico 0 (Real.pi / 2)) ∧
+    HasAngularDensity (oppositeSurfaceData B).1
+      (fun t ↦ inner ℝ (-(deriv (fun s ↦ paperGerverContacts s 1) t))
+        (tangentVector (t : Real.Angle)))
+      (Set.Ico (gerverStageTimes 3) (gerverStageTimes 5)) ∧
+    HasAngularDensity (surfaceAreaMeasure K.val.val)
+      (fun t ↦ inner ℝ (-(deriv (fun s ↦ paperGerverContacts s 2) (t - Real.pi / 2)))
+        (normalVector ((t - Real.pi / 2 : ℝ) : Real.Angle)))
+      (Set.Ioc (Real.pi / 2) Real.pi) ∧
+    HasAngularDensity (oppositeSurfaceData D).1
+      (fun t ↦ inner ℝ (deriv (fun s ↦ paperGerverContacts s 3) (t - Real.pi / 2))
+        (normalVector ((t - Real.pi / 2 : ℝ) : Real.Angle)))
+      (Set.Ioc (Real.pi / 2 + gerverStageTimes 0)
+        (Real.pi / 2 + gerverStageTimes 2)) := by
+  exact ⟨gerver_angular_density_outer_first K hK,
+    gerver_angular_density_right_tail K B D hK hB hD,
+    gerver_angular_density_outer_second K hK,
+    gerver_angular_density_left_tail K B D hK hB hD⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The eight phase identities of Gerver's cap
+
+The ten Gerver phase intervals cut the angular window `[0, π]` at the five stage times and at
+their reflections.  On each of them the surface-area measure of the cap `K = C(G)` of Gerver's
+sofa is one of: zero, the inner-corner measure `ι_K`, the opposite surface measure of one of the
+two tail bodies, or a sum of the last two (`gerver_phaseMeasures`).
+
+Every clause is proved set-wise.  The translated-measure proposition `gerver_measureTranslation`
+presents the surface measure of the cap and the two opposite tail measures as angular densities
+on the two half-windows, so each of them gives the angular image of a measurable subset of a
+phase interval the Lebesgue integral of the corresponding contact derivative
+(`HasAngularDensity.angleImage_eq_setLIntegral`), and the corner measure does the same with its
+own density (`capCornerAngleMeasure_angleImage_eq_setLIntegral`).  The five stagewise contact
+equations `gerver_stageODEs` identify those densities on the interior of each stage, which is
+almost all of it, and `Real.Angle.measure_restrict_image_congr_of_ae_eq` turns an
+almost-everywhere identity of densities into an equality of the restricted measures.  No atom
+computation at the included stage endpoints is needed: a single parameter is Lebesgue-null and
+the four measures are only ever evaluated through their densities.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- The angular image of one of the ten phase intervals. -/
+def gerverPhaseAngles (j : Fin 10) : Set Real.Angle :=
+  (fun t : ℝ ↦ (t : Real.Angle)) '' gerverPhaseIntervals j
+
+/-- The phase-by-phase surface-measure identities relating the cap, corner and two tail
+bodies. -/
+def GerverPhaseMeasureIdentities (K : SpecialCapSpace) (B D : ConvexBody Point) : Prop :=
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 0) = 0 ∧
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 1 ∪ gerverPhaseAngles 2) =
+    (capCornerAngleMeasure K).restrict (gerverPhaseAngles 1 ∪ gerverPhaseAngles 2) ∧
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 3) =
+    ((oppositeSurfaceData B).1 + capCornerAngleMeasure K).restrict (gerverPhaseAngles 3) ∧
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 4) =
+    (oppositeSurfaceData B).1.restrict (gerverPhaseAngles 4) ∧
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 5) =
+    (oppositeSurfaceData D).1.restrict (gerverPhaseAngles 5) ∧
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 6) =
+    ((oppositeSurfaceData D).1 + capCornerAngleMeasure K).restrict (gerverPhaseAngles 6) ∧
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 7 ∪ gerverPhaseAngles 8) =
+    (capCornerAngleMeasure K).restrict (gerverPhaseAngles 7 ∪ gerverPhaseAngles 8) ∧
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 9) = 0
+
+/-! ### The corner density of the cap of Gerver's sofa
+
+On each of the two open quarter turns the inner corner of the cap is the direct Gerver path
+(`derivWithin_capInnerCorner_eq_deriv_paperGerverPath`), so the two branches of
+`capCornerDensity` are the two velocity components `β` and `-α` of that path. -/
+
+/-- On the first open quarter turn the corner density of the cap of Gerver's sofa is the
+tangential velocity component of the direct Gerver path. -/
+theorem capCornerDensity_eq_velocity_tangential (K : SpecialCapSpace)
+    (hK : (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2)) {s : ℝ}
+    (hs : s ∈ Set.Ioo (0 : ℝ) (Real.pi / 2)) :
+    capCornerDensity K s = (paperGerverVelocityComponents s).2 := by
+  simp only [capCornerDensity]
+  split_ifs with h1 h2
+  · simp only [capVelocityCoefficients, paperGerverVelocityComponents,
+      derivWithin_capInnerCorner_eq_deriv_paperGerverPath K.val hK hs]
+  · exact absurd ⟨hs.1, hs.2.le⟩ h1
+  · exact absurd ⟨hs.1, hs.2.le⟩ h1
+
+/-- On the second open quarter turn the corner density of the cap of Gerver's sofa is minus the
+normal velocity component of the direct Gerver path, at the parameter shifted by a quarter
+turn. -/
+theorem capCornerDensity_eq_neg_velocity_normal (K : SpecialCapSpace)
+    (hK : (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2)) {s : ℝ}
+    (hs : s ∈ Set.Ioo (Real.pi / 2) Real.pi) :
+    capCornerDensity K s = -(paperGerverVelocityComponents (s - Real.pi / 2)).1 := by
+  have hs' : s - Real.pi / 2 ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) :=
+    ⟨by linarith [hs.1], by linarith [hs.2]⟩
+  simp only [capCornerDensity]
+  split_ifs with h1 h2
+  · exact absurd h1.2 (not_le.2 hs.1)
+  · simp only [capVelocityCoefficients, paperGerverVelocityComponents,
+      derivWithin_capInnerCorner_eq_deriv_paperGerverPath K.val hK hs']
+  · exact absurd ⟨hs.1, hs.2.le⟩ h2
+
+private theorem gerver_contact_density_stage_identities
+    (fA fB fC fD : ℝ → ℝ)
+    (hfAdef : fA = fun t ↦ inner ℝ
+      (deriv (fun s ↦ paperGerverContacts s 0) t) (tangentVector (t : Real.Angle)))
+    (hfBdef : fB = fun t ↦ inner ℝ
+      (-(deriv (fun s ↦ paperGerverContacts s 1) t))
+      (tangentVector (t : Real.Angle)))
+    (hfCdef : fC = fun t ↦ inner ℝ
+      (-(deriv (fun s ↦ paperGerverContacts s 2) (t - Real.pi / 2)))
+      (normalVector ((t - Real.pi / 2 : ℝ) : Real.Angle)))
+    (hfDdef : fD = fun t ↦ inner ℝ
+      (deriv (fun s ↦ paperGerverContacts s 3) (t - Real.pi / 2))
+      (normalVector ((t - Real.pi / 2 : ℝ) : Real.Angle)))
+    : (∀ t ∈ Set.Ioo (0 : ℝ) GerversSofa.φ,
+     fA t = 0 ∧ fC (t + Real.pi / 2) = fD (t + Real.pi / 2)) ∧
+    (∀ t ∈ Set.Ioo GerversSofa.φ GerversSofa.θ,
+     fA t = (paperGerverVelocityComponents t).2 ∧
+     fC (t + Real.pi / 2) =
+       fD (t + Real.pi / 2) - (paperGerverVelocityComponents t).1) ∧
+    (∀ t ∈ Set.Ioo GerversSofa.θ (Real.pi / 2 - GerversSofa.θ),
+     fA t = (paperGerverVelocityComponents t).2 ∧
+     fC (t + Real.pi / 2) = -(paperGerverVelocityComponents t).1) ∧
+    (∀ t ∈ Set.Ioo (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ),
+     fA t = fB t + (paperGerverVelocityComponents t).2 ∧
+     fC (t + Real.pi / 2) = -(paperGerverVelocityComponents t).1) ∧
+    (∀ t ∈ Set.Ioo (Real.pi / 2 - GerversSofa.φ) (Real.pi / 2),
+     fA t = fB t ∧ fC (t + Real.pi / 2) = 0) := by
+  -- ### The five closed stage intervals
+  have hI0 : gerverStageIntervals 0 = Set.Icc 0 GerversSofa.φ := by
+    simp [gerverStageIntervals, gerverStageTimes]
+  have hI1 : gerverStageIntervals 1 = Set.Icc GerversSofa.φ GerversSofa.θ := by
+    simp [gerverStageIntervals, gerverStageTimes]
+  have hI2 : gerverStageIntervals 2 =
+      Set.Icc GerversSofa.θ (Real.pi / 2 - GerversSofa.θ) := by
+    simp [gerverStageIntervals, gerverStageTimes]
+  have hI3 : gerverStageIntervals 3 =
+      Set.Icc (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ) := by
+    simp [gerverStageIntervals, gerverStageTimes]
+  have hI4 : gerverStageIntervals 4 =
+      Set.Icc (Real.pi / 2 - GerversSofa.φ) (Real.pi / 2) := by
+    simp [gerverStageIntervals, gerverStageTimes]
+  -- ### Inside an open stage the stagewise derivatives are the global ones
+  have hconv : ∀ (i : Fin 5) (a b t : ℝ), gerverStageIntervals i = Set.Icc a b →
+      a < t → t < b →
+      gerverStageContactDerivatives i t 0 = fA t ∧
+      gerverStageContactDerivatives i t 1 = -fB t ∧
+      gerverStageContactDerivatives i t 2 = fC (t + Real.pi / 2) ∧
+      gerverStageContactDerivatives i t 3 = fD (t + Real.pi / 2) := by
+    intro i a b t hI ha hb
+    have hn : gerverStageIntervals i ∈ nhds t := by rw [hI]; exact Icc_mem_nhds ha hb
+    have e : ∀ k : Fin 4,
+        derivWithin (fun s ↦ paperGerverContacts s k) (gerverStageIntervals i) t =
+          deriv (fun s ↦ paperGerverContacts s k) t := fun _ ↦ derivWithin_of_mem_nhds hn
+    refine ⟨?_, ?_, ?_, ?_⟩
+    · change inner ℝ (derivWithin (fun s ↦ paperGerverContacts s 0) (gerverStageIntervals i) t)
+        (tangentVector (t : Real.Angle)) = fA t
+      rw [e 0, hfAdef]
+    · change inner ℝ (derivWithin (fun s ↦ paperGerverContacts s 1) (gerverStageIntervals i) t)
+        (tangentVector (t : Real.Angle)) = -fB t
+      rw [e 1, hfBdef]
+      simp [inner_neg_left]
+    · change inner ℝ (-derivWithin (fun s ↦ paperGerverContacts s 2) (gerverStageIntervals i) t)
+        (normalVector (t : Real.Angle)) = fC (t + Real.pi / 2)
+      rw [e 2, hfCdef]
+      simp
+    · change inner ℝ (derivWithin (fun s ↦ paperGerverContacts s 3) (gerverStageIntervals i) t)
+        (normalVector (t : Real.Angle)) = fD (t + Real.pi / 2)
+      rw [e 3, hfDdef]
+      simp
+  -- ### The five stage coefficient identities, at interior parameters
+  have hode : ∀ (i : Fin 5) (a b t : ℝ), gerverStageIntervals i = Set.Icc a b →
+      a < t → t < b →
+      (gerverStageContactDerivatives i t 0, gerverStageContactDerivatives i t 2) =
+        ![(0, gerverStageContactDerivatives i t 3),
+          ((paperGerverVelocityComponents t).2,
+            gerverStageContactDerivatives i t 3 - (paperGerverVelocityComponents t).1),
+          ((paperGerverVelocityComponents t).2, -(paperGerverVelocityComponents t).1),
+          (-gerverStageContactDerivatives i t 1 + (paperGerverVelocityComponents t).2,
+            -(paperGerverVelocityComponents t).1),
+          (-gerverStageContactDerivatives i t 1, 0)] i :=
+    fun i a b t hI ha hb ↦ gerver_stageODEs i t (by rw [hI]; exact ⟨ha.le, hb.le⟩)
+  have hs0 : ∀ t ∈ Set.Ioo (0 : ℝ) GerversSofa.φ,
+      fA t = 0 ∧ fC (t + Real.pi / 2) = fD (t + Real.pi / 2) := by
+    intro t ht
+    obtain ⟨e0, -, e2, e3⟩ := hconv 0 _ _ t hI0 ht.1 ht.2
+    have h := hode 0 _ _ t hI0 ht.1 ht.2
+    simp only [Matrix.cons_val, Prod.mk.injEq] at h
+    exact ⟨e0.symm.trans h.1, e2.symm.trans (h.2.trans e3)⟩
+  have hs1 : ∀ t ∈ Set.Ioo GerversSofa.φ GerversSofa.θ,
+      fA t = (paperGerverVelocityComponents t).2 ∧
+      fC (t + Real.pi / 2) =
+        fD (t + Real.pi / 2) - (paperGerverVelocityComponents t).1 := by
+    intro t ht
+    obtain ⟨e0, -, e2, e3⟩ := hconv 1 _ _ t hI1 ht.1 ht.2
+    have h := hode 1 _ _ t hI1 ht.1 ht.2
+    simp only [Matrix.cons_val, Prod.mk.injEq] at h
+    exact ⟨e0.symm.trans h.1, e2.symm.trans (h.2.trans (by rw [e3]))⟩
+  have hs2 : ∀ t ∈ Set.Ioo GerversSofa.θ (Real.pi / 2 - GerversSofa.θ),
+      fA t = (paperGerverVelocityComponents t).2 ∧
+      fC (t + Real.pi / 2) = -(paperGerverVelocityComponents t).1 := by
+    intro t ht
+    obtain ⟨e0, -, e2, -⟩ := hconv 2 _ _ t hI2 ht.1 ht.2
+    have h := hode 2 _ _ t hI2 ht.1 ht.2
+    simp only [Matrix.cons_val, Prod.mk.injEq] at h
+    exact ⟨e0.symm.trans h.1, e2.symm.trans h.2⟩
+  have hs3 : ∀ t ∈ Set.Ioo (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ),
+      fA t = fB t + (paperGerverVelocityComponents t).2 ∧
+      fC (t + Real.pi / 2) = -(paperGerverVelocityComponents t).1 := by
+    intro t ht
+    obtain ⟨e0, e1, e2, -⟩ := hconv 3 _ _ t hI3 ht.1 ht.2
+    have h := hode 3 _ _ t hI3 ht.1 ht.2
+    simp only [Matrix.cons_val, Prod.mk.injEq] at h
+    refine ⟨e0.symm.trans (h.1.trans ?_), e2.symm.trans h.2⟩
+    rw [e1, neg_neg]
+  have hs4 : ∀ t ∈ Set.Ioo (Real.pi / 2 - GerversSofa.φ) (Real.pi / 2),
+      fA t = fB t ∧ fC (t + Real.pi / 2) = 0 := by
+    intro t ht
+    obtain ⟨e0, e1, e2, -⟩ := hconv 4 _ _ t hI4 ht.1 ht.2
+    have h := hode 4 _ _ t hI4 ht.1 ht.2
+    simp only [Matrix.cons_val, Prod.mk.injEq] at h
+    refine ⟨e0.symm.trans (h.1.trans ?_), e2.symm.trans h.2⟩
+    rw [e1, neg_neg]
+  exact ⟨hs0, hs1, hs2, hs3, hs4⟩
+
+private theorem gerver_phaseMeasures_first_half (K : SpecialCapSpace) (B D : ConvexBody Point)
+    (hK : (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2))
+    (hB : (B : Set Point) = (canonicalTailSets K).1)
+    (hD : (D : Set Point) = (canonicalTailSets K).2) :
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 0) = 0 ∧
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 1 ∪ gerverPhaseAngles 2) =
+    (capCornerAngleMeasure K).restrict (gerverPhaseAngles 1 ∪ gerverPhaseAngles 2) ∧
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 3) =
+    ((oppositeSurfaceData B).1 + capCornerAngleMeasure K).restrict (gerverPhaseAngles 3) ∧
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 4) =
+    (oppositeSurfaceData B).1.restrict (gerverPhaseAngles 4) := by
+  -- ### The six stage endpoints, written out
+  have hmono := gerverStageTimes_strictMono
+  have h01 := hmono (show (0 : Fin 6) < 1 by decide)
+  have h12 := hmono (show (1 : Fin 6) < 2 by decide)
+  have h23 := hmono (show (2 : Fin 6) < 3 by decide)
+  have h34 := hmono (show (3 : Fin 6) < 4 by decide)
+  have h45 := hmono (show (4 : Fin 6) < 5 by decide)
+  simp only [gerverStageTimes, Matrix.cons_val] at h01 h12 h23 h34 h45
+  have hpi : (3 : ℝ) < Real.pi := Real.pi_gt_three
+  have hturn : Real.pi ≤ -1 + 2 * Real.pi := by linarith
+  -- ### The five relevant phase intervals
+  have hJ0 : gerverPhaseIntervals 0 = Set.Ico 0 GerversSofa.φ :=
+    gerverPhaseIntervals_explicit 0
+  have hJ1 : gerverPhaseIntervals 1 = Set.Ico GerversSofa.φ GerversSofa.θ :=
+    gerverPhaseIntervals_explicit 1
+  have hJ2 : gerverPhaseIntervals 2 =
+      Set.Ico GerversSofa.θ (Real.pi / 2 - GerversSofa.θ) :=
+    gerverPhaseIntervals_explicit 2
+  have hJ3 : gerverPhaseIntervals 3 =
+      Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ) :=
+    gerverPhaseIntervals_explicit 3
+  have hJ4 : gerverPhaseIntervals 4 =
+      Set.Ico (Real.pi / 2 - GerversSofa.φ) (Real.pi / 2) :=
+    gerverPhaseIntervals_explicit 4
+  -- ### The four contact densities of the translated-measure proposition
+  have hd := gerver_measureTranslation K B D hK hB hD
+  obtain ⟨fA, hfAdef⟩ : ∃ f : ℝ → ℝ, f = fun t ↦ inner ℝ
+      (deriv (fun s ↦ paperGerverContacts s 0) t) (tangentVector (t : Real.Angle)) := ⟨_, rfl⟩
+  obtain ⟨fB, hfBdef⟩ : ∃ f : ℝ → ℝ, f = fun t ↦ inner ℝ
+      (-(deriv (fun s ↦ paperGerverContacts s 1) t))
+      (tangentVector (t : Real.Angle)) := ⟨_, rfl⟩
+  obtain ⟨fC, hfCdef⟩ : ∃ f : ℝ → ℝ, f = fun t ↦ inner ℝ
+      (-(deriv (fun s ↦ paperGerverContacts s 2) (t - Real.pi / 2)))
+      (normalVector ((t - Real.pi / 2 : ℝ) : Real.Angle)) := ⟨_, rfl⟩
+  obtain ⟨fD, hfDdef⟩ : ∃ f : ℝ → ℝ, f = fun t ↦ inner ℝ
+      (deriv (fun s ↦ paperGerverContacts s 3) (t - Real.pi / 2))
+      (normalVector ((t - Real.pi / 2 : ℝ) : Real.Angle)) := ⟨_, rfl⟩
+  rw [← hfAdef, ← hfBdef, ← hfCdef, ← hfDdef,
+    show Set.Ico (gerverStageTimes 3) (gerverStageTimes 5) =
+      Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2) from by simp [gerverStageTimes],
+    show Set.Ioc (Real.pi / 2 + gerverStageTimes 0) (Real.pi / 2 + gerverStageTimes 2) =
+      Set.Ioc (Real.pi / 2) (Real.pi / 2 + GerversSofa.θ) from by simp [gerverStageTimes]] at hd
+  obtain ⟨hdA, hdB, -, -⟩ := hd
+  -- ### The two relevant density windows sit inside a single turn
+  have hsubA : Set.Ico (0 : ℝ) (Real.pi / 2) ⊆ Set.Ioc (-1 : ℝ) Real.pi :=
+    fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+  have hsubB : Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2) ⊆
+      Set.Ioc (-1 : ℝ) Real.pi := fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+  obtain ⟨hs0, hs1, hs2, hs3, hs4⟩ :=
+    gerver_contact_density_stage_identities fA fB fC fD hfAdef hfBdef hfCdef hfDdef
+  -- ### The strict interior signs of the injectivity condition of the given special cap
+  have hβ : ∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+      0 ≤ (paperGerverVelocityComponents t).2 := by
+    intro t ht
+    have h := (K.2.1.2.2 t ht).2
+    rw [derivWithin_capInnerCorner_eq_deriv_paperGerverPath K.val hK ht] at h
+    exact h.le
+  refine ⟨?_, ?_, ?_, ?_⟩
+  · have hsub : Set.Ico (0 : ℝ) GerversSofa.φ ⊆ Set.Ico (0 : ℝ) (Real.pi / 2) :=
+      fun x hx ↦ ⟨hx.1, by linarith [hx.2]⟩
+    simp only [gerverPhaseAngles, hJ0]
+    rw [Measure.restrict_eq_zero,
+      hdA.angleImage_eq_setLIntegral hturn hsubA measurableSet_Ico hsub]
+    refine (lintegral_congr_ae ?_).trans lintegral_zero
+    filter_upwards [ae_restrict_Ico_mem_Ioo (μ := volume) 0 GerversSofa.φ] with t ht
+    rw [(hs0 t ht).1, ENNReal.ofReal_zero]
+  · have hsub : Set.Ico GerversSofa.φ (Real.pi / 2 - GerversSofa.θ) ⊆
+        Set.Ico (0 : ℝ) (Real.pi / 2) :=
+      fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+    have hsubI : Set.Ico GerversSofa.φ (Real.pi / 2 - GerversSofa.θ) ⊆
+        Set.Icc (0 : ℝ) Real.pi :=
+      fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+    have hU : gerverPhaseAngles 1 ∪ gerverPhaseAngles 2 =
+        (fun s : ℝ ↦ (s : Real.Angle)) ''
+          Set.Ico GerversSofa.φ (Real.pi / 2 - GerversSofa.θ) := by
+      simp only [gerverPhaseAngles, hJ1, hJ2, ← Set.image_union,
+        Set.Ico_union_Ico_eq_Ico h12.le h23.le]
+    rw [hU]
+    refine Real.Angle.measure_restrict_image_congr_of_ae_eq measurableSet_Ico
+      (fun T hT hTsub ↦ hdA.angleImage_eq_setLIntegral hturn hsubA hT (hTsub.trans hsub))
+      (fun T hT hTsub ↦ capCornerAngleMeasure_angleImage_eq_setLIntegral K hT
+        (hTsub.trans hsubI)) ?_
+    filter_upwards [ae_restrict_Ico_mem_Ioo_union_Ioo (μ := volume) GerversSofa.φ GerversSofa.θ
+      (Real.pi / 2 - GerversSofa.θ)] with t ht
+    rcases ht with ht | ht
+    · rw [(hs1 t ht).1,
+        capCornerDensity_eq_velocity_tangential K hK ⟨by linarith [ht.1], by linarith [ht.2]⟩]
+    · rw [(hs2 t ht).1,
+        capCornerDensity_eq_velocity_tangential K hK ⟨by linarith [ht.1], by linarith [ht.2]⟩]
+  · have hsub : Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ) ⊆
+        Set.Ico (0 : ℝ) (Real.pi / 2) :=
+      fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+    have hsubB' : Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ) ⊆
+        Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2) :=
+      fun x hx ↦ ⟨hx.1, by linarith [hx.2]⟩
+    have hsubI : Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ) ⊆
+        Set.Icc (0 : ℝ) Real.pi :=
+      fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+    have hsubO : Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ) ⊆
+        Set.Ioo (0 : ℝ) (Real.pi / 2) :=
+      fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+    have hcorner : ∀ T, MeasurableSet T →
+        T ⊆ Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ) →
+        capCornerAngleMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' T) =
+          ∫⁻ t in T, ENNReal.ofReal ((paperGerverVelocityComponents t).2) ∂volume := by
+      intro T hT hTsub
+      rw [capCornerAngleMeasure_angleImage_eq_setLIntegral K hT (hTsub.trans hsubI)]
+      exact setLIntegral_congr_fun hT fun t ht ↦ by
+        rw [capCornerDensity_eq_velocity_tangential K hK (hsubO (hTsub ht))]
+    have hf0 : ∀ᵐ t ∂volume.restrict
+        (Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ)), 0 ≤ fB t :=
+      ae_restrict_of_ae_restrict_of_subset hsubB' hdB.2.1
+    have hg0 : ∀ᵐ t ∂volume.restrict
+        (Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ)),
+        0 ≤ (paperGerverVelocityComponents t).2 := by
+      filter_upwards [ae_restrict_mem measurableSet_Ico] with t ht using hβ t (hsubO ht)
+    have hsum : ∀ᵐ t ∂volume.restrict
+        (Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ)),
+        fA t = fB t + (paperGerverVelocityComponents t).2 := by
+      filter_upwards [ae_restrict_Ico_mem_Ioo (μ := volume) (Real.pi / 2 - GerversSofa.θ)
+        (Real.pi / 2 - GerversSofa.φ)] with t ht using (hs3 t ht).1
+    simp only [gerverPhaseAngles, hJ3]
+    exact Real.Angle.measure_restrict_image_congr_add measurableSet_Ico
+      (fun T hT hTsub ↦ hdA.angleImage_eq_setLIntegral hturn hsubA hT (hTsub.trans hsub))
+      (fun T hT hTsub ↦ hdB.angleImage_eq_setLIntegral hturn hsubB hT (hTsub.trans hsubB'))
+      hcorner (hdB.1.aemeasurable.mono_measure (Measure.restrict_mono hsubB' le_rfl))
+      hf0 hg0 hsum
+  · have hsub : Set.Ico (Real.pi / 2 - GerversSofa.φ) (Real.pi / 2) ⊆
+        Set.Ico (0 : ℝ) (Real.pi / 2) := fun x hx ↦ ⟨by linarith [hx.1], hx.2⟩
+    have hsubB' : Set.Ico (Real.pi / 2 - GerversSofa.φ) (Real.pi / 2) ⊆
+        Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2) :=
+      fun x hx ↦ ⟨by linarith [hx.1], hx.2⟩
+    simp only [gerverPhaseAngles, hJ4]
+    refine Real.Angle.measure_restrict_image_congr_of_ae_eq measurableSet_Ico
+      (fun T hT hTsub ↦ hdA.angleImage_eq_setLIntegral hturn hsubA hT (hTsub.trans hsub))
+      (fun T hT hTsub ↦ hdB.angleImage_eq_setLIntegral hturn hsubB hT (hTsub.trans hsubB')) ?_
+    filter_upwards [ae_restrict_Ico_mem_Ioo (μ := volume) (Real.pi / 2 - GerversSofa.φ)
+      (Real.pi / 2)] with t ht
+    rw [(hs4 t ht).1]
+
+private theorem gerver_phaseMeasures_second_half (K : SpecialCapSpace) (B D : ConvexBody Point)
+    (hK : (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2))
+    (hB : (B : Set Point) = (canonicalTailSets K).1)
+    (hD : (D : Set Point) = (canonicalTailSets K).2) :
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 5) =
+    (oppositeSurfaceData D).1.restrict (gerverPhaseAngles 5) ∧
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 6) =
+    ((oppositeSurfaceData D).1 + capCornerAngleMeasure K).restrict (gerverPhaseAngles 6) ∧
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 7 ∪ gerverPhaseAngles 8) =
+    (capCornerAngleMeasure K).restrict (gerverPhaseAngles 7 ∪ gerverPhaseAngles 8) ∧
+  (surfaceAreaMeasure K.val.val).restrict (gerverPhaseAngles 9) = 0 := by
+  -- ### The six stage endpoints, written out
+  have hmono := gerverStageTimes_strictMono
+  have h01 := hmono (show (0 : Fin 6) < 1 by decide)
+  have h12 := hmono (show (1 : Fin 6) < 2 by decide)
+  have h23 := hmono (show (2 : Fin 6) < 3 by decide)
+  have h34 := hmono (show (3 : Fin 6) < 4 by decide)
+  have h45 := hmono (show (4 : Fin 6) < 5 by decide)
+  simp only [gerverStageTimes, Matrix.cons_val] at h01 h12 h23 h34 h45
+  have hpi : (3 : ℝ) < Real.pi := Real.pi_gt_three
+  have hturn : Real.pi ≤ -1 + 2 * Real.pi := by linarith
+  -- ### The five relevant phase intervals
+  have hJ5 : gerverPhaseIntervals 5 =
+      Set.Ioc (Real.pi / 2) (Real.pi / 2 + GerversSofa.φ) :=
+    gerverPhaseIntervals_explicit 5
+  have hJ6 : gerverPhaseIntervals 6 =
+      Set.Ioc (Real.pi / 2 + GerversSofa.φ) (Real.pi / 2 + GerversSofa.θ) :=
+    gerverPhaseIntervals_explicit 6
+  have hJ7 : gerverPhaseIntervals 7 =
+      Set.Ioc (Real.pi / 2 + GerversSofa.θ) (Real.pi - GerversSofa.θ) :=
+    gerverPhaseIntervals_explicit 7
+  have hJ8 : gerverPhaseIntervals 8 =
+      Set.Ioc (Real.pi - GerversSofa.θ) (Real.pi - GerversSofa.φ) :=
+    gerverPhaseIntervals_explicit 8
+  have hJ9 : gerverPhaseIntervals 9 = Set.Ioc (Real.pi - GerversSofa.φ) Real.pi :=
+    gerverPhaseIntervals_explicit 9
+  -- ### The four contact densities of the translated-measure proposition
+  have hd := gerver_measureTranslation K B D hK hB hD
+  obtain ⟨fA, hfAdef⟩ : ∃ f : ℝ → ℝ, f = fun t ↦ inner ℝ
+      (deriv (fun s ↦ paperGerverContacts s 0) t) (tangentVector (t : Real.Angle)) := ⟨_, rfl⟩
+  obtain ⟨fB, hfBdef⟩ : ∃ f : ℝ → ℝ, f = fun t ↦ inner ℝ
+      (-(deriv (fun s ↦ paperGerverContacts s 1) t))
+      (tangentVector (t : Real.Angle)) := ⟨_, rfl⟩
+  obtain ⟨fC, hfCdef⟩ : ∃ f : ℝ → ℝ, f = fun t ↦ inner ℝ
+      (-(deriv (fun s ↦ paperGerverContacts s 2) (t - Real.pi / 2)))
+      (normalVector ((t - Real.pi / 2 : ℝ) : Real.Angle)) := ⟨_, rfl⟩
+  obtain ⟨fD, hfDdef⟩ : ∃ f : ℝ → ℝ, f = fun t ↦ inner ℝ
+      (deriv (fun s ↦ paperGerverContacts s 3) (t - Real.pi / 2))
+      (normalVector ((t - Real.pi / 2 : ℝ) : Real.Angle)) := ⟨_, rfl⟩
+  rw [← hfAdef, ← hfBdef, ← hfCdef, ← hfDdef,
+    show Set.Ico (gerverStageTimes 3) (gerverStageTimes 5) =
+      Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2) from by simp [gerverStageTimes],
+    show Set.Ioc (Real.pi / 2 + gerverStageTimes 0) (Real.pi / 2 + gerverStageTimes 2) =
+      Set.Ioc (Real.pi / 2) (Real.pi / 2 + GerversSofa.θ) from by simp [gerverStageTimes]] at hd
+  obtain ⟨-, -, hdC, hdD⟩ := hd
+  -- ### The two relevant density windows sit inside a single turn
+  have hsubC : Set.Ioc (Real.pi / 2) Real.pi ⊆ Set.Ioc (-1 : ℝ) Real.pi :=
+    fun x hx ↦ ⟨by linarith [hx.1], hx.2⟩
+  have hsubD : Set.Ioc (Real.pi / 2) (Real.pi / 2 + GerversSofa.θ) ⊆
+      Set.Ioc (-1 : ℝ) Real.pi := fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+  obtain ⟨hs0, hs1, hs2, hs3, hs4⟩ :=
+    gerver_contact_density_stage_identities fA fB fC fD hfAdef hfBdef hfCdef hfDdef
+  -- ### The strict interior signs of the injectivity condition of the given special cap
+  have hα : ∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+      0 ≤ -(paperGerverVelocityComponents t).1 := by
+    intro t ht
+    have h := (K.2.1.2.2 t ht).1
+    rw [derivWithin_capInnerCorner_eq_deriv_paperGerverPath K.val hK ht] at h
+    exact neg_nonneg.2 h.le
+  refine ⟨?_, ?_, ?_, ?_⟩
+  · have hsub : Set.Ioc (Real.pi / 2) (Real.pi / 2 + GerversSofa.φ) ⊆
+        Set.Ioc (Real.pi / 2) Real.pi := fun x hx ↦ ⟨hx.1, by linarith [hx.2]⟩
+    have hsubD' : Set.Ioc (Real.pi / 2) (Real.pi / 2 + GerversSofa.φ) ⊆
+        Set.Ioc (Real.pi / 2) (Real.pi / 2 + GerversSofa.θ) :=
+      fun x hx ↦ ⟨hx.1, by linarith [hx.2]⟩
+    simp only [gerverPhaseAngles, hJ5]
+    refine Real.Angle.measure_restrict_image_congr_of_ae_eq measurableSet_Ioc
+      (fun T hT hTsub ↦ hdC.angleImage_eq_setLIntegral hturn hsubC hT (hTsub.trans hsub))
+      (fun T hT hTsub ↦ hdD.angleImage_eq_setLIntegral hturn hsubD hT (hTsub.trans hsubD')) ?_
+    filter_upwards [ae_restrict_Ioc_mem_Ioo (μ := volume) (Real.pi / 2)
+      (Real.pi / 2 + GerversSofa.φ)] with u hu
+    have h := (hs0 (u - Real.pi / 2) ⟨by linarith [hu.1], by linarith [hu.2]⟩).2
+    rw [show u - Real.pi / 2 + Real.pi / 2 = u from by ring] at h
+    rw [h]
+  · have hsub : Set.Ioc (Real.pi / 2 + GerversSofa.φ) (Real.pi / 2 + GerversSofa.θ) ⊆
+        Set.Ioc (Real.pi / 2) Real.pi :=
+      fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+    have hsubD' : Set.Ioc (Real.pi / 2 + GerversSofa.φ) (Real.pi / 2 + GerversSofa.θ) ⊆
+        Set.Ioc (Real.pi / 2) (Real.pi / 2 + GerversSofa.θ) :=
+      fun x hx ↦ ⟨by linarith [hx.1], hx.2⟩
+    have hsubI : Set.Ioc (Real.pi / 2 + GerversSofa.φ) (Real.pi / 2 + GerversSofa.θ) ⊆
+        Set.Icc (0 : ℝ) Real.pi :=
+      fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+    have hsubO : Set.Ioc (Real.pi / 2 + GerversSofa.φ) (Real.pi / 2 + GerversSofa.θ) ⊆
+        Set.Ioo (Real.pi / 2) Real.pi :=
+      fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+    have hcorner : ∀ T, MeasurableSet T →
+        T ⊆ Set.Ioc (Real.pi / 2 + GerversSofa.φ) (Real.pi / 2 + GerversSofa.θ) →
+        capCornerAngleMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' T) =
+          ∫⁻ u in T,
+            ENNReal.ofReal (-(paperGerverVelocityComponents (u - Real.pi / 2)).1) ∂volume := by
+      intro T hT hTsub
+      rw [capCornerAngleMeasure_angleImage_eq_setLIntegral K hT (hTsub.trans hsubI)]
+      exact setLIntegral_congr_fun hT fun u hu ↦ by
+        rw [capCornerDensity_eq_neg_velocity_normal K hK (hsubO (hTsub hu))]
+    have hf0 : ∀ᵐ u ∂volume.restrict
+        (Set.Ioc (Real.pi / 2 + GerversSofa.φ) (Real.pi / 2 + GerversSofa.θ)), 0 ≤ fD u :=
+      ae_restrict_of_ae_restrict_of_subset hsubD' hdD.2.1
+    have hg0 : ∀ᵐ u ∂volume.restrict
+        (Set.Ioc (Real.pi / 2 + GerversSofa.φ) (Real.pi / 2 + GerversSofa.θ)),
+        0 ≤ -(paperGerverVelocityComponents (u - Real.pi / 2)).1 := by
+      filter_upwards [ae_restrict_mem measurableSet_Ioc] with u hu
+      exact hα _ ⟨by linarith [hu.1], by linarith [hu.2]⟩
+    have hsum : ∀ᵐ u ∂volume.restrict
+        (Set.Ioc (Real.pi / 2 + GerversSofa.φ) (Real.pi / 2 + GerversSofa.θ)),
+        fC u = fD u + -(paperGerverVelocityComponents (u - Real.pi / 2)).1 := by
+      filter_upwards [ae_restrict_Ioc_mem_Ioo (μ := volume) (Real.pi / 2 + GerversSofa.φ)
+        (Real.pi / 2 + GerversSofa.θ)] with u hu
+      have h := (hs1 (u - Real.pi / 2) ⟨by linarith [hu.1], by linarith [hu.2]⟩).2
+      rw [show u - Real.pi / 2 + Real.pi / 2 = u from by ring] at h
+      rw [h]
+      ring
+    simp only [gerverPhaseAngles, hJ6]
+    exact Real.Angle.measure_restrict_image_congr_add measurableSet_Ioc
+      (fun T hT hTsub ↦ hdC.angleImage_eq_setLIntegral hturn hsubC hT (hTsub.trans hsub))
+      (fun T hT hTsub ↦ hdD.angleImage_eq_setLIntegral hturn hsubD hT (hTsub.trans hsubD'))
+      hcorner (hdD.1.aemeasurable.mono_measure (Measure.restrict_mono hsubD' le_rfl))
+      hf0 hg0 hsum
+  · have hsub : Set.Ioc (Real.pi / 2 + GerversSofa.θ) (Real.pi - GerversSofa.φ) ⊆
+        Set.Ioc (Real.pi / 2) Real.pi :=
+      fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+    have hsubI : Set.Ioc (Real.pi / 2 + GerversSofa.θ) (Real.pi - GerversSofa.φ) ⊆
+        Set.Icc (0 : ℝ) Real.pi :=
+      fun x hx ↦ ⟨by linarith [hx.1], by linarith [hx.2]⟩
+    have hU : gerverPhaseAngles 7 ∪ gerverPhaseAngles 8 =
+        (fun s : ℝ ↦ (s : Real.Angle)) ''
+          Set.Ioc (Real.pi / 2 + GerversSofa.θ) (Real.pi - GerversSofa.φ) := by
+      simp only [gerverPhaseAngles, hJ7, hJ8, ← Set.image_union,
+        Set.Ioc_union_Ioc_eq_Ioc (by linarith : Real.pi / 2 + GerversSofa.θ ≤
+          Real.pi - GerversSofa.θ) (by linarith : Real.pi - GerversSofa.θ ≤
+          Real.pi - GerversSofa.φ)]
+    rw [hU]
+    refine Real.Angle.measure_restrict_image_congr_of_ae_eq measurableSet_Ioc
+      (fun T hT hTsub ↦ hdC.angleImage_eq_setLIntegral hturn hsubC hT (hTsub.trans hsub))
+      (fun T hT hTsub ↦ capCornerAngleMeasure_angleImage_eq_setLIntegral K hT
+        (hTsub.trans hsubI)) ?_
+    filter_upwards [ae_restrict_Ioc_mem_Ioo_union_Ioo (μ := volume)
+      (Real.pi / 2 + GerversSofa.θ) (Real.pi - GerversSofa.θ)
+      (Real.pi - GerversSofa.φ)] with u hu
+    have hmem : u ∈ Set.Ioo (Real.pi / 2) Real.pi := by
+      rcases hu with hu | hu
+      · exact ⟨by linarith [hu.1], by linarith [hu.2]⟩
+      · exact ⟨by linarith [hu.1], by linarith [hu.2]⟩
+    rw [capCornerDensity_eq_neg_velocity_normal K hK hmem]
+    rcases hu with hu | hu
+    · have h := (hs2 (u - Real.pi / 2) ⟨by linarith [hu.1], by linarith [hu.2]⟩).2
+      rw [show u - Real.pi / 2 + Real.pi / 2 = u from by ring] at h
+      rw [h]
+    · have h := (hs3 (u - Real.pi / 2) ⟨by linarith [hu.1], by linarith [hu.2]⟩).2
+      rw [show u - Real.pi / 2 + Real.pi / 2 = u from by ring] at h
+      rw [h]
+  · have hsub : Set.Ioc (Real.pi - GerversSofa.φ) Real.pi ⊆
+        Set.Ioc (Real.pi / 2) Real.pi := fun x hx ↦ ⟨by linarith [hx.1], hx.2⟩
+    simp only [gerverPhaseAngles, hJ9]
+    rw [Measure.restrict_eq_zero,
+      hdC.angleImage_eq_setLIntegral hturn hsubC measurableSet_Ioc hsub]
+    refine (lintegral_congr_ae ?_).trans lintegral_zero
+    filter_upwards [ae_restrict_Ioc_mem_Ioo (μ := volume) (Real.pi - GerversSofa.φ)
+      Real.pi] with u hu
+    have h := (hs4 (u - Real.pi / 2) ⟨by linarith [hu.1], by linarith [hu.2]⟩).2
+    rw [show u - Real.pi / 2 + Real.pi / 2 = u from by ring] at h
+    rw [h, ENNReal.ofReal_zero]
+
+theorem gerver_phaseMeasures (K : SpecialCapSpace) (B D : ConvexBody Point)
+    (hK : (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2))
+    (hB : (B : Set Point) = (canonicalTailSets K).1)
+    (hD : (D : Set Point) = (canonicalTailSets K).2) :
+    GerverPhaseMeasureIdentities K B D := by
+  obtain ⟨h0, h1, h2, h3⟩ := gerver_phaseMeasures_first_half K B D hK hB hD
+  obtain ⟨h4, h5, h6, h7⟩ := gerver_phaseMeasures_second_half K B D hK hB hD
+  exact ⟨h0, h1, h2, h3, h4, h5, h6, h7⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Integrating the variation of `𝒬` over the Gerver phases
+
+The ten phase windows `gerverPhaseAngles j` partition the angular window `[0, π]` except for the
+single angle `π/2`, and six of them cover the domain of the inner-corner measure except for two
+angles.  Since the cap surface measure integrates a continuous integrand and the corner measure
+has no atoms, `qVariationIntegral_eq_phaseSum` rewrites the four integrals of the variation
+formula as the eight contributions of the source table (`gerverPhaseContribution`), grouped so
+that `gerver_phaseMeasures` applies to each of them.
+
+`gerverPhaseContribution_nonpos` then signs each contribution.  Six of the eight vanish or cancel
+outright against the corner measure; on the two active right windows and the two active left
+windows the phase identities turn the remaining cap measure into the tail measure, and the
+resulting integrand is the difference between the competitor's support sum and the base support
+sum, which is `≤ 1 - 1 = 0` by the cap-tail constraints.
+-/
+
+/-! ### The angular phase windows are measurable -/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- Each of the ten angular phase windows of Gerver's cap is a Borel set. -/
+theorem measurableSet_gerverPhaseAngles (j : Fin 10) : MeasurableSet (gerverPhaseAngles j) := by
+  have hmono := gerverStageTimes_strictMono
+  have h01 := hmono (show (0 : Fin 6) < 1 by decide)
+  have h12 := hmono (show (1 : Fin 6) < 2 by decide)
+  have h23 := hmono (show (2 : Fin 6) < 3 by decide)
+  have h34 := hmono (show (3 : Fin 6) < 4 by decide)
+  have h45 := hmono (show (4 : Fin 6) < 5 by decide)
+  simp only [gerverStageTimes, Matrix.cons_val] at h01 h12 h23 h34 h45
+  have hpi : (3 : ℝ) < Real.pi := Real.pi_gt_three
+  have hturn : Real.pi ≤ -1 + 2 * Real.pi := by linarith
+  simp only [gerverPhaseAngles, gerverPhaseIntervals_explicit]
+  fin_cases j <;>
+    refine Real.Angle.measurableSet_image_of_subset_Ioc hturn
+      (by first | exact measurableSet_Ico | exact measurableSet_Ioc) ?_ <;>
+    exact fun x hx ↦ ⟨by linarith [hx.1, hx.2], by linarith [hx.1, hx.2]⟩
+
+/-! ### The phase windows partition the upper half-circle -/
+
+private theorem injOn_angleCoe_Icc_zero_pi :
+    Set.InjOn (fun t : ℝ ↦ (t : Real.Angle)) (Set.Icc 0 Real.pi) := by
+  have hsub : Set.Icc (0 : ℝ) Real.pi ⊆ Set.Ioc (-1 : ℝ) Real.pi :=
+    fun x hx ↦ ⟨by linarith [hx.1, Real.pi_pos], hx.2⟩
+  exact (Real.Angle.injOn_coe_Ioc (by linarith [Real.pi_gt_three])).mono hsub
+
+private theorem iUnion_gerverPhaseIntervals :
+    (⋃ j, gerverPhaseIntervals j) =
+      Set.Ico 0 (Real.pi / 2) ∪ Set.Ioc (Real.pi / 2) Real.pi := by
+  have horder := gerverStageTimes_strictMono
+  have h01 := horder (show (0 : Fin 6) < 1 by decide)
+  have h12 := horder (show (1 : Fin 6) < 2 by decide)
+  have h23 := horder (show (2 : Fin 6) < 3 by decide)
+  have h34 := horder (show (3 : Fin 6) < 4 by decide)
+  have h45 := horder (show (4 : Fin 6) < 5 by decide)
+  simp only [gerverStageTimes, Matrix.cons_val] at h01 h12 h23 h34 h45
+  simp only [gerverPhaseIntervals_explicit]
+  simp only [Set.iUnion_fin_add_one_eq_iUnion_succ, Function.comp_def,
+    Matrix.cons_val_zero, Matrix.cons_val_succ, Set.iUnion_of_empty, Set.union_empty]
+  have hlo : (((Set.Ico 0 GerversSofa.φ ∪ Set.Ico GerversSofa.φ GerversSofa.θ) ∪
+      Set.Ico GerversSofa.θ (Real.pi / 2 - GerversSofa.θ)) ∪
+      Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ)) ∪
+      Set.Ico (Real.pi / 2 - GerversSofa.φ) (Real.pi / 2) =
+      Set.Ico 0 (Real.pi / 2) := by
+    rw [Set.Ico_union_Ico_eq_Ico h01.le h12.le,
+      Set.Ico_union_Ico_eq_Ico (by linarith) h23.le,
+      Set.Ico_union_Ico_eq_Ico (by linarith) h34.le,
+      Set.Ico_union_Ico_eq_Ico (by linarith) h45.le]
+  have hhi : (((Set.Ioc (Real.pi / 2) (Real.pi / 2 + GerversSofa.φ) ∪
+      Set.Ioc (Real.pi / 2 + GerversSofa.φ) (Real.pi / 2 + GerversSofa.θ)) ∪
+      Set.Ioc (Real.pi / 2 + GerversSofa.θ) (Real.pi - GerversSofa.θ)) ∪
+      Set.Ioc (Real.pi - GerversSofa.θ) (Real.pi - GerversSofa.φ)) ∪
+      Set.Ioc (Real.pi - GerversSofa.φ) Real.pi = Set.Ioc (Real.pi / 2) Real.pi := by
+    rw [Set.Ioc_union_Ioc_eq_Ioc (by linarith) (by linarith),
+      Set.Ioc_union_Ioc_eq_Ioc (by linarith) (by linarith),
+      Set.Ioc_union_Ioc_eq_Ioc (by linarith) (by linarith),
+      Set.Ioc_union_Ioc_eq_Ioc (by linarith) (by linarith)]
+  rw [← hlo, ← hhi]
+  ac_rfl
+
+private theorem pairwise_disjoint_gerverPhaseIntervals :
+    Pairwise (fun i j ↦ Disjoint (gerverPhaseIntervals i) (gerverPhaseIntervals j)) := by
+  have horder := gerverStageTimes_strictMono
+  have h01 := horder (show (0 : Fin 6) < 1 by decide)
+  have h12 := horder (show (1 : Fin 6) < 2 by decide)
+  have h23 := horder (show (2 : Fin 6) < 3 by decide)
+  have h34 := horder (show (3 : Fin 6) < 4 by decide)
+  have h45 := horder (show (4 : Fin 6) < 5 by decide)
+  simp only [gerverStageTimes, Matrix.cons_val] at h01 h12 h23 h34 h45
+  intro i j hij
+  apply Set.disjoint_left.mpr
+  intro t hi hj
+  rw [gerverPhaseIntervals_explicit] at hi hj
+  fin_cases i <;> fin_cases j <;> simp_all only [ne_eq, not_true_eq_false]
+  all_goals linarith [hi.1, hi.2, hj.1, hj.2]
+
+private theorem gerverPhaseIntervals_subset_Icc (k : Fin 10) :
+    gerverPhaseIntervals k ⊆ Set.Icc 0 Real.pi := by
+  intro z hz
+  have hz' := Set.mem_iUnion.mpr ⟨k, hz⟩
+  rw [iUnion_gerverPhaseIntervals] at hz'
+  rcases hz' with h | h
+  · exact ⟨h.1, by linarith [h.2, Real.pi_pos]⟩
+  · exact ⟨by linarith [h.1, Real.pi_pos], h.2⟩
+
+private theorem iUnion_gerverPhaseAngles_union_top :
+    (⋃ j, gerverPhaseAngles j) ∪ {((Real.pi / 2 : ℝ) : Real.Angle)} =
+      (fun t : ℝ ↦ (t : Real.Angle)) '' Set.Icc 0 Real.pi := by
+  have hreal : (⋃ j, gerverPhaseIntervals j) ∪ {Real.pi / 2} = Set.Icc 0 Real.pi := by
+    rw [iUnion_gerverPhaseIntervals]
+    ext t
+    simp only [Set.mem_union, Set.mem_Ico, Set.mem_Ioc, Set.mem_singleton_iff, Set.mem_Icc]
+    constructor
+    · rintro ((h | h) | rfl) <;> constructor <;> linarith [Real.pi_pos]
+    · intro ht
+      rcases lt_trichotomy t (Real.pi / 2) with h | h | h
+      · exact Or.inl (Or.inl ⟨ht.1, h⟩)
+      · exact Or.inr h
+      · exact Or.inl (Or.inr ⟨h, ht.2⟩)
+  rw [← hreal, Set.image_union, Set.image_iUnion]
+  simp only [gerverPhaseAngles, Set.image_singleton]
+
+private theorem pairwise_disjoint_gerverPhaseAngles :
+    Pairwise (fun i j ↦ Disjoint (gerverPhaseAngles i) (gerverPhaseAngles j)) := by
+  intro i j hij
+  apply Set.disjoint_left.mpr
+  rintro t ⟨x, hx, rfl⟩ ⟨y, hy, heq⟩
+  have hxy := injOn_angleCoe_Icc_zero_pi (gerverPhaseIntervals_subset_Icc j hy)
+    (gerverPhaseIntervals_subset_Icc i hx) heq
+  subst y
+  exact Set.disjoint_left.mp (pairwise_disjoint_gerverPhaseIntervals hij) hx hy
+
+/-- The two active right-tail phases make up the angular window `[π/2 - θ, π/2)`. -/
+theorem gerverPhaseAngles_three_union_four :
+    gerverPhaseAngles 3 ∪ gerverPhaseAngles 4 =
+      (fun t : ℝ ↦ (t : Real.Angle)) ''
+        Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2) := by
+  have horder := gerverStageTimes_strictMono
+  have h34 := horder (show (3 : Fin 6) < 4 by decide)
+  have h45 := horder (show (4 : Fin 6) < 5 by decide)
+  simp only [gerverStageTimes, Matrix.cons_val] at h34 h45
+  simp only [gerverPhaseAngles, ← Set.image_union, gerverPhaseIntervals_explicit,
+    Matrix.cons_val]
+  rw [Set.Ico_union_Ico_eq_Ico h34.le h45.le]
+
+/-- The two active left-tail phases make up the angular window `(π/2, π/2 + θ]`. -/
+theorem gerverPhaseAngles_five_union_six :
+    gerverPhaseAngles 5 ∪ gerverPhaseAngles 6 =
+      (fun t : ℝ ↦ (t : Real.Angle)) ''
+        Set.Ioc (Real.pi / 2) (Real.pi / 2 + GerversSofa.θ) := by
+  have horder := gerverStageTimes_strictMono
+  have h01 := horder (show (0 : Fin 6) < 1 by decide)
+  have h12 := horder (show (1 : Fin 6) < 2 by decide)
+  simp only [gerverStageTimes, Matrix.cons_val] at h01 h12
+  simp only [gerverPhaseAngles, ← Set.image_union, gerverPhaseIntervals_explicit,
+    Matrix.cons_val]
+  rw [Set.Ioc_union_Ioc_eq_Ioc (by linarith) (by linarith)]
+
+/-! ### Splitting the four variation integrals over the phase windows -/
+
+private theorem setIntegral_union_eq_left_of_measure_eq_zero (μ : Measure Real.Angle)
+    (f : Real.Angle → ℝ) {S T : Set Real.Angle} (hT : μ T = 0) :
+    (∫ t in S ∪ T, f t ∂μ) = ∫ t in S, f t ∂μ := by
+  apply integral_union_eq_left_of_ae
+  rw [Measure.restrict_eq_zero.mpr hT]
+  simp
+
+private theorem setIntegral_angleImage_Icc_eq_sum (μ : Measure Real.Angle)
+    (f : Real.Angle → ℝ) (hf : Integrable f μ)
+    (htop : f ((Real.pi / 2 : ℝ) : Real.Angle) = 0) :
+    (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Icc 0 Real.pi, f t ∂μ) =
+      ∑ j, ∫ t in gerverPhaseAngles j, f t ∂μ := by
+  rw [← iUnion_gerverPhaseAngles_union_top,
+    integral_union_eq_left_of_forall (measurableSet_singleton _) (by
+      intro t ht
+      obtain rfl := Set.mem_singleton_iff.mp ht
+      exact htop)]
+  exact integral_iUnion_fintype measurableSet_gerverPhaseAngles
+    pairwise_disjoint_gerverPhaseAngles fun _ ↦ hf.restrict
+
+private theorem setIntegral_right_window_eq_add (μ : Measure Real.Angle) (f : Real.Angle → ℝ)
+    (hf : Integrable f μ)
+    (hinactive : μ ((fun s : ℝ ↦ (s : Real.Angle)) ''
+      Set.Ioo paperGerverConstants.2.1 (Real.pi / 2 - GerversSofa.θ)) = 0) :
+    (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) ''
+      Set.Ioo paperGerverConstants.2.1 (Real.pi / 2), f t ∂μ) =
+      (∫ t in gerverPhaseAngles 3, f t ∂μ) + (∫ t in gerverPhaseAngles 4, f t ∂μ) := by
+  have horder := gerverStageTimes_strictMono
+  have h13 := horder (show (1 : Fin 6) < 3 by decide)
+  have h35 := horder (show (3 : Fin 6) < 5 by decide)
+  simp only [gerverStageTimes, Matrix.cons_val] at h13 h35
+  have hreal : Set.Ioo paperGerverConstants.2.1 (Real.pi / 2) =
+      Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2) ∪
+        Set.Ioo paperGerverConstants.2.1 (Real.pi / 2 - GerversSofa.θ) := by
+    ext t
+    change (_ < t ∧ t < _) ↔ (_ ≤ t ∧ t < _) ∨ (_ < t ∧ t < _)
+    change (GerversSofa.φ < t ∧ t < Real.pi / 2) ↔ _
+    constructor
+    · intro ht
+      by_cases h : Real.pi / 2 - GerversSofa.θ ≤ t
+      · exact Or.inl ⟨h, ht.2⟩
+      · exact Or.inr ⟨ht.1, lt_of_not_ge h⟩
+    · rintro (h | h) <;> (try simp only [paperGerverConstants] at h) <;>
+        constructor <;> linarith [h.1, h.2]
+  rw [hreal, Set.image_union, ← gerverPhaseAngles_three_union_four,
+    setIntegral_union_eq_left_of_measure_eq_zero μ f hinactive]
+  exact setIntegral_union (pairwise_disjoint_gerverPhaseAngles (by decide : (3 : Fin 10) ≠ 4))
+    (measurableSet_gerverPhaseAngles 4) hf.restrict hf.restrict
+
+private theorem setIntegral_left_window_eq_add (μ : Measure Real.Angle) (f : Real.Angle → ℝ)
+    (hf : Integrable f μ)
+    (hinactive : μ ((fun s : ℝ ↦ (s : Real.Angle)) ''
+      Set.Ioo (Real.pi / 2 + GerversSofa.θ)
+        (Real.pi / 2 + paperGerverConstants.2.2)) = 0) :
+    (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) ''
+      Set.Ioo (Real.pi / 2) (Real.pi / 2 + paperGerverConstants.2.2), f t ∂μ) =
+      (∫ t in gerverPhaseAngles 5, f t ∂μ) + (∫ t in gerverPhaseAngles 6, f t ∂μ) := by
+  have horder := gerverStageTimes_strictMono
+  have h02 := horder (show (0 : Fin 6) < 2 by decide)
+  have h24 := horder (show (2 : Fin 6) < 4 by decide)
+  simp only [gerverStageTimes, Matrix.cons_val] at h02 h24
+  have hreal : Set.Ioo (Real.pi / 2) (Real.pi / 2 + paperGerverConstants.2.2) =
+      Set.Ioc (Real.pi / 2) (Real.pi / 2 + GerversSofa.θ) ∪
+        Set.Ioo (Real.pi / 2 + GerversSofa.θ)
+          (Real.pi / 2 + paperGerverConstants.2.2) := by
+    ext t
+    change (_ < t ∧ t < _) ↔ (_ < t ∧ t ≤ _) ∨ (_ < t ∧ t < _)
+    change (Real.pi / 2 < t ∧ t < Real.pi / 2 + (Real.pi / 2 - GerversSofa.φ)) ↔ _
+    constructor
+    · intro ht
+      by_cases h : t ≤ Real.pi / 2 + GerversSofa.θ
+      · exact Or.inl ⟨ht.1, h⟩
+      · exact Or.inr ⟨lt_of_not_ge h, ht.2⟩
+    · rintro (h | h) <;> (try simp only [paperGerverConstants] at h) <;>
+        constructor <;> linarith [h.1, h.2]
+  rw [hreal, Set.image_union, ← gerverPhaseAngles_five_union_six,
+    setIntegral_union_eq_left_of_measure_eq_zero μ f hinactive]
+  exact setIntegral_union (pairwise_disjoint_gerverPhaseAngles (by decide : (5 : Fin 10) ≠ 6))
+    (measurableSet_gerverPhaseAngles 6) hf.restrict hf.restrict
+
+/-- The six phases on which the inner-corner measure is active. -/
+private def cornerPhaseIndex : Fin 6 → Fin 10 := ![1, 2, 3, 6, 7, 8]
+
+private theorem iUnion_gerverPhaseIntervals_cornerPhaseIndex :
+    (⋃ i, gerverPhaseIntervals (cornerPhaseIndex i)) =
+      Set.Ico paperGerverConstants.2.1 paperGerverConstants.2.2 ∪
+        Set.Ioc (Real.pi / 2 + paperGerverConstants.2.1)
+          (Real.pi / 2 + paperGerverConstants.2.2) := by
+  have horder := gerverStageTimes_strictMono
+  have h12 := horder (show (1 : Fin 6) < 2 by decide)
+  have h23 := horder (show (2 : Fin 6) < 3 by decide)
+  have h34 := horder (show (3 : Fin 6) < 4 by decide)
+  simp only [gerverStageTimes, Matrix.cons_val] at h12 h23 h34
+  simp only [Set.iUnion_fin_add_one_eq_iUnion_succ, Function.comp_def,
+    cornerPhaseIndex, Matrix.cons_val_zero, Matrix.cons_val_succ,
+    Set.iUnion_of_empty, Set.union_empty, gerverPhaseIntervals_explicit]
+  simp only [Matrix.cons_val, paperGerverConstants]
+  have hlo : (Set.Ico GerversSofa.φ GerversSofa.θ ∪
+      Set.Ico GerversSofa.θ (Real.pi / 2 - GerversSofa.θ)) ∪
+      Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ) =
+      Set.Ico GerversSofa.φ (Real.pi / 2 - GerversSofa.φ) := by
+    rw [Set.Ico_union_Ico_eq_Ico h12.le h23.le, Set.Ico_union_Ico_eq_Ico (by linarith) h34.le]
+  have hhi : (Set.Ioc (Real.pi / 2 + GerversSofa.φ) (Real.pi / 2 + GerversSofa.θ) ∪
+      Set.Ioc (Real.pi / 2 + GerversSofa.θ) (Real.pi - GerversSofa.θ)) ∪
+      Set.Ioc (Real.pi - GerversSofa.θ) (Real.pi - GerversSofa.φ) =
+      Set.Ioc (Real.pi / 2 + GerversSofa.φ) (Real.pi - GerversSofa.φ) := by
+    rw [Set.Ioc_union_Ioc_eq_Ioc (by linarith) (by linarith),
+      Set.Ioc_union_Ioc_eq_Ioc (by linarith) (by linarith)]
+  have hend : Real.pi / 2 + (Real.pi / 2 - GerversSofa.φ) = Real.pi - GerversSofa.φ := by ring
+  rw [hend, ← hlo, ← hhi]
+  ac_rfl
+
+private theorem iUnion_gerverPhaseAngles_cornerPhaseIndex_union_ends :
+    (⋃ i, gerverPhaseAngles (cornerPhaseIndex i)) ∪
+      {((paperGerverConstants.2.2 : ℝ) : Real.Angle),
+        ((Real.pi / 2 + paperGerverConstants.2.1 : ℝ) : Real.Angle)} =
+      (fun t : ℝ ↦ (t : Real.Angle)) ''
+        (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2 ∪
+          Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+            (Real.pi / 2 + paperGerverConstants.2.2)) := by
+  have horder := gerverStageTimes_strictMono
+  have h14 := horder (show (1 : Fin 6) < 4 by decide)
+  change paperGerverConstants.2.1 < paperGerverConstants.2.2 at h14
+  have hreal : (⋃ i, gerverPhaseIntervals (cornerPhaseIndex i)) ∪
+      {paperGerverConstants.2.2, Real.pi / 2 + paperGerverConstants.2.1} =
+      Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2 ∪
+        Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+          (Real.pi / 2 + paperGerverConstants.2.2) := by
+    rw [iUnion_gerverPhaseIntervals_cornerPhaseIndex]
+    ext t
+    simp only [Set.mem_union, Set.mem_Ico, Set.mem_Ioc, Set.mem_insert_iff,
+      Set.mem_singleton_iff, Set.mem_Icc]
+    constructor
+    · rintro ((h | h) | (rfl | rfl))
+      · exact Or.inl ⟨h.1, h.2.le⟩
+      · exact Or.inr ⟨h.1.le, h.2⟩
+      · exact Or.inl ⟨h14.le, le_rfl⟩
+      · exact Or.inr ⟨le_rfl, by linarith⟩
+    · rintro (h | h)
+      · rcases lt_or_eq_of_le h.2 with hlt | heq
+        · exact Or.inl (Or.inl ⟨h.1, hlt⟩)
+        · exact Or.inr (Or.inl heq)
+      · rcases lt_or_eq_of_le h.1 with hlt | heq
+        · exact Or.inl (Or.inr ⟨hlt, h.2⟩)
+        · exact Or.inr (Or.inr heq.symm)
+  rw [← hreal, Set.image_union, Set.image_iUnion]
+  simp only [gerverPhaseAngles, Set.image_pair]
+
+private theorem setIntegral_corner_window_eq_sum (μ : Measure Real.Angle)
+    [NullSingletonClass μ] (f : Real.Angle → ℝ) (hf : Integrable f μ) :
+    (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) ''
+      (Set.Icc paperGerverConstants.2.1 paperGerverConstants.2.2 ∪
+        Set.Icc (Real.pi / 2 + paperGerverConstants.2.1)
+          (Real.pi / 2 + paperGerverConstants.2.2)), f t ∂μ) =
+      ∑ i, ∫ t in gerverPhaseAngles (cornerPhaseIndex i), f t ∂μ := by
+  rw [← iUnion_gerverPhaseAngles_cornerPhaseIndex_union_ends,
+    setIntegral_union_eq_left_of_measure_eq_zero μ f
+      (((Set.finite_singleton _).insert _).measure_zero μ)]
+  have hinj : Function.Injective cornerPhaseIndex := by
+    intro i j h
+    fin_cases i <;> fin_cases j <;> simp_all [cornerPhaseIndex]
+  exact integral_iUnion_fintype (fun i ↦ measurableSet_gerverPhaseAngles (cornerPhaseIndex i))
+    (fun i j hij ↦ pairwise_disjoint_gerverPhaseAngles fun h ↦ hij (hinj h)) fun _ ↦ hf.restrict
+
+/-! ### The eight phase contributions of the variation integral -/
+
+/-- The eight contributions of the source table of the variation of `𝒬`: on each group of phases,
+the cap surface integral of the support difference, minus the active inner-corner integral, plus
+the active tail integral of the tail support difference. -/
+def gerverPhaseContribution (X Y : CapTailSpace) : Fin 8 → ℝ :=
+  let f := fun t ↦ supportValue Y.cap.val.val t - supportValue X.cap.val.val t
+  let g := fun t ↦ (oppositeSurfaceData Y.rightBody).2 t -
+    (oppositeSurfaceData X.rightBody).2 t
+  let h := fun t ↦ (oppositeSurfaceData Y.leftBody).2 t -
+    (oppositeSurfaceData X.leftBody).2 t
+  let a := fun S ↦ ∫ t in S, f t ∂surfaceAreaMeasure X.cap.val.val
+  let c := fun S ↦ ∫ t in S, f t ∂capCornerAngleMeasure X.cap
+  let b := fun S ↦ ∫ t in S, g t ∂(oppositeSurfaceData X.rightBody).1
+  let d := fun S ↦ ∫ t in S, h t ∂(oppositeSurfaceData X.leftBody).1
+  ![a (gerverPhaseAngles 0),
+    a (gerverPhaseAngles 1 ∪ gerverPhaseAngles 2) -
+      c (gerverPhaseAngles 1 ∪ gerverPhaseAngles 2),
+    a (gerverPhaseAngles 3) - c (gerverPhaseAngles 3) + b (gerverPhaseAngles 3),
+    a (gerverPhaseAngles 4) + b (gerverPhaseAngles 4),
+    a (gerverPhaseAngles 5) + d (gerverPhaseAngles 5),
+    a (gerverPhaseAngles 6) - c (gerverPhaseAngles 6) + d (gerverPhaseAngles 6),
+    a (gerverPhaseAngles 7 ∪ gerverPhaseAngles 8) -
+      c (gerverPhaseAngles 7 ∪ gerverPhaseAngles 8),
+    a (gerverPhaseAngles 9)]
+
+private theorem setIntegral_add_of_restrict_eq_add {S : Set Real.Angle}
+    {μ ν κ : Measure Real.Angle} {f g : Real.Angle → ℝ}
+    (h : μ.restrict S = (ν + κ).restrict S)
+    (hfν : Integrable f ν) (hfκ : Integrable f κ) (hg : Integrable g ν) :
+    (∫ t in S, f t ∂μ) - (∫ t in S, f t ∂κ) + (∫ t in S, g t ∂ν) =
+      ∫ t in S, (f t + g t) ∂ν := by
+  rw [h, Measure.restrict_add, integral_add_measure hfν.restrict hfκ.restrict,
+    integral_add hfν.restrict hg.restrict]
+  ring
+
+private theorem setIntegral_add_of_restrict_eq {S : Set Real.Angle}
+    {μ ν : Measure Real.Angle} {f g : Real.Angle → ℝ}
+    (h : μ.restrict S = ν.restrict S)
+    (hf : Integrable f ν) (hg : Integrable g ν) :
+    (∫ t in S, f t ∂μ) + (∫ t in S, g t ∂ν) =
+      ∫ t in S, (f t + g t) ∂ν := by
+  rw [h, integral_add hf.restrict hg.restrict]
+
+private theorem right_support_difference_nonpos (X Y : CapTailSpace) (t : ℝ)
+    (ht : t ∈ Set.Icc paperGerverConstants.2.1 (Real.pi / 2))
+    (hX : supportValue X.cap.val.val (t : Real.Angle) +
+      (oppositeSurfaceData X.rightBody).2 (t : Real.Angle) = 1) :
+    (supportValue Y.cap.val.val (t : Real.Angle) -
+      supportValue X.cap.val.val (t : Real.Angle)) +
+      ((oppositeSurfaceData Y.rightBody).2 (t : Real.Angle) -
+        (oppositeSurfaceData X.rightBody).2 (t : Real.Angle)) ≤ 0 := by
+  have hY := Y.right_bound t ht
+  simp only [oppositeSurfaceData] at hX ⊢
+  simp only [Real.Angle.coe_add, add_comm (Real.pi : Real.Angle)] at hY
+  linarith
+
+private theorem left_support_difference_nonpos (X Y : CapTailSpace) (t : ℝ)
+    (ht : t ∈ Set.Icc (0 : ℝ) paperGerverConstants.2.2)
+    (hX : supportValue X.cap.val.val ((Real.pi / 2 + t : ℝ) : Real.Angle) +
+      (oppositeSurfaceData X.leftBody).2 ((Real.pi / 2 + t : ℝ) : Real.Angle) = 1) :
+    (supportValue Y.cap.val.val ((Real.pi / 2 + t : ℝ) : Real.Angle) -
+      supportValue X.cap.val.val ((Real.pi / 2 + t : ℝ) : Real.Angle)) +
+      ((oppositeSurfaceData Y.leftBody).2 ((Real.pi / 2 + t : ℝ) : Real.Angle) -
+        (oppositeSurfaceData X.leftBody).2 ((Real.pi / 2 + t : ℝ) : Real.Angle)) ≤ 0 := by
+  have hY := Y.left_bound t ht
+  have heq : (((Real.pi / 2 + t : ℝ) : Real.Angle) + (Real.pi : Real.Angle)) =
+      ((3 * Real.pi / 2 + t : ℝ) : Real.Angle) := by
+    rw [← Real.Angle.coe_add]
+    congr 1
+    ring
+  simp only [oppositeSurfaceData, heq] at hX ⊢
+  linarith
+
+private theorem exists_mem_right_tail_domain {t : Real.Angle}
+    (ht : t ∈ gerverPhaseAngles 3 ∪ gerverPhaseAngles 4) :
+    ∃ s ∈ Set.Icc paperGerverConstants.2.1 (Real.pi / 2), (s : Real.Angle) = t := by
+  have horder := gerverStageTimes_strictMono
+  have h13 := horder (show (1 : Fin 6) < 3 by decide)
+  have h35 := horder (show (3 : Fin 6) < 5 by decide)
+  simp only [gerverStageTimes, Matrix.cons_val] at h13 h35
+  rw [gerverPhaseAngles_three_union_four] at ht
+  obtain ⟨s, hs, rfl⟩ := ht
+  exact ⟨s, ⟨le_trans h13.le hs.1, hs.2.le⟩, rfl⟩
+
+private theorem exists_mem_left_tail_domain {t : Real.Angle}
+    (ht : t ∈ gerverPhaseAngles 5 ∪ gerverPhaseAngles 6) :
+    ∃ s ∈ Set.Icc (0 : ℝ) paperGerverConstants.2.2,
+      ((Real.pi / 2 + s : ℝ) : Real.Angle) = t := by
+  have horder := gerverStageTimes_strictMono
+  have h02 := horder (show (0 : Fin 6) < 2 by decide)
+  have h24 := horder (show (2 : Fin 6) < 4 by decide)
+  simp only [gerverStageTimes, Matrix.cons_val] at h02 h24
+  rw [gerverPhaseAngles_five_union_six] at ht
+  obtain ⟨s, hs, rfl⟩ := ht
+  refine ⟨s - Real.pi / 2, ⟨by linarith [hs.1], ?_⟩, ?_⟩
+  · change s - Real.pi / 2 ≤ Real.pi / 2 - GerversSofa.φ
+    linarith [hs.2]
+  · congr 1
+    ring
+
+/-- Each of the eight phase contributions is nonpositive: the phase measure identities cancel the
+cap measure against the active corner and tail measures, and the cap-tail constraints sign the
+remaining tail integrands. -/
+theorem gerverPhaseContribution_nonpos (X Y : CapTailSpace)
+    (hphase : GerverPhaseMeasureIdentities X.cap X.rightBody X.leftBody)
+    (hbaseB : ∀ t ∈ gerverPhaseAngles 3 ∪ gerverPhaseAngles 4,
+      supportValue X.cap.val.val t + (oppositeSurfaceData X.rightBody).2 t = 1)
+    (hbaseD : ∀ t ∈ gerverPhaseAngles 5 ∪ gerverPhaseAngles 6,
+      supportValue X.cap.val.val t + (oppositeSurfaceData X.leftBody).2 t = 1)
+    (i : Fin 8) : gerverPhaseContribution X Y i ≤ 0 := by
+  obtain ⟨h0, h12, h3, h4, h5, h6, h78, h9⟩ := hphase
+  have hfB : Integrable (fun t ↦ supportValue Y.cap.val.val t -
+      supportValue X.cap.val.val t) (oppositeSurfaceData X.rightBody).1 :=
+    Real.Angle.integrable_of_continuous
+      ((continuous_supportValue _).sub (continuous_supportValue _))
+  have hfD : Integrable (fun t ↦ supportValue Y.cap.val.val t -
+      supportValue X.cap.val.val t) (oppositeSurfaceData X.leftBody).1 :=
+    Real.Angle.integrable_of_continuous
+      ((continuous_supportValue _).sub (continuous_supportValue _))
+  have hfC : Integrable (fun t ↦ supportValue Y.cap.val.val t -
+      supportValue X.cap.val.val t) (capCornerAngleMeasure X.cap) :=
+    Real.Angle.integrable_of_continuous
+      ((continuous_supportValue _).sub (continuous_supportValue _))
+  have hgB : Integrable (fun t ↦ (oppositeSurfaceData Y.rightBody).2 t -
+      (oppositeSurfaceData X.rightBody).2 t) (oppositeSurfaceData X.rightBody).1 :=
+    Real.Angle.integrable_of_continuous
+      ((continuous_oppositeSurfaceData_snd _).sub (continuous_oppositeSurfaceData_snd _))
+  have hgD : Integrable (fun t ↦ (oppositeSurfaceData Y.leftBody).2 t -
+      (oppositeSurfaceData X.leftBody).2 t) (oppositeSurfaceData X.leftBody).1 :=
+    Real.Angle.integrable_of_continuous
+      ((continuous_oppositeSurfaceData_snd _).sub (continuous_oppositeSurfaceData_snd _))
+  have hB : ∀ t ∈ gerverPhaseAngles 3 ∪ gerverPhaseAngles 4,
+      (supportValue Y.cap.val.val t - supportValue X.cap.val.val t) +
+        ((oppositeSurfaceData Y.rightBody).2 t -
+          (oppositeSurfaceData X.rightBody).2 t) ≤ 0 := by
+    intro t ht
+    obtain ⟨s, hs, rfl⟩ := exists_mem_right_tail_domain ht
+    exact right_support_difference_nonpos X Y s hs (hbaseB _ ht)
+  have hD : ∀ t ∈ gerverPhaseAngles 5 ∪ gerverPhaseAngles 6,
+      (supportValue Y.cap.val.val t - supportValue X.cap.val.val t) +
+        ((oppositeSurfaceData Y.leftBody).2 t -
+          (oppositeSurfaceData X.leftBody).2 t) ≤ 0 := by
+    intro t ht
+    obtain ⟨s, hs, rfl⟩ := exists_mem_left_tail_domain ht
+    exact left_support_difference_nonpos X Y s hs (hbaseD _ ht)
+  fin_cases i
+  · simp [gerverPhaseContribution, h0]
+  · simp [gerverPhaseContribution, h12]
+  · change (_ - _ + _) ≤ 0
+    rw [setIntegral_add_of_restrict_eq_add h3 hfB hfC hgB]
+    exact setIntegral_nonpos (measurableSet_gerverPhaseAngles 3) fun t ht ↦ hB t (Or.inl ht)
+  · change (_ + _) ≤ 0
+    rw [setIntegral_add_of_restrict_eq h4 hfB hgB]
+    exact setIntegral_nonpos (measurableSet_gerverPhaseAngles 4) fun t ht ↦ hB t (Or.inr ht)
+  · change (_ + _) ≤ 0
+    rw [setIntegral_add_of_restrict_eq h5 hfD hgD]
+    exact setIntegral_nonpos (measurableSet_gerverPhaseAngles 5) fun t ht ↦ hD t (Or.inl ht)
+  · change (_ - _ + _) ≤ 0
+    rw [setIntegral_add_of_restrict_eq_add h6 hfD hfC hgD]
+    exact setIntegral_nonpos (measurableSet_gerverPhaseAngles 6) fun t ht ↦ hD t (Or.inr ht)
+  · simp [gerverPhaseContribution, h78]
+  · simp [gerverPhaseContribution, h9]
+
+/-! ### The four variation integrals as the eight phase contributions -/
+
+/-- The four integrals of the variation formula regroup into the eight contributions of the source
+table, given that the two tails carry no surface measure between their active windows. -/
+theorem qVariationIntegral_eq_phaseSum (X Y : CapTailSpace)
+    (hinactiveB : (oppositeSurfaceData X.rightBody).1 ((fun s : ℝ ↦ (s : Real.Angle)) ''
+      Set.Ioo paperGerverConstants.2.1 (Real.pi / 2 - GerversSofa.θ)) = 0)
+    (hinactiveD : (oppositeSurfaceData X.leftBody).1 ((fun s : ℝ ↦ (s : Real.Angle)) ''
+      Set.Ioo (Real.pi / 2 + GerversSofa.θ) (Real.pi / 2 + paperGerverConstants.2.2)) = 0) :
+    qVariationIntegral X Y = ∑ i, gerverPhaseContribution X Y i := by
+  have hfK : Integrable (fun t ↦ supportValue Y.cap.val.val t -
+      supportValue X.cap.val.val t) (surfaceAreaMeasure X.cap.val.val) :=
+    Real.Angle.integrable_of_continuous
+      ((continuous_supportValue _).sub (continuous_supportValue _))
+  have hfC : Integrable (fun t ↦ supportValue Y.cap.val.val t -
+      supportValue X.cap.val.val t) (capCornerAngleMeasure X.cap) :=
+    Real.Angle.integrable_of_continuous
+      ((continuous_supportValue _).sub (continuous_supportValue _))
+  have hgB : Integrable (fun t ↦ (oppositeSurfaceData Y.rightBody).2 t -
+      (oppositeSurfaceData X.rightBody).2 t) (oppositeSurfaceData X.rightBody).1 :=
+    Real.Angle.integrable_of_continuous
+      ((continuous_oppositeSurfaceData_snd _).sub (continuous_oppositeSurfaceData_snd _))
+  have hgD : Integrable (fun t ↦ (oppositeSurfaceData Y.leftBody).2 t -
+      (oppositeSurfaceData X.leftBody).2 t) (oppositeSurfaceData X.leftBody).1 :=
+    Real.Angle.integrable_of_continuous
+      ((continuous_oppositeSurfaceData_snd _).sub (continuous_oppositeSurfaceData_snd _))
+  rw [qVariationIntegral,
+    setIntegral_angleImage_Icc_eq_sum _ _ hfK (by
+      rw [Y.cap.val.property.2.2.2.1, X.cap.val.property.2.2.2.1, sub_self]),
+    setIntegral_corner_window_eq_sum _ _ hfC,
+    setIntegral_right_window_eq_add _ _ hgB hinactiveB,
+    setIntegral_left_window_eq_add _ _ hgD hinactiveD]
+  have h12 := pairwise_disjoint_gerverPhaseAngles (by decide : (1 : Fin 10) ≠ 2)
+  have h78 := pairwise_disjoint_gerverPhaseAngles (by decide : (7 : Fin 10) ≠ 8)
+  have hK12 := setIntegral_union h12 (measurableSet_gerverPhaseAngles 2)
+    hfK.restrict hfK.restrict
+  have hC12 := setIntegral_union h12 (measurableSet_gerverPhaseAngles 2)
+    hfC.restrict hfC.restrict
+  have hK78 := setIntegral_union h78 (measurableSet_gerverPhaseAngles 8)
+    hfK.restrict hfK.restrict
+  have hC78 := setIntegral_union h78 (measurableSet_gerverPhaseAngles 8)
+    hfC.restrict hfC.restrict
+  simp only [gerverPhaseContribution, Fin.sum_univ_succ, Fin.sum_univ_zero,
+    Matrix.cons_val_zero, Matrix.cons_val_succ, cornerPhaseIndex,
+    hK12, hC12, hK78, hC78]
+  ring!
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver's cap attains the upper bound `Q`
+
+`gerver_upperBoundQ_matches` evaluates the upper-bound functional `Q` at the cap of Gerver's
+sofa together with its two canonical tails, and finds the sofa area functional `A` there.
+
+The niche area is read off the four-piece counterclockwise traversal of the niche boundary
+(`gerver_niche_orientation`): additivity of the curve area functional over that traversal
+(`curveArea_concatenation`) expresses the niche area as the curve area of the middle corner
+path minus the curve areas of the two inner contact curves, because the two contact pieces are
+traversed backwards and the base segment lies on the line `y = 0`.  The tail geometry
+(`gerver_tailGeometry`) identifies the two contact curves with the two convex tail arcs, so
+their curve areas are the two `convexArcArea` terms of `Q`, and it identifies the two corners
+of `Q`'s connector segments, whose signed areas therefore vanish.
+-/
+
+/-! ## The four-piece traversal of the niche boundary -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The niche traversal is the concatenation of four pieces, each reparametrized by `[0, 1]`
+through an increasing surjection: the second contact curve reversed, the inner corner path over
+`[t₁, t₄]`, the fourth contact curve reversed, and the base segment.  The two reversed contact
+pieces enter as bounded-variation paths `y₁`, `y₃` on `[0, 1]`, described by the two affine
+parameter changes of the traversal. -/
+private theorem isPathConcatenation_gerverNicheBoundary (K : SpecialCapSpace)
+    (hcorner : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      capInnerCorner K.val t = paperGerverPath t)
+    {Γ : ContinuousBVPaths 0 4} (hΓ : Γ.val = gerverNicheBoundary)
+    {y₁ y₃ : ContinuousBVPaths 0 1}
+    (hy₁ : ∀ u : Set.Icc (0 : ℝ) 1, y₁.val u =
+      paperGerverContacts (Real.pi / 2 - (Real.pi / 2 - gerverStageTimes 3) * (u : ℝ)) 1)
+    (hy₃ : ∀ u : Set.Icc (0 : ℝ) 1, y₃.val u =
+      paperGerverContacts (gerverStageTimes 2 * (1 - (u : ℝ))) 3) :
+    IsPathConcatenation (⟨0, 4, by norm_num, Γ⟩ : RectifiablePathData)
+      ![(⟨0, 1, zero_le_one, y₁⟩ : RectifiablePathData),
+        ⟨gerverStageTimes 1, gerverStageTimes 4,
+          (gerverStageTimes_strictMono (show (1 : Fin 6) < 4 by decide)).le, capMiddleBV K⟩,
+        ⟨0, 1, zero_le_one, y₃⟩,
+        ⟨0, 1, zero_le_one, lineSegmentBVPath (paperGerverContacts 0 3)
+          (paperGerverContacts (Real.pi / 2) 1)⟩] := by
+  have h0 : gerverStageTimes 0 = 0 := gerverStageTimes_zero
+  have h5 : gerverStageTimes 5 = Real.pi / 2 := rfl
+  have h01 : gerverStageTimes 0 < gerverStageTimes 1 :=
+    gerverStageTimes_strictMono (show (0 : Fin 6) < 1 by decide)
+  have h45 : gerverStageTimes 4 < gerverStageTimes 5 :=
+    gerverStageTimes_strictMono (show (4 : Fin 6) < 5 by decide)
+  obtain ⟨ψ₂, hψ₂c, hψ₂m, hψ₂s, hψ₂v⟩ := Set.Icc.exists_affine_monotone_surjection zero_lt_one
+    (gerverStageTimes_strictMono (show (1 : Fin 6) < 4 by decide))
+  refine ⟨by norm_num, ![⟨0, by norm_num⟩, ⟨1, by norm_num⟩, ⟨2, by norm_num⟩,
+    ⟨3, by norm_num⟩, ⟨4, by norm_num⟩], ?_, rfl, rfl, ?_⟩
+  · intro i j hij
+    fin_cases i <;> fin_cases j <;>
+      first
+        | rfl
+        | (exact absurd hij (by decide))
+        | (refine Subtype.mk_le_mk.mpr ?_; norm_num)
+  · intro i
+    fin_cases i
+    -- ### `[0, 1]`: the second contact curve, reversed
+    · obtain ⟨ϕ, hϕc, hϕm, hϕs, hϕv⟩ :
+          ∃ ϕ : Set.Icc (0 : ℝ) 1 → Set.Icc (0 : ℝ) 1, Continuous ϕ ∧ Monotone ϕ ∧
+            Function.Surjective ϕ ∧ ∀ u : Set.Icc (0 : ℝ) 1, (ϕ u : ℝ) = 0 + (u : ℝ) :=
+        Set.Icc.exists_translation_surjection (by norm_num)
+      refine ⟨ϕ, id, hϕc, hϕm, hϕs, continuous_id, monotone_id, Function.surjective_id,
+        fun u ↦ ?_⟩
+      have hv := hϕv u
+      have hmem : (ϕ u : ℝ) ∈ Set.Icc (0 : ℝ) 4 :=
+        ⟨by rw [hv]; linarith [u.2.1], by rw [hv]; linarith [u.2.2]⟩
+      have harg : Real.pi / 2 - (Real.pi / 2 - gerverStageTimes 3) * (ϕ u : ℝ) =
+          Real.pi / 2 - (Real.pi / 2 - gerverStageTimes 3) * (u : ℝ) := by rw [hv]; ring
+      rw [hΓ]
+      refine (gerverNicheBoundary_of_le_one hmem (by rw [hv]; linarith [u.2.2])).trans ?_
+      rw [harg]
+      exact (hy₁ u).symm
+    -- ### `[1, 2]`: the inner corner path, forwards
+    · obtain ⟨ϕ, hϕc, hϕm, hϕs, hϕv⟩ :
+          ∃ ϕ : Set.Icc (0 : ℝ) 1 → Set.Icc (1 : ℝ) 2, Continuous ϕ ∧ Monotone ϕ ∧
+            Function.Surjective ϕ ∧ ∀ u : Set.Icc (0 : ℝ) 1, (ϕ u : ℝ) = 1 + (u : ℝ) :=
+        Set.Icc.exists_translation_surjection (by norm_num)
+      refine ⟨ϕ, ψ₂, hϕc, hϕm, hϕs, hψ₂c, hψ₂m, hψ₂s, fun u ↦ ?_⟩
+      have hv := hϕv u
+      have hmem : (ϕ u : ℝ) ∈ Set.Icc (0 : ℝ) 4 :=
+        ⟨by rw [hv]; linarith [u.2.1], by rw [hv]; linarith [u.2.2]⟩
+      have hψmem : (ψ₂ u : ℝ) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+        ⟨by linarith [(ψ₂ u).2.1], by linarith [(ψ₂ u).2.2]⟩
+      have harg : gerverStageTimes 1 +
+          (gerverStageTimes 4 - gerverStageTimes 1) * ((ϕ u : ℝ) - 1) = (ψ₂ u : ℝ) := by
+        rw [hv, hψ₂v u]; ring
+      rw [hΓ]
+      refine (gerverNicheBoundary_mid hmem (by rw [hv]; linarith [u.2.1])
+        (by rw [hv]; linarith [u.2.2])).trans ?_
+      rw [harg]
+      exact (hcorner _ hψmem).symm
+    -- ### `[2, 3]`: the fourth contact curve, reversed
+    · obtain ⟨ϕ, hϕc, hϕm, hϕs, hϕv⟩ :
+          ∃ ϕ : Set.Icc (0 : ℝ) 1 → Set.Icc (2 : ℝ) 3, Continuous ϕ ∧ Monotone ϕ ∧
+            Function.Surjective ϕ ∧ ∀ u : Set.Icc (0 : ℝ) 1, (ϕ u : ℝ) = 2 + (u : ℝ) :=
+        Set.Icc.exists_translation_surjection (by norm_num)
+      refine ⟨ϕ, id, hϕc, hϕm, hϕs, continuous_id, monotone_id, Function.surjective_id,
+        fun u ↦ ?_⟩
+      have hv := hϕv u
+      have hmem : (ϕ u : ℝ) ∈ Set.Icc (0 : ℝ) 4 :=
+        ⟨by rw [hv]; linarith [u.2.1], by rw [hv]; linarith [u.2.2]⟩
+      have harg : gerverStageTimes 2 * (3 - (ϕ u : ℝ)) =
+          gerverStageTimes 2 * (1 - (u : ℝ)) := by rw [hv]; ring
+      rw [hΓ]
+      refine (gerverNicheBoundary_third hmem (by rw [hv]; linarith [u.2.1])
+        (by rw [hv]; linarith [u.2.2])).trans ?_
+      rw [harg]
+      exact (hy₃ u).symm
+    -- ### `[3, 4]`: the base segment
+    · obtain ⟨ϕ, hϕc, hϕm, hϕs, hϕv⟩ :
+          ∃ ϕ : Set.Icc (0 : ℝ) 1 → Set.Icc (3 : ℝ) 4, Continuous ϕ ∧ Monotone ϕ ∧
+            Function.Surjective ϕ ∧ ∀ u : Set.Icc (0 : ℝ) 1, (ϕ u : ℝ) = 3 + (u : ℝ) :=
+        Set.Icc.exists_translation_surjection (by norm_num)
+      refine ⟨ϕ, id, hϕc, hϕm, hϕs, continuous_id, monotone_id, Function.surjective_id,
+        fun u ↦ ?_⟩
+      have hv := hϕv u
+      have hmem : (ϕ u : ℝ) ∈ Set.Icc (0 : ℝ) 4 :=
+        ⟨by rw [hv]; linarith [u.2.1], by rw [hv]; linarith [u.2.2]⟩
+      rw [hΓ]
+      refine (gerverNicheBoundary_base hmem (by rw [hv]; linarith [u.2.1])).trans ?_
+      change _ = (lineSegmentBVPath (paperGerverContacts 0 3)
+        (paperGerverContacts (Real.pi / 2) 1)).val u
+      rw [lineSegmentBVPath_apply, hv, show (4 : ℝ) - (3 + (u : ℝ)) = 1 - (u : ℝ) from by ring,
+        show (3 : ℝ) + (u : ℝ) - 3 = (u : ℝ) from by ring]
+
+/-- Additivity of the curve area functional over the four-piece traversal of the niche
+boundary.  The base segment contributes nothing because both of its endpoints lie on the line
+`y = 0`, and the two contact pieces are traversed backwards, so their reparametrizations
+contribute the negatives of their curve areas. -/
+private theorem curveAreaFunctional_gerverNicheBoundary (K : SpecialCapSpace)
+    (hcorner : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      capInnerCorner K.val t = paperGerverPath t)
+    {Γ : ContinuousBVPaths 0 4} (hΓ : Γ.val = gerverNicheBoundary) :
+    curveAreaFunctional Γ = curveAreaFunctional (capMiddleBV K) -
+      curveAreaFunctional gerverRightContactBV - curveAreaFunctional gerverLeftContactBV := by
+  have h0 : gerverStageTimes 0 = 0 := gerverStageTimes_zero
+  have h5 : gerverStageTimes 5 = Real.pi / 2 := rfl
+  have h35 : gerverStageTimes 3 < gerverStageTimes 5 :=
+    gerverStageTimes_strictMono (show (3 : Fin 6) < 5 by decide)
+  have h02 : gerverStageTimes 0 < gerverStageTimes 2 :=
+    gerverStageTimes_strictMono (show (0 : Fin 6) < 2 by decide)
+  obtain ⟨φ₁, hφ₁c, hφ₁a, hφ₁s, hφ₁v⟩ :=
+    Set.Icc.exists_affine_antitone_surjection zero_lt_one h35
+  obtain ⟨φ₃, hφ₃c, hφ₃a, hφ₃s, hφ₃v⟩ :=
+    Set.Icc.exists_affine_antitone_surjection zero_lt_one h02
+  obtain ⟨y₁, hy₁val, -, hy₁area⟩ := curveArea_reparametrization.1 _ _ 0 1 h35 zero_lt_one
+    gerverRightContactBV φ₁ hφ₁c hφ₁s (Or.inr hφ₁a)
+  obtain ⟨y₃, hy₃val, -, hy₃area⟩ := curveArea_reparametrization.1 _ _ 0 1 h02 zero_lt_one
+    gerverLeftContactBV φ₃ hφ₃c hφ₃s (Or.inr hφ₃a)
+  have hy₁ : ∀ u : Set.Icc (0 : ℝ) 1, y₁.val u =
+      paperGerverContacts (Real.pi / 2 - (Real.pi / 2 - gerverStageTimes 3) * (u : ℝ)) 1 := by
+    intro u
+    rw [hy₁val]
+    exact congrArg (fun r : ℝ ↦ paperGerverContacts r 1) (by rw [hφ₁v u, h5]; ring)
+  have hy₃ : ∀ u : Set.Icc (0 : ℝ) 1, y₃.val u =
+      paperGerverContacts (gerverStageTimes 2 * (1 - (u : ℝ))) 3 := by
+    intro u
+    rw [hy₃val]
+    exact congrArg (fun r : ℝ ↦ paperGerverContacts r 3) (by rw [hφ₃v u, h0]; ring)
+  have hsum : curveAreaFunctional Γ = curveAreaFunctional y₁ +
+      curveAreaFunctional (capMiddleBV K) + curveAreaFunctional y₃ +
+      curveAreaFunctional (lineSegmentBVPath (paperGerverContacts 0 3)
+        (paperGerverContacts (Real.pi / 2) 1)) := by
+    have h := curveArea_concatenation (⟨0, 4, by norm_num, Γ⟩ : RectifiablePathData) _
+      (isPathConcatenation_gerverNicheBoundary K hcorner hΓ hy₁ hy₃)
+    rw [Fin.sum_univ_four] at h
+    exact h
+  have hseg : curveAreaFunctional (lineSegmentBVPath (paperGerverContacts 0 3)
+      (paperGerverContacts (Real.pi / 2) 1)) = 0 := by
+    rw [curveAreaFunctional_lineSegmentBVPath, segmentArea, planeCrossProduct,
+      gerver_niche_piece_endpoints.2.2.1, gerver_niche_piece_endpoints.2.2.2]
+    ring
+  rw [hsum, hseg, hy₁area hφ₁a, hy₃area hφ₃a]
+  ring
+
+/-! ## The niche area of the Gerver cap -/
+
+/-- The niche area of the Gerver cap, in terms of the signed curve areas of the three
+nondegenerate pieces of its boundary traversal: the middle corner path, and the two inner
+contact curves, which the traversal runs backwards. -/
+theorem gerver_capNiche_area_eq (K : SpecialCapSpace)
+    (hK : (K.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2)) :
+    ClassicalResults.area (capNiche K.val) =
+      curveAreaFunctional (capMiddleBV K) - curveAreaFunctional gerverRightContactBV -
+        curveAreaFunctional gerverLeftContactBV := by
+  obtain ⟨-, hGeq, -, -, hcapeq, -, -⟩ := gerver_capSupport_identification
+  have hcarrier : (K.val.val : Set Point) = gerverOuterCap := by rw [hK, hGeq, hcapeq]
+  obtain ⟨K₃, hK₃set, Γ, hΓ, -, -, -, harea, -, -, -⟩ := gerver_niche_orientation
+  have h3eq : K₃ = K.val := Subtype.ext (SetLike.coe_injective (hK₃set.trans hcarrier.symm))
+  subst h3eq
+  rw [← harea]
+  exact curveAreaFunctional_gerverNicheBoundary K
+    (fun _ ht ↦ capInnerCorner_eq_paperGerverPath K.val hK ht) hΓ
+
+/-! ## The value of `Q` at Gerver's cap -/
+
+theorem gerver_upperBoundQ_matches (X : CapTailSpace)
+    (hK : (X.cap.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2))
+    (hB : (X.rightBody : Set Point) = (canonicalTailSets X.cap).1)
+    (hD : (X.leftBody : Set Point) = (canonicalTailSets X.cap).2) :
+    rightAngleAreaFunctional X.cap.val = upperBoundQ X := by
+  obtain ⟨-, -, hxL, hYD, -, hparamD, hxR, hXB, -, hparamB, -, -⟩ :=
+    gerver_tailGeometry X.cap X.rightBody X.leftBody hK hB hD
+  obtain ⟨hltB, -, hinjB, himgB, hstartB, hendB⟩ := hparamB
+  obtain ⟨hltD, -, hinjD, himgD, hstartD, hendD⟩ := hparamD
+  -- ### The two tail arcs carry the curve areas of the two inner contact curves
+  have hareaB : convexArcArea X.rightBody (Real.pi + paperGerverConstants.2.1)
+      (3 * Real.pi / 2) = curveAreaFunctional gerverRightContactBV :=
+    convexArcArea_eq_curveAreaFunctional_of_injOn gerverRightContactBV hltB
+      gerverRightContactBV_apply hinjB himgB hstartB hendB
+  have hareaD : convexArcArea X.leftBody (3 * Real.pi / 2)
+      (3 * Real.pi / 2 + paperGerverConstants.2.2) = curveAreaFunctional gerverLeftContactBV :=
+    convexArcArea_eq_curveAreaFunctional_of_injOn gerverLeftContactBV hltD
+      gerverLeftContactBV_apply hinjD himgD hstartD hendD
+  -- ### Both connector segments have coincident endpoints, hence vanishing signed area
+  have hsegL : segmentArea (rightLeftTailArcs X.rightBody X.leftBody).2.endPoint
+      (distinguishedCapSides X.cap.val).2.corner = 0 := by
+    rw [hYD, hxL, segmentArea, planeCrossProduct]; ring
+  have hsegR : segmentArea (distinguishedCapSides X.cap.val).1.corner
+      (rightLeftTailArcs X.rightBody X.leftBody).1.startPoint = 0 := by
+    rw [hxR, hXB, segmentArea, planeCrossProduct]; ring
+  simp only [rightAngleAreaFunctional, capAreaFunctional, upperBoundQ]
+  rw [hareaB, hareaD, hsegL, hsegR, gerver_capNiche_area_eq X.cap hK]
+  ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The variation of `𝒬` at Gerver's cap is nonpositive
+
+At the Gerver base triple the tail geometry identifies the two tail endpoints with the two cap
+corners, so `upperBoundQ_variation` presents the derivative as `qVariationIntegral`.  Two of its
+four integration windows are larger than the two active tail windows, and the excess is null:
+on `(φᴿ, π/2 - θ)` one and the same contact point supports the right tail at both endpoint
+normals, and likewise for the left tail on `(π/2 + θ, π/2 + φᴸ)`, so
+`oppositeSurfaceData_angleImage_Ioo_eq_zero_of_mem_exposedEdge` applies.
+
+The variation integral is therefore the sum of the eight phase contributions
+(`qVariationIntegral_eq_phaseSum`), and each of them is nonpositive
+(`gerverPhaseContribution_nonpos`) because the tail support sums are exactly `1` on the two active
+windows while the competitor's are at most `1`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+theorem gerver_upperBoundQ_variation (X Y : CapTailSpace)
+    (hK : (X.cap.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2))
+    (hB : (X.rightBody : Set Point) = (canonicalTailSets X.cap).1)
+    (hD : (X.leftBody : Set Point) = (canonicalTailSets X.cap).2) :
+    convexDirectionalDerivative capTailCombination upperBoundQ X Y ≤ 0 := by
+  have horder := gerverStageTimes_strictMono
+  have h01 := horder (show (0 : Fin 6) < 1 by decide)
+  have h02 := horder (show (0 : Fin 6) < 2 by decide)
+  have h13 := horder (show (1 : Fin 6) < 3 by decide)
+  have h24 := horder (show (2 : Fin 6) < 4 by decide)
+  have hpi : (3 : ℝ) < Real.pi := Real.pi_gt_three
+  have hr : paperGerverConstants.2.1 = GerversSofa.φ := rfl
+  have ht2 : gerverStageTimes 2 = GerversSofa.θ := by simp [gerverStageTimes]
+  have ht3 : gerverStageTimes 3 = Real.pi / 2 - GerversSofa.θ := by simp [gerverStageTimes]
+  simp only [gerverStageTimes, Matrix.cons_val] at h01 h02 h13 h24
+  obtain ⟨-, -, hLcorner, hLend, hDvert, -, hRcorner, hRstart, hBvert, -, hsumD, hsumB⟩ :=
+    gerver_tailGeometry X.cap X.rightBody X.leftBody hK hB hD
+  -- ### The general variation formula applies at this base triple
+  rw [upperBoundQ_variation X Y (hRstart.trans hRcorner.symm) (hLend.trans hLcorner.symm)]
+  -- ### The right tail carries no surface measure between its two active windows
+  have hinactiveB : (oppositeSurfaceData X.rightBody).1 ((fun s : ℝ ↦ (s : Real.Angle)) ''
+      Set.Ioo paperGerverConstants.2.1 (Real.pi / 2 - GerversSofa.θ)) = 0 := by
+    refine oppositeSurfaceData_angleImage_Ioo_eq_zero_of_mem_exposedEdge X.rightBody
+      (p := paperGerverContacts (gerverStageTimes 3) 1) h13 (by rw [hr]; linarith) ?_ ?_
+    · have h : paperGerverContacts (gerverStageTimes 3) 1 =
+          (edgeVertices X.rightBody
+            ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle)).1 := hRstart.symm
+      rw [h, show ((paperGerverConstants.2.1 + Real.pi : ℝ) : Real.Angle) =
+        ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle) from by congr 1; ring]
+      exact edgeVertices_fst_mem _ _
+    · have h : paperGerverContacts (gerverStageTimes 3) 1 =
+          (edgeVertices X.rightBody
+            ((Real.pi + gerverStageTimes 3 : ℝ) : Real.Angle)).2 := by rw [hBvert]
+      rw [h, show ((Real.pi / 2 - GerversSofa.θ + Real.pi : ℝ) : Real.Angle) =
+        ((Real.pi + gerverStageTimes 3 : ℝ) : Real.Angle) from by rw [ht3]; congr 1; ring]
+      exact edgeVertices_snd_mem _ _
+  -- ### The left tail carries no surface measure between its two active windows
+  have hinactiveD : (oppositeSurfaceData X.leftBody).1 ((fun s : ℝ ↦ (s : Real.Angle)) ''
+      Set.Ioo (Real.pi / 2 + GerversSofa.θ)
+        (Real.pi / 2 + paperGerverConstants.2.2)) = 0 := by
+    have hlt : Real.pi / 2 + GerversSofa.θ < Real.pi / 2 + paperGerverConstants.2.2 := by
+      change Real.pi / 2 + GerversSofa.θ < Real.pi / 2 + (Real.pi / 2 - GerversSofa.φ)
+      linarith
+    refine oppositeSurfaceData_angleImage_Ioo_eq_zero_of_mem_exposedEdge X.leftBody
+      (p := paperGerverContacts (gerverStageTimes 2) 3) hlt (by
+        change Real.pi / 2 + (Real.pi / 2 - GerversSofa.φ) <
+          Real.pi / 2 + GerversSofa.θ + Real.pi
+        linarith) ?_ ?_
+    · have h : paperGerverContacts (gerverStageTimes 2) 3 =
+          (edgeVertices X.leftBody
+            ((3 * Real.pi / 2 + gerverStageTimes 2 : ℝ) : Real.Angle)).1 := by rw [hDvert]
+      rw [h, show ((Real.pi / 2 + GerversSofa.θ + Real.pi : ℝ) : Real.Angle) =
+        ((3 * Real.pi / 2 + gerverStageTimes 2 : ℝ) : Real.Angle) from by
+          rw [ht2]; congr 1; ring]
+      exact edgeVertices_fst_mem _ _
+    · have h : paperGerverContacts (gerverStageTimes 2) 3 =
+          (edgeVertices X.leftBody
+            ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle)).2 := hLend.symm
+      rw [h, show ((Real.pi / 2 + paperGerverConstants.2.2 + Real.pi : ℝ) : Real.Angle) =
+        ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle) from by congr 1; ring]
+      exact edgeVertices_snd_mem _ _
+  -- ### On the two active windows the base support sums are exactly one
+  have hbaseB : ∀ t ∈ gerverPhaseAngles 3 ∪ gerverPhaseAngles 4,
+      supportValue X.cap.val.val t + (oppositeSurfaceData X.rightBody).2 t = 1 := by
+    intro t ht
+    rw [gerverPhaseAngles_three_union_four] at ht
+    obtain ⟨s, hs, rfl⟩ := ht
+    have h := hsumB s (by rw [ht3]; exact ⟨hs.1, hs.2.le⟩)
+    simp only [oppositeSurfaceData]
+    rw [show ((s : Real.Angle) + ((Real.pi : ℝ) : Real.Angle)) =
+      ((Real.pi + s : ℝ) : Real.Angle) from by rw [← Real.Angle.coe_add]; congr 1; ring]
+    exact h
+  have hbaseD : ∀ t ∈ gerverPhaseAngles 5 ∪ gerverPhaseAngles 6,
+      supportValue X.cap.val.val t + (oppositeSurfaceData X.leftBody).2 t = 1 := by
+    intro t ht
+    rw [gerverPhaseAngles_five_union_six] at ht
+    obtain ⟨u, hu, rfl⟩ := ht
+    have h := hsumD (u - Real.pi / 2) (by
+      rw [gerverStageTimes_zero, ht2]
+      exact ⟨by linarith [hu.1], by linarith [hu.2]⟩)
+    rw [show ((Real.pi / 2 + (u - Real.pi / 2) : ℝ) : Real.Angle) = (u : Real.Angle) from by
+      congr 1; ring] at h
+    simp only [oppositeSurfaceData]
+    rw [show ((u : Real.Angle) + ((Real.pi : ℝ) : Real.Angle)) =
+      ((3 * Real.pi / 2 + (u - Real.pi / 2) : ℝ) : Real.Angle) from by
+      rw [← Real.Angle.coe_add]; congr 1; ring]
+    exact h
+  -- ### Each of the eight phase contributions is nonpositive
+  rw [qVariationIntegral_eq_phaseSum X Y hinactiveB hinactiveD]
+  exact Finset.sum_nonpos fun i _ ↦ gerverPhaseContribution_nonpos X Y
+    (gerver_phaseMeasures X.cap X.rightBody X.leftBody hK hB hD) hbaseB hbaseD i
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Bounds.Upper.Properties`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Bounds / Upper / Properties
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem upperBoundQ_quadratic_concave :
+    IsQuadraticFunctional capTailCombination upperBoundQ ∧
+      IsConvexFunctional capTailCombination upperBoundQ true := by
+  obtain ⟨hcomb, -⟩ := capTail_isConvexDomain
+  obtain ⟨hmidConv, hmidQuad, hrConv, hrQuad, hlConv, hlQuad⟩ := sofaMamikon_quadratic_convex
+  -- ### The three coordinate projections preserve the interpolation
+  have hcapval : ∀ t X Y, (capTailCombination t X Y).cap = specialCapCombination t X.cap Y.cap :=
+    fun t X Y ↦ Subtype.ext (Subtype.ext
+      ((hcomb t X Y).1.trans (specialCap_isConvexDomain.1 t X.cap Y.cap).symm))
+  have hcap : IsConvexLinear capTailCombination specialCapCombination
+      fun X : CapTailSpace ↦ X.cap := hcapval
+  have hright : IsConvexLinear capTailCombination convexBodyCombination
+      fun X : CapTailSpace ↦ X.rightBody := fun t X Y ↦ (hcomb t X Y).2.1
+  have hleft : IsConvexLinear capTailCombination convexBodyCombination
+      fun X : CapTailSpace ↦ X.leftBody := fun t X Y ↦ (hcomb t X Y).2.2
+  -- ### The middle equivalence exhibits the affine part of the cap term
+  have hAlin : IsConvexLinear capTailCombination realCombination
+      fun X : CapTailSpace ↦ middleMamikon X.cap + upperBoundMiddle X.cap := by
+    intro t X Y
+    have h := middleMamikon_equivalent_neg_upperBoundMiddle t X.cap Y.cap
+    simp only [sub_neg_eq_add, realCombination] at h
+    change middleMamikon (capTailCombination t X Y).cap +
+      upperBoundMiddle (capTailCombination t X Y).cap = realCombination t _ _
+    rw [hcapval t X Y]
+    simp only [realCombination]
+    linarith
+  -- ### The exact decomposition, with the cap term split as affine minus convex
+  have hQ : upperBoundQ = fun X : CapTailSpace ↦
+      ((middleMamikon X.cap + upperBoundMiddle X.cap + -middleMamikon X.cap) +
+        -rightTailMamikon X.rightBody) + -leftTailMamikon X.leftBody := by
+    funext X
+    rw [upperBoundQ_decomposition X]
+    ring
+  rw [hQ]
+  exact ⟨((hAlin.isQuadraticFunctional.add (hmidQuad.comp_isConvexLinear hcap).neg).add
+        (hrQuad.comp_isConvexLinear hright).neg).add (hlQuad.comp_isConvexLinear hleft).neg,
+    (((hAlin.isConvexFunctional true).add (hmidConv.comp_isConvexLinear hcap).neg).add
+        (hrConv.comp_isConvexLinear hright).neg).add (hlConv.comp_isConvexLinear hleft).neg⟩
+
+/-- The right tail's initial vertex, the right fan point and the right inner corner all lie on the
+right distinguished inner wall `b_K(φᴿ)`, so the two signed segment areas along it add. -/
+private theorem segmentArea_add_segmentArea_rightWall (X : CapTailSpace) :
+    segmentArea (rightLeftTailArcs X.rightBody X.leftBody).1.startPoint
+        (distinguishedCapSides X.cap.val).1.fanPoint +
+      segmentArea (distinguishedCapSides X.cap.val).1.fanPoint
+        (distinguishedCapSides X.cap.val).1.corner =
+    segmentArea (rightLeftTailArcs X.rightBody X.leftBody).1.startPoint
+      (distinguishedCapSides X.cap.val).1.corner := by
+  obtain ⟨hrIoo, -, -⟩ := paperGerverConstants_snd_mem_Ioo
+  have hcosr : 0 < Real.cos paperGerverConstants.2.1 :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos, hrIoo.1], hrIoo.2⟩
+  have hXBdef : (rightLeftTailArcs X.rightBody X.leftBody).1.startPoint =
+      (edgeVertices X.rightBody ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle)).1 := rfl
+  have hWdef : (distinguishedCapSides X.cap.val).1.fanPoint =
+      (wedgeEndpoints X.cap.val paperGerverConstants.2.1).1 := rfl
+  have hxRdef : (distinguishedCapSides X.cap.val).1.corner =
+      capInnerCorner X.cap.val paperGerverConstants.2.1 := rfl
+  have hXB : inner ℝ
+      (edgeVertices X.rightBody ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle)).1
+      (normalVector ((paperGerverConstants.2.1 : ℝ) : Real.Angle)) =
+      supportValue (X.cap.val.val : Set Point)
+        ((paperGerverConstants.2.1 : ℝ) : Real.Angle) - 1 := by
+    have hmem : inner ℝ
+        (edgeVertices X.rightBody ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle)).1
+        (normalVector ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle)) =
+        supportValue (X.rightBody : Set Point)
+          ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle) :=
+      (edgeVertices_fst_mem X.rightBody _).2
+    have hneg : normalVector ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle) =
+        -normalVector ((paperGerverConstants.2.1 : ℝ) : Real.Angle) := by
+      rw [show (Real.pi + paperGerverConstants.2.1 : ℝ) =
+        paperGerverConstants.2.1 + Real.pi from by ring]
+      exact normalVector_add_pi _
+    rw [hneg, inner_neg_right] at hmem
+    have heq := X.right_eq paperGerverConstants.2.1 (by simp)
+    linarith
+  have hW : inner ℝ (wedgeEndpoints X.cap.val paperGerverConstants.2.1).1
+      (normalVector ((paperGerverConstants.2.1 : ℝ) : Real.Angle)) =
+      supportValue (X.cap.val.val : Set Point)
+        ((paperGerverConstants.2.1 : ℝ) : Real.Angle) - 1 := by
+    change inner ℝ (((supportValue (X.cap.val.val : Set Point)
+        ((paperGerverConstants.2.1 : ℝ) : Real.Angle) - 1) /
+        Real.cos paperGerverConstants.2.1) • normalVector 0)
+        (normalVector ((paperGerverConstants.2.1 : ℝ) : Real.Angle)) = _
+    have h0 : normalVector (0 : Real.Angle) 0 = 1 := by simp [normalVector, frame]
+    have h1 : normalVector (0 : Real.Angle) 1 = 0 := by simp [normalVector, frame]
+    rw [real_inner_smul_left, inner_normalVector_real, h0, h1, one_mul, zero_mul, add_zero,
+      div_mul_cancel₀ _ hcosr.ne']
+  rw [hXBdef, hWdef, hxRdef]
+  linarith [segmentArea_sub_segmentArea_of_inner_normalVector_eq hXB hW
+    (inner_capInnerCorner_normalVector X.cap.val paperGerverConstants.2.1)]
+
+/-- The left inner corner, the left fan point and the left tail's final vertex all lie on the
+left distinguished inner wall `d_K(φᴸ)`, so the two signed segment areas along it add. -/
+private theorem segmentArea_add_segmentArea_leftWall (X : CapTailSpace) :
+    segmentArea (distinguishedCapSides X.cap.val).2.corner
+        (distinguishedCapSides X.cap.val).2.fanPoint +
+      segmentArea (distinguishedCapSides X.cap.val).2.fanPoint
+        (rightLeftTailArcs X.rightBody X.leftBody).2.endPoint =
+    segmentArea (distinguishedCapSides X.cap.val).2.corner
+      (rightLeftTailArcs X.rightBody X.leftBody).2.endPoint := by
+  obtain ⟨-, hlIoo, -⟩ := paperGerverConstants_snd_mem_Ioo
+  have hsinl : 0 < Real.sin paperGerverConstants.2.2 :=
+    Real.sin_pos_of_pos_of_lt_pi hlIoo.1 (by linarith [Real.pi_pos, hlIoo.2])
+  have hZdef : (distinguishedCapSides X.cap.val).2.fanPoint =
+      (wedgeEndpoints X.cap.val paperGerverConstants.2.2).2 := rfl
+  have hxLdef : (distinguishedCapSides X.cap.val).2.corner =
+      capInnerCorner X.cap.val paperGerverConstants.2.2 := rfl
+  have hYDdef : (rightLeftTailArcs X.rightBody X.leftBody).2.endPoint =
+      (edgeVertices X.leftBody
+        ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle)).2 := rfl
+  have hxL : inner ℝ (capInnerCorner X.cap.val paperGerverConstants.2.2)
+      (normalVector ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle)) =
+      supportValue (X.cap.val.val : Set Point)
+        ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+    rw [normalVector_add_pi_div_two_real]
+    exact inner_capInnerCorner_tangentVector X.cap.val paperGerverConstants.2.2
+  have hZ : inner ℝ (wedgeEndpoints X.cap.val paperGerverConstants.2.2).2
+      (normalVector ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle)) =
+      supportValue (X.cap.val.val : Set Point)
+        ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+    change inner ℝ (((supportValue (X.cap.val.val : Set Point)
+        ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+        Real.cos (Real.pi / 2 - paperGerverConstants.2.2)) •
+        tangentVector ((Real.pi / 2 : ℝ) : Real.Angle))
+        (normalVector ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle)) = _
+    rw [real_inner_smul_left, inner_tangentVector_normalVector_real,
+      show paperGerverConstants.2.2 + Real.pi / 2 - Real.pi / 2 = paperGerverConstants.2.2 from
+        by ring, Real.cos_pi_div_two_sub, div_mul_cancel₀ _ hsinl.ne']
+  have hYD : inner ℝ (edgeVertices X.leftBody
+      ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle)).2
+      (normalVector ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle)) =
+      supportValue (X.cap.val.val : Set Point)
+        ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+    have hmem : inner ℝ (edgeVertices X.leftBody
+        ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle)).2
+        (normalVector ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle)) =
+        supportValue (X.leftBody : Set Point)
+          ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle) :=
+      (edgeVertices_snd_mem X.leftBody _).2
+    have hneg : normalVector ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle) =
+        -normalVector ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) := by
+      rw [show (3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) =
+        paperGerverConstants.2.2 + Real.pi / 2 + Real.pi from by ring]
+      exact normalVector_add_pi _
+    rw [hneg, inner_neg_right] at hmem
+    have heq := X.left_eq paperGerverConstants.2.2 (by simp)
+    rw [show (Real.pi / 2 + paperGerverConstants.2.2 : ℝ) =
+      paperGerverConstants.2.2 + Real.pi / 2 from by ring] at heq
+    linarith
+  rw [hZdef, hxLdef, hYDdef]
+  linarith [segmentArea_sub_segmentArea_of_inner_normalVector_eq hxL hZ hYD]
+
+theorem areaFunctional_le_upperBoundQ (X : CapTailSpace)
+    (hB : (X.rightBody : Set Point) = (canonicalTailSets X.cap).1)
+    (hD : (X.leftBody : Set Point) = (canonicalTailSets X.cap).2) :
+    rightAngleAreaFunctional X.cap.val ≤ upperBoundQ X := by
+  have hsplit := (niche_three_regions_area X.cap _ _ _ rfl rfl rfl).2.2.2.2.2.2.2.2.2.2
+  have hmid := capMiddle_area_lower_bound X.cap
+  rw [Set.sdiff_sdiff] at hmid
+  obtain ⟨hrightTail, hleftTail⟩ :=
+    canonicalTail_niche_area_lower_bounds X.cap X.rightBody X.leftBody hB hD
+  have hwallR := segmentArea_add_segmentArea_rightWall X
+  have hwallL := segmentArea_add_segmentArea_leftWall X
+  have hswapR := segmentArea_swap (rightLeftTailArcs X.rightBody X.leftBody).1.startPoint
+    (distinguishedCapSides X.cap.val).1.corner
+  have hswapL := segmentArea_swap (distinguishedCapSides X.cap.val).2.corner
+    (rightLeftTailArcs X.rightBody X.leftBody).2.endPoint
+  simp only [rightAngleAreaFunctional, capAreaFunctional, upperBoundQ]
+  linarith
+
+theorem gerver_upperBoundQ_maximum :
+    ∃ X : CapTailSpace,
+      (X.cap.val.val : Set Point) = capOfSofa paperGerverSofa (Real.pi / 2) ∧
+      (X.rightBody : Set Point) = (canonicalTailSets X.cap).1 ∧
+      (X.leftBody : Set Point) = (canonicalTailSets X.cap).2 ∧
+      ∀ Y : CapTailSpace, upperBoundQ Y ≤ upperBoundQ X := by
+  -- ### The special-cap space contains Gerver's cap, and the canonical tails extend it to `𝓛`
+  obtain ⟨K, hK⟩ := specialCap_isConvexDomain.2.2.2
+  obtain ⟨X, hXcap, hB, hD⟩ := exists_canonicalCapTail K
+  subst K
+  refine ⟨X, hK, hB, hD, ?_⟩
+  -- ### `𝒬` is quadratic and concave, so nonpositive variation is exactly maximality
+  obtain ⟨hquad, hconcave⟩ := upperBoundQ_quadratic_concave
+  exact (quadratic_maximum_iff capTailCombination capTail_isConvexDomain.2 upperBoundQ hquad
+    hconcave X).mpr fun Y ↦ gerver_upperBoundQ_variation X Y hK hB hD
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Sofa.Balanced`.
+* `Sofa.BalancedConsumed`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Sofa / Balanced
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A monotonized standard-position sofa whose associated cap is a balanced maximum cap. -/
+def IsBalancedMaximumSofa (s : Set Point) (ω : ℝ) : Prop :=
+  (∃ s₀ : Set Point, IsStandardPosition s₀ ω ∧ s = monotonization s₀ ω) ∧
+    ∃ K : CapSpace ω, (K.val : Set Point) = capOfSofa s ω ∧ IsBalancedMaximumCap K
+
+theorem exists_balancedMaximumSofa (ω : ℝ) (hω : 0 < ω)
+    (hω' : ω ≤ Real.pi / 2) :
+    ∃ K : CapSpace ω, IsBalancedMaximumCap K ∧
+      IsBalancedMaximumSofa ((K.val : Set Point) \ capNiche K) ω ∧
+      ∀ s : Set Point, HasRotationAngle s ω →
+        ClassicalResults.area s ≤ ClassicalResults.area ((K.val : Set Point) \ capNiche K) := by
+  obtain ⟨K, hK⟩ := exists_balancedMaximumCap ω hω hω'
+  obtain ⟨s₀, hs₀, hcap⟩ := (cap_isMonotoneCap_iff_niche_subset K).mpr
+    (balancedMaximumCap_niche_subset K hK)
+  have hshape := monotoneSofa_structure (monotonization s₀ ω) ω ⟨s₀, hs₀, rfl⟩ K hcap
+  have hmonoS : ∃ s₀ : Set Point, IsStandardPosition s₀ ω ∧
+      (K.val : Set Point) \ capNiche K = monotonization s₀ ω := ⟨s₀, hs₀, hshape.symm⟩
+  have hcapS : (K.val : Set Point) = capOfSofa ((K.val : Set Point) \ capNiche K) ω := by
+    rwa [← hshape]
+  have hareaK := capAreaFunctional_eq_sofaArea _ ω hmonoS K hcapS
+  refine ⟨K, hK, ⟨hmonoS, K, hcapS, hK⟩, ?_⟩
+  intro s hs
+  obtain ⟨v, hv⟩ := (exists_standardPosition_translation s ω hs ⟨hω, hω'⟩).1
+  let s₁ := (fun p ↦ p + v) '' s
+  have hstd := standardPosition_monotonization_standard s₁ ω hv
+  obtain ⟨L, hL⟩ := standardPosition_cap (monotonization s₁ ω) ω hstd.1
+  calc
+    ClassicalResults.area s = ClassicalResults.area s₁ :=
+      (ClassicalResults.area_image_add s v).symm
+    _ ≤ ClassicalResults.area (monotonization s₁ ω) :=
+      MeasureTheory.measureReal_mono hstd.2 hstd.1.1.measure_lt_top.ne
+    _ = capAreaFunctional L :=
+      (capAreaFunctional_eq_sofaArea _ ω ⟨s₁, hv, rfl⟩ L hL).symm
+    _ ≤ capAreaFunctional K := balancedMaximumCap_maximizes_area K hK L
+    _ = _ := hareaK
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Sofa / Balanced Consumed
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The right wedge gap is bounded below by the calculation variable `g`. -/
+private theorem le_wedgeGaps_fst_of_calculationVariables {ω : ℝ} (K : CapSpace ω)
+    (hω' : ω < Real.pi / 2) {c d r g : ℝ}
+    (hccos : c * Real.cos ω = 1 - Real.sin ω)
+    (hsupp : supportValue K.val (0 : Real.Angle) = c + d)
+    (hrsin : r * Real.sin ω = Real.sin ω - d * Real.cos ω)
+    (hgsq : g ^ 2 = 1 - r ^ 2)
+    {t : ℝ} (ht : t ∈ Set.Ioo 0 ω) :
+    g ≤ (wedgeGaps K t).1 := by
+  have hpi := Real.pi_pos
+  have hω0 : 0 < ω := K.property.1
+  have hcos : 0 < Real.cos ω := Real.cos_pos_of_mem_Ioo ⟨by linarith, hω'⟩
+  have hsin : 0 < Real.sin ω := Real.sin_pos_of_pos_of_lt_pi hω0 (by linarith)
+  have hcost : 0 < Real.cos t :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith [ht.1], by linarith [ht.2]⟩
+  have hsint : 0 ≤ Real.sin t := (Real.sin_pos_of_pos_of_lt_pi ht.1 (by linarith [ht.2])).le
+  have hsinδ : 0 ≤ Real.sin ω * Real.cos t - Real.cos ω * Real.sin t := by
+    rw [← Real.sin_sub]
+    exact (Real.sin_pos_of_pos_of_lt_pi (by linarith [ht.2]) (by linarith [ht.1])).le
+  have hcs : r * Real.sin t + g * Real.cos t ≤ 1 := by
+    nlinarith [sq_nonneg (r - Real.sin t), sq_nonneg (g - Real.cos t),
+      Real.sin_sq_add_cos_sq t]
+  -- the support value at `t` is bounded by the value at the intersection point `R`
+  have hbound : Real.sin ω * supportValue K.val (t : Real.Angle) ≤
+      Real.sin (ω - t) * (c + d) + Real.sin t := by
+    rw [mul_comm, ← le_div_iff₀ hsin]
+    apply csSup_le (K.val.nonempty.image _)
+    rintro _ ⟨p, hp, rfl⟩
+    have hx : p 0 ≤ c + d := by
+      rw [← hsupp]
+      simpa [normalVector, frame, PiLp.inner_apply] using
+        inner_le_supportValue K.val hp (0 : Real.Angle)
+    have hu : Real.cos ω * p 0 + Real.sin ω * p 1 ≤ 1 := by
+      have hle := inner_le_supportValue K.val hp (ω : Real.Angle)
+      rw [K.property.2.2.1] at hle
+      simpa [normalVector, frame, PiLp.inner_apply] using hle
+    have hval : inner ℝ p (normalVector (t : Real.Angle)) =
+        Real.cos t * p 0 + Real.sin t * p 1 := by
+      simp [normalVector, frame, PiLp.inner_apply]
+    simp only [le_div_iff₀ hsin, hval, Real.sin_sub]
+    nlinarith [mul_nonneg hsinδ (sub_nonneg.2 hx), mul_nonneg hsint (sub_nonneg.2 hu)]
+  have hRu : Real.sin ω * ((c + d) * Real.cos t + r * Real.sin t) =
+      Real.sin (ω - t) * (c + d) + Real.sin t := by
+    rw [Real.sin_sub]
+    linear_combination Real.sin t * hrsin + Real.sin t * hccos
+  have hSle : supportValue K.val (t : Real.Angle) ≤ (c + d) * Real.cos t + r * Real.sin t := by
+    rw [← hRu] at hbound
+    exact le_of_mul_le_mul_left hbound hsin
+  have hkey : supportValue K.val (t : Real.Angle) - 1 ≤ (c + d - g) * Real.cos t := by
+    nlinarith
+  rw [wedgeGaps_fst_eq_supportValue, hsupp, ← sub_nonneg]
+  have := (div_le_iff₀ hcost).2 hkey
+  linarith
+
+/-- The exposed top face of a cap contains the horizontal segment of its own atomic
+surface mass, measured leftwards from the upper distinguished point. -/
+private theorem topFace_sub_smul_normalVector_mem {ω : ℝ} (K : CapSpace ω)
+    (ho : (stripParallelogram ω).2.2 ∈ (K.val : Set Point)) {c g : ℝ}
+    (hccos : c * Real.cos ω = 1 - Real.sin ω)
+    (hcoord : (stripParallelogram ω).2.2 = !₂[c, 1]) (hcos : 0 < Real.cos ω)
+    (hgnn : 0 ≤ g)
+    (hg : g ≤ (surfaceAreaMeasure K.val {((Real.pi / 2 : ℝ) : Real.Angle)}).toReal) :
+    (stripParallelogram ω).2.2 - g • normalVector (0 : Real.Angle) ∈ (K.val : Set Point) := by
+  have hT1 : supportValue K.val ((Real.pi / 2 : ℝ) : Real.Angle) = 1 := K.property.2.2.2.1
+  have htan : tangentVector ((Real.pi / 2 : ℝ) : Real.Angle) = -normalVector (0 : Real.Angle) := by
+    ext i
+    fin_cases i <;> simp [tangentVector, normalVector, frame]
+  have hface : ∀ p ∈ exposedEdge K.val ((Real.pi / 2 : ℝ) : Real.Angle), p 1 = 1 ∧ p 0 ≤ c := by
+    intro p hp
+    have hpe : inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) =
+        supportValue K.val ((Real.pi / 2 : ℝ) : Real.Angle) := hp.2
+    rw [hT1] at hpe
+    have h1 : p 1 = 1 := by simpa [normalVector, frame, PiLp.inner_apply] using hpe
+    refine ⟨h1, ?_⟩
+    have hle := inner_le_supportValue K.val hp.1 (ω : Real.Angle)
+    rw [K.property.2.2.1] at hle
+    simp [normalVector, frame, PiLp.inner_apply, h1] at hle
+    nlinarith
+  have hoedge : (stripParallelogram ω).2.2 ∈
+      exposedEdge K.val ((Real.pi / 2 : ℝ) : Real.Angle) := by
+    refine ⟨ho, ?_⟩
+    change inner ℝ (stripParallelogram ω).2.2 (normalVector _) = _
+    rw [hT1, hcoord]
+    simp [normalVector, frame, PiLp.inner_apply]
+  have hsnd := edgeVertices_snd_mem K.val ((Real.pi / 2 : ℝ) : Real.Angle)
+  have hsndle : inner ℝ (edgeVertices K.val ((Real.pi / 2 : ℝ) : Real.Angle)).2
+      (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) ≤
+      inner ℝ (stripParallelogram ω).2.2 (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) := by
+    rw [inner_edgeVertices_snd_tangent]
+    refine csInf_le ?_ ⟨_, hoedge, rfl⟩
+    exact ((isCompact_exposedEdge K.val _).image (continuous_id.inner continuous_const)).bddBelow
+  have hsndeq : (edgeVertices K.val ((Real.pi / 2 : ℝ) : Real.Angle)).2 =
+      (stripParallelogram ω).2.2 := by
+    obtain ⟨hy, hx⟩ := hface _ hsnd
+    have hocx : inner ℝ (stripParallelogram ω).2.2
+        (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) = -c := by
+      rw [htan, inner_neg_right, hcoord]
+      simp [normalVector, frame, PiLp.inner_apply]
+    have hvx : inner ℝ (edgeVertices K.val ((Real.pi / 2 : ℝ) : Real.Angle)).2
+        (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) =
+        -(edgeVertices K.val ((Real.pi / 2 : ℝ) : Real.Angle)).2 0 := by
+      rw [htan, inner_neg_right]
+      simp [normalVector, frame, PiLp.inner_apply]
+    rw [hocx, hvx] at hsndle
+    have hx0 : (edgeVertices K.val ((Real.pi / 2 : ℝ) : Real.Angle)).2 0 = c := by linarith
+    rw [hcoord]
+    ext i
+    fin_cases i
+    · simpa using hx0
+    · simpa using hy
+  obtain ⟨L, hL⟩ : ∃ L, (surfaceAreaMeasure K.val {((Real.pi / 2 : ℝ) : Real.Angle)}).toReal = L :=
+    ⟨_, rfl⟩
+  rw [hL] at hg
+  have hatom := surfaceAreaMeasure_atom_length K.val ((Real.pi / 2 : ℝ) : Real.Angle)
+  rw [hL] at hatom
+  have hfstK : (stripParallelogram ω).2.2 - L • normalVector (0 : Real.Angle) ∈
+      (K.val : Set Point) := by
+    rw [show (stripParallelogram ω).2.2 - L • normalVector (0 : Real.Angle) =
+        (edgeVertices K.val ((Real.pi / 2 : ℝ) : Real.Angle)).1 from by
+      rw [hatom.2.2, hsndeq, htan, smul_neg, ← sub_eq_add_neg]]
+    exact (edgeVertices_fst_mem K.val _).1
+  rcases eq_or_lt_of_le (hgnn.trans hg) with hL0 | hLpos
+  · have hg0 : g = 0 := le_antisymm (hg.trans hL0.symm.le) hgnn
+    simpa [hg0] using ho
+  · have hkey := K.val.convex ho hfstK (a := 1 - g / L) (b := g / L)
+      (by
+        have : g / L ≤ 1 := (div_le_one hLpos).2 hg
+        linarith)
+      (div_nonneg hgnn hLpos.le) (by ring)
+    have hsmul : (1 - g / L) • (stripParallelogram ω).2.2 +
+        (g / L) • ((stripParallelogram ω).2.2 - L • normalVector (0 : Real.Angle)) =
+        (stripParallelogram ω).2.2 - g • normalVector (0 : Real.Angle) := by
+      rw [smul_sub, smul_smul, div_mul_cancel₀ _ hLpos.ne']
+      module
+    rwa [hsmul] at hkey
+
+/-- The three distinguished points lie in the inward quadrant at the angle `π / 2 - ω`
+once the two contact points realize the strict calculation inequalities. -/
+private theorem consumed_points_subset_innerQuadrant {ω : ℝ} (K : CapSpace ω)
+    (hω4 : Real.pi / 4 < ω) (hω' : ω < Real.pi / 2) {c : ℝ} (hcpos : 0 < c)
+    (hc0 : (stripParallelogram ω).2.2 - tangentVector 0 = c • normalVector (0 : Real.Angle))
+    (hcω : (stripParallelogram ω).2.2 - normalVector (ω : Real.Angle) =
+      c • tangentVector (ω : Real.Angle))
+    {p q : Point} (hp : p ∈ (K.val : Set Point)) (hq : q ∈ (K.val : Set Point))
+    (h1 : 1 < inner ℝ (p - ((stripParallelogram ω).2.2 - tangentVector 0))
+      (normalVector ((Real.pi / 2 - ω : ℝ) : Real.Angle)))
+    (h2 : 1 < inner ℝ (q - ((stripParallelogram ω).2.2 - normalVector (ω : Real.Angle)))
+      (tangentVector ((Real.pi / 2 - ω : ℝ) : Real.Angle))) :
+    ({0, (stripParallelogram ω).2.2 - tangentVector 0,
+      (stripParallelogram ω).2.2 - normalVector (ω : Real.Angle)} : Set Point) ⊆
+      innerQuadrant (K.val : Set Point) (Real.pi / 2 - ω) := by
+  have hpi := Real.pi_pos
+  have hcos : 0 < Real.cos ω := Real.cos_pos_of_mem_Ioo ⟨by linarith, hω'⟩
+  have hsin : 0 < Real.sin ω := Real.sin_pos_of_pos_of_lt_pi (by linarith) (by linarith)
+  have hcslt : Real.cos ω < Real.sin ω := by
+    rw [← Real.sin_pi_div_two_sub]
+    exact Real.strictMonoOn_sin ⟨by linarith, by linarith⟩ ⟨by linarith, by linarith⟩ (by linarith)
+  have hnt : normalVector ((Real.pi / 2 - ω + Real.pi / 2 : ℝ) : Real.Angle) =
+      tangentVector ((Real.pi / 2 - ω : ℝ) : Real.Angle) := by
+    ext i
+    fin_cases i <;>
+      simp [normalVector, tangentVector, frame, Real.cos_add, Real.sin_add,
+        -Real.Angle.coe_sub, -Real.Angle.coe_add]
+  have hnn : inner ℝ (normalVector (0 : Real.Angle))
+      (normalVector ((Real.pi / 2 - ω : ℝ) : Real.Angle)) = Real.sin ω := by
+    simp [normalVector, frame, PiLp.inner_apply, Real.cos_pi_div_two_sub, -Real.Angle.coe_sub]
+  have hnt' : inner ℝ (normalVector (0 : Real.Angle))
+      (tangentVector ((Real.pi / 2 - ω : ℝ) : Real.Angle)) = -Real.cos ω := by
+    simp [normalVector, tangentVector, frame, PiLp.inner_apply, Real.sin_pi_div_two_sub,
+      -Real.Angle.coe_sub]
+  have htn : inner ℝ (tangentVector (ω : Real.Angle))
+      (normalVector ((Real.pi / 2 - ω : ℝ) : Real.Angle)) =
+      Real.cos ω ^ 2 - Real.sin ω ^ 2 := by
+    simp [tangentVector, normalVector, frame, PiLp.inner_apply, Real.cos_pi_div_two_sub,
+      Real.sin_pi_div_two_sub, -Real.Angle.coe_sub]
+    ring
+  have htt : inner ℝ (tangentVector (ω : Real.Angle))
+      (tangentVector ((Real.pi / 2 - ω : ℝ) : Real.Angle)) =
+      2 * (Real.sin ω * Real.cos ω) := by
+    simp [tangentVector, frame, PiLp.inner_apply, Real.cos_pi_div_two_sub,
+      Real.sin_pi_div_two_sub, -Real.Angle.coe_sub]
+    ring
+  have hA := inner_le_supportValue K.val hp ((Real.pi / 2 - ω : ℝ) : Real.Angle)
+  have hB := inner_le_supportValue K.val hq ((Real.pi / 2 - ω + Real.pi / 2 : ℝ) : Real.Angle)
+  rw [hnt] at hB
+  have hstrict1 : c * Real.sin ω <
+      supportValue K.val ((Real.pi / 2 - ω : ℝ) : Real.Angle) - 1 := by
+    rw [inner_sub_left, hc0, real_inner_smul_left, hnn] at h1
+    linarith
+  have hstrict2 : c * (2 * (Real.sin ω * Real.cos ω)) <
+      supportValue K.val ((Real.pi / 2 - ω + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+    rw [inner_sub_left, hcω, real_inner_smul_left, htt] at h2
+    linarith
+  have hmem : ∀ x : Point,
+      inner ℝ x (normalVector ((Real.pi / 2 - ω : ℝ) : Real.Angle)) <
+        supportValue K.val ((Real.pi / 2 - ω : ℝ) : Real.Angle) - 1 →
+      inner ℝ x (tangentVector ((Real.pi / 2 - ω : ℝ) : Real.Angle)) <
+        supportValue K.val ((Real.pi / 2 - ω + Real.pi / 2 : ℝ) : Real.Angle) - 1 →
+      x ∈ innerQuadrant (K.val : Set Point) (Real.pi / 2 - ω) := by
+    intro x hx1 hx2
+    refine ⟨hx1, ?_⟩
+    change inner ℝ x (normalVector ((Real.pi / 2 - ω + Real.pi / 2 : ℝ) : Real.Angle)) < _
+    rwa [hnt]
+  have hpos1 : 0 < c * Real.sin ω := mul_pos hcpos hsin
+  have hpos2 : 0 < c * (2 * (Real.sin ω * Real.cos ω)) := by
+    have := mul_pos hsin hcos
+    nlinarith
+  intro x hx
+  simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hx
+  rcases hx with rfl | rfl | rfl
+  · exact hmem _ (by rw [inner_zero_left]; linarith) (by rw [inner_zero_left]; linarith)
+  · refine hmem _ ?_ ?_
+    · rw [hc0, real_inner_smul_left, hnn]
+      exact hstrict1
+    · rw [hc0, real_inner_smul_left, hnt']
+      nlinarith
+  · refine hmem _ ?_ ?_
+    · rw [hcω, real_inner_smul_left, htn]
+      have hneg : c * (Real.cos ω ^ 2 - Real.sin ω ^ 2) < 0 :=
+        mul_neg_of_pos_of_neg hcpos (by nlinarith)
+      linarith
+    · rw [hcω, real_inner_smul_left, htt]
+      exact hstrict2
+
+/-- Auxiliary form of the consumption theorem under the right-hand support bound. -/
+private theorem consumed_of_le_supportValue_zero {ω : ℝ}
+    (hω : Real.arccos (5 / 11 : ℝ) ≤ ω) (hω' : ω < Real.pi / 2)
+    (K : CapSpace ω) (hBalanced : IsBalancedMaximumCap K)
+    (ho : (stripParallelogram ω).2.2 ∈ (K.val : Set Point))
+    (hd : rotationCalculationMinimum ⟨ω, hω, hω'⟩ + Real.tan ((Real.pi / 2 - ω) / 2) ≤
+      supportValue K.val (0 : Real.Angle)) :
+    ({0, (stripParallelogram ω).2.2 - tangentVector 0,
+      (stripParallelogram ω).2.2 - normalVector (ω : Real.Angle)} : Set Point) ⊆
+      innerQuadrant (K.val : Set Point) (Real.pi / 2 - ω) := by
+  have hpi := Real.pi_pos
+  have hω4 : Real.pi / 4 < ω := rotationCalculation_angle_bounds.1.trans_le hω
+  have hω0 : 0 < ω := by linarith
+  have hcos : 0 < Real.cos ω := Real.cos_pos_of_mem_Ioo ⟨by linarith, hω'⟩
+  have hsin : 0 < Real.sin ω := Real.sin_pos_of_pos_of_lt_pi hω0 (by linarith)
+  have hgap := parallelogram_gap ω ⟨hω0.le, hω'⟩
+  have hcpos : 0 < Real.tan ((Real.pi / 2 - ω) / 2) :=
+    Real.tan_pos_of_pos_of_lt_pi_div_two (by linarith) (by linarith)
+  have hccos : Real.tan ((Real.pi / 2 - ω) / 2) * Real.cos ω = 1 - Real.sin ω := by
+    rw [hgap.2.2.2.2, Real.tan_eq_sin_div_cos]
+    field_simp
+  obtain ⟨d, hsupp⟩ : ∃ d, supportValue K.val (0 : Real.Angle) =
+      Real.tan ((Real.pi / 2 - ω) / 2) + d :=
+    ⟨supportValue K.val (0 : Real.Angle) - Real.tan ((Real.pi / 2 - ω) / 2), by ring⟩
+  have hdmin : rotationCalculationMinimum ⟨ω, hω, hω'⟩ ≤ d := by
+    rw [hsupp] at hd; linarith
+  have hdpos : 0 < d := (rotationCalculationMinimum_pos _).trans_le hdmin
+  have hq0K : supportValue K.val (0 : Real.Angle) • normalVector (0 : Real.Angle) ∈
+      (K.val : Set Point) := supportValue_zero_smul_normalVector_mem K
+  have hdtan : d ≤ Real.tan ω := by
+    have hle := inner_le_supportValue K.val hq0K (ω : Real.Angle)
+    rw [K.property.2.2.1, real_inner_smul_left] at hle
+    have hinner : inner ℝ (normalVector (0 : Real.Angle))
+        (normalVector (ω : Real.Angle)) = Real.cos ω := by
+      simp [normalVector, frame, PiLp.inner_apply]
+    rw [hinner, hsupp] at hle
+    rw [Real.tan_eq_sin_div_cos, le_div_iff₀ hcos]
+    linarith
+  have hrnn : 0 ≤ (rotationCalculationValues ⟨ω, hω, hω'⟩ ⟨d, hdmin, hdtan⟩).1 := by
+    change 0 ≤ 1 - d * (Real.cos ω / Real.sin ω)
+    rw [sub_nonneg, mul_div_assoc', div_le_one hsin]
+    rw [Real.tan_eq_sin_div_cos, le_div_iff₀ hcos] at hdtan
+    exact hdtan
+  obtain ⟨r, hr⟩ :
+      ∃ r, (rotationCalculationValues ⟨ω, hω, hω'⟩ ⟨d, hdmin, hdtan⟩).1 = r := ⟨_, rfl⟩
+  obtain ⟨g, hg⟩ :
+      ∃ g, (rotationCalculationValues ⟨ω, hω, hω'⟩ ⟨d, hdmin, hdtan⟩).2.1 = g := ⟨_, rfl⟩
+  have hrval : r = 1 - d * (Real.cos ω / Real.sin ω) := by rw [← hr]; rfl
+  have hgval : g = Real.sqrt (1 - r ^ 2) := by rw [← hg, ← hr]; rfl
+  have hrnn' : 0 ≤ r := by rw [← hr]; exact hrnn
+  have hrle : r ≤ 1 := by
+    rw [hrval]
+    have := mul_nonneg hdpos.le (div_nonneg hcos.le hsin.le)
+    linarith
+  have hgnn : 0 ≤ g := by rw [hgval]; exact Real.sqrt_nonneg _
+  have hgsq : g ^ 2 = 1 - r ^ 2 := by
+    rw [hgval]
+    exact Real.sq_sqrt (by nlinarith)
+  have hrsin : r * Real.sin ω = Real.sin ω - d * Real.cos ω := by
+    rw [hrval]
+    field_simp
+  -- the wedge-gap infimum, hence the vertical surface atom, is at least `g`
+  have hinf : g ≤ (wedgeGapInfimum K).1 := by
+    apply le_csInf ((Set.nonempty_Ioo.mpr hω0).image _)
+    rintro _ ⟨t, ht, rfl⟩
+    exact le_wedgeGaps_fst_of_calculationVariables K hω' hccos hsupp hrsin hgsq ht
+  have hsurf := hinf.trans (balancedMaximumCap_gap_le_surface K hBalanced hω').1
+  have hcoord : (stripParallelogram ω).2.2 = !₂[Real.tan ((Real.pi / 2 - ω) / 2), 1] := by
+    have harg : Real.pi / 4 - ω / 2 = (Real.pi / 2 - ω) / 2 := by ring
+    simp only [stripParallelogram, harg]
+  have hq1K := topFace_sub_smul_normalVector_mem K ho hccos hcoord hcos hgnn hsurf
+  have hineq := rotationCalculation_inequalities ⟨ω, hω, hω'⟩ ⟨d, hdmin, hdtan⟩ hrnn
+  have hP0 : (rotationCalculationValues ⟨ω, hω, hω'⟩ ⟨d, hdmin, hdtan⟩).2.2.1 ∈
+      (K.val : Set Point) := by
+    have hq : (rotationCalculationValues ⟨ω, hω, hω'⟩ ⟨d, hdmin, hdtan⟩).2.2.1 =
+        (stripParallelogram ω).2.2 - tangentVector 0 + d • normalVector (0 : Real.Angle) := rfl
+    rw [hq, hgap.1, ← add_smul, ← hsupp]
+    exact hq0K
+  have hP1 : (rotationCalculationValues ⟨ω, hω, hω'⟩ ⟨d, hdmin, hdtan⟩).2.2.2 ∈
+      (K.val : Set Point) := by
+    have hq : (rotationCalculationValues ⟨ω, hω, hω'⟩ ⟨d, hdmin, hdtan⟩).2.2.2 =
+        (stripParallelogram ω).2.2 -
+          (rotationCalculationValues ⟨ω, hω, hω'⟩ ⟨d, hdmin, hdtan⟩).2.1 •
+            normalVector (0 : Real.Angle) := rfl
+    rw [hq, hg]
+    exact hq1K
+  exact consumed_points_subset_innerQuadrant K hω4 hω' hcpos hgap.1 hgap.2.1 hP0 hP1
+    hineq.1 hineq.2
+
+theorem balancedMaximumCap_consumed {ω : ℝ} (K : CapSpace ω)
+    (hω : Real.arccos (5 / 11 : ℝ) ≤ ω) (hω' : ω < Real.pi / 2)
+    (hBalanced : IsBalancedMaximumCap K) (hArea : 11 / 5 ≤ capAreaFunctional K) :
+    ∃ t ∈ Set.Ioo 0 ω,
+      ({0, (stripParallelogram ω).2.2 - tangentVector 0,
+        (stripParallelogram ω).2.2 - normalVector (ω : Real.Angle)} : Set Point) ⊆
+        (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerQuadrant := by
+  have hpi := Real.pi_pos
+  have hω4 : Real.pi / 4 < ω := rotationCalculation_angle_bounds.1.trans_le hω
+  have hω0 : 0 < ω := by linarith
+  have hcos : 0 < Real.cos ω := Real.cos_pos_of_mem_Ioo ⟨by linarith, hω'⟩
+  have hsin : 0 < Real.sin ω := Real.sin_pos_of_pos_of_lt_pi hω0 (by linarith)
+  have hgap := parallelogram_gap ω ⟨hω0.le, hω'⟩
+  have hccos : Real.tan ((Real.pi / 2 - ω) / 2) * Real.cos ω = 1 - Real.sin ω := by
+    rw [hgap.2.2.2.2, Real.tan_eq_sin_div_cos]
+    field_simp
+  have hcoord : (stripParallelogram ω).2.2 = !₂[Real.tan ((Real.pi / 2 - ω) / 2), 1] := by
+    have harg : Real.pi / 4 - ω / 2 = (Real.pi / 2 - ω) / 2 := by ring
+    simp only [stripParallelogram, harg]
+  -- the upper distinguished point belongs to the limiting cap
+  have ho : (stripParallelogram ω).2.2 ∈ (K.val : Set Point) := by
+    obtain ⟨n, hn, -, -, P, hmax, hlim⟩ := hBalanced
+    exact ConvexBody.mem_of_tendsto_hausdorffDist _ (fun i ↦ (P i).val.val) K.val
+      (Filter.Eventually.of_forall fun i ↦ (hmax i).1) hlim
+  by_cases hcase : rotationCalculationMinimum ⟨ω, hω, hω'⟩ +
+      Real.tan ((Real.pi / 2 - ω) / 2) ≤ supportValue K.val (0 : Real.Angle)
+  · refine ⟨Real.pi / 2 - ω, ⟨by linarith, by linarith⟩, ?_⟩
+    rw [rotatingHallwayParts_innerQuadrant]
+    exact consumed_of_le_supportValue_zero hω hω' K hBalanced ho hcase
+  push Not at hcase
+  -- otherwise the clipped-cap area bound forces the left support value to be large
+  have hother : rotationCalculationMinimum ⟨ω, hω, hω'⟩ + Real.tan ((Real.pi / 2 - ω) / 2) ≤
+      supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) := by
+    by_contra hcon
+    push Not at hcon
+    have hsub : (K.val : Set Point) ⊆
+        clippedCap ω (rotationCalculationMinimum ⟨ω, hω, hω'⟩) := by
+      intro p hp
+      exact ⟨⟨K.subset_stripParallelogram hp,
+        (inner_le_supportValue K.val hp (0 : Real.Angle)).trans hcase.le⟩,
+        (inner_le_supportValue K.val hp ((ω + Real.pi / 2 : ℝ) : Real.Angle)).trans hcon.le⟩
+    have hbdd : Bornology.IsBounded
+        (clippedCap ω (rotationCalculationMinimum ⟨ω, hω, hω'⟩)) := by
+      refine (EuclideanSpace.isBounded_coordinate_rectangle
+        (-((rotationCalculationMinimum ⟨ω, hω, hω'⟩ +
+            Real.tan ((Real.pi / 2 - ω) / 2)) / Real.sin ω))
+        (rotationCalculationMinimum ⟨ω, hω, hω'⟩ + Real.tan ((Real.pi / 2 - ω) / 2))
+        0 1).subset ?_
+      intro p hp
+      obtain ⟨⟨hy0, hy1⟩, -, hx, hz⟩ := (mem_clippedCap_iff _ _ p).1 hp
+      refine ⟨?_, hx, hy0, hy1⟩
+      have h2 : -(rotationCalculationMinimum ⟨ω, hω, hω'⟩ +
+          Real.tan ((Real.pi / 2 - ω) / 2)) ≤ p 0 * Real.sin ω := by
+        nlinarith [mul_nonneg hcos.le hy0]
+      have h3 := (div_le_iff₀ hsin).2 h2
+      rwa [neg_div] at h3
+    have harea : ClassicalResults.area (K.val : Set Point) ≤
+        ClassicalResults.area (clippedCap ω (rotationCalculationMinimum ⟨ω, hω, hω'⟩)) :=
+      ENNReal.toReal_mono hbdd.measure_lt_top.ne (MeasureTheory.measure_mono hsub)
+    have hmin : ClassicalResults.area
+        (clippedCap ω (rotationCalculationMinimum ⟨ω, hω, hω'⟩)) < 11 / 5 :=
+      clippedCap_minimum_area ⟨ω, hω, hω'⟩
+    have hniche : 0 ≤ ClassicalResults.area (capNiche K) := ENNReal.toReal_nonneg
+    rw [capAreaFunctional] at hArea
+    linarith
+  -- pass to the mirror cap, whose zero-angle support value is the left one
+  have hMeq : mirrorReflection ω = ((capReflection ω : Point ≃ₗᵢ[ℝ] Point) : Point → Point) :=
+    stripTopReflection_eq_capReflection ω hω0 K.property.2.1
+  have htanω : normalVector ((ω + Real.pi / 2 : ℝ) : Real.Angle) =
+      tangentVector (ω : Real.Angle) := by
+    ext i
+    fin_cases i <;>
+      simp [normalVector, tangentVector, frame, Real.cos_add, Real.sin_add, -Real.Angle.coe_add]
+  have htanT : tangentVector ((Real.pi / 2 : ℝ) : Real.Angle) = -normalVector (0 : Real.Angle) := by
+    ext i
+    fin_cases i <;> simp [tangentVector, normalVector, frame]
+  have hu0 : capReflection ω (normalVector (0 : Real.Angle)) = tangentVector (ω : Real.Angle) := by
+    rw [capReflection_normalVector_angle,
+      show reflectedAngle ω (0 : Real.Angle) = ((ω + Real.pi / 2 : ℝ) : Real.Angle) from by
+        simp [reflectedAngle]]
+    exact htanω
+  have hvω : capReflection ω (tangentVector (ω : Real.Angle)) = normalVector (0 : Real.Angle) := by
+    rw [capReflection_tangentVector_angle,
+      show reflectedAngle ω (ω : Real.Angle) = ((Real.pi / 2 : ℝ) : Real.Angle) from by
+        simp [reflectedAngle, Real.Angle.coe_add],
+      htanT, neg_neg]
+  have hc1 : Real.tan ((Real.pi / 2 - ω) / 2) * (1 + Real.sin ω) = Real.cos ω := by
+    apply mul_right_cancel₀ hcos.ne'
+    nlinarith [hccos, Real.sin_sq_add_cos_sq ω]
+  have hfix : capReflection ω (stripParallelogram ω).2.2 = (stripParallelogram ω).2.2 := by
+    have hx0 : capReflection ω (stripParallelogram ω).2.2 0 = (stripParallelogram ω).2.2 0 := by
+      rw [capReflection_apply_zero, hcoord]
+      simp only [Matrix.cons_val_zero, Matrix.cons_val_one]
+      linarith
+    have hx1 : capReflection ω (stripParallelogram ω).2.2 1 = (stripParallelogram ω).2.2 1 := by
+      rw [capReflection_apply_one, hcoord]
+      simp only [Matrix.cons_val_zero, Matrix.cons_val_one]
+      linarith
+    ext i
+    fin_cases i
+    · exact hx0
+    · exact hx1
+  have hQbal : IsBalancedMaximumCap
+      (⟨reflectedBody ω K.val, reflectedBody_isCap K⟩ : CapSpace ω) := by
+    obtain ⟨P, hPset, hPbal⟩ := balancedMaximumCap_mirror K hBalanced
+    have hPQ : P = (⟨reflectedBody ω K.val, reflectedBody_isCap K⟩ : CapSpace ω) := by
+      apply Subtype.ext
+      apply ConvexBody.ext
+      rw [hPset, hMeq]
+      rfl
+    exact hPQ ▸ hPbal
+  have hoQ : (stripParallelogram ω).2.2 ∈ (reflectedBody ω K.val : Set Point) :=
+    ⟨_, ho, hfix⟩
+  have hQsupp : supportValue (reflectedBody ω K.val) (0 : Real.Angle) =
+      supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) := by
+    rw [supportValue_reflectedBody]
+    congr 1
+    simp [reflectedAngle]
+  have hmain : ({0, (stripParallelogram ω).2.2 - tangentVector 0,
+      (stripParallelogram ω).2.2 - normalVector (ω : Real.Angle)} : Set Point) ⊆
+      innerQuadrant (reflectedBody ω K.val : Set Point) (Real.pi / 2 - ω) :=
+    consumed_of_le_supportValue_zero hω hω' ⟨reflectedBody ω K.val, reflectedBody_isCap K⟩
+      hQbal hoQ (by rw [hQsupp]; exact hother)
+  have hR1 : capReflection ω ((stripParallelogram ω).2.2 - tangentVector 0) =
+      (stripParallelogram ω).2.2 - normalVector (ω : Real.Angle) := by
+    rw [hgap.1, hgap.2.1, map_smul, hu0]
+  have hR2 : capReflection ω ((stripParallelogram ω).2.2 - normalVector (ω : Real.Angle)) =
+      (stripParallelogram ω).2.2 - tangentVector 0 := by
+    rw [hgap.1, hgap.2.1, map_smul, hvω]
+  refine ⟨ω - (Real.pi / 2 - ω), ⟨by linarith, by linarith⟩, ?_⟩
+  rw [rotatingHallwayParts_innerQuadrant]
+  intro y hy
+  simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hy
+  have hyR : capReflection ω y ∈ ({0, (stripParallelogram ω).2.2 - tangentVector 0,
+      (stripParallelogram ω).2.2 - normalVector (ω : Real.Angle)} : Set Point) := by
+    rcases hy with rfl | rfl | rfl
+    · simp
+    · rw [hR1]; simp
+    · rw [hR2]; simp
+  have h1 := hmain hyR
+  rw [innerQuadrant_reflection, ← capReflection_preimage_eq_image] at h1
+  simpa [capReflection_involutive] using h1
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Motion.RotationAngle`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Motion / Rotation Angle
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A nonnegative combination of two points of a convex set containing the origin, with
+coefficient sum at most one, lies in the set. -/
+private theorem smul_add_smul_mem_of_convex {U : Set Point} (hU : Convex ℝ U)
+    (h0 : (0 : Point) ∈ U) {x y : Point} (hx : x ∈ U) (hy : y ∈ U)
+    {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b ≤ 1) :
+    a • x + b • y ∈ U := by
+  have hw : ∀ i ∈ (Finset.univ : Finset (Fin 3)), 0 ≤ (![a, b, 1 - a - b] : Fin 3 → ℝ) i := by
+    intro i _
+    fin_cases i
+    · exact ha
+    · exact hb
+    · change (0 : ℝ) ≤ 1 - a - b
+      linarith
+  have hsum : ∑ i ∈ (Finset.univ : Finset (Fin 3)), (![a, b, 1 - a - b] : Fin 3 → ℝ) i = 1 := by
+    simp [Fin.sum_univ_three]
+  have hz : ∀ i ∈ (Finset.univ : Finset (Fin 3)),
+      (![x, y, (0 : Point)] : Fin 3 → Point) i ∈ U := by
+    intro i _
+    fin_cases i
+    · exact hx
+    · exact hy
+    · exact h0
+  simpa [Fin.sum_univ_three] using hU.sum_mem hw hsum hz
+
+/-- A point of the strip parallelogram on or below the line through `o_ω - v₀` and
+`o_ω - u_ω` lies in every convex set containing the origin and those two points. -/
+private theorem mem_of_inner_add_le_one_sub_sin {ω : ℝ} (hω0 : 0 < ω) (hω : ω < Real.pi / 2)
+    {U : Set Point} (hU : Convex ℝ U) (h0 : (0 : Point) ∈ U)
+    (ha : (stripParallelogram ω).2.2 - tangentVector 0 ∈ U)
+    (hb : (stripParallelogram ω).2.2 - normalVector (ω : Real.Angle) ∈ U)
+    {p : Point} (hy : 0 ≤ p 1) (hx : 0 ≤ inner ℝ p (normalVector (ω : Real.Angle)))
+    (hsum : inner ℝ p (normalVector (ω : Real.Angle)) + p 1 ≤ 1 - Real.sin ω) :
+    p ∈ U := by
+  have hpi := Real.pi_pos
+  have hcos : 0 < Real.cos ω := Real.cos_pos_of_mem_Ioo ⟨by linarith, hω⟩
+  have hsinlt : Real.sin ω < 1 := by nlinarith [Real.sin_sq_add_cos_sq ω]
+  have hgap := parallelogram_gap ω ⟨hω0.le, hω⟩
+  have hccos : Real.tan ((Real.pi / 2 - ω) / 2) * Real.cos ω = 1 - Real.sin ω := by
+    rw [hgap.2.2.2.2, Real.tan_eq_sin_div_cos]
+    field_simp
+  have hi1 : inner ℝ (normalVector (0 : Real.Angle)) (normalVector (ω : Real.Angle)) =
+      Real.cos ω := by
+    rw [← Real.Angle.coe_zero, inner_normalVector_normalVector, zero_sub, Real.cos_neg]
+  have hi2 : inner ℝ (normalVector (0 : Real.Angle))
+      (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := by
+    rw [← Real.Angle.coe_zero, inner_normalVector_normalVector, zero_sub, Real.cos_neg,
+      Real.cos_pi_div_two]
+  have hi3 : inner ℝ (tangentVector (ω : Real.Angle)) (normalVector (ω : Real.Angle)) = 0 := by
+    rw [inner_tangentVector_normalVector_real, sub_self, Real.sin_zero]
+  have hi4 : inner ℝ (tangentVector (ω : Real.Angle))
+      (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = Real.cos ω := by
+    rw [inner_tangentVector_normalVector_real, Real.sin_pi_div_two_sub]
+  have hsne : Real.sin (ω - Real.pi / 2) = -Real.cos ω := by
+    rw [Real.sin_sub, Real.sin_pi_div_two, Real.cos_pi_div_two]
+    ring
+  have hden : (1 : ℝ) - Real.sin ω ≠ 0 := by linarith
+  have hkey : p = (inner ℝ p (normalVector (ω : Real.Angle)) / (1 - Real.sin ω)) •
+        ((stripParallelogram ω).2.2 - tangentVector 0) +
+      (p 1 / (1 - Real.sin ω)) •
+        ((stripParallelogram ω).2.2 - normalVector (ω : Real.Angle)) := by
+    refine eq_of_inner_normalVector_eq (s := ω) (t := Real.pi / 2) ?_ ?_ ?_
+    · rw [hsne]
+      exact neg_ne_zero.mpr hcos.ne'
+    · rw [hgap.1, hgap.2.1, inner_add_left, real_inner_smul_left, real_inner_smul_left,
+        real_inner_smul_left, real_inner_smul_left, hi1, hi3, hccos]
+      field_simp
+      ring
+    · rw [inner_normalVector_pi_div_two, hgap.1, hgap.2.1, inner_add_left,
+        real_inner_smul_left, real_inner_smul_left, real_inner_smul_left,
+        real_inner_smul_left, hi2, hi4, hccos]
+      field_simp
+      ring
+  rw [hkey]
+  refine smul_add_smul_mem_of_convex hU h0 ha hb (div_nonneg hx (by linarith))
+    (div_nonneg hy (by linarith)) ?_
+  rw [← add_div, div_le_one (by linarith)]
+  linarith
+
+/-- The width of a clipped strip parallelogram is at most one in every direction between
+its rotation angle and a right angle. -/
+private theorem inner_sub_inner_le_one_of_clipped {ω : ℝ} (hω0 : 0 ≤ ω)
+    (hcos : 0 < Real.cos ω) {p q : Point}
+    (hpA : inner ℝ p (normalVector (ω : Real.Angle)) ≤ 1) (hpB : p 1 ≤ 1)
+    (hqA : 0 ≤ inner ℝ q (normalVector (ω : Real.Angle))) (hqB : 0 ≤ q 1)
+    (hq : 1 - Real.sin ω ≤ inner ℝ q (normalVector (ω : Real.Angle)) + q 1)
+    {t : ℝ} (ht : ω ≤ t) (ht' : t ≤ Real.pi / 2) :
+    inner ℝ p (normalVector (t : Real.Angle)) -
+      inner ℝ q (normalVector (t : Real.Angle)) ≤ 1 := by
+  have hpi := Real.pi_pos
+  have hct : 0 ≤ Real.cos t := Real.cos_nonneg_of_mem_Icc ⟨by linarith, ht'⟩
+  have hst : 0 ≤ Real.sin (t - ω) :=
+    Real.sin_nonneg_of_nonneg_of_le_pi (by linarith) (by linarith)
+  have hsub : Real.sin (t - ω) = Real.sin t * Real.cos ω - Real.cos t * Real.sin ω :=
+    Real.sin_sub t ω
+  have hid : ∀ z : Point, Real.cos ω * inner ℝ z (normalVector (t : Real.Angle)) =
+      Real.cos t * inner ℝ z (normalVector (ω : Real.Angle)) + Real.sin (t - ω) * z 1 := by
+    intro z
+    rw [inner_normalVector_real, inner_normalVector_real, hsub]
+    ring
+  have hmain : Real.cos ω * (inner ℝ p (normalVector (t : Real.Angle)) -
+      inner ℝ q (normalVector (t : Real.Angle))) ≤ Real.cos ω := by
+    rw [mul_sub, hid p, hid q]
+    have h1 : Real.cos t * inner ℝ p (normalVector (ω : Real.Angle)) ≤ Real.cos t :=
+      mul_le_of_le_one_right hct hpA
+    have h2 : Real.sin (t - ω) * p 1 ≤ Real.sin (t - ω) :=
+      mul_le_of_le_one_right hst hpB
+    rcases le_total (Real.cos t) (Real.sin (t - ω)) with hle | hle
+    · have e1 : Real.cos t * (1 - Real.sin ω) ≤
+          Real.cos t * (inner ℝ q (normalVector (ω : Real.Angle)) + q 1) :=
+        mul_le_mul_of_nonneg_left hq hct
+      have e2 : 0 ≤ (Real.sin (t - ω) - Real.cos t) * q 1 := mul_nonneg (by linarith) hqB
+      have h3 : Real.cos t * (1 - Real.sin ω) ≤
+          Real.cos t * inner ℝ q (normalVector (ω : Real.Angle)) + Real.sin (t - ω) * q 1 := by
+        nlinarith [e1, e2]
+      have h4 : Real.sin (t - ω) + Real.cos t * Real.sin ω = Real.sin t * Real.cos ω := by
+        rw [hsub]; ring
+      have h5 : Real.sin t * Real.cos ω ≤ Real.cos ω := by
+        nlinarith [Real.sin_le_one t]
+      linarith
+    · have e1 : Real.sin (t - ω) * (1 - Real.sin ω) ≤
+          Real.sin (t - ω) * (inner ℝ q (normalVector (ω : Real.Angle)) + q 1) :=
+        mul_le_mul_of_nonneg_left hq hst
+      have e2 : 0 ≤ (Real.cos t - Real.sin (t - ω)) *
+          inner ℝ q (normalVector (ω : Real.Angle)) := mul_nonneg (by linarith) hqA
+      have h3 : Real.sin (t - ω) * (1 - Real.sin ω) ≤
+          Real.cos t * inner ℝ q (normalVector (ω : Real.Angle)) + Real.sin (t - ω) * q 1 := by
+        nlinarith [e1, e2]
+      have h4 : Real.cos t + Real.sin (t - ω) * Real.sin ω =
+          Real.cos ω * Real.cos (t - ω) := by
+        rw [hsub, Real.cos_sub]
+        linear_combination (-Real.cos t) * Real.sin_sq_add_cos_sq ω
+      have h5 : Real.cos ω * Real.cos (t - ω) ≤ Real.cos ω := by
+        nlinarith [Real.cos_le_one (t - ω)]
+      linarith
+  nlinarith [hmain]
+
+/-- Normalizing data for the rotated copies of a sofa of width at most one: a continuous
+minimal height and a uniform horizontal bound. -/
+private theorem exists_normalization_of_width {s : Set Point} {ω : ℝ}
+    (hcpt : IsCompact s) (hne : s.Nonempty)
+    (hwidth : ∀ t : ℝ, ω ≤ t → t ≤ Real.pi / 2 → ∀ p ∈ s, ∀ q ∈ s,
+      inner ℝ p (normalVector (t : Real.Angle)) -
+        inner ℝ q (normalVector (t : Real.Angle)) ≤ 1) :
+    ∃ (low : ℝ → ℝ) (M : ℝ), Continuous low ∧
+      (∀ d : ℝ, ∀ p ∈ s, (rotationMap ((d : ℝ) : Real.Angle) p) 0 ≤ M) ∧
+      (∀ d : ℝ, ∀ p ∈ s, low d ≤ (rotationMap ((d : ℝ) : Real.Angle) p) 1) ∧
+      (∀ d : ℝ, 0 ≤ d → d ≤ Real.pi / 2 - ω → ∀ p ∈ s,
+        (rotationMap ((d : ℝ) : Real.Angle) p) 1 - low d ≤ 1) := by
+  have hcoord1 : ∀ (d : ℝ) (p : Point), (rotationMap ((d : ℝ) : Real.Angle) p) 1 =
+      inner ℝ p (normalVector ((Real.pi / 2 - d : ℝ) : Real.Angle)) := by
+    intro d p
+    rw [rotationMap_apply_one, inner_normalVector_real, Real.Angle.cos_coe, Real.Angle.sin_coe,
+      Real.cos_pi_div_two_sub, Real.sin_pi_div_two_sub]
+    ring
+  obtain ⟨R, hR⟩ := hcpt.isBounded.subset_closedBall (0 : Point)
+  refine ⟨fun d ↦ -supportValue s ((Real.pi / 2 - d + Real.pi : ℝ) : Real.Angle),
+    max R 0, ?_, ?_, ?_, ?_⟩
+  · exact ((compactSet_support_continuity s s hne hcpt hne hcpt).2.2.1.comp
+      (Real.Angle.continuous_coe.comp (by fun_prop))).neg
+  · intro d p hp
+    have hnp : ‖p‖ ≤ max R 0 := by
+      have h := hR hp
+      rw [Metric.mem_closedBall, dist_zero_right] at h
+      exact h.trans (le_max_left _ _)
+    have hn : ‖rotationMap ((d : ℝ) : Real.Angle) p‖ = ‖p‖ := by
+      change ‖(EuclideanGeometry.o.rotation ((d : ℝ) : Real.Angle)) p‖ = ‖p‖
+      exact LinearIsometryEquiv.norm_map _ p
+    calc (rotationMap ((d : ℝ) : Real.Angle) p) 0
+        = inner ℝ (rotationMap ((d : ℝ) : Real.Angle) p)
+            (normalVector ((0 : ℝ) : Real.Angle)) := (inner_normalVector_zero _).symm
+      _ ≤ ‖rotationMap ((d : ℝ) : Real.Angle) p‖ *
+            ‖normalVector ((0 : ℝ) : Real.Angle)‖ := real_inner_le_norm _ _
+      _ = ‖p‖ := by rw [norm_normalVector_real, hn, mul_one]
+      _ ≤ max R 0 := hnp
+  · intro d p hp
+    rw [hcoord1]
+    have h := inner_le_supportValue_of_isCompact hcpt hp
+      ((Real.pi / 2 - d + Real.pi : ℝ) : Real.Angle)
+    rw [normalVector_add_pi, inner_neg_right] at h
+    change -supportValue s ((Real.pi / 2 - d + Real.pi : ℝ) : Real.Angle) ≤ _
+    linarith
+  · intro d hd0 hd1 p hp
+    have ht : ω ≤ Real.pi / 2 - d := by linarith
+    have ht' : Real.pi / 2 - d ≤ Real.pi / 2 := by linarith
+    have hstep : supportValue s ((Real.pi / 2 - d + Real.pi : ℝ) : Real.Angle) ≤
+        1 - inner ℝ p (normalVector ((Real.pi / 2 - d : ℝ) : Real.Angle)) := by
+      refine csSup_le (hne.image _) ?_
+      rintro _ ⟨q, hq, rfl⟩
+      simp only [normalVector_add_pi, inner_neg_right]
+      have h := hwidth (Real.pi / 2 - d) ht ht' p hp q hq
+      linarith
+    rw [hcoord1]
+    change inner ℝ p (normalVector ((Real.pi / 2 - d : ℝ) : Real.Angle)) -
+      -supportValue s ((Real.pi / 2 - d + Real.pi : ℝ) : Real.Angle) ≤ 1
+    linarith
+
+/-- A sofa of rotation angle `ω` that can be placed at height between zero and one in every
+intermediate rotated position admits, after rotating by `π/2 - ω`, a motion of rotation
+angle `π/2`. -/
+private theorem hasRotationAngle_rotationMap_of_normalization {s : Set Point} {ω : ℝ}
+    (hlt : ω < Real.pi / 2) (hconn : IsConnected s) (hcpt : IsCompact s)
+    (hrot : HasRotationAngle s ω) (low : ℝ → ℝ) (M : ℝ) (hlowcont : Continuous low)
+    (hhoriz : ∀ d : ℝ, ∀ p ∈ s, (rotationMap ((d : ℝ) : Real.Angle) p) 0 ≤ M)
+    (hloleq : ∀ d : ℝ, ∀ p ∈ s, low d ≤ (rotationMap ((d : ℝ) : Real.Angle) p) 1)
+    (hupleq : ∀ d : ℝ, 0 ≤ d → d ≤ Real.pi / 2 - ω → ∀ p ∈ s,
+      (rotationMap ((d : ℝ) : Real.Angle) p) 1 - low d ≤ 1) :
+    HasRotationAngle (rotationMap ((Real.pi / 2 - ω : ℝ) : Real.Angle) '' s)
+      (Real.pi / 2) := by
+  have hrot0 : ∀ p : Point, rotationMap ((0 : ℝ) : Real.Angle) p = p := by
+    intro p
+    rw [rotationMap, Real.Angle.coe_zero, Orientation.rotation_zero]
+    rfl
+  set δ₀ : ℝ := Real.pi / 2 - ω with hδ₀def
+  have hδ₀pos : 0 < δ₀ := by rw [hδ₀def]; linarith
+  -- the original motion witnessing rotation angle `ω`
+  obtain ⟨m₀, hm₀, α₀, hα₀c, hα₀0, hα₀1, hm₀lift⟩ := hrot
+  obtain ⟨v, hv⟩ := hm₀.2.2.2.1
+  have hm₀0 : m₀ 0 0 = v := by rw [hv]; exact zero_add v
+  have hinit : ∀ p ∈ s, p 0 + v 0 ≤ 1 ∧ 0 ≤ p 1 + v 1 ∧ p 1 + v 1 ≤ 1 := by
+    intro p hp
+    exact mem_horizontalHallway_coordinates (hm₀.2.2.2.2.2.1 ⟨p, hp, hv p⟩)
+  have hmcam : Continuous
+      (fun t : unitInterval ↦ (m₀ t).toAffineIsometry.toContinuousAffineMap) :=
+    continuous_induced_dom.comp hm₀.2.2.1
+  -- the reparametrized motion of the rotated sofa
+  have hτmem : ∀ r : unitInterval, max (3 * (r : ℝ) - 2) 0 ∈ unitInterval :=
+    fun r ↦ ⟨le_max_right _ _, max_le (by linarith [r.2.2]) (by norm_num)⟩
+  set τ : unitInterval → unitInterval := fun r ↦ ⟨max (3 * (r : ℝ) - 2) 0, hτmem r⟩ with hτdef
+  set Θ : unitInterval → ℝ := fun r ↦ -δ₀ * min (3 * (r : ℝ)) 1 + α₀ (τ r) with hΘdef
+  set dd : unitInterval → ℝ := fun r ↦ δ₀ * (1 - min (3 * (r : ℝ)) 1) with hdddef
+  set σ : unitInterval → ℝ := fun r ↦ min (max (3 * (r : ℝ) - 1) 0) 1 with hσdef
+  set C : unitInterval → Point := fun r ↦
+    (1 - σ r) • (!₂[-M, -low (dd r)] : Point) + σ r • v + (m₀ (τ r) 0 - v) with hCdef
+  set n : unitInterval → Point ≃ᵃⁱ[ℝ] Point := fun r ↦
+    (EuclideanGeometry.o.rotation ((Θ r : ℝ) : Real.Angle)).toAffineIsometryEquiv.trans
+      (AffineIsometryEquiv.vaddConst ℝ (C r)) with hndef
+  have hnapp : ∀ (r : unitInterval) (p : Point),
+      n r p = rotationMap ((Θ r : ℝ) : Real.Angle) p + C r := fun _ _ ↦ rfl
+  have hn0eq : ∀ r : unitInterval, n r 0 = C r := by
+    intro r
+    rw [hnapp, rotationMap]
+    simp
+  have hcompose : ∀ (r : unitInterval) (p : Point),
+      n r (rotationMap ((δ₀ : ℝ) : Real.Angle) p) =
+        rotationMap ((dd r + α₀ (τ r) : ℝ) : Real.Angle) p + C r := by
+    intro r p
+    have hΘsum : Θ r + δ₀ = dd r + α₀ (τ r) := by
+      change -δ₀ * min (3 * (r : ℝ)) 1 + α₀ (τ r) + δ₀ =
+        δ₀ * (1 - min (3 * (r : ℝ)) 1) + α₀ (τ r)
+      ring
+    rw [hnapp]
+    congr 1
+    change (EuclideanGeometry.o.rotation ((Θ r : ℝ) : Real.Angle))
+        ((EuclideanGeometry.o.rotation ((δ₀ : ℝ) : Real.Angle)) p) =
+      (EuclideanGeometry.o.rotation ((dd r + α₀ (τ r) : ℝ) : Real.Angle)) p
+    rw [Orientation.rotation_rotation, ← Real.Angle.coe_add, hΘsum]
+  have hτzero : ∀ r : unitInterval, 3 * (r : ℝ) ≤ 2 → τ r = 0 := by
+    intro r hr
+    apply Subtype.ext
+    change max (3 * (r : ℝ) - 2) 0 = 0
+    exact max_eq_right (by linarith)
+  -- phase one: rotating down to the horizontal position
+  have hphaseA : ∀ r : unitInterval, 3 * (r : ℝ) ≤ 1 → ∀ p ∈ s,
+      n r (rotationMap ((δ₀ : ℝ) : Real.Angle) p) ∈ horizontalHallway := by
+    intro r hr p hp
+    have hr0 : (0 : ℝ) ≤ (r : ℝ) := r.2.1
+    have hτ0 : τ r = 0 := hτzero r (by linarith)
+    have hσ0 : σ r = 0 := by
+      change min (max (3 * (r : ℝ) - 1) 0) 1 = 0
+      rw [max_eq_right (by linarith)]
+      exact min_eq_left (by norm_num)
+    have hdd : dd r = δ₀ * (1 - 3 * (r : ℝ)) := by
+      change δ₀ * (1 - min (3 * (r : ℝ)) 1) = _
+      rw [min_eq_left hr]
+    have hdd0 : 0 ≤ dd r := by
+      rw [hdd]
+      exact mul_nonneg hδ₀pos.le (by linarith)
+    have hdd1 : dd r ≤ δ₀ := by
+      have h := mul_le_mul_of_nonneg_left (show 1 - 3 * (r : ℝ) ≤ 1 by linarith) hδ₀pos.le
+      rw [hdd]
+      linarith [h]
+    have hC : C r = (!₂[-M, -low (dd r)] : Point) := by
+      change (1 - σ r) • (!₂[-M, -low (dd r)] : Point) + σ r • v + (m₀ (τ r) 0 - v) = _
+      rw [hσ0, hτ0, hm₀0]
+      simp
+    have heval : n r (rotationMap ((δ₀ : ℝ) : Real.Angle) p) =
+        rotationMap ((dd r : ℝ) : Real.Angle) p + (!₂[-M, -low (dd r)] : Point) := by
+      rw [hcompose, hτ0, hα₀0, add_zero, hC]
+    rw [heval]
+    refine mem_horizontalHallway_of_coordinates _ ?_ ⟨?_, ?_⟩
+    · change (rotationMap ((dd r : ℝ) : Real.Angle) p) 0 + -M ≤ 1
+      have h := hhoriz (dd r) p hp
+      linarith
+    · change (0 : ℝ) ≤ (rotationMap ((dd r : ℝ) : Real.Angle) p) 1 + -low (dd r)
+      have h := hloleq (dd r) p hp
+      linarith
+    · change (rotationMap ((dd r : ℝ) : Real.Angle) p) 1 + -low (dd r) ≤ 1
+      have h := hupleq (dd r) hdd0 hdd1 p hp
+      linarith
+  -- phase two: translating to the initial placement of the original motion
+  have hphaseB : ∀ r : unitInterval, 1 ≤ 3 * (r : ℝ) → 3 * (r : ℝ) ≤ 2 → ∀ p ∈ s,
+      n r (rotationMap ((δ₀ : ℝ) : Real.Angle) p) ∈ horizontalHallway := by
+    intro r hr hr' p hp
+    have hτ0 : τ r = 0 := hτzero r hr'
+    have hσ : σ r = 3 * (r : ℝ) - 1 := by
+      change min (max (3 * (r : ℝ) - 1) 0) 1 = _
+      rw [max_eq_left (by linarith)]
+      exact min_eq_left (by linarith)
+    have hσ0 : 0 ≤ σ r := by rw [hσ]; linarith
+    have hσ1 : σ r ≤ 1 := by rw [hσ]; linarith
+    have hdd : dd r = 0 := by
+      change δ₀ * (1 - min (3 * (r : ℝ)) 1) = 0
+      rw [min_eq_right hr]
+      ring
+    have hC : C r = (1 - σ r) • (!₂[-M, -low 0] : Point) + σ r • v := by
+      change (1 - σ r) • (!₂[-M, -low (dd r)] : Point) + σ r • v + (m₀ (τ r) 0 - v) = _
+      rw [hdd, hτ0, hm₀0]
+      simp
+    have heval : n r (rotationMap ((δ₀ : ℝ) : Real.Angle) p) =
+        p + ((1 - σ r) • (!₂[-M, -low 0] : Point) + σ r • v) := by
+      rw [hcompose, hτ0, hα₀0, add_zero, hdd, hrot0, hC]
+    have h1 := hloleq 0 p hp
+    have h2 := hupleq 0 le_rfl hδ₀pos.le p hp
+    have hhz := hhoriz 0 p hp
+    rw [hrot0] at h1 h2 hhz
+    obtain ⟨hi1, hi2, hi3⟩ := hinit p hp
+    rw [heval]
+    refine mem_horizontalHallway_of_coordinates _ ?_ ⟨?_, ?_⟩
+    · change p 0 + ((1 - σ r) * -M + σ r * v 0) ≤ 1
+      have k1 : 0 ≤ (1 - σ r) * (M - p 0) := mul_nonneg (by linarith) (by linarith)
+      have k2 : 0 ≤ σ r * (1 - (p 0 + v 0)) := mul_nonneg hσ0 (by linarith)
+      nlinarith only [k1, k2, hσ1]
+    · change (0 : ℝ) ≤ p 1 + ((1 - σ r) * -low 0 + σ r * v 1)
+      have k3 : 0 ≤ (1 - σ r) * (p 1 - low 0) := mul_nonneg (by linarith) (by linarith)
+      have k4 : 0 ≤ σ r * (p 1 + v 1) := mul_nonneg hσ0 (by linarith)
+      nlinarith only [k3, k4]
+    · change p 1 + ((1 - σ r) * -low 0 + σ r * v 1) ≤ 1
+      have k5 : 0 ≤ (1 - σ r) * (1 - (p 1 - low 0)) := mul_nonneg (by linarith) (by linarith)
+      have k6 : 0 ≤ σ r * (1 - (p 1 + v 1)) := mul_nonneg hσ0 (by linarith)
+      nlinarith only [k5, k6]
+  -- phase three: the original motion
+  have hphaseC : ∀ r : unitInterval, 2 ≤ 3 * (r : ℝ) → ∀ p : Point,
+      n r (rotationMap ((δ₀ : ℝ) : Real.Angle) p) = m₀ (τ r) p := by
+    intro r hr p
+    have hσ1 : σ r = 1 := by
+      change min (max (3 * (r : ℝ) - 1) 0) 1 = 1
+      rw [max_eq_left (by linarith)]
+      exact min_eq_right (by linarith)
+    have hdd : dd r = 0 := by
+      change δ₀ * (1 - min (3 * (r : ℝ)) 1) = 0
+      rw [min_eq_right (by linarith)]
+      ring
+    have hC : C r = m₀ (τ r) 0 := by
+      change (1 - σ r) • (!₂[-M, -low (dd r)] : Point) + σ r • v + (m₀ (τ r) 0 - v) = _
+      rw [hσ1]
+      simp
+    rw [hcompose, hC, hdd, zero_add, hm₀lift (τ r) p]
+  -- continuity of the reparametrized motion
+  have hτcont : Continuous τ := by
+    rw [hτdef]
+    exact Continuous.subtype_mk (by fun_prop) _
+  have hσcont : Continuous σ := by rw [hσdef]; fun_prop
+  have hddcont : Continuous dd := by rw [hdddef]; fun_prop
+  have hΘcont : Continuous Θ := by
+    rw [hΘdef]
+    exact (continuous_const.mul (by fun_prop)).add (hα₀c.comp hτcont)
+  have hbasecont : Continuous (fun r : unitInterval ↦ (!₂[-M, -low (dd r)] : Point)) := by
+    have h : Continuous (fun r : unitInterval ↦ -low (dd r)) := (hlowcont.comp hddcont).neg
+    fun_prop
+  have hCcont : Continuous C := by
+    rw [hCdef]
+    exact (((continuous_const.sub hσcont).smul hbasecont).add
+        (hσcont.smul continuous_const)).add
+      (((hmcam.eval_const 0).comp hτcont).sub continuous_const)
+  have hncont : Continuous n := by
+    rw [hndef]
+    exact continuous_rotation_trans_vaddConst (Real.Angle.continuous_coe.comp hΘcont) hCcont
+  -- endpoint values
+  have hτ1 : τ 1 = 1 := by
+    apply Subtype.ext
+    change max (3 * ((1 : unitInterval) : ℝ) - 2) 0 = ((1 : unitInterval) : ℝ)
+    norm_num
+  have hΘ0 : Θ 0 = 0 := by
+    change -δ₀ * min (3 * ((0 : unitInterval) : ℝ)) 1 + α₀ (τ 0) = 0
+    rw [hτzero 0 (by norm_num), hα₀0]
+    norm_num
+  have hΘ1 : Θ 1 = -(Real.pi / 2) := by
+    change -δ₀ * min (3 * ((1 : unitInterval) : ℝ)) 1 + α₀ (τ 1) = -(Real.pi / 2)
+    rw [hτ1, hα₀1, show (3 : ℝ) * ((1 : unitInterval) : ℝ) = 3 by norm_num,
+      min_eq_right (by norm_num : (1 : ℝ) ≤ 3), hδ₀def]
+    ring
+  have hrotcont : Continuous (rotationMap ((δ₀ : ℝ) : Real.Angle)) :=
+    (EuclideanGeometry.o.rotation ((δ₀ : ℝ) : Real.Angle)).continuous
+  refine ⟨n, ⟨hconn.image _ hrotcont.continuousOn, (hcpt.image hrotcont).isClosed, hncont,
+    ⟨C 0, ?_⟩, ?_, ?_, ?_, ?_⟩, Θ, hΘcont, hΘ0, hΘ1, ?_⟩
+  · intro p
+    rw [hnapp, hΘ0, hrot0]
+  · intro t
+    exact ⟨((Θ t : ℝ) : Real.Angle), fun p ↦ by rw [hnapp, hn0eq]⟩
+  · rintro _ ⟨_, ⟨p, hp, rfl⟩, rfl⟩
+    exact hphaseA 0 (by norm_num) p hp
+  · intro r
+    rintro _ ⟨_, ⟨p, hp, rfl⟩, rfl⟩
+    rcases le_total (3 * (r : ℝ)) 1 with h | h
+    · exact Set.subset_union_left (hphaseA r h p hp)
+    · rcases le_total (3 * (r : ℝ)) 2 with h' | h'
+      · exact Set.subset_union_left (hphaseB r h h' p hp)
+      · rw [hphaseC r h' p]
+        exact hm₀.2.2.2.2.2.2.1 (τ r) ⟨p, hp, rfl⟩
+  · rintro _ ⟨_, ⟨p, hp, rfl⟩, rfl⟩
+    rw [hphaseC 1 (by norm_num) p, hτ1]
+    exact hm₀.2.2.2.2.2.2.2 ⟨p, hp, rfl⟩
+  · intro r p
+    rw [hnapp, hn0eq]
+
+theorem balancedMaximumSofa_rightAngle (s : Set Point) (ω : ℝ)
+    (hs : IsBalancedMaximumSofa s ω)
+    (hω : ω ∈ Set.Icc (Real.arccos (5 / 11 : ℝ)) (Real.pi / 2))
+    (hArea : 11 / 5 ≤ ClassicalResults.area s) :
+    ∃ α : Real.Angle, HasRotationAngle (rotationMap α '' s) (Real.pi / 2) := by
+  have hpi := Real.pi_pos
+  obtain ⟨⟨s₀, hs₀, hsdef⟩, K, hKcap, hKbal⟩ := hs
+  have hstd : IsStandardPosition s ω := by
+    rw [hsdef]
+    exact (standardPosition_monotonization_standard s₀ ω hs₀).1
+  have hω0 : 0 < ω := hstd.2.2.1
+  have hconn : IsConnected s := by
+    rw [hsdef]
+    exact standardPosition_monotonization_connected s₀ ω hs₀
+  have hcpt : IsCompact s := hstd.1
+  have hrot : HasRotationAngle s ω := hstd.2.1
+  rcases eq_or_lt_of_le hω.2 with heq | hlt
+  · refine ⟨((0 : ℝ) : Real.Angle), ?_⟩
+    have hrot0 : ∀ p : Point, rotationMap ((0 : ℝ) : Real.Angle) p = p := by
+      intro p
+      rw [rotationMap, Real.Angle.coe_zero, Orientation.rotation_zero]
+      rfl
+    rw [show rotationMap ((0 : ℝ) : Real.Angle) = id from funext hrot0, Set.image_id]
+    exact heq ▸ hrot
+  -- the consumed corner triangle of the balanced maximum cap
+  have hcos : 0 < Real.cos ω := Real.cos_pos_of_mem_Ioo ⟨by linarith, hlt⟩
+  have hstrip : s ⊆ (stripParallelogram ω).1 := by
+    rw [hsdef]
+    exact Set.inter_subset_left
+  have hareaK : (11 : ℝ) / 5 ≤ capAreaFunctional K := by
+    rw [capAreaFunctional_eq_sofaArea s ω ⟨s₀, hs₀, hsdef⟩ K hKcap]
+    exact hArea
+  have hsK : s = (K.val : Set Point) \ capNiche K :=
+    monotoneSofa_structure s ω ⟨s₀, hs₀, hsdef⟩ K hKcap
+  obtain ⟨t₀, ht₀, hsub3⟩ := balancedMaximumCap_consumed K hω.1 hlt hKbal hareaK
+  rw [rotatingHallwayParts_innerQuadrant] at hsub3
+  have hgap := parallelogram_gap ω ⟨hω0.le, hlt⟩
+  have hccos : Real.tan ((Real.pi / 2 - ω) / 2) * Real.cos ω = 1 - Real.sin ω := by
+    rw [hgap.2.2.2.2, Real.tan_eq_sin_div_cos]
+    field_simp
+  have hsinlt : Real.sin ω < 1 := by nlinarith [Real.sin_sq_add_cos_sq ω]
+  have hcpos : 0 < Real.tan ((Real.pi / 2 - ω) / 2) := by nlinarith
+  have hmem_capFan : ∀ z : Point, 0 ≤ inner ℝ z (normalVector (ω : Real.Angle)) →
+      0 ≤ inner ℝ z (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) → z ∈ capFan ω :=
+    fun _ h1 h2 ↦ ⟨h1, h2⟩
+  have hfan0 : (0 : Point) ∈ capFan ω := hmem_capFan 0 (by simp) (by simp)
+  have hfana : (stripParallelogram ω).2.2 - tangentVector 0 ∈ capFan ω := by
+    rw [hgap.1]
+    refine hmem_capFan _ ?_ ?_
+    · rw [real_inner_smul_left, ← Real.Angle.coe_zero, inner_normalVector_normalVector,
+        zero_sub, Real.cos_neg]
+      exact mul_nonneg hcpos.le hcos.le
+    · rw [real_inner_smul_left, ← Real.Angle.coe_zero, inner_normalVector_normalVector,
+        zero_sub, Real.cos_neg, Real.cos_pi_div_two, mul_zero]
+  have hfanb : (stripParallelogram ω).2.2 - normalVector (ω : Real.Angle) ∈ capFan ω := by
+    rw [hgap.2.1]
+    refine hmem_capFan _ ?_ ?_
+    · rw [real_inner_smul_left, inner_tangentVector_normalVector_real, sub_self,
+        Real.sin_zero, mul_zero]
+    · rw [real_inner_smul_left, inner_tangentVector_normalVector_real,
+        Real.sin_pi_div_two_sub]
+      exact mul_nonneg hcpos.le hcos.le
+  have hUconv : Convex ℝ (capFan ω ∩ innerQuadrant (K.val : Set Point) t₀) :=
+    (convex_capFan ω).inter (convex_innerQuadrant _ _)
+  have hUniche : capFan ω ∩ innerQuadrant (K.val : Set Point) t₀ ⊆ capNiche K :=
+    fun z hz ↦ ⟨hz.1, Set.mem_iUnion₂.mpr ⟨t₀, ht₀, hz.2⟩⟩
+  have h0U : (0 : Point) ∈ capFan ω ∩ innerQuadrant (K.val : Set Point) t₀ :=
+    ⟨hfan0, hsub3 (by simp)⟩
+  have haU : (stripParallelogram ω).2.2 - tangentVector 0 ∈
+      capFan ω ∩ innerQuadrant (K.val : Set Point) t₀ := ⟨hfana, hsub3 (by simp)⟩
+  have hbU : (stripParallelogram ω).2.2 - normalVector (ω : Real.Angle) ∈
+      capFan ω ∩ innerQuadrant (K.val : Set Point) t₀ := ⟨hfanb, hsub3 (by simp)⟩
+  have hlowbd : ∀ p ∈ s, 1 - Real.sin ω ≤ inner ℝ p (normalVector (ω : Real.Angle)) + p 1 := by
+    intro p hp
+    by_contra hcon
+    push Not at hcon
+    obtain ⟨hy, hx⟩ := (mem_stripParallelogram_iff ω p).1 (hstrip hp)
+    have hmem := hUniche (mem_of_inner_add_le_one_sub_sin hω0 hlt hUconv h0U haU hbU
+      hy.1 hx.1 hcon.le)
+    rw [hsK] at hp
+    exact hp.2 hmem
+  have hwidth : ∀ t : ℝ, ω ≤ t → t ≤ Real.pi / 2 → ∀ p ∈ s, ∀ q ∈ s,
+      inner ℝ p (normalVector (t : Real.Angle)) -
+        inner ℝ q (normalVector (t : Real.Angle)) ≤ 1 := by
+    intro t ht ht' p hp q hq
+    obtain ⟨hp1, hp2⟩ := (mem_stripParallelogram_iff ω p).1 (hstrip hp)
+    obtain ⟨hq1, hq2⟩ := (mem_stripParallelogram_iff ω q).1 (hstrip hq)
+    exact inner_sub_inner_le_one_of_clipped hω0.le hcos hp2.2 hp1.2 hq2.1 hq1.1
+      (hlowbd q hq) ht ht'
+  obtain ⟨low, M, hlowcont, hhoriz, hloleq, hupleq⟩ :=
+    exists_normalization_of_width hcpt hconn.nonempty hwidth
+  exact ⟨((Real.pi / 2 - ω : ℝ) : Real.Angle),
+    hasRotationAngle_rotationMap_of_normalization hlt hconn hcpt hrot low M hlowcont
+      hhoriz hloleq hupleq⟩
+
+private theorem hasRotationAngle_right_of_reaches_right (S : Set Point)
+    (m : unitInterval → Point ≃ᵃⁱ[ℝ] Point) (hm : IsMovingSofa S m)
+    (α : unitInterval → ℝ) (e : unitInterval → Point)
+    (hαc : Continuous α) (hα0 : α 0 = 0)
+    (hlift : ∀ t p, m t p = rotationMap (α t : Real.Angle) p + e t)
+    (ts : unitInterval) (hts : α ts = -(Real.pi / 2)) :
+    HasRotationAngle S (Real.pi / 2) := by
+  have hcoord : ∀ (u v : Point) (i : Fin 2), (u + v) i = u i + v i := fun _ _ _ ↦ rfl
+  have hH : ∀ p ∈ S, 0 ≤ p 1 ∧ p 1 ≤ 1 := by
+    intro p hp
+    obtain ⟨-, h1, h2⟩ := mem_horizontalHallway_coordinates (hm.initial hp)
+    exact ⟨h1, h2⟩
+  have hLt : ∀ (t : unitInterval) (p : Point), p ∈ S →
+      (m t p) 0 ≤ 1 ∧ (m t p) 1 ≤ 1 ∧ (0 ≤ (m t p) 0 ∨ 0 ≤ (m t p) 1) := by
+    intro t p hp
+    obtain ⟨⟨h1, h2⟩, h3⟩ := (mem_hallway_iff _).mp (hm.subset_hallway t ⟨p, hp, rfl⟩)
+    exact ⟨h1, h2, h3⟩
+  have hcs : ((α ts : ℝ) : Real.Angle).cos = 0 := by
+    rw [Real.Angle.cos_coe, hts]
+    simp
+  have hsns : ((α ts : ℝ) : Real.Angle).sin = -1 := by
+    rw [Real.Angle.sin_coe, hts]
+    simp
+  have hmts0 : ∀ p : Point, (m ts p) 0 = p 1 + (e ts) 0 := by
+    intro p
+    rw [hlift ts p, hcoord, rotationMap_apply_zero, hcs, hsns]
+    ring
+  have hmts1 : ∀ p : Point, (m ts p) 1 = -p 0 + (e ts) 1 := by
+    intro p
+    rw [hlift ts p, hcoord, rotationMap_apply_one, hcs, hsns]
+    ring
+  -- the supremum of the vertical coordinates of the initial placement
+  obtain ⟨p₀, hp₀⟩ := hm.isConnected.nonempty
+  have hTne : ((fun p : Point ↦ p 1) '' S).Nonempty := ⟨p₀ 1, p₀, hp₀, rfl⟩
+  have hTbdd : BddAbove ((fun p : Point ↦ p 1) '' S) := by
+    refine ⟨1, ?_⟩
+    rintro _ ⟨p, hp, rfl⟩
+    exact (hH p hp).2
+  set N : ℝ := sSup ((fun p : Point ↦ p 1) '' S) with hNdef
+  have hNub : ∀ p ∈ S, p 1 ≤ N := fun p hp ↦ le_csSup hTbdd ⟨p, hp, rfl⟩
+  have hNle : N ≤ 1 := by
+    refine csSup_le hTne ?_
+    rintro _ ⟨p, hp, rfl⟩
+    exact (hH p hp).2
+  have hNd : N + (e ts) 0 ≤ 1 := by
+    have hle : N ≤ 1 - (e ts) 0 := by
+      refine csSup_le hTne ?_
+      rintro _ ⟨p, hp, rfl⟩
+      have hx := (hLt ts p hp).1
+      rw [hmts0 p] at hx
+      linarith
+    linarith
+  set δ : ℝ := 1 - N - (e ts) 0 with hδdef
+  have hδ : 0 ≤ δ := by
+    rw [hδdef]
+    linarith
+  -- the reparametrized motion, followed by the horizontal translation
+  have hφmem : ∀ r : unitInterval, min (2 * (r : ℝ)) 1 * (ts : ℝ) ∈ unitInterval := by
+    intro r
+    constructor
+    · exact mul_nonneg (le_min (by linarith [r.2.1]) (by norm_num)) ts.2.1
+    · calc min (2 * (r : ℝ)) 1 * (ts : ℝ) ≤ 1 * 1 :=
+            mul_le_mul (min_le_right _ _) ts.2.2 ts.2.1 (by norm_num)
+      _ = 1 := one_mul 1
+  set φ : unitInterval → unitInterval :=
+    fun r ↦ ⟨min (2 * (r : ℝ)) 1 * (ts : ℝ), hφmem r⟩ with hφdef
+  set ψ : unitInterval → ℝ := fun r ↦ max (2 * (r : ℝ) - 1) 0 with hψdef
+  set w : Point := !₂[δ, 0] with hwdef
+  have hw0 : w 0 = δ := rfl
+  have hw1 : w 1 = 0 := rfl
+  set n : unitInterval → Point ≃ᵃⁱ[ℝ] Point :=
+    fun r ↦ (m (φ r)).trans (AffineIsometryEquiv.vaddConst ℝ (ψ r • w)) with hndef
+  have hnapp : ∀ (r : unitInterval) (p : Point), n r p = m (φ r) p + ψ r • w := fun _ _ ↦ rfl
+  have hφcont : Continuous φ := by
+    rw [hφdef]
+    exact Continuous.subtype_mk (by fun_prop) _
+  have hψcont : Continuous ψ := by
+    rw [hψdef]
+    fun_prop
+  have hφ0 : φ 0 = 0 := by
+    apply Subtype.ext
+    change min (2 * ((0 : unitInterval) : ℝ)) 1 * (ts : ℝ) = 0
+    norm_num
+  have hφ1 : φ 1 = ts := by
+    apply Subtype.ext
+    change min (2 * ((1 : unitInterval) : ℝ)) 1 * (ts : ℝ) = (ts : ℝ)
+    norm_num
+  have hψ0 : ψ 0 = 0 := by
+    change max (2 * ((0 : unitInterval) : ℝ) - 1) 0 = 0
+    norm_num
+  have hψ1 : ψ 1 = 1 := by
+    change max (2 * ((1 : unitInterval) : ℝ) - 1) 0 = 1
+    norm_num
+  have hψnn : ∀ r : unitInterval, 0 ≤ ψ r := fun r ↦ le_max_right _ _
+  have hψle : ∀ r : unitInterval, ψ r ≤ 1 := by
+    intro r
+    exact max_le (by linarith [r.2.2]) (by norm_num)
+  have hψpos : ∀ r : unitInterval, ψ r ≠ 0 → φ r = ts := by
+    intro r hr
+    have hlt : 1 < 2 * (r : ℝ) := by
+      by_contra hcon
+      push Not at hcon
+      apply hr
+      change max (2 * (r : ℝ) - 1) 0 = 0
+      exact max_eq_right (by linarith)
+    apply Subtype.ext
+    change min (2 * (r : ℝ)) 1 * (ts : ℝ) = (ts : ℝ)
+    rw [min_eq_right hlt.le, one_mul]
+  -- the hallway containment of the new motion
+  have hnhall : ∀ (r : unitInterval), n r '' S ⊆ hallway := by
+    rintro r _ ⟨p, hp, rfl⟩
+    obtain ⟨hy0, hy1⟩ := hH p hp
+    obtain ⟨hx1, hx2, hx3⟩ := hLt (φ r) p hp
+    by_cases hr : ψ r = 0
+    · have heq : n r p = m (φ r) p := by
+        rw [hnapp, hr, zero_smul, add_zero]
+      rw [heq]
+      exact (mem_hallway_iff _).mpr ⟨⟨hx1, hx2⟩, hx3⟩
+    · have hφr := hψpos r hr
+      obtain ⟨hz1, hz2, hz3⟩ := hLt ts p hp
+      have hc0 : (n r p) 0 = p 1 + (e ts) 0 + ψ r * δ := by
+        rw [hnapp, hcoord, hφr, hmts0 p]
+        change p 1 + (e ts) 0 + ψ r * w 0 = _
+        rw [hw0]
+      have hc1 : (n r p) 1 = -p 0 + (e ts) 1 := by
+        rw [hnapp, hcoord, hφr, hmts1 p]
+        change -p 0 + (e ts) 1 + ψ r * w 1 = _
+        rw [hw1]
+        ring
+      rw [hmts0 p] at hz1 hz3
+      rw [hmts1 p] at hz2 hz3
+      refine (mem_hallway_iff _).mpr ⟨⟨?_, ?_⟩, ?_⟩
+      · rw [hc0]
+        have hmul : ψ r * δ ≤ 1 * δ := mul_le_mul_of_nonneg_right (hψle r) hδ
+        have := hNub p hp
+        rw [hδdef] at hmul
+        linarith
+      · rw [hc1]
+        linarith
+      · rcases hz3 with h | h
+        · left
+          rw [hc0]
+          have : 0 ≤ ψ r * δ := mul_nonneg (hψnn r) hδ
+          linarith
+        · right
+          rw [hc1]
+          linarith
+  have hnfinal : n 1 '' S ⊆ verticalHallway := by
+    rintro _ ⟨p, hp, rfl⟩
+    obtain ⟨hy0, hy1⟩ := hH p hp
+    obtain ⟨hz1, hz2, hz3⟩ := hLt ts p hp
+    rw [hmts1 p] at hz2
+    have hc0 : (n 1 p) 0 = p 1 + (e ts) 0 + δ := by
+      rw [hnapp, hcoord, hφ1, hmts0 p, hψ1]
+      change p 1 + (e ts) 0 + 1 * w 0 = _
+      rw [hw0]
+      ring
+    have hc1 : (n 1 p) 1 = -p 0 + (e ts) 1 := by
+      rw [hnapp, hcoord, hφ1, hmts1 p, hψ1]
+      change -p 0 + (e ts) 1 + 1 * w 1 = _
+      rw [hw1]
+      ring
+    refine mem_verticalHallway_of_coordinates _ ⟨?_, ?_⟩ ?_
+    · rw [hc0, hδdef]
+      linarith
+    · rw [hc0, hδdef]
+      have := hNub p hp
+      linarith
+    · rw [hc1]
+      linarith
+  have hmcam : Continuous (fun t : unitInterval ↦ (m t).toAffineIsometry.toContinuousAffineMap) :=
+    continuous_induced_dom.comp hm.continuous
+  have hncont : Continuous n := by
+    rw [continuous_induced_rng]
+    refine ContinuousAffineMap.continuous_rng (fun p ↦ ?_) ?_
+    · change Continuous fun r : unitInterval ↦ m (φ r) p + ψ r • w
+      exact ((hmcam.comp hφcont).eval_const p).add (hψcont.smul continuous_const)
+    · change Continuous fun r : unitInterval ↦
+        ((m (φ r)).toAffineIsometry.toContinuousAffineMap).contLinear
+      exact ContinuousAffineMap.continuous_contLinear.comp (hmcam.comp hφcont)
+  refine ⟨n, ⟨hm.isConnected, hm.isClosed, hncont, ⟨0, ?_⟩, ?_, ?_, hnhall, hnfinal⟩,
+    fun r ↦ α (φ r), hαc.comp hφcont, ?_, ?_, ?_⟩
+  · intro p
+    rw [hnapp, hφ0, hψ0, zero_smul, add_zero, hm.zero]
+    simp
+  · intro r
+    exact ⟨((α (φ r) : ℝ) : Real.Angle), fun p ↦ by
+      rw [hnapp, hnapp, hlift (φ r) p, hlift (φ r) 0]
+      simp [rotationMap]
+      abel⟩
+  · rintro _ ⟨p, hp, rfl⟩
+    have hid : n 0 p = p := by
+      rw [hnapp, hφ0, hψ0, zero_smul, add_zero, hm.zero]
+      rfl
+    rw [hid]
+    exact hm.initial hp
+  · change α (φ 0) = 0
+    rw [hφ0, hα0]
+  · change α (φ 1) = -(Real.pi / 2)
+    rw [hφ1, hts]
+  · intro r p
+    rw [hnapp, hnapp, hlift (φ r) p, hlift (φ r) 0]
+    simp [rotationMap]
+    abel
+
+theorem movingSofa_rotationAngle_bound (s : Set Point)
+    (hs : IsPaperMovingSofa s) (hArea : 11 / 5 ≤ ClassicalResults.area s) :
+    ∃ ω ∈ Set.Icc (Real.arccos (5 / 11 : ℝ)) (Real.pi / 2),
+      HasRotationAngle s ω := by
+  obtain ⟨q, m, hm, hvol⟩ := canonical_paper_motion_bridge.1 s hs
+  set S : Set Point := (fun p ↦ p + q) '' s with hSdef
+  obtain ⟨α, e, hαc, hec, hα0, he0, hlift⟩ :=
+    exists_continuous_motion_angle_lift m hm.continuous hm.zero
+  have hcoord : ∀ (u v : Point) (i : Fin 2), (u + v) i = u i + v i := fun _ _ _ ↦ rfl
+  have htransfer : ∀ ω : ℝ, HasRotationAngle S ω → HasRotationAngle s ω := by
+    intro ω hω
+    have h := hasRotationAngle_image_add S (-q) ω hω
+    have himg : (fun p : Point ↦ p + -q) '' S = s := by
+      rw [hSdef, Set.image_image]
+      simp
+    rwa [himg] at h
+  have harea : (11 : ℝ) / 5 ≤ (MeasureTheory.volume S).toReal := by
+    rw [hSdef, hvol]
+    exact hArea
+  have hH : ∀ p ∈ S, 0 ≤ p 1 ∧ p 1 ≤ 1 := by
+    intro p hp
+    obtain ⟨-, h1, h2⟩ := mem_horizontalHallway_coordinates (hm.initial hp)
+    exact ⟨h1, h2⟩
+  have hLt : ∀ (t : unitInterval) (p : Point), p ∈ S →
+      (m t p) 0 ≤ 1 ∧ (m t p) 1 ≤ 1 ∧ (0 ≤ (m t p) 0 ∨ 0 ≤ (m t p) 1) := by
+    intro t p hp
+    obtain ⟨⟨h1, h2⟩, h3⟩ := (mem_hallway_iff _).mp (hm.subset_hallway t ⟨p, hp, rfl⟩)
+    exact ⟨h1, h2, h3⟩
+  have hVf : ∀ p ∈ S, 0 ≤ (m 1 p) 0 ∧ (m 1 p) 0 ≤ 1 ∧ (m 1 p) 1 ≤ 1 :=
+    fun p hp ↦ mem_verticalHallway_coordinates (hm.final ⟨p, hp, rfl⟩)
+  -- the clockwise angle never reaches a counterclockwise quarter of a right angle
+  have hquarter : ∀ t : unitInterval, α t ≠ Real.pi / 4 := by
+    intro t ht
+    set r := Real.sqrt 2 with hrdef
+    have hrsq : r * r = 2 := Real.mul_self_sqrt (by norm_num)
+    have hrpos : 0 < r := by rw [hrdef]; positivity
+    have hc : ((α t : ℝ) : Real.Angle).cos = r / 2 := by
+      rw [Real.Angle.cos_coe, ht, Real.cos_pi_div_four]
+    have hsn : ((α t : ℝ) : Real.Angle).sin = r / 2 := by
+      rw [Real.Angle.sin_coe, ht, Real.sin_pi_div_four]
+    have hmul : ∀ x : ℝ, r * (r / 2 * x) = x := by
+      intro x
+      linear_combination x / 2 * hrsq
+    have hbound : MeasureTheory.volume S ≤ ENNReal.ofReal r := by
+      refine volume_le_of_subset_horizontalBand
+        (f := fun y ↦ min (y + (r - r * (e t) 0)) (-y + (r - r * (e t) 1)) - r)
+        (c := r) (by fun_prop) hrpos.le ?_
+      intro p hp
+      obtain ⟨hy0, hy1⟩ := hH p hp
+      obtain ⟨hx1, hx2, hx3⟩ := hLt t p hp
+      have hm0 : (m t p) 0 = r / 2 * p 0 - r / 2 * p 1 + (e t) 0 := by
+        rw [hlift t p, hcoord, rotationMap_apply_zero, hc, hsn]
+      have hm1 : (m t p) 1 = r / 2 * p 0 + r / 2 * p 1 + (e t) 1 := by
+        rw [hlift t p, hcoord, rotationMap_apply_one, hc, hsn]
+      rw [hm0] at hx1
+      rw [hm1] at hx2
+      rw [hm0, hm1] at hx3
+      have g1 : p 0 - p 1 + r * (e t) 0 ≤ r := by
+        have h := mul_le_mul_of_nonneg_left hx1 hrpos.le
+        rwa [mul_add, mul_sub, hmul, hmul, mul_one] at h
+      have g2 : p 0 + p 1 + r * (e t) 1 ≤ r := by
+        have h := mul_le_mul_of_nonneg_left hx2 hrpos.le
+        rwa [mul_add, mul_add, hmul, hmul, mul_one] at h
+      have hmin := min_le_left (p 1 + (r - r * (e t) 0)) (-p 1 + (r - r * (e t) 1))
+      have hmin' := min_le_right (p 1 + (r - r * (e t) 0)) (-p 1 + (r - r * (e t) 1))
+      refine ⟨⟨hy0, hy1⟩, ?_, ?_⟩
+      · simp only
+        rcases hx3 with h | h
+        · have h' := mul_le_mul_of_nonneg_left h hrpos.le
+          rw [mul_zero, mul_add, mul_sub, hmul, hmul] at h'
+          linarith
+        · have h' := mul_le_mul_of_nonneg_left h hrpos.le
+          rw [mul_zero, mul_add, mul_add, hmul, hmul] at h'
+          linarith
+      · simp only
+        have hle' : p 0 ≤ min (p 1 + (r - r * (e t) 0)) (-p 1 + (r - r * (e t) 1)) :=
+          le_min (by linarith) (by linarith)
+        linarith
+    have hle : (MeasureTheory.volume S).toReal ≤ r := ENNReal.toReal_le_of_le_ofReal hrpos.le hbound
+    nlinarith [hrsq, hrpos, harea, hle, sq_nonneg (r - 11 / 5)]
+  have hquarter' : ∀ t : unitInterval, α t < Real.pi / 4 := by
+    intro t
+    rcases lt_trichotomy (α t) (Real.pi / 4) with h | h | h
+    · exact h
+    · exact absurd h (hquarter t)
+    · exfalso
+      have hmem : Real.pi / 4 ∈ Set.range α := by
+        refine intermediate_value_univ 0 t hαc ?_
+        rw [hα0]
+        exact ⟨by positivity, h.le⟩
+      obtain ⟨u, hu⟩ := hmem
+      exact hquarter u hu
+  -- the final clockwise angle has small cosine
+  have hcosbound : |Real.cos (α 1)| ≤ 5 / 11 := by
+    by_contra hcon
+    push Not at hcon
+    have habs : (0 : ℝ) < |Real.cos (α 1)| := by linarith
+    have hne : Real.cos (α 1) ≠ 0 := by
+      intro h0
+      rw [h0] at habs
+      simp at habs
+    have hbound : MeasureTheory.volume S ≤ ENNReal.ofReal (1 / |Real.cos (α 1)|) := by
+      refine le_trans (MeasureTheory.measure_mono ?_)
+        (volume_horizontalBand_inter_le (Real.cos (α 1)) (-Real.sin (α 1)) (-(e 1) 0) hne)
+      intro p hp
+      obtain ⟨hy0, hy1⟩ := hH p hp
+      obtain ⟨hv0, hv1, -⟩ := hVf p hp
+      have hm0 : (m 1 p) 0 = Real.cos (α 1) * p 0 - Real.sin (α 1) * p 1 + (e 1) 0 := by
+        rw [hlift 1 p, hcoord, rotationMap_apply_zero, Real.Angle.cos_coe, Real.Angle.sin_coe]
+      rw [hm0] at hv0 hv1
+      exact ⟨⟨hy0, hy1⟩, by linarith, by linarith⟩
+    have hle : (MeasureTheory.volume S).toReal ≤ 1 / |Real.cos (α 1)| :=
+      ENNReal.toReal_le_of_le_ofReal (by positivity) hbound
+    have hlt : 1 / |Real.cos (α 1)| < 11 / 5 := by
+      rw [div_lt_iff₀ habs]
+      linarith
+    linarith
+  -- the angle constants
+  set A := Real.arccos (5 / 11 : ℝ) with hAdef
+  have hcosA : Real.cos A = 5 / 11 := Real.cos_arccos (by norm_num) (by norm_num)
+  have hApi2 : A ≤ Real.pi / 2 := Real.arccos_le_pi_div_two.mpr (by norm_num)
+  have hApi : A ≤ Real.pi := Real.arccos_le_pi _
+  have hAquarter : Real.pi / 4 < A := by
+    have hs2 : Real.sqrt 2 * Real.sqrt 2 = 2 := Real.mul_self_sqrt (by norm_num)
+    have hs2pos : (0 : ℝ) < Real.sqrt 2 := by positivity
+    have hlt : (5 : ℝ) / 11 < Real.sqrt 2 / 2 := by nlinarith
+    have h4 : Real.arccos (Real.sqrt 2 / 2) = Real.pi / 4 := by
+      rw [← Real.cos_pi_div_four, Real.arccos_cos (by positivity) (by linarith [Real.pi_pos])]
+    rw [hAdef, ← h4]
+    exact Real.arccos_lt_arccos (by norm_num) hlt (by nlinarith)
+  have hfinal : α 1 ≤ -A := by
+    by_contra hcon
+    push Not at hcon
+    have habs : |α 1| < A := by
+      rw [abs_lt]
+      exact ⟨by linarith, lt_trans (hquarter' 1) hAquarter⟩
+    have hmono : Real.cos A < Real.cos |α 1| :=
+      Real.cos_lt_cos_of_nonneg_of_le_pi (abs_nonneg _) hApi habs
+    rw [Real.cos_abs] at hmono
+    have := le_abs_self (Real.cos (α 1))
+    linarith
+  have hm0e : ∀ t : unitInterval, m t 0 = e t := by
+    intro t
+    rw [hlift t 0]
+    simp [rotationMap]
+  have hrot : HasRotationAngle S (-α 1) := by
+    refine ⟨m, hm.isPaperMotion, α, hαc, hα0, by ring, fun t p ↦ ?_⟩
+    rw [hlift t p, hm0e t]
+  by_cases hbig : -α 1 ≤ Real.pi / 2
+  · exact ⟨-α 1, ⟨by linarith, hbig⟩, htransfer _ hrot⟩
+  · push Not at hbig
+    refine ⟨Real.pi / 2, ⟨hApi2, le_refl _⟩, htransfer _ ?_⟩
+    -- the intermediate right-angle time
+    have hmemr : -(Real.pi / 2) ∈ Set.range α := by
+      refine intermediate_value_univ 1 0 hαc ?_
+      rw [hα0]
+      exact ⟨by linarith, by linarith [Real.pi_pos]⟩
+    obtain ⟨ts, hts⟩ := hmemr
+    exact hasRotationAngle_right_of_reaches_right S m hm α e hαc hα0 hlift ts hts
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Sofa.BalancedRightAngle`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Sofa / Balanced Right Angle
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem exists_dominating_balancedRightAngleSofa (s : Set Point)
+    (hs : IsPaperMovingSofa s) (harea : (11 : ℝ) / 5 ≤ ClassicalResults.area s) :
+    ∃ s' : Set Point, IsBalancedMaximumSofa s' (Real.pi / 2) ∧
+      ClassicalResults.area s ≤ ClassicalResults.area s' := by
+  obtain ⟨ω, hω, hrot⟩ := movingSofa_rotationAngle_bound s hs harea
+  have hω0 : 0 < ω := lt_of_lt_of_le (Real.arccos_pos.2 (by norm_num)) hω.1
+  obtain ⟨K, -, hBsofa, hBmax⟩ := exists_balancedMaximumSofa ω hω0 hω.2
+  have hsB : ClassicalResults.area s ≤
+      ClassicalResults.area ((K.val : Set Point) \ capNiche K) := hBmax s hrot
+  obtain ⟨α, hrotB⟩ := balancedMaximumSofa_rightAngle _ ω hBsofa hω (harea.trans hsB)
+  obtain ⟨K', -, hS'sofa, hS'max⟩ :=
+    exists_balancedMaximumSofa (Real.pi / 2) (by positivity) le_rfl
+  refine ⟨(K'.val : Set Point) \ capNiche K', hS'sofa, hsB.trans ?_⟩
+  calc ClassicalResults.area ((K.val : Set Point) \ capNiche K)
+      = ClassicalResults.area (rotationMap α '' ((K.val : Set Point) \ capNiche K)) :=
+        (area_image_rotationMap α _).symm
+    _ ≤ _ := hS'max _ hrotB
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Sofa.Maximum`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver's sofa attains the maximum moving-sofa area
+
+The paper's main theorem: Gerver's paper-frame set `paperGerverSofa` is a moving
+sofa, and every moving sofa has area at most its area.
+
+The proof splits on the certified lower bound `11 / 5`.  A sofa below that bound
+is dominated outright.  A sofa at or above it is dominated by a balanced maximum
+sofa of angle `π / 2`, whose cap lies in the special domain and extends to a
+triple in the cap-tail space; the area functional is then bounded by `Q`, which
+is maximized by Gerver's triple and matches the area functional there.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem paperGerverSofa_maximum :
+    IsPaperMovingSofa paperGerverSofa ∧
+      ∀ s : Set Point, IsPaperMovingSofa s →
+        ClassicalResults.area s ≤ ClassicalResults.area paperGerverSofa := by
+  -- `G` is a moving sofa in standard position at angle `π / 2`.
+  have hstd : IsStandardPosition paperGerverSofa (Real.pi / 2) := by
+    rw [gerver_capSupport_identification.2.1]
+    exact gerver_capSupport_identification.2.2.1
+  -- `G` is monotone, so the sofa-area identity applies to it.
+  have hmono : ∃ s₀ : Set Point, IsStandardPosition s₀ (Real.pi / 2) ∧
+      paperGerverSofa = monotonization s₀ (Real.pi / 2) :=
+    ⟨paperGerverSofa, hstd, (gerver_paperNiche_identification.2.2.1).symm⟩
+  -- The certified lower bound, transported along the path identification.
+  have hG : (11 : ℝ) / 5 ≤ ClassicalResults.area paperGerverSofa := by
+    rw [← gerver_canonical_paper_literal.1]
+    exact gerver_area_lower_bound.2
+  refine ⟨?_, ?_⟩
+  · obtain ⟨m, hm, -⟩ := hstd.2.1
+    exact ⟨m, hm⟩
+  intro s hs
+  by_cases harea : (11 : ℝ) / 5 ≤ ClassicalResults.area s
+  · -- `K_G` maximizes `Q`, and `Q (K_G, B, D) = A (K_G) = |G|`.
+    obtain ⟨X, hK, hB, hD, hmax⟩ := gerver_upperBoundQ_maximum
+    have hGarea : rightAngleAreaFunctional X.cap.val = ClassicalResults.area paperGerverSofa :=
+      capAreaFunctional_eq_sofaArea paperGerverSofa (Real.pi / 2) hmono X.cap.val hK
+    -- Dominate `S` by a balanced maximum sofa of angle `π / 2`.
+    obtain ⟨s', ⟨hmono', L, hL, hbalanced⟩, hdom⟩ :=
+      exists_dominating_balancedRightAngleSofa s hs harea
+    obtain ⟨M, hM⟩ := specialCap_isConvexDomain.2.2.1 L hbalanced
+    obtain ⟨Y, rfl, hYR, hYL⟩ := exists_canonicalCapTail M
+    calc
+      ClassicalResults.area s ≤ ClassicalResults.area s' := hdom
+      _ = rightAngleAreaFunctional L :=
+        (capAreaFunctional_eq_sofaArea s' (Real.pi / 2) hmono' L hL).symm
+      _ = rightAngleAreaFunctional Y.cap.val := by rw [hM]
+      _ ≤ upperBoundQ Y := areaFunctional_le_upperBoundQ Y hYR hYL
+      _ ≤ upperBoundQ X := hmax Y
+      _ = rightAngleAreaFunctional X.cap.val := (gerver_upperBoundQ_matches X hK hB hD).symm
+      _ = ClassicalResults.area paperGerverSofa := hGarea
+  · exact (le_of_not_ge harea).trans hG
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Motion.CanonicalUpperBound`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The paper's upper bound transported to the canonical target
+
+The paper's main theorem bounds the real area of every moving sofa in the paper normalization by
+the area of the paper Gerver sofa. The canonical and paper Gerver sets coincide, every admissible
+set is a paper moving sofa, and admissible sets are compact, so the comparison of real areas is a
+comparison of Lebesgue measures.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open MeasureTheory
+open scoped unitInterval
+
+/-- Every admissible set has Lebesgue measure at most that of Gerver's sofa. -/
+theorem areaUpperBound : AreaUpperBound := by
+  intro s m hs
+  have harea := paperGerverSofa_maximum.2 s (canonical_paper_motion_bridge.2 s m hs)
+  rw [← gerver_canonical_paper_literal.1] at harea
+  obtain ⟨-, -, hfinS⟩ := canonical_motion_compactness s m hs
+  obtain ⟨mG, hmG⟩ := isMovingSofa_gerversSofa
+  obtain ⟨-, -, hfinG⟩ := canonical_motion_compactness gerversSofa mG hmG
+  exact (real_area_le_iff_volume_le s gerversSofa hfinS hfinG).mp harea
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The moving sofa problem: proofs
+
+The imported development proves uniqueness of Gerver's defining parameters in
+`MovingSofa.GerversSofa.ABφθSpec.existsUnique` and admissibility of the resulting shape in
+`MovingSofa.isMovingSofa_gerversSofa`. The main theorem combines this admissibility result
+with the proved area upper bound `MovingSofa.areaUpperBound`.
+-/
+
+@[expose] public section
+
+namespace MovingSofa
+
+open MeasureTheory
+
+/-- Gerver's sofa attains the sofa constant (Baek, arXiv:2411.19826). -/
+theorem sofaConstant_eq_volume_gerversSofa : sofaConstant = volume gerversSofa := by
+  apply optimality_of_areaUpperBound
+  exact areaUpperBound
+
+end MovingSofa
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/Development/Geometry/Foundations/Development001.lean b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development001.lean
new file mode 100644
index 0000000000..95d69b668c
--- /dev/null
+++ b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development001.lean
@@ -0,0 +1,11648 @@
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+module
+
+public import Mathlib.Algebra.BigOperators.Fin
+public import Mathlib.Algebra.BigOperators.Group.Finset.Basic
+public import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
+public import Mathlib.Algebra.Order.Archimedean.Real.Basic
+public import Mathlib.Analysis.Calculus.ContDiff.Deriv
+public import Mathlib.Analysis.Calculus.Deriv.Prod
+public import Mathlib.Analysis.Calculus.Deriv.Shift
+public import Mathlib.Analysis.Calculus.Deriv.Slope
+public import Mathlib.Analysis.Calculus.LocalExtr.Basic
+public import Mathlib.Analysis.Calculus.MeanValue
+public import Mathlib.Analysis.Calculus.Rademacher
+public import Mathlib.Analysis.Complex.MeanValue
+public import Mathlib.Analysis.Convex.Basic
+public import Mathlib.Analysis.Convex.Body
+public import Mathlib.Analysis.Convex.Combination
+public import Mathlib.Analysis.Convex.Deriv
+public import Mathlib.Analysis.Convex.Function
+public import Mathlib.Analysis.Convex.Gauge
+public import Mathlib.Analysis.Convex.GaugeRescale
+public import Mathlib.Analysis.Convex.Segment
+public import Mathlib.Analysis.Convex.Topology
+public import Mathlib.Analysis.InnerProductSpace.Basic
+public import Mathlib.Analysis.InnerProductSpace.Calculus
+public import Mathlib.Analysis.InnerProductSpace.Continuous
+public import Mathlib.Analysis.InnerProductSpace.PiL2
+public import Mathlib.Analysis.InnerProductSpace.Projection.Minimal
+public import Mathlib.Analysis.Normed.Affine.AddTorsorBases
+public import Mathlib.Analysis.Normed.Affine.ContinuousAffineMap
+public import Mathlib.Analysis.Normed.Group.Bounded
+public import Mathlib.Analysis.Normed.Group.Uniform
+public import Mathlib.Analysis.Normed.Lp.MeasurableSpace
+public import Mathlib.Analysis.Normed.Module.Basic
+public import Mathlib.Analysis.Real.Pi.Bounds
+public import Mathlib.Analysis.SpecialFunctions.Complex.Arg
+public import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
+public import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
+public import Mathlib.Analysis.SpecialFunctions.Pow.Real
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
+public import Mathlib.Analysis.SpecificLimits.Basic
+public import Mathlib.Analysis.SpecificLimits.Normed
+public import Mathlib.Basic.Real.Basic
+public import Mathlib.Data.Fin.Basic
+public import Mathlib.Data.Fin.SuccPredOrder
+public import Mathlib.Data.Finset.Sort
+public import Mathlib.Data.Fintype.EquivFin
+public import Mathlib.Data.List.MinMax
+public import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
+public import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
+public import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
+public import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
+public import Mathlib.MeasureTheory.Function.AbsolutelyContinuous
+public import Mathlib.MeasureTheory.Integral.Bochner.Basic
+public import Mathlib.MeasureTheory.Integral.Bochner.Set
+public import Mathlib.MeasureTheory.Integral.Bochner.SumMeasure
+public import Mathlib.MeasureTheory.Integral.DominatedConvergence
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
+public import Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
+public import Mathlib.MeasureTheory.Measure.FiniteMeasure
+public import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
+public import Mathlib.MeasureTheory.Measure.Hausdorff
+public import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
+public import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
+public import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
+public import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
+public import Mathlib.MeasureTheory.Measure.Map
+public import Mathlib.MeasureTheory.Measure.Portmanteau
+public import Mathlib.MeasureTheory.Measure.WithDensity
+public import Mathlib.MeasureTheory.SetSemiring
+public import Mathlib.MeasureTheory.VectorMeasure.BoundedVariation
+public import Mathlib.MeasureTheory.VectorMeasure.IntegrationByParts
+public import Mathlib.MeasureTheory.VectorMeasure.WithDensity
+public import Mathlib.Order.ConditionallyCompleteLattice.Basic
+public import Mathlib.Order.Fin.Basic
+public import Mathlib.Order.Hom.Set
+public import Mathlib.Order.SuccPred.IntervalSucc
+public import Mathlib.Order.SuccPred.LinearLocallyFinite
+public import Mathlib.Tactic
+public import Mathlib.Tactic.FieldSimp
+public import Mathlib.Tactic.FinCases
+public import Mathlib.Tactic.FunProp
+public import Mathlib.Tactic.GCongr
+public import Mathlib.Tactic.Linarith
+public import Mathlib.Tactic.LinearCombination
+public import Mathlib.Tactic.NormNum
+public import Mathlib.Tactic.Positivity
+public import Mathlib.Tactic.Ring
+public import Mathlib.Tactic.SplitIfs
+public import Mathlib.Topology.Algebra.Group.Quotient
+public import Mathlib.Topology.Algebra.Ring.Real
+public import Mathlib.Topology.Covering.AddCircle
+public import Mathlib.Topology.EMetricSpace.BoundedVariation
+public import Mathlib.Topology.EMetricSpace.VariationOnFromTo
+public import Mathlib.Topology.Homotopy.Lifting
+public import Mathlib.Topology.Instances.AddCircle.Real
+public import Mathlib.Topology.Instances.Matrix
+public import Mathlib.Topology.Instances.Real.Lemmas
+public import Mathlib.Topology.MetricSpace.Closeds
+public import Mathlib.Topology.MetricSpace.HausdorffDistance
+public import Mathlib.Topology.Order.Compact
+public import Mathlib.Topology.Order.IntermediateValue
+public import Mathlib.Topology.Order.LeftRightNhds
+public import Mathlib.Topology.Order.MonotoneContinuity
+public import Mathlib.Topology.Order.ProjIcc
+public import Mathlib.Topology.Separation.Lemmas
+public import Mathlib.Topology.Sequences
+public import Mathlib.Topology.UniformSpace.Compact
+
+/-!
+# Moving sofa: related mathematical developments
+
+* `Infrastructure.Analysis.Foundations.Development001`.
+* `Infrastructure.Geometry.Foundations.Development001`.
+* `Infrastructure.MathlibExtensions.Foundations.Development001`.
+* `Infrastructure.Geometry.Foundations.Development002`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Analysis.Foundations.Development001`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Analysis.BoundedDensity`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Bounded Density
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+open scoped NNReal ENNReal
+
+namespace MovingSofa
+
+theorem boundedDensity_of_domination (a b : ℝ) (μ : Measure ℝ) [IsFiniteMeasure μ]
+    (q : ℝ → ℝ) (hq : Measurable q)
+    (hq_nonneg : ∀ t ∈ Set.Icc a b, 0 ≤ q t)
+    (hq_bound : ∃ C : ℝ, ∀ t ∈ Set.Icc a b, q t ≤ C)
+    (hdom : ∀ E : Set ℝ, MeasurableSet E →
+      μ E ≤ ENNReal.ofReal (∫ t in E, q t ∂volume.restrict (Set.Icc a b))) :
+    ∃ r : ℝ → ℝ, Measurable r ∧ Integrable r (volume.restrict (Set.Icc a b)) ∧
+      (∀ᵐ t ∂volume.restrict (Set.Icc a b), 0 ≤ r t ∧ r t ≤ q t) ∧
+      μ = (volume.restrict (Set.Icc a b)).withDensity (fun t ↦ ENNReal.ofReal (r t)) ∧
+      (∃ C : ℝ, ∀ᵐ t ∂volume.restrict (Set.Icc a b), |r t| ≤ C) := by
+  classical
+  obtain ⟨C, hC⟩ := hq_bound
+  let lam : Measure ℝ := volume.restrict (Set.Icc a b)
+  let nu : Measure ℝ := lam.withDensity (fun t ↦ ENNReal.ofReal (q t))
+  have hq_ae : 0 ≤ᵐ[lam] q := by
+    filter_upwards [ae_restrict_mem measurableSet_Icc] with t ht
+    exact hq_nonneg t ht
+  have hq_bound_ae : ∀ᵐ t ∂lam, ‖q t‖ ≤ max C 0 := by
+    filter_upwards [ae_restrict_mem measurableSet_Icc] with t ht
+    rw [Real.norm_eq_abs, abs_of_nonneg (hq_nonneg t ht)]
+    exact le_max_of_le_left (hC t ht)
+  have hq_int : Integrable q lam := by
+    exact Measure.integrableOn_of_bounded (μ := volume)
+      (s := Set.Icc a b) (M := max C 0) (by simp) hq.aestronglyMeasurable hq_bound_ae
+  have hν_apply : ∀ E : Set ℝ, MeasurableSet E →
+      nu E = ENNReal.ofReal (∫ t in E, q t ∂lam) := by
+    intro E hE
+    change (lam.withDensity (fun t ↦ ENNReal.ofReal (q t))) E = _
+    rw [withDensity_apply _ hE]
+    exact (ofReal_integral_eq_lintegral_ofReal
+      (μ := lam.restrict E) hq_int.integrableOn
+      (ae_restrict_of_ae hq_ae)).symm
+  have hμν : μ ≤ nu := by
+    rw [Measure.le_iff]
+    intro E hE
+    rw [hν_apply E hE]
+    exact hdom E hE
+  have hνlam : nu.AbsolutelyContinuous lam :=
+    withDensity_absolutelyContinuous lam _
+  have hμlam : μ.AbsolutelyContinuous lam :=
+    (Measure.absolutelyContinuous_of_le hμν).trans hνlam
+  let r : ℝ → ℝ := fun t ↦ (μ.rnDeriv lam t).toReal
+  have hr_meas : Measurable r := by
+    exact (Measure.measurable_rnDeriv μ lam).ennreal_toReal
+  have hr_int : Integrable r lam := by
+    simpa [r] using
+      (Measure.integrableOn_toReal_rnDeriv
+        (μ := μ) (ν := lam) (s := Set.univ) (by simp))
+  have hr_eq : lam.withDensity (fun t ↦ μ.rnDeriv lam t) = μ :=
+    Measure.withDensity_rnDeriv_eq μ lam hμlam
+  have hr_le_q_enn : μ.rnDeriv lam ≤ᵐ[lam] (fun t ↦ ENNReal.ofReal (q t)) := by
+    apply ae_le_of_forall_setLIntegral_le_of_sigmaFinite
+      (Measure.measurable_rnDeriv μ lam)
+    intro E hE hEfin
+    rw [Measure.setLIntegral_rnDeriv hμlam E]
+    calc
+      μ E ≤ nu E := hμν E
+      _ = ∫⁻ t in E, ENNReal.ofReal (q t) ∂lam := by
+        exact withDensity_apply' _ _
+  have hr_bounds : ∀ᵐ t ∂lam, 0 ≤ r t ∧ r t ≤ q t := by
+    filter_upwards [hr_le_q_enn, Measure.rnDeriv_ne_top μ lam,
+      ae_restrict_mem measurableSet_Icc] with t hle htop ht
+    constructor
+    · exact ENNReal.toReal_nonneg
+    · have hto := (ENNReal.toReal_le_toReal htop ENNReal.ofReal_ne_top).mpr hle
+      simpa [r, ENNReal.toReal_ofReal (hq_nonneg t ht)] using hto
+  have hfun : (fun t ↦ ENNReal.ofReal (r t)) =ᵐ[lam] μ.rnDeriv lam := by
+    filter_upwards [Measure.rnDeriv_ne_top μ lam] with t htop
+    simpa [r] using ENNReal.ofReal_toReal htop
+  have hr_eq' : lam.withDensity (fun t ↦ ENNReal.ofReal (r t)) = μ := by
+    rw [← hr_eq]
+    exact withDensity_congr_ae hfun
+  have hr_bound : ∃ K : ℝ, ∀ᵐ t ∂lam, |r t| ≤ K := by
+    refine ⟨max C 0, ?_⟩
+    filter_upwards [hr_bounds, ae_restrict_mem measurableSet_Icc] with t ⟨h₀, hq⟩ ht
+    rw [abs_of_nonneg h₀]
+    exact hq.trans ((hC t ht).trans (le_max_left _ _))
+  refine ⟨r, hr_meas, hr_int, hr_bounds, ?_, hr_bound⟩
+  simpa [lam] using hr_eq'.symm
+
+/-- A finite measure carried by `Set.Icc a b` and dominated on that interval by the integral of
+a bounded, almost everywhere strongly measurable nonnegative function `f` has an integrable real
+density with respect to the Lebesgue measure on `Set.Icc a b`, and that density is bounded above
+by `f` almost everywhere. -/
+theorem exists_density_le_of_domination {a b : ℝ} (μ : Measure ℝ) [IsFiniteMeasure μ]
+    (hμ : μ (Set.Icc a b)ᶜ = 0) (f : ℝ → ℝ)
+    (hf : AEStronglyMeasurable f (volume.restrict (Set.Icc a b)))
+    (hf_nonneg : ∀ t ∈ Set.Icc a b, 0 ≤ f t)
+    {M : ℝ} (hf_bound : ∀ t ∈ Set.Icc a b, f t ≤ M)
+    (hdom : ∀ E : Set ℝ, MeasurableSet E → E ⊆ Set.Icc a b →
+      μ E ≤ ENNReal.ofReal (∫ t in E, f t)) :
+    ∃ r : ℝ → ℝ, Measurable r ∧ IntegrableOn r (Set.Icc a b) volume ∧
+      (∀ᵐ t ∂volume.restrict (Set.Icc a b), 0 ≤ r t ∧ r t ≤ f t) ∧
+      μ = (volume.restrict (Set.Icc a b)).withDensity (fun t ↦ ENNReal.ofReal (r t)) := by
+  classical
+  obtain ⟨g, hg, hgf⟩ := hf
+  set q : ℝ → ℝ := fun t ↦ max 0 (min M (g t)) with hq
+  have hq_meas : Measurable q := measurable_const.max (measurable_const.min hg.measurable)
+  have hq_nonneg : ∀ t, 0 ≤ q t := fun t ↦ le_max_left _ _
+  have hq_bound : ∀ t, q t ≤ max M 0 := fun t ↦
+    max_le (le_max_right M 0) ((min_le_left _ _).trans (le_max_left _ _))
+  have hqf : f =ᵐ[volume.restrict (Set.Icc a b)] q := by
+    filter_upwards [hgf, ae_restrict_mem measurableSet_Icc] with t hgt ht
+    simp only [hq, ← hgt, min_eq_right (hf_bound t ht), max_eq_right (hf_nonneg t ht)]
+  have hdom' : ∀ E : Set ℝ, MeasurableSet E →
+      μ E ≤ ENNReal.ofReal (∫ t in E, q t ∂volume.restrict (Set.Icc a b)) := by
+    intro E hE
+    have hnull : μ (E \ Set.Icc a b) = 0 := measure_mono_null (fun x hx ↦ hx.2) hμ
+    have hEsub : μ E ≤ μ (E ∩ Set.Icc a b) :=
+      calc μ E ≤ μ (E ∩ Set.Icc a b) + μ (E \ Set.Icc a b) := measure_le_inter_add_sdiff μ E _
+        _ = μ (E ∩ Set.Icc a b) := by rw [hnull, add_zero]
+    refine hEsub.trans ((hdom _ (hE.inter measurableSet_Icc) Set.inter_subset_right).trans ?_)
+    rw [Measure.restrict_restrict hE]
+    exact le_of_eq (congrArg ENNReal.ofReal (integral_congr_ae
+      (ae_restrict_of_ae_restrict_of_subset Set.inter_subset_right hqf)))
+  obtain ⟨r, hr_meas, hr_int, hr_bounds, hr_eq, -⟩ :=
+    boundedDensity_of_domination a b μ q hq_meas (fun t _ ↦ hq_nonneg t)
+      ⟨max M 0, fun t _ ↦ hq_bound t⟩ hdom'
+  refine ⟨r, hr_meas, hr_int, ?_, hr_eq⟩
+  filter_upwards [hr_bounds, hqf] with t ht htq
+  exact ⟨ht.1, htq ▸ ht.2⟩
+
+/-- A finite measure carried by `Set.Icc a b` and dominated on that interval by the integral of
+a bounded, almost everywhere strongly measurable nonnegative function has a measurable
+`ℝ≥0`-valued density with respect to the Lebesgue measure on `Set.Icc a b`. -/
+theorem exists_nnreal_density_of_domination {a b : ℝ} (μ : Measure ℝ) [IsFiniteMeasure μ]
+    (hμ : μ (Set.Icc a b)ᶜ = 0) (f : ℝ → ℝ)
+    (hf : AEStronglyMeasurable f (volume.restrict (Set.Icc a b)))
+    (hf_nonneg : ∀ t ∈ Set.Icc a b, 0 ≤ f t)
+    {M : ℝ} (hf_bound : ∀ t ∈ Set.Icc a b, f t ≤ M)
+    (hdom : ∀ E : Set ℝ, MeasurableSet E → E ⊆ Set.Icc a b →
+      μ E ≤ ENNReal.ofReal (∫ t in E, f t)) :
+    ∃ r : ℝ → ℝ≥0, Measurable r ∧
+      μ = (volume.restrict (Set.Icc a b)).withDensity (fun t ↦ (r t : ℝ≥0∞)) := by
+  obtain ⟨r, hr_meas, -, -, hr_eq⟩ :=
+    exists_density_le_of_domination μ hμ f hf hf_nonneg hf_bound hdom
+  exact ⟨fun t ↦ (r t).toNNReal, hr_meas.real_toNNReal, hr_eq⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Classical.Foundations.Development001`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Classical.Area`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Planar area
+
+The real-valued Lebesgue area of a planar set, and its invariance under translation.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+
+namespace MovingSofa.ClassicalResults
+
+/-- The Euclidean plane. -/
+abbrev Plane := EuclideanSpace ℝ (Fin 2)
+
+/-- The real-valued Lebesgue area of a planar set. -/
+def area (s : Set Plane) : ℝ := (volume s).toReal
+
+/-- Planar area is invariant under translation. -/
+theorem area_image_add (S : Set Plane) (v : Plane) :
+    area ((fun p ↦ p + v) '' S) = area S := by
+  change (MeasureTheory.volume ((fun p ↦ p + v) '' S)).toReal =
+    (MeasureTheory.volume S).toReal
+  rw [Set.image_add_right]
+  have hf : (fun x : Plane ↦ x + -v) = fun x ↦ -v + x := by
+    funext x
+    exact add_comm x (-v)
+  rw [hf, MeasureTheory.measure_preimage_add]
+
+end MovingSofa.ClassicalResults
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `ForMathlib.Algebra.Foundations.Development001`.
+* `ForMathlib.Analysis.Foundations.Development001`.
+* `ForMathlib.BoundedVariation.Foundations.Development001`.
+* `ForMathlib.Convex.Foundations.Development001`.
+* `ForMathlib.Geometry.Foundations.Development001`.
+* `ForMathlib.MeasureTheory.Foundations.Development001`.
+* `ForMathlib.Order.Foundations.Development001`.
+* `ForMathlib.Topology.Foundations.Development001`.
+* `ForMathlib.MeasureTheory.Foundations.Development002`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `ForMathlib.Algebra.BigOperators.Triangle`.
+* `ForMathlib.Algebra.Order.Fin`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Triangular rearrangement of a double sum over a `Finset`
+
+A double sum over `s ×ˢ s` splits into the closed lower triangle `{(u, t) | u ≤ t}` and its
+transpose. The two pieces overlap exactly on the diagonal, so for a kernel vanishing there the
+lower-triangular sum plus its transpose recovers the whole double sum.
+
+`Finset.sum_sum_Ioi_add_eq_sum_sum_off_diag` is the `Fintype` and `LocallyFiniteOrder` analogue,
+and `Fin.sum_sum_eq_sum_triangle_add` the `Fin` analogue; neither applies to a general `Finset`
+of a plain `LinearOrder`.
+-/
+
+@[expose] public section
+
+namespace Finset
+
+variable {ι M : Type*} [LinearOrder ι] [AddCommMonoid M]
+
+/-- For a kernel vanishing on the diagonal, the closed lower-triangular double sum plus the same
+sum with the two arguments swapped is the full double sum. -/
+theorem sum_filter_le_add_sum_filter_le_swap (s : Finset ι) (f : ι → ι → M)
+    (hdiag : ∀ x, f x x = 0) :
+    ((∑ t ∈ s, ∑ u ∈ s.filter (fun u ↦ u ≤ t), f u t) +
+        ∑ t ∈ s, ∑ u ∈ s.filter (fun u ↦ u ≤ t), f t u) =
+      ∑ x ∈ s, ∑ y ∈ s, f x y := by
+  classical
+  calc ((∑ t ∈ s, ∑ u ∈ s.filter (fun u ↦ u ≤ t), f u t) +
+          ∑ t ∈ s, ∑ u ∈ s.filter (fun u ↦ u ≤ t), f t u)
+      = (∑ t ∈ s, ∑ u ∈ s.filter (fun u ↦ t < u), f t u) +
+          ∑ t ∈ s, ∑ u ∈ s.filter (fun u ↦ u ≤ t), f t u := by
+        rw [Finset.sum_comm' (t' := s) (s' := fun u ↦ s.filter (fun t ↦ u ≤ t))
+          (by simp; tauto)]
+        refine congrArg₂ _ (Finset.sum_congr rfl fun t _ ↦ (Finset.sum_subset
+          (monotone_filter_right _ fun _ _ ↦ le_of_lt) fun u hu hu' ↦ ?_).symm) rfl
+        simp only [mem_filter, not_and, not_lt] at hu hu'
+        rw [le_antisymm (hu' hu.1) hu.2, hdiag]
+    _ = ∑ x ∈ s, ∑ y ∈ s, f x y := by
+        rw [add_comm, ← Finset.sum_add_distrib]
+        exact Finset.sum_congr rfl fun x _ ↦ by
+          simpa using Finset.sum_filter_add_sum_filter_not s (· ≤ x) (f x)
+
+end Finset
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Algebra / Order / Fin
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace Fin
+
+/-- A finite sequence is bounded by its final value plus its positive backward increments. -/
+theorem apply_le_last_add_sum_max_sub {n : ℕ} (f : Fin (n + 1) → ℝ) (i : Fin (n + 1)) :
+    f i ≤ f (Fin.last n) + ∑ j : Fin n, max (f j.castSucc - f j.succ) 0 := by
+  induction n with
+  | zero => fin_cases i; simp
+  | succ n ih =>
+    rw [Fin.sum_univ_succ]
+    have htail (k : Fin (n + 1)) := ih (fun j ↦ f j.succ) k
+    have hnonneg : 0 ≤ max (f 0 - f (Fin.succ 0)) 0 := le_max_right _ _
+    refine Fin.cases ?_ (fun k ↦ ?_) i
+    · have h := htail 0
+      have hmax := le_max_left (f 0 - f (Fin.succ 0)) 0
+      simpa only [Fin.succ_last, Fin.succ_castSucc, Fin.castSucc_zero] using
+        (show f 0 ≤ f (Fin.last (n + 1)) +
+          (max (f 0 - f (Fin.succ 0)) 0 +
+            ∑ j : Fin n, max (f j.castSucc.succ - f j.succ.succ) 0) by
+          simp only [Fin.succ_last] at h
+          linarith)
+    · have h := htail k
+      simpa only [Fin.succ_last, Fin.succ_castSucc, Fin.castSucc_zero] using
+        (show f k.succ ≤ f (Fin.last (n + 1)) +
+          (max (f 0 - f (Fin.succ 0)) 0 +
+            ∑ j : Fin n, max (f j.castSucc.succ - f j.succ.succ) 0) by
+          simp only [Fin.succ_last] at h
+          linarith)
+
+end Fin
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `ForMathlib.Analysis.Calculus.Deriv.Shift`.
+* `ForMathlib.Analysis.Calculus.FirstReturn`.
+* `ForMathlib.Analysis.Calculus.Interval`.
+* `ForMathlib.Analysis.Calculus.LocalExtr.OneSided`.
+* `ForMathlib.Analysis.Complex.DiskIntegral`.
+* `ForMathlib.Analysis.Convex.Basic`.
+* `ForMathlib.Analysis.Convex.Deriv`.
+* `ForMathlib.Analysis.Convex.Gauge`.
+* `ForMathlib.Analysis.Convex.GaugeRescale`.
+* `ForMathlib.Analysis.Convex.Radial`.
+* `ForMathlib.Analysis.FiniteEnvelope`.
+* `ForMathlib.Analysis.InnerProductSpace.Box`.
+* `ForMathlib.Analysis.InnerProductSpace.Linear`.
+* `ForMathlib.Analysis.Normed.Affine.ContinuousAffineMap`.
+* `ForMathlib.Analysis.SpecialFunctions.Angle`.
+* `ForMathlib.Analysis.SpecialFunctions.AngleLift`.
+* `ForMathlib.Analysis.SpecialFunctions.Arctan`.
+* `ForMathlib.Analysis.SpecificLimits.AlternatingBracket`.
+* `ForMathlib.Analysis.SpecialFunctions.Trigonometric`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Shifting the base point of a derivative
+
+`HasDerivAt.comp_add_const` transports a derivative at a shifted base point to the shifted
+function; this file records the converse implication, packaged as an `Iff`.
+-/
+
+@[expose] public section
+
+/-- A derivative at a shifted base point is the derivative of the shifted function. -/
+theorem hasDerivAt_comp_add_const_iff {𝕜 : Type*} [NontriviallyNormedField 𝕜] {F : Type*}
+    [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : 𝕜 → F} {f' : F} (x a : 𝕜) :
+    HasDerivAt (fun u ↦ f (u + a)) f' x ↔ HasDerivAt f f' (x + a) := by
+  refine ⟨fun h ↦ ?_, fun h ↦ h.comp_add_const x a⟩
+  have h' : HasDerivAt (fun u ↦ f (u + a)) f' (x + a + -a) := by simpa using h
+  simpa using h'.comp_add_const (x + a) (-a)
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# First return to a level
+
+A continuous real function that starts strictly below a level and reaches it has a smallest
+time at which it attains that level, and its derivative there is nonnegative.
+-/
+
+@[expose] public section
+
+/-- A continuous function below a level at the left end has a first return to that level. -/
+theorem exists_first_return {g : ℝ → ℝ} {a b c : ℝ} (hab : a < b)
+    (hcont : ContinuousOn g (Set.Icc a b)) (hga : g a < c) (hgb : c ≤ g b) :
+    ∃ t ∈ Set.Ioc a b, g t = c ∧ ∀ u ∈ Set.Ico a t, g u < c := by
+  set E : Set ℝ := Set.Icc a b ∩ g ⁻¹' Set.Ici c with hE
+  have hEcl : IsClosed E := hcont.preimage_isClosed_of_isClosed isClosed_Icc isClosed_Ici
+  have hEne : E.Nonempty := ⟨b, ⟨hab.le, le_refl b⟩, hgb⟩
+  have hEbdd : BddBelow E := ⟨a, fun u hu => hu.1.1⟩
+  set t : ℝ := sInf E with ht
+  have htE : t ∈ E := hEcl.csInf_mem hEne hEbdd
+  have htlb : ∀ u ∈ E, t ≤ u := fun u hu => csInf_le hEbdd hu
+  have hta : a < t := by
+    rcases eq_or_lt_of_le htE.1.1 with h | h
+    · exact absurd htE.2 (by rw [← h]; exact not_le.mpr hga)
+    · exact h
+  have hleft : ∀ u ∈ Set.Ico a t, g u < c := by
+    intro u hu
+    by_contra hcc
+    push Not at hcc
+    exact absurd (htlb u ⟨⟨hu.1, le_trans hu.2.le htE.1.2⟩, hcc⟩) (not_le.mpr hu.2)
+  have hIco : Set.Ico a t ∈ nhdsWithin t (Set.Iio t) := Ico_mem_nhdsLT hta
+  have hsub : Set.Ico a t ⊆ Set.Icc a b := fun u hu => ⟨hu.1, le_trans hu.2.le htE.1.2⟩
+  have htend : Filter.Tendsto g (nhdsWithin t (Set.Iio t)) (nhds (g t)) :=
+    ((hcont t htE.1).mono hsub).tendsto.mono_left (nhdsWithin_le_of_mem hIco)
+  refine ⟨t, ⟨hta, htE.1.2⟩, le_antisymm ?_ htE.2, hleft⟩
+  exact le_of_tendsto htend (Filter.eventually_of_mem hIco fun u hu => (hleft u hu).le)
+
+/-- At a first return from below, the derivative is nonnegative. -/
+theorem nonneg_of_first_return {g : ℝ → ℝ} {a t c d : ℝ} (hat : a < t)
+    (hd : HasDerivAt g d t) (hgt : g t = c) (hleft : ∀ u ∈ Set.Ico a t, g u < c) :
+    0 ≤ d := by
+  have hIco : Set.Ico a t ∈ nhdsWithin t (Set.Iio t) := Ico_mem_nhdsLT hat
+  have hsl := (hasDerivAt_iff_tendsto_slope.mp hd).mono_left
+    (nhdsWithin_mono t (fun x hx => ne_of_lt hx) :
+      nhdsWithin t (Set.Iio t) ≤ nhdsWithin t {t}ᶜ)
+  refine ge_of_tendsto hsl (Filter.eventually_of_mem hIco fun u hu => ?_)
+  rw [slope_def_field, hgt]
+  exact le_of_lt (div_pos_of_neg_of_neg (by linarith only [hleft u hu])
+    (by linarith only [hu.2]))
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Analysis / Calculus / Interval
+-/
+
+@[expose] public section
+
+open Set
+
+/-- Glue one-sided derivatives on a closed interval, including its endpoints. -/
+theorem hasDerivWithinAt_Icc_of_oneSided {E : Type*}
+    [NormedAddCommGroup E] [NormedSpace ℝ E] {a b t : ℝ} (hab : a < b)
+    (ht : t ∈ Icc a b) {f : ℝ → E} {d : E}
+    (hr : t < b → HasDerivWithinAt f d (Ici t) t)
+    (hl : a < t → HasDerivWithinAt f d (Iic t) t) :
+    HasDerivWithinAt f d (Icc a b) t := by
+  rcases eq_or_lt_of_le ht.1 with rfl | hat
+  · exact (hr hab).mono Icc_subset_Ici_self
+  · rcases lt_or_eq_of_le ht.2 with htb | rfl
+    · have h := (hl hat).union (hr htb)
+      rw [Iic_union_Ici] at h
+      exact h.mono (subset_univ _)
+    · exact (hl hat).mono Icc_subset_Iic_self
+
+/-- A continuous derivative field on a uniquely differentiable set gives order-one regularity. -/
+theorem contDiffOn_one_of_continuous_derivative {E : Type*}
+    [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set ℝ}
+    (hs : UniqueDiffOn ℝ s) (f : ℝ → E) (d : s → E) (hd : Continuous d)
+    (hf : ∀ t : s, HasDerivWithinAt f (d t) s t) : ContDiffOn ℝ 1 f s := by
+  apply (contDiffOn_one_iff_derivWithin hs).mpr
+  refine ⟨fun t ht ↦ (hf ⟨t, ht⟩).differentiableWithinAt, ?_⟩
+  apply continuousOn_iff_continuous_domRestrict.mpr
+  apply hd.congr
+  intro t
+  exact ((hf t).derivWithin (hs t t.property)).symm
+
+/-- Glue two everywhere differentiable functions along the switch `{s ≤ c}`.  If the values
+and the derivatives agree at `c`, the glued function is differentiable everywhere and its
+derivative is the analogous glue of the two derivative fields.  At `c` itself the statement
+is genuine two-sided differentiability, obtained from the two one-sided derivatives. -/
+theorem hasDerivAt_if_le {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {f g f' g' : ℝ → E} {c : ℝ}
+    (hf : ∀ t, HasDerivAt f (f' t) t) (hg : ∀ t, HasDerivAt g (g' t) t)
+    (hfg : f c = g c) (hfg' : f' c = g' c) (t : ℝ) :
+    HasDerivAt (fun s => if s ≤ c then f s else g s) (if t ≤ c then f' t else g' t) t := by
+  rcases lt_trichotomy t c with ht | ht | ht
+  · rw [ite_eq_left ht.le]
+    refine (hf t).congr_of_eventuallyEq ?_
+    filter_upwards [Iio_mem_nhds ht] with s hs
+    exact ite_eq_left (le_of_lt hs)
+  · subst ht
+    rw [ite_eq_left le_rfl]
+    have hl : HasDerivWithinAt (fun s => if s ≤ t then f s else g s) (f' t) (Iic t) t :=
+      (hf t).hasDerivWithinAt.congr (fun _ hs => ite_eq_left hs) (ite_eq_left le_rfl)
+    have hr : HasDerivWithinAt (fun s => if s ≤ t then f s else g s) (f' t) (Ici t) t := by
+      rw [hfg']
+      refine (hg t).hasDerivWithinAt.congr (fun s hs => ?_) (by rw [ite_eq_left le_rfl, hfg])
+      rcases eq_or_lt_of_le (mem_Ici.mp hs) with rfl | hlt
+      · rw [ite_eq_left le_rfl, hfg]
+      · rw [ite_eq_right (not_le.mpr hlt)]
+    have hu := hl.union hr
+    rw [Iic_union_Ici] at hu
+    exact hasDerivWithinAt_univ.mp hu
+  · rw [ite_eq_right (not_le.mpr ht)]
+    refine (hg t).congr_of_eventuallyEq ?_
+    filter_upwards [Ioi_mem_nhds ht] with s hs
+    exact ite_eq_right (not_le.mpr hs)
+
+/-- A globally defined continuous derivative field witnesses continuous differentiability. -/
+theorem contDiff_one_of_hasDerivAt {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {f f' : ℝ → E} (hf : ∀ t, HasDerivAt f (f' t) t) (hf' : Continuous f') :
+    ContDiff ℝ 1 f :=
+  contDiff_one_iff_deriv.2 ⟨fun t => (hf t).differentiableAt, deriv_eq hf ▸ hf'⟩
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Fermat's theorem for one-sided derivatives on the real line
+
+`IsLocalMinOn.hasFDerivWithinAt_nonneg` signs a derivative within a set against the positive
+tangent cone of that set.  On the real line the positive tangent cones of the two half-lines
+at `a` are generated by `1` and by `-1`, which turns that statement into the two familiar
+one-sided derivative tests at a one-sided minimum.
+-/
+
+@[expose] public section
+
+/-- The forward direction belongs to the positive tangent cone of a right half-line. -/
+theorem one_mem_posTangentConeAt_Ici (a : ℝ) : (1 : ℝ) ∈ posTangentConeAt (Set.Ici a) a :=
+  mem_posTangentConeAt_of_segment_subset
+    ((convex_Ici a).segment_subset (Set.mem_Ici.2 le_rfl) (Set.mem_Ici.2 (by linarith)))
+
+/-- The backward direction belongs to the positive tangent cone of a left half-line. -/
+theorem neg_one_mem_posTangentConeAt_Iic (a : ℝ) :
+    (-1 : ℝ) ∈ posTangentConeAt (Set.Iic a) a :=
+  mem_posTangentConeAt_of_segment_subset
+    ((convex_Iic a).segment_subset (Set.mem_Iic.2 le_rfl) (Set.mem_Iic.2 (by linarith)))
+
+/-- A right derivative at a minimum over a right half-line is nonnegative. -/
+theorem IsLocalMinOn.hasDerivWithinAt_Ici_nonneg {f : ℝ → ℝ} {f' a : ℝ}
+    (h : IsLocalMinOn f (Set.Ici a) a) (hf : HasDerivWithinAt f f' (Set.Ici a) a) : 0 ≤ f' := by
+  simpa using h.hasFDerivWithinAt_nonneg hf (one_mem_posTangentConeAt_Ici a)
+
+/-- A left derivative at a minimum over a left half-line is nonpositive. -/
+theorem IsLocalMinOn.hasDerivWithinAt_Iic_nonpos {f : ℝ → ℝ} {f' a : ℝ}
+    (h : IsLocalMinOn f (Set.Iic a) a) (hf : HasDerivWithinAt f f' (Set.Iic a) a) : f' ≤ 0 :=
+  neg_nonneg.1 <| by
+    simpa using h.hasFDerivWithinAt_nonneg hf (neg_one_mem_posTangentConeAt_Iic a)
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Analysis / Complex / Disk Integral
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace Complex
+
+private theorem integrableOn_polarCoord_target
+    {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
+    (g : ℝ × ℝ → F) (hg : Integrable g) :
+    IntegrableOn (fun p : ℝ × ℝ ↦ p.1 • g (polarCoord.symm p))
+      polarCoord.target := by
+  have hinj : Set.InjOn (polarCoord.symm : ℝ × ℝ → ℝ × ℝ)
+      polarCoord.target := by
+    rw [← polarCoord.symm_source]
+    exact polarCoord.symm.injOn
+  have himage : polarCoord.symm '' polarCoord.target =
+      polarCoord.source := by
+    rw [← polarCoord.symm_source, ← polarCoord.symm_target]
+    exact polarCoord.symm.image_source_eq_target
+  have hj := (integrableOn_image_iff_integrableOn_abs_det_fderiv_smul
+    (s := polarCoord.target) volume polarCoord.open_target.measurableSet
+    (fun p _ ↦ (hasFDerivAt_polarCoord_symm p).hasFDerivWithinAt)
+    hinj g).mp (by
+      rw [himage]
+      exact hg.integrableOn)
+  refine hj.congr_fun ?_ polarCoord.open_target.measurableSet
+  intro p hp
+  dsimp only
+  rw [det_fderivPolarCoordSymm, abs_of_pos hp.1]
+
+private theorem integrableOn_complex_polarCoord_target
+    {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
+    (f : ℂ → F) (hf : Integrable f) :
+    IntegrableOn (fun p : ℝ × ℝ ↦ p.1 • f (Complex.polarCoord.symm p))
+      Complex.polarCoord.target := by
+  rw [Complex.polarCoord_target]
+  let g : ℝ × ℝ → F := f ∘ Complex.measurableEquivRealProd.symm
+  have hg : Integrable g :=
+    ((Complex.volume_preserving_equiv_real_prod.symm).integrable_comp_emb
+      Complex.measurableEquivRealProd.symm.measurableEmbedding).mpr hf
+  have h := integrableOn_polarCoord_target g hg
+  change IntegrableOn
+    (fun p ↦ p.1 • (f ∘ Complex.measurableEquivRealProd.symm)
+      (polarCoord.symm p)) (Set.Ioi 0 ×ˢ Set.Ioo (-Real.pi) Real.pi) at h
+  simpa only [Function.comp_apply,
+    Complex.measurableEquivRealProd_symm_polarCoord_symm_apply] using h
+
+private theorem integral_circleMap_Ioo_eq_circleAverage
+    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (f : ℂ → E) (r : ℝ) :
+    (∫ θ in Set.Ioo (-Real.pi) Real.pi, f (circleMap 0 r θ)) =
+      (2 * Real.pi) • Real.circleAverage f 0 r := by
+  rw [Real.circleAverage_eq_integral_add (-Real.pi), smul_smul]
+  have hp : (2 * Real.pi) * (2 * Real.pi)⁻¹ = 1 := by
+    field_simp [Real.pi_ne_zero]
+  rw [hp, one_smul, ← MeasureTheory.integral_Ioc_eq_integral_Ioo,
+    ← intervalIntegral.integral_of_le (le_of_lt (neg_lt_self Real.pi_pos))]
+  have hshift := intervalIntegral.integral_comp_add_right
+    (f := fun θ ↦ f (circleMap 0 r θ)) (a := 0) (b := 2 * Real.pi) (-Real.pi)
+  simpa [two_mul] using hshift.symm
+
+/-- The radial profile of the inverse distance truncated to a disk of radius `S`. -/
+private def invNormCutoff (S r : ℝ) : ℝ := if r < S then r⁻¹ else 0
+
+private theorem indicator_ball_inv_norm_eq (S : ℝ) :
+    (fun p : ℂ ↦ (Metric.ball 0 S).indicator (fun p ↦ ‖p‖⁻¹) p) =
+      fun p ↦ invNormCutoff S ‖p‖ := by
+  funext p
+  rw [Set.indicator]
+  simp only [Metric.mem_ball, dist_zero_right, invNormCutoff]
+
+private theorem radial_invNormCutoff_ae (S : ℝ) :
+    (fun y : ℝ ↦ y ^ (Module.finrank ℝ ℂ - 1) • invNormCutoff S y)
+      =ᵐ[volume.restrict (Set.Ioi 0)]
+      (Set.Iio S).indicator (fun _ ↦ (1 : ℝ)) := by
+  filter_upwards [ae_restrict_mem measurableSet_Ioi] with y hy
+  have hy0 : y ≠ 0 := ne_of_gt hy
+  simp [invNormCutoff, Set.indicator, smul_eq_mul, hy0]
+
+/-- Inverse distance from zero is integrable on a complex disk centred at zero. -/
+theorem integrableOn_inv_norm_ball (S : ℝ) :
+    IntegrableOn (fun p : ℂ ↦ ‖p‖⁻¹) (Metric.ball 0 S) := by
+  rw [← integrable_indicator_iff measurableSet_ball]
+  change Integrable (fun p : ℂ ↦ (Metric.ball 0 S).indicator (fun p ↦ ‖p‖⁻¹) p)
+  rw [indicator_ball_inv_norm_eq,
+    MeasureTheory.integrable_fun_norm_addHaar volume (f := invNormCutoff S)]
+  refine IntegrableOn.congr_fun_ae ?_ (radial_invNormCutoff_ae S).symm
+  rw [integrableOn_indicator_iff measurableSet_Iio, Set.Iio_inter_Ioi]
+  exact integrableOn_const (by simp)
+
+/-- The integral of inverse distance from zero over a complex disk centred at zero. -/
+theorem setIntegral_inv_norm_ball (S : ℝ) (hS : 0 < S) :
+    (∫ p : ℂ in Metric.ball 0 S, ‖p‖⁻¹) = 2 * Real.pi * S := by
+  rw [← integral_indicator (f := fun p : ℂ ↦ ‖p‖⁻¹) measurableSet_ball,
+    indicator_ball_inv_norm_eq,
+    MeasureTheory.integral_fun_norm_addHaar volume (invNormCutoff S),
+    integral_congr_ae (radial_invNormCutoff_ae S),
+    setIntegral_indicator measurableSet_Iio, Set.Ioi_inter_Iio]
+  simp [hS.le, Measure.real, Complex.volume_ball, mul_assoc]
+
+private theorem integrableOn_inv_sub_ball (R : ℝ) (a : ℂ) (ha : ‖a‖ < R) :
+    IntegrableOn (fun q : ℂ ↦ (a - q)⁻¹) (Metric.ball 0 R) := by
+  rw [← integrable_indicator_iff measurableSet_ball]
+  let g : ℂ → ℝ :=
+    (Metric.ball a (2 * R)).indicator (fun q ↦ ‖a - q‖⁻¹)
+  have hg : Integrable g := by
+    have hcenter : (fun q : ℂ ↦
+        (Metric.ball a (2 * R)).indicator (fun q ↦ ‖a - q‖⁻¹) q) =
+        fun q ↦ (Metric.ball 0 (2 * R)).indicator (fun w ↦ ‖w‖⁻¹) (a - q) := by
+      funext q
+      have hm : q ∈ Metric.ball a (2 * R) ↔ a - q ∈ Metric.ball 0 (2 * R) := by
+        simp only [Metric.mem_ball, dist_eq_norm]
+        simp [sub_zero, norm_sub_rev]
+      by_cases hq : q ∈ Metric.ball a (2 * R)
+      · simp [hq, hm.mp hq]
+      · simp [hq, mt hm.mpr hq]
+    change Integrable (fun q : ℂ ↦
+      (Metric.ball a (2 * R)).indicator (fun q ↦ ‖a - q‖⁻¹) q)
+    rw [hcenter, MeasureTheory.integrable_comp_sub_left]
+    exact (integrable_indicator_iff measurableSet_ball).mpr
+      (integrableOn_inv_norm_ball (2 * R))
+  refine Integrable.mono' hg ?_ ?_
+  · exact ((measurable_const.sub measurable_id).inv.indicator
+      measurableSet_ball).aestronglyMeasurable
+  · filter_upwards with q
+    by_cases hq : q ∈ Metric.ball (0 : ℂ) R
+    · have hqa : q ∈ Metric.ball a (2 * R) := by
+        rw [Metric.mem_ball, dist_eq_norm]
+        have hqR : ‖q‖ < R := by simpa [Metric.mem_ball, dist_eq_norm] using hq
+        calc
+          ‖q - a‖ ≤ ‖q‖ + ‖a‖ := norm_sub_le q a
+          _ < R + R := add_lt_add hqR ha
+          _ = 2 * R := by ring
+      simp only [Set.indicator_of_mem hq, norm_inv]
+      simp [g, hqa]
+    · simp only [Set.indicator, hq, ↓reduceIte, norm_zero]
+      exact Set.indicator_apply_nonneg fun _ ↦ by positivity
+
+private theorem circleAverage_inv_sub_eq_inv_of_radius_lt_norm
+    (a : ℂ) (r : ℝ) (hr : 0 ≤ r) (hout : r < ‖a‖) :
+    Real.circleAverage (fun w ↦ (a - w)⁻¹) 0 r = a⁻¹ := by
+  have hne : ∀ w ∈ Metric.closedBall (0 : ℂ) |r|, a - w ≠ 0 := by
+    intro w hw hwa
+    have hwa' : w = a := (sub_eq_zero.mp hwa).symm
+    subst a
+    have : ‖w‖ ≤ r := by simpa [abs_of_nonneg hr, dist_eq_norm] using hw
+    linarith
+  have hcont : ContinuousOn (fun w : ℂ ↦ (a - w)⁻¹) (Metric.closedBall 0 |r|) :=
+    ((continuous_const.sub continuous_id).continuousOn).inv₀ hne
+  have hdiff : DiffContOnCl ℂ (fun w : ℂ ↦ (a - w)⁻¹) (Metric.ball 0 |r|) :=
+    DiffContOnCl.mk_ball
+      (fun w hw ↦ (((differentiableAt_const a).sub differentiableAt_id).inv
+        (hne w (Metric.ball_subset_closedBall hw))).differentiableWithinAt) hcont
+  simpa using hdiff.circleAverage
+
+private theorem partialFraction_circleIntegral_congr (a : ℂ) (r : ℝ) (hr : 0 < r)
+    (ha : a ≠ 0) (hra : ‖a‖ ≠ r) :
+    (∮ w in C(0, r), (w - 0)⁻¹ * (a - w)⁻¹) =
+      ∮ w in C(0, r), a⁻¹ * ((w - 0)⁻¹ - (w - a)⁻¹) := by
+  refine circleIntegral.integral_congr hr.le fun w hw ↦ ?_
+  have hw0 : w ≠ 0 := by
+    intro h
+    subst w
+    have hz : (0 : ℝ) = r := by simpa using (Metric.mem_sphere.mp hw)
+    linarith
+  have hwa : w ≠ a := by
+    intro h
+    subst w
+    exact hra (by simpa [dist_eq_norm] using (Metric.mem_sphere.mp hw))
+  field_simp
+  ring
+
+private theorem circleIntegrable_sub_inv_of_norm_ne (a : ℂ) (r : ℝ) (hr : 0 < r)
+    (hra : ‖a‖ ≠ r) : CircleIntegrable (fun w : ℂ ↦ (w - a)⁻¹) 0 r :=
+  circleIntegrable_sub_inv_iff.mpr (Or.inr fun h ↦
+    hra (by simpa [dist_eq_norm, abs_of_pos hr] using (Metric.mem_sphere.mp h)))
+
+private theorem circleAverage_inv_sub_eq_zero_of_norm_lt_radius
+    (a : ℂ) (r : ℝ) (hr : 0 < r) (ha : a ≠ 0) (hinner : ‖a‖ < r) :
+    Real.circleAverage (fun w ↦ (a - w)⁻¹) 0 r = 0 := by
+  rw [Real.circleAverage_eq_circleIntegral hr.ne']
+  change (2 * (Real.pi : ℂ) * Complex.I)⁻¹ *
+      (∮ z in C(0, r), (z - 0)⁻¹ * (a - z)⁻¹) = 0
+  have hmem : a ∈ Metric.ball (0 : ℂ) r := by
+    simpa [Metric.mem_ball, dist_eq_norm] using hinner
+  rw [partialFraction_circleIntegral_congr a r hr ha hinner.ne,
+    circleIntegral.integral_const_mul,
+    circleIntegral.integral_sub
+      (circleIntegrable_sub_inv_of_norm_ne 0 r hr (by simpa using hr.ne))
+      (circleIntegrable_sub_inv_of_norm_ne a r hr hinner.ne),
+    circleIntegral.integral_sub_center_inv 0 hr.ne',
+    circleIntegral.integral_sub_inv_of_mem_ball hmem]
+  simp
+
+private theorem setIntegral_inv_sub_ball_of_ne_zero (R : ℝ) (a : ℂ) (ha0 : a ≠ 0)
+    (ha : ‖a‖ < R) :
+    (∫ q in Metric.ball 0 R, (a - q)⁻¹) = Real.pi * starRingEnd ℂ a := by
+  let f : ℂ → ℂ := (Metric.ball 0 R).indicator (fun q ↦ (a - q)⁻¹)
+  have hf : Integrable f :=
+    (integrable_indicator_iff measurableSet_ball).mpr (integrableOn_inv_sub_ball R a ha)
+  have hpInt := integrableOn_complex_polarCoord_target f hf
+  have hpolar := Complex.integral_comp_polarCoord_symm f
+  have hprod :
+      (∫ p in polarCoord.target,
+        p.1 • f (Complex.polarCoord.symm p)) =
+        ∫ r in Set.Ioi (0 : ℝ), ∫ θ in Set.Ioo (-Real.pi) Real.pi,
+          r • f (Complex.polarCoord.symm (r, θ)) := by
+    change (∫ p in Set.Ioi (0 : ℝ) ×ˢ Set.Ioo (-Real.pi) Real.pi,
+      p.1 • f (Complex.polarCoord.symm p)) = _
+    exact setIntegral_prod _ (by
+      rw [← Measure.volume_eq_prod ℝ ℝ]
+      simpa only [Complex.polarCoord_target] using hpInt)
+  rw [← integral_indicator measurableSet_ball]
+  change (∫ q, f q) = _
+  rw [← hpolar, hprod]
+  have hinner : ∀ r ∈ Set.Ioi (0 : ℝ),
+      (∫ θ in Set.Ioo (-Real.pi) Real.pi,
+          r • f (Complex.polarCoord.symm (r, θ))) =
+        if r < R then r • ((2 * Real.pi) •
+          Real.circleAverage (fun q ↦ (a - q)⁻¹) 0 r) else 0 := by
+    intro r hr
+    have hr0 : 0 < r := hr
+    have hpoint : ∀ θ, Complex.polarCoord.symm (r, θ) = circleMap 0 r θ := by
+      intro θ
+      rw [Complex.polarCoord_symm_apply, circleMap_zero, Complex.exp_mul_I]
+      simp
+    by_cases hrR : r < R
+    · simp only [hrR, ↓reduceIte]
+      have hmem : ∀ θ, Complex.polarCoord.symm (r, θ) ∈ Metric.ball (0 : ℂ) R := by
+        intro θ
+        rw [hpoint]
+        simpa [Metric.mem_ball, dist_eq_norm, abs_of_pos hr0] using hrR
+      simp_rw [f, Set.indicator_of_mem (hmem _), hpoint]
+      rw [integral_smul]
+      rw [integral_circleMap_Ioo_eq_circleAverage (fun q ↦ (a - q)⁻¹) r]
+    · simp only [hrR, ↓reduceIte]
+      have hnotmem : ∀ θ, Complex.polarCoord.symm (r, θ) ∉ Metric.ball (0 : ℂ) R := by
+        intro θ hθ
+        rw [hpoint] at hθ
+        have : r < R := by
+          simpa [Metric.mem_ball, dist_eq_norm, abs_of_pos hr0] using hθ
+        exact hrR this
+      simp_rw [f, Set.indicator, hnotmem]
+      simp
+  rw [setIntegral_congr_fun measurableSet_Ioi hinner]
+  have hradial :
+      (fun r : ℝ ↦ if r < R then r • ((2 * Real.pi) •
+        Real.circleAverage (fun q ↦ (a - q)⁻¹) 0 r) else 0)
+        =ᵐ[volume.restrict (Set.Ioi 0)]
+      fun r ↦ if r < ‖a‖ then r • ((2 * Real.pi) • a⁻¹) else 0 := by
+    filter_upwards [ae_restrict_mem measurableSet_Ioi,
+      ae_restrict_of_ae (Measure.ae_ne volume ‖a‖)] with r hr hra
+    have hr0 : 0 < r := hr
+    by_cases hra' : r < ‖a‖
+    · have hrR : r < R := hra'.trans ha
+      simp only [hrR, hra', ↓reduceIte]
+      rw [circleAverage_inv_sub_eq_inv_of_radius_lt_norm a r hr0.le hra']
+    · have har : ‖a‖ < r := lt_of_le_of_ne (le_of_not_gt hra') (Ne.symm hra)
+      by_cases hrR : r < R
+      · simp only [hrR, hra', ↓reduceIte]
+        rw [circleAverage_inv_sub_eq_zero_of_norm_lt_radius a r hr0 ha0 har, smul_zero]
+        simp
+      · simp only [hrR, hra', ↓reduceIte]
+  rw [integral_congr_ae hradial]
+  rw [show (fun r : ℝ ↦ if r < ‖a‖ then r • ((2 * Real.pi) • a⁻¹) else 0) =
+      (Set.Iio ‖a‖).indicator (fun r ↦ r • ((2 * Real.pi) • a⁻¹)) by
+    funext r
+    simp [Set.indicator]]
+  rw [setIntegral_indicator measurableSet_Iio, Set.Ioi_inter_Iio]
+  rw [← integral_Ioc_eq_integral_Ioo,
+    ← intervalIntegral.integral_of_le (norm_nonneg a)]
+  rw [intervalIntegral.integral_smul_const]
+  rw [integral_id]
+  rw [Complex.inv_def]
+  simp only [starRingEnd_apply, Complex.normSq_eq_norm_sq, Complex.real_smul]
+  have hnorm : ‖a‖ ≠ 0 := norm_ne_zero_iff.mpr ha0
+  have hnormc : (‖a‖ : ℂ) ≠ 0 := by exact_mod_cast hnorm
+  push_cast
+  field_simp [hnormc]
+  ring
+
+private theorem setIntegral_inv_sub_ball_zero (R : ℝ) :
+    (∫ q in Metric.ball 0 R, ((0 : ℂ) - q)⁻¹) = 0 := by
+  rw [← integral_indicator (f := fun q : ℂ ↦ ((0 : ℂ) - q)⁻¹) measurableSet_ball]
+  let f : ℂ → ℂ := (Metric.ball 0 R).indicator (fun q ↦ ((0 : ℂ) - q)⁻¹)
+  have hodd : ∀ q, f (-q) = -f q := by
+    intro q
+    have hm : -q ∈ Metric.ball (0 : ℂ) R ↔ q ∈ Metric.ball 0 R := by
+      simp [Metric.mem_ball, dist_eq_norm]
+    by_cases hq : q ∈ Metric.ball (0 : ℂ) R
+    · simp [f, hq, hm.mpr hq]
+    · simp [f, hq, mt hm.mp hq]
+  have hneg : (∫ q, f (-q)) = -(∫ q, f q) := by
+    rw [integral_congr_ae (Filter.Eventually.of_forall hodd), integral_neg]
+  have hself : (∫ q, f q) = -(∫ q, f q) :=
+    (MeasureTheory.integral_neg_eq_self f volume).symm.trans hneg
+  have ht : (2 : ℝ) • (∫ q, f q) = 0 := by
+    rw [two_smul]
+    calc
+      (∫ q, f q) + ∫ q, f q = -(∫ q, f q) + ∫ q, f q :=
+        congrArg (fun x ↦ x + ∫ q, f q) hself
+      _ = 0 := neg_add_cancel _
+  exact (smul_eq_zero.mp ht).resolve_left (by norm_num)
+
+/-- The integral of `q ↦ (a - q)⁻¹` over a disk centred at zero containing `a`. -/
+theorem setIntegral_inv_sub_ball (R : ℝ) (a : ℂ) (ha : ‖a‖ < R) :
+    (∫ q in Metric.ball 0 R, (a - q)⁻¹) = Real.pi * starRingEnd ℂ a := by
+  by_cases ha0 : a = 0
+  · subst a
+    simpa using setIntegral_inv_sub_ball_zero R
+  · exact setIntegral_inv_sub_ball_of_ne_zero R a ha0 ha
+
+end Complex
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Analysis / Convex / Basic
+-/
+
+@[expose] public section
+
+namespace Convex
+
+/-- Every nonnegative scalar between zero and a known scalar multiple stays in a convex set. -/
+theorem smul_mem_of_nonneg_of_le {E : Type*} [AddCommGroup E] [Module ℝ E]
+    {s : Set E} (hs : Convex ℝ s) {v : E} {x a : ℝ}
+    (hzero : (0 : E) ∈ s) (ha : a • v ∈ s) (hx : 0 ≤ x) (hxa : x ≤ a) :
+    x • v ∈ s := by
+  by_cases ha0 : a = 0
+  · have hx0 : x = 0 := by linarith
+    simpa [hx0] using hzero
+  · have hapos : 0 < a := lt_of_le_of_ne (hx.trans hxa) (Ne.symm ha0)
+    have hratio : x / a ∈ Set.Icc (0 : ℝ) 1 :=
+      ⟨div_nonneg hx hapos.le, (div_le_one hapos).2 hxa⟩
+    have hmem := hs.smul_mem_of_zero_mem hzero ha hratio
+    simpa only [smul_smul, div_mul_cancel₀ x ha0] using hmem
+
+end Convex
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Analysis / Convex / Deriv
+-/
+
+@[expose] public section
+
+open Filter Set
+open scoped Topology
+
+namespace ConcaveOn
+
+/-- A differentiable concave real function lies below its tangent line. -/
+theorem le_add_deriv_mul_sub {s : Set ℝ} {g : ℝ → ℝ}
+    (hg : ConcaveOn ℝ s g) {x : ℝ} (hx : x ∈ s)
+    (hdiff : DifferentiableAt ℝ g x) {y : ℝ} (hy : y ∈ s) :
+    g y ≤ g x + deriv g x * (y - x) := by
+  rcases lt_trichotomy y x with hyx | rfl | hxy
+  · have hslope := hg.deriv_le_slope hy hx hyx hdiff
+    rw [slope] at hslope
+    have hpos : 0 < x - y := sub_pos.mpr hyx
+    have hs : deriv g x * (x - y) ≤ g x - g y := by
+      apply (le_div_iff₀ hpos).mp
+      simpa [smul_eq_mul, inv_mul_eq_div] using hslope
+    nlinarith
+  · simp
+  · have hslope := hg.slope_le_of_hasDerivAt hx hy hxy hdiff.hasDerivAt
+    rw [slope] at hslope
+    have hpos : 0 < y - x := sub_pos.mpr hxy
+    have hs : g y - g x ≤ deriv g x * (y - x) := by
+      apply (div_le_iff₀ hpos).mp
+      simpa [smul_eq_mul, inv_mul_eq_div] using hslope
+    linarith
+
+/-- Pointwise convergence of concave functions forces derivative convergence at a common
+differentiability point in the interior of an interval. -/
+theorem tendsto_deriv_of_tendsto_Ioo {ι : Type*} {F : Filter ι}
+    {f : ι → ℝ → ℝ} {g : ℝ → ℝ}
+    {a b x : ℝ} (hx : x ∈ Ioo a b)
+    (hf : ∀ᶠ n in F, ConcaveOn ℝ (Ioo a b) (f n))
+    (hlim : ∀ y ∈ Ioo a b, Tendsto (fun n ↦ f n y) F (𝓝 (g y)))
+    (hdf : ∀ᶠ n in F, DifferentiableAt ℝ (f n) x)
+    (hdg : DifferentiableAt ℝ g x) :
+    Tendsto (fun n ↦ deriv (f n) x) F (𝓝 (deriv g x)) := by
+  have hslope (y : ℝ) (hy : y ∈ Ioo a b) :
+      Tendsto (fun n ↦ slope (f n) x y) F (𝓝 (slope g x y)) := by
+    simpa only [slope, smul_eq_mul, vsub_eq_sub] using
+      ((hlim y hy).sub (hlim x hx)).const_mul (y - x)⁻¹
+  have hnear : ∀ᶠ y in 𝓝 x, y ∈ Ioo a b := isOpen_Ioo.mem_nhds hx
+  apply tendsto_order.mpr
+  constructor
+  · intro l hl
+    have hright := hdg.hasDerivAt.tendsto_slope.mono_left (nhdsGT_le_nhdsNE x)
+    have hgood : ∀ᶠ y in 𝓝[>] x, y ∈ Ioo a b ∧ x < y ∧ l < slope g x y := by
+      filter_upwards [hnear.filter_mono nhdsWithin_le_nhds, self_mem_nhdsWithin,
+        hright.eventually (lt_mem_nhds hl)] with y hy hxy hsy
+      exact ⟨hy, hxy, hsy⟩
+    obtain ⟨y, hy, hxy, hly⟩ := hgood.exists
+    filter_upwards [hf, hdf, (hslope y hy).eventually (lt_mem_nhds hly)] with n hfn hdn hn
+    exact hn.trans_le (hfn.slope_le_of_hasDerivAt hx hy hxy hdn.hasDerivAt)
+  · intro u hu
+    have hleft := hdg.hasDerivAt.tendsto_slope.mono_left (nhdsLT_le_nhdsNE x)
+    have hgood : ∀ᶠ y in 𝓝[<] x, y ∈ Ioo a b ∧ y < x ∧ slope g x y < u := by
+      filter_upwards [hnear.filter_mono nhdsWithin_le_nhds, self_mem_nhdsWithin,
+        hleft.eventually (gt_mem_nhds hu)] with y hy hyx hsy
+      exact ⟨hy, hyx, hsy⟩
+    obtain ⟨y, hy, hyx, hyu⟩ := hgood.exists
+    filter_upwards [hf, hdf, (hslope y hy).eventually (gt_mem_nhds hyu)] with n hfn hdn hn
+    have hle := hfn.deriv_le_slope hy hx hyx hdn
+    rw [slope_comm] at hle
+    exact hle.trans_lt hn
+
+end ConcaveOn
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Analysis / Convex / Gauge
+-/
+
+@[expose] public section
+
+open Set
+open scoped Topology NNReal
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+/-- A direction of positive gauge has a unique positive multiple on the frontier. -/
+lemma Convex.existsUnique_pos_smul_mem_frontier_of_gauge_pos {s : Set E}
+    (hs : Convex ℝ s) (h₀ : s ∈ nhds (0 : E)) {u : E} (hu : 0 < gauge s u) :
+    ∃! r : ℝ, 0 < r ∧ r • u ∈ frontier s := by
+  refine ⟨(gauge s u)⁻¹, ⟨inv_pos.mpr hu, ?_⟩, ?_⟩
+  · apply (gauge_eq_one_iff_mem_frontier hs h₀).mp
+    rw [gauge_smul_of_nonneg (inv_nonneg.mpr hu.le), smul_eq_mul, inv_mul_cancel₀ hu.ne']
+  · intro r hr
+    have h := (gauge_eq_one_iff_mem_frontier hs h₀).mpr hr.2
+    rw [gauge_smul_of_nonneg hr.1.le, smul_eq_mul] at h
+    simpa only [one_div] using (eq_div_iff hu.ne').mpr h
+
+/-- Every nonzero direction has a unique positive multiple on the frontier of a bounded convex
+neighborhood of zero. -/
+lemma Convex.existsUnique_pos_smul_mem_frontier {s : Set E}
+    (hs : Convex ℝ s) (h₀ : s ∈ nhds (0 : E))
+    (hb : Bornology.IsVonNBounded ℝ s) {u : E} (hu : u ≠ 0) :
+    ∃! r : ℝ, 0 < r ∧ r • u ∈ frontier s :=
+  hs.existsUnique_pos_smul_mem_frontier_of_gauge_pos h₀
+    ((gauge_pos (absorbent_nhds_zero h₀) hb).mpr hu)
+
+/-- Reciprocation is Lipschitz on real numbers bounded below by a positive constant. -/
+lemma Real.dist_inv_le_of_pos_lower_bound {a b c : ℝ} (hc : 0 < c)
+    (ha : c ≤ a) (hb : c ≤ b) :
+    dist a⁻¹ b⁻¹ ≤ c⁻¹ ^ 2 * dist a b := by
+  have ha₀ := hc.trans_le ha
+  have hb₀ := hc.trans_le hb
+  rw [dist_inv_inv₀ ha₀.ne' hb₀.ne', Real.norm_eq_abs, Real.norm_eq_abs,
+    abs_of_pos ha₀, abs_of_pos hb₀]
+  calc
+    dist a b / (a * b) ≤ dist a b / (c * c) := by gcongr
+    _ = c⁻¹ ^ 2 * dist a b := by ring
+
+/-- The reciprocal of a positive uniformly bounded-below Lipschitz function is Lipschitz. -/
+lemma LipschitzWith.inv_of_pos_lower_bound {X : Type*} [PseudoMetricSpace X]
+    {f : X → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) {c : ℝ}
+    (hc : 0 < c) (hbound : ∀ x, c ≤ f x) :
+    LipschitzWith (Real.toNNReal (c⁻¹ ^ 2) * K) (fun x ↦ (f x)⁻¹) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro x y
+  calc
+    dist (f x)⁻¹ (f y)⁻¹ ≤ c⁻¹ ^ 2 * dist (f x) (f y) :=
+      Real.dist_inv_le_of_pos_lower_bound hc (hbound x) (hbound y)
+    _ ≤ c⁻¹ ^ 2 * ((K : ℝ) * dist x y) :=
+      mul_le_mul_of_nonneg_left (hf.dist_le_mul x y) (sq_nonneg _)
+    _ = _ := by
+      rw [NNReal.coe_mul, Real.coe_toNNReal _ (sq_nonneg _)]
+      ring
+
+/-- The reciprocal gauge is Lipschitz on the unit sphere of a bounded convex neighborhood of zero.
+-/
+lemma Convex.exists_lipschitzWith_inv_gauge_sphere {s : Set E}
+    (hs : Convex ℝ s) (h₀ : s ∈ nhds (0 : E)) {R : ℝ} (hR : 0 < R)
+    (hbound : s ⊆ Metric.closedBall 0 R) :
+    ∃ C, LipschitzWith C (fun u : {u : E | ‖u‖ = 1} ↦ (gauge s u.val)⁻¹) := by
+  obtain ⟨K, hK⟩ := hs.lipschitz_gauge h₀
+  have hf : LipschitzWith K (fun u : {u : E | ‖u‖ = 1} ↦ gauge s u.val) :=
+    by
+      apply LipschitzWith.of_dist_le_mul
+      intro u v
+      exact hK.dist_le_mul u.val v.val
+  have hlower (u : {u : E | ‖u‖ = 1}) : 1 / R ≤ gauge s u.val := by
+    have hu : ‖(u : E)‖ = 1 := u.property
+    calc
+      1 / R = ‖u.val‖ / R := congrArg (fun r ↦ r / R) hu.symm
+      _ ≤ gauge s u.val :=
+        le_gauge_of_subset_closedBall (absorbent_nhds_zero h₀) hR.le hbound
+  exact ⟨_, hf.inv_of_pos_lower_bound (one_div_pos.mpr hR) hlower⟩
+
+/-- Radial gauge rescaling is Lipschitz on the unit sphere. -/
+lemma Convex.exists_lipschitzWith_radial_gauge_sphere {s : Set E}
+    (hs : Convex ℝ s) (h₀ : s ∈ nhds (0 : E)) {R : ℝ} (hR : 0 < R)
+    (hbound : s ⊆ Metric.closedBall 0 R) :
+    ∃ C, LipschitzWith C
+      (fun u : {u : E | ‖u‖ = 1} ↦ (gauge s u.val)⁻¹ • u.val) := by
+  obtain ⟨C, hC⟩ := hs.exists_lipschitzWith_inv_gauge_sphere h₀ hR hbound
+  have hr (u : {u : E | ‖u‖ = 1}) : 0 ≤ (gauge s u.val)⁻¹ ∧
+      (gauge s u.val)⁻¹ ≤ R := by
+    have hu : ‖u.val‖ = 1 := u.property
+    have hlower : 1 / R ≤ gauge s u.val := by
+      calc
+        1 / R = ‖u.val‖ / R := congrArg (fun r ↦ r / R) hu.symm
+        _ ≤ _ := le_gauge_of_subset_closedBall (absorbent_nhds_zero h₀) hR.le hbound
+    refine ⟨inv_nonneg.mpr (gauge_nonneg _), ?_⟩
+    simpa only [one_div, inv_inv] using
+      one_div_le_one_div_of_le (one_div_pos.mpr hR) hlower
+  refine ⟨Real.toNNReal R + C, LipschitzWith.of_dist_le_mul fun u v ↦ ?_⟩
+  have hv : ‖v.val‖ = 1 := v.property
+  calc
+    dist ((gauge s u.val)⁻¹ • u.val) ((gauge s v.val)⁻¹ • v.val) ≤
+        dist ((gauge s u.val)⁻¹ • u.val) ((gauge s u.val)⁻¹ • v.val) +
+        dist ((gauge s u.val)⁻¹ • v.val) ((gauge s v.val)⁻¹ • v.val) :=
+      dist_triangle _ _ _
+    _ = (gauge s u.val)⁻¹ * dist u v +
+        dist (gauge s u.val)⁻¹ (gauge s v.val)⁻¹ := by
+      rw [dist_smul₀, Real.norm_eq_abs, abs_of_nonneg (hr u).1]
+      congr 1
+      rw [dist_eq_norm, ← sub_smul, norm_smul, hv, mul_one, dist_eq_norm]
+    _ ≤ R * dist u v + (C : ℝ) * dist u v :=
+      add_le_add (mul_le_mul_of_nonneg_right (hr u).2 dist_nonneg) (hC.dist_le_mul u v)
+    _ = _ := by rw [NNReal.coe_add, Real.coe_toNNReal _ hR.le]; ring
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Analysis / Convex / Gauge Rescale
+-/
+
+@[expose] public section
+
+open Set
+open scoped Topology NNReal Pointwise
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+/-- Gauge rescaling identifies the unit sphere with the frontier of a bounded convex neighborhood.
+-/
+noncomputable def radialGaugeHomeomorph {s : Set E}
+    (hs : Convex ℝ s) (h₀ : s ∈ nhds (0 : E))
+    (hb : Bornology.IsVonNBounded ℝ s) :
+    {u : E | ‖u‖ = 1} ≃ₜ ↥(frontier s) := by
+  let e := gaugeRescaleHomeomorph (Metric.ball (0 : E) 1) s
+    (convex_ball 0 1) (Metric.ball_mem_nhds _ (by norm_num))
+    (NormedSpace.isVonNBounded_ball ℝ E 1) hs h₀ hb
+  have he : e '' frontier (Metric.ball (0 : E) 1) = frontier s := by
+    rw [← closure_sdiff_interior, Set.image_sdiff e.injective]
+    rw [image_gaugeRescaleHomeomorph_closure, image_gaugeRescaleHomeomorph_interior]
+    exact closure_sdiff_interior s
+  have hunit : {u : E | ‖u‖ = 1} = frontier (Metric.ball (0 : E) 1) := by
+    rw [frontier_ball _ one_ne_zero]
+    ext u
+    simp
+  exact (Homeomorph.setCongr hunit).trans ((e.image _).trans (Homeomorph.setCongr he))
+
+/-- The radial gauge homeomorphism sends a unit vector to its reciprocal-gauge multiple. -/
+lemma radialGaugeHomeomorph_apply {s : Set E}
+    (hs : Convex ℝ s) (h₀ : s ∈ nhds (0 : E))
+    (hb : Bornology.IsVonNBounded ℝ s) (u : {u : E | ‖u‖ = 1}) :
+    (radialGaugeHomeomorph hs h₀ hb u : E) = (gauge s u.val)⁻¹ • u.val := by
+  change (gauge (Metric.ball (0 : E) 1) u.val / gauge s u.val) • u.val = _
+  rw [gauge_ball (by norm_num), div_one]
+  have hu : ‖u.val‖ = 1 := u.property
+  rw [hu, one_div]
+
+open scoped Pointwise
+
+/-- Translate the radial gauge homeomorphism by a fixed vector. -/
+noncomputable def translatedRadialGaugeHomeomorph {s : Set E}
+    (hs : Convex ℝ s) (h₀ : s ∈ nhds (0 : E))
+    (hb : Bornology.IsVonNBounded ℝ s) (o : E) :
+    {u : E | ‖u‖ = 1} ≃ₜ ↥(frontier (o +ᵥ s)) := by
+  let e : E ≃ₜ E := Homeomorph.addLeft o
+  have he : e '' frontier s = frontier (o +ᵥ s) := by
+    rw [e.image_frontier]
+    rfl
+  exact (radialGaugeHomeomorph hs h₀ hb).trans
+    ((e.image _).trans (Homeomorph.setCongr he))
+
+/-- The translated radial gauge homeomorphism has the expected affine formula. -/
+lemma translatedRadialGaugeHomeomorph_apply {s : Set E}
+    (hs : Convex ℝ s) (h₀ : s ∈ nhds (0 : E))
+    (hb : Bornology.IsVonNBounded ℝ s) (o : E) (u : {u : E | ‖u‖ = 1}) :
+    (translatedRadialGaugeHomeomorph hs h₀ hb o u : E) =
+      o + (gauge s u.val)⁻¹ • u.val := by
+  change o + (radialGaugeHomeomorph hs h₀ hb u : E) = _
+  rw [radialGaugeHomeomorph_apply]
+
+/-- The translated radial gauge homeomorphism is Lipschitz on the unit sphere. -/
+lemma exists_lipschitzWith_translatedRadialGaugeHomeomorph {s : Set E}
+    (hs : Convex ℝ s) (h₀ : s ∈ nhds (0 : E))
+    (hb : Bornology.IsVonNBounded ℝ s) (o : E) {R : ℝ} (hR : 0 < R)
+    (hbound : s ⊆ Metric.closedBall 0 R) :
+    ∃ C, LipschitzWith C (fun u ↦ (translatedRadialGaugeHomeomorph hs h₀ hb o u : E)) := by
+  obtain ⟨C, hC⟩ := hs.exists_lipschitzWith_radial_gauge_sphere h₀ hR hbound
+  refine ⟨C, LipschitzWith.of_dist_le_mul fun u v ↦ ?_⟩
+  rw [translatedRadialGaugeHomeomorph_apply, translatedRadialGaugeHomeomorph_apply,
+    dist_add_left]
+  exact hC.dist_le_mul u v
+
+/-- A bounded convex set with an interior basepoint admits a positive Lipschitz radial
+parametrization of its frontier. -/
+lemma exists_radial_homeomorph {s : Set E}
+    (hs : Convex ℝ s) (hb : Bornology.IsBounded s) (o : E) (ho : o ∈ interior s) :
+    ∃ (ρ : E → ℝ) (e : {u : E | ‖u‖ = 1} ≃ₜ ↥(frontier s)) (C : NNReal),
+      (∀ u, ‖u‖ = 1 → 0 < ρ u ∧ o + ρ u • u ∈ frontier s ∧
+        ∀ r : ℝ, 0 < r → o + r • u ∈ frontier s → r = ρ u) ∧
+      (∀ u, (e u : E) = o + ρ u.val • u.val) ∧
+      LipschitzWith C (fun u ↦ (e u : E)) := by
+  let t : Set E := -o +ᵥ s
+  have htconv : Convex ℝ t := hs.vadd (-o)
+  have htzero : t ∈ nhds (0 : E) := by
+    simpa [t, ← mem_interior_iff_mem_nhds, interior_vadd,
+      mem_vadd_set_iff_neg_vadd_mem] using ho
+  have htbound : Bornology.IsBounded t := hb.vadd (-o)
+  have htbounded := NormedSpace.isVonNBounded_of_isBounded ℝ htbound
+  have htranslate : o +ᵥ t = s := by simp [t]
+  let e := (translatedRadialGaugeHomeomorph htconv htzero htbounded o).trans
+    (Homeomorph.setCongr (congrArg frontier htranslate))
+  obtain ⟨R, hR, hnorm⟩ := htbound.exists_pos_norm_le
+  have hball : t ⊆ Metric.closedBall 0 R := by
+    intro x hx
+    simpa using hnorm x hx
+  obtain ⟨C, hC⟩ := exists_lipschitzWith_translatedRadialGaugeHomeomorph
+    htconv htzero htbounded o hR hball
+  refine ⟨fun u ↦ (gauge t u)⁻¹, e, C, ?_, ?_, ?_⟩
+  · intro u hu
+    have hune : u ≠ 0 := by intro h; simp [h] at hu
+    have hg := (gauge_pos (absorbent_nhds_zero htzero) htbounded).mpr hune
+    refine ⟨inv_pos.mpr hg, ?_, ?_⟩
+    · have hm := (e ⟨u, hu⟩).property
+      change (translatedRadialGaugeHomeomorph htconv htzero htbounded o ⟨u, hu⟩ : E)
+        ∈ frontier s at hm
+      rwa [translatedRadialGaugeHomeomorph_apply] at hm
+    · intro r hr hru
+      have hz : r • u ∈ frontier t := by
+        change r • u ∈ frontier (-o +ᵥ s)
+        rw [frontier, closure_vadd, interior_vadd]
+        simpa only [frontier, Set.mem_sdiff, mem_vadd_set_iff_neg_vadd_mem, neg_neg, vadd_eq_add]
+          using hru
+      have hgauge := (gauge_eq_one_iff_mem_frontier htconv htzero).mpr hz
+      rw [gauge_smul_of_nonneg hr.le, smul_eq_mul] at hgauge
+      simpa only [one_div] using (eq_div_iff hg.ne').mpr hgauge
+  · intro u
+    exact translatedRadialGaugeHomeomorph_apply htconv htzero htbounded o u
+  · exact hC
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Radial exit points of a convex set
+
+From an interior base point of a compact convex set, every point of the set lies on a segment
+ending at a boundary point, and that boundary point is unique. These are the facts behind a
+radial decomposition of a convex body into cones over its boundary.
+-/
+
+@[expose] public section
+
+open Set
+open scoped Topology Pointwise
+
+open Set
+open scoped Topology
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+/-- A point lying a fixed fraction of the way, less than all of it, from an interior point of a
+convex set towards a point of the set is itself an interior point. -/
+theorem Convex.mem_interior_of_sub_eq_smul_sub {s : Set E} (hs : Convex ℝ s) {o w z : E}
+    {γ : ℝ} (ho : o ∈ interior s) (hw : w ∈ s) (hγ₀ : 0 ≤ γ) (hγ₁ : γ < 1)
+    (h : z - o = γ • (w - o)) : z ∈ interior s := by
+  have hz : z = (1 - γ) • o + γ • w := by
+    have hzo : z = o + γ • (w - o) := by rw [← h]; abel
+    rw [hzo, smul_sub, sub_smul, one_smul]
+    abel
+  rw [hz]
+  exact hs.combo_interior_closure_mem_interior ho (subset_closure hw) (by linarith) hγ₀ (by ring)
+
+/-- From an interior base point, every point of a compact convex set lies on a segment ending at
+a boundary point: the radial exit point of its direction. -/
+theorem Convex.exists_mem_frontier_mem_segment [Nontrivial E] {s : Set E} (hs : Convex ℝ s)
+    (hcomp : IsCompact s) {o : E} (ho : o ∈ interior s) {x : E} (hx : x ∈ s) :
+    ∃ p ∈ frontier s, x ∈ segment ℝ o p := by
+  have himage : ((-o) +ᵥ s) = (fun z : E ↦ -o + z) '' s := (Set.image_vadd).symm
+  have hsconv : Convex ℝ ((-o) +ᵥ s) := by
+    rw [himage]; exact hs.translate (-o)
+  have hscompact : IsCompact ((-o) +ᵥ s) := by
+    rw [himage]; exact hcomp.image (continuous_const.add continuous_id)
+  have h0 : (0 : E) ∈ interior ((-o) +ᵥ s) := by
+    rw [interior_vadd]
+    exact ⟨o, ho, by simp only [vadd_eq_add]; abel⟩
+  have hnhds : ((-o) +ᵥ s) ∈ 𝓝 (0 : E) := mem_interior_iff_mem_nhds.mp h0
+  have hgbound := NormedSpace.isVonNBounded_of_isBounded ℝ hscompact.isBounded
+  have hfrontier : ∀ {z : E}, z ∈ frontier ((-o) +ᵥ s) → o + z ∈ frontier s := by
+    intro z hz
+    have hzs : z ∈ ((-o) +ᵥ s) := by
+      have h := frontier_subset_closure hz
+      rwa [hscompact.isClosed.closure_eq] at h
+    obtain ⟨k, hk, hkeq⟩ := hzs
+    have hok : o + z = k := by rw [← hkeq]; simp only [vadd_eq_add]; abel
+    rw [hok, mem_frontier_iff_notMem_interior hk]
+    intro hkint
+    exact hz.2 (by rw [interior_vadd]; exact ⟨k, hkint, hkeq⟩)
+  by_cases hxo : x = o
+  · obtain ⟨u, hu⟩ := exists_ne (0 : E)
+    obtain ⟨r, hr, -⟩ := hsconv.existsUnique_pos_smul_mem_frontier hnhds hgbound hu
+    exact ⟨o + r • u, hfrontier hr.2, hxo ▸ left_mem_segment ℝ o _⟩
+  have hxs : x - o ∈ ((-o) +ᵥ s) := ⟨x, hx, by simp only [vadd_eq_add]; abel⟩
+  have hgpos : 0 < gauge ((-o) +ᵥ s) (x - o) :=
+    (gauge_pos (absorbent_nhds_zero hnhds) hgbound).mpr (sub_ne_zero.mpr hxo)
+  have hgle : gauge ((-o) +ᵥ s) (x - o) ≤ 1 := gauge_le_one_of_mem hxs
+  have hfr : (gauge ((-o) +ᵥ s) (x - o))⁻¹ • (x - o) ∈ frontier ((-o) +ᵥ s) := by
+    apply (gauge_eq_one_iff_mem_frontier hsconv hnhds).mp
+    rw [gauge_smul_of_nonneg (inv_nonneg.mpr hgpos.le), smul_eq_mul, inv_mul_cancel₀ hgpos.ne']
+  refine ⟨o + (gauge ((-o) +ᵥ s) (x - o))⁻¹ • (x - o), hfrontier hfr, ?_⟩
+  rw [segment_eq_image']
+  refine ⟨gauge ((-o) +ᵥ s) (x - o), ⟨hgpos.le, hgle⟩, ?_⟩
+  change o + gauge ((-o) +ᵥ s) (x - o) • (o + (gauge ((-o) +ᵥ s) (x - o))⁻¹ • (x - o) - o) = x
+  rw [add_sub_cancel_left, smul_smul, mul_inv_cancel₀ hgpos.ne', one_smul]
+  abel
+
+/-- The radial exit point from an interior base point is unique. -/
+theorem Convex.eq_of_mem_frontier_of_mem_segment {s : Set E} (hs : Convex ℝ s)
+    (hclosed : IsClosed s) {o x z w : E} (ho : o ∈ interior s) (hxo : x ≠ o)
+    (hz : z ∈ frontier s) (hw : w ∈ frontier s) (hxz : x ∈ segment ℝ o z)
+    (hxw : x ∈ segment ℝ o w) : z = w := by
+  have hzs : z ∈ s := hclosed.frontier_subset hz
+  have hws : w ∈ s := hclosed.frontier_subset hw
+  have hzint : z ∉ interior s := (mem_frontier_iff_notMem_interior hzs).mp hz
+  have hwint : w ∉ interior s := (mem_frontier_iff_notMem_interior hws).mp hw
+  rw [segment_eq_image'] at hxz hxw
+  obtain ⟨α, hα, hxα⟩ := hxz
+  obtain ⟨β, hβ, hxβ⟩ := hxw
+  have hxα' : o + α • (z - o) = x := hxα
+  have hxβ' : o + β • (w - o) = x := hxβ
+  have hαpos : 0 < α := hα.1.lt_or_eq.resolve_right fun h ↦ hxo (by rw [← hxα', ← h]; simp)
+  have hβpos : 0 < β := hβ.1.lt_or_eq.resolve_right fun h ↦ hxo (by rw [← hxβ', ← h]; simp)
+  have hkey : α • (z - o) = β • (w - o) := add_left_cancel (hxα'.trans hxβ'.symm)
+  have hzw : z - o = (α⁻¹ * β) • (w - o) := by
+    rw [mul_smul, ← hkey, smul_smul, inv_mul_cancel₀ hαpos.ne', one_smul]
+  have hwz : w - o = (α⁻¹ * β)⁻¹ • (z - o) := by
+    rw [hzw, smul_smul, inv_mul_cancel₀ (by positivity), one_smul]
+  rcases lt_trichotomy (α⁻¹ * β) 1 with hlt | heq | hgt
+  · exact absurd (hs.mem_interior_of_sub_eq_smul_sub ho hws (by positivity) hlt hzw) hzint
+  · have hzo : z - o = w - o := by rw [hzw, heq, one_smul]
+    simpa using congrArg (fun p : E ↦ p + o) hzo
+  · refine absurd (hs.mem_interior_of_sub_eq_smul_sub ho hzs (by positivity) ?_ hwz) hwint
+    rw [inv_lt_one_iff₀]
+    exact Or.inr hgt
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Analysis / Finite Envelope
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace List
+
+open Filter Topology
+
+lemma abs_foldr_min_apply_sub_le (l : List (ℝ → ℝ)) (r L x z : ℝ) (hL : 0 ≤ L)
+    (hl : ∀ f ∈ l, |f x - f z| ≤ L * |x - z|) :
+    |l.foldr (fun f s ↦ min (f x) s) r -
+        l.foldr (fun f s ↦ min (f z) s) r| ≤ L * |x - z| := by
+  induction l with
+  | nil => simpa using mul_nonneg hL (abs_nonneg (x - z))
+  | cons f l ih =>
+      simp only [List.foldr_cons]
+      refine (abs_min_sub_min_le_max _ _ _ _).trans (max_le ?_ ?_)
+      · exact hl f (by simp)
+      · exact ih (fun g hg ↦ hl g (by simp [hg]))
+
+lemma abs_foldr_max_apply_sub_le (l : List (ℝ → ℝ)) (r L x z : ℝ) (hL : 0 ≤ L)
+    (hl : ∀ f ∈ l, |f x - f z| ≤ L * |x - z|) :
+    |l.foldr (fun f s ↦ max (f x) s) r -
+        l.foldr (fun f s ↦ max (f z) s) r| ≤ L * |x - z| := by
+  induction l with
+  | nil => simpa using mul_nonneg hL (abs_nonneg (x - z))
+  | cons f l ih =>
+      simp only [List.foldr_cons]
+      refine (abs_max_sub_max_le_max _ _ _ _).trans (max_le ?_ ?_)
+      · exact hl f (by simp)
+      · exact ih (fun g hg ↦ hl g (by simp [hg]))
+
+lemma le_foldr_min_apply_iff (l : List (ℝ → ℝ)) (r x y : ℝ) :
+    y ≤ l.foldr (fun f s ↦ min (f x) s) r ↔ y ≤ r ∧ ∀ f ∈ l, y ≤ f x := by
+  induction l with
+  | nil => simp
+  | cons f l ih => simp [ih, and_left_comm]
+
+lemma foldr_max_apply_le_iff (l : List (ℝ → ℝ)) (r x y : ℝ) :
+    l.foldr (fun f s ↦ max (f x) s) r ≤ y ↔ r ≤ y ∧ ∀ f ∈ l, f x ≤ y := by
+  induction l with
+  | nil => simp
+  | cons f l ih => simp [ih, and_left_comm]
+
+end List
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Analysis / Inner Product Space / Box
+-/
+
+@[expose] public section
+
+namespace EuclideanSpace
+
+/-- A rectangle with finite coordinate bounds is bounded in the Euclidean plane. -/
+theorem isBounded_coordinate_rectangle (l r a b : ℝ) :
+    Bornology.IsBounded {p : EuclideanSpace ℝ (Fin 2) | l ≤ p 0 ∧ p 0 ≤ r ∧ a ≤ p 1 ∧ p 1 ≤ b} := by
+  apply isBounded_iff_forall_norm_le.mpr
+  refine ⟨(|l| + |r|) + (|a| + |b|), ?_⟩
+  rintro p ⟨hl, hr, ha, hb⟩
+  have hx : |p 0| ≤ |l| + |r| := abs_le.mpr
+    ⟨by linarith [neg_abs_le l, abs_nonneg r],
+      by linarith [le_abs_self r, abs_nonneg l]⟩
+  have hy : |p 1| ≤ |a| + |b| := abs_le.mpr
+    ⟨by linarith [neg_abs_le a, abs_nonneg b],
+      by linarith [le_abs_self b, abs_nonneg a]⟩
+  have hx2 := (sq_le_sq₀ (abs_nonneg (p 0)) (by positivity)).mpr hx
+  have hy2 := (sq_le_sq₀ (abs_nonneg (p 1)) (by positivity)).mpr hy
+  have hn := EuclideanSpace.norm_sq_eq p
+  simp only [Fin.sum_univ_two, Real.norm_eq_abs, sq_abs] at hn hx2 hy2
+  nlinarith [mul_nonneg (by positivity : 0 ≤ |l| + |r|)
+    (by positivity : 0 ≤ |a| + |b|), norm_nonneg p,
+    abs_nonneg l, abs_nonneg r, abs_nonneg a, abs_nonneg b]
+
+end EuclideanSpace
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Linearity of a real inner product in its left argument
+
+`Mathlib.Analysis.InnerProductSpace.Basic` provides the continuous linear map
+`innerSL ℝ v = fun x ↦ ⟪v, x⟫`; the bundled form of the symmetric slot is what the
+half-space convexity lemmas `convex_halfSpace_le` and `convex_halfSpace_ge` consume.
+-/
+
+@[expose] public section
+
+/-- `x ↦ ⟪x, v⟫` is a linear map of a real inner product space. -/
+theorem isLinearMap_inner_left {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+    (v : E) : IsLinearMap ℝ fun x : E ↦ inner ℝ x v :=
+  ⟨fun a b ↦ inner_add_left a b v, fun c a ↦ real_inner_smul_left a v c⟩
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Analysis / Normed / Affine / Continuous Affine Map
+-/
+
+@[expose] public section
+
+namespace ContinuousAffineMap
+
+variable {E F G : Type*} [NormedAddCommGroup E] [NormedAddCommGroup F]
+  [NormedAddCommGroup G] [NormedSpace ℝ E] [NormedSpace ℝ F] [NormedSpace ℝ G]
+
+/-- Precomposition by a fixed continuous affine map is continuous. -/
+theorem continuous_comp_right (g : E →ᴬ[ℝ] F) :
+    Continuous (fun f : F →ᴬ[ℝ] G ↦ f.comp g) := by
+  let C : NNReal := ⟨‖g‖ + 1, by positivity⟩
+  apply (LipschitzWith.of_dist_le_mul (K := C) fun f h ↦ ?_).continuous
+  rw [dist_eq_norm, dist_eq_norm]
+  have heq : f.comp g - h.comp g = (f - h).comp g := by
+    ext x
+    simp
+  rw [heq]
+  calc
+    ‖(f - h).comp g‖ ≤ ‖f - h‖ * ‖g‖ + ‖(f - h) 0‖ :=
+      ContinuousAffineMap.norm_comp_le _ _
+    _ ≤ ‖f - h‖ * ‖g‖ + ‖f - h‖ := by
+      gcongr
+      exact ContinuousAffineMap.norm_image_zero_le _
+    _ = (C : ℝ) * ‖f - h‖ := by
+      change _ = (‖g‖ + 1) * ‖f - h‖
+      ring
+
+end ContinuousAffineMap
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Analysis / Special Functions / Angle
+-/
+
+@[expose] public section
+
+namespace Real.Angle
+
+/-- Distinct reals at distance less than a full turn have distinct angles. -/
+theorem coe_ne_coe_of_abs_sub_lt {x y : ℝ} (hne : x ≠ y) (h : |x - y| < 2 * Real.pi) :
+    ((x : ℝ) : Real.Angle) ≠ ((y : ℝ) : Real.Angle) := by
+  intro he
+  rw [Real.Angle.angle_eq_iff_two_pi_dvd_sub] at he
+  obtain ⟨k, hk⟩ := he
+  have hpi := Real.pi_pos
+  rw [abs_lt] at h
+  rcases lt_trichotomy k 0 with hk0 | rfl | hk0
+  · have hk' : (k : ℝ) ≤ -1 := by exact_mod_cast (by omega : k ≤ -1)
+    nlinarith [h.1]
+  · simp only [Int.cast_zero, mul_zero] at hk
+    exact hne (by linarith)
+  · have hk' : (1 : ℝ) ≤ (k : ℝ) := by exact_mod_cast (by omega : (1 : ℤ) ≤ k)
+    nlinarith [h.2]
+
+end Real.Angle
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Analysis / Special Functions / Angle Lift
+-/
+
+@[expose] public section
+
+namespace Real
+
+/-- Continuous real angle lifts of the same circle-valued map differ by a constant. -/
+theorem sub_eq_sub_of_cos_eq_cos_of_sin_eq_sin {X : Type*} [TopologicalSpace X] [PreconnectedSpace
+  X]
+    {θ ψ : X → ℝ} (hθ : Continuous θ) (hψ : Continuous ψ)
+    (hcos : ∀ x, Real.cos (θ x) = Real.cos (ψ x))
+    (hsin : ∀ x, Real.sin (θ x) = Real.sin (ψ x)) (x y : X) :
+    θ x - ψ x = θ y - ψ y := by
+  have hsub : Set.range (fun z ↦ θ z - ψ z) ⊆ Set.range (fun n : ℤ ↦ 2 * Real.pi * n) := by
+    rintro _ ⟨z, rfl⟩
+    obtain ⟨n, hn⟩ := Real.Angle.angle_eq_iff_two_pi_dvd_sub.mp
+      (Real.Angle.cos_sin_inj (hcos z) (hsin z))
+    exact ⟨n, hn.symm⟩
+  have hconst := (Set.countable_range (fun n : ℤ ↦ 2 * Real.pi * n)).isTotallyDisconnected
+    _ hsub (isPreconnected_range (hθ.sub hψ))
+  exact hconst (Set.mem_range_self x) (Set.mem_range_self y)
+
+end Real
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# A cubic remainder bound for the arctangent
+
+`Real.abs_arctan_sub_self_le` complements `Real.abs_arctan_le_abs` by quantifying the
+first-order approximation `arctan s ≈ s` near the origin.
+-/
+
+@[expose] public section
+
+namespace Real
+
+/-- The arctangent differs from the identity by at most a cubic error. -/
+theorem abs_arctan_sub_self_le (s : ℝ) : |arctan s - s| ≤ |s| ^ 3 := by
+  have hderiv : ∀ u ∈ Set.uIcc (0 : ℝ) s,
+      HasDerivWithinAt (fun u : ℝ ↦ arctan u - u) (1 / (1 + u ^ 2) - 1)
+        (Set.uIcc (0 : ℝ) s) u := fun u _ ↦
+    ((hasDerivAt_arctan u).sub (hasDerivAt_id u)).hasDerivWithinAt
+  have hbound : ∀ u ∈ Set.uIcc (0 : ℝ) s, ‖1 / (1 + u ^ 2) - 1‖ ≤ s ^ 2 := by
+    intro u hu
+    have hu' : |u| ≤ |s| := by
+      rcases Set.mem_uIcc.mp hu with ⟨h1, h2⟩ | ⟨h1, h2⟩ <;>
+        rcases abs_cases u with ⟨e1, e2⟩ | ⟨e1, e2⟩ <;>
+          rcases abs_cases s with ⟨f1, f2⟩ | ⟨f1, f2⟩ <;> linarith
+    have hpos : (0 : ℝ) < 1 + u ^ 2 := by positivity
+    have heq : 1 / (1 + u ^ 2) - 1 = -(u ^ 2 / (1 + u ^ 2)) := by field_simp; ring
+    rw [heq, norm_neg, Real.norm_eq_abs, abs_of_nonneg (by positivity), div_le_iff₀ hpos]
+    nlinarith [sq_abs u, sq_abs s, abs_nonneg u, abs_nonneg s, sq_nonneg u]
+  have hmain := (convex_uIcc (0 : ℝ) s).norm_image_sub_le_of_norm_hasDerivWithin_le
+    hderiv hbound Set.left_mem_uIcc Set.right_mem_uIcc
+  have hmain' : |arctan s - s| ≤ s ^ 2 * |s| := by simpa using hmain
+  calc |arctan s - s| ≤ s ^ 2 * |s| := hmain'
+    _ = |s| ^ 3 := by rw [← sq_abs s]; ring
+
+/-- First-order angle estimate. If `W = (w0, w1)` has norm `nw > 0` and the displacement
+`d = (d0, d1)` has norm `nd` with `2 * nd ≤ nw`, then the principal angle between `W` and
+`W + d`, in the arctangent form given by their cross and dot products, differs from its
+linearization `(w0 * d1 - w1 * d0) / nw ^ 2` by at most `6 * nd ^ 2 / nw ^ 2`. -/
+theorem abs_arctan_div_sub_le_of_small {w0 w1 d0 d1 nw nd : ℝ}
+    (hnw : 0 < nw) (hnw2 : nw ^ 2 = w0 ^ 2 + w1 ^ 2)
+    (hnd : 0 ≤ nd) (hnd2 : nd ^ 2 = d0 ^ 2 + d1 ^ 2)
+    (hsmall : 2 * nd ≤ nw) :
+    |arctan ((w0 * d1 - w1 * d0) / (nw ^ 2 + (w0 * d0 + w1 * d1))) -
+        (w0 * d1 - w1 * d0) / nw ^ 2| ≤ 6 * nd ^ 2 / nw ^ 2 := by
+  set N := w0 * d1 - w1 * d0 with hN
+  set P := w0 * d0 + w1 * d1 with hP
+  have hlag : N ^ 2 + P ^ 2 = nw ^ 2 * nd ^ 2 := by
+    rw [hN, hP, hnw2, hnd2]; ring
+  have hprod : 0 ≤ nw * nd := mul_nonneg hnw.le hnd
+  have hCS : |N| ≤ nw * nd := by
+    nlinarith [sq_abs N, abs_nonneg N, sq_nonneg P]
+  have hCS' : |P| ≤ nw * nd := by
+    nlinarith [sq_abs P, abs_nonneg P, sq_nonneg N]
+  set D := nw ^ 2 + P with hD
+  have hPlow : -(nw * nd) ≤ P := neg_le_of_abs_le hCS'
+  have hDlow : nw ^ 2 / 2 ≤ D := by nlinarith
+  have hD0 : 0 < D := lt_of_lt_of_le (by positivity) hDlow
+  set s := N / D with hs
+  have hsle : |s| ≤ 2 * nd / nw := by
+    rw [hs, abs_div, abs_of_pos hD0, div_le_div_iff₀ hD0 hnw]
+    nlinarith [abs_nonneg N]
+  have hs1 : |s| ≤ 1 := by
+    refine hsle.trans ?_
+    rw [div_le_one hnw]
+    linarith
+  have h1 : |arctan s - s| ≤ 4 * nd ^ 2 / nw ^ 2 := by
+    refine (abs_arctan_sub_self_le s).trans ?_
+    have hstep : |s| ^ 3 ≤ (2 * nd / nw) ^ 2 := by
+      nlinarith [abs_nonneg s, hsle, hs1, sq_nonneg (|s|)]
+    refine hstep.trans_eq ?_
+    field_simp
+    ring
+  have h2 : |s - N / nw ^ 2| ≤ 2 * nd ^ 2 / nw ^ 2 := by
+    have heq : s - N / nw ^ 2 = -(N * P) / (D * nw ^ 2) := by
+      rw [hs, hD]
+      rw [div_sub_div _ _ (ne_of_gt hD0) (by positivity : (nw : ℝ) ^ 2 ≠ 0)]
+      rw [hD]
+      ring_nf
+    rw [heq, abs_div, abs_neg, abs_mul, abs_of_pos (by positivity : (0 : ℝ) < D * nw ^ 2),
+      div_le_div_iff₀ (by positivity) (by positivity : (0 : ℝ) < nw ^ 2)]
+    nlinarith [mul_le_mul hCS hCS' (abs_nonneg P) hprod, abs_nonneg N, abs_nonneg P,
+      mul_nonneg (abs_nonneg N) (abs_nonneg P), sq_nonneg nd, sq_nonneg nw]
+  calc |arctan s - N / nw ^ 2| ≤ |arctan s - s| + |s - N / nw ^ 2| :=
+        abs_sub_le _ _ _
+    _ ≤ 4 * nd ^ 2 / nw ^ 2 + 2 * nd ^ 2 / nw ^ 2 := add_le_add h1 h2
+    _ = 6 * nd ^ 2 / nw ^ 2 := by ring
+
+end Real
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Two-sided bracketing of an alternating series with an antitone tail
+
+`alternating_series_bracket_of_antitone_shift` upgrades the one-sided Leibniz estimates
+`Antitone.alternating_series_le_tendsto` and `Antitone.tendsto_le_alternating_series` to a
+two-sided bracket for an alternating series whose term magnitudes are antitone only from an
+even index `2 * N` onwards: the sum then lies between the partial sums of `2 * N` and of
+`2 * N + 1` terms.  `antitone_pow_div_factorial_two_mul_add` is the antitonicity criterion
+for the stride-two factorial quotients `x ^ (2 * n + a) / (2 * n + a)!` that the Taylor
+series of the trigonometric functions produce.
+-/
+
+@[expose] public section
+
+/-- The stride-two factorial quotients `x ^ (2 * n + a) / (2 * n + a)!` are antitone as soon
+as `x ^ 2 ≤ (a + 1) * (a + 2)`, i.e. as soon as the first term ratio is at most one. -/
+theorem antitone_pow_div_factorial_two_mul_add {x : ℝ} (hx0 : 0 ≤ x) {a : ℕ}
+    (hx : x ^ 2 ≤ ((a + 1) * (a + 2) : ℕ)) :
+    Antitone fun n : ℕ ↦ x ^ (2 * n + a) / (Nat.factorial (2 * n + a) : ℝ) := by
+  refine antitone_nat_of_succ_le fun n ↦ ?_
+  set m := 2 * n + a with hm
+  have ham : a ≤ m := by omega
+  have hstep : 2 * (n + 1) + a = m + 1 + 1 := by omega
+  have hfac : (0 : ℝ) < (Nat.factorial m : ℝ) := by positivity
+  have hle : x ^ 2 ≤ ((m : ℝ) + 1) * ((m : ℝ) + 2) := by
+    refine hx.trans ?_
+    have hcast : (a : ℝ) ≤ (m : ℝ) := by exact_mod_cast ham
+    have h0 : (0 : ℝ) ≤ (a : ℝ) := Nat.cast_nonneg a
+    push_cast
+    nlinarith
+  have hpow : x ^ (m + 1 + 1) = x ^ m * x * x := by ring
+  rw [hstep, Nat.factorial_succ, Nat.factorial_succ, hpow,
+    div_le_div_iff₀ (by positivity) hfac]
+  push_cast
+  nlinarith [mul_nonneg (mul_nonneg (pow_nonneg hx0 m) hfac.le) (sub_nonneg.2 hle)]
+
+/-- Leibniz bracketing for an alternating series whose term magnitudes are antitone only
+from index `2 * N` onwards: the limit lies between the partial sums of `2 * N` and of
+`2 * N + 1` terms. -/
+theorem alternating_series_bracket_of_antitone_shift {f : ℕ → ℝ} {l : ℝ} (N : ℕ)
+    (hfl : Filter.Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1 : ℝ) ^ i * f i)
+      Filter.atTop (nhds l))
+    (hfa : Antitone fun n ↦ f (2 * N + n)) :
+    (∑ i ∈ Finset.range (2 * N), (-1 : ℝ) ^ i * f i) ≤ l ∧
+      l ≤ ∑ i ∈ Finset.range (2 * N + 1), (-1 : ℝ) ^ i * f i := by
+  set S := ∑ i ∈ Finset.range (2 * N), (-1 : ℝ) ^ i * f i with hS
+  have key : ∀ n : ℕ, (∑ i ∈ Finset.range n, (-1 : ℝ) ^ i * f (2 * N + i))
+      = (∑ i ∈ Finset.range (2 * N + n), (-1 : ℝ) ^ i * f i) - S := by
+    intro n
+    rw [hS, Finset.sum_range_add]
+    simp [pow_add, pow_mul]
+  have hshift : Filter.Tendsto
+      (fun n ↦ ∑ i ∈ Finset.range (2 * N + n), (-1 : ℝ) ^ i * f i)
+      Filter.atTop (nhds l) := by
+    have h := hfl.comp (Filter.tendsto_add_atTop_nat (2 * N))
+    simpa [Function.comp_def, Nat.add_comm] using h
+  have htend : Filter.Tendsto
+      (fun n ↦ ∑ i ∈ Finset.range n, (-1 : ℝ) ^ i * f (2 * N + i))
+      Filter.atTop (nhds (l - S)) := by
+    simp only [key]
+    exact hshift.sub_const S
+  have hlow := Antitone.alternating_series_le_tendsto htend hfa 0
+  have hhigh := Antitone.tendsto_le_alternating_series htend hfa 0
+  simp only [Nat.mul_zero, Finset.range_zero, Finset.sum_empty, Nat.zero_add,
+    Finset.sum_range_one, pow_zero, one_mul, Nat.add_zero] at hlow hhigh
+  refine ⟨by linarith, ?_⟩
+  rw [Finset.sum_range_succ, ← hS, pow_mul]
+  simp only [neg_one_sq, one_pow, one_mul]
+  linarith
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Analysis / Special Functions / Trigonometric
+-/
+
+@[expose] public section
+
+/-- The tangent of half a complementary angle is secant minus tangent on the first quadrant. -/
+theorem Real.tan_pi_div_two_sub_div_two (ω : ℝ) (hω : ω ∈ Set.Ico 0 (Real.pi / 2)) :
+    Real.tan ((Real.pi / 2 - ω) / 2) = (Real.cos ω)⁻¹ - Real.tan ω := by
+  let u := (Real.pi / 2 - ω) / 2
+  have hu : 0 < Real.cos u := Real.cos_pos_of_mem_Ioo ⟨by
+    dsimp [u]
+    linarith [Real.pi_pos, hω.1, hω.2], by
+    dsimp [u]
+    linarith [Real.pi_pos, hω.1, hω.2]⟩
+  have hw : 0 < Real.cos ω :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos, hω.1, hω.2], hω.2⟩
+  have h2 : 2 * u = Real.pi / 2 - ω := by
+    dsimp [u]
+    ring
+  have hs : 2 * Real.sin u * Real.cos u = Real.cos ω := by
+    rw [← Real.sin_two_mul, h2, Real.sin_pi_div_two_sub]
+  have hc : 2 * Real.cos u ^ 2 - 1 = Real.sin ω := by
+    rw [← Real.cos_two_mul, h2, Real.cos_pi_div_two_sub]
+  change Real.tan u = _
+  rw [Real.tan_eq_sin_div_cos, Real.tan_eq_sin_div_cos]
+  field_simp [hu.ne', hw.ne']
+  nlinarith [Real.sin_sq_add_cos_sq u]
+
+/-- Cotangent is antitone between its consecutive poles at negative pi and zero. -/
+theorem Real.antitoneOn_cos_div_sin_Ioo_neg_pi_zero :
+    AntitoneOn (fun x : ℝ ↦ Real.cos x / Real.sin x) (Set.Ioo (-Real.pi) 0) := by
+  intro x hx y hy hxy
+  have hxsin := Real.sin_neg_of_neg_of_neg_pi_lt hx.2 hx.1
+  have hysin := Real.sin_neg_of_neg_of_neg_pi_lt hy.2 hy.1
+  change Real.cos y / Real.sin y ≤ Real.cos x / Real.sin x
+  rw [← neg_div_neg_eq (Real.cos y) (Real.sin y),
+    ← neg_div_neg_eq (Real.cos x) (Real.sin x)]
+  apply (div_le_div_iff₀ (neg_pos.mpr hysin) (neg_pos.mpr hxsin)).mpr
+  have hsin : 0 ≤ Real.sin (y - x) := Real.sin_nonneg_of_nonneg_of_le_pi
+    (by linarith) (by linarith [hx.1, hy.2])
+  rw [Real.sin_sub] at hsin
+  nlinarith
+
+/-- A cubic upper bound for the tangent on `[0, 1]`, companion to `Real.sin_gt_sub_cube`. -/
+theorem Real.tan_le_self_add_cube {x : ℝ} (hx : 0 ≤ x) (hx1 : x ≤ 1) :
+    Real.tan x ≤ x + 4 / 3 * x ^ 3 := by
+  have hpi : (3 : ℝ) < Real.pi := Real.pi_gt_three
+  have hcos : 0 < Real.cos x :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith, by linarith⟩
+  have hcl : 1 - x ^ 2 / 2 ≤ Real.cos x := Real.one_sub_sq_div_two_le_cos
+  have hs : Real.sin x ≤ x := Real.sin_le hx
+  rw [Real.tan_eq_sin_div_cos, div_le_iff₀ hcos]
+  have h1 : (0 : ℝ) ≤ x + 4 / 3 * x ^ 3 := by positivity
+  have h2 : (x + 4 / 3 * x ^ 3) * (1 - x ^ 2 / 2) ≤ (x + 4 / 3 * x ^ 3) * Real.cos x :=
+    mul_le_mul_of_nonneg_left hcl h1
+  have h3 : x ≤ (x + 4 / 3 * x ^ 3) * (1 - x ^ 2 / 2) := by
+    nlinarith [mul_nonneg (pow_nonneg hx 3) (show (0 : ℝ) ≤ 5 / 6 - 2 / 3 * x ^ 2 by nlinarith)]
+  linarith
+
+/-- Leibniz bracketing of the sine by its Maclaurin partial sums: for `0 ≤ x` with
+`x ^ 2 ≤ (4 * N + 2) * (4 * N + 3)` the value `Real.sin x` lies between the partial sums of
+`2 * N` and of `2 * N + 1` terms, the first of which undershoots and the second overshoots. -/
+theorem Real.sin_mem_Icc_taylor_sums (N : ℕ) {x : ℝ} (hx0 : 0 ≤ x)
+    (hx : x ^ 2 ≤ ((4 * N + 2) * (4 * N + 3) : ℕ)) :
+    Real.sin x ∈ Set.Icc
+      (∑ k ∈ Finset.range (2 * N),
+        (-1 : ℝ) ^ k * x ^ (2 * k + 1) / (Nat.factorial (2 * k + 1) : ℝ))
+      (∑ k ∈ Finset.range (2 * N + 1),
+        (-1 : ℝ) ^ k * x ^ (2 * k + 1) / (Nat.factorial (2 * k + 1) : ℝ)) := by
+  have hsum : ∀ n : ℕ, (∑ i ∈ Finset.range n,
+        (-1 : ℝ) ^ i * (x ^ (2 * i + 1) / (Nat.factorial (2 * i + 1) : ℝ)))
+      = ∑ k ∈ Finset.range n,
+        (-1 : ℝ) ^ k * x ^ (2 * k + 1) / (Nat.factorial (2 * k + 1) : ℝ) :=
+    fun n ↦ Finset.sum_congr rfl fun k _ ↦ (mul_div_assoc _ _ _).symm
+  have htend : Filter.Tendsto (fun n ↦ ∑ i ∈ Finset.range n,
+      (-1 : ℝ) ^ i * (x ^ (2 * i + 1) / (Nat.factorial (2 * i + 1) : ℝ)))
+      Filter.atTop (nhds (Real.sin x)) := by
+    have h := (Real.hasSum_sin x).tendsto_sum_nat
+    simpa only [mul_div_assoc] using h
+  have hfa : Antitone fun n : ℕ ↦
+      x ^ (2 * (2 * N + n) + 1) / (Nat.factorial (2 * (2 * N + n) + 1) : ℝ) := by
+    have e : ∀ n : ℕ, 2 * (2 * N + n) + 1 = 2 * n + (4 * N + 1) := fun n ↦ by omega
+    simp only [e]
+    refine antitone_pow_div_factorial_two_mul_add (a := 4 * N + 1) hx0 ?_
+    refine hx.trans (le_of_eq ?_)
+    push_cast
+    ring
+  obtain ⟨hlow, hhigh⟩ :=
+    alternating_series_bracket_of_antitone_shift
+      (f := fun k : ℕ ↦ x ^ (2 * k + 1) / (Nat.factorial (2 * k + 1) : ℝ)) N htend hfa
+  rw [hsum] at hlow hhigh
+  exact ⟨hlow, hhigh⟩
+
+/-- Leibniz bracketing of the cosine by its Maclaurin partial sums: for `0 ≤ x` with
+`x ^ 2 ≤ (4 * N + 1) * (4 * N + 2)` the value `Real.cos x` lies between the partial sums of
+`2 * N` and of `2 * N + 1` terms, the first of which undershoots and the second overshoots. -/
+theorem Real.cos_mem_Icc_taylor_sums (N : ℕ) {x : ℝ} (hx0 : 0 ≤ x)
+    (hx : x ^ 2 ≤ ((4 * N + 1) * (4 * N + 2) : ℕ)) :
+    Real.cos x ∈ Set.Icc
+      (∑ k ∈ Finset.range (2 * N),
+        (-1 : ℝ) ^ k * x ^ (2 * k) / (Nat.factorial (2 * k) : ℝ))
+      (∑ k ∈ Finset.range (2 * N + 1),
+        (-1 : ℝ) ^ k * x ^ (2 * k) / (Nat.factorial (2 * k) : ℝ)) := by
+  have hsum : ∀ n : ℕ, (∑ i ∈ Finset.range n,
+        (-1 : ℝ) ^ i * (x ^ (2 * i) / (Nat.factorial (2 * i) : ℝ)))
+      = ∑ k ∈ Finset.range n,
+        (-1 : ℝ) ^ k * x ^ (2 * k) / (Nat.factorial (2 * k) : ℝ) :=
+    fun n ↦ Finset.sum_congr rfl fun k _ ↦ (mul_div_assoc _ _ _).symm
+  have htend : Filter.Tendsto (fun n ↦ ∑ i ∈ Finset.range n,
+      (-1 : ℝ) ^ i * (x ^ (2 * i) / (Nat.factorial (2 * i) : ℝ)))
+      Filter.atTop (nhds (Real.cos x)) := by
+    have h := (Real.hasSum_cos x).tendsto_sum_nat
+    simpa only [mul_div_assoc] using h
+  have hfa : Antitone fun n : ℕ ↦
+      x ^ (2 * (2 * N + n)) / (Nat.factorial (2 * (2 * N + n)) : ℝ) := by
+    have e : ∀ n : ℕ, 2 * (2 * N + n) = 2 * n + 4 * N := fun n ↦ by omega
+    simp only [e]
+    refine antitone_pow_div_factorial_two_mul_add (a := 4 * N) hx0 ?_
+    refine hx.trans (le_of_eq ?_)
+    push_cast
+    ring
+  obtain ⟨hlow, hhigh⟩ :=
+    alternating_series_bracket_of_antitone_shift
+      (f := fun k : ℕ ↦ x ^ (2 * k) / (Nat.factorial (2 * k) : ℝ)) N htend hfa
+  rw [hsum] at hlow hhigh
+  exact ⟨hlow, hhigh⟩
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `ForMathlib.BoundedVariation`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Bounded Variation
+-/
+
+@[expose] public section
+
+/-- The variation of a sum is at most the sum of the variations. -/
+theorem eVariationOn.add_le {α E : Type*} [LinearOrder α]
+    [SeminormedAddCommGroup E] (f g : α → E) (s : Set α) :
+    eVariationOn (f + g) s ≤ eVariationOn f s + eVariationOn g s := by
+  apply iSup_le
+  rintro ⟨n, u, hu, hus⟩
+  calc
+    _ ≤ ∑ i ∈ Finset.range n,
+        (edist (f (u (i + 1))) (f (u i)) + edist (g (u (i + 1))) (g (u i))) :=
+      Finset.sum_le_sum fun _ _ ↦ edist_add_add_le _ _ _ _
+    _ = _ := Finset.sum_add_distrib
+    _ ≤ _ := add_le_add (eVariationOn.sum_le hu hus) (eVariationOn.sum_le hu hus)
+
+/-- A sum of two functions of bounded variation has bounded variation. -/
+theorem BoundedVariationOn.add {α E : Type*} [LinearOrder α]
+    [SeminormedAddCommGroup E] {f g : α → E} {s : Set α}
+    (hf : BoundedVariationOn f s) (hg : BoundedVariationOn g s) :
+    BoundedVariationOn (f + g) s := by
+  refine ne_top_of_le_ne_top (ENNReal.add_ne_top.mpr ⟨hf, hg⟩) ?_
+  exact eVariationOn.add_le f g s
+
+/-- Variation bounds the increments along a finite monotone partition. -/
+theorem eVariationOn.sum_edist_fin_le {α E : Type*} [LinearOrder α] [PseudoEMetricSpace E]
+    {f : α → E} {s : Set α} {n : ℕ} (u : Fin (n + 1) → α)
+    (hu : Monotone u) (hus : ∀ i, u i ∈ s) :
+    ∑ i : Fin n, edist (f (u i.succ)) (f (u i.castSucc)) ≤ eVariationOn f s := by
+  let v : ℕ → α := fun k ↦ u ⟨min k n, by omega⟩
+  have hv : Monotone v := fun i j hij ↦ hu (min_le_min_right n hij)
+  have h := eVariationOn.sum_le (f := f) (n := n) hv
+    (fun k ↦ hus ⟨min k n, by omega⟩)
+  rw [← Fin.sum_univ_eq_sum_range] at h
+  convert h using 1
+  apply Finset.sum_congr rfl
+  intro i _
+  simp only [v]
+  have h₁ : min (i.val + 1) n = i.val + 1 := Nat.min_eq_left (by omega)
+  have h₀ : min i.val n = i.val := Nat.min_eq_left (by omega)
+  congr 2 <;> apply congrArg u <;> apply Fin.ext
+  · exact h₁.symm
+  · exact h₀.symm
+
+/-- Total variation bounds the sum of norms of increments on a finite partition. -/
+theorem BoundedVariationOn.sum_norm_sub_fin_le {α E : Type*} [LinearOrder α]
+    [NormedAddCommGroup E] {f : α → E} {s : Set α} (hf : BoundedVariationOn f s)
+    {n : ℕ} (u : Fin (n + 1) → α) (hu : Monotone u) (hus : ∀ i, u i ∈ s) :
+    ∑ i : Fin n, ‖f (u i.succ) - f (u i.castSucc)‖ ≤ (eVariationOn f s).toReal := by
+  have h := ENNReal.toReal_mono hf (eVariationOn.sum_edist_fin_le u hu hus)
+  rw [ENNReal.toReal_sum (fun i _ ↦ edist_ne_top _ _)] at h
+  simpa only [edist_dist, dist_eq_norm, ENNReal.toReal_ofReal (norm_nonneg _)] using h
+
+/-- Squared increments are bounded by the largest increment times the total variation. -/
+theorem BoundedVariationOn.sum_norm_sub_sq_fin_le {α E : Type*} [LinearOrder α]
+    [NormedAddCommGroup E] {f : α → E} {s : Set α} (hf : BoundedVariationOn f s)
+    {n : ℕ} (u : Fin (n + 1) → α) (hu : Monotone u) (hus : ∀ i, u i ∈ s)
+    {δ : ℝ} (hδ : 0 ≤ δ) (hbound : ∀ i : Fin n, ‖f (u i.succ) - f (u i.castSucc)‖ ≤ δ) :
+    ∑ i : Fin n, ‖f (u i.succ) - f (u i.castSucc)‖ ^ 2 ≤ δ * (eVariationOn f s).toReal := by
+  calc
+    _ ≤ ∑ i : Fin n, δ * ‖f (u i.succ) - f (u i.castSucc)‖ := by
+      apply Finset.sum_le_sum
+      intro i _
+      nlinarith [hbound i, norm_nonneg (f (u i.succ) - f (u i.castSucc))]
+    _ = δ * ∑ i : Fin n, ‖f (u i.succ) - f (u i.castSucc)‖ := by
+      rw [Finset.mul_sum]
+    _ ≤ _ := mul_le_mul_of_nonneg_left (hf.sum_norm_sub_fin_le u hu hus) hδ
+
+/-- A telescoping sum of increments with quadratic remainders is controlled by the
+largest increment times the total variation. -/
+theorem BoundedVariationOn.abs_sub_sum_le_of_param_quadratic_remainder
+    {α E : Type*} [LinearOrder α] [NormedAddCommGroup E]
+    {f : α → E} {s : Set α} (hf : BoundedVariationOn f s)
+    {n : ℕ} (u : Fin (n + 1) → α) (hu : Monotone u) (hus : ∀ i, u i ∈ s)
+    (A : α → ℝ) (L : Fin n → ℝ) {C δ : ℝ} (hC : 0 ≤ C) (hδ : 0 ≤ δ)
+    (hbound : ∀ i : Fin n, ‖f (u i.succ) - f (u i.castSucc)‖ ≤ δ)
+    (herror : ∀ i : Fin n,
+      |A (u i.succ) - A (u i.castSucc) - L i| ≤
+        C * ‖f (u i.succ) - f (u i.castSucc)‖ ^ 2) :
+    |A (u (Fin.last n)) - A (u 0) - ∑ i, L i| ≤ C * δ * (eVariationOn f s).toReal := by
+  have htel : ∑ i : Fin n, (A (u i.succ) - A (u i.castSucc)) =
+      A (u (Fin.last n)) - A (u 0) := by
+    have hfirst := Fin.sum_univ_succ (fun i ↦ A (u i))
+    have hlast := Fin.sum_univ_castSucc (fun i ↦ A (u i))
+    rw [Finset.sum_sub_distrib]
+    linarith
+  rw [← htel, ← Finset.sum_sub_distrib]
+  calc
+    _ ≤ ∑ i : Fin n, |A (u i.succ) - A (u i.castSucc) - L i| :=
+      Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ i : Fin n, C * ‖f (u i.succ) - f (u i.castSucc)‖ ^ 2 :=
+      Finset.sum_le_sum (fun i _ ↦ herror i)
+    _ = C * ∑ i : Fin n, ‖f (u i.succ) - f (u i.castSucc)‖ ^ 2 := by
+      rw [Finset.mul_sum]
+    _ ≤ C * (δ * (eVariationOn f s).toReal) :=
+      mul_le_mul_of_nonneg_left (hf.sum_norm_sub_sq_fin_le u hu hus hδ hbound) hC
+    _ = _ := by ring
+
+/-- If the increments of a real function `A` agree with a linear functional of the
+increments of a bounded variation path up to a quadratic remainder on all parameter
+pairs closer than a fixed positive threshold, then the linear sums along any family of
+partitions whose mesh tends to zero converge to the endpoint increment of `A`. -/
+theorem BoundedVariationOn.tendsto_sum_of_local_quadratic_remainder
+    {E : Type*} [NormedAddCommGroup E] {a b : ℝ} (hab : a ≤ b)
+    {f : Set.Icc a b → E} (hf : BoundedVariationOn f Set.univ) (hfc : Continuous f)
+    (A : Set.Icc a b → ℝ) (L : Set.Icc a b → E → ℝ) {C : ℝ} (hC : 0 ≤ C)
+    {δ₀ : ℝ} (hδ₀ : 0 < δ₀)
+    (herror : ∀ t u : Set.Icc a b, (t : ℝ) ≤ (u : ℝ) → (u : ℝ) - (t : ℝ) < δ₀ →
+      |A u - A t - L t (f u - f t)| ≤ C * ‖f u - f t‖ ^ 2)
+    (N : ℕ → ℕ) (cuts : ∀ k, Fin (N k + 1) → Set.Icc a b)
+    (hcuts : ∀ k, Monotone (cuts k))
+    (hzero : ∀ k, (cuts k 0 : ℝ) = a)
+    (hlast : ∀ k, (cuts k (Fin.last (N k)) : ℝ) = b)
+    (hmesh : ∀ δ > 0, ∀ᶠ k in Filter.atTop, ∀ i : Fin (N k),
+      (cuts k i.succ : ℝ) - (cuts k i.castSucc : ℝ) < δ) :
+    Filter.Tendsto (fun k ↦ ∑ i : Fin (N k),
+      L (cuts k i.castSucc) (f (cuts k i.succ) - f (cuts k i.castSucc)))
+      Filter.atTop (nhds (A ⟨b, hab, le_rfl⟩ - A ⟨a, le_rfl, hab⟩)) := by
+  rw [Metric.tendsto_nhds]
+  intro ε hε
+  set V := (eVariationOn f Set.univ).toReal with hVdef
+  have hV : 0 ≤ V := ENNReal.toReal_nonneg
+  set η := ε / (C * V + 1) with hηdef
+  have hη : 0 < η := div_pos hε (by positivity)
+  obtain ⟨δ, hδ, hmod⟩ := Metric.uniformContinuous_iff.mp
+    (CompactSpace.uniformContinuous_of_continuous hfc) η hη
+  filter_upwards [hmesh (min δ δ₀) (lt_min hδ hδ₀)] with k hk
+  have hinc (i : Fin (N k)) :
+      ‖f (cuts k i.succ) - f (cuts k i.castSucc)‖ ≤ η := by
+    have hle : (cuts k i.castSucc : ℝ) ≤ (cuts k i.succ : ℝ) :=
+      hcuts k (Fin.castSucc_le_succ i)
+    have hd : dist (cuts k i.succ) (cuts k i.castSucc) < δ := by
+      rw [Subtype.dist_eq, Real.dist_eq, abs_of_nonneg (sub_nonneg.mpr hle)]
+      exact lt_of_lt_of_le (hk i) (min_le_left _ _)
+    simpa only [dist_eq_norm] using (hmod hd).le
+  have hbound := hf.abs_sub_sum_le_of_param_quadratic_remainder (cuts k) (hcuts k)
+    (fun _ ↦ Set.mem_univ _) A
+    (fun i ↦ L (cuts k i.castSucc) (f (cuts k i.succ) - f (cuts k i.castSucc)))
+    hC hη.le hinc
+    (fun i ↦ herror (cuts k i.castSucc) (cuts k i.succ) (hcuts k (Fin.castSucc_le_succ i))
+      (lt_of_lt_of_le (hk i) (min_le_right _ _)))
+  have hleft : cuts k 0 = ⟨a, le_rfl, hab⟩ := Subtype.ext (hzero k)
+  have hright : cuts k (Fin.last (N k)) = ⟨b, hab, le_rfl⟩ := Subtype.ext (hlast k)
+  rw [hleft, hright] at hbound
+  rw [Real.dist_eq, abs_sub_comm]
+  apply hbound.trans_lt
+  have hcancel : η * (C * V + 1) = ε := div_mul_cancel₀ ε (by positivity)
+  change C * η * V < ε
+  nlinarith
+
+/-- Bounded variation on two adjacent closed intervals gives bounded variation on their
+union. -/
+theorem BoundedVariationOn.Icc_union_Icc {α E : Type*} [LinearOrder α]
+    [PseudoEMetricSpace E] {f : α → E} {a b c : α} (hab : a ≤ b) (hbc : b ≤ c)
+    (h₁ : BoundedVariationOn f (Set.Icc a b)) (h₂ : BoundedVariationOn f (Set.Icc b c)) :
+    BoundedVariationOn f (Set.Icc a c) := by
+  have h := eVariationOn.Icc_add_Icc f hab hbc (Set.mem_univ b)
+  simp only [Set.univ_inter] at h
+  change eVariationOn f (Set.Icc a c) ≠ ⊤
+  rw [← h]
+  exact ENNReal.add_ne_top.mpr ⟨h₁, h₂⟩
+
+/-- A function on the order subtype `Set.Icc a b` has bounded variation everywhere as soon as
+it has bounded variation on the closed interval between the two endpoints of that subtype. -/
+theorem BoundedVariationOn.univ_of_Icc_endpoints {α E : Type*} [LinearOrder α]
+    [PseudoEMetricSpace E] {a b : α} (hab : a ≤ b) {g : Set.Icc a b → E}
+    (h : BoundedVariationOn g (Set.Icc ⟨a, le_rfl, hab⟩ ⟨b, hab, le_rfl⟩)) :
+    BoundedVariationOn g Set.univ := by
+  have huniv : Set.Icc (⟨a, le_rfl, hab⟩ : Set.Icc a b) ⟨b, hab, le_rfl⟩ = Set.univ := by
+    ext x
+    simp only [Set.mem_Icc, Set.mem_univ, iff_true]
+    exact ⟨x.2.1, x.2.2⟩
+  rwa [huniv] at h
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `ForMathlib.Convex.Body.BoundaryMeasure`.
+* `ForMathlib.Convex.Body.Hausdorff`.
+* `ForMathlib.Convex.Collinear`.
+* `ForMathlib.Convex.Body.Segment`.
+* `ForMathlib.Convex.Function`.
+* `ForMathlib.Convex.Hausdorff`.
+* `ForMathlib.Convex.Support`.
+* `ForMathlib.Convex.Translation`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Convex / Body / Boundary Measure
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+open scoped ENNReal Pointwise Topology
+
+namespace ConvexBody
+
+private abbrev Plane_m58ecd66 := EuclideanSpace ℝ (Fin 2)
+
+private def unitCircleParam (t : ℝ) : Plane_m58ecd66 :=
+  WithLp.toLp 2 (![Real.cos t, Real.sin t] : Fin 2 → ℝ)
+
+private theorem lipschitzWith_unitCircleParam :
+    LipschitzWith 2 unitCircleParam := by
+  rw [lipschitzWith_iff_dist_le_mul]
+  intro x y
+  rw [dist_eq_norm]
+  rw [Real.dist_eq]
+  apply (sq_le_sq₀ (norm_nonneg _) (mul_nonneg (by positivity) (abs_nonneg _))).mp
+  rw [EuclideanSpace.norm_sq_eq]
+  simp only [unitCircleParam, PiLp.sub_apply, Fin.sum_univ_two, Real.norm_eq_abs,
+    Matrix.cons_val_zero, Matrix.cons_val_one]
+  have hc : |Real.cos x - Real.cos y| ≤ |x - y| := by
+    exact Real.abs_cos_sub_cos_le x y
+  have hs : |Real.sin x - Real.sin y| ≤ |x - y| := by
+    exact Real.abs_sin_sub_sin_le x y
+  have hxy : 0 ≤ |x - y| := abs_nonneg _
+  have hc_sq := sq_le_sq₀ (abs_nonneg _) hxy |>.mpr hc
+  have hs_sq := sq_le_sq₀ (abs_nonneg _) hxy |>.mpr hs
+  norm_num
+  nlinarith [sq_abs (Real.cos x - Real.cos y), sq_abs (Real.sin x - Real.sin y)]
+
+private theorem hausdorffMeasure_one_sphere_lt_top :
+    Measure.hausdorffMeasure 1 (Metric.sphere (0 : Plane_m58ecd66) 1) < ⊤ := by
+  have hsphere : Metric.sphere (0 : Plane_m58ecd66) 1 ⊆
+      unitCircleParam '' Set.Icc (-Real.pi) Real.pi := by
+    intro p hp
+    have hnorm : ‖p‖ = 1 := by simpa [Metric.mem_sphere, dist_zero_right] using hp
+    let z : ℂ := ⟨p 0, p 1⟩
+    have hzNorm : ‖z‖ = 1 := by
+      rw [Complex.norm_def, show Complex.normSq z = ‖p‖ ^ 2 by
+        rw [EuclideanSpace.norm_sq_eq]
+        simp [z, Complex.normSq_apply, Fin.sum_univ_two, pow_two, Real.norm_eq_abs]]
+      simp [hnorm]
+    have hz : z ≠ 0 := by
+      intro hz
+      simp [hz] at hzNorm
+    refine ⟨z.arg, ⟨(Complex.neg_pi_lt_arg z).le, Complex.arg_le_pi z⟩, ?_⟩
+    ext i
+    fin_cases i
+    · simpa [unitCircleParam, z, hzNorm] using Complex.cos_arg hz
+    · simpa [unitCircleParam, z, hzNorm] using Complex.sin_arg z
+  calc
+    Measure.hausdorffMeasure 1 (Metric.sphere (0 : Plane_m58ecd66) 1) ≤
+        Measure.hausdorffMeasure 1 (unitCircleParam '' Set.Icc (-Real.pi) Real.pi) :=
+      measure_mono hsphere
+    _ ≤ ((2 : NNReal) : ENNReal) ^ (1 : ℝ) *
+        Measure.hausdorffMeasure 1 (Set.Icc (-Real.pi) Real.pi) :=
+      lipschitzWith_unitCircleParam.hausdorffMeasure_image_le (by positivity) _
+    _ < ⊤ := by
+      rw [hausdorffMeasure_real, Real.volume_Icc]
+      finiteness
+
+private theorem image_gaugeRescale_unitSphere_eq_frontier {s : Set Plane_m58ecd66}
+    (hconv : Convex ℝ s) (h₀ : s ∈ nhds (0 : Plane_m58ecd66)) (hbounded : Bornology.IsBounded s)
+    (hclosed : IsClosed s) :
+    gaugeRescale (Metric.ball 0 1) s '' Metric.sphere 0 1 = frontier s := by
+  have hvnb : Bornology.IsVonNBounded ℝ s :=
+    NormedSpace.isVonNBounded_of_isBounded ℝ hbounded
+  let h := gaugeRescaleHomeomorph (Metric.ball (0 : Plane_m58ecd66) 1) s
+    (convex_ball 0 1) (Metric.ball_mem_nhds 0 zero_lt_one)
+    (NormedSpace.isVonNBounded_ball ℝ Plane_m58ecd66 1) hconv h₀ hvnb
+  change h '' Metric.sphere 0 1 = frontier s
+  rw [← frontier_ball, ← closure_sdiff_interior, Set.image_sdiff h.injective]
+  · rw [image_gaugeRescaleHomeomorph_closure, image_gaugeRescaleHomeomorph_interior]
+    rw [hclosed.closure_eq]
+    simpa [hclosed.closure_eq] using closure_sdiff_interior s
+  · norm_num
+
+private theorem exists_lipschitzOnWith_gaugeRescale_unitSphere {s : Set Plane_m58ecd66}
+    (hconv : Convex ℝ s) (h₀ : s ∈ nhds (0 : Plane_m58ecd66)) (hbounded : Bornology.IsBounded s) :
+    ∃ C, LipschitzOnWith C (gaugeRescale (Metric.ball 0 1) s) (Metric.sphere 0 1) := by
+  obtain ⟨r, hr, hrs⟩ := Metric.mem_nhds_iff.mp h₀
+  obtain ⟨B, hB⟩ := hbounded.exists_norm_le
+  let R : ℝ := |B| + 1
+  have hR : 0 < R := by dsimp [R]; positivity
+  have hsR : s ⊆ Metric.closedBall (0 : Plane_m58ecd66) R := by
+    intro x hx
+    rw [Metric.mem_closedBall, dist_zero_right]
+    exact (hB x hx).trans (by dsimp [R]; linarith [le_abs_self B])
+  have habs : Absorbent ℝ s := absorbent_nhds_zero h₀
+  let rn : NNReal := ⟨r, hr.le⟩
+  have hg : LipschitzWith rn⁻¹ (gauge s) := hconv.lipschitzWith_gauge hr hrs
+  let C : NNReal := ⟨R + (rn⁻¹ : ℝ) * R ^ 2, by positivity⟩
+  refine ⟨C, ?_⟩
+  rw [lipschitzOnWith_iff_dist_le_mul]
+  intro x hx y hy
+  have hnx : ‖x‖ = 1 := by simpa [Metric.mem_sphere, dist_zero_right] using hx
+  have hny : ‖y‖ = 1 := by simpa [Metric.mem_sphere, dist_zero_right] using hy
+  have hgx : 1 / R ≤ gauge s x := by
+    simpa [hnx] using (le_gauge_of_subset_closedBall habs hR.le hsR : ‖x‖ / R ≤ gauge s x)
+  have hgy : 1 / R ≤ gauge s y := by
+    simpa [hny] using (le_gauge_of_subset_closedBall habs hR.le hsR : ‖y‖ / R ≤ gauge s y)
+  have hgxpos : 0 < gauge s x := (div_pos one_pos hR).trans_le hgx
+  have hgypos : 0 < gauge s y := (div_pos one_pos hR).trans_le hgy
+  have hinvx : (gauge s x)⁻¹ ≤ R := by
+    rw [inv_le_comm₀ hgxpos hR]
+    simpa [div_eq_inv_mul] using hgx
+  have hinvy : (gauge s y)⁻¹ ≤ R := by
+    rw [inv_le_comm₀ hgypos hR]
+    simpa [div_eq_inv_mul] using hgy
+  have hginv : dist (gauge s x)⁻¹ (gauge s y)⁻¹ ≤
+      (rn⁻¹ : ℝ) * R ^ 2 * dist x y := by
+    rw [dist_inv_inv₀ hgxpos.ne' hgypos.ne']
+    have hdist := hg.dist_le_mul x y
+    have hprod : 1 / R ^ 2 ≤ ‖gauge s x‖ * ‖gauge s y‖ := by
+      rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_of_pos hgxpos, abs_of_pos hgypos]
+      calc
+        1 / R ^ 2 = (1 / R) * (1 / R) := by ring_nf
+        _ ≤ gauge s x * gauge s y :=
+          mul_le_mul hgx hgy (div_nonneg one_pos.le hR.le) hgxpos.le
+    calc
+      dist (gauge s x) (gauge s y) / (‖gauge s x‖ * ‖gauge s y‖) ≤
+          dist (gauge s x) (gauge s y) / (1 / R ^ 2) := by
+        exact div_le_div_of_nonneg_left (dist_nonneg) (by positivity) hprod
+      _ ≤ ((rn⁻¹ : ℝ) * dist x y) / (1 / R ^ 2) := by
+        apply div_le_div_of_nonneg_right _ (by positivity)
+        simpa [NNReal.coe_inv] using hdist
+      _ = (rn⁻¹ : ℝ) * R ^ 2 * dist x y := by field_simp
+  change dist ((gauge (Metric.ball 0 1) x / gauge s x) • x)
+      ((gauge (Metric.ball 0 1) y / gauge s y) • y) ≤ (C : ℝ) * dist x y
+  simp_rw [gauge_ball (by positivity : (0 : ℝ) ≤ 1)]
+  rw [hnx, hny]
+  simp only [inv_one, one_div]
+  rw [dist_eq_norm]
+  calc
+    ‖(gauge s x)⁻¹ • x - (gauge s y)⁻¹ • y‖ ≤
+        ‖(gauge s x)⁻¹ • (x - y)‖ +
+          ‖((gauge s x)⁻¹ - (gauge s y)⁻¹) • y‖ := by
+      have heq : (gauge s x)⁻¹ • x - (gauge s y)⁻¹ • y =
+          (gauge s x)⁻¹ • (x - y) +
+            ((gauge s x)⁻¹ - (gauge s y)⁻¹) • y := by module
+      rw [heq]
+      exact norm_add_le _ _
+    _ ≤ R * dist x y + ((rn⁻¹ : ℝ) * R ^ 2 * dist x y) := by
+      simp only [norm_smul, Real.norm_eq_abs, dist_eq_norm]
+      have hix : |(gauge s x)⁻¹| ≤ R := by rw [abs_of_pos (inv_pos.mpr hgxpos)]; exact hinvx
+      have hid : |(gauge s x)⁻¹ - (gauge s y)⁻¹| ≤
+          (rn⁻¹ : ℝ) * R ^ 2 * ‖x - y‖ := by simpa [dist_eq_norm] using hginv
+      rw [hny, mul_one]
+      exact add_le_add (mul_le_mul_of_nonneg_right hix (norm_nonneg _)) hid
+    _ = (C : ℝ) * dist x y := by
+      change _ = (R + (rn⁻¹ : ℝ) * R ^ 2) * dist x y
+      ring
+
+private theorem hausdorffMeasure_frontier_lt_top_of_mem_nhds_zero {s : Set Plane_m58ecd66}
+    (hconv : Convex ℝ s) (h₀ : s ∈ nhds (0 : Plane_m58ecd66)) (hcompact : IsCompact s) :
+    Measure.hausdorffMeasure 1 (frontier s) < ⊤ := by
+  obtain ⟨C, hC⟩ :=
+    exists_lipschitzOnWith_gaugeRescale_unitSphere hconv h₀ hcompact.isBounded
+  rw [← image_gaugeRescale_unitSphere_eq_frontier hconv h₀ hcompact.isBounded hcompact.isClosed]
+  refine (hC.hausdorffMeasure_image_le (by positivity)).trans_lt ?_
+  exact ENNReal.mul_lt_top (ENNReal.rpow_lt_top_of_nonneg (by positivity) (by simp))
+    hausdorffMeasure_one_sphere_lt_top
+
+/-- The frontier of a planar convex body with nonempty interior has finite
+one-dimensional Hausdorff measure. -/
+theorem hausdorffMeasure_frontier_lt_top
+    (K : ConvexBody (EuclideanSpace ℝ (Fin 2)))
+    (hint : (interior (K : Set (EuclideanSpace ℝ (Fin 2)))).Nonempty) :
+    Measure.hausdorffMeasure 1
+      (frontier (K : Set (EuclideanSpace ℝ (Fin 2)))) < ⊤ := by
+  obtain ⟨c, hc⟩ := hint
+  let s : Set Plane_m58ecd66 := -c +ᵥ (K : Set Plane_m58ecd66)
+  have hsconv : Convex ℝ s := K.convex.vadd (-c)
+  have hscompact : IsCompact s := by
+    change IsCompact ((fun x : Plane_m58ecd66 ↦ -c + x) '' (K : Set Plane_m58ecd66))
+    exact K.isCompact.image (continuous_const.add continuous_id)
+  have hs₀ : s ∈ nhds (0 : Plane_m58ecd66) := by
+    rw [← mem_interior_iff_mem_nhds]
+    change (0 : Plane_m58ecd66) ∈ interior (-c +ᵥ (K : Set Plane_m58ecd66))
+    rw [interior_vadd]
+    simpa [Set.mem_vadd_set_iff_neg_vadd_mem] using hc
+  have hsfinite := hausdorffMeasure_frontier_lt_top_of_mem_nhds_zero hsconv hs₀ hscompact
+  have himage : (fun z : Plane_m58ecd66 ↦ c + z) '' frontier s = frontier (K : Set Plane_m58ecd66)
+    := by
+    calc
+      (fun z : Plane_m58ecd66 ↦ c + z) '' frontier s =
+          frontier ((fun z : Plane_m58ecd66 ↦ c + z) '' s) :=
+        (Homeomorph.addLeft c).image_frontier s
+      _ = frontier (K : Set Plane_m58ecd66) := by
+        congr 1
+        ext x
+        simp [s, Set.mem_vadd_set_iff_neg_vadd_mem]
+  have hisom : Isometry (fun z : Plane_m58ecd66 ↦ c + z) := by
+    intro x y
+    simp
+  rw [← himage, hisom.hausdorffMeasure_image (Or.inl zero_le_one)]
+  exact hsfinite
+
+end ConvexBody
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Convex / Body / Hausdorff
+-/
+
+@[expose] public section
+
+open Filter
+open scoped Topology
+
+namespace ConvexBody
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+/-- Eventual membership persists in a Hausdorff limit of convex bodies. -/
+theorem mem_of_tendsto_hausdorffDist (q : E) (K : ℕ → ConvexBody E) (L : ConvexBody E)
+    (hev : ∀ᶠ n in atTop, q ∈ (K n : Set E))
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set E) (L : Set E))
+      atTop (𝓝 0)) : q ∈ (L : Set E) := by
+  have hle : (fun _ : ℕ ↦ Metric.infDist q (L : Set E)) ≤ᶠ[atTop]
+      (fun n ↦ Metric.hausdorffDist (K n : Set E) (L : Set E)) := by
+    filter_upwards [hev] with n hn
+    exact Metric.infDist_le_hausdorffDist_of_mem hn
+      (Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded
+        (K n).nonempty L.nonempty (K n).isCompact.isBounded L.isCompact.isBounded)
+  have hz : Metric.infDist q (L : Set E) ≤ 0 :=
+    le_of_tendsto_of_tendsto tendsto_const_nhds hlim hle
+  exact (IsClosed.mem_iff_infDist_zero L.isClosed L.nonempty).mpr
+    (le_antisymm hz Metric.infDist_nonneg)
+
+end ConvexBody
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Convex / Collinear
+-/
+
+@[expose] public section
+
+open Module
+
+/-- A convex set with empty interior in dimension at most two is collinear. -/
+theorem Convex.collinear_of_interior_eq_empty {E : Type*}
+    [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+    {s : Set E} (hs : Convex ℝ s) (hdim : finrank ℝ E ≤ 2)
+    (hint : interior s = ∅) : Collinear ℝ s := by
+  rcases s.eq_empty_or_nonempty with rfl | hne
+  · exact collinear_empty ℝ E
+  have hspan : vectorSpan ℝ s ≠ ⊤ := by
+    intro h
+    have ha : affineSpan ℝ s = ⊤ :=
+      (AffineSubspace.direction_eq_top_iff_of_nonempty
+        (hne.mono (subset_affineSpan ℝ s))).mp (by simpa only [direction_affineSpan] using h)
+    have hi := hs.interior_nonempty_iff_affineSpan_eq_top.mpr ha
+    simp [hint] at hi
+  apply collinear_iff_finrank_le_one.mpr
+  have hlt := Submodule.finrank_lt_finrank_of_lt (lt_top_iff_ne_top.mpr hspan)
+  simp only [finrank_top] at hlt
+  omega
+
+/-- A separating linear functional orders points on a line into a segment. -/
+theorem Collinear.mem_segment_of_apply_le {E : Type*} [AddCommGroup E] [Module ℝ E]
+    {s : Set E} (hs : Collinear ℝ s) {a p x : E}
+    (ha : a ∈ s) (hp : p ∈ s) (hx : x ∈ s) (f : E →ₗ[ℝ] ℝ)
+    (hpos : 0 < f (p - a)) (hle : f p ≤ f x) : p ∈ segment ℝ a x := by
+  have hne : a ≠ p := by
+    intro h
+    simp [h] at hpos
+  obtain ⟨r, hr⟩ := mem_affineSpan_pair_iff_exists_lineMap_eq.mp
+    (hs.mem_affineSpan_of_mem_of_ne ha hp hx hne)
+  have hrle : 1 ≤ r := by
+    rw [← hr, AffineMap.lineMap_apply_module', map_add, map_smul] at hle
+    have hsub := f.map_sub p a
+    change f p ≤ r * f (p - a) + f a at hle
+    nlinarith
+  have hrpos : 0 < r := lt_of_lt_of_le zero_lt_one hrle
+  have hinv : r⁻¹ ∈ Set.Icc (0 : ℝ) 1 :=
+    ⟨inv_nonneg.mpr hrpos.le, inv_le_one_of_one_le₀ hrle⟩
+  have hmem := lineMap_mem_segment ℝ a x hinv
+  rw [← hr, AffineMap.lineMap_lineMap_right, inv_mul_cancel₀ hrpos.ne',
+    AffineMap.lineMap_apply_one] at hmem
+  simpa only [hr] using hmem
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Convex / Body / Segment
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped Pointwise
+
+namespace ConvexBody
+
+private abbrev Plane_mea511e5 := EuclideanSpace ℝ (Fin 2)
+
+/-- A nonsingleton planar convex body with empty interior is a nontrivial segment. -/
+theorem exists_eq_segment_of_interior_empty
+    (K : ConvexBody (EuclideanSpace ℝ (Fin 2)))
+    (hsub : ¬(K : Set (EuclideanSpace ℝ (Fin 2))).Subsingleton)
+    (hint : interior (K : Set (EuclideanSpace ℝ (Fin 2))) = ∅) :
+    ∃ a b, a ≠ b ∧
+      (K : Set (EuclideanSpace ℝ (Fin 2))) = segment ℝ a b := by
+  have hcol : Collinear ℝ (K : Set Plane_mea511e5) :=
+    K.convex.collinear_of_interior_eq_empty
+      (by simp) hint
+  obtain ⟨p₀, v, hv⟩ := (collinear_iff_exists_forall_eq_smul_vadd
+    (k := ℝ) (K : Set Plane_mea511e5)).mp hcol
+  have hv₀ : v ≠ 0 := by
+    intro hz
+    apply hsub
+    intro p hp q hq
+    obtain ⟨r, hr⟩ := hv p hp
+    obtain ⟨s, hs⟩ := hv q hq
+    simpa [hz] using hr.trans (by simpa [hz] using hs.symm)
+  let f : Plane_mea511e5 → ℝ := fun p ↦ inner ℝ v (p - p₀) / inner ℝ v v
+  have hvv : inner ℝ v v ≠ 0 := inner_self_ne_zero.mpr hv₀
+  have hrepr : ∀ p ∈ (K : Set Plane_mea511e5), p = f p • v + p₀ := by
+    intro p hp
+    obtain ⟨r, rfl⟩ := hv p hp
+    have hf : f (r • v + p₀) = r := by
+      dsimp [f]
+      rw [add_sub_cancel_right, inner_smul_right]
+      field_simp [hvv]
+    change r • v + p₀ = f (r • v + p₀) • v + p₀
+    rw [hf]
+  have hfcont : Continuous f := by fun_prop
+  obtain ⟨a, ha, hamin⟩ := K.isCompact.exists_isMinOn K.nonempty hfcont.continuousOn
+  obtain ⟨b, hb, hbmax⟩ := K.isCompact.exists_isMaxOn K.nonempty hfcont.continuousOn
+  have habf : f a < f b := by
+    apply lt_of_le_of_ne (hamin hb)
+    intro heq
+    apply hsub
+    intro p hp q hq
+    rw [hrepr p hp, hrepr q hq]
+    have hpfa : f p = f a :=
+      le_antisymm ((hbmax hp).trans_eq heq.symm) (hamin hp)
+    have hqfa : f q = f a :=
+      le_antisymm ((hbmax hq).trans_eq heq.symm) (hamin hq)
+    rw [hpfa, hqfa]
+  refine ⟨a, b, ?_, Set.Subset.antisymm ?_ ?_⟩
+  · intro hab
+    rw [hab] at habf
+    exact habf.false
+  · intro p hp
+    rw [segment_eq_image']
+    let t := (f p - f a) / (f b - f a)
+    have ht : t ∈ Set.Icc (0 : ℝ) 1 := ⟨
+      div_nonneg (sub_nonneg.mpr (hamin hp)) (sub_nonneg.mpr habf.le),
+      (div_le_one (sub_pos.mpr habf)).mpr (sub_le_sub_right (hbmax hp) _)⟩
+    refine ⟨t, ht, ?_⟩
+    rw [hrepr p hp, hrepr a ha, hrepr b hb]
+    have hden : f b - f a ≠ 0 := sub_ne_zero.mpr habf.ne'
+    have htalg : f a + t * (f b - f a) = f p := by
+      dsimp [t]
+      field_simp [hden]
+      ring
+    rw [← htalg]
+    module
+  · exact K.convex.segment_subset ha hb
+
+end ConvexBody
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Convex / Function
+-/
+
+@[expose] public section
+
+/-- A convex function stays below a bound before the right endpoint if the left
+endpoint is strictly below the bound and the right endpoint is at most the bound. -/
+theorem ConvexOn.lt_on_Ico_of_lt_of_le {f : ℝ → ℝ} {a b t c : ℝ}
+    (hf : ConvexOn ℝ (Set.Icc a b) f) (ha : f a < c) (hb : f b ≤ c)
+    (hat : a ≤ t) (htb : t < b) : f t < c := by
+  have hab : a < b := lt_of_le_of_lt hat htb
+  let u := (b - t) / (b - a)
+  let v := (t - a) / (b - a)
+  have hu : 0 < u := div_pos (sub_pos.mpr htb) (sub_pos.mpr hab)
+  have hv : 0 ≤ v := div_nonneg (sub_nonneg.mpr hat) (sub_nonneg.mpr hab.le)
+  have huv : u + v = 1 := by dsimp [u, v]; field_simp [ne_of_gt (sub_pos.mpr hab)]; ring
+  have ht : u * a + v * b = t := by dsimp [u, v]; field_simp [ne_of_gt (sub_pos.mpr hab)]; ring
+  have h := hf.2 ⟨le_rfl, hab.le⟩ ⟨hab.le, le_rfl⟩ hu.le hv huv
+  simp only [smul_eq_mul, ht] at h
+  have hleft : u * f a < u * c := mul_lt_mul_of_pos_left ha hu
+  have hright : v * f b ≤ v * c := mul_le_mul_of_nonneg_left hb hv
+  calc
+    f t ≤ u * f a + v * f b := h
+    _ < u * c + v * c := add_lt_add_of_lt_of_le hleft hright
+    _ = c := by rw [← add_mul, huv, one_mul]
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Convex / Hausdorff
+-/
+
+@[expose] public section
+
+open Filter TopologicalSpace
+open scoped Topology
+
+/-- Hausdorff limits of nonempty compact convex sets are convex. -/
+theorem TopologicalSpace.NonemptyCompacts.convex_of_tendsto {E ι : Type*}
+    [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {l : Filter ι} [l.NeBot] {K : ι → NonemptyCompacts E} {L : NonemptyCompacts E}
+    (hK : ∀ n, Convex ℝ (K n : Set E)) (hlim : Tendsto K l (𝓝 L)) :
+    Convex ℝ (L : Set E) := by
+  have happrox : ∀ x ∈ (L : Set E), ∃ p : ι → E,
+      (∀ n, p n ∈ (K n : Set E)) ∧ Tendsto p l (𝓝 x) := by
+    intro x hx
+    choose p hp hd using fun n ↦ (K n).isCompact.exists_infDist_eq_dist (K n).nonempty x
+    refine ⟨p, hp, ?_⟩
+    have hinf := (NonemptyCompacts.lipschitz_infDist_const x).continuous.tendsto L |>.comp hlim
+    have hz : Metric.infDist x (L : Set E) = 0 := Metric.infDist_zero_of_mem hx
+    rw [hz] at hinf
+    change Tendsto (fun n ↦ Metric.infDist x (K n : Set E)) l (𝓝 0) at hinf
+    apply tendsto_iff_dist_tendsto_zero.mpr
+    simpa only [Function.comp_apply, hd, dist_comm] using hinf
+  intro x hx y hy a b ha hb hab
+  obtain ⟨p, hp, hpx⟩ := happrox x hx
+  obtain ⟨q, hq, hqy⟩ := happrox y hy
+  have hz := (hpx.const_smul a).add (hqy.const_smul b)
+  have hi := NonemptyCompacts.uniformContinuous_infDist.continuous.tendsto
+    (a • x + b • y, L) |>.comp (hz.prodMk_nhds hlim)
+  have hzero : (fun n ↦ Metric.infDist (a • p n + b • q n) (K n : Set E)) = fun _ ↦ 0 := by
+    funext n
+    exact Metric.infDist_zero_of_mem (hK n (hp n) (hq n) ha hb hab)
+  change Tendsto (fun n ↦ Metric.infDist (a • p n + b • q n) (K n : Set E))
+    l (𝓝 (Metric.infDist (a • x + b • y) (L : Set E))) at hi
+  rw [hzero] at hi
+  exact (L.isCompact.isClosed.mem_iff_infDist_zero L.nonempty).mpr
+    (tendsto_nhds_unique hi tendsto_const_nhds)
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Convex / Support
+-/
+
+@[expose] public section
+
+open scoped ENNReal
+
+/-- Support values of compact sets differ by at most their Hausdorff distance times the norm of
+the direction. -/
+theorem IsCompact.abs_csSup_inner_sub_le_hausdorffDist
+    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+    {S T : Set E} (hS : IsCompact S) (hneS : S.Nonempty)
+    (hT : IsCompact T) (hneT : T.Nonempty) (u : E) :
+    |sSup ((fun x ↦ inner ℝ x u) '' S) - sSup ((fun x ↦ inner ℝ x u) '' T)| ≤
+      Metric.hausdorffDist S T * ‖u‖ := by
+  obtain ⟨x, hxS, hx, hxmax⟩ := hS.exists_sSup_image_eq_and_ge
+    (f := fun x : E ↦ inner ℝ x u) hneS
+    (continuous_inner.comp (continuous_id.prodMk continuous_const)).continuousOn
+  obtain ⟨y, hyT, hy⟩ := hT.exists_infDist_eq_dist hneT x
+  obtain ⟨z, hzT, hz, hzmax⟩ := hT.exists_sSup_image_eq_and_ge
+    (f := fun x : E ↦ inner ℝ x u) hneT
+    (continuous_inner.comp (continuous_id.prodMk continuous_const)).continuousOn
+  have hfin : Metric.hausdorffEDist S T ≠ ⊤ :=
+    Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded hneS hneT
+      hS.isBounded hT.isBounded
+  have hdist : ‖x - y‖ ≤ Metric.hausdorffDist S T := by
+    calc
+      ‖x - y‖ = dist x y := (dist_eq_norm x y).symm
+      _ = Metric.infDist x T := hy.symm
+      _ ≤ Metric.hausdorffDist S T := Metric.infDist_le_hausdorffDist_of_mem hxS hfin
+  have hupper : sSup ((fun x ↦ inner ℝ x u) '' S) -
+      sSup ((fun x ↦ inner ℝ x u) '' T) ≤ Metric.hausdorffDist S T * ‖u‖ := by
+    rw [hx, hz]
+    calc
+      inner ℝ x u - inner ℝ z u ≤ inner ℝ x u - inner ℝ y u :=
+        sub_le_sub_left (hzmax y hyT) _
+      _ = inner ℝ (x - y) u := by rw [inner_sub_left]
+      _ ≤ |inner ℝ (x - y) u| := le_abs_self _
+      _ ≤ ‖x - y‖ * ‖u‖ := by
+        simpa only [Real.norm_eq_abs] using norm_inner_le_norm (𝕜 := ℝ) (x - y) u
+      _ ≤ Metric.hausdorffDist S T * ‖u‖ := by gcongr
+  have hreverse : sSup ((fun x ↦ inner ℝ x u) '' T) -
+      sSup ((fun x ↦ inner ℝ x u) '' S) ≤ Metric.hausdorffDist S T * ‖u‖ := by
+    have hfin' : Metric.hausdorffEDist T S ≠ ⊤ :=
+      Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded hneT hneS
+        hT.isBounded hS.isBounded
+    obtain ⟨x', hx'S, hx'⟩ := hS.exists_infDist_eq_dist hneS z
+    have hdist' : ‖z - x'‖ ≤ Metric.hausdorffDist S T := by
+      calc
+        ‖z - x'‖ = dist z x' := (dist_eq_norm z x').symm
+        _ = Metric.infDist z S := hx'.symm
+        _ ≤ Metric.hausdorffDist T S :=
+          Metric.infDist_le_hausdorffDist_of_mem hzT hfin'
+        _ = Metric.hausdorffDist S T := Metric.hausdorffDist_comm
+    rw [hz, hx]
+    calc
+      inner ℝ z u - inner ℝ x u ≤ inner ℝ z u - inner ℝ x' u :=
+        sub_le_sub_left (hxmax x' hx'S) _
+      _ = inner ℝ (z - x') u := by rw [inner_sub_left]
+      _ ≤ |inner ℝ (z - x') u| := le_abs_self _
+      _ ≤ ‖z - x'‖ * ‖u‖ := by
+        simpa only [Real.norm_eq_abs] using norm_inner_le_norm (𝕜 := ℝ) (z - x') u
+      _ ≤ Metric.hausdorffDist S T * ‖u‖ := by gcongr
+  rw [abs_le]
+  constructor <;> linarith
+
+/-- The support values of a compact set are Lipschitz in the direction, with constant equal to the
+maximum norm of a point in the set. -/
+theorem IsCompact.abs_csSup_inner_sub_le
+    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+    {S : Set E} (hS : IsCompact S) (hneS : S.Nonempty) (u v : E) :
+    |sSup ((fun x ↦ inner ℝ x u) '' S) - sSup ((fun x ↦ inner ℝ x v) '' S)| ≤
+      sSup (norm '' S) * ‖u - v‖ := by
+  obtain ⟨x, hxS, hxu, hxu'⟩ := hS.exists_sSup_image_eq_and_ge
+    (f := fun x : E ↦ inner ℝ x u) hneS
+    (continuous_inner.comp (continuous_id.prodMk continuous_const)).continuousOn
+  obtain ⟨y, hyS, hyv, hyv'⟩ := hS.exists_sSup_image_eq_and_ge
+    (f := fun x : E ↦ inner ℝ x v) hneS
+    (continuous_inner.comp (continuous_id.prodMk continuous_const)).continuousOn
+  have hbound (z : E) (hz : z ∈ S) (w : E) :
+      |inner ℝ z w| ≤ sSup (norm '' S) * ‖w‖ := by
+    calc
+      |inner ℝ z w| ≤ ‖z‖ * ‖w‖ := by
+        simpa only [Real.norm_eq_abs] using norm_inner_le_norm (𝕜 := ℝ) z w
+      _ ≤ sSup (norm '' S) * ‖w‖ := by
+        gcongr
+        exact le_csSup (hS.bddAbove_image continuous_norm.continuousOn) ⟨z, hz, rfl⟩
+  have huv : sSup ((fun x ↦ inner ℝ x u) '' S) -
+      sSup ((fun x ↦ inner ℝ x v) '' S) ≤ sSup (norm '' S) * ‖u - v‖ := by
+    rw [hxu, hyv]
+    calc
+      inner ℝ x u - inner ℝ y v ≤ inner ℝ x u - inner ℝ x v :=
+        sub_le_sub_left (hyv' x hxS) _
+      _ = inner ℝ x (u - v) := by rw [inner_sub_right]
+      _ ≤ |inner ℝ x (u - v)| := le_abs_self _
+      _ ≤ sSup (norm '' S) * ‖u - v‖ := hbound x hxS (u - v)
+  have hvu : sSup ((fun x ↦ inner ℝ x v) '' S) -
+      sSup ((fun x ↦ inner ℝ x u) '' S) ≤ sSup (norm '' S) * ‖u - v‖ := by
+    rw [hyv, hxu]
+    calc
+      inner ℝ y v - inner ℝ x u ≤ inner ℝ y v - inner ℝ y u :=
+        sub_le_sub_left (hxu' y hyS) _
+      _ = inner ℝ y (v - u) := by rw [inner_sub_right]
+      _ ≤ |inner ℝ y (v - u)| := le_abs_self _
+      _ ≤ sSup (norm '' S) * ‖v - u‖ := hbound y hyS (v - u)
+      _ = sSup (norm '' S) * ‖u - v‖ := by rw [norm_sub_rev]
+  rw [abs_le]
+  constructor <;> linarith
+
+/-- A convex body in a real inner product space is the intersection of its supporting
+half-spaces. -/
+theorem ConvexBody.eq_iInter_halfSpaces
+    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+    (K : ConvexBody E) :
+    (K : Set E) = ⋂ u ∈ {u : E | ‖u‖ = 1},
+      {p : E | inner ℝ p u ≤ sSup ((fun x ↦ inner ℝ x u) '' (K : Set E))} := by
+  apply Set.Subset.antisymm
+  · intro p hp
+    simp only [Set.mem_iInter, Set.mem_ofPred_eq]
+    intro u _
+    obtain ⟨x, hx, hxmax, hmax⟩ := K.isCompact.exists_sSup_image_eq_and_ge
+      (f := fun x : E ↦ inner ℝ x u) K.nonempty
+      (continuous_inner.comp (continuous_id.prodMk continuous_const)).continuousOn
+    rw [hxmax]
+    exact hmax p hp
+  · intro p hp
+    by_contra hpK
+    obtain ⟨q, hq, hqmin⟩ :=
+      exists_norm_eq_iInf_of_complete_convex K.nonempty K.isCompact.isComplete K.convex p
+    have hpq : p - q ≠ 0 := sub_ne_zero.mpr (fun h ↦ hpK (h ▸ hq))
+    let u : E := ‖p - q‖⁻¹ • (p - q)
+    have hnorm : 0 < ‖p - q‖ := norm_pos_iff.mpr hpq
+    have hu : ‖u‖ = 1 := by simp [u, norm_smul, hnorm.ne']
+    have hproj : ∀ y ∈ (K : Set E), inner ℝ (p - q) (y - q) ≤ 0 :=
+      (norm_eq_iInf_iff_real_inner_le_zero K.convex hq).mp hqmin
+    have hqmax : sSup ((fun x ↦ inner ℝ x u) '' (K : Set E)) = inner ℝ q u := by
+      obtain ⟨x, hx, hxmax, hmax⟩ := K.isCompact.exists_sSup_image_eq_and_ge
+        (f := fun x : E ↦ inner ℝ x u) K.nonempty
+        (continuous_inner.comp (continuous_id.prodMk continuous_const)).continuousOn
+      rw [hxmax]
+      apply le_antisymm
+      · dsimp [u]
+        rw [inner_smul_right, inner_smul_right]
+        apply mul_le_mul_of_nonneg_left _ (inv_nonneg.mpr hnorm.le)
+        have h := hproj x hx
+        have h' : inner ℝ (x - q) (p - q) ≤ 0 := by rwa [real_inner_comm]
+        rw [inner_sub_left] at h'
+        linarith
+      · exact hmax q hq
+    have hpu := Set.mem_iInter.mp (Set.mem_iInter.mp hp u) hu
+    rw [hqmax] at hpu
+    have hstrict : inner ℝ q u < inner ℝ p u := by
+      dsimp [u]
+      rw [inner_smul_right, inner_smul_right]
+      have hself : 0 < inner ℝ (p - q) (p - q) := by
+        rw [real_inner_self_eq_norm_sq]
+        positivity
+      rw [← sub_pos, ← mul_sub, ← inner_sub_left]
+      exact mul_pos (inv_pos.mpr hnorm) hself
+    exact (not_lt_of_ge hpu) hstrict
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Convex / Translation
+-/
+
+@[expose] public section
+
+/-- Translation of a convex body by a vector. -/
+def ConvexBody.translate {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (K : ConvexBody E) (v : E) : ConvexBody E where
+  carrier := (fun p ↦ p + v) '' (K : Set E)
+  convex' := by simpa only [add_comm] using K.convex.translate v
+  isCompact' := K.isCompact.image (continuous_id.add continuous_const)
+  nonempty' := K.nonempty.image _
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `ForMathlib.Geometry.Euclidean.Segment`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Geometry / Euclidean / Segment
+-/
+
+@[expose] public section
+
+open MeasureTheory
+
+namespace EuclideanGeometry
+
+/-- Two unit vectors perpendicular to the same nonzero planar vector agree up to sign. -/
+theorem eq_or_eq_neg_of_unit_orthogonal
+    (o : Orientation ℝ (EuclideanSpace ℝ (Fin 2)) (Fin 2))
+    {v n m : EuclideanSpace ℝ (Fin 2)} (hv : v ≠ 0)
+    (hn : ‖n‖ = 1) (hm : ‖m‖ = 1)
+    (hvn : inner ℝ v n = 0) (hvm : inner ℝ v m = 0) : n = m ∨ n = -m := by
+  let _ : Fact (Module.finrank ℝ (EuclideanSpace ℝ (Fin 2)) = 2) :=
+    ⟨by simp [finrank_euclideanSpace]⟩
+  obtain ⟨r, hr⟩ :=
+    ((o.inner_eq_zero_iff_eq_zero_or_eq_smul_rotation_pi_div_two).mp hvn).resolve_left hv
+  obtain ⟨s, hs⟩ :=
+    ((o.inner_eq_zero_iff_eq_zero_or_eq_smul_rotation_pi_div_two).mp hvm).resolve_left hv
+  have hvnorm : 0 < ‖v‖ := norm_pos_iff.mpr hv
+  have hrabs : |r| * ‖v‖ = 1 := by
+    calc
+      |r| * ‖v‖ = ‖r • o.rotation (Real.pi / 2 : ℝ) v‖ := by simp [norm_smul]
+      _ = ‖n‖ := congrArg norm hr
+      _ = 1 := hn
+  have hsabs : |s| * ‖v‖ = 1 := by
+    calc
+      |s| * ‖v‖ = ‖s • o.rotation (Real.pi / 2 : ℝ) v‖ := by simp [norm_smul]
+      _ = ‖m‖ := congrArg norm hs
+      _ = 1 := hm
+  have habs : |r| = |s| := mul_right_cancel₀ (ne_of_gt hvnorm) (hrabs.trans hsabs.symm)
+  rcases (abs_eq_abs.mp habs) with hrs | hrs
+  · left
+    rw [← hr, ← hs, hrs]
+  · right
+    rw [← hr, ← hs, hrs, neg_smul]
+
+/-- Equal segments have the same orthogonal directions. -/
+theorem inner_direction_eq_zero_of_segment_eq
+    {a b c d n : (EuclideanSpace ℝ (Fin 2))} (hseg : segment ℝ a b = segment ℝ c d)
+    (hcdn : inner ℝ (d - c) n = 0) : inner ℝ (b - a) n = 0 := by
+  have ha : a ∈ segment ℝ c d := hseg ▸ left_mem_segment ℝ a b
+  have hb : b ∈ segment ℝ c d := hseg ▸ right_mem_segment ℝ a b
+  rw [segment_eq_image'] at ha hb
+  obtain ⟨r, _, hr⟩ := ha
+  obtain ⟨s, _, hs⟩ := hb
+  rw [← hr, ← hs]
+  have heq : c + s • (d - c) - (c + r • (d - c)) = (s - r) • (d - c) := by
+    module
+  rw [heq]
+  rw [inner_smul_left, hcdn, mul_zero]
+
+/-- Equal segments have equal endpoint distances. -/
+theorem dist_eq_of_segment_eq {a b c d : (EuclideanSpace ℝ (Fin 2))}
+    (hseg : segment ℝ a b = segment ℝ c d) : dist a b = dist c d := by
+  have h := congrArg (Measure.hausdorffMeasure 1) hseg
+  simpa only [MeasureTheory.hausdorffMeasure_segment, edist_dist,
+    ENNReal.ofReal_eq_ofReal_iff, dist_nonneg] using h
+
+/-- A segment transverse to a hyperplane meets it in at most one point. -/
+lemma segment_inter_hyperplane_subsingleton {a b n : EuclideanSpace ℝ (Fin 2)} {c : ℝ}
+    (h : inner ℝ (b - a) n ≠ 0) :
+    (segment ℝ a b ∩ {q | inner ℝ q n = c}).Subsingleton := by
+  rintro x ⟨hx, hxc⟩ y ⟨hy, hyc⟩
+  rw [segment_eq_image'] at hx hy
+  obtain ⟨r, hr, rfl⟩ := hx
+  obtain ⟨s, hs, rfl⟩ := hy
+  have heq : r * inner ℝ (b - a) n = s * inner ℝ (b - a) n := by
+    have hx : inner ℝ a n + r * inner ℝ (b - a) n = c := by
+      simpa only [Set.mem_ofPred_eq, inner_add_left, real_inner_smul_left] using hxc
+    have hy : inner ℝ a n + s * inner ℝ (b - a) n = c := by
+      simpa only [Set.mem_ofPred_eq, inner_add_left, real_inner_smul_left] using hyc
+    linarith
+  rw [mul_right_cancel₀ h heq]
+
+/-- A transverse segment has zero length on a hyperplane. -/
+lemma hausdorffMeasure_segment_inter_hyperplane_eq_zero {a b n : EuclideanSpace ℝ (Fin 2)} {c : ℝ}
+    (h : inner ℝ (b - a) n ≠ 0) :
+    Measure.hausdorffMeasure 1 (segment ℝ a b ∩ {q | inner ℝ q n = c}) = 0 := by
+  have := Measure.nullSingletonClass_hausdorff (EuclideanSpace ℝ (Fin 2)) (by norm_num : (0 : ℝ) <
+    1)
+  exact (segment_inter_hyperplane_subsingleton h).countable.measure_zero _
+
+/-- A segment not contained in a hyperplane meets it in at most one point. -/
+lemma segment_inter_hyperplane_subsingleton_of_not_subset {a b n : EuclideanSpace ℝ (Fin 2)} {c : ℝ}
+    (hnot : ¬ segment ℝ a b ⊆ {q | inner ℝ q n = c}) :
+    (segment ℝ a b ∩ {q | inner ℝ q n = c}).Subsingleton := by
+  by_cases hdir : inner ℝ (b - a) n = 0
+  · intro x hx y hy
+    exfalso
+    apply hnot
+    have hconstant {q : EuclideanSpace ℝ (Fin 2)} (hq : q ∈ segment ℝ a b) :
+        inner ℝ q n = inner ℝ a n := by
+      rw [segment_eq_image'] at hq
+      obtain ⟨r, hr, rfl⟩ := hq
+      simp only [inner_add_left, real_inner_smul_left, hdir, mul_zero, add_zero]
+    intro q hq
+    exact (hconstant hq).trans ((hconstant hx.1).symm.trans hx.2)
+  · exact segment_inter_hyperplane_subsingleton hdir
+
+/-- A nondegenerate segment covered by finitely many hyperplanes lies in one of them. -/
+lemma exists_segment_subset_hyperplane_of_finite_cover {ι : Type*}
+    (I : Finset ι) (n : ι → EuclideanSpace ℝ (Fin 2)) (c : ι → ℝ) {a b : EuclideanSpace ℝ (Fin 2)}
+      (hab : a ≠ b)
+    (hcover : segment ℝ a b ⊆ ⋃ i ∈ I, {q | inner ℝ q (n i) = c i}) :
+    ∃ i ∈ I, segment ℝ a b ⊆ {q | inner ℝ q (n i) = c i} := by
+  classical
+  have := Measure.nullSingletonClass_hausdorff (EuclideanSpace ℝ (Fin 2)) (by norm_num : (0 : ℝ) <
+    1)
+  by_contra hnot
+  push Not at hnot
+  have hnull (i : ι) (hi : i ∈ I) :
+      Measure.hausdorffMeasure 1 (segment ℝ a b ∩ {q | inner ℝ q (n i) = c i}) = 0 :=
+    (segment_inter_hyperplane_subsingleton_of_not_subset (hnot i hi)).countable.measure_zero _
+  have hsub : segment ℝ a b ⊆
+      ⋃ i ∈ I, segment ℝ a b ∩ {q | inner ℝ q (n i) = c i} := by
+    intro q hq
+    obtain ⟨i, hi, hqi⟩ := Set.mem_iUnion₂.mp (hcover hq)
+    exact Set.mem_iUnion₂.mpr ⟨i, hi, hq, hqi⟩
+  have hm := measure_mono_null hsub
+    ((measure_biUnion_null_iff I.finite_toSet.countable).mpr hnull)
+  rw [hausdorffMeasure_segment] at hm
+  exact hab (edist_eq_zero.mp hm)
+
+end EuclideanGeometry
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `ForMathlib.MeasureTheory.EuclideanSpace`.
+* `ForMathlib.MeasureTheory.FiniteMeasure.Portmanteau`.
+* `ForMathlib.MeasureTheory.FiniteMeasure.Restriction`.
+* `ForMathlib.MeasureTheory.Angle`.
+* `ForMathlib.MeasureTheory.Hausdorff.Arclength`.
+* `ForMathlib.MeasureTheory.Hausdorff.Graph`.
+* `ForMathlib.MeasureTheory.Hausdorff.PlanarGraph`.
+* `ForMathlib.MeasureTheory.Integral.AtomicBounds`.
+* `ForMathlib.MeasureTheory.Integral.IntervalExhaustion`.
+* `ForMathlib.MeasureTheory.Integral.MovingIntervals`.
+* `ForMathlib.MeasureTheory.Integral.Translation`.
+* `ForMathlib.MeasureTheory.Measure.Atoms`.
+* `ForMathlib.MeasureTheory.Measure.HaarNullSets`.
+* `ForMathlib.MeasureTheory.Measure.Map`.
+* `ForMathlib.MeasureTheory.Measure.WithDensity`.
+* `ForMathlib.MeasureTheory.RegionBetween`.
+* `ForMathlib.MeasureTheory.Measure.PlanarTrapezoid`.
+* `ForMathlib.MeasureTheory.Measure.PlanarTriangle`.
+* `ForMathlib.MeasureTheory.StieltjesDensity`.
+* `ForMathlib.MeasureTheory.VectorMeasure.Interval`.
+* `ForMathlib.MeasureTheory.VectorMeasure.WithDensity`.
+* `ForMathlib.MeasureTheory.Volume`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Euclidean Space
+-/
+
+@[expose] public section
+
+open MeasureTheory
+
+namespace EuclideanSpace
+
+theorem volume_preserving_finTwoCoordinates :
+    MeasurePreserving (fun p : EuclideanSpace ℝ (Fin 2) ↦ (p 0, p 1)) volume volume := by
+  exact (volume_preserving_finTwoArrow ℝ).comp
+    (EuclideanSpace.volume_preserving_symm_measurableEquiv_toLp (Fin 2))
+
+/-- The volume of a closed coordinate box of the Euclidean plane. -/
+theorem volume_setOf_apply_mem_Icc (l r b t : ℝ) :
+    volume {p : EuclideanSpace ℝ (Fin 2) | p 0 ∈ Set.Icc l r ∧ p 1 ∈ Set.Icc b t} =
+      ENNReal.ofReal (r - l) * ENNReal.ofReal (t - b) := by
+  have h := volume_preserving_finTwoCoordinates.measure_preimage
+    ((measurableSet_Icc.prod measurableSet_Icc).nullMeasurableSet :
+      NullMeasurableSet (Set.Icc l r ×ˢ Set.Icc b t) (volume : Measure (ℝ × ℝ)))
+  calc
+    volume {p : EuclideanSpace ℝ (Fin 2) | p 0 ∈ Set.Icc l r ∧ p 1 ∈ Set.Icc b t} =
+        volume (Set.Icc l r ×ˢ Set.Icc b t) := by
+      convert h using 1
+      congr 1
+    _ = _ := by rw [Measure.volume_eq_prod, Measure.prod_prod, Real.volume_Icc, Real.volume_Icc]
+
+/-- The coordinates of a point of the Euclidean plane, listed second coordinate first. -/
+def finTwoCoordinatesSwap (p : EuclideanSpace ℝ (Fin 2)) : ℝ × ℝ :=
+  (p 1, p 0)
+
+/-- Reading the plane coordinates in the reversed order is measure preserving. -/
+theorem volume_preserving_finTwoCoordinatesSwap :
+    MeasurePreserving finTwoCoordinatesSwap volume
+      ((volume : Measure ℝ).prod (volume : Measure ℝ)) := by
+  have hswap : MeasurePreserving Prod.swap
+      ((volume : Measure ℝ).prod (volume : Measure ℝ))
+      ((volume : Measure ℝ).prod (volume : Measure ℝ)) :=
+    Measure.measurePreserving_swap
+  convert hswap.comp volume_preserving_finTwoCoordinates using 1
+  funext p
+  rfl
+
+end EuclideanSpace
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Portmanteau for finite measures: the open-set inequality
+
+Mathlib proves the open-set portmanteau inequality
+`MeasureTheory.ProbabilityMeasure.le_liminf_measure_open_of_tendsto` for probability measures and
+only the closed-set inequality `MeasureTheory.FiniteMeasure.limsup_measure_closed_le_of_tendsto`
+for finite measures. This file supplies the missing open-set inequality for finite measures, by
+combining the closed-set inequality on the complement with the convergence of the total masses.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter Set
+open scoped Topology
+
+namespace MeasureTheory.FiniteMeasure
+
+/-- Portmanteau for finite measures: weak convergence bounds the mass of an open set by the
+lower limit of the approximating masses. -/
+theorem le_liminf_measure_open_of_tendsto
+    {Ω ι : Type*} {L : Filter ι}
+    [MeasurableSpace Ω] [TopologicalSpace Ω] [HasOuterApproxClosed Ω]
+    [OpensMeasurableSpace Ω] {μ : FiniteMeasure Ω} {μs : ι → FiniteMeasure Ω}
+    (hlim : Tendsto μs L (𝓝 μ)) {G : Set Ω} (hG : IsOpen G) :
+    (μ : Measure Ω) G ≤ L.liminf (fun i ↦ (μs i : Measure Ω) G) := by
+  have hclosed := MeasureTheory.FiniteMeasure.limsup_measure_closed_le_of_tendsto hlim
+    hG.isClosed_compl
+  rw [le_liminf_iff (by isBoundedDefault) (by isBoundedDefault)]
+  intro y hy
+  have hμGtop : (μ : Measure Ω) G ≠ ⊤ := measure_ne_top _ _
+  have hyTop : y ≠ ⊤ := ne_top_of_lt (hy.trans_le (le_top))
+  have hyR : y.toReal < ((μ : Measure Ω) G).toReal :=
+    (ENNReal.toReal_lt_toReal hyTop hμGtop).mpr hy
+  let d : ℝ := (((μ : Measure Ω) G).toReal - y.toReal) / 3
+  have hd : 0 < d := by
+    dsimp [d]
+    linarith
+  have hmass : Tendsto (fun i ↦ ((μs i).mass : ℝ)) L (𝓝 (μ.mass : ℝ)) :=
+    (NNReal.continuous_coe.tendsto μ.mass).comp
+      (MeasureTheory.FiniteMeasure.continuous_mass.tendsto μ |>.comp hlim)
+  have hevmass : ∀ᶠ i in L, (μ.mass : ℝ) - d < ((μs i).mass : ℝ) :=
+    hmass.eventually_const_lt (sub_lt_self _ hd)
+  let q : ENNReal := ENNReal.ofReal (((μ : Measure Ω) Gᶜ).toReal + d)
+  have hqd : 0 ≤ ((μ : Measure Ω) Gᶜ).toReal + d := by
+    have hcR : 0 ≤ ((μ : Measure Ω) Gᶜ).toReal := ENNReal.toReal_nonneg
+    linarith
+  have hcompq : (μ : Measure Ω) Gᶜ < q := by
+    rw [← ENNReal.toReal_lt_toReal (measure_ne_top _ _) ENNReal.ofReal_ne_top]
+    simp only [ENNReal.toReal_ofReal hqd]
+    linarith
+  have hlsq : L.limsup (fun i ↦ (μs i : Measure Ω) Gᶜ) < q :=
+    hclosed.trans_lt hcompq
+  have hevcomp : ∀ᶠ i in L, (μs i : Measure Ω) Gᶜ < q :=
+    eventually_lt_of_limsup_lt hlsq
+  filter_upwards [hevmass, hevcomp] with i him hic
+  have hmassEq : ((μs i : Measure Ω) univ).toReal = ((μs i).mass : ℝ) := by
+    simp [← MeasureTheory.FiniteMeasure.ennreal_mass]
+  have hcompR : ((μs i : Measure Ω) Gᶜ).toReal <
+      ((μ : Measure Ω) Gᶜ).toReal + d := by
+    have h := (ENNReal.toReal_lt_toReal (measure_ne_top _ _)
+      ENNReal.ofReal_ne_top).mpr hic
+    simpa [q, ENNReal.toReal_ofReal hqd] using h
+  have hsplit : ((μs i : Measure Ω) G).toReal =
+      ((μs i).mass : ℝ) - ((μs i : Measure Ω) Gᶜ).toReal := by
+    rw [show (μs i : Measure Ω) G = (μs i : Measure Ω) univ -
+      (μs i : Measure Ω) Gᶜ by
+        rw [measure_compl hG.measurableSet (measure_ne_top _ _),
+          ENNReal.sub_sub_cancel (measure_ne_top _ _)
+            (measure_mono (subset_univ G))]]
+    rw [ENNReal.toReal_sub_of_le (measure_mono (subset_univ _))
+      (measure_ne_top _ _), hmassEq]
+  have hμsplit : ((μ : Measure Ω) G).toReal =
+      (μ.mass : ℝ) - ((μ : Measure Ω) Gᶜ).toReal := by
+    rw [show (μ : Measure Ω) G = (μ : Measure Ω) univ -
+      (μ : Measure Ω) Gᶜ by
+        rw [measure_compl hG.measurableSet (measure_ne_top _ _),
+          ENNReal.sub_sub_cancel (measure_ne_top _ _)
+            (measure_mono (subset_univ G))]]
+    rw [ENNReal.toReal_sub_of_le (measure_mono (subset_univ _))
+      (measure_ne_top _ _)]
+    simp [← MeasureTheory.FiniteMeasure.ennreal_mass]
+  apply (ENNReal.toReal_lt_toReal hyTop (measure_ne_top _ _)).mp
+  rw [hsplit]
+  dsimp [d] at him hcompR hd
+  linarith [hyR, hμsplit]
+
+end MeasureTheory.FiniteMeasure
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Finite Measure / Restriction
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter Set
+open scoped BoundedContinuousFunction ENNReal NNReal Topology
+
+namespace MeasureTheory.FiniteMeasure
+
+/-- Weak convergence of finite measures and their restrictions implies weak
+convergence of the complementary restrictions. -/
+theorem tendsto_restrict_compl_of_tendsto_restrict
+    {X ι : Type*} [TopologicalSpace X] [MeasurableSpace X] [OpensMeasurableSpace X]
+    {F : Filter ι} {μs : ι → FiniteMeasure X} {μ : FiniteMeasure X} {s : Set X}
+    (hs : MeasurableSet s) (hμ : Tendsto μs F (𝓝 μ))
+    (hsμ : Tendsto (fun n ↦ (μs n).restrict s) F (𝓝 (μ.restrict s))) :
+    Tendsto (fun n ↦ (μs n).restrict sᶜ) F (𝓝 (μ.restrict sᶜ)) := by
+  apply tendsto_iff_forall_integral_tendsto.mpr
+  intro f
+  have hi (ν : FiniteMeasure X) :
+      (∫ x, f x ∂(ν.restrict sᶜ : Measure X)) =
+        (∫ x, f x ∂(ν : Measure X)) - (∫ x, f x ∂(ν.restrict s : Measure X)) := by
+    have h := integral_add_compl hs (f.integrable (μ := (ν : Measure X)))
+    change (∫ x in sᶜ, f x ∂(ν : Measure X)) =
+      (∫ x, f x ∂(ν : Measure X)) - (∫ x in s, f x ∂(ν : Measure X))
+    linarith
+  simpa only [hi] using
+    ((tendsto_iff_forall_integral_tendsto.mp hμ) f).sub
+      ((tendsto_iff_forall_integral_tendsto.mp hsμ) f)
+
+end MeasureTheory.FiniteMeasure
+
+namespace MeasureTheory.FiniteMeasure
+
+/-- A continuity set has convergent masses under weak convergence of finite measures. -/
+theorem tendsto_apply_of_null_frontier
+    {X ι : Type*} [TopologicalSpace X] [MeasurableSpace X] [OpensMeasurableSpace X]
+    [HasOuterApproxClosed X] [Nonempty X]
+    {F : Filter ι} {μs : ι → FiniteMeasure X} {μ : FiniteMeasure X} {s : Set X}
+    (hμ : Tendsto μs F (𝓝 μ)) (hs : μ (frontier s) = 0) :
+    Tendsto (fun n ↦ μs n s) F (𝓝 (μ s)) := by
+  by_cases hzero : μ = 0
+  · subst μ
+    have hm := hμ.mass
+    simp only [zero_mass] at hm
+    have hlim := tendsto_of_tendsto_of_tendsto_of_le_of_le
+      (tendsto_const_nhds : Tendsto (fun _ : ι ↦ (0 : ℝ≥0)) F (𝓝 0)) hm
+      (fun _ ↦ zero_le) (fun n ↦ (μs n).apply_le_mass s)
+    simpa using hlim
+  · have hn := μ.tendsto_normalize_of_tendsto hμ hzero
+    have hs' : μ.normalize (frontier s) = 0 := by
+      rw [μ.normalize_eq_of_nonzero hzero, hs, mul_zero]
+    have hset := ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto hn hs'
+    simpa only [← self_eq_mass_mul_normalize] using hμ.mass.mul hset
+
+end MeasureTheory.FiniteMeasure
+
+namespace MeasureTheory.Measure
+
+/-- Equal atoms on a finite set give equal restrictions to that set. -/
+theorem restrict_finset_eq_of_singleton_eq
+    {X : Type*} [MeasurableSpace X] [MeasurableSingletonClass X]
+    {μ ν : Measure X} (s : Finset X) (h : ∀ x ∈ s, μ {x} = ν {x}) :
+    μ.restrict (s : Set X) = ν.restrict (s : Set X) := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simp
+  | @insert a s ha ih =>
+    have hd : Disjoint ({a} : Set X) (s : Set X) := by simpa using ha
+    have heq := ih (fun x hx ↦ h x (Finset.mem_insert_of_mem hx))
+    rw [Finset.coe_insert, ← Set.singleton_union]
+    rw [restrict_union hd s.measurableSet, restrict_union hd s.measurableSet]
+    rw [restrict_singleton, restrict_singleton, h a (Finset.mem_insert_self _ _), heq]
+
+end MeasureTheory.Measure
+
+namespace MeasureTheory.FiniteMeasure
+
+/-- Removing finitely many fixed atoms preserves weak convergence. -/
+theorem tendsto_restrict_compl_finset_of_singleton_eq
+    {X ι : Type*} [TopologicalSpace X] [MeasurableSpace X] [OpensMeasurableSpace X]
+    [MeasurableSingletonClass X] {F : Filter ι}
+    {μs : ι → FiniteMeasure X} {μ : FiniteMeasure X} (s : Finset X)
+    (hμ : Tendsto μs F (𝓝 μ))
+    (h : ∀ n x, x ∈ s → (μs n : Measure X) {x} = (μ : Measure X) {x}) :
+    Tendsto (fun n ↦ (μs n).restrict (s : Set X)ᶜ) F
+      (𝓝 (μ.restrict (s : Set X)ᶜ)) := by
+  apply tendsto_restrict_compl_of_tendsto_restrict s.measurableSet hμ
+  have heq (n : ι) : (μs n).restrict (s : Set X) = μ.restrict (s : Set X) :=
+    Subtype.ext (Measure.restrict_finset_eq_of_singleton_eq s (h n))
+  simpa only [heq] using
+    (tendsto_const_nhds : Tendsto (fun _ : ι ↦ μ.restrict (s : Set X)) F
+      (𝓝 (μ.restrict (s : Set X))))
+
+/-- A finite measure weighted by a bounded continuous nonnegative function. -/
+private def withDensityNN {X : Type*} [TopologicalSpace X] [MeasurableSpace X]
+    (μ : FiniteMeasure X) (g : X →ᵇ ℝ≥0) : FiniteMeasure X :=
+  ⟨(μ : Measure X).withDensity (fun x ↦ g x),
+    isFiniteMeasure_withDensity (g.lintegral_lt_top_of_nnreal μ).ne⟩
+
+/-- Weighting by a fixed bounded continuous nonnegative function preserves weak convergence. -/
+private theorem tendsto_withDensityNN
+    {X ι : Type*} [TopologicalSpace X] [MeasurableSpace X] [OpensMeasurableSpace X]
+    {F : Filter ι} {μs : ι → FiniteMeasure X} {μ : FiniteMeasure X}
+    (hμ : Tendsto μs F (𝓝 μ)) (g : X →ᵇ ℝ≥0) :
+    Tendsto (fun n ↦ withDensityNN (μs n) g) F (𝓝 (withDensityNN μ g)) := by
+  apply tendsto_iff_forall_lintegral_tendsto.mpr
+  intro f
+  have hprod := (tendsto_iff_forall_lintegral_tendsto.mp hμ) (g * f)
+  have h_lintegral (ν : FiniteMeasure X) :
+      (∫⁻ x, (f x : ℝ≥0∞) ∂(withDensityNN ν g : Measure X)) =
+        ∫⁻ x, ((g * f) x : ℝ≥0∞) ∂(ν : Measure X) := by
+    rw [withDensityNN, toMeasure_mk, lintegral_withDensity_eq_lintegral_mul]
+    · simp
+    · exact (ENNReal.continuous_coe.comp g.continuous).measurable
+    · exact (ENNReal.continuous_coe.comp f.continuous).measurable
+  simpa only [h_lintegral] using hprod
+
+/-- Weak convergence is preserved by restriction to a measurable continuity set. -/
+theorem tendsto_restrict_of_null_frontier
+    {X ι : Type*} [TopologicalSpace X] [MeasurableSpace X] [OpensMeasurableSpace X]
+    [HasOuterApproxClosed X] [Nonempty X]
+    {F : Filter ι} {μs : ι → FiniteMeasure X} {μ : FiniteMeasure X} {s : Set X}
+    (hs : MeasurableSet s) (hμ : Tendsto μs F (𝓝 μ))
+    (hfrontier : μ (frontier s) = 0) :
+    Tendsto (fun n ↦ (μs n).restrict s) F (𝓝 (μ.restrict s)) := by
+  apply tendsto_iff_forall_lintegral_tendsto.mpr
+  intro g
+  have hweighted := tendsto_withDensityNN hμ g
+  have hfrontier' : (withDensityNN μ g) (frontier s) = 0 := by
+    apply ENNReal.coe_eq_zero.mp
+    rw [ennreal_coeFn_eq_coeFn_toMeasure]
+    apply withDensity_absolutelyContinuous (μ : Measure X) (fun x ↦ g x)
+    simpa only [← ennreal_coeFn_eq_coeFn_toMeasure, ENNReal.coe_zero] using
+      congrArg ((↑) : ℝ≥0 → ℝ≥0∞) hfrontier
+  have hmass := tendsto_apply_of_null_frontier hweighted hfrontier'
+  have hmass_ennreal := (ENNReal.continuous_coe.tendsto _).comp hmass
+  change Tendsto (fun n ↦ (withDensityNN (μs n) g s : ℝ≥0∞)) F
+    (𝓝 (withDensityNN μ g s : ℝ≥0∞)) at hmass_ennreal
+  have h_lintegral (ν : FiniteMeasure X) :
+      (withDensityNN ν g s : ℝ≥0∞) =
+        ∫⁻ x in s, (g x : ℝ≥0∞) ∂(ν : Measure X) := by
+    rw [ennreal_coeFn_eq_coeFn_toMeasure, withDensityNN, toMeasure_mk, withDensity_apply _ hs]
+  simpa only [h_lintegral, restrict_measure_eq] using hmass_ennreal
+
+/-- Fixed atoms on a finite set containing the frontier allow weak restriction convergence. -/
+theorem tendsto_restrict_of_frontier_subset_finset
+    {X ι : Type*} [TopologicalSpace X] [MeasurableSpace X] [OpensMeasurableSpace X]
+    [HasOuterApproxClosed X] [Nonempty X] [MeasurableSingletonClass X]
+    {F : Filter ι} {μs : ι → FiniteMeasure X} {μ : FiniteMeasure X} {s : Set X}
+    (hs : MeasurableSet s) (t : Finset X) (hfrontier : frontier s ⊆ (t : Set X))
+    (hμ : Tendsto μs F (𝓝 μ))
+    (hatoms : ∀ n x, x ∈ t → (μs n : Measure X) {x} = (μ : Measure X) {x}) :
+    Tendsto (fun n ↦ (μs n).restrict s) F (𝓝 (μ.restrict s)) := by
+  have hremoved := tendsto_restrict_compl_finset_of_singleton_eq t hμ hatoms
+  have hzero : (μ.restrict (t : Set X)ᶜ) (frontier s) = 0 := by
+    apply (null_iff_toMeasure_null _ _).mpr
+    rw [restrict_measure_eq, Measure.restrict_apply measurableSet_frontier]
+    have he : frontier s ∩ (t : Set X)ᶜ = ∅ := by
+      apply Set.eq_empty_iff_forall_notMem.mpr
+      intro x hx
+      exact hx.2 (hfrontier hx.1)
+    rw [he, measure_empty]
+  have hrestricted := tendsto_restrict_of_null_frontier hs hremoved hzero
+  have hfixed (n : ι) : ((μs n).restrict (t : Set X)).restrict s =
+      (μ.restrict (t : Set X)).restrict s := by
+    congr 1
+    exact Subtype.ext (Measure.restrict_finset_eq_of_singleton_eq t (hatoms n))
+  have hsplit (ν : FiniteMeasure X) : ν.restrict s =
+      (ν.restrict (t : Set X)ᶜ).restrict s + (ν.restrict (t : Set X)).restrict s := by
+    apply Subtype.ext
+    change (ν : Measure X).restrict s =
+      ((ν : Measure X).restrict (t : Set X)ᶜ).restrict s +
+        ((ν : Measure X).restrict (t : Set X)).restrict s
+    rw [← Measure.restrict_add]
+    congr 1
+    rw [add_comm, Measure.restrict_add_restrict_compl t.measurableSet]
+  have hconst : Tendsto (fun n ↦ ((μs n).restrict (t : Set X)).restrict s) F
+      (𝓝 ((μ.restrict (t : Set X)).restrict s)) := by
+    simpa only [hfixed] using
+      (tendsto_const_nhds : Tendsto (fun _ : ι ↦ (μ.restrict (t : Set X)).restrict s) F _)
+  simpa only [← hsplit] using hrestricted.add hconst
+
+end MeasureTheory.FiniteMeasure
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Angle
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter Set MeasureTheory
+open scoped Topology BoundedContinuousFunction
+
+namespace Real.Angle
+
+/-- A half-open interval of at most one turn has distinct angular representatives. -/
+theorem injOn_coe_Ioc {a b : ℝ} (h : b ≤ a + 2 * Real.pi) :
+    Set.InjOn (fun t : ℝ ↦ (t : Angle)) (Ioc a b) := by
+  let : Fact (0 < 2 * Real.pi) := ⟨mul_pos (by norm_num) Real.pi_pos⟩
+  intro x hx y hy hxy
+  exact (AddCircle.coe_eq_coe_iff_of_mem_Ioc
+    ⟨hx.1, hx.2.trans h⟩ ⟨hy.1, hy.2.trans h⟩).mp hxy
+
+/-- Every fibre of the angular projection is countable, being a full residue class. -/
+theorem countable_preimage_coe_singleton (x : Angle) :
+    ((fun s : ℝ ↦ (s : Angle)) ⁻¹' {x}).Countable := by
+  refine Set.Countable.mono ?_
+    (Set.countable_range fun k : ℤ ↦ x.toReal + 2 * Real.pi * (k : ℝ))
+  intro y hy
+  have hy' : ((y : ℝ) : Angle) = ((x.toReal : ℝ) : Angle) := by
+    rw [mem_preimage, mem_singleton_iff] at hy
+    rw [hy, coe_toReal]
+  obtain ⟨k, hk⟩ := angle_eq_iff_two_pi_dvd_sub.mp hy'
+  exact ⟨k, by linarith [hk]⟩
+
+/-- The quotient map sends an open real interval to an open angular arc. -/
+theorem isOpen_image_Ioo (a b : ℝ) :
+    IsOpen ((fun t : ℝ ↦ (t : Angle)) '' Ioo a b) := by
+  exact QuotientAddGroup.isOpenMap_coe _ isOpen_Ioo
+
+/-- The frontier of an angular arc is contained in its two endpoint angles. -/
+theorem frontier_image_Ioo_subset (a b : ℝ) :
+    frontier ((fun t : ℝ ↦ (t : Angle)) '' Ioo a b) ⊆
+      {(a : Angle), (b : Angle)} := by
+  intro x hx
+  have hc : IsClosed ((fun t : ℝ ↦ (t : Angle)) '' Icc a b) :=
+    (isCompact_Icc.image continuous_coe).isClosed
+  have hx' : x ∈ (fun t : ℝ ↦ (t : Angle)) '' Icc a b :=
+    closure_minimal (image_mono Ioo_subset_Icc_self) hc hx.1
+  obtain ⟨t, ht, rfl⟩ := hx'
+  have hout : (t : Angle) ∉ (fun t : ℝ ↦ (t : Angle)) '' Ioo a b := by
+    simpa only [(isOpen_image_Ioo a b).interior_eq] using hx.2
+  have hends : t = a ∨ t = b := by
+    by_contra h
+    push Not at h
+    exact hout ⟨t, ⟨lt_of_le_of_ne ht.1 (Ne.symm h.1),
+      lt_of_le_of_ne ht.2 h.2⟩, rfl⟩
+  rcases hends with rfl | rfl <;> simp
+
+/-- A nonempty half-open arc consists of its open arc and its terminal angle. -/
+theorem image_Ioc_eq_image_Ioo_union {a b : ℝ} (hab : a < b) :
+    (fun t : ℝ ↦ (t : Angle)) '' Ioc a b =
+      (fun t : ℝ ↦ (t : Angle)) '' Ioo a b ∪ {(b : Angle)} := by
+  rw [← Ioo_union_right hab, image_union, image_singleton]
+
+/-- The frontier of a nonempty half-open angular arc lies in its two endpoint angles. -/
+theorem frontier_image_Ioc_subset {a b : ℝ} (hab : a < b) :
+    frontier ((fun t : ℝ ↦ (t : Angle)) '' Ioc a b) ⊆
+      {(a : Angle), (b : Angle)} := by
+  rw [image_Ioc_eq_image_Ioo_union hab]
+  refine (frontier_union_subset _ _).trans ?_
+  refine union_subset ?_ ?_
+  · exact inter_subset_left.trans (frontier_image_Ioo_subset a b)
+  · refine inter_subset_right.trans (frontier_subset_closure.trans ?_)
+    simp
+
+/-- Half-open real intervals have measurable angular images. -/
+theorem measurableSet_image_Ioc [MeasurableSpace Angle] [BorelSpace Angle] (a b : ℝ) :
+    MeasurableSet ((fun t : ℝ ↦ (t : Angle)) '' Ioc a b) := by
+  by_cases hab : a < b
+  · rw [image_Ioc_eq_image_Ioo_union hab]
+    exact (isOpen_image_Ioo a b).measurableSet.union (measurableSet_singleton _)
+  · simp [Ioc_eq_empty_of_le (le_of_not_gt hab)]
+
+/-- A Borel subset of a real interval of at most one turn has a Borel angular image. -/
+theorem measurableSet_image_of_subset_Ioc [MeasurableSpace Angle] [BorelSpace Angle]
+    {a b : ℝ} (hab : b ≤ a + 2 * Real.pi) {E : Set ℝ} (hE : MeasurableSet E)
+    (hEab : E ⊆ Ioc a b) :
+    MeasurableSet ((fun t : ℝ ↦ (t : Angle)) '' E) :=
+  hE.image_of_continuousOn_injOn continuous_coe.continuousOn ((injOn_coe_Ioc hab).mono hEab)
+
+/-- The preimage of an open angular arc under the half-turn shift is the shifted arc. -/
+theorem preimage_sub_pi_image_Ioo (a b : ℝ) :
+    (fun t : Angle ↦ t - ((Real.pi : ℝ) : Angle)) ⁻¹'
+        ((fun s : ℝ ↦ (s : Angle)) '' Ioo a b) =
+      (fun s : ℝ ↦ (s : Angle)) '' Ioo (a + Real.pi) (b + Real.pi) := by
+  ext x
+  simp only [mem_preimage, mem_image, mem_Ioo]
+  constructor
+  · rintro ⟨s, hs, hsx⟩
+    refine ⟨s + Real.pi, ⟨by linarith [hs.1], by linarith [hs.2]⟩, ?_⟩
+    rw [coe_add, hsx]
+    abel
+  · rintro ⟨r, hr, hrx⟩
+    refine ⟨r - Real.pi, ⟨by linarith [hr.1], by linarith [hr.2]⟩, ?_⟩
+    rw [coe_sub, hrx]
+
+/-- The terminal atom is disjoint from the open arc, even for a full turn. -/
+theorem disjoint_image_Ioo_singleton {a b : ℝ} (h : b ≤ a + 2 * Real.pi) :
+    Disjoint ((fun t : ℝ ↦ (t : Angle)) '' Ioo a b) {(b : Angle)} := by
+  rw [Set.disjoint_singleton_right]
+  rintro ⟨t, ht, heq⟩
+  have htb := injOn_coe_Ioc h ⟨ht.1, ht.2.le⟩ ⟨ht.1.trans ht.2, le_rfl⟩ heq
+  exact ht.2.ne htb
+
+/-- Passing from an open arc to a half-open arc restores exactly its terminal atom. -/
+theorem integral_image_Ioc [MeasurableSpace Angle] [BorelSpace Angle]
+    (μ : MeasureTheory.FiniteMeasure Angle) (f : Angle →ᵇ ℝ)
+    {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi) :
+    (∫ x in (fun t : ℝ ↦ (t : Angle)) '' Ioc a b, f x ∂(μ : MeasureTheory.Measure Angle)) =
+      (∫ x in (fun t : ℝ ↦ (t : Angle)) '' Ioo a b, f x ∂(μ : MeasureTheory.Measure Angle)) +
+        (μ : MeasureTheory.Measure Angle).real {(b : Angle)} * f (b : Angle) := by
+  rw [image_Ioc_eq_image_Ioo_union hab, MeasureTheory.setIntegral_union
+    (disjoint_image_Ioo_singleton hturn) (measurableSet_singleton _)
+    (f.integrable (μ := (μ : Measure Angle))).integrableOn
+    (f.integrable (μ := (μ : Measure Angle))).integrableOn]
+  rw [MeasureTheory.integral_singleton]
+  rfl
+
+/-- Weak convergence with fixed endpoint atoms preserves integrals on half-open angular arcs. -/
+theorem tendsto_integral_image_Ioc_of_fixed_endpoint_atoms
+    [MeasurableSpace Angle] [BorelSpace Angle]
+    {ι : Type*} {F : Filter ι} {μs : ι → FiniteMeasure Angle} {μ : FiniteMeasure Angle}
+    {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi)
+    (hμ : Tendsto μs F (𝓝 μ))
+    (ha : ∀ n, (μs n : Measure Angle) {(a : Angle)} = (μ : Measure Angle) {(a : Angle)})
+    (hb : ∀ n, (μs n : Measure Angle) {(b : Angle)} = (μ : Measure Angle) {(b : Angle)})
+    (f : Angle →ᵇ ℝ) :
+    Tendsto (fun n ↦ ∫ x in (fun t : ℝ ↦ (t : Angle)) '' Ioc a b,
+      f x ∂(μs n : Measure Angle)) F
+      (𝓝 (∫ x in (fun t : ℝ ↦ (t : Angle)) '' Ioc a b, f x ∂(μ : Measure Angle))) := by
+  classical
+  have hatoms : ∀ n x, x ∈ ({(a : Angle), (b : Angle)} : Finset Angle) →
+      (μs n : Measure Angle) {x} = (μ : Measure Angle) {x} := by
+    intro n x hx
+    simp only [Finset.mem_insert, Finset.mem_singleton] at hx
+    rcases hx with rfl | rfl
+    · exact ha n
+    · exact hb n
+  have hres := FiniteMeasure.tendsto_restrict_of_frontier_subset_finset
+    (isOpen_image_Ioo a b).measurableSet ({(a : Angle), (b : Angle)} : Finset Angle)
+    (by simpa using frontier_image_Ioo_subset a b) hμ hatoms
+  have hint := FiniteMeasure.tendsto_iff_forall_integral_tendsto.mp hres f
+  have hatom (n : ι) : (μs n : Measure Angle).real {(b : Angle)} =
+      (μ : Measure Angle).real {(b : Angle)} := by
+    exact congrArg ENNReal.toReal (hb n)
+  simpa only [integral_image_Ioc _ _ hab hturn, hatom,
+    FiniteMeasure.restrict_measure_eq] using
+      hint.add (tendsto_const_nhds (x := (μ : Measure Angle).real {(b : Angle)} * f (b : Angle)))
+
+/-! ### Integration over the circle of directions -/
+
+/-- A continuous function of a direction is integrable for every finite angular measure, the
+circle of directions being compact. -/
+theorem integrable_of_continuous [MeasurableSpace Angle] [BorelSpace Angle] {E : Type*}
+    [NormedAddCommGroup E] {μ : Measure Angle} [IsFiniteMeasure μ] {f : Angle → E}
+    (hf : Continuous f) : Integrable f μ := by
+  have : Fact (0 < 2 * Real.pi) := ⟨mul_pos (by norm_num) Real.pi_pos⟩
+  have : CompactSpace Angle := inferInstanceAs (CompactSpace (AddCircle (2 * Real.pi)))
+  exact hf.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace f)
+
+/-! ### Angular measures reading a real density -/
+
+/-- On a real window of at most one turn, the pushforward of a weighted measure along the
+angular projection gives the angular image of a measurable subset the integral of the weight
+over that subset. -/
+theorem map_coe_withDensity_image_eq_setLIntegral [MeasurableSpace Angle] [BorelSpace Angle]
+    {ν : Measure ℝ} {w : ℝ → ENNReal} {S T : Set ℝ} {a b : ℝ}
+    (hturn : b ≤ a + 2 * Real.pi) (hS : S ⊆ Ioc a b) (hT : MeasurableSet T) (hTS : T ⊆ S) :
+    Measure.map (fun t : ℝ ↦ (t : Angle)) ((ν.restrict S).withDensity w)
+        ((fun t : ℝ ↦ (t : Angle)) '' T) = ∫⁻ t in T, w t ∂ν := by
+  have hmeas : Measurable fun t : ℝ ↦ (t : Angle) := continuous_coe.measurable
+  have himage : MeasurableSet ((fun t : ℝ ↦ (t : Angle)) '' T) :=
+    measurableSet_image_of_subset_Ioc hturn hT (hTS.trans hS)
+  have hpre : MeasurableSet ((fun t : ℝ ↦ (t : Angle)) ⁻¹'
+      ((fun t : ℝ ↦ (t : Angle)) '' T)) := himage.preimage hmeas
+  have hback : (fun t : ℝ ↦ (t : Angle)) ⁻¹' ((fun t : ℝ ↦ (t : Angle)) '' T) ∩ S = T := by
+    refine Subset.antisymm ?_ fun x hx ↦ ⟨mem_image_of_mem _ hx, hTS hx⟩
+    rintro x ⟨⟨y, hy, hxy⟩, hxS⟩
+    exact injOn_coe_Ioc hturn (hS (hTS hy)) (hS hxS) hxy ▸ hy
+  rw [Measure.map_apply hmeas himage, withDensity_apply _ hpre, Measure.restrict_restrict hpre,
+    hback]
+
+/-- Two angular measures that agree on the angular image of every measurable subset of a real
+window agree after restriction to that window's angular image. -/
+theorem measure_restrict_image_congr [MeasurableSpace Angle] [BorelSpace Angle]
+    {μ ν : Measure Angle} {J : Set ℝ} (hJ : MeasurableSet J)
+    (h : ∀ T, MeasurableSet T → T ⊆ J →
+      μ ((fun t : ℝ ↦ (t : Angle)) '' T) = ν ((fun t : ℝ ↦ (t : Angle)) '' T)) :
+    μ.restrict ((fun t : ℝ ↦ (t : Angle)) '' J) =
+      ν.restrict ((fun t : ℝ ↦ (t : Angle)) '' J) := by
+  have hmeas : Measurable fun t : ℝ ↦ (t : Angle) := continuous_coe.measurable
+  ext A hA
+  have hpre : MeasurableSet ((fun t : ℝ ↦ (t : Angle)) ⁻¹' A) := hA.preimage hmeas
+  rw [Measure.restrict_apply hA, Measure.restrict_apply hA, inter_comm,
+    ← image_inter_preimage, h _ (hJ.inter hpre) inter_subset_left]
+
+/-- Two angular measures reading extended-real weights on a real window agree after restriction
+to its angular image as soon as the weights agree almost everywhere on the window. -/
+theorem measure_restrict_image_congr_of_ae_eq [MeasurableSpace Angle] [BorelSpace Angle]
+    {μ ν : Measure Angle} {ρ : Measure ℝ} {w w' : ℝ → ENNReal} {J : Set ℝ}
+    (hJ : MeasurableSet J)
+    (hμ : ∀ T, MeasurableSet T → T ⊆ J →
+      μ ((fun t : ℝ ↦ (t : Angle)) '' T) = ∫⁻ t in T, w t ∂ρ)
+    (hν : ∀ T, MeasurableSet T → T ⊆ J →
+      ν ((fun t : ℝ ↦ (t : Angle)) '' T) = ∫⁻ t in T, w' t ∂ρ)
+    (hw : ∀ᵐ t ∂ρ.restrict J, w t = w' t) :
+    μ.restrict ((fun t : ℝ ↦ (t : Angle)) '' J) =
+      ν.restrict ((fun t : ℝ ↦ (t : Angle)) '' J) := by
+  refine measure_restrict_image_congr hJ fun T hT hTJ ↦ ?_
+  rw [hμ T hT hTJ, hν T hT hTJ]
+  exact lintegral_congr_ae (ae_restrict_of_ae_restrict_of_subset hTJ hw)
+
+/-- An angular measure reading a real density on a window agrees, after restriction to the
+window's angular image, with a sum of two angular measures whose densities add up to it almost
+everywhere. -/
+theorem measure_restrict_image_congr_add [MeasurableSpace Angle] [BorelSpace Angle]
+    {μ ν₁ ν₂ : Measure Angle} {ρ : Measure ℝ} {f g h : ℝ → ℝ} {J : Set ℝ}
+    (hJ : MeasurableSet J)
+    (hμ : ∀ T, MeasurableSet T → T ⊆ J →
+      μ ((fun t : ℝ ↦ (t : Angle)) '' T) = ∫⁻ t in T, ENNReal.ofReal (h t) ∂ρ)
+    (hν₁ : ∀ T, MeasurableSet T → T ⊆ J →
+      ν₁ ((fun t : ℝ ↦ (t : Angle)) '' T) = ∫⁻ t in T, ENNReal.ofReal (f t) ∂ρ)
+    (hν₂ : ∀ T, MeasurableSet T → T ⊆ J →
+      ν₂ ((fun t : ℝ ↦ (t : Angle)) '' T) = ∫⁻ t in T, ENNReal.ofReal (g t) ∂ρ)
+    (hf : AEMeasurable f (ρ.restrict J)) (hf0 : ∀ᵐ t ∂ρ.restrict J, 0 ≤ f t)
+    (hg0 : ∀ᵐ t ∂ρ.restrict J, 0 ≤ g t) (hfg : ∀ᵐ t ∂ρ.restrict J, h t = f t + g t) :
+    μ.restrict ((fun t : ℝ ↦ (t : Angle)) '' J) =
+      (ν₁ + ν₂).restrict ((fun t : ℝ ↦ (t : Angle)) '' J) := by
+  refine measure_restrict_image_congr hJ fun T hT hTJ ↦ ?_
+  have hsplit : ∀ᵐ t ∂ρ.restrict T,
+      ENNReal.ofReal (h t) = ENNReal.ofReal (f t) + ENNReal.ofReal (g t) := by
+    filter_upwards [ae_restrict_of_ae_restrict_of_subset hTJ hf0,
+      ae_restrict_of_ae_restrict_of_subset hTJ hg0,
+      ae_restrict_of_ae_restrict_of_subset hTJ hfg] with t h1 h2 h3
+    rw [h3, ENNReal.ofReal_add h1 h2]
+  have hfT : AEMeasurable (fun t ↦ ENNReal.ofReal (f t)) (ρ.restrict T) :=
+    ENNReal.measurable_ofReal.comp_aemeasurable
+      (hf.mono_measure (Measure.restrict_mono hTJ le_rfl))
+  rw [hμ T hT hTJ, Measure.add_apply, hν₁ T hT hTJ, hν₂ T hT hTJ, lintegral_congr_ae hsplit,
+    lintegral_add_left' hfT]
+
+end Real.Angle
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Hausdorff / Arclength
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter
+open scoped Topology
+
+open MeasureTheory
+
+namespace MeasureTheory
+
+private theorem dist_le_hausdorffMeasure_image_Icc_of_continuousOn
+    (γ : ℝ → (EuclideanSpace ℝ (Fin 2))) {a b : ℝ} (hab : a ≤ b) (hγ : ContinuousOn γ (Set.Icc a
+      b)) :
+    ENNReal.ofReal (dist (γ a) (γ b)) ≤
+      Measure.hausdorffMeasure 1 (γ '' Set.Icc a b) := by
+  by_cases heq : γ a = γ b
+  · simp [heq]
+  let n : (EuclideanSpace ℝ (Fin 2)) := ‖γ b - γ a‖⁻¹ • (γ b - γ a)
+  let f : (EuclideanSpace ℝ (Fin 2)) → ℝ := fun p ↦ inner ℝ (p - γ a) n
+  have hnorm : ‖n‖ = 1 := by
+    rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_norm,
+      inv_mul_cancel₀ (norm_ne_zero_iff.mpr (sub_ne_zero.mpr (Ne.symm heq)))]
+  have hf : LipschitzWith 1 f := by
+    apply LipschitzWith.of_dist_le_mul
+    intro p q
+    rw [NNReal.coe_one, one_mul, Real.dist_eq]
+    change |inner ℝ (p - γ a) n - inner ℝ (q - γ a) n| ≤ dist p q
+    rw [← inner_sub_left]
+    have h := abs_real_inner_le_norm ((p - γ a) - (q - γ a)) n
+    rw [hnorm, mul_one, show (p - γ a) - (q - γ a) = p - q by module,
+      ← dist_eq_norm] at h
+    simpa [sub_eq_add_neg, add_assoc, add_left_comm, add_comm] using h
+  have hfa : f (γ a) = 0 := by simp [f]
+  have hfb : f (γ b) = dist (γ a) (γ b) := by
+    dsimp only [f, n]
+    rw [inner_smul_right, real_inner_self_eq_norm_sq, dist_eq_norm, inv_mul_eq_div,
+      norm_sub_rev]
+    field_simp [norm_ne_zero_iff.mpr (sub_ne_zero.mpr (Ne.symm heq))]
+  have hinterval : Set.Icc 0 (dist (γ a) (γ b)) ⊆ f '' (γ '' Set.Icc a b) := by
+    rw [← hfa, ← hfb]
+    intro y hy
+    obtain ⟨x, hx, hxy⟩ := intermediate_value_Icc hab (hf.continuous.comp_continuousOn hγ) hy
+    exact ⟨γ x, ⟨x, hx, rfl⟩, hxy⟩
+  calc
+    ENNReal.ofReal (dist (γ a) (γ b)) =
+        Measure.hausdorffMeasure 1 (Set.Icc 0 (dist (γ a) (γ b))) := by
+      rw [MeasureTheory.hausdorffMeasure_real, Real.volume_Icc]
+      simp
+    _ ≤ Measure.hausdorffMeasure 1 (f '' (γ '' Set.Icc a b)) := measure_mono hinterval
+    _ ≤ ((1 : NNReal) : ENNReal) ^ (1 : ℝ) *
+        Measure.hausdorffMeasure 1 (γ '' Set.Icc a b) :=
+      hf.hausdorffMeasure_image_le (by norm_num) _
+    _ = Measure.hausdorffMeasure 1 (γ '' Set.Icc a b) := by simp
+
+private theorem hausdorffMeasure_image_Icc_add_of_injectiveOn
+    (γ : ℝ → (EuclideanSpace ℝ (Fin 2))) {a b c : ℝ} (hac : a ≤ c) (hcb : c ≤ b)
+    (hγ : ContinuousOn γ (Set.Icc a b))
+    (hinj : Set.InjOn γ (Set.Icc a b)) :
+    Measure.hausdorffMeasure 1 (γ '' Set.Icc a b) =
+      Measure.hausdorffMeasure 1 (γ '' Set.Icc a c) +
+        Measure.hausdorffMeasure 1 (γ '' Set.Icc c b) := by
+  let μ : Measure (EuclideanSpace ℝ (Fin 2)) := Measure.hausdorffMeasure 1
+  have hleft : Set.Icc a c ⊆ Set.Icc a b := Set.Icc_subset_Icc_right hcb
+  have hright : Set.Icc c b ⊆ Set.Icc a b := Set.Icc_subset_Icc_left hac
+  have hinter : γ '' Set.Icc a c ∩ γ '' Set.Icc c b ⊆ {γ c} := by
+    rintro p ⟨⟨x, hx, rfl⟩, ⟨y, hy, hxy⟩⟩
+    have hxeq : x = y := hinj (hleft hx) (hright hy) hxy.symm
+    have : x = c := le_antisymm hx.2 (hxeq ▸ hy.1)
+    simp [this]
+  have hnull : μ (γ '' Set.Icc a c ∩ γ '' Set.Icc c b) = 0 := by
+    let _ := Measure.nullSingletonClass_hausdorff (EuclideanSpace ℝ (Fin 2))
+      (by norm_num : (0 : ℝ) < 1)
+    exact measure_mono_null hinter (measure_singleton (γ c))
+  have hmeas : MeasurableSet (γ '' Set.Icc c b) :=
+    (isCompact_Icc.image_of_continuousOn (hγ.mono hright)).measurableSet
+  have hunion : γ '' Set.Icc a b = γ '' Set.Icc a c ∪ γ '' Set.Icc c b := by
+    rw [← Set.image_union, Set.Icc_union_Icc_eq_Icc hac hcb]
+  rw [hunion]
+  exact measure_union₀ hmeas.nullMeasurableSet hnull
+
+private theorem hausdorff_partition_sum_le (γ : ℝ → (EuclideanSpace ℝ (Fin 2))) (u : ℕ → ℝ) (n : ℕ)
+    (hu : Monotone u) (hγ : ContinuousOn γ (Set.Icc (u 0) (u n)))
+    (hinj : Set.InjOn γ (Set.Icc (u 0) (u n))) :
+    (∑ i ∈ Finset.range n, edist (γ (u (i + 1))) (γ (u i))) ≤
+      Measure.hausdorffMeasure 1 (γ '' Set.Icc (u 0) (u n)) := by
+  induction n generalizing u with
+  | zero => simp
+  | succ n ih =>
+      let v : ℕ → ℝ := fun i ↦ u (i + 1)
+      have huv : u 0 ≤ u 1 := hu (Nat.zero_le 1)
+      have hvn : u 1 ≤ u (n + 1) := hu (Nat.succ_le_succ (Nat.zero_le n))
+      have hwhole : u 0 ≤ u (n + 1) := huv.trans hvn
+      have hleft : Set.Icc (u 0) (u 1) ⊆ Set.Icc (u 0) (u (n + 1)) :=
+        Set.Icc_subset_Icc_right hvn
+      have hright : Set.Icc (u 1) (u (n + 1)) ⊆ Set.Icc (u 0) (u (n + 1)) :=
+        Set.Icc_subset_Icc_left huv
+      rw [Finset.sum_range_succ']
+      have htail : (∑ i ∈ Finset.range n,
+          edist (γ (u (i + 1 + 1))) (γ (u (i + 1)))) ≤
+          Measure.hausdorffMeasure 1 (γ '' Set.Icc (u 1) (u (n + 1))) := by
+        simpa [v, Nat.add_assoc] using ih v (fun _ _ hij ↦ hu (Nat.add_le_add_right hij 1))
+          (hγ.mono hright) (hinj.mono hright)
+      have hfirst := dist_le_hausdorffMeasure_image_Icc_of_continuousOn γ huv
+        (hγ.mono hleft)
+      rw [edist_dist]
+      rw [hausdorffMeasure_image_Icc_add_of_injectiveOn γ huv hvn hγ hinj]
+      simpa [add_comm, dist_comm] using add_le_add htail hfirst
+
+private theorem eVariationOn_le_hausdorffMeasure_image_Icc_of_injectiveOn
+    (γ : ℝ → (EuclideanSpace ℝ (Fin 2))) {a b : ℝ} (hγ : ContinuousOn γ (Set.Icc a b))
+    (hinj : Set.InjOn γ (Set.Icc a b)) :
+    eVariationOn γ (Set.Icc a b) ≤
+      Measure.hausdorffMeasure 1 (γ '' Set.Icc a b) := by
+  rw [eVariationOn]
+  refine iSup_le fun p ↦ ?_
+  rcases p with ⟨n, u, hu, hus⟩
+  have hu0 := hus 0
+  have hun := hus n
+  have hsub : Set.Icc (u 0) (u n) ⊆ Set.Icc a b :=
+    Set.Icc_subset_Icc hu0.1 hun.2
+  calc
+    (∑ i ∈ Finset.range n, edist (γ (u (i + 1))) (γ (u i))) ≤
+        Measure.hausdorffMeasure 1 (γ '' Set.Icc (u 0) (u n)) :=
+      hausdorff_partition_sum_le γ u n hu (hγ.mono hsub) (hinj.mono hsub)
+    _ ≤ Measure.hausdorffMeasure 1 (γ '' Set.Icc a b) :=
+      measure_mono (Set.image_mono hsub)
+
+private theorem hausdorffMeasure_image_Icc_le_eVariationOn_of_injectiveOn
+    (γ : ℝ → (EuclideanSpace ℝ (Fin 2))) {a b : ℝ} (hab : a ≤ b)
+    (hinj : Set.InjOn γ (Set.Icc a b))
+    (hBV : BoundedVariationOn γ (Set.Icc a b)) :
+    Measure.hausdorffMeasure 1 (γ '' Set.Icc a b) ≤
+      eVariationOn γ (Set.Icc a b) := by
+  let ℓ : ℝ → ℝ := variationOnFromTo γ (Set.Icc a b) a
+  have hℓstrict : StrictMonoOn ℓ (Set.Icc a b) := by
+    intro x hx y hy hxy
+    have hnonneg : 0 ≤ variationOnFromTo γ (Set.Icc a b) x y :=
+      variationOnFromTo.nonneg_of_le γ _ hxy.le
+    have hadd := variationOnFromTo.add hBV.locallyBoundedVariationOn
+      (Set.left_mem_Icc.mpr hab) hx hy
+    have hle : ℓ x ≤ ℓ y := by dsimp only [ℓ]; linarith
+    refine lt_of_le_of_ne hle ?_
+    intro heq
+    have hzero : variationOnFromTo γ (Set.Icc a b) x y = 0 := by
+      dsimp only [ℓ] at heq
+      linarith
+    have hed := variationOnFromTo.edist_zero_of_eq_zero hBV.locallyBoundedVariationOn
+      hx hy hzero
+    have hxy' : γ x = γ y := edist_eq_zero.mp hed
+    exact hxy.ne (hinj hx hy hxy')
+  let iso := hℓstrict.orderIso ℓ (Set.Icc a b)
+  let Γ : (ℓ '' Set.Icc a b) → (EuclideanSpace ℝ (Fin 2)) := fun z ↦ γ (iso.symm z)
+  have hΓle : ∀ z w : (ℓ '' Set.Icc a b), z ≤ w →
+      dist (Γ z) (Γ w) ≤ dist z w := by
+    intro z w hzw
+    have hxy : (iso.symm z : ℝ) ≤ iso.symm w := iso.symm.monotone hzw
+    have hvar : dist (γ (iso.symm z)) (γ (iso.symm w)) ≤
+        variationOnFromTo γ (Set.Icc a b) (iso.symm z) (iso.symm w) := by
+      rw [variationOnFromTo.eq_of_le _ _ hxy, dist_edist]
+      apply ENNReal.toReal_mono
+        (hBV.locallyBoundedVariationOn _ _ (iso.symm z).property (iso.symm w).property)
+      exact eVariationOn.edist_le γ
+        ⟨(iso.symm z).property, le_rfl, hxy⟩
+        ⟨(iso.symm w).property, hxy, le_rfl⟩
+    have hadd := variationOnFromTo.add hBV.locallyBoundedVariationOn
+      (Set.left_mem_Icc.mpr hab) (iso.symm z).property (iso.symm w).property
+    change dist (γ (iso.symm z)) (γ (iso.symm w)) ≤ dist (z : ℝ) (w : ℝ)
+    rw [Real.dist_eq, abs_of_nonpos (sub_nonpos.mpr (show (z : ℝ) ≤ (w : ℝ) from hzw))]
+    have hz : ℓ (iso.symm z) = z := congrArg Subtype.val (iso.apply_symm_apply z)
+    have hw : ℓ (iso.symm w) = w := congrArg Subtype.val (iso.apply_symm_apply w)
+    dsimp only [ℓ] at hz hw
+    linarith
+  have hΓ : LipschitzWith 1 Γ := by
+    apply LipschitzWith.of_dist_le_mul
+    intro z w
+    simp only [NNReal.coe_one, one_mul]
+    rcases le_total z w with hzw | hwz
+    · exact hΓle z w hzw
+    · simpa only [dist_comm] using hΓle w z hwz
+  have himage : Γ '' Set.univ = γ '' Set.Icc a b := by
+    ext p
+    constructor
+    · rintro ⟨z, _, rfl⟩
+      exact ⟨iso.symm z, (iso.symm z).property, rfl⟩
+    · rintro ⟨x, hx, rfl⟩
+      refine ⟨iso ⟨x, hx⟩, Set.mem_univ _, ?_⟩
+      simp [Γ]
+  have hval : (Subtype.val : (ℓ '' Set.Icc a b) → ℝ) '' Set.univ =
+      ℓ '' Set.Icc a b := by
+    ext x
+    constructor
+    · rintro ⟨z, _, rfl⟩; exact z.property
+    · intro hx; exact ⟨⟨x, hx⟩, Set.mem_univ _, rfl⟩
+  have hdom : Measure.hausdorffMeasure 1 (Set.univ : Set (ℓ '' Set.Icc a b)) =
+      Measure.hausdorffMeasure 1 (ℓ '' Set.Icc a b) := by
+    conv_rhs => rw [← hval]
+    have hi : Isometry (Subtype.val : (ℓ '' Set.Icc a b) → ℝ) := fun _ _ ↦ rfl
+    exact (hi.hausdorffMeasure_image (Or.inl (by norm_num : (0 : ℝ) ≤ 1)) _).symm
+  have hsub : ℓ '' Set.Icc a b ⊆
+      Set.Icc 0 (eVariationOn γ (Set.Icc a b)).toReal := by
+    rintro _ ⟨x, hx, rfl⟩
+    exact ⟨variationOnFromTo.nonneg_of_le γ _ hx.1,
+      (le_abs_self _).trans (variationOnFromTo.abs_le_eVariationOn hBV)⟩
+  calc
+    Measure.hausdorffMeasure 1 (γ '' Set.Icc a b) =
+        Measure.hausdorffMeasure 1 (Γ '' Set.univ) := congrArg _ himage.symm
+    _ ≤ Measure.hausdorffMeasure 1 (Set.univ : Set (ℓ '' Set.Icc a b)) := by
+      simpa using hΓ.hausdorffMeasure_image_le (by norm_num : (0 : ℝ) ≤ 1) Set.univ
+    _ = Measure.hausdorffMeasure 1 (ℓ '' Set.Icc a b) := hdom
+    _ ≤ Measure.hausdorffMeasure 1
+        (Set.Icc 0 (eVariationOn γ (Set.Icc a b)).toReal) := measure_mono hsub
+    _ = eVariationOn γ (Set.Icc a b) := by
+      rw [hausdorffMeasure_real, Real.volume_Icc, sub_zero, ENNReal.ofReal_toReal hBV]
+
+/-- Hausdorff length of a continuous injective arc is its total variation. -/
+theorem hausdorffMeasure_image_Icc_eq_eVariationOn
+    (γ : ℝ → (EuclideanSpace ℝ (Fin 2))) {a b : ℝ} (hab : a ≤ b)
+    (hγ : ContinuousOn γ (Set.Icc a b))
+    (hinj : Set.InjOn γ (Set.Icc a b))
+    (hBV : BoundedVariationOn γ (Set.Icc a b)) :
+    Measure.hausdorffMeasure 1 (γ '' Set.Icc a b) =
+      eVariationOn γ (Set.Icc a b) :=
+  le_antisymm (hausdorffMeasure_image_Icc_le_eVariationOn_of_injectiveOn γ hab hinj hBV)
+    (eVariationOn_le_hausdorffMeasure_image_Icc_of_injectiveOn γ hγ hinj)
+
+private theorem eVariationOn_Icc_le_of_lipschitzOn
+    {γ : ℝ → (EuclideanSpace ℝ (Fin 2))} {C : NNReal} {a b : ℝ}
+    (hγ : LipschitzOnWith C γ (Set.Icc a b)) :
+    eVariationOn γ (Set.Icc a b) ≤ C * ENNReal.ofReal (b - a) := by
+  simpa using hγ.comp_eVariationOn_le (g := id) (s := Set.Icc a b) (fun _ hx ↦ hx)
+
+/-- Accumulated variation preserves a curve's Lipschitz bound. -/
+private theorem lipschitzOnWith_variationOnFromTo
+    {γ : ℝ → (EuclideanSpace ℝ (Fin 2))} {C : NNReal} {a b : ℝ} (hab : a ≤ b)
+    (hγ : LipschitzOnWith C γ (Set.Icc a b)) :
+    LipschitzOnWith C (variationOnFromTo γ (Set.Icc a b) a) (Set.Icc a b) := by
+  let ℓ := variationOnFromTo γ (Set.Icc a b) a
+  have hBV : BoundedVariationOn γ (Set.Icc a b) :=
+    ne_top_of_le_ne_top (by finiteness) (eVariationOn_Icc_le_of_lipschitzOn hγ)
+  have hbound : ∀ x ∈ Set.Icc a b, ∀ y ∈ Set.Icc a b, x ≤ y →
+      dist (ℓ x) (ℓ y) ≤ C * dist x y := by
+    intro x hx y hy hxy
+    have hsub : Set.Icc x y ⊆ Set.Icc a b := Set.Icc_subset_Icc hx.1 hy.2
+    have hvar := eVariationOn_Icc_le_of_lipschitzOn (hγ.mono hsub)
+    have hreal := ENNReal.toReal_mono (by finiteness) hvar
+    rw [ENNReal.toReal_mul, ENNReal.coe_toReal,
+      ENNReal.toReal_ofReal (sub_nonneg.mpr hxy)] at hreal
+    have hadd := variationOnFromTo.add hBV.locallyBoundedVariationOn
+      (Set.left_mem_Icc.mpr hab) hx hy
+    have hm := variationOnFromTo.monotoneOn hBV.locallyBoundedVariationOn
+      (Set.left_mem_Icc.mpr hab) hx hy hxy
+    rw [variationOnFromTo.eq_of_le _ _ hxy, Set.inter_eq_right.mpr hsub] at hadd
+    change dist (variationOnFromTo γ (Set.Icc a b) a x)
+      (variationOnFromTo γ (Set.Icc a b) a y) ≤ _
+    rw [Real.dist_eq, abs_of_nonpos (sub_nonpos.mpr hm), Real.dist_eq,
+      abs_of_nonpos (sub_nonpos.mpr hxy)]
+    linarith
+  apply LipschitzOnWith.of_dist_le_mul
+  intro x hx y hy
+  rcases le_total x y with hxy | hyx
+  · exact hbound x hx y hy hxy
+  · simpa only [dist_comm] using hbound y hy x hx hyx
+
+/-- Total variation of a Lipschitz curve is the integral of its accumulated variation derivative. -/
+private theorem integral_deriv_variationOnFromTo
+    {γ : ℝ → (EuclideanSpace ℝ (Fin 2))} {C : NNReal} {a b : ℝ} (hab : a ≤ b)
+    (hγ : LipschitzOnWith C γ (Set.Icc a b)) :
+    (∫ x in a..b, deriv (variationOnFromTo γ (Set.Icc a b) a) x) =
+      (eVariationOn γ (Set.Icc a b)).toReal := by
+  have hL := lipschitzOnWith_variationOnFromTo hab hγ
+  have hAC := (show LipschitzOnWith C (variationOnFromTo γ (Set.Icc a b) a)
+    (Set.uIcc a b) by simpa only [Set.uIcc_of_le hab] using hL).absolutelyContinuousOnInterval
+  rw [hAC.integral_deriv_eq_sub, variationOnFromTo.self,
+    variationOnFromTo.eq_of_le _ _ hab, Set.inter_self, sub_zero]
+
+private theorem norm_deriv_le_abs_deriv_of_eventually_dist_le
+    {γ : ℝ → (EuclideanSpace ℝ (Fin 2))} {ℓ : ℝ → ℝ} {x : ℝ}
+    (hγ : DifferentiableAt ℝ γ x) (hℓ : DifferentiableAt ℝ ℓ x)
+    (hbound : ∀ᶠ y in nhds x, dist (γ y) (γ x) ≤ dist (ℓ y) (ℓ x)) :
+    ‖deriv γ x‖ ≤ |deriv ℓ x| := by
+  have hγlim := hγ.hasDerivAt.tendsto_slope.norm
+  have hℓlim := hℓ.hasDerivAt.tendsto_slope.norm
+  change Filter.Tendsto _ _ (nhds |deriv ℓ x|) at hℓlim
+  apply le_of_tendsto_of_tendsto hγlim hℓlim
+  filter_upwards [hbound.filter_mono nhdsWithin_le_nhds] with y hy
+  simp only [slope, norm_smul]
+  rw [dist_eq_norm, dist_eq_norm] at hy
+  exact mul_le_mul_of_nonneg_left hy (norm_nonneg _)
+
+private theorem dist_le_dist_variationOnFromTo
+    {γ : ℝ → (EuclideanSpace ℝ (Fin 2))} {a b x y : ℝ} (hab : a ≤ b)
+    (hBV : BoundedVariationOn γ (Set.Icc a b))
+    (hx : x ∈ Set.Icc a b) (hy : y ∈ Set.Icc a b) :
+    dist (γ x) (γ y) ≤
+      dist (variationOnFromTo γ (Set.Icc a b) a x)
+        (variationOnFromTo γ (Set.Icc a b) a y) := by
+  wlog hxy : x ≤ y generalizing x y
+  · simpa only [dist_comm] using this hy hx (le_of_not_ge hxy)
+  have hvar : dist (γ x) (γ y) ≤ variationOnFromTo γ (Set.Icc a b) x y := by
+    rw [variationOnFromTo.eq_of_le _ _ hxy, dist_edist]
+    apply ENNReal.toReal_mono (hBV.locallyBoundedVariationOn _ _ hx hy)
+    exact eVariationOn.edist_le γ ⟨hx, le_rfl, hxy⟩ ⟨hy, hxy, le_rfl⟩
+  have hadd := variationOnFromTo.add hBV.locallyBoundedVariationOn
+    (Set.left_mem_Icc.mpr hab) hx hy
+  have hm := variationOnFromTo.monotoneOn hBV.locallyBoundedVariationOn
+    (Set.left_mem_Icc.mpr hab) hx hy hxy
+  rw [Real.dist_eq, abs_of_nonpos (sub_nonpos.mpr hm)]
+  linarith
+
+/-- The derivative norm of a curve is bounded by the derivative of accumulated variation. -/
+private theorem norm_deriv_le_deriv_variationOnFromTo
+    {γ : ℝ → (EuclideanSpace ℝ (Fin 2))} {a b x : ℝ}
+    (hBV : BoundedVariationOn γ (Set.Icc a b)) (hx : x ∈ Set.Ioo a b)
+    (hγ : DifferentiableAt ℝ γ x)
+    (hℓ : DifferentiableAt ℝ (variationOnFromTo γ (Set.Icc a b) a) x) :
+    ‖deriv γ x‖ ≤ deriv (variationOnFromTo γ (Set.Icc a b) a) x := by
+  have hab := hx.1.le.trans hx.2.le
+  have hnhds := Icc_mem_nhds hx.1 hx.2
+  have hbound : ∀ᶠ y in nhds x, dist (γ y) (γ x) ≤
+      dist (variationOnFromTo γ (Set.Icc a b) a y)
+        (variationOnFromTo γ (Set.Icc a b) a x) := by
+    filter_upwards [hnhds] with y hy
+    exact dist_le_dist_variationOnFromTo hab hBV hy ⟨hx.1.le, hx.2.le⟩
+  have h := norm_deriv_le_abs_deriv_of_eventually_dist_le hγ hℓ hbound
+  have hm := variationOnFromTo.monotoneOn hBV.locallyBoundedVariationOn
+    (Set.left_mem_Icc.mpr hab)
+  have hnonneg := hm.derivWithin_nonneg (x := x)
+  rw [derivWithin_of_mem_nhds hnhds] at hnonneg
+  rwa [abs_of_nonneg hnonneg] at h
+
+/-- The derivative of a planar Lipschitz curve is integrable on its interval. -/
+theorem integrableOn_deriv_of_lipschitzOn
+    {γ : ℝ → (EuclideanSpace ℝ (Fin 2))} {C : NNReal} {a b : ℝ}
+    (hγ : LipschitzOnWith C γ (Set.Icc a b)) :
+    IntegrableOn (deriv γ) (Set.Icc a b) := by
+  rw [IntegrableOn, ← restrict_Ioo_eq_restrict_Icc]
+  apply (integrable_const (C : ℝ)).mono' (aestronglyMeasurable_deriv γ _)
+  filter_upwards [ae_restrict_mem measurableSet_Ioo] with x hx
+  exact norm_deriv_le_of_lipschitzOn (Icc_mem_nhds hx.1 hx.2) hγ
+
+/-- The integral of speed is bounded by the total variation of a Lipschitz curve. -/
+private theorem integral_norm_deriv_le_eVariationOn
+    {γ : ℝ → (EuclideanSpace ℝ (Fin 2))} {C : NNReal} {a b : ℝ} (hab : a ≤ b)
+    (hγ : LipschitzOnWith C γ (Set.Icc a b)) :
+    (∫ x in a..b, ‖deriv γ x‖) ≤ (eVariationOn γ (Set.Icc a b)).toReal := by
+  let ℓ := variationOnFromTo γ (Set.Icc a b) a
+  have hL := lipschitzOnWith_variationOnFromTo hab hγ
+  have hAC := (show LipschitzOnWith C ℓ (Set.uIcc a b) by
+    simpa only [Set.uIcc_of_le hab] using hL).absolutelyContinuousOnInterval
+  have hBV : BoundedVariationOn γ (Set.Icc a b) :=
+    ne_top_of_le_ne_top (by finiteness) (eVariationOn_Icc_le_of_lipschitzOn hγ)
+  rw [← integral_deriv_variationOnFromTo hab hγ]
+  apply intervalIntegral.integral_mono_ae_restrict hab
+    ((intervalIntegrable_iff_integrableOn_Icc_of_le hab).mpr
+      (integrableOn_deriv_of_lipschitzOn hγ).norm) hAC.intervalIntegrable_deriv
+  rw [Filter.EventuallyLE, ae_restrict_iff' measurableSet_Icc]
+  filter_upwards [hγ.ae_differentiableWithinAt_of_mem,
+    hL.ae_differentiableWithinAt_of_mem,
+    show ∀ᵐ x : ℝ, x ≠ a by simp [ae_iff, measure_singleton],
+    show ∀ᵐ x : ℝ, x ≠ b by simp [ae_iff, measure_singleton]] with x hxγ hxL hxa hxb hx
+  have hx' : x ∈ Set.Ioo a b := ⟨lt_of_le_of_ne hx.1 (Ne.symm hxa), lt_of_le_of_ne hx.2 hxb⟩
+  have hxcc : x ∈ Set.Icc a b := ⟨hx'.1.le, hx'.2.le⟩
+  exact norm_deriv_le_deriv_variationOnFromTo hBV hx'
+    ((hxγ hxcc).differentiableAt (Icc_mem_nhds hx'.1 hx'.2))
+    ((hxL hxcc).differentiableAt (Icc_mem_nhds hx'.1 hx'.2))
+
+/-- Fundamental theorem of calculus for a planar Lipschitz curve. -/
+private theorem integral_deriv_eq_sub_of_lipschitzOn
+    {γ : ℝ → (EuclideanSpace ℝ (Fin 2))} {C : NNReal} {a b : ℝ} (hab : a ≤ b)
+    (hγ : LipschitzOnWith C γ (Set.Icc a b)) :
+    (∫ x in a..b, deriv γ x) = γ b - γ a := by
+  have hint : IntervalIntegrable (deriv γ) volume a b :=
+    (intervalIntegrable_iff_integrableOn_Icc_of_le hab).mpr
+      (integrableOn_deriv_of_lipschitzOn hγ)
+  ext i
+  let L : (EuclideanSpace ℝ (Fin 2)) →L[ℝ] ℝ := PiLp.proj 2 (fun _ : Fin 2 ↦ ℝ) i
+  have hcoord := L.lipschitzWith.comp_lipschitzOnWith hγ
+  have hAC := (show LipschitzOnWith _ (fun x ↦ L (γ x)) (Set.uIcc a b) by
+    simpa only [Set.uIcc_of_le hab, LipschitzOnWith, Function.comp_apply] using
+      hcoord).absolutelyContinuousOnInterval
+  change L (∫ x in a..b, deriv γ x) = L (γ b - γ a)
+  rw [← L.intervalIntegral_comp_comm hint, map_sub, ← hAC.integral_deriv_eq_sub]
+  apply intervalIntegral.integral_congr_ae
+  filter_upwards [hγ.ae_differentiableWithinAt_of_mem,
+    show ∀ᵐ x : ℝ, x ≠ b by simp [ae_iff, measure_singleton]] with x hx hxb hxI
+  have hx' : x ∈ Set.Ioo a b := by
+    rw [Set.uIoc_of_le hab] at hxI
+    exact ⟨hxI.1, lt_of_le_of_ne hxI.2 hxb⟩
+  have hd := (hx ⟨hx'.1.le, hx'.2.le⟩).differentiableAt (Icc_mem_nhds hx'.1 hx'.2)
+  exact ((L.hasFDerivAt.comp_hasDerivAt x hd.hasDerivAt).deriv).symm
+
+private theorem edist_le_ofReal_integral_norm_deriv
+    {γ : ℝ → (EuclideanSpace ℝ (Fin 2))} {C : NNReal} {a b : ℝ} (hab : a ≤ b)
+    (hγ : LipschitzOnWith C γ (Set.Icc a b)) :
+    edist (γ b) (γ a) ≤ ENNReal.ofReal (∫ x in a..b, ‖deriv γ x‖) := by
+  rw [edist_dist, dist_eq_norm, ← integral_deriv_eq_sub_of_lipschitzOn hab hγ]
+  exact ENNReal.ofReal_le_ofReal (intervalIntegral.norm_integral_le_integral_norm hab)
+
+/-- Total variation is bounded by the integral of speed for a Lipschitz curve. -/
+private theorem eVariationOn_le_ofReal_integral_norm_deriv
+    {γ : ℝ → (EuclideanSpace ℝ (Fin 2))} {C : NNReal} {a b : ℝ} (hab : a ≤ b)
+    (hγ : LipschitzOnWith C γ (Set.Icc a b)) :
+    eVariationOn γ (Set.Icc a b) ≤ ENNReal.ofReal (∫ x in a..b, ‖deriv γ x‖) := by
+  have hint : IntervalIntegrable (fun x ↦ ‖deriv γ x‖) volume a b :=
+    (intervalIntegrable_iff_integrableOn_Icc_of_le hab).mpr
+      (integrableOn_deriv_of_lipschitzOn hγ).norm
+  rw [eVariationOn]
+  refine iSup_le fun p ↦ ?_
+  rcases p with ⟨n, u, hu, hus⟩
+  have hsub (i : ℕ) : Set.Icc (u i) (u (i + 1)) ⊆ Set.Icc a b :=
+    Set.Icc_subset_Icc (hus i).1 (hus (i + 1)).2
+  have hints (i : ℕ) : IntervalIntegrable (fun x ↦ ‖deriv γ x‖) volume (u i) (u (i + 1)) :=
+    (intervalIntegrable_iff_integrableOn_Icc_of_le (hu (Nat.le_succ i))).mpr
+      (IntegrableOn.mono_set (integrableOn_deriv_of_lipschitzOn hγ).norm (hsub i))
+  calc
+    (∑ i ∈ Finset.range n, edist (γ (u (i + 1))) (γ (u i))) ≤
+        ∑ i ∈ Finset.range n, ENNReal.ofReal (∫ x in u i..u (i + 1), ‖deriv γ x‖) :=
+      Finset.sum_le_sum fun i _ ↦
+        edist_le_ofReal_integral_norm_deriv (hu (Nat.le_succ i)) (hγ.mono (hsub i))
+    _ = ENNReal.ofReal (∑ i ∈ Finset.range n, ∫ x in u i..u (i + 1), ‖deriv γ x‖) := by
+      symm
+      exact ENNReal.ofReal_sum_of_nonneg fun i _ ↦
+        intervalIntegral.integral_nonneg (hu (Nat.le_succ i)) (fun _ _ ↦ norm_nonneg _)
+    _ = ENNReal.ofReal (∫ x in u 0..u n, ‖deriv γ x‖) := by
+      rw [intervalIntegral.sum_integral_adjacent_intervals (fun i _ ↦ hints i)]
+    _ ≤ ENNReal.ofReal (∫ x in a..b, ‖deriv γ x‖) := by
+      apply ENNReal.ofReal_le_ofReal
+      exact intervalIntegral.integral_mono_interval (hus 0).1 (hu (Nat.zero_le n)) (hus n).2
+        (Filter.Eventually.of_forall fun _ ↦ norm_nonneg _) hint
+
+/-- Total variation of a planar Lipschitz curve is the integral of its speed. -/
+theorem eVariationOn_eq_ofReal_integral_norm_deriv
+    {γ : ℝ → (EuclideanSpace ℝ (Fin 2))} {C : NNReal} {a b : ℝ} (hab : a ≤ b)
+    (hγ : LipschitzOnWith C γ (Set.Icc a b)) :
+    eVariationOn γ (Set.Icc a b) = ENNReal.ofReal (∫ x in a..b, ‖deriv γ x‖) := by
+  apply le_antisymm (eVariationOn_le_ofReal_integral_norm_deriv hab hγ)
+  have hBV : eVariationOn γ (Set.Icc a b) ≠ ⊤ :=
+    ne_top_of_le_ne_top (by finiteness) (eVariationOn_Icc_le_of_lipschitzOn hγ)
+  rw [ENNReal.ofReal_le_iff_le_toReal hBV]
+  exact integral_norm_deriv_le_eVariationOn hab hγ
+
+/-- Hausdorff length of a planar injective Lipschitz curve is the integral of its speed. -/
+theorem hausdorffMeasure_image_Icc_eq_ofReal_integral_norm_deriv
+    {γ : ℝ → (EuclideanSpace ℝ (Fin 2))} {C : NNReal} {a b : ℝ} (hab : a ≤ b)
+    (hγ : LipschitzOnWith C γ (Set.Icc a b)) (hinj : Set.InjOn γ (Set.Icc a b)) :
+    Measure.hausdorffMeasure 1 (γ '' Set.Icc a b) =
+      ENNReal.ofReal (∫ x in a..b, ‖deriv γ x‖) := by
+  rw [hausdorffMeasure_image_Icc_eq_eVariationOn γ hab hγ.continuousOn hinj
+    (ne_top_of_le_ne_top (by finiteness) (eVariationOn_Icc_le_of_lipschitzOn hγ)),
+    eVariationOn_eq_ofReal_integral_norm_deriv hab hγ]
+
+/-- The image of a continuous injective planar arc of bounded variation is Lebesgue null. -/
+theorem volume_image_Icc_eq_zero_of_boundedVariationOn
+    (γ : ℝ → (EuclideanSpace ℝ (Fin 2))) {a b : ℝ} (hab : a ≤ b)
+    (hγ : ContinuousOn γ (Set.Icc a b)) (hinj : Set.InjOn γ (Set.Icc a b))
+    (hBV : BoundedVariationOn γ (Set.Icc a b)) :
+    volume (γ '' Set.Icc a b) = 0 := by
+  have hlength : Measure.hausdorffMeasure 1 (γ '' Set.Icc a b) ≠ ⊤ := by
+    rw [hausdorffMeasure_image_Icc_eq_eVariationOn γ hab hγ hinj hBV]
+    exact hBV
+  have harea : Measure.hausdorffMeasure 2 (γ '' Set.Icc a b) = 0 :=
+    (Measure.hausdorffMeasure_zero_or_top (by norm_num : (1 : ℝ) < 2) _).resolve_right hlength
+  have habs := MeasureTheory.Measure.absolutelyContinuous_isAddHaarMeasure
+    (volume : Measure (EuclideanSpace ℝ (Fin 2)))
+    (Measure.hausdorffMeasure (Module.finrank ℝ (EuclideanSpace ℝ (Fin 2))))
+  apply habs
+  simpa [finrank_euclideanSpace_fin] using harea
+
+end MeasureTheory
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Hausdorff / Graph
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter
+open scoped Topology
+
+namespace MeasureTheory
+
+private theorem map_firstCoordinate_restrict_curve_Icc
+    (γ : ℝ → (EuclideanSpace ℝ (Fin 2))) (a b c d : ℝ)
+    (hcoord : ∀ x ∈ Set.Icc a b, γ x 0 = x) :
+    Measure.map (fun p : (EuclideanSpace ℝ (Fin 2)) ↦ p 0)
+      ((Measure.hausdorffMeasure 1).restrict (γ '' Set.Icc a b)) (Set.Icc c d) =
+      Measure.hausdorffMeasure 1 (γ '' Set.Icc (max a c) (min b d)) := by
+  have hm : Measurable (fun p : (EuclideanSpace ℝ (Fin 2)) ↦ p 0) :=
+    (PiLp.proj (𝕜 := ℝ) 2 (fun _ : Fin 2 ↦ ℝ) 0).continuous.measurable
+  rw [Measure.map_apply hm measurableSet_Icc,
+    Measure.restrict_apply (hm measurableSet_Icc)]
+  congr 1
+  ext p
+  constructor
+  · rintro ⟨hp, x, hx, rfl⟩
+    refine ⟨x, ?_, rfl⟩
+    change γ x 0 ∈ Set.Icc c d at hp
+    rw [hcoord x hx] at hp
+    exact ⟨max_le hx.1 hp.1, le_min hx.2 hp.2⟩
+  · rintro ⟨x, hx, rfl⟩
+    refine ⟨?_, x, ⟨(le_max_left _ _).trans hx.1, hx.2.trans (min_le_left _ _)⟩, rfl⟩
+    change γ x 0 ∈ Set.Icc c d
+    rw [hcoord x ⟨(le_max_left _ _).trans hx.1, hx.2.trans (min_le_left _ _)⟩]
+    exact ⟨(le_max_right _ _).trans hx.1, hx.2.trans (min_le_right _ _)⟩
+
+/-- Projected Hausdorff measure on a Lipschitz graph has speed as its density. -/
+theorem map_firstCoordinate_restrict_curve_eq_withDensity
+    {γ : ℝ → (EuclideanSpace ℝ (Fin 2))} {C : NNReal} {a b : ℝ} (hab : a ≤ b)
+    (hγ : LipschitzOnWith C γ (Set.Icc a b)) (hcoord : ∀ x ∈ Set.Icc a b, γ x 0 = x) :
+    Measure.map (fun p : (EuclideanSpace ℝ (Fin 2)) ↦ p 0)
+      ((Measure.hausdorffMeasure 1).restrict (γ '' Set.Icc a b)) =
+      (volume.restrict (Set.Icc a b)).withDensity (fun x ↦ ENNReal.ofReal ‖deriv γ x‖) := by
+  have hinj : Set.InjOn γ (Set.Icc a b) := by
+    intro x hx y hy hxy
+    simpa only [hcoord x hx, hcoord y hy] using congrArg (fun p : (EuclideanSpace ℝ (Fin 2)) ↦ p
+      0) hxy
+  have hfinite : Measure.hausdorffMeasure 1 (γ '' Set.Icc a b) ≠ ⊤ := by
+    rw [hausdorffMeasure_image_Icc_eq_ofReal_integral_norm_deriv hab hγ hinj]
+    exact ENNReal.ofReal_ne_top
+  let _ : IsFiniteMeasure ((Measure.hausdorffMeasure 1).restrict (γ '' Set.Icc a b)) :=
+    (isFiniteMeasure_restrict).mpr hfinite
+  apply Measure.ext_of_Icc
+  intro c d _
+  rw [map_firstCoordinate_restrict_curve_Icc γ a b c d hcoord,
+    withDensity_apply _ measurableSet_Icc,
+    Measure.restrict_restrict measurableSet_Icc]
+  have hinter : Set.Icc c d ∩ Set.Icc a b = Set.Icc (max a c) (min b d) := by
+    rw [Set.Icc_inter_Icc]
+    simp only [max_comm, min_comm]
+  rw [hinter]
+  by_cases hcd : max a c ≤ min b d
+  · have hsub : Set.Icc (max a c) (min b d) ⊆ Set.Icc a b :=
+      Set.Icc_subset_Icc (le_max_left _ _) (min_le_left _ _)
+    rw [hausdorffMeasure_image_Icc_eq_ofReal_integral_norm_deriv hcd
+      (hγ.mono hsub) (hinj.mono hsub)]
+    rw [intervalIntegral.integral_of_le hcd, ← integral_Icc_eq_integral_Ioc]
+    exact ofReal_integral_eq_lintegral_ofReal
+      (IntegrableOn.mono_set (integrableOn_deriv_of_lipschitzOn hγ).norm hsub)
+      (Filter.Eventually.of_forall fun _ ↦ norm_nonneg _)
+  · rw [Set.Icc_eq_empty_of_lt (lt_of_not_ge hcd)]
+    simp
+
+/-- Weighted arclength formula for a planar Lipschitz graph on a compact interval. -/
+theorem integral_restrict_curve_eq_integral_norm_deriv_mul
+    {γ : ℝ → (EuclideanSpace ℝ (Fin 2))} {C : NNReal} {a b : ℝ} (hab : a ≤ b)
+    (hγ : LipschitzOnWith C γ (Set.Icc a b)) (hcoord : ∀ x ∈ Set.Icc a b, γ x 0 = x)
+    {φ : (EuclideanSpace ℝ (Fin 2)) → ℝ} (hφ : Measurable φ) :
+    (∫ p, φ p ∂(Measure.hausdorffMeasure 1).restrict (γ '' Set.Icc a b)) =
+      ∫ x in a..b, ‖deriv γ x‖ * φ (γ x) := by
+  let γc : ℝ → (EuclideanSpace ℝ (Fin 2)) := fun x ↦ γ (max a (min x b))
+  have hc : Continuous γc := hγ.continuousOn.comp_continuous
+    (continuous_const.max (continuous_id.min continuous_const))
+    (fun x ↦ ⟨le_max_left _ _, max_le hab (min_le_right _ _)⟩)
+  have hceq (x : ℝ) (hx : x ∈ Set.Icc a b) : γc x = γ x := by
+    simp only [γc, min_eq_left hx.2, max_eq_right hx.1]
+  have hm : Measurable (fun p : (EuclideanSpace ℝ (Fin 2)) ↦ p 0) :=
+    (PiLp.proj (𝕜 := ℝ) 2 (fun _ : Fin 2 ↦ ℝ) 0).continuous.measurable
+  have hset : MeasurableSet (γ '' Set.Icc a b) :=
+    (isCompact_Icc.image_of_continuousOn hγ.continuousOn).measurableSet
+  have heq : (fun p ↦ φ (γc (p 0))) =ᵐ[
+      (Measure.hausdorffMeasure 1).restrict (γ '' Set.Icc a b)] φ := by
+    filter_upwards [ae_restrict_mem hset] with p hp
+    obtain ⟨x, hx, rfl⟩ := hp
+    rw [hcoord x hx, hceq x hx]
+  rw [← integral_congr_ae heq,
+    ← integral_map (μ := (Measure.hausdorffMeasure 1).restrict (γ '' Set.Icc a b))
+      (f := fun x : ℝ ↦ φ (γc x)) hm.aemeasurable
+      (hφ.comp hc.measurable).aestronglyMeasurable,
+    map_firstCoordinate_restrict_curve_eq_withDensity hab hγ hcoord,
+    integral_withDensity_eq_integral_toReal_smul
+      (measurable_deriv γ |>.norm |>.ennreal_ofReal)
+      (Filter.Eventually.of_forall fun _ ↦ ENNReal.ofReal_lt_top)]
+  simp only [ENNReal.toReal_ofReal (norm_nonneg _), smul_eq_mul]
+  rw [intervalIntegral.integral_of_le hab, ← integral_Icc_eq_integral_Ioc]
+  apply setIntegral_congr_fun measurableSet_Icc
+  intro x hx
+  dsimp only
+  rw [hceq x hx]
+
+end MeasureTheory
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Hausdorff / Planar Graph
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter
+open scoped ENNReal Topology
+
+namespace MeasureTheory
+
+private theorem lipschitzOnWith_planarGraph {g : ℝ → ℝ} {s : Set ℝ} {C : NNReal}
+    (hg : LipschitzOnWith C g s) :
+    LipschitzOnWith (C + 1) (fun x ↦ !₂[x, g x] : ℝ → EuclideanSpace ℝ (Fin 2)) s := by
+  rw [lipschitzOnWith_iff_dist_le_mul]
+  intro x hx y hy
+  have hgxy := hg.dist_le_mul x hx y hy
+  rw [dist_eq_norm, EuclideanSpace.norm_eq]
+  simp only [PiLp.sub_apply, Fin.sum_univ_two, Real.norm_eq_abs, pow_two]
+  have hxy : dist x y = |x - y| := Real.dist_eq x y
+  have hgy : |g x - g y| ≤ (C : ℝ) * |x - y| := by
+    simpa [Real.dist_eq] using hgxy
+  have hnonneg : 0 ≤ ((C : ℝ) + 1) * |x - y| := mul_nonneg (by positivity) (abs_nonneg _)
+  simp only [Matrix.cons_val_zero, Matrix.cons_val_one, NNReal.coe_add, NNReal.coe_one,
+    Real.dist_eq]
+  rw [Real.sqrt_le_iff]
+  constructor
+  · exact hnonneg
+  · have hsq := mul_self_le_mul_self (abs_nonneg (g x - g y)) hgy
+    nlinarith [sq_nonneg ((C : ℝ) * |x - y|), NNReal.coe_nonneg C]
+
+private theorem hausdorffMeasure_planarGraph_eq_zero {g : ℝ → ℝ} {s t : Set ℝ}
+    {C : NNReal} (hg : LipschitzOnWith C g s) (hts : t ⊆ s) (ht : volume t = 0) :
+    Measure.hausdorffMeasure 1
+      ((fun x ↦ !₂[x, g x] : ℝ → EuclideanSpace ℝ (Fin 2)) '' t) = 0 := by
+  have hgraph := (lipschitzOnWith_planarGraph hg).mono hts
+  apply le_zero_iff.mp
+  calc
+    Measure.hausdorffMeasure 1
+        ((fun x ↦ !₂[x, g x] : ℝ → EuclideanSpace ℝ (Fin 2)) '' t) ≤
+        ((C + 1 : NNReal) : ENNReal) ^ (1 : ℝ) * Measure.hausdorffMeasure 1 t :=
+      hgraph.hausdorffMeasure_image_le (by positivity)
+    _ = 0 := by rw [MeasureTheory.hausdorffMeasure_real, ht, mul_zero]
+
+private theorem hausdorffMeasure_planarGraph_nondifferentiableOn_Icc_eq_zero
+    {g : ℝ → ℝ} {a b : ℝ} {C : NNReal}
+    (hg : LipschitzOnWith C g (Set.Icc a b)) :
+    Measure.hausdorffMeasure 1
+      ((fun x ↦ !₂[x, g x] : ℝ → EuclideanSpace ℝ (Fin 2)) ''
+        {x | x ∈ Set.Icc a b ∧ ¬ DifferentiableAt ℝ g x}) = 0 := by
+  apply hausdorffMeasure_planarGraph_eq_zero hg (by
+    intro x hx
+    exact hx.1)
+  rcases le_total a b with hab | hba
+  · rw [MeasureTheory.measure_eq_zero_iff_ae_notMem]
+    have hg' : LipschitzOnWith C g (Set.uIcc a b) := by
+      rwa [Set.uIcc_of_le hab]
+    have hBV := hg'.absolutelyContinuousOnInterval.boundedVariationOn
+    have hdiff := hBV.ae_differentiableAt_of_mem_uIcc
+    rw [Set.uIcc_of_le hab] at hdiff
+    filter_upwards [hdiff] with x hx hbad
+    exact hbad.2 (hx hbad.1)
+  · rcases hba.eq_or_lt with rfl | hba
+    · apply measure_mono_null (by aesop) (MeasureTheory.measure_singleton b)
+    · rw [Set.Icc_eq_empty (not_le_of_gt hba)]
+      simp
+
+private theorem hausdorffMeasure_planarGraph_nondifferentiableOn_Ioo_eq_zero
+    {g : ℝ → ℝ} {a b : ℝ} (hg : LocallyLipschitzOn (Set.Ioo a b) g) :
+    Measure.hausdorffMeasure 1
+      ((fun x ↦ !₂[x, g x] : ℝ → EuclideanSpace ℝ (Fin 2)) ''
+        {x | x ∈ Set.Ioo a b ∧ ¬ DifferentiableAt ℝ g x}) = 0 := by
+  let s : ℕ → Set ℝ := fun n ↦
+    Set.Icc (a + 1 / (n + 1 : ℝ)) (b - 1 / (n + 1 : ℝ))
+  let bad : ℕ → Set ℝ := fun n ↦ {x | x ∈ s n ∧ ¬ DifferentiableAt ℝ g x}
+  have hs (n : ℕ) : s n ⊆ Set.Ioo a b := by
+    intro x hx
+    dsimp only [s] at hx
+    have hpos : 0 < 1 / (n + 1 : ℝ) := by positivity
+    exact ⟨lt_of_lt_of_le (lt_add_of_pos_right a hpos) hx.1,
+      lt_of_le_of_lt hx.2 (sub_lt_self b hpos)⟩
+  have hzero (n : ℕ) : Measure.hausdorffMeasure 1
+      ((fun x ↦ !₂[x, g x] : ℝ → EuclideanSpace ℝ (Fin 2)) '' bad n) = 0 := by
+    obtain ⟨C, hC⟩ := LocallyLipschitzOn.exists_lipschitzOnWith_of_compact
+      isCompact_Icc (hg.mono (hs n))
+    exact hausdorffMeasure_planarGraph_nondifferentiableOn_Icc_eq_zero hC
+  apply measure_mono_null
+    (t := ⋃ n, (fun x ↦ !₂[x, g x] : ℝ → EuclideanSpace ℝ (Fin 2)) '' bad n) ?_
+    (MeasureTheory.measure_iUnion_null hzero)
+  rintro p ⟨x, hx, rfl⟩
+  have hδ : 0 < min (x - a) (b - x) := lt_min (sub_pos.mpr hx.1.1) (sub_pos.mpr hx.1.2)
+  obtain ⟨n, hn⟩ := exists_nat_one_div_lt hδ
+  rw [Set.mem_iUnion]
+  refine ⟨n, x, ?_, rfl⟩
+  refine ⟨?_, hx.2⟩
+  dsimp only [s]
+  constructor <;> linarith [lt_of_lt_of_le hn (min_le_left _ _),
+    lt_of_lt_of_le hn (min_le_right _ _)]
+
+/-- The graph image of the nondifferentiability set of a locally Lipschitz real
+function has zero one-dimensional Hausdorff measure, in isometric coordinates. -/
+theorem hausdorffMeasure_coordinateGraph_nondifferentiable_eq_zero
+    {g : ℝ → ℝ} {a b : ℝ} (hg : LocallyLipschitzOn (Set.Ioo a b) g)
+    (o : EuclideanSpace ℝ (Fin 2))
+    (e : EuclideanSpace ℝ (Fin 2) ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin 2)) :
+    Measure.hausdorffMeasure 1
+      ((fun x ↦ o + e.symm !₂[x, g x]) ''
+        {x | x ∈ Set.Ioo a b ∧ ¬ DifferentiableAt ℝ g x}) = 0 := by
+  let graph : ℝ → EuclideanSpace ℝ (Fin 2) := fun x ↦ !₂[x, g x]
+  let transform : EuclideanSpace ℝ (Fin 2) → EuclideanSpace ℝ (Fin 2) :=
+    fun p ↦ o + e.symm p
+  have htransform : LipschitzWith 1 transform := by
+    apply LipschitzWith.of_dist_le_mul
+    intro p q
+    simp only [transform, NNReal.coe_one, one_mul, dist_add_left]
+    rw [e.symm.dist_map]
+  have hzero := hausdorffMeasure_planarGraph_nondifferentiableOn_Ioo_eq_zero hg
+  apply le_zero_iff.mp
+  rw [show (fun x ↦ o + e.symm !₂[x, g x]) ''
+      {x | x ∈ Set.Ioo a b ∧ ¬ DifferentiableAt ℝ g x} =
+      transform '' (graph '' {x | x ∈ Set.Ioo a b ∧ ¬ DifferentiableAt ℝ g x}) by
+    simp only [transform, graph, Set.image_image]]
+  calc
+    Measure.hausdorffMeasure 1
+        (transform '' (graph '' {x | x ∈ Set.Ioo a b ∧ ¬ DifferentiableAt ℝ g x})) ≤
+      ((1 : NNReal) : ENNReal) ^ (1 : ℝ) * Measure.hausdorffMeasure 1
+        (graph '' {x | x ∈ Set.Ioo a b ∧ ¬ DifferentiableAt ℝ g x}) :=
+      htransform.hausdorffMeasure_image_le (by positivity) _
+    _ = 0 := by rw [hzero, mul_zero]
+
+private theorem norm_coordinateGraph_deriv (g : ℝ → ℝ) (x : ℝ) :
+    ‖(!₂[1, deriv g x] : EuclideanSpace ℝ (Fin 2))‖ = Real.sqrt (1 + (deriv g x) ^ 2) := by
+  rw [EuclideanSpace.norm_eq]
+  congr 1
+  simp [Fin.sum_univ_two, Real.norm_eq_abs, pow_two]
+
+private theorem hasDerivAt_coordinateGraph {g : ℝ → ℝ} {x : ℝ}
+    (hg : DifferentiableAt ℝ g x) :
+    HasDerivAt (fun y ↦ !₂[y, g y] : ℝ → EuclideanSpace ℝ (Fin 2))
+      !₂[1, deriv g x] x := by
+  let L := (PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 2 ↦ ℝ)).symm.toContinuousLinearMap
+  have hpi : HasDerivAt (fun y i ↦ !₂[y, g y].ofLp i)
+      (fun i ↦ !₂[1, deriv g x].ofLp i) x := by
+    rw [hasDerivAt_pi]
+    intro i
+    fin_cases i
+    · exact hasDerivAt_id x
+    · exact hg.hasDerivAt
+  have hcomp := L.hasFDerivAt.comp x hpi
+  have hfun : (L ∘ fun y i ↦ !₂[y, g y].ofLp i) =
+      (fun y ↦ !₂[y, g y] : ℝ → EuclideanSpace ℝ (Fin 2)) := rfl
+  have hder : L.comp (ContinuousLinearMap.toSpanSingleton ℝ
+      (fun i ↦ !₂[1, deriv g x].ofLp i)) =
+      ContinuousLinearMap.toSpanSingleton ℝ
+        (!₂[1, deriv g x] : EuclideanSpace ℝ (Fin 2)) := by
+    apply ContinuousLinearMap.ext
+    intro r
+    rw [WithLp.ext_iff]
+    funext i
+    fin_cases i <;> simp [L, ContinuousLinearMap.comp_apply]
+  rw [hfun, hder] at hcomp
+  exact hcomp
+
+private theorem integral_restrict_planarGraph_eq_integral_sqrt_mul
+    {g : ℝ → ℝ} {C : NNReal} {a b : ℝ} (hab : a ≤ b)
+    (hg : LipschitzOnWith C g (Set.Icc a b))
+    {φ : EuclideanSpace ℝ (Fin 2) → ℝ} (hφ : Measurable φ) :
+    (∫ p, φ p ∂(Measure.hausdorffMeasure 1).restrict
+      ((fun x ↦ !₂[x, g x] : ℝ → EuclideanSpace ℝ (Fin 2)) '' Set.Icc a b)) =
+      ∫ x in a..b, Real.sqrt (1 + (deriv g x) ^ 2) * φ !₂[x, g x] := by
+  let γ : ℝ → EuclideanSpace ℝ (Fin 2) := fun x ↦ !₂[x, g x]
+  rw [MeasureTheory.integral_restrict_curve_eq_integral_norm_deriv_mul hab
+    (lipschitzOnWith_planarGraph hg) (fun _ _ ↦ rfl) hφ]
+  apply intervalIntegral.integral_congr_ae
+  have hg' : LipschitzOnWith C g (Set.uIcc a b) := by
+    rw [Set.uIcc_of_le hab]
+    exact hg
+  have hdiff := hg'.absolutelyContinuousOnInterval.boundedVariationOn
+    |>.ae_differentiableAt_of_mem_uIcc
+  filter_upwards [hdiff] with x hx hxi
+  have hderiv := (hasDerivAt_coordinateGraph (hx ⟨le_of_lt hxi.1, hxi.2⟩)).deriv
+  rw [hderiv, norm_coordinateGraph_deriv]
+
+/-- Weighted Hausdorff integration over an isometric planar graph equals the
+parameter integral weighted by its almost-everywhere speed. -/
+theorem integral_restrict_coordinateGraph_eq_integral_sqrt_mul
+    {g : ℝ → ℝ} {C : NNReal} {a b : ℝ} (hab : a ≤ b)
+    (hg : LipschitzOnWith C g (Set.Icc a b)) (o : EuclideanSpace ℝ (Fin 2))
+    (e : EuclideanSpace ℝ (Fin 2) ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin 2))
+    {φ : EuclideanSpace ℝ (Fin 2) → ℝ} (hφ : Measurable φ) :
+    (∫ p, φ p ∂(Measure.hausdorffMeasure 1).restrict
+      ((fun x ↦ o + e.symm !₂[x, g x] : ℝ → EuclideanSpace ℝ (Fin 2)) '' Set.Icc a b)) =
+      ∫ x in a..b, Real.sqrt (1 + (deriv g x) ^ 2) * φ (o + e.symm !₂[x, g x]) := by
+  let F : EuclideanSpace ℝ (Fin 2) ≃ᵢ EuclideanSpace ℝ (Fin 2) :=
+    e.symm.toIsometryEquiv.trans
+      (AffineIsometryEquiv.vaddConst (V := EuclideanSpace ℝ (Fin 2))
+        (P := EuclideanSpace ℝ (Fin 2)) ℝ o).toIsometryEquiv
+  let γ : ℝ → EuclideanSpace ℝ (Fin 2) := fun x ↦ !₂[x, g x]
+  have hF : ∀ p, F p = o + e.symm p := by
+    intro p
+    simp [F, add_comm]
+  rw [show (fun x ↦ o + e.symm !₂[x, g x] : ℝ → EuclideanSpace ℝ (Fin 2)) '' Set.Icc a b =
+      F '' (γ '' Set.Icc a b) by
+    rw [Set.image_image]
+    congr 1
+    funext x
+    exact (hF _).symm]
+  rw [(F.measurePreserving_hausdorffMeasure 1).setIntegral_image_emb
+    F.toHomeomorph.measurableEmbedding φ (γ '' Set.Icc a b)]
+  have hbase := integral_restrict_planarGraph_eq_integral_sqrt_mul hab hg
+    (hφ.comp F.continuous.measurable)
+  simpa only [γ, Function.comp_apply, hF] using hbase
+
+end MeasureTheory
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Integral / Atomic Bounds
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MeasureTheory
+
+/-- A finite sum of atomic contributions is bounded by the integral of a nonnegative function. -/
+theorem sum_measureReal_mul_le_setIntegral {α : Type*} [MeasurableSpace α]
+    [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) (D : Finset α)
+    {E : Set α} {f : α → ℝ} (hE : MeasurableSet E)
+    (hDE : (D : Set α) ⊆ E) (hf : MeasureTheory.IntegrableOn f E μ)
+    (hnonneg : ∀ x ∈ E, 0 ≤ f x) :
+    ∑ x ∈ D, (μ {x}).toReal * f x ≤ ∫ x in E, f x ∂μ := by
+  have hfinite := MeasureTheory.setIntegral_finset D (hf.mono_set hDE)
+  simp only [smul_eq_mul, MeasureTheory.measureReal_def] at hfinite
+  rw [← hfinite]
+  apply MeasureTheory.setIntegral_mono_set hf
+  · exact (MeasureTheory.ae_restrict_iff' hE).mpr (Filter.Eventually.of_forall hnonneg)
+  · exact Filter.Eventually.of_forall hDE
+
+end MeasureTheory
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Integral / Interval Exhaustion
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter
+open scoped Topology
+
+namespace MeasureTheory
+
+/-- Every interior point eventually belongs to the standard inner exhaustion of an interval. -/
+theorem eventually_mem_innerIcc_of_mem_Ioo {a b x : ℝ}
+    (hx : x ∈ Set.Ioo a b) :
+    ∀ᶠ n : ℕ in atTop,
+      x ∈ Set.Icc (a + 1 / ((n : ℝ) + 1)) (b - 1 / ((n : ℝ) + 1)) := by
+  have hleft := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).eventually
+    (Iio_mem_nhds (sub_pos.mpr hx.1))
+  have hright := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).eventually
+    (Iio_mem_nhds (sub_pos.mpr hx.2))
+  exact (hleft.and hright).mono fun n hn ↦ by
+    constructor <;> linarith [hn.1, hn.2]
+
+/-- Each interval in the standard inner exhaustion lies in the open interval. -/
+theorem innerIcc_subset_Ioo (a b : ℝ) (n : ℕ) :
+    Set.Icc (a + 1 / ((n : ℝ) + 1)) (b - 1 / ((n : ℝ) + 1)) ⊆
+      Set.Ioo a b := by
+  intro x hx
+  have hn : 0 < 1 / ((n : ℝ) + 1) := by positivity
+  constructor <;> linarith [hx.1, hx.2]
+
+/-- Integrals over compact intervals exhausting the interior converge to the
+integral over the full compact interval. -/
+theorem tendsto_integral_innerIcc
+    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {f : ℝ → E} {a b : ℝ} (hf : Integrable f (volume.restrict (Set.Icc a b))) :
+    Tendsto (fun n : ℕ ↦ ∫ x in Set.Icc (a + 1 / ((n : ℝ) + 1))
+        (b - 1 / ((n : ℝ) + 1)), f x)
+      atTop (nhds (∫ x in Set.Icc a b, f x)) := by
+  let t := Set.Icc a b
+  let s : ℕ → Set ℝ := fun n ↦
+    Set.Icc (a + 1 / ((n : ℝ) + 1)) (b - 1 / ((n : ℝ) + 1))
+  have hs (n : ℕ) : MeasurableSet (s n) := measurableSet_Icc
+  have hft : Integrable (t.indicator f) volume :=
+    IntegrableOn.integrable_indicator hf measurableSet_Icc
+  have hmeas (n : ℕ) : AEStronglyMeasurable ((s n).indicator f) volume := by
+    apply (hft.1.indicator (hs n)).congr
+    filter_upwards [] with x
+    by_cases hx : x ∈ s n
+    · have hxi := innerIcc_subset_Ioo a b n hx
+      have hxt : x ∈ t := ⟨hxi.1.le, hxi.2.le⟩
+      simp [Set.indicator, hx, hxt]
+    · simp [Set.indicator, hx]
+  have hbound (n : ℕ) : ∀ᵐ x ∂volume,
+      ‖(s n).indicator f x‖ ≤ ‖t.indicator f x‖ :=
+    Filter.Eventually.of_forall fun x ↦ by
+      by_cases hx : x ∈ s n
+      · have hxi := innerIcc_subset_Ioo a b n hx
+        have hxt : x ∈ t := ⟨hxi.1.le, hxi.2.le⟩
+        simp [Set.indicator, hx, hxt]
+      · simp only [Set.indicator, hx, ↓reduceIte, norm_zero]
+        exact norm_nonneg _
+  have hIooVolume : Set.Ioo a b =ᵐ[volume] Set.Icc a b := Ioo_ae_eq_Icc
+  have hlim : ∀ᵐ x ∂volume, Tendsto (fun n ↦ (s n).indicator f x)
+      atTop (nhds (t.indicator f x)) := by
+    filter_upwards [hIooVolume] with x hxeq
+    by_cases hx : x ∈ Set.Ioo a b
+    · have hev := eventually_mem_innerIcc_of_mem_Ioo hx
+      apply tendsto_const_nhds.congr'
+      exact hev.mono fun n hn ↦
+        (Set.indicator_of_mem (hxeq.mp hx) f).trans
+          (Set.indicator_of_mem hn f).symm
+    · have hxt : x ∉ t := fun h ↦ hx (hxeq.mpr h)
+      apply tendsto_const_nhds.congr'
+      filter_upwards [] with n
+      have hxn : x ∉ s n := fun h ↦ by
+        have hxi := innerIcc_subset_Ioo a b n h
+        exact hxt ⟨hxi.1.le, hxi.2.le⟩
+      simp only [Set.indicator, hxn, hxt, ↓reduceIte]
+  have ht := tendsto_integral_of_dominated_convergence (μ := volume)
+    (fun x ↦ ‖t.indicator f x‖) hmeas hft.norm hbound hlim
+  simpa only [s, t, integral_indicator, hs, measurableSet_Icc] using ht
+
+end MeasureTheory
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Integral / Moving Intervals
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter Set
+open scoped Topology
+
+namespace MeasureTheory
+
+/-- Moving interval cutoffs preserve almost-everywhere convergence on the limiting interior. -/
+theorem ae_tendsto_indicator_Icc_of_tendsto_endpoints
+    {E : Type*} [NormedAddCommGroup E] {a b : ℕ → ℝ} {c d : ℝ}
+    {f : ℕ → ℝ → E} {g : ℝ → E}
+    (ha : Tendsto a atTop (𝓝 c)) (hb : Tendsto b atTop (𝓝 d))
+    (hf : ∀ᵐ x ∂volume, x ∈ Ioo c d → Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))) :
+    ∀ᵐ x ∂volume, Tendsto (fun n ↦ (Icc (a n) (b n)).indicator (f n) x)
+      atTop (𝓝 ((Icc c d).indicator g x)) := by
+  have hne (y : ℝ) : ∀ᵐ x ∂volume, x ≠ y := by
+    simpa using (measure_eq_zero_iff_ae_notMem.mp (measure_singleton y :
+      volume ({y} : Set ℝ) = 0))
+  filter_upwards [hf, hne c, hne d] with x hx hxc hxd
+  by_cases hmem : x ∈ Icc c d
+  · have hxi : x ∈ Ioo c d :=
+      ⟨lt_of_le_of_ne hmem.1 (Ne.symm hxc), lt_of_le_of_ne hmem.2 hxd⟩
+    rw [indicator_of_mem hmem]
+    apply (hx hxi).congr'
+    filter_upwards [ha.eventually (gt_mem_nhds hxi.1),
+      hb.eventually (lt_mem_nhds hxi.2)] with n hn hn'
+    exact (indicator_of_mem (show x ∈ Icc (a n) (b n) from ⟨hn.le, hn'.le⟩) _).symm
+  · rw [indicator_of_notMem hmem]
+    apply tendsto_const_nhds.congr'
+    have hout : x < c ∨ d < x := by
+      simpa only [mem_Icc, not_and_or, not_le] using hmem
+    rcases hout with hleft | hright
+    · filter_upwards [ha.eventually (lt_mem_nhds hleft)] with n hn
+      exact (indicator_of_notMem (show x ∉ Icc (a n) (b n) from
+        fun h ↦ (not_le.mpr hn) h.1) _).symm
+    · filter_upwards [hb.eventually (gt_mem_nhds hright)] with n hn
+      exact (indicator_of_notMem (show x ∉ Icc (a n) (b n) from
+        fun h ↦ (not_le.mpr hn) h.2) _).symm
+
+/-- Dominated convergence on intervals whose endpoints converge, using convergence
+only in the interior of the limiting interval. -/
+theorem tendsto_integral_Icc_of_tendsto_endpoints
+    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {a b : ℕ → ℝ} {c d : ℝ} {f : ℕ → ℝ → E} {g : ℝ → E} {bound : ℝ → ℝ}
+    (ha : Tendsto a atTop (𝓝 c)) (hb : Tendsto b atTop (𝓝 d))
+    (hmeas : ∀ᶠ n in atTop, AEStronglyMeasurable (f n) (volume.restrict (Icc (a n) (b n))))
+    (hbound : Integrable bound) (hbound_nonneg : ∀ᵐ x ∂volume, 0 ≤ bound x)
+    (hdom : ∀ᶠ n in atTop, ∀ᵐ x ∂volume, x ∈ Icc (a n) (b n) → ‖f n x‖ ≤ bound x)
+    (hf : ∀ᵐ x ∂volume, x ∈ Ioo c d → Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))) :
+    Tendsto (fun n ↦ ∫ x in Icc (a n) (b n), f n x) atTop
+      (𝓝 (∫ x in Icc c d, g x)) := by
+  have hm : ∀ᶠ n in atTop,
+      AEStronglyMeasurable ((Icc (a n) (b n)).indicator (f n)) volume := by
+    filter_upwards [hmeas] with n hn
+    exact (aestronglyMeasurable_indicator_iff measurableSet_Icc).mpr hn
+  have hd : ∀ᶠ n in atTop, ∀ᵐ x ∂volume,
+      ‖(Icc (a n) (b n)).indicator (f n) x‖ ≤ bound x := by
+    filter_upwards [hdom] with n hn
+    filter_upwards [hn, hbound_nonneg] with x hx hnonneg
+    by_cases hmem : x ∈ Icc (a n) (b n)
+    · simpa only [indicator_of_mem hmem] using hx hmem
+    · simpa only [indicator_of_notMem hmem, norm_zero] using hnonneg
+  have h := tendsto_integral_filter_of_dominated_convergence bound hm hd hbound
+    (ae_tendsto_indicator_Icc_of_tendsto_endpoints ha hb hf)
+  simpa only [integral_indicator measurableSet_Icc] using h
+
+/-- A uniform bound on moving finite intervals suffices for dominated convergence. -/
+theorem tendsto_integral_Icc_of_tendsto_endpoints_of_bound
+    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {a b : ℕ → ℝ} {c d M : ℝ} {f : ℕ → ℝ → E} {g : ℝ → E}
+    (ha : Tendsto a atTop (𝓝 c)) (hb : Tendsto b atTop (𝓝 d))
+    (hmeas : ∀ᶠ n in atTop, AEStronglyMeasurable (f n) (volume.restrict (Icc (a n) (b n))))
+    (hM : 0 ≤ M)
+    (hdom : ∀ᶠ n in atTop, ∀ᵐ x ∂volume, x ∈ Icc (a n) (b n) → ‖f n x‖ ≤ M)
+    (hf : ∀ᵐ x ∂volume, x ∈ Ioo c d → Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))) :
+    Tendsto (fun n ↦ ∫ x in Icc (a n) (b n), f n x) atTop
+      (𝓝 (∫ x in Icc c d, g x)) := by
+  let s := Icc (c - 1) (d + 1)
+  have hs : MeasurableSet s := measurableSet_Icc
+  have hi : Integrable (s.indicator (fun _ : ℝ ↦ M)) volume := by
+    apply IntegrableOn.integrable_indicator _ hs
+    exact integrableOn_const isCompact_Icc.measure_lt_top.ne
+  apply tendsto_integral_Icc_of_tendsto_endpoints ha hb hmeas hi
+  · exact Eventually.of_forall fun x ↦ by
+      by_cases hx : x ∈ s <;> simp [indicator, hx, hM]
+  · filter_upwards [hdom, ha.eventually (lt_mem_nhds (show c - 1 < c by linarith)),
+      hb.eventually (gt_mem_nhds (show d < d + 1 by linarith))] with n hn hna hnb
+    filter_upwards [hn] with x hx
+    intro hmem
+    have hxs : x ∈ s := ⟨hna.le.trans hmem.1, hmem.2.trans hnb.le⟩
+    simpa only [indicator_of_mem hxs] using hx hmem
+  · exact hf
+
+end MeasureTheory
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Integral / Translation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped ENNReal Topology
+
+namespace MeasureTheory
+
+/-- Translation of a measurable real set preserves integrals. -/
+theorem integral_image_add_right_eq (c : ℝ) (E : Set ℝ) (hE : MeasurableSet E)
+    (f : ℝ → ℝ) :
+    (∫ y in (fun x : ℝ ↦ x + c) '' E, f y) = ∫ x in E, f (x + c) := by
+  let φ : ℝ → ℝ := fun x ↦ x + c
+  have hφ : MeasurableEmbedding φ :=
+    (Homeomorph.addRight c).isClosedEmbedding.measurableEmbedding
+  have hpres : MeasurePreserving φ volume volume :=
+    ⟨hφ.measurable, map_add_right_eq_self volume c⟩
+  have himage : MeasurableSet (φ '' E) := hφ.measurableSet_image' hE
+  have hpre : φ ⁻¹' (φ '' E) = E := Set.preimage_image_eq E hφ.injective
+  have hr := hpres.restrict_preimage himage
+  rw [hpre] at hr
+  exact (hr.integral_comp hφ f).symm
+
+/-- Translating both endpoints of a closed real interval translates the integrand. -/
+theorem integral_Icc_const_add_eq (c a b : ℝ) (f : ℝ → ℝ) :
+    (∫ y in Icc (c + a) (c + b), f y) = ∫ x in Icc a b, f (x + c) := by
+  rw [show c + a = a + c from add_comm c a, show c + b = b + c from add_comm c b,
+    ← Set.image_add_const_Icc, integral_image_add_right_eq c _ measurableSet_Icc]
+
+/-- Integrability on a translated measurable set is preserved by translation. -/
+theorem integrableOn_comp_add_right_iff (c : ℝ) (E : Set ℝ) (hE : MeasurableSet E)
+    (f : ℝ → ℝ) :
+    IntegrableOn (fun x ↦ f (x + c)) E ↔
+      IntegrableOn f ((fun x : ℝ ↦ x + c) '' E) := by
+  let φ : ℝ → ℝ := fun x ↦ x + c
+  have hφ : MeasurableEmbedding φ :=
+    (Homeomorph.addRight c).isClosedEmbedding.measurableEmbedding
+  have hpres : MeasurePreserving φ volume volume :=
+    ⟨hφ.measurable, map_add_right_eq_self volume c⟩
+  have himage : MeasurableSet (φ '' E) := hφ.measurableSet_image' hE
+  have hpre : φ ⁻¹' (φ '' E) = E := Set.preimage_image_eq E hφ.injective
+  have hr := hpres.restrict_preimage himage
+  rw [hpre] at hr
+  exact hr.integrable_comp_emb hφ
+
+/-- A lower Lebesgue integral of a right-translated function is the integral of the function
+itself over the translated set. -/
+theorem setLIntegral_comp_sub_right (w : ℝ → ℝ≥0∞) (c : ℝ) (A : Set ℝ) :
+    ∫⁻ u in A, w (u - c) = ∫⁻ t in (fun t : ℝ ↦ t + c) ⁻¹' A, w t := by
+  have hemb : MeasurableEmbedding fun t : ℝ ↦ t + c :=
+    (MeasurableEquiv.addRight c).measurableEmbedding
+  simpa using ((measurePreserving_add_right (volume : Measure ℝ) c).setLIntegral_comp_preimage_emb
+    hemb (fun u ↦ w (u - c)) A).symm
+
+/-- Pushing a weighted restriction of Lebesgue measure forward along `t ↦ t + c` translates both
+the set and the weight. -/
+theorem map_add_right_restrict_withDensity (c : ℝ) (S : Set ℝ) (w : ℝ → ℝ≥0∞) :
+    Measure.map (fun t : ℝ ↦ t + c) ((volume.restrict S).withDensity w) =
+      (volume.restrict ((fun t : ℝ ↦ t + c) '' S)).withDensity (fun u ↦ w (u - c)) := by
+  have hemb : MeasurableEmbedding fun t : ℝ ↦ t + c :=
+    (MeasurableEquiv.addRight c).measurableEmbedding
+  ext A hA
+  have hpre : MeasurableSet ((fun t : ℝ ↦ t + c) ⁻¹' A) := hA.preimage hemb.measurable
+  rw [Measure.map_apply hemb.measurable hA, withDensity_apply _ hpre,
+    Measure.restrict_restrict hpre, withDensity_apply _ hA, Measure.restrict_restrict hA,
+    setLIntegral_comp_sub_right, Set.preimage_inter, hemb.injective.preimage_image]
+
+end MeasureTheory
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Measure / Atoms
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Set
+
+namespace MeasureTheory
+
+/-- A membership that fails only on a countable set holds almost everywhere on the restriction
+of a measure with null singletons. -/
+theorem ae_restrict_mem_of_countable_diff {α : Type*} [MeasurableSpace α] {μ : Measure α}
+    [NullSingletonClass μ] {S A N : Set α} (hS : MeasurableSet S) (hN : N.Countable)
+    (hsub : S \ A ⊆ N) : ∀ᵐ x ∂μ.restrict S, x ∈ A := by
+  rw [ae_iff, Measure.restrict_apply' hS]
+  exact measure_mono_null (fun x hx ↦ hsub ⟨hx.2, hx.1⟩) (hN.measure_zero μ)
+
+/-- Along an injective parametrization, a finite-measure atom occurs only almost nowhere. -/
+theorem ae_measure_singleton_comp_eq_zero_of_injOn
+    {X : Type*} [MeasurableSpace X] [MeasurableSingletonClass X]
+    (μ : Measure X) [SFinite μ] {s : Set ℝ} (hs : MeasurableSet s)
+    {f : ℝ → X} (hf : Set.InjOn f s) :
+    ∀ᵐ t ∂volume.restrict s, μ {f t} = 0 := by
+  let A : Set X := {x | 0 < μ {x}}
+  have hA : A.Countable := μ.countable_meas_level_set_pos measurable_id
+  let B : Set ℝ := {t | t ∈ s ∧ f t ∈ A}
+  have hmaps : MapsTo f B A := fun _ h ↦ h.2
+  have hB : B.Countable := hmaps.countable_of_injOn (hf.mono fun _ h ↦ h.1) hA
+  filter_upwards [ae_restrict_mem hs, hB.ae_notMem (volume.restrict s)] with t hts htB
+  by_contra hne
+  exact htB ⟨hts, pos_iff_ne_zero.mpr hne⟩
+
+/-- A measure carried by a finite set is the sum of its atomic contributions. -/
+theorem measure_eq_sum_singleton_inter_of_compl_eq_zero
+    {α : Type*} [MeasurableSpace α] [MeasurableSingletonClass α]
+    (μ : Measure α) (S : Finset α) (hS : μ (S : Set α)ᶜ = 0)
+    (E : Set α) (hE : MeasurableSet E) :
+    μ E = ∑ x ∈ S, μ ({x} ∩ E) := by
+  classical
+  have hae : ∀ᵐ x ∂μ, x ∈ (S : Set α) := by
+    rw [ae_iff]
+    exact hS
+  have hrestrict : μ.restrict (S : Set α) = μ :=
+    Measure.restrict_eq_self_of_ae_mem hae
+  have hinter : μ E = μ (E ∩ (S : Set α)) := by
+    calc
+      μ E = (μ.restrict (S : Set α)) E := by rw [hrestrict]
+      _ = μ (E ∩ (S : Set α)) := Measure.restrict_apply hE
+  rw [hinter]
+  have heq : E ∩ (S : Set α) = ⋃ x ∈ S, ({x} ∩ E : Set α) := by
+    ext x
+    simp [and_comm]
+  rw [heq, measure_biUnion_finset]
+  · intro i _ j _ hij
+    simp only [Set.disjoint_left]
+    intro x hxi hxj
+    exact hij (hxi.1.symm.trans hxj.1)
+  · intro b _
+    exact (measurableSet_singleton b).inter hE
+
+/-- A measure carried by the image of a finite index set is bounded by the sum of atomic bounds
+over any index subset containing every index whose atom meets the measured set. -/
+theorem measure_le_sum_of_measure_compl_image_eq_zero
+    {α ι : Type*} [MeasurableSpace α] [MeasurableSingletonClass α]
+    (μ : Measure α) (S S' : Finset ι) (f : ι → α) (b : ι → ENNReal)
+    (hf : Set.InjOn f (S : Set ι)) (hS : μ (f '' (S : Set ι))ᶜ = 0)
+    (hb : ∀ i ∈ S, μ {f i} ≤ b i) (E : Set α) (hE : MeasurableSet E)
+    (hactive : ∀ i ∈ S, f i ∈ E → i ∈ S') :
+    μ E ≤ ∑ i ∈ S', b i := by
+  classical
+  have himage : ((S.image f : Finset α) : Set α) = f '' (S : Set ι) := by
+    ext x
+    simp
+  rw [measure_eq_sum_singleton_inter_of_compl_eq_zero μ (S.image f)
+    (by rw [himage]; exact hS) E hE]
+  calc
+    ∑ x ∈ S.image f, μ ({x} ∩ E) = ∑ i ∈ S, μ ({f i} ∩ E) :=
+      Finset.sum_image fun i hi j hj h ↦ hf (Finset.mem_coe.2 hi) (Finset.mem_coe.2 hj) h
+    _ ≤ ∑ i ∈ S, (if f i ∈ E then b i else 0) := by
+      refine Finset.sum_le_sum fun i hi ↦ ?_
+      by_cases hfi : f i ∈ E
+      · have hinter : ({f i} : Set α) ∩ E = {f i} := by
+          ext y
+          simp [hfi]
+        simp only [hinter, hfi, ite_true]
+        exact hb i hi
+      · have hinter : ({f i} : Set α) ∩ E = ∅ := by
+          ext y
+          simp [hfi]
+        simp [hinter, hfi]
+    _ = ∑ i ∈ S.filter (fun i ↦ f i ∈ E), b i := (Finset.sum_filter _ _).symm
+    _ ≤ ∑ i ∈ S', b i := by
+      refine Finset.sum_le_sum_of_subset fun i hi ↦ ?_
+      obtain ⟨hiS, hiE⟩ := Finset.mem_filter.1 hi
+      exact hactive i hiS hiE
+
+/-! ### Almost every point of a half-open interval is interior -/
+
+/-- Almost every point of a half-open interval, for a measure with null singletons, lies in the
+corresponding open interval. -/
+theorem ae_restrict_Ico_mem_Ioo {μ : Measure ℝ} [NullSingletonClass μ] (a b : ℝ) :
+    ∀ᵐ t ∂μ.restrict (Ico a b), t ∈ Ioo a b :=
+  ae_restrict_mem_of_countable_diff measurableSet_Ico (countable_singleton a)
+    fun _ ⟨hx, hx'⟩ ↦ mem_singleton_iff.2 (le_antisymm (not_lt.1 fun h ↦ hx' ⟨h, hx.2⟩) hx.1)
+
+/-- Almost every point of a half-open interval, for a measure with null singletons, lies in the
+corresponding open interval. -/
+theorem ae_restrict_Ioc_mem_Ioo {μ : Measure ℝ} [NullSingletonClass μ] (a b : ℝ) :
+    ∀ᵐ t ∂μ.restrict (Ioc a b), t ∈ Ioo a b :=
+  ae_restrict_mem_of_countable_diff measurableSet_Ioc (countable_singleton b)
+    fun _ ⟨hx, hx'⟩ ↦ mem_singleton_iff.2 (le_antisymm hx.2 (not_lt.1 fun h ↦ hx' ⟨hx.1, h⟩))
+
+/-- Almost every point of a half-open interval, for a measure with null singletons, lies in one
+of the two open intervals cut out by an arbitrary intermediate point. -/
+theorem ae_restrict_Ico_mem_Ioo_union_Ioo {μ : Measure ℝ} [NullSingletonClass μ] (a b c : ℝ) :
+    ∀ᵐ t ∂μ.restrict (Ico a c), t ∈ Ioo a b ∪ Ioo b c := by
+  refine ae_restrict_mem_of_countable_diff measurableSet_Ico
+    ((countable_singleton b).insert a) ?_
+  rintro x ⟨hx, hx'⟩
+  rcases lt_trichotomy x b with h | h | h
+  · exact Or.inl (le_antisymm (not_lt.1 fun hlt ↦ hx' (Or.inl ⟨hlt, h⟩)) hx.1)
+  · exact Or.inr (mem_singleton_iff.2 h)
+  · exact absurd (Or.inr ⟨h, hx.2⟩) hx'
+
+/-- Almost every point of a half-open interval, for a measure with null singletons, lies in one
+of the two open intervals cut out by an arbitrary intermediate point. -/
+theorem ae_restrict_Ioc_mem_Ioo_union_Ioo {μ : Measure ℝ} [NullSingletonClass μ] (a b c : ℝ) :
+    ∀ᵐ t ∂μ.restrict (Ioc a c), t ∈ Ioo a b ∪ Ioo b c := by
+  refine ae_restrict_mem_of_countable_diff measurableSet_Ioc
+    ((countable_singleton c).insert b) ?_
+  rintro x ⟨hx, hx'⟩
+  rcases lt_trichotomy x b with h | h | h
+  · exact absurd (Or.inl ⟨hx.1, h⟩) hx'
+  · exact Or.inl h
+  · exact Or.inr (mem_singleton_iff.2
+      (le_antisymm hx.2 (not_lt.1 fun hlt ↦ hx' (Or.inr ⟨h, hlt⟩))))
+
+end MeasureTheory
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Haar-null lines and inner-product level sets
+
+Lines and hyperplanes of a finite-dimensional real normed space carry no additive Haar mass.
+This file records the two convenient forms used when a planar region is exhausted by triangles
+up to the rays through finitely many vertices.
+-/
+
+@[expose] public section
+
+open MeasureTheory
+open scoped Pointwise
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+/-- A line through the origin is a proper subspace once the ambient dimension exceeds one. -/
+theorem Submodule.span_singleton_ne_top (h : 1 < Module.finrank ℝ E) (v : E) :
+    (ℝ ∙ v) ≠ ⊤ := by
+  rcases eq_or_ne v 0 with rfl | hv
+  · have : Nontrivial E := Module.nontrivial_of_finrank_pos (R := ℝ) (M := E) (by omega)
+    rw [Submodule.span_zero_singleton]
+    exact bot_ne_top
+  · intro htop
+    have hfin : Module.finrank ℝ E = 1 := (finrank_eq_one_iff_of_nonzero v hv).2 htop
+    omega
+
+variable [MeasurableSpace E] [BorelSpace E] [FiniteDimensional ℝ E]
+
+/-- Any line in a space of dimension at least two is null for an additive Haar measure. -/
+theorem MeasureTheory.Measure.addHaar_vadd_span_singleton (μ : Measure E)
+    [μ.IsAddHaarMeasure] (h : 1 < Module.finrank ℝ E) (o v : E) :
+    μ (o +ᵥ (ℝ ∙ v : Set E)) = 0 := by
+  rw [measure_vadd]
+  exact μ.addHaar_submodule _ (Submodule.span_singleton_ne_top h v)
+
+/-- A countable union of lines through a common point is null. -/
+theorem MeasureTheory.Measure.addHaar_iUnion_vadd_span_singleton {ι : Type*} [Countable ι]
+    (μ : Measure E) [μ.IsAddHaarMeasure] (h : 1 < Module.finrank ℝ E) (o : E) (v : ι → E) :
+    μ (⋃ i, o +ᵥ (ℝ ∙ v i : Set E)) = 0 :=
+  measure_iUnion_null fun i ↦ μ.addHaar_vadd_span_singleton h o (v i)
+
+/-- A level set of a nonzero inner-product functional is null for an additive Haar measure. -/
+theorem MeasureTheory.Measure.addHaar_setOf_real_inner_eq {F : Type*} [NormedAddCommGroup F]
+    [InnerProductSpace ℝ F] [MeasurableSpace F] [BorelSpace F] [FiniteDimensional ℝ F]
+    (μ : Measure F) [μ.IsAddHaarMeasure] {u : F} (hu : u ≠ 0) (c : ℝ) :
+    μ {p : F | inner ℝ p u = c} = 0 := by
+  have hu2 : (0 : ℝ) < ‖u‖ ^ 2 := by positivity
+  set f : F →ᵃ[ℝ] ℝ := (innerSL ℝ u).toLinearMap.toAffineMap with hf
+  set A := (AffineSubspace.mk' c (⊥ : Submodule ℝ ℝ)).comap f with hA
+  have hAset : (A : Set F) = {p : F | inner ℝ p u = c} := by
+    ext p
+    simp [hA, hf, AffineSubspace.mem_mk', real_inner_comm, sub_eq_zero]
+  rw [← hAset]
+  refine μ.addHaar_affineSubspace _ fun htop ↦ ?_
+  have hp : ((c + ‖u‖ ^ 2) / ‖u‖ ^ 2) • u ∈ A := by rw [htop]; trivial
+  rw [← SetLike.mem_coe, hAset, Set.mem_ofPred_eq] at hp
+  have : inner ℝ (((c + ‖u‖ ^ 2) / ‖u‖ ^ 2) • u) u = c + ‖u‖ ^ 2 := by
+    rw [real_inner_smul_left, real_inner_self_eq_norm_sq]
+    field_simp
+  rw [this] at hp
+  linarith
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Measure / Map
+-/
+
+@[expose] public section
+
+namespace MeasureTheory.Measure
+
+variable {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α}
+
+/-- If `g` is almost everywhere a left inverse of `f`, then pushing a measure forward along `f`
+and then along `g` recovers it. This is the almost-everywhere form of
+`MeasurableEquiv.map_symm_map`. -/
+theorem map_map_of_ae_leftInverse {f : α → β} (hf : Measurable f) {g : β → α}
+    (hg : Measurable g) (h : ∀ᵐ a ∂μ, g (f a) = a) :
+    (μ.map f).map g = μ := by
+  rw [map_map hg hf, show μ.map (g ∘ f) = μ.map id from map_congr h, map_id]
+
+end MeasureTheory.Measure
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Measure / With Density
+-/
+
+@[expose] public section
+
+open Set
+open scoped ENNReal
+
+namespace MeasureTheory.Measure
+
+/-- An injective image of a weighted restriction preserves null singleton masses. -/
+theorem map_restrict_withDensity_singleton {α β : Type*}
+    [MeasurableSpace α] [MeasurableSpace β] [MeasurableSingletonClass β]
+    (μ : Measure α) [NullSingletonClass μ] (f : α → β) (hf : Measurable f)
+    (s : Set α) (hinj : Set.InjOn f s) (w : α → ℝ≥0∞) {x : α} (hx : x ∈ s) :
+    Measure.map f ((μ.restrict s).withDensity w) {f x} = 0 := by
+  rw [Measure.map_apply hf (measurableSet_singleton _)]
+  apply withDensity_absolutelyContinuous
+  rw [Measure.restrict_apply (hf (measurableSet_singleton _))]
+  have heq : f ⁻¹' {f x} ∩ s = {x} := by
+    ext y
+    simp only [mem_inter_iff, mem_preimage, mem_singleton_iff]
+    exact ⟨fun h ↦ hinj h.2 hx h.1, fun h ↦ by subst y; exact ⟨rfl, hx⟩⟩
+  rw [heq, measure_singleton]
+
+/-- Pushing a weighted measure forward along a measurable map is linear in the weight: if `w` is
+the pointwise combination `a * w₁ + b * w₂`, then the pushforward of `μ.withDensity w` is the
+same combination of the pushforwards of `μ.withDensity w₁` and `μ.withDensity w₂`. -/
+theorem map_withDensity_eq_smul_add_smul {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]
+    (μ : Measure α) {φ : α → β} (hφ : Measurable φ) (a b : ℝ≥0∞) {w w₁ w₂ : α → ℝ≥0∞}
+    (h₁ : Measurable w₁) (h₂ : Measurable w₂) (hw : ∀ x, w x = a * w₁ x + b * w₂ x) :
+    Measure.map φ (μ.withDensity w) =
+      a • Measure.map φ (μ.withDensity w₁) + b • Measure.map φ (μ.withDensity w₂) := by
+  have hwfun : w = a • w₁ + b • w₂ := funext hw
+  rw [hwfun, withDensity_add_left (h₁.const_smul a), withDensity_smul _ h₁,
+    withDensity_smul _ h₂, Measure.map_add _ _ hφ, Measure.map_smul _ hφ.aemeasurable,
+    Measure.map_smul _ hφ.aemeasurable]
+
+end MeasureTheory.Measure
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Region Between
+-/
+
+@[expose] public section
+
+open EuclideanSpace MeasureTheory Set
+
+theorem volume_setOf_mem_Icc_eq_volume_regionBetween {f g : ℝ → ℝ} {s : Set ℝ}
+    (hf : Measurable f) (hg : Measurable g) (hs : MeasurableSet s) :
+    volume {p : ℝ × ℝ | p.1 ∈ s ∧ p.2 ∈ Icc (f p.1) (g p.1)} =
+      volume (regionBetween f g s) := by
+  change (volume.prod volume) _ = (volume.prod volume) _
+  rw [Measure.prod_apply (measurableSet_region_between_cc hf hg hs),
+    Measure.prod_apply (measurableSet_regionBetween hf hg hs)]
+  apply lintegral_congr
+  intro x
+  by_cases hx : x ∈ s
+  · simp only [regionBetween, Set.preimage_ofPred_eq, hx, true_and]
+    change volume (Icc (f x) (g x)) = volume (Ioo (f x) (g x))
+    rw [Real.volume_Icc, Real.volume_Ioo]
+  · simp [regionBetween, hx]
+
+theorem volume_regionBetween_triangle {b h : ℝ} (hb : 0 < b) (hh : 0 ≤ h) :
+    volume (regionBetween (fun _ : ℝ ↦ 0) (fun x ↦ h - h / b * x) (Ioc 0 b)) =
+      ENNReal.ofReal (b * h / 2) := by
+  have hc : Continuous (fun x : ℝ ↦ h - h / b * x) := by fun_prop
+  have hi := hc.intervalIntegrable (μ := volume) 0 b
+  have hz := (continuous_const : Continuous (fun _ : ℝ ↦ (0 : ℝ))).intervalIntegrable
+    (μ := volume) 0 b
+  change (volume.prod volume) _ = _
+  rw [volume_regionBetween_eq_integral
+    ((intervalIntegrable_iff_integrableOn_Ioc_of_le hb.le).mp hz)
+    ((intervalIntegrable_iff_integrableOn_Ioc_of_le hb.le).mp hi) measurableSet_Ioc]
+  · congr 1
+    simp only [Pi.sub_apply, sub_zero]
+    rw [← intervalIntegral.integral_of_le hb.le]
+    have hm : IntervalIntegrable (fun x : ℝ ↦ h / b * x) volume 0 b :=
+      (continuous_const.mul continuous_id).intervalIntegrable 0 b
+    have hh' : IntervalIntegrable (fun _ : ℝ ↦ h) volume 0 b :=
+      continuous_const.intervalIntegrable 0 b
+    rw [intervalIntegral.integral_sub hh' hm,
+      intervalIntegral.integral_const,
+      intervalIntegral.integral_const_mul, integral_id]
+    simp only [sub_zero, smul_eq_mul]
+    field_simp
+    ring
+  · intro x hx
+    have := mul_le_mul_of_nonneg_left hx.2 (div_nonneg hh hb.le)
+    have heq : h / b * b = h := div_mul_cancel₀ h hb.ne'
+    linarith
+
+theorem volume_setOf_mem_Ioc_eq_volume_regionBetween
+    {f g : ℝ → ℝ} {s : Set ℝ}
+    (hf : Measurable f) (hg : Measurable g) (hs : MeasurableSet s) :
+    volume {p : ℝ × ℝ | p.1 ∈ s ∧ p.2 ∈ Ioc (f p.1) (g p.1)} =
+      volume (regionBetween f g s) := by
+  change (volume.prod volume) _ = (volume.prod volume) _
+  rw [Measure.prod_apply (measurableSet_region_between_oc hf hg hs),
+    Measure.prod_apply (measurableSet_regionBetween hf hg hs)]
+  apply lintegral_congr
+  intro x
+  by_cases hx : x ∈ s
+  · simp only [regionBetween, Set.preimage_ofPred_eq, hx, true_and]
+    change volume (Ioc (f x) (g x)) = volume (Ioo (f x) (g x))
+    rw [Real.volume_Ioc, Real.volume_Ioo]
+  · simp [regionBetween, hx]
+
+/-- The planar volume of the horizontal band between the graphs of `f` and `g` over `s`,
+where the first coordinate is the one squeezed between the two graphs. -/
+theorem volume_horizontalIcc {f g : ℝ → ℝ} {s : Set ℝ}
+    (hf : Measurable f) (hg : Measurable g) (hs : MeasurableSet s)
+    (hfi : IntegrableOn f s) (hgi : IntegrableOn g s)
+    (hfg : ∀ x ∈ s, f x ≤ g x) :
+    volume {p : EuclideanSpace ℝ (Fin 2) | p 1 ∈ s ∧ p 0 ∈ Icc (f (p 1)) (g (p 1))} =
+      ENNReal.ofReal (∫ x in s, (g - f) x) := by
+  let T : Set (ℝ × ℝ) :=
+    {p | p.1 ∈ s ∧ p.2 ∈ Icc (f p.1) (g p.1)}
+  have hT : MeasurableSet T := measurableSet_region_between_cc hf hg hs
+  rw [show {p : EuclideanSpace ℝ (Fin 2) | p 1 ∈ s ∧ p 0 ∈ Icc (f (p 1)) (g (p 1))} =
+      finTwoCoordinatesSwap ⁻¹' T by rfl,
+    volume_preserving_finTwoCoordinatesSwap.measure_preimage hT.nullMeasurableSet]
+  change volume T = _
+  rw [volume_setOf_mem_Icc_eq_volume_regionBetween hf hg hs]
+  change (volume.prod volume) (regionBetween f g s) = _
+  rw [volume_regionBetween_eq_integral hfi hgi hs hfg]
+
+/-- The half-open variant of `volume_horizontalIcc`. -/
+theorem volume_horizontalIoc {f g : ℝ → ℝ} {s : Set ℝ}
+    (hf : Measurable f) (hg : Measurable g) (hs : MeasurableSet s)
+    (hfi : IntegrableOn f s) (hgi : IntegrableOn g s)
+    (hfg : ∀ x ∈ s, f x ≤ g x) :
+    volume {p : EuclideanSpace ℝ (Fin 2) | p 1 ∈ s ∧ p 0 ∈ Ioc (f (p 1)) (g (p 1))} =
+      ENNReal.ofReal (∫ x in s, (g - f) x) := by
+  let T : Set (ℝ × ℝ) :=
+    {p | p.1 ∈ s ∧ p.2 ∈ Ioc (f p.1) (g p.1)}
+  have hT : MeasurableSet T := measurableSet_region_between_oc hf hg hs
+  rw [show {p : EuclideanSpace ℝ (Fin 2) | p 1 ∈ s ∧ p 0 ∈ Ioc (f (p 1)) (g (p 1))} =
+      finTwoCoordinatesSwap ⁻¹' T by rfl,
+    volume_preserving_finTwoCoordinatesSwap.measure_preimage hT.nullMeasurableSet]
+  change volume T = _
+  rw [volume_setOf_mem_Ioc_eq_volume_regionBetween hf hg hs]
+  change (volume.prod volume) (regionBetween f g s) = _
+  rw [volume_regionBetween_eq_integral hfi hgi hs hfg]
+
+/-- A planar set whose second coordinate lies in `[0, 1]` and whose first coordinate is
+squeezed between a continuous graph and its horizontal translate by `c` has volume at
+most `c`. Only the upper bound is asserted, so no measurability of the set is needed. -/
+theorem volume_le_of_subset_horizontalBand {X : Set (EuclideanSpace ℝ (Fin 2))}
+    {f : ℝ → ℝ} {c : ℝ} (hf : Continuous f) (hc : 0 ≤ c)
+    (hX : ∀ p ∈ X, (0 ≤ p 1 ∧ p 1 ≤ 1) ∧ f (p 1) ≤ p 0 ∧ p 0 ≤ f (p 1) + c) :
+    volume X ≤ ENNReal.ofReal c := by
+  have hg : Continuous (fun y ↦ f y + c) := hf.add continuous_const
+  have hfi : IntegrableOn f (Icc (0 : ℝ) 1) :=
+    (intervalIntegrable_iff_integrableOn_Icc_of_le (by norm_num)).mp
+      (hf.intervalIntegrable 0 1)
+  have hgi : IntegrableOn (fun y ↦ f y + c) (Icc (0 : ℝ) 1) :=
+    (intervalIntegrable_iff_integrableOn_Icc_of_le (by norm_num)).mp
+      (hg.intervalIntegrable 0 1)
+  have key := volume_horizontalIcc (f := f) (g := fun y ↦ f y + c)
+    (s := Icc (0 : ℝ) 1) hf.measurable hg.measurable measurableSet_Icc hfi hgi
+    (fun x _ ↦ by simp [hc])
+  have hint : ∫ x in Icc (0 : ℝ) 1, ((fun y ↦ f y + c) - f) x = c := by
+    have hdiff : ((fun y ↦ f y + c) - f) = fun _ ↦ c := by funext y; simp
+    rw [hdiff, setIntegral_const]
+    simp
+  rw [hint] at key
+  refine le_trans (measure_mono ?_) key.le
+  intro p hp
+  obtain ⟨h1, h2, h3⟩ := hX p hp
+  exact ⟨h1, h2, h3⟩
+
+/-- The horizontal unit strip meets the band `c ≤ a * x + b * y ≤ c + 1` in a
+parallelogram of area `1 / |a|`. Only the upper bound is asserted. -/
+theorem volume_horizontalBand_inter_le (a b c : ℝ) (ha : a ≠ 0) :
+    volume {p : EuclideanSpace ℝ (Fin 2) | (0 ≤ p 1 ∧ p 1 ≤ 1) ∧
+        (c ≤ a * p 0 + b * p 1 ∧ a * p 0 + b * p 1 ≤ c + 1)} ≤
+      ENNReal.ofReal (1 / |a|) := by
+  have habs : 0 < |a| := abs_pos.mpr ha
+  rcases lt_or_gt_of_ne ha with hneg | hpos
+  · refine volume_le_of_subset_horizontalBand (f := fun y ↦ (c + 1 - b * y) / a)
+      (c := 1 / |a|) (by fun_prop) (by positivity) ?_
+    rintro p ⟨h1, h2, h3⟩
+    refine ⟨h1, ?_, ?_⟩
+    · rw [div_le_iff_of_neg hneg]
+      linarith
+    · have hrw : (c + 1 - b * p 1) / a + 1 / |a| = (c - b * p 1) / a := by
+        rw [abs_of_neg hneg]
+        field_simp
+        ring
+      rw [hrw, le_div_iff_of_neg hneg]
+      linarith
+  · refine volume_le_of_subset_horizontalBand (f := fun y ↦ (c - b * y) / a)
+      (c := 1 / |a|) (by fun_prop) (by positivity) ?_
+    rintro p ⟨h1, h2, h3⟩
+    refine ⟨h1, ?_, ?_⟩
+    · rw [div_le_iff₀ hpos]
+      linarith
+    · have hrw : (c - b * p 1) / a + 1 / |a| = (c + 1 - b * p 1) / a := by
+        rw [abs_of_pos hpos]
+        field_simp
+        ring
+      rw [hrw, le_div_iff₀ hpos]
+      linarith
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Planar trapezoids with horizontal parallel sides
+-/
+
+@[expose] public section
+
+namespace EuclideanSpace
+
+/-- A convex planar set containing the four vertices of a trapezoid whose parallel sides are
+the horizontal segments at heights `0` and `1` contains the whole trapezoid. -/
+theorem horizontalTrapezoid_subset_of_convex {S : Set (EuclideanSpace ℝ (Fin 2))}
+    (hS : Convex ℝ S) {l₀ r₀ l₁ r₁ : ℝ}
+    (h₀l : (!₂[l₀, 0] : EuclideanSpace ℝ (Fin 2)) ∈ S)
+    (h₀r : (!₂[r₀, 0] : EuclideanSpace ℝ (Fin 2)) ∈ S)
+    (h₁l : (!₂[l₁, 1] : EuclideanSpace ℝ (Fin 2)) ∈ S)
+    (h₁r : (!₂[r₁, 1] : EuclideanSpace ℝ (Fin 2)) ∈ S) :
+    {q : EuclideanSpace ℝ (Fin 2) | q 1 ∈ Set.Icc (0 : ℝ) 1 ∧
+        q 0 ∈ Set.Icc ((1 - q 1) * l₀ + q 1 * l₁) ((1 - q 1) * r₀ + q 1 * r₁)} ⊆ S := by
+  rintro q ⟨⟨hy0, hy1⟩, hql, hqr⟩
+  have hleft : ((1 - q 1) • (!₂[l₀, 0] : EuclideanSpace ℝ (Fin 2)) + q 1 • !₂[l₁, 1]) ∈ S :=
+    hS h₀l h₁l (by linarith) hy0 (by ring)
+  have hright : ((1 - q 1) • (!₂[r₀, 0] : EuclideanSpace ℝ (Fin 2)) + q 1 • !₂[r₁, 1]) ∈ S :=
+    hS h₀r h₁r (by linarith) hy0 (by ring)
+  obtain ⟨s, hs0, hs1, hs⟩ : ∃ s : ℝ, 0 ≤ s ∧ s ≤ 1 ∧
+      q 0 = (1 - s) * ((1 - q 1) * l₀ + q 1 * l₁) + s * ((1 - q 1) * r₀ + q 1 * r₁) := by
+    rcases (hql.trans hqr).eq_or_lt with hLR | hLR
+    · exact ⟨0, le_rfl, zero_le_one, by rw [sub_zero, one_mul, zero_mul, add_zero]; linarith⟩
+    · have hne : (1 - q 1) * r₀ + q 1 * r₁ - ((1 - q 1) * l₀ + q 1 * l₁) ≠ 0 := by linarith
+      refine ⟨(q 0 - ((1 - q 1) * l₀ + q 1 * l₁)) /
+          ((1 - q 1) * r₀ + q 1 * r₁ - ((1 - q 1) * l₀ + q 1 * l₁)),
+        div_nonneg (by linarith) (by linarith), by rw [div_le_one (by linarith)]; linarith, ?_⟩
+      field_simp
+      ring
+  have hcoord : ∀ z w : EuclideanSpace ℝ (Fin 2), z 0 = w 0 → z 1 = w 1 → z = w := by
+    intro z w h0 h1
+    ext i
+    fin_cases i
+    · exact h0
+    · exact h1
+  have heq : q = (1 - s) • ((1 - q 1) • (!₂[l₀, 0] : EuclideanSpace ℝ (Fin 2)) + q 1 • !₂[l₁, 1]) +
+      s • ((1 - q 1) • (!₂[r₀, 0] : EuclideanSpace ℝ (Fin 2)) + q 1 • !₂[r₁, 1]) := by
+    refine hcoord _ _ ?_ ?_
+    · simp only [PiLp.add_apply, PiLp.smul_apply, smul_eq_mul, Matrix.cons_val_zero]
+      linear_combination hs
+    · simp only [PiLp.add_apply, PiLp.smul_apply, smul_eq_mul, Matrix.cons_val_one,
+        Matrix.cons_val_fin_one]
+      ring
+  rw [heq]
+  exact hS hleft hright (by linarith) hs0 (by ring)
+
+/-- The planar area of the trapezoid cut from the horizontal band `y₀ ≤ y ≤ y₁` by the two lines
+`x = a + b * y` and `x = c + d * y`: the height of the band times the mean of the lengths of its
+two horizontal sides. -/
+theorem volume_horizontalTrapezoid_band {y₀ y₁ a b c d : ℝ} (hy : y₀ ≤ y₁)
+    (h₀ : a + b * y₀ ≤ c + d * y₀) (h₁ : a + b * y₁ ≤ c + d * y₁) :
+    MeasureTheory.volume {q : EuclideanSpace ℝ (Fin 2) | q 1 ∈ Set.Icc y₀ y₁ ∧
+        q 0 ∈ Set.Icc (a + b * q 1) (c + d * q 1)} =
+      ENNReal.ofReal ((y₁ - y₀) *
+        ((c + d * y₀ - (a + b * y₀)) + (c + d * y₁ - (a + b * y₁))) / 2) := by
+  have hfc : Continuous fun y : ℝ ↦ a + b * y := by fun_prop
+  have hgc : Continuous fun y : ℝ ↦ c + d * y := by fun_prop
+  rw [volume_horizontalIcc (f := fun y : ℝ ↦ a + b * y) (g := fun y : ℝ ↦ c + d * y)
+    (s := Set.Icc y₀ y₁) hfc.measurable hgc.measurable measurableSet_Icc
+    ((intervalIntegrable_iff_integrableOn_Icc_of_le hy).mp (hfc.intervalIntegrable y₀ y₁))
+    ((intervalIntegrable_iff_integrableOn_Icc_of_le hy).mp (hgc.intervalIntegrable y₀ y₁))
+    (fun y hy' ↦ by
+      rcases hy.eq_or_lt with rfl | hylt
+      · have : y = y₀ := le_antisymm hy'.2 hy'.1
+        rw [this]
+        exact h₀
+      · have hcoef : (y₁ - y) * (c + d * y₀ - (a + b * y₀)) +
+            (y - y₀) * (c + d * y₁ - (a + b * y₁)) =
+            (y₁ - y₀) * (c + d * y - (a + b * y)) := by ring
+        nlinarith [mul_nonneg (by linarith [hy'.2] : (0 : ℝ) ≤ y₁ - y) (by linarith : (0 : ℝ) ≤
+            c + d * y₀ - (a + b * y₀)),
+          mul_nonneg (by linarith [hy'.1] : (0 : ℝ) ≤ y - y₀) (by linarith : (0 : ℝ) ≤
+            c + d * y₁ - (a + b * y₁))])]
+  congr 1
+  have hrw : ((fun y : ℝ ↦ c + d * y) - fun y : ℝ ↦ a + b * y) =
+      fun y : ℝ ↦ (c - a) + (d - b) * y := by
+    funext y
+    simp only [Pi.sub_apply]
+    ring
+  have hii : IntervalIntegrable (fun x : ℝ ↦ (d - b) * x) MeasureTheory.volume y₀ y₁ :=
+    (by fun_prop : Continuous fun x : ℝ ↦ (d - b) * x).intervalIntegrable y₀ y₁
+  rw [hrw, MeasureTheory.integral_Icc_eq_integral_Ioc,
+    ← intervalIntegral.integral_of_le hy,
+    intervalIntegral.integral_add intervalIntegrable_const hii,
+    intervalIntegral.integral_const, intervalIntegral.integral_const_mul, integral_id]
+  simp only [smul_eq_mul]
+  ring
+
+/-- The planar area of a trapezoid whose parallel sides are the horizontal segments
+`[l₀, r₀] × {0}` and `[l₁, r₁] × {1}` is the mean of their lengths. -/
+theorem volume_horizontalTrapezoid {l₀ r₀ l₁ r₁ : ℝ} (h₀ : l₀ ≤ r₀) (h₁ : l₁ ≤ r₁) :
+    MeasureTheory.volume {q : EuclideanSpace ℝ (Fin 2) | q 1 ∈ Set.Icc (0 : ℝ) 1 ∧
+        q 0 ∈ Set.Icc ((1 - q 1) * l₀ + q 1 * l₁) ((1 - q 1) * r₀ + q 1 * r₁)} =
+      ENNReal.ofReal ((r₀ - l₀ + (r₁ - l₁)) / 2) := by
+  have hset : {q : EuclideanSpace ℝ (Fin 2) | q 1 ∈ Set.Icc (0 : ℝ) 1 ∧
+        q 0 ∈ Set.Icc ((1 - q 1) * l₀ + q 1 * l₁) ((1 - q 1) * r₀ + q 1 * r₁)} =
+      {q : EuclideanSpace ℝ (Fin 2) | q 1 ∈ Set.Icc (0 : ℝ) 1 ∧
+        q 0 ∈ Set.Icc (l₀ + (l₁ - l₀) * q 1) (r₀ + (r₁ - r₀) * q 1)} := by
+    refine Set.ext fun q ↦ ?_
+    simp only [Set.mem_ofPred_eq,
+      show (1 - q 1) * l₀ + q 1 * l₁ = l₀ + (l₁ - l₀) * q 1 from by ring,
+      show (1 - q 1) * r₀ + q 1 * r₁ = r₀ + (r₁ - r₀) * q 1 from by ring]
+  rw [hset, volume_horizontalTrapezoid_band zero_le_one (by simpa using h₀) (by simpa using h₁)]
+  congr 1
+  ring
+
+end EuclideanSpace
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The area of a planar triangle
+
+The convex hull of three points of the Euclidean plane has area one half of the absolute
+determinant of the two edge vectors emanating from the first point. The proof transports the
+standard right triangle, whose area is computed by integration, along the linear map sending
+the coordinate basis to the two edge vectors.
+-/
+
+@[expose] public section
+
+open MeasureTheory Set
+open scoped Pointwise
+
+namespace EuclideanSpace
+
+private def standardTriangleProd : Set (ℝ × ℝ) :=
+  {p | p.1 ∈ Ioc 0 1 ∧ p.2 ∈ Icc 0 (1 - p.1)}
+
+private theorem measurableSet_standardTriangleProd : MeasurableSet standardTriangleProd := by
+  have h : MeasurableSet {p : ℝ × ℝ | (0 < p.1 ∧ p.1 ≤ 1) ∧ 0 ≤ p.2 ∧ p.2 ≤ 1 - p.1} :=
+    ((measurableSet_lt measurable_const measurable_fst).inter
+        (measurableSet_le measurable_fst measurable_const)).inter
+      ((measurableSet_le measurable_const measurable_snd).inter
+        (measurableSet_le measurable_snd (measurable_const.sub measurable_fst)))
+  simpa only [standardTriangleProd, Set.mem_Ioc, Set.mem_Icc] using h
+
+private theorem volume_standardTriangleProd :
+    volume standardTriangleProd = ENNReal.ofReal (1 / 2) := by
+  rw [show standardTriangleProd =
+      {p : ℝ × ℝ | p.1 ∈ Ioc 0 1 ∧ p.2 ∈ Icc (0 : ℝ) (1 - p.1)} from rfl,
+    volume_setOf_mem_Icc_eq_volume_regionBetween
+      (f := fun _ : ℝ ↦ 0) (g := fun x ↦ 1 - x) (s := Ioc 0 1)
+      measurable_const (measurable_const.sub measurable_id) measurableSet_Ioc]
+  simpa using
+    (volume_regionBetween_triangle (b := 1) (h := 1)
+      (by norm_num : (0 : ℝ) < 1) (by norm_num : (0 : ℝ) ≤ 1))
+
+private def standardTriangle : Set (EuclideanSpace ℝ (Fin 2)) :=
+  {p | p 0 ∈ Ioc 0 1 ∧ p 1 ∈ Icc 0 (1 - p 0)}
+
+private theorem volume_standardTriangle : volume standardTriangle = ENNReal.ofReal (1 / 2) := by
+  have hpre := volume_preserving_finTwoCoordinates.measure_preimage
+    measurableSet_standardTriangleProd.nullMeasurableSet
+  rw [show (fun p : EuclideanSpace ℝ (Fin 2) ↦ (p 0, p 1)) ⁻¹' standardTriangleProd =
+      standardTriangle from rfl, volume_standardTriangleProd] at hpre
+  exact hpre
+
+private def closedStandardTriangle : Set (EuclideanSpace ℝ (Fin 2)) :=
+  {p | 0 ≤ p 0 ∧ 0 ≤ p 1 ∧ p 1 ≤ 1 - p 0}
+
+private theorem volume_closedStandardTriangle :
+    volume closedStandardTriangle = ENNReal.ofReal (1 / 2) := by
+  rw [← volume_standardTriangle]
+  refine (measure_eq_measure_of_null_sdiff (μ := volume) ?_ ?_).symm
+  · exact fun p hp ↦ ⟨hp.1.1.le, hp.2⟩
+  · refine measure_mono_null
+      (t := {p : EuclideanSpace ℝ (Fin 2) | inner ℝ p (EuclideanSpace.single 0 (1 : ℝ)) = 0})
+      (fun p hp ↦ ?_) ?_
+    · have hp0 : p 0 = 0 := by
+        refine le_antisymm (le_of_not_gt fun hpos ↦ hp.2 ⟨⟨hpos, ?_⟩, hp.1.2⟩) hp.1.1
+        linarith [hp.1.2.1, hp.1.2.2]
+      simpa [EuclideanSpace.inner_single_right] using hp0
+    · refine volume.addHaar_setOf_real_inner_eq (fun h ↦ ?_) 0
+      have hone : (1 : ℝ) = 0 := by
+        simpa [PiLp.single_apply] using congrArg (fun p : EuclideanSpace ℝ (Fin 2) ↦ p 0) h
+      exact one_ne_zero hone
+
+private def standardTriangleVertices : Set (EuclideanSpace ℝ (Fin 2)) :=
+  {0, EuclideanSpace.single 0 (1 : ℝ), EuclideanSpace.single 1 (1 : ℝ)}
+
+private theorem convex_closedStandardTriangle : Convex ℝ closedStandardTriangle := by
+  intro p hp q hq a b ha hb hab
+  simp only [closedStandardTriangle, Set.mem_ofPred_eq] at hp hq ⊢
+  refine ⟨?_, ?_, ?_⟩ <;>
+    simp only [PiLp.add_apply, PiLp.smul_apply, smul_eq_mul] <;>
+    nlinarith [hp.1, hq.1, hp.2.1, hq.2.1, hp.2.2, hq.2.2]
+
+private theorem convexHull_standardTriangleVertices :
+    convexHull ℝ standardTriangleVertices = closedStandardTriangle := by
+  refine Set.Subset.antisymm (convexHull_min (fun p hp ↦ ?_) convex_closedStandardTriangle)
+    (fun p hp ↦ ?_)
+  · simp only [standardTriangleVertices, Set.mem_insert_iff, Set.mem_singleton_iff] at hp
+    rcases hp with rfl | rfl | rfl <;>
+      norm_num [closedStandardTriangle, PiLp.single_apply]
+  · simp only [closedStandardTriangle, Set.mem_ofPred_eq] at hp
+    refine mem_convexHull_of_exists_fintype ![1 - p 0 - p 1, p 0, p 1]
+      ![0, EuclideanSpace.single 0 (1 : ℝ), EuclideanSpace.single 1 (1 : ℝ)] ?_ ?_ ?_ ?_
+    · intro i
+      fin_cases i <;> simp <;> linarith
+    · simp [Fin.sum_univ_succ]
+    · intro i
+      fin_cases i <;> simp [standardTriangleVertices]
+    · ext i
+      fin_cases i <;> simp [Fin.sum_univ_succ]
+
+private def triangleMap (u v : EuclideanSpace ℝ (Fin 2)) :
+    EuclideanSpace ℝ (Fin 2) →ₗ[ℝ] EuclideanSpace ℝ (Fin 2) where
+  toFun p := p 0 • u + p 1 • v
+  map_add' p q := by
+    ext i
+    simp [add_smul, add_assoc, add_left_comm]
+  map_smul' c p := by
+    ext i
+    simp [mul_smul]
+
+private theorem det_triangleMap (u v : EuclideanSpace ℝ (Fin 2)) :
+    LinearMap.det (triangleMap u v) = u 0 * v 1 - v 0 * u 1 := by
+  rw [← LinearMap.det_toMatrix (PiLp.basisFun 2 ℝ (Fin 2)), Matrix.det_fin_two]
+  simp only [LinearMap.toMatrix_apply, PiLp.basisFun_repr, triangleMap,
+    PiLp.basisFun_apply, Fin.isValue]
+  norm_num
+
+private theorem convexHull_triple_eq_image (u v : EuclideanSpace ℝ (Fin 2)) :
+    convexHull ℝ ({0, u, v} : Set (EuclideanSpace ℝ (Fin 2))) =
+      triangleMap u v '' closedStandardTriangle := by
+  have himage : triangleMap u v '' standardTriangleVertices =
+      ({0, u, v} : Set (EuclideanSpace ℝ (Fin 2))) := by
+    ext p
+    constructor
+    · rintro ⟨q, hq, rfl⟩
+      simp only [standardTriangleVertices, Set.mem_insert_iff, Set.mem_singleton_iff] at hq
+      rcases hq with rfl | rfl | rfl <;> simp [triangleMap]
+    · intro hp
+      simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hp
+      rcases hp with rfl | rfl | rfl
+      · exact ⟨0, by simp [standardTriangleVertices], by simp [triangleMap]⟩
+      · exact ⟨EuclideanSpace.single 0 (1 : ℝ), by simp [standardTriangleVertices],
+          by simp [triangleMap]⟩
+      · exact ⟨EuclideanSpace.single 1 (1 : ℝ), by simp [standardTriangleVertices],
+          by simp [triangleMap]⟩
+  rw [← himage, ← LinearMap.image_convexHull, convexHull_standardTriangleVertices]
+
+/-- The triangle spanned by the origin and two vectors of the Euclidean plane has area one
+half of the absolute determinant of their coordinates. -/
+theorem volume_convexHull_zero_pair (u v : EuclideanSpace ℝ (Fin 2)) :
+    volume (convexHull ℝ ({0, u, v} : Set (EuclideanSpace ℝ (Fin 2)))) =
+      ENNReal.ofReal (|u 0 * v 1 - v 0 * u 1| / 2) := by
+  rw [convexHull_triple_eq_image, Measure.addHaar_image_linearMap, det_triangleMap,
+    volume_closedStandardTriangle, ← ENNReal.ofReal_mul (abs_nonneg _)]
+  congr 1
+  ring
+
+/-- The area of a planar triangle is one half of the absolute coordinate determinant of its
+two edge vectors. -/
+theorem volume_convexHull_triple (a b c : EuclideanSpace ℝ (Fin 2)) :
+    volume (convexHull ℝ ({a, b, c} : Set (EuclideanSpace ℝ (Fin 2)))) =
+      ENNReal.ofReal (|(b - a) 0 * (c - a) 1 - (c - a) 0 * (b - a) 1| / 2) := by
+  have hvertices : ({a, b, c} : Set (EuclideanSpace ℝ (Fin 2))) =
+      a +ᵥ ({0, b - a, c - a} : Set (EuclideanSpace ℝ (Fin 2))) := by
+    ext p
+    simp only [Set.mem_insert_iff, Set.mem_singleton_iff, Set.mem_vadd_set]
+    constructor
+    · intro hp
+      rcases hp with hp | hp | hp
+      · exact ⟨0, by simp, by simp [hp]⟩
+      · refine ⟨b - a, by simp, ?_⟩
+        rw [hp]
+        change a + (b - a) = b
+        abel
+      · refine ⟨c - a, by simp, ?_⟩
+        rw [hp]
+        change a + (c - a) = c
+        abel
+    · rintro ⟨q, hq, hp⟩
+      rcases hq with hq | hq | hq
+      · subst hq
+        simp only [vadd_eq_add, add_zero] at hp
+        exact hp ▸ Or.inl rfl
+      · refine Or.inr (Or.inl ?_)
+        rw [← hp, hq]
+        change a + (b - a) = b
+        abel
+      · refine Or.inr (Or.inr ?_)
+        rw [← hp, hq]
+        change a + (c - a) = c
+        abel
+  rw [hvertices, convexHull_vadd, measure_vadd, volume_convexHull_zero_pair]
+
+end EuclideanSpace
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Stieltjes Density
+-/
+
+@[expose] public section
+
+open MeasureTheory Set
+
+/-- Equality of interval increments identifies a continuous BV function's vector measure. -/
+theorem BoundedVariationOn.vectorMeasure_eq_withDensity_of_integral_Icc
+    {α E : Type*} [LinearOrder α] [DenselyOrdered α] [TopologicalSpace α] [OrderTopology α]
+    [SecondCountableTopology α] [MeasurableSpace α] [BorelSpace α]
+    [CompactIccSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+    {f r : α → E} {μ : Measure α} (hf : BoundedVariationOn f univ)
+    (hcont : Continuous f) (hr : Integrable r μ)
+    (hinc : ∀ a b, a ≤ b → f b - f a = ∫ x in Icc a b, r x ∂μ) :
+    hf.vectorMeasure = μ.withDensityᵥ r := by
+  apply VectorMeasure.ext_of_Icc
+  intro a b hab
+  rw [hf.vectorMeasure_Icc hab, (hcont.continuousAt (x := b)).continuousWithinAt.rightLim_eq,
+    (hcont.continuousAt (x := a)).continuousWithinAt.leftLim_eq,
+    withDensityᵥ_apply hr measurableSet_Icc]
+  exact hinc a b hab
+
+/-- Integrating on a subtype and a measurable preimage agrees with restricting both sets. -/
+theorem MeasureTheory.integral_subtype_preimage {α E : Type*} [MeasurableSpace α] {μ : Measure α}
+    [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {s t : Set α} (hs : MeasurableSet s) (ht : MeasurableSet t) (f : α → E) :
+    (∫ x in {x : s | (x : α) ∈ t}, f (x : α) ∂μ.comap Subtype.val) =
+      ∫ x in t, f x ∂μ.restrict s := by
+  have hpre : MeasurableSet {x : s | (x : α) ∈ t} := ht.preimage measurable_subtype_coe
+  rw [← integral_indicator (μ := μ.comap (Subtype.val : s → α))
+    (f := fun x : s ↦ f (x : α)) hpre]
+  change (∫ x : s, (t.indicator f) (x : α) ∂μ.comap Subtype.val) = _
+  rw [integral_subtype_comap hs, integral_indicator ht]
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Vector Measure / Interval
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Set MeasureTheory
+
+namespace MeasureTheory.VectorMeasure
+
+/-- Agreement on half-open intervals and the whole space determines a vector measure. -/
+theorem ext_of_Ioc {α E : Type*} [LinearOrder α] [TopologicalSpace α] [OrderTopology α]
+    [SecondCountableTopology α] [MeasurableSpace α] [BorelSpace α]
+    [NormedAddCommGroup E] (μ ν : VectorMeasure α E)
+    (h : ∀ a b, a < b → μ (Ioc a b) = ν (Ioc a b))
+    (huniv : μ univ = ν univ) : μ = ν := by
+  apply ext_of_generateFrom {s | ∃ a b, a < b ∧ Ioc a b = s} _
+    (BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ioc α))
+    (isPiSystem_Ioc id id) huniv
+  rintro s ⟨a, b, hab, rfl⟩
+  exact h a b hab
+
+end MeasureTheory.VectorMeasure
+
+/-- Pulling an integrable density back along a measurable embedding gives its image integrals. -/
+theorem MeasurableEmbedding.exists_vectorMeasure_image_integral
+    {α β E : Type*} [MeasurableSpace α] [MeasurableSpace β]
+    [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {f : α → β} (hf : MeasurableEmbedding f) (μ : Measure β) (g : β → E)
+    (hg : Integrable g μ) :
+    ∃ ν : VectorMeasure α E, ∀ s, MeasurableSet s → ν s = ∫ x in f '' s, g x ∂μ := by
+  have hi : Integrable (fun x ↦ g (f x)) (μ.comap f) := by
+    apply hf.integrable_map_iff.mp
+    rw [hf.map_comap]
+    exact hg.integrableOn
+  refine ⟨(μ.comap f).withDensityᵥ (fun x ↦ g (f x)), ?_⟩
+  intro s hs
+  rw [withDensityᵥ_apply hi hs]
+  have h := hf.setIntegral_map (μ := μ.comap f) g (f '' s)
+  rw [hf.map_comap, hf.injective.preimage_image,
+    Measure.restrict_restrict_of_subset (image_subset_range _ _)] at h
+  exact h.symm
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Vector Measure / With Density
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped Topology
+
+namespace MeasureTheory
+
+/-- Integrating a bounded function against a real signed density multiplies the ordinary
+integrand by that density. -/
+theorem VectorMeasure.setIntegral_withDensity_mul_of_bounded
+    {X : Type*} [MeasurableSpace X]
+    {μ : Measure X} {r q : X → ℝ}
+    (hr : Integrable r μ) (hq : AEStronglyMeasurable q μ) (C : ℝ)
+    (hq_bound : ∀ x, ‖q x‖ ≤ C) (E : Set X) (hE : MeasurableSet E) :
+    (∫ᵛ x in E, q x ∂[ContinuousLinearMap.mul ℝ ℝ; μ.withDensityᵥ r]) =
+      ∫ x in E, q x * r x ∂μ := by
+  rw [withDensityᵥ_eq_withDensity_pos_part_sub_withDensity_neg_part hr]
+  let μp := μ.withDensity fun x ↦ ENNReal.ofReal (r x)
+  let μn := μ.withDensity fun x ↦ ENNReal.ofReal (-r x)
+  let _ : IsFiniteMeasure μp := isFiniteMeasure_withDensity_ofReal hr.2
+  let _ : IsFiniteMeasure μn := isFiniteMeasure_withDensity_ofReal hr.neg.2
+  have hqp : (μp.toSignedMeasure : SignedMeasure X).Integrable q := by
+    rw [VectorMeasure.Integrable, Measure.variation_toSignedMeasure]
+    exact Integrable.of_bound (hq.mono_ac (withDensity_absolutelyContinuous _ _)) C
+      (ae_of_all _ hq_bound)
+  have hqn : (μn.toSignedMeasure : SignedMeasure X).Integrable q := by
+    rw [VectorMeasure.Integrable, Measure.variation_toSignedMeasure]
+    exact Integrable.of_bound (hq.mono_ac (withDensity_absolutelyContinuous _ _)) C
+      (ae_of_all _ hq_bound)
+  have hmul : ContinuousLinearMap.mul ℝ ℝ =
+      (ContinuousLinearMap.lsmul ℝ ℝ).flip := by
+    ext; simp
+  change ∫ᵛ x, q x ∂[ContinuousLinearMap.mul ℝ ℝ;
+      ((μp.toSignedMeasure : SignedMeasure X) - μn.toSignedMeasure).restrict E] = _
+  rw [VectorMeasure.restrict_sub,
+    VectorMeasure.integral_sub_vectorMeasure hqp.restrict hqn.restrict]
+  have hp : (∫ᵛ x in E, q x ∂[ContinuousLinearMap.mul ℝ ℝ; μp.toSignedMeasure]) =
+      ∫ x in E, q x ∂μp := by
+    rw [hmul]
+    exact VectorMeasure.setIntegral_toSignedMeasure hE
+  have hn : (∫ᵛ x in E, q x ∂[ContinuousLinearMap.mul ℝ ℝ; μn.toSignedMeasure]) =
+      ∫ x in E, q x ∂μn := by
+    rw [hmul]
+    exact VectorMeasure.setIntegral_toSignedMeasure hE
+  have hdp : ∫ x in E, q x ∂μp =
+      ∫ x in E, (ENNReal.ofReal (r x)).toReal • q x ∂μ := by
+    apply setIntegral_withDensity_eq_setIntegral_toReal_smul₀
+    · exact hr.aestronglyMeasurable.aemeasurable.ennreal_ofReal.restrict
+    · filter_upwards with x
+      simp
+    · exact hE
+  have hdn : ∫ x in E, q x ∂μn =
+      ∫ x in E, (ENNReal.ofReal (-r x)).toReal • q x ∂μ := by
+    apply setIntegral_withDensity_eq_setIntegral_toReal_smul₀
+    · exact hr.neg.aestronglyMeasurable.aemeasurable.ennreal_ofReal.restrict
+    · filter_upwards with x
+      simp
+    · exact hE
+  rw [hp, hn, hdp, hdn]
+  have hip : Integrable (fun x ↦ (ENNReal.ofReal (r x)).toReal • q x) μ := by
+    simpa only [ENNReal.toReal_ofReal', smul_eq_mul, mul_comm] using
+      hr.pos_part.bdd_mul hq (ae_of_all _ hq_bound)
+  have hin : Integrable (fun x ↦ (ENNReal.ofReal (-r x)).toReal • q x) μ := by
+    simpa only [ENNReal.toReal_ofReal', smul_eq_mul, mul_comm] using
+      hr.neg_part.bdd_mul hq (ae_of_all _ hq_bound)
+  rw [← MeasureTheory.integral_sub hip.integrableOn hin.integrableOn]
+  apply MeasureTheory.integral_congr_ae
+  filter_upwards with x
+  simp only [smul_eq_mul]
+  rcases le_total 0 (r x) with hx | hx
+  · rw [ENNReal.toReal_ofReal', ENNReal.toReal_ofReal']
+    simp only [max_eq_left hx, max_eq_right (neg_nonpos.mpr hx)]
+    ring
+  · rw [ENNReal.toReal_ofReal', ENNReal.toReal_ofReal']
+    simp only [max_eq_right hx, max_eq_left (neg_nonneg.mpr hx)]
+    ring
+
+/-- Integrating a continuous real function on a compact space against a signed density
+multiplies the ordinary integrand by that density. -/
+theorem VectorMeasure.setIntegral_withDensity_mul
+    {X : Type*} [MeasurableSpace X] [TopologicalSpace X] [BorelSpace X] [CompactSpace X]
+    {μ : Measure X} {r q : X → ℝ}
+    (hr : Integrable r μ) (hq : Continuous q) (E : Set X) (hE : MeasurableSet E) :
+    (∫ᵛ x in E, q x ∂[ContinuousLinearMap.mul ℝ ℝ; μ.withDensityᵥ r]) =
+      ∫ x in E, q x * r x ∂μ := by
+  let q' : BoundedContinuousFunction X ℝ :=
+    ContinuousMap.equivBoundedOfCompact X ℝ ⟨q, hq⟩
+  exact VectorMeasure.setIntegral_withDensity_mul_of_bounded hr hq.aestronglyMeasurable ‖q'‖
+    (fun x ↦ BoundedContinuousFunction.norm_coe_le_norm q' x) E hE
+
+end MeasureTheory
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Volume
+-/
+
+@[expose] public section
+
+open scoped Pointwise
+
+namespace MeasureTheory
+
+/-- The square root of planar volume scales linearly under nonnegative dilations. -/
+theorem volume_smul_rpow_half (S : Set (EuclideanSpace ℝ (Fin 2))) (a : ℝ) (ha : 0 ≤ a) :
+    volume (a • S) ^ (2 : ℝ)⁻¹ = ENNReal.ofReal a * volume S ^ (2 : ℝ)⁻¹ := by
+  rw [MeasureTheory.Measure.addHaar_smul_of_nonneg (μ := volume) ha]
+  simp only [finrank_euclideanSpace_fin]
+  rw [ENNReal.mul_rpow_of_nonneg _ _ (by positivity)]
+  congr 1
+  rw [ENNReal.ofReal_pow ha]
+  simpa using ENNReal.pow_rpow_inv_natCast (by norm_num : (2 : ℕ) ≠ 0) (ENNReal.ofReal a)
+
+end MeasureTheory
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `ForMathlib.Order.Infimum`.
+* `ForMathlib.Order.IntervalPartition`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Order / Infimum
+-/
+
+@[expose] public section
+
+open Set
+
+/-- Uniformly close real functions have uniformly close infima on a nonempty set. -/
+theorem abs_sInf_image_sub_sInf_image_le {ι : Type*} {s : Set ι}
+    (hs : s.Nonempty) (f g : ι → ℝ) (hf : BddBelow (f '' s)) (hg : BddBelow (g '' s))
+    {C : ℝ} (h : ∀ x ∈ s, |f x - g x| ≤ C) :
+    |sInf (f '' s) - sInf (g '' s)| ≤ C := by
+  have hfg : sInf (f '' s) - C ≤ sInf (g '' s) := by
+    apply le_csInf (hs.image g)
+    rintro _ ⟨x, hx, rfl⟩
+    have hfx : sInf (f '' s) ≤ f x := csInf_le hf ⟨x, hx, rfl⟩
+    have hpoint := (abs_le.mp (h x hx)).2
+    linarith
+  have hgf : sInf (g '' s) - C ≤ sInf (f '' s) := by
+    apply le_csInf (hs.image f)
+    rintro _ ⟨x, hx, rfl⟩
+    have hgx : sInf (g '' s) ≤ g x := csInf_le hg ⟨x, hx, rfl⟩
+    have hpoint := (abs_le.mp (h x hx)).1
+    linarith
+  rw [abs_le]
+  constructor <;> linarith
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Order / Interval Partition
+-/
+
+@[expose] public section
+
+open Set
+
+/-- Adjacent half-open intervals of a finite monotone sequence cover its endpoint interval. -/
+theorem Monotone.iUnion_Ioc_fin {α : Type*} [LinearOrder α] {n : ℕ} (cuts : Fin (n + 1) → α)
+    (hcuts : Monotone cuts) :
+    (⋃ i : Fin n, Ioc (cuts i.castSucc) (cuts i.succ)) = Ioc (cuts 0) (cuts (Fin.last n)) := by
+  rw [← hcuts.biUnion_Ico_Ioc_map_succ 0 (Fin.last n)]
+  ext x
+  simp only [mem_iUnion]
+  constructor
+  · rintro ⟨i, hi⟩
+    refine ⟨i.castSucc, ⟨Fin.zero_le _, i.castSucc_lt_last⟩, ?_⟩
+    simpa only [Fin.orderSucc_castSucc] using hi
+  · rintro ⟨i, hi, hx⟩
+    have hn : i.val < n := hi.2
+    refine ⟨⟨i, hn⟩, ?_⟩
+    change x ∈ Ioc (cuts (⟨i, hn⟩ : Fin n).castSucc)
+      (cuts (Order.succ (⟨i, hn⟩ : Fin n).castSucc)) at hx
+    simpa only [Fin.orderSucc_castSucc] using hx
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `ForMathlib.Topology.Angle`.
+* `ForMathlib.Topology.Order.Compact`.
+* `ForMathlib.Topology.Order.Concatenation`.
+* `ForMathlib.Topology.Order.Interval`.
+* `ForMathlib.Topology.Order.IntervalExtension`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Topology / Angle
+-/
+
+@[expose] public section
+
+open scoped unitInterval
+
+namespace Real.Angle
+
+/-- A continuous angle path starting at zero has a continuous real lift starting at zero. -/
+theorem exists_continuous_lift_zero (θ : I → Real.Angle) (hθ : Continuous θ)
+    (hzero : θ 0 = 0) :
+    ∃ α : I → ℝ, Continuous α ∧ α 0 = 0 ∧ ∀ t, (α t : Real.Angle) = θ t := by
+  have hcov : IsCoveringMap ((↑) : ℝ → Real.Angle) := AddCircle.isCoveringMap_coe (2 * Real.pi)
+  obtain ⟨α, hα, hα0⟩ := hcov.exists_path_lifts
+    (⟨θ, hθ⟩ : C(I, Real.Angle)) (0 : ℝ) (by simpa using hzero)
+  exact ⟨α, α.continuous, hα0, fun t ↦ congrFun hα t⟩
+
+end Real.Angle
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Topology / Order / Compact
+-/
+
+@[expose] public section
+
+/-- A positive continuous function has a positive uniform lower bound on a nonempty compact set. -/
+theorem IsCompact.exists_pos_forall_le {X : Type*} [TopologicalSpace X]
+    {s : Set X} (hs : IsCompact s) (hne : s.Nonempty) {f : X → ℝ}
+    (hf : ContinuousOn f s) (hpos : ∀ x ∈ s, 0 < f x) :
+    ∃ m > 0, ∀ x ∈ s, m ≤ f x := by
+  obtain ⟨x, hx, hxmin⟩ := hs.exists_isMinOn hne hf
+  exact ⟨f x, hpos x hx, fun y hy ↦ hxmin hy⟩
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Topology / Order / Concatenation
+-/
+
+@[expose] public section
+
+namespace Function
+
+/-- Concatenate two functions on `[0, 1]` on the interval `[0, 2]`. -/
+noncomputable def concatUnitIntervals {E : Type*} (x y : Set.Icc (0 : ℝ) 1 → E)
+    (t : Set.Icc (0 : ℝ) 2) : E :=
+  if (t : ℝ) ≤ 1 then x (Set.projIcc 0 1 (by norm_num) t)
+  else y (Set.projIcc 0 1 (by norm_num) ((t : ℝ) - 1))
+
+/-- Concatenation is continuous when the endpoint values agree. -/
+theorem continuous_concatUnitIntervals {E : Type*} [TopologicalSpace E]
+    {x y : Set.Icc (0 : ℝ) 1 → E} (hx : Continuous x) (hy : Continuous y)
+    (hjoin : x ⟨1, by norm_num⟩ = y ⟨0, by norm_num⟩) :
+    Continuous (concatUnitIntervals x y) := by
+  unfold concatUnitIntervals
+  apply continuous_if_le continuous_subtype_val continuous_const
+  · simpa only [Function.comp_def] using
+      (hx.comp (continuous_projIcc.comp continuous_subtype_val)).continuousOn
+  · simpa only [Function.comp_def, Pi.sub_apply] using
+      (hy.comp
+        (continuous_projIcc.comp (continuous_subtype_val.sub continuous_const))).continuousOn
+  intro t ht
+  simpa [ht] using hjoin
+
+@[simp] theorem concatUnitIntervals_zero {E : Type*} (x y : Set.Icc (0 : ℝ) 1 → E) :
+    concatUnitIntervals x y ⟨0, by norm_num⟩ = x ⟨0, by norm_num⟩ := by
+  simp [concatUnitIntervals]
+
+@[simp] theorem concatUnitIntervals_one {E : Type*} (x y : Set.Icc (0 : ℝ) 1 → E) :
+    concatUnitIntervals x y ⟨1, by norm_num⟩ = x ⟨1, by norm_num⟩ := by
+  simp [concatUnitIntervals]
+
+@[simp] theorem concatUnitIntervals_two {E : Type*} (x y : Set.Icc (0 : ℝ) 1 → E) :
+    concatUnitIntervals x y ⟨2, by norm_num⟩ = y ⟨1, by norm_num⟩ := by
+  norm_num [concatUnitIntervals]
+
+/-- Joined paths have precisely the union of the two original ranges. -/
+theorem range_concatUnitIntervals {E : Type*} (x y : Set.Icc (0 : ℝ) 1 → E)
+    (hjoin : x ⟨1, by norm_num⟩ = y ⟨0, by norm_num⟩) :
+    Set.range (concatUnitIntervals x y) = Set.range x ∪ Set.range y := by
+  ext z
+  constructor
+  · rintro ⟨t, rfl⟩
+    unfold concatUnitIntervals
+    split_ifs
+    · exact Or.inl (Set.mem_range_self _)
+    · exact Or.inr (Set.mem_range_self _)
+  · rintro (⟨t, rfl⟩ | ⟨t, rfl⟩)
+    · refine ⟨⟨t, t.property.1, t.property.2.trans (by norm_num)⟩, ?_⟩
+      simp [concatUnitIntervals, t.property.2, Set.projIcc_of_mem (by norm_num) t.property]
+    · by_cases ht : (t : ℝ) = 0
+      · refine ⟨⟨1, by norm_num⟩, ?_⟩
+        rw [concatUnitIntervals_one, hjoin]
+        congr 1
+        exact Subtype.ext ht.symm
+      · have htpos : 0 < (t : ℝ) := lt_of_le_of_ne t.property.1 (Ne.symm ht)
+        refine ⟨⟨(t : ℝ) + 1, by constructor <;> linarith [t.property.1, t.property.2]⟩, ?_⟩
+        have hnot : ¬(t : ℝ) + 1 ≤ 1 := by linarith
+        simp [concatUnitIntervals, hnot, Set.projIcc_of_mem (by norm_num) t.property]
+
+/-- A strict cyclic cut preserves injectivity away from the final endpoint. -/
+theorem injOn_concatUnitIntervals_comp_of_cyclic_endpoints
+    {E : Type*} {a b : ℝ} (hab : a ≤ b) {x : Set.Icc a b → E}
+    (hinj : Set.InjOn x {t | (t : ℝ) < b})
+    (hclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (s : Set.Icc a b) (has : a < s) (hsb : (s : ℝ) < b)
+    (φ ψ : Set.Icc (0 : ℝ) 1 → Set.Icc a b)
+    (hφ : StrictMono φ) (hψ : StrictMono ψ)
+    (hφ₀ : φ ⟨0, by norm_num⟩ = s)
+    (hψ₀ : ψ ⟨0, by norm_num⟩ = ⟨a, le_rfl, hab⟩)
+    (hψ₁ : ψ ⟨1, by norm_num⟩ = s) :
+    Set.InjOn (concatUnitIntervals (x ∘ φ) (x ∘ ψ)) {t | (t : ℝ) < 2} := by
+  have hab' : a < b := has.trans hsb
+  have hinj' : Set.InjOn x {t | a < (t : ℝ)} := by
+    intro p hp q hq hpq
+    by_cases hpb : (p : ℝ) < b
+    · by_cases hqb : (q : ℝ) < b
+      · exact hinj hpb hqb hpq
+      · have hqeq : q = (⟨b, hab, le_rfl⟩ : Set.Icc a b) := by
+          apply Subtype.ext
+          exact le_antisymm q.property.2 (le_of_not_gt hqb)
+        have hpa : p = (⟨a, le_rfl, hab⟩ : Set.Icc a b) := by
+          apply hinj hpb hab'
+          rw [hqeq] at hpq
+          exact hpq.trans hclosed.symm
+        exact False.elim ((ne_of_gt hp) (congrArg Subtype.val hpa))
+    · have hpeq : p = (⟨b, hab, le_rfl⟩ : Set.Icc a b) := by
+        apply Subtype.ext
+        exact le_antisymm p.property.2 (le_of_not_gt hpb)
+      by_cases hqb : (q : ℝ) < b
+      · have hqa : q = (⟨a, le_rfl, hab⟩ : Set.Icc a b) := by
+          apply hinj hqb hab'
+          rw [hpeq] at hpq
+          exact hpq.symm.trans hclosed.symm
+        exact False.elim ((ne_of_gt hq) (congrArg Subtype.val hqa))
+      · have hqeq : q = (⟨b, hab, le_rfl⟩ : Set.Icc a b) := by
+          apply Subtype.ext
+          exact le_antisymm q.property.2 (le_of_not_gt hqb)
+        exact hpeq.trans hqeq.symm
+  intro u hu v hv huv
+  have hu0 : 0 ≤ (u : ℝ) := u.property.1
+  have hv0 : 0 ≤ (v : ℝ) := v.property.1
+  have hu2 : (u : ℝ) < 2 := hu
+  have hv2 : (v : ℝ) < 2 := hv
+  by_cases hu1 : (u : ℝ) ≤ 1
+  · let pu : Set.Icc (0 : ℝ) 1 := ⟨u, hu0, hu1⟩
+    have hueval : concatUnitIntervals (x ∘ φ) (x ∘ ψ) u = x (φ pu) := by
+      simp [concatUnitIntervals, hu1, pu,
+        Set.projIcc_of_mem (by norm_num) ⟨hu0, hu1⟩]
+    have hφu : (s : ℝ) ≤ φ pu := by
+      rw [← hφ₀]
+      exact hφ.monotone pu.property.1
+    by_cases hv1 : (v : ℝ) ≤ 1
+    · let pv : Set.Icc (0 : ℝ) 1 := ⟨v, hv0, hv1⟩
+      have hveval : concatUnitIntervals (x ∘ φ) (x ∘ ψ) v = x (φ pv) := by
+        simp [concatUnitIntervals, hv1, pv,
+          Set.projIcc_of_mem (by norm_num) ⟨hv0, hv1⟩]
+      have hφv : (s : ℝ) ≤ φ pv := by
+        rw [← hφ₀]
+        exact hφ.monotone pv.property.1
+      have hp : φ pu = φ pv := hinj' (has.trans_le hφu) (has.trans_le hφv)
+        (hueval ▸ huv ▸ hveval)
+      apply Subtype.ext
+      simpa [pu, pv] using congrArg Subtype.val (hφ.injective hp)
+    · have hv1' : 1 < (v : ℝ) := lt_of_not_ge hv1
+      have hvp : 0 ≤ (v : ℝ) - 1 ∧ (v : ℝ) - 1 ≤ 1 := ⟨by linarith, by linarith⟩
+      let pv : Set.Icc (0 : ℝ) 1 := ⟨(v : ℝ) - 1, hvp⟩
+      have hveval : concatUnitIntervals (x ∘ φ) (x ∘ ψ) v = x (ψ pv) := by
+        simp [concatUnitIntervals, hv1, pv,
+          Set.projIcc_of_mem (by norm_num) hvp]
+      have hψvpos : a < (ψ pv : ℝ) := by
+        have h := hψ (show (⟨0, by norm_num⟩ : Set.Icc (0 : ℝ) 1) < pv by
+          change 0 < (v : ℝ) - 1
+          linarith)
+        rw [hψ₀] at h
+        exact h
+      have hp : φ pu = ψ pv := hinj' (has.trans_le hφu) hψvpos
+        (hueval ▸ huv ▸ hveval)
+      have hψvlt : (ψ pv : ℝ) < s := by
+        have h := hψ (show pv < (⟨1, by norm_num⟩ : Set.Icc (0 : ℝ) 1) by
+          change (v : ℝ) - 1 < 1
+          linarith)
+        rw [hψ₁] at h
+        exact h
+      exact False.elim ((not_le_of_gt hψvlt) (hp ▸ hφu))
+  · have hu1' : 1 < (u : ℝ) := lt_of_not_ge hu1
+    have hup : 0 ≤ (u : ℝ) - 1 ∧ (u : ℝ) - 1 ≤ 1 := ⟨by linarith, by linarith⟩
+    let pu : Set.Icc (0 : ℝ) 1 := ⟨(u : ℝ) - 1, hup⟩
+    have hueval : concatUnitIntervals (x ∘ φ) (x ∘ ψ) u = x (ψ pu) := by
+      simp [concatUnitIntervals, hu1, pu,
+        Set.projIcc_of_mem (by norm_num) hup]
+    have hψupos : a < (ψ pu : ℝ) := by
+      have h := hψ (show (⟨0, by norm_num⟩ : Set.Icc (0 : ℝ) 1) < pu by
+        change 0 < (u : ℝ) - 1
+        linarith)
+      rw [hψ₀] at h
+      exact h
+    by_cases hv1 : (v : ℝ) ≤ 1
+    · let pv : Set.Icc (0 : ℝ) 1 := ⟨v, hv0, hv1⟩
+      have hveval : concatUnitIntervals (x ∘ φ) (x ∘ ψ) v = x (φ pv) := by
+        simp [concatUnitIntervals, hv1, pv,
+          Set.projIcc_of_mem (by norm_num) ⟨hv0, hv1⟩]
+      have hφv : (s : ℝ) ≤ φ pv := by
+        rw [← hφ₀]
+        exact hφ.monotone pv.property.1
+      have hp : ψ pu = φ pv := hinj' hψupos (has.trans_le hφv)
+        (hueval ▸ huv ▸ hveval)
+      have hψult : (ψ pu : ℝ) < s := by
+        have h := hψ (show pu < (⟨1, by norm_num⟩ : Set.Icc (0 : ℝ) 1) by
+          change (u : ℝ) - 1 < 1
+          linarith)
+        rw [hψ₁] at h
+        exact h
+      exact False.elim ((not_le_of_gt hψult) (hp ▸ hφv))
+    · have hvp : 0 ≤ (v : ℝ) - 1 ∧ (v : ℝ) - 1 ≤ 1 := ⟨by linarith, by linarith⟩
+      let pv : Set.Icc (0 : ℝ) 1 := ⟨(v : ℝ) - 1, hvp⟩
+      have hveval : concatUnitIntervals (x ∘ φ) (x ∘ ψ) v = x (ψ pv) := by
+        simp [concatUnitIntervals, hv1, pv,
+          Set.projIcc_of_mem (by norm_num) hvp]
+      have hψvpos : a < (ψ pv : ℝ) := by
+        have h := hψ (show (⟨0, by norm_num⟩ : Set.Icc (0 : ℝ) 1) < pv by
+          change 0 < (v : ℝ) - 1
+          linarith)
+        rw [hψ₀] at h
+        exact h
+      have hp : ψ pu = ψ pv := hinj' hψupos hψvpos (hueval ▸ huv ▸ hveval)
+      have hp' := congrArg Subtype.val (hψ.injective hp)
+      apply Subtype.ext
+      simp [pu, pv] at hp'
+      linarith
+
+end Function
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Topology / Order / Interval
+-/
+
+@[expose] public section
+
+namespace Set
+
+instance instCompactIccSpaceIcc (a b : ℝ) : CompactIccSpace (Icc a b) :=
+  ⟨fun {_ _} ↦ isClosed_Icc.isCompact⟩
+
+end Set
+
+/-- Reverse a closed real interval about its midpoint. -/
+def Set.Icc.reverse {a b : ℝ} (_hab : a ≤ b) :
+    Set.Icc a b → Set.Icc a b := fun t ↦
+  ⟨a + b - t, by linarith [t.property.2], by linarith [t.property.1]⟩
+
+/-- Interval reversal is continuous. -/
+theorem Set.Icc.continuous_reverse {a b : ℝ} (hab : a ≤ b) :
+    Continuous (Set.Icc.reverse hab) :=
+  (continuous_const.add continuous_const |>.sub continuous_subtype_val).subtype_mk _
+
+/-- Interval reversal is antitone. -/
+theorem Set.Icc.antitone_reverse {a b : ℝ} (hab : a ≤ b) :
+    Antitone (Set.Icc.reverse hab) := by
+  intro x y hxy
+  have hxy' : (x : ℝ) ≤ y := hxy
+  change a + b - (y : ℝ) ≤ a + b - (x : ℝ)
+  linarith
+
+/-- Reversing a closed interval twice is the identity. -/
+theorem Set.Icc.involutive_reverse {a b : ℝ} (hab : a ≤ b) :
+    Function.Involutive (Set.Icc.reverse hab) := by
+  intro x
+  apply Subtype.ext
+  simp [Set.Icc.reverse]
+
+/-- Interval reversal is surjective. -/
+theorem Set.Icc.surjective_reverse {a b : ℝ} (hab : a ≤ b) :
+    Function.Surjective (Set.Icc.reverse hab) :=
+  (Set.Icc.involutive_reverse hab).surjective
+
+open Set Filter
+open scoped Topology
+
+/-- Replace the terminal value of a one-sided continuous function by its left limit. -/
+theorem continuousOn_replace_right_endpoint {E : Type*} [TopologicalSpace E]
+    {a b : ℝ} (hab : a < b) (f : ℝ → E) (z : E)
+    (hr : ∀ t ∈ Ico a b, Tendsto f (𝓝[>] t) (𝓝 (f t)))
+    (hl : ∀ t ∈ Ioo a b, Tendsto f (𝓝[<] t) (𝓝 (f t)))
+    (hb : Tendsto f (𝓝[<] b) (𝓝 z)) :
+    ContinuousOn (fun t ↦ if t = b then z else f t) (Icc a b) := by
+  let g : ℝ → E := fun t ↦ if t = b then z else f t
+  have heql {t : ℝ} (ht : t ≤ b) : g =ᶠ[𝓝[<] t] f := by
+    filter_upwards [self_mem_nhdsWithin] with u hu
+    have hne : u ≠ b := (lt_of_lt_of_le hu ht).ne
+    simp [g, hne]
+  have heqr {t : ℝ} (ht : t < b) : g =ᶠ[𝓝[>] t] f := by
+    filter_upwards [nhdsWithin_le_nhds (Iio_mem_nhds ht)] with u hu
+    simp [g, (show u ≠ b from hu.ne)]
+  intro t ht
+  change ContinuousWithinAt g (Icc a b) t
+  rcases eq_or_lt_of_le ht.1 with rfl | hat
+  · have h : ContinuousWithinAt g (Ioi a) a := by
+      change Tendsto g _ _
+      simpa only [g, ite_eq_right hab.ne] using (hr a ⟨le_rfl, hab⟩).congr' (heqr hab).symm
+    exact (continuousWithinAt_Ioi_iff_Ici.mp h).mono Icc_subset_Ici_self
+  · rcases lt_or_eq_of_le ht.2 with htb | rfl
+    · apply ContinuousAt.continuousWithinAt
+      apply continuousAt_iff_continuous_left'_right'.mpr
+      constructor
+      · change Tendsto g _ _
+        simpa only [g, ite_eq_right htb.ne] using (hl t ⟨hat, htb⟩).congr' (heql htb.le).symm
+      · change Tendsto g _ _
+        simpa only [g, ite_eq_right htb.ne] using (hr t ⟨hat.le, htb⟩).congr' (heqr htb).symm
+    · have h : ContinuousWithinAt g (Iio t) t := by
+        change Tendsto g _ _
+        simpa only [g, ite_eq_left rfl] using hb.congr' (heql le_rfl).symm
+      exact (continuousWithinAt_Iio_iff_Iic.mp h).mono Icc_subset_Iic_self
+
+/-- Every compact real interval carries finite monotone partitions, with prescribed
+endpoints, whose mesh eventually falls below any positive threshold. -/
+theorem Set.Icc.exists_partitions_mesh_tendsto_zero {a b : ℝ} (hab : a ≤ b) :
+    ∃ cuts : ∀ k : ℕ, Fin (k + 2) → Set.Icc a b,
+      (∀ k, Monotone (cuts k)) ∧ (∀ k, (cuts k 0 : ℝ) = a) ∧
+      (∀ k, (cuts k (Fin.last (k + 1)) : ℝ) = b) ∧
+      ∀ δ > 0, ∀ᶠ k in Filter.atTop, ∀ i : Fin (k + 1),
+        (cuts k i.succ : ℝ) - (cuts k i.castSucc : ℝ) < δ := by
+  let cuts (k : ℕ) (i : Fin (k + 2)) : Set.Icc a b :=
+    ⟨a + (b - a) * (i : ℝ) / (k + 1), by
+      have hk : (0 : ℝ) < k + 1 := by positivity
+      have hi : (i : ℝ) ≤ k + 1 := by exact_mod_cast Nat.le_of_lt_succ i.isLt
+      constructor
+      · exact le_add_of_nonneg_right (div_nonneg
+          (mul_nonneg (sub_nonneg.mpr hab) (Nat.cast_nonneg _)) hk.le)
+      · have hmul := mul_le_mul_of_nonneg_left hi (sub_nonneg.mpr hab)
+        have hdiv : (b - a) * (i : ℝ) / (k + 1) ≤ b - a :=
+          (div_le_iff₀ hk).mpr hmul
+        linarith⟩
+  refine ⟨cuts, ?_, ?_, ?_, ?_⟩
+  · intro k i j hij
+    change a + (b - a) * (i : ℝ) / (k + 1) ≤ a + (b - a) * (j : ℝ) / (k + 1)
+    gcongr
+    exact_mod_cast hij
+  · intro k
+    simp [cuts]
+  · intro k
+    dsimp [cuts]
+    push_cast
+    rw [mul_div_cancel_right₀ _ (by positivity)]
+    ring
+  · intro δ hδ
+    have ht : Filter.Tendsto (fun k : ℕ ↦ (b - a) / (k + 1))
+        Filter.atTop (nhds 0) :=
+      tendsto_const_nhds.div_atTop (Filter.tendsto_atTop_add_const_right _ 1
+        tendsto_natCast_atTop_atTop)
+    filter_upwards [ht.eventually (gt_mem_nhds hδ)] with k hk
+    intro i
+    convert hk using 1
+    dsimp [cuts]
+    push_cast
+    ring
+
+/-- The increasing affine surjection between two nondegenerate closed intervals. -/
+theorem Set.Icc.exists_affine_monotone_surjection {a b c d : ℝ} (hab : a < b) (hcd : c < d) :
+    ∃ φ : Set.Icc a b → Set.Icc c d, Continuous φ ∧ Monotone φ ∧ Function.Surjective φ ∧
+      ∀ t : Set.Icc a b, (φ t : ℝ) = c + ((t : ℝ) - a) / (b - a) * (d - c) := by
+  have hba : (0 : ℝ) < b - a := by linarith
+  have hdc : (0 : ℝ) < d - c := by linarith
+  have hfrac : ∀ t : Set.Icc a b,
+      0 ≤ ((t : ℝ) - a) / (b - a) ∧ ((t : ℝ) - a) / (b - a) ≤ 1 := by
+    intro t
+    refine ⟨div_nonneg (by linarith [t.property.1]) hba.le, ?_⟩
+    rw [div_le_one hba]
+    linarith [t.property.2]
+  refine ⟨fun t ↦ ⟨c + ((t : ℝ) - a) / (b - a) * (d - c), ?_, ?_⟩, ?_, ?_, ?_, fun _ ↦ rfl⟩
+  · nlinarith [(hfrac t).1]
+  · nlinarith [(hfrac t).2]
+  · exact Continuous.subtype_mk (by fun_prop) _
+  · intro s t hst
+    have hst' : (s : ℝ) ≤ (t : ℝ) := hst
+    have hdiv : ((s : ℝ) - a) / (b - a) ≤ ((t : ℝ) - a) / (b - a) := by gcongr
+    change c + ((s : ℝ) - a) / (b - a) * (d - c) ≤ c + ((t : ℝ) - a) / (b - a) * (d - c)
+    nlinarith
+  · intro y
+    have hyl : 0 ≤ ((y : ℝ) - c) / (d - c) := div_nonneg (by linarith [y.property.1]) hdc.le
+    have hyr : ((y : ℝ) - c) / (d - c) ≤ 1 := by
+      rw [div_le_one hdc]
+      linarith [y.property.2]
+    refine ⟨⟨a + ((y : ℝ) - c) / (d - c) * (b - a), by nlinarith, by nlinarith⟩, ?_⟩
+    apply Subtype.ext
+    change c + (a + ((y : ℝ) - c) / (d - c) * (b - a) - a) / (b - a) * (d - c) = (y : ℝ)
+    field_simp
+    ring
+
+/-- The decreasing affine surjection between two nondegenerate closed intervals: the increasing
+one reflected in the midpoint of its codomain. -/
+theorem Set.Icc.exists_affine_antitone_surjection {a b c d : ℝ} (hab : a < b) (hcd : c < d) :
+    ∃ φ : Set.Icc a b → Set.Icc c d, Continuous φ ∧ Antitone φ ∧ Function.Surjective φ ∧
+      ∀ t : Set.Icc a b, (φ t : ℝ) = d - ((t : ℝ) - a) / (b - a) * (d - c) := by
+  obtain ⟨φ, hφc, hφm, hφs, hφv⟩ := Set.Icc.exists_affine_monotone_surjection hab hcd
+  refine ⟨fun t ↦ ⟨c + d - (φ t : ℝ), ?_, ?_⟩, ?_, ?_, ?_, fun t ↦ ?_⟩
+  · linarith [(φ t).property.2]
+  · linarith [(φ t).property.1]
+  · exact (continuous_const.sub (continuous_subtype_val.comp hφc)).subtype_mk _
+  · intro s t hst
+    exact Subtype.mk_le_mk.mpr (by linarith [show (φ s : ℝ) ≤ (φ t : ℝ) from hφm hst])
+  · intro y
+    have hy : c + d - (y : ℝ) ∈ Set.Icc c d :=
+      ⟨by linarith [y.property.2], by linarith [y.property.1]⟩
+    obtain ⟨t, ht⟩ := hφs ⟨c + d - (y : ℝ), hy⟩
+    refine ⟨t, Subtype.ext ?_⟩
+    change c + d - (φ t : ℝ) = (y : ℝ)
+    rw [show (φ t : ℝ) = c + d - (y : ℝ) from congrArg Subtype.val ht]
+    ring
+  · change c + d - (φ t : ℝ) = _
+    rw [hφv t]
+    ring
+
+/-- The translation of the unit interval onto a closed interval of length one. -/
+theorem Set.Icc.exists_translation_surjection {c d : ℝ} (hcd : d = c + 1) :
+    ∃ φ : Set.Icc (0 : ℝ) 1 → Set.Icc c d, Continuous φ ∧ Monotone φ ∧
+      Function.Surjective φ ∧ ∀ u : Set.Icc (0 : ℝ) 1, (φ u : ℝ) = c + (u : ℝ) := by
+  obtain ⟨φ, hφc, hφm, hφs, hφv⟩ :=
+    Set.Icc.exists_affine_monotone_surjection zero_lt_one (show c < d by rw [hcd]; linarith)
+  exact ⟨φ, hφc, hφm, hφs, fun u ↦ by rw [hφv u, hcd]; ring⟩
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Topology / Order / Interval Extension
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace OrderIso
+
+/-- Extend an order isomorphism of open real intervals by matching the endpoints. -/
+def extendIoo {a b c d : ℝ} (hcd : c < d)
+    (e : Set.Ioo a b ≃o Set.Ioo c d) (x : Set.Icc a b) : Set.Icc c d :=
+  if hxa : (x : ℝ) = a then ⟨c, le_rfl, hcd.le⟩
+  else if hxb : (x : ℝ) = b then ⟨d, hcd.le, le_rfl⟩
+  else let t := e ⟨x, lt_of_le_of_ne x.property.1 (Ne.symm hxa),
+      lt_of_le_of_ne x.property.2 hxb⟩
+    ⟨t, t.property.1.le, t.property.2.le⟩
+
+/-- The extension maps the left endpoint to the left endpoint. -/
+theorem extendIoo_left {a b c d : ℝ} (hab : a < b) (hcd : c < d)
+    (e : Set.Ioo a b ≃o Set.Ioo c d) :
+    extendIoo hcd e ⟨a, le_rfl, hab.le⟩ = ⟨c, le_rfl, hcd.le⟩ := by
+  simp [extendIoo]
+
+/-- The extension maps the right endpoint to the right endpoint. -/
+theorem extendIoo_right {a b c d : ℝ} (hab : a < b) (hcd : c < d)
+    (e : Set.Ioo a b ≃o Set.Ioo c d) :
+    extendIoo hcd e ⟨b, hab.le, le_rfl⟩ = ⟨d, hcd.le, le_rfl⟩ := by
+  simp [extendIoo, ne_of_gt hab]
+
+/-- The extension agrees with the original order isomorphism on the interior. -/
+theorem extendIoo_interior {a b c d : ℝ} (hcd : c < d)
+    (e : Set.Ioo a b ≃o Set.Ioo c d) (x : Set.Ioo a b) :
+    (extendIoo hcd e ⟨x, x.property.1.le, x.property.2.le⟩ : ℝ) = e x := by
+  simp [extendIoo, ne_of_gt x.property.1, ne_of_lt x.property.2]
+
+/-- The endpoint extension is monotone. -/
+theorem monotone_extendIoo {a b c d : ℝ} (hab : a < b) (hcd : c < d)
+    (e : Set.Ioo a b ≃o Set.Ioo c d) : Monotone (extendIoo hcd e) := by
+  intro x y hxy
+  by_cases hx : (x : ℝ) = a
+  · have he : x = ⟨a, le_rfl, hab.le⟩ := Subtype.ext hx
+    rw [he, extendIoo_left hab]
+    exact (extendIoo hcd e y).property.1
+  by_cases hy : (y : ℝ) = b
+  · have he : y = ⟨b, hab.le, le_rfl⟩ := Subtype.ext hy
+    rw [he, extendIoo_right hab]
+    exact (extendIoo hcd e x).property.2
+  have hxlo : a < (x : ℝ) := lt_of_le_of_ne x.property.1 (Ne.symm hx)
+  have hyhi : (y : ℝ) < b := lt_of_le_of_ne y.property.2 hy
+  have hxhi : (x : ℝ) < b := lt_of_le_of_lt hxy hyhi
+  have hylo : a < (y : ℝ) := lt_of_lt_of_le hxlo hxy
+  change (extendIoo hcd e x : ℝ) ≤ extendIoo hcd e y
+  rw [extendIoo_interior hcd e ⟨x, hxlo, hxhi⟩,
+    extendIoo_interior hcd e ⟨y, hylo, hyhi⟩]
+  exact e.monotone hxy
+
+/-- The endpoint extension is surjective. -/
+theorem surjective_extendIoo {a b c d : ℝ} (hab : a < b) (hcd : c < d)
+    (e : Set.Ioo a b ≃o Set.Ioo c d) : Function.Surjective (extendIoo hcd e) := by
+  intro y
+  by_cases hyc : (y : ℝ) = c
+  · refine ⟨⟨a, le_rfl, hab.le⟩, ?_⟩
+    rw [extendIoo_left hab]
+    exact Subtype.ext hyc.symm
+  by_cases hyd : (y : ℝ) = d
+  · refine ⟨⟨b, hab.le, le_rfl⟩, ?_⟩
+    rw [extendIoo_right hab]
+    exact Subtype.ext hyd.symm
+  let y' : Set.Ioo c d := ⟨y, lt_of_le_of_ne y.property.1 (Ne.symm hyc),
+    lt_of_le_of_ne y.property.2 hyd⟩
+  let x := e.symm y'
+  refine ⟨⟨x, x.property.1.le, x.property.2.le⟩, Subtype.ext ?_⟩
+  rw [extendIoo_interior]
+  exact congrArg Subtype.val (e.apply_symm_apply y')
+
+/-- The endpoint extension is continuous. -/
+theorem continuous_extendIoo {a b c d : ℝ} (hab : a < b) (hcd : c < d)
+    (e : Set.Ioo a b ≃o Set.Ioo c d) : Continuous (extendIoo hcd e) :=
+  (monotone_extendIoo hab hcd e).continuous_of_surjective (surjective_extendIoo hab hcd e)
+
+end OrderIso
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `ForMathlib.MeasureTheory.StieltjesTransport`.
+* `ForMathlib.MeasureTheory.VectorMeasure.LocallyConstant`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Stieltjes Transport
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+
+/-- A monotone surjection of compact intervals preserves bounded variation. -/
+theorem BoundedVariationOn.comp_monotone_surjective_Icc
+    {a b c d : ℝ} (_hab : a ≤ b) {f : Set.Icc a b → ℝ}
+    (hf : BoundedVariationOn f Set.univ)
+    {φ : Set.Icc c d → Set.Icc a b} (hφ : Monotone φ) (hφs : Function.Surjective φ) :
+    BoundedVariationOn (f ∘ φ) Set.univ := by
+  let _ : Fact (a ≤ b) := ⟨_hab⟩
+  rw [BoundedVariationOn,
+    eVariationOn.comp_eq_of_monotoneOn (t := Set.univ) f φ
+      (fun _ _ _ _ hxy ↦ hφ hxy)]
+  rw [Set.image_univ, hφs.range_eq]
+  exact hf
+
+/-- The Stieltjes measure is preserved by continuous monotone surjective reparametrization. -/
+theorem BoundedVariationOn.vectorMeasure_map_comp_monotone_surjective_Icc
+    {a b c d : ℝ} (hab : a ≤ b) (hcd : c ≤ d) {f : Set.Icc a b → ℝ}
+    (hf : BoundedVariationOn f Set.univ)
+    (hfc : Continuous f) {φ : Set.Icc c d → Set.Icc a b} (hφc : Continuous φ)
+    (hφ : Monotone φ) (hφs : Function.Surjective φ) :
+    (BoundedVariationOn.comp_monotone_surjective_Icc hab hf hφ hφs).vectorMeasure.map φ =
+      hf.vectorMeasure := by
+  let _ : Fact (a ≤ b) := ⟨hab⟩
+  let _ : Fact (c ≤ d) := ⟨hcd⟩
+  let hfφ := BoundedVariationOn.comp_monotone_surjective_Icc hab hf hφ hφs
+  have fiberGreatest (y : Set.Icc a b) :
+      ∃ x, IsGreatest (φ ⁻¹' {y}) x := by
+    apply (isClosed_singleton.preimage hφc).isCompact.exists_isGreatest
+    rcases hφs y with ⟨x, rfl⟩
+    exact ⟨x, rfl⟩
+  have preimage_Ioc (u v : Set.Icc a b) :
+      ∃ x y, φ x = u ∧ φ y = v ∧ (u ≤ v → x ≤ y) ∧
+        φ ⁻¹' Ioc u v = Ioc x y := by
+    obtain ⟨x, hxmem, hxmax⟩ := fiberGreatest u
+    obtain ⟨y, hymem, hymax⟩ := fiberGreatest v
+    change φ x = u at hxmem
+    change φ y = v at hymem
+    have hxy : u ≤ v → x ≤ y := by
+      intro huv
+      by_contra hxy
+      have hyx : y ≤ x := le_of_not_ge hxy
+      have huv' : u = v := le_antisymm huv (hxmem ▸ hymem ▸ hφ hyx)
+      exact hxy (hymax (by change φ x = v; rw [hxmem, huv']))
+    refine ⟨x, y, hxmem, hymem, hxy, Set.ext fun z ↦ ?_⟩
+    simp only [mem_preimage, mem_Ioc]
+    constructor <;> intro hz
+    · constructor
+      · by_contra hzx
+        have hzx' : z ≤ x := le_of_not_gt hzx
+        exact (not_le_of_gt hz.1) (hxmem ▸ hφ hzx')
+      · by_contra hyz
+        have hyz' : y < z := lt_of_not_ge hyz
+        have hvz : v ≤ φ z := hymem ▸ hφ hyz'.le
+        have : φ z = v := le_antisymm hz.2 hvz
+        exact (not_le_of_gt hyz') (hymax this)
+    · constructor
+      · have hux : u ≤ φ z := hxmem ▸ hφ hz.1.le
+        exact hux.lt_of_ne fun hzu ↦ (not_le_of_gt hz.1) (hxmax hzu.symm)
+      · exact hymem ▸ hφ hz.2
+  have hright (x : Set.Icc a b) : f.rightLim x = f x :=
+    hfc.continuousAt.continuousWithinAt.rightLim_eq
+  have hright_comp (x : Set.Icc c d) : (f ∘ φ).rightLim x = (f ∘ φ) x :=
+    (hfc.comp hφc).continuousAt.continuousWithinAt.rightLim_eq
+  have hinterval (u v : Set.Icc a b) (huv : u ≤ v) :
+      (hfφ.vectorMeasure.map φ) (Ioc u v) = hf.vectorMeasure (Ioc u v) := by
+    obtain ⟨x, y, hx, hy, hxy, hpre⟩ := preimage_Ioc u v
+    rw [VectorMeasure.map_apply _ hφc.measurable measurableSet_Ioc, hpre,
+      hfφ.vectorMeasure_Ioc (hxy huv), hf.vectorMeasure_Ioc huv, hright_comp, hright_comp,
+      hright, hright, Function.comp_apply, Function.comp_apply, hx, hy]
+  apply VectorMeasure.ext_of_generateFrom
+      {s | ∃ u v : Set.Icc a b, u ≤ v ∧ s = Ioc u v}
+  · rintro s ⟨u, v, huv, rfl⟩
+    exact hinterval u v huv
+  · exact BorelSpace.measurable_eq.trans <| by
+      rw [borel_eq_generateFrom_Ioc_le]
+      congr 1
+      ext s
+      simp only [Set.mem_ofPred_eq]
+      constructor
+      · rintro ⟨u, v, huv, rfl⟩
+        exact ⟨u, v, huv, rfl⟩
+      · rintro ⟨u, v, huv, rfl⟩
+        exact ⟨u, v, huv, rfl⟩
+  · exact IsSetSemiring.isPiSystem IsSetSemiring.Ioc
+  · have hφtop : φ ⊤ = ⊤ := by
+      obtain ⟨x, hx⟩ := hφs ⊤
+      exact top_unique (hx ▸ hφ le_top)
+    have hφbot : φ ⊥ = ⊥ := by
+      obtain ⟨x, hx⟩ := hφs ⊥
+      exact bot_unique (hx ▸ hφ bot_le)
+    have htopφ : Filter.atTop.limUnder (f ∘ φ) = (f ∘ φ) ⊤ := by
+      apply tendsto_nhds_unique hfφ.tendsto_atTop_limUnder
+      rw [Filter.atTop_eq_pure_of_isTop isTop_top]
+      exact tendsto_pure_nhds _ _
+    have hbotφ : Filter.atBot.limUnder (f ∘ φ) = (f ∘ φ) ⊥ := by
+      apply tendsto_nhds_unique hfφ.tendsto_atBot_limUnder
+      rw [Filter.atBot_eq_pure_of_isBot isBot_bot]
+      exact tendsto_pure_nhds _ _
+    have htop : Filter.atTop.limUnder f = f ⊤ := by
+      apply tendsto_nhds_unique hf.tendsto_atTop_limUnder
+      rw [Filter.atTop_eq_pure_of_isTop isTop_top]
+      exact tendsto_pure_nhds _ _
+    have hbot : Filter.atBot.limUnder f = f ⊥ := by
+      apply tendsto_nhds_unique hf.tendsto_atBot_limUnder
+      rw [Filter.atBot_eq_pure_of_isBot isBot_bot]
+      exact tendsto_pure_nhds _ _
+    rw [VectorMeasure.map_apply _ hφc.measurable MeasurableSet.univ, preimage_univ,
+      hfφ.vectorMeasure_univ, hf.vectorMeasure_univ, htopφ, hbotφ, htop, hbot,
+      Function.comp_apply, Function.comp_apply, hφtop, hφbot]
+
+/-- The Stieltjes measure of a continuous BV function has no point masses. -/
+theorem BoundedVariationOn.vectorMeasure_singleton_eq_zero_of_continuous
+    {a b : ℝ} {f : Set.Icc a b → ℝ} (hf : BoundedVariationOn f Set.univ)
+    (hfc : Continuous f) (x : Set.Icc a b) : hf.vectorMeasure {x} = 0 := by
+  rw [hf.vectorMeasure_singleton,
+    hfc.continuousAt.continuousWithinAt.rightLim_eq,
+    hfc.continuousAt.continuousWithinAt.leftLim_eq, sub_self]
+
+/-- The Stieltjes measure on a closed subinterval pushes forward to the half-open restriction. -/
+theorem BoundedVariationOn.vectorMeasure_map_Icc_inclusion
+    {a b : ℝ} {f : Set.Icc a b → ℝ} (hf : BoundedVariationOn f Set.univ)
+    (hfc : Continuous f) (l u : Set.Icc a b) (hlu : l ≤ u) :
+    let ι : Set.Icc (l : ℝ) u → Set.Icc a b := fun x ↦
+      ⟨x, le_trans l.property.1 x.property.1, le_trans x.property.2 u.property.2⟩
+    let hfr : BoundedVariationOn (f ∘ ι) Set.univ :=
+      ne_top_of_le_ne_top hf (eVariationOn.comp_le_of_monotoneOn f ι
+        (fun _ _ _ _ h ↦ h) (mapsTo_univ ι _))
+    hfr.vectorMeasure.map ι = hf.vectorMeasure.restrict (Ioc l u) := by
+  dsimp only
+  let ι : Set.Icc (l : ℝ) u → Set.Icc a b := fun x ↦
+    ⟨x, le_trans l.property.1 x.property.1, le_trans x.property.2 u.property.2⟩
+  let hfr : BoundedVariationOn (f ∘ ι) Set.univ :=
+    ne_top_of_le_ne_top hf (eVariationOn.comp_le_of_monotoneOn f ι
+      (fun _ _ _ _ h ↦ h) (mapsTo_univ ι _))
+  let _ : Fact ((l : ℝ) ≤ u) := ⟨hlu⟩
+  change hfr.vectorMeasure.map ι = hf.vectorMeasure.restrict (Ioc l u)
+  have hι : Continuous ι := continuous_subtype_val.subtype_mk _
+  have hright (x : Set.Icc a b) : f.rightLim x = f x :=
+    hfc.continuousAt.continuousWithinAt.rightLim_eq
+  have hright_comp (x : Set.Icc (l : ℝ) u) : (f ∘ ι).rightLim x = (f ∘ ι) x :=
+    (hfc.comp hι).continuousAt.continuousWithinAt.rightLim_eq
+  have htop : Filter.atTop.limUnder (f ∘ ι) = (f ∘ ι) ⊤ := by
+    apply tendsto_nhds_unique hfr.tendsto_atTop_limUnder
+    rw [Filter.atTop_eq_pure_of_isTop isTop_top]
+    exact tendsto_pure_nhds _ _
+  have hbot : Filter.atBot.limUnder (f ∘ ι) = (f ∘ ι) ⊥ := by
+    apply tendsto_nhds_unique hfr.tendsto_atBot_limUnder
+    rw [Filter.atBot_eq_pure_of_isBot isBot_bot]
+    exact tendsto_pure_nhds _ _
+  apply VectorMeasure.ext_of_generateFrom (Set.range Iic)
+  · rintro s ⟨x, rfl⟩
+    rw [VectorMeasure.map_apply _ hι.measurable measurableSet_Iic,
+      VectorMeasure.restrict_apply _ measurableSet_Ioc measurableSet_Iic]
+    by_cases hxl : x < l
+    · have hp : ι ⁻¹' Iic x = ∅ := by
+        apply eq_empty_iff_forall_notMem.mpr
+        intro y hy
+        exact (not_le_of_gt hxl) (le_trans y.property.1 hy)
+      have hi : Iic x ∩ Ioc l u = ∅ := by ext y; simp; grind
+      simp [hp, hi]
+    · by_cases hxu : u ≤ x
+      · have hp : ι ⁻¹' Iic x = Set.univ := by
+          apply eq_univ_of_forall
+          intro y
+          exact le_trans y.property.2 hxu
+        have hi : Iic x ∩ Ioc l u = Ioc l u := by ext y; simp; grind
+        rw [hp, hi, hfr.vectorMeasure_univ, hf.vectorMeasure_Ioc hlu]
+        rw [htop, hbot, hright, hright]
+        rfl
+      · have hlx : l ≤ x := le_of_not_gt hxl
+        have hxu' : x ≤ u := le_of_not_ge hxu
+        let y : Set.Icc (l : ℝ) u := ⟨x, hlx, hxu'⟩
+        have hp : ι ⁻¹' Iic x = Iic y := by ext z; rfl
+        have hi : Iic x ∩ Ioc l u = Ioc l x := by ext z; simp; grind
+        rw [hp, hi, hfr.vectorMeasure_Iic, hf.vectorMeasure_Ioc hlx,
+          hright_comp, hbot, hright, hright]
+        rfl
+  · exact BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Iic _)
+  · exact isPiSystem_Iic
+  · rw [VectorMeasure.map_apply _ hι.measurable MeasurableSet.univ,
+      preimage_univ, VectorMeasure.restrict_apply_univ,
+      hfr.vectorMeasure_univ, hf.vectorMeasure_Ioc hlu, htop, hbot, hright, hright]
+    rfl
+
+/-- An antitone surjection of compact intervals preserves bounded variation. -/
+theorem BoundedVariationOn.comp_antitone_surjective_Icc
+    {a b c d : ℝ} (_hab : a ≤ b) {f : Set.Icc a b → ℝ}
+    (hf : BoundedVariationOn f Set.univ)
+    {φ : Set.Icc c d → Set.Icc a b} (hφ : Antitone φ)
+    (hφs : Function.Surjective φ) :
+    BoundedVariationOn (f ∘ φ) Set.univ := by
+  let _ : Fact (a ≤ b) := ⟨_hab⟩
+  rw [BoundedVariationOn,
+    eVariationOn.comp_eq_of_antitoneOn (t := Set.univ) f φ
+      (fun _ _ _ _ hxy ↦ hφ hxy)]
+  rw [Set.image_univ, hφs.range_eq]
+  exact hf
+
+/-- Reversing a continuous BV function negates its transported Stieltjes measure. -/
+theorem BoundedVariationOn.vectorMeasure_map_reverse_Icc
+    {a b : ℝ} (hab : a ≤ b) {f : Set.Icc a b → ℝ}
+    (hf : BoundedVariationOn f Set.univ) (hfc : Continuous f) :
+    let r := Set.Icc.reverse hab
+    let hfr := BoundedVariationOn.comp_antitone_surjective_Icc hab hf
+      (Set.Icc.antitone_reverse hab) (Set.Icc.surjective_reverse hab)
+    hfr.vectorMeasure.map r = -hf.vectorMeasure := by
+  dsimp only
+  let _ : Fact (a ≤ b) := ⟨hab⟩
+  let r := Set.Icc.reverse hab
+  let hfr := BoundedVariationOn.comp_antitone_surjective_Icc hab hf
+    (Set.Icc.antitone_reverse hab) (Set.Icc.surjective_reverse hab)
+  have hr : Continuous r := Set.Icc.continuous_reverse hab
+  have hleft (x : Set.Icc a b) : f.leftLim x = f x :=
+    hfc.continuousAt.continuousWithinAt.leftLim_eq
+  have hleft_comp (x : Set.Icc a b) : (f ∘ r).leftLim x = (f ∘ r) x :=
+    (hfc.comp hr).continuousAt.continuousWithinAt.leftLim_eq
+  have hright (x : Set.Icc a b) : f.rightLim x = f x :=
+    hfc.continuousAt.continuousWithinAt.rightLim_eq
+  have hpre (u v : Set.Icc a b) :
+      r ⁻¹' Ioc u v = Ico (r v) (r u) := by
+    ext x
+    simp only [mem_preimage, mem_Ioc, mem_Ico]
+    change (u : ℝ) < a + b - x ∧ a + b - x ≤ v ↔
+      a + b - v ≤ x ∧ x < a + b - u
+    constructor <;> intro hx <;> constructor <;> linarith
+  have hinterval (u v : Set.Icc a b) (huv : u ≤ v) :
+      (hfr.vectorMeasure.map r) (Ioc u v) = (-hf.vectorMeasure) (Ioc u v) := by
+    rw [MeasureTheory.VectorMeasure.map_apply _ hr.measurable measurableSet_Ioc, hpre,
+      hfr.vectorMeasure_Ico ((Set.Icc.antitone_reverse hab) huv)]
+    change _ = -(hf.vectorMeasure (Ioc u v))
+    rw [hf.vectorMeasure_Ioc huv, hleft_comp, hleft_comp, hright, hright]
+    simp [r, Set.Icc.reverse, Function.comp_apply]
+  apply MeasureTheory.VectorMeasure.ext_of_generateFrom
+      {s | ∃ u v : Set.Icc a b, u ≤ v ∧ s = Ioc u v}
+  · rintro s ⟨u, v, huv, rfl⟩
+    exact hinterval u v huv
+  · exact BorelSpace.measurable_eq.trans <| by
+      rw [borel_eq_generateFrom_Ioc_le]
+      congr 1
+      ext s
+      simp only [Set.mem_ofPred_eq]
+      constructor
+      · rintro ⟨u, v, huv, rfl⟩
+        exact ⟨u, v, huv, rfl⟩
+      · rintro ⟨u, v, huv, rfl⟩
+        exact ⟨u, v, huv, rfl⟩
+  · exact IsSetSemiring.isPiSystem IsSetSemiring.Ioc
+  · rw [MeasureTheory.VectorMeasure.map_apply _ hr.measurable MeasurableSet.univ,
+      preimage_univ, hfr.vectorMeasure_univ]
+    change _ = -(hf.vectorMeasure Set.univ)
+    rw [hf.vectorMeasure_univ]
+    have htop : Filter.atTop.limUnder (f ∘ r) = f ⊥ := by
+      apply tendsto_nhds_unique hfr.tendsto_atTop_limUnder
+      rw [Filter.atTop_eq_pure_of_isTop isTop_top]
+      convert tendsto_pure_nhds (f ∘ r) ⊤ using 1
+      apply congrArg nhds
+      apply congrArg f
+      apply Subtype.ext
+      change a = a + b - b
+      ring
+    have hbot : Filter.atBot.limUnder (f ∘ r) = f ⊤ := by
+      apply tendsto_nhds_unique hfr.tendsto_atBot_limUnder
+      rw [Filter.atBot_eq_pure_of_isBot isBot_bot]
+      convert tendsto_pure_nhds (f ∘ r) ⊥ using 1
+      apply congrArg nhds
+      apply congrArg f
+      apply Subtype.ext
+      change b = a + b - a
+      ring
+    have htopf : Filter.atTop.limUnder f = f ⊤ := by
+      apply tendsto_nhds_unique hf.tendsto_atTop_limUnder
+      rw [Filter.atTop_eq_pure_of_isTop isTop_top]
+      exact tendsto_pure_nhds _ _
+    have hbotf : Filter.atBot.limUnder f = f ⊥ := by
+      apply tendsto_nhds_unique hf.tendsto_atBot_limUnder
+      rw [Filter.atBot_eq_pure_of_isBot isBot_bot]
+      exact tendsto_pure_nhds _ _
+    rw [htop, hbot, htopf, hbotf]
+    simp
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# For Mathlib / Measure Theory / Vector Measure / Locally Constant
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter MeasureTheory
+open scoped Topology
+
+/-- Local constancy away from the initial endpoint gives a variation-null neighborhood. -/
+theorem BoundedVariationOn.exists_nhds_variation_vectorMeasure_eq_zero_of_eventuallyEq_const
+    {a b : ℝ} (hab : a < b) {g : Set.Icc a b → ℝ}
+    (hg : BoundedVariationOn g Set.univ)
+    (hr : ∀ y, ContinuousWithinAt g (Set.Ici y) y) (x : Set.Icc a b)
+    (hxbot : a < (x : ℝ))
+    (hx : g =ᶠ[𝓝 x] fun _ ↦ g x) :
+    ∃ U ∈ 𝓝 x, hg.vectorMeasure.variation U = 0 := by
+  let _ : Fact (a ≤ b) := ⟨hab.le⟩
+  let _ : Nontrivial (Set.Icc a b) :=
+    ⟨⟨⟨a, le_rfl, hab.le⟩, ⟨b, hab.le, le_rfl⟩, fun h ↦
+      hab.ne (congrArg Subtype.val h)⟩⟩
+  have hright (y : Set.Icc a b) : Function.rightLim g y = g y :=
+    (hr y).rightLim_eq
+  by_cases hxtop : x = ⊤
+  · subst x
+    obtain ⟨c, hc, hsub⟩ := (nhds_top_basis.mem_iff.mp hx)
+    refine ⟨Set.Ioi c, nhds_top_basis.mem_iff.mpr ⟨c, hc, subset_rfl⟩, ?_⟩
+    rw [hg.variation_vectorMeasure_Ioi]
+    rw [show Function.rightLim g = g from funext hright]
+    apply eVariationOn.constant_on
+    rintro _ ⟨y, hy, rfl⟩ _ ⟨z, hz, rfl⟩
+    exact (hsub hy).trans (hsub hz).symm
+  · have hxlt : x < ⊤ := lt_top_iff_ne_top.mpr hxtop
+    have hxgt : ⊥ < x := hxbot
+    obtain ⟨c, d, hxcd, hsub⟩ :=
+      (mem_nhds_iff_exists_Ioo_subset' ⟨⊥, hxgt⟩ ⟨⊤, hxlt⟩).mp hx
+    refine ⟨Set.Ioo c d, Ioo_mem_nhds hxcd.1 hxcd.2, ?_⟩
+    rw [hg.variation_vectorMeasure_Ioo_right]
+    rw [show Function.rightLim g = g from funext hright]
+    apply eVariationOn.constant_on
+    rintro _ ⟨y, hy, rfl⟩ _ ⟨z, hz, rfl⟩
+    exact (hsub hy).trans (hsub hz).symm
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Geometry.Foundations.Development001`.
+* `Cap.Foundations.Development001`.
+* `Polygon.Foundations.Development001`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Geometry.Plane`.
+* `Geometry.Basic`.
+* `Geometry.Frame`.
+* `Geometry.NormalLines`.
+* `Geometry.QuadrantBounds`.
+* `Geometry.Subgraph`.
+* `Geometry.Support`.
+* `Geometry.Contacts`.
+* `Geometry.FrameCalculus`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Plane
+-/
+
+@[expose] public section
+
+namespace MovingSofa
+
+/-- The two-dimensional real Euclidean space used for sofa geometry. -/
+abbrev Point := EuclideanSpace ℝ (Fin 2)
+
+/-- The squared norm of a planar point in coordinates. -/
+theorem Point.norm_sq_eq (z : Point) : ‖z‖ ^ 2 = z 0 ^ 2 + z 1 ^ 2 := by
+  rw [EuclideanSpace.norm_sq_eq]
+  simp [Fin.sum_univ_two, Real.norm_eq_abs, sq_abs]
+
+/-- Cauchy-Schwarz for the coordinate form of the planar inner product. -/
+theorem Point.abs_inner_coords_le (w d : Point) :
+    |w 0 * d 0 + w 1 * d 1| ≤ ‖w‖ * ‖d‖ := by
+  have hinner : (inner ℝ w d : ℝ) = w 0 * d 0 + w 1 * d 1 := by
+    simp [inner, Fin.sum_univ_two]
+    ring
+  rw [← hinner]
+  exact abs_real_inner_le_norm w d
+
+/-- Each coordinate of a planar point is bounded by its norm. -/
+theorem Point.abs_apply_le_norm (z : Point) (i : Fin 2) : |z i| ≤ ‖z‖ := by
+  simpa [PiLp.norm_single, EuclideanSpace.inner_single_right] using
+    abs_real_inner_le_norm z (EuclideanSpace.single i (1 : ℝ))
+
+end MovingSofa
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Basic
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The normal and positively oriented tangent at an angular direction. -/
+def frame (t : Real.Angle) : Point × Point :=
+  (!₂[t.cos, t.sin], !₂[-t.sin, t.cos])
+
+/-- The unit normal of the angular frame. -/
+abbrev normalVector (t : Real.Angle) : Point := (frame t).1
+
+/-- The counterclockwise unit tangent of the angular frame. -/
+abbrev tangentVector (t : Real.Angle) : Point := (frame t).2
+
+/-- The support value; geometric results require a nonempty compact set. -/
+def supportValue (s : Set Point) (t : Real.Angle) : ℝ :=
+  sSup ((fun p ↦ inner ℝ p (normalVector t)) '' s)
+
+/-- The line with the given unit normal and signed offset. -/
+def normalLine (t : Real.Angle) (h : ℝ) : Set Point :=
+  {p | inner ℝ p (normalVector t) = h}
+
+/-- A normal half-plane: `upper` chooses the greater side, `strict` its open version. -/
+def normalHalfPlane (t : Real.Angle) (h : ℝ) (upper strict : Bool) : Set Point :=
+  {p | if upper then
+    if strict then h < inner ℝ p (normalVector t) else h ≤ inner ℝ p (normalVector t)
+  else
+    if strict then inner ℝ p (normalVector t) < h else inner ℝ p (normalVector t) ≤ h}
+
+/-- Closed normal half-planes are closed, on either side of the boundary line. -/
+theorem isClosed_normalHalfPlane (t : Real.Angle) (h : ℝ) (upper : Bool) :
+    IsClosed (normalHalfPlane t h upper false) := by
+  cases upper
+  · exact isClosed_le (continuous_id.inner continuous_const) continuous_const
+  · exact isClosed_le continuous_const (continuous_id.inner continuous_const)
+
+/-- Closed normal half-planes are convex, on either side of the boundary line. -/
+theorem convex_normalHalfPlane (t : Real.Angle) (h : ℝ) (upper : Bool) :
+    Convex ℝ (normalHalfPlane t h upper false) := by
+  have hlin : IsLinearMap ℝ fun p : Point ↦ inner ℝ p (normalVector t) :=
+    isLinearMap_inner_left _
+  cases upper
+  · exact convex_halfSpace_le hlin h
+  · exact convex_halfSpace_ge hlin h
+
+/-- Open normal half-planes are convex, on either side of the boundary line. -/
+theorem convex_normalHalfPlane_open (t : Real.Angle) (h : ℝ) (upper : Bool) :
+    Convex ℝ (normalHalfPlane t h upper true) := by
+  have hlin : IsLinearMap ℝ fun p : Point ↦ inner ℝ p (normalVector t) :=
+    isLinearMap_inner_left _
+  cases upper
+  · exact convex_halfSpace_lt hlin h
+  · exact convex_halfSpace_gt hlin h
+
+/-- The supporting line and closed containing half-plane of a nonempty compact set. -/
+def supportingLineHalfPlane (s : Set Point) (t : Real.Angle) : Set Point × Set Point :=
+  (normalLine t (supportValue s t), normalHalfPlane t (supportValue s t) false false)
+
+/-- A closed supporting half-plane, read through the opposite normal direction. -/
+theorem supportingLineHalfPlane_snd_eq_normalHalfPlane {s : Set Point} {t u : Real.Angle}
+    {h : ℝ} (hn : normalVector u = -normalVector t) (hs : supportValue s u = -h) :
+    (supportingLineHalfPlane s u).2 = normalHalfPlane t h true false := by
+  ext q
+  change inner ℝ q (normalVector u) ≤ supportValue s u ↔ h ≤ inner ℝ q (normalVector t)
+  rw [hn, inner_neg_right, hs]
+  constructor <;> intro hq <;> linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Frame
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Every unit vector is the normal vector of an angular frame. -/
+theorem exists_angle_normalVector_eq {u : Point} (hu : ‖u‖ = 1) :
+    ∃ t : Real.Angle, normalVector t = u := by
+  let z : ℂ := ⟨u 0, u 1⟩
+  have hzNorm : ‖z‖ = 1 := by
+    rw [Complex.norm_def, show Complex.normSq z = ‖u‖ ^ 2 by
+      rw [EuclideanSpace.norm_sq_eq]
+      simp [z, Complex.normSq_apply, Fin.sum_univ_two, pow_two, Real.norm_eq_abs]]
+    simp [hu]
+  have hz : z ≠ 0 := by
+    intro hz
+    simp [hz] at hzNorm
+  refine ⟨(z.arg : ℝ), ?_⟩
+  ext i
+  fin_cases i
+  · simpa [normalVector, frame, z, hzNorm] using Complex.cos_arg hz
+  · simpa [normalVector, frame, z, hzNorm] using Complex.sin_arg z
+
+theorem continuous_normalVector_real :
+    Continuous (fun t : ℝ ↦ normalVector (t : Real.Angle)) := by
+  let c : ℝ → (i : Fin 2) → ℝ := fun t i ↦
+    Fin.cases (Real.cos t) (fun _ ↦ Real.sin t) i
+  have hc : Continuous c := by
+    apply continuous_pi
+    intro i
+    fin_cases i
+    · exact Real.continuous_cos
+    · exact Real.continuous_sin
+  have heq : (fun t : ℝ ↦ normalVector (t : Real.Angle)) =
+      (fun t ↦ WithLp.toLp 2 (c t)) := by
+    funext t
+    ext i
+    fin_cases i <;> rfl
+  rw [heq]
+  exact (PiLp.continuous_toLp (2 : ENNReal) (fun _ : Fin 2 ↦ ℝ)).comp hc
+
+/-- Opposite real angles give opposite normal vectors. -/
+theorem normalVector_add_pi (t : ℝ) :
+    normalVector ((t + Real.pi : ℝ) : Real.Angle) = -normalVector (t : Real.Angle) := by
+  ext i
+  fin_cases i <;> simp [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe]
+
+/-- Opposite real angles give opposite tangent vectors. -/
+theorem tangentVector_add_pi (t : ℝ) :
+    tangentVector ((t + Real.pi : ℝ) : Real.Angle) = -tangentVector (t : Real.Angle) := by
+  ext i
+  fin_cases i <;> simp [tangentVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe]
+
+/-- The projection onto the normal direction of a real angle, in coordinates. -/
+theorem inner_normalVector_real (p : Point) (t : ℝ) :
+    inner ℝ p (normalVector (t : Real.Angle)) = p 0 * Real.cos t + p 1 * Real.sin t := by
+  simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+  ring
+
+/-- The projection onto the tangent direction of a real angle, in coordinates. -/
+theorem inner_tangentVector_real (p : Point) (t : ℝ) :
+    inner ℝ p (tangentVector (t : Real.Angle)) = -(p 0 * Real.sin t) + p 1 * Real.cos t := by
+  simp [tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+  ring
+
+/-- The inner product of two unit normals is the cosine of their angle difference. -/
+theorem inner_normalVector_normalVector (s t : ℝ) :
+    inner ℝ (normalVector (s : Real.Angle)) (normalVector (t : Real.Angle)) =
+      Real.cos (s - t) := by
+  simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two, Real.cos_sub]
+  ring
+
+/-- A normal vector has squared length one. -/
+theorem inner_normalVector_self (t : ℝ) :
+    inner ℝ (normalVector (t : Real.Angle)) (normalVector (t : Real.Angle)) = 1 := by
+  rw [inner_normalVector_normalVector]
+  simp
+
+/-- A normal vector at an angle has squared length one. -/
+theorem inner_normalVector_self_angle (t : Real.Angle) :
+    inner ℝ (normalVector t) (normalVector t) = 1 := by
+  induction t using Real.Angle.induction_on with
+  | _ r => exact inner_normalVector_self r
+
+/-- A normal vector at a real angle has length one. -/
+theorem norm_normalVector_real (t : ℝ) : ‖normalVector (t : Real.Angle)‖ = 1 := by
+  rw [← sq_eq_sq₀ (norm_nonneg _) (by norm_num : (0 : ℝ) ≤ 1)]
+  rw [EuclideanSpace.norm_sq_eq]
+  simp [normalVector, frame, Fin.sum_univ_two]
+
+theorem inner_normalVector_smul_add_inner_tangentVector_smul (p : Point) (t : Real.Angle) :
+    inner ℝ p (normalVector t) • normalVector t +
+      inner ℝ p (tangentVector t) • tangentVector t = p := by
+  ext i
+  fin_cases i <;>
+    simp [normalVector, tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two] <;>
+    nlinarith [congrArg (fun r : ℝ ↦ r * p 0) t.cos_sq_add_sin_sq,
+      congrArg (fun r : ℝ ↦ r * p 1) t.cos_sq_add_sin_sq]
+
+theorem normalVector_add_real (t δ : ℝ) :
+    normalVector ((t + δ : ℝ) : Real.Angle) =
+      Real.cos δ • normalVector (t : Real.Angle) +
+      Real.sin δ • tangentVector (t : Real.Angle) := by
+  ext i
+  fin_cases i <;> simp [normalVector, tangentVector, frame, -Real.Angle.coe_add, Real.cos_add,
+    Real.sin_add] <;> ring
+
+theorem inner_normalVector_tangentVector (t : ℝ) :
+    inner ℝ (normalVector (t : Real.Angle)) (tangentVector (t : Real.Angle)) = 0 := by
+  simp [normalVector, tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+  ring
+
+theorem inner_tangentVector_self (t : ℝ) :
+    inner ℝ (tangentVector (t : Real.Angle)) (tangentVector (t : Real.Angle)) = 1 := by
+  rw [PiLp.inner_apply]
+  simp [tangentVector, frame, Fin.sum_univ_two, Real.sin_sq_add_cos_sq]
+
+theorem tangentVector_add_pi_div_two (t : ℝ) :
+    tangentVector ((t + Real.pi / 2 : ℝ) : Real.Angle) = -normalVector (t : Real.Angle) := by
+  ext i
+  fin_cases i <;> simp [normalVector, tangentVector, frame,
+    Real.Angle.sin_add_pi_div_two, Real.Angle.cos_add_pi_div_two]
+
+/-- Adding pi to an angle reverses its normal vector. -/
+theorem normalVector_add_pi_angle (a : Real.Angle) :
+    normalVector (a + ((Real.pi : ℝ) : Real.Angle)) = -normalVector a := by
+  induction a using Real.Angle.induction_on with
+  | _ a => simpa only [Real.Angle.coe_add] using normalVector_add_pi a
+
+/-- The sine convolution kernel is the negative normal projection of the moving tangent. -/
+theorem sin_sub_eq_neg_inner_normalVector_tangentVector (t u : Real.Angle) :
+    (u - t).sin = -inner ℝ (normalVector t) (tangentVector u) := by
+  induction t using Real.Angle.induction_on with
+  | _ t =>
+    induction u using Real.Angle.induction_on with
+    | _ u =>
+      change Real.sin (u - t) = _
+      simp [normalVector, tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two,
+        Real.sin_sub]
+      ring
+
+/-- The normal projection of a tangent vector is the sine of the angle difference. -/
+theorem inner_tangentVector_normalVector_real (t s : ℝ) :
+    inner ℝ (tangentVector (t : Real.Angle)) (normalVector (s : Real.Angle)) =
+      Real.sin (s - t) := by
+  simp [tangentVector, normalVector, frame, PiLp.inner_apply, Real.sin_sub]
+  ring
+
+/-- The inner product of two unit tangents is the cosine of their angle difference. -/
+theorem inner_tangentVector_tangentVector (s t : ℝ) :
+    inner ℝ (tangentVector (s : Real.Angle)) (tangentVector (t : Real.Angle)) =
+      Real.cos (s - t) := by
+  simp [tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two, Real.cos_sub]
+  ring
+
+/-- The normal coordinate of a vector in the frame at another angle. -/
+theorem inner_normalVector_eq_frame_rotate (w : Point) (s t : ℝ) :
+    inner ℝ w (normalVector (t : Real.Angle)) =
+      inner ℝ w (normalVector (s : Real.Angle)) * Real.cos (s - t) -
+        inner ℝ w (tangentVector (s : Real.Angle)) * Real.sin (s - t) := by
+  nth_rewrite 1 [← inner_normalVector_smul_add_inner_tangentVector_smul w (s : Real.Angle)]
+  rw [inner_add_left, real_inner_smul_left, real_inner_smul_left,
+    inner_normalVector_normalVector, inner_tangentVector_normalVector_real,
+    show t - s = -(s - t) by ring, Real.sin_neg]
+  ring
+
+/-- The tangent coordinate of a vector in the frame at another angle. -/
+theorem inner_tangentVector_eq_frame_rotate (w : Point) (s t : ℝ) :
+    inner ℝ w (tangentVector (t : Real.Angle)) =
+      inner ℝ w (normalVector (s : Real.Angle)) * Real.sin (s - t) +
+        inner ℝ w (tangentVector (s : Real.Angle)) * Real.cos (s - t) := by
+  have hcross : inner ℝ (normalVector (s : Real.Angle)) (tangentVector (t : Real.Angle)) =
+      Real.sin (s - t) := by
+    rw [real_inner_comm]
+    exact inner_tangentVector_normalVector_real t s
+  nth_rewrite 1 [← inner_normalVector_smul_add_inner_tangentVector_smul w (s : Real.Angle)]
+  rw [inner_add_left, real_inner_smul_left, real_inner_smul_left,
+    inner_tangentVector_tangentVector, hcross]
+
+/-- Two points whose normal coordinates agree at two transverse angles coincide. -/
+theorem eq_of_inner_normalVector_eq {p q : Point} {s t : ℝ}
+    (hst : Real.sin (s - t) ≠ 0)
+    (hs : inner ℝ p (normalVector (s : Real.Angle)) =
+      inner ℝ q (normalVector (s : Real.Angle)))
+    (ht : inner ℝ p (normalVector (t : Real.Angle)) =
+      inner ℝ q (normalVector (t : Real.Angle))) : p = q := by
+  have hp := inner_normalVector_eq_frame_rotate p s t
+  have hq := inner_normalVector_eq_frame_rotate q s t
+  have hmul : (inner ℝ p (tangentVector (s : Real.Angle)) -
+      inner ℝ q (tangentVector (s : Real.Angle))) * Real.sin (s - t) = 0 := by
+    linear_combination hp - hq - ht + Real.cos (s - t) * hs
+  have hv : inner ℝ p (tangentVector (s : Real.Angle)) =
+      inner ℝ q (tangentVector (s : Real.Angle)) :=
+    sub_eq_zero.mp ((mul_eq_zero.mp hmul).resolve_right hst)
+  calc p = inner ℝ p (normalVector (s : Real.Angle)) • normalVector (s : Real.Angle) +
+        inner ℝ p (tangentVector (s : Real.Angle)) • tangentVector (s : Real.Angle) :=
+        (inner_normalVector_smul_add_inner_tangentVector_smul p (s : Real.Angle)).symm
+    _ = inner ℝ q (normalVector (s : Real.Angle)) • normalVector (s : Real.Angle) +
+        inner ℝ q (tangentVector (s : Real.Angle)) • tangentVector (s : Real.Angle) := by
+        rw [hs, hv]
+    _ = q := inner_normalVector_smul_add_inner_tangentVector_smul q (s : Real.Angle)
+
+/-- Each coordinate of the frame tangent is continuous in the angle. -/
+theorem continuous_tangentVector_coordinate (i : Fin 2) :
+    Continuous fun u : Real.Angle ↦ tangentVector u i := by
+  fin_cases i
+  · exact Real.Angle.continuous_sin.neg
+  · exact Real.Angle.continuous_cos
+
+/-- The frame tangent is a unit vector. -/
+theorem norm_tangentVector (t : Real.Angle) : ‖tangentVector t‖ = 1 := by
+  rw [← sq_eq_sq₀ (norm_nonneg _) (by norm_num : (0 : ℝ) ≤ 1), Point.norm_sq_eq]
+  simp [tangentVector, frame, add_comm]
+
+/-- Each coordinate of the frame tangent is bounded by one. -/
+theorem norm_tangentVector_coordinate_le_one (t : Real.Angle) (i : Fin 2) :
+    ‖tangentVector t i‖ ≤ 1 := by
+  simpa [Real.norm_eq_abs, norm_tangentVector t] using
+    Point.abs_apply_le_norm (tangentVector t) i
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Normal Lines
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A nontrivial segment has at most one normal direction strictly between zero and pi. -/
+theorem eq_of_inner_sub_normalVector_eq_zero_of_ne {a b : Point} {s t : ℝ}
+    (hs : s ∈ Set.Ioo 0 Real.pi) (ht : t ∈ Set.Ioo 0 Real.pi) (hab : a ≠ b)
+    (horths : inner ℝ (b - a) (normalVector (s : Real.Angle)) = 0)
+    (hortht : inner ℝ (b - a) (normalVector (t : Real.Angle)) = 0) : s = t := by
+  have hs' : (b 0 - a 0) * Real.cos s + (b 1 - a 1) * Real.sin s = 0 := by
+    simpa [normalVector, frame, PiLp.inner_apply, mul_comm] using horths
+  have ht' : (b 0 - a 0) * Real.cos t + (b 1 - a 1) * Real.sin t = 0 := by
+    simpa [normalVector, frame, PiLp.inner_apply, mul_comm] using hortht
+  have hxprod : (b 0 - a 0) * Real.sin (s - t) = 0 := by
+    rw [Real.sin_sub]
+    linear_combination Real.sin s * ht' - Real.sin t * hs'
+  have hyprod : (b 1 - a 1) * Real.sin (s - t) = 0 := by
+    rw [Real.sin_sub]
+    linear_combination Real.cos t * hs' - Real.cos s * ht'
+  have hsin : Real.sin (s - t) = 0 := by
+    by_contra hne
+    have hx : b 0 = a 0 := by
+      exact sub_eq_zero.mp ((mul_eq_zero.mp hxprod).resolve_right hne)
+    have hy : b 1 = a 1 := by
+      exact sub_eq_zero.mp ((mul_eq_zero.mp hyprod).resolve_right hne)
+    apply hab
+    ext i
+    fin_cases i
+    · exact hx.symm
+    · exact hy.symm
+  have hz := (Real.sin_eq_zero_iff_of_lt_of_lt
+    (by linarith [hs.1, ht.2]) (by linarith [hs.2, ht.1])).mp hsin
+  linarith
+
+/-- Normal lines with angles strictly between zero and pi agree exactly when their data agree. -/
+theorem normalLine_eq_iff_of_mem_Ioo {s t c d : ℝ}
+    (hs : s ∈ Set.Ioo 0 Real.pi) (ht : t ∈ Set.Ioo 0 Real.pi) :
+    normalLine (s : Real.Angle) c = normalLine (t : Real.Angle) d ↔
+      s = t ∧ c = d := by
+  constructor
+  · intro hline
+    let p := c • normalVector (s : Real.Angle)
+    let q := p + tangentVector (s : Real.Angle)
+    have hp : p ∈ normalLine (s : Real.Angle) c := by
+      change inner ℝ p (normalVector (s : Real.Angle)) = c
+      rw [show p = c • normalVector (s : Real.Angle) by rfl,
+        real_inner_smul_left, inner_normalVector_self]
+      simp
+    have hq : q ∈ normalLine (s : Real.Angle) c := by
+      change inner ℝ q (normalVector (s : Real.Angle)) = c
+      rw [show q = p + tangentVector (s : Real.Angle) by rfl, inner_add_left]
+      rw [show inner ℝ p (normalVector (s : Real.Angle)) = c by exact hp]
+      rw [real_inner_comm, inner_normalVector_tangentVector, add_zero]
+    have hp' : p ∈ normalLine (t : Real.Angle) d := hline ▸ hp
+    have hq' : q ∈ normalLine (t : Real.Angle) d := hline ▸ hq
+    have hv : tangentVector (s : Real.Angle) ≠ 0 := by
+      intro hz
+      have hinner := inner_tangentVector_self s
+      rw [hz, inner_zero_left] at hinner
+      norm_num at hinner
+    have hsorth : inner ℝ (tangentVector (s : Real.Angle))
+        (normalVector (s : Real.Angle)) = 0 := by
+      rw [real_inner_comm, inner_normalVector_tangentVector]
+    have htorth : inner ℝ (tangentVector (s : Real.Angle))
+        (normalVector (t : Real.Angle)) = 0 := by
+      rw [show tangentVector (s : Real.Angle) = q - p by simp [q]]
+      rw [inner_sub_left, show inner ℝ q (normalVector (t : Real.Angle)) = d by exact hq',
+        show inner ℝ p (normalVector (t : Real.Angle)) = d by exact hp', sub_self]
+    have hst := eq_of_inner_sub_normalVector_eq_zero_of_ne
+      (a := 0) (b := tangentVector (s : Real.Angle)) hs ht hv.symm
+      (by simpa using hsorth) (by simpa using htorth)
+    subst t
+    refine ⟨rfl, ?_⟩
+    change inner ℝ p (normalVector (s : Real.Angle)) = d at hp'
+    rw [show inner ℝ p (normalVector (s : Real.Angle)) = c by exact hp] at hp'
+    exact hp'
+  · rintro ⟨rfl, rfl⟩
+    rfl
+
+/-- Opposite normal angles describe the same line with the opposite offset. -/
+theorem normalLine_eq_of_cut {a b c : ℝ}
+    (hab : ((b : ℝ) : Real.Angle) = ((a + Real.pi : ℝ) : Real.Angle)) :
+    normalLine ((b : ℝ) : Real.Angle) c = normalLine ((a : ℝ) : Real.Angle) (-c) := by
+  rw [hab]
+  ext p
+  simp only [normalLine, Set.mem_ofPred_eq, normalVector_add_pi, inner_neg_right]
+  constructor <;> intro h <;> linarith only [h]
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Quadrant Bounds
+-/
+
+@[expose] public section
+
+namespace MovingSofa
+
+/-- A horizontal lower bound cuts a first-quadrant pair of upper projection bounds to a bounded set.
+-/
+theorem isBounded_setOf_le_snd_and_inner_lt (a b c t : ℝ) (ht : t ∈ Set.Ioo 0 (Real.pi / 2)) :
+    Bornology.IsBounded {p : Point | a ≤ p 1 ∧
+      inner ℝ p (normalVector (t : Real.Angle)) < b ∧
+      inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) < c} := by
+  have hs : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi ht.1
+    (ht.2.trans (by linarith [Real.pi_pos]))
+  have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [ht.1, Real.pi_pos], ht.2⟩
+  apply (EuclideanSpace.isBounded_coordinate_rectangle ((Real.cos t * a - c) / Real.sin t)
+    ((b - Real.sin t * a) / Real.cos t) a (Real.sin t * b + Real.cos t * c)).subset
+  rintro p ⟨hy, h₁, h₂⟩
+  have hb : Real.cos t * p 0 + Real.sin t * p 1 < b := by
+    simpa [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two, mul_comm] using h₁
+  have hd : -Real.sin t * p 0 + Real.cos t * p 1 < c := by
+    simpa [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two,
+      Real.cos_add, Real.sin_add, -Real.Angle.coe_add, mul_comm] using h₂
+  refine ⟨?_, ?_, hy, ?_⟩
+  · apply (div_le_iff₀ hs).2
+    nlinarith only [hd, mul_le_mul_of_nonneg_left hy hc.le]
+  · apply (le_div_iff₀ hc).2
+    nlinarith only [hb, mul_le_mul_of_nonneg_left hy hs.le]
+  · have hyid : (Real.sin t ^ 2 + Real.cos t ^ 2) * p 1 = p 1 := by
+      rw [Real.sin_sq_add_cos_sq, one_mul]
+    have hb' := mul_lt_mul_of_pos_left hb hs
+    have hd' := mul_lt_mul_of_pos_left hd hc
+    nlinarith only [hyid, hb', hd']
+
+end MovingSofa
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The planar region under the graph of a function
+
+For `a b : ℝ` and `f : ℝ → ℝ` this file describes the three planar regions between the
+horizontal axis and the graph of `f` over `[a, b]`: `closedSubgraph`, which contains both the
+base segment and the graph, `strictSubgraph`, which contains the base but not the graph, and
+`openSubgraph`, which contains neither.
+
+For a function continuous on `[a, b]`, vanishing at `a` and `b` and positive in between, the
+closed region is the closure of either smaller region (`closure_openSubgraph`,
+`closure_strictSubgraph`) and the open region is its interior (`interior_closedSubgraph`).
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A planar point is recovered from its two coordinates. -/
+theorem Point.eq_vecNotation (z : Point) : z = !₂[z 0, z 1] := by
+  ext i
+  fin_cases i <;> simp
+
+/-- The closed region between the base and the graph of `f` over `[a, b]`. -/
+def closedSubgraph (a b : ℝ) (f : ℝ → ℝ) : Set Point :=
+  {q | a ≤ q 0 ∧ q 0 ≤ b ∧ 0 ≤ q 1 ∧ q 1 ≤ f (q 0)}
+
+/-- The region under the graph of `f` over `[a, b]`, including the base but not the graph. -/
+def strictSubgraph (a b : ℝ) (f : ℝ → ℝ) : Set Point :=
+  {q | a ≤ q 0 ∧ q 0 ≤ b ∧ 0 ≤ q 1 ∧ q 1 < f (q 0)}
+
+/-- The open region strictly between the base and the graph of `f` over `(a, b)`. -/
+def openSubgraph (a b : ℝ) (f : ℝ → ℝ) : Set Point :=
+  {q | a < q 0 ∧ q 0 < b ∧ 0 < q 1 ∧ q 1 < f (q 0)}
+
+section Subgraph
+
+variable {a b : ℝ} {f : ℝ → ℝ}
+
+/-- The open subgraph omits the base, which the strict subgraph contains. -/
+theorem openSubgraph_subset_strictSubgraph : openSubgraph a b f ⊆ strictSubgraph a b f :=
+  fun _ h ↦ ⟨h.1.le, h.2.1.le, h.2.2.1.le, h.2.2.2⟩
+
+/-- The strict subgraph omits the graph, which the closed subgraph contains. -/
+theorem strictSubgraph_subset_closedSubgraph : strictSubgraph a b f ⊆ closedSubgraph a b f :=
+  fun _ h ↦ ⟨h.1, h.2.1, h.2.2.1, h.2.2.2.le⟩
+
+/-- The open subgraph is contained in the closed one. -/
+theorem openSubgraph_subset_closedSubgraph : openSubgraph a b f ⊆ closedSubgraph a b f :=
+  openSubgraph_subset_strictSubgraph.trans strictSubgraph_subset_closedSubgraph
+
+/-- The vertical line through a fixed horizontal coordinate is continuous. -/
+private theorem continuous_verticalLine (c : ℝ) :
+    Continuous fun y : ℝ ↦ (!₂[c, y] : Point) :=
+  (PiLp.continuous_toLp (2 : ENNReal) fun _ : Fin 2 ↦ ℝ).comp
+    (continuous_const.matrixVecCons (continuous_id.matrixVecCons continuous_const))
+
+/-- The closed subgraph of a function continuous on the base interval is closed. -/
+theorem isClosed_closedSubgraph (hf : ContinuousOn f (Set.Icc a b)) :
+    IsClosed (closedSubgraph a b f) := by
+  have hc0 : Continuous fun q : Point ↦ q 0 := PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 0
+  have hc1 : Continuous fun q : Point ↦ q 1 := PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 1
+  have hSclosed : IsClosed {q : Point | q 0 ∈ Set.Icc a b} := isClosed_Icc.preimage hc0
+  have hgS : ContinuousOn (fun q : Point ↦ f (q 0) - q 1) {q : Point | q 0 ∈ Set.Icc a b} :=
+    (hf.comp hc0.continuousOn fun q hq ↦ hq).sub hc1.continuousOn
+  have hone := hgS.preimage_isClosed_of_isClosed hSclosed (isClosed_Ici (a := (0 : ℝ)))
+  have htwo : IsClosed {q : Point | 0 ≤ q 1} := isClosed_Ici.preimage hc1
+  have hsplit : closedSubgraph a b f = ({q : Point | q 0 ∈ Set.Icc a b} ∩
+      (fun q : Point ↦ f (q 0) - q 1) ⁻¹' Set.Ici 0) ∩ {q : Point | 0 ≤ q 1} := by
+    ext q
+    simp only [closedSubgraph, Set.mem_ofPred_eq, Set.mem_inter_iff, Set.mem_Icc,
+      Set.mem_preimage, Set.mem_Ici, sub_nonneg]
+    tauto
+  rw [hsplit]
+  exact hone.inter htwo
+
+/-- The open subgraph of a function continuous on the base interval is open. -/
+theorem isOpen_openSubgraph (hf : ContinuousOn f (Set.Icc a b)) :
+    IsOpen (openSubgraph a b f) := by
+  have hc0 : Continuous fun q : Point ↦ q 0 := PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 0
+  have hc1 : Continuous fun q : Point ↦ q 1 := PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 1
+  rw [isOpen_iff_mem_nhds]
+  rintro p ⟨h1, h2, h3, h4⟩
+  have hfat : ContinuousAt f (p 0) :=
+    (hf.mono Set.Ioo_subset_Icc_self).continuousAt (Ioo_mem_nhds h1 h2)
+  have hgap : ContinuousAt (fun q : Point ↦ f (q 0) - q 1) p :=
+    (ContinuousAt.comp (g := f) (f := fun q : Point ↦ q 0) (x := p) hfat
+      hc0.continuousAt).sub hc1.continuousAt
+  filter_upwards [hc0.continuousAt (isOpen_Ioo.mem_nhds (⟨h1, h2⟩ : p 0 ∈ Set.Ioo a b)),
+    hc1.continuousAt (isOpen_Ioi.mem_nhds (show p 1 ∈ Set.Ioi (0 : ℝ) from h3)),
+    hgap (isOpen_Ioi.mem_nhds (show f (p 0) - p 1 ∈ Set.Ioi (0 : ℝ) from sub_pos.mpr h4))]
+    with q hq1 hq2 hq3
+  exact ⟨hq1.1, hq1.2, hq2, by simpa using sub_pos.mp hq3⟩
+
+/-- The closed subgraph is the closure of the open one: interior graph points are limits from
+below, base points at interior horizontal coordinates are limits from above, and the two
+corners are limits of half-height points over the open base interval. -/
+theorem closure_openSubgraph (hab : a < b) (hf : ContinuousOn f (Set.Icc a b))
+    (ha : f a = 0) (hb : f b = 0) (hpos : ∀ x ∈ Set.Ioo a b, 0 < f x) :
+    closure (openSubgraph a b f) = closedSubgraph a b f := by
+  refine Set.Subset.antisymm ((isClosed_closedSubgraph hf).closure_subset_iff.mpr ?_) ?_
+  · rintro q ⟨h1, h2, h3, h4⟩
+    exact ⟨h1.le, h2.le, h3.le, h4.le⟩
+  · have hF : ContinuousOn (fun c ↦ (!₂[c, f c / 2] : Point)) (Set.Icc a b) :=
+      (PiLp.continuous_toLp (2 : ENNReal) fun _ : Fin 2 ↦ ℝ).comp_continuousOn
+        (continuous_id.continuousOn.matrixVecCons
+          ((hf.div_const 2).matrixVecCons continuousOn_const))
+    have hFsub : (fun c ↦ (!₂[c, f c / 2] : Point)) '' Set.Ioo a b ⊆ openSubgraph a b f := by
+      rintro _ ⟨c, hc, rfl⟩
+      have hfc := hpos c hc
+      exact ⟨by simpa using hc.1, by simpa using hc.2, by simpa using by linarith,
+        by simpa using by linarith⟩
+    have hcl : closure (Set.Ioo a b) = Set.Icc a b := closure_Ioo hab.ne
+    have hF' : ContinuousOn (fun c ↦ (!₂[c, f c / 2] : Point)) (closure (Set.Ioo a b)) := by
+      rw [hcl]; exact hF
+    have hcorner : ∀ c ∈ Set.Icc a b, (!₂[c, f c / 2] : Point) ∈ closure (openSubgraph a b f) :=
+      fun c hc ↦ closure_mono hFsub (hF'.image_closure ⟨c, by rw [hcl]; exact hc, rfl⟩)
+    rintro q ⟨h1, h2, h3, h4⟩
+    rcases eq_or_lt_of_le h1 with hqa | hqa
+    · have : q 1 = 0 := le_antisymm (by rw [← hqa, ha] at h4; exact h4) h3
+      have hq : q = !₂[a, f a / 2] := by
+        rw [Point.eq_vecNotation q, ← hqa, this, ha]
+        norm_num
+      rw [hq]
+      exact hcorner a ⟨le_rfl, hab.le⟩
+    rcases eq_or_lt_of_le h2 with hqb | hqb
+    · have : q 1 = 0 := le_antisymm (by rw [hqb, hb] at h4; exact h4) h3
+      have hq : q = !₂[b, f b / 2] := by
+        rw [Point.eq_vecNotation q, hqb, this, hb]
+        norm_num
+      rw [hq]
+      exact hcorner b ⟨hab.le, le_rfl⟩
+    · have hfc : 0 < f (q 0) := hpos _ ⟨hqa, hqb⟩
+      have hsub : (fun y : ℝ ↦ (!₂[q 0, y] : Point)) '' Set.Ioo 0 (f (q 0)) ⊆
+          openSubgraph a b f := by
+        rintro _ ⟨y, hy, rfl⟩
+        exact ⟨by simpa using hqa, by simpa using hqb, by simpa using hy.1,
+          by simpa using hy.2⟩
+      refine closure_mono hsub ?_
+      have := image_closure_subset_closure_image (continuous_verticalLine (q 0))
+        (s := Set.Ioo 0 (f (q 0)))
+        ⟨q 1, by rw [closure_Ioo hfc.ne]; exact ⟨h3, h4⟩, rfl⟩
+      have heq : (!₂[q 0, q 1] : Point) = q := (Point.eq_vecNotation q).symm
+      simpa only [heq] using this
+
+/-- A point of the closed subgraph which is not in the open one lies on the base or on the
+graph, hence is a limit of points outside the closed subgraph. -/
+private theorem mem_closure_compl_closedSubgraph (ha : f a = 0) (hb : f b = 0)
+    {q : Point} (hq : q ∈ closedSubgraph a b f)
+    (hq' : q ∉ openSubgraph a b f) : q ∈ closure (closedSubgraph a b f)ᶜ := by
+  obtain ⟨h1, h2, h3, h4⟩ := hq
+  have hbelow : q 1 = 0 → q ∈ closure (closedSubgraph a b f)ᶜ := by
+    intro hz
+    have hsub : (fun y : ℝ ↦ (!₂[q 0, y] : Point)) '' Set.Iio 0 ⊆
+        (closedSubgraph a b f)ᶜ := by
+      rintro _ ⟨y, hy, rfl⟩
+      intro hmem
+      have h := hmem.2.2.1
+      simp only [Matrix.cons_val_one, Matrix.cons_val_fin_one] at h
+      exact absurd h (not_le.mpr hy)
+    refine closure_mono hsub ?_
+    have := image_closure_subset_closure_image (continuous_verticalLine (q 0)) (s := Set.Iio 0)
+      ⟨q 1, by rw [closure_Iio]; exact hz.le, rfl⟩
+    have heq : (!₂[q 0, q 1] : Point) = q := (Point.eq_vecNotation q).symm
+    simpa only [heq] using this
+  rcases eq_or_lt_of_le h3 with hz | hz
+  · exact hbelow hz.symm
+  rcases eq_or_lt_of_le h4 with htop | htop
+  · -- the point sits on the graph, which is strictly above the base here
+    have hfpos : 0 < f (q 0) := htop ▸ hz
+    have hsub : (fun y : ℝ ↦ (!₂[q 0, y] : Point)) '' Set.Ioi (f (q 0)) ⊆
+        (closedSubgraph a b f)ᶜ := by
+      rintro _ ⟨y, hy, rfl⟩
+      intro hmem
+      have h := hmem.2.2.2
+      simp only [Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one] at h
+      exact absurd h (not_le.mpr hy)
+    refine closure_mono hsub ?_
+    have := image_closure_subset_closure_image (continuous_verticalLine (q 0))
+      (s := Set.Ioi (f (q 0))) ⟨q 1, by rw [closure_Ioi]; exact htop.ge, rfl⟩
+    have heq : (!₂[q 0, q 1] : Point) = q := (Point.eq_vecNotation q).symm
+    simpa only [heq] using this
+  · -- strictly between base and graph: the horizontal coordinate must be interior
+    exfalso
+    have hfpos : 0 < f (q 0) := hz.trans htop
+    have hqa : a < q 0 := by
+      rcases eq_or_lt_of_le h1 with h | h
+      · rw [← h, ha] at hfpos; exact absurd hfpos (lt_irrefl 0)
+      · exact h
+    have hqb : q 0 < b := by
+      rcases eq_or_lt_of_le h2 with h | h
+      · rw [h, hb] at hfpos; exact absurd hfpos (lt_irrefl 0)
+      · exact h
+    exact hq' ⟨hqa, hqb, hz, htop⟩
+
+/-- The open subgraph is the interior of the closed one. -/
+theorem interior_closedSubgraph (hf : ContinuousOn f (Set.Icc a b))
+    (ha : f a = 0) (hb : f b = 0) :
+    interior (closedSubgraph a b f) = openSubgraph a b f := by
+  refine Set.Subset.antisymm ?_ (interior_maximal openSubgraph_subset_closedSubgraph
+    (isOpen_openSubgraph hf))
+  intro q hq
+  by_contra hqU
+  have h2 := mem_closure_compl_closedSubgraph ha hb (interior_subset hq) hqU
+  rw [closure_compl] at h2
+  exact h2 hq
+
+/-- The closed subgraph is also the closure of the strict subgraph. -/
+theorem closure_strictSubgraph (hab : a < b) (hf : ContinuousOn f (Set.Icc a b))
+    (ha : f a = 0) (hb : f b = 0) (hpos : ∀ x ∈ Set.Ioo a b, 0 < f x) :
+    closure (strictSubgraph a b f) = closedSubgraph a b f := by
+  refine Set.Subset.antisymm
+    ((isClosed_closedSubgraph hf).closure_subset_iff.mpr
+      strictSubgraph_subset_closedSubgraph) ?_
+  rw [← closure_openSubgraph hab hf ha hb hpos]
+  exact closure_mono openSubgraph_subset_strictSubgraph
+
+end Subgraph
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Support
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A compact convex body attains its support value in every normal direction. -/
+theorem exists_mem_inner_eq_supportValue (K : ConvexBody Point) (t : Real.Angle) :
+    ∃ p ∈ (K : Set Point), inner ℝ p (normalVector t) = supportValue K t := by
+  obtain ⟨p, hp, hmax, _⟩ := K.isCompact.exists_sSup_image_eq_and_ge
+    (f := fun p : Point ↦ inner ℝ p (normalVector t)) K.nonempty
+    (continuous_inner.comp (continuous_id.prodMk continuous_const)).continuousOn
+  exact ⟨p, hp, by simpa only [supportValue] using hmax.symm⟩
+
+/-- Translating a nonempty compact set adds the normal component to its support value. -/
+theorem supportValue_image_add_of_isCompact {s : Set Point}
+    (hs : IsCompact s) (hne : s.Nonempty) (v : Point) (t : Real.Angle) :
+    supportValue ((fun p ↦ p + v) '' s) t =
+      supportValue s t + inner ℝ v (normalVector t) := by
+  obtain ⟨x, hx, hmax, hbound⟩ := hs.exists_sSup_image_eq_and_ge
+    (f := fun p : Point ↦ inner ℝ p (normalVector t)) hne
+    (continuous_inner.comp (continuous_id.prodMk continuous_const)).continuousOn
+  apply IsGreatest.csSup_eq
+  constructor
+  · refine ⟨x + v, ⟨x, hx, rfl⟩, ?_⟩
+    simpa only [inner_add_left, supportValue] using congrArg
+      (fun r ↦ r + inner ℝ v (normalVector t)) hmax.symm
+  · rintro z ⟨y, ⟨p, hp, rfl⟩, rfl⟩
+    simp only [inner_add_left]
+    simpa only [supportValue, hmax] using
+      add_le_add (hbound p hp) (le_refl (inner ℝ v (normalVector t)))
+
+/-- Translating a compact convex body adds the normal component of the translation to support. -/
+theorem supportValue_image_add (K : ConvexBody Point) (v : Point) (t : Real.Angle) :
+    supportValue ((fun p ↦ p + v) '' (K : Set Point)) t =
+      supportValue K t + inner ℝ v (normalVector t) :=
+  supportValue_image_add_of_isCompact K.isCompact K.nonempty v t
+
+/-- Every point of a compact set lies below its support value. -/
+theorem inner_le_supportValue_of_isCompact {s : Set Point}
+    (hs : IsCompact s) {p : Point} (hp : p ∈ s) (a : Real.Angle) :
+    inner ℝ p (normalVector a) ≤ supportValue s a := by
+  apply le_csSup (hs.image
+    (continuous_inner.comp (continuous_id.prodMk continuous_const))).bddAbove
+  exact ⟨p, hp, rfl⟩
+
+/-- A point of a convex body lies below each supporting line. -/
+theorem inner_le_supportValue (K : ConvexBody Point) {p : Point}
+    (hp : p ∈ K) (t : Real.Angle) : inner ℝ p (normalVector t) ≤ supportValue K t :=
+  inner_le_supportValue_of_isCompact K.isCompact hp t
+
+/-- Containment in a closed normal half-plane bounds the support value. -/
+theorem supportValue_le_of_subset_normalHalfPlane (K : ConvexBody Point)
+    (t : Real.Angle) (c : ℝ)
+    (hK : (K : Set Point) ⊆ normalHalfPlane t c false false) :
+    supportValue K t ≤ c := by
+  apply csSup_le (K.nonempty.image _)
+  rintro _ ⟨p, hp, rfl⟩
+  exact hK hp
+
+/-- Enlarging a nonempty set without increasing its directional upper bound preserves support. -/
+theorem supportValue_eq_of_subset_of_inner_le {s u : Set Point}
+    (hs : s.Nonempty) (hsu : s ⊆ u) (a : Real.Angle)
+    (hu : ∀ p ∈ u, inner ℝ p (normalVector a) ≤ supportValue s a) :
+    supportValue u a = supportValue s a := by
+  have hbounded : BddAbove ((fun p ↦ inner ℝ p (normalVector a)) '' u) :=
+    ⟨supportValue s a, by rintro _ ⟨p, hp, rfl⟩; exact hu p hp⟩
+  apply le_antisymm
+  · exact csSup_le ((hs.mono hsu).image _) (by rintro _ ⟨p, hp, rfl⟩; exact hu p hp)
+  · exact csSup_le_csSup hbounded (hs.image _) (Set.image_mono hsu)
+
+/-- A convex set meets every intermediate level of a linear functional. -/
+theorem exists_mem_inner_eq_of_convex {s : Set Point} (hconv : Convex ℝ s)
+    {A C : Point} (hA : A ∈ s) (hC : C ∈ s) {u : Point} {c : ℝ}
+    (hCle : inner ℝ C u ≤ c) (hAge : c ≤ inner ℝ A u) :
+    ∃ q ∈ s, inner ℝ q u = c := by
+  rcases eq_or_lt_of_le (hCle.trans hAge) with heq | hlt
+  · exact ⟨C, hC, le_antisymm hCle (heq ▸ hAge)⟩
+  · set d : ℝ := inner ℝ A u - inner ℝ C u with hd
+    have hdpos : 0 < d := by simp only [hd]; linarith
+    set lam : ℝ := (c - inner ℝ C u) / d with hlam
+    have hlam0 : 0 ≤ lam := div_nonneg (by linarith) hdpos.le
+    have hlam1 : lam ≤ 1 := by
+      rw [hlam, div_le_one hdpos]
+      simp only [hd]; linarith
+    refine ⟨lam • A + (1 - lam) • C, hconv hA hC hlam0 (by linarith) (by ring), ?_⟩
+    rw [inner_add_left, real_inner_smul_left, real_inner_smul_left, hlam]
+    field_simp
+    simp only [hd]
+    ring
+
+/-- Support bound in the direction opposite to a cut line lying below the set. -/
+theorem supportValue_le_of_cut {s : Set Point} (hne : s.Nonempty) {a b c : ℝ}
+    (hab : ((b : ℝ) : Real.Angle) = ((a + Real.pi : ℝ) : Real.Angle))
+    (hs : ∀ p ∈ s, c ≤ inner ℝ p (normalVector (a : Real.Angle))) :
+    supportValue s ((b : ℝ) : Real.Angle) ≤ -c := by
+  rw [hab]
+  refine csSup_le (hne.image _) ?_
+  rintro _ ⟨p, hp, rfl⟩
+  dsimp only
+  rw [normalVector_add_pi, inner_neg_right]
+  linarith only [hs p hp]
+
+/-- A contact point on a cut line attains the opposite support value. -/
+theorem le_supportValue_of_cut {s : Set Point} (hcomp : IsCompact s) {a b c : ℝ} {p : Point}
+    (hab : ((b : ℝ) : Real.Angle) = ((a + Real.pi : ℝ) : Real.Angle)) (hp : p ∈ s)
+    (hpc : inner ℝ p (normalVector (a : Real.Angle)) = c) :
+    -c ≤ supportValue s ((b : ℝ) : Real.Angle) := by
+  rw [hab]
+  have h := inner_le_supportValue_of_isCompact hcomp hp ((a + Real.pi : ℝ) : Real.Angle)
+  rw [normalVector_add_pi, inner_neg_right, hpc] at h
+  linarith only [h]
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Contacts
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The exposed edge at a normal direction, including singleton edges. -/
+def exposedEdge (K : ConvexBody Point) (t : Real.Angle) : Set Point :=
+  (K : Set Point) ∩ (supportingLineHalfPlane K t).1
+
+/-- The positive and negative tangent endpoints of an exposed edge. -/
+def edgeVertices (K : ConvexBody Point) (t : Real.Angle) : Point × Point :=
+  let heights := (fun p ↦ inner ℝ p (tangentVector t)) '' exposedEdge K t
+  (supportValue K t • normalVector t + sSup heights • tangentVector t,
+    supportValue K t • normalVector t + sInf heights • tangentVector t)
+
+/-- The intersection point of two supporting lines at nonparallel normal directions. -/
+def supportingIntersection (K : ConvexBody Point) (a b : Real.Angle) : Point :=
+  supportValue K a • normalVector a +
+    ((supportValue K b - supportValue K a * (b - a).cos) / (b - a).sin) • tangentVector a
+
+theorem exposedEdge_nonempty (K : ConvexBody Point) (t : Real.Angle) :
+    (exposedEdge K t).Nonempty := by
+  obtain ⟨x, hx, hxmax, _⟩ := K.isCompact.exists_sSup_image_eq_and_ge
+    (f := fun x : Point ↦ inner ℝ x (normalVector t)) K.nonempty
+    (continuous_inner.comp (continuous_id.prodMk continuous_const)).continuousOn
+  refine ⟨x, hx, ?_⟩
+  simpa [supportingLineHalfPlane, normalLine, supportValue] using hxmax.symm
+
+theorem convex_exposedEdge (K : ConvexBody Point) (t : Real.Angle) :
+    Convex ℝ (exposedEdge K t) := by
+  apply K.convex.inter
+  intro x hx y hy a b ha hb hab
+  change inner ℝ x (normalVector t) = supportValue K t at hx
+  change inner ℝ y (normalVector t) = supportValue K t at hy
+  change inner ℝ (a • x + b • y) (normalVector t) = supportValue K t
+  rw [inner_add_left, inner_smul_left, inner_smul_left, hx, hy]
+  simp only [RCLike.conj_to_real]
+  rw [← add_mul, hab, one_mul]
+
+theorem isConnected_exposedEdge (K : ConvexBody Point) (t : Real.Angle) :
+    IsConnected (exposedEdge K t) :=
+  (convex_exposedEdge K t).isConnected (exposedEdge_nonempty K t)
+
+theorem supportingIntersection_inner_left (K : ConvexBody Point) (s t : ℝ) :
+    inner ℝ (supportingIntersection K (s : Real.Angle) (t : Real.Angle))
+      (normalVector (s : Real.Angle)) = supportValue K (s : Real.Angle) := by
+  have htn : inner ℝ (tangentVector (s : Real.Angle)) (normalVector (s : Real.Angle)) = 0 := by
+    rw [real_inner_comm, inner_normalVector_tangentVector]
+  simp only [supportingIntersection, inner_add_left, real_inner_smul_left,
+    inner_normalVector_self, htn, mul_one, mul_zero, add_zero]
+
+theorem supportingIntersection_inner_right (K : ConvexBody Point) (s t : ℝ)
+    (h : Real.sin (t - s) ≠ 0) :
+    inner ℝ (supportingIntersection K (s : Real.Angle) (t : Real.Angle))
+      (normalVector (t : Real.Angle)) = supportValue K (t : Real.Angle) := by
+  have hn : normalVector (t : Real.Angle) =
+      Real.cos (t - s) • normalVector (s : Real.Angle) +
+      Real.sin (t - s) • tangentVector (s : Real.Angle) := by
+    simpa only [add_sub_cancel] using normalVector_add_real s (t - s)
+  rw [hn, inner_add_right, inner_smul_right, inner_smul_right,
+    supportingIntersection_inner_left]
+  simp only [supportingIntersection, inner_add_left, real_inner_smul_left,
+    inner_normalVector_tangentVector, inner_tangentVector_self, mul_zero, mul_one, zero_add,
+    ← Real.Angle.coe_sub, Real.Angle.cos_coe, Real.Angle.sin_coe]
+  field_simp
+  ring
+
+theorem supportingIntersection_comm (K : ConvexBody Point) (s t : ℝ)
+    (h : Real.sin (t - s) ≠ 0) :
+    supportingIntersection K (s : Real.Angle) (t : Real.Angle) =
+      supportingIntersection K (t : Real.Angle) (s : Real.Angle) := by
+  have h' : Real.sin (s - t) ≠ 0 := by
+    rw [show s - t = -(t - s) by ring, Real.sin_neg]
+    exact neg_ne_zero.mpr h
+  let p := supportingIntersection K (t : Real.Angle) (s : Real.Angle)
+  have hp := supportingIntersection_inner_right K t s h'
+  have hq := supportingIntersection_inner_left K t s
+  change inner ℝ p (normalVector (s : Real.Angle)) = _ at hp
+  change inner ℝ p (normalVector (t : Real.Angle)) = _ at hq
+  have hn : normalVector (t : Real.Angle) =
+      Real.cos (t - s) • normalVector (s : Real.Angle) +
+      Real.sin (t - s) • tangentVector (s : Real.Angle) := by
+    simpa only [add_sub_cancel] using normalVector_add_real s (t - s)
+  rw [hn, inner_add_right, inner_smul_right, inner_smul_right, hp] at hq
+  have ht : inner ℝ p (tangentVector (s : Real.Angle)) =
+      (supportValue K (t : Real.Angle) - supportValue K (s : Real.Angle) *
+        Real.cos (t - s)) / Real.sin (t - s) := by
+    apply (eq_div_iff h).mpr
+    linarith
+  have heq := inner_normalVector_smul_add_inner_tangentVector_smul p (s : Real.Angle)
+  rw [hp, ht] at heq
+  simpa only [p, supportingIntersection, ← Real.Angle.coe_sub, Real.Angle.cos_coe,
+    Real.Angle.sin_coe] using heq
+
+/-- Every exposed edge of a compact convex body is compact. -/
+theorem isCompact_exposedEdge (K : ConvexBody Point) (t : Real.Angle) :
+    IsCompact (exposedEdge K t) :=
+  K.isCompact.inter_right (isClosed_eq
+    (continuous_id.inner continuous_const) continuous_const)
+
+/-- The positive tangent endpoint belongs to its exposed edge. -/
+theorem edgeVertices_fst_mem (K : ConvexBody Point) (t : Real.Angle) :
+    (edgeVertices K t).1 ∈ exposedEdge K t := by
+  obtain ⟨p, hp, hmax⟩ := (isCompact_exposedEdge K t).exists_sSup_image_eq
+    (exposedEdge_nonempty K t)
+    (f := fun p : Point ↦ inner ℝ p (tangentVector t))
+    (continuous_id.inner continuous_const).continuousOn
+  have hpnormal : inner ℝ p (normalVector t) = supportValue K t := hp.2
+  have heq : (edgeVertices K t).1 = p := by
+    change supportValue K t • normalVector t +
+      sSup ((fun q ↦ inner ℝ q (tangentVector t)) '' exposedEdge K t) • tangentVector t = p
+    rw [hmax, ← hpnormal]
+    exact inner_normalVector_smul_add_inner_tangentVector_smul p t
+  exact heq ▸ hp
+
+/-- The negative tangent endpoint belongs to its exposed edge. -/
+theorem edgeVertices_snd_mem (K : ConvexBody Point) (t : Real.Angle) :
+    (edgeVertices K t).2 ∈ exposedEdge K t := by
+  obtain ⟨p, hp, hmin⟩ := (isCompact_exposedEdge K t).exists_sInf_image_eq
+    (exposedEdge_nonempty K t)
+    (f := fun p : Point ↦ inner ℝ p (tangentVector t))
+    (continuous_id.inner continuous_const).continuousOn
+  have hpnormal : inner ℝ p (normalVector t) = supportValue K t := hp.2
+  have heq : (edgeVertices K t).2 = p := by
+    change supportValue K t • normalVector t +
+      sInf ((fun q ↦ inner ℝ q (tangentVector t)) '' exposedEdge K t) • tangentVector t = p
+    rw [hmin, ← hpnormal]
+    exact inner_normalVector_smul_add_inner_tangentVector_smul p t
+  exact heq ▸ hp
+
+/-- A singleton exposed face is both of its tangent endpoints. -/
+theorem edgeVertices_eq_of_exposedEdge_singleton {K : ConvexBody Point} {t : Real.Angle}
+    {p : Point} (h : exposedEdge K t = {p}) : edgeVertices K t = (p, p) := by
+  have h1 := edgeVertices_fst_mem K t
+  have h2 := edgeVertices_snd_mem K t
+  rw [h, Set.mem_singleton_iff] at h1 h2
+  exact Prod.ext h1 h2
+
+/-- The positive face vertex attains the largest tangent coordinate. -/
+theorem inner_edgeVertices_fst_tangent (K : ConvexBody Point) (t : Real.Angle) :
+    inner ℝ (edgeVertices K t).1 (tangentVector t) =
+      sSup ((fun p ↦ inner ℝ p (tangentVector t)) '' exposedEdge K t) := by
+  induction t using Real.Angle.induction_on with
+  | _ t =>
+    simp only [edgeVertices, inner_add_left, real_inner_smul_left,
+      inner_normalVector_tangentVector, inner_tangentVector_self]
+    ring
+
+/-- The negative face vertex attains the smallest tangent coordinate. -/
+theorem inner_edgeVertices_snd_tangent (K : ConvexBody Point) (t : Real.Angle) :
+    inner ℝ (edgeVertices K t).2 (tangentVector t) =
+      sInf ((fun p ↦ inner ℝ p (tangentVector t)) '' exposedEdge K t) := by
+  induction t using Real.Angle.induction_on with
+  | _ t =>
+    simp only [edgeVertices, inner_add_left, real_inner_smul_left,
+      inner_normalVector_tangentVector, inner_tangentVector_self]
+    ring
+
+private theorem exposedEdge_subset_segment_edgeVertices (K : ConvexBody Point)
+    (t : Real.Angle) :
+    exposedEdge K t ⊆ segment ℝ (edgeVertices K t).2 (edgeVertices K t).1 := by
+  intro p hp
+  let lo := (edgeVertices K t).2
+  let hi := (edgeVertices K t).1
+  let z := inner ℝ p (tangentVector t)
+  let zlo := inner ℝ lo (tangentVector t)
+  let zhi := inner ℝ hi (tangentVector t)
+  have hbddBelow : BddBelow ((fun q ↦ inner ℝ q (tangentVector t)) '' exposedEdge K t) :=
+    (isCompact_exposedEdge K t).image
+      (continuous_id.inner continuous_const) |>.bddBelow
+  have hbddAbove : BddAbove ((fun q ↦ inner ℝ q (tangentVector t)) '' exposedEdge K t) :=
+    (isCompact_exposedEdge K t).image
+      (continuous_id.inner continuous_const) |>.bddAbove
+  have hzmem : z ∈ (fun q ↦ inner ℝ q (tangentVector t)) '' exposedEdge K t :=
+    ⟨p, hp, rfl⟩
+  have hzlo : zlo ≤ z := by
+    dsimp only [zlo, z, lo]
+    rw [inner_edgeVertices_snd_tangent]
+    exact csInf_le hbddBelow hzmem
+  have hzhi : z ≤ zhi := by
+    dsimp only [zhi, z, hi]
+    rw [inner_edgeVertices_fst_tangent]
+    exact le_csSup hbddAbove hzmem
+  by_cases hzh : zlo = zhi
+  · have hz : z = zlo := le_antisymm (hzhi.trans_eq hzh.symm) hzlo
+    have hpnormal : inner ℝ p (normalVector t) = supportValue K t := hp.2
+    have hlonormal : inner ℝ lo (normalVector t) = supportValue K t :=
+      (edgeVertices_snd_mem K t).2
+    have hpl : p = lo := by
+      rw [← inner_normalVector_smul_add_inner_tangentVector_smul p t,
+        ← inner_normalVector_smul_add_inner_tangentVector_smul lo t,
+        hpnormal, hlonormal]
+      change inner ℝ p (tangentVector t) = inner ℝ lo (tangentVector t) at hz
+      rw [hz]
+    simpa only [lo, hi, hpl] using left_mem_segment ℝ lo hi
+  · have hlt : zlo < zhi := lt_of_le_of_ne (hzlo.trans hzhi) hzh
+    let u := (z - zlo) / (zhi - zlo)
+    have hu : u ∈ Set.Icc (0 : ℝ) 1 := by
+      constructor
+      · exact div_nonneg (sub_nonneg.mpr hzlo) (sub_nonneg.mpr hlt.le)
+      · rw [div_le_one (sub_pos.mpr hlt)]
+        linarith
+    rw [segment_eq_image]
+    refine ⟨u, hu, ?_⟩
+    change (1 - u) • lo + u • hi = p
+    have hpnormal : inner ℝ p (normalVector t) = supportValue K t := hp.2
+    have hlonormal : inner ℝ lo (normalVector t) = supportValue K t :=
+      (edgeVertices_snd_mem K t).2
+    have hhinormal : inner ℝ hi (normalVector t) = supportValue K t :=
+      (edgeVertices_fst_mem K t).2
+    have hnormal : inner ℝ ((1 - u) • lo + u • hi) (normalVector t) =
+        inner ℝ p (normalVector t) := by
+      simp only [inner_add_left, real_inner_smul_left, hlonormal, hhinormal, hpnormal]
+      ring
+    have htangent : inner ℝ ((1 - u) • lo + u • hi) (tangentVector t) =
+        inner ℝ p (tangentVector t) := by
+      simp only [inner_add_left, real_inner_smul_left]
+      change (1 - u) * zlo + u * zhi = z
+      dsimp [u]
+      field_simp [sub_ne_zero.mpr (ne_of_gt hlt)]
+      ring
+    rw [← inner_normalVector_smul_add_inner_tangentVector_smul
+      ((1 - u) • lo + u • hi) t,
+      ← inner_normalVector_smul_add_inner_tangentVector_smul p t,
+      hnormal, htangent]
+
+/-- An exposed face is the segment joining its two tangent-extreme vertices. -/
+theorem exposedEdge_eq_segment_edgeVertices (K : ConvexBody Point)
+    (t : Real.Angle) :
+    exposedEdge K t = segment ℝ (edgeVertices K t).2 (edgeVertices K t).1 := by
+  apply Set.Subset.antisymm (exposedEdge_subset_segment_edgeVertices K t)
+  exact (convex_exposedEdge K t).segment_subset
+    (edgeVertices_snd_mem K t) (edgeVertices_fst_mem K t)
+
+/-- An exposed face determines the support value and both of its endpoint vertices. -/
+theorem edgeVertices_eq_of_exposedEdge_eq (K L : ConvexBody Point) (u : Real.Angle)
+    (h : exposedEdge K u = exposedEdge L u) : edgeVertices K u = edgeVertices L u := by
+  obtain ⟨p, hp⟩ := exposedEdge_nonempty K u
+  have hpL : p ∈ exposedEdge L u := h ▸ hp
+  have hK : inner ℝ p (normalVector u) = supportValue K u := hp.2
+  have hL : inner ℝ p (normalVector u) = supportValue L u := hpL.2
+  simp only [edgeVertices, hK.symm.trans hL, h]
+
+/-- A support-line intersection in the body is the negative endpoint at the later normal. -/
+theorem supportingIntersection_eq_edgeVertices_snd_of_mem (K : ConvexBody Point) {a b : ℝ}
+    (hab : 0 < b - a) (hpi : b - a < Real.pi)
+    (hp : supportingIntersection K (a : Real.Angle) (b : Real.Angle) ∈ K) :
+    supportingIntersection K (a : Real.Angle) (b : Real.Angle) =
+      (edgeVertices K (b : Real.Angle)).2 := by
+  let p := supportingIntersection K (a : Real.Angle) (b : Real.Angle)
+  let e := (edgeVertices K (b : Real.Angle)).2
+  have hsin : 0 < Real.sin (b - a) := Real.sin_pos_of_pos_of_lt_pi hab hpi
+  have hsina : Real.sin (a - b) < 0 := by
+    rw [show a - b = -(b - a) by ring, Real.sin_neg]
+    exact neg_neg_of_pos hsin
+  have hpa : inner ℝ p (normalVector (a : Real.Angle)) =
+      supportValue K (a : Real.Angle) := supportingIntersection_inner_left K a b
+  have hpb : inner ℝ p (normalVector (b : Real.Angle)) =
+      supportValue K (b : Real.Angle) :=
+    supportingIntersection_inner_right K a b hsin.ne'
+  have hpedge : p ∈ exposedEdge K (b : Real.Angle) := ⟨hp, hpb⟩
+  have heedge := edgeVertices_snd_mem K (b : Real.Angle)
+  have hen : inner ℝ e (normalVector (b : Real.Angle)) =
+      supportValue K (b : Real.Angle) := heedge.2
+  have hpt : inner ℝ p (tangentVector (b : Real.Angle)) =
+      inner ℝ e (tangentVector (b : Real.Angle)) := by
+    rw [inner_edgeVertices_snd_tangent]
+    apply le_antisymm
+    · apply le_csInf ((exposedEdge_nonempty K _).image _)
+      rintro _ ⟨q, hq, rfl⟩
+      have hqa := inner_le_supportValue K hq.1 (a : Real.Angle)
+      have hna : normalVector (a : Real.Angle) =
+          Real.cos (a - b) • normalVector (b : Real.Angle) +
+            Real.sin (a - b) • tangentVector (b : Real.Angle) := by
+        simpa only [add_sub_cancel] using normalVector_add_real b (a - b)
+      rw [← hpa, hna] at hqa
+      simp only [inner_add_right, inner_smul_right] at hqa
+      rw [hpb, hq.2] at hqa
+      nlinarith
+    · exact csInf_le ((isCompact_exposedEdge K _).image
+        (continuous_id.inner continuous_const)).bddBelow ⟨p, hpedge, rfl⟩
+  change p = e
+  rw [← inner_normalVector_smul_add_inner_tangentVector_smul p (b : Real.Angle),
+    ← inner_normalVector_smul_add_inner_tangentVector_smul e (b : Real.Angle), hpb, hen,
+    hpt]
+
+/-- A support-line intersection in the body is the positive endpoint at the earlier normal. -/
+theorem supportingIntersection_eq_edgeVertices_fst_of_mem (K : ConvexBody Point) {a b : ℝ}
+    (hab : 0 < b - a) (hpi : b - a < Real.pi)
+    (hp : supportingIntersection K (a : Real.Angle) (b : Real.Angle) ∈ K) :
+    supportingIntersection K (a : Real.Angle) (b : Real.Angle) =
+      (edgeVertices K (a : Real.Angle)).1 := by
+  let p := supportingIntersection K (a : Real.Angle) (b : Real.Angle)
+  let e := (edgeVertices K (a : Real.Angle)).1
+  have hsin : 0 < Real.sin (b - a) := Real.sin_pos_of_pos_of_lt_pi hab hpi
+  have hpa : inner ℝ p (normalVector (a : Real.Angle)) =
+      supportValue K (a : Real.Angle) := supportingIntersection_inner_left K a b
+  have hpb : inner ℝ p (normalVector (b : Real.Angle)) =
+      supportValue K (b : Real.Angle) :=
+    supportingIntersection_inner_right K a b hsin.ne'
+  have hpedge : p ∈ exposedEdge K (a : Real.Angle) := ⟨hp, hpa⟩
+  have heedge := edgeVertices_fst_mem K (a : Real.Angle)
+  have hen : inner ℝ e (normalVector (a : Real.Angle)) =
+      supportValue K (a : Real.Angle) := heedge.2
+  have hpt : inner ℝ p (tangentVector (a : Real.Angle)) =
+      inner ℝ e (tangentVector (a : Real.Angle)) := by
+    rw [inner_edgeVertices_fst_tangent]
+    apply le_antisymm
+    · exact le_csSup ((isCompact_exposedEdge K _).image
+        (continuous_id.inner continuous_const)).bddAbove ⟨p, hpedge, rfl⟩
+    · apply csSup_le ((exposedEdge_nonempty K _).image _)
+      rintro _ ⟨q, hq, rfl⟩
+      have hqb := inner_le_supportValue K hq.1 (b : Real.Angle)
+      have hnb : normalVector (b : Real.Angle) =
+          Real.cos (b - a) • normalVector (a : Real.Angle) +
+            Real.sin (b - a) • tangentVector (a : Real.Angle) := by
+        simpa only [add_sub_cancel] using normalVector_add_real a (b - a)
+      rw [← hpb, hnb] at hqb
+      simp only [inner_add_right, inner_smul_right] at hqb
+      rw [hpa, hq.2] at hqb
+      nlinarith
+  change p = e
+  rw [← inner_normalVector_smul_add_inner_tangentVector_smul p (a : Real.Angle),
+    ← inner_normalVector_smul_add_inner_tangentVector_smul e (a : Real.Angle), hpa, hen,
+    hpt]
+
+/-- The first endpoint reaches the adjacent supporting-line intersection along a positive
+tangent ray. -/
+theorem supportingIntersection_eq_fst_add_pos_tangent
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hne : (edgeVertices K (a : Real.Angle)).1 ≠
+      (edgeVertices K (b : Real.Angle)).2) :
+    ∃ d : ℝ, 0 < d ∧ supportingIntersection K a b =
+      (edgeVertices K (a : Real.Angle)).1 + d • tangentVector (a : Real.Angle) := by
+  let O := supportingIntersection K (a : Real.Angle) (b : Real.Angle)
+  let P := (edgeVertices K (a : Real.Angle)).1
+  let d := inner ℝ (O - P) (tangentVector (a : Real.Angle))
+  have hsin : 0 < Real.sin (b - a) :=
+    Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hab) (by linarith)
+  have hOn : inner ℝ O (normalVector (a : Real.Angle)) = supportValue K a :=
+    supportingIntersection_inner_left K a b
+  have hPn : inner ℝ P (normalVector (a : Real.Angle)) = supportValue K a :=
+    (edgeVertices_fst_mem K (a : Real.Angle)).2
+  have hnormal : inner ℝ (O - P) (normalVector (a : Real.Angle)) = 0 := by
+    rw [inner_sub_left, hOn, hPn, sub_self]
+  have hdir : O = P + d • tangentVector (a : Real.Angle) := by
+    rw [show O = P + (O - P) by abel, ← inner_normalVector_smul_add_inner_tangentVector_smul
+      (O - P) (a : Real.Angle), hnormal, zero_smul, zero_add]
+  have hdnonneg : 0 ≤ d := by
+    have hOb : inner ℝ O (normalVector (b : Real.Angle)) = supportValue K b :=
+      supportingIntersection_inner_right K a b hsin.ne'
+    have hPb := inner_le_supportValue K (edgeVertices_fst_mem K (a : Real.Angle)).1
+      (b : Real.Angle)
+    change inner ℝ P (normalVector (b : Real.Angle)) ≤ supportValue K b at hPb
+    have hnb : normalVector (b : Real.Angle) =
+        Real.cos (b - a) • normalVector (a : Real.Angle) +
+          Real.sin (b - a) • tangentVector (a : Real.Angle) := by
+      simpa only [add_sub_cancel] using normalVector_add_real a (b - a)
+    rw [← hOb, hdir, hnb, inner_add_left, inner_add_right, inner_smul_right,
+      inner_smul_right, real_inner_smul_left, hPn] at hPb
+    have htn : inner ℝ (tangentVector (a : Real.Angle))
+        (normalVector (a : Real.Angle)) = 0 := by
+      rw [real_inner_comm, inner_normalVector_tangentVector]
+    simp only [inner_add_right, inner_smul_right, htn,
+      inner_tangentVector_self, mul_zero, mul_one, zero_add] at hPb
+    nlinarith
+  have hdne : d ≠ 0 := by
+    intro hd
+    have hOP : O = P := by simpa [hd] using hdir
+    have hOmem : O ∈ K := hOP ▸ (edgeVertices_fst_mem K _).1
+    have hOQ := supportingIntersection_eq_edgeVertices_snd_of_mem K
+      (sub_pos.mpr hab) (by linarith) hOmem
+    exact hne (hOP.symm.trans hOQ)
+  exact ⟨d, lt_of_le_of_ne hdnonneg (Ne.symm hdne), hdir⟩
+
+/-- The second endpoint reaches the adjacent supporting-line intersection against a positive
+tangent ray. -/
+theorem supportingIntersection_eq_snd_sub_pos_tangent
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hne : (edgeVertices K (a : Real.Angle)).1 ≠
+      (edgeVertices K (b : Real.Angle)).2) :
+    ∃ d : ℝ, 0 < d ∧ supportingIntersection K a b =
+      (edgeVertices K (b : Real.Angle)).2 - d • tangentVector (b : Real.Angle) := by
+  let O := supportingIntersection K (a : Real.Angle) (b : Real.Angle)
+  let Q := (edgeVertices K (b : Real.Angle)).2
+  let d := inner ℝ (Q - O) (tangentVector (b : Real.Angle))
+  have hsin : 0 < Real.sin (b - a) :=
+    Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hab) (by linarith)
+  have hOn : inner ℝ O (normalVector (b : Real.Angle)) = supportValue K b :=
+    supportingIntersection_inner_right K a b hsin.ne'
+  have hQn : inner ℝ Q (normalVector (b : Real.Angle)) = supportValue K b :=
+    (edgeVertices_snd_mem K (b : Real.Angle)).2
+  have hnormal : inner ℝ (Q - O) (normalVector (b : Real.Angle)) = 0 := by
+    rw [inner_sub_left, hQn, hOn, sub_self]
+  have hdir : O = Q - d • tangentVector (b : Real.Angle) := by
+    rw [show O = Q - (Q - O) by abel, ← inner_normalVector_smul_add_inner_tangentVector_smul
+      (Q - O) (b : Real.Angle), hnormal, zero_smul, zero_add]
+  have hdnonneg : 0 ≤ d := by
+    have hOa : inner ℝ O (normalVector (a : Real.Angle)) = supportValue K a :=
+      supportingIntersection_inner_left K a b
+    have hQa := inner_le_supportValue K (edgeVertices_snd_mem K (b : Real.Angle)).1
+      (a : Real.Angle)
+    change inner ℝ Q (normalVector (a : Real.Angle)) ≤ supportValue K a at hQa
+    have hna : normalVector (a : Real.Angle) =
+        Real.cos (a - b) • normalVector (b : Real.Angle) +
+          Real.sin (a - b) • tangentVector (b : Real.Angle) := by
+      simpa only [add_sub_cancel] using normalVector_add_real b (a - b)
+    rw [← hOa, hdir, hna, inner_sub_left, inner_add_right, inner_smul_right,
+      inner_smul_right, real_inner_smul_left, hQn] at hQa
+    have htn : inner ℝ (tangentVector (b : Real.Angle))
+        (normalVector (b : Real.Angle)) = 0 := by
+      rw [real_inner_comm, inner_normalVector_tangentVector]
+    simp only [inner_add_right, inner_smul_right, htn,
+      inner_tangentVector_self, mul_zero, mul_one, zero_add] at hQa
+    have hsneg : Real.sin (a - b) < 0 := by
+      rw [show a - b = -(b - a) by ring, Real.sin_neg]
+      exact neg_neg_of_pos hsin
+    nlinarith
+  have hdne : d ≠ 0 := by
+    intro hd
+    have hOQ : O = Q := by simpa [hd] using hdir
+    have hOmem : O ∈ K := hOQ ▸ (edgeVertices_snd_mem K _).1
+    have hOP := supportingIntersection_eq_edgeVertices_fst_of_mem K
+      (sub_pos.mpr hab) (by linarith) hOmem
+    exact hne (hOP.symm.trans hOQ)
+  exact ⟨d, lt_of_le_of_ne hdnonneg (Ne.symm hdne), hdir⟩
+
+/-! ### Faces at a reversed normal direction -/
+
+/-- A body bounded below in a normal direction, with the bound attained, has the opposite
+support value in the reversed direction. -/
+theorem supportValue_add_pi_eq_neg_of_forall_le {L : ConvexBody Point} {c t : ℝ} {p : Point}
+    (hle : ∀ q ∈ (L : Set Point), c ≤ inner ℝ q (normalVector (t : Real.Angle)))
+    (hp : p ∈ (L : Set Point)) (hpc : inner ℝ p (normalVector (t : Real.Angle)) = c) :
+    supportValue L ((t + Real.pi : ℝ) : Real.Angle) = -c :=
+  le_antisymm (supportValue_le_of_cut L.nonempty rfl hle)
+    (le_supportValue_of_cut L.isCompact rfl hp hpc)
+
+/-- The face at a reversed normal direction consists of the points attaining the attained
+lower bound. -/
+theorem exposedEdge_add_pi_eq_of_forall_le {L : ConvexBody Point} {c t : ℝ} {p : Point}
+    (hle : ∀ q ∈ (L : Set Point), c ≤ inner ℝ q (normalVector (t : Real.Angle)))
+    (hp : p ∈ (L : Set Point)) (hpc : inner ℝ p (normalVector (t : Real.Angle)) = c) :
+    exposedEdge L ((t + Real.pi : ℝ) : Real.Angle) =
+      {q | q ∈ (L : Set Point) ∧ inner ℝ q (normalVector (t : Real.Angle)) = c} := by
+  have hsup := supportValue_add_pi_eq_neg_of_forall_le hle hp hpc
+  ext q
+  constructor
+  · intro hq
+    have h2 : inner ℝ q (normalVector ((t + Real.pi : ℝ) : Real.Angle)) =
+      supportValue L ((t + Real.pi : ℝ) : Real.Angle) := hq.2
+    rw [normalVector_add_pi, inner_neg_right, hsup] at h2
+    exact ⟨hq.1, by linarith only [h2]⟩
+  · rintro ⟨hqL, hqc⟩
+    refine ⟨hqL, ?_⟩
+    change inner ℝ q (normalVector ((t + Real.pi : ℝ) : Real.Angle)) = _
+    rw [normalVector_add_pi, inner_neg_right, hsup, hqc]
+
+/-! ### Faces between two supporting normals with a common contact point -/
+
+/-- A point on two transverse supporting lines is their intersection. -/
+theorem eq_supportingIntersection_of_mem_exposedEdge {L : ConvexBody Point} {a b : ℝ}
+    {p : Point} (hsin : Real.sin (a - b) ≠ 0) (hpa : p ∈ exposedEdge L (a : Real.Angle))
+    (hpb : p ∈ exposedEdge L (b : Real.Angle)) :
+    p = supportingIntersection L (a : Real.Angle) (b : Real.Angle) := by
+  have hsin' : Real.sin (b - a) ≠ 0 := by
+    rw [show b - a = -(a - b) by ring, Real.sin_neg]
+    exact neg_ne_zero.mpr hsin
+  refine eq_of_inner_normalVector_eq hsin ?_ ?_
+  · rw [supportingIntersection_inner_left]
+    exact hpa.2
+  · rw [supportingIntersection_inner_right L a b hsin']
+    exact hpb.2
+
+/-- Between two supporting normals less than a straight angle apart with a common contact
+point, every intervening face is that point. -/
+theorem exposedEdge_eq_singleton_of_mem_exposedEdge_of_mem_Ioo {L : ConvexBody Point}
+    {a b s : ℝ} {p : Point} (hba : b < a + Real.pi) (hs : s ∈ Set.Ioo a b)
+    (hpa : p ∈ exposedEdge L (a : Real.Angle)) (hpb : p ∈ exposedEdge L (b : Real.Angle)) :
+    exposedEdge L (s : Real.Angle) = {p} := by
+  have hab : a < b := hs.1.trans hs.2
+  have hsba : 0 < Real.sin (b - a) :=
+    Real.sin_pos_of_pos_of_lt_pi (by linarith only [hab]) (by linarith only [hba])
+  have hsbs : 0 < Real.sin (b - s) :=
+    Real.sin_pos_of_pos_of_lt_pi (by linarith only [hs.2]) (by linarith only [hba, hs.1])
+  have hssa : 0 < Real.sin (s - a) :=
+    Real.sin_pos_of_pos_of_lt_pi (by linarith only [hs.1]) (by linarith only [hba, hs.2])
+  -- the positive combination of the two endpoint normals
+  have hcomb : ∀ q : Point, Real.sin (b - s) * inner ℝ q (normalVector (a : Real.Angle)) +
+      Real.sin (s - a) * inner ℝ q (normalVector (b : Real.Angle)) =
+      Real.sin (b - a) * inner ℝ q (normalVector (s : Real.Angle)) := by
+    intro q
+    rw [inner_normalVector_real, inner_normalVector_real, inner_normalVector_real,
+      Real.sin_sub, Real.sin_sub, Real.sin_sub]
+    ring
+  have hqa : ∀ q ∈ (L : Set Point), inner ℝ q (normalVector (a : Real.Angle)) ≤
+      inner ℝ p (normalVector (a : Real.Angle)) := by
+    intro q hq
+    rw [hpa.2]
+    exact inner_le_supportValue L hq _
+  have hqb : ∀ q ∈ (L : Set Point), inner ℝ q (normalVector (b : Real.Angle)) ≤
+      inner ℝ p (normalVector (b : Real.Angle)) := by
+    intro q hq
+    rw [hpb.2]
+    exact inner_le_supportValue L hq _
+  have hps : ∀ q ∈ (L : Set Point), inner ℝ q (normalVector (s : Real.Angle)) ≤
+      inner ℝ p (normalVector (s : Real.Angle)) := by
+    intro q hq
+    have h1 := mul_le_mul_of_nonneg_left (hqa q hq) hsbs.le
+    have h2 := mul_le_mul_of_nonneg_left (hqb q hq) hssa.le
+    have h := add_le_add h1 h2
+    rw [hcomb q, hcomb p] at h
+    exact le_of_mul_le_mul_left (by linarith only [h]) hsba
+  have hpsval : inner ℝ p (normalVector (s : Real.Angle)) = supportValue L (s : Real.Angle) :=
+    le_antisymm (inner_le_supportValue L hpa.1 _) (csSup_le (L.nonempty.image _)
+      (by rintro _ ⟨q, hq, rfl⟩; exact hps q hq))
+  ext q
+  simp only [Set.mem_singleton_iff]
+  refine ⟨fun hq ↦ ?_, fun hq ↦ hq ▸ ⟨hpa.1, hpsval⟩⟩
+  have hqs : inner ℝ q (normalVector (s : Real.Angle)) =
+      inner ℝ p (normalVector (s : Real.Angle)) := by rw [hq.2, hpsval]
+  have hsum : Real.sin (b - s) *
+        (inner ℝ p (normalVector (a : Real.Angle)) - inner ℝ q (normalVector (a : Real.Angle))) +
+      Real.sin (s - a) *
+        (inner ℝ p (normalVector (b : Real.Angle)) - inner ℝ q (normalVector (b : Real.Angle)))
+      = 0 := by
+    have h1 := hcomb p
+    have h2 := hcomb q
+    rw [hqs] at h2
+    linear_combination h1 - h2
+  have hza : inner ℝ q (normalVector (a : Real.Angle)) =
+      inner ℝ p (normalVector (a : Real.Angle)) := by
+    nlinarith only [hsum, hsbs, hssa, hqa q hq.1, hqb q hq.1]
+  have hzb : inner ℝ q (normalVector (b : Real.Angle)) =
+      inner ℝ p (normalVector (b : Real.Angle)) := by
+    nlinarith only [hsum, hsbs, hssa, hqa q hq.1, hqb q hq.1]
+  have hsinab : Real.sin (a - b) ≠ 0 := by
+    rw [show a - b = -(b - a) by ring, Real.sin_neg]
+    exact neg_ne_zero.mpr hsba.ne'
+  exact eq_of_inner_normalVector_eq hsinab hza hzb
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Frame Calculus
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem hasDerivAt_normalVector (t : ℝ) :
+    HasDerivAt (fun s : ℝ ↦ normalVector (s : Real.Angle))
+      (tangentVector (t : Real.Angle)) t := by
+  have h : HasDerivAt (fun s : ℝ ↦ (![Real.cos s, Real.sin s] : Fin 2 → ℝ))
+      (![-Real.sin t, Real.cos t] : Fin 2 → ℝ) t := by
+    apply hasDerivAt_pi.mpr
+    intro i
+    fin_cases i
+    · exact Real.hasDerivAt_cos t
+    · exact Real.hasDerivAt_sin t
+  exact (PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 2 ↦ ℝ)).symm.hasFDerivAt.comp_hasDerivAt t h
+
+theorem hasDerivAt_tangentVector (t : ℝ) :
+    HasDerivAt (fun s : ℝ ↦ tangentVector (s : Real.Angle))
+      (-normalVector (t : Real.Angle)) t := by
+  have h : HasDerivAt (fun s : ℝ ↦ (![-Real.sin s, Real.cos s] : Fin 2 → ℝ))
+      (![-Real.cos t, -Real.sin t] : Fin 2 → ℝ) t := by
+    apply hasDerivAt_pi.mpr
+    intro i
+    fin_cases i
+    · exact (Real.hasDerivAt_sin t).neg
+    · exact Real.hasDerivAt_cos t
+  convert (PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 2 ↦ ℝ)).symm.hasFDerivAt.comp_hasDerivAt t
+    h using 1
+  · rfl
+  · ext i
+    fin_cases i <;> rfl
+
+/-- The angular unit-normal parametrization is one-Lipschitz. -/
+theorem lipschitzWith_normalVector_real :
+    LipschitzWith 1 (fun t : ℝ ↦ normalVector (t : Real.Angle)) := by
+  apply lipschitzWith_of_nnnorm_deriv_le
+    (fun t ↦ (hasDerivAt_normalVector t).differentiableAt)
+  intro t
+  rw [(hasDerivAt_normalVector t).deriv]
+  change ‖tangentVector (t : Real.Angle)‖ ≤ (1 : ℝ)
+  rw [norm_tangentVector]
+
+/-- The angular unit-tangent parametrization is one-Lipschitz. -/
+theorem lipschitzWith_tangentVector_real :
+    LipschitzWith 1 (fun t : ℝ ↦ tangentVector (t : Real.Angle)) := by
+  apply lipschitzWith_of_nnnorm_deriv_le
+    (fun t ↦ (hasDerivAt_tangentVector t).differentiableAt)
+  intro t
+  rw [(hasDerivAt_tangentVector t).deriv]
+  change ‖-normalVector (t : Real.Angle)‖ ≤ (1 : ℝ)
+  rw [norm_neg, norm_normalVector_real]
+
+/-- A combination of the rotating frame whose coefficients are globally Lipschitz and bounded on a
+set is Lipschitz on that set. -/
+theorem lipschitzOnWith_frameCombination {f g : ℝ → ℝ} {C M : ℝ} {D : NNReal} {s : Set ℝ}
+    (hD : 2 * (C + M) ≤ (D : ℝ)) (hM : 0 ≤ M)
+    (hf : ∀ x y : ℝ, |f x - f y| ≤ C * |x - y|) (hg : ∀ x y : ℝ, |g x - g y| ≤ C * |x - y|)
+    (hfM : ∀ x ∈ s, |f x| ≤ M) (hgM : ∀ x ∈ s, |g x| ≤ M) :
+    LipschitzOnWith D (fun x : ℝ ↦ f x • normalVector (x : Real.Angle) +
+      g x • tangentVector (x : Real.Angle)) s := by
+  refine LipschitzOnWith.of_dist_le_mul fun x hx y hy ↦ ?_
+  have hframe (u v : ℝ) : dist (normalVector (u : Real.Angle)) (normalVector (v : Real.Angle)) ≤
+      |u - v| ∧ dist (tangentVector (u : Real.Angle)) (tangentVector (v : Real.Angle)) ≤
+        |u - v| := by
+    constructor
+    · simpa [Real.dist_eq] using lipschitzWith_normalVector_real.dist_le_mul u v
+    · simpa [Real.dist_eq] using lipschitzWith_tangentVector_real.dist_le_mul u v
+  have hpair (p q : ℝ) (u w : Point) (hu : ‖u‖ = 1) (hq : |q| ≤ M)
+      (huw : dist u w ≤ |x - y|) : dist (p • u) (q • w) ≤ |p - q| + M * |x - y| := by
+    have h1 : dist (p • u) (q • u) ≤ |p - q| := by
+      simpa [hu, Real.dist_eq] using dist_pair_smul p q u
+    have h2 : dist (q • u) (q • w) ≤ M * |x - y| := by
+      refine (dist_smul_pair q u w).trans ?_
+      rw [Real.dist_eq, sub_zero]
+      exact mul_le_mul hq huw dist_nonneg hM
+    exact (dist_triangle _ _ _).trans (add_le_add h1 h2)
+  have h1 := hpair (f x) (f y) (normalVector (x : Real.Angle)) (normalVector (y : Real.Angle))
+    (norm_normalVector_real x) (hfM y hy) (hframe x y).1
+  have h2 := hpair (g x) (g y) (tangentVector (x : Real.Angle)) (tangentVector (y : Real.Angle))
+    (norm_tangentVector _) (hgM y hy) (hframe x y).2
+  have htri := dist_add_add_le (f x • normalVector (x : Real.Angle))
+    (g x • tangentVector (x : Real.Angle)) (f y • normalVector (y : Real.Angle))
+    (g y • tangentVector (y : Real.Angle))
+  have hfxy := hf x y
+  have hgxy := hg x y
+  have hd : 2 * (C + M) * |x - y| ≤ (D : ℝ) * |x - y| :=
+    mul_le_mul_of_nonneg_right hD (abs_nonneg _)
+  rw [Real.dist_eq]
+  linarith
+
+/-- The angular derivative of a fixed normal projection is its tangent projection. -/
+theorem hasDerivAt_inner_normalVector (A : Point) (t : ℝ) :
+    HasDerivAt (fun u : ℝ ↦ inner ℝ A (normalVector (u : Real.Angle)))
+      (inner ℝ A (tangentVector (t : Real.Angle))) t := by
+  simpa using (hasDerivAt_const t A).inner ℝ (hasDerivAt_normalVector t)
+
+/-- A contact point in a normal direction has the support derivative as tangent coordinate. -/
+theorem exists_contact_of_hasDerivAt {s : Set Point} (hcomp : IsCompact s)
+    (hne : s.Nonempty) {t d : ℝ}
+    (hd : HasDerivAt (fun u : ℝ ↦ supportValue s (u : Real.Angle)) d t) :
+    ∃ A ∈ s, inner ℝ A (normalVector (t : Real.Angle)) = supportValue s (t : Real.Angle) ∧
+      inner ℝ A (tangentVector (t : Real.Angle)) = d := by
+  obtain ⟨A, hA, hmax, -⟩ := hcomp.exists_sSup_image_eq_and_ge
+    (f := fun p : Point ↦ inner ℝ p (normalVector (t : Real.Angle))) hne
+    (continuous_inner.comp (continuous_id.prodMk continuous_const)).continuousOn
+  have hAt : inner ℝ A (normalVector (t : Real.Angle)) = supportValue s (t : Real.Angle) := by
+    simpa only [supportValue] using hmax.symm
+  refine ⟨A, hA, hAt, ?_⟩
+  set F : ℝ → ℝ := fun u ↦ supportValue s (u : Real.Angle) -
+    inner ℝ A (normalVector (u : Real.Angle)) with hF
+  have hFmin : IsLocalMin F t := by
+    filter_upwards with u
+    have h1 : inner ℝ A (normalVector (u : Real.Angle)) ≤ supportValue s (u : Real.Angle) :=
+      inner_le_supportValue_of_isCompact hcomp hA _
+    simp only [hF, hAt, sub_self]
+    linarith
+  have hFderiv : HasDerivAt F (d - inner ℝ A (tangentVector (t : Real.Angle))) t :=
+    hd.sub (hasDerivAt_inner_normalVector A t)
+  have := hFmin.hasDerivAt_eq_zero hFderiv
+  linarith
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Cap.Basic`.
+* `Cap.AngleDomain`.
+* `Cap.Area`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Basic
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The two real intervals of upper cap normals. -/
+def capUpperAngles (ω : ℝ) : Set ℝ :=
+  Set.Icc 0 ω ∪ Set.Icc (Real.pi / 2) (ω + Real.pi / 2)
+
+/-- The normal directions of the two lower strip boundaries. -/
+def capLowerNormals (ω : ℝ) : Set Real.Angle :=
+  {((ω + Real.pi : ℝ) : Real.Angle), ((3 * Real.pi / 2 : ℝ) : Real.Angle)}
+
+/-- A set represented by closed lower half-planes with allowed normal directions. -/
+def HasHalfPlaneRepresentation (s : Set Point) (normals : Set Real.Angle) : Prop :=
+  ∃ constraints : Set (Real.Angle × ℝ),
+    (∀ c ∈ constraints, c.1 ∈ normals) ∧
+      s = ⋂ c ∈ constraints, normalHalfPlane c.1 c.2 false false
+
+/-- The normalized cap conditions, including its nonzero rotation-angle domain. -/
+def IsCap (ω : ℝ) (K : ConvexBody Point) : Prop :=
+  0 < ω ∧ ω ≤ Real.pi / 2 ∧
+    supportValue K (ω : Real.Angle) = 1 ∧
+    supportValue K ((Real.pi / 2 : ℝ) : Real.Angle) = 1 ∧
+    supportValue K ((ω + Real.pi : ℝ) : Real.Angle) = 0 ∧
+    supportValue K ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 ∧
+    HasHalfPlaneRepresentation K
+      (((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles ω) ∪ capLowerNormals ω)
+
+/-- The space of caps at a fixed rotation angle. -/
+def CapSpace (ω : ℝ) := {K : ConvexBody Point // IsCap ω K}
+
+/-- A nonempty finite set of angles strictly between zero and its rotation angle. -/
+structure AngleSet where
+  /-- The terminal rotation angle of the polygonal approximation. -/
+  angle : ℝ
+  angle_pos : 0 < angle
+  angle_le : angle ≤ Real.pi / 2
+  /-- The finite nonempty set of interior wall directions. -/
+  directions : Finset ℝ
+  nonempty : directions.Nonempty
+  interior : ∀ t ∈ directions, t ∈ Set.Ioo 0 angle
+
+/-- The finite upper-normal domain associated with an angle set. -/
+def angleDomain (Θ : AngleSet) : Set ℝ :=
+  (Θ.directions : Set ℝ) ∪
+    ((fun t ↦ t + Real.pi / 2) '' (Θ.directions : Set ℝ)) ∪
+    {Θ.angle, Real.pi / 2}
+
+/-- Polygon caps whose upper normals belong to the specified angle domain. -/
+def PolygonCapSpace (Θ : AngleSet) :=
+  {K : CapSpace Θ.angle // HasHalfPlaneRepresentation (K.1 : Set Point)
+    (((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain Θ) ∪ capLowerNormals Θ.angle)}
+
+/-- The fan above both lower strip boundaries. -/
+def capFan (ω : ℝ) : Set Point :=
+  normalHalfPlane (ω : Real.Angle) 0 true false ∩
+    normalHalfPlane ((Real.pi / 2 : ℝ) : Real.Angle) 0 true false
+
+/-- The fan of a rotation angle is convex. -/
+theorem convex_capFan (ω : ℝ) : Convex ℝ (capFan ω) :=
+  (convex_normalHalfPlane _ _ _).inter (convex_normalHalfPlane _ _ _)
+
+/-- The open inward quadrant of the supporting hallway, in support coordinates. -/
+def innerQuadrant (s : Set Point) (t : ℝ) : Set Point :=
+  normalHalfPlane (t : Real.Angle) (supportValue s (t : Real.Angle) - 1) false true ∩
+    normalHalfPlane ((t + Real.pi / 2 : ℝ) : Real.Angle)
+      (supportValue s ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) false true
+
+/-- The open inward quadrant of a supporting hallway is convex. -/
+theorem convex_innerQuadrant (s : Set Point) (t : ℝ) : Convex ℝ (innerQuadrant s t) :=
+  (convex_normalHalfPlane_open _ _ _).inter (convex_normalHalfPlane_open _ _ _)
+
+/-- The niche is the union of inward quadrants clipped by the fan. -/
+def capNiche {ω : ℝ} (K : CapSpace ω) : Set Point :=
+  capFan ω ∩ ⋃ t ∈ Set.Ioo 0 ω, innerQuadrant (K.1 : Set Point) t
+
+/-- The finite-angle niche uses the same fan clipping as the continuous niche. -/
+def polygonNiche (Θ : AngleSet) (K : CapSpace Θ.angle) : Set Point :=
+  capFan Θ.angle ∩ ⋃ t ∈ Θ.directions, innerQuadrant (K.1 : Set Point) t
+
+/-- Avoiding an inward quadrant means lying above one of its two inner walls. -/
+theorem notMem_innerQuadrant_iff (S : Set Point) (u : ℝ) (p : Point) :
+    p ∉ innerQuadrant S u ↔
+      p ∈ normalHalfPlane (u : Real.Angle) (supportValue S (u : Real.Angle) - 1) true false ∨
+        p ∈ normalHalfPlane ((u + Real.pi / 2 : ℝ) : Real.Angle)
+          (supportValue S ((u + Real.pi / 2 : ℝ) : Real.Angle) - 1) true false := by
+  have h1 : (p ∈ innerQuadrant S u) ↔
+      (inner ℝ p (normalVector (u : Real.Angle)) < supportValue S (u : Real.Angle) - 1 ∧
+        inner ℝ p (normalVector ((u + Real.pi / 2 : ℝ) : Real.Angle)) <
+          supportValue S ((u + Real.pi / 2 : ℝ) : Real.Angle) - 1) := Iff.rfl
+  have h2 : (p ∈ normalHalfPlane (u : Real.Angle)
+      (supportValue S (u : Real.Angle) - 1) true false) ↔
+      (supportValue S (u : Real.Angle) - 1 ≤ inner ℝ p (normalVector (u : Real.Angle))) := Iff.rfl
+  have h3 : (p ∈ normalHalfPlane ((u + Real.pi / 2 : ℝ) : Real.Angle)
+      (supportValue S ((u + Real.pi / 2 : ℝ) : Real.Angle) - 1) true false) ↔
+      (supportValue S ((u + Real.pi / 2 : ℝ) : Real.Angle) - 1 ≤
+        inner ℝ p (normalVector ((u + Real.pi / 2 : ℝ) : Real.Angle))) := Iff.rfl
+  rw [h1, h2, h3, not_and_or, not_lt, not_lt]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Angle Domain
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Every upper normal of an angle set has strictly positive sine. -/
+lemma angleDomain_subset_Ioo (Θ : AngleSet) : angleDomain Θ ⊆ Set.Ioo 0 Real.pi := by
+  intro t ht
+  rcases ht with (ht | ⟨s, hs, rfl⟩) | ht
+  · have h := Θ.interior t ht
+    constructor <;> linarith [h.1, h.2, Θ.angle_le, Real.pi_pos]
+  · have h := Θ.interior s hs
+    constructor <;> linarith [h.1, h.2, Θ.angle_le, Real.pi_pos]
+  · rcases ht with rfl | ht
+    · constructor <;> linarith [Θ.angle_pos, Θ.angle_le, Real.pi_pos]
+    · have ht : t = Real.pi / 2 := ht
+      rw [ht]
+      constructor <;> linarith [Real.pi_pos]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Area
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Cap area minus niche area, with real Lebesgue area as in the classical interface. -/
+def capAreaFunctional {ω : ℝ} (K : CapSpace ω) : ℝ :=
+  ClassicalResults.area (K.1 : Set Point) - ClassicalResults.area (capNiche K)
+
+/-- The cap space at a right angle. -/
+abbrev RightAngleCapSpace := CapSpace (Real.pi / 2)
+
+/-- The sofa area functional on the right-angle cap space. -/
+def rightAngleAreaFunctional (K : RightAngleCapSpace) : ℝ := capAreaFunctional K
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Polygon.AngleSet`.
+* `Polygon.BooleanFunctions`.
+* `Polygon.Height.Space`.
+* `Polygon.Height.Bounds`.
+* `Polygon.Height.RaisedSupport`.
+* `Polygon.Nef.Basic`.
+* `Polygon.Nef.CapConstruction`.
+* `Polygon.Nef.Cells`.
+* `Polygon.Nef.Height`.
+* `Polygon.PerturbationBounds`.
+* `Polygon.Polyline.Basic`.
+* `Polygon.Polyline.Displacement`.
+* `Polygon.Polyline.Graph`.
+* `Polygon.Polyline.Measure`.
+* `Polygon.Polyline.Projection`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Angle Set
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter
+open scoped Topology
+
+namespace MovingSofa
+
+/-- The interior points of the uniform angular grid. -/
+def uniformAngleSet (ω : ℝ) (hω : 0 < ω) (hω' : ω ≤ Real.pi / 2)
+    (n : ℕ) (hn : 2 ≤ n) : AngleSet where
+  angle := ω
+  angle_pos := hω
+  angle_le := hω'
+  directions := (Finset.Ioo 0 n).image (fun i : ℕ ↦ (i : ℝ) / n * ω)
+  nonempty := by
+    apply Finset.Nonempty.image
+    exact ⟨1, by simp; omega⟩
+  interior := by
+    intro t ht
+    obtain ⟨i, hi, rfl⟩ := Finset.mem_image.mp ht
+    obtain ⟨hi, hin⟩ := Finset.mem_Ioo.mp hi
+    have hn' : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
+    have hi' : (0 : ℝ) < i := by exact_mod_cast hi
+    have hin' : (i : ℝ) < n := by exact_mod_cast hin
+    refine ⟨mul_pos (div_pos hi' hn') hω, ?_⟩
+    exact (mul_lt_mul_of_pos_right ((div_lt_one hn').2 hin') hω).trans_eq (one_mul ω)
+
+/-- A divisible grid refines the original grid. -/
+theorem uniformAngleSet_directions_mono_of_dvd (ω : ℝ) (hω : 0 < ω)
+    (hω' : ω ≤ Real.pi / 2) {m n : ℕ} (hm : 2 ≤ m) (hn : 2 ≤ n)
+    (hmn : m ∣ n) :
+    (uniformAngleSet ω hω hω' m hm).directions ⊆
+      (uniformAngleSet ω hω hω' n hn).directions := by
+  obtain ⟨k, rfl⟩ := hmn
+  intro t ht
+  obtain ⟨j, hj, rfl⟩ := Finset.mem_image.mp ht
+  obtain ⟨hj0, hjm⟩ := Finset.mem_Ioo.mp hj
+  have hk : 0 < k := by nlinarith
+  apply Finset.mem_image.mpr
+  refine ⟨j * k, Finset.mem_Ioo.mpr ⟨Nat.mul_pos hj0 hk,
+    Nat.mul_lt_mul_of_pos_right hjm hk⟩, ?_⟩
+  have hk' : (k : ℝ) ≠ 0 := by exact_mod_cast hk.ne'
+  push_cast
+  rw [mul_div_mul_right _ _ hk']
+
+/-- Monotone dyadic grids have nested directions. -/
+theorem uniformAngleSet_directions_mono_of_dyadic (ω : ℝ) (hω : 0 < ω)
+    (hω' : ω ≤ Real.pi / 2) (n : ℕ → ℕ) (hn : ∀ i, 2 ≤ n i)
+    (hmono : Monotone n) (hdyadic : ∀ i, ∃ k : ℕ, n i = 2 ^ k)
+    {i j : ℕ} (hij : i ≤ j) :
+    (uniformAngleSet ω hω hω' (n i) (hn i)).directions ⊆
+      (uniformAngleSet ω hω hω' (n j) (hn j)).directions := by
+  apply uniformAngleSet_directions_mono_of_dvd
+  obtain ⟨a, ha⟩ := hdyadic i
+  obtain ⟨b, hb⟩ := hdyadic j
+  have hab : a ≤ b := by
+    apply (pow_le_pow_iff_right₀ (by norm_num : 1 < (2 : ℕ))).mp
+    simpa only [← ha, ← hb] using hmono hij
+  rw [ha, hb]
+  exact pow_dvd_pow 2 hab
+
+/-- An interval longer than the mesh contains a grid direction. -/
+theorem exists_uniformAngleSet_mem_Ioo (ω : ℝ) (hω : 0 < ω)
+    (hω' : ω ≤ Real.pi / 2) (n : ℕ) (hn : 2 ≤ n)
+    {a b : ℝ} (ha : 0 ≤ a) (hb : b ≤ ω) (hmesh : ω / n < b - a) :
+    ∃ t ∈ (uniformAngleSet ω hω hω' n hn).directions, t ∈ Set.Ioo a b := by
+  have hnpos : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega)
+  let j := ⌊a * n / ω⌋₊ + 1
+  have hjpos : 0 < j := by dsimp [j]; omega
+  have hfloor := Nat.floor_le (show 0 ≤ a * n / ω by positivity)
+  have hfloor' := Nat.lt_floor_add_one (a * n / ω)
+  have hjlo : a < (j : ℝ) / n * ω := by
+    have h : a * n < (j : ℝ) * ω := by
+      apply (div_lt_iff₀ hω).mp
+      simpa only [j, Nat.cast_add, Nat.cast_one] using hfloor'
+    rw [div_mul_eq_mul_div]
+    exact (lt_div_iff₀ hnpos).mpr h
+  have hjhi : (j : ℝ) / n * ω ≤ a + ω / n := by
+    have h : ((j : ℝ) - 1) * ω ≤ a * n := by
+      apply (le_div_iff₀ hω).mp
+      simpa only [j, Nat.cast_add, Nat.cast_one, add_sub_cancel_right] using hfloor
+    rw [div_mul_eq_mul_div]
+    apply (div_le_iff₀ hnpos).mpr
+    rw [add_mul, div_mul_cancel₀ _ hnpos.ne']
+    nlinarith
+  have hjb : (j : ℝ) / n * ω < b := hjhi.trans_lt (by linarith)
+  have hjn : j < n := by
+    have h : (j : ℝ) / n < 1 := (mul_lt_mul_iff_left₀ hω).mp (by simpa using hjb.trans_le hb)
+    exact_mod_cast (div_lt_one hnpos).mp h
+  refine ⟨(j : ℝ) / n * ω, Finset.mem_image.mpr ⟨j, Finset.mem_Ioo.mpr ⟨hjpos, hjn⟩, rfl⟩,
+    hjlo, hjb⟩
+
+/-- An increasing sequence of grids eventually meets each interior open interval. -/
+theorem eventually_exists_uniformAngleSet_mem_Ioo (ω : ℝ) (hω : 0 < ω)
+    (hω' : ω ≤ Real.pi / 2) (n : ℕ → ℕ) (hn : ∀ i, 2 ≤ n i)
+    (hmono : StrictMono n) {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) (hb : b ≤ ω) :
+    ∀ᶠ i in atTop, ∃ t ∈ (uniformAngleSet ω hω hω' (n i) (hn i)).directions,
+      t ∈ Set.Ioo a b := by
+  have hlim : Tendsto (fun i ↦ ω / (n i : ℝ)) atTop (𝓝 0) :=
+    (tendsto_const_div_atTop_nhds_zero_nat ω).comp hmono.tendsto_atTop
+  filter_upwards [hlim.eventually_lt_const (sub_pos.mpr hab)] with i hi
+  exact exists_uniformAngleSet_mem_Ioo ω hω hω' (n i) (hn i) ha hb hi
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Boolean Functions
+-/
+
+@[expose] public section
+
+namespace MovingSofa
+
+/-- Boolean functions of finitely many Boolean variables. -/
+abbrev BooleanFunction (n : ℕ) := (Fin n → Bool) → Bool
+
+/-- Changing inputs from false to true cannot change a true output to false. -/
+def IsMonotoneBooleanFunction {n : ℕ} (E : BooleanFunction n) : Prop :=
+  ∀ P Q, (∀ i, P i = true → Q i = true) → E P = true → E Q = true
+
+/-- Formulas built from variables using only conjunction and disjunction. -/
+inductive PositiveBooleanFormula (n : ℕ) where
+  | variable (i : Fin n)
+  | conjunction (left right : PositiveBooleanFormula n)
+  | disjunction (left right : PositiveBooleanFormula n)
+
+/-- Evaluate a positive formula under a Boolean assignment. -/
+def PositiveBooleanFormula.eval {n : ℕ} : PositiveBooleanFormula n → BooleanFunction n
+  | .variable i => fun P ↦ P i
+  | .conjunction left right => fun P ↦ left.eval P && right.eval P
+  | .disjunction left right => fun P ↦ left.eval P || right.eval P
+
+theorem positiveBooleanFormula_monotone {n : ℕ} (E : PositiveBooleanFormula n) :
+    IsMonotoneBooleanFunction E.eval := by
+  induction E <;>
+    simp_all [IsMonotoneBooleanFunction, PositiveBooleanFormula.eval,
+      Bool.and_eq_true, Bool.or_eq_true] <;>
+    aesop
+
+end MovingSofa
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Height / Space
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Point sets obtained by translating a polygonal cap with the fixed angle data. -/
+def PolygonCapTranslateSpace (Θ : AngleSet) :=
+  {S : Set Point // ∃ (K : PolygonCapSpace Θ) (q : Point),
+    S = (fun p ↦ p + q) '' (K.val.val : Set Point)}
+
+/-- Real wall heights indexed by the finite angle domain. -/
+def PolygonHeightSpace (Θ : AngleSet) := angleDomain Θ → ℝ
+
+/-- Extend a wall-height function by zero outside its angle domain. -/
+def polygonHeightValue {Θ : AngleSet} (h : PolygonHeightSpace Θ) (t : ℝ) : ℝ := by
+  classical
+  exact if ht : t ∈ angleDomain Θ then h ⟨t, ht⟩ else 0
+
+/-- Intersect the two endpoint strips of unit width determined by the height data. -/
+def polygonHeightParallelogram {Θ : AngleSet} (h : PolygonHeightSpace Θ) : Set Point :=
+  ⋂ t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+    normalHalfPlane (t : Real.Angle) (polygonHeightValue h t) false false ∩
+    normalHalfPlane (t : Real.Angle) (polygonHeightValue h t - 1) true false
+
+/-- Cut the endpoint parallelogram by all upper wall half-planes. -/
+def polygonHeightCap {Θ : AngleSet} (h : PolygonHeightSpace Θ) : Set Point :=
+  polygonHeightParallelogram h ∩
+    ⋂ t ∈ (Θ.directions : Set ℝ) ∪ ((fun t : ℝ ↦ t + Real.pi / 2) '' Θ.directions),
+      normalHalfPlane (t : Real.Angle) (polygonHeightValue h t) false false
+
+/-- Intersect the lower half-planes at the two endpoint directions. -/
+def polygonHeightFan {Θ : AngleSet} (h : PolygonHeightSpace Θ) : Set Point :=
+  ⋂ t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+    normalHalfPlane (t : Real.Angle) (polygonHeightValue h t - 1) true false
+
+/-- Intersect the endpoint fan with the union of forbidden inner corners. -/
+def polygonHeightNiche {Θ : AngleSet} (h : PolygonHeightSpace Θ) : Set Point :=
+  polygonHeightFan h ∩ ⋃ t ∈ Θ.directions,
+    normalHalfPlane (t : Real.Angle) (polygonHeightValue h t - 1) false true ∩
+    normalHalfPlane ((t + Real.pi / 2 : ℝ) : Real.Angle)
+      (polygonHeightValue h (t + Real.pi / 2) - 1) false true
+
+/-- The real area of the cap minus the real area of its niche. -/
+def polygonHeightArea {Θ : AngleSet} (h : PolygonHeightSpace Θ) : ℝ :=
+  ClassicalResults.area (polygonHeightCap h) - ClassicalResults.area (polygonHeightNiche h)
+
+/-- Bundle the parallelogram, cap, fan, niche and area constructed from wall heights. -/
+def polygonHeightExtensions {Θ : AngleSet} (h : PolygonHeightSpace Θ) :
+    Set Point × Set Point × Set Point × Set Point × ℝ :=
+  (polygonHeightParallelogram h, polygonHeightCap h, polygonHeightFan h,
+    polygonHeightNiche h, polygonHeightArea h)
+
+/-- Take support values of a translated cap in the prescribed directions. -/
+def polygonTranslateHeight {Θ : AngleSet} (K : PolygonCapTranslateSpace Θ) :
+    PolygonHeightSpace Θ := fun t ↦ supportValue K.val (t.val : Real.Angle)
+
+/-- The niche and area functional associated with a translated cap. -/
+def polygonTranslateExtensions {Θ : AngleSet} (K : PolygonCapTranslateSpace Θ) :
+    Set Point × ℝ :=
+  (polygonHeightNiche (polygonTranslateHeight K), polygonHeightArea (polygonTranslateHeight K))
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Height / Bounds
+-/
+
+@[expose] public section
+
+namespace MovingSofa
+
+/-- A height cap satisfies each selected upper half-plane constraint. -/
+theorem polygonHeightCap_subset_normalHalfPlane {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    {t : ℝ} (ht : t ∈ angleDomain Θ) :
+    polygonHeightCap h ⊆ normalHalfPlane (t : Real.Angle)
+      (polygonHeightValue h t) false false := by
+  intro p hp
+  rcases ht with ht | ht
+  · exact Set.mem_iInter₂.mp hp.2 t ht
+  · exact (Set.mem_iInter₂.mp hp.1 t ht).1
+
+/-- The support of a reconstructed height cap is bounded by each defining height. -/
+theorem supportValue_le_polygonHeightValue {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    (K : ConvexBody Point) (hK : polygonHeightCap h = (K : Set Point))
+    {t : ℝ} (ht : t ∈ angleDomain Θ) :
+    supportValue K (t : Real.Angle) ≤ polygonHeightValue h t := by
+  apply supportValue_le_of_subset_normalHalfPlane
+  rw [← hK]
+  exact polygonHeightCap_subset_normalHalfPlane h ht
+
+/-- Unit width forces equality with the height of a distinguished strip. -/
+theorem supportValue_eq_polygonHeightValue_of_width_one {Θ : AngleSet}
+    (h : PolygonHeightSpace Θ) (K : ConvexBody Point)
+    (hK : polygonHeightCap h = (K : Set Point)) {t : ℝ}
+    (ht : t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ))
+    (hw : supportValue K (t : Real.Angle) +
+      supportValue K ((t + Real.pi : ℝ) : Real.Angle) = 1) :
+    supportValue K (t : Real.Angle) = polygonHeightValue h t := by
+  apply le_antisymm (supportValue_le_polygonHeightValue h K hK (Or.inr ht))
+  have hopp : supportValue K ((t + Real.pi : ℝ) : Real.Angle) ≤
+      1 - polygonHeightValue h t := by
+    apply supportValue_le_of_subset_normalHalfPlane
+    intro p hp
+    have hp' : p ∈ polygonHeightCap h := hK.symm ▸ hp
+    have hl := (Set.mem_iInter₂.mp hp'.1 t ht).2
+    change polygonHeightValue h t - 1 ≤ inner ℝ p (normalVector (t : Real.Angle)) at hl
+    change inner ℝ p (normalVector ((t + Real.pi : ℝ) : Real.Angle)) ≤ _
+    rw [normalVector_add_pi, inner_neg_right]
+    linarith
+  linarith
+
+/-- Niches grow with their heights when the fan remains fixed. -/
+theorem polygonHeightNiche_mono_of_eq_endpoints {Θ : AngleSet}
+    {h g : PolygonHeightSpace Θ} (hle : ∀ t, h t ≤ g t)
+    (heq : ∀ t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+      polygonHeightValue h t = polygonHeightValue g t) :
+    polygonHeightNiche h ⊆ polygonHeightNiche g := by
+  have hval (t : ℝ) (ht : t ∈ angleDomain Θ) :
+      polygonHeightValue h t ≤ polygonHeightValue g t := by
+    simpa only [polygonHeightValue, dite_eq_left ht] using hle ⟨t, ht⟩
+  rintro p ⟨hpF, hpN⟩
+  constructor
+  · apply Set.mem_iInter₂.mpr
+    intro t ht
+    have hp := Set.mem_iInter₂.mp hpF t ht
+    rwa [heq t ht] at hp
+  · obtain ⟨t, ht, hp⟩ := Set.mem_iUnion₂.mp hpN
+    apply Set.mem_iUnion₂.mpr
+    refine ⟨t, ht, ?_, ?_⟩
+    · change inner ℝ p (normalVector (t : Real.Angle)) < polygonHeightValue g t - 1
+      have hp₁ : inner ℝ p (normalVector (t : Real.Angle)) <
+          polygonHeightValue h t - 1 := hp.1
+      exact hp₁.trans_le (sub_le_sub_right (hval t (Or.inl (Or.inl ht))) 1)
+    · change inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) <
+        polygonHeightValue g (t + Real.pi / 2) - 1
+      have hp₂ : inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) <
+          polygonHeightValue h (t + Real.pi / 2) - 1 := hp.2
+      exact hp₂.trans_le (sub_le_sub_right
+        (hval (t + Real.pi / 2) (Or.inl (Or.inr ⟨t, ht, rfl⟩))) 1)
+
+/-- Every polygon height niche is bounded. -/
+theorem isBounded_polygonHeightNiche {Θ : AngleSet} (h : PolygonHeightSpace Θ) :
+    Bornology.IsBounded (polygonHeightNiche h) := by
+  let a := polygonHeightValue h (Real.pi / 2) - 1
+  let Q (t : ℝ) : Set Point := {p | a ≤ p 1 ∧
+    inner ℝ p (normalVector (t : Real.Angle)) < polygonHeightValue h t - 1 ∧
+    inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) <
+      polygonHeightValue h (t + Real.pi / 2) - 1}
+  have hQ : Bornology.IsBounded (⋃ t ∈ Θ.directions, Q t) := by
+    apply (Bornology.isBounded_biUnion_finset Θ.directions).2
+    intro t ht
+    exact isBounded_setOf_le_snd_and_inner_lt _ _ _ t
+      ⟨(Θ.interior t ht).1, (Θ.interior t ht).2.trans_le Θ.angle_le⟩
+  apply hQ.subset
+  rintro p ⟨hpF, hpN⟩
+  obtain ⟨t, ht, hp⟩ := Set.mem_iUnion₂.mp hpN
+  apply Set.mem_iUnion₂.mpr
+  refine ⟨t, ht, ?_, hp⟩
+  have hpT := Set.mem_iInter₂.mp hpF (Real.pi / 2) (by simp)
+  change a ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at hpT
+  simpa [normalVector, frame, PiLp.inner_apply] using hpT
+
+end MovingSofa
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Height / Raised Support
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Increase one selected support height by the prescribed amount. -/
+def raisedPolygonSupport {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (t : angleDomain Θ) (ε : ℝ) : PolygonHeightSpace Θ := by
+  classical
+  exact fun s ↦ supportValue (K.val.val : Set Point) (s.val : Real.Angle) +
+    if s = t then ε else 0
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Nef / Basic
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Normal direction, height and boundary conventions specifying a planar half-plane. -/
+structure PlanarHalfPlaneData where
+  /-- The angle of the half-plane’s normal vector. -/
+  angle : Real.Angle
+  /-- The scalar-product threshold defining the boundary line. -/
+  height : ℝ
+  /-- Select the side above the threshold when true, or below it when false. -/
+  upper : Bool
+  /-- Exclude the boundary line when true. -/
+  strict : Bool
+
+/-- The half-plane defined by the recorded normal, threshold and conventions. -/
+def PlanarHalfPlaneData.carrier (H : PlanarHalfPlaneData) : Set Point :=
+  normalHalfPlane H.angle H.height H.upper H.strict
+
+/-- The line where the normal scalar product equals the recorded height. -/
+def PlanarHalfPlaneData.boundaryLine (H : PlanarHalfPlaneData) : Set Point :=
+  normalLine H.angle H.height
+
+/-- Evaluate a Boolean function on the point’s memberships in a finite family of sets. -/
+def booleanSet {n : ℕ} (E : BooleanFunction n) (H : Fin n → Set Point) : Set Point := by
+  classical
+  exact {p | E (fun i ↦ decide (p ∈ H i)) = true}
+
+/-- The set is a finite Boolean combination of planar half-planes. -/
+def IsNefPolygon (X : Set Point) : Prop :=
+  ∃ (n : ℕ) (E : BooleanFunction n) (H : Fin n → PlanarHalfPlaneData),
+    X = booleanSet E (fun i ↦ (H i).carrier)
+
+/-- A monotone Boolean representation uses the supplied walls with distinct boundary lines. -/
+def IsSimpleNefPolygonWith {n : ℕ} (X : Set Point)
+    (H : Fin n → PlanarHalfPlaneData) : Prop :=
+  Function.Injective (fun i ↦ (H i).boundaryLine) ∧
+    ∃ E : BooleanFunction n, IsMonotoneBooleanFunction E ∧
+      X = booleanSet E (fun i ↦ (H i).carrier)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Nef / Cap Construction
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The upper walls and two lower endpoint walls defining the polygonal cap. -/
+def polygonCapWalls {Θ : AngleSet} (h : PolygonHeightSpace Θ) : Set PlanarHalfPlaneData :=
+  (fun t : ℝ ↦ (⟨(t : Real.Angle), polygonHeightValue h t, false, false⟩ :
+    PlanarHalfPlaneData)) '' angleDomain Θ ∪
+  (fun t : ℝ ↦ (⟨(t : Real.Angle), polygonHeightValue h t - 1, true, false⟩ :
+    PlanarHalfPlaneData)) '' {Θ.angle, Real.pi / 2}
+
+/-- The strict inner walls and two lower endpoint walls defining the polygonal niche. -/
+def polygonNicheWalls {Θ : AngleSet} (h : PolygonHeightSpace Θ) : Set PlanarHalfPlaneData :=
+  (fun t : ℝ ↦ (⟨(t : Real.Angle), polygonHeightValue h t - 1, false, true⟩ :
+    PlanarHalfPlaneData)) ''
+      ((Θ.directions : Set ℝ) ∪ ((fun t : ℝ ↦ t + Real.pi / 2) '' Θ.directions)) ∪
+  (fun t : ℝ ↦ (⟨(t : Real.Angle), polygonHeightValue h t - 1, true, false⟩ :
+    PlanarHalfPlaneData)) '' {Θ.angle, Real.pi / 2}
+
+private theorem angleDomain_finite (Θ : AngleSet) : (angleDomain Θ).Finite := by
+  unfold angleDomain
+  exact (Θ.directions.finite_toSet.union
+    (Θ.directions.finite_toSet.image (fun t : ℝ ↦ t + Real.pi / 2))).union
+      (Set.finite_singleton (Real.pi / 2) |>.insert Θ.angle)
+
+private theorem polygonInnerAngles_finite (Θ : AngleSet) :
+    ((Θ.directions : Set ℝ) ∪
+      ((fun t : ℝ ↦ t + Real.pi / 2) '' Θ.directions)).Finite :=
+  Θ.directions.finite_toSet.union
+    (Θ.directions.finite_toSet.image (fun t : ℝ ↦ t + Real.pi / 2))
+
+private theorem polygonInnerAngles_subset_angleDomain (Θ : AngleSet) :
+    (Θ.directions : Set ℝ) ∪
+      ((fun t : ℝ ↦ t + Real.pi / 2) '' Θ.directions) ⊆ angleDomain Θ := by
+  intro t ht
+  exact Or.inl ht
+
+private theorem polygonInnerAngle_ne_endpoint {Θ : AngleSet} {s t : ℝ}
+    (hs : s ∈ (Θ.directions : Set ℝ) ∪
+      ((fun r : ℝ ↦ r + Real.pi / 2) '' Θ.directions))
+    (ht : t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) : s ≠ t := by
+  rcases hs with hs | ⟨r, hr, rfl⟩
+  · have hsI := Θ.interior s hs
+    rcases ht with ht | ht
+    · intro hst
+      exact hsI.2.ne (hst.trans ht)
+    · have ht' : t = Real.pi / 2 := by simpa using ht
+      intro hst
+      exact (ne_of_lt (hsI.2.trans_le Θ.angle_le)) (hst.trans ht')
+  · have hrI := Θ.interior r hr
+    rcases ht with ht | ht
+    · intro hst
+      have hgt : Θ.angle < r + Real.pi / 2 := by
+        linarith [Θ.angle_le, hrI.1, Real.pi_pos]
+      exact hgt.ne' (hst.trans ht)
+    · have ht' : t = Real.pi / 2 := by simpa using ht
+      intro hst
+      have hgt : Real.pi / 2 < r + Real.pi / 2 := by linarith [hrI.1]
+      exact hgt.ne' (hst.trans ht')
+
+private theorem polygonCapWalls_finite {Θ : AngleSet} (h : PolygonHeightSpace Θ) :
+    (polygonCapWalls h).Finite := by
+  unfold polygonCapWalls
+  exact ((angleDomain_finite Θ).image _).union
+    ((Set.finite_singleton (Real.pi / 2) |>.insert Θ.angle).image _)
+
+private theorem polygonNicheWalls_finite {Θ : AngleSet} (h : PolygonHeightSpace Θ) :
+    (polygonNicheWalls h).Finite := by
+  unfold polygonNicheWalls
+  exact ((polygonInnerAngles_finite Θ).image _).union
+    ((Set.finite_singleton (Real.pi / 2) |>.insert Θ.angle).image _)
+
+private def polygonUpperWall {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    (t : ℝ) : PlanarHalfPlaneData :=
+  ⟨(t : Real.Angle), polygonHeightValue h t, false, false⟩
+
+private def polygonLowerWall {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    (t : ℝ) : PlanarHalfPlaneData :=
+  ⟨(t : Real.Angle), polygonHeightValue h t - 1, true, false⟩
+
+private def polygonInnerWall {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    (t : ℝ) : PlanarHalfPlaneData :=
+  ⟨(t : Real.Angle), polygonHeightValue h t - 1, false, true⟩
+
+private theorem polygonCapWalls_eq {Θ : AngleSet} (h : PolygonHeightSpace Θ) :
+    polygonCapWalls h =
+      polygonUpperWall h '' angleDomain Θ ∪
+        polygonLowerWall h '' ({Θ.angle, Real.pi / 2} : Set ℝ) := by
+  rfl
+
+private theorem polygonNicheWalls_eq {Θ : AngleSet} (h : PolygonHeightSpace Θ) :
+    polygonNicheWalls h =
+      polygonInnerWall h ''
+          ((Θ.directions : Set ℝ) ∪
+            ((fun t : ℝ ↦ t + Real.pi / 2) '' Θ.directions)) ∪
+        polygonLowerWall h '' ({Θ.angle, Real.pi / 2} : Set ℝ) := by
+  rfl
+
+private theorem boundaryLine_injOn_polygonCapWalls {Θ : AngleSet}
+    (h : PolygonHeightSpace Θ) :
+    Set.InjOn PlanarHalfPlaneData.boundaryLine (polygonCapWalls h) := by
+  intro A hA B hB hline
+  rw [polygonCapWalls_eq] at hA hB
+  rcases hA with ⟨s, hs, rfl⟩ | ⟨s, hs, rfl⟩ <;>
+    rcases hB with ⟨t, ht, rfl⟩ | ⟨t, ht, rfl⟩
+  · have hst := (normalLine_eq_iff_of_mem_Ioo
+      (angleDomain_subset_Ioo Θ hs) (angleDomain_subset_Ioo Θ ht)).mp hline
+    simp [hst.1]
+  · have htD : t ∈ angleDomain Θ := Or.inr ht
+    have hst := (normalLine_eq_iff_of_mem_Ioo
+      (angleDomain_subset_Ioo Θ hs) (angleDomain_subset_Ioo Θ htD)).mp hline
+    have hfalse := hst.2
+    simp [polygonUpperWall, polygonLowerWall, hst.1] at hfalse
+    exfalso
+    linarith
+  · have hsD : s ∈ angleDomain Θ := Or.inr hs
+    have hst := (normalLine_eq_iff_of_mem_Ioo
+      (angleDomain_subset_Ioo Θ hsD) (angleDomain_subset_Ioo Θ ht)).mp hline
+    have hfalse := hst.2
+    simp [polygonUpperWall, polygonLowerWall, hst.1] at hfalse
+  · have hsD : s ∈ angleDomain Θ := Or.inr hs
+    have htD : t ∈ angleDomain Θ := Or.inr ht
+    have hst := (normalLine_eq_iff_of_mem_Ioo
+      (angleDomain_subset_Ioo Θ hsD) (angleDomain_subset_Ioo Θ htD)).mp hline
+    simp [hst.1]
+
+private theorem boundaryLine_injOn_polygonNicheWalls {Θ : AngleSet}
+    (h : PolygonHeightSpace Θ) :
+    Set.InjOn PlanarHalfPlaneData.boundaryLine (polygonNicheWalls h) := by
+  intro A hA B hB hline
+  rw [polygonNicheWalls_eq] at hA hB
+  rcases hA with ⟨s, hs, rfl⟩ | ⟨s, hs, rfl⟩ <;>
+    rcases hB with ⟨t, ht, rfl⟩ | ⟨t, ht, rfl⟩
+  · have hsD := polygonInnerAngles_subset_angleDomain Θ hs
+    have htD := polygonInnerAngles_subset_angleDomain Θ ht
+    have hst := (normalLine_eq_iff_of_mem_Ioo
+      (angleDomain_subset_Ioo Θ hsD) (angleDomain_subset_Ioo Θ htD)).mp hline
+    simp [hst.1]
+  · have hsD := polygonInnerAngles_subset_angleDomain Θ hs
+    have htD : t ∈ angleDomain Θ := Or.inr ht
+    have hst := (normalLine_eq_iff_of_mem_Ioo
+      (angleDomain_subset_Ioo Θ hsD) (angleDomain_subset_Ioo Θ htD)).mp hline
+    exact (polygonInnerAngle_ne_endpoint hs ht hst.1).elim
+  · have hsD : s ∈ angleDomain Θ := Or.inr hs
+    have htD := polygonInnerAngles_subset_angleDomain Θ ht
+    have hst := (normalLine_eq_iff_of_mem_Ioo
+      (angleDomain_subset_Ioo Θ hsD) (angleDomain_subset_Ioo Θ htD)).mp hline
+    exact (polygonInnerAngle_ne_endpoint ht hs hst.1.symm).elim
+  · have hsD : s ∈ angleDomain Θ := Or.inr hs
+    have htD : t ∈ angleDomain Θ := Or.inr ht
+    have hst := (normalLine_eq_iff_of_mem_Ioo
+      (angleDomain_subset_Ioo Θ hsD) (angleDomain_subset_Ioo Θ htD)).mp hline
+    simp [hst.1]
+
+private def PositiveBooleanFormula.conjunctions {n : ℕ}
+    (default : PositiveBooleanFormula n) :
+    List (PositiveBooleanFormula n) → PositiveBooleanFormula n
+  | [] => default
+  | E :: Es => .conjunction E (conjunctions default Es)
+
+private def PositiveBooleanFormula.disjunctions {n : ℕ}
+    (default : PositiveBooleanFormula n) :
+    List (PositiveBooleanFormula n) → PositiveBooleanFormula n
+  | [] => default
+  | E :: Es => .disjunction E (disjunctions default Es)
+
+private theorem PositiveBooleanFormula.conjunctions_eval_eq_true {n : ℕ}
+    (default : PositiveBooleanFormula n) (Es : List (PositiveBooleanFormula n))
+    (P : Fin n → Bool) :
+    (default.conjunctions Es).eval P = true ↔
+      default.eval P = true ∧ ∀ E ∈ Es, E.eval P = true := by
+  induction Es with
+  | nil => simp [PositiveBooleanFormula.conjunctions]
+  | cons E Es ih =>
+      simp [PositiveBooleanFormula.conjunctions, PositiveBooleanFormula.eval, ih,
+        and_left_comm]
+
+private theorem PositiveBooleanFormula.disjunctions_eval_eq_true {n : ℕ}
+    (default : PositiveBooleanFormula n) (Es : List (PositiveBooleanFormula n))
+    (P : Fin n → Bool) :
+    (default.disjunctions Es).eval P = true ↔
+      default.eval P = true ∨ ∃ E ∈ Es, E.eval P = true := by
+  induction Es with
+  | nil => simp [PositiveBooleanFormula.disjunctions]
+  | cons E Es ih =>
+      simp [PositiveBooleanFormula.disjunctions, PositiveBooleanFormula.eval, ih,
+        or_left_comm]
+
+private theorem mem_polygonHeightCap_iff_walls {Θ : AngleSet}
+    (h : PolygonHeightSpace Θ) (p : Point) :
+    p ∈ polygonHeightCap h ↔
+      ∀ W ∈ polygonCapWalls h, p ∈ W.carrier := by
+  simp only [polygonHeightCap, polygonHeightParallelogram, polygonCapWalls,
+    Set.mem_inter_iff, Set.mem_iInter, Set.mem_union, Set.mem_image]
+  constructor
+  · rintro ⟨hendpoint, hinner⟩ W
+    rintro (⟨t, ht, rfl⟩ | ⟨t, ht, rfl⟩)
+    · change p ∈ normalHalfPlane (t : Real.Angle) (polygonHeightValue h t) false false
+      rcases ht with ht | ht
+      · exact hinner t ht
+      · exact (hendpoint t ht).1
+    · change p ∈ normalHalfPlane (t : Real.Angle)
+          (polygonHeightValue h t - 1) true false
+      exact (hendpoint t ht).2
+  · intro hall
+    refine ⟨?_, ?_⟩
+    · intro t ht
+      constructor
+      · exact hall (polygonUpperWall h t) (Or.inl ⟨t, Or.inr ht, rfl⟩)
+      · exact hall (polygonLowerWall h t) (Or.inr ⟨t, ht, rfl⟩)
+    · intro t ht
+      exact hall (polygonUpperWall h t) (Or.inl ⟨t, Or.inl ht, rfl⟩)
+
+private theorem mem_polygonHeightNiche_iff_walls {Θ : AngleSet}
+    (h : PolygonHeightSpace Θ) (p : Point) :
+    p ∈ polygonHeightNiche h ↔
+      (∀ t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+        p ∈ (polygonLowerWall h t).carrier) ∧
+      ∃ t ∈ Θ.directions,
+        p ∈ (polygonInnerWall h t).carrier ∧
+        p ∈ (polygonInnerWall h (t + Real.pi / 2)).carrier := by
+  unfold polygonHeightNiche polygonHeightFan
+  simp only [Set.mem_inter_iff, Set.mem_iInter]
+  constructor
+  · rintro ⟨hfan, hniche⟩
+    refine ⟨?_, ?_⟩
+    · intro t ht
+      exact hfan t ht
+    · obtain ⟨t, ht, hp⟩ := Set.mem_iUnion₂.mp hniche
+      exact ⟨t, ht, hp⟩
+  · rintro ⟨hfan, t, ht, hp⟩
+    refine ⟨hfan, Set.mem_iUnion₂.mpr ⟨t, ht, ?_⟩⟩
+    exact hp
+
+private theorem exists_polygonHeightCap_simpleNef (Θ : AngleSet)
+    (h : PolygonHeightSpace Θ) :
+    ∃ (n : ℕ) (H : Fin n → PlanarHalfPlaneData),
+      Set.range H = polygonCapWalls h ∧
+        IsSimpleNefPolygonWith (polygonHeightCap h) H := by
+  classical
+  let S : Set PlanarHalfPlaneData := polygonCapWalls h
+  have hSfinite : S.Finite := polygonCapWalls_finite h
+  let _ : Fintype S := hSfinite.fintype
+  let n := Fintype.card S
+  let e : Fin n ≃ S := (Fintype.equivFin S).symm
+  let H : Fin n → PlanarHalfPlaneData := fun i ↦ (e i).1
+  have hrange : Set.range H = polygonCapWalls h := by
+    ext W
+    constructor
+    · rintro ⟨i, rfl⟩
+      exact (e i).2
+    · intro hW
+      obtain ⟨i, hi⟩ := e.surjective ⟨W, hW⟩
+      exact ⟨i, congrArg Subtype.val hi⟩
+  have hHinj : Function.Injective (fun i ↦ (H i).boundaryLine) := by
+    intro i j hij
+    apply e.injective
+    apply Subtype.ext
+    exact boundaryLine_injOn_polygonCapWalls h (e i).2 (e j).2 hij
+  have hupperω : polygonUpperWall h Θ.angle ∈ S := by
+    exact Or.inl ⟨Θ.angle, by simp [angleDomain], rfl⟩
+  let i₀ : Fin n := e.symm ⟨polygonUpperWall h Θ.angle, hupperω⟩
+  let varList : List (PositiveBooleanFormula n) :=
+    List.ofFn fun i : Fin n ↦ PositiveBooleanFormula.variable i
+  let formula : PositiveBooleanFormula n :=
+    (PositiveBooleanFormula.variable i₀).conjunctions varList
+  refine ⟨n, H, hrange, hHinj, formula.eval,
+    positiveBooleanFormula_monotone formula, ?_⟩
+  ext p
+  rw [mem_polygonHeightCap_iff_walls]
+  change (∀ W ∈ polygonCapWalls h, p ∈ W.carrier) ↔
+    formula.eval (fun i ↦ decide (p ∈ (H i).carrier)) = true
+  have hformula (P : Fin n → Bool) :
+      formula.eval P = true ↔ ∀ i, P i = true := by
+    rw [show formula = (PositiveBooleanFormula.variable i₀).conjunctions varList by rfl,
+      PositiveBooleanFormula.conjunctions_eval_eq_true]
+    simp [varList, PositiveBooleanFormula.eval]
+    aesop
+  rw [hformula]
+  constructor
+  · intro hall i
+    apply decide_eq_true
+    exact hall (H i) (e i).2
+  · intro hall W hW
+    obtain ⟨i, hi⟩ := e.surjective ⟨W, hW⟩
+    have hi' : H i = W := congrArg Subtype.val hi
+    subst W
+    exact of_decide_eq_true (hall i)
+
+private theorem exists_polygonHeightNiche_simpleNef (Θ : AngleSet)
+    (h : PolygonHeightSpace Θ) :
+    ∃ (n : ℕ) (H : Fin n → PlanarHalfPlaneData),
+      Set.range H = polygonNicheWalls h ∧
+        IsSimpleNefPolygonWith (polygonHeightNiche h) H := by
+  classical
+  let S : Set PlanarHalfPlaneData := polygonNicheWalls h
+  have hSfinite : S.Finite := polygonNicheWalls_finite h
+  let _ : Fintype S := hSfinite.fintype
+  let n := Fintype.card S
+  let e : Fin n ≃ S := (Fintype.equivFin S).symm
+  let H : Fin n → PlanarHalfPlaneData := fun i ↦ (e i).1
+  have hrange : Set.range H = polygonNicheWalls h := by
+    ext W
+    constructor
+    · rintro ⟨i, rfl⟩
+      exact (e i).2
+    · intro hW
+      obtain ⟨i, hi⟩ := e.surjective ⟨W, hW⟩
+      exact ⟨i, congrArg Subtype.val hi⟩
+  have hHinj : Function.Injective (fun i ↦ (H i).boundaryLine) := by
+    intro i j hij
+    apply e.injective
+    apply Subtype.ext
+    exact boundaryLine_injOn_polygonNicheWalls h (e i).2 (e j).2 hij
+  have hlowerω : polygonLowerWall h Θ.angle ∈ S := by
+    exact Or.inr ⟨Θ.angle, by simp, rfl⟩
+  have hlowerT : polygonLowerWall h (Real.pi / 2) ∈ S := by
+    exact Or.inr ⟨Real.pi / 2, by simp, rfl⟩
+  have hinner (t : ℝ) (ht : t ∈ Θ.directions) :
+      polygonInnerWall h t ∈ S := by
+    exact Or.inl ⟨t, Or.inl ht, rfl⟩
+  have hinnerShift (t : ℝ) (ht : t ∈ Θ.directions) :
+      polygonInnerWall h (t + Real.pi / 2) ∈ S := by
+    exact Or.inl ⟨t + Real.pi / 2, Or.inr ⟨t, ht, rfl⟩, rfl⟩
+  let iω : Fin n := e.symm ⟨polygonLowerWall h Θ.angle, hlowerω⟩
+  let iT : Fin n := e.symm ⟨polygonLowerWall h (Real.pi / 2), hlowerT⟩
+  let innerIndex : Θ.directions → Fin n :=
+    fun t ↦ e.symm ⟨polygonInnerWall h t, hinner t t.property⟩
+  let shiftedIndex : Θ.directions → Fin n :=
+    fun t ↦ e.symm ⟨polygonInnerWall h (t + Real.pi / 2), hinnerShift t t.property⟩
+  let pairFormula : Θ.directions → PositiveBooleanFormula n :=
+    fun t ↦ .conjunction (.variable (innerIndex t)) (.variable (shiftedIndex t))
+  let t₀ : Θ.directions := ⟨Θ.nonempty.choose, Θ.nonempty.choose_spec⟩
+  let pairList : List (PositiveBooleanFormula n) :=
+    Θ.directions.attach.toList.map pairFormula
+  let endpointFormula : PositiveBooleanFormula n :=
+    .conjunction (.variable iω) (.variable iT)
+  let interiorFormula : PositiveBooleanFormula n :=
+    (pairFormula t₀).disjunctions pairList
+  let formula : PositiveBooleanFormula n :=
+    .conjunction endpointFormula interiorFormula
+  have hHiω : H iω = polygonLowerWall h Θ.angle := by
+    exact congrArg Subtype.val (e.apply_symm_apply
+      ⟨polygonLowerWall h Θ.angle, hlowerω⟩)
+  have hHiT : H iT = polygonLowerWall h (Real.pi / 2) := by
+    exact congrArg Subtype.val (e.apply_symm_apply
+      ⟨polygonLowerWall h (Real.pi / 2), hlowerT⟩)
+  have hHinner (t : Θ.directions) :
+      H (innerIndex t) = polygonInnerWall h t := by
+    exact congrArg Subtype.val (e.apply_symm_apply
+      ⟨polygonInnerWall h t, hinner t t.property⟩)
+  have hHshift (t : Θ.directions) :
+      H (shiftedIndex t) = polygonInnerWall h (t + Real.pi / 2) := by
+    exact congrArg Subtype.val (e.apply_symm_apply
+      ⟨polygonInnerWall h (t + Real.pi / 2), hinnerShift t t.property⟩)
+  refine ⟨n, H, hrange, hHinj, formula.eval,
+    positiveBooleanFormula_monotone formula, ?_⟩
+  ext p
+  rw [mem_polygonHeightNiche_iff_walls]
+  change ((∀ t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+      p ∈ (polygonLowerWall h t).carrier) ∧
+    ∃ t ∈ Θ.directions,
+      p ∈ (polygonInnerWall h t).carrier ∧
+      p ∈ (polygonInnerWall h (t + Real.pi / 2)).carrier) ↔
+    formula.eval (fun i ↦ decide (p ∈ (H i).carrier)) = true
+  have hpair (t : Θ.directions) (P : Fin n → Bool) :
+      (pairFormula t).eval P = true ↔
+        P (innerIndex t) = true ∧ P (shiftedIndex t) = true := by
+    simp [pairFormula, PositiveBooleanFormula.eval]
+  have hinterior (P : Fin n → Bool) :
+      interiorFormula.eval P = true ↔
+        ∃ t : Θ.directions,
+          P (innerIndex t) = true ∧ P (shiftedIndex t) = true := by
+    rw [show interiorFormula = (pairFormula t₀).disjunctions pairList by rfl,
+      PositiveBooleanFormula.disjunctions_eval_eq_true]
+    simp only [hpair]
+    constructor
+    · rintro (ht₀ | ⟨E, hE, hEval⟩)
+      · exact ⟨t₀, ht₀⟩
+      · rw [show pairList = Θ.directions.attach.toList.map pairFormula by rfl] at hE
+        obtain ⟨t, ht, rfl⟩ := List.mem_map.mp hE
+        exact ⟨t, (hpair t P).mp hEval⟩
+    · rintro ⟨t, hEval⟩
+      right
+      refine ⟨pairFormula t, ?_, (hpair t P).mpr hEval⟩
+      rw [show pairList = Θ.directions.attach.toList.map pairFormula by rfl]
+      exact List.mem_map.mpr ⟨t, by simp, rfl⟩
+  have hformula (P : Fin n → Bool) :
+      formula.eval P = true ↔
+        (P iω = true ∧ P iT = true) ∧
+          ∃ t : Θ.directions,
+            P (innerIndex t) = true ∧ P (shiftedIndex t) = true := by
+    rw [show formula = .conjunction endpointFormula interiorFormula by rfl]
+    simp only [PositiveBooleanFormula.eval, Bool.and_eq_true, hinterior]
+    simp [endpointFormula, PositiveBooleanFormula.eval]
+  rw [hformula]
+  constructor
+  · rintro ⟨hendpoint, t, ht, htinner, htshift⟩
+    let ts : Θ.directions := ⟨t, ht⟩
+    refine ⟨⟨?_, ?_⟩, ts, ?_, ?_⟩
+    · apply decide_eq_true
+      rw [hHiω]
+      exact hendpoint Θ.angle (by simp)
+    · apply decide_eq_true
+      rw [hHiT]
+      exact hendpoint (Real.pi / 2) (by simp)
+    · apply decide_eq_true
+      rw [hHinner ts]
+      exact htinner
+    · apply decide_eq_true
+      rw [hHshift ts]
+      exact htshift
+  · rintro ⟨⟨hω, hT⟩, t, htinner, htshift⟩
+    refine ⟨?_, t, t.property, ?_, ?_⟩
+    · intro s hs
+      rcases hs with hs | hs
+      · have hsval : s = Θ.angle := hs
+        subst s
+        rw [← hHiω]
+        exact of_decide_eq_true hω
+      · have hsval : s = Real.pi / 2 := by simpa using hs
+        subst s
+        rw [← hHiT]
+        exact of_decide_eq_true hT
+    · rw [← hHinner t]
+      exact of_decide_eq_true htinner
+    · rw [← hHshift t]
+      exact of_decide_eq_true htshift
+
+theorem polygonCap_niche_simpleNef (Θ : AngleSet) (h : PolygonHeightSpace Θ) :
+    (∃ (n : ℕ) (H : Fin n → PlanarHalfPlaneData),
+      Set.range H = polygonCapWalls h ∧ IsSimpleNefPolygonWith (polygonHeightCap h) H) ∧
+    (∃ (n : ℕ) (H : Fin n → PlanarHalfPlaneData),
+      Set.range H = polygonNicheWalls h ∧ IsSimpleNefPolygonWith (polygonHeightNiche h) H) := by
+  exact ⟨exists_polygonHeightCap_simpleNef Θ h, exists_polygonHeightNiche_simpleNef Θ h⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Nef / Cells
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa.Nef
+
+open Filter Topology
+
+lemma monotoneBoolean_update_false {n : ℕ} {E : BooleanFunction n}
+    (hE : IsMonotoneBooleanFunction E) (P : Fin n → Bool) (i : Fin n) :
+    E (Function.update P i false) = true → E P = true := by
+  apply hE
+  intro j hj
+  by_cases hji : j = i
+  · subst j
+    simp at hj
+  · simpa [Function.update_of_ne hji] using hj
+
+lemma monotoneBoolean_update_true {n : ℕ} {E : BooleanFunction n}
+    (hE : IsMonotoneBooleanFunction E) (P : Fin n → Bool) (i : Fin n) :
+    E P = true → E (Function.update P i true) = true := by
+  apply hE
+  intro j hj
+  by_cases hji : j = i
+  · subst j
+    simp
+  · simpa [Function.update_of_ne hji] using hj
+
+lemma monotoneBoolean_eval_iff {n : ℕ} {E : BooleanFunction n}
+    (hE : IsMonotoneBooleanFunction E) (P : Fin n → Bool) (i : Fin n) :
+    E P = true ↔ E (Function.update P i false) = true ∨
+      (P i = true ∧ E (Function.update P i true) = true) := by
+  constructor
+  · intro h
+    cases hp : P i
+    · left
+      rwa [← hp, Function.update_eq_self]
+    · exact Or.inr ⟨rfl, monotoneBoolean_update_true hE P i h⟩
+  · rintro (h | ⟨hp, h⟩)
+    · exact monotoneBoolean_update_false hE P i h
+    · rwa [(Function.update_eq_self_iff).mpr hp.symm] at h
+
+lemma booleanSet_update_eq {n : ℕ} {E : BooleanFunction n}
+    (hE : IsMonotoneBooleanFunction E) (H : Fin n → Set Point) (i : Fin n)
+    (S : Set Point) :
+    booleanSet E (Function.update H i S) =
+      booleanSet E (Function.update H i ∅) ∪
+        (S ∩ booleanSet E (Function.update H i Set.univ)) := by
+  classical
+  ext p
+  have hfun (T : Set Point) :
+      (fun j ↦ decide (p ∈ Function.update H i T j)) =
+        Function.update (fun j ↦ decide (p ∈ H j)) i (decide (p ∈ T)) := by
+    funext j
+    by_cases hji : j = i <;> simp [hji, Function.update_of_ne]
+  simp only [booleanSet, Set.mem_ofPred_eq, Set.mem_union, Set.mem_inter_iff, hfun]
+  simp only [Set.mem_empty_iff_false, decide_false, Set.mem_univ, decide_true]
+  have h := monotoneBoolean_eval_iff hE
+    (Function.update (fun j ↦ decide (p ∈ H j)) i (decide (p ∈ S))) i
+  simpa only [Function.update_idem, Function.update_self, decide_eq_true_eq] using h
+
+lemma booleanSet_update_sdiff {n : ℕ} {E : BooleanFunction n}
+    (hE : IsMonotoneBooleanFunction E) (H : Fin n → Set Point) (i : Fin n)
+    (S T : Set Point) :
+    booleanSet E (Function.update H i S) \ booleanSet E (Function.update H i T) =
+      (S \ T) ∩ (booleanSet E (Function.update H i Set.univ) \
+        booleanSet E (Function.update H i ∅)) := by
+  rw [booleanSet_update_eq hE H i S, booleanSet_update_eq hE H i T]
+  ext p
+  simp only [Set.mem_sdiff, Set.mem_union, Set.mem_inter_iff]
+  tauto
+
+lemma booleanSet_mono {n : ℕ} {E : BooleanFunction n}
+    (hE : IsMonotoneBooleanFunction E) {H G : Fin n → Set Point}
+    (hHG : ∀ i, H i ⊆ G i) : booleanSet E H ⊆ booleanSet E G := by
+  classical
+  intro p hp
+  apply hE _ _ ?_ hp
+  intro i hi
+  simp only [decide_eq_true_eq] at hi ⊢
+  exact hHG i hi
+
+/-- The points realizing exactly the specified Boolean membership pattern. -/
+def booleanCell {n : ℕ} (H : Fin n → Set Point) (P : Fin n → Bool) : Set Point :=
+  ⋂ j, if P j then H j else (H j)ᶜ
+
+/-- The vector of membership decisions for a point in a finite set family. -/
+def setMembershipPattern {n : ℕ} (H : Fin n → Set Point) (p : Point) : Fin n → Bool := by
+  classical
+  exact fun j ↦ decide (p ∈ H j)
+
+lemma mem_booleanCell_iff {n : ℕ} (H : Fin n → Set Point) (P : Fin n → Bool)
+    (p : Point) :
+    p ∈ booleanCell H P ↔ setMembershipPattern H p = P := by
+  classical
+  simp only [booleanCell, Set.mem_iInter]
+  constructor
+  · intro hp
+    funext j
+    have hj := hp j
+    cases hP : P j <;> simp only [hP, Bool.false_eq_true, ↓reduceIte, Set.mem_compl_iff,
+      setMembershipPattern,
+      decide_eq_false_iff_not, decide_eq_true_eq] at hj ⊢ <;> exact hj
+  · intro hp j
+    have hj := congrFun hp j
+    cases hP : P j <;> simp only [setMembershipPattern, hP, decide_eq_false_iff_not,
+      Bool.false_eq_true, ↓reduceIte,
+      Set.mem_compl_iff, decide_eq_true_eq] at hj ⊢ <;> exact hj
+
+lemma booleanSet_eq_iUnion_booleanCell {n : ℕ} (E : BooleanFunction n)
+    (H : Fin n → Set Point) :
+    booleanSet E H = ⋃ P : Fin n → Bool, if E P = true then booleanCell H P else ∅ := by
+  classical
+  ext p
+  simp only [booleanSet, Set.mem_ofPred_eq, Set.mem_iUnion, Set.mem_ite_empty_right]
+  constructor
+  · intro hp
+    change E (setMembershipPattern H p) = true at hp
+    refine ⟨setMembershipPattern H p, ?_, (mem_booleanCell_iff H _ p).mpr rfl⟩
+    exact hp
+  · rintro ⟨P, hEP, hp⟩
+    have hpat := (mem_booleanCell_iff H P p).mp hp
+    calc
+      E (fun j ↦ decide (p ∈ H j)) = E (setMembershipPattern H p) := by
+        congr 1
+      _ = E P := congrArg E hpat
+      _ = true := hEP
+
+/-- Intersect the closed sides of each wall selected by the Boolean pattern. -/
+def closedBooleanCell {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (P : Fin n → Bool) : Set Point :=
+  ⋂ j, normalHalfPlane (H j).angle (H j).height
+    (if P j then (H j).upper else !(H j).upper) false
+
+lemma mem_booleanCell_iff_mem_closedBooleanCell_of_ne {n : ℕ}
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) (p : Point)
+    (hp : ∀ j, inner ℝ p (normalVector (H j).angle) ≠ (H j).height) :
+    p ∈ booleanCell (fun j ↦ (H j).carrier) P ↔ p ∈ closedBooleanCell H P := by
+  simp only [booleanCell, closedBooleanCell, Set.mem_iInter]
+  constructor <;> intro h j
+  · have hj := h j
+    have hne := hp j
+    rcases lt_or_gt_of_ne hne with hlt | hgt <;>
+      cases hP : P j <;> cases hu : (H j).upper <;> cases hs : (H j).strict <;>
+      simp only [hP, hu, hs, PlanarHalfPlaneData.carrier, normalHalfPlane,
+        Bool.false_eq_true, Bool.not_false, Bool.not_true, ↓reduceIte,
+        Set.mem_compl_iff, Set.mem_ofPred_eq] at hj ⊢ <;> linarith
+  · have hj := h j
+    have hne := hp j
+    rcases lt_or_gt_of_ne hne with hlt | hgt <;>
+      cases hP : P j <;> cases hu : (H j).upper <;> cases hs : (H j).strict <;>
+      simp only [hP, hu, hs, PlanarHalfPlaneData.carrier, normalHalfPlane,
+        Bool.false_eq_true, Bool.not_false, Bool.not_true, ↓reduceIte,
+        Set.mem_compl_iff, Set.mem_ofPred_eq] at hj ⊢ <;> linarith
+
+/-- Switching the selected input from false to true switches the output to true. -/
+def IsActiveBooleanPattern {n : ℕ} (E : BooleanFunction n) (i : Fin n)
+    (P : Fin n → Bool) : Prop :=
+  E (Function.update P i false) = false ∧
+    E (Function.update P i true) = true
+
+/-- All input patterns for which the selected Boolean variable is active. -/
+def activeBooleanPatterns {n : ℕ} (E : BooleanFunction n) (i : Fin n) :
+    Finset (Fin n → Bool) := by
+  classical
+  exact Finset.univ.filter (IsActiveBooleanPattern E i)
+
+/-- The set gained by making the selected wall universally true rather than false. -/
+def activeBooleanRegion {n : ℕ} (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n) : Set Point :=
+  booleanSet E (Function.update (fun j ↦ (H j).carrier) i Set.univ) \
+    booleanSet E (Function.update (fun j ↦ (H j).carrier) i ∅)
+
+lemma mem_activeBooleanRegion_iff {n : ℕ} (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n) (p : Point) :
+    p ∈ activeBooleanRegion E H i ↔
+      IsActiveBooleanPattern E i (setMembershipPattern (fun j ↦ (H j).carrier) p) := by
+  classical
+  unfold activeBooleanRegion IsActiveBooleanPattern
+  simp only [Set.mem_sdiff, booleanSet, Set.mem_ofPred_eq]
+  have huniv : (fun j ↦ decide (p ∈ Function.update
+      (fun j ↦ (H j).carrier) i Set.univ j)) =
+      Function.update (setMembershipPattern (fun j ↦ (H j).carrier) p) i true := by
+    funext j
+    by_cases hji : j = i <;>
+      simp [hji, Function.update_of_ne, setMembershipPattern]
+  have hempty : (fun j ↦ decide (p ∈ Function.update
+      (fun j ↦ (H j).carrier) i ∅ j)) =
+      Function.update (setMembershipPattern (fun j ↦ (H j).carrier) p) i false := by
+    funext j
+    by_cases hji : j = i <;>
+      simp [hji, Function.update_of_ne, setMembershipPattern]
+  rw [huniv, hempty]
+  cases hfalse : E (Function.update (setMembershipPattern
+      (fun j ↦ (H j).carrier) p) i false) <;> simp
+
+lemma measurableSet_planarHalfPlane_carrier (H : PlanarHalfPlaneData) :
+    MeasurableSet H.carrier := by
+  cases hu : H.upper <;> cases hs : H.strict <;>
+    simp only [PlanarHalfPlaneData.carrier, normalHalfPlane, hu, hs,
+      Bool.false_eq_true, ↓reduceIte] <;> measurability
+
+lemma eventually_mem_carrier_iff_of_not_mem_boundaryLine (H : PlanarHalfPlaneData)
+    (p : Point) (hp : p ∉ H.boundaryLine) :
+    ∀ᶠ q in 𝓝 p, (q ∈ H.carrier ↔ p ∈ H.carrier) := by
+  have hpne : inner ℝ p (normalVector H.angle) ≠ H.height := hp
+  have hcont : Continuous (fun q : Point ↦ inner ℝ q (normalVector H.angle)) := by
+    fun_prop
+  rcases lt_or_gt_of_ne hpne with hplt | hpgt
+  · have hev : ∀ᶠ q in 𝓝 p, inner ℝ q (normalVector H.angle) < H.height :=
+      hcont.continuousAt.eventually_lt continuousAt_const hplt
+    filter_upwards [hev] with q hq
+    change (q ∈ normalHalfPlane H.angle H.height H.upper H.strict ↔
+      p ∈ normalHalfPlane H.angle H.height H.upper H.strict)
+    simp only [normalHalfPlane, Set.mem_ofPred_eq]
+    cases H.upper <;> cases H.strict <;>
+      simp only [Bool.false_eq_true, ↓reduceIte] <;>
+      constructor <;> intro hm <;> linarith
+  · have hev : ∀ᶠ q in 𝓝 p, H.height < inner ℝ q (normalVector H.angle) :=
+      continuousAt_const.eventually_lt hcont.continuousAt hpgt
+    filter_upwards [hev] with q hq
+    change (q ∈ normalHalfPlane H.angle H.height H.upper H.strict ↔
+      p ∈ normalHalfPlane H.angle H.height H.upper H.strict)
+    simp only [normalHalfPlane, Set.mem_ofPred_eq]
+    cases H.upper <;> cases H.strict <;>
+      simp only [Bool.false_eq_true, ↓reduceIte] <;>
+      constructor <;> intro hm <;> linarith
+
+lemma eventually_setMembershipPattern_eq_of_ne {n : ℕ}
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n) (p : Point)
+    (hp : ∀ j, j ≠ i → p ∉ (H j).boundaryLine) :
+    ∀ᶠ q in 𝓝 p, ∀ j, j ≠ i →
+      setMembershipPattern (fun k ↦ (H k).carrier) q j =
+        setMembershipPattern (fun k ↦ (H k).carrier) p j := by
+  suffices ∀ᶠ q in 𝓝 p, ∀ j ∈ (Set.univ : Set (Fin n)), j ≠ i →
+      setMembershipPattern (fun k ↦ (H k).carrier) q j =
+        setMembershipPattern (fun k ↦ (H k).carrier) p j by
+    exact this.mono fun q hq j ↦ hq j (Set.mem_univ j)
+  apply (Filter.eventually_all_finite Set.finite_univ).2
+  intro j _
+  by_cases hji : j = i
+  · exact Filter.Eventually.of_forall fun _ hj ↦ (hj hji).elim
+  · filter_upwards [eventually_mem_carrier_iff_of_not_mem_boundaryLine (H j) p
+      (hp j hji)] with q hq
+    intro _
+    simp only [setMembershipPattern]
+    apply Bool.eq_iff_iff.mpr
+    simpa only [decide_eq_true_eq] using hq
+
+lemma measurableSet_booleanCell {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (P : Fin n → Bool) :
+    MeasurableSet (booleanCell (fun j ↦ (H j).carrier) P) := by
+  apply MeasurableSet.iInter
+  intro j
+  cases hP : P j
+  · simpa [booleanCell, hP] using (measurableSet_planarHalfPlane_carrier (H j)).compl
+  · simpa [booleanCell, hP] using measurableSet_planarHalfPlane_carrier (H j)
+
+lemma measurableSet_booleanSet {n : ℕ} (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) :
+    MeasurableSet (booleanSet E (fun j ↦ (H j).carrier)) := by
+  rw [booleanSet_eq_iUnion_booleanCell]
+  apply MeasurableSet.iUnion
+  intro P
+  by_cases hP : E P = true
+  · simpa [hP] using measurableSet_booleanCell H P
+  · simp [hP]
+
+lemma monotoneBoolean_update_false_true {n : ℕ} {E : BooleanFunction n}
+    (hE : IsMonotoneBooleanFunction E) (i : Fin n) (P : Fin n → Bool) :
+    E (Function.update P i false) = true →
+      E (Function.update P i true) = true := by
+  intro htrue
+  apply hE _ _ _ htrue
+  intro j hj
+  by_cases hji : j = i
+  · subst j
+    simp
+  · simpa [Function.update_of_ne hji] using hj
+
+lemma not_isActiveBooleanPattern_iff {n : ℕ} {E : BooleanFunction n}
+    (hE : IsMonotoneBooleanFunction E) (i : Fin n) (P : Fin n → Bool) :
+    ¬IsActiveBooleanPattern E i P ↔
+      E (Function.update P i false) = E (Function.update P i true) := by
+  unfold IsActiveBooleanPattern
+  constructor
+  · intro hnot
+    cases hf : E (Function.update P i false) <;>
+      cases ht : E (Function.update P i true)
+    · rfl
+    · exact (hnot ⟨hf, ht⟩).elim
+    · have hcontra := monotoneBoolean_update_false_true hE i P hf
+      rw [ht] at hcontra
+      exact (Bool.false_ne_true hcontra).elim
+    · rfl
+  · intro heq hactive
+    rw [hactive.1, hactive.2] at heq
+    exact Bool.false_ne_true heq
+
+end MovingSofa.Nef
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Nef / Height
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Change one wall’s height by `δ` in a Boolean half-plane representation. -/
+def perturbNefHeight {n : ℕ} (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n) (δ : ℝ) : Set Point :=
+  booleanSet E (fun j ↦
+    (if j = i then { H j with height := (H j).height + δ } else H j).carrier)
+
+end MovingSofa
+
+namespace MovingSofa.Nef
+
+open Filter Topology
+
+lemma perturbNefHeight_eq_update {n : ℕ} (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n) (δ : ℝ) :
+    perturbNefHeight E H i δ = booleanSet E
+      (Function.update (fun j ↦ (H j).carrier) i
+        ({ H i with height := (H i).height + δ }).carrier) := by
+  unfold perturbNefHeight
+  congr 1
+  funext j
+  by_cases hji : j = i <;> simp [hji, Function.update_of_ne]
+
+lemma perturbNefHeight_sdiff {n : ℕ} {E : BooleanFunction n}
+    (hE : IsMonotoneBooleanFunction E) (H : Fin n → PlanarHalfPlaneData)
+    (i : Fin n) (δ ε : ℝ) :
+    perturbNefHeight E H i δ \ perturbNefHeight E H i ε =
+      (({ H i with height := (H i).height + δ }).carrier \
+        ({ H i with height := (H i).height + ε }).carrier) ∩
+        (booleanSet E (Function.update (fun j ↦ (H j).carrier) i Set.univ) \
+          booleanSet E (Function.update (fun j ↦ (H j).carrier) i ∅)) := by
+  rw [perturbNefHeight_eq_update, perturbNefHeight_eq_update]
+  exact booleanSet_update_sdiff hE _ _ _ _
+
+/-- The half-open strip-height interval matching the wall’s strictness convention. -/
+def heightInterval (strict : Bool) (h δ ε : ℝ) : Set ℝ :=
+  if strict then Set.Ico (h + ε) (h + δ) else Set.Ioc (h + ε) (h + δ)
+
+lemma measurableSet_heightInterval (strict : Bool) (h δ ε : ℝ) :
+    MeasurableSet (heightInterval strict h δ ε) := by
+  cases strict <;> simp [heightInterval]
+
+lemma integral_heightInterval_eq_intervalIntegral (strict : Bool)
+    (h δ ε : ℝ) (f : ℝ → ℝ) (hεδ : ε ≤ δ) :
+    (∫ x in heightInterval strict h δ ε, f x) =
+      ∫ x in h + ε..h + δ, f x := by
+  have hle : h + ε ≤ h + δ := by linarith
+  cases strict
+  · simp only [heightInterval, Bool.false_eq_true, ↓reduceIte]
+    exact (intervalIntegral.integral_of_le hle).symm
+  · simp only [heightInterval, ↓reduceIte]
+    rw [MeasureTheory.integral_Ico_eq_integral_Ioc]
+    exact (intervalIntegral.integral_of_le hle).symm
+
+lemma mem_heightInterval_bounds {strict : Bool} {h δ ε x : ℝ}
+    (hx : x ∈ heightInterval strict h δ ε) :
+    h + ε ≤ x ∧ x ≤ h + δ := by
+  cases strict <;> simp [heightInterval] at hx <;>
+    constructor <;> linarith [hx.1, hx.2]
+
+lemma carrier_sdiff_carrier_eq_heightInterval (H : PlanarHalfPlaneData)
+    (hSide : H.upper = false) (δ ε : ℝ) :
+    ({ H with height := H.height + δ }).carrier \
+        ({ H with height := H.height + ε }).carrier =
+      {p | inner ℝ p (normalVector H.angle) ∈
+        heightInterval H.strict H.height δ ε} := by
+  ext p
+  cases hs : H.strict <;>
+    simp [PlanarHalfPlaneData.carrier, normalHalfPlane, hSide,
+      heightInterval] <;> tauto
+
+lemma perturbNefHeight_sdiff_eq_heightInterval {n : ℕ}
+    {E : BooleanFunction n} (hE : IsMonotoneBooleanFunction E)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (hSide : (H i).upper = false) (δ ε : ℝ) :
+    perturbNefHeight E H i δ \ perturbNefHeight E H i ε =
+      {p | inner ℝ p (normalVector (H i).angle) ∈
+        heightInterval (H i).strict (H i).height δ ε} ∩
+        (booleanSet E (Function.update (fun j ↦ (H j).carrier) i Set.univ) \
+          booleanSet E (Function.update (fun j ↦ (H j).carrier) i ∅)) := by
+  rw [perturbNefHeight_sdiff hE H i δ ε,
+    carrier_sdiff_carrier_eq_heightInterval (H i) hSide]
+
+lemma perturbNefHeight_mono {n : ℕ} {E : BooleanFunction n}
+    (hE : IsMonotoneBooleanFunction E) (H : Fin n → PlanarHalfPlaneData)
+    (i : Fin n) (hSide : (H i).upper = false) :
+    Monotone (perturbNefHeight E H i) := by
+  intro δ ε hδε
+  apply booleanSet_mono hE
+  intro j p hp
+  by_cases hji : j = i
+  · subst j
+    simp only at hp ⊢
+    change p ∈ normalHalfPlane (H i).angle ((H i).height + δ) (H i).upper
+      (H i).strict at hp
+    change p ∈ normalHalfPlane (H i).angle ((H i).height + ε) (H i).upper
+      (H i).strict
+    cases hstrict : (H i).strict <;>
+      simp only [normalHalfPlane, hSide, hstrict, Bool.false_eq_true, ite_false,
+        ite_true, Set.mem_ofPred_eq] at hp ⊢
+    · exact hp.trans (add_le_add_right hδε _)
+    · exact hp.trans_le (add_le_add_right hδε _)
+  · simpa only [ite_eq_right hji] using hp
+
+lemma measurableSet_perturbNefHeight {n : ℕ} (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n) (δ : ℝ) :
+    MeasurableSet (perturbNefHeight E H i δ) := by
+  unfold perturbNefHeight
+  apply measurableSet_booleanSet
+
+lemma area_perturb_sub_eq_volume_sdiff {n : ℕ} {E : BooleanFunction n}
+    (hE : IsMonotoneBooleanFunction E) (H : Fin n → PlanarHalfPlaneData)
+    (i : Fin n) (hSide : (H i).upper = false) (R ε₀ δ ε : ℝ)
+    (hεδ : ε ≤ δ) (hδ : |δ| ≤ ε₀) (hε : |ε| ≤ ε₀)
+    (hBound : ∀ z : ℝ, |z| ≤ ε₀ →
+      perturbNefHeight E H i z ⊆ Metric.closedBall 0 R) :
+    ClassicalResults.area (perturbNefHeight E H i δ) -
+        ClassicalResults.area (perturbNefHeight E H i ε) =
+      (MeasureTheory.volume
+        (perturbNefHeight E H i δ \ perturbNefHeight E H i ε)).toReal := by
+  have hmono := perturbNefHeight_mono hE H i hSide hεδ
+  have hfiniteδ : MeasureTheory.volume (perturbNefHeight E H i δ) ≠ ⊤ := by
+    apply ne_of_lt
+    exact lt_of_le_of_lt (MeasureTheory.measure_mono (hBound δ hδ))
+      MeasureTheory.measure_closedBall_lt_top
+  have hfiniteε : MeasureTheory.volume (perturbNefHeight E H i ε) ≠ ⊤ := by
+    apply ne_of_lt
+    exact lt_of_le_of_lt (MeasureTheory.measure_mono (hBound ε hε))
+      MeasureTheory.measure_closedBall_lt_top
+  rw [ClassicalResults.area, ClassicalResults.area,
+    ← ENNReal.toReal_sub_of_le (MeasureTheory.measure_mono hmono) hfiniteδ,
+    ← MeasureTheory.measure_sdiff hmono
+      (measurableSet_perturbNefHeight E H i ε).nullMeasurableSet hfiniteε]
+
+lemma perturbNefHeight_zero {n : ℕ} (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n) :
+    perturbNefHeight E H i 0 = booleanSet E (fun j ↦ (H j).carrier) := by
+  ext p
+  simp [perturbNefHeight, PlanarHalfPlaneData.carrier]
+
+end MovingSofa.Nef
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Perturbation Bounds
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The polygonal cap with independently specified upper and lower wall heights. -/
+def independentWallCap {Θ : AngleSet} (upper lower : PolygonHeightSpace Θ) : Set Point :=
+  (⋂ t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+    normalHalfPlane (t : Real.Angle) (polygonHeightValue upper t) false false ∩
+    normalHalfPlane (t : Real.Angle) (polygonHeightValue lower t) true false) ∩
+  ⋂ t ∈ (Θ.directions : Set ℝ) ∪ ((fun t : ℝ ↦ t + Real.pi / 2) '' Θ.directions),
+    normalHalfPlane (t : Real.Angle) (polygonHeightValue upper t) false false
+
+/-- The polygonal niche determined by independently specified lower wall heights. -/
+def independentWallNiche {Θ : AngleSet} (lower : PolygonHeightSpace Θ) : Set Point :=
+  (⋂ t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+    normalHalfPlane (t : Real.Angle) (polygonHeightValue lower t) true false) ∩
+  ⋃ t ∈ Θ.directions,
+    normalHalfPlane (t : Real.Angle) (polygonHeightValue lower t) false true ∩
+    normalHalfPlane ((t + Real.pi / 2 : ℝ) : Real.Angle)
+      (polygonHeightValue lower (t + Real.pi / 2)) false true
+
+theorem polygonPerturbation_uniform_bounds (Θ : AngleSet)
+    (h : PolygonHeightSpace Θ) :
+    ∃ R ε₀ : ℝ, 0 < R ∧ 0 < ε₀ ∧
+      ∀ upper lower : PolygonHeightSpace Θ,
+        (∀ t, |upper t - h t| ≤ ε₀) →
+        (∀ t, |lower t - (h t - 1)| ≤ ε₀) →
+        independentWallCap upper lower ⊆ Metric.closedBall 0 R ∧
+        independentWallNiche lower ⊆ Metric.closedBall 0 R := by
+  let a := polygonHeightValue h (Real.pi / 2) - 2
+  let Q (t : ℝ) : Set Point := {p | a ≤ p 1 ∧
+    inner ℝ p (normalVector (t : Real.Angle)) < polygonHeightValue h t + 2 ∧
+    inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) <
+      polygonHeightValue h (t + Real.pi / 2) + 2}
+  have hQ : Bornology.IsBounded (⋃ t ∈ Θ.directions, Q t) := by
+    apply (Bornology.isBounded_biUnion_finset Θ.directions).2
+    intro t ht
+    exact isBounded_setOf_le_snd_and_inner_lt _ _ _ t
+      ⟨(Θ.interior t ht).1, (Θ.interior t ht).2.trans_le Θ.angle_le⟩
+  obtain ⟨R, hR, hbound⟩ := hQ.exists_pos_norm_le
+  refine ⟨R, 1, hR, zero_lt_one, ?_⟩
+  intro upper lower hu hl
+  have huv (t : ℝ) (ht : t ∈ angleDomain Θ) :
+      polygonHeightValue upper t ≤ polygonHeightValue h t + 1 := by
+    have hh := (abs_le.mp (hu ⟨t, ht⟩)).2
+    simp only [polygonHeightValue, dite_eq_left ht]
+    linarith
+  have hlv (t : ℝ) (ht : t ∈ angleDomain Θ) :
+      polygonHeightValue h t - 2 ≤ polygonHeightValue lower t ∧
+      polygonHeightValue lower t ≤ polygonHeightValue h t := by
+    have hh := abs_le.mp (hl ⟨t, ht⟩)
+    simp only [polygonHeightValue, dite_eq_left ht]
+    constructor <;> linarith [hh.1, hh.2]
+  have hy {p : Point}
+      (hp : p ∈ ⋂ t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+        normalHalfPlane (t : Real.Angle) (polygonHeightValue lower t) true false) :
+      a ≤ p 1 := by
+    have hp' := Set.mem_iInter₂.mp hp (Real.pi / 2) (by simp)
+    have hb := (hlv (Real.pi / 2) (Or.inr (by simp))).1
+    have hp'' : polygonHeightValue lower (Real.pi / 2) ≤ p 1 := by
+      simpa [normalHalfPlane, normalVector, frame, PiLp.inner_apply] using hp'
+    exact hb.trans hp''
+  have hball {p : Point} (hp : p ∈ ⋃ t ∈ Θ.directions, Q t) :
+      p ∈ Metric.closedBall 0 R := by
+    simpa only [Metric.mem_closedBall, dist_zero_right] using hbound p hp
+  constructor
+  · intro p hp
+    apply hball
+    obtain ⟨t, ht⟩ := Θ.nonempty
+    apply Set.mem_iUnion₂.mpr
+    refine ⟨t, ht, ?_, ?_, ?_⟩
+    · apply hy
+      exact Set.mem_iInter₂.mpr fun s hs ↦ (Set.mem_iInter₂.mp hp.1 s hs).2
+    · have hb := Set.mem_iInter₂.mp hp.2 t (Or.inl ht)
+      change inner ℝ p (normalVector (t : Real.Angle)) ≤ polygonHeightValue upper t at hb
+      have hu' := huv t (Or.inl (Or.inl ht))
+      linarith
+    · have hb := Set.mem_iInter₂.mp hp.2 (t + Real.pi / 2) (Or.inr ⟨t, ht, rfl⟩)
+      change inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) ≤
+        polygonHeightValue upper (t + Real.pi / 2) at hb
+      have hu' := huv (t + Real.pi / 2) (Or.inl (Or.inr ⟨t, ht, rfl⟩))
+      linarith
+  · rintro p ⟨hpF, hpN⟩
+    apply hball
+    obtain ⟨t, ht, hp⟩ := Set.mem_iUnion₂.mp hpN
+    apply Set.mem_iUnion₂.mpr
+    refine ⟨t, ht, hy hpF, ?_, ?_⟩
+    · have hb := (hlv t (Or.inl (Or.inl ht))).2
+      have hp₁ : inner ℝ p (normalVector (t : Real.Angle)) <
+          polygonHeightValue lower t := hp.1
+      exact hp₁.trans_le (by linarith)
+    · have hb := (hlv (t + Real.pi / 2) (Or.inl (Or.inr ⟨t, ht, rfl⟩))).2
+      exact hp.2.trans_le (by linarith)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Polyline / Basic
+-/
+
+@[expose] public section
+
+namespace MovingSofa
+
+/-- A finite vertex sequence whose horizontal coordinates strictly increase. -/
+structure XMonotonePolylineData where
+  /-- The number of consecutive line segments in the polyline. -/
+  edges : ℕ
+  /-- The ordered sequence of polyline vertices. -/
+  vertices : Fin (edges + 1) → Point
+  increasing : StrictMono (fun i ↦ vertices i 0)
+
+/-- The union of the segments between consecutive vertices. -/
+def XMonotonePolylineData.carrier (p : XMonotonePolylineData) : Set Point :=
+  ⋃ i : Fin p.edges, segment ℝ (p.vertices i.castSucc) (p.vertices i.succ)
+
+/-- The set is the carrier of a polyline with strictly increasing horizontal coordinates. -/
+def IsXMonotonePolyline (S : Set Point) : Prop :=
+  ∃ p : XMonotonePolylineData, S = p.carrier
+
+end MovingSofa
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Polyline / Displacement
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The horizontal displacement of a rightward edge is its length times the sine of its normal. -/
+lemma dist_mul_sin_eq_fst_sub_of_inner_sub_eq_zero {a b : Point} {t : ℝ}
+    (ht : t ∈ Set.Ioo 0 Real.pi) (hab : a 0 < b 0)
+    (horth : inner ℝ (b - a) (normalVector (t : Real.Angle)) = 0) :
+    dist a b * Real.sin t = b 0 - a 0 := by
+  let r := inner ℝ (b - a) (tangentVector (t : Real.Angle))
+  have hvec : r • tangentVector (t : Real.Angle) = b - a := by
+    simpa only [horth, zero_smul, zero_add] using
+      inner_normalVector_smul_add_inner_tangentVector_smul (b - a) (t : Real.Angle)
+  have hcoord : -(r * Real.sin t) = b 0 - a 0 := by
+    have h := congrArg (fun p : Point ↦ p 0) hvec
+    simpa [tangentVector, frame] using h
+  have hsin := Real.sin_pos_of_pos_of_lt_pi ht.1 ht.2
+  have hr : r < 0 := by nlinarith
+  have hn : ‖tangentVector (t : Real.Angle)‖ = 1 := by
+    rw [← sq_eq_sq₀ (norm_nonneg _) (by norm_num : (0 : ℝ) ≤ 1)]
+    rw [EuclideanSpace.norm_sq_eq]
+    simp [tangentVector, frame, Fin.sum_univ_two, Real.sin_sq_add_cos_sq]
+  have hdist : dist a b = -r := by
+    rw [dist_comm, dist_eq_norm, ← hvec, norm_smul, hn, mul_one,
+      Real.norm_eq_abs, abs_of_neg hr]
+  rw [hdist]
+  linarith
+
+/-- The sine-weighted edge lengths telescope to the horizontal endpoint displacement. -/
+lemma XMonotonePolylineData.sum_dist_mul_sin
+    (p : XMonotonePolylineData) (t : Fin p.edges → ℝ)
+    (ht : ∀ i, t i ∈ Set.Ioo 0 Real.pi)
+    (horth : ∀ i, inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+      (normalVector (t i : Real.Angle)) = 0) :
+    ∑ i : Fin p.edges,
+      dist (p.vertices i.castSucc) (p.vertices i.succ) * Real.sin (t i) =
+        p.vertices (Fin.last p.edges) 0 - p.vertices 0 0 := by
+  have hedge (i : Fin p.edges) := dist_mul_sin_eq_fst_sub_of_inner_sub_eq_zero
+    (ht i) (p.increasing i.castSucc_lt_succ) (horth i)
+  simp_rw [hedge]
+  rw [Finset.sum_sub_distrib]
+  have hfirst := Fin.sum_univ_succ (fun i : Fin (p.edges + 1) ↦ p.vertices i 0)
+  have hlast := Fin.sum_univ_castSucc (fun i : Fin (p.edges + 1) ↦ p.vertices i 0)
+  linarith
+
+/-- A nonvertical segment has at most one normal angle strictly between zero and pi. -/
+lemma eq_of_inner_sub_normalVector_eq_zero {a b : Point} {s t : ℝ}
+    (hs : s ∈ Set.Ioo 0 Real.pi) (ht : t ∈ Set.Ioo 0 Real.pi)
+    (hab : a 0 < b 0)
+    (horths : inner ℝ (b - a) (normalVector (s : Real.Angle)) = 0)
+    (hortht : inner ℝ (b - a) (normalVector (t : Real.Angle)) = 0) : s = t := by
+  have hs' : (b 0 - a 0) * Real.cos s + (b 1 - a 1) * Real.sin s = 0 := by
+    simpa [normalVector, frame, PiLp.inner_apply, mul_comm] using horths
+  have ht' : (b 0 - a 0) * Real.cos t + (b 1 - a 1) * Real.sin t = 0 := by
+    simpa [normalVector, frame, PiLp.inner_apply, mul_comm] using hortht
+  have hprod : (b 0 - a 0) * Real.sin (s - t) = 0 := by
+    rw [Real.sin_sub]
+    linear_combination Real.sin s * ht' - Real.sin t * hs'
+  have hz : Real.sin (s - t) = 0 :=
+    (mul_eq_zero.mp hprod).resolve_left (sub_pos.mpr hab).ne'
+  have := (Real.sin_eq_zero_iff_of_lt_of_lt
+    (by linarith [hs.1, ht.2]) (by linarith [hs.2, ht.1])).mp hz
+  linarith
+
+/-- Grouping edge lengths by normal preserves the horizontal displacement identity. -/
+lemma XMonotonePolylineData.sum_normal_lengths_mul_sin
+    (p : XMonotonePolylineData) (D : Finset ℝ)
+    (hD : ∀ t ∈ D, t ∈ Set.Ioo 0 Real.pi)
+    (hlabels : ∀ i : Fin p.edges, ∃ t ∈ D,
+      inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+        (normalVector (t : Real.Angle)) = 0) :
+    ∑ t ∈ D, (∑ i : Fin p.edges,
+      if inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+        (normalVector (t : Real.Angle)) = 0 then
+        dist (p.vertices i.castSucc) (p.vertices i.succ) else 0) * Real.sin t =
+      p.vertices (Fin.last p.edges) 0 - p.vertices 0 0 := by
+  classical
+  choose t ht horth using hlabels
+  calc
+    _ = ∑ i : Fin p.edges, ∑ u ∈ D,
+        (if inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+          (normalVector (u : Real.Angle)) = 0 then
+          dist (p.vertices i.castSucc) (p.vertices i.succ) else 0) * Real.sin u := by
+      simp_rw [Finset.sum_mul]
+      rw [Finset.sum_comm]
+    _ = ∑ i : Fin p.edges,
+        dist (p.vertices i.castSucc) (p.vertices i.succ) * Real.sin (t i) := by
+      apply Finset.sum_congr rfl
+      intro i _
+      rw [Finset.sum_eq_single (t i)]
+      · rw [ite_eq_left (horth i)]
+      · intro u hu hut
+        have hne : inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+            (normalVector (u : Real.Angle)) ≠ 0 := by
+          intro hzero
+          exact hut (eq_of_inner_sub_normalVector_eq_zero (hD u hu) (hD _ (ht i))
+            (p.increasing i.castSucc_lt_succ) hzero (horth i))
+        simp [hne]
+      · exact fun hnot ↦ (hnot (ht i)).elim
+    _ = _ := p.sum_dist_mul_sin t (fun i ↦ hD _ (ht i)) horth
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Polyline / Graph
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The point on the graph of a real-valued function above a given abscissa. -/
+def pointOnGraph (f : ℝ → ℝ) (x : ℝ) : Point := !₂[x, f x]
+
+/-- Evaluation of the affine function with slope-intercept pair `c`. -/
+def affineValue (c : ℝ × ℝ) (x : ℝ) : ℝ := c.1 * x + c.2
+
+/-- An affine real-valued function is continuous. -/
+theorem continuous_affineValue (c : ℝ × ℝ) : Continuous (affineValue c) := by
+  unfold affineValue
+  fun_prop
+
+private def affineCrossings (L : Finset (ℝ × ℝ)) : Set ℝ :=
+  ⋃ c ∈ L, ⋃ d ∈ L.erase c, {x | affineValue c x = affineValue d x}
+
+private theorem finite_affineValue_eq_of_ne {c d : ℝ × ℝ} (hcd : c ≠ d) :
+    {x | affineValue c x = affineValue d x}.Finite := by
+  apply Set.Subsingleton.finite
+  intro x hx y hy
+  simp only [Set.mem_ofPred_eq, affineValue] at hx hy
+  by_cases hm : c.1 = d.1
+  · rw [hm] at hx
+    have hb : c.2 = d.2 := by linarith
+    exact (hcd (Prod.ext hm hb)).elim
+  · have hs : c.1 - d.1 ≠ 0 := sub_ne_zero.mpr hm
+    have hprod : (c.1 - d.1) * (x - y) = 0 := by nlinarith
+    exact sub_eq_zero.mp ((mul_eq_zero.mp hprod).resolve_left hs)
+
+private theorem finite_affineCrossings (L : Finset (ℝ × ℝ)) :
+    (affineCrossings L).Finite := by
+  apply L.finite_toSet.biUnion
+  intro c hc
+  apply (L.erase c).finite_toSet.biUnion
+  intro d hd
+  exact finite_affineValue_eq_of_ne (Finset.ne_of_mem_erase hd).symm
+
+@[simp] theorem pointOnGraph_apply_zero (f : ℝ → ℝ) (x : ℝ) :
+    pointOnGraph f x 0 = x := by
+  rfl
+
+@[simp] theorem pointOnGraph_apply_one (f : ℝ → ℝ) (x : ℝ) :
+    pointOnGraph f x 1 = f x := by
+  rfl
+
+/-- The closed vertical epigraph of a real-valued function. -/
+def verticalEpigraph (f : ℝ → ℝ) : Set Point :=
+  {p | f (p 0) ≤ p 1}
+
+/-- A continuous function has a closed vertical epigraph. -/
+theorem isClosed_verticalEpigraph {f : ℝ → ℝ} (hf : Continuous f) :
+    IsClosed (verticalEpigraph f) := by
+  exact isClosed_le
+    (hf.comp (PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 0))
+    (PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 1)
+
+private theorem pointOnGraph_not_mem_interior_verticalEpigraph
+    (f : ℝ → ℝ) (x : ℝ) :
+    pointOnGraph f x ∉ interior (verticalEpigraph f) := by
+  intro hx
+  let q : ℕ → Point := fun n ↦ !₂[x, f x - (1 : ℝ) / (n + 1)]
+  have hq : Filter.Tendsto q Filter.atTop (nhds (pointOnGraph f x)) := by
+    apply (PiLp.homeomorph 2 (fun _ : Fin 2 ↦ ℝ)).isInducing.tendsto_nhds_iff.mpr
+    apply tendsto_pi_nhds.mpr
+    intro i
+    fin_cases i
+    · change Filter.Tendsto (fun _ : ℕ ↦ x) Filter.atTop (nhds x)
+      exact tendsto_const_nhds
+    · change Filter.Tendsto (fun n : ℕ ↦ f x - (1 : ℝ) / (n + 1))
+        Filter.atTop (nhds (f x))
+      simpa using (tendsto_const_nhds.sub
+        (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)))
+  have hev : ∀ᶠ n in Filter.atTop, q n ∈ interior (verticalEpigraph f) :=
+    hq (isOpen_interior.mem_nhds hx)
+  obtain ⟨n, hn⟩ := hev.exists
+  have hn' := interior_subset hn
+  change f x ≤ f x - (1 : ℝ) / (n + 1) at hn'
+  have : 0 < (1 : ℝ) / (n + 1) := by positivity
+  linarith
+
+/-- The frontier of a continuous vertical epigraph is its graph. -/
+theorem frontier_verticalEpigraph {f : ℝ → ℝ} (hf : Continuous f) :
+    frontier (verticalEpigraph f) = Set.range (pointOnGraph f) := by
+  have hclosed := isClosed_verticalEpigraph hf
+  ext p
+  constructor
+  · intro hp
+    have hpE : p ∈ verticalEpigraph f := by
+      exact hclosed.closure_eq ▸ frontier_subset_closure hp
+    have hpNotInt : p ∉ interior (verticalEpigraph f) :=
+      (mem_frontier_iff_notMem_interior hpE).mp hp
+    have heq : f (p 0) = p 1 := by
+      apply le_antisymm hpE
+      apply le_of_not_gt
+      intro hlt
+      apply hpNotInt
+      have hopen : IsOpen {q : Point | f (q 0) < q 1} :=
+        isOpen_lt
+          (hf.comp (PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 0))
+          (PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 1)
+      exact interior_maximal (fun q hq ↦ by
+        change f (q 0) ≤ q 1
+        exact hq.le) hopen hlt
+    refine ⟨p 0, ?_⟩
+    ext i
+    fin_cases i
+    · simp [pointOnGraph]
+    · simpa [pointOnGraph] using heq
+  · rintro ⟨x, rfl⟩
+    apply (mem_frontier_iff_notMem_interior (s := verticalEpigraph f)
+      (x := pointOnGraph f x) (by
+        change f x ≤ f x
+        exact le_rfl)).mpr
+    exact pointOnGraph_not_mem_interior_verticalEpigraph f x
+
+private def graphPolyline (n : ℕ) (x : Fin (n + 1) → ℝ)
+    (hx : StrictMono x) (f : ℝ → ℝ) : XMonotonePolylineData where
+  edges := n
+  vertices i := pointOnGraph f (x i)
+  increasing := by
+    simpa only [pointOnGraph_apply_zero] using hx
+
+@[simp] private theorem graphPolyline_vertices (n : ℕ) (x : Fin (n + 1) → ℝ)
+    (hx : StrictMono x) (f : ℝ → ℝ) (i : Fin (n + 1)) :
+    (graphPolyline n x hx f).vertices i = pointOnGraph f (x i) := by
+  rfl
+
+private theorem segment_pointOnGraph_affine {f : ℝ → ℝ} {m c a b : ℝ}
+    (hab : a ≤ b) (hf : ∀ z ∈ Set.Icc a b, f z = m * z + c) :
+    segment ℝ (pointOnGraph f a) (pointOnGraph f b) =
+      pointOnGraph f '' Set.Icc a b := by
+  ext p
+  constructor
+  · rintro ⟨u, v, hu, hv, huv, rfl⟩
+    let z := u * a + v * b
+    have hz : z ∈ Set.Icc a b := by
+      constructor
+      · calc
+          a = (u + v) * a := by rw [huv]; ring
+          _ = u * a + v * a := by ring
+          _ ≤ u * a + v * b := add_le_add (le_refl _) (mul_le_mul_of_nonneg_left hab hv)
+      · calc
+          u * a + v * b ≤ u * b + v * b :=
+            add_le_add (mul_le_mul_of_nonneg_left hab hu) (le_refl _)
+          _ = (u + v) * b := by ring
+          _ = b := by rw [huv]; ring
+    refine ⟨z, hz, ?_⟩
+    have hfa := hf a ⟨le_rfl, hab⟩
+    have hfb := hf b ⟨hab, le_rfl⟩
+    have hfz := hf z hz
+    ext i
+    fin_cases i
+    · simp [pointOnGraph, z]
+    · simp only [pointOnGraph, hfz, Fin.mk_one, Fin.isValue, Matrix.cons_val_one,
+      Matrix.cons_val_fin_one, hfa, hfb, PiLp.add_apply, PiLp.smul_apply, smul_eq_mul, z]
+      calc
+        m * (u * a + v * b) + c =
+            u * (m * a) + v * (m * b) + (u + v) * c := by rw [huv]; ring
+        _ = u * (m * a + c) + v * (m * b + c) := by ring
+  · rintro ⟨z, hz, rfl⟩
+    by_cases hab' : a = b
+    · subst b
+      have : z = a := by exact le_antisymm hz.2 hz.1
+      subst z
+      simp
+    · have hlt : a < b := lt_of_le_of_ne hab hab'
+      let u := (b - z) / (b - a)
+      let v := (z - a) / (b - a)
+      have hden : 0 < b - a := sub_pos.mpr hlt
+      have hu : 0 ≤ u := div_nonneg (sub_nonneg.mpr hz.2) hden.le
+      have hv : 0 ≤ v := div_nonneg (sub_nonneg.mpr hz.1) hden.le
+      have huv : u + v = 1 := by
+        dsimp [u, v]
+        field_simp
+        ring
+      refine ⟨u, v, hu, hv, huv, ?_⟩
+      have hfa := hf a ⟨le_rfl, hab⟩
+      have hfb := hf b ⟨hab, le_rfl⟩
+      have hfz := hf z hz
+      ext i
+      fin_cases i
+      · simp [pointOnGraph]
+        dsimp [u, v]
+        field_simp
+        ring
+      · simp [pointOnGraph, hfz, hfa, hfb]
+        dsimp [u, v]
+        field_simp
+        ring
+
+/-- A graph segment of slope `m` is orthogonal to each normal annihilating `(1,m)`. -/
+theorem inner_pointOnGraph_sub_normalVector_eq_zero {f : ℝ → ℝ}
+    {m c a b t : ℝ}
+    (hf : Set.EqOn f (fun x ↦ m * x + c) (Set.Icc a b))
+    (hab : a ≤ b) (horth : Real.cos t + m * Real.sin t = 0) :
+    inner ℝ (pointOnGraph f b - pointOnGraph f a)
+      (normalVector (t : Real.Angle)) = 0 := by
+  have hfa := hf ⟨le_rfl, hab⟩
+  have hfb := hf ⟨hab, le_rfl⟩
+  simp [pointOnGraph, normalVector, frame, PiLp.inner_apply, hfa, hfb]
+  nlinarith
+
+private theorem graphPolyline_carrier_eq_image_Icc {n : ℕ} {x : Fin (n + 1) → ℝ}
+    (hx : StrictMono x) {f : ℝ → ℝ}
+    (hpiece : ∀ i : Fin n, ∃ m c : ℝ, ∀ z ∈ Set.Icc (x i.castSucc) (x i.succ),
+      f z = m * z + c)
+    (hcover : Set.Icc (x 0) (x (Fin.last n)) =
+      ⋃ i : Fin n, Set.Icc (x i.castSucc) (x i.succ)) :
+    (graphPolyline n x hx f).carrier =
+      pointOnGraph f '' Set.Icc (x 0) (x (Fin.last n)) := by
+  rw [hcover, Set.image_iUnion]
+  apply Set.iUnion_congr
+  intro i
+  obtain ⟨m, c, hi⟩ := hpiece i
+  exact segment_pointOnGraph_affine (hx i.castSucc_lt_succ).le hi
+
+private theorem iUnion_Icc_fin {n : ℕ} (hn : 0 < n) {x : Fin (n + 1) → ℝ}
+    (hx : StrictMono x) :
+    Set.Icc (x 0) (x (Fin.last n)) =
+      ⋃ i : Fin n, Set.Icc (x i.castSucc) (x i.succ) := by
+  induction n with
+  | zero => simp at hn
+  | succ n ih =>
+      by_cases hn0 : n = 0
+      · subst n
+        ext z
+        constructor
+        · intro hz
+          exact Set.mem_iUnion.mpr ⟨0, by simpa using hz⟩
+        · intro hz
+          obtain ⟨i, hi⟩ := Set.mem_iUnion.mp hz
+          fin_cases i
+          simpa using hi
+      · have hnpos : 0 < n := Nat.pos_of_ne_zero hn0
+        let y : Fin (n + 1) → ℝ := fun i ↦ x i.castSucc
+        have hy : StrictMono y := hx.comp Fin.strictMono_castSucc
+        have hprefix := ih hnpos hy
+        have hprefix' : Set.Icc (x 0) (x (Fin.last n).castSucc) =
+            ⋃ i : Fin n, Set.Icc (x i.castSucc.castSucc) (x i.succ.castSucc) := by
+          simpa [y] using hprefix
+        have hprefix'' : Set.Icc (x 0) (x (Fin.last n).castSucc) =
+            Set.iUnion ((fun i : Fin (n + 1) ↦
+              Set.Icc (x i.castSucc) (x i.succ)) ∘ Fin.castSucc) := by
+          rw [hprefix']
+          apply Set.iUnion_congr
+          intro i
+          congr 2
+        rw [Set.iUnion_fin_add_one_eq_iUnion_castSucc]
+        rw [← hprefix'']
+        exact (Set.Icc_union_Icc_eq_Icc
+          (hx.monotone (Fin.zero_le _))
+          (hx (Fin.last n).castSucc_lt_succ).le).symm
+
+private theorem exists_affineValue_eqOn_of_finite_selector {I : Set ℝ}
+    (hI : IsPreconnected I) (L : Finset (ℝ × ℝ)) {f : ℝ → ℝ}
+    (hf : ContinuousOn f I)
+    (hsel : ∀ x ∈ I, ∃ c ∈ L, f x = affineValue c x)
+    (hsep : ∀ c ∈ L, ∀ d ∈ L, c ≠ d →
+      ∀ x ∈ I, affineValue c x ≠ affineValue d x)
+    (hIne : I.Nonempty) :
+    ∃ c ∈ L, Set.EqOn f (affineValue c) I := by
+  classical
+  let _ : PreconnectedSpace I := Subtype.preconnectedSpace hI
+  obtain ⟨x₀, hx₀⟩ := hIne
+  obtain ⟨c₀, hc₀L, hc₀⟩ := hsel x₀ hx₀
+  let A : Set I := {x | f x = affineValue c₀ x}
+  have hAclosed : IsClosed A := by
+    apply isClosed_eq
+    · change Continuous (I.domRestrict f)
+      exact continuousOn_iff_continuous_domRestrict.mp hf
+    · exact (continuous_affineValue c₀).comp continuous_subtype_val
+  have hAc : Aᶜ = ⋃ d ∈ L.erase c₀,
+      {x : I | f x = affineValue d x} := by
+    ext x
+    constructor
+    · intro hx
+      have hxne : f x ≠ affineValue c₀ x := by simpa [A] using hx
+      obtain ⟨d, hdL, hd⟩ := hsel x x.2
+      have hdc : d ≠ c₀ := by
+        intro h
+        subst d
+        exact hxne hd
+      exact Set.mem_iUnion₂.mpr ⟨d, Finset.mem_erase.mpr ⟨hdc, hdL⟩, hd⟩
+    · intro hx
+      obtain ⟨d, hdL, hd⟩ := Set.mem_iUnion₂.mp hx
+      have hdc := (Finset.mem_erase.mp hdL).1
+      intro hxc
+      exact hsep d (Finset.mem_of_mem_erase hdL) c₀ hc₀L hdc x x.2 (hd.symm.trans hxc)
+  have hAcclosed : IsClosed Aᶜ := by
+    rw [hAc]
+    apply isClosed_biUnion_finset
+    intro d hd
+    apply isClosed_eq
+    · change Continuous (I.domRestrict f)
+      exact continuousOn_iff_continuous_domRestrict.mp hf
+    · exact (continuous_affineValue d).comp continuous_subtype_val
+  have hAclopen : IsClopen A := ⟨hAclosed, isClosed_compl_iff.mp hAcclosed⟩
+  have hAnonempty : A.Nonempty := ⟨⟨x₀, hx₀⟩, hc₀⟩
+  have hAuniv : A = Set.univ := hAclopen.eq_univ hAnonempty
+  refine ⟨c₀, hc₀L, ?_⟩
+  intro x hx
+  have : (⟨x, hx⟩ : I) ∈ A := hAuniv.symm ▸ Set.mem_univ _
+  exact this
+
+private theorem exists_affineValue_eqOn_Icc_of_avoids_crossings {a b : ℝ}
+    (hab : a < b) (L : Finset (ℝ × ℝ)) {f : ℝ → ℝ}
+    (hf : ContinuousOn f (Set.Icc a b))
+    (hsel : ∀ x ∈ Set.Icc a b, ∃ c ∈ L, f x = affineValue c x)
+    (hcross : Set.Ioo a b ∩ affineCrossings L = ∅) :
+    ∃ c ∈ L, Set.EqOn f (affineValue c) (Set.Icc a b) := by
+  classical
+  have hf' : ContinuousOn f (Set.Ioo a b) :=
+    hf.mono (Set.Ioo_subset_Icc_self)
+  have hsel' : ∀ x ∈ Set.Ioo a b, ∃ c ∈ L, f x = affineValue c x :=
+    fun x hx ↦ hsel x (Set.Ioo_subset_Icc_self hx)
+  have hsep : ∀ c ∈ L, ∀ d ∈ L, c ≠ d →
+      ∀ x ∈ Set.Ioo a b, affineValue c x ≠ affineValue d x := by
+    intro c hc d hd hcd x hx hEq
+    have hxCross : x ∈ affineCrossings L := by
+      exact Set.mem_iUnion₂.mpr ⟨c, hc, Set.mem_iUnion₂.mpr
+        ⟨d, Finset.mem_erase.mpr ⟨hcd.symm, hd⟩, hEq⟩⟩
+    have : x ∈ Set.Ioo a b ∩ affineCrossings L := ⟨hx, hxCross⟩
+    simp [hcross] at this
+  obtain ⟨c, hcL, hc⟩ := exists_affineValue_eqOn_of_finite_selector
+    isPreconnected_Ioo L hf' hsel' hsep (Set.nonempty_Ioo.mpr hab)
+  refine ⟨c, hcL, ?_⟩
+  exact hc.of_subset_closure hf (continuous_affineValue c).continuousOn
+    Set.Ioo_subset_Icc_self (by simp [closure_Ioo hab.ne])
+
+/-- A continuous finite selector of affine functions on a compact interval is a polyline. -/
+theorem exists_graphPolyline_of_finite_affine_selector {a b : ℝ}
+    (hab : a < b) (L : Finset (ℝ × ℝ)) {f : ℝ → ℝ}
+    (hf : ContinuousOn f (Set.Icc a b))
+    (hsel : ∀ x ∈ Set.Icc a b, ∃ c ∈ L, f x = affineValue c x) :
+    ∃ p : XMonotonePolylineData,
+      0 < p.edges ∧
+      p.vertices 0 = pointOnGraph f a ∧
+      p.vertices (Fin.last p.edges) = pointOnGraph f b ∧
+      p.carrier = pointOnGraph f '' Set.Icc a b ∧
+      (∀ i : Fin (p.edges + 1),
+        p.vertices i = pointOnGraph f (p.vertices i 0)) ∧
+      ∀ i : Fin p.edges, ∃ c ∈ L,
+        Set.EqOn f (affineValue c)
+          (Set.Icc (p.vertices i.castSucc 0) (p.vertices i.succ 0)) := by
+  classical
+  let C := (finite_affineCrossings L).toFinset.filter (· ∈ Set.Icc a b)
+  let B := insert a (insert b C)
+  have haB : a ∈ B := by simp [B]
+  have hbB : b ∈ B := by simp [B]
+  have hBsub : ∀ z ∈ B, z ∈ Set.Icc a b := by
+    intro z hz
+    simp only [B, Finset.mem_insert] at hz
+    rcases hz with rfl | rfl | hz
+    · exact ⟨le_rfl, hab.le⟩
+    · exact ⟨hab.le, le_rfl⟩
+    · exact (Finset.mem_filter.mp hz).2
+  have hcard2 : 2 ≤ B.card := by
+    have hsub : ({a, b} : Finset ℝ) ⊆ B := by
+      intro z hz
+      simp only [Finset.mem_insert, Finset.mem_singleton] at hz
+      rcases hz with rfl | rfl
+      · exact haB
+      · exact hbB
+    have hcard := Finset.card_le_card hsub
+    simpa [hab.ne] using hcard
+  let n := B.card - 1
+  have hnpos : 0 < n := by dsimp [n]; omega
+  have hcard : B.card = n + 1 := by dsimp [n]; omega
+  let x : Fin (n + 1) → ℝ := B.orderEmbOfFin hcard
+  have hx : StrictMono x := (B.orderEmbOfFin hcard).strictMono
+  have hxmem (i : Fin (n + 1)) : x i ∈ B := by
+    exact B.orderEmbOfFin_mem hcard i
+  have hxrange : Set.range x = B := by
+    exact B.range_orderEmbOfFin hcard
+  have hxzero : x 0 = a := by
+    apply le_antisymm
+    · have haRange : a ∈ Set.range x := by simpa [hxrange] using haB
+      obtain ⟨i, hi⟩ := haRange
+      rw [← hi]
+      exact hx.monotone (Fin.zero_le i)
+    · exact (hBsub _ (hxmem 0)).1
+  have hxlast : x (Fin.last n) = b := by
+    apply le_antisymm
+    · exact (hBsub _ (hxmem (Fin.last n))).2
+    · have hbRange : b ∈ Set.range x := by simpa [hxrange] using hbB
+      obtain ⟨i, hi⟩ := hbRange
+      rw [← hi]
+      exact hx.monotone (Fin.le_last i)
+  have hnocross (i : Fin n) :
+      Set.Ioo (x i.castSucc) (x i.succ) ∩ affineCrossings L = ∅ := by
+    ext z
+    constructor
+    · intro hz
+      have hzab : z ∈ Set.Icc a b := by
+        rw [← hxzero, ← hxlast]
+        exact ⟨(hx.monotone (Fin.zero_le i.castSucc)).trans hz.1.1.le,
+          hz.1.2.le.trans (hx.monotone (Fin.le_last i.succ))⟩
+      have hzB : z ∈ B := by
+        apply Finset.mem_insert.mpr
+        right
+        apply Finset.mem_insert.mpr
+        right
+        apply Finset.mem_filter.mpr
+        exact ⟨(finite_affineCrossings L).mem_toFinset.mpr hz.2, hzab⟩
+      have hzRange : z ∈ Set.range x := by simpa [hxrange] using hzB
+      obtain ⟨j, hj⟩ := hzRange
+      rw [← hj] at hz
+      have hij : i.castSucc < j := hx.lt_iff_lt.mp hz.1.1
+      have hji : j < i.succ := hx.lt_iff_lt.mp hz.1.2
+      change i.val < j.val at hij
+      change j.val < i.val + 1 at hji
+      omega
+    · intro hz
+      exact hz.elim
+  have hpiece : ∀ i : Fin n, ∃ c ∈ L,
+      Set.EqOn f (affineValue c) (Set.Icc (x i.castSucc) (x i.succ)) := by
+    intro i
+    exact exists_affineValue_eqOn_Icc_of_avoids_crossings
+      (hx i.castSucc_lt_succ) L
+      (hf.mono (Set.Icc_subset_Icc
+        (by rw [← hxzero]; exact hx.monotone (Fin.zero_le _))
+        (by rw [← hxlast]; exact hx.monotone (Fin.le_last _))))
+      (fun z hz ↦ hsel z (Set.Icc_subset_Icc
+        (by rw [← hxzero]; exact hx.monotone (Fin.zero_le _))
+        (by rw [← hxlast]; exact hx.monotone (Fin.le_last _)) hz))
+      (hnocross i)
+  let p := graphPolyline n x hx f
+  refine ⟨p, hnpos, ?_, ?_, ?_, ?_, ?_⟩
+  · change pointOnGraph f (x 0) = pointOnGraph f a
+    rw [hxzero]
+  · change pointOnGraph f (x (Fin.last n)) = pointOnGraph f b
+    rw [hxlast]
+  · rw [graphPolyline_carrier_eq_image_Icc hx]
+    · simp [hxzero, hxlast]
+    · intro i
+      obtain ⟨c, -, hc⟩ := hpiece i
+      exact ⟨c.1, c.2, hc⟩
+    · exact iUnion_Icc_fin hnpos hx
+  · intro i
+    rfl
+  · intro i
+    change ∃ c ∈ L, Set.EqOn f (affineValue c)
+      (Set.Icc (x i.castSucc) (x i.succ))
+    exact hpiece i
+
+/-- Injective graph parametrizations preserve disjointness of parameter sets. -/
+theorem pointOnGraph_image_disjoint
+    (f : ℝ → ℝ) {s t : Set ℝ} (hst : Disjoint s t) :
+    Disjoint (pointOnGraph f '' s) (pointOnGraph f '' t) := by
+  rw [Set.disjoint_left]
+  rintro q ⟨x, hx, rfl⟩ ⟨y, hy, hxy⟩
+  have heq : x = y := by
+    have := congrArg (fun p : Point ↦ p 0) hxy
+    simpa [pointOnGraph] using this.symm
+  exact Set.disjoint_left.mp hst hx (heq ▸ hy)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Polyline / Measure
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private lemma fst_mem_Icc_of_mem_segment_mbe8a367 {a b q : Point}
+    (hab : a 0 ≤ b 0) (hq : q ∈ segment ℝ a b) : q 0 ∈ Set.Icc (a 0) (b 0) := by
+  rw [segment_eq_image'] at hq
+  obtain ⟨r, hr, rfl⟩ := hq
+  change a 0 + r * (b 0 - a 0) ∈ Set.Icc (a 0) (b 0)
+  constructor <;> nlinarith [hr.1, hr.2]
+
+private lemma eq_right_of_mem_segment_of_fst_eq {a b q : Point}
+    (hab : a 0 < b 0) (hq : q ∈ segment ℝ a b) (heq : q 0 = b 0) : q = b := by
+  rw [segment_eq_image'] at hq
+  obtain ⟨r, hr, rfl⟩ := hq
+  have hcoord : a 0 + r * (b 0 - a 0) = b 0 := heq
+  have hrone : r = 1 := by nlinarith
+  simp [hrone]
+
+/-- Distinct increasing polyline segments meet only at a possible common endpoint. -/
+lemma XMonotonePolylineData.segment_inter_subset_singleton
+    (p : XMonotonePolylineData) {i j : Fin p.edges} (hij : i < j) :
+    segment ℝ (p.vertices i.castSucc) (p.vertices i.succ) ∩
+        segment ℝ (p.vertices j.castSucc) (p.vertices j.succ) ⊆
+      {p.vertices i.succ} := by
+  intro q hq
+  have hi := fst_mem_Icc_of_mem_segment_mbe8a367 (p.increasing i.castSucc_lt_succ).le hq.1
+  have hj := fst_mem_Icc_of_mem_segment_mbe8a367 (p.increasing j.castSucc_lt_succ).le hq.2
+  have hindex : i.succ ≤ j.castSucc := by
+    change i.val + 1 ≤ j.val
+    exact hij
+  have horder := p.increasing.monotone hindex
+  apply Set.mem_singleton_iff.mpr
+  exact eq_right_of_mem_segment_of_fst_eq (p.increasing i.castSucc_lt_succ) hq.1
+    (le_antisymm hi.2 (horder.trans hj.1))
+
+open MeasureTheory
+
+/-- The segments of an increasing polyline are almost disjoint for length measure. -/
+lemma XMonotonePolylineData.pairwise_aedisjoint_segments
+    (p : XMonotonePolylineData) :
+    Pairwise (fun i j : Fin p.edges ↦ AEDisjoint (Measure.hausdorffMeasure 1)
+      (segment ℝ (p.vertices i.castSucc) (p.vertices i.succ))
+      (segment ℝ (p.vertices j.castSucc) (p.vertices j.succ))) := by
+  have := Measure.nullSingletonClass_hausdorff Point (by norm_num : (0 : ℝ) < 1)
+  have hlt (i j : Fin p.edges) (hij : i < j) :
+      AEDisjoint (Measure.hausdorffMeasure 1)
+        (segment ℝ (p.vertices i.castSucc) (p.vertices i.succ))
+        (segment ℝ (p.vertices j.castSucc) (p.vertices j.succ)) := by
+    exact measure_mono_null (p.segment_inter_subset_singleton hij) (measure_singleton _)
+  intro i j hij
+  rcases lt_or_gt_of_ne hij with h | h
+  · exact hlt i j h
+  · exact (hlt j i h).symm
+
+/-- A supporting-line slice has the sum of the lengths of its parallel segments. -/
+lemma XMonotonePolylineData.hausdorffMeasure_carrier_inter_hyperplane
+    (p : XMonotonePolylineData) (n : Point) (c : ℝ)
+    (hparallel : ∀ i : Fin p.edges,
+      inner ℝ (p.vertices i.succ - p.vertices i.castSucc) n = 0 →
+      segment ℝ (p.vertices i.castSucc) (p.vertices i.succ) ⊆
+        {q | inner ℝ q n = c}) :
+    Measure.hausdorffMeasure 1 (p.carrier ∩ {q | inner ℝ q n = c}) =
+      ∑ i : Fin p.edges,
+        if inner ℝ (p.vertices i.succ - p.vertices i.castSucc) n = 0 then
+          ENNReal.ofReal (dist (p.vertices i.castSucc) (p.vertices i.succ)) else 0 := by
+  classical
+  have hd : Pairwise (fun i j : Fin p.edges ↦ AEDisjoint (Measure.hausdorffMeasure 1)
+      (segment ℝ (p.vertices i.castSucc) (p.vertices i.succ) ∩ {q | inner ℝ q n = c})
+      (segment ℝ (p.vertices j.castSucc) (p.vertices j.succ) ∩ {q | inner ℝ q n = c})) := by
+    intro i j hij
+    exact (p.pairwise_aedisjoint_segments hij).mono Set.inter_subset_left Set.inter_subset_left
+  rw [XMonotonePolylineData.carrier, Set.iUnion_inter, measure_iUnion₀ hd, tsum_fintype]
+  · apply Finset.sum_congr rfl
+    intro i _
+    by_cases hi : inner ℝ (p.vertices i.succ - p.vertices i.castSucc) n = 0
+    · rw [ite_eq_left hi, Set.inter_eq_left.mpr (hparallel i hi), hausdorffMeasure_segment,
+        edist_dist]
+    · rw [ite_eq_right hi]
+      exact EuclideanGeometry.hausdorffMeasure_segment_inter_hyperplane_eq_zero hi
+  · intro i
+    have hc : IsCompact (segment ℝ (p.vertices i.castSucc) (p.vertices i.succ)) := by
+      rw [segment_eq_image']
+      exact isCompact_Icc.image (by fun_prop)
+    exact (hc.isClosed.measurableSet.inter
+      (isClosed_eq (by fun_prop) continuous_const).measurableSet).nullMeasurableSet
+
+/-- The real length of a supporting-line slice is the sum of its parallel segment lengths. -/
+lemma XMonotonePolylineData.toReal_hausdorffMeasure_carrier_inter_hyperplane
+    (p : XMonotonePolylineData) (n : Point) (c : ℝ)
+    (hparallel : ∀ i : Fin p.edges,
+      inner ℝ (p.vertices i.succ - p.vertices i.castSucc) n = 0 →
+      segment ℝ (p.vertices i.castSucc) (p.vertices i.succ) ⊆
+        {q | inner ℝ q n = c}) :
+    (Measure.hausdorffMeasure 1 (p.carrier ∩ {q | inner ℝ q n = c})).toReal =
+      ∑ i : Fin p.edges,
+        if inner ℝ (p.vertices i.succ - p.vertices i.castSucc) n = 0 then
+          dist (p.vertices i.castSucc) (p.vertices i.succ) else 0 := by
+  classical
+  rw [p.hausdorffMeasure_carrier_inter_hyperplane n c hparallel,
+    ENNReal.toReal_sum (by intro i _; split_ifs <;> simp)]
+  apply Finset.sum_congr rfl
+  intro i _
+  split_ifs <;> simp [dist_nonneg]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Polyline / Projection
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A leftward edge displacement is its length times its oriented tangent. -/
+theorem sub_eq_dist_smul_tangentVector {a b : Point} {t : ℝ}
+    (ht : t ∈ Set.Ioo 0 Real.pi) (hab : a 0 < b 0)
+    (horth : inner ℝ (b - a) (normalVector (t : Real.Angle)) = 0) :
+    a - b = dist a b • tangentVector (t : Real.Angle) := by
+  let r := inner ℝ (b - a) (tangentVector (t : Real.Angle))
+  have hvec : r • tangentVector (t : Real.Angle) = b - a := by
+    simpa only [horth, zero_smul, zero_add] using
+      inner_normalVector_smul_add_inner_tangentVector_smul (b - a) (t : Real.Angle)
+  have hcoord : -(r * Real.sin t) = b 0 - a 0 := by
+    have h := congrArg (fun p : Point ↦ p 0) hvec
+    simpa [tangentVector, frame] using h
+  have hdist := dist_mul_sin_eq_fst_sub_of_inner_sub_eq_zero ht hab horth
+  have hr : r = -dist a b := by
+    have hs := Real.sin_pos_of_pos_of_lt_pi ht.1 ht.2
+    nlinarith
+  rw [hr, neg_smul] at hvec
+  simpa only [neg_neg, neg_sub] using (congrArg Neg.neg hvec).symm
+
+/-- Every point of a polyline satisfies the bound by positive projected edge increments. -/
+theorem XMonotonePolylineData.inner_le_endpoint_add_sum_pos
+    (p : XMonotonePolylineData) (u : Point) {q : Point} (hq : q ∈ p.carrier) :
+    inner ℝ q u ≤ inner ℝ (p.vertices (Fin.last p.edges)) u +
+      ∑ i : Fin p.edges, max (inner ℝ (p.vertices i.castSucc - p.vertices i.succ) u) 0 := by
+  have hvertex (i : Fin (p.edges + 1)) :=
+    Fin.apply_le_last_add_sum_max_sub (fun i ↦ inner ℝ (p.vertices i) u) i
+  simp only [← inner_sub_left] at hvertex
+  obtain ⟨i, hqi⟩ := Set.mem_iUnion.mp hq
+  obtain ⟨a, b, ha, hb, hab, rfl⟩ := hqi
+  rw [inner_add_left, real_inner_smul_left, real_inner_smul_left]
+  have h₁ := mul_le_mul_of_nonneg_left (hvertex i.castSucc) ha
+  have h₂ := mul_le_mul_of_nonneg_left (hvertex i.succ) hb
+  have h := add_le_add h₁ h₂
+  rwa [← add_mul, hab, one_mul] at h
+
+/-- Grouping edge lengths by their unique normal preserves every weighted sum. -/
+theorem XMonotonePolylineData.sum_normal_lengths_mul
+    (p : XMonotonePolylineData) (D : Finset ℝ) (g : ℝ → ℝ)
+    (hD : ∀ t ∈ D, t ∈ Set.Ioo 0 Real.pi)
+    (t : Fin p.edges → ℝ) (ht : ∀ i, t i ∈ D)
+    (horth : ∀ i, inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+      (normalVector (t i : Real.Angle)) = 0) :
+    ∑ u ∈ D, (∑ i : Fin p.edges,
+      if inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+        (normalVector (u : Real.Angle)) = 0 then
+        dist (p.vertices i.castSucc) (p.vertices i.succ) else 0) * g u =
+      ∑ i : Fin p.edges, dist (p.vertices i.castSucc) (p.vertices i.succ) * g (t i) := by
+  classical
+  simp_rw [Finset.sum_mul]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [Finset.sum_eq_single (t i)]
+  · rw [ite_eq_left (horth i)]
+  · intro u hu hut
+    have hne : inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+        (normalVector (u : Real.Angle)) ≠ 0 := by
+      intro hzero
+      exact hut (eq_of_inner_sub_normalVector_eq_zero (hD u hu) (hD _ (ht i))
+        (p.increasing i.castSucc_lt_succ) hzero (horth i))
+    simp [hne]
+  · exact fun hnot ↦ (hnot (ht i)).elim
+
+/-- Positive normal projections bound every point of a polyline by its right endpoint. -/
+theorem XMonotonePolylineData.inner_le_endpoint_add_sum_normal_lengths
+    (p : XMonotonePolylineData) (D : Finset ℝ)
+    (hD : ∀ t ∈ D, t ∈ Set.Ioo 0 Real.pi)
+    (hlabels : ∀ i : Fin p.edges, ∃ t ∈ D,
+      inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+        (normalVector (t : Real.Angle)) = 0)
+    (s : ℝ) {q : Point} (hq : q ∈ p.carrier) :
+    inner ℝ q (normalVector (s : Real.Angle)) ≤
+      inner ℝ (p.vertices (Fin.last p.edges)) (normalVector (s : Real.Angle)) +
+      ∑ u ∈ D, (∑ i : Fin p.edges,
+        if inner ℝ (p.vertices i.succ - p.vertices i.castSucc)
+          (normalVector (u : Real.Angle)) = 0 then
+          dist (p.vertices i.castSucc) (p.vertices i.succ) else 0) *
+        max (Real.sin (s - u)) 0 := by
+  classical
+  choose t ht horth using hlabels
+  rw [p.sum_normal_lengths_mul D (fun u ↦ max (Real.sin (s - u)) 0) hD t ht horth]
+  have h := p.inner_le_endpoint_add_sum_pos (normalVector (s : Real.Angle)) hq
+  convert h using 2
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [sub_eq_dist_smul_tangentVector (hD _ (ht i))
+    (p.increasing i.castSucc_lt_succ) (horth i), real_inner_smul_left,
+    inner_tangentVector_normalVector_real, mul_max_of_nonneg _ _ dist_nonneg, mul_zero]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/Development/Geometry/Foundations/Development002.lean b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development002.lean
new file mode 100644
index 0000000000..d4a5780132
--- /dev/null
+++ b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development002.lean
@@ -0,0 +1,9903 @@
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+module
+
+public import LeanPool.Isoperimetric.BrunnMinkowski
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development001
+
+
+public import Mathlib.Algebra.Category.ModuleCat.Basic
+public import Mathlib.Analysis.BoundedVariation
+public import Mathlib.Analysis.Calculus.ContDiff.Deriv
+public import Mathlib.Analysis.Calculus.ContDiff.RCLike
+public import Mathlib.Analysis.Calculus.Deriv.Add
+public import Mathlib.Analysis.Calculus.Deriv.Basic
+public import Mathlib.Analysis.Calculus.Deriv.Mul
+public import Mathlib.Analysis.Calculus.Deriv.Polynomial
+public import Mathlib.Analysis.Calculus.Deriv.Prod
+public import Mathlib.Analysis.Calculus.LocalExtr.Basic
+public import Mathlib.Analysis.Calculus.MeanValue
+public import Mathlib.Analysis.Complex.Circle
+public import Mathlib.Analysis.Convex.Measure
+public import Mathlib.Analysis.InnerProductSpace.Dual
+public import Mathlib.Analysis.LocallyConvex.Separation
+public import Mathlib.Analysis.Normed.Lp.MeasurableSpace
+public import Mathlib.Analysis.Normed.Lp.PiLp
+public import Mathlib.Analysis.Normed.Operator.BoundedLinearMaps
+public import Mathlib.Analysis.SpecialFunctions.Complex.Arg
+public import Mathlib.Analysis.SpecialFunctions.Complex.Circle
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
+public import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
+public import Mathlib.MeasureTheory.Function.AbsolutelyContinuous
+public import Mathlib.MeasureTheory.Function.LocallyIntegrable
+public import Mathlib.MeasureTheory.Integral.Bochner.Basic
+public import Mathlib.MeasureTheory.Integral.DominatedConvergence
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
+public import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
+public import Mathlib.MeasureTheory.Measure.Hausdorff
+public import Mathlib.MeasureTheory.Measure.Map
+public import Mathlib.MeasureTheory.VectorMeasure.BoundedVariation
+public import Mathlib.MeasureTheory.VectorMeasure.Decomposition.RadonNikodym
+public import Mathlib.MeasureTheory.VectorMeasure.IntegrationByParts
+public import Mathlib.MeasureTheory.VectorMeasure.SetIntegral
+public import Mathlib.MeasureTheory.VectorMeasure.WithDensityVec
+public import Mathlib.Tactic
+public import Mathlib.Tactic.Linarith
+public import Mathlib.Topology.Connected.Basic
+public import Mathlib.Topology.ContinuousMap.Algebra
+public import Mathlib.Topology.EMetricSpace.BoundedVariation
+public import Mathlib.Topology.Homeomorph.Lemmas
+public import Mathlib.Topology.LocallyConstant.Basic
+public import Mathlib.Topology.Maps.Proper.Basic
+public import Mathlib.Topology.MetricSpace.HausdorffDistance
+public import Mathlib.Topology.MetricSpace.Thickening
+public import Mathlib.Topology.Order.IntermediateValue
+/-!
+# Moving sofa: related mathematical developments
+
+* `Infrastructure.Analysis.Foundations.Development002`.
+* `Infrastructure.Curves.Foundations.Development001`.
+* `Infrastructure.Analysis.Foundations.Development004`.
+* `Infrastructure.Geometry.Foundations.Development003`.
+* `Infrastructure.Analysis.Foundations.Development003`.
+* `Infrastructure.Geometry.Foundations.Development004`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Analysis.Foundations.Development002`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Analysis.BoundedVariation`.
+* `Analysis.MeasureProducts`.
+* `Analysis.Stieltjes.Integral`.
+* `Analysis.Stieltjes.AbsoluteContinuity`.
+* `Analysis.Stieltjes.Calculus`.
+* `Analysis.Stieltjes.DensityIntegration`.
+* `Analysis.Stieltjes.Linearity`.
+* `Analysis.Stieltjes.InnerProduct`.
+* `Analysis.Stieltjes.Smooth`.
+* `Analysis.Stieltjes.Frame`.
+* `Analysis.Stieltjes.Transport`.
+* `Analysis.Stieltjes.Continuous`.
+* `Analysis.Stieltjes.Affine`.
+* `Analysis.Stieltjes.RiemannSums`.
+* `Analysis.Stieltjes.Shift`.
+* `Analysis.SurfaceMeasure.Basic`.
+* `Analysis.SurfaceMeasure.ExteriorNormal`.
+* `Analysis.SurfaceMeasure.GraphDefinitions`.
+* `Analysis.SurfaceMeasure.Regularity`.
+* `Analysis.SurfaceMeasure.Segment`.
+* `Analysis.SurfaceMeasure.SegmentFaces`.
+* `Analysis.SurfaceMeasure.SegmentGraph`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Bounded Variation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Bounded variation for a real function on its actual closed-interval domain. -/
+def IsIntervalBoundedVariation (a b : ℝ) (f : Set.Icc a b → ℝ) : Prop :=
+  BoundedVariationOn f Set.univ
+
+private theorem intervalBV_add {a b : ℝ} {f g : Set.Icc a b → ℝ}
+    (hf : IsIntervalBoundedVariation a b f) (hg : IsIntervalBoundedVariation a b g) :
+    IsIntervalBoundedVariation a b (f + g) := by
+  refine ne_top_of_le_ne_top (ENNReal.add_ne_top.mpr ⟨hf, hg⟩) ?_
+  apply iSup_le
+  rintro ⟨n, u, hu, hus⟩
+  calc
+    _ ≤ ∑ i ∈ Finset.range n,
+        (edist (f (u (i + 1))) (f (u i)) + edist (g (u (i + 1))) (g (u i))) :=
+      Finset.sum_le_sum fun _ _ ↦ edist_add_add_le _ _ _ _
+    _ = _ := Finset.sum_add_distrib
+    _ ≤ _ := add_le_add (eVariationOn.sum_le hu hus) (eVariationOn.sum_le hu hus)
+
+/-- The submodule of continuous planar interval functions with coordinatewise finite variation. -/
+def continuousBVSubmodule (a b : ℝ) : Submodule ℝ (Set.Icc a b → Point) where
+  carrier := {f | Continuous f ∧
+    ∀ i : Fin 2, IsIntervalBoundedVariation a b (fun t ↦ f t i)}
+  zero_mem' := by
+    refine ⟨continuous_const, fun i ↦ ?_⟩
+    simp [IsIntervalBoundedVariation, BoundedVariationOn, eVariationOn]
+  add_mem' := by
+    intro f g hf hg
+    exact ⟨hf.1.add hg.1, fun i ↦ intervalBV_add (hf.2 i) (hg.2 i)⟩
+  smul_mem' := by
+    intro c f hf
+    refine ⟨continuous_const.smul hf.1, fun i ↦ ?_⟩
+    exact (ContinuousLinearMap.lsmul ℝ ℝ c).lipschitzWith.comp_boundedVariationOn (hf.2 i)
+
+/-- The real vector space of continuous planar BV paths on a closed interval. -/
+abbrev ContinuousBVPaths (a b : ℝ) : Type := continuousBVSubmodule a b
+
+/-- A continuous planar BV path has finite vector variation on its whole domain. -/
+theorem ContinuousBVPaths.boundedVariationOn {a b : ℝ} (x : ContinuousBVPaths a b) :
+    BoundedVariationOn x.val Set.univ := by
+  have hdist (p q : Point) : edist p q ≤ edist (p 0) (q 0) + edist (p 1) (q 1) := by
+    have hnorm (z : Point) : ‖z‖ ≤ |z 0| + |z 1| := by
+      nlinarith [Point.norm_sq_eq z, sq_abs (z 0), sq_abs (z 1), abs_nonneg (z 0),
+        abs_nonneg (z 1), norm_nonneg z, mul_nonneg (abs_nonneg (z 0)) (abs_nonneg (z 1))]
+    simp only [edist_dist, dist_eq_norm, Real.norm_eq_abs]
+    rw [← ENNReal.ofReal_add (abs_nonneg _) (abs_nonneg _)]
+    exact ENNReal.ofReal_le_ofReal (by simpa using hnorm (p - q))
+  refine ne_top_of_le_ne_top (ENNReal.add_ne_top.mpr ⟨x.property.2 0, x.property.2 1⟩) ?_
+  apply iSup_le
+  rintro ⟨n, u, hu, hus⟩
+  calc
+    _ ≤ ∑ i ∈ Finset.range n,
+        (edist (x.val (u (i + 1)) 0) (x.val (u i) 0) +
+          edist (x.val (u (i + 1)) 1) (x.val (u i) 1)) :=
+      Finset.sum_le_sum fun _ _ ↦ hdist _ _
+    _ = _ := Finset.sum_add_distrib
+    _ ≤ _ := add_le_add (eVariationOn.sum_le (f := fun t ↦ x.val t 0) (n := n) hu hus)
+      (eVariationOn.sum_le (f := fun t ↦ x.val t 1) (n := n) hu hus)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Measure Products
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- Multiply a signed measure by a real density using the continuous multiplication map. -/
+def functionMeasureMul {X : Type*} [MeasurableSpace X]
+    (f : X → ℝ) (μ : SignedMeasure X) : SignedMeasure X :=
+  μ.withDensity f (ContinuousLinearMap.mul ℝ ℝ)
+
+/-- Multiply one signed measure by each of two scalar densities. -/
+def pairFunctionMeasureMul {X : Type*} [MeasurableSpace X]
+    (f : (X → ℝ) × (X → ℝ)) (μ : SignedMeasure X) :
+    SignedMeasure X × SignedMeasure X :=
+  (functionMeasureMul f.1 μ, functionMeasureMul f.2 μ)
+
+/-- Multiply both components of a signed-measure pair by one scalar density. -/
+def functionPairMeasureMul {X : Type*} [MeasurableSpace X]
+    (f : X → ℝ) (μ : SignedMeasure X × SignedMeasure X) :
+    SignedMeasure X × SignedMeasure X :=
+  (functionMeasureMul f μ.1, functionMeasureMul f μ.2)
+
+/-- The sum of the coordinatewise density products of a function pair and measure pair. -/
+def functionMeasureDot {X : Type*} [MeasurableSpace X]
+    (f : (X → ℝ) × (X → ℝ)) (μ : SignedMeasure X × SignedMeasure X) :
+    SignedMeasure X :=
+  functionMeasureMul f.1 μ.1 + functionMeasureMul f.2 μ.2
+
+/-- The oriented planar determinant `p₀ q₁ - p₁ q₀`. -/
+def planeCrossProduct (p q : Point) : ℝ :=
+  p 0 * q 1 - p 1 * q 0
+
+/-- The signed-measure determinant of a function pair and a measure pair. -/
+def functionMeasureCross {X : Type*} [MeasurableSpace X]
+    (f : (X → ℝ) × (X → ℝ)) (μ : SignedMeasure X × SignedMeasure X) :
+    SignedMeasure X :=
+  functionMeasureMul f.1 μ.2 - functionMeasureMul f.2 μ.1
+
+/-- Bundle the scalar, vector and dot-product operations on signed measures. -/
+def functionMeasureProducts (X : Type*) [MeasurableSpace X] :
+    ((X → ℝ) → SignedMeasure X → SignedMeasure X) ×
+    (((X → ℝ) × (X → ℝ)) → SignedMeasure X → SignedMeasure X × SignedMeasure X) ×
+    ((X → ℝ) → (SignedMeasure X × SignedMeasure X) → SignedMeasure X × SignedMeasure X) ×
+    (((X → ℝ) × (X → ℝ)) → (SignedMeasure X × SignedMeasure X) → SignedMeasure X) :=
+  (functionMeasureMul, pairFunctionMeasureMul, functionPairMeasureMul, functionMeasureDot)
+
+/-- Bundle the planar determinant and its signed-measure counterpart. -/
+def planeCrossProducts (X : Type*) [MeasurableSpace X] :
+    (Point → Point → ℝ) ×
+    (((X → ℝ) × (X → ℝ)) → (SignedMeasure X × SignedMeasure X) → SignedMeasure X) :=
+  (planeCrossProduct, functionMeasureCross)
+
+/-! ### The oriented planar determinant
+
+Basic algebra of `planeCrossProduct`, the transitivity of the order it induces on the closed
+first quadrant, and the two shapes of level set used when a planar region is fanned into
+triangles over a base point.
+-/
+
+@[simp] theorem planeCrossProduct_self (p : Point) : planeCrossProduct p p = 0 := by
+  simp only [planeCrossProduct]
+  ring
+
+@[simp] theorem planeCrossProduct_zero_left (p : Point) : planeCrossProduct 0 p = 0 := by
+  simp [planeCrossProduct]
+
+@[simp] theorem planeCrossProduct_zero_right (p : Point) : planeCrossProduct p 0 = 0 := by
+  simp [planeCrossProduct]
+
+/-- The oriented determinant is antisymmetric. -/
+theorem planeCrossProduct_swap (p q : Point) :
+    planeCrossProduct p q = -planeCrossProduct q p := by
+  simp only [planeCrossProduct]
+  ring
+
+/-- The oriented determinant is the inner product against the quarter turn of its first
+argument. -/
+theorem planeCrossProduct_eq_inner (p q : Point) :
+    planeCrossProduct p q = inner ℝ q !₂[-p 1, p 0] := by
+  simp [planeCrossProduct, PiLp.inner_apply, Fin.sum_univ_two]
+  ring
+
+/-- In the closed first quadrant the oriented determinant order is transitive: if `v` is
+nonzero and both `u × v` and `v × w` are nonnegative, then so is `u × w`. -/
+theorem planeCrossProduct_nonneg_trans {u v w : Point} (hv : v ≠ 0)
+    (hu0 : 0 ≤ u 0) (hu1 : 0 ≤ u 1) (hv0 : 0 ≤ v 0) (hv1 : 0 ≤ v 1)
+    (hw0 : 0 ≤ w 0) (hw1 : 0 ≤ w 1)
+    (huv : 0 ≤ planeCrossProduct u v) (hvw : 0 ≤ planeCrossProduct v w) :
+    0 ≤ planeCrossProduct u w := by
+  have hvpos : 0 < v 0 ^ 2 + v 1 ^ 2 := by
+    rcases eq_or_lt_of_le (by positivity : (0 : ℝ) ≤ v 0 ^ 2 + v 1 ^ 2) with h | h
+    · refine absurd ?_ hv
+      have h0 : v 0 = 0 := by nlinarith [sq_nonneg (v 0), sq_nonneg (v 1)]
+      have h1 : v 1 = 0 := by nlinarith [sq_nonneg (v 0), sq_nonneg (v 1)]
+      ext i
+      fin_cases i
+      · simpa using h0
+      · simpa using h1
+    · exact h
+  have hid : planeCrossProduct u w * (v 0 ^ 2 + v 1 ^ 2)
+      = (w 0 * v 0 + w 1 * v 1) * planeCrossProduct u v
+        + planeCrossProduct v w * (u 0 * v 0 + u 1 * v 1) := by
+    simp only [planeCrossProduct]
+    ring
+  nlinarith [mul_nonneg (add_nonneg (mul_nonneg hw0 hv0) (mul_nonneg hw1 hv1)) huv,
+    mul_nonneg hvw (add_nonneg (mul_nonneg hu0 hv0) (mul_nonneg hu1 hv1))]
+
+/-- The closed angular sector between two rays through a base point is convex. -/
+theorem convex_setOf_planeCrossProduct_fan (u v L : Point) :
+    Convex ℝ {x : Point | 0 ≤ planeCrossProduct u (x - L) ∧
+      0 ≤ planeCrossProduct (x - L) v} := by
+  intro x hx y hy a b ha hb hab
+  obtain rfl : b = 1 - a := by linarith
+  simp only [Set.mem_ofPred_eq, planeCrossProduct, PiLp.sub_apply, PiLp.add_apply,
+    PiLp.smul_apply, smul_eq_mul] at hx hy ⊢
+  constructor
+  · nlinarith [mul_nonneg ha hx.1, mul_nonneg hb hy.1]
+  · nlinarith [mul_nonneg ha hx.2, mul_nonneg hb hy.2]
+
+/-- The line through a base point in a nonzero direction carries no planar area. -/
+theorem volume_setOf_planeCrossProduct_sub_eq_zero {w : Point} (hw : w ≠ 0) (L : Point) :
+    volume {x : Point | planeCrossProduct w (x - L) = 0} = 0 := by
+  have hrot : (!₂[-w 1, w 0] : Point) ≠ 0 := by
+    intro h
+    refine hw ?_
+    have h0 : w 0 = 0 := by simpa using congrArg (fun v : Point ↦ v 1) h
+    have h1 : w 1 = 0 := by simpa using congrArg (fun v : Point ↦ v 0) h
+    ext i
+    fin_cases i
+    · simpa using h0
+    · simpa using h1
+  have hset : {x : Point | planeCrossProduct w (x - L) = 0}
+      = {x : Point | inner ℝ x (!₂[-w 1, w 0] : Point) =
+          inner ℝ L (!₂[-w 1, w 0] : Point)} := by
+    ext x
+    simp only [Set.mem_ofPred_eq, planeCrossProduct_eq_inner, inner_sub_left, sub_eq_zero]
+  rw [hset]
+  exact volume.addHaar_setOf_real_inner_eq hrot _
+
+/-- The planar cross product is the frame determinant at every angle. -/
+theorem planeCrossProduct_eq_inner_frame (a b : Point) (t : ℝ) :
+    planeCrossProduct a b =
+      inner ℝ a (normalVector (t : Real.Angle)) * inner ℝ b (tangentVector (t : Real.Angle)) -
+        inner ℝ a (tangentVector (t : Real.Angle)) *
+          inner ℝ b (normalVector (t : Real.Angle)) := by
+  simp only [planeCrossProduct, normalVector, tangentVector, frame, PiLp.inner_apply,
+    Fin.sum_univ_two, Real.Angle.cos_coe, Real.Angle.sin_coe, RCLike.inner_apply,
+    conj_trivial, Matrix.cons_val_zero, Matrix.cons_val_one]
+  linear_combination (a.ofLp 1 * b.ofLp 0 - a.ofLp 0 * b.ofLp 1) * (Real.sin_sq_add_cos_sq t)
+
+/-- The oriented determinant of a point against the frame tangent is its normal coordinate. -/
+theorem planeCrossProduct_tangentVector (p : Point) (t : Real.Angle) :
+    planeCrossProduct p (tangentVector t) = inner ℝ p (normalVector t) := by
+  simp [planeCrossProduct, normalVector, tangentVector, frame, PiLp.inner_apply,
+    Fin.sum_univ_two]
+  ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Stieltjes / Integral
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- A right-continuous real BV function on its actual closed-interval domain. -/
+structure RightContinuousIntervalBV (a b : ℝ) where
+  /-- The real-valued function on the closed parameter interval. -/
+  toFun : Set.Icc a b → ℝ
+  boundedVariation : IsIntervalBoundedVariation a b toFun
+  right_continuous : ∀ t, ContinuousWithinAt toFun (Set.Ici t) t
+
+/-- A right-continuous interval-BV function is determined by its underlying function. -/
+theorem RightContinuousIntervalBV.toFun_injective {a b : ℝ} :
+    Function.Injective (@RightContinuousIntervalBV.toFun a b) := by
+  rintro ⟨f, hf, hfr⟩ ⟨g, hg, hgr⟩ hfg
+  cases hfg
+  rfl
+
+/-- A right-continuous interval-BV function on a nonempty interval is bounded, by its value at
+the left endpoint plus the total variation. -/
+theorem RightContinuousIntervalBV.exists_norm_bound {a b : ℝ} (hab : a ≤ b)
+    (f : RightContinuousIntervalBV a b) : ∃ C : ℝ, ∀ t, ‖f.toFun t‖ ≤ C := by
+  refine ⟨‖f.toFun ⟨a, le_rfl, hab⟩‖ + (eVariationOn f.toFun Set.univ).toReal, fun t ↦ ?_⟩
+  calc ‖f.toFun t‖
+      ≤ ‖f.toFun ⟨a, le_rfl, hab⟩‖ + ‖f.toFun t - f.toFun ⟨a, le_rfl, hab⟩‖ :=
+        norm_le_norm_add_norm_sub' _ _
+    _ ≤ _ := by
+        gcongr
+        simpa [dist_eq_norm_sub] using
+          f.boundedVariation.dist_le (Set.mem_univ t) (Set.mem_univ ⟨a, le_rfl, hab⟩)
+
+/-- The finite signed Stieltjes measure on the interval, with zero initial atom. -/
+def intervalStieltjesMeasure {a b : ℝ} (f : RightContinuousIntervalBV a b) :
+    SignedMeasure (Set.Icc a b) :=
+  f.boundedVariation.vectorMeasure
+
+/-- The Stieltjes integral, used for bounded measurable integrands and Borel subsets. -/
+def intervalStieltjesIntegral {a b : ℝ} (f : RightContinuousIntervalBV a b)
+    (g : Set.Icc a b → ℝ) (X : Set (Set.Icc a b)) : ℝ :=
+  ∫ᵛ t in X, g t ∂[ContinuousLinearMap.mul ℝ ℝ; intervalStieltjesMeasure f]
+
+/-- Over the whole parameter interval, the interval Stieltjes integral is the unrestricted
+vector-measure integral. -/
+theorem intervalStieltjesIntegral_univ {a b : ℝ} (f : RightContinuousIntervalBV a b)
+    (g : Set.Icc a b → ℝ) :
+    intervalStieltjesIntegral f g Set.univ =
+      VectorMeasure.integral (intervalStieltjesMeasure f) g (ContinuousLinearMap.mul ℝ ℝ) := by
+  simp only [intervalStieltjesIntegral, VectorMeasure.restrict_univ]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Stieltjes / Absolute Continuity
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+
+namespace MovingSofa
+
+/-- Extend an interval function to the real line by zero outside its domain. -/
+def stieltjesScalarExtension {a b : ℝ} (f : RightContinuousIntervalBV a b)
+    (t : ℝ) : ℝ := by
+  classical
+  exact if h : t ∈ Set.Icc a b then f.toFun ⟨t, h⟩ else 0
+
+/-- An integrable density represents the interval Stieltjes measure on measurable sets. -/
+def HasIntervalStieltjesDensity {a b : ℝ} (f : RightContinuousIntervalBV a b)
+    (r : ℝ → ℝ) : Prop :=
+  Integrable r (volume.restrict (Set.Icc a b)) ∧
+    ∀ E : Set ℝ, MeasurableSet E →
+      intervalStieltjesMeasure f {t | (t : ℝ) ∈ E} =
+        ∫ t in E, r t ∂volume.restrict (Set.Icc a b)
+
+theorem intervalStieltjesMeasure_Ioc {a b : ℝ}
+    (f : RightContinuousIntervalBV a b) (c d : Set.Icc a b) (hcd : c ≤ d) :
+    intervalStieltjesMeasure f (Set.Ioc c d) = f.toFun d - f.toFun c := by
+  rw [intervalStieltjesMeasure, f.boundedVariation.vectorMeasure_Ioc hcd]
+  rw [f.right_continuous d |>.rightLim_eq, f.right_continuous c |>.rightLim_eq]
+
+private theorem stieltjesScalarExtension_eq_add_integral {a b : ℝ} (hab : a ≤ b)
+    (f : RightContinuousIntervalBV a b) (r : ℝ → ℝ)
+    (hr : HasIntervalStieltjesDensity f r) (t : Set.Icc a b) :
+    stieltjesScalarExtension f (t : ℝ) =
+      f.toFun ⟨a, le_rfl, hab⟩ + ∫ x in a..(t : ℝ), r x := by
+  have hinc := hr.2 (Ioc a (t : ℝ)) measurableSet_Ioc
+  have hset : {x : Icc a b | (x : ℝ) ∈ Ioc a (t : ℝ)} =
+      Ioc (⟨a, le_rfl, hab⟩ : Icc a b) t := rfl
+  rw [hset, intervalStieltjesMeasure_Ioc f _ _ t.property.1] at hinc
+  have hmeasure :
+      (volume.restrict (Set.Icc a b)).restrict (Set.Ioc a (t : ℝ)) =
+        volume.restrict (Set.Ioc a (t : ℝ)) := by
+    exact Measure.restrict_restrict_of_subset
+      (fun x hx ↦ ⟨hx.1.le, hx.2.trans t.property.2⟩)
+  have hrestrict :
+      (∫ x in Set.Ioc a (t : ℝ), r x ∂volume.restrict (Set.Icc a b)) =
+        ∫ x in Set.Ioc a (t : ℝ), r x := by
+    rw [hmeasure]
+  rw [stieltjesScalarExtension]
+  simp only [dite_eq_left t.2]
+  change f.toFun t = _
+  rw [intervalIntegral.integral_of_le t.2.1]
+  have hinc' := hinc.trans hrestrict
+  linarith
+
+private theorem absolutelyContinuousOnInterval_of_stieltjesDensity {a b : ℝ} (hab : a ≤ b)
+    (f : RightContinuousIntervalBV a b) (r : ℝ → ℝ)
+    (hr : HasIntervalStieltjesDensity f r) :
+    AbsolutelyContinuousOnInterval (stieltjesScalarExtension f) a b := by
+  have hri : IntervalIntegrable r volume a b :=
+    (intervalIntegrable_iff_integrableOn_Icc_of_le hab).2 hr.1
+  have hp := hri.absolutelyContinuousOnInterval_intervalIntegral
+    (c := a) (by simp [hab])
+  have hc : AbsolutelyContinuousOnInterval (fun _ : ℝ ↦ f.toFun ⟨a, le_rfl, hab⟩) a b :=
+    (LipschitzWith.const _).lipschitzOnWith.absolutelyContinuousOnInterval
+  have hs := hc.add hp
+  apply hs.congr
+  intro x hx
+  have hx' : x ∈ Set.Icc a b := by
+    simpa [Set.uIcc_of_le hab] using hx
+  have hprim := stieltjesScalarExtension_eq_add_integral hab f r hr ⟨x, hx'⟩
+  simpa [stieltjesScalarExtension, hx'] using hprim.symm
+
+private theorem ae_hasDerivAt_of_stieltjesDensity {a b : ℝ} (hab : a ≤ b)
+    (f : RightContinuousIntervalBV a b) (r : ℝ → ℝ)
+    (hr : HasIntervalStieltjesDensity f r) :
+    ∀ᵐ t ∂volume.restrict (Set.Ioo a b),
+      HasDerivAt (stieltjesScalarExtension f) (r t) t := by
+  have hri : IntervalIntegrable r volume a b :=
+    (intervalIntegrable_iff_integrableOn_Icc_of_le hab).2 hr.1
+  have hder := IntervalIntegrable.ae_hasDerivAt_integral hri
+  refine (ae_restrict_iff' measurableSet_Ioo).2 ?_
+  filter_upwards [hder] with x hx hxo
+  have hxu : x ∈ Set.uIcc a b := by
+    simpa [Set.uIcc_of_le hab] using ⟨hxo.1.le, hxo.2.le⟩
+  have hd := hx hxu a (by simp [hab])
+  have hd' := hd.const_add (f.toFun ⟨a, le_rfl, hab⟩)
+  apply hd'.congr_of_eventuallyEq
+  filter_upwards [isOpen_Ioo.mem_nhds hxo] with y hyo
+  have hycc : y ∈ Set.Icc a b := ⟨hyo.1.le, hyo.2.le⟩
+  have hprim := stieltjesScalarExtension_eq_add_integral hab f r hr ⟨y, hycc⟩
+  simpa [stieltjesScalarExtension, hycc] using hprim
+
+private theorem intervalStieltjes_hasDensity_deriv {a b : ℝ} (hab : a ≤ b)
+    (f : RightContinuousIntervalBV a b)
+    (hf : AbsolutelyContinuousOnInterval (stieltjesScalarExtension f) a b) :
+    HasIntervalStieltjesDensity f (deriv (stieltjesScalarExtension f)) := by
+  let g := stieltjesScalarExtension f
+  have hr : IntegrableOn (deriv g) (Icc a b) :=
+    (intervalIntegrable_iff_integrableOn_Icc_of_le hab).mp hf.intervalIntegrable_deriv
+  have hsub : Continuous f.toFun := by
+    have h := (continuousOn_iff_continuous_domRestrict).mp
+      (show ContinuousOn g (Icc a b) by simpa only [uIcc_of_le hab] using hf.continuousOn)
+    convert h using 1
+    funext x
+    simp [g, stieltjesScalarExtension, x.property]
+  have hri : Integrable (fun x : Icc a b ↦ deriv g x)
+      (volume.comap (Subtype.val : Icc a b → ℝ)) :=
+    (integrableOn_iff_comap_subtypeVal measurableSet_Icc).mp hr
+  have hmeasure : intervalStieltjesMeasure f =
+      (volume.comap (Subtype.val : Icc a b → ℝ)).withDensityᵥ
+        (fun x : Icc a b ↦ deriv g x) := by
+    apply f.boundedVariation.vectorMeasure_eq_withDensity_of_integral_Icc hsub hri
+    intro c d hcd
+    have hinc := (hf.mono (show uIcc (c : ℝ) (d : ℝ) ⊆ uIcc a b by
+      rw [uIcc_of_le (show (c : ℝ) ≤ d from hcd), uIcc_of_le hab]
+      exact Icc_subset_Icc c.property.1 d.property.2)).integral_deriv_eq_sub
+    have hset : Icc c d = {x : Icc a b | (x : ℝ) ∈ Icc (c : ℝ) (d : ℝ)} := rfl
+    rw [hset, integral_subtype_preimage measurableSet_Icc measurableSet_Icc]
+    rw [Measure.restrict_restrict_of_subset (Icc_subset_Icc c.property.1 d.property.2),
+      integral_Icc_eq_integral_Ioc, ← intervalIntegral.integral_of_le hcd, hinc]
+    simp [stieltjesScalarExtension, c.property, d.property]
+  refine ⟨hr, ?_⟩
+  intro E hE
+  have hpre : MeasurableSet {t : Icc a b | (t : ℝ) ∈ E} :=
+    hE.preimage measurable_subtype_coe
+  rw [hmeasure, withDensityᵥ_apply hri hpre,
+    integral_subtype_preimage measurableSet_Icc hE]
+
+theorem intervalStieltjes_absoluteContinuity (a b : ℝ) (hab : a ≤ b)
+    (f : RightContinuousIntervalBV a b) :
+    (AbsolutelyContinuousOnInterval (stieltjesScalarExtension f) a b ↔
+      ∃ r : ℝ → ℝ, HasIntervalStieltjesDensity f r) ∧
+    (∀ r : ℝ → ℝ, HasIntervalStieltjesDensity f r →
+      ∀ᵐ t ∂volume.restrict (Set.Ioo a b),
+        HasDerivAt (stieltjesScalarExtension f) (r t) t) := by
+  refine ⟨⟨?_, ?_⟩, fun r hr ↦ ae_hasDerivAt_of_stieltjesDensity hab f r hr⟩
+  · intro hf
+    exact ⟨deriv (stieltjesScalarExtension f), intervalStieltjes_hasDensity_deriv hab f hf⟩
+  · rintro ⟨r, hr⟩
+    exact absolutelyContinuousOnInterval_of_stieltjesDensity hab f r hr
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Stieltjes / Calculus
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter MeasureTheory
+open scoped Topology
+
+namespace MovingSofa
+
+theorem intervalStieltjes_integration_by_parts (a b : ℝ) (hab : a ≤ b)
+    (f g : RightContinuousIntervalBV a b) :
+    ∃ left : Set.Icc a b → ℝ,
+      (∀ t : Set.Icc a b, a < (t : ℝ) →
+        Tendsto f.toFun (nhdsWithin t (Set.Iio t)) (𝓝 (left t))) ∧
+      intervalStieltjesIntegral f g.toFun {t | a < (t : ℝ)} +
+        intervalStieltjesIntegral g left {t | a < (t : ℝ)} =
+      f.toFun ⟨b, hab, le_rfl⟩ * g.toFun ⟨b, hab, le_rfl⟩ -
+        f.toFun ⟨a, le_rfl, hab⟩ * g.toFun ⟨a, le_rfl, hab⟩ := by
+  let a' : Set.Icc a b := ⟨a, le_rfl, hab⟩
+  let b' : Set.Icc a b := ⟨b, hab, le_rfl⟩
+  refine ⟨Function.leftLim f.toFun, fun t _ ↦ f.boundedVariation.tendsto_leftLim t, ?_⟩
+  have hfa (t : Set.Icc a b) : Function.rightLim f.toFun t = f.toFun t :=
+    (f.right_continuous t).rightLim_eq
+  have hga (t : Set.Icc a b) : Function.rightLim g.toFun t = g.toFun t :=
+    (g.right_continuous t).rightLim_eq
+  have hparts := BoundedVariationOn.setIntegral_Ioc_leftLim_vectorMeasure_eq_sub
+    (B := ContinuousLinearMap.mul ℝ ℝ) f.boundedVariation g.boundedVariation (show a' ≤ b' by
+      exact hab)
+  have hset : {t : Set.Icc a b | a < (t : ℝ)} = Set.Ioc a' b' := by
+    ext t
+    constructor
+    · intro ht
+      exact ⟨ht, t.property.2⟩
+    · exact fun ht ↦ ht.1
+  have hmulflip : (ContinuousLinearMap.mul ℝ ℝ).flip = ContinuousLinearMap.mul ℝ ℝ := by
+    ext
+    simp
+  rw [hmulflip] at hparts
+  rw [hset]
+  unfold intervalStieltjesIntegral intervalStieltjesMeasure
+  change (∫ᵛ t in Set.Ioc a' b', g.toFun t
+      ∂[ContinuousLinearMap.mul ℝ ℝ; f.boundedVariation.vectorMeasure]) +
+    (∫ᵛ t in Set.Ioc a' b', Function.leftLim f.toFun t
+      ∂[ContinuousLinearMap.mul ℝ ℝ; g.boundedVariation.vectorMeasure]) = _
+  rw [hparts]
+  simp only [hfa, hga]
+  change _ + (f.toFun b' * g.toFun b' - f.toFun a' * g.toFun a' - _) = _
+  dsimp [a', b']
+  ring
+
+/-- Stieltjes integration by parts on the open interval `(a, b)`: no endpoint atom is included
+at `b`, and none is introduced at `a`. The left limit in the first integrand accounts for
+simultaneous jumps. -/
+theorem intervalStieltjes_integration_by_parts_Ioo (a b : ℝ) (hab : a < b)
+    (f g : RightContinuousIntervalBV a b) :
+    intervalStieltjesIntegral g (Function.leftLim f.toFun)
+        {t | a < (t : ℝ) ∧ (t : ℝ) < b} +
+      intervalStieltjesIntegral f g.toFun {t | a < (t : ℝ) ∧ (t : ℝ) < b} =
+      Function.leftLim f.toFun ⟨b, hab.le, le_rfl⟩ *
+          Function.leftLim g.toFun ⟨b, hab.le, le_rfl⟩ -
+        f.toFun ⟨a, le_rfl, hab.le⟩ * g.toFun ⟨a, le_rfl, hab.le⟩ := by
+  have hfa (t : Set.Icc a b) : Function.rightLim f.toFun t = f.toFun t :=
+    (f.right_continuous t).rightLim_eq
+  have hga (t : Set.Icc a b) : Function.rightLim g.toFun t = g.toFun t :=
+    (g.right_continuous t).rightLim_eq
+  have hparts := BoundedVariationOn.setIntegral_Ioo_leftLim_vectorMeasure_eq_sub
+    (B := ContinuousLinearMap.mul ℝ ℝ) f.boundedVariation g.boundedVariation
+    (show (⟨a, le_rfl, hab.le⟩ : Set.Icc a b) < ⟨b, hab.le, le_rfl⟩ from hab)
+  have hset : {t : Set.Icc a b | a < (t : ℝ) ∧ (t : ℝ) < b} =
+      Set.Ioo (⟨a, le_rfl, hab.le⟩ : Set.Icc a b) ⟨b, hab.le, le_rfl⟩ := by
+    ext t
+    exact ⟨fun ht ↦ ⟨ht.1, ht.2⟩, fun ht ↦ ⟨ht.1, ht.2⟩⟩
+  have hmulflip : (ContinuousLinearMap.mul ℝ ℝ).flip = ContinuousLinearMap.mul ℝ ℝ := by
+    ext
+    simp
+  rw [hmulflip] at hparts
+  rw [hset]
+  unfold intervalStieltjesIntegral intervalStieltjesMeasure
+  rw [hparts]
+  simp only [hfa, hga, ContinuousLinearMap.mul_apply']
+  ring
+
+theorem intervalStieltjes_product (a b : ℝ)
+    (f g : RightContinuousIntervalBV a b)
+    (hcont : Continuous f.toFun ∨ Continuous g.toFun) :
+    ∃ h : RightContinuousIntervalBV a b,
+      (∀ t, h.toFun t = f.toFun t * g.toFun t) ∧
+      ∀ E : Set (Set.Icc a b), MeasurableSet E →
+        intervalStieltjesMeasure h E =
+          intervalStieltjesIntegral f g.toFun E + intervalStieltjesIntegral g f.toFun E := by
+  let h : RightContinuousIntervalBV a b :=
+    ⟨fun t ↦ f.toFun t * g.toFun t,
+      f.boundedVariation.bilinear_comp g.boundedVariation (ContinuousLinearMap.mul ℝ ℝ),
+      fun t ↦ (f.right_continuous t).mul (g.right_continuous t)⟩
+  refine ⟨h, fun _ ↦ rfl, fun E _ ↦ ?_⟩
+  have hflip : (ContinuousLinearMap.mul ℝ ℝ).flip = ContinuousLinearMap.mul ℝ ℝ := by
+    ext
+    simp
+  have hfr : Function.rightLim f.toFun = f.toFun :=
+    funext fun t ↦ (f.right_continuous t).rightLim_eq
+  have hgr : Function.rightLim g.toFun = g.toFun :=
+    funext fun t ↦ (g.right_continuous t).rightLim_eq
+  rcases hcont with hf | hg
+  · have hfl : Function.leftLim f.toFun = f.toFun :=
+      funext fun t ↦ hf.continuousAt.continuousWithinAt.leftLim_eq
+    have hp := f.boundedVariation.setIntegral_leftLim_vectorMeasure_eq_sub
+      (B := ContinuousLinearMap.mul ℝ ℝ) g.boundedVariation (s := E)
+    rw [hflip, hfl, hgr] at hp
+    change _ = _ + _
+    unfold intervalStieltjesIntegral intervalStieltjesMeasure
+    change (f.boundedVariation.bilinear_comp g.boundedVariation
+      (ContinuousLinearMap.mul ℝ ℝ)).vectorMeasure E = _
+    linarith
+  · have hgl : Function.leftLim g.toFun = g.toFun :=
+      funext fun t ↦ hg.continuousAt.continuousWithinAt.leftLim_eq
+    have hp := f.boundedVariation.setIntegral_rightLim_vectorMeasure_eq_sub
+      (B := ContinuousLinearMap.mul ℝ ℝ) g.boundedVariation (s := E)
+    rw [hflip, hfr, hgl] at hp
+    unfold intervalStieltjesIntegral intervalStieltjesMeasure
+    change (f.boundedVariation.bilinear_comp g.boundedVariation
+      (ContinuousLinearMap.mul ℝ ℝ)).vectorMeasure E = _
+    linarith
+
+/-- The integrated Stieltjes product rule: a bounded measurable weight distributes over the
+Lebesgue–Stieltjes measure of a product one factor of which is continuous. -/
+theorem intervalStieltjesIntegral_product_of_bounded {a b : ℝ} (hab : a ≤ b)
+    (f g h : RightContinuousIntervalBV a b) (hgc : Continuous g.toFun)
+    (hh : ∀ t, h.toFun t = f.toFun t * g.toFun t)
+    (φ : Set.Icc a b → ℝ) (hφ : Measurable φ) (C : ℝ) (hφb : ∀ t, ‖φ t‖ ≤ C)
+    (E : Set (Set.Icc a b)) (hE : MeasurableSet E) :
+    intervalStieltjesIntegral h φ E =
+      intervalStieltjesIntegral f (fun t ↦ φ t * g.toFun t) E +
+        intervalStieltjesIntegral g (fun t ↦ φ t * f.toFun t) E := by
+  obtain ⟨h', hh', hh'm⟩ := intervalStieltjes_product a b f g (Or.inr hgc)
+  have hheq : h = h' :=
+    RightContinuousIntervalBV.toFun_injective (funext fun t ↦ (hh t).trans (hh' t).symm)
+  obtain ⟨Cf, hCf⟩ := f.exists_norm_bound hab
+  obtain ⟨Cg, hCg⟩ := g.exists_norm_bound hab
+  have hfm : Measurable f.toFun := f.boundedVariation.measurable
+  have hgm : Measurable g.toFun := g.boundedVariation.measurable
+  -- both Stieltjes measures are absolutely continuous over the sum of their total variations
+  set ρ : Measure (Set.Icc a b) := (intervalStieltjesMeasure f).totalVariation +
+    (intervalStieltjesMeasure g).totalVariation with hρ
+  have _ : IsFiniteMeasure ρ := by rw [hρ]; infer_instance
+  have hac : ∀ μ : SignedMeasure (Set.Icc a b), μ.totalVariation ≤ ρ →
+      μ ≪ᵥ ρ.toENNRealVectorMeasure := fun μ hle ↦ by
+    rw [SignedMeasure.absolutelyContinuous_ennreal_iff,
+      VectorMeasure.ennrealToMeasure_toENNRealVectorMeasure]
+    exact Measure.absolutelyContinuous_of_le hle
+  have hacf := hac (intervalStieltjesMeasure f) (by rw [hρ]; exact Measure.le_add_right le_rfl)
+  have hacg := hac (intervalStieltjesMeasure g) (by rw [hρ]; exact Measure.le_add_left le_rfl)
+  set rf : Set.Icc a b → ℝ := (intervalStieltjesMeasure f).rnDeriv ρ with hrf
+  set rg : Set.Icc a b → ℝ := (intervalStieltjesMeasure g).rnDeriv ρ with hrg
+  have hrfi : Integrable rf ρ := SignedMeasure.integrable_rnDeriv _ ρ
+  have hrgi : Integrable rg ρ := SignedMeasure.integrable_rnDeriv _ ρ
+  have hμf : ρ.withDensityᵥ rf = intervalStieltjesMeasure f :=
+    SignedMeasure.withDensityᵥ_rnDeriv_eq _ ρ hacf
+  have hμg : ρ.withDensityᵥ rg = intervalStieltjesMeasure g :=
+    SignedMeasure.withDensityᵥ_rnDeriv_eq _ ρ hacg
+  have hgrf : Integrable (fun x ↦ g.toFun x * rf x) ρ :=
+    hrfi.bdd_mul hgm.aestronglyMeasurable (Filter.Eventually.of_forall hCg)
+  have hfrg : Integrable (fun x ↦ f.toFun x * rg x) ρ :=
+    hrgi.bdd_mul hfm.aestronglyMeasurable (Filter.Eventually.of_forall hCf)
+  have hsumi : Integrable (fun x ↦ g.toFun x * rf x + f.toFun x * rg x) ρ := hgrf.add hfrg
+  have hφg : Measurable fun t ↦ φ t * g.toFun t := hφ.mul hgm
+  have hφf : Measurable fun t ↦ φ t * f.toFun t := hφ.mul hfm
+  -- the product measure is the density of the two-term Radon–Nikodym sum
+  have hsum : intervalStieltjesMeasure h =
+      ρ.withDensityᵥ (fun x ↦ g.toFun x * rf x + f.toFun x * rg x) := by
+    rw [hheq]
+    ext S hS
+    rw [withDensityᵥ_apply hsumi hS, hh'm S hS]
+    have h1 : intervalStieltjesIntegral f g.toFun S = ∫ x in S, g.toFun x * rf x ∂ρ := by
+      rw [intervalStieltjesIntegral, ← hμf,
+        VectorMeasure.setIntegral_withDensity_mul_of_bounded hrfi
+          hgm.aestronglyMeasurable Cg hCg S hS]
+    have h2 : intervalStieltjesIntegral g f.toFun S = ∫ x in S, f.toFun x * rg x ∂ρ := by
+      rw [intervalStieltjesIntegral, ← hμg,
+        VectorMeasure.setIntegral_withDensity_mul_of_bounded hrgi
+          hfm.aestronglyMeasurable Cf hCf S hS]
+    rw [h1, h2, integral_add hgrf.integrableOn hfrg.integrableOn]
+  have hb1 : ∀ x, ‖φ x * g.toFun x‖ ≤ C * Cg := fun x ↦ by
+    rw [norm_mul]
+    exact mul_le_mul (hφb x) (hCg x) (norm_nonneg _) ((norm_nonneg _).trans (hφb x))
+  have hb2 : ∀ x, ‖φ x * f.toFun x‖ ≤ C * Cf := fun x ↦ by
+    rw [norm_mul]
+    exact mul_le_mul (hφb x) (hCf x) (norm_nonneg _) ((norm_nonneg _).trans (hφb x))
+  have hi1 : Integrable (fun x ↦ φ x * g.toFun x * rf x) ρ :=
+    hrfi.bdd_mul hφg.aestronglyMeasurable (Filter.Eventually.of_forall hb1)
+  have hi2 : Integrable (fun x ↦ φ x * f.toFun x * rg x) ρ :=
+    hrgi.bdd_mul hφf.aestronglyMeasurable (Filter.Eventually.of_forall hb2)
+  rw [show intervalStieltjesIntegral h φ E =
+      ∫ x in E, φ x * (g.toFun x * rf x + f.toFun x * rg x) ∂ρ by
+    rw [intervalStieltjesIntegral, hsum,
+      VectorMeasure.setIntegral_withDensity_mul_of_bounded hsumi
+        hφ.aestronglyMeasurable C hφb E hE],
+    show intervalStieltjesIntegral f (fun t ↦ φ t * g.toFun t) E =
+      ∫ x in E, φ x * g.toFun x * rf x ∂ρ by
+    rw [intervalStieltjesIntegral, ← hμf,
+      VectorMeasure.setIntegral_withDensity_mul_of_bounded hrfi
+        hφg.aestronglyMeasurable (C * Cg) hb1 E hE],
+    show intervalStieltjesIntegral g (fun t ↦ φ t * f.toFun t) E =
+      ∫ x in E, φ x * f.toFun x * rg x ∂ρ by
+    rw [intervalStieltjesIntegral, ← hμg,
+      VectorMeasure.setIntegral_withDensity_mul_of_bounded hrgi
+        hφf.aestronglyMeasurable (C * Cf) hb2 E hE],
+    ← integral_add hi1.integrableOn hi2.integrableOn]
+  exact setIntegral_congr_fun hE fun x _ ↦ by ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Stieltjes / Density Integration
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped Topology
+
+namespace MovingSofa
+
+/-- A continuous integrand against an interval Stieltjes measure with an ordinary density is
+the corresponding weighted Lebesgue integral on the interval subtype. -/
+theorem intervalStieltjesIntegral_eq_integral_mul_of_density
+    {a b : ℝ} (F : RightContinuousIntervalBV a b) {r : ℝ → ℝ}
+    (hr : HasIntervalStieltjesDensity F r) {q : Icc a b → ℝ} (hq : Continuous q)
+    (E : Set (Icc a b)) (hE : MeasurableSet E) :
+    intervalStieltjesIntegral F q E =
+      ∫ t in E, q t * r t ∂volume.comap (Subtype.val : Icc a b → ℝ) := by
+  have hri : Integrable (fun t : Icc a b ↦ r t)
+      (volume.comap (Subtype.val : Icc a b → ℝ)) :=
+    (integrableOn_iff_comap_subtypeVal measurableSet_Icc).mp hr.1
+  have hmeasure : intervalStieltjesMeasure F =
+      (volume.comap (Subtype.val : Icc a b → ℝ)).withDensityᵥ
+        (fun t : Icc a b ↦ r t) := by
+    ext S hS
+    have himage : MeasurableSet ((Subtype.val : Icc a b → ℝ) '' S) :=
+      (MeasurableEmbedding.subtype_coe measurableSet_Icc).measurableSet_image' hS
+    have hpre : {t : Icc a b | (t : ℝ) ∈ (Subtype.val : Icc a b → ℝ) '' S} = S :=
+      Set.preimage_image_eq S Subtype.val_injective
+    rw [withDensityᵥ_apply hri hS, ← hpre,
+      integral_subtype_preimage measurableSet_Icc himage]
+    exact hr.2 _ himage
+  let _ : IsFiniteMeasure (volume.comap (Subtype.val : Icc a b → ℝ)) :=
+    ⟨by
+      rw [comap_subtype_coe_apply measurableSet_Icc]
+      simp only [image_univ, Subtype.range_val]
+      exact measure_Icc_lt_top⟩
+  unfold intervalStieltjesIntegral
+  rw [hmeasure]
+  exact VectorMeasure.setIntegral_withDensity_mul hri hq E hE
+
+/-- The density formula also applies to a bounded-variation integrand on a nonempty compact
+interval. -/
+theorem intervalStieltjesIntegral_eq_integral_mul_of_density_bv
+    {a b : ℝ} (hab : a ≤ b) (F : RightContinuousIntervalBV a b) {r : ℝ → ℝ}
+    (hr : HasIntervalStieltjesDensity F r) {q : Icc a b → ℝ}
+    (hq : BoundedVariationOn q univ) (E : Set (Icc a b)) (hE : MeasurableSet E) :
+    intervalStieltjesIntegral F q E =
+      ∫ t in E, q t * r t ∂volume.comap (Subtype.val : Icc a b → ℝ) := by
+  have hri : Integrable (fun t : Icc a b ↦ r t)
+      (volume.comap (Subtype.val : Icc a b → ℝ)) :=
+    (integrableOn_iff_comap_subtypeVal measurableSet_Icc).mp hr.1
+  have hmeasure : intervalStieltjesMeasure F =
+      (volume.comap (Subtype.val : Icc a b → ℝ)).withDensityᵥ
+        (fun t : Icc a b ↦ r t) := by
+    ext S hS
+    have himage : MeasurableSet ((Subtype.val : Icc a b → ℝ) '' S) :=
+      (MeasurableEmbedding.subtype_coe measurableSet_Icc).measurableSet_image' hS
+    have hpre : {t : Icc a b | (t : ℝ) ∈ (Subtype.val : Icc a b → ℝ) '' S} = S :=
+      Set.preimage_image_eq S Subtype.val_injective
+    rw [withDensityᵥ_apply hri hS, ← hpre,
+      integral_subtype_preimage measurableSet_Icc himage]
+    exact hr.2 _ himage
+  let _ : IsFiniteMeasure (volume.comap (Subtype.val : Icc a b → ℝ)) :=
+    ⟨by
+      rw [comap_subtype_coe_apply measurableSet_Icc]
+      simp only [image_univ, Subtype.range_val]
+      exact measure_Icc_lt_top⟩
+  let t₀ : Icc a b := ⟨a, le_rfl, hab⟩
+  let C := ‖q t₀‖ + (eVariationOn q univ).toReal
+  have hq_bound : ∀ t, ‖q t‖ ≤ C := by
+    intro t
+    calc
+      ‖q t‖ ≤ ‖q t₀‖ + ‖q t - q t₀‖ := norm_le_norm_add_norm_sub' _ _
+      _ ≤ ‖q t₀‖ + (eVariationOn q univ).toReal := by
+        gcongr
+        simpa [dist_eq_norm_sub] using hq.dist_le (mem_univ t) (mem_univ t₀)
+      _ = C := rfl
+  unfold intervalStieltjesIntegral
+  rw [hmeasure]
+  exact VectorMeasure.setIntegral_withDensity_mul_of_bounded hri
+    hq.stronglyMeasurable.aestronglyMeasurable C hq_bound E hE
+
+/-- A real function agreeing with a BV representative is integrable on its interval. -/
+theorem RightContinuousIntervalBV.integrableOn_Icc_of_eq
+    {a b : ℝ} (f : RightContinuousIntervalBV a b) (q : ℝ → ℝ)
+    (hq : ∀ t : Icc a b, q t = f.toFun t) : IntegrableOn q (Icc a b) := by
+  have hfinite : IsFiniteMeasure (volume.comap (Subtype.val : Icc a b → ℝ)) :=
+    ⟨by
+      rw [comap_subtype_coe_apply measurableSet_Icc]
+      simp only [image_univ, Subtype.range_val]
+      exact measure_Icc_lt_top⟩
+  let _ := hfinite
+  rw [IntegrableOn, ← map_comap_subtype_coe measurableSet_Icc,
+    (MeasurableEmbedding.subtype_coe measurableSet_Icc).integrable_map_iff
+      (μ := volume.comap (Subtype.val : Icc a b → ℝ))]
+  convert f.boundedVariation.integrable
+      (μ := volume.comap (Subtype.val : Icc a b → ℝ)) using 1
+  funext t
+  exact hq t
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Stieltjes / Linearity
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped Topology
+
+namespace MovingSofa
+
+theorem intervalStieltjes_linear_combination (a b : ℝ)
+    (f g : RightContinuousIntervalBV a b) (r s : ℝ) :
+    ∃ h : RightContinuousIntervalBV a b,
+      (∀ t, h.toFun t = r * f.toFun t + s * g.toFun t) ∧
+      intervalStieltjesMeasure h =
+        r • intervalStieltjesMeasure f + s • intervalStieltjesMeasure g := by
+  let hfun : Set.Icc a b → ℝ := fun t ↦ r * f.toFun t + s * g.toFun t
+  have hBV : IsIntervalBoundedVariation a b hfun := by
+    have hfr := (ContinuousLinearMap.lsmul ℝ ℝ r).lipschitzWith.comp_boundedVariationOn
+      f.boundedVariation
+    have hgr := (ContinuousLinearMap.lsmul ℝ ℝ s).lipschitzWith.comp_boundedVariationOn
+      g.boundedVariation
+    exact BoundedVariationOn.add hfr hgr
+  change BoundedVariationOn hfun Set.univ at hBV
+  let h : RightContinuousIntervalBV a b :=
+    { toFun := hfun
+      boundedVariation := hBV
+      right_continuous := fun t ↦ by
+        exact (f.right_continuous t).const_mul r |>.add ((g.right_continuous t).const_mul s) }
+  refine ⟨h, ?_, ?_⟩
+  · intro t
+    rfl
+  · apply MeasureTheory.VectorMeasure.ext_of_Icc
+    intro x y hxy
+    have hbv : BoundedVariationOn h.toFun Set.univ := h.boundedVariation
+    have hleft (z : Set.Icc a b) :
+        Function.leftLim h.toFun z =
+          r * Function.leftLim f.toFun z + s * Function.leftLim g.toFun z := by
+      rcases Filter.eq_or_neBot (𝓝[<] z) with hz | hz
+      · simp [leftLim_eq_of_eq_bot _ hz, h, hfun]
+      · exact tendsto_nhds_unique (hbv.tendsto_leftLim z)
+          ((f.boundedVariation.tendsto_leftLim z).const_smul r |>.add
+            ((g.boundedVariation.tendsto_leftLim z).const_smul s))
+    have hfxy : Function.rightLim f.toFun y = f.toFun y :=
+      (f.right_continuous y).rightLim_eq
+    have gfxy : Function.rightLim g.toFun y = g.toFun y :=
+      (g.right_continuous y).rightLim_eq
+    have hxy' : Function.rightLim h.toFun y = h.toFun y :=
+      (h.right_continuous y).rightLim_eq
+    simp only [smul_apply, add_apply]
+    rw [show intervalStieltjesMeasure h (Set.Icc x y) =
+        Function.rightLim h.toFun y - Function.leftLim h.toFun x by
+          exact BoundedVariationOn.vectorMeasure_Icc hbv hxy]
+    rw [show intervalStieltjesMeasure f (Set.Icc x y) =
+        Function.rightLim f.toFun y - Function.leftLim f.toFun x by
+          exact BoundedVariationOn.vectorMeasure_Icc f.boundedVariation hxy]
+    rw [show intervalStieltjesMeasure g (Set.Icc x y) =
+        Function.rightLim g.toFun y - Function.leftLim g.toFun x by
+          exact BoundedVariationOn.vectorMeasure_Icc g.boundedVariation hxy]
+    rw [hleft x]
+    rw [hxy', hfxy, gfxy]
+    simp [h, hfun, sub_eq_add_neg]
+    ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Stieltjes / Inner Product
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped Topology
+
+namespace MovingSofa
+
+/-- Componentwise product rule for the Euclidean pairing of two planar interval-BV functions,
+when the second function is continuous. -/
+theorem intervalStieltjes_inner_fin_two {a b : ℝ}
+    (f g : Fin 2 → RightContinuousIntervalBV a b)
+    (hg : ∀ i, Continuous (g i).toFun) :
+    ∃ h : RightContinuousIntervalBV a b,
+      (∀ t, h.toFun t = ∑ i, (f i).toFun t * (g i).toFun t) ∧
+      ∀ E : Set (Icc a b), MeasurableSet E →
+        intervalStieltjesMeasure h E =
+          ∑ i, (intervalStieltjesIntegral (f i) (g i).toFun E +
+            intervalStieltjesIntegral (g i) (f i).toFun E) := by
+  obtain ⟨p0, hp0, hp0m⟩ := intervalStieltjes_product a b (f 0) (g 0) (Or.inr (hg 0))
+  obtain ⟨p1, hp1, hp1m⟩ := intervalStieltjes_product a b (f 1) (g 1) (Or.inr (hg 1))
+  obtain ⟨h, hh, hhm⟩ := intervalStieltjes_linear_combination a b p0 p1 1 1
+  refine ⟨h, ?_, ?_⟩
+  · intro t
+    rw [hh, hp0, hp1, Fin.sum_univ_two]
+    ring
+  · intro E hE
+    rw [show intervalStieltjesMeasure h E =
+        intervalStieltjesMeasure p0 E + intervalStieltjesMeasure p1 E by
+      rw [hhm]
+      simp]
+    rw [hp0m E hE, hp1m E hE, Fin.sum_univ_two]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Stieltjes / Smooth
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped Topology
+
+namespace MovingSofa
+
+/-- A continuously differentiable real function restricted to a compact interval, together with
+its Stieltjes density. -/
+theorem exists_intervalBV_of_hasDerivAt {a b : ℝ} (hab : a ≤ b)
+    (φ φ' : ℝ → ℝ) (hderiv : ∀ t, HasDerivAt φ (φ' t) t) (hφ' : Continuous φ') :
+    ∃ F : RightContinuousIntervalBV a b,
+      (∀ t, F.toFun t = φ t) ∧ HasIntervalStieltjesDensity F φ' := by
+  have hdiff : Differentiable ℝ φ := fun t ↦ (hderiv t).differentiableAt
+  have hcontDiff : ContDiff ℝ 1 φ := by
+    rw [contDiff_one_iff_deriv]
+    refine ⟨hdiff, ?_⟩
+    convert hφ' using 1
+    funext t
+    exact (hderiv t).deriv
+  have hBVreal : BoundedVariationOn φ (Icc a b) := by
+    simpa only [uIcc_of_le hab] using
+      (hcontDiff.contDiffOn.absolutelyContinuousOnInterval (a := a) (b := b)).boundedVariationOn
+  let ι : Icc a b → ℝ := (↑)
+  have hBV : BoundedVariationOn (φ ∘ ι) Set.univ :=
+    ne_top_of_le_ne_top hBVreal
+      (eVariationOn.comp_le_of_monotoneOn φ ι
+        (fun _ _ _ _ h ↦ h) (fun t _ ↦ t.property))
+  let F : RightContinuousIntervalBV a b :=
+    { toFun := φ ∘ ι
+      boundedVariation := hBV
+      right_continuous := fun t ↦
+        (hderiv (t : ℝ)).continuousAt.comp_continuousWithinAt
+          continuous_subtype_val.continuousWithinAt }
+  have hφ'int : Integrable φ' (volume.restrict (Icc a b)) :=
+    hφ'.continuousOn.integrableOn_compact isCompact_Icc
+  have hsubtype : Integrable (fun t : Icc a b ↦ φ' t)
+      (volume.comap (Subtype.val : Icc a b → ℝ)) :=
+    (integrableOn_iff_comap_subtypeVal measurableSet_Icc).mp hφ'int
+  have hmeasure : intervalStieltjesMeasure F =
+      (volume.comap (Subtype.val : Icc a b → ℝ)).withDensityᵥ
+        (fun t : Icc a b ↦ φ' t) := by
+    apply F.boundedVariation.vectorMeasure_eq_withDensity_of_integral_Icc
+    · exact (continuous_iff_continuousAt.2 fun t ↦ (hderiv t).continuousAt).comp
+        continuous_subtype_val
+    · exact hsubtype
+    · intro c d hcd
+      have hftc : ∫ x in (c : ℝ)..(d : ℝ), φ' x = φ d - φ c :=
+        intervalIntegral.integral_eq_sub_of_hasDerivAt
+          (fun x _ ↦ hderiv x) (hφ'.intervalIntegrable (c : ℝ) (d : ℝ))
+      have hset : Icc c d = {x : Icc a b | (x : ℝ) ∈ Icc (c : ℝ) (d : ℝ)} := rfl
+      rw [hset, integral_subtype_preimage measurableSet_Icc measurableSet_Icc,
+        Measure.restrict_restrict_of_subset
+          (Icc_subset_Icc c.property.1 d.property.2)]
+      rw [integral_Icc_eq_integral_Ioc, ← intervalIntegral.integral_of_le hcd, hftc]
+      simp [F, ι]
+  refine ⟨F, fun _ ↦ rfl, hφ'int, ?_⟩
+  intro E hE
+  have hpre : MeasurableSet {t : Icc a b | (t : ℝ) ∈ E} :=
+    hE.preimage measurable_subtype_coe
+  rw [hmeasure, withDensityᵥ_apply hsubtype hpre,
+    integral_subtype_preimage measurableSet_Icc hE]
+
+/-- A right-continuous interval bounded-variation function that agrees on its interval with an
+absolutely continuous function differentiable on the open interval has that derivative as its
+Stieltjes density. Unlike `exists_intervalBV_of_hasDerivAt` this identifies the density of a
+*given* bounded-variation function, and asks for differentiability only in the interior. -/
+theorem hasIntervalStieltjesDensity_of_hasDerivAt {a b : ℝ} (hab : a ≤ b)
+    (F : RightContinuousIntervalBV a b) (φ φ' : ℝ → ℝ)
+    (hF : ∀ t : Icc a b, F.toFun t = φ t)
+    (hac : AbsolutelyContinuousOnInterval φ a b)
+    (hderiv : ∀ t ∈ Ioo a b, HasDerivAt φ (φ' t) t)
+    (hint : IntegrableOn φ' (Icc a b)) :
+    HasIntervalStieltjesDensity F φ' := by
+  have hext : ∀ t ∈ Icc a b, stieltjesScalarExtension F t = φ t := by
+    intro t ht
+    rw [stieltjesScalarExtension, dite_eq_left ht]
+    exact hF ⟨t, ht⟩
+  have hac' : AbsolutelyContinuousOnInterval (stieltjesScalarExtension F) a b :=
+    hac.congr fun t ht ↦ (hext t (by simpa only [uIcc_of_le hab] using ht)).symm
+  obtain ⟨ρ, hρ⟩ := (intervalStieltjes_absoluteContinuity a b hab F).1.1 hac'
+  have hIoo : ρ =ᵐ[volume.restrict (Ioo a b)] φ' := by
+    filter_upwards [(intervalStieltjes_absoluteContinuity a b hab F).2 ρ hρ,
+      ae_restrict_mem measurableSet_Ioo] with t ht htmem
+    refine ht.unique ((hderiv t htmem).congr_of_eventuallyEq ?_)
+    filter_upwards [isOpen_Ioo.mem_nhds htmem] with u hu
+    exact hext u (Ioo_subset_Icc_self hu)
+  have hIcc : ρ =ᵐ[volume.restrict (Icc a b)] φ' := by
+    rwa [Measure.restrict_congr_set (MeasureTheory.Ioo_ae_eq_Icc (μ := volume))] at hIoo
+  refine ⟨hint, fun E hE ↦ ?_⟩
+  rw [hρ.2 E hE]
+  exact integral_congr_ae (hIcc.filter_mono (ae_mono Measure.restrict_le_self))
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Stieltjes / Frame
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped Topology
+
+namespace MovingSofa
+
+/-- A coordinate of the rotating normal frame is a continuous interval-BV function whose
+Stieltjes density is the corresponding tangent coordinate. -/
+theorem exists_normalVector_coordinate_intervalBV {a b : ℝ} (hab : a ≤ b) (i : Fin 2) :
+    ∃ F : RightContinuousIntervalBV a b,
+      (∀ t, F.toFun t = normalVector (((t : ℝ) : Real.Angle)) i) ∧
+      HasIntervalStieltjesDensity F
+        (fun t ↦ tangentVector ((t : ℝ) : Real.Angle) i) := by
+  refine exists_intervalBV_of_hasDerivAt hab
+    (fun t : ℝ ↦ normalVector (t : Real.Angle) i)
+    (fun t : ℝ ↦ tangentVector (t : Real.Angle) i) ?_ ?_
+  · intro t
+    fin_cases i
+    · change HasDerivAt Real.cos (-Real.sin t) t
+      exact Real.hasDerivAt_cos t
+    · change HasDerivAt Real.sin (Real.cos t) t
+      exact Real.hasDerivAt_sin t
+  · fin_cases i
+    · change Continuous (fun t : ℝ ↦ -Real.sin t)
+      fun_prop
+    · change Continuous Real.cos
+      fun_prop
+
+/-- A coordinate of the rotating tangent frame is a continuous interval-BV function whose
+Stieltjes density is the negative normal coordinate. -/
+theorem exists_tangentVector_coordinate_intervalBV {a b : ℝ} (hab : a ≤ b) (i : Fin 2) :
+    ∃ F : RightContinuousIntervalBV a b,
+      (∀ t, F.toFun t = tangentVector (((t : ℝ) : Real.Angle)) i) ∧
+      HasIntervalStieltjesDensity F
+        (fun t ↦ -normalVector ((t : ℝ) : Real.Angle) i) := by
+  refine exists_intervalBV_of_hasDerivAt hab
+    (fun t : ℝ ↦ tangentVector (t : Real.Angle) i)
+    (fun t : ℝ ↦ -normalVector (t : Real.Angle) i) ?_ ?_
+  · intro t
+    fin_cases i
+    · change HasDerivAt (fun t : ℝ ↦ -Real.sin t) (-Real.cos t) t
+      convert (Real.hasDerivAt_sin t).neg using 1
+    · change HasDerivAt Real.cos (-Real.sin t) t
+      exact Real.hasDerivAt_cos t
+  · fin_cases i
+    · change Continuous (fun t : ℝ ↦ -Real.cos t)
+      fun_prop
+    · change Continuous (fun t : ℝ ↦ -Real.sin t)
+      fun_prop
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Stieltjes / Transport
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped Function
+
+namespace MovingSofa
+
+/-- Continuous monotone surjective reparametrization preserves the Stieltjes integral. -/
+theorem intervalStieltjesIntegral_comp_monotone_surjective
+    {a b c d : ℝ} (hab : a ≤ b) (hcd : c ≤ d) (F : RightContinuousIntervalBV a b)
+    (hFc : Continuous F.toFun) (g : Set.Icc a b → ℝ) (hgc : Continuous g)
+    (φ : Set.Icc c d → Set.Icc a b) (hφc : Continuous φ) (hφ : Monotone φ)
+    (hφs : Function.Surjective φ) :
+    intervalStieltjesIntegral F g Set.univ =
+      intervalStieltjesIntegral
+        { toFun := F.toFun ∘ φ
+          boundedVariation := BoundedVariationOn.comp_monotone_surjective_Icc
+            hab F.boundedVariation hφ hφs
+          right_continuous := fun _ ↦
+            (hFc.comp hφc).continuousAt.continuousWithinAt }
+        (g ∘ φ) Set.univ := by
+  let Fφ : RightContinuousIntervalBV c d :=
+    { toFun := F.toFun ∘ φ
+      boundedVariation := BoundedVariationOn.comp_monotone_surjective_Icc
+        hab F.boundedVariation hφ hφs
+      right_continuous := fun _ ↦
+        (hFc.comp hφc).continuousAt.continuousWithinAt }
+  have hmap : (intervalStieltjesMeasure Fφ).map φ = intervalStieltjesMeasure F :=
+    BoundedVariationOn.vectorMeasure_map_comp_monotone_surjective_Icc
+      hab hcd F.boundedVariation hFc hφc hφ hφs
+  let _ : IsFiniteMeasure (intervalStieltjesMeasure Fφ).variation := by
+    change IsFiniteMeasure Fφ.boundedVariation.vectorMeasure.variation
+    exact BoundedVariationOn.instIsFiniteMeasureVariationVectorMeasure Fφ.boundedVariation
+  have hgint : (intervalStieltjesMeasure Fφ).Integrable (g ∘ φ) :=
+    (hgc.comp hφc).integrable_of_hasCompactSupport
+      (isCompact_univ.of_isClosed_subset isClosed_closure (subset_univ _))
+  have hgm : AEStronglyMeasurable g
+      ((intervalStieltjesMeasure Fφ).variation.map φ) :=
+    hgc.aestronglyMeasurable
+  unfold intervalStieltjesIntegral
+  simp only [VectorMeasure.restrict_univ]
+  rw [← hmap, MeasureTheory.VectorMeasure.integral_map hφc.measurable hgm hgint]
+  rfl
+
+/-- Restricting a continuous Stieltjes integral agrees with integration on the subinterval. -/
+theorem intervalStieltjesIntegral_Ioc_eq_inclusion
+    {a b : ℝ} (F : RightContinuousIntervalBV a b) (hFc : Continuous F.toFun)
+    (g : Set.Icc a b → ℝ) (hgc : Continuous g) (l u : Set.Icc a b) (hlu : l ≤ u) :
+    let ι : Set.Icc (l : ℝ) u → Set.Icc a b := fun x ↦
+      ⟨x, le_trans l.property.1 x.property.1, le_trans x.property.2 u.property.2⟩
+    let hfr : BoundedVariationOn (F.toFun ∘ ι) Set.univ :=
+      ne_top_of_le_ne_top F.boundedVariation (eVariationOn.comp_le_of_monotoneOn F.toFun ι
+        (fun _ _ _ _ h ↦ h) (mapsTo_univ ι _))
+    intervalStieltjesIntegral F g (Ioc l u) =
+      intervalStieltjesIntegral
+        { toFun := F.toFun ∘ ι
+          boundedVariation := hfr
+          right_continuous := fun _ ↦
+            (hFc.comp (continuous_subtype_val.subtype_mk _)).continuousAt.continuousWithinAt }
+        (g ∘ ι) Set.univ := by
+  dsimp only
+  let ι : Set.Icc (l : ℝ) u → Set.Icc a b := fun x ↦
+    ⟨x, le_trans l.property.1 x.property.1, le_trans x.property.2 u.property.2⟩
+  let hfr : BoundedVariationOn (F.toFun ∘ ι) Set.univ :=
+    ne_top_of_le_ne_top F.boundedVariation (eVariationOn.comp_le_of_monotoneOn F.toFun ι
+      (fun _ _ _ _ h ↦ h) (mapsTo_univ ι _))
+  let Fr : RightContinuousIntervalBV (l : ℝ) u :=
+    { toFun := F.toFun ∘ ι
+      boundedVariation := hfr
+      right_continuous := fun _ ↦
+        (hFc.comp (continuous_subtype_val.subtype_mk _)).continuousAt.continuousWithinAt }
+  have hmap : (intervalStieltjesMeasure Fr).map ι =
+      (intervalStieltjesMeasure F).restrict (Ioc l u) :=
+    BoundedVariationOn.vectorMeasure_map_Icc_inclusion F.boundedVariation hFc l u hlu
+  let _ : IsFiniteMeasure (intervalStieltjesMeasure Fr).variation := by
+    change IsFiniteMeasure hfr.vectorMeasure.variation
+    exact BoundedVariationOn.instIsFiniteMeasureVariationVectorMeasure hfr
+  have hgint : (intervalStieltjesMeasure Fr).Integrable (g ∘ ι) :=
+    (hgc.comp (continuous_subtype_val.subtype_mk _)).integrable_of_hasCompactSupport
+      (isCompact_univ.of_isClosed_subset isClosed_closure (subset_univ _))
+  have hgm : AEStronglyMeasurable g
+      ((intervalStieltjesMeasure Fr).variation.map ι) :=
+    hgc.aestronglyMeasurable
+  unfold intervalStieltjesIntegral
+  simp only [VectorMeasure.restrict_univ]
+  rw [← hmap, MeasureTheory.VectorMeasure.integral_map
+    (continuous_subtype_val.subtype_mk _).measurable hgm hgint]
+  rfl
+
+/-- A continuous Stieltjes integral splits over a finite monotone partition. -/
+theorem intervalStieltjesIntegral_eq_sum_Ioc
+    {a b : ℝ} (F : RightContinuousIntervalBV a b) (hFc : Continuous F.toFun)
+    (g : Set.Icc a b → ℝ) (hgc : Continuous g) {n : ℕ}
+    (cuts : Fin (n + 1) → Set.Icc a b) (hcuts : Monotone cuts)
+    (hzero : (cuts 0 : ℝ) = a) (hlast : (cuts (Fin.last n) : ℝ) = b) :
+    intervalStieltjesIntegral F g Set.univ =
+      ∑ i : Fin n, intervalStieltjesIntegral F g (Ioc (cuts i.castSucc) (cuts i.succ)) := by
+  have hab : a ≤ b := by
+    calc
+      a = cuts 0 := hzero.symm
+      _ ≤ cuts (Fin.last n) := hcuts (Fin.zero_le _)
+      _ = b := hlast
+  let _ : Fact (a ≤ b) := ⟨hab⟩
+  let _ : IsFiniteMeasure (intervalStieltjesMeasure F).variation := by
+    exact BoundedVariationOn.instIsFiniteMeasureVariationVectorMeasure F.boundedVariation
+  unfold intervalStieltjesIntegral
+  have hmeas (i : Fin n) : MeasurableSet (Ioc (cuts i.castSucc) (cuts i.succ)) :=
+    measurableSet_Ioc
+  have hall : (intervalStieltjesMeasure F).Integrable g :=
+    hgc.integrable_of_hasCompactSupport
+      (isCompact_univ.of_isClosed_subset isClosed_closure (subset_univ _))
+  have hint (i : Fin n) :
+      (intervalStieltjesMeasure F).IntegrableOn g (Ioc (cuts i.castSucc) (cuts i.succ)) :=
+    hall.integrableOn
+  have hpair : Set.Pairwise ((Finset.univ : Finset (Fin n)) : Set (Fin n))
+      (Disjoint on fun i ↦ Ioc (cuts i.castSucc) (cuts i.succ)) := by
+    intro i _ j _ hij
+    change Disjoint (Ioc (cuts i.castSucc) (cuts i.succ))
+      (Ioc (cuts j.castSucc) (cuts j.succ))
+    rw [Set.disjoint_left]
+    intro x hxi hxj
+    rcases lt_or_gt_of_ne hij with hij' | hji'
+    · have hij_fin : i.succ ≤ j.castSucc := by
+        simpa using Nat.succ_le_of_lt hij'
+      exact (not_lt_of_ge (le_trans hxi.2 (hcuts hij_fin))) hxj.1
+    · have hji_fin : j.succ ≤ i.castSucc := by
+        simpa using Nat.succ_le_of_lt hji'
+      exact (not_lt_of_ge (le_trans hxj.2 (hcuts hji_fin))) hxi.1
+  rw [← MeasureTheory.VectorMeasure.setIntegral_biUnion_finset Finset.univ
+    (fun i _ ↦ hmeas i) hpair (fun i _ ↦ hint i)]
+  have hunion :
+      (⋃ i : Fin n, Ioc (cuts i.castSucc) (cuts i.succ)) = Set.univ \ {cuts 0} := by
+    rw [hcuts.iUnion_Ioc_fin]
+    ext x
+    simp only [mem_Ioc, mem_sdiff, mem_univ, true_and, mem_singleton_iff]
+    constructor
+    · rintro ⟨hx0, _⟩ hxe
+      exact hx0.ne' hxe
+    · intro hxe
+      exact ⟨lt_of_le_of_ne (by
+        change (cuts 0 : ℝ) ≤ (x : ℝ)
+        simpa [hzero] using x.property.1) (fun hx ↦ hxe hx.symm),
+        by
+          change (x : ℝ) ≤ (cuts (Fin.last n) : ℝ)
+          simpa [hlast] using x.property.2⟩
+  simp only [Finset.mem_univ, iUnion_true]
+  rw [hunion]
+  rw [MeasureTheory.VectorMeasure.setIntegral_sdiff (s := Set.univ) (t := {cuts 0})
+    MeasurableSet.univ (measurableSet_singleton _) hall.integrableOn (subset_univ _)]
+  rw [MeasureTheory.VectorMeasure.integral_singleton]
+  have hz : intervalStieltjesMeasure F {cuts 0} = 0 :=
+    BoundedVariationOn.vectorMeasure_singleton_eq_zero_of_continuous F.boundedVariation hFc _
+  rw [hz]
+  simp
+
+/-- Reversing both continuous integrand and BV integrator negates their Stieltjes integral. -/
+theorem intervalStieltjesIntegral_comp_reverse
+    {a b : ℝ} (hab : a ≤ b) (F : RightContinuousIntervalBV a b)
+    (hFc : Continuous F.toFun) (g : Set.Icc a b → ℝ) (hgc : Continuous g) :
+    let r := Set.Icc.reverse hab
+    let hfr := BoundedVariationOn.comp_antitone_surjective_Icc hab F.boundedVariation
+      (Set.Icc.antitone_reverse hab) (Set.Icc.surjective_reverse hab)
+    intervalStieltjesIntegral
+        { toFun := F.toFun ∘ r
+          boundedVariation := hfr
+          right_continuous := fun _ ↦
+            (hFc.comp (Set.Icc.continuous_reverse hab)).continuousAt.continuousWithinAt }
+        (g ∘ r) Set.univ =
+      -intervalStieltjesIntegral F g Set.univ := by
+  dsimp only
+  let r := Set.Icc.reverse hab
+  let hfr := BoundedVariationOn.comp_antitone_surjective_Icc hab F.boundedVariation
+    (Set.Icc.antitone_reverse hab) (Set.Icc.surjective_reverse hab)
+  let Fr : RightContinuousIntervalBV a b :=
+    { toFun := F.toFun ∘ r
+      boundedVariation := hfr
+      right_continuous := fun _ ↦
+        (hFc.comp (Set.Icc.continuous_reverse hab)).continuousAt.continuousWithinAt }
+  have hmap : (intervalStieltjesMeasure Fr).map r = -intervalStieltjesMeasure F :=
+    BoundedVariationOn.vectorMeasure_map_reverse_Icc hab F.boundedVariation hFc
+  let _ : IsFiniteMeasure (intervalStieltjesMeasure Fr).variation := by
+    change IsFiniteMeasure hfr.vectorMeasure.variation
+    exact BoundedVariationOn.instIsFiniteMeasureVariationVectorMeasure hfr
+  have hgint : (intervalStieltjesMeasure Fr).Integrable (g ∘ r) :=
+    (hgc.comp (Set.Icc.continuous_reverse hab)).integrable_of_hasCompactSupport
+      (isCompact_univ.of_isClosed_subset isClosed_closure (subset_univ _))
+  have hgm : AEStronglyMeasurable g
+      ((intervalStieltjesMeasure Fr).variation.map r) :=
+    hgc.aestronglyMeasurable
+  unfold intervalStieltjesIntegral
+  simp only [MeasureTheory.VectorMeasure.restrict_univ]
+  rw [← MeasureTheory.VectorMeasure.integral_neg_vectorMeasure, ← hmap,
+    MeasureTheory.VectorMeasure.integral_map
+      (Set.Icc.continuous_reverse hab).measurable hgm hgint]
+  rfl
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Stieltjes / Continuous
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+
+namespace MovingSofa
+
+/-- A continuous function is integrable against a BV Stieltjes measure on a compact interval. -/
+theorem RightContinuousIntervalBV.integrable_of_continuous {a b : ℝ} (F :
+  RightContinuousIntervalBV a b)
+    {g : Set.Icc a b → ℝ} (hg : Continuous g) : (intervalStieltjesMeasure F).Integrable g := by
+  let _ : IsFiniteMeasure (intervalStieltjesMeasure F).variation :=
+    BoundedVariationOn.instIsFiniteMeasureVariationVectorMeasure F.boundedVariation
+  exact hg.integrable_of_hasCompactSupport
+    (isCompact_univ.of_isClosed_subset isClosed_closure (subset_univ _))
+
+private theorem intervalStieltjesIntegral_univ_eq_Ioc_of_continuous
+    {a b : ℝ} (hab : a ≤ b) (F : RightContinuousIntervalBV a b)
+    (hF : Continuous F.toFun) (g : Set.Icc a b → ℝ) (hg : Continuous g) :
+    intervalStieltjesIntegral F g Set.univ =
+      intervalStieltjesIntegral F g {t | a < (t : ℝ)} := by
+  let a' : Set.Icc a b := ⟨a, le_rfl, hab⟩
+  let b' : Set.Icc a b := ⟨b, hab, le_rfl⟩
+  let cuts : Fin 2 → Set.Icc a b := Fin.cases a' (fun _ ↦ b')
+  have hcuts : Monotone cuts := by
+    intro i j hij
+    fin_cases i <;> fin_cases j <;> simp only [Fin.zero_eta, Fin.isValue, Std.le_refl,
+      Nat.reduceAdd, Fin.cases_zero, cuts, a', b',
+      Fin.mk_one, zero_le, nonpos_iff_eq_zero, one_ne_zero] at hij ⊢
+    exact hab
+  have hparts := intervalStieltjesIntegral_eq_sum_Ioc F hF g hg cuts hcuts rfl rfl
+  have hset : Set.Ioc a' b' = {t : Set.Icc a b | a < (t : ℝ)} := by
+    ext t
+    simp only [Set.mem_Ioc, Set.mem_ofPred_eq]
+    constructor
+    · exact fun ht ↦ ht.1
+    · exact fun ht ↦ ⟨ht, t.property.2⟩
+  dsimp [cuts] at hparts
+  simpa [Fin.sum_univ_succ, hset] using hparts
+
+/-- For a continuous driver and a continuous integrand, the closed-interval Stieltjes integral
+agrees with the open-interval one: neither endpoint carries an atom. -/
+theorem intervalStieltjesIntegral_univ_eq_Ioo_of_continuous {a b : ℝ} (hab : a ≤ b)
+    (F : RightContinuousIntervalBV a b) (hF : Continuous F.toFun)
+    (g : Set.Icc a b → ℝ) (hg : Continuous g) :
+    intervalStieltjesIntegral F g Set.univ =
+      intervalStieltjesIntegral F g (Set.Ioo ⟨a, le_rfl, hab⟩ ⟨b, hab, le_rfl⟩) := by
+  set a' : Set.Icc a b := ⟨a, le_rfl, hab⟩
+  set b' : Set.Icc a b := ⟨b, hab, le_rfl⟩
+  have hdiff : (Set.univ : Set (Set.Icc a b)) \ Set.Ioo a' b' ⊆ {a'} ∪ {b'} := by
+    intro t ht
+    rcases lt_or_ge a' t with hlt | hle
+    · exact Or.inr (le_antisymm t.property.2 (not_lt.mp fun h ↦ ht.2 ⟨hlt, h⟩))
+    · exact Or.inl (le_antisymm hle t.property.1)
+  have hzero : intervalStieltjesIntegral F g
+      ((Set.univ : Set (Set.Icc a b)) \ Set.Ioo a' b') = 0 := by
+    refine VectorMeasure.setIntegral_of_variation_apply_eq_zero _ (measure_mono_null hdiff ?_)
+    refine measure_union_null ?_ ?_ <;>
+      · rw [intervalStieltjesMeasure, F.boundedVariation.variation_vectorMeasure_singleton,
+          hF.continuousAt.continuousWithinAt.rightLim_eq,
+          hF.continuousAt.continuousWithinAt.leftLim_eq]
+        simp
+  have hsplit := VectorMeasure.setIntegral_inter_add_sdiff
+    (μ := intervalStieltjesMeasure F) (B := ContinuousLinearMap.mul ℝ ℝ) (f := g)
+    (s := (Set.univ : Set (Set.Icc a b))) (t := Set.Ioo a' b')
+    MeasurableSet.univ measurableSet_Ioo (F.integrable_of_continuous hg).integrableOn
+  rw [Set.univ_inter] at hsplit
+  change intervalStieltjesIntegral F g (Set.Ioo a' b') +
+    intervalStieltjesIntegral F g ((Set.univ : Set (Set.Icc a b)) \ Set.Ioo a' b') =
+    intervalStieltjesIntegral F g Set.univ at hsplit
+  rw [hzero, add_zero] at hsplit
+  exact hsplit.symm
+
+/-- Integration by parts for continuous BV functions on the full compact interval. -/
+theorem intervalStieltjes_integration_by_parts_of_continuous
+    (a b : ℝ) (hab : a ≤ b) (F G : RightContinuousIntervalBV a b)
+    (hF : Continuous F.toFun) (hG : Continuous G.toFun) :
+    intervalStieltjesIntegral F G.toFun Set.univ +
+        intervalStieltjesIntegral G F.toFun Set.univ =
+      F.toFun ⟨b, hab, le_rfl⟩ * G.toFun ⟨b, hab, le_rfl⟩ -
+        F.toFun ⟨a, le_rfl, hab⟩ * G.toFun ⟨a, le_rfl, hab⟩ := by
+  obtain ⟨left, hleft, hparts⟩ := intervalStieltjes_integration_by_parts a b hab F G
+  have hleft_eq : Set.EqOn left F.toFun {t | a < (t : ℝ)} := by
+    intro t ht
+    change a < (t : ℝ) at ht
+    have hnonempty : (Set.Iio t).Nonempty := by
+      let s : Set.Icc a b := ⟨(a + (t : ℝ)) / 2, by constructor <;> linarith [t.property.2]⟩
+      exact ⟨s, by change (a + (t : ℝ)) / 2 < (t : ℝ); linarith⟩
+    have hne : (nhdsWithin t (Set.Iio t)).NeBot :=
+      nhdsWithin_Iio_neBot' hnonempty le_rfl
+    let _ : (nhdsWithin t (Set.Iio t)).NeBot := hne
+    have hcont : Filter.Tendsto F.toFun (nhdsWithin t (Set.Iio t))
+        (nhds (F.toFun t)) := hF.continuousAt.continuousWithinAt
+    exact tendsto_nhds_unique (hleft t ht) hcont
+  have hintegrand : intervalStieltjesIntegral G left {t | a < (t : ℝ)} =
+      intervalStieltjesIntegral G F.toFun {t | a < (t : ℝ)} := by
+    unfold intervalStieltjesIntegral
+    exact MeasureTheory.VectorMeasure.setIntegral_congr_fun hleft_eq
+  rw [intervalStieltjesIntegral_univ_eq_Ioc_of_continuous hab F hF G.toFun hG,
+    intervalStieltjesIntegral_univ_eq_Ioc_of_continuous hab G hG F.toFun hF,
+    ← hintegrand]
+  exact hparts
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Stieltjes / Affine
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open MeasureTheory Set
+
+private def unitIntervalIdentity : RightContinuousIntervalBV 0 1 where
+  toFun t := t
+  boundedVariation := by
+    apply ((show Monotone ((↑) : Set.Icc (0 : ℝ) 1 → ℝ) from fun _ _ h ↦ h).monotoneOn
+      Set.univ).boundedVariationOn (C := 1)
+    intro t _
+    rw [abs_of_nonneg t.property.1]
+    exact t.property.2
+  right_continuous t := continuous_subtype_val.continuousAt.continuousWithinAt
+
+private theorem continuous_unitIntervalIdentity :
+    Continuous unitIntervalIdentity.toFun :=
+  continuous_subtype_val
+
+/-- A continuous BV driver carries total Stieltjes mass equal to its increment. -/
+theorem intervalStieltjesMeasure_univ_of_continuous {a b : ℝ} (hab : a ≤ b)
+    (Q : RightContinuousIntervalBV a b) (hQ : Continuous Q.toFun) :
+    intervalStieltjesMeasure Q univ =
+      Q.toFun ⟨b, hab, le_rfl⟩ - Q.toFun ⟨a, le_rfl, hab⟩ := by
+  have hu : (univ : Set (Icc a b)) = Icc ⟨a, le_rfl, hab⟩ ⟨b, hab, le_rfl⟩ := by
+    ext x
+    simp only [mem_univ, mem_Icc, true_iff]
+    exact x.property
+  unfold intervalStieltjesMeasure
+  rw [hu, Q.boundedVariation.vectorMeasure_Icc (show
+      (⟨a, le_rfl, hab⟩ : Icc a b) ≤ ⟨b, hab, le_rfl⟩ from hab),
+    hQ.continuousAt.continuousWithinAt.rightLim_eq,
+    hQ.continuousAt.continuousWithinAt.leftLim_eq]
+
+/-- An affine function of a continuous BV driver has that driver's Stieltjes measure, scaled by
+the affine map's slope. -/
+theorem intervalStieltjesMeasure_affine {a b : ℝ}
+    (Q F : RightContinuousIntervalBV a b) (hQ : Continuous Q.toFun) (u v : ℝ)
+    (hF : ∀ t, F.toFun t = u + v * Q.toFun t) :
+    intervalStieltjesMeasure F = v • intervalStieltjesMeasure Q := by
+  have hFc : Continuous F.toFun := by
+    rw [show F.toFun = fun t ↦ u + v * Q.toFun t from funext hF]
+    exact continuous_const.add (continuous_const.mul hQ)
+  apply VectorMeasure.ext_of_Icc
+  intro x y hxy
+  simp only [smul_apply]
+  unfold intervalStieltjesMeasure
+  rw [F.boundedVariation.vectorMeasure_Icc hxy, Q.boundedVariation.vectorMeasure_Icc hxy,
+    hFc.continuousAt.continuousWithinAt.rightLim_eq,
+    hFc.continuousAt.continuousWithinAt.leftLim_eq,
+    hQ.continuousAt.continuousWithinAt.rightLim_eq,
+    hQ.continuousAt.continuousWithinAt.leftLim_eq, hF x, hF y]
+  ring
+
+/-- Shifting a continuous BV driver and its integrand by constants shifts the interval Stieltjes
+integral over the whole parameter interval by the integrand's shift times the driver's increment;
+the driver's own shift has no effect. -/
+theorem intervalStieltjesIntegral_add_const {a b : ℝ} (hab : a ≤ b)
+    (F G : RightContinuousIntervalBV a b) (hF : Continuous F.toFun) (c : ℝ)
+    (hFG : ∀ t, G.toFun t = F.toFun t + c) (g : Icc a b → ℝ) (hg : Continuous g) (d : ℝ) :
+    intervalStieltjesIntegral G (fun t ↦ g t + d) univ =
+      intervalStieltjesIntegral F g univ +
+        d * (F.toFun ⟨b, hab, le_rfl⟩ - F.toFun ⟨a, le_rfl, hab⟩) := by
+  let _ : IsFiniteMeasure (intervalStieltjesMeasure F).variation :=
+    BoundedVariationOn.instIsFiniteMeasureVariationVectorMeasure F.boundedVariation
+  have hmeas : intervalStieltjesMeasure G = intervalStieltjesMeasure F := by
+    rw [intervalStieltjesMeasure_affine F G hF c 1 fun t ↦ by rw [hFG t]; ring, one_smul]
+  have hsum : (fun t ↦ g t + d) = g + fun _ ↦ d := rfl
+  rw [intervalStieltjesIntegral_univ, intervalStieltjesIntegral_univ, hmeas, hsum,
+    VectorMeasure.integral_add (F.integrable_of_continuous hg)
+      (F.integrable_of_continuous continuous_const),
+    VectorMeasure.integral_const, ← intervalStieltjesMeasure_univ_of_continuous hab F hF]
+  simp
+
+/-- The signed cross integral of two affine functions of one continuous BV driver. -/
+theorem intervalStieltjesIntegral_affine_driver_cross {a b : ℝ} (hab : a ≤ b)
+    (Q F G : RightContinuousIntervalBV a b) (hQ : Continuous Q.toFun)
+    (u v w z : ℝ) (hF : ∀ t, F.toFun t = u + v * Q.toFun t)
+    (hG : ∀ t, G.toFun t = w + z * Q.toFun t) :
+    intervalStieltjesIntegral G F.toFun univ - intervalStieltjesIntegral F G.toFun univ =
+      (u * z - w * v) * (Q.toFun ⟨b, hab, le_rfl⟩ - Q.toFun ⟨a, le_rfl, hab⟩) := by
+  let _ : IsFiniteMeasure (intervalStieltjesMeasure Q).variation :=
+    BoundedVariationOn.instIsFiniteMeasureVariationVectorMeasure Q.boundedVariation
+  have hFm := intervalStieltjesMeasure_affine Q F hQ u v hF
+  have hGm := intervalStieltjesMeasure_affine Q G hQ w z hG
+  have hi (H : RightContinuousIntervalBV a b) (c d : ℝ)
+      (hH : ∀ t, H.toFun t = c + d * Q.toFun t) :
+      intervalStieltjesIntegral Q H.toFun univ =
+        c * (Q.toFun ⟨b, hab, le_rfl⟩ - Q.toFun ⟨a, le_rfl, hab⟩) +
+          d * intervalStieltjesIntegral Q Q.toFun univ := by
+    have heq : H.toFun = (fun _ ↦ c) + d • Q.toFun := by
+      funext t
+      exact hH t
+    have hc := Q.integrable_of_continuous (g := fun _ ↦ c) continuous_const
+    have hd := Q.integrable_of_continuous (g := d • Q.toFun) (continuous_const.smul hQ)
+    unfold intervalStieltjesIntegral
+    simp only [VectorMeasure.restrict_univ, heq]
+    rw [VectorMeasure.integral_add hc hd, VectorMeasure.integral_smul,
+      VectorMeasure.integral_const, intervalStieltjesMeasure_univ_of_continuous hab Q hQ]
+    simp
+  have hFI := hi F u v hF
+  have hGI := hi G w z hG
+  unfold intervalStieltjesIntegral at hFI hGI ⊢
+  simp only [VectorMeasure.restrict_univ] at hFI hGI ⊢
+  rw [hFm, hGm, VectorMeasure.integral_smul_vectorMeasure,
+    VectorMeasure.integral_smul_vectorMeasure, hFI, hGI]
+  ring
+
+theorem intervalStieltjesIntegral_affine_cross
+    (F G : RightContinuousIntervalBV 0 1) (a b c d : ℝ)
+    (hF : ∀ t, F.toFun t = a + b * (t : ℝ))
+    (hG : ∀ t, G.toFun t = c + d * (t : ℝ)) :
+    intervalStieltjesIntegral G F.toFun Set.univ -
+        intervalStieltjesIntegral F G.toFun Set.univ = a * d - c * b := by
+  simpa only [unitIntervalIdentity, sub_zero, mul_one] using
+    intervalStieltjesIntegral_affine_driver_cross (by norm_num : (0 : ℝ) ≤ 1)
+      unitIntervalIdentity F G continuous_unitIntervalIdentity a b c d hF hG
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Riemann-Stieltjes sums for continuous integrands
+
+Left-endpoint sums over finite monotone partitions converge to the interval Stieltjes
+integral of a continuous integrand against a continuous BV integrator, uniformly in the
+mesh of the partition.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+
+namespace MovingSofa
+
+/-- Two Stieltjes integrals differ by at most the uniform distance of their integrands
+times the total variation of the integrator. -/
+theorem intervalStieltjesIntegral_sub_le_of_abs_sub_le
+    {a b : ℝ} (F : RightContinuousIntervalBV a b) {g h : Set.Icc a b → ℝ}
+    (hg : (intervalStieltjesMeasure F).Integrable g)
+    (hh : (intervalStieltjesMeasure F).Integrable h)
+    {ε : ℝ} (hε : 0 ≤ ε)
+    (hbound : ∀ᵐ t ∂(intervalStieltjesMeasure F).variation, |g t - h t| ≤ ε) :
+    |intervalStieltjesIntegral F g univ - intervalStieltjesIntegral F h univ| ≤
+      ε * (intervalStieltjesMeasure F).variation.real univ := by
+  let _ : IsFiniteMeasure (intervalStieltjesMeasure F).variation :=
+    BoundedVariationOn.instIsFiniteMeasureVariationVectorMeasure F.boundedVariation
+  unfold intervalStieltjesIntegral
+  simp only [VectorMeasure.restrict_univ]
+  rw [← VectorMeasure.integral_sub hg hh, ← Real.norm_eq_abs]
+  have h := VectorMeasure.norm_integral_le_of_norm_le_const
+    (B := ContinuousLinearMap.mul ℝ ℝ) (by simpa only [Real.norm_eq_abs] using hbound)
+  refine h.trans ?_
+  calc
+    ε * ‖ContinuousLinearMap.mul ℝ ℝ‖ * (intervalStieltjesMeasure F).variation.real univ ≤
+        ε * 1 * (intervalStieltjesMeasure F).variation.real univ := by
+      gcongr
+      exact ContinuousLinearMap.opNorm_mul_le ℝ ℝ
+    _ = _ := by ring
+
+private theorem intervalStieltjesIntegral_sum_indicator_Ioc
+    {a b : ℝ} (F : RightContinuousIntervalBV a b) {ι : Type*}
+    (s : Finset ι) (l u : ι → Set.Icc a b) (c : ι → ℝ)
+    (hlu : ∀ i ∈ s, l i ≤ u i) :
+    intervalStieltjesIntegral F
+        (fun t ↦ ∑ i ∈ s, (Ioc (l i) (u i)).indicator (fun _ ↦ c i) t) univ =
+      ∑ i ∈ s, c i * (F.toFun (u i) - F.toFun (l i)) := by
+  classical
+  let _ : IsFiniteMeasure (intervalStieltjesMeasure F).variation :=
+    BoundedVariationOn.instIsFiniteMeasureVariationVectorMeasure F.boundedVariation
+  unfold intervalStieltjesIntegral
+  rw [VectorMeasure.restrict_univ, VectorMeasure.integral_finsetSum]
+  · apply Finset.sum_congr rfl
+    intro i hi
+    rw [VectorMeasure.integral_indicator_const _ measurableSet_Ioc]
+    change c i * F.boundedVariation.vectorMeasure (Ioc (l i) (u i)) = _
+    rw [F.boundedVariation.vectorMeasure_Ioc (hlu i hi),
+      (F.right_continuous (u i)).rightLim_eq, (F.right_continuous (l i)).rightLim_eq]
+  · intro i _
+    exact (MeasureTheory.integrable_const (c i)).indicator measurableSet_Ioc
+
+/-- The error in a left-endpoint Stieltjes sum is controlled by the cell oscillation of the
+integrand times the total variation of the integrator. -/
+theorem intervalStieltjesIntegral_sub_sum_le_of_oscillation
+    {a b : ℝ} (F : RightContinuousIntervalBV a b) (hF : Continuous F.toFun)
+    {g : Set.Icc a b → ℝ} (hg : Continuous g) {n : ℕ}
+    (cuts : Fin (n + 1) → Set.Icc a b) (hcuts : Monotone cuts)
+    (hzero : (cuts 0 : ℝ) = a) (hlast : (cuts (Fin.last n) : ℝ) = b)
+    {ε : ℝ} (hε : 0 ≤ ε)
+    (hosc : ∀ (i : Fin n) t, t ∈ Ioc (cuts i.castSucc) (cuts i.succ) →
+      |g t - g (cuts i.castSucc)| ≤ ε) :
+    |intervalStieltjesIntegral F g univ -
+        ∑ i : Fin n, g (cuts i.castSucc) *
+          (F.toFun (cuts i.succ) - F.toFun (cuts i.castSucc))| ≤
+      ε * (intervalStieltjesMeasure F).variation.real univ := by
+  classical
+  let _ : IsFiniteMeasure (intervalStieltjesMeasure F).variation :=
+    BoundedVariationOn.instIsFiniteMeasureVariationVectorMeasure F.boundedVariation
+  let step : Set.Icc a b → ℝ := fun t ↦ ∑ i : Fin n,
+    (Ioc (cuts i.castSucc) (cuts i.succ)).indicator (fun _ ↦ g (cuts i.castSucc)) t
+  have hstep : (intervalStieltjesMeasure F).Integrable step := by
+    apply MeasureTheory.integrable_finsetSum
+    intro i _
+    exact (MeasureTheory.integrable_const _).indicator measurableSet_Ioc
+  have heval : intervalStieltjesIntegral F step univ =
+      ∑ i : Fin n, g (cuts i.castSucc) *
+        (F.toFun (cuts i.succ) - F.toFun (cuts i.castSucc)) :=
+    intervalStieltjesIntegral_sum_indicator_Ioc (ι := Fin n) F Finset.univ
+      (fun i ↦ cuts i.castSucc) (fun i ↦ cuts i.succ)
+      (fun i ↦ g (cuts i.castSucc)) (fun i _ ↦ hcuts (Fin.castSucc_le_succ i))
+  rw [← heval]
+  apply intervalStieltjesIntegral_sub_le_of_abs_sub_le F
+    (F.integrable_of_continuous hg) hstep hε
+  have hatom : (intervalStieltjesMeasure F).variation {cuts 0} = 0 := by
+    rw [VectorMeasure.variation_apply_singleton]
+    have hz : intervalStieltjesMeasure F {cuts 0} = 0 :=
+      BoundedVariationOn.vectorMeasure_singleton_eq_zero_of_continuous
+        F.boundedVariation hF _
+    simp [hz]
+  have hae : ∀ᵐ t ∂(intervalStieltjesMeasure F).variation, t ≠ cuts 0 := by
+    rw [ae_iff]
+    have hset : {x : Set.Icc a b | ¬x ≠ cuts 0} = {cuts 0} := by
+      ext x
+      simp
+    rw [hset]
+    exact hatom
+  filter_upwards [hae] with t ht
+  have htcell : t ∈ ⋃ i : Fin n, Ioc (cuts i.castSucc) (cuts i.succ) := by
+    rw [hcuts.iUnion_Ioc_fin]
+    refine ⟨lt_of_le_of_ne ?_ (Ne.symm ht), ?_⟩
+    · change (cuts 0 : ℝ) ≤ (t : ℝ)
+      simpa [hzero] using t.property.1
+    · change (t : ℝ) ≤ (cuts (Fin.last n) : ℝ)
+      simpa [hlast] using t.property.2
+  obtain ⟨i, hi⟩ := Set.mem_iUnion.mp htcell
+  have hvalue : step t = g (cuts i.castSucc) := by
+    dsimp [step]
+    rw [Finset.sum_eq_single i]
+    · exact Set.indicator_of_mem hi _
+    · intro j _ hji
+      apply Set.indicator_of_notMem
+      intro hj
+      rcases lt_or_gt_of_ne hji with hji | hij
+      · have hji' : j.succ ≤ i.castSucc := by simpa using Nat.succ_le_of_lt hji
+        exact (not_lt_of_ge (hj.2.trans (hcuts hji'))) hi.1
+      · have hij' : i.succ ≤ j.castSucc := by simpa using Nat.succ_le_of_lt hij
+        exact (not_lt_of_ge (hi.2.trans (hcuts hij'))) hj.1
+    · simp
+  rw [hvalue]
+  exact hosc i t hi
+
+/-- Sufficiently fine partitions approximate a continuous Stieltjes integrand uniformly. -/
+theorem exists_mesh_bound_intervalStieltjesIntegral_sub_sum
+    {a b : ℝ} (F : RightContinuousIntervalBV a b) (hF : Continuous F.toFun)
+    {g : Set.Icc a b → ℝ} (hg : Continuous g) {ε : ℝ} (hε : 0 < ε) :
+    ∃ δ > 0, ∀ {n : ℕ} (cuts : Fin (n + 1) → Set.Icc a b),
+      Monotone cuts → (cuts 0 : ℝ) = a → (cuts (Fin.last n) : ℝ) = b →
+      (∀ i : Fin n, (cuts i.succ : ℝ) - (cuts i.castSucc : ℝ) < δ) →
+      |intervalStieltjesIntegral F g univ -
+          ∑ i : Fin n, g (cuts i.castSucc) *
+            (F.toFun (cuts i.succ) - F.toFun (cuts i.castSucc))| ≤
+        ε * (intervalStieltjesMeasure F).variation.real univ := by
+  obtain ⟨δ, hδ, hmod⟩ := Metric.uniformContinuous_iff.mp
+    (CompactSpace.uniformContinuous_of_continuous hg) ε hε
+  refine ⟨δ, hδ, fun cuts hcuts hzero hlast hmesh ↦ ?_⟩
+  apply intervalStieltjesIntegral_sub_sum_le_of_oscillation F hF hg cuts hcuts hzero hlast
+    hε.le
+  intro i t ht
+  have hdist : dist t (cuts i.castSucc) < δ := by
+    rw [Subtype.dist_eq, Real.dist_eq, abs_of_nonneg
+      (sub_nonneg.mpr (show (cuts i.castSucc : ℝ) ≤ (t : ℝ) from ht.1.le))]
+    exact (sub_le_sub_right (show (t : ℝ) ≤ (cuts i.succ : ℝ) from ht.2) _).trans_lt
+      (hmesh i)
+  simpa only [Real.dist_eq] using (hmod hdist).le
+
+/-- Left-endpoint Stieltjes sums converge along any family of partitions whose mesh
+tends to zero. -/
+theorem tendsto_stieltjesSum_of_mesh_tendsto_zero
+    {a b : ℝ} (F : RightContinuousIntervalBV a b) (hF : Continuous F.toFun)
+    {g : Set.Icc a b → ℝ} (hg : Continuous g) (N : ℕ → ℕ)
+    (cuts : ∀ k, Fin (N k + 1) → Set.Icc a b)
+    (hcuts : ∀ k, Monotone (cuts k))
+    (hzero : ∀ k, (cuts k 0 : ℝ) = a)
+    (hlast : ∀ k, (cuts k (Fin.last (N k)) : ℝ) = b)
+    (hmesh : ∀ δ > 0, ∀ᶠ k in Filter.atTop, ∀ i : Fin (N k),
+      (cuts k i.succ : ℝ) - (cuts k i.castSucc : ℝ) < δ) :
+    Filter.Tendsto (fun k ↦ ∑ i : Fin (N k), g (cuts k i.castSucc) *
+      (F.toFun (cuts k i.succ) - F.toFun (cuts k i.castSucc)))
+      Filter.atTop (nhds (intervalStieltjesIntegral F g univ)) := by
+  rw [Metric.tendsto_nhds]
+  intro ε hε
+  set V := (intervalStieltjesMeasure F).variation.real univ with hVdef
+  have hV : 0 ≤ V := measureReal_nonneg
+  have hpos : 0 < ε / (V + 1) := div_pos hε (by positivity)
+  obtain ⟨δ, hδ, happrox⟩ :=
+    exists_mesh_bound_intervalStieltjesIntegral_sub_sum F hF hg hpos
+  filter_upwards [hmesh δ hδ] with k hk
+  have hbound := happrox (cuts k) (hcuts k) (hzero k) (hlast k) hk
+  rw [Real.dist_eq, abs_sub_comm]
+  apply hbound.trans_lt
+  have hcancel : ε / (V + 1) * (V + 1) = ε :=
+    div_mul_cancel₀ ε (by positivity)
+  change ε / (V + 1) * V < ε
+  nlinarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Stieltjes / Shift
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped Topology
+
+namespace MovingSofa
+
+/-- Translate a continuous interval-BV representative and its Stieltjes measure. -/
+theorem exists_intervalBV_shift_add {a b c : ℝ} (hab : a ≤ b)
+    (f : RightContinuousIntervalBV (a + c) (b + c)) (hfc : Continuous f.toFun) :
+    ∃ g : RightContinuousIntervalBV a b,
+      (∀ t, g.toFun t = f.toFun
+        ⟨(t : ℝ) + c, by constructor <;> linarith [t.property.1, t.property.2]⟩) ∧
+      ∀ E : Set (Icc a b), MeasurableSet E →
+        intervalStieltjesMeasure g E = intervalStieltjesMeasure f
+          ((fun t : Icc a b ↦
+            ⟨(t : ℝ) + c, by constructor <;> linarith [t.property.1, t.property.2]⟩) '' E) := by
+  let φ : Icc a b → Icc (a + c) (b + c) := fun t ↦
+    ⟨(t : ℝ) + c, by constructor <;> linarith [t.property.1, t.property.2]⟩
+  let ψ : Icc (a + c) (b + c) → Icc a b := fun t ↦
+    ⟨(t : ℝ) - c, by constructor <;> linarith [t.property.1, t.property.2]⟩
+  have hφc : Continuous φ := (continuous_subtype_val.add_const c).subtype_mk _
+  have hψc : Continuous ψ := (continuous_subtype_val.sub continuous_const).subtype_mk _
+  have hφm : Monotone φ := fun x y hxy ↦ by
+    change (x : ℝ) + c ≤ (y : ℝ) + c
+    linarith [show (x : ℝ) ≤ (y : ℝ) from hxy]
+  have hφs : Function.Surjective φ := by
+    intro y
+    refine ⟨ψ y, ?_⟩
+    apply Subtype.ext
+    simp [φ, ψ]
+  have hφi : Function.Injective φ := by
+    intro x y hxy
+    apply Subtype.ext
+    simpa [φ] using congrArg Subtype.val hxy
+  let e : Icc a b ≃ₜ Icc (a + c) (b + c) :=
+    { toFun := φ
+      invFun := ψ
+      left_inv := fun x ↦ by apply Subtype.ext; simp [φ, ψ]
+      right_inv := fun x ↦ by apply Subtype.ext; simp [φ, ψ]
+      continuous_toFun := hφc
+      continuous_invFun := hψc }
+  let hgbv := BoundedVariationOn.comp_monotone_surjective_Icc
+    (by linarith : a + c ≤ b + c)
+    f.boundedVariation hφm hφs
+  let g : RightContinuousIntervalBV a b :=
+    { toFun := f.toFun ∘ φ
+      boundedVariation := hgbv
+      right_continuous := fun t ↦
+        (hfc.comp hφc).continuousAt.continuousWithinAt }
+  refine ⟨g, fun _ ↦ rfl, ?_⟩
+  intro E hE
+  have hmap := BoundedVariationOn.vectorMeasure_map_comp_monotone_surjective_Icc
+    (by linarith : a + c ≤ b + c) hab f.boundedVariation hfc hφc hφm hφs
+  have hφE : MeasurableSet (φ '' E) := e.measurableEmbedding.measurableSet_image' hE
+  change hgbv.vectorMeasure E = f.boundedVariation.vectorMeasure (φ '' E)
+  rw [← hmap, VectorMeasure.map_apply _ hφc.measurable hφE,
+    hφi.preimage_image]
+
+/-- Specialize interval translation to an interval starting at zero. -/
+theorem exists_intervalBV_shift_from_zero {b c : ℝ} (hb : 0 ≤ b)
+    (f : RightContinuousIntervalBV (0 + c) (b + c)) (hfc : Continuous f.toFun) :
+    ∃ g : RightContinuousIntervalBV 0 b,
+      (∀ t, g.toFun t = f.toFun
+        ⟨(t : ℝ) + c, by constructor <;> linarith [t.property.1, t.property.2]⟩) ∧
+      ∀ E : Set (Icc 0 b), MeasurableSet E →
+        intervalStieltjesMeasure g E = intervalStieltjesMeasure f
+          ((fun t : Icc 0 b ↦
+            ⟨(t : ℝ) + c, by constructor <;> linarith [t.property.1, t.property.2]⟩) '' E) := by
+  exact exists_intervalBV_shift_add (a := 0) (b := b) (c := c) hb f hfc
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Basic
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+instance angleMeasurableSpace : MeasurableSpace Real.Angle := borel Real.Angle
+
+instance angleBorelSpace : BorelSpace Real.Angle := ⟨rfl⟩
+
+/-- An exterior normal direction at a point of a convex body. -/
+def IsExteriorNormal (K : ConvexBody Point) (p : Point) (a : Real.Angle) : Prop :=
+  ∀ q ∈ (K : Set Point), inner ℝ (q - p) (normalVector a) ≤ 0
+
+/-- Boundary points with exactly one exterior unit normal. -/
+def regularBoundary (K : ConvexBody Point) : Set Point :=
+  {p | p ∈ frontier (K : Set Point) ∧ ∃! a, IsExteriorNormal K p a}
+
+/-- The unique exterior normal at regular points, extended by zero elsewhere. -/
+def exteriorNormalAngle (K : ConvexBody Point) (p : Point) : Real.Angle := by
+  classical
+  exact if h : ∃! a, IsExteriorNormal K p a then h.exists.choose else 0
+
+/-- A nontrivial segment presentation and a perpendicular angular direction. -/
+def IsSegmentPresentation (K : ConvexBody Point) (d : Point × Point × Real.Angle) : Prop :=
+  d.1 ≠ d.2.1 ∧ (K : Set Point) = segment ℝ d.1 d.2.1 ∧
+    inner ℝ (d.2.1 - d.1) (normalVector d.2.2) = 0
+
+/-- Surface measure in angular coordinates, including point and segment bodies. -/
+def surfaceAreaMeasure (K : ConvexBody Point) : Measure Real.Angle := by
+  classical
+  exact if (K : Set Point).Subsingleton then 0
+  else if h : ∃ d, IsSegmentPresentation K d then
+    let d := h.choose
+    ENNReal.ofReal (dist d.1 d.2.1) •
+      (Measure.dirac d.2.2 + Measure.dirac (d.2.2 + ((Real.pi : ℝ) : Real.Angle)))
+  else
+    Measure.map (exteriorNormalAngle K)
+      ((Measure.hausdorffMeasure 1).restrict (regularBoundary K))
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Exterior Normal
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open Set
+
+/-- Every frontier point of a convex body with nonempty interior admits an exterior unit normal. -/
+theorem exists_isExteriorNormal_of_mem_frontier
+    (K : ConvexBody Point) (hK : (interior (K : Set Point)).Nonempty)
+    {p : Point} (hp : p ∈ frontier (K : Set Point)) :
+    ∃ a, IsExteriorNormal K p a := by
+  have hpnot : p ∉ interior (K : Set Point) := by
+    have hpK : p ∈ K := by
+      change p ∈ (K : Set Point)
+      rw [← K.isClosed.closure_eq]
+      exact frontier_subset_closure hp
+    exact (mem_frontier_iff_notMem_interior hpK).mp hp
+  obtain ⟨f, hf⟩ := geometric_hahn_banach_point_open K.convex.interior isOpen_interior hpnot
+  let fc : Point →L[ℝ] ℝ := f.toContinuousLinearMap
+  have hfne : fc ≠ 0 := by
+    obtain ⟨q, hq⟩ := hK
+    intro hzero
+    have h := hf q hq
+    have hfp : fc p = 0 := by rw [hzero]; rfl
+    have hfq : fc q = 0 := by rw [hzero]; rfl
+    change fc p < fc q at h
+    linarith
+  let w : Point := -(InnerProductSpace.toDual ℝ Point).symm fc
+  have hw : w ≠ 0 := by
+    intro hzero
+    apply hfne
+    apply (InnerProductSpace.toDual ℝ Point).symm.injective
+    simpa using neg_eq_zero.mp hzero
+  obtain ⟨a, ha⟩ := exists_angle_normalVector_eq (u := ‖w‖⁻¹ • w) (by
+    rw [norm_smul, norm_inv, Real.norm_eq_abs, abs_norm,
+      inv_mul_cancel₀ (norm_ne_zero_iff.mpr hw)])
+  refine ⟨a, ?_⟩
+  rw [IsExteriorNormal, ha]
+  intro q hq
+  have hclosed : IsClosed {q : Point | fc p ≤ fc q} :=
+    isClosed_le continuous_const fc.continuous
+  have hinterior : interior (K : Set Point) ⊆ {q : Point | fc p ≤ fc q} :=
+    fun q hq ↦ (hf q hq).le
+  have hsubset : (K : Set Point) ⊆ {q : Point | fc p ≤ fc q} := by
+    rw [← K.isClosed.closure_eq,
+      ← K.convex.closure_interior_eq_closure_of_nonempty_interior hK]
+    exact closure_minimal hinterior hclosed
+  have hfpq := hsubset hq
+  rw [inner_smul_right, inner_neg_right, real_inner_comm,
+    InnerProductSpace.toDual_symm_apply]
+  dsimp only [fc] at hfpq ⊢
+  rw [map_sub]
+  exact mul_nonpos_of_nonneg_of_nonpos (inv_nonneg.mpr (norm_nonneg w))
+    (neg_nonpos.mpr (sub_nonneg.mpr hfpq))
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Graph Definitions
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Project the convex body horizontally after translating and changing orthonormal frame. -/
+def horizontalProjection (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) : Set ℝ :=
+  (fun p : Point ↦ e (p - o) 0) '' (K : Set Point)
+
+/-- The infimum and supremum of the body’s horizontal projection in the chosen frame. -/
+def horizontalBounds (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) : ℝ × ℝ :=
+  (sInf (horizontalProjection K o e), sSup (horizontalProjection K o e))
+
+/-- The supremum of the body’s vertical section at a given horizontal coordinate. -/
+def upperGraphHeight (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) (x : ℝ) : ℝ :=
+  sSup {y : ℝ | o + e.symm !₂[x, y] ∈ (K : Set Point)}
+
+/-- The argument of a planar vector, viewed as an angle modulo a full turn. -/
+def vectorNormalAngle (p : Point) : Real.Angle :=
+  (Complex.arg ⟨p 0, p 1⟩ : ℝ)
+
+/-- The normal vector associated to a nonzero planar vector is its normalization. -/
+theorem normalVector_vectorNormalAngle {p : Point} (hp : p ≠ 0) :
+    normalVector (vectorNormalAngle p) = ‖p‖⁻¹ • p := by
+  let z : ℂ := ⟨p 0, p 1⟩
+  have hz : z ≠ 0 := by
+    intro hz
+    apply hp
+    ext i
+    fin_cases i
+    · exact congrArg Complex.re hz
+    · exact congrArg Complex.im hz
+  have hnorm : ‖z‖ = ‖p‖ := by
+    rw [Complex.norm_def, EuclideanSpace.norm_eq]
+    congr 1
+    simp only [z, Complex.normSq_apply, Fin.sum_univ_two]
+    simp [Real.norm_eq_abs, pow_two]
+  ext i
+  fin_cases i
+  · simpa [vectorNormalAngle, normalVector, frame, z, hnorm, div_eq_inv_mul] using
+      Complex.cos_arg hz
+  · simpa [vectorNormalAngle, normalVector, frame, z, hnorm, div_eq_inv_mul] using
+      Complex.sin_arg z
+
+/-- Weight the upper graph by its normal direction and arc-length Jacobian. -/
+def upperGraphSurfaceIntegrand (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) (ψ : Real.Angle → ℝ) (x : ℝ) : ℝ :=
+  ψ (vectorNormalAngle (e.symm !₂[-deriv (upperGraphHeight K o e) x, 1])) *
+    Real.sqrt (1 + (deriv (upperGraphHeight K o e) x) ^ 2)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Regularity
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+instance angleCompactSpace : CompactSpace Real.Angle :=
+  AddCircle.homeomorphCircle'.symm.compactSpace
+
+/-- The unit normal depends continuously on its angle. -/
+theorem continuous_normalVector_angle :
+    Continuous normalVector := by
+  let c : Real.Angle → (i : Fin 2) → ℝ := fun t i ↦
+    Fin.cases t.cos (fun _ ↦ t.sin) i
+  have hc : Continuous c := by
+    apply continuous_pi
+    intro i
+    fin_cases i
+    · exact Real.Angle.continuous_cos
+    · exact Real.Angle.continuous_sin
+  have heq : normalVector = (fun t ↦ WithLp.toLp 2 (c t)) := by
+    funext t
+    ext i
+    fin_cases i <;> rfl
+  rw [heq]
+  exact (PiLp.continuous_toLp (2 : ENNReal) (fun _ : Fin 2 ↦ ℝ)).comp hc
+
+private theorem isClosed_isExteriorNormal (K : ConvexBody Point) :
+    IsClosed {z : Point × Real.Angle | IsExteriorNormal K z.1 z.2} := by
+  change IsClosed {z : Point × Real.Angle |
+    ∀ q ∈ (K : Set Point), inner ℝ (q - z.1) (normalVector z.2) ≤ 0}
+  simp only [Set.ofPred_forall]
+  apply isClosed_iInter
+  intro q
+  apply isClosed_iInter
+  intro _
+  apply isClosed_le
+  · exact (continuous_const.sub continuous_fst).inner
+      (continuous_normalVector_angle.comp continuous_snd)
+  · exact continuous_const
+
+private def normalGraph (K : ConvexBody Point) : Set (Point × Real.Angle) :=
+  {z | z.1 ∈ frontier (K : Set Point) ∧ IsExteriorNormal K z.1 z.2}
+
+private theorem isCompact_normalGraph (K : ConvexBody Point) :
+    IsCompact (normalGraph K) := by
+  have hfront : IsCompact (frontier (K : Set Point)) := by
+    apply K.isCompact.of_isClosed_subset isClosed_frontier
+    simpa only [K.isClosed.closure_eq] using
+      (frontier_subset_closure : frontier (K : Set Point) ⊆ closure (K : Set Point))
+  have hclosed : IsClosed (normalGraph K) := by
+    exact (isClosed_frontier.preimage continuous_fst).inter (isClosed_isExteriorNormal K)
+  apply (hfront.prod isCompact_univ).of_isClosed_subset
+  · exact hclosed
+  · rintro ⟨p, a⟩ ⟨hp, _⟩
+    exact ⟨hp, Set.mem_univ a⟩
+
+private def normalDomain (K : ConvexBody Point) : Set Point :=
+  Prod.fst '' normalGraph K
+
+private theorem isCompact_normalDomain (K : ConvexBody Point) :
+    IsCompact (normalDomain K) :=
+  (isCompact_normalGraph K).image continuous_fst
+
+private def separatedNormalPairs (K : ConvexBody Point) (n : ℕ) :
+    Set (Point × (Real.Angle × Real.Angle)) :=
+  {z | z.1 ∈ frontier (K : Set Point) ∧
+    IsExteriorNormal K z.1 z.2.1 ∧ IsExteriorNormal K z.1 z.2.2 ∧
+      (1 : ℝ) / (n + 1) ≤ dist z.2.1 z.2.2}
+
+private theorem isCompact_separatedNormalPairs (K : ConvexBody Point) (n : ℕ) :
+    IsCompact (separatedNormalPairs K n) := by
+  have hfront : IsCompact (frontier (K : Set Point)) := by
+    apply K.isCompact.of_isClosed_subset isClosed_frontier
+    simpa only [K.isClosed.closure_eq] using
+      (frontier_subset_closure : frontier (K : Set Point) ⊆ closure (K : Set Point))
+  have hrel₁ : IsClosed {z : Point × (Real.Angle × Real.Angle) |
+      IsExteriorNormal K z.1 z.2.1} :=
+    (isClosed_isExteriorNormal K).preimage
+      (continuous_fst.prodMk (continuous_fst.comp continuous_snd))
+  have hrel₂ : IsClosed {z : Point × (Real.Angle × Real.Angle) |
+      IsExteriorNormal K z.1 z.2.2} :=
+    (isClosed_isExteriorNormal K).preimage
+      (continuous_fst.prodMk (continuous_snd.comp continuous_snd))
+  have hsep : IsClosed {z : Point × (Real.Angle × Real.Angle) |
+      (1 : ℝ) / (n + 1) ≤ dist z.2.1 z.2.2} := by
+    exact isClosed_le continuous_const
+      ((continuous_fst.comp continuous_snd).dist (continuous_snd.comp continuous_snd))
+  have hfclosed : IsClosed {z : Point × (Real.Angle × Real.Angle) |
+      z.1 ∈ frontier (K : Set Point)} :=
+    isClosed_frontier.preimage continuous_fst
+  have hclosed : IsClosed (separatedNormalPairs K n) := by
+    have h := ((hfclosed.inter hrel₁).inter hrel₂).inter hsep
+    simpa only [separatedNormalPairs, Set.inter_def, Set.mem_ofPred_eq, and_assoc] using h
+  apply (hfront.prod (isCompact_univ.prod isCompact_univ)).of_isClosed_subset hclosed
+  rintro ⟨p, a, b⟩ ⟨hp, _⟩
+  exact ⟨hp, Set.mem_univ _, Set.mem_univ _⟩
+
+private def separatedNormalPoints (K : ConvexBody Point) (n : ℕ) : Set Point :=
+  Prod.fst '' separatedNormalPairs K n
+
+private theorem isCompact_separatedNormalPoints (K : ConvexBody Point) (n : ℕ) :
+    IsCompact (separatedNormalPoints K n) :=
+  (isCompact_separatedNormalPairs K n).image continuous_fst
+
+private theorem regularBoundary_eq_normalDomain_sdiff_iUnion (K : ConvexBody Point) :
+    regularBoundary K = normalDomain K \ ⋃ n, separatedNormalPoints K n := by
+  ext p
+  constructor
+  · rintro ⟨hp, a, ha, hua⟩
+    refine ⟨⟨(p, a), ⟨hp, ha⟩, rfl⟩, ?_⟩
+    intro hnonunique
+    obtain ⟨n, hn⟩ := Set.mem_iUnion.mp hnonunique
+    obtain ⟨⟨q, b, c⟩, ⟨_, hb, hc, hdist⟩, rfl⟩ := hn
+    rw [hua b hb, hua c hc, dist_self] at hdist
+    have hpos : 0 < (1 : ℝ) / (n + 1) := by positivity
+    linarith
+  · rintro ⟨hdom, hnot⟩
+    obtain ⟨⟨q, a⟩, ⟨hp, ha⟩, hqp⟩ := hdom
+    change q = p at hqp
+    subst q
+    refine ⟨hp, a, ha, ?_⟩
+    intro b hb
+    by_contra hba
+    have hdpos : 0 < dist a b := dist_pos.mpr (Ne.symm hba)
+    obtain ⟨n, hn⟩ := exists_nat_one_div_lt hdpos
+    apply hnot
+    apply Set.mem_iUnion.mpr
+    refine ⟨n, ⟨(p, (a, b)), ?_, rfl⟩⟩
+    exact ⟨hp, ha, hb, hn.le⟩
+
+/-- Points with a unique exterior normal form a Borel set. -/
+theorem measurableSet_regularBoundary (K : ConvexBody Point) :
+    MeasurableSet (regularBoundary K) := by
+  rw [regularBoundary_eq_normalDomain_sdiff_iUnion]
+  exact (isCompact_normalDomain K).measurableSet.diff
+    (MeasurableSet.iUnion fun n ↦ (isCompact_separatedNormalPoints K n).measurableSet)
+
+/-- The exterior normal varies continuously on the regular boundary. -/
+theorem continuousOn_exteriorNormalAngle_regularBoundary (K : ConvexBody Point) :
+    ContinuousOn (exteriorNormalAngle K) (regularBoundary K) := by
+  have hnormal (p : Point) (hp : p ∈ regularBoundary K) :
+      IsExteriorNormal K p (exteriorNormalAngle K p) := by
+    rcases hp.2 with ⟨a, ha, hua⟩
+    dsimp only [exteriorNormalAngle]
+    split
+    · rename_i h
+      exact h.exists.choose_spec
+    · rename_i h
+      exact (h ⟨a, ha, hua⟩).elim
+  have hunique (p : Point) (hp : p ∈ regularBoundary K) (a : Real.Angle)
+      (ha : IsExteriorNormal K p a) : a = exteriorNormalAngle K p := by
+    rcases hp.2 with ⟨b, hb, hub⟩
+    exact (hub a ha).trans (hub _ (hnormal p hp)).symm
+  rw [continuousOn_iff_continuous_domRestrict]
+  apply continuous_of_isClosed_graph
+  have hgraph : Function.graph ((regularBoundary K).domRestrict (exteriorNormalAngle K)) =
+      {z : (regularBoundary K) × Real.Angle | IsExteriorNormal K z.1 z.2} := by
+    ext z
+    constructor
+    · intro hz
+      change exteriorNormalAngle K z.1 = z.2 at hz
+      change IsExteriorNormal K z.1 z.2
+      rw [← hz]
+      exact hnormal z.1 z.1.2
+    · intro hz
+      change exteriorNormalAngle K z.1 = z.2
+      exact (hunique z.1 z.1.2 z.2 hz).symm
+  rw [hgraph]
+  exact (isClosed_isExteriorNormal K).preimage
+    ((continuous_subtype_val.comp continuous_fst).prodMk continuous_snd)
+
+/-- The exterior normal is measurable for any measure restricted to the regular boundary. -/
+theorem aemeasurable_exteriorNormalAngle_restrict_regularBoundary
+    (K : ConvexBody Point) (μ : Measure Point) :
+    AEMeasurable (exteriorNormalAngle K) (μ.restrict (regularBoundary K)) :=
+  aemeasurable_restrict_of_measurable_subtype (measurableSet_regularBoundary K)
+    (continuousOn_exteriorNormalAngle_regularBoundary K).domRestrict.measurable
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Segment
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- Two angular unit normals perpendicular to a nonzero planar vector are equal or antipodal. -/
+theorem normalVector_eq_or_eq_add_pi_of_orthogonal
+    {v : Point} (hv : v ≠ 0) {s t : Real.Angle}
+    (hvs : inner ℝ v (normalVector s) = 0)
+    (hvt : inner ℝ v (normalVector t) = 0) : s = t ∨ s = t + (Real.pi : Real.Angle) := by
+  let o : Orientation ℝ Point (Fin 2) :=
+    (EuclideanSpace.basisFun (Fin 2) ℝ).toBasis.orientation
+  have hnorm (a : Real.Angle) : ‖normalVector a‖ = 1 := by
+    induction a using Real.Angle.induction_on with
+    | _ a =>
+      rw [EuclideanSpace.norm_eq]
+      simp [normalVector, frame, Fin.sum_univ_two]
+  rcases EuclideanGeometry.eq_or_eq_neg_of_unit_orthogonal o hv (hnorm s) (hnorm t) hvs hvt with
+    h | h
+  · left
+    induction s using Real.Angle.induction_on with
+    | _ s =>
+      induction t using Real.Angle.induction_on with
+      | _ t =>
+        apply Real.Angle.cos_sin_inj
+        · exact congrFun (congrArg WithLp.ofLp h) 0
+        · exact congrFun (congrArg WithLp.ofLp h) 1
+  · right
+    induction s using Real.Angle.induction_on with
+    | _ s =>
+      induction t using Real.Angle.induction_on with
+      | _ t =>
+        apply Real.Angle.cos_sin_inj
+        · have h0 := congrFun (congrArg WithLp.ofLp h) 0
+          simpa [normalVector, frame, Real.cos_add_pi] using h0
+        · have h1 := congrFun (congrArg WithLp.ofLp h) 1
+          simpa [normalVector, frame, Real.sin_add_pi] using h1
+
+private theorem dirac_add_antipode_eq_of_orthogonal
+    {v : Point} (hv : v ≠ 0) {s t : Real.Angle}
+    (hvs : inner ℝ v (normalVector s) = 0)
+    (hvt : inner ℝ v (normalVector t) = 0) :
+    Measure.dirac s + Measure.dirac (s + (Real.pi : Real.Angle)) =
+      Measure.dirac t + Measure.dirac (t + (Real.pi : Real.Angle)) := by
+  rcases normalVector_eq_or_eq_add_pi_of_orthogonal hv hvs hvt with h | h
+  · rw [h]
+  · rw [h]
+    have hpi : (t + (Real.pi : Real.Angle)) + (Real.pi : Real.Angle) = t := by
+      rw [add_assoc, Real.Angle.coe_pi_add_coe_pi, add_zero]
+    rw [hpi, add_comm]
+
+/-- The atomic segment measure is independent of its segment presentation. -/
+theorem segmentPresentation_measure_eq (K : ConvexBody Point)
+    {d c : Point × Point × Real.Angle}
+    (hd : IsSegmentPresentation K d) (hc : IsSegmentPresentation K c) :
+    ENNReal.ofReal (dist d.1 d.2.1) •
+        (Measure.dirac d.2.2 + Measure.dirac (d.2.2 + (Real.pi : Real.Angle))) =
+      ENNReal.ofReal (dist c.1 c.2.1) •
+        (Measure.dirac c.2.2 + Measure.dirac (c.2.2 + (Real.pi : Real.Angle))) := by
+  have hseg : segment ℝ d.1 d.2.1 = segment ℝ c.1 c.2.1 := hd.2.1.symm.trans hc.2.1
+  have hlen : dist d.1 d.2.1 = dist c.1 c.2.1 := EuclideanGeometry.dist_eq_of_segment_eq hseg
+  have hcn : inner ℝ (d.2.1 - d.1) (normalVector c.2.2) = 0 :=
+    EuclideanGeometry.inner_direction_eq_zero_of_segment_eq hseg hc.2.2
+  rw [hlen, dirac_add_antipode_eq_of_orthogonal (sub_ne_zero.mpr hd.1.symm) hd.2.2 hcn]
+
+/-- A singleton convex body has zero surface measure. -/
+theorem surfaceAreaMeasure_eq_zero_of_subsingleton (K : ConvexBody Point)
+    (hK : (K : Set Point).Subsingleton) :
+    surfaceAreaMeasure K = 0 := by
+  simp [surfaceAreaMeasure, hK]
+
+private theorem surfaceAreaMeasure_eq_chosenSegment (K : ConvexBody Point)
+    (hK : ¬(K : Set Point).Subsingleton) (hseg : ∃ d, IsSegmentPresentation K d) :
+    surfaceAreaMeasure K =
+      ENNReal.ofReal (dist hseg.choose.1 hseg.choose.2.1) •
+        (Measure.dirac hseg.choose.2.2 +
+          Measure.dirac (hseg.choose.2.2 + ((Real.pi : ℝ) : Real.Angle))) := by
+  simp [surfaceAreaMeasure, hK, hseg]
+
+/-- A nondegenerate segment presentation makes the represented convex body nonsingleton. -/
+theorem not_subsingleton_of_isSegmentPresentation (K : ConvexBody Point)
+    {d : Point × Point × Real.Angle} (hd : IsSegmentPresentation K d) :
+    ¬(K : Set Point).Subsingleton := by
+  intro hK
+  apply hd.1
+  apply hK
+  · rw [hd.2.1]
+    exact left_mem_segment ℝ _ _
+  · rw [hd.2.1]
+    exact right_mem_segment ℝ _ _
+
+/-- The surface measure of a segment is its length times the two normal atoms. -/
+theorem surfaceAreaMeasure_eq_segmentPresentation (K : ConvexBody Point)
+    (d : Point × Point × Real.Angle) (hd : IsSegmentPresentation K d) :
+    surfaceAreaMeasure K = ENNReal.ofReal (dist d.1 d.2.1) •
+      (Measure.dirac d.2.2 + Measure.dirac (d.2.2 + (Real.pi : Real.Angle))) := by
+  let hseg : ∃ c, IsSegmentPresentation K c := ⟨d, hd⟩
+  rw [surfaceAreaMeasure_eq_chosenSegment K
+    (not_subsingleton_of_isSegmentPresentation K hd) hseg]
+  exact segmentPresentation_measure_eq K hseg.choose_spec hd
+
+/-- The surface area measure of a singleton convex body is finite. -/
+theorem isFiniteMeasure_surfaceAreaMeasure_of_subsingleton (K : ConvexBody Point)
+    (hK : (K : Set Point).Subsingleton) :
+    IsFiniteMeasure (surfaceAreaMeasure K) := by
+  rw [surfaceAreaMeasure_eq_zero_of_subsingleton K hK]
+  infer_instance
+
+/-- The surface area measure of a convex body with a segment presentation is finite. -/
+theorem isFiniteMeasure_surfaceAreaMeasure_of_segmentPresentation (K : ConvexBody Point)
+    {d : Point × Point × Real.Angle} (hd : IsSegmentPresentation K d) :
+    IsFiniteMeasure (surfaceAreaMeasure K) := by
+  rw [surfaceAreaMeasure_eq_segmentPresentation K d hd]
+  exact (Measure.dirac d.2.2 +
+    Measure.dirac (d.2.2 + ((Real.pi : ℝ) : Real.Angle))).smul_finite
+      ENNReal.ofReal_ne_top
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Segment Faces
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+private theorem not_both_mem_short_angleArc {E : Set Real.Angle} {a b : ℝ}
+    (hab : a ≤ b) (hshort : b < a + Real.pi)
+    (hE : E ⊆ (fun t : ℝ ↦ (t : Real.Angle)) '' Set.Icc a b)
+    (t : Real.Angle) : ¬(t ∈ E ∧ t + (Real.pi : Real.Angle) ∈ E) := by
+  rintro ⟨ht, htpi⟩
+  obtain ⟨s, hs, hst⟩ := hE ht
+  obtain ⟨u, hu, hut⟩ := hE htpi
+  have hang : (u : Real.Angle) = ((s + Real.pi : ℝ) : Real.Angle) := by
+    calc
+      (u : Real.Angle) = t + (Real.pi : Real.Angle) := hut
+      _ = (s : Real.Angle) + (Real.pi : Real.Angle) := by rw [← hst]
+      _ = ((s + Real.pi : ℝ) : Real.Angle) := Real.Angle.coe_add _ _ |>.symm
+  obtain ⟨k, hk⟩ := Real.Angle.angle_eq_iff_two_pi_dvd_sub.mp hang
+  have hlow : -(2 * Real.pi) < u - (s + Real.pi) := by
+    linarith [hs.1, hs.2, hu.1, hu.2]
+  have hupp : u - (s + Real.pi) < 0 := by
+    linarith [hs.1, hs.2, hu.1, hu.2]
+  rw [hk] at hlow hupp
+  have hkneg : (k : ℝ) < 0 := by nlinarith [Real.pi_pos]
+  have hkgt : (-1 : ℝ) < k := by nlinarith [Real.pi_pos]
+  rw [← Int.cast_zero] at hkneg
+  have hcast_neg_one : (((-1 : ℤ) : ℝ)) = -1 := by norm_num
+  rw [← hcast_neg_one] at hkgt
+  have hkneg' : k < 0 := Int.cast_lt.mp hkneg
+  have hkgt' : (-1 : ℤ) < k := Int.cast_lt.mp hkgt
+  omega
+
+/-- A normal perpendicular to a segment exposes the whole segment. -/
+theorem exposedEdge_eq_segment_of_orthogonal (K : ConvexBody Point)
+    (d : Point × Point × Real.Angle) (hd : IsSegmentPresentation K d)
+    (t : Real.Angle) (ht : inner ℝ (d.2.1 - d.1) (normalVector t) = 0) :
+    exposedEdge K t = (K : Set Point) := by
+  have hfst : d.1 ∈ K := by
+    change d.1 ∈ (K : Set Point)
+    rw [hd.2.1]
+    exact left_mem_segment ℝ _ _
+  have hconst : ∀ p ∈ (K : Set Point),
+      inner ℝ p (normalVector t) = inner ℝ d.1 (normalVector t) := by
+    intro p hp
+    rw [hd.2.1, segment_eq_image'] at hp
+    obtain ⟨u, hu, rfl⟩ := hp
+    rw [inner_add_left, inner_smul_left, ht, mul_zero, add_zero]
+  have hsupport : supportValue K t = inner ℝ d.1 (normalVector t) := by
+    apply le_antisymm
+    · apply csSup_le (K.nonempty.image _)
+      rintro _ ⟨p, hp, rfl⟩
+      exact (hconst p hp).le
+    · exact inner_le_supportValue K hfst t
+  ext p
+  constructor
+  · exact fun hp ↦ hp.1
+  · intro hp
+    refine ⟨hp, ?_⟩
+    change inner ℝ p (normalVector t) = supportValue K t
+    rw [hconst p hp, hsupport]
+
+private theorem exposedEdge_subset_endpoints_of_not_orthogonal (K : ConvexBody Point)
+    (d : Point × Point × Real.Angle) (hd : IsSegmentPresentation K d)
+    (t : Real.Angle) (ht : inner ℝ (d.2.1 - d.1) (normalVector t) ≠ 0) :
+    exposedEdge K t ⊆ {d.1, d.2.1} := by
+  have hfst : d.1 ∈ K := by
+    change d.1 ∈ (K : Set Point)
+    rw [hd.2.1]
+    exact left_mem_segment ℝ _ _
+  have hsnd : d.2.1 ∈ K := by
+    change d.2.1 ∈ (K : Set Point)
+    rw [hd.2.1]
+    exact right_mem_segment ℝ _ _
+  intro p hp
+  have hpK : p ∈ segment ℝ d.1 d.2.1 := by
+    rw [← hd.2.1]
+    exact hp.1
+  rw [segment_eq_image'] at hpK
+  obtain ⟨u, hu, rfl⟩ := hpK
+  have hx := inner_le_supportValue K hfst t
+  have hy := inner_le_supportValue K hsnd t
+  rw [← hp.2] at hx hy
+  simp only [inner_add_left, inner_smul_left] at hx hy
+  let c := inner ℝ (d.2.1 - d.1) (normalVector t)
+  change inner ℝ d.1 (normalVector t) ≤
+    inner ℝ d.1 (normalVector t) + u * c at hx
+  change inner ℝ d.2.1 (normalVector t) ≤
+    inner ℝ d.1 (normalVector t) + u * c at hy
+  have hdiff : inner ℝ d.2.1 (normalVector t) =
+      inner ℝ d.1 (normalVector t) + c := by
+    dsimp only [c]
+    rw [inner_sub_left]
+    ring
+  rw [hdiff] at hy
+  have hc : c ≠ 0 := ht
+  rcases lt_or_gt_of_ne hc with hcneg | hcpos
+  · have hu0 : u = 0 := by
+      apply le_antisymm _ hu.1
+      by_contra hnot
+      have hupos : 0 < u := lt_of_not_ge hnot
+      nlinarith
+    simp [hu0]
+  · have hu1 : u = 1 := by
+      apply le_antisymm hu.2
+      by_contra hnot
+      have hult : u < 1 := lt_of_not_ge hnot
+      nlinarith
+    simp [hu1]
+
+/-- The face-union formula for a segment on an angular arc shorter than a half-turn. -/
+theorem surfaceAreaMeasure_face_union_of_segmentPresentation
+    (K : ConvexBody Point) (d : Point × Point × Real.Angle)
+    (hd : IsSegmentPresentation K d) (E : Set Real.Angle) (hE : MeasurableSet E)
+    {a b : ℝ} (hab : a ≤ b) (hshort : b < a + Real.pi)
+    (hsubset : E ⊆ (fun t : ℝ ↦ (t : Real.Angle)) '' Set.Icc a b) :
+    surfaceAreaMeasure K E =
+      Measure.hausdorffMeasure 1 (⋃ t ∈ E, exposedEdge K t) := by
+  have hnotboth := not_both_mem_short_angleArc hab hshort hsubset d.2.2
+  have hnormalPi :
+      normalVector (d.2.2 + (Real.pi : Real.Angle)) = -normalVector d.2.2 := by
+    induction d.2.2 using Real.Angle.induction_on with
+    | _ t => simpa only [← Real.Angle.coe_add] using normalVector_add_pi t
+  have hpiorth :
+      inner ℝ (d.2.1 - d.1)
+        (normalVector (d.2.2 + (Real.pi : Real.Angle))) = 0 := by
+    rw [hnormalPi, inner_neg_right, hd.2.2, neg_zero]
+  by_cases hn : d.2.2 ∈ E
+  · have hnpi : d.2.2 + (Real.pi : Real.Angle) ∉ E := fun hnpi ↦ hnotboth ⟨hn, hnpi⟩
+    have hunion : (⋃ t ∈ E, exposedEdge K t) = (K : Set Point) := by
+      apply Set.Subset.antisymm
+      · intro p hp
+        obtain ⟨t, ht, hpt⟩ := Set.mem_iUnion₂.mp hp
+        exact hpt.1
+      · intro p hp
+        exact Set.mem_iUnion₂.mpr ⟨d.2.2, hn, by
+          rw [exposedEdge_eq_segment_of_orthogonal K d hd d.2.2 hd.2.2]
+          exact hp⟩
+    rw [surfaceAreaMeasure_eq_segmentPresentation K d hd, hunion, hd.2.1]
+    simp [hE, hn, hnpi, edist_dist]
+  · by_cases hnpi : d.2.2 + (Real.pi : Real.Angle) ∈ E
+    · have hunion : (⋃ t ∈ E, exposedEdge K t) = (K : Set Point) := by
+        apply Set.Subset.antisymm
+        · intro p hp
+          obtain ⟨t, ht, hpt⟩ := Set.mem_iUnion₂.mp hp
+          exact hpt.1
+        · intro p hp
+          exact Set.mem_iUnion₂.mpr ⟨d.2.2 + (Real.pi : Real.Angle), hnpi, by
+            rw [exposedEdge_eq_segment_of_orthogonal K d hd _ hpiorth]
+            exact hp⟩
+      rw [surfaceAreaMeasure_eq_segmentPresentation K d hd, hunion, hd.2.1]
+      simp [hE, hn, hnpi, edist_dist]
+    · have hunion : (⋃ t ∈ E, exposedEdge K t) ⊆ {d.1, d.2.1} := by
+        intro p hp
+        obtain ⟨t, ht, hpt⟩ := Set.mem_iUnion₂.mp hp
+        have htorth : inner ℝ (d.2.1 - d.1) (normalVector t) ≠ 0 := by
+          intro hortho
+          rcases normalVector_eq_or_eq_add_pi_of_orthogonal
+              (sub_ne_zero.mpr hd.1.symm) hortho hd.2.2 with h | h
+          · exact hn (h ▸ ht)
+          · exact hnpi (h ▸ ht)
+        exact exposedEdge_subset_endpoints_of_not_orthogonal K d hd t htorth hpt
+      rw [surfaceAreaMeasure_eq_segmentPresentation K d hd]
+      have hzero : Measure.hausdorffMeasure 1 (⋃ t ∈ E, exposedEdge K t) = 0 := by
+        let _ := Measure.nullSingletonClass_hausdorff Point (by norm_num : (0 : ℝ) < 1)
+        exact measure_mono_null hunion
+          ((Set.finite_singleton d.2.1).insert d.1 |>.measure_zero _)
+      rw [hzero]
+      simp [hE, hn, hnpi]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Segment Graph
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- A singleton convex body has equal horizontal projection bounds. -/
+theorem horizontalBounds_eq_of_subsingleton (K : ConvexBody Point)
+    (hK : (K : Set Point).Subsingleton) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) :
+    (horizontalBounds K o e).1 = (horizontalBounds K o e).2 := by
+  obtain ⟨p, hp⟩ := K.nonempty
+  have hset : (K : Set Point) = {p} :=
+    Set.eq_singleton_iff_unique_mem.mpr ⟨hp, fun x hx ↦ hK hx hp⟩
+  simp [horizontalBounds, horizontalProjection, hset]
+
+/-- Every surface-area integral of a singleton convex body vanishes. -/
+theorem surfaceAreaMeasure_integral_eq_zero_of_subsingleton (K : ConvexBody Point)
+    (hK : (K : Set Point).Subsingleton) (ψ : Real.Angle → ℝ) :
+    (∫ t, ψ t ∂surfaceAreaMeasure K) = 0 := by
+  rw [surfaceAreaMeasure_eq_zero_of_subsingleton K hK]
+  simp
+
+private theorem horizontalProjection_segment (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) {a b : Point}
+    (hK : (K : Set Point) = segment ℝ a b) :
+    horizontalProjection K o e = Set.uIcc (e (a - o) 0) (e (b - o) 0) := by
+  rw [horizontalProjection, hK, segment_eq_image']
+  calc
+    (fun p : Point ↦ e (p - o) 0) ''
+        ((fun t : ℝ ↦ a + t • (b - a)) '' Set.Icc 0 1) =
+        (fun t : ℝ ↦ e (a - o) 0 + t * (e (b - o) 0 - e (a - o) 0)) ''
+          Set.Icc 0 1 := by
+      rw [Set.image_image]
+      congr 1
+      funext t
+      simp only [map_sub, map_add, map_smul, PiLp.add_apply,
+        PiLp.sub_apply, PiLp.smul_apply, smul_eq_mul]
+      ring
+    _ = segment ℝ (e (a - o) 0) (e (b - o) 0) :=
+      (segment_eq_image' ℝ (e (a - o) 0) (e (b - o) 0)).symm
+    _ = Set.uIcc (e (a - o) 0) (e (b - o) 0) := segment_eq_uIcc _ _
+
+private theorem horizontalBounds_segment (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) {a b : Point}
+    (hK : (K : Set Point) = segment ℝ a b) :
+    horizontalBounds K o e =
+      (min (e (a - o) 0) (e (b - o) 0), max (e (a - o) 0) (e (b - o) 0)) := by
+  rw [horizontalBounds, horizontalProjection_segment K o e hK]
+  simp [Set.uIcc, csInf_Icc, csSup_Icc]
+
+private theorem coordinate_normal_second_eq_zero_of_horizontalBounds_eq
+    (K : ConvexBody Point) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point)
+    (d : Point × Point × Real.Angle) (hd : IsSegmentPresentation K d)
+    (hbounds : (horizontalBounds K o e).1 = (horizontalBounds K o e).2) :
+    e (normalVector d.2.2) 1 = 0 := by
+  have hb := horizontalBounds_segment K o e hd.2.1
+  rw [hb] at hbounds
+  dsimp only at hbounds
+  have hx : e (d.1 - o) 0 = e (d.2.1 - o) 0 := by
+    have hle₁ := min_le_left (e (d.1 - o) 0) (e (d.2.1 - o) 0)
+    have hle₂ := min_le_right (e (d.1 - o) 0) (e (d.2.1 - o) 0)
+    have hge₁ := le_max_left (e (d.1 - o) 0) (e (d.2.1 - o) 0)
+    have hge₂ := le_max_right (e (d.1 - o) 0) (e (d.2.1 - o) 0)
+    linarith
+  have hdir0 : e (d.2.1 - d.1) 0 = 0 := by
+    simp only [map_sub, PiLp.sub_apply] at hx ⊢
+    linarith
+  have hdir_ne : e (d.2.1 - d.1) 1 ≠ 0 := by
+    intro h
+    have hz : d.2.1 - d.1 = 0 := e.injective (by
+      simp only [map_zero]
+      ext i
+      fin_cases i
+      · exact hdir0
+      · exact h)
+    exact hd.1.symm (sub_eq_zero.mp hz)
+  have hinner : inner ℝ (e (d.2.1 - d.1)) (e (normalVector d.2.2)) = 0 := by
+    rw [e.inner_map_map]
+    exact hd.2.2
+  simp only [PiLp.inner_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two,
+    hdir0, mul_zero, zero_add] at hinner
+  exact (mul_eq_zero.mp hinner).resolve_right hdir_ne
+
+/-- A segment with zero horizontal width contributes zero against weights supported on
+normals with positive vertical coordinate. -/
+theorem segment_integral_eq_zero_of_horizontalBounds_eq
+    (K : ConvexBody Point) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point)
+    (ψ : Real.Angle → ℝ) (ε : ℝ) (hε : 0 < ε)
+    (hsupport : ∀ t, e (normalVector t) 1 < ε → ψ t = 0)
+    (d : Point × Point × Real.Angle) (hd : IsSegmentPresentation K d)
+    (hbounds : (horizontalBounds K o e).1 = (horizontalBounds K o e).2) :
+    (∫ t, ψ t ∂surfaceAreaMeasure K) = 0 := by
+  have hn := coordinate_normal_second_eq_zero_of_horizontalBounds_eq K o e d hd hbounds
+  have hψn : ψ d.2.2 = 0 := hsupport d.2.2 (by linarith)
+  have hnpi : e (normalVector (d.2.2 + (Real.pi : Real.Angle))) 1 = 0 := by
+    have hv : normalVector (d.2.2 + (Real.pi : Real.Angle)) = -normalVector d.2.2 := by
+      induction d.2.2 using Real.Angle.induction_on with
+      | _ t => simpa only [← Real.Angle.coe_add] using normalVector_add_pi t
+    rw [hv, map_neg, PiLp.neg_apply, hn, neg_zero]
+  have hψnpi : ψ (d.2.2 + (Real.pi : Real.Angle)) = 0 :=
+    hsupport _ (by rw [hnpi]; exact hε)
+  rw [surfaceAreaMeasure_eq_segmentPresentation K d hd]
+  rw [MeasureTheory.integral_smul_measure]
+  rw [MeasureTheory.integral_add_measure (MeasureTheory.integrable_dirac (by finiteness))
+    (MeasureTheory.integrable_dirac (by finiteness))]
+  simp [hψn, hψnpi]
+
+private theorem upperGraphHeight_segment_of_lt (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) {a b : Point}
+    (hK : (K : Set Point) = segment ℝ a b)
+    (hab : e (a - o) 0 < e (b - o) 0) {x : ℝ}
+    (hx : x ∈ Set.Icc (e (a - o) 0) (e (b - o) 0)) :
+    upperGraphHeight K o e x = e (a - o) 1 +
+      ((x - e (a - o) 0) / (e (b - o) 0 - e (a - o) 0)) *
+        (e (b - o) 1 - e (a - o) 1) := by
+  let t := (x - e (a - o) 0) / (e (b - o) 0 - e (a - o) 0)
+  let y := e (a - o) 1 + t * (e (b - o) 1 - e (a - o) 1)
+  have ht : t ∈ Set.Icc (0 : ℝ) 1 := by
+    constructor
+    · exact div_nonneg (sub_nonneg.mpr hx.1) (sub_nonneg.mpr hab.le)
+    · exact (div_le_one (sub_pos.mpr hab)).mpr (by linarith [hx.2])
+  have hfiber : {z : ℝ | o + e.symm !₂[x, z] ∈ (K : Set Point)} = {y} := by
+    ext z
+    rw [Set.mem_ofPred_eq, hK, segment_eq_image', Set.mem_image, Set.mem_singleton_iff]
+    constructor
+    · rintro ⟨r, hr, heq⟩
+      have heq' := congrArg (fun p : Point ↦ e (p - o)) heq
+      have heq0 := congrFun (congrArg WithLp.ofLp heq') 0
+      have heq1 := congrFun (congrArg WithLp.ofLp heq') 1
+      have hrt : r = t := by
+        dsimp [t]
+        simp only [map_sub, map_add, map_smul, LinearIsometryEquiv.apply_symm_apply,
+          PiLp.add_apply, PiLp.sub_apply, PiLp.smul_apply, Matrix.cons_val_zero,
+          smul_eq_mul] at heq0
+        simp at heq0
+        apply (eq_div_iff (sub_ne_zero.mpr hab.ne')).mpr
+        simp only [map_sub, PiLp.sub_apply]
+        linarith
+      dsimp [y]
+      rw [← hrt]
+      simp only [map_sub, map_add, map_smul, LinearIsometryEquiv.apply_symm_apply,
+          PiLp.add_apply, PiLp.sub_apply, PiLp.smul_apply, Matrix.cons_val_one,
+          smul_eq_mul] at heq1
+      simp at heq1
+      simp only [map_sub, PiLp.sub_apply]
+      linarith
+    · intro hzy
+      subst z
+      refine ⟨t, ht, ?_⟩
+      apply e.injective
+      ext i
+      fin_cases i
+      · have hdx : e b 0 - e a 0 ≠ 0 := by
+          simp only [map_sub, PiLp.sub_apply] at hab
+          linarith
+        simp only [map_add, map_sub, LinearIsometryEquiv.apply_symm_apply, map_smul,
+          PiLp.add_apply, PiLp.sub_apply, PiLp.smul_apply, smul_eq_mul]
+        simp
+        dsimp [t]
+        simp only [map_sub, PiLp.sub_apply]
+        field_simp [hdx]
+        ring
+      · simp only [map_add, map_sub, LinearIsometryEquiv.apply_symm_apply, map_smul,
+          PiLp.add_apply, PiLp.sub_apply, PiLp.smul_apply, smul_eq_mul]
+        simp
+        dsimp [y]
+        simp only [map_sub, PiLp.sub_apply]
+        ring
+  rw [upperGraphHeight, hfiber, csSup_singleton]
+
+private theorem deriv_upperGraphHeight_segment_of_lt (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) {a b : Point}
+    (hK : (K : Set Point) = segment ℝ a b)
+    (hab : e (a - o) 0 < e (b - o) 0) {x : ℝ}
+    (hx : x ∈ Set.Ioo (e (a - o) 0) (e (b - o) 0)) :
+    deriv (upperGraphHeight K o e) x =
+      (e (b - o) 1 - e (a - o) 1) / (e (b - o) 0 - e (a - o) 0) := by
+  let g : ℝ → ℝ := fun y ↦ e (a - o) 1 +
+    ((y - e (a - o) 0) / (e (b - o) 0 - e (a - o) 0)) *
+      (e (b - o) 1 - e (a - o) 1)
+  have hg : HasDerivAt g
+      ((e (b - o) 1 - e (a - o) 1) / (e (b - o) 0 - e (a - o) 0)) x := by
+    dsimp [g]
+    have hsub : HasDerivAt (fun y : ℝ ↦ y - e (a - o) 0) 1 x :=
+      (hasDerivAt_id x).sub_const _
+    have hdiv := HasDerivAt.div_const hsub (e (b - o) 0 - e (a - o) 0)
+    have hmul := HasDerivAt.mul_const hdiv (e (b - o) 1 - e (a - o) 1)
+    convert hmul.const_add (e (a - o) 1) using 1
+    all_goals ring
+  have heq : upperGraphHeight K o e =ᶠ[nhds x] g := by
+    filter_upwards [isOpen_Ioo.mem_nhds hx] with y hy
+    exact upperGraphHeight_segment_of_lt K o e hK hab ⟨hy.1.le, hy.2.le⟩
+  exact (hg.congr_of_eventuallyEq heq).deriv
+
+private theorem segment_graph_normal (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) {a b : Point}
+    (hab : e (a - o) 0 < e (b - o) 0) :
+    let m := (e (b - o) 1 - e (a - o) 1) / (e (b - o) 0 - e (a - o) 0)
+    let q := e.symm !₂[-m, 1]
+    q ≠ 0 ∧ inner ℝ (b - a) q = 0 ∧
+      e (normalVector (vectorNormalAngle q)) 1 = ‖q‖⁻¹ ∧
+      dist a b = (e (b - o) 0 - e (a - o) 0) * ‖q‖ := by
+  dsimp only
+  let m := (e (b - o) 1 - e (a - o) 1) / (e (b - o) 0 - e (a - o) 0)
+  let q := e.symm !₂[-m, 1]
+  have hq : q ≠ 0 := by
+    intro h
+    have h1 := congrFun (congrArg WithLp.ofLp (congrArg e h)) 1
+    simp [q] at h1
+  have hdx : 0 < e (b - o) 0 - e (a - o) 0 := sub_pos.mpr hab
+  have hden : e b 0 - e a 0 ≠ 0 := by
+    simp only [map_sub, PiLp.sub_apply] at hab
+    linarith
+  have horth : inner ℝ (b - a) q = 0 := by
+    rw [← e.inner_map_map]
+    simp only [q, m, LinearIsometryEquiv.apply_symm_apply, PiLp.inner_apply,
+      RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, map_sub, PiLp.sub_apply]
+    simp only [Matrix.cons_val_zero, Matrix.cons_val_one]
+    field_simp [hden]
+    ring
+  have hup : e (normalVector (vectorNormalAngle q)) 1 = ‖q‖⁻¹ := by
+    rw [normalVector_vectorNormalAngle hq, map_smul, PiLp.smul_apply]
+    simp [q]
+  have hlen : dist a b = (e (b - o) 0 - e (a - o) 0) * ‖q‖ := by
+    rw [dist_eq_norm, ← e.norm_map]
+    have hv : e (b - a) = (e (b - o) 0 - e (a - o) 0) • !₂[1, m] := by
+      ext i
+      fin_cases i
+      · simp [m]
+      · simp [m]
+        field_simp [hden]
+    rw [show a - b = -(b - a) by module, map_neg, norm_neg, hv, norm_smul,
+      Real.norm_eq_abs, abs_of_pos hdx]
+    congr 1
+    rw [← e.norm_map q, LinearIsometryEquiv.apply_symm_apply]
+    rw [EuclideanSpace.norm_eq, EuclideanSpace.norm_eq]
+    congr 1
+    simp [m, Real.norm_eq_abs, pow_two]
+    ring
+  exact ⟨hq, horth, hup, hlen⟩
+
+private theorem segment_graph_integrand_ae (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) (ψ : Real.Angle → ℝ) {a b : Point}
+    (hK : (K : Set Point) = segment ℝ a b)
+    (hab : e (a - o) 0 < e (b - o) 0) :
+    let m := (e (b - o) 1 - e (a - o) 1) / (e (b - o) 0 - e (a - o) 0)
+    let q := e.symm !₂[-m, 1]
+    upperGraphSurfaceIntegrand K o e ψ =ᵐ[
+        volume.restrict (Set.Icc (e (a - o) 0) (e (b - o) 0))]
+      fun _ ↦ ψ (vectorNormalAngle q) * ‖q‖ := by
+  dsimp only
+  let m := (e (b - o) 1 - e (a - o) 1) / (e (b - o) 0 - e (a - o) 0)
+  let q := e.symm !₂[-m, 1]
+  have hnormq : Real.sqrt (1 + m ^ 2) = ‖q‖ := by
+    rw [← e.norm_map q, LinearIsometryEquiv.apply_symm_apply, EuclideanSpace.norm_eq]
+    congr 1
+    simp [Fin.sum_univ_two, Real.norm_eq_abs, pow_two]
+    ring
+  rw [← restrict_Ioo_eq_restrict_Icc]
+  refine ae_restrict_of_forall_mem measurableSet_Ioo fun x hx ↦ ?_
+  rw [upperGraphSurfaceIntegrand,
+    deriv_upperGraphHeight_segment_of_lt K o e hK hab hx]
+  change ψ (vectorNormalAngle q) * Real.sqrt (1 + m ^ 2) = _
+  rw [hnormq]
+
+private theorem integrable_segment_graph (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) (ψ : Real.Angle → ℝ) {a b : Point}
+    (hK : (K : Set Point) = segment ℝ a b)
+    (hab : e (a - o) 0 < e (b - o) 0) :
+    Integrable (upperGraphSurfaceIntegrand K o e ψ)
+      (volume.restrict (Set.Icc (e (a - o) 0) (e (b - o) 0))) := by
+  have hae := segment_graph_integrand_ae K o e ψ hK hab
+  have hc : Integrable (fun _ : ℝ ↦ ψ (vectorNormalAngle
+      (e.symm !₂[-((e (b - o) 1 - e (a - o) 1) /
+        (e (b - o) 0 - e (a - o) 0)), 1])) *
+      ‖e.symm !₂[-((e (b - o) 1 - e (a - o) 1) /
+        (e (b - o) 0 - e (a - o) 0)), 1]‖)
+      (volume.restrict (Set.Icc (e (a - o) 0) (e (b - o) 0))) := integrable_const _
+  exact hc.congr hae.symm
+
+private theorem segment_graph_integral (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) (ψ : Real.Angle → ℝ) {a b : Point}
+    (hK : (K : Set Point) = segment ℝ a b)
+    (hab : e (a - o) 0 < e (b - o) 0) {ε : ℝ} (hε : 0 < ε)
+    (hsupport : ∀ t, e (normalVector t) 1 < ε → ψ t = 0) :
+    (∫ t, ψ t ∂surfaceAreaMeasure K) =
+      ∫ x in Set.Icc (e (a - o) 0) (e (b - o) 0),
+        upperGraphSurfaceIntegrand K o e ψ x := by
+  let m := (e (b - o) 1 - e (a - o) 1) / (e (b - o) 0 - e (a - o) 0)
+  let q := e.symm !₂[-m, 1]
+  let t := vectorNormalAngle q
+  obtain ⟨hq, horth, hup, hlen⟩ := segment_graph_normal o e hab
+  have hd : IsSegmentPresentation K (a, b, t) := by
+    refine ⟨?_, hK, ?_⟩
+    · intro heq
+      have := hab
+      simp only at heq
+      rw [heq] at this
+      exact this.false
+    · dsimp [t]
+      rw [normalVector_vectorNormalAngle hq, inner_smul_right, horth, mul_zero]
+  have hqnorm : 0 < ‖q‖ := norm_pos_iff.mpr hq
+  have hdowncoord : e (normalVector (t + (Real.pi : Real.Angle))) 1 = -‖q‖⁻¹ := by
+    have hv : normalVector (t + (Real.pi : Real.Angle)) = -normalVector t := by
+      induction t using Real.Angle.induction_on with
+      | _ t => simpa only [← Real.Angle.coe_add] using normalVector_add_pi t
+    rw [hv, map_neg, PiLp.neg_apply]
+    exact congrArg Neg.neg hup
+  have hdown : ψ (t + (Real.pi : Real.Angle)) = 0 :=
+    hsupport _ (by
+      rw [hdowncoord]
+      have hinv : 0 < ‖q‖⁻¹ := inv_pos.mpr hqnorm
+      linarith)
+  have hlhs : (∫ u, ψ u ∂surfaceAreaMeasure K) = dist a b * ψ t := by
+    rw [surfaceAreaMeasure_eq_segmentPresentation K (a, b, t) hd]
+    rw [MeasureTheory.integral_smul_measure]
+    rw [MeasureTheory.integral_add_measure (MeasureTheory.integrable_dirac (by finiteness))
+      (MeasureTheory.integrable_dirac (by finiteness))]
+    simp [hdown, ENNReal.toReal_ofReal (dist_nonneg : 0 ≤ dist a b)]
+  rw [hlhs, hlen]
+  have hae := segment_graph_integrand_ae K o e ψ hK hab
+  rw [integral_congr_ae hae]
+  have hdx : 0 ≤ e (b - o) 0 - e (a - o) 0 := sub_nonneg.mpr hab.le
+  have hdx' : 0 ≤ e b 0 - e a 0 := by
+    simp only [map_sub, PiLp.sub_apply] at hdx
+    linarith
+  simp only [Fin.isValue, map_sub, PiLp.sub_apply, sub_sub_sub_cancel_right, norm_map,
+    integral_const, MeasurableSet.univ, measureReal_restrict_apply, Set.univ_inter,
+    Real.volume_real_Icc, smul_eq_mul, t, q, m]
+  rw [max_eq_left hdx']
+  ring
+
+/-- The surface-area integral of a nonvertical segment agrees with its upper
+graph integral. -/
+theorem segment_graph_formula (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) (ψ : Real.Angle → ℝ) {ε : ℝ} (hε : 0 < ε)
+    (hsupport : ∀ t, e (normalVector t) 1 < ε → ψ t = 0)
+    (d : Point × Point × Real.Angle) (hd : IsSegmentPresentation K d)
+    (hbounds : (horizontalBounds K o e).1 < (horizontalBounds K o e).2) :
+    Integrable (upperGraphSurfaceIntegrand K o e ψ)
+        (volume.restrict (Set.Icc (horizontalBounds K o e).1
+          (horizontalBounds K o e).2)) ∧
+      (∫ t, ψ t ∂surfaceAreaMeasure K) =
+        ∫ x in Set.Icc (horizontalBounds K o e).1 (horizontalBounds K o e).2,
+          upperGraphSurfaceIntegrand K o e ψ x := by
+  rw [horizontalBounds_segment K o e hd.2.1] at hbounds ⊢
+  dsimp only at hbounds ⊢
+  have hne : e (d.1 - o) 0 ≠ e (d.2.1 - o) 0 := by
+    intro heq
+    simp only [map_sub, PiLp.sub_apply] at heq hbounds
+    rw [heq, min_self, max_self] at hbounds
+    exact hbounds.false
+  rcases lt_or_gt_of_ne hne with hab | hba
+  · rw [min_eq_left hab.le, max_eq_right hab.le] at hbounds ⊢
+    exact ⟨integrable_segment_graph K o e ψ hd.2.1 hbounds,
+      segment_graph_integral K o e ψ hd.2.1 hbounds hε hsupport⟩
+  · rw [min_eq_right hba.le, max_eq_left hba.le] at hbounds ⊢
+    have hK' : (K : Set Point) = segment ℝ d.2.1 d.1 := by
+      rw [segment_symm]
+      exact hd.2.1
+    exact ⟨integrable_segment_graph K o e ψ hK' hbounds,
+      segment_graph_integral K o e ψ hK' hbounds hε hsupport⟩
+
+/-- A nonsingleton planar convex body with empty interior has a segment presentation. -/
+theorem exists_segmentPresentation_of_interior_empty (K : ConvexBody Point)
+    (hsub : ¬(K : Set Point).Subsingleton) (hint : interior (K : Set Point) = ∅) :
+    ∃ d, IsSegmentPresentation K d := by
+  obtain ⟨a, b, hab, hK⟩ := K.exists_eq_segment_of_interior_empty hsub hint
+  let orient : Orientation ℝ Point (Fin 2) :=
+    (EuclideanSpace.basisFun (Fin 2) ℝ).toBasis.orientation
+  let _ : Fact (Module.finrank ℝ Point = 2) := ⟨by simp [Point]⟩
+  let n := ‖b - a‖⁻¹ • orient.rotation (Real.pi / 2 : ℝ) (b - a)
+  have hvnorm : 0 < ‖b - a‖ := norm_pos_iff.mpr (sub_ne_zero.mpr hab.symm)
+  have hn : ‖n‖ = 1 := by
+    rw [norm_smul, (orient.rotation (Real.pi / 2 : ℝ)).norm_map]
+    simp [hvnorm.ne']
+  obtain ⟨t, ht⟩ := exists_angle_normalVector_eq hn
+  refine ⟨(a, b, t), hab, hK, ?_⟩
+  rw [ht]
+  exact orient.inner_smul_rotation_pi_div_two_right (b - a) ‖b - a‖⁻¹
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Curve.Foundations.Development001`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Curve.Area`.
+* `Curve.AreaTransport`.
+* `Curve.Concatenation`.
+* `Curve.AreaAdditivity`.
+* `Curve.CyclicRotation`.
+* `Curve.Jordan.Basic`.
+* `Curve.Jordan.Orientation`.
+* `Curve.Jordan.Area`.
+* `Curve.Jordan.ArcArea`.
+* `Curve.Jordan.Parametrization`.
+* `Curve.Jordan.UnitSphere`.
+* `Curve.Jordan.Winding`.
+* `Curve.Jordan.RadialWinding`.
+* `Curve.Jordan.WindingConcatenation`.
+* `Curve.Jordan.CyclicRotation`.
+* `Curve.Jordan.WindingKernel`.
+* `Curve.Jordan.WindingLocalConstancy`.
+* `Curve.Jordan.WindingLifts`.
+* `Curve.NullRange`.
+* `Curve.SegmentArea`.
+* `Curve.SegmentArea.Parametrization`.
+* `Curve.SmoothIntervalPaths`.
+* `Curve.Jordan.RadialLoop`.
+* `Curve.StieltjesChainRule`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Area
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The constant planar path, as a continuous path of bounded variation. -/
+def constBVPath (a b : ℝ) (p : Point) : ContinuousBVPaths a b :=
+  ⟨fun _ ↦ p, continuous_const, fun i ↦ by
+    simp [IsIntervalBoundedVariation, BoundedVariationOn, eVariationOn]⟩
+
+/-- View one coordinate of a continuous BV path as a right-continuous BV function. -/
+def continuousBVCoordinate {a b : ℝ} (x : ContinuousBVPaths a b) (i : Fin 2) :
+    RightContinuousIntervalBV a b where
+  toFun t := x.val t i
+  boundedVariation := x.property.2 i
+  right_continuous _ :=
+    ((PiLp.continuous_apply 2 _ i).comp x.property.1).continuousAt.continuousWithinAt
+
+/-- Half the difference of the two coordinate Stieltjes integrals, giving signed area. -/
+def curveAreaFunctional {a b : ℝ} (x : ContinuousBVPaths a b) : ℝ :=
+  (intervalStieltjesIntegral (continuousBVCoordinate x 1) (fun t ↦ x.val t 0) Set.univ -
+    intervalStieltjesIntegral (continuousBVCoordinate x 0) (fun t ↦ x.val t 1) Set.univ) / 2
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Area Transport
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open MeasureTheory Set
+
+/-- A continuous monotone surjection preserves continuous bounded variation. -/
+theorem continuousBVPaths_comp_monotone_surjective
+    {a b c d : ℝ} (hab : a ≤ b) (x : ContinuousBVPaths a b)
+    (φ : Set.Icc c d → Set.Icc a b) (hφc : Continuous φ)
+    (hφm : Monotone φ) (hφs : Function.Surjective φ) :
+    ∃ y : ContinuousBVPaths c d, y.val = x.val ∘ φ := by
+  let y : ContinuousBVPaths c d :=
+    ⟨x.val ∘ φ, x.property.1.comp hφc, fun i ↦
+      BoundedVariationOn.comp_monotone_surjective_Icc hab (x.property.2 i) hφm hφs⟩
+  exact ⟨y, rfl⟩
+
+/-- A continuous monotone surjection preserves signed path area. -/
+theorem curveArea_comp_monotone_surjective
+    {a b c d : ℝ} (hab : a ≤ b) (hcd : c ≤ d) (x : ContinuousBVPaths a b)
+    (φ : Set.Icc c d → Set.Icc a b) (hφc : Continuous φ)
+    (hφm : Monotone φ) (hφs : Function.Surjective φ) :
+    ∃ y : ContinuousBVPaths c d,
+      y.val = x.val ∘ φ ∧ curveAreaFunctional y = curveAreaFunctional x := by
+  let y : ContinuousBVPaths c d :=
+    ⟨x.val ∘ φ, x.property.1.comp hφc, fun i ↦
+      BoundedVariationOn.comp_monotone_surjective_Icc hab (x.property.2 i) hφm hφs⟩
+  refine ⟨y, rfl, ?_⟩
+  unfold curveAreaFunctional
+  rw [intervalStieltjesIntegral_comp_monotone_surjective hab hcd
+      (continuousBVCoordinate x 1)
+      ((PiLp.continuous_apply 2 _ 1).comp x.property.1)
+      (fun t ↦ x.val t 0) ((PiLp.continuous_apply 2 _ 0).comp x.property.1)
+      φ hφc hφm hφs,
+    intervalStieltjesIntegral_comp_monotone_surjective hab hcd
+      (continuousBVCoordinate x 0)
+      ((PiLp.continuous_apply 2 _ 0).comp x.property.1)
+      (fun t ↦ x.val t 1) ((PiLp.continuous_apply 2 _ 1).comp x.property.1)
+      φ hφc hφm hφs]
+  rfl
+
+/-- A continuous monotone surjective reparametrisation identifies the signed areas of two given
+paths whenever one is the composite of the other with it. -/
+theorem curveAreaFunctional_eq_of_comp_monotone_surjective {a b c d : ℝ}
+    (hab : a ≤ b) (hcd : c ≤ d) (x : ContinuousBVPaths a b) (y : ContinuousBVPaths c d)
+    (φ : Set.Icc c d → Set.Icc a b) (hφc : Continuous φ) (hφm : Monotone φ)
+    (hφs : Function.Surjective φ) (hy : y.val = x.val ∘ φ) :
+    curveAreaFunctional y = curveAreaFunctional x := by
+  obtain ⟨z, hz, hza⟩ := curveArea_comp_monotone_surjective hab hcd x φ hφc hφm hφs
+  rw [show y = z from Subtype.ext (hy.trans hz.symm)]
+  exact hza
+
+/-- Translating a continuous BV path shifts its signed area by the cross product of the
+translation vector with the path's total displacement, halved. -/
+theorem curveAreaFunctional_add_constBVPath {a b : ℝ} (hab : a ≤ b)
+    (x : ContinuousBVPaths a b) (v : Point) :
+    curveAreaFunctional (x + constBVPath a b v) =
+      curveAreaFunctional x +
+        (v 0 * (x.val ⟨b, hab, le_rfl⟩ 1 - x.val ⟨a, le_rfl, hab⟩ 1) -
+          v 1 * (x.val ⟨b, hab, le_rfl⟩ 0 - x.val ⟨a, le_rfl, hab⟩ 0)) / 2 := by
+  have hcont : ∀ i : Fin 2, Continuous (continuousBVCoordinate x i).toFun :=
+    fun i ↦ (PiLp.continuous_apply 2 _ i).comp x.property.1
+  have hval : ∀ (i : Fin 2) (t : Set.Icc a b),
+      (continuousBVCoordinate (x + constBVPath a b v) i).toFun t =
+        (continuousBVCoordinate x i).toFun t + v i := by
+    intro i t
+    change (x.val t + v) i = x.val t i + v i
+    simp
+  have hint : ∀ i j : Fin 2,
+      intervalStieltjesIntegral (continuousBVCoordinate (x + constBVPath a b v) i)
+          (fun t ↦ (x + constBVPath a b v).val t j) univ =
+        intervalStieltjesIntegral (continuousBVCoordinate x i) (fun t ↦ x.val t j) univ +
+          v j * (x.val ⟨b, hab, le_rfl⟩ i - x.val ⟨a, le_rfl, hab⟩ i) := by
+    intro i j
+    rw [show (fun t ↦ (x + constBVPath a b v).val t j) =
+      fun t ↦ (continuousBVCoordinate x j).toFun t + v j from funext (hval j)]
+    exact intervalStieltjesIntegral_add_const hab (continuousBVCoordinate x i) _ (hcont i)
+      (v i) (hval i) _ (hcont j) (v j)
+  unfold curveAreaFunctional
+  rw [hint 1 0, hint 0 1]
+  ring
+
+/-- Reversing a continuous BV path negates its signed area. -/
+theorem curveArea_comp_reverse
+    {a b : ℝ} (hab : a ≤ b) (x : ContinuousBVPaths a b) :
+    let r := Set.Icc.reverse hab
+    let y : ContinuousBVPaths a b :=
+      ⟨x.val ∘ r, x.property.1.comp (Set.Icc.continuous_reverse hab), fun i ↦
+        BoundedVariationOn.comp_antitone_surjective_Icc hab (x.property.2 i)
+          (Set.Icc.antitone_reverse hab) (Set.Icc.surjective_reverse hab)⟩
+    curveAreaFunctional y = -curveAreaFunctional x := by
+  dsimp only
+  let r := Set.Icc.reverse hab
+  let y : ContinuousBVPaths a b :=
+    ⟨x.val ∘ r, x.property.1.comp (Set.Icc.continuous_reverse hab), fun i ↦
+      BoundedVariationOn.comp_antitone_surjective_Icc hab (x.property.2 i)
+        (Set.Icc.antitone_reverse hab) (Set.Icc.surjective_reverse hab)⟩
+  have hcoord (p q : Fin 2) :
+      intervalStieltjesIntegral (continuousBVCoordinate y p)
+          (fun t ↦ y.val t q) Set.univ =
+        -intervalStieltjesIntegral (continuousBVCoordinate x p)
+          (fun t ↦ x.val t q) Set.univ := by
+    have hrev := intervalStieltjesIntegral_comp_reverse hab
+      (continuousBVCoordinate x p)
+      ((PiLp.continuous_apply 2 _ p).comp x.property.1)
+      (fun t ↦ x.val t q) ((PiLp.continuous_apply 2 _ q).comp x.property.1)
+    change intervalStieltjesIntegral _ _ Set.univ = -intervalStieltjesIntegral _ _ Set.univ
+    exact hrev
+  unfold curveAreaFunctional
+  rw [hcoord 1 0, hcoord 0 1]
+  ring
+
+/-- A continuous antitone surjection negates signed path area. -/
+theorem curveArea_comp_antitone_surjective
+    {a b c d : ℝ} (hab : a ≤ b) (hcd : c ≤ d) (x : ContinuousBVPaths a b)
+    (φ : Set.Icc c d → Set.Icc a b) (hφc : Continuous φ)
+    (hφa : Antitone φ) (hφs : Function.Surjective φ) :
+    ∃ y : ContinuousBVPaths c d,
+      y.val = x.val ∘ φ ∧ curveAreaFunctional y = -curveAreaFunctional x := by
+  let y : ContinuousBVPaths c d :=
+    ⟨x.val ∘ φ, x.property.1.comp hφc, fun i ↦
+      BoundedVariationOn.comp_antitone_surjective_Icc hab (x.property.2 i) hφa hφs⟩
+  let r := Set.Icc.reverse hcd
+  let ψ : Set.Icc c d → Set.Icc a b := φ ∘ r
+  have hψc : Continuous ψ := hφc.comp (Set.Icc.continuous_reverse hcd)
+  have hψm : Monotone ψ := fun _ _ hst ↦
+    hφa ((Set.Icc.antitone_reverse hcd) hst)
+  have hψs : Function.Surjective ψ :=
+    hφs.comp (Set.Icc.surjective_reverse hcd)
+  obtain ⟨z, hz, hzarea⟩ :=
+    curveArea_comp_monotone_surjective hab hcd x ψ hψc hψm hψs
+  have hyrev : curveAreaFunctional z = -curveAreaFunctional y := by
+    have hz' : z.val = y.val ∘ r := by
+      rw [hz]
+      rfl
+    let yr : ContinuousBVPaths c d :=
+      ⟨y.val ∘ r, y.property.1.comp (Set.Icc.continuous_reverse hcd), fun i ↦
+        BoundedVariationOn.comp_antitone_surjective_Icc hcd (y.property.2 i)
+          (Set.Icc.antitone_reverse hcd) (Set.Icc.surjective_reverse hcd)⟩
+    have hyr := curveArea_comp_reverse hcd y
+    change curveAreaFunctional yr = -curveAreaFunctional y at hyr
+    have hzy : z = yr := by
+      apply Subtype.ext
+      exact hz'
+    simpa [hzy] using hyr
+  refine ⟨y, rfl, ?_⟩
+  linarith
+
+/-- A constant path has zero signed area, including on an empty parameter interval. -/
+theorem curveAreaFunctional_eq_zero_of_constant
+    {a b : ℝ} (x : ContinuousBVPaths a b) (p : Point)
+    (hx : ∀ t, x.val t = p) : curveAreaFunctional x = 0 := by
+  have hfun : x.val = fun _ ↦ p := funext hx
+  have hcoord (i : Fin 2) : (fun t ↦ x.val t i) = fun _ ↦ p i := by
+    funext t
+    rw [hx t]
+  let _ : MeasureTheory.IsFiniteMeasure
+      (intervalStieltjesMeasure (continuousBVCoordinate x 0)).variation :=
+    BoundedVariationOn.instIsFiniteMeasureVariationVectorMeasure (x.property.2 0)
+  let _ : MeasureTheory.IsFiniteMeasure
+      (intervalStieltjesMeasure (continuousBVCoordinate x 1)).variation :=
+    BoundedVariationOn.instIsFiniteMeasureVariationVectorMeasure (x.property.2 1)
+  unfold curveAreaFunctional
+  unfold intervalStieltjesIntegral
+  rw [MeasureTheory.VectorMeasure.setIntegral_congr_fun
+      (s := Set.univ) (f := fun t ↦ x.val t 0) (g := fun _ ↦ p 0)
+      (fun _ _ ↦ congrFun (hcoord 0) _),
+    MeasureTheory.VectorMeasure.setIntegral_congr_fun
+      (s := Set.univ) (f := fun t ↦ x.val t 1) (g := fun _ ↦ p 1)
+      (fun _ _ ↦ congrFun (hcoord 1) _)]
+  rw [MeasureTheory.VectorMeasure.setIntegral_const,
+    MeasureTheory.VectorMeasure.setIntegral_const]
+  by_cases hab : a ≤ b
+  · let _ : Fact (a ≤ b) := ⟨hab⟩
+    simp [intervalStieltjesMeasure, continuousBVCoordinate, hfun, Filter.limUnder,
+      Filter.map_const]
+  · let _ : IsEmpty (Set.Icc a b) :=
+      ⟨fun t ↦ hab (le_trans t.property.1 t.property.2)⟩
+    have hfilters : (Filter.atTop : Filter (Set.Icc a b)) = Filter.atBot :=
+      Subsingleton.elim _ _
+    simp [intervalStieltjesMeasure, continuousBVCoordinate, hfun, Filter.limUnder, hfilters]
+
+/-- A continuous monotone or antitone surjection transports BV paths and signed area. -/
+theorem curveArea_comp_monotone_or_antitone_surjective
+    {a b c d : ℝ} (hab : a ≤ b) (hcd : c ≤ d) (x : ContinuousBVPaths a b)
+    (φ : Set.Icc c d → Set.Icc a b) (hφc : Continuous φ)
+    (hφs : Function.Surjective φ) (hφ : Monotone φ ∨ Antitone φ) :
+    ∃ y : ContinuousBVPaths c d, y.val = x.val ∘ φ ∧
+      (Monotone φ → curveAreaFunctional y = curveAreaFunctional x) ∧
+      (Antitone φ → curveAreaFunctional y = -curveAreaFunctional x) := by
+  rcases hφ with hm | ha
+  · obtain ⟨y, hy, harea⟩ := curveArea_comp_monotone_surjective hab hcd x φ hφc hm hφs
+    refine ⟨y, hy, fun _ ↦ harea, ?_⟩
+    intro ha
+    obtain ⟨z, hz, hzarea⟩ := curveArea_comp_antitone_surjective hab hcd x φ hφc ha hφs
+    have hzy : z = y := Subtype.ext (hz.trans hy.symm)
+    simpa only [hzy] using hzarea
+  · obtain ⟨y, hy, harea⟩ := curveArea_comp_antitone_surjective hab hcd x φ hφc ha hφs
+    refine ⟨y, hy, ?_, fun _ ↦ harea⟩
+    intro hm
+    obtain ⟨z, hz, hzarea⟩ := curveArea_comp_monotone_surjective hab hcd x φ hφc hm hφs
+    have hzy : z = y := Subtype.ext (hz.trans hy.symm)
+    simpa only [hzy] using hzarea
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Concatenation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A continuous path of bounded variation with an ordered real parameter interval. -/
+structure RectifiablePathData where
+  /-- The initial real parameter. -/
+  a : ℝ
+  /-- The terminal real parameter. -/
+  b : ℝ
+  ordered : a ≤ b
+  /-- The continuous parametrization of bounded variation. -/
+  path : ContinuousBVPaths a b
+
+/-- An ordered interval partition identifies the path with a nonempty list of parametrized
+pieces. -/
+def IsPathConcatenation (Γ : RectifiablePathData) {n : ℕ}
+    (pieces : Fin n → RectifiablePathData) : Prop :=
+  0 < n ∧ ∃ cuts : Fin (n + 1) → Set.Icc Γ.a Γ.b,
+    Monotone cuts ∧ (cuts 0 : ℝ) = Γ.a ∧ (cuts (Fin.last n) : ℝ) = Γ.b ∧
+    ∀ i : Fin n,
+      ∃ (φ : Set.Icc (0 : ℝ) 1 →
+          Set.Icc (cuts i.castSucc : ℝ) (cuts i.succ : ℝ))
+        (ψ : Set.Icc (0 : ℝ) 1 → Set.Icc (pieces i).a (pieces i).b),
+        Continuous φ ∧ Monotone φ ∧ Function.Surjective φ ∧
+        Continuous ψ ∧ Monotone ψ ∧ Function.Surjective ψ ∧
+        ∀ s, Γ.path.val ⟨(φ s : ℝ),
+          le_trans (cuts i.castSucc).property.1 (φ s).property.1,
+          le_trans (φ s).property.2 (cuts i.succ).property.2⟩ = (pieces i).path.val (ψ s)
+
+/-- A concatenated path stays inside the union of the ranges of its pieces. -/
+theorem IsPathConcatenation.range_subset_iUnion {Γ : RectifiablePathData} {n : ℕ}
+    {pieces : Fin n → RectifiablePathData} (h : IsPathConcatenation Γ pieces) :
+    Set.range Γ.path.val ⊆ ⋃ i, Set.range (pieces i).path.val := by
+  classical
+  obtain ⟨hn, cuts, hmono, hfirst, hlast, hconc⟩ := h
+  rintro _ ⟨t, rfl⟩
+  obtain ⟨i, hi₁, hi₂⟩ : ∃ i : Fin n,
+      (cuts i.castSucc : ℝ) ≤ (t : ℝ) ∧ (t : ℝ) ≤ (cuts i.succ : ℝ) := by
+    set T := Finset.univ.filter fun i : Fin (n + 1) ↦ (t : ℝ) ≤ (cuts i : ℝ) with hT
+    have hmemT : ∀ i, i ∈ T ↔ (t : ℝ) ≤ (cuts i : ℝ) := fun i ↦ by simp [hT]
+    have hTne : T.Nonempty :=
+      ⟨Fin.last n, (hmemT _).mpr (by rw [hlast]; exact t.property.2)⟩
+    have hmin : (t : ℝ) ≤ (cuts (T.min' hTne) : ℝ) := (hmemT _).mp (T.min'_mem hTne)
+    rcases Fin.eq_zero_or_eq_succ (T.min' hTne) with hzero | ⟨j, hj⟩
+    · refine ⟨⟨0, hn⟩, ?_, ?_⟩
+      · rw [show ((⟨0, hn⟩ : Fin n).castSucc) = 0 from rfl, hfirst]
+        exact t.property.1
+      · exact (hzero ▸ hmin).trans
+          (Subtype.coe_le_coe.mpr (hmono (Fin.zero_le ((⟨0, hn⟩ : Fin n).succ))))
+    · refine ⟨j, ?_, by rw [← hj]; exact hmin⟩
+      by_contra hlt
+      have hle := T.min'_le _ ((hmemT j.castSucc).mpr (not_le.mp hlt).le)
+      rw [hj] at hle
+      exact absurd (lt_of_lt_of_le (Fin.castSucc_lt_succ (i := j)) hle) (lt_irrefl _)
+  obtain ⟨φ, ψ, -, -, hφs, -, -, -, heq⟩ := hconc i
+  obtain ⟨s, hs⟩ := hφs ⟨(t : ℝ), hi₁, hi₂⟩
+  have hval : ((φ s : ℝ)) = (t : ℝ) := congrArg Subtype.val hs
+  refine Set.mem_iUnion.mpr ⟨i, ψ s, ?_⟩
+  rw [← heq s]
+  exact congrArg _ (Subtype.ext hval)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Area Additivity
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped Function
+
+namespace MovingSofa
+
+theorem curveArea_concatenation (Γ : RectifiablePathData) {n : ℕ}
+    (pieces : Fin n → RectifiablePathData) (h : IsPathConcatenation Γ pieces) :
+    curveAreaFunctional Γ.path = ∑ i, curveAreaFunctional (pieces i).path := by
+  rcases h with ⟨_, cuts, hcuts, hcuts_zero, hcuts_last, hpieces⟩
+  have hcoord (i : Fin n) (p q : Fin 2) :
+      intervalStieltjesIntegral (continuousBVCoordinate Γ.path p)
+          (fun t ↦ Γ.path.val t q) (Ioc (cuts i.castSucc) (cuts i.succ)) =
+        intervalStieltjesIntegral (continuousBVCoordinate (pieces i).path p)
+          (fun t ↦ (pieces i).path.val t q) Set.univ := by
+    obtain ⟨φ, ψ, hφc, hφm, hφs, hψc, hψm, hψs, heq⟩ := hpieces i
+    let l := cuts i.castSucc
+    let u := cuts i.succ
+    let ι : Set.Icc (l : ℝ) u → Set.Icc Γ.a Γ.b := fun x ↦
+      ⟨x, le_trans l.property.1 x.property.1, le_trans x.property.2 u.property.2⟩
+    let hp : BoundedVariationOn ((continuousBVCoordinate Γ.path p).toFun ∘ ι) Set.univ :=
+      ne_top_of_le_ne_top (continuousBVCoordinate Γ.path p).boundedVariation
+        (eVariationOn.comp_le_of_monotoneOn _ ι
+          (fun _ _ _ _ h ↦ h) (mapsTo_univ ι _))
+    let Fp : RightContinuousIntervalBV (l : ℝ) u :=
+      { toFun := (continuousBVCoordinate Γ.path p).toFun ∘ ι
+        boundedVariation := hp
+        right_continuous := fun _ ↦
+          (((PiLp.continuous_apply 2 _ p).comp Γ.path.property.1).comp
+            (continuous_subtype_val.subtype_mk _)).continuousAt.continuousWithinAt }
+    rw [intervalStieltjesIntegral_Ioc_eq_inclusion
+      (continuousBVCoordinate Γ.path p)
+      ((PiLp.continuous_apply 2 _ p).comp Γ.path.property.1)
+      (fun t ↦ Γ.path.val t q)
+      ((PiLp.continuous_apply 2 _ q).comp Γ.path.property.1) l u
+      (hcuts (Fin.castSucc_le_succ i))]
+    change intervalStieltjesIntegral Fp
+        ((fun t ↦ Γ.path.val t q) ∘ ι) Set.univ =
+      intervalStieltjesIntegral (continuousBVCoordinate (pieces i).path p)
+        (fun t ↦ (pieces i).path.val t q) Set.univ
+    rw [intervalStieltjesIntegral_comp_monotone_surjective
+      (show (l : ℝ) ≤ u from hcuts (Fin.castSucc_le_succ i)) (by norm_num) Fp
+      (((PiLp.continuous_apply 2 _ p).comp Γ.path.property.1).comp
+        (continuous_subtype_val.subtype_mk _))
+      ((fun t ↦ Γ.path.val t q) ∘ ι)
+      (((PiLp.continuous_apply 2 _ q).comp Γ.path.property.1).comp
+        (continuous_subtype_val.subtype_mk _))
+      φ hφc hφm hφs,
+      intervalStieltjesIntegral_comp_monotone_surjective
+        (pieces i).ordered (by norm_num) (continuousBVCoordinate (pieces i).path p)
+        ((PiLp.continuous_apply 2 _ p).comp (pieces i).path.property.1)
+        (fun t ↦ (pieces i).path.val t q)
+        ((PiLp.continuous_apply 2 _ q).comp (pieces i).path.property.1)
+        ψ hψc hψm hψs]
+    congr 1
+    · congr 1
+      funext s
+      simpa [Fp, continuousBVCoordinate, ι, Function.comp_apply] using
+        congrArg (fun z ↦ z p) (heq s)
+    · funext s
+      simpa [continuousBVCoordinate, ι, Function.comp_apply] using
+        congrArg (fun z ↦ z q) (heq s)
+  unfold curveAreaFunctional
+  rw [intervalStieltjesIntegral_eq_sum_Ioc (continuousBVCoordinate Γ.path 1)
+      ((PiLp.continuous_apply 2 _ 1).comp Γ.path.property.1)
+      (fun t ↦ Γ.path.val t 0) ((PiLp.continuous_apply 2 _ 0).comp Γ.path.property.1)
+      cuts hcuts hcuts_zero hcuts_last,
+    intervalStieltjesIntegral_eq_sum_Ioc (continuousBVCoordinate Γ.path 0)
+      ((PiLp.continuous_apply 2 _ 0).comp Γ.path.property.1)
+      (fun t ↦ Γ.path.val t 1) ((PiLp.continuous_apply 2 _ 1).comp Γ.path.property.1)
+      cuts hcuts hcuts_zero hcuts_last]
+  simp_rw [hcoord]
+  rw [← Finset.sum_sub_distrib]
+  simp_rw [div_eq_mul_inv, Finset.sum_mul]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Cyclic Rotation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open MeasureTheory Set
+
+/-- Restrict a continuous BV path to a closed subinterval. -/
+def ContinuousBVPaths.restrict {a b : ℝ} (x : ContinuousBVPaths a b)
+    (l u : Set.Icc a b) (_hlu : l ≤ u) : ContinuousBVPaths (l : ℝ) u :=
+  let ι : Set.Icc (l : ℝ) u → Set.Icc a b := fun t ↦
+    ⟨t, le_trans l.property.1 t.property.1, le_trans t.property.2 u.property.2⟩
+  ⟨x.val ∘ ι, x.property.1.comp (continuous_subtype_val.subtype_mk _), fun i ↦
+    ne_top_of_le_ne_top (x.property.2 i)
+      (eVariationOn.comp_le_of_monotoneOn (fun t ↦ x.val t i) ι
+        (fun _ _ _ _ h ↦ h) (mapsTo_univ ι _))⟩
+
+private theorem curveArea_restrict_Ioc
+    {a b : ℝ} (x : ContinuousBVPaths a b) (l u : Set.Icc a b) (hlu : l ≤ u) :
+    (intervalStieltjesIntegral (continuousBVCoordinate x 1)
+        (fun t ↦ x.val t 0) (Ioc l u) -
+      intervalStieltjesIntegral (continuousBVCoordinate x 0)
+        (fun t ↦ x.val t 1) (Ioc l u)) / 2 =
+      curveAreaFunctional (ContinuousBVPaths.restrict x l u hlu) := by
+  unfold curveAreaFunctional
+  rw [intervalStieltjesIntegral_Ioc_eq_inclusion
+      (continuousBVCoordinate x 1)
+      ((PiLp.continuous_apply 2 _ 1).comp x.property.1)
+      (fun t ↦ x.val t 0) ((PiLp.continuous_apply 2 _ 0).comp x.property.1)
+      l u hlu,
+    intervalStieltjesIntegral_Ioc_eq_inclusion
+      (continuousBVCoordinate x 0)
+      ((PiLp.continuous_apply 2 _ 0).comp x.property.1)
+      (fun t ↦ x.val t 1) ((PiLp.continuous_apply 2 _ 1).comp x.property.1)
+      l u hlu]
+  rfl
+
+/-- Signed path area is additive along any monotone chain of cuts of the parameter interval. -/
+theorem curveArea_eq_sum_restrict {a b : ℝ} (x : ContinuousBVPaths a b)
+    {n : ℕ} (cuts : Fin (n + 1) → Set.Icc a b) (hcuts : Monotone cuts)
+    (hzero : (cuts 0 : ℝ) = a) (hlast : (cuts (Fin.last n) : ℝ) = b) :
+    curveAreaFunctional x =
+      ∑ i : Fin n, curveAreaFunctional (ContinuousBVPaths.restrict x (cuts i.castSucc)
+        (cuts i.succ) (hcuts (Fin.castSucc_le_succ i))) := by
+  simp_rw [← curveArea_restrict_Ioc x]
+  unfold curveAreaFunctional
+  rw [intervalStieltjesIntegral_eq_sum_Ioc (continuousBVCoordinate x 1)
+      ((PiLp.continuous_apply 2 _ 1).comp x.property.1)
+      (fun t ↦ x.val t 0) ((PiLp.continuous_apply 2 _ 0).comp x.property.1)
+      cuts hcuts hzero hlast,
+    intervalStieltjesIntegral_eq_sum_Ioc (continuousBVCoordinate x 0)
+      ((PiLp.continuous_apply 2 _ 0).comp x.property.1)
+      (fun t ↦ x.val t 1) ((PiLp.continuous_apply 2 _ 1).comp x.property.1)
+      cuts hcuts hzero hlast, ← Finset.sum_sub_distrib]
+  simp_rw [div_eq_mul_inv, Finset.sum_mul]
+
+/-- Split the signed area of a continuous BV path at any parameter value. -/
+theorem curveArea_eq_restriction_add_restriction
+    {a b : ℝ} (hab : a ≤ b) (x : ContinuousBVPaths a b) (s : Set.Icc a b) :
+    let a' : Set.Icc a b := ⟨a, le_rfl, hab⟩
+    let b' : Set.Icc a b := ⟨b, hab, le_rfl⟩
+    curveAreaFunctional x =
+      curveAreaFunctional (ContinuousBVPaths.restrict x a' s s.property.1) +
+      curveAreaFunctional (ContinuousBVPaths.restrict x s b' s.property.2) := by
+  let a' : Set.Icc a b := ⟨a, le_rfl, hab⟩
+  let b' : Set.Icc a b := ⟨b, hab, le_rfl⟩
+  have hs1 : a ≤ (s : ℝ) := s.property.1
+  have hs2 : (s : ℝ) ≤ b := s.property.2
+  change curveAreaFunctional x =
+    curveAreaFunctional (ContinuousBVPaths.restrict x a' s hs1) +
+      curveAreaFunctional (ContinuousBVPaths.restrict x s b' hs2)
+  have hcuts : Monotone (![a', s, b'] : Fin 3 → Set.Icc a b) := by
+    refine Fin.monotone_iff_le_succ.mpr fun i ↦ ?_
+    fin_cases i
+    · exact hs1
+    · exact hs2
+  have hsum := curveArea_eq_sum_restrict x ![a', s, b'] hcuts rfl rfl
+  rw [Fin.sum_univ_two] at hsum
+  exact hsum
+
+/-- The tail and head restrictions have signed areas summing to the original area. -/
+theorem curveArea_cyclic_cut_sum
+    {a b : ℝ} (hab : a ≤ b) (x : ContinuousBVPaths a b) (s : Set.Icc a b) :
+    let a' : Set.Icc a b := ⟨a, le_rfl, hab⟩
+    let b' : Set.Icc a b := ⟨b, hab, le_rfl⟩
+    curveAreaFunctional x =
+      curveAreaFunctional (ContinuousBVPaths.restrict x s b' s.property.2) +
+      curveAreaFunctional (ContinuousBVPaths.restrict x a' s s.property.1) := by
+  rw [curveArea_eq_restriction_add_restriction hab x s, add_comm]
+
+/-- Package a restricted continuous BV path with its interval endpoints. -/
+def ContinuousBVPaths.restrictionData {a b : ℝ} (x : ContinuousBVPaths a b)
+    (l u : Set.Icc a b) (hlu : l ≤ u) : RectifiablePathData where
+  a := l
+  b := u
+  ordered := hlu
+  path := ContinuousBVPaths.restrict x l u hlu
+
+/-- A tail-then-head concatenation preserves the signed area. -/
+theorem curveArea_cyclic_rotation
+    {a b c d : ℝ} (hab : a ≤ b) (x : ContinuousBVPaths a b)
+    (s : Set.Icc a b) (rotated : ContinuousBVPaths c d) (hcd : c ≤ d)
+    (hrot : IsPathConcatenation
+      { a := c, b := d, ordered := hcd, path := rotated }
+      ![ContinuousBVPaths.restrictionData x s ⟨b, hab, le_rfl⟩ s.property.2,
+        ContinuousBVPaths.restrictionData x ⟨a, le_rfl, hab⟩ s s.property.1]) :
+    curveAreaFunctional rotated = curveAreaFunctional x := by
+  rw [curveArea_concatenation _ _ hrot]
+  simp only [Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one,
+    ContinuousBVPaths.restrictionData]
+  exact (curveArea_cyclic_cut_sum hab x s).symm
+
+/-- Concatenating continuous paths with matching endpoints preserves coordinatewise variation. -/
+theorem boundedVariation_concatUnitIntervals_coordinate
+    (p q : ContinuousBVPaths 0 1)
+    (hjoin : p.val ⟨1, by norm_num⟩ = q.val ⟨0, by norm_num⟩) (i : Fin 2) :
+    BoundedVariationOn (fun t ↦ Function.concatUnitIntervals p.val q.val t i) Set.univ := by
+  let z : Set.Icc (0 : ℝ) 2 := ⟨0, by norm_num⟩
+  let o : Set.Icc (0 : ℝ) 2 := ⟨1, by norm_num⟩
+  let w : Set.Icc (0 : ℝ) 2 := ⟨2, by norm_num⟩
+  let f := fun t ↦ Function.concatUnitIntervals p.val q.val t i
+  have hzo : z ≤ o := by change (0 : ℝ) ≤ 1; norm_num
+  have how : o ≤ w := by change (1 : ℝ) ≤ 2; norm_num
+  have hsplit := eVariationOn.Icc_add_Icc f hzo how (Set.mem_univ o)
+  simp only [Set.univ_inter] at hsplit
+  have hwhole : Set.Icc z w = Set.univ := by
+    ext t
+    change ((0 : ℝ) ≤ t ∧ (t : ℝ) ≤ 2) ↔ True
+    exact iff_true_intro t.property
+  rw [hwhole] at hsplit
+  change eVariationOn f Set.univ ≠ ⊤
+  rw [← hsplit]
+  apply ENNReal.add_ne_top.mpr
+  constructor
+  · refine ne_top_of_le_ne_top (p.property.2 i) ?_
+    calc
+      eVariationOn f (Set.Icc z o) =
+          eVariationOn (fun t : Set.Icc (0 : ℝ) 2 ↦
+            p.val (Set.projIcc 0 1 (by norm_num) (t : ℝ)) i)
+            (Set.Icc z o) := by
+              apply eVariationOn.congr
+              intro t ht
+              have ht' : (t : ℝ) ≤ 1 := by exact ht.2
+              simp [f, Function.concatUnitIntervals, ht']
+      _ ≤ eVariationOn (fun t ↦ p.val t i) Set.univ :=
+        by
+          simpa only [Function.comp_def] using
+            (eVariationOn.comp_le_of_monotoneOn (fun t ↦ p.val t i)
+              (t := Set.Icc z o)
+              (fun t : Set.Icc (0 : ℝ) 2 ↦ Set.projIcc 0 1 (by norm_num) (t : ℝ))
+              (fun _ _ _ _ hxy ↦ Set.monotone_projIcc (by norm_num) hxy)
+              (mapsTo_univ _ _))
+  · refine ne_top_of_le_ne_top (q.property.2 i) ?_
+    calc
+      eVariationOn f (Set.Icc o w) =
+          eVariationOn
+            (fun t : Set.Icc (0 : ℝ) 2 ↦
+              q.val (Set.projIcc 0 1 (by norm_num) ((t : ℝ) - 1)) i)
+            (Set.Icc o w) := by
+              apply eVariationOn.congr
+              intro t ht
+              have hleft : (1 : ℝ) ≤ t := by exact ht.1
+              have ht' : ¬(t : ℝ) ≤ 1 ∨ (t : ℝ) = 1 := by
+                rcases lt_or_eq_of_le hleft with h | h
+                · exact Or.inl (not_le_of_gt h)
+                · exact Or.inr h.symm
+              rcases ht' with ht' | htEq
+              · simp [f, Function.concatUnitIntervals, ht']
+              · simpa [f, Function.concatUnitIntervals, htEq] using congrArg (fun z ↦ z i) hjoin
+      _ ≤ eVariationOn (fun t ↦ q.val t i) Set.univ :=
+        by
+          simpa only [Function.comp_def] using
+            (eVariationOn.comp_le_of_monotoneOn (fun t ↦ q.val t i)
+              (t := Set.Icc o w)
+              (fun t : Set.Icc (0 : ℝ) 2 ↦
+                Set.projIcc 0 1 (by norm_num) ((t : ℝ) - 1))
+              (fun _ _ _ _ hxy ↦ Set.monotone_projIcc (by norm_num)
+                (sub_le_sub_right (show (_ : ℝ) ≤ _ from hxy) 1))
+              (mapsTo_univ _ _))
+
+private def concatUnitPaths (p q : ContinuousBVPaths 0 1)
+    (hjoin : p.val ⟨1, by norm_num⟩ = q.val ⟨0, by norm_num⟩) :
+    ContinuousBVPaths 0 2 :=
+  ⟨Function.concatUnitIntervals p.val q.val,
+    Function.continuous_concatUnitIntervals p.property.1 q.property.1 hjoin,
+    boundedVariation_concatUnitIntervals_coordinate p q hjoin⟩
+
+private theorem isPathConcatenation_concatUnitPaths
+    (p q : ContinuousBVPaths 0 1)
+    (hjoin : p.val ⟨1, by norm_num⟩ = q.val ⟨0, by norm_num⟩) :
+    IsPathConcatenation
+      { a := 0, b := 2, ordered := by norm_num, path := concatUnitPaths p q hjoin }
+      ![{ a := 0, b := 1, ordered := by norm_num, path := p },
+        { a := 0, b := 1, ordered := by norm_num, path := q }] := by
+  let cuts : Fin 3 → Set.Icc (0 : ℝ) 2 :=
+    ![⟨0, by norm_num⟩, ⟨1, by norm_num⟩, ⟨2, by norm_num⟩]
+  refine ⟨by norm_num, cuts, ?_, rfl, rfl, ?_⟩
+  · intro i j hij
+    fin_cases i <;> fin_cases j <;> simp [cuts, Matrix.cons_val_zero,
+      Matrix.cons_val_one] at hij ⊢
+  intro i
+  fin_cases i
+  · let φ : Set.Icc (0 : ℝ) 1 → Set.Icc (cuts (Fin.castSucc 0) : ℝ)
+        (cuts (Fin.succ 0) : ℝ) := fun t ↦ ⟨t, by simp [cuts]⟩
+    let ψ : Set.Icc (0 : ℝ) 1 → Set.Icc (0 : ℝ) 1 := id
+    refine ⟨φ, ψ, ?_, ?_, ?_, continuous_id, monotone_id, Function.surjective_id, ?_⟩
+    · exact Continuous.subtype_mk continuous_subtype_val _
+    · intro x y hxy
+      exact hxy
+    · intro y
+      exact ⟨⟨y, by simpa [cuts] using y.property⟩, Subtype.ext rfl⟩
+    · intro t
+      have ht : (t : ℝ) ≤ 1 := t.property.2
+      simp [φ, ψ, concatUnitPaths, Function.concatUnitIntervals, ht]
+  · let φ : Set.Icc (0 : ℝ) 1 → Set.Icc (cuts (Fin.castSucc 1) : ℝ)
+        (cuts (Fin.succ 1) : ℝ) := fun t ↦ ⟨(t : ℝ) + 1, by
+          change (1 : ℝ) ≤ (t : ℝ) + 1 ∧ (t : ℝ) + 1 ≤ 2
+          constructor <;> linarith [t.property.1, t.property.2]⟩
+    let ψ : Set.Icc (0 : ℝ) 1 → Set.Icc (0 : ℝ) 1 := id
+    refine ⟨φ, ψ, ?_, ?_, ?_, continuous_id, monotone_id, Function.surjective_id, ?_⟩
+    · exact Continuous.subtype_mk (continuous_subtype_val.add continuous_const) _
+    · intro x y hxy
+      change (x : ℝ) + 1 ≤ (y : ℝ) + 1
+      linarith [show (x : ℝ) ≤ y from hxy]
+    · intro y
+      refine ⟨⟨(y : ℝ) - 1, ?_⟩, Subtype.ext ?_⟩
+      · have hy1 : (1 : ℝ) ≤ y := by exact y.property.1
+        have hy2 : (y : ℝ) ≤ 2 := by exact y.property.2
+        constructor <;> linarith [y.property.1, y.property.2]
+      · simp [φ]
+    · intro t
+      by_cases ht : (t : ℝ) = 0
+      · have htSub : t = (⟨0, by norm_num⟩ : Set.Icc (0 : ℝ) 1) := Subtype.ext ht
+        rw [htSub]
+        simpa [φ, ψ, concatUnitPaths, Function.concatUnitIntervals] using hjoin
+      · have htpos : 0 < (t : ℝ) := lt_of_le_of_ne t.property.1 (Ne.symm ht)
+        simp [φ, ψ, concatUnitPaths, Function.concatUnitIntervals, htpos]
+
+private def unitIntervalParam (a b : ℝ) (hab : a ≤ b) :
+    Set.Icc (0 : ℝ) 1 → Set.Icc a b :=
+  Set.Icc.convexComb ⟨a, le_rfl, hab⟩ ⟨b, hab, le_rfl⟩
+
+private theorem continuous_unitIntervalParam (a b : ℝ) (hab : a ≤ b) :
+    Continuous (unitIntervalParam a b hab) := Set.Icc.continuous_convexComb _ _
+
+private theorem monotone_unitIntervalParam (a b : ℝ) (hab : a ≤ b) :
+    Monotone (unitIntervalParam a b hab) := by
+  intro s t hst
+  change (1 - (s : ℝ)) * a + (s : ℝ) * b ≤ (1 - (t : ℝ)) * a + (t : ℝ) * b
+  have hst' : (s : ℝ) ≤ t := hst
+  nlinarith
+
+private theorem surjective_unitIntervalParam (a b : ℝ) (hab : a ≤ b) :
+    Function.Surjective (unitIntervalParam a b hab) := by
+  intro x
+  rcases hab.eq_or_lt with rfl | hab
+  · refine ⟨⟨0, by norm_num⟩, Subtype.ext ?_⟩
+    change (1 - (0 : ℝ)) * a + 0 * a = (x : ℝ)
+    have hx : (x : ℝ) = a := le_antisymm x.property.2 x.property.1
+    simp [hx]
+  · refine ⟨⟨((x : ℝ) - a) / (b - a), ?_⟩, Subtype.ext ?_⟩
+    · constructor
+      · exact div_nonneg (sub_nonneg.mpr x.property.1) (sub_nonneg.mpr hab.le)
+      · exact (div_le_one (sub_pos.mpr hab)).2 (sub_le_sub_right x.property.2 a)
+    · change (1 - ((x : ℝ) - a) / (b - a)) * a + ((x : ℝ) - a) / (b - a) * b = (x : ℝ)
+      field_simp [ne_of_gt (sub_pos.mpr hab)]
+      ring
+
+private theorem unitIntervalParam_zero (a b : ℝ) (hab : a ≤ b) :
+    unitIntervalParam a b hab ⟨0, by norm_num⟩ = ⟨a, le_rfl, hab⟩ := by
+  exact Set.Icc.convexComb_zero _ _
+
+private theorem unitIntervalParam_one (a b : ℝ) (hab : a ≤ b) :
+    unitIntervalParam a b hab ⟨1, by norm_num⟩ = ⟨b, hab, le_rfl⟩ := by
+  exact Set.Icc.convexComb_one _ _
+
+/-- Rotate a closed continuous BV path by concatenating its tail and head. -/
+theorem exists_cyclic_rotation_path {a b : ℝ} (hab : a ≤ b) (x : ContinuousBVPaths a b)
+    (s : Set.Icc a b) (hx : x.val ⟨b, hab, le_rfl⟩ = x.val ⟨a, le_rfl, hab⟩) :
+    ∃ rotated : ContinuousBVPaths 0 2,
+      rotated.val = Function.concatUnitIntervals
+        (x.val ∘ Set.Icc.convexComb s ⟨b, hab, le_rfl⟩)
+        (x.val ∘ Set.Icc.convexComb ⟨a, le_rfl, hab⟩ s) ∧
+      IsPathConcatenation { a := 0, b := 2, ordered := by norm_num, path := rotated }
+        ![ContinuousBVPaths.restrictionData x s ⟨b, hab, le_rfl⟩ s.property.2,
+          ContinuousBVPaths.restrictionData x ⟨a, le_rfl, hab⟩ s s.property.1] := by
+  let tail := ContinuousBVPaths.restrict x s ⟨b, hab, le_rfl⟩ s.property.2
+  let head := ContinuousBVPaths.restrict x ⟨a, le_rfl, hab⟩ s s.property.1
+  obtain ⟨p, hp⟩ := continuousBVPaths_comp_monotone_surjective
+    s.property.2 tail
+    (unitIntervalParam s b s.property.2) (continuous_unitIntervalParam _ _ _)
+    (monotone_unitIntervalParam _ _ _) (surjective_unitIntervalParam _ _ _)
+  obtain ⟨q, hq⟩ := continuousBVPaths_comp_monotone_surjective
+    s.property.1 head
+    (unitIntervalParam a s s.property.1) (continuous_unitIntervalParam _ _ _)
+    (monotone_unitIntervalParam _ _ _) (surjective_unitIntervalParam _ _ _)
+  have hjoin : p.val ⟨1, by norm_num⟩ = q.val ⟨0, by norm_num⟩ := by
+    rw [hp, hq]
+    simp only [Function.comp_apply, unitIntervalParam_one, unitIntervalParam_zero]
+    exact hx
+  let rotated := concatUnitPaths p q hjoin
+  have hrot : IsPathConcatenation
+      { a := 0, b := 2, ordered := by norm_num, path := rotated }
+      ![ContinuousBVPaths.restrictionData x s ⟨b, hab, le_rfl⟩ s.property.2,
+        ContinuousBVPaths.restrictionData x ⟨a, le_rfl, hab⟩ s s.property.1] := by
+    obtain ⟨hn, cuts, hm, hz, ho, hc⟩ := isPathConcatenation_concatUnitPaths p q hjoin
+    refine ⟨hn, cuts, hm, hz, ho, ?_⟩
+    intro i
+    fin_cases i
+    · obtain ⟨φ, ψ, hφc, hφm, hφs, hψc, hψm, hψs, heq⟩ := hc 0
+      refine ⟨φ, unitIntervalParam s b s.property.2 ∘ ψ, hφc, hφm, hφs,
+        (continuous_unitIntervalParam _ _ _).comp hψc,
+        (monotone_unitIntervalParam _ _ _).comp hψm,
+        (surjective_unitIntervalParam _ _ _).comp hψs, ?_⟩
+      intro t
+      have h := heq t
+      change rotated.val _ = p.val (ψ t) at h
+      rw [hp] at h
+      exact h
+    · obtain ⟨φ, ψ, hφc, hφm, hφs, hψc, hψm, hψs, heq⟩ := hc 1
+      refine ⟨φ, unitIntervalParam a s s.property.1 ∘ ψ, hφc, hφm, hφs,
+        (continuous_unitIntervalParam _ _ _).comp hψc,
+        (monotone_unitIntervalParam _ _ _).comp hψm,
+        (surjective_unitIntervalParam _ _ _).comp hψs, ?_⟩
+      intro t
+      have h := heq t
+      change rotated.val _ = q.val (ψ t) at h
+      rw [hq] at h
+      exact h
+  refine ⟨rotated, ?_, hrot⟩
+  change Function.concatUnitIntervals p.val q.val = _
+  rw [hp, hq]
+  rfl
+
+/-- Construct an area-preserving cyclic rotation of a closed continuous BV path. -/
+theorem exists_cyclic_rotation_eq_concat {a b : ℝ} (hab : a ≤ b) (x : ContinuousBVPaths a b)
+    (s : Set.Icc a b) (hx : x.val ⟨b, hab, le_rfl⟩ = x.val ⟨a, le_rfl, hab⟩) :
+    ∃ rotated : ContinuousBVPaths 0 2,
+      rotated.val = Function.concatUnitIntervals
+        (x.val ∘ Set.Icc.convexComb s ⟨b, hab, le_rfl⟩)
+        (x.val ∘ Set.Icc.convexComb ⟨a, le_rfl, hab⟩ s) ∧
+      IsPathConcatenation { a := 0, b := 2, ordered := by norm_num, path := rotated }
+        ![ContinuousBVPaths.restrictionData x s ⟨b, hab, le_rfl⟩ s.property.2,
+          ContinuousBVPaths.restrictionData x ⟨a, le_rfl, hab⟩ s s.property.1] ∧
+      curveAreaFunctional rotated = curveAreaFunctional x := by
+  obtain ⟨rotated, hr, hrot⟩ := exists_cyclic_rotation_path hab x s hx
+  exact ⟨rotated, hr, hrot,
+    curveArea_cyclic_rotation hab x s rotated (by norm_num) hrot⟩
+
+private theorem mem_range_convexComb {a b : ℝ} (l u z : Set.Icc a b)
+    (hlz : l ≤ z) (hzu : z ≤ u) : z ∈ Set.range (Set.Icc.convexComb l u) := by
+  obtain ⟨t, ht⟩ := surjective_unitIntervalParam (l : ℝ) u (hlz.trans hzu)
+    ⟨z, hlz, hzu⟩
+  refine ⟨t, Subtype.ext ?_⟩
+  have h := congrArg (fun y : Set.Icc (l : ℝ) u ↦ (y : ℝ)) ht
+  exact h
+
+/-- Cutting and rejoining a closed path does not change its carrier. -/
+theorem range_cyclic_concat {a b : ℝ} (hab : a ≤ b)
+    (x : Set.Icc a b → Point) (s : Set.Icc a b)
+    (hx : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩) :
+    Set.range (Function.concatUnitIntervals
+      (x ∘ Set.Icc.convexComb s ⟨b, hab, le_rfl⟩)
+      (x ∘ Set.Icc.convexComb ⟨a, le_rfl, hab⟩ s)) = Set.range x := by
+  have hjoin : (x ∘ Set.Icc.convexComb s ⟨b, hab, le_rfl⟩) ⟨1, by norm_num⟩ =
+      (x ∘ Set.Icc.convexComb ⟨a, le_rfl, hab⟩ s) ⟨0, by norm_num⟩ := by
+    simpa using hx.symm
+  rw [Function.range_concatUnitIntervals _ _ hjoin]
+  apply Set.Subset.antisymm
+  · exact Set.union_subset (Set.range_comp_subset_range _ _) (Set.range_comp_subset_range _ _)
+  · rintro _ ⟨t, rfl⟩
+    by_cases hst : s ≤ t
+    · obtain ⟨r, hr⟩ := mem_range_convexComb s ⟨b, hab, le_rfl⟩ t hst t.property.2
+      exact Or.inl ⟨r, congrArg x hr⟩
+    · obtain ⟨r, hr⟩ := mem_range_convexComb ⟨a, le_rfl, hab⟩ s t t.property.1
+        (le_of_not_ge hst)
+      exact Or.inr ⟨r, congrArg x hr⟩
+
+/-- Every point of a nondegenerate closed path occurs before its terminal parameter. -/
+theorem exists_param_lt_top_of_mem_range {a b : ℝ} (hab : a < b)
+    (x : Set.Icc a b → Point)
+    (hx : x ⟨a, le_rfl, hab.le⟩ = x ⟨b, hab.le, le_rfl⟩)
+    {p : Point} (hp : p ∈ Set.range x) :
+    ∃ s : Set.Icc a b, (s : ℝ) < b ∧ x s = p := by
+  obtain ⟨s, hs⟩ := hp
+  by_cases hsb : (s : ℝ) < b
+  · exact ⟨s, hsb, hs⟩
+  · have hst : s = ⟨b, hab.le, le_rfl⟩ :=
+      Subtype.ext (le_antisymm s.property.2 (le_of_not_gt hsb))
+    exact ⟨⟨a, le_rfl, hab.le⟩, hab, hx.trans (hst ▸ hs)⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Basic
+-/
+
+@[expose] public section
+
+namespace MovingSofa
+
+/-- The set is the range of an injective continuous map from a closed real interval. -/
+def IsJordanArc (Γ : Set Point) : Prop :=
+  ∃ (a b : ℝ) (_ : a ≤ b) (x : Set.Icc a b → Point),
+    Continuous x ∧ Function.Injective x ∧ Set.range x = Γ
+
+/-- The set is the range of an injective continuous map from the circle. -/
+def IsJordanCurve (Γ : Set Point) : Prop :=
+  ∃ x : Circle → Point, Continuous x ∧ Function.Injective x ∧ Set.range x = Γ
+
+/-- Bundle the predicates for Jordan arcs and Jordan curves. -/
+def jordanSets : (Set Point → Prop) × (Set Point → Prop) :=
+  (IsJordanArc, IsJordanCurve)
+
+/-- A Jordan arc admits a parametrization with the specified starting and ending points. -/
+def IsOrientedJordanArc (Γ : Set Point) (p q : Point) : Prop :=
+  ∃ (a b : ℝ) (hab : a ≤ b) (x : Set.Icc a b → Point),
+    Continuous x ∧ Function.Injective x ∧ Set.range x = Γ ∧
+    x ⟨a, le_rfl, hab⟩ = p ∧ x ⟨b, hab, le_rfl⟩ = q
+
+/-- A Jordan arc equipped with ordered endpoints. -/
+structure OrientedJordanArc where
+  /-- The point set traced by the arc. -/
+  carrier : Set Point
+  /-- The initial endpoint of the oriented arc. -/
+  startPoint : Point
+  /-- The terminal endpoint of the oriented arc. -/
+  endPoint : Point
+  parametrizable : IsOrientedJordanArc carrier startPoint endPoint
+
+end MovingSofa
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Orientation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Points outside the curve whose connected component in its complement is bounded. -/
+def jordanInterior (Γ : Set Point) : Set Point :=
+  {p | p ∉ Γ ∧ Bornology.IsBounded (connectedComponentIn Γᶜ p)}
+
+/-- A continuous real angle lifts the normalized displacement from the specified point. -/
+def IsCurveAngleLift {a b : ℝ} (x : Set.Icc a b → Point) (p : Point)
+    (θ : Set.Icc a b → ℝ) : Prop :=
+  Continuous θ ∧ ∀ t,
+    Real.cos (θ t) = (x t - p) 0 / ‖x t - p‖ ∧
+    Real.sin (θ t) = (x t - p) 1 / ‖x t - p‖
+
+/-- The angle-lift increment divided by `2π`, with value zero if no lift exists. -/
+def curveWinding {a b : ℝ} (hab : a ≤ b) (x : Set.Icc a b → Point) (p : Point) : ℝ := by
+  classical
+  exact if h : ∃ θ, IsCurveAngleLift x p θ then
+    (h.choose ⟨b, hab, le_rfl⟩ - h.choose ⟨a, le_rfl, hab⟩) / (2 * Real.pi)
+  else 0
+
+/-- A simple closed parametrization with the specified winding sign on the interior. -/
+def IsOrientedJordanParametrization {a b : ℝ} (hab : a ≤ b)
+    (Γ : Set Point) (counterclockwise : Bool) (x : Set.Icc a b → Point) : Prop :=
+  a < b ∧ IsJordanCurve Γ ∧ Continuous x ∧ Set.range x = Γ ∧
+    x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩ ∧
+    Set.InjOn x {t | (t : ℝ) < b} ∧
+    ∀ p ∈ jordanInterior Γ, curveWinding hab x p = if counterclockwise then 1 else -1
+
+/-- A Jordan curve together with a choice of clockwise or counterclockwise orientation. -/
+structure OrientedJordanCurve where
+  /-- The point set traced by the Jordan curve. -/
+  carrier : Set Point
+  isJordan : IsJordanCurve carrier
+  /-- Select positive winding orientation when true and negative orientation when false. -/
+  counterclockwise : Bool
+
+/-- Bundle the winding functional and the oriented-parametrization predicate. -/
+def jordanCurveOrientation :
+    (∀ (a b : ℝ), a ≤ b → (Set.Icc a b → Point) → Point → ℝ) ×
+    (∀ (a b : ℝ), a ≤ b → Set Point → Bool → (Set.Icc a b → Point) → Prop) :=
+  (fun _ _ hab ↦ curveWinding hab, fun _ _ hab ↦ IsOrientedJordanParametrization hab)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Area
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- An injective continuous BV parametrization respecting an oriented arc’s endpoints. -/
+structure ArcBVParametrization (Γ : OrientedJordanArc) where
+  /-- The initial real parameter. -/
+  a : ℝ
+  /-- The terminal real parameter. -/
+  b : ℝ
+  ordered : a ≤ b
+  /-- The continuous parametrization of bounded variation. -/
+  path : ContinuousBVPaths a b
+  injective : Function.Injective path.val
+  range_eq : Set.range path.val = Γ.carrier
+  start_eq : path.val ⟨a, le_rfl, ordered⟩ = Γ.startPoint
+  end_eq : path.val ⟨b, ordered, le_rfl⟩ = Γ.endPoint
+
+/-- A continuous BV parametrization respecting a Jordan curve’s orientation. -/
+structure ClosedBVParametrization (Γ : OrientedJordanCurve) where
+  /-- The initial real parameter. -/
+  a : ℝ
+  /-- The terminal real parameter. -/
+  b : ℝ
+  ordered : a ≤ b
+  /-- The continuous parametrization of bounded variation. -/
+  path : ContinuousBVPaths a b
+  oriented : IsOrientedJordanParametrization ordered Γ.carrier Γ.counterclockwise path.val
+
+/-- Oriented Jordan arcs admitting a continuous BV parametrization. -/
+abbrev RectifiableOrientedArc :=
+  {Γ : OrientedJordanArc // Nonempty (ArcBVParametrization Γ)}
+
+/-- Oriented Jordan curves admitting a continuous BV parametrization. -/
+abbrev RectifiableOrientedCurve :=
+  {Γ : OrientedJordanCurve // Nonempty (ClosedBVParametrization Γ)}
+
+/-- The signed Stieltjes area of a chosen BV parametrization of the oriented arc. -/
+def jordanArcArea (Γ : RectifiableOrientedArc) : ℝ :=
+  curveAreaFunctional (Classical.choice Γ.property).path
+
+/-- The signed Stieltjes area of a chosen BV parametrization of the oriented curve. -/
+def jordanClosedCurveArea (Γ : RectifiableOrientedCurve) : ℝ :=
+  curveAreaFunctional (Classical.choice Γ.property).path
+
+/-- Bundle the signed area functionals for oriented arcs and closed curves. -/
+def jordanArea : (RectifiableOrientedArc → ℝ) × (RectifiableOrientedCurve → ℝ) :=
+  (jordanArcArea, jordanClosedCurveArea)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Arc Area
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open MeasureTheory Set
+
+/-- Same-carrier Jordan arcs have equal or opposite signed areas according to their endpoints. -/
+theorem curveArea_arc_same_carrier
+    (Γ Δ : OrientedJordanArc) (x : ArcBVParametrization Γ)
+    (y : ArcBVParametrization Δ) (hcarrier : Γ.carrier = Δ.carrier) :
+    (Γ.startPoint = Δ.startPoint → Γ.endPoint = Δ.endPoint →
+      curveAreaFunctional x.path = curveAreaFunctional y.path) ∧
+    (Γ.startPoint = Δ.endPoint → Γ.endPoint = Δ.startPoint →
+      curveAreaFunctional x.path = -curveAreaFunctional y.path) := by
+  let _ : Fact (x.a ≤ x.b) := ⟨x.ordered⟩
+  let _ : Fact (y.a ≤ y.b) := ⟨y.ordered⟩
+  have hxemb : Topology.IsEmbedding x.path.val :=
+    (x.path.property.1.isClosedEmbedding x.injective).isEmbedding
+  let φ : Set.Icc y.a y.b → Set.Icc x.a x.b := fun t ↦
+    hxemb.toHomeomorph.symm ⟨y.path.val t, by
+      rw [x.range_eq, hcarrier, ← y.range_eq]
+      exact Set.mem_range_self t⟩
+  have hφc : Continuous φ :=
+    hxemb.toHomeomorph.symm.continuous.comp <|
+      y.path.property.1.subtype_mk _
+  have hxy (t : Set.Icc y.a y.b) : x.path.val (φ t) = y.path.val t := by
+    exact congrArg Subtype.val (hxemb.toHomeomorph.apply_symm_apply
+      ⟨y.path.val t, by
+        rw [x.range_eq, hcarrier, ← y.range_eq]
+        exact Set.mem_range_self t⟩)
+  have hφi : Function.Injective φ := fun s t hst ↦
+    y.injective <| by rw [← hxy s, ← hxy t, hst]
+  have hφs : Function.Surjective φ := by
+    intro s
+    have hmem : x.path.val s ∈ Set.range y.path.val := by
+      rw [y.range_eq, ← hcarrier, ← x.range_eq]
+      exact Set.mem_range_self s
+    obtain ⟨t, ht⟩ := hmem
+    refine ⟨t, ?_⟩
+    apply x.injective
+    rw [hxy t, ht]
+  have hxbot : x.path.val (⊥ : Set.Icc x.a x.b) = Γ.startPoint := by
+    convert x.start_eq using 1
+  have hxtop : x.path.val (⊤ : Set.Icc x.a x.b) = Γ.endPoint := by
+    convert x.end_eq using 1
+  have hybot : y.path.val (⊥ : Set.Icc y.a y.b) = Δ.startPoint := by
+    convert y.start_eq using 1
+  have hytop : y.path.val (⊤ : Set.Icc y.a y.b) = Δ.endPoint := by
+    convert y.end_eq using 1
+  have hmono (hstart : Γ.startPoint = Δ.startPoint)
+      (hend : Γ.endPoint = Δ.endPoint) : Monotone φ := by
+    apply (hφc.strictMono_of_inj_boundedOrder ?_ hφi).monotone
+    have hbot : φ ⊥ = ⊥ := by
+      apply x.injective
+      rw [hxy ⊥, hybot, ← hstart, hxbot]
+    have htop : φ ⊤ = ⊤ := by
+      apply x.injective
+      rw [hxy ⊤, hytop, ← hend, hxtop]
+    rw [hbot, htop]
+    exact bot_le
+  have hanti (hstart : Γ.startPoint = Δ.endPoint)
+      (hend : Γ.endPoint = Δ.startPoint) : Antitone φ := by
+    apply (hφc.strictAnti_of_inj_boundedOrder ?_ hφi).antitone
+    have htop : φ ⊤ = ⊥ := by
+      apply x.injective
+      rw [hxy ⊤, hytop, ← hstart, hxbot]
+    have hbot : φ ⊥ = ⊤ := by
+      apply x.injective
+      rw [hxy ⊥, hybot, ← hend, hxtop]
+    rw [htop, hbot]
+    exact bot_le
+  constructor
+  · intro hstart hend
+    obtain ⟨z, hz, hzarea⟩ := curveArea_comp_monotone_surjective
+      x.ordered y.ordered x.path φ hφc (hmono hstart hend) hφs
+    have hzy : z = y.path := by
+      apply Subtype.ext
+      rw [hz]
+      funext t
+      exact hxy t
+    simpa [hzy] using hzarea.symm
+  · intro hstart hend
+    obtain ⟨z, hz, hzarea⟩ := curveArea_comp_antitone_surjective
+      x.ordered y.ordered x.path φ hφc (hanti hstart hend) hφs
+    have hzy : z = y.path := by
+      apply Subtype.ext
+      rw [hz]
+      funext t
+      exact hxy t
+    rw [hzy] at hzarea
+    linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Parametrization
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Set Function
+
+namespace MovingSofa
+
+/-- Identify an open interval with the corresponding subset of the closed interval. -/
+def openIntervalToIntervalInterior {a b : ℝ} (_hab : a < b) :
+    Set.Ioo a b ≃ₜ {t : Set.Icc a b | a < (t : ℝ) ∧ (t : ℝ) < b} where
+  toFun t := ⟨⟨t, t.property.1.le, t.property.2.le⟩, t.property⟩
+  invFun t := ⟨t, t.property⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  continuous_toFun :=
+    (continuous_subtype_val.subtype_mk fun t ↦ ⟨t.property.1.le, t.property.2.le⟩).subtype_mk _
+  continuous_invFun := continuous_subtype_val.comp continuous_subtype_val |>.subtype_mk _
+
+/-- Regard a parametrized point as an element of the path’s range. -/
+def intervalToRange {a b : ℝ} (x : Set.Icc a b → Point) :
+    Set.Icc a b → Set.range x := fun t ↦ ⟨x t, ⟨t, rfl⟩⟩
+
+theorem intervalToRange_continuous {a b : ℝ} (x : Set.Icc a b → Point)
+    (hx : Continuous x) : Continuous (intervalToRange x) :=
+  hx.subtype_mk fun t ↦ ⟨t, rfl⟩
+
+theorem intervalToRange_surjective {a b : ℝ} (x : Set.Icc a b → Point) :
+    Function.Surjective (intervalToRange x) := by
+  rintro ⟨p, t, rfl⟩
+  exact ⟨t, rfl⟩
+
+/-- The carrier of a closed parametrized loop with its basepoint removed. -/
+def puncturedLoopRangeSet {a b : ℝ} (hab : a ≤ b)
+    (x : Set.Icc a b → Point) : Set (Set.range x) :=
+  {p | (p : Point) ≠ x ⟨a, le_rfl, hab⟩}
+
+theorem puncturedLoopRangeSet_isOpen {a b : ℝ} (hab : a ≤ b)
+    (x : Set.Icc a b → Point) : IsOpen (puncturedLoopRangeSet hab x) := by
+  let p : Set.range x := ⟨x ⟨a, le_rfl, hab⟩, ⟨⟨a, le_rfl, hab⟩, rfl⟩⟩
+  rw [show puncturedLoopRangeSet hab x = ({p}ᶜ : Set (Set.range x)) by
+    ext q
+    change ((q : Point) ≠ (p : Point)) ↔ q ≠ p
+    exact not_congr Subtype.ext_iff.symm]
+  exact isOpen_compl_singleton
+
+theorem intervalToRange_preimage_punctured {a b : ℝ} (hab : a ≤ b)
+    (hab' : a < b) (x : Set.Icc a b → Point)
+    (hclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (hinj : Set.InjOn x {t | (t : ℝ) < b}) :
+    intervalToRange x ⁻¹' puncturedLoopRangeSet hab x =
+      {t : Set.Icc a b | a < (t : ℝ) ∧ (t : ℝ) < b} := by
+  ext t
+  simp only [Set.mem_preimage, puncturedLoopRangeSet, Set.mem_ofPred_eq, intervalToRange]
+  constructor
+  · intro ht
+    constructor
+    · exact lt_of_le_of_ne t.property.1 fun hta ↦ by
+        apply ht
+        exact congrArg x (Subtype.ext hta.symm)
+    · exact lt_of_le_of_ne t.property.2 fun htb ↦ by
+        apply ht
+        have ht_top : t = ⟨b, hab, le_rfl⟩ := Subtype.ext htb
+        rw [ht_top]
+        exact hclosed.symm
+  · rintro ⟨hat, htb⟩ htx
+    have : t = ⟨a, le_rfl, hab⟩ := hinj htb hab' htx
+    exact hat.ne (congrArg Subtype.val this).symm
+
+theorem intervalToPuncturedRange_injective {a b : ℝ} (hab : a ≤ b)
+    (_hab' : a < b) (x : Set.Icc a b → Point)
+    (hclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (hinj : Set.InjOn x {t | (t : ℝ) < b}) :
+    Function.Injective
+      ((puncturedLoopRangeSet hab x).restrictPreimage (intervalToRange x)) := by
+  intro s t hst
+  apply Subtype.ext
+  apply hinj (x₁ := s.val) (x₂ := t.val)
+  · have hsne : x s.val ≠ x ⟨a, le_rfl, hab⟩ := s.property
+    exact lt_of_le_of_ne s.val.property.2 fun hsb ↦ by
+      apply hsne
+      have hs_top : s.val = ⟨b, hab, le_rfl⟩ := Subtype.ext hsb
+      rw [hs_top]
+      exact hclosed.symm
+  · have htne : x t.val ≠ x ⟨a, le_rfl, hab⟩ := t.property
+    exact lt_of_le_of_ne t.val.property.2 fun htb ↦ by
+      apply htne
+      have ht_top : t.val = ⟨b, hab, le_rfl⟩ := Subtype.ext htb
+      rw [ht_top]
+      exact hclosed.symm
+  · exact congrArg (fun p : puncturedLoopRangeSet hab x ↦ (p : Point)) hst
+
+/-- Removing the basepoint turns a closed once-traversal into a homeomorphism from the
+open parameter interval onto the punctured carrier. -/
+noncomputable def openIntervalHomeomorphPuncturedRange {a b : ℝ} (hab : a ≤ b)
+    (hab' : a < b) (x : Set.Icc a b → Point) (hx : Continuous x)
+    (hclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (hinj : Set.InjOn x {t | (t : ℝ) < b}) :
+    Set.Ioo a b ≃ₜ puncturedLoopRangeSet hab x := by
+  let f := intervalToRange x
+  let s := puncturedLoopRangeSet hab x
+  have hfq : Topology.IsQuotientMap f :=
+    (intervalToRange_continuous x hx).isClosedMap.isQuotientMap
+      (intervalToRange_continuous x hx) (intervalToRange_surjective x)
+  have hgq : Topology.IsQuotientMap (s.restrictPreimage f) :=
+    hfq.restrictPreimage_isOpen (puncturedLoopRangeSet_isOpen hab x)
+  have hgh : IsHomeomorph (s.restrictPreimage f) :=
+    isHomeomorph_iff_isQuotientMap_injective.2
+      ⟨hgq, intervalToPuncturedRange_injective hab hab' x hclosed hinj⟩
+  exact (openIntervalToIntervalInterior hab').trans <|
+    (Homeomorph.setCongr
+      (intervalToRange_preimage_punctured hab hab' x hclosed hinj).symm).trans
+      (hgh.homeomorph _)
+
+/-- The punctured-loop homeomorphism agrees pointwise with the original path. -/
+theorem openIntervalHomeomorphPuncturedRange_coe {a b : ℝ} (hab : a ≤ b)
+    (hab' : a < b) (x : Set.Icc a b → Point) (hx : Continuous x)
+    (hclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (hinj : Set.InjOn x {t | (t : ℝ) < b}) (t : Set.Ioo a b) :
+    ((openIntervalHomeomorphPuncturedRange hab hab' x hx hclosed hinj t :
+        puncturedLoopRangeSet hab x) : Point) =
+      x ⟨t, t.property.1.le, t.property.2.le⟩ := by
+  rfl
+
+private def puncturedLoopRangeHomeomorphOfEq
+    {a b c d : ℝ} (hab : a ≤ b) (hcd : c ≤ d)
+    (x : Set.Icc a b → Point) (y : Set.Icc c d → Point)
+    (hrange : Set.range x = Set.range y)
+    (hstart : x ⟨a, le_rfl, hab⟩ = y ⟨c, le_rfl, hcd⟩) :
+    puncturedLoopRangeSet hcd y ≃ₜ puncturedLoopRangeSet hab x where
+  toFun q :=
+    ⟨⟨q, by rw [hrange]; exact q.val.property⟩, fun h ↦ q.property (h.trans hstart)⟩
+  invFun q :=
+    ⟨⟨q, by rw [← hrange]; exact q.val.property⟩, fun h ↦ q.property (h.trans hstart.symm)⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  continuous_toFun :=
+    (continuous_subtype_val.comp continuous_subtype_val |>.subtype_mk _).subtype_mk _
+  continuous_invFun :=
+    (continuous_subtype_val.comp continuous_subtype_val |>.subtype_mk _).subtype_mk _
+
+/-- The transition between two endpoint-matched once-traversals, restricted to their open
+parameter intervals. -/
+private noncomputable def openIntervalClosedCurveTransition
+    {a b c d : ℝ} (hab : a ≤ b) (hab' : a < b) (hcd : c ≤ d) (hcd' : c < d)
+    (x : Set.Icc a b → Point) (y : Set.Icc c d → Point)
+    (hxc : Continuous x) (hyc : Continuous y)
+    (hxclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (hyclosed : y ⟨c, le_rfl, hcd⟩ = y ⟨d, hcd, le_rfl⟩)
+    (hxinj : Set.InjOn x {t | (t : ℝ) < b})
+    (hyinj : Set.InjOn y {t | (t : ℝ) < d})
+    (hrange : Set.range x = Set.range y)
+    (hstart : x ⟨a, le_rfl, hab⟩ = y ⟨c, le_rfl, hcd⟩) :
+    Set.Ioo c d ≃ₜ Set.Ioo a b :=
+  (openIntervalHomeomorphPuncturedRange hcd hcd' y hyc hyclosed hyinj).trans
+    ((puncturedLoopRangeHomeomorphOfEq hab hcd x y hrange hstart).trans
+      (openIntervalHomeomorphPuncturedRange hab hab' x hxc hxclosed hxinj).symm)
+
+private theorem openIntervalClosedCurveTransition_point
+    {a b c d : ℝ} (hab : a ≤ b) (hab' : a < b) (hcd : c ≤ d) (hcd' : c < d)
+    (x : Set.Icc a b → Point) (y : Set.Icc c d → Point)
+    (hxc : Continuous x) (hyc : Continuous y)
+    (hxclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (hyclosed : y ⟨c, le_rfl, hcd⟩ = y ⟨d, hcd, le_rfl⟩)
+    (hxinj : Set.InjOn x {t | (t : ℝ) < b})
+    (hyinj : Set.InjOn y {t | (t : ℝ) < d})
+    (hrange : Set.range x = Set.range y)
+    (hstart : x ⟨a, le_rfl, hab⟩ = y ⟨c, le_rfl, hcd⟩)
+    (t : Set.Ioo c d) :
+    x ⟨openIntervalClosedCurveTransition hab hab' hcd hcd' x y hxc hyc hxclosed
+          hyclosed hxinj hyinj hrange hstart t,
+        (openIntervalClosedCurveTransition hab hab' hcd hcd' x y hxc hyc hxclosed
+          hyclosed hxinj hyinj hrange hstart t).property.1.le,
+        (openIntervalClosedCurveTransition hab hab' hcd hcd' x y hxc hyc hxclosed
+          hyclosed hxinj hyinj hrange hstart t).property.2.le⟩ =
+      y ⟨t, t.property.1.le, t.property.2.le⟩ := by
+  let ex := openIntervalHomeomorphPuncturedRange hab hab' x hxc hxclosed hxinj
+  let ey := openIntervalHomeomorphPuncturedRange hcd hcd' y hyc hyclosed hyinj
+  let ec := puncturedLoopRangeHomeomorphOfEq hab hcd x y hrange hstart
+  change x ⟨ex.symm (ec (ey t)), (ex.symm (ec (ey t))).property.1.le,
+      (ex.symm (ec (ey t))).property.2.le⟩ =
+    y ⟨t, t.property.1.le, t.property.2.le⟩
+  rw [← openIntervalHomeomorphPuncturedRange_coe hab hab' x hxc hxclosed hxinj
+      (ex.symm (ec (ey t))),
+    ← openIntervalHomeomorphPuncturedRange_coe hcd hcd' y hyc hyclosed hyinj t]
+  exact congrArg (fun q : puncturedLoopRangeSet hab x ↦ (q : Point))
+    (ex.apply_symm_apply (ec (ey t)))
+
+private theorem homeomorph_Ioo_strictMono_or_strictAnti
+    {a b c d : ℝ} (hcd : c < d) (e : Set.Ioo c d ≃ₜ Set.Ioo a b) :
+    StrictMono e ∨ StrictAnti e := by
+  let f : ℝ → ℝ := Function.extend ((↑) : Set.Ioo c d → ℝ)
+    (fun t ↦ (e t : ℝ)) 0
+  have hf_apply (t : Set.Ioo c d) : f t = (e t : ℝ) :=
+    Subtype.val_injective.extend_apply _ _ t
+  have hfc : ContinuousOn f (Set.Ioo c d) := by
+    rw [continuousOn_iff_continuous_domRestrict]
+    convert continuous_subtype_val.comp e.continuous using 1
+    funext t
+    exact hf_apply t
+  have hfi : Set.InjOn f (Set.Ioo c d) := by
+    intro s hs t ht hst
+    have heq : e ⟨s, hs⟩ = e ⟨t, ht⟩ := by
+      apply Subtype.ext
+      calc
+        (e ⟨s, hs⟩ : ℝ) = f s := (hf_apply ⟨s, hs⟩).symm
+        _ = f t := hst
+        _ = (e ⟨t, ht⟩ : ℝ) := hf_apply ⟨t, ht⟩
+    exact congrArg Subtype.val (e.injective heq)
+  rcases ContinuousOn.strictMonoOn_of_injOn_Ioo hcd hfc hfi with hm | ha
+  · left
+    intro s t hst
+    have := hm s.property t.property hst
+    change (e s : ℝ) < (e t : ℝ)
+    simpa only [hf_apply] using this
+  · right
+    intro s t hst
+    have := ha s.property t.property hst
+    change (e t : ℝ) < (e s : ℝ)
+    simpa only [hf_apply] using this
+
+private def closedIntervalReverse {a b : ℝ} (_hab : a ≤ b) :
+    Set.Icc a b ≃ₜ Set.Icc a b where
+  toFun t := ⟨a + b - t, by linarith [t.property.2], by linarith [t.property.1]⟩
+  invFun t := ⟨a + b - t, by linarith [t.property.2], by linarith [t.property.1]⟩
+  left_inv t := by apply Subtype.ext; dsimp; ring
+  right_inv t := by apply Subtype.ext; dsimp; ring
+  continuous_toFun :=
+    (continuous_const.add continuous_const |>.sub continuous_subtype_val).subtype_mk _
+  continuous_invFun :=
+    (continuous_const.add continuous_const |>.sub continuous_subtype_val).subtype_mk _
+
+private theorem closedIntervalReverse_antitone {a b : ℝ} (_hab : a ≤ b) :
+    Antitone (closedIntervalReverse _hab) := by
+  intro s t hst
+  have hst' : (s : ℝ) ≤ (t : ℝ) := hst
+  change a + b - (t : ℝ) ≤ a + b - (s : ℝ)
+  linarith
+
+private theorem closedIntervalReverse_left {a b : ℝ} (hab : a ≤ b) :
+    closedIntervalReverse hab ⟨a, le_rfl, hab⟩ = ⟨b, hab, le_rfl⟩ := by
+  apply Subtype.ext
+  dsimp [closedIntervalReverse]
+  ring
+
+private theorem closedIntervalReverse_right {a b : ℝ} (hab : a ≤ b) :
+    closedIntervalReverse hab ⟨b, hab, le_rfl⟩ = ⟨a, le_rfl, hab⟩ := by
+  apply Subtype.ext
+  dsimp [closedIntervalReverse]
+  ring
+
+private def openIntervalReverse {a b : ℝ} (_hab : a < b) :
+    Set.Ioo a b ≃ₜ Set.Ioo a b where
+  toFun t := ⟨a + b - t, by linarith [t.property.2], by linarith [t.property.1]⟩
+  invFun t := ⟨a + b - t, by linarith [t.property.2], by linarith [t.property.1]⟩
+  left_inv t := by apply Subtype.ext; dsimp; ring
+  right_inv t := by apply Subtype.ext; dsimp; ring
+  continuous_toFun :=
+    (continuous_const.add continuous_const |>.sub continuous_subtype_val).subtype_mk _
+  continuous_invFun :=
+    (continuous_const.add continuous_const |>.sub continuous_subtype_val).subtype_mk _
+
+private theorem openIntervalReverse_strictAnti {a b : ℝ} (_hab : a < b) :
+    StrictAnti (openIntervalReverse _hab) := by
+  intro s t hst
+  have hst' : (s : ℝ) < (t : ℝ) := hst
+  change a + b - (t : ℝ) < a + b - (s : ℝ)
+  linarith
+
+private theorem openIntervalClosedCurveTransition_strictMono_or_strictAnti
+    {a b c d : ℝ} (hab : a ≤ b) (hab' : a < b) (hcd : c ≤ d) (hcd' : c < d)
+    (x : Set.Icc a b → Point) (y : Set.Icc c d → Point)
+    (hxc : Continuous x) (hyc : Continuous y)
+    (hxclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (hyclosed : y ⟨c, le_rfl, hcd⟩ = y ⟨d, hcd, le_rfl⟩)
+    (hxinj : Set.InjOn x {t | (t : ℝ) < b})
+    (hyinj : Set.InjOn y {t | (t : ℝ) < d})
+    (hrange : Set.range x = Set.range y)
+    (hstart : x ⟨a, le_rfl, hab⟩ = y ⟨c, le_rfl, hcd⟩) :
+    StrictMono (openIntervalClosedCurveTransition hab hab' hcd hcd' x y hxc hyc
+      hxclosed hyclosed hxinj hyinj hrange hstart) ∨
+    StrictAnti (openIntervalClosedCurveTransition hab hab' hcd hcd' x y hxc hyc
+      hxclosed hyclosed hxinj hyinj hrange hstart) :=
+  homeomorph_Ioo_strictMono_or_strictAnti hcd'
+    (openIntervalClosedCurveTransition hab hab' hcd hcd' x y hxc hyc hxclosed
+      hyclosed hxinj hyinj hrange hstart)
+
+/-- An endpoint-matched pair of continuous once-traversals of the same carrier differ by a
+continuous surjective monotone or antitone reparametrization of the closed intervals. -/
+private theorem exists_reparametrization_with_endpoints
+    {a b c d : ℝ} (hab : a ≤ b) (hab' : a < b) (hcd : c ≤ d) (hcd' : c < d)
+    (x : Set.Icc a b → Point) (y : Set.Icc c d → Point)
+    (hxc : Continuous x) (hyc : Continuous y)
+    (hxclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (hyclosed : y ⟨c, le_rfl, hcd⟩ = y ⟨d, hcd, le_rfl⟩)
+    (hxinj : Set.InjOn x {t | (t : ℝ) < b})
+    (hyinj : Set.InjOn y {t | (t : ℝ) < d})
+    (hrange : Set.range x = Set.range y)
+    (hstart : x ⟨a, le_rfl, hab⟩ = y ⟨c, le_rfl, hcd⟩) :
+    ∃ φ : Set.Icc c d → Set.Icc a b,
+      Continuous φ ∧ Function.Surjective φ ∧
+      ((Monotone φ ∧ φ ⟨c, le_rfl, hcd⟩ = ⟨a, le_rfl, hab⟩ ∧
+          φ ⟨d, hcd, le_rfl⟩ = ⟨b, hab, le_rfl⟩) ∨
+        (Antitone φ ∧ φ ⟨c, le_rfl, hcd⟩ = ⟨b, hab, le_rfl⟩ ∧
+          φ ⟨d, hcd, le_rfl⟩ = ⟨a, le_rfl, hab⟩)) ∧
+      ∀ t, x (φ t) = y t := by
+  let e := openIntervalClosedCurveTransition hab hab' hcd hcd' x y hxc hyc
+    hxclosed hyclosed hxinj hyinj hrange hstart
+  rcases openIntervalClosedCurveTransition_strictMono_or_strictAnti hab hab' hcd hcd'
+      x y hxc hyc hxclosed hyclosed hxinj hyinj hrange hstart with hm | ha
+  · let eo : Set.Ioo c d ≃o Set.Ioo a b :=
+      StrictMono.orderIsoOfRightInverse e hm e.symm e.apply_symm_apply
+    let φ := OrderIso.extendIoo hab' eo
+    refine ⟨φ, OrderIso.continuous_extendIoo hcd' hab' eo,
+      OrderIso.surjective_extendIoo hcd' hab' eo, Or.inl ⟨?_, ?_, ?_⟩, ?_⟩
+    · exact OrderIso.monotone_extendIoo hcd' hab' eo
+    · exact OrderIso.extendIoo_left hcd' hab' eo
+    · exact OrderIso.extendIoo_right hcd' hab' eo
+    · intro t
+      by_cases htc : (t : ℝ) = c
+      · have ht : t = ⟨c, le_rfl, hcd⟩ := Subtype.ext htc
+        rw [ht, show φ ⟨c, le_rfl, hcd⟩ = ⟨a, le_rfl, hab⟩ from
+          OrderIso.extendIoo_left hcd' hab' eo]
+        exact hstart
+      by_cases htd : (t : ℝ) = d
+      · have ht : t = ⟨d, hcd, le_rfl⟩ := Subtype.ext htd
+        rw [ht, show φ ⟨d, hcd, le_rfl⟩ = ⟨b, hab, le_rfl⟩ from
+          OrderIso.extendIoo_right hcd' hab' eo]
+        exact hxclosed.symm.trans (hstart.trans hyclosed)
+      · let ti : Set.Ioo c d :=
+          ⟨t, lt_of_le_of_ne t.property.1 (Ne.symm htc),
+            lt_of_le_of_ne t.property.2 htd⟩
+        have hφ : φ t = ⟨e ti, (e ti).property.1.le, (e ti).property.2.le⟩ := by
+          apply Subtype.ext
+          have ht : t = ⟨ti, ti.property.1.le, ti.property.2.le⟩ := Subtype.ext rfl
+          rw [ht]
+          exact OrderIso.extendIoo_interior hab' eo ti
+        rw [hφ]
+        exact openIntervalClosedCurveTransition_point hab hab' hcd hcd' x y hxc hyc
+          hxclosed hyclosed hxinj hyinj hrange hstart ti
+  · let r := openIntervalReverse hab'
+    let g : Set.Ioo c d ≃ₜ Set.Ioo a b := e.trans r
+    have hgm : StrictMono g := by
+      intro s t hst
+      exact openIntervalReverse_strictAnti hab' (ha hst)
+    let go : Set.Ioo c d ≃o Set.Ioo a b :=
+      StrictMono.orderIsoOfRightInverse g hgm g.symm g.apply_symm_apply
+    let ψ := OrderIso.extendIoo hab' go
+    let φ : Set.Icc c d → Set.Icc a b := fun t ↦ closedIntervalReverse hab (ψ t)
+    refine ⟨φ, (closedIntervalReverse hab).continuous.comp
+        (OrderIso.continuous_extendIoo hcd' hab' go),
+      (closedIntervalReverse hab).surjective.comp
+        (OrderIso.surjective_extendIoo hcd' hab' go), Or.inr ⟨?_, ?_, ?_⟩, ?_⟩
+    · exact (closedIntervalReverse_antitone hab).comp_monotone
+        (OrderIso.monotone_extendIoo hcd' hab' go)
+    · rw [show φ ⟨c, le_rfl, hcd⟩ =
+          closedIntervalReverse hab (ψ ⟨c, le_rfl, hcd⟩) from rfl,
+        show ψ ⟨c, le_rfl, hcd⟩ = ⟨a, le_rfl, hab⟩ from
+          OrderIso.extendIoo_left hcd' hab' go]
+      apply Subtype.ext
+      dsimp [closedIntervalReverse]
+      ring
+    · rw [show φ ⟨d, hcd, le_rfl⟩ =
+          closedIntervalReverse hab (ψ ⟨d, hcd, le_rfl⟩) from rfl,
+        show ψ ⟨d, hcd, le_rfl⟩ = ⟨b, hab, le_rfl⟩ from
+          OrderIso.extendIoo_right hcd' hab' go]
+      apply Subtype.ext
+      dsimp [closedIntervalReverse]
+      ring
+    · intro t
+      by_cases htc : (t : ℝ) = c
+      · have ht : t = ⟨c, le_rfl, hcd⟩ := Subtype.ext htc
+        rw [ht]
+        change x (closedIntervalReverse hab (ψ ⟨c, le_rfl, hcd⟩)) = _
+        rw [show ψ ⟨c, le_rfl, hcd⟩ = ⟨a, le_rfl, hab⟩ from
+          OrderIso.extendIoo_left hcd' hab' go]
+        rw [closedIntervalReverse_left hab]
+        exact hxclosed.symm.trans hstart
+      by_cases htd : (t : ℝ) = d
+      · have ht : t = ⟨d, hcd, le_rfl⟩ := Subtype.ext htd
+        rw [ht]
+        change x (closedIntervalReverse hab (ψ ⟨d, hcd, le_rfl⟩)) = _
+        rw [show ψ ⟨d, hcd, le_rfl⟩ = ⟨b, hab, le_rfl⟩ from
+          OrderIso.extendIoo_right hcd' hab' go]
+        rw [closedIntervalReverse_right hab]
+        exact hstart.trans hyclosed
+      · let ti : Set.Ioo c d :=
+          ⟨t, lt_of_le_of_ne t.property.1 (Ne.symm htc),
+            lt_of_le_of_ne t.property.2 htd⟩
+        have hφ : φ t = ⟨e ti, (e ti).property.1.le, (e ti).property.2.le⟩ := by
+          apply Subtype.ext
+          have hψ := OrderIso.extendIoo_interior hab' go ti
+          dsimp [go, g, r] at hψ
+          change a + b - (ψ t : ℝ) = (e ti : ℝ)
+          rw [hψ]
+          change a + b - (a + b - (e ti : ℝ)) = (e ti : ℝ)
+          ring
+        rw [hφ]
+        exact openIntervalClosedCurveTransition_point hab hab' hcd hcd' x y hxc hyc
+          hxclosed hyclosed hxinj hyinj hrange hstart ti
+
+/-- Equal-start simple closed paths with the same range admit a monotone or antitone transition. -/
+theorem exists_reparametrization_of_range_eq_of_start_eq
+    {a b c d : ℝ} (hab : a ≤ b) (hab' : a < b) (hcd : c ≤ d) (hcd' : c < d)
+    (x : Set.Icc a b → Point) (y : Set.Icc c d → Point)
+    (hxc : Continuous x) (hyc : Continuous y)
+    (hxclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (hyclosed : y ⟨c, le_rfl, hcd⟩ = y ⟨d, hcd, le_rfl⟩)
+    (hxinj : Set.InjOn x {t | (t : ℝ) < b})
+    (hyinj : Set.InjOn y {t | (t : ℝ) < d})
+    (hrange : Set.range x = Set.range y)
+    (hstart : x ⟨a, le_rfl, hab⟩ = y ⟨c, le_rfl, hcd⟩) :
+    ∃ φ : Set.Icc c d → Set.Icc a b,
+      Continuous φ ∧ Function.Surjective φ ∧ (Monotone φ ∨ Antitone φ) ∧
+        y = x ∘ φ := by
+  rcases exists_reparametrization_with_endpoints hab hab' hcd hcd' x y hxc hyc
+      hxclosed hyclosed hxinj hyinj hrange hstart with ⟨φ, hφc, hφs, hφo, hφxy⟩
+  refine ⟨φ, hφc, hφs, hφo.imp (fun h ↦ h.1) (fun h ↦ h.1), ?_⟩
+  funext t
+  exact (hφxy t).symm
+
+/-- Equal-start closed Jordan parametrizations admit a monotone or antitone transition. -/
+theorem ClosedBVParametrization.exists_reparametrization_of_start_eq
+    {Γ Δ : OrientedJordanCurve} (x : ClosedBVParametrization Γ)
+    (y : ClosedBVParametrization Δ) (hcarrier : Γ.carrier = Δ.carrier)
+    (hstart : x.path.val ⟨x.a, le_rfl, x.ordered⟩ =
+      y.path.val ⟨y.a, le_rfl, y.ordered⟩) :
+    ∃ φ : Set.Icc y.a y.b → Set.Icc x.a x.b,
+      Continuous φ ∧ Function.Surjective φ ∧ (Monotone φ ∨ Antitone φ) ∧
+        y.path.val = x.path.val ∘ φ := by
+  have hrange : Set.range x.path.val = Set.range y.path.val := by
+    rw [x.oriented.2.2.2.1, y.oriented.2.2.2.1]
+    exact hcarrier
+  exact exists_reparametrization_of_range_eq_of_start_eq x.ordered x.oriented.1 y.ordered
+    y.oriented.1
+    x.path.val y.path.val x.path.property.1 y.path.property.1
+    x.oriented.2.2.2.2.1 y.oriented.2.2.2.2.1
+    x.oriented.2.2.2.2.2.1 y.oriented.2.2.2.2.2.1 hrange hstart
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Unit Sphere
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace MovingSofa
+
+private def circlePoint (z : Circle) : Point := WithLp.toLp 2 ![(z : ℂ).re, (z : ℂ).im]
+
+private lemma norm_circlePoint (z : Circle) : ‖circlePoint z‖ = 1 := by
+  rw [← sq_eq_sq₀ (norm_nonneg _) (by norm_num : (0 : ℝ) ≤ 1)]
+  rw [EuclideanSpace.norm_sq_eq]
+  simp only [circlePoint, Fin.sum_univ_two, WithLp.ofLp_toLp, Matrix.cons_val_zero,
+    Matrix.cons_val_one, Matrix.cons_val_fin_one, one_pow, Real.norm_eq_abs, sq_abs]
+  have h := Complex.sq_norm (z : ℂ)
+  simp only [Complex.normSq_apply, Circle.norm_coe] at h
+  nlinarith
+
+private lemma continuous_circlePoint : Continuous circlePoint := by
+  apply (PiLp.continuous_toLp 2 (fun _ : Fin 2 ↦ ℝ)).comp
+  apply continuous_pi
+  intro i
+  fin_cases i
+  · exact Complex.continuous_re.comp continuous_subtype_val
+  · exact Complex.continuous_im.comp continuous_subtype_val
+
+/-- A planar set homeomorphic to the unit sphere is a Jordan curve. -/
+lemma isJordanCurve_of_unitSphere_homeomorph {Γ : Set Point}
+    (e : {u : Point | ‖u‖ = 1} ≃ₜ ↥Γ) : IsJordanCurve Γ := by
+  let f : Circle → {u : Point | ‖u‖ = 1} := fun z ↦ ⟨circlePoint z, norm_circlePoint z⟩
+  refine ⟨fun z ↦ (e (f z) : Point), continuous_subtype_val.comp
+    (e.continuous.comp (continuous_circlePoint.subtype_mk _)), ?_, ?_⟩
+  · intro z w h
+    have he := e.injective (Subtype.ext h)
+    have hpoint := congrArg Subtype.val he
+    apply Subtype.ext
+    apply Complex.ext
+    · exact congrFun (congrArg WithLp.ofLp hpoint) 0
+    · exact congrFun (congrArg WithLp.ofLp hpoint) 1
+  · apply Set.Subset.antisymm
+    · rintro p ⟨z, rfl⟩
+      exact (e (f z)).property
+    · intro p hp
+      let u := e.symm ⟨p, hp⟩
+      let z : ℂ := ⟨u.val 0, u.val 1⟩
+      have hz : ‖z‖ = 1 := by
+        rw [← sq_eq_sq₀ (norm_nonneg _) (by norm_num : (0 : ℝ) ≤ 1)]
+        rw [Complex.sq_norm]
+        have hu : ‖u.val‖ = 1 := u.property
+        have hsq := congrArg (fun r : ℝ ↦ r ^ 2) hu
+        rw [EuclideanSpace.norm_sq_eq] at hsq
+        simpa [z, Fin.sum_univ_two, sq] using hsq
+      let ζ : Circle := ⟨z, by
+        change z ∈ Metric.sphere 0 1
+        simpa only [Metric.mem_sphere, dist_zero_right] using hz⟩
+      refine ⟨ζ, ?_⟩
+      have hf : f ζ = u := by
+        apply Subtype.ext
+        ext i
+        fin_cases i <;> rfl
+      change (e (f ζ) : Point) = p
+      rw [hf]
+      exact congrArg Subtype.val (e.apply_symm_apply ⟨p, hp⟩)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Winding
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Two continuous angle lifts have the same endpoint increment. -/
+theorem IsCurveAngleLift.endpoint_increment_eq {a b : ℝ} (hab : a ≤ b)
+    {x : Set.Icc a b → Point} {p : Point} {θ ψ : Set.Icc a b → ℝ}
+    (hθ : IsCurveAngleLift x p θ) (hψ : IsCurveAngleLift x p ψ) :
+    θ ⟨b, hab, le_rfl⟩ - θ ⟨a, le_rfl, hab⟩ =
+      ψ ⟨b, hab, le_rfl⟩ - ψ ⟨a, le_rfl, hab⟩ := by
+  let _ : PreconnectedSpace (Set.Icc a b) := Subtype.preconnectedSpace isPreconnected_Icc
+  have h := Real.sub_eq_sub_of_cos_eq_cos_of_sin_eq_sin hθ.1 hψ.1
+    (fun t ↦ (hθ.2 t).1.trans (hψ.2 t).1.symm)
+    (fun t ↦ (hθ.2 t).2.trans (hψ.2 t).2.symm)
+    ⟨b, hab, le_rfl⟩ ⟨a, le_rfl, hab⟩
+  linarith
+
+/-- Compute the winding value from any continuous angle lift. -/
+theorem IsCurveAngleLift.curveWinding_eq {a b : ℝ} (hab : a ≤ b)
+    {x : Set.Icc a b → Point} {p : Point} {θ : Set.Icc a b → ℝ}
+    (hθ : IsCurveAngleLift x p θ) :
+    curveWinding hab x p = (θ ⟨b, hab, le_rfl⟩ - θ ⟨a, le_rfl, hab⟩) / (2 * Real.pi) := by
+  unfold curveWinding
+  rw [dite_eq_left ⟨θ, hθ⟩]
+  congr 1
+  exact IsCurveAngleLift.endpoint_increment_eq hab (Exists.choose_spec ⟨θ, hθ⟩) hθ
+
+/-- A nonzero winding value provides a continuous angle lift. -/
+theorem exists_curveAngleLift_of_curveWinding_ne_zero {a b : ℝ} (hab : a ≤ b)
+    {x : Set.Icc a b → Point} {p : Point} (h : curveWinding hab x p ≠ 0) :
+    ∃ θ, IsCurveAngleLift x p θ := by
+  by_contra hn
+  apply h
+  simp only [curveWinding, dite_eq_right hn]
+
+/-- Pull back an angle lift along a continuous parameter map. -/
+theorem IsCurveAngleLift.comp {a b c d : ℝ}
+    {x : Set.Icc a b → Point} {p : Point} {θ : Set.Icc a b → ℝ}
+    (hθ : IsCurveAngleLift x p θ) {φ : Set.Icc c d → Set.Icc a b} (hφ : Continuous φ) :
+    IsCurveAngleLift (x ∘ φ) p (θ ∘ φ) :=
+  ⟨hθ.1.comp hφ, fun t ↦ hθ.2 (φ t)⟩
+
+/-- Compute winding after continuous reparametrization from lifted endpoint values. -/
+theorem IsCurveAngleLift.curveWinding_comp {a b c d : ℝ} (hcd : c ≤ d)
+    {x : Set.Icc a b → Point} {p : Point} {θ : Set.Icc a b → ℝ}
+    (hθ : IsCurveAngleLift x p θ) {φ : Set.Icc c d → Set.Icc a b} (hφ : Continuous φ) :
+    curveWinding hcd (x ∘ φ) p =
+      (θ (φ ⟨d, hcd, le_rfl⟩) - θ (φ ⟨c, le_rfl, hcd⟩)) / (2 * Real.pi) :=
+  (hθ.comp hφ).curveWinding_eq hcd
+
+/-- A continuous reparametrization preserving endpoints preserves winding. -/
+theorem curveWinding_comp_of_endpoints {a b c d : ℝ} (hab : a ≤ b) (hcd : c ≤ d)
+    {x : Set.Icc a b → Point} {p : Point}
+    (hx : ∃ θ, IsCurveAngleLift x p θ) {φ : Set.Icc c d → Set.Icc a b} (hφ : Continuous φ)
+    (hφa : φ ⟨c, le_rfl, hcd⟩ = ⟨a, le_rfl, hab⟩)
+    (hφb : φ ⟨d, hcd, le_rfl⟩ = ⟨b, hab, le_rfl⟩) :
+    curveWinding hcd (x ∘ φ) p = curveWinding hab x p := by
+  obtain ⟨θ, hθ⟩ := hx
+  rw [hθ.curveWinding_comp hcd hφ, hφa, hφb, hθ.curveWinding_eq hab]
+
+/-- A continuous reparametrization exchanging endpoints negates winding. -/
+theorem curveWinding_comp_of_reversed_endpoints {a b c d : ℝ}
+    (hab : a ≤ b) (hcd : c ≤ d) {x : Set.Icc a b → Point} {p : Point}
+    (hx : ∃ θ, IsCurveAngleLift x p θ) {φ : Set.Icc c d → Set.Icc a b} (hφ : Continuous φ)
+    (hφa : φ ⟨c, le_rfl, hcd⟩ = ⟨b, hab, le_rfl⟩)
+    (hφb : φ ⟨d, hcd, le_rfl⟩ = ⟨a, le_rfl, hab⟩) :
+    curveWinding hcd (x ∘ φ) p = -curveWinding hab x p := by
+  obtain ⟨θ, hθ⟩ := hx
+  rw [hθ.curveWinding_comp hcd hφ, hφa, hφb, hθ.curveWinding_eq hab]
+  ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Radial Winding
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace MovingSofa
+
+/-- The parameter angle lifts a positive radial loop about its center. -/
+lemma isCurveAngleLift_radial {a b : ℝ} (o : Point) (r : Set.Icc a b → ℝ)
+    (hr : ∀ t, 0 < r t) :
+    IsCurveAngleLift (fun t ↦ o + r t • normalVector ((t : ℝ) : Real.Angle)) o
+      (fun t ↦ (t : ℝ)) := by
+  refine ⟨continuous_subtype_val, fun t ↦ ?_⟩
+  have hn : ‖r t • normalVector ((t : ℝ) : Real.Angle)‖ = r t := by
+    rw [norm_smul, Real.norm_eq_abs, abs_of_pos (hr t), norm_normalVector_real, mul_one]
+  simp only [add_sub_cancel_left, hn]
+  constructor
+  · change Real.cos (t : ℝ) = r t * Real.cos (t : ℝ) / r t
+    field_simp [(hr t).ne']
+  · change Real.sin (t : ℝ) = r t * Real.sin (t : ℝ) / r t
+    field_simp [(hr t).ne']
+
+/-- A positive radial loop winds once around its center. -/
+lemma curveWinding_radial_center (o : Point) (r : Set.Icc (0 : ℝ) (2 * Real.pi) → ℝ)
+    (hr : ∀ t, 0 < r t) :
+    curveWinding (by positivity)
+      (fun t ↦ o + r t • normalVector ((t : ℝ) : Real.Angle)) o = 1 := by
+  rw [(isCurveAngleLift_radial o r hr).curveWinding_eq]
+  simp
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Winding Concatenation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private theorem cos_sin_add_sub {α β : ℝ}
+    (hc : Real.cos α = Real.cos β) (hs : Real.sin α = Real.sin β) (t : ℝ) :
+    Real.cos (t + (α - β)) = Real.cos t ∧
+      Real.sin (t + (α - β)) = Real.sin t := by
+  have hcd : Real.cos (α - β) = 1 := by
+    rw [Real.cos_sub, hc, hs]
+    nlinarith [Real.cos_sq_add_sin_sq β]
+  have hsd : Real.sin (α - β) = 0 := by
+    rw [Real.sin_sub, hc, hs]
+    ring
+  simp [Real.cos_add, Real.sin_add, hcd, hsd]
+
+/-- Winding is additive for two paths joined at a common endpoint. -/
+theorem curveWinding_concatUnitIntervals
+    {x y : Set.Icc (0 : ℝ) 1 → Point} {p : Point}
+    {θ ψ : Set.Icc (0 : ℝ) 1 → ℝ}
+    (hθ : IsCurveAngleLift x p θ) (hψ : IsCurveAngleLift y p ψ)
+    (hjoin : x ⟨1, by norm_num⟩ = y ⟨0, by norm_num⟩) :
+    curveWinding (by norm_num) (Function.concatUnitIntervals x y) p =
+      curveWinding (by norm_num) x p + curveWinding (by norm_num) y p := by
+  let δ := θ ⟨1, by norm_num⟩ - ψ ⟨0, by norm_num⟩
+  have htrig (t : Set.Icc (0 : ℝ) 1) :
+      Real.cos (ψ t + δ) = Real.cos (ψ t) ∧
+      Real.sin (ψ t + δ) = Real.sin (ψ t) := by
+    apply cos_sin_add_sub
+    · rw [(hθ.2 _).1, (hψ.2 _).1, hjoin]
+    · rw [(hθ.2 _).2, (hψ.2 _).2, hjoin]
+  let η := Function.concatUnitIntervals θ (fun t ↦ ψ t + δ)
+  have hη : IsCurveAngleLift (Function.concatUnitIntervals x y) p η := by
+    refine ⟨Function.continuous_concatUnitIntervals hθ.1
+      (hψ.1.add continuous_const) (by dsimp [δ]; ring), ?_⟩
+    intro t
+    dsimp [η, Function.concatUnitIntervals]
+    split_ifs with ht
+    · exact hθ.2 _
+    · exact ⟨(htrig _).1.trans (hψ.2 _).1, (htrig _).2.trans (hψ.2 _).2⟩
+  rw [hη.curveWinding_eq, hθ.curveWinding_eq, hψ.curveWinding_eq]
+  simp only [η, Function.concatUnitIntervals]
+  norm_num
+  dsimp [δ]
+  change (ψ ⟨1, _⟩ + (θ ⟨1, _⟩ - ψ ⟨0, _⟩) - θ ⟨0, _⟩) / (2 * Real.pi) =
+    (θ ⟨1, _⟩ - θ ⟨0, _⟩) / (2 * Real.pi) +
+      (ψ ⟨1, _⟩ - ψ ⟨0, _⟩) / (2 * Real.pi)
+  ring
+
+/-- Moving a closed path's cut point preserves winding. -/
+theorem curveWinding_concat_of_cyclic_endpoints {a b : ℝ} (hab : a ≤ b)
+    {x : Set.Icc a b → Point} {p : Point} {θ : Set.Icc a b → ℝ}
+    (hθ : IsCurveAngleLift x p θ)
+    (hx : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (s : Set.Icc a b) (φ ψ : Set.Icc (0 : ℝ) 1 → Set.Icc a b)
+    (hφ : Continuous φ) (hψ : Continuous ψ)
+    (hφ₀ : φ ⟨0, by norm_num⟩ = s)
+    (hφ₁ : φ ⟨1, by norm_num⟩ = ⟨b, hab, le_rfl⟩)
+    (hψ₀ : ψ ⟨0, by norm_num⟩ = ⟨a, le_rfl, hab⟩)
+    (hψ₁ : ψ ⟨1, by norm_num⟩ = s) :
+    curveWinding (by norm_num) (Function.concatUnitIntervals (x ∘ φ) (x ∘ ψ)) p =
+      curveWinding hab x p := by
+  have hjoin : (x ∘ φ) ⟨1, by norm_num⟩ = (x ∘ ψ) ⟨0, by norm_num⟩ := by
+    simp only [Function.comp_apply, hφ₁, hψ₀]
+    exact hx.symm
+  rw [curveWinding_concatUnitIntervals (hθ.comp hφ) (hθ.comp hψ) hjoin,
+    hθ.curveWinding_comp (by norm_num) hφ, hθ.curveWinding_comp (by norm_num) hψ,
+    hφ₀, hφ₁, hψ₀, hψ₁, hθ.curveWinding_eq hab]
+  ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Cyclic Rotation
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace MovingSofa
+
+private theorem strictMono_convexComb {a b : ℝ} (l u : Set.Icc a b) (hlu : l < u) :
+    StrictMono (Set.Icc.convexComb l u) := by
+  intro s t hst
+  have hlu' : (l : ℝ) < u := hlu
+  have hst' : (s : ℝ) < t := hst
+  change (1 - (s : ℝ)) * l + (s : ℝ) * u <
+    (1 - (t : ℝ)) * l + (t : ℝ) * u
+  nlinarith [mul_pos (sub_pos.mpr hst') (sub_pos.mpr hlu')]
+
+/-- Moving an oriented Jordan path's start to an interior parameter preserves area and
+orientation. -/
+theorem exists_oriented_cyclic_rotation {a b : ℝ} (hab : a ≤ b)
+    {Γ : Set Point} {ccw : Bool} (x : ContinuousBVPaths a b)
+    (hx : IsOrientedJordanParametrization hab Γ ccw x.val)
+    (s : Set.Icc a b) (has : a < s) (hsb : (s : ℝ) < b) :
+    ∃ r : ContinuousBVPaths 0 2,
+      IsOrientedJordanParametrization (by norm_num) Γ ccw r.val ∧
+      r.val ⟨0, by norm_num⟩ = x.val s ∧
+      curveAreaFunctional r = curveAreaFunctional x := by
+  have hclosed := hx.2.2.2.2.1
+  obtain ⟨r, hr, _, harea⟩ := exists_cyclic_rotation_eq_concat hab x s hclosed.symm
+  refine ⟨r, ?_, ?_, harea⟩
+  · refine ⟨by norm_num, hx.2.1, r.property.1, ?_, ?_, ?_, ?_⟩
+    · rw [hr, range_cyclic_concat hab x.val s hclosed]
+      exact hx.2.2.2.1
+    · rw [hr]
+      simp
+    · rw [hr]
+      exact Function.injOn_concatUnitIntervals_comp_of_cyclic_endpoints hab
+        hx.2.2.2.2.2.1 hclosed s has hsb
+        (Set.Icc.convexComb s ⟨b, hab, le_rfl⟩)
+        (Set.Icc.convexComb ⟨a, le_rfl, hab⟩ s)
+        (strictMono_convexComb _ _ hsb) (strictMono_convexComb _ _ has)
+        (by simp) (by simp) (by simp)
+    · intro p hp
+      have hxw := hx.2.2.2.2.2.2 p hp
+      have hn : curveWinding hab x.val p ≠ 0 := by
+        rw [hxw]
+        cases ccw <;> norm_num
+      obtain ⟨θ, hθ⟩ := exists_curveAngleLift_of_curveWinding_ne_zero hab hn
+      rw [hr, curveWinding_concat_of_cyclic_endpoints hab hθ hclosed s
+        (Set.Icc.convexComb s ⟨b, hab, le_rfl⟩)
+        (Set.Icc.convexComb ⟨a, le_rfl, hab⟩ s)
+        (Set.Icc.continuous_convexComb _ _) (Set.Icc.continuous_convexComb _ _)
+        (by simp) (by simp) (by simp) (by simp)]
+      exact hxw
+  · rw [hr]
+    simp
+
+/-- A cyclic rotation together with its literal tail-then-head formula. -/
+theorem exists_oriented_cyclic_rotation_eq_concat {a b : ℝ} (hab : a ≤ b)
+    {Γ : Set Point} {ccw : Bool} (x : ContinuousBVPaths a b)
+    (hx : IsOrientedJordanParametrization hab Γ ccw x.val)
+    (s : Set.Icc a b) (has : a < s) (hsb : (s : ℝ) < b) :
+    ∃ r : ContinuousBVPaths 0 2,
+      IsOrientedJordanParametrization (by norm_num) Γ ccw r.val ∧
+      r.val = Function.concatUnitIntervals
+        (x.val ∘ Set.Icc.convexComb s ⟨b, hab, le_rfl⟩)
+        (x.val ∘ Set.Icc.convexComb ⟨a, le_rfl, hab⟩ s) := by
+  have hclosed := hx.2.2.2.2.1
+  obtain ⟨r, hr, _⟩ :=
+    exists_cyclic_rotation_path hab x s hclosed.symm
+  refine ⟨r, ?_, hr⟩
+  refine ⟨by norm_num, hx.2.1, r.property.1, ?_, ?_, ?_, ?_⟩
+  · rw [hr, range_cyclic_concat hab x.val s hclosed]
+    exact hx.2.2.2.1
+  · rw [hr]
+    simp
+  · rw [hr]
+    exact Function.injOn_concatUnitIntervals_comp_of_cyclic_endpoints hab
+      hx.2.2.2.2.2.1 hclosed s has hsb
+      (Set.Icc.convexComb s ⟨b, hab, le_rfl⟩)
+      (Set.Icc.convexComb ⟨a, le_rfl, hab⟩ s)
+      (strictMono_convexComb _ _ hsb)
+      (strictMono_convexComb _ _ has)
+      (by simp) (by simp) (by simp)
+  · intro p hp
+    have hxw := hx.2.2.2.2.2.2 p hp
+    have hn : curveWinding hab x.val p ≠ 0 := by
+      rw [hxw]
+      cases ccw <;> norm_num
+    obtain ⟨θ, hθ⟩ := exists_curveAngleLift_of_curveWinding_ne_zero hab hn
+    rw [hr, curveWinding_concat_of_cyclic_endpoints hab hθ hclosed s
+      (Set.Icc.convexComb s ⟨b, hab, le_rfl⟩)
+      (Set.Icc.convexComb ⟨a, le_rfl, hab⟩ s)
+      (Set.Icc.continuous_convexComb _ _) (Set.Icc.continuous_convexComb _ _)
+      (by simp) (by simp) (by simp) (by simp)]
+    exact hxw
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Winding Kernel
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- The inverse-distance vector kernel based at `z`. -/
+def windingKernel (z p : Point) : Point :=
+  (‖z - p‖ ^ 2)⁻¹ • (z - p)
+
+private def pointComplex : Point ≃ₗᵢ[ℝ] ℂ :=
+  Complex.orthonormalBasisOneI.repr.symm
+
+private theorem pointComplex_windingKernel (z p : Point) :
+    pointComplex (windingKernel z p) =
+      (starRingEnd ℂ (pointComplex z - pointComplex p))⁻¹ := by
+  have hsub : pointComplex (z - p) = pointComplex z - pointComplex p :=
+    map_sub pointComplex z p
+  rw [windingKernel, map_smul, hsub]
+  let v : ℂ := pointComplex z - pointComplex p
+  change (‖z - p‖ ^ 2)⁻¹ • v = (starRingEnd ℂ v)⁻¹
+  rw [Complex.inv_def]
+  simp only [starRingEnd_apply, star_star, Complex.normSq_eq_norm_sq, norm_star]
+  have hn : ‖v‖ = ‖z - p‖ := by
+    rw [show v = pointComplex (z - p) by simp [v, hsub]]
+    exact LinearIsometryEquiv.norm_map pointComplex (z - p)
+  rw [hn, Complex.real_smul, Complex.ofReal_inv, Complex.ofReal_pow]
+  ring
+
+private theorem norm_windingKernel (z p : Point) :
+    ‖windingKernel z p‖ = if p = z then 0 else ‖z - p‖⁻¹ := by
+  by_cases hp : p = z
+  · subst p
+    simp [windingKernel]
+  · have hnorm : ‖z - p‖ ≠ 0 := by
+      simpa [norm_eq_zero, sub_eq_zero] using Ne.symm hp
+    rw [windingKernel, norm_smul, Real.norm_eq_abs]
+    simp only [abs_inv, abs_pow, abs_norm]
+    simp only [hp, ↓reduceIte]
+    field_simp
+
+private theorem ball_zero_subset_ball_two_mul (R : ℝ) (z : Point)
+    (hz : ‖z‖ < R) : Metric.ball 0 R ⊆ Metric.ball z (2 * R) := by
+  intro p hp
+  rw [Metric.mem_ball, dist_eq_norm] at hp ⊢
+  have hp' : ‖p‖ < R := by simpa using hp
+  calc
+    ‖p - z‖ ≤ ‖p‖ + ‖z‖ := norm_sub_le p z
+    _ < R + R := add_lt_add hp' hz
+    _ = 2 * R := by ring
+
+private theorem measurable_windingKernel (z : Point) : Measurable (windingKernel z) := by
+  unfold windingKernel
+  exact (((measurable_const.sub measurable_id).norm.pow_const 2).inv.smul
+    (measurable_const.sub measurable_id))
+
+private theorem indicator_inv_norm_pointComplex (S : ℝ) (p : Point) :
+    (Metric.ball (0 : Point) S).indicator (fun p ↦ ‖p‖⁻¹) p =
+      (Metric.ball (0 : ℂ) S).indicator (fun q ↦ ‖q‖⁻¹) (pointComplex p) := by
+  have hm : pointComplex p ∈ Metric.ball (0 : ℂ) S ↔
+      p ∈ Metric.ball (0 : Point) S := by
+    simp [Metric.mem_ball, dist_eq_norm]
+  by_cases hp : p ∈ Metric.ball (0 : Point) S
+  · simp [hp, hm.mpr hp]
+  · simp [hp, mt hm.mp hp]
+
+private theorem integrableOn_inv_norm_ball (S : ℝ) :
+    IntegrableOn (fun p : Point ↦ ‖p‖⁻¹) (Metric.ball 0 S) := by
+  rw [← integrable_indicator_iff measurableSet_ball]
+  have hf : Integrable ((Metric.ball (0 : ℂ) S).indicator fun q ↦ ‖q‖⁻¹) :=
+    (integrable_indicator_iff measurableSet_ball).mpr (Complex.integrableOn_inv_norm_ball S)
+  refine (((LinearIsometryEquiv.measurePreserving pointComplex).integrable_comp_emb
+    pointComplex.toHomeomorph.measurableEmbedding).mpr hf).congr ?_
+  filter_upwards with p
+  exact (indicator_inv_norm_pointComplex S p).symm
+
+private theorem setIntegral_inv_norm_ball (S : ℝ) (hS : 0 < S) :
+    (∫ p : Point in Metric.ball 0 S, ‖p‖⁻¹) = 2 * Real.pi * S := by
+  rw [← integral_indicator (f := fun p : Point ↦ ‖p‖⁻¹) measurableSet_ball]
+  calc
+    (∫ p : Point, (Metric.ball 0 S).indicator (fun p ↦ ‖p‖⁻¹) p) =
+        ∫ p : Point, (Metric.ball (0 : ℂ) S).indicator (fun q ↦ ‖q‖⁻¹) (pointComplex p) :=
+      integral_congr_ae (Filter.Eventually.of_forall (indicator_inv_norm_pointComplex S))
+    _ = ∫ q : ℂ, (Metric.ball (0 : ℂ) S).indicator (fun q ↦ ‖q‖⁻¹) q :=
+      (LinearIsometryEquiv.measurePreserving pointComplex).integral_comp
+        pointComplex.toHomeomorph.measurableEmbedding _
+    _ = 2 * Real.pi * S := by
+      rw [integral_indicator measurableSet_ball]
+      exact Complex.setIntegral_inv_norm_ball S hS
+
+private theorem indicator_centered_inv_norm_eq (z : Point) (S : ℝ) :
+    (Metric.ball z S).indicator (fun p ↦ ‖z - p‖⁻¹) =
+      fun p ↦ (Metric.ball 0 S).indicator (fun q ↦ ‖q‖⁻¹) (z - p) := by
+  funext p
+  have hm : p ∈ Metric.ball z S ↔ z - p ∈ Metric.ball 0 S := by
+    simp only [Metric.mem_ball, dist_eq_norm]
+    simp [sub_zero, norm_sub_rev]
+  by_cases hp : p ∈ Metric.ball z S
+  · simp [hp, hm.mp hp]
+  · simp [hp, mt hm.mpr hp]
+
+private theorem indicator_centered_inv_norm_nonneg (z : Point) (S : ℝ) (p : Point) :
+    0 ≤ (Metric.ball z S).indicator (fun p ↦ ‖z - p‖⁻¹) p :=
+  Set.indicator_apply_nonneg fun _ ↦ by positivity
+
+private theorem integrable_indicator_centered_inv_norm (z : Point) (S : ℝ) :
+    Integrable ((Metric.ball z S).indicator fun p ↦ ‖z - p‖⁻¹) := by
+  rw [indicator_centered_inv_norm_eq, MeasureTheory.integrable_comp_sub_left]
+  exact (integrable_indicator_iff measurableSet_ball).mpr (integrableOn_inv_norm_ball S)
+
+private theorem integral_indicator_centered_inv_norm (z : Point) (S : ℝ) (hS : 0 < S) :
+    (∫ p : Point, (Metric.ball z S).indicator (fun p ↦ ‖z - p‖⁻¹) p) = 2 * Real.pi * S := by
+  rw [indicator_centered_inv_norm_eq,
+    MeasureTheory.integral_sub_left_eq_self _ volume z, integral_indicator measurableSet_ball]
+  exact setIntegral_inv_norm_ball S hS
+
+private theorem norm_windingKernel_le_indicator (R : ℝ) (z : Point) (hz : ‖z‖ < R) :
+    (fun p ↦ ‖windingKernel z p‖) ≤ᵐ[volume.restrict (Metric.ball 0 R)]
+      (Metric.ball z (2 * R)).indicator (fun p ↦ ‖z - p‖⁻¹) := by
+  filter_upwards [ae_restrict_mem measurableSet_ball] with p hp
+  have hpz : p ∈ Metric.ball z (2 * R) := ball_zero_subset_ball_two_mul R z hz hp
+  rw [norm_windingKernel]
+  by_cases heq : p = z
+  · simp only [heq, ↓reduceIte]
+    exact indicator_centered_inv_norm_nonneg z (2 * R) z
+  · simp [heq, hpz]
+
+/-- The winding kernel is integrable on a disk containing its base point. -/
+theorem integrableOn_windingKernel_ball (R : ℝ) (z : Point) (hz : ‖z‖ < R) :
+    IntegrableOn (windingKernel z) (Metric.ball 0 R) :=
+  Integrable.mono' (integrable_indicator_centered_inv_norm z (2 * R)).integrableOn
+    (measurable_windingKernel z).aestronglyMeasurable
+    (norm_windingKernel_le_indicator R z hz)
+
+/-- The norm of the winding kernel has an integrable uniform majorant on an interior disk. -/
+theorem setIntegral_norm_windingKernel_ball_le
+    (R : ℝ) (hR : 0 < R) (z : Point) (hz : ‖z‖ < R) :
+    (∫ p in Metric.ball 0 R, ‖windingKernel z p‖) ≤ 4 * Real.pi * R := by
+  have hg : Integrable ((Metric.ball z (2 * R)).indicator fun p ↦ ‖z - p‖⁻¹) :=
+    integrable_indicator_centered_inv_norm z (2 * R)
+  calc
+    (∫ p in Metric.ball 0 R, ‖windingKernel z p‖) ≤
+        ∫ p in Metric.ball 0 R, (Metric.ball z (2 * R)).indicator (fun p ↦ ‖z - p‖⁻¹) p :=
+      setIntegral_mono_ae_restrict (integrableOn_windingKernel_ball R z hz).norm
+        hg.integrableOn (norm_windingKernel_le_indicator R z hz)
+    _ ≤ ∫ p, (Metric.ball z (2 * R)).indicator (fun p ↦ ‖z - p‖⁻¹) p :=
+      setIntegral_le_integral hg
+        (Filter.Eventually.of_forall (indicator_centered_inv_norm_nonneg z (2 * R)))
+    _ = 2 * Real.pi * (2 * R) :=
+      integral_indicator_centered_inv_norm z (2 * R) (by linarith)
+    _ = 4 * Real.pi * R := by ring
+
+/-- The integral of the winding kernel over a disk is `π` times its base point. -/
+theorem setIntegral_windingKernel_ball (R : ℝ) (z : Point) (hz : ‖z‖ < R) :
+    (∫ p in Metric.ball 0 R, windingKernel z p) = Real.pi • z := by
+  have hInt := integrableOn_windingKernel_ball R z hz
+  let T : Point ≃ₗᵢ[ℝ] ℂ := pointComplex.trans Complex.conjLIE
+  let a : ℂ := T z
+  have ha : ‖a‖ < R := by simpa [a, T] using hz
+  have hc := Complex.setIntegral_inv_sub_ball R a ha
+  let G : ℂ → ℂ := (Metric.ball 0 R).indicator (fun q ↦ (a - q)⁻¹)
+  have hpres := (LinearIsometryEquiv.measurePreserving T).integral_comp
+    T.toHomeomorph.measurableEmbedding G
+  have hfun : ∀ p : Point,
+      G (T p) = (Metric.ball 0 R).indicator
+        (fun p ↦ pointComplex (windingKernel z p)) p := by
+    intro p
+    have hm : T p ∈ Metric.ball (0 : ℂ) R ↔ p ∈ Metric.ball (0 : Point) R := by
+      simp [Metric.mem_ball, dist_eq_norm, T]
+    by_cases hp : p ∈ Metric.ball (0 : Point) R
+    · rw [show G (T p) = (a - T p)⁻¹ by simp [G, hm.mpr hp]]
+      rw [show (Metric.ball (0 : Point) R).indicator
+          (fun p ↦ pointComplex (windingKernel z p)) p =
+          pointComplex (windingKernel z p) by simp [hp]]
+      rw [pointComplex_windingKernel]
+      simp only [a, T, LinearIsometryEquiv.trans_apply, Complex.conjLIE_apply,
+        map_sub]
+    · rw [show G (T p) = 0 by simp [G, mt hm.mp hp]]
+      simp [hp]
+  have htransport :
+      (∫ q in Metric.ball 0 R, (a - q)⁻¹) =
+        pointComplex (∫ p in Metric.ball 0 R, windingKernel z p) := by
+    rw [← integral_indicator measurableSet_ball]
+    change (∫ q, G q) = _
+    rw [← hpres]
+    rw [integral_congr_ae (Filter.Eventually.of_forall hfun)]
+    rw [integral_indicator measurableSet_ball]
+    simpa using
+      (pointComplex.toContinuousLinearEquiv.toContinuousLinearMap.integral_comp_comm hInt)
+  apply pointComplex.injective
+  rw [← htransport, hc]
+  simp [a, T, Complex.real_smul]
+
+/-- Off the range of a continuous interval path, each coordinate of the winding kernel is
+a bounded continuous function of the parameter. -/
+theorem windingKernel_coord_continuous_bounded
+    {a b : ℝ} {x : Set.Icc a b → Point} (hx : Continuous x) {p : Point}
+    (hp : p ∉ Set.range x) (i : Fin 2) :
+    Continuous (fun t ↦ windingKernel (x t) p i) ∧
+      ∃ C : ℝ, ∀ t, |windingKernel (x t) p i| ≤ C := by
+  have hsub : Continuous (fun t ↦ x t - p) := hx.sub continuous_const
+  have hne : ∀ t, ‖x t - p‖ ^ 2 ≠ 0 := by
+    intro t
+    apply pow_ne_zero
+    rw [norm_ne_zero_iff]
+    intro h
+    exact hp ⟨t, sub_eq_zero.mp h⟩
+  have hkernel : Continuous (fun t ↦ windingKernel (x t) p) := by
+    unfold windingKernel
+    exact ((hsub.norm.pow 2).inv₀ hne).smul hsub
+  have hcoord : Continuous (fun t ↦ windingKernel (x t) p i) :=
+    (PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) i).comp hkernel
+  refine ⟨hcoord, ?_⟩
+  have hbdd : BddAbove ((fun t ↦ |windingKernel (x t) p i|) '' Set.univ) :=
+    isCompact_univ.bddAbove_image hcoord.abs.continuousOn
+  obtain ⟨C, hC⟩ := hbdd
+  exact ⟨C, fun t ↦ hC ⟨t, Set.mem_univ t, rfl⟩⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Winding Local Constancy
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval
+
+namespace MovingSofa
+
+private lemma pointComplex_re_m85d0e79 (v : Point) : (Complex.orthonormalBasisOneI.repr.symm v).re =
+    v 0 := by
+  simp [Complex.orthonormalBasisOneI_repr_symm_apply]
+
+private lemma pointComplex_im_m85d0e79 (v : Point) : (Complex.orthonormalBasisOneI.repr.symm v).im =
+    v 1 := by
+  simp [Complex.orthonormalBasisOneI_repr_symm_apply]
+
+private lemma conj_pointComplex_mul_re_m85d0e79 (v w : Point) :
+    (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm v) *
+        Complex.orthonormalBasisOneI.repr.symm w).re = inner ℝ v w := by
+  simp [Complex.orthonormalBasisOneI_repr_symm_apply, inner, Fin.sum_univ_two]
+  ring
+
+/-- The principal relative argument varies continuously while the relative dot product is
+positive. This is the branch needed for a small displacement of the basepoint. -/
+lemma continuous_arg_conj_mul_of_re_pos {α : Type*} [TopologicalSpace α]
+    {z w : α → ℂ} (hz : Continuous z) (hw : Continuous w)
+    (hpos : ∀ t, 0 < (starRingEnd ℂ (z t) * w t).re) :
+    Continuous (fun t ↦ Complex.arg (starRingEnd ℂ (z t) * w t)) := by
+  have hf : Continuous (fun u ↦ starRingEnd ℂ (z u) * w u) :=
+    (continuous_star.comp hz).mul hw
+  change Continuous (Complex.arg ∘ fun u ↦ starRingEnd ℂ (z u) * w u)
+  exact Complex.continuousOn_arg.comp_continuous hf (fun t ↦ Or.inl (hpos t))
+
+/-- The principal relative argument rotates the normalized coordinates of `z` to those of `w`. -/
+lemma normalized_complex_rotation_by_arg {z w : ℂ}
+    (hz : z ≠ 0) (hw : w ≠ 0) :
+    Real.cos (Complex.arg z + Complex.arg (starRingEnd ℂ z * w)) = w.re / ‖w‖ ∧
+      Real.sin (Complex.arg z + Complex.arg (starRingEnd ℂ z * w)) = w.im / ‖w‖ := by
+  have hzw : starRingEnd ℂ z * w ≠ 0 :=
+    mul_ne_zero ((map_ne_zero (starRingEnd ℂ)).2 hz) hw
+  rw [Real.cos_add, Real.sin_add, Complex.cos_arg hz, Complex.sin_arg,
+    Complex.cos_arg hzw, Complex.sin_arg]
+  simp only [Complex.norm_mul, Complex.mul_re, Complex.mul_im,
+    Complex.conj_re, Complex.conj_im]
+  rw [Complex.norm_conj]
+  have hn : z.re ^ 2 + z.im ^ 2 = ‖z‖ ^ 2 := by
+    rw [Complex.sq_norm, Complex.normSq_apply]
+    ring
+  have hnz : ‖z‖ ≠ 0 := norm_ne_zero_iff.mpr hz
+  constructor
+  · field_simp [hnz]
+    linear_combination w.re * hn
+  · field_simp [hnz]
+    linear_combination w.im * hn
+
+/-- Coordinate form of the preceding rotation identity for any chosen real lift of `z`. -/
+lemma normalized_complex_rotation_by_relative_arg {z w : ℂ} {θ : ℝ}
+    (hz : z ≠ 0) (hw : w ≠ 0)
+    (hc : Real.cos θ = z.re / ‖z‖) (hs : Real.sin θ = z.im / ‖z‖) :
+    Real.cos (θ + Complex.arg (starRingEnd ℂ z * w)) = w.re / ‖w‖ ∧
+      Real.sin (θ + Complex.arg (starRingEnd ℂ z * w)) = w.im / ‖w‖ := by
+  have hzw : starRingEnd ℂ z * w ≠ 0 :=
+    mul_ne_zero ((map_ne_zero (starRingEnd ℂ)).2 hz) hw
+  rw [Real.cos_add, Real.sin_add, hc, hs, Complex.cos_arg hzw, Complex.sin_arg,
+    Complex.norm_mul, Complex.norm_conj]
+  simp only [Complex.mul_re, Complex.mul_im, Complex.conj_re, Complex.conj_im]
+  have hn : z.re ^ 2 + z.im ^ 2 = ‖z‖ ^ 2 := by
+    rw [Complex.sq_norm, Complex.normSq_apply]
+    ring
+  have hnz : ‖z‖ ≠ 0 := norm_ne_zero_iff.mpr hz
+  constructor
+  · field_simp [hnz]
+    linear_combination w.re * hn
+  · field_simp [hnz]
+    linear_combination w.im * hn
+
+/-- A positive relative dot product supplies the continuous principal correction between two
+basepoints. -/
+lemma IsCurveAngleLift.add_principal_basepoint_correction {a b : ℝ}
+    {x : Set.Icc a b → Point} {p q : Point} {θ : Set.Icc a b → ℝ}
+    (hx : Continuous x) (hθ : IsCurveAngleLift x p θ)
+    (hpos : ∀ u, 0 < (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm (x u - p)) *
+      Complex.orthonormalBasisOneI.repr.symm (x u - q)).re) :
+    IsCurveAngleLift x q (fun u ↦ θ u +
+      Complex.arg (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm (x u - p)) *
+          Complex.orthonormalBasisOneI.repr.symm (x u - q))) := by
+  let z := fun u ↦ Complex.orthonormalBasisOneI.repr.symm (x u - p)
+  let w := fun u ↦ Complex.orthonormalBasisOneI.repr.symm (x u - q)
+  have hz : Continuous z := Complex.orthonormalBasisOneI.repr.symm.continuous.comp (hx.sub
+      continuous_const)
+  have hw : Continuous w := Complex.orthonormalBasisOneI.repr.symm.continuous.comp (hx.sub
+      continuous_const)
+  have hδ : Continuous (fun u ↦ Complex.arg (starRingEnd ℂ (z u) * w u)) :=
+    continuous_arg_conj_mul_of_re_pos hz hw hpos
+  refine ⟨hθ.1.add hδ, fun u ↦ ?_⟩
+  have hprod : starRingEnd ℂ (z u) * w u ≠ 0 := by
+    intro hzero
+    have := hpos u
+    rw [hzero] at this
+    simp at this
+  have hzne : z u ≠ 0 := fun hzero ↦ hprod (by simp [hzero])
+  have hwne : w u ≠ 0 := fun hzero ↦ hprod (by simp [hzero])
+  have hc : Real.cos (θ u) = (z u).re / ‖z u‖ := by
+    simpa only [z, pointComplex_re_m85d0e79, Complex.orthonormalBasisOneI.repr.symm.norm_map]
+      using (hθ.2
+        u).1
+  have hs : Real.sin (θ u) = (z u).im / ‖z u‖ := by
+    simpa only [z, pointComplex_im_m85d0e79, Complex.orthonormalBasisOneI.repr.symm.norm_map]
+      using (hθ.2
+        u).2
+  simpa only [z, w, pointComplex_re_m85d0e79, pointComplex_im_m85d0e79,
+      Complex.orthonormalBasisOneI.repr.symm.norm_map] using
+    normalized_complex_rotation_by_relative_arg hzne hwne hc hs
+
+/-- Moving the basepoint through the positive-relative-dot neighborhood preserves winding. -/
+theorem curveWinding_eq_of_relative_dot_pos {a b : ℝ} (hab : a ≤ b)
+    {x : Set.Icc a b → Point} (hx : Continuous x)
+    (hclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    {p q : Point} {θ : Set.Icc a b → ℝ} (hθ : IsCurveAngleLift x p θ)
+    (hpos : ∀ u, 0 < (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm (x u - p)) *
+      Complex.orthonormalBasisOneI.repr.symm (x u - q)).re) :
+    curveWinding hab x q = curveWinding hab x p := by
+  let δ := fun u ↦ Complex.arg (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm (x u - p)) *
+    Complex.orthonormalBasisOneI.repr.symm (x u - q))
+  have hθδ : IsCurveAngleLift x q (fun u ↦ θ u + δ u) :=
+    hθ.add_principal_basepoint_correction hx hpos
+  have hδ : δ ⟨a, le_rfl, hab⟩ = δ ⟨b, hab, le_rfl⟩ := by
+    simp only [δ, hclosed]
+  rw [hθδ.curveWinding_eq hab, hθ.curveWinding_eq hab]
+  rw [hδ]
+  ring
+
+/-- Off a compact continuous loop, every sufficiently nearby basepoint has positive relative dot
+product with the original radial vectors. -/
+theorem exists_ball_relative_dot_pos {a b : ℝ} (hab : a ≤ b)
+    {x : Set.Icc a b → Point} (hx : Continuous x) {p : Point}
+    (hp : p ∉ Set.range x) :
+    ∃ r > 0, ∀ q, dist q p < r → ∀ u,
+      0 < (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm (x u - p)) *
+          Complex.orthonormalBasisOneI.repr.symm (x u - q)).re := by
+  let _ : Nonempty (Set.Icc a b) := ⟨⟨a, le_rfl, hab⟩⟩
+  let f : Set.Icc a b → ℝ := fun u ↦ ‖x u - p‖
+  have hf : Continuous f := (hx.sub continuous_const).norm
+  obtain ⟨u₀, -, hu₀⟩ := isCompact_univ.exists_isMinOn Set.univ_nonempty hf.continuousOn
+  have hne (u : Set.Icc a b) : x u - p ≠ 0 := by
+    intro hzero
+    apply hp
+    refine ⟨u, ?_⟩
+    exact sub_eq_zero.mp hzero
+  have hfu₀ : 0 < f u₀ := norm_pos_iff.mpr (hne u₀)
+  refine ⟨f u₀ / 2, half_pos hfu₀, fun q hq u ↦ ?_⟩
+  have hmin : f u₀ ≤ f u := hu₀ (Set.mem_univ u)
+  have hd : ‖p - q‖ < f u₀ / 2 := by
+    simpa [dist_eq_norm, norm_sub_rev] using hq
+  have hd' : ‖p - q‖ < ‖x u - p‖ := lt_of_lt_of_le hd (by linarith)
+  rw [conj_pointComplex_mul_re_m85d0e79]
+  have hw : x u - q = (x u - p) + (p - q) := by abel
+  rw [hw, inner_add_right, real_inner_self_eq_norm_sq]
+  have hcs := abs_real_inner_le_norm (x u - p) (p - q)
+  have hlower : -(‖x u - p‖ * ‖p - q‖) ≤ inner ℝ (x u - p) (p - q) :=
+    (neg_le_of_abs_le hcs)
+  nlinarith [norm_pos_iff.mpr (hne u)]
+
+/-- Winding of a continuous closed loop is locally constant away from its range. -/
+theorem curveWinding_locally_constant_off_range {a b : ℝ} (hab : a ≤ b)
+    {x : Set.Icc a b → Point} (hx : Continuous x)
+    (hclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    {p : Point} (hp : p ∉ Set.range x) :
+    ∃ U : Set Point, IsOpen U ∧ p ∈ U ∧
+      ∀ q ∈ U, curveWinding hab x q = curveWinding hab x p := by
+  obtain ⟨r, hr, hdot⟩ := exists_ball_relative_dot_pos hab hx hp
+  refine ⟨Metric.ball p r, Metric.isOpen_ball, Metric.mem_ball_self hr, fun q hq ↦ ?_⟩
+  have hqp : dist q p < r := by simpa [dist_comm] using hq
+  have hpos := hdot q hqp
+  by_cases hpLift : ∃ θ, IsCurveAngleLift x p θ
+  · obtain ⟨θ, hθ⟩ := hpLift
+    exact curveWinding_eq_of_relative_dot_pos hab hx hclosed hθ hpos
+  · have hqLift : ¬∃ ψ, IsCurveAngleLift x q ψ := by
+      rintro ⟨ψ, hψ⟩
+      have hrev : ∀ u, 0 < (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm (x u - q)) *
+          Complex.orthonormalBasisOneI.repr.symm (x u - p)).re := by
+        intro u
+        rw [conj_pointComplex_mul_re_m85d0e79, real_inner_comm, ← conj_pointComplex_mul_re_m85d0e79]
+        exact hpos u
+      exact hpLift ⟨_, hψ.add_principal_basepoint_correction hx hrev⟩
+    unfold curveWinding
+    rw [dite_eq_right hqLift, dite_eq_right hpLift]
+
+/-- While the relative dot product with the initial radius vector stays positive, the
+increment of a continuous angle lift is the principal relative argument, computed as the
+arctangent of the ratio of the relative cross product to the relative dot product. -/
+theorem IsCurveAngleLift.sub_eq_arctan_of_dot_pos {a b : ℝ}
+    {x : Set.Icc a b → Point} {p : Point} {θ : Set.Icc a b → ℝ}
+    (hx : Continuous x) (hθ : IsCurveAngleLift x p θ) {t u : Set.Icc a b}
+    (htu : (t : ℝ) ≤ (u : ℝ))
+    (hpos : ∀ s : Set.Icc a b, (t : ℝ) ≤ (s : ℝ) → (s : ℝ) ≤ (u : ℝ) →
+      0 < (x t - p) 0 * (x s - p) 0 + (x t - p) 1 * (x s - p) 1) :
+    θ u - θ t = Real.arctan
+      (((x t - p) 0 * (x u - p) 1 - (x t - p) 1 * (x u - p) 0) /
+        ((x t - p) 0 * (x u - p) 0 + (x t - p) 1 * (x u - p) 1)) := by
+  have hsub : ∀ s : Set.Icc (t : ℝ) (u : ℝ), (s : ℝ) ∈ Set.Icc a b := fun s ↦
+    ⟨le_trans t.property.1 s.property.1, le_trans s.property.2 u.property.2⟩
+  let ι : Set.Icc (t : ℝ) (u : ℝ) → Set.Icc a b := fun s ↦ ⟨s.val, hsub s⟩
+  have hι : Continuous ι := continuous_subtype_val.subtype_mk _
+  set y : Set.Icc (t : ℝ) (u : ℝ) → Point := x ∘ ι with hy
+  have hycont : Continuous y := hx.comp hι
+  have hιu : ι ⟨(u : ℝ), htu, le_rfl⟩ = u := Subtype.ext rfl
+  have hιt : ι ⟨(t : ℝ), le_rfl, htu⟩ = t := Subtype.ext rfl
+  set z : ℂ := Complex.orthonormalBasisOneI.repr.symm (x t - p) with hz
+  set w : Set.Icc (t : ℝ) (u : ℝ) → ℂ :=
+    fun s ↦ Complex.orthonormalBasisOneI.repr.symm (y s - p) with hw
+  have hwcont : Continuous w :=
+    Complex.orthonormalBasisOneI.repr.symm.continuous.comp (hycont.sub continuous_const)
+  have hre (s : Set.Icc (t : ℝ) (u : ℝ)) :
+      (starRingEnd ℂ z * w s).re =
+        (x t - p) 0 * (x (ι s) - p) 0 + (x t - p) 1 * (x (ι s) - p) 1 := by
+    simp only [hz, hw, hy, Complex.mul_re, Complex.conj_re, Complex.conj_im,
+      pointComplex_re_m85d0e79, pointComplex_im_m85d0e79, Function.comp_apply]
+    ring
+  have him (s : Set.Icc (t : ℝ) (u : ℝ)) :
+      (starRingEnd ℂ z * w s).im =
+        (x t - p) 0 * (x (ι s) - p) 1 - (x t - p) 1 * (x (ι s) - p) 0 := by
+    simp only [hz, hw, hy, Complex.mul_im, Complex.conj_re, Complex.conj_im,
+      pointComplex_re_m85d0e79, pointComplex_im_m85d0e79, Function.comp_apply]
+    ring
+  have hposC : ∀ s : Set.Icc (t : ℝ) (u : ℝ), 0 < (starRingEnd ℂ z * w s).re := by
+    intro s
+    rw [hre s]
+    exact hpos (ι s) s.property.1 s.property.2
+  have hzne : z ≠ 0 := by
+    intro h0
+    have := hposC ⟨(t : ℝ), le_rfl, htu⟩
+    rw [h0] at this
+    simp at this
+  have hwne : ∀ s, w s ≠ 0 := by
+    intro s h0
+    have := hposC s
+    rw [h0] at this
+    simp at this
+  have hznorm : ‖z‖ = ‖x t - p‖ := Complex.orthonormalBasisOneI.repr.symm.norm_map _
+  have hc : Real.cos (θ t) = z.re / ‖z‖ := by
+    rw [hznorm, hz, pointComplex_re_m85d0e79]; exact (hθ.2 t).1
+  have hsn : Real.sin (θ t) = z.im / ‖z‖ := by
+    rw [hznorm, hz, pointComplex_im_m85d0e79]; exact (hθ.2 t).2
+  have hlift1 : IsCurveAngleLift y p (θ ∘ ι) := hθ.comp hι
+  have hlift2 : IsCurveAngleLift y p
+      (fun s ↦ θ t + Complex.arg (starRingEnd ℂ z * w s)) := by
+    refine ⟨continuous_const.add
+      (continuous_arg_conj_mul_of_re_pos continuous_const hwcont hposC), fun s ↦ ?_⟩
+    have hrot := normalized_complex_rotation_by_relative_arg hzne (hwne s) hc hsn
+    have hwnorm : ‖w s‖ = ‖y s - p‖ :=
+      Complex.orthonormalBasisOneI.repr.symm.norm_map _
+    rw [hwnorm] at hrot
+    simpa only [hw, pointComplex_re_m85d0e79, pointComplex_im_m85d0e79] using hrot
+  have hincr := IsCurveAngleLift.endpoint_increment_eq htu hlift1 hlift2
+  simp only [Function.comp_apply, hιu, hιt] at hincr
+  have hwt : w ⟨(t : ℝ), le_rfl, htu⟩ = z := by
+    simp only [hw, hy, Function.comp_apply, hιt, hz]
+  have hargt : Complex.arg (starRingEnd ℂ z * w ⟨(t : ℝ), le_rfl, htu⟩) = 0 := by
+    rw [hwt, mul_comm, Complex.mul_conj]
+    exact Complex.arg_ofReal_of_nonneg (Complex.normSq_nonneg z)
+  rw [hargt] at hincr
+  have hre' : (starRingEnd ℂ z * w ⟨(u : ℝ), htu, le_rfl⟩).re =
+      (x t - p) 0 * (x u - p) 0 + (x t - p) 1 * (x u - p) 1 := by
+    rw [hre ⟨(u : ℝ), htu, le_rfl⟩, hιu]
+  have him' : (starRingEnd ℂ z * w ⟨(u : ℝ), htu, le_rfl⟩).im =
+      (x t - p) 0 * (x u - p) 1 - (x t - p) 1 * (x u - p) 0 := by
+    rw [him ⟨(u : ℝ), htu, le_rfl⟩, hιu]
+  have hargu : Complex.arg (starRingEnd ℂ z * w ⟨(u : ℝ), htu, le_rfl⟩) =
+      Real.arctan
+        (((x t - p) 0 * (x u - p) 1 - (x t - p) 1 * (x u - p) 0) /
+          ((x t - p) 0 * (x u - p) 0 + (x t - p) 1 * (x u - p) 1)) := by
+    rw [← hre', ← him']
+    have hlt : |Complex.arg (starRingEnd ℂ z * w ⟨(u : ℝ), htu, le_rfl⟩)| < Real.pi / 2 :=
+      Complex.abs_arg_lt_pi_div_two_iff.mpr (Or.inl (hposC _))
+    refine (Real.arctan_eq_of_tan_eq (Complex.tan_arg _) ?_).symm
+    exact ⟨neg_lt_of_abs_lt hlt, lt_of_abs_lt hlt⟩
+  rw [hargu] at hincr
+  linarith [hincr]
+
+/-- Winding is constant on an open neighbourhood of any point off the range of a closed
+continuous loop, and that neighbourhood avoids the range. -/
+theorem curveWinding_locally_constant_on_compl_range
+    {a b : ℝ} (hab : a ≤ b) {x : Set.Icc a b → Point} (hx : Continuous x)
+    (hclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    {p : Point} (hp : p ∉ Set.range x) :
+    ∃ U : Set Point, IsOpen U ∧ p ∈ U ∧ ∀ q ∈ U,
+      q ∉ Set.range x ∧ curveWinding hab x q = curveWinding hab x p := by
+  obtain ⟨U, hU, hpU, heq⟩ := curveWinding_locally_constant_off_range hab hx hclosed hp
+  have hclosedRange : IsClosed (Set.range x) := by
+    simpa only [Set.image_univ] using (isCompact_univ.image hx).isClosed
+  exact ⟨U ∩ (Set.range x)ᶜ, hU.inter hclosedRange.isOpen_compl, ⟨hpU, hp⟩,
+    fun q hq ↦ ⟨hq.2, heq q hq.1⟩⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Winding Lifts
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval
+
+namespace MovingSofa
+
+private lemma pointComplex_re_m8588f5d (v : Point) : (Complex.orthonormalBasisOneI.repr.symm v).re =
+    v 0 := by
+  simp [Complex.orthonormalBasisOneI_repr_symm_apply]
+
+private lemma pointComplex_im_m8588f5d (v : Point) : (Complex.orthonormalBasisOneI.repr.symm v).im =
+    v 1 := by
+  simp [Complex.orthonormalBasisOneI_repr_symm_apply]
+
+private lemma conj_pointComplex_mul_re_m8588f5d (v w : Point) :
+    (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm v) *
+        Complex.orthonormalBasisOneI.repr.symm w).re = inner ℝ v w := by
+  simp [Complex.orthonormalBasisOneI_repr_symm_apply, inner, Fin.sum_univ_two]
+  ring
+
+/-- Every continuous interval path avoiding a point admits a continuous angle lift. -/
+theorem exists_curveAngleLift_of_avoids {a b : ℝ} (hab : a < b)
+    {x : Set.Icc a b → Point} (hx : Continuous x) {p : Point}
+    (hp : p ∉ Set.range x) : ∃ α, IsCurveAngleLift x p α := by
+  let A : Set.Icc a b := ⟨a, le_rfl, hab.le⟩
+  let B : Set.Icc a b := ⟨b, hab.le, le_rfl⟩
+  let φ : I → Set.Icc a b := Set.Icc.convexComb A B
+  let z : I → ℂ := fun u ↦ Complex.orthonormalBasisOneI.repr.symm (x (φ u) - p)
+  have hz : Continuous z :=
+    Complex.orthonormalBasisOneI.repr.symm.continuous.comp ((hx.comp (Set.Icc.continuous_convexComb
+        A B)).sub
+      continuous_const)
+  have hzne (u : I) : z u ≠ 0 := by
+    rw [← norm_ne_zero_iff, Complex.orthonormalBasisOneI.repr.symm.norm_map, norm_ne_zero_iff,
+        sub_ne_zero]
+    intro heq
+    exact hp ⟨φ u, heq⟩
+  let ϑ : I → Real.Angle := fun u ↦ (Complex.arg (z u) : Real.Angle)
+  have hϑ : Continuous ϑ := by
+    rw [continuous_iff_continuousAt]
+    intro u
+    exact (Complex.continuousAt_arg_coe_angle (hzne u)).comp hz.continuousAt
+  let ϑ₀ := ϑ 0
+  obtain ⟨β, hβ, hβ0, hβϑ⟩ := Real.Angle.exists_continuous_lift_zero
+    (fun u ↦ ϑ u - ϑ₀) (hϑ.sub continuous_const) (by simp [ϑ₀])
+  let ψ : Set.Icc a b → I := fun u ↦
+    ⟨((u : ℝ) - a) / (b - a), by
+      constructor
+      · exact div_nonneg (sub_nonneg.mpr u.property.1) (sub_nonneg.mpr hab.le)
+      · exact (div_le_one (sub_pos.mpr hab)).2 (by linarith [u.property.2])⟩
+  have hψ : Continuous ψ := by
+    apply Continuous.subtype_mk
+    fun_prop
+  have hφψ (u : Set.Icc a b) : φ (ψ u) = u := by
+    apply Subtype.ext
+    simp [φ, ψ, A, B]
+    field_simp [sub_ne_zero.mpr hab.ne']
+    ring
+  let c := Complex.arg (z 0)
+  refine ⟨fun u ↦ β (ψ u) + c, hβ.comp hψ |>.add continuous_const, fun u ↦ ?_⟩
+  have hangle : ((β (ψ u) + c : ℝ) : Real.Angle) = Complex.arg
+      (Complex.orthonormalBasisOneI.repr.symm (x u - p)) := by
+    rw [Real.Angle.coe_add, hβϑ]
+    simp only [ϑ₀, c, ϑ]
+    change (Complex.arg (Complex.orthonormalBasisOneI.repr.symm (x (φ (ψ u)) - p)) : Real.Angle) -
+        (Complex.arg (z 0) : Real.Angle) + (Complex.arg (z 0) : Real.Angle) = _
+    rw [hφψ]
+    simp
+  have hne : Complex.orthonormalBasisOneI.repr.symm (x u - p) ≠ 0 := by
+    rw [← norm_ne_zero_iff, Complex.orthonormalBasisOneI.repr.symm.norm_map, norm_ne_zero_iff,
+        sub_ne_zero]
+    intro heq
+    exact hp ⟨u, heq⟩
+  constructor
+  · have := congrArg Real.Angle.cos hangle
+    rw [Real.Angle.cos_coe, Real.Angle.cos_coe, Complex.cos_arg hne] at this
+    simpa only [pointComplex_re_m8588f5d, Complex.orthonormalBasisOneI.repr.symm.norm_map] using
+      this
+  · have := congrArg Real.Angle.sin hangle
+    rw [Real.Angle.sin_coe, Real.Angle.sin_coe, Complex.sin_arg] at this
+    simpa only [pointComplex_im_m8588f5d, Complex.orthonormalBasisOneI.repr.symm.norm_map] using
+      this
+
+/-- A closed path contained in a strict half-plane about a point has winding zero there. -/
+theorem curveWinding_eq_zero_of_inner_pos {a b : ℝ} (hab : a ≤ b)
+    {x : Set.Icc a b → Point} (hx : Continuous x)
+    (hclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (p v : Point) (hv : v ≠ 0) (hpos : ∀ u, 0 < inner ℝ v (x u - p)) :
+    curveWinding hab x p = 0 := by
+  let z := Complex.orthonormalBasisOneI.repr.symm v
+  let w := fun u ↦ Complex.orthonormalBasisOneI.repr.symm (x u - p)
+  have hz : z ≠ 0 := by
+    rw [← norm_ne_zero_iff, Complex.orthonormalBasisOneI.repr.symm.norm_map, norm_ne_zero_iff]
+    exact hv
+  have hpositive (u : Set.Icc a b) : 0 < (starRingEnd ℂ z * w u).re := by
+    rw [conj_pointComplex_mul_re_m8588f5d]
+    exact hpos u
+  have harg : Continuous (fun u ↦ Complex.arg (starRingEnd ℂ z * w u)) :=
+    continuous_arg_conj_mul_of_re_pos continuous_const
+      (Complex.orthonormalBasisOneI.repr.symm.continuous.comp (hx.sub continuous_const)) hpositive
+  let α := fun u ↦ Complex.arg z + Complex.arg (starRingEnd ℂ z * w u)
+  have hα : IsCurveAngleLift x p α := by
+    refine ⟨continuous_const.add harg, fun u ↦ ?_⟩
+    have hw : w u ≠ 0 := by
+      intro heq
+      simpa [heq] using hpositive u
+    simpa only [α, w, pointComplex_re_m8588f5d, pointComplex_im_m8588f5d,
+        Complex.orthonormalBasisOneI.repr.symm.norm_map] using
+      normalized_complex_rotation_by_arg hz hw
+  rw [hα.curveWinding_eq hab]
+  have hend : α ⟨b, hab, le_rfl⟩ = α ⟨a, le_rfl, hab⟩ := by
+    simp only [α, w, hclosed]
+  rw [hend, sub_self, zero_div]
+
+/-- A closed continuous loop has winding zero at every point of an unbounded connected
+component of the complement of its range. -/
+theorem curveWinding_eq_zero_of_unbounded_component
+    {a b : ℝ} (hab : a ≤ b) {x : Set.Icc a b → Point} (hx : Continuous x)
+    (hclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    {p : Point} (hp : p ∉ Set.range x)
+    (hunbounded : ¬Bornology.IsBounded (connectedComponentIn (Set.range x)ᶜ p)) :
+    curveWinding hab x p = 0 := by
+  let V := connectedComponentIn (Set.range x)ᶜ p
+  let _ : PreconnectedSpace V :=
+    Subtype.preconnectedSpace isPreconnected_connectedComponentIn
+  have hlc : IsLocallyConstant (fun z : V ↦ curveWinding hab x z.val) := by
+    apply (IsLocallyConstant.iff_exists_open _).mpr
+    intro z
+    have hzout : z.val ∉ Set.range x := connectedComponentIn_subset _ _ z.property
+    obtain ⟨W, hWopen, hzW, hWeq⟩ :=
+      curveWinding_locally_constant_off_range hab hx hclosed hzout
+    exact ⟨Subtype.val ⁻¹' W, hWopen.preimage continuous_subtype_val, hzW,
+      fun z' hz' ↦ hWeq z'.val hz'⟩
+  have hrangeBounded : Bornology.IsBounded (Set.range x) := by
+    simpa only [Set.image_univ] using (isCompact_univ.image hx).isBounded
+  obtain ⟨R, hR⟩ := (Metric.isBounded_iff_subset_closedBall (0 : Point)).mp hrangeBounded
+  have hRnonneg : 0 ≤ R := by
+    have h := hR (Set.mem_range_self ⟨a, le_rfl, hab⟩)
+    have hnorm : ‖x ⟨a, le_rfl, hab⟩‖ ≤ R := by
+      simpa [Metric.mem_closedBall, dist_zero_right] using h
+    exact (norm_nonneg _).trans hnorm
+  have hvfar : ∃ v ∈ V, R + 1 < ‖v‖ := by
+    by_contra hn
+    push Not at hn
+    apply hunbounded
+    refine (Metric.isBounded_iff_subset_closedBall (0 : Point)).2 ⟨R + 1, ?_⟩
+    intro v hv
+    simpa [Metric.mem_closedBall, dist_zero_right] using hn v hv
+  obtain ⟨v, hvV, hvnorm⟩ := hvfar
+  have hvzero : curveWinding hab x v = 0 := by
+    apply curveWinding_eq_zero_of_inner_pos hab hx hclosed v (-v)
+    · exact neg_ne_zero.mpr (by
+        intro hv0
+        rw [hv0, norm_zero] at hvnorm
+        linarith)
+    · intro u
+      rw [inner_neg_left, inner_sub_right, real_inner_self_eq_norm_sq]
+      rw [← real_inner_comm v (x u)]
+      have hxu := hR (Set.mem_range_self u)
+      have hxnorm : ‖x u‖ ≤ R := by
+        simpa [Metric.mem_closedBall, dist_zero_right] using hxu
+      have hvpos : 0 < ‖v‖ := lt_of_le_of_lt hRnonneg (lt_add_one R) |>.trans hvnorm
+      have hvlarge : R < ‖v‖ := lt_trans (lt_add_one R) hvnorm
+      have hinner := abs_real_inner_le_norm (x u) v
+      have hlower : inner ℝ (x u) v ≤ ‖x u‖ * ‖v‖ := le_trans (le_abs_self _) hinner
+      have hprod : ‖x u‖ * ‖v‖ < ‖v‖ * ‖v‖ :=
+        lt_of_le_of_lt (mul_le_mul_of_nonneg_right hxnorm (norm_nonneg v))
+          (mul_lt_mul_of_pos_right hvlarge hvpos)
+      rw [pow_two]
+      linarith
+  exact (hlc.apply_eq_of_preconnectedSpace ⟨p, mem_connectedComponentIn hp⟩ ⟨v, hvV⟩).trans
+    hvzero
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The range of an almost injective continuous BV path is null
+
+A continuous planar BV path that is injective on `[a, b)` sweeps a Lebesgue null set: each
+compact initial subarc has finite Hausdorff length, hence vanishing Hausdorff `2`-measure,
+and a rational exhaustion together with the terminal point covers the whole range.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+
+namespace MovingSofa
+
+/-- The range of a continuous planar BV path injective on `[a, b)` is Lebesgue null. -/
+theorem ContinuousBVPaths.volume_range_eq_zero_of_injOn
+    {a b : ℝ} (hab : a ≤ b) (x : ContinuousBVPaths a b)
+    (hinj : Set.InjOn x.val {t | (t : ℝ) < b}) : volume (Set.range x.val) = 0 := by
+  let γ : ℝ → Point := x.val ∘ Set.projIcc a b hab
+  have hγcont : Continuous γ := x.property.1.comp continuous_projIcc
+  have hγBV (q : ℝ) : BoundedVariationOn γ (Icc a q) := by
+    apply ne_top_of_le_ne_top x.boundedVariationOn
+    exact eVariationOn.comp_le_of_monotoneOn x.val (Set.projIcc a b hab)
+      ((Set.monotone_projIcc hab).monotoneOn _) (fun _ _ ↦ mem_univ _)
+  have hnull (q : {q : ℚ // a ≤ (q : ℝ) ∧ (q : ℝ) < b}) :
+      volume (γ '' Icc a (q : ℝ)) = 0 := by
+    apply volume_image_Icc_eq_zero_of_boundedVariationOn γ q.property.1 hγcont.continuousOn
+      _ (hγBV _)
+    intro u hu v hv huv
+    have huab : u ∈ Icc a b := ⟨hu.1, hu.2.trans q.property.2.le⟩
+    have hvab : v ∈ Icc a b := ⟨hv.1, hv.2.trans q.property.2.le⟩
+    have heq := hinj (show (Set.projIcc a b hab u : ℝ) < b by
+        rw [Set.projIcc_of_mem hab huab]; exact hu.2.trans_lt q.property.2)
+      (show (Set.projIcc a b hab v : ℝ) < b by
+        rw [Set.projIcc_of_mem hab hvab]; exact hv.2.trans_lt q.property.2) huv
+    have hval := congrArg Subtype.val heq
+    simpa only [Set.projIcc_of_mem hab huab, Set.projIcc_of_mem hab hvab] using hval
+  apply measure_mono_null (t := {x.val ⟨b, hab, le_rfl⟩} ∪
+    ⋃ q : {q : ℚ // a ≤ (q : ℝ) ∧ (q : ℝ) < b}, γ '' Icc a (q : ℝ))
+  · rintro y ⟨t, rfl⟩
+    by_cases ht : (t : ℝ) = b
+    · exact Or.inl (congrArg x.val (show t = ⟨b, hab, le_rfl⟩ from Subtype.ext ht))
+    · have htb : (t : ℝ) < b := lt_of_le_of_ne t.property.2 ht
+      obtain ⟨q, htq, hqb⟩ := exists_rat_btwn htb
+      apply Or.inr
+      apply mem_iUnion.2
+      refine ⟨⟨q, t.property.1.trans htq.le, hqb⟩, (t : ℝ), ⟨t.property.1, htq.le⟩, ?_⟩
+      dsimp [γ]
+      rw [Set.projIcc_of_mem hab t.property]
+  · exact measure_union_null (measure_singleton _) (measure_iUnion_null hnull)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Segment Area
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Half the oriented determinant of the two endpoints. -/
+def segmentArea (p q : Point) : ℝ := planeCrossProduct p q / 2
+
+/-- The signed segment area is antisymmetric in its two endpoints. -/
+theorem segmentArea_swap (p q : Point) : segmentArea p q = -segmentArea q p := by
+  rw [segmentArea, segmentArea, planeCrossProduct_swap p q]
+  ring
+
+/-- Translating both endpoints of a segment shifts its signed area by a boundary term. -/
+theorem segmentArea_add_right (p q v : Point) :
+    segmentArea (p + v) (q + v) =
+      segmentArea p q + (v 0 * (q 1 - p 1) - v 1 * (q 0 - p 0)) / 2 := by
+  have hadd : ∀ (x y : Point) (i : Fin 2), (x + y) i = x i + y i := fun _ _ _ ↦ by simp
+  simp only [segmentArea, planeCrossProduct, hadd]
+  ring
+
+/-- The signed area of the segment joining two convex combinations of endpoints is the same
+combination of the two signed areas, provided the two endpoints of each pair have a common height:
+the mixed terms then cancel. -/
+theorem segmentArea_combination_of_apply_one_eq (c : ℝ) {p₁ p₂ q₁ q₂ : Point}
+    (hp : p₁ 1 = p₂ 1) (hq : q₁ 1 = q₂ 1) :
+    segmentArea ((1 - c) • p₁ + c • p₂) ((1 - c) • q₁ + c • q₂) =
+      (1 - c) * segmentArea p₁ q₁ + c * segmentArea p₂ q₂ := by
+  simp only [segmentArea, planeCrossProduct, PiLp.add_apply, PiLp.smul_apply, smul_eq_mul, hp, hq]
+  ring
+
+/-- Two points on a normal line through the origin span no signed area. -/
+theorem segmentArea_eq_zero_of_inner_normalVector_eq_zero {p q : Point} {t : ℝ}
+    (hp : inner ℝ p (normalVector (t : Real.Angle)) = 0)
+    (hq : inner ℝ q (normalVector (t : Real.Angle)) = 0) : segmentArea p q = 0 := by
+  rw [segmentArea, planeCrossProduct_eq_inner_frame p q t, hp, hq]
+  ring
+
+/-- Collinear additivity of the signed segment area on a common normal line. -/
+theorem segmentArea_sub_segmentArea_of_inner_normalVector_eq {p q s : Point} {t c : ℝ}
+    (hp : inner ℝ p (normalVector (t : Real.Angle)) = c)
+    (hq : inner ℝ q (normalVector (t : Real.Angle)) = c)
+    (hs : inner ℝ s (normalVector (t : Real.Angle)) = c) :
+    segmentArea p s - segmentArea q s = segmentArea p q := by
+  rw [segmentArea, segmentArea, segmentArea, planeCrossProduct_eq_inner_frame p s t,
+    planeCrossProduct_eq_inner_frame q s t, planeCrossProduct_eq_inner_frame p q t,
+    hp, hq, hs]
+  ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Segment Area / Parametrization
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open MeasureTheory Set
+
+/-- The affine segment from `p` to `q`, bundled as a continuous BV path. -/
+def lineSegmentBVPath (p q : Point) : ContinuousBVPaths 0 1 where
+  val := Path.segment p q
+  property := by
+    constructor
+    · exact (Path.segment p q).continuous
+    · intro i
+      let C : NNReal := ⟨|q i - p i|, abs_nonneg _⟩
+      have hLip : LipschitzWith C (fun r : ℝ ↦ p i + r * (q i - p i)) := by
+        apply LipschitzWith.of_dist_le_mul
+        intro x y
+        simp only [Real.dist_eq]
+        change |(p i + x * (q i - p i)) - (p i + y * (q i - p i))| ≤
+          |q i - p i| * |x - y|
+        rw [show (p i + x * (q i - p i)) - (p i + y * (q i - p i)) =
+          (x - y) * (q i - p i) by ring, abs_mul, mul_comm]
+      have hid : BoundedVariationOn ((↑) : Set.Icc (0 : ℝ) 1 → ℝ) Set.univ := by
+        apply ((show Monotone ((↑) : Set.Icc (0 : ℝ) 1 → ℝ) from fun _ _ h ↦ h).monotoneOn
+          Set.univ).boundedVariationOn (C := 1)
+        intro t _
+        rw [abs_of_nonneg t.property.1]
+        exact t.property.2
+      have hbv := hLip.comp_boundedVariationOn hid
+      rw [show (fun t ↦ (Path.segment p q t) i) =
+          (fun r : ℝ ↦ p i + r * (q i - p i)) ∘ ((↑) : Set.Icc (0 : ℝ) 1 → ℝ) by
+        funext t
+        simp only [Path.segment_apply, AffineMap.lineMap_apply_module', PiLp.add_apply,
+          PiLp.smul_apply,
+          PiLp.sub_apply, smul_eq_mul, Function.comp_apply]
+        change (t : ℝ) * (q i - p i) + p i =
+          p i + (t : ℝ) * (q i - p i)
+        ring]
+      exact hbv
+
+/-- The affine parametrization of an oriented segment, evaluated. -/
+theorem lineSegmentBVPath_apply (p q : Point) (s : Set.Icc (0 : ℝ) 1) :
+    (lineSegmentBVPath p q).val s = (1 - (s : ℝ)) • p + (s : ℝ) • q := by
+  change Path.segment p q s = _
+  simp [Path.segment_apply, AffineMap.lineMap_apply_module']
+  module
+
+theorem curveAreaFunctional_lineSegmentBVPath (p q : Point) :
+    curveAreaFunctional (lineSegmentBVPath p q) = segmentArea p q := by
+  have hcoord (i : Fin 2) (t : Set.Icc (0 : ℝ) 1) :
+      (continuousBVCoordinate (lineSegmentBVPath p q) i).toFun t =
+        p i + (q i - p i) * (t : ℝ) := by
+    change (Path.segment p q t) i = _
+    simp [Path.segment_apply, AffineMap.lineMap_apply_module']
+    ring
+  unfold curveAreaFunctional segmentArea
+  change
+    (intervalStieltjesIntegral
+        (continuousBVCoordinate (lineSegmentBVPath p q) 1)
+        (continuousBVCoordinate (lineSegmentBVPath p q) 0).toFun Set.univ -
+      intervalStieltjesIntegral
+        (continuousBVCoordinate (lineSegmentBVPath p q) 0)
+        (continuousBVCoordinate (lineSegmentBVPath p q) 1).toFun Set.univ) / 2 =
+      planeCrossProduct p q / 2
+  rw [intervalStieltjesIntegral_affine_cross
+    (continuousBVCoordinate (lineSegmentBVPath p q) 0)
+    (continuousBVCoordinate (lineSegmentBVPath p q) 1)
+    (p 0) (q 0 - p 0) (p 1) (q 1 - p 1) (hcoord 0) (hcoord 1)]
+  simp only [planeCrossProduct]
+  ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Smooth Interval Paths
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A continuously differentiable planar path on a compact interval has continuous
+bounded-variation coordinates. -/
+def continuousBVOfContDiffOn {a b : ℝ} (f : ℝ → Point)
+    (hf : ContDiffOn ℝ 1 f (Set.Icc a b)) : ContinuousBVPaths a b := by
+  refine ⟨fun t ↦ f t, continuousOn_iff_continuous_domRestrict.mp hf.continuousOn, ?_⟩
+  obtain ⟨C, hC⟩ := hf.exists_lipschitzOnWith (by norm_num) (convex_Icc a b) isCompact_Icc
+  have hid : BoundedVariationOn (fun t : Set.Icc a b ↦ (t : ℝ)) Set.univ := by
+    apply MonotoneOn.boundedVariationOn (C := |a| + |b|)
+      (show MonotoneOn (fun t : Set.Icc a b ↦ (t : ℝ)) Set.univ from
+        fun _ _ _ _ h ↦ h)
+    intro t _
+    apply abs_le.mpr
+    constructor
+    · linarith [t.property.1, neg_abs_le a, abs_nonneg b]
+    · linarith [t.property.2, le_abs_self b, abs_nonneg a]
+  have hpath := hC.comp_boundedVariationOn (fun t _ ↦ t.property) hid
+  intro i
+  exact (EuclideanSpace.proj (𝕜 := ℝ) i).lipschitzWith.comp_boundedVariationOn hpath
+
+/-- A Lipschitz planar path on a compact interval has continuous bounded-variation coordinates. -/
+def continuousBVOfLipschitz {a b : ℝ} (f : Set.Icc a b → Point)
+    {C : NNReal} (hf : LipschitzWith C f) : ContinuousBVPaths a b := by
+  refine ⟨f, hf.continuous, ?_⟩
+  have hid : BoundedVariationOn (fun t : Set.Icc a b ↦ (t : ℝ)) Set.univ := by
+    apply MonotoneOn.boundedVariationOn (C := |a| + |b|)
+      (show MonotoneOn (fun t : Set.Icc a b ↦ (t : ℝ)) Set.univ from
+        fun _ _ _ _ h ↦ h)
+    intro t _
+    apply abs_le.mpr
+    constructor
+    · linarith [t.property.1, neg_abs_le a, abs_nonneg b]
+    · linarith [t.property.2, le_abs_self b, abs_nonneg a]
+  have hpath : BoundedVariationOn f Set.univ := by
+    exact hf.comp_boundedVariationOn
+      (show BoundedVariationOn (id : Set.Icc a b → Set.Icc a b) Set.univ from hid)
+  intro i
+  exact (EuclideanSpace.proj (𝕜 := ℝ) i).lipschitzWith.comp_boundedVariationOn hpath
+
+/-- A coordinate of an interval path has bounded variation on a closed subinterval on which the
+path agrees with a continuously differentiable function. -/
+theorem boundedVariationOn_coord_Icc_of_contDiffOn {a b : ℝ} (g : Set.Icc a b → Point)
+    {l r : ℝ} (hl : a ≤ l) (hlr : l ≤ r) (hr : r ≤ b) {F : ℝ → Point}
+    (hF : ContDiffOn ℝ 1 F (Set.Icc l r))
+    (hgF : ∀ t : Set.Icc l r, g ⟨t.val, hl.trans t.2.1, t.2.2.trans hr⟩ = F t.val)
+    (i : Fin 2) :
+    BoundedVariationOn (fun t : Set.Icc a b ↦ g t i)
+      (Set.Icc ⟨l, hl, hlr.trans hr⟩ ⟨r, hl.trans hlr, hr⟩) := by
+  set ι : Set.Icc l r → Set.Icc a b := fun t ↦ ⟨t.val, hl.trans t.2.1, t.2.2.trans hr⟩
+  have hmono : MonotoneOn ι Set.univ := fun _ _ _ _ h ↦ h
+  have himg : ι '' Set.univ =
+      Set.Icc (⟨l, hl, hlr.trans hr⟩ : Set.Icc a b) ⟨r, hl.trans hlr, hr⟩ := by
+    ext x
+    constructor
+    · rintro ⟨t, -, rfl⟩
+      exact ⟨t.2.1, t.2.2⟩
+    · intro hx
+      exact ⟨⟨x.val, hx.1, hx.2⟩, Set.mem_univ _, Subtype.ext rfl⟩
+  change eVariationOn _ _ ≠ ⊤
+  rw [← himg, ← eVariationOn.comp_eq_of_monotoneOn _ ι hmono]
+  have hfun : ((fun t : Set.Icc a b ↦ g t i) ∘ ι) = fun t : Set.Icc l r ↦ F t.val i := by
+    funext t
+    exact congrArg (fun p : Point ↦ p i) (hgF t)
+  rw [hfun]
+  exact (continuousBVOfContDiffOn F hF).property.2 i
+
+/-- Gluing two continuously differentiable pieces along a shared endpoint gives a continuous
+path of bounded variation. -/
+def continuousBVOfContDiffOnIccUnionIcc {a b c : ℝ} (f : ℝ → Point) (hab : a ≤ b) (hbc : b ≤ c)
+    (h₁ : ContDiffOn ℝ 1 f (Set.Icc a b)) (h₂ : ContDiffOn ℝ 1 f (Set.Icc b c)) :
+    ContinuousBVPaths a c := by
+  have hf : ContinuousOn f (Set.Icc a c) := by
+    rw [← Set.Icc_union_Icc_eq_Icc hab hbc]
+    exact h₁.continuousOn.union_of_isClosed h₂.continuousOn isClosed_Icc isClosed_Icc
+  refine ⟨fun t ↦ f t, continuousOn_iff_continuous_domRestrict.mp hf, fun i ↦ ?_⟩
+  refine BoundedVariationOn.univ_of_Icc_endpoints (hab.trans hbc)
+    (BoundedVariationOn.Icc_union_Icc
+      (show (⟨a, le_rfl, hab.trans hbc⟩ : Set.Icc a c) ≤ ⟨b, hab, hbc⟩ from hab)
+      (show (⟨b, hab, hbc⟩ : Set.Icc a c) ≤ ⟨c, hab.trans hbc, le_rfl⟩ from hbc)
+      (boundedVariationOn_coord_Icc_of_contDiffOn _ le_rfl hab hbc h₁ (fun _ ↦ rfl) i)
+      (boundedVariationOn_coord_Icc_of_contDiffOn _ hab hbc le_rfl h₂ (fun _ ↦ rfl) i))
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Radial Loop
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace MovingSofa
+
+/-- A Lipschitz map on the unit circle induces a continuous BV loop in increasing angular order. -/
+def radialBVLoop (f : {u : Point | ‖u‖ = 1} → Point) {C : NNReal}
+    (hf : LipschitzWith C f) : ContinuousBVPaths 0 (2 * Real.pi) := by
+  let n : Set.Icc (0 : ℝ) (2 * Real.pi) → {u : Point | ‖u‖ = 1} :=
+    fun t ↦ ⟨normalVector ((t : ℝ) : Real.Angle), norm_normalVector_real t⟩
+  have hn : LipschitzWith 1 n := by
+    apply LipschitzWith.of_dist_le_mul
+    intro t u
+    exact lipschitzWith_normalVector_real.dist_le_mul t.val u.val
+  exact continuousBVOfLipschitz (f ∘ n) (hf.comp hn)
+
+/-- The radial BV loop evaluates by applying the circle map to the angular normal. -/
+lemma radialBVLoop_apply (f : {u : Point | ‖u‖ = 1} → Point) {C : NNReal}
+    (hf : LipschitzWith C f) (t : Set.Icc (0 : ℝ) (2 * Real.pi)) :
+    (radialBVLoop f hf).val t =
+      f ⟨normalVector ((t : ℝ) : Real.Angle), norm_normalVector_real t⟩ := rfl
+
+/-- The radial BV loop has equal endpoints. -/
+lemma radialBVLoop_closed (f : {u : Point | ‖u‖ = 1} → Point) {C : NNReal}
+    (hf : LipschitzWith C f) :
+    (radialBVLoop f hf).val ⟨0, le_rfl, by positivity⟩ =
+      (radialBVLoop f hf).val ⟨2 * Real.pi, by positivity, le_rfl⟩ := by
+  rw [radialBVLoop_apply, radialBVLoop_apply]
+  apply congrArg f
+  apply Subtype.ext
+  ext i
+  fin_cases i <;> simp [normalVector, frame]
+
+/-- An injective circle map gives a radial loop injective before its final endpoint. -/
+lemma radialBVLoop_injOn (f : {u : Point | ‖u‖ = 1} → Point) {C : NNReal}
+    (hf : LipschitzWith C f) (hinj : Function.Injective f) :
+    Set.InjOn (radialBVLoop f hf).val {t | (t : ℝ) < 2 * Real.pi} := by
+  intro t ht u hu h
+  let : Fact (0 < 2 * Real.pi) := ⟨by positivity⟩
+  rw [radialBVLoop_apply, radialBVLoop_apply] at h
+  have hn := congrArg Subtype.val (hinj h)
+  have ha : (t.val : Real.Angle) = (u.val : Real.Angle) := by
+    apply Real.Angle.cos_sin_inj
+    · exact congrFun (congrArg WithLp.ofLp hn) 0
+    · exact congrFun (congrArg WithLp.ofLp hn) 1
+  apply Subtype.ext
+  exact (AddCircle.coe_eq_coe_iff_of_mem_Ico
+    (show t.val ∈ Set.Ico 0 (0 + 2 * Real.pi) from ⟨t.property.1, by simpa using ht⟩)
+    (show u.val ∈ Set.Ico 0 (0 + 2 * Real.pi) from ⟨u.property.1, by simpa using hu⟩)).mp ha
+
+/-- The radial BV loop has the same range as the underlying unit-circle map. -/
+lemma range_radialBVLoop (f : {u : Point | ‖u‖ = 1} → Point) {C : NNReal}
+    (hf : LipschitzWith C f) : Set.range (radialBVLoop f hf).val = Set.range f := by
+  apply Set.Subset.antisymm
+  · rintro p ⟨t, rfl⟩
+    exact ⟨_, (radialBVLoop_apply f hf t).symm⟩
+  · rintro p ⟨u, rfl⟩
+    obtain ⟨θ, hθ⟩ := exists_angle_normalVector_eq u.property
+    let : Fact (0 < 2 * Real.pi) := ⟨by positivity⟩
+    let t := AddCircle.equivIco (2 * Real.pi) 0 θ
+    have ht : (t.val : Real.Angle) = θ := AddCircle.coe_equivIco
+    refine ⟨⟨t.val, t.property.1, ?_⟩, ?_⟩
+    · simpa using t.property.2.le
+    · rw [radialBVLoop_apply]
+      apply congrArg f
+      apply Subtype.ext
+      exact (congrArg normalVector ht).trans hθ
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# A Stieltjes chain rule with a local quadratic remainder
+
+The hypothesis of `ContinuousBVPaths.stieltjes_chain_rule_of_local_quadratic_remainder`
+quantifies the remainder only over parameter pairs closer than a fixed positive threshold.
+This is what a locally defined argument branch supplies: no single plane function has to be
+named, and no constant is needed across a branch cut.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Set
+
+namespace MovingSofa
+
+/-- If a real parameter function `A` has increments matching the signed pair
+`D₁ dγ₁ - D₀ dγ₀` up to a quadratic remainder on all parameter pairs closer than a fixed
+positive threshold, then its endpoint increment is the corresponding difference of
+coordinate Stieltjes integrals. -/
+theorem ContinuousBVPaths.stieltjes_chain_rule_of_local_quadratic_remainder
+    {a b : ℝ} (hab : a ≤ b) (x : ContinuousBVPaths a b)
+    (hBV : BoundedVariationOn x.val univ) (A : Set.Icc a b → ℝ) (D : Point → Fin 2 → ℝ)
+    (hD : ∀ i, Continuous (fun t ↦ D (x.val t) i)) {C : ℝ} (hC : 0 ≤ C)
+    {δ₀ : ℝ} (hδ₀ : 0 < δ₀)
+    (herror : ∀ t u : Set.Icc a b, (t : ℝ) ≤ (u : ℝ) → (u : ℝ) - (t : ℝ) < δ₀ →
+      |A u - A t -
+        (D (x.val t) 1 * (x.val u 1 - x.val t 1) -
+          D (x.val t) 0 * (x.val u 0 - x.val t 0))| ≤ C * ‖x.val u - x.val t‖ ^ 2) :
+    A ⟨b, hab, le_rfl⟩ - A ⟨a, le_rfl, hab⟩ =
+      intervalStieltjesIntegral (continuousBVCoordinate x 1) (fun t ↦ D (x.val t) 1) univ -
+      intervalStieltjesIntegral (continuousBVCoordinate x 0) (fun t ↦ D (x.val t) 0) univ := by
+  obtain ⟨cuts, hcuts, hzero, hlast, hmesh⟩ :=
+    Set.Icc.exists_partitions_mesh_tendsto_zero hab
+  let N : ℕ → ℕ := fun k ↦ k + 1
+  have htaylor := hBV.tendsto_sum_of_local_quadratic_remainder hab x.property.1 A
+    (fun t v ↦ D (x.val t) 1 * v 1 - D (x.val t) 0 * v 0) hC hδ₀ herror
+    N cuts hcuts hzero hlast hmesh
+  have hcoord (i : Fin 2) := tendsto_stieltjesSum_of_mesh_tendsto_zero
+    (continuousBVCoordinate x i) ((PiLp.continuous_apply 2 _ i).comp x.property.1)
+    (hD i) N cuts hcuts hzero hlast hmesh
+  have hsum := (hcoord 1).sub (hcoord 0)
+  simp only [← Finset.sum_sub_distrib, continuousBVCoordinate] at hsum
+  exact tendsto_nhds_unique htaylor hsum
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Area.Foundations.Development002`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Area.MonotoneRoof`.
+* `Area.ThreePieceRoof`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Area / Monotone Roof
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Recover the vertical coordinate using a chosen inverse of the path’s horizontal
+coordinate. -/
+def monotoneRoofHeight {a b : ℝ} (hab : a < b) (γ : ContinuousBVPaths a b)
+    (s : ℝ) : ℝ := by
+  classical
+  letI : Nonempty (Set.Icc a b) := ⟨⟨a, le_rfl, hab.le⟩⟩
+  exact γ.val (Function.invFun (fun t ↦ γ.val t 0) s) 1
+
+/-- The closed region between the horizontal axis and the path’s roof graph. -/
+def monotoneRoofRegion {a b : ℝ} (hab : a < b) (γ : ContinuousBVPaths a b) : Set Point :=
+  {p | γ.val ⟨a, le_rfl, hab.le⟩ 0 ≤ p 0 ∧
+    p 0 ≤ γ.val ⟨b, hab.le, le_rfl⟩ 0 ∧ 0 ≤ p 1 ∧
+    p 1 ≤ monotoneRoofHeight hab γ (p 0)}
+
+/-- Traverse the roof path backwards and return along its horizontal base. -/
+def monotoneRoofLoop {a b : ℝ} (hab : a < b) (γ : ContinuousBVPaths a b)
+    (s : Set.Icc (0 : ℝ) 2) : Point :=
+  if hs : s.val ≤ 1 then
+    γ.val ⟨b - (b - a) * s.val, by
+      constructor
+      · nlinarith [s.property.1]
+      · nlinarith [s.property.1]⟩
+  else
+    (2 - s.val) • γ.val ⟨a, le_rfl, hab.le⟩ +
+      (s.val - 1) • γ.val ⟨b, hab.le, le_rfl⟩
+
+private theorem monotoneRoofLoop_concatenation {a b : ℝ} (hab : a < b)
+    (γ γrev : ContinuousBVPaths a b)
+    (hγrev : γrev.val = γ.val ∘ Set.Icc.reverse hab.le)
+    (Γ : ContinuousBVPaths 0 2) (hΓval : Γ.val = monotoneRoofLoop hab γ) :
+    IsPathConcatenation (⟨0, 2, by norm_num, Γ⟩ : RectifiablePathData)
+      ![(⟨a, b, hab.le, γrev⟩ : RectifiablePathData),
+        ⟨0, 1, zero_le_one, lineSegmentBVPath
+          (γ.val ⟨a, le_rfl, hab.le⟩) (γ.val ⟨b, hab.le, le_rfl⟩)⟩] := by
+  let A : Set.Icc a b := ⟨a, le_rfl, hab.le⟩
+  let B : Set.Icc a b := ⟨b, hab.le, le_rfl⟩
+  have hba : (0 : ℝ) < b - a := sub_pos.mpr hab
+  refine ⟨by norm_num,
+    ![⟨0, by norm_num⟩, ⟨1, by norm_num⟩, ⟨2, by norm_num⟩], ?_, ?_, ?_, ?_⟩
+  · intro i j hij
+    fin_cases i <;> fin_cases j <;>
+      first
+        | rfl
+        | (exact absurd hij (by decide))
+        | (refine Subtype.mk_le_mk.mpr ?_; norm_num)
+  · rfl
+  · rfl
+  · intro i
+    fin_cases i
+    · change ∃ (ϕ : Set.Icc (0 : ℝ) 1 → Set.Icc (0 : ℝ) 1)
+          (ψ : Set.Icc (0 : ℝ) 1 → Set.Icc a b),
+          Continuous ϕ ∧ Monotone ϕ ∧ Function.Surjective ϕ ∧
+            Continuous ψ ∧ Monotone ψ ∧ Function.Surjective ψ ∧
+            ∀ u, Γ.val ⟨(ϕ u : ℝ), (ϕ u).property.1,
+              le_trans (ϕ u).property.2 (by norm_num)⟩ = γrev.val (ψ u)
+      refine ⟨fun u ↦ ⟨(u : ℝ), u.property.1, u.property.2⟩,
+        fun u ↦ ⟨a + (b - a) * (u : ℝ),
+          by linarith only [mul_nonneg hba.le u.property.1],
+          by linarith only [mul_le_mul_of_nonneg_left u.property.2 hba.le]⟩,
+        ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
+      · exact continuous_subtype_val.subtype_mk _
+      · exact fun x y hxy ↦ hxy
+      · exact fun z ↦ ⟨⟨(z : ℝ), z.property.1, z.property.2⟩, Subtype.ext rfl⟩
+      · exact (continuous_const.add
+          (continuous_const.mul continuous_subtype_val)).subtype_mk _
+      · intro x y hxy
+        have hxy' : (x : ℝ) ≤ (y : ℝ) := hxy
+        change a + (b - a) * (x : ℝ) ≤ a + (b - a) * (y : ℝ)
+        linarith only [mul_le_mul_of_nonneg_left hxy' hba.le]
+      · intro t
+        refine ⟨⟨((t : ℝ) - a) / (b - a), ?_, ?_⟩, ?_⟩
+        · exact div_nonneg (by linarith only [t.property.1]) hba.le
+        · rw [div_le_one hba]
+          linarith only [t.property.2]
+        · apply Subtype.ext
+          change a + (b - a) * (((t : ℝ) - a) / (b - a)) = (t : ℝ)
+          field_simp
+          ring
+      · intro u
+        have h0 : (0 : ℝ) ≤ (u : ℝ) := u.property.1
+        have h1 : (u : ℝ) ≤ 1 := u.property.2
+        rw [hΓval, hγrev]
+        change monotoneRoofLoop hab γ ⟨(u : ℝ), h0, by linarith⟩ =
+          γ.val (Set.Icc.reverse hab.le ⟨a + (b - a) * (u : ℝ),
+            by linarith only [mul_nonneg hba.le h0],
+            by linarith only [mul_le_mul_of_nonneg_left h1 hba.le]⟩)
+        unfold monotoneRoofLoop
+        rw [dite_eq_left (show ((⟨(u : ℝ), h0, by linarith⟩ : Set.Icc (0 : ℝ) 2) : ℝ) ≤ 1
+          from h1)]
+        exact congrArg (fun t ↦ γ.val t) (Subtype.ext (by
+          change b - (b - a) * (u : ℝ) = a + b - (a + (b - a) * (u : ℝ))
+          ring))
+    · change ∃ (ϕ : Set.Icc (0 : ℝ) 1 → Set.Icc (1 : ℝ) 2)
+          (ψ : Set.Icc (0 : ℝ) 1 → Set.Icc (0 : ℝ) 1),
+          Continuous ϕ ∧ Monotone ϕ ∧ Function.Surjective ϕ ∧
+            Continuous ψ ∧ Monotone ψ ∧ Function.Surjective ψ ∧
+            ∀ u, Γ.val ⟨(ϕ u : ℝ), le_trans (by norm_num) (ϕ u).property.1,
+              (ϕ u).property.2⟩ =
+              (lineSegmentBVPath (γ.val A) (γ.val B)).val (ψ u)
+      refine ⟨fun u ↦ ⟨1 + (u : ℝ), by linarith [u.property.1], by
+          linarith [u.property.2]⟩,
+        fun u ↦ u, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
+      · exact (continuous_const.add continuous_subtype_val).subtype_mk _
+      · intro x y hxy
+        have hxy' : (x : ℝ) ≤ (y : ℝ) := hxy
+        change (1 : ℝ) + (x : ℝ) ≤ 1 + (y : ℝ)
+        linarith
+      · intro z
+        refine ⟨⟨(z : ℝ) - 1, by linarith [z.property.1], by
+          linarith [z.property.2]⟩, ?_⟩
+        apply Subtype.ext
+        change (1 : ℝ) + ((z : ℝ) - 1) = (z : ℝ)
+        ring
+      · exact continuous_id
+      · exact fun x y hxy ↦ hxy
+      · exact fun z ↦ ⟨z, rfl⟩
+      · intro u
+        have h0 : (0 : ℝ) ≤ (u : ℝ) := u.property.1
+        have h1 : (u : ℝ) ≤ 1 := u.property.2
+        rw [hΓval]
+        change monotoneRoofLoop hab γ ⟨1 + (u : ℝ), by linarith, by linarith⟩ =
+          Path.segment (γ.val A) (γ.val B) u
+        unfold monotoneRoofLoop
+        rcases eq_or_lt_of_le h0 with hu0 | hu0
+        · rw [dite_eq_left (show ((⟨1 + (u : ℝ), by linarith, by linarith⟩ :
+            Set.Icc (0 : ℝ) 2) : ℝ) ≤ 1 from by simp [← hu0])]
+          have huval : (u : ℝ) = 0 := hu0.symm
+          have hseg : Path.segment (γ.val A) (γ.val B) u = γ.val A := by
+            rw [Path.segment_apply]
+            rw [show (u : ℝ) = 0 from huval]
+            simp
+          rw [hseg]
+          exact congrArg (fun t ↦ γ.val t) (Subtype.ext (by
+            change b - (b - a) * (1 + (u : ℝ)) = a
+            rw [huval]
+            ring))
+        · rw [dite_eq_right (show ¬ ((⟨1 + (u : ℝ), by linarith, by linarith⟩ :
+            Set.Icc (0 : ℝ) 2) : ℝ) ≤ 1 from by
+              change ¬ (1 + (u : ℝ) ≤ 1)
+              linarith)]
+          change (2 - (1 + (u : ℝ))) • γ.val A + ((1 + (u : ℝ)) - 1) • γ.val B =
+            Path.segment (γ.val A) (γ.val B) u
+          rw [Path.segment_apply, AffineMap.lineMap_apply_module']
+          module
+
+private theorem monotoneRoof_area {a b : ℝ} (hab : a < b)
+    (γ : ContinuousBVPaths a b) (hk : StrictMono (fun t ↦ γ.val t 0))
+    (hh : ∀ t, 0 ≤ γ.val t 1)
+    (ha : γ.val ⟨a, le_rfl, hab.le⟩ 1 = 0)
+    (hb : γ.val ⟨b, hab.le, le_rfl⟩ 1 = 0) :
+    ClassicalResults.area (monotoneRoofRegion hab γ) = -curveAreaFunctional γ ∧
+      ClassicalResults.area (monotoneRoofRegion hab γ) =
+        ∫ s in γ.val ⟨a, le_rfl, hab.le⟩ 0..γ.val ⟨b, hab.le, le_rfl⟩ 0,
+          monotoneRoofHeight hab γ s := by
+  classical
+  set A : Set.Icc a b := ⟨a, le_rfl, hab.le⟩ with hAdef
+  set B : Set.Icc a b := ⟨b, hab.le, le_rfl⟩ with hBdef
+  set c : ℝ := γ.val A 0 with hcdef
+  set d : ℝ := γ.val B 0 with hddef
+  have hAB : A < B := hab
+  have hcd : c < d := hk hAB
+  -- the horizontal coordinate as a map onto `[c, d]`
+  have hγ0 : Continuous fun t : Set.Icc a b ↦ γ.val t 0 :=
+    (PiLp.continuous_apply 2 _ 0).comp γ.property.1
+  have hγ1 : Continuous fun t : Set.Icc a b ↦ γ.val t 1 :=
+    (PiLp.continuous_apply 2 _ 1).comp γ.property.1
+  have hmem : ∀ t : Set.Icc a b, γ.val t 0 ∈ Set.Icc c d := fun t ↦
+    ⟨hk.monotone (show A ≤ t from t.property.1),
+      hk.monotone (show t ≤ B from t.property.2)⟩
+  set kmap : Set.Icc a b → Set.Icc c d := fun t ↦ ⟨γ.val t 0, hmem t⟩ with hkmapdef
+  have hkc : Continuous kmap := hγ0.subtype_mk _
+  have hkinj : Function.Injective kmap := fun x y h ↦ hk.injective (congrArg Subtype.val h)
+  have : PreconnectedSpace (Set.Icc a b) := Subtype.preconnectedSpace isPreconnected_Icc
+  have hksurj : Function.Surjective kmap := by
+    intro s
+    obtain ⟨t, ht⟩ := intermediate_value_univ A B hγ0 s.property
+    exact ⟨t, Subtype.ext ht⟩
+  -- the continuous monotone inverse `φ = k⁻¹`
+  set khom : Set.Icc a b ≃ₜ Set.Icc c d :=
+    Continuous.homeoOfEquivCompactToT2 (f := Equiv.ofBijective kmap ⟨hkinj, hksurj⟩) hkc
+    with hkhomdef
+  set φ : Set.Icc c d → Set.Icc a b := ⇑khom.symm with hφdef
+  have hkhom_apply : ∀ t, khom t = kmap t := fun _ ↦ rfl
+  have hφk : ∀ s : Set.Icc c d, γ.val (φ s) 0 = (s : ℝ) := by
+    intro s
+    have := khom.apply_symm_apply s
+    rw [hkhom_apply] at this
+    exact congrArg Subtype.val this
+  have hφc : Continuous φ := khom.symm.continuous
+  have hφs : Function.Surjective φ := khom.symm.surjective
+  have hφm : Monotone φ := by
+    intro s t hst
+    by_contra hcon
+    have hlt : φ t < φ s := lt_of_not_ge hcon
+    have hlt2 := hk hlt
+    simp only [hφk] at hlt2
+    exact absurd hst (not_le.mpr hlt2)
+  -- the roof height, as a globally continuous function on the line
+  set g : ℝ → ℝ := fun s ↦ γ.val (φ (Set.projIcc c d hcd.le s)) 1 with hgdef
+  have hgc : Continuous g := hγ1.comp (hφc.comp continuous_projIcc)
+  have hgnonneg : ∀ s, 0 ≤ g s := fun s ↦ hh _
+  have hheight : ∀ s ∈ Set.Icc c d, monotoneRoofHeight hab γ s = g s := by
+    intro s hs
+    have hex : ∃ t : Set.Icc a b, γ.val t 0 = s := ⟨φ ⟨s, hs⟩, hφk ⟨s, hs⟩⟩
+    have hinv := @Function.invFun_eq _ _ ⟨A⟩ (fun t ↦ γ.val t 0) s hex
+    have hpt : @Function.invFun _ _ ⟨A⟩ (fun t ↦ γ.val t 0) s = φ ⟨s, hs⟩ :=
+      hk.injective (by simpa using hinv.trans (hφk ⟨s, hs⟩).symm)
+    change γ.val (@Function.invFun _ _ ⟨A⟩ (fun t ↦ γ.val t 0) s) 1 = g s
+    rw [hpt]
+    simp [hgdef, Set.projIcc_of_mem _ hs]
+  -- the roof region is the closed subgraph of `g` over `[c, d]`
+  have hregion : monotoneRoofRegion hab γ =
+      {p : Point | p 0 ∈ Set.Icc c d ∧
+        p 1 ∈ Set.Icc ((fun _ : ℝ ↦ (0 : ℝ)) (p 0)) (g (p 0))} := by
+    ext p
+    constructor
+    · rintro ⟨h1, h2, h3, h4⟩
+      exact ⟨⟨h1, h2⟩, h3, by rwa [hheight (p 0) ⟨h1, h2⟩] at h4⟩
+    · rintro ⟨⟨h1, h2⟩, h3, h4⟩
+      exact ⟨h1, h2, h3, by rwa [hheight (p 0) ⟨h1, h2⟩]⟩
+  -- the planar volume of a closed vertical subgraph
+  have hvolVertical : ∀ (f₀ g₀ : ℝ → ℝ) (S : Set ℝ), Measurable f₀ → Measurable g₀ →
+      MeasurableSet S → MeasureTheory.IntegrableOn f₀ S → MeasureTheory.IntegrableOn g₀ S →
+      (∀ x ∈ S, f₀ x ≤ g₀ x) →
+      MeasureTheory.volume {p : Point | p 0 ∈ S ∧ p 1 ∈ Set.Icc (f₀ (p 0)) (g₀ (p 0))} =
+        ENNReal.ofReal (∫ x in S, (g₀ - f₀) x) := by
+    intro f₀ g₀ S hf hg hS hfi hgi hfg
+    have hT : MeasurableSet {q : ℝ × ℝ | q.1 ∈ S ∧ q.2 ∈ Set.Icc (f₀ q.1) (g₀ q.1)} :=
+      measurableSet_region_between_cc hf hg hS
+    have hpre : {p : Point | p 0 ∈ S ∧ p 1 ∈ Set.Icc (f₀ (p 0)) (g₀ (p 0))} =
+        (fun p : Point ↦ (p 0, p 1)) ⁻¹'
+          {q : ℝ × ℝ | q.1 ∈ S ∧ q.2 ∈ Set.Icc (f₀ q.1) (g₀ q.1)} := rfl
+    rw [hpre, EuclideanSpace.volume_preserving_finTwoCoordinates.measure_preimage
+      hT.nullMeasurableSet]
+    rw [volume_setOf_mem_Icc_eq_volume_regionBetween hf hg hS]
+    change (MeasureTheory.volume.prod MeasureTheory.volume) (regionBetween f₀ g₀ S) = _
+    rw [volume_regionBetween_eq_integral hfi hgi hS hfg]
+  have hgint : MeasureTheory.IntegrableOn g (Set.Icc c d) :=
+    hgc.continuousOn.integrableOn_compact isCompact_Icc
+  have hIcc : (∫ x in Set.Icc c d, g x) = ∫ s in c..d, g s := by
+    rw [intervalIntegral.integral_of_le hcd.le, MeasureTheory.integral_Icc_eq_integral_Ioc]
+  have harea : ClassicalResults.area (monotoneRoofRegion hab γ) = ∫ s in c..d, g s := by
+    rw [ClassicalResults.area, hregion,
+      hvolVertical (fun _ ↦ 0) g (Set.Icc c d) measurable_const hgc.measurable
+        measurableSet_Icc (MeasureTheory.integrableOn_zero) hgint
+        (fun x _ ↦ hgnonneg x)]
+    rw [ENNReal.toReal_ofReal]
+    · simp only [Pi.sub_apply, sub_zero]
+      exact hIcc
+    · exact MeasureTheory.integral_nonneg (fun x ↦ by simp [hgnonneg x])
+  -- the two coordinate Stieltjes drivers
+  have hBVext : ∀ {u v : ℝ} (F G : RightContinuousIntervalBV u v),
+      F.toFun = G.toFun → F = G := by
+    intro u v F G hFG
+    cases F
+    cases G
+    simp only at hFG
+    subst hFG
+    rfl
+  set K := continuousBVCoordinate γ 0 with hKdef
+  set H := continuousBVCoordinate γ 1 with hHdef
+  have hKcont : Continuous K.toFun := hγ0
+  have hHcont : Continuous H.toFun := hγ1
+  -- the identity driver on `[c, d]`, with Lebesgue density one
+  obtain ⟨Q, hQfun, hQdens⟩ := exists_intervalBV_of_hasDerivAt hcd.le (fun x ↦ x)
+    (fun _ ↦ (1 : ℝ)) (fun t ↦ hasDerivAt_id t) continuous_const
+  have hstieltjes : intervalStieltjesIntegral K H.toFun Set.univ =
+      intervalStieltjesIntegral Q (H.toFun ∘ φ) Set.univ := by
+    rw [intervalStieltjesIntegral_comp_monotone_surjective hab.le hcd.le K hKcont
+      H.toFun hHcont φ hφc hφm hφs]
+    congr 1
+    apply hBVext
+    funext s
+    change γ.val (φ s) 0 = Q.toFun s
+    rw [hQfun s, hφk s]
+  have hroofint : intervalStieltjesIntegral K H.toFun Set.univ = ∫ s in c..d, g s := by
+    rw [hstieltjes, intervalStieltjesIntegral_eq_integral_mul_of_density Q hQdens
+      (hHcont.comp hφc) Set.univ MeasurableSet.univ, MeasureTheory.Measure.restrict_univ]
+    have hfun : (fun t : Set.Icc c d ↦ (H.toFun ∘ φ) t * (1 : ℝ)) =
+        fun t : Set.Icc c d ↦ g (t : ℝ) := by
+      funext t
+      simp only [Function.comp_apply, mul_one, hgdef, Set.projIcc_val]
+      rfl
+    rw [hfun, MeasureTheory.integral_subtype_comap measurableSet_Icc g, hIcc]
+  -- integration by parts identifies the curve area with the roof integral
+  have hparts := intervalStieltjes_integration_by_parts_of_continuous a b hab.le K H
+    hKcont hHcont
+  have hends : K.toFun B * H.toFun B - K.toFun A * H.toFun A = 0 := by
+    change γ.val B 0 * γ.val B 1 - γ.val A 0 * γ.val A 1 = 0
+    rw [ha, hb]
+    ring
+  have hJγ : curveAreaFunctional γ = -intervalStieltjesIntegral K H.toFun Set.univ := by
+    have hform : curveAreaFunctional γ =
+        (intervalStieltjesIntegral H K.toFun Set.univ -
+          intervalStieltjesIntegral K H.toFun Set.univ) / 2 := rfl
+    rw [hform]
+    rw [hends] at hparts
+    linarith
+  have hheightint : (∫ s in c..d, monotoneRoofHeight hab γ s) = ∫ s in c..d, g s := by
+    apply intervalIntegral.integral_congr
+    intro s hs
+    exact hheight s (by rwa [Set.uIcc_of_le hcd.le] at hs)
+  constructor
+  · rw [hJγ, neg_neg, hroofint, harea]
+  · exact harea.trans hheightint.symm
+
+theorem monotone_roof_signed_area {a b : ℝ} (hab : a < b)
+    (γ : ContinuousBVPaths a b) (hk : StrictMono (fun t ↦ γ.val t 0))
+    (hh : ∀ t, 0 ≤ γ.val t 1)
+    (ha : γ.val ⟨a, le_rfl, hab.le⟩ 1 = 0)
+    (hb : γ.val ⟨b, hab.le, le_rfl⟩ 1 = 0) :
+    ∃ Γ : ContinuousBVPaths 0 2, Γ.val = monotoneRoofLoop hab γ ∧
+      curveAreaFunctional Γ = ClassicalResults.area (monotoneRoofRegion hab γ) ∧
+      ClassicalResults.area (monotoneRoofRegion hab γ) =
+        ∫ s in γ.val ⟨a, le_rfl, hab.le⟩ 0..γ.val ⟨b, hab.le, le_rfl⟩ 0,
+          monotoneRoofHeight hab γ s := by
+  classical
+  let A : Set.Icc a b := ⟨a, le_rfl, hab.le⟩
+  let B : Set.Icc a b := ⟨b, hab.le, le_rfl⟩
+  obtain ⟨harea, hheight⟩ := monotoneRoof_area hab γ hk hh ha hb
+  -- the reversed roof, as a path on `[a, b]`
+  have hJrev := curveArea_comp_reverse hab.le γ
+  set γrev : ContinuousBVPaths a b :=
+    ⟨γ.val ∘ Set.Icc.reverse hab.le,
+      γ.property.1.comp (Set.Icc.continuous_reverse hab.le), fun i ↦
+        BoundedVariationOn.comp_antitone_surjective_Icc hab.le (γ.property.2 i)
+          (Set.Icc.antitone_reverse hab.le) (Set.Icc.surjective_reverse hab.le)⟩ with hγrevdef
+  change curveAreaFunctional γrev = -curveAreaFunctional γ at hJrev
+  -- the antitone reparametrization of `[0, 2]` onto `[a, b]`, pausing on `[1, 2]`
+  set revmap : Set.Icc (0 : ℝ) 2 → Set.Icc a b :=
+    fun s ↦ Set.projIcc a b hab.le (b - (b - a) * (s : ℝ)) with hrevmapdef
+  have hrevc : Continuous revmap :=
+    continuous_projIcc.comp (by fun_prop)
+  have hrevanti : Antitone revmap := by
+    intro s t hst
+    have hst' : (s : ℝ) ≤ (t : ℝ) := hst
+    have hmul := mul_le_mul_of_nonneg_left hst' (sub_nonneg.mpr hab.le)
+    change Set.projIcc a b hab.le (b - (b - a) * (t : ℝ)) ≤
+      Set.projIcc a b hab.le (b - (b - a) * (s : ℝ))
+    exact Set.monotone_projIcc hab.le (by linarith)
+  have hrevsurj : Function.Surjective revmap := by
+    intro t
+    have hba : (0 : ℝ) < b - a := by linarith
+    refine ⟨⟨(b - (t : ℝ)) / (b - a), ?_, ?_⟩, ?_⟩
+    · exact div_nonneg (by linarith [t.property.2]) hba.le
+    · rw [div_le_iff₀ hba]
+      nlinarith [t.property.1]
+    · have hval : b - (b - a) * ((b - (t : ℝ)) / (b - a)) = (t : ℝ) := by
+        field_simp
+        ring
+      change Set.projIcc a b hab.le (b - (b - a) * ((b - (t : ℝ)) / (b - a))) = t
+      rw [hval, Set.projIcc_val]
+  -- the base segment piece, paused on `[0, 1]`
+  set w : Point := γ.val B - γ.val A with hwdef
+  set ρ : Set.Icc (0 : ℝ) 2 → Set.Icc (0 : ℝ) 1 :=
+    fun s ↦ Set.projIcc 0 1 zero_le_one ((s : ℝ) - 1) with hρdef
+  have hρc : Continuous ρ := continuous_projIcc.comp (by fun_prop)
+  have hρm : Monotone ρ := by
+    intro s t hst
+    have hst' : (s : ℝ) ≤ (t : ℝ) := hst
+    change Set.projIcc 0 1 zero_le_one ((s : ℝ) - 1) ≤
+      Set.projIcc 0 1 zero_le_one ((t : ℝ) - 1)
+    exact Set.monotone_projIcc zero_le_one (by linarith)
+  have hρs : Function.Surjective ρ := by
+    intro u
+    refine ⟨⟨(u : ℝ) + 1, ?_, ?_⟩, ?_⟩
+    · linarith [u.property.1]
+    · linarith [u.property.2]
+    · change Set.projIcc 0 1 zero_le_one ((u : ℝ) + 1 - 1) = u
+      rw [show ((u : ℝ) + 1 - 1) = (u : ℝ) from by ring, Set.projIcc_val]
+  obtain ⟨Bpath, hBpathval⟩ := continuousBVPaths_comp_monotone_surjective
+    (by norm_num : (0 : ℝ) ≤ 1) (lineSegmentBVPath 0 w) ρ hρc hρm hρs
+  set Apath : ContinuousBVPaths 0 2 :=
+    ⟨γ.val ∘ revmap, γ.property.1.comp hrevc, fun i ↦
+      BoundedVariationOn.comp_antitone_surjective_Icc hab.le (γ.property.2 i) hrevanti
+        hrevsurj⟩ with hApathdef
+  set Γ : ContinuousBVPaths 0 2 := Apath + Bpath with hΓdef
+  have hΓval : Γ.val = monotoneRoofLoop hab γ := by
+    funext s
+    have h0 : (0 : ℝ) ≤ (s : ℝ) := s.property.1
+    have h2 : (s : ℝ) ≤ 2 := s.property.2
+    have hval : Γ.val s = γ.val (revmap s) + Path.segment (0 : Point) w (ρ s) := by
+      change Apath.val s + Bpath.val s = _
+      rw [hBpathval]
+      rfl
+    rw [hval]
+    unfold monotoneRoofLoop
+    split_ifs with hs
+    · have hrevs : revmap s = ⟨b - (b - a) * (s : ℝ),
+          by constructor <;> nlinarith⟩ := by
+        change Set.projIcc a b hab.le (b - (b - a) * (s : ℝ)) = _
+        exact Set.projIcc_of_mem hab.le _
+      have hρ0 : ρ s = ⟨0, by norm_num⟩ := by
+        change Set.projIcc 0 1 zero_le_one ((s : ℝ) - 1) = _
+        exact Set.projIcc_of_le_left zero_le_one (by linarith)
+      rw [hrevs, hρ0]
+      simp
+    · rw [not_le] at hs
+      have hrevs : revmap s = A := by
+        change Set.projIcc a b hab.le (b - (b - a) * (s : ℝ)) = _
+        rw [Set.projIcc_of_le_left hab.le (by nlinarith)]
+      have hρ1 : ρ s = ⟨(s : ℝ) - 1, by constructor <;> linarith⟩ := by
+        change Set.projIcc 0 1 zero_le_one ((s : ℝ) - 1) = _
+        exact Set.projIcc_of_mem zero_le_one _
+      rw [hrevs, hρ1]
+      change γ.val A + AffineMap.lineMap (0 : Point) w ((s : ℝ) - 1) =
+        (2 - (s : ℝ)) • γ.val A + ((s : ℝ) - 1) • γ.val B
+      rw [AffineMap.lineMap_apply_module', hwdef]
+      module
+  -- additivity of the curve area along the cut at `s = 1`
+  have hΓ2 : (0 : ℝ) ≤ 2 := by norm_num
+  have hba : (0 : ℝ) < b - a := by linarith
+  have hconcat : IsPathConcatenation (⟨0, 2, hΓ2, Γ⟩ : RectifiablePathData)
+      ![(⟨a, b, hab.le, γrev⟩ : RectifiablePathData),
+        ⟨0, 1, zero_le_one, lineSegmentBVPath (γ.val A) (γ.val B)⟩] :=
+    monotoneRoofLoop_concatenation hab γ γrev rfl Γ hΓval
+  -- the two pieces and the final assembly
+  have hsum := curveArea_concatenation (⟨0, 2, hΓ2, Γ⟩ : RectifiablePathData)
+    ![(⟨a, b, hab.le, γrev⟩ : RectifiablePathData),
+      ⟨0, 1, zero_le_one, lineSegmentBVPath (γ.val A) (γ.val B)⟩] hconcat
+  have hbase : curveAreaFunctional (lineSegmentBVPath (γ.val A) (γ.val B)) = 0 := by
+    rw [curveAreaFunctional_lineSegmentBVPath, segmentArea, planeCrossProduct, ha, hb]
+    ring
+  have hJΓ : curveAreaFunctional Γ = -curveAreaFunctional γ := by
+    have hΓsum : curveAreaFunctional Γ =
+        curveAreaFunctional γrev +
+          curveAreaFunctional (lineSegmentBVPath (γ.val A) (γ.val B)) := by
+      rw [show curveAreaFunctional Γ =
+        curveAreaFunctional (⟨0, 2, hΓ2, Γ⟩ : RectifiablePathData).path from rfl, hsum,
+        Fin.sum_univ_two]
+      rfl
+    rw [hΓsum, hbase, hJrev, add_zero]
+  exact ⟨Γ, hΓval, hJΓ.trans harea.symm, hheight⟩
+
+/-- Above a parameter's abscissa, a strictly monotone roof has that parameter's ordinate. -/
+theorem monotoneRoofHeight_apply {a b : ℝ} (hab : a < b) (γ : ContinuousBVPaths a b)
+    (hk : StrictMono fun t ↦ γ.val t 0) (s : Set.Icc a b) :
+    monotoneRoofHeight hab γ (γ.val s 0) = γ.val s 1 := by
+  classical
+  have hne : Nonempty (Set.Icc a b) := ⟨⟨a, le_rfl, hab.le⟩⟩
+  have hinv : @Function.invFun _ _ hne (fun t ↦ γ.val t 0) (γ.val s 0) = s :=
+    hk.injective (@Function.invFun_eq _ _ hne (fun t ↦ γ.val t 0) _ ⟨s, rfl⟩)
+  change γ.val (@Function.invFun _ _ hne (fun t ↦ γ.val t 0) (γ.val s 0)) 1 = γ.val s 1
+  rw [hinv]
+
+/-- Every abscissa between a roof's endpoints is the abscissa of a roof parameter. -/
+theorem exists_eq_monotoneRoof_fst {a b : ℝ} (hab : a < b) (γ : ContinuousBVPaths a b)
+    {c : ℝ} (hc : c ∈ Set.Icc (γ.val ⟨a, le_rfl, hab.le⟩ 0) (γ.val ⟨b, hab.le, le_rfl⟩ 0)) :
+    ∃ s : Set.Icc a b, γ.val s 0 = c := by
+  have : PreconnectedSpace (Set.Icc a b) := Subtype.preconnectedSpace isPreconnected_Icc
+  exact intermediate_value_univ _ _ ((PiLp.continuous_apply 2 _ 0).comp γ.property.1) hc
+
+/-- The closed region under a strictly monotone roof, described parametrically: it consists of the
+points on or below the roof on the vertical line through some roof parameter. -/
+theorem monotoneRoofRegion_eq_param {a b : ℝ} (hab : a < b) (γ : ContinuousBVPaths a b)
+    (hk : StrictMono fun t ↦ γ.val t 0) :
+    monotoneRoofRegion hab γ =
+      {p : Point | ∃ s : Set.Icc a b, p 0 = γ.val s 0 ∧ 0 ≤ p 1 ∧ p 1 ≤ γ.val s 1} := by
+  ext p
+  constructor
+  · rintro ⟨h1, h2, h3, h4⟩
+    obtain ⟨s, hs⟩ := exists_eq_monotoneRoof_fst hab γ ⟨h1, h2⟩
+    refine ⟨s, hs.symm, h3, ?_⟩
+    rwa [← hs, monotoneRoofHeight_apply hab γ hk s] at h4
+  · rintro ⟨s, hs0, hs1, hs2⟩
+    refine ⟨?_, ?_, hs1, ?_⟩
+    · rw [hs0]
+      exact hk.monotone (show (⟨a, le_rfl, hab.le⟩ : Set.Icc a b) ≤ s from s.property.1)
+    · rw [hs0]
+      exact hk.monotone (show s ≤ (⟨b, hab.le, le_rfl⟩ : Set.Icc a b) from s.property.2)
+    · rw [hs0, monotoneRoofHeight_apply hab γ hk s]
+      exact hs2
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-! # Three-piece monotone roofs
+
+A *three-piece roof* over `[r, l]` is the continuous BV path on `[0, 3]` that runs up a straight
+segment from a base point `P` to `f l`, traverses an arc `f` backwards from `l` to `r`, and runs
+down a straight segment from `f r` to a base point `Q`.  When the arc's first coordinate is
+strictly decreasing and `P`, `Q` lie strictly to the left of `f l` resp. to the right of `f r`, the
+resulting path has strictly increasing first coordinate, so it is a monotone roof in the sense of
+`MovingSofa.monotoneRoofRegion`.
+
+`area_region_under_strictMono_roof` computes the area of the closed region under such a roof, above
+the base line `y = -h` carrying `P` and `Q`, as the signed area of the four-piece closed loop that
+bounds it, and `area_region_under_roof_le` compares that area with the areas of a base set and of a
+set containing the rest of the open region.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A segment path, read as its base point plus a multiple of its displacement. -/
+private theorem segment_eq_add_smul (p q : Point) (u : Set.Icc (0 : ℝ) 1) :
+    Path.segment p q u = p + (u : ℝ) • (q - p) := by
+  rw [Path.segment_apply, AffineMap.lineMap_apply_module']
+  module
+
+/-- Clamping to a nondegenerate interval is strictly monotone away from both clamps. -/
+private theorem projIcc_lt_projIcc {a b : ℝ} (hab : a < b) {s t : ℝ} (hst : s < t)
+    (hs : s < b) (ht : a < t) :
+    Set.projIcc a b hab.le s < Set.projIcc a b hab.le t := by
+  change max a (min b s) < max a (min b t)
+  rw [min_eq_right hs.le, max_eq_right (le_min hab.le ht.le)]
+  exact max_lt (lt_min hab ht) (lt_min hs hst)
+
+/-- A convex combination of two numbers bounded by `c` is bounded by `c`. -/
+private theorem affine_le_of_le {A B c s : ℝ} (hA : A ≤ c) (hB : B ≤ c)
+    (hs0 : 0 ≤ s) (hs1 : s ≤ 1) : A + s * (B - A) ≤ c := by nlinarith
+
+/-- The middle arc's parameter times run over the arc's own interval. -/
+theorem mem_Icc_roofArcTime {r l : ℝ} (hrl : r < l) {s : ℝ} (hs1 : 1 ≤ s) (hs2 : s ≤ 2) :
+    l + (s - 1) * (r - l) ∈ Set.Icc r l := by
+  refine ⟨?_, ?_⟩
+  · have h : (0 : ℝ) ≤ (1 - (s - 1)) * (l - r) :=
+      mul_nonneg (by linarith only [hs2]) (by linarith only [hrl])
+    linarith only [h]
+  · have h : (s - 1) * (r - l) ≤ 0 :=
+      mul_nonpos_of_nonneg_of_nonpos (by linarith only [hs1]) (by linarith only [hrl])
+    linarith only [h]
+
+/-- Three pieces — a rising segment into the left end of a strictly decreasing arc, that arc
+traversed backwards, and a rising segment out of its right end — assemble into a continuous BV
+path on `[0, 3]` whose first coordinate is strictly increasing. -/
+theorem exists_strictMono_roof_of_strictAntiOn {r l : ℝ} (hrl : r < l)
+    (c : ContinuousBVPaths r l) (f : ℝ → Point) (P Q : Point)
+    (hf : ∀ (t : ℝ) (ht : t ∈ Set.Icc r l), c.val ⟨t, ht⟩ = f t)
+    (hanti : StrictAntiOn (fun t ↦ f t 0) (Set.Icc r l))
+    (hP : P 0 < f l 0) (hQ : f r 0 < Q 0) :
+    ∃ γ : ContinuousBVPaths 0 3,
+      (∀ s : Set.Icc (0 : ℝ) 3, (s : ℝ) ≤ 1 → γ.val s = P + (s : ℝ) • (f l - P)) ∧
+        (∀ s : Set.Icc (0 : ℝ) 3, 1 ≤ (s : ℝ) → (s : ℝ) ≤ 2 →
+          γ.val s = f (l + ((s : ℝ) - 1) * (r - l))) ∧
+        (∀ s : Set.Icc (0 : ℝ) 3, 2 ≤ (s : ℝ) →
+          γ.val s = f r + ((s : ℝ) - 2) • (Q - f r)) ∧
+        StrictMono fun s ↦ γ.val s 0 := by
+  have hlr : r - l < 0 := by linarith
+  have hcf : ∀ u : Set.Icc r l, c.val u = f (u : ℝ) := fun u ↦ hf u u.property
+  have hcanti : ∀ u v : Set.Icc r l, u ≤ v → f (v : ℝ) 0 ≤ f (u : ℝ) 0 := by
+    intro u v huv
+    rcases eq_or_lt_of_le huv with h | h
+    · rw [h]
+    · exact (hanti u.property v.property h).le
+  -- the antitone reparametrisation of `[0, 3]` onto the arc's domain
+  set μ : Set.Icc (0 : ℝ) 3 → Set.Icc r l :=
+    fun s ↦ Set.projIcc r l hrl.le (l + ((s : ℝ) - 1) * (r - l)) with hμdef
+  have hμc : Continuous μ := continuous_projIcc.comp (by fun_prop)
+  have hμa : Antitone μ := by
+    intro s t hst
+    have hst' : (s : ℝ) ≤ t := hst
+    exact Set.monotone_projIcc hrl.le (by nlinarith)
+  have hμs : Function.Surjective μ := by
+    intro u
+    have hu1 : r ≤ (u : ℝ) := u.property.1
+    have hu2 : (u : ℝ) ≤ l := u.property.2
+    have hne : l - r ≠ 0 := by linarith
+    have hq : 0 ≤ (l - (u : ℝ)) / (l - r) := div_nonneg (by linarith) (by linarith)
+    have hq1 : (l - (u : ℝ)) / (l - r) ≤ 1 := by
+      rw [div_le_one (by linarith)]
+      linarith
+    refine ⟨⟨1 + (l - (u : ℝ)) / (l - r), by linarith, by linarith⟩, ?_⟩
+    change Set.projIcc r l hrl.le (l + (1 + (l - (u : ℝ)) / (l - r) - 1) * (r - l)) = u
+    rw [show l + (1 + (l - (u : ℝ)) / (l - r) - 1) * (r - l) = (u : ℝ) by
+      field_simp; ring]
+    exact Set.projIcc_of_mem hrl.le u.property
+  have hμleft : ∀ s : Set.Icc (0 : ℝ) 3, (s : ℝ) ≤ 1 → μ s = ⟨l, hrl.le, le_rfl⟩ := by
+    intro s hs
+    change Set.projIcc r l hrl.le (l + ((s : ℝ) - 1) * (r - l)) = _
+    exact Set.projIcc_of_right_le hrl.le
+      (le_add_of_nonneg_right (show (0 : ℝ) ≤ ((s : ℝ) - 1) * (r - l) by nlinarith))
+  have hμmid : ∀ s : Set.Icc (0 : ℝ) 3, 1 ≤ (s : ℝ) → (s : ℝ) ≤ 2 →
+      (μ s : ℝ) = l + ((s : ℝ) - 1) * (r - l) := by
+    intro s hs1 hs2
+    change (Set.projIcc r l hrl.le (l + ((s : ℝ) - 1) * (r - l)) : ℝ) = _
+    rw [Set.projIcc_of_mem hrl.le ⟨by nlinarith, by nlinarith⟩]
+  have hμright : ∀ s : Set.Icc (0 : ℝ) 3, 2 ≤ (s : ℝ) → μ s = ⟨r, le_rfl, hrl.le⟩ := by
+    intro s hs
+    change Set.projIcc r l hrl.le (l + ((s : ℝ) - 1) * (r - l)) = _
+    exact Set.projIcc_of_le_left hrl.le (by nlinarith)
+  have hμstrict : ∀ s t : Set.Icc (0 : ℝ) 3, 1 ≤ (s : ℝ) → (s : ℝ) < (t : ℝ) →
+      (t : ℝ) ≤ 2 → μ t < μ s := by
+    intro s t hs hst ht
+    exact projIcc_lt_projIcc hrl (by nlinarith) (by nlinarith) (by nlinarith)
+  -- the two clamped affine reparametrisations of the side segments
+  set ρ₀ : Set.Icc (0 : ℝ) 3 → Set.Icc (0 : ℝ) 1 :=
+    fun s ↦ Set.projIcc 0 1 zero_le_one (s : ℝ) with hρ₀def
+  set ρ₂ : Set.Icc (0 : ℝ) 3 → Set.Icc (0 : ℝ) 1 :=
+    fun s ↦ Set.projIcc 0 1 zero_le_one ((s : ℝ) - 2) with hρ₂def
+  have hρ₀c : Continuous ρ₀ := continuous_projIcc.comp (by fun_prop)
+  have hρ₂c : Continuous ρ₂ := continuous_projIcc.comp (by fun_prop)
+  have hρ₀m : Monotone ρ₀ := fun s t hst ↦ Set.monotone_projIcc zero_le_one hst
+  have hρ₂m : Monotone ρ₂ := fun s t hst ↦
+    Set.monotone_projIcc zero_le_one (by linarith [show (s : ℝ) ≤ t from hst])
+  have hρ₀s : Function.Surjective ρ₀ := fun u ↦
+    ⟨⟨(u : ℝ), u.property.1, le_trans u.property.2 (by norm_num)⟩,
+      Set.projIcc_of_mem zero_le_one u.property⟩
+  have hρ₂s : Function.Surjective ρ₂ := by
+    intro u
+    refine ⟨⟨(u : ℝ) + 2, by linarith [u.property.1], by linarith [u.property.2]⟩, ?_⟩
+    change Set.projIcc 0 1 zero_le_one ((u : ℝ) + 2 - 2) = u
+    rw [show (u : ℝ) + 2 - 2 = (u : ℝ) by ring]
+    exact Set.projIcc_of_mem zero_le_one u.property
+  have hρ₀left : ∀ s : Set.Icc (0 : ℝ) 3, (s : ℝ) ≤ 1 → (ρ₀ s : ℝ) = (s : ℝ) := by
+    intro s hs
+    change (Set.projIcc 0 1 zero_le_one (s : ℝ) : ℝ) = _
+    rw [Set.projIcc_of_mem zero_le_one ⟨s.property.1, hs⟩]
+  have hρ₀right : ∀ s : Set.Icc (0 : ℝ) 3, 1 ≤ (s : ℝ) → (ρ₀ s : ℝ) = 1 := by
+    intro s hs
+    change (Set.projIcc 0 1 zero_le_one (s : ℝ) : ℝ) = _
+    rw [Set.projIcc_of_right_le zero_le_one hs]
+  have hρ₂left : ∀ s : Set.Icc (0 : ℝ) 3, (s : ℝ) ≤ 2 → (ρ₂ s : ℝ) = 0 := by
+    intro s hs
+    change (Set.projIcc 0 1 zero_le_one ((s : ℝ) - 2) : ℝ) = _
+    rw [Set.projIcc_of_le_left zero_le_one (by linarith)]
+  have hρ₂right : ∀ s : Set.Icc (0 : ℝ) 3, 2 ≤ (s : ℝ) → (ρ₂ s : ℝ) = (s : ℝ) - 2 := by
+    intro s hs
+    change (Set.projIcc 0 1 zero_le_one ((s : ℝ) - 2) : ℝ) = _
+    rw [Set.projIcc_of_mem zero_le_one ⟨by linarith, by linarith [s.property.2]⟩]
+  -- the three summands of the roof
+  obtain ⟨A₀, hA₀⟩ := continuousBVPaths_comp_monotone_surjective zero_le_one
+    (lineSegmentBVPath P (f l)) ρ₀ hρ₀c hρ₀m hρ₀s
+  obtain ⟨A₂, hA₂⟩ := continuousBVPaths_comp_monotone_surjective zero_le_one
+    (lineSegmentBVPath 0 (Q - f r)) ρ₂ hρ₂c hρ₂m hρ₂s
+  set A₁ : ContinuousBVPaths 0 3 :=
+    ⟨c.val ∘ μ, c.property.1.comp hμc, fun i ↦
+      BoundedVariationOn.comp_antitone_surjective_Icc hrl.le (c.property.2 i) hμa hμs⟩
+    with hA₁def
+  have hval : ∀ s : Set.Icc (0 : ℝ) 3,
+      (A₀ + (A₁ - constBVPath 0 3 (f l)) + A₂).val s =
+        (P + (ρ₀ s : ℝ) • (f l - P)) + (f (μ s : ℝ) - f l) +
+          (ρ₂ s : ℝ) • (Q - f r) := by
+    intro s
+    change A₀.val s + (A₁.val s - f l) + A₂.val s = _
+    rw [hA₀, hA₂]
+    change Path.segment P (f l) (ρ₀ s) + (c.val (μ s) - f l) +
+      Path.segment (0 : Point) (Q - f r) (ρ₂ s) = _
+    rw [segment_eq_add_smul, segment_eq_add_smul, hcf (μ s)]
+    simp
+  have hcoord : ∀ s : Set.Icc (0 : ℝ) 3,
+      (A₀ + (A₁ - constBVPath 0 3 (f l)) + A₂).val s 0 =
+        P 0 + (ρ₀ s : ℝ) * (f l 0 - P 0) + (f (μ s : ℝ) 0 - f l 0) +
+          (ρ₂ s : ℝ) * (Q 0 - f r 0) := by
+    intro s
+    rw [hval s]
+    simp
+  refine ⟨A₀ + (A₁ - constBVPath 0 3 (f l)) + A₂, ?_, ?_, ?_, ?_⟩
+  · intro s hs
+    rw [hval s, hρ₀left s hs, hμleft s hs]
+    simp [hρ₂left s (by linarith)]
+  · intro s hs1 hs2
+    rw [hval s, hρ₀right s hs1, hμmid s hs1 hs2, hρ₂left s hs2]
+    simp
+  · intro s hs
+    rw [hval s, hρ₀right s (by linarith), hμright s hs, hρ₂right s hs]
+    simp
+  · intro s t hst
+    have hst' : (s : ℝ) < (t : ℝ) := hst
+    have hd0 : (0 : ℝ) < f l 0 - P 0 := by linarith
+    have hd2 : (0 : ℝ) < Q 0 - f r 0 := by linarith
+    have m0 : (ρ₀ s : ℝ) * (f l 0 - P 0) ≤ (ρ₀ t : ℝ) * (f l 0 - P 0) :=
+      mul_le_mul_of_nonneg_right (hρ₀m hst.le) hd0.le
+    have m2 : (ρ₂ s : ℝ) * (Q 0 - f r 0) ≤ (ρ₂ t : ℝ) * (Q 0 - f r 0) :=
+      mul_le_mul_of_nonneg_right (hρ₂m hst.le) hd2.le
+    have m1 : f (μ s : ℝ) 0 ≤ f (μ t : ℝ) 0 := hcanti (μ t) (μ s) (hμa hst.le)
+    simp only [hcoord]
+    rcases lt_or_ge (s : ℝ) 1 with hc | hc
+    · have hs0 : (ρ₀ s : ℝ) < (ρ₀ t : ℝ) :=
+        projIcc_lt_projIcc zero_lt_one hst' hc (by linarith [s.property.1])
+      have := mul_lt_mul_of_pos_right hs0 hd0
+      linarith only [this, m1, m2]
+    · rcases le_or_gt (t : ℝ) 2 with hc2 | hc2
+      · have hmid : f (μ s : ℝ) 0 < f (μ t : ℝ) 0 :=
+          hanti (μ t).property (μ s).property (hμstrict s t hc hst' hc2)
+        linarith only [hmid, m0, m2]
+      · have hs2 : (ρ₂ s : ℝ) < (ρ₂ t : ℝ) :=
+          projIcc_lt_projIcc zero_lt_one (by linarith)
+            (by linarith [t.property.2]) (by linarith)
+        have := mul_lt_mul_of_pos_right hs2 hd2
+        linarith only [this, m0, m1]
+
+/-- A linear functional bounded at the two base corners of a three-piece roof and along its
+middle arc is bounded along the whole roof. -/
+theorem le_of_threePieceRoof {r l α β cc : ℝ} (hrl : r < l) (f : ℝ → Point) (P Q : Point)
+    (γ : ContinuousBVPaths 0 3)
+    (hγ0 : ∀ s : Set.Icc (0 : ℝ) 3, (s : ℝ) ≤ 1 → γ.val s = P + (s : ℝ) • (f l - P))
+    (hγ1 : ∀ s : Set.Icc (0 : ℝ) 3, 1 ≤ (s : ℝ) → (s : ℝ) ≤ 2 →
+      γ.val s = f (l + ((s : ℝ) - 1) * (r - l)))
+    (hγ2 : ∀ s : Set.Icc (0 : ℝ) 3, 2 ≤ (s : ℝ) → γ.val s = f r + ((s : ℝ) - 2) • (Q - f r))
+    (hP : α * P 0 + β * P 1 ≤ cc) (hQ : α * Q 0 + β * Q 1 ≤ cc)
+    (harc : ∀ t ∈ Set.Icc r l, α * f t 0 + β * f t 1 ≤ cc) (s : Set.Icc (0 : ℝ) 3) :
+    α * γ.val s 0 + β * γ.val s 1 ≤ cc := by
+  have hcoord : ∀ (p q : Point) (u : ℝ) (i : Fin 2),
+      (p + u • (q - p)) i = p i + u * (q i - p i) := fun _ _ _ _ ↦ by simp
+  rcases le_or_gt (s : ℝ) 1 with hs | hs
+  · rw [hγ0 s hs, hcoord, hcoord]
+    have hmid := affine_le_of_le (A := α * P 0 + β * P 1) (B := α * f l 0 + β * f l 1)
+      (c := cc) (s := (s : ℝ)) hP (harc l ⟨hrl.le, le_rfl⟩) s.property.1 hs
+    calc α * (P 0 + (s : ℝ) * (f l 0 - P 0)) + β * (P 1 + (s : ℝ) * (f l 1 - P 1))
+        = (α * P 0 + β * P 1) +
+          (s : ℝ) * ((α * f l 0 + β * f l 1) - (α * P 0 + β * P 1)) := by ring
+      _ ≤ cc := hmid
+  · rcases le_or_gt (s : ℝ) 2 with hs2 | hs2
+    · rw [hγ1 s hs.le hs2]
+      exact harc _ (mem_Icc_roofArcTime hrl hs.le hs2)
+    · rw [hγ2 s hs2.le, hcoord, hcoord]
+      have hmid := affine_le_of_le (A := α * f r 0 + β * f r 1) (B := α * Q 0 + β * Q 1)
+        (c := cc) (s := (s : ℝ) - 2) (harc r ⟨le_rfl, hrl.le⟩) hQ
+        (by linarith only [hs2]) (by linarith only [s.property.2])
+      calc α * (f r 0 + ((s : ℝ) - 2) * (Q 0 - f r 0)) +
+            β * (f r 1 + ((s : ℝ) - 2) * (Q 1 - f r 1))
+          = (α * f r 0 + β * f r 1) +
+            ((s : ℝ) - 2) * ((α * Q 0 + β * Q 1) - (α * f r 0 + β * f r 1)) := by ring
+        _ ≤ cc := hmid
+
+/-- The closed region under a strictly monotone three-piece roof whose two side segments begin
+and end on the line `y = -h` has area equal to the signed area of the four-piece loop that
+bounds it: the base segment, the two side segments and the arc. -/
+theorem area_region_under_strictMono_roof {r l h : ℝ} (hrl : r < l)
+    (c : ContinuousBVPaths r l) (f : ℝ → Point) (P Q : Point) (γ : ContinuousBVPaths 0 3)
+    (hf : ∀ (t : ℝ) (ht : t ∈ Set.Icc r l), c.val ⟨t, ht⟩ = f t)
+    (hP : P 1 = -h) (hQ : Q 1 = -h)
+    (hγ0 : ∀ s : Set.Icc (0 : ℝ) 3, (s : ℝ) ≤ 1 → γ.val s = P + (s : ℝ) • (f l - P))
+    (hγ1 : ∀ s : Set.Icc (0 : ℝ) 3, 1 ≤ (s : ℝ) → (s : ℝ) ≤ 2 →
+      γ.val s = f (l + ((s : ℝ) - 1) * (r - l)))
+    (hγ2 : ∀ s : Set.Icc (0 : ℝ) 3, 2 ≤ (s : ℝ) → γ.val s = f r + ((s : ℝ) - 2) • (Q - f r))
+    (hγmono : StrictMono fun s ↦ γ.val s 0)
+    (hγlow : ∀ s : Set.Icc (0 : ℝ) 3, -h ≤ γ.val s 1) :
+    ClassicalResults.area {p : Point | ∃ s : Set.Icc (0 : ℝ) 3,
+        p 0 = γ.val s 0 ∧ -h ≤ p 1 ∧ p 1 ≤ γ.val s 1} =
+      segmentArea Q (f r) + curveAreaFunctional c + segmentArea (f l) P + segmentArea P Q := by
+  have h03 : (0 : ℝ) < 3 := by norm_num
+  have h02 : (0 : ℝ) ≤ 2 := by norm_num
+  have hadd : ∀ (x y : Point) (i : Fin 2), (x + y) i = x i + y i := fun _ _ _ ↦ by simp
+  have hcf : ∀ u : Set.Icc r l, c.val u = f (u : ℝ) := fun u ↦ hf u u.property
+  -- the upward shift taking the base line to the horizontal axis
+  set v : Point := !₂[0, h] with hvdef
+  have hv0 : v 0 = 0 := rfl
+  have hv1 : v 1 = h := rfl
+  set γ' : ContinuousBVPaths 0 3 := γ + constBVPath 0 3 v with hγ'def
+  have hγ'val : ∀ s, γ'.val s = γ.val s + v := fun _ ↦ rfl
+  have hγ'coord : ∀ (s : Set.Icc (0 : ℝ) 3) (i : Fin 2), γ'.val s i = γ.val s i + v i := by
+    intro s i
+    rw [hγ'val s, hadd]
+  have hγ'mono : StrictMono fun s ↦ γ'.val s 0 := by
+    intro s t hst
+    have hm : γ.val s 0 < γ.val t 0 := hγmono hst
+    change γ'.val s 0 < γ'.val t 0
+    rw [hγ'coord s 0, hγ'coord t 0]
+    linarith only [hm]
+  have hγstart : γ.val ⟨0, le_rfl, h03.le⟩ = P := by
+    rw [hγ0 _ (by norm_num)]
+    simp
+  have hγend : γ.val ⟨3, h03.le, le_rfl⟩ = Q := by
+    rw [hγ2 _ (by norm_num)]
+    norm_num
+  have hγ'zero : γ'.val ⟨0, le_rfl, h03.le⟩ 1 = 0 := by
+    rw [hγ'coord, hγstart, hP, hv1]
+    ring
+  have hγ'three : γ'.val ⟨3, h03.le, le_rfl⟩ 1 = 0 := by
+    rw [hγ'coord, hγend, hQ, hv1]
+    ring
+  have hγ'nonneg : ∀ s, 0 ≤ γ'.val s 1 := by
+    intro s
+    rw [hγ'coord, hv1]
+    linarith only [hγlow s]
+  -- the region is the roof region of the shifted path, translated back down
+  have hregion : {p : Point | ∃ s : Set.Icc (0 : ℝ) 3,
+        p 0 = γ.val s 0 ∧ -h ≤ p 1 ∧ p 1 ≤ γ.val s 1} =
+      (fun q : Point ↦ q + v) ⁻¹' monotoneRoofRegion h03 γ' := by
+    rw [monotoneRoofRegion_eq_param h03 γ' hγ'mono]
+    ext p
+    simp only [Set.mem_preimage, Set.mem_ofPred_eq, hadd, hγ'coord, hv0, hv1, add_zero]
+    constructor
+    · rintro ⟨s, e1, e2, e3⟩
+      exact ⟨s, e1, by linarith only [e2], by linarith only [e3]⟩
+    · rintro ⟨s, e1, e2, e3⟩
+      exact ⟨s, e1, by linarith only [e2], by linarith only [e3]⟩
+  have harea : ClassicalResults.area {p : Point | ∃ s : Set.Icc (0 : ℝ) 3,
+        p 0 = γ.val s 0 ∧ -h ≤ p 1 ∧ p 1 ≤ γ.val s 1} =
+      ClassicalResults.area (monotoneRoofRegion h03 γ') := by
+    rw [ClassicalResults.area, ClassicalResults.area, hregion,
+      MeasureTheory.measure_preimage_add_right]
+  obtain ⟨Γ, hΓval, hΓarea, -⟩ :=
+    monotone_roof_signed_area h03 γ' hγ'mono hγ'nonneg hγ'zero hγ'three
+  -- reading the closed loop on its two branches
+  have hloopLow : ∀ (t : Set.Icc (0 : ℝ) 2) (u : Set.Icc (0 : ℝ) 3), (t : ℝ) ≤ 1 →
+      (u : ℝ) = 3 - 3 * (t : ℝ) → Γ.val t = γ.val u + v := by
+    intro t u ht hu
+    rw [hΓval]
+    unfold monotoneRoofLoop
+    split_ifs with hs
+    · rw [hγ'val]
+      exact congrArg (fun z : Point ↦ z + v) (congrArg (fun z ↦ γ.val z)
+        (Subtype.ext (by rw [hu]; ring)))
+    · exact absurd ht hs
+  have hloopHigh : ∀ t : Set.Icc (0 : ℝ) 2, 1 ≤ (t : ℝ) →
+      Γ.val t = (2 - (t : ℝ)) • (P + v) + ((t : ℝ) - 1) • (Q + v) := by
+    intro t ht
+    rcases eq_or_lt_of_le ht with heq | hlt
+    · rw [hloopLow t ⟨0, le_rfl, h03.le⟩ (le_of_eq heq.symm) (by rw [← heq]; norm_num),
+        hγstart, ← heq]
+      module
+    · rw [hΓval]
+      unfold monotoneRoofLoop
+      split_ifs with hs
+      · exact absurd hs (not_le.mpr hlt)
+      · simp only [hγ'val, hγstart, hγend]
+  -- the four cuts of the loop's parameter interval
+  set e0 : Set.Icc (0 : ℝ) 2 := ⟨0, le_rfl, h02⟩ with he0
+  set e1 : Set.Icc (0 : ℝ) 2 := ⟨1 / 3, by norm_num⟩ with he1
+  set e2 : Set.Icc (0 : ℝ) 2 := ⟨2 / 3, by norm_num⟩ with he2
+  set e3 : Set.Icc (0 : ℝ) 2 := ⟨1, by norm_num⟩ with he3
+  set e4 : Set.Icc (0 : ℝ) 2 := ⟨2, h02, le_rfl⟩ with he4
+  have hle01 : e0 ≤ e1 := by change (0 : ℝ) ≤ 1 / 3; norm_num
+  have hle12 : e1 ≤ e2 := by change (1 : ℝ) / 3 ≤ 2 / 3; norm_num
+  have hle23 : e2 ≤ e3 := by change (2 : ℝ) / 3 ≤ 1; norm_num
+  have hle34 : e3 ≤ e4 := by change (1 : ℝ) ≤ 2; norm_num
+  have hsplit : curveAreaFunctional Γ =
+      curveAreaFunctional (ContinuousBVPaths.restrict Γ e0 e1 hle01) +
+        curveAreaFunctional (ContinuousBVPaths.restrict Γ e1 e2 hle12) +
+        curveAreaFunctional (ContinuousBVPaths.restrict Γ e2 e3 hle23) +
+        curveAreaFunctional (ContinuousBVPaths.restrict Γ e3 e4 hle34) := by
+    have hcm : Monotone (![e0, e1, e2, e3, e4] : Fin 5 → Set.Icc (0 : ℝ) 2) := by
+      refine Fin.monotone_iff_le_succ.mpr fun i ↦ ?_
+      fin_cases i
+      · exact hle01
+      · exact hle12
+      · exact hle23
+      · exact hle34
+    have hsum := curveArea_eq_sum_restrict Γ ![e0, e1, e2, e3, e4] hcm rfl rfl
+    rw [Fin.sum_univ_four] at hsum
+    exact hsum
+  -- each restriction, read off through an affine reparametrisation
+  have hres : ∀ (u1 u2 : Set.Icc (0 : ℝ) 2) (hu : u1 ≤ u2)
+      (t : Set.Icc ((u1 : ℝ)) ((u2 : ℝ))) (wp : Set.Icc (0 : ℝ) 2), (wp : ℝ) = (t : ℝ) →
+      (ContinuousBVPaths.restrict Γ u1 u2 hu).val t = Γ.val wp :=
+    fun _ _ _ _ _ hw ↦ congrArg (fun z ↦ Γ.val z) (Subtype.ext hw.symm)
+  have hseg : ∀ (u1 u2 : Set.Icc (0 : ℝ) 2) (hu : u1 ≤ u2) (p q : Point)
+      (φ : Set.Icc ((u1 : ℝ)) ((u2 : ℝ)) → Set.Icc (0 : ℝ) 1), Continuous φ → Monotone φ →
+      Function.Surjective φ →
+      (ContinuousBVPaths.restrict Γ u1 u2 hu).val = (lineSegmentBVPath p q).val ∘ φ →
+      curveAreaFunctional (ContinuousBVPaths.restrict Γ u1 u2 hu) = segmentArea p q := by
+    intro u1 u2 hu p q φ hφc hφm hφs hcomp
+    rw [curveAreaFunctional_eq_of_comp_monotone_surjective zero_le_one hu
+        (lineSegmentBVPath p q) _ φ hφc hφm hφs hcomp,
+      curveAreaFunctional_lineSegmentBVPath]
+  obtain ⟨φ₁, hφ₁c, hφ₁m, hφ₁s, hφ₁v⟩ :=
+    Set.Icc.exists_affine_monotone_surjection (show ((e0 : ℝ)) < ((e1 : ℝ)) by
+      change (0 : ℝ) < 1 / 3; norm_num) (show (0 : ℝ) < 1 by norm_num)
+  obtain ⟨φ₂, hφ₂c, hφ₂m, hφ₂s, hφ₂v⟩ :=
+    Set.Icc.exists_affine_monotone_surjection (show ((e1 : ℝ)) < ((e2 : ℝ)) by
+      change (1 : ℝ) / 3 < 2 / 3; norm_num) hrl
+  obtain ⟨φ₃, hφ₃c, hφ₃m, hφ₃s, hφ₃v⟩ :=
+    Set.Icc.exists_affine_monotone_surjection (show ((e2 : ℝ)) < ((e3 : ℝ)) by
+      change (2 : ℝ) / 3 < 1; norm_num) (show (0 : ℝ) < 1 by norm_num)
+  obtain ⟨φ₄, hφ₄c, hφ₄m, hφ₄s, hφ₄v⟩ :=
+    Set.Icc.exists_affine_monotone_surjection (show ((e3 : ℝ)) < ((e4 : ℝ)) by
+      change (1 : ℝ) < 2; norm_num) (show (0 : ℝ) < 1 by norm_num)
+  have hp1 : curveAreaFunctional (ContinuousBVPaths.restrict Γ e0 e1 hle01) =
+      segmentArea (Q + v) (f r + v) := by
+    refine hseg e0 e1 hle01 _ _ φ₁ hφ₁c hφ₁m hφ₁s ?_
+    funext t
+    have ht0 : (0 : ℝ) ≤ (t : ℝ) := t.property.1
+    have ht1 : (t : ℝ) ≤ 1 / 3 := t.property.2
+    rw [hres e0 e1 hle01 t ⟨(t : ℝ), by linarith, by linarith⟩ rfl,
+      hloopLow _ ⟨3 - 3 * (t : ℝ), by constructor <;> linarith⟩ (by linarith) rfl,
+      hγ2 _ (by change (2 : ℝ) ≤ 3 - 3 * (t : ℝ); linarith)]
+    change _ = Path.segment (Q + v) (f r + v) (φ₁ t)
+    rw [segment_eq_add_smul, hφ₁v t,
+      show (0 : ℝ) + ((t : ℝ) - ((e0 : ℝ))) / (((e1 : ℝ)) - ((e0 : ℝ))) * (1 - 0) =
+        3 * (t : ℝ) from by change (0 : ℝ) + ((t : ℝ) - 0) / (1 / 3 - 0) * (1 - 0) = _; ring,
+      show ((3 : ℝ) - 3 * (t : ℝ)) - 2 = 1 - 3 * (t : ℝ) from by ring]
+    module
+  have hp2 : curveAreaFunctional (ContinuousBVPaths.restrict Γ e1 e2 hle12) =
+      curveAreaFunctional (c + constBVPath r l v) := by
+    refine curveAreaFunctional_eq_of_comp_monotone_surjective hrl.le hle12
+      (c + constBVPath r l v) _ φ₂ hφ₂c hφ₂m hφ₂s ?_
+    funext t
+    have ht0 : (1 : ℝ) / 3 ≤ (t : ℝ) := t.property.1
+    have ht1 : (t : ℝ) ≤ 2 / 3 := t.property.2
+    rw [hres e1 e2 hle12 t ⟨(t : ℝ), by linarith, by linarith⟩ rfl,
+      hloopLow _ ⟨3 - 3 * (t : ℝ), by constructor <;> linarith⟩ (by linarith) rfl,
+      hγ1 _ (by change (1 : ℝ) ≤ 3 - 3 * (t : ℝ); linarith)
+        (by change (3 : ℝ) - 3 * (t : ℝ) ≤ 2; linarith),
+      show l + (((3 : ℝ) - 3 * (t : ℝ)) - 1) * (r - l) = ((φ₂ t : ℝ)) from by
+        rw [hφ₂v t]
+        change _ = r + ((t : ℝ) - 1 / 3) / (2 / 3 - 1 / 3) * (l - r)
+        ring]
+    change f ((φ₂ t : ℝ)) + v = c.val (φ₂ t) + v
+    rw [hcf (φ₂ t)]
+  have hp3 : curveAreaFunctional (ContinuousBVPaths.restrict Γ e2 e3 hle23) =
+      segmentArea (f l + v) (P + v) := by
+    refine hseg e2 e3 hle23 _ _ φ₃ hφ₃c hφ₃m hφ₃s ?_
+    funext t
+    have ht0 : (2 : ℝ) / 3 ≤ (t : ℝ) := t.property.1
+    have ht1 : (t : ℝ) ≤ 1 := t.property.2
+    rw [hres e2 e3 hle23 t ⟨(t : ℝ), by linarith, by linarith⟩ rfl,
+      hloopLow _ ⟨3 - 3 * (t : ℝ), by constructor <;> linarith⟩ ht1 rfl,
+      hγ0 _ (by change (3 : ℝ) - 3 * (t : ℝ) ≤ 1; linarith)]
+    change _ = Path.segment (f l + v) (P + v) (φ₃ t)
+    rw [segment_eq_add_smul, hφ₃v t,
+      show (0 : ℝ) + ((t : ℝ) - ((e2 : ℝ))) / (((e3 : ℝ)) - ((e2 : ℝ))) * (1 - 0) =
+        3 * (t : ℝ) - 2 from by
+          change (0 : ℝ) + ((t : ℝ) - 2 / 3) / (1 - 2 / 3) * (1 - 0) = _
+          ring]
+    module
+  have hp4 : curveAreaFunctional (ContinuousBVPaths.restrict Γ e3 e4 hle34) =
+      segmentArea (P + v) (Q + v) := by
+    refine hseg e3 e4 hle34 _ _ φ₄ hφ₄c hφ₄m hφ₄s ?_
+    funext t
+    have ht0 : (1 : ℝ) ≤ (t : ℝ) := t.property.1
+    have ht1 : (t : ℝ) ≤ 2 := t.property.2
+    rw [hres e3 e4 hle34 t ⟨(t : ℝ), by linarith, by linarith⟩ rfl,
+      hloopHigh _ ht0]
+    change _ = Path.segment (P + v) (Q + v) (φ₄ t)
+    rw [segment_eq_add_smul, hφ₄v t,
+      show (0 : ℝ) + ((t : ℝ) - ((e3 : ℝ))) / (((e4 : ℝ)) - ((e3 : ℝ))) * (1 - 0) =
+        (t : ℝ) - 1 from by change (0 : ℝ) + ((t : ℝ) - 1) / (2 - 1) * (1 - 0) = _; ring]
+    module
+  -- assembling the four signed areas and undoing the shift
+  rw [harea, ← hΓarea, hsplit, hp1, hp2, hp3, hp4,
+    curveAreaFunctional_add_constBVPath hrl.le c v, hcf ⟨l, hrl.le, le_rfl⟩,
+    hcf ⟨r, le_rfl, hrl.le⟩]
+  simp only [segmentArea_add_right, hv0, hv1]
+  ring
+
+/-- Comparing the closed region under a strictly monotone roof with a base set that absorbs the
+region's bottom edge and a set that absorbs the rest of the open region: the roof itself is a
+null set, so the three areas satisfy the expected inequality. -/
+theorem area_region_under_roof_le {h : ℝ} (γ : ContinuousBVPaths 0 3)
+    (hγmono : StrictMono fun s ↦ γ.val s 0) (R T : Set Point)
+    (hRfin : MeasureTheory.volume R ≠ ⊤) (hTfin : MeasureTheory.volume T ≠ ⊤)
+    (hbase : ∀ p : Point, (∃ s : Set.Icc (0 : ℝ) 3, p 0 = γ.val s 0) → p 1 = -h → p ∈ R)
+    (hincl : {p : Point | ∃ s : Set.Icc (0 : ℝ) 3,
+        p 0 = γ.val s 0 ∧ -h < p 1 ∧ p 1 < γ.val s 1} \ R ⊆ T) :
+    ClassicalResults.area {p : Point | ∃ s : Set.Icc (0 : ℝ) 3,
+        p 0 = γ.val s 0 ∧ -h ≤ p 1 ∧ p 1 ≤ γ.val s 1} ≤
+      ClassicalResults.area R + ClassicalResults.area T := by
+  have h03 : (0 : ℝ) < 3 := by norm_num
+  set Pu : Set Point :=
+    {p : Point | ∃ s : Set.Icc (0 : ℝ) 3, p 0 = γ.val s 0 ∧ -h ≤ p 1 ∧ p 1 ≤ γ.val s 1}
+    with hPudef
+  set Pu' : Set Point :=
+    {p : Point | ∃ s : Set.Icc (0 : ℝ) 3, p 0 = γ.val s 0 ∧ -h < p 1 ∧ p 1 < γ.val s 1}
+    with hPu'def
+  have hsplit : Pu \ R ⊆ (Pu' \ R) ∪ Set.range γ.val := by
+    rintro p ⟨hpP, hpR⟩
+    obtain ⟨s, hs0, hs1, hs2⟩ := hpP
+    rcases eq_or_lt_of_le hs1 with heq | hlt
+    · exact absurd (hbase p ⟨s, hs0⟩ heq.symm) hpR
+    · rcases eq_or_lt_of_le hs2 with heq2 | hlt2
+      · refine Or.inr ⟨s, ?_⟩
+        ext i
+        fin_cases i
+        · exact hs0.symm
+        · exact heq2.symm
+      · exact Or.inl ⟨⟨s, hs0, hlt, hlt2⟩, hpR⟩
+  have hnull : MeasureTheory.volume (Set.range γ.val) = 0 := by
+    refine ContinuousBVPaths.volume_range_eq_zero_of_injOn h03.le γ ?_
+    intro a _ b _ hab
+    exact hγmono.injective (congrArg (fun p : Point ↦ p 0) hab)
+  have hkey : MeasureTheory.volume Pu ≤ MeasureTheory.volume R + MeasureTheory.volume T := by
+    refine le_trans (MeasureTheory.measure_le_inter_add_sdiff _ _ R) ?_
+    refine add_le_add (MeasureTheory.measure_mono Set.inter_subset_right) ?_
+    refine le_trans (MeasureTheory.measure_mono hsplit) ?_
+    refine le_trans (MeasureTheory.measure_union_le _ _) ?_
+    rw [hnull, add_zero]
+    exact MeasureTheory.measure_mono hincl
+  have hPufin : MeasureTheory.volume Pu ≠ ⊤ :=
+    ne_top_of_le_ne_top (ENNReal.add_ne_top.mpr ⟨hRfin, hTfin⟩) hkey
+  simp only [ClassicalResults.area]
+  rw [← ENNReal.toReal_add hRfin hTfin]
+  exact (ENNReal.toReal_le_toReal hPufin (ENNReal.add_ne_top.mpr ⟨hRfin, hTfin⟩)).mpr hkey
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Convex.Foundations.Development001`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Convex.BoundaryVariation`.
+* `Convex.BoundaryApproximation`.
+* `Convex.Combination`.
+* `Convex.AreaSuperlevel`.
+* `Convex.CombinationPullback`.
+* `Convex.ContactFanArea`.
+* `Convex.ExposedFaces`.
+* `Convex.Limits`.
+* `Convex.SupportEmbedding`.
+* `Convex.CombinationProperties`.
+* `Convex.EnvelopeFace`.
+* `Convex.Space`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Boundary Variation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private theorem tangent_cone_decomposition (p : Point) (s t : ℝ)
+    (h : Real.sin (t - s) ≠ 0) :
+    p = (inner ℝ p (normalVector (t : Real.Angle)) / Real.sin (t - s)) •
+        tangentVector (s : Real.Angle) +
+      (-inner ℝ p (normalVector (s : Real.Angle)) / Real.sin (t - s)) •
+        tangentVector (t : Real.Angle) := by
+  ext i
+  fin_cases i <;>
+    simp only [normalVector, tangentVector, frame, PiLp.inner_apply,
+      Fin.sum_univ_two, PiLp.add_apply, PiLp.smul_apply, smul_eq_mul,
+      Matrix.cons_val_zero, Matrix.cons_val_one, Real.Angle.cos_coe,
+      Real.Angle.sin_coe, Real.inner_apply] <;>
+    norm_num <;> field_simp [h] <;> rw [Real.sin_sub] <;> ring
+
+private theorem exists_nonneg_tangent_coefficients (p q : Point) (s t : ℝ)
+    (hst : s < t) (hts : t < s + Real.pi)
+    (hs : inner ℝ q (normalVector (s : Real.Angle)) ≤
+      inner ℝ p (normalVector (s : Real.Angle)))
+    (ht : inner ℝ p (normalVector (t : Real.Angle)) ≤
+      inner ℝ q (normalVector (t : Real.Angle))) :
+    ∃ α β : ℝ, 0 ≤ α ∧ 0 ≤ β ∧
+      q - p = α • tangentVector (s : Real.Angle) +
+        β • tangentVector (t : Real.Angle) := by
+  have hsin : 0 < Real.sin (t - s) :=
+    Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hst) (by linarith)
+  refine ⟨inner ℝ (q - p) (normalVector (t : Real.Angle)) / Real.sin (t - s),
+    -inner ℝ (q - p) (normalVector (s : Real.Angle)) / Real.sin (t - s),
+    ?_, ?_, tangent_cone_decomposition (q - p) s t hsin.ne'⟩
+  · apply div_nonneg _ hsin.le
+    rw [inner_sub_left]
+    exact sub_nonneg.mpr ht
+  · apply div_nonneg _ hsin.le
+    rw [inner_sub_left]
+    linarith
+
+private theorem inner_tangent_normal (r a : ℝ) :
+    inner ℝ (tangentVector (r : Real.Angle)) (normalVector (a : Real.Angle)) =
+      -Real.sin (r - a) := by
+  simp [tangentVector, normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two,
+    Real.sin_sub]
+  ring
+
+private theorem inner_tangent_tangent (r a : ℝ) :
+    inner ℝ (tangentVector (r : Real.Angle)) (tangentVector (a : Real.Angle)) =
+      Real.cos (r - a) := by
+  simp [tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two, Real.cos_sub]
+  ring
+
+private theorem tangent_projection_signs {a r : ℝ}
+    (hr : r ∈ Set.Icc a (a + Real.pi / 2)) :
+    inner ℝ (tangentVector (r : Real.Angle)) (normalVector (a : Real.Angle)) ≤ 0 ∧
+    0 ≤ inner ℝ (tangentVector (r : Real.Angle)) (tangentVector (a : Real.Angle)) := by
+  rw [inner_tangent_normal, inner_tangent_tangent]
+  constructor
+  · exact neg_nonpos.mpr (Real.sin_nonneg_of_nonneg_of_le_pi (by linarith [hr.1])
+      (by linarith [hr.2, Real.pi_pos]))
+  · exact Real.cos_nonneg_of_mem_Icc ⟨by linarith [hr.1, Real.pi_pos],
+      by linarith [hr.2]⟩
+
+private theorem support_selection_projection_order (K : ConvexBody Point)
+    {a s t : ℝ} (hs : s ∈ Set.Icc a (a + Real.pi / 2))
+    (ht : t ∈ Set.Icc a (a + Real.pi / 2)) (hst : s < t)
+    {p q : Point} (hp : p ∈ exposedEdge K (s : Real.Angle))
+    (hq : q ∈ exposedEdge K (t : Real.Angle)) :
+    inner ℝ q (normalVector (a : Real.Angle)) ≤
+        inner ℝ p (normalVector (a : Real.Angle)) ∧
+      inner ℝ p (tangentVector (a : Real.Angle)) ≤
+        inner ℝ q (tangentVector (a : Real.Angle)) := by
+  have hsp := inner_le_supportValue K hq.1 (s : Real.Angle)
+  have htp := inner_le_supportValue K hp.1 (t : Real.Angle)
+  rw [← hp.2] at hsp
+  rw [← hq.2] at htp
+  obtain ⟨α, β, hα, hβ, hrepr⟩ := exists_nonneg_tangent_coefficients p q s t hst
+    (by linarith [hs.1, ht.2, Real.pi_pos]) hsp htp
+  have hsigns := tangent_projection_signs hs
+  have hsignt := tangent_projection_signs ht
+  constructor
+  · have h := congrArg (fun p ↦ inner ℝ p (normalVector (a : Real.Angle))) hrepr
+    simp only [inner_sub_left, inner_add_left, real_inner_smul_left] at h
+    linarith [mul_nonpos_of_nonneg_of_nonpos hα hsigns.1,
+      mul_nonpos_of_nonneg_of_nonpos hβ hsignt.1]
+  · have h := congrArg (fun p ↦ inner ℝ p (tangentVector (a : Real.Angle))) hrepr
+    simp only [inner_sub_left, inner_add_left, real_inner_smul_left] at h
+    linarith [mul_nonneg hα hsigns.2, mul_nonneg hβ hsignt.2]
+
+private theorem support_selection_boundedVariationOn_quarter (K : ConvexBody Point)
+    (f : ℝ → Point) (a : ℝ)
+    (hf : ∀ t ∈ Set.Icc a (a + Real.pi / 2), f t ∈ exposedEdge K (t : Real.Angle)) :
+    BoundedVariationOn f (Set.Icc a (a + Real.pi / 2)) := by
+  let x : ℝ → ℝ := fun t ↦ -inner ℝ (f t) (normalVector (a : Real.Angle))
+  let y : ℝ → ℝ := fun t ↦ inner ℝ (f t) (tangentVector (a : Real.Angle))
+  have hx : MonotoneOn x (Set.Icc a (a + Real.pi / 2)) := by
+    intro s hs t ht hst
+    rcases hst.eq_or_lt with rfl | hst
+    · rfl
+    · exact neg_le_neg (support_selection_projection_order K hs ht hst (hf s hs) (hf t ht)).1
+  have hy : MonotoneOn y (Set.Icc a (a + Real.pi / 2)) := by
+    intro s hs t ht hst
+    rcases hst.eq_or_lt with rfl | hst
+    · rfl
+    · exact (support_selection_projection_order K hs ht hst (hf s hs) (hf t ht)).2
+  have ha : a ≤ a + Real.pi / 2 := by linarith [Real.pi_pos]
+  have hxv : BoundedVariationOn x (Set.Icc a (a + Real.pi / 2)) := by
+    simpa only [Set.inter_self] using
+      hx.locallyBoundedVariationOn a (a + Real.pi / 2) ⟨le_rfl, ha⟩ ⟨ha, le_rfl⟩
+  have hyv : BoundedVariationOn y (Set.Icc a (a + Real.pi / 2)) := by
+    simpa only [Set.inter_self] using
+      hy.locallyBoundedVariationOn a (a + Real.pi / 2) ⟨le_rfl, ha⟩ ⟨ha, le_rfl⟩
+  let U := ContinuousLinearMap.toSpanSingleton ℝ (-normalVector (a : Real.Angle))
+  let V := ContinuousLinearMap.toSpanSingleton ℝ (tangentVector (a : Real.Angle))
+  have hsum := (U.lipschitzWith.comp_boundedVariationOn hxv).add
+    (V.lipschitzWith.comp_boundedVariationOn hyv)
+  have heq : (U ∘ x) + (V ∘ y) = f := by
+    funext t
+    simpa [U, V, x, y, Function.comp_def] using
+      inner_normalVector_smul_add_inner_tangentVector_smul (f t) (a : Real.Angle)
+  rwa [heq] at hsum
+
+private theorem boundedVariationOn_Icc_trans {f : ℝ → Point} {a b c : ℝ}
+    (hab : a ≤ b) (hbc : b ≤ c) (hf : BoundedVariationOn f (Set.Icc a b))
+    (hg : BoundedVariationOn f (Set.Icc b c)) :
+    BoundedVariationOn f (Set.Icc a c) := by
+  have h := eVariationOn.Icc_add_Icc f (s := Set.univ) hab hbc (Set.mem_univ b)
+  simp only [Set.univ_inter] at h
+  change eVariationOn f (Set.Icc a c) ≠ ⊤
+  rw [← h]
+  exact ENNReal.add_ne_top.mpr ⟨hf, hg⟩
+
+/-- Every selection of points from the exposed edges has bounded variation on bounded intervals. -/
+theorem boundedVariationOn_of_mem_exposedEdge (K : ConvexBody Point)
+    (f : ℝ → Point) (hf : ∀ t : ℝ, f t ∈ exposedEdge K (t : Real.Angle)) (a b : ℝ) :
+    BoundedVariationOn f (Set.Icc a b) := by
+  have hN : ∀ n : ℕ, BoundedVariationOn f (Set.Icc a (a + n * (Real.pi / 2))) := by
+    intro n
+    induction n with
+    | zero =>
+      simp only [Nat.cast_zero, zero_mul, add_zero]
+      exact BoundedVariationOn.of_subsingleton (Set.subsingleton_Icc_of_ge (le_refl a))
+    | succ n ih =>
+      have hn : a ≤ a + n * (Real.pi / 2) :=
+        le_add_of_nonneg_right (mul_nonneg (Nat.cast_nonneg n) (by positivity))
+      have hn' : a + n * (Real.pi / 2) ≤ a + (n + 1) * (Real.pi / 2) := by
+        linarith [Real.pi_pos]
+      have hpiece := support_selection_boundedVariationOn_quarter K f
+        (a + n * (Real.pi / 2)) (fun t _ ↦ hf t)
+      have hstep := boundedVariationOn_Icc_trans hn hn' ih
+        (by convert hpiece using 1; congr 1; ring)
+      simpa only [Nat.cast_add, Nat.cast_one] using hstep
+  obtain ⟨n, hn⟩ := exists_nat_gt ((b - a) / (Real.pi / 2))
+  have hb : b ≤ a + n * (Real.pi / 2) := by
+    have h := (div_lt_iff₀ (by positivity : 0 < Real.pi / 2)).mp hn
+    linarith
+  exact (hN n).mono (Set.Icc_subset_Icc_right hb)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Boundary Approximation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem positiveVertex_boundedVariation (K : ConvexBody Point) (a b : ℝ) (hab : a ≤ b) :
+    BoundedVariationOn (fun t : ℝ ↦ (edgeVertices K (t : Real.Angle)).1) (Set.Icc a b) := by
+  rcases hab.eq_or_lt with rfl | hab
+  · exact BoundedVariationOn.of_subsingleton (Set.subsingleton_Icc_of_ge le_rfl)
+  · exact boundedVariationOn_of_mem_exposedEdge K _ (fun t ↦ edgeVertices_fst_mem K _) a b
+
+private theorem exists_facePreserving_polygon (K : ConvexBody Point)
+    (F : Finset Real.Angle) {ε : ℝ} (hε : 0 < ε) :
+    ∃ (V : Finset Point) (P : ConvexBody Point),
+      V.Nonempty ∧
+      (P : Set Point) = convexHull ℝ (V : Set Point) ∧
+      (P : Set Point) ⊆ (K : Set Point) ∧
+      (∀ t ∈ F, exposedEdge P t = exposedEdge K t) ∧
+      Metric.hausdorffDist (P : Set Point) (K : Set Point) ≤ ε := by
+  obtain ⟨S, hSK, hSfinite, hcover⟩ :=
+    Metric.finite_approx_of_totallyBounded K.isCompact.totallyBounded ε hε
+  let N : Finset Point := hSfinite.toFinset
+  let E : Finset Point :=
+    F.image (fun t ↦ (edgeVertices K t).1) ∪
+      F.image (fun t ↦ (edgeVertices K t).2)
+  let p₀ : Point := Classical.choose K.nonempty
+  let V : Finset Point := insert p₀ (N ∪ E)
+  have hp₀ : p₀ ∈ K := Classical.choose_spec K.nonempty
+  have hVnonempty : V.Nonempty := by
+    exact ⟨p₀, Finset.mem_insert_self p₀ _⟩
+  have hVsubset : (V : Set Point) ⊆ (K : Set Point) := by
+    intro p hp
+    simp only [V, E, Finset.mem_coe, Finset.mem_insert, Finset.mem_union,
+      Finset.mem_image] at hp
+    rcases hp with rfl | hpN | hpE
+    · exact hp₀
+    · exact hSK (hSfinite.mem_toFinset.mp hpN)
+    · rcases hpE with hpE | hpE
+      · obtain ⟨t, ht, rfl⟩ := hpE
+        exact (edgeVertices_fst_mem K t).1
+      · obtain ⟨t, ht, rfl⟩ := hpE
+        exact (edgeVertices_snd_mem K t).1
+  let P : ConvexBody Point :=
+    { carrier := convexHull ℝ (V : Set Point)
+      convex' := convex_convexHull ℝ _
+      isCompact' := V.finite_toSet.isCompact_convexHull ℝ
+      nonempty' := ⟨p₀, subset_convexHull ℝ _ (Finset.mem_coe.mpr
+        (Finset.mem_insert_self p₀ _))⟩ }
+  have hPcarrier : (P : Set Point) = convexHull ℝ (V : Set Point) := rfl
+  have hPK : (P : Set Point) ⊆ (K : Set Point) := by
+    change convexHull ℝ (V : Set Point) ⊆ (K : Set Point)
+    exact convexHull_min hVsubset K.convex
+  have hfaces : ∀ t ∈ F, exposedEdge P t = exposedEdge K t := by
+    intro t ht
+    have hfstV : (edgeVertices K t).1 ∈ V := by
+      simp only [V, E, Finset.mem_insert, Finset.mem_union, Finset.mem_image]
+      exact Or.inr (Or.inr (Or.inl ⟨t, ht, rfl⟩))
+    have hsndV : (edgeVertices K t).2 ∈ V := by
+      simp only [V, E, Finset.mem_insert, Finset.mem_union, Finset.mem_image]
+      exact Or.inr (Or.inr (Or.inr ⟨t, ht, rfl⟩))
+    have hfstP : (edgeVertices K t).1 ∈ P := by
+      exact subset_convexHull ℝ (V : Set Point) hfstV
+    have hsndP : (edgeVertices K t).2 ∈ P := by
+      exact subset_convexHull ℝ (V : Set Point) hsndV
+    have hsupport : supportValue P t = supportValue K t := by
+      apply le_antisymm
+      · apply csSup_le (P.nonempty.image _)
+        rintro _ ⟨p, hp, rfl⟩
+        exact inner_le_supportValue K (hPK hp) t
+      · have h := inner_le_supportValue P hfstP t
+        rw [(edgeVertices_fst_mem K t).2] at h
+        exact h
+    ext p
+    constructor
+    · intro hp
+      exact ⟨hPK hp.1, hp.2.trans hsupport⟩
+    · intro hp
+      have hpseg : p ∈ segment ℝ (edgeVertices K t).2 (edgeVertices K t).1 := by
+        rw [← exposedEdge_eq_segment_edgeVertices]
+        exact hp
+      have hpP : p ∈ P := P.convex.segment_subset hsndP hfstP hpseg
+      exact ⟨hpP, hp.2.trans hsupport.symm⟩
+  have hdist : Metric.hausdorffDist (P : Set Point) (K : Set Point) ≤ ε := by
+    apply Metric.hausdorffDist_le_of_mem_dist hε.le
+    · intro p hp
+      exact ⟨p, hPK hp, dist_self p ▸ hε.le⟩
+    · intro p hp
+      have hpcover := hcover hp
+      simp only [Set.mem_iUnion, Metric.mem_ball] at hpcover
+      obtain ⟨q, hqS, hpq⟩ := hpcover
+      have hqN : q ∈ N := hSfinite.mem_toFinset.mpr hqS
+      have hqV : q ∈ V := by
+        simp only [V, Finset.mem_insert, Finset.mem_union]
+        exact Or.inr (Or.inl hqN)
+      have hqP : q ∈ P := subset_convexHull ℝ (V : Set Point) hqV
+      exact ⟨q, hqP, hpq.le⟩
+  exact ⟨V, P, hVnonempty, hPcarrier, hPK, hfaces, hdist⟩
+
+theorem exists_facePreserving_polygonApproximation (K : ConvexBody Point)
+    (F : Finset Real.Angle) :
+    ∃ (V : ℕ → Finset Point) (P : ℕ → ConvexBody Point),
+      (∀ n, (V n).Nonempty ∧
+        (P n : Set Point) = convexHull ℝ (V n : Set Point) ∧
+        (P n : Set Point) ⊆ (K : Set Point) ∧
+        ∀ t ∈ F, exposedEdge (P n) t = exposedEdge K t) ∧
+      ∀ n : ℕ, 1 ≤ n → Metric.hausdorffDist (P n : Set Point) (K : Set Point) ≤ 1 / (n : ℝ) := by
+  let m : ℕ → ℕ := fun n ↦ max n 1
+  let V : ℕ → Finset Point := fun n ↦
+    Classical.choose (exists_facePreserving_polygon K F
+      (show 0 < 1 / (m n : ℝ) by positivity))
+  let P : ℕ → ConvexBody Point := fun n ↦
+    Classical.choose (Classical.choose_spec (exists_facePreserving_polygon K F
+      (show 0 < 1 / (m n : ℝ) by positivity)))
+  have hdata (n : ℕ) := Classical.choose_spec (Classical.choose_spec
+    (exists_facePreserving_polygon K F (show 0 < 1 / (m n : ℝ) by positivity)))
+  refine ⟨V, P, ?_, ?_⟩
+  · intro n
+    exact ⟨(hdata n).1, (hdata n).2.1, (hdata n).2.2.1, (hdata n).2.2.2.1⟩
+  · intro n hn
+    have hmn : m n = n := max_eq_left hn
+    simpa only [P, m, hmn] using (hdata n).2.2.2.2
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Combination
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval
+
+universe u v w
+
+namespace MovingSofa
+
+/-- A barycentric operation has an injective realization as convex combinations in a real space. -/
+def IsConvexDomain {α : Type u} (c : I → α → α → α) : Prop :=
+  ∃ (V : ModuleCat.{v} ℝ) (e : α → V), Function.Injective e ∧
+    Convex ℝ (Set.range e) ∧
+    ∀ (t : I) x y, e (c t x y) = (1 - (t : ℝ)) • e x + (t : ℝ) • e y
+
+/-- Preservation of the specified barycentric operations. -/
+def IsConvexLinear {α : Type u} {β : Type v}
+    (cα : I → α → α → α) (cβ : I → β → β → β) (f : α → β) : Prop :=
+  ∀ t x y, f (cα t x y) = cβ t (f x) (f y)
+
+/-- Separate preservation of barycentric combinations in both variables. -/
+def IsConvexBilinear {α : Type u} {β : Type v} {γ : Type w}
+    (cα : I → α → α → α) (cβ : I → β → β → β) (cγ : I → γ → γ → γ)
+    (g : α → β → γ) : Prop :=
+  (∀ x, IsConvexLinear cβ cγ (g x)) ∧
+    ∀ y, IsConvexLinear cα cγ (fun x ↦ g x y)
+
+/-- The usual barycentric combination of real numbers. -/
+def realCombination (t : I) (x y : ℝ) : ℝ := (1 - (t : ℝ)) * x + (t : ℝ) * y
+
+/-- A quadratic functional is the diagonal of a separately convex-linear real map. -/
+def IsQuadraticFunctional {α : Type u} (c : I → α → α → α) (f : α → ℝ) : Prop :=
+  ∃ g : α → α → ℝ, IsConvexBilinear c c realCombination g ∧ ∀ x, f x = g x x
+
+/-- Concavity or convexity according to the direction of the barycentric inequality. -/
+def IsConvexFunctional {α : Type u} (c : I → α → α → α) (f : α → ℝ)
+    (concave : Bool) : Prop :=
+  ∀ t x y, if concave then realCombination t (f x) (f y) ≤ f (c t x y)
+    else f (c t x y) ≤ realCombination t (f x) (f y)
+
+/-- The segment function, extended by zero outside its parameter interval. -/
+def segmentFunctional {α : Type u} (c : I → α → α → α) (f : α → ℝ)
+    (x y : α) (t : ℝ) : ℝ :=
+  if ht : t ∈ Set.Icc (0 : ℝ) 1 then f (c ⟨t, ht⟩ x y) else 0
+
+/-- The right derivative along the barycentric segment; used for quadratic functionals. -/
+def convexDirectionalDerivative {α : Type u} (c : I → α → α → α)
+    (f : α → ℝ) (x y : α) : ℝ :=
+  derivWithin (segmentFunctional c f x y) (Set.Icc 0 1) 0
+
+/-- Minkowski interpolation of nonempty compact convex bodies, including both endpoints. -/
+def convexBodyCombination (t : I) (K L : ConvexBody Point) : ConvexBody Point :=
+  (1 - (t : ℝ)) • K + (t : ℝ) • L
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Area Superlevel
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped Pointwise unitInterval
+
+namespace MovingSofa
+
+theorem convexBody_area_superlevel (K L : ConvexBody Point)
+    (hK : (11 : ℝ) / 5 ≤ ClassicalResults.area (K : Set Point))
+    (hL : (11 : ℝ) / 5 ≤ ClassicalResults.area (L : Set Point)) (t : I) :
+    (11 : ℝ) / 5 ≤ ClassicalResults.area (convexBodyCombination t K L : Set Point) := by
+  let a : ℝ := 1 - (t : ℝ)
+  let b : ℝ := t
+  let A : Set Point := a • (K : Set Point)
+  let B : Set Point := b • (L : Set Point)
+  have ha : 0 ≤ a := by dsimp [a]; exact sub_nonneg.mpr t.2.2
+  have hb : 0 ≤ b := t.2.1
+  have hab : a + b = 1 := by dsimp [a, b]; ring
+  have hAcomp : IsCompact A := K.isCompact.smul a
+  have hBcomp : IsCompact B := L.isCompact.smul b
+  have hABcomp : IsCompact (A + B) := hAcomp.add hBcomp
+  have hbm :
+      volume A ^ (2 : ℝ)⁻¹ + volume B ^ (2 : ℝ)⁻¹ ≤
+        volume (A + B) ^ (2 : ℝ)⁻¹ := by
+    convert brunn_minkowski_euclideanSpace (d := 1) (A := A) (B := B)
+      (K.nonempty.smul_set (a := a)) hAcomp.measurableSet
+      (L.nonempty.smul_set (a := b)) hBcomp.measurableSet hABcomp.measurableSet using 1 <;>
+      norm_num
+  have hKfin : volume (K : Set Point) ≠ ⊤ := K.isCompact.measure_lt_top.ne
+  have hLfin : volume (L : Set Point) ≠ ⊤ := L.isCompact.measure_lt_top.ne
+  have hABfin : volume (A + B) ≠ ⊤ := hABcomp.measure_lt_top.ne
+  have hKvol : ENNReal.ofReal ((11 : ℝ) / 5) ≤ volume (K : Set Point) :=
+    (ENNReal.ofReal_le_iff_le_toReal hKfin).2 hK
+  have hLvol : ENNReal.ofReal ((11 : ℝ) / 5) ≤ volume (L : Set Point) :=
+    (ENNReal.ofReal_le_iff_le_toReal hLfin).2 hL
+  have hKroot := ENNReal.monotone_rpow_of_nonneg (by positivity : 0 ≤ (2 : ℝ)⁻¹) hKvol
+  have hLroot := ENNReal.monotone_rpow_of_nonneg (by positivity : 0 ≤ (2 : ℝ)⁻¹) hLvol
+  have hroot :
+      ENNReal.ofReal ((11 : ℝ) / 5) ^ (2 : ℝ)⁻¹ ≤
+        volume (A + B) ^ (2 : ℝ)⁻¹ := by
+    rw [volume_smul_rpow_half (K : Set Point) a ha,
+      volume_smul_rpow_half (L : Set Point) b hb] at hbm
+    calc
+      _ = (ENNReal.ofReal a + ENNReal.ofReal b) *
+          ENNReal.ofReal ((11 : ℝ) / 5) ^ (2 : ℝ)⁻¹ := by
+        rw [← ENNReal.ofReal_add ha hb, hab, ENNReal.ofReal_one, one_mul]
+      _ ≤ ENNReal.ofReal a * volume (K : Set Point) ^ (2 : ℝ)⁻¹ +
+          ENNReal.ofReal b * volume (L : Set Point) ^ (2 : ℝ)⁻¹ := by
+        rw [add_mul]
+        exact add_le_add
+          (by simpa [mul_comm] using mul_le_mul_left hKroot (ENNReal.ofReal a))
+          (by simpa [mul_comm] using mul_le_mul_left hLroot (ENNReal.ofReal b))
+      _ ≤ _ := hbm
+  have hvol : ENNReal.ofReal ((11 : ℝ) / 5) ≤ volume (A + B) := by
+    calc
+      _ = (ENNReal.ofReal ((11 : ℝ) / 5) ^ (2 : ℝ)⁻¹) ^ (2 : ℕ) :=
+        (ENNReal.rpow_inv_natCast_pow (by norm_num) _).symm
+      _ ≤ (volume (A + B) ^ (2 : ℝ)⁻¹) ^ (2 : ℕ) := pow_le_pow_left' hroot 2
+      _ = _ := ENNReal.rpow_inv_natCast_pow (by norm_num) _
+  have hcomb : (convexBodyCombination t K L : Set Point) = A + B := by
+    rfl
+  rw [hcomb]
+  exact (ENNReal.ofReal_le_iff_le_toReal hABfin).1 hvol
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Pulling barycentric functionals back along convex-linear maps
+
+A convex-linear map transports one barycentric operation to another, so every notion defined
+from that operation pulls back along it: a quadratic functional stays quadratic, and its
+directional derivative along a segment is the directional derivative between the images of the
+two endpoints. Neither statement needs any topology or differentiability: the two segment
+functions are literally equal.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval
+
+universe u v
+
+namespace MovingSofa
+
+/-- A quadratic functional pulled back along a convex-linear map is quadratic. -/
+theorem IsQuadraticFunctional.comp_isConvexLinear {α : Type u} {β : Type v}
+    {cα : I → α → α → α} {cβ : I → β → β → β} {F : α → β}
+    (hF : IsConvexLinear cα cβ F) {f : β → ℝ} (hf : IsQuadraticFunctional cβ f) :
+    IsQuadraticFunctional cα fun x ↦ f (F x) := by
+  obtain ⟨g, hg, hfg⟩ := hf
+  refine ⟨fun x y ↦ g (F x) (F y), ⟨fun x s y z ↦ ?_, fun y s x z ↦ ?_⟩, fun x ↦ hfg (F x)⟩
+  · change g (F x) (F (cα s y z)) = _
+    rw [hF]
+    exact hg.1 (F x) s (F y) (F z)
+  · change g (F (cα s x z)) (F y) = _
+    rw [hF]
+    exact hg.2 (F y) s (F x) (F z)
+
+/-- The directional derivative of a functional pulled back along a convex-linear map is the
+directional derivative of the functional between the images. -/
+theorem convexDirectionalDerivative_comp_isConvexLinear {α : Type u} {β : Type v}
+    {cα : I → α → α → α} {cβ : I → β → β → β} {F : α → β}
+    (hF : IsConvexLinear cα cβ F) (f : β → ℝ) (x y : α) :
+    convexDirectionalDerivative cα (fun z ↦ f (F z)) x y =
+      convexDirectionalDerivative cβ f (F x) (F y) := by
+  unfold convexDirectionalDerivative
+  congr 1
+  funext s
+  simp only [segmentFunctional]
+  split_ifs with hs
+  · exact congrArg f (hF _ x y)
+  · rfl
+
+/-- A convex or concave functional pulled back along a convex-linear map keeps its direction of
+convexity. -/
+theorem IsConvexFunctional.comp_isConvexLinear {α : Type u} {β : Type v}
+    {cα : I → α → α → α} {cβ : I → β → β → β} {F : α → β}
+    (hF : IsConvexLinear cα cβ F) {f : β → ℝ} {concave : Bool}
+    (hf : IsConvexFunctional cβ f concave) :
+    IsConvexFunctional cα (fun x ↦ f (F x)) concave := by
+  intro s x y
+  have h := hf s (F x) (F y)
+  rw [← hF s x y] at h
+  exact h
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Area lower bound from ordered support contacts
+
+A compact convex set `K` of the plane that contains a base point `L = (ℓ, 0)` and lies in the
+closed quadrant `{(a, b) | ℓ ≤ a, 0 ≤ b}` has area at least the shoelace expression of any
+finite fan `L, P₀, …, P_n` of support contacts taken at strictly increasing normal angles in
+`[0, π)`.
+
+Two ordered contacts span a nonnegatively oriented determinant over `L`: expanding the
+support inequalities in coordinates turns `sin (β - α) · (P - L) × (Q - L)` into a sum of two
+products of nonnegative factors. The fan triangles `convexHull ℝ {L, Pᵢ, Pᵢ₊₁}` therefore lie
+in `K` with area one half of their determinant, and two of them can meet only along the line
+through `L` and the contact they share, which is planar null. Finite additivity of the volume
+off that null set and monotonicity give the bound. Repeated contacts, zero contact vectors and
+collinear consecutive rays are allowed: a degenerate triangle simply has vanishing area.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- Two ordered support contacts of a set lying in the closed quadrant above a base point
+have nonnegative oriented determinant over that base point. -/
+theorem planeCrossProduct_sub_nonneg_of_support {K : Set Point} {L P Q : Point}
+    (hL : L ∈ K) (hquadrant : ∀ p ∈ K, L 0 ≤ p 0 ∧ 0 ≤ p 1)
+    {α β : ℝ} (hα : 0 ≤ α) (hαβ : α < β) (hβ : β < Real.pi)
+    (hP : P ∈ K) (hQ : Q ∈ K)
+    (hPs : ∀ x ∈ K, inner ℝ x (normalVector (α : Real.Angle)) ≤
+      inner ℝ P (normalVector (α : Real.Angle)))
+    (hQs : ∀ x ∈ K, inner ℝ x (normalVector (β : Real.Angle)) ≤
+      inner ℝ Q (normalVector (β : Real.Angle))) :
+    0 ≤ planeCrossProduct (P - L) (Q - L) := by
+  have hsβ : 0 < Real.sin β := Real.sin_pos_of_pos_of_lt_pi (hα.trans_lt hαβ) hβ
+  have hsα : 0 ≤ Real.sin α := Real.sin_nonneg_of_nonneg_of_le_pi hα (by linarith)
+  have hsδ : 0 < Real.sin β * Real.cos α - Real.cos β * Real.sin α := by
+    have h := Real.sin_pos_of_pos_of_lt_pi (x := β - α) (by linarith) (by linarith)
+    rwa [Real.sin_sub] at h
+  have hz : 0 ≤ (P 0 - Q 0) * Real.cos α + (P 1 - Q 1) * Real.sin α := by
+    have h := hPs Q hQ
+    rw [inner_normalVector_real, inner_normalVector_real] at h
+    linarith
+  have hk : 0 ≤ (Q 0 - P 0) * Real.cos β + (Q 1 - P 1) * Real.sin β := by
+    have h := hQs P hP
+    rw [inner_normalVector_real, inner_normalVector_real] at h
+    linarith
+  have hB : 0 ≤ (Q 0 - L 0) * Real.cos β + (Q 1 - L 1) * Real.sin β := by
+    have h := hQs L hL
+    rw [inner_normalVector_real, inner_normalVector_real] at h
+    linarith
+  have hq0 : 0 ≤ Q 0 - L 0 := sub_nonneg.mpr (hquadrant Q hQ).1
+  have hc : 0 ≤ (Q 0 - L 0) * Real.cos α + (Q 1 - L 1) * Real.sin α := by
+    have hid : Real.sin β * ((Q 0 - L 0) * Real.cos α + (Q 1 - L 1) * Real.sin α)
+        = Real.sin α * ((Q 0 - L 0) * Real.cos β + (Q 1 - L 1) * Real.sin β)
+          + (Real.sin β * Real.cos α - Real.cos β * Real.sin α) * (Q 0 - L 0) := by
+      ring
+    nlinarith [mul_nonneg hsα hB, mul_nonneg hsδ.le hq0]
+  have hcross : planeCrossProduct (P - L) (Q - L)
+      = (P 0 - L 0) * (Q 1 - L 1) - (P 1 - L 1) * (Q 0 - L 0) := by
+    simp [planeCrossProduct]
+  have hid2 : (Real.sin β * Real.cos α - Real.cos β * Real.sin α) *
+      ((P 0 - L 0) * (Q 1 - L 1) - (P 1 - L 1) * (Q 0 - L 0))
+      = ((Q 0 - L 0) * Real.cos α + (Q 1 - L 1) * Real.sin α) *
+          ((Q 0 - P 0) * Real.cos β + (Q 1 - P 1) * Real.sin β)
+        + ((P 0 - Q 0) * Real.cos α + (P 1 - Q 1) * Real.sin α) *
+          ((Q 0 - L 0) * Real.cos β + (Q 1 - L 1) * Real.sin β) := by
+    ring
+  rw [hcross]
+  nlinarith [mul_nonneg hc hk, mul_nonneg hz hB]
+
+theorem supportContact_fan_area (K : Set Point)
+    (hcK : IsCompact K) (hvK : Convex ℝ K) (L : Point) (hL : L ∈ K)
+    (hLy : L 1 = 0) (hquadrant : ∀ p ∈ K, L 0 ≤ p 0 ∧ 0 ≤ p 1)
+    (n : ℕ) (θ : Fin (n + 1) → ℝ)
+    (hθ : StrictMono θ) (hθrange : ∀ i, 0 ≤ θ i ∧ θ i < Real.pi)
+    (P : Fin (n + 1) → Point) (hP : ∀ i, P i ∈ K)
+    (hsupport : ∀ i q, q ∈ K →
+      inner ℝ q (normalVector (θ i : Real.Angle)) ≤
+        inner ℝ (P i) (normalVector (θ i : Real.Angle))) :
+    0 ≤ (1 / 2 : ℝ) * ∑ i : Fin n,
+      planeCrossProduct (P i.castSucc - L) (P i.succ - L) ∧
+    (1 / 2 : ℝ) * (∑ i : Fin n,
+      planeCrossProduct (P i.castSucc - L) (P i.succ - L)) ≤ ClassicalResults.area K := by
+  have hquad : ∀ x ∈ K, 0 ≤ (x - L) 0 ∧ 0 ≤ (x - L) 1 := by
+    intro x hx
+    obtain ⟨h1, h2⟩ := hquadrant x hx
+    exact ⟨by simpa using sub_nonneg.mpr h1, by simpa [hLy] using h2⟩
+  have hmono : ∀ i j : Fin (n + 1), i ≤ j → 0 ≤ planeCrossProduct (P i - L) (P j - L) := by
+    intro i j hij
+    rcases eq_or_lt_of_le hij with rfl | hlt
+    · simp
+    · exact planeCrossProduct_sub_nonneg_of_support hL hquadrant (hθrange i).1 (hθ hlt)
+        (hθrange j).2 (hP i) (hP j) (fun x hx ↦ hsupport i x hx) (fun x hx ↦ hsupport j x hx)
+  have hcnn : ∀ i : Fin n, 0 ≤ planeCrossProduct (P i.castSucc - L) (P i.succ - L) :=
+    fun i ↦ hmono _ _ Fin.castSucc_lt_succ.le
+  have hsumnn : 0 ≤ ∑ i : Fin n, planeCrossProduct (P i.castSucc - L) (P i.succ - L) :=
+    Finset.sum_nonneg fun i _ ↦ hcnn i
+  refine ⟨by linarith, ?_⟩
+  set T : Fin n → Set Point := fun i ↦ convexHull ℝ {L, P i.castSucc, P i.succ} with hTdef
+  have hTsub : ∀ i, T i ⊆ K := by
+    intro i
+    refine convexHull_min ?_ hvK
+    intro x hx
+    simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hx
+    rcases hx with rfl | rfl | rfl
+    exacts [hL, hP _, hP _]
+  have hTvol : ∀ i, volume (T i) = ENNReal.ofReal
+      ((1 / 2 : ℝ) * planeCrossProduct (P i.castSucc - L) (P i.succ - L)) := by
+    intro i
+    have hpos := hcnn i
+    simp only [planeCrossProduct] at hpos
+    simp only [hTdef, EuclideanSpace.volume_convexHull_triple]
+    congr 1
+    rw [abs_of_nonneg (by linarith)]
+    simp only [planeCrossProduct]
+    ring
+  have hTmeas : ∀ i, NullMeasurableSet (T i) volume := by
+    intro i
+    refine (IsCompact.measurableSet ?_).nullMeasurableSet
+    exact (((Set.finite_singleton _).insert _).insert _).isCompact_convexHull ℝ
+  have hsector : ∀ i : Fin n, T i ⊆ {x : Point |
+      0 ≤ planeCrossProduct (P i.castSucc - L) (x - L) ∧
+      0 ≤ planeCrossProduct (x - L) (P i.succ - L)} := by
+    intro i
+    refine convexHull_min ?_ (convex_setOf_planeCrossProduct_fan _ _ _)
+    intro x hx
+    simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hx
+    rcases hx with rfl | rfl | rfl
+    · exact ⟨by simp, by simp⟩
+    · exact ⟨by simp, hcnn i⟩
+    · exact ⟨hcnn i, by simp⟩
+  have hdisj : Pairwise (Function.onFun (AEDisjoint volume) T) := by
+    have hkey : ∀ i j : Fin n, i < j → volume (T i ∩ T j) = 0 := by
+      intro i j hij
+      rcases eq_or_lt_of_le (hcnn i) with h0 | h0
+      · exact measure_mono_null Set.inter_subset_left (by rw [hTvol i, ← h0]; simp)
+      rcases eq_or_lt_of_le (hcnn j) with h1 | h1
+      · exact measure_mono_null Set.inter_subset_right (by rw [hTvol j, ← h1]; simp)
+      have hne2 : P j.castSucc - L ≠ 0 := by
+        intro h
+        rw [h] at h1
+        simp [planeCrossProduct] at h1
+      have hne1 : P i.succ - L ≠ 0 := by
+        intro h
+        rw [h] at h0
+        simp [planeCrossProduct] at h0
+      refine measure_mono_null ?_ (volume_setOf_planeCrossProduct_sub_eq_zero hne1 L)
+      rintro x ⟨hxi, hxj⟩
+      have hxK : x ∈ K := hTsub i hxi
+      have h2 := (hsector i hxi).2
+      have h3 := (hsector j hxj).1
+      have hle : i.succ ≤ j.castSucc := by
+        have hij' : (i : ℕ) < (j : ℕ) := hij
+        rw [Fin.le_def]
+        simp only [Fin.val_succ, Fin.val_castSucc]
+        omega
+      have h5 : 0 ≤ planeCrossProduct (P i.succ - L) (x - L) :=
+        planeCrossProduct_nonneg_trans hne2
+          (hquad _ (hP _)).1 (hquad _ (hP _)).2 (hquad _ (hP _)).1 (hquad _ (hP _)).2
+          (hquad x hxK).1 (hquad x hxK).2 (hmono _ _ hle) h3
+      change planeCrossProduct (P i.succ - L) (x - L) = 0
+      rw [planeCrossProduct_swap (x - L)] at h2
+      linarith
+    intro i j hij
+    rcases lt_or_gt_of_ne hij with h | h
+    · exact hkey i j h
+    · change volume (T i ∩ T j) = 0
+      rw [Set.inter_comm]
+      exact hkey j i h
+  have hunion : ∑ i : Fin n, volume (T i) ≤ volume K := by
+    have h := measure_iUnion₀ (μ := volume) hdisj hTmeas
+    rw [tsum_fintype] at h
+    rw [← h]
+    exact measure_mono (Set.iUnion_subset hTsub)
+  have hgoal : ENNReal.ofReal ((1 / 2 : ℝ) * ∑ i : Fin n,
+      planeCrossProduct (P i.castSucc - L) (P i.succ - L)) ≤ volume K := by
+    rw [Finset.mul_sum, ENNReal.ofReal_sum_of_nonneg (fun i _ ↦ by linarith [hcnn i])]
+    simpa only [hTvol] using hunion
+  exact (ENNReal.ofReal_le_iff_le_toReal hcK.measure_lt_top.ne).mp hgoal
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Exposed Faces
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped Pointwise unitInterval
+
+namespace MovingSofa
+
+private theorem support_bound (K : ConvexBody Point) (a : Real.Angle)
+    {x : Point} (hx : x ∈ K) : inner ℝ x (normalVector a) ≤ supportValue K a := by
+  apply le_csSup
+  · exact (K.isCompact.image (continuous_id.inner continuous_const)).bddAbove
+  · exact ⟨x, hx, rfl⟩
+
+private theorem edge_nonempty (K : ConvexBody Point) (a : Real.Angle) :
+    (exposedEdge K a).Nonempty := by
+  obtain ⟨x, hx, he⟩ := (K.isCompact.image (continuous_id.inner continuous_const)).sSup_mem
+    (K.nonempty.image (fun x ↦ inner ℝ x (normalVector a)))
+  exact ⟨x, hx, he⟩
+
+private theorem mem_edge_iff (K : ConvexBody Point) (a : Real.Angle) (x : Point) :
+    x ∈ exposedEdge K a ↔ x ∈ K ∧ ∀ y ∈ K,
+      inner ℝ y (normalVector a) ≤ inner ℝ x (normalVector a) := by
+  change (x ∈ K ∧ inner ℝ x (normalVector a) = supportValue K a) ↔ _
+  constructor
+  · rintro ⟨hx, he⟩
+    exact ⟨hx, fun y hy ↦ he ▸ support_bound K a hy⟩
+  · rintro ⟨hx, hm⟩
+    refine ⟨hx, le_antisymm (support_bound K a hx) ?_⟩
+    exact csSup_le (K.nonempty.image _) (by rintro _ ⟨y, hy, rfl⟩; exact hm y hy)
+
+private theorem combination_zero (K L : ConvexBody Point) :
+    convexBodyCombination 0 K L = K := by
+  apply SetLike.coe_injective
+  simp [convexBodyCombination, Set.zero_smul_set L.nonempty]
+
+private theorem combination_one (K L : ConvexBody Point) :
+    convexBodyCombination 1 K L = L := by
+  apply SetLike.coe_injective
+  simp [convexBodyCombination, Set.zero_smul_set K.nonempty]
+
+/-- Exposed faces commute with convex interpolation, including zero and unit weights. -/
+theorem exposedEdge_convexBodyCombination (K L : ConvexBody Point) (a : Real.Angle)
+    (t : unitInterval) :
+    exposedEdge (convexBodyCombination t K L) a =
+      (1 - (t : ℝ)) • exposedEdge K a + (t : ℝ) • exposedEdge L a := by
+  by_cases h0 : t = 0
+  · subst t
+    simp [combination_zero, Set.zero_smul_set (edge_nonempty L a)]
+  by_cases h1 : t = 1
+  · subst t
+    simp [combination_one, Set.zero_smul_set (edge_nonempty K a)]
+  have ht : 0 < (t : ℝ) := lt_of_le_of_ne t.property.1 (by
+    intro h; exact h0 (Subtype.ext h.symm))
+  have hu : 0 < 1 - (t : ℝ) := sub_pos.mpr (lt_of_le_of_ne t.property.2 (by
+    intro h; exact h1 (Subtype.ext h)))
+  have mem_combo (x y : Point) (hx : x ∈ K) (hy : y ∈ L) :
+      (1 - (t : ℝ)) • x + (t : ℝ) • y ∈ convexBodyCombination t K L := by
+    exact Set.add_mem_add (Set.smul_mem_smul_set hx) (Set.smul_mem_smul_set hy)
+  ext p
+  constructor
+  · intro hp
+    obtain ⟨hp, hm⟩ := (mem_edge_iff _ a p).mp hp
+    obtain ⟨u, huK, v, hvL, rfl⟩ := hp
+    obtain ⟨x, hx, rfl⟩ := huK
+    obtain ⟨y, hy, rfl⟩ := hvL
+    have hxedge : x ∈ exposedEdge K a := (mem_edge_iff K a x).mpr ⟨hx, by
+      intro z hz
+      have h := hm _ (mem_combo z y hz hy)
+      simp only [inner_add_left, inner_smul_left, RCLike.conj_to_real] at h
+      nlinarith⟩
+    have hyedge : y ∈ exposedEdge L a := (mem_edge_iff L a y).mpr ⟨hy, by
+      intro z hz
+      have h := hm _ (mem_combo x z hx hz)
+      simp only [inner_add_left, inner_smul_left, RCLike.conj_to_real] at h
+      nlinarith⟩
+    exact Set.add_mem_add (Set.smul_mem_smul_set hxedge) (Set.smul_mem_smul_set hyedge)
+  · rintro ⟨u, huK, v, hvL, rfl⟩
+    obtain ⟨x, hx, rfl⟩ := huK
+    obtain ⟨y, hy, rfl⟩ := hvL
+    obtain ⟨hx, hmx⟩ := (mem_edge_iff K a x).mp hx
+    obtain ⟨hy, hmy⟩ := (mem_edge_iff L a y).mp hy
+    apply (mem_edge_iff _ a _).mpr
+    refine ⟨mem_combo x y hx hy, ?_⟩
+    rintro p ⟨u, huK, v, hvL, rfl⟩
+    obtain ⟨z, hz, rfl⟩ := huK
+    obtain ⟨w, hw, rfl⟩ := hvL
+    have hxz := hmx z hz
+    have hyw := hmy w hw
+    simp only [inner_add_left, inner_smul_left, RCLike.conj_to_real]
+    nlinarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Limits
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter
+open scoped Topology
+
+namespace MovingSofa
+
+/-- The supremum of scalar products with a specified direction vector. -/
+def vectorSupport (S : Set Point) (u : Point) : ℝ := sSup ((fun x ↦ inner ℝ x u) '' S)
+
+/-- Support values in a unit direction vary by at most the Hausdorff distance. -/
+theorem abs_vectorSupport_sub_le_hausdorffDist {S T : Set Point}
+    (hS : S.Nonempty) (hcS : IsCompact S) (hT : T.Nonempty) (hcT : IsCompact T)
+    {u : Point} (hu : ‖u‖ = 1) :
+    |vectorSupport S u - vectorSupport T u| ≤ Metric.hausdorffDist S T := by
+  simpa only [vectorSupport, hu, mul_one] using
+    hcS.abs_csSup_inner_sub_le_hausdorffDist hS hcT hT u
+
+theorem convexBody_selection (A : Set Point) (hA : IsCompact A)
+    (K : ℕ → ConvexBody Point) (hK : ∀ n, (K n : Set Point) ⊆ A) :
+    ∃ (φ : ℕ → ℕ) (L : ConvexBody Point), StrictMono φ ∧
+      Tendsto (fun n ↦ Metric.hausdorffDist (K (φ n) : Set Point) (L : Set Point))
+        atTop (𝓝 0) := by
+  let C : ℕ → TopologicalSpace.NonemptyCompacts Point := fun n ↦
+    ⟨⟨(K n : Set Point), (K n).isCompact⟩, (K n).nonempty⟩
+  obtain ⟨L, hLA, φ, hφ, hlim⟩ :=
+    (TopologicalSpace.NonemptyCompacts.isCompact_subsets_of_isCompact hA).tendsto_subseq
+      (show ∀ n, C n ∈ {L : TopologicalSpace.NonemptyCompacts Point | (L : Set Point) ⊆ A}
+        from hK)
+  have hconv : Convex ℝ (L : Set Point) :=
+    TopologicalSpace.NonemptyCompacts.convex_of_tendsto (fun n ↦ (K (φ n)).convex) hlim
+  refine ⟨φ, ⟨(L : Set Point), hconv, L.isCompact, L.nonempty⟩, hφ, ?_⟩
+  simpa only [C, Function.comp_apply, TopologicalSpace.NonemptyCompacts.dist_eq,
+    TopologicalSpace.NonemptyCompacts.coe_mk, TopologicalSpace.Compacts.coe_mk, ConvexBody.coe_mk]
+      using
+    (tendsto_iff_dist_tendsto_zero.mp hlim)
+
+theorem compactSet_support_continuity (S T : Set Point)
+    (hS : S.Nonempty) (hcS : IsCompact S) (hT : T.Nonempty) (hcT : IsCompact T) :
+    (∀ u v : Point, ‖u‖ = 1 → ‖v‖ = 1 →
+      |vectorSupport S u - vectorSupport S v| ≤ sSup (norm '' S) * ‖u - v‖) ∧
+    (∀ u : Point, ‖u‖ = 1 →
+      |vectorSupport S u - vectorSupport T u| ≤ Metric.hausdorffDist S T) ∧
+    Continuous (fun t : Real.Angle ↦ supportValue S t) ∧
+    (∀ (K : ℕ → Set Point), (∀ n, (K n).Nonempty ∧ IsCompact (K n)) →
+      Tendsto (fun n ↦ Metric.hausdorffDist (K n) S) atTop (𝓝 0) →
+      TendstoUniformlyOn (fun n u ↦ vectorSupport (K n) u) (vectorSupport S)
+        atTop {u | ‖u‖ = 1}) := by
+  constructor
+  · intro u v _ _
+    exact hcS.abs_csSup_inner_sub_le hS u v
+  constructor
+  · intro u hu
+    exact abs_vectorSupport_sub_le_hausdorffDist hS hcS hT hcT hu
+  constructor
+  · unfold supportValue
+    apply hcS.continuous_sSup
+    change Continuous (fun q : Real.Angle × Point ↦ inner ℝ q.2 (normalVector q.1))
+    have hn : Continuous (fun q : Real.Angle × Point ↦ normalVector q.1) := by
+      let c : Real.Angle × Point → (i : Fin 2) → ℝ :=
+        fun q i ↦ Fin.cases q.1.cos (fun _ ↦ q.1.sin) i
+      have hc : Continuous c := by
+        apply continuous_pi
+        intro i
+        fin_cases i
+        · exact Real.Angle.continuous_cos.comp continuous_fst
+        · exact Real.Angle.continuous_sin.comp continuous_fst
+      have heq : (fun q : Real.Angle × Point ↦ normalVector q.1) =
+          (fun q ↦ WithLp.toLp 2 (c q)) := by
+        funext q
+        ext i
+        fin_cases i <;> rfl
+      rw [heq]
+      exact (PiLp.continuous_toLp (2 : ENNReal) (fun _ : Fin 2 ↦ ℝ)).comp hc
+    simpa [Function.comp_def] using
+      (continuous_inner (𝕜 := ℝ) (E := Point)).comp (continuous_snd.prodMk hn)
+  · intro K hK hlim
+    apply Metric.tendstoUniformlyOn_iff.mpr
+    intro ε hε
+    filter_upwards [hlim.eventually (gt_mem_nhds hε)] with n hn
+    intro u hu
+    rw [Real.dist_eq]
+    exact (abs_vectorSupport_sub_le_hausdorffDist hS hcS
+      (hK n).1 (hK n).2 hu).trans_lt (by simpa [Metric.hausdorffDist_comm] using hn)
+
+private theorem exists_inner_pos_of_eq_iInter
+    (U : Set Point) (K : Set Point) (hne : K.Nonempty) (hbounded : Bornology.IsBounded K)
+    (hK : K = ⋂ u ∈ U, {x | inner ℝ x u ≤ vectorSupport K u}) :
+    ∀ d : Point, d ≠ 0 → ∃ u ∈ U, 0 < inner ℝ d u := by
+  intro d hd
+  by_contra hno
+  push Not at hno
+  obtain ⟨z, hz⟩ := hne
+  obtain ⟨R, hR⟩ := Metric.isBounded_iff_subset_closedBall (0 : Point) |>.1 hbounded
+  have hzI : z ∈ ⋂ u ∈ U, {x | inner ℝ x u ≤ vectorSupport K u} := by
+    rw [← hK]
+    exact hz
+  have hmem : ∀ t : ℝ, 0 ≤ t → z + t • d ∈ K := by
+    intro t ht
+    rw [hK]
+    simp only [Set.mem_iInter]
+    intro u hu
+    have hz' := (Set.mem_iInter.mp (Set.mem_iInter.mp hzI u) hu)
+    dsimp at hz' ⊢
+    rw [inner_add_left, inner_smul_left]
+    change inner ℝ z u + t * inner ℝ d u ≤ vectorSupport K u
+    have hdu : inner ℝ d u ≤ 0 := hno u hu
+    have htd : t * inner ℝ d u ≤ 0 := mul_nonpos_of_nonneg_of_nonpos ht hdu
+    nlinarith [hz', htd]
+  have hnorm : ∀ t : ℝ, 0 ≤ t → ‖z + t • d‖ ≤ R := by
+    intro t ht
+    simpa [dist_eq_norm] using (Metric.mem_closedBall.mp (hR (hmem t ht)))
+  have hd' : 0 < ‖d‖ := norm_pos_iff.mpr hd
+  have hR0 : 0 ≤ R := dist_nonneg.trans (Metric.mem_closedBall.mp (hR hz))
+  let t : ℝ := (R + ‖z‖ + 1) / ‖d‖
+  have ht : 0 ≤ t := by dsimp [t]; exact div_nonneg (by positivity) (le_of_lt hd')
+  have hn := hnorm t ht
+  have hts : ‖t • d‖ = R + ‖z‖ + 1 := by
+    dsimp [t]
+    rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg ht]
+    exact div_mul_cancel₀ _ (ne_of_gt hd')
+  have htriangle' : ‖t • d‖ ≤ ‖z + t • d‖ + ‖z‖ := by
+    have h := norm_sub_le (z + t • d) z
+    simpa [sub_eq_add_neg, add_assoc] using h
+  rw [hts] at htriangle'
+  nlinarith
+
+private theorem ball_support_margin
+    (U : Set Point) (L : Set Point) (q u : Point) (r : ℝ) (hr : 0 ≤ r)
+    (hu : u ∈ U) (hunit : ‖u‖ = 1)
+    (hball : Metric.closedBall q r ⊆ ⋂ u ∈ U,
+      {x | inner ℝ x u ≤ vectorSupport L u}) :
+    inner ℝ q u + r ≤ vectorSupport L u := by
+  have hp : q + r • u ∈ Metric.closedBall q r := by
+    rw [Metric.mem_closedBall, dist_eq_norm]
+    simp only [add_sub_cancel_left]
+    rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg hr, hunit]
+    simp
+  have hp' := hball hp
+  have hpu := (Set.mem_iInter.mp (Set.mem_iInter.mp hp' u) hu)
+  dsimp at hpu
+  rw [inner_add_left, inner_smul_left] at hpu
+  have huu : inner ℝ u u = 1 := by
+    rw [real_inner_self_eq_norm_sq, hunit]
+    norm_num
+  simp only [huu] at hpu
+  simpa using hpu
+
+private theorem eventually_ball_center_mem
+    (U : Set Point) (q : Point) (r : ℝ) (hr : 0 < r)
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hK : ∀ n, (K n : Set Point) = ⋂ u ∈ U,
+      {x | inner ℝ x u ≤ vectorSupport (K n) u})
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point) (L : Set Point))
+      atTop (𝓝 0))
+    (hq : ∀ u ∈ U, inner ℝ q u + r ≤ vectorSupport (L : Set Point) u)
+    (hU : ∀ u ∈ U, ‖u‖ = 1) :
+    ∀ᶠ n in atTop, q ∈ (K n : Set Point) := by
+  filter_upwards [hlim.eventually (gt_mem_nhds hr)] with n hn
+  rw [hK n]
+  simp only [Set.mem_iInter]
+  intro u hu
+  have habs := (compactSet_support_continuity (K n : Set Point) (L : Set Point)
+    (K n).nonempty (K n).isCompact L.nonempty L.isCompact).2.1 u (hU u hu)
+  have hlower : vectorSupport (L : Set Point) u -
+      Metric.hausdorffDist (K n : Set Point) (L : Set Point) ≤
+      vectorSupport (K n : Set Point) u := by
+    linarith [neg_le_of_abs_le habs]
+  have hqr := hq u hu
+  change inner ℝ q u ≤ vectorSupport (K n : Set Point) u
+  calc
+    inner ℝ q u ≤ vectorSupport (L : Set Point) u - r := by linarith
+    _ ≤ vectorSupport (L : Set Point) u - Metric.hausdorffDist (K n : Set Point) L := by
+      exact le_of_lt (by linarith)
+    _ ≤ vectorSupport (K n : Set Point) u := hlower
+
+private theorem ball_center_mem_limit
+    (U : Set Point) (hU : ∀ u ∈ U, ‖u‖ = 1) (q : Point) (r : ℝ) (hr : 0 < r)
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hK : ∀ n, (K n : Set Point) = ⋂ u ∈ U,
+      {x | inner ℝ x u ≤ vectorSupport (K n) u})
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point) (L : Set Point))
+      atTop (𝓝 0))
+    (hball : Metric.closedBall q r ⊆ ⋂ u ∈ U,
+      {x | inner ℝ x u ≤ vectorSupport L u}) :
+    q ∈ (L : Set Point) := by
+  exact ConvexBody.mem_of_tendsto_hausdorffDist q K L
+    (eventually_ball_center_mem U q r hr K L hK hlim
+      (fun u hu ↦ ball_support_margin U L q u r hr.le hu (hU u hu) hball) hU) hlim
+
+theorem fixedNormalBody_closed (U : Set Point) (hU : ∀ u ∈ U, ‖u‖ = 1)
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hK : ∀ n, (K n : Set Point) = ⋂ u ∈ U, {x | inner ℝ x u ≤ vectorSupport (K n) u})
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point) (L : Set Point))
+      atTop (𝓝 0)) :
+    (L : Set Point) = ⋂ u ∈ U, {x | inner ℝ x u ≤ vectorSupport L u} := by
+  let H : Set Point := ⋂ u ∈ U, {x | inner ℝ x u ≤ vectorSupport L u}
+  have hLH : (L : Set Point) ⊆ H := by
+    intro x hx
+    simp only [H, Set.mem_iInter, Set.mem_ofPred_eq]
+    intro u hu
+    exact le_csSup (L.isCompact.bddAbove_image
+      (continuous_inner.comp (continuous_id.prodMk continuous_const)).continuousOn)
+      ⟨x, hx, rfl⟩
+  have hH : Convex ℝ H := by
+    intro x hx y hy a b ha hb hab
+    simp only [H, Set.mem_iInter, Set.mem_ofPred_eq] at hx hy ⊢
+    intro u hu
+    rw [inner_add_left, real_inner_smul_left, real_inner_smul_left]
+    calc
+      a * inner ℝ x u + b * inner ℝ y u ≤
+          a * vectorSupport L u + b * vectorSupport L u :=
+        add_le_add (mul_le_mul_of_nonneg_left (hx u hu) ha)
+          (mul_le_mul_of_nonneg_left (hy u hu) hb)
+      _ = vectorSupport L u := by rw [← add_mul, hab, one_mul]
+  have hint : interior H ⊆ (L : Set Point) := by
+    intro x hx
+    obtain ⟨r, hr, hball⟩ := Metric.nhds_basis_closedBall.mem_iff.mp
+      (mem_interior_iff_mem_nhds.mp hx)
+    exact ball_center_mem_limit U hU x r hr K L hK hlim hball
+  apply Set.Subset.antisymm hLH
+  rcases (interior H).eq_empty_or_nonempty with hempty | hnonempty
+  · have hcol : Collinear ℝ H := hH.collinear_of_interior_eq_empty (by
+      simp [Point, finrank_euclideanSpace]) hempty
+    intro p hp
+    obtain ⟨a, ha⟩ := L.nonempty
+    by_cases hpa : p = a
+    · simpa only [hpa] using ha
+    obtain ⟨u, hu, hpos⟩ := exists_inner_pos_of_eq_iInter U (K 0) (K 0).nonempty
+      (K 0).isCompact.isBounded (hK 0)
+      (p - a) (sub_ne_zero.mpr hpa)
+    have hmax : ∃ x ∈ (L : Set Point), vectorSupport L u = inner ℝ x u ∧
+        ∀ y ∈ (L : Set Point), inner ℝ y u ≤ inner ℝ x u := by
+      simpa only [vectorSupport, Function.comp_apply, id_eq] using
+        L.isCompact.exists_sSup_image_eq_and_ge (α := ℝ) (β := Point)
+          (f := fun x : Point ↦ inner ℝ x u) L.nonempty
+          (continuous_inner.comp (continuous_id.prodMk continuous_const)).continuousOn
+    obtain ⟨x, hx, hxu, _⟩ := hmax
+    have hle : inner ℝ p u ≤ inner ℝ x u := by
+      rw [← hxu]
+      exact Set.mem_iInter.mp (Set.mem_iInter.mp hp u) hu
+    have hsegment := hcol.mem_segment_of_apply_le (hLH ha) hp (hLH hx)
+      (innerSL ℝ u).toLinearMap (by change 0 < inner ℝ u (p - a); rwa [real_inner_comm])
+      (by change inner ℝ u p ≤ inner ℝ u x; simpa only [real_inner_comm u] using hle)
+    exact L.convex.segment_subset ha hx hsegment
+  · have hclosure : closure (interior H) = closure H :=
+      hH.closure_interior_eq_closure_of_nonempty_interior hnonempty
+    exact subset_closure.trans (hclosure ▸ L.isClosed.closure_subset_iff.mpr hint)
+
+theorem convexArea_hausdorff_continuity (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point) (L : Set Point))
+      atTop (𝓝 0)) :
+    Tendsto (fun n ↦ ClassicalResults.area (K n)) atTop (𝓝 (ClassicalResults.area L)) := by
+  let U : Set Point := {u | ‖u‖ = 1}
+  have hrepK : ∀ n, (K n : Set Point) = ⋂ u ∈ U,
+      {x | inner ℝ x u ≤ vectorSupport (K n) u} := by
+    intro n
+    simpa only [U, vectorSupport] using ConvexBody.eq_iInter_halfSpaces (K n)
+  have hrepL : (L : Set Point) = ⋂ u ∈ U,
+      {x | inner ℝ x u ≤ vectorSupport L u} := by
+    simpa only [U, vectorSupport] using ConvexBody.eq_iInter_halfSpaces L
+  obtain ⟨R, hR, hcompact⟩ := L.isCompact.exists_isCompact_cthickening
+  let C : Set Point := Metric.cthickening R (L : Set Point)
+  have hsubset : ∀ᶠ n in atTop, (K n : Set Point) ⊆ C := by
+    filter_upwards [hlim.eventually (gt_mem_nhds hR)] with n hn p hp
+    obtain ⟨q, hq, hpq⟩ := Metric.exists_dist_lt_of_hausdorffDist_lt hp hn
+      (Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded
+        (K n).nonempty L.nonempty (K n).isCompact.isBounded L.isCompact.isBounded)
+    exact Metric.mem_cthickening_of_dist_le p q R L hq hpq.le
+  have hpointwise : ∀ p ∉ frontier (L : Set Point),
+      Tendsto (fun n ↦ (K n : Set Point).indicator (fun _ ↦ (1 : ℝ)) p) atTop
+        (𝓝 ((L : Set Point).indicator (fun _ ↦ (1 : ℝ)) p)) := by
+    intro p hpfrontier
+    by_cases hpL : p ∈ (L : Set Point)
+    · have hpint : p ∈ interior (L : Set Point) := by
+        by_contra hp
+        exact hpfrontier ((mem_frontier_iff_notMem_interior hpL).mpr hp)
+      obtain ⟨r, hr, hball⟩ := Metric.nhds_basis_closedBall.mem_iff.mp
+        (mem_interior_iff_mem_nhds.mp hpint)
+      have hmargin : ∀ u ∈ U, inner ℝ p u + r ≤ vectorSupport L u := by
+        intro u hu
+        rw [hrepL] at hball
+        exact ball_support_margin U L p u r hr.le hu hu hball
+      have hev := eventually_ball_center_mem U p r hr K L hrepK hlim hmargin
+        (fun _ hu ↦ hu)
+      apply tendsto_nhds_of_eventually_eq
+      filter_upwards [hev] with n hn
+      simp [hn, hpL]
+    · have hdist : 0 < Metric.infDist p (L : Set Point) := by
+        exact (Metric.infDist_pos_iff_notMem_closure L.nonempty).mp (by
+          rwa [L.isClosed.closure_eq])
+      have hev : ∀ᶠ n in atTop, p ∉ (K n : Set Point) := by
+        filter_upwards [hlim.eventually (gt_mem_nhds hdist)] with n hn hpn
+        have hle := Metric.infDist_le_hausdorffDist_of_mem hpn
+          (Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded
+            (K n).nonempty L.nonempty (K n).isCompact.isBounded L.isCompact.isBounded)
+        linarith
+      apply tendsto_nhds_of_eventually_eq
+      filter_upwards [hev] with n hn
+      simp [hn, hpL]
+  have hfrontier : MeasureTheory.volume (frontier (L : Set Point)) = 0 :=
+    L.convex.addHaar_frontier MeasureTheory.volume
+  have hmeas : ∀ n, MeasurableSet (K n : Set Point) := fun n ↦ (K n).isCompact.measurableSet
+  have hCmeas : MeasurableSet C := hcompact.measurableSet
+  have hCint : MeasureTheory.Integrable (C.indicator fun _ ↦ (1 : ℝ)) := by
+    exact (MeasureTheory.integrableOn_const hcompact.measure_lt_top.ne).integrable_indicator hCmeas
+  have htendsto := MeasureTheory.tendsto_integral_filter_of_dominated_convergence
+    (μ := MeasureTheory.volume)
+    (F := fun n ↦ (K n : Set Point).indicator fun _ ↦ (1 : ℝ))
+    (f := (L : Set Point).indicator fun _ ↦ (1 : ℝ))
+    (C.indicator fun _ ↦ (1 : ℝ))
+    (Filter.Eventually.of_forall fun n ↦
+      (measurable_const.indicator (hmeas n)).aestronglyMeasurable)
+    (by
+      filter_upwards [hsubset] with n hn
+      filter_upwards [] with p
+      by_cases hp : p ∈ (K n : Set Point)
+      · have hpC := hn hp
+        simp [hp, hpC]
+      · by_cases hpC : p ∈ C <;> simp [hp, hpC])
+    hCint
+    (by
+      filter_upwards [MeasureTheory.compl_mem_ae_iff.mpr hfrontier] with p hp
+      exact hpointwise p hp)
+  have heqK : ∀ n, (∫ p, (K n : Set Point).indicator (fun _ ↦ (1 : ℝ)) p) =
+      MeasureTheory.volume.real (K n : Set Point) := by
+    intro n
+    change (∫ p, (K n : Set Point).indicator 1 p) = _
+    exact MeasureTheory.integral_indicator_one (μ := MeasureTheory.volume) (hmeas n)
+  have heqL : (∫ p, (L : Set Point).indicator (fun _ ↦ (1 : ℝ)) p) =
+      MeasureTheory.volume.real (L : Set Point) := by
+    change (∫ p, (L : Set Point).indicator 1 p) = _
+    exact MeasureTheory.integral_indicator_one (μ := MeasureTheory.volume)
+      L.isCompact.measurableSet
+  simp_rw [heqK] at htendsto
+  rw [heqL] at htendsto
+  simpa [ClassicalResults.area, MeasureTheory.Measure.real] using htendsto
+
+/-- The support function of a convex body is continuous in the normal direction. -/
+theorem continuous_supportValue (K : ConvexBody Point) :
+    Continuous fun t : Real.Angle ↦ supportValue (K : Set Point) t :=
+  (compactSet_support_continuity K K K.nonempty K.isCompact K.nonempty K.isCompact).2.2.1
+
+theorem continuous_supportValue_real (K : ConvexBody Point) :
+    Continuous (fun t : ℝ ↦ supportValue K (t : Real.Angle)) :=
+  (continuous_supportValue K).comp Real.Angle.continuous_coe
+
+/-- Support values of a nonempty compact set are Lipschitz in the real angle, with the largest
+norm of a point of the set as Lipschitz constant. -/
+theorem abs_supportValue_sub_le_of_isCompact {s : Set Point} (hne : s.Nonempty)
+    (hc : IsCompact s) (x y : ℝ) :
+    |supportValue s (x : Real.Angle) - supportValue s (y : Real.Angle)| ≤
+      sSup (norm '' s) * |x - y| := by
+  have hdir := (compactSet_support_continuity s s hne hc hne hc).1
+    (normalVector (x : Real.Angle)) (normalVector (y : Real.Angle))
+    (norm_normalVector_real x) (norm_normalVector_real y)
+  have hframe : ‖normalVector (x : Real.Angle) - normalVector (y : Real.Angle)‖ ≤ |x - y| := by
+    simpa [dist_eq_norm, Real.dist_eq] using lipschitzWith_normalVector_real.dist_le_mul x y
+  have h0 : 0 ≤ sSup (norm '' s) :=
+    Real.sSup_nonneg (by rintro _ ⟨p, -, rfl⟩; exact norm_nonneg p)
+  exact hdir.trans (mul_le_mul_of_nonneg_left hframe h0)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Support Embedding
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem supportFunction_minkowski_embedding (K L : ConvexBody Point) :
+    (∀ (a b : ℝ), 0 ≤ a → 0 ≤ b → ∀ t : Real.Angle,
+      supportValue {z : Point | ∃ x ∈ (K : Set Point), ∃ y ∈ (L : Set Point),
+        z = a • x + b • y} t =
+        a * supportValue K t + b * supportValue L t) ∧
+    ((∀ t : Real.Angle, supportValue K t = supportValue L t) → K = L) ∧
+    Continuous (fun t : Real.Angle ↦ supportValue K t) ∧
+    (K : Set Point) = ⋂ u ∈ {u : Point | ‖u‖ = 1},
+      {p : Point | inner ℝ p u ≤ vectorSupport K u} := by
+  refine ⟨?_, ?_, (compactSet_support_continuity K K K.nonempty K.isCompact
+    K.nonempty K.isCompact).2.2.1, K.eq_iInter_halfSpaces⟩
+  · intro a b ha hb t
+    obtain ⟨x, hx, hxmax, hxle⟩ := K.isCompact.exists_sSup_image_eq_and_ge
+      (f := fun x : Point ↦ inner ℝ x (normalVector t)) K.nonempty
+      (continuous_inner.comp (continuous_id.prodMk continuous_const)).continuousOn
+    obtain ⟨y, hy, hymax, hyle⟩ := L.isCompact.exists_sSup_image_eq_and_ge
+      (f := fun x : Point ↦ inner ℝ x (normalVector t)) L.nonempty
+      (continuous_inner.comp (continuous_id.prodMk continuous_const)).continuousOn
+    change sSup ((fun p ↦ inner ℝ p (normalVector t)) '' K) = _ at hxmax
+    change sSup ((fun p ↦ inner ℝ p (normalVector t)) '' L) = _ at hymax
+    unfold supportValue
+    apply IsGreatest.csSup_eq
+    constructor
+    · refine ⟨a • x + b • y, ?_, ?_⟩
+      · exact ⟨x, hx, y, hy, rfl⟩
+      · change inner ℝ (a • x + b • y) (normalVector t) = _
+        rw [inner_add_left, inner_smul_left, inner_smul_left, ← hxmax, ← hymax]
+        simp
+    · intro r hr
+      obtain ⟨z, ⟨x', hx', y', hy', rfl⟩, rfl⟩ := hr
+      change inner ℝ (a • x' + b • y') (normalVector t) ≤ _
+      rw [inner_add_left, inner_smul_left, inner_smul_left]
+      rw [hxmax, hymax]
+      exact add_le_add (mul_le_mul_of_nonneg_left (hxle x' hx') ha)
+        (mul_le_mul_of_nonneg_left (hyle y' hy') hb)
+  · intro hKL
+    have hvector (u : Point) (hu : ‖u‖ = 1) : vectorSupport K u = vectorSupport L u := by
+      obtain ⟨t, rfl⟩ := exists_angle_normalVector_eq hu
+      exact hKL t
+    simp only [vectorSupport] at hvector
+    apply ConvexBody.ext
+    rw [K.eq_iInter_halfSpaces,
+      L.eq_iInter_halfSpaces]
+    ext p
+    simp only [Set.mem_iInter, Set.mem_ofPred_eq]
+    constructor <;> intro hp u hu
+    · simpa [hvector u hu] using hp u hu
+    · simpa [hvector u hu] using hp u hu
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Combination Properties
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped Pointwise unitInterval
+
+namespace MovingSofa
+
+/-- Pointwise description of the Minkowski interpolation of two convex bodies. -/
+theorem mem_convexBodyCombination_iff (t : I) (K L : ConvexBody Point) (z : Point) :
+    z ∈ (convexBodyCombination t K L : Set Point) ↔
+      ∃ x ∈ (K : Set Point), ∃ y ∈ (L : Set Point),
+        z = (1 - (t : ℝ)) • x + (t : ℝ) • y := by
+  change z ∈ (1 - (t : ℝ)) • (K : Set Point) + (t : ℝ) • (L : Set Point) ↔ _
+  constructor
+  · rintro ⟨_, ⟨x, hx, rfl⟩, _, ⟨y, hy, rfl⟩, rfl⟩
+    exact ⟨x, hx, y, hy, rfl⟩
+  · rintro ⟨x, hx, y, hy, rfl⟩
+    exact ⟨_, ⟨x, hx, rfl⟩, _, ⟨y, hy, rfl⟩, rfl⟩
+
+/-- The first edge vertex maximizes tangent coordinate on its exposed edge. -/
+theorem inner_le_edgeVertices_fst_tangent (K : ConvexBody Point)
+    (a : Real.Angle) {p : Point} (hp : p ∈ exposedEdge K a) :
+    inner ℝ p (tangentVector a) ≤ inner ℝ (edgeVertices K a).1 (tangentVector a) := by
+  rw [inner_edgeVertices_fst_tangent]
+  exact le_csSup ((isCompact_exposedEdge K a).image
+    (continuous_id.inner continuous_const) |>.bddAbove) ⟨p, hp, rfl⟩
+
+/-- A tangent-coordinate maximizer on an exposed edge is its first vertex. -/
+theorem edgeVertices_fst_eq_of_tangent_isGreatest (K : ConvexBody Point)
+    (a : Real.Angle) {p : Point} (hp : p ∈ exposedEdge K a)
+    (hmax : ∀ q ∈ exposedEdge K a,
+      inner ℝ q (tangentVector a) ≤ inner ℝ p (tangentVector a)) :
+    (edgeVertices K a).1 = p := by
+  have ht := le_antisymm (hmax _ (edgeVertices_fst_mem K a))
+    (inner_le_edgeVertices_fst_tangent K a hp)
+  rw [← inner_normalVector_smul_add_inner_tangentVector_smul (edgeVertices K a).1 a,
+    ← inner_normalVector_smul_add_inner_tangentVector_smul p a,
+    (edgeVertices_fst_mem K a).2, hp.2, ht]
+
+/-- A tangent-coordinate minimizer on an exposed edge is its second vertex. -/
+theorem edgeVertices_snd_eq_of_tangent_isLeast (K : ConvexBody Point)
+    (a : Real.Angle) {p : Point} (hp : p ∈ exposedEdge K a)
+    (hmin : ∀ q ∈ exposedEdge K a,
+      inner ℝ p (tangentVector a) ≤ inner ℝ q (tangentVector a)) :
+    (edgeVertices K a).2 = p := by
+  have hbdd : BddBelow
+      ((fun q : Point ↦ inner ℝ q (tangentVector a)) '' exposedEdge K a) :=
+    ((isCompact_exposedEdge K a).image
+      (continuous_id.inner continuous_const)).bddBelow
+  have ht : inner ℝ (edgeVertices K a).2 (tangentVector a) =
+      inner ℝ p (tangentVector a) := by
+    rw [inner_edgeVertices_snd_tangent]
+    apply le_antisymm (csInf_le hbdd ⟨p, hp, rfl⟩)
+    rw [← inner_edgeVertices_snd_tangent]
+    exact hmin _ (edgeVertices_snd_mem K a)
+  rw [← inner_normalVector_smul_add_inner_tangentVector_smul (edgeVertices K a).2 a,
+    ← inner_normalVector_smul_add_inner_tangentVector_smul p a,
+    (edgeVertices_snd_mem K a).2, hp.2, ht]
+
+private theorem inner_edgeVertices_snd_tangent_le_point (K : ConvexBody Point)
+    (a : Real.Angle) {p : Point} (hp : p ∈ exposedEdge K a) :
+    inner ℝ (edgeVertices K a).2 (tangentVector a) ≤ inner ℝ p (tangentVector a) := by
+  rw [inner_edgeVertices_snd_tangent]
+  exact csInf_le ((isCompact_exposedEdge K a).image
+    (continuous_id.inner continuous_const) |>.bddBelow) ⟨p, hp, rfl⟩
+
+/-- The extreme points of exposed edges commute with convex combinations. -/
+theorem edgeVertices_convexBodyCombination (t : I) (K L : ConvexBody Point)
+    (a : Real.Angle) :
+    (edgeVertices (convexBodyCombination t K L) a).1 =
+        (1 - (t : ℝ)) • (edgeVertices K a).1 + (t : ℝ) • (edgeVertices L a).1 ∧
+      (edgeVertices (convexBodyCombination t K L) a).2 =
+        (1 - (t : ℝ)) • (edgeVertices K a).2 + (t : ℝ) • (edgeVertices L a).2 := by
+  let M := convexBodyCombination t K L
+  have hedge := exposedEdge_convexBodyCombination K L a t
+  have hnonneg : 0 ≤ 1 - (t : ℝ) := sub_nonneg.mpr t.property.2
+  have htnonneg : 0 ≤ (t : ℝ) := t.property.1
+  constructor
+  · apply edgeVertices_fst_eq_of_tangent_isGreatest M a
+    · rw [hedge]
+      exact Set.add_mem_add (Set.smul_mem_smul_set (edgeVertices_fst_mem K a))
+        (Set.smul_mem_smul_set (edgeVertices_fst_mem L a))
+    · intro q hq
+      rw [hedge] at hq
+      obtain ⟨_, ⟨x, hx, rfl⟩, _, ⟨y, hy, rfl⟩, rfl⟩ := hq
+      simp only [inner_add_left, inner_smul_left, RCLike.conj_to_real]
+      exact add_le_add
+        (mul_le_mul_of_nonneg_left (inner_le_edgeVertices_fst_tangent K a hx) hnonneg)
+        (mul_le_mul_of_nonneg_left (inner_le_edgeVertices_fst_tangent L a hy) htnonneg)
+  · apply edgeVertices_snd_eq_of_tangent_isLeast M a
+    · rw [hedge]
+      exact Set.add_mem_add (Set.smul_mem_smul_set (edgeVertices_snd_mem K a))
+        (Set.smul_mem_smul_set (edgeVertices_snd_mem L a))
+    · intro q hq
+      rw [hedge] at hq
+      obtain ⟨_, ⟨x, hx, rfl⟩, _, ⟨y, hy, rfl⟩, rfl⟩ := hq
+      simp only [inner_add_left, inner_smul_left, RCLike.conj_to_real]
+      exact add_le_add
+        (mul_le_mul_of_nonneg_left (inner_edgeVertices_snd_tangent_le_point K a hx) hnonneg)
+        (mul_le_mul_of_nonneg_left (inner_edgeVertices_snd_tangent_le_point L a hy) htnonneg)
+
+/-- Support values commute with convex combinations. -/
+theorem supportValue_convexBodyCombination (t : I) (K L : ConvexBody Point)
+    (a : Real.Angle) :
+    supportValue (convexBodyCombination t K L) a =
+      (1 - (t : ℝ)) * supportValue K a + (t : ℝ) * supportValue L a := by
+  rw [show (convexBodyCombination t K L : Set Point) =
+      {z : Point | ∃ x ∈ (K : Set Point), ∃ y ∈ (L : Set Point),
+        z = (1 - (t : ℝ)) • x + (t : ℝ) • y} from
+    Set.ext (mem_convexBodyCombination_iff t K L)]
+  exact (supportFunction_minkowski_embedding K L).1
+    (1 - (t : ℝ)) t (sub_nonneg.mpr t.property.2) t.property.1 a
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Faces of a body cut out by a differentiable family of half-planes
+
+Let `L` be a convex body lying in every half-plane `{q | m s ≤ ⟪q, u_s⟫}` of a family indexed
+by a real angle parameter `s`, and let `p ∈ L` attain the bound at `s = t`.  If `m` is
+differentiable at `t` with `m' t = ⟪p, v_t⟫` — that is, if `p` is the first-order contact point
+of the family — then the reversed face of `L` at `t + π` is pinned down by the sign of the
+one-sided derivatives of `s ↦ ⟪z, u_s⟫ - m s` at `t`:
+`exposedEdge_add_pi_eq_singleton_of_mem_Ioo` at an interior parameter gives the singleton
+`{p}`, while `edgeVertices_add_pi_fst_eq_of_lt` and `edgeVertices_add_pi_snd_eq_of_lt` identify
+`p` with one endpoint vertex of the face at the two boundary parameters.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The tangent coordinate of a point whose normal coordinate dominates a differentiable family
+to the left of the touching parameter is at most that of the contact point. -/
+theorem inner_tangentVector_le_of_forall_le_Ioo {m : ℝ → ℝ} {p z : Point} {a t : ℝ}
+    (hat : a < t) (hle : ∀ s ∈ Set.Ioo a t, m s ≤ inner ℝ z (normalVector (s : Real.Angle)))
+    (hzm : inner ℝ z (normalVector (t : Real.Angle)) = m t)
+    (hd : HasDerivAt m (inner ℝ p (tangentVector (t : Real.Angle))) t) :
+    inner ℝ z (tangentVector (t : Real.Angle)) ≤
+      inner ℝ p (tangentVector (t : Real.Angle)) := by
+  have hmin : IsLocalMinOn (fun s : ℝ ↦ inner ℝ z (normalVector (s : Real.Angle)) - m s)
+      (Set.Iic t) t := by
+    filter_upwards [nhdsWithin_le_nhds (Ioi_mem_nhds hat), self_mem_nhdsWithin] with s hs hst
+    rcases (Set.mem_Iic.1 hst).lt_or_eq with h | rfl
+    · simpa only [hzm, sub_self] using sub_nonneg.2 (hle s ⟨hs, h⟩)
+    · exact le_rfl
+  have hF := (hasDerivAt_inner_normalVector z t).sub hd
+  linarith only [hmin.hasDerivWithinAt_Iic_nonpos hF.hasDerivWithinAt]
+
+/-- The tangent coordinate of a point whose normal coordinate dominates a differentiable family
+to the right of the touching parameter is at least that of the contact point. -/
+theorem le_inner_tangentVector_of_forall_le_Ioo {m : ℝ → ℝ} {p z : Point} {t b : ℝ}
+    (htb : t < b) (hle : ∀ s ∈ Set.Ioo t b, m s ≤ inner ℝ z (normalVector (s : Real.Angle)))
+    (hzm : inner ℝ z (normalVector (t : Real.Angle)) = m t)
+    (hd : HasDerivAt m (inner ℝ p (tangentVector (t : Real.Angle))) t) :
+    inner ℝ p (tangentVector (t : Real.Angle)) ≤
+      inner ℝ z (tangentVector (t : Real.Angle)) := by
+  have hmin : IsLocalMinOn (fun s : ℝ ↦ inner ℝ z (normalVector (s : Real.Angle)) - m s)
+      (Set.Ici t) t := by
+    filter_upwards [nhdsWithin_le_nhds (Iio_mem_nhds htb), self_mem_nhdsWithin] with s hs hst
+    rcases (Set.mem_Ici.1 hst).lt_or_eq with h | rfl
+    · simpa only [hzm, sub_self] using sub_nonneg.2 (hle s ⟨h, hs⟩)
+    · exact le_rfl
+  have hF := (hasDerivAt_inner_normalVector z t).sub hd
+  linarith only [hmin.hasDerivWithinAt_Ici_nonneg hF.hasDerivWithinAt]
+
+/-- At an interior touching parameter of a differentiable family of supporting half-planes the
+reversed face is the singleton contact point. -/
+theorem exposedEdge_add_pi_eq_singleton_of_mem_Ioo {L : ConvexBody Point} {m : ℝ → ℝ}
+    {p : Point} {a b t : ℝ} (ht : t ∈ Set.Ioo a b)
+    (hle : ∀ q ∈ (L : Set Point), ∀ s ∈ Set.Ioo a b,
+      m s ≤ inner ℝ q (normalVector (s : Real.Angle)))
+    (hp : p ∈ (L : Set Point))
+    (hpm : inner ℝ p (normalVector (t : Real.Angle)) = m t)
+    (hd : HasDerivAt m (inner ℝ p (tangentVector (t : Real.Angle))) t) :
+    exposedEdge L ((t + Real.pi : ℝ) : Real.Angle) = {p} := by
+  have hlet : ∀ q ∈ (L : Set Point), m t ≤ inner ℝ q (normalVector (t : Real.Angle)) :=
+    fun q hq ↦ hle q hq t ht
+  rw [exposedEdge_add_pi_eq_of_forall_le hlet hp hpm]
+  ext q
+  simp only [Set.mem_ofPred_eq, Set.mem_singleton_iff]
+  refine ⟨fun hq ↦ ?_, fun hq ↦ ⟨hq ▸ hp, hq ▸ hpm⟩⟩
+  have h1 : inner ℝ q (tangentVector (t : Real.Angle)) ≤
+      inner ℝ p (tangentVector (t : Real.Angle)) :=
+    inner_tangentVector_le_of_forall_le_Ioo ht.1
+      (fun s hs ↦ hle q hq.1 s ⟨hs.1, hs.2.trans ht.2⟩) hq.2 hd
+  have h2 : inner ℝ p (tangentVector (t : Real.Angle)) ≤
+      inner ℝ q (tangentVector (t : Real.Angle)) :=
+    le_inner_tangentVector_of_forall_le_Ioo ht.2
+      (fun s hs ↦ hle q hq.1 s ⟨ht.1.trans hs.1, hs.2⟩) hq.2 hd
+  rw [← inner_normalVector_smul_add_inner_tangentVector_smul q (t : Real.Angle),
+    ← inner_normalVector_smul_add_inner_tangentVector_smul p (t : Real.Angle),
+    hq.2, hpm, le_antisymm h1 h2]
+
+/-- At the left endpoint parameter of a differentiable family of supporting half-planes the
+contact point is the positive vertex of the reversed face. -/
+theorem edgeVertices_add_pi_fst_eq_of_lt {L : ConvexBody Point} {m : ℝ → ℝ} {p : Point}
+    {a b : ℝ} (hab : a < b)
+    (hle : ∀ q ∈ (L : Set Point), ∀ s ∈ Set.Ioo a b,
+      m s ≤ inner ℝ q (normalVector (s : Real.Angle)))
+    (hlea : ∀ q ∈ (L : Set Point), m a ≤ inner ℝ q (normalVector (a : Real.Angle)))
+    (hp : p ∈ (L : Set Point))
+    (hpm : inner ℝ p (normalVector (a : Real.Angle)) = m a)
+    (hd : HasDerivAt m (inner ℝ p (tangentVector (a : Real.Angle))) a) :
+    (edgeVertices L ((a + Real.pi : ℝ) : Real.Angle)).1 = p := by
+  have hface := exposedEdge_add_pi_eq_of_forall_le hlea hp hpm
+  refine edgeVertices_fst_eq_of_tangent_isGreatest L _ (hface ▸ ⟨hp, hpm⟩) ?_
+  intro q hq
+  rw [hface] at hq
+  have h1 : inner ℝ p (tangentVector (a : Real.Angle)) ≤
+      inner ℝ q (tangentVector (a : Real.Angle)) :=
+    le_inner_tangentVector_of_forall_le_Ioo hab (fun s hs ↦ hle q hq.1 s hs) hq.2 hd
+  rw [tangentVector_add_pi, inner_neg_right, inner_neg_right]
+  linarith only [h1]
+
+/-- At the right endpoint parameter of a differentiable family of supporting half-planes the
+contact point is the negative vertex of the reversed face. -/
+theorem edgeVertices_add_pi_snd_eq_of_lt {L : ConvexBody Point} {m : ℝ → ℝ} {p : Point}
+    {a b : ℝ} (hab : a < b)
+    (hle : ∀ q ∈ (L : Set Point), ∀ s ∈ Set.Ioo a b,
+      m s ≤ inner ℝ q (normalVector (s : Real.Angle)))
+    (hleb : ∀ q ∈ (L : Set Point), m b ≤ inner ℝ q (normalVector (b : Real.Angle)))
+    (hp : p ∈ (L : Set Point))
+    (hpm : inner ℝ p (normalVector (b : Real.Angle)) = m b)
+    (hd : HasDerivAt m (inner ℝ p (tangentVector (b : Real.Angle))) b) :
+    (edgeVertices L ((b + Real.pi : ℝ) : Real.Angle)).2 = p := by
+  have hface := exposedEdge_add_pi_eq_of_forall_le hleb hp hpm
+  refine edgeVertices_snd_eq_of_tangent_isLeast L _ (hface ▸ ⟨hp, hpm⟩) ?_
+  intro q hq
+  rw [hface] at hq
+  have h1 : inner ℝ q (tangentVector (b : Real.Angle)) ≤
+      inner ℝ p (tangentVector (b : Real.Angle)) :=
+    inner_tangentVector_le_of_forall_le_Ioo hab (fun s hs ↦ hle q hq.1 s hs) hq.2 hd
+  rw [tangentVector_add_pi, inner_neg_right, inner_neg_right]
+  linarith only [h1]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Space
+-/
+
+@[expose] public section
+
+open scoped unitInterval Pointwise
+
+namespace MovingSofa
+
+/-- Convex bodies form a convex domain under Minkowski interpolation. -/
+theorem convexBody_isConvexDomain : IsConvexDomain.{0, 0} convexBodyCombination := by
+  let e : ConvexBody Point → C(Real.Angle, ℝ) := fun K ↦
+    ⟨fun t ↦ supportValue K t, (supportFunction_minkowski_embedding K K).2.2.1⟩
+  have he (t : I) (K L : ConvexBody Point) :
+      e (convexBodyCombination t K L) = (1 - (t : ℝ)) • e K + (t : ℝ) • e L := by
+    ext u
+    change supportValue (convexBodyCombination t K L) u = _
+    exact supportValue_convexBodyCombination t K L u
+  refine ⟨ModuleCat.of ℝ C(Real.Angle, ℝ), e, ?_, ?_, he⟩
+  · intro K L h
+    apply (supportFunction_minkowski_embedding K L).2.1
+    intro t
+    exact congrArg (fun f : C(Real.Angle, ℝ) ↦ f t) h
+  · rintro _ ⟨K, rfl⟩ _ ⟨L, rfl⟩ a b ha hb hab
+    refine ⟨convexBodyCombination ⟨b, hb, by linarith⟩ K L, ?_⟩
+    have h : 1 - b = a := by linarith
+    simpa only [h] using he ⟨b, hb, by linarith⟩ K L
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Area.Foundations.Development001`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Area.ModuloLinear`.
+* `Area.Quadratic`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Area / Modulo Linear
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval
+
+universe u
+
+namespace MovingSofa
+
+/-- The difference of two functionals preserves the specified convex combinations. -/
+def EquivalentModuloConvexLinear {α : Type u} (c : I → α → α → α)
+    (f g : α → ℝ) : Prop :=
+  IsConvexLinear c realCombination (fun x ↦ f x - g x)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Area / Quadratic
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval
+
+universe u v
+
+namespace MovingSofa
+
+/-- The directional derivative along a barycentric segment is computed by any derivative of the
+segment function at the base point. -/
+theorem convexDirectionalDerivative_eq_of_hasDerivWithinAt {α : Type u} (c : I → α → α → α)
+    (f : α → ℝ) (x y : α) {d : ℝ}
+    (h : HasDerivWithinAt (segmentFunctional c f x y) d (Set.Icc 0 1) 0) :
+    convexDirectionalDerivative c f x y = d :=
+  h.derivWithin (uniqueDiffOn_Icc_zero_one.uniqueDiffWithinAt (by norm_num))
+
+/-- A quadratic diagonal has the stated segment derivative, affine in its destination. -/
+theorem quadratic_directional_derivative {α : Type u} (c : I → α → α → α) (f : α → ℝ) (h : α → α →
+  ℝ)
+    (hh : IsConvexBilinear c c realCombination h) (hf : ∀ x, f x = h x x) :
+    (∀ x y, HasDerivWithinAt (segmentFunctional c f x y)
+      (h x y + h y x - 2 * h x x) (Set.Icc 0 1) 0) ∧
+    (∀ x y, convexDirectionalDerivative c f x y = h x y + h y x - 2 * h x x) ∧
+    ∀ x, IsConvexLinear c realCombination (fun y ↦ convexDirectionalDerivative c f x y) := by
+  have hquad (t : I) (x y : α) :
+      f (c t x y) =
+        (1 - (t : ℝ)) ^ 2 * h x x +
+          (t : ℝ) * (1 - (t : ℝ)) * (h x y + h y x) + (t : ℝ) ^ 2 * h y y := by
+    rw [hf]
+    have hleft := hh.2 (c t x y) t x y
+    change h (c t x y) (c t x y) =
+      realCombination t (h x (c t x y)) (h y (c t x y)) at hleft
+    rw [hleft, hh.1 x t x y, hh.1 y t x y]
+    simp only [realCombination]
+    ring
+  have hderiv (x y : α) : HasDerivAt
+      (fun t : ℝ ↦ (1 - t) ^ 2 * h x x +
+        t * (1 - t) * (h x y + h y x) + t ^ 2 * h y y)
+      (h x y + h y x - 2 * h x x) 0 := by
+    have hid : HasDerivAt (fun t : ℝ ↦ t) 1 0 := hasDerivAt_id 0
+    have hsub : HasDerivAt (fun t : ℝ ↦ 1 - t) (-1) 0 := hid.const_sub 1
+    convert (((hsub.pow 2).mul_const (h x x)).add
+      ((hid.mul hsub).mul_const (h x y + h y x))).add
+      ((hid.pow 2).mul_const (h y y)) using 1
+    all_goals ring
+  have hfirst : ∀ x y, HasDerivWithinAt (segmentFunctional c f x y)
+      (h x y + h y x - 2 * h x x) (Set.Icc 0 1) 0 := by
+    intro x y
+    apply (hderiv x y).hasDerivWithinAt.congr
+    · intro t ht
+      simp only [segmentFunctional, ht, ↓reduceDIte]
+      exact hquad ⟨t, ht⟩ x y
+    · simp [segmentFunctional, hquad]
+  refine ⟨hfirst, ?_, ?_⟩
+  · exact fun x y ↦ convexDirectionalDerivative_eq_of_hasDerivWithinAt c f x y (hfirst x y)
+  · intro x t y z
+    unfold convexDirectionalDerivative
+    have huniq : UniqueDiffWithinAt ℝ (Set.Icc (0 : ℝ) 1) 0 :=
+      uniqueDiffOn_Icc_zero_one.uniqueDiffWithinAt (by norm_num)
+    change derivWithin (segmentFunctional c f x (c t y z)) (Set.Icc 0 1) 0 =
+      realCombination t
+        (derivWithin (segmentFunctional c f x y) (Set.Icc 0 1) 0)
+        (derivWithin (segmentFunctional c f x z) (Set.Icc 0 1) 0)
+    rw [(hfirst x (c t y z)).derivWithin huniq,
+      (hfirst x y).derivWithin huniq, (hfirst x z).derivWithin huniq]
+    simp only [realCombination]
+    have hright := hh.2 x t y z
+    change h (c t y z) x = realCombination t (h y x) (h z x) at hright
+    rw [hh.1 x t y z, hright]
+    simp only [realCombination]
+    ring
+
+/-- The segment function of a quadratic functional is differentiable at the base point, with the
+directional derivative as its derivative. -/
+theorem IsQuadraticFunctional.hasDerivWithinAt_segmentFunctional {α : Type u}
+    {c : I → α → α → α} {f : α → ℝ} (hf : IsQuadraticFunctional c f)
+    (x y : α) :
+    HasDerivWithinAt (segmentFunctional c f x y)
+      (convexDirectionalDerivative c f x y) (Set.Icc 0 1) 0 := by
+  obtain ⟨g, hg, hfg⟩ := hf
+  obtain ⟨hderiv, hderiv_eq, -⟩ := quadratic_directional_derivative c f g hg hfg
+  rw [hderiv_eq x y]
+  exact hderiv x y
+
+/-- A concave quadratic functional is maximized exactly where all directional derivatives are
+nonpositive. -/
+theorem quadratic_maximum_iff {α : Type u} (c : I → α → α → α)
+    (hc : IsConvexDomain.{u, v} c) (f : α → ℝ)
+    (hq : IsQuadraticFunctional c f) (hconcave : IsConvexFunctional c f true) (x : α) :
+    (∀ y, f y ≤ f x) ↔ ∀ y, convexDirectionalDerivative c f x y ≤ 0 := by
+  obtain ⟨g, hg, hfg⟩ := hq
+  obtain ⟨hderiv, hderiv_eq, -⟩ :=
+    quadratic_directional_derivative c f g hg hfg
+  obtain ⟨V, e, he, -, hcmap⟩ := hc
+  have hc0 (z : α) : c 0 x z = x := by
+    apply he
+    rw [hcmap]
+    simp
+  have hquad (t : I) (y : α) :
+      f (c t x y) =
+        (1 - (t : ℝ)) ^ 2 * g x x +
+          (t : ℝ) * (1 - (t : ℝ)) * (g x y + g y x) +
+            (t : ℝ) ^ 2 * g y y := by
+    rw [hfg]
+    have hleft := hg.2 (c t x y) t x y
+    change g (c t x y) (c t x y) =
+      realCombination t (g x (c t x y)) (g y (c t x y)) at hleft
+    rw [hleft, hg.1 x t x y, hg.1 y t x y]
+    simp only [realCombination]
+    ring
+  constructor
+  · intro hmax y
+    have hlocal : IsLocalMaxOn (segmentFunctional c f x y) (Set.Icc 0 1) 0 :=
+      by
+        filter_upwards [self_mem_nhdsWithin] with t ht
+        simp only [segmentFunctional, ht, ↓reduceDIte, Set.mem_Icc, Std.le_refl, zero_le_one,
+          and_self,
+          Set.Icc.mk_zero, hc0]
+        exact hmax _
+    have htangent : (1 : ℝ) ∈ posTangentConeAt (Set.Icc 0 1) 0 := by
+      simpa using sub_mem_posTangentConeAt_of_segment_subset
+        (show segment ℝ (0 : ℝ) 1 ⊆ Set.Icc 0 1 by rw [segment_eq_Icc (by norm_num)])
+    have := hlocal.hasFDerivWithinAt_nonpos (hderiv x y).hasFDerivWithinAt htangent
+    simpa [hderiv_eq] using this
+  · intro hd y
+    have hmid := hconcave (⟨1 / 2, by constructor <;> norm_num⟩ : I) x y
+    simp only [↓reduceIte, realCombination] at hmid
+    rw [hquad, hfg x, hfg y] at hmid
+    have hcoef : g x x - g x y - g y x + g y y ≤ 0 := by
+      norm_num at hmid ⊢
+      linarith
+    have := hd y
+    rw [hderiv_eq x y] at this
+    rw [hfg y, hfg x]
+    linarith
+
+/-- A sum of quadratic functionals is quadratic. -/
+theorem IsQuadraticFunctional.add {α : Type u} {c : I → α → α → α} {f g : α → ℝ}
+    (hf : IsQuadraticFunctional c f) (hg : IsQuadraticFunctional c g) :
+    IsQuadraticFunctional c (fun x ↦ f x + g x) := by
+  obtain ⟨F, ⟨hFright, hFleft⟩, hFf⟩ := hf
+  obtain ⟨G, ⟨hGright, hGleft⟩, hGg⟩ := hg
+  refine ⟨fun x y ↦ F x y + G x y, ⟨fun x t y z ↦ ?_, fun z t x y ↦ ?_⟩, fun x ↦ ?_⟩
+  · change F x (c t y z) + G x (c t y z) = realCombination t (F x y + G x y) (F x z + G x z)
+    rw [hFright x t y z, hGright x t y z]
+    simp only [realCombination]
+    ring
+  · have hF : F (c t x y) z = realCombination t (F x z) (F y z) := hFleft z t x y
+    have hG : G (c t x y) z = realCombination t (G x z) (G y z) := hGleft z t x y
+    change F (c t x y) z + G (c t x y) z = realCombination t (F x z + G x z) (F y z + G y z)
+    rw [hF, hG]
+    simp only [realCombination]
+    ring
+  · change f x + g x = F x x + G x x
+    rw [hFf x, hGg x]
+
+/-- A sum of convex functionals is convex, and a sum of concave functionals is concave. -/
+theorem IsConvexFunctional.add {α : Type u} {c : I → α → α → α} {f g : α → ℝ} {concave : Bool}
+    (hf : IsConvexFunctional c f concave) (hg : IsConvexFunctional c g concave) :
+    IsConvexFunctional c (fun x ↦ f x + g x) concave := by
+  intro t x y
+  have hsum : ∀ u v w z : ℝ, realCombination t u v + realCombination t w z =
+      realCombination t (u + w) (v + z) := by
+    intro u v w z
+    simp only [realCombination]
+    ring
+  cases concave
+  · have hfxy : f (c t x y) ≤ realCombination t (f x) (f y) := hf t x y
+    have hgxy : g (c t x y) ≤ realCombination t (g x) (g y) := hg t x y
+    change f (c t x y) + g (c t x y) ≤ realCombination t (f x + g x) (f y + g y)
+    rw [← hsum]
+    exact add_le_add hfxy hgxy
+  · have hfxy : realCombination t (f x) (f y) ≤ f (c t x y) := hf t x y
+    have hgxy : realCombination t (g x) (g y) ≤ g (c t x y) := hg t x y
+    change realCombination t (f x + g x) (f y + g y) ≤ f (c t x y) + g (c t x y)
+    rw [← hsum]
+    exact add_le_add hfxy hgxy
+
+/-- A convex-linear real functional satisfies both barycentric inequalities, with equality. -/
+theorem IsConvexLinear.isConvexFunctional {α : Type u} {c : I → α → α → α} {f : α → ℝ}
+    (hf : IsConvexLinear c realCombination f) (concave : Bool) :
+    IsConvexFunctional c f concave := by
+  intro t x y
+  cases concave
+  · change f (c t x y) ≤ realCombination t (f x) (f y)
+    exact (hf t x y).le
+  · change realCombination t (f x) (f y) ≤ f (c t x y)
+    exact (hf t x y).ge
+
+/-- A convex-linear real functional is quadratic: it is the diagonal of the mean of its values. -/
+theorem IsConvexLinear.isQuadraticFunctional {α : Type u} {c : I → α → α → α} {f : α → ℝ}
+    (hf : IsConvexLinear c realCombination f) : IsQuadraticFunctional c f := by
+  refine ⟨fun x y ↦ (f x + f y) / 2, ⟨fun x t y z ↦ ?_, fun z t x y ↦ ?_⟩, fun x ↦ ?_⟩
+  · change (f x + f (c t y z)) / 2 = realCombination t ((f x + f y) / 2) ((f x + f z) / 2)
+    rw [hf t y z]
+    simp only [realCombination]
+    ring
+  · change (f (c t x y) + f z) / 2 = realCombination t ((f x + f z) / 2) ((f y + f z) / 2)
+    rw [hf t x y]
+    simp only [realCombination]
+    ring
+  · change f x = (f x + f x) / 2
+    ring
+
+/-- Negation exchanges convexity and concavity. -/
+theorem IsConvexFunctional.neg {α : Type u} {c : I → α → α → α} {f : α → ℝ} {concave : Bool}
+    (hf : IsConvexFunctional c f concave) : IsConvexFunctional c (fun x ↦ -f x) (!concave) := by
+  intro t x y
+  have hneg : realCombination t (-f x) (-f y) = -realCombination t (f x) (f y) := by
+    simp only [realCombination]
+    ring
+  cases concave
+  · have h : f (c t x y) ≤ realCombination t (f x) (f y) := hf t x y
+    change realCombination t (-f x) (-f y) ≤ -f (c t x y)
+    rw [hneg]
+    exact neg_le_neg h
+  · have h : realCombination t (f x) (f y) ≤ f (c t x y) := hf t x y
+    change -f (c t x y) ≤ realCombination t (-f x) (-f y)
+    rw [hneg]
+    exact neg_le_neg h
+
+/-- The negative of a quadratic functional is quadratic. -/
+theorem IsQuadraticFunctional.neg {α : Type u} {c : I → α → α → α} {f : α → ℝ}
+    (hf : IsQuadraticFunctional c f) : IsQuadraticFunctional c (fun x ↦ -f x) := by
+  obtain ⟨F, ⟨hFright, hFleft⟩, hFf⟩ := hf
+  have hneg : ∀ (t : I) (u v : ℝ), -realCombination t u v = realCombination t (-u) (-v) := by
+    intro t u v
+    simp only [realCombination]
+    ring
+  refine ⟨fun x y ↦ -F x y, ⟨fun x t y z ↦ ?_, fun z t x y ↦ ?_⟩, fun x ↦ ?_⟩
+  · change -F x (c t y z) = realCombination t (-F x y) (-F x z)
+    rw [hFright x t y z, hneg]
+  · have hF : F (c t x y) z = realCombination t (F x z) (F y z) := hFleft z t x y
+    change -F (c t x y) z = realCombination t (-F x z) (-F y z)
+    rw [hF, hneg]
+  · change -f x = -F x x
+    rw [hFf x]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Geometry.Foundations.Development002`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Geometry.Convex.FrontierInterior`.
+* `Geometry.RadialBoundary`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Convex / Frontier Interior
+-/
+
+@[expose] public section
+
+open Set
+open scoped Topology
+
+namespace MovingSofa
+
+/-- A bounded component of the frontier complement that meets the set lies in its interior. -/
+lemma connectedComponentIn_compl_frontier_subset_interior {s : Set Point}
+    (hs : IsClosed s) {p : Point} (hp : p ∈ interior s) :
+    connectedComponentIn (frontier s)ᶜ p ⊆ interior s := by
+  have hsub : connectedComponentIn (frontier s)ᶜ p ⊆ interior s ∪ sᶜ := by
+    intro q hq
+    have hq' := connectedComponentIn_subset (frontier s)ᶜ p hq
+    simp only [frontier, hs.closure_eq, mem_compl_iff, mem_sdiff, not_and_or,
+      not_not] at hq'
+    exact hq'.symm
+  rcases isPreconnected_connectedComponentIn.subset_or_subset isOpen_interior
+      hs.isOpen_compl (disjoint_left.mpr (fun _ hx hy ↦ hy (interior_subset hx))) hsub with h | h
+  · exact h
+  · have hp' : p ∈ (frontier s)ᶜ := by
+      simp only [frontier, hs.closure_eq, mem_compl_iff, mem_sdiff, not_and_or,
+        not_not]
+      exact Or.inr hp
+    exact False.elim ((h (mem_connectedComponentIn hp')) (interior_subset hp))
+
+/-- The interior of a bounded closed set lies in the bounded component of its frontier complement.
+-/
+lemma interior_subset_jordanInterior_frontier {s : Set Point}
+    (hs : IsClosed s) (hb : Bornology.IsBounded s) :
+    interior s ⊆ jordanInterior (frontier s) := by
+  intro p hp
+  refine ⟨?_, hb.subset ((connectedComponentIn_compl_frontier_subset_interior hs hp).trans
+    interior_subset)⟩
+  intro hfront
+  exact hfront.2 hp
+
+/-- The outward ray from a point outside a convex set remains outside the set. -/
+lemma ray_smul_sub_notMem {s : Set Point} (hs : Convex ℝ s)
+    {o p : Point} (ho : o ∈ s) (hp : p ∉ s) {r : ℝ} (hr : 1 ≤ r) :
+    o + r • (p - o) ∉ s := by
+  intro hq
+  have hrpos : 0 < r := lt_of_lt_of_le zero_lt_one hr
+  have h := (hs.starConvex ho).add_smul_sub_mem hq
+    (inv_nonneg.mpr hrpos.le) (inv_le_one_of_one_le₀ hr)
+  have heq : o + r⁻¹ • (o + r • (p - o) - o) = p := by
+    rw [add_sub_cancel_left, smul_smul, inv_mul_cancel₀ hrpos.ne', one_smul]
+    abel
+  exact hp (heq ▸ h)
+
+/-- Every exterior point of a nonempty bounded closed convex set has an unbounded component outside
+its frontier. -/
+lemma exterior_component_unbounded {s : Set Point} (hs : Convex ℝ s)
+    (hclosed : IsClosed s) {o p : Point} (ho : o ∈ s) (hp : p ∉ s) :
+    ¬ Bornology.IsBounded (connectedComponentIn (frontier s)ᶜ p) := by
+  let f : ℝ → Point := fun r ↦ o + r • (p - o)
+  have hf : Continuous f := continuous_const.add (continuous_id.smul continuous_const)
+  have hpre : IsPreconnected (f '' Set.Ici 1) :=
+    isPreconnected_Ici.image f hf.continuousOn
+  have hsub : f '' Set.Ici 1 ⊆ (frontier s)ᶜ := by
+    rintro q ⟨r, hr, rfl⟩ hq
+    exact ray_smul_sub_notMem hs ho hp hr (hclosed.closure_eq ▸ hq.1)
+  have hpmem : p ∈ f '' Set.Ici 1 := ⟨1, by simp, by simp [f]⟩
+  have hcomp := hpre.subset_connectedComponentIn hpmem hsub
+  intro hb
+  obtain ⟨C, hC⟩ := hb.exists_norm_le
+  have hnorm : 0 < ‖p - o‖ := norm_pos_iff.mpr (sub_ne_zero.mpr (by
+    intro h; exact hp (h ▸ ho)))
+  let r := max 1 ((C + ‖o‖ + 1) / ‖p - o‖)
+  have hr : 1 ≤ r := le_max_left _ _
+  have hq := hC (f r) (hcomp ⟨r, hr, rfl⟩)
+  have hdiff : ‖r • (p - o)‖ ≤ C + ‖o‖ := by
+    calc
+      ‖r • (p - o)‖ = ‖f r - o‖ := by simp [f]
+      _ ≤ ‖f r‖ + ‖o‖ := norm_sub_le _ _
+      _ ≤ C + ‖o‖ := by linarith
+  rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (by linarith : 0 ≤ r)] at hdiff
+  have hlower := (div_le_iff₀ hnorm).mp (le_max_right 1 ((C + ‖o‖ + 1) / ‖p - o‖))
+  change C + ‖o‖ + 1 ≤ r * ‖p - o‖ at hlower
+  linarith
+
+/-- The bounded complementary region of a nonempty compact convex set frontier is its interior. -/
+lemma jordanInterior_frontier_eq_interior {s : Set Point} (hs : Convex ℝ s)
+    (hclosed : IsClosed s) (hb : Bornology.IsBounded s) (hne : s.Nonempty) :
+    jordanInterior (frontier s) = interior s := by
+  apply Set.Subset.antisymm
+  · intro p hp
+    have hps : p ∈ s := by
+      by_contra hnot
+      obtain ⟨o, ho⟩ := hne
+      exact exterior_component_unbounded hs hclosed ho hnot hp.2
+    by_contra hnot
+    exact hp.1 ⟨subset_closure hps, hnot⟩
+  · exact interior_subset_jordanInterior_frontier hclosed hb
+
+end MovingSofa
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Radial Boundary
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem convexBody_radial_boundary (K : ConvexBody Point) (o : Point)
+    (ho : o ∈ interior (K : Set Point)) :
+    ∃ (ρ : Point → ℝ)
+      (e : {u : Point | ‖u‖ = 1} ≃ₜ ↥(frontier (K : Set Point)))
+      (C : NNReal) (γ : ContinuousBVPaths 0 (2 * Real.pi)),
+      (∀ u : Point, ‖u‖ = 1 →
+        0 < ρ u ∧ o + ρ u • u ∈ frontier (K : Set Point) ∧
+        ∀ r : ℝ, 0 < r → o + r • u ∈ frontier (K : Set Point) → r = ρ u) ∧
+      (∀ u : {u : Point | ‖u‖ = 1}, (e u : Point) = o + ρ u • (u : Point)) ∧
+      LipschitzWith C (fun u ↦ (e u : Point)) ∧
+      (∀ t : Set.Icc (0 : ℝ) (2 * Real.pi),
+        γ.val t = o + ρ (normalVector ((t : ℝ) : Real.Angle)) •
+          normalVector ((t : ℝ) : Real.Angle)) ∧
+      IsOrientedJordanParametrization (by positivity)
+        (frontier (K : Set Point)) true γ.val ∧
+      jordanInterior (frontier (K : Set Point)) = interior (K : Set Point) := by
+  obtain ⟨ρ, e, C, hρ, he, hC⟩ := exists_radial_homeomorph K.convex K.isCompact.isBounded o ho
+  let f := fun u ↦ (e u : Point)
+  let γ := radialBVLoop f hC
+  have hrange : Set.range γ.val = frontier (K : Set Point) := by
+    rw [range_radialBVLoop]
+    ext p
+    constructor
+    · rintro ⟨u, rfl⟩
+      exact (e u).property
+    · intro hp
+      obtain ⟨u, hu⟩ := e.surjective ⟨p, hp⟩
+      exact ⟨u, congrArg Subtype.val hu⟩
+  have hclosed := radialBVLoop_closed f hC
+  have hinj := radialBVLoop_injOn f hC (Subtype.val_injective.comp e.injective)
+  have hformula (t : Set.Icc (0 : ℝ) (2 * Real.pi)) : γ.val t =
+      o + ρ (normalVector ((t : ℝ) : Real.Angle)) • normalVector ((t : ℝ) : Real.Angle) := by
+    rw [radialBVLoop_apply]
+    exact he _
+  have hinter := jordanInterior_frontier_eq_interior K.convex K.isCompact.isClosed
+    K.isCompact.isBounded K.nonempty
+  have hwcenter : curveWinding (by positivity) γ.val o = 1 := by
+    have hx := funext hformula
+    rw [hx]
+    exact curveWinding_radial_center o _ (fun t ↦ (hρ _ (norm_normalVector_real t)).1)
+  let : PreconnectedSpace ↥(interior (K : Set Point)) :=
+    Subtype.preconnectedSpace K.convex.interior.isPreconnected
+  have hlc : IsLocallyConstant
+      (fun p : ↥(interior (K : Set Point)) ↦ curveWinding (by positivity) γ.val p.val) := by
+    apply (IsLocallyConstant.iff_exists_open _).mpr
+    intro p
+    have hp : p.val ∉ Set.range γ.val := by
+      rw [hrange]
+      exact fun h ↦ h.2 p.property
+    obtain ⟨U, hU, hpU, hUeq⟩ :=
+      curveWinding_locally_constant_off_range (by positivity) γ.property.1 hclosed hp
+    exact ⟨Subtype.val ⁻¹' U, hU.preimage continuous_subtype_val, hpU,
+      fun q hq ↦ hUeq q.val hq⟩
+  refine ⟨ρ, e, C, γ, hρ, he, hC, hformula, ?_, hinter⟩
+  refine ⟨by positivity, isJordanCurve_of_unitSphere_homeomorph e,
+    γ.property.1, hrange, hclosed, hinj, ?_⟩
+  intro p hp
+  rw [hinter] at hp
+  exact (hlc.apply_eq_of_preconnectedSpace ⟨p, hp⟩ ⟨o, ho⟩).trans hwcenter
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/Development/Geometry/Foundations/Development003.lean b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development003.lean
new file mode 100644
index 0000000000..088d3191f6
--- /dev/null
+++ b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development003.lean
@@ -0,0 +1,9077 @@
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+module
+
+public import LeanPool.Isoperimetric.BrunnMinkowski
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development002
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development001
+
+
+public import Mathlib.Algebra.Module.LinearMap.DivisionRing
+public import Mathlib.Analysis.Calculus.MeanValue
+public import Mathlib.Analysis.Complex.Isometry
+public import Mathlib.Analysis.Complex.Tietze
+public import Mathlib.Analysis.Convex.Basic
+public import Mathlib.Analysis.Convex.GaugeRescale
+public import Mathlib.Analysis.InnerProductSpace.PiL2
+public import Mathlib.Analysis.LocallyConvex.Separation
+public import Mathlib.Analysis.LocallyConvex.WithSeminorms
+public import Mathlib.Analysis.Normed.Module.Basic
+public import Mathlib.Analysis.Normed.Module.Connected
+public import Mathlib.Analysis.Normed.Module.Convex
+public import Mathlib.Analysis.SpecialFunctions.Complex.Circle
+public import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
+public import Mathlib.MeasureTheory.Integral.Bochner.Basic
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.ContDiff
+public import Mathlib.MeasureTheory.SetSemiring
+public import Mathlib.MeasureTheory.VectorMeasure.BoundedVariation
+public import Mathlib.MeasureTheory.VectorMeasure.IntegrationByParts
+public import Mathlib.MeasureTheory.VectorMeasure.WithDensityVec
+public import Mathlib.SetTheory.Cardinal.Finite
+public import Mathlib.Tactic
+public import Mathlib.Topology.Bornology.Basic
+public import Mathlib.Topology.Connected.Basic
+public import Mathlib.Topology.Connected.Clopen
+public import Mathlib.Topology.Connected.LocallyConnected
+public import Mathlib.Topology.EMetricSpace.BoundedVariation
+public import Mathlib.Topology.Homeomorph.Lemmas
+public import Mathlib.Topology.Homotopy.Lifting
+public import Mathlib.Topology.Instances.AddCircle.Real
+public import Mathlib.Topology.Maps.Basic
+public import Mathlib.Topology.Separation.Hausdorff
+/-!
+# Moving sofa: related mathematical developments
+
+* `Infrastructure.Topology.Foundations.Development001`.
+* `Infrastructure.Curves.Foundations.Development002`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `JordanPick.JordanCurve.Arcs`.
+* `JordanPick.JordanCurve.Brouwer`.
+* `JordanPick.JordanCurve.Counting`.
+* `JordanPick.JordanCurve`.
+* `TauCeti.Analysis.Calculus.MetricVariation`.
+* `TauCeti.Analysis.Normed.Module.Ball.Exterior`.
+* `TauCeti.Analysis.Normed.Module.HalfSpace`.
+* `TauCeti.Topology.EMetricSpace.BoundedVariation`.
+* `External`.
+* `TauCeti.Topology.Frontier`.
+* `TauCeti.Topology.FilledHull`.
+* `TauCeti.Analysis.Normed.Module.FilledHull`.
+* `TauCeti.Topology.LocallyConnected`.
+* `TauCeti.Topology.JordanCurve.Basic`.
+* `TauCeti.Topology.JordanCurve.Path`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Rado Kirov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Rado Kirov
+-/
+/-!
+# Arc scaffolding for the (continuous) Jordan curve theorem
+
+Reusable topology lemmas that split a topological circle into closed arcs, each
+homeomorphic to the unit interval.  These feed Maehara's proof of the Jordan
+curve theorem.
+
+`Plane := EuclideanSpace ℝ (Fin 2)` and the circle is `Metric.sphere (0:Plane) 1`.
+
+Main deliverables:
+* `circleHomeoSphere` / `spherePlaneHomeoCircle` — the bridge between Mathlib's
+  `Circle` (unit circle in `ℂ`) and `sphere (0:Plane) 1`.
+* `param` — the angle parametrization `ℝ → sphere (0:Plane) 1`, continuous,
+  `2π`-periodic, with explicit fibers and surjective.
+* `arcHomeoUnitInterval` — a closed arc `param '' Icc a b` (with `a < b`,
+  `b - a < 2π`) is homeomorphic to `unitInterval`.
+* `sphere_split` — two distinct points cut the circle into two closed arcs, each
+  `≃ₜ unitInterval`, with union the whole circle and intersection the two points.
+* `jordanCurve_split` — transport of `sphere_split` across a homeomorphism
+  `sphere (0:Plane) 1 ≃ₜ K`.
+* `exists_proper_arc` — a proper closed subset of the circle sits inside a proper
+  closed arc `≃ₜ unitInterval`.
+-/
+
+@[expose] public section
+
+namespace JordanCurve.Arcs
+
+open Metric Set Function Real
+
+/-- The plane `ℝ²`. -/
+abbrev Plane := EuclideanSpace ℝ (Fin 2)
+
+/-! ## 1. The circle model bridge -/
+
+/-- A linear isometry equivalence `ℂ ≃ₗᵢ[ℝ] Plane`, from the standard orthonormal
+basis of `EuclideanSpace ℝ (Fin 2)`. -/
+noncomputable def complexLIE : ℂ ≃ₗᵢ[ℝ] Plane :=
+  Complex.isometryOfOrthonormal (EuclideanSpace.basisFun (Fin 2) ℝ)
+
+/-- The underlying `Equiv` between Mathlib's `Circle` and the unit sphere of the
+plane, induced by `complexLIE`. -/
+noncomputable def circleEquivSphere : Circle ≃ sphere (0 : Plane) 1 where
+  toFun z := ⟨complexLIE z, by
+    rw [mem_sphere_zero_iff_norm, complexLIE.norm_map]; exact z.norm_coe⟩
+  invFun w := ⟨complexLIE.symm w, by
+    change complexLIE.symm w ∈ sphere (0 : ℂ) 1
+    rw [mem_sphere_zero_iff_norm, complexLIE.symm.norm_map, ← mem_sphere_zero_iff_norm]
+    exact w.2⟩
+  left_inv z := by ext; simp [complexLIE.symm_apply_apply]
+  right_inv w := by ext; simp [complexLIE.apply_symm_apply]
+
+/-- **Circle model bridge.** `Circle ≃ₜ sphere (0:Plane) 1`. -/
+noncomputable def circleHomeoSphere : Circle ≃ₜ sphere (0 : Plane) 1 :=
+  Continuous.homeoOfEquivCompactToT2 (f := circleEquivSphere) <| by
+    apply Continuous.subtype_mk
+    exact complexLIE.continuous.comp continuous_subtype_val
+
+/-- **Circle model bridge** (as requested): `sphere (0:Plane) 1 ≃ₜ Circle`. -/
+noncomputable def spherePlaneHomeoCircle : sphere (0 : Plane) 1 ≃ₜ Circle :=
+  circleHomeoSphere.symm
+
+@[simp] lemma circleHomeoSphere_coe (z : Circle) :
+    (circleHomeoSphere z : Plane) = complexLIE z := rfl
+
+/-! ## 2. The angle parametrization -/
+
+/-- The angle parametrization `ℝ → sphere (0:Plane) 1`, `θ ↦` the plane point at
+angle `θ` on the unit circle. -/
+noncomputable def param (θ : ℝ) : sphere (0 : Plane) 1 := circleHomeoSphere (Circle.exp θ)
+
+@[continuity, fun_prop]
+lemma continuous_param : Continuous param :=
+  circleHomeoSphere.continuous.comp Circle.exp.continuous
+
+/-- Two angles give the same point iff they differ by an integer multiple of `2π`. -/
+lemma param_eq_iff {s t : ℝ} : param s = param t ↔ ∃ m : ℤ, s = t + m * (2 * π) := by
+  unfold param
+  rw [circleHomeoSphere.injective.eq_iff, Circle.exp_eq_exp]
+
+/-- The parametrization is surjective. -/
+lemma param_surjective : Surjective param :=
+  circleHomeoSphere.surjective.comp Circle.exp_surjective
+
+/-- The parametrization is `2π`-periodic. -/
+lemma param_periodic : Function.Periodic param (2 * π) := by
+  intro θ
+  rw [param_eq_iff]
+  exact ⟨1, by push_cast; ring⟩
+
+/-! ## 2. Closed arc ≃ₜ unitInterval -/
+
+/-- On a closed interval shorter than a full turn, `param` is injective. -/
+lemma param_injOn {a b : ℝ} (h : b - a < 2 * π) : InjOn param (Icc a b) := by
+  intro s hs t ht hst
+  obtain ⟨m, hm⟩ := param_eq_iff.1 hst
+  have hp : (0 : ℝ) < 2 * π := by positivity
+  obtain ⟨hαs, hsβ⟩ := hs
+  obtain ⟨hat, htb⟩ := ht
+  have e : s - t = (m : ℝ) * (2 * π) := by linarith
+  have hlt : (m : ℝ) * (2 * π) < 1 * (2 * π) := by rw [one_mul, ← e]; linarith
+  have hgt : (-1 : ℝ) * (2 * π) < (m : ℝ) * (2 * π) := by rw [neg_one_mul, ← e]; linarith
+  have u1 : (m : ℝ) < 1 := lt_of_mul_lt_mul_right hlt hp.le
+  have u2 : (-1 : ℝ) < (m : ℝ) := lt_of_mul_lt_mul_right hgt hp.le
+  have hm0 : m = 0 := by
+    have b1 : m < (1 : ℤ) := by exact_mod_cast u1
+    have b2 : (-1 : ℤ) < m := by exact_mod_cast u2
+    omega
+  rw [hm0] at hm; push_cast at hm; linarith
+
+/-- Image of a closed interval of angles under `param` (a "closed arc"). -/
+lemma isClosed_arc (a b : ℝ) : IsClosed (param '' Icc a b) :=
+  (isCompact_Icc.image continuous_param).isClosed
+
+/-- On a short closed interval `param` restricts to a homeomorphism onto its
+image (the arc). -/
+noncomputable def arcHomeoIcc {a b : ℝ} (h : b - a < 2 * π) :
+    (Icc a b) ≃ₜ (param '' Icc a b) :=
+  Continuous.homeoOfEquivCompactToT2
+    (f := Equiv.Set.imageOfInjOn param (Icc a b) (param_injOn h))
+    (continuous_induced_rng.2 (continuous_param.comp continuous_subtype_val))
+
+/-- **Closed arc ≃ₜ unitInterval.** A closed arc `param '' Icc a b` with
+`a < b` and `b - a < 2π` is homeomorphic to the unit interval. -/
+noncomputable def arcHomeoUnitInterval {a b : ℝ} (hab : a < b) (h : b - a < 2 * π) :
+    (param '' Icc a b) ≃ₜ unitInterval :=
+  (arcHomeoIcc h).symm.trans (iccHomeoI a b hab)
+
+/-- The left endpoint `param a` of the arc maps to `0` under `arcHomeoUnitInterval`. -/
+lemma arcHomeoUnitInterval_apply_left {a b : ℝ} (hab : a < b) (h : b - a < 2 * π)
+    (hmem : param a ∈ param '' Icc a b) :
+    arcHomeoUnitInterval hab h ⟨param a, hmem⟩ = 0 := by
+  have ha : a ∈ Icc a b := left_mem_Icc.2 hab.le
+  have key : (arcHomeoIcc h) ⟨a, ha⟩ = ⟨param a, hmem⟩ := Subtype.ext rfl
+  have hsymm : (arcHomeoIcc h).symm ⟨param a, hmem⟩ = ⟨a, ha⟩ := by
+    rw [← key, Homeomorph.symm_apply_apply]
+  apply Subtype.ext
+  change ((arcHomeoIcc h).symm.trans (iccHomeoI a b hab) ⟨param a, hmem⟩ : ℝ)
+      = ((0 : unitInterval) : ℝ)
+  rw [Homeomorph.trans_apply, hsymm, iccHomeoI_apply_coe, Set.Icc.coe_zero]
+  change (a - a) / (b - a) = 0
+  rw [sub_self, zero_div]
+
+/-- The right endpoint `param b` of the arc maps to `1` under `arcHomeoUnitInterval`. -/
+lemma arcHomeoUnitInterval_apply_right {a b : ℝ} (hab : a < b) (h : b - a < 2 * π)
+    (hmem : param b ∈ param '' Icc a b) :
+    arcHomeoUnitInterval hab h ⟨param b, hmem⟩ = 1 := by
+  have hb : b ∈ Icc a b := right_mem_Icc.2 hab.le
+  have key : (arcHomeoIcc h) ⟨b, hb⟩ = ⟨param b, hmem⟩ := Subtype.ext rfl
+  have hsymm : (arcHomeoIcc h).symm ⟨param b, hmem⟩ = ⟨b, hb⟩ := by
+    rw [← key, Homeomorph.symm_apply_apply]
+  apply Subtype.ext
+  change ((arcHomeoIcc h).symm.trans (iccHomeoI a b hab) ⟨param b, hmem⟩ : ℝ)
+      = ((1 : unitInterval) : ℝ)
+  rw [Homeomorph.trans_apply, hsymm, iccHomeoI_apply_coe, Set.Icc.coe_one]
+  change (b - a) / (b - a) = 1
+  rw [div_self (by linarith : b - a ≠ 0)]
+
+/-! ## 2b. Interior of an arc is path-connected -/
+
+/-- **Interior of an arc is path-connected.** If `A ≃ₜ unitInterval` via `e` and two
+points `x, y ∈ A` are the endpoints (`{e x, e y} = {0, 1}`), then `A \ {x, y}` — the
+arc with its endpoints removed — is path-connected. -/
+theorem arc_interior_isPathConnected {X : Type*} [TopologicalSpace X] {A : Set X}
+    (e : A ≃ₜ unitInterval) {x y : X} (hx : x ∈ A) (hy : y ∈ A)
+    (he : ({e ⟨x, hx⟩, e ⟨y, hy⟩} : Set unitInterval) = {0, 1}) :
+    IsPathConnected (A \ {x, y}) := by
+  -- endpoint values, extracted from the set equality
+  have hmx : e ⟨x, hx⟩ ∈ ({0, 1} : Set unitInterval) := by
+    rw [← he]; exact Set.mem_insert _ _
+  have hmy : e ⟨y, hy⟩ ∈ ({0, 1} : Set unitInterval) := by
+    rw [← he]; exact Set.mem_insert_of_mem _ rfl
+  have h0 : (0 : unitInterval) ∈ ({e ⟨x, hx⟩, e ⟨y, hy⟩} : Set unitInterval) := by
+    rw [he]; exact Set.mem_insert _ _
+  have h1 : (1 : unitInterval) ∈ ({e ⟨x, hx⟩, e ⟨y, hy⟩} : Set unitInterval) := by
+    rw [he]; exact Set.mem_insert_of_mem _ rfl
+  -- the parametrizing map from `Ioo 0 1 ⊆ ℝ`
+  set g : ℝ → X := fun s => ((e.symm (Set.projIcc 0 1 (by norm_num) s) : A) : X) with hg
+  have hgcont : Continuous g :=
+    continuous_subtype_val.comp (e.symm.continuous.comp continuous_projIcc)
+  have hIoo : IsPathConnected (Set.Ioo (0 : ℝ) 1) :=
+    (convex_Ioo (0 : ℝ) 1).isPathConnected ⟨1 / 2, by norm_num⟩
+  have himg : g '' Set.Ioo (0 : ℝ) 1 = A \ {x, y} := by
+    ext p
+    simp only [Set.mem_image, Set.mem_sdiff, Set.mem_insert_iff, Set.mem_singleton_iff]
+    constructor
+    · rintro ⟨s, hs, rfl⟩
+      have hsI : s ∈ Set.Icc (0 : ℝ) 1 := ⟨hs.1.le, hs.2.le⟩
+      have hproj : Set.projIcc (0 : ℝ) 1 (by norm_num) s = ⟨s, hsI⟩ :=
+        Set.projIcc_of_mem _ hsI
+      refine ⟨(e.symm _).2, ?_⟩
+      rintro (hgx | hgy)
+      · -- g s = x is impossible
+        have heq : e.symm (Set.projIcc (0 : ℝ) 1 (by norm_num) s) = ⟨x, hx⟩ :=
+          Subtype.ext hgx
+        have hval : Set.projIcc (0 : ℝ) 1 (by norm_num) s = e ⟨x, hx⟩ := by
+          rw [← heq, Homeomorph.apply_symm_apply]
+        rw [hproj] at hval
+        rcases hmx with hxe | hxe
+        · rw [hxe] at hval
+          exact absurd (congrArg Subtype.val hval) (by simpa using hs.1.ne')
+        · rw [Set.mem_singleton_iff] at hxe; rw [hxe] at hval
+          exact absurd (congrArg Subtype.val hval) (by simpa using hs.2.ne)
+      · -- g s = y is impossible
+        have heq : e.symm (Set.projIcc (0 : ℝ) 1 (by norm_num) s) = ⟨y, hy⟩ :=
+          Subtype.ext hgy
+        have hval : Set.projIcc (0 : ℝ) 1 (by norm_num) s = e ⟨y, hy⟩ := by
+          rw [← heq, Homeomorph.apply_symm_apply]
+        rw [hproj] at hval
+        rcases hmy with hye | hye
+        · rw [hye] at hval
+          exact absurd (congrArg Subtype.val hval) (by simpa using hs.1.ne')
+        · rw [Set.mem_singleton_iff] at hye; rw [hye] at hval
+          exact absurd (congrArg Subtype.val hval) (by simpa using hs.2.ne)
+    · rintro ⟨hpA, hpxy⟩
+      have hpx : p ≠ x := fun h => hpxy (Or.inl h)
+      have hpy : p ≠ y := fun h => hpxy (Or.inr h)
+      set t : unitInterval := e ⟨p, hpA⟩ with ht
+      have htne0 : t ≠ 0 := by
+        intro h; apply hpx
+        rcases h0 with h0e | h0e
+        · have : e ⟨p, hpA⟩ = e ⟨x, hx⟩ := by rw [← ht, h, h0e]
+          exact congrArg Subtype.val (e.injective this)
+        · rw [Set.mem_singleton_iff] at h0e
+          have : e ⟨p, hpA⟩ = e ⟨y, hy⟩ := by rw [← ht, h, h0e]
+          exact absurd (congrArg Subtype.val (e.injective this)) hpy
+      have htne1 : t ≠ 1 := by
+        intro h; apply hpx
+        rcases h1 with h1e | h1e
+        · have : e ⟨p, hpA⟩ = e ⟨x, hx⟩ := by rw [← ht, h, h1e]
+          exact congrArg Subtype.val (e.injective this)
+        · rw [Set.mem_singleton_iff] at h1e
+          have : e ⟨p, hpA⟩ = e ⟨y, hy⟩ := by rw [← ht, h, h1e]
+          exact absurd (congrArg Subtype.val (e.injective this)) hpy
+      have htmem : (t : ℝ) ∈ Set.Ioo (0 : ℝ) 1 := by
+        refine ⟨lt_of_le_of_ne t.2.1 ?_, lt_of_le_of_ne t.2.2 ?_⟩
+        · intro h; exact htne0 (Subtype.ext h.symm)
+        · intro h; exact htne1 (Subtype.ext h)
+      refine ⟨(t : ℝ), htmem, ?_⟩
+      have hproj : Set.projIcc (0 : ℝ) 1 (by norm_num) (t : ℝ) = t :=
+        Set.projIcc_val _ t
+      change ((e.symm (Set.projIcc (0 : ℝ) 1 (by norm_num) (t : ℝ)) : A) : X) = p
+      rw [hproj, ht, Homeomorph.symm_apply_apply]
+  rw [← himg]
+  exact hIoo.image hgcont
+
+/-- **`arc_interior_joinedIn`.** For an arc `A ≃ₜ unitInterval` via `e` whose two
+endpoints are `x, y` (`{e x, e y} = {0, 1}`), any two interior points `u, v ∈ A`
+(neither equal to `x` nor `y`) are joined by a path inside `A \ {x, y}`. -/
+theorem arc_interior_joinedIn {X : Type*} [TopologicalSpace X] {A : Set X}
+    (e : A ≃ₜ unitInterval) {x y : X} (hx : x ∈ A) (hy : y ∈ A)
+    (he : ({e ⟨x, hx⟩, e ⟨y, hy⟩} : Set unitInterval) = {0, 1})
+    {u v : X} (hu : u ∈ A) (hv : v ∈ A)
+    (hux : u ∉ ({x, y} : Set X)) (hvx : v ∉ ({x, y} : Set X)) :
+    JoinedIn (A \ {x, y}) u v :=
+  (arc_interior_isPathConnected e hx hy he).joinedIn u ⟨hu, hux⟩ v ⟨hv, hvx⟩
+
+/-! ## 3. The two-point split -/
+
+/-- **Two-point split.** Two distinct points `x`, `y` on the circle cut it into
+two closed arcs `A₁`, `A₂`, each homeomorphic to the unit interval, whose union
+is the whole circle and whose intersection is exactly `{x, y}`. -/
+theorem sphere_split {x y : sphere (0 : Plane) 1} (hxy : x ≠ y) :
+    ∃ A₁ A₂ : Set (sphere (0 : Plane) 1),
+      IsClosed A₁ ∧ IsClosed A₂ ∧ A₁ ∪ A₂ = univ ∧ A₁ ∩ A₂ = {x, y} ∧
+      Nonempty (A₁ ≃ₜ unitInterval) ∧ Nonempty (A₂ ≃ₜ unitInterval) ∧
+      IsPathConnected (A₁ \ {x, y}) ∧ IsPathConnected (A₂ \ {x, y}) := by
+  have hp : (0 : ℝ) < 2 * π := by positivity
+  obtain ⟨α, hα⟩ := param_surjective x
+  obtain ⟨β₀, hβ₀⟩ := param_surjective y
+  set β := toIocMod hp α β₀ with hβdef
+  have hmemβ : β ∈ Ioc α (α + 2 * π) := toIocMod_mem_Ioc hp α β₀
+  -- `param β = y`
+  have hpar_β : param β = y := by
+    have hz : β₀ - β = (toIocDiv hp α β₀ : ℤ) • (2 * π) := self_sub_toIocMod hp α β₀
+    rw [zsmul_eq_mul] at hz
+    have hpp : param β = param β₀ := by
+      rw [param_eq_iff]; exact ⟨-(toIocDiv hp α β₀), by push_cast; linarith⟩
+    rw [hpp, hβ₀]
+  -- `β < α + 2π` (else `y = x`)
+  have hβlt : β < α + 2 * π := by
+    rcases lt_or_eq_of_le hmemβ.2 with h | h
+    · exact h
+    · exact absurd (by rw [← hpar_β, h, param_periodic, hα] : y = x).symm hxy
+  have hlen1 : β - α < 2 * π := by linarith [hβlt]
+  have hlen2 : (α + 2 * π) - β < 2 * π := by linarith [hmemβ.1]
+  -- union is everything
+  have hunion : (param '' Icc α β) ∪ (param '' Icc β (α + 2 * π)) = univ := by
+    rw [← image_union, Icc_union_Icc_eq_Icc hmemβ.1.le hβlt.le,
+      param_periodic.image_Icc hp α, param_surjective.range_eq]
+  -- intersection is `{x, y}`
+  have hinter : (param '' Icc α β) ∩ (param '' Icc β (α + 2 * π)) = {x, y} := by
+    ext z
+    simp only [mem_inter_iff, mem_image, mem_insert_iff, mem_singleton_iff]
+    constructor
+    · rintro ⟨⟨s, hs, hsz⟩, ⟨t, ht, htz⟩⟩
+      have hst : param s = param t := hsz.trans htz.symm
+      obtain ⟨m, hm⟩ := param_eq_iff.1 hst
+      obtain ⟨hαs, hsβ⟩ := hs
+      obtain ⟨hβt, htα⟩ := ht
+      have e : s - t = (m : ℝ) * (2 * π) := by linarith
+      have hm_ub : (m : ℝ) ≤ 0 := by
+        have h2 : (m : ℝ) * (2 * π) ≤ 0 * (2 * π) := by rw [zero_mul, ← e]; linarith
+        exact le_of_mul_le_mul_right h2 hp
+      have hm_lb : (-1 : ℝ) ≤ (m : ℝ) := by
+        have h2 : (-1 : ℝ) * (2 * π) ≤ (m : ℝ) * (2 * π) := by
+          rw [neg_one_mul, ← e]; linarith
+        exact le_of_mul_le_mul_right h2 hp
+      have hcase : m = 0 ∨ m = -1 := by
+        have a0 : m ≤ (0 : ℤ) := by exact_mod_cast hm_ub
+        have a1 : (-1 : ℤ) ≤ m := by exact_mod_cast hm_lb
+        omega
+      rcases hcase with h0 | h1
+      · right
+        have hst2 : s = t := by rw [h0] at hm; push_cast at hm; linarith
+        have hsβeq : s = β := le_antisymm hsβ (by rw [hst2]; exact hβt)
+        rw [← hsz, hsβeq]; exact hpar_β
+      · left
+        have hst2 : s = t - 2 * π := by rw [h1] at hm; push_cast at hm; linarith
+        have hsαeq : s = α := le_antisymm (by rw [hst2]; linarith) hαs
+        rw [← hsz, hsαeq]; exact hα
+    · rintro (rfl | rfl)
+      · exact ⟨⟨α, ⟨le_refl α, hmemβ.1.le⟩, hα⟩,
+          ⟨α + 2 * π, ⟨hβlt.le, le_refl _⟩, by rw [param_periodic]; exact hα⟩⟩
+      · exact ⟨⟨β, ⟨hmemβ.1.le, le_refl β⟩, hpar_β⟩,
+          ⟨β, ⟨le_refl β, hβlt.le⟩, hpar_β⟩⟩
+  -- path-connectedness of the two arcs with their shared endpoints removed
+  have hmem_α₁ : param α ∈ param '' Icc α β := ⟨α, left_mem_Icc.2 hmemβ.1.le, rfl⟩
+  have hmem_β₁ : param β ∈ param '' Icc α β := ⟨β, right_mem_Icc.2 hmemβ.1.le, rfl⟩
+  have hxA1 : x ∈ param '' Icc α β := hα ▸ hmem_α₁
+  have hyA1 : y ∈ param '' Icc α β := hpar_β ▸ hmem_β₁
+  have hpc1 : IsPathConnected (param '' Icc α β \ {x, y}) := by
+    refine arc_interior_isPathConnected (arcHomeoUnitInterval hmemβ.1 hlen1) hxA1 hyA1 ?_
+    have hex : arcHomeoUnitInterval hmemβ.1 hlen1 ⟨x, hxA1⟩ = 0 := by
+      rw [show (⟨x, hxA1⟩ : ↥(param '' Icc α β)) = ⟨param α, hmem_α₁⟩ from
+        Subtype.ext hα.symm]
+      exact arcHomeoUnitInterval_apply_left hmemβ.1 hlen1 hmem_α₁
+    have hey : arcHomeoUnitInterval hmemβ.1 hlen1 ⟨y, hyA1⟩ = 1 := by
+      rw [show (⟨y, hyA1⟩ : ↥(param '' Icc α β)) = ⟨param β, hmem_β₁⟩ from
+        Subtype.ext hpar_β.symm]
+      exact arcHomeoUnitInterval_apply_right hmemβ.1 hlen1 hmem_β₁
+    rw [hex, hey]
+  have hmem_β₂ : param β ∈ param '' Icc β (α + 2 * π) :=
+    ⟨β, left_mem_Icc.2 hβlt.le, rfl⟩
+  have hmem_α₂ : param (α + 2 * π) ∈ param '' Icc β (α + 2 * π) :=
+    ⟨α + 2 * π, right_mem_Icc.2 hβlt.le, rfl⟩
+  have hxeq : param (α + 2 * π) = x := by rw [param_periodic]; exact hα
+  have hxA2 : x ∈ param '' Icc β (α + 2 * π) := hxeq ▸ hmem_α₂
+  have hyA2 : y ∈ param '' Icc β (α + 2 * π) := hpar_β ▸ hmem_β₂
+  have hpc2 : IsPathConnected (param '' Icc β (α + 2 * π) \ {x, y}) := by
+    refine arc_interior_isPathConnected (arcHomeoUnitInterval hβlt hlen2) hxA2 hyA2 ?_
+    have hex : arcHomeoUnitInterval hβlt hlen2 ⟨x, hxA2⟩ = 1 := by
+      rw [show (⟨x, hxA2⟩ : ↥(param '' Icc β (α + 2 * π))) = ⟨param (α + 2 * π), hmem_α₂⟩
+        from Subtype.ext hxeq.symm]
+      exact arcHomeoUnitInterval_apply_right hβlt hlen2 hmem_α₂
+    have hey : arcHomeoUnitInterval hβlt hlen2 ⟨y, hyA2⟩ = 0 := by
+      rw [show (⟨y, hyA2⟩ : ↥(param '' Icc β (α + 2 * π))) = ⟨param β, hmem_β₂⟩ from
+        Subtype.ext hpar_β.symm]
+      exact arcHomeoUnitInterval_apply_left hβlt hlen2 hmem_β₂
+    rw [hex, hey]; exact Set.pair_comm 1 0
+  exact ⟨param '' Icc α β, param '' Icc β (α + 2 * π),
+    isClosed_arc α β, isClosed_arc β (α + 2 * π), hunion, hinter,
+    ⟨arcHomeoUnitInterval hmemβ.1 hlen1⟩, ⟨arcHomeoUnitInterval hβlt hlen2⟩, hpc1, hpc2⟩
+
+/-! ## 4. Transport across a homeomorphism to a Jordan curve -/
+
+/-- **Transport to a Jordan curve.** Given a homeomorphism `f` from the circle to
+a space `K` and two distinct points, the images of the two arcs split `K` into two
+closed arcs `≃ₜ unitInterval` meeting exactly at `{f x, f y}`. -/
+theorem jordanCurve_split {K : Type*} [TopologicalSpace K]
+    (f : sphere (0 : Plane) 1 ≃ₜ K) {x y : sphere (0 : Plane) 1} (hxy : x ≠ y) :
+    ∃ A₁ A₂ : Set K,
+      IsClosed A₁ ∧ IsClosed A₂ ∧ A₁ ∪ A₂ = univ ∧ A₁ ∩ A₂ = {f x, f y} ∧
+      Nonempty (A₁ ≃ₜ unitInterval) ∧ Nonempty (A₂ ≃ₜ unitInterval) ∧
+      IsPathConnected (A₁ \ {f x, f y}) ∧ IsPathConnected (A₂ \ {f x, f y}) := by
+  obtain ⟨A₁, A₂, hc1, hc2, hu, hi, ⟨e1⟩, ⟨e2⟩, hpc1, hpc2⟩ := sphere_split hxy
+  have himg1 : f '' (A₁ \ {x, y}) = f '' A₁ \ {f x, f y} := by
+    rw [Set.image_sdiff f.injective, Set.image_insert_eq, Set.image_singleton]
+  have himg2 : f '' (A₂ \ {x, y}) = f '' A₂ \ {f x, f y} := by
+    rw [Set.image_sdiff f.injective, Set.image_insert_eq, Set.image_singleton]
+  refine ⟨f '' A₁, f '' A₂, f.isClosedMap _ hc1, f.isClosedMap _ hc2, ?_, ?_,
+    ⟨(f.image A₁).symm.trans e1⟩, ⟨(f.image A₂).symm.trans e2⟩,
+    himg1 ▸ hpc1.image f.continuous, himg2 ▸ hpc2.image f.continuous⟩
+  · rw [← image_union, hu, image_univ, f.surjective.range_eq]
+  · rw [← Set.image_inter f.injective, hi, Set.image_insert_eq, Set.image_singleton]
+
+/-! ## 5. A proper closed arc containing a proper closed set -/
+
+/-- **Proper arc containing a set.** A proper closed subset `C` of the circle is
+contained in a proper closed arc `A` (homeomorphic to the unit interval). -/
+theorem exists_proper_arc {C : Set (sphere (0 : Plane) 1)}
+    (hC : IsClosed C) (hCne : C ≠ univ) :
+    ∃ A : Set (sphere (0 : Plane) 1),
+      C ⊆ A ∧ A ≠ univ ∧ IsClosed A ∧ Nonempty (A ≃ₜ unitInterval) := by
+  have hp : (0 : ℝ) < 2 * π := by positivity
+  obtain ⟨z, hz⟩ := (Set.ne_univ_iff_exists_notMem C).1 hCne
+  obtain ⟨γ, hγ⟩ := param_surjective z
+  have hopen : IsOpen (param ⁻¹' Cᶜ) := hC.isOpen_compl.preimage continuous_param
+  have hmemγ : γ ∈ param ⁻¹' Cᶜ := by rw [mem_preimage, hγ]; exact hz
+  obtain ⟨ε, hεpos, hball⟩ := Metric.isOpen_iff.1 hopen γ hmemγ
+  set δ := min (ε / 2) (π / 2) with hδdef
+  have hδpos : 0 < δ := lt_min (by linarith) (by positivity)
+  have hδπ : δ < π := lt_of_le_of_lt (min_le_right _ _) (by linarith [pi_pos])
+  have hδε : δ ≤ ε := le_trans (min_le_left _ _) (by linarith)
+  have hsub : Ioo (γ - δ) (γ + δ) ⊆ param ⁻¹' Cᶜ := by
+    intro w hw
+    apply hball
+    rw [Metric.mem_ball, Real.dist_eq, abs_lt]
+    obtain ⟨hw1, hw2⟩ := hw
+    constructor <;> linarith
+  -- `z ∉ A`, so `A ≠ univ`
+  have hzA : z ∉ param '' Icc (γ + δ) (γ + 2 * π - δ) := by
+    rintro ⟨θ, hθ, hθz⟩
+    have hpe : param θ = param γ := by rw [hθz, hγ]
+    obtain ⟨m, hm⟩ := param_eq_iff.1 hpe
+    obtain ⟨hθ1, hθ2⟩ := hθ
+    have e : (m : ℝ) * (2 * π) = θ - γ := by linarith
+    have hpos : (0 : ℝ) < (m : ℝ) * (2 * π) := by rw [e]; linarith
+    have hlt : (m : ℝ) * (2 * π) < 2 * π := by rw [e]; linarith
+    have m1 : (0 : ℝ) < (m : ℝ) :=
+      lt_of_mul_lt_mul_right (by rw [zero_mul]; exact hpos) hp.le
+    have m2 : (m : ℝ) < 1 :=
+      lt_of_mul_lt_mul_right (by rw [one_mul]; exact hlt) hp.le
+    have a0 : (0 : ℤ) < m := by exact_mod_cast m1
+    have a1 : m < (1 : ℤ) := by exact_mod_cast m2
+    omega
+  refine ⟨param '' Icc (γ + δ) (γ + 2 * π - δ), ?_,
+    (Set.ne_univ_iff_exists_notMem _).2 ⟨z, hzA⟩, isClosed_arc _ _,
+    ⟨arcHomeoUnitInterval (by linarith) (by linarith)⟩⟩
+  -- `C ⊆ A`
+  intro c hc
+  obtain ⟨θc, hθc⟩ := param_surjective c
+  set θ' := toIcoMod hp (γ + δ) θc with hθ'def
+  have hmem' : θ' ∈ Ico (γ + δ) (γ + δ + 2 * π) := toIcoMod_mem_Ico hp (γ + δ) θc
+  have hpar' : param θ' = c := by
+    have hz2 : θc - θ' = (toIcoDiv hp (γ + δ) θc : ℤ) • (2 * π) :=
+      self_sub_toIcoMod hp (γ + δ) θc
+    rw [zsmul_eq_mul] at hz2
+    have : param θ' = param θc := by
+      rw [param_eq_iff]; exact ⟨-(toIcoDiv hp (γ + δ) θc), by push_cast; linarith⟩
+    rw [this, hθc]
+  have hθ'ub : θ' ≤ γ + 2 * π - δ := by
+    by_contra h
+    rw [not_le] at h
+    have hmem2 : θ' - 2 * π ∈ Ioo (γ - δ) (γ + δ) := by
+      constructor
+      · linarith
+      · linarith [hmem'.2]
+    have hpre : (θ' - 2 * π) ∈ param ⁻¹' Cᶜ := hsub hmem2
+    rw [mem_preimage] at hpre
+    have hpc : param (θ' - 2 * π) = c := by
+      rw [show θ' = (θ' - 2 * π) + 2 * π by ring, param_periodic] at hpar'
+      exact hpar'
+    rw [hpc] at hpre
+    exact hpre hc
+  exact ⟨θ', ⟨hmem'.1, hθ'ub⟩, hpar'⟩
+
+end JordanCurve.Arcs
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Rado Kirov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Rado Kirov
+-/
+/-!
+# Toward the 2D Brouwer fixed point theorem
+
+We build the classical topological proof of the two-dimensional Brouwer fixed
+point theorem, in four phases:
+
+* **Phase 1** — the once-around loop on `AddCircle 1` is not homotopic (rel
+  endpoints) to the constant loop.  This is the mathematical heart: it is proved
+  from the covering `ℝ → AddCircle 1` via unique path lifting
+  (`IsCoveringMap.liftPath_apply_one_eq_of_homotopicRel`).
+* **Phase 1.5** — transport of Phase 1 across `AddCircle 1 ≃ₜ Circle ≃ₜ
+  sphere (0 : ℝ²) 1` to obtain a non-nullhomotopic loop on the geometric circle.
+* **Phase 2** — no retraction of the disk onto its boundary circle: a retraction
+  would give a null-homotopy of the Phase 1.5 loop.
+* **Phase 3** — Brouwer for the closed unit disk (`brouwer_disk`): a
+  fixed-point-free self-map yields a retraction (ray construction).
+* **Phase 4** — the general convex/compact/nonempty statement `brouwerFPT`.
+-/
+
+@[expose] public section
+
+namespace JordanCurve.Brouwer
+
+open Metric Set Function unitInterval Topology
+open scoped RealInnerProductSpace
+
+/-- The plane `ℝ²`. -/
+abbrev Plane := EuclideanSpace ℝ (Fin 2)
+
+/-! ## Phase 1 — the once-around loop on `AddCircle 1` is not nullhomotopic -/
+
+/-- The covering map `ℝ → AddCircle 1`. -/
+theorem cover : IsCoveringMap ((↑) : ℝ → AddCircle (1 : ℝ)) :=
+  AddCircle.isCoveringMap_coe 1
+
+/-- The once-around loop `t ↦ ↑t` in `AddCircle 1`. -/
+noncomputable def acLoop : C(I, AddCircle (1 : ℝ)) :=
+  ⟨fun t => ((t : ℝ) : AddCircle (1 : ℝ)), cover.continuous.comp continuous_subtype_val⟩
+
+/-- The lift of `acLoop` to `ℝ` starting at `0`: the identity `t ↦ ↑t`. -/
+noncomputable def idLift : C(I, ℝ) := ⟨fun t => (t : ℝ), continuous_subtype_val⟩
+
+@[simp] lemma acLoop_apply (t : I) : acLoop t = ((t : ℝ) : AddCircle (1 : ℝ)) := rfl
+@[simp] lemma idLift_apply (t : I) : idLift t = (t : ℝ) := rfl
+
+/-- **Phase 1.** The once-around loop is not homotopic rel endpoints to the
+constant loop. -/
+theorem acLoop_not_homotopic :
+    ¬ acLoop.HomotopicRel (ContinuousMap.const I (0 : AddCircle (1 : ℝ))) {0, 1} := by
+  intro h
+  have h0 : acLoop 0 = ((0 : ℝ) : AddCircle (1 : ℝ)) := by simp
+  have h1 : (ContinuousMap.const I (0 : AddCircle (1 : ℝ))) 0 = ((0 : ℝ) : AddCircle (1 : ℝ)) := by
+    simp
+  have key := cover.liftPath_apply_one_eq_of_homotopicRel h (0 : ℝ) h0 h1
+  -- Identify the two lifts.
+  have e1 : cover.liftPath acLoop (0 : ℝ) h0 = idLift := by
+    refine ((cover.eq_liftPath_iff' h0).mpr ⟨?_, ?_⟩).symm
+    · ext t; simp
+    · simp
+  have e2 : cover.liftPath (ContinuousMap.const I (0 : AddCircle (1 : ℝ))) (0 : ℝ) h1
+      = ContinuousMap.const I (0 : ℝ) := cover.liftPath_const h1
+  rw [e1, e2] at key
+  simp at key
+
+/-! ## Phase 1.5 — transport to the geometric circle `sphere (0 : ℝ²) 1` -/
+
+/-- The homeomorphism `AddCircle 1 ≃ₜ sphere (0 : ℝ²) 1`, via `Circle`. -/
+noncomputable def acToSphere : AddCircle (1 : ℝ) ≃ₜ sphere (0 : Plane) 1 :=
+  (AddCircle.homeomorphCircle (one_ne_zero)).trans Arcs.circleHomeoSphere
+
+/-- The base point of the sphere loop. -/
+noncomputable def sBase : sphere (0 : Plane) 1 := acToSphere 0
+
+/-- The once-around loop on the geometric circle `sphere (0 : ℝ²) 1`. -/
+noncomputable def sLoop : C(I, sphere (0 : Plane) 1) :=
+  (⟨acToSphere, acToSphere.continuous⟩ : C(AddCircle (1 : ℝ), sphere (0 : Plane) 1)).comp acLoop
+
+@[simp] lemma sLoop_apply (t : I) : sLoop t = acToSphere (acLoop t) := rfl
+
+lemma sLoop_zero : sLoop 0 = sBase := by simp [sBase]
+
+/-- **Phase 1.5.** The once-around loop on the geometric circle is not homotopic
+rel endpoints to the constant loop. -/
+theorem sLoop_not_homotopic :
+    ¬ sLoop.HomotopicRel (ContinuousMap.const I sBase) {0, 1} := by
+  intro h
+  apply acLoop_not_homotopic
+  have hh := h.comp_continuousMap
+    (⟨acToSphere.symm, acToSphere.symm.continuous⟩ : C(sphere (0 : Plane) 1, AddCircle (1 : ℝ)))
+  have e1 : (⟨acToSphere.symm, acToSphere.symm.continuous⟩ :
+      C(sphere (0 : Plane) 1, AddCircle (1 : ℝ))).comp sLoop = acLoop := by
+    ext t; simp [sLoop, acToSphere.symm_apply_apply]
+  have e2 : (⟨acToSphere.symm, acToSphere.symm.continuous⟩ :
+      C(sphere (0 : Plane) 1, AddCircle (1 : ℝ))).comp (ContinuousMap.const I sBase)
+      = ContinuousMap.const I (0 : AddCircle (1 : ℝ)) := by
+    ext t; simp [sBase, acToSphere.symm_apply_apply]
+  rwa [e1, e2] at hh
+
+lemma sLoop_one : sLoop 1 = sBase := by
+  have : acLoop 1 = (0 : AddCircle (1 : ℝ)) := by
+    simp only [acLoop_apply]
+    have : ((1 : I) : ℝ) = (1 : ℝ) := rfl
+    rw [this]; exact AddCircle.coe_period 1
+  simp [sBase, this]
+
+/-! ## Phase 2 — no retraction of the disk onto its boundary circle -/
+
+/-- The straight-line contraction point `(1-t)·(loop s) + t·base` in the disk. -/
+noncomputable def diskPt (t s : I) : Plane :=
+  (1 - (t : ℝ)) • (sLoop s : Plane) + (t : ℝ) • (sBase : Plane)
+
+lemma diskPt_mem (t s : I) : diskPt t s ∈ closedBall (0 : Plane) 1 := by
+  have hv : (sLoop s : Plane) ∈ closedBall (0 : Plane) 1 :=
+    sphere_subset_closedBall (sLoop s).2
+  have hw : (sBase : Plane) ∈ closedBall (0 : Plane) 1 :=
+    sphere_subset_closedBall sBase.2
+  exact convex_closedBall 0 1 hv hw (by unit_interval) (by unit_interval) (by ring)
+
+lemma continuous_diskPt : Continuous (fun p : I × I => diskPt p.1 p.2) := by
+  unfold diskPt
+  fun_prop
+
+/-- The contraction as a continuous map into the disk. -/
+noncomputable def diskMap : C(I × I, closedBall (0 : Plane) 1) :=
+  ⟨fun p => ⟨diskPt p.1 p.2, diskPt_mem p.1 p.2⟩,
+    (continuous_diskPt).subtype_mk _⟩
+
+/-- **Phase 2.** There is no retraction of the closed disk onto its boundary
+circle. -/
+theorem no_retraction (ρ : C(closedBall (0 : Plane) 1, closedBall (0 : Plane) 1))
+    (hrange : ∀ x, (ρ x : Plane) ∈ sphere (0 : Plane) 1)
+    (hid : ∀ x : closedBall (0 : Plane) 1,
+      (x : Plane) ∈ sphere (0 : Plane) 1 → (ρ x : Plane) = (x : Plane)) :
+    False := by
+  apply sLoop_not_homotopic
+  -- The homotopy `H t s = ρ ((1-t)·loop s + t·base)`, valued in the sphere.
+  set H : C(I × I, sphere (0 : Plane) 1) :=
+    ⟨fun p => ⟨(ρ (diskMap p) : Plane), hrange _⟩,
+      (map_continuous ρ |>.comp (map_continuous diskMap)).subtype_val.subtype_mk _⟩ with hH
+  refine ⟨{
+    toContinuousMap := H
+    map_zero_left := ?_
+    map_one_left := ?_
+    prop' := ?_ }⟩
+  · -- H (0, s) = sLoop s
+    intro s
+    have hmem : (diskMap (0, s) : Plane) ∈ sphere (0 : Plane) 1 := by
+      change diskPt 0 s ∈ _
+      simp only [diskPt, Set.Icc.coe_zero, sub_zero, one_smul, zero_smul, add_zero]
+      exact (sLoop s).2
+    apply Subtype.ext
+    change (ρ (diskMap (0, s)) : Plane) = (sLoop s : Plane)
+    rw [hid _ hmem]
+    show (diskMap (0, s) : Plane) = (sLoop s : Plane)
+    change diskPt 0 s = (sLoop s : Plane)
+    simp [diskPt]
+  · -- H (1, s) = base
+    intro s
+    have hmem : (diskMap (1, s) : Plane) ∈ sphere (0 : Plane) 1 := by
+      change diskPt 1 s ∈ _
+      simp only [diskPt, Set.Icc.coe_one, sub_self, zero_smul, one_smul, zero_add]
+      exact sBase.2
+    apply Subtype.ext
+    change (ρ (diskMap (1, s)) : Plane) = (sBase : Plane)
+    rw [hid _ hmem]
+    show (diskMap (1, s) : Plane) = (sBase : Plane)
+    change diskPt 1 s = (sBase : Plane)
+    simp [diskPt]
+  · -- rel endpoints: for s ∈ {0,1}, H t s = sLoop s = base
+    intro t s hs
+    have hs' : (sLoop s : Plane) = (sBase : Plane) := by
+      rcases hs with h | h
+      · rw [show s = (0 : I) from h, sLoop_zero]
+      · rw [show s = (1 : I) from h, sLoop_one]
+    have hmem : (diskMap (t, s) : Plane) ∈ sphere (0 : Plane) 1 := by
+      change diskPt t s ∈ _
+      simp only [diskPt, hs', ← add_smul, sub_add_cancel, one_smul]
+      exact sBase.2
+    apply Subtype.ext
+    change (ρ (diskMap (t, s)) : Plane) = (sLoop s : Plane)
+    rw [hid _ hmem, hs']
+    show (diskMap (t, s) : Plane) = (sBase : Plane)
+    change diskPt t s = (sBase : Plane)
+    simp only [diskPt, hs', ← add_smul, sub_add_cancel, one_smul]
+
+/-! ## Phase 3 — Brouwer for the closed unit disk
+
+Given a fixed-point-free self-map `f` of the disk, the ray from `f x` through `x`
+exits the boundary circle at a point `ρ x`; this `ρ` is a retraction, forbidden
+by Phase 2. -/
+
+section Disk
+
+variable (f : C(closedBall (0 : Plane) 1, closedBall (0 : Plane) 1))
+
+/-- The direction vector `x - f x` of the ray. -/
+noncomputable def dvec (x : closedBall (0 : Plane) 1) : Plane := (x : Plane) - (f x : Plane)
+
+/-- Quadratic coefficient `A = ‖x - f x‖²`. -/
+noncomputable def Acoef (x : closedBall (0 : Plane) 1) : ℝ := ⟪dvec f x, dvec f x⟫
+/-- Coefficient `B = ⟪f x, x - f x⟫`. -/
+noncomputable def Bcoef (x : closedBall (0 : Plane) 1) : ℝ := ⟪(f x : Plane), dvec f x⟫
+/-- Coefficient `C = ‖f x‖² - 1 ≤ 0`. -/
+noncomputable def Ccoef (x : closedBall (0 : Plane) 1) : ℝ := ‖(f x : Plane)‖ ^ 2 - 1
+/-- Discriminant `B² - A·C ≥ 0`. -/
+noncomputable def discr (x : closedBall (0 : Plane) 1) : ℝ := (Bcoef f x) ^ 2 - Acoef f x * Ccoef
+  f x
+/-- The (larger) root parameter `t = (-B + √disc)/A`. -/
+noncomputable def tparam (x : closedBall (0 : Plane) 1) : ℝ :=
+  (- Bcoef f x + Real.sqrt (discr f x)) / Acoef f x
+/-- The exit point `ρ x = f x + t·(x - f x)` on the boundary circle. -/
+noncomputable def rhoPt (x : closedBall (0 : Plane) 1) : Plane :=
+  (f x : Plane) + tparam f x • dvec f x
+
+variable (hf : ∀ x, (f x : Plane) ≠ (x : Plane))
+include hf
+
+lemma dvec_ne (x) : dvec f x ≠ 0 := by
+  simp only [dvec, sub_ne_zero]; exact fun h => hf x h.symm
+
+lemma Acoef_pos (x) : 0 < Acoef f x := by
+  rw [Acoef, real_inner_self_eq_norm_sq]
+  exact pow_pos (norm_pos_iff.mpr (dvec_ne f hf x)) 2
+
+omit hf in
+lemma Ccoef_nonpos (x) : Ccoef f x ≤ 0 := by
+  rw [Ccoef, sub_nonpos]
+  have : ‖(f x : Plane)‖ ≤ 1 := by
+    have := (f x).2; rw [mem_closedBall, dist_zero_right] at this; exact this
+  nlinarith [norm_nonneg (f x : Plane)]
+
+lemma discr_nonneg (x) : 0 ≤ discr f x := by
+  have hA := (Acoef_pos f hf x).le
+  have hC := Ccoef_nonpos f x
+  have : 0 ≤ Acoef f x * (- Ccoef f x) := mul_nonneg hA (by linarith)
+  rw [discr]; nlinarith [sq_nonneg (Bcoef f x)]
+
+/-- The exit point lies on the unit circle. -/
+lemma norm_rhoPt (x) : ‖rhoPt f x‖ = 1 := by
+  have hA := Acoef_pos f hf x
+  have hsq : (Real.sqrt (discr f x)) ^ 2 = discr f x :=
+    Real.sq_sqrt (discr_nonneg f hf x)
+  have hnormsq : ‖rhoPt f x‖ ^ 2 = 1 := by
+    rw [rhoPt, norm_add_sq_real, norm_smul, real_inner_smul_right]
+    have hAe : ⟪dvec f x, dvec f x⟫ = Acoef f x := rfl
+    have hBe : ⟪(f x : Plane), dvec f x⟫ = Bcoef f x := rfl
+    have hCe : ‖(f x : Plane)‖ ^ 2 = Ccoef f x + 1 := by rw [Ccoef]; ring
+    rw [hBe, hCe]
+    have hnd : ‖dvec f x‖ ^ 2 = Acoef f x := by
+      rw [← real_inner_self_eq_norm_sq]; rfl
+    rw [Real.norm_eq_abs, mul_pow, sq_abs, hnd]
+    -- Now: (C+1) + 2*(t*B) + t^2 * A = 1, with t = (-B+√disc)/A
+    rw [tparam]
+    field_simp
+    rw [discr] at hsq ⊢
+    nlinarith [hsq, hA]
+  have := norm_nonneg (rhoPt f x)
+  nlinarith [hnormsq, this]
+
+/-- On the boundary circle, `ρ` is the identity. -/
+lemma rhoPt_of_mem_sphere (x : closedBall (0 : Plane) 1)
+    (hx : (x : Plane) ∈ sphere (0 : Plane) 1) :
+    rhoPt f x = (x : Plane) := by
+  have hxn : ‖(x : Plane)‖ = 1 := by rwa [mem_sphere_zero_iff_norm] at hx
+  have hA := Acoef_pos f hf x
+  have eA : Acoef f x = ‖dvec f x‖ ^ 2 := real_inner_self_eq_norm_sq _
+  have eC : Ccoef f x = ‖(f x : Plane)‖ ^ 2 - 1 := rfl
+  have hAdd : dvec f x + (f x : Plane) = (x : Plane) := by rw [dvec]; abel
+  have hcomm : ⟪dvec f x, (f x : Plane)⟫ = Bcoef f x := by rw [Bcoef, real_inner_comm]
+  have hexp : ‖(x : Plane)‖ ^ 2
+      = ‖dvec f x‖ ^ 2 + 2 * ⟪dvec f x, (f x : Plane)⟫ + ‖(f x : Plane)‖ ^ 2 := by
+    rw [← hAdd, norm_add_sq_real]
+  rw [hxn, hcomm] at hexp
+  -- A + 2B + C = 0
+  have hkey : Acoef f x + 2 * Bcoef f x + Ccoef f x = 0 := by
+    rw [eA, eC]; linear_combination -hexp
+  -- A + B = ⟪x, x - f x⟫ = 1 - ⟪x, f x⟫ ≥ 0
+  have hApB : Acoef f x + Bcoef f x = ⟪(x : Plane), dvec f x⟫ := by
+    rw [Acoef, Bcoef, ← inner_add_left, hAdd]
+  have hxd : ⟪(x : Plane), dvec f x⟫ = 1 - ⟪(x : Plane), (f x : Plane)⟫ := by
+    rw [dvec, inner_sub_right, real_inner_self_eq_norm_sq, hxn]; norm_num
+  have hcs : ⟪(x : Plane), (f x : Plane)⟫ ≤ 1 := by
+    have hfn : ‖(f x : Plane)‖ ≤ 1 := by
+      have := (f x).2; rwa [mem_closedBall, dist_zero_right] at this
+    calc ⟪(x : Plane), (f x : Plane)⟫ ≤ ‖(x : Plane)‖ * ‖(f x : Plane)‖ := real_inner_le_norm _ _
+      _ ≤ 1 * 1 := by rw [hxn]; exact mul_le_mul le_rfl hfn (norm_nonneg _) (by norm_num)
+      _ = 1 := by norm_num
+  have hAB : 0 ≤ Acoef f x + Bcoef f x := by rw [hApB, hxd]; linarith
+  -- disc = (A+B)², so √disc = A+B, t = 1
+  have hdisc : discr f x = (Acoef f x + Bcoef f x) ^ 2 := by
+    rw [discr]; linear_combination (-Acoef f x) * hkey
+  have hsqrt : Real.sqrt (discr f x) = Acoef f x + Bcoef f x := by
+    rw [hdisc, Real.sqrt_sq hAB]
+  have ht1 : tparam f x = 1 := by
+    have hnum : -Bcoef f x + (Acoef f x + Bcoef f x) = Acoef f x := by ring
+    rw [tparam, hsqrt, hnum, div_self hA.ne']
+  rw [rhoPt, ht1, one_smul, dvec]; abel
+
+/-! ### Continuity of the retraction -/
+
+omit hf in
+lemma continuous_fval : Continuous fun x : closedBall (0 : Plane) 1 => (f x : Plane) :=
+  continuous_subtype_val.comp (map_continuous f)
+
+omit hf in
+lemma continuous_dvec : Continuous (dvec f) :=
+  continuous_subtype_val.sub (continuous_fval f)
+
+omit hf in
+lemma continuous_Acoef : Continuous (Acoef f) :=
+  (continuous_dvec f).inner (continuous_dvec f)
+
+omit hf in
+lemma continuous_Bcoef : Continuous (Bcoef f) :=
+  (continuous_fval f).inner (continuous_dvec f)
+
+omit hf in
+lemma continuous_Ccoef : Continuous (Ccoef f) :=
+  ((continuous_fval f).norm.pow 2).sub continuous_const
+
+omit hf in
+lemma continuous_discr : Continuous (discr f) :=
+  ((continuous_Bcoef f).pow 2).sub ((continuous_Acoef f).mul (continuous_Ccoef f))
+
+lemma continuous_tparam : Continuous (tparam f) :=
+  Continuous.div (((continuous_Bcoef f).neg).add (continuous_discr f).sqrt)
+    (continuous_Acoef f) (fun x => (Acoef_pos f hf x).ne')
+
+lemma continuous_rhoPt : Continuous (rhoPt f) :=
+  (continuous_fval f).add ((continuous_tparam f hf).smul (continuous_dvec f))
+
+end Disk
+
+/-- **Phase 3.** Brouwer's fixed point theorem for the closed unit disk. -/
+theorem brouwer_disk (f : C(closedBall (0 : Plane) 1, closedBall (0 : Plane) 1)) :
+    ∃ x, f x = x := by
+  by_contra hcon
+  push Not at hcon
+  have hf : ∀ x, (f x : Plane) ≠ (x : Plane) := fun x h => hcon x (Subtype.ext h)
+  have hρmem : ∀ x, rhoPt f x ∈ closedBall (0 : Plane) 1 := fun x => by
+    rw [mem_closedBall, dist_zero_right, norm_rhoPt f hf x]
+  refine no_retraction ⟨fun x => ⟨rhoPt f x, hρmem x⟩, (continuous_rhoPt f hf).subtype_mk _⟩
+    (fun x => ?_) (fun x hx => ?_)
+  · change rhoPt f x ∈ sphere (0 : Plane) 1
+    rw [mem_sphere_zero_iff_norm]; exact norm_rhoPt f hf x
+  · change rhoPt f x = (x : Plane)
+    exact rhoPt_of_mem_sphere f hf x hx
+
+/-! ## Phase 4 — general nonempty compact convex sets -/
+
+/-- Transfer of the fixed-point property along a homeomorphism. -/
+theorem fixedPoint_transfer {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
+    (φ : X ≃ₜ Y) (hY : ∀ g : C(Y, Y), ∃ y, g y = y) (f : C(X, X)) : ∃ x, f x = x := by
+  obtain ⟨y, hy⟩ := hY ((⟨φ, φ.continuous⟩ : C(X, Y)).comp
+      (f.comp (⟨φ.symm, φ.symm.continuous⟩ : C(Y, X))))
+  refine ⟨φ.symm y, ?_⟩
+  have hfy : φ (f (φ.symm y)) = y := hy
+  calc f (φ.symm y) = φ.symm (φ (f (φ.symm y))) := (φ.symm_apply_apply _).symm
+    _ = φ.symm y := by rw [hfy]
+
+/-! ### Brouwer on an arbitrary closed ball (by rescaling) -/
+
+/-- Rescaling homeomorphism `closedBall 0 R ≃ₜ closedBall 0 1` (`x ↦ R⁻¹ • x`). -/
+noncomputable def ballScale (R : ℝ) (hR : 0 < R) :
+    closedBall (0 : Plane) R ≃ₜ closedBall (0 : Plane) 1 where
+  toFun x := ⟨R⁻¹ • (x : Plane), by
+    rw [mem_closedBall, dist_zero_right, norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hR)]
+    have hx : ‖(x : Plane)‖ ≤ R := by have := x.2; rwa [mem_closedBall, dist_zero_right] at this
+    rw [inv_mul_le_iff₀ hR]; simpa using hx⟩
+  invFun y := ⟨R • (y : Plane), by
+    rw [mem_closedBall, dist_zero_right, norm_smul, Real.norm_eq_abs, abs_of_pos hR]
+    have hy : ‖(y : Plane)‖ ≤ 1 := by have := y.2; rwa [mem_closedBall, dist_zero_right] at this
+    nlinarith [norm_nonneg (y : Plane)]⟩
+  left_inv x := by ext; simp [smul_smul, mul_inv_cancel₀ hR.ne']
+  right_inv y := by ext; simp [smul_smul, inv_mul_cancel₀ hR.ne']
+  -- `fun_prop` rather than term-mode `Continuous.subtype_mk`: since Mathlib
+  -- `905b9581`, unifying `Continuous.subtype_mk _ ?hp` against the `continuous_invFun`
+  -- field of this `where` block diverges (it exhausts even a 1M heartbeat budget in
+  -- `isDefEq`). Not a proof-size problem — hoisting the membership obligations into
+  -- standalone lemmas does not help; only avoiding that unification does.
+  continuous_toFun := by fun_prop
+  continuous_invFun := by fun_prop
+
+/-- **Brouwer for a closed ball of arbitrary positive radius.** -/
+theorem brouwer_ball (R : ℝ) (hR : 0 < R)
+    (f : C(closedBall (0 : Plane) R, closedBall (0 : Plane) R)) : ∃ x, f x = x :=
+  fixedPoint_transfer (ballScale R hR) (fun g => brouwer_disk g) f
+
+/-! ### Nearest-point projection onto a nonempty compact convex set
+
+Any nonempty compact convex set `s ⊆ ℝ²` is a retract of any closed ball
+containing it, via the (nonexpansive, hence continuous) nearest-point
+projection.  This yields Brouwer for all such `s` uniformly — in particular the
+degenerate empty-interior case is handled without any dimension reduction. -/
+
+section Projection
+
+variable {s : Set Plane} (hconv : Convex ℝ s) (hcomp : IsCompact s) (hne : s.Nonempty)
+
+/-- Nearest-point projection of `u` onto the nonempty compact convex set `s`. -/
+noncomputable def projFun (u : Plane) : Plane :=
+  (exists_norm_eq_iInf_of_complete_convex hne hcomp.isComplete hconv u).choose
+
+lemma projFun_mem (u : Plane) : projFun hconv hcomp hne u ∈ s :=
+  (exists_norm_eq_iInf_of_complete_convex hne hcomp.isComplete hconv u).choose_spec.1
+
+/-- The variational characterization of the projection. -/
+lemma projFun_inner_le (u : Plane) {w : Plane} (hw : w ∈ s) :
+    ⟪u - projFun hconv hcomp hne u, w - projFun hconv hcomp hne u⟫ ≤ 0 :=
+  (norm_eq_iInf_iff_real_inner_le_zero hconv (projFun_mem hconv hcomp hne u)).mp
+    (exists_norm_eq_iInf_of_complete_convex hne hcomp.isComplete hconv u).choose_spec.2 w hw
+
+/-- The projection fixes the points of `s`. -/
+lemma projFun_eq_self {u : Plane} (hu : u ∈ s) : projFun hconv hcomp hne u = u := by
+  have h := projFun_inner_le hconv hcomp hne u hu
+  rw [real_inner_self_eq_norm_sq] at h
+  have hz : u - projFun hconv hcomp hne u = 0 := by
+    have hn := norm_nonneg (u - projFun hconv hcomp hne u)
+    have : ‖u - projFun hconv hcomp hne u‖ = 0 := by nlinarith
+    rwa [norm_eq_zero] at this
+  rw [sub_eq_zero] at hz; exact hz.symm
+
+/-- The projection is nonexpansive. -/
+lemma projFun_dist_le (u₁ u₂ : Plane) :
+    ‖projFun hconv hcomp hne u₁ - projFun hconv hcomp hne u₂‖ ≤ ‖u₁ - u₂‖ := by
+  set v₁ := projFun hconv hcomp hne u₁
+  set v₂ := projFun hconv hcomp hne u₂
+  have hb1 : 0 ≤ ⟪u₁ - v₁, v₁ - v₂⟫ := by
+    have h := projFun_inner_le hconv hcomp hne u₁ (projFun_mem hconv hcomp hne u₂)
+    rw [show v₂ - v₁ = -(v₁ - v₂) by abel, inner_neg_right] at h
+    linarith
+  have hb2 : ⟪u₂ - v₂, v₁ - v₂⟫ ≤ 0 :=
+    projFun_inner_le hconv hcomp hne u₂ (projFun_mem hconv hcomp hne u₁)
+  have decomp : ⟪u₁ - u₂, v₁ - v₂⟫
+      = ⟪u₁ - v₁, v₁ - v₂⟫ - ⟪u₂ - v₂, v₁ - v₂⟫ + ⟪v₁ - v₂, v₁ - v₂⟫ := by
+    have hsum : u₁ - u₂ = (u₁ - v₁ - (u₂ - v₂)) + (v₁ - v₂) := by abel
+    rw [← inner_sub_left, ← inner_add_left, ← hsum]
+  have key : ‖v₁ - v₂‖ ^ 2 ≤ ⟪u₁ - u₂, v₁ - v₂⟫ := by
+    rw [decomp, ← real_inner_self_eq_norm_sq]; linarith
+  have hcs : ⟪u₁ - u₂, v₁ - v₂⟫ ≤ ‖u₁ - u₂‖ * ‖v₁ - v₂‖ := real_inner_le_norm _ _
+  rcases (norm_nonneg (v₁ - v₂)).eq_or_lt with h0 | hpos
+  · rw [← h0]; exact norm_nonneg _
+  · have hsq : ‖v₁ - v₂‖ * ‖v₁ - v₂‖ ≤ ‖u₁ - u₂‖ * ‖v₁ - v₂‖ := by
+      rw [← pow_two]; exact le_trans key hcs
+    exact le_of_mul_le_mul_right hsq hpos
+
+lemma continuous_projFun : Continuous (projFun hconv hcomp hne) :=
+  (LipschitzWith.mk_one (fun u₁ u₂ => by
+    rw [dist_eq_norm, dist_eq_norm]; exact projFun_dist_le hconv hcomp hne u₁ u₂)).continuous
+
+end Projection
+
+/-- **The 2D Brouwer fixed point theorem.** Every continuous self-map of a
+nonempty compact convex subset of `ℝ²` has a fixed point.  This matches the
+`JordanCurve.BrouwerFPT` interface used to discharge the Jordan curve theorem.
+
+The set `s` is contained in a closed ball `closedBall 0 R`; the nearest-point
+projection `r : closedBall 0 R → s` is a continuous retraction, so the self-map
+`incl ∘ f ∘ r` of the ball has (by `brouwer_ball`) a fixed point `x`, whose
+coordinates lie in `s`, whence `r x` is a fixed point of `f`. -/
+theorem brouwerFPT : ∀ s : Set Plane, Convex ℝ s → IsCompact s → s.Nonempty →
+    ∀ f : C(s, s), ∃ x, f x = x := by
+  intro s hconv hcomp hne f
+  obtain ⟨R, hR, hsub⟩ := hcomp.isBounded.subset_closedBall_lt 0 0
+  let incl : C(s, closedBall (0 : Plane) R) :=
+    ⟨fun y => ⟨(y : Plane), hsub y.2⟩, continuous_subtype_val.subtype_mk _⟩
+  let r : C(closedBall (0 : Plane) R, s) :=
+    ⟨fun x => ⟨projFun hconv hcomp hne (x : Plane), projFun_mem hconv hcomp hne (x : Plane)⟩,
+      ((continuous_projFun hconv hcomp hne).comp continuous_subtype_val).subtype_mk _⟩
+  obtain ⟨x, hx⟩ := brouwer_ball R hR (incl.comp (f.comp r))
+  refine ⟨r x, ?_⟩
+  apply Subtype.ext
+  have hval : (f (r x) : Plane) = (x : Plane) := congrArg Subtype.val hx
+  have hxs : (x : Plane) ∈ s := by rw [← hval]; exact (f (r x)).2
+  have hrx : ((r x : s) : Plane) = (x : Plane) := projFun_eq_self hconv hcomp hne hxs
+  change (f (r x) : Plane) = ((r x : s) : Plane)
+  rw [hval, hrx]
+
+end Brouwer
+
+end JordanCurve
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Rado Kirov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Rado Kirov
+-/
+/-!
+# Counting plumbing for the Jordan curve theorem
+
+Pure-topology lemmas, independent of any geometry, used to turn the statement
+"the complement of the curve has exactly two connected components" into the
+numerical fact `Nat.card (ConnectedComponents …) = 2`.
+
+The geometry side works with `connectedComponentIn S x` for `S : Set Plane` the
+(open, nonempty) complement of the curve, and with two distinguished points in
+the two components.  This file provides:
+
+* `nat_card_connectedComponents_eq_two` — from two points in distinct components
+  together covering everything, conclude `Nat.card (ConnectedComponents X) = 2`.
+* `connectedComponents_subtype_eq_iff` — the bridge identifying equality of
+  classes of subtype points with equality of `connectedComponentIn`.
+-/
+
+@[expose] public section
+
+namespace JordanCurve.Counting
+
+open Set
+
+variable {X : Type*} [TopologicalSpace X]
+
+/-- If a space `X` has two points `a`, `b` lying in distinct connected components
+and every point's component is one of those two, then `X` has exactly two
+connected components. -/
+theorem nat_card_connectedComponents_eq_two
+    (a b : X) (hab : ConnectedComponents.mk a ≠ ConnectedComponents.mk b)
+    (hcover : ∀ x : X, ConnectedComponents.mk x = ConnectedComponents.mk a
+                     ∨ ConnectedComponents.mk x = ConnectedComponents.mk b) :
+    Nat.card (ConnectedComponents X) = 2 := by
+  rw [Nat.card_eq_two_iff]
+  refine ⟨ConnectedComponents.mk a, ConnectedComponents.mk b, hab, ?_⟩
+  rw [eq_univ_iff_forall]
+  intro z
+  obtain ⟨x, rfl⟩ := ConnectedComponents.surjective_coe z
+  rcases hcover x with h | h
+  · exact mem_insert_iff.mpr (Or.inl h)
+  · exact mem_insert_iff.mpr (Or.inr (mem_singleton_iff.mpr h))
+
+/-- Bridge lemma.  For two points `x`, `y` of a subset `S`, the connected
+components of the corresponding subtype points agree iff their
+`connectedComponentIn S` subsets agree.  This lets the geometry side, which
+phrases things via `connectedComponentIn S`, feed
+`nat_card_connectedComponents_eq_two`. -/
+theorem connectedComponents_subtype_eq_iff {S : Set X} {x y : X}
+    (hx : x ∈ S) (hy : y ∈ S) :
+    ConnectedComponents.mk (⟨x, hx⟩ : S) = ConnectedComponents.mk (⟨y, hy⟩ : S)
+      ↔ connectedComponentIn S x = connectedComponentIn S y := by
+  rw [connectedComponentIn_eq_image hx, connectedComponentIn_eq_image hy,
+    (image_injective.mpr Subtype.coe_injective).eq_iff,
+    ← ConnectedComponents.coe_eq_coe]
+
+end JordanCurve.Counting
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Rado Kirov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Rado Kirov
+-/
+/-!
+# The (continuous) Jordan Curve Theorem, via Brouwer — Maehara's proof
+
+Target (lean-eval `jordan_curve`, pure Mathlib): a continuous injection
+`r : S¹ → ℝ²` has a complement with exactly two connected components.
+
+**Strategy (Maehara, *The Jordan curve theorem via the Brouwer fixed point
+theorem*, Amer. Math. Monthly 1984).** Reduce to the Brouwer fixed point theorem,
+taken here as an explicit interface `BrouwerFPT` (to be discharged from upstream
+Mathlib — PR #36770 is landing Brouwer — or built separately). The reduction uses:
+
+* **Lemma 2 (crossing)** — two transversal paths in a rectangle must meet; proved
+  directly from Brouwer via an explicit map of the square to its boundary.
+* **Lemma 1** — if `ℝ²∖J` is disconnected, each component has `J` as its
+  boundary; via the Tietze extension theorem + Brouwer.
+* **Main construction** — using the farthest pair `a,b ∈ J` and the points
+  `l,m,p,q` on a vertical segment, show `ℝ²∖J` has exactly one bounded component;
+  with the unique unbounded component that gives exactly two.
+
+This file is imported by the top-level `JordanPick` module and is complete:
+sorry-free, with `#print axioms JordanCurve.jordan_curve` reporting only
+`[propext, Classical.choice, Quot.sound]`.
+-/
+
+@[expose] public section
+
+namespace JordanCurve
+
+open Metric Set Function Bornology
+
+/-- The plane `ℝ²` as used by the eval problem. -/
+abbrev Plane := EuclideanSpace ℝ (Fin 2)
+
+/-- **Brouwer fixed point theorem** (interface). Every continuous self-map of a
+nonempty convex compact subset of the plane has a fixed point. Discharged
+separately (upstream Mathlib PR #36770, or a standalone build). All of Maehara's
+argument is `BrouwerFPT → …`. -/
+def BrouwerFPT : Prop :=
+  ∀ s : Set Plane, Convex ℝ s → IsCompact s → s.Nonempty →
+    ∀ f : C(s, s), ∃ x, f x = x
+
+/-- **Lemma 2 (crossing lemma, Maehara).** Two continuous paths in a rectangle
+`[a,b]×[c,d]`, one running from the left edge to the right edge (`h`), the other
+from the bottom edge to the top edge (`v`), must meet. Proved directly from
+Brouwer: if they were disjoint, the explicit normalized map
+`F(s,t) = ((v₁ t − h₁ s)/N, (h₂ s − v₂ t)/N)` (with `N` the sup-norm of
+`h s − v t`) sends the parameter square `[-1,1]²` to its boundary with no fixed
+point. Coordinates are `p 0`, `p 1` of `p : Plane`. -/
+theorem crossing (hbr : BrouwerFPT) {a b c d : ℝ} (_hab : a ≤ b) (_hcd : c ≤ d)
+    (h v : ℝ → Plane)
+    (hh : ContinuousOn h (Icc (-1) 1)) (hv : ContinuousOn v (Icc (-1) 1))
+    (hhE : ∀ t ∈ Icc (-1 : ℝ) 1, h t 0 ∈ Icc a b ∧ h t 1 ∈ Icc c d)
+    (hvE : ∀ t ∈ Icc (-1 : ℝ) 1, v t 0 ∈ Icc a b ∧ v t 1 ∈ Icc c d)
+    (hh1 : h (-1) 0 = a) (hh2 : h 1 0 = b)
+    (hv1 : v (-1) 1 = c) (hv2 : v 1 1 = d) :
+    ∃ s ∈ Icc (-1 : ℝ) 1, ∃ t ∈ Icc (-1 : ℝ) 1, h s = v t := by
+  by_contra hcon
+  push Not at hcon
+  -- `hcon : ∀ s ∈ Icc (-1) 1, ∀ t ∈ Icc (-1) 1, h s ≠ v t`
+  -- The parameter square `Q = [-1,1]²`.
+  set Q : Set Plane := {p | p 0 ∈ Icc (-1 : ℝ) 1 ∧ p 1 ∈ Icc (-1 : ℝ) 1} with hQdef
+  have hQ0 : ∀ p ∈ Q, p 0 ∈ Icc (-1 : ℝ) 1 := fun p hp => hp.1
+  have hQ1 : ∀ p ∈ Q, p 1 ∈ Icc (-1 : ℝ) 1 := fun p hp => hp.2
+  -- `Q` is convex.
+  have hconv : Convex ℝ Q := by
+    have c0 : Convex ℝ {p : Plane | p 0 ∈ Icc (-1 : ℝ) 1} := by
+      exact (convex_Icc (-1 : ℝ) 1).is_linear_preimage (EuclideanSpace.proj (0:Fin 2)).isLinear
+    have c1 : Convex ℝ {p : Plane | p 1 ∈ Icc (-1 : ℝ) 1} := by
+      exact (convex_Icc (-1 : ℝ) 1).is_linear_preimage (EuclideanSpace.proj (1:Fin 2)).isLinear
+    exact c0.inter c1
+  -- `Q` is compact (closed and bounded in a finite-dimensional space).
+  have hcomp : IsCompact Q := by
+    apply Metric.isCompact_of_isClosed_isBounded
+    · have d0 : IsClosed {p : Plane | p 0 ∈ Icc (-1 : ℝ) 1} :=
+        isClosed_Icc.preimage (EuclideanSpace.proj (0:Fin 2)).continuous
+      have d1 : IsClosed {p : Plane | p 1 ∈ Icc (-1 : ℝ) 1} :=
+        isClosed_Icc.preimage (EuclideanSpace.proj (1:Fin 2)).continuous
+      exact d0.inter d1
+    · apply (Metric.isBounded_closedBall (x := (0:Plane)) (r := 2)).subset
+      intro p hp
+      simp only [mem_closedBall, dist_zero_right]
+      have h0 : |p 0| ≤ 1 := abs_le.2 ⟨hp.1.1, hp.1.2⟩
+      have h1 : |p 1| ≤ 1 := abs_le.2 ⟨hp.2.1, hp.2.2⟩
+      rw [EuclideanSpace.norm_eq, Fin.sum_univ_two, Real.norm_eq_abs, Real.norm_eq_abs,
+        show (2 : ℝ) = Real.sqrt 4 by rw [show (4 : ℝ) = 2 ^ 2 by ring, Real.sqrt_sq]; norm_num]
+      apply Real.sqrt_le_sqrt
+      nlinarith [abs_nonneg (p 0), abs_nonneg (p 1)]
+  have hne0 : Q.Nonempty := by
+    refine ⟨0, ?_, ?_⟩ <;>
+      · simp only [mem_Icc, show (0:Plane) 0 = 0 from rfl, show (0:Plane) 1 = 0 from rfl]
+        norm_num
+  -- The sup-norm distance and the normalized boundary map.
+  let N : Plane → ℝ := fun p => max |h (p 0) 0 - v (p 1) 0| |h (p 0) 1 - v (p 1) 1|
+  let n0 : Plane → ℝ := fun p => (v (p 1) 0 - h (p 0) 0) / N p
+  let n1 : Plane → ℝ := fun p => (h (p 0) 1 - v (p 1) 1) / N p
+  let F : Plane → Plane := fun p => !₂[n0 p, n1 p]
+  have hNnn : ∀ p, 0 ≤ N p := fun p => le_trans (abs_nonneg _) (le_max_left _ _)
+  -- `N > 0` on `Q`: otherwise `h (p 0) = v (p 1)`, contradicting `hcon`.
+  have hNpos : ∀ p ∈ Q, 0 < N p := by
+    intro p hp
+    rcases (hNnn p).lt_or_eq with hlt | heq
+    · exact hlt
+    · exfalso
+      have hNz : N p = 0 := heq.symm
+      have hA0 : |h (p 0) 0 - v (p 1) 0| ≤ N p := le_max_left _ _
+      have hB0 : |h (p 0) 1 - v (p 1) 1| ≤ N p := le_max_right _ _
+      rw [hNz] at hA0 hB0
+      have eA : h (p 0) 0 = v (p 1) 0 := by
+        have := le_antisymm hA0 (abs_nonneg _); rwa [abs_eq_zero, sub_eq_zero] at this
+      have eB : h (p 0) 1 = v (p 1) 1 := by
+        have := le_antisymm hB0 (abs_nonneg _); rwa [abs_eq_zero, sub_eq_zero] at this
+      refine hcon (p 0) (hQ0 p hp) (p 1) (hQ1 p hp) ?_
+      ext i; fin_cases i
+      · exact eA
+      · exact eB
+  -- Continuity ingredients.
+  have hc0 : Continuous (fun p : Plane => p 0) := by fun_prop
+  have hc1 : Continuous (fun p : Plane => p 1) := by fun_prop
+  have mt0 : MapsTo (fun p : Plane => p 0) Q (Icc (-1 : ℝ) 1) := fun p hp => hQ0 p hp
+  have mt1 : MapsTo (fun p : Plane => p 1) Q (Icc (-1 : ℝ) 1) := fun p hp => hQ1 p hp
+  have hHx0 : ContinuousOn (fun p : Plane => h (p 0) 0) Q :=
+    (hc0.comp_continuousOn hh).comp hc0.continuousOn mt0
+  have hHx1 : ContinuousOn (fun p : Plane => h (p 0) 1) Q :=
+    (hc1.comp_continuousOn hh).comp hc0.continuousOn mt0
+  have hVx0 : ContinuousOn (fun p : Plane => v (p 1) 0) Q :=
+    (hc0.comp_continuousOn hv).comp hc1.continuousOn mt1
+  have hVx1 : ContinuousOn (fun p : Plane => v (p 1) 1) Q :=
+    (hc1.comp_continuousOn hv).comp hc1.continuousOn mt1
+  have hNcont : ContinuousOn N Q :=
+    ContinuousOn.sup ((hHx0.sub hVx0).abs) ((hHx1.sub hVx1).abs)
+  have hn0c : ContinuousOn n0 Q :=
+    (hVx0.sub hHx0).div hNcont (fun p hp => (hNpos p hp).ne')
+  have hn1c : ContinuousOn n1 Q :=
+    (hHx1.sub hVx1).div hNcont (fun p hp => (hNpos p hp).ne')
+  have hFcont : ContinuousOn F Q := by
+    have hG : ContinuousOn (fun p : Plane => (![n0 p, n1 p] : Fin 2 → ℝ)) Q := by
+      rw [continuousOn_pi]
+      intro i; fin_cases i
+      · simpa using hn0c
+      · simpa using hn1c
+    exact (PiLp.continuous_toLp 2 (fun _ : Fin 2 => ℝ)).comp_continuousOn hG
+  -- `F` maps `Q` into `Q`: each coordinate has absolute value `≤ 1`.
+  have hbound : ∀ p ∈ Q, |n0 p| ≤ 1 ∧ |n1 p| ≤ 1 := by
+    intro p hp
+    have hN := hNpos p hp
+    refine ⟨?_, ?_⟩
+    · change |(v (p 1) 0 - h (p 0) 0) / N p| ≤ 1
+      rw [abs_div, abs_of_pos hN, div_le_one hN]
+      calc |v (p 1) 0 - h (p 0) 0| = |h (p 0) 0 - v (p 1) 0| := abs_sub_comm _ _
+        _ ≤ N p := le_max_left _ _
+    · change |(h (p 0) 1 - v (p 1) 1) / N p| ≤ 1
+      rw [abs_div, abs_of_pos hN, div_le_one hN]
+      exact le_max_right _ _
+  have hmaps : MapsTo F Q Q := by
+    intro p hp
+    obtain ⟨hb0, hb1⟩ := hbound p hp
+    exact ⟨mem_Icc.2 (abs_le.1 hb0), mem_Icc.2 (abs_le.1 hb1)⟩
+  -- Package as a self-map of `Q` and apply Brouwer.
+  let f : C(Q, Q) := ⟨fun p => ⟨F p.1, hmaps p.2⟩, by
+    apply Continuous.subtype_mk; exact hFcont.domRestrict⟩
+  obtain ⟨x, hx⟩ := hbr Q hconv hcomp hne0 f
+  set p₀ : Plane := (x : Plane) with hp0def
+  have hp0Q : p₀ ∈ Q := x.2
+  have hN := hNpos p₀ hp0Q
+  have hfix : F p₀ = p₀ := congrArg Subtype.val hx
+  have hfix0 : p₀ 0 = (v (p₀ 1) 0 - h (p₀ 0) 0) / N p₀ := by
+    have e : (F p₀) 0 = (v (p₀ 1) 0 - h (p₀ 0) 0) / N p₀ := rfl
+    rw [← e, hfix]
+  have hfix1 : p₀ 1 = (h (p₀ 0) 1 - v (p₀ 1) 1) / N p₀ := by
+    have e : (F p₀) 1 = (h (p₀ 0) 1 - v (p₀ 1) 1) / N p₀ := rfl
+    rw [← e, hfix]
+  -- The fixed point lies on the boundary: `|p₀ 0| = 1` or `|p₀ 1| = 1`.
+  have a0 : |p₀ 0| = |h (p₀ 0) 0 - v (p₀ 1) 0| / N p₀ := by
+    conv_lhs => rw [hfix0]
+    rw [abs_div, abs_of_pos hN, abs_sub_comm]
+  have a1 : |p₀ 1| = |h (p₀ 0) 1 - v (p₀ 1) 1| / N p₀ := by
+    conv_lhs => rw [hfix1]
+    rw [abs_div, abs_of_pos hN]
+  have hbdry : |p₀ 0| = 1 ∨ |p₀ 1| = 1 := by
+    rcases le_total |h (p₀ 0) 1 - v (p₀ 1) 1| |h (p₀ 0) 0 - v (p₀ 1) 0| with hle | hle
+    · left
+      have hNA : N p₀ = |h (p₀ 0) 0 - v (p₀ 1) 0| := max_eq_left hle
+      rw [a0, hNA]; exact div_self (by rw [← hNA]; exact hN.ne')
+    · right
+      have hNB : N p₀ = |h (p₀ 0) 1 - v (p₀ 1) 1| := max_eq_right hle
+      rw [a1, hNB]; exact div_self (by rw [← hNB]; exact hN.ne')
+  -- Boundary contradiction in each of the four cases.
+  rcases hbdry with hb | hb
+  · rcases (abs_eq (by norm_num : (0 : ℝ) ≤ 1)).1 hb with hpos | hneg
+    · -- `p₀ 0 = 1`: then `h (p₀ 0) 0 = b ≥ v (p₀ 1) 0`, so `p₀ 0 ≤ 0`.
+      have hh0b : h (p₀ 0) 0 = b := by rw [hpos]; exact hh2
+      have hvle : v (p₀ 1) 0 ≤ b := (hvE (p₀ 1) (hQ1 p₀ hp0Q)).1.2
+      have : p₀ 0 ≤ 0 := by
+        rw [hfix0]; exact div_nonpos_of_nonpos_of_nonneg (by linarith) hN.le
+      linarith
+    · -- `p₀ 0 = -1`: then `h (p₀ 0) 0 = a ≤ v (p₀ 1) 0`, so `p₀ 0 ≥ 0`.
+      have hh0a : h (p₀ 0) 0 = a := by rw [hneg]; exact hh1
+      have hvge : a ≤ v (p₀ 1) 0 := (hvE (p₀ 1) (hQ1 p₀ hp0Q)).1.1
+      have : 0 ≤ p₀ 0 := by rw [hfix0]; exact div_nonneg (by linarith) hN.le
+      linarith
+  · rcases (abs_eq (by norm_num : (0 : ℝ) ≤ 1)).1 hb with hpos | hneg
+    · -- `p₀ 1 = 1`: then `v (p₀ 1) 1 = d ≥ h (p₀ 0) 1`, so `p₀ 1 ≤ 0`.
+      have hv1d : v (p₀ 1) 1 = d := by rw [hpos]; exact hv2
+      have hhle : h (p₀ 0) 1 ≤ d := (hhE (p₀ 0) (hQ0 p₀ hp0Q)).2.2
+      have : p₀ 1 ≤ 0 := by
+        rw [hfix1]; exact div_nonpos_of_nonpos_of_nonneg (by linarith) hN.le
+      linarith
+    · -- `p₀ 1 = -1`: then `v (p₀ 1) 1 = c ≤ h (p₀ 0) 1`, so `p₀ 1 ≥ 0`.
+      have hv1c : v (p₀ 1) 1 = c := by rw [hneg]; exact hv1
+      have hhge : c ≤ h (p₀ 0) 1 := (hhE (p₀ 0) (hQ0 p₀ hp0Q)).2.1
+      have : 0 ≤ p₀ 1 := by rw [hfix1]; exact div_nonneg (by linarith) hN.le
+      linarith
+
+/-! ### Foundational topology of a Jordan curve `J = range r`
+
+A Jordan curve is `J = range r` for `r : S¹ → ℝ²` continuous and injective. Here we
+collect the purely topological facts about `J` and its complement that Maehara's
+argument needs, independent of Brouwer:
+
+* `J` is compact and closed, its complement `Jᶜ` is open (tasks 1-2);
+* `r` is a (closed) embedding, so `J ≃ₜ S¹` (task 3);
+* `Jᶜ` has exactly one unbounded connected component (PRELIM (a));
+* each connected component of `Jᶜ` is open and path-connected (PRELIM (b)).
+-/
+
+/-- The plane has rank `2 > 1`; the engine for connectivity of ball-complements. -/
+theorem one_lt_rank_plane : 1 < Module.rank ℝ Plane := by
+  have h : Module.rank ℝ Plane = 2 := by
+    rw [← Module.finrank_eq_rank, finrank_euclideanSpace_fin]; norm_num
+  rw [h]; norm_num
+
+/-- The exterior `{x | R < ‖x‖}` of a closed ball is connected in the plane
+(dimension `2 > 1`). Proved as the continuous image of the connected set
+`sphere 0 1 ×ˢ Ioi R` under `(s, t) ↦ t • s`. -/
+theorem isConnected_compl_closedBall {R : ℝ} (hR : 0 ≤ R) :
+    IsConnected {x : Plane | R < ‖x‖} := by
+  have hconn : IsConnected ((sphere (0 : Plane) 1) ×ˢ (Ioi R)) :=
+    (isConnected_sphere one_lt_rank_plane 0 zero_le_one).prod isConnected_Ioi
+  have himg := hconn.image (fun p : Plane × ℝ => p.2 • p.1)
+    (continuous_snd.smul continuous_fst).continuousOn
+  have hset : {x : Plane | R < ‖x‖}
+      = (fun p : Plane × ℝ => p.2 • p.1) '' ((sphere (0 : Plane) 1) ×ˢ (Ioi R)) := by
+    ext x
+    simp only [mem_ofPred_eq, mem_image, mem_prod, mem_Ioi, Prod.exists]
+    constructor
+    · intro hx
+      have hxpos : 0 < ‖x‖ := lt_of_le_of_lt hR hx
+      refine ⟨‖x‖⁻¹ • x, ‖x‖, ⟨?_, hx⟩, ?_⟩
+      · rw [mem_sphere_zero_iff_norm, norm_smul, Real.norm_eq_abs, abs_inv, abs_norm,
+          inv_mul_cancel₀ (ne_of_gt hxpos)]
+      · rw [smul_smul, mul_inv_cancel₀ (ne_of_gt hxpos), one_smul]
+    · rintro ⟨s, t, ⟨hs, ht⟩, rfl⟩
+      rw [mem_sphere_zero_iff_norm] at hs
+      rw [norm_smul, hs, mul_one, Real.norm_eq_abs, abs_of_pos (lt_of_le_of_lt hR ht)]
+      exact ht
+  rw [hset]; exact himg
+
+section JordanCurveData
+
+variable (r : sphere (0 : Plane) 1 → Plane)
+
+/-- **Task 1.** The Jordan curve `J = range r` is compact (continuous image of the
+compact circle `S¹`). -/
+theorem jordanCurve_isCompact (hcont : Continuous r) : IsCompact (range r) :=
+  isCompact_range hcont
+
+/-- **Task 1.** The Jordan curve `J = range r` is closed (compact in a Hausdorff
+space). -/
+theorem jordanCurve_isClosed (hcont : Continuous r) : IsClosed (range r) :=
+  (jordanCurve_isCompact r hcont).isClosed
+
+/-- **Task 2.** The complement of the Jordan curve is open. -/
+theorem isOpen_compl_jordanCurve (hcont : Continuous r) : IsOpen (range r)ᶜ :=
+  (jordanCurve_isClosed r hcont).isOpen_compl
+
+/-- **Task 3.** A continuous injection of the compact circle into the plane is a
+closed embedding. -/
+theorem r_isClosedEmbedding (hcont : Continuous r) (hinj : Injective r) :
+    Topology.IsClosedEmbedding r :=
+  hcont.isClosedEmbedding hinj
+
+/-- **Task 3.** `r` is a topological embedding. -/
+theorem r_isEmbedding (hcont : Continuous r) (hinj : Injective r) :
+    Topology.IsEmbedding r :=
+  (r_isClosedEmbedding r hcont hinj).isEmbedding
+
+/-- **Task 3.** The Jordan curve `J = range r` is homeomorphic to the circle `S¹`. -/
+noncomputable def jordanCurveHomeo (hcont : Continuous r) (hinj : Injective r) :
+    sphere (0 : Plane) 1 ≃ₜ (range r) :=
+  (r_isEmbedding r hcont hinj).toHomeomorph
+
+/-- A radius `R ≥ 0` with `range r ⊆ closedBall 0 R` (compact ⇒ bounded). -/
+private theorem exists_jordanCurve_bound (hcont : Continuous r) :
+    ∃ R : ℝ, 0 ≤ R ∧ range r ⊆ closedBall (0 : Plane) R := by
+  obtain ⟨R, hR⟩ := (jordanCurve_isCompact r hcont).isBounded.subset_closedBall 0
+  exact ⟨max R 0, le_max_right R 0,
+    hR.trans (closedBall_subset_closedBall (le_max_left R 0))⟩
+
+/-- **PRELIM (a).** `Jᶜ` has an unbounded connected component: pick any point in the
+cobounded exterior `{x | R < ‖x‖}` of a ball containing `J`; that exterior is
+connected, lies in `Jᶜ`, and is unbounded, so the component containing it is
+unbounded. -/
+theorem exists_unbounded_component (hcont : Continuous r) :
+    ∃ x ∈ (range r)ᶜ, ¬ IsBounded (connectedComponentIn (range r)ᶜ x) := by
+  obtain ⟨R, hR0, hsub⟩ := exists_jordanCurve_bound r hcont
+  have hEconn : IsConnected {z : Plane | R < ‖z‖} := isConnected_compl_closedBall hR0
+  have hEsub : {z : Plane | R < ‖z‖} ⊆ (range r)ᶜ := by
+    intro z hz hzr
+    have := hsub hzr
+    rw [mem_closedBall_zero_iff] at this
+    exact absurd this (not_le.2 hz)
+  have hEnb : ¬ IsBounded {z : Plane | R < ‖z‖} := by
+    intro hb
+    obtain ⟨M, hM⟩ := hb.subset_closedBall 0
+    obtain ⟨z, hz⟩ := NormedSpace.exists_lt_norm ℝ Plane (max M R)
+    have hzE : z ∈ {z : Plane | R < ‖z‖} := lt_of_le_of_lt (le_max_right M R) hz
+    have hzM := hM hzE
+    rw [mem_closedBall_zero_iff] at hzM
+    exact absurd (le_max_left M R) (not_le.2 (lt_of_lt_of_le hz hzM))
+  obtain ⟨x, hxE⟩ := hEconn.nonempty
+  refine ⟨x, hEsub hxE, ?_⟩
+  have hEcomp : {z : Plane | R < ‖z‖} ⊆ connectedComponentIn (range r)ᶜ x :=
+    hEconn.isPreconnected.subset_connectedComponentIn hxE hEsub
+  exact fun hbdd => hEnb (hbdd.subset hEcomp)
+
+/-- **PRELIM (a).** Any two unbounded components of `Jᶜ` coincide: each unbounded
+component must meet the connected cobounded exterior `{x | R < ‖x‖}`, which therefore
+lies in a single component. -/
+theorem unbounded_component_unique (hcont : Continuous r) {x y : Plane}
+    (hxu : ¬ IsBounded (connectedComponentIn (range r)ᶜ x))
+    (hyu : ¬ IsBounded (connectedComponentIn (range r)ᶜ y)) :
+    connectedComponentIn (range r)ᶜ x = connectedComponentIn (range r)ᶜ y := by
+  obtain ⟨R, hR0, hsub⟩ := exists_jordanCurve_bound r hcont
+  set E : Set Plane := {z : Plane | R < ‖z‖} with hE
+  have hEconn : IsConnected E := isConnected_compl_closedBall hR0
+  have hEsub : E ⊆ (range r)ᶜ := by
+    intro z hz hzr
+    have := hsub hzr
+    rw [mem_closedBall_zero_iff] at this
+    exact absurd this (not_le.2 hz)
+  -- An unbounded component meets `E`.
+  have meetsE : ∀ z : Plane,
+      ¬ IsBounded (connectedComponentIn (range r)ᶜ z) →
+      ∃ w ∈ E, w ∈ connectedComponentIn (range r)ᶜ z := by
+    intro z hz
+    by_contra hcon
+    push Not at hcon
+    apply hz
+    have hsubBall : connectedComponentIn (range r)ᶜ z ⊆ closedBall (0 : Plane) R := by
+      intro w hw
+      rw [mem_closedBall_zero_iff]
+      by_contra hwn
+      exact hcon w (not_le.1 hwn) hw
+    exact isBounded_closedBall.subset hsubBall
+  obtain ⟨p, hpE⟩ := hEconn.nonempty
+  have hEcomp : E ⊆ connectedComponentIn (range r)ᶜ p :=
+    hEconn.isPreconnected.subset_connectedComponentIn hpE hEsub
+  -- Both components equal the component of `p`.
+  obtain ⟨wx, hwxE, hwxc⟩ := meetsE x hxu
+  obtain ⟨wy, hwyE, hwyc⟩ := meetsE y hyu
+  have hx : connectedComponentIn (range r)ᶜ x = connectedComponentIn (range r)ᶜ p := by
+    rw [connectedComponentIn_eq hwxc, ← connectedComponentIn_eq (hEcomp hwxE)]
+  have hy : connectedComponentIn (range r)ᶜ y = connectedComponentIn (range r)ᶜ p := by
+    rw [connectedComponentIn_eq hwyc, ← connectedComponentIn_eq (hEcomp hwyE)]
+  rw [hx, hy]
+
+/-- **PRELIM (b).** Each connected component of `Jᶜ` is open (`Jᶜ` is open and the
+plane is locally connected). -/
+theorem isOpen_component (hcont : Continuous r) (x : Plane) :
+    IsOpen (connectedComponentIn (range r)ᶜ x) :=
+  (isOpen_compl_jordanCurve r hcont).connectedComponentIn
+
+/-- **PRELIM (b).** Each connected component of `Jᶜ` is path-connected (it is open
+and connected in the locally path-connected plane). -/
+theorem isPathConnected_component (hcont : Continuous r) {x : Plane}
+    (hx : x ∈ (range r)ᶜ) :
+    IsPathConnected (connectedComponentIn (range r)ᶜ x) :=
+  ((isOpen_component r hcont x).isConnected_iff_isPathConnected).mp
+    (isConnected_connectedComponentIn_iff.mpr hx)
+
+end JordanCurveData
+
+/-- **Maehara's Lemma 1 core: "an arc does not separate the plane".** If `A ⊆ ℝ²`
+is an arc (homeomorphic to the unit interval `[0,1]`) then its complement `Aᶜ` is
+connected.
+
+Proof (Brouwer + Tietze). If `Aᶜ` were disconnected it would have a bounded
+component (there is a unique unbounded one, containing the cobounded exterior of a
+disc through `A`); pick a point `o` in it. Since `A ≃ₜ [0,1]` and `[0,1]` is an
+absolute retract (`TietzeExtension`), the identity `A → A` extends to a retraction
+`ρ : ℝ² → A`. Glue `ρ` on the component `K ∋ o` with the identity elsewhere: the two
+pieces agree on `frontier K ⊆ A` where `ρ = id`, giving a continuous
+`Q : ℝ² → ℝ²∖{o}` that is the identity outside `K`. On a large disc `D = closedBall o R`
+containing `A` and `K`, the map `z ↦ o - R·(Q z - o)/‖Q z - o‖` (antipodal radial
+projection about `o`) is a fixed-point-free continuous self-map of `D`, contradicting
+Brouwer. -/
+theorem arc_not_separates (hbr : BrouwerFPT) {A : Set Plane}
+    (φ : A ≃ₜ unitInterval) : IsConnected Aᶜ := by
+  classical
+  -- `A` is compact, hence closed, bounded, and `Aᶜ` is open.
+  have : CompactSpace ↥unitInterval := inferInstance
+  have hcsA : CompactSpace ↥A := φ.symm.compactSpace
+  have hAcpt : IsCompact A := isCompact_iff_compactSpace.mpr hcsA
+  have hAcl : IsClosed A := hAcpt.isClosed
+  have hAbd : IsBounded A := hAcpt.isBounded
+  have hAopen : IsOpen Aᶜ := hAcl.isOpen_compl
+  -- Tietze retraction `ρ : ℝ² → ℝ²`, `range ρ ⊆ A`, `ρ = id` on `A`.
+  have : TietzeExtension ↥A := TietzeExtension.of_homeo φ
+  obtain ⟨g, hg⟩ := ContinuousMap.exists_restrict_eq hAcl (ContinuousMap.id ↥A)
+  set ρ : Plane → Plane := fun x => (g x : Plane) with hρdef
+  have hρcont : Continuous ρ := continuous_subtype_val.comp g.continuous
+  have hρmem : ∀ x, ρ x ∈ A := fun x => (g x).2
+  have hρid : ∀ a ∈ A, ρ a = a := by
+    intro a ha
+    have h1 := DFunLike.congr_fun hg ⟨a, ha⟩
+    simp only [ContinuousMap.restrict_apply, ContinuousMap.id_apply] at h1
+    change (g a : Plane) = a
+    exact congrArg Subtype.val h1
+  -- A radius `R0 ≥ 0` with `A ⊆ closedBall 0 R0`; the cobounded exterior `E`.
+  obtain ⟨R0, hR0, hAsub⟩ : ∃ R0 : ℝ, 0 ≤ R0 ∧ A ⊆ closedBall (0 : Plane) R0 := by
+    obtain ⟨R, hR⟩ := hAbd.subset_closedBall (0 : Plane)
+    exact ⟨max R 0, le_max_right R 0,
+      hR.trans (closedBall_subset_closedBall (le_max_left R 0))⟩
+  set E : Set Plane := {z : Plane | R0 < ‖z‖} with hEdef
+  have hEconn : IsConnected E := isConnected_compl_closedBall hR0
+  have hEsub : E ⊆ Aᶜ := by
+    intro z hz hzA
+    have := hAsub hzA
+    rw [mem_closedBall_zero_iff] at this
+    exact absurd this (not_le.2 hz)
+  obtain ⟨p, hpE⟩ := hEconn.nonempty
+  have hpAc : p ∈ Aᶜ := hEsub hpE
+  have hEcomp : E ⊆ connectedComponentIn Aᶜ p :=
+    hEconn.isPreconnected.subset_connectedComponentIn hpE hEsub
+  -- An unbounded component meets `E`.
+  have meetsE : ∀ z, z ∈ Aᶜ → ¬ IsBounded (connectedComponentIn Aᶜ z) →
+      ∃ w ∈ E, w ∈ connectedComponentIn Aᶜ z := by
+    intro z _ hzu
+    by_contra hcon
+    push Not at hcon
+    apply hzu
+    have hsubBall : connectedComponentIn Aᶜ z ⊆ closedBall (0 : Plane) R0 := by
+      intro w hw
+      rw [mem_closedBall_zero_iff]
+      by_contra hwn
+      exact hcon w (not_le.1 hwn) hw
+    exact isBounded_closedBall.subset hsubBall
+  refine ⟨⟨p, hpAc⟩, ?_⟩
+  -- `IsPreconnected Aᶜ`: either every component is unbounded (⇒ connected), or a
+  -- bounded one exists (⇒ Brouwer contradiction).
+  by_cases hbdd : ∃ o ∈ Aᶜ, IsBounded (connectedComponentIn Aᶜ o)
+  · -- Bounded component: derive a contradiction via Brouwer.
+    exfalso
+    obtain ⟨o, hoAc, hKbd⟩ := hbdd
+    set K : Set Plane := connectedComponentIn Aᶜ o with hKdef
+    have hoK : o ∈ K := mem_connectedComponentIn hoAc
+    have hKopen : IsOpen K := hAopen.connectedComponentIn
+    have hoA : o ∉ A := hoAc
+    -- `frontier K ⊆ A`.
+    have hfrontier : frontier K ⊆ A := by
+      intro x hx
+      by_contra hxA
+      have hxAc : x ∈ Aᶜ := hxA
+      have hxcl : x ∈ closure K := frontier_subset_closure hx
+      have hxnK : x ∉ K := by
+        intro hxK
+        have : x ∈ K ∩ frontier K := ⟨hxK, hx⟩
+        rw [hKopen.inter_frontier_eq] at this
+        exact absurd this (Set.notMem_empty x)
+      have hCxopen : IsOpen (connectedComponentIn Aᶜ x) := hAopen.connectedComponentIn
+      have hxCx : x ∈ connectedComponentIn Aᶜ x := mem_connectedComponentIn hxAc
+      rw [_root_.mem_closure_iff] at hxcl
+      obtain ⟨w, hwCx, hwK⟩ := hxcl _ hCxopen hxCx
+      have e1 : connectedComponentIn Aᶜ x = connectedComponentIn Aᶜ w :=
+        connectedComponentIn_eq hwCx
+      rw [hKdef] at hwK
+      have e2 : connectedComponentIn Aᶜ o = connectedComponentIn Aᶜ w :=
+        connectedComponentIn_eq hwK
+      rw [e1, ← e2] at hxCx
+      rw [← hKdef] at hxCx
+      exact hxnK hxCx
+    -- A radius `R > 0` with `A ⊆ ball o R` and `K ⊆ ball o R`.
+    obtain ⟨R, hRpos, hAR, hKR⟩ :
+        ∃ R : ℝ, 0 < R ∧ A ⊆ ball o R ∧ K ⊆ ball o R := by
+      obtain ⟨rA, hrA⟩ := hAbd.subset_closedBall o
+      obtain ⟨rK, hrK⟩ := hKbd.subset_closedBall o
+      refine ⟨max (max rA rK) 0 + 1, by positivity, ?_, ?_⟩
+      · exact hrA.trans (closedBall_subset_ball
+          (lt_of_le_of_lt ((le_max_left rA rK).trans (le_max_left _ 0)) (lt_add_one _)))
+      · exact hrK.trans (closedBall_subset_ball
+          (lt_of_le_of_lt ((le_max_right rA rK).trans (le_max_left _ 0)) (lt_add_one _)))
+    -- The glued map `Q` and its properties.
+    set Q : Plane → Plane := K.piecewise ρ id with hQdef
+    have hQcont : Continuous Q := by
+      refine Continuous.piecewise ?_ hρcont continuous_id
+      intro a ha
+      show ρ a = id a
+      simpa using hρid a (hfrontier ha)
+    have hQA : ∀ z ∈ K, Q z = ρ z := fun z hz => by
+      rw [hQdef]; exact Set.piecewise_eq_of_mem _ _ _ hz
+    have hQid : ∀ z, z ∉ K → Q z = z := fun z hz => by
+      rw [hQdef]; exact Set.piecewise_eq_of_notMem _ _ _ hz
+    have hQne : ∀ z, Q z ≠ o := by
+      intro z
+      by_cases hz : z ∈ K
+      · rw [hQA z hz]; intro h; exact hoA (h ▸ hρmem z)
+      · rw [hQid z hz]; intro h; apply hz; rw [h]; exact hoK
+    -- The antipodal radial projection about `o`.
+    set Φ : Plane → Plane := fun z => o - R • (‖Q z - o‖⁻¹ • (Q z - o)) with hΦdef
+    have hΦcont : Continuous Φ := by
+      have h1 : Continuous (fun z => Q z - o) := hQcont.sub continuous_const
+      have h2 : Continuous (fun z => (‖Q z - o‖)⁻¹) :=
+        h1.norm.inv₀ (fun z => by
+          simp only [ne_eq, norm_eq_zero, sub_eq_zero]; exact hQne z)
+      -- Each step is ascribed explicitly, and the scalar is passed to `const_smul`
+      -- rather than inferred: since Mathlib `905b9581`, `continuous_const.smul _`
+      -- leaves the scalar an undetermined metavariable, because unifying the `Pi`-level
+      -- `f • g` of `Continuous.smul` against `fun z ↦ R • …` no longer solves for it.
+      have h3 : Continuous (fun z => (‖Q z - o‖)⁻¹ • (Q z - o)) := h2.smul h1
+      have h4 : Continuous (fun z => R • ((‖Q z - o‖)⁻¹ • (Q z - o))) := h3.const_smul R
+      exact continuous_const.sub h4
+    have hΦsphere : ∀ z, ‖Φ z - o‖ = R := by
+      intro z
+      have hne : Q z - o ≠ 0 := sub_ne_zero.mpr (hQne z)
+      have hw1 : ‖‖Q z - o‖⁻¹ • (Q z - o)‖ = 1 := norm_smul_inv_norm hne
+      have he : Φ z - o = -(R • (‖Q z - o‖⁻¹ • (Q z - o))) := by
+        simp only [hΦdef]; abel
+      rw [he, norm_neg, norm_smul, Real.norm_eq_abs, abs_of_nonneg hRpos.le, hw1,
+        mul_one]
+    -- `Φ` maps `D = closedBall o R` into `D`.
+    set D : Set Plane := closedBall o R with hDdef
+    have hΦD : ∀ z ∈ D, Φ z ∈ D := by
+      intro z _
+      rw [hDdef, mem_closedBall]
+      exact le_of_eq (by rw [dist_eq_norm]; exact hΦsphere z)
+    have hDconv : Convex ℝ D := convex_closedBall o R
+    have hDcomp : IsCompact D := isCompact_closedBall o R
+    have hDne : D.Nonempty := ⟨o, by rw [hDdef]; exact mem_closedBall_self hRpos.le⟩
+    -- Package as a self-map of `D` and apply Brouwer.
+    let F : C(↥D, ↥D) :=
+      ⟨fun z => ⟨Φ z.1, hΦD z.1 z.2⟩,
+        Continuous.subtype_mk (hΦcont.comp continuous_subtype_val) _⟩
+    obtain ⟨x, hx⟩ := hbr D hDconv hDcomp hDne F
+    set x0 : Plane := (x : Plane) with hx0
+    have hfix : Φ x0 = x0 := congrArg Subtype.val hx
+    have hx0R : ‖x0 - o‖ = R := by rw [← hfix]; exact hΦsphere x0
+    have hx0nK : x0 ∉ K := by
+      intro hk
+      have hin := hKR hk
+      rw [mem_ball, dist_eq_norm, hx0R] at hin
+      exact lt_irrefl R hin
+    have hQx0 : Q x0 = x0 := hQid x0 hx0nK
+    have hcompute : Φ x0 = o - (x0 - o) := by
+      simp only [hΦdef]
+      rw [hQx0, hx0R, smul_smul, mul_inv_cancel₀ hRpos.ne', one_smul]
+    rw [hcompute] at hfix
+    have hz : x0 - o = 0 := by
+      have e : (2 : ℝ) • (x0 - o) = x0 - (o - (x0 - o)) := by rw [two_smul]; abel
+      have h20 : (2 : ℝ) • (x0 - o) = 0 := by rw [e, hfix]; abel
+      exact (smul_eq_zero.mp h20).resolve_left (by norm_num)
+    rw [hz, norm_zero] at hx0R
+    exact absurd hx0R.symm hRpos.ne'
+  · -- No bounded component: every point's component equals that of `p`.
+    push Not at hbdd
+    have hAeq : Aᶜ ⊆ connectedComponentIn Aᶜ p := by
+      intro x hx
+      obtain ⟨w, hwE, hwx⟩ := meetsE x hx (hbdd x hx)
+      have e1 : connectedComponentIn Aᶜ x = connectedComponentIn Aᶜ w :=
+        connectedComponentIn_eq hwx
+      have e2 : connectedComponentIn Aᶜ p = connectedComponentIn Aᶜ w :=
+        connectedComponentIn_eq (hEcomp hwE)
+      have hxc : x ∈ connectedComponentIn Aᶜ x := mem_connectedComponentIn hx
+      rw [e1, ← e2] at hxc
+      exact hxc
+    have hsup : connectedComponentIn Aᶜ p ⊆ Aᶜ := connectedComponentIn_subset Aᶜ p
+    rw [Set.Subset.antisymm hAeq hsup]
+    exact (isConnected_connectedComponentIn_iff.mpr hpAc).isPreconnected
+
+/-- **Maehara's Lemma 1.** If `ℝ²∖J` is disconnected, each component has the whole
+Jordan curve `J = range r` as its boundary. We state (and prove) the sharper
+unconditional form: for any `x` in the complement, the frontier of its connected
+component equals `range r`.
+
+Proof (arc argument, via `arc_not_separates`). Write `U` for the component of `x`.
+Always `frontier U ⊆ range r` (a boundary point outside the curve would lie in some
+component of the open complement, forcing it into `U` — impossible for a frontier
+point). If `frontier U ⊊ range r`, transport the proper closed set
+`C = r⁻¹(frontier U) ⊊ S¹` to a proper closed arc `A₀` (via `exists_proper_arc`) and
+push forward to `A = r '' A₀`, a proper arc with `frontier U ⊆ A ⊊ range r`. Then
+`Aᶜ` is connected (`arc_not_separates`), yet `U` and `(closure U)ᶜ` split it into two
+nonempty relatively open pieces (`x ∈ U`; a point of `range r ∖ A` lies in
+`(closure U)ᶜ`) — contradicting connectedness. -/
+theorem component_boundary_eq (hbr : BrouwerFPT)
+    {r : sphere (0 : Plane) 1 → Plane} (hcont : Continuous r) (hinj : Injective r)
+    {x : Plane} (hx : x ∈ (range r)ᶜ)
+    (_hy : ∃ y ∈ (range r)ᶜ,
+      connectedComponentIn (range r)ᶜ y ≠ connectedComponentIn (range r)ᶜ x) :
+    frontier (connectedComponentIn (range r)ᶜ x) = range r := by
+  classical
+  set U := connectedComponentIn (range r)ᶜ x with hUdef
+  have hUopen : IsOpen U := isOpen_component r hcont x
+  have hxU : x ∈ U := mem_connectedComponentIn hx
+  -- **Step 1: `frontier U ⊆ range r`.**
+  have hsub : frontier U ⊆ range r := by
+    intro w hw
+    by_contra hwr
+    have hwc : w ∈ (range r)ᶜ := hwr
+    have hwcl : w ∈ closure U := frontier_subset_closure hw
+    have hWopen : IsOpen (connectedComponentIn (range r)ᶜ w) :=
+      isOpen_component r hcont w
+    have hwW : w ∈ connectedComponentIn (range r)ᶜ w := mem_connectedComponentIn hwc
+    rw [_root_.mem_closure_iff] at hwcl
+    obtain ⟨v, hvW, hvU⟩ := hwcl _ hWopen hwW
+    have e1 : connectedComponentIn (range r)ᶜ w = connectedComponentIn (range r)ᶜ v :=
+      connectedComponentIn_eq hvW
+    have e2 : U = connectedComponentIn (range r)ᶜ v := connectedComponentIn_eq hvU
+    have hwU : w ∈ U := by rw [e2, ← e1]; exact hwW
+    have : w ∈ U ∩ frontier U := ⟨hwU, hw⟩
+    rw [hUopen.inter_frontier_eq] at this
+    exact absurd this (Set.notMem_empty w)
+  -- **Step 2: `frontier U = range r`.**
+  by_contra hne
+  have hssub : frontier U ⊂ range r := hsub.ssubset_of_ne hne
+  -- Pull `frontier U` back to a proper closed subset `C ⊊ S¹`.
+  set C : Set (sphere (0 : Plane) 1) := r ⁻¹' frontier U with hCdef
+  have hCclosed : IsClosed C := isClosed_frontier.preimage hcont
+  have hCne : C ≠ univ := by
+    intro h
+    apply hne
+    refine Set.Subset.antisymm hsub ?_
+    intro w hw
+    obtain ⟨s, rfl⟩ := hw
+    have hsC : s ∈ C := h.symm ▸ mem_univ s
+    exact hsC
+  obtain ⟨A₀, hCA0, hA0ne, hA0closed, ⟨e0⟩⟩ :=
+    JordanCurve.Arcs.exists_proper_arc hCclosed hCne
+  set A : Set Plane := r '' A₀ with hAdef
+  -- Compactness of the sphere subtype ⇒ of `A₀`, giving `A ≃ₜ unitInterval`.
+  have hcsSphere : CompactSpace (sphere (0 : Plane) 1) :=
+    JordanCurve.Arcs.circleHomeoSphere.compactSpace
+  have hcsA0 : CompactSpace ↥A₀ := isCompact_iff_compactSpace.mp hA0closed.isCompact
+  have eImg : (↥A₀) ≃ₜ (↥A) :=
+    Continuous.homeoOfEquivCompactToT2 (f := Equiv.Set.image r A₀ hinj)
+      (continuous_induced_rng.2 (hcont.comp continuous_subtype_val))
+  have eA : A ≃ₜ unitInterval := eImg.symm.trans e0
+  -- `A ⊆ range r`, `frontier U ⊆ A`, and a point of `range r ∖ A`.
+  have hAr : A ⊆ range r := by
+    rw [hAdef]; rintro _ ⟨s, _, rfl⟩; exact ⟨s, rfl⟩
+  have hfrA : frontier U ⊆ A := by
+    rw [hAdef]
+    intro w hw
+    obtain ⟨s, rfl⟩ := hsub hw
+    exact ⟨s, hCA0 hw, rfl⟩
+  obtain ⟨s₀, hs₀⟩ := (Set.ne_univ_iff_exists_notMem A₀).1 hA0ne
+  have hq'r : r s₀ ∈ range r := ⟨s₀, rfl⟩
+  have hq'A : r s₀ ∉ A := by
+    rw [hAdef]
+    rintro ⟨s, hsA, hss⟩
+    exact hs₀ (hinj hss ▸ hsA)
+  -- `Aᶜ` is connected (an arc does not separate the plane).
+  have hAconn : IsConnected Aᶜ := arc_not_separates hbr eA
+  -- The clopen (preconnectedness) contradiction.
+  have hUsub : U ⊆ (range r)ᶜ := connectedComponentIn_subset (range r)ᶜ x
+  have hUAc : U ⊆ Aᶜ := hUsub.trans (compl_subset_compl.mpr hAr)
+  have hxAc : x ∈ Aᶜ := hUAc hxU
+  have hVopen : IsOpen (closure U)ᶜ := isClosed_closure.isOpen_compl
+  have hcover : Aᶜ ⊆ U ∪ (closure U)ᶜ := by
+    intro w hwAc
+    by_cases hwcl : w ∈ closure U
+    · left
+      by_contra hwU
+      have hfr : w ∈ frontier U := by
+        refine ⟨hwcl, ?_⟩
+        rw [hUopen.interior_eq]; exact hwU
+      exact hwAc (hfrA hfr)
+    · right; exact hwcl
+  have hqV : r s₀ ∈ (closure U)ᶜ := by
+    intro hcl
+    have hnU : r s₀ ∉ U := fun h => (hUsub h) hq'r
+    have hfr : r s₀ ∈ frontier U := by
+      refine ⟨hcl, ?_⟩
+      rw [hUopen.interior_eq]; exact hnU
+    exact hq'A (hfrA hfr)
+  obtain ⟨w, _, hwU, hwV⟩ :=
+    hAconn.isPreconnected U (closure U)ᶜ hUopen hVopen hcover
+      ⟨x, hxAc, hxU⟩ ⟨r s₀, hq'A, hqV⟩
+  exact hwV (subset_closure hwU)
+
+/-! ## Normalization foundation (Maehara farthest-pair setup)
+
+We prepare Maehara's WLOG normalization: the diameter of `J = range r` is realized
+by a farthest pair `a, b`, and a similarity homeomorphism `T` moves `a ↦ !₂[-1,0]`,
+`b ↦ !₂[1,0]`, scaling every distance by the fixed factor `2 / dist a b`.  This
+reduces the geometric core (`step_A`, `step_B`) to *normalized* statements in which
+the farthest pair sits at `(±1, 0)`. -/
+
+/-- `complexLIE 1 = !₂[1,0]`. -/
+private lemma complexLIE_one : Arcs.complexLIE (1 : ℂ) = !₂[(1 : ℝ), 0] := by
+  have h := Complex.isometryOfOrthonormal_apply (EuclideanSpace.basisFun (Fin 2) ℝ) (1 : ℂ)
+  rw [show Arcs.complexLIE
+        = Complex.isometryOfOrthonormal (EuclideanSpace.basisFun (Fin 2) ℝ) from rfl, h]
+  simp only [Complex.one_re, Complex.one_im, one_smul, zero_smul, add_zero,
+    EuclideanSpace.basisFun_apply]
+  ext i; fin_cases i <;>
+    simp
+
+/-- `complexLIE (-1) = !₂[-1,0]`. -/
+private lemma complexLIE_negOne : Arcs.complexLIE (-1 : ℂ) = !₂[(-1 : ℝ), 0] := by
+  have h := Complex.isometryOfOrthonormal_apply (EuclideanSpace.basisFun (Fin 2) ℝ) (-1 : ℂ)
+  rw [show Arcs.complexLIE
+        = Complex.isometryOfOrthonormal (EuclideanSpace.basisFun (Fin 2) ℝ) from rfl, h]
+  simp only [Complex.neg_re, Complex.one_re, Complex.neg_im, Complex.one_im, neg_zero,
+    zero_smul, add_zero, EuclideanSpace.basisFun_apply]
+  ext i; fin_cases i <;>
+    simp
+
+/-- **Task 1 (farthest pair).** The compact curve `range r` contains a pair `a, b`
+realizing the diameter of `J`, and `a ≠ b` (as `r` is injective on the sphere, which
+has at least two points). -/
+theorem exists_farthest_pair {r : sphere (0 : Plane) 1 → Plane}
+    (hcont : Continuous r) (hinj : Injective r) :
+    ∃ a ∈ range r, ∃ b ∈ range r,
+      (∀ z ∈ range r, ∀ w ∈ range r, dist z w ≤ dist a b) ∧ a ≠ b := by
+  have hcpt : IsCompact (range r) := jordanCurve_isCompact r hcont
+  -- two explicit distinct points on the sphere, hence in `range r`
+  have hs1 : (!₂[(1 : ℝ), 0] : Plane) ∈ sphere (0 : Plane) 1 := by
+    rw [mem_sphere_zero_iff_norm, EuclideanSpace.norm_eq, Fin.sum_univ_two]
+    simp
+  have hs2 : (!₂[(-1 : ℝ), 0] : Plane) ∈ sphere (0 : Plane) 1 := by
+    rw [mem_sphere_zero_iff_norm, EuclideanSpace.norm_eq, Fin.sum_univ_two]
+    simp
+  set s1 : sphere (0 : Plane) 1 := ⟨!₂[(1 : ℝ), 0], hs1⟩ with hs1def
+  set s2 : sphere (0 : Plane) 1 := ⟨!₂[(-1 : ℝ), 0], hs2⟩ with hs2def
+  have hs12 : s1 ≠ s2 := by
+    intro h
+    rw [hs1def, hs2def, Subtype.mk_eq_mk] at h
+    have hh : (!₂[(1 : ℝ), 0] : Plane) 0 = (!₂[(-1 : ℝ), 0] : Plane) 0 := by rw [h]
+    norm_num [PiLp.toLp_apply] at hh
+  have hne : (range r).Nonempty := ⟨r s1, mem_range_self _⟩
+  have hprod : IsCompact ((range r) ×ˢ (range r)) := hcpt.prod hcpt
+  have hpne : ((range r) ×ˢ (range r)).Nonempty := hne.prod hne
+  obtain ⟨⟨a, b⟩, hmem, hmax⟩ :=
+    hprod.exists_isMaxOn hpne
+      (continuous_dist.continuousOn (s := (range r) ×ˢ (range r)))
+  rw [mem_prod] at hmem
+  refine ⟨a, hmem.1, b, hmem.2, ?_, ?_⟩
+  · intro z hz w hw
+    exact isMaxOn_iff.mp hmax (z, w) (mem_prod.mpr ⟨hz, hw⟩)
+  · intro hEq
+    have hd0 : dist a b = 0 := by rw [hEq, dist_self]
+    have hle : dist (r s1) (r s2) ≤ dist a b :=
+      isMaxOn_iff.mp hmax (r s1, r s2) (mem_prod.mpr ⟨mem_range_self _, mem_range_self _⟩)
+    rw [hd0] at hle
+    exact hs12 (hinj (eq_of_dist_eq_zero (le_antisymm hle dist_nonneg)))
+
+/-- **Task 2 (similarity normalization).** Given `a ≠ b` on the plane there is a
+similarity homeomorphism `T : Plane ≃ₜ Plane` scaling every distance by the fixed
+positive factor `2 / dist a b`, with `T a = !₂[-1,0]` and `T b = !₂[1,0]`. -/
+theorem exists_similarity {a b : Plane} (hab : a ≠ b) :
+    ∃ T : Plane ≃ₜ Plane,
+      (∀ z w, dist (T z) (T w) = (2 / dist a b) * dist z w) ∧
+        T a = !₂[(-1 : ℝ), 0] ∧ T b = !₂[(1 : ℝ), 0] := by
+  set L := Arcs.complexLIE with hL
+  set A : ℂ := L.symm a with hA
+  set B : ℂ := L.symm b with hB
+  have hBAne : B ≠ A := by
+    intro h; exact hab (L.symm.injective h).symm
+  have hABne : B - A ≠ 0 := sub_ne_zero.mpr hBAne
+  set α : ℂ := 2 / (B - A) with hα
+  have hαne : α ≠ 0 := by rw [hα]; exact div_ne_zero two_ne_zero hABne
+  set β : ℂ := -(A + B) / (B - A) with hβ
+  set e : ℂ ≃ₜ ℂ := (Homeomorph.mulLeft₀ α hαne).trans (Homeomorph.addRight β) with he
+  have he_apply : ∀ w : ℂ, e w = α * w + β := by
+    intro w
+    rw [he]
+    simp only [Homeomorph.trans_apply, Homeomorph.coe_mulLeft₀, Homeomorph.coe_addRight]
+  set T : Plane ≃ₜ Plane := (L.symm.toHomeomorph).trans (e.trans L.toHomeomorph) with hT
+  have hT_apply : ∀ w : Plane, T w = L (α * L.symm w + β) := by
+    intro w
+    rw [hT]
+    simp only [Homeomorph.trans_apply, LinearIsometryEquiv.coe_toHomeomorph, he_apply]
+  refine ⟨T, ?_, ?_, ?_⟩
+  · -- distance scaling
+    intro z w
+    rw [hT_apply, hT_apply, L.dist_map, dist_eq_norm,
+      show (α * L.symm z + β) - (α * L.symm w + β) = α * (L.symm z - L.symm w) by ring,
+      norm_mul, ← dist_eq_norm, L.symm.dist_map]
+    have hαnorm : ‖α‖ = 2 / dist a b := by
+      rw [hα, norm_div]
+      rw [show ‖(2:ℂ)‖ = 2 by norm_num]
+      congr 1
+      rw [hB, hA, ← dist_eq_norm, L.symm.dist_map, dist_comm]
+    rw [hαnorm]
+  · -- T a = !₂[-1,0]
+    have hval : α * A + β = -1 := by rw [hα, hβ]; field_simp; ring
+    rw [hT_apply, ← hA, hval, hL]; exact complexLIE_negOne
+  · -- T b = !₂[1,0]
+    have hval : α * B + β = 1 := by rw [hα, hβ]; field_simp; ring
+    rw [hT_apply, ← hB, hval, hL]; exact complexLIE_one
+
+/-- Boundedness is preserved and reflected by a map scaling all distances by a fixed
+positive factor. -/
+theorem isBounded_image_scaling {T : Plane → Plane} {c : ℝ} (hc : 0 < c)
+    (hscale : ∀ z w, dist (T z) (T w) = c * dist z w) (K : Set Plane) :
+    IsBounded (T '' K) ↔ IsBounded K := by
+  simp only [Metric.isBounded_iff]
+  constructor
+  · rintro ⟨C, hC⟩
+    refine ⟨C / c, fun x hx y hy => ?_⟩
+    have h := hC (mem_image_of_mem T hx) (mem_image_of_mem T hy)
+    rw [hscale] at h
+    rw [le_div_iff₀ hc, mul_comm]; exact h
+  · rintro ⟨C, hC⟩
+    refine ⟨c * C, ?_⟩
+    rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩
+    rw [hscale]
+    exact mul_le_mul_of_nonneg_left (hC hx hy) hc.le
+
+/-! ### Setup lemmas for the normalized frame (Steps A, B) -/
+
+/-- Squared Euclidean distance in `Plane` expanded in coordinates. -/
+private lemma dist_sq_coord (z w : Plane) :
+    dist z w ^ 2 = (z 0 - w 0) ^ 2 + (z 1 - w 1) ^ 2 := by
+  rw [EuclideanSpace.dist_eq, Real.sq_sqrt (by positivity)]
+  simp [Fin.sum_univ_two, Real.dist_eq, sq_abs]
+
+/-- **Setup 1 (the "lens").** In the normalized frame with the farthest pair at
+`a = !₂[-1,0]`, `b = !₂[1,0]` and diameter `2` (`hfar`), the whole curve lies in
+the rectangle `[-1,1] × [-2,2]`.  For `z ∈ range r`, both `dist z a ≤ 2` and
+`dist z b ≤ 2`, i.e. `(z 0+1)²+(z 1)² ≤ 4` and `(z 0-1)²+(z 1)² ≤ 4`; these force
+`z 0 ∈ [-1,1]` and (adding them) `(z 0)²+(z 1)² ≤ 3`, hence `z 1 ∈ [-2,2]`. -/
+theorem normalized_subset_rectangle
+    {r : sphere (0 : Plane) 1 → Plane}
+    (hm : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r) (hp : (!₂[(1 : ℝ), 0] : Plane) ∈ range r)
+    (hfar : ∀ z ∈ range r, ∀ w ∈ range r, dist z w ≤ 2) :
+    range r ⊆ {p : Plane | p 0 ∈ Icc (-1 : ℝ) 1 ∧ p 1 ∈ Icc (-2 : ℝ) 2} := by
+  intro z hz
+  have hza : dist z (!₂[(-1 : ℝ), 0]) ≤ 2 := hfar z hz _ hm
+  have hzb : dist z (!₂[(1 : ℝ), 0]) ≤ 2 := hfar z hz _ hp
+  have hza2 : dist z (!₂[(-1 : ℝ), 0]) ^ 2 ≤ 4 := by
+    nlinarith [dist_nonneg (x := z) (y := (!₂[(-1 : ℝ), 0] : Plane))]
+  have hzb2 : dist z (!₂[(1 : ℝ), 0]) ^ 2 ≤ 4 := by
+    nlinarith [dist_nonneg (x := z) (y := (!₂[(1 : ℝ), 0] : Plane))]
+  rw [dist_sq_coord] at hza2 hzb2
+  have ea0 : (!₂[(-1 : ℝ), 0] : Plane) 0 = -1 := by simp
+  have ea1 : (!₂[(-1 : ℝ), 0] : Plane) 1 = 0 := by simp
+  have eb0 : (!₂[(1 : ℝ), 0] : Plane) 0 = 1 := by simp
+  have eb1 : (!₂[(1 : ℝ), 0] : Plane) 1 = 0 := by simp
+  rw [ea0, ea1] at hza2
+  rw [eb0, eb1] at hzb2
+  simp only [mem_ofPred_eq, mem_Icc]
+  refine ⟨⟨?_, ?_⟩, ⟨?_, ?_⟩⟩
+  · nlinarith [sq_nonneg (z 1)]
+  · nlinarith [sq_nonneg (z 1)]
+  · nlinarith [sq_nonneg (z 0), sq_nonneg (z 1 - 2), sq_nonneg (z 1 + 2)]
+  · nlinarith [sq_nonneg (z 0), sq_nonneg (z 1 - 2), sq_nonneg (z 1 + 2)]
+
+/-- **`arc_path` (reusable reparametrization).** A path in a set `S ⊆ Plane`
+joining `a` to `b` can be reparametrized to a map `h : ℝ → Plane` on the interval
+`[-1,1]`, with `h (-1) = a`, `h 1 = b`, continuous on `[-1,1]`, and staying in `S`.
+This is exactly the shape required by the `crossing` lemma for the horizontal path
+(endpoints on the left/right edges of a rectangle). -/
+theorem arc_path {S : Set Plane} {a b : Plane} (hj : JoinedIn S a b) :
+    ∃ h : ℝ → Plane, ContinuousOn h (Icc (-1 : ℝ) 1) ∧ h (-1) = a ∧ h 1 = b ∧
+      ∀ t ∈ Icc (-1 : ℝ) 1, h t ∈ S := by
+  refine ⟨fun t => hj.somePath (Set.projIcc 0 1 (by norm_num) ((t + 1) / 2)),
+    ?_, ?_, ?_, ?_⟩
+  · exact (hj.somePath.continuous.comp
+      (continuous_projIcc.comp (by fun_prop))).continuousOn
+  · change hj.somePath _ = a
+    rw [show Set.projIcc (0 : ℝ) 1 (by norm_num) (((-1 : ℝ) + 1) / 2) = 0 from
+      Subtype.ext (by rw [Set.coe_projIcc]; norm_num)]
+    exact hj.somePath.source
+  · change hj.somePath _ = b
+    rw [show Set.projIcc (0 : ℝ) 1 (by norm_num) (((1 : ℝ) + 1) / 2) = 1 from
+      Subtype.ext (by rw [Set.coe_projIcc]; norm_num)]
+    exact hj.somePath.target
+  · intro t _; exact hj.somePath_mem _
+
+/-- **Setup 2 (segment meets curve).** The vertical segment from `s = !₂[0,-2]` to
+`n = !₂[0,2]` meets the curve.  Proof via the `crossing` lemma with horizontal path
+`h` a curve-path from `a = !₂[-1,0]` to `b = !₂[1,0]` (the curve is path-connected,
+reparametrized by `arc_path`) staying in the rectangle `[-1,1]×[-2,2]`
+(`normalized_subset_rectangle`), and vertical path `v t = !₂[0, 2t]` running from the
+bottom edge `y=-2` to the top edge `y=2`.  Delivers a curve point on the vertical
+axis. -/
+theorem segment_meets_curve (hbr : BrouwerFPT)
+    {r : sphere (0 : Plane) 1 → Plane} (hcont : Continuous r) (_hinj : Injective r)
+    (hm : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r) (hp : (!₂[(1 : ℝ), 0] : Plane) ∈ range r)
+    (hfar : ∀ z ∈ range r, ∀ w ∈ range r, dist z w ≤ 2) :
+    ∃ p ∈ range r, p 0 = 0 ∧ p 1 ∈ Icc (-2 : ℝ) 2 := by
+  -- the curve is path-connected (continuous image of the path-connected circle)
+  have hsp : IsPathConnected (sphere (0:Plane) 1) :=
+    isPathConnected_sphere one_lt_rank_plane 0 (by norm_num)
+  have : PathConnectedSpace ↥(sphere (0:Plane) 1) :=
+    isPathConnected_iff_pathConnectedSpace.mp hsp
+  have h1 : IsPathConnected (univ : Set ↥(sphere (0:Plane) 1)) :=
+    pathConnectedSpace_iff_univ.mp ‹_›
+  have hpc : IsPathConnected (range r) := by
+    have := h1.image hcont; rwa [image_univ] at this
+  have hjoin : JoinedIn (range r) (!₂[(-1 : ℝ), 0]) (!₂[(1 : ℝ), 0]) :=
+    hpc.joinedIn _ hm _ hp
+  obtain ⟨h, hcontOn, hh1, hh2, hhmem⟩ := arc_path hjoin
+  have hrect := normalized_subset_rectangle hm hp hfar
+  -- vertical path `v t = !₂[0, 2t]`
+  set v : ℝ → Plane := fun t => !₂[(0 : ℝ), 2 * t] with hvdef
+  have hvcont : ContinuousOn v (Icc (-1 : ℝ) 1) := by
+    apply Continuous.continuousOn
+    apply (PiLp.continuous_toLp 2 (fun _ : Fin 2 => ℝ)).comp
+    refine continuous_pi (fun i => ?_)
+    fin_cases i <;> simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.zero_eta, Fin.isValue,
+      Matrix.cons_val_zero,
+      Fin.mk_one, Matrix.cons_val_one, Matrix.cons_val_fin_one]<;> fun_prop
+  have hv0 : ∀ t : ℝ, v t 0 = 0 := by intro t; simp [hvdef]
+  have hv1 : ∀ t : ℝ, v t 1 = 2 * t := by intro t; simp [hvdef]
+  -- edge/rectangle conditions
+  have hhE : ∀ t ∈ Icc (-1 : ℝ) 1, h t 0 ∈ Icc (-1 : ℝ) 1 ∧ h t 1 ∈ Icc (-2 : ℝ) 2 :=
+    fun t ht => hrect (hhmem t ht)
+  have hvE : ∀ t ∈ Icc (-1 : ℝ) 1, v t 0 ∈ Icc (-1 : ℝ) 1 ∧ v t 1 ∈ Icc (-2 : ℝ) 2 := by
+    intro t ht
+    rw [mem_Icc] at ht
+    refine ⟨?_, ?_⟩
+    · rw [hv0]; simp
+    · rw [hv1, mem_Icc]; constructor <;> nlinarith [ht.1, ht.2]
+  have hh1' : h (-1) 0 = -1 := by rw [hh1]; simp
+  have hh2' : h 1 0 = 1 := by rw [hh2]; simp
+  have hv1e : v (-1) 1 = -2 := by rw [hv1]; norm_num
+  have hv2e : v 1 1 = 2 := by rw [hv1]; norm_num
+  obtain ⟨s, hs, t, ht, heq⟩ := crossing hbr (by norm_num : (-1 : ℝ) ≤ 1)
+    (by norm_num : (-2 : ℝ) ≤ 2) h v hcontOn hvcont hhE hvE hh1' hh2' hv1e hv2e
+  refine ⟨h s, hhmem s hs, ?_, ?_⟩
+  · rw [heq, hv0]
+  · rw [heq]; exact (hvE t ht).2
+
+/-- **Setup 3 (top of the axis).** The intersection of the curve with the vertical
+axis `Jax = {p ∈ range r | p 0 = 0}` is compact (closed subset of the compact
+curve) and nonempty (`segment_meets_curve`), hence attains its `y`-maximum at some
+point `l ∈ range r` with `l 0 = 0`. -/
+theorem exists_ymax_on_axis (hbr : BrouwerFPT)
+    {r : sphere (0 : Plane) 1 → Plane} (hcont : Continuous r) (hinj : Injective r)
+    (hm : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r) (hp : (!₂[(1 : ℝ), 0] : Plane) ∈ range r)
+    (hfar : ∀ z ∈ range r, ∀ w ∈ range r, dist z w ≤ 2) :
+    ∃ l ∈ range r, l 0 = 0 ∧ ∀ p ∈ range r, p 0 = 0 → p 1 ≤ l 1 := by
+  set Jax : Set Plane := {p ∈ range r | p 0 = 0} with hJax
+  have hsub : Jax ⊆ range r := fun p hp => hp.1
+  have hJaxcl : IsClosed Jax :=
+    (jordanCurve_isClosed r hcont).inter (isClosed_eq (by fun_prop) continuous_const)
+  have hJaxcpt : IsCompact Jax :=
+    (jordanCurve_isCompact r hcont).of_isClosed_subset hJaxcl hsub
+  have hne : Jax.Nonempty := by
+    obtain ⟨p, hpr, hp0, _⟩ := segment_meets_curve hbr hcont hinj hm hp hfar
+    exact ⟨p, hpr, hp0⟩
+  have hcy : Continuous (fun p : Plane => p 1) := by fun_prop
+  obtain ⟨l, hlJax, hlmax⟩ :=
+    hJaxcpt.exists_isMaxOn hne (f := fun p : Plane => p 1) hcy.continuousOn
+  refine ⟨l, hlJax.1, hlJax.2, ?_⟩
+  intro p hpr hp0
+  exact hlmax (show p ∈ Jax from ⟨hpr, hp0⟩)
+
+/-- **Image of an arc under the subtype coercion is again an arc.** If `A ⊆ ↥K`
+(a subset of a subtype of `Plane`) is homeomorphic to `unitInterval`, then its image
+`(↑) '' A ⊆ Plane` is homeomorphic to `unitInterval`.  The composite coercion
+`↥A → Plane` is a continuous injection from a compact space to a Hausdorff space,
+hence a (closed) embedding, so `↥A ≃ₜ ↥(range …) = ↥((↑) '' A)`. -/
+private lemma planar_arc_homeo {K : Set Plane} {A : Set ↥K}
+    (e : A ≃ₜ unitInterval) : Nonempty (((↑) '' A : Set Plane) ≃ₜ unitInterval) := by
+  have : CompactSpace ↥A := e.symm.compactSpace
+  let k : ↥A → Plane := fun a => ((a : ↥K) : Plane)
+  have hkc : Continuous k := continuous_subtype_val.comp continuous_subtype_val
+  have hki : Injective k := Subtype.coe_injective.comp Subtype.coe_injective
+  have hemb : Topology.IsEmbedding k := (hkc.isClosedEmbedding hki).isEmbedding
+  have hrange : Set.range k = ((↑) '' A : Set Plane) := by
+    ext y
+    constructor
+    · rintro ⟨a, rfl⟩; exact ⟨a.1, a.2, rfl⟩
+    · rintro ⟨x, hx, rfl⟩; exact ⟨⟨x, hx⟩, rfl⟩
+  exact ⟨((Homeomorph.setCongr hrange).symm.trans hemb.toHomeomorph.symm).trans e⟩
+
+/-- **Setup 4 (arc split at `a`, `b`).** The curve `range r` splits at the farthest
+pair `a = !₂[-1,0]`, `b = !₂[1,0]` into two closed arcs `J_n`, `J_s ⊆ Plane`, each
+`≃ₜ unitInterval`, with `J_n ∪ J_s = range r` and `J_n ∩ J_s = {a, b}`, labelled so
+that the `y`-topmost axis point `l` lies in `J_n`. -/
+theorem jordan_arcs (hbr : BrouwerFPT)
+    {r : sphere (0 : Plane) 1 → Plane} (hcont : Continuous r) (hinj : Injective r)
+    (hm : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r) (hp : (!₂[(1 : ℝ), 0] : Plane) ∈ range r)
+    (hfar : ∀ z ∈ range r, ∀ w ∈ range r, dist z w ≤ 2) :
+    ∃ J_n J_s : Set Plane,
+      IsClosed J_n ∧ IsClosed J_s ∧
+      J_n ∪ J_s = range r ∧
+      J_n ∩ J_s = {!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]} ∧
+      Nonempty (J_n ≃ₜ unitInterval) ∧ Nonempty (J_s ≃ₜ unitInterval) ∧
+      IsPathConnected (J_n \ {!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]}) ∧
+      IsPathConnected (J_s \ {!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]}) ∧
+      ∃ l ∈ range r, l 0 = 0 ∧ (∀ p ∈ range r, p 0 = 0 → p 1 ≤ l 1) ∧ l ∈ J_n := by
+  obtain ⟨l, hlr, hl0, hlmax⟩ := exists_ymax_on_axis hbr hcont hinj hm hp hfar
+  set J := jordanCurveHomeo r hcont hinj with hJdef
+  have hab : (!₂[(-1 : ℝ), 0] : Plane) ≠ !₂[(1 : ℝ), 0] := by
+    intro h
+    have h0 : (!₂[(-1 : ℝ), 0] : Plane) 0 = (!₂[(1 : ℝ), 0] : Plane) 0 := by rw [h]
+    rw [show (!₂[(-1 : ℝ), 0] : Plane) 0 = -1 from by simp,
+        show (!₂[(1 : ℝ), 0] : Plane) 0 = 1 from by simp] at h0
+    norm_num at h0
+  set x : sphere (0 : Plane) 1 := J.symm ⟨_, hm⟩ with hxdef
+  set y : sphere (0 : Plane) 1 := J.symm ⟨_, hp⟩ with hydef
+  have hxy : x ≠ y := by
+    intro h
+    exact hab (Subtype.ext_iff.mp (J.symm.injective h))
+  obtain ⟨A₁, A₂, hc1, hc2, hu, hi, ⟨e1⟩, ⟨e2⟩, hpcA1, hpcA2⟩ := Arcs.jordanCurve_split J hxy
+  -- transported endpoints: `J x = ⟨a,hm⟩`, `J y = ⟨b,hp⟩`
+  have hJx : J x = ⟨_, hm⟩ := J.apply_symm_apply _
+  have hJy : J y = ⟨_, hp⟩ := J.apply_symm_apply _
+  -- planar images of the two arcs
+  set B₁ : Set Plane := (↑) '' A₁ with hB1
+  set B₂ : Set Plane := (↑) '' A₂ with hB2
+  have hBu : B₁ ∪ B₂ = range r := by
+    rw [hB1, hB2, ← Set.image_union, hu, Set.image_univ, Subtype.range_coe]
+  have hBi : B₁ ∩ B₂ = {!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]} := by
+    rw [hB1, hB2, ← Set.image_inter Subtype.coe_injective, hi, hJx, hJy,
+      Set.image_pair]
+  have hn1 : Nonempty (B₁ ≃ₜ unitInterval) := planar_arc_homeo e1
+  have hn2 : Nonempty (B₂ ≃ₜ unitInterval) := planar_arc_homeo e2
+  have hcl1 : IsClosed B₁ :=
+    (isCompact_iff_compactSpace.mpr hn1.some.symm.compactSpace).isClosed
+  have hcl2 : IsClosed B₂ :=
+    (isCompact_iff_compactSpace.mpr hn2.some.symm.compactSpace).isClosed
+  -- transport path-connectedness of the arc interiors down to the plane
+  have hpcB1 : IsPathConnected (B₁ \ {!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]}) := by
+    have h := hpcA1.image (continuous_subtype_val (p := fun z => z ∈ range r))
+    rwa [Set.image_sdiff Subtype.coe_injective, Set.image_insert_eq, Set.image_singleton,
+      hJx, hJy] at h
+  have hpcB2 : IsPathConnected (B₂ \ {!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]}) := by
+    have h := hpcA2.image (continuous_subtype_val (p := fun z => z ∈ range r))
+    rwa [Set.image_sdiff Subtype.coe_injective, Set.image_insert_eq, Set.image_singleton,
+      hJx, hJy] at h
+  -- `l` lies in `range r = B₁ ∪ B₂`; label the containing arc `J_n`
+  have hlB : l ∈ B₁ ∪ B₂ := by rw [hBu]; exact hlr
+  rcases hlB with hl | hl
+  · exact ⟨B₁, B₂, hcl1, hcl2, hBu, hBi, hn1, hn2, hpcB1, hpcB2, l, hlr, hl0, hlmax, hl⟩
+  · refine ⟨B₂, B₁, hcl2, hcl1, ?_, ?_, hn2, hn1, hpcB2, hpcB1, l, hlr, hl0, hlmax, hl⟩
+    · rw [Set.union_comm]; exact hBu
+    · rw [Set.inter_comm]; exact hBi
+
+/-! ### Path-concatenation infrastructure and Maehara's interior points
+
+The tools below package repeated use of the `crossing` lemma: any bottom-to-top
+path in the normalized rectangle meets each arc (`vertical_meets_arc`), together with
+the plumbing (`concatPath2`, `concatPath3`) that glues path pieces into one
+continuous path on `[-1,1]` whose image is the union of the pieces. -/
+
+/-- An arc `J ⊆ Plane` homeomorphic to `unitInterval` is path-connected. -/
+theorem arc_isPathConnected {J : Set Plane} (e : ↥J ≃ₜ unitInterval) :
+    IsPathConnected J := by
+  have : PathConnectedSpace unitInterval :=
+    isPathConnected_iff_pathConnectedSpace.mp
+      ((convex_Icc (0 : ℝ) 1).isPathConnected (Set.nonempty_Icc.mpr (by norm_num)))
+  have h1 : IsPathConnected (univ : Set unitInterval) := isPathConnected_univ
+  have h2 := h1.image
+    (show Continuous (fun x : unitInterval => ((e.symm x : ↥J) : Plane)) from
+      continuous_subtype_val.comp e.symm.continuous)
+  rw [Set.image_univ] at h2
+  have hrange : (Set.range fun x : unitInterval => ((e.symm x : ↥J) : Plane)) = J := by
+    ext p
+    simp only [Set.mem_range]
+    constructor
+    · rintro ⟨x, rfl⟩; exact (e.symm x).2
+    · intro hp; exact ⟨e ⟨p, hp⟩, by simp⟩
+  rwa [hrange] at h2
+
+/-- An arc `J ≃ₜ unitInterval` joins any two of its points inside `J`. -/
+theorem arc_joinedIn {J : Set Plane} (e : ↥J ≃ₜ unitInterval)
+    {a b : Plane} (ha : a ∈ J) (hb : b ∈ J) : JoinedIn J a b :=
+  (arc_isPathConnected e).joinedIn a ha b hb
+
+/-- **Key reusable tool.** Any path `v` running from the bottom edge (`v (-1) 1 = -2`)
+to the top edge (`v 1 1 = 2`) of the normalized rectangle `[-1,1]×[-2,2]` must meet an
+arc `J` that joins the left point `a = !₂[-1,0]` to the right point `b = !₂[1,0]` inside
+the rectangle.  Immediate from `crossing`, using `arc_path` for the horizontal path. -/
+theorem vertical_meets_arc (hbr : BrouwerFPT) {J : Set Plane}
+    (hJsub : J ⊆ {p : Plane | p 0 ∈ Icc (-1 : ℝ) 1 ∧ p 1 ∈ Icc (-2 : ℝ) 2})
+    (hJoin : JoinedIn J (!₂[(-1 : ℝ), 0]) (!₂[(1 : ℝ), 0]))
+    {v : ℝ → Plane} (hvcont : ContinuousOn v (Icc (-1 : ℝ) 1))
+    (hvE : ∀ t ∈ Icc (-1 : ℝ) 1, v t 0 ∈ Icc (-1 : ℝ) 1 ∧ v t 1 ∈ Icc (-2 : ℝ) 2)
+    (hv1 : v (-1) 1 = -2) (hv2 : v 1 1 = 2) :
+    ∃ t ∈ Icc (-1 : ℝ) 1, v t ∈ J := by
+  obtain ⟨h, hcontOn, hh1, hh2, hhmem⟩ := arc_path hJoin
+  have hhE : ∀ t ∈ Icc (-1 : ℝ) 1, h t 0 ∈ Icc (-1 : ℝ) 1 ∧ h t 1 ∈ Icc (-2 : ℝ) 2 :=
+    fun t ht => hJsub (hhmem t ht)
+  have hh1' : h (-1) 0 = -1 := by rw [hh1]; simp
+  have hh2' : h 1 0 = 1 := by rw [hh2]; simp
+  obtain ⟨s, hs, t, ht, heq⟩ := crossing hbr (by norm_num : (-1 : ℝ) ≤ 1)
+    (by norm_num : (-2 : ℝ) ≤ 2) h v hcontOn hvcont hhE hvE hh1' hh2' hv1 hv2
+  exact ⟨t, ht, heq ▸ hhmem s hs⟩
+
+/-- A path piece `f : ℝ → Plane`, continuous on `[-1,1]`, as a `Path (f (-1)) (f 1)`
+over `unitInterval` (reparametrized by `s ↦ 2s-1`). -/
+private noncomputable def pathOfPiece {f : ℝ → Plane}
+    (hf : ContinuousOn f (Icc (-1 : ℝ) 1)) : Path (f (-1)) (f 1) :=
+  Path.mk ⟨fun s : unitInterval => f (2 * (s : ℝ) - 1), by
+      refine hf.comp_continuous (by fun_prop) (fun s => ?_)
+      have hs := s.2; rw [Set.mem_Icc] at hs ⊢
+      constructor <;> [linarith [hs.1]; linarith [hs.2]]⟩
+    (by norm_num [Set.Icc.coe_zero]) (by norm_num [Set.Icc.coe_one])
+
+private lemma pathOfPiece_range {f : ℝ → Plane}
+    (hf : ContinuousOn f (Icc (-1 : ℝ) 1)) :
+    Set.range (pathOfPiece hf) = f '' Icc (-1 : ℝ) 1 := by
+  ext p
+  simp only [Set.mem_range, Set.mem_image, Set.mem_Icc]
+  constructor
+  · rintro ⟨s, rfl⟩
+    have hs := s.2; rw [Set.mem_Icc] at hs
+    exact ⟨2 * (s : ℝ) - 1, ⟨by linarith [hs.1], by linarith [hs.2]⟩, rfl⟩
+  · rintro ⟨y, ⟨hy1, hy2⟩, rfl⟩
+    refine ⟨⟨(y + 1) / 2, by rw [Set.mem_Icc]; constructor <;> linarith⟩, ?_⟩
+    change f (2 * ((y + 1) / 2) - 1) = f y
+    congr 1; ring
+
+/-- **Path concatenation (2 pieces).** Two path pieces on `[-1,1]` that chain up
+(`f 1 = g (-1)`) glue to one path `c` on `[-1,1]` from `f (-1)` to `g 1`, whose image
+is contained in the union of the pieces' images. -/
+theorem concatPath2 {f g : ℝ → Plane}
+    (hf : ContinuousOn f (Icc (-1 : ℝ) 1)) (hg : ContinuousOn g (Icc (-1 : ℝ) 1))
+    (hchain : f 1 = g (-1)) :
+    ∃ c : ℝ → Plane, ContinuousOn c (Icc (-1 : ℝ) 1) ∧ c (-1) = f (-1) ∧ c 1 = g 1 ∧
+      ∀ t ∈ Icc (-1 : ℝ) 1, c t ∈ f '' Icc (-1 : ℝ) 1 ∪ g '' Icc (-1 : ℝ) 1 := by
+  set P : Path (f (-1)) (g 1) :=
+    (pathOfPiece hf).trans ((pathOfPiece hg).cast hchain rfl) with hP
+  refine ⟨fun t => P (Set.projIcc 0 1 (by norm_num) ((t + 1) / 2)), ?_, ?_, ?_, ?_⟩
+  · exact (P.continuous.comp (continuous_projIcc.comp (by fun_prop))).continuousOn
+  · change P _ = f (-1)
+    rw [show Set.projIcc (0 : ℝ) 1 (by norm_num) (((-1 : ℝ) + 1) / 2) = 0 from
+      Subtype.ext (by rw [Set.coe_projIcc]; norm_num)]
+    exact P.source
+  · change P _ = g 1
+    rw [show Set.projIcc (0 : ℝ) 1 (by norm_num) (((1 : ℝ) + 1) / 2) = 1 from
+      Subtype.ext (by rw [Set.coe_projIcc]; norm_num)]
+    exact P.target
+  · intro t _
+    have hmem : P (Set.projIcc 0 1 (by norm_num) ((t + 1) / 2)) ∈ Set.range P := ⟨_, rfl⟩
+    rw [hP, Path.trans_range] at hmem
+    have hr1 : Set.range (pathOfPiece hf) = f '' Icc (-1 : ℝ) 1 := pathOfPiece_range hf
+    have hr2 : Set.range ((pathOfPiece hg).cast hchain rfl) = g '' Icc (-1 : ℝ) 1 := by
+      simp only [Path.cast_coe]; exact pathOfPiece_range hg
+    rw [hr1, hr2] at hmem
+    exact hmem
+
+/-- **Path concatenation (3 pieces).** Three chained path pieces on `[-1,1]` glue to
+one path `c` on `[-1,1]` from `f (-1)` to `h 1`, with image in the union of the three
+pieces' images. -/
+theorem concatPath3 {f g h : ℝ → Plane}
+    (hf : ContinuousOn f (Icc (-1 : ℝ) 1)) (hg : ContinuousOn g (Icc (-1 : ℝ) 1))
+    (hh : ContinuousOn h (Icc (-1 : ℝ) 1))
+    (hfg : f 1 = g (-1)) (hgh : g 1 = h (-1)) :
+    ∃ c : ℝ → Plane, ContinuousOn c (Icc (-1 : ℝ) 1) ∧ c (-1) = f (-1) ∧ c 1 = h 1 ∧
+      ∀ t ∈ Icc (-1 : ℝ) 1,
+        c t ∈ f '' Icc (-1 : ℝ) 1 ∪ g '' Icc (-1 : ℝ) 1 ∪ h '' Icc (-1 : ℝ) 1 := by
+  obtain ⟨c2, hc2cont, hc2a, hc2b, hc2mem⟩ := concatPath2 hf hg hfg
+  obtain ⟨c3, hc3cont, hc3a, hc3b, hc3mem⟩ :=
+    concatPath2 hc2cont hh (by rw [hc2b]; exact hgh)
+  refine ⟨c3, hc3cont, by rw [hc3a, hc2a], hc3b, fun t ht => ?_⟩
+  rcases hc3mem t ht with hin | hin
+  · obtain ⟨u, hu, huc⟩ := hin
+    exact Set.mem_union_left _ (huc ▸ hc2mem u hu)
+  · exact Set.mem_union_right _ hin
+
+/-- **Path concatenation (5 pieces).** Five chained path pieces on `[-1,1]` glue to one
+path `c` on `[-1,1]` from `f1 (-1)` to `f5 1`, with image in the union of the five
+pieces' images. -/
+theorem concatPath5 {f1 f2 f3 f4 f5 : ℝ → Plane}
+    (h1 : ContinuousOn f1 (Icc (-1 : ℝ) 1)) (h2 : ContinuousOn f2 (Icc (-1 : ℝ) 1))
+    (h3 : ContinuousOn f3 (Icc (-1 : ℝ) 1)) (h4 : ContinuousOn f4 (Icc (-1 : ℝ) 1))
+    (h5 : ContinuousOn f5 (Icc (-1 : ℝ) 1))
+    (c12 : f1 1 = f2 (-1)) (c23 : f2 1 = f3 (-1)) (c34 : f3 1 = f4 (-1))
+    (c45 : f4 1 = f5 (-1)) :
+    ∃ c : ℝ → Plane, ContinuousOn c (Icc (-1 : ℝ) 1) ∧ c (-1) = f1 (-1) ∧ c 1 = f5 1 ∧
+      ∀ t ∈ Icc (-1 : ℝ) 1,
+        c t ∈ ((((f1 '' Icc (-1 : ℝ) 1 ∪ f2 '' Icc (-1 : ℝ) 1) ∪ f3 '' Icc (-1 : ℝ) 1)
+          ∪ f4 '' Icc (-1 : ℝ) 1) ∪ f5 '' Icc (-1 : ℝ) 1) := by
+  obtain ⟨c123, hc123cont, hc123a, hc123b, hc123mem⟩ := concatPath3 h1 h2 h3 c12 c23
+  obtain ⟨c45, hc45cont, hc45a, hc45b, hc45mem⟩ := concatPath2 h4 h5 c45
+  obtain ⟨c, hccont, hca, hcb, hcmem⟩ :=
+    concatPath2 hc123cont hc45cont (by rw [hc123b, hc45a]; exact c34)
+  refine ⟨c, hccont, by rw [hca, hc123a], by rw [hcb, hc45b], fun t ht => ?_⟩
+  rcases hcmem t ht with hin | hin
+  · obtain ⟨u, hu, hue⟩ := hin
+    have hmem := hc123mem u hu
+    rw [hue] at hmem
+    exact Or.inl (Or.inl hmem)
+  · obtain ⟨u, hu, hue⟩ := hin
+    have hmem := hc45mem u hu
+    rw [hue] at hmem
+    rcases hmem with h | h
+    · exact Or.inl (Or.inr h)
+    · exact Or.inr h
+
+/-- **Interior point `m`.** The intersection `J_n ∩ axis` (`axis = {p | p 0 = 0}`) is
+compact and nonempty (it contains `l`), hence attains its `y`-minimum at a point `m`. -/
+theorem exists_ymin_Jn_axis {J_n : Set Plane} (hJncl : IsClosed J_n)
+    (hcpt : IsCompact J_n) {l : Plane} (hlJn : l ∈ J_n) (hl0 : l 0 = 0) :
+    ∃ m ∈ J_n, m 0 = 0 ∧ ∀ p ∈ J_n, p 0 = 0 → m 1 ≤ p 1 := by
+  set Jm : Set Plane := {p ∈ J_n | p 0 = 0} with hJm
+  have hsub : Jm ⊆ J_n := fun p hp => hp.1
+  have hcl : IsClosed Jm :=
+    hJncl.inter (isClosed_eq (by fun_prop) continuous_const)
+  have hJmcpt : IsCompact Jm := hcpt.of_isClosed_subset hcl hsub
+  have hne : Jm.Nonempty := ⟨l, hlJn, hl0⟩
+  have hcy : Continuous (fun p : Plane => p 1) := by fun_prop
+  obtain ⟨m, hmJm, hmmin⟩ :=
+    hJmcpt.exists_isMinOn hne (f := fun p : Plane => p 1) hcy.continuousOn
+  exact ⟨m, hmJm.1, hmJm.2, fun p hpJn hp0 => hmmin (show p ∈ Jm from ⟨hpJn, hp0⟩)⟩
+
+/-- **The vertical segment `m → s` meets `J_s`.** With `s = !₂[0,-2]` (below the
+rectangle's bottom edge), the axis segment from the interior point `m ∈ J_n` down to
+`s` must cross the southern arc `J_s`.  Proof by contradiction: were it disjoint from
+`J_s`, the concatenation `s → m` (axis), `m → l` (a path inside `J_n \ {a,b}`),
+`l → n` (axis, `n = !₂[0,2]`) would give a bottom-to-top path in the rectangle
+avoiding `J_s`, contradicting `vertical_meets_arc` (`J_s` joins `a` to `b`).  The
+crossing point lies on the segment, so it sits on the axis at height `≤ m 1`. -/
+theorem ms_meets_Js (hbr : BrouwerFPT)
+    {r : sphere (0 : Plane) 1 → Plane} (_hcont : Continuous r) (_hinj : Injective r)
+    (hmr : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r) (hpr : (!₂[(1 : ℝ), 0] : Plane) ∈ range r)
+    (hfar : ∀ z ∈ range r, ∀ w ∈ range r, dist z w ≤ 2)
+    {J_n J_s : Set Plane}
+    (hUnion : J_n ∪ J_s = range r)
+    (hInter : J_n ∩ J_s = {!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]})
+    (eJs : ↥J_s ≃ₜ unitInterval)
+    (hpcJn : IsPathConnected (J_n \ {!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]}))
+    {l : Plane} (hlJn : l ∈ J_n) (hl0 : l 0 = 0)
+    (hlmax : ∀ q ∈ range r, q 0 = 0 → q 1 ≤ l 1)
+    {m : Plane} (hmJn : m ∈ J_n) (hm0 : m 0 = 0) :
+    ∃ w ∈ J_s, w 0 = 0 ∧ w 1 ≤ m 1 := by
+  set a : Plane := !₂[(-1 : ℝ), 0] with ha_def
+  set b : Plane := !₂[(1 : ℝ), 0] with hb_def
+  -- opaque bottom / top axis points (avoid unfolding `!₂` in later arithmetic)
+  obtain ⟨s, hs0, hs1⟩ : ∃ s : Plane, s 0 = 0 ∧ s 1 = -2 :=
+    ⟨!₂[(0 : ℝ), -2], by simp, by simp⟩
+  obtain ⟨nn, hn0, hn1⟩ : ∃ nn : Plane, nn 0 = 0 ∧ nn 1 = 2 :=
+    ⟨!₂[(0 : ℝ), 2], by simp, by simp⟩
+  have ha0 : a 0 = -1 := by simp [ha_def]
+  have hb0 : b 0 = 1 := by simp [hb_def]
+  -- basic containments and the rectangle bound
+  have hJnr : J_n ⊆ range r := by rw [← hUnion]; exact subset_union_left
+  have hJsr : J_s ⊆ range r := by rw [← hUnion]; exact subset_union_right
+  have hrect := normalized_subset_rectangle hmr hpr hfar
+  have hmrange : m ∈ range r := hJnr hmJn
+  have hlrange : l ∈ range r := hJnr hlJn
+  have hm1lb : (-2 : ℝ) ≤ m 1 := (hrect hmrange).2.1
+  have hm1ub : m 1 ≤ 2 := (hrect hmrange).2.2
+  have hl1lb : (-2 : ℝ) ≤ l 1 := (hrect hlrange).2.1
+  have hl1ub : l 1 ≤ 2 := (hrect hlrange).2.2
+  -- `a, b ∈ J_s`; `J_s` joins them
+  have haJs : a ∈ J_s := by
+    have : a ∈ J_n ∩ J_s := by rw [hInter]; exact Set.mem_insert _ _
+    exact this.2
+  have hbJs : b ∈ J_s := by
+    have : b ∈ J_n ∩ J_s := by rw [hInter]; exact Set.mem_insert_of_mem _ rfl
+    exact this.2
+  have hJoinJs : JoinedIn J_s a b := arc_joinedIn eJs haJs hbJs
+  have hJsrect : J_s ⊆ {p : Plane | p 0 ∈ Icc (-1 : ℝ) 1 ∧ p 1 ∈ Icc (-2 : ℝ) 2} :=
+    hJsr.trans hrect
+  -- `m, l ∉ {a,b}` (they are on the axis `x = 0`, while `a 0 = -1`, `b 0 = 1`)
+  have hmab : m ∉ ({a, b} : Set Plane) := by
+    simp only [Set.mem_insert_iff, Set.mem_singleton_iff]
+    rintro (h | h) <;>
+      (have hc := congrArg (fun p : Plane => p 0) h;
+       simp only [hm0, ha0, hb0] at hc; norm_num at hc)
+  have hlab : l ∉ ({a, b} : Set Plane) := by
+    simp only [Set.mem_insert_iff, Set.mem_singleton_iff]
+    rintro (h | h) <;>
+      (have hc := congrArg (fun p : Plane => p 0) h;
+       simp only [hl0, ha0, hb0] at hc; norm_num at hc)
+  by_contra hcon
+  push Not at hcon
+  -- `hcon : ∀ w ∈ J_s, w 0 = 0 → m 1 < w 1`
+  -- three path pieces on `[-1,1]`
+  set f1 : ℝ → Plane := fun t => s + ((t + 1) / 2) • (m - s) with hf1def
+  set f3 : ℝ → Plane := fun t => l + ((t + 1) / 2) • (nn - l) with hf3def
+  obtain ⟨f2, hf2cont, hf2m, hf2l, hf2mem⟩ :=
+    arc_path (hpcJn.joinedIn m ⟨hmJn, hmab⟩ l ⟨hlJn, hlab⟩)
+  have hf1cont : ContinuousOn f1 (Icc (-1 : ℝ) 1) := (by fun_prop : Continuous f1).continuousOn
+  have hf3cont : ContinuousOn f3 (Icc (-1 : ℝ) 1) := (by fun_prop : Continuous f3).continuousOn
+  have hf1_1 : f1 1 = m := by simp [hf1def]
+  have hf1_m1 : f1 (-1) = s := by simp [hf1def]
+  have hf3_1 : f3 1 = nn := by simp [hf3def]
+  have hf3_m1 : f3 (-1) = l := by simp [hf3def]
+  have hf1f2 : f1 1 = f2 (-1) := by rw [hf1_1, hf2m]
+  have hf2f3 : f2 1 = f3 (-1) := by rw [hf2l, hf3_m1]
+  obtain ⟨c, hccont, hcs, hcn, hcmem⟩ := concatPath3 hf1cont hf2cont hf3cont hf1f2 hf2f3
+  -- endpoints of the concatenation are on the bottom / top edges
+  have hcm1 : c (-1) 1 = -2 := by rw [hcs, hf1_m1]; exact hs1
+  have hcn1 : c 1 1 = 2 := by rw [hcn, hf3_1]; exact hn1
+  -- the three pieces stay in the rectangle
+  have hf1x : ∀ t, f1 t 0 = 0 := fun t => by simp [hf1def, hs0, hm0]
+  have hf3x : ∀ t, f3 t 0 = 0 := fun t => by simp [hf3def, hl0, hn0]
+  have hf1y_le : ∀ t ∈ Icc (-1 : ℝ) 1, f1 t 1 ≤ m 1 := by
+    intro t ht; rw [mem_Icc] at ht
+    simp only [hf1def, PiLp.add_apply, PiLp.smul_apply, PiLp.sub_apply, smul_eq_mul, hs1]
+    nlinarith [ht.1, ht.2, hm1lb]
+  have hf1y_ge : ∀ t ∈ Icc (-1 : ℝ) 1, (-2 : ℝ) ≤ f1 t 1 := by
+    intro t ht; rw [mem_Icc] at ht
+    simp only [hf1def, PiLp.add_apply, PiLp.smul_apply, PiLp.sub_apply, smul_eq_mul, hs1]
+    nlinarith [ht.1, ht.2, hm1lb]
+  have hf3y_ge : ∀ t ∈ Icc (-1 : ℝ) 1, l 1 ≤ f3 t 1 := by
+    intro t ht; rw [mem_Icc] at ht
+    simp only [hf3def, PiLp.add_apply, PiLp.smul_apply, PiLp.sub_apply, smul_eq_mul, hn1]
+    nlinarith [ht.1, ht.2, hl1ub]
+  have hf3y_le : ∀ t ∈ Icc (-1 : ℝ) 1, f3 t 1 ≤ 2 := by
+    intro t ht; rw [mem_Icc] at ht
+    simp only [hf3def, PiLp.add_apply, PiLp.smul_apply, PiLp.sub_apply, smul_eq_mul, hn1]
+    nlinarith [ht.1, ht.2, hl1lb, hl1ub]
+  have hbound : ∀ z ∈ (f1 '' Icc (-1 : ℝ) 1 ∪ f2 '' Icc (-1 : ℝ) 1) ∪ f3 '' Icc (-1 : ℝ) 1,
+      z 0 ∈ Icc (-1 : ℝ) 1 ∧ z 1 ∈ Icc (-2 : ℝ) 2 := by
+    rintro z ((⟨t1, ht1, rfl⟩ | ⟨t2, ht2, rfl⟩) | ⟨t3, ht3, rfl⟩)
+    · exact ⟨by rw [hf1x]; norm_num,
+        mem_Icc.2 ⟨hf1y_ge t1 ht1, le_trans (hf1y_le t1 ht1) hm1ub⟩⟩
+    · exact hrect (hJnr (hf2mem t2 ht2).1)
+    · exact ⟨by rw [hf3x]; norm_num,
+        mem_Icc.2 ⟨le_trans hl1lb (hf3y_ge t3 ht3), hf3y_le t3 ht3⟩⟩
+  have hcE : ∀ t ∈ Icc (-1 : ℝ) 1, c t 0 ∈ Icc (-1 : ℝ) 1 ∧ c t 1 ∈ Icc (-2 : ℝ) 2 :=
+    fun t ht => hbound (c t) (hcmem t ht)
+  -- the concatenation is a bottom-to-top path, so it must meet `J_s`
+  obtain ⟨t, ht, hctJs⟩ :=
+    vertical_meets_arc hbr hJsrect hJoinJs hccont hcE hcm1 hcn1
+  rcases hcmem t ht with (⟨t1, ht1, ht1e⟩ | ⟨t2, ht2, ht2e⟩) | ⟨t3, ht3, ht3e⟩
+  · -- on the lower axis segment: contradicts `hcon` (height `≤ m 1`)
+    have hct0 : c t 0 = 0 := by rw [← ht1e]; exact hf1x t1
+    have hcty : c t 1 ≤ m 1 := by rw [← ht1e]; exact hf1y_le t1 ht1
+    exact absurd (hcon (c t) hctJs hct0) (not_lt.2 hcty)
+  · -- on the middle piece: lands in `J_n \ {a,b}`, disjoint from `J_s`
+    have hctJn : c t ∈ J_n := ht2e ▸ (hf2mem t2 ht2).1
+    have hctnab : c t ∉ ({a, b} : Set Plane) := ht2e ▸ (hf2mem t2 ht2).2
+    have : c t ∈ J_n ∩ J_s := ⟨hctJn, hctJs⟩
+    rw [hInter] at this
+    exact hctnab this
+  · -- on the upper axis segment: forced to equal `l`, but `l ∉ J_s`
+    have hct0 : c t 0 = 0 := by rw [← ht3e]; exact hf3x t3
+    have hcty_ge : l 1 ≤ c t 1 := by rw [← ht3e]; exact hf3y_ge t3 ht3
+    have hcty_le : c t 1 ≤ l 1 := hlmax (c t) (hJsr hctJs) hct0
+    have hcty : c t 1 = l 1 := le_antisymm hcty_le hcty_ge
+    have hctl : c t = l := by ext i; fin_cases i <;> simp [hct0, hl0, hcty]
+    have hlnJs : l ∉ J_s := by
+      intro h
+      have : l ∈ J_n ∩ J_s := ⟨hlJn, h⟩
+      rw [hInter] at this; exact hlab this
+    exact hlnJs (hctl ▸ hctJs)
+
+/-- **Maehara's construction points.** Bundles the five points of Maehara's figure
+on the vertical axis `x = 0`:
+* `l` — the topmost intersection of the axis with the curve (in `J_n`);
+* `m` — the lowest point of `J_n` on the axis;
+* `p` — the highest point of `J_s` on the axis lying at or below `m` (the top of the
+  southern arc on the segment `m → s`, via `ms_meets_Js`);
+* `q` — the lowest point of `J_s` on the axis;
+* `z₀` — the midpoint `!₂[0,(m 1 + p 1)/2]` of `m` and `p`.
+
+The established `y`-ordering is `q 1 ≤ p 1 ≤ z₀ 1 ≤ m 1 ≤ l 1`.  (Note `p` is the
+`J_s`-max **restricted to heights `≤ m 1`**; this is what makes `p 1 ≤ m 1` — hence
+the placement of `z₀` between `p` and `m` — provable at this stage.) -/
+theorem exists_construction_points (hbr : BrouwerFPT)
+    {r : sphere (0 : Plane) 1 → Plane} (hcont : Continuous r) (hinj : Injective r)
+    (hm : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r) (hp : (!₂[(1 : ℝ), 0] : Plane) ∈ range r)
+    (hfar : ∀ z ∈ range r, ∀ w ∈ range r, dist z w ≤ 2) :
+    ∃ (J_n J_s : Set Plane) (l m p q z₀ : Plane),
+      J_n ∪ J_s = range r ∧
+      J_n ∩ J_s = {!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]} ∧
+      IsPathConnected (J_n \ {!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]}) ∧
+      (l ∈ J_n ∧ l 0 = 0 ∧ ∀ w ∈ range r, w 0 = 0 → w 1 ≤ l 1) ∧
+      (m ∈ J_n ∧ m 0 = 0 ∧ ∀ w ∈ J_n, w 0 = 0 → m 1 ≤ w 1) ∧
+      (p ∈ J_s ∧ p 0 = 0 ∧ p 1 ≤ m 1 ∧ ∀ w ∈ J_s, w 0 = 0 → w 1 ≤ m 1 → w 1 ≤ p 1) ∧
+      (q ∈ J_s ∧ q 0 = 0 ∧ ∀ w ∈ J_s, w 0 = 0 → q 1 ≤ w 1) ∧
+      z₀ = !₂[(0 : ℝ), (m 1 + p 1) / 2] ∧
+      m 1 ≤ l 1 ∧ q 1 ≤ p 1 ∧ p 1 ≤ z₀ 1 ∧ z₀ 1 ≤ m 1 ∧
+      IsPathConnected (J_s \ {!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]}) ∧
+      JoinedIn J_n (!₂[(-1 : ℝ), 0]) (!₂[(1 : ℝ), 0]) ∧
+      JoinedIn J_s (!₂[(-1 : ℝ), 0]) (!₂[(1 : ℝ), 0]) := by
+  obtain ⟨J_n, J_s, hJncl, hJscl, hUnion, hInter, ⟨eJn⟩, ⟨eJs⟩, hpcJn, hpcJs,
+    l, hlr, hl0, hlmax, hlJn⟩ := jordan_arcs hbr hcont hinj hm hp hfar
+  have hcptr : IsCompact (range r) := jordanCurve_isCompact r hcont
+  have hJnr : J_n ⊆ range r := by rw [← hUnion]; exact subset_union_left
+  have hJsr : J_s ⊆ range r := by rw [← hUnion]; exact subset_union_right
+  have hJncpt : IsCompact J_n := hcptr.of_isClosed_subset hJncl hJnr
+  have hJscpt : IsCompact J_s := hcptr.of_isClosed_subset hJscl hJsr
+  -- `m` : lowest point of `J_n` on the axis
+  obtain ⟨m, hmJn, hm0, hmmin⟩ := exists_ymin_Jn_axis hJncl hJncpt hlJn hl0
+  -- `J_s` meets the axis below `m`
+  obtain ⟨w₀, hw₀Js, hw₀0, hw₀m⟩ :=
+    ms_meets_Js hbr hcont hinj hm hp hfar hUnion hInter eJs hpcJn hlJn hl0 hlmax hmJn hm0
+  -- `p` : highest point of `J_s` on the axis at or below `m`
+  set Sp : Set Plane := {w ∈ J_s | w 0 = 0 ∧ w 1 ≤ m 1} with hSpdef
+  have hSpcl : IsClosed Sp := by
+    have hEq : Sp = (J_s ∩ {w : Plane | w 0 = 0}) ∩ {w : Plane | w 1 ≤ m 1} := by
+      ext w
+      constructor
+      · rintro ⟨hJ, h0, hle⟩; exact ⟨⟨hJ, h0⟩, hle⟩
+      · rintro ⟨⟨hJ, h0⟩, hle⟩; exact ⟨hJ, h0, hle⟩
+    rw [hEq]
+    exact (hJscl.inter (isClosed_eq (by fun_prop) continuous_const)).inter
+      (isClosed_le (by fun_prop) continuous_const)
+  have hSpcpt : IsCompact Sp := hJscpt.of_isClosed_subset hSpcl (fun w hw => hw.1)
+  obtain ⟨p, hpSp, hpmax⟩ := hSpcpt.exists_isMaxOn ⟨w₀, hw₀Js, hw₀0, hw₀m⟩
+    (f := fun w : Plane => w 1) (by fun_prop : Continuous (fun w : Plane => w 1)).continuousOn
+  -- `q` : lowest point of `J_s` on the axis
+  set Sq : Set Plane := {w ∈ J_s | w 0 = 0} with hSqdef
+  have hSqcl : IsClosed Sq :=
+    hJscl.inter (isClosed_eq (by fun_prop) continuous_const)
+  have hSqcpt : IsCompact Sq := hJscpt.of_isClosed_subset hSqcl (fun w hw => hw.1)
+  obtain ⟨q, hqSq, hqmin⟩ := hSqcpt.exists_isMinOn ⟨w₀, hw₀Js, hw₀0⟩
+    (f := fun w : Plane => w 1) (by fun_prop : Continuous (fun w : Plane => w 1)).continuousOn
+  -- extract the packaged facts
+  have hpm : p 1 ≤ m 1 := hpSp.2.2
+  have hml : m 1 ≤ l 1 := hlmax m (hJnr hmJn) hm0
+  have hqp : q 1 ≤ p 1 := hqmin ⟨hpSp.1, hpSp.2.1⟩
+  have haJn : (!₂[(-1 : ℝ), 0] : Plane) ∈ J_n := by
+    have : (!₂[(-1 : ℝ), 0] : Plane) ∈ J_n ∩ J_s := by rw [hInter]; exact mem_insert _ _
+    exact this.1
+  have hbJn : (!₂[(1 : ℝ), 0] : Plane) ∈ J_n := by
+    have : (!₂[(1 : ℝ), 0] : Plane) ∈ J_n ∩ J_s := by rw [hInter]; exact mem_insert_of_mem _ rfl
+    exact this.1
+  have haJs : (!₂[(-1 : ℝ), 0] : Plane) ∈ J_s := by
+    have : (!₂[(-1 : ℝ), 0] : Plane) ∈ J_n ∩ J_s := by rw [hInter]; exact mem_insert _ _
+    exact this.2
+  have hbJs : (!₂[(1 : ℝ), 0] : Plane) ∈ J_s := by
+    have : (!₂[(1 : ℝ), 0] : Plane) ∈ J_n ∩ J_s := by rw [hInter]; exact mem_insert_of_mem _ rfl
+    exact this.2
+  have hJoinJn : JoinedIn J_n (!₂[(-1 : ℝ), 0]) (!₂[(1 : ℝ), 0]) := arc_joinedIn eJn haJn hbJn
+  have hJoinJs : JoinedIn J_s (!₂[(-1 : ℝ), 0]) (!₂[(1 : ℝ), 0]) := arc_joinedIn eJs haJs hbJs
+  refine ⟨J_n, J_s, l, m, p, q, !₂[(0 : ℝ), (m 1 + p 1) / 2], hUnion, hInter, hpcJn,
+    ⟨hlJn, hl0, hlmax⟩, ⟨hmJn, hm0, hmmin⟩,
+    ⟨hpSp.1, hpSp.2.1, hpm, fun w hwJs hw0 hwm => hpmax ⟨hwJs, hw0, hwm⟩⟩,
+    ⟨hqSq.1, hqSq.2, fun w hwJs hw0 => hqmin ⟨hwJs, hw0⟩⟩, rfl, hml, hqp, ?_, ?_,
+    hpcJs, hJoinJn, hJoinJs⟩
+  · rw [show (!₂[(0 : ℝ), (m 1 + p 1) / 2] : Plane) 1 = (m 1 + p 1) / 2 from by simp]
+    linarith [hpm]
+  · rw [show (!₂[(0 : ℝ), (m 1 + p 1) / 2] : Plane) 1 = (m 1 + p 1) / 2 from by simp]
+    linarith [hpm]
+
+/-- **First-exit lemma (A.1).** A path `α` on `[0,1]` starting inside an open set `O`
+and ending outside a closed superset `C ⊇ O` has a *first exit time* `tw ∈ (0,1]`:
+`α tw` lies in the frontier layer `C \ O`, and `α u ∈ O` for all `u < tw`. -/
+theorem exists_first_exit {α : ℝ → Plane} {O C : Set Plane}
+    (hO : IsOpen O) (hC : IsClosed C) (hOC : O ⊆ C)
+    (hcont : ContinuousOn α (Icc (0 : ℝ) 1))
+    (h0 : α 0 ∈ O) (h1 : α 1 ∉ C) :
+    ∃ tw ∈ Ioc (0 : ℝ) 1, α tw ∈ C ∧ α tw ∉ O ∧ ∀ u ∈ Ico (0 : ℝ) tw, α u ∈ O := by
+  set S : Set ℝ := {t ∈ Icc (0 : ℝ) 1 | α t ∉ O} with hSdef
+  -- `S` is closed: it is `Icc 0 1 ∩ α⁻¹(Oᶜ)`
+  have hScl : IsClosed S := by
+    have hEq : S = Icc (0 : ℝ) 1 ∩ α ⁻¹' Oᶜ := by
+      ext t; simp only [hSdef, mem_ofPred_eq, mem_inter_iff, mem_preimage, mem_compl_iff]
+    rw [hEq]; exact hcont.preimage_isClosed_of_isClosed isClosed_Icc hO.isClosed_compl
+  have hSne : S.Nonempty :=
+    ⟨1, ⟨by norm_num, fun h => h1 (hOC h)⟩⟩
+  have hSbdd : BddBelow S := ⟨0, fun t ht => ht.1.1⟩
+  set tw := sInf S with htw
+  have htwS : tw ∈ S := hScl.csInf_mem hSne hSbdd
+  have htw01 : tw ∈ Icc (0 : ℝ) 1 := htwS.1
+  have htwO : α tw ∉ O := htwS.2
+  -- `tw > 0` (since `0 ∉ S`)
+  have h0S : (0 : ℝ) ∉ S := fun h => h.2 h0
+  have htwpos : 0 < tw := lt_of_le_of_ne htw01.1 (fun h => h0S (h ▸ htwS))
+  -- points strictly below `tw` are still in `O`
+  have hbelow : ∀ u ∈ Ico (0 : ℝ) tw, α u ∈ O := by
+    intro u hu
+    by_contra hnot
+    have huS : u ∈ S := ⟨⟨hu.1, le_trans hu.2.le htw01.2⟩, hnot⟩
+    exact absurd (csInf_le hSbdd huS) (not_le.2 hu.2)
+  -- `α tw ∈ C` by closure of the pre-exit segment
+  have htwC : α tw ∈ C := by
+    have hT : IsClosed (Icc (0 : ℝ) 1 ∩ α ⁻¹' C) :=
+      hcont.preimage_isClosed_of_isClosed isClosed_Icc hC
+    have hsub : Ico (0 : ℝ) tw ⊆ Icc (0 : ℝ) 1 ∩ α ⁻¹' C := by
+      intro u hu
+      exact ⟨⟨hu.1, le_trans hu.2.le htw01.2⟩, hOC (hbelow u hu)⟩
+    have : tw ∈ Icc (0 : ℝ) 1 ∩ α ⁻¹' C := by
+      apply hT.closure_subset_iff.mpr hsub
+      rw [closure_Ico (ne_of_lt htwpos)]
+      exact ⟨htwpos.le, le_refl tw⟩
+    exact this.2
+  exact ⟨tw, ⟨htwpos, htw01.2⟩, htwC, htwO, hbelow⟩
+
+/-- A vertical segment `{p | p 0 = c ∧ p 1 ∈ T}` (with `T` path-connected) is
+path-connected: it is the image of `T` under `y ↦ !₂[c, y]`. -/
+theorem isPathConnected_vertSeg (c : ℝ) {T : Set ℝ} (hT : IsPathConnected T) :
+    IsPathConnected {p : Plane | p 0 = c ∧ p 1 ∈ T} := by
+  have hfcont : Continuous (fun y : ℝ => (!₂[c, y] : Plane)) := by
+    apply (PiLp.continuous_toLp 2 (fun _ : Fin 2 => ℝ)).comp
+    refine continuous_pi (fun i => ?_)
+    fin_cases i <;> simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.zero_eta, Fin.isValue,
+      Matrix.cons_val_zero,
+      Fin.mk_one, Matrix.cons_val_one, Matrix.cons_val_fin_one]<;> fun_prop
+  have himg := hT.image hfcont
+  have hEq : (fun y : ℝ => (!₂[c, y] : Plane)) '' T = {p : Plane | p 0 = c ∧ p 1 ∈ T} := by
+    ext p
+    simp only [mem_image, mem_ofPred_eq]
+    constructor
+    · rintro ⟨y, hy, rfl⟩; exact ⟨by simp, by simpa using hy⟩
+    · rintro ⟨h0, h1⟩
+      exact ⟨p 1, h1, by ext i; fin_cases i <;> simp [h0]⟩
+  rwa [hEq] at himg
+
+/-- A horizontal segment `{p | p 1 = c ∧ p 0 ∈ T}` is path-connected. -/
+theorem isPathConnected_horizSeg (c : ℝ) {T : Set ℝ} (hT : IsPathConnected T) :
+    IsPathConnected {p : Plane | p 1 = c ∧ p 0 ∈ T} := by
+  have hfcont : Continuous (fun x : ℝ => (!₂[x, c] : Plane)) := by
+    apply (PiLp.continuous_toLp 2 (fun _ : Fin 2 => ℝ)).comp
+    refine continuous_pi (fun i => ?_)
+    fin_cases i <;> simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.zero_eta, Fin.isValue,
+      Matrix.cons_val_zero,
+      Fin.mk_one, Matrix.cons_val_one, Matrix.cons_val_fin_one]<;> fun_prop
+  have himg := hT.image hfcont
+  have hEq : (fun x : ℝ => (!₂[x, c] : Plane)) '' T = {p : Plane | p 1 = c ∧ p 0 ∈ T} := by
+    ext p
+    simp only [mem_image, mem_ofPred_eq]
+    constructor
+    · rintro ⟨y, hy, rfl⟩; exact ⟨by simp, by simpa using hy⟩
+    · rintro ⟨h0, h1⟩
+      exact ⟨p 0, h1, by ext i; fin_cases i <;> simp [h0]⟩
+  rwa [hEq] at himg
+
+/-- **Curve meets the rectangle boundary only on the axis (A.2).** Any curve point on
+the frontier of the normalized rectangle `[-1,1]×[-2,2]` has `y = 0` (so it is `a` or
+`b`).  The "lens" bound `(p 0±1)²+(p 1)² ≤ 4` rules out the top/bottom edges outright
+and pins the left/right edges to `y = 0`. -/
+theorem curve_boundary_axis
+    {r : sphere (0 : Plane) 1 → Plane}
+    (hm : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r) (hp : (!₂[(1 : ℝ), 0] : Plane) ∈ range r)
+    (hfar : ∀ z ∈ range r, ∀ w ∈ range r, dist z w ≤ 2) :
+    ∀ p ∈ range r, (p 0 = -1 ∨ p 0 = 1 ∨ p 1 = -2 ∨ p 1 = 2) → p 1 = 0 := by
+  intro p hp' hbdry
+  have hza : dist p (!₂[(-1 : ℝ), 0]) ^ 2 ≤ 4 := by
+    have := hfar p hp' _ hm; nlinarith [dist_nonneg (x := p) (y := (!₂[(-1 : ℝ),0] : Plane))]
+  have hzb : dist p (!₂[(1 : ℝ), 0]) ^ 2 ≤ 4 := by
+    have := hfar p hp' _ hp; nlinarith [dist_nonneg (x := p) (y := (!₂[(1 : ℝ),0] : Plane))]
+  rw [dist_sq_coord] at hza hzb
+  simp only [Matrix.cons_val_zero, Matrix.cons_val_one] at hza hzb
+  rcases hbdry with h | h | h | h
+  · have h2 : (p 1)^2 = 0 := le_antisymm (by nlinarith [hzb]) (sq_nonneg _)
+    exact pow_eq_zero_iff (by norm_num) |>.mp h2
+  · have h2 : (p 1)^2 = 0 := le_antisymm (by nlinarith [hza]) (sq_nonneg _)
+    exact pow_eq_zero_iff (by norm_num) |>.mp h2
+  · exfalso; nlinarith [hza, hzb, sq_nonneg (p 0 + 1), sq_nonneg (p 0 - 1), h, sq_nonneg (p 1 + 2)]
+  · exfalso; nlinarith [hza, hzb, sq_nonneg (p 0 + 1), sq_nonneg (p 0 - 1), h, sq_nonneg (p 1 - 2)]
+
+/-- **Lower boundary arc (A.2).** The part of the rectangle frontier with `y < 0` is
+path-connected, contains the bottom axis point `!₂[0,-2]`, lies in the rectangle, and
+avoids the curve; moreover any frontier point with `y < 0` lies in it. -/
+theorem exists_lower_boundary_path
+    {r : sphere (0 : Plane) 1 → Plane}
+    (hm : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r) (hp : (!₂[(1 : ℝ), 0] : Plane) ∈ range r)
+    (hfar : ∀ z ∈ range r, ∀ w ∈ range r, dist z w ≤ 2) :
+    ∃ G : Set Plane,
+      IsPathConnected G ∧
+      (!₂[(0 : ℝ), -2] : Plane) ∈ G ∧
+      G ⊆ {p : Plane | p 0 ∈ Icc (-1 : ℝ) 1 ∧ p 1 ∈ Icc (-2 : ℝ) 2} ∧
+      (∀ p ∈ G, p ∉ range r) ∧
+      (∀ p : Plane, p 0 ∈ Icc (-1 : ℝ) 1 → p 1 ∈ Icc (-2 : ℝ) 2 →
+        ¬(p 0 ∈ Ioo (-1 : ℝ) 1 ∧ p 1 ∈ Ioo (-2 : ℝ) 2) → p 1 < 0 → p ∈ G) := by
+  set L : Set Plane := {p : Plane | p 0 = -1 ∧ p 1 ∈ Ico (-2 : ℝ) 0} with hLdef
+  set Rt : Set Plane := {p : Plane | p 0 = 1 ∧ p 1 ∈ Ico (-2 : ℝ) 0} with hRdef
+  set Bot : Set Plane := {p : Plane | p 1 = -2 ∧ p 0 ∈ Icc (-1 : ℝ) 1} with hBdef
+  have hLpc : IsPathConnected L :=
+    isPathConnected_vertSeg (-1) ((convex_Ico _ _).isPathConnected (nonempty_Ico.mpr (by norm_num)))
+  have hRpc : IsPathConnected Rt :=
+    isPathConnected_vertSeg 1 ((convex_Ico _ _).isPathConnected (nonempty_Ico.mpr (by norm_num)))
+  have hBpc : IsPathConnected Bot :=
+    isPathConnected_horizSeg (-2) ((convex_Icc _ _).isPathConnected (nonempty_Icc.mpr
+      (by norm_num)))
+  have memL : (!₂[(-1 : ℝ), -2] : Plane) ∈ L := ⟨by simp, by rw [mem_Ico]; exact ⟨by simp, by
+    norm_num⟩⟩
+  have memBotL : (!₂[(-1 : ℝ), -2] : Plane) ∈ Bot := ⟨by simp, by rw [mem_Icc]; exact ⟨by norm_num,
+    by norm_num⟩⟩
+  have memBotR : (!₂[(1 : ℝ), -2] : Plane) ∈ Bot := ⟨by simp, by rw [mem_Icc]; exact ⟨by norm_num,
+    by norm_num⟩⟩
+  have memR : (!₂[(1 : ℝ), -2] : Plane) ∈ Rt := ⟨by simp, by rw [mem_Ico]; exact ⟨by simp, by
+    norm_num⟩⟩
+  have hLB : IsPathConnected (L ∪ Bot) := hLpc.union hBpc ⟨_, memL, memBotL⟩
+  have hG : IsPathConnected ((L ∪ Bot) ∪ Rt) := hLB.union hRpc ⟨_, Or.inr memBotR, memR⟩
+  refine ⟨(L ∪ Bot) ∪ Rt, hG,
+    Or.inl (Or.inr ⟨by simp, by rw [mem_Icc]; exact ⟨by norm_num, by norm_num⟩⟩), ?_, ?_, ?_⟩
+  · rintro p ((hpp | hpp) | hpp)
+    · obtain ⟨h0, h1⟩ := hpp; rw [mem_Ico] at h1
+      exact ⟨by rw [mem_Icc, h0]; norm_num, by rw [mem_Icc]; exact ⟨h1.1, by linarith [h1.2]⟩⟩
+    · obtain ⟨h1, h0⟩ := hpp
+      exact ⟨h0, by rw [mem_Icc, h1]; norm_num⟩
+    · obtain ⟨h0, h1⟩ := hpp; rw [mem_Ico] at h1
+      exact ⟨by rw [mem_Icc, h0]; norm_num, by rw [mem_Icc]; exact ⟨h1.1, by linarith [h1.2]⟩⟩
+  · rintro p ((hpp | hpp) | hpp) hpr
+    · have := curve_boundary_axis hm hp hfar p hpr (Or.inl hpp.1); linarith [hpp.2.2]
+    · have := curve_boundary_axis hm hp hfar p hpr (Or.inr (Or.inr (Or.inl hpp.1))); linarith
+      [hpp.1]
+    · have := curve_boundary_axis hm hp hfar p hpr (Or.inr (Or.inl hpp.1)); linarith [hpp.2.2]
+  · intro p hp0 hp1 hnint hpneg
+    rcases not_and_or.mp hnint with hn0 | hn1
+    · rw [mem_Ioo, not_and_or] at hn0
+      rcases hn0 with h | h
+      · exact Or.inl (Or.inl ⟨le_antisymm (not_lt.mp h) hp0.1, mem_Ico.mpr ⟨hp1.1, hpneg⟩⟩)
+      · exact Or.inr ⟨le_antisymm hp0.2 (not_lt.mp h), mem_Ico.mpr ⟨hp1.1, hpneg⟩⟩
+    · rw [mem_Ioo, not_and_or] at hn1
+      rcases hn1 with h | h
+      · exact Or.inl (Or.inr ⟨le_antisymm (not_lt.mp h) hp1.1, hp0⟩)
+      · exfalso; linarith [not_lt.mp h]
+
+/-- **Upper boundary arc (A.2).** The part of the rectangle frontier with `y > 0` is
+path-connected, contains the top axis point `!₂[0,2]`, lies in the rectangle, and
+avoids the curve; moreover any frontier point with `y > 0` lies in it. -/
+theorem exists_upper_boundary_path
+    {r : sphere (0 : Plane) 1 → Plane}
+    (hm : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r) (hp : (!₂[(1 : ℝ), 0] : Plane) ∈ range r)
+    (hfar : ∀ z ∈ range r, ∀ w ∈ range r, dist z w ≤ 2) :
+    ∃ G : Set Plane,
+      IsPathConnected G ∧
+      (!₂[(0 : ℝ), 2] : Plane) ∈ G ∧
+      G ⊆ {p : Plane | p 0 ∈ Icc (-1 : ℝ) 1 ∧ p 1 ∈ Icc (-2 : ℝ) 2} ∧
+      (∀ p ∈ G, p ∉ range r) ∧
+      (∀ p : Plane, p 0 ∈ Icc (-1 : ℝ) 1 → p 1 ∈ Icc (-2 : ℝ) 2 →
+        ¬(p 0 ∈ Ioo (-1 : ℝ) 1 ∧ p 1 ∈ Ioo (-2 : ℝ) 2) → 0 < p 1 → p ∈ G) := by
+  set L : Set Plane := {p : Plane | p 0 = -1 ∧ p 1 ∈ Ioc (0 : ℝ) 2} with hLdef
+  set Rt : Set Plane := {p : Plane | p 0 = 1 ∧ p 1 ∈ Ioc (0 : ℝ) 2} with hRdef
+  set Top : Set Plane := {p : Plane | p 1 = 2 ∧ p 0 ∈ Icc (-1 : ℝ) 1} with hTdef
+  have hLpc : IsPathConnected L :=
+    isPathConnected_vertSeg (-1) ((convex_Ioc _ _).isPathConnected (nonempty_Ioc.mpr (by norm_num)))
+  have hRpc : IsPathConnected Rt :=
+    isPathConnected_vertSeg 1 ((convex_Ioc _ _).isPathConnected (nonempty_Ioc.mpr (by norm_num)))
+  have hTpc : IsPathConnected Top :=
+    isPathConnected_horizSeg 2 ((convex_Icc _ _).isPathConnected (nonempty_Icc.mpr (by norm_num)))
+  have memL : (!₂[(-1 : ℝ), 2] : Plane) ∈ L := ⟨by simp, by rw [mem_Ioc]; exact ⟨by norm_num, by
+    simp⟩⟩
+  have memTopL : (!₂[(-1 : ℝ), 2] : Plane) ∈ Top := ⟨by simp, by rw [mem_Icc]; exact ⟨by norm_num,
+    by norm_num⟩⟩
+  have memTopR : (!₂[(1 : ℝ), 2] : Plane) ∈ Top := ⟨by simp, by rw [mem_Icc]; exact ⟨by norm_num, by
+    norm_num⟩⟩
+  have memR : (!₂[(1 : ℝ), 2] : Plane) ∈ Rt := ⟨by simp, by rw [mem_Ioc]; exact ⟨by norm_num, by
+    simp⟩⟩
+  have hLT : IsPathConnected (L ∪ Top) := hLpc.union hTpc ⟨_, memL, memTopL⟩
+  have hG : IsPathConnected ((L ∪ Top) ∪ Rt) := hLT.union hRpc ⟨_, Or.inr memTopR, memR⟩
+  refine ⟨(L ∪ Top) ∪ Rt, hG,
+    Or.inl (Or.inr ⟨by simp, by rw [mem_Icc]; exact ⟨by norm_num, by norm_num⟩⟩), ?_, ?_, ?_⟩
+  · rintro p ((hpp | hpp) | hpp)
+    · obtain ⟨h0, h1⟩ := hpp; rw [mem_Ioc] at h1
+      exact ⟨by rw [mem_Icc, h0]; norm_num, by rw [mem_Icc]; exact ⟨by linarith [h1.1], h1.2⟩⟩
+    · obtain ⟨h1, h0⟩ := hpp
+      exact ⟨h0, by rw [mem_Icc, h1]; norm_num⟩
+    · obtain ⟨h0, h1⟩ := hpp; rw [mem_Ioc] at h1
+      exact ⟨by rw [mem_Icc, h0]; norm_num, by rw [mem_Icc]; exact ⟨by linarith [h1.1], h1.2⟩⟩
+  · rintro p ((hpp | hpp) | hpp) hpr
+    · have := curve_boundary_axis hm hp hfar p hpr (Or.inl hpp.1); linarith [hpp.2.1]
+    · have := curve_boundary_axis hm hp hfar p hpr (Or.inr (Or.inr (Or.inr hpp.1))); linarith
+      [hpp.1]
+    · have := curve_boundary_axis hm hp hfar p hpr (Or.inr (Or.inl hpp.1)); linarith [hpp.2.1]
+  · intro p hp0 hp1 hnint hppos
+    rcases not_and_or.mp hnint with hn0 | hn1
+    · rw [mem_Ioo, not_and_or] at hn0
+      rcases hn0 with h | h
+      · exact Or.inl (Or.inl ⟨le_antisymm (not_lt.mp h) hp0.1, mem_Ioc.mpr ⟨hppos, hp1.2⟩⟩)
+      · exact Or.inr ⟨le_antisymm hp0.2 (not_lt.mp h), mem_Ioc.mpr ⟨hppos, hp1.2⟩⟩
+    · rw [mem_Ioo, not_and_or] at hn1
+      rcases hn1 with h | h
+      · exfalso; linarith [not_lt.mp h]
+      · exact Or.inl (Or.inr ⟨le_antisymm hp1.2 (not_lt.mp h), hp0⟩)
+
+/-- Points on the vertical axis are neither `!₂[-1,0]` nor `!₂[1,0]`. -/
+private lemma off_axis_endpoints {x : Plane} (hx : x 0 = 0) :
+    x ∉ ({!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]} : Set Plane) := by
+  simp only [Set.mem_insert_iff, Set.mem_singleton_iff]
+  rintro (h | h) <;>
+    (have hc := congrArg (fun w : Plane => w 0) h; simp only [hx] at hc; norm_num at hc)
+
+/-- `tw * u ∈ [0, tw]` for a unit-interval scalar `u`. -/
+private lemma mul_unit_mem_Icc {tw u : ℝ} (htw : 0 ≤ tw) (h0 : 0 ≤ u) (h1 : u ≤ 1) :
+    tw * u ∈ Icc (0 : ℝ) tw := by
+  rw [mem_Icc]
+  refine ⟨mul_nonneg htw h0, ?_⟩
+  calc tw * u ≤ tw * 1 := mul_le_mul_of_nonneg_left h1 htw
+    _ = tw := mul_one tw
+
+/-- The vertical segment from `x` up to `y` on the axis, parametrized over `[-1,1]`. -/
+private lemma axis_segment {x y : Plane} (hx0 : x 0 = 0) (hy0 : y 0 = 0) (hxy : x 1 ≤ y 1) :
+    ∃ g : ℝ → Plane, ContinuousOn g (Icc (-1 : ℝ) 1) ∧ g (-1) = x ∧ g 1 = y ∧
+      (∀ t, g t 0 = 0) ∧ ∀ t ∈ Icc (-1 : ℝ) 1, g t 1 ∈ Icc (x 1) (y 1) := by
+  refine ⟨fun t => x + ((t + 1) / 2) • (y - x),
+    (by fun_prop : Continuous fun t : ℝ => x + ((t + 1) / 2) • (y - x)).continuousOn,
+    by change x + (((-1 : ℝ) + 1) / 2) • (y - x) = x; module,
+    by change x + (((1 : ℝ) + 1) / 2) • (y - x) = y; module,
+    fun t => by simp [PiLp.add_apply, PiLp.smul_apply, PiLp.sub_apply, hx0, hy0], ?_⟩
+  intro t ht; rw [mem_Icc] at ht ⊢
+  simp only [PiLp.add_apply, PiLp.smul_apply, PiLp.sub_apply, smul_eq_mul]
+  have h0 : (0 : ℝ) ≤ (t + 1) / 2 := by linarith [ht.1]
+  have h1 : (t + 1) / 2 ≤ 1 := by linarith [ht.2]
+  have hd : (0 : ℝ) ≤ y 1 - x 1 := by linarith
+  constructor
+  · linarith [mul_nonneg h0 hd]
+  · linarith [mul_le_mul_of_nonneg_right h1 hd]
+
+/-- The closed rectangle `[-1,1] × [-2,2]` is bounded. -/
+private lemma isBounded_rectE :
+    IsBounded {w : Plane | w 0 ∈ Icc (-1 : ℝ) 1 ∧ w 1 ∈ Icc (-2 : ℝ) 2} := by
+  apply (Metric.isBounded_closedBall (x := (0:Plane)) (r := 3)).subset
+  intro z hz
+  rw [Metric.mem_closedBall]
+  have hz1 := hz.1; have hz2 := hz.2; rw [mem_Icc] at hz1 hz2
+  have hsq : dist z (0:Plane) ^ 2 ≤ 9 := by
+    rw [dist_sq_coord, show (0:Plane) 0 = 0 from by simp, show (0:Plane) 1 = 0 from by simp]
+    nlinarith [hz1.1, hz1.2, hz2.1, hz2.2]
+  nlinarith [dist_nonneg (x := z) (y := (0:Plane)), hsq]
+
+/-- The open rectangle `(-1,1) × (-2,2)` is open. -/
+private lemma isOpen_rectO :
+    IsOpen {w : Plane | w 0 ∈ Ioo (-1 : ℝ) 1 ∧ w 1 ∈ Ioo (-2 : ℝ) 2} := by
+  have hEq : {w : Plane | w 0 ∈ Ioo (-1 : ℝ) 1 ∧ w 1 ∈ Ioo (-2 : ℝ) 2}
+      = (fun w : Plane => w 0) ⁻¹' Ioo (-1 : ℝ) 1 ∩ (fun w : Plane => w 1) ⁻¹' Ioo (-2 : ℝ) 2 := by
+    ext w; simp only [mem_ofPred_eq, mem_inter_iff, mem_preimage]
+  rw [hEq]
+  exact (isOpen_Ioo.preimage (by fun_prop)).inter (isOpen_Ioo.preimage (by fun_prop))
+
+/-- The closed rectangle `[-1,1] × [-2,2]` is closed. -/
+private lemma isClosed_rectE :
+    IsClosed {w : Plane | w 0 ∈ Icc (-1 : ℝ) 1 ∧ w 1 ∈ Icc (-2 : ℝ) 2} := by
+  have hEq : {w : Plane | w 0 ∈ Icc (-1 : ℝ) 1 ∧ w 1 ∈ Icc (-2 : ℝ) 2}
+      = (fun w : Plane => w 0) ⁻¹' Icc (-1 : ℝ) 1 ∩ (fun w : Plane => w 1) ⁻¹' Icc (-2 : ℝ) 2 := by
+    ext w; simp only [mem_ofPred_eq, mem_inter_iff, mem_preimage]
+  rw [hEq]
+  exact (isClosed_Icc.preimage (by fun_prop)).inter (isClosed_Icc.preimage (by fun_prop))
+
+/-- A point of the frontier `E ∖ O` on the horizontal axis is one of the endpoints
+`!₂[±1, 0]`, hence on the curve. -/
+private lemma axis_zero_mem_range {r : sphere (0 : Plane) 1 → Plane}
+    (hm : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r) (hp : (!₂[(1 : ℝ), 0] : Plane) ∈ range r)
+    {x : Plane} (hxIcc : x 0 ∈ Icc (-1 : ℝ) 1)
+    (hxO : ¬(x 0 ∈ Ioo (-1 : ℝ) 1 ∧ x 1 ∈ Ioo (-2 : ℝ) 2))
+    (hx1 : x 1 = 0) : x ∈ range r := by
+  have hnotIoo0 : x 0 ∉ Ioo (-1 : ℝ) 1 := fun hIoo =>
+    hxO ⟨hIoo, by rw [hx1, mem_Ioo]; constructor <;> norm_num⟩
+  rw [mem_Ioo, not_and_or] at hnotIoo0
+  rcases hnotIoo0 with h | h
+  · have hx : x 0 = -1 := le_antisymm (not_lt.mp h) hxIcc.1
+    rw [show x = !₂[(-1 : ℝ), 0] from by ext i; fin_cases i <;> simp [hx, hx1]]; exact hm
+  · have hx : x 0 = 1 := le_antisymm hxIcc.2 (not_lt.mp h)
+    rw [show x = !₂[(1 : ℝ), 0] from by ext i; fin_cases i <;> simp [hx, hx1]]; exact hp
+
+/-- **Step A, low exit case.** If the escaping path first meets the frontier at a point
+below the axis, the concatenation `s → w → z₀ → m → l → n` (lower boundary region,
+escaping path reversed, axis, north arc, axis) misses `J_s`, contradicting
+`vertical_meets_arc`. -/
+private lemma step_A_case_low (hbr : BrouwerFPT)
+    {r : sphere (0 : Plane) 1 → Plane}
+    (hm : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r) (hp : (!₂[(1 : ℝ), 0] : Plane) ∈ range r)
+    (hfar : ∀ z ∈ range r, ∀ w ∈ range r, dist z w ≤ 2)
+    {J_n J_s : Set Plane} (hJnr : J_n ⊆ range r) (hJsr : J_s ⊆ range r)
+    (hInter : J_n ∩ J_s = ({!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]} : Set Plane))
+    (hpcJn : IsPathConnected (J_n \ ({!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]} : Set Plane)))
+    (hJoinJs : JoinedIn J_s (!₂[(-1 : ℝ), 0]) (!₂[(1 : ℝ), 0]))
+    {l m p z₀ : Plane}
+    (hlJn : l ∈ J_n) (hl0 : l 0 = 0) (hlmax : ∀ w ∈ range r, w 0 = 0 → w 1 ≤ l 1)
+    (hmJn : m ∈ J_n) (hm0 : m 0 = 0)
+    (hpJs : p ∈ J_s) (hpmax : ∀ w ∈ J_s, w 0 = 0 → w 1 ≤ m 1 → w 1 ≤ p 1)
+    (hz₀0 : z₀ 0 = 0) (hpz₀_lt : p 1 < z₀ 1) (hz₀m_lt : z₀ 1 < m 1)
+    {α : ℝ → Plane} {tw : ℝ}
+    (hαcont : Continuous α) (hα0 : α 0 = z₀) (htw0 : 0 ≤ tw)
+    (hαE : ∀ u ∈ Icc (0 : ℝ) tw, α u 0 ∈ Icc (-1 : ℝ) 1 ∧ α u 1 ∈ Icc (-2 : ℝ) 2)
+    (hαnr : ∀ u : ℝ, α u ∉ range r)
+    (hwC0 : α tw 0 ∈ Icc (-1 : ℝ) 1) (hwC1 : α tw 1 ∈ Icc (-2 : ℝ) 2)
+    (hwO : ¬(α tw 0 ∈ Ioo (-1 : ℝ) 1 ∧ α tw 1 ∈ Ioo (-2 : ℝ) 2))
+    (hwneg : α tw 1 < 0) : False := by
+  have hErect := normalized_subset_rectangle hm hp hfar
+  have hmab := off_axis_endpoints hm0
+  have hlab := off_axis_endpoints hl0
+  have hlE1 : l 1 ∈ Icc (-2 : ℝ) 2 := (hErect (hJnr hlJn)).2
+  have hmE1 : m 1 ∈ Icc (-2 : ℝ) 2 := (hErect (hJnr hmJn)).2
+  have hpE1 : p 1 ∈ Icc (-2 : ℝ) 2 := (hErect (hJsr hpJs)).2
+  have hz₀1lb : (-2 : ℝ) ≤ z₀ 1 := le_of_lt (lt_of_le_of_lt hpE1.1 hpz₀_lt)
+  have hs1 : (!₂[(0 : ℝ), -2] : Plane) 1 = -2 := by simp
+  have hn1 : (!₂[(0 : ℝ), 2] : Plane) 1 = 2 := by simp
+  obtain ⟨Glo, hGloPC, hsGlo, hGloSub, hGloAvoid, hGloFrom⟩ :=
+    exists_lower_boundary_path hm hp hfar
+  have hwGlo : α tw ∈ Glo := hGloFrom (α tw) hwC0 hwC1 hwO hwneg
+  -- piece 1 : `s → w` inside `Glo`
+  obtain ⟨g1, hg1cont, hg1a, hg1b, hg1mem⟩ :=
+    arc_path (hGloPC.joinedIn (!₂[(0 : ℝ), -2]) hsGlo (α tw) hwGlo)
+  -- piece 2 : `w → z₀` along the escaping path
+  set g2 : ℝ → Plane := fun t => α (tw * (1 - t) / 2) with hg2def
+  have hg2cont : ContinuousOn g2 (Icc (-1 : ℝ) 1) :=
+    (hαcont.comp (by fun_prop : Continuous fun t : ℝ => tw * (1 - t) / 2)).continuousOn
+  have hg2a : g2 (-1) = α tw := by change α (tw * (1 - (-1)) / 2) = α tw; congr 1; ring
+  have hg2b : g2 1 = z₀ := by
+    change α (tw * (1 - 1) / 2) = z₀; rw [show tw * (1 - 1) / 2 = (0 : ℝ) from by ring, hα0]
+  have hg2arg : ∀ t ∈ Icc (-1 : ℝ) 1, tw * (1 - t) / 2 ∈ Icc (0 : ℝ) tw := by
+    intro t ht; rw [mem_Icc] at ht
+    rw [mul_div_assoc]
+    exact mul_unit_mem_Icc htw0 (by linarith [ht.2]) (by linarith [ht.1])
+  -- piece 3 : `z₀ → m` along the axis
+  obtain ⟨g3, hg3cont, hg3a, hg3b, hg3x, hg3y⟩ := axis_segment hz₀0 hm0 hz₀m_lt.le
+  have hg3missJs : ∀ t ∈ Icc (-1 : ℝ) 1, g3 t ∉ J_s := by
+    intro t ht hJs
+    have hy := hg3y t ht; rw [mem_Icc] at hy
+    have := hpmax (g3 t) hJs (hg3x t) hy.2
+    linarith [hy.1, hpz₀_lt]
+  -- piece 4 : `m → l` inside `J_n \ {a,b}`
+  obtain ⟨g4, hg4cont, hg4a, hg4b, hg4mem⟩ :=
+    arc_path (hpcJn.joinedIn m ⟨hmJn, hmab⟩ l ⟨hlJn, hlab⟩)
+  have hg4missJs : ∀ t ∈ Icc (-1 : ℝ) 1, g4 t ∉ J_s := by
+    intro t ht hJs
+    have hg4 := hg4mem t ht
+    have hmem : g4 t ∈ J_n ∩ J_s := ⟨hg4.1, hJs⟩
+    rw [hInter] at hmem; exact hg4.2 hmem
+  -- piece 5 : `l → n` along the axis
+  obtain ⟨g5, hg5cont, hg5a, hg5b, hg5x, hg5y⟩ :=
+    axis_segment hl0 (show (!₂[(0 : ℝ), 2] : Plane) 0 = 0 from by simp) (by rw [hn1]; exact hlE1.2)
+  have hg5missJs : ∀ t ∈ Icc (-1 : ℝ) 1, g5 t ∉ J_s := by
+    intro t ht hJs
+    have hy := hg5y t ht; rw [mem_Icc] at hy
+    have hle := hlmax (g5 t) (hJsr hJs) (hg5x t)
+    have heq : g5 t 1 = l 1 := le_antisymm hle hy.1
+    have hgl : g5 t = l := by ext i; fin_cases i <;> simp [hg5x t, hl0, heq]
+    rw [hgl] at hJs
+    have hmem : l ∈ J_n ∩ J_s := ⟨hlJn, hJs⟩
+    rw [hInter] at hmem; exact hlab hmem
+  -- concatenate `s → w → z₀ → m → l → n`
+  obtain ⟨V, hVcont, hVa, hVb, hVmem⟩ :=
+    concatPath5 hg1cont hg2cont hg3cont hg4cont hg5cont
+      (by rw [hg1b, hg2a]) (by rw [hg2b, hg3a]) (by rw [hg3b, hg4a]) (by rw [hg4b, hg5a])
+  have hVm1 : V (-1) 1 = -2 := by rw [hVa, hg1a]; exact hs1
+  have hVn1 : V 1 1 = 2 := by rw [hVb, hg5b]; exact hn1
+  have hVE : ∀ t ∈ Icc (-1 : ℝ) 1, V t 0 ∈ Icc (-1 : ℝ) 1 ∧ V t 1 ∈ Icc (-2 : ℝ) 2 := by
+    intro t ht
+    rcases hVmem t ht with ((((h | h) | h) | h) | h)
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]; exact hGloSub (hg1mem u hu)
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]; exact hαE _ (hg2arg u hu)
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]
+      have hy := hg3y u hu; rw [mem_Icc] at hy
+      exact ⟨by rw [hg3x u, mem_Icc]; constructor <;> norm_num,
+        mem_Icc.2 ⟨le_trans hz₀1lb hy.1, le_trans hy.2 hmE1.2⟩⟩
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]; exact hErect (hJnr (hg4mem u hu).1)
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]
+      have hy := hg5y u hu; rw [mem_Icc] at hy
+      exact ⟨by rw [hg5x u, mem_Icc]; constructor <;> norm_num,
+        mem_Icc.2 ⟨le_trans hlE1.1 hy.1, hy.2⟩⟩
+  have hVmiss : ∀ t ∈ Icc (-1 : ℝ) 1, V t ∉ J_s := by
+    intro t ht hJs
+    rcases hVmem t ht with ((((h | h) | h) | h) | h)
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue] at hJs
+      exact hGloAvoid (g1 u) (hg1mem u hu) (hJsr hJs)
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue] at hJs
+      exact hαnr (tw * (1 - u) / 2) (hJsr hJs)
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue] at hJs; exact hg3missJs u hu hJs
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue] at hJs; exact hg4missJs u hu hJs
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue] at hJs; exact hg5missJs u hu hJs
+  have hJsrect : J_s ⊆ {w : Plane | w 0 ∈ Icc (-1 : ℝ) 1 ∧ w 1 ∈ Icc (-2 : ℝ) 2} :=
+    fun w hw => hErect (hJsr hw)
+  obtain ⟨t, ht, htJs⟩ := vertical_meets_arc hbr hJsrect hJoinJs hVcont hVE hVm1 hVn1
+  exact hVmiss t ht htJs
+
+/-- **Step A, high exit case.** Mirror of `step_A_case_low`: if the escaping path first
+meets the frontier above the axis, the concatenation `s → q → p → z₀ → w → n` misses
+`J_n`, contradicting `vertical_meets_arc`. -/
+private lemma step_A_case_high (hbr : BrouwerFPT)
+    {r : sphere (0 : Plane) 1 → Plane}
+    (hm : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r) (hp : (!₂[(1 : ℝ), 0] : Plane) ∈ range r)
+    (hfar : ∀ z ∈ range r, ∀ w ∈ range r, dist z w ≤ 2)
+    {J_n J_s : Set Plane} (hJnr : J_n ⊆ range r) (hJsr : J_s ⊆ range r)
+    (hInter : J_n ∩ J_s = ({!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]} : Set Plane))
+    (hpcJs : IsPathConnected (J_s \ ({!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]} : Set Plane)))
+    (hJoinJn : JoinedIn J_n (!₂[(-1 : ℝ), 0]) (!₂[(1 : ℝ), 0]))
+    {m p q z₀ : Plane}
+    (hmJn : m ∈ J_n) (_hm0 : m 0 = 0) (hmmin : ∀ w ∈ J_n, w 0 = 0 → m 1 ≤ w 1)
+    (hpJs : p ∈ J_s) (hp0 : p 0 = 0)
+    (hqJs : q ∈ J_s) (hq0 : q 0 = 0) (hqm : q 1 < m 1)
+    (hz₀0 : z₀ 0 = 0) (hpz₀_lt : p 1 < z₀ 1) (hz₀m_lt : z₀ 1 < m 1)
+    {α : ℝ → Plane} {tw : ℝ}
+    (hαcont : Continuous α) (hα0 : α 0 = z₀) (htw0 : 0 ≤ tw)
+    (hαE : ∀ u ∈ Icc (0 : ℝ) tw, α u 0 ∈ Icc (-1 : ℝ) 1 ∧ α u 1 ∈ Icc (-2 : ℝ) 2)
+    (hαnr : ∀ u : ℝ, α u ∉ range r)
+    (hwC0 : α tw 0 ∈ Icc (-1 : ℝ) 1) (hwC1 : α tw 1 ∈ Icc (-2 : ℝ) 2)
+    (hwO : ¬(α tw 0 ∈ Ioo (-1 : ℝ) 1 ∧ α tw 1 ∈ Ioo (-2 : ℝ) 2))
+    (hwpos : 0 < α tw 1) : False := by
+  have hErect := normalized_subset_rectangle hm hp hfar
+  have hqab := off_axis_endpoints hq0
+  have hpab := off_axis_endpoints hp0
+  have hqE1 : q 1 ∈ Icc (-2 : ℝ) 2 := (hErect (hJsr hqJs)).2
+  have hpE1 : p 1 ∈ Icc (-2 : ℝ) 2 := (hErect (hJsr hpJs)).2
+  have hmE1 : m 1 ∈ Icc (-2 : ℝ) 2 := (hErect (hJnr hmJn)).2
+  have hz₀1ub : z₀ 1 ≤ 2 := le_of_lt (lt_of_lt_of_le hz₀m_lt hmE1.2)
+  have hs1 : (!₂[(0 : ℝ), -2] : Plane) 1 = -2 := by simp
+  have hn1 : (!₂[(0 : ℝ), 2] : Plane) 1 = 2 := by simp
+  obtain ⟨Gup, hGupPC, hnGup, hGupSub, hGupAvoid, hGupFrom⟩ :=
+    exists_upper_boundary_path hm hp hfar
+  have hwGup : α tw ∈ Gup := hGupFrom (α tw) hwC0 hwC1 hwO hwpos
+  -- piece 1 : `s → q` along the axis
+  obtain ⟨g1, hg1cont, hg1a, hg1b, hg1x, hg1y⟩ :=
+    axis_segment (show (!₂[(0 : ℝ), -2] : Plane) 0 = 0 from by simp) hq0 (by rw [hs1]; exact hqE1.1)
+  have hg1missJn : ∀ t ∈ Icc (-1 : ℝ) 1, g1 t ∉ J_n := by
+    intro t ht hJn
+    have hy := hg1y t ht; rw [mem_Icc] at hy
+    have := hmmin (g1 t) hJn (hg1x t)
+    linarith [hy.2, hqm]
+  -- piece 2 : `q → p` inside `J_s \ {a,b}`
+  obtain ⟨g2, hg2cont, hg2a, hg2b, hg2mem⟩ :=
+    arc_path (hpcJs.joinedIn q ⟨hqJs, hqab⟩ p ⟨hpJs, hpab⟩)
+  have hg2missJn : ∀ t ∈ Icc (-1 : ℝ) 1, g2 t ∉ J_n := by
+    intro t ht hJn
+    have hg2' := hg2mem t ht
+    have hmem : g2 t ∈ J_n ∩ J_s := ⟨hJn, hg2'.1⟩
+    rw [hInter] at hmem; exact hg2'.2 hmem
+  -- piece 3 : `p → z₀` along the axis
+  obtain ⟨g3, hg3cont, hg3a, hg3b, hg3x, hg3y⟩ := axis_segment hp0 hz₀0 hpz₀_lt.le
+  have hg3missJn : ∀ t ∈ Icc (-1 : ℝ) 1, g3 t ∉ J_n := by
+    intro t ht hJn
+    have hy := hg3y t ht; rw [mem_Icc] at hy
+    have := hmmin (g3 t) hJn (hg3x t)
+    linarith [hy.2, hz₀m_lt]
+  -- piece 4 : `z₀ → w` along the escaping path
+  set g4 : ℝ → Plane := fun t => α (tw * (t + 1) / 2) with hg4def
+  have hg4cont : ContinuousOn g4 (Icc (-1 : ℝ) 1) :=
+    (hαcont.comp (by fun_prop : Continuous fun t : ℝ => tw * (t + 1) / 2)).continuousOn
+  have hg4a : g4 (-1) = z₀ := by
+    change α (tw * ((-1) + 1) / 2) = z₀; rw [show tw * ((-1) + 1) / 2 = (0 : ℝ) from by ring, hα0]
+  have hg4b : g4 1 = α tw := by change α (tw * (1 + 1) / 2) = α tw; congr 1; ring
+  have hg4arg : ∀ t ∈ Icc (-1 : ℝ) 1, tw * (t + 1) / 2 ∈ Icc (0 : ℝ) tw := by
+    intro t ht; rw [mem_Icc] at ht
+    rw [mul_div_assoc]
+    exact mul_unit_mem_Icc htw0 (by linarith [ht.1]) (by linarith [ht.2])
+  -- piece 5 : `w → n` inside `Gup`
+  obtain ⟨g5, hg5cont, hg5a, hg5b, hg5mem⟩ :=
+    arc_path (hGupPC.joinedIn (α tw) hwGup (!₂[(0 : ℝ), 2]) hnGup)
+  have hg5missJn : ∀ t ∈ Icc (-1 : ℝ) 1, g5 t ∉ J_n := by
+    intro t ht hJn; exact hGupAvoid (g5 t) (hg5mem t ht) (hJnr hJn)
+  -- concatenate `s → q → p → z₀ → w → n`
+  obtain ⟨V, hVcont, hVa, hVb, hVmem⟩ :=
+    concatPath5 hg1cont hg2cont hg3cont hg4cont hg5cont
+      (by rw [hg1b, hg2a]) (by rw [hg2b, hg3a]) (by rw [hg3b, hg4a]) (by rw [hg4b, hg5a])
+  have hVm1 : V (-1) 1 = -2 := by rw [hVa, hg1a]; exact hs1
+  have hVn1 : V 1 1 = 2 := by rw [hVb, hg5b]; exact hn1
+  have hVE : ∀ t ∈ Icc (-1 : ℝ) 1, V t 0 ∈ Icc (-1 : ℝ) 1 ∧ V t 1 ∈ Icc (-2 : ℝ) 2 := by
+    intro t ht
+    rcases hVmem t ht with ((((h | h) | h) | h) | h)
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]
+      have hy := hg1y u hu; rw [mem_Icc] at hy
+      exact ⟨by rw [hg1x u, mem_Icc]; constructor <;> norm_num,
+        mem_Icc.2 ⟨hy.1, le_trans hy.2 hqE1.2⟩⟩
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]; exact hErect (hJsr (hg2mem u hu).1)
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]
+      have hy := hg3y u hu; rw [mem_Icc] at hy
+      exact ⟨by rw [hg3x u, mem_Icc]; constructor <;> norm_num,
+        mem_Icc.2 ⟨le_trans hpE1.1 hy.1, le_trans hy.2 hz₀1ub⟩⟩
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]; exact hαE _ (hg4arg u hu)
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]; exact hGupSub (hg5mem u hu)
+  have hVmiss : ∀ t ∈ Icc (-1 : ℝ) 1, V t ∉ J_n := by
+    intro t ht hJn
+    rcases hVmem t ht with ((((h | h) | h) | h) | h)
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue] at hJn; exact hg1missJn u hu hJn
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue] at hJn; exact hg2missJn u hu hJn
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue] at hJn; exact hg3missJn u hu hJn
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue] at hJn
+      exact hαnr (tw * (u + 1) / 2) (hJnr hJn)
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue] at hJn; exact hg5missJn u hu hJn
+  have hJnrect : J_n ⊆ {w : Plane | w 0 ∈ Icc (-1 : ℝ) 1 ∧ w 1 ∈ Icc (-2 : ℝ) 2} :=
+    fun w hw => hErect (hJnr hw)
+  obtain ⟨t, ht, htJn⟩ := vertical_meets_arc hbr hJnrect hJoinJn hVcont hVE hVm1 hVn1
+  exact hVmiss t ht htJn
+
+/-- **Maehara Step A (normalized).** The geometric core in normalized coordinates:
+the farthest pair sits at `!₂[-1,0]`, `!₂[1,0]` (so the diameter is `2`). -/
+theorem step_A_normalized (hbr : BrouwerFPT)
+    {r : sphere (0 : Plane) 1 → Plane} (hcont : Continuous r) (hinj : Injective r)
+    (hm : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r) (hp : (!₂[(1 : ℝ), 0] : Plane) ∈ range r)
+    (hfar : ∀ z ∈ range r, ∀ w ∈ range r, dist z w ≤ 2) :
+    ∃ x ∈ (range r)ᶜ, IsBounded (connectedComponentIn (range r)ᶜ x) := by
+  classical
+  obtain ⟨J_n, J_s, l, m, p, q, z₀, hUnion, hInter, hpcJn,
+    ⟨hlJn, hl0, hlmax⟩, ⟨hmJn, hm0, hmmin⟩, ⟨hpJs, hp0, hpm, hpmax⟩,
+    ⟨hqJs, hq0, hqmin⟩, hz₀def, hml, hqp, hpz₀, hz₀m,
+    hpcJs, hJoinJn, hJoinJs⟩ := exists_construction_points hbr hcont hinj hm hp hfar
+  -- basic containments, coordinates, and rectangle bounds
+  have hJnr : J_n ⊆ range r := hUnion ▸ subset_union_left
+  have hJsr : J_s ⊆ range r := hUnion ▸ subset_union_right
+  have hErect := normalized_subset_rectangle hm hp hfar
+  have hz₀0 : z₀ 0 = 0 := by rw [hz₀def]; simp
+  have hz₀1e : z₀ 1 = (m 1 + p 1) / 2 := by rw [hz₀def]; simp
+  have hmE1 : m 1 ∈ Icc (-2 : ℝ) 2 := (hErect (hJnr hmJn)).2
+  have hpE1 : p 1 ∈ Icc (-2 : ℝ) 2 := (hErect (hJsr hpJs)).2
+  -- `p 1 < m 1` (else `p = m` would lie in `J_n ∩ J_s = {a,b}` off the axis)
+  have hpm_lt : p 1 < m 1 := by
+    rcases lt_or_eq_of_le hpm with h | h
+    · exact h
+    · exfalso
+      have hpm_eq : p = m := by ext i; fin_cases i <;> simp [hp0, hm0, h]
+      have hmem : m ∈ J_n ∩ J_s := ⟨hmJn, hpm_eq ▸ hpJs⟩
+      rw [hInter] at hmem
+      rcases hmem with h' | h' <;>
+        (have hc := congrArg (fun x : Plane => x 0) h'; simp only [hm0] at hc; norm_num at hc)
+  have hqm : q 1 < m 1 := lt_of_le_of_lt hqp hpm_lt
+  have hpz₀_lt : p 1 < z₀ 1 := by rw [hz₀1e]; linarith
+  have hz₀m_lt : z₀ 1 < m 1 := by rw [hz₀1e]; linarith
+  -- **A.0** : `z₀ ∉ range r`
+  have hz₀c : z₀ ∈ (range r)ᶜ := by
+    intro hz₀r
+    rw [← hUnion] at hz₀r
+    rcases hz₀r with hJn | hJs
+    · exact absurd (hmmin z₀ hJn hz₀0) (not_le.2 hz₀m_lt)
+    · exact absurd (hpmax z₀ hJs hz₀0 hz₀m_lt.le) (not_le.2 hpz₀_lt)
+  refine ⟨z₀, hz₀c, ?_⟩
+  -- the rectangle `E`, its open interior `O`
+  set E : Set Plane := {p : Plane | p 0 ∈ Icc (-1 : ℝ) 1 ∧ p 1 ∈ Icc (-2 : ℝ) 2} with hEdef
+  set O : Set Plane := {p : Plane | p 0 ∈ Ioo (-1 : ℝ) 1 ∧ p 1 ∈ Ioo (-2 : ℝ) 2} with hOdef
+  have hEbdd : IsBounded E := by rw [hEdef]; exact isBounded_rectE
+  have hOopen : IsOpen O := by rw [hOdef]; exact isOpen_rectO
+  have hEclosed : IsClosed E := by rw [hEdef]; exact isClosed_rectE
+  have hOE : O ⊆ E := fun w hw => ⟨Ioo_subset_Icc_self hw.1, Ioo_subset_Icc_self hw.2⟩
+  have hz₀O : z₀ ∈ O := by
+    refine ⟨?_, ?_⟩
+    · rw [mem_Ioo, hz₀0]; constructor <;> norm_num
+    · rw [mem_Ioo, hz₀1e]; exact ⟨by linarith [hpE1.1, hpm_lt], by linarith [hmE1.2, hpm_lt]⟩
+  -- assume the component is unbounded and derive a contradiction
+  by_contra hUunb
+  set U : Set Plane := connectedComponentIn (range r)ᶜ z₀ with hUeq
+  have hUsub : U ⊆ (range r)ᶜ := connectedComponentIn_subset _ _
+  have hz₀U : z₀ ∈ U := mem_connectedComponentIn hz₀c
+  have hUpc : IsPathConnected U := isPathConnected_component r hcont hz₀c
+  have hUnotsub : ¬ U ⊆ E := fun h => hUunb (hEbdd.subset h)
+  obtain ⟨y, hyU, hyE⟩ := Set.not_subset.mp hUnotsub
+  -- reparametrized escaping path `α : ℝ → Plane`
+  have hJoinU : JoinedIn U z₀ y := hUpc.joinedIn z₀ hz₀U y hyU
+  set α : ℝ → Plane := fun t => hJoinU.somePath (Set.projIcc (0 : ℝ) 1 (by norm_num) t) with hαdef
+  have hαcont : Continuous α := hJoinU.somePath.continuous.comp continuous_projIcc
+  have hα0 : α 0 = z₀ := by
+    change hJoinU.somePath _ = z₀
+    rw [show Set.projIcc (0 : ℝ) 1 (by norm_num) 0 = 0 from Subtype.ext
+      (by rw [Set.coe_projIcc]; norm_num)]
+    exact hJoinU.somePath.source
+  have hα1 : α 1 = y := by
+    change hJoinU.somePath _ = y
+    rw [show Set.projIcc (0 : ℝ) 1 (by norm_num) 1 = 1 from Subtype.ext
+      (by rw [Set.coe_projIcc]; norm_num)]
+    exact hJoinU.somePath.target
+  have hαU : ∀ t : ℝ, α t ∈ U := fun t => hJoinU.somePath_mem _
+  -- first exit through the frontier `E \ O`
+  obtain ⟨tw, htw, hwC, hwO, hbelow⟩ :=
+    exists_first_exit hOopen hEclosed hOE hαcont.continuousOn (by rw [hα0]; exact hz₀O)
+      (by rw [hα1]; exact hyE)
+  have hαE : ∀ u ∈ Icc (0 : ℝ) tw, α u ∈ E := by
+    intro u hu
+    rcases eq_or_lt_of_le hu.2 with h | h
+    · rw [h]; exact hwC
+    · exact hOE (hbelow u ⟨hu.1, h⟩)
+  have hwnr : α tw ∉ range r := hUsub (hαU tw)
+  -- `α tw` is off the axis
+  have hw1ne : α tw 1 ≠ 0 := fun hzero =>
+    hwnr (axis_zero_mem_range hm hp hwC.1 hwO hzero)
+  rcases lt_or_gt_of_ne hw1ne with hwneg | hwpos
+  · exact step_A_case_low hbr hm hp hfar hJnr hJsr hInter hpcJn hJoinJs
+      hlJn hl0 hlmax hmJn hm0 hpJs hpmax hz₀0 hpz₀_lt hz₀m_lt
+      hαcont hα0 htw.1.le (fun u hu => hαE u hu) (fun u => hUsub (hαU u))
+      hwC.1 hwC.2 hwO hwneg
+  · exact step_A_case_high hbr hm hp hfar hJnr hJsr hInter hpcJs hJoinJn
+      hmJn hm0 hmmin hpJs hp0 hqJs hq0 hqm hz₀0 hpz₀_lt hz₀m_lt
+      hαcont hα0 htw.1.le (fun u hu => hαE u hu) (fun u => hUsub (hαU u))
+      hwC.1 hwC.2 hwO hwpos
+
+/-- **Maehara Step A (geometric core).** The complement `ℝ²∖J` of a Jordan curve
+has at least one *bounded* connected component.  Reduced to the normalized version
+`step_A_normalized` via the farthest-pair similarity `T`. -/
+theorem step_A_exists_bounded (hbr : BrouwerFPT)
+    {r : sphere (0 : Plane) 1 → Plane} (hcont : Continuous r) (hinj : Injective r) :
+    ∃ x ∈ (range r)ᶜ, IsBounded (connectedComponentIn (range r)ᶜ x) := by
+  obtain ⟨a, ha, b, hb, hfar, hab⟩ := exists_farthest_pair hcont hinj
+  obtain ⟨T, hscale, hTa, hTb⟩ := exists_similarity hab
+  have hdpos : 0 < dist a b := dist_pos.mpr hab
+  have hcpos : (0 : ℝ) < 2 / dist a b := by positivity
+  -- the transported curve `r' = T ∘ r`
+  set r' : sphere (0 : Plane) 1 → Plane := fun p => T (r p) with hr'
+  have hr'cont : Continuous r' := T.continuous.comp hcont
+  have hr'inj : Injective r' := T.injective.comp hinj
+  have hrange : range r' = T '' range r := by
+    rw [hr', show (fun p => T (r p)) = (T : Plane → Plane) ∘ r from rfl, Set.range_comp]
+  -- normalized hypotheses for `r'`
+  have hm' : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r' := by
+    rw [hrange, ← hTa]; exact mem_image_of_mem _ ha
+  have hp' : (!₂[(1 : ℝ), 0] : Plane) ∈ range r' := by
+    rw [hrange, ← hTb]; exact mem_image_of_mem _ hb
+  have hfar' : ∀ z ∈ range r', ∀ w ∈ range r', dist z w ≤ 2 := by
+    rw [hrange]
+    rintro _ ⟨z, hz, rfl⟩ _ ⟨w, hw, rfl⟩
+    rw [hscale]
+    calc 2 / dist a b * dist z w
+        ≤ 2 / dist a b * dist a b := by
+          exact mul_le_mul_of_nonneg_left (hfar z hz w hw) hcpos.le
+      _ = 2 := by field_simp
+  obtain ⟨x, hx, hxb⟩ := step_A_normalized hbr hr'cont hr'inj hm' hp' hfar'
+  -- transport the bounded component back by `T⁻¹`
+  rw [hrange] at hx hxb
+  -- `x ∈ (T '' range r)ᶜ = T '' (range r)ᶜ`
+  have himg : (T '' range r)ᶜ = T '' (range r)ᶜ :=
+    (Set.image_compl_eq (f := (T : Plane → Plane)) T.bijective).symm
+  rw [himg] at hx hxb
+  obtain ⟨x₀, hx₀, rfl⟩ := hx
+  refine ⟨x₀, hx₀, ?_⟩
+  -- component of `x = T x₀` corresponds to component of `x₀` under `T`
+  have hcc : connectedComponentIn ((T : Plane → Plane) '' (range r)ᶜ) (T x₀)
+      = T '' connectedComponentIn (range r)ᶜ x₀ :=
+    (T.image_connectedComponentIn hx₀).symm
+  rw [hcc] at hxb
+  exact (isBounded_image_scaling hcpos hscale _).mp hxb
+
+/-- The Euclidean norm of an axis point `!₂[0,y]` equals `|y|`. -/
+private lemma norm_axis_pt (y : ℝ) : ‖(!₂[(0 : ℝ), y] : Plane)‖ = |y| := by
+  have e0 : (!₂[(0 : ℝ), y] : Plane) 0 = 0 := by simp
+  have e1 : (!₂[(0 : ℝ), y] : Plane) 1 = y := by simp
+  rw [EuclideanSpace.norm_eq, Fin.sum_univ_two, e0, e1]
+  simp [Real.sqrt_sq_eq_abs]
+
+/-- The upward axis ray `{p | p 0 = 0 ∧ c < p 1}` is unbounded. -/
+private lemma axis_ray_up_unbounded (c : ℝ) :
+    ¬ IsBounded {p : Plane | p 0 = 0 ∧ p 1 ∈ Ioi c} := by
+  intro hb
+  obtain ⟨M, hM⟩ := hb.subset_closedBall (0 : Plane)
+  have hy1 : (!₂[(0 : ℝ), |c| + |M| + 1] : Plane) 1 = |c| + |M| + 1 := by simp
+  have hmem : (!₂[(0 : ℝ), |c| + |M| + 1] : Plane) ∈ {p : Plane | p 0 = 0 ∧ p 1 ∈ Ioi c} := by
+    refine ⟨by simp, ?_⟩
+    rw [mem_Ioi, hy1]; linarith [le_abs_self c, abs_nonneg M]
+  have hin := hM hmem
+  rw [mem_closedBall_zero_iff, norm_axis_pt, abs_of_nonneg (by positivity : (0 : ℝ) ≤ |c| + |M| +
+    1)] at hin
+  linarith [le_abs_self M, abs_nonneg c]
+
+/-- The downward axis ray `{p | p 0 = 0 ∧ p 1 < c}` is unbounded. -/
+private lemma axis_ray_down_unbounded (c : ℝ) :
+    ¬ IsBounded {p : Plane | p 0 = 0 ∧ p 1 ∈ Iio c} := by
+  intro hb
+  obtain ⟨M, hM⟩ := hb.subset_closedBall (0 : Plane)
+  have hy1 : (!₂[(0 : ℝ), -(|c| + |M| + 1)] : Plane) 1 = -(|c| + |M| + 1) := by simp
+  have hmem : (!₂[(0 : ℝ), -(|c| + |M| + 1)] : Plane) ∈ {p : Plane | p 0 = 0 ∧ p 1 ∈ Iio c} := by
+    refine ⟨by simp, ?_⟩
+    rw [mem_Iio, hy1]; linarith [neg_abs_le c, abs_nonneg M]
+  have hin := hM hmem
+  rw [mem_closedBall_zero_iff, norm_axis_pt, abs_neg,
+    abs_of_nonneg (by positivity : (0 : ℝ) ≤ |c| + |M| + 1)] at hin
+  linarith [le_abs_self M, abs_nonneg c]
+
+/-- The complement of the normalized rectangle `[-1,1]×[-2,2]` is preconnected
+(it is the union of four half-planes chained around the rectangle). -/
+private lemma isPreconnected_rect_compl :
+    IsPreconnected {v : Plane | ¬(v 0 ∈ Icc (-1 : ℝ) 1 ∧ v 1 ∈ Icc (-2 : ℝ) 2)} := by
+  have cA : Convex ℝ {v : Plane | v 0 ∈ Iio (-1 : ℝ)} :=
+    (convex_Iio _).is_linear_preimage (EuclideanSpace.proj (0:Fin 2)).isLinear
+  have cB : Convex ℝ {v : Plane | v 0 ∈ Ioi (1 : ℝ)} :=
+    (convex_Ioi _).is_linear_preimage (EuclideanSpace.proj (0:Fin 2)).isLinear
+  have cC : Convex ℝ {v : Plane | v 1 ∈ Iio (-2 : ℝ)} :=
+    (convex_Iio _).is_linear_preimage (EuclideanSpace.proj (1:Fin 2)).isLinear
+  have cD : Convex ℝ {v : Plane | v 1 ∈ Ioi (2 : ℝ)} :=
+    (convex_Ioi _).is_linear_preimage (EuclideanSpace.proj (1:Fin 2)).isLinear
+  -- membership witnesses at the four "corners"
+  have mAD_A : (!₂[(-2 : ℝ), 3] : Plane) ∈ {v : Plane | v 0 ∈ Iio (-1 : ℝ)} := by
+    simp only [Set.mem_ofPred_eq, mem_Iio]; norm_num
+  have mAD_D : (!₂[(-2 : ℝ), 3] : Plane) ∈ {v : Plane | v 1 ∈ Ioi (2 : ℝ)} := by
+    simp only [Set.mem_ofPred_eq, mem_Ioi]; norm_num
+  have mB_D : (!₂[(2 : ℝ), 3] : Plane) ∈ {v : Plane | v 1 ∈ Ioi (2 : ℝ)} := by
+    simp only [Set.mem_ofPred_eq, mem_Ioi]; norm_num
+  have mB_B : (!₂[(2 : ℝ), 3] : Plane) ∈ {v : Plane | v 0 ∈ Ioi (1 : ℝ)} := by
+    simp only [Set.mem_ofPred_eq, mem_Ioi]; norm_num
+  have mC_B : (!₂[(2 : ℝ), -3] : Plane) ∈ {v : Plane | v 0 ∈ Ioi (1 : ℝ)} := by
+    simp only [Set.mem_ofPred_eq, mem_Ioi]; norm_num
+  have mC_C : (!₂[(2 : ℝ), -3] : Plane) ∈ {v : Plane | v 1 ∈ Iio (-2 : ℝ)} := by
+    simp only [Set.mem_ofPred_eq, mem_Iio]; norm_num
+  have hAD : IsPreconnected ({v : Plane | v 0 ∈ Iio (-1 : ℝ)} ∪ {v : Plane | v 1 ∈ Ioi (2 : ℝ)}) :=
+    cA.isPreconnected.union _ mAD_A mAD_D cD.isPreconnected
+  have hADB : IsPreconnected (({v : Plane | v 0 ∈ Iio (-1 : ℝ)} ∪ {v : Plane | v 1 ∈ Ioi (2 : ℝ)})
+      ∪ {v : Plane | v 0 ∈ Ioi (1 : ℝ)}) :=
+    hAD.union _ (Or.inr mB_D) mB_B cB.isPreconnected
+  have hADBC : IsPreconnected ((({v : Plane | v 0 ∈ Iio (-1 : ℝ)} ∪ {v : Plane | v 1 ∈ Ioi (2 : ℝ)})
+      ∪ {v : Plane | v 0 ∈ Ioi (1 : ℝ)}) ∪ {v : Plane | v 1 ∈ Iio (-2 : ℝ)}) :=
+    hADB.union _ (Or.inr mC_B) mC_C cC.isPreconnected
+  have hEq : {v : Plane | ¬(v 0 ∈ Icc (-1 : ℝ) 1 ∧ v 1 ∈ Icc (-2 : ℝ) 2)}
+      = ((({v : Plane | v 0 ∈ Iio (-1 : ℝ)} ∪ {v : Plane | v 1 ∈ Ioi (2 : ℝ)})
+      ∪ {v : Plane | v 0 ∈ Ioi (1 : ℝ)}) ∪ {v : Plane | v 1 ∈ Iio (-2 : ℝ)}) := by
+    ext v
+    simp only [mem_ofPred_eq, mem_union, mem_Icc, mem_Iio, mem_Ioi, not_and_or, not_le]
+    tauto
+  rw [hEq]; exact hADBC
+
+/-- Straight segment from `P` to `Q`, reparametrized on `[-1,1]`. -/
+private noncomputable def segP (P Q : Plane) : ℝ → Plane := fun t => P + ((t + 1) / 2) • (Q - P)
+
+private lemma segP_cont (P Q : Plane) : Continuous (segP P Q) := by unfold segP; fun_prop
+
+private lemma segP_neg1 (P Q : Plane) : segP P Q (-1) = P := by simp [segP]
+
+private lemma segP_one (P Q : Plane) : segP P Q 1 = Q := by simp [segP]
+
+private lemma segP_coord (P Q : Plane) (t : ℝ) (i : Fin 2) :
+    segP P Q t i = P i + ((t + 1) / 2) * (Q i - P i) := by
+  simp only [segP, PiLp.add_apply, PiLp.smul_apply, PiLp.sub_apply, smul_eq_mul]
+
+private lemma segP_dist_le (P Q : Plane) {t : ℝ} (ht : t ∈ Icc (-1 : ℝ) 1) :
+    dist (segP P Q t) P ≤ dist Q P := by
+  have hs : |(t + 1) / 2| ≤ 1 := by
+    rw [mem_Icc] at ht; rw [abs_le]; constructor <;> linarith [ht.1, ht.2]
+  have heq : dist (segP P Q t) P = |(t + 1) / 2| * dist Q P := by
+    rw [segP, dist_eq_norm, add_sub_cancel_left, norm_smul, Real.norm_eq_abs, ← dist_eq_norm]
+  rw [heq]
+  calc |(t + 1) / 2| * dist Q P ≤ 1 * dist Q P :=
+        mul_le_mul_of_nonneg_right hs dist_nonneg
+    _ = dist Q P := one_mul _
+
+private lemma segP_dist_le' (P Q : Plane) {t : ℝ} (ht : t ∈ Icc (-1 : ℝ) 1) :
+    dist (segP P Q t) Q ≤ dist P Q := by
+  have hs : |1 - (t + 1) / 2| ≤ 1 := by
+    rw [mem_Icc] at ht; rw [abs_le]; constructor <;> linarith [ht.1, ht.2]
+  have heq : dist (segP P Q t) Q = |1 - (t + 1) / 2| * dist P Q := by
+    rw [segP, dist_eq_norm,
+      show P + ((t + 1) / 2) • (Q - P) - Q = (1 - (t + 1) / 2) • (P - Q) from by module,
+      norm_smul, Real.norm_eq_abs, ← dist_eq_norm]
+  rw [heq]
+  calc |1 - (t + 1) / 2| * dist P Q ≤ 1 * dist P Q :=
+        mul_le_mul_of_nonneg_right hs dist_nonneg
+    _ = dist P Q := one_mul _
+
+/-- A coordinate of a segment point lies in any interval containing both endpoints'
+corresponding coordinates. -/
+private lemma segP_coord_mem_Icc {P Q : Plane} (i : Fin 2) {lo hi : ℝ}
+    (hP : P i ∈ Icc lo hi) (hQ : Q i ∈ Icc lo hi) {t : ℝ} (ht : t ∈ Icc (-1 : ℝ) 1) :
+    segP P Q t i ∈ Icc lo hi := by
+  rw [mem_Icc] at hP hQ ht
+  rw [segP_coord, mem_Icc]
+  have hs0 : (0 : ℝ) ≤ (t + 1) / 2 := by linarith [ht.1]
+  have hs1 : (t + 1) / 2 ≤ 1 := by linarith [ht.2]
+  constructor
+  · nlinarith [mul_nonneg hs0 (by linarith [hQ.1] : (0 : ℝ) ≤ Q i - lo),
+      mul_nonneg (by linarith [hs1] : (0 : ℝ) ≤ 1 - (t + 1) / 2) (by linarith [hP.1] : (0 : ℝ) ≤ P
+        i -
+        lo)]
+  · nlinarith [mul_nonneg hs0 (by linarith [hQ.2] : (0 : ℝ) ≤ hi - Q i),
+      mul_nonneg (by linarith [hs1] : (0 : ℝ) ≤ 1 - (t + 1) / 2) (by linarith [hP.2] : (0 : ℝ) ≤
+        hi -
+        P i)]
+
+/-- **Maehara Step B (normalized).** Uniqueness of the bounded component in
+normalized coordinates (farthest pair at `!₂[-1,0]`, `!₂[1,0]`). Any bounded
+component equals the `z₀`-component `U`, hence all bounded components coincide. -/
+private theorem bounded_component_subset_normalized_rectangle
+    {r : sphere (0 : Plane) 1 → Plane}
+    (hErect : range r ⊆ {p | p 0 ∈ Icc (-1 : ℝ) 1 ∧ p 1 ∈ Icc (-2 : ℝ) 2})
+    (w : Plane) (hwb : IsBounded (connectedComponentIn (range r)ᶜ w)) :
+    connectedComponentIn (range r)ᶜ w ⊆
+      {p | p 0 ∈ Icc (-1 : ℝ) 1 ∧ p 1 ∈ Icc (-2 : ℝ) 2} := by
+  let W := connectedComponentIn (range r)ᶜ w
+  have hbounded : IsBounded W := hwb
+  have compW : ∀ {v : Plane}, v ∈ W → connectedComponentIn (range r)ᶜ v = W := by
+    intro v hv
+    exact (connectedComponentIn_eq hv).symm
+  intro v hvW
+  by_contra hvE
+  change ¬(v 0 ∈ Icc (-1 : ℝ) 1 ∧ v 1 ∈ Icc (-2 : ℝ) 2) at hvE
+  have hvEc : v ∈ {u : Plane | ¬(u 0 ∈ Icc (-1 : ℝ) 1 ∧ u 1 ∈ Icc (-2 : ℝ) 2)} := hvE
+  have hEcc : {u : Plane | ¬(u 0 ∈ Icc (-1 : ℝ) 1 ∧ u 1 ∈ Icc (-2 : ℝ) 2)} ⊆ (range r)ᶜ :=
+    fun u hu hur => hu (hErect hur)
+  have h1 := isPreconnected_rect_compl.subset_connectedComponentIn hvEc hEcc
+  have hup : {p : Plane | p 0 = 0 ∧ p 1 ∈ Ioi (2 : ℝ)}
+      ⊆ {u : Plane | ¬(u 0 ∈ Icc (-1 : ℝ) 1 ∧ u 1 ∈ Icc (-2 : ℝ) 2)} := by
+    rintro u ⟨_, hu1⟩ ⟨_, hu2⟩
+    rw [mem_Icc] at hu2; rw [mem_Ioi] at hu1; linarith [hu2.2]
+  exact axis_ray_up_unbounded 2 (hbounded.subset (subset_trans hup (compW hvW ▸ h1)))
+
+private theorem path_component_crossing_contradiction (hbr : BrouwerFPT)
+    (W : Set Plane) (hWpc : IsPathConnected W)
+    (hWE : W ⊆ {p | p 0 ∈ Icc (-1 : ℝ) 1 ∧ p 1 ∈ Icc (-2 : ℝ) 2})
+    (β : ℝ → Plane) (hβcont : ContinuousOn β (Icc (-1 : ℝ) 1))
+    (hβE : ∀ t ∈ Icc (-1 : ℝ) 1, β t 0 ∈ Icc (-1 : ℝ) 1 ∧ β t 1 ∈ Icc (-2 : ℝ) 2)
+    (hβa : β (-1) = !₂[(0 : ℝ), 2]) (hβb : β 1 = !₂[(0 : ℝ), -2])
+    (hβmiss : ∀ t ∈ Icc (-1 : ℝ) 1, β t ∉ W)
+    (haclW : (!₂[(-1 : ℝ), 0] : Plane) ∈ closure W)
+    (hbclW : (!₂[(1 : ℝ), 0] : Plane) ∈ closure W)
+    (ha_notβ : (!₂[(-1 : ℝ), 0] : Plane) ∉ β '' Icc (-1 : ℝ) 1)
+    (hb_notβ : (!₂[(1 : ℝ), 0] : Plane) ∉ β '' Icc (-1 : ℝ) 1) : False := by
+  classical
+  let βimg := β '' Icc (-1 : ℝ) 1
+  have hβimg_closed : IsClosed βimg := (isCompact_Icc.image_of_continuousOn hβcont).isClosed
+  have ha0 : (!₂[(-1 : ℝ), 0] : Plane) 0 = -1 := by simp
+  have ha1 : (!₂[(-1 : ℝ), 0] : Plane) 1 = 0 := by simp
+  have hb0 : (!₂[(1 : ℝ), 0] : Plane) 0 = 1 := by simp
+  have hb1 : (!₂[(1 : ℝ), 0] : Plane) 1 = 0 := by simp
+  have hs0 : (!₂[(0 : ℝ), -2] : Plane) 0 = 0 := by simp
+  have hs1 : (!₂[(0 : ℝ), -2] : Plane) 1 = -2 := by simp
+  have hn0 : (!₂[(0 : ℝ), 2] : Plane) 0 = 0 := by simp
+  have hn1 : (!₂[(0 : ℝ), 2] : Plane) 1 = 2 := by simp
+  have hIcc0x : (0 : ℝ) ∈ Icc (-1 : ℝ) 1 := by rw [mem_Icc]; norm_num
+  have hIcc0y : (0 : ℝ) ∈ Icc (-2 : ℝ) 2 := by rw [mem_Icc]; norm_num
+  have hIcc2y : (2 : ℝ) ∈ Icc (-2 : ℝ) 2 := by rw [mem_Icc]; norm_num
+  have hIccm2y : (-2 : ℝ) ∈ Icc (-2 : ℝ) 2 := by rw [mem_Icc]; norm_num
+  have hIccm1x : (-1 : ℝ) ∈ Icc (-1 : ℝ) 1 := by rw [mem_Icc]; norm_num
+  have hIcc1x : (1 : ℝ) ∈ Icc (-1 : ℝ) 1 := by rw [mem_Icc]; norm_num
+  obtain ⟨εa, hεa, hballa⟩ := Metric.isOpen_iff.mp hβimg_closed.isOpen_compl _ ha_notβ
+  obtain ⟨εb, hεb, hballb⟩ := Metric.isOpen_iff.mp hβimg_closed.isOpen_compl _ hb_notβ
+  -- points of `W` near `a, b`
+  rw [Metric.mem_closure_iff] at haclW hbclW
+  obtain ⟨a₁, ha₁W, ha₁d⟩ := haclW εa hεa
+  obtain ⟨b₁, hb₁W, hb₁d⟩ := hbclW εb hεb
+  have ha₁E : a₁ 0 ∈ Icc (-1 : ℝ) 1 ∧ a₁ 1 ∈ Icc (-2 : ℝ) 2 := hWE ha₁W
+  have hb₁E : b₁ 0 ∈ Icc (-1 : ℝ) 1 ∧ b₁ 1 ∈ Icc (-2 : ℝ) 2 := hWE hb₁W
+  -- **path** `H : a → b`, left to right, inside `E`, missing `β`
+  have hHp1_cont : ContinuousOn (segP (!₂[(-1 : ℝ), 0]) a₁) (Icc (-1 : ℝ) 1) := (segP_cont _
+    _).continuousOn
+  obtain ⟨Harc, hHarc_cont, hHarc_a, hHarc_b, hHarc_mem⟩ :=
+    arc_path (hWpc.joinedIn a₁ ha₁W b₁ hb₁W)
+  have hHp3_cont : ContinuousOn (segP b₁ (!₂[(1 : ℝ), 0])) (Icc (-1 : ℝ) 1) := (segP_cont _
+    _).continuousOn
+  obtain ⟨H, hHcont, hHa, hHb, hHmem⟩ :=
+    concatPath3 hHp1_cont hHarc_cont hHp3_cont
+      (by rw [segP_one, hHarc_a]) (by rw [hHarc_b, segP_neg1])
+  have hHa0 : H (-1) 0 = -1 := by rw [hHa, segP_neg1, ha0]
+  have hHb0 : H 1 0 = 1 := by rw [hHb, segP_one, hb0]
+  have hHE : ∀ t ∈ Icc (-1 : ℝ) 1, H t 0 ∈ Icc (-1 : ℝ) 1 ∧ H t 1 ∈ Icc (-2 : ℝ) 2 := by
+    intro t ht
+    rcases hHmem t ht with (h | h) | h
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]
+      exact ⟨segP_coord_mem_Icc 0 (by rw [ha0]; exact hIccm1x) ha₁E.1 hu,
+        segP_coord_mem_Icc 1 (by rw [ha1]; exact hIcc0y) ha₁E.2 hu⟩
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]; exact hWE (hHarc_mem u hu)
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]
+      exact ⟨segP_coord_mem_Icc 0 hb₁E.1 (by rw [hb0]; exact hIcc1x) hu,
+        segP_coord_mem_Icc 1 hb₁E.2 (by rw [hb1]; exact hIcc0y) hu⟩
+  have hHmiss : ∀ t ∈ Icc (-1 : ℝ) 1, H t ∉ βimg := by
+    intro t ht
+    rcases hHmem t ht with (h | h) | h
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]; intro hmemβ
+      have hin : segP (!₂[(-1 : ℝ), 0]) a₁ u ∈ Metric.ball (!₂[(-1 : ℝ), 0] : Plane) εa := by
+        rw [Metric.mem_ball]
+        calc dist (segP (!₂[(-1 : ℝ), 0]) a₁ u) (!₂[(-1 : ℝ), 0])
+              ≤ dist a₁ (!₂[(-1 : ℝ), 0]) := segP_dist_le _ _ hu
+          _ < εa := by rw [dist_comm]; exact ha₁d
+      exact hballa hin hmemβ
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]; intro hmemβ
+      obtain ⟨t', ht', ht'e⟩ := hmemβ
+      exact hβmiss t' ht' (by rw [ht'e]; exact hHarc_mem u hu)
+    · obtain ⟨u, hu, hue⟩ := h; rw [← hue]; intro hmemβ
+      have hin : segP b₁ (!₂[(1 : ℝ), 0]) u ∈ Metric.ball (!₂[(1 : ℝ), 0] : Plane) εb := by
+        rw [Metric.mem_ball]
+        calc dist (segP b₁ (!₂[(1 : ℝ), 0]) u) (!₂[(1 : ℝ), 0])
+              ≤ dist b₁ (!₂[(1 : ℝ), 0]) := segP_dist_le' _ _ hu
+          _ < εb := by rw [dist_comm]; exact hb₁d
+      exact hballb hin hmemβ
+  -- reorient `β` bottom-to-top and cross with `H`
+  set v : ℝ → Plane := fun t => β (-t) with hvdef
+  have hvcont : ContinuousOn v (Icc (-1 : ℝ) 1) :=
+    hβcont.comp continuous_neg.continuousOn (fun t ht => by
+      rw [mem_Icc] at ht ⊢; exact ⟨by linarith [ht.2], by linarith [ht.1]⟩)
+  have hvE : ∀ t ∈ Icc (-1 : ℝ) 1, v t 0 ∈ Icc (-1 : ℝ) 1 ∧ v t 1 ∈ Icc (-2 : ℝ) 2 := by
+    intro t ht; rw [mem_Icc] at ht
+    exact hβE (-t) (by rw [mem_Icc]; exact ⟨by linarith [ht.2], by linarith [ht.1]⟩)
+  have hv1 : v (-1) 1 = -2 := by change β (-(-1)) 1 = -2; rw [neg_neg, hβb, hs1]
+  have hv2 : v 1 1 = 2 := by change β (-1) 1 = 2; rw [hβa, hn1]
+  obtain ⟨s, hs, t, ht, heq⟩ := crossing hbr (by norm_num : (-1 : ℝ) ≤ 1) (by norm_num : (-2 : ℝ)
+    ≤ 2)
+    H v hHcont hvcont hHE hvE hHa0 hHb0 hv1 hv2
+  apply hHmiss s hs
+  rw [heq]
+  exact Set.mem_image_of_mem β (show (-t) ∈ Icc (-1 : ℝ) 1 by
+    rw [mem_Icc] at ht ⊢; exact ⟨by linarith [ht.2], by linarith [ht.1]⟩)
+
+private theorem spine_image_avoids_horizontal_endpoint
+    (l m p q : Plane) (hl0 : l 0 = 0) (hm0 : m 0 = 0) (hp0 : p 0 = 0) (hq0 : q 0 = 0)
+    (lm pq β : ℝ → Plane)
+    (hβmem : ∀ t ∈ Icc (-1 : ℝ) 1,
+      β t ∈ ((((segP !₂[(0 : ℝ), 2] l '' Icc (-1 : ℝ) 1 ∪
+        lm '' Icc (-1 : ℝ) 1) ∪ segP m p '' Icc (-1 : ℝ) 1) ∪
+        pq '' Icc (-1 : ℝ) 1) ∪ segP q !₂[(0 : ℝ), -2] '' Icc (-1 : ℝ) 1))
+    (u : Plane) (hu0 : u 0 ≠ 0)
+    (hLm : u ∉ lm '' Icc (-1 : ℝ) 1) (hPq : u ∉ pq '' Icc (-1 : ℝ) 1) :
+    u ∉ β '' Icc (-1 : ℝ) 1 := by
+  have hseg (a b : Plane) (ha : a 0 = 0) (hb : b 0 = 0) :
+      u ∉ segP a b '' Icc (-1 : ℝ) 1 := by
+    rintro ⟨t, _, htu⟩
+    apply hu0
+    rw [← htu, segP_coord, ha, hb]
+    ring
+  rintro ⟨t, ht, htu⟩
+  have h := hβmem t ht
+  rw [htu] at h
+  rcases h with (((h | h) | h) | h) | h
+  · exact hseg _ _ (by simp) hl0 h
+  · exact hLm h
+  · exact hseg _ _ hm0 hp0 h
+  · exact hPq h
+  · exact hseg _ _ hq0 (by simp) h
+
+private theorem bounded_component_avoids_axis_up
+    (S W : Set Plane) (hWsub : W ⊆ Sᶜ) (hwb : IsBounded W)
+    (compW : ∀ {v : Plane}, v ∈ W → connectedComponentIn Sᶜ v = W)
+    (l : Plane) (hl0 : l 0 = 0) (hlr : l ∈ S)
+    (hRayNc : {v : Plane | v 0 = 0 ∧ v 1 ∈ Ioi (l 1)} ⊆ Sᶜ) :
+    ∀ v : Plane, v 0 = 0 → l 1 ≤ v 1 → v ∉ W := by
+  let RayN : Set Plane := {v : Plane | v 0 = 0 ∧ v 1 ∈ Ioi (l 1)}
+  have hRayNPC : IsPathConnected RayN :=
+    isPathConnected_vertSeg 0 ((convex_Ioi (l 1)).isPathConnected Set.nonempty_Ioi)
+  intro v hv0 hvl hvW
+  rcases eq_or_lt_of_le hvl with h | h
+  · have hveq : v = l := by ext i; fin_cases i <;> simp [hv0, hl0, h]
+    exact hWsub hvW (by rw [hveq]; exact hlr)
+  · have h1 := hRayNPC.isConnected.isPreconnected.subset_connectedComponentIn
+      (show v ∈ RayN from ⟨hv0, h⟩) hRayNc
+    exact axis_ray_up_unbounded (l 1) (hwb.subset (compW hvW ▸ h1))
+
+private theorem bounded_component_avoids_axis_down
+    (S W : Set Plane) (hWsub : W ⊆ Sᶜ) (hwb : IsBounded W)
+    (compW : ∀ {v : Plane}, v ∈ W → connectedComponentIn Sᶜ v = W)
+    (q : Plane) (hq0 : q 0 = 0) (hqr : q ∈ S)
+    (hRaySc : {v : Plane | v 0 = 0 ∧ v 1 ∈ Iio (q 1)} ⊆ Sᶜ) :
+    ∀ v : Plane, v 0 = 0 → v 1 ≤ q 1 → v ∉ W := by
+  let RayS : Set Plane := {v : Plane | v 0 = 0 ∧ v 1 ∈ Iio (q 1)}
+  have hRaySPC : IsPathConnected RayS :=
+    isPathConnected_vertSeg 0 ((convex_Iio (q 1)).isPathConnected Set.nonempty_Iio)
+  intro v hv0 hvq hvW
+  rcases eq_or_lt_of_le hvq with h | h
+  · have hveq : v = q := by ext i; fin_cases i <;> simp [hv0, hq0, h]
+    exact hWsub hvW (by rw [hveq]; exact hqr)
+  · have h1 := hRaySPC.isConnected.isPreconnected.subset_connectedComponentIn
+      (show v ∈ RayS from ⟨hv0, h⟩) hRaySc
+    exact axis_ray_down_unbounded (q 1) (hwb.subset (compW hvW ▸ h1))
+
+private theorem disjoint_component_avoids_axis_middle
+    (S W U : Set Plane) (hWsub : W ⊆ Sᶜ) (hWU_disj : Disjoint W U)
+    (p m : Plane) (hp0 : p 0 = 0) (hm0 : m 0 = 0) (hpr : p ∈ S) (hmr : m ∈ S)
+    (hMidU : {v : Plane | v 0 = 0 ∧ v 1 ∈ Ioo (p 1) (m 1)} ⊆ U) :
+    ∀ v : Plane, v 0 = 0 → p 1 ≤ v 1 → v 1 ≤ m 1 → v ∉ W := by
+  intro v hv0 hvp hvm hvW
+  rcases eq_or_lt_of_le hvp with h | h
+  · have hveq : v = p := by ext i; fin_cases i <;> simp [hv0, hp0, h]
+    exact hWsub hvW (by rw [hveq]; exact hpr)
+  · rcases eq_or_lt_of_le hvm with h' | h'
+    · have hveq : v = m := by ext i; fin_cases i <;> simp [hv0, hm0, h']
+      exact hWsub hvW (by rw [hveq]; exact hmr)
+    · exact (Set.disjoint_left.mp hWU_disj hvW) (hMidU ⟨hv0, ⟨h, h'⟩⟩)
+
+private theorem axis_point_avoids_horizontal_endpoints (p : Plane) (hp : p 0 = 0) :
+    p ∉ ({!₂[(-1 : ℝ), 0], !₂[(1 : ℝ), 0]} : Set Plane) := by
+  simp only [Set.mem_insert_iff, Set.mem_singleton_iff]
+  rintro (h | h) <;>
+    (have hc := congrArg (fun x : Plane => x 0) h; simp only [hp] at hc; norm_num at hc)
+
+theorem step_B_normalized (hbr : BrouwerFPT)
+    {r : sphere (0 : Plane) 1 → Plane} (hcont : Continuous r) (hinj : Injective r)
+    (hm : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r) (hp : (!₂[(1 : ℝ), 0] : Plane) ∈ range r)
+    (hfar : ∀ z ∈ range r, ∀ w ∈ range r, dist z w ≤ 2) :
+    ∀ x ∈ (range r)ᶜ, ∀ y ∈ (range r)ᶜ,
+      IsBounded (connectedComponentIn (range r)ᶜ x) →
+      IsBounded (connectedComponentIn (range r)ᶜ y) →
+      connectedComponentIn (range r)ᶜ x = connectedComponentIn (range r)ᶜ y := by
+  classical
+  obtain ⟨J_n, J_s, l, m, p, q, z₀, hUnion, hInter, hpcJn,
+    ⟨hlJn, hl0, hlmax⟩, ⟨hmJn, hm0, hmmin⟩, ⟨hpJs, hp0, hpm, hpmax⟩,
+    ⟨hqJs, hq0, hqmin⟩, hz₀def, hml, hqp, hpz₀, hz₀m,
+    hpcJs, hJoinJn, hJoinJs⟩ := exists_construction_points hbr hcont hinj hm hp hfar
+  -- basic containments, coordinates, and rectangle bounds
+  have hJnr : J_n ⊆ range r := hUnion ▸ subset_union_left
+  have hJsr : J_s ⊆ range r := hUnion ▸ subset_union_right
+  have hErect := normalized_subset_rectangle hm hp hfar
+  have hs0 : (!₂[(0 : ℝ), -2] : Plane) 0 = 0 := by simp
+  have hs1 : (!₂[(0 : ℝ), -2] : Plane) 1 = -2 := by simp
+  have hn0 : (!₂[(0 : ℝ), 2] : Plane) 0 = 0 := by simp
+  have hn1 : (!₂[(0 : ℝ), 2] : Plane) 1 = 2 := by simp
+  have hz₀0 : z₀ 0 = 0 := by rw [hz₀def]; simp
+  have hz₀1e : z₀ 1 = (m 1 + p 1) / 2 := by rw [hz₀def]; simp
+  have hmr : m ∈ range r := hJnr hmJn
+  have hlr : l ∈ range r := hJnr hlJn
+  have hpr : p ∈ range r := hJsr hpJs
+  have hqr : q ∈ range r := hJsr hqJs
+  have hmE1 : m 1 ∈ Icc (-2 : ℝ) 2 := (hErect hmr).2
+  have hlE1 : l 1 ∈ Icc (-2 : ℝ) 2 := (hErect hlr).2
+  have hpE1 : p 1 ∈ Icc (-2 : ℝ) 2 := (hErect hpr).2
+  have hqE1 : q 1 ∈ Icc (-2 : ℝ) 2 := (hErect hqr).2
+  have hpm_lt : p 1 < m 1 := by
+    rcases lt_or_eq_of_le hpm with h | h
+    · exact h
+    · exfalso
+      have hpm_eq : p = m := by ext i; fin_cases i <;> simp [hp0, hm0, h]
+      have hmem : m ∈ J_n ∩ J_s := ⟨hmJn, hpm_eq ▸ hpJs⟩
+      rw [hInter] at hmem
+      rcases hmem with h' | h' <;>
+        (have hc := congrArg (fun x : Plane => x 0) h'; simp only [hm0] at hc; norm_num at hc)
+  have hpz₀_lt : p 1 < z₀ 1 := by rw [hz₀1e]; linarith
+  have hz₀m_lt : z₀ 1 < m 1 := by rw [hz₀1e]; linarith
+  have hmab := axis_point_avoids_horizontal_endpoints m hm0
+  have hlab := axis_point_avoids_horizontal_endpoints l hl0
+  have hqab := axis_point_avoids_horizontal_endpoints q hq0
+  have hpab := axis_point_avoids_horizontal_endpoints p hp0
+  have hz₀c : z₀ ∈ (range r)ᶜ := by
+    intro hz₀r
+    rw [← hUnion] at hz₀r
+    rcases hz₀r with hJn | hJs
+    · exact absurd (hmmin z₀ hJn hz₀0) (not_le.2 hz₀m_lt)
+    · exact absurd (hpmax z₀ hJs hz₀0 hz₀m_lt.le) (not_le.2 hpz₀_lt)
+  -- interval facts
+  have hIcc0x : (0 : ℝ) ∈ Icc (-1 : ℝ) 1 := by rw [mem_Icc]; norm_num
+  have hIcc2y : (2 : ℝ) ∈ Icc (-2 : ℝ) 2 := by rw [mem_Icc]; norm_num
+  have hIccm2y : (-2 : ℝ) ∈ Icc (-2 : ℝ) 2 := by rw [mem_Icc]; norm_num
+  -- the `z₀`-component `U`, the rectangle `E`
+  set U : Set Plane := connectedComponentIn (range r)ᶜ z₀ with hUdef
+  set E : Set Plane := {p : Plane | p 0 ∈ Icc (-1 : ℝ) 1 ∧ p 1 ∈ Icc (-2 : ℝ) 2} with hEdef
+  -- the open middle axis segment `(p,m)` lies in `U`
+  set Mid : Set Plane := {v : Plane | v 0 = 0 ∧ v 1 ∈ Ioo (p 1) (m 1)} with hMiddef
+  have hMidPC : IsPathConnected Mid :=
+    isPathConnected_vertSeg 0 ((convex_Ioo (p 1) (m 1)).isPathConnected (Set.nonempty_Ioo.mpr
+      hpm_lt))
+  have hMidc : Mid ⊆ (range r)ᶜ := by
+    rintro v ⟨hv0, hv1⟩ hvr
+    rw [← hUnion] at hvr
+    rcases hvr with h | h
+    · exact absurd (hmmin v h hv0) (not_le.2 hv1.2)
+    · exact absurd (hpmax v h hv0 hv1.2.le) (not_le.2 hv1.1)
+  have hz₀Mid : z₀ ∈ Mid :=
+    ⟨hz₀0, by rw [hz₀1e]; exact ⟨by linarith [hpm_lt], by linarith [hpm_lt]⟩⟩
+  have hMidU : Mid ⊆ U :=
+    hMidPC.isConnected.isPreconnected.subset_connectedComponentIn hz₀Mid hMidc
+  -- upward and downward axis rays lie in the complement
+  set RayN : Set Plane := {v : Plane | v 0 = 0 ∧ v 1 ∈ Ioi (l 1)} with hRayNdef
+  have hRayNc : RayN ⊆ (range r)ᶜ := by
+    rintro v ⟨hv0, hv1⟩ hvr; exact absurd (hlmax v hvr hv0) (not_le.2 hv1)
+  set RayS : Set Plane := {v : Plane | v 0 = 0 ∧ v 1 ∈ Iio (q 1)} with hRaySdef
+  have hRaySc : RayS ⊆ (range r)ᶜ := by
+    rintro v ⟨hv0, hv1⟩ hvr
+    rw [← hUnion] at hvr
+    rcases hvr with h | h
+    · exact absurd (le_trans (le_trans hqp hpm) (hmmin v h hv0)) (not_le.2 hv1)
+    · exact absurd (hqmin v h hv0) (not_le.2 hv1)
+  -- **KEY**: every bounded component equals `U`.
+  have KEY : ∀ w ∈ (range r)ᶜ, IsBounded (connectedComponentIn (range r)ᶜ w) →
+      connectedComponentIn (range r)ᶜ w = U := by
+    intro w hw hwb
+    by_contra hWne
+    set W : Set Plane := connectedComponentIn (range r)ᶜ w with hWdef
+    have hWsub : W ⊆ (range r)ᶜ := connectedComponentIn_subset _ _
+    have hWpc : IsPathConnected W := isPathConnected_component r hcont hw
+    -- `W` and `U` are disjoint
+    have hWU_disj : Disjoint W U := by
+      rw [Set.disjoint_left]
+      intro v hvW hvU
+      apply hWne
+      rw [hWdef, connectedComponentIn_eq (hWdef ▸ hvW), hUdef]
+      exact (connectedComponentIn_eq (hUdef ▸ hvU)).symm
+    -- component-equation helper: any `v ∈ W` has component `= W`
+    have compW : ∀ {v : Plane}, v ∈ W → connectedComponentIn (range r)ᶜ v = W := by
+      intro v hvW; rw [hWdef]; exact (connectedComponentIn_eq (hWdef ▸ hvW)).symm
+    -- avoid lemmas
+    have avoid_up := bounded_component_avoids_axis_up
+      (range r) W hWsub hwb compW l hl0 hlr hRayNc
+    have avoid_down := bounded_component_avoids_axis_down
+      (range r) W hWsub hwb compW q hq0 hqr hRaySc
+    have avoid_mid := disjoint_component_avoids_axis_middle
+      (range r) W U hWsub hWU_disj p m hp0 hm0 hpr hmr hMidU
+    -- `W ⊆ E` (a bounded component lies in the rectangle)
+    have hWE : W ⊆ E := bounded_component_subset_normalized_rectangle hErect w hwb
+    -- **spine path** `β : n → s`, top to bottom, missing `W`, inside `E`
+    have hnl_cont : ContinuousOn (segP (!₂[(0 : ℝ), 2]) l) (Icc (-1 : ℝ) 1) := (segP_cont _
+      _).continuousOn
+    obtain ⟨lm, hlm_cont, hlm_a, hlm_b, hlm_mem⟩ :=
+      arc_path (hpcJn.joinedIn l ⟨hlJn, hlab⟩ m ⟨hmJn, hmab⟩)
+    have hmp_cont : ContinuousOn (segP m p) (Icc (-1 : ℝ) 1) := (segP_cont _ _).continuousOn
+    obtain ⟨pq, hpq_cont, hpq_a, hpq_b, hpq_mem⟩ :=
+      arc_path (hpcJs.joinedIn p ⟨hpJs, hpab⟩ q ⟨hqJs, hqab⟩)
+    have hqs_cont : ContinuousOn (segP q (!₂[(0 : ℝ), -2])) (Icc (-1 : ℝ) 1) := (segP_cont _
+      _).continuousOn
+    obtain ⟨β, hβcont, hβa, hβb, hβmem⟩ :=
+      concatPath5 hnl_cont hlm_cont hmp_cont hpq_cont hqs_cont
+        (by rw [segP_one, hlm_a]) (by rw [hlm_b, segP_neg1]) (by rw [segP_one, hpq_a])
+        (by rw [hpq_b, segP_neg1])
+    have hβE : ∀ t ∈ Icc (-1 : ℝ) 1, β t 0 ∈ Icc (-1 : ℝ) 1 ∧ β t 1 ∈ Icc (-2 : ℝ) 2 := by
+      intro t ht
+      rcases hβmem t ht with ((((h | h) | h) | h) | h)
+      · obtain ⟨u, hu, hue⟩ := h; rw [← hue]
+        exact ⟨segP_coord_mem_Icc 0 (by rw [hn0]; exact hIcc0x) (by rw [hl0]; exact hIcc0x) hu,
+          segP_coord_mem_Icc 1 (by rw [hn1]; exact hIcc2y) hlE1 hu⟩
+      · obtain ⟨u, hu, hue⟩ := h; rw [← hue]; exact hErect (hJnr (hlm_mem u hu).1)
+      · obtain ⟨u, hu, hue⟩ := h; rw [← hue]
+        exact ⟨segP_coord_mem_Icc 0 (by rw [hm0]; exact hIcc0x) (by rw [hp0]; exact hIcc0x) hu,
+          segP_coord_mem_Icc 1 hmE1 hpE1 hu⟩
+      · obtain ⟨u, hu, hue⟩ := h; rw [← hue]; exact hErect (hJsr (hpq_mem u hu).1)
+      · obtain ⟨u, hu, hue⟩ := h; rw [← hue]
+        exact ⟨segP_coord_mem_Icc 0 (by rw [hq0]; exact hIcc0x) (by rw [hs0]; exact hIcc0x) hu,
+          segP_coord_mem_Icc 1 hqE1 (by rw [hs1]; exact hIccm2y) hu⟩
+    have hβmiss : ∀ t ∈ Icc (-1 : ℝ) 1, β t ∉ W := by
+      intro t ht
+      rcases hβmem t ht with ((((h | h) | h) | h) | h)
+      · obtain ⟨u, hu, hue⟩ := h; rw [← hue]
+        refine avoid_up _ ?_ ?_
+        · rw [segP_coord, hn0, hl0]; ring
+        · rw [segP_coord, hn1]; rw [mem_Icc] at hu; nlinarith [hu.1, hu.2, hlE1.2]
+      · obtain ⟨u, hu, hue⟩ := h; rw [← hue]
+        exact fun hW => hWsub hW (hJnr (hlm_mem u hu).1)
+      · obtain ⟨u, hu, hue⟩ := h; rw [← hue]
+        refine avoid_mid _ ?_ ?_ ?_
+        · rw [segP_coord, hm0, hp0]; ring
+        · rw [segP_coord]; rw [mem_Icc] at hu; nlinarith [hu.1, hu.2, hpm_lt.le]
+        · rw [segP_coord]; rw [mem_Icc] at hu; nlinarith [hu.1, hu.2, hpm_lt.le]
+      · obtain ⟨u, hu, hue⟩ := h; rw [← hue]
+        exact fun hW => hWsub hW (hJsr (hpq_mem u hu).1)
+      · obtain ⟨u, hu, hue⟩ := h; rw [← hue]
+        refine avoid_down _ ?_ ?_
+        · rw [segP_coord, hq0, hs0]; ring
+        · rw [segP_coord, hs1]; rw [mem_Icc] at hu; nlinarith [hu.1, hu.2, hqE1.1]
+    -- `a, b ∈ closure W` (frontier of a component is the whole curve)
+    have hfrW : frontier W = range r := by
+      have hexists : ∃ y ∈ (range r)ᶜ,
+          connectedComponentIn (range r)ᶜ y ≠ connectedComponentIn (range r)ᶜ w := by
+        refine ⟨z₀, hz₀c, ?_⟩
+        rw [← hUdef, ← hWdef]; exact fun h => hWne h.symm
+      have hcbe := component_boundary_eq hbr hcont hinj hw hexists
+      rwa [← hWdef] at hcbe
+    have hafrW : (!₂[(-1 : ℝ), 0] : Plane) ∈ frontier W := by rw [hfrW]; exact hm
+    have hbfrW : (!₂[(1 : ℝ), 0] : Plane) ∈ frontier W := by rw [hfrW]; exact hp
+    have haclW : (!₂[(-1 : ℝ), 0] : Plane) ∈ closure W := frontier_subset_closure hafrW
+    have hbclW : (!₂[(1 : ℝ), 0] : Plane) ∈ closure W := frontier_subset_closure hbfrW
+    -- The crossing contradiction uses horizontal endpoints outside the spine.
+    set βimg : Set Plane := β '' Icc (-1 : ℝ) 1 with hβimgdef
+    have ha_notβ : (!₂[(-1 : ℝ), 0] : Plane) ∉ βimg := by
+      apply spine_image_avoids_horizontal_endpoint l m p q hl0 hm0 hp0 hq0 lm pq β hβmem
+        (!₂[(-1 : ℝ), 0]) (by norm_num)
+      · rintro ⟨u, hu, hue⟩
+        exact (hlm_mem u hu).2 (by rw [hue]; exact Set.mem_insert _ _)
+      · rintro ⟨u, hu, hue⟩
+        exact (hpq_mem u hu).2 (by rw [hue]; exact Set.mem_insert _ _)
+    have hb_notβ : (!₂[(1 : ℝ), 0] : Plane) ∉ βimg := by
+      apply spine_image_avoids_horizontal_endpoint l m p q hl0 hm0 hp0 hq0 lm pq β hβmem
+        (!₂[(1 : ℝ), 0]) (by norm_num)
+      · rintro ⟨u, hu, hue⟩
+        exact (hlm_mem u hu).2 (by rw [hue]; exact Set.mem_insert_of_mem _ rfl)
+      · rintro ⟨u, hu, hue⟩
+        exact (hpq_mem u hu).2 (by rw [hue]; exact Set.mem_insert_of_mem _ rfl)
+    exact path_component_crossing_contradiction hbr W hWpc hWE β hβcont hβE
+      (by simpa only [segP_neg1] using hβa) (by simpa only [segP_one] using hβb)
+      hβmiss haclW hbclW ha_notβ hb_notβ
+  -- assemble: bounded components of `x, y` both equal `U`
+  intro x hx y hy hxb hyb
+  rw [KEY x hx hxb, KEY y hy hyb]
+
+/-- **Maehara Step B (geometric core).** Bounded components of `ℝ²∖J` are unique.
+Reduced to the normalized version `step_B_normalized` via the farthest-pair
+similarity `T`. -/
+theorem step_B_bounded_unique (hbr : BrouwerFPT)
+    {r : sphere (0 : Plane) 1 → Plane} (hcont : Continuous r) (hinj : Injective r) :
+    ∀ x ∈ (range r)ᶜ, ∀ y ∈ (range r)ᶜ,
+      IsBounded (connectedComponentIn (range r)ᶜ x) →
+      IsBounded (connectedComponentIn (range r)ᶜ y) →
+      connectedComponentIn (range r)ᶜ x = connectedComponentIn (range r)ᶜ y := by
+  intro x hx y hy hxb hyb
+  obtain ⟨a, ha, b, hb, hfar0, hab⟩ := exists_farthest_pair hcont hinj
+  obtain ⟨T, hscale, hTa, hTb⟩ := exists_similarity hab
+  have hdpos : 0 < dist a b := dist_pos.mpr hab
+  have hcpos : (0 : ℝ) < 2 / dist a b := by positivity
+  set r' : sphere (0 : Plane) 1 → Plane := fun p => T (r p) with hr'
+  have hr'cont : Continuous r' := T.continuous.comp hcont
+  have hr'inj : Injective r' := T.injective.comp hinj
+  have hrange : range r' = T '' range r := by
+    rw [hr', show (fun p => T (r p)) = (T : Plane → Plane) ∘ r from rfl, Set.range_comp]
+  have himg : (T '' range r)ᶜ = T '' (range r)ᶜ :=
+    (Set.image_compl_eq (f := (T : Plane → Plane)) T.bijective).symm
+  have hrc : (range r')ᶜ = (T : Plane → Plane) '' (range r)ᶜ := by rw [hrange, himg]
+  have hm' : (!₂[(-1 : ℝ), 0] : Plane) ∈ range r' := by
+    rw [hrange, ← hTa]; exact mem_image_of_mem _ ha
+  have hp' : (!₂[(1 : ℝ), 0] : Plane) ∈ range r' := by
+    rw [hrange, ← hTb]; exact mem_image_of_mem _ hb
+  have hfar' : ∀ z ∈ range r', ∀ w ∈ range r', dist z w ≤ 2 := by
+    rw [hrange]
+    rintro _ ⟨z, hz, rfl⟩ _ ⟨w, hw, rfl⟩
+    rw [hscale]
+    calc 2 / dist a b * dist z w
+        ≤ 2 / dist a b * dist a b :=
+          mul_le_mul_of_nonneg_left (hfar0 z hz w hw) hcpos.le
+      _ = 2 := by field_simp
+  -- transport the two components forward by `T`
+  have hTx : T x ∈ (range r')ᶜ := by rw [hrc]; exact mem_image_of_mem _ hx
+  have hTy : T y ∈ (range r')ᶜ := by rw [hrc]; exact mem_image_of_mem _ hy
+  have hccx : connectedComponentIn (range r')ᶜ (T x)
+      = T '' connectedComponentIn (range r)ᶜ x := by
+    rw [hrc]; exact (T.image_connectedComponentIn hx).symm
+  have hccy : connectedComponentIn (range r')ᶜ (T y)
+      = T '' connectedComponentIn (range r)ᶜ y := by
+    rw [hrc]; exact (T.image_connectedComponentIn hy).symm
+  have hTxb : IsBounded (connectedComponentIn (range r')ᶜ (T x)) := by
+    rw [hccx]; exact (isBounded_image_scaling hcpos hscale _).mpr hxb
+  have hTyb : IsBounded (connectedComponentIn (range r')ᶜ (T y)) := by
+    rw [hccy]; exact (isBounded_image_scaling hcpos hscale _).mpr hyb
+  have hEq := step_B_normalized hbr hr'cont hr'inj hm' hp' hfar'
+    (T x) hTx (T y) hTy hTxb hTyb
+  rw [hccx, hccy] at hEq
+  exact Set.image_injective.mpr T.injective hEq
+
+/-- The Jordan curve theorem reduced to Brouwer (Maehara). The eval `jordan_curve`
+follows by discharging `BrouwerFPT`. Given the two geometric core facts — existence
+(`step_A_exists_bounded`) and uniqueness (`step_B_bounded_unique`) of a bounded
+component — together with the unique unbounded component (`exists_unbounded_component`,
+`unbounded_component_unique`), the complement has exactly two components. -/
+theorem jordan_curve_of_brouwer (hbr : BrouwerFPT)
+    (r : sphere (0 : Plane) 1 → Plane)
+    (hcont : Continuous r) (hinj : Injective r) :
+    Nat.card (ConnectedComponents ((range r)ᶜ : Set Plane)) = 2 := by
+  -- The unbounded component `x₀` and a bounded component `x₁`.
+  obtain ⟨x₀, hx₀S, hx₀u⟩ := exists_unbounded_component r hcont
+  obtain ⟨x₁, hx₁S, hx₁b⟩ := step_A_exists_bounded hbr hcont hinj
+  set a : ↥((range r)ᶜ) := ⟨x₀, hx₀S⟩ with ha
+  set b : ↥((range r)ᶜ) := ⟨x₁, hx₁S⟩ with hb
+  -- The two components are distinct: one unbounded, one bounded.
+  have hne : connectedComponentIn (range r)ᶜ x₀ ≠ connectedComponentIn (range r)ᶜ x₁ := by
+    intro heq
+    exact hx₀u (heq ▸ hx₁b)
+  have hab : ConnectedComponents.mk a ≠ ConnectedComponents.mk b :=
+    (Counting.connectedComponents_subtype_eq_iff hx₀S hx₁S).not.mpr hne
+  -- Every component is either the bounded one (`= b`) or the unbounded one (`= a`).
+  have hcover : ∀ z : ↥((range r)ᶜ),
+      ConnectedComponents.mk z = ConnectedComponents.mk a
+        ∨ ConnectedComponents.mk z = ConnectedComponents.mk b := by
+    intro z
+    by_cases hzb : IsBounded (connectedComponentIn (range r)ᶜ z.val)
+    · right
+      exact (Counting.connectedComponents_subtype_eq_iff z.2 hx₁S).mpr
+        (step_B_bounded_unique hbr hcont hinj z.val z.2 x₁ hx₁S hzb hx₁b)
+    · left
+      exact (Counting.connectedComponents_subtype_eq_iff z.2 hx₀S).mpr
+        (unbounded_component_unique r hcont hzb hx₀u)
+  exact Counting.nat_card_connectedComponents_eq_two a b hab hcover
+
+/-- **Jordan curve theorem** (the eval statement). -/
+theorem jordan_curve
+    (r : sphere (0 : EuclideanSpace ℝ (Fin 2)) 1 → EuclideanSpace ℝ (Fin 2))
+    (_hcont : Continuous r) (_hinj : Injective r) :
+    Nat.card (ConnectedComponents ((range r)ᶜ : Set (EuclideanSpace ℝ (Fin 2)))) = 2 := by
+  -- Brouwer FPT for the plane, built from scratch in `JordanCurve.Brouwer`
+  -- (circle non-nullhomotopy via Mathlib's covering-space lifting → no-retraction
+  -- → disk Brouwer → convex-compact).
+  have hbr : BrouwerFPT := Brouwer.brouwerFPT
+  exact jordan_curve_of_brouwer hbr r _hcont _hinj
+
+end JordanCurve
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 The Tau Ceti contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: The Tau Ceti contributors
+-/
+/-!
+# Metric variation of a continuously differentiable curve
+
+For a continuously differentiable curve in a complete real normed space, its
+metric total variation equals the integral of the norm of its derivative.
+
+The upper bound sums the fundamental theorem of calculus over finite
+partitions. For the reverse bound, the curve is clamped to the compact
+parameter interval. The resulting globally Lipschitz curve has bounded
+variation, and its associated vector measure has density given by the
+derivative on the interval.
+
+## Main result
+
+* `TauCeti.eVariationOn_eq_lintegral_enorm_derivWithin`: metric variation
+  equals the integral of the derivative norm.
+
+## Roadmap alignment
+
+This module advances the `Regular reparametrization and limits` target under
+`Layer 0: the reconciled Riemannian distance` in
+`roadmap/HopfRinow/README.md`. It supplies the vector metric-variation
+identity used as an analytic prerequisite by the lower-semicontinuity
+comparison; this file does not claim the broader
+Hopf--Rinow dependency path.
+
+## Provenance
+
+The partition-comparison architecture follows
+`DoCarmoLib/Riemannian/Geodesic/HopfRinow/EVariationLePathELength.lean` in the
+Apache-2.0 `frenzymath/Poincare-Conjecture` source at revision
+`24f32e4d600878bfaac6bc2f2f9324175571c321`. That source proves the forward
+comparison with Riemannian path length. The reverse derivative-integral
+comparison here is a Tau Ceti proof using Mathlib's clamped-curve and
+vector-measure APIs; it is not asserted to be present in that source.
+
+## References
+
+* M. P. do Carmo, *Riemannian Geometry*, Chapter 7, Section 2.
+* The source-first Lean snapshot above, revision
+  `24f32e4d600878bfaac6bc2f2f9324175571c321`, supplies the partition architecture
+  via `eVariationOn_le_pathELength`; the reverse inequality is original to this
+  module.
+
+Original authors: The Tau Ceti contributors, Archon Horizon (claude+codex), Axel Delaval,
+Chunlei Liu, Jinxuan Chen, Wanxu Yang, Zekun Sheng, Yuxuan Liao, Jie Xu.
+-/
+
+@[expose] public section
+
+open Filter Set MeasureTheory
+open scoped ENNReal NNReal Topology
+
+namespace TauCeti
+
+noncomputable section
+
+variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
+
+/-- For a `C¹` curve in a real normed space, metric total variation is bounded
+by the integral of the norm of its derivative. -/
+theorem eVariationOn_le_lintegral_enorm_derivWithin {f : ℝ → F} {a b : ℝ}
+    (hf : ContDiffOn ℝ 1 f (Icc a b)) :
+    eVariationOn f (Icc a b) ≤ ∫⁻ t in Icc a b, ‖derivWithin f (Icc a b) t‖ₑ := by
+  let L : ℝ → ℝ → ℝ≥0∞ := fun c d ↦
+    ∫⁻ t in Icc c d, ‖derivWithin f (Icc a b) t‖ₑ
+  have hL_add {c d e : ℝ} (hcd : c ≤ d) (hde : d ≤ e) : L c d + L d e = L c e := by
+    have hset : Icc c e = Icc c d ∪ Ioc d e := (Icc_union_Ioc_eq_Icc hcd hde).symm
+    simp only [L, hset]
+    rw [lintegral_union measurableSet_Ioc]
+    · simp only [restrict_Ioc_eq_restrict_Icc]
+    · exact disjoint_iff_forall_ne.mpr (fun u hu v hv ↦ (hu.2.trans_lt hv.1).ne)
+  apply iSup_le
+  rintro ⟨n, u, hu, hus⟩
+  have hsegment : ∀ i, edist (f (u (i + 1))) (f (u i)) ≤ L (u i) (u (i + 1)) := by
+    intro i
+    have hlocal := enorm_sub_le_lintegral_derivWithin_Icc_of_contDiffOn_Icc
+      (hf.mono (Icc_subset_Icc (hus i).1 (hus (i + 1)).2)) (hu (Nat.le_succ i))
+    calc
+      edist (f (u (i + 1))) (f (u i)) = ‖f (u (i + 1)) - f (u i)‖ₑ := by
+        rw [edist_eq_enorm_sub, enorm_sub_rev]
+      _ ≤ ∫⁻ t in Icc (u i) (u (i + 1)),
+          ‖derivWithin f (Icc (u i) (u (i + 1))) t‖ₑ := hlocal
+      _ = L (u i) (u (i + 1)) := by
+        simp only [L]
+        rw [← restrict_Ioo_eq_restrict_Icc]
+        apply setLIntegral_congr_fun measurableSet_Ioo
+        intro t ht
+        have hderiv :
+            derivWithin f (Icc (u i) (u (i + 1))) t = derivWithin f (Icc a b) t := by
+          rw [derivWithin_of_mem_nhds (Icc_mem_nhds ht.1 ht.2),
+            derivWithin_of_mem_nhds
+              (Icc_mem_nhds ((hus i).1.trans_lt ht.1) (ht.2.trans_le (hus (i + 1)).2))]
+        simp only [hderiv]
+  have htelescoping : ∀ m, ∑ i ∈ Finset.range m, L (u i) (u (i + 1)) = L (u 0) (u m) := by
+    intro m
+    induction m with
+    | zero => simp [L]
+    | succ k ih => rw [Finset.sum_range_succ, ih, hL_add
+        (hu (Nat.zero_le k)) (hu (Nat.le_succ k))]
+  calc
+    ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))
+        ≤ ∑ i ∈ Finset.range n, L (u i) (u (i + 1)) :=
+      Finset.sum_le_sum fun i _ ↦ hsegment i
+    _ = L (u 0) (u n) := htelescoping n
+    _ ≤ L a b := lintegral_mono_set (Icc_subset_Icc (hus 0).1 (hus n).2)
+    _ = ∫⁻ t in Icc a b, ‖derivWithin f (Icc a b) t‖ₑ := rfl
+
+variable [CompleteSpace F]
+
+/-- For a `C¹` curve in a complete real normed space, the integral of the norm
+of its derivative is bounded by its metric total variation. -/
+theorem lintegral_enorm_derivWithin_le_eVariationOn {f : ℝ → F} {a b : ℝ}
+    (hf : ContDiffOn ℝ 1 f (Icc a b)) :
+    ∫⁻ t in Icc a b, ‖derivWithin f (Icc a b) t‖ₑ ≤ eVariationOn f (Icc a b) := by
+  by_cases hab : a ≤ b
+  swap
+  · rw [Icc_eq_empty hab]
+    simp
+  rcases hab.eq_or_lt with rfl | hab
+  · simp
+  let p : ℝ → ℝ := fun t ↦ (Set.projIcc a b hab.le t).val
+  let g : ℝ → F := f ∘ p
+  let D : ℝ → F := (Icc a b).indicator (derivWithin f (Icc a b))
+  have hp_mono : Monotone p := fun x y hxy ↦ by
+    exact_mod_cast Set.monotone_projIcc hab.le hxy
+  have hp_mem : MapsTo p univ (Icc a b) := fun t _ ↦ (Set.projIcc a b hab.le t).2
+  have hD_cont : ContinuousOn (derivWithin f (Icc a b)) (Icc a b) :=
+    hf.continuousOn_derivWithin (uniqueDiffOn_Icc hab) (by norm_num)
+  obtain ⟨C, hCpos, hC⟩ :=
+    (isCompact_Icc.image_of_continuousOn hD_cont).isBounded.exists_pos_norm_le
+  let K : ℝ≥0 := ⟨C, hCpos.le⟩
+  have hK : LipschitzOnWith K f (Icc a b) := by
+    apply (convex_Icc a b).lipschitzOnWith_of_nnnorm_derivWithin_le
+      hf.differentiableOn_one
+    intro t ht
+    exact_mod_cast hC _ (mem_image_of_mem _ ht)
+  have hg_lip : LipschitzWith K g := by
+    have h := hK.comp (s := univ) (LipschitzWith.projIcc hab.le).lipschitzOnWith hp_mem
+    rw [mul_one, lipschitzOnWith_univ] at h
+    exact h
+  have hp_bv : BoundedVariationOn p univ := by
+    apply MonotoneOn.boundedVariationOn (hp_mono.monotoneOn univ) (C := max |a| |b|)
+    intro t _
+    exact (abs_le_max_abs_abs (Set.projIcc a b hab.le t).2.1
+      (Set.projIcc a b hab.le t).2.2)
+  have hg_bv : BoundedVariationOn g univ := hK.comp_boundedVariationOn hp_mem hp_bv
+  have hg_cont : Continuous g := hg_lip.continuous
+  have hright : Function.rightLim g = g := by
+    funext t
+    exact hg_cont.continuousWithinAt.rightLim_eq
+  have hD_int : Integrable D :=
+    hD_cont.integrableOn_Icc.integrable_indicator measurableSet_Icc
+  have hprimitive : ∀ x : ℝ, ∫ t in a..x, D t = g x - g a := by
+    intro x
+    rcases le_total x a with hxa | hax
+    · have hz : ∫ t in x..a, D t = 0 := by
+        apply intervalIntegral.integral_zero_ae
+        filter_upwards [volume.ae_ne a] with t hne hmem
+        rw [uIoc_of_le hxa] at hmem
+        have hlt : t < a := lt_of_le_of_ne hmem.2 hne
+        have hnot : t ∉ Icc a b := fun hmem ↦ (not_le_of_gt hlt) hmem.1
+        simp [D, hnot]
+      rw [intervalIntegral.integral_symm, hz, neg_zero]
+      have hpa : p a = a := by simp [p]
+      have hpx : p x = a := by simp [p, Set.projIcc_of_le_left hab.le hxa]
+      simp [g, hpa, hpx]
+    · rcases le_total x b with hxb | hbx
+      · have hcongr : ∫ t in a..x, D t = ∫ t in a..x, deriv f t := by
+          apply intervalIntegral.integral_congr_Ioo_of_le hax
+          intro t ht
+          have htab : t ∈ Icc a b := ⟨ht.1.le, ht.2.le.trans hxb⟩
+          simp only [D, Set.indicator_of_mem htab]
+          exact derivWithin_of_mem_nhds (Icc_mem_nhds ht.1 (ht.2.trans_le hxb))
+        rw [hcongr, intervalIntegral.integral_deriv_of_contDiffOn_Icc
+          (hf.mono (Icc_subset_Icc_right hxb)) hax]
+        have hpa : p a = a := by simp [p]
+        have hpx : p x = x := by simp [p, Set.projIcc_of_mem hab.le ⟨hax, hxb⟩]
+        simp [g, hpa, hpx]
+      · have hab_int : ∫ t in a..b, D t = f b - f a := by
+          have hcongr : ∫ t in a..b, D t = ∫ t in a..b, deriv f t := by
+            apply intervalIntegral.integral_congr_Ioo_of_le hab.le
+            intro t ht
+            simp only [D, Set.indicator_of_mem (Ioo_subset_Icc_self ht)]
+            exact derivWithin_of_mem_nhds (Icc_mem_nhds ht.1 ht.2)
+          rw [hcongr, intervalIntegral.integral_deriv_of_contDiffOn_Icc hf hab.le]
+        have hzero : ∫ t in b..x, D t = 0 := by
+          apply intervalIntegral.integral_zero_ae
+          filter_upwards with t ht
+          rw [uIoc_of_le hbx] at ht
+          have hnot : t ∉ Icc a b := fun hmem ↦ (not_le_of_gt ht.1) hmem.2
+          simp [D, hnot]
+        rw [← intervalIntegral.integral_add_adjacent_intervals
+          hD_int.intervalIntegrable hD_int.intervalIntegrable, hab_int, hzero, add_zero]
+        have hpa : p a = a := by simp [p]
+        have hpx : p x = b := by simp [p, Set.projIcc_of_right_le hab.le hbx]
+        simp [g, hpa, hpx]
+  have hinterval : ∀ {c d : ℝ}, c ≤ d → ∫ t in c..d, D t = g d - g c := by
+    intro c d hcd
+    have hac := intervalIntegral.integral_add_adjacent_intervals
+      hD_int.intervalIntegrable hD_int.intervalIntegrable (a := a) (b := c) (c := d)
+    rw [hprimitive c, hprimitive d] at hac
+    apply eq_sub_iff_add_eq.mpr
+    calc
+      (∫ t in c..d, D t) + g c = (g c - g a + ∫ t in c..d, D t) + g a := by abel
+      _ = (g d - g a) + g a := by rw [hac]
+      _ = g d := by abel
+  have hmeasure : hg_bv.vectorMeasure = volume.withDensityᵥ D := by
+    apply VectorMeasure.ext_of_generateFrom
+      {s : Set ℝ | ∃ c d, c ≤ d ∧ s = Ioc c d}
+    · rintro s ⟨c, d, hcd, rfl⟩
+      rw [hg_bv.vectorMeasure_Ioc hcd, hright, withDensityᵥ_apply hD_int measurableSet_Ioc,
+        ← intervalIntegral.integral_of_le hcd, hinterval hcd]
+    · convert! BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ioc_le ℝ) using 2;
+        grind only
+    · exact IsSetSemiring.Ioc.isPiSystem
+    · rw [hg_bv.vectorMeasure_univ, withDensityᵥ_apply hD_int MeasurableSet.univ]
+      simp only [Measure.restrict_univ, D]
+      rw [MeasureTheory.integral_indicator measurableSet_Icc,
+        MeasureTheory.integral_Icc_eq_integral_Ioc,
+        ← intervalIntegral.integral_of_le hab.le,
+        intervalIntegral.integral_derivWithin_Icc_of_contDiffOn_Icc hf hab.le]
+      have htop : Tendsto g atTop (𝓝 (f b)) := by
+        apply tendsto_const_nhds.congr'
+        filter_upwards [Ici_mem_atTop b] with t ht
+        simp [g, p, Set.projIcc_of_right_le hab.le ht]
+      have hbot : Tendsto g atBot (𝓝 (f a)) := by
+        apply tendsto_const_nhds.congr'
+        filter_upwards [Iic_mem_atBot a] with t ht
+        simp [g, p, Set.projIcc_of_le_left hab.le ht]
+      rw [tendsto_nhds_unique hg_bv.tendsto_atTop_limUnder htop,
+        tendsto_nhds_unique hg_bv.tendsto_atBot_limUnder hbot]
+  have hvariation : eVariationOn g univ = eVariationOn f (Icc a b) := by
+    apply le_antisymm
+    · apply iSup_le
+      rintro ⟨n, u, hu, -⟩
+      simpa only [g, Function.comp_apply] using
+        (eVariationOn.sum_le (f := f) (hp_mono.comp hu)
+          (fun i ↦ hp_mem (mem_univ _)))
+    · calc
+        eVariationOn f (Icc a b) = eVariationOn g (Icc a b) := by
+          apply eVariationOn.eq_of_eqOn
+          intro t ht
+          simp [g, p, Set.projIcc_of_mem hab.le ht]
+        _ ≤ eVariationOn g univ := eVariationOn.mono g (subset_univ _)
+  calc
+    ∫⁻ t in Icc a b, ‖derivWithin f (Icc a b) t‖ₑ
+        = (volume.withDensity fun t ↦ ‖D t‖ₑ) univ := by
+          rw [withDensity_apply _ MeasurableSet.univ]
+          simp only [Measure.restrict_univ]
+          rw [← lintegral_indicator measurableSet_Icc]
+          apply lintegral_congr
+          intro t
+          by_cases ht : t ∈ Icc a b <;> simp [D, ht]
+    _ = (volume.withDensityᵥ D).variation univ := by
+      rw [Measure.variation_withDensityᵥ hD_int]
+    _ = hg_bv.vectorMeasure.variation univ := by rw [hmeasure]
+    _ ≤ eVariationOn g univ := hg_bv.variation_vectorMeasure_univ_le
+    _ = eVariationOn f (Icc a b) := hvariation
+
+/-- The metric total variation of a `C¹` curve in a complete real normed space
+equals the integral of the norm of its derivative. -/
+theorem eVariationOn_eq_lintegral_enorm_derivWithin {f : ℝ → F} {a b : ℝ}
+    (hf : ContDiffOn ℝ 1 f (Icc a b)) :
+    eVariationOn f (Icc a b) = ∫⁻ t in Icc a b, ‖derivWithin f (Icc a b) t‖ₑ :=
+  le_antisymm (eVariationOn_le_lintegral_enorm_derivWithin hf)
+    (lintegral_enorm_derivWithin_le_eVariationOn hf)
+
+end
+
+end TauCeti
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 The Tau Ceti contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: The Tau Ceti contributors
+-/
+/-!
+# The exterior of a closed ball is preconnected
+
+In a real normed space of dimension at least two the complement of a closed ball is preconnected:
+it is the union, over the radii `M` exceeding the ball's, of the spheres of radius `M`, strung
+together along a single ray from the centre.
+
+The consequences for bounded sets — uniqueness of the unbounded component and the filled-hull
+alternative — are in `TauCeti/Analysis/Normed/Module/FilledHull.lean`.
+
+## Main results
+
+* `TauCeti.isPreconnected_compl_closedBall` — the exterior of a closed ball is preconnected in a
+  real normed space of dimension at least two.
+
+This is a prerequisite of the planar-separation step of the `ConformalMapping` roadmap (L5).
+
+## References
+
+* Ch. Pommerenke, *Boundary Behaviour of Conformal Maps*, Ch. 2.
+-/
+
+@[expose] public section
+
+namespace TauCeti
+
+open Bornology Metric Set
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+/-- **The exterior of a closed ball is preconnected** in a real normed space of dimension at least
+two. -/
+theorem isPreconnected_compl_closedBall (h : 1 < Module.rank ℝ E) (x : E) (r : ℝ) :
+    IsPreconnected (closedBall x r)ᶜ := by
+  rcases lt_or_ge r 0 with hr | hr
+  · rw [closedBall_eq_empty.mpr hr, compl_empty]
+    exact isPreconnected_univ
+  have : Nontrivial E := rank_pos_iff_nontrivial.mp (zero_lt_one.trans h)
+  obtain ⟨u, hu⟩ := exists_norm_eq E zero_le_one
+  set L : Set E := (fun t : ℝ => x + t • u) '' Ioi r
+  have hLc : IsPreconnected L :=
+    isPreconnected_Ioi.image _ (by fun_prop : Continuous fun t : ℝ => x + t • u).continuousOn
+  have hLmem : ∀ t, r < t → x + t • u ∈ L := fun t ht => ⟨t, ht, rfl⟩
+  have hdist : ∀ t : ℝ, 0 ≤ t → dist (x + t • u) x = t := by
+    intro t ht
+    simp [dist_eq_norm, norm_smul, hu, abs_of_nonneg ht]
+  have hmem : ∀ w, w ∈ (closedBall x r)ᶜ ↔ r < dist w x := by
+    simp [mem_closedBall, not_le]
+  have hLsub : L ⊆ (closedBall x r)ᶜ := by
+    rintro _ ⟨t, ht, rfl⟩
+    rw [hmem, hdist t (hr.trans ht.le)]
+    exact ht
+  have key : (closedBall x r)ᶜ = ⋃ M : Ioi r, (sphere x M ∪ L) := by
+    ext w
+    constructor
+    · intro hw
+      exact mem_iUnion.mpr ⟨⟨dist w x, (hmem w).mp hw⟩, Or.inl (mem_sphere.mpr rfl)⟩
+    · intro hw
+      obtain ⟨⟨M, hM⟩, hwM⟩ := mem_iUnion.mp hw
+      rcases hwM with hwM | hwM
+      · rw [hmem]; rw [mem_sphere, dist_comm] at hwM; rw [dist_comm, hwM]; exact hM
+      · exact hLsub hwM
+  rw [key]
+  refine isPreconnected_iUnion ⟨x + (r + 1) • u, ?_⟩ fun ⟨M, hM⟩ => ?_
+  · simp only [mem_iInter]
+    exact fun _ => Or.inr (hLmem _ (by linarith))
+  · refine IsPreconnected.union (x + M • u) ?_ (hLmem M hM) (isPreconnected_sphere h x M) hLc
+    rw [mem_sphere, hdist M (hr.trans hM.le)]
+
+end TauCeti
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 The Tau Ceti contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: The Tau Ceti contributors
+-/
+/-!
+# Strict half-spaces of a real normed space are unbounded
+
+A strict half-space `{y | φ y < u}` cut out by a nonzero linear functional holds points of
+arbitrarily large norm, and is therefore unbounded. Linearity alone suffices: `φ` need not be
+continuous, so the results apply to a discontinuous functional on an infinite-dimensional space.
+For such a `φ` the set need not be topologically open, which is why it is called strict rather
+than open here.
+
+## Main results
+
+* `TauCeti.exists_apply_lt_and_lt_norm` and `TauCeti.exists_lt_apply_and_lt_norm` — either side
+  of a nonzero linear functional holds points of arbitrarily large norm.
+* `TauCeti.not_isBounded_halfSpace_lt` and `TauCeti.not_isBounded_halfSpace_gt` — either strict
+  half-space is unbounded.
+-/
+
+@[expose] public section
+
+namespace TauCeti
+
+open Bornology
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+/-- **A strict half-space contains points of arbitrarily large norm.** For a nonzero linear
+functional `φ`, every bound `u` and every radius `R` admit a `y` with `φ y < u` and `R < ‖y‖`.
+Linearity suffices; `φ` need not be continuous. -/
+theorem exists_apply_lt_and_lt_norm {φ : E →ₗ[ℝ] ℝ} (hφ : φ ≠ 0) (u R : ℝ) :
+    ∃ y : E, φ y < u ∧ R < ‖y‖ := by
+  obtain ⟨v, hφv⟩ := LinearMap.surjective_iff_ne_zero.mpr hφ 1
+  have hvnorm : 0 < ‖v‖ := norm_pos_iff.mpr fun h => by simp [h] at hφv
+  -- Walk to `-t • v` for a `t` large enough to break both the bound `u` and the radius `R`.
+  obtain ⟨t, ht1, ht2⟩ : ∃ t : ℝ, (R + 1) / ‖v‖ ≤ t ∧ |u| + 1 ≤ t :=
+    ⟨_, le_max_left _ _, le_max_right _ _⟩
+  have ht0 : 0 ≤ t := le_trans (by positivity) ht2
+  refine ⟨(-t) • v, ?_, ?_⟩
+  · rw [map_smul, hφv, smul_eq_mul, mul_one]
+    linarith [neg_abs_le u]
+  · rw [norm_smul, norm_neg, Real.norm_eq_abs, abs_of_nonneg ht0]
+    linarith [(div_le_iff₀ hvnorm).mp ht1]
+
+/-- **The other side of a nonzero linear functional also contains points of arbitrarily large
+norm**: every bound `u` and radius `R` admit a `y` with `u < φ y` and `R < ‖y‖`. -/
+theorem exists_lt_apply_and_lt_norm {φ : E →ₗ[ℝ] ℝ} (hφ : φ ≠ 0) (u R : ℝ) :
+    ∃ y : E, u < φ y ∧ R < ‖y‖ := by
+  obtain ⟨y, hy, hn⟩ := exists_apply_lt_and_lt_norm (φ := -φ) (neg_ne_zero.mpr hφ) (-u) R
+  exact ⟨y, by simpa using hy, hn⟩
+
+/-- **A strict half-space cut out by a nonzero linear functional is unbounded.** No radius bounds
+`{y | φ y < u}`. Linearity suffices; `φ` need not be continuous. -/
+theorem not_isBounded_halfSpace_lt {φ : E →ₗ[ℝ] ℝ} (hφ : φ ≠ 0) (u : ℝ) :
+    ¬ IsBounded {y | φ y < u} := by
+  intro hbdd
+  obtain ⟨R, hR⟩ := isBounded_iff_forall_norm_le.mp hbdd
+  obtain ⟨y, hy, hn⟩ := exists_apply_lt_and_lt_norm hφ u R
+  exact absurd (hR y (by simpa using hy)) (not_le.mpr hn)
+
+/-- **The half-space on the other side of a nonzero linear functional is unbounded.** No radius
+bounds `{y | u < φ y}` either. Linearity suffices; `φ` need not be continuous. -/
+theorem not_isBounded_halfSpace_gt {φ : E →ₗ[ℝ] ℝ} (hφ : φ ≠ 0) (u : ℝ) :
+    ¬ IsBounded {y | u < φ y} := by
+  intro hbdd
+  obtain ⟨R, hR⟩ := isBounded_iff_forall_norm_le.mp hbdd
+  obtain ⟨y, hy, hn⟩ := exists_lt_apply_and_lt_norm hφ u R
+  exact absurd (hR y (by simpa using hy)) (not_le.mpr hn)
+
+end TauCeti
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 The Tau Ceti contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: The Tau Ceti contributors
+-/
+/-!
+# Limits of total variation bounds
+
+This file transfers eventual upper bounds on the total variations of a family of maps to a
+`liminf` bound on the total variation of a pointwise limit.
+
+## Main results
+
+* `TauCeti.eVariationOn_le_liminf_of_eventually_le`: an eventual bound on the total variations of
+  a family of maps bounds the total variation of a pointwise limit by the `liminf` of the bounds.
+-/
+
+@[expose] public section
+
+open Filter
+open scoped ENNReal
+
+namespace TauCeti
+
+variable {α : Type*} [LinearOrder α] {X : Type*} [PseudoEMetricSpace X]
+
+/-- The metric variation of a path in a subtype is unchanged by applying its coercion. -/
+@[simp] theorem eVariationOn_subtypeVal_comp {s : Set X} {f : α → s} {t : Set α} :
+    eVariationOn f t = eVariationOn ((Subtype.val : s → X) ∘ f) t := by
+  rfl
+
+/-- If the total variations of the maps `F i` on `s` are eventually bounded by `u i`, then the
+total variation on `s` of a pointwise limit of the `F i` is at most `liminf u`. -/
+theorem eVariationOn_le_liminf_of_eventually_le {ι : Type*} {l : Filter ι} {s : Set α}
+    {f : α → X} {F : ι → α → X} {u : ι → ℝ≥0∞}
+    (hu : ∀ᶠ i in l, eVariationOn (F i) s ≤ u i)
+    (hf : ∀ x ∈ s, Tendsto (fun i ↦ F i x) l (nhds (f x))) :
+    eVariationOn f s ≤ liminf u l := by
+  rw [le_liminf_iff]
+  intro v hv
+  filter_upwards [eVariationOn.lowerSemicontinuous_aux hf hv, hu] with i hi hui
+  exact hi.trans_le hui
+
+end TauCeti
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# External libraries
+
+The modules from Mathlib, lean-pool, TauCeti and jordan_pick that the development uses beyond its
+own files, collected in one place.
+-/
+
+@[expose] public section
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 The Tau Ceti contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: The Tau Ceti contributors
+-/
+/-!
+# Three frontier lemmas: straddling, splitting a domain in two, and clinging to it from inside
+
+Three elementary facts about `frontier`, each the topological core of a step that a boundary
+argument would otherwise carry out inside a concrete space.
+
+## A connected set that straddles a set meets its frontier
+
+A preconnected set that meets both a set `V` and its complement must meet `frontier V`: it cannot
+cross from the inside of `V` to the outside without touching the boundary. This is the
+intermediate-value principle in its purely topological form, and it is the mechanism by which a
+*path* leaving a set produces a *boundary point* of that set.
+
+Mathlib records the two extreme cases — `frontier_eq_empty_iff` and `nonempty_frontier_iff` say
+that in a preconnected *space* the frontier of `V` is empty exactly when `V` is `∅` or `univ` — but
+not this relative form, which is the one an argument along a segment or a path needs. No hypothesis
+is placed on `V`; only preconnectedness of the straddling set is used.
+
+The proof is the standard clopen argument: the complement of `frontier V` is the disjoint union of
+the two open sets `interior V` and `interior Vᶜ` (`compl_frontier_eq_union_interior`), so a
+preconnected set avoiding the frontier lies inside one of them, and then it misses `V` entirely or
+is contained in `V` entirely.
+
+## Where the boundary of the image of one side of a split domain can lie
+
+Split a set `U` into two pieces `s` and `t` that a map `f` sends to *disjoint open* sets, plus a
+remainder `u`. Then `frontier (f '' s) ⊆ f '' u ∪ frontier (f '' U)`
+(`TauCeti.frontier_image_subset_image_union_frontier_image`): the boundary of the image of one side
+consists of images of the remainder — the cut — and of boundary points of the whole image, and of
+nothing else.
+
+The proof is a three-way case split. A point `p` of `frontier (f '' s)` lies in `closure (f '' U)`,
+so if it is not on `frontier (f '' U)` it is a value `f w` with `w` in one of the three covering
+sets. It cannot come from `s`, since `f '' s` is open and therefore disjoint from its own frontier;
+and it cannot come from `t`, since `f '' t` is then an open neighbourhood of `p`, which must meet
+`f '' s`, contradicting disjointness of the two images. So `w ∈ u`.
+
+The source carries no topology; the sides enter topologically only through their images, which are
+asked to be open and disjoint, and that is all the argument uses of them. What is asked of the
+sides themselves is purely set-theoretic: `s ⊆ U` and the covering `U ⊆ s ∪ t ∪ u`. In particular
+`t` need not lie in `U`, and neither side need be open or disjoint from the other. A consumer whose
+map is open and injective on two disjoint open sides supplies both image hypotheses, as the
+conformal one below does through the open mapping theorem and `Disjoint.image`.
+
+## What a set's frontier sees of a subset
+
+A subset `A` of a set `V` cannot reach `frontier V` except through its own frontier:
+
+> `frontier V ∩ closure A = frontier V ∩ frontier A`
+
+(`TauCeti.frontier_inter_closure_eq_frontier_inter_frontier`). The reason is that `closure A` is
+`A ∪ frontier A`, and a point of `A` on `frontier V` is already on `frontier A`: it is adherent to
+`A` and, since `interior A ⊆ interior V`, it is not interior to `A`. So the part of `frontier V`
+that `A` clings to has two interchangeable descriptions — as the reach of `closure A`, and as the
+meeting of two frontiers. The first is the one an argument about limits of points of `A` produces;
+the second is the one a diameter estimate consumes, `frontier` being where the estimates of a
+domain-splitting argument live.
+
+## Consumers
+
+All three lemmas serve layer **L5** of `TauCetiRoadmap/ConformalMapping/README.md`, Carathéodory's
+boundary correspondence. The first does so through
+`TauCeti/Analysis/Normed/Module/DiamFrontier.lean`: a ray leaving a bounded set crosses its
+frontier, which is what makes the frontier of such a set as wide as the set itself. The second is
+the splitting step of `TauCeti/Analysis/Complex/Conformal/CutDiameter.lean`, where `s` and `t` are
+the two sides of a circular crosscut of a domain and `u` is the crosscut arc. The third is what
+lets `TauCeti/Analysis/Complex/Conformal/ClusterSet.lean` identify the boundary piece that one
+side of such a crosscut cuts off, whose description as a union of cluster sets is naturally a
+statement about a closure. Nothing here is specific to those uses; no lemma mentions a metric, let
+alone a holomorphic map.
+
+## Main results
+
+* `IsPreconnected.inter_frontier_nonempty` — a preconnected set meeting both a set and its
+  complement meets the frontier of that set.
+* `TauCeti.frontier_image_subset_image_union_frontier_image` — for a set split into two sides with
+  disjoint open images plus a remainder, the frontier of the image of the side lying in that set
+  lies on the image of the remainder and on the frontier of the image of the whole.
+* `TauCeti.frontier_inter_closure_eq_frontier_inter_frontier` — the frontier of a set meets the
+  closure of a subset exactly where it meets that subset's frontier.
+-/
+
+@[expose] public section
+
+namespace TauCeti
+
+open Set
+
+section Straddle
+
+variable {X : Type*} [TopologicalSpace X] {S V : Set X}
+
+/-- **A preconnected set that meets both a set and its complement meets its frontier.** If `S` is
+preconnected and contains a point of `V` and a point outside `V`, then `S` meets `frontier V`.
+
+Nothing is assumed about `V`; the argument is that `(frontier V)ᶜ` is the union of the two disjoint
+open sets `interior V` and `interior Vᶜ`, so a preconnected set missing the frontier is confined to
+one of them and therefore cannot straddle `V`. -/
+theorem _root_.IsPreconnected.inter_frontier_nonempty (hS : IsPreconnected S)
+    (h₁ : (S ∩ V).Nonempty) (h₂ : (S \ V).Nonempty) : (S ∩ frontier V).Nonempty := by
+  by_contra hcon
+  have hsub : S ⊆ interior V ∪ interior Vᶜ := by
+    rw [← compl_frontier_eq_union_interior]
+    exact fun x hx hxf => hcon ⟨x, hx, hxf⟩
+  have hdisj : Disjoint (interior V) (interior Vᶜ) :=
+    disjoint_compl_right.mono interior_subset interior_subset
+  rcases hS.subset_or_subset isOpen_interior isOpen_interior hdisj hsub with h | h
+  · obtain ⟨x, hxS, hxV⟩ := h₂
+    exact hxV (interior_subset (h hxS))
+  · obtain ⟨x, hxS, hxV⟩ := h₁
+    exact interior_subset (h hxS) hxV
+
+end Straddle
+
+section ImageSplit
+
+variable {X Y : Type*} [TopologicalSpace Y] {f : X → Y} {U s t u : Set X}
+
+/-- **The boundary of the image of one side of a split domain lies on the image of the remainder
+and on the boundary of the image of the domain.** If `U` is covered by two sets `s`, `t` with
+disjoint open images together with a third set `u`, and `s ⊆ U`, then
+`frontier (f '' s) ⊆ f '' u ∪ frontier (f '' U)`.
+
+The conclusion is about `s`, the side that is asked to lie in `U`. The argument treats the two
+sides alike apart from that, so a consumer with `t ⊆ U` in hand bounds `t` as well by swapping
+their roles.
+
+A frontier point of `f '' s` lies in `closure (f '' U)`, so if it is not a frontier point of
+`f '' U` it is a value `f w` with `w` in one of the three covering sets: `w ∈ s` would place it
+inside the open set `f '' s`, which is disjoint from its own frontier, and `w ∈ t` inside the open
+set `f '' t`, which meets `f '' s` because the point is in its closure, contradicting disjointness
+of the two images. So `w ∈ u`.
+
+The source carries no topology at all: `U` and the three sets covering it are constrained only by
+the two set-theoretic hypotheses `hsU : s ⊆ U` and `hcov : U ⊆ s ∪ t ∪ u`, while every topological
+hypothesis, and the conclusion, lives in the target. In particular `t` need not lie in `U`, and
+the two sides need be neither open nor disjoint nor separated by injectivity — only their images
+need be open and disjoint, which is what the argument uses and what an injective open map on two
+disjoint open sides supplies. -/
+theorem frontier_image_subset_image_union_frontier_image (hfs : IsOpen (f '' s))
+    (hft : IsOpen (f '' t)) (hst : Disjoint (f '' s) (f '' t)) (hsU : s ⊆ U)
+    (hcov : U ⊆ s ∪ t ∪ u) :
+    frontier (f '' s) ⊆ f '' u ∪ frontier (f '' U) := by
+  intro p hp
+  have hpΩ : p ∈ closure (f '' U) := closure_mono (image_mono hsU) hp.1
+  rw [closure_eq_self_union_frontier] at hpΩ
+  rcases hpΩ with hpin | hpfr
+  · obtain ⟨w, hw, rfl⟩ := hpin
+    rcases hcov hw with (hws | hwt) | hwu
+    · exact absurd ⟨mem_image_of_mem f hws, hp⟩
+        (eq_empty_iff_forall_notMem.mp hfs.inter_frontier_eq (f w))
+    · obtain ⟨q, hqt, hqs⟩ := mem_closure_iff.mp hp.1 _ hft (mem_image_of_mem f hwt)
+      exact absurd hqs (disjoint_right.mp hst hqt)
+    · exact Or.inl (mem_image_of_mem f hwu)
+  · exact Or.inr hpfr
+
+end ImageSplit
+
+section Inside
+
+variable {X : Type*} [TopologicalSpace X] {A V : Set X}
+
+/-- **The frontier of a set meets the closure of a subset exactly where it meets that subset's
+frontier.** For any `A ⊆ V`,
+
+> `frontier V ∩ closure A = frontier V ∩ frontier A`.
+
+Writing `closure A` as `A ∪ frontier A`, the first piece brings nothing new: a point of `A` on
+`frontier V` is adherent to `A` and is kept out of `interior A` by `interior A ⊆ interior V`, so it
+already lies on `frontier A`. Only the inclusion `A ⊆ V` is used — `V` need be neither open nor
+closed, and `A` is arbitrary.
+
+So the part of `frontier V` that `A` clings to may be described either as its meeting with
+`closure A` or as its meeting with `frontier A`. An argument about limits of points of `A` produces
+the first; a diameter estimate obtained by splitting `V` into pieces consumes the second. -/
+theorem frontier_inter_closure_eq_frontier_inter_frontier (hAV : A ⊆ V) :
+    frontier V ∩ closure A = frontier V ∩ frontier A := by
+  rw [closure_eq_self_union_frontier, inter_union_distrib_left]
+  refine union_eq_right.mpr fun x hx =>
+    ⟨hx.1, (mem_frontier_iff_notMem_interior hx.2).mpr fun hint => ?_⟩
+  exact (mem_frontier_iff_notMem_interior (hAV hx.2)).mp hx.1 (interior_mono hAV hint)
+
+end Inside
+
+end TauCeti
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 The Tau Ceti contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: The Tau Ceti contributors
+-/
+/-!
+# Filling in the bounded complementary components of a set
+
+The **filled hull** `TauCeti.filledHull K` of a subset `K` of a topological space with a bornology
+is `K` together with the bounded connected components of its complement: the points whose component
+in `Kᶜ` is bounded. Points of `K` qualify vacuously, their component in `Kᶜ` being empty. Filling a
+circle gives the closed disc it bounds; filling a segment, or any set whose complement is connected
+and unbounded, changes nothing.
+
+This file is the topological layer: the definition and the structural facts, which ask only for a
+topology and a bornology, being about components and boundedness and nothing else. That filling
+does not make a set wider needs a real normed space and lives in
+`TauCeti/Analysis/Normed/Module/FilledHull.lean`.
+
+The shape in which the structural side is spent is `IsPreconnected.subset_filledHull`: a
+preconnected set disjoint from `K` is trapped inside the filled hull as soon as it meets it, since
+it then lies in a single bounded component. Together with the width bound of the normed file it
+says that *a connected set that a small `K` cuts off from infinity is itself small*, with no
+regularity asked of `K`; that composite is `IsPreconnected.diam_le_diam_of_disjoint` there.
+
+The negation of membership — that the component of a point in the complement of `K` is *unbounded*
+— already occurs, unfolded, in the winding-number layer: it is the hypothesis of
+`TauCeti.Contour.windingNumber_eq_zero_of_unbounded_component` in
+`TauCeti/Analysis/Contour/Winding/UnboundedComponent.lean` and of its cycle form
+`TauCeti.Contour.Cycle.windingNumber_eq_zero_of_unbounded_component` in
+`TauCeti/Analysis/Contour/Cycle/Winding.lean`, both of which say that the winding number vanishes
+off the filled hull of the trace. Those statements are left as they stand: they are about the
+unbounded side, which needs no name, whereas everything here is about the filled side.
+
+The hull is deliberately *not* claimed to be closed, connected, or idempotent — none of which is
+needed downstream, and the first two of which fail without hypotheses on `K`.
+
+## Roadmap role
+
+Plane separation for Jordan curves was the open frontier item of layer **L5** of
+`TauCetiRoadmap/ConformalMapping/README.md`, the Carathéodory boundary correspondence. The
+enclosure step now runs through `IsPreconnected (K \ {f z₀})` and the winding-number two-sidedness
+theorem
+(`TauCeti.image_inter_ball_subset_filledHull_of_diam_lt_of_isPreconnected_sdiff_singleton`),
+which `IsJordanCurve.isPathConnected_sdiff_singleton` discharges; `Caratheodory.lean` is
+unconditional.
+The inside of `J` is `filledHull J \ J` in the vocabulary defined here. Nothing here assumes
+separation, or any other regularity of `K`.
+
+## Main results
+
+* `TauCeti.filledHull` — the filled hull, and `TauCeti.subset_filledHull`,
+  `TauCeti.filledHull_mono` its two structural properties.
+* `TauCeti.filledHull_eq_self` — filling a set whose complement is preconnected and unbounded
+  changes nothing.
+* `IsPreconnected.subset_filledHull` — a preconnected set disjoint from `K` that meets the
+  filled hull lies in it.
+* `TauCeti.subset_filledHull_of_frontier_subset` — a bounded set whose frontier `K` swallows
+  lies in the filled hull, with no connectivity asked of it.
+-/
+
+@[expose] public section
+
+namespace TauCeti
+
+open Bornology Set
+
+variable {E : Type*} [TopologicalSpace E] [Bornology E] {K L S : Set E} {x : E}
+
+/-- The **filled hull** of a set `K`: the points whose connected component in the complement of `K`
+is bounded. Equivalently, `K` together with the bounded connected components of `Kᶜ`; a point of
+`K` belongs because its component in `Kᶜ` is empty. -/
+def filledHull (K : Set E) : Set E := {x | IsBounded (connectedComponentIn Kᶜ x)}
+
+@[simp]
+theorem mem_filledHull_iff : x ∈ filledHull K ↔ IsBounded (connectedComponentIn Kᶜ x) := Iff.rfl
+
+/-- **A set lies in its filled hull.** For `x ∈ K` the component of `x` in `Kᶜ` is empty, and the
+empty set is bounded. -/
+theorem subset_filledHull : K ⊆ filledHull K := by
+  intro x hx
+  have hxc : x ∉ Kᶜ := by simpa using hx
+  simp [mem_filledHull_iff, connectedComponentIn_eq_empty hxc]
+
+/-- **Filling is monotone.** Enlarging `K` shrinks the complement, hence shrinks each component of
+it, hence can only turn unbounded components into bounded ones. -/
+@[gcongr]
+theorem filledHull_mono (h : K ⊆ L) : filledHull K ⊆ filledHull L := fun _ hx =>
+  mem_filledHull_iff.mpr <|
+    (mem_filledHull_iff.mp hx).subset (connectedComponentIn_mono _ (compl_subset_compl.mpr h))
+
+/-- **Filling changes nothing when the complement is connected and unbounded.** The complement is
+then a single component and that component is unbounded, so no point outside `K` is filled in. This
+is the case of a segment in the plane, and of any set that does not separate the space. -/
+theorem filledHull_eq_self (h : IsPreconnected Kᶜ) (hu : ¬ IsBounded Kᶜ) : filledHull K = K := by
+  refine Subset.antisymm (fun x hx => ?_) subset_filledHull
+  by_contra hxK
+  exact hu ((mem_filledHull_iff.mp hx).subset
+    (h.subset_connectedComponentIn (mem_compl hxK) subset_rfl))
+
+/-- **A preconnected set that a set cuts off from infinity lies in its filled hull.** If `S` is
+preconnected and disjoint from `K`, then `S` lies in a single connected component of `Kᶜ`; meeting
+the filled hull says that component is bounded, so all of `S` is in the hull. -/
+theorem _root_.IsPreconnected.subset_filledHull (hS : IsPreconnected S) (hSK : Disjoint S K)
+    (hne : (S ∩ filledHull K).Nonempty) : S ⊆ filledHull K := by
+  obtain ⟨x, hxS, hxH⟩ := hne
+  have hScompl : S ⊆ Kᶜ := fun y hy => Set.disjoint_left.mp hSK hy
+  have hScomp : S ⊆ connectedComponentIn Kᶜ x := hS.subset_connectedComponentIn hxS hScompl
+  intro y hy
+  rw [mem_filledHull_iff, ← connectedComponentIn_eq (hScomp hy)]
+  exact mem_filledHull_iff.mp hxH
+
+/-- **A bounded set whose frontier lies in `K` is cut off from infinity by `K`.** A point of
+`S \ K` lies in `interior S`, since every non-interior point of `S` lies on `frontier S ⊆ K`. Its
+connected component in `Kᶜ` cannot leave `interior S`: were it to, it would meet
+`frontier (interior S) ⊆ frontier S ⊆ K` by `IsPreconnected.inter_frontier_nonempty`,
+while lying in `Kᶜ`. So that component is bounded because `S` is. Points of `S ∩ K` lie in the
+filled hull directly.
+
+Unlike `IsPreconnected.subset_filledHull` this asks nothing of the connectivity of `S` and
+nothing about the hull being met, at the price of asking `K` to swallow the whole frontier — the
+same trade as between `TauCeti.diam_le_diam_of_frontier_subset` and
+`IsPreconnected.diam_le_diam_of_disjoint`. -/
+theorem subset_filledHull_of_frontier_subset (hSb : IsBounded S) (hfr : frontier S ⊆ K) :
+    S ⊆ filledHull K := by
+  intro x hx
+  by_cases hxK : x ∈ K
+  · exact subset_filledHull hxK
+  have hxKc : x ∈ Kᶜ := hxK
+  have hxi : x ∈ interior S := (mem_interior_iff_notMem_frontier hx).2 fun h => hxK (hfr h)
+  have hcomp : connectedComponentIn Kᶜ x ⊆ interior S := by
+    by_contra h
+    obtain ⟨y, hy, hyi⟩ := not_subset.mp h
+    obtain ⟨z, hz, hzf⟩ := isPreconnected_connectedComponentIn.inter_frontier_nonempty
+      ⟨x, mem_connectedComponentIn hxKc, hxi⟩ ⟨y, hy, hyi⟩
+    exact connectedComponentIn_subset _ _ hz (hfr (frontier_interior_subset hzf))
+  exact mem_filledHull_iff.mpr (hSb.subset (hcomp.trans interior_subset))
+
+end TauCeti
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 The Tau Ceti contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: The Tau Ceti contributors
+-/
+-- `NormedSpace.toLocallyConvexSpace`, needed to apply `geometric_hahn_banach_point_closed`.
+
+/-!
+# The width of a filled hull
+
+The filled hull `TauCeti.filledHull K` — `K` together with the bounded connected components of its
+complement — is defined in `TauCeti/Topology/FilledHull.lean`, where it needs only a topology and a
+bornology. In a real normed space the one substantial fact is that filling does not make a set
+wider:
+
+> `TauCeti.filledHull_subset_closedConvexHull` — `filledHull K ⊆ closedConvexHull ℝ K`,
+
+whence `TauCeti.diam_filledHull`: a set and its filled hull have the same diameter. The
+mechanism is separation: a point `x` outside the closed convex hull of `K` is cut off
+from it by a continuous linear functional (`geometric_hahn_banach_point_closed`), and the open
+half-space `{y | φ y < u}` so produced is a convex — hence preconnected — subset of `Kᶜ` containing
+`x`, and it is unbounded (`TauCeti.not_isBounded_halfSpace_lt`). So the component of `x` in `Kᶜ` is
+unbounded and `x` is not in the filled hull. Nonemptiness of `K` is needed only to know that
+`φ ≠ 0`; for `K = ∅` and a zero-dimensional space the convex-hull statement is false, the hull then
+being everything and the convex hull empty. The diameter statements survive that case
+unhypothesised, because `filledHull ∅` is empty in a nontrivial space
+(`TauCeti.filledHull_empty`) and the single point of the zero space otherwise, of diameter `0`
+either way.
+
+Because the width of a filled hull is controlled, so is that of anything inside it, and the shape
+in which this is spent is `IsPreconnected.subset_filledHull`: a preconnected set disjoint from `K`
+is trapped inside the filled hull as soon as it meets it. Their composite,
+`IsPreconnected.diam_le_diam_of_disjoint`, says that *a connected set that a small `K` cuts off
+from infinity is itself small*, with no regularity asked of `K`.
+
+## Roadmap role
+
+The filled hull is the vocabulary in which the enclosure step of layer **L5** of
+`TauCetiRoadmap/ConformalMapping/README.md` is stated. That step is now unconditional: the
+preconnectedness/winding-number route in `TauCeti/Analysis/Complex/Conformal/Crosscut/Inside.lean`
+places one image piece of a crosscut in the filled hull without plane separation. In the diameter
+bound that follows, `TauCeti/Analysis/Complex/Conformal/Crosscut/SmallJordanCurve.lean` encloses a
+short image crosscut in an arbitrarily small Jordan curve `J`, and
+`IsPreconnected.diam_le_diam_of_disjoint` makes the cut-off piece no wider than `J`.
+
+This is a different route to a diameter bound from `TauCeti.diam_le_diam_of_frontier_subset` of
+`TauCeti/Analysis/Normed/Module/DiamFrontier.lean`, which bounds a set by *any* bounded set
+containing its frontier: there the enclosing set must be known to contain the whole frontier, here
+only that the set is cut off from infinity. The frontier route is the special case of the enclosure
+route obtained from `TauCeti.subset_filledHull_of_frontier_subset`; the enclosure route does not
+require the whole frontier to be caught.
+
+## Generality
+
+The width statements are stated for an arbitrary real normed space — nothing about the plane is
+used, and the separation argument is the general Hahn–Banach one.
+
+## Main results
+
+* `TauCeti.filledHull_subset_closedConvexHull` — a filled hull lies in the closed convex hull.
+* `TauCeti.diam_filledHull` and `TauCeti.isBounded_filledHull` — filling preserves the diameter, and
+  a filled hull is bounded exactly when the set filled is.
+* `TauCeti.diam_le_diam_of_subset_filledHull` and
+  `IsPreconnected.diam_le_diam_of_disjoint` — a set inside the filled hull of a bounded `K`,
+  in particular a preconnected set that `K` cuts off from infinity, is no wider than `K`.
+* `TauCeti.isBounded_closedConvexHull`,
+  `TauCeti.diam_closedConvexHull` — the closed forms of the two convex-hull facts the width
+  argument runs on.
+* `TauCeti.connectedComponentIn_compl_eq_of_unbounded_component` — the unbounded connected
+  component of the complement of a bounded set is unique (dimension at least two).
+* `TauCeti.mem_filledHull_or_mem_filledHull_of_notMem_connectedComponentIn` — of two points in
+  different components, at least one lies in the filled hull (dimension at least two).
+-/
+
+@[expose] public section
+
+namespace TauCeti
+
+open Bornology Metric Set
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {K S : Set E}
+
+/-- **A closed convex hull is bounded exactly when the set is.** The closed form of
+`isBounded_convexHull`, the closure adding nothing. -/
+@[simp]
+theorem isBounded_closedConvexHull : IsBounded (closedConvexHull ℝ K) ↔ IsBounded K := by
+  rw [closedConvexHull_eq_closure_convexHull, isBounded_closure_iff, isBounded_convexHull]
+
+/-- **Taking the closed convex hull preserves the diameter.** The closed form of `convexHull_diam`,
+the closure adding nothing by `Metric.diam_closure`. -/
+@[simp]
+theorem diam_closedConvexHull : diam (closedConvexHull ℝ K) = diam K := by
+  rw [closedConvexHull_eq_closure_convexHull, diam_closure, convexHull_diam]
+
+/-- **The filled hull lies in the closed convex hull.** A point outside the closed convex hull of a
+nonempty `K` is separated from it by a continuous linear functional; the open half-space this
+produces is convex, avoids `K`, and is unbounded, so the component of the point in `Kᶜ` is
+unbounded.
+
+Nonemptiness of `K` is what forces the separating functional to be nonzero, and so the half-space to
+be unbounded; without it the statement fails in the zero space, where `filledHull ∅ = univ`. -/
+theorem filledHull_subset_closedConvexHull (hK : K.Nonempty) :
+    filledHull K ⊆ closedConvexHull ℝ K := by
+  intro x hx
+  rw [mem_filledHull_iff] at hx
+  by_contra hxC
+  obtain ⟨φ, u, hφx, hφC⟩ := geometric_hahn_banach_point_closed convex_closedConvexHull
+    isClosed_closedConvexHull hxC
+  -- The open half-space cut off by `φ` is a preconnected subset of `Kᶜ` containing `x`.
+  have hHK : {y | φ y < u} ⊆ Kᶜ :=
+    fun y hy hyK => absurd (hφC y (subset_closedConvexHull hyK)) (not_lt.mpr hy.le)
+  have hsub : {y | φ y < u} ⊆ connectedComponentIn Kᶜ x :=
+    (convex_halfSpace_lt φ.toLinearMap.isLinear u).isPreconnected.subset_connectedComponentIn
+      hφx hHK
+  -- It is unbounded, because a nonempty `K` forces `φ` to be nonzero.
+  obtain ⟨b, hb⟩ := hK
+  have hφne : (φ : E →ₗ[ℝ] ℝ) ≠ 0 := by
+    intro h
+    have hzero : ∀ y : E, φ y = 0 := fun y =>
+      (LinearMap.congr_fun h y).trans (LinearMap.zero_apply y)
+    have hb' := hφC b (subset_closedConvexHull hb)
+    rw [hzero] at hφx hb'
+    linarith
+  exact not_isBounded_halfSpace_lt (φ := (φ : E →ₗ[ℝ] ℝ)) hφne u (hx.subset hsub)
+
+/-- **The filled hull of the empty set is empty** in a nontrivial space: the whole space is
+connected and unbounded, so every component of `∅ᶜ = univ` is unbounded. -/
+@[simp]
+theorem filledHull_empty [Nontrivial E] : filledHull (∅ : Set E) = ∅ := by
+  apply filledHull_eq_self
+  · rw [compl_empty]
+    exact isPreconnected_univ
+  · rw [compl_empty]
+    exact NormedSpace.unbounded_univ ℝ E
+
+/-- The filled hull of the empty set is a subsingleton: empty in a nontrivial space by
+`TauCeti.filledHull_empty`, and the whole zero space, a single point, otherwise. Either way it is as
+wide as `∅`, which is why the diameter statements below need no nonemptiness hypothesis. -/
+private theorem subsingleton_filledHull_empty : (filledHull (∅ : Set E)).Subsingleton := by
+  rcases subsingleton_or_nontrivial E with _ | _
+  · exact fun a _ b _ => Subsingleton.elim a b
+  · rw [filledHull_empty]
+    exact subsingleton_empty
+
+/-- **A filled hull is bounded exactly when the set filled is.** One direction is
+`TauCeti.subset_filledHull`; the other holds because the hull lies in the closed convex hull. -/
+@[simp]
+theorem isBounded_filledHull : IsBounded (filledHull K) ↔ IsBounded K := by
+  refine ⟨fun h => h.subset subset_filledHull, fun hKb => ?_⟩
+  rcases K.eq_empty_or_nonempty with rfl | hK
+  · exact subsingleton_filledHull_empty.finite.isBounded
+  · exact (isBounded_closedConvexHull.mpr hKb).subset (filledHull_subset_closedConvexHull hK)
+
+/-- **Filling preserves the diameter.** The hull contains `K`, and for nonempty `K` it is contained
+in the closed convex hull of `K`, which by `TauCeti.diam_closedConvexHull` is exactly as wide as
+`K`; `filledHull ∅` is a subsingleton, of diameter `0` like `∅` itself. An unbounded `K` has an
+unbounded hull by `TauCeti.isBounded_filledHull`, and both diameters are then `0`. -/
+@[simp]
+theorem diam_filledHull : diam (filledHull K) = diam K := by
+  by_cases hKb : IsBounded K
+  · rcases K.eq_empty_or_nonempty with rfl | hK
+    · rw [diam_subsingleton subsingleton_filledHull_empty, diam_empty]
+    refine le_antisymm ?_ (diam_mono subset_filledHull (isBounded_filledHull.mpr hKb))
+    calc diam (filledHull K) ≤ diam (closedConvexHull ℝ K) :=
+          diam_mono (filledHull_subset_closedConvexHull hK) (isBounded_closedConvexHull.mpr hKb)
+      _ = diam K := diam_closedConvexHull
+  · rw [diam_eq_zero_of_unbounded (mt isBounded_filledHull.mp hKb), diam_eq_zero_of_unbounded hKb]
+
+/-- **Anything a bounded `K` cuts off from infinity is no wider than `K`.** A set inside the filled
+hull is no wider than the hull by `Metric.diam_mono`, and the hull is no wider than `K` by
+`TauCeti.diam_filledHull`. -/
+theorem diam_le_diam_of_subset_filledHull (hK : IsBounded K) (h : S ⊆ filledHull K) :
+    diam S ≤ diam K :=
+  (diam_mono h (isBounded_filledHull.mpr hK)).trans_eq diam_filledHull
+
+/-- **A preconnected set that a bounded `K` cuts off from infinity is no wider than `K`.** If `S` is
+preconnected, disjoint from `K`, and meets the filled hull of `K`, then it lies inside that hull by
+`IsPreconnected.subset_filledHull`, which is no wider than `K` by
+`TauCeti.diam_filledHull`. No regularity is asked of `K`. -/
+theorem _root_.IsPreconnected.diam_le_diam_of_disjoint (hS : IsPreconnected S) (hSK : Disjoint S K)
+    (hne : (S ∩ filledHull K).Nonempty) (hK : IsBounded K) : diam S ≤ diam K :=
+  diam_le_diam_of_subset_filledHull hK (hS.subset_filledHull hSK hne)
+
+variable {x y : E}
+
+/-- **The unbounded component of the complement of a bounded set is unique** in a real normed space
+of dimension at least two. -/
+theorem connectedComponentIn_compl_eq_of_unbounded_component (h : 1 < Module.rank ℝ E)
+    (hK : IsBounded K) (hx : ¬ IsBounded (connectedComponentIn Kᶜ x))
+    (hy : ¬ IsBounded (connectedComponentIn Kᶜ y)) :
+    connectedComponentIn Kᶜ x = connectedComponentIn Kᶜ y := by
+  obtain ⟨R, hR⟩ := hK.subset_closedBall (0 : E)
+  have hext : (closedBall (0 : E) R)ᶜ ⊆ Kᶜ := compl_subset_compl.mpr hR
+  have hesc : ∀ z : E, ¬ IsBounded (connectedComponentIn Kᶜ z) →
+      ∃ z' ∈ connectedComponentIn Kᶜ z, z' ∈ (closedBall (0 : E) R)ᶜ := fun z hz => by
+    by_contra hcon
+    push Not at hcon
+    exact hz ((isBounded_closedBall (x := (0 : E)) (r := R)).subset fun w hw =>
+      notMem_compl_iff.mp (hcon w hw))
+  obtain ⟨x', hx'c, hx'R⟩ := hesc x hx
+  obtain ⟨y', hy'c, hy'R⟩ := hesc y hy
+  have h1 : y' ∈ connectedComponentIn Kᶜ x' :=
+    (isPreconnected_compl_closedBall h 0 R).subset_connectedComponentIn hx'R hext hy'R
+  rw [connectedComponentIn_eq hx'c, connectedComponentIn_eq h1, ← connectedComponentIn_eq hy'c]
+
+/-- **Two points in different components of the complement of a bounded set cannot both lie outside
+the filled hull** in a real normed space of dimension at least two. -/
+theorem mem_filledHull_or_mem_filledHull_of_notMem_connectedComponentIn (h : 1 < Module.rank ℝ E)
+    (hK : IsBounded K) (hxy : y ∉ connectedComponentIn Kᶜ x) :
+    x ∈ filledHull K ∨ y ∈ filledHull K := by
+  by_cases hy : y ∈ K
+  · exact Or.inr (subset_filledHull hy)
+  · by_contra hcon
+    push Not at hcon
+    simp only [mem_filledHull_iff] at hcon
+    exact hxy ((connectedComponentIn_compl_eq_of_unbounded_component h hK hcon.1 hcon.2).symm ▸
+      mem_connectedComponentIn (mem_compl hy))
+
+end TauCeti
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 The Tau Ceti contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: The Tau Ceti contributors
+-/
+/-!
+# Local connectedness of continuous images of compact spaces
+
+Local connectedness is not preserved by continuous images in general — every metric space is a
+continuous image of a discrete one — but it *is* preserved by quotient maps, and hence by the
+continuous images that are automatically quotient maps: those of a compact space in a Hausdorff
+one. This file proves that, in the type-level form and in the set-level form
+`TauCeti.locallyConnectedSpace_image_of_isCompact` that a subset of a topological space needs.
+
+The quotient step itself is Mathlib's. `Topology.IsCoinducing.locallyConnectedSpace` states that
+a topology coinduced by a locally connected one is locally connected, which reaches quotient maps
+through `IsQuotientMap.isCoinducing` and is strictly more general, since coinducing does not ask
+for surjectivity. What is added here is the passage from that to a continuous surjection out of a
+compact space, which is closed and therefore a quotient map.
+
+The intended consumer is layer **L5** of the conformal-mapping roadmap, Carathéodory's boundary
+correspondence: a conformal map that extends continuously to the closure of its domain carries a
+locally connected boundary to a locally connected boundary, which is the necessary half of
+Carathéodory's continuity theorem. That application is in
+`TauCeti/Analysis/Complex/Conformal/LocallyConnectedBoundary.lean`; nothing here is specific to it.
+
+## Main results
+
+* `TauCeti.locallyConnectedSpace_of_continuous_surjective` — the continuous image of a compact
+  locally connected space in a Hausdorff space is locally connected.
+* `TauCeti.locallyConnectedSpace_image_of_isCompact` — the set-level form: `f '' s` is locally
+  connected for a compact, locally connected `s` on which `f` is continuous.
+
+## References
+
+* Mathlib, `Mathlib.Topology.Connected.LocallyConnected`,
+  `Topology.IsCoinducing.locallyConnectedSpace`: a topology coinduced by a locally connected
+  topology is locally connected. This is the quotient-map result applied below.
+-/
+
+@[expose] public section
+
+namespace TauCeti
+
+open Set Topology
+
+variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
+
+/-- **The continuous image of a compact locally connected space in a Hausdorff space is locally
+connected.** A continuous surjection out of a compact space onto a Hausdorff one is closed, hence
+a quotient map, hence coinducing, so `Topology.IsCoinducing.locallyConnectedSpace` applies. -/
+theorem locallyConnectedSpace_of_continuous_surjective [LocallyConnectedSpace X] [CompactSpace X]
+    [T2Space Y] {f : X → Y} (hf : Continuous f) (hsurj : Function.Surjective f) :
+    LocallyConnectedSpace Y :=
+  (hf.isClosedMap.isQuotientMap hf hsurj).isCoinducing.locallyConnectedSpace
+
+/-- **The set-level form: a compact, locally connected set has locally connected continuous
+images.** Stated with the subtype topologies on `s` and on `f '' s`, which is how a boundary or a
+closure of a subset of a normed space is met in practice. -/
+theorem locallyConnectedSpace_image_of_isCompact [T2Space Y] {s : Set X} {f : X → Y}
+    [LocallyConnectedSpace s] (hs : IsCompact s) (hf : ContinuousOn f s) :
+    LocallyConnectedSpace (f '' s) := by
+  have : CompactSpace s := isCompact_iff_compactSpace.mp hs
+  exact locallyConnectedSpace_of_continuous_surjective (hf.mapsToRestrict (mapsTo_image f s))
+    (surjective_mapsTo_image_restrict f s)
+
+end TauCeti
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 The Tau Ceti contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: The Tau Ceti contributors
+-/
+/-!
+# Jordan curves
+
+A **Jordan curve** — a simple closed curve — is a subset of a topological space homeomorphic to
+the circle. This file introduces the predicate `TauCeti.IsJordanCurve` and its basic API.
+
+The circle is Mathlib's `Circle`, the unit circle of `ℂ` as a topological group; the notion itself
+is purely topological, so `TauCeti.IsJordanCurve` is stated for a subset of an arbitrary
+topological space. `ℂ` is mentioned only by the two concrete curves towards the end of the file:
+the model curve, a circle `Metric.sphere c r` of positive radius, and the frontier of a bounded
+convex set with nonempty interior. The model is built from the complex affine change of coordinates
+`w ↦ (w - c) / r`, so it uses the field structure of `ℂ` and not only its metric, and the convex
+frontier is obtained from the model by transport.
+
+Phrasing the predicate as *the set is homeomorphic to the circle*, rather than *the set is the
+range of a continuous map on `[0, 1]` that is injective except for matching endpoints*, is what
+makes it usable: over a Hausdorff ambient space the two agree, because there a continuous injection
+out of a compact space is an embedding, but the parametrized form buries that argument in every
+use. Passing from a parametrization to the predicate is `TauCeti.IsJordanCurve.image`, which turns
+a continuous injective map defined on a set already known to be a Jordan curve — a circle in `ℂ`,
+say — into a proof that its image is one; `TauCeti.IsJordanCurve.of_image` runs the other way,
+transporting the property back to a compact set from an image already known to be a Jordan curve.
+
+## Main definitions
+
+* `TauCeti.IsJordanCurve` — a set homeomorphic to the circle.
+* `TauCeti.jordanParam` — the parametrization of a Jordan curve by the circle underlying a
+  homeomorphism of the curve with `Circle`.
+
+## Main results
+
+* `TauCeti.IsJordanCurve.isCompact`, `TauCeti.IsJordanCurve.isPathConnected`,
+  `TauCeti.IsJordanCurve.nonempty` and `TauCeti.IsJordanCurve.not_subsingleton` — a Jordan curve is
+  a nonempty compact path-connected set with more than one point.
+* `TauCeti.IsJordanCurve.image` and `TauCeti.IsJordanCurve.of_image` — being a Jordan curve
+  transfers in both directions along a map that is continuous and injective on the set, provided
+  the codomain is Hausdorff (and, in the direction that transports the property back from the
+  image, the source set is known to be compact).
+* `TauCeti.IsJordanCurve.image_homeomorph` and `TauCeti.isJordanCurve_image_homeomorph_iff` — being
+  a Jordan curve is invariant under a homeomorphism of the ambient spaces; no separation axiom is
+  needed.
+* `TauCeti.continuous_jordanParam`, `TauCeti.jordanParam_injective`,
+  `TauCeti.isInducing_jordanParam`, `TauCeti.range_jordanParam`, `TauCeti.jordanParam_apply` and
+  `TauCeti.jordanParam_apply_apply` — the parametrization of a Jordan curve by the circle is a
+  continuous injection, is inducing, traces out exactly the curve, and undoes `e`; this is what
+  carries a statement about the circle to one about an arbitrary Jordan curve.
+* `TauCeti.sphereCircleHomeomorph` and `TauCeti.isJordanCurve_sphere` — a circle of positive radius
+  in `ℂ` is a Jordan curve, by the affine change of coordinates `w ↦ (w - c) / r`.
+* `TauCeti.isJordanCurve_frontier_of_convex` — the frontier of a bounded convex subset of `ℂ` with
+  nonempty interior is a Jordan curve. This is `TauCeti.isJordanCurve_sphere` with the disc
+  weakened to an arbitrary convex body, the affine change of coordinates being replaced by Mathlib's
+  gauge rescaling.
+* `TauCeti.locallyConnectedSpace_sphere` and `TauCeti.IsJordanCurve.locallyConnectedSpace` — a
+  circle in `ℂ`, and hence every Jordan curve, is locally connected.
+
+## Motivation
+
+This is the vocabulary layer **L5** of the conformal-mapping roadmap
+(`TauCetiRoadmap/ConformalMapping/README.md`) is stated in: its milestone, the Carathéodory
+boundary correspondence, is about the Riemann map of a *Jordan domain*, and the roadmap records
+that the pinned Mathlib has no Jordan-curve vocabulary to state it against. The complex-analytic
+half — Jordan domains, the discs among them, and the boundary of a domain that a conformal map
+carries onto a disc — is in `TauCeti/Analysis/Complex/Conformal/Jordan/Domain.lean`.
+
+Local connectedness of a Jordan curve is what that milestone needs of the *hypothesis* side: the
+route to the extension theorem for a Jordan domain `Ω` runs through Carathéodory's continuity
+theorem, whose hypothesis is that `frontier Ω` be locally connected, and
+`TauCeti.IsJordanCurve.locallyConnectedSpace` is what discharges it (as
+`TauCeti.IsJordanDomain.locallyConnectedSpace_frontier`). Mathlib knows the circle is compact,
+connected and path connected, but records no local connectedness for it, and the property is not
+preserved by continuous images, so it is proved here from
+`TauCeti.locallyConnectedSpace_image_of_isCompact`.
+
+## References
+
+* C. Jordan, *Cours d'analyse de l'École Polytechnique*, vol. 3 (1887).
+* Ch. Pommerenke, *Boundary Behaviour of Conformal Maps*, Ch. 2.
+-/
+
+@[expose] public section
+
+namespace TauCeti
+
+open Metric Set Topology
+
+variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {C : Set X} {p : X} {r : ℝ}
+
+/-- A **Jordan curve**, or simple closed curve, in a topological space: a subset homeomorphic to
+the circle.
+
+The predicate is `Nonempty (C ≃ₜ Circle)` rather than a chosen homeomorphism, so that it is a
+`Prop`; `TauCeti.isJordanCurve_iff` recovers the homeomorphism from another module, where the
+definition itself is not exposed. -/
+def IsJordanCurve (C : Set X) : Prop := Nonempty (C ≃ₜ Circle)
+
+/-- A set is a Jordan curve exactly when it is homeomorphic to the circle. This is the interface
+to `TauCeti.IsJordanCurve` outside its defining module. -/
+theorem isJordanCurve_iff : IsJordanCurve C ↔ Nonempty (C ≃ₜ Circle) := Iff.rfl
+
+/-- A continuous injection defined on a compact set is a homeomorphism onto its image, the image
+carrying the subspace topology of a Hausdorff space. This is the set-level form of
+`Continuous.homeoOfEquivCompactToT2`, and the engine of both transfer lemmas below. -/
+private noncomputable def imageHomeomorphOfIsCompact [T2Space Y] (hC : IsCompact C) {g : X → Y}
+    (hg : ContinuousOn g C) (hgi : InjOn g C) : C ≃ₜ g '' C :=
+  haveI : CompactSpace C := isCompact_iff_compactSpace.mp hC
+  Continuous.homeoOfEquivCompactToT2 (f := hgi.bijOn_image.equiv g)
+    (hg.mapsToRestrict hgi.bijOn_image.mapsTo)
+
+/-- A Jordan curve is compact: the circle is. -/
+theorem IsJordanCurve.isCompact (h : IsJordanCurve C) : IsCompact C := by
+  obtain ⟨e⟩ := h
+  exact isCompact_iff_compactSpace.mpr e.symm.compactSpace
+
+/-- A Jordan curve in a Hausdorff space is closed. -/
+theorem IsJordanCurve.isClosed [T2Space X] (h : IsJordanCurve C) : IsClosed C :=
+  h.isCompact.isClosed
+
+/-- A Jordan curve is path connected: the circle is. -/
+theorem IsJordanCurve.isPathConnected (h : IsJordanCurve C) : IsPathConnected C := by
+  obtain ⟨e⟩ := h
+  exact isPathConnected_iff_pathConnectedSpace.mpr
+    (e.symm.surjective.pathConnectedSpace e.symm.continuous)
+
+/-- A Jordan curve is connected. -/
+theorem IsJordanCurve.isConnected (h : IsJordanCurve C) : IsConnected C :=
+  h.isPathConnected.isConnected
+
+/-- A Jordan curve is nonempty. -/
+theorem IsJordanCurve.nonempty (h : IsJordanCurve C) : C.Nonempty := h.isConnected.nonempty
+
+/-- A Jordan curve has more than one point: the circle contains both `1` and `-1`. Together with
+`TauCeti.IsJordanCurve.isConnected` this rules out the degenerate curves, so a Jordan curve is a
+nondegenerate continuum. -/
+theorem IsJordanCurve.not_subsingleton (h : IsJordanCurve C) : ¬ C.Subsingleton := by
+  obtain ⟨e⟩ := h
+  intro hsub
+  exact Circle.neg_ne_self 1
+    (e.symm.injective (Subtype.ext (hsub (e.symm (-1)).2 (e.symm 1).2)))
+
+/-- **A Jordan curve is carried to a Jordan curve by a continuous injection.** Only continuity and
+injectivity *on the curve* are needed, the curve supplying the compactness that upgrades them to a
+homeomorphism onto the image. -/
+theorem IsJordanCurve.image [T2Space Y] (h : IsJordanCurve C) {g : X → Y} (hg : ContinuousOn g C)
+    (hgi : InjOn g C) : IsJordanCurve (g '' C) := by
+  have hC := h.isCompact
+  obtain ⟨e⟩ := h
+  exact ⟨(imageHomeomorphOfIsCompact hC hg hgi).symm.trans e⟩
+
+/-- **A compact set carried onto a Jordan curve by a continuous injection is a Jordan curve.**
+This is the converse of `TauCeti.IsJordanCurve.image`; compactness of the source has to be assumed
+here, since it is no longer inherited from the curve. It is the form in which the predicate is
+verified when the curve is the *unknown* rather than the parameter: one exhibits a continuous
+injective map of the set onto a set already known to be a Jordan curve, as the boundary
+correspondence does with the boundary of a disc. -/
+theorem IsJordanCurve.of_image [T2Space Y] (hC : IsCompact C) {g : X → Y} (hg : ContinuousOn g C)
+    (hgi : InjOn g C) (h : IsJordanCurve (g '' C)) : IsJordanCurve C := by
+  obtain ⟨e⟩ := h
+  exact ⟨(imageHomeomorphOfIsCompact hC hg hgi).trans e⟩
+
+/-- The image of a Jordan curve under a homeomorphism of the ambient spaces is a Jordan curve.
+Unlike `TauCeti.IsJordanCurve.image` this needs no separation axiom on the codomain, because
+`Homeomorph.image` supplies the homeomorphism onto the image outright. -/
+theorem IsJordanCurve.image_homeomorph (h : IsJordanCurve C) (e : X ≃ₜ Y) :
+    IsJordanCurve (e '' C) := by
+  obtain ⟨f⟩ := h
+  exact ⟨(e.image C).symm.trans f⟩
+
+/-- Being a Jordan curve is invariant under a homeomorphism of the ambient spaces. This is the
+characteristic form of `TauCeti.IsJordanCurve.image_homeomorph`: the backward direction is that
+lemma applied to `e.symm`, which no consumer then has to spell out. -/
+@[simp]
+theorem isJordanCurve_image_homeomorph_iff (e : X ≃ₜ Y) :
+    IsJordanCurve (e '' C) ↔ IsJordanCurve C :=
+  ⟨fun h => e.symm_image_image C ▸ h.image_homeomorph e.symm, fun h => h.image_homeomorph e⟩
+
+/-! ## The parametrization by the circle
+
+Every transport of a statement about `Circle` to a Jordan curve goes through the parametrization
+`jordanParam e` attached to a homeomorphism `e`, so its properties — continuity, injectivity,
+range, and that it is inducing — are collected here rather than rebuilt at each use, both by the
+cutting of a curve at one or two of its points
+(`TauCeti/Topology/JordanCurve/Separation.lean`) and by the quantitative form of that cutting
+(`TauCeti/Topology/JordanCurve/SmallArc.lean`). -/
+
+/-- The parametrization of a Jordan curve by the circle underlying a homeomorphism `e`: the
+composite of `e.symm` with the inclusion of the curve into the ambient space. -/
+noncomputable def jordanParam (e : C ≃ₜ Circle) : Circle → X :=
+  fun u => ((e.symm u : C) : X)
+
+/-- The parametrization `TauCeti.jordanParam` of a Jordan curve by the circle is continuous. -/
+lemma continuous_jordanParam (e : C ≃ₜ Circle) : Continuous (jordanParam e) :=
+  continuous_subtype_val.comp e.symm.continuous
+
+/-- The parametrization `TauCeti.jordanParam` of a Jordan curve by the circle is injective: this is
+the simplicity of the curve. -/
+lemma jordanParam_injective (e : C ≃ₜ Circle) : Function.Injective (jordanParam e) :=
+  Subtype.val_injective.comp e.symm.injective
+
+/-- The parametrization `TauCeti.jordanParam` of a Jordan curve by the circle is inducing, so
+preconnectedness of a subset of the curve may be tested on its preimage of parameters. -/
+lemma isInducing_jordanParam (e : C ≃ₜ Circle) : Topology.IsInducing (jordanParam e) :=
+  Topology.IsInducing.subtypeVal.comp e.symm.isInducing
+
+/-- The parametrization `TauCeti.jordanParam` of a Jordan curve by the circle traces out exactly the
+curve. -/
+@[simp]
+lemma range_jordanParam (e : C ≃ₜ Circle) : range (jordanParam e) = C := by
+  refine subset_antisymm ?_ fun x hx => ⟨e ⟨x, hx⟩, by simp [jordanParam]⟩
+  rintro _ ⟨u, rfl⟩
+  exact (e.symm u).2
+
+/-- The defining equation of `TauCeti.jordanParam`: the parameter `u` names the point `e.symm u` of
+the curve, read in the ambient space. This is the general application lemma, so a consumer never has
+to unfold the definition. -/
+@[simp]
+lemma jordanParam_apply (e : C ≃ₜ Circle) (u : Circle) :
+    jordanParam e u = ((e.symm u : C) : X) := by
+  simp [jordanParam]
+
+/-- The parametrization `TauCeti.jordanParam` of a Jordan curve by the circle undoes `e`: it sends
+the parameter `e ⟨p, hp⟩` of a point `p` of the curve back to `p`. Simp proves this from
+`TauCeti.jordanParam_apply`; it is stated for the `rw` steps that produce the point `p` itself
+rather than a coerced parameter. -/
+lemma jordanParam_apply_apply (e : C ≃ₜ Circle) (hp : p ∈ C) :
+    jordanParam e (e ⟨p, hp⟩) = p := by
+  simp
+
+/-! ## The model curve: a circle in `ℂ` -/
+
+/-- `Circle` is *by definition* the unit sphere of `ℂ`, but it is a `def` rather than an
+abbreviation, so its topology is only definitionally the subtype topology. This identification is
+the one place where that unfolding happens; the lemmas below then compute with the unit sphere
+alone. -/
+noncomputable def unitSphereCircleHomeomorph : sphere (0 : ℂ) 1 ≃ₜ Circle :=
+  Homeomorph.refl _
+
+private lemma coe_unitSphereCircleHomeomorph (z : sphere (0 : ℂ) 1) :
+    (unitSphereCircleHomeomorph z : ℂ) = (z : ℂ) := rfl
+
+private lemma coe_unitSphereCircleHomeomorph_symm (z : Circle) :
+    ((unitSphereCircleHomeomorph.symm z : sphere (0 : ℂ) 1) : ℂ) = (z : ℂ) := rfl
+
+/-- The affine parametrization `w ↦ (w - c) / r` of a circle of centre `c` and positive radius `r`
+in `ℂ` by the unit circle. It is the restriction to the spheres of the inverse of Mathlib's ambient
+`affineHomeomorph r c`, so only the membership equivalence is proved here. -/
+noncomputable def sphereCircleHomeomorph (c : ℂ) (hr : 0 < r) : sphere c r ≃ₜ Circle :=
+  haveI hr0 : (r : ℂ) ≠ 0 := by exact_mod_cast hr.ne'
+  (((affineHomeomorph (r : ℂ) c hr0).symm).subtype fun w => by
+      rw [affineHomeomorph_symm_apply]
+      simp [mem_sphere_iff_norm, abs_of_pos hr, div_eq_one_iff_eq, hr.ne']).trans
+    unitSphereCircleHomeomorph
+
+/-- The parametrization of `sphere c r` by the unit circle divides out the affine change of
+coordinates. -/
+@[simp]
+lemma coe_sphereCircleHomeomorph_apply (c : ℂ) (hr : 0 < r) (w : sphere c r) :
+    (sphereCircleHomeomorph c hr w : ℂ) = ((w : ℂ) - c) / r := by
+  simp [sphereCircleHomeomorph, coe_unitSphereCircleHomeomorph]
+
+/-- The inverse parametrization of `sphere c r` by the unit circle is the affine change of
+coordinates. -/
+@[simp]
+lemma coe_sphereCircleHomeomorph_symm_apply (c : ℂ) (hr : 0 < r) (z : Circle) :
+    (((sphereCircleHomeomorph c hr).symm z : sphere c r) : ℂ) = c + r * (z : ℂ) := by
+  simp [sphereCircleHomeomorph, coe_unitSphereCircleHomeomorph_symm, add_comm]
+
+/-- A circle of positive radius in `ℂ` is a Jordan curve. -/
+theorem isJordanCurve_sphere (c : ℂ) (hr : 0 < r) : IsJordanCurve (sphere c r) :=
+  isJordanCurve_iff.mpr ⟨sphereCircleHomeomorph c hr⟩
+
+/-- **The frontier of a bounded convex subset of `ℂ` with nonempty interior is a Jordan curve.**
+
+This is `TauCeti.isJordanCurve_sphere` with the disc weakened to an arbitrary convex body, which it
+recovers at `s = Metric.ball c r`. The model curve transports because Mathlib's gauge rescaling
+supplies an *ambient* homeomorphism `e : ℂ ≃ₜ ℂ` carrying `frontier s` onto the unit circle
+(`exists_homeomorph_image_interior_closure_frontier_eq_unitBall`), so the affine change of
+coordinates above is simply replaced by a nonlinear one and no further topology is needed.
+
+The set is asked neither to be open nor to be nonempty: what a convex set needs in order to have a
+one-dimensional frontier is that it be *solid*, and that is `(interior s).Nonempty`. Without it the
+statement fails — a segment is convex and bounded, and is its own frontier. -/
+theorem isJordanCurve_frontier_of_convex {s : Set ℂ} (hs : Convex ℝ s)
+    (hne : (interior s).Nonempty) (hb : Bornology.IsBounded s) : IsJordanCurve (frontier s) := by
+  obtain ⟨e, -, -, hfr⟩ := exists_homeomorph_image_interior_closure_frontier_eq_unitBall hs hne hb
+  refine (isJordanCurve_image_homeomorph_iff e).mp ?_
+  rw [hfr]
+  exact isJordanCurve_sphere 0 one_pos
+
+/-! ## Local connectedness -/
+
+open Complex in
+/-- **A circle in `ℂ` is locally connected.** It is the image of the compact interval `[-π, π]`,
+which is convex and hence locally connected, under the continuous `θ ↦ c + r * exp (θ * I)`, so
+`TauCeti.locallyConnectedSpace_image_of_isCompact` applies. A sphere of negative radius is empty,
+and vacuously locally connected. -/
+instance locallyConnectedSpace_sphere (c : ℂ) (r : ℝ) : LocallyConnectedSpace (sphere c r) := by
+  rcases lt_or_ge r 0 with hr | hr
+  · have hempty : sphere c r = (∅ : Set ℂ) := by
+      ext z
+      simp only [mem_sphere_iff_norm, mem_empty_iff_false, iff_false]
+      intro h
+      have : (0 : ℝ) ≤ r := h ▸ norm_nonneg (z - c)
+      linarith
+    rw [hempty]
+    exact ⟨fun x => absurd x.2 (notMem_empty _)⟩
+  · have hparam :
+        sphere c r = (fun θ : ℝ => c + r * exp (θ * I)) '' Icc (-Real.pi) Real.pi := by
+      ext z
+      simp only [mem_sphere_iff_norm, mem_image, mem_Icc]
+      constructor
+      · intro hz
+        refine ⟨arg (z - c), ⟨(neg_pi_lt_arg (z - c)).le, arg_le_pi (z - c)⟩, ?_⟩
+        rw [← hz, norm_mul_exp_arg_mul_I (z - c)]
+        ring
+      · rintro ⟨θ, -, rfl⟩
+        simp [abs_of_nonneg hr]
+    have := (convex_Icc (-Real.pi) Real.pi).locallyPathConnectedSpace
+    rw [hparam]
+    exact locallyConnectedSpace_image_of_isCompact isCompact_Icc
+      (Continuous.continuousOn (by fun_prop))
+
+/-- **The circle is locally connected.** `Circle` is the unit circle of `ℂ`, so this is the unit
+case of `TauCeti.locallyConnectedSpace_sphere` transported along `TauCeti.sphereCircleHomeomorph`.
+Mathlib records the circle as compact, connected and path connected, but not as locally
+connected. -/
+instance locallyConnectedSpace_circle : LocallyConnectedSpace Circle :=
+  (sphereCircleHomeomorph (0 : ℂ) one_pos).symm.locallyConnectedSpace
+
+/-- **A Jordan curve is locally connected**, being homeomorphic to the circle.
+
+This is the form in which the hypothesis of Carathéodory's continuity theorem — that the boundary
+of the domain be locally connected — is met by a Jordan domain, which is what layer **L5** of the
+conformal-mapping roadmap is about; see `TauCeti.IsJordanDomain.locallyConnectedSpace_frontier`. -/
+theorem IsJordanCurve.locallyConnectedSpace (h : IsJordanCurve C) : LocallyConnectedSpace C :=
+  (isJordanCurve_iff.mp h).elim fun e => e.locallyConnectedSpace
+
+end TauCeti
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 The Tau Ceti contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: The Tau Ceti contributors
+-/
+/-!
+# Jordan curves traced by paths
+
+A path whose two endpoints agree is a parametrised closed curve, but its range need not be a
+Jordan curve: the path may pause, retrace an arc, or cross itself. This file supplies the exact
+criterion needed to exclude those degeneracies. If a closed path has no repeated values except
+for its two endpoint parameters, its range is a Jordan curve
+(`TauCeti.isJordanCurve_range_of_eq_or_eq_endpoints`).
+
+The proof uses the quotient model of the circle already in Mathlib. The extension of a path
+`γ : Path x x` to `ℝ` has equal values at `0` and `1`, so
+`AddCircle.liftIco 1 0 γ.extend` factors it through the additive circle `ℝ / ℤ`. The hypothesis on
+repetitions says precisely that this factor is injective, and its range is the range of `γ`. The
+additive circle is itself a Jordan curve, `AddCircle.homeomorphCircle` identifying it with
+`Circle`, so `TauCeti.IsJordanCurve.image` carries that along the factor: the compactness argument
+upgrading a continuous injection to a homeomorphism onto its image is already packaged there and is
+not repeated here.
+
+The condition is stated directly rather than bundled as a new notion of simple closed path. This
+is the only operation needed here, and keeping it as a theorem hypothesis avoids introducing a
+second simplicity vocabulary alongside Mathlib's path API.
+
+## Gluing two arcs
+
+The criterion has one immediate use that is worth naming on its own: two *arcs* — ranges of
+injective paths — that share their two endpoints and meet nowhere else glue to a Jordan curve
+(`TauCeti.isJordanCurve_range_union_range_of_inter_eq_pair`). The closed path traversed is
+`γ.trans δ.symm`, whose range is `range γ ∪ range δ`; the meeting hypothesis is what turns a
+coincidence between a value of `γ` and a value of `δ` into a coincidence of endpoints, and
+injectivity of each of the two paths handles the coincidences internal to one of them. The
+endpoints are not asked to be distinct: injectivity of `δ` already forces that, since a path with
+equal endpoints repeats the value at the two distinct parameters `0` and `1`.
+
+## Main results
+
+* `TauCeti.isJordanCurve_range_of_eq_or_eq_endpoints` — the range of a closed path whose only
+  possible repetition is its pair of endpoints is a Jordan curve.
+* `TauCeti.isJordanCurve_range_union_range_of_inter_eq_pair` — two arcs with the same two
+  endpoints, meeting exactly there, glue to a Jordan curve.
+
+## Roadmap role
+
+This is the topological gluing step used by layer **L5** of
+`TauCetiRoadmap/ConformalMapping/README.md`, the Carathéodory boundary correspondence. A
+finite-length image crosscut is already packaged as a path with exactly this simplicity property in
+`TauCeti/Analysis/Complex/Conformal/Crosscut/Path.lean`; when its two boundary ends coincide, the
+first result below identifies the closure of that crosscut as a Jordan curve, and when they are
+distinct the second closes that crosscut up with an arc of the boundary of the image domain. The
+coincident-end specialization is in `TauCeti/Analysis/Complex/Conformal/Crosscut/Jordan.lean` and
+the distinct-end one in `TauCeti/Analysis/Complex/Conformal/Crosscut/Arc.lean`.
+-/
+
+@[expose] public section
+
+namespace TauCeti
+
+open Set
+
+variable {X : Type*} [TopologicalSpace X] [T2Space X] {x : X}
+
+omit [T2Space X] in
+/-- **A simple closed path lifts injectively to the circle.** If equality `γ s = γ t` forces `s = t`
+or the unordered pair of parameters to be `{0, 1}`, then `AddCircle.liftIco 1 0 γ.extend` is
+injective. -/
+private theorem liftIco_extend_injective (γ : Path x x)
+    (hγ : ∀ ⦃s t : unitInterval⦄, γ s = γ t →
+      s = t ∨ (s = 0 ∧ t = 1) ∨ (s = 1 ∧ t = 0)) :
+    Function.Injective (AddCircle.liftIco 1 0 γ.extend) := by
+  intro q q' hqq'
+  -- representatives in `[0, 1)` cannot form the exceptional endpoint pair
+  obtain ⟨s, hs, rfl⟩ := AddCircle.eq_coe_Ico q
+  obtain ⟨t, ht, rfl⟩ := AddCircle.eq_coe_Ico q'
+  rw [(AddCircle.liftIco_zero_coe_apply hs).trans (γ.extend_extends' ⟨s, hs.1, hs.2.le⟩),
+    (AddCircle.liftIco_zero_coe_apply ht).trans (γ.extend_extends' ⟨t, ht.1, ht.2.le⟩)] at hqq'
+  rcases hγ hqq' with hst | hends | hends
+  · exact congrArg (fun u : unitInterval => ((u : ℝ) : AddCircle (1 : ℝ))) hst
+  · exact absurd (congrArg ((↑) : unitInterval → ℝ) hends.2) ht.2.ne
+  · exact absurd (congrArg ((↑) : unitInterval → ℝ) hends.1) hs.2.ne
+
+/-- **The range of a simple closed path is a Jordan curve.** Let `γ : Path x x` be a closed path.
+If equality `γ s = γ t` forces either `s = t` or the unordered pair of parameters to be `{0, 1}`,
+then `range γ` is homeomorphic to the circle.
+
+The disjunction records both orientations of the exceptional endpoint pair explicitly. No local
+injectivity or embedding hypothesis is needed, and no separation assumption on the ambient space
+beyond the Hausdorffness that `TauCeti.IsJordanCurve.image` asks for. -/
+theorem isJordanCurve_range_of_eq_or_eq_endpoints (γ : Path x x)
+    (hγ : ∀ ⦃s t : unitInterval⦄, γ s = γ t →
+      s = t ∨ (s = 0 ∧ t = 1) ∨ (s = 1 ∧ t = 0)) :
+    IsJordanCurve (range γ) := by
+  -- Factor the extended path through `[0, 1]` with its endpoints identified; the additive circle
+  -- is a Jordan curve, and the factor carries it onto the range of `γ`.
+  have hg0 : γ.extend 0 = γ.extend 1 := by rw [γ.extend_zero, γ.extend_one]
+  have hgc : Continuous (AddCircle.liftIco 1 0 γ.extend) :=
+    AddCircle.liftIco_zero_continuous hg0 γ.continuous_extend.continuousOn
+  have huniv : IsJordanCurve (univ : Set (AddCircle (1 : ℝ))) :=
+    isJordanCurve_iff.mpr
+      ⟨(Homeomorph.Set.univ (AddCircle (1 : ℝ))).trans (AddCircle.homeomorphCircle one_ne_zero)⟩
+  have himage := huniv.image hgc.continuousOn (liftIco_extend_injective γ hγ).injOn
+  -- present the lift as a quotient map off `[0, 1]` and read its range through the quotient
+  rwa [image_univ, AddCircle.liftIco_eq_lift_Icc (by simp), (Equiv.surjective _).range_comp,
+    Set.range_quot_lift, Set.range_domRestrict, γ.image_extend_of_subset (by norm_num)] at himage
+
+/-! ### Gluing two arcs along their endpoints -/
+
+omit [T2Space X] in
+/-- **A value shared by two arcs with the same endpoints is attained at a shared parameter.** If
+the ranges of `γ δ : Path x y` meet exactly in `{x, y}`, then `γ a = δ b` forces both parameters to
+be `0` or both to be `1`.
+
+The shared value lies in `range γ ∩ range δ = {x, y}`, and injectivity of each path identifies the
+parameter at which it takes the endpoint value. -/
+private theorem eq_zero_and_eq_zero_or_eq_one_and_eq_one_of_apply_eq {y : X} {γ δ : Path x y}
+    (hγ : Function.Injective γ) (hδ : Function.Injective δ)
+    (hmeet : range γ ∩ range δ = {x, y}) {a b : unitInterval} (hab : γ a = δ b) :
+    (a = 0 ∧ b = 0) ∨ (a = 1 ∧ b = 1) := by
+  have hmem : γ a ∈ ({x, y} : Set X) :=
+    hmeet ▸ ⟨mem_range_self a, hab ▸ mem_range_self b⟩
+  rcases hmem with h | h
+  · exact Or.inl ⟨hγ (h.trans γ.source.symm), hδ (hab.symm.trans (h.trans δ.source.symm))⟩
+  · exact Or.inr ⟨hγ (h.trans γ.target.symm), hδ (hab.symm.trans (h.trans δ.target.symm))⟩
+
+/-- **Two arcs meeting exactly at their common endpoints glue to a Jordan curve.** Let
+`γ δ : Path x y` be injective and let their ranges meet in exactly the two endpoints,
+`range γ ∩ range δ = {x, y}`. Then `range γ ∪ range δ` is a Jordan curve.
+
+The curve traversed is `γ.trans δ.symm`, a closed path at `x` whose range is `range γ ∪ range δ`
+by `Path.trans_range` and `Path.symm_range`. Its only repetitions are the ones
+`TauCeti.isJordanCurve_range_of_eq_or_eq_endpoints` allows: a coincidence between two parameters on
+the same half is excluded by injectivity of that half, and one between the two halves lands in
+`{x, y}`, so it is either the pair `{0, 1}` of endpoint parameters or the single parameter `1 / 2`
+at which the two halves are joined.
+
+Distinctness of `x` and `y` is a consequence rather than a hypothesis: `δ 0 = δ 1` would contradict
+injectivity of `δ`. -/
+theorem isJordanCurve_range_union_range_of_inter_eq_pair {y : X} {γ δ : Path x y}
+    (hγ : Function.Injective γ) (hδ : Function.Injective δ)
+    (hmeet : range γ ∩ range δ = {x, y}) :
+    IsJordanCurve (range γ ∪ range δ) := by
+  have hrange : range (γ.trans δ.symm) = range γ ∪ range δ := by
+    rw [Path.trans_range, Path.symm_range]
+  rw [← hrange]
+  refine isJordanCurve_range_of_eq_or_eq_endpoints _ fun s t hst => ?_
+  rw [Path.trans_apply, Path.trans_apply] at hst
+  split_ifs at hst with hs ht ht
+  · -- Both parameters on the first half: injectivity of `γ`.
+    have h : (2 : ℝ) * s = 2 * t := congrArg Subtype.val (hγ hst)
+    exact Or.inl (Subtype.ext (show (s : ℝ) = (t : ℝ) by linarith))
+  · -- The first parameter on `γ`, the second on `δ` read backwards.
+    simp only [Path.symm_apply, Function.comp_apply] at hst
+    rcases eq_zero_and_eq_zero_or_eq_one_and_eq_one_of_apply_eq hγ hδ hmeet hst with ⟨ha, hb⟩ | h
+    · have h1 : (2 : ℝ) * s = 0 := congrArg Subtype.val ha
+      have h2 : 1 - (2 * (t : ℝ) - 1) = 0 := congrArg Subtype.val hb
+      exact Or.inr (Or.inl ⟨Subtype.ext (show (s : ℝ) = 0 by linarith),
+        Subtype.ext (show (t : ℝ) = 1 by linarith)⟩)
+    · have h1 : (2 : ℝ) * s = 1 := congrArg Subtype.val h.1
+      have h2 : 1 - (2 * (t : ℝ) - 1) = 1 := congrArg Subtype.val h.2
+      exact Or.inl (Subtype.ext (show (s : ℝ) = (t : ℝ) by linarith))
+  · -- The mirror image of the previous case.
+    simp only [Path.symm_apply, Function.comp_apply] at hst
+    rcases eq_zero_and_eq_zero_or_eq_one_and_eq_one_of_apply_eq hγ hδ hmeet hst.symm with
+      ⟨ha, hb⟩ | h
+    · have h1 : (2 : ℝ) * t = 0 := congrArg Subtype.val ha
+      have h2 : 1 - (2 * (s : ℝ) - 1) = 0 := congrArg Subtype.val hb
+      exact Or.inr (Or.inr ⟨Subtype.ext (show (s : ℝ) = 1 by linarith),
+        Subtype.ext (show (t : ℝ) = 0 by linarith)⟩)
+    · have h1 : (2 : ℝ) * t = 1 := congrArg Subtype.val h.1
+      have h2 : 1 - (2 * (s : ℝ) - 1) = 1 := congrArg Subtype.val h.2
+      exact Or.inl (Subtype.ext (show (s : ℝ) = (t : ℝ) by linarith))
+  · -- Both parameters on the second half: injectivity of `δ` read backwards.
+    simp only [Path.symm_apply, Function.comp_apply] at hst
+    have h : 1 - (2 * (s : ℝ) - 1) = 1 - (2 * (t : ℝ) - 1) := congrArg Subtype.val (hδ hst)
+    exact Or.inl (Subtype.ext (show (s : ℝ) = (t : ℝ) by linarith))
+
+end TauCeti
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Curve.Foundations.Development002`.
+* `Curve.Foundations.Development003`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Curve.Jordan.Separation`.
+* `Curve.Jordan.Interior`.
+* `Curve.Jordan.OrientationTransport`.
+* `Curve.Jordan.AreaTransport`.
+* `Curve.Jordan.ClosedArea`.
+* `Curve.Jordan.SignedArea`.
+* `Curve.Jordan.SupportingOrientation`.
+* `Curve.Jordan.Subarc`.
+* `Curve.Reparametrization`.
+* `Curve.SegmentAreaProperties`.
+* `Curve.Jordan.SubarcArea`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Separation
+-/
+
+@[expose] public section
+
+namespace MovingSofa
+
+private theorem IsJordanCurve.exists_sphere_parametrization {Γ : Set Point}
+    (hΓ : IsJordanCurve Γ) :
+    ∃ r : Metric.sphere (0 : Point) 1 → Point,
+      Continuous r ∧ Function.Injective r ∧ Set.range r = Γ := by
+  obtain ⟨f, hf, hfi, hfr⟩ := hΓ
+  let e := Complex.orthonormalBasisOneI.repr.symm
+  let φ : Metric.sphere (0 : Point) 1 → Circle := fun p ↦
+    ⟨e p.val, mem_sphere_zero_iff_norm.mpr (by
+      rw [e.norm_map]
+      exact mem_sphere_zero_iff_norm.mp p.property)⟩
+  have hφc : Continuous φ := (e.continuous.comp continuous_subtype_val).subtype_mk _
+  have hφi : Function.Injective φ := by
+    intro p q h
+    apply Subtype.ext
+    exact e.injective (congrArg Subtype.val h)
+  have hφs : Function.Surjective φ := by
+    intro z
+    refine ⟨⟨e.symm z.val, mem_sphere_zero_iff_norm.mpr ?_⟩, Subtype.ext ?_⟩
+    · rw [e.symm.norm_map]
+      exact Circle.norm_coe z
+    · exact e.apply_symm_apply z.val
+  refine ⟨f ∘ φ, hf.comp hφc, hfi.comp hφi, ?_⟩
+  rw [Set.range_comp, hφs.range_eq, Set.image_univ, hfr]
+
+theorem jordan_separation {Γ : Set Point} (hΓ : IsJordanCurve Γ) :
+    ∃ U V : Set Point,
+      IsOpen U ∧ IsOpen V ∧ IsConnected U ∧ IsConnected V ∧
+      Bornology.IsBounded U ∧ ¬Bornology.IsBounded V ∧
+      Disjoint U V ∧ U ∪ V = Γᶜ ∧ frontier U = Γ ∧ frontier V = Γ ∧
+      (∀ p ∈ U, connectedComponentIn Γᶜ p = U) ∧
+      (∀ p ∈ V, connectedComponentIn Γᶜ p = V) := by
+  obtain ⟨r, hr, hri, hrΓ⟩ := hΓ.exists_sphere_parametrization
+  rw [← hrΓ]
+  obtain ⟨u, hu, hub⟩ := JordanCurve.step_A_exists_bounded JordanCurve.Brouwer.brouwerFPT hr hri
+  obtain ⟨v, hv, hvu⟩ := JordanCurve.exists_unbounded_component r hr
+  let U := connectedComponentIn (Set.range r)ᶜ u
+  let V := connectedComponentIn (Set.range r)ᶜ v
+  have hne : U ≠ V := by
+    intro h
+    change connectedComponentIn (Set.range r)ᶜ u = connectedComponentIn (Set.range r)ᶜ v at h
+    exact hvu (h ▸ hub)
+  have hdis : Disjoint U V := by
+    apply Set.disjoint_left.mpr
+    intro p hpU hpV
+    exact hne ((connectedComponentIn_eq hpU).trans (connectedComponentIn_eq hpV).symm)
+  have hcover : U ∪ V = (Set.range r)ᶜ := by
+    apply Set.Subset.antisymm
+    · exact Set.union_subset (connectedComponentIn_subset _ _) (connectedComponentIn_subset _ _)
+    · intro p hp
+      by_cases hpb : Bornology.IsBounded (connectedComponentIn (Set.range r)ᶜ p)
+      · left
+        have h := JordanCurve.step_B_bounded_unique JordanCurve.Brouwer.brouwerFPT hr hri
+          p hp u hu hpb hub
+        simpa only [U, ← h] using mem_connectedComponentIn hp
+      · right
+        have h := JordanCurve.unbounded_component_unique r hr hpb hvu
+        simpa only [V, ← h] using mem_connectedComponentIn hp
+  refine ⟨U, V, JordanCurve.isOpen_component r hr u, JordanCurve.isOpen_component r hr v,
+    isConnected_connectedComponentIn_iff.mpr hu, isConnected_connectedComponentIn_iff.mpr hv,
+    hub, hvu, hdis, hcover, ?_, ?_, ?_, ?_⟩
+  · exact JordanCurve.component_boundary_eq JordanCurve.Brouwer.brouwerFPT hr hri hu
+      ⟨v, hv, hne.symm⟩
+  · exact JordanCurve.component_boundary_eq JordanCurve.Brouwer.brouwerFPT hr hri hv
+      ⟨u, hu, hne⟩
+  · intro p hp
+    exact (connectedComponentIn_eq hp).symm
+  · intro p hp
+    exact (connectedComponentIn_eq hp).symm
+
+end MovingSofa
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Interior
+-/
+
+@[expose] public section
+
+namespace MovingSofa
+
+/-- A Jordan curve has a point in its bounded complementary component. -/
+theorem IsJordanCurve.jordanInterior_nonempty {Γ : Set Point}
+    (hΓ : IsJordanCurve Γ) : (jordanInterior Γ).Nonempty := by
+  obtain ⟨U, V, _, _, hU, _, hUb, _, _, hcover, _, _, hcomp, _⟩ :=
+    jordan_separation hΓ
+  obtain ⟨p, hp⟩ := hU.nonempty
+  refine ⟨p, ?_, ?_⟩
+  · have : p ∈ Γᶜ := hcover ▸ Set.mem_union_left V hp
+    exact this
+  · rw [hcomp p hp]
+    exact hUb
+
+/-- The frontier of the bounded complementary component of a Jordan curve is the curve
+itself. -/
+theorem IsJordanCurve.frontier_jordanInterior {Γ : Set Point} (hΓ : IsJordanCurve Γ) :
+    frontier (jordanInterior Γ) = Γ := by
+  obtain ⟨U, V, -, -, -, -, hUbdd, hVunbdd, -, hcover, hfrontU, -, hUcomp, hVcomp⟩ :=
+    jordan_separation hΓ
+  have hUeq : jordanInterior Γ = U := by
+    ext p
+    simp only [jordanInterior, Set.mem_ofPred_eq]
+    constructor
+    · rintro ⟨hpΓ, hpb⟩
+      rcases hcover.symm.subset hpΓ with h | h
+      · exact h
+      · rw [hVcomp p h] at hpb
+        exact absurd hpb hVunbdd
+    · intro hpU
+      refine ⟨?_, ?_⟩
+      · have hmem : p ∈ Γᶜ := by rw [← hcover]; exact Or.inl hpU
+        exact hmem
+      · rw [hUcomp p hpU]
+        exact hUbdd
+  rw [hUeq]
+  exact hfrontU
+
+end MovingSofa
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Orientation Transport
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace MovingSofa
+
+/-- Endpoint-preserving continuous parameter changes preserve Jordan orientation. -/
+theorem IsOrientedJordanParametrization.orientation_eq_of_comp
+    {a b c d : ℝ} {hab : a ≤ b} {hcd : c ≤ d} {Γ : Set Point}
+    {ccw₁ ccw₂ : Bool} {x : Set.Icc a b → Point} {y : Set.Icc c d → Point}
+    (hx : IsOrientedJordanParametrization hab Γ ccw₁ x)
+    (hy : IsOrientedJordanParametrization hcd Γ ccw₂ y)
+    (φ : Set.Icc c d → Set.Icc a b) (hφ : Continuous φ)
+    (hφa : φ ⟨c, le_rfl, hcd⟩ = ⟨a, le_rfl, hab⟩)
+    (hφb : φ ⟨d, hcd, le_rfl⟩ = ⟨b, hab, le_rfl⟩)
+    (hxy : y = x ∘ φ) : ccw₂ = ccw₁ := by
+  obtain ⟨p, hp⟩ := hx.2.1.jordanInterior_nonempty
+  have hxw := hx.2.2.2.2.2.2 p hp
+  have hyw := hy.2.2.2.2.2.2 p hp
+  have hn : curveWinding hab x p ≠ 0 := by
+    rw [hxw]
+    cases ccw₁ <;> norm_num
+  have hw := curveWinding_comp_of_endpoints hab hcd
+    (exists_curveAngleLift_of_curveWinding_ne_zero hab hn) hφ hφa hφb
+  rw [← hxy, hxw, hyw] at hw
+  cases ccw₁ <;> cases ccw₂ <;> first | rfl | norm_num at hw
+
+/-- Exchanging the parameter endpoints reverses Jordan orientation. -/
+theorem IsOrientedJordanParametrization.orientation_eq_not_of_comp
+    {a b c d : ℝ} {hab : a ≤ b} {hcd : c ≤ d} {Γ : Set Point}
+    {ccw₁ ccw₂ : Bool} {x : Set.Icc a b → Point} {y : Set.Icc c d → Point}
+    (hx : IsOrientedJordanParametrization hab Γ ccw₁ x)
+    (hy : IsOrientedJordanParametrization hcd Γ ccw₂ y)
+    (φ : Set.Icc c d → Set.Icc a b) (hφ : Continuous φ)
+    (hφa : φ ⟨c, le_rfl, hcd⟩ = ⟨b, hab, le_rfl⟩)
+    (hφb : φ ⟨d, hcd, le_rfl⟩ = ⟨a, le_rfl, hab⟩)
+    (hxy : y = x ∘ φ) : ccw₂ = !ccw₁ := by
+  obtain ⟨p, hp⟩ := hx.2.1.jordanInterior_nonempty
+  have hxw := hx.2.2.2.2.2.2 p hp
+  have hyw := hy.2.2.2.2.2.2 p hp
+  have hn : curveWinding hab x p ≠ 0 := by
+    rw [hxw]
+    cases ccw₁ <;> norm_num
+  have hw := curveWinding_comp_of_reversed_endpoints hab hcd
+    (exists_curveAngleLift_of_curveWinding_ne_zero hab hn) hφ hφa hφb
+  rw [← hxy, hxw, hyw] at hw
+  cases ccw₁ <;> cases ccw₂ <;> first | rfl | norm_num at hw
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Area Transport
+-/
+
+@[expose] public section
+
+namespace MovingSofa
+
+/-- A monotone or antitone transition between oriented Jordan paths determines their
+signed areas. -/
+theorem curveArea_eq_of_oriented_reparametrization
+    {a b c d : ℝ} (hab : a ≤ b) (hcd : c ≤ d)
+    {Γ : Set Point} {ccw₁ ccw₂ : Bool}
+    (x : ContinuousBVPaths a b) (y : ContinuousBVPaths c d)
+    (hx : IsOrientedJordanParametrization hab Γ ccw₁ x.val)
+    (hy : IsOrientedJordanParametrization hcd Γ ccw₂ y.val)
+    (φ : Set.Icc c d → Set.Icc a b) (hφc : Continuous φ)
+    (hφs : Function.Surjective φ) (hφ : Monotone φ ∨ Antitone φ)
+    (hcomp : y.val = x.val ∘ φ) :
+    (ccw₁ = ccw₂ → curveAreaFunctional x = curveAreaFunctional y) ∧
+    (ccw₁ ≠ ccw₂ → curveAreaFunctional x = -curveAreaFunctional y) := by
+  obtain ⟨z, hz, hmarea, haarea⟩ :=
+    curveArea_comp_monotone_or_antitone_surjective hab hcd x φ hφc hφs hφ
+  have hzy : z = y := Subtype.ext (hz.trans hcomp.symm)
+  subst z
+  obtain ⟨u, hu⟩ := hφs ⟨a, le_rfl, hab⟩
+  obtain ⟨v, hv⟩ := hφs ⟨b, hab, le_rfl⟩
+  rcases hφ with hm | ha
+  · have hleft : φ ⟨c, le_rfl, hcd⟩ = ⟨a, le_rfl, hab⟩ := by
+      apply le_antisymm
+      · simpa only [hu] using
+          hm (show (⟨c, le_rfl, hcd⟩ : Set.Icc c d) ≤ u from u.property.1)
+      · exact (φ ⟨c, le_rfl, hcd⟩).property.1
+    have hright : φ ⟨d, hcd, le_rfl⟩ = ⟨b, hab, le_rfl⟩ := by
+      apply le_antisymm
+      · exact (φ ⟨d, hcd, le_rfl⟩).property.2
+      · simpa only [hv] using
+          hm (show v ≤ (⟨d, hcd, le_rfl⟩ : Set.Icc c d) from v.property.2)
+    have horient := hx.orientation_eq_of_comp hy φ hφc hleft hright hcomp
+    exact ⟨fun _ ↦ (hmarea hm).symm, fun hn ↦ (hn horient.symm).elim⟩
+  · have hleft : φ ⟨c, le_rfl, hcd⟩ = ⟨b, hab, le_rfl⟩ := by
+      apply le_antisymm
+      · exact (φ ⟨c, le_rfl, hcd⟩).property.2
+      · simpa only [hv] using
+          ha (show (⟨c, le_rfl, hcd⟩ : Set.Icc c d) ≤ v from v.property.1)
+    have hright : φ ⟨d, hcd, le_rfl⟩ = ⟨a, le_rfl, hab⟩ := by
+      apply le_antisymm
+      · simpa only [hu] using
+          ha (show u ≤ (⟨d, hcd, le_rfl⟩ : Set.Icc c d) from u.property.2)
+      · exact (φ ⟨d, hcd, le_rfl⟩).property.1
+    have horient := hx.orientation_eq_not_of_comp hy φ hφc hleft hright hcomp
+    constructor
+    · intro heq
+      have hf : ccw₁ = !ccw₁ := heq.trans horient
+      cases ccw₁ <;> contradiction
+    · intro _
+      have heq := haarea ha
+      linarith
+
+end MovingSofa
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Closed Area
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Same-carrier closed Jordan parametrizations have signed areas determined by orientation. -/
+theorem curveArea_closed_same_carrier (Γ Δ : OrientedJordanCurve)
+    (x : ClosedBVParametrization Γ) (y : ClosedBVParametrization Δ)
+    (hcarrier : Γ.carrier = Δ.carrier) :
+    (Γ.counterclockwise = Δ.counterclockwise →
+      curveAreaFunctional x.path = curveAreaFunctional y.path) ∧
+    (Γ.counterclockwise ≠ Δ.counterclockwise →
+      curveAreaFunctional x.path = -curveAreaFunctional y.path) := by
+  have hy : IsOrientedJordanParametrization y.ordered Γ.carrier Δ.counterclockwise y.path.val := by
+    rw [hcarrier]
+    exact y.oriented
+  have hcompare (z : ClosedBVParametrization Γ)
+      (hstart : z.path.val ⟨z.a, le_rfl, z.ordered⟩ =
+        y.path.val ⟨y.a, le_rfl, y.ordered⟩) :
+      (Γ.counterclockwise = Δ.counterclockwise →
+        curveAreaFunctional z.path = curveAreaFunctional y.path) ∧
+      (Γ.counterclockwise ≠ Δ.counterclockwise →
+        curveAreaFunctional z.path = -curveAreaFunctional y.path) := by
+    obtain ⟨φ, hφc, hφs, hφo, hcomp⟩ :=
+      z.exists_reparametrization_of_start_eq y hcarrier hstart
+    exact curveArea_eq_of_oriented_reparametrization z.ordered y.ordered z.path y.path
+      z.oriented hy φ hφc hφs hφo hcomp
+  by_cases hstart : x.path.val ⟨x.a, le_rfl, x.ordered⟩ =
+      y.path.val ⟨y.a, le_rfl, y.ordered⟩
+  · exact hcompare x hstart
+  have hmem : y.path.val ⟨y.a, le_rfl, y.ordered⟩ ∈ Set.range x.path.val := by
+    rw [x.oriented.2.2.2.1, hcarrier, ← y.oriented.2.2.2.1]
+    exact Set.mem_range_self _
+  obtain ⟨s, hsb, hs⟩ := exists_param_lt_top_of_mem_range x.oriented.1 x.path.val
+    x.oriented.2.2.2.2.1 hmem
+  have has : x.a < (s : ℝ) := by
+    apply lt_of_le_of_ne s.property.1
+    intro heq
+    have hsa : s = ⟨x.a, le_rfl, x.ordered⟩ := Subtype.ext heq.symm
+    apply hstart
+    simpa only [hsa] using hs
+  obtain ⟨r, hr, hrstart, hrarea⟩ :=
+    exists_oriented_cyclic_rotation x.ordered x.path x.oriented s has hsb
+  let z : ClosedBVParametrization Γ :=
+    { a := 0, b := 2, ordered := by norm_num, path := r, oriented := hr }
+  have hzstart : z.path.val ⟨z.a, le_rfl, z.ordered⟩ =
+      y.path.val ⟨y.a, le_rfl, y.ordered⟩ := hrstart.trans hs
+  simpa only [show z.path = r from rfl, hrarea] using hcompare z hzstart
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Signed Area
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+theorem windingKernel_disk_integral (R : ℝ) (hR : 0 < R) (z : Point) (hz : ‖z‖ < R) :
+    Integrable (windingKernel z) (volume.restrict (Metric.ball 0 R)) ∧
+    (∀ i : Fin 2, Integrable (fun p ↦ windingKernel z p i)
+      (volume.restrict (Metric.ball 0 R))) ∧
+    (∫ p in Metric.ball 0 R, ‖windingKernel z p‖) ≤ 4 * Real.pi * R ∧
+    (∫ p in Metric.ball 0 R, windingKernel z p) = Real.pi • z := by
+  have hInt := integrableOn_windingKernel_ball R z hz
+  exact ⟨hInt, fun i ↦ by
+    simpa using (EuclideanSpace.proj (𝕜 := ℝ) i).integrable_comp hInt,
+    setIntegral_norm_windingKernel_ball_le R hR z hz,
+    setIntegral_windingKernel_ball R z hz⟩
+
+theorem jordanBV_winding_integral (a b : ℝ) (hab : a < b) (x : ContinuousBVPaths a b)
+    (hclosed : x.val ⟨a, le_rfl, hab.le⟩ = x.val ⟨b, hab.le, le_rfl⟩)
+    (hinj : Set.InjOn x.val {t | (t : ℝ) < b}) :
+    volume (Set.range x.val) = 0 ∧
+    (∀ p ∉ Set.range x.val,
+      (∀ i : Fin 2, Continuous (fun t ↦ windingKernel (x.val t) p i) ∧
+        ∃ C : ℝ, ∀ t, |windingKernel (x.val t) p i| ≤ C) ∧
+      2 * Real.pi * curveWinding hab.le x.val p =
+        intervalStieltjesIntegral (continuousBVCoordinate x 1)
+          (fun t ↦ windingKernel (x.val t) p 0) Set.univ -
+        intervalStieltjesIntegral (continuousBVCoordinate x 0)
+          (fun t ↦ windingKernel (x.val t) p 1) Set.univ) ∧
+    (∀ p ∉ Set.range x.val, ∃ U : Set Point, IsOpen U ∧ p ∈ U ∧
+      ∀ q ∈ U, q ∉ Set.range x.val ∧
+        curveWinding hab.le x.val q = curveWinding hab.le x.val p) ∧
+    (∀ p ∉ Set.range x.val,
+      ¬Bornology.IsBounded (connectedComponentIn (Set.range x.val)ᶜ p) →
+      curveWinding hab.le x.val p = 0) := by
+  refine ⟨x.volume_range_eq_zero_of_injOn hab.le hinj, ?_, ?_, ?_⟩
+  · intro p hp
+    refine ⟨fun i ↦ windingKernel_coord_continuous_bounded x.property.1 hp i, ?_⟩
+    obtain ⟨θ, hθ⟩ := exists_curveAngleLift_of_avoids hab x.property.1 hp
+    have hnormpos : ∀ t, 0 < ‖x.val t - p‖ := by
+      intro t
+      rw [norm_pos_iff, sub_ne_zero]
+      intro h
+      exact hp ⟨t, h⟩
+    have : Nonempty (Set.Icc a b) := ⟨⟨a, le_rfl, hab.le⟩⟩
+    obtain ⟨t₀, -, ht₀⟩ := isCompact_univ.exists_isMinOn (Set.univ_nonempty)
+      ((x.property.1.sub continuous_const).norm).continuousOn
+    set m : ℝ := ‖x.val t₀ - p‖ with hmdef
+    have hmpos : 0 < m := hnormpos t₀
+    have hmle : ∀ t, m ≤ ‖x.val t - p‖ := fun t ↦ ht₀ (Set.mem_univ t)
+    obtain ⟨δ₀, hδ₀, hmod⟩ := Metric.uniformContinuous_iff.mp
+      (CompactSpace.uniformContinuous_of_continuous x.property.1) (m / 2) (by linarith)
+    have hclose : ∀ t u : Set.Icc a b, |(u : ℝ) - (t : ℝ)| < δ₀ →
+        ‖x.val u - x.val t‖ < m / 2 := by
+      intro t u hlt
+      have hd : dist u t < δ₀ := by
+        rw [Subtype.dist_eq, Real.dist_eq]; exact hlt
+      simpa only [dist_eq_norm] using hmod hd
+    set D : Point → Fin 2 → ℝ := fun z ↦ ![windingKernel z p 1, windingKernel z p 0] with hDdef
+    have hD0 : ∀ z, D z 0 = windingKernel z p 1 := fun z ↦ rfl
+    have hD1 : ∀ z, D z 1 = windingKernel z p 0 := fun z ↦ rfl
+    have hDcont : ∀ i, Continuous (fun t ↦ D (x.val t) i) := by
+      intro i
+      fin_cases i
+      · change Continuous (fun t ↦ D (x.val t) 0)
+        simpa only [hD0] using (windingKernel_coord_continuous_bounded x.property.1 hp 1).1
+      · change Continuous (fun t ↦ D (x.val t) 1)
+        simpa only [hD1] using (windingKernel_coord_continuous_bounded x.property.1 hp 0).1
+    have hcoordsub : ∀ (v w : Point) (i : Fin 2), (v - w) i = v i - w i := by
+      intro v w i; simp
+    have hkey : ∀ t u : Set.Icc a b, (t : ℝ) ≤ (u : ℝ) → (u : ℝ) - (t : ℝ) < δ₀ →
+        |θ u - θ t -
+          (D (x.val t) 1 * (x.val u 1 - x.val t 1) -
+            D (x.val t) 0 * (x.val u 0 - x.val t 0))| ≤
+          6 / m ^ 2 * ‖x.val u - x.val t‖ ^ 2 := by
+      intro t u htu hlt
+      have hnwpos : 0 < ‖x.val t - p‖ := hnormpos t
+      have hnw2 : ‖x.val t - p‖ ^ 2 = (x.val t - p) 0 ^ 2 + (x.val t - p) 1 ^ 2 :=
+        Point.norm_sq_eq _
+      have hnd2 : ‖x.val u - x.val t‖ ^ 2 =
+          (x.val u - x.val t) 0 ^ 2 + (x.val u - x.val t) 1 ^ 2 := Point.norm_sq_eq _
+      have hndlt : ‖x.val u - x.val t‖ < m / 2 :=
+        hclose t u (by rw [abs_of_nonneg (by linarith)]; exact hlt)
+      have hsmall : 2 * ‖x.val u - x.val t‖ ≤ ‖x.val t - p‖ := by
+        have := hmle t; linarith
+      have hposdot : ∀ s : Set.Icc a b, (t : ℝ) ≤ (s : ℝ) → (s : ℝ) ≤ (u : ℝ) →
+          0 < (x.val t - p) 0 * (x.val s - p) 0 + (x.val t - p) 1 * (x.val s - p) 1 := by
+        intro s hts hsu
+        have hds : ‖x.val s - x.val t‖ < m / 2 :=
+          hclose t s (by rw [abs_of_nonneg (by linarith)]; linarith)
+        have hdot := Point.abs_inner_coords_le (x.val t - p) (x.val s - x.val t)
+        have he0 : (x.val s - p) 0 = (x.val t - p) 0 + (x.val s - x.val t) 0 := by
+          rw [hcoordsub, hcoordsub, hcoordsub]; ring
+        have he1 : (x.val s - p) 1 = (x.val t - p) 1 + (x.val s - x.val t) 1 := by
+          rw [hcoordsub, hcoordsub, hcoordsub]; ring
+        rw [he0, he1]
+        have hmt := hmle t
+        have hnn := norm_nonneg (x.val s - x.val t)
+        nlinarith [abs_le.mp hdot, hnw2]
+      have hangle := hθ.sub_eq_arctan_of_dot_pos x.property.1 htu hposdot
+      have hest := Real.abs_arctan_div_sub_le_of_small (w0 := (x.val t - p) 0)
+        (w1 := (x.val t - p) 1) (d0 := (x.val u - x.val t) 0)
+        (d1 := (x.val u - x.val t) 1) (nw := ‖x.val t - p‖)
+        (nd := ‖x.val u - x.val t‖) hnwpos hnw2 (norm_nonneg _) hnd2 hsmall
+      have he0 : (x.val u - p) 0 = (x.val t - p) 0 + (x.val u - x.val t) 0 := by
+        rw [hcoordsub, hcoordsub, hcoordsub]; ring
+      have he1 : (x.val u - p) 1 = (x.val t - p) 1 + (x.val u - x.val t) 1 := by
+        rw [hcoordsub, hcoordsub, hcoordsub]; ring
+      have harg :
+          ((x.val t - p) 0 * (x.val u - x.val t) 1 -
+              (x.val t - p) 1 * (x.val u - x.val t) 0) /
+            (‖x.val t - p‖ ^ 2 +
+              ((x.val t - p) 0 * (x.val u - x.val t) 0 +
+                (x.val t - p) 1 * (x.val u - x.val t) 1)) =
+          ((x.val t - p) 0 * (x.val u - p) 1 - (x.val t - p) 1 * (x.val u - p) 0) /
+            ((x.val t - p) 0 * (x.val u - p) 0 + (x.val t - p) 1 * (x.val u - p) 1) := by
+        have hquot : ∀ A B C E : ℝ,
+            (A * E - B * C) / (A ^ 2 + B ^ 2 + (A * C + B * E)) =
+              (A * (B + E) - B * (A + C)) / (A * (A + C) + B * (B + E)) := by
+          intro A B C E
+          congr 1 <;> ring
+        rw [hnw2, he0, he1]
+        exact hquot _ _ _ _
+      rw [harg, ← hangle] at hest
+      have hlin : D (x.val t) 1 * (x.val u 1 - x.val t 1) -
+          D (x.val t) 0 * (x.val u 0 - x.val t 0) =
+          ((x.val t - p) 0 * (x.val u - x.val t) 1 -
+            (x.val t - p) 1 * (x.val u - x.val t) 0) / ‖x.val t - p‖ ^ 2 := by
+        have hne : ‖x.val t - p‖ ≠ 0 := ne_of_gt hnwpos
+        rw [hD0, hD1]
+        simp only [windingKernel, PiLp.smul_apply, smul_eq_mul, hcoordsub]
+        field_simp
+      rw [hlin]
+      refine hest.trans ?_
+      have hmsq : m ^ 2 ≤ ‖x.val t - p‖ ^ 2 := by
+        have := hmle t; nlinarith
+      rw [div_mul_eq_mul_div]
+      apply div_le_div_of_nonneg_left (by positivity) (by positivity) hmsq
+    have hchain := ContinuousBVPaths.stieltjes_chain_rule_of_local_quadratic_remainder
+      hab.le x x.boundedVariationOn θ D hDcont (by positivity) hδ₀ hkey
+    simp only [hD0, hD1] at hchain
+    rw [hθ.curveWinding_eq hab.le, ← hchain]
+    have hpi : (2 : ℝ) * Real.pi ≠ 0 := by positivity
+    field_simp
+  · intro p hp
+    exact curveWinding_locally_constant_on_compl_range hab.le x.property.1 hclosed hp
+  · intro p hp hub
+    exact curveWinding_eq_zero_of_unbounded_component hab.le x.property.1 hclosed hp hub
+
+/-- Joint measurability of the winding kernel along a continuous path. -/
+private theorem measurable_windingKernel_prod {a b : ℝ} (x : ContinuousBVPaths a b) :
+    Measurable fun q : Point × Set.Icc a b ↦ windingKernel (x.val q.2) q.1 := by
+  have h1 : Measurable fun q : Point × Set.Icc a b ↦ x.val q.2 - q.1 :=
+    (x.property.1.measurable.comp measurable_snd).sub measurable_fst
+  have h2 : Measurable fun q : Point × Set.Icc a b ↦
+      (‖x.val q.2 - q.1‖ ^ 2)⁻¹ • (x.val q.2 - q.1) :=
+    Measurable.smul ((h1.norm.pow_const 2).inv) h1
+  exact h2
+
+/-- The winding kernel of a path is product-integrable against area on an enclosing disk
+and any finite measure in the time variable. -/
+private theorem integrable_windingKernel_prod {a b : ℝ} (x : ContinuousBVPaths a b)
+    {R : ℝ} (hR : 0 < R) (hxR : ∀ t, ‖x.val t‖ < R)
+    (ν : Measure (Set.Icc a b)) [IsFiniteMeasure ν] :
+    Integrable (fun q : Point × Set.Icc a b ↦ windingKernel (x.val q.2) q.1)
+      ((volume.restrict (Metric.ball (0 : Point) R)).prod ν) := by
+  have hfinball : IsFiniteMeasure (volume.restrict (Metric.ball (0 : Point) R)) :=
+    isFiniteMeasure_restrict.2 measure_ball_lt_top.ne
+  have hmeas := measurable_windingKernel_prod x
+  refine ⟨hmeas.aestronglyMeasurable, ?_⟩
+  have hbound : ∀ t : Set.Icc a b,
+      (∫⁻ p, ‖windingKernel (x.val t) p‖ₑ ∂(volume.restrict (Metric.ball (0 : Point) R)))
+        ≤ ENNReal.ofReal (4 * Real.pi * R) := by
+    intro t
+    obtain ⟨hint, -, hnorm, -⟩ := windingKernel_disk_integral R hR (x.val t) (hxR t)
+    rw [← ofReal_integral_norm_eq_lintegral_enorm hint]
+    exact ENNReal.ofReal_le_ofReal hnorm
+  rw [HasFiniteIntegral, lintegral_prod_symm _ hmeas.enorm.aemeasurable]
+  calc
+    (∫⁻ t, ∫⁻ p, ‖windingKernel (x.val t) p‖ₑ
+        ∂(volume.restrict (Metric.ball (0 : Point) R)) ∂ν)
+        ≤ ∫⁻ _, ENNReal.ofReal (4 * Real.pi * R) ∂ν := lintegral_mono hbound
+    _ < ⊤ := by
+        rw [lintegral_const]
+        exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top (measure_lt_top ν Set.univ)
+
+/-- Fubini exchange between the coordinate Stieltjes measure of a continuous BV path and
+planar area on an enclosing disk, evaluated through the disk mean of the winding kernel. -/
+private theorem setIntegral_ball_stieltjes_windingKernel {a b : ℝ} (x : ContinuousBVPaths a b)
+    {R : ℝ} (hR : 0 < R) (hxR : ∀ t, ‖x.val t‖ < R) (i j : Fin 2) :
+    Integrable (fun p ↦ intervalStieltjesIntegral (continuousBVCoordinate x j)
+        (fun t ↦ windingKernel (x.val t) p i) Set.univ)
+      (volume.restrict (Metric.ball (0 : Point) R)) ∧
+    (∫ p in Metric.ball (0 : Point) R, intervalStieltjesIntegral (continuousBVCoordinate x j)
+        (fun t ↦ windingKernel (x.val t) p i) Set.univ) =
+      Real.pi * intervalStieltjesIntegral (continuousBVCoordinate x j)
+        (fun t ↦ x.val t i) Set.univ := by
+  have hfinball : IsFiniteMeasure (volume.restrict (Metric.ball (0 : Point) R)) :=
+    isFiniteMeasure_restrict.2 measure_ball_lt_top.ne
+  have hfinvar : IsFiniteMeasure
+      (intervalStieltjesMeasure (continuousBVCoordinate x j)).variation :=
+    BoundedVariationOn.instIsFiniteMeasureVariationVectorMeasure
+      (continuousBVCoordinate x j).boundedVariation
+  have hfinmu : IsFiniteMeasure
+      ((volume.restrict (Metric.ball (0 : Point) R)).toSignedMeasure).variation := by
+    rw [Measure.variation_toSignedMeasure]
+    infer_instance
+  simp only [intervalStieltjesIntegral_univ]
+  have hprodK : Integrable (fun q : Point × Set.Icc a b ↦ windingKernel (x.val q.2) q.1 i)
+      ((volume.restrict (Metric.ball (0 : Point) R)).prod
+        (intervalStieltjesMeasure (continuousBVCoordinate x j)).variation) := by
+    simpa using (EuclideanSpace.proj (𝕜 := ℝ) i).integrable_comp
+      (integrable_windingKernel_prod x hR hxR
+        (intervalStieltjesMeasure (continuousBVCoordinate x j)).variation)
+  have hintI : Integrable (fun p ↦ VectorMeasure.integral
+      (intervalStieltjesMeasure (continuousBVCoordinate x j))
+      (fun t ↦ windingKernel (x.val t) p i) (ContinuousLinearMap.mul ℝ ℝ))
+      (volume.restrict (Metric.ball (0 : Point) R)) := by
+    have hh := Integrable.integral_vectorMeasure_prod_left
+      (B := ContinuousLinearMap.mul ℝ ℝ) hprodK
+    simpa using hh
+  refine ⟨hintI, ?_⟩
+  have hinner : ∀ t : Set.Icc a b,
+      (∫ p in Metric.ball (0 : Point) R, windingKernel (x.val t) p i) =
+        Real.pi * x.val t i := by
+    intro t
+    obtain ⟨hint, -, -, heq⟩ := windingKernel_disk_integral R hR (x.val t) (hxR t)
+    have hc := (EuclideanSpace.proj (𝕜 := ℝ) i).integral_comp_comm hint
+    rw [heq] at hc
+    simpa using hc
+  have hfub := VectorMeasure.integral_integral_swap
+    (μ := (volume.restrict (Metric.ball (0 : Point) R)).toSignedMeasure)
+    (ν := intervalStieltjesMeasure (continuousBVCoordinate x j))
+    (B := ContinuousLinearMap.mul ℝ ℝ)
+    (C := (ContinuousLinearMap.lsmul ℝ ℝ).flip)
+    (A := (ContinuousLinearMap.lsmul ℝ ℝ).flip)
+    (D := ContinuousLinearMap.mul ℝ ℝ)
+    (f := fun (p : Point) (t : Set.Icc a b) ↦ windingKernel (x.val t) p i)
+    (by rw [Measure.variation_toSignedMeasure]; exact hprodK)
+    (by intro u v w; simp; ring)
+  simp only [VectorMeasure.integral_toSignedMeasure] at hfub
+  rw [hfub]
+  simp only [hinner]
+  simpa [smul_eq_mul] using VectorMeasure.integral_fun_smul
+    (μ := intervalStieltjesMeasure (continuousBVCoordinate x j))
+    (B := ContinuousLinearMap.mul ℝ ℝ) Real.pi (fun t ↦ x.val t i)
+
+theorem curveArea_eq_jordanInterior_area (a b : ℝ) (hab : a ≤ b)
+    (Γ : Set Point) (x : ContinuousBVPaths a b)
+    (hx : IsOrientedJordanParametrization hab Γ true x.val) :
+    curveAreaFunctional x = ClassicalResults.area (jordanInterior Γ) := by
+  obtain ⟨hlt, hΓ, hcont, hrange, hclosed, hinj, hwind⟩ := hx
+  subst hrange
+  obtain ⟨U, V, hUopen, -, -, -, hUbdd, hVunbdd, -, hcover, -, -, hUcomp, hVcomp⟩ :=
+    jordan_separation hΓ
+  obtain ⟨hnull, hwindint, -, hext⟩ := jordanBV_winding_integral a b hlt x hclosed hinj
+  have hUeq : jordanInterior (Set.range x.val) = U := by
+    ext p
+    simp only [jordanInterior, Set.mem_ofPred_eq]
+    constructor
+    · rintro ⟨hpΓ, hpb⟩
+      have hpc : p ∈ U ∪ V := by rw [hcover]; exact hpΓ
+      rcases hpc with h | h
+      · exact h
+      · rw [hVcomp p h] at hpb
+        exact absurd hpb hVunbdd
+    · intro hpU
+      refine ⟨?_, ?_⟩
+      · have hmem : p ∈ (Set.range x.val)ᶜ := by rw [← hcover]; exact Or.inl hpU
+        exact hmem
+      · rw [hUcomp p hpU]
+        exact hUbdd
+  obtain ⟨R, hR, hxR, hUR⟩ :
+      ∃ R : ℝ, 0 < R ∧ (∀ t, ‖x.val t‖ < R) ∧ U ⊆ Metric.ball (0 : Point) R := by
+    have hcpt : IsCompact (Set.range x.val) := isCompact_range hcont
+    obtain ⟨r, hr⟩ := (hcpt.isBounded.union hUbdd).subset_closedBall (0 : Point)
+    refine ⟨max r 0 + 1, by have := le_max_right r 0; linarith, ?_, ?_⟩
+    · intro t
+      have h1 : x.val t ∈ Metric.closedBall (0 : Point) r :=
+        hr (Set.mem_union_left _ ⟨t, rfl⟩)
+      rw [Metric.mem_closedBall, dist_zero_right] at h1
+      have := le_max_left r 0
+      linarith
+    · intro p hp
+      have h1 : p ∈ Metric.closedBall (0 : Point) r := hr (Set.mem_union_right _ hp)
+      rw [Metric.mem_closedBall, dist_zero_right] at h1
+      rw [Metric.mem_ball, dist_zero_right]
+      have := le_max_left r 0
+      linarith
+  have hae : ∀ᵐ p ∂(volume : Measure Point), p ∉ Set.range x.val :=
+    measure_eq_zero_iff_ae_notMem.mp hnull
+  -- winding equals the indicator of the bounded component off the curve
+  have hwind_eq : ∀ p ∉ Set.range x.val,
+      curveWinding hab x.val p = U.indicator (fun _ ↦ (1 : ℝ)) p := by
+    intro p hp
+    by_cases hpU : p ∈ U
+    · rw [Set.indicator_of_mem hpU]
+      have hw := hwind p (by rw [hUeq]; exact hpU)
+      simpa using hw
+    · rw [Set.indicator_of_notMem hpU]
+      have hpV : p ∈ V := by
+        have hpc : p ∈ U ∪ V := by rw [hcover]; exact hp
+        rcases hpc with h | h
+        · exact absurd h hpU
+        · exact h
+      refine hext p hp ?_
+      rw [hVcomp p hpV]
+      exact hVunbdd
+  have hArea : (∫ p in Metric.ball (0 : Point) R, curveWinding hab x.val p) =
+      ClassicalResults.area (jordanInterior (Set.range x.val)) := by
+    have h1 : (∫ p in Metric.ball (0 : Point) R, curveWinding hab x.val p) =
+        ∫ p in Metric.ball (0 : Point) R, U.indicator (fun _ ↦ (1 : ℝ)) p := by
+      refine setIntegral_congr_ae measurableSet_ball ?_
+      filter_upwards [hae] with p hp _
+      exact hwind_eq p hp
+    rw [h1, setIntegral_indicator hUopen.measurableSet,
+      Set.inter_eq_self_of_subset_right hUR, setIntegral_const, hUeq]
+    simp [ClassicalResults.area, measureReal_def]
+  have hFunctional : (∫ p in Metric.ball (0 : Point) R, curveWinding hab x.val p) =
+      curveAreaFunctional x := by
+    obtain ⟨hint01, heq01⟩ := setIntegral_ball_stieltjes_windingKernel x hR hxR 0 1
+    obtain ⟨hint10, heq10⟩ := setIntegral_ball_stieltjes_windingKernel x hR hxR 1 0
+    have hpi : (0 : ℝ) < 2 * Real.pi := by positivity
+    have hL : (∫ p in Metric.ball (0 : Point) R, 2 * Real.pi * curveWinding hab x.val p) =
+        2 * Real.pi * ∫ p in Metric.ball (0 : Point) R, curveWinding hab x.val p :=
+      integral_const_mul _ _
+    have hcong : (∫ p in Metric.ball (0 : Point) R, 2 * Real.pi * curveWinding hab x.val p)
+        = ∫ p in Metric.ball (0 : Point) R,
+            (intervalStieltjesIntegral (continuousBVCoordinate x 1)
+              (fun t ↦ windingKernel (x.val t) p 0) Set.univ -
+             intervalStieltjesIntegral (continuousBVCoordinate x 0)
+              (fun t ↦ windingKernel (x.val t) p 1) Set.univ) := by
+      refine setIntegral_congr_ae measurableSet_ball ?_
+      filter_upwards [hae] with p hp _
+      exact (hwindint p hp).2
+    rw [hcong, integral_sub hint01 hint10, heq01, heq10] at hL
+    refine mul_left_cancel₀ (ne_of_gt hpi) ?_
+    rw [← hL, curveAreaFunctional]
+    ring
+  rw [← hFunctional, hArea]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Supporting Orientation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval
+
+namespace MovingSofa
+
+private lemma pointComplex_re_me89eded (v : Point) : (Complex.orthonormalBasisOneI.repr.symm v).re =
+    v 0 := by
+  simp [Complex.orthonormalBasisOneI_repr_symm_apply]
+
+private lemma pointComplex_im_me89eded (v : Point) : (Complex.orthonormalBasisOneI.repr.symm v).im =
+    v 1 := by
+  simp [Complex.orthonormalBasisOneI_repr_symm_apply]
+
+private lemma conj_pointComplex_mul_re_me89eded (v w : Point) :
+    (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm v) *
+        Complex.orthonormalBasisOneI.repr.symm w).re = inner ℝ v w := by
+  simp [Complex.orthonormalBasisOneI_repr_symm_apply, inner, Fin.sum_univ_two]
+  ring
+
+private theorem continuous_arg_comp_of_re_nonneg {A : Type*} [TopologicalSpace A]
+    {z : A → ℂ} (hz : Continuous z) (hre : ∀ u, 0 ≤ (z u).re) (hne : ∀ u, z u ≠ 0) :
+    Continuous (fun u ↦ Complex.arg (z u)) := by
+  rw [continuous_iff_continuousAt]
+  intro u
+  have hmem : z u ∈ Complex.slitPlane := by
+    rw [Complex.mem_slitPlane_iff_arg]
+    constructor
+    · intro harg
+      have hneg := (Complex.arg_eq_pi_iff.mp harg).1
+      linarith [hre u]
+    · exact hne u
+  exact (Complex.continuousAt_arg hmem).comp_of_eq hz.continuousAt rfl
+
+private def frameComplex (t : Real.Angle) (v : Point) : ℂ :=
+  inner ℝ v (normalVector t) + inner ℝ v (tangentVector t) * Complex.I
+
+@[simp] private theorem frameComplex_re (t : Real.Angle) (v : Point) :
+    (frameComplex t v).re = inner ℝ v (normalVector t) := by
+  simp [frameComplex]
+
+@[simp] private theorem frameComplex_im (t : Real.Angle) (v : Point) :
+    (frameComplex t v).im = inner ℝ v (tangentVector t) := by
+  simp [frameComplex]
+
+private theorem norm_frameComplex (t : Real.Angle) (v : Point) :
+    ‖frameComplex t v‖ = ‖v‖ := by
+  rw [← sq_eq_sq₀ (norm_nonneg _) (norm_nonneg _)]
+  simp [frameComplex, Complex.sq_norm, Complex.normSq_apply, EuclideanSpace.norm_sq_eq,
+    normalVector, tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+  ring_nf
+  linear_combination (v 0 ^ 2 + v 1 ^ 2) * t.cos_sq_add_sin_sq
+
+private theorem conj_pointComplex_mul_eq_conj_frameComplex_mul (t : Real.Angle)
+    (v w : Point) :
+    starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm v) *
+        Complex.orthonormalBasisOneI.repr.symm w =
+      starRingEnd ℂ (frameComplex t v) * frameComplex t w := by
+  apply Complex.ext <;>
+    simp [Complex.orthonormalBasisOneI_repr_symm_apply, frameComplex, normalVector, tangentVector,
+        frame,
+      PiLp.inner_apply, Fin.sum_univ_two] <;> ring_nf
+  · linear_combination -(v 0 * w 0 + v 1 * w 1) * t.cos_sq_add_sin_sq
+  · linear_combination -(v 0 * w 1 - v 1 * w 0) * t.cos_sq_add_sin_sq
+
+private theorem frameComplex_ne_zero {t : Real.Angle} {v : Point} (hv : v ≠ 0) :
+    frameComplex t v ≠ 0 := by
+  rw [← norm_ne_zero_iff, norm_frameComplex, norm_ne_zero_iff]
+  exact hv
+
+private theorem normalized_frameComplex_arg (t : Real.Angle) (v : Point) (hv : v ≠ 0) :
+    Real.cos (t.toReal + Complex.arg (frameComplex t v)) = v 0 / ‖v‖ ∧
+      Real.sin (t.toReal + Complex.arg (frameComplex t v)) = v 1 / ‖v‖ := by
+  have hf := frameComplex_ne_zero (t := t) hv
+  rw [Real.cos_add, Real.sin_add, Complex.cos_arg hf, Complex.sin_arg,
+    norm_frameComplex]
+  simp [frameComplex, normalVector, tangentVector, frame, PiLp.inner_apply,
+    Fin.sum_univ_two]
+  constructor
+  · field_simp
+    ring_nf
+    linear_combination (v 0) * t.cos_sq_add_sin_sq
+  · field_simp
+    ring_nf
+    linear_combination (v 1) * t.cos_sq_add_sin_sq
+
+/-- The frame argument gives an angle lift when a path lies on the nonnegative side of a normal
+through its basepoint. -/
+private theorem isCurveAngleLift_frameArg_of_inner_nonneg {a b : ℝ}
+    {x : Set.Icc a b → Point} {p : Point} (t : Real.Angle)
+    (hx : Continuous x) (hre : ∀ u, 0 ≤ inner ℝ (x u - p) (normalVector t))
+    (hne : ∀ u, x u ≠ p) :
+    IsCurveAngleLift x p
+      (fun u ↦ t.toReal + Complex.arg (frameComplex t (x u - p))) := by
+  have hz : Continuous (fun u ↦ frameComplex t (x u - p)) := by
+    unfold frameComplex
+    fun_prop
+  have hzne (u) : frameComplex t (x u - p) ≠ 0 :=
+    frameComplex_ne_zero (sub_ne_zero.mpr (hne u))
+  refine ⟨continuous_const.add (continuous_arg_comp_of_re_nonneg hz
+    (fun u ↦ by simpa using hre u) hzne),
+    fun u ↦ ?_⟩
+  exact normalized_frameComplex_arg t (x u - p) (sub_ne_zero.mpr (hne u))
+
+/-- Negating frame coordinates and adding `π` gives the corresponding lift on the nonpositive
+side of a normal through the basepoint. -/
+private theorem isCurveAngleLift_frameArg_neg_add_pi_of_inner_nonpos {a b : ℝ}
+    {x : Set.Icc a b → Point} {p : Point} (t : Real.Angle)
+    (hx : Continuous x) (hre : ∀ u, inner ℝ (x u - p) (normalVector t) ≤ 0)
+    (hne : ∀ u, x u ≠ p) :
+    IsCurveAngleLift x p (fun u ↦
+      t.toReal + Complex.arg (-frameComplex t (x u - p)) + Real.pi) := by
+  have hz : Continuous (fun u ↦ -frameComplex t (x u - p)) := by
+    unfold frameComplex
+    fun_prop
+  have hzne (u) : frameComplex t (x u - p) ≠ 0 :=
+    frameComplex_ne_zero (sub_ne_zero.mpr (hne u))
+  have harg : Continuous (fun u ↦ Complex.arg (-frameComplex t (x u - p))) :=
+    continuous_arg_comp_of_re_nonneg hz (by
+      intro u
+      simpa using neg_nonneg.mpr (hre u)) (fun u ↦ neg_ne_zero.mpr (hzne u))
+  refine ⟨(continuous_const.add harg).add continuous_const, fun u ↦ ?_⟩
+  have hbase := normalized_frameComplex_arg t (x u - p) (sub_ne_zero.mpr (hne u))
+  have hneg :
+      Real.cos (Complex.arg (-frameComplex t (x u - p)) + Real.pi) =
+          (frameComplex t (x u - p)).re / ‖frameComplex t (x u - p)‖ ∧
+        Real.sin (Complex.arg (-frameComplex t (x u - p)) + Real.pi) =
+          (frameComplex t (x u - p)).im / ‖frameComplex t (x u - p)‖ := by
+    rw [Real.cos_add_pi, Real.sin_add_pi,
+      Complex.cos_arg (neg_ne_zero.mpr (hzne u)), Complex.sin_arg, norm_neg]
+    simp only [Complex.neg_re, Complex.neg_im, neg_div, neg_neg]
+    exact ⟨trivial, trivial⟩
+  dsimp only
+  rw [add_assoc, Real.cos_add t.toReal, Real.sin_add t.toReal, hneg.1, hneg.2,
+    norm_frameComplex]
+  rw [Real.cos_add, Real.sin_add, Complex.cos_arg (hzne u), Complex.sin_arg,
+    norm_frameComplex] at hbase
+  exact hbase
+
+private def cyclicComplement {a b : ℝ} (hab : a ≤ b) (x : Set.Icc a b → Point)
+    (s t : Set.Icc a b) : Set.Icc (0 : ℝ) 2 → Point :=
+  Function.concatUnitIntervals
+    (x ∘ Set.Icc.convexComb t ⟨b, hab, le_rfl⟩)
+    (x ∘ Set.Icc.convexComb ⟨a, le_rfl, hab⟩ s)
+
+private theorem continuous_cyclicComplement {a b : ℝ} (hab : a ≤ b)
+    {x : Set.Icc a b → Point} (hx : Continuous x)
+    (hclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (s t : Set.Icc a b) : Continuous (cyclicComplement hab x s t) := by
+  apply Function.continuous_concatUnitIntervals
+  · exact hx.comp (Set.Icc.continuous_convexComb _ _)
+  · exact hx.comp (Set.Icc.continuous_convexComb _ _)
+  · simpa [Function.comp_apply] using hclosed.symm
+
+@[simp] private theorem cyclicComplement_zero {a b : ℝ} (hab : a ≤ b)
+    (x : Set.Icc a b → Point) (s t : Set.Icc a b) :
+    cyclicComplement hab x s t ⟨0, by norm_num⟩ = x t := by
+  simp [cyclicComplement]
+
+@[simp] private theorem cyclicComplement_two {a b : ℝ} (hab : a ≤ b)
+    (x : Set.Icc a b → Point) (s t : Set.Icc a b) :
+    cyclicComplement hab x s t ⟨2, by norm_num⟩ = x s := by
+  simp [cyclicComplement]
+
+private theorem tangentVector_ne_zero (t : Real.Angle) : tangentVector t ≠ 0 := by
+  intro h
+  have hone : inner ℝ (tangentVector t) (tangentVector t) = 1 := by
+    rw [← t.coe_toReal]
+    exact inner_tangentVector_self t.toReal
+  rw [h] at hone
+  simp at hone
+
+private theorem convexComb_between {a b : ℝ} {l u : Set.Icc a b} (hlu : l ≤ u)
+    (r : Set.Icc (0 : ℝ) 1) :
+    l ≤ Set.Icc.convexComb l u r ∧ Set.Icc.convexComb l u r ≤ u := by
+  change (l : ℝ) ≤ (1 - (r : ℝ)) * l + (r : ℝ) * u ∧
+    (1 - (r : ℝ)) * l + (r : ℝ) * u ≤ u
+  constructor <;> nlinarith [r.property.1, r.property.2, show (l : ℝ) ≤ u from hlu]
+
+private theorem range_comp_convexComb {a b : ℝ} {l u : Set.Icc a b} (hlu : l ≤ u)
+    (x : Set.Icc a b → Point) :
+    Set.range (x ∘ Set.Icc.convexComb l u) = x '' Set.Icc l u := by
+  by_cases heq : l = u
+  · subst u
+    simp [Set.Icc_self]
+  have hneval : (l : ℝ) ≠ (u : ℝ) := fun h ↦ heq (Subtype.ext h)
+  have hlt : (l : ℝ) < (u : ℝ) := lt_of_le_of_ne hlu hneval
+  ext p
+  constructor
+  · rintro ⟨r, rfl⟩
+    exact ⟨_, convexComb_between hlu r, rfl⟩
+  · rintro ⟨v, hv, rfl⟩
+    let r : I := ⟨((v : ℝ) - l) / ((u : ℝ) - l), by
+      constructor
+      · exact div_nonneg (sub_nonneg.mpr hv.1)
+          (sub_nonneg.mpr (show (l : ℝ) ≤ u from hlu))
+      · exact (div_le_one (sub_pos.mpr hlt)).2
+          (by
+            have hv₂ : (v : ℝ) ≤ (u : ℝ) := hv.2
+            linarith)⟩
+    refine ⟨r, congrArg x ?_⟩
+    apply Subtype.ext
+    simp [r]
+    field_simp [sub_ne_zero.mpr hlt.ne']
+    ring
+
+private theorem inner_eq_of_mem_segment {P Q z v : Point} {c : ℝ}
+    (hP : inner ℝ P v = c) (hQ : inner ℝ Q v = c)
+    (hz : z ∈ segment ℝ P Q) : inner ℝ z v = c := by
+  rw [segment_eq_image] at hz
+  obtain ⟨r, hr, rfl⟩ := hz
+  rw [inner_add_left, real_inner_smul_left, real_inner_smul_left, hP, hQ]
+  nlinarith [hr.1, hr.2]
+
+private theorem arg_conj_I_mul_of_re_pos {z : ℂ} (hz : 0 < z.re) :
+    Complex.arg (starRingEnd ℂ Complex.I * z) = Complex.arg z - Real.pi / 2 := by
+  have hzne : z ≠ 0 := fun h ↦ by simp [h] at hz
+  have harg := Complex.abs_arg_lt_pi_div_two_iff.mpr (Or.inl hz)
+  rw [Complex.arg_mul (by simp) hzne]
+  · simp only [Complex.conj_I, Complex.arg_neg_I]
+    ring
+  · simp only [Complex.conj_I, Complex.arg_neg_I]
+    constructor <;> have h := abs_lt.mp harg <;> linarith [Real.pi_pos]
+
+private theorem arg_conj_neg_I_mul_of_re_pos {z : ℂ} (hz : 0 < z.re) :
+    Complex.arg (starRingEnd ℂ (-Complex.I) * z) = Complex.arg z + Real.pi / 2 := by
+  have hzne : z ≠ 0 := fun h ↦ by simp [h] at hz
+  have harg := Complex.abs_arg_lt_pi_div_two_iff.mpr (Or.inl hz)
+  rw [Complex.arg_mul (by simp) hzne]
+  · simp only [map_neg, Complex.conj_I, neg_neg, Complex.arg_I]
+    ring
+  · simp only [map_neg, Complex.conj_I, neg_neg, Complex.arg_I]
+    constructor <;> have h := abs_lt.mp harg <;> linarith [Real.pi_pos]
+
+private theorem relative_arg_of_frame_eq_pos_I {t : Real.Angle} {v w : Point}
+    {D : ℝ} (hD : 0 < D) (hv : frameComplex t v = (D : ℂ) * Complex.I)
+    (hw : 0 < (frameComplex t w).re) :
+    Complex.arg (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm v) *
+        Complex.orthonormalBasisOneI.repr.symm w) =
+      Complex.arg (frameComplex t w) - Real.pi / 2 := by
+  rw [conj_pointComplex_mul_eq_conj_frameComplex_mul, hv]
+  have heq : starRingEnd ℂ ((D : ℂ) * Complex.I) * frameComplex t w =
+      (D : ℂ) * (starRingEnd ℂ Complex.I * frameComplex t w) := by
+    simp [mul_assoc]
+  rw [heq, Complex.arg_real_mul _ hD]
+  exact arg_conj_I_mul_of_re_pos hw
+
+private theorem relative_arg_of_frame_eq_neg_I {t : Real.Angle} {v w : Point}
+    {D : ℝ} (hD : 0 < D) (hv : frameComplex t v = -(D : ℂ) * Complex.I)
+    (hw : 0 < (frameComplex t w).re) :
+    Complex.arg (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm v) *
+        Complex.orthonormalBasisOneI.repr.symm w) =
+      Complex.arg (frameComplex t w) + Real.pi / 2 := by
+  rw [conj_pointComplex_mul_eq_conj_frameComplex_mul, hv]
+  have heq : starRingEnd ℂ (-(D : ℂ) * Complex.I) * frameComplex t w =
+      (D : ℂ) * (starRingEnd ℂ (-Complex.I) * frameComplex t w) := by
+    simp [mul_assoc]
+  rw [heq, Complex.arg_real_mul _ hD]
+  exact arg_conj_neg_I_mul_of_re_pos hw
+
+/-- The midpoint of a nontrivial segment traversed on `[s,t]` is avoided by the cyclic
+complementary path. -/
+private theorem cyclicComplement_ne_midpoint {a b : ℝ} (hab : a < b)
+    {x : Set.Icc a b → Point}
+    (hclosed : x ⟨a, le_rfl, hab.le⟩ = x ⟨b, hab.le, le_rfl⟩)
+    (hinj : Set.InjOn x {u | (u : ℝ) < b})
+    (s t : Set.Icc a b) (hst : s < t) (hxt : x s ≠ x t)
+    (hsegment : x '' Set.Icc s t = segment ℝ (x s) (x t)) :
+    ∀ u, cyclicComplement hab.le x s t u ≠ midpoint ℝ (x s) (x t) := by
+  have hmseg : midpoint ℝ (x s) (x t) ∈ x '' Set.Icc s t := by
+    rw [hsegment]
+    exact midpoint_mem_segment _ _
+  obtain ⟨v, hvst, hv⟩ := hmseg
+  have hmvP : midpoint ℝ (x s) (x t) ≠ x s := by
+    intro hm
+    exact hxt ((midpoint_eq_left_iff ℝ).mp hm)
+  have hmvQ : midpoint ℝ (x s) (x t) ≠ x t := by
+    intro hm
+    exact hxt ((midpoint_eq_right_iff ℝ).mp hm)
+  have hsv : s < v := lt_of_le_of_ne hvst.1 (fun h ↦ hmvP (hv ▸ congrArg x h.symm))
+  have hvt : v < t := lt_of_le_of_ne hvst.2 (fun h ↦ hmvQ (hv ▸ congrArg x h))
+  have hsv' : (s : ℝ) < v := hsv
+  have hvt' : (v : ℝ) < t := hvt
+  have hvb : (v : ℝ) < b := lt_of_lt_of_le hvt t.property.2
+  intro u hu
+  have heval :
+      (x ∘ Set.Icc.convexComb t ⟨b, hab.le, le_rfl⟩)
+          (Set.projIcc 0 1 (by norm_num) (u : ℝ)) =
+          midpoint ℝ (x s) (x t) ∨
+        (x ∘ Set.Icc.convexComb ⟨a, le_rfl, hab.le⟩ s)
+            (Set.projIcc 0 1 (by norm_num) ((u : ℝ) - 1)) =
+              midpoint ℝ (x s) (x t) := by
+    unfold cyclicComplement Function.concatUnitIntervals at hu
+    split_ifs at hu with h
+    · exact Or.inl hu
+    · exact Or.inr hu
+  rcases heval with htail | hhead
+  · let w := Set.Icc.convexComb t ⟨b, hab.le, le_rfl⟩
+      (Set.projIcc 0 1 (by norm_num) (u : ℝ))
+    have htw : t ≤ w := (convexComb_between t.property.2 _).1
+    have hxwv : x w = x v := htail.trans hv.symm
+    by_cases hwb : (w : ℝ) = b
+    · have hwtop : w = (⟨b, hab.le, le_rfl⟩ : Set.Icc a b) := Subtype.ext hwb
+      have hxav : x ⟨a, le_rfl, hab.le⟩ = x v := hclosed.trans (hwtop ▸ hxwv)
+      have hav := hinj (by simp [hab]) (by exact hvb) hxav
+      have hav' := congrArg Subtype.val hav
+      linarith [s.property.1]
+    · have hwb' : (w : ℝ) < b := lt_of_le_of_ne w.property.2 hwb
+      have hwv := hinj hwb' hvb hxwv
+      have hwv' := congrArg Subtype.val hwv
+      have htw' : (t : ℝ) ≤ w := htw
+      linarith
+  · let w := Set.Icc.convexComb ⟨a, le_rfl, hab.le⟩ s
+      (Set.projIcc 0 1 (by norm_num) ((u : ℝ) - 1))
+    have hws : w ≤ s := (convexComb_between s.property.1 _).2
+    have hxwv : x w = x v := hhead.trans hv.symm
+    have hwb : (w : ℝ) < b := lt_of_le_of_lt hws (lt_of_lt_of_le hst t.property.2)
+    have hwv := hinj hwb hvb hxwv
+    have hwv' := congrArg Subtype.val hwv
+    have hws' : (w : ℝ) ≤ s := hws
+    linarith
+
+private theorem jordan_counterclockwise_of_winding_one
+    (a b : ℝ) (hab : a < b) (x : Set.Icc a b → Point)
+    (hx : Continuous x) (hΓ : IsJordanCurve (Set.range x))
+    (hclosed : x ⟨a, le_rfl, hab.le⟩ = x ⟨b, hab.le, le_rfl⟩)
+    (hinj : Set.InjOn x {t | (t : ℝ) < b})
+    (q : Point) (hqrange : q ∉ Set.range x)
+    (hwinding_q : curveWinding hab.le x q = 1) :
+    IsOrientedJordanParametrization hab.le (Set.range x) true x := by
+  refine ⟨hab, hΓ, hx, rfl, hclosed, hinj, ?_⟩
+  intro p hp
+  obtain ⟨U, V, hUopen, hVopen, hUconn, hVconn, hUbounded, hVunbounded, hdis,
+    hcover, hfrontU, hfrontV, hcompU, hcompV⟩ := jordan_separation hΓ
+  have hqcompl : q ∈ (Set.range x)ᶜ := hqrange
+  have hqUV : q ∈ U ∪ V := hcover.symm.subset hqcompl
+  have hqU : q ∈ U := by
+    rcases hqUV with hqU | hqV
+    · exact hqU
+    · exfalso
+      let _ : PreconnectedSpace V := Subtype.preconnectedSpace hVconn.isPreconnected
+      have hlc : IsLocallyConstant
+          (fun z : V ↦ curveWinding hab.le x z.val) := by
+        apply (IsLocallyConstant.iff_exists_open _).mpr
+        intro z
+        have hzout : z.val ∉ Set.range x := by
+          have hzcompl : z.val ∈ (Set.range x)ᶜ := hcover ▸ Or.inr z.property
+          exact hzcompl
+        obtain ⟨W, hWopen, hzW, hWeq⟩ :=
+          curveWinding_locally_constant_off_range hab.le hx hclosed hzout
+        exact ⟨Subtype.val ⁻¹' W, hWopen.preimage continuous_subtype_val, hzW,
+          fun z' hz' ↦ hWeq z'.val hz'⟩
+      have hrangeBounded : Bornology.IsBounded (Set.range x) :=
+        by simpa only [Set.image_univ] using (isCompact_univ.image hx).isBounded
+      obtain ⟨R, hR⟩ := (Metric.isBounded_iff_subset_closedBall (0 : Point)).mp
+        hrangeBounded
+      have hRnonneg : 0 ≤ R := by
+        have := hR (Set.mem_range_self ⟨a, le_rfl, hab.le⟩)
+        have hnorm : ‖x ⟨a, le_rfl, hab.le⟩‖ ≤ R := by
+          simpa [Metric.mem_closedBall, dist_zero_right] using this
+        exact (norm_nonneg _).trans hnorm
+      have hvfar : ∃ v ∈ V, R + 1 < ‖v‖ := by
+        by_contra hn
+        push Not at hn
+        apply hVunbounded
+        refine (Metric.isBounded_iff_subset_closedBall (0 : Point)).2 ⟨R + 1, ?_⟩
+        intro v hv
+        simpa [Metric.mem_closedBall, dist_zero_right] using hn v hv
+      obtain ⟨v, hvV, hvnorm⟩ := hvfar
+      have hvzero : curveWinding hab.le x v = 0 := by
+        apply curveWinding_eq_zero_of_inner_pos hab.le hx hclosed v (-v)
+        · exact neg_ne_zero.mpr (by
+            intro hv0
+            rw [hv0, norm_zero] at hvnorm
+            linarith)
+        · intro u
+          rw [inner_neg_left, inner_sub_right, real_inner_self_eq_norm_sq]
+          rw [← real_inner_comm v (x u)]
+          have hxu := hR (Set.mem_range_self u)
+          have hinner := abs_real_inner_le_norm (x u) v
+          have hxnorm : ‖x u‖ ≤ R := by
+            simpa [Metric.mem_closedBall, dist_zero_right] using hxu
+          have hvpos : 0 < ‖v‖ := lt_of_le_of_lt hRnonneg (lt_add_one R) |>.trans hvnorm
+          have hvlarge : R < ‖v‖ := lt_trans (lt_add_one R) hvnorm
+          have hlower : inner ℝ (x u) v ≤ ‖x u‖ * ‖v‖ := le_trans (le_abs_self _) hinner
+          have hprod : ‖x u‖ * ‖v‖ < ‖v‖ * ‖v‖ :=
+            lt_of_le_of_lt (mul_le_mul_of_nonneg_right hxnorm (norm_nonneg v))
+              (mul_lt_mul_of_pos_right hvlarge hvpos)
+          rw [pow_two]
+          linarith
+      have heq := hlc.apply_eq_of_preconnectedSpace ⟨q, hqV⟩ ⟨v, hvV⟩
+      rw [hwinding_q, hvzero] at heq
+      norm_num at heq
+  have hpUV : p ∈ U ∪ V := hcover.symm.subset hp.1
+  have hpU : p ∈ U := by
+    rcases hpUV with hpU | hpV
+    · exact hpU
+    · exfalso
+      apply hVunbounded
+      rw [← hcompV p hpV]
+      exact hp.2
+  let _ : PreconnectedSpace U := Subtype.preconnectedSpace hUconn.isPreconnected
+  have hlcU : IsLocallyConstant (fun z : U ↦ curveWinding hab.le x z.val) := by
+    apply (IsLocallyConstant.iff_exists_open _).mpr
+    intro z
+    have hzout : z.val ∉ Set.range x := by
+      have : z.val ∈ (Set.range x)ᶜ := hcover ▸ Or.inl z.property
+      exact this
+    obtain ⟨W, hWopen, hzW, hWeq⟩ :=
+      curveWinding_locally_constant_off_range hab.le hx hclosed hzout
+    exact ⟨Subtype.val ⁻¹' W, hWopen.preimage continuous_subtype_val, hzW,
+      fun z' hz' ↦ hWeq z'.val hz'⟩
+  exact (hlcU.apply_eq_of_preconnectedSpace ⟨p, hpU⟩ ⟨q, hqU⟩).trans hwinding_q
+
+private theorem supporting_segment_argument_corrections (P Q : Point) (θ : Real.Angle)
+    (d ε : ℝ) (hd : 0 < d) (hε : 0 < ε) (hdirection : Q = P + d • tangentVector θ) :
+    let m := midpoint ℝ P Q
+    let q := m - ε • normalVector θ
+    (Complex.arg (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm (Q - m)) *
+    Complex.orthonormalBasisOneI.repr.symm (Q - q)) =
+  Complex.arg (frameComplex θ (Q - q)) - Real.pi / 2) ∧
+    (Complex.arg (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm (P - m)) *
+    Complex.orthonormalBasisOneI.repr.symm (P - q)) =
+  Complex.arg (frameComplex θ (P - q)) + Real.pi / 2) ∧
+    (Complex.arg ((d / 2 : ℝ) * Complex.I) = Real.pi / 2) ∧
+    (Complex.arg (-((d / 2 : ℝ) * Complex.I)) = -Real.pi / 2) ∧
+    (-frameComplex θ (P - m) = (d / 2 : ℝ) * Complex.I) ∧
+    (-frameComplex θ (Q - m) = -((d / 2 : ℝ) * Complex.I)) := by
+  let m := midpoint ℝ P Q
+  let q := m - ε • normalVector θ
+  have htn : inner ℝ (tangentVector θ) (normalVector θ) = 0 := by
+    rw [← θ.coe_toReal, real_inner_comm]
+    exact inner_normalVector_tangentVector θ.toReal
+  have htm : Q - m = (d / 2) • tangentVector θ := by
+    dsimp [m]
+    rw [midpoint_eq_smul_add, hdirection]
+    norm_num
+    module
+  have hsm : P - m = -(d / 2) • tangentVector θ := by
+    dsimp [m]
+    rw [midpoint_eq_smul_add, hdirection]
+    norm_num
+    module
+  have hTT : inner ℝ (tangentVector θ) (tangentVector θ) = 1 := by
+    rw [← θ.coe_toReal]
+    exact inner_tangentVector_self θ.toReal
+  have hNN : inner ℝ (normalVector θ) (normalVector θ) = 1 := by
+    rw [← θ.coe_toReal]
+    exact inner_normalVector_self θ.toReal
+  have hNT : inner ℝ (normalVector θ) (tangentVector θ) = 0 := by
+    rw [real_inner_comm]
+    exact htn
+  have hframe_tm : frameComplex θ (Q - m) = (d / 2 : ℝ) * Complex.I := by
+    rw [htm]
+    apply Complex.ext
+    · rw [frameComplex_re, real_inner_smul_left, htn]
+      simp
+    · rw [frameComplex_im, real_inner_smul_left, hTT]
+      simp
+  have hframe_sm : frameComplex θ (P - m) = -(d / 2 : ℝ) * Complex.I := by
+    rw [hsm]
+    apply Complex.ext
+    · rw [frameComplex_re, real_inner_smul_left, htn]
+      simp
+    · rw [frameComplex_im, real_inner_smul_left, hTT]
+      simp
+  have htq : Q - q = ε • normalVector θ + (d / 2) • tangentVector θ := by
+    calc
+      Q - q = (Q - m) + ε • normalVector θ := by dsimp [q]; abel
+      _ = _ := by rw [htm]; abel
+  have hsq : P - q = ε • normalVector θ - (d / 2) • tangentVector θ := by
+    calc
+      P - q = (P - m) + ε • normalVector θ := by dsimp [q]; abel
+      _ = _ := by rw [hsm]; module
+  have hframe_tq : frameComplex θ (Q - q) = ε + (d / 2 : ℝ) * Complex.I := by
+    rw [htq]
+    apply Complex.ext
+    · rw [frameComplex_re, inner_add_left, real_inner_smul_left,
+        real_inner_smul_left, hNN, htn]
+      simp
+    · rw [frameComplex_im, inner_add_left, real_inner_smul_left,
+        real_inner_smul_left, hNT, hTT]
+      simp
+  have hframe_sq : frameComplex θ (P - q) = ε - (d / 2 : ℝ) * Complex.I := by
+    rw [hsq]
+    apply Complex.ext
+    · rw [frameComplex_re, inner_sub_left, real_inner_smul_left,
+        real_inner_smul_left, hNN, htn]
+      simp
+    · rw [frameComplex_im, inner_sub_left, real_inner_smul_left,
+        real_inner_smul_left, hNT, hTT]
+      simp
+  have hcorr_t :
+      Complex.arg (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm (Q - m)) *
+          Complex.orthonormalBasisOneI.repr.symm (Q - q)) =
+        Complex.arg (frameComplex θ (Q - q)) - Real.pi / 2 := by
+    apply relative_arg_of_frame_eq_pos_I (half_pos hd) hframe_tm
+    rw [hframe_tq]
+    simpa using hε
+  have hcorr_s :
+      Complex.arg (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm (P - m)) *
+          Complex.orthonormalBasisOneI.repr.symm (P - q)) =
+        Complex.arg (frameComplex θ (P - q)) + Real.pi / 2 := by
+    apply relative_arg_of_frame_eq_neg_I (half_pos hd) hframe_sm
+    rw [hframe_sq]
+    simpa using hε
+  have harg_tm : Complex.arg ((d / 2 : ℝ) * Complex.I) = Real.pi / 2 := by
+    rw [Complex.arg_real_mul _ (half_pos hd), Complex.arg_I]
+  have harg_sm : Complex.arg (-((d / 2 : ℝ) * Complex.I)) = -Real.pi / 2 := by
+    rw [show -((d / 2 : ℝ) * Complex.I) = (d / 2 : ℝ) * (-Complex.I) by ring,
+      Complex.arg_real_mul _ (half_pos hd), Complex.arg_neg_I]
+    ring
+  have hneg_frame_sm : -frameComplex θ (P - m) = (d / 2 : ℝ) * Complex.I := by
+    rw [hframe_sm]
+    ring
+  have hneg_frame_tm : -frameComplex θ (Q - m) = -((d / 2 : ℝ) * Complex.I) := by
+    rw [hframe_tm]
+  exact ⟨hcorr_t, hcorr_s, harg_tm, harg_sm, hneg_frame_sm, hneg_frame_tm⟩
+
+private theorem cyclic_partition_range {a b : ℝ} (hab : a < b)
+    (x : Set.Icc a b → Point)
+    (hclosed : x ⟨a, le_rfl, hab.le⟩ = x ⟨b, hab.le, le_rfl⟩)
+    (s t : Set.Icc a b) (hst : s < t) :
+    let y := cyclicComplement hab.le x s t
+    let z : Set.Icc (0 : ℝ) 1 → Point := fun u ↦ x (Set.Icc.convexComb s t u)
+    let doubleParam : Set.Icc (0 : ℝ) 1 → Set.Icc (0 : ℝ) 2 := fun u ↦
+    ⟨2 * (u : ℝ), by constructor <;> nlinarith [u.property.1, u.property.2]⟩
+    let y₁ : Set.Icc (0 : ℝ) 1 → Point := y ∘ doubleParam
+    Set.range (Function.concatUnitIntervals z y₁) = Set.range x := by
+  let y := cyclicComplement hab.le x s t
+  let z : Set.Icc (0 : ℝ) 1 → Point := fun u ↦ x (Set.Icc.convexComb s t u)
+  let doubleParam : Set.Icc (0 : ℝ) 1 → Set.Icc (0 : ℝ) 2 := fun u ↦
+    ⟨2 * (u : ℝ), by constructor <;> nlinarith [u.property.1, u.property.2]⟩
+  let y₁ : Set.Icc (0 : ℝ) 1 → Point := y ∘ doubleParam
+  have hjoin : z ⟨1, by norm_num⟩ = y₁ ⟨0, by norm_num⟩ := by
+    simp [z, y₁, doubleParam, y]
+  have hdouble_surj : Function.Surjective doubleParam := by
+    intro u
+    refine ⟨⟨(u : ℝ) / 2, by constructor <;> nlinarith [u.property.1, u.property.2]⟩, ?_⟩
+    apply Subtype.ext
+    dsimp [doubleParam]
+    ring
+  have hrange_y₁ : Set.range y₁ = Set.range y := by
+    change Set.range (y ∘ doubleParam) = Set.range y
+    rw [Set.range_comp, hdouble_surj.range_eq, Set.image_univ]
+  have hrange_w : Set.range (Function.concatUnitIntervals z y₁) = Set.range x := by
+    rw [Function.range_concatUnitIntervals z y₁ hjoin, hrange_y₁]
+    change Set.range z ∪ Set.range (Function.concatUnitIntervals
+      (x ∘ Set.Icc.convexComb t ⟨b, hab.le, le_rfl⟩)
+      (x ∘ Set.Icc.convexComb ⟨a, le_rfl, hab.le⟩ s)) = Set.range x
+    rw [Function.range_concatUnitIntervals _ _ (by simpa using hclosed.symm),
+      show Set.range z = x '' Set.Icc s t by
+        change Set.range (x ∘ Set.Icc.convexComb s t) = _
+        exact range_comp_convexComb hst.le x,
+      range_comp_convexComb t.property.2 x, range_comp_convexComb s.property.1 x]
+    ext p
+    constructor
+    · rintro (⟨u, hu, rfl⟩ | ⟨u, hu, rfl⟩ | ⟨u, hu, rfl⟩) <;>
+        exact Set.mem_range_self u
+    · rintro ⟨u, rfl⟩
+      by_cases hus : u ≤ s
+      · exact Or.inr (Or.inr ⟨u, ⟨u.property.1, hus⟩, rfl⟩)
+      by_cases hut : u ≤ t
+      · exact Or.inl ⟨u, ⟨le_of_not_ge hus, hut⟩, rfl⟩
+      · exact Or.inr (Or.inl ⟨u, ⟨le_of_not_ge hut, u.property.2⟩, rfl⟩)
+  exact hrange_w
+
+theorem jordan_counterclockwise_of_supporting_segment
+    (a b : ℝ) (hab : a < b) (x : Set.Icc a b → Point)
+    (hx : Continuous x) (hΓ : IsJordanCurve (Set.range x))
+    (hclosed : x ⟨a, le_rfl, hab.le⟩ = x ⟨b, hab.le, le_rfl⟩)
+    (hinj : Set.InjOn x {t | (t : ℝ) < b})
+    (θ : Real.Angle) (h : ℝ)
+    (hhalf : ∀ t, x t ∈ normalHalfPlane θ h false false)
+    (s t : Set.Icc a b) (hst : s < t)
+    (hline : x s ∈ normalLine θ h)
+    (d : ℝ) (hd : 0 < d) (hdirection : x t = x s + d • tangentVector θ)
+    (hsegment : x '' Set.Icc s t = segment ℝ (x s) (x t)) :
+    IsOrientedJordanParametrization hab.le (Set.range x) true x := by
+  have hxt : x s ≠ x t := by
+    intro heq
+    have hz : d • tangentVector θ = 0 := by
+      calc
+        d • tangentVector θ = x t - x s := by rw [hdirection]; abel
+        _ = 0 := by rw [← heq, sub_self]
+    exact tangentVector_ne_zero θ ((smul_eq_zero.mp hz).resolve_left hd.ne')
+  let m := midpoint ℝ (x s) (x t)
+  let y := cyclicComplement hab.le x s t
+  have hy : Continuous y := continuous_cyclicComplement hab.le hx hclosed s t
+  have hym (u : Set.Icc (0 : ℝ) 2) : y u ≠ m := by
+    exact cyclicComplement_ne_midpoint hab hclosed hinj s t hst hxt hsegment u
+  obtain ⟨r, hr, hdot⟩ := exists_ball_relative_dot_pos (by norm_num) hy (by
+    intro hm
+    obtain ⟨u, hu⟩ := hm
+    exact hym u hu)
+  have hxsN : inner ℝ (x s) (normalVector θ) = h := hline
+  have htn : inner ℝ (tangentVector θ) (normalVector θ) = 0 := by
+    rw [← θ.coe_toReal, real_inner_comm]
+    exact inner_normalVector_tangentVector θ.toReal
+  have hxtN : inner ℝ (x t) (normalVector θ) = h := by
+    rw [hdirection, inner_add_left, real_inner_smul_left, hxsN, htn, mul_zero, add_zero]
+  have hmN : inner ℝ m (normalVector θ) = h := by
+    dsimp [m]
+    rw [midpoint_eq_smul_add, real_inner_smul_left, inner_add_left, hxsN, hxtN]
+    norm_num
+    ring
+  have hyhalf (u : Set.Icc (0 : ℝ) 2) :
+      inner ℝ (y u - m) (normalVector θ) ≤ 0 := by
+    unfold y cyclicComplement Function.concatUnitIntervals
+    split_ifs
+    · rw [inner_sub_left, hmN]
+      have hh := hhalf (Set.Icc.convexComb t ⟨b, hab.le, le_rfl⟩
+        (Set.projIcc 0 1 (by norm_num) (u : ℝ)))
+      simp only [normalHalfPlane, Bool.false_eq_true, ↓reduceIte, Set.mem_ofPred_eq] at hh
+      simpa only [Function.comp_apply] using sub_nonpos.mpr hh
+    · rw [inner_sub_left, hmN]
+      have hh := hhalf (Set.Icc.convexComb ⟨a, le_rfl, hab.le⟩ s
+        (Set.projIcc 0 1 (by norm_num) ((u : ℝ) - 1)))
+      simp only [normalHalfPlane, Bool.false_eq_true, ↓reduceIte, Set.mem_ofPred_eq] at hh
+      simpa only [Function.comp_apply] using sub_nonpos.mpr hh
+  have hymLift : IsCurveAngleLift y m (fun u ↦
+      θ.toReal + Complex.arg (-frameComplex θ (y u - m)) + Real.pi) :=
+    isCurveAngleLift_frameArg_neg_add_pi_of_inner_nonpos θ hy hyhalf hym
+  let ε := min (r / 2) (d / 4)
+  have hε : 0 < ε := lt_min (half_pos hr) (by positivity)
+  have hεr : ε < r := lt_of_le_of_lt (min_le_left _ _) (half_lt_self hr)
+  let q := m - ε • normalVector θ
+  have hnormN : ‖normalVector θ‖ = 1 := by
+    rw [← θ.coe_toReal]
+    exact norm_normalVector_real θ.toReal
+  have hqm : dist q m = ε := by
+    rw [dist_eq_norm]
+    simp only [q, sub_sub_cancel_left, norm_neg, norm_smul, Real.norm_eq_abs, hnormN,
+      mul_one, abs_of_pos hε]
+  have hqclose : dist q m < r := hqm.trans_lt hεr
+  have hyqLift := hymLift.add_principal_basepoint_correction hy (hdot q hqclose)
+  let z : Set.Icc (0 : ℝ) 1 → Point := fun u ↦ x (Set.Icc.convexComb s t u)
+  have hz : Continuous z := hx.comp (Set.Icc.continuous_convexComb _ _)
+  have hzN (u : Set.Icc (0 : ℝ) 1) : inner ℝ (z u) (normalVector θ) = h := by
+    apply inner_eq_of_mem_segment hxsN hxtN
+    rw [← hsegment]
+    exact ⟨Set.Icc.convexComb s t u, convexComb_between hst.le u, rfl⟩
+  have hqN : inner ℝ q (normalVector θ) = h - ε := by
+    dsimp [q]
+    rw [inner_sub_left, real_inner_smul_left, hmN]
+    have hNN : inner ℝ (normalVector θ) (normalVector θ) = 1 := by
+      rw [← θ.coe_toReal]
+      exact inner_normalVector_self θ.toReal
+    rw [hNN, mul_one]
+  have hzright (u : Set.Icc (0 : ℝ) 1) :
+      0 ≤ inner ℝ (z u - q) (normalVector θ) := by
+    rw [inner_sub_left, hzN, hqN]
+    linarith
+  have hzq (u : Set.Icc (0 : ℝ) 1) : z u ≠ q := by
+    intro heq
+    have := congrArg (fun w : Point ↦ inner ℝ w (normalVector θ)) heq
+    rw [hzN, hqN] at this
+    linarith
+  have hzqLift : IsCurveAngleLift z q
+      (fun u ↦ θ.toReal + Complex.arg (frameComplex θ (z u - q))) :=
+    isCurveAngleLift_frameArg_of_inner_nonneg θ hz hzright hzq
+  let doubleParam : Set.Icc (0 : ℝ) 1 → Set.Icc (0 : ℝ) 2 := fun u ↦
+    ⟨2 * (u : ℝ), by constructor <;> nlinarith [u.property.1, u.property.2]⟩
+  have hdouble : Continuous doubleParam := by
+    exact Continuous.subtype_mk (continuous_const.mul continuous_subtype_val) _
+  let y₁ : Set.Icc (0 : ℝ) 1 → Point := y ∘ doubleParam
+  have hy₁ : Continuous y₁ := hy.comp hdouble
+  have hy₁qLift : IsCurveAngleLift y₁ q
+      ((fun u ↦ θ.toReal + Complex.arg (-frameComplex θ (y u - m)) + Real.pi +
+        Complex.arg (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm (y u - m)) *
+            Complex.orthonormalBasisOneI.repr.symm (y u - q))) ∘
+          doubleParam) := hyqLift.comp hdouble
+  have hjoin : z ⟨1, by norm_num⟩ = y₁ ⟨0, by norm_num⟩ := by
+    simp [z, y₁, doubleParam, y]
+  obtain ⟨hcorr_t, hcorr_s, harg_tm, harg_sm, hneg_frame_sm, hneg_frame_tm⟩ :=
+    supporting_segment_argument_corrections (x s) (x t) θ d ε hd hε hdirection
+  change Complex.arg (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm (x t - m)) *
+          Complex.orthonormalBasisOneI.repr.symm (x t - q)) =
+        Complex.arg (frameComplex θ (x t - q)) - Real.pi / 2 at hcorr_t
+  change Complex.arg (starRingEnd ℂ (Complex.orthonormalBasisOneI.repr.symm (x s - m)) *
+          Complex.orthonormalBasisOneI.repr.symm (x s - q)) =
+        Complex.arg (frameComplex θ (x s - q)) + Real.pi / 2 at hcorr_s
+  have hz0 : z ⟨0, by norm_num⟩ = x s := by simp [z]
+  have hz1 : z ⟨1, by norm_num⟩ = x t := by simp [z]
+  have hy₁0 : y₁ ⟨0, by norm_num⟩ = x t := by
+    simp [y₁, doubleParam, y]
+  have hy₁1 : y₁ ⟨1, by norm_num⟩ = x s := by
+    simp [y₁, doubleParam, y]
+  have hwinding : curveWinding (by norm_num)
+      (Function.concatUnitIntervals z y₁) q = 1 := by
+    rw [curveWinding_concatUnitIntervals hzqLift hy₁qLift hjoin,
+      hzqLift.curveWinding_eq, hy₁qLift.curveWinding_eq]
+    rw [hz1, hz0]
+    simp only [Function.comp_apply]
+    simp only [add_sub_add_left_eq_sub, cyclicComplement, mul_one, Function.concatUnitIntervals_two,
+      Set.Icc.mk_one, Function.comp_apply, Set.Icc.convexComb_one,
+      Complex.orthonormalBasisOneI_repr_symm_apply, Fin.isValue, map_add, map_sub,
+        Complex.conj_ofReal, map_mul, Complex.conj_I,
+      mul_neg, mul_zero, Function.concatUnitIntervals_zero, Set.Icc.mk_zero,
+      Set.Icc.convexComb_zero, y, doubleParam]
+    simp only [Complex.orthonormalBasisOneI_repr_symm_apply, Fin.isValue, map_add, map_sub,
+      Complex.conj_ofReal, map_mul, Complex.conj_I,
+      mul_neg] at hcorr_t hcorr_s
+    rw [hneg_frame_sm, hneg_frame_tm, harg_tm, harg_sm, hcorr_t, hcorr_s]
+    field_simp [Real.pi_ne_zero]
+    ring
+  have hrange_w := cyclic_partition_range hab x hclosed s t hst
+  have hwinding_eq_x (p : Point) (hp : p ∉ Set.range x) :
+      curveWinding (by norm_num) (Function.concatUnitIntervals z y₁) p =
+        curveWinding hab.le x p := by
+    obtain ⟨α, hα⟩ := exists_curveAngleLift_of_avoids hab hx hp
+    let φst := Set.Icc.convexComb s t
+    let φtb := Set.Icc.convexComb t ⟨b, hab.le, le_rfl⟩
+    let φas := Set.Icc.convexComb ⟨a, le_rfl, hab.le⟩ s
+    have hzα : IsCurveAngleLift z p (α ∘ φst) := hα.comp (Set.Icc.continuous_convexComb _ _)
+    have htbα : IsCurveAngleLift (x ∘ φtb) p (α ∘ φtb) :=
+      hα.comp (Set.Icc.continuous_convexComb _ _)
+    have hasα : IsCurveAngleLift (x ∘ φas) p (α ∘ φas) :=
+      hα.comp (Set.Icc.continuous_convexComb _ _)
+    have htailJoin : (x ∘ φtb) ⟨1, by norm_num⟩ = (x ∘ φas) ⟨0, by norm_num⟩ := by
+      simpa [φtb, φas] using hclosed.symm
+    have hyp : p ∉ Set.range y := by
+      rintro ⟨u, rfl⟩
+      apply hp
+      unfold y cyclicComplement Function.concatUnitIntervals
+      split_ifs <;> exact ⟨_, rfl⟩
+    obtain ⟨β, hβ⟩ := exists_curveAngleLift_of_avoids (by norm_num) hy hyp
+    have hy₁wind : curveWinding (by norm_num) y₁ p = curveWinding (by norm_num) y p := by
+      apply curveWinding_comp_of_endpoints (by norm_num) (by norm_num) ⟨β, hβ⟩ hdouble
+      · apply Subtype.ext
+        simp [doubleParam]
+      · apply Subtype.ext
+        simp [doubleParam]
+    rw [curveWinding_concatUnitIntervals hzα (hβ.comp hdouble) hjoin, hy₁wind]
+    change curveWinding (by norm_num) z p +
+      curveWinding (by norm_num) (Function.concatUnitIntervals (x ∘ φtb) (x ∘ φas)) p = _
+    rw [curveWinding_concatUnitIntervals htbα hasα htailJoin]
+    rw [hzα.curveWinding_eq, htbα.curveWinding_eq, hasα.curveWinding_eq,
+      hα.curveWinding_eq]
+    have hst0 : α (φst ⟨0, by norm_num⟩) = α s := congrArg α (by simp [φst])
+    have hst1 : α (φst ⟨1, by norm_num⟩) = α t := congrArg α (by simp [φst])
+    have htb0 : α (φtb ⟨0, by norm_num⟩) = α t := congrArg α (by simp [φtb])
+    have htb1 : α (φtb ⟨1, by norm_num⟩) = α ⟨b, hab.le, le_rfl⟩ :=
+      congrArg α (by simp [φtb])
+    have has0 : α (φas ⟨0, by norm_num⟩) = α ⟨a, le_rfl, hab.le⟩ :=
+      congrArg α (by simp [φas])
+    have has1 : α (φas ⟨1, by norm_num⟩) = α s := congrArg α (by simp [φas])
+    simp only [Function.comp_apply]
+    rw [hst0, hst1, htb0, htb1, has0, has1]
+    ring
+  have hqrange : q ∉ Set.range x := by
+    rw [← hrange_w]
+    intro hq
+    obtain ⟨α, hα⟩ := exists_curveAngleLift_of_curveWinding_ne_zero
+      (a := 0) (b := 2) (by norm_num)
+      (by rw [hwinding]; norm_num)
+    obtain ⟨u, hu⟩ := hq
+    have hc := (hα.2 u).1
+    have hs := (hα.2 u).2
+    rw [hu, sub_self] at hc hs
+    simp at hc hs
+    nlinarith only [hc, hs, Real.cos_sq_add_sin_sq (α u)]
+  have hwinding_q : curveWinding hab.le x q = 1 := by
+    rw [← hwinding_eq_x q hqrange, hwinding]
+  exact jordan_counterclockwise_of_winding_one a b hab x hx hΓ hclosed hinj q
+    hqrange hwinding_q
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Subarc
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open Set
+
+/-- Reversing the parameter of a closed path preserves injectivity away from the
+identified terminal endpoint. -/
+theorem injOn_comp_reverse_of_closed_injOn
+    {α : Type*} {a b : ℝ} (hab : a < b) (x : Set.Icc a b → α)
+    (hclosed : x ⟨a, le_rfl, hab.le⟩ = x ⟨b, hab.le, le_rfl⟩)
+    (hinj : Set.InjOn x {t | (t : ℝ) < b}) :
+    Set.InjOn (x ∘ Set.Icc.reverse hab.le) {t | (t : ℝ) < b} := by
+  intro s hs t ht hst
+  change (s : ℝ) < b at hs
+  change (t : ℝ) < b at ht
+  have hrs_pos : a < (Set.Icc.reverse hab.le s : ℝ) := by
+    change a < a + b - (s : ℝ)
+    linarith
+  have hrt_pos : a < (Set.Icc.reverse hab.le t : ℝ) := by
+    change a < a + b - (t : ℝ)
+    linarith
+  by_cases hrs : (Set.Icc.reverse hab.le s : ℝ) < b
+  · by_cases hrt : (Set.Icc.reverse hab.le t : ℝ) < b
+    · exact (Set.Icc.involutive_reverse hab.le).injective (hinj hrs hrt hst)
+    · have hrteq : Set.Icc.reverse hab.le t = ⟨b, hab.le, le_rfl⟩ := by
+        apply Subtype.ext
+        exact le_antisymm (Set.Icc.reverse hab.le t).property.2 (le_of_not_gt hrt)
+      have hra : x (Set.Icc.reverse hab.le s) = x ⟨a, le_rfl, hab.le⟩ := by
+        rw [hclosed, ← hrteq]
+        exact hst
+      have heq := hinj hrs hab hra
+      exfalso
+      have := congrArg Subtype.val heq
+      linarith
+  · have hrseq : Set.Icc.reverse hab.le s = ⟨b, hab.le, le_rfl⟩ := by
+      apply Subtype.ext
+      exact le_antisymm (Set.Icc.reverse hab.le s).property.2 (le_of_not_gt hrs)
+    by_cases hrt : (Set.Icc.reverse hab.le t : ℝ) < b
+    · have hra : x ⟨a, le_rfl, hab.le⟩ = x (Set.Icc.reverse hab.le t) := by
+        rw [hclosed, ← hrseq]
+        exact hst
+      have heq := hinj hab hrt hra
+      exfalso
+      have := congrArg Subtype.val heq
+      linarith
+    · have hrteq : Set.Icc.reverse hab.le t = ⟨b, hab.le, le_rfl⟩ := by
+        apply Subtype.ext
+        exact le_antisymm (Set.Icc.reverse hab.le t).property.2 (le_of_not_gt hrt)
+      exact (Set.Icc.involutive_reverse hab.le).injective (hrseq.trans hrteq.symm)
+
+/-- Parameter reversal sends the reversed closed interval to the original interval image. -/
+theorem image_Icc_reverse_interval {α : Type*} {a b : ℝ} (hab : a ≤ b)
+    (x : Set.Icc a b → α) (l u : Set.Icc a b) (_hlu : l ≤ u) :
+    (x ∘ Set.Icc.reverse hab) ''
+        Set.Icc (Set.Icc.reverse hab u) (Set.Icc.reverse hab l) =
+      x '' Set.Icc l u := by
+  ext z
+  constructor
+  · rintro ⟨s, hs, rfl⟩
+    refine ⟨Set.Icc.reverse hab s, ?_, rfl⟩
+    constructor
+    · simpa only [Set.Icc.involutive_reverse hab l] using
+        Set.Icc.antitone_reverse hab hs.2
+    · simpa only [Set.Icc.involutive_reverse hab u] using
+        Set.Icc.antitone_reverse hab hs.1
+  · rintro ⟨s, hs, rfl⟩
+    refine ⟨Set.Icc.reverse hab s, ?_, ?_⟩
+    · constructor
+      · exact Set.Icc.antitone_reverse hab hs.2
+      · exact Set.Icc.antitone_reverse hab hs.1
+    · simp only [Function.comp_apply, Set.Icc.involutive_reverse hab s]
+
+-- Duplicate of the current private UpperGraph helper; promote to Geometry.Support.
+private theorem exists_Icc_coe_image_of_compact_connected_Ioo {a b : ℝ}
+    {S : Set (Set.Ioo a b)} (hS : IsCompact S) (hconn : IsConnected S) :
+    ∃ l u : ℝ, ((fun t : Set.Ioo a b ↦ (t : ℝ)) '' S) = Set.Icc l u := by
+  let T := (fun t : Set.Ioo a b ↦ (t : ℝ)) '' S
+  have hTc : IsCompact T := hS.image continuous_subtype_val
+  have hTconn : IsConnected T := hconn.image _ continuous_subtype_val.continuousOn
+  exact ⟨sInf T, sSup T, eq_Icc_of_connected_compact hTconn hTc⟩
+
+private theorem exists_Icc_pullback_of_compact_connected_puncturedLoop
+    {a b : ℝ} {x : Set.Icc a b → Point} (hab : a ≤ b) (hab' : a < b)
+    (hx : Continuous x)
+    (hclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (hinj : Set.InjOn x {t | (t : ℝ) < b})
+    (C : Set (puncturedLoopRangeSet hab x)) (hC : IsCompact C)
+    (hconn : IsConnected C) :
+    ∃ l u : ℝ,
+      (fun t : Set.Ioo a b ↦ (t : ℝ)) ''
+          ((openIntervalHomeomorphPuncturedRange hab hab' x hx hclosed hinj).symm '' C) =
+        Set.Icc l u := by
+  let e := openIntervalHomeomorphPuncturedRange hab hab' x hx hclosed hinj
+  exact exists_Icc_coe_image_of_compact_connected_Ioo
+    (hC.image e.symm.continuous) (hconn.image e.symm e.symm.continuous.continuousOn)
+
+private theorem exists_Icc_parameters_of_compact_pathConnected_subset_loop
+    {a b : ℝ} {x : Set.Icc a b → Point} (hab : a ≤ b) (hab' : a < b)
+    (hx : Continuous x)
+    (hclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (hinj : Set.InjOn x {t | (t : ℝ) < b})
+    (C : Set Point) (hC : IsCompact C) (hpath : IsPathConnected C)
+    (hCrange : C ⊆ Set.range x) (hbase : x ⟨a, le_rfl, hab⟩ ∉ C) :
+    ∃ l u : ℝ,
+      (fun t : Set.Ioo a b ↦ (t : ℝ)) ''
+          ((openIntervalHomeomorphPuncturedRange hab hab' x hx hclosed hinj).symm ''
+            {q : puncturedLoopRangeSet hab x | (q : Point) ∈ C}) = Set.Icc l u := by
+  let D : Set (puncturedLoopRangeSet hab x) := {q | (q : Point) ∈ C}
+  let f : puncturedLoopRangeSet hab x → Point := fun q ↦ q
+  have hf_ind : Topology.IsInducing f :=
+    Topology.IsInducing.comp Topology.IsEmbedding.subtypeVal.isInducing
+      Topology.IsEmbedding.subtypeVal.isInducing
+  have hCrange' : C ⊆ Set.range f := by
+    intro p hp
+    obtain ⟨t, ht⟩ := hCrange hp
+    let q : Set.range x := ⟨p, ⟨t, ht⟩⟩
+    have hq : (q : Point) ≠ x ⟨a, le_rfl, hab⟩ := by
+      intro heq
+      exact hbase (heq ▸ hp)
+    exact ⟨⟨q, hq⟩, rfl⟩
+  have hDcompact : IsCompact D := by
+    change IsCompact (f ⁻¹' C)
+    exact hf_ind.isCompact_preimage' hC hCrange'
+  have hDrange : {q : Set.range x | (q : Point) ∈ C} ⊆ puncturedLoopRangeSet hab x := by
+    intro q hq
+    exact fun heq ↦ hbase (heq ▸ hq)
+  have hDpath : IsPathConnected D := by
+    have h₁ := hpath.preimage_coe hCrange
+    exact h₁.preimage_coe hDrange
+  exact exists_Icc_pullback_of_compact_connected_puncturedLoop hab hab' hx hclosed hinj
+    D hDcompact hDpath.isConnected
+
+private theorem exists_rectifiableOrientedArc_restrict_closedJordan_of_lt_top
+    {a b : ℝ} {x : ContinuousBVPaths a b} {Γ : Set Point}
+    (hab : a ≤ b) (hx : IsOrientedJordanParametrization hab Γ true x.val)
+    (l u : Set.Icc a b) (hlu : l ≤ u) (hub : (u : ℝ) < b) :
+    ∃ A : RectifiableOrientedArc,
+      A.val.carrier = Set.range (ContinuousBVPaths.restrict x l u hlu).val ∧
+      A.val.startPoint = x.val l ∧ A.val.endPoint = x.val u := by
+  let y := ContinuousBVPaths.restrict x l u hlu
+  have hyinj : Function.Injective y.val := by
+    intro s t hst
+    apply Subtype.ext
+    have hs : (s : ℝ) < b := lt_of_le_of_lt s.property.2 hub
+    have ht : (t : ℝ) < b := lt_of_le_of_lt t.property.2 hub
+    have h := hx.2.2.2.2.2.1 hs ht hst
+    exact congrArg (fun z : Set.Icc a b ↦ (z : ℝ)) h
+  let A0 : OrientedJordanArc :=
+    { carrier := Set.range y.val
+      startPoint := x.val l
+      endPoint := x.val u
+      parametrizable := ⟨l, u, hlu, y.val, y.property.1, hyinj, rfl, rfl, rfl⟩ }
+  let p : ArcBVParametrization A0 :=
+    { a := l, b := u, ordered := hlu, path := y, injective := hyinj,
+      range_eq := rfl, start_eq := rfl, end_eq := rfl }
+  exact ⟨⟨A0, ⟨p⟩⟩, rfl, rfl, rfl⟩
+
+/-- Convex interpolation between ordered interval points is strictly increasing. -/
+theorem strictMono_convexComb_of_lt {a b : ℝ}
+    (l u : Set.Icc a b) (hlu : l < u) : StrictMono (Set.Icc.convexComb l u) := by
+  intro s t hst
+  have hlu' : (l : ℝ) < u := hlu
+  have hst' : (s : ℝ) < t := hst
+  change (1 - (s : ℝ)) * l + (s : ℝ) * u <
+    (1 - (t : ℝ)) * l + (t : ℝ) * u
+  nlinarith [mul_pos (sub_pos.mpr hst') (sub_pos.mpr hlu')]
+
+/-- The two closed pieces outside an interior parameter interval trace exactly the
+closed curve with the open interval image removed. -/
+theorem image_complement_interval_of_closed_injOn
+    {α : Type*} {a b : ℝ} (hab : a ≤ b) (x : Set.Icc a b → α)
+    (hclosed : x ⟨a, le_rfl, hab⟩ = x ⟨b, hab, le_rfl⟩)
+    (hinj : Set.InjOn x {t | (t : ℝ) < b})
+    (l u : Set.Icc a b) (hal : a < l) (hub : (u : ℝ) < b) :
+    x '' {t | t ≤ l ∨ u ≤ t} =
+      Set.range x \ x '' {t | l < t ∧ t < u} := by
+  ext p
+  constructor
+  · rintro ⟨t, ht, rfl⟩
+    refine ⟨Set.mem_range_self t, ?_⟩
+    rintro ⟨s, hs, heq⟩
+    have hsb : (s : ℝ) < b := lt_trans hs.2 hub
+    by_cases htb : (t : ℝ) < b
+    · have hst := hinj hsb htb heq
+      subst t
+      exact ht.elim (not_le_of_gt hs.1) (not_le_of_gt hs.2)
+    · have hte : t = ⟨b, hab, le_rfl⟩ := by
+        apply Subtype.ext
+        exact le_antisymm t.2.2 (le_of_not_gt htb)
+      have hab' : a < b := hal.trans_le l.2.2
+      have hsa : s = ⟨a, le_rfl, hab⟩ :=
+        hinj hsb hab' (heq.trans ((congrArg x hte).trans hclosed.symm))
+      have : (l : ℝ) < a := by
+        have h := hs.1
+        rw [hsa] at h
+        exact h
+      exact (not_lt_of_ge hal.le) this
+  · rintro ⟨⟨t, rfl⟩, hp⟩
+    refine ⟨t, ?_, rfl⟩
+    by_contra ht
+    have hs : l < t ∧ t < u := ⟨lt_of_not_ge (fun h ↦ ht (Or.inl h)),
+      lt_of_not_ge (fun h ↦ ht (Or.inr h))⟩
+    exact hp ⟨t, hs, rfl⟩
+
+/-- Convex interpolation between the interval endpoints covers the interval. -/
+theorem surjective_convexComb_endpoints (a b : ℝ) (hab : a ≤ b) :
+    Function.Surjective
+      (Set.Icc.convexComb (⟨a, le_rfl, hab⟩ : Set.Icc a b) ⟨b, hab, le_rfl⟩) := by
+  intro x
+  rcases hab.eq_or_lt with rfl | hab
+  · refine ⟨⟨0, by norm_num⟩, Subtype.ext ?_⟩
+    change (1 - (0 : ℝ)) * a + 0 * a = (x : ℝ)
+    have hx : (x : ℝ) = a := le_antisymm x.property.2 x.property.1
+    simp [hx]
+  · refine ⟨⟨((x : ℝ) - a) / (b - a), ?_⟩, Subtype.ext ?_⟩
+    · constructor
+      · exact div_nonneg (sub_nonneg.mpr x.property.1) (sub_nonneg.mpr hab.le)
+      · exact (div_le_one (sub_pos.mpr hab)).2 (sub_le_sub_right x.property.2 a)
+    · change (1 - ((x : ℝ) - a) / (b - a)) * a +
+        ((x : ℝ) - a) / (b - a) * b = (x : ℝ)
+      field_simp [ne_of_gt (sub_pos.mpr hab)]
+      ring
+
+/-- The initial restriction of a cyclically concatenated closed path traces the
+complement of an interior parameter interval. -/
+theorem range_cyclicConcat_restrict_to_complement
+    {α : Type*} {a b : ℝ} (hab : a ≤ b) (f : Set.Icc a b → α)
+    (hclosed : f ⟨a, le_rfl, hab⟩ = f ⟨b, hab, le_rfl⟩)
+    (l u : Set.Icc a b) (hal : a < l) (hlu : l < u) (hub : (u : ℝ) < b)
+    (θ : Set.Icc (0 : ℝ) 1)
+    (hθ : Set.Icc.convexComb ⟨a, le_rfl, hab⟩ u θ = l) :
+    let v : ℝ := 1 + θ
+    Set.range (fun z : Set.Icc (0 : ℝ) v ↦
+      Function.concatUnitIntervals
+        (f ∘ Set.Icc.convexComb u ⟨b, hab, le_rfl⟩)
+        (f ∘ Set.Icc.convexComb ⟨a, le_rfl, hab⟩ u)
+        ⟨z, z.property.1, z.property.2.trans (by
+          have hθle : (θ : ℝ) ≤ 1 := θ.property.2
+          dsimp only [v]
+          linarith)⟩) = f '' {z | z ≤ l ∨ u ≤ z} := by
+  dsimp only
+  ext p
+  constructor
+  · rintro ⟨z, rfl⟩
+    by_cases hz : (z : ℝ) ≤ 1
+    · let q : Set.Icc (0 : ℝ) 1 := ⟨z, z.property.1, hz⟩
+      let y := Set.Icc.convexComb u ⟨b, hab, le_rfl⟩ q
+      refine ⟨y, Or.inr ?_, ?_⟩
+      · change (u : ℝ) ≤ (1 - (q : ℝ)) * u + (q : ℝ) * b
+        nlinarith [q.property.1, q.property.2, u.property.2]
+      · have hvnonneg : 0 ≤ 1 + (θ : ℝ) := by linarith [θ.property.1]
+        have hzmem : (z : ℝ) ∈ Set.Icc (0 : ℝ) 1 := ⟨z.property.1, hz⟩
+        change f y = Function.concatUnitIntervals _ _
+          (⟨z, z.property.1, z.property.2.trans (by linarith [θ.property.2])⟩ :
+            Set.Icc (0 : ℝ) 2)
+        simp [Function.concatUnitIntervals, hz, q, y,
+          Set.projIcc_of_mem (by norm_num) hzmem]
+    · have hz1 : 1 < (z : ℝ) := lt_of_not_ge hz
+      have hzsub0 : 0 ≤ (z : ℝ) - 1 := by linarith
+      have hzsub1 : (z : ℝ) - 1 ≤ 1 := by
+        have := z.property.2
+        nlinarith [θ.property.2]
+      let q : Set.Icc (0 : ℝ) 1 := ⟨(z : ℝ) - 1, hzsub0, hzsub1⟩
+      let y := Set.Icc.convexComb ⟨a, le_rfl, hab⟩ u q
+      refine ⟨y, Or.inl ?_, ?_⟩
+      · have hzθ : (q : ℝ) ≤ θ := by
+          dsimp only [q]
+          linarith [z.property.2]
+        change (1 - (q : ℝ)) * a + (q : ℝ) * u ≤ (l : ℝ)
+        have hθval := congrArg Subtype.val hθ
+        change (1 - (θ : ℝ)) * a + (θ : ℝ) * u = (l : ℝ) at hθval
+        nlinarith [hzθ, le_of_lt (lt_trans hal hlu)]
+      · simp [Function.concatUnitIntervals, hz, q, y,
+          Set.projIcc_of_mem (by norm_num) ⟨hzsub0, hzsub1⟩]
+  · rintro ⟨y, hy, rfl⟩
+    rcases hy with hyl | huy
+    · let y' : Set.Icc a (u : ℝ) := ⟨y, y.property.1, hyl.trans hlu.le⟩
+      obtain ⟨q, hq'⟩ := surjective_convexComb_endpoints a u
+        (le_trans l.property.1 hlu.le) y'
+      have hq : Set.Icc.convexComb ⟨a, le_rfl, hab⟩ u q = y := by
+        apply Subtype.ext
+        have h := congrArg (fun z : Set.Icc a (u : ℝ) ↦ (z : ℝ)) hq'
+        simpa only [Set.Icc.coe_convexComb] using h
+      have hqθ : (q : ℝ) ≤ θ := by
+        have hqval := congrArg Subtype.val hq
+        have hθval := congrArg Subtype.val hθ
+        change (1 - (q : ℝ)) * a + (q : ℝ) * u = (y : ℝ) at hqval
+        change (1 - (θ : ℝ)) * a + (θ : ℝ) * u = (l : ℝ) at hθval
+        have hau : a < (u : ℝ) := hal.trans hlu
+        by_contra hn
+        have hθq : (θ : ℝ) < q := lt_of_not_ge hn
+        have hprod : 0 < ((q : ℝ) - θ) * ((u : ℝ) - a) :=
+          mul_pos (sub_pos.mpr hθq) (sub_pos.mpr hau)
+        have : (l : ℝ) < y := by nlinarith [hqval, hθval, hprod]
+        exact (not_lt_of_ge hyl) this
+      by_cases hq0 : (q : ℝ) = 0
+      · let z : Set.Icc (0 : ℝ) (1 + (θ : ℝ)) := ⟨1, by
+          constructor
+          · norm_num
+          · linarith [θ.property.1]⟩
+        refine ⟨z, ?_⟩
+        have hyA : y = ⟨a, le_rfl, hab⟩ := by
+          apply Subtype.ext
+          have hqval := congrArg (fun z : Set.Icc a b ↦ (z : ℝ)) hq
+          change (1 - (q : ℝ)) * a + (q : ℝ) * u = (y : ℝ) at hqval
+          simp [hq0] at hqval
+          exact hqval.symm
+        rw [hyA, hclosed]
+        simp [z, Function.comp_def]
+      · have hqpos : 0 < (q : ℝ) := lt_of_le_of_ne q.property.1 (Ne.symm hq0)
+        let z : Set.Icc (0 : ℝ) (1 + (θ : ℝ)) := ⟨(q : ℝ) + 1, by
+          constructor <;> linarith⟩
+        refine ⟨z, ?_⟩
+        have hznot : ¬(z : ℝ) ≤ 1 := by dsimp only [z]; linarith
+        rw [← hq]
+        simp [z, Function.concatUnitIntervals, hznot,
+          Set.projIcc_of_mem (by norm_num) q.property]
+    · let y' : Set.Icc (u : ℝ) b := ⟨y, huy, y.property.2⟩
+      obtain ⟨q, hq'⟩ := surjective_convexComb_endpoints u b hub.le y'
+      have hq : Set.Icc.convexComb u ⟨b, hab, le_rfl⟩ q = y := by
+        apply Subtype.ext
+        have h := congrArg (fun z : Set.Icc (u : ℝ) b ↦ (z : ℝ)) hq'
+        simpa only [Set.Icc.coe_convexComb] using h
+      let z : Set.Icc (0 : ℝ) (1 + (θ : ℝ)) := ⟨q, by
+        constructor
+        · exact q.property.1
+        · linarith [q.property.2, θ.property.1]⟩
+      refine ⟨z, ?_⟩
+      have hzle : (z : ℝ) ≤ 1 := q.property.2
+      rw [← hq]
+      simp [z, Function.concatUnitIntervals, hzle,
+        Set.projIcc_of_mem (by norm_num) q.property]
+
+private theorem exists_complementary_rectifiableOrientedArc
+    {a b : ℝ} {x : ContinuousBVPaths a b} {Γ : Set Point}
+    (hab : a ≤ b) (hx : IsOrientedJordanParametrization hab Γ true x.val)
+    (l u : Set.Icc a b) (hal : a < l) (hlu : l < u) (hub : (u : ℝ) < b)
+    (P Q : Point) (hlQ : x.val l = Q) (huP : x.val u = P) :
+    ∃ A : RectifiableOrientedArc,
+      A.val.carrier = Set.range x.val \ x.val '' {z | l < z ∧ z < u} ∧
+      A.val.startPoint = P ∧ A.val.endPoint = Q := by
+  let l' : Set.Icc a (u : ℝ) := ⟨l, l.property.1, hlu.le⟩
+  obtain ⟨θ, hθ'⟩ := surjective_convexComb_endpoints a u
+    (le_trans l.property.1 hlu.le) l'
+  have hθ : Set.Icc.convexComb ⟨a, le_rfl, hab⟩ u θ = l := by
+    apply Subtype.ext
+    have h := congrArg (fun z : Set.Icc a (u : ℝ) ↦ (z : ℝ)) hθ'
+    simpa only [Set.Icc.coe_convexComb] using h
+  have hθpos : 0 < (θ : ℝ) := by
+    by_contra hn
+    have hθzero : (θ : ℝ) = 0 := le_antisymm (le_of_not_gt hn) θ.property.1
+    have h := congrArg (fun z : Set.Icc a b ↦ (z : ℝ)) hθ
+    change (1 - (θ : ℝ)) * a + (θ : ℝ) * u = (l : ℝ) at h
+    simp [hθzero] at h
+    linarith
+  have hθlt : (θ : ℝ) < 1 := by
+    by_contra hn
+    have hθone : (θ : ℝ) = 1 := le_antisymm θ.property.2 (le_of_not_gt hn)
+    have h := congrArg (fun z : Set.Icc a b ↦ (z : ℝ)) hθ
+    change (1 - (θ : ℝ)) * a + (θ : ℝ) * u = (l : ℝ) at h
+    simp [hθone] at h
+    exact hlu.ne (Subtype.ext h.symm)
+  obtain ⟨r, hrJordan, hr⟩ :=
+    exists_oriented_cyclic_rotation_eq_concat hab x hx u
+      (hal.trans hlu) hub
+  let z : Set.Icc (0 : ℝ) 2 := ⟨0, by norm_num⟩
+  let v : Set.Icc (0 : ℝ) 2 := ⟨1 + (θ : ℝ), by
+    constructor <;> linarith⟩
+  have hzv : z ≤ v := by change (0 : ℝ) ≤ 1 + θ; linarith
+  have hvtop : (v : ℝ) < 2 := by dsimp only [v]; linarith
+  obtain ⟨A, hAcarrier, hAstart, hAend⟩ :=
+    exists_rectifiableOrientedArc_restrict_closedJordan_of_lt_top
+      (a := 0) (b := 2) (x := r) (Γ := Γ) (by norm_num) hrJordan z v hzv hvtop
+  have hrzero : r.val z = P := by
+    rw [hr]
+    simp [z, Function.comp_def, huP]
+  have hrv : r.val v = Q := by
+    rw [hr]
+    have hvnot : ¬(v : ℝ) ≤ 1 := by dsimp only [v]; linarith
+    have hθmem : (θ : ℝ) ∈ Set.Icc (0 : ℝ) 1 := θ.property
+    have hproj : Set.projIcc (0 : ℝ) 1 (by norm_num) ((v : ℝ) - 1) = θ := by
+      apply Subtype.ext
+      simp [v]
+    simp [Function.concatUnitIntervals, hvnot, hproj, hθ, hlQ]
+  refine ⟨A, ?_, hAstart.trans hrzero, hAend.trans hrv⟩
+  rw [hAcarrier]
+  have hrange : Set.range (ContinuousBVPaths.restrict r z v hzv).val =
+      Set.range (fun w : Set.Icc (0 : ℝ) (1 + (θ : ℝ)) ↦
+        Function.concatUnitIntervals
+          (x.val ∘ Set.Icc.convexComb u ⟨b, hab, le_rfl⟩)
+          (x.val ∘ Set.Icc.convexComb ⟨a, le_rfl, hab⟩ u)
+          ⟨w, w.property.1, w.property.2.trans (by linarith [θ.property.2])⟩) := by
+    change Set.range (r.val ∘ fun w : Set.Icc (0 : ℝ) (v : ℝ) ↦
+      (⟨w, w.property.1, w.property.2.trans v.property.2⟩ : Set.Icc (0 : ℝ) 2)) = _
+    rw [hr]
+    rfl
+  rw [hrange, range_cyclicConcat_restrict_to_complement hab x.val
+    hx.2.2.2.2.1 l u hal hlu hub θ hθ]
+  exact image_complement_interval_of_closed_injOn hab x.val hx.2.2.2.2.1
+    hx.2.2.2.2.2.1 l u hal hub
+
+/-- If a closed parameter interval traces a nondegenerate segment injectively, its
+open interval traces the segment with its endpoints removed. -/
+theorem image_Ioo_eq_segment_diff_endpoints_of_image_Icc
+    {a b : ℝ} {x : Set.Icc a b → Point}
+    (hinj : Set.InjOn x {t | (t : ℝ) < b})
+    (l u : Set.Icc a b) (hlu : l < u) (hub : (u : ℝ) < b)
+    (P Q : Point) (hl : x l = Q) (hu : x u = P)
+    (himage : x '' Set.Icc l u = segment ℝ P Q) :
+    x '' {z | l < z ∧ z < u} = segment ℝ P Q \ {P, Q} := by
+  apply Set.Subset.antisymm
+  · rintro p ⟨z, hz, rfl⟩
+    refine ⟨himage ▸ ⟨z, ⟨hz.1.le, hz.2.le⟩, rfl⟩, ?_⟩
+    intro hp
+    rcases hp with hp | hp
+    · have hzu := hinj (show (z : ℝ) < b from lt_trans hz.2 hub) hub
+        (hp.trans hu.symm)
+      exact (ne_of_lt hz.2) hzu
+    · have hzl := hinj (show (z : ℝ) < b from lt_trans hz.2 hub)
+        (show (l : ℝ) < b from lt_trans hlu hub) (hp.trans hl.symm)
+      exact (ne_of_gt hz.1) hzl
+  · rintro p ⟨hpseg, hpends⟩
+    rw [← himage] at hpseg
+    obtain ⟨z, hz, rfl⟩ := hpseg
+    refine ⟨z, ⟨?_, ?_⟩, rfl⟩
+    · refine lt_of_le_of_ne hz.1 ?_
+      intro h
+      apply hpends
+      exact Or.inr ((congrArg x h).symm.trans hl)
+    · refine lt_of_le_of_ne hz.2 ?_
+      intro h
+      apply hpends
+      exact Or.inl ((congrArg x h).trans hu)
+
+private theorem exists_rectifiableOrientedArc_of_cut_parametrization
+    {a b : ℝ} {x : ContinuousBVPaths a b} {Γ U : Set Point}
+    (hab : a ≤ b) (hx : IsOrientedJordanParametrization hab Γ true x.val)
+    (l u : Set.Icc a b) (hal : a < l) (hlu : l < u) (hub : (u : ℝ) < b)
+    (P Q : Point) (hl : x.val l = Q) (hu : x.val u = P)
+    (hfrontier : Γ = U ∪ segment ℝ P Q)
+    (hinter : U ∩ segment ℝ P Q = {P, Q})
+    (himage : x.val '' Set.Icc l u = segment ℝ P Q) :
+    ∃ A : RectifiableOrientedArc,
+      A.val.carrier = U ∧ A.val.startPoint = P ∧ A.val.endPoint = Q := by
+  obtain ⟨A, hA, hstart, hend⟩ :=
+    exists_complementary_rectifiableOrientedArc hab hx l u hal hlu hub P Q hl hu
+  refine ⟨A, ?_, hstart, hend⟩
+  rw [hA, hx.2.2.2.1, hfrontier,
+    image_Ioo_eq_segment_diff_endpoints_of_image_Icc
+      hx.2.2.2.2.2.1 l u hlu hub P Q hl hu himage]
+  ext p
+  constructor
+  · rintro ⟨hp, hpnot⟩
+    rcases hp with hpU | hpseg
+    · exact hpU
+    · have hpends : p ∈ ({P, Q} : Set Point) := by
+        by_contra hn
+        exact hpnot ⟨hpseg, hn⟩
+      rw [← hinter] at hpends
+      exact hpends.1
+  · intro hpU
+    refine ⟨Or.inl hpU, ?_⟩
+    rintro ⟨hpseg, hpnotends⟩
+    apply hpnotends
+    rw [← hinter]
+    exact ⟨hpU, hpseg⟩
+
+/-- A nondegenerate segment in a closed Jordan curve, away from the base point,
+is traced by an interior parameter interval. -/
+theorem exists_parameter_interval_of_segment_subset_closedJordan
+    {a b : ℝ} {x : ContinuousBVPaths a b} {Γ : Set Point}
+    (hab : a < b) (hx : IsOrientedJordanParametrization hab.le Γ true x.val)
+    (P Q : Point) (hPQ : P ≠ Q) (hsegment : segment ℝ P Q ⊆ Γ)
+    (hbase : x.val ⟨a, le_rfl, hab.le⟩ ∉ segment ℝ P Q) :
+    ∃ l u : Set.Icc a b, a < l ∧ l < u ∧ (u : ℝ) < b ∧
+      x.val '' Set.Icc l u = segment ℝ P Q := by
+  have hcompact : IsCompact (segment ℝ P Q) := by
+    rw [segment_eq_image]
+    exact isCompact_Icc.image (by fun_prop)
+  have hpath : IsPathConnected (segment ℝ P Q) :=
+    (convex_segment P Q).isPathConnected ⟨P, left_mem_segment ℝ P Q⟩
+  have hCrange : segment ℝ P Q ⊆ Set.range x.val := by
+    rw [hx.2.2.2.1]
+    exact hsegment
+  obtain ⟨l, u, hlu⟩ := exists_Icc_parameters_of_compact_pathConnected_subset_loop
+    hab.le hab x.property.1 hx.2.2.2.2.1 hx.2.2.2.2.2.1
+    (segment ℝ P Q) hcompact hpath hCrange hbase
+  have hpre_nonempty :
+      ((openIntervalHomeomorphPuncturedRange hab.le hab x.val x.property.1
+        hx.2.2.2.2.1 hx.2.2.2.2.2.1).symm ''
+        {q : puncturedLoopRangeSet hab.le x.val | (q : Point) ∈ segment ℝ P Q}).Nonempty := by
+    let q : Set.range x.val := ⟨P, hCrange (left_mem_segment ℝ P Q)⟩
+    have hq : (q : Point) ≠ x.val ⟨a, le_rfl, hab.le⟩ := by
+      intro heq
+      exact hbase (heq ▸ left_mem_segment ℝ P Q)
+    let q' : puncturedLoopRangeSet hab.le x.val := ⟨q, hq⟩
+    refine ⟨(openIntervalHomeomorphPuncturedRange hab.le hab x.val x.property.1
+      hx.2.2.2.2.1 hx.2.2.2.2.2.1).symm q', ?_⟩
+    exact ⟨q', left_mem_segment ℝ P Q, rfl⟩
+  have hIcc_nonempty : (Set.Icc l u).Nonempty := by
+    rw [← hlu]
+    exact hpre_nonempty.image _
+  have hlu_order : l ≤ u := Set.nonempty_Icc.mp hIcc_nonempty
+  have hlmem : l ∈ Set.Icc l u := ⟨le_rfl, hlu_order⟩
+  have humem : u ∈ Set.Icc l u := ⟨hlu_order, le_rfl⟩
+  rw [← hlu] at hlmem humem
+  obtain ⟨sl, hsl, hslval⟩ := hlmem
+  obtain ⟨su, hsu, hsuval⟩ := humem
+  change (sl : ℝ) = l at hslval
+  change (su : ℝ) = u at hsuval
+  let l' : Set.Icc a b := ⟨sl, sl.property.1.le, sl.property.2.le⟩
+  let u' : Set.Icc a b := ⟨su, su.property.1.le, su.property.2.le⟩
+  have hlstrict : a < (l' : ℝ) := sl.property.1
+  have hustricttop : (u' : ℝ) < b := su.property.2
+  have hlu' : l' ≤ u' := by
+    change (sl : ℝ) ≤ su
+    linarith
+  have himage : x.val '' Set.Icc l' u' = segment ℝ P Q := by
+    apply Set.Subset.antisymm
+    · rintro p ⟨s, hs, rfl⟩
+      let si : Set.Ioo a b := ⟨s, lt_of_lt_of_le hlstrict hs.1,
+        lt_of_le_of_lt hs.2 hustricttop⟩
+      have hscoord : (s : ℝ) ∈
+          (fun z : Set.Ioo a b ↦ (z : ℝ)) ''
+            ((openIntervalHomeomorphPuncturedRange hab.le hab x.val x.property.1
+              hx.2.2.2.2.1 hx.2.2.2.2.2.1).symm ''
+              {q : puncturedLoopRangeSet hab.le x.val |
+                (q : Point) ∈ segment ℝ P Q}) := by
+        rw [hlu]
+        change l ≤ (s : ℝ) ∧ (s : ℝ) ≤ u
+        constructor
+        · calc l = (sl : ℝ) := hslval.symm
+               _ ≤ s := hs.1
+        · calc (s : ℝ) ≤ su := hs.2
+               _ = u := hsuval
+      obtain ⟨z, hz, hzval⟩ := hscoord
+      obtain ⟨q, hqseg, hzq⟩ := hz
+      have hzcoe := openIntervalHomeomorphPuncturedRange_coe hab.le hab x.val
+        x.property.1 hx.2.2.2.2.1 hx.2.2.2.2.2.1 z
+      have hzs : z = si := Subtype.ext hzval
+      rw [hzs] at hzcoe hzq
+      let e := openIntervalHomeomorphPuncturedRange hab.le hab x.val x.property.1
+        hx.2.2.2.2.1 hx.2.2.2.2.2.1
+      have heq : e si = q := by
+        exact e.eq_symm_apply.mp hzq.symm
+      have heq' := congrArg (fun z : puncturedLoopRangeSet hab.le x.val ↦ (z : Point)) heq
+      rw [hzcoe] at heq'
+      change (q : Point) ∈ segment ℝ P Q at hqseg
+      have heq'' : x.val s = (q : Point) := by simpa [si] using heq'
+      rw [heq'']
+      exact hqseg
+    · intro p hp
+      let q : Set.range x.val := ⟨p, hCrange hp⟩
+      have hqbase : (q : Point) ≠ x.val ⟨a, le_rfl, hab.le⟩ := by
+        intro heq
+        exact hbase (heq ▸ hp)
+      let q' : puncturedLoopRangeSet hab.le x.val := ⟨q, hqbase⟩
+      let s := (openIntervalHomeomorphPuncturedRange hab.le hab x.val x.property.1
+        hx.2.2.2.2.1 hx.2.2.2.2.2.1).symm q'
+      have hscoord : (s : ℝ) ∈ Set.Icc l u := by
+        rw [← hlu]
+        exact ⟨s, ⟨q', hp, rfl⟩, rfl⟩
+      let s' : Set.Icc a b := ⟨s, s.property.1.le, s.property.2.le⟩
+      refine ⟨s', ?_, ?_⟩
+      · change l' ≤ s' ∧ s' ≤ u'
+        constructor <;> change (_ : ℝ) ≤ _ <;> dsimp only [l', u', s'] <;>
+          linarith [hscoord.1, hscoord.2]
+      have hscoe := openIntervalHomeomorphPuncturedRange_coe hab.le hab x.val
+        x.property.1 hx.2.2.2.2.1 hx.2.2.2.2.2.1 s
+      let e := openIntervalHomeomorphPuncturedRange hab.le hab x.val x.property.1
+        hx.2.2.2.2.1 hx.2.2.2.2.2.1
+      have heq : e s = q' := e.apply_symm_apply q'
+      have heq' := congrArg (fun z : puncturedLoopRangeSet hab.le x.val ↦ (z : Point)) heq
+      exact hscoe.symm.trans heq'
+  have hlune : l' ≠ u' := by
+    intro heq
+    have hsingle : segment ℝ P Q = {x.val l'} := by
+      rw [← himage, heq, Set.Icc_self, Set.image_singleton]
+    have hPm : P ∈ ({x.val l'} : Set Point) := hsingle ▸ left_mem_segment ℝ P Q
+    have hQm : Q ∈ ({x.val l'} : Set Point) := hsingle ▸ right_mem_segment ℝ P Q
+    exact hPQ (by simpa using hPm.trans hQm.symm)
+  exact ⟨l', u', hlstrict, lt_of_le_of_ne hlu' hlune,
+    hustricttop, himage⟩
+
+private theorem inner_sub_right_injOn_segment (P Q : Point) (hPQ : P ≠ Q) :
+    Set.InjOn (fun p : Point ↦ inner ℝ (p - P) (Q - P)) (segment ℝ P Q) := by
+  intro p hp q hq hpq
+  rw [segment_eq_image' ℝ P Q] at hp hq
+  obtain ⟨s, hs, rfl⟩ := hp
+  obtain ⟨t, ht, rfl⟩ := hq
+  have hd : Q - P ≠ 0 := sub_ne_zero.mpr hPQ.symm
+  have hnorm : 0 < ‖Q - P‖ ^ 2 := sq_pos_of_pos (norm_pos_iff.mpr hd)
+  have hst : s = t := by
+    simp only [add_sub_cancel_left, inner_smul_left, real_inner_self_eq_norm_sq] at hpq
+    have hpq' : s * ‖Q - P‖ ^ 2 = t * ‖Q - P‖ ^ 2 := by simpa using hpq
+    nlinarith
+  rw [hst]
+
+private theorem exists_rectifiableOrientedArc_reverse (A : RectifiableOrientedArc) :
+    ∃ B : RectifiableOrientedArc,
+      B.val.carrier = A.val.carrier ∧ B.val.startPoint = A.val.endPoint ∧
+        B.val.endPoint = A.val.startPoint := by
+  let p := Classical.choice A.property
+  let r := Set.Icc.reverse p.ordered
+  let y : ContinuousBVPaths p.a p.b :=
+    ⟨p.path.val ∘ r, p.path.property.1.comp (Set.Icc.continuous_reverse p.ordered),
+      fun i ↦ BoundedVariationOn.comp_antitone_surjective_Icc p.ordered
+        (p.path.property.2 i) (Set.Icc.antitone_reverse p.ordered)
+          (Set.Icc.surjective_reverse p.ordered)⟩
+  have hyinj : Function.Injective y.val :=
+    p.injective.comp (Set.Icc.involutive_reverse p.ordered).injective
+  have hyrange : Set.range y.val = A.val.carrier := by
+    have hrange : Set.range y.val = Set.range p.path.val := by
+      apply Set.Subset.antisymm
+      · rintro z ⟨t, rfl⟩
+        exact ⟨r t, rfl⟩
+      · rintro z ⟨t, rfl⟩
+        obtain ⟨s, hs⟩ := Set.Icc.surjective_reverse p.ordered t
+        refine ⟨s, ?_⟩
+        change p.path.val (r s) = p.path.val t
+        rw [show r s = t by exact hs]
+    rw [hrange]
+    exact p.range_eq
+  have hystart : y.val ⟨p.a, le_rfl, p.ordered⟩ = A.val.endPoint := by
+    simpa [y, r, Set.Icc.reverse] using p.end_eq
+  have hyend : y.val ⟨p.b, p.ordered, le_rfl⟩ = A.val.startPoint := by
+    simpa [y, r, Set.Icc.reverse] using p.start_eq
+  let B0 : OrientedJordanArc :=
+    { carrier := A.val.carrier
+      startPoint := A.val.endPoint
+      endPoint := A.val.startPoint
+      parametrizable := ⟨p.a, p.b, p.ordered, y.val, y.property.1, hyinj,
+        hyrange, hystart, hyend⟩ }
+  let q : ArcBVParametrization B0 :=
+    { a := p.a
+      b := p.b
+      ordered := p.ordered
+      path := y
+      injective := hyinj
+      range_eq := hyrange
+      start_eq := hystart
+      end_eq := hyend }
+  exact ⟨⟨B0, ⟨q⟩⟩, rfl, rfl, rfl⟩
+
+/-- The endpoints of an injectively parametrized nondegenerate segment are the
+parameter-interval endpoints, in one of the two possible orders. -/
+theorem endpoints_eq_or_eq_swap_of_image_Icc_eq_segment
+    {a b : ℝ} {x : ContinuousBVPaths a b}
+    (hab : a ≤ b) (hinj : Set.InjOn x.val {t | (t : ℝ) < b})
+    (l u : Set.Icc a b) (hlu : l < u) (hub : (u : ℝ) < b)
+    (P Q : Point) (hPQ : P ≠ Q)
+    (himage : x.val '' Set.Icc l u = segment ℝ P Q) :
+    (x.val l = P ∧ x.val u = Q) ∨ (x.val l = Q ∧ x.val u = P) := by
+  let f : ℝ → ℝ := fun t ↦
+    inner ℝ (x.val (Set.projIcc a b hab t) - P) (Q - P)
+  have hf_cont : Continuous f := by
+    exact (x.property.1.comp continuous_projIcc).sub continuous_const |>.inner
+      continuous_const
+  have hproj (t : ℝ) (ht : t ∈ Set.Icc (l : ℝ) u) :
+      Set.projIcc a b hab t =
+        ⟨t, l.property.1.trans ht.1, ht.2.trans u.property.2⟩ := by
+    exact Set.projIcc_of_mem hab ⟨l.property.1.trans ht.1, ht.2.trans u.property.2⟩
+  have hparam (t : ℝ) (ht : t ∈ Set.Icc (l : ℝ) u) :
+      x.val (Set.projIcc a b hab t) ∈ segment ℝ P Q := by
+    rw [← himage]
+    refine ⟨⟨t, l.property.1.trans ht.1, ht.2.trans u.property.2⟩, ?_, ?_⟩
+    · exact ht
+    · apply congrArg x.val
+      apply Subtype.ext
+      exact congrArg Subtype.val (hproj t ht).symm
+  have hf_inj : Set.InjOn f (Set.Icc (l : ℝ) u) := by
+    intro s hs t ht hst
+    have hxs := inner_sub_right_injOn_segment P Q hPQ
+      (hparam s hs) (hparam t ht) hst
+    have hst' := hinj
+      (show ((Set.projIcc a b hab s : Set.Icc a b) : ℝ) < b by
+        simpa [hproj s hs] using lt_of_le_of_lt hs.2 hub)
+      (show ((Set.projIcc a b hab t : Set.Icc a b) : ℝ) < b by
+        simpa [hproj t ht] using lt_of_le_of_lt ht.2 hub)
+      hxs
+    simpa [hproj s hs, hproj t ht] using congrArg Subtype.val hst'
+  have hmono := ContinuousOn.strictMonoOn_of_injOn_Icc'
+    (show (l : ℝ) ≤ u from hlu.le) hf_cont.continuousOn hf_inj
+  have hP : P ∈ x.val '' Set.Icc l u := himage ▸ left_mem_segment ℝ P Q
+  have hQ : Q ∈ x.val '' Set.Icc l u := himage ▸ right_mem_segment ℝ P Q
+  obtain ⟨s, hs, hsP⟩ := hP
+  obtain ⟨t, ht, htQ⟩ := hQ
+  have hs' : (s : ℝ) ∈ Set.Icc (l : ℝ) u := hs
+  have ht' : (t : ℝ) ∈ Set.Icc (l : ℝ) u := ht
+  have hl' : (l : ℝ) ∈ Set.Icc (l : ℝ) u := ⟨le_rfl, hlu.le⟩
+  have hu' : (u : ℝ) ∈ Set.Icc (l : ℝ) u := ⟨hlu.le, le_rfl⟩
+  rcases hmono with hmono | hanti
+  · left
+    constructor
+    · have hfls : f l ≤ f s := hmono.monotoneOn hl' hs' hs.1
+      have hflP : f l = f s := by
+        have hnonneg : 0 ≤ f l := by
+          have hmem := hparam l hl'
+          rw [segment_eq_image' ℝ P Q] at hmem
+          obtain ⟨r, hr, hrl⟩ := hmem
+          dsimp only [f]
+          rw [← hrl]
+          simp only [add_sub_cancel_left, inner_smul_left,
+            real_inner_self_eq_norm_sq]
+          exact mul_nonneg hr.1 (sq_nonneg _)
+        have hfs : f s = 0 := by simp [f, hproj s hs', hsP]
+        linarith
+      simpa [hproj l hl'] using
+        (inner_sub_right_injOn_segment P Q hPQ
+          (hparam l hl') (hparam s hs') hflP |>.trans
+            (by simpa [hproj s hs'] using hsP))
+    · have hftu : f t ≤ f u := hmono.monotoneOn ht' hu' ht.2
+      have hfuQ : f u = f t := by
+        have hupper : f u ≤ ‖Q - P‖ ^ 2 := by
+          have hmem := hparam u hu'
+          rw [segment_eq_image' ℝ P Q] at hmem
+          obtain ⟨r, hr, hru⟩ := hmem
+          dsimp only [f]
+          rw [← hru]
+          simp only [add_sub_cancel_left, inner_smul_left,
+            real_inner_self_eq_norm_sq]
+          exact mul_le_of_le_one_left (sq_nonneg _) hr.2
+        have hft : f t = ‖Q - P‖ ^ 2 := by
+          simp [f, hproj t ht', htQ]
+        linarith
+      simpa [hproj u hu'] using
+        (inner_sub_right_injOn_segment P Q hPQ
+          (hparam u hu') (hparam t ht') hfuQ |>.trans
+            (by simpa [hproj t ht'] using htQ))
+  · right
+    constructor
+    · have hftl : f t ≤ f l := hanti.antitoneOn hl' ht' ht.1
+      have hflQ : f l = f t := by
+        have hupper : f l ≤ ‖Q - P‖ ^ 2 := by
+          have hmem := hparam l hl'
+          rw [segment_eq_image' ℝ P Q] at hmem
+          obtain ⟨r, hr, hrl⟩ := hmem
+          dsimp only [f]
+          rw [← hrl]
+          simp only [add_sub_cancel_left, inner_smul_left,
+            real_inner_self_eq_norm_sq]
+          exact mul_le_of_le_one_left (sq_nonneg _) hr.2
+        have hft : f t = ‖Q - P‖ ^ 2 := by
+          simp [f, hproj t ht', htQ]
+        linarith
+      simpa [hproj l hl'] using
+        (inner_sub_right_injOn_segment P Q hPQ
+          (hparam l hl') (hparam t ht') hflQ |>.trans
+            (by simpa [hproj t ht'] using htQ))
+    · have hfus : f u ≤ f s := hanti.antitoneOn hs' hu' hs.2
+      have hfuP : f u = f s := by
+        have hnonneg : 0 ≤ f u := by
+          have hmem := hparam u hu'
+          rw [segment_eq_image' ℝ P Q] at hmem
+          obtain ⟨r, hr, hru⟩ := hmem
+          dsimp only [f]
+          rw [← hru]
+          simp only [add_sub_cancel_left, inner_smul_left,
+            real_inner_self_eq_norm_sq]
+          exact mul_nonneg hr.1 (sq_nonneg _)
+        have hfs : f s = 0 := by simp [f, hproj s hs', hsP]
+        linarith
+      simpa [hproj u hu'] using
+        (inner_sub_right_injOn_segment P Q hPQ
+          (hparam u hu') (hparam s hs') hfuP |>.trans
+            (by simpa [hproj s hs'] using hsP))
+
+/-- Removing a supporting chord from a counterclockwise closed Jordan path yields the oriented
+complementary arc. -/
+theorem exists_rectifiableOrientedArc_of_closedJordan_cut
+    {a b : ℝ} {x : ContinuousBVPaths a b} {Γ U : Set Point}
+    (hab : a < b) (hx : IsOrientedJordanParametrization hab.le Γ true x.val)
+    (P Q : Point) (hPQ : P ≠ Q)
+    (hbase : x.val ⟨a, le_rfl, hab.le⟩ ∉ segment ℝ P Q)
+    (hfrontier : Γ = U ∪ segment ℝ P Q)
+    (hinter : U ∩ segment ℝ P Q = {P, Q}) :
+    ∃ A : RectifiableOrientedArc,
+      A.val.carrier = U ∧ A.val.startPoint = P ∧ A.val.endPoint = Q := by
+  have hsegment : segment ℝ P Q ⊆ Γ := by
+    rw [hfrontier]
+    exact Set.subset_union_right
+  obtain ⟨l, u, hal, hlu, hub, himage⟩ :=
+    exists_parameter_interval_of_segment_subset_closedJordan hab hx P Q hPQ hsegment hbase
+  rcases endpoints_eq_or_eq_swap_of_image_Icc_eq_segment hab.le
+      hx.2.2.2.2.2.1 l u hlu hub P Q hPQ himage with hend | hend
+  · have hfrontier' : Γ = U ∪ segment ℝ Q P := by
+      simpa only [segment_symm ℝ Q P] using hfrontier
+    have hinter' : U ∩ segment ℝ Q P = {Q, P} := by
+      rw [segment_symm ℝ Q P, hinter]
+      exact Set.pair_comm P Q
+    have himage' : x.val '' Set.Icc l u = segment ℝ Q P := by
+      simpa only [segment_symm ℝ Q P] using himage
+    obtain ⟨A, hA, hstart, hend'⟩ :=
+      exists_rectifiableOrientedArc_of_cut_parametrization hab.le hx l u hal hlu hub
+        Q P hend.1 hend.2 hfrontier' hinter' himage'
+    obtain ⟨B, hB, hBstart, hBend⟩ := exists_rectifiableOrientedArc_reverse A
+    exact ⟨B, hB.trans hA, hBstart.trans hend', hBend.trans hstart⟩
+  · exact exists_rectifiableOrientedArc_of_cut_parametrization hab.le hx l u hal hlu hub
+      P Q hend.1 hend.2 hfrontier hinter himage
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Reparametrization
+-/
+
+@[expose] public section
+
+namespace MovingSofa
+
+theorem curveArea_reparametrization :
+    (∀ (a b c d : ℝ), a < b → c < d →
+      ∀ (x : ContinuousBVPaths a b) (φ : Set.Icc c d → Set.Icc a b),
+      Continuous φ → Function.Surjective φ → (Monotone φ ∨ Antitone φ) →
+      ∃ y : ContinuousBVPaths c d, y.val = x.val ∘ φ ∧
+        (Monotone φ → curveAreaFunctional y = curveAreaFunctional x) ∧
+        (Antitone φ → curveAreaFunctional y = -curveAreaFunctional x)) ∧
+    (∀ (Γ Δ : OrientedJordanArc) (x : ArcBVParametrization Γ)
+      (y : ArcBVParametrization Δ), Γ.carrier = Δ.carrier →
+      (Γ.startPoint = Δ.startPoint → Γ.endPoint = Δ.endPoint →
+        curveAreaFunctional x.path = curveAreaFunctional y.path) ∧
+      (Γ.startPoint = Δ.endPoint → Γ.endPoint = Δ.startPoint →
+        curveAreaFunctional x.path = -curveAreaFunctional y.path)) ∧
+    (∀ (Γ Δ : OrientedJordanCurve) (x : ClosedBVParametrization Γ)
+      (y : ClosedBVParametrization Δ), Γ.carrier = Δ.carrier →
+      (Γ.counterclockwise = Δ.counterclockwise →
+        curveAreaFunctional x.path = curveAreaFunctional y.path) ∧
+      (Γ.counterclockwise ≠ Δ.counterclockwise →
+        curveAreaFunctional x.path = -curveAreaFunctional y.path)) ∧
+    (∀ (a b : ℝ) (x : ContinuousBVPaths a b) (p : Point),
+      (∀ t, x.val t = p) → curveAreaFunctional x = 0) := by
+  refine ⟨?_, curveArea_arc_same_carrier, ?_, ?_⟩
+  · intro a b c d hab hcd x φ hφc hφs hφ
+    exact curveArea_comp_monotone_or_antitone_surjective hab.le hcd.le x φ hφc hφs hφ
+  · exact curveArea_closed_same_carrier
+  · intro a b x p hx
+    exact curveAreaFunctional_eq_zero_of_constant x p hx
+
+end MovingSofa
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Segment Area Properties
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open MeasureTheory Set
+
+theorem segmentArea_jordan_and_frame (p q : Point) :
+    (∃ A : RectifiableOrientedArc,
+      A.val.carrier = segment ℝ p q ∧ A.val.startPoint = p ∧ A.val.endPoint = q ∧
+      jordanArcArea A = segmentArea p q) ∧
+    (∀ (t : Real.Angle) (h d : ℝ), p ∈ normalLine t h → q ∈ normalLine t h →
+      q - p = d • tangentVector t → segmentArea p q = h * d / 2) := by
+  constructor
+  · by_cases hpq : p = q
+    · subst q
+      let Γ : OrientedJordanArc :=
+        { carrier := segment ℝ p p
+          startPoint := p
+          endPoint := p
+          parametrizable := by
+            refine ⟨0, 0, le_rfl, (constBVPath 0 0 p).val,
+              (constBVPath 0 0 p).property.1, ?_, ?_, rfl, rfl⟩
+            · intro s t _
+              apply Subtype.ext
+              exact le_antisymm (s.property.2.trans t.property.1)
+                (t.property.2.trans s.property.1)
+            · simp [constBVPath] }
+      let z : ArcBVParametrization Γ :=
+        { a := 0
+          b := 0
+          ordered := le_rfl
+          path := constBVPath 0 0 p
+          injective := by
+            intro s t _
+            apply Subtype.ext
+            exact le_antisymm (s.property.2.trans t.property.1)
+              (t.property.2.trans s.property.1)
+          range_eq := by simp [constBVPath, Γ]
+          start_eq := rfl
+          end_eq := rfl }
+      let A : RectifiableOrientedArc := ⟨Γ, ⟨z⟩⟩
+      refine ⟨A, rfl, rfl, rfl, ?_⟩
+      change curveAreaFunctional (Classical.choice A.property).path = segmentArea p p
+      calc
+        _ = curveAreaFunctional z.path :=
+          (curveArea_reparametrization.2.1 Γ Γ
+            (Classical.choice A.property) z rfl).1 rfl rfl
+        _ = 0 := curveArea_reparametrization.2.2.2 0 0 z.path p (by intro t; rfl)
+        _ = segmentArea p p := by simp [segmentArea, planeCrossProduct]; ring
+    · let Γ : OrientedJordanArc :=
+        { carrier := segment ℝ p q
+          startPoint := p
+          endPoint := q
+          parametrizable := by
+            refine ⟨0, 1, by norm_num, Path.segment p q,
+              (Path.segment p q).continuous, Path.segment_injective_of_ne hpq,
+              Path.range_segment p q, ?_, ?_⟩
+            · simp
+            · simp }
+      let z : ArcBVParametrization Γ :=
+        { a := 0
+          b := 1
+          ordered := by norm_num
+          path := lineSegmentBVPath p q
+          injective := Path.segment_injective_of_ne hpq
+          range_eq := Path.range_segment p q
+          start_eq := by simp [lineSegmentBVPath, Γ]
+          end_eq := by simp [lineSegmentBVPath, Γ] }
+      let A : RectifiableOrientedArc := ⟨Γ, ⟨z⟩⟩
+      refine ⟨A, rfl, rfl, rfl, ?_⟩
+      change curveAreaFunctional (Classical.choice A.property).path = segmentArea p q
+      calc
+        _ = curveAreaFunctional z.path :=
+          (curveArea_reparametrization.2.1 Γ Γ
+            (Classical.choice A.property) z rfl).1 rfl rfl
+        _ = segmentArea p q := curveAreaFunctional_lineSegmentBVPath p q
+  · intro t h d hp hq hd
+    simp only [normalLine, Set.mem_ofPred_eq, segmentArea, planeCrossProduct] at hp hq ⊢
+    have hd0 := congrArg (fun x : Point => x 0) hd
+    have hd1 := congrArg (fun x : Point => x 1) hd
+    simp only [normalVector, frame, Fin.isValue, PiLp.sub_apply, tangentVector, PiLp.smul_apply,
+      Matrix.cons_val_zero, smul_eq_mul, mul_neg, Matrix.cons_val_one, Matrix.cons_val_fin_one,
+      ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, div_left_inj'] at hp hq hd0 hd1 ⊢
+    rw [PiLp.inner_apply, Fin.sum_univ_two, Real.inner_apply, Real.inner_apply] at hp hq
+    simp only [Fin.isValue, Matrix.cons_val_zero, Matrix.cons_val_one,
+      Matrix.cons_val_fin_one] at hp hq
+    have hq0 : q.ofLp 0 = p.ofLp 0 - d * t.sin := by linarith [hd0]
+    have hq1 : q.ofLp 1 = p.ofLp 1 + d * t.cos := by linarith [hd1]
+    rw [hq0, hq1]
+    calc
+      p.ofLp 0 * (p.ofLp 1 + d * t.cos) - p.ofLp 1 * (p.ofLp 0 - d * t.sin) =
+          d * (p.ofLp 0 * t.cos + p.ofLp 1 * t.sin) := by ring
+      _ = d * h := by rw [hp]
+      _ = h * d := by ring
+
+theorem segmentArea_collinear_origin (p q : Point)
+    (h : Collinear ℝ ({0, p, q} : Set Point)) : segmentArea p q = 0 := by
+  obtain ⟨v, hv⟩ := (collinear_iff_of_mem (by simp : (0 : Point) ∈ ({0, p, q} : Set Point))).mp h
+  obtain ⟨a, ha⟩ := hv p (by simp)
+  obtain ⟨b, hb⟩ := hv q (by simp)
+  simp only [vadd_eq_add, add_zero] at ha hb
+  rw [ha, hb]
+  simp [segmentArea, planeCrossProduct]
+  ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Curve / Jordan / Subarc Area
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open Set
+/-- A proper restriction of a closed BV Jordan parametrization realizes an arc with the same
+signed area as the restricted path. -/
+theorem exists_rectifiableOrientedArc_restrict_closedJordan_with_area
+    {a b : ℝ} {x : ContinuousBVPaths a b} {Γ : Set Point}
+    (hab : a ≤ b) (hx : IsOrientedJordanParametrization hab Γ true x.val)
+    (l u : Set.Icc a b) (hlu : l ≤ u) (hub : (u : ℝ) < b) :
+    ∃ A : RectifiableOrientedArc,
+      A.val.carrier = Set.range (ContinuousBVPaths.restrict x l u hlu).val ∧
+      A.val.startPoint = x.val l ∧ A.val.endPoint = x.val u ∧
+      jordanArcArea A =
+        curveAreaFunctional (ContinuousBVPaths.restrict x l u hlu) := by
+  let y := ContinuousBVPaths.restrict x l u hlu
+  have hyinj : Function.Injective y.val := by
+    intro s t hst
+    apply Subtype.ext
+    have hs : (s : ℝ) < b := lt_of_le_of_lt s.property.2 hub
+    have ht : (t : ℝ) < b := lt_of_le_of_lt t.property.2 hub
+    have h := hx.2.2.2.2.2.1 hs ht hst
+    exact congrArg (fun z : Set.Icc a b ↦ (z : ℝ)) h
+  let A0 : OrientedJordanArc :=
+    { carrier := Set.range y.val
+      startPoint := x.val l
+      endPoint := x.val u
+      parametrizable := ⟨l, u, hlu, y.val, y.property.1, hyinj, rfl, rfl, rfl⟩ }
+  let p : ArcBVParametrization A0 :=
+    { a := l, b := u, ordered := hlu, path := y, injective := hyinj,
+      range_eq := rfl, start_eq := rfl, end_eq := rfl }
+  let A : RectifiableOrientedArc := ⟨A0, ⟨p⟩⟩
+  refine ⟨A, rfl, rfl, rfl, ?_⟩
+  change curveAreaFunctional (Classical.choice A.property).path = curveAreaFunctional y
+  exact (curveArea_reparametrization.2.1 A0 A0
+    (Classical.choice A.property) p rfl).1 rfl rfl
+
+/-- The suffix of the cyclic rotation, after the complementary arc, is an increasing
+reparametrization of the removed interval. -/
+theorem curveArea_cyclicSuffix_eq_restriction
+    {a b : ℝ} (hab : a ≤ b) (x : ContinuousBVPaths a b)
+    (l u : Set.Icc a b) (hal : a < l) (hlu : l < u)
+    (θ : Set.Icc (0 : ℝ) 1)
+    (hθ : Set.Icc.convexComb ⟨a, le_rfl, hab⟩ u θ = l)
+    (r : ContinuousBVPaths 0 2)
+    (hr : r.val = Function.concatUnitIntervals
+      (x.val ∘ Set.Icc.convexComb u ⟨b, hab, le_rfl⟩)
+      (x.val ∘ Set.Icc.convexComb ⟨a, le_rfl, hab⟩ u)) :
+    let v : Set.Icc (0 : ℝ) 2 := ⟨1 + (θ : ℝ), by
+      constructor <;> linarith [θ.property.1, θ.property.2]⟩
+    curveAreaFunctional (ContinuousBVPaths.restrict r v ⟨2, by norm_num⟩ v.property.2) =
+      curveAreaFunctional (ContinuousBVPaths.restrict x l u hlu.le) := by
+  let v : Set.Icc (0 : ℝ) 2 := ⟨1 + (θ : ℝ), by
+    constructor <;> linarith [θ.property.1, θ.property.2]⟩
+  change curveAreaFunctional (ContinuousBVPaths.restrict r v ⟨2, by norm_num⟩
+      v.property.2) =
+    curveAreaFunctional (ContinuousBVPaths.restrict x l u hlu.le)
+  have hθlt : (θ : ℝ) < 1 := by
+    by_contra hn
+    have hθone : (θ : ℝ) = 1 := le_antisymm θ.property.2 (le_of_not_gt hn)
+    have h := congrArg Subtype.val hθ
+    change (1 - (θ : ℝ)) * a + (θ : ℝ) * u = (l : ℝ) at h
+    simp [hθone] at h
+    exact hlu.ne (Subtype.ext h.symm)
+  have hθpos_suffix : 0 < (θ : ℝ) := by
+    by_contra hn
+    have hθzero : (θ : ℝ) = 0 := le_antisymm (le_of_not_gt hn) θ.property.1
+    have h := congrArg Subtype.val hθ
+    change (1 - (θ : ℝ)) * a + (θ : ℝ) * u = (l : ℝ) at h
+    simp [hθzero] at h
+    linarith
+  have hvlt : (v : ℝ) < 2 := by dsimp only [v]; linarith
+  let ψ : Set.Icc (v : ℝ) 2 → Set.Icc (0 : ℝ) 1 := fun w ↦
+    ⟨((w : ℝ) - (v : ℝ)) / (2 - (v : ℝ)), by
+      constructor
+      · exact div_nonneg (sub_nonneg.mpr w.property.1) (sub_nonneg.mpr hvlt.le)
+      · exact (div_le_one (sub_pos.mpr hvlt)).2
+          (sub_le_sub_right w.property.2 (v : ℝ))⟩
+  let φ : Set.Icc (v : ℝ) 2 → Set.Icc (l : ℝ) u := fun w ↦
+    Set.Icc.convexComb (⟨l, le_rfl, hlu.le⟩ : Set.Icc (l : ℝ) u)
+      ⟨u, hlu.le, le_rfl⟩ (ψ w)
+  have hψc : Continuous ψ := by
+    apply Continuous.subtype_mk
+    fun_prop
+  have hψm : Monotone ψ := by
+    intro s t hst
+    apply Subtype.coe_le_coe.mp
+    dsimp only [ψ]
+    exact div_le_div_of_nonneg_right
+      (sub_le_sub_right (show (s : ℝ) ≤ t from hst) _) (sub_nonneg.mpr hvlt.le)
+  have hψs : Function.Surjective ψ := by
+    intro q
+    let w : Set.Icc (v : ℝ) 2 := ⟨(v : ℝ) + (2 - (v : ℝ)) * (q : ℝ), by
+      constructor
+      · nlinarith [q.property.1, sub_pos.mpr hvlt]
+      · nlinarith [q.property.2, sub_pos.mpr hvlt]⟩
+    refine ⟨w, Subtype.ext ?_⟩
+    dsimp only [ψ, w]
+    field_simp [ne_of_gt (sub_pos.mpr hvlt)]
+    ring
+  have hφc : Continuous φ :=
+    (Set.Icc.continuous_convexComb _ _).comp hψc
+  have hφm : Monotone φ := by
+    intro s t hst
+    change (1 - (ψ s : ℝ)) * (l : ℝ) + (ψ s : ℝ) * (u : ℝ) ≤
+      (1 - (ψ t : ℝ)) * (l : ℝ) + (ψ t : ℝ) * (u : ℝ)
+    have hψ := show (ψ s : ℝ) ≤ ψ t from hψm hst
+    nlinarith [show (l : ℝ) < u from hlu]
+  have hφs : Function.Surjective φ :=
+    (surjective_convexComb_endpoints (l : ℝ) u hlu.le).comp hψs
+  let xu := ContinuousBVPaths.restrict x l u hlu.le
+  obtain ⟨y, hy, hyarea⟩ := curveArea_comp_monotone_surjective
+    hlu.le v.property.2 xu φ hφc hφm hφs
+  have hyval : y.val = (ContinuousBVPaths.restrict r v ⟨2, by norm_num⟩
+      v.property.2).val := by
+    rw [hy]
+    funext w
+    simp only [xu, ContinuousBVPaths.restrict, Function.comp_apply]
+    rw [hr]
+    have hwlow : 1 + (θ : ℝ) ≤ (w : ℝ) := w.property.1
+    have hwsub : (w : ℝ) - 1 ∈ Set.Icc (0 : ℝ) 1 := by
+      constructor
+      · linarith
+      · linarith [w.property.2]
+    have hwnle : ¬(w : ℝ) ≤ 1 := by
+      linarith
+    simp only [Function.concatUnitIntervals, hwnle, ↓reduceIte,
+      Set.projIcc_of_mem (by norm_num) hwsub]
+    apply congrArg x.val
+    apply Subtype.ext
+    have hθval := congrArg Subtype.val hθ
+    change (1 - (θ : ℝ)) * a + (θ : ℝ) * u = (l : ℝ) at hθval
+    dsimp only [φ, ψ]
+    simp only [Set.Icc.coe_convexComb]
+    dsimp only [v]
+    rw [← hθval]
+    have hdenne : 2 - (1 + (θ : ℝ)) ≠ 0 := by linarith
+    field_simp [hdenne]
+    ring
+  have hyEq : y = ContinuousBVPaths.restrict r v ⟨2, by norm_num⟩ v.property.2 := by
+    ext w i
+    exact congrArg (fun z : Point ↦ z i) (congrFun hyval w)
+  rw [← hyEq, hyarea]
+
+private theorem exists_complementary_rectifiableOrientedArc_with_area
+    {a b : ℝ} {x : ContinuousBVPaths a b} {Γ : Set Point}
+    (hab : a ≤ b) (hx : IsOrientedJordanParametrization hab Γ true x.val)
+    (l u : Set.Icc a b) (hal : a < l) (hlu : l < u) (hub : (u : ℝ) < b)
+    (P Q : Point) (hlQ : x.val l = Q) (huP : x.val u = P) :
+    ∃ A : RectifiableOrientedArc,
+      A.val.carrier = Set.range x.val \ x.val '' {z | l < z ∧ z < u} ∧
+      A.val.startPoint = P ∧ A.val.endPoint = Q ∧
+      curveAreaFunctional x = jordanArcArea A +
+        curveAreaFunctional (ContinuousBVPaths.restrict x l u hlu.le) := by
+  let l' : Set.Icc a (u : ℝ) := ⟨l, l.property.1, hlu.le⟩
+  obtain ⟨θ, hθ'⟩ := surjective_convexComb_endpoints a u
+    (le_trans l.property.1 hlu.le) l'
+  have hθ : Set.Icc.convexComb ⟨a, le_rfl, hab⟩ u θ = l := by
+    apply Subtype.ext
+    have h := congrArg (fun z : Set.Icc a (u : ℝ) ↦ (z : ℝ)) hθ'
+    simpa only [Set.Icc.coe_convexComb] using h
+  have hθpos : 0 < (θ : ℝ) := by
+    by_contra hn
+    have hθzero : (θ : ℝ) = 0 := le_antisymm (le_of_not_gt hn) θ.property.1
+    have h := congrArg (fun z : Set.Icc a b ↦ (z : ℝ)) hθ
+    change (1 - (θ : ℝ)) * a + (θ : ℝ) * u = (l : ℝ) at h
+    simp [hθzero] at h
+    linarith
+  have hθlt : (θ : ℝ) < 1 := by
+    by_contra hn
+    have hθone : (θ : ℝ) = 1 := le_antisymm θ.property.2 (le_of_not_gt hn)
+    have h := congrArg (fun z : Set.Icc a b ↦ (z : ℝ)) hθ
+    change (1 - (θ : ℝ)) * a + (θ : ℝ) * u = (l : ℝ) at h
+    simp [hθone] at h
+    exact hlu.ne (Subtype.ext h.symm)
+  obtain ⟨r, hrJordan, hr⟩ :=
+    exists_oriented_cyclic_rotation_eq_concat hab x hx u
+      (hal.trans hlu) hub
+  let z : Set.Icc (0 : ℝ) 2 := ⟨0, by norm_num⟩
+  let v : Set.Icc (0 : ℝ) 2 := ⟨1 + (θ : ℝ), by
+    constructor <;> linarith⟩
+  have hzv : z ≤ v := by change (0 : ℝ) ≤ 1 + θ; linarith
+  have hvtop : (v : ℝ) < 2 := by dsimp only [v]; linarith
+  obtain ⟨A, hAcarrier, hAstart, hAend, hAarea⟩ :=
+    exists_rectifiableOrientedArc_restrict_closedJordan_with_area
+      (a := 0) (b := 2) (x := r) (Γ := Γ) (by norm_num) hrJordan z v hzv hvtop
+  have hrzero : r.val z = P := by
+    rw [hr]
+    simp [z, Function.comp_def, huP]
+  have hrv : r.val v = Q := by
+    rw [hr]
+    have hvnot : ¬(v : ℝ) ≤ 1 := by dsimp only [v]; linarith
+    have hθmem : (θ : ℝ) ∈ Set.Icc (0 : ℝ) 1 := θ.property
+    have hproj : Set.projIcc (0 : ℝ) 1 (by norm_num) ((v : ℝ) - 1) = θ := by
+      apply Subtype.ext
+      simp [v]
+    simp [Function.concatUnitIntervals, hvnot, hproj, hθ, hlQ]
+  have hcarrier : A.val.carrier =
+      Set.range x.val \ x.val '' {z | l < z ∧ z < u} := by
+    rw [hAcarrier]
+    have hrange : Set.range (ContinuousBVPaths.restrict r z v hzv).val =
+        Set.range (fun w : Set.Icc (0 : ℝ) (1 + (θ : ℝ)) ↦
+          Function.concatUnitIntervals
+            (x.val ∘ Set.Icc.convexComb u ⟨b, hab, le_rfl⟩)
+            (x.val ∘ Set.Icc.convexComb ⟨a, le_rfl, hab⟩ u)
+            ⟨w, w.property.1, w.property.2.trans (by linarith [θ.property.2])⟩) := by
+      change Set.range (r.val ∘ fun w : Set.Icc (0 : ℝ) (v : ℝ) ↦
+        (⟨w, w.property.1, w.property.2.trans v.property.2⟩ : Set.Icc (0 : ℝ) 2)) = _
+      rw [hr]
+      rfl
+    rw [hrange, range_cyclicConcat_restrict_to_complement hab x.val
+      hx.2.2.2.2.1 l u hal hlu hub θ hθ]
+    exact image_complement_interval_of_closed_injOn hab x.val hx.2.2.2.2.1
+      hx.2.2.2.2.2.1 l u hal hub
+  have hsuffix : curveAreaFunctional
+      (ContinuousBVPaths.restrict r v ⟨2, by norm_num⟩ v.property.2) =
+      curveAreaFunctional (ContinuousBVPaths.restrict x l u hlu.le) :=
+    curveArea_cyclicSuffix_eq_restriction hab x l u hal hlu θ hθ r hr
+  obtain ⟨r', hr', _, hrotate'⟩ :=
+    exists_cyclic_rotation_eq_concat hab x u hx.2.2.2.2.1.symm
+  have hrr : r = r' := by
+    ext w i
+    exact congrArg (fun z : Point ↦ z i) (congrFun (hr.trans hr'.symm) w)
+  have hrotate : curveAreaFunctional r = curveAreaFunctional x := by
+    rw [hrr]
+    exact hrotate'
+  have hsplit := curveArea_eq_restriction_add_restriction (by norm_num) r v
+  dsimp only at hsplit
+  refine ⟨A, hcarrier, hAstart.trans hrzero, hAend.trans hrv, ?_⟩
+  rw [← hrotate, hsplit, ← hAarea, hsuffix]
+
+private theorem exists_rectifiableOrientedArc_of_cut_parametrization_with_area
+    {a b : ℝ} {x : ContinuousBVPaths a b} {Γ U : Set Point}
+    (hab : a ≤ b) (hx : IsOrientedJordanParametrization hab Γ true x.val)
+    (l u : Set.Icc a b) (hal : a < l) (hlu : l < u) (hub : (u : ℝ) < b)
+    (P Q : Point) (hl : x.val l = Q) (hu : x.val u = P)
+    (hfrontier : Γ = U ∪ segment ℝ P Q)
+    (hinter : U ∩ segment ℝ P Q = {P, Q})
+    (himage : x.val '' Set.Icc l u = segment ℝ P Q) :
+    ∃ A : RectifiableOrientedArc,
+      A.val.carrier = U ∧ A.val.startPoint = P ∧ A.val.endPoint = Q ∧
+      curveAreaFunctional x = jordanArcArea A +
+        curveAreaFunctional (ContinuousBVPaths.restrict x l u hlu.le) := by
+  obtain ⟨A, hA, hstart, hend, harea⟩ :=
+    exists_complementary_rectifiableOrientedArc_with_area
+      hab hx l u hal hlu hub P Q hl hu
+  refine ⟨A, ?_, hstart, hend, harea⟩
+  rw [hA, hx.2.2.2.1, hfrontier,
+    image_Ioo_eq_segment_diff_endpoints_of_image_Icc
+      hx.2.2.2.2.2.1 l u hlu hub P Q hl hu himage]
+  ext p
+  constructor
+  · rintro ⟨hp, hpnot⟩
+    rcases hp with hpU | hpseg
+    · exact hpU
+    · have hpends : p ∈ ({P, Q} : Set Point) := by
+        by_contra hn
+        exact hpnot ⟨hpseg, hn⟩
+      rw [← hinter] at hpends
+      exact hpends.1
+  · intro hpU
+    refine ⟨Or.inl hpU, ?_⟩
+    rintro ⟨hpseg, hpnotends⟩
+    apply hpnotends
+    rw [← hinter]
+    exact ⟨hpU, hpseg⟩
+
+theorem endpoints_eq_of_counterclockwise_supporting_chord
+    {a b : ℝ} {x : ContinuousBVPaths a b} {Γ : Set Point}
+    (hab : a < b) (hx : IsOrientedJordanParametrization hab.le Γ true x.val)
+    (l u : Set.Icc a b) (hlu : l < u) (hub : (u : ℝ) < b)
+    (P Q : Point) (hPQ : P ≠ Q)
+    (himage : x.val '' Set.Icc l u = segment ℝ P Q)
+    (θ : Real.Angle) (h : ℝ)
+    (hhalf : Γ ⊆ normalHalfPlane θ h false false)
+    (hQline : Q ∈ normalLine θ h) (d : ℝ) (hd : 0 < d)
+    (hdir : P = Q + d • tangentVector θ) :
+    x.val l = Q ∧ x.val u = P := by
+  rcases endpoints_eq_or_eq_swap_of_image_Icc_eq_segment hab.le
+      hx.2.2.2.2.2.1 l u hlu hub P Q hPQ himage with hbad | hgood
+  · exfalso
+    let r := Set.Icc.reverse hab.le
+    let y : Set.Icc a b → Point := x.val ∘ r
+    let l' := r u
+    let u' := r l
+    have hl'u' : l' < u' := by
+      have hlu' : (l : ℝ) < u := hlu
+      change a + b - (u : ℝ) < a + b - (l : ℝ)
+      linarith
+    have hyrange : Set.range y = Γ := by
+      rw [show Set.range y = Set.range x.val by
+        apply Set.Subset.antisymm
+        · rintro z ⟨s, rfl⟩; exact ⟨r s, rfl⟩
+        · rintro z ⟨s, rfl⟩
+          obtain ⟨q, hq⟩ := Set.Icc.surjective_reverse hab.le s
+          exact ⟨q, by simpa [y, r] using congrArg x.val hq⟩]
+      exact hx.2.2.2.1
+    have hyl : y l' = Q := by
+      change x.val (r (r u)) = Q
+      rw [show r (r u) = u by exact Set.Icc.involutive_reverse hab.le u, hbad.2]
+    have hyu : y u' = P := by
+      change x.val (r (r l)) = P
+      rw [show r (r l) = l by exact Set.Icc.involutive_reverse hab.le l, hbad.1]
+    have hytrue := jordan_counterclockwise_of_supporting_segment a b hab y
+      (hx.2.2.1.comp (Set.Icc.continuous_reverse hab.le))
+      (by rw [hyrange]; exact hx.2.1)
+      (by simpa [y, r, Set.Icc.reverse] using hx.2.2.2.2.1.symm)
+      (injOn_comp_reverse_of_closed_injOn hab x.val hx.2.2.2.2.1
+        hx.2.2.2.2.2.1) θ h
+      (fun z ↦ hhalf (by rw [← hyrange]; exact Set.mem_range_self z)) l' u' hl'u'
+      (by rw [hyl]; exact hQline) d hd
+      (by rw [hyl, hyu, hdir])
+      (by rw [image_Icc_reverse_interval hab.le x.val l u hlu.le, himage,
+        segment_symm ℝ P Q, hyl, hyu])
+    rw [hyrange] at hytrue
+    have horient := hx.orientation_eq_not_of_comp hytrue r
+      (Set.Icc.continuous_reverse hab.le) (by simp [r, Set.Icc.reverse])
+      (by simp [r, Set.Icc.reverse]) rfl
+    norm_num at horient
+  · exact hgood
+
+/-- Removing a supporting chord from a counterclockwise closed BV Jordan path gives the
+complementary arc, and closed signed area splits into arc area and the oriented chord area. -/
+theorem exists_rectifiableOrientedArc_of_supportingChord_with_area
+    {a b : ℝ} {x : ContinuousBVPaths a b} {Γ U : Set Point}
+    (hab : a < b) (hx : IsOrientedJordanParametrization hab.le Γ true x.val)
+    (P Q : Point) (hPQ : P ≠ Q)
+    (hbase : x.val ⟨a, le_rfl, hab.le⟩ ∉ segment ℝ P Q)
+    (hfrontier : Γ = U ∪ segment ℝ P Q)
+    (hinter : U ∩ segment ℝ P Q = {P, Q})
+    (θ : Real.Angle) (h : ℝ)
+    (hhalf : Γ ⊆ normalHalfPlane θ h false false)
+    (hQline : Q ∈ normalLine θ h) (d : ℝ) (hd : 0 < d)
+    (hdir : P = Q + d • tangentVector θ) :
+    ∃ A : RectifiableOrientedArc,
+      A.val.carrier = U ∧ A.val.startPoint = P ∧ A.val.endPoint = Q ∧
+      curveAreaFunctional x = jordanArcArea A + segmentArea Q P := by
+  have hsegment : segment ℝ P Q ⊆ Γ := by
+    rw [hfrontier]
+    exact Set.subset_union_right
+  obtain ⟨l, u, hal, hlu, hub, himage⟩ :=
+    exists_parameter_interval_of_segment_subset_closedJordan
+      hab hx P Q hPQ hsegment hbase
+  have hend := endpoints_eq_of_counterclockwise_supporting_chord hab hx l u hlu hub
+    P Q hPQ himage θ h hhalf hQline d hd hdir
+  obtain ⟨A, hA, hAstart, hAend, harea⟩ :=
+    exists_rectifiableOrientedArc_of_cut_parametrization_with_area hab.le hx l u hal hlu hub
+      P Q hend.1 hend.2 hfrontier hinter himage
+  obtain ⟨B, hBcarrier, hBstart, hBend, hBarea⟩ :=
+    exists_rectifiableOrientedArc_restrict_closedJordan_with_area
+      hab.le hx l u hlu.le hub
+  obtain ⟨S, hScarrier, hSstart, hSend, hSarea⟩ :=
+    (segmentArea_jordan_and_frame Q P).1
+  have hBcarrier' : B.val.carrier = segment ℝ Q P := by
+    rw [hBcarrier]
+    change Set.range (x.val ∘ fun z : Set.Icc (l : ℝ) u ↦
+      (⟨z, le_trans l.property.1 z.property.1,
+        le_trans z.property.2 u.property.2⟩ : Set.Icc a b)) = _
+    rw [segment_symm ℝ Q P, ← himage]
+    ext p
+    constructor
+    · rintro ⟨z, rfl⟩
+      exact ⟨⟨z, le_trans l.property.1 z.property.1,
+        le_trans z.property.2 u.property.2⟩, z.property, rfl⟩
+    · rintro ⟨z, hz, rfl⟩
+      exact ⟨⟨z, hz⟩, rfl⟩
+  have hBstart' : B.val.startPoint = Q := hBstart.trans hend.1
+  have hBend' : B.val.endPoint = P := hBend.trans hend.2
+  have hBSarea : jordanArcArea B = jordanArcArea S := by
+    change curveAreaFunctional (Classical.choice B.property).path =
+      curveAreaFunctional (Classical.choice S.property).path
+    exact (curveArea_reparametrization.2.1 B.val S.val
+      (Classical.choice B.property) (Classical.choice S.property)
+      (hBcarrier'.trans hScarrier.symm)).1
+        (hBstart'.trans hSstart.symm) (hBend'.trans hSend.symm)
+  refine ⟨A, hA, hAstart, hAend, ?_⟩
+  rw [harea, ← hBarea, hBSarea, hSarea]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Curve.PositiveGraphOrientation`.
+* `Curve.PositiveGraphRegion`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The counterclockwise loop around a positive graph
+
+For `a < b` and a continuous `f : ℝ → ℝ` vanishing at `a` and `b` and positive on `(a, b)`,
+`MovingSofa.positiveGraphLoop` traverses the graph of `f` from `(b, 0)` to `(a, 0)` and then the
+base segment from `(a, 0)` back to `(b, 0)`. The main result
+`MovingSofa.positiveGraphLoop_counterclockwise` shows that this is a counterclockwise once-traversed
+Jordan parametrization whose bounded complementary component is the open subgraph
+`{p | a < p 0 ∧ p 0 < b ∧ 0 < p 1 ∧ p 1 < f (p 0)}`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Traverse the graph from right to left and return along the horizontal axis. -/
+def positiveGraphLoop (a b : ℝ) (f : ℝ → ℝ) (t : Set.Icc (0 : ℝ) 2) : Point :=
+  if t.val ≤ 1 then !₂[b - (b - a) * t.val, f (b - (b - a) * t.val)]
+  else !₂[a + (b - a) * (t.val - 1), 0]
+
+/-- Every planar point is built from its two coordinates. -/
+private lemma point_eq_vecNotation_coords (p : Point) : p = !₂[p 0, p 1] := by
+  ext i
+  fin_cases i <;> simp
+
+section PositiveGraph
+
+variable {a b : ℝ} {f : ℝ → ℝ}
+
+/-- A positive graph over `[a, b]` vanishing at the ends is nonnegative there. -/
+private lemma nonneg_of_pos_on_Ioo (ha : f a = 0) (hb : f b = 0)
+    (hpos : ∀ x ∈ Set.Ioo a b, 0 < f x) {c : ℝ} (hc : c ∈ Set.Icc a b) : 0 ≤ f c := by
+  rcases eq_or_lt_of_le hc.1 with h | h
+  · rw [← h, ha]
+  · rcases eq_or_lt_of_le hc.2 with h' | h'
+    · rw [h', hb]
+    · exact (hpos c ⟨h, h'⟩).le
+
+/-- Such a graph vanishes only at the two endpoints. -/
+private lemma eq_endpoint_of_apply_eq_zero (hpos : ∀ x ∈ Set.Ioo a b, 0 < f x)
+    {c : ℝ} (hc : c ∈ Set.Icc a b) (h : f c = 0) : c = a ∨ c = b := by
+  by_contra hcon
+  push Not at hcon
+  exact (hpos c ⟨lt_of_le_of_ne hc.1 (Ne.symm hcon.1), lt_of_le_of_ne hc.2 hcon.2⟩).ne' h
+
+/-- The reversed affine reparametrization of `[0, 1]` lands in the base interval `[a, b]`. -/
+private lemma sub_mul_mem_Icc (hab : a < b) {s : ℝ} (hs : s ∈ Set.Icc (0 : ℝ) 1) :
+    b - (b - a) * s ∈ Set.Icc a b := by
+  obtain ⟨h0, h1⟩ := hs
+  constructor <;> nlinarith [sub_pos.mpr hab]
+
+/-- On the first half of the parameter interval the loop traverses the graph right to left. -/
+lemma positiveGraphLoop_apply_of_le {s : ℝ} (hs : s ∈ Set.Icc (0 : ℝ) 2) (h : s ≤ 1) :
+    positiveGraphLoop a b f ⟨s, hs⟩ = !₂[b - (b - a) * s, f (b - (b - a) * s)] :=
+  ite_eq_left h
+
+/-- On the second half of the parameter interval the loop traverses the base left to right. -/
+lemma positiveGraphLoop_apply_of_not_le {s : ℝ} (hs : s ∈ Set.Icc (0 : ℝ) 2)
+    (h : ¬ s ≤ 1) : positiveGraphLoop a b f ⟨s, hs⟩ = !₂[a + (b - a) * (s - 1), 0] :=
+  ite_eq_right h
+
+/-- Values of the loop on the base half, including the joining parameter. -/
+private lemma positiveGraphLoop_apply_base (ha : f a = 0) {u : ℝ} (hu : u ∈ Set.Icc (0 : ℝ) 2)
+    (h1 : 1 ≤ u) : positiveGraphLoop a b f ⟨u, hu⟩ = !₂[a + (b - a) * (u - 1), 0] := by
+  by_cases h : u ≤ 1
+  · have hu1 : u = 1 := le_antisymm h h1
+    rw [positiveGraphLoop_apply_of_le hu h, show b - (b - a) * u = a by rw [hu1]; ring, ha,
+      show a + (b - a) * (u - 1) = a by rw [hu1]; ring]
+  · exact positiveGraphLoop_apply_of_not_le hu h
+
+/-- The two halves agree at the joining parameter, so the loop is continuous. -/
+private lemma continuous_positiveGraphLoop (hab : a < b) (hf : ContinuousOn f (Set.Icc a b))
+    (ha : f a = 0) : Continuous (positiveGraphLoop a b f) := by
+  unfold positiveGraphLoop
+  apply continuous_if_le continuous_subtype_val continuous_const
+  · refine (PiLp.continuous_toLp (2 : ENNReal) fun _ : Fin 2 ↦ ℝ).comp_continuousOn
+      (ContinuousOn.matrixVecCons (by fun_prop)
+        (ContinuousOn.matrixVecCons (hf.comp (by fun_prop) ?_) continuousOn_const))
+    intro t ht
+    exact sub_mul_mem_Icc hab ⟨t.property.1, ht⟩
+  · exact (PiLp.continuous_toLp (2 : ENNReal) fun _ : Fin 2 ↦ ℝ).comp_continuousOn
+      (ContinuousOn.matrixVecCons (by fun_prop)
+        (ContinuousOn.matrixVecCons continuousOn_const continuousOn_const))
+  · intro t ht
+    have h1 : b - (b - a) * (t : ℝ) = a := by rw [ht]; ring
+    have h2 : a + (b - a) * ((t : ℝ) - 1) = a := by rw [ht]; ring
+    rw [h1, h2, ha]
+
+/-- The loop traces exactly the graph of `f` together with the base segment. -/
+private lemma range_positiveGraphLoop (hab : a < b) (ha : f a = 0) :
+    Set.range (positiveGraphLoop a b f) =
+      (fun c ↦ (!₂[c, f c] : Point)) '' Set.Icc a b ∪
+        (fun c ↦ (!₂[c, (0 : ℝ)] : Point)) '' Set.Icc a b := by
+  have hL : (0 : ℝ) < b - a := sub_pos.mpr hab
+  apply Set.Subset.antisymm
+  · rintro _ ⟨⟨s, hs⟩, rfl⟩
+    obtain ⟨h0, h2⟩ := hs
+    by_cases h : s ≤ 1
+    · exact Or.inl ⟨_, sub_mul_mem_Icc hab ⟨h0, h⟩, (positiveGraphLoop_apply_of_le _ h).symm⟩
+    · push Not at h
+      exact Or.inr ⟨_, ⟨by nlinarith, by nlinarith⟩,
+        (positiveGraphLoop_apply_of_not_le _ (not_le.mpr h)).symm⟩
+  · rintro p (⟨c, hc, rfl⟩ | ⟨c, hc, rfl⟩)
+    · have h1 : (b - c) / (b - a) ≤ 1 := (div_le_one hL).mpr (by linarith [hc.1])
+      have h0 : 0 ≤ (b - c) / (b - a) := div_nonneg (by linarith [hc.2]) hL.le
+      have hmem : (b - c) / (b - a) ∈ Set.Icc (0 : ℝ) 2 := ⟨h0, by linarith⟩
+      refine ⟨⟨(b - c) / (b - a), hmem⟩, ?_⟩
+      rw [positiveGraphLoop_apply_of_le hmem h1,
+        show b - (b - a) * ((b - c) / (b - a)) = c by field_simp; ring]
+    · by_cases hca : c = a
+      · have hmem : (1 : ℝ) ∈ Set.Icc (0 : ℝ) 2 := by norm_num
+        refine ⟨⟨1, hmem⟩, ?_⟩
+        rw [positiveGraphLoop_apply_of_le hmem le_rfl,
+          show b - (b - a) * (1 : ℝ) = a by ring, ha, hca]
+      · have hac : a < c := lt_of_le_of_ne hc.1 (Ne.symm hca)
+        have h0 : 0 < (c - a) / (b - a) := div_pos (by linarith) hL
+        have h1 : (c - a) / (b - a) ≤ 1 := (div_le_one hL).mpr (by linarith [hc.2])
+        have hmem : 1 + (c - a) / (b - a) ∈ Set.Icc (0 : ℝ) 2 := ⟨by linarith, by linarith⟩
+        refine ⟨⟨1 + (c - a) / (b - a), hmem⟩, ?_⟩
+        rw [positiveGraphLoop_apply_of_not_le hmem (by simp; linarith),
+          show a + (b - a) * (1 + (c - a) / (b - a) - 1) = c by field_simp; ring]
+
+/-- The loop is injective before its final parameter: the two halves meet only at the ends. -/
+private lemma injOn_positiveGraphLoop (hab : a < b)
+    (hpos : ∀ x ∈ Set.Ioo a b, 0 < f x) :
+    Set.InjOn (positiveGraphLoop a b f) {t | (t : ℝ) < 2} := by
+  have hL : (0 : ℝ) < b - a := sub_pos.mpr hab
+  rintro ⟨s, hs⟩ hs2 ⟨u, hu⟩ hu2 heq
+  simp only [Set.mem_ofPred_eq] at hs2 hu2
+  refine Subtype.ext (show s = u from ?_)
+  by_cases hsl : s ≤ 1 <;> by_cases hul : u ≤ 1
+  · rw [positiveGraphLoop_apply_of_le hs hsl, positiveGraphLoop_apply_of_le hu hul] at heq
+    simp only [WithLp.toLp.injEq, Matrix.vecCons_inj, and_true] at heq
+    have h := heq.1
+    nlinarith
+  · exfalso
+    rw [positiveGraphLoop_apply_of_le hs hsl, positiveGraphLoop_apply_of_not_le hu hul] at heq
+    simp only [WithLp.toLp.injEq, Matrix.vecCons_inj, and_true] at heq
+    push Not at hul
+    rcases eq_endpoint_of_apply_eq_zero hpos (sub_mul_mem_Icc hab ⟨hs.1, hsl⟩) heq.2 with h | h
+    · rw [h] at heq
+      nlinarith [heq.1]
+    · rw [h] at heq
+      nlinarith [heq.1]
+  · exfalso
+    rw [positiveGraphLoop_apply_of_not_le hs hsl, positiveGraphLoop_apply_of_le hu hul] at heq
+    simp only [WithLp.toLp.injEq, Matrix.vecCons_inj, and_true] at heq
+    push Not at hsl
+    rcases eq_endpoint_of_apply_eq_zero hpos (sub_mul_mem_Icc hab ⟨hu.1, hul⟩) heq.2.symm with h | h
+    · rw [h] at heq
+      nlinarith [heq.1]
+    · rw [h] at heq
+      nlinarith [heq.1]
+  · rw [positiveGraphLoop_apply_of_not_le hs hsl, positiveGraphLoop_apply_of_not_le hu hul] at heq
+    simp only [WithLp.toLp.injEq, Matrix.vecCons_inj, and_true] at heq
+    nlinarith [heq]
+
+/-- The graph arc and the base arc share exactly their endpoints, so they glue to a Jordan
+curve. -/
+private lemma isJordanCurve_range_positiveGraphLoop (hab : a < b)
+    (hf : ContinuousOn f (Set.Icc a b)) (ha : f a = 0) (hb : f b = 0)
+    (hpos : ∀ x ∈ Set.Ioo a b, 0 < f x) :
+    IsJordanCurve (Set.range (positiveGraphLoop a b f)) := by
+  have hL : (0 : ℝ) < b - a := sub_pos.mpr hab
+  have hmapsto : ∀ u : unitInterval, b - (b - a) * (u : ℝ) ∈ Set.Icc a b :=
+    fun u ↦ sub_mul_mem_Icc hab u.property
+  have hcontf : Continuous fun u : unitInterval ↦ f (b - (b - a) * (u : ℝ)) :=
+    continuousOn_univ.mp (hf.comp (by fun_prop) fun u _ ↦ hmapsto u)
+  have hcontx : Continuous fun u : unitInterval ↦ b - (b - a) * (u : ℝ) := by fun_prop
+  let γ : Path (!₂[b, (0 : ℝ)] : Point) (!₂[a, (0 : ℝ)] : Point) :=
+    { toFun := fun u ↦ !₂[b - (b - a) * (u : ℝ), f (b - (b - a) * (u : ℝ))]
+      continuous_toFun := (PiLp.continuous_toLp (2 : ENNReal) fun _ : Fin 2 ↦ ℝ).comp
+        (hcontx.matrixVecCons (hcontf.matrixVecCons continuous_const))
+      source' := by norm_num [hb]
+      target' := by norm_num [ha] }
+  let δ : Path (!₂[b, (0 : ℝ)] : Point) (!₂[a, (0 : ℝ)] : Point) :=
+    { toFun := fun u ↦ !₂[b - (b - a) * (u : ℝ), (0 : ℝ)]
+      continuous_toFun := (PiLp.continuous_toLp (2 : ENNReal) fun _ : Fin 2 ↦ ℝ).comp
+        (hcontx.matrixVecCons (continuous_const.matrixVecCons continuous_const))
+      source' := by norm_num
+      target' := by norm_num }
+  have hparam : ∀ {u v : unitInterval},
+      b - (b - a) * (u : ℝ) = b - (b - a) * (v : ℝ) → u = v :=
+    fun h ↦ Subtype.ext (by nlinarith)
+  have hcoords : ∀ {x y u v : ℝ}, (!₂[x, y] : Point) = !₂[u, v] → x = u ∧ y = v := by
+    intro x y u v h
+    simpa only [WithLp.toLp.injEq, Matrix.vecCons_inj, and_true] using h
+  have hγinj : Function.Injective γ := fun u v h ↦ hparam (hcoords h).1
+  have hδinj : Function.Injective δ := fun u v h ↦ hparam (hcoords h).1
+  have hparam_surj : ∀ c ∈ Set.Icc a b, ∃ u : unitInterval, b - (b - a) * (u : ℝ) = c := by
+    intro c hc
+    have h0 : 0 ≤ (b - c) / (b - a) := div_nonneg (by linarith [hc.2]) hL.le
+    have h1 : (b - c) / (b - a) ≤ 1 := (div_le_one hL).mpr (by linarith [hc.1])
+    exact ⟨⟨(b - c) / (b - a), h0, h1⟩, by field_simp; ring⟩
+  have hrangeγ : Set.range γ = (fun c ↦ (!₂[c, f c] : Point)) '' Set.Icc a b := by
+    apply Set.Subset.antisymm
+    · rintro _ ⟨u, rfl⟩
+      exact ⟨_, hmapsto u, rfl⟩
+    · rintro _ ⟨c, hc, rfl⟩
+      obtain ⟨u, hu⟩ := hparam_surj c hc
+      exact ⟨u, by change (!₂[_, f _] : Point) = _; rw [hu]⟩
+  have hrangeδ : Set.range δ = (fun c ↦ (!₂[c, (0 : ℝ)] : Point)) '' Set.Icc a b := by
+    apply Set.Subset.antisymm
+    · rintro _ ⟨u, rfl⟩
+      exact ⟨_, hmapsto u, rfl⟩
+    · rintro _ ⟨c, hc, rfl⟩
+      obtain ⟨u, hu⟩ := hparam_surj c hc
+      exact ⟨u, by change (!₂[_, (0 : ℝ)] : Point) = _; rw [hu]⟩
+  have hmeet : Set.range γ ∩ Set.range δ =
+      {(!₂[b, (0 : ℝ)] : Point), (!₂[a, (0 : ℝ)] : Point)} := by
+    apply Set.Subset.antisymm
+    · rintro z ⟨⟨u, rfl⟩, ⟨v, hv⟩⟩
+      have hcoord := hcoords
+        (show (!₂[b - (b - a) * (u : ℝ), f (b - (b - a) * (u : ℝ))] : Point) =
+          !₂[b - (b - a) * (v : ℝ), (0 : ℝ)] from hv.symm)
+      rcases eq_endpoint_of_apply_eq_zero hpos (hmapsto u) hcoord.2 with h | h
+      · have hu1 : u = 1 := Subtype.ext (by simpa using show (u : ℝ) = 1 by nlinarith)
+        exact Or.inr (by rw [show (γ u : Point) = γ 1 from congrArg _ hu1, γ.target]; rfl)
+      · have hu0 : u = 0 := Subtype.ext (by simpa using show (u : ℝ) = 0 by nlinarith)
+        exact Or.inl (by rw [show (γ u : Point) = γ 0 from congrArg _ hu0, γ.source])
+    · rintro z (rfl | rfl)
+      · exact ⟨⟨0, γ.source⟩, ⟨0, δ.source⟩⟩
+      · exact ⟨⟨1, γ.target⟩, ⟨1, δ.target⟩⟩
+  have htau := TauCeti.isJordanCurve_range_union_range_of_inter_eq_pair hγinj hδinj hmeet
+  rw [range_positiveGraphLoop hab ha, ← hrangeγ, ← hrangeδ]
+  obtain ⟨e⟩ := htau
+  refine ⟨fun z ↦ (e.symm z : Point), continuous_subtype_val.comp e.symm.continuous,
+    fun z w h ↦ e.symm.injective (Subtype.ext h), Set.Subset.antisymm ?_ ?_⟩
+  · rintro z ⟨w, rfl⟩
+    exact (e.symm w).property
+  · intro z hz
+    exact ⟨e ⟨z, hz⟩, congrArg Subtype.val (e.symm_apply_apply ⟨z, hz⟩)⟩
+
+/-- The loop lies in the closed upper half-plane and traverses the supporting base segment in the
+positive tangent direction, so it is counterclockwise. -/
+private lemma isOrientedJordanParametrization_positiveGraphLoop (hab : a < b)
+    (hf : ContinuousOn f (Set.Icc a b)) (ha : f a = 0) (hb : f b = 0)
+    (hpos : ∀ x ∈ Set.Ioo a b, 0 < f x) :
+    IsOrientedJordanParametrization (by norm_num : (0 : ℝ) ≤ 2)
+      (Set.range (positiveGraphLoop a b f)) true (positiveGraphLoop a b f) := by
+  have hL : (0 : ℝ) < b - a := sub_pos.mpr hab
+  set θ : Real.Angle := ((-(Real.pi / 2) : ℝ) : Real.Angle) with hθ
+  have hN : normalVector θ = !₂[(0 : ℝ), -1] := by
+    ext i
+    fin_cases i <;> simp [normalVector, frame, hθ, Real.Angle.cos_coe, Real.Angle.sin_coe]
+  have hT : tangentVector θ = !₂[(1 : ℝ), 0] := by
+    ext i
+    fin_cases i <;> simp [tangentVector, frame, hθ, Real.Angle.cos_coe, Real.Angle.sin_coe]
+  have hinner : ∀ p : Point, inner ℝ p (normalVector θ) = -p 1 := by
+    intro p
+    rw [hN]
+    simp [PiLp.inner_apply, Fin.sum_univ_two]
+  have hone : positiveGraphLoop a b f ⟨1, by norm_num⟩ = !₂[a, (0 : ℝ)] := by
+    rw [positiveGraphLoop_apply_base ha (by norm_num) le_rfl]
+    norm_num
+  have htwo : positiveGraphLoop a b f ⟨2, by norm_num⟩ = !₂[b, (0 : ℝ)] := by
+    rw [positiveGraphLoop_apply_base ha (by norm_num) (by norm_num)]
+    norm_num
+  refine jordan_counterclockwise_of_supporting_segment 0 2 (by norm_num) _
+    (continuous_positiveGraphLoop hab hf ha)
+    (isJordanCurve_range_positiveGraphLoop hab hf ha hb hpos) ?_
+    (injOn_positiveGraphLoop hab hpos) θ 0 ?_ ⟨1, by norm_num⟩ ⟨2, by norm_num⟩
+    (Subtype.mk_lt_mk.mpr (by norm_num)) ?_ (b - a) hL ?_ ?_
+  · rw [positiveGraphLoop_apply_of_le (by norm_num) (by norm_num),
+      positiveGraphLoop_apply_base ha (by norm_num) (by norm_num)]
+    norm_num [hb]
+  · intro t
+    simp only [normalHalfPlane, Bool.false_eq_true, ↓reduceIte, Set.mem_ofPred_eq, hinner,
+      neg_nonpos]
+    obtain ⟨s, hs⟩ := t
+    by_cases h : s ≤ 1
+    · rw [positiveGraphLoop_apply_of_le hs h]
+      exact nonneg_of_pos_on_Ioo ha hb hpos (sub_mul_mem_Icc hab ⟨hs.1, h⟩)
+    · rw [positiveGraphLoop_apply_of_not_le hs h]
+      exact le_rfl
+  · simp only [normalLine, Set.mem_ofPred_eq, hinner, hone]
+    norm_num
+  · rw [hone, htwo, hT]
+    ext i
+    fin_cases i <;> simp
+  · rw [hone, htwo, segment_eq_image]
+    apply Set.Subset.antisymm
+    · rintro _ ⟨⟨u, hu⟩, hmem, rfl⟩
+      have h1 : (1 : ℝ) ≤ u := hmem.1
+      have h2 : u ≤ 2 := hmem.2
+      refine ⟨u - 1, ⟨by linarith, by linarith⟩, ?_⟩
+      rw [positiveGraphLoop_apply_base ha hu h1]
+      ext i
+      fin_cases i
+      · simp
+        ring
+      · simp
+    · rintro _ ⟨c, hc, rfl⟩
+      have hmem : 1 + c ∈ Set.Icc (0 : ℝ) 2 := ⟨by linarith [hc.1], by linarith [hc.2]⟩
+      refine ⟨⟨1 + c, hmem⟩, ⟨Subtype.mk_le_mk.mpr (by linarith [hc.1]),
+        Subtype.mk_le_mk.mpr (by linarith [hc.2])⟩, ?_⟩
+      rw [positiveGraphLoop_apply_base ha hmem (by linarith [hc.1])]
+      ext i
+      fin_cases i
+      · simp
+        ring
+      · simp
+
+/-- The positive subgraph is open, connected, bounded and relatively closed in the complement of
+the loop, hence it is the bounded complementary component. -/
+private lemma jordanInterior_range_positiveGraphLoop_of_components (hab : a < b)
+    (hf : ContinuousOn f (Set.Icc a b)) (ha : f a = 0) (hb : f b = 0)
+    (hpos : ∀ x ∈ Set.Ioo a b, 0 < f x) :
+    jordanInterior (Set.range (positiveGraphLoop a b f)) =
+      {p : Point | a < p 0 ∧ p 0 < b ∧ 0 < p 1 ∧ p 1 < f (p 0)} := by
+  set Γ := Set.range (positiveGraphLoop a b f) with hΓ
+  set U : Set Point := {p : Point | a < p 0 ∧ p 0 < b ∧ 0 < p 1 ∧ p 1 < f (p 0)} with hU
+  set K : Set Point := {p : Point | a ≤ p 0 ∧ p 0 ≤ b ∧ 0 ≤ p 1 ∧ p 1 ≤ f (p 0)} with hK
+  have hrange : Γ = (fun c ↦ (!₂[c, f c] : Point)) '' Set.Icc a b ∪
+      (fun c ↦ (!₂[c, (0 : ℝ)] : Point)) '' Set.Icc a b := range_positiveGraphLoop hab ha
+  have hUK : U ⊆ K := fun p hp ↦ ⟨hp.1.le, hp.2.1.le, hp.2.2.1.le, hp.2.2.2.le⟩
+  have hUΓ : U ⊆ Γᶜ := by
+    intro p hp hpΓ
+    rw [hrange] at hpΓ
+    rcases hpΓ with ⟨c, _, hc⟩ | ⟨c, _, hc⟩
+    · exact absurd hp.2.2.2 (by rw [← hc]; simp)
+    · exact absurd hp.2.2.1 (by rw [← hc]; simp)
+  have hKΓU : ∀ p ∈ K, p ∉ Γ → p ∈ U := by
+    intro p hp hpΓ
+    rw [hrange] at hpΓ
+    have hpc : p = !₂[p 0, p 1] := point_eq_vecNotation_coords p
+    have h1 : p 1 ≠ 0 := fun h0 ↦ hpΓ (Or.inr ⟨p 0, ⟨hp.1, hp.2.1⟩,
+      show (!₂[p 0, (0 : ℝ)] : Point) = p by rw [← h0]; exact hpc.symm⟩)
+    have h2 : p 1 ≠ f (p 0) := fun heq ↦ hpΓ (Or.inl ⟨p 0, ⟨hp.1, hp.2.1⟩,
+      show (!₂[p 0, f (p 0)] : Point) = p by rw [← heq]; exact hpc.symm⟩)
+    have h3 : 0 < p 1 := lt_of_le_of_ne hp.2.2.1 (Ne.symm h1)
+    have h4 : p 1 < f (p 0) := lt_of_le_of_ne hp.2.2.2 h2
+    refine ⟨?_, ?_, h3, h4⟩
+    · rcases eq_or_lt_of_le hp.1 with h | h
+      · rw [← h, ha] at h4
+        linarith
+      · exact h
+    · rcases eq_or_lt_of_le hp.2.1 with h | h
+      · rw [h, hb] at h4
+        linarith
+      · exact h
+  have hcoord0 : Continuous fun q : Point ↦ q 0 := by fun_prop
+  have hcoord1 : Continuous fun q : Point ↦ q 1 := by fun_prop
+  have hUopen : IsOpen U := by
+    rw [isOpen_iff_mem_nhds]
+    rintro p ⟨h1, h2, h3, h4⟩
+    have hfat : ContinuousAt f (p 0) :=
+      (hf.mono Set.Ioo_subset_Icc_self).continuousAt (Ioo_mem_nhds h1 h2)
+    have hfcoord : ContinuousAt (fun q : Point ↦ f (q 0)) p :=
+      ContinuousAt.comp (g := f) (f := fun q : Point ↦ q 0) (x := p) hfat hcoord0.continuousAt
+    have hgap : ContinuousAt (fun q : Point ↦ f (q 0) - q 1) p :=
+      hfcoord.sub hcoord1.continuousAt
+    filter_upwards [hcoord0.continuousAt (isOpen_Ioo.mem_nhds (⟨h1, h2⟩ : p 0 ∈ Set.Ioo a b)),
+      hcoord1.continuousAt (isOpen_Ioi.mem_nhds (show p 1 ∈ Set.Ioi (0 : ℝ) from h3)),
+      hgap (isOpen_Ioi.mem_nhds (show f (p 0) - p 1 ∈ Set.Ioi (0 : ℝ) from sub_pos.mpr h4))]
+      with q hq1 hq2 hq3
+    exact ⟨hq1.1, hq1.2, hq2, by simpa using sub_pos.mp hq3⟩
+  have hSclosed : IsClosed {q : Point | q 0 ∈ Set.Icc a b} := isClosed_Icc.preimage hcoord0
+  have hKclosed : IsClosed K := by
+    have hgS : ContinuousOn (fun q : Point ↦ f (q 0) - q 1) {q : Point | q 0 ∈ Set.Icc a b} :=
+      (hf.comp hcoord0.continuousOn fun q hq ↦ hq).sub hcoord1.continuousOn
+    have hone := hgS.preimage_isClosed_of_isClosed hSclosed (isClosed_Ici (a := (0 : ℝ)))
+    have htwo : IsClosed {q : Point | 0 ≤ q 1} := isClosed_Ici.preimage hcoord1
+    have hsplit : K = ({q : Point | q 0 ∈ Set.Icc a b} ∩
+        (fun q : Point ↦ f (q 0) - q 1) ⁻¹' Set.Ici 0) ∩ {q : Point | 0 ≤ q 1} := by
+      ext q
+      simp only [hK, Set.mem_ofPred_eq, Set.mem_inter_iff, Set.mem_Icc, Set.mem_preimage,
+        Set.mem_Ici, sub_nonneg]
+      tauto
+    rw [hsplit]
+    exact hone.inter htwo
+  have hUconn : IsConnected U := by
+    have hfcomp : ContinuousOn (fun q : ℝ × ℝ ↦ f q.1) (Set.Ioo a b ×ˢ Set.Ioo (0 : ℝ) 1) :=
+      (hf.mono Set.Ioo_subset_Icc_self).comp continuous_fst.continuousOn fun q hq ↦ hq.1
+    have hmap : ContinuousOn (fun q : ℝ × ℝ ↦ (!₂[q.1, q.2 * f q.1] : Point))
+        (Set.Ioo a b ×ˢ Set.Ioo (0 : ℝ) 1) :=
+      (PiLp.continuous_toLp (2 : ENNReal) fun _ : Fin 2 ↦ ℝ).comp_continuousOn
+        (continuous_fst.continuousOn.matrixVecCons
+          ((continuous_snd.continuousOn.mul hfcomp).matrixVecCons continuousOn_const))
+    have hset : U = (fun q : ℝ × ℝ ↦ (!₂[q.1, q.2 * f q.1] : Point)) ''
+        (Set.Ioo a b ×ˢ Set.Ioo (0 : ℝ) 1) := by
+      apply Set.Subset.antisymm
+      · rintro p ⟨h1, h2, h3, h4⟩
+        have hfpos : 0 < f (p 0) := h3.trans h4
+        refine ⟨(p 0, p 1 / f (p 0)), ⟨⟨h1, h2⟩, div_pos h3 hfpos,
+          (div_lt_one hfpos).mpr h4⟩, ?_⟩
+        change (!₂[p 0, p 1 / f (p 0) * f (p 0)] : Point) = p
+        rw [div_mul_cancel₀ _ hfpos.ne']
+        exact (point_eq_vecNotation_coords p).symm
+      · rintro _ ⟨⟨c, r⟩, ⟨⟨hc1, hc2⟩, hr1, hr2⟩, rfl⟩
+        have hfc : 0 < f c := hpos c ⟨hc1, hc2⟩
+        refine ⟨by simpa using hc1, by simpa using hc2, by simpa using mul_pos hr1 hfc, ?_⟩
+        change r * f c < f c
+        nlinarith
+    rw [hset]
+    exact ((isConnected_Ioo hab).prod (isConnected_Ioo (by norm_num))).image _ hmap
+  obtain ⟨c₀, -, hmax⟩ := isCompact_Icc.exists_isMaxOn (Set.nonempty_Icc.mpr hab.le) hf
+  have hKbdd : Bornology.IsBounded K :=
+    (EuclideanSpace.isBounded_coordinate_rectangle a b 0 (f c₀)).subset (by
+      rintro p ⟨h1, h2, h3, h4⟩
+      exact ⟨h1, h2, h3, h4.trans (hmax ⟨h1, h2⟩)⟩)
+  have hUbdd : Bornology.IsBounded U := hKbdd.subset hUK
+  have hcompU : ∀ p ∈ U, connectedComponentIn Γᶜ p = U := by
+    intro p hp
+    have hsub : connectedComponentIn Γᶜ p ⊆ Γᶜ := connectedComponentIn_subset _ _
+    refine Set.Subset.antisymm ?_ (hUconn.isPreconnected.subset_connectedComponentIn hp hUΓ)
+    have hcover : connectedComponentIn Γᶜ p ⊆ U ∪ Kᶜ := by
+      intro q hq
+      by_cases hqK : q ∈ K
+      · exact Or.inl (hKΓU q hqK (hsub hq))
+      · exact Or.inr hqK
+    have hmeet : (connectedComponentIn Γᶜ p ∩ U).Nonempty :=
+      ⟨p, mem_connectedComponentIn (hUΓ hp), hp⟩
+    have hempty : ¬ (connectedComponentIn Γᶜ p ∩ Kᶜ).Nonempty := by
+      intro hk
+      obtain ⟨z, hz⟩ := isPreconnected_connectedComponentIn U Kᶜ hUopen
+        hKclosed.isOpen_compl hcover hmeet hk
+      exact hz.2.2 (hUK hz.2.1)
+    intro q hq
+    by_contra hqU
+    exact hempty ⟨q, hq, fun hqK ↦ hqU (hKΓU q hqK (hsub hq))⟩
+  obtain ⟨p₀, hp₀⟩ := hUconn.nonempty
+  obtain ⟨U', V', -, -, -, -, -, hV'ub, -, hcover, -, -, hcU, hcV⟩ :=
+    jordan_separation (isJordanCurve_range_positiveGraphLoop hab hf ha hb hpos)
+  have hp₀U' : p₀ ∈ U' := by
+    rcases hcover.symm.subset (hUΓ hp₀) with h | h
+    · exact h
+    · exact absurd (by rw [← hcV p₀ h, hcompU p₀ hp₀]; exact hUbdd) hV'ub
+  have hU'eq : U' = U := by rw [← hcU p₀ hp₀U', hcompU p₀ hp₀]
+  ext p
+  constructor
+  · rintro ⟨hpΓ, hpb⟩
+    rcases hcover.symm.subset hpΓ with h | h
+    · rwa [← hU'eq]
+    · exact absurd (by rwa [← hcV p h]) hV'ub
+  · intro hp
+    exact ⟨hUΓ hp, by rw [hcompU p hp]; exact hUbdd⟩
+
+end PositiveGraph
+
+theorem positiveGraphLoop_counterclockwise (a b : ℝ) (hab : a < b) (f : ℝ → ℝ)
+    (hf : ContinuousOn f (Set.Icc a b)) (ha : f a = 0) (hb : f b = 0)
+    (hpos : ∀ x ∈ Set.Ioo a b, 0 < f x) :
+    IsOrientedJordanParametrization (by norm_num : (0 : ℝ) ≤ 2)
+      (Set.range (positiveGraphLoop a b f)) true (positiveGraphLoop a b f) ∧
+    jordanInterior (Set.range (positiveGraphLoop a b f)) =
+      {p : Point | a < p 0 ∧ p 0 < b ∧ 0 < p 1 ∧ p 1 < f (p 0)} :=
+  ⟨isOrientedJordanParametrization_positiveGraphLoop hab hf ha hb hpos,
+    jordanInterior_range_positiveGraphLoop_of_components hab hf ha hb hpos⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The region enclosed by the counterclockwise loop around a positive graph
+
+`MovingSofa.positiveGraphLoop_counterclockwise` identifies the bounded complementary component
+of the loop around the graph of a positive `f` with the open subgraph of `f`.  This file records
+the resulting description of the closed subgraph: the loop traces exactly the frontier of either
+region, and the closed subgraph is the open one together with that frontier.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+variable {a b : ℝ} {f : ℝ → ℝ} (hab : a < b) (hf : ContinuousOn f (Set.Icc a b))
+  (ha : f a = 0) (hb : f b = 0) (hpos : ∀ x ∈ Set.Ioo a b, 0 < f x)
+
+include hab hf ha hb hpos
+
+/-- The bounded complementary component of the positive-graph loop is the open subgraph. -/
+theorem jordanInterior_range_positiveGraphLoop :
+    jordanInterior (Set.range (positiveGraphLoop a b f)) = openSubgraph a b f :=
+  (positiveGraphLoop_counterclockwise a b hab f hf ha hb hpos).2
+
+/-- The positive-graph loop traces the frontier of the open subgraph. -/
+theorem frontier_openSubgraph :
+    frontier (openSubgraph a b f) = Set.range (positiveGraphLoop a b f) := by
+  rw [← jordanInterior_range_positiveGraphLoop hab hf ha hb hpos]
+  exact (positiveGraphLoop_counterclockwise a b hab f hf ha hb
+    hpos).1.2.1.frontier_jordanInterior
+
+/-- The part of the closed subgraph outside the open one is exactly the loop. -/
+theorem closedSubgraph_sdiff_openSubgraph :
+    closedSubgraph a b f \ openSubgraph a b f = Set.range (positiveGraphLoop a b f) := by
+  rw [← frontier_openSubgraph hab hf ha hb hpos, frontier, (isOpen_openSubgraph hf).interior_eq,
+    closure_openSubgraph hab hf ha hb hpos]
+
+/-- The closed subgraph is the open subgraph together with the loop. -/
+theorem closedSubgraph_eq_openSubgraph_union_range :
+    closedSubgraph a b f = openSubgraph a b f ∪ Set.range (positiveGraphLoop a b f) := by
+  rw [← closedSubgraph_sdiff_openSubgraph hab hf ha hb hpos]
+  exact (Set.union_sdiff_cancel openSubgraph_subset_closedSubgraph).symm
+
+/-- The positive-graph loop also traces the frontier of the closed subgraph. -/
+theorem frontier_closedSubgraph :
+    frontier (closedSubgraph a b f) = Set.range (positiveGraphLoop a b f) := by
+  rw [frontier, (isClosed_closedSubgraph hf).closure_eq, interior_closedSubgraph hf ha hb,
+    closedSubgraph_sdiff_openSubgraph hab hf ha hb hpos]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/Development/Geometry/Foundations/Development004.lean b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development004.lean
new file mode 100644
index 0000000000..a88cdcdf21
--- /dev/null
+++ b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development004.lean
@@ -0,0 +1,849 @@
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development001
+/-!
+# Moving sofa: related mathematical developments
+
+* `Gerver.Foundations.Development001`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Gerver.AreaCertificate`.
+* `Gerver.Area.TrigEnclosure`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Exact rational certificate for the Gerver cap and niche areas
+
+This module carries the finite, kernel-checkable part of the rational area bounds for
+Gerver's sofa: integer interval arithmetic at scale `M = 10 ^ 30`, certified enclosures of
+the twenty-two direct parameters and of `π`, interval sine and cosine by the Taylor
+recurrence, the grid of rotation angles, the five phase formulas, the support-contact fan
+of the cap and the rectangle cover of the niche.
+
+Everything here is computable and `decide +kernel`-checkable; the two closed numeric
+conclusions `capOK_true` and `nicheOK_true` are the only facts the analytic layer in
+`MovingSofa.Gerver.Area` needs from this module.
+-/
+
+/-! ### Integer interval arithmetic at scale `10 ^ 30`
+
+The kernel accelerates `Int` arithmetic but not `Rat` arithmetic, so the whole finite
+calculation is carried out on pairs of integers denoting the interval `[lo / M, hi / M]`.
+-/
+
+@[expose] public section
+
+namespace MovingSofa.GerverAreaCert
+
+/-- The common denominator of every interval endpoint. -/
+def M : ℤ := 10 ^ 30
+
+/-- `⟨lo, hi⟩` denotes the real interval `[lo / M, hi / M]`. -/
+structure SI where
+  /-- The scaled integer lower endpoint. -/
+  lo : ℤ
+  /-- The scaled integer upper endpoint. -/
+  hi : ℤ
+  deriving DecidableEq
+
+namespace SI
+
+/-- Kernel-cheap `min` on `ℤ`. -/
+def imin (a b : ℤ) : ℤ := if a ≤ b then a else b
+/-- Kernel-cheap `max` on `ℤ`. -/
+def imax (a b : ℤ) : ℤ := if a ≤ b then b else a
+
+/-- Outward enclosure of `a / b` for `0 < b`. -/
+def ratI (a b : ℤ) : SI := ⟨(a * M) / b, -((-(a * M)) / b)⟩
+
+/-- The exact interval `[0, 0]`. -/
+def zero : SI := ⟨0, 0⟩
+/-- The exact interval `[1, 1]`. -/
+def one : SI := ⟨M, M⟩
+
+/-- Interval addition. -/
+def add (x y : SI) : SI := ⟨x.lo + y.lo, x.hi + y.hi⟩
+/-- Interval negation. -/
+def neg (x : SI) : SI := ⟨-x.hi, -x.lo⟩
+/-- Interval subtraction. -/
+def sub (x y : SI) : SI := ⟨x.lo - y.hi, x.hi - y.lo⟩
+
+/-- Interval multiplication: the extreme endpoint products, rounded outward. -/
+def mul (x y : SI) : SI :=
+  let p₁ := x.lo * y.lo
+  let p₂ := x.lo * y.hi
+  let p₃ := x.hi * y.lo
+  let p₄ := x.hi * y.hi
+  ⟨imin (imin p₁ p₂) (imin p₃ p₄) / M,
+    -((-(imax (imax p₁ p₂) (imax p₃ p₄))) / M)⟩
+
+/-- Division by a positive integer. -/
+def divn (x : SI) (b : ℤ) : SI := ⟨x.lo / b, -((-x.hi) / b)⟩
+
+/-- Multiplication by an integer. -/
+def imul (k : ℤ) (x : SI) : SI :=
+  ⟨imin (k * x.lo) (k * x.hi), imax (k * x.lo) (k * x.hi)⟩
+
+/-! #### Real semantics and soundness -/
+
+/-- `z` denotes the real interval `[z.lo / M, z.hi / M]`. -/
+def Contains (z : SI) (x : ℝ) : Prop :=
+  (z.lo : ℝ) ≤ (M : ℝ) * x ∧ (M : ℝ) * x ≤ (z.hi : ℝ)
+
+/-- `imin` is `min`. -/
+theorem imin_eq_min (a b : ℤ) : imin a b = min a b := by
+  simp only [imin, min_def]
+
+/-- `imax` is `max`. -/
+theorem imax_eq_max (a b : ℤ) : imax a b = max a b := by
+  rcases le_total a b with h | h
+  · simp [imax, h]
+  · rcases eq_or_lt_of_le h with rfl | hlt
+    · simp [imax]
+    · simp [imax, not_le.mpr hlt, max_eq_left h]
+
+/-- The scale is positive. -/
+theorem Mpos : (0 : ℝ) < (M : ℝ) := by
+  have : (M : ℤ) = 10 ^ 30 := rfl
+  rw [this]
+  norm_num
+
+private theorem mul_ediv_le (p : ℤ) {b : ℤ} (hb : 0 < b) : b * (p / b) ≤ p := by
+  have h := Int.mul_ediv_add_emod p b
+  have h2 : 0 ≤ p % b := Int.emod_nonneg p hb.ne'
+  linarith
+
+private theorem le_mul_neg_ediv (p : ℤ) {b : ℤ} (hb : 0 < b) : p ≤ b * -((-p) / b) := by
+  have h := mul_ediv_le (-p) hb
+  rw [mul_neg]
+  linarith
+
+/-- Integer floor division rounds down in the reals. -/
+theorem ediv_le_of_le {p : ℤ} {b : ℤ} (hb : 0 < b) {r : ℝ} (h : (p : ℝ) ≤ (b : ℝ) * r) :
+    ((p / b : ℤ) : ℝ) ≤ r := by
+  have hbR : (0 : ℝ) < (b : ℝ) := by exact_mod_cast hb
+  have h1 : ((b : ℝ)) * ((p / b : ℤ) : ℝ) ≤ (p : ℝ) := by
+    exact_mod_cast mul_ediv_le p hb
+  have := h1.trans h
+  exact le_of_mul_le_mul_left this hbR
+
+/-- Integer ceiling division rounds up in the reals. -/
+theorem le_neg_ediv_of_le {p : ℤ} {b : ℤ} (hb : 0 < b) {r : ℝ} (h : (b : ℝ) * r ≤ (p : ℝ)) :
+    r ≤ ((-((-p) / b) : ℤ) : ℝ) := by
+  have hbR : (0 : ℝ) < (b : ℝ) := by exact_mod_cast hb
+  have h1 : (p : ℝ) ≤ (b : ℝ) * ((-((-p) / b) : ℤ) : ℝ) := by
+    exact_mod_cast le_mul_neg_ediv p hb
+  exact le_of_mul_le_mul_left (h.trans h1) hbR
+
+private theorem min_mul_le_mul {l h u v : ℝ} (hl : l ≤ u) (hu : u ≤ h) :
+    min (l * v) (h * v) ≤ u * v := by
+  rcases le_total 0 v with hv | hv
+  · exact (min_le_left _ _).trans (by nlinarith)
+  · exact (min_le_right _ _).trans (by nlinarith)
+
+private theorem mul_le_max_mul {l h u v : ℝ} (hl : l ≤ u) (hu : u ≤ h) :
+    u * v ≤ max (l * v) (h * v) := by
+  rcases le_total 0 v with hv | hv
+  · exact le_trans (by nlinarith) (le_max_right _ _)
+  · exact le_trans (by nlinarith) (le_max_left _ _)
+
+private theorem min_four_le_mul {l₁ h₁ l₂ h₂ u v : ℝ}
+    (h1 : l₁ ≤ u) (h2 : u ≤ h₁) (h3 : l₂ ≤ v) (h4 : v ≤ h₂) :
+    min (min (l₁ * l₂) (l₁ * h₂)) (min (h₁ * l₂) (h₁ * h₂)) ≤ u * v := by
+  have hA : min (min (l₁ * l₂) (l₁ * h₂)) (min (h₁ * l₂) (h₁ * h₂)) ≤ min (l₁ * l₂) (h₁ * l₂) :=
+    le_min ((min_le_left _ _).trans (min_le_left _ _))
+      ((min_le_right _ _).trans (min_le_left _ _))
+  have hB : min (min (l₁ * l₂) (l₁ * h₂)) (min (h₁ * l₂) (h₁ * h₂)) ≤ min (l₁ * h₂) (h₁ * h₂) :=
+    le_min ((min_le_left _ _).trans (min_le_right _ _))
+      ((min_le_right _ _).trans (min_le_right _ _))
+  have h5 : min (min (l₁ * l₂) (l₁ * h₂)) (min (h₁ * l₂) (h₁ * h₂)) ≤ u * l₂ :=
+    hA.trans (min_mul_le_mul h1 h2)
+  have h6 : min (min (l₁ * l₂) (l₁ * h₂)) (min (h₁ * l₂) (h₁ * h₂)) ≤ u * h₂ :=
+    hB.trans (min_mul_le_mul h1 h2)
+  have h7 : min (u * l₂) (u * h₂) ≤ u * v := by
+    simpa [mul_comm] using min_mul_le_mul (l := l₂) (h := h₂) (u := v) (v := u) h3 h4
+  exact (le_min h5 h6).trans h7
+
+private theorem mul_le_max_four {l₁ h₁ l₂ h₂ u v : ℝ}
+    (h1 : l₁ ≤ u) (h2 : u ≤ h₁) (h3 : l₂ ≤ v) (h4 : v ≤ h₂) :
+    u * v ≤ max (max (l₁ * l₂) (l₁ * h₂)) (max (h₁ * l₂) (h₁ * h₂)) := by
+  have hA : max (l₁ * l₂) (h₁ * l₂) ≤ max (max (l₁ * l₂) (l₁ * h₂)) (max (h₁ * l₂) (h₁ * h₂)) :=
+    max_le ((le_max_left _ _).trans (le_max_left _ _))
+      ((le_max_left _ _).trans (le_max_right _ _))
+  have hB : max (l₁ * h₂) (h₁ * h₂) ≤ max (max (l₁ * l₂) (l₁ * h₂)) (max (h₁ * l₂) (h₁ * h₂)) :=
+    max_le ((le_max_right _ _).trans (le_max_left _ _))
+      ((le_max_right _ _).trans (le_max_right _ _))
+  have h7 : u * v ≤ max (u * l₂) (u * h₂) := by
+    simpa [mul_comm] using mul_le_max_mul (l := l₂) (h := h₂) (u := v) (v := u) h3 h4
+  exact h7.trans (max_le ((mul_le_max_mul h1 h2).trans hA) ((mul_le_max_mul h1 h2).trans hB))
+
+/-- Interval addition is sound. -/
+theorem contains_add {x y : SI} {a b : ℝ} (hx : Contains x a) (hy : Contains y b) :
+    Contains (add x y) (a + b) := by
+  obtain ⟨hx1, hx2⟩ := hx
+  obtain ⟨hy1, hy2⟩ := hy
+  refine ⟨?_, ?_⟩ <;> push_cast [add] <;> nlinarith [Mpos]
+
+/-- Interval negation is sound. -/
+theorem contains_neg {x : SI} {a : ℝ} (hx : Contains x a) : Contains (neg x) (-a) := by
+  obtain ⟨hx1, hx2⟩ := hx
+  refine ⟨?_, ?_⟩ <;> push_cast [neg] <;> nlinarith [Mpos]
+
+/-- Interval subtraction is sound. -/
+theorem contains_sub {x y : SI} {a b : ℝ} (hx : Contains x a) (hy : Contains y b) :
+    Contains (sub x y) (a - b) := by
+  obtain ⟨hx1, hx2⟩ := hx
+  obtain ⟨hy1, hy2⟩ := hy
+  refine ⟨?_, ?_⟩ <;> push_cast [sub] <;> nlinarith [Mpos]
+
+/-- Interval multiplication is sound. -/
+theorem contains_mul {x y : SI} {a b : ℝ} (hx : Contains x a) (hy : Contains y b) :
+    Contains (mul x y) (a * b) := by
+  obtain ⟨hx1, hx2⟩ := hx
+  obtain ⟨hy1, hy2⟩ := hy
+  have hM := Mpos
+  have hprod : (M : ℝ) * ((M : ℝ) * (a * b)) = ((M : ℝ) * a) * ((M : ℝ) * b) := by ring
+  refine ⟨?_, ?_⟩
+  · simp only [mul, imin_eq_min, imax_eq_max]
+    refine ediv_le_of_le (b := M) (by exact_mod_cast hM) ?_
+    push_cast
+    rw [hprod]
+    exact min_four_le_mul hx1 hx2 hy1 hy2
+  · simp only [mul, imin_eq_min, imax_eq_max]
+    refine le_neg_ediv_of_le (b := M) (by exact_mod_cast hM) ?_
+    push_cast
+    rw [hprod]
+    exact mul_le_max_four hx1 hx2 hy1 hy2
+
+/-- Interval division by a positive integer is sound. -/
+theorem contains_divn {x : SI} {a : ℝ} {b : ℤ} (hb : 0 < b) (hx : Contains x a) :
+    Contains (divn x b) (a / (b : ℝ)) := by
+  obtain ⟨hx1, hx2⟩ := hx
+  have hbR : (0 : ℝ) < (b : ℝ) := by exact_mod_cast hb
+  have hkey : (b : ℝ) * ((M : ℝ) * (a / (b : ℝ))) = (M : ℝ) * a := by
+    field_simp
+  refine ⟨ediv_le_of_le hb ?_, le_neg_ediv_of_le hb ?_⟩
+  · rw [hkey]; exact hx1
+  · rw [hkey]; exact hx2
+
+/-- Interval multiplication by an integer is sound. -/
+theorem contains_imul {x : SI} {a : ℝ} (k : ℤ) (hx : Contains x a) :
+    Contains (imul k x) ((k : ℝ) * a) := by
+  obtain ⟨hx1, hx2⟩ := hx
+  have hkey : (M : ℝ) * ((k : ℝ) * a) = (k : ℝ) * ((M : ℝ) * a) := by ring
+  refine ⟨?_, ?_⟩ <;> simp only [imul, imin_eq_min, imax_eq_max] <;> push_cast <;> rw [hkey]
+  · simpa [mul_comm] using min_mul_le_mul (l := (x.lo : ℝ)) (h := (x.hi : ℝ))
+      (u := (M : ℝ) * a) (v := (k : ℝ)) hx1 hx2
+  · simpa [mul_comm] using mul_le_max_mul (l := (x.lo : ℝ)) (h := (x.hi : ℝ))
+      (u := (M : ℝ) * a) (v := (k : ℝ)) hx1 hx2
+
+/-- The outward enclosure of a rational is sound. -/
+theorem contains_ratI {a b : ℤ} (hb : 0 < b) : Contains (ratI a b) ((a : ℝ) / (b : ℝ)) := by
+  have hbR : (0 : ℝ) < (b : ℝ) := by exact_mod_cast hb
+  have hkey : (b : ℝ) * ((M : ℝ) * ((a : ℝ) / (b : ℝ))) = ((a * M : ℤ) : ℝ) := by
+    push_cast
+    field_simp
+  refine ⟨ediv_le_of_le hb ?_, le_neg_ediv_of_le hb ?_⟩
+  · rw [hkey]
+  · rw [hkey]
+
+/-- `zero` encloses `0`. -/
+theorem contains_zero : Contains zero 0 := by
+  constructor <;> simp [zero]
+
+/-- `one` encloses `1`. -/
+theorem contains_one : Contains one 1 := by
+  constructor <;> simp [one]
+
+/-- A left fold of exact interval additions over an initial segment encloses the
+corresponding real sum. -/
+theorem contains_foldl_range {f : ℕ → SI} {g : ℕ → ℝ} (n : ℕ)
+    (hfg : ∀ i < n, Contains (f i) (g i)) :
+    Contains ((List.range n).foldl (fun acc i => add acc (f i)) zero)
+      (∑ i ∈ Finset.range n, g i) := by
+  induction n with
+  | zero => simpa using contains_zero
+  | succ n ih =>
+    rw [List.range_succ, List.foldl_append, Finset.sum_range_succ]
+    exact contains_add (ih fun i hi => hfg i (by omega)) (hfg n (by omega))
+
+/-- A left fold of integer additions over an initial segment is the corresponding sum. -/
+theorem foldl_range_int (f : ℕ → ℤ) (n : ℕ) :
+    (List.range n).foldl (fun acc i => acc + f i) 0 = ∑ i ∈ Finset.range n, f i := by
+  induction n with
+  | zero => simp
+  | succ n ih => rw [List.range_succ, List.foldl_append, Finset.sum_range_succ, ← ih]; simp
+
+end SI
+
+open SI
+
+/-! ### Certified enclosures of the twenty-two parameters and of `π` -/
+
+/-- Enclosure of the direct parameter `k₁₁`. -/
+def k11Z : SI := ⟨-210322422072688751416285718490, -210322422072688751416085718490⟩
+/-- Enclosure of the direct parameter `k₁₂`. -/
+def k12Z : SI := ⟨249999999999999999999900000000, 250000000000000000000100000000⟩
+/-- Enclosure of the direct parameter `k₂₁`. -/
+def k21Z : SI := ⟨-919179292771593322274796102890, -919179292771593322274596102890⟩
+/-- Enclosure of the direct parameter `k₂₂`. -/
+def k22Z : SI := ⟨472406619750805465181660762512, 472406619750805465181860762512⟩
+/-- Enclosure of the direct parameter `k₃₁`. -/
+def k31Z : SI := ⟨-613763229430251668555014291320, -613763229430251668554814291320⟩
+/-- Enclosure of the direct parameter `k₃₂`. -/
+def k32Z : SI := ⟨889626479003221860726943050050, 889626479003221860727143050050⟩
+/-- Enclosure of the direct parameter `k₄₁`. -/
+def k41Z : SI := ⟨-308347166088910014835232479740, -308347166088910014835032479740⟩
+/-- Enclosure of the direct parameter `k₄₂`. -/
+def k42Z : SI := ⟨472406619750805465181660762512, 472406619750805465181860762512⟩
+/-- Enclosure of the direct parameter `k₅₁`. -/
+def k51Z : SI := ⟨-1017204036787814585693742864150, -1017204036787814585693542864150⟩
+/-- Enclosure of the direct parameter `k₅₂`. -/
+def k52Z : SI := ⟨249999999999999999999900000000, 250000000000000000000100000000⟩
+/-- Enclosure of the direct parameter `a₁`. -/
+def a1Z : SI := ⟨1210322422072688751416085718500, 1210322422072688751416285718500⟩
+/-- Enclosure of the direct parameter `a₂`. -/
+def a2Z : SI := ⟨-250000000000000000000100000000, -249999999999999999999900000000⟩
+/-- Enclosure of the direct parameter `b₁`. -/
+def b1Z : SI := ⟨-527624598026784624160603809370, -527624598026784624160403809370⟩
+/-- Enclosure of the direct parameter `b₂`. -/
+def b2Z : SI := ⟨920258385160637622893605795010, 920258385160637622893805795010⟩
+/-- Enclosure of the direct parameter `c₁`. -/
+def c1Z : SI := ⟨626045522848465867552229310386, 626045522848465867552429310386⟩
+/-- Enclosure of the direct parameter `c₂`. -/
+def c2Z : SI := ⟨-944750803946430751679092381250, -944750803946430751678892381250⟩
+/-- Enclosure of the direct parameter `d₁`. -/
+def d1Z : SI := ⟨1313022761424232933776064655200, 1313022761424232933776264655200⟩
+/-- Enclosure of the direct parameter `d₂`. -/
+def d2Z : SI := ⟨-525382670414554437202936294305, -525382670414554437202736294305⟩
+/-- Enclosure of the direct parameter `e₁`. -/
+def e1Z : SI := ⟨1210322422072688751416085718500, 1210322422072688751416285718500⟩
+/-- Enclosure of the direct parameter `e₂`. -/
+def e2Z : SI := ⟨249999999999999999999900000000, 250000000000000000000100000000⟩
+/-- Enclosure of the first stage angle `φ`. -/
+def phiZ : SI := ⟨39177364790083641863217874980, 39177364790083641863217875500⟩
+/-- Enclosure of the second stage angle `θ`. -/
+def thetaZ : SI := ⟨681301509382724894473855754540, 681301509382724894473855759660⟩
+
+/-- The integer enclosure of `π`, from the certified Machin arctangent sums. -/
+def piZ : SI := ⟨3141592653589793238462643383279, 3141592653589793238462643383280⟩
+
+/-- The integer enclosure of `π / 2`. -/
+def piHalfZ : SI := divn piZ 2
+
+/-! #### The parameter enclosures are the certified box rows -/
+
+private theorem le_M_mul {c n : ℤ} {d : ℕ} (hd : 0 < (d : ℤ)) (hc : c * (d : ℤ) = n * M)
+    {x : ℝ} (h : (n : ℝ) / (d : ℝ) ≤ x) : (c : ℝ) ≤ (M : ℝ) * x := by
+  have hdR : (0 : ℝ) < (d : ℝ) := by exact_mod_cast hd
+  have hcR : (c : ℝ) * (d : ℝ) = (n : ℝ) * (M : ℝ) := by exact_mod_cast hc
+  rw [div_le_iff₀ hdR] at h
+  nlinarith [SI.Mpos, h, hcR]
+
+private theorem M_mul_le {c n : ℤ} {d : ℕ} (hd : 0 < (d : ℤ)) (hc : c * (d : ℤ) = n * M)
+    {x : ℝ} (h : x ≤ (n : ℝ) / (d : ℝ)) : (M : ℝ) * x ≤ (c : ℝ) := by
+  have hdR : (0 : ℝ) < (d : ℝ) := by exact_mod_cast hd
+  have hcR : (c : ℝ) * (d : ℝ) = (n : ℝ) * (M : ℝ) := by exact_mod_cast hc
+  rw [le_div_iff₀ hdR] at h
+  nlinarith [SI.Mpos, h, hcR]
+
+/-- Every one of the twenty-two integer enclosures contains the corresponding certified
+Gerver parameter. -/
+theorem contains_params :
+      SI.Contains k11Z GerverSofa.PartB.params.k11 ∧
+      SI.Contains k12Z GerverSofa.PartB.params.k12 ∧
+      SI.Contains k21Z GerverSofa.PartB.params.k21 ∧
+      SI.Contains k22Z GerverSofa.PartB.params.k22 ∧
+      SI.Contains k31Z GerverSofa.PartB.params.k31 ∧
+      SI.Contains k32Z GerverSofa.PartB.params.k32 ∧
+      SI.Contains k41Z GerverSofa.PartB.params.k41 ∧
+      SI.Contains k42Z GerverSofa.PartB.params.k42 ∧
+      SI.Contains k51Z GerverSofa.PartB.params.k51 ∧
+      SI.Contains k52Z GerverSofa.PartB.params.k52 ∧
+      SI.Contains a1Z GerverSofa.PartB.params.a1 ∧
+      SI.Contains a2Z GerverSofa.PartB.params.a2 ∧
+      SI.Contains b1Z GerverSofa.PartB.params.b1 ∧
+      SI.Contains b2Z GerverSofa.PartB.params.b2 ∧
+      SI.Contains c1Z GerverSofa.PartB.params.c1 ∧
+      SI.Contains c2Z GerverSofa.PartB.params.c2 ∧
+      SI.Contains d1Z GerverSofa.PartB.params.d1 ∧
+      SI.Contains d2Z GerverSofa.PartB.params.d2 ∧
+      SI.Contains e1Z GerverSofa.PartB.params.e1 ∧
+      SI.Contains e2Z GerverSofa.PartB.params.e2 ∧
+      SI.Contains phiZ GerverSofa.PartB.params.phi ∧
+      SI.Contains thetaZ GerverSofa.PartB.params.theta := by
+  have hbox := GerverSofa.PartB.params_mem
+  dsimp only [GerverSofa.Romik.box, GerverSofa.qR, Set.mem_ofPred_eq] at hbox
+  obtain ⟨hk11lo, hk11hi, hk12lo, hk12hi, hk21lo, hk21hi, hk22lo, hk22hi, hk31lo, hk31hi,
+      hk32lo, hk32hi, hk41lo, hk41hi, hk42lo, hk42hi, hk51lo, hk51hi, hk52lo, hk52hi,
+      ha1lo, ha1hi, ha2lo, ha2hi, hb1lo, hb1hi, hb2lo, hb2hi, hc1lo, hc1hi, hc2lo, hc2hi,
+      hd1lo, hd1hi, hd2lo, hd2hi, he1lo, he1hi, he2lo, he2hi, hphilo, hphihi, hthetalo,
+      hthetahi⟩ := hbox
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, k11Z]) hk11lo,
+    M_mul_le (by norm_num) (by norm_num [M, k11Z]) hk11hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, k12Z]) hk12lo,
+    M_mul_le (by norm_num) (by norm_num [M, k12Z]) hk12hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, k21Z]) hk21lo,
+    M_mul_le (by norm_num) (by norm_num [M, k21Z]) hk21hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, k22Z]) hk22lo,
+    M_mul_le (by norm_num) (by norm_num [M, k22Z]) hk22hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, k31Z]) hk31lo,
+    M_mul_le (by norm_num) (by norm_num [M, k31Z]) hk31hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, k32Z]) hk32lo,
+    M_mul_le (by norm_num) (by norm_num [M, k32Z]) hk32hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, k41Z]) hk41lo,
+    M_mul_le (by norm_num) (by norm_num [M, k41Z]) hk41hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, k42Z]) hk42lo,
+    M_mul_le (by norm_num) (by norm_num [M, k42Z]) hk42hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, k51Z]) hk51lo,
+    M_mul_le (by norm_num) (by norm_num [M, k51Z]) hk51hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, k52Z]) hk52lo,
+    M_mul_le (by norm_num) (by norm_num [M, k52Z]) hk52hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, a1Z]) ha1lo,
+    M_mul_le (by norm_num) (by norm_num [M, a1Z]) ha1hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, a2Z]) ha2lo,
+    M_mul_le (by norm_num) (by norm_num [M, a2Z]) ha2hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, b1Z]) hb1lo,
+    M_mul_le (by norm_num) (by norm_num [M, b1Z]) hb1hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, b2Z]) hb2lo,
+    M_mul_le (by norm_num) (by norm_num [M, b2Z]) hb2hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, c1Z]) hc1lo,
+    M_mul_le (by norm_num) (by norm_num [M, c1Z]) hc1hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, c2Z]) hc2lo,
+    M_mul_le (by norm_num) (by norm_num [M, c2Z]) hc2hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, d1Z]) hd1lo,
+    M_mul_le (by norm_num) (by norm_num [M, d1Z]) hd1hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, d2Z]) hd2lo,
+    M_mul_le (by norm_num) (by norm_num [M, d2Z]) hd2hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, e1Z]) he1lo,
+    M_mul_le (by norm_num) (by norm_num [M, e1Z]) he1hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, e2Z]) he2lo,
+    M_mul_le (by norm_num) (by norm_num [M, e2Z]) he2hi⟩, ?_⟩
+  refine ⟨⟨le_M_mul (by norm_num) (by norm_num [M, phiZ]) hphilo,
+    M_mul_le (by norm_num) (by norm_num [M, phiZ]) hphihi⟩, ?_⟩
+  exact ⟨le_M_mul (by norm_num) (by norm_num [M, thetaZ]) hthetalo,
+    M_mul_le (by norm_num) (by norm_num [M, thetaZ]) hthetahi⟩
+
+/-- The integer enclosure of `π`. -/
+theorem contains_piZ : SI.Contains piZ Real.pi := by
+  obtain ⟨h1, h2⟩ := GerverSofa.ExactReplay.piI_contains_pi
+  have hM := SI.Mpos
+  refine ⟨?_, ?_⟩
+  · have hnum : ((piZ.lo : ℤ) : ℝ) ≤ (M : ℝ) * ((GerverSofa.ExactReplay.piI.lo : ℚ) : ℝ) := by
+      norm_num [piZ, M, GerverSofa.ExactReplay.piI, GerverSofa.ExactReplay.q]
+    nlinarith [h1]
+  · have hnum : (M : ℝ) * ((GerverSofa.ExactReplay.piI.hi : ℚ) : ℝ) ≤ ((piZ.hi : ℤ) : ℝ) := by
+      norm_num [piZ, M, GerverSofa.ExactReplay.piI, GerverSofa.ExactReplay.q]
+    nlinarith [h2]
+
+/-- The integer enclosure of `π / 2`. -/
+theorem contains_piHalfZ : SI.Contains piHalfZ (Real.pi / 2) := by
+  change SI.Contains (SI.divn piZ 2) (Real.pi / 2)
+  simpa using SI.contains_divn (b := 2) (by norm_num) contains_piZ
+
+/-! ### Sine and cosine by the degree-41/40 Taylor recurrence -/
+
+/-- `trigIter z t n = (Sₙ, Cₙ, ∑_{k ≤ n} Sₖ, ∑_{k ≤ n} Cₖ)` where `Sₖ` and `Cₖ` enclose the
+`k`-th signed sine and cosine Taylor terms and `z` encloses `t ^ 2`. -/
+def trigIter (z t : SI) : ℕ → SI × SI × SI × SI
+  | 0 => (t, one, t, one)
+  | n + 1 =>
+    let st := trigIter z t n
+    let sterm := neg (divn (mul st.1 z) (((2 * n + 2) * (2 * n + 3) : ℕ) : ℤ))
+    let cterm := neg (divn (mul st.2.1 z) (((2 * n + 1) * (2 * n + 2) : ℕ) : ℤ))
+    (sterm, cterm, add st.2.2.1 sterm, add st.2.2.2 cterm)
+
+/-- Sine and cosine enclosures of every real number in the argument interval.
+Valid for arguments in `[0, 2]`. -/
+def trigZ (t : SI) : SI × SI :=
+  let st := trigIter (mul t t) t 20
+  (⟨st.2.2.1.lo - 1, st.2.2.1.hi + 1⟩, ⟨st.2.2.2.lo - 1, st.2.2.2.hi + 1⟩)
+
+/-! ### Stage endpoints and the uniform grid -/
+
+/-- Number of subintervals per analytic stage. -/
+def NN : ℕ := 64
+
+/-- Integer enclosures of the six stage endpoints `0, φ, θ, η, τ, π / 2`, constant past `5`. -/
+def endZ : ℕ → SI
+  | 0 => zero
+  | 1 => phiZ
+  | 2 => thetaZ
+  | 3 => sub piHalfZ thetaZ
+  | 4 => sub piHalfZ phiZ
+  | _ => piHalfZ
+
+/-- The `m`-th grid angle, `0 ≤ m ≤ 5 * NN`. -/
+def ttZ (m : ℕ) : SI :=
+  divn (add (imul ((NN - m % NN : ℕ) : ℤ) (endZ (m / NN)))
+    (imul ((m % NN : ℕ) : ℤ) (endZ (m / NN + 1)))) (NN : ℤ)
+
+/-- The analytic branch that the piecewise definitions select at the `m`-th grid angle. -/
+def stageOf (m : ℕ) : ℕ := if m % NN = 0 then max 1 (m / NN) else m / NN + 1
+
+/-! ### The five phase formulas and the four contact curves -/
+
+/-- `kind`: `0` the path, `1` `A`, `2` `B`, `3` `C`, `4` `D`. -/
+def evalZ (stage : ℕ) (t : SI) (kind : ℕ) : SI × SI :=
+  let sc := trigZ t
+  let s := sc.1
+  let c := sc.2
+  let data : SI × SI × SI × SI × SI × SI :=
+    if stage = 1 then
+      (add (add (mul a1Z c) (mul a2Z s)) (ratI (-1) 1),
+        add (add (mul (neg a2Z) c) (mul a1Z s)) (ratI (-1) 2),
+        add (add (mul (imul (-2) a1Z) s) (mul (imul 2 a2Z) c)) (ratI 1 2),
+        add (add (mul (imul 2 a1Z) c) (mul (imul 2 a2Z) s)) (ratI (-1) 1),
+        k11Z, k12Z)
+    else if stage = 2 then
+      let f := add (add (divn (neg (mul t t)) 4) (mul b1Z t)) b2Z
+      (f, add (add (divn t 2) (neg b1Z)) (ratI (-1) 1),
+        add (add (ratI 1 1) (imul 2 b1Z)) (neg t),
+        add f (ratI 1 2), k21Z, k22Z)
+    else if stage = 3 then
+      let f := sub c1Z t
+      let g := add c2Z t
+      (f, g, add (ratI (-1) 1) (neg g), add (ratI 1 1) f, k31Z, k32Z)
+    else if stage = 4 then
+      let g := add (add (divn (neg (mul t t)) 4) (mul d1Z t)) d2Z
+      (add (add (neg (divn t 2)) d1Z) (ratI (-1) 1), g,
+        add (neg g) (ratI (-1) 2),
+        add (add (imul 2 d1Z) (ratI (-1) 1)) (neg t), k41Z, k42Z)
+    else
+      (add (add (mul e1Z c) (mul e2Z s)) (ratI (-1) 2),
+        add (add (mul (neg e2Z) c) (mul e1Z s)) (ratI (-1) 1),
+        add (add (ratI 1 1) (mul (imul (-2) e1Z) s)) (mul (imul 2 e2Z) c),
+        add (add (mul (imul 2 e1Z) c) (mul (imul 2 e2Z) s)) (ratI (-1) 2),
+        k51Z, k52Z)
+  let f := data.1
+  let g := data.2.1
+  let al := data.2.2.1
+  let be := data.2.2.2.1
+  let kx := data.2.2.2.2.1
+  let ky := data.2.2.2.2.2
+  let x0 := add (sub (mul c f) (mul s g)) kx
+  let y0 := add (add (mul s f) (mul c g)) ky
+  let x1 := if kind = 1 ∨ kind = 2 then sub x0 (mul al s) else x0
+  let y1 := if kind = 1 ∨ kind = 2 then add y0 (mul al c) else y0
+  let x2 := if kind = 1 then add x1 c else x1
+  let y2 := if kind = 1 then add y1 s else y1
+  let x3 := if kind = 3 ∨ kind = 4 then sub x2 (mul be c) else x2
+  let y3 := if kind = 3 ∨ kind = 4 then sub y2 (mul be s) else y2
+  (if kind = 3 then sub x3 s else x3, if kind = 3 then add y3 c else y3)
+
+/-- The contact point at grid index `m`, evaluated on the branch the definitions select. -/
+def contactZ (m kind : ℕ) : SI × SI := evalZ (stageOf m) (ttZ m) kind
+
+/-! ### Cap: the ordered support contacts and their fan shoelace sum -/
+
+/-- Number of listed support contacts feeding the fan, excluding the anchor. -/
+def fanCount : ℕ := 10 * NN
+
+/-- Contact kind at fan position `i`: `A` for `i ≤ 5 * NN`, otherwise `C`. -/
+def fanKind (i : ℕ) : ℕ := if i ≤ 5 * NN then 1 else 3
+
+/-- Grid index at fan position `i`. -/
+def fanIdx (i : ℕ) : ℕ := if i ≤ 5 * NN then i else i - 5 * NN
+
+/-- Both coordinates of the contact listed at fan position `i`. -/
+def fanZ (i : ℕ) : SI × SI := contactZ (fanIdx i) (fanKind i)
+
+/-- The fan anchor `L = C (π / 2)`. -/
+def anchorZ : SI × SI := contactZ (5 * NN) 3
+
+/-- The `i`-th fan determinant over the anchor. -/
+def crossZ (i : ℕ) : SI :=
+  sub (mul (sub (fanZ i).1 anchorZ.1) (sub (fanZ (i + 1)).2 anchorZ.2))
+    (mul (sub (fanZ i).2 anchorZ.2) (sub (fanZ (i + 1)).1 anchorZ.1))
+
+/-- Twice the signed area of the fan polygon: the shoelace sum of the listed contacts over
+the anchor. -/
+def capDoubledZ : SI :=
+  (List.range (fanCount - 1)).foldl (fun acc i => add acc (crossZ i)) zero
+
+/-! ### Niche: the seven roof pieces and their covering rectangles -/
+
+/-- The seven roof pieces as `(kind, stage, traversed forwards?)`. -/
+def rowData : ℕ → ℕ × ℕ × Bool
+  | 0 => (4, 1, true)
+  | 1 => (4, 2, true)
+  | 2 => (0, 4, false)
+  | 3 => (0, 3, false)
+  | 4 => (0, 2, false)
+  | 5 => (2, 4, true)
+  | _ => (2, 5, true)
+
+/-- Contact kind traced by roof piece `r`. -/
+def rowKind (r : ℕ) : ℕ := (rowData r).1
+/-- Analytic stage traced by roof piece `r`. -/
+def rowStage (r : ℕ) : ℕ := (rowData r).2.1
+/-- Whether roof piece `r` is traversed in increasing time. -/
+def rowFwd (r : ℕ) : Bool := (rowData r).2.2
+
+/-- Grid index of the left end of the `j`-th subinterval of row `r`. -/
+def rowBase (r j : ℕ) : ℕ := (rowStage r - 1) * NN + j
+
+/-- Left horizontal bound of the covering rectangle of `(r, j)`. -/
+def rectLoZ (r j : ℕ) : ℤ :=
+  if rowFwd r then (contactZ (rowBase r j) (rowKind r)).1.lo
+  else (contactZ (rowBase r j + 1) (rowKind r)).1.lo
+
+/-- Right horizontal bound of the covering rectangle of `(r, j)`. -/
+def rectHiZ (r j : ℕ) : ℤ :=
+  if rowFwd r then (contactZ (rowBase r j + 1) (rowKind r)).1.hi
+  else (contactZ (rowBase r j) (rowKind r)).1.hi
+
+/-- Height bound of the covering rectangle of `(r, j)`: the branch evaluation over the
+whole subinterval, widened to cover the left endpoint, which sits on the previous branch. -/
+def rectHZ (r j : ℕ) : ℤ :=
+  imax (evalZ (stageOf (rowBase r j + 1))
+      ⟨(ttZ (rowBase r j)).lo, (ttZ (rowBase r j + 1)).hi⟩ (rowKind r)).2.hi
+    (contactZ (rowBase r j) (rowKind r)).2.hi
+
+/-- Area of the covering rectangle of `(r, j)`, in units of `1 / M ^ 2`. -/
+def rectAreaZ (r j : ℕ) : ℤ :=
+  imax 0 (rectHiZ r j - rectLoZ r j) * imax 0 (rectHZ r j)
+
+/-- Total covering area of row `r`, in units of `1 / M ^ 2`. -/
+def rowSumZ (r : ℕ) : ℤ := (List.range NN).foldl (fun acc j => acc + rectAreaZ r j) 0
+
+/-- Total covering area of the `7 * NN` rectangles, in units of `1 / M ^ 2`. -/
+def nicheSumZ : ℤ := (List.range 7).foldl (fun acc r => acc + rowSumZ r) 0
+
+/-! ### The two decidable numeric conclusions -/
+
+/-- The closed numeric cap check: `2 * 28609 / 10000 ≤ capDoubledZ.lo / M`. -/
+def capOK : Bool := 2 * 28609 * M ≤ 10000 * capDoubledZ.lo
+/-- The closed numeric niche check: `nicheSumZ / M ^ 2 ≤ 3301 / 5000`. -/
+def nicheOK : Bool := 5000 * nicheSumZ ≤ 3301 * M * M
+
+/-- The niche check passes, by kernel reduction. -/
+theorem nicheOK_true : nicheOK = true := by decide +kernel
+
+/-- The cap check passes, by kernel reduction. -/
+theorem capOK_true : capOK = true := by decide +kernel
+
+end MovingSofa.GerverAreaCert
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Interval enclosures of the sine and cosine at scale `M`
+
+The certificate evaluates the sine and the cosine by the interval Taylor recurrence
+`GerverAreaCert.trigIter`, truncated at degree 41 and 40 and widened by one unit at scale
+`M = 10 ^ 30`.  `trigZ_sound` is the soundness statement of that evaluator on `[0, 2]`: the
+recurrence encloses the signed Taylor terms and their partial sums (`contains_trigIter`),
+the alternating brackets of `Real.sin` and `Real.cos` control the truncation error by the
+first omitted term, and `2 ^ 41 * 10 ^ 30 ≤ 41!` and `2 ^ 40 * 10 ^ 30 ≤ 40!` justify the
+one-unit widening.
+-/
+
+/-! ### Sine and cosine enclosures -/
+
+@[expose] public section
+
+namespace MovingSofa
+
+open GerverAreaCert
+
+/-- The integer term recurrence encloses the signed Taylor terms and their partial
+sums.  Pure interval arithmetic: an induction on `n` using the `SI` soundness lemmas. -/
+theorem contains_trigIter {z t : SI} {x : ℝ} (hz : SI.Contains z (x * x))
+    (ht : SI.Contains t x) (n : ℕ) :
+    SI.Contains (trigIter z t n).1
+        ((-1 : ℝ) ^ n * x ^ (2 * n + 1) / (Nat.factorial (2 * n + 1) : ℝ)) ∧
+      SI.Contains (trigIter z t n).2.1
+        ((-1 : ℝ) ^ n * x ^ (2 * n) / (Nat.factorial (2 * n) : ℝ)) ∧
+      SI.Contains (trigIter z t n).2.2.1
+        (∑ k ∈ Finset.range (n + 1),
+          (-1 : ℝ) ^ k * x ^ (2 * k + 1) / (Nat.factorial (2 * k + 1) : ℝ)) ∧
+      SI.Contains (trigIter z t n).2.2.2
+        (∑ k ∈ Finset.range (n + 1),
+          (-1 : ℝ) ^ k * x ^ (2 * k) / (Nat.factorial (2 * k) : ℝ)) := by
+  induction n with
+  | zero =>
+    refine ⟨by simpa [trigIter] using ht, by simpa [trigIter] using SI.contains_one,
+      by simpa [trigIter] using ht, ?_⟩
+    simpa [trigIter] using SI.contains_one
+  | succ n ih =>
+    obtain ⟨ih1, ih2, ih3, ih4⟩ := ih
+    have hfac1 : (Nat.factorial (2 * n + 1) : ℝ) ≠ 0 :=
+      Nat.cast_ne_zero.2 (Nat.factorial_ne_zero _)
+    have hfac2 : (Nat.factorial (2 * n) : ℝ) ≠ 0 :=
+      Nat.cast_ne_zero.2 (Nat.factorial_ne_zero _)
+    have hd1 : (0 : ℤ) < (((2 * n + 2) * (2 * n + 3) : ℕ) : ℤ) := by positivity
+    have hd2 : (0 : ℤ) < (((2 * n + 1) * (2 * n + 2) : ℕ) : ℤ) := by positivity
+    have hd1' : ((((2 * n + 2) * (2 * n + 3) : ℕ) : ℤ) : ℝ) ≠ 0 := by
+      have : (0 : ℝ) < ((((2 * n + 2) * (2 * n + 3) : ℕ) : ℤ) : ℝ) := by exact_mod_cast hd1
+      exact ne_of_gt this
+    have hd2' : ((((2 * n + 1) * (2 * n + 2) : ℕ) : ℤ) : ℝ) ≠ 0 := by
+      have : (0 : ℝ) < ((((2 * n + 1) * (2 * n + 2) : ℕ) : ℤ) : ℝ) := by exact_mod_cast hd2
+      exact ne_of_gt this
+    have e1 : (Nat.factorial (2 * (n + 1) + 1) : ℝ) =
+        (Nat.factorial (2 * n + 1) : ℝ) * ((((2 * n + 2) * (2 * n + 3) : ℕ) : ℤ) : ℝ) := by
+      have h : 2 * (n + 1) + 1 = 2 * n + 1 + 1 + 1 := by ring
+      rw [h, Nat.factorial_succ, Nat.factorial_succ]
+      push_cast
+      ring
+    have e2 : (Nat.factorial (2 * (n + 1)) : ℝ) =
+        (Nat.factorial (2 * n) : ℝ) * ((((2 * n + 1) * (2 * n + 2) : ℕ) : ℤ) : ℝ) := by
+      have h : 2 * (n + 1) = 2 * n + 1 + 1 := by ring
+      rw [h, Nat.factorial_succ, Nat.factorial_succ]
+      push_cast
+      ring
+    have v1 : -((((-1 : ℝ) ^ n * x ^ (2 * n + 1) / (Nat.factorial (2 * n + 1) : ℝ)) * (x * x)) /
+          ((((2 * n + 2) * (2 * n + 3) : ℕ) : ℤ) : ℝ)) =
+        (-1 : ℝ) ^ (n + 1) * x ^ (2 * (n + 1) + 1) /
+          (Nat.factorial (2 * (n + 1) + 1) : ℝ) := by
+      rw [e1, show x ^ (2 * (n + 1) + 1) = x ^ (2 * n + 1) * (x * x) by ring]
+      field_simp
+      ring
+    have v2 : -((((-1 : ℝ) ^ n * x ^ (2 * n) / (Nat.factorial (2 * n) : ℝ)) * (x * x)) /
+          ((((2 * n + 1) * (2 * n + 2) : ℕ) : ℤ) : ℝ)) =
+        (-1 : ℝ) ^ (n + 1) * x ^ (2 * (n + 1)) / (Nat.factorial (2 * (n + 1)) : ℝ) := by
+      rw [e2, show x ^ (2 * (n + 1)) = x ^ (2 * n) * (x * x) by ring]
+      field_simp
+      ring
+    have hs : SI.Contains
+        (SI.neg (SI.divn (SI.mul (trigIter z t n).1 z) (((2 * n + 2) * (2 * n + 3) : ℕ) : ℤ)))
+        ((-1 : ℝ) ^ (n + 1) * x ^ (2 * (n + 1) + 1) /
+          (Nat.factorial (2 * (n + 1) + 1) : ℝ)) := by
+      rw [← v1]
+      exact SI.contains_neg (SI.contains_divn hd1 (SI.contains_mul ih1 hz))
+    have hc : SI.Contains
+        (SI.neg (SI.divn (SI.mul (trigIter z t n).2.1 z) (((2 * n + 1) * (2 * n + 2) : ℕ) : ℤ)))
+        ((-1 : ℝ) ^ (n + 1) * x ^ (2 * (n + 1)) / (Nat.factorial (2 * (n + 1)) : ℝ)) := by
+      rw [← v2]
+      exact SI.contains_neg (SI.contains_divn hd2 (SI.contains_mul ih2 hz))
+    refine ⟨hs, hc, ?_, ?_⟩
+    · rw [Finset.sum_range_succ]
+      exact SI.contains_add ih3 hs
+    · rw [Finset.sum_range_succ]
+      exact SI.contains_add ih4 hc
+
+/-- The degree-41 sine Taylor polynomial overshoots by at most `1 / M` on `[0, 2]`. -/
+private theorem sin_taylor_bracket {x : ℝ} (hx0 : 0 ≤ x) (hx2 : x ≤ 2) :
+    (∑ k ∈ Finset.range 21,
+        (-1 : ℝ) ^ k * x ^ (2 * k + 1) / (Nat.factorial (2 * k + 1) : ℝ)) -
+        1 / (M : ℝ) ≤ Real.sin x ∧
+      Real.sin x ≤ ∑ k ∈ Finset.range 21,
+        (-1 : ℝ) ^ k * x ^ (2 * k + 1) / (Nat.factorial (2 * k + 1) : ℝ) := by
+  obtain ⟨hlow, hhigh⟩ := Real.sin_mem_Icc_taylor_sums 10 hx0 (by push_cast; nlinarith)
+  simp only [show (2 * 10 + 1 : ℕ) = 21 from rfl, show (2 * 10 : ℕ) = 20 from rfl] at hlow hhigh
+  refine ⟨?_, hhigh⟩
+  have hsplit : (∑ k ∈ Finset.range 21,
+      (-1 : ℝ) ^ k * x ^ (2 * k + 1) / (Nat.factorial (2 * k + 1) : ℝ))
+      = (∑ k ∈ Finset.range 20,
+        (-1 : ℝ) ^ k * x ^ (2 * k + 1) / (Nat.factorial (2 * k + 1) : ℝ))
+        + x ^ 41 / (Nat.factorial 41 : ℝ) := by
+    rw [show (21 : ℕ) = 20 + 1 from rfl, Finset.sum_range_succ]
+    norm_num
+  have htail : x ^ 41 / (Nat.factorial 41 : ℝ) ≤ 1 / (M : ℝ) := by
+    have hMv : ((M : ℤ) : ℝ) = 10 ^ 30 := by norm_num [M]
+    rw [hMv]
+    calc x ^ 41 / (Nat.factorial 41 : ℝ) ≤ (2 : ℝ) ^ 41 / (Nat.factorial 41 : ℝ) := by gcongr
+      _ ≤ 1 / 10 ^ 30 := by norm_num [Nat.factorial]
+  rw [hsplit]
+  linarith
+
+/-- The degree-40 cosine Taylor polynomial overshoots by at most `1 / M` on `[0, 2]`. -/
+private theorem cos_taylor_bracket {x : ℝ} (hx0 : 0 ≤ x) (hx2 : x ≤ 2) :
+    (∑ k ∈ Finset.range 21,
+        (-1 : ℝ) ^ k * x ^ (2 * k) / (Nat.factorial (2 * k) : ℝ)) -
+        1 / (M : ℝ) ≤ Real.cos x ∧
+      Real.cos x ≤ ∑ k ∈ Finset.range 21,
+        (-1 : ℝ) ^ k * x ^ (2 * k) / (Nat.factorial (2 * k) : ℝ) := by
+  obtain ⟨hlow, hhigh⟩ := Real.cos_mem_Icc_taylor_sums 10 hx0 (by push_cast; nlinarith)
+  simp only [show (2 * 10 + 1 : ℕ) = 21 from rfl, show (2 * 10 : ℕ) = 20 from rfl] at hlow hhigh
+  refine ⟨?_, hhigh⟩
+  have hsplit : (∑ k ∈ Finset.range 21,
+      (-1 : ℝ) ^ k * x ^ (2 * k) / (Nat.factorial (2 * k) : ℝ))
+      = (∑ k ∈ Finset.range 20,
+        (-1 : ℝ) ^ k * x ^ (2 * k) / (Nat.factorial (2 * k) : ℝ))
+        + x ^ 40 / (Nat.factorial 40 : ℝ) := by
+    rw [show (21 : ℕ) = 20 + 1 from rfl, Finset.sum_range_succ]
+    norm_num
+  have htail : x ^ 40 / (Nat.factorial 40 : ℝ) ≤ 1 / (M : ℝ) := by
+    have hMv : ((M : ℤ) : ℝ) = 10 ^ 30 := by norm_num [M]
+    rw [hMv]
+    calc x ^ 40 / (Nat.factorial 40 : ℝ) ≤ (2 : ℝ) ^ 40 / (Nat.factorial 40 : ℝ) := by gcongr
+      _ ≤ 1 / 10 ^ 30 := by norm_num [Nat.factorial]
+  rw [hsplit]
+  linarith
+
+/-- Soundness of the executable sine/cosine enclosure on `[0, 2]`. -/
+theorem trigZ_sound {t : SI} {x : ℝ} (ht : SI.Contains t x) (hx0 : 0 ≤ x) (hx2 : x ≤ 2) :
+    SI.Contains (trigZ t).1 (Real.sin x) ∧ SI.Contains (trigZ t).2 (Real.cos x) := by
+  have hM : (0 : ℝ) < (M : ℝ) := SI.Mpos
+  have hMinv : (M : ℝ) * (1 / (M : ℝ)) = 1 := by field_simp
+  have hwiden : ∀ (S : SI) (y v : ℝ), SI.Contains S v → v - 1 / (M : ℝ) ≤ y → y ≤ v →
+      SI.Contains (⟨S.lo - 1, S.hi + 1⟩ : SI) y := by
+    intro S y v hS h1 h2
+    obtain ⟨ha, hb⟩ := hS
+    have hlow : (M : ℝ) * v - 1 ≤ (M : ℝ) * y := by
+      have h := mul_le_mul_of_nonneg_left h1 hM.le
+      nlinarith [hMinv]
+    have hhigh : (M : ℝ) * y ≤ (M : ℝ) * v := mul_le_mul_of_nonneg_left h2 hM.le
+    refine ⟨?_, ?_⟩ <;> push_cast <;> linarith
+  have h20 : (20 : ℕ) + 1 = 21 := by norm_num
+  have hz : SI.Contains (SI.mul t t) (x * x) := SI.contains_mul ht ht
+  obtain ⟨-, -, hS, hC⟩ := contains_trigIter hz ht 20
+  rw [h20] at hS hC
+  obtain ⟨hs1, hs2⟩ := sin_taylor_bracket hx0 hx2
+  obtain ⟨hc1, hc2⟩ := cos_taylor_bracket hx0 hx2
+  exact ⟨hwiden _ (Real.sin x) _ hS hs1 hs2, hwiden _ (Real.cos x) _ hC hc1 hc2⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/Development/Geometry/Foundations/Development005.lean b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development005.lean
new file mode 100644
index 0000000000..af9f766413
--- /dev/null
+++ b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development005.lean
@@ -0,0 +1,11356 @@
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+module
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development004
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle008
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development002
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development003
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development001
+public import Mathlib.Analysis.Convex.Continuous
+public import Mathlib.Analysis.Convex.Segment
+public import Mathlib.Analysis.Normed.Affine.Isometry
+public import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
+public import Mathlib.LinearAlgebra.AffineSpace.Independent
+public import Mathlib.MeasureTheory.Constructions.Polish.Basic
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
+public import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
+public import Mathlib.Tactic.FunProp
+public import Mathlib.Tactic.Positivity
+/-!
+# Moving sofa: related mathematical developments
+
+* `Canonical.Foundations.Development001`.
+* `Geometry.Foundations.Development003`.
+* `Analysis.Foundations.Development003`.
+* `Cap.Foundations.Development002`.
+* `Analysis.Foundations.Development004`.
+* `Convex.Foundations.Development002`.
+* `Geometry.Foundations.Development004`.
+* `Gerver.Foundations.Development002`.
+* `Cap.Foundations.Development003`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Canonical.Definitions`.
+* `Canonical.GerverDefinitions`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 The Formal Conjectures Authors and Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: The Formal Conjectures Authors, Dean Cureton
+-/
+/-
+Copyright 2026 The Formal Conjectures Authors.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    https://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+-/
+
+
+/-!
+# Definitions for the moving sofa problem
+
+The definitions of `MovingSofaSubmission/Challenge.lean`, copied from
+`FormalConjectures/Wikipedia/MovingSofa.lean` in google-deepmind/formal-conjectures at commit
+`ddfbaf90f4482030d88aae5233fe933874296a23`. Here `ABφθSpec.existsUnique` is proved, from the
+certificate in GerverSofaLean, so that Gerver's constants are defined from a proved statement.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+/-- The standard two-dimensional Euclidean space over the reals. -/
+scoped[EuclideanGeometry] notation "ℝ²" => EuclideanSpace ℝ (Fin 2)
+
+open scoped EuclideanGeometry
+
+/-- The plane `ℝ²` with the orientation of its standard basis, so that rotations are
+counterclockwise. -/
+noncomputable instance Module.orientedEuclideanSpaceFinTwo : Module.Oriented ℝ ℝ² (Fin 2) :=
+  ⟨Module.Basis.orientation <| PiLp.basisFun 2 _ _⟩
+
+/-- The plane `ℝ²` has dimension two. -/
+instance fact_finrank_euclideanSpace_fin_two : Fact (Module.finrank ℝ ℝ² = 2) :=
+  ⟨finrank_euclideanSpace_fin⟩
+
+namespace MovingSofa
+
+open Topology
+open scoped Real unitInterval EuclideanGeometry
+
+/-- The **horizontal side** of the hallway is $(-\infty, 1] \times [0, 1]$. -/
+def horizontalHallway : Set ℝ² := {!₂[x, y] | (x) (y) (_ : x ≤ 1 ∧ 0 ≤ y ∧ y ≤ 1)}
+
+/-- The **vertical side** of the hallway is $[0, 1] \times (-\infty, 1]$. -/
+def verticalHallway : Set ℝ² := {!₂[x, y] | (x) (y) (_ : 0 ≤ x ∧ x ≤ 1 ∧ y ≤ 1)}
+
+/-- The **hallway** is the union of its horizontal and vertical sides. -/
+def hallway : Set ℝ² := horizontalHallway ∪ verticalHallway
+
+/-- The affine isometry group of the Euclidean plane. -/
+scoped notation "E(2)" => ℝ² ≃ᵃⁱ[ℝ] ℝ²
+
+/-- The topology on the isometry group `E(2)`, induced from the continuous affine maps of the
+plane. -/
+instance rigidMotionTopology : TopologicalSpace E(2) :=
+  .induced (·.toAffineIsometry.toContinuousAffineMap) inferInstance
+
+/--
+A connected closed set $s$ is a **moving sofa** according to a rigid motion $m:I\to\mathrm{SE}(2)$,
+if the sofa is initially in the horizontal side of the hallway and ends up in the vertical side.
+Here, since $\mathrm{SE}(2)$ is not in Mathlib yet, we use $\mathrm{E}(2)$ and rely on continuity
+and $m(0) = \mathrm{id}$ to ensure $m$ is in $\mathrm{SE}(2)$.
+-/
+structure IsMovingSofa (s : Set ℝ²) (m : I → E(2)) : Prop where
+  isConnected : IsConnected s
+  isClosed : IsClosed s
+  continuous : Continuous m
+  zero : m 0 = .refl ℝ ℝ²
+  initial : s ⊆ horizontalHallway
+  subset_hallway : ∀ t, m t '' s ⊆ hallway
+  final : m 1 '' s ⊆ verticalHallway
+
+/--
+The rigid motion that translates by $p$ and then rotates counterclockwise by $\alpha$.
+Note that [Ge92] used this definition while [Ro18] used rotation first and then translation.
+-/
+def rotateTranslate (α : Real.Angle) (p : ℝ²) : E(2) :=
+  (AffineIsometryEquiv.vaddConst ℝ p).trans
+    (EuclideanGeometry.o.rotation α).toAffineIsometryEquiv
+
+/--
+The sofa according to a rotation path $p : [0, \pi/2] \to \mathbb{R}^2$ as in [Ge92] is the
+intersection over $\alpha \in [0, \pi/2]$ of hallways each translated by $p(\alpha)$ and then
+rotated by $\alpha$, with the special cases that the hallway at $0$ is the horizontal side
+and the hallway at $\pi/2$ is the vertical side.
+-/
+def sofaOfRotateTranslatePath (p : ℝ → ℝ²) : Set ℝ² :=
+  rotateTranslate 0 (p 0) '' horizontalHallway ∩
+  rotateTranslate ↑(π / 2) (p (π / 2)) '' verticalHallway ∩
+  ⋂ α ∈ Set.Icc 0 (π / 2), rotateTranslate α (p α) '' hallway
+
+namespace GerversSofa
+
+/-
+Gerver's constants defining the sofa.
+
+This section follows Theorem 2 of Gerver's paper [Ge92].
+-/
+
+/--
+Eq. 1-4 of [Ro18], which specifies the constants $A$, $B$, $\varphi$, and $\theta$ of [Ge92].
+-/
+def ABφθSpec (A B φ θ : ℝ) : Prop :=
+  0 ≤ φ ∧ φ ≤ θ ∧ θ ≤ π / 4 ∧ 0 ≤ A ∧ 0 ≤ B ∧
+  A * (θ.cos - φ.cos) - 2 * B * φ.sin
+    + (θ - φ - 1) * θ.cos - θ.sin + φ.cos + φ.sin = 0 ∧
+  A * (3 * θ.sin + φ.sin) - 2 * B * φ.cos
+    + 3 * (θ - φ - 1) * θ.sin + 3 * θ.cos - φ.sin + φ.cos = 0 ∧
+  A * φ.cos - (φ.sin + 1 / 2 - φ.cos / 2 + B * φ.sin) = 0 ∧
+  (A + π / 2 - φ - θ) - (B - (θ - φ) * (1 + A) / 2 - (θ - φ)^2 / 4) = 0
+
+/-- There exist unique constants $A$, $B$, $\varphi$, and $\theta$ satisfying the spec. -/
+theorem ABφθSpec.existsUnique : ∃! ABφθ : ℝ × ℝ × ℝ × ℝ,
+    ABφθSpec ABφθ.1 ABφθ.2.1 ABφθ.2.2.1 ABφθ.2.2.2 :=
+  GerverSofa.PartF.Parameters.existsUnique
+
+/-- Gerver's constant $A$: the first component of the unique solution of `ABφθSpec`. -/
+def A : ℝ := ABφθSpec.existsUnique.choose.1
+/-- Gerver's constant $B$: the second component of the unique solution of `ABφθSpec`. -/
+def B : ℝ := ABφθSpec.existsUnique.choose.2.1
+/-- Gerver's angle $\varphi$: the third component of the unique solution of `ABφθSpec`. -/
+def φ : ℝ := ABφθSpec.existsUnique.choose.2.2.1
+/-- Gerver's angle $\theta$: the fourth component of the unique solution of `ABφθSpec`. -/
+def θ : ℝ := ABφθSpec.existsUnique.choose.2.2.2
+
+/-- The integral-path auxiliary function $r$, with break points $\varphi$, $\theta$,
+$\pi/2 - \theta$ and $\pi/2 - \varphi$. The functions `x` and `y` are integrals of it. -/
+def r (α : ℝ) : ℝ :=
+  if α ≤ φ then
+    1 / 2
+  else if α ≤ θ then
+    (1 + A + α - φ) / 2
+  else if α ≤ π / 2 - θ then
+    A + α - φ
+  else if α ≤ π / 2 - φ then
+    B - (π / 2 - α - φ) * (1 + A) / 2 - (π / 2 - α - φ) ^ 2 / 4
+  else
+    0
+
+/-- $y(\alpha) = \int_\alpha^{\pi/2 - \varphi} r(t) \sin t \, dt$, used in the canonical
+integral-path definition. -/
+def y (α : ℝ) : ℝ :=
+  ∫ t in α..π / 2 - φ, r t * t.sin
+
+/-- $x(\alpha) = 1 - \int_\alpha^{\pi/2 - \varphi} r(t) \cos t \, dt$, used in the canonical
+integral-path definition. -/
+def x (α : ℝ) : ℝ :=
+  1 - ∫ t in α..π / 2 - φ, r t * t.cos
+
+/-- The rotation path of Gerver's sofa: `p α` is the translation applied to the hallway before it
+is rotated by the angle $\alpha \in [0, \pi/2]$, in the convention of `rotateTranslate`. -/
+def p (α : ℝ) : ℝ² :=
+  !₂[if α ≤ φ
+      then α.cos - 1
+      else x (π / 2 - α) * α.cos + y (π / 2 - α) * α.sin - 1,
+    if α ≤ π / 2 - φ
+      then y α * α.cos - (4 * x 0 - 2 - x α) * α.sin - 1
+      else -(4 * x 0 - 3) * α.sin - 1]
+
+end GerversSofa
+
+/-- Gerver's sofa is the sofa according to the rotation path `GerversSofa.p`. -/
+def gerversSofa : Set ℝ² :=
+  sofaOfRotateTranslatePath GerversSofa.p
+
+open MeasureTheory
+open scoped ENNReal
+
+/-- The **sofa constant** is the maximal area of a moving sofa. -/
+def sofaConstant : ℝ≥0∞ := ⨆ (s : Set ℝ²) (_ : ∃ m, IsMovingSofa s m), volume s
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-
+MIT License
+
+Copyright (c) 2026 Dawid Trela
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.
+-/
+
+
+
+/-!
+# Gerver's definitions in GerverSofaLean agree with the ones used here
+
+Adapted from `F07UpstreamAdapter` in GerverSofaLean v1.1.0 (MIT). GerverSofaLean states its results
+for its own copy of the moving sofa definitions. This file identifies that copy with the
+definitions of `MovingSofa.Canonical.Definitions`.
+-/
+
+@[expose] public section
+
+noncomputable section
+open scoped unitInterval
+
+namespace GerverSofa.PartF.ProjectAdapter
+
+open Coordinates
+
+theorem horizontalHallway_eq_model :
+    MovingSofa.horizontalHallway = Model.horizontalHallway := by
+  ext q
+  constructor
+  · rintro ⟨x, y, hxy, rfl⟩
+    exact hxy
+  · intro hq
+    exact ⟨q 0, q 1, hq, plane_ext rfl rfl⟩
+
+theorem verticalHallway_eq_model :
+    MovingSofa.verticalHallway = Model.verticalHallway := by
+  ext q
+  constructor
+  · rintro ⟨x, y, hxy, rfl⟩
+    exact hxy
+  · intro hq
+    exact ⟨q 0, q 1, hq, plane_ext rfl rfl⟩
+
+theorem hallway_eq_model : MovingSofa.hallway = Model.hallway := by
+  unfold MovingSofa.hallway Model.hallway
+  rw [horizontalHallway_eq_model, verticalHallway_eq_model]
+
+theorem isMovingSofa_iff_model (s : Set Plane) (m : I → Rigid) :
+    MovingSofa.IsMovingSofa s m ↔ Model.IsMovingSofa s m := by
+  constructor
+  · intro h
+    refine ⟨h.isConnected, h.isClosed, h.continuous, h.zero, ?_, ?_, ?_⟩
+    · simpa only [horizontalHallway_eq_model] using h.initial
+    · intro t
+      simpa only [hallway_eq_model] using h.subset_hallway t
+    · simpa only [verticalHallway_eq_model] using h.final
+  · intro h
+    refine ⟨h.isConnected, h.isClosed, h.continuous, h.zero, ?_, ?_, ?_⟩
+    · simpa only [horizontalHallway_eq_model] using h.initial
+    · intro t
+      simpa only [hallway_eq_model] using h.subset_hallway t
+    · simpa only [verticalHallway_eq_model] using h.final
+
+theorem sofaOfRotateTranslatePath_eq_bodySofa (p : ℝ → Plane) :
+    MovingSofa.sofaOfRotateTranslatePath p =
+      bodySofa p Model.horizontalHallway Model.verticalHallway Model.hallway := by
+  unfold MovingSofa.sofaOfRotateTranslatePath bodySofa angleIntersection
+  rw [horizontalHallway_eq_model, verticalHallway_eq_model, hallway_eq_model]
+  rfl
+
+/-- The reduced parameter tuple corresponding to the canonical choice of the unique angle
+solution. -/
+def selected : Reduced.Params :=
+  PartE.tupleEquiv MovingSofa.GerversSofa.ABφθSpec.existsUnique.choose
+
+theorem selected_eq_certified : selected = Integrals.certified := by
+  have h : MovingSofa.GerversSofa.ABφθSpec.existsUnique.choose = Parameters.certified :=
+    Parameters.choice_independent MovingSofa.GerversSofa.ABφθSpec.existsUnique
+  exact (congrArg PartE.tupleEquiv h).trans (PartE.tupleEquiv.apply_symm_apply _)
+
+theorem p_eq_selected (t : ℝ) :
+    MovingSofa.GerversSofa.p t = toPlane (Integrals.path selected t) := rfl
+
+theorem p_eq_certified : MovingSofa.GerversSofa.p =
+    fun t => toPlane (Integrals.path Integrals.certified t) := by
+  funext t
+  rw [p_eq_selected, selected_eq_certified]
+
+theorem integral_rotation_to_full (t : ℝ) (ht : t ∈ Set.Icc 0 (Real.pi / 2)) :
+    rotation t (MovingSofa.GerversSofa.p t) = toPlane (Romik.path PartC.params t) := by
+  rw [p_eq_certified]
+  exact Integrals.certified_integral_rotation_to_full t ht
+
+theorem gerversSofa_eq_certified : MovingSofa.gerversSofa = EuclideanMotion.sofa := by
+  unfold MovingSofa.gerversSofa
+  rw [sofaOfRotateTranslatePath_eq_bodySofa, p_eq_certified]
+  exact Integrals.integral_sofa_eq_certified
+
+end GerverSofa.PartF.ProjectAdapter
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Geometry.Hallway`.
+* `Geometry.HallwayParts`.
+* `Geometry.HallwayPartsProperties`.
+* `Geometry.HallwayRay`.
+* `Geometry.HallwaySupport`.
+* `Geometry.Parallelogram`.
+* `Geometry.ParallelogramGap`.
+* `Geometry.PathHalfPlaneCap`.
+* `Geometry.Reflection`.
+* `Geometry.SupportingHallway`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Hallway
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Counterclockwise rotation about the origin. -/
+def rotationMap (t : Real.Angle) (p : Point) : Point :=
+  (EuclideanGeometry.o.rotation t) p
+
+/-- The horizontal, vertical and rotated vertical strips. -/
+def strips (ω : ℝ) : Set Point × Set Point × Set Point :=
+  let H : Set Point := {p | 0 ≤ p 1 ∧ p 1 ≤ 1}
+  let V : Set Point := {p | 0 ≤ p 0 ∧ p 0 ≤ 1}
+  (H, V, rotationMap (ω : Real.Angle) '' V)
+
+/-- The intersection of strips and its two distinguished points. -/
+def stripParallelogram (ω : ℝ) : Set Point × Point × Point :=
+  ((strips ω).1 ∩ (strips ω).2.2, 0, !₂[Real.tan (Real.pi / 4 - ω / 2), 1])
+
+/-- Corners, walls, rays and quadrants of a hallway. -/
+structure HallwayParts where
+  /-- The inner reentrant corner of the hallway. -/
+  innerCorner : Point
+  /-- The outer corner opposite the inner corner. -/
+  outerCorner : Point
+  /-- The outer wall corresponding to the fixed hallway’s line `x = 1`. -/
+  a : Set Point
+  /-- The inner wall corresponding to the fixed hallway’s line `x = 0`. -/
+  b : Set Point
+  /-- The outer wall corresponding to the fixed hallway’s line `y = 1`. -/
+  c : Set Point
+  /-- The inner wall corresponding to the fixed hallway’s line `y = 0`. -/
+  d : Set Point
+  /-- The ray on the inner B wall extending away from the corner. -/
+  bRay : Set Point
+  /-- The ray on the inner D wall extending away from the corner. -/
+  dRay : Set Point
+  /-- The closed quadrant cut out by the two outer walls. -/
+  outerQuadrant : Set Point
+  /-- The open forbidden quadrant behind the inner corner. -/
+  innerQuadrant : Set Point
+
+/-- The named parts of the fixed hallway. -/
+def hallwayParts : HallwayParts where
+  innerCorner := 0
+  outerCorner := !₂[1, 1]
+  a := {p | p 0 = 1}
+  b := {p | p 0 = 0}
+  c := {p | p 1 = 1}
+  d := {p | p 1 = 0}
+  bRay := {p | p 0 = 0 ∧ p 1 ≤ 0}
+  dRay := {p | p 0 ≤ 0 ∧ p 1 = 0}
+  outerQuadrant := {p | p 0 ≤ 1 ∧ p 1 ≤ 1}
+  innerQuadrant := {p | p 0 < 0 ∧ p 1 < 0}
+
+/-- Rotation followed by the support-determined translation. -/
+def supportingPlacement (s : Set Point) (t : Real.Angle) (p : Point) : Point :=
+  rotationMap t p + (supportValue s t - 1) • normalVector t +
+    (supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) - 1) • tangentVector t
+
+/-- The supporting hallway of a nonempty compact set. -/
+def supportingHallway (s : Set Point) (t : Real.Angle) : Set Point :=
+  supportingPlacement s t '' hallway
+
+/-- The images of all named hallway parts under its supporting placement. -/
+def rotatingHallwayParts (s : Set Point) (t : Real.Angle) : HallwayParts where
+  innerCorner := supportingPlacement s t hallwayParts.innerCorner
+  outerCorner := supportingPlacement s t hallwayParts.outerCorner
+  a := supportingPlacement s t '' hallwayParts.a
+  b := supportingPlacement s t '' hallwayParts.b
+  c := supportingPlacement s t '' hallwayParts.c
+  d := supportingPlacement s t '' hallwayParts.d
+  bRay := supportingPlacement s t '' hallwayParts.bRay
+  dRay := supportingPlacement s t '' hallwayParts.dRay
+  outerQuadrant := supportingPlacement s t '' hallwayParts.outerQuadrant
+  innerQuadrant := supportingPlacement s t '' hallwayParts.innerQuadrant
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Hallway Parts
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem rightAngleRotation_apply (p : Point) :
+    EuclideanGeometry.o.rightAngleRotation p = !₂[-p 1, p 0] := by
+  apply (ext_inner_left (𝕜 := ℝ))
+  intro y
+  rw [Orientation.inner_rightAngleRotation_right, Orientation.areaForm_to_volumeForm,
+    EuclideanGeometry.o.volumeForm_robust (EuclideanSpace.basisFun (Fin 2) ℝ) rfl,
+    Module.Basis.det_apply]
+  simp [Matrix.det_fin_two, Module.Basis.toMatrix, PiLp.inner_apply]
+  ring
+
+theorem inner_rotationMap_normalVector (p : Point) (t : Real.Angle) :
+    inner ℝ (rotationMap t p) (normalVector t) = p 0 := by
+  rw [rotationMap, Orientation.rotation_apply, rightAngleRotation_apply]
+  simp [normalVector, frame, PiLp.inner_apply]
+  linear_combination p 0 * Real.Angle.cos_sq_add_sin_sq t
+
+theorem inner_rotationMap_tangentVector (p : Point) (t : Real.Angle) :
+    inner ℝ (rotationMap t p) (tangentVector t) = p 1 := by
+  rw [rotationMap, Orientation.rotation_apply, rightAngleRotation_apply]
+  simp [tangentVector, frame, PiLp.inner_apply]
+  linear_combination p 1 * Real.Angle.cos_sq_add_sin_sq t
+
+theorem normalVector_add_pi_div_two (t : Real.Angle) :
+    normalVector (t + ((Real.pi / 2 : ℝ) : Real.Angle)) = tangentVector t := by
+  simp [normalVector, tangentVector, frame, Real.Angle.cos_add_pi_div_two,
+    Real.Angle.sin_add_pi_div_two]
+
+/-- Rotating a real normal direction by a right angle gives the tangent direction. -/
+theorem normalVector_add_pi_div_two_real (t : ℝ) :
+    normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle) = tangentVector (t : Real.Angle) := by
+  rw [Real.Angle.coe_add, normalVector_add_pi_div_two]
+
+theorem inner_supportingPlacement_normalVector (s : Set Point) (t : Real.Angle)
+    (p : Point) :
+    inner ℝ (supportingPlacement s t p) (normalVector t) =
+      p 0 + supportValue s t - 1 := by
+  simp only [supportingPlacement, inner_add_left, real_inner_smul_left,
+    inner_rotationMap_normalVector]
+  simp [normalVector, tangentVector, frame, PiLp.inner_apply,
+    PiLp.norm_sq_eq_of_L2, Real.Angle.cos_sq_add_sin_sq]
+  ring
+
+theorem inner_supportingPlacement_tangentVector (s : Set Point) (t : Real.Angle)
+    (p : Point) :
+    inner ℝ (supportingPlacement s t p) (tangentVector t) =
+      p 1 + supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) - 1 := by
+  simp only [supportingPlacement, inner_add_left, real_inner_smul_left,
+    inner_rotationMap_tangentVector]
+  simp [normalVector, tangentVector, frame, PiLp.inner_apply,
+    PiLp.norm_sq_eq_of_L2, Real.Angle.cos_sq_add_sin_sq, add_comm]
+  ring
+
+theorem hallway_eq_outerQuadrant_sdiff_innerQuadrant :
+    hallway = hallwayParts.outerQuadrant \ hallwayParts.innerQuadrant := by
+  ext p
+  constructor
+  · rintro (⟨a, b, h, rfl⟩ | ⟨a, b, h, rfl⟩)
+    · change (a ≤ 1 ∧ b ≤ 1) ∧ ¬(a < 0 ∧ b < 0)
+      exact ⟨⟨h.1, h.2.2⟩, fun hn ↦ (not_lt_of_ge h.2.1) hn.2⟩
+    · change (a ≤ 1 ∧ b ≤ 1) ∧ ¬(a < 0 ∧ b < 0)
+      exact ⟨⟨h.2.1, h.2.2⟩, fun hn ↦ (not_lt_of_ge h.1) hn.1⟩
+  · intro hp
+    change (p 0 ≤ 1 ∧ p 1 ≤ 1) ∧ ¬(p 0 < 0 ∧ p 1 < 0) at hp
+    have heq : (!₂[p 0, p 1] : Point) = p := by
+      ext i
+      fin_cases i <;> rfl
+    by_cases h : 0 ≤ p 0
+    · exact Or.inr ⟨p 0, p 1, ⟨h, hp.1.1, hp.1.2⟩, heq⟩
+    · exact Or.inl ⟨p 0, p 1,
+        ⟨hp.1.1, le_of_not_gt (fun hy ↦ hp.2 ⟨lt_of_not_ge h, hy⟩), hp.1.2⟩, heq⟩
+
+/-- The first coordinate of a rotated point. -/
+theorem rotationMap_apply_zero (θ : Real.Angle) (x : Point) :
+    rotationMap θ x 0 = θ.cos * x 0 - θ.sin * x 1 := by
+  rw [rotationMap, Orientation.rotation_apply, rightAngleRotation_apply]
+  simp
+  ring
+
+/-- The second coordinate of a rotated point. -/
+theorem rotationMap_apply_one (θ : Real.Angle) (x : Point) :
+    rotationMap θ x 1 = θ.sin * x 0 + θ.cos * x 1 := by
+  rw [rotationMap, Orientation.rotation_apply, rightAngleRotation_apply]
+  simp
+  ring
+
+/-- Coordinate bounds for a point of the horizontal side of the hallway. -/
+theorem mem_horizontalHallway_coordinates {p : Point} (hp : p ∈ horizontalHallway) :
+    p 0 ≤ 1 ∧ 0 ≤ p 1 ∧ p 1 ≤ 1 := by
+  obtain ⟨x, y, h, rfl⟩ := hp
+  simpa using h
+
+/-- Coordinate bounds for a point of the vertical side of the hallway. -/
+theorem mem_verticalHallway_coordinates {p : Point} (hp : p ∈ verticalHallway) :
+    0 ≤ p 0 ∧ p 0 ≤ 1 ∧ p 1 ≤ 1 := by
+  obtain ⟨x, y, h, rfl⟩ := hp
+  simpa using h
+
+/-- A point lies in the hallway exactly when it lies in the outer quadrant and is not
+strictly inside the inner one. -/
+theorem mem_hallway_iff (q : Point) : q ∈ hallway ↔
+    (q 0 ≤ 1 ∧ q 1 ≤ 1) ∧ (0 ≤ q 0 ∨ 0 ≤ q 1) := by
+  constructor
+  · rintro (⟨a, b, h, rfl⟩ | ⟨a, b, h, rfl⟩)
+    · exact ⟨⟨h.1, h.2.2⟩, Or.inr h.2.1⟩
+    · exact ⟨⟨h.2.1, h.2.2⟩, Or.inl h.1⟩
+  · rintro ⟨⟨hx, hy⟩, h⟩
+    have heq : (!₂[q 0, q 1] : Point) = q := by
+      ext i
+      fin_cases i <;> rfl
+    rcases h with hx0 | hy0
+    · exact Or.inr ⟨q 0, q 1, ⟨hx0, hx, hy⟩, heq⟩
+    · exact Or.inl ⟨q 0, q 1, ⟨hx, hy0, hy⟩, heq⟩
+
+/-- Coordinate bounds place a point in the horizontal side of the hallway. -/
+theorem mem_horizontalHallway_of_coordinates (p : Point)
+    (hx : p 0 ≤ 1) (hy : p 1 ∈ Set.Icc (0 : ℝ) 1) : p ∈ horizontalHallway := by
+  refine ⟨p 0, p 1, ⟨hx, hy⟩, ?_⟩
+  ext i
+  fin_cases i <;> rfl
+
+/-- Coordinate bounds place a point in the vertical side of the hallway. -/
+theorem mem_verticalHallway_of_coordinates (p : Point)
+    (hx : p 0 ∈ Set.Icc (0 : ℝ) 1) (hy : p 1 ≤ 1) : p ∈ verticalHallway := by
+  refine ⟨p 0, p 1, ⟨hx.1, hx.2, hy⟩, ?_⟩
+  ext i
+  fin_cases i <;> rfl
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Hallway Parts Properties
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A supporting hallway lies in its outer quadrant. -/
+theorem supportingHallway_subset_outerQuadrant (s : Set Point) (t : Real.Angle) :
+    supportingHallway s t ⊆ (rotatingHallwayParts s t).outerQuadrant := by
+  change supportingPlacement s t '' hallway ⊆
+    supportingPlacement s t '' hallwayParts.outerQuadrant
+  apply Set.image_mono
+  rw [hallway_eq_outerQuadrant_sdiff_innerQuadrant]
+  exact Set.sdiff_subset
+
+theorem rotatingHallwayParts_formulas (s : Set Point) (t : Real.Angle) :
+    supportingHallway s t = (rotatingHallwayParts s t).outerQuadrant \ (rotatingHallwayParts s
+      t).innerQuadrant ∧
+    (rotatingHallwayParts s t).innerCorner = (supportValue s t - 1) • normalVector t +
+      (supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) - 1) • tangentVector t ∧
+    (rotatingHallwayParts s t).outerCorner = supportValue s t • normalVector t + supportValue s (t
+      + ((Real.pi / 2 : ℝ) : Real.Angle)) • tangentVector t ∧
+    (rotatingHallwayParts s t).a = normalLine t (supportValue s t) ∧
+    (rotatingHallwayParts s t).b = normalLine t (supportValue s t - 1) ∧
+    (rotatingHallwayParts s t).c = normalLine (t + ((Real.pi / 2 : ℝ) : Real.Angle)) (supportValue
+      s (t + ((Real.pi / 2 : ℝ) : Real.Angle))) ∧
+    (rotatingHallwayParts s t).d = normalLine (t + ((Real.pi / 2 : ℝ) : Real.Angle)) (supportValue
+      s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) - 1) ∧
+    (rotatingHallwayParts s t).outerQuadrant =
+      normalHalfPlane t (supportValue s t) false false ∩
+        normalHalfPlane (t + ((Real.pi / 2 : ℝ) : Real.Angle)) (supportValue s (t + ((Real.pi / 2
+          : ℝ) : Real.Angle))) false false ∧
+    (rotatingHallwayParts s t).innerQuadrant =
+      normalHalfPlane t (supportValue s t - 1) false true ∩
+        normalHalfPlane (t + ((Real.pi / 2 : ℝ) : Real.Angle)) (supportValue s (t + ((Real.pi / 2
+          : ℝ) : Real.Angle)) - 1) false true := by
+  have hinj : Function.Injective (supportingPlacement s t) := by
+    intro p q h
+    apply (EuclideanGeometry.o.rotation t).injective
+    simpa only [supportingPlacement, rotationMap, add_left_inj] using h
+  have hsurj : Function.Surjective (supportingPlacement s t) := by
+    intro q
+    obtain ⟨p, hp⟩ := (EuclideanGeometry.o.rotation t).surjective
+      (q - (supportValue s t - 1) • normalVector t -
+        (supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) - 1) • tangentVector t)
+    refine ⟨p, ?_⟩
+    simp only [supportingPlacement, rotationMap, hp]
+    abel
+  have himage (A B : Set Point)
+      (h : ∀ p, p ∈ A ↔ supportingPlacement s t p ∈ B) :
+      supportingPlacement s t '' A = B := by
+    apply (Set.preimage_eq_preimage hsurj).mp
+    rw [Set.preimage_image_eq A hinj]
+    exact Set.ext h
+  refine ⟨?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
+  · apply (Set.preimage_eq_preimage hsurj).mp
+    change supportingPlacement s t ⁻¹' (supportingPlacement s t '' hallway) =
+      supportingPlacement s t ⁻¹' (supportingPlacement s t '' hallwayParts.outerQuadrant \
+        supportingPlacement s t '' hallwayParts.innerQuadrant)
+    rw [Set.preimage_sdiff, Set.preimage_image_eq _ hinj,
+      Set.preimage_image_eq _ hinj, Set.preimage_image_eq _ hinj]
+    exact hallway_eq_outerQuadrant_sdiff_innerQuadrant
+  · simp [rotatingHallwayParts, supportingPlacement, hallwayParts, rotationMap]
+  · simp only [rotatingHallwayParts, supportingPlacement, hallwayParts,
+      rotationMap, Orientation.rotation_apply, rightAngleRotation_apply]
+    ext i
+    fin_cases i <;> simp [normalVector, tangentVector, frame] <;> ring
+  · apply himage
+    intro p
+    change p 0 = 1 ↔ inner ℝ (supportingPlacement s t p) (normalVector t) = _
+    rw [inner_supportingPlacement_normalVector]
+    constructor <;> intro h <;> linarith
+  · apply himage
+    intro p
+    change p 0 = 0 ↔ inner ℝ (supportingPlacement s t p) (normalVector t) = _
+    rw [inner_supportingPlacement_normalVector]
+    constructor <;> intro h <;> linarith
+  · apply himage
+    intro p
+    change p 1 = 1 ↔ inner ℝ (supportingPlacement s t p)
+      (normalVector (t + ((Real.pi / 2 : ℝ) : Real.Angle))) = _
+    rw [normalVector_add_pi_div_two, inner_supportingPlacement_tangentVector]
+    constructor <;> intro h <;> linarith
+  · apply himage
+    intro p
+    change p 1 = 0 ↔ inner ℝ (supportingPlacement s t p)
+      (normalVector (t + ((Real.pi / 2 : ℝ) : Real.Angle))) = _
+    rw [normalVector_add_pi_div_two, inner_supportingPlacement_tangentVector]
+    constructor <;> intro h <;> linarith
+  · apply himage
+    intro p
+    change (p 0 ≤ 1 ∧ p 1 ≤ 1) ↔
+      (inner ℝ (supportingPlacement s t p) (normalVector t) ≤ _) ∧
+      (inner ℝ (supportingPlacement s t p)
+        (normalVector (t + ((Real.pi / 2 : ℝ) : Real.Angle))) ≤ _)
+    rw [normalVector_add_pi_div_two, inner_supportingPlacement_normalVector,
+      inner_supportingPlacement_tangentVector]
+    constructor <;> rintro ⟨h₀, h₁⟩ <;> constructor <;> linarith
+  · apply himage
+    intro p
+    change (p 0 < 0 ∧ p 1 < 0) ↔
+      (inner ℝ (supportingPlacement s t p) (normalVector t) < _) ∧
+      (inner ℝ (supportingPlacement s t p)
+        (normalVector (t + ((Real.pi / 2 : ℝ) : Real.Angle))) < _)
+    rw [normalVector_add_pi_div_two, inner_supportingPlacement_normalVector,
+      inner_supportingPlacement_tangentVector]
+    constructor <;> rintro ⟨h₀, h₁⟩ <;> constructor <;> linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Hallway Ray
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Set MeasureTheory
+
+namespace MovingSofa
+
+/-- Coordinate description of the downward ray of a supporting hallway. -/
+theorem mem_rotatingHallwayParts_bRay_iff (s : Set Point) (t : Real.Angle) (p : Point) :
+    p ∈ (rotatingHallwayParts s t).bRay ↔
+      inner ℝ p (normalVector t) = supportValue s t - 1 ∧
+      inner ℝ p (tangentVector t) ≤
+        supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) - 1 := by
+  constructor
+  · rintro ⟨q, hq, rfl⟩
+    change q 0 = 0 ∧ q 1 ≤ 0 at hq
+    rw [inner_supportingPlacement_normalVector, inner_supportingPlacement_tangentVector]
+    exact ⟨by linarith [hq.1], by linarith [hq.2]⟩
+  · intro hp
+    obtain ⟨q, rfl⟩ := (show Function.Surjective (supportingPlacement s t) by
+      intro p
+      obtain ⟨q, hq⟩ := (EuclideanGeometry.o.rotation t).surjective
+        (p - (supportValue s t - 1) • normalVector t -
+          (supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) - 1) • tangentVector t)
+      refine ⟨q, ?_⟩
+      simp only [supportingPlacement, rotationMap, hq]
+      abel) p
+    rw [inner_supportingPlacement_normalVector, inner_supportingPlacement_tangentVector] at hp
+    change supportingPlacement s t q ∈ supportingPlacement s t '' hallwayParts.bRay
+    refine ⟨q, ?_, rfl⟩
+    change q 0 = 0 ∧ q 1 ≤ 0
+    exact ⟨by linarith [hp.1], by linarith [hp.2]⟩
+
+/-- Coordinate description of the leftward ray of a supporting hallway. -/
+theorem mem_rotatingHallwayParts_dRay_iff (s : Set Point) (t : Real.Angle) (p : Point) :
+    p ∈ (rotatingHallwayParts s t).dRay ↔
+      inner ℝ p (normalVector t) ≤ supportValue s t - 1 ∧
+      inner ℝ p (tangentVector t) =
+        supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) - 1 := by
+  constructor
+  · rintro ⟨q, hq, rfl⟩
+    change q 0 ≤ 0 ∧ q 1 = 0 at hq
+    rw [inner_supportingPlacement_normalVector, inner_supportingPlacement_tangentVector]
+    exact ⟨by linarith [hq.1], by linarith [hq.2]⟩
+  · intro hp
+    obtain ⟨q, rfl⟩ := (show Function.Surjective (supportingPlacement s t) by
+      intro p
+      obtain ⟨q, hq⟩ := (EuclideanGeometry.o.rotation t).surjective
+        (p - (supportValue s t - 1) • normalVector t -
+          (supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) - 1) • tangentVector t)
+      refine ⟨q, ?_⟩
+      simp only [supportingPlacement, rotationMap, hq]
+      abel) p
+    rw [inner_supportingPlacement_normalVector, inner_supportingPlacement_tangentVector] at hp
+    change supportingPlacement s t q ∈ supportingPlacement s t '' hallwayParts.dRay
+    refine ⟨q, ?_, rfl⟩
+    change q 0 ≤ 0 ∧ q 1 = 0
+    exact ⟨by linarith [hp.1], by linarith [hp.2]⟩
+
+/-- The part of a supporting hallway ray above a transverse line has the expected length. -/
+theorem hausdorffMeasure_bRay_inter_normalHalfPlane (s : Set Point) (t u h : ℝ)
+    (htrans : 0 < inner ℝ (tangentVector (t : Real.Angle))
+      (normalVector (u : Real.Angle))) :
+    Measure.hausdorffMeasure 1
+        ((rotatingHallwayParts s (t : Real.Angle)).bRay ∩
+          normalHalfPlane (u : Real.Angle) h true false) =
+      ENNReal.ofReal (max 0
+        ((inner ℝ (rotatingHallwayParts s (t : Real.Angle)).innerCorner
+          (normalVector (u : Real.Angle)) - h) /
+          inner ℝ (tangentVector (t : Real.Angle)) (normalVector (u : Real.Angle)))) := by
+  let x := (rotatingHallwayParts s (t : Real.Angle)).innerCorner
+  let v := tangentVector (t : Real.Angle)
+  let d := inner ℝ v (normalVector (u : Real.Angle))
+  let α := (inner ℝ x (normalVector (u : Real.Angle)) - h) / d
+  have hd : 0 < d := htrans
+  have hxn : inner ℝ x (normalVector (t : Real.Angle)) =
+      supportValue s (t : Real.Angle) - 1 := by
+    change inner ℝ (supportingPlacement s (t : Real.Angle) (0 : Point))
+      (normalVector (t : Real.Angle)) = _
+    rw [inner_supportingPlacement_normalVector]
+    simp
+  have hxt : inner ℝ x v =
+      supportValue s ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle)) - 1 := by
+    change inner ℝ (supportingPlacement s (t : Real.Angle) (0 : Point))
+      (tangentVector (t : Real.Angle)) = _
+    rw [inner_supportingPlacement_tangentVector]
+    simp only [PiLp.zero_apply]
+    ring
+  have hvn : inner ℝ v (normalVector (t : Real.Angle)) = 0 := by
+    rw [real_inner_comm, inner_normalVector_tangentVector]
+  have hvv : inner ℝ v v = 1 := inner_tangentVector_self t
+  have hpoint (p : Point) (lam : ℝ)
+      (hn : inner ℝ p (normalVector (t : Real.Angle)) =
+        inner ℝ x (normalVector (t : Real.Angle)))
+      (ht : inner ℝ p v = inner ℝ x v - lam) : p = x - lam • v := by
+    rw [← inner_normalVector_smul_add_inner_tangentVector_smul p (t : Real.Angle),
+      ← inner_normalVector_smul_add_inner_tangentVector_smul x (t : Real.Angle)]
+    rw [hn, ht]
+    module
+  by_cases hα : 0 ≤ α
+  · have hset : (rotatingHallwayParts s (t : Real.Angle)).bRay ∩
+        normalHalfPlane (u : Real.Angle) h true false = segment ℝ x (x - α • v) := by
+      ext p
+      change (p ∈ (rotatingHallwayParts s (t : Real.Angle)).bRay ∧
+        h ≤ inner ℝ p (normalVector (u : Real.Angle))) ↔ _
+      constructor
+      · rintro ⟨hray, hu⟩
+        rw [mem_rotatingHallwayParts_bRay_iff] at hray
+        let lam := inner ℝ x v - inner ℝ p v
+        have hlam0 : 0 ≤ lam := by dsimp [lam]; rw [hxt]; linarith [hray.2]
+        have hp : p = x - lam • v := by
+          apply hpoint p lam
+          · rw [hray.1, hxn]
+          · dsimp [lam]
+            ring
+        have hlamα : lam ≤ α := by
+          rw [hp, inner_sub_left, real_inner_smul_left] at hu
+          apply (le_div_iff₀ hd).2
+          dsimp [d]
+          nlinarith
+        by_cases hα0 : α = 0
+        · have : lam = 0 := le_antisymm (hα0 ▸ hlamα) hlam0
+          simp [hα0, hp, this]
+        · rw [segment_eq_image]
+          refine ⟨lam / α,
+            ⟨div_nonneg hlam0 hα, (div_le_one (lt_of_le_of_ne hα ?_)).2 hlamα⟩, ?_⟩
+          · exact Ne.symm hα0
+          · change (1 - lam / α) • x + (lam / α) • (x - α • v) = p
+            rw [hp]
+            rw [smul_sub, smul_smul, div_mul_cancel₀ _ hα0]
+            module
+      · intro hp
+        rw [segment_eq_image] at hp
+        obtain ⟨r, hr, rfl⟩ := hp
+        have heq : (1 - r) • x + r • (x - α • v) = x - (r * α) • v := by module
+        rw [show (fun θ : ℝ ↦ (1 - θ) • x + θ • (x - α • v)) r =
+          x - (r * α) • v from heq]
+        constructor
+        · rw [mem_rotatingHallwayParts_bRay_iff]
+          constructor
+          · simp only [inner_sub_left, real_inner_smul_left, hvn, mul_zero, sub_zero, hxn]
+          · simp only [inner_sub_left, real_inner_smul_left]
+            change inner ℝ x v - r * α * inner ℝ v v ≤ _
+            rw [hvv, mul_one, hxt]
+            nlinarith [mul_nonneg hr.1 hα]
+        · simp only [inner_sub_left, real_inner_smul_left]
+          have hrα : r * α ≤ α := by nlinarith [hr.1, hr.2, hα]
+          have hαeq : inner ℝ x (normalVector (u : Real.Angle)) - h = α * d := by
+            dsimp [α]
+            rw [div_mul_cancel₀ _ hd.ne']
+          dsimp [d] at hαeq ⊢
+          nlinarith
+    rw [hset, MeasureTheory.hausdorffMeasure_segment, edist_dist, dist_eq_norm]
+    have hvnorm : ‖v‖ = 1 := by
+      dsimp [v]
+      rw [← sq_eq_sq₀ (norm_nonneg _) (by norm_num : (0 : ℝ) ≤ 1)]
+      rw [EuclideanSpace.norm_sq_eq]
+      simp [tangentVector, frame, Fin.sum_univ_two, Real.sin_sq_add_cos_sq]
+    have hsub : x - (x - α • v) = α • v := by module
+    rw [hsub, norm_smul, hvnorm, mul_one]
+    simp only [Real.norm_eq_abs, abs_of_nonneg hα]
+    simp only [x, v, d, α, max_eq_right hα]
+  · have hset : (rotatingHallwayParts s (t : Real.Angle)).bRay ∩
+        normalHalfPlane (u : Real.Angle) h true false = ∅ := by
+      ext p
+      simp only [Set.mem_empty_iff_false, iff_false]
+      intro hp
+      change p ∈ (rotatingHallwayParts s (t : Real.Angle)).bRay ∧
+        h ≤ inner ℝ p (normalVector (u : Real.Angle)) at hp
+      rw [mem_rotatingHallwayParts_bRay_iff] at hp
+      let lam := inner ℝ x v - inner ℝ p v
+      have hlam0 : 0 ≤ lam := by dsimp [lam]; rw [hxt]; linarith [hp.1.2]
+      have hp' : p = x - lam • v := by
+        apply hpoint p lam
+        · rw [hp.1.1, hxn]
+        · dsimp [lam]
+          ring
+      have hlamα : lam ≤ α := by
+        have hu := hp.2
+        rw [hp', inner_sub_left, real_inner_smul_left] at hu
+        apply (le_div_iff₀ hd).2
+        dsimp [d]
+        nlinarith
+      linarith
+    rw [hset]
+    have hα' : α ≤ 0 := le_of_not_ge hα
+    simp only [measure_empty, ENNReal.ofReal_zero, x, v, d, α, max_eq_left hα']
+
+/-- A planar set on one line of a moving frame, with bounded tangent coordinate, is short. -/
+theorem hausdorffMeasure_le_of_frame_bounds {S : Set Point} {t e lo hi : ℝ}
+    (hS : ∀ p ∈ S, inner ℝ p (normalVector (t : Real.Angle)) = e ∧
+      lo ≤ inner ℝ p (tangentVector (t : Real.Angle)) ∧
+      inner ℝ p (tangentVector (t : Real.Angle)) ≤ hi) :
+    Measure.hausdorffMeasure 1 S ≤ ENNReal.ofReal (max 0 (hi - lo)) := by
+  have hvv : inner ℝ (tangentVector (t : Real.Angle)) (tangentVector (t : Real.Angle)) = 1 :=
+    inner_tangentVector_self t
+  have hnorm : ‖tangentVector (t : Real.Angle)‖ = 1 := by
+    have h := real_inner_self_eq_norm_mul_norm (tangentVector (t : Real.Angle))
+    rw [hvv] at h
+    nlinarith [norm_nonneg (tangentVector (t : Real.Angle))]
+  have hsub : S ⊆ segment ℝ
+      (e • normalVector (t : Real.Angle) + lo • tangentVector (t : Real.Angle))
+      (e • normalVector (t : Real.Angle) + max lo hi • tangentVector (t : Real.Angle)) := by
+    intro p hp
+    obtain ⟨hpn, hplo, hphi⟩ := hS p hp
+    obtain ⟨y, hy⟩ : ∃ y : ℝ, inner ℝ p (tangentVector (t : Real.Angle)) = y := ⟨_, rfl⟩
+    rw [hy] at hplo hphi
+    have hdecomp : p = e • normalVector (t : Real.Angle) + y • tangentVector (t : Real.Angle) := by
+      conv_lhs => rw [← inner_normalVector_smul_add_inner_tangentVector_smul p (t : Real.Angle)]
+      rw [hpn, hy]
+    have hym : y ≤ max lo hi := hphi.trans (le_max_right lo hi)
+    rcases le_or_gt (max lo hi) lo with hml | hml
+    · have hylo : y = lo := le_antisymm (hym.trans hml) hplo
+      rw [hdecomp, hylo]
+      exact left_mem_segment ℝ _ _
+    · rw [segment_eq_image]
+      refine ⟨(y - lo) / (max lo hi - lo),
+        ⟨div_nonneg (by linarith) (by linarith), ?_⟩, ?_⟩
+      · rw [div_le_one (by linarith)]
+        linarith
+      · have hθ : lo + (y - lo) / (max lo hi - lo) * (max lo hi - lo) = y := by
+          field_simp
+          ring
+        have hexp : (1 - (y - lo) / (max lo hi - lo)) •
+              (e • normalVector (t : Real.Angle) + lo • tangentVector (t : Real.Angle)) +
+            ((y - lo) / (max lo hi - lo)) •
+              (e • normalVector (t : Real.Angle) +
+                max lo hi • tangentVector (t : Real.Angle)) =
+            e • normalVector (t : Real.Angle) +
+              (lo + (y - lo) / (max lo hi - lo) * (max lo hi - lo)) •
+                tangentVector (t : Real.Angle) := by
+          module
+        change (1 - (y - lo) / (max lo hi - lo)) •
+              (e • normalVector (t : Real.Angle) + lo • tangentVector (t : Real.Angle)) +
+            ((y - lo) / (max lo hi - lo)) •
+              (e • normalVector (t : Real.Angle) +
+                max lo hi • tangentVector (t : Real.Angle)) = p
+        rw [hexp, hθ]
+        exact hdecomp.symm
+  refine (measure_mono hsub).trans (le_of_eq ?_)
+  rw [MeasureTheory.hausdorffMeasure_segment, edist_dist, dist_eq_norm]
+  congr 1
+  have hdiff : e • normalVector (t : Real.Angle) + lo • tangentVector (t : Real.Angle) -
+      (e • normalVector (t : Real.Angle) + max lo hi • tangentVector (t : Real.Angle)) =
+      (lo - max lo hi) • tangentVector (t : Real.Angle) := by module
+  rw [hdiff, norm_smul, hnorm, mul_one, Real.norm_eq_abs,
+    abs_of_nonpos (sub_nonpos.mpr (le_max_left lo hi)), neg_sub]
+  rcases le_total hi lo with hle | hle
+  · rw [max_eq_left hle, max_eq_left (by linarith), sub_self]
+  · rw [max_eq_right hle, max_eq_right (by linarith)]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Hallway Support
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem outerCorner_eq_support_sum (K : ConvexBody Point) (t : ℝ) :
+    (rotatingHallwayParts (K : Set Point) (t : Real.Angle)).outerCorner =
+      supportValue K (t : Real.Angle) • normalVector (t : Real.Angle) +
+      supportValue K ((t + Real.pi / 2 : ℝ) : Real.Angle) • tangentVector (t : Real.Angle) := by
+  have hrot : rotationMap (t : Real.Angle) (!₂[1, 1] : Point) =
+      normalVector (t : Real.Angle) + tangentVector (t : Real.Angle) := by
+    unfold rotationMap
+    rw [Orientation.rotation_apply, rightAngleRotation_apply]
+    ext i
+    fin_cases i <;> simp [normalVector, tangentVector, frame]
+    ring
+  simp only [rotatingHallwayParts, supportingPlacement, hallwayParts, hrot, Real.Angle.coe_add]
+  module
+
+/-- Two supporting lines at angular difference exactly `π / 2` meet at the outer corner of the
+rotating supporting hallway. -/
+theorem supportingIntersection_add_pi_div_two_eq_outerCorner (K : ConvexBody Point) (t : ℝ) :
+    supportingIntersection K (t : Real.Angle) ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+      (rotatingHallwayParts (K : Set Point) (t : Real.Angle)).outerCorner := by
+  have hd : ((t + Real.pi / 2 : ℝ) : Real.Angle) - (t : Real.Angle) =
+      ((Real.pi / 2 : ℝ) : Real.Angle) := by
+    rw [← Real.Angle.coe_sub, show t + Real.pi / 2 - t = Real.pi / 2 from by ring]
+  rw [supportingIntersection, hd, Real.Angle.cos_coe, Real.Angle.sin_coe, Real.cos_pi_div_two,
+    Real.sin_pi_div_two, outerCorner_eq_support_sum]
+  simp only [mul_zero, sub_zero, div_one]
+
+/-- The outer corner of the rotating supporting hallway is its inner corner translated by the
+frame sum `u_t + v_t`. -/
+theorem outerCorner_eq_innerCorner_add (K : ConvexBody Point) (t : ℝ) :
+    (rotatingHallwayParts (K : Set Point) (t : Real.Angle)).outerCorner =
+      (rotatingHallwayParts (K : Set Point) (t : Real.Angle)).innerCorner +
+        (normalVector (t : Real.Angle) + tangentVector (t : Real.Angle)) := by
+  rw [outerCorner_eq_support_sum]
+  simp only [rotatingHallwayParts, hallwayParts, supportingPlacement, rotationMap,
+    map_zero, zero_add, Real.Angle.coe_add]
+  module
+
+theorem continuous_outerCorner (K : ConvexBody Point) :
+    Continuous
+      (fun s : ℝ ↦ (rotatingHallwayParts (K : Set Point) (s : Real.Angle)).outerCorner) := by
+  simp_rw [outerCorner_eq_support_sum]
+  exact ((continuous_supportValue_real K).smul continuous_normalVector_real).add
+    (((continuous_supportValue_real K).comp (continuous_id.add_const _)).smul
+      (continuous_iff_continuousAt.mpr fun t ↦ (hasDerivAt_tangentVector t).continuousAt))
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Parallelogram
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The strip intersection is described by its vertical and rotated coordinates. -/
+theorem mem_stripParallelogram_iff (ω : ℝ) (p : Point) :
+    p ∈ (stripParallelogram ω).1 ↔
+      (0 ≤ p 1 ∧ p 1 ≤ 1) ∧
+      (0 ≤ inner ℝ p (normalVector (ω : Real.Angle)) ∧
+        inner ℝ p (normalVector (ω : Real.Angle)) ≤ 1) := by
+  change ((0 ≤ p 1 ∧ p 1 ≤ 1) ∧
+    p ∈ rotationMap (ω : Real.Angle) '' {q : Point | 0 ≤ q 0 ∧ q 0 ≤ 1}) ↔ _
+  apply and_congr_right
+  intro _
+  constructor
+  · rintro ⟨q, hq, rfl⟩
+    simpa only [inner_rotationMap_normalVector, Set.mem_ofPred_eq] using hq
+  · intro hp
+    obtain ⟨q, rfl⟩ := (EuclideanGeometry.o.rotation (ω : Real.Angle)).surjective p
+    change 0 ≤ inner ℝ (rotationMap (ω : Real.Angle) q)
+        (normalVector (ω : Real.Angle)) ∧
+      inner ℝ (rotationMap (ω : Real.Angle) q) (normalVector (ω : Real.Angle)) ≤ 1 at hp
+    exact ⟨q, by
+      simpa only [inner_rotationMap_normalVector, Set.mem_ofPred_eq] using hp, rfl⟩
+
+/-- Projection on the downward unit normal negates the vertical coordinate. -/
+theorem inner_normalVector_three_pi_div_two (p : Point) :
+    inner ℝ p (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) = -p 1 := by
+  simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two,
+    show 3 * Real.pi / 2 = Real.pi + Real.pi / 2 by ring, Real.cos_add, Real.sin_add,
+    -Real.Angle.coe_add]
+
+/-- The horizontal coordinate is the projection on the normal at angle zero. -/
+theorem inner_normalVector_zero (p : Point) :
+    inner ℝ p (normalVector ((0 : ℝ) : Real.Angle)) = p 0 := by
+  simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+
+/-- The vertical coordinate is the projection on the tangent at angle zero. -/
+theorem inner_tangentVector_zero (p : Point) :
+    inner ℝ p (tangentVector ((0 : ℝ) : Real.Angle)) = p 1 := by
+  simp [tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+
+/-- The vertical coordinate is the projection on the upward unit normal. -/
+theorem inner_normalVector_pi_div_two (p : Point) :
+    inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = p 1 := by
+  simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+
+/-- Projection on the leftward tangent at a right angle negates the horizontal coordinate. -/
+theorem inner_tangentVector_pi_div_two (p : Point) :
+    inner ℝ p (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) = -p 0 := by
+  simp [tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+
+/-- Projection on the leftward tangent at the straight angle negates the vertical coordinate. -/
+theorem inner_tangentVector_pi (p : Point) :
+    inner ℝ p (tangentVector ((Real.pi : ℝ) : Real.Angle)) = -p 1 := by
+  simp [tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+
+/-- The projection of a multiple of the horizontal normal on another unit normal. -/
+theorem inner_smul_normalVector_zero (x t : ℝ) :
+    inner ℝ (x • normalVector (0 : Real.Angle)) (normalVector (t : Real.Angle)) =
+      x * Real.cos t := by
+  simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two, mul_comm]
+
+/-- The horizontal bottom of the strip intersection has zero downward support. -/
+theorem supportValue_stripParallelogram_bottom (ω : ℝ) :
+    supportValue (stripParallelogram ω).1 ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := by
+  have hzero : (0 : Point) ∈ (stripParallelogram ω).1 := by
+    rw [mem_stripParallelogram_iff]
+    simp
+  have hbound : ∀ y ∈ (fun p ↦ inner ℝ p (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle))) ''
+      (stripParallelogram ω).1, y ≤ 0 := by
+    rintro y ⟨p, hp, rfl⟩
+    dsimp only
+    rw [inner_normalVector_three_pi_div_two]
+    exact neg_nonpos.mpr ((mem_stripParallelogram_iff ω p).1 hp).1.1
+  have hmem : (0 : ℝ) ∈ (fun p ↦ inner ℝ p (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle))) ''
+      (stripParallelogram ω).1 := ⟨0, hzero, by simp⟩
+  exact le_antisymm (csSup_le ⟨0, hmem⟩ hbound) (le_csSup ⟨0, hbound⟩ hmem)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Parallelogram Gap
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem parallelogram_gap (ω : ℝ) (hω : ω ∈ Set.Ico 0 (Real.pi / 2)) :
+    (stripParallelogram ω).2.2 - tangentVector 0 =
+      Real.tan ((Real.pi / 2 - ω) / 2) • normalVector 0 ∧
+    (stripParallelogram ω).2.2 - normalVector (ω : Real.Angle) =
+      Real.tan ((Real.pi / 2 - ω) / 2) • tangentVector (ω : Real.Angle) ∧
+    (stripParallelogram ω).1 ∩
+        (supportingLineHalfPlane (stripParallelogram ω).1
+          ((3 * Real.pi / 2 : ℝ) : Real.Angle)).1 =
+      segment ℝ (0 : Point) ((Real.cos ω)⁻¹ • normalVector 0) ∧
+    normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0 ∩
+        normalLine (ω : Real.Angle) 1 =
+      {((Real.cos ω)⁻¹ • normalVector 0 : Point)} ∧
+    Real.tan ((Real.pi / 2 - ω) / 2) = (Real.cos ω)⁻¹ - Real.tan ω := by
+  have hcos : 0 < Real.cos ω :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos, hω.1], hω.2⟩
+  have hgap := Real.tan_pi_div_two_sub_div_two ω hω
+  have hfirst : (stripParallelogram ω).2.2 - tangentVector 0 =
+      Real.tan ((Real.pi / 2 - ω) / 2) • normalVector 0 := by
+    ext i
+    fin_cases i <;> simp [stripParallelogram, normalVector, tangentVector, frame]
+    ring_nf
+  have hsecond : (stripParallelogram ω).2.2 - normalVector (ω : Real.Angle) =
+      Real.tan ((Real.pi / 2 - ω) / 2) • tangentVector (ω : Real.Angle) := by
+    have harg : Real.pi / 4 - ω / 2 = (Real.pi / 2 - ω) / 2 := by ring
+    have hc : Real.tan ((Real.pi / 2 - ω) / 2) * Real.cos ω = 1 - Real.sin ω := by
+      rw [hgap, Real.tan_eq_sin_div_cos]
+      field_simp [hcos.ne']
+    have hc' : Real.tan ((Real.pi / 2 - ω) / 2) * (1 + Real.sin ω) =
+        Real.cos ω := by
+      rw [hgap, Real.tan_eq_sin_div_cos]
+      field_simp [hcos.ne']
+      nlinarith [Real.sin_sq_add_cos_sq ω]
+    ext i
+    fin_cases i <;>
+      simp [stripParallelogram, normalVector, tangentVector, frame, harg] <;> nlinarith
+  have hedge : (stripParallelogram ω).1 ∩
+        (supportingLineHalfPlane (stripParallelogram ω).1
+          ((3 * Real.pi / 2 : ℝ) : Real.Angle)).1 =
+      segment ℝ (0 : Point) ((Real.cos ω)⁻¹ • normalVector 0) := by
+    rw [supportingLineHalfPlane, supportValue_stripParallelogram_bottom]
+    ext p
+    simp only [Set.mem_inter_iff, mem_stripParallelogram_iff, normalLine,
+      Set.mem_ofPred_eq, segment_eq_image, Set.mem_image, Set.mem_Icc]
+    rw [inner_normalVector_three_pi_div_two]
+    constructor
+    · rintro ⟨⟨⟨hp₀, hp₁⟩, hq₀, hq₁⟩, hbottom⟩
+      have hpzero : p 1 = 0 := by linarith
+      have hx₀ : 0 ≤ p 0 * Real.cos ω := by
+        simpa [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two, hpzero,
+          mul_comm] using hq₀
+      have hx₁ : p 0 * Real.cos ω ≤ 1 := by
+        simpa [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two, hpzero,
+          mul_comm] using hq₁
+      refine ⟨p 0 * Real.cos ω, ⟨hx₀, hx₁⟩, ?_⟩
+      ext i
+      fin_cases i <;> simp [normalVector, frame, hpzero, hcos.ne']
+    · rintro ⟨x, hx, rfl⟩
+      have hinner : inner ℝ (x • (Real.cos ω)⁻¹ • normalVector 0)
+          (normalVector (ω : Real.Angle)) = x := by
+        simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+        field_simp [hcos.ne']
+      constructor
+      · exact ⟨by simp [normalVector, frame], by simpa [hinner] using hx⟩
+      · simp [normalVector, frame]
+  have hintersection : normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0 ∩
+        normalLine (ω : Real.Angle) 1 =
+      {((Real.cos ω)⁻¹ • normalVector 0 : Point)} := by
+    ext p
+    simp only [Set.mem_inter_iff, normalLine, Set.mem_ofPred_eq, Set.mem_singleton_iff]
+    constructor
+    · rintro ⟨hvertical, hrotated⟩
+      have hpone : p 1 = 0 := by
+        simpa [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two] using hvertical
+      have hpzero : p 0 * Real.cos ω = 1 := by
+        simpa [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two, hpone,
+          mul_comm] using hrotated
+      ext i
+      fin_cases i
+      · simp [normalVector, frame]
+        field_simp [hcos.ne']
+        exact hpzero
+      · simp [normalVector, frame, hpone]
+    · intro hp
+      rw [hp]
+      constructor
+      · simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+      · simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+        field_simp [hcos.ne']
+  exact ⟨hfirst, hsecond, hedge, hintersection, hgap⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Path Half Plane Cap
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The set satisfying the horizontal base and all upper support constraints of a path. -/
+def outerPathConstraintSet (x : Set.Icc (0 : ℝ) (Real.pi / 2) → Point) : Set Point :=
+  {q | 0 ≤ q 1 ∧ ∀ t,
+    inner ℝ q (normalVector (t.val : Real.Angle)) ≤
+      inner ℝ (x t) (normalVector (t.val : Real.Angle)) + 1 ∧
+    inner ℝ q (tangentVector (t.val : Real.Angle)) ≤
+      inner ℝ (x t) (tangentVector (t.val : Real.Angle)) + 1}
+
+theorem outerPathConstraintSet_isCap
+    (x : Set.Icc (0 : ℝ) (Real.pi / 2) → Point)
+    (hx : x ⟨0, le_rfl, by positivity⟩ = 0)
+    (hbottom : ∃ p ∈ outerPathConstraintSet x, p 1 = 0)
+    (htop : ∃ p ∈ outerPathConstraintSet x, p 1 = 1) :
+    ∃ K : ConvexBody Point, (K : Set Point) = outerPathConstraintSet x ∧
+      IsCap (Real.pi / 2) K := by
+  have hpi : (0 : ℝ) ≤ Real.pi / 2 := by positivity
+  have hxO : x ⟨0, le_rfl, hpi⟩ = 0 := hx
+  -- half-plane representation with cap-admissible normals
+  obtain ⟨C, hCnormals, hrep⟩ :
+      ∃ C : Set (Real.Angle × ℝ),
+        (∀ c ∈ C, c.1 ∈ ((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles (Real.pi / 2)) ∪
+          capLowerNormals (Real.pi / 2)) ∧
+        outerPathConstraintSet x = ⋂ c ∈ C, normalHalfPlane c.1 c.2 false false := by
+    refine ⟨{(((3 * Real.pi / 2 : ℝ) : Real.Angle), (0 : ℝ))} ∪
+      ((Set.range fun t : Set.Icc (0 : ℝ) (Real.pi / 2) ↦
+          (((t.val : ℝ) : Real.Angle),
+            inner ℝ (x t) (normalVector ((t.val : ℝ) : Real.Angle)) + 1)) ∪
+        (Set.range fun t : Set.Icc (0 : ℝ) (Real.pi / 2) ↦
+          (((t.val + Real.pi / 2 : ℝ) : Real.Angle),
+            inner ℝ (x t) (tangentVector ((t.val : ℝ) : Real.Angle)) + 1))), ?_, ?_⟩
+    · rintro c (rfl | ⟨t, rfl⟩ | ⟨t, rfl⟩)
+      · exact Or.inr (Or.inr rfl)
+      · exact Or.inl ⟨t.val, Or.inl ⟨t.2.1, t.2.2⟩, rfl⟩
+      · exact Or.inl ⟨t.val + Real.pi / 2,
+          Or.inr ⟨by linarith [t.2.1], by linarith [t.2.2]⟩, rfl⟩
+    · ext q
+      simp only [Set.mem_iInter, Set.mem_union, Set.mem_singleton_iff, Set.mem_range]
+      constructor
+      · rintro ⟨hq0, hqt⟩ c (rfl | ⟨t, rfl⟩ | ⟨t, rfl⟩)
+        · change inner ℝ q (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) ≤ 0
+          rw [inner_normalVector_three_pi_div_two]
+          linarith
+        · exact (hqt t).1
+        · change inner ℝ q (normalVector ((t.val + Real.pi / 2 : ℝ) : Real.Angle)) ≤ _
+          rw [normalVector_add_pi_div_two_real]
+          exact (hqt t).2
+      · intro hq
+        refine ⟨?_, fun t ↦ ⟨?_, ?_⟩⟩
+        · have h : inner ℝ q (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) ≤ 0 :=
+            hq _ (Or.inl rfl)
+          rw [inner_normalVector_three_pi_div_two] at h
+          linarith
+        · exact hq _ (Or.inr (Or.inl ⟨t, rfl⟩))
+        · have h : inner ℝ q (normalVector ((t.val + Real.pi / 2 : ℝ) : Real.Angle)) ≤
+              inner ℝ (x t) (tangentVector ((t.val : ℝ) : Real.Angle)) + 1 :=
+            hq _ (Or.inr (Or.inr ⟨t, rfl⟩))
+          rwa [normalVector_add_pi_div_two_real] at h
+  -- the constraints at the two endpoints confine the set to a bounded rectangle
+  have hxbox : ∀ q ∈ outerPathConstraintSet x,
+      x ⟨Real.pi / 2, hpi, le_rfl⟩ 0 - 1 ≤ q 0 ∧ q 0 ≤ 1 ∧ 0 ≤ q 1 ∧ q 1 ≤ 1 := by
+    rintro q ⟨hq0, hqt⟩
+    have h0n : inner ℝ q (normalVector ((0 : ℝ) : Real.Angle)) ≤
+        inner ℝ (x ⟨0, le_rfl, hpi⟩) (normalVector ((0 : ℝ) : Real.Angle)) + 1 :=
+      (hqt ⟨0, le_rfl, hpi⟩).1
+    have h0t : inner ℝ q (tangentVector ((0 : ℝ) : Real.Angle)) ≤
+        inner ℝ (x ⟨0, le_rfl, hpi⟩) (tangentVector ((0 : ℝ) : Real.Angle)) + 1 :=
+      (hqt ⟨0, le_rfl, hpi⟩).2
+    have hTt : inner ℝ q (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) ≤
+        inner ℝ (x ⟨Real.pi / 2, hpi, le_rfl⟩)
+          (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) + 1 :=
+      (hqt ⟨Real.pi / 2, hpi, le_rfl⟩).2
+    rw [hxO, inner_zero_left, inner_normalVector_zero] at h0n
+    rw [hxO, inner_zero_left, inner_tangentVector_zero] at h0t
+    rw [inner_tangentVector_pi_div_two, inner_tangentVector_pi_div_two] at hTt
+    exact ⟨by linarith, by linarith, hq0, by linarith⟩
+  obtain ⟨pb, hpb, hpb1⟩ := hbottom
+  obtain ⟨pt, hpt, hpt1⟩ := htop
+  obtain ⟨K, hKset⟩ : ∃ K : ConvexBody Point, (K : Set Point) = outerPathConstraintSet x := by
+    refine ⟨{ carrier := outerPathConstraintSet x
+              convex' := ?_
+              isCompact' := ?_
+              nonempty' := ⟨pb, hpb⟩ }, rfl⟩
+    · rw [hrep]
+      exact convex_iInter fun c ↦ convex_iInter fun _ ↦ convex_normalHalfPlane c.1 c.2 false
+    · refine Metric.isCompact_iff_isClosed_bounded.2 ⟨?_, ?_⟩
+      · rw [hrep]
+        exact isClosed_iInter fun c ↦ isClosed_iInter fun _ ↦
+          isClosed_normalHalfPlane c.1 c.2 false
+      · refine (EuclideanSpace.isBounded_coordinate_rectangle
+          (x ⟨Real.pi / 2, hpi, le_rfl⟩ 0 - 1) 1 0 1).subset fun q hq ↦ ?_
+        obtain ⟨h1, h2, h3, h4⟩ := hxbox q hq
+        exact ⟨h1, h2, h3, h4⟩
+  -- the strip inclusion and the two contacts fix the four normalized support values
+  have hsvTop : supportValue (K : Set Point) ((Real.pi / 2 : ℝ) : Real.Angle) = 1 := by
+    refine le_antisymm (supportValue_le_of_subset_normalHalfPlane K _ 1 fun q hq ↦ ?_) ?_
+    · change inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) ≤ 1
+      rw [inner_normalVector_pi_div_two]
+      rw [hKset] at hq
+      exact (hxbox q hq).2.2.2
+    · have h := inner_le_supportValue K (hKset ▸ hpt) ((Real.pi / 2 : ℝ) : Real.Angle)
+      rwa [inner_normalVector_pi_div_two, hpt1] at h
+  have hsvBot : supportValue (K : Set Point) ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := by
+    refine le_antisymm (supportValue_le_of_subset_normalHalfPlane K _ 0 fun q hq ↦ ?_) ?_
+    · change inner ℝ q (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) ≤ 0
+      rw [inner_normalVector_three_pi_div_two]
+      rw [hKset] at hq
+      linarith [(hxbox q hq).2.2.1]
+    · have h := inner_le_supportValue K (hKset ▸ hpb) ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+      rwa [inner_normalVector_three_pi_div_two, hpb1, neg_zero] at h
+  have hang : ((Real.pi / 2 + Real.pi : ℝ) : Real.Angle) =
+      ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+    congr 1
+    ring
+  exact ⟨K, hKset, by positivity, le_rfl, hsvTop, hsvTop, by rw [hang]; exact hsvBot, hsvBot,
+    C, hCnormals, hKset.trans hrep⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Reflection
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Exchange the two coordinates of the Euclidean plane. -/
+def coordinateSwap : Point ≃ₗᵢ[ℝ] Point :=
+  LinearIsometryEquiv.piLpCongrLeft 2 ℝ ℝ (Equiv.swap 0 1)
+
+/-- Reflection exchanging the normals at angles zero and `ω + π / 2`. -/
+def capReflection (ω : ℝ) : Point ≃ₗᵢ[ℝ] Point :=
+  coordinateSwap.trans (EuclideanGeometry.o.rotation (ω : Real.Angle))
+
+/-- First coordinate of the cap reflection. -/
+theorem capReflection_apply_zero (ω : ℝ) (p : Point) :
+    capReflection ω p 0 = -Real.sin ω * p 0 + Real.cos ω * p 1 := by
+  rw [capReflection, LinearIsometryEquiv.trans_apply, Orientation.rotation_apply,
+    rightAngleRotation_apply]
+  simp [coordinateSwap]
+  ring
+
+/-- Second coordinate of the cap reflection. -/
+theorem capReflection_apply_one (ω : ℝ) (p : Point) :
+    capReflection ω p 1 = Real.cos ω * p 0 + Real.sin ω * p 1 := by
+  rw [capReflection, LinearIsometryEquiv.trans_apply, Orientation.rotation_apply,
+    rightAngleRotation_apply]
+  simp [coordinateSwap]
+
+/-- Reflection across the line through the upper vertex of the cap strip. -/
+def stripTopReflection (ω : ℝ) (p : Point) : Point :=
+  (2 * inner ℝ p (stripParallelogram ω).2.2 /
+    inner ℝ (stripParallelogram ω).2.2 (stripParallelogram ω).2.2) •
+      (stripParallelogram ω).2.2 - p
+
+/-- The strip-top reflection is the explicit cap reflection. -/
+theorem stripTopReflection_eq_capReflection (ω : ℝ)
+    (hω0 : 0 < ω) (hωle : ω ≤ Real.pi / 2) :
+    stripTopReflection ω = capReflection ω := by
+  funext p
+  by_cases hωtop : ω = Real.pi / 2
+  · subst ω
+    have harg : Real.pi / 4 - (Real.pi / 2) / 2 = 0 := by ring
+    ext i
+    fin_cases i
+    · change stripTopReflection (Real.pi / 2) p 0 = capReflection (Real.pi / 2) p 0
+      rw [capReflection_apply_zero]
+      simp [stripTopReflection, stripParallelogram, harg, PiLp.inner_apply,
+        Fin.sum_univ_two]
+    · change stripTopReflection (Real.pi / 2) p 1 = capReflection (Real.pi / 2) p 1
+      rw [capReflection_apply_one]
+      have hnorm : ‖(!₂[0, 1] : Point)‖ ^ 2 = 1 := by
+        simpa [Fin.sum_univ_two] using
+          (EuclideanSpace.norm_sq_eq (!₂[0, 1] : Point))
+      simp [stripTopReflection, stripParallelogram, harg, PiLp.inner_apply,
+        Fin.sum_univ_two, hnorm]
+      ring
+  · have hωlt : ω < Real.pi / 2 := lt_of_le_of_ne hωle hωtop
+    let C := Real.cos ω
+    let S := Real.sin ω
+    let c := Real.tan (Real.pi / 4 - ω / 2)
+    have hC : 0 < C :=
+      Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos, hω0], hωlt⟩
+    have hgap : c = C⁻¹ - Real.tan ω := by
+      simpa [c, show Real.pi / 4 - ω / 2 = (Real.pi / 2 - ω) / 2 by ring]
+        using Real.tan_pi_div_two_sub_div_two ω ⟨hω0.le, hωlt⟩
+    have htan : Real.tan ω = S / C := Real.tan_eq_sin_div_cos ω
+    have hcC : c * C = 1 - S := by
+      rw [hgap, htan]
+      field_simp [hC.ne']
+    have hc : c * (1 + S) = C := by
+      rw [hgap, htan]
+      field_simp [hC.ne']
+      nlinarith [hcC, Real.sin_sq_add_cos_sq ω]
+    have hcsq : c * c * (1 + S) = 1 - S := by
+      calc
+        c * c * (1 + S) = c * (c * (1 + S)) := by ring
+        _ = c * C := by rw [hc]
+        _ = 1 - S := hcC
+    have hcoef₀ : c * c - 1 = -S * (c * c + 1) := by
+      nlinarith [hcsq]
+    have hbase : 2 = (1 + S) * (c * c + 1) := by
+      nlinarith [hcoef₀]
+    have hcoef₁ : 2 * c = C * (c * c + 1) := by
+      rw [← hc]
+      linear_combination c * hbase
+    have hcoef₂ : 1 - c * c = S * (c * c + 1) := by
+      nlinarith [hcsq]
+    have hden : c * c + 1 ≠ 0 := by nlinarith [sq_nonneg c]
+    ext i
+    fin_cases i
+    · change stripTopReflection ω p 0 = capReflection ω p 0
+      rw [capReflection_apply_zero]
+      simp only [stripTopReflection, stripParallelogram, PiLp.inner_apply,
+        Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one,
+        Real.inner_apply, PiLp.smul_apply, PiLp.sub_apply, smul_eq_mul]
+      simp only [mul_one]
+      change 2 * (p 0 * c + p 1) / (c * c + 1) * c - p 0 =
+        -S * p 0 + C * p 1
+      field_simp [hden]
+      linear_combination (p 0) * hcoef₀ + (p 1) * hcoef₁
+    · change stripTopReflection ω p 1 = capReflection ω p 1
+      rw [capReflection_apply_one]
+      simp only [stripTopReflection, stripParallelogram, PiLp.inner_apply,
+        Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one,
+        Real.inner_apply, PiLp.smul_apply, PiLp.sub_apply, smul_eq_mul]
+      simp only [mul_one]
+      change 2 * (p 0 * c + p 1) / (c * c + 1) - p 1 =
+        C * p 0 + S * p 1
+      field_simp [hden]
+      linear_combination (p 0) * hcoef₁ + (p 1) * hcoef₂
+
+/-- The strip-top reflection fixes the upper strip vertex. -/
+theorem stripTopReflection_stripTop (ω : ℝ) :
+    stripTopReflection ω (stripParallelogram ω).2.2 =
+      (stripParallelogram ω).2.2 := by
+  have ho : (stripParallelogram ω).2.2 ≠ (0 : Point) := by
+    intro h
+    have h1 := congrArg (fun p : Point ↦ p 1) h
+    simp [stripParallelogram] at h1
+  unfold stripTopReflection
+  have hden : inner ℝ (stripParallelogram ω).2.2
+      (stripParallelogram ω).2.2 ≠ 0 := inner_self_ne_zero.mpr ho
+  rw [mul_div_assoc, div_self hden]
+  module
+
+/-- The cap reflection is an involution. -/
+theorem capReflection_involutive (ω : ℝ) (p : Point) :
+    capReflection ω (capReflection ω p) = p := by
+  ext i
+  fin_cases i
+  · change capReflection ω (capReflection ω p) 0 = p 0
+    rw [capReflection_apply_zero, capReflection_apply_zero, capReflection_apply_one]
+    linear_combination (p 0) * (Real.sin_sq_add_cos_sq ω)
+  · change capReflection ω (capReflection ω p) 1 = p 1
+    rw [capReflection_apply_one, capReflection_apply_zero, capReflection_apply_one]
+    linear_combination (p 1) * (Real.sin_sq_add_cos_sq ω)
+
+/-- Reflection of a normal angle across the cap-reflection axis. -/
+def reflectedAngle (ω : ℝ) (a : Real.Angle) : Real.Angle :=
+  ((ω + Real.pi / 2 : ℝ) : Real.Angle) - a
+
+/-- Reflection of normal angles is involutive. -/
+theorem reflectedAngle_involutive (ω : ℝ) (a : Real.Angle) :
+    reflectedAngle ω (reflectedAngle ω a) = a := by
+  simp only [reflectedAngle]
+  abel
+
+/-- Coercion formula for a reflected real angle. -/
+theorem reflectedAngle_coe (ω t : ℝ) :
+    reflectedAngle ω (t : Real.Angle) =
+      ((ω + Real.pi / 2 - t : ℝ) : Real.Angle) := by
+  simp only [reflectedAngle, Real.Angle.coe_sub, Real.Angle.coe_add]
+
+/-- The cap reflection transports normal vectors at real angles. -/
+theorem capReflection_normalVector (ω a : ℝ) :
+    capReflection ω (normalVector (a : Real.Angle)) =
+      normalVector ((ω + Real.pi / 2 - a : ℝ) : Real.Angle) := by
+  have hrho : (ω : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) -
+      (a : Real.Angle) = ((ω + Real.pi / 2 - a : ℝ) : Real.Angle) := by
+    rw [← Real.Angle.coe_add, ← Real.Angle.coe_sub]
+  ext i
+  fin_cases i
+  · change capReflection ω (normalVector (a : Real.Angle)) 0 =
+      normalVector ((ω + Real.pi / 2 - a : ℝ) : Real.Angle) 0
+    rw [capReflection_apply_zero, ← hrho]
+    simp only [Fin.isValue, normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe,
+      Matrix.cons_val_zero, neg_mul, Matrix.cons_val_one, Matrix.cons_val_fin_one]
+    rw [hrho, Real.Angle.cos_coe, Real.cos_sub, Real.sin_add, Real.cos_add]
+    simp
+  · change capReflection ω (normalVector (a : Real.Angle)) 1 =
+      normalVector ((ω + Real.pi / 2 - a : ℝ) : Real.Angle) 1
+    rw [capReflection_apply_one, ← hrho]
+    simp only [Fin.isValue, normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe,
+      Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one]
+    rw [hrho, Real.Angle.sin_coe, Real.sin_sub, Real.sin_add, Real.cos_add]
+    simp
+
+/-- The cap reflection reverses tangent vectors at real angles. -/
+theorem capReflection_tangentVector (ω a : ℝ) :
+    capReflection ω (tangentVector (a : Real.Angle)) =
+      -tangentVector ((ω + Real.pi / 2 - a : ℝ) : Real.Angle) := by
+  have hrho : (ω : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) -
+      (a : Real.Angle) = ((ω + Real.pi / 2 - a : ℝ) : Real.Angle) := by
+    rw [← Real.Angle.coe_add, ← Real.Angle.coe_sub]
+  ext i
+  fin_cases i
+  · change capReflection ω (tangentVector (a : Real.Angle)) 0 =
+      (-tangentVector ((ω + Real.pi / 2 - a : ℝ) : Real.Angle)) 0
+    rw [capReflection_apply_zero, ← hrho]
+    simp only [Fin.isValue, tangentVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe,
+      Matrix.cons_val_zero, mul_neg, neg_mul, neg_neg, Matrix.cons_val_one,
+      Matrix.cons_val_fin_one, PiLp.neg_apply]
+    rw [hrho, Real.Angle.sin_coe, Real.sin_sub, Real.sin_add, Real.cos_add]
+    simp
+    ring
+  · change capReflection ω (tangentVector (a : Real.Angle)) 1 =
+      (-tangentVector ((ω + Real.pi / 2 - a : ℝ) : Real.Angle)) 1
+    rw [capReflection_apply_one, ← hrho]
+    simp only [Fin.isValue, tangentVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe,
+      Matrix.cons_val_zero, mul_neg, Matrix.cons_val_one, Matrix.cons_val_fin_one,
+      PiLp.neg_apply]
+    rw [hrho, Real.Angle.cos_coe, Real.cos_sub, Real.sin_add, Real.cos_add]
+    simp
+
+/-- The cap reflection transports normal vectors by reflected angles. -/
+theorem capReflection_normalVector_angle (ω : ℝ) (a : Real.Angle) :
+    capReflection ω (normalVector a) = normalVector (reflectedAngle ω a) := by
+  calc
+    capReflection ω (normalVector a) =
+        capReflection ω (normalVector (a.toReal : Real.Angle)) := by rw [a.coe_toReal]
+    _ = normalVector ((ω + Real.pi / 2 - a.toReal : ℝ) : Real.Angle) :=
+      capReflection_normalVector ω a.toReal
+    _ = normalVector (reflectedAngle ω a) := by
+      congr 1
+      simp only [reflectedAngle, Real.Angle.coe_sub, Real.Angle.coe_add, a.coe_toReal]
+
+/-- The cap reflection reverses tangent vectors at reflected angles. -/
+theorem capReflection_tangentVector_angle (ω : ℝ) (a : Real.Angle) :
+    capReflection ω (tangentVector a) = -tangentVector (reflectedAngle ω a) := by
+  calc
+    capReflection ω (tangentVector a) =
+        capReflection ω (tangentVector (a.toReal : Real.Angle)) := by rw [a.coe_toReal]
+    _ = -tangentVector ((ω + Real.pi / 2 - a.toReal : ℝ) : Real.Angle) :=
+      capReflection_tangentVector ω a.toReal
+    _ = -tangentVector (reflectedAngle ω a) := by
+      congr 2
+      simp only [reflectedAngle, Real.Angle.coe_sub, Real.Angle.coe_add, a.coe_toReal]
+
+/-- Inner products with normals transform under the cap reflection. -/
+theorem inner_capReflection_normalVector (ω : ℝ) (p : Point)
+    (a : Real.Angle) :
+    inner ℝ (capReflection ω p) (normalVector a) =
+      inner ℝ p (normalVector (reflectedAngle ω a)) := by
+  have hn : capReflection ω (normalVector (reflectedAngle ω a)) =
+      normalVector a := by
+    rw [capReflection_normalVector_angle, reflectedAngle_involutive]
+  rw [← hn]
+  exact (capReflection ω).inner_map_map p (normalVector (reflectedAngle ω a))
+
+/-- Inner products with tangents transform under the cap reflection. -/
+theorem inner_capReflection_tangentVector (ω : ℝ) (p : Point)
+    (a : Real.Angle) :
+    inner ℝ (capReflection ω p) (tangentVector a) =
+      -inner ℝ p (tangentVector (reflectedAngle ω a)) := by
+  have ht : capReflection ω (-tangentVector (reflectedAngle ω a)) =
+      tangentVector a := by
+    rw [map_neg, capReflection_tangentVector_angle, reflectedAngle_involutive,
+      neg_neg]
+  rw [← ht, (capReflection ω).inner_map_map, inner_neg_right]
+
+/-- The cap reflection transports every open or closed normal half-plane. -/
+theorem capReflection_image_normalHalfPlane (ω h : ℝ)
+    (a : Real.Angle) (upper strict : Bool) :
+    capReflection ω '' normalHalfPlane a h upper strict =
+      normalHalfPlane (reflectedAngle ω a) h upper strict := by
+  ext p
+  constructor
+  · rintro ⟨q, hq, rfl⟩
+    cases upper <;> cases strict <;>
+      simp only [normalHalfPlane, Bool.false_eq_true, ↓reduceIte,
+        Set.mem_ofPred_eq] at hq ⊢ <;>
+      rwa [inner_capReflection_normalVector, reflectedAngle_involutive]
+  · intro hp
+    refine ⟨capReflection ω p, ?_, capReflection_involutive ω p⟩
+    cases upper <;> cases strict <;>
+      simp only [normalHalfPlane, Bool.false_eq_true, ↓reduceIte,
+        Set.mem_ofPred_eq] at hp ⊢ <;>
+      rwa [inner_capReflection_normalVector]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Supporting Hallway
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private theorem supportValue_le_of_subset_rotatedHallway (s : Set Point) (t : Real.Angle)
+    (hs : s.Nonempty) (v : Point)
+    (hv : s ⊆ (fun p ↦ rotationMap t p + v) '' hallway) :
+    supportValue s t ≤ 1 + inner ℝ v (normalVector t) ∧
+      supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) ≤
+        1 + inner ℝ v (tangentVector t) := by
+  have hbound₁ : supportValue s t ≤ 1 + inner ℝ v (normalVector t) := by
+    apply csSup_le (hs.image _)
+    rintro _ ⟨p, hp, rfl⟩
+    obtain ⟨q, hq, rfl⟩ := hv hp
+    dsimp only
+    rw [inner_add_left, inner_rotationMap_normalVector]
+    linarith [(mem_hallway_iff q).mp hq |>.1.1]
+  have hbound₂ : supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) ≤
+      1 + inner ℝ v (tangentVector t) := by
+    apply csSup_le (hs.image _)
+    rintro _ ⟨p, hp, rfl⟩
+    obtain ⟨q, hq, rfl⟩ := hv hp
+    dsimp only
+    rw [normalVector_add_pi_div_two, inner_add_left, inner_rotationMap_tangentVector]
+    linarith [(mem_hallway_iff q).mp hq |>.1.2]
+  exact ⟨hbound₁, hbound₂⟩
+
+theorem subset_supportingHallway (s : Set Point) (t : Real.Angle)
+    (hs : s.Nonempty) (hc : IsCompact s)
+    (hL : ∃ v : Point, s ⊆ (fun p ↦ rotationMap t p + v) '' hallway) :
+    s ⊆ supportingHallway s t := by
+  rcases hL with ⟨v, hv⟩
+  obtain ⟨hbound₁, hbound₂⟩ := supportValue_le_of_subset_rotatedHallway s t hs v hv
+  intro p hp
+  obtain ⟨x, hx⟩ := (EuclideanGeometry.o.rotation t).surjective
+    (p - (supportValue s t - 1) • normalVector t -
+      (supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) - 1) • tangentVector t)
+  have hxp : supportingPlacement s t x = p := by
+    simp only [supportingPlacement, rotationMap, hx]
+    abel
+  have hx₀ := inner_supportingPlacement_normalVector s t x
+  have hx₁ := inner_supportingPlacement_tangentVector s t x
+  rw [hxp] at hx₀ hx₁
+  have hsup (w : Real.Angle) : inner ℝ p (normalVector w) ≤ supportValue s w :=
+    le_csSup (hc.bddAbove_image (continuous_id.inner continuous_const).continuousOn)
+      ⟨p, hp, rfl⟩
+  have hupper₀ := hsup t
+  have hupper₁ := hsup (t + ((Real.pi / 2 : ℝ) : Real.Angle))
+  rw [normalVector_add_pi_div_two] at hupper₁
+  refine ⟨x, (mem_hallway_iff x).mpr ⟨⟨by linarith, by linarith⟩, ?_⟩, hxp⟩
+  obtain ⟨q, hq, hqp⟩ := hv hp
+  rcases (mem_hallway_iff q).mp hq |>.2 with hq₀ | hq₁
+  · have hcoord : inner ℝ p (normalVector t) = q 0 + inner ℝ v (normalVector t) := by
+      rw [← hqp, inner_add_left, inner_rotationMap_normalVector]
+    exact Or.inl (by linarith)
+  · have hcoord : inner ℝ p (tangentVector t) = q 1 + inner ℝ v (tangentVector t) := by
+      rw [← hqp, inner_add_left, inner_rotationMap_tangentVector]
+    exact Or.inr (by linarith)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Analysis.SurfaceMeasure.UpperGraph`.
+* `Analysis.SurfaceMeasure.ExposedFrontier`.
+* `Analysis.SurfaceMeasure.RegularBoundaryHelpers`.
+* `Analysis.SurfaceMeasure.GraphIntegral`.
+* `Analysis.SurfaceMeasure.Construction`.
+* `Analysis.SurfaceMeasure.GraphConvergence`.
+* `Analysis.SurfaceMeasure.Properties`.
+* `Analysis.SurfaceMeasure.BoundaryExtension`.
+* `Analysis.SurfaceMeasure.Opposite`.
+* `Analysis.SurfaceMeasure.WeakConvergence`.
+* `Analysis.SurfaceMeasure.AtomLimits`.
+* `Analysis.SurfaceMeasure.WeightedBoundary`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Upper Graph
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+open scoped Pointwise
+
+namespace MovingSofa
+
+private def negCoordinates (e : Point ≃ₗᵢ[ℝ] Point) : Point ≃ₗᵢ[ℝ] Point :=
+  e.trans (LinearIsometryEquiv.neg ℝ)
+
+private def swapCoordinates (e : Point ≃ₗᵢ[ℝ] Point) : Point ≃ₗᵢ[ℝ] Point :=
+  e.trans coordinateSwap
+
+@[simp] private theorem negCoordinates_apply (e : Point ≃ₗᵢ[ℝ] Point) (p : Point) (i : Fin 2) :
+    negCoordinates e p i = -e p i := by
+  simp [negCoordinates]
+
+@[simp] private theorem swapCoordinates_apply_zero (e : Point ≃ₗᵢ[ℝ] Point)
+    (p : Point) : swapCoordinates e p 0 = e p 1 := by
+  simp [swapCoordinates, coordinateSwap]
+
+@[simp] private theorem swapCoordinates_apply_one (e : Point ≃ₗᵢ[ℝ] Point)
+    (p : Point) : swapCoordinates e p 1 = e p 0 := by
+  simp [swapCoordinates, coordinateSwap]
+
+/-- The upper graph height is the greatest vertical coordinate in its fiber. -/
+theorem upperGraphHeight_isGreatest (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) {x : ℝ} (hx : x ∈ horizontalProjection K o e) :
+    IsGreatest {y : ℝ | o + e.symm !₂[x, y] ∈ (K : Set Point)}
+      (upperGraphHeight K o e x) := by
+  let F : Set ℝ := {y | o + e.symm !₂[x, y] ∈ (K : Set Point)}
+  let Y : Set ℝ := (fun p : Point ↦ e (p - o) 1) '' (K : Set Point)
+  have hY : IsCompact Y := K.isCompact.image (by fun_prop)
+  have hFc : IsClosed F := K.isClosed.preimage (by fun_prop)
+  have hFY : F ⊆ Y := by
+    intro y hy
+    refine ⟨o + e.symm !₂[x, y], hy, ?_⟩
+    simp
+  have hF : IsCompact F := hY.of_isClosed_subset hFc hFY
+  have hFne : F.Nonempty := by
+    obtain ⟨p, hp, hpx⟩ := hx
+    refine ⟨e (p - o) 1, ?_⟩
+    dsimp [F]
+    convert hp using 1
+    apply sub_eq_zero.mp
+    apply e.injective
+    ext i
+    fin_cases i
+    · simp only [map_sub, PiLp.sub_apply] at hpx
+      simp
+      linarith
+    · simp
+  change IsGreatest F (sSup F)
+  exact hF.isGreatest_sSup hFne
+
+/-- An attained upper graph point belongs to the convex body. -/
+theorem upperGraphHeight_mem (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) {x : ℝ} (hx : x ∈ horizontalProjection K o e) :
+    o + e.symm !₂[x, upperGraphHeight K o e x] ∈ (K : Set Point) :=
+  (upperGraphHeight_isGreatest K o e hx).1
+
+private theorem upperGraphHeight_eq_coordinate_of_isExteriorNormal_of_pos
+    (K : ConvexBody Point) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) {p : Point} (hp : p ∈ K)
+    {a : Real.Angle} (ha : IsExteriorNormal K p a) (hpos : 0 < e (normalVector a) 1) :
+    upperGraphHeight K o e (e (p - o) 0) = e (p - o) 1 := by
+  have hx : e (p - o) 0 ∈ horizontalProjection K o e := ⟨p, hp, rfl⟩
+  apply le_antisymm
+  · apply le_of_not_gt
+    intro hgt
+    have hmem := upperGraphHeight_mem K o e hx
+    have hs := ha (o + e.symm !₂[e (p - o) 0,
+      upperGraphHeight K o e (e (p - o) 0)]) hmem
+    rw [← e.inner_map_map] at hs
+    simp only [map_sub, map_add, LinearIsometryEquiv.apply_symm_apply, PiLp.inner_apply,
+      RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Matrix.cons_val_zero,
+      Matrix.cons_val_one, PiLp.sub_apply, PiLp.add_apply] at hs
+    have hzero : e (p - o) 0 = e p 0 - e o 0 := by simp
+    have hone : e (p - o) 1 = e p 1 - e o 1 := by simp
+    ring_nf at hs
+    rw [hzero, hone] at hgt
+    rw [show -(e o 0) + e p 0 = e p 0 - e o 0 by ring] at hs
+    have hnonpos : e (normalVector a) 1 *
+        (upperGraphHeight K o e (e p 0 - e o 0) - (e p 1 - e o 1)) ≤ 0 := by
+      nlinarith [hs]
+    exact (not_lt_of_ge hnonpos) (mul_pos hpos (sub_pos.mpr hgt))
+  · exact (upperGraphHeight_isGreatest K o e hx).2 (by
+      change o + e.symm !₂[e (p - o) 0, e (p - o) 1] ∈ (K : Set Point)
+      rw [show o + e.symm !₂[e (p - o) 0, e (p - o) 1] = p by
+        apply e.injective
+        ext i
+        fin_cases i <;> simp [map_sub]]
+      exact hp)
+
+/-- A boundary point whose exterior normal has positive vertical coordinate lies
+on the upper coordinate graph. -/
+theorem eq_upperCoordinateGraph_of_isExteriorNormal_of_pos
+    (K : ConvexBody Point) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) {p : Point} (hp : p ∈ K)
+    {a : Real.Angle} (ha : IsExteriorNormal K p a) (hpos : 0 < e (normalVector a) 1) :
+    p = o + e.symm !₂[e (p - o) 0, upperGraphHeight K o e (e (p - o) 0)] := by
+  rw [upperGraphHeight_eq_coordinate_of_isExteriorNormal_of_pos K o e hp ha hpos]
+  apply e.injective
+  ext i
+  fin_cases i <;> simp [map_sub]
+
+private theorem eq_upperCoordinateGraph_of_maximal_verticalCoordinate
+    (K : ConvexBody Point) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) {p : Point} (hp : p ∈ K)
+    (hmax : ∀ q ∈ K, e (q - o) 1 ≤ e (p - o) 1) :
+    p = o + e.symm !₂[e (p - o) 0, upperGraphHeight K o e (e (p - o) 0)] := by
+  have hx : e (p - o) 0 ∈ horizontalProjection K o e := ⟨p, hp, rfl⟩
+  have hpGreatest : IsGreatest
+      {y : ℝ | o + e.symm !₂[e (p - o) 0, y] ∈ (K : Set Point)} (e (p - o) 1) := by
+    constructor
+    · change o + e.symm !₂[e (p - o) 0, e (p - o) 1] ∈ (K : Set Point)
+      rw [show o + e.symm !₂[e (p - o) 0, e (p - o) 1] = p by
+        apply e.injective
+        ext i
+        fin_cases i <;> simp [map_sub]]
+      exact hp
+    · intro y hy
+      have := hmax (o + e.symm !₂[e (p - o) 0, y]) hy
+      simpa using this
+  have heq := (upperGraphHeight_isGreatest K o e hx).unique hpGreatest
+  rw [heq]
+  apply e.injective
+  ext i
+  fin_cases i <;> simp [map_sub]
+
+private theorem horizontalCoordinate_le_rightBound (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) {p : Point} (hp : p ∈ K) :
+    e (p - o) 0 ≤ (horizontalBounds K o e).2 := by
+  exact (K.isCompact.image (by fun_prop)).isGreatest_sSup (K.nonempty.image _)
+    |>.2 ⟨p, hp, rfl⟩
+
+private theorem horizontalProjection_negCoordinates (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) :
+    horizontalProjection K o (negCoordinates e) = -(horizontalProjection K o e) := by
+  ext x
+  simp only [horizontalProjection, negCoordinates, LinearIsometryEquiv.trans_apply, map_sub,
+    LinearIsometryEquiv.coe_neg, Fin.isValue, PiLp.sub_apply, PiLp.neg_apply, Set.mem_image,
+    SetLike.mem_coe, Set.mem_neg]
+  constructor
+  · rintro ⟨p, hp, h⟩
+    exact ⟨p, hp, by linarith⟩
+  · rintro ⟨p, hp, h⟩
+    exact ⟨p, hp, by linarith⟩
+
+private theorem horizontalBounds_negCoordinates (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) :
+    horizontalBounds K o (negCoordinates e) =
+      (-(horizontalBounds K o e).2, -(horizontalBounds K o e).1) := by
+  simp only [horizontalBounds, horizontalProjection_negCoordinates, Real.sInf_neg,
+    Real.sSup_neg]
+
+private theorem leftBound_le_horizontalCoordinate (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) {p : Point} (hp : p ∈ K) :
+    (horizontalBounds K o e).1 ≤ e (p - o) 0 := by
+  exact (K.isCompact.image (by fun_prop)).isLeast_sInf (K.nonempty.image _)
+    |>.2 ⟨p, hp, rfl⟩
+
+private theorem eq_swapUpperGraph_of_horizontalCoordinate_eq_rightBound
+    (K : ConvexBody Point) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) {p : Point} (hp : p ∈ K)
+    (hpr : e (p - o) 0 = (horizontalBounds K o e).2) :
+    p = o + (swapCoordinates e).symm
+      !₂[swapCoordinates e (p - o) 0,
+        upperGraphHeight K o (swapCoordinates e) (swapCoordinates e (p - o) 0)] := by
+  apply eq_upperCoordinateGraph_of_maximal_verticalCoordinate K o (swapCoordinates e) hp
+  intro q hq
+  have hqle := horizontalCoordinate_le_rightBound K o e hq
+  simp only [swapCoordinates_apply_one]
+  simp only [map_sub, PiLp.sub_apply] at hpr hqle ⊢
+  linarith
+
+private theorem eq_negSwapUpperGraph_of_horizontalCoordinate_eq_leftBound
+    (K : ConvexBody Point) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) {p : Point} (hp : p ∈ K)
+    (hpl : e (p - o) 0 = (horizontalBounds K o e).1) :
+    p = o + (negCoordinates (swapCoordinates e)).symm
+      !₂[negCoordinates (swapCoordinates e) (p - o) 0,
+        upperGraphHeight K o (negCoordinates (swapCoordinates e))
+          (negCoordinates (swapCoordinates e) (p - o) 0)] := by
+  apply eq_upperCoordinateGraph_of_maximal_verticalCoordinate K o
+    (negCoordinates (swapCoordinates e)) hp
+  intro q hq
+  have hle := leftBound_le_horizontalCoordinate K o e hq
+  simp only [negCoordinates_apply, swapCoordinates_apply_one]
+  simp only [map_sub, PiLp.sub_apply] at hpl hle ⊢
+  linarith
+
+private def coordinateCornerSet (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) : Set Point :=
+  {p | e (p - o) 0 ∈ ({(horizontalBounds K o e).1,
+      (horizontalBounds K o e).2} : Set ℝ) ∧
+    e (p - o) 1 ∈ ({(horizontalBounds K o (swapCoordinates e)).1,
+      (horizontalBounds K o (swapCoordinates e)).2} : Set ℝ)}
+
+private theorem coordinateCornerSet_finite (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) : (coordinateCornerSet K o e).Finite := by
+  let coords : Point → ℝ × ℝ := fun p ↦ (e (p - o) 0, e (p - o) 1)
+  let xs : Set ℝ := {(horizontalBounds K o e).1, (horizontalBounds K o e).2}
+  let ys : Set ℝ := {(horizontalBounds K o (swapCoordinates e)).1,
+    (horizontalBounds K o (swapCoordinates e)).2}
+  have hcoords : Function.Injective coords := by
+    intro p q hpq
+    apply sub_left_injective (b := o)
+    apply e.injective
+    ext i
+    fin_cases i
+    · exact congrArg Prod.fst hpq
+    · exact congrArg Prod.snd hpq
+  have hfinite : (xs ×ˢ ys).Finite := Set.toFinite xs |>.prod (Set.toFinite ys)
+  have hpre : (coords ⁻¹' (xs ×ˢ ys)).Finite := hfinite.preimage hcoords.injOn
+  have heq : coordinateCornerSet K o e = coords ⁻¹' (xs ×ˢ ys) := by
+    ext p
+    simp only [coordinateCornerSet, coords, xs, ys, Set.mem_ofPred_eq, Set.mem_preimage,
+      Set.mem_prod, Set.mem_insert_iff, Set.mem_singleton_iff]
+  rw [heq]
+  exact hpre
+
+private theorem hausdorffMeasure_coordinateCornerSet_eq_zero (K : ConvexBody Point)
+    (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) :
+    Measure.hausdorffMeasure 1 (coordinateCornerSet K o e) = 0 := by
+  let _ := MeasureTheory.Measure.nullSingletonClass_hausdorff Point (by norm_num : (0 : ℝ) < 1)
+  exact (coordinateCornerSet_finite K o e).measure_zero _
+
+/-- The upper height function of a convex body is concave on its projection. -/
+theorem concaveOn_upperGraphHeight (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) :
+    ConcaveOn ℝ (horizontalProjection K o e) (upperGraphHeight K o e) := by
+  have hproj : Convex ℝ (horizontalProjection K o e) := by
+    intro x hx y hy a b ha hb hab
+    obtain ⟨p, hp, hpx⟩ := hx
+    obtain ⟨q, hq, hqx⟩ := hy
+    refine ⟨a • p + b • q, K.convex hp hq ha hb hab, ?_⟩
+    simp only [map_sub, map_add, map_smul, PiLp.sub_apply, PiLp.add_apply,
+      PiLp.smul_apply] at hpx hqx ⊢
+    linear_combination a * hpx + b * hqx + e o 0 * hab
+  refine ⟨hproj, ?_⟩
+  · intro x hx y hy a b ha hb hab
+    apply (upperGraphHeight_isGreatest K o e (by
+      exact hproj hx hy ha hb hab)).2
+    have hpx := upperGraphHeight_mem K o e hx
+    have hpy := upperGraphHeight_mem K o e hy
+    have hconv := K.convex hpx hpy ha hb hab
+    change o + e.symm !₂[a • x + b • y,
+      a • upperGraphHeight K o e x + b • upperGraphHeight K o e y] ∈ (K : Set Point)
+    rw [show o + e.symm !₂[a • x + b • y,
+        a • upperGraphHeight K o e x + b • upperGraphHeight K o e y] =
+        a • (o + e.symm !₂[x, upperGraphHeight K o e x]) +
+          b • (o + e.symm !₂[y, upperGraphHeight K o e y]) by
+      apply e.injective
+      ext i
+      fin_cases i <;> simp [map_add, map_smul] <;>
+        linear_combination -(e o _) * hab]
+    exact hconv
+
+/-- The horizontal projection is the interval between its compact extrema. -/
+theorem horizontalProjection_eq_Icc (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) :
+    horizontalProjection K o e =
+      Set.Icc (horizontalBounds K o e).1 (horizontalBounds K o e).2 := by
+  let s := horizontalProjection K o e
+  have hscompact : IsCompact s := K.isCompact.image (by fun_prop)
+  have hsne : s.Nonempty := K.nonempty.image _
+  have hsconv : Convex ℝ s := (concaveOn_upperGraphHeight K o e).1
+  change s = Set.Icc (sInf s) (sSup s)
+  apply Set.Subset.antisymm
+  · exact hscompact.isBounded.subset_Icc_sInf_sSup
+  · have hle := (hscompact.isLeast_sInf hsne).2 (hscompact.sSup_mem hsne)
+    rw [← Set.uIcc_of_le hle, ← segment_eq_uIcc]
+    exact hsconv.segment_subset (hscompact.sInf_mem hsne) (hscompact.sSup_mem hsne)
+
+/-- The upper boundary height is locally Lipschitz inside its projection interval. -/
+theorem locallyLipschitzOn_upperGraphHeight (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) :
+    LocallyLipschitzOn
+      (Set.Ioo (horizontalBounds K o e).1 (horizontalBounds K o e).2)
+      (upperGraphHeight K o e) := by
+  have hlip := (concaveOn_upperGraphHeight K o e).locallyLipschitzOn_interior
+  rw [horizontalProjection_eq_Icc K o e, interior_Icc] at hlip
+  exact hlip
+
+/-- The upward normal determined by the derivative of an upper graph is exterior. -/
+theorem upperGraph_deriv_isExteriorNormal (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) {x : ℝ}
+    (hx : x ∈ Set.Ioo (horizontalBounds K o e).1 (horizontalBounds K o e).2)
+    (hdiff : DifferentiableAt ℝ (upperGraphHeight K o e) x) :
+    IsExteriorNormal K (o + e.symm !₂[x, upperGraphHeight K o e x])
+      (vectorNormalAngle (e.symm !₂[-deriv (upperGraphHeight K o e) x, 1])) := by
+  let g := upperGraphHeight K o e
+  let q : Point := e.symm !₂[-deriv g x, 1]
+  have hq : q ≠ 0 := by
+    intro hzero
+    have := congrFun (congrArg WithLp.ofLp (congrArg e hzero)) 1
+    simp [q] at this
+  change IsExteriorNormal K (o + e.symm !₂[x, g x]) (vectorNormalAngle q)
+  rw [IsExteriorNormal]
+  rw [normalVector_vectorNormalAngle hq]
+  intro p hp
+  have hxp : e (p - o) 0 ∈ horizontalProjection K o e := ⟨p, hp, rfl⟩
+  have hxproj : x ∈ horizontalProjection K o e := by
+    rw [horizontalProjection_eq_Icc K o e]
+    exact ⟨hx.1.le, hx.2.le⟩
+  have hvertical : e (p - o) 1 ≤ g (e (p - o) 0) :=
+    (upperGraphHeight_isGreatest K o e hxp).2 (by
+      change o + e.symm !₂[e (p - o) 0, e (p - o) 1] ∈ (K : Set Point)
+      rw [show o + e.symm !₂[e (p - o) 0, e (p - o) 1] = p by
+        apply e.injective
+        ext i
+        fin_cases i <;> simp [map_sub]]
+      exact hp)
+  have htangent := ConcaveOn.le_add_deriv_mul_sub
+    (concaveOn_upperGraphHeight K o e) hxproj hdiff hxp
+  have hvertical' : e p 1 - e o 1 ≤ g (e p 0 - e o 0) := by
+    simpa only [map_sub, PiLp.sub_apply] using hvertical
+  have htangent' : g (e p 0 - e o 0) ≤
+      g x + deriv g x * ((e p 0 - e o 0) - x) := by
+    simpa only [map_sub, PiLp.sub_apply] using htangent
+  rw [inner_smul_right]
+  apply mul_nonpos_of_nonneg_of_nonpos (inv_nonneg.mpr (norm_nonneg q))
+  rw [← e.inner_map_map]
+  simp only [map_sub, map_add, LinearIsometryEquiv.apply_symm_apply, q, PiLp.inner_apply,
+    RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Matrix.cons_val_zero,
+    Matrix.cons_val_one, PiLp.sub_apply, PiLp.add_apply]
+  dsimp only [g] at hvertical' htangent' ⊢
+  nlinarith [hvertical', htangent']
+
+/-- Angular normal vectors have norm one. -/
+theorem norm_normalVector (a : Real.Angle) : ‖normalVector a‖ = 1 := by
+  induction a using Real.Angle.induction_on with
+  | _ a =>
+    rw [EuclideanSpace.norm_eq]
+    simp [normalVector, frame, Fin.sum_univ_two]
+
+private theorem horizontalCoordinate_eq_endpoint_of_isExteriorNormal_of_vertical_eq_zero
+    (K : ConvexBody Point) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) {p : Point} (hp : p ∈ K)
+    {a : Real.Angle} (ha : IsExteriorNormal K p a)
+    (hzero : e (normalVector a) 1 = 0) :
+    e (p - o) 0 = (horizontalBounds K o e).1 ∨
+      e (p - o) 0 = (horizontalBounds K o e).2 := by
+  have hxne : e (normalVector a) 0 ≠ 0 := by
+    intro hx
+    have he : e (normalVector a) = 0 := by
+      ext i
+      fin_cases i <;> simp [hx, hzero]
+    have hn : ‖e (normalVector a)‖ = 1 := by rw [e.norm_map, norm_normalVector]
+    simp [he] at hn
+  rcases lt_or_gt_of_ne hxne with hneg | hpos
+  · left
+    have hleast : IsLeast (horizontalProjection K o e) (e (p - o) 0) := by
+      refine ⟨⟨p, hp, rfl⟩, ?_⟩
+      rintro x ⟨q, hq, rfl⟩
+      have hs := ha q hq
+      rw [← e.inner_map_map] at hs
+      simp only [map_sub, PiLp.inner_apply, RCLike.inner_apply, conj_trivial,
+        Fin.sum_univ_two, PiLp.sub_apply, hzero] at hs
+      have hxcoord : e (q - o) 0 - e (p - o) 0 = e q 0 - e p 0 := by simp
+      nlinarith
+    exact hleast.unique ((K.isCompact.image (by fun_prop)).isLeast_sInf (K.nonempty.image _))
+  · right
+    have hgreatest : IsGreatest (horizontalProjection K o e) (e (p - o) 0) := by
+      refine ⟨⟨p, hp, rfl⟩, ?_⟩
+      rintro x ⟨q, hq, rfl⟩
+      have hs := ha q hq
+      rw [← e.inner_map_map] at hs
+      simp only [map_sub, PiLp.inner_apply, RCLike.inner_apply, conj_trivial,
+        Fin.sum_univ_two, PiLp.sub_apply, hzero] at hs
+      have hxcoord : e (q - o) 0 - e (p - o) 0 = e q 0 - e p 0 := by simp
+      nlinarith
+    exact hgreatest.unique ((K.isCompact.image (by fun_prop)).isGreatest_sSup
+      (K.nonempty.image _))
+
+/-- A point of a convex body admitting an exterior unit normal belongs to its frontier. -/
+theorem mem_frontier_of_mem_of_isExteriorNormal (K : ConvexBody Point)
+    {p : Point} (hp : p ∈ K) {a : Real.Angle} (ha : IsExteriorNormal K p a) :
+    p ∈ frontier (K : Set Point) := by
+  rw [mem_frontier_iff_notMem_interior hp]
+  intro hpint
+  obtain ⟨ε, hε, hball⟩ := Metric.isOpen_iff.mp isOpen_interior p hpint
+  let q := p + (ε / 2) • normalVector a
+  have hqp : q ∈ K := interior_subset (hball (by
+    rw [Metric.mem_ball, dist_eq_norm]
+    rw [show q - p = (ε / 2) • normalVector a by simp [q], norm_smul,
+      norm_normalVector]
+    rw [Real.norm_eq_abs, abs_of_pos (div_pos hε (by norm_num))]
+    linarith))
+  have := ha q hqp
+  rw [show q - p = (ε / 2) • normalVector a by simp [q], inner_smul_left,
+    real_inner_self_eq_norm_sq, norm_normalVector a] at this
+  have hnonpos : ε / 2 ≤ 0 := by simpa using this
+  linarith
+
+private theorem upperGraph_tangent_orthogonal_of_isExteriorNormal
+    (K : ConvexBody Point) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) {x : ℝ}
+    (hx : x ∈ Set.Ioo (horizontalBounds K o e).1 (horizontalBounds K o e).2)
+    (hdiff : DifferentiableAt ℝ (upperGraphHeight K o e) x) {a : Real.Angle}
+    (ha : IsExteriorNormal K
+      (o + e.symm !₂[x, upperGraphHeight K o e x]) a) :
+    inner ℝ (e.symm !₂[1, deriv (upperGraphHeight K o e) x]) (normalVector a) = 0 := by
+  let g := upperGraphHeight K o e
+  let n := e (normalVector a)
+  let f : ℝ → ℝ := fun y ↦ (y - x) * n 0 + (g y - g x) * n 1
+  have hlocal : IsLocalMax f x := by
+    filter_upwards [Ioo_mem_nhds hx.1 hx.2] with y hy
+    have hyproj : y ∈ horizontalProjection K o e := by
+      rw [horizontalProjection_eq_Icc K o e]
+      exact ⟨hy.1.le, hy.2.le⟩
+    have hmem := upperGraphHeight_mem K o e hyproj
+    have hsupport := ha (o + e.symm !₂[y, g y]) hmem
+    rw [← e.inner_map_map] at hsupport
+    simp only [map_sub, map_add, LinearIsometryEquiv.apply_symm_apply, PiLp.inner_apply,
+      RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Matrix.cons_val_zero,
+      Matrix.cons_val_one, PiLp.sub_apply, PiLp.add_apply] at hsupport
+    dsimp only [f, g, n]
+    dsimp only [g] at hsupport
+    nlinarith [hsupport]
+  have hfderiv : HasDerivAt f (n 0 + deriv g x * n 1) x := by
+    convert (((hasDerivAt_id x).sub_const x).mul_const (n 0)).add
+      ((hdiff.hasDerivAt.sub_const (g x)).mul_const (n 1)) using 1
+    · funext y
+      rfl
+    · simp [g]
+  have hzero := hlocal.hasDerivAt_eq_zero hfderiv
+  rw [← e.inner_map_map]
+  simp only [LinearIsometryEquiv.apply_symm_apply, PiLp.inner_apply, RCLike.inner_apply,
+    conj_trivial, Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one]
+  dsimp only [g, n] at hzero ⊢
+  nlinarith [hzero]
+
+private theorem normalVector_injective : Function.Injective normalVector := by
+  intro a b hab
+  induction a using Real.Angle.induction_on with
+  | _ a =>
+    induction b using Real.Angle.induction_on with
+    | _ b =>
+      apply Real.Angle.cos_sin_inj
+      · exact congrFun (congrArg WithLp.ofLp hab) 0
+      · exact congrFun (congrArg WithLp.ofLp hab) 1
+
+private theorem exteriorNormal_eq_of_orthogonal_of_interior_nonempty
+    (K : ConvexBody Point) (hK : (interior (K : Set Point)).Nonempty)
+    {p v : Point} (hv : v ≠ 0) {a b : Real.Angle}
+    (ha : IsExteriorNormal K p a) (hb : IsExteriorNormal K p b)
+    (hva : inner ℝ v (normalVector a) = 0)
+    (hvb : inner ℝ v (normalVector b) = 0) : a = b := by
+  let orientation : Orientation ℝ Point (Fin 2) :=
+    (EuclideanSpace.basisFun (Fin 2) ℝ).toBasis.orientation
+  rcases EuclideanGeometry.eq_or_eq_neg_of_unit_orthogonal orientation hv (norm_normalVector a)
+      (norm_normalVector b) hva hvb with hab | hab
+  · exact normalVector_injective hab
+  · exfalso
+    obtain ⟨z, hz⟩ := hK
+    obtain ⟨ε, hε, hball⟩ := Metric.isOpen_iff.mp isOpen_interior z hz
+    let q := z + (ε / 2) • normalVector a
+    have hq : q ∈ K := interior_subset (hball (by
+      rw [Metric.mem_ball, dist_eq_norm]
+      rw [show q - z = (ε / 2) • normalVector a by simp [q], norm_smul,
+        norm_normalVector, Real.norm_eq_abs, abs_of_pos (div_pos hε (by norm_num))]
+      norm_num
+      linarith))
+    have haz := ha z (interior_subset hz)
+    have hbz := hb z (interior_subset hz)
+    have hba : normalVector b = -normalVector a := by rw [hab]; simp
+    rw [hba, inner_neg_right] at hbz
+    have heq : inner ℝ (z - p) (normalVector a) = 0 := by linarith
+    have haq := ha q hq
+    have hqp : q - p = (z - p) + (ε / 2) • normalVector a := by
+      dsimp only [q]
+      module
+    rw [hqp,
+      inner_add_left, inner_smul_left, real_inner_self_eq_norm_sq,
+      norm_normalVector a, heq, zero_add] at haq
+    have : ε / 2 ≤ 0 := by simpa using haq
+    linarith
+
+/-- A differentiable interior point of an upper boundary graph is regular. -/
+theorem upperGraph_mem_regularBoundary (K : ConvexBody Point)
+    (hK : (interior (K : Set Point)).Nonempty) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point)
+    {x : ℝ} (hx : x ∈ Set.Ioo (horizontalBounds K o e).1 (horizontalBounds K o e).2)
+    (hdiff : DifferentiableAt ℝ (upperGraphHeight K o e) x) :
+    o + e.symm !₂[x, upperGraphHeight K o e x] ∈ regularBoundary K := by
+  let p := o + e.symm !₂[x, upperGraphHeight K o e x]
+  let a := vectorNormalAngle
+    (e.symm !₂[-deriv (upperGraphHeight K o e) x, 1])
+  have hxproj : x ∈ horizontalProjection K o e := by
+    rw [horizontalProjection_eq_Icc K o e]
+    exact ⟨hx.1.le, hx.2.le⟩
+  have hp : p ∈ K := upperGraphHeight_mem K o e hxproj
+  have ha : IsExteriorNormal K p a := upperGraph_deriv_isExteriorNormal K o e hx hdiff
+  refine ⟨mem_frontier_of_mem_of_isExteriorNormal K hp ha, a, ha, ?_⟩
+  intro b hb
+  symm
+  apply exteriorNormal_eq_of_orthogonal_of_interior_nonempty K hK
+    (v := e.symm !₂[1, deriv (upperGraphHeight K o e) x])
+  · intro hv
+    have := congrFun (congrArg WithLp.ofLp (congrArg e hv)) 0
+    simp at this
+  · exact ha
+  · exact hb
+  · rw [normalVector_vectorNormalAngle]
+    · rw [← e.inner_map_map]
+      simp [PiLp.inner_apply, Fin.sum_univ_two]
+    · intro hv
+      have := congrFun (congrArg WithLp.ofLp (congrArg e hv)) 1
+      simp at this
+  · exact upperGraph_tangent_orthogonal_of_isExteriorNormal K o e hx hdiff (a := b) hb
+
+/-- Points of an upper graph lying above nondifferentiability parameters. -/
+def irregularUpperGraph (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) : Set Point :=
+  (fun x ↦ o + e.symm !₂[x, upperGraphHeight K o e x]) ''
+    {x | x ∈ Set.Ioo (horizontalBounds K o e).1 (horizontalBounds K o e).2 ∧
+      ¬ DifferentiableAt ℝ (upperGraphHeight K o e) x}
+
+/-- An irregular upper graph has zero one-dimensional Hausdorff measure. -/
+theorem hausdorffMeasure_irregularUpperGraph_eq_zero (K : ConvexBody Point)
+    (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) :
+    Measure.hausdorffMeasure 1 (irregularUpperGraph K o e) = 0 := by
+  exact MeasureTheory.hausdorffMeasure_coordinateGraph_nondifferentiable_eq_zero
+    (locallyLipschitzOn_upperGraphHeight K o e) o e
+
+private theorem eq_left_or_eq_right_or_mem_Ioo {a b x : ℝ} (hx : x ∈ Set.Icc a b) :
+    x = a ∨ x = b ∨ x ∈ Set.Ioo a b := by
+  rcases hx.1.eq_or_lt with h | h
+  · exact Or.inl h.symm
+  rcases hx.2.eq_or_lt with h' | h'
+  · exact Or.inr (Or.inl h')
+  · exact Or.inr (Or.inr ⟨h, h'⟩)
+
+private theorem irregularBoundary_subset_graphs_union_corners (K : ConvexBody Point)
+    (hK : (interior (K : Set Point)).Nonempty) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) :
+    frontier (K : Set Point) \ regularBoundary K ⊆
+      irregularUpperGraph K o e ∪ (irregularUpperGraph K o (negCoordinates e) ∪
+      (irregularUpperGraph K o (swapCoordinates e) ∪
+      (irregularUpperGraph K o (negCoordinates (swapCoordinates e)) ∪
+      coordinateCornerSet K o e))) := by
+  intro p hp
+  have hpK : p ∈ K := by
+    change p ∈ (K : Set Point)
+    rw [← K.isClosed.closure_eq]
+    exact frontier_subset_closure hp.1
+  let x := e (p - o) 0
+  let y := e (p - o) 1
+  have hxIcc : x ∈ Set.Icc (horizontalBounds K o e).1
+      (horizontalBounds K o e).2 := by
+    rw [← horizontalProjection_eq_Icc K o e]
+    exact ⟨p, hpK, rfl⟩
+  have hyIcc : y ∈ Set.Icc (horizontalBounds K o (swapCoordinates e)).1
+      (horizontalBounds K o (swapCoordinates e)).2 := by
+    rw [← horizontalProjection_eq_Icc K o (swapCoordinates e)]
+    refine ⟨p, hpK, ?_⟩
+    simp [y, swapCoordinates_apply_zero]
+  rcases eq_left_or_eq_right_or_mem_Ioo hxIcc with hxleft | hxright | hxint
+  · rcases eq_left_or_eq_right_or_mem_Ioo hyIcc with hyleft | hyright | hyint
+    · right; right; right; right
+      exact ⟨Or.inl (by simpa [x] using hxleft), Or.inl (by simpa [y] using hyleft)⟩
+    · right; right; right; right
+      exact ⟨Or.inl (by simpa [x] using hxleft), Or.inr (by simpa [y] using hyright)⟩
+    · right; right; right; left
+      let e' := negCoordinates (swapCoordinates e)
+      let z := e' (p - o) 0
+      have hzint : z ∈ Set.Ioo (horizontalBounds K o e').1
+          (horizontalBounds K o e').2 := by
+        rw [horizontalBounds_negCoordinates]
+        change -y ∈ Set.Ioo (-(horizontalBounds K o (swapCoordinates e)).2)
+          (-(horizontalBounds K o (swapCoordinates e)).1)
+        exact ⟨neg_lt_neg hyint.2, neg_lt_neg hyint.1⟩
+      have hgraph := eq_negSwapUpperGraph_of_horizontalCoordinate_eq_leftBound
+        K o e hpK (by simpa [x] using hxleft)
+      refine ⟨z, ⟨hzint, ?_⟩, hgraph.symm⟩
+      intro hdiff
+      apply hp.2
+      rw [hgraph]
+      exact upperGraph_mem_regularBoundary K hK o e' hzint hdiff
+  · rcases eq_left_or_eq_right_or_mem_Ioo hyIcc with hyleft | hyright | hyint
+    · right; right; right; right
+      exact ⟨Or.inr (by simpa [x] using hxright), Or.inl (by simpa [y] using hyleft)⟩
+    · right; right; right; right
+      exact ⟨Or.inr (by simpa [x] using hxright), Or.inr (by simpa [y] using hyright)⟩
+    · right; right; left
+      let e' := swapCoordinates e
+      let z := e' (p - o) 0
+      have hzint : z ∈ Set.Ioo (horizontalBounds K o e').1
+          (horizontalBounds K o e').2 := by simpa [e', z, y] using hyint
+      have hgraph := eq_swapUpperGraph_of_horizontalCoordinate_eq_rightBound
+        K o e hpK (by simpa [x] using hxright)
+      refine ⟨z, ⟨hzint, ?_⟩, hgraph.symm⟩
+      intro hdiff
+      apply hp.2
+      rw [hgraph]
+      exact upperGraph_mem_regularBoundary K hK o e' hzint hdiff
+  · obtain ⟨a, ha⟩ := exists_isExteriorNormal_of_mem_frontier K hK hp.1
+    have hvertical : e (normalVector a) 1 ≠ 0 := by
+      intro hzero
+      rcases horizontalCoordinate_eq_endpoint_of_isExteriorNormal_of_vertical_eq_zero
+        K o e hpK ha hzero with hleft | hright
+      · exact hxint.1.ne' (by simpa [x] using hleft)
+      · exact hxint.2.ne (by simpa [x] using hright)
+    rcases lt_or_gt_of_ne hvertical with hneg | hpos
+    · right; left
+      let e' := negCoordinates e
+      let z := e' (p - o) 0
+      have hzint : z ∈ Set.Ioo (horizontalBounds K o e').1
+          (horizontalBounds K o e').2 := by
+        rw [horizontalBounds_negCoordinates]
+        change -x ∈ Set.Ioo (-(horizontalBounds K o e).2)
+          (-(horizontalBounds K o e).1)
+        exact ⟨neg_lt_neg hxint.2, neg_lt_neg hxint.1⟩
+      have hgraph := eq_upperCoordinateGraph_of_isExteriorNormal_of_pos K o e'
+        hpK ha (by simpa [e'] using neg_pos.mpr hneg)
+      refine ⟨z, ⟨hzint, ?_⟩, hgraph.symm⟩
+      intro hdiff
+      apply hp.2
+      rw [hgraph]
+      exact upperGraph_mem_regularBoundary K hK o e' hzint hdiff
+    · left
+      have hgraph := eq_upperCoordinateGraph_of_isExteriorNormal_of_pos K o e hpK ha hpos
+      refine ⟨x, ⟨hxint, ?_⟩, ?_⟩
+      · intro hdiff
+        apply hp.2
+        rw [hgraph]
+        exact upperGraph_mem_regularBoundary K hK o e hxint hdiff
+      · simpa [x] using hgraph.symm
+
+/-- The irregular boundary of a planar convex body with nonempty interior has
+zero one-dimensional Hausdorff measure. -/
+theorem hausdorffMeasure_irregularBoundary_eq_zero (K : ConvexBody Point)
+    (hK : (interior (K : Set Point)).Nonempty) :
+    Measure.hausdorffMeasure 1
+      (frontier (K : Set Point) \ regularBoundary K) = 0 := by
+  let e : Point ≃ₗᵢ[ℝ] Point := LinearIsometryEquiv.refl ℝ Point
+  let o : Point := 0
+  apply measure_mono_null
+    (irregularBoundary_subset_graphs_union_corners K hK o e)
+  exact MeasureTheory.measure_union_null
+    (hausdorffMeasure_irregularUpperGraph_eq_zero K o e)
+    (MeasureTheory.measure_union_null
+      (hausdorffMeasure_irregularUpperGraph_eq_zero K o (negCoordinates e))
+      (MeasureTheory.measure_union_null
+        (hausdorffMeasure_irregularUpperGraph_eq_zero K o (swapCoordinates e))
+        (MeasureTheory.measure_union_null
+          (hausdorffMeasure_irregularUpperGraph_eq_zero K o
+            (negCoordinates (swapCoordinates e)))
+          (hausdorffMeasure_coordinateCornerSet_eq_zero K o e))))
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Exposed faces and the boundary of a convex body
+
+The exposed faces of a convex body are exactly the sets of boundary points realizing a support
+value: each exposed face lies on the boundary, and, when the body has interior, every boundary
+point lies on some exposed face.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Every boundary point of a convex body with nonempty interior lies on a supporting
+exposed face. -/
+theorem exists_mem_exposedEdge_of_mem_frontier
+    (K : ConvexBody Point) (hK : (interior (K : Set Point)).Nonempty)
+    {p : Point} (hp : p ∈ frontier (K : Set Point)) :
+    ∃ a : Real.Angle, p ∈ exposedEdge K a := by
+  have hpK : p ∈ K := by
+    change p ∈ (K : Set Point)
+    rw [← K.isClosed.closure_eq]
+    exact frontier_subset_closure hp
+  obtain ⟨a, ha⟩ := exists_isExteriorNormal_of_mem_frontier K hK hp
+  refine ⟨a, hpK, ?_⟩
+  change inner ℝ p (normalVector a) = supportValue K a
+  apply le_antisymm
+  · exact inner_le_supportValue K hpK a
+  · obtain ⟨q, hq, hqeq⟩ := exists_mem_inner_eq_supportValue K a
+    rw [← hqeq]
+    have := ha q hq
+    rw [inner_sub_left] at this
+    linarith
+
+/-- Every exposed face of a convex body lies on its boundary. -/
+theorem exposedEdge_subset_frontier
+    (K : ConvexBody Point) (a : Real.Angle) :
+    exposedEdge K a ⊆ frontier (K : Set Point) := by
+  intro p hp
+  rw [mem_frontier_iff_notMem_interior hp.1]
+  intro hpint
+  obtain ⟨ε, hε, hball⟩ := Metric.isOpen_iff.mp isOpen_interior p hpint
+  let z := p + (ε / 2) • normalVector a
+  have hzball : z ∈ Metric.ball p ε := by
+    simp only [Metric.mem_ball, dist_eq_norm, z, add_sub_cancel_left, norm_smul,
+      norm_normalVector, mul_one, Real.norm_eq_abs, abs_of_pos (half_pos hε)]
+    linarith
+  have hzK : z ∈ K := interior_subset (hball hzball)
+  have hzle := inner_le_supportValue K hzK a
+  have hpEq := hp.2
+  change inner ℝ p (normalVector a) = supportValue K a at hpEq
+  have hnorm : inner ℝ (normalVector a) (normalVector a) = 1 := by
+    induction a using Real.Angle.induction_on with
+    | _ a => exact inner_normalVector_self a
+  rw [show z = p + (ε / 2) • normalVector a from rfl, inner_add_left,
+    real_inner_smul_left, hnorm, hpEq] at hzle
+  linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Regular Boundary Helpers
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- A planar convex body with nonempty interior has no segment presentation. -/
+theorem not_exists_segmentPresentation_of_interior_nonempty (K : ConvexBody Point)
+    (hint : (interior (K : Set Point)).Nonempty) : ¬ ∃ d, IsSegmentPresentation K d := by
+  rintro ⟨d, hd⟩
+  have htop : affineSpan ℝ (K : Set Point) = ⊤ :=
+    K.convex.interior_nonempty_iff_affineSpan_eq_top.mp hint
+  have hle : affineSpan ℝ (K : Set Point) ≤ affineSpan ℝ ({d.1, d.2.1} : Set Point) := by
+    rw [hd.2.1]
+    apply affineSpan_le.2
+    exact (affineSpan ℝ ({d.1, d.2.1} : Set Point)).convex.segment_subset
+      (subset_affineSpan ℝ _ (by simp)) (subset_affineSpan ℝ _ (by simp))
+  rw [htop] at hle
+  have heq : affineSpan ℝ ({d.1, d.2.1} : Set Point) = ⊤ := top_unique hle
+  have hfin := (collinear_iff_finrank_le_one.mp (collinear_pair ℝ d.1 d.2.1))
+  rw [← direction_affineSpan, heq] at hfin
+  rw [AffineSubspace.direction_top ℝ Point Point, finrank_top] at hfin
+  norm_num [Point, finrank_euclideanSpace_fin] at hfin
+
+/-- A convex body with nonempty interior is not a singleton. -/
+theorem not_subsingleton_of_interior_nonempty (K : ConvexBody Point)
+    (hint : (interior (K : Set Point)).Nonempty) : ¬ (K : Set Point).Subsingleton := by
+  intro hsub
+  obtain ⟨p, hp⟩ := K.nonempty
+  have hK : (K : Set Point) = {p} :=
+    Set.eq_singleton_iff_unique_mem.mpr ⟨hp, fun x hx ↦ hsub hx hp⟩
+  rw [hK, interior_singleton] at hint
+  exact hint.ne_empty rfl
+
+private theorem surfaceAreaMeasure_eq_map_of_interior_nonempty (K : ConvexBody Point)
+    (hint : (interior (K : Set Point)).Nonempty) :
+    surfaceAreaMeasure K = Measure.map (exteriorNormalAngle K)
+      ((Measure.hausdorffMeasure 1).restrict (regularBoundary K)) := by
+  have hsub := not_subsingleton_of_interior_nonempty K hint
+  have hseg := not_exists_segmentPresentation_of_interior_nonempty K hint
+  simp [surfaceAreaMeasure, hsub, hseg]
+
+/-- Surface area measure is finite when the convex body has nonempty interior. -/
+theorem isFiniteMeasure_surfaceAreaMeasure_of_interior_nonempty (K : ConvexBody Point)
+    (hint : (interior (K : Set Point)).Nonempty) : IsFiniteMeasure (surfaceAreaMeasure K) := by
+  rw [surfaceAreaMeasure_eq_map_of_interior_nonempty K hint]
+  let μ := (Measure.hausdorffMeasure 1).restrict (regularBoundary K)
+  have hμ : μ Set.univ < ⊤ := by
+    rw [Measure.restrict_apply_univ]
+    exact lt_of_le_of_lt (measure_mono fun p hp ↦ hp.1)
+      (K.hausdorffMeasure_frontier_lt_top hint)
+  let _ : IsFiniteMeasure μ := ⟨hμ⟩
+  exact Measure.isFiniteMeasure_map μ (exteriorNormalAngle K)
+
+/-- Any extension of the exterior-normal angle from the regular boundary gives
+the surface area measure as a pushforward of frontier length. -/
+theorem surfaceAreaMeasure_eq_map_frontier_of_eq_on_regularBoundary
+    (K : ConvexBody Point) (hK : (interior (K : Set Point)).Nonempty)
+    (ν : Point → Real.Angle)
+    (hν : ∀ p ∈ regularBoundary K, ν p = exteriorNormalAngle K p) :
+    surfaceAreaMeasure K = Measure.map ν
+      ((Measure.hausdorffMeasure 1).restrict (frontier (K : Set Point))) := by
+  let μ : Measure Point := Measure.hausdorffMeasure 1
+  have hregular : regularBoundary K ⊆ frontier (K : Set Point) := fun _ hp ↦ hp.1
+  have hae : frontier (K : Set Point) =ᵐ[μ] regularBoundary K := by
+    rw [ae_eq_set]
+    exact ⟨hausdorffMeasure_irregularBoundary_eq_zero K hK,
+      measure_mono_null (fun _ hp ↦ (hp.2 (hregular hp.1)).elim) measure_empty⟩
+  have hrestrict : μ.restrict (frontier (K : Set Point)) =
+      μ.restrict (regularBoundary K) := Measure.restrict_congr_set hae
+  rw [surfaceAreaMeasure_eq_map_of_interior_nonempty K hK, hrestrict]
+  apply Measure.map_congr
+  filter_upwards [ae_restrict_mem (measurableSet_regularBoundary K)] with p hp
+  exact (hν p hp).symm
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Graph Integral
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter MeasureTheory
+open scoped Topology
+
+namespace MovingSofa
+
+private theorem exteriorNormalAngle_upperGraph_eq (K : ConvexBody Point)
+    (hK : (interior (K : Set Point)).Nonempty) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point)
+    {x : ℝ} (hx : x ∈ Set.Ioo (horizontalBounds K o e).1 (horizontalBounds K o e).2)
+    (hdiff : DifferentiableAt ℝ (upperGraphHeight K o e) x) :
+    exteriorNormalAngle K (o + e.symm !₂[x, upperGraphHeight K o e x]) =
+      vectorNormalAngle (e.symm !₂[-deriv (upperGraphHeight K o e) x, 1]) := by
+  have hp := upperGraph_mem_regularBoundary K hK o e hx hdiff
+  have hu := hp.2
+  have hn : IsExteriorNormal K (o + e.symm !₂[x, upperGraphHeight K o e x])
+      (exteriorNormalAngle K (o + e.symm !₂[x, upperGraphHeight K o e x])) := by
+    unfold exteriorNormalAngle
+    rw [dite_eq_left hu]
+    exact hu.exists.choose_spec
+  exact hu.unique hn (upperGraph_deriv_isExteriorNormal K o e hx hdiff)
+
+private theorem coordinate_normalVector_vectorNormalAngle_second
+    (e : Point ≃ₗᵢ[ℝ] Point) (m : ℝ) :
+    e (normalVector (vectorNormalAngle (e.symm !₂[-m, 1]))) 1 =
+      (Real.sqrt (1 + m ^ 2))⁻¹ := by
+  let q : Point := e.symm !₂[-m, 1]
+  have hq : q ≠ 0 := by
+    intro h
+    have h1 := congrFun (congrArg WithLp.ofLp (congrArg e h)) 1
+    simp [q] at h1
+  rw [normalVector_vectorNormalAngle hq, map_smul, PiLp.smul_apply]
+  have hnorm : ‖q‖ = Real.sqrt (1 + m ^ 2) := by
+    rw [← e.norm_map q, LinearIsometryEquiv.apply_symm_apply, EuclideanSpace.norm_eq]
+    congr 1
+    simp [Fin.sum_univ_two, Real.norm_eq_abs, pow_two]
+    ring
+  rw [hnorm]
+  simp [q]
+
+private theorem upperGraphSurfaceIntegrand_eq_zero_of_speed_gt
+    (K : ConvexBody Point) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point)
+    (ψ : Real.Angle → ℝ) {ε : ℝ} (hε : 0 < ε)
+    (hsupport : ∀ t, e (normalVector t) 1 < ε → ψ t = 0) {x : ℝ}
+    (hx : ε⁻¹ < Real.sqrt (1 + (deriv (upperGraphHeight K o e) x) ^ 2)) :
+    upperGraphSurfaceIntegrand K o e ψ x = 0 := by
+  rw [upperGraphSurfaceIntegrand, hsupport, zero_mul]
+  rw [coordinate_normalVector_vectorNormalAngle_second]
+  exact (inv_lt_comm₀ (Real.sqrt_pos.2 (by positivity)) hε).2 hx
+
+private theorem measurable_vectorNormalAngle : Measurable vectorNormalAngle := by
+  unfold vectorNormalAngle
+  apply Real.Angle.continuous_coe.measurable.comp
+  apply Complex.measurable_arg.comp
+  have hp0 : Measurable (fun p : Point ↦ p 0) :=
+    (PiLp.proj (𝕜 := ℝ) 2 (fun _ : Fin 2 ↦ ℝ) 0).continuous.measurable
+  have hp1 : Measurable (fun p : Point ↦ p 1) :=
+    (PiLp.proj (𝕜 := ℝ) 2 (fun _ : Fin 2 ↦ ℝ) 1).continuous.measurable
+  have h0 : Measurable (fun p : Point ↦ (p 0 : ℂ)) :=
+    Complex.measurable_ofReal.comp hp0
+  have h1 : Measurable (fun p : Point ↦ (p 1 : ℂ)) :=
+    Complex.measurable_ofReal.comp hp1
+  convert h0.add (h1.mul_const Complex.I) using 1
+  funext p
+  apply Complex.ext <;> simp
+
+private theorem measurable_upperGraphSurfaceIntegrand (K : ConvexBody Point)
+    (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) (ψ : Real.Angle → ℝ) (hψ : Continuous ψ) :
+    Measurable (upperGraphSurfaceIntegrand K o e ψ) := by
+  unfold upperGraphSurfaceIntegrand
+  have hd : Measurable (deriv (upperGraphHeight K o e)) := measurable_deriv _
+  have hv : Measurable (fun x ↦ e.symm !₂[-deriv (upperGraphHeight K o e) x, 1]) := by
+    fun_prop
+  apply (hψ.measurable.comp (measurable_vectorNormalAngle.comp ?_)).mul
+    ((measurable_const.add (hd.pow_const 2)).sqrt)
+  exact hv
+
+/-- A normal support cutoff gives a uniform bound on the weighted graph density. -/
+theorem norm_upperGraphSurfaceIntegrand_le
+    (K : ConvexBody Point) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point)
+    (ψ : Real.Angle → ℝ) {ε M : ℝ} (hε : 0 < ε)
+    (hsupport : ∀ t, e (normalVector t) 1 < ε → ψ t = 0)
+    (hM : ∀ t, ‖ψ t‖ ≤ M) (x : ℝ) :
+    ‖upperGraphSurfaceIntegrand K o e ψ x‖ ≤ M * ε⁻¹ := by
+  have hM0 : 0 ≤ M := (norm_nonneg (ψ 0)).trans (hM 0)
+  let s := Real.sqrt (1 + (deriv (upperGraphHeight K o e) x) ^ 2)
+  by_cases hs : s ≤ ε⁻¹
+  · rw [upperGraphSurfaceIntegrand, Real.norm_eq_abs, abs_mul]
+    rw [abs_of_nonneg (Real.sqrt_nonneg _)]
+    exact mul_le_mul (hM _) hs (Real.sqrt_nonneg _) hM0
+  · rw [upperGraphSurfaceIntegrand_eq_zero_of_speed_gt K o e ψ hε hsupport
+      (lt_of_not_ge hs), norm_zero]
+    exact mul_nonneg hM0 (le_of_lt (inv_pos.mpr hε))
+
+/-- The weighted surface integrand of an upper graph is integrable on its
+horizontal projection interval. -/
+theorem integrable_upperGraphSurfaceIntegrand (K : ConvexBody Point)
+    (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) (ψ : Real.Angle → ℝ) (hψ : Continuous ψ)
+    {ε : ℝ} (hε : 0 < ε)
+    (hsupport : ∀ t, e (normalVector t) 1 < ε → ψ t = 0) (a b : ℝ) :
+    Integrable (upperGraphSurfaceIntegrand K o e ψ) (volume.restrict (Set.Icc a b)) := by
+  obtain ⟨t, -, ht⟩ := isCompact_univ.exists_isMaxOn Set.univ_nonempty hψ.norm.continuousOn
+  let M := ‖ψ t‖
+  have hM (u : Real.Angle) : ‖ψ u‖ ≤ M := ht trivial
+  apply (integrable_const (μ := volume.restrict (Set.Icc a b))
+    (c := M * ε⁻¹)).mono'
+    (measurable_upperGraphSurfaceIntegrand K o e ψ hψ).aestronglyMeasurable
+  exact Eventually.of_forall fun x ↦
+    norm_upperGraphSurfaceIntegrand_le K o e ψ hε hsupport hM x
+
+private theorem upperCoordinateGraph_mem_frontier (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) {x : ℝ}
+    (hx : x ∈ Set.Icc (horizontalBounds K o e).1 (horizontalBounds K o e).2) :
+    o + e.symm !₂[x, upperGraphHeight K o e x] ∈ frontier (K : Set Point) := by
+  have hxproj : x ∈ horizontalProjection K o e := by
+    rw [horizontalProjection_eq_Icc]
+    exact hx
+  let p := o + e.symm !₂[x, upperGraphHeight K o e x]
+  have hpK : p ∈ K := upperGraphHeight_mem K o e hxproj
+  rw [mem_frontier_iff_notMem_interior hpK]
+  intro hpint
+  obtain ⟨r, hr, hball⟩ := Metric.isOpen_iff.mp isOpen_interior p hpint
+  let q := p + (r / 2) • e.symm !₂[0, 1]
+  have hqp : q ∈ K := interior_subset (hball (by
+    rw [Metric.mem_ball, dist_eq_norm]
+    rw [show q - p = (r / 2) • e.symm !₂[0, 1] by simp [q], norm_smul,
+      e.symm.norm_map, EuclideanSpace.norm_eq]
+    simp only [norm_div, Real.norm_eq_abs, sq_abs, Fin.sum_univ_two, Fin.isValue,
+      Matrix.cons_val_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow,
+      Matrix.cons_val_one, Matrix.cons_val_fin_one, one_pow, zero_add, Real.sqrt_one, mul_one]
+    rw [abs_of_pos hr]
+    linarith))
+  have hqfiber : o + e.symm !₂[x, upperGraphHeight K o e x + r / 2] ∈ K := by
+    convert hqp using 1
+    apply e.injective
+    ext i
+    fin_cases i
+    · simp [p, q, map_add, map_smul]
+    · simp [p, q, map_add, map_smul]
+      ring
+  have hle := (upperGraphHeight_isGreatest K o e hxproj).2 hqfiber
+  linarith
+
+/-- Evaluate an angular weight at the upper graph’s normal direction above a point’s abscissa. -/
+def upperGraphWeight (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) (ψ : Real.Angle → ℝ) (p : Point) : ℝ :=
+  ψ (vectorNormalAngle (e.symm !₂[
+    -deriv (upperGraphHeight K o e) (e (p - o) 0), 1]))
+
+private theorem measurable_upperGraphWeight (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) (ψ : Real.Angle → ℝ) (hψ : Continuous ψ) :
+    Measurable (upperGraphWeight K o e ψ) := by
+  unfold upperGraphWeight
+  apply hψ.measurable.comp (measurable_vectorNormalAngle.comp ?_)
+  have hx : Measurable (fun p : Point ↦ e (p - o) 0) := by fun_prop
+  have hd : Measurable (fun p : Point ↦
+      deriv (upperGraphHeight K o e) (e (p - o) 0)) :=
+    (measurable_deriv _).comp hx
+  fun_prop
+
+private theorem integral_restrict_upperCoordinateGraph_eq_integral
+    (K : ConvexBody Point) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point)
+    (ψ : Real.Angle → ℝ) (hψ : Continuous ψ) {a b : ℝ}
+    (hab : a ≤ b)
+    (hsub : Set.Icc a b ⊆
+      Set.Ioo (horizontalBounds K o e).1 (horizontalBounds K o e).2) :
+    (∫ p, upperGraphWeight K o e ψ p ∂(Measure.hausdorffMeasure 1).restrict
+      ((fun x ↦ o + e.symm !₂[x, upperGraphHeight K o e x] : ℝ → Point) ''
+        Set.Icc a b)) =
+      ∫ x in a..b, upperGraphSurfaceIntegrand K o e ψ x := by
+  have hlocal : LocallyLipschitzOn (Set.Icc a b) (upperGraphHeight K o e) :=
+    (locallyLipschitzOn_upperGraphHeight K o e).mono hsub
+  obtain ⟨C, hC⟩ :=
+    LocallyLipschitzOn.exists_lipschitzOnWith_of_compact isCompact_Icc hlocal
+  rw [MeasureTheory.integral_restrict_coordinateGraph_eq_integral_sqrt_mul hab hC o e
+    (measurable_upperGraphWeight K o e ψ hψ)]
+  apply intervalIntegral.integral_congr
+  intro x _
+  unfold upperGraphWeight upperGraphSurfaceIntegrand
+  simp only [add_sub_cancel_left, LinearIsometryEquiv.apply_symm_apply, Fin.isValue,
+    Matrix.cons_val_zero]
+  apply mul_comm
+
+private theorem integral_restrict_upperCoordinateGraph_eq_setIntegral
+    (K : ConvexBody Point) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point)
+    (ψ : Real.Angle → ℝ) (hψ : Continuous ψ) {a b : ℝ}
+    (hab : a ≤ b)
+    (hsub : Set.Icc a b ⊆
+      Set.Ioo (horizontalBounds K o e).1 (horizontalBounds K o e).2) :
+    (∫ p, upperGraphWeight K o e ψ p ∂(Measure.hausdorffMeasure 1).restrict
+      ((fun x ↦ o + e.symm !₂[x, upperGraphHeight K o e x] : ℝ → Point) ''
+        Set.Icc a b)) =
+      ∫ x in Set.Icc a b, upperGraphSurfaceIntegrand K o e ψ x := by
+  rw [integral_restrict_upperCoordinateGraph_eq_integral K o e ψ hψ hab hsub,
+    intervalIntegral.integral_of_le hab, ← integral_Icc_eq_integral_Ioc]
+
+private theorem upperCoordinateGraph_image_subset_frontier (K : ConvexBody Point)
+    (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) {s : Set ℝ}
+    (hs : s ⊆ Set.Icc (horizontalBounds K o e).1 (horizontalBounds K o e).2) :
+    (fun x ↦ o + e.symm !₂[x, upperGraphHeight K o e x] : ℝ → Point) '' s ⊆
+      frontier (K : Set Point) := by
+  rintro _ ⟨x, hx, rfl⟩
+  exact upperCoordinateGraph_mem_frontier K o e (hs hx)
+
+private theorem hausdorffMeasure_upperGraph_endpoints_eq_zero (K : ConvexBody Point)
+    (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) :
+    Measure.hausdorffMeasure 1
+      ((fun x ↦ o + e.symm !₂[x, upperGraphHeight K o e x] : ℝ → Point) ''
+        ({(horizontalBounds K o e).1, (horizontalBounds K o e).2} : Set ℝ)) = 0 := by
+  let _ := MeasureTheory.Measure.nullSingletonClass_hausdorff Point (by norm_num : (0 : ℝ) < 1)
+  apply Set.Finite.measure_zero
+  exact (Set.toFinite _).image _
+
+private theorem eventually_innerIcc_nonempty {a b : ℝ} (hab : a < b) :
+    ∀ᶠ n : ℕ in Filter.atTop,
+      a + 1 / ((n : ℝ) + 1) ≤ b - 1 / ((n : ℝ) + 1) := by
+  have hevent := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).eventually
+    (Iio_mem_nhds (half_pos (sub_pos.mpr hab)))
+  exact hevent.mono fun n hn ↦ by linarith
+
+/-- Integrals over the inner upper graph equal integrals over its coordinate interval. -/
+theorem eventually_integral_innerUpperGraph_eq_innerIcc
+    (K : ConvexBody Point) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point)
+    (ψ : Real.Angle → ℝ) (hψ : Continuous ψ)
+    (hwidth : (horizontalBounds K o e).1 < (horizontalBounds K o e).2) :
+    ∀ᶠ n : ℕ in Filter.atTop,
+      (∫ p, upperGraphWeight K o e ψ p ∂(Measure.hausdorffMeasure 1).restrict
+        ((fun x ↦ o + e.symm !₂[x, upperGraphHeight K o e x] : ℝ → Point) ''
+          Set.Icc ((horizontalBounds K o e).1 + 1 / ((n : ℝ) + 1))
+            ((horizontalBounds K o e).2 - 1 / ((n : ℝ) + 1)))) =
+        ∫ x in Set.Icc ((horizontalBounds K o e).1 + 1 / ((n : ℝ) + 1))
+          ((horizontalBounds K o e).2 - 1 / ((n : ℝ) + 1)),
+          upperGraphSurfaceIntegrand K o e ψ x := by
+  filter_upwards [eventually_innerIcc_nonempty hwidth] with n hn
+  exact integral_restrict_upperCoordinateGraph_eq_setIntegral K o e ψ hψ hn
+    (innerIcc_subset_Ioo _ _ n)
+
+private theorem measurableSet_innerUpperGraph (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) (n : ℕ) :
+    MeasurableSet
+      ((fun x ↦ o + e.symm !₂[x, upperGraphHeight K o e x] : ℝ → Point) ''
+        Set.Icc ((horizontalBounds K o e).1 + 1 / ((n : ℝ) + 1))
+          ((horizontalBounds K o e).2 - 1 / ((n : ℝ) + 1))) := by
+  let s := Set.Icc ((horizontalBounds K o e).1 + 1 / ((n : ℝ) + 1))
+    ((horizontalBounds K o e).2 - 1 / ((n : ℝ) + 1))
+  have hs : s ⊆ Set.Ioo (horizontalBounds K o e).1
+      (horizontalBounds K o e).2 := innerIcc_subset_Ioo _ _ n
+  have hcont : ContinuousOn (upperGraphHeight K o e) s :=
+    ((locallyLipschitzOn_upperGraphHeight K o e).mono hs).continuousOn
+  exact (isCompact_Icc.image_of_continuousOn (by fun_prop)).measurableSet
+
+private theorem tendsto_innerUpperGraph_indicator_at_regularPoint
+    (K : ConvexBody Point) (hK : (interior (K : Set Point)).Nonempty)
+    (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) (ψ : Real.Angle → ℝ)
+    {ε : ℝ} (hε : 0 < ε)
+    (hsupport : ∀ t, e (normalVector t) 1 < ε → ψ t = 0) {p : Point}
+    (hp : p ∈ regularBoundary K)
+    (hpend : p ∉ (fun x ↦ o + e.symm !₂[x, upperGraphHeight K o e x] : ℝ → Point) ''
+      ({(horizontalBounds K o e).1, (horizontalBounds K o e).2} : Set ℝ))
+    (hpirr : p ∉ irregularUpperGraph K o e) :
+    Filter.Tendsto (fun n : ℕ ↦
+      (((fun x ↦ o + e.symm !₂[x, upperGraphHeight K o e x] : ℝ → Point) ''
+        Set.Icc ((horizontalBounds K o e).1 + 1 / ((n : ℝ) + 1))
+          ((horizontalBounds K o e).2 - 1 / ((n : ℝ) + 1))).indicator
+            (upperGraphWeight K o e ψ) p)) Filter.atTop
+      (nhds (ψ (exteriorNormalAngle K p))) := by
+  let a := exteriorNormalAngle K p
+  have ha : IsExteriorNormal K p a := by
+    dsimp only [a]
+    unfold exteriorNormalAngle
+    rw [dite_eq_left hp.2]
+    exact hp.2.exists.choose_spec
+  have hpK : p ∈ K := by
+    change p ∈ (K : Set Point)
+    rw [← K.isClosed.closure_eq]
+    exact frontier_subset_closure hp.1
+  by_cases hψa : ψ a = 0
+  · apply tendsto_const_nhds.congr'
+    filter_upwards [] with n
+    by_cases hpn : p ∈ (fun x ↦ o + e.symm !₂[x, upperGraphHeight K o e x] : ℝ → Point) ''
+        Set.Icc ((horizontalBounds K o e).1 + 1 / ((n : ℝ) + 1))
+          ((horizontalBounds K o e).2 - 1 / ((n : ℝ) + 1))
+    · obtain ⟨x, hx, rfl⟩ := hpn
+      have hxint := innerIcc_subset_Ioo (horizontalBounds K o e).1
+        (horizontalBounds K o e).2 n hx
+      have hdiff : DifferentiableAt ℝ (upperGraphHeight K o e) x := by
+        by_contra hndiff
+        exact hpirr ⟨x, ⟨hxint, hndiff⟩, rfl⟩
+      have hmem : (o + e.symm !₂[x, upperGraphHeight K o e x]) ∈
+          (fun y ↦ o + e.symm !₂[y, upperGraphHeight K o e y] : ℝ → Point) ''
+            Set.Icc ((horizontalBounds K o e).1 + 1 / ((n : ℝ) + 1))
+              ((horizontalBounds K o e).2 - 1 / ((n : ℝ) + 1)) := ⟨x, hx, rfl⟩
+      rw [Set.indicator_of_mem hmem]
+      unfold upperGraphWeight
+      simp only [add_sub_cancel_left, LinearIsometryEquiv.apply_symm_apply, Fin.isValue,
+        Matrix.cons_val_zero]
+      rw [← exteriorNormalAngle_upperGraph_eq K hK o e hxint hdiff]
+    · change ψ a = _
+      rw [Set.indicator]
+      simp only [hpn, ↓reduceIte]
+      exact hψa
+  · have hcoord : ε ≤ e (normalVector a) 1 := by
+      apply le_of_not_gt
+      exact fun h ↦ hψa (hsupport a h)
+    have hpos : 0 < e (normalVector a) 1 := hε.trans_le hcoord
+    have hgraph := eq_upperCoordinateGraph_of_isExteriorNormal_of_pos K o e hpK ha hpos
+    let x := e (p - o) 0
+    have hxproj : x ∈ horizontalProjection K o e := ⟨p, hpK, rfl⟩
+    have hxIcc : x ∈ Set.Icc (horizontalBounds K o e).1
+        (horizontalBounds K o e).2 := by
+      rw [← horizontalProjection_eq_Icc]
+      exact hxproj
+    have hxneleft : x ≠ (horizontalBounds K o e).1 := by
+      intro hx
+      apply hpend
+      refine ⟨x, by simp [hx], ?_⟩
+      exact hgraph.symm
+    have hxneright : x ≠ (horizontalBounds K o e).2 := by
+      intro hx
+      apply hpend
+      refine ⟨x, by simp [hx], ?_⟩
+      exact hgraph.symm
+    have hxint : x ∈ Set.Ioo (horizontalBounds K o e).1
+        (horizontalBounds K o e).2 :=
+      ⟨lt_of_le_of_ne hxIcc.1 (Ne.symm hxneleft), lt_of_le_of_ne hxIcc.2 hxneright⟩
+    have hdiff : DifferentiableAt ℝ (upperGraphHeight K o e) x := by
+      by_contra hndiff
+      apply hpirr
+      refine ⟨x, ⟨hxint, hndiff⟩, ?_⟩
+      exact hgraph.symm
+    have hweight : upperGraphWeight K o e ψ p = ψ a := by
+      unfold upperGraphWeight
+      dsimp only [x]
+      have haeq : a = vectorNormalAngle
+          (e.symm !₂[-deriv (upperGraphHeight K o e) (e (p - o) 0), 1]) := by
+        have he := exteriorNormalAngle_upperGraph_eq K hK o e hxint hdiff
+        dsimp only [x] at he
+        calc
+          a = exteriorNormalAngle K p := rfl
+          _ = exteriorNormalAngle K
+              (o + e.symm !₂[e (p - o) 0, upperGraphHeight K o e (e (p - o) 0)]) :=
+            congrArg (exteriorNormalAngle K) hgraph
+          _ = _ := he
+      rw [haeq]
+    have hev := eventually_mem_innerIcc_of_mem_Ioo hxint
+    apply tendsto_const_nhds.congr'
+    exact hev.mono fun n hn ↦ by
+      have hmem : p ∈ (fun y ↦ o + e.symm !₂[y, upperGraphHeight K o e y] : ℝ → Point) ''
+          Set.Icc ((horizontalBounds K o e).1 + 1 / ((n : ℝ) + 1))
+            ((horizontalBounds K o e).2 - 1 / ((n : ℝ) + 1)) := ⟨x, hn, hgraph.symm⟩
+      change ψ (exteriorNormalAngle K p) = _
+      simp only [Set.indicator, hmem, ↓reduceIte, hweight]
+      rfl
+
+/-- Weighted integrals over compact inner upper graphs converge to the weighted
+integral over the full frontier. -/
+theorem tendsto_integral_innerUpperGraph
+    (K : ConvexBody Point) (hK : (interior (K : Set Point)).Nonempty)
+    (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) (ψ : Real.Angle → ℝ) (hψ : Continuous ψ)
+    {ε : ℝ} (hε : 0 < ε)
+    (hsupport : ∀ t, e (normalVector t) 1 < ε → ψ t = 0) :
+    Filter.Tendsto (fun n : ℕ ↦
+      ∫ p, upperGraphWeight K o e ψ p ∂(Measure.hausdorffMeasure 1).restrict
+        ((fun x ↦ o + e.symm !₂[x, upperGraphHeight K o e x] : ℝ → Point) ''
+          Set.Icc ((horizontalBounds K o e).1 + 1 / ((n : ℝ) + 1))
+            ((horizontalBounds K o e).2 - 1 / ((n : ℝ) + 1)))) Filter.atTop
+      (nhds (∫ p in frontier (K : Set Point), ψ (exteriorNormalAngle K p)
+        ∂Measure.hausdorffMeasure 1)) := by
+  let μ : Measure Point := Measure.hausdorffMeasure 1
+  let t := frontier (K : Set Point)
+  let g : ℕ → Set Point := fun n ↦
+    (fun x ↦ o + e.symm !₂[x, upperGraphHeight K o e x] : ℝ → Point) ''
+      Set.Icc ((horizontalBounds K o e).1 + 1 / ((n : ℝ) + 1))
+        ((horizontalBounds K o e).2 - 1 / ((n : ℝ) + 1))
+  obtain ⟨u, -, hu⟩ := isCompact_univ.exists_isMaxOn Set.univ_nonempty hψ.norm.continuousOn
+  let M := ‖ψ u‖
+  have hM (v : Real.Angle) : ‖ψ v‖ ≤ M := hu trivial
+  have hM0 : 0 ≤ M := norm_nonneg _
+  have hgt (n : ℕ) : g n ⊆ t :=
+    upperCoordinateGraph_image_subset_frontier K o e
+      ((innerIcc_subset_Ioo _ _ n).trans Set.Ioo_subset_Icc_self)
+  have hgmeas (n : ℕ) : MeasurableSet (g n) := measurableSet_innerUpperGraph K o e n
+  have hFmeas (n : ℕ) : AEStronglyMeasurable
+      ((g n).indicator (upperGraphWeight K o e ψ)) μ :=
+    (measurable_upperGraphWeight K o e ψ hψ).aestronglyMeasurable.indicator (hgmeas n)
+  have hbound (n : ℕ) : ∀ᵐ p ∂μ,
+      ‖(g n).indicator (upperGraphWeight K o e ψ) p‖ ≤ t.indicator (fun _ ↦ M) p :=
+    Filter.Eventually.of_forall fun p ↦ by
+      by_cases hp : p ∈ g n
+      · have hpt := hgt n hp
+        rw [Set.indicator_of_mem hp, Set.indicator_of_mem hpt]
+        exact hM _
+      · rw [Set.indicator]
+        simp only [hp, ↓reduceIte, norm_zero]
+        by_cases hpt : p ∈ t <;> simp [hpt, hM0]
+  have hboundInt : Integrable (t.indicator fun _ ↦ M) μ := by
+    apply MeasureTheory.IntegrableOn.integrable_indicator
+    · exact MeasureTheory.integrableOn_const
+        (lt_top_iff_ne_top.mp (K.hausdorffMeasure_frontier_lt_top hK))
+    · exact measurableSet_frontier
+  have hae : t =ᵐ[μ] regularBoundary K := by
+    rw [ae_eq_set]
+    exact ⟨hausdorffMeasure_irregularBoundary_eq_zero K hK,
+      measure_mono_null (fun _ hp ↦ (hp.2 hp.1.1).elim) measure_empty⟩
+  have hpreg : ∀ᵐ p ∂μ, p ∈ t → p ∈ regularBoundary K := by
+    filter_upwards [hae] with p hp
+    exact fun hpt ↦ hp.mp hpt
+  have hpend : ∀ᵐ p ∂μ, p ∉
+      (fun x ↦ o + e.symm !₂[x, upperGraphHeight K o e x] : ℝ → Point) ''
+        ({(horizontalBounds K o e).1, (horizontalBounds K o e).2} : Set ℝ) :=
+    measure_eq_zero_iff_ae_notMem.mp (hausdorffMeasure_upperGraph_endpoints_eq_zero K o e)
+  have hpirr : ∀ᵐ p ∂μ, p ∉ irregularUpperGraph K o e :=
+    measure_eq_zero_iff_ae_notMem.mp (hausdorffMeasure_irregularUpperGraph_eq_zero K o e)
+  have hlim : ∀ᵐ p ∂μ, Filter.Tendsto
+      (fun n ↦ (g n).indicator (upperGraphWeight K o e ψ) p) Filter.atTop
+      (nhds (t.indicator (fun p ↦ ψ (exteriorNormalAngle K p)) p)) := by
+    filter_upwards [hpreg, hpend, hpirr] with p hpreg hpend hpirr
+    by_cases hpt : p ∈ t
+    · rw [Set.indicator_of_mem hpt]
+      exact tendsto_innerUpperGraph_indicator_at_regularPoint K hK o e ψ hε hsupport
+        (hpreg hpt) hpend hpirr
+    · simp only [Set.indicator, hpt, ↓reduceIte]
+      apply tendsto_const_nhds.congr'
+      filter_upwards [] with n
+      have hpgn : p ∉ g n := fun h ↦ hpt (hgt n h)
+      simp [hpgn]
+  have ht := MeasureTheory.tendsto_integral_of_dominated_convergence (μ := μ)
+    (t.indicator fun _ ↦ M) hFmeas hboundInt hbound hlim
+  have htarget : (∫ p, t.indicator (fun p ↦ ψ (exteriorNormalAngle K p)) p ∂μ) =
+      ∫ p in t, ψ (exteriorNormalAngle K p) ∂μ := by
+    rw [MeasureTheory.integral_indicator measurableSet_frontier]
+  rw [← htarget]
+  simpa only [μ, g, MeasureTheory.integral_indicator, hgmeas] using ht
+
+/-- A planar convex body with nonempty interior has distinct horizontal bounds
+in every isometric coordinate frame. -/
+theorem horizontalBounds_lt_of_interior_nonempty (K : ConvexBody Point)
+    (hK : (interior (K : Set Point)).Nonempty) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) :
+    (horizontalBounds K o e).1 < (horizontalBounds K o e).2 := by
+  obtain ⟨z, hz⟩ := hK
+  obtain ⟨ε, hε, hball⟩ := Metric.isOpen_iff.mp isOpen_interior z hz
+  let v := e.symm (normalVector 0)
+  let q := z + (ε / 2) • v
+  have hq : q ∈ K := interior_subset (hball (by
+    rw [Metric.mem_ball, dist_eq_norm]
+    rw [show q - z = (ε / 2) • v by simp [q], norm_smul]
+    simp only [v, e.symm.norm_map, norm_normalVector, mul_one, Real.norm_eq_abs,
+      abs_of_pos (half_pos hε)]
+    linarith))
+  have hzI : e (z - o) 0 ∈ Set.Icc (horizontalBounds K o e).1
+      (horizontalBounds K o e).2 := by
+    rw [← horizontalProjection_eq_Icc]
+    exact ⟨z, interior_subset hz, rfl⟩
+  have hqI : e (q - o) 0 ∈ Set.Icc (horizontalBounds K o e).1
+      (horizontalBounds K o e).2 := by
+    rw [← horizontalProjection_eq_Icc]
+    exact ⟨q, hq, rfl⟩
+  have hcoord : e (q - o) 0 = e (z - o) 0 + ε / 2 := by
+    rw [show q - o = (z - o) + (ε / 2) • v by simp [q]; module]
+    simp [v, normalVector, frame]
+  linarith [hzI.1, hqI.2]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Construction
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+theorem surfaceAreaMeasure_construction (K : ConvexBody Point) :
+    IsFiniteMeasure (surfaceAreaMeasure K) ∧
+    ((K : Set Point).Subsingleton → surfaceAreaMeasure K = 0) ∧
+    (∀ d, IsSegmentPresentation K d → surfaceAreaMeasure K =
+      ENNReal.ofReal (dist d.1 d.2.1) •
+        (Measure.dirac d.2.2 + Measure.dirac (d.2.2 + ((Real.pi : ℝ) : Real.Angle)))) ∧
+    ((interior (K : Set Point)).Nonempty →
+      Measure.hausdorffMeasure 1 (frontier (K : Set Point)) < ⊤ ∧
+      Measure.hausdorffMeasure 1 (frontier (K : Set Point) \ regularBoundary K) = 0 ∧
+      ∀ ν : Point → Real.Angle, (∀ p ∈ regularBoundary K, ν p = exteriorNormalAngle K p) →
+        surfaceAreaMeasure K = Measure.map ν
+          ((Measure.hausdorffMeasure 1).restrict (frontier (K : Set Point)))) ∧
+    (∀ (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) (ψ : Real.Angle → ℝ), Continuous ψ →
+      ∀ ε : ℝ, 0 < ε → (∀ t, e (normalVector t) 1 < ε → ψ t = 0) →
+      ((horizontalBounds K o e).1 < (horizontalBounds K o e).2 →
+        Integrable (upperGraphSurfaceIntegrand K o e ψ)
+          (volume.restrict (Set.Icc (horizontalBounds K o e).1 (horizontalBounds K o e).2)) ∧
+        (∫ t, ψ t ∂surfaceAreaMeasure K) =
+          ∫ x in Set.Icc (horizontalBounds K o e).1 (horizontalBounds K o e).2,
+            upperGraphSurfaceIntegrand K o e ψ x) ∧
+      ((horizontalBounds K o e).1 = (horizontalBounds K o e).2 →
+        (∫ t, ψ t ∂surfaceAreaMeasure K) = 0)) := by
+  by_cases hsub : (K : Set Point).Subsingleton
+  · refine ⟨isFiniteMeasure_surfaceAreaMeasure_of_subsingleton K hsub,
+      fun _ ↦ surfaceAreaMeasure_eq_zero_of_subsingleton K hsub, ?_, ?_, ?_⟩
+    · intro d hd
+      exact (not_subsingleton_of_isSegmentPresentation K hd hsub).elim
+    · intro hint
+      obtain ⟨p, hp⟩ := K.nonempty
+      have hset : (K : Set Point) = {p} :=
+        Set.eq_singleton_iff_unique_mem.mpr ⟨hp, fun x hx ↦ hsub hx hp⟩
+      rw [hset, interior_singleton] at hint
+      exact hint.ne_empty rfl |>.elim
+    · intro o e ψ _ ε _ _
+      have hbounds := horizontalBounds_eq_of_subsingleton K hsub o e
+      constructor
+      · intro hlt
+        exact (hlt.ne hbounds).elim
+      · intro _
+        exact surfaceAreaMeasure_integral_eq_zero_of_subsingleton K hsub ψ
+  · by_cases hint : (interior (K : Set Point)).Nonempty
+    · refine ⟨isFiniteMeasure_surfaceAreaMeasure_of_interior_nonempty K hint,
+        fun hs ↦ (not_subsingleton_of_interior_nonempty K hint hs).elim, ?_, ?_, ?_⟩
+      · intro d hd
+        exact (not_exists_segmentPresentation_of_interior_nonempty K hint ⟨d, hd⟩).elim
+      · intro _
+        refine ⟨K.hausdorffMeasure_frontier_lt_top hint, ?_, ?_⟩
+        · exact hausdorffMeasure_irregularBoundary_eq_zero K hint
+        · intro ν hν
+          exact surfaceAreaMeasure_eq_map_frontier_of_eq_on_regularBoundary K hint ν hν
+      · intro o e ψ hψ ε hε hsupport
+        constructor
+        · intro hwidth
+          have hintg := integrable_upperGraphSurfaceIntegrand K o e ψ hψ hε hsupport
+            (horizontalBounds K o e).1 (horizontalBounds K o e).2
+          refine ⟨hintg, ?_⟩
+          rw [surfaceAreaMeasure_eq_map_frontier_of_eq_on_regularBoundary K hint
+            (exteriorNormalAngle K) (fun _ _ ↦ rfl)]
+          rw [MeasureTheory.integral_map]
+          · apply tendsto_nhds_unique
+              (tendsto_integral_innerUpperGraph K hint o e ψ hψ hε hsupport)
+            apply (tendsto_integral_innerIcc hintg).congr'
+            filter_upwards [eventually_integral_innerUpperGraph_eq_innerIcc
+              K o e ψ hψ hwidth] with n hn
+            exact hn.symm
+          · have hae : frontier (K : Set Point) =ᵐ[Measure.hausdorffMeasure 1]
+                regularBoundary K := by
+              rw [ae_eq_set]
+              exact ⟨hausdorffMeasure_irregularBoundary_eq_zero K hint,
+                measure_mono_null (fun _ hp ↦ (hp.2 hp.1.1).elim) measure_empty⟩
+            have hrestrict : (Measure.hausdorffMeasure 1).restrict
+                (frontier (K : Set Point)) =
+                (Measure.hausdorffMeasure 1).restrict (regularBoundary K) :=
+              Measure.restrict_congr_set hae
+            rw [hrestrict]
+            exact aemeasurable_exteriorNormalAngle_restrict_regularBoundary K
+              (Measure.hausdorffMeasure 1)
+          · exact hψ.aestronglyMeasurable
+        · intro hwidth
+          exact ((horizontalBounds_lt_of_interior_nonempty K hint o e).ne hwidth).elim
+    · have hinterior : interior (K : Set Point) = ∅ := Set.not_nonempty_iff_eq_empty.mp hint
+      have hseg := exists_segmentPresentation_of_interior_empty K hsub hinterior
+      refine ⟨isFiniteMeasure_surfaceAreaMeasure_of_segmentPresentation K hseg.choose_spec,
+        fun hs ↦ (hsub hs).elim, ?_, ?_, ?_⟩
+      · intro d hd
+        exact surfaceAreaMeasure_eq_segmentPresentation K d hd
+      · exact fun hi ↦ (hint hi).elim
+      · intro o e ψ _ ε hε hsupport
+        constructor
+        · intro hbounds
+          exact segment_graph_formula K o e ψ hε hsupport hseg.choose hseg.choose_spec hbounds
+        · intro hbounds
+          exact segment_integral_eq_zero_of_horizontalBounds_eq K o e ψ ε hε hsupport
+            hseg.choose hseg.choose_spec hbounds
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Graph Convergence
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter
+open MeasureTheory
+open scoped Topology
+
+namespace MovingSofa
+
+/-- The angle of a planar vector varies continuously away from zero. -/
+theorem continuousAt_vectorNormalAngle {p : Point} (hp : p ≠ 0) :
+    ContinuousAt vectorNormalAngle p := by
+  have hz : (⟨p 0, p 1⟩ : ℂ) ≠ 0 := by
+    intro h
+    apply hp
+    ext i
+    fin_cases i
+    · exact congrArg Complex.re h
+    · exact congrArg Complex.im h
+  have hc : Continuous (fun q : Point ↦ (⟨q 0, q 1⟩ : ℂ)) := by
+    have h : Continuous (fun q : Point ↦ (q 0 : ℂ) + (q 1 : ℂ) * Complex.I) := by
+      fun_prop
+    convert h using 1
+    funext q
+    apply Complex.ext <;> simp
+  exact (Complex.continuousAt_arg_coe_angle hz).comp
+    (f := fun q : Point ↦ (⟨q 0, q 1⟩ : ℂ)) hc.continuousAt
+
+/-- The weighted upper-graph surface density is continuous as a function of slope. -/
+theorem continuous_surfaceDensity_of_slope (e : Point ≃ₗᵢ[ℝ] Point)
+    {ψ : Real.Angle → ℝ} (hψ : Continuous ψ) :
+    Continuous (fun r : ℝ ↦ ψ (vectorNormalAngle (e.symm !₂[-r, 1])) *
+      Real.sqrt (1 + r ^ 2)) := by
+  have hv : Continuous (fun r : ℝ ↦ vectorNormalAngle (e.symm !₂[-r, 1])) := by
+    apply continuous_iff_continuousAt.mpr
+    intro r
+    have hne : e.symm !₂[-r, 1] ≠ 0 := by
+      intro h
+      have h1 := congrArg (fun p : Point ↦ e p 1) h
+      simp at h1
+    apply (continuousAt_vectorNormalAngle hne).comp
+      (f := fun r : ℝ ↦ e.symm !₂[-r, 1])
+    fun_prop
+  exact (hψ.comp hv).mul (by fun_prop)
+
+private theorem ae_differentiableAt_upperGraphHeight (K : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) :
+    ∀ᵐ x : ℝ, x ∈ Set.Ioo (horizontalBounds K o e).1 (horizontalBounds K o e).2 →
+      DifferentiableAt ℝ (upperGraphHeight K o e) x := by
+  let a := (horizontalBounds K o e).1
+  let b := (horizontalBounds K o e).2
+  let δ : ℕ → ℝ := fun m ↦ 1 / ((m : ℝ) + 1)
+  have hδpos (m : ℕ) : 0 < δ m := by positivity
+  have hcompact (m : ℕ) : IsCompact (Set.Icc (a + δ m) (b - δ m)) := isCompact_Icc
+  have hsub (m : ℕ) : Set.Icc (a + δ m) (b - δ m) ⊆ Set.Ioo a b := by
+    intro x hx
+    exact ⟨lt_of_lt_of_le (lt_add_of_pos_right a (hδpos m)) hx.1,
+      lt_of_le_of_lt hx.2 (sub_lt_self b (hδpos m))⟩
+  have hae (m : ℕ) : ∀ᵐ x : ℝ,
+      x ∈ Set.Ioo (a + δ m) (b - δ m) →
+        DifferentiableAt ℝ (upperGraphHeight K o e) x := by
+    obtain ⟨C, hC⟩ :=
+      ((locallyLipschitzOn_upperGraphHeight K o e).mono (hsub m)).exists_lipschitzOnWith_of_compact
+        (hcompact m)
+    filter_upwards [hC.ae_differentiableWithinAt_of_mem_of_real] with x hdiff hx
+    exact (hdiff ⟨hx.1.le, hx.2.le⟩).differentiableAt (Icc_mem_nhds hx.1 hx.2)
+  have hall : ∀ᵐ x : ℝ, ∀ m, x ∈ Set.Ioo (a + δ m) (b - δ m) →
+      DifferentiableAt ℝ (upperGraphHeight K o e) x :=
+    MeasureTheory.ae_all_iff.mpr hae
+  filter_upwards [hall] with x hxall hx
+  have hδ : Tendsto δ atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
+  have hevent : ∀ᶠ m in atTop, δ m < x - a ∧ δ m < b - x :=
+    ((hδ.eventually_lt_const (sub_pos.mpr hx.1)).and
+      (hδ.eventually_lt_const (sub_pos.mpr hx.2)))
+  obtain ⟨m, hm⟩ := hevent.exists
+  apply hxall m
+  constructor <;> linarith [hm.1, hm.2]
+
+private theorem exists_tendsto_points_of_mem_of_hausdorffDist
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point)
+      (L : Set Point)) atTop (𝓝 0)) {p : Point} (hp : p ∈ L) :
+    ∃ q : ℕ → Point, (∀ n, q n ∈ K n) ∧ Tendsto q atTop (𝓝 p) := by
+  have hnear (n : ℕ) : ∃ q ∈ (K n : Set Point),
+      dist p q < Metric.hausdorffDist (K n : Set Point) (L : Set Point) +
+        1 / ((n : ℝ) + 1) := by
+    obtain ⟨q, hq, hd⟩ := Metric.exists_dist_lt_of_hausdorffDist_lt'
+      (r := Metric.hausdorffDist (K n : Set Point) (L : Set Point) +
+        1 / ((n : ℝ) + 1)) hp
+      (by linarith [show 0 < 1 / ((n : ℝ) + 1) by positivity])
+      (Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded
+        (K n).nonempty L.nonempty (K n).isCompact.isBounded L.isCompact.isBounded)
+    exact ⟨q, hq, by simpa [dist_comm] using hd⟩
+  choose q hqK hqdist using hnear
+  refine ⟨q, hqK, ?_⟩
+  have hsum : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point)
+      (L : Set Point) + 1 / ((n : ℝ) + 1)) atTop (𝓝 0) := by
+    simpa only [add_zero] using hlim.add tendsto_one_div_add_atTop_nhds_zero_nat
+  apply tendsto_iff_dist_tendsto_zero.mpr
+  apply squeeze_zero (fun _ ↦ dist_nonneg) _ hsum
+  intro n
+  simpa [dist_comm] using (hqdist n).le
+
+private theorem exists_horizontal_side_points (L : ConvexBody Point) (o : Point)
+    (e : Point ≃ₗᵢ[ℝ] Point) {x : ℝ}
+    (hx : x ∈ Set.Ioo (horizontalBounds L o e).1 (horizontalBounds L o e).2) :
+    ∃ l ∈ (L : Set Point), ∃ r ∈ (L : Set Point),
+      e (l - o) 0 < x ∧ x < e (r - o) 0 := by
+  let a := (horizontalBounds L o e).1
+  let b := (horizontalBounds L o e).2
+  have hlcoord : (a + x) / 2 ∈ horizontalProjection L o e := by
+    rw [horizontalProjection_eq_Icc]
+    dsimp [a, b]
+    constructor <;> linarith [hx.1, hx.2]
+  have hrcoord : (x + b) / 2 ∈ horizontalProjection L o e := by
+    rw [horizontalProjection_eq_Icc]
+    dsimp [a, b]
+    constructor <;> linarith [hx.1, hx.2]
+  obtain ⟨l, hl, hleq⟩ := hlcoord
+  obtain ⟨r, hr, hreq⟩ := hrcoord
+  refine ⟨l, hl, r, hr, ?_, ?_⟩
+  · change e (l - o) 0 = (a + x) / 2 at hleq
+    rw [hleq]
+    dsimp [a]
+    linarith [hx.1]
+  · change e (r - o) 0 = (x + b) / 2 at hreq
+    rw [hreq]
+    dsimp [b]
+    linarith [hx.2]
+
+/-- Points on an interior horizontal fiber can be approximated within the same fibers. -/
+theorem exists_tendsto_points_with_horizontal_coordinate
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point)
+      (L : Set Point)) atTop (𝓝 0)) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point)
+    {p : Point} (hp : p ∈ L)
+    (hpint : e (p - o) 0 ∈ Set.Ioo (horizontalBounds L o e).1
+      (horizontalBounds L o e).2) :
+    ∃ q : ℕ → Point, (∀ n, q n ∈ K n) ∧ Tendsto q atTop (𝓝 p) ∧
+      ∀ᶠ n in atTop, e (q n - o) 0 = e (p - o) 0 := by
+  let c : Point → ℝ := fun z ↦ e (z - o) 0
+  have hc : Continuous c := by
+    exact (PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 0).comp
+      (e.continuous.comp (continuous_id.sub continuous_const))
+  have hc_lineMap (a b : Point) (t : ℝ) :
+      c (AffineMap.lineMap a b t) = (1 - t) * c a + t * c b := by
+    dsimp [c]
+    rw [AffineMap.lineMap_apply_module]
+    simp only [map_sub, map_add, map_smul, PiLp.sub_apply, PiLp.add_apply,
+      PiLp.smul_apply, smul_eq_mul]
+    ring
+  obtain ⟨l, hlL, r, hrL, hl, hr⟩ := exists_horizontal_side_points L o e hpint
+  obtain ⟨pn, hpnK, hpn⟩ := exists_tendsto_points_of_mem_of_hausdorffDist K L hlim hp
+  obtain ⟨ln, hlnK, hln⟩ := exists_tendsto_points_of_mem_of_hausdorffDist K L hlim hlL
+  obtain ⟨rn, hrnK, hrn⟩ := exists_tendsto_points_of_mem_of_hausdorffDist K L hlim hrL
+  let x := c p
+  let tR : ℕ → ℝ := fun n ↦ (x - c (pn n)) / (c (rn n) - c (pn n))
+  let tL : ℕ → ℝ := fun n ↦ (c (pn n) - x) / (c (pn n) - c (ln n))
+  let qR : ℕ → Point := fun n ↦ AffineMap.lineMap (pn n) (rn n) (tR n)
+  let qL : ℕ → Point := fun n ↦ AffineMap.lineMap (pn n) (ln n) (tL n)
+  let corrected : ℕ → Point := fun n ↦ if c (pn n) ≤ x then qR n else qL n
+  let good : ℕ → Prop := fun n ↦ c (ln n) < x ∧ x < c (rn n)
+  let q : ℕ → Point := fun n ↦ if good n then corrected n else pn n
+  have hcp : Tendsto (fun n ↦ c (pn n)) atTop (𝓝 (c p)) := (hc.tendsto p).comp hpn
+  have hcl : Tendsto (fun n ↦ c (ln n)) atTop (𝓝 (c l)) := (hc.tendsto l).comp hln
+  have hcr : Tendsto (fun n ↦ c (rn n)) atTop (𝓝 (c r)) := (hc.tendsto r).comp hrn
+  have htR : Tendsto tR atTop (𝓝 0) := by
+    have hden : c r - c p ≠ 0 := by
+      dsimp [c] at hr
+      linarith
+    have hxlim : Tendsto (fun _ : ℕ ↦ x) atTop (𝓝 x) := tendsto_const_nhds
+    change Tendsto ((fun n ↦ x - c (pn n)) /
+      fun n ↦ c (rn n) - c (pn n)) atTop (𝓝 0)
+    simpa [x] using (hxlim.sub hcp).div (hcr.sub hcp) hden
+  have htL : Tendsto tL atTop (𝓝 0) := by
+    have hden : c p - c l ≠ 0 := by
+      dsimp [c] at hl
+      linarith
+    have hxlim : Tendsto (fun _ : ℕ ↦ x) atTop (𝓝 x) := tendsto_const_nhds
+    change Tendsto ((fun n ↦ c (pn n) - x) /
+      fun n ↦ c (pn n) - c (ln n)) atTop (𝓝 0)
+    simpa [x] using (hcp.sub hxlim).div (hcp.sub hcl) hden
+  have hqR : Tendsto qR atTop (𝓝 p) := by
+    simpa [qR] using hpn.lineMap hrn htR
+  have hqL : Tendsto qL atTop (𝓝 p) := by
+    simpa [qL] using hpn.lineMap hln htL
+  have hcorrected : Tendsto corrected atTop (𝓝 p) := by
+    exact hqR.if' hqL
+  have hq : Tendsto q atTop (𝓝 p) := by
+    exact hcorrected.if' hpn
+  have hgood : ∀ᶠ n in atTop, good n := by
+    filter_upwards [hcl.eventually_lt tendsto_const_nhds hl,
+      tendsto_const_nhds.eventually_lt hcr hr] with n hnl hnr
+    exact ⟨hnl, hnr⟩
+  refine ⟨q, ?_, hq, ?_⟩
+  · intro n
+    by_cases hgn : good n
+    · simp only [q, hgn, ite_true, corrected]
+      dsimp [good] at hgn
+      by_cases hpnx : c (pn n) ≤ x
+      · simp only [hpnx, ite_true, qR]
+        apply (K n).convex.lineMap_mem (hpnK n) (hrnK n)
+        constructor
+        · exact div_nonneg (sub_nonneg.mpr hpnx) (sub_nonneg.mpr (le_trans hpnx hgn.2.le))
+        · apply (div_le_one (sub_pos.mpr (lt_of_le_of_lt hpnx hgn.2))).mpr
+          linarith
+      · simp only [hpnx, ite_false, qL]
+        apply (K n).convex.lineMap_mem (hpnK n) (hlnK n)
+        constructor
+        · exact div_nonneg (sub_nonneg.mpr (le_of_not_ge hpnx))
+            (sub_nonneg.mpr (le_trans hgn.1.le (le_of_not_ge hpnx)))
+        · apply (div_le_one (sub_pos.mpr (lt_of_lt_of_le hgn.1 (le_of_not_ge hpnx)))).mpr
+          linarith
+    · simp [q, hgn, hpnK n]
+  · filter_upwards [hgood] with n hgn
+    simp only [q, hgn, ite_true, corrected]
+    dsimp [good] at hgn
+    by_cases hpnx : c (pn n) ≤ x
+    · simp only [hpnx, ite_true, qR, tR]
+      rw [show e (AffineMap.lineMap (pn n) (rn n)
+          ((x - c (pn n)) / (c (rn n) - c (pn n))) - o) 0 =
+          c (AffineMap.lineMap (pn n) (rn n)
+            ((x - c (pn n)) / (c (rn n) - c (pn n)))) by rfl]
+      rw [hc_lineMap]
+      change _ = x
+      field_simp [ne_of_gt (sub_pos.mpr (lt_of_le_of_lt hpnx hgn.2))]
+      ring
+    · simp only [hpnx, ite_false, qL, tL]
+      rw [show e (AffineMap.lineMap (pn n) (ln n)
+          ((c (pn n) - x) / (c (pn n) - c (ln n))) - o) 0 =
+          c (AffineMap.lineMap (pn n) (ln n)
+            ((c (pn n) - x) / (c (pn n) - c (ln n)))) by rfl]
+      rw [hc_lineMap]
+      change _ = x
+      field_simp [ne_of_gt (sub_pos.mpr (lt_of_lt_of_le hgn.1 (le_of_not_ge hpnx)))]
+      ring
+
+/-- A varying point of the approximating bodies is asymptotically close to the limit body. -/
+private theorem exists_points_in_limit_tendsto_dist_zero
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point)
+      (L : Set Point)) atTop (𝓝 0)) (p : ℕ → Point) (hp : ∀ n, p n ∈ K n) :
+    ∃ q : ℕ → Point, (∀ n, q n ∈ L) ∧
+      Tendsto (fun n ↦ dist (p n) (q n)) atTop (𝓝 0) := by
+  have hnear (n : ℕ) : ∃ q ∈ (L : Set Point),
+      dist (p n) q < Metric.hausdorffDist (K n : Set Point) (L : Set Point) +
+        1 / ((n : ℝ) + 1) := by
+    obtain ⟨q, hq, hd⟩ := Metric.exists_dist_lt_of_hausdorffDist_lt
+      (r := Metric.hausdorffDist (K n : Set Point) (L : Set Point) +
+        1 / ((n : ℝ) + 1)) (hp n)
+      (by linarith [show 0 < 1 / ((n : ℝ) + 1) by positivity])
+      (Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded
+        (K n).nonempty L.nonempty (K n).isCompact.isBounded L.isCompact.isBounded)
+    exact ⟨q, hq, hd⟩
+  choose q hqL hqdist using hnear
+  refine ⟨q, hqL, ?_⟩
+  have hsum : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point)
+      (L : Set Point) + 1 / ((n : ℝ) + 1)) atTop (𝓝 0) := by
+    simpa only [add_zero] using hlim.add tendsto_one_div_add_atTop_nhds_zero_nat
+  apply squeeze_zero (fun _ ↦ dist_nonneg) _ hsum
+  exact fun n ↦ (hqdist n).le
+
+/-- Upper graph heights converge at every interior point of the limit projection. -/
+theorem tendsto_upperGraphHeight_of_hausdorffDist
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point)
+      (L : Set Point)) atTop (𝓝 0)) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point)
+    {x : ℝ} (hx : x ∈ Set.Ioo (horizontalBounds L o e).1
+      (horizontalBounds L o e).2) :
+    Tendsto (fun n ↦ upperGraphHeight (K n) o e x) atTop
+      (𝓝 (upperGraphHeight L o e x)) := by
+  classical
+  let cx : Point → ℝ := fun p ↦ e (p - o) 0
+  let cy : Point → ℝ := fun p ↦ e (p - o) 1
+  have reconstruct (p : Point) : o + e.symm !₂[cx p, cy p] = p := by
+    dsimp [cx, cy]
+    rw [← e.injective.eq_iff]
+    ext i
+    fin_cases i <;> simp
+  have hxproj : x ∈ horizontalProjection L o e := by
+    rw [horizontalProjection_eq_Icc]
+    exact ⟨hx.1.le, hx.2.le⟩
+  let pTop := o + e.symm !₂[x, upperGraphHeight L o e x]
+  have hpTop : pTop ∈ L := upperGraphHeight_mem L o e hxproj
+  have hcxTop : cx pTop = x := by simp [cx, pTop]
+  have hcyTop : cy pTop = upperGraphHeight L o e x := by simp [cy, pTop]
+  have hpTopInt : cx pTop ∈ Set.Ioo (horizontalBounds L o e).1
+      (horizontalBounds L o e).2 := by simpa [hcxTop] using hx
+  obtain ⟨q, hqK, hq, hqx⟩ :=
+    exists_tendsto_points_with_horizontal_coordinate K L hlim o e hpTop hpTopInt
+  have hcyq : Tendsto (fun n ↦ cy (q n)) atTop
+      (𝓝 (upperGraphHeight L o e x)) := by
+    have hcy : Continuous cy := by
+      exact (PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 1).comp
+        (e.continuous.comp (continuous_id.sub continuous_const))
+    change Tendsto (cy ∘ q) atTop (𝓝 (upperGraphHeight L o e x))
+    simpa [hcyTop] using (hcy.tendsto pTop).comp hq
+  have hqcx : ∀ᶠ n in atTop, cx (q n) = x := by
+    filter_upwards [hqx] with n hnx
+    dsimp [cx]
+    rw [hnx]
+    exact hcxTop
+  have hqproj : ∀ᶠ n in atTop, x ∈ horizontalProjection (K n) o e := by
+    filter_upwards [hqcx] with n hnx
+    refine ⟨q n, hqK n, ?_⟩
+    exact hnx
+  let z : ℕ → Point := fun n ↦ if hn : x ∈ horizontalProjection (K n) o e then
+    o + e.symm !₂[x, upperGraphHeight (K n) o e x] else q n
+  have hzK (n : ℕ) : z n ∈ K n := by
+    dsimp [z]
+    split_ifs with hn
+    · exact upperGraphHeight_mem (K n) o e hn
+    · exact hqK n
+  have hzeq : z =ᶠ[atTop]
+      fun n ↦ o + e.symm !₂[x, upperGraphHeight (K n) o e x] := by
+    filter_upwards [hqproj] with n hn
+    simp [z, hn]
+  obtain ⟨w, hwL, hd⟩ := exists_points_in_limit_tendsto_dist_zero K L hlim z hzK
+  have hsub : Tendsto (fun n ↦ z n - w n) atTop (𝓝 0) := by
+    apply tendsto_iff_dist_tendsto_zero.mpr
+    simpa [dist_eq_norm] using hd
+  have hdx : Tendsto (fun n ↦ cx (z n) - cx (w n)) atTop (𝓝 0) := by
+    have heval : Continuous (fun v : Point ↦ e v 0) :=
+      (PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 0).comp e.continuous
+    rw [show (fun n ↦ cx (z n) - cx (w n)) =
+        fun n ↦ e (z n - w n) 0 by
+      funext n
+      simp [cx]]
+    change Tendsto ((fun v : Point ↦ e v 0) ∘ fun n ↦ z n - w n) atTop (𝓝 0)
+    simpa only [map_zero, PiLp.zero_apply] using (heval.tendsto 0).comp hsub
+  have hdy : Tendsto (fun n ↦ cy (z n) - cy (w n)) atTop (𝓝 0) := by
+    have heval : Continuous (fun v : Point ↦ e v 1) :=
+      (PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 1).comp e.continuous
+    rw [show (fun n ↦ cy (z n) - cy (w n)) =
+        fun n ↦ e (z n - w n) 1 by
+      funext n
+      simp [cy]]
+    change Tendsto ((fun v : Point ↦ e v 1) ∘ fun n ↦ z n - w n) atTop (𝓝 0)
+    simpa only [map_zero, PiLp.zero_apply] using (heval.tendsto 0).comp hsub
+  have hcxz : Tendsto (fun n ↦ cx (z n)) atTop (𝓝 x) := by
+    apply tendsto_const_nhds.congr'
+    filter_upwards [hzeq] with n hn
+    rw [hn]
+    simp [cx]
+  have hcxw : Tendsto (fun n ↦ cx (w n)) atTop (𝓝 x) := by
+    have h := hcxz.sub hdx
+    convert h using 1 <;> simp
+  have hgL : Tendsto (fun n ↦ upperGraphHeight L o e (cx (w n))) atTop
+      (𝓝 (upperGraphHeight L o e x)) := by
+    have hcont : ContinuousAt (upperGraphHeight L o e) x :=
+      (locallyLipschitzOn_upperGraphHeight L o e).continuousOn.continuousAt
+        (isOpen_Ioo.mem_nhds hx)
+    exact hcont.tendsto.comp hcxw
+  have hlower : ∀ᶠ n in atTop,
+      cy (q n) ≤ upperGraphHeight (K n) o e x := by
+    filter_upwards [hqcx] with n hnx
+    apply (upperGraphHeight_isGreatest (K n) o e (by
+      exact ⟨q n, hqK n, hnx⟩)).2
+    change o + e.symm !₂[x, cy (q n)] ∈ K n
+    rw [show o + e.symm !₂[x, cy (q n)] = q n by
+      calc
+        o + e.symm !₂[x, cy (q n)] = o + e.symm !₂[cx (q n), cy (q n)] := by
+          rw [hnx]
+        _ = q n := reconstruct (q n)]
+    exact hqK n
+  have hupper : ∀ᶠ n in atTop,
+      upperGraphHeight (K n) o e x ≤
+        upperGraphHeight L o e (cx (w n)) + (cy (z n) - cy (w n)) := by
+    filter_upwards [hzeq] with n hzn
+    have hwproj : cx (w n) ∈ horizontalProjection L o e :=
+      ⟨w n, hwL n, rfl⟩
+    have hwy : cy (w n) ≤ upperGraphHeight L o e (cx (w n)) := by
+      apply (upperGraphHeight_isGreatest L o e hwproj).2
+      change o + e.symm !₂[cx (w n), cy (w n)] ∈ L
+      rw [reconstruct (w n)]
+      exact hwL n
+    have hzy : cy (z n) = upperGraphHeight (K n) o e x := by
+      rw [hzn]
+      simp [cy]
+    rw [← hzy]
+    linarith
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' hcyq
+    (by simpa using hgL.add hdy) hlower hupper
+
+private theorem eventually_mem_horizontalProjection_of_hausdorffDist
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point)
+      (L : Set Point)) atTop (𝓝 0)) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point)
+    {x : ℝ} (hx : x ∈ Set.Ioo (horizontalBounds L o e).1
+      (horizontalBounds L o e).2) :
+    ∀ᶠ n in atTop, x ∈ horizontalProjection (K n) o e := by
+  have hxproj : x ∈ horizontalProjection L o e := by
+    rw [horizontalProjection_eq_Icc]
+    exact ⟨hx.1.le, hx.2.le⟩
+  let p := o + e.symm !₂[x, upperGraphHeight L o e x]
+  have hp : p ∈ L := upperGraphHeight_mem L o e hxproj
+  have hpcoord : e (p - o) 0 = x := by simp [p]
+  have hpint : e (p - o) 0 ∈ Set.Ioo (horizontalBounds L o e).1
+      (horizontalBounds L o e).2 := by
+    rw [hpcoord]
+    exact hx
+  obtain ⟨q, hqK, -, hqx⟩ :=
+    exists_tendsto_points_with_horizontal_coordinate K L hlim o e hp hpint
+  filter_upwards [hqx] with n hn
+  exact ⟨q n, hqK n, hn.trans hpcoord⟩
+
+/-- Upper graph surface densities converge almost everywhere on the interior limit projection. -/
+theorem ae_tendsto_upperGraphSurfaceIntegrand_of_hausdorffDist
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point)
+      (L : Set Point)) atTop (𝓝 0)) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point)
+    {ψ : Real.Angle → ℝ} (hψ : Continuous ψ) :
+    ∀ᵐ x : ℝ, x ∈ Set.Ioo (horizontalBounds L o e).1
+      (horizontalBounds L o e).2 →
+        Tendsto (fun n ↦ upperGraphSurfaceIntegrand (K n) o e ψ x) atTop
+          (𝓝 (upperGraphSurfaceIntegrand L o e ψ x)) := by
+  have haeK : ∀ᵐ x : ℝ, ∀ n,
+      x ∈ Set.Ioo (horizontalBounds (K n) o e).1 (horizontalBounds (K n) o e).2 →
+        DifferentiableAt ℝ (upperGraphHeight (K n) o e) x :=
+    MeasureTheory.ae_all_iff.mpr fun n ↦ ae_differentiableAt_upperGraphHeight (K n) o e
+  have haeL := ae_differentiableAt_upperGraphHeight L o e
+  filter_upwards [haeK, haeL] with x hdiffK hdiffL hx
+  let left := (horizontalBounds L o e).1
+  let right := (horizontalBounds L o e).2
+  let a := (left + x) / 2
+  let b := (x + right) / 2
+  have hax : a < x := by dsimp [a, left]; linarith [hx.1]
+  have hxb : x < b := by dsimp [b, right]; linarith [hx.2]
+  have haL : a ∈ Set.Ioo (horizontalBounds L o e).1
+      (horizontalBounds L o e).2 := by
+    dsimp [a, left]
+    constructor <;> linarith [hx.1, hx.2]
+  have hbL : b ∈ Set.Ioo (horizontalBounds L o e).1
+      (horizontalBounds L o e).2 := by
+    dsimp [b, right]
+    constructor <;> linarith [hx.1, hx.2]
+  have haK := eventually_mem_horizontalProjection_of_hausdorffDist K L hlim o e haL
+  have hbK := eventually_mem_horizontalProjection_of_hausdorffDist K L hlim o e hbL
+  have hconc : ∀ᶠ n in atTop,
+      ConcaveOn ℝ (Set.Ioo a b) (upperGraphHeight (K n) o e) := by
+    filter_upwards [haK, hbK] with n han hbn
+    apply (concaveOn_upperGraphHeight (K n) o e).subset _ (convex_Ioo a b)
+    intro y hy
+    rw [horizontalProjection_eq_Icc] at han hbn ⊢
+    exact ⟨han.1.trans hy.1.le, hy.2.le.trans hbn.2⟩
+  have hdiffEventually : ∀ᶠ n in atTop,
+      DifferentiableAt ℝ (upperGraphHeight (K n) o e) x := by
+    filter_upwards [haK, hbK] with n han hbn
+    apply hdiffK n
+    rw [horizontalProjection_eq_Icc] at han hbn
+    exact ⟨han.1.trans_lt hax, hxb.trans_le hbn.2⟩
+  have hpointwise (y : ℝ) (hy : y ∈ Set.Ioo a b) :
+      Tendsto (fun n ↦ upperGraphHeight (K n) o e y) atTop
+        (𝓝 (upperGraphHeight L o e y)) := by
+    apply tendsto_upperGraphHeight_of_hausdorffDist K L hlim o e
+    exact ⟨haL.1.trans hy.1, hy.2.trans hbL.2⟩
+  have hderiv : Tendsto (fun n ↦ deriv (upperGraphHeight (K n) o e) x) atTop
+      (𝓝 (deriv (upperGraphHeight L o e) x)) :=
+    ConcaveOn.tendsto_deriv_of_tendsto_Ioo ⟨hax, hxb⟩ hconc hpointwise hdiffEventually
+      (hdiffL hx)
+  have hdensity := (continuous_surfaceDensity_of_slope e hψ).continuousAt.tendsto.comp hderiv
+  change Tendsto ((fun r : ℝ ↦ ψ (vectorNormalAngle (e.symm !₂[-r, 1])) *
+    Real.sqrt (1 + r ^ 2)) ∘ fun n ↦ deriv (upperGraphHeight (K n) o e) x)
+      atTop (𝓝 (upperGraphSurfaceIntegrand L o e ψ x))
+  simpa only [upperGraphSurfaceIntegrand] using hdensity
+
+private theorem tendsto_vectorSupport_of_hausdorffDist
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point)
+      (L : Set Point)) atTop (𝓝 0)) {u : Point} (hu : ‖u‖ = 1) :
+    Tendsto (fun n ↦ vectorSupport (K n) u) atTop (𝓝 (vectorSupport L u)) := by
+  apply tendsto_iff_dist_tendsto_zero.mpr
+  apply squeeze_zero (fun _ ↦ dist_nonneg) _ hlim
+  intro n
+  simpa only [Real.dist_eq] using
+    (compactSet_support_continuity (K n) L (K n).nonempty
+      (K n).isCompact L.nonempty L.isCompact).2.1 u hu
+
+private theorem horizontalBounds_snd_eq_vectorSupport (K : ConvexBody Point)
+    (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) :
+    (horizontalBounds K o e).2 =
+      vectorSupport K (e.symm !₂[1, 0]) - inner ℝ o (e.symm !₂[1, 0]) := by
+  let u : Point := e.symm !₂[1, 0]
+  have hu (p : Point) : e (p - o) 0 = inner ℝ p u - inner ℝ o u := by
+    rw [← inner_sub_left, ← e.inner_map_map]
+    simp [u, PiLp.inner_apply, Fin.sum_univ_two]
+  obtain ⟨p, hp, hsup, hge⟩ := K.isCompact.exists_sSup_image_eq_and_ge K.nonempty
+    (show Continuous (fun p : Point ↦ inner ℝ p u) from
+      continuous_id.inner continuous_const).continuousOn
+  change sSup ((fun p : Point ↦ e (p - o) 0) '' (K : Set Point)) = _
+  rw [show vectorSupport K u = inner ℝ p u by exact hsup]
+  apply le_antisymm
+  · apply csSup_le (K.nonempty.image _)
+    rintro _ ⟨q, hq, rfl⟩
+    change e (q - o) 0 ≤ inner ℝ p u - inner ℝ o u
+    rw [hu q]
+    exact sub_le_sub_right (hge q hq) _
+  · apply le_csSup
+    · have hc : Continuous (fun p : Point ↦ e (p - o)) :=
+        e.continuous.comp (continuous_id.sub continuous_const)
+      exact K.isCompact.bddAbove_image
+        ((PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 0).comp hc).continuousOn
+    · exact ⟨p, hp, hu p⟩
+
+private theorem horizontalBounds_fst_eq_vectorSupport (K : ConvexBody Point)
+    (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) :
+    (horizontalBounds K o e).1 =
+      -vectorSupport K (-(e.symm !₂[1, 0])) - inner ℝ o (e.symm !₂[1, 0]) := by
+  let u : Point := e.symm !₂[1, 0]
+  have hu (p : Point) : e (p - o) 0 = inner ℝ p u - inner ℝ o u := by
+    rw [← inner_sub_left, ← e.inner_map_map]
+    simp [u, PiLp.inner_apply, Fin.sum_univ_two]
+  obtain ⟨p, hp, hsup, hge⟩ := K.isCompact.exists_sSup_image_eq_and_ge K.nonempty
+    (show Continuous (fun p : Point ↦ inner ℝ p (-u)) from
+      continuous_id.inner continuous_const).continuousOn
+  change sInf ((fun p : Point ↦ e (p - o) 0) '' (K : Set Point)) = _
+  rw [show vectorSupport K (-u) = inner ℝ p (-u) by exact hsup]
+  apply le_antisymm
+  · apply csInf_le
+    · have hc : Continuous (fun p : Point ↦ e (p - o)) :=
+        e.continuous.comp (continuous_id.sub continuous_const)
+      exact K.isCompact.bddBelow_image
+        ((PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 0).comp hc).continuousOn
+    · refine ⟨p, hp, ?_⟩
+      change e (p - o) 0 = -inner ℝ p (-u) - inner ℝ o u
+      rw [hu p]
+      simp
+  · apply le_csInf (K.nonempty.image _)
+    rintro _ ⟨q, hq, rfl⟩
+    change -inner ℝ p (-u) - inner ℝ o u ≤ e (q - o) 0
+    rw [hu q]
+    have h := hge q hq
+    simp only [inner_neg_right] at h ⊢
+    linarith
+
+/-- Horizontal projection endpoints converge with their convex bodies. -/
+private theorem tendsto_horizontalBounds_of_hausdorffDist
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point)
+      (L : Set Point)) atTop (𝓝 0)) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point) :
+    Tendsto (fun n ↦ horizontalBounds (K n) o e) atTop
+      (𝓝 (horizontalBounds L o e)) := by
+  let u : Point := e.symm !₂[1, 0]
+  have hu : ‖u‖ = 1 := by
+    rw [show ‖u‖ = ‖(!₂[1, 0] : Point)‖ by exact e.symm.norm_map _]
+    rw [EuclideanSpace.norm_eq]
+    norm_num [Fin.sum_univ_two]
+  rw [show (fun n ↦ horizontalBounds (K n) o e) =
+      fun n ↦ (-vectorSupport (K n) (-u) - inner ℝ o u,
+        vectorSupport (K n) u - inner ℝ o u) by
+      funext n
+      ext <;> simp [horizontalBounds_fst_eq_vectorSupport,
+        horizontalBounds_snd_eq_vectorSupport, u]]
+  rw [show horizontalBounds L o e =
+      (-vectorSupport L (-u) - inner ℝ o u,
+        vectorSupport L u - inner ℝ o u) by
+      ext <;> simp [horizontalBounds_fst_eq_vectorSupport,
+        horizontalBounds_snd_eq_vectorSupport, u]]
+  apply Tendsto.prodMk_nhds
+  · exact (tendsto_vectorSupport_of_hausdorffDist K L hlim (by simpa using hu)).neg.sub_const _
+  · exact (tendsto_vectorSupport_of_hausdorffDist K L hlim hu).sub_const _
+
+private theorem weightedSurfaceIntegral_eq_upperGraphIntegral
+    (K : ConvexBody Point) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point)
+    (ψ : Real.Angle → ℝ) (hψ : Continuous ψ) {ε : ℝ} (hε : 0 < ε)
+    (hsupport : ∀ t, e (normalVector t) 1 < ε → ψ t = 0) :
+    (∫ t, ψ t ∂surfaceAreaMeasure K) =
+      ∫ x in Set.Icc (horizontalBounds K o e).1 (horizontalBounds K o e).2,
+        upperGraphSurfaceIntegrand K o e ψ x := by
+  have hformula := (surfaceAreaMeasure_construction K).2.2.2.2 o e ψ hψ ε hε hsupport
+  have hle : (horizontalBounds K o e).1 ≤ (horizontalBounds K o e).2 := by
+    obtain ⟨x, hx⟩ := K.nonempty.image (fun p : Point ↦ e (p - o) 0)
+    change x ∈ horizontalProjection K o e at hx
+    rw [horizontalProjection_eq_Icc] at hx
+    exact hx.1.trans hx.2
+  rcases hle.lt_or_eq with hlt | heq
+  · exact (hformula.1 hlt).2
+  · rw [hformula.2 heq, heq]
+    simp
+
+/-- Weighted upper-normal surface integrals are continuous under Hausdorff convergence. -/
+theorem tendsto_integral_surfaceAreaMeasure_of_hausdorffDist
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point)
+      (L : Set Point)) atTop (𝓝 0)) (o : Point) (e : Point ≃ₗᵢ[ℝ] Point)
+    {ψ : Real.Angle → ℝ} (hψ : Continuous ψ) {ε : ℝ} (hε : 0 < ε)
+    (hsupport : ∀ t, e (normalVector t) 1 < ε → ψ t = 0) :
+    Tendsto (fun n ↦ ∫ t, ψ t ∂surfaceAreaMeasure (K n)) atTop
+      (𝓝 (∫ t, ψ t ∂surfaceAreaMeasure L)) := by
+  obtain ⟨t, -, ht⟩ := isCompact_univ.exists_isMaxOn Set.univ_nonempty hψ.norm.continuousOn
+  let M := ‖ψ t‖
+  have hM (u : Real.Angle) : ‖ψ u‖ ≤ M := ht trivial
+  have hM0 : 0 ≤ M := norm_nonneg _
+  have hb := tendsto_horizontalBounds_of_hausdorffDist K L hlim o e
+  have ha : Tendsto (fun n ↦ (horizontalBounds (K n) o e).1) atTop
+      (𝓝 (horizontalBounds L o e).1) := continuousAt_fst.tendsto.comp hb
+  have hb' : Tendsto (fun n ↦ (horizontalBounds (K n) o e).2) atTop
+      (𝓝 (horizontalBounds L o e).2) := continuousAt_snd.tendsto.comp hb
+  have hgraph := tendsto_integral_Icc_of_tendsto_endpoints_of_bound ha hb'
+    (Eventually.of_forall fun n ↦
+      (integrable_upperGraphSurfaceIntegrand (K n) o e ψ hψ hε hsupport _ _).1)
+    (mul_nonneg hM0 (inv_nonneg.mpr hε.le))
+    (Eventually.of_forall fun n ↦ Eventually.of_forall fun x _ ↦
+      norm_upperGraphSurfaceIntegrand_le (K n) o e ψ hε hsupport hM x)
+    (ae_tendsto_upperGraphSurfaceIntegrand_of_hausdorffDist K L hlim o e hψ)
+  rw [weightedSurfaceIntegral_eq_upperGraphIntegral L o e ψ hψ hε hsupport]
+  apply hgraph.congr'
+  filter_upwards [] with n
+  exact (weightedSurfaceIntegral_eq_upperGraphIntegral (K n) o e ψ hψ hε hsupport).symm
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Properties
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+private theorem mem_exposedEdge_iff_isExteriorNormal (K : ConvexBody Point)
+    {p : Point} (hp : p ∈ K) (t : Real.Angle) :
+    p ∈ exposedEdge K t ↔ IsExteriorNormal K p t := by
+  constructor
+  · intro h q hq
+    have he : inner ℝ p (normalVector t) = supportValue K t := h.2
+    rw [inner_sub_left, he]
+    exact sub_nonpos.mpr (inner_le_supportValue K hq t)
+  · intro h
+    refine ⟨hp, le_antisymm (inner_le_supportValue K hp t) ?_⟩
+    apply csSup_le (K.nonempty.image _)
+    rintro _ ⟨q, hq, rfl⟩
+    exact sub_nonpos.mp (by simpa only [inner_sub_left] using h q hq)
+
+private theorem mem_exposedEdge_iff_exteriorNormalAngle_eq (K : ConvexBody Point)
+    {p : Point} (hp : p ∈ regularBoundary K) (t : Real.Angle) :
+    p ∈ exposedEdge K t ↔ exteriorNormalAngle K p = t := by
+  have hpK : p ∈ K := by
+    change p ∈ (K : Set Point)
+    simpa only [K.isCompact.isClosed.closure_eq] using frontier_subset_closure hp.1
+  rw [mem_exposedEdge_iff_isExteriorNormal K hpK]
+  have hn : IsExteriorNormal K p (exteriorNormalAngle K p) := by
+    simp only [exteriorNormalAngle, dite_eq_left hp.2]
+    exact hp.2.exists.choose_spec
+  exact ⟨fun ht ↦ hp.2.unique hn ht, fun ht ↦ ht ▸ hn⟩
+
+private theorem faceUnion_inter_regularBoundary (K : ConvexBody Point) (E : Set Real.Angle) :
+    (⋃ t ∈ E, exposedEdge K t) ∩ regularBoundary K =
+      exteriorNormalAngle K ⁻¹' E ∩ regularBoundary K := by
+  ext p
+  constructor
+  · rintro ⟨hp, hreg⟩
+    obtain ⟨t, ht, hp⟩ := Set.mem_iUnion₂.mp hp
+    refine ⟨?_, hreg⟩
+    change exteriorNormalAngle K p ∈ E
+    rwa [(mem_exposedEdge_iff_exteriorNormalAngle_eq K hreg t).mp hp]
+  · rintro ⟨hp, hreg⟩
+    exact ⟨Set.mem_iUnion₂.mpr ⟨exteriorNormalAngle K p, hp,
+      (mem_exposedEdge_iff_exteriorNormalAngle_eq K hreg _).mpr rfl⟩, hreg⟩
+
+private theorem surfaceAreaMeasure_face_union_of_interior_nonempty (K : ConvexBody Point)
+    (hK : (interior (K : Set Point)).Nonempty)
+    (E : Set Real.Angle) (hE : MeasurableSet E) :
+    surfaceAreaMeasure K E = Measure.hausdorffMeasure 1 (⋃ t ∈ E, exposedEdge K t) := by
+  simp only [surfaceAreaMeasure,
+    ite_eq_right (not_subsingleton_of_interior_nonempty K hK),
+    dite_eq_right (not_exists_segmentPresentation_of_interior_nonempty K hK)]
+  rw [Measure.map_apply_of_aemeasurable
+      (aemeasurable_exteriorNormalAngle_restrict_regularBoundary K _) hE,
+    Measure.restrict_apply' (measurableSet_regularBoundary K),
+    ← faceUnion_inter_regularBoundary]
+  apply measure_congr
+  rw [ae_eq_set]
+  constructor
+  · apply measure_mono_null (fun p hp ↦ (hp.2 hp.1.1).elim) measure_empty
+  · apply measure_mono_null ?_ (hausdorffMeasure_irregularBoundary_eq_zero K hK)
+    rintro p ⟨hp, hnot⟩
+    refine ⟨?_, fun hreg ↦ hnot ⟨hp, hreg⟩⟩
+    obtain ⟨t, ht, hpt⟩ := Set.mem_iUnion₂.mp hp
+    exact mem_frontier_of_mem_of_isExteriorNormal K hpt.1
+      ((mem_exposedEdge_iff_isExteriorNormal K hpt.1 t).mp hpt)
+
+private theorem surfaceAreaMeasure_face_union_of_subsingleton (K : ConvexBody Point)
+    (hK : (K : Set Point).Subsingleton) (E : Set Real.Angle) :
+    surfaceAreaMeasure K E = Measure.hausdorffMeasure 1 (⋃ t ∈ E, exposedEdge K t) := by
+  rw [surfaceAreaMeasure_eq_zero_of_subsingleton K hK]
+  change 0 = _
+  symm
+  let _ := Measure.nullSingletonClass_hausdorff Point (by norm_num : (0 : ℝ) < 1)
+  obtain ⟨p, hp⟩ := K.nonempty
+  apply measure_mono_null (t := {p})
+  · intro q hq
+    obtain ⟨t, ht, hqt⟩ := Set.mem_iUnion₂.mp hq
+    exact hK hqt.1 hp
+  · simp
+
+theorem surfaceAreaMeasure_face_union (K : ConvexBody Point) :
+    IsFiniteMeasure (surfaceAreaMeasure K) ∧
+    ((K : Set Point).Subsingleton → surfaceAreaMeasure K = 0) ∧
+    (∀ d, IsSegmentPresentation K d → surfaceAreaMeasure K =
+      ENNReal.ofReal (dist d.1 d.2.1) •
+        (Measure.dirac d.2.2 + Measure.dirac (d.2.2 + ((Real.pi : ℝ) : Real.Angle)))) ∧
+    ((interior (K : Set Point)).Nonempty → surfaceAreaMeasure K =
+      Measure.map (exteriorNormalAngle K)
+        ((Measure.hausdorffMeasure 1).restrict (regularBoundary K))) ∧
+    (∀ E : Set Real.Angle, MeasurableSet E →
+      ((interior (K : Set Point)).Nonempty ∨
+        ∃ a b : ℝ, a ≤ b ∧ b < a + Real.pi ∧
+          E ⊆ (fun t : ℝ ↦ (t : Real.Angle)) '' Set.Icc a b) →
+      surfaceAreaMeasure K E =
+        Measure.hausdorffMeasure 1 (⋃ t ∈ E, exposedEdge K t)) := by
+  obtain ⟨hfinite, hpoint, hsegment, _, _⟩ := surfaceAreaMeasure_construction K
+  refine ⟨hfinite, hpoint, hsegment, ?_, ?_⟩
+  · intro hK
+    simp only [surfaceAreaMeasure,
+      ite_eq_right (not_subsingleton_of_interior_nonempty K hK),
+      dite_eq_right (not_exists_segmentPresentation_of_interior_nonempty K hK)]
+  · intro E hE hdomain
+    by_cases hsub : (K : Set Point).Subsingleton
+    · exact surfaceAreaMeasure_face_union_of_subsingleton K hsub E
+    by_cases hint : (interior (K : Set Point)).Nonempty
+    · exact surfaceAreaMeasure_face_union_of_interior_nonempty K hint E hE
+    obtain ⟨a, b, hab, hwidth, hsubset⟩ := hdomain.resolve_left hint
+    have hseg := exists_segmentPresentation_of_interior_empty K hsub
+      (Set.not_nonempty_iff_eq_empty.mp hint)
+    obtain ⟨d, hd⟩ := hseg
+    exact surfaceAreaMeasure_face_union_of_segmentPresentation K d hd E hE hab hwidth hsubset
+
+instance isFiniteMeasure_surfaceAreaMeasure (K : ConvexBody Point) :
+    IsFiniteMeasure (surfaceAreaMeasure K) := (surfaceAreaMeasure_face_union K).1
+
+/-- A measurable set of normal directions lying in an angular interval of width less than `π`,
+on which every face of `K` degenerates to one and the same point, is null for the surface area
+measure of `K`. -/
+theorem surfaceAreaMeasure_null_of_exposedEdge_subset_singleton (K : ConvexBody Point)
+    {E : Set Real.Angle} (hE : MeasurableSet E) {a b : ℝ} (hab : a ≤ b) (hba : b < a + Real.pi)
+    (hEab : E ⊆ (fun t : ℝ ↦ (t : Real.Angle)) '' Set.Icc a b) (p : Point)
+    (hsub : ∀ t ∈ E, exposedEdge K t ⊆ {p}) :
+    surfaceAreaMeasure K E = 0 := by
+  let _ := Measure.nullSingletonClass_hausdorff Point (by norm_num : (0 : ℝ) < 1)
+  rw [(surfaceAreaMeasure_face_union K).2.2.2.2 E hE (Or.inr ⟨a, b, hab, hba, hEab⟩)]
+  refine measure_mono_null (t := {p}) ?_ (by simp)
+  intro x hx
+  obtain ⟨t, ht, hxt⟩ := Set.mem_iUnion₂.mp hx
+  exact hsub t ht hxt
+
+/-- The surface measure of a convex body vanishes on the open angular window strictly between two
+normal angles less than a half turn apart at which one and the same point attains the support. -/
+theorem surfaceAreaMeasure_angleImage_Ioo_eq_zero_of_mem_exposedEdge (K : ConvexBody Point)
+    {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi) {p : Point}
+    (hpa : p ∈ exposedEdge K (a : Real.Angle))
+    (hpb : p ∈ exposedEdge K (b : Real.Angle)) :
+    surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b) = 0 := by
+  refine surfaceAreaMeasure_null_of_exposedEdge_subset_singleton K
+    (Real.Angle.isOpen_image_Ioo a b).measurableSet hab.le hba
+    (Set.image_mono Set.Ioo_subset_Icc_self) p fun t ht ↦ ?_
+  obtain ⟨s, hs, rfl⟩ := ht
+  exact (exposedEdge_eq_singleton_of_mem_exposedEdge_of_mem_Ioo hba hs hpa hpb).subset
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Boundary Extension
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Set MeasureTheory
+open scoped Topology
+
+namespace MovingSofa
+
+/-- Agreement of positive-vertex increments with tangent-coordinate surface integrals extends
+from half-open subintervals to every measurable set avoiding the initial endpoint. -/
+theorem intervalStieltjesMeasure_eq_surfaceIntegral_of_increment
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi)
+    (f : Fin 2 → RightContinuousIntervalBV a b)
+    (hf : ∀ i t, (f i).toFun t =
+      (edgeVertices K (((t : Set.Icc a b) : ℝ) : Real.Angle)).1 i)
+    (hinc : ∀ (c d : Set.Icc a b), c < d → ∀ i : Fin 2,
+      (edgeVertices K (((d : Set.Icc a b) : ℝ) : Real.Angle)).1 i -
+          (edgeVertices K (((c : Set.Icc a b) : ℝ) : Real.Angle)).1 i =
+        ∫ u in (fun t : ℝ ↦ (t : Real.Angle)) '' Ioc (c : ℝ) d,
+          tangentVector u i ∂surfaceAreaMeasure K) :
+    ∀ (i : Fin 2) (E : Set (Set.Icc a b)), MeasurableSet E →
+      (∀ t ∈ E, a < (t : ℝ)) →
+      intervalStieltjesMeasure (f i) E =
+        ∫ u in (fun t : Set.Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+          tangentVector u i ∂surfaceAreaMeasure K := by
+  let A := Set.Ioc a b
+  let c : A → Real.Angle := fun t ↦ ((t : ℝ) : Real.Angle)
+  let j : A → Set.Icc a b := fun t ↦ ⟨t, t.property.1.le, t.property.2⟩
+  have hc : MeasurableEmbedding c := by
+    refine ⟨?_, (Real.Angle.continuous_coe.comp continuous_subtype_val).measurable, ?_⟩
+    · intro x y hxy
+      exact Subtype.ext (Real.Angle.injOn_coe_Ioc hturn x.property y.property hxy)
+    · intro s hs
+      let S : Set ℝ := Subtype.val '' s
+      have hS : MeasurableSet S :=
+        (MeasurableEmbedding.subtype_coe measurableSet_Ioc).measurableSet_image' hs
+      have hSI : Set.InjOn (fun t : ℝ ↦ (t : Real.Angle)) S := by
+        intro x hx y hy hxy
+        obtain ⟨tx, htx, rfl⟩ := hx
+        obtain ⟨ty, hty, rfl⟩ := hy
+        exact Real.Angle.injOn_coe_Ioc hturn tx.property ty.property hxy
+      have hm := hS.image_of_continuousOn_injOn Real.Angle.continuous_coe.continuousOn hSI
+      convert hm using 1
+      ext u
+      simp [S, c]
+  have hj : Measurable j := by fun_prop
+  intro i E hE hEa
+  let _ : IsFiniteMeasure (surfaceAreaMeasure K) :=
+    (surfaceAreaMeasure_face_union K).1
+  have hgi : Integrable (fun u : Real.Angle ↦ tangentVector u i) (surfaceAreaMeasure K) := by
+    rw [← integrableOn_univ]
+    exact ContinuousOn.integrableOn_compact isCompact_univ (by
+      fin_cases i
+      · exact Real.Angle.continuous_sin.neg.continuousOn
+      · exact Real.Angle.continuous_cos.continuousOn)
+  obtain ⟨ν, hν⟩ :=
+    MeasurableEmbedding.exists_vectorMeasure_image_integral hc (surfaceAreaMeasure K)
+      (fun u ↦ tangentVector u i) hgi
+  have hmeasure : intervalStieltjesMeasure (f i) = ν.map j := by
+    apply MeasureTheory.VectorMeasure.ext_of_Ioc
+    · intro x y hxy
+      have hpre : j ⁻¹' Ioc x y = {t : A | (x : ℝ) < (t : ℝ) ∧ (t : ℝ) ≤ (y : ℝ)} := rfl
+      have hpremeas : MeasurableSet (j ⁻¹' Ioc x y) :=
+        measurableSet_Ioc.preimage hj
+      have himage : c '' (j ⁻¹' Ioc x y) =
+          (fun t : ℝ ↦ (t : Real.Angle)) '' Ioc (x : ℝ) y := by
+        ext u
+        constructor
+        · rintro ⟨t, ht, rfl⟩
+          exact ⟨t, ht, rfl⟩
+        · rintro ⟨t, ht, rfl⟩
+          exact ⟨⟨t, x.property.1.trans_lt ht.1, ht.2.trans y.property.2⟩, ht, rfl⟩
+      rw [intervalStieltjesMeasure_Ioc (f i) x y hxy.le,
+        VectorMeasure.map_apply _ hj measurableSet_Ioc, hν _ hpremeas, himage,
+        hf i y, hf i x, hinc x y hxy i]
+    · let aa : Set.Icc a b := ⟨a, le_rfl, hab.le⟩
+      let bb : Set.Icc a b := ⟨b, hab.le, le_rfl⟩
+      have hpre : j ⁻¹' univ = univ := preimage_univ
+      have hunivmeas : MeasurableSet (univ : Set A) := MeasurableSet.univ
+      have himage : c '' (univ : Set A) =
+          (fun t : ℝ ↦ (t : Real.Angle)) '' Ioc a b := by
+        ext u
+        simp [c, A]
+      have hzero : intervalStieltjesMeasure (f i) {aa} = 0 := by
+        rw [intervalStieltjesMeasure, (f i).boundedVariation.vectorMeasure_singleton,
+          (f i).right_continuous aa |>.rightLim_eq]
+        have hbot : 𝓝[<] aa = ⊥ := by
+          have hIio : Iio aa = ∅ := by
+            ext t
+            simp only [mem_Iio, mem_empty_iff_false, iff_false]
+            exact not_lt_of_ge t.property.1
+          rw [hIio]
+          exact nhdsWithin_empty aa
+        rw [leftLim_eq_of_eq_bot _ hbot, sub_self]
+      have hsplit : (univ : Set (Set.Icc a b)) = Ioc aa bb ∪ {aa} := by
+        ext t
+        simp only [mem_univ, true_iff, mem_union, mem_Ioc, mem_singleton_iff]
+        by_cases hta : aa = t
+        · exact Or.inr hta.symm
+        · exact Or.inl ⟨lt_of_le_of_ne t.property.1 hta, t.property.2⟩
+      have hlhs : intervalStieltjesMeasure (f i) univ =
+          intervalStieltjesMeasure (f i) (Ioc aa bb) := by
+        rw [hsplit, VectorMeasure.of_union (disjoint_singleton_right.2 (by simp))
+          measurableSet_Ioc (measurableSet_singleton aa),
+          hzero, add_zero]
+      rw [hlhs, intervalStieltjesMeasure_Ioc (f i) aa bb hab.le,
+        VectorMeasure.map_apply _ hj MeasurableSet.univ, hpre, hν _ hunivmeas, himage,
+        hf i bb, hf i aa, hinc aa bb hab i]
+  have himageE : c '' (j ⁻¹' E) =
+      (fun t : Set.Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E := by
+    ext u
+    constructor
+    · rintro ⟨t, ht, rfl⟩
+      exact ⟨j t, ht, rfl⟩
+    · rintro ⟨t, htE, rfl⟩
+      exact ⟨⟨t, hEa t htE, t.property.2⟩, htE, rfl⟩
+  rw [hmeasure, VectorMeasure.map_apply _ hj hE, hν _ (hE.preimage hj), himageE]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Opposite
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- The opposite-angle surface measure and support function. -/
+def oppositeSurfaceData (K : ConvexBody Point) :
+    Measure Real.Angle × (Real.Angle → ℝ) :=
+  (Measure.map (fun t ↦ t - ((Real.pi : ℝ) : Real.Angle)) (surfaceAreaMeasure K),
+    fun t ↦ supportValue K (t + ((Real.pi : ℝ) : Real.Angle)))
+
+/-- The opposite surface measure is the half-turn translate of the surface measure. -/
+theorem oppositeSurfaceData_fst (K : ConvexBody Point) :
+    (oppositeSurfaceData K).1 =
+      Measure.map (fun t : Real.Angle ↦ t - ((Real.pi : ℝ) : Real.Angle))
+        (surfaceAreaMeasure K) := rfl
+
+instance isFiniteMeasure_oppositeSurfaceData (K : ConvexBody Point) :
+    IsFiniteMeasure (oppositeSurfaceData K).1 := by
+  rw [oppositeSurfaceData_fst]
+  exact Measure.isFiniteMeasure_map _ _
+
+/-- The opposite support function of a convex body is continuous in the normal direction. -/
+theorem continuous_oppositeSurfaceData_snd (K : ConvexBody Point) :
+    Continuous (oppositeSurfaceData K).2 :=
+  (continuous_supportValue K).comp (continuous_id.add continuous_const)
+
+/-- Integrating a `π`-shifted integrand against the opposite surface measure over an open angular
+window is integrating the integrand itself against the surface-area measure over the `π`-translated
+window.  The endpoints of the translated window are given as hypotheses so that call sites may
+normalize them arithmetically. -/
+theorem setIntegral_oppositeSurfaceData_angleImage_Ioo (K : ConvexBody Point) {a b a' b' : ℝ}
+    (ha : a' = a + Real.pi) (hb : b' = b + Real.pi) {f : Real.Angle → ℝ} (hf : Measurable f) :
+    (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        f (t + ((Real.pi : ℝ) : Real.Angle)) ∂(oppositeSurfaceData K).1) =
+      ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a' b',
+        f t ∂surfaceAreaMeasure K := by
+  subst ha hb
+  have hshift : Measurable fun t : Real.Angle ↦ t - ((Real.pi : ℝ) : Real.Angle) :=
+    (continuous_id.sub continuous_const).measurable
+  have hfshift : Measurable fun t : Real.Angle ↦ f (t + ((Real.pi : ℝ) : Real.Angle)) :=
+    hf.comp (continuous_id.add continuous_const).measurable
+  have hE : MeasurableSet ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b) :=
+    (Real.Angle.isOpen_image_Ioo a b).measurableSet
+  rw [oppositeSurfaceData_fst, setIntegral_map hE hfshift.aestronglyMeasurable hshift.aemeasurable,
+    Real.Angle.preimage_sub_pi_image_Ioo]
+  simp only [sub_add_cancel]
+
+/-- The opposite surface measure vanishes on an open angular window whose half-turn translate
+carries a single support point. -/
+theorem oppositeSurfaceData_angleImage_Ioo_eq_zero_of_mem_exposedEdge (K : ConvexBody Point)
+    {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi) {p : Point}
+    (hpa : p ∈ exposedEdge K ((a + Real.pi : ℝ) : Real.Angle))
+    (hpb : p ∈ exposedEdge K ((b + Real.pi : ℝ) : Real.Angle)) :
+    (oppositeSurfaceData K).1 ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b) = 0 := by
+  have hshift : Measurable fun t : Real.Angle ↦ t - ((Real.pi : ℝ) : Real.Angle) :=
+    (continuous_id.sub continuous_const).measurable
+  rw [oppositeSurfaceData_fst,
+    Measure.map_apply hshift (Real.Angle.isOpen_image_Ioo a b).measurableSet,
+    Real.Angle.preimage_sub_pi_image_Ioo]
+  exact surfaceAreaMeasure_angleImage_Ioo_eq_zero_of_mem_exposedEdge K (by linarith) (by linarith)
+    hpa hpb
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Weak Convergence
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped BigOperators
+open Filter MeasureTheory
+open scoped Topology
+
+namespace MovingSofa
+
+private def normalCoordinateFrames : Fin 4 → Point ≃ₗᵢ[ℝ] Point :=
+  ![LinearIsometryEquiv.refl ℝ Point, LinearIsometryEquiv.neg ℝ,
+    coordinateSwap, coordinateSwap.trans (LinearIsometryEquiv.neg ℝ)]
+
+private theorem exists_normalCoordinateFrame (t : Real.Angle) :
+    ∃ i, (1 / 2 : ℝ) < normalCoordinateFrames i (normalVector t) 1 := by
+  by_contra h
+  push Not at h
+  have h0 := h 0
+  have h1 := h 1
+  have h2 := h 2
+  have h3 := h 3
+  simp [normalCoordinateFrames, coordinateSwap, normalVector, frame] at h0 h1 h2 h3
+  have hs : t.sin ^ 2 ≤ 1 / 4 := by nlinarith
+  have hc : t.cos ^ 2 ≤ 1 / 4 := by nlinarith
+  nlinarith [t.cos_sq_add_sin_sq]
+
+/-- Four coordinate half-circles admit a continuous partition of unity supported
+where the upward component of the normal is at least one half. -/
+theorem exists_normalCoordinate_partition :
+    ∃ (e : Fin 4 → Point ≃ₗᵢ[ℝ] Point) (χ : Fin 4 → Real.Angle → ℝ),
+      (∀ i, Continuous (χ i)) ∧ (∀ t, ∑ i, χ i t = 1) ∧
+        (∀ i t, e i (normalVector t) 1 < 1 / 2 → χ i t = 0) := by
+  let ρ : Fin 4 → Real.Angle → ℝ := fun i t ↦
+    max 0 (normalCoordinateFrames i (normalVector t) 1 - 1 / 2)
+  have hρ : ∀ i, Continuous (ρ i) := by
+    intro i
+    dsimp [ρ]
+    exact continuous_const.max
+      (((PiLp.continuous_apply 2 (fun _ : Fin 2 ↦ ℝ) 1).comp
+        ((normalCoordinateFrames i).continuous.comp continuous_normalVector_angle)).sub
+          continuous_const)
+  have hpos (t : Real.Angle) : 0 < ∑ i, ρ i t := by
+    obtain ⟨i, hi⟩ := exists_normalCoordinateFrame t
+    apply Finset.sum_pos' (fun j _ ↦ le_max_left _ _)
+    refine ⟨i, Finset.mem_univ _, ?_⟩
+    exact (sub_pos.mpr hi).trans_le (le_max_right _ _)
+  refine ⟨normalCoordinateFrames, fun i t ↦ ρ i t / ∑ j, ρ j t, ?_, ?_, ?_⟩
+  · intro i
+    exact (hρ i).div (continuous_finsetSum _ fun j _ ↦ hρ j)
+      (fun t ↦ (hpos t).ne')
+  · intro t
+    rw [← Finset.sum_div]
+    exact div_self (hpos t).ne'
+  · intro i t ht
+    have hz : ρ i t = 0 := max_eq_left (sub_nonpos.mpr ht.le)
+    dsimp only
+    rw [hz, zero_div]
+
+/-- Convergence for test functions supported in upward normal patches implies
+convergence for all continuous test functions. -/
+theorem tendsto_surfaceIntegral_of_normal_patches
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hpatch : ∀ (e : Point ≃ₗᵢ[ℝ] Point) (ψ : Real.Angle → ℝ), Continuous ψ →
+      (∀ t, e (normalVector t) 1 < 1 / 2 → ψ t = 0) →
+      Tendsto (fun n ↦ ∫ t, ψ t ∂surfaceAreaMeasure (K n)) atTop
+        (𝓝 (∫ t, ψ t ∂surfaceAreaMeasure L)))
+    (φ : Real.Angle → ℝ) (hφ : Continuous φ) :
+    Tendsto (fun n ↦ ∫ t, φ t ∂surfaceAreaMeasure (K n)) atTop
+      (𝓝 (∫ t, φ t ∂surfaceAreaMeasure L)) := by
+  obtain ⟨e, χ, hχ, hsum, hsupp⟩ := exists_normalCoordinate_partition
+  let ψ : Fin 4 → Real.Angle → ℝ := fun i t ↦ φ t * χ i t
+  have hψ (i : Fin 4) : Continuous (ψ i) := hφ.mul (hχ i)
+  have hψsum (t : Real.Angle) : ∑ i, ψ i t = φ t := by
+    simp only [ψ, ← Finset.mul_sum, hsum, mul_one]
+  have hint (C : ConvexBody Point) :
+      ∑ i, (∫ t, ψ i t ∂surfaceAreaMeasure C) = ∫ t, φ t ∂surfaceAreaMeasure C := by
+    let _ := (surfaceAreaMeasure_construction C).1
+    rw [← integral_finsetSum]
+    · simp only [hψsum]
+    · intro i _
+      exact integrableOn_univ.mp ((hψ i).continuousOn.integrableOn_compact isCompact_univ)
+  have ht (i : Fin 4) := hpatch (e i) (ψ i) (hψ i) (by
+    intro t ht
+    simp only [ψ, hsupp i t ht, mul_zero])
+  have h := tendsto_finsetSum Finset.univ (fun i _ ↦ ht i)
+  simpa only [hint] using h
+
+theorem surfaceAreaMeasure_weak_continuity (K : ℕ → ConvexBody Point)
+    (L : ConvexBody Point)
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point) (L : Set Point))
+      atTop (𝓝 0)) (φ : Real.Angle → ℝ) (hφ : Continuous φ) :
+    Tendsto (fun n ↦ ∫ u, φ u ∂surfaceAreaMeasure (K n))
+      atTop (𝓝 (∫ u, φ u ∂surfaceAreaMeasure L)) := by
+  apply tendsto_surfaceIntegral_of_normal_patches K L _ φ hφ
+  intro e ψ hψ hsupp
+  exact tendsto_integral_surfaceAreaMeasure_of_hausdorffDist K L hlim 0 e hψ
+    (by norm_num : (0 : ℝ) < 1 / 2) hsupp
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Atom Limits
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter MeasureTheory Set
+open scoped Topology
+
+namespace MovingSofa
+
+/-- An upper bound by a moving surface atom passes to a Hausdorff limit. -/
+theorem le_surfaceAreaMeasure_atom_of_tendsto {K : ℕ → ConvexBody Point}
+    {L : ConvexBody Point}
+    (hK : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point) (L : Set Point))
+      atTop (𝓝 0)) {u : Real.Angle} {x : ℕ → ℝ} {a : ℝ}
+    (hx : Tendsto x atTop (𝓝 a))
+    (hle : ∀ n, x n ≤ (surfaceAreaMeasure (K n) {u}).toReal) :
+    a ≤ (surfaceAreaMeasure L {u}).toReal := by
+  let μs : ℕ → FiniteMeasure Real.Angle := fun n ↦
+    ⟨surfaceAreaMeasure (K n), (surfaceAreaMeasure_face_union (K n)).1⟩
+  let μ : FiniteMeasure Real.Angle :=
+    ⟨surfaceAreaMeasure L, (surfaceAreaMeasure_face_union L).1⟩
+  have hμ : Tendsto μs atTop (𝓝 μ) := by
+    apply FiniteMeasure.tendsto_iff_forall_integral_tendsto.mpr
+    intro f
+    exact surfaceAreaMeasure_weak_continuity K L hK f f.continuous
+  have hclosed :
+      Filter.limsup (fun n ↦ surfaceAreaMeasure (K n) {u}) atTop ≤
+        surfaceAreaMeasure L {u} := by
+    have h := FiniteMeasure.limsup_measure_closed_le_of_tendsto hμ
+      (isClosed_singleton : IsClosed {u})
+    change Filter.limsup (fun n ↦ surfaceAreaMeasure (K n) {u}) atTop ≤
+      surfaceAreaMeasure L {u} at h
+    exact h
+  have hpoint : ∀ n, ENNReal.ofReal (x n) ≤ surfaceAreaMeasure (K n) {u} := by
+    intro n
+    exact ENNReal.ofReal_le_of_le_toReal (hle n)
+  have hlimsup :
+      Filter.limsup (fun n ↦ ENNReal.ofReal (x n)) atTop ≤
+        Filter.limsup (fun n ↦ surfaceAreaMeasure (K n) {u}) atTop :=
+    Filter.limsup_le_limsup (Eventually.of_forall hpoint)
+  have hmain : ENNReal.ofReal a ≤ surfaceAreaMeasure L {u} := by
+    calc
+      ENNReal.ofReal a = Filter.limsup (fun n ↦ ENNReal.ofReal (x n)) atTop :=
+        (ENNReal.tendsto_ofReal hx).limsup_eq.symm
+      _ ≤ Filter.limsup (fun n ↦ surfaceAreaMeasure (K n) {u}) atTop := hlimsup
+      _ ≤ surfaceAreaMeasure L {u} := hclosed
+  let _ : IsFiniteMeasure (surfaceAreaMeasure L) := (surfaceAreaMeasure_face_union L).1
+  exact (ENNReal.ofReal_le_iff_le_toReal (measure_ne_top _ _)).mp hmain
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Weighted Boundary
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped Topology
+
+namespace MovingSofa
+
+/-- On a half-open parameter interval of at most one turn, the angular projection is a
+measurable embedding. -/
+theorem measurableEmbedding_angleCoe_Ioc {a b : ℝ} (hturn : b ≤ a + 2 * Real.pi) :
+    MeasurableEmbedding fun t : Ioc a b ↦ ((t : ℝ) : Real.Angle) := by
+  refine ⟨?_, (Real.Angle.continuous_coe.comp continuous_subtype_val).measurable, ?_⟩
+  · intro x y hxy
+    exact Subtype.ext (Real.Angle.injOn_coe_Ioc hturn x.property y.property hxy)
+  · intro s hs
+    let S : Set ℝ := Subtype.val '' s
+    have hS : MeasurableSet S :=
+      (MeasurableEmbedding.subtype_coe measurableSet_Ioc).measurableSet_image' hs
+    have hSI : Set.InjOn (fun t : ℝ ↦ (t : Real.Angle)) S := by
+      intro x hx y hy hxy
+      obtain ⟨tx, htx, rfl⟩ := hx
+      obtain ⟨ty, hty, rfl⟩ := hy
+      exact Real.Angle.injOn_coe_Ioc hturn tx.property ty.property hxy
+    have hm := hS.image_of_continuousOn_injOn Real.Angle.continuous_coe.continuousOn hSI
+    convert hm using 1
+    ext u
+    simp [S]
+
+/-- A bounded measurable function on a half-open parameter interval of at most one full turn
+extends to a bounded measurable function of the angle. -/
+theorem exists_bounded_measurable_angle_extension {a b : ℝ} (hturn : b ≤ a + 2 * Real.pi)
+    (ψ : Ioc a b → ℝ) (hψm : Measurable ψ) (C : ℝ) (hC : 0 ≤ C) (hψb : ∀ t, ‖ψ t‖ ≤ C) :
+    ∃ Q : Real.Angle → ℝ, Measurable Q ∧ (∀ u, ‖Q u‖ ≤ C) ∧
+      ∀ t : Ioc a b, Q ((t : ℝ) : Real.Angle) = ψ t := by
+  have hc : MeasurableEmbedding fun t : Ioc a b ↦ ((t : ℝ) : Real.Angle) :=
+    measurableEmbedding_angleCoe_Ioc hturn
+  refine ⟨Function.extend (fun t : Ioc a b ↦ ((t : ℝ) : Real.Angle)) ψ (fun _ ↦ 0),
+    hc.measurable_extend hψm measurable_const, fun u ↦ ?_,
+    fun t ↦ hc.injective.extend_apply ψ _ t⟩
+  by_cases hu : ∃ t : Ioc a b, ((t : ℝ) : Real.Angle) = u
+  · obtain ⟨t, rfl⟩ := hu
+    rw [hc.injective.extend_apply]
+    exact hψb t
+  · rw [Function.extend_apply' _ _ _ hu]
+    simpa using hC
+
+/-- A bounded measurable angular weight may be transported through the positive-vertex
+Stieltjes identity, coordinate by coordinate. -/
+theorem intervalStieltjesIntegral_positiveVertex_coordinate_of_measurable
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi)
+    (f : Fin 2 → RightContinuousIntervalBV a b)
+    (hf : ∀ (i : Fin 2) (E : Set (Icc a b)), MeasurableSet E →
+      (∀ t ∈ E, a < (t : ℝ)) →
+      intervalStieltjesMeasure (f i) E =
+        ∫ u in (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+          tangentVector u i ∂surfaceAreaMeasure K)
+    (i : Fin 2) (Q : Real.Angle → ℝ) (C : ℝ)
+    (hQm : Measurable fun t : Ioc a b ↦ Q (((t : ℝ) : Real.Angle)))
+    (hQb : ∀ u, ‖Q u‖ ≤ C)
+    (E : Set (Icc a b)) (hE : MeasurableSet E) (hEa : ∀ t ∈ E, a < (t : ℝ)) :
+    intervalStieltjesIntegral (f i)
+        (fun t ↦ Q (((t : ℝ) : Real.Angle))) E =
+      ∫ u in (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+        Q u * tangentVector u i ∂surfaceAreaMeasure K := by
+  let A := Ioc a b
+  let c : A → Real.Angle := fun t ↦ ((t : ℝ) : Real.Angle)
+  let j : A → Icc a b := fun t ↦ ⟨t, t.property.1.le, t.property.2⟩
+  have hc : MeasurableEmbedding c := measurableEmbedding_angleCoe_Ioc hturn
+  have hj : MeasurableEmbedding j := by
+    refine ⟨fun x y h ↦ Subtype.ext (congrArg (fun z : Icc a b ↦ (z : ℝ)) h), ?_, ?_⟩
+    · fun_prop
+    · intro s hs
+      have hsval : MeasurableSet ((Subtype.val : A → ℝ) '' s) :=
+        (MeasurableEmbedding.subtype_coe measurableSet_Ioc).measurableSet_image' hs
+      have hsj : j '' s = {t : Icc a b | (t : ℝ) ∈ (Subtype.val : A → ℝ) '' s} := by
+        ext t
+        constructor
+        · rintro ⟨x, hx, rfl⟩
+          exact ⟨x, hx, rfl⟩
+        · rintro ⟨x, hx, hxt⟩
+          exact ⟨x, hx, Subtype.ext hxt⟩
+      rw [hsj]
+      exact hsval.preimage measurable_subtype_coe
+  let μA := (surfaceAreaMeasure K).comap c
+  let _ : IsFiniteMeasure (surfaceAreaMeasure K) := (surfaceAreaMeasure_face_union K).1
+  let _ : IsFiniteMeasure μA :=
+    ⟨by
+      rw [hc.comap_apply]
+      exact measure_lt_top _ _⟩
+  have htancont : Continuous (fun t : A ↦ tangentVector (c t) i) :=
+    (continuous_tangentVector_coordinate i).comp
+      (Real.Angle.continuous_coe.comp continuous_subtype_val)
+  have htan : Integrable (fun t : A ↦ tangentVector (c t) i) μA :=
+    Integrable.of_bound htancont.aestronglyMeasurable 1
+      (ae_of_all _ fun t ↦ norm_tangentVector_coordinate_le_one (c t) i)
+  let ν : SignedMeasure A := μA.withDensityᵥ (fun t ↦ tangentVector (c t) i)
+  have hν (S : Set A) (hS : MeasurableSet S) :
+      ν S = ∫ u in c '' S, tangentVector u i ∂surfaceAreaMeasure K := by
+    rw [show ν S = ∫ t in S, tangentVector (c t) i ∂μA by
+      exact withDensityᵥ_apply htan hS]
+    have hm := hc.setIntegral_map (μ := μA) (fun u ↦ tangentVector u i) (c '' S)
+    rw [hc.map_comap, hc.injective.preimage_image,
+      Measure.restrict_restrict_of_subset (image_subset_range c S)] at hm
+    exact hm.symm
+  have hmeasure : intervalStieltjesMeasure (f i) = ν.map j := by
+    apply MeasureTheory.VectorMeasure.ext_of_Ioc
+    · intro x y hxy
+      have hpremeas : MeasurableSet (j ⁻¹' Ioc x y) :=
+        measurableSet_Ioc.preimage hj.measurable
+      have himage : c '' (j ⁻¹' Ioc x y) =
+          (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' Ioc x y := by
+        ext u
+        constructor
+        · rintro ⟨t, ht, rfl⟩
+          exact ⟨j t, ht, rfl⟩
+        · rintro ⟨t, ht, rfl⟩
+          exact ⟨⟨t, x.property.1.trans_lt ht.1, t.property.2⟩, ht, rfl⟩
+      rw [VectorMeasure.map_apply _ hj.measurable measurableSet_Ioc, hν _ hpremeas,
+        himage, hf i (Ioc x y) measurableSet_Ioc]
+      intro t ht
+      exact x.property.1.trans_lt ht.1
+    · let aa : Icc a b := ⟨a, le_rfl, hab.le⟩
+      let bb : Icc a b := ⟨b, hab.le, le_rfl⟩
+      have hzero : intervalStieltjesMeasure (f i) {aa} = 0 := by
+        rw [intervalStieltjesMeasure, (f i).boundedVariation.vectorMeasure_singleton,
+          (f i).right_continuous aa |>.rightLim_eq]
+        have hbot : 𝓝[<] aa = ⊥ := by
+          have hIio : Iio aa = ∅ := by
+            ext t
+            simp only [mem_Iio, mem_empty_iff_false, iff_false]
+            exact not_lt_of_ge t.property.1
+          rw [hIio]
+          exact nhdsWithin_empty aa
+        rw [leftLim_eq_of_eq_bot _ hbot, sub_self]
+      have hsplit : (univ : Set (Icc a b)) = Ioc aa bb ∪ {aa} := by
+        ext t
+        simp only [mem_univ, true_iff, mem_union, mem_Ioc, mem_singleton_iff]
+        by_cases hta : aa = t
+        · exact Or.inr hta.symm
+        · exact Or.inl ⟨lt_of_le_of_ne t.property.1 hta, t.property.2⟩
+      have hlhs : intervalStieltjesMeasure (f i) univ =
+          intervalStieltjesMeasure (f i) (Ioc aa bb) := by
+        rw [hsplit, VectorMeasure.of_union (disjoint_singleton_right.2 (by simp))
+          measurableSet_Ioc (measurableSet_singleton aa), hzero, add_zero]
+      have himage : c '' (univ : Set A) =
+          (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' Ioc aa bb := by
+        ext u
+        constructor
+        · rintro ⟨t, -, rfl⟩
+          exact ⟨j t, t.property, rfl⟩
+        · rintro ⟨t, ht, rfl⟩
+          exact ⟨⟨t, ht⟩, mem_univ _, rfl⟩
+      rw [hlhs, VectorMeasure.map_apply _ hj.measurable MeasurableSet.univ,
+        preimage_univ, hν _ MeasurableSet.univ, himage,
+        hf i (Ioc aa bb) measurableSet_Ioc]
+      intro t ht
+      exact ht.1
+  unfold intervalStieltjesIntegral
+  rw [hmeasure, hj.setIntegral_map_vectorMeasure hE]
+  change (∫ᵛ t in j ⁻¹' E, Q (c t)
+      ∂[ContinuousLinearMap.mul ℝ ℝ; μA.withDensityᵥ fun t ↦ tangentVector (c t) i]) = _
+  rw [VectorMeasure.setIntegral_withDensity_mul_of_bounded (q := fun t : A ↦ Q (c t)) htan
+    hQm.aestronglyMeasurable C (fun t ↦ hQb (c t))
+    (j ⁻¹' E) (hE.preimage hj.measurable)]
+  have hm := hc.setIntegral_map (μ := μA)
+    (fun u ↦ Q u * tangentVector u i) (c '' (j ⁻¹' E))
+  rw [hc.map_comap, hc.injective.preimage_image,
+    Measure.restrict_restrict_of_subset (image_subset_range c (j ⁻¹' E))] at hm
+  have himageE : c '' (j ⁻¹' E) =
+      (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E := by
+    ext u
+    constructor
+    · rintro ⟨t, ht, rfl⟩
+      exact ⟨j t, ht, rfl⟩
+    · rintro ⟨t, htE, rfl⟩
+      exact ⟨⟨t, hEa t htE, t.property.2⟩, htE, rfl⟩
+  rw [← himageE]
+  exact hm.symm
+
+/-- A continuous angular weight may be transported through the positive-vertex Stieltjes
+identity, coordinate by coordinate. -/
+theorem intervalStieltjesIntegral_positiveVertex_coordinate
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi)
+    (f : Fin 2 → RightContinuousIntervalBV a b)
+    (hf : ∀ (i : Fin 2) (E : Set (Icc a b)), MeasurableSet E →
+      (∀ t ∈ E, a < (t : ℝ)) →
+      intervalStieltjesMeasure (f i) E =
+        ∫ u in (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+          tangentVector u i ∂surfaceAreaMeasure K)
+    (i : Fin 2) (Q : Real.Angle → ℝ) (hQ : Continuous Q)
+    (E : Set (Icc a b)) (hE : MeasurableSet E) (hEa : ∀ t ∈ E, a < (t : ℝ)) :
+    intervalStieltjesIntegral (f i)
+        (fun t ↦ Q (((t : ℝ) : Real.Angle))) E =
+      ∫ u in (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+        Q u * tangentVector u i ∂surfaceAreaMeasure K :=
+  intervalStieltjesIntegral_positiveVertex_coordinate_of_measurable K hab hturn f hf i Q
+    ‖ContinuousMap.equivBoundedOfCompact Real.Angle ℝ ⟨Q, hQ⟩‖
+    (hQ.comp (Real.Angle.continuous_coe.comp continuous_subtype_val)).measurable
+    (fun u ↦ BoundedContinuousFunction.norm_coe_le_norm
+      (ContinuousMap.equivBoundedOfCompact Real.Angle ℝ ⟨Q, hQ⟩) u) E hE hEa
+
+/-- A bounded measurable angular weight times a frame tangent coordinate is integrable against
+the surface measure on every measurable set of angles represented in the half-open parameter
+interval. -/
+theorem integrableOn_mul_tangentVector_of_bounded
+    (K : ConvexBody Point) {a b : ℝ} (hturn : b ≤ a + 2 * Real.pi)
+    (i : Fin 2) (Q : Real.Angle → ℝ) (C : ℝ)
+    (hQm : Measurable fun t : Ioc a b ↦ Q (((t : ℝ) : Real.Angle)))
+    (hQb : ∀ u, ‖Q u‖ ≤ C)
+    (S : Set Real.Angle) (hS : MeasurableSet S)
+    (hSsub : S ⊆ Set.range fun t : Ioc a b ↦ ((t : ℝ) : Real.Angle)) :
+    IntegrableOn (fun u ↦ Q u * tangentVector u i) S (surfaceAreaMeasure K) := by
+  let _ : IsFiniteMeasure (surfaceAreaMeasure K) := (surfaceAreaMeasure_face_union K).1
+  have hC : 0 ≤ C := le_trans (norm_nonneg (Q 0)) (hQb 0)
+  obtain ⟨Q', hQ'meas, hQ'b, hQ'eq⟩ := exists_bounded_measurable_angle_extension hturn
+    (fun t : Ioc a b ↦ Q (((t : ℝ) : Real.Angle))) hQm C hC (fun t ↦ hQb _)
+  have hglobal : Integrable (fun u ↦ Q' u * tangentVector u i) (surfaceAreaMeasure K) := by
+    refine Integrable.of_bound
+      (hQ'meas.mul (continuous_tangentVector_coordinate i).measurable).aestronglyMeasurable C
+      (ae_of_all _ fun u ↦ ?_)
+    calc ‖Q' u * tangentVector u i‖ = ‖Q' u‖ * ‖tangentVector u i‖ := norm_mul _ _
+      _ ≤ C * 1 :=
+        mul_le_mul (hQ'b u) (norm_tangentVector_coordinate_le_one u i) (norm_nonneg _) hC
+      _ = C := mul_one C
+  refine hglobal.integrableOn.congr_fun (fun u hu ↦ ?_) hS
+  obtain ⟨t, rfl⟩ := hSsub hu
+  simp only [hQ'eq t]
+
+/-- Pairing the positive-vertex Stieltjes measures with a bounded measurable planar weight is the
+surface integral of the pointwise contraction of that weight with the tangent frame. -/
+theorem sum_intervalStieltjesIntegral_positiveVertex_dot
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi)
+    (f : Fin 2 → RightContinuousIntervalBV a b)
+    (hf : ∀ (i : Fin 2) (E : Set (Icc a b)), MeasurableSet E →
+      (∀ t ∈ E, a < (t : ℝ)) →
+      intervalStieltjesMeasure (f i) E =
+        ∫ u in (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+          tangentVector u i ∂surfaceAreaMeasure K)
+    (φ : Fin 2 → Icc a b → ℝ) (C : ℝ) (hC : 0 ≤ C)
+    (hφm : ∀ i, Measurable (φ i)) (hφb : ∀ i t, ‖φ i t‖ ≤ C)
+    (g : Real.Angle → ℝ)
+    (hg : ∀ t : Icc a b, a < (t : ℝ) →
+      ∑ i : Fin 2, φ i t * tangentVector (((t : ℝ) : Real.Angle)) i =
+        g (((t : ℝ) : Real.Angle)))
+    (E : Set (Icc a b)) (hE : MeasurableSet E) (hEa : ∀ t ∈ E, a < (t : ℝ)) :
+    (∑ i : Fin 2, intervalStieltjesIntegral (f i) (φ i) E) =
+      ∫ u in (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E, g u ∂surfaceAreaMeasure K := by
+  set incl : Ioc a b → Icc a b := fun t ↦ ⟨(t : ℝ), t.property.1.le, t.property.2⟩
+  have hinclm : Measurable incl := (Continuous.subtype_mk continuous_subtype_val _).measurable
+  have himg : (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E =
+      (fun t : Ioc a b ↦ ((t : ℝ) : Real.Angle)) '' (incl ⁻¹' E) := by
+    ext u
+    constructor
+    · rintro ⟨t, htE, rfl⟩
+      exact ⟨⟨(t : ℝ), hEa t htE, t.property.2⟩, htE, rfl⟩
+    · rintro ⟨t, htE, rfl⟩
+      exact ⟨incl t, htE, rfl⟩
+  have hEmeas : MeasurableSet ((fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E) := by
+    rw [himg]
+    exact (measurableEmbedding_angleCoe_Ioc hturn).measurableSet_image' (hE.preimage hinclm)
+  have hsub : (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E ⊆
+      Set.range fun t : Ioc a b ↦ ((t : ℝ) : Real.Angle) := by
+    rintro u ⟨t, htE, rfl⟩
+    exact ⟨⟨(t : ℝ), hEa t htE, t.property.2⟩, rfl⟩
+  choose Q hQm hQb hQeq using fun i : Fin 2 ↦ exists_bounded_measurable_angle_extension hturn
+    (fun t : Ioc a b ↦ φ i (incl t)) ((hφm i).comp hinclm) C hC fun t ↦ hφb i (incl t)
+  have hQmIoc (i : Fin 2) : Measurable fun t : Ioc a b ↦ Q i (((t : ℝ) : Real.Angle)) :=
+    (hQm i).comp (Real.Angle.continuous_coe.comp continuous_subtype_val).measurable
+  have hQφ (i : Fin 2) (t : Icc a b) (ht : a < (t : ℝ)) :
+      Q i (((t : ℝ) : Real.Angle)) = φ i t := hQeq i ⟨(t : ℝ), ht, t.property.2⟩
+  have hterm (i : Fin 2) : intervalStieltjesIntegral (f i) (φ i) E =
+      ∫ u in (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+        Q i u * tangentVector u i ∂surfaceAreaMeasure K := by
+    rw [show intervalStieltjesIntegral (f i) (φ i) E =
+        intervalStieltjesIntegral (f i) (fun t ↦ Q i (((t : ℝ) : Real.Angle))) E from
+      VectorMeasure.setIntegral_congr_fun fun t htE ↦ (hQφ i t (hEa t htE)).symm]
+    exact intervalStieltjesIntegral_positiveVertex_coordinate_of_measurable K hab hturn f hf i
+      (Q i) C (hQmIoc i) (hQb i) E hE hEa
+  have hint (i : Fin 2) : IntegrableOn (fun u ↦ Q i u * tangentVector u i)
+      ((fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E) (surfaceAreaMeasure K) :=
+    integrableOn_mul_tangentVector_of_bounded K hturn i (Q i) C (hQmIoc i) (hQb i) _ hEmeas hsub
+  rw [Fin.sum_univ_two, hterm 0, hterm 1, ← integral_add (hint 0) (hint 1)]
+  refine setIntegral_congr_fun hEmeas ?_
+  rintro u ⟨t, htE, rfl⟩
+  have ht := hEa t htE
+  simpa only [hQφ 0 t ht, hQφ 1 t ht, Fin.sum_univ_two] using hg t ht
+
+/-- Pairing the positive-vertex Stieltjes measure with the tangent frame gives surface mass. -/
+theorem sum_intervalStieltjesIntegral_positiveVertex_tangent
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi)
+    (f : Fin 2 → RightContinuousIntervalBV a b)
+    (hf : ∀ (i : Fin 2) (E : Set (Icc a b)), MeasurableSet E →
+      (∀ t ∈ E, a < (t : ℝ)) →
+      intervalStieltjesMeasure (f i) E =
+        ∫ u in (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+          tangentVector u i ∂surfaceAreaMeasure K)
+    (E : Set (Icc a b)) (hE : MeasurableSet E) (hEa : ∀ t ∈ E, a < (t : ℝ)) :
+    (∑ i : Fin 2, intervalStieltjesIntegral (f i)
+      (fun t ↦ tangentVector (((t : ℝ) : Real.Angle)) i) E) =
+      (surfaceAreaMeasure K
+        ((fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E)).toReal := by
+  let _ : IsFiniteMeasure (surfaceAreaMeasure K) := (surfaceAreaMeasure_face_union K).1
+  have hcoord (i : Fin 2) :
+      intervalStieltjesIntegral (f i)
+          (fun t ↦ tangentVector (((t : ℝ) : Real.Angle)) i) E =
+        ∫ u in (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+          tangentVector u i * tangentVector u i ∂surfaceAreaMeasure K := by
+    have hQi : Continuous (fun u : Real.Angle ↦ tangentVector u i) := by
+      fin_cases i
+      · exact Real.Angle.continuous_sin.neg
+      · exact Real.Angle.continuous_cos
+    exact intervalStieltjesIntegral_positiveVertex_coordinate K hab hturn f hf i
+      (fun u ↦ tangentVector u i) hQi E hE hEa
+  rw [Fin.sum_univ_two, hcoord 0, hcoord 1]
+  have h0 : Integrable (fun u : Real.Angle ↦ tangentVector u 0 * tangentVector u 0)
+      (surfaceAreaMeasure K) := by
+    apply Continuous.integrable_of_hasCompactSupport
+    · convert Real.Angle.continuous_sin.mul Real.Angle.continuous_sin using 1
+      funext u
+      simp [tangentVector, frame]
+    · exact isCompact_univ.of_isClosed_subset isClosed_closure (subset_univ _)
+  have h1 : Integrable (fun u : Real.Angle ↦ tangentVector u 1 * tangentVector u 1)
+      (surfaceAreaMeasure K) := by
+    apply Continuous.integrable_of_hasCompactSupport
+    · convert Real.Angle.continuous_cos.mul Real.Angle.continuous_cos using 1
+      funext u
+      simp [tangentVector, frame]
+    · exact isCompact_univ.of_isClosed_subset isClosed_closure (subset_univ _)
+  rw [← integral_add h0.integrableOn h1.integrableOn]
+  have hone : (fun u : Real.Angle ↦
+      tangentVector u 0 * tangentVector u 0 + tangentVector u 1 * tangentVector u 1) = 1 := by
+    funext u
+    simpa [tangentVector, frame, pow_two, add_comm] using u.cos_sq_add_sin_sq
+  rw [hone]
+  let S := (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E
+  calc
+    integral ((surfaceAreaMeasure K).restrict S) 1 =
+        ((surfaceAreaMeasure K).restrict S).real univ := by
+          change (∫ _ : Real.Angle, (1 : ℝ) ∂(surfaceAreaMeasure K).restrict S) = _
+          rw [integral_const (μ := (surfaceAreaMeasure K).restrict S) (1 : ℝ),
+            smul_eq_mul, mul_one]
+    _ = (surfaceAreaMeasure K).real S :=
+      measureReal_restrict_apply_univ (μ := surfaceAreaMeasure K) S
+    _ = (surfaceAreaMeasure K S).toReal := measureReal_def _ _
+
+/-- Pairing the positive-vertex Stieltjes measure with the normal frame vanishes. -/
+theorem sum_intervalStieltjesIntegral_positiveVertex_normal
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi)
+    (f : Fin 2 → RightContinuousIntervalBV a b)
+    (hf : ∀ (i : Fin 2) (E : Set (Icc a b)), MeasurableSet E →
+      (∀ t ∈ E, a < (t : ℝ)) →
+      intervalStieltjesMeasure (f i) E =
+        ∫ u in (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+          tangentVector u i ∂surfaceAreaMeasure K)
+    (E : Set (Icc a b)) (hE : MeasurableSet E) (hEa : ∀ t ∈ E, a < (t : ℝ)) :
+    (∑ i : Fin 2, intervalStieltjesIntegral (f i)
+      (fun t ↦ normalVector (((t : ℝ) : Real.Angle)) i) E) = 0 := by
+  let _ : IsFiniteMeasure (surfaceAreaMeasure K) := (surfaceAreaMeasure_face_union K).1
+  have hcoord (i : Fin 2) :
+      intervalStieltjesIntegral (f i)
+          (fun t ↦ normalVector (((t : ℝ) : Real.Angle)) i) E =
+        ∫ u in (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+          normalVector u i * tangentVector u i ∂surfaceAreaMeasure K := by
+    have hQi : Continuous (fun u : Real.Angle ↦ normalVector u i) := by
+      fin_cases i
+      · exact Real.Angle.continuous_cos
+      · exact Real.Angle.continuous_sin
+    exact intervalStieltjesIntegral_positiveVertex_coordinate K hab hturn f hf i
+      (fun u ↦ normalVector u i) hQi E hE hEa
+  rw [Fin.sum_univ_two, hcoord 0, hcoord 1]
+  have h0 : Integrable (fun u : Real.Angle ↦ normalVector u 0 * tangentVector u 0)
+      (surfaceAreaMeasure K) := by
+    apply Continuous.integrable_of_hasCompactSupport
+    · convert (Real.Angle.continuous_cos.mul Real.Angle.continuous_sin).neg using 1
+      funext u
+      simp [normalVector, tangentVector, frame]
+    · exact isCompact_univ.of_isClosed_subset isClosed_closure (subset_univ _)
+  have h1 : Integrable (fun u : Real.Angle ↦ normalVector u 1 * tangentVector u 1)
+      (surfaceAreaMeasure K) := by
+    apply Continuous.integrable_of_hasCompactSupport
+    · convert Real.Angle.continuous_sin.mul Real.Angle.continuous_cos using 1
+      funext u
+      simp [normalVector, tangentVector, frame]
+    · exact isCompact_univ.of_isClosed_subset isClosed_closure (subset_univ _)
+  rw [← integral_add h0.integrableOn h1.integrableOn]
+  apply integral_eq_zero_of_ae
+  filter_upwards with u
+  simp [normalVector, tangentVector, frame]
+  ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Cap.Contacts`.
+* `Cap.ArmCoordinates`.
+* `Cap.ContactIdentities`.
+* `Cap.HalfPlanes`.
+* `Cap.FanProjection`.
+* `Cap.HallwayQuadrant`.
+* `Cap.LowerNormalMeasure`.
+* `Cap.ReflectionGeometry`.
+* `Cap.SupportIntersections`.
+* `Cap.TopCorner`.
+* `Cap.Vertical`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Contacts
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Width in a normal direction, for geometric use on nonempty compact sets. -/
+def directionalWidth (s : Set Point) (t : Real.Angle) : ℝ :=
+  supportValue s t + supportValue s (t + ((Real.pi : ℝ) : Real.Angle))
+
+/-- The positive/negative contacts at the two outer supporting walls. -/
+def capVertices {ω : ℝ} (K : CapSpace ω) (t : ℝ) : (Point × Point) × (Point × Point) :=
+  (edgeVertices K.1 (t : Real.Angle),
+    edgeVertices K.1 ((t + Real.pi / 2 : ℝ) : Real.Angle))
+
+/-- The positive/negative right and left tangent arm lengths of a right-angle cap. -/
+def tangentArmLengths (K : RightAngleCapSpace) (t : ℝ) : (ℝ × ℝ) × (ℝ × ℝ) :=
+  let y := (rotatingHallwayParts (K.1 : Set Point) (t : Real.Angle)).outerCorner
+  let v := capVertices K t
+  ((inner ℝ (y - v.1.1) (tangentVector (t : Real.Angle)),
+    inner ℝ (y - v.1.2) (tangentVector (t : Real.Angle))),
+    (inner ℝ (y - v.2.1) (normalVector (t : Real.Angle)),
+      inner ℝ (y - v.2.2) (normalVector (t : Real.Angle))))
+
+/-- The open inner quadrant clipped by the fan. -/
+def capWedge {ω : ℝ} (K : CapSpace ω) (t : ℝ) : Set Point :=
+  capFan ω ∩ (rotatingHallwayParts (K.1 : Set Point) (t : Real.Angle)).innerQuadrant
+
+/-- The two inner-wall intersections with the lower fan boundary lines. -/
+def wedgeEndpoints {ω : ℝ} (K : CapSpace ω) (t : ℝ) : Point × Point :=
+  (((supportValue K.1 (t : Real.Angle) - 1) / Real.cos t) • normalVector 0,
+    ((supportValue K.1 ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+      Real.cos (ω - t)) • tangentVector (ω : Real.Angle))
+
+/-- Signed right and left gaps between the wedge endpoints and bottom cap contacts. -/
+def wedgeGaps {ω : ℝ} (K : CapSpace ω) (t : ℝ) : ℝ × ℝ :=
+  (inner ℝ ((capVertices K 0).1.2 - (wedgeEndpoints K t).1) (normalVector 0),
+    inner ℝ ((capVertices K ω).2.1 - (wedgeEndpoints K t).2)
+      (tangentVector (ω : Real.Angle)))
+
+/-- The selected short boundary arc, including only the specified endpoints. -/
+def convexBoundaryArc (K : ConvexBody Point) (a b : ℝ) : Set Point :=
+  {(edgeVertices K (a : Real.Angle)).1} ∪
+    (⋃ t ∈ Set.Ioo a b, exposedEdge K (t : Real.Angle)) ∪
+    {(edgeVertices K (b : Real.Angle)).2}
+
+/-- The right wedge gap in support-function coordinates. -/
+theorem wedgeGaps_fst_eq_supportValue {ω : ℝ} (K : CapSpace ω) (t : ℝ) :
+    (wedgeGaps K t).1 = supportValue K.val (0 : Real.Angle) -
+      (supportValue K.val (t : Real.Angle) - 1) / Real.cos t := by
+  have hA := (edgeVertices_snd_mem K.val (0 : Real.Angle)).2
+  change inner ℝ (capVertices K 0).1.2 (normalVector (0 : Real.Angle)) =
+    supportValue K.val (0 : Real.Angle) at hA
+  simp only [wedgeGaps, wedgeEndpoints, inner_sub_left, real_inner_smul_left, hA]
+  have hu0 : inner ℝ (normalVector (0 : Real.Angle))
+      (normalVector (0 : Real.Angle)) = 1 := by
+    rw [← Real.Angle.coe_zero]
+    exact inner_normalVector_self 0
+  rw [hu0]
+  ring
+
+/-- The left wedge gap in support-function coordinates. -/
+theorem wedgeGaps_snd_eq_supportValue {ω : ℝ} (K : CapSpace ω)
+    (t : ℝ) :
+    (wedgeGaps K t).2 =
+      supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) -
+        (supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+          Real.cos (ω - t) := by
+  have hC := (edgeVertices_fst_mem K.val
+    ((ω + Real.pi / 2 : ℝ) : Real.Angle)).2
+  change inner ℝ (capVertices K ω).2.1
+      (normalVector ((ω + Real.pi / 2 : ℝ) : Real.Angle)) =
+    supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) at hC
+  simp only [normalVector, frame, Real.Angle.coe_add, Real.Angle.cos_add_pi_div_two,
+    Real.Angle.sin_coe, Real.Angle.sin_add_pi_div_two, Real.Angle.cos_coe] at hC
+  change inner ℝ (capVertices K ω).2.1 (tangentVector (ω : Real.Angle)) =
+    supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) at hC
+  simp only [wedgeGaps, wedgeEndpoints, inner_sub_left,
+    real_inner_smul_left, hC]
+  rw [inner_tangentVector_self]
+  ring
+
+/-- The right wedge endpoint lies on the horizontal axis, at the horizontal intercept of the right
+inner wall. -/
+theorem wedgeEndpoints_fst_coords {ω : ℝ} (K : CapSpace ω) (t : ℝ) :
+    (wedgeEndpoints K t).1 0 = (supportValue K.val (t : Real.Angle) - 1) / Real.cos t ∧
+      (wedgeEndpoints K t).1 1 = 0 := by
+  constructor <;> simp [wedgeEndpoints, normalVector, frame]
+
+/-- For a right-angle cap the left wedge endpoint also lies on the horizontal axis, at the
+horizontal intercept of the left inner wall. -/
+theorem wedgeEndpoints_snd_coords (K : RightAngleCapSpace) (t : ℝ) :
+    (wedgeEndpoints K t).2 0 =
+        -((supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) / Real.sin t) ∧
+      (wedgeEndpoints K t).2 1 = 0 := by
+  constructor <;> simp [wedgeEndpoints, tangentVector, frame, Real.cos_pi_div_two_sub]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Arm Coordinates
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped Topology
+
+namespace MovingSofa
+
+/-- Express the two positive arm lengths through support and moving-frame coordinates. -/
+theorem tangentArmLengths_positive_frame (K : RightAngleCapSpace) (t : ℝ) :
+    (tangentArmLengths K t).1.1 =
+        supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) -
+          inner ℝ (edgeVertices K.val (t : Real.Angle)).1
+            (tangentVector (t : Real.Angle)) ∧
+    (tangentArmLengths K t).2.1 =
+        inner ℝ (edgeVertices K.val (t : Real.Angle)).1
+            (normalVector (t : Real.Angle)) +
+          inner ℝ (edgeVertices K.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).1
+            (tangentVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) := by
+  have hnn := inner_normalVector_self t
+  have htt := inner_tangentVector_self t
+  have hnt := inner_normalVector_tangentVector t
+  have htn : inner ℝ (tangentVector (t : Real.Angle))
+      (normalVector (t : Real.Angle)) = 0 := by
+    rw [real_inner_comm, hnt]
+  have hA := (edgeVertices_fst_mem K.val (t : Real.Angle)).2
+  simp only [tangentArmLengths, capVertices, outerCorner_eq_support_sum,
+    inner_sub_left, inner_add_left, real_inner_smul_left, hnn, htt, hnt, htn]
+  rw [hA, tangentVector_add_pi_div_two]
+  simp [real_inner_comm]
+  ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Contact Identities
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+theorem capTangentArm_identities (K : RightAngleCapSpace) (t : ℝ) :
+    (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).outerCorner =
+      (capVertices K t).1.1 + (tangentArmLengths K t).1.1 • tangentVector (t : Real.Angle) ∧
+    (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).outerCorner =
+      (capVertices K t).1.2 + (tangentArmLengths K t).1.2 • tangentVector (t : Real.Angle) ∧
+    (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).outerCorner =
+      (capVertices K t).2.1 + (tangentArmLengths K t).2.1 • normalVector (t : Real.Angle) ∧
+    (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).outerCorner =
+      (capVertices K t).2.2 + (tangentArmLengths K t).2.2 • normalVector (t : Real.Angle) := by
+  have hrot : rotationMap (t : Real.Angle) (!₂[1, 1] : Point) =
+      normalVector (t : Real.Angle) + tangentVector (t : Real.Angle) := by
+    unfold rotationMap
+    rw [Orientation.rotation_apply, rightAngleRotation_apply]
+    ext i
+    fin_cases i <;> simp [normalVector, tangentVector, frame]
+    ring
+  have hy : (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).outerCorner =
+      supportValue K.val (t : Real.Angle) • normalVector (t : Real.Angle) +
+      supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) • tangentVector (t : Real.Angle) := by
+    simp only [rotatingHallwayParts, supportingPlacement, hallwayParts, hrot, Real.Angle.coe_add]
+    module
+  have hnn : inner ℝ (normalVector (t : Real.Angle)) (normalVector (t : Real.Angle)) = 1 := by
+    rw [PiLp.inner_apply]
+    simp [normalVector, frame, Fin.sum_univ_two, Real.cos_sq_add_sin_sq]
+  have htt : inner ℝ (tangentVector (t : Real.Angle)) (tangentVector (t : Real.Angle)) = 1 := by
+    rw [PiLp.inner_apply]
+    simp [tangentVector, frame, Fin.sum_univ_two, Real.sin_sq_add_cos_sq]
+  have hnt : inner ℝ (normalVector (t : Real.Angle)) (tangentVector (t : Real.Angle)) = 0 := by
+    simp [normalVector, tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+    ring
+  have htn : inner ℝ (tangentVector (t : Real.Angle)) (normalVector (t : Real.Angle)) = 0 := by
+    rw [real_inner_comm, hnt]
+  have hA (a b c : ℝ) :
+      a • normalVector (t : Real.Angle) + b • tangentVector (t : Real.Angle) =
+      (a • normalVector (t : Real.Angle) + c • tangentVector (t : Real.Angle)) +
+        inner ℝ ((a • normalVector (t : Real.Angle) + b • tangentVector (t : Real.Angle)) -
+          (a • normalVector (t : Real.Angle) + c • tangentVector (t : Real.Angle)))
+          (tangentVector (t : Real.Angle)) • tangentVector (t : Real.Angle) := by
+    simp only [inner_sub_left, inner_add_left, real_inner_smul_left, hnt, htt]
+    module
+  have hC (a b c : ℝ) :
+      a • normalVector (t : Real.Angle) + b • tangentVector (t : Real.Angle) =
+      (b • tangentVector (t : Real.Angle) + c • -normalVector (t : Real.Angle)) +
+        inner ℝ ((a • normalVector (t : Real.Angle) + b • tangentVector (t : Real.Angle)) -
+          (b • tangentVector (t : Real.Angle) + c • -normalVector (t : Real.Angle)))
+          (normalVector (t : Real.Angle)) • normalVector (t : Real.Angle) := by
+    simp only [inner_sub_left, inner_add_left, real_inner_smul_left, inner_neg_left, hnn, htn]
+    module
+  have hv : tangentVector ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle)) =
+      -normalVector (t : Real.Angle) := by
+    ext i
+    fin_cases i <;> simp [normalVector, tangentVector, frame,
+      Real.Angle.sin_add_pi_div_two, Real.Angle.cos_add_pi_div_two]
+  simp only [tangentArmLengths, hy, capVertices, edgeVertices]
+  refine ⟨hA _ _ _, hA _ _ _, ?_, ?_⟩ <;>
+    simp only [Real.Angle.coe_add, normalVector_add_pi_div_two, hv] <;>
+    exact hC _ _ _
+
+theorem surfaceAreaMeasure_atom_length (K : ConvexBody Point) (t : Real.Angle) :
+    surfaceAreaMeasure K {t} = Measure.hausdorffMeasure 1 (exposedEdge K t) ∧
+    surfaceAreaMeasure K {t} = ENNReal.ofReal (dist (edgeVertices K t).1 (edgeVertices K t).2) ∧
+    (edgeVertices K t).1 = (edgeVertices K t).2 +
+      (surfaceAreaMeasure K {t}).toReal • tangentVector t := by
+  induction t using Real.Angle.induction_on with
+  | _ t =>
+    have hface : surfaceAreaMeasure K {(t : Real.Angle)} =
+        Measure.hausdorffMeasure 1 (exposedEdge K (t : Real.Angle)) := by
+      have h := (surfaceAreaMeasure_face_union K).2.2.2.2
+        {(t : Real.Angle)} (measurableSet_singleton _) (Or.inr
+          ⟨t, t, le_rfl, by linarith [Real.pi_pos], by simp⟩)
+      simpa using h
+    have hlength : surfaceAreaMeasure K {(t : Real.Angle)} =
+        ENNReal.ofReal (dist (edgeVertices K (t : Real.Angle)).1
+          (edgeVertices K (t : Real.Angle)).2) := by
+      rw [hface, exposedEdge_eq_segment_edgeVertices, hausdorffMeasure_segment,
+        edist_dist, dist_comm]
+    refine ⟨hface, hlength, ?_⟩
+    let S := (fun p ↦ inner ℝ p (tangentVector (t : Real.Angle))) ''
+      exposedEdge K (t : Real.Angle)
+    have hcompact : IsCompact S := (isCompact_exposedEdge K _).image
+      (continuous_id.inner continuous_const)
+    have hnonneg : 0 ≤ sSup S - sInf S := sub_nonneg.mpr
+      (csInf_le_csSup ((exposedEdge_nonempty K _).image _) hcompact.bddBelow hcompact.bddAbove)
+    have hdiff : (edgeVertices K (t : Real.Angle)).1 -
+        (edgeVertices K (t : Real.Angle)).2 =
+        (sSup S - sInf S) • tangentVector (t : Real.Angle) := by
+      simp only [edgeVertices, S]
+      module
+    have hnorm : ‖tangentVector (t : Real.Angle)‖ = 1 := by
+      have h := inner_tangentVector_self t
+      rw [real_inner_self_eq_norm_sq] at h
+      nlinarith [norm_nonneg (tangentVector (t : Real.Angle))]
+    have hdist : dist (edgeVertices K (t : Real.Angle)).1
+        (edgeVertices K (t : Real.Angle)).2 = sSup S - sInf S := by
+      rw [dist_eq_norm, hdiff, norm_smul, Real.norm_eq_abs, abs_of_nonneg hnonneg,
+        hnorm, mul_one]
+    rw [hlength, ENNReal.toReal_ofReal dist_nonneg, hdist]
+    exact (sub_eq_iff_eq_add.mp hdiff).trans (add_comm _ _)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Half Planes
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A finite set of allowed normals gives a finite supporting-half-plane representation. -/
+theorem HasHalfPlaneRepresentation.finite_constraints
+    {K : ConvexBody Point} {N : Set Real.Angle} (hN : N.Finite)
+    (hK : HasHalfPlaneRepresentation K N) :
+    ∃ constraints : Set (Real.Angle × ℝ), constraints.Finite ∧
+      (∀ c ∈ constraints, c.1 ∈ N) ∧
+      (K : Set Point) = ⋂ c ∈ constraints, normalHalfPlane c.1 c.2 false false := by
+  obtain ⟨C, hC, hKC⟩ := hK
+  refine ⟨(fun t ↦ (t, supportValue K t)) '' N, hN.image _, ?_, ?_⟩
+  · rintro _ ⟨t, ht, rfl⟩
+    exact ht
+  · ext x
+    simp only [Set.mem_iInter]
+    constructor
+    · intro hx c hc
+      obtain ⟨t, ht, rfl⟩ := hc
+      change inner ℝ x (normalVector t) ≤ supportValue K t
+      exact le_csSup ((K.isCompact.image
+        (continuous_inner.comp (continuous_id.prodMk continuous_const))).bddAbove)
+        ⟨x, hx, rfl⟩
+    · intro hx
+      rw [hKC]
+      simp only [Set.mem_iInter]
+      intro c hc
+      have hxs := hx (c.1, supportValue K c.1) ⟨c.1, hC c hc, rfl⟩
+      change inner ℝ x (normalVector c.1) ≤ supportValue K c.1 at hxs
+      change inner ℝ x (normalVector c.1) ≤ c.2
+      apply hxs.trans
+      apply csSup_le (K.nonempty.image _)
+      rintro z ⟨p, hp, rfl⟩
+      rw [hKC] at hp
+      exact Set.mem_iInter.mp (Set.mem_iInter.mp hp c) hc
+
+/-- Translation preserves a half-plane representation's allowed normals. -/
+theorem HasHalfPlaneRepresentation.translate {K : ConvexBody Point}
+    {N : Set Real.Angle} (hK : HasHalfPlaneRepresentation K N) (v : Point) :
+    HasHalfPlaneRepresentation (ConvexBody.translate K v) N := by
+  obtain ⟨C, hC, hKC⟩ := hK
+  refine ⟨(fun c ↦ (c.1, c.2 + inner ℝ v (normalVector c.1))) '' C, ?_, ?_⟩
+  · rintro _ ⟨c, hc, rfl⟩
+    exact hC c hc
+  · ext x
+    simp only [ConvexBody.translate, Set.mem_image, Set.mem_iInter]
+    constructor
+    · rintro ⟨y, hy, rfl⟩ c ⟨d, hd, rfl⟩
+      change inner ℝ (y + v) (normalVector d.1) ≤
+        d.2 + inner ℝ v (normalVector d.1)
+      rw [inner_add_left]
+      have hmem := Set.mem_iInter.mp (Set.mem_iInter.mp (hKC ▸ hy) d) hd
+      change inner ℝ y (normalVector d.1) ≤ d.2 at hmem
+      simpa [add_comm] using add_le_add_right hmem (inner ℝ v (normalVector d.1))
+    · intro hx
+      refine ⟨x - v, ?_, by simp⟩
+      rw [hKC]
+      simp only [Set.mem_iInter]
+      intro c hc
+      have h := hx (c.1, c.2 + inner ℝ v (normalVector c.1)) ⟨c, hc, rfl⟩
+      change inner ℝ x (normalVector c.1) ≤
+        c.2 + inner ℝ v (normalVector c.1) at h
+      change inner ℝ (x - v) (normalVector c.1) ≤ c.2
+      rw [inner_sub_left]
+      linarith
+
+/-- A normalized cap is contained in its lower fan. -/
+theorem CapSpace.subset_capFan {ω : ℝ} (K : CapSpace ω) :
+    (K.val : Set Point) ⊆ capFan ω := by
+  intro p hp
+  constructor
+  · change 0 ≤ inner ℝ p (normalVector (ω : Real.Angle))
+    have h := inner_le_supportValue K.val hp ((ω + Real.pi : ℝ) : Real.Angle)
+    rw [K.property.2.2.2.2.1, normalVector_add_pi, inner_neg_right] at h
+    linarith
+  · change 0 ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+    have h := inner_le_supportValue K.val hp ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+    have hang : (((Real.pi / 2 + Real.pi : ℝ) : Real.Angle)) =
+        ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+      congr 1
+      ring
+    rw [K.property.2.2.2.2.2.1, ← hang] at h
+    rw [normalVector_add_pi, inner_neg_right] at h
+    linarith
+
+/-- Fan membership and strict upper support inequalities imply cap membership. -/
+theorem CapSpace.mem_of_mem_capFan_of_lt_supportValue {ω : ℝ}
+    (K : CapSpace ω) {p : Point} (hp : p ∈ capFan ω)
+    (hupper : ∀ t ∈ Set.Icc 0 (ω + Real.pi / 2),
+      inner ℝ p (normalVector (t : Real.Angle)) < supportValue K.val (t : Real.Angle)) :
+    p ∈ (K.val : Set Point) := by
+  obtain ⟨C, hCN, hKC⟩ := K.property.2.2.2.2.2.2
+  rw [hKC]
+  simp only [Set.mem_iInter]
+  intro c hc
+  have hsupport : supportValue K.val c.1 ≤ c.2 := by
+    apply supportValue_le_of_subset_normalHalfPlane
+    intro q hq
+    rw [hKC] at hq
+    exact Set.mem_iInter.mp (Set.mem_iInter.mp hq c) hc
+  rcases hCN c hc with hcupper | hclower
+  · obtain ⟨t, ht, heq⟩ := hcupper
+    rw [← heq] at hsupport ⊢
+    have htI : t ∈ Set.Icc 0 (ω + Real.pi / 2) := by
+      rcases ht with ht | ht
+      · exact ⟨ht.1, ht.2.trans (le_add_of_nonneg_right (by positivity))⟩
+      · exact ⟨(by positivity : 0 ≤ Real.pi / 2).trans ht.1, ht.2⟩
+    change inner ℝ p (normalVector (t : Real.Angle)) ≤ c.2
+    exact (hupper t htI).le.trans hsupport
+  · rcases hclower with hclower | hclower
+    · rw [hclower] at hsupport ⊢
+      change inner ℝ p (normalVector ((ω + Real.pi : ℝ) : Real.Angle)) ≤ c.2
+      rw [normalVector_add_pi, inner_neg_right]
+      have hpω := hp.1
+      change 0 ≤ inner ℝ p (normalVector (ω : Real.Angle)) at hpω
+      rw [K.property.2.2.2.2.1] at hsupport
+      linarith [hsupport]
+    · rw [hclower] at hsupport ⊢
+      have hang : (((Real.pi / 2 + Real.pi : ℝ) : Real.Angle)) =
+          ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+        congr 1
+        ring
+      change inner ℝ p (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) ≤ c.2
+      rw [← hang, normalVector_add_pi, inner_neg_right]
+      have hpT := hp.2
+      change 0 ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at hpT
+      rw [K.property.2.2.2.2.2.1] at hsupport
+      linarith [hsupport]
+
+/-- A normalized cap lies in the intersection of its two unit strips. -/
+theorem CapSpace.subset_stripParallelogram {ω : ℝ} (K : CapSpace ω) :
+    (K.val : Set Point) ⊆ (stripParallelogram ω).1 := by
+  intro p hp
+  have hlower := inner_le_supportValue K.val hp ((ω + Real.pi : ℝ) : Real.Angle)
+  have hbottom := inner_le_supportValue K.val hp ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+  rw [K.property.2.2.2.2.1, normalVector_add_pi, inner_neg_right] at hlower
+  rw [K.property.2.2.2.2.2.1, inner_normalVector_three_pi_div_two] at hbottom
+  have hupper := inner_le_supportValue K.val hp ((Real.pi / 2 : ℝ) : Real.Angle)
+  have hright := inner_le_supportValue K.val hp (ω : Real.Angle)
+  rw [K.property.2.2.2.1] at hupper
+  rw [K.property.2.2.1] at hright
+  rw [mem_stripParallelogram_iff]
+  have hlo : 0 ≤ p 1 := by linarith
+  have hhi : p 1 ≤ 1 := by
+    simpa [normalVector, frame, PiLp.inner_apply] using hupper
+  exact ⟨⟨hlo, hhi⟩, by linarith, hright⟩
+
+/-- A half-plane representation can be tightened at every allowed normal. -/
+theorem HasHalfPlaneRepresentation.eq_iInter_supportValue {K : ConvexBody Point}
+    {N : Set Real.Angle} (hK : HasHalfPlaneRepresentation K N) :
+    (K : Set Point) = ⋂ t ∈ N, normalHalfPlane t (supportValue K t) false false := by
+  obtain ⟨C, hCN, hKC⟩ := hK
+  ext p
+  simp only [Set.mem_iInter]
+  constructor
+  · intro hp t _
+    exact inner_le_supportValue K hp t
+  · intro hp
+    rw [hKC]
+    simp only [Set.mem_iInter]
+    intro c hc
+    have ht := hp c.1 (hCN c hc)
+    change inner ℝ p (normalVector c.1) ≤ c.2
+    apply ht.trans
+    apply csSup_le (K.nonempty.image _)
+    rintro _ ⟨q, hq, rfl⟩
+    rw [hKC] at hq
+    exact Set.mem_iInter.mp (Set.mem_iInter.mp hq c) hc
+
+/-- A fan point satisfying all upper supporting inequalities belongs to the cap. -/
+theorem CapSpace.mem_of_mem_capFan_of_le_supportValue {ω : ℝ}
+    (K : CapSpace ω) {p : Point} (hp : p ∈ capFan ω)
+    (hupper : ∀ t ∈ capUpperAngles ω,
+      inner ℝ p (normalVector (t : Real.Angle)) ≤ supportValue K.val (t : Real.Angle)) :
+    p ∈ (K.val : Set Point) := by
+  rw [K.property.2.2.2.2.2.2.eq_iInter_supportValue]
+  simp only [Set.mem_iInter]
+  intro a ha
+  rcases ha with ⟨t, ht, rfl⟩ | ha
+  · exact hupper t ht
+  rcases ha with rfl | rfl
+  · change inner ℝ p (normalVector ((ω + Real.pi : ℝ) : Real.Angle)) ≤
+      supportValue K.val ((ω + Real.pi : ℝ) : Real.Angle)
+    rw [K.property.2.2.2.2.1]
+    rw [normalVector_add_pi, inner_neg_right]
+    exact neg_nonpos.mpr hp.1
+  · have hang : ((3 * Real.pi / 2 : ℝ) : Real.Angle) =
+        (((Real.pi / 2 + Real.pi : ℝ) : Real.Angle)) := by
+      congr 1
+      ring
+    change inner ℝ p (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) ≤
+      supportValue K.val ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+    rw [K.property.2.2.2.2.2.1]
+    rw [hang, normalVector_add_pi, inner_neg_right]
+    exact neg_nonpos.mpr hp.2
+
+/-- Upper support bounds supplied by the two unit-height constraints of a cap. -/
+theorem CapSpace.supportValue_upper_bounds {ω t : ℝ} (K : CapSpace ω)
+    (ht : t ∈ Set.Ioo 0 ω) :
+    supportValue K.val (t : Real.Angle) ≤
+        Real.cos t * supportValue K.val (0 : Real.Angle) + Real.sin t ∧
+      supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) ≤
+        Real.sin (ω - t) + Real.cos (ω - t) *
+          supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) := by
+  have hcost : 0 ≤ Real.cos t := (Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, ht.1], ht.2.trans_le K.property.2.1⟩).le
+  have hsint : 0 ≤ Real.sin t := (Real.sin_pos_of_pos_of_lt_pi ht.1
+    (by linarith [ht.2, K.property.2.1, Real.pi_pos])).le
+  have hcosδ : 0 ≤ Real.cos (ω - t) := (Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [sub_pos.mpr ht.2, Real.pi_pos],
+      by linarith [ht.1, K.property.2.1]⟩).le
+  have hsinδ : 0 ≤ Real.sin (ω - t) := (Real.sin_pos_of_pos_of_lt_pi
+    (sub_pos.mpr ht.2) (by linarith [ht.1, K.property.2.1, Real.pi_pos])).le
+  constructor
+  · apply csSup_le (K.val.nonempty.image _)
+    rintro _ ⟨p, hp, rfl⟩
+    have hx := inner_le_supportValue K.val hp (0 : Real.Angle)
+    have hy := inner_le_supportValue K.val hp ((Real.pi / 2 : ℝ) : Real.Angle)
+    rw [K.property.2.2.2.1] at hy
+    simp [normalVector, frame, PiLp.inner_apply] at hx hy ⊢
+    nlinarith
+  · apply csSup_le (K.val.nonempty.image _)
+    rintro _ ⟨p, hp, rfl⟩
+    have hu := inner_le_supportValue K.val hp (ω : Real.Angle)
+    have hv := inner_le_supportValue K.val hp
+      ((ω + Real.pi / 2 : ℝ) : Real.Angle)
+    rw [K.property.2.2.1] at hu
+    have hvec : normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+        Real.sin (ω - t) • normalVector (ω : Real.Angle) +
+          Real.cos (ω - t) • tangentVector (ω : Real.Angle) := by
+      have h := normalVector_add_real ω (Real.pi / 2 - (ω - t))
+      rw [Real.cos_pi_div_two_sub, Real.sin_pi_div_two_sub] at h
+      simpa only [show ω + (Real.pi / 2 - (ω - t)) = t + Real.pi / 2 by ring] using h
+    change inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) ≤ _
+    rw [hvec, inner_add_right, inner_smul_right, inner_smul_right]
+    have hv' : inner ℝ p (tangentVector (ω : Real.Angle)) ≤
+        supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) := by
+      simpa [normalVector, tangentVector, frame, Real.Angle.cos_add_pi_div_two,
+        Real.Angle.sin_add_pi_div_two] using hv
+    nlinarith
+
+/-- Every point of a cap lies above the horizontal base line. -/
+theorem CapSpace.inner_normalVector_pi_div_two_nonneg {ω : ℝ} (K : CapSpace ω) {q : Point}
+    (hq : q ∈ (K.1 : Set Point)) :
+    0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) := (K.subset_capFan hq).2
+
+/-- Lowering a point of a right-angle cap onto the base line keeps it inside the cap: every
+upper normal of such a cap has nonnegative vertical component, so no upper constraint is
+tightened, and the two base constraints of the fan coincide here and hold with equality. -/
+theorem CapSpace.base_projection_mem (K : CapSpace (Real.pi / 2)) {q : Point}
+    (hq : q ∈ (K.val : Set Point)) :
+    q - q 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle) ∈ (K.val : Set Point) := by
+  have hq1 : 0 ≤ q 1 := by
+    simpa only [inner_normalVector_pi_div_two] using K.inner_normalVector_pi_div_two_nonneg hq
+  have hinner (s : ℝ) : inner ℝ (q - q 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+      (normalVector (s : Real.Angle)) =
+      inner ℝ q (normalVector (s : Real.Angle)) - q 1 * Real.sin s := by
+    rw [inner_sub_left, real_inner_smul_left, inner_normalVector_normalVector,
+      Real.cos_pi_div_two_sub]
+  have hbase : inner ℝ (q - q 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+      (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := by
+    rw [hinner, inner_normalVector_pi_div_two, Real.sin_pi_div_two, mul_one, sub_self]
+  refine K.mem_of_mem_capFan_of_le_supportValue ⟨?_, ?_⟩ ?_
+  · change 0 ≤ inner ℝ _ (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+    rw [hbase]
+  · change 0 ≤ inner ℝ _ (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+    rw [hbase]
+  · intro s hs
+    have hs' : 0 ≤ s ∧ s ≤ Real.pi := by
+      rcases hs with hs | hs
+      · exact ⟨hs.1, by linarith [hs.2, Real.pi_pos]⟩
+      · exact ⟨by linarith [hs.1, Real.pi_pos], by linarith [hs.2]⟩
+    have hsin : 0 ≤ Real.sin s := Real.sin_nonneg_of_nonneg_of_le_pi hs'.1 hs'.2
+    rw [hinner]
+    linarith [mul_nonneg hq1 hsin, inner_le_supportValue K.val hq (s : Real.Angle)]
+
+/-- The top supporting line of a right-angle cap is the horizontal line of height one, so every
+point attaining the support value at the vertical normal has height one. -/
+theorem CapSpace.apply_one_eq_one (K : CapSpace (Real.pi / 2)) {p : Point}
+    (hp : inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) =
+      supportValue (K.val : Set Point) ((Real.pi / 2 : ℝ) : Real.Angle)) : p 1 = 1 := by
+  rwa [inner_normalVector_pi_div_two, K.property.2.2.2.1] at hp
+
+/-- A singleton extreme face of a right-angle cap at a horizontal normal lies on the base line.
+Lowering its unique point onto the base line keeps it in the cap, and a horizontal normal does not
+see that vertical displacement, so the lowered point lies in the same face; the face being a
+singleton, the displacement vanishes. -/
+theorem CapSpace.edgeVertices_fst_apply_one_eq_zero (K : CapSpace (Real.pi / 2)) {s : ℝ}
+    (hs : Real.sin s = 0)
+    (hface : (edgeVertices K.val (s : Real.Angle)).1 = (edgeVertices K.val (s : Real.Angle)).2) :
+    (edgeVertices K.val (s : Real.Angle)).1 1 = 0 := by
+  have hmem := edgeVertices_fst_mem K.val (s : Real.Angle)
+  have hkey : inner ℝ ((edgeVertices K.val (s : Real.Angle)).1 -
+      (edgeVertices K.val (s : Real.Angle)).1 1 •
+        normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) (normalVector (s : Real.Angle)) =
+      supportValue (K.val : Set Point) (s : Real.Angle) := by
+    rw [inner_sub_left, real_inner_smul_left, inner_normalVector_normalVector,
+      Real.cos_pi_div_two_sub, hs, mul_zero, sub_zero]
+    exact hmem.2
+  have hlow : (edgeVertices K.val (s : Real.Angle)).1 -
+      (edgeVertices K.val (s : Real.Angle)).1 1 •
+        normalVector ((Real.pi / 2 : ℝ) : Real.Angle) ∈
+      exposedEdge K.val (s : Real.Angle) := ⟨K.base_projection_mem hmem.1, hkey⟩
+  rw [exposedEdge_eq_segment_edgeVertices, ← hface, segment_same, Set.mem_singleton_iff,
+    sub_eq_self] at hlow
+  have h := congrArg (fun p : Point ↦ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)))
+    hlow
+  rwa [real_inner_smul_left, inner_normalVector_self, mul_one, inner_zero_left] at h
+
+/-- A convex body with vanishing base support value that is stable under vertical projection to
+the base line is cut out by upper half-planes together with the base half-plane. -/
+theorem hasHalfPlaneRepresentation_of_base_projection (M : ConvexBody Point)
+    (hbase : supportValue M ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0)
+    (hproj : ∀ q ∈ (M : Set Point),
+      q - q 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle) ∈ (M : Set Point)) :
+    HasHalfPlaneRepresentation M
+      (((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles (Real.pi / 2)) ∪
+        capLowerNormals (Real.pi / 2)) := by
+  -- At a right angle the allowed upper normals are exactly the angles of `[0, π]`.
+  have hangle : ∀ s ∈ Set.Icc (0 : ℝ) Real.pi,
+      ((s : ℝ) : Real.Angle) ∈
+        (((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles (Real.pi / 2)) ∪
+          capLowerNormals (Real.pi / 2)) := by
+    intro s hs
+    refine Or.inl ⟨s, ?_, rfl⟩
+    rcases le_or_gt s (Real.pi / 2) with h | h
+    · exact Or.inl ⟨hs.1, h⟩
+    · exact Or.inr ⟨h.le, by linarith [hs.2]⟩
+  -- The vertical projection keeps the horizontal coordinate and kills the vertical one.
+  have hflat : ∀ q : Point,
+      (q - q 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) 0 = q 0 ∧
+        (q - q 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) 1 = 0 := by
+    intro q
+    constructor <;> simp [normalVector, frame]
+  -- The two attained horizontal extrema, lowered onto the base line.
+  obtain ⟨qR, hqR, hqRval⟩ := exists_mem_inner_eq_supportValue M ((0 : ℝ) : Real.Angle)
+  obtain ⟨qL, hqL, hqLval⟩ := exists_mem_inner_eq_supportValue M ((Real.pi : ℝ) : Real.Angle)
+  rw [inner_normalVector_zero] at hqRval
+  rw [inner_normalVector_real, Real.cos_pi, Real.sin_pi] at hqLval
+  refine ⟨(fun a : Real.Angle ↦ (a, supportValue M a)) ''
+    ((((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles (Real.pi / 2)) ∪
+      capLowerNormals (Real.pi / 2))), ?_, ?_⟩
+  · rintro _ ⟨a, ha, rfl⟩
+    exact ha
+  · ext p
+    simp only [Set.mem_iInter]
+    refine ⟨fun hp c hc ↦ ?_, fun hp ↦ ?_⟩
+    · obtain ⟨a, -, rfl⟩ := hc
+      exact inner_le_supportValue M hp a
+    · have hpupper : ∀ s ∈ Set.Icc (0 : ℝ) Real.pi,
+          inner ℝ p (normalVector ((s : ℝ) : Real.Angle)) ≤
+            supportValue M ((s : ℝ) : Real.Angle) :=
+        fun s hs ↦ hp _ ⟨_, hangle s hs, rfl⟩
+      have hpbase := hp _ ⟨((3 * Real.pi / 2 : ℝ) : Real.Angle),
+        Or.inr (Set.mem_insert_iff.2 (Or.inr rfl)), rfl⟩
+      change inner ℝ p (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) ≤
+        supportValue M ((3 * Real.pi / 2 : ℝ) : Real.Angle) at hpbase
+      rw [inner_normalVector_three_pi_div_two, hbase] at hpbase
+      have hp1 : 0 ≤ p 1 := by linarith
+      have hpR : p 0 ≤ qR 0 := by
+        have h := hpupper 0 ⟨le_rfl, Real.pi_pos.le⟩
+        rw [inner_normalVector_zero, ← hqRval] at h
+        exact h
+      have hpL : qL 0 ≤ p 0 := by
+        have h := hpupper Real.pi ⟨Real.pi_pos.le, le_rfl⟩
+        rw [inner_normalVector_real, Real.cos_pi, Real.sin_pi, ← hqLval] at h
+        linarith
+      rw [M.eq_iInter_halfSpaces]
+      simp only [Set.mem_iInter]
+      intro u hu
+      obtain ⟨θ, rfl⟩ := exists_angle_normalVector_eq hu
+      change inner ℝ p (normalVector θ) ≤ supportValue M θ
+      rw [← Real.Angle.coe_toReal θ]
+      set s : ℝ := θ.toReal
+      rcases le_or_gt 0 s with hs0 | hs0
+      · exact hpupper s ⟨hs0, Real.Angle.toReal_le_pi θ⟩
+      · have hsin : Real.sin s < 0 :=
+          Real.sin_neg_of_neg_of_neg_pi_lt hs0 (Real.Angle.neg_pi_lt_toReal θ)
+        rw [inner_normalVector_real]
+        rcases le_or_gt 0 (Real.cos s) with hc | hc
+        · have h := inner_le_supportValue M (hproj qR hqR) ((s : ℝ) : Real.Angle)
+          rw [inner_normalVector_real, (hflat qR).1, (hflat qR).2] at h
+          nlinarith [mul_le_mul_of_nonneg_right hpR hc,
+            mul_nonpos_of_nonneg_of_nonpos hp1 hsin.le]
+        · have h := inner_le_supportValue M (hproj qL hqL) ((s : ℝ) : Real.Angle)
+          rw [inner_normalVector_real, (hflat qL).1, (hflat qL).2] at h
+          nlinarith [mul_le_mul_of_nonpos_right hpL hc.le,
+            mul_nonpos_of_nonneg_of_nonpos hp1 hsin.le]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Fan Projection
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Support values in the upper angular range of a non-right cap are nonnegative. -/
+theorem supportValue_nonneg_of_mem_capUpperAngles {ω : ℝ} (K : CapSpace ω)
+    (hω : ω < Real.pi / 2) {φ : ℝ} (hφ : φ ∈ capUpperAngles ω) :
+    0 ≤ supportValue K.val (φ : Real.Angle) := by
+  have hcosω : 0 < Real.cos ω := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [K.property.1, Real.pi_pos], hω⟩
+  rcases hφ with hφ | hφ
+  · obtain ⟨p, hpK, hpnormal⟩ := exists_mem_inner_eq_supportValue K.val
+      ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+    have hpFan := K.subset_capFan hpK
+    have hpy : p 1 = 0 := by
+      change inner ℝ p (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) =
+        supportValue K.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) at hpnormal
+      rw [K.property.2.2.2.2.2.1] at hpnormal
+      have hang : ((3 * Real.pi / 2 : ℝ) : Real.Angle) =
+          (((Real.pi / 2 + Real.pi : ℝ) : Real.Angle)) := by
+        congr 1
+        ring
+      rw [hang, normalVector_add_pi, inner_neg_right] at hpnormal
+      simpa [normalVector, frame, PiLp.inner_apply] using hpnormal
+    have hpx : 0 ≤ p 0 := by
+      have hpω := hpFan.1
+      change 0 ≤ inner ℝ p (normalVector (ω : Real.Angle)) at hpω
+      simp [normalVector, frame, PiLp.inner_apply, hpy] at hpω
+      by_contra hneg
+      have := mul_neg_of_pos_of_neg hcosω (lt_of_not_ge hneg)
+      linarith
+    have hcosφ : 0 ≤ Real.cos φ := Real.cos_nonneg_of_mem_Icc
+      ⟨(neg_nonpos.mpr (by positivity : 0 ≤ Real.pi / 2)).trans hφ.1,
+        hφ.2.trans hω.le⟩
+    have hinner : 0 ≤ inner ℝ p (normalVector (φ : Real.Angle)) := by
+      simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply,
+        RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue, Matrix.cons_val_zero,
+        Matrix.cons_val_one, Matrix.cons_val_fin_one, hpy, mul_zero, add_zero]
+      exact mul_nonneg hcosφ hpx
+    exact hinner.trans (inner_le_supportValue K.val hpK (φ : Real.Angle))
+  · obtain ⟨p, hpK, hpnormal⟩ := exists_mem_inner_eq_supportValue K.val
+      ((ω + Real.pi : ℝ) : Real.Angle)
+    have hpFan := K.subset_capFan hpK
+    have hpu : inner ℝ p (normalVector (ω : Real.Angle)) = 0 := by
+      change inner ℝ p (normalVector ((ω + Real.pi : ℝ) : Real.Angle)) =
+        supportValue K.val ((ω + Real.pi : ℝ) : Real.Angle) at hpnormal
+      rw [K.property.2.2.2.2.1, normalVector_add_pi, inner_neg_right] at hpnormal
+      linarith
+    let μ := inner ℝ p (tangentVector (ω : Real.Angle))
+    have hp : μ • tangentVector (ω : Real.Angle) = p := by
+      have hframe := inner_normalVector_smul_add_inner_tangentVector_smul
+        p (ω : Real.Angle)
+      rw [hpu, zero_smul, zero_add] at hframe
+      exact hframe
+    have hμ : 0 ≤ μ := by
+      have hpy := hpFan.2
+      change 0 ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at hpy
+      have hp1 := congrArg (fun q : Point ↦ q 1) hp
+      change μ * Real.cos ω = p 1 at hp1
+      have hpy' : 0 ≤ p 1 := by
+        simpa [normalVector, frame, PiLp.inner_apply] using hpy
+      rw [← hp1] at hpy'
+      by_contra hneg
+      have := mul_neg_of_neg_of_pos (lt_of_not_ge hneg) hcosω
+      linarith
+    have hsin : 0 ≤ Real.sin (φ - ω) := Real.sin_nonneg_of_nonneg_of_le_pi
+      (by linarith [hφ.1]) (by linarith [hφ.2, Real.pi_pos])
+    have hinner : 0 ≤ inner ℝ p (normalVector (φ : Real.Angle)) := by
+      have htv : inner ℝ (tangentVector (ω : Real.Angle))
+          (normalVector (φ : Real.Angle)) = Real.sin (φ - ω) := by
+        simp [tangentVector, normalVector, frame, PiLp.inner_apply, Real.sin_sub]
+        ring
+      have heq : inner ℝ p (normalVector (φ : Real.Angle)) = μ * Real.sin (φ - ω) := by
+        rw [← hp, real_inner_smul_left, htv]
+      rw [heq]
+      exact mul_nonneg hμ hsin
+    exact hinner.trans (inner_le_supportValue K.val hpK (φ : Real.Angle))
+
+/-- The origin belongs to every cap of angle strictly below a right angle. -/
+theorem zero_mem_cap_of_lt {ω : ℝ} (K : CapSpace ω)
+    (hω : ω < Real.pi / 2) : (0 : Point) ∈ (K.val : Set Point) := by
+  apply K.mem_of_mem_capFan_of_le_supportValue
+  · simp [capFan, normalHalfPlane]
+  · intro φ hφ
+    simpa using supportValue_nonneg_of_mem_capUpperAngles K hω hφ
+
+/-- The normal projection of the zero-angle support value belongs to the cap. -/
+theorem supportValue_zero_smul_normalVector_mem {ω : ℝ} (K : CapSpace ω) :
+    supportValue K.val (0 : Real.Angle) • normalVector (0 : Real.Angle) ∈
+      (K.val : Set Point) := by
+  obtain ⟨p, hpK, hpnormal⟩ :=
+    exists_mem_inner_eq_supportValue K.val (0 : Real.Angle)
+  have hpFan := K.subset_capFan hpK
+  have hpx : p 0 = supportValue K.val (0 : Real.Angle) := by
+    change inner ℝ p (normalVector (0 : Real.Angle)) = supportValue K.val (0 : Real.Angle)
+      at hpnormal
+    simpa [normalVector, frame, PiLp.inner_apply] using hpnormal
+  have hpy : 0 ≤ p 1 := by
+    have := hpFan.2
+    change 0 ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at this
+    simpa [normalVector, frame, PiLp.inner_apply] using this
+  apply K.mem_of_mem_capFan_of_le_supportValue
+  · constructor
+    · change 0 ≤ inner ℝ (supportValue K.val (0 : Real.Angle) •
+          normalVector (0 : Real.Angle)) (normalVector (ω : Real.Angle))
+      by_cases hω : ω = Real.pi / 2
+      · simp [normalVector, frame, PiLp.inner_apply, hω]
+      · have hωlt : ω < Real.pi / 2 := K.property.2.1.lt_of_ne hω
+        have hs : 0 ≤ supportValue K.val (0 : Real.Angle) :=
+          supportValue_nonneg_of_mem_capUpperAngles K hωlt
+            (Or.inl ⟨le_rfl, K.property.1.le⟩)
+        simp only [real_inner_smul_left]
+        simp only [normalVector, frame, Real.Angle.cos_zero, Real.Angle.sin_zero, neg_zero,
+          Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply, RCLike.inner_apply,
+          conj_trivial, Fin.sum_univ_two, Fin.isValue, Matrix.cons_val_zero, mul_one,
+          Matrix.cons_val_one, Matrix.cons_val_fin_one, mul_zero, add_zero, ge_iff_le]
+        exact mul_nonneg hs (Real.cos_nonneg_of_mem_Icc
+          ⟨by linarith [K.property.1, Real.pi_pos], K.property.2.1⟩)
+    · change 0 ≤ inner ℝ (supportValue K.val (0 : Real.Angle) •
+          normalVector (0 : Real.Angle))
+        (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+      simp [normalVector, frame, PiLp.inner_apply]
+  · intro φ hφ
+    have hφI : φ ∈ Set.Icc 0 Real.pi := by
+      rcases hφ with hφ | hφ
+      · exact ⟨hφ.1, hφ.2.trans
+          (K.property.2.1.trans (by linarith [Real.pi_pos]))⟩
+      · exact ⟨(by positivity : 0 ≤ Real.pi / 2).trans hφ.1,
+          hφ.2.trans (by linarith [K.property.2.1, Real.pi_pos])⟩
+    have hsinφ : 0 ≤ Real.sin φ := Real.sin_nonneg_of_mem_Icc hφI
+    have hbound := inner_le_supportValue K.val hpK (φ : Real.Angle)
+    have hinner : inner ℝ
+        (supportValue K.val (0 : Real.Angle) • normalVector (0 : Real.Angle))
+        (normalVector (φ : Real.Angle)) ≤ inner ℝ p (normalVector (φ : Real.Angle)) := by
+      simp only [normalVector, frame, Real.Angle.cos_zero, Real.Angle.sin_zero, neg_zero,
+        Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply, PiLp.smul_apply, smul_eq_mul,
+        RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue, Matrix.cons_val_zero,
+        mul_one, Matrix.cons_val_one, Matrix.cons_val_fin_one, mul_zero, add_zero, hpx,
+        le_add_iff_nonneg_right]
+      exact mul_nonneg hsinφ hpy
+    exact hinner.trans hbound
+
+/-- At a right angle, the opposite normal projection belongs to the cap. -/
+theorem supportValue_pi_smul_normalVector_mem_of_eq {ω : ℝ} (K : CapSpace ω)
+    (hω : ω = Real.pi / 2) :
+    supportValue K.val (Real.pi : Real.Angle) • normalVector (Real.pi : Real.Angle) ∈
+      (K.val : Set Point) := by
+  obtain ⟨p, hpK, hpnormal⟩ :=
+    exists_mem_inner_eq_supportValue K.val (Real.pi : Real.Angle)
+  have hpFan := K.subset_capFan hpK
+  have hpx : -p 0 = supportValue K.val (Real.pi : Real.Angle) := by
+    change inner ℝ p (normalVector (Real.pi : Real.Angle)) =
+      supportValue K.val (Real.pi : Real.Angle) at hpnormal
+    simpa [normalVector, frame, PiLp.inner_apply] using hpnormal
+  have hpy : 0 ≤ p 1 := by
+    have := hpFan.2
+    change 0 ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at this
+    simpa [normalVector, frame, PiLp.inner_apply] using this
+  apply K.mem_of_mem_capFan_of_le_supportValue
+  · constructor <;>
+      simp [normalHalfPlane, normalVector, frame, PiLp.inner_apply, hω]
+  · intro φ hφ
+    have hφI : φ ∈ Set.Icc 0 Real.pi := by
+      rcases hφ with hφ | hφ
+      · exact ⟨hφ.1, hφ.2.trans (by rw [hω]; linarith [Real.pi_pos])⟩
+      · exact ⟨(by positivity : 0 ≤ Real.pi / 2).trans hφ.1,
+          hφ.2.trans (by rw [hω]; linarith)⟩
+    have hsinφ : 0 ≤ Real.sin φ := Real.sin_nonneg_of_mem_Icc hφI
+    have hbound := inner_le_supportValue K.val hpK (φ : Real.Angle)
+    have hinner : inner ℝ
+        (supportValue K.val (Real.pi : Real.Angle) • normalVector (Real.pi : Real.Angle))
+        (normalVector (φ : Real.Angle)) ≤ inner ℝ p (normalVector (φ : Real.Angle)) := by
+      rw [← hpx]
+      simp only [Fin.isValue, normalVector, frame, Real.Angle.cos_coe, Real.cos_pi,
+        Real.Angle.sin_coe, Real.sin_pi, neg_zero, neg_smul, inner_neg_left, PiLp.inner_apply,
+        PiLp.smul_apply, smul_eq_mul, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two,
+        Matrix.cons_val_zero, mul_neg, mul_one, Matrix.cons_val_one, Matrix.cons_val_fin_one,
+        mul_zero, add_zero, neg_neg, le_add_iff_nonneg_right]
+      exact mul_nonneg hsinφ hpy
+    exact hinner.trans hbound
+
+/-- For a non-right cap, the terminal tangent projection belongs to the cap. -/
+theorem supportValue_add_pi_div_two_smul_tangentVector_mem_of_lt {ω : ℝ}
+    (K : CapSpace ω) (hω : ω < Real.pi / 2) :
+    supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) •
+        tangentVector (ω : Real.Angle) ∈ (K.val : Set Point) := by
+  have hcosω : 0 < Real.cos ω := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [K.property.1, Real.pi_pos], hω⟩
+  let L := supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle)
+  obtain ⟨p, hpK, hpnormal⟩ := exists_mem_inner_eq_supportValue K.val
+    ((ω + Real.pi / 2 : ℝ) : Real.Angle)
+  have hpFan := K.subset_capFan hpK
+  have hpv : inner ℝ p (tangentVector (ω : Real.Angle)) = L := by
+    change inner ℝ p (normalVector ((ω + Real.pi / 2 : ℝ) : Real.Angle)) = L at hpnormal
+    rw [show ((ω + Real.pi / 2 : ℝ) : Real.Angle) =
+      (ω : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) by rfl,
+      normalVector_add_pi_div_two] at hpnormal
+    exact hpnormal
+  let a := inner ℝ p (normalVector (ω : Real.Angle))
+  have ha : 0 ≤ a := hpFan.1
+  have hp : a • normalVector (ω : Real.Angle) + L • tangentVector (ω : Real.Angle) = p := by
+    simpa only [a, hpv] using
+      inner_normalVector_smul_add_inner_tangentVector_smul p (ω : Real.Angle)
+  have hL : 0 ≤ L := supportValue_nonneg_of_mem_capUpperAngles K hω
+    (Or.inr ⟨by linarith [K.property.1, Real.pi_pos], le_rfl⟩)
+  apply K.mem_of_mem_capFan_of_le_supportValue
+  · constructor
+    · change 0 ≤ inner ℝ (L • tangentVector (ω : Real.Angle))
+        (normalVector (ω : Real.Angle))
+      rw [real_inner_smul_left, real_inner_comm, inner_normalVector_tangentVector]
+      simp
+    · change 0 ≤ inner ℝ (L • tangentVector (ω : Real.Angle))
+        (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+      simp only [tangentVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, normalVector,
+        Real.cos_pi_div_two, Real.sin_pi_div_two, PiLp.inner_apply, PiLp.smul_apply,
+        smul_eq_mul, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue,
+        Matrix.cons_val_zero, mul_neg, zero_mul, neg_zero, Matrix.cons_val_one,
+        Matrix.cons_val_fin_one, one_mul, zero_add]
+      exact mul_nonneg hL hcosω.le
+  · intro φ hφ
+    have hdiff : φ - ω ∈ Set.Icc (-(Real.pi / 2)) (Real.pi / 2) := by
+      rcases hφ with hφ | hφ
+      · exact ⟨by linarith [hφ.1, K.property.2.1],
+          by linarith [hφ.2, Real.pi_pos]⟩
+      · exact ⟨by linarith [hφ.1, K.property.2.1, Real.pi_pos],
+          by linarith [hφ.2]⟩
+    have hcos : 0 ≤ Real.cos (φ - ω) := Real.cos_nonneg_of_mem_Icc hdiff
+    have hnu : inner ℝ (normalVector (ω : Real.Angle))
+        (normalVector (φ : Real.Angle)) = Real.cos (φ - ω) := by
+      rw [inner_normalVector_normalVector]
+      rw [show ω - φ = -(φ - ω) by ring, Real.cos_neg]
+    have hbound := inner_le_supportValue K.val hpK (φ : Real.Angle)
+    have hinner : inner ℝ (L • tangentVector (ω : Real.Angle))
+        (normalVector (φ : Real.Angle)) ≤ inner ℝ p (normalVector (φ : Real.Angle)) := by
+      rw [← hp, inner_add_left, real_inner_smul_left, real_inner_smul_left, hnu]
+      exact le_add_of_nonneg_left (mul_nonneg ha hcos)
+    exact hinner.trans hbound
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Hallway Quadrant
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The inward quadrant of the supporting hallway at angle `t` is the inward quadrant of `s`
+written in support-value coordinates. -/
+theorem rotatingHallwayParts_innerQuadrant (s : Set Point) (t : ℝ) :
+    (rotatingHallwayParts s (t : Real.Angle)).innerQuadrant = innerQuadrant s t := by
+  rw [(rotatingHallwayParts_formulas s (t : Real.Angle)).2.2.2.2.2.2.2.2]
+  simp only [innerQuadrant, Real.Angle.coe_add]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The surface area measure of a right-angle cap at its lower normals
+
+A right-angle cap lies above its base line and is stable under vertical projection onto it, so
+at a strictly downward normal direction the support value is attained only on the base line, at
+whichever horizontal extremum the sign of the horizontal normal component selects.  Both open
+quarter arcs of lower normals therefore carry faces that degenerate to a single base corner,
+and the surface area measure vanishes on them
+(`MovingSofa.CapSpace.surfaceAreaMeasure_image_Ioo_lower_eq_zero`).  Outside the closed upper
+semicircle only the bottom normal `3π / 2` is left, so an integrand vanishing there integrates
+to zero (`MovingSofa.CapSpace.setIntegral_compl_image_Icc_zero_pi_eq_zero`).
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- At a strictly downward normal direction the face of a right-angle cap degenerates to the
+base point below a horizontal extremum: positive height strictly lowers the normal coordinate,
+and on the base line the sign of `Real.cos t` selects a horizontal extremum. -/
+theorem CapSpace.exposedEdge_subset_singleton_of_isMaxOn (K : CapSpace (Real.pi / 2))
+    {q : Point} (hq : q ∈ (K.val : Set Point)) {t : ℝ} (hcos : Real.cos t ≠ 0)
+    (hsin : Real.sin t < 0)
+    (hmax : IsMaxOn (fun p : Point ↦ p 0 * Real.cos t) (K.val : Set Point) q) :
+    exposedEdge K.val (t : Real.Angle) ⊆
+      {q - q 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle)} := by
+  have hflat0 : (q - q 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) 0 = q 0 := by
+    simp [normalVector, frame]
+  have hflat1 : (q - q 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) 1 = 0 := by
+    simp [normalVector, frame]
+  intro p hp
+  have hpK : p ∈ (K.val : Set Point) := hp.1
+  have hp1 : 0 ≤ p 1 := by
+    simpa only [inner_normalVector_pi_div_two] using K.inner_normalVector_pi_div_two_nonneg hpK
+  have hpsup : inner ℝ p (normalVector (t : Real.Angle)) =
+    supportValue K.val (t : Real.Angle) := hp.2
+  have hple : inner ℝ (q - q 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+      (normalVector (t : Real.Angle)) ≤ inner ℝ p (normalVector (t : Real.Angle)) := by
+    rw [hpsup]
+    exact inner_le_supportValue K.val (K.base_projection_mem hq) _
+  rw [inner_normalVector_real, inner_normalVector_real, hflat0, hflat1] at hple
+  have h1 : p 0 * Real.cos t ≤ q 0 * Real.cos t := isMaxOn_iff.mp hmax p hpK
+  have h2 : p 1 * Real.sin t ≤ 0 := mul_nonpos_of_nonneg_of_nonpos hp1 hsin.le
+  have he1 : p 0 * Real.cos t = q 0 * Real.cos t := by linarith
+  have he2 : p 1 = 0 := by
+    have h3 : p 1 * Real.sin t = 0 := by linarith
+    exact (mul_eq_zero.mp h3).resolve_right hsin.ne
+  have g0 : p 0 = (q - q 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) 0 := by
+    rw [hflat0]; exact mul_right_cancel₀ hcos he1
+  have g1 : p 1 = (q - q 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) 1 := by
+    rw [hflat1]; exact he2
+  have hpt : p = q - q 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle) := by
+    ext i
+    fin_cases i
+    · exact g0
+    · exact g1
+  simpa only [Set.mem_singleton_iff] using hpt
+
+/-- The two base corners of a right-angle cap: on the open lower left quarter of normals every
+face degenerates to the base point below a leftmost point of the cap, and on the open lower
+right quarter to the base point below a rightmost one. -/
+theorem CapSpace.exists_exposedEdge_subset_singleton (K : CapSpace (Real.pi / 2)) :
+    ∃ pl pr : Point,
+      (∀ t : ℝ, Real.cos t < 0 → Real.sin t < 0 →
+        exposedEdge K.val (t : Real.Angle) ⊆ {pl}) ∧
+      ∀ t : ℝ, 0 < Real.cos t → Real.sin t < 0 →
+        exposedEdge K.val (t : Real.Angle) ⊆ {pr} := by
+  have hcont : Continuous fun q : Point ↦ q 0 := PiLp.continuous_apply 2 _ 0
+  obtain ⟨ql, hql, hqlmin⟩ :=
+    K.val.isCompact.exists_isMinOn K.val.nonempty hcont.continuousOn
+  obtain ⟨qr, hqr, hqrmax⟩ :=
+    K.val.isCompact.exists_isMaxOn K.val.nonempty hcont.continuousOn
+  refine ⟨ql - ql 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle),
+    qr - qr 1 • normalVector ((Real.pi / 2 : ℝ) : Real.Angle), ?_, ?_⟩
+  · exact fun t hcos hsin ↦ K.exposedEdge_subset_singleton_of_isMaxOn hql hcos.ne hsin
+      (isMaxOn_iff.mpr fun p hp ↦
+        mul_le_mul_of_nonpos_right (isMinOn_iff.mp hqlmin p hp) hcos.le)
+  · exact fun t hcos hsin ↦ K.exposedEdge_subset_singleton_of_isMaxOn hqr hcos.ne' hsin
+      (isMaxOn_iff.mpr fun p hp ↦
+        mul_le_mul_of_nonneg_right (isMaxOn_iff.mp hqrmax p hp) hcos.le)
+
+/-- The surface area measure of a right-angle cap vanishes on both open quarter arcs of lower
+normals, because there every face degenerates to a single base corner. -/
+theorem CapSpace.surfaceAreaMeasure_image_Ioo_lower_eq_zero (K : CapSpace (Real.pi / 2)) :
+    surfaceAreaMeasure K.val
+        ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo Real.pi (3 * Real.pi / 2)) = 0 ∧
+      surfaceAreaMeasure K.val
+        ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo (3 * Real.pi / 2) (2 * Real.pi)) = 0 := by
+  have hpi := Real.pi_pos
+  obtain ⟨pl, pr, hl, hr⟩ := K.exists_exposedEdge_subset_singleton
+  have hsin : ∀ s : ℝ, Real.pi < s → s < 2 * Real.pi → Real.sin s < 0 := by
+    intro s h1 h2
+    have h := Real.sin_pos_of_pos_of_lt_pi (x := s - Real.pi) (by linarith) (by linarith)
+    rw [Real.sin_sub_pi] at h
+    linarith
+  have main : ∀ (a b : ℝ) (p : Point), a ≤ b → b < a + Real.pi →
+      (∀ s ∈ Set.Ioo a b, exposedEdge K.val (s : Real.Angle) ⊆ {p}) →
+      surfaceAreaMeasure K.val ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b) = 0 := by
+    intro a b p hab hba hsub
+    refine surfaceAreaMeasure_null_of_exposedEdge_subset_singleton K.val
+      (Real.Angle.isOpen_image_Ioo a b).measurableSet hab hba
+      (Set.image_mono Set.Ioo_subset_Icc_self) p ?_
+    intro u hu
+    obtain ⟨s, hs, rfl⟩ := hu
+    exact hsub s hs
+  refine ⟨main Real.pi (3 * Real.pi / 2) pl (by linarith) (by linarith) fun s hs ↦ ?_,
+    main (3 * Real.pi / 2) (2 * Real.pi) pr (by linarith) (by linarith) fun s hs ↦ ?_⟩
+  · exact hl s (Real.cos_neg_of_pi_div_two_lt_of_lt (by linarith [hs.1]) (by linarith [hs.2]))
+      (hsin s hs.1 (by linarith [hs.2]))
+  · refine hr s ?_ (hsin s (by linarith [hs.1]) hs.2)
+    have h := Real.cos_pos_of_mem_Ioo
+      (x := s - 2 * Real.pi) ⟨by linarith [hs.1], by linarith [hs.2]⟩
+    rwa [Real.cos_sub_two_pi] at h
+
+/-- Outside the closed upper semicircle the surface area measure of a right-angle cap is carried
+by the single downward normal `3π / 2`, so any integrand vanishing there integrates to zero. -/
+theorem CapSpace.setIntegral_compl_image_Icc_zero_pi_eq_zero (K : CapSpace (Real.pi / 2))
+    (f : Real.Angle → ℝ) (hf : f ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0) :
+    ∫ t in ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Icc 0 Real.pi)ᶜ,
+      f t ∂surfaceAreaMeasure K.val = 0 := by
+  obtain ⟨hA, hB⟩ := K.surfaceAreaMeasure_image_Ioo_lower_eq_zero
+  have hS : MeasurableSet ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Icc 0 Real.pi) :=
+    (isCompact_Icc.image Real.Angle.continuous_coe).isClosed.measurableSet
+  have haenot : ∀ᵐ t ∂surfaceAreaMeasure K.val,
+      t ∉ ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo Real.pi (3 * Real.pi / 2)) ∪
+        ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo (3 * Real.pi / 2) (2 * Real.pi)) := by
+    simpa only [ae_iff, not_not, Set.ofPred_mem_eq] using measure_union_null hA hB
+  have hcongr : ∫ t in ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Icc 0 Real.pi)ᶜ,
+      f t ∂surfaceAreaMeasure K.val =
+      ∫ _t in ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Icc 0 Real.pi)ᶜ,
+        (0 : ℝ) ∂surfaceAreaMeasure K.val := by
+    refine setIntegral_congr_ae hS.compl ?_
+    filter_upwards [haenot] with t ht hts
+    have hcoe : ((t.toReal + 2 * Real.pi : ℝ) : Real.Angle) = t := by
+      rw [Real.Angle.coe_add, Real.Angle.coe_two_pi, add_zero, Real.Angle.coe_toReal]
+    have hneg : t.toReal < 0 := by
+      by_contra hge
+      exact hts ⟨t.toReal, ⟨not_lt.mp hge, Real.Angle.toReal_le_pi t⟩, t.coe_toReal⟩
+    have hlow : Real.pi < t.toReal + 2 * Real.pi :=
+      by linarith [Real.Angle.neg_pi_lt_toReal t]
+    have hmid : t = ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+      rcases lt_trichotomy (t.toReal + 2 * Real.pi) (3 * Real.pi / 2) with h | h | h
+      · exact absurd (Or.inl ⟨t.toReal + 2 * Real.pi, ⟨hlow, h⟩, hcoe⟩) ht
+      · rw [← hcoe, h]
+      · exact absurd (Or.inr ⟨t.toReal + 2 * Real.pi, ⟨h, by linarith⟩, hcoe⟩) ht
+    rw [hmid, hf]
+  rw [hcongr, integral_zero]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Reflection Geometry
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Image of a convex body under the cap reflection. -/
+def reflectedBody (ω : ℝ) (K : ConvexBody Point) : ConvexBody Point where
+  carrier := capReflection ω '' (K : Set Point)
+  convex' := K.convex.linear_image (capReflection ω).toLinearEquiv.toLinearMap
+  isCompact' := K.isCompact.image (capReflection ω).continuous
+  nonempty' := K.nonempty.image _
+
+/-- Support values of a reflected body are indexed by reflected normal angles. -/
+theorem supportValue_reflectedBody (ω : ℝ) (K : ConvexBody Point)
+    (a : Real.Angle) :
+    supportValue (reflectedBody ω K) a =
+      supportValue K (reflectedAngle ω a) := by
+  unfold supportValue
+  congr 1
+  ext x
+  constructor
+  · rintro ⟨p, ⟨q, hq, rfl⟩, rfl⟩
+    exact ⟨q, hq, (inner_capReflection_normalVector ω q a).symm⟩
+  · rintro ⟨q, hq, rfl⟩
+    exact ⟨capReflection ω q, ⟨q, hq, rfl⟩,
+      inner_capReflection_normalVector ω q a⟩
+
+/-- A reflected normal remains among the allowed cap normals. -/
+theorem reflectedAngle_mem_capNormals {ω : ℝ} (a : Real.Angle)
+    (ha : a ∈ ((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles ω) ∪
+      capLowerNormals ω) :
+    reflectedAngle ω a ∈
+      ((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles ω) ∪
+        capLowerNormals ω := by
+  rcases ha with ha | ha
+  · obtain ⟨t, ht, rfl⟩ := ha
+    left
+    refine ⟨ω + Real.pi / 2 - t, ?_, (reflectedAngle_coe ω t).symm⟩
+    rcases ht with ht | ht
+    · right
+      exact ⟨by linarith [ht.2], by linarith [ht.1]⟩
+    · left
+      exact ⟨by linarith [ht.2], by linarith [ht.1]⟩
+  · right
+    simp only [capLowerNormals, Set.mem_insert_iff, Set.mem_singleton_iff] at ha ⊢
+    rcases ha with rfl | rfl
+    · right
+      rw [reflectedAngle_coe]
+      have hperiod :
+          (((-Real.pi / 2 : ℝ) : Real.Angle)) =
+            (((3 * Real.pi / 2 : ℝ) : Real.Angle)) := by
+        calc
+          (((-Real.pi / 2 : ℝ) : Real.Angle)) =
+              (((-Real.pi / 2 + 2 * Real.pi : ℝ) : Real.Angle)) := by
+                rw [Real.Angle.coe_add, Real.Angle.coe_two_pi, add_zero]
+          _ = (((3 * Real.pi / 2 : ℝ) : Real.Angle)) := by congr 1; ring
+      convert hperiod using 1
+      all_goals ring_nf
+    · left
+      rw [reflectedAngle_coe]
+      have hperiod :
+          (((ω - Real.pi : ℝ) : Real.Angle)) =
+            (((ω + Real.pi : ℝ) : Real.Angle)) := by
+        calc
+          (((ω - Real.pi : ℝ) : Real.Angle)) =
+              (((ω - Real.pi + 2 * Real.pi : ℝ) : Real.Angle)) := by
+                rw [Real.Angle.coe_add, Real.Angle.coe_two_pi, add_zero]
+          _ = (((ω + Real.pi : ℝ) : Real.Angle)) := by congr 1; ring
+      convert hperiod using 1
+      all_goals ring_nf
+
+/-- Reflect a half-plane presentation when its allowed normals are transported. -/
+theorem HasHalfPlaneRepresentation.reflectedBody {ω : ℝ}
+    {K : ConvexBody Point} {N N' : Set Real.Angle}
+    (hK : HasHalfPlaneRepresentation K N)
+    (hN : ∀ a ∈ N, reflectedAngle ω a ∈ N') :
+    HasHalfPlaneRepresentation (reflectedBody ω K) N' := by
+  obtain ⟨C, hCN, hKC⟩ := hK
+  let ρ : Real.Angle × ℝ → Real.Angle × ℝ :=
+    fun c ↦ (reflectedAngle ω c.1, c.2)
+  refine ⟨ρ '' C, ?_, ?_⟩
+  · rintro _ ⟨c, hc, rfl⟩
+    exact hN c.1 (hCN c hc)
+  · ext p
+    constructor
+    · rintro ⟨q, hq, rfl⟩
+      simp only [Set.mem_iInter]
+      intro c hc
+      obtain ⟨d, hd, rfl⟩ := hc
+      change inner ℝ (capReflection ω q)
+        (normalVector (reflectedAngle ω d.1)) ≤ d.2
+      rw [inner_capReflection_normalVector, reflectedAngle_involutive]
+      rw [hKC] at hq
+      exact Set.mem_iInter.mp (Set.mem_iInter.mp hq d) hd
+    · intro hp
+      refine ⟨capReflection ω p, ?_, capReflection_involutive ω p⟩
+      rw [hKC]
+      simp only [Set.mem_iInter]
+      intro c hc
+      have hpc := Set.mem_iInter.mp
+        (Set.mem_iInter.mp hp (ρ c)) ⟨c, hc, rfl⟩
+      change inner ℝ p (normalVector (reflectedAngle ω c.1)) ≤ c.2 at hpc
+      change inner ℝ (capReflection ω p) (normalVector c.1) ≤ c.2
+      rw [inner_capReflection_normalVector]
+      exact hpc
+
+/-- Reflection preserves the standard cap half-plane presentation. -/
+theorem reflectedBody_halfPlaneRepresentation {ω : ℝ}
+    (K : ConvexBody Point)
+    (hK : HasHalfPlaneRepresentation K
+      (((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles ω) ∪
+        capLowerNormals ω)) :
+    HasHalfPlaneRepresentation (reflectedBody ω K)
+      (((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles ω) ∪
+        capLowerNormals ω) :=
+  hK.reflectedBody fun a ha ↦ reflectedAngle_mem_capNormals a ha
+
+/-- Reflection exchanges the normalized upper normal at `ω` with the vertical normal. -/
+theorem reflectedAngle_at_omega (ω : ℝ) :
+    reflectedAngle ω (ω : Real.Angle) =
+      ((Real.pi / 2 : ℝ) : Real.Angle) := by
+  rw [reflectedAngle_coe]
+  congr 1
+  ring
+
+/-- Reflection exchanges the vertical normal with the normalized upper normal at `ω`. -/
+theorem reflectedAngle_at_pi_div_two (ω : ℝ) :
+    reflectedAngle ω ((Real.pi / 2 : ℝ) : Real.Angle) =
+      (ω : Real.Angle) := by
+  rw [reflectedAngle_coe]
+  congr 1
+  ring
+
+/-- Reflection exchanges the two lower cap normals. -/
+theorem reflectedAngle_at_omega_add_pi (ω : ℝ) :
+    reflectedAngle ω ((ω + Real.pi : ℝ) : Real.Angle) =
+      ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+  rw [reflectedAngle_coe]
+  have hperiod :
+      (((-Real.pi / 2 : ℝ) : Real.Angle)) =
+        (((3 * Real.pi / 2 : ℝ) : Real.Angle)) := by
+    calc
+      (((-Real.pi / 2 : ℝ) : Real.Angle)) =
+          (((-Real.pi / 2 + 2 * Real.pi : ℝ) : Real.Angle)) := by
+            rw [Real.Angle.coe_add, Real.Angle.coe_two_pi, add_zero]
+      _ = (((3 * Real.pi / 2 : ℝ) : Real.Angle)) := by congr 1; ring
+  convert hperiod using 1
+  all_goals ring_nf
+
+/-- Reflection exchanges the two lower cap normals. -/
+theorem reflectedAngle_at_three_pi_div_two (ω : ℝ) :
+    reflectedAngle ω ((3 * Real.pi / 2 : ℝ) : Real.Angle) =
+      ((ω + Real.pi : ℝ) : Real.Angle) := by
+  rw [reflectedAngle_coe]
+  have hperiod :
+      (((ω - Real.pi : ℝ) : Real.Angle)) =
+        (((ω + Real.pi : ℝ) : Real.Angle)) := by
+    calc
+      (((ω - Real.pi : ℝ) : Real.Angle)) =
+          (((ω - Real.pi + 2 * Real.pi : ℝ) : Real.Angle)) := by
+            rw [Real.Angle.coe_add, Real.Angle.coe_two_pi, add_zero]
+      _ = (((ω + Real.pi : ℝ) : Real.Angle)) := by congr 1; ring
+  convert hperiod using 1
+  all_goals ring_nf
+
+/-- Reflection preserves the normalized cap conditions. -/
+theorem reflectedBody_isCap {ω : ℝ} (K : CapSpace ω) :
+    IsCap ω (reflectedBody ω K.val) := by
+  rcases K.property with ⟨hω0, hωle, hω, hpi, hlowω, hlowpi, hrepr⟩
+  refine ⟨hω0, hωle, ?_, ?_, ?_, ?_,
+    reflectedBody_halfPlaneRepresentation K.val hrepr⟩
+  · rw [supportValue_reflectedBody, reflectedAngle_at_omega, hpi]
+  · rw [supportValue_reflectedBody, reflectedAngle_at_pi_div_two, hω]
+  · rw [supportValue_reflectedBody, reflectedAngle_at_omega_add_pi, hlowpi]
+  · rw [supportValue_reflectedBody, reflectedAngle_at_three_pi_div_two, hlowω]
+
+/-- The cap reflection preserves the lower fan. -/
+theorem capReflection_image_capFan (ω : ℝ) :
+    capReflection ω '' capFan ω = capFan ω := by
+  unfold capFan
+  rw [Set.image_inter (capReflection ω).injective,
+    capReflection_image_normalHalfPlane,
+    capReflection_image_normalHalfPlane,
+    reflectedAngle_at_omega, reflectedAngle_at_pi_div_two,
+    Set.inter_comm]
+
+/-- Reflection sends the complementary normal to its paired normal. -/
+theorem reflectedAngle_sub (ω t : ℝ) :
+    reflectedAngle ω ((ω - t : ℝ) : Real.Angle) =
+      ((t + Real.pi / 2 : ℝ) : Real.Angle) := by
+  rw [reflectedAngle_coe]
+  congr 1
+  ring
+
+/-- Reflection sends the complementary paired normal back to the original normal. -/
+theorem reflectedAngle_sub_add_pi_div_two (ω t : ℝ) :
+    reflectedAngle ω ((ω - t + Real.pi / 2 : ℝ) : Real.Angle) =
+      (t : Real.Angle) := by
+  rw [reflectedAngle_coe]
+  congr 1
+  ring
+
+/-- Reflection sends a paired normal to the complementary normal. -/
+theorem reflectedAngle_add_pi_div_two (ω t : ℝ) :
+    reflectedAngle ω ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+      ((ω - t : ℝ) : Real.Angle) := by
+  rw [reflectedAngle_coe]
+  congr 1
+  ring
+
+/-- Reflection transports inward quadrants at complementary angles. -/
+theorem innerQuadrant_reflection (ω t : ℝ) (K : ConvexBody Point) :
+    innerQuadrant (reflectedBody ω K) t =
+      capReflection ω '' innerQuadrant K (ω - t) := by
+  unfold innerQuadrant
+  rw [Set.image_inter (capReflection ω).injective,
+    capReflection_image_normalHalfPlane,
+    capReflection_image_normalHalfPlane,
+    supportValue_reflectedBody, supportValue_reflectedBody,
+    reflectedAngle_sub, reflectedAngle_sub_add_pi_div_two,
+    reflectedAngle_add_pi_div_two, reflectedAngle_coe, Set.inter_comm]
+  congr 2
+  congr 1
+  ring_nf
+
+/-- The preimage and image of a set agree under the involutive cap reflection. -/
+theorem capReflection_preimage_eq_image (ω : ℝ) (S : Set Point) :
+    capReflection ω ⁻¹' S = capReflection ω '' S := by
+  ext p
+  constructor
+  · intro hp
+    exact ⟨capReflection ω p, hp, capReflection_involutive ω p⟩
+  · rintro ⟨q, hq, rfl⟩
+    simpa only [Set.mem_preimage, capReflection_involutive] using hq
+
+/-- The cap reflection preserves the real Lebesgue area of measurable sets. -/
+theorem area_image_capReflection (ω : ℝ) (S : Set Point)
+    (hS : MeasurableSet S) :
+    ClassicalResults.area (capReflection ω '' S) = ClassicalResults.area S := by
+  change (MeasureTheory.volume (capReflection ω '' S)).toReal =
+    (MeasureTheory.volume S).toReal
+  rw [← capReflection_preimage_eq_image]
+  congr 1
+  exact (LinearIsometryEquiv.measurePreserving (capReflection ω)).measure_preimage
+    hS.nullMeasurableSet
+
+/-- The vertical reflection of a body has mirrored support values. -/
+theorem supportValue_reflectedBody_pi_div_two (K : ConvexBody Point) (u : ℝ) :
+    supportValue (reflectedBody (Real.pi / 2) K : Set Point) (u : Real.Angle) =
+      supportValue (K : Set Point) ((Real.pi - u : ℝ) : Real.Angle) := by
+  have hang : ((Real.pi / 2 + Real.pi / 2 - u : ℝ) : Real.Angle) =
+      ((Real.pi - u : ℝ) : Real.Angle) := by congr 1; ring
+  rw [supportValue_reflectedBody, reflectedAngle_coe, hang]
+
+/-- The vertical reflection mirrors normal projections. -/
+theorem inner_capReflection_pi_div_two (p : Point) (u : ℝ) :
+    inner ℝ (capReflection (Real.pi / 2) p) (normalVector (u : Real.Angle)) =
+      inner ℝ p (normalVector ((Real.pi - u : ℝ) : Real.Angle)) := by
+  have hang : ((Real.pi / 2 + Real.pi / 2 - u : ℝ) : Real.Angle) =
+      ((Real.pi - u : ℝ) : Real.Angle) := by congr 1; ring
+  rw [inner_capReflection_normalVector, reflectedAngle_coe, hang]
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Support Intersections
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Set MeasureTheory
+
+namespace MovingSofa
+
+private theorem supportingIntersection_le_supportValue_of_mem_Icc_left
+    (K : ConvexBody Point) {a b r : ℝ} (hab : 0 < b - a) (hpi : b - a < Real.pi)
+    (hr : r ∈ Set.Icc (a - Real.pi) a) :
+    inner ℝ (supportingIntersection K (a : Real.Angle) (b : Real.Angle))
+        (normalVector (r : Real.Angle)) ≤ supportValue K (r : Real.Angle) := by
+  let p := supportingIntersection K (a : Real.Angle) (b : Real.Angle)
+  let q := (edgeVertices K (a : Real.Angle)).1
+  have hsin : 0 < Real.sin (b - a) :=
+    Real.sin_pos_of_pos_of_lt_pi hab hpi
+  have hpa : inner ℝ p (normalVector (a : Real.Angle)) =
+      supportValue K (a : Real.Angle) := supportingIntersection_inner_left K a b
+  have hpb : inner ℝ p (normalVector (b : Real.Angle)) =
+      supportValue K (b : Real.Angle) :=
+    supportingIntersection_inner_right K a b hsin.ne'
+  have hqmem := edgeVertices_fst_mem K (a : Real.Angle)
+  have hqa : inner ℝ q (normalVector (a : Real.Angle)) =
+      supportValue K (a : Real.Angle) := hqmem.2
+  have hqb : inner ℝ q (normalVector (b : Real.Angle)) ≤
+      supportValue K (b : Real.Angle) := inner_le_supportValue K hqmem.1 _
+  have hnormal : inner ℝ (p - q) (normalVector (a : Real.Angle)) = 0 := by
+    rw [inner_sub_left, hpa, hqa, sub_self]
+  have htangent : 0 ≤ inner ℝ (p - q) (tangentVector (a : Real.Angle)) := by
+    have hdiff : 0 ≤ inner ℝ (p - q) (normalVector (b : Real.Angle)) := by
+      rw [inner_sub_left, hpb]
+      linarith
+    have hnb : normalVector (b : Real.Angle) =
+        Real.cos (b - a) • normalVector (a : Real.Angle) +
+          Real.sin (b - a) • tangentVector (a : Real.Angle) := by
+      simpa only [add_sub_cancel] using normalVector_add_real a (b - a)
+    rw [hnb, inner_add_right, inner_smul_right, inner_smul_right, hnormal,
+      mul_zero, zero_add] at hdiff
+    nlinarith
+  have hqr : inner ℝ q (normalVector (r : Real.Angle)) ≤
+      supportValue K (r : Real.Angle) := inner_le_supportValue K hqmem.1 _
+  have hnr : normalVector (r : Real.Angle) =
+      Real.cos (r - a) • normalVector (a : Real.Angle) +
+        Real.sin (r - a) • tangentVector (a : Real.Angle) := by
+    simpa only [add_sub_cancel] using normalVector_add_real a (r - a)
+  have hsinr : Real.sin (r - a) ≤ 0 :=
+    Real.sin_nonpos_of_nonpos_of_neg_pi_le (by linarith [hr.2]) (by linarith [hr.1])
+  have hdiff : inner ℝ (p - q) (normalVector (r : Real.Angle)) ≤ 0 := by
+    rw [hnr, inner_add_right, inner_smul_right, inner_smul_right, hnormal,
+      mul_zero, zero_add]
+    exact mul_nonpos_of_nonpos_of_nonneg hsinr htangent
+  rw [inner_sub_left] at hdiff
+  linarith
+
+private theorem supportingIntersection_le_supportValue_of_mem_Icc_right
+    (K : ConvexBody Point) {a b r : ℝ} (hab : 0 < b - a) (hpi : b - a < Real.pi)
+    (hr : r ∈ Set.Icc b (b + Real.pi)) :
+    inner ℝ (supportingIntersection K (a : Real.Angle) (b : Real.Angle))
+        (normalVector (r : Real.Angle)) ≤ supportValue K (r : Real.Angle) := by
+  let p := supportingIntersection K (a : Real.Angle) (b : Real.Angle)
+  let q := (edgeVertices K (b : Real.Angle)).2
+  have hsin : 0 < Real.sin (b - a) :=
+    Real.sin_pos_of_pos_of_lt_pi hab hpi
+  have hsina : Real.sin (a - b) < 0 := by
+    rw [show a - b = -(b - a) by ring, Real.sin_neg]
+    exact neg_neg_of_pos hsin
+  have hpa : inner ℝ p (normalVector (a : Real.Angle)) =
+      supportValue K (a : Real.Angle) := supportingIntersection_inner_left K a b
+  have hpb : inner ℝ p (normalVector (b : Real.Angle)) =
+      supportValue K (b : Real.Angle) :=
+    supportingIntersection_inner_right K a b hsin.ne'
+  have hqmem := edgeVertices_snd_mem K (b : Real.Angle)
+  have hqb : inner ℝ q (normalVector (b : Real.Angle)) =
+      supportValue K (b : Real.Angle) := hqmem.2
+  have hqa : inner ℝ q (normalVector (a : Real.Angle)) ≤
+      supportValue K (a : Real.Angle) := inner_le_supportValue K hqmem.1 _
+  have hnormal : inner ℝ (p - q) (normalVector (b : Real.Angle)) = 0 := by
+    rw [inner_sub_left, hpb, hqb, sub_self]
+  have htangent : inner ℝ (p - q) (tangentVector (b : Real.Angle)) ≤ 0 := by
+    have hdiff : 0 ≤ inner ℝ (p - q) (normalVector (a : Real.Angle)) := by
+      rw [inner_sub_left, hpa]
+      linarith
+    have hna : normalVector (a : Real.Angle) =
+        Real.cos (a - b) • normalVector (b : Real.Angle) +
+          Real.sin (a - b) • tangentVector (b : Real.Angle) := by
+      simpa only [add_sub_cancel] using normalVector_add_real b (a - b)
+    rw [hna, inner_add_right, inner_smul_right, inner_smul_right, hnormal,
+      mul_zero, zero_add] at hdiff
+    nlinarith
+  have hqr : inner ℝ q (normalVector (r : Real.Angle)) ≤
+      supportValue K (r : Real.Angle) := inner_le_supportValue K hqmem.1 _
+  have hnr : normalVector (r : Real.Angle) =
+      Real.cos (r - b) • normalVector (b : Real.Angle) +
+        Real.sin (r - b) • tangentVector (b : Real.Angle) := by
+    simpa only [add_sub_cancel] using normalVector_add_real b (r - b)
+  have hsinr : 0 ≤ Real.sin (r - b) :=
+    Real.sin_nonneg_of_nonneg_of_le_pi (by linarith [hr.1]) (by linarith [hr.2])
+  have hdiff : inner ℝ (p - q) (normalVector (r : Real.Angle)) ≤ 0 := by
+    rw [hnr, inner_add_right, inner_smul_right, inner_smul_right, hnormal,
+      mul_zero, zero_add]
+    exact mul_nonpos_of_nonneg_of_nonpos hsinr htangent
+  rw [inner_sub_left] at hdiff
+  linarith
+
+/-- Adjacent allowed support normals meet in the represented convex body. -/
+theorem HasHalfPlaneRepresentation.supportingIntersection_mem_of_gap
+    (K : ConvexBody Point) (R : Set ℝ) {a b : ℝ}
+    (hab : 0 < b - a) (hpi : b - a < Real.pi)
+    (hK : HasHalfPlaneRepresentation K ((fun r : ℝ ↦ (r : Real.Angle)) '' R))
+    (hR : ∀ r ∈ R, r ∈ Set.Icc (a - Real.pi) a ∨ r ∈ Set.Icc b (b + Real.pi)) :
+    supportingIntersection K (a : Real.Angle) (b : Real.Angle) ∈ K := by
+  change supportingIntersection K (a : Real.Angle) (b : Real.Angle) ∈ (K : Set Point)
+  rw [hK.eq_iInter_supportValue]
+  simp only [Set.mem_iInter]
+  intro u hu
+  obtain ⟨r, hr, rfl⟩ := hu
+  rcases hR r hr with hr | hr
+  · exact supportingIntersection_le_supportValue_of_mem_Icc_left K hab hpi hr
+  · exact supportingIntersection_le_supportValue_of_mem_Icc_right K hab hpi hr
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Top Corner
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Every polygon cap of angle less than pi/2 contains the top parallelogram corner. -/
+theorem stripParallelogram_top_mem_of_angle_lt (Θ : AngleSet)
+    (hΘ : Θ.angle < Real.pi / 2) (K : PolygonCapSpace Θ) :
+    (stripParallelogram Θ.angle).2.2 ∈ (K.val.val : Set Point) := by
+  let o := (stripParallelogram Θ.angle).2.2
+  have hc : 0 < Real.cos Θ.angle := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Θ.angle_pos, Real.pi_pos], hΘ⟩
+  obtain ⟨pω, hpω, hpωeq⟩ := (K.val.val.isCompact.image
+    (continuous_inner.comp (continuous_id.prodMk continuous_const))).sSup_mem
+      (K.val.val.nonempty.image (fun p ↦ inner ℝ p (normalVector (Θ.angle : Real.Angle))))
+  obtain ⟨pT, hpT, hpTeq⟩ := (K.val.val.isCompact.image
+    (continuous_inner.comp (continuous_id.prodMk continuous_const))).sSup_mem
+      (K.val.val.nonempty.image
+        (fun p ↦ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))))
+  have hpω' : inner ℝ pω (normalVector (Θ.angle : Real.Angle)) = 1 := by
+    change inner ℝ pω (normalVector (Θ.angle : Real.Angle)) =
+      supportValue K.val.val (Θ.angle : Real.Angle) at hpωeq
+    simpa only [K.val.property.2.2.1] using hpωeq
+  have hpT' : inner ℝ pT (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 1 := by
+    change inner ℝ pT (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) =
+      supportValue K.val.val ((Real.pi / 2 : ℝ) : Real.Angle) at hpTeq
+    simpa only [K.val.property.2.2.2.1] using hpTeq
+  have hpωT : inner ℝ pω (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) ≤ 1 := by
+    exact (inner_le_supportValue K.val.val hpω _).trans_eq K.val.property.2.2.2.1
+  have hpTω : inner ℝ pT (normalVector (Θ.angle : Real.Angle)) ≤ 1 := by
+    exact (inner_le_supportValue K.val.val hpT _).trans_eq K.val.property.2.2.1
+  have hoω : inner ℝ o (normalVector (Θ.angle : Real.Angle)) = 1 := by
+    have hgap : Real.tan (Real.pi / 4 - Θ.angle / 2) =
+        (Real.cos Θ.angle)⁻¹ - Real.tan Θ.angle := by
+      simpa [show Real.pi / 4 - Θ.angle / 2 =
+          (Real.pi / 2 - Θ.angle) / 2 by ring]
+        using Real.tan_pi_div_two_sub_div_two Θ.angle ⟨Θ.angle_pos.le, hΘ⟩
+    have htan := Real.tan_eq_sin_div_cos Θ.angle
+    simp [o, stripParallelogram, normalVector, frame, PiLp.inner_apply, hgap, htan]
+    field_simp [hc.ne']
+    ring
+  have hoT : inner ℝ o (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 1 := by
+    simp [o, stripParallelogram, normalVector, frame, PiLp.inner_apply]
+  rw [K.property.eq_iInter_supportValue]
+  simp only [Set.mem_iInter]
+  intro a ha
+  change inner ℝ o (normalVector a) ≤ supportValue K.val.val a
+  rcases ha with ⟨t, ht, rfl⟩ | ha
+  · have htupper : t ∈ capUpperAngles Θ.angle := by
+      rcases ht with (ht | ⟨s, hs, rfl⟩) | ht
+      · exact Or.inl ⟨(Θ.interior t ht).1.le, (Θ.interior t ht).2.le⟩
+      · exact Or.inr ⟨by dsimp; linarith [(Θ.interior s hs).1],
+          by dsimp; linarith [(Θ.interior s hs).2]⟩
+      · rcases ht with rfl | rfl
+        · exact Or.inl ⟨Θ.angle_pos.le, le_rfl⟩
+        · exact Or.inr ⟨le_rfl, by linarith [Θ.angle_pos]⟩
+    rcases htupper with ht | ht
+    · let c := Real.cos t / Real.cos Θ.angle
+      let s := Real.sin (Θ.angle - t) / Real.cos Θ.angle
+      have hc0 : 0 ≤ c := div_nonneg
+        (Real.cos_nonneg_of_mem_Icc
+          ⟨(neg_nonpos.mpr (by positivity)).trans ht.1, ht.2.trans Θ.angle_le⟩) hc.le
+      have hs0 : 0 ≤ s := div_nonneg
+        (Real.sin_nonneg_of_nonneg_of_le_pi (sub_nonneg.mpr ht.2)
+          (by linarith [ht.1, Θ.angle_le, Real.pi_pos])) hc.le
+      have hdecomp : normalVector (t : Real.Angle) =
+          c • normalVector (Θ.angle : Real.Angle) -
+            s • normalVector ((Real.pi / 2 : ℝ) : Real.Angle) := by
+        ext i
+        fin_cases i
+        · simp [c, s, normalVector, frame, Real.sin_sub]
+          field_simp [hc.ne']
+        · simp [c, s, normalVector, frame, Real.sin_sub]
+          field_simp [hc.ne']
+          ring
+      rw [hdecomp, inner_sub_right, inner_smul_right, inner_smul_right, hoω, hoT]
+      apply (inner_le_supportValue K.val.val hpω (t : Real.Angle)).trans'
+      rw [hdecomp, inner_sub_right, inner_smul_right, inner_smul_right, hpω']
+      simpa only [mul_one] using
+        sub_le_sub_left (mul_le_mul_of_nonneg_left hpωT hs0) c
+    · let c := Real.sin (t - Θ.angle) / Real.cos Θ.angle
+      let s := -Real.cos t / Real.cos Θ.angle
+      have hc0 : 0 ≤ c := div_nonneg
+        (Real.sin_nonneg_of_nonneg_of_le_pi (by linarith [ht.1, Θ.angle_le])
+          (by linarith [ht.2, Θ.angle_pos, Real.pi_pos])) hc.le
+      have hs0 : 0 ≤ s := div_nonneg
+        (neg_nonneg.mpr (Real.cos_nonpos_of_pi_div_two_le_of_le ht.1
+          (by linarith [ht.2, Θ.angle_le, Real.pi_pos]))) hc.le
+      have hdecomp : normalVector (t : Real.Angle) =
+          c • normalVector ((Real.pi / 2 : ℝ) : Real.Angle) -
+            s • normalVector (Θ.angle : Real.Angle) := by
+        ext i
+        fin_cases i
+        · simp [c, s, normalVector, frame, Real.sin_sub]
+          field_simp [hc.ne']
+        · simp [c, s, normalVector, frame, Real.sin_sub]
+          field_simp [hc.ne']
+          ring
+      rw [hdecomp, inner_sub_right, inner_smul_right, inner_smul_right, hoT, hoω]
+      apply (inner_le_supportValue K.val.val hpT (t : Real.Angle)).trans'
+      rw [hdecomp, inner_sub_right, inner_smul_right, inner_smul_right, hpT']
+      simpa only [mul_one] using
+        sub_le_sub_left (mul_le_mul_of_nonneg_left hpTω hs0) c
+  · rcases ha with rfl | rfl
+    · rw [normalVector_add_pi, inner_neg_right, hoω, K.val.property.2.2.2.2.1]
+      norm_num
+    · rw [inner_normalVector_three_pi_div_two, K.val.property.2.2.2.2.2.1]
+      simp [o, stripParallelogram]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Vertical
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A cap is closed under downward vertical movement that remains inside its fan. -/
+theorem CapSpace.mem_of_fst_eq_of_snd_le {ω : ℝ} (K : CapSpace ω)
+    {p q : Point} (hp : p ∈ (K.val : Set Point)) (hq : q ∈ capFan ω)
+    (hx : q 0 = p 0) (hy : q 1 ≤ p 1) : q ∈ (K.val : Set Point) := by
+  apply K.mem_of_mem_capFan_of_le_supportValue hq
+  intro t ht
+  apply le_trans _ (inner_le_supportValue K.val hp (t : Real.Angle))
+  have htI : t ∈ Set.Icc 0 Real.pi := by
+    rcases ht with ht | ht
+    · exact ⟨ht.1, by linarith [ht.2, K.property.2.1, Real.pi_pos]⟩
+    · exact ⟨by linarith [ht.1, Real.pi_pos], by linarith [ht.2, K.property.2.1]⟩
+  have h := mul_le_mul_of_nonneg_left hy (Real.sin_nonneg_of_mem_Icc htI)
+  simp [normalVector, frame, PiLp.inner_apply, hx]
+  linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Analysis.SurfaceMeasure.ArcConvergence`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Arc Convergence
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter MeasureTheory Set
+open scoped Topology BoundedContinuousFunction
+
+namespace MovingSofa
+
+/-- Hausdorff convergence with fixed endpoint atoms preserves half-open surface integrals. -/
+theorem tendsto_integral_surfaceAreaMeasure_Ioc_of_fixed_atoms
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point) (L : Set Point))
+      atTop (𝓝 0)) {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi)
+    (ha : ∀ n, surfaceAreaMeasure (K n) {(a : Real.Angle)} =
+      surfaceAreaMeasure L {(a : Real.Angle)})
+    (hb : ∀ n, surfaceAreaMeasure (K n) {(b : Real.Angle)} =
+      surfaceAreaMeasure L {(b : Real.Angle)})
+    (f : Real.Angle → ℝ) (hf : Continuous f) :
+    Tendsto (fun n ↦ ∫ u in (fun t : ℝ ↦ (t : Real.Angle)) '' Ioc a b,
+      f u ∂surfaceAreaMeasure (K n)) atTop
+      (𝓝 (∫ u in (fun t : ℝ ↦ (t : Real.Angle)) '' Ioc a b, f u ∂surfaceAreaMeasure L)) := by
+  let μs : ℕ → FiniteMeasure Real.Angle := fun n ↦
+    ⟨surfaceAreaMeasure (K n), (surfaceAreaMeasure_face_union (K n)).1⟩
+  let μ : FiniteMeasure Real.Angle :=
+    ⟨surfaceAreaMeasure L, (surfaceAreaMeasure_face_union L).1⟩
+  have hμ : Tendsto μs atTop (𝓝 μ) := by
+    apply FiniteMeasure.tendsto_iff_forall_integral_tendsto.mpr
+    intro g
+    exact surfaceAreaMeasure_weak_continuity K L hlim g g.continuous
+  exact Real.Angle.tendsto_integral_image_Ioc_of_fixed_endpoint_atoms
+    hab hturn hμ ha hb (BoundedContinuousFunction.mkOfCompact ⟨f, hf⟩)
+
+/-- Preserving the two endpoint faces supplies the atomic hypotheses for arc convergence. -/
+theorem tendsto_integral_surfaceAreaMeasure_Ioc_of_preserves_faces
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point) (L : Set Point))
+      atTop (𝓝 0)) {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi)
+    (ha : ∀ n, exposedEdge (K n) (a : Real.Angle) = exposedEdge L (a : Real.Angle))
+    (hb : ∀ n, exposedEdge (K n) (b : Real.Angle) = exposedEdge L (b : Real.Angle))
+    (f : Real.Angle → ℝ) (hf : Continuous f) :
+    Tendsto (fun n ↦ ∫ u in (fun t : ℝ ↦ (t : Real.Angle)) '' Ioc a b,
+      f u ∂surfaceAreaMeasure (K n)) atTop
+      (𝓝 (∫ u in (fun t : ℝ ↦ (t : Real.Angle)) '' Ioc a b, f u ∂surfaceAreaMeasure L)) := by
+  apply tendsto_integral_surfaceAreaMeasure_Ioc_of_fixed_atoms K L hlim hab hturn
+    (f := f) (hf := hf)
+  · intro n
+    rw [(surfaceAreaMeasure_atom_length (K n) _).1,
+      (surfaceAreaMeasure_atom_length L _).1, ha n]
+  · intro n
+    rw [(surfaceAreaMeasure_atom_length (K n) _).1,
+      (surfaceAreaMeasure_atom_length L _).1, hb n]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Convex.CurveCut`.
+* `Convex.ArcCutArea`.
+* `Convex.OuterCornerPath`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Curve Cut
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private theorem normalVector_eq_positive_combo {a b t : ℝ}
+    (hab : a < b) (hba : b < a + Real.pi) (_hat : a < t) (_htb : t < b) :
+    normalVector (t : Real.Angle) =
+      (Real.sin (b - t) / Real.sin (b - a)) • normalVector (a : Real.Angle) +
+        (Real.sin (t - a) / Real.sin (b - a)) • normalVector (b : Real.Angle) := by
+  have hs : Real.sin (b - a) ≠ 0 :=
+    (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hab) (by linarith)).ne'
+  have hs' : Real.sin b * Real.cos a - Real.cos b * Real.sin a ≠ 0 := by
+    simpa only [Real.sin_sub] using hs
+  apply PiLp.ext
+  intro i
+  fin_cases i <;>
+    simp [normalVector, frame, Real.sin_sub] <;>
+    field_simp [hs'] <;> ring
+
+private theorem eq_supportingIntersection_of_mem_endpoint_faces
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (p : Point) (hpa : p ∈ exposedEdge K (a : Real.Angle))
+    (hpb : p ∈ exposedEdge K (b : Real.Angle)) :
+    supportingIntersection K a b = p := by
+  let O := supportingIntersection K a b
+  have hs : Real.sin (b - a) ≠ 0 :=
+    (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hab) (by linarith)).ne'
+  have hna : inner ℝ (O - p) (normalVector (a : Real.Angle)) = 0 := by
+    rw [inner_sub_left, supportingIntersection_inner_left, hpa.2, sub_self]
+  have hnb : inner ℝ (O - p) (normalVector (b : Real.Angle)) = 0 := by
+    rw [inner_sub_left, supportingIntersection_inner_right K a b hs, hpb.2, sub_self]
+  have hrot := normalVector_add_real a (b - a)
+  have htan : inner ℝ (O - p) (tangentVector (a : Real.Angle)) = 0 := by
+    have := hnb
+    rw [show normalVector (b : Real.Angle) =
+      Real.cos (b - a) • normalVector (a : Real.Angle) +
+        Real.sin (b - a) • tangentVector (a : Real.Angle) by
+          simpa only [add_sub_cancel] using hrot,
+      inner_add_right, inner_smul_right, inner_smul_right, hna] at this
+    simp only [mul_zero, zero_add] at this
+    exact (mul_eq_zero.mp this).resolve_left hs
+  change O = p
+  rw [show O = p + (O - p) by abel,
+    ← inner_normalVector_smul_add_inner_tangentVector_smul
+      (O - p) (a : Real.Angle), hna, htan, zero_smul, zero_smul, add_zero, add_zero]
+
+private theorem exposedEdge_eq_singleton_of_common_endpoint
+    (K : ConvexBody Point) {a b t : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hat : a < t) (htb : t < b) (p : Point)
+    (hpa : p ∈ exposedEdge K (a : Real.Angle))
+    (hpb : p ∈ exposedEdge K (b : Real.Angle)) :
+    exposedEdge K (t : Real.Angle) = {p} := by
+  have hsa : 0 < Real.sin (b - t) / Real.sin (b - a) := div_pos
+    (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr htb) (by linarith))
+    (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hab) (by linarith))
+  have hsb : 0 < Real.sin (t - a) / Real.sin (b - a) := div_pos
+    (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hat) (by linarith))
+    (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hab) (by linarith))
+  have hn := normalVector_eq_positive_combo hab hba hat htb
+  have hpt : inner ℝ p (normalVector (t : Real.Angle)) = supportValue K t := by
+    rw [hn, inner_add_right, inner_smul_right, inner_smul_right, hpa.2, hpb.2]
+    apply le_antisymm
+    · have hp_le := inner_le_supportValue K hpa.1 (t : Real.Angle)
+      rw [hn, inner_add_right, inner_smul_right, inner_smul_right,
+        hpa.2, hpb.2] at hp_le
+      exact hp_le
+    · apply csSup_le (K.nonempty.image _)
+      rintro _ ⟨z, hzK, rfl⟩
+      change inner ℝ z (normalVector (t : Real.Angle)) ≤ _
+      rw [hn, inner_add_right, inner_smul_right, inner_smul_right]
+      exact add_le_add
+        (mul_le_mul_of_nonneg_left (inner_le_supportValue K hzK _) hsa.le)
+        (mul_le_mul_of_nonneg_left (inner_le_supportValue K hzK _) hsb.le)
+  apply Set.Subset.antisymm
+  · intro z hz
+    have hza := inner_le_supportValue K hz.1 (a : Real.Angle)
+    have hzb := inner_le_supportValue K hz.1 (b : Real.Angle)
+    have hzt := hz.2
+    change inner ℝ z (normalVector (t : Real.Angle)) = supportValue K t at hzt
+    rw [← hpt, hn, inner_add_right, inner_smul_right, inner_smul_right] at hzt
+    rw [inner_add_right, inner_smul_right, inner_smul_right, hpa.2, hpb.2] at hzt
+    have hzea : inner ℝ z (normalVector (a : Real.Angle)) = supportValue K a := by
+      by_contra hne
+      have hlt := lt_of_le_of_ne hza hne
+      nlinarith
+    have hzeb : inner ℝ z (normalVector (b : Real.Angle)) = supportValue K b := by
+      by_contra hne
+      have hlt := lt_of_le_of_ne hzb hne
+      nlinarith
+    have hzpa : inner ℝ (z - p) (normalVector (a : Real.Angle)) = 0 := by
+      rw [inner_sub_left, hzea, hpa.2, sub_self]
+    have hzpb : inner ℝ (z - p) (normalVector (b : Real.Angle)) = 0 := by
+      rw [inner_sub_left, hzeb, hpb.2, sub_self]
+    have hs : Real.sin (b - a) ≠ 0 :=
+      (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hab) (by linarith)).ne'
+    have htan : inner ℝ (z - p) (tangentVector (a : Real.Angle)) = 0 := by
+      have hrot := normalVector_add_real a (b - a)
+      rw [show normalVector (b : Real.Angle) =
+        Real.cos (b - a) • normalVector (a : Real.Angle) +
+          Real.sin (b - a) • tangentVector (a : Real.Angle) by
+            simpa only [add_sub_cancel] using hrot,
+        inner_add_right, inner_smul_right, inner_smul_right, hzpa] at hzpb
+      simp only [mul_zero, zero_add] at hzpb
+      exact (mul_eq_zero.mp hzpb).resolve_left hs
+    have hzp : z = p := by
+      rw [show z = p + (z - p) by abel,
+        ← inner_normalVector_smul_add_inner_tangentVector_smul
+          (z - p) (a : Real.Angle), hzpa, htan,
+        zero_smul, zero_smul, add_zero, add_zero]
+    simp [hzp]
+  · intro z hz
+    simp only [Set.mem_singleton_iff] at hz
+    subst z
+    exact ⟨hpa.1, hpt⟩
+
+private theorem exists_cut_angle
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hne : (edgeVertices K (a : Real.Angle)).1 ≠
+      (edgeVertices K (b : Real.Angle)).2) :
+    ∃ t d : ℝ, a < t ∧ t < b ∧ 0 < d ∧
+      (edgeVertices K (b : Real.Angle)).2 - (edgeVertices K (a : Real.Angle)).1 =
+        d • tangentVector (t : Real.Angle) := by
+  let P := (edgeVertices K (a : Real.Angle)).1
+  let Q := (edgeVertices K (b : Real.Angle)).2
+  let O := supportingIntersection K a b
+  obtain ⟨α, hα, hOa⟩ := supportingIntersection_eq_fst_add_pos_tangent K hab hba hne
+  obtain ⟨β, hβ, hOb⟩ := supportingIntersection_eq_snd_sub_pos_tangent K hab hba hne
+  have hchord : Q - P = α • tangentVector (a : Real.Angle) +
+      β • tangentVector (b : Real.Angle) := by
+    change O = P + α • tangentVector (a : Real.Angle) at hOa
+    change O = Q - β • tangentVector (b : Real.Angle) at hOb
+    have hQO : Q = O + β • tangentVector (b : Real.Angle) := by
+      rw [hOb]
+      module
+    rw [hQO, hOa]
+    module
+  let f : ℝ → ℝ := fun s ↦ inner ℝ (Q - P) (normalVector (s : Real.Angle))
+  have hf : Continuous f := by
+    dsimp only [f]
+    rw [show (fun s : ℝ ↦ inner ℝ (Q - P) (normalVector (s : Real.Angle))) =
+      fun s ↦ (Q - P) 0 * Real.cos s + (Q - P) 1 * Real.sin s by
+        funext s
+        simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+        ring]
+    fun_prop
+  have hsin : 0 < Real.sin (b - a) :=
+    Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hab) (by linarith)
+  have hfa : f a < 0 := by
+    dsimp only [f]
+    rw [hchord, inner_add_left, real_inner_smul_left, real_inner_smul_left]
+    simp only [tangentVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, normalVector,
+      PiLp.inner_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue,
+      Matrix.cons_val_zero, mul_neg, Matrix.cons_val_one, Matrix.cons_val_fin_one]
+    have hid : -(Real.cos a * Real.sin b) + Real.sin a * Real.cos b =
+        -Real.sin (b - a) := by rw [Real.sin_sub]; ring
+    rw [hid]
+    nlinarith
+  have hfb : 0 < f b := by
+    dsimp only [f]
+    rw [hchord, inner_add_left, real_inner_smul_left, real_inner_smul_left]
+    simp only [tangentVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, normalVector,
+      PiLp.inner_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue,
+      Matrix.cons_val_zero, mul_neg, Matrix.cons_val_one, Matrix.cons_val_fin_one]
+    have hid : -(Real.cos b * Real.sin a) + Real.sin b * Real.cos a =
+        Real.sin (b - a) := by rw [Real.sin_sub]; ring
+    rw [hid]
+    nlinarith
+  have hz : (0 : ℝ) ∈ Set.Icc (f a) (f b) := ⟨hfa.le, hfb.le⟩
+  obtain ⟨t, ht, hft⟩ := intermediate_value_Icc hab.le hf.continuousOn hz
+  have hat : a < t := lt_of_le_of_ne ht.1 fun h ↦ by subst t; linarith
+  have htb : t < b := lt_of_le_of_ne ht.2 fun h ↦ by subst t; linarith
+  let d := inner ℝ (Q - P) (tangentVector (t : Real.Angle))
+  have hdecomp : Q - P = d • tangentVector (t : Real.Angle) := by
+    rw [← inner_normalVector_smul_add_inner_tangentVector_smul
+      (Q - P) (t : Real.Angle)]
+    change inner ℝ (Q - P) (normalVector (t : Real.Angle)) = 0 at hft
+    rw [hft, zero_smul, zero_add]
+  have hd : 0 < d := by
+    have hqa : inner ℝ (Q - P) (normalVector (a : Real.Angle)) < 0 := hfa
+    rw [hdecomp, real_inner_smul_left] at hqa
+    have hsinat : 0 < Real.sin (t - a) :=
+      Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hat) (by linarith)
+    have hinner : inner ℝ (tangentVector (t : Real.Angle))
+        (normalVector (a : Real.Angle)) = -Real.sin (t - a) := by
+      rw [real_inner_comm]
+      have h := sin_sub_eq_neg_inner_normalVector_tangentVector
+        (a : Real.Angle) (t : Real.Angle)
+      change Real.sin (t - a) =
+        -inner ℝ (normalVector (a : Real.Angle)) (tangentVector (t : Real.Angle)) at h
+      have h' := h
+      linarith
+    rw [hinner] at hqa
+    nlinarith
+  exact ⟨t, d, hat, htb, hd, hdecomp⟩
+
+private theorem mem_segment_of_mem_body_of_eq_cut_inner
+    (K : ConvexBody Point) {a b t c d : ℝ} (hat : a < t) (htb : t < b)
+    (hba : b < a + Real.pi)
+    (P Q z : Point) (hP : P = (edgeVertices K (a : Real.Angle)).1)
+    (hQ : Q = (edgeVertices K (b : Real.Angle)).2)
+    (hd : 0 < d) (hdir : Q - P = d • tangentVector (t : Real.Angle))
+    (hPc : inner ℝ P (normalVector (t : Real.Angle)) = c)
+    (hzc : inner ℝ z (normalVector (t : Real.Angle)) = c) (hzK : z ∈ K) :
+    z ∈ segment ℝ P Q := by
+  let r := inner ℝ (z - P) (tangentVector (t : Real.Angle))
+  have hnormal : inner ℝ (z - P) (normalVector (t : Real.Angle)) = 0 := by
+    rw [inner_sub_left, hzc, hPc, sub_self]
+  have hzrepr : z = P + r • tangentVector (t : Real.Angle) := by
+    rw [show z = P + (z - P) by abel,
+      ← inner_normalVector_smul_add_inner_tangentVector_smul
+        (z - P) (t : Real.Angle), hnormal, zero_smul, zero_add]
+  have hza := inner_le_supportValue K hzK (a : Real.Angle)
+  have hPa : inner ℝ P (normalVector (a : Real.Angle)) = supportValue K a := by
+    rw [hP]
+    exact (edgeVertices_fst_mem K (a : Real.Angle)).2
+  have hsinA : 0 < Real.sin (t - a) :=
+    Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hat) (by linarith)
+  have hta : inner ℝ (tangentVector (t : Real.Angle))
+      (normalVector (a : Real.Angle)) = -Real.sin (t - a) := by
+    rw [real_inner_comm]
+    have h := sin_sub_eq_neg_inner_normalVector_tangentVector
+      (a : Real.Angle) (t : Real.Angle)
+    change Real.sin (t - a) =
+      -inner ℝ (normalVector (a : Real.Angle)) (tangentVector (t : Real.Angle)) at h
+    linarith
+  have hr0 : 0 ≤ r := by
+    rw [hzrepr, inner_add_left, real_inner_smul_left, hta, hPa] at hza
+    nlinarith
+  have hzb := inner_le_supportValue K hzK (b : Real.Angle)
+  have hQb : inner ℝ Q (normalVector (b : Real.Angle)) = supportValue K b := by
+    rw [hQ]
+    exact (edgeVertices_snd_mem K (b : Real.Angle)).2
+  have hsinB : 0 < Real.sin (b - t) :=
+    Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr htb) (by linarith)
+  have htbinner : inner ℝ (tangentVector (t : Real.Angle))
+      (normalVector (b : Real.Angle)) = Real.sin (b - t) := by
+    have h := sin_sub_eq_neg_inner_normalVector_tangentVector
+      (b : Real.Angle) (t : Real.Angle)
+    change Real.sin (t - b) =
+      -inner ℝ (normalVector (b : Real.Angle)) (tangentVector (t : Real.Angle)) at h
+    rw [real_inner_comm]
+    rw [show t - b = -(b - t) by ring, Real.sin_neg] at h
+    linarith
+  have hQrepr : Q = P + d • tangentVector (t : Real.Angle) := by
+    rw [show Q = P + (Q - P) by abel, hdir]
+  have hrd : r ≤ d := by
+    rw [hzrepr, inner_add_left, real_inner_smul_left, htbinner] at hzb
+    have hQsupport := hQb
+    rw [hQrepr, inner_add_left, real_inner_smul_left, htbinner] at hQsupport
+    rw [← hQsupport] at hzb
+    nlinarith
+  rw [segment_eq_image']
+  refine ⟨r / d, ⟨div_nonneg hr0 hd.le, (div_le_one hd).2 hrd⟩, ?_⟩
+  rw [hdir]
+  dsimp only
+  rw [smul_smul]
+  rw [div_mul_cancel₀ r hd.ne', hzrepr]
+
+private theorem exposedEdge_subset_upper_cut
+    (K : ConvexBody Point) {a b t c s : ℝ} (hat : a < t) (htb : t < b)
+    (hba : b < a + Real.pi) (hs : s ∈ Set.Ioo a b)
+    (P Q : Point) (hP : P = (edgeVertices K (a : Real.Angle)).1)
+    (hQ : Q = (edgeVertices K (b : Real.Angle)).2)
+    (hPc : inner ℝ P (normalVector (t : Real.Angle)) = c)
+    (hQc : inner ℝ Q (normalVector (t : Real.Angle)) = c) :
+    exposedEdge K (s : Real.Angle) ⊆ normalHalfPlane (t : Real.Angle) c true false := by
+  intro z hz
+  change c ≤ inner ℝ z (normalVector (t : Real.Angle))
+  by_cases hst : s = t
+  · subst s
+    have hp_le := inner_le_supportValue K (edgeVertices_fst_mem K (a : Real.Angle)).1
+      (t : Real.Angle)
+    rw [← hP, hPc, ← hz.2] at hp_le
+    exact hp_le
+  rcases lt_or_gt_of_ne hst with hst | hts
+  · have hn := normalVector_eq_positive_combo hat (by linarith) hs.1 hst
+    have hA : 0 < Real.sin (t - s) / Real.sin (t - a) := div_pos
+      (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hst) (by linarith [hba, hs.1]))
+      (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hat) (by linarith))
+    have hB : 0 < Real.sin (s - a) / Real.sin (t - a) := div_pos
+      (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hs.1) (by linarith))
+      (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hat) (by linarith))
+    by_contra hnot
+    have hzt : inner ℝ z (normalVector (t : Real.Angle)) < c := lt_of_not_ge hnot
+    have hza := inner_le_supportValue K hz.1 (a : Real.Angle)
+    have hPa : inner ℝ P (normalVector (a : Real.Angle)) = supportValue K a := by
+      rw [hP]; exact (edgeVertices_fst_mem K _).2
+    have hpK : P ∈ K := by rw [hP]; exact (edgeVertices_fst_mem K _).1
+    have hp_le := inner_le_supportValue K hpK (s : Real.Angle)
+    rw [← hz.2, hn, inner_add_right, inner_smul_right, inner_smul_right,
+      inner_add_right, inner_smul_right, inner_smul_right, hPa, hPc] at hp_le
+    nlinarith
+  · have hn := normalVector_eq_positive_combo htb (by linarith) hts hs.2
+    have hA : 0 < Real.sin (b - s) / Real.sin (b - t) := div_pos
+      (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hs.2) (by linarith))
+      (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr htb) (by linarith))
+    have hB : 0 < Real.sin (s - t) / Real.sin (b - t) := div_pos
+      (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hts) (by linarith [hba, hat, hs.2]))
+      (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr htb) (by linarith))
+    by_contra hnot
+    have hzt : inner ℝ z (normalVector (t : Real.Angle)) < c := lt_of_not_ge hnot
+    have hzb := inner_le_supportValue K hz.1 (b : Real.Angle)
+    have hQb : inner ℝ Q (normalVector (b : Real.Angle)) = supportValue K b := by
+      rw [hQ]; exact (edgeVertices_snd_mem K _).2
+    have hqK : Q ∈ K := by rw [hQ]; exact (edgeVertices_snd_mem K _).1
+    have hq_le := inner_le_supportValue K hqK (s : Real.Angle)
+    rw [← hz.2, hn, inner_add_right, inner_smul_right, inner_smul_right,
+      inner_add_right, inner_smul_right, inner_smul_right, hQc, hQb] at hq_le
+    nlinarith
+
+private theorem not_collinear_of_inner_eq_eq_lt
+    (P Q O n : Point) (hPQ : P ≠ Q)
+    (hPQn : inner ℝ P n = inner ℝ Q n) (hO : inner ℝ P n < inner ℝ O n) :
+    ¬Collinear ℝ ({P, Q, O} : Set Point) := by
+  intro hcol
+  obtain ⟨p₀, v, hv⟩ := (collinear_iff_exists_forall_eq_smul_vadd
+    (k := ℝ) ({P, Q, O} : Set Point)).mp hcol
+  obtain ⟨rP, hrP⟩ := hv P (by simp)
+  obtain ⟨rQ, hrQ⟩ := hv Q (by simp)
+  obtain ⟨rO, hrO⟩ := hv O (by simp)
+  have hrne : rP ≠ rQ := by
+    intro h
+    apply hPQ
+    rw [hrP, hrQ, h]
+  have hvn : inner ℝ v n = 0 := by
+    rw [hrP, hrQ, vadd_eq_add, vadd_eq_add, inner_add_left, inner_add_left,
+      real_inner_smul_left, real_inner_smul_left] at hPQn
+    have : (rP - rQ) * inner ℝ v n = 0 := by linarith
+    exact (mul_eq_zero.mp this).resolve_left (sub_ne_zero.mpr hrne)
+  rw [hrP, hrO, vadd_eq_add, vadd_eq_add, inner_add_left, inner_add_left,
+    real_inner_smul_left, real_inner_smul_left, hvn] at hO
+  simp only [mul_zero, zero_add] at hO
+  exact hO.false
+
+private theorem exposedEdge_upper_cut_first (K K' : ConvexBody Point)
+    (a b t c : ℝ) (hat : a < t) (htb : t < b) (hba : b < a + Real.pi)
+    (hsub : (K' : Set Point) ⊆ (K : Set Point))
+    (hcut : ∀ z ∈ (K' : Set Point), c ≤ inner ℝ z (normalVector (t : Real.Angle)))
+    (hPmem : (edgeVertices K (a : Real.Angle)).1 ∈ K')
+    (hPc : inner ℝ (edgeVertices K (a : Real.Angle)).1
+      (normalVector (t : Real.Angle)) = c) :
+    exposedEdge K' (a : Real.Angle) =
+      {(edgeVertices K (a : Real.Angle)).1} := by
+  let P := (edgeVertices K (a : Real.Angle)).1
+  have hsin : 0 < Real.sin (t - a) :=
+    Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hat) (by linarith)
+  have hsupport : supportValue K' (a : Real.Angle) = supportValue K (a : Real.Angle) := by
+    apply le_antisymm
+    · apply csSup_le (K'.nonempty.image _)
+      rintro _ ⟨z, hz, rfl⟩
+      exact inner_le_supportValue K (hsub hz) (a : Real.Angle)
+    · rw [← (edgeVertices_fst_mem K (a : Real.Angle)).2]
+      exact inner_le_supportValue K' hPmem (a : Real.Angle)
+  apply Set.Subset.antisymm
+  · intro z hz
+    have hzKedge : z ∈ exposedEdge K (a : Real.Angle) := by
+      refine ⟨hsub hz.1, ?_⟩
+      change inner ℝ z (normalVector (a : Real.Angle)) = supportValue K a
+      rw [← hsupport]
+      exact hz.2
+    have hnormal : inner ℝ (z - P) (normalVector (a : Real.Angle)) = 0 := by
+      rw [inner_sub_left, hzKedge.2]
+      change supportValue K (a : Real.Angle) -
+        inner ℝ (edgeVertices K (a : Real.Angle)).1
+          (normalVector (a : Real.Angle)) = 0
+      rw [(edgeVertices_fst_mem K (a : Real.Angle)).2, sub_self]
+    have htangent : inner ℝ (z - P) (tangentVector (a : Real.Angle)) ≤ 0 := by
+      rw [inner_sub_left, sub_nonpos]
+      change inner ℝ z (tangentVector (a : Real.Angle)) ≤
+        inner ℝ (edgeVertices K (a : Real.Angle)).1
+          (tangentVector (a : Real.Angle))
+      rw [inner_edgeVertices_fst_tangent]
+      exact le_csSup ((isCompact_exposedEdge K (a : Real.Angle)).image
+        (continuous_id.inner continuous_const) |>.bddAbove) ⟨z, hzKedge, rfl⟩
+    have hcut : 0 ≤ inner ℝ (z - P) (normalVector (t : Real.Angle)) := by
+      rw [inner_sub_left, hPc]
+      exact sub_nonneg.mpr (hcut z hz.1)
+    have hrotate : normalVector (t : Real.Angle) =
+        Real.cos (t - a) • normalVector (a : Real.Angle) +
+          Real.sin (t - a) • tangentVector (a : Real.Angle) := by
+      simpa only [add_sub_cancel] using normalVector_add_real a (t - a)
+    rw [hrotate, inner_add_right, inner_smul_right, inner_smul_right, hnormal,
+      mul_zero, zero_add] at hcut
+    have htangent0 : inner ℝ (z - P) (tangentVector (a : Real.Angle)) = 0 := by
+      nlinarith
+    have hzP : z = P := by
+      rw [show z = P + (z - P) by abel,
+        ← inner_normalVector_smul_add_inner_tangentVector_smul
+          (z - P) (a : Real.Angle), hnormal, htangent0,
+        zero_smul, zero_smul, add_zero, add_zero]
+    simp [hzP, P]
+  · rintro z rfl
+    refine ⟨hPmem, ?_⟩
+    change inner ℝ P (normalVector (a : Real.Angle)) = supportValue K' a
+    rw [hsupport]
+    exact (edgeVertices_fst_mem K (a : Real.Angle)).2
+
+private theorem exposedEdge_upper_cut_last (K K' : ConvexBody Point)
+    (a b t c : ℝ) (hat : a < t) (htb : t < b) (hba : b < a + Real.pi)
+    (hsub : (K' : Set Point) ⊆ (K : Set Point))
+    (hcut : ∀ z ∈ (K' : Set Point), c ≤ inner ℝ z (normalVector (t : Real.Angle)))
+    (hQmem : (edgeVertices K (b : Real.Angle)).2 ∈ K')
+    (hQc : inner ℝ (edgeVertices K (b : Real.Angle)).2
+      (normalVector (t : Real.Angle)) = c) :
+    exposedEdge K' (b : Real.Angle) =
+      {(edgeVertices K (b : Real.Angle)).2} := by
+  let Q := (edgeVertices K (b : Real.Angle)).2
+  have hsupport : supportValue K' (b : Real.Angle) = supportValue K (b : Real.Angle) := by
+    apply le_antisymm
+    · apply csSup_le (K'.nonempty.image _)
+      rintro _ ⟨z, hz, rfl⟩
+      exact inner_le_supportValue K (hsub hz) (b : Real.Angle)
+    · rw [← (edgeVertices_snd_mem K (b : Real.Angle)).2]
+      exact inner_le_supportValue K' hQmem (b : Real.Angle)
+  apply Set.Subset.antisymm
+  · intro z hz
+    have hzKedge : z ∈ exposedEdge K (b : Real.Angle) := by
+      refine ⟨hsub hz.1, ?_⟩
+      change inner ℝ z (normalVector (b : Real.Angle)) = supportValue K b
+      rw [← hsupport]
+      exact hz.2
+    have hnormal : inner ℝ (z - Q) (normalVector (b : Real.Angle)) = 0 := by
+      rw [inner_sub_left, hzKedge.2]
+      change supportValue K (b : Real.Angle) -
+        inner ℝ (edgeVertices K (b : Real.Angle)).2
+          (normalVector (b : Real.Angle)) = 0
+      rw [(edgeVertices_snd_mem K (b : Real.Angle)).2, sub_self]
+    have htangent : 0 ≤ inner ℝ (z - Q) (tangentVector (b : Real.Angle)) := by
+      rw [inner_sub_left, sub_nonneg]
+      change inner ℝ (edgeVertices K (b : Real.Angle)).2
+          (tangentVector (b : Real.Angle)) ≤
+        inner ℝ z (tangentVector (b : Real.Angle))
+      rw [inner_edgeVertices_snd_tangent]
+      exact csInf_le ((isCompact_exposedEdge K (b : Real.Angle)).image
+        (continuous_id.inner continuous_const) |>.bddBelow) ⟨z, hzKedge, rfl⟩
+    have hcut : 0 ≤ inner ℝ (z - Q) (normalVector (t : Real.Angle)) := by
+      rw [inner_sub_left, hQc]
+      exact sub_nonneg.mpr (hcut z hz.1)
+    have hrotate : normalVector (t : Real.Angle) =
+        Real.cos (t - b) • normalVector (b : Real.Angle) +
+          Real.sin (t - b) • tangentVector (b : Real.Angle) := by
+      simpa only [add_sub_cancel] using normalVector_add_real b (t - b)
+    have hsinb : Real.sin (t - b) < 0 := by
+      rw [show t - b = -(b - t) by ring, Real.sin_neg, neg_lt_zero]
+      exact Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr htb) (by linarith)
+    rw [hrotate, inner_add_right, inner_smul_right, inner_smul_right, hnormal,
+      mul_zero, zero_add] at hcut
+    have htangent0 : inner ℝ (z - Q) (tangentVector (b : Real.Angle)) = 0 := by
+      nlinarith
+    have hzQ : z = Q := by
+      rw [show z = Q + (z - Q) by abel,
+        ← inner_normalVector_smul_add_inner_tangentVector_smul
+          (z - Q) (b : Real.Angle), hnormal, htangent0,
+        zero_smul, zero_smul, add_zero, add_zero]
+    simp [hzQ, Q]
+  · rintro z rfl
+    refine ⟨hQmem, ?_⟩
+    change inner ℝ Q (normalVector (b : Real.Angle)) = supportValue K' b
+    rw [hsupport]
+    exact (edgeVertices_snd_mem K (b : Real.Angle)).2
+
+theorem convexBoundaryArc_cut (K : ConvexBody Point) (a b : ℝ)
+    (hab : a < b) (hba : b < a + Real.pi)
+    (P Q O : Point) (hP : P = (edgeVertices K a).1)
+    (hQ : Q = (edgeVertices K b).2) (hO : O = supportingIntersection K a b) :
+    (P = Q → O = P ∧ convexBoundaryArc K a b = {P}) ∧
+    (P ≠ Q → ¬ Collinear ℝ ({P, Q, O} : Set Point) ∧
+      ∃ (t c : ℝ) (K' : ConvexBody Point),
+        a < t ∧ t < b ∧
+        inner ℝ P (normalVector (t : Real.Angle)) = c ∧
+        inner ℝ Q (normalVector (t : Real.Angle)) = c ∧
+        c < inner ℝ O (normalVector (t : Real.Angle)) ∧
+        (K' : Set Point) = (K : Set Point) ∩ normalHalfPlane t c true false ∧
+        (∀ s ∈ Set.Ioc (t - Real.pi) a, exposedEdge K' s = {P}) ∧
+        (∀ s ∈ Set.Ioo a b, exposedEdge K' s = exposedEdge K s) ∧
+        (∀ s ∈ Set.Ico b (t + Real.pi), exposedEdge K' s = {Q}) ∧
+        exposedEdge K' (t + Real.pi) = segment ℝ Q P) := by
+  subst P
+  subst Q
+  subst O
+  constructor
+  · intro hPQ
+    have hPa := edgeVertices_fst_mem K (a : Real.Angle)
+    have hPb : (edgeVertices K (a : Real.Angle)).1 ∈ exposedEdge K (b : Real.Angle) := by
+      rw [hPQ]
+      exact edgeVertices_snd_mem K (b : Real.Angle)
+    have hO := eq_supportingIntersection_of_mem_endpoint_faces K hab hba
+      (edgeVertices K (a : Real.Angle)).1 hPa hPb
+    refine ⟨hO, ?_⟩
+    have hfaces : ∀ t ∈ Set.Ioo a b, exposedEdge K (t : Real.Angle) =
+        {(edgeVertices K (a : Real.Angle)).1} := by
+      intro t ht
+      exact exposedEdge_eq_singleton_of_common_endpoint K hab hba ht.1 ht.2 _ hPa hPb
+    rw [convexBoundaryArc, hPQ]
+    apply Set.Subset.antisymm
+    · refine Set.union_subset (Set.union_subset (Set.Subset.rfl) ?_) Set.Subset.rfl
+      exact Set.iUnion₂_subset fun t ht ↦ by simpa [← hPQ] using (hfaces t ht).le
+    · exact Set.subset_union_left.trans Set.subset_union_left
+  · intro hPQ
+    let P := (edgeVertices K (a : Real.Angle)).1
+    let Q := (edgeVertices K (b : Real.Angle)).2
+    let O := supportingIntersection K a b
+    obtain ⟨t, d, hat, htb, hd, hdir⟩ := exists_cut_angle K hab hba hPQ
+    let c := inner ℝ P (normalVector (t : Real.Angle))
+    have hPc : inner ℝ P (normalVector (t : Real.Angle)) = c := rfl
+    have hQc : inner ℝ Q (normalVector (t : Real.Angle)) = c := by
+      have hdiff : inner ℝ (Q - P) (normalVector (t : Real.Angle)) = 0 := by
+        rw [hdir, real_inner_smul_left, real_inner_comm,
+          inner_normalVector_tangentVector, mul_zero]
+      rw [inner_sub_left, hPc] at hdiff
+      linarith
+    have hPK : P ∈ K := edgeVertices_fst_mem K (a : Real.Angle) |>.1
+    have hQK : Q ∈ K := edgeVertices_snd_mem K (b : Real.Angle) |>.1
+    let K' : ConvexBody Point :=
+      { carrier := (K : Set Point) ∩ normalHalfPlane (t : Real.Angle) c true false
+        convex' := K.convex.inter (convex_normalHalfPlane _ _ true)
+        isCompact' := K.isCompact.inter_right (isClosed_normalHalfPlane _ _ true)
+        nonempty' := ⟨P, hPK, by change c ≤ _; exact hPc.ge⟩ }
+    have hK' : (K' : Set Point) = (K : Set Point) ∩
+        normalHalfPlane (t : Real.Angle) c true false := rfl
+    obtain ⟨α, hα, hOa⟩ :=
+      supportingIntersection_eq_fst_add_pos_tangent K hab hba hPQ
+    have hsin : 0 < Real.sin (t - a) :=
+      Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hat) (by linarith)
+    have hta : inner ℝ (tangentVector (a : Real.Angle))
+        (normalVector (t : Real.Angle)) = Real.sin (t - a) := by
+      have h := sin_sub_eq_neg_inner_normalVector_tangentVector
+        (t : Real.Angle) (a : Real.Angle)
+      change Real.sin (a - t) =
+        -inner ℝ (normalVector (t : Real.Angle)) (tangentVector (a : Real.Angle)) at h
+      rw [real_inner_comm, show a - t = -(t - a) by ring, Real.sin_neg] at h
+      linarith
+    have hcO : c < inner ℝ O (normalVector (t : Real.Angle)) := by
+      change O = P + α • tangentVector (a : Real.Angle) at hOa
+      rw [hOa, inner_add_left, real_inner_smul_left, hta, ← hPc]
+      nlinarith
+    have hncol : ¬Collinear ℝ ({P, Q, O} : Set Point) :=
+      not_collinear_of_inner_eq_eq_lt P Q O (normalVector (t : Real.Angle))
+        hPQ (hPc.trans hQc.symm) hcO
+    have hPmem : P ∈ K' := ⟨hPK, by change c ≤ _; exact hPc.ge⟩
+    have hQmem : Q ∈ K' := ⟨hQK, by change c ≤ _; exact hQc.ge⟩
+    have hsupportTerminal :
+        supportValue K' ((t + Real.pi : ℝ) : Real.Angle) = -c := by
+      apply le_antisymm
+      · apply csSup_le (K'.nonempty.image _)
+        rintro _ ⟨z, hz, rfl⟩
+        have hzcut := hz.2
+        change c ≤ inner ℝ z (normalVector (t : Real.Angle)) at hzcut
+        rw [normalVector_add_pi]
+        change inner ℝ z (-normalVector (t : Real.Angle)) ≤ -c
+        rw [inner_neg_right]
+        linarith
+      · have hp := inner_le_supportValue K' hPmem ((t + Real.pi : ℝ) : Real.Angle)
+        rw [normalVector_add_pi, inner_neg_right, hPc] at hp
+        exact hp
+    have haFace := exposedEdge_upper_cut_first K K' a b t c hat htb hba
+      (fun _ hz ↦ hz.1) (fun _ hz ↦ hz.2) hPmem hPc
+    have hbFace := exposedEdge_upper_cut_last K K' a b t c hat htb hba
+      (fun _ hz ↦ hz.1) (fun _ hz ↦ hz.2) hQmem hQc
+    have hangleTerminal :
+        ((t - Real.pi : ℝ) : Real.Angle) = ((t + Real.pi : ℝ) : Real.Angle) := by
+      rw [Real.Angle.angle_eq_iff_two_pi_dvd_sub]
+      refine ⟨-1, ?_⟩
+      norm_num
+      ring
+    have hPterminalPlus : P ∈ exposedEdge K' ((t + Real.pi : ℝ) : Real.Angle) := by
+      refine ⟨hPmem, ?_⟩
+      change inner ℝ P (normalVector ((t + Real.pi : ℝ) : Real.Angle)) =
+        supportValue K' ((t + Real.pi : ℝ) : Real.Angle)
+      rw [normalVector_add_pi, inner_neg_right, hsupportTerminal, hPc]
+    have hQterminalPlus : Q ∈ exposedEdge K' ((t + Real.pi : ℝ) : Real.Angle) := by
+      refine ⟨hQmem, ?_⟩
+      change inner ℝ Q (normalVector ((t + Real.pi : ℝ) : Real.Angle)) =
+        supportValue K' ((t + Real.pi : ℝ) : Real.Angle)
+      rw [normalVector_add_pi, inner_neg_right, hsupportTerminal, hQc]
+    have hPterminalMinus : P ∈ exposedEdge K' ((t - Real.pi : ℝ) : Real.Angle) := by
+      rw [hangleTerminal]
+      exact hPterminalPlus
+    refine ⟨hncol, t, c, K', hat, htb, hPc, hQc, hcO, hK', ?_, ?_, ?_, ?_⟩
+    · intro s hs
+      rcases eq_or_lt_of_le hs.2 with hsa | hsa
+      · subst s
+        exact haFace
+      · exact exposedEdge_eq_singleton_of_common_endpoint K'
+          (by linarith) (by linarith) hs.1 hsa P hPterminalMinus
+          (by rw [haFace]; rfl)
+    · intro s hs
+      have hsub := exposedEdge_subset_upper_cut K hat htb hba hs P Q rfl rfl hPc hQc
+      obtain ⟨z₀, hz₀⟩ := exposedEdge_nonempty K (s : Real.Angle)
+      have hz₀K' : z₀ ∈ K' := ⟨hz₀.1, hsub hz₀⟩
+      have hsupport : supportValue K' (s : Real.Angle) = supportValue K (s : Real.Angle) := by
+        apply le_antisymm
+        · apply csSup_le (K'.nonempty.image _)
+          rintro _ ⟨z, hz, rfl⟩
+          exact inner_le_supportValue K hz.1 (s : Real.Angle)
+        · rw [← hz₀.2]
+          exact inner_le_supportValue K' hz₀K' (s : Real.Angle)
+      ext z
+      constructor
+      · intro hz
+        refine ⟨hz.1.1, ?_⟩
+        change inner ℝ z (normalVector (s : Real.Angle)) = supportValue K s
+        rw [← hsupport]
+        exact hz.2
+      · intro hz
+        refine ⟨⟨hz.1, hsub hz⟩, ?_⟩
+        change inner ℝ z (normalVector (s : Real.Angle)) = supportValue K' s
+        rw [hsupport]
+        exact hz.2
+    · intro s hs
+      rcases eq_or_lt_of_le hs.1 with hbs | hbs
+      · subst s
+        exact hbFace
+      · exact exposedEdge_eq_singleton_of_common_endpoint K'
+          (by linarith) (by linarith) hbs hs.2 Q
+          (by rw [hbFace]; rfl) hQterminalPlus
+    · have hsupport := hsupportTerminal
+      apply Set.Subset.antisymm
+      · intro z hz
+        have hzcut := hz.1.2
+        change c ≤ inner ℝ z (normalVector (t : Real.Angle)) at hzcut
+        have hzeq := hz.2
+        change inner ℝ z (normalVector ((t + Real.pi : ℝ) : Real.Angle)) =
+          supportValue K' ((t + Real.pi : ℝ) : Real.Angle) at hzeq
+        rw [normalVector_add_pi, inner_neg_right, hsupport] at hzeq
+        have hzc : inner ℝ z (normalVector (t : Real.Angle)) = c := by linarith
+        rw [segment_symm ℝ]
+        exact mem_segment_of_mem_body_of_eq_cut_inner K hat htb hba P Q z rfl rfl
+          hd hdir hPc hzc hz.1.1
+      · intro z hz
+        have hzK : z ∈ K := K.convex.segment_subset hQK hPK hz
+        have hzc : inner ℝ z (normalVector (t : Real.Angle)) = c := by
+          have hlin := (convex_normalHalfPlane (t : Real.Angle) c true).segment_subset
+            (show P ∈ normalHalfPlane (t : Real.Angle) c true false by
+              change c ≤ _; exact hPc.ge)
+            (show Q ∈ normalHalfPlane (t : Real.Angle) c true false by
+              change c ≤ _; exact hQc.ge)
+          have hzcut := hlin (show z ∈ segment ℝ P Q by simpa [segment_symm ℝ] using hz)
+          rw [segment_eq_image'] at hz
+          obtain ⟨u, hu, rfl⟩ := hz
+          rw [inner_add_left, inner_smul_left, inner_sub_left, hPc, hQc]
+          ring
+        refine ⟨⟨hzK, by change c ≤ _; exact hzc.ge⟩, ?_⟩
+        change inner ℝ z (normalVector ((t + Real.pi : ℝ) : Real.Angle)) =
+          supportValue K' ((t + Real.pi : ℝ) : Real.Angle)
+        rw [normalVector_add_pi, inner_neg_right, hsupport, hzc]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Arc Cut Area
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open Set
+
+/-- A supporting chord of a convex-body frontier cuts off a rectifiable oriented arc, and the
+closed signed area is the sum of the arc area and the oppositely oriented chord area. -/
+theorem exists_rectifiableOrientedArc_of_cut_with_area
+    (K : ConvexBody Point) {t c d α β : ℝ} {P Q : Point} {U : Set Point}
+    {x : ContinuousBVPaths α β}
+    (hαβ : α < β)
+    (hx : IsOrientedJordanParametrization hαβ.le (frontier (K : Set Point)) true x.val)
+    (hPQ : P ≠ Q) (hbase : x.val ⟨α, le_rfl, hαβ.le⟩ ∉ segment ℝ P Q)
+    (hfrontier : frontier (K : Set Point) = U ∪ segment ℝ P Q)
+    (hinter : U ∩ segment ℝ P Q = {P, Q})
+    (hPt : inner ℝ P (normalVector (t : Real.Angle)) = c)
+    (hQt : inner ℝ Q (normalVector (t : Real.Angle)) = c)
+    (hterminal : exposedEdge K (t + Real.pi) = segment ℝ Q P)
+    (hd : 0 < d) (hdir : Q - P = d • tangentVector (t : Real.Angle)) :
+    ∃ A : RectifiableOrientedArc,
+      A.val.carrier = U ∧
+      A.val.startPoint = P ∧ A.val.endPoint = Q ∧
+      curveAreaFunctional x = jordanArcArea A + segmentArea Q P := by
+  let θ : Real.Angle := (t + Real.pi : ℝ)
+  have hhalf : frontier (K : Set Point) ⊆ normalHalfPlane θ (-c) false false := by
+    intro z hz
+    have hzK : z ∈ K := K.isClosed.frontier_subset hz
+    have hzle := inner_le_supportValue K hzK θ
+    have hPterm : P ∈ exposedEdge K θ := by
+      rw [show exposedEdge K θ = segment ℝ Q P by
+        simpa only [θ, Real.Angle.coe_add] using hterminal]
+      exact right_mem_segment ℝ Q P
+    have hsupp : supportValue K θ = -c := by
+      have h := hPterm.2
+      change inner ℝ P (normalVector θ) = supportValue K θ at h
+      change inner ℝ P (normalVector (((t + Real.pi : ℝ) : Real.Angle))) = _ at h
+      rw [normalVector_add_pi, inner_neg_right, hPt] at h
+      linarith
+    change inner ℝ z (normalVector θ) ≤ -c
+    rwa [hsupp] at hzle
+  have hQline : Q ∈ normalLine θ (-c) := by
+    change inner ℝ Q (normalVector θ) = -c
+    change inner ℝ Q (normalVector (((t + Real.pi : ℝ) : Real.Angle))) = -c
+    rw [normalVector_add_pi, inner_neg_right, hQt]
+  have hdir' : P = Q + d • tangentVector θ := by
+    have htangent : tangentVector θ = -tangentVector (t : Real.Angle) := by
+      ext i
+      fin_cases i <;> simp [θ, tangentVector, frame]
+    rw [htangent]
+    calc
+      P = Q - (Q - P) := by module
+      _ = Q - d • tangentVector (t : Real.Angle) := by rw [hdir]
+      _ = Q + d • -tangentVector (t : Real.Angle) := by module
+  exact exists_rectifiableOrientedArc_of_supportingChord_with_area
+    hαβ hx P Q hPQ
+    hbase hfrontier hinter θ (-c) hhalf hQline d hd hdir'
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The outer corner of a convex body as a path of bounded variation
+
+The outer corner of the rotating supporting hallway of a convex body `K` is
+`h_K(t) • u_t + h_K(t + π/2) • v_t`.  Support values are Lipschitz in the angle and bounded on a
+compact interval, so this expression is a Lipschitz, hence continuous bounded-variation, path on
+every compact interval of angles; and it is convex-linear in `K` because support values are.
+These are the two hypotheses of Mamikon convexity for the middle summand of the sofa area.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval
+
+namespace MovingSofa
+
+/-- The outer corner of a convex body traces a continuous path of bounded variation over every
+compact interval of angles. -/
+theorem exists_outerCornerBV (K : ConvexBody Point) (a b : ℝ) :
+    ∃ γ : ContinuousBVPaths a b, ∀ s : Set.Icc a b,
+      γ.val s = (rotatingHallwayParts (K : Set Point) ((s : ℝ) : Real.Angle)).outerCorner := by
+  set C := sSup (norm '' (K : Set Point))
+  have hC0 : 0 ≤ C := Real.sSup_nonneg (by rintro _ ⟨p, -, rfl⟩; exact norm_nonneg p)
+  have hlipn : ∀ x y : ℝ, |supportValue (K : Set Point) (x : Real.Angle) -
+      supportValue (K : Set Point) (y : Real.Angle)| ≤ C * |x - y| :=
+    abs_supportValue_sub_le_of_isCompact K.nonempty K.isCompact
+  have hlipt : ∀ x y : ℝ,
+      |supportValue (K : Set Point) ((x + Real.pi / 2 : ℝ) : Real.Angle) -
+        supportValue (K : Set Point) ((y + Real.pi / 2 : ℝ) : Real.Angle)| ≤ C * |x - y| := by
+    intro x y
+    simpa only [add_sub_add_right_eq_sub] using hlipn (x + Real.pi / 2) (y + Real.pi / 2)
+  have hcont : Continuous fun x : ℝ ↦
+      supportValue (K : Set Point) ((x + Real.pi / 2 : ℝ) : Real.Angle) :=
+    (continuous_supportValue_real K).comp (continuous_id.add_const _)
+  obtain ⟨Mn, hMn⟩ := isCompact_Icc.exists_bound_of_continuousOn (s := Set.Icc a b)
+    (continuous_supportValue_real K).continuousOn
+  obtain ⟨Mt, hMt⟩ := isCompact_Icc.exists_bound_of_continuousOn (s := Set.Icc a b)
+    hcont.continuousOn
+  set M := max (max Mn Mt) 0
+  have hM0 : 0 ≤ M := le_max_right _ _
+  have hMn' : ∀ x ∈ Set.Icc a b, |supportValue (K : Set Point) (x : Real.Angle)| ≤ M := by
+    intro x hx
+    simpa only [Real.norm_eq_abs] using
+      (hMn x hx).trans ((le_max_left Mn Mt).trans (le_max_left _ 0))
+  have hMt' : ∀ x ∈ Set.Icc a b,
+      |supportValue (K : Set Point) ((x + Real.pi / 2 : ℝ) : Real.Angle)| ≤ M := by
+    intro x hx
+    simpa only [Real.norm_eq_abs] using
+      (hMt x hx).trans ((le_max_right Mn Mt).trans (le_max_left _ 0))
+  exact ⟨continuousBVOfLipschitz _ (lipschitzOnWith_frameCombination
+      (D := ⟨2 * (C + M), by linarith⟩) le_rfl hM0 hlipn hlipt hMn' hMt').to_restrict,
+    fun s ↦ (outerCorner_eq_support_sum K (s : ℝ)).symm⟩
+
+/-- The outer-corner paths of a convex body may be chosen convex-linearly in the body.  The
+barycentric operation on paths is `bvPathCombination`, spelled out here because it is defined
+downstream of this module. -/
+theorem exists_outerCornerBV_convexLinear (a b : ℝ) :
+    ∃ γ : ConvexBody Point → ContinuousBVPaths a b,
+      (∀ (K : ConvexBody Point) (s : Set.Icc a b), (γ K).val s =
+        (rotatingHallwayParts (K : Set Point) ((s : ℝ) : Real.Angle)).outerCorner) ∧
+      IsConvexLinear convexBodyCombination
+        (fun r x y ↦ (1 - (r : ℝ)) • x + (r : ℝ) • y) γ := by
+  choose γ hγ using fun K : ConvexBody Point ↦ exists_outerCornerBV K a b
+  refine ⟨γ, hγ, fun t K L ↦ Subtype.ext (funext fun s ↦ ?_)⟩
+  change (γ (convexBodyCombination t K L)).val s =
+    ((1 - (t : ℝ)) • (γ K).val + (t : ℝ) • (γ L).val) s
+  rw [hγ, show ((1 - (t : ℝ)) • (γ K).val + (t : ℝ) • (γ L).val) s =
+      (1 - (t : ℝ)) • (γ K).val s + (t : ℝ) • (γ L).val s from rfl, hγ, hγ]
+  simp only [outerCorner_eq_support_sum, supportValue_convexBodyCombination]
+  module
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Geometry.Convex.HorizontalExtrema`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Convex / Horizontal Extrema
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A horizontal extreme slice is a singleton when every defining normal has nonzero sine. -/
+lemma ConvexBody.eq_of_mem_of_fst_eq_of_fst_extremal
+    (K : ConvexBody Point) (N : Set Real.Angle) (hN : N.Finite)
+    (hK : HasHalfPlaneRepresentation K N) (hsin : ∀ t ∈ N, t.sin ≠ 0)
+    {a b : Point} (ha : a ∈ (K : Set Point)) (hb : b ∈ (K : Set Point))
+    (hab : a 0 = b 0)
+    (hext : (∀ q ∈ (K : Set Point), q 0 ≤ a 0) ∨
+      (∀ q ∈ (K : Set Point), a 0 ≤ q 0)) : a = b := by
+  by_contra hne
+  have hy : a 1 ≠ b 1 := by
+    intro h
+    apply hne
+    ext i
+    fin_cases i
+    · exact hab
+    · exact h
+  let m : Point := (2 : ℝ)⁻¹ • (a + b)
+  let U : Set Point := ⋂ t ∈ N,
+    {q | inner ℝ q (normalVector t) < supportValue K t}
+  have hU : IsOpen U := by
+    apply hN.isOpen_biInter
+    intro t ht
+    exact isOpen_lt (by fun_prop) continuous_const
+  have hm : m ∈ U := by
+    simp only [U, Set.mem_iInter, Set.mem_ofPred_eq]
+    intro t ht
+    have ha' := inner_le_supportValue K ha t
+    have hb' := inner_le_supportValue K hb t
+    have hneq : inner ℝ a (normalVector t) ≠ inner ℝ b (normalVector t) := by
+      intro heq
+      have hdiff : t.sin * (a 1 - b 1) = 0 := by
+        simp only [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two] at heq
+        simp only [Fin.isValue, Matrix.cons_val_zero, RCLike.inner_apply, conj_trivial,
+          Matrix.cons_val_one, Matrix.cons_val_fin_one] at heq
+        rw [hab] at heq
+        nlinarith
+      exact hy (sub_eq_zero.mp ((mul_eq_zero.mp hdiff).resolve_left (hsin t ht)))
+    have hmid : inner ℝ m (normalVector t) =
+        (inner ℝ a (normalVector t) + inner ℝ b (normalVector t)) / 2 := by
+      simp only [m, inner_add_left, real_inner_smul_left]
+      ring
+    rw [hmid]
+    rcases lt_or_gt_of_ne hneq with h | h <;> linarith
+  obtain ⟨ε, hε, hball⟩ := (Metric.isOpen_iff.mp hU) m hm
+  have hmem (r : ℝ) (hr : |r| < ε) :
+      m + r • normalVector (0 : Real.Angle) ∈ (K : Set Point) := by
+    have hqm : m + r • normalVector (0 : Real.Angle) ∈ Metric.ball m ε := by
+      rw [Metric.mem_ball, dist_eq_norm]
+      simp only [add_sub_cancel_left, norm_smul]
+      have hn : ‖normalVector (0 : Real.Angle)‖ = 1 := by
+        simpa only [Real.Angle.coe_zero] using norm_normalVector_real 0
+      simpa only [hn, mul_one, Real.norm_eq_abs] using hr
+    rw [hK.eq_iInter_supportValue]
+    simp only [Set.mem_iInter]
+    intro t ht
+    have h := Set.mem_iInter.mp (Set.mem_iInter.mp (hball hqm) t) ht
+    change inner ℝ (m + r • normalVector (0 : Real.Angle)) (normalVector t) <
+      supportValue K t at h
+    exact h.le
+  have hcoord (r : ℝ) : (m + r • normalVector (0 : Real.Angle)) 0 = a 0 + r := by
+    simp [m, normalVector, frame, ← hab]
+    ring
+  rcases hext with hmax | hmin
+  · have hq := hmem (ε / 2) (by rw [abs_of_pos (half_pos hε)]; linarith)
+    have h := hmax _ hq
+    rw [hcoord] at h
+    linarith
+  · have hq := hmem (-ε / 2) (by rw [abs_of_neg (by linarith : -ε / 2 < 0)]; linarith)
+    have h := hmin _ hq
+    rw [hcoord] at h
+    linarith
+
+/-- Every horizontal coordinate between two points of a convex body is attained. -/
+lemma ConvexBody.exists_mem_fst_eq_of_mem_Icc (K : ConvexBody Point) {a b : Point}
+    (ha : a ∈ (K : Set Point)) (hb : b ∈ (K : Set Point)) (hab : a 0 < b 0)
+    {x : ℝ} (hx : x ∈ Set.Icc (a 0) (b 0)) :
+    ∃ q ∈ (K : Set Point), q 0 = x := by
+  let r := (x - a 0) / (b 0 - a 0)
+  have hd : 0 < b 0 - a 0 := sub_pos.mpr hab
+  have hr : 0 ≤ r := div_nonneg (sub_nonneg.mpr hx.1) hd.le
+  have hr1 : r ≤ 1 := by
+    dsimp [r]
+    rw [div_le_one hd]
+    linarith [hx.2]
+  refine ⟨(1 - r) • a + r • b,
+    K.convex ha hb (sub_nonneg.mpr hr1) hr (by ring), ?_⟩
+  simp only [PiLp.add_apply, PiLp.smul_apply, smul_eq_mul]
+  dsimp [r]
+  field_simp
+  ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Gerver.Motion`.
+* `Gerver.PaperPath`.
+* `Gerver.PaperSet`.
+* `Gerver.Parameters`.
+* `Gerver.ParameterDictionary`.
+* `Gerver.Partition`.
+* `Gerver.ReversePhysicalDomain`.
+* `Gerver.DirectRegularity`.
+* `Gerver.LiteralSets`.
+* `Gerver.LiteralConnected`.
+* `Gerver.ParameterIdentification`.
+* `Gerver.Contacts`.
+* `Gerver.Niche.Roof`.
+* `Gerver.ODEs`.
+* `Gerver.OuterContacts`.
+* `Gerver.StageRegularity`.
+* `Gerver.Area.Grid`.
+* `Gerver.Niche.RoofProperties`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-
+MIT License
+
+Copyright (c) 2026 Dawid Trela
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.
+-/
+
+
+/-! Adapted from GerverSofaLean v1.1.0, F07UpstreamMotion (MIT). -/
+
+@[expose] public section
+
+noncomputable section
+open MeasureTheory
+open scoped unitInterval ENNReal
+
+namespace MovingSofa
+
+open GerverSofa.PartF
+
+/-- The certified continuous rigid motion carrying Gerver’s sofa through the hallway. -/
+def gerversSofaMotion : I → Rigid := EuclideanMotion.motion
+
+theorem isMovingSofa_gerversSofa_motion : IsMovingSofa gerversSofa gerversSofaMotion := by
+  apply (ProjectAdapter.isMovingSofa_iff_model _ _).2
+  rw [ProjectAdapter.gerversSofa_eq_certified]
+  exact EuclideanMotion.concrete_movingSofa
+
+theorem isMovingSofa_gerversSofa : ∃ m, IsMovingSofa gerversSofa m :=
+  ⟨gerversSofaMotion, isMovingSofa_gerversSofa_motion⟩
+
+theorem volume_gerversSofa_le_sofaConstant : volume gerversSofa ≤ sofaConstant := by
+  unfold sofaConstant
+  exact le_iSup_of_le gerversSofa (le_iSup_of_le isMovingSofa_gerversSofa le_rfl)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The paper's Gerver path and the vendor coordinate transport
+
+`paperGerverPath` is the certified direct five-phase Gerver path read through the vendor
+coordinate identification `GerverSofa.PartF.Coordinates.toPlane`.  This file records the
+transport lemmas for that identification (derivatives, smoothness, continuity) and the
+frame readers that express the moving frame `normalVector`/`tangentVector` in the same
+coordinates.
+
+The frame readers live here rather than in `MovingSofa/Geometry/Basic.lean` because their
+statements mention `toPlane`, which `Geometry/Basic.lean` does not import; this is the
+lowest module that sees both the vendor coordinates and `MovingSofa.frame`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped ContDiff
+
+namespace MovingSofa
+
+/-- The certified direct five-phase Gerver path, taking the earlier branch at each switch. -/
+def paperGerverPath (t : ℝ) : Point :=
+  GerverSofa.PartF.Coordinates.toPlane (GerverSofa.Romik.path GerverSofa.PartB.params t)
+
+theorem canonical_path_rotation_eq_paper (t : ℝ) (ht : t ∈ Set.Icc 0 (Real.pi / 2)) :
+    rotationMap (t : Real.Angle) (GerversSofa.p t) = paperGerverPath t :=
+  GerverSofa.PartF.ProjectAdapter.integral_rotation_to_full t ht
+
+section Transport
+
+open GerverSofa.Romik GerverSofa.PartF.Coordinates
+
+/-! ### Transport along the coordinate identification -/
+
+/-- The coordinate identification `toPlane` is a continuous linear map, so it transports
+derivatives of plane-valued curves. -/
+theorem hasDerivAt_toPlane {q : ℝ → GerverSofa.Point} {q' : GerverSofa.Point} {t : ℝ}
+    (hq : HasDerivAt q q' t) : HasDerivAt (fun s => toPlane (q s)) (toPlane q') t := by
+  obtain ⟨L, hL⟩ : ∃ L : GerverSofa.Point →L[ℝ] Point, ⇑L = toPlane :=
+    ⟨{ toLinearMap := linearEquiv.toLinearMap, cont := continuous_toPlane }, rfl⟩
+  rw [← hL]
+  simpa [Function.comp_def] using L.hasFDerivAt.comp_hasDerivAt t hq
+
+/-- The coordinate identification `toPlane` preserves smoothness of plane-valued curves. -/
+theorem contDiff_toPlane {q : ℝ → GerverSofa.Point} (hq : ContDiff ℝ ∞ q) :
+    ContDiff ℝ ∞ fun t => toPlane (q t) := by
+  rw [contDiff_euclidean]
+  intro i
+  fin_cases i
+  · exact hq.fst
+  · exact hq.snd
+
+/-! ### Frame readers -/
+
+/-- The angular frame normal is the coordinate image of the standard trigonometric pair. -/
+theorem normalVector_coe_eq_toPlane (t : ℝ) :
+    normalVector (t : Real.Angle) = toPlane (Real.cos t, Real.sin t) := rfl
+
+/-- The angular frame tangent is the coordinate image of the rotated trigonometric pair. -/
+theorem tangentVector_coe_eq_toPlane (t : ℝ) :
+    tangentVector (t : Real.Angle) = toPlane (-Real.sin t, Real.cos t) := rfl
+
+/-- The normal component of a body-frame vector rotated by `t` is its first coordinate. -/
+theorem inner_toPlane_rot_normalVector (t : ℝ) (z : GerverSofa.Point) :
+    inner ℝ (toPlane (rot t z)) (normalVector (t : Real.Angle)) = z.1 := by
+  have h := Real.sin_sq_add_cos_sq t
+  simp only [normalVector, MovingSofa.frame, PiLp.inner_apply, RCLike.inner_apply,
+    Fin.sum_univ_two, rot, toPlane_zero_coord, toPlane_one_coord, conj_trivial,
+    Matrix.cons_val_zero, Matrix.cons_val_one, Real.Angle.cos_coe, Real.Angle.sin_coe]
+  linear_combination z.1 * h
+
+/-- The tangential component of a body-frame vector rotated by `t` is its second
+coordinate. -/
+theorem inner_toPlane_rot_tangentVector (t : ℝ) (z : GerverSofa.Point) :
+    inner ℝ (toPlane (rot t z)) (tangentVector (t : Real.Angle)) = z.2 := by
+  have h := Real.sin_sq_add_cos_sq t
+  simp only [tangentVector, MovingSofa.frame, PiLp.inner_apply, RCLike.inner_apply,
+    Fin.sum_univ_two, rot, toPlane_zero_coord, toPlane_one_coord, conj_trivial,
+    Matrix.cons_val_zero, Matrix.cons_val_one, Real.Angle.cos_coe, Real.Angle.sin_coe]
+  linear_combination z.2 * h
+
+/-- The normal frame component of a plane point is the vendor scalar product of its
+coordinate pair with the vendor normal `u`. -/
+theorem inner_normalVector_eq_dot (q : Point) (s : ℝ) :
+    inner ℝ q (normalVector (s : Real.Angle)) =
+      GerverSofa.dot (fromPlane q) (GerverSofa.u s) := by
+  simp only [normalVector, MovingSofa.frame, PiLp.inner_apply, RCLike.inner_apply, conj_trivial,
+    Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one, Real.Angle.cos_coe,
+    Real.Angle.sin_coe, GerverSofa.dot, GerverSofa.u, fromPlane]
+  ring
+
+/-- The tangent frame component of a plane point is the vendor scalar product of its
+coordinate pair with the vendor tangent `v`. -/
+theorem inner_tangentVector_eq_dot (q : Point) (s : ℝ) :
+    inner ℝ q (tangentVector (s : Real.Angle)) =
+      GerverSofa.dot (fromPlane q) (GerverSofa.v s) := by
+  simp only [tangentVector, MovingSofa.frame, PiLp.inner_apply, RCLike.inner_apply, conj_trivial,
+    Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one, Real.Angle.cos_coe,
+    Real.Angle.sin_coe, GerverSofa.dot, GerverSofa.v, fromPlane]
+  ring
+
+/-- The angular frame normal is a smooth function of the angle. -/
+theorem contDiff_normalVector : ContDiff ℝ ∞ fun t : ℝ => normalVector (t : Real.Angle) := by
+  simp only [normalVector_coe_eq_toPlane]
+  exact contDiff_toPlane (Real.contDiff_cos.prodMk Real.contDiff_sin)
+
+/-- The angular frame tangent is a smooth function of the angle. -/
+theorem contDiff_tangentVector :
+    ContDiff ℝ ∞ fun t : ℝ => tangentVector (t : Real.Angle) := by
+  simp only [tangentVector_coe_eq_toPlane]
+  exact contDiff_toPlane (Real.contDiff_sin.neg.prodMk Real.contDiff_cos)
+
+/-! ### The paper path as a transported direct path
+
+The hypothesis `ContDiff ℝ 1 (path GerverSofa.PartB.params)` in the lemmas below is the
+first conjunct of `MovingSofa.gerver_direct_path_regularity`, which lives in a later
+module; passing it as a hypothesis keeps this file free of that dependency. -/
+
+/-- The paper path is the coordinate image of the certified direct path. -/
+theorem paperGerverPath_eq_toPlane :
+    paperGerverPath = fun t => toPlane (path GerverSofa.PartB.params t) := rfl
+
+/-- The paper path differentiates by transporting the derivative of the direct path. -/
+theorem hasDerivAt_paperGerverPath (hC1 : ContDiff ℝ 1 (path GerverSofa.PartB.params))
+    (t : ℝ) :
+    HasDerivAt paperGerverPath (toPlane (deriv (path GerverSofa.PartB.params) t)) t :=
+  hasDerivAt_toPlane ((hC1.differentiable one_ne_zero t).hasDerivAt)
+
+/-- The derivative of the paper path is the transported derivative of the direct path. -/
+theorem deriv_paperGerverPath (hC1 : ContDiff ℝ 1 (path GerverSofa.PartB.params)) (t : ℝ) :
+    deriv paperGerverPath t = toPlane (deriv (path GerverSofa.PartB.params) t) :=
+  (hasDerivAt_paperGerverPath hC1 t).deriv
+
+/-- The paper path is continuous. -/
+theorem continuous_paperGerverPath (hC1 : ContDiff ℝ 1 (path GerverSofa.PartB.params)) :
+    Continuous paperGerverPath := by
+  rw [paperGerverPath_eq_toPlane]
+  exact continuous_toPlane.comp hC1.continuous
+
+/-- The paper path is continuously differentiable. -/
+theorem continuous_deriv_paperGerverPath
+    (hC1 : ContDiff ℝ 1 (path GerverSofa.PartB.params)) :
+    Continuous (deriv paperGerverPath) := by
+  rw [funext (deriv_paperGerverPath hC1)]
+  exact continuous_toPlane.comp (hC1.continuous_deriv le_rfl)
+
+end Transport
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver / Paper Set
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Gerver's paper-frame set, cut out by the explicit path of rotated hallways. -/
+def paperGerverSofa : Set Point :=
+  (strips (Real.pi / 2)).1 ∩ (strips (Real.pi / 2)).2.2 ∩
+    ⋂ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      (fun p ↦ rotationMap (t : Real.Angle) p + paperGerverPath t) '' hallway
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver / Parameters
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The two Gerver switching angles and the right/left distinguished angles. -/
+def paperGerverConstants : (ℝ × ℝ) × (ℝ × ℝ) :=
+  ((GerversSofa.φ, GerversSofa.θ), (GerversSofa.φ, Real.pi / 2 - GerversSofa.φ))
+
+/-- The six endpoints of the five Gerver stages. -/
+def gerverStageTimes : Fin 6 → ℝ :=
+  ![0, GerversSofa.φ, GerversSofa.θ, Real.pi / 2 - GerversSofa.θ,
+    Real.pi / 2 - GerversSofa.φ, Real.pi / 2]
+
+/-- The five closed stage intervals in their source order. -/
+def gerverStageIntervals (i : Fin 5) : Set ℝ :=
+  Set.Icc (gerverStageTimes i.castSucc) (gerverStageTimes i.succ)
+
+/-- The exact direct 22-equation Gerver system, with the certified phase-map convention. -/
+def gerverDirectEquations (p : GerverSofa.Romik.Params) : Prop :=
+  GerverSofa.Romik.Equations p
+
+/-- The exact closed rational box for the direct Gerver parameters. -/
+def gerverDirectBox : Set GerverSofa.Romik.Params := GerverSofa.Romik.box
+
+/-- The exact box bounds the second-stage linear coefficient `b₁` from below by `-53/100`. -/
+theorem gerverDirectBox_b1_lower_bound {p : GerverSofa.Romik.Params} (hp : p ∈ gerverDirectBox) :
+    (-53 : ℝ) / 100 ≤ p.b1 := by
+  dsimp only [gerverDirectBox, GerverSofa.Romik.box, GerverSofa.qR, Set.mem_ofPred_eq] at hp
+  obtain ⟨-, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -,
+      -, -, -, -, hb1, -⟩ := hp
+  exact le_trans (by norm_num) hb1
+
+/-- The exact box bounds the fourth-stage linear coefficient `d₁` from above by `33/25`. -/
+theorem gerverDirectBox_d1_upper_bound {p : GerverSofa.Romik.Params} (hp : p ∈ gerverDirectBox) :
+    p.d1 ≤ (33 : ℝ) / 25 := by
+  dsimp only [gerverDirectBox, GerverSofa.Romik.box, GerverSofa.qR, Set.mem_ofPred_eq] at hp
+  obtain ⟨-, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -,
+      -, -, -, -, -, -, -, -, -, -, -, -, -, hd1, -⟩ := hp
+  exact le_trans hd1 (by norm_num)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver / Parameter Dictionary
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Bundle the forward and reverse dictionaries between reduced and full Romik parameters. -/
+def gerverParameterDictionary :
+    (GerverSofa.Reduced.Params → GerverSofa.Romik.Params) ×
+      (GerverSofa.Romik.Params → GerverSofa.Reduced.Params) :=
+  (GerverSofa.PartF.Phases.dictionary, GerverSofa.PartF.Phases.undictionary)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver / Partition
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The ten half-open Gerver phase intervals, indexed from zero. -/
+def gerverPhaseIntervals (j : Fin 10) : Set ℝ :=
+  let lower (i : Fin 5) := Set.Ico (gerverStageTimes i.castSucc) (gerverStageTimes i.succ)
+  Fin.addCases (motive := fun _ ↦ Set ℝ) lower
+    (fun i ↦ (fun t ↦ Real.pi - t) '' lower i.rev) j
+
+/-- The ten Gerver phase intervals, written out as explicit half-open real intervals. -/
+theorem gerverPhaseIntervals_explicit (j : Fin 10) :
+    gerverPhaseIntervals j =
+      ![Set.Ico 0 GerversSofa.φ,
+        Set.Ico GerversSofa.φ GerversSofa.θ,
+        Set.Ico GerversSofa.θ (Real.pi / 2 - GerversSofa.θ),
+        Set.Ico (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ),
+        Set.Ico (Real.pi / 2 - GerversSofa.φ) (Real.pi / 2),
+        Set.Ioc (Real.pi / 2) (Real.pi / 2 + GerversSofa.φ),
+        Set.Ioc (Real.pi / 2 + GerversSofa.φ) (Real.pi / 2 + GerversSofa.θ),
+        Set.Ioc (Real.pi / 2 + GerversSofa.θ) (Real.pi - GerversSofa.θ),
+        Set.Ioc (Real.pi - GerversSofa.θ) (Real.pi - GerversSofa.φ),
+        Set.Ioc (Real.pi - GerversSofa.φ) Real.pi] j := by
+  fin_cases j
+  all_goals simp [gerverPhaseIntervals, gerverStageTimes, Fin.addCases, Fin.rev]
+  all_goals congr 1 <;> ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver / Reverse Physical Domain
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem gerver_reverse_physical_domain (p : GerverSofa.Romik.Params)
+    (hp : p ∈ gerverDirectBox) :
+    0 < (gerverParameterDictionary.2 p).phi ∧
+    (gerverParameterDictionary.2 p).phi < (gerverParameterDictionary.2 p).theta ∧
+    (gerverParameterDictionary.2 p).theta < Real.pi / 4 ∧
+    0 < (gerverParameterDictionary.2 p).a ∧ 0 < (gerverParameterDictionary.2 p).b ∧
+    (39 : ℝ) / 1000 ≤ (gerverParameterDictionary.2 p).phi ∧
+    (gerverParameterDictionary.2 p).phi ≤ (40 : ℝ) / 1000 := by
+  dsimp [gerverDirectBox, GerverSofa.Romik.box, GerverSofa.qR] at hp
+  rcases hp with ⟨_, _, _, _, _, _, _, _,
+    _, _, _, _, _, _, _, _,
+    _, _, _, _, _, _, _, _,
+    hb1lo, hb1hi, hb2lo, _, _, _, _, _,
+    _, _, _, _, _, _, _, _,
+    hphilo, hphihi, hthetalo, hthetahi⟩
+  norm_num at hb1lo hb1hi hb2lo hphilo hphihi hthetalo hthetahi
+  have hphiLo : (39 : ℝ) / 1000 ≤ p.phi := by linarith only [hphilo]
+  have hphiHi : p.phi ≤ (1 : ℝ) / 25 := by linarith only [hphihi]
+  have hphiPos : (0 : ℝ) < p.phi := by linarith only [hphiLo]
+  have hthetaLo : (3 : ℝ) / 5 ≤ p.theta := by linarith only [hthetalo]
+  have hthetaHi : p.theta ≤ (7 : ℝ) / 10 := by linarith only [hthetahi]
+  have hb1Lo : (-66 : ℝ) / 125 ≤ p.b1 := by linarith only [hb1lo]
+  have hb1Hi : p.b1 ≤ (-527 : ℝ) / 1000 := by linarith only [hb1hi]
+  have hb2Lo : (23 : ℝ) / 25 ≤ p.b2 := by linarith only [hb2lo]
+  have hAPos : (0 : ℝ) < p.phi - 1 - 2 * p.b1 := by linarith only [hphiLo, hb1Hi]
+  have hBPos :
+      (0 : ℝ) <
+        p.b2 + 1 / 2 - (1 + (p.phi - 1 - 2 * p.b1)) * p.phi / 2 + p.phi ^ 2 / 4 := by
+    have hprod : (0 : ℝ) ≤ (p.b1 + 66 / 125) * p.phi :=
+      mul_nonneg (by linarith only [hb1Lo]) hphiPos.le
+    have hsq : (0 : ℝ) ≤ p.phi * (1 / 25 - p.phi) :=
+      mul_nonneg hphiPos.le (by linarith only [hphiHi])
+    nlinarith only [hb2Lo, hprod, hsq, hphiPos, hphiHi]
+  refine ⟨hphiPos, show p.phi < p.theta from ?_,
+    show p.theta < Real.pi / 4 from ?_, hAPos, hBPos, hphiLo,
+    show p.phi ≤ (40 : ℝ) / 1000 from ?_⟩
+  · linarith only [hphiHi, hthetaLo]
+  · nlinarith only [hthetaHi, Real.pi_gt_three]
+  · linarith only [hphiHi]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Matching and `C¹` regularity of the direct Gerver path
+
+The vendor library assembles the Gerver path `GerverSofa.Romik.path` from five smooth
+branches `path1, …, path5` switched at `φ < θ < π/2 - θ < π/2 - φ`.  This file supplies the
+differential interface for those branches — each `pathᵢ` has derivative
+`rot t (alphaBetaᵢ p t)` — and deduces from the direct equations that values and derivatives
+agree at all four switches, hence that `path p` is continuously differentiable with
+`path p 0 = 0`.
+
+It also records, for each stage, that the glued path agrees with its analytic branch on the
+*closed* stage interval and that `deriv (path p)` is the corresponding `rot t (alphaBetaᵢ p t)`
+there, endpoints included.
+-/
+
+/-! ### Differential interface for the five direct branches -/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped ContDiff
+
+namespace GerverSofa.Romik
+
+/-- Rotating a differentiable body-frame curve and translating it differentiates by the
+product rule, contributing the infinitesimal rotation `(z₁, z₂) ↦ (-z₂, z₁)`. -/
+theorem hasDerivAt_addK_rot {z₁ z₂ : ℝ → ℝ} {z₁' z₂' k₁ k₂ t : ℝ}
+    (hz₁ : HasDerivAt z₁ z₁' t) (hz₂ : HasDerivAt z₂ z₂' t) :
+    HasDerivAt (fun s => addK (rot s (z₁ s, z₂ s)) k₁ k₂)
+      (rot t (z₁' - z₂ t, z₂' + z₁ t)) t := by
+  apply HasDerivAt.prodMk
+  · change HasDerivAt (fun s => Real.cos s * z₁ s - Real.sin s * z₂ s + k₁)
+      (Real.cos t * (z₁' - z₂ t) - Real.sin t * (z₂' + z₁ t)) t
+    exact ((((Real.hasDerivAt_cos t).fun_mul hz₁).fun_sub
+        ((Real.hasDerivAt_sin t).fun_mul hz₂)).fun_add
+      (hasDerivAt_const t k₁)).congr_deriv (by ring)
+  · change HasDerivAt (fun s => Real.sin s * z₁ s + Real.cos s * z₂ s + k₂)
+      (Real.sin t * (z₁' - z₂ t) + Real.cos t * (z₂' + z₁ t)) t
+    exact ((((Real.hasDerivAt_sin t).fun_mul hz₁).fun_add
+        ((Real.hasDerivAt_cos t).fun_mul hz₂)).fun_add
+      (hasDerivAt_const t k₂)).congr_deriv (by ring)
+
+private theorem hasDerivAt_mul_self (t : ℝ) : HasDerivAt (fun s : ℝ => s * s) (t + t) t := by
+  simpa only [id_eq, one_mul, mul_one] using (hasDerivAt_id t).fun_mul (hasDerivAt_id t)
+
+/-- The first direct branch has body-frame velocity `alphaBeta1`. -/
+theorem hasDerivAt_path1 (p : Params) (t : ℝ) :
+    HasDerivAt (path1 p) (rot t (alphaBeta1 p t)) t := by
+  have hz₁ : HasDerivAt (fun s => p.a1 * Real.cos s + p.a2 * Real.sin s - 1)
+      (-p.a1 * Real.sin t + p.a2 * Real.cos t) t :=
+    ((((Real.hasDerivAt_cos t).const_mul p.a1).fun_add
+      ((Real.hasDerivAt_sin t).const_mul p.a2)).fun_sub
+      (hasDerivAt_const t (1 : ℝ))).congr_deriv (by ring)
+  have hz₂ : HasDerivAt (fun s => -p.a2 * Real.cos s + p.a1 * Real.sin s - 1 / 2)
+      (p.a2 * Real.sin t + p.a1 * Real.cos t) t :=
+    ((((Real.hasDerivAt_cos t).const_mul (-p.a2)).fun_add
+      ((Real.hasDerivAt_sin t).const_mul p.a1)).fun_sub
+      (hasDerivAt_const t (1 / 2 : ℝ))).congr_deriv (by ring)
+  refine (hasDerivAt_addK_rot (k₁ := p.k11) (k₂ := p.k12) hz₁ hz₂).congr_deriv ?_
+  apply congrArg (rot t)
+  ext <;> simp [alphaBeta1] <;> ring
+
+/-- The second direct branch has body-frame velocity `alphaBeta2`. -/
+theorem hasDerivAt_path2 (p : Params) (t : ℝ) :
+    HasDerivAt (path2 p) (rot t (alphaBeta2 p t)) t := by
+  have hz₁ : HasDerivAt (fun s => -(1 / 4 : ℝ) * s * s + p.b1 * s + p.b2)
+      (-(1 / 2 : ℝ) * t + p.b1) t := by
+    have h := (((hasDerivAt_mul_self t).const_mul (-(1 / 4 : ℝ))).fun_add
+      ((hasDerivAt_id t).const_mul p.b1)).fun_add (hasDerivAt_const t p.b2)
+    simpa only [id_eq, mul_assoc] using h.congr_deriv (by ring)
+  have hz₂ : HasDerivAt (fun s => (1 / 2 : ℝ) * s - p.b1 - 1) (1 / 2 : ℝ) t :=
+    ((((hasDerivAt_id t).const_mul (1 / 2 : ℝ)).fun_sub (hasDerivAt_const t p.b1)).fun_sub
+      (hasDerivAt_const t (1 : ℝ))).congr_deriv (by simp)
+  refine (hasDerivAt_addK_rot (k₁ := p.k21) (k₂ := p.k22) hz₁ hz₂).congr_deriv ?_
+  apply congrArg (rot t)
+  ext <;> simp [alphaBeta2] <;> ring
+
+/-- The third direct branch has body-frame velocity `alphaBeta3`. -/
+theorem hasDerivAt_path3 (p : Params) (t : ℝ) :
+    HasDerivAt (path3 p) (rot t (alphaBeta3 p t)) t := by
+  have hz₁ : HasDerivAt (fun s => p.c1 - s) (-1) t :=
+    ((hasDerivAt_const t p.c1).fun_sub (hasDerivAt_id t)).congr_deriv (by simp)
+  have hz₂ : HasDerivAt (fun s => p.c2 + s) 1 t :=
+    ((hasDerivAt_const t p.c2).fun_add (hasDerivAt_id t)).congr_deriv (by simp)
+  refine (hasDerivAt_addK_rot (k₁ := p.k31) (k₂ := p.k32) hz₁ hz₂).congr_deriv ?_
+  apply congrArg (rot t)
+  ext <;> simp [alphaBeta3] <;> ring
+
+/-- The fourth direct branch has body-frame velocity `alphaBeta4`. -/
+theorem hasDerivAt_path4 (p : Params) (t : ℝ) :
+    HasDerivAt (path4 p) (rot t (alphaBeta4 p t)) t := by
+  have hz₁ : HasDerivAt (fun s => -(1 / 2 : ℝ) * s + p.d1 - 1) (-(1 / 2 : ℝ)) t :=
+    ((((hasDerivAt_id t).const_mul (-(1 / 2 : ℝ))).fun_add (hasDerivAt_const t p.d1)).fun_sub
+      (hasDerivAt_const t (1 : ℝ))).congr_deriv (by simp)
+  have hz₂ : HasDerivAt (fun s => -(1 / 4 : ℝ) * s * s + p.d1 * s + p.d2)
+      (-(1 / 2 : ℝ) * t + p.d1) t := by
+    have h := (((hasDerivAt_mul_self t).const_mul (-(1 / 4 : ℝ))).fun_add
+      ((hasDerivAt_id t).const_mul p.d1)).fun_add (hasDerivAt_const t p.d2)
+    simpa only [id_eq, mul_assoc] using h.congr_deriv (by ring)
+  refine (hasDerivAt_addK_rot (k₁ := p.k41) (k₂ := p.k42) hz₁ hz₂).congr_deriv ?_
+  apply congrArg (rot t)
+  ext <;> simp [alphaBeta4] <;> ring
+
+/-- The fifth direct branch has body-frame velocity `alphaBeta5`. -/
+theorem hasDerivAt_path5 (p : Params) (t : ℝ) :
+    HasDerivAt (path5 p) (rot t (alphaBeta5 p t)) t := by
+  have hz₁ : HasDerivAt (fun s => p.e1 * Real.cos s + p.e2 * Real.sin s - 1 / 2)
+      (-p.e1 * Real.sin t + p.e2 * Real.cos t) t :=
+    ((((Real.hasDerivAt_cos t).const_mul p.e1).fun_add
+      ((Real.hasDerivAt_sin t).const_mul p.e2)).fun_sub
+      (hasDerivAt_const t (1 / 2 : ℝ))).congr_deriv (by ring)
+  have hz₂ : HasDerivAt (fun s => -p.e2 * Real.cos s + p.e1 * Real.sin s - 1)
+      (p.e2 * Real.sin t + p.e1 * Real.cos t) t :=
+    ((((Real.hasDerivAt_cos t).const_mul (-p.e2)).fun_add
+      ((Real.hasDerivAt_sin t).const_mul p.e1)).fun_sub
+      (hasDerivAt_const t (1 : ℝ))).congr_deriv (by ring)
+  refine (hasDerivAt_addK_rot (k₁ := p.k51) (k₂ := p.k52) hz₁ hz₂).congr_deriv ?_
+  apply congrArg (rot t)
+  ext <;> simp [alphaBeta5] <;> ring
+
+/-- Continuity of the body-frame velocity field on each branch, transported to the world
+frame by the rotation. -/
+theorem continuous_rot_alphaBeta1 (p : Params) :
+    Continuous fun t : ℝ => rot t (alphaBeta1 p t) := by
+  unfold rot alphaBeta1; fun_prop
+
+@[inherit_doc continuous_rot_alphaBeta1]
+theorem continuous_rot_alphaBeta2 (p : Params) :
+    Continuous fun t : ℝ => rot t (alphaBeta2 p t) := by
+  unfold rot alphaBeta2; fun_prop
+
+@[inherit_doc continuous_rot_alphaBeta1]
+theorem continuous_rot_alphaBeta3 (p : Params) :
+    Continuous fun t : ℝ => rot t (alphaBeta3 p t) := by
+  unfold rot alphaBeta3; fun_prop
+
+@[inherit_doc continuous_rot_alphaBeta1]
+theorem continuous_rot_alphaBeta4 (p : Params) :
+    Continuous fun t : ℝ => rot t (alphaBeta4 p t) := by
+  unfold rot alphaBeta4; fun_prop
+
+@[inherit_doc continuous_rot_alphaBeta1]
+theorem continuous_rot_alphaBeta5 (p : Params) :
+    Continuous fun t : ℝ => rot t (alphaBeta5 p t) := by
+  unfold rot alphaBeta5; fun_prop
+
+/-- Equation 32 of the direct system.  The remaining scalar consequences used here are
+already extracted by the vendor in `GerverSofa/KernelOnly/EndpointSymmetry.lean`. -/
+theorem k11_eq_one_sub_a1_of_equations {p : Params} (heq : Equations p) :
+    p.k11 = 1 - p.a1 := by
+  have h5 := congrFun heq (5 : Fin 22)
+  simp [system] at h5
+  linarith
+
+/-! ### Reflection identities for the body-frame velocities
+
+With `S (r, s) = (-s, -r)` the direct equations give `w₃(π/2 - t) = S w₃(t)`,
+`w₄(π/2 - t) = S w₂(t)` and `w₅(π/2 - t) = S w₁(t)`. -/
+
+/-- The middle branch velocity is anti-symmetric about `π/4`. -/
+theorem alphaBeta3_pi_div_two_sub {p : Params} (heq : Equations p) (t : ℝ) :
+    alphaBeta3 p (Real.pi / 2 - t) = (-(alphaBeta3 p t).2, -(alphaBeta3 p t).1) := by
+  have hc2 := c2_eq_c1_sub_halfPi_of_equations heq
+  simp only [alphaBeta3, Prod.mk.injEq]
+  constructor <;> rw [hc2] <;> ring
+
+/-- The fourth branch velocity reflects onto the second. -/
+theorem alphaBeta4_pi_div_two_sub {p : Params} (heq : Equations p) (t : ℝ) :
+    alphaBeta4 p (Real.pi / 2 - t) = (-(alphaBeta2 p t).2, -(alphaBeta2 p t).1) := by
+  have hd1 := d1_eq_quarterPi_sub_b1_of_equations heq
+  have hd2 := d2_eq_b2_add_quarterPi_correction_of_equations heq
+  simp only [alphaBeta4, alphaBeta2, Prod.mk.injEq]
+  constructor <;> simp only [hd1, hd2] <;> ring
+
+/-- The fifth branch velocity reflects onto the first. -/
+theorem alphaBeta5_pi_div_two_sub {p : Params} (heq : Equations p) (t : ℝ) :
+    alphaBeta5 p (Real.pi / 2 - t) = (-(alphaBeta1 p t).2, -(alphaBeta1 p t).1) := by
+  have he1 := e1_eq_a1_of_equations heq
+  have he2 := e2_eq_neg_a2_of_equations heq
+  simp only [alphaBeta5, alphaBeta1, Prod.mk.injEq, Real.sin_pi_div_two_sub,
+    Real.cos_pi_div_two_sub]
+  constructor <;> simp only [he1, he2] <;> ring
+
+/-- Velocity matching at the third switch `π/2 - θ`, by reflecting the match at `θ`. -/
+theorem alphaBeta34_eq {p : Params} (heq : Equations p) :
+    alphaBeta3 p (Real.pi / 2 - p.theta) = alphaBeta4 p (Real.pi / 2 - p.theta) := by
+  rw [alphaBeta3_pi_div_two_sub heq, alphaBeta4_pi_div_two_sub heq,
+    GerverSofa.PartF.Phases.alphaBeta23_eq heq]
+
+/-- Velocity matching at the fourth switch `π/2 - φ`, by reflecting the match at `φ`. -/
+theorem alphaBeta45_eq {p : Params} (heq : Equations p) :
+    alphaBeta4 p (Real.pi / 2 - p.phi) = alphaBeta5 p (Real.pi / 2 - p.phi) := by
+  rw [alphaBeta4_pi_div_two_sub heq, alphaBeta5_pi_div_two_sub heq,
+    GerverSofa.PartF.Phases.alphaBeta12_eq heq]
+
+/-! ### Branch selection on the closed stages
+
+Each stage interval is closed, so the two stages adjacent to a switch both contain it.  The
+glued `path` picks the *earlier* branch there, and the certified value-matching equations
+say that this is also the later branch's value. -/
+
+/-- The glued direct path unfolds to the nested selection of its five analytic branches. -/
+theorem path_eq_ite (p : Params) (t : ℝ) :
+    path p t =
+      if t ≤ p.phi then path1 p t
+      else if t ≤ p.theta then path2 p t
+      else if t ≤ Real.pi / 2 - p.theta then path3 p t
+      else if t ≤ Real.pi / 2 - p.phi then path4 p t else path5 p t := rfl
+
+/-- On the closed first stage `[0, φ]` the glued path is the first branch. -/
+theorem path_eq_path1_of_mem_Icc (p : Params) {s : ℝ} (hs : s ∈ Set.Icc (0 : ℝ) p.phi) :
+    path p s = path1 p s := by
+  rw [path_eq_ite, ite_eq_left hs.2]
+
+/-- On the closed second stage `[φ, θ]` the glued path is the second branch; at the left
+endpoint this is the certified value match `match_path12_of_equations`. -/
+theorem path_eq_path2_of_mem_Icc {p : Params} (heq : Equations p) {s : ℝ}
+    (hs : s ∈ Set.Icc p.phi p.theta) : path p s = path2 p s := by
+  rw [path_eq_ite]
+  by_cases hc : s ≤ p.phi
+  · obtain rfl : s = p.phi := le_antisymm hc hs.1
+    rw [ite_eq_left hc]
+    exact match_path12_of_equations heq
+  · rw [ite_eq_right hc, ite_eq_left hs.2]
+
+/-- On the closed third stage `[θ, π/2 - θ]` the glued path is the third branch; at the left
+endpoint this is the certified value match `match_path23_of_equations`. -/
+theorem path_eq_path3_of_mem_Icc {p : Params} (heq : Equations p) (hpt : p.phi < p.theta)
+    {s : ℝ} (hs : s ∈ Set.Icc p.theta (Real.pi / 2 - p.theta)) : path p s = path3 p s := by
+  have hb := hs.1
+  rw [path_eq_ite, ite_eq_right (by linarith : ¬ s ≤ p.phi)]
+  by_cases hc : s ≤ p.theta
+  · obtain rfl : s = p.theta := le_antisymm hc hs.1
+    rw [ite_eq_left hc]
+    exact match_path23_of_equations heq
+  · rw [ite_eq_right hc, ite_eq_left hs.2]
+
+/-- On the closed fourth stage `[π/2 - θ, π/2 - φ]` the glued path is the fourth branch; at
+the left endpoint this is the certified value match `match_path34_of_equations`. -/
+theorem path_eq_path4_of_mem_Icc {p : Params} (heq : Equations p) (hpt : p.phi < p.theta)
+    (hq : p.theta < Real.pi / 4) {s : ℝ}
+    (hs : s ∈ Set.Icc (Real.pi / 2 - p.theta) (Real.pi / 2 - p.phi)) :
+    path p s = path4 p s := by
+  have hb := hs.1
+  rw [path_eq_ite, ite_eq_right (by linarith : ¬ s ≤ p.phi),
+    ite_eq_right (by linarith : ¬ s ≤ p.theta)]
+  by_cases hc : s ≤ Real.pi / 2 - p.theta
+  · obtain rfl : s = Real.pi / 2 - p.theta := le_antisymm hc hs.1
+    rw [ite_eq_left hc]
+    exact match_path34_of_equations heq
+  · rw [ite_eq_right hc, ite_eq_left hs.2]
+
+/-- On the closed fifth stage `[π/2 - φ, π/2]` the glued path is the fifth branch; at the
+left endpoint this is the certified value match `match_path45_of_equations`. -/
+theorem path_eq_path5_of_mem_Icc {p : Params} (heq : Equations p) (hpt : p.phi < p.theta)
+    (hq : p.theta < Real.pi / 4) {s : ℝ}
+    (hs : s ∈ Set.Icc (Real.pi / 2 - p.phi) (Real.pi / 2)) : path p s = path5 p s := by
+  have hb := hs.1
+  rw [path_eq_ite, ite_eq_right (by linarith : ¬ s ≤ p.phi),
+    ite_eq_right (by linarith : ¬ s ≤ p.theta),
+    ite_eq_right (by linarith : ¬ s ≤ Real.pi / 2 - p.theta)]
+  by_cases hc : s ≤ Real.pi / 2 - p.phi
+  · obtain rfl : s = Real.pi / 2 - p.phi := le_antisymm hc hs.1
+    rw [ite_eq_left hc]
+    exact match_path45_of_equations heq
+  · rw [ite_eq_right hc]
+
+/-! ### Smoothness of the branches and of their body-frame velocities -/
+
+/-- The first analytic branch is smooth on all of `ℝ`. -/
+theorem contDiff_path1 (p : Params) : ContDiff ℝ ∞ (path1 p) := by
+  unfold path1 addK rot; fun_prop
+
+@[inherit_doc contDiff_path1]
+theorem contDiff_path2 (p : Params) : ContDiff ℝ ∞ (path2 p) := by
+  unfold path2 addK rot; fun_prop
+
+@[inherit_doc contDiff_path1]
+theorem contDiff_path3 (p : Params) : ContDiff ℝ ∞ (path3 p) := by
+  unfold path3 addK rot; fun_prop
+
+@[inherit_doc contDiff_path1]
+theorem contDiff_path4 (p : Params) : ContDiff ℝ ∞ (path4 p) := by
+  unfold path4 addK rot; fun_prop
+
+@[inherit_doc contDiff_path1]
+theorem contDiff_path5 (p : Params) : ContDiff ℝ ∞ (path5 p) := by
+  unfold path5 addK rot; fun_prop
+
+/-- The first body-frame velocity pair is smooth on all of `ℝ`. -/
+theorem contDiff_alphaBeta1 (p : Params) : ContDiff ℝ ∞ (alphaBeta1 p) := by
+  unfold alphaBeta1; fun_prop
+
+@[inherit_doc contDiff_alphaBeta1]
+theorem contDiff_alphaBeta2 (p : Params) : ContDiff ℝ ∞ (alphaBeta2 p) := by
+  unfold alphaBeta2; fun_prop
+
+@[inherit_doc contDiff_alphaBeta1]
+theorem contDiff_alphaBeta3 (p : Params) : ContDiff ℝ ∞ (alphaBeta3 p) := by
+  unfold alphaBeta3; fun_prop
+
+@[inherit_doc contDiff_alphaBeta1]
+theorem contDiff_alphaBeta4 (p : Params) : ContDiff ℝ ∞ (alphaBeta4 p) := by
+  unfold alphaBeta4; fun_prop
+
+@[inherit_doc contDiff_alphaBeta1]
+theorem contDiff_alphaBeta5 (p : Params) : ContDiff ℝ ∞ (alphaBeta5 p) := by
+  unfold alphaBeta5; fun_prop
+
+/-! ### The stage derivatives, endpoints included
+
+A nondegenerate closed interval has a unique tangent direction at each of its points,
+including its endpoints, so a `C¹` function agreeing there with a differentiable curve
+already has that curve's derivative at every point of the interval. -/
+
+/-- If the `C¹` glued path agrees on a nondegenerate closed interval with a curve whose
+derivative is `rot s (W s)`, then that is its derivative everywhere on the interval,
+endpoints included. -/
+theorem deriv_path_of_eqOn_Icc {p : Params} (hC1 : ContDiff ℝ 1 (path p))
+    {X W : ℝ → Point} {a b t : ℝ} (hab : a < b)
+    (hX : ∀ s, HasDerivAt X (rot s (W s)) s) (hXe : ∀ s ∈ Set.Icc a b, path p s = X s)
+    (ht : t ∈ Set.Icc a b) : deriv (path p) t = rot t (W t) := by
+  have h1 : HasDerivWithinAt (path p) (deriv (path p) t) (Set.Icc a b) t :=
+    ((hC1.differentiable one_ne_zero t).hasDerivAt).hasDerivWithinAt
+  have h2 : HasDerivWithinAt (path p) (rot t (W t)) (Set.Icc a b) t :=
+    ((hX t).hasDerivWithinAt).congr hXe (hXe t ht)
+  exact (uniqueDiffOn_Icc hab t ht).eq_deriv _ h1 h2
+
+/-- On the closed first stage the glued path has body-frame velocity `alphaBeta1`. -/
+theorem deriv_path_eq_rot_alphaBeta1 {p : Params} (hC1 : ContDiff ℝ 1 (path p))
+    (hphi : 0 < p.phi) {t : ℝ} (ht : t ∈ Set.Icc (0 : ℝ) p.phi) :
+    deriv (path p) t = rot t (alphaBeta1 p t) :=
+  deriv_path_of_eqOn_Icc hC1 hphi (hasDerivAt_path1 p)
+    (fun _ hs => path_eq_path1_of_mem_Icc p hs) ht
+
+/-- On the closed second stage the glued path has body-frame velocity `alphaBeta2`. -/
+theorem deriv_path_eq_rot_alphaBeta2 {p : Params} (hC1 : ContDiff ℝ 1 (path p))
+    (heq : Equations p) (hpt : p.phi < p.theta) {t : ℝ} (ht : t ∈ Set.Icc p.phi p.theta) :
+    deriv (path p) t = rot t (alphaBeta2 p t) :=
+  deriv_path_of_eqOn_Icc hC1 hpt (hasDerivAt_path2 p)
+    (fun _ hs => path_eq_path2_of_mem_Icc heq hs) ht
+
+/-- On the closed third stage the glued path has body-frame velocity `alphaBeta3`. -/
+theorem deriv_path_eq_rot_alphaBeta3 {p : Params} (hC1 : ContDiff ℝ 1 (path p))
+    (heq : Equations p) (hpt : p.phi < p.theta) (hq : p.theta < Real.pi / 4) {t : ℝ}
+    (ht : t ∈ Set.Icc p.theta (Real.pi / 2 - p.theta)) :
+    deriv (path p) t = rot t (alphaBeta3 p t) :=
+  deriv_path_of_eqOn_Icc hC1 (by linarith) (hasDerivAt_path3 p)
+    (fun _ hs => path_eq_path3_of_mem_Icc heq hpt hs) ht
+
+/-- On the closed fourth stage the glued path has body-frame velocity `alphaBeta4`. -/
+theorem deriv_path_eq_rot_alphaBeta4 {p : Params} (hC1 : ContDiff ℝ 1 (path p))
+    (heq : Equations p) (hpt : p.phi < p.theta) (hq : p.theta < Real.pi / 4) {t : ℝ}
+    (ht : t ∈ Set.Icc (Real.pi / 2 - p.theta) (Real.pi / 2 - p.phi)) :
+    deriv (path p) t = rot t (alphaBeta4 p t) :=
+  deriv_path_of_eqOn_Icc hC1 (by linarith) (hasDerivAt_path4 p)
+    (fun _ hs => path_eq_path4_of_mem_Icc heq hpt hq hs) ht
+
+/-- On the closed fifth stage the glued path has body-frame velocity `alphaBeta5`. -/
+theorem deriv_path_eq_rot_alphaBeta5 {p : Params} (hC1 : ContDiff ℝ 1 (path p))
+    (heq : Equations p) (hphi : 0 < p.phi) (hpt : p.phi < p.theta)
+    (hq : p.theta < Real.pi / 4) {t : ℝ}
+    (ht : t ∈ Set.Icc (Real.pi / 2 - p.phi) (Real.pi / 2)) :
+    deriv (path p) t = rot t (alphaBeta5 p t) :=
+  deriv_path_of_eqOn_Icc hC1 (by linarith) (hasDerivAt_path5 p)
+    (fun _ hs => path_eq_path5_of_mem_Icc heq hpt hq hs) ht
+
+end GerverSofa.Romik
+
+namespace MovingSofa
+
+open GerverSofa.Romik
+
+/-- The five explicit Romik path branches for a parameter tuple. -/
+def gerverDirectBranches (p : GerverSofa.Romik.Params) : Fin 5 → ℝ → GerverSofa.Point :=
+  ![GerverSofa.Romik.path1 p, GerverSofa.Romik.path2 p, GerverSofa.Romik.path3 p,
+    GerverSofa.Romik.path4 p, GerverSofa.Romik.path5 p]
+
+/-- The four switching times separating the explicit Romik path branches. -/
+def gerverDirectSwitches (p : GerverSofa.Romik.Params) : Fin 4 → ℝ :=
+  ![p.phi, p.theta, Real.pi / 2 - p.theta, Real.pi / 2 - p.phi]
+
+theorem gerver_direct_path_regularity (p : GerverSofa.Romik.Params)
+    (hp : p ∈ gerverDirectBox) (heq : gerverDirectEquations p) :
+    ContDiff ℝ 1 (GerverSofa.Romik.path p) ∧ GerverSofa.Romik.path p 0 = 0 ∧
+    ∀ i : Fin 4,
+      gerverDirectBranches p i.castSucc (gerverDirectSwitches p i) =
+        gerverDirectBranches p i.succ (gerverDirectSwitches p i) ∧
+      deriv (gerverDirectBranches p i.castSucc) (gerverDirectSwitches p i) =
+        deriv (gerverDirectBranches p i.succ) (gerverDirectSwitches p i) := by
+  -- The switching angles are strictly ordered by the reverse-domain bounds.
+  have hdom := gerver_reverse_physical_domain p hp
+  have hphi_pos : 0 < p.phi := hdom.1
+  have hphi_theta : p.phi < p.theta := hdom.2.1
+  have htheta_quarter : p.theta < Real.pi / 4 := hdom.2.2.1
+  have htheta_eta : p.theta < Real.pi / 2 - p.theta := by linarith
+  have heta_tau : Real.pi / 2 - p.theta < Real.pi / 2 - p.phi := by linarith
+  -- The body-frame velocity matches at the four switches.
+  have hw12 := GerverSofa.PartF.Phases.alphaBeta12_eq (p := p) heq
+  have hw23 := GerverSofa.PartF.Phases.alphaBeta23_eq (p := p) heq
+  have hw34 := alphaBeta34_eq (p := p) heq
+  have hw45 := alphaBeta45_eq (p := p) heq
+  -- The four positional matches, from the certified vendor extraction.
+  have hv12 := match_path12_of_equations (p := p) heq
+  have hv23 := match_path23_of_equations (p := p) heq
+  have hv34 := match_path34_of_equations (p := p) heq
+  have hv45 := match_path45_of_equations (p := p) heq
+  -- Glue the branches from the last switch backwards.
+  have hD45 : ∀ t : ℝ, HasDerivAt
+      (fun s => if s ≤ Real.pi / 2 - p.phi then path4 p s else path5 p s)
+      (if t ≤ Real.pi / 2 - p.phi then rot t (alphaBeta4 p t)
+        else rot t (alphaBeta5 p t)) t :=
+    hasDerivAt_if_le (hasDerivAt_path4 p) (hasDerivAt_path5 p) hv45
+      (congrArg (rot (Real.pi / 2 - p.phi)) hw45)
+  have hC45 : Continuous fun t : ℝ =>
+      if t ≤ Real.pi / 2 - p.phi then rot t (alphaBeta4 p t) else rot t (alphaBeta5 p t) :=
+    (continuous_rot_alphaBeta4 p).if_le (continuous_rot_alphaBeta5 p)
+      continuous_id continuous_const (by
+        intro t ht
+        subst ht
+        exact congrArg (rot (Real.pi / 2 - p.phi)) hw45)
+  have hD345 : ∀ t : ℝ, HasDerivAt
+      (fun s => if s ≤ Real.pi / 2 - p.theta then path3 p s
+        else if s ≤ Real.pi / 2 - p.phi then path4 p s else path5 p s)
+      (if t ≤ Real.pi / 2 - p.theta then rot t (alphaBeta3 p t)
+        else if t ≤ Real.pi / 2 - p.phi then rot t (alphaBeta4 p t)
+        else rot t (alphaBeta5 p t)) t :=
+    hasDerivAt_if_le (hasDerivAt_path3 p) hD45
+      (by rw [ite_eq_left heta_tau.le]; exact hv34)
+      (by rw [ite_eq_left heta_tau.le]
+          exact congrArg (rot (Real.pi / 2 - p.theta)) hw34)
+  have hC345 : Continuous fun t : ℝ =>
+      if t ≤ Real.pi / 2 - p.theta then rot t (alphaBeta3 p t)
+      else if t ≤ Real.pi / 2 - p.phi then rot t (alphaBeta4 p t)
+      else rot t (alphaBeta5 p t) :=
+    (continuous_rot_alphaBeta3 p).if_le hC45 continuous_id continuous_const (by
+      intro t ht
+      subst ht
+      rw [ite_eq_left heta_tau.le]
+      exact congrArg (rot (Real.pi / 2 - p.theta)) hw34)
+  have hD2345 : ∀ t : ℝ, HasDerivAt
+      (fun s => if s ≤ p.theta then path2 p s
+        else if s ≤ Real.pi / 2 - p.theta then path3 p s
+        else if s ≤ Real.pi / 2 - p.phi then path4 p s else path5 p s)
+      (if t ≤ p.theta then rot t (alphaBeta2 p t)
+        else if t ≤ Real.pi / 2 - p.theta then rot t (alphaBeta3 p t)
+        else if t ≤ Real.pi / 2 - p.phi then rot t (alphaBeta4 p t)
+        else rot t (alphaBeta5 p t)) t :=
+    hasDerivAt_if_le (hasDerivAt_path2 p) hD345
+      (by rw [ite_eq_left htheta_eta.le]; exact hv23)
+      (by rw [ite_eq_left htheta_eta.le]
+          exact congrArg (rot p.theta) hw23)
+  have hC2345 : Continuous fun t : ℝ =>
+      if t ≤ p.theta then rot t (alphaBeta2 p t)
+      else if t ≤ Real.pi / 2 - p.theta then rot t (alphaBeta3 p t)
+      else if t ≤ Real.pi / 2 - p.phi then rot t (alphaBeta4 p t)
+      else rot t (alphaBeta5 p t) :=
+    (continuous_rot_alphaBeta2 p).if_le hC345 continuous_id continuous_const (by
+      intro t ht
+      subst ht
+      rw [ite_eq_left htheta_eta.le]
+      exact congrArg (rot p.theta) hw23)
+  have hDpath : ∀ t : ℝ, HasDerivAt (path p)
+      (if t ≤ p.phi then rot t (alphaBeta1 p t)
+        else if t ≤ p.theta then rot t (alphaBeta2 p t)
+        else if t ≤ Real.pi / 2 - p.theta then rot t (alphaBeta3 p t)
+        else if t ≤ Real.pi / 2 - p.phi then rot t (alphaBeta4 p t)
+        else rot t (alphaBeta5 p t)) t :=
+    hasDerivAt_if_le (hasDerivAt_path1 p) hD2345
+      (by rw [ite_eq_left hphi_theta.le]; exact hv12)
+      (by rw [ite_eq_left hphi_theta.le]
+          exact congrArg (rot p.phi) hw12)
+  have hCpath : Continuous fun t : ℝ =>
+      if t ≤ p.phi then rot t (alphaBeta1 p t)
+      else if t ≤ p.theta then rot t (alphaBeta2 p t)
+      else if t ≤ Real.pi / 2 - p.theta then rot t (alphaBeta3 p t)
+      else if t ≤ Real.pi / 2 - p.phi then rot t (alphaBeta4 p t)
+      else rot t (alphaBeta5 p t) :=
+    (continuous_rot_alphaBeta1 p).if_le hC2345 continuous_id continuous_const (by
+      intro t ht
+      subst ht
+      rw [ite_eq_left hphi_theta.le]
+      exact congrArg (rot p.phi) hw12)
+  -- The four derivative matches.
+  have hdv12 : deriv (path1 p) p.phi = deriv (path2 p) p.phi := by
+    rw [(hasDerivAt_path1 p p.phi).deriv, (hasDerivAt_path2 p p.phi).deriv, hw12]
+  have hdv23 : deriv (path2 p) p.theta = deriv (path3 p) p.theta := by
+    rw [(hasDerivAt_path2 p p.theta).deriv, (hasDerivAt_path3 p p.theta).deriv, hw23]
+  have hdv34 : deriv (path3 p) (Real.pi / 2 - p.theta) =
+      deriv (path4 p) (Real.pi / 2 - p.theta) := by
+    rw [(hasDerivAt_path3 p _).deriv, (hasDerivAt_path4 p _).deriv, hw34]
+  have hdv45 : deriv (path4 p) (Real.pi / 2 - p.phi) =
+      deriv (path5 p) (Real.pi / 2 - p.phi) := by
+    rw [(hasDerivAt_path4 p _).deriv, (hasDerivAt_path5 p _).deriv, hw45]
+  refine ⟨contDiff_one_of_hasDerivAt hDpath hCpath, ?_, ?_⟩
+  · -- `0 < φ` selects the first branch, and the normalizations make it vanish at `0`.
+    have hk11 := k11_eq_one_sub_a1_of_equations (p := p) heq
+    have hk12 := k12_eq_quarter_of_equations (p := p) heq
+    have ha2 := a2_eq_neg_quarter_of_equations (p := p) heq
+    have h0 : path p 0 = path1 p 0 := by
+      unfold GerverSofa.Romik.path
+      exact ite_eq_left hphi_pos.le
+    rw [h0]
+    simp only [path1, addK, rot, Real.cos_zero, Real.sin_zero, Prod.ext_iff, Prod.fst_zero,
+      Prod.snd_zero]
+    constructor <;> linarith
+  · intro i
+    fin_cases i
+    · exact ⟨hv12, hdv12⟩
+    · exact ⟨hv23, hdv23⟩
+    · exact ⟨hv34, hdv34⟩
+    · exact ⟨hv45, hdv45⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver / Literal Sets
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The outer cap defined by all upper path support constraints and the horizontal base. -/
+def gerverOuterCap : Set Point :=
+  {q | 0 ≤ q 1 ∧ ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+    inner ℝ q (normalVector (t : Real.Angle)) ≤
+      inner ℝ (paperGerverPath t) (normalVector (t : Real.Angle)) + 1 ∧
+    inner ℝ q (tangentVector (t : Real.Angle)) ≤
+      inner ℝ (paperGerverPath t) (tangentVector (t : Real.Angle)) + 1}
+
+/-- The union of strict forbidden inner corners above the horizontal base along the Gerver
+path. -/
+def gerverLiteralNiche : Set Point :=
+  {q | 0 ≤ q 1 ∧ ∃ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+    inner ℝ (q - paperGerverPath t) (normalVector (t : Real.Angle)) < 0 ∧
+    inner ℝ (q - paperGerverPath t) (tangentVector (t : Real.Angle)) < 0}
+
+/-- The literal Gerver sofa obtained by removing its niche from its outer cap. -/
+def gerverLiteralSofa : Set Point := gerverOuterCap \ gerverLiteralNiche
+
+/-- Bundle the literal niche and sofa sets for comparison with the canonical definitions. -/
+def gerverLiteralSets : Set Point × Set Point := (gerverLiteralNiche, gerverLiteralSofa)
+
+/-- The paper outer cap is the coordinate transport of the certified Romik cap `K₀`. -/
+theorem mem_gerverOuterCap_iff (q : Point) :
+    q ∈ gerverOuterCap ↔
+      GerverSofa.PartF.Coordinates.fromPlane q ∈
+        GerverSofa.Romik.K0 GerverSofa.PartB.params := by
+  simp only [gerverOuterCap, Set.mem_ofPred_eq, GerverSofa.Romik.mem_K0, Set.mem_inter_iff,
+    GerverSofa.Romik.mem_supportHalfU, GerverSofa.Romik.mem_supportHalfV,
+    inner_normalVector_eq_dot, inner_tangentVector_eq_dot]
+  exact Iff.rfl
+
+/-- A point whose coordinate pair lies in the certified Romik cap `K₀` lies in the paper
+outer cap. -/
+theorem toPlane_mem_gerverOuterCap {x : GerverSofa.Point}
+    (hx : x ∈ GerverSofa.Romik.K0 GerverSofa.PartB.params) :
+    GerverSofa.PartF.Coordinates.toPlane x ∈ gerverOuterCap :=
+  (mem_gerverOuterCap_iff _).2 (by rwa [GerverSofa.PartF.Coordinates.fromPlane_toPlane])
+
+/-- The paper literal niche is the coordinate transport of the certified Romik niche. -/
+theorem mem_gerverLiteralNiche_iff (q : Point) :
+    q ∈ gerverLiteralNiche ↔
+      GerverSofa.PartF.Coordinates.fromPlane q ∈
+        GerverSofa.Romik.niche GerverSofa.PartB.params := by
+  have hsub : ∀ (r : Point) (t : ℝ),
+      GerverSofa.PartF.Coordinates.fromPlane (r - paperGerverPath t) =
+        ((GerverSofa.PartF.Coordinates.fromPlane r).1 -
+            (GerverSofa.Romik.path GerverSofa.PartB.params t).1,
+          (GerverSofa.PartF.Coordinates.fromPlane r).2 -
+            (GerverSofa.Romik.path GerverSofa.PartB.params t).2) := fun _ _ => rfl
+  constructor
+  · rintro ⟨hy, t, ht, h1, h2⟩
+    rw [inner_normalVector_eq_dot, hsub] at h1
+    rw [inner_tangentVector_eq_dot, hsub] at h2
+    exact ⟨hy, t, ht, h1, h2⟩
+  · rintro ⟨hy, t, ht, h1, h2⟩
+    refine ⟨hy, t, ht, ?_, ?_⟩
+    · rw [inner_normalVector_eq_dot, hsub]; exact h1
+    · rw [inner_tangentVector_eq_dot, hsub]; exact h2
+
+theorem paperGerverSofa_eq_literal : paperGerverSofa = gerverLiteralSofa := by
+  have hTpos : (0 : ℝ) < Real.pi / 2 := by positivity
+  have hmemhall : ∀ (t : Real.Angle) (v q : Point),
+      q ∈ (fun p ↦ rotationMap t p + v) '' hallway ↔
+        (inner ℝ (q - v) (normalVector t) ≤ 1 ∧
+          inner ℝ (q - v) (tangentVector t) ≤ 1) ∧
+        (0 ≤ inner ℝ (q - v) (normalVector t) ∨
+          0 ≤ inner ℝ (q - v) (tangentVector t)) := by
+    intro t v q
+    constructor
+    · rintro ⟨p, hp, rfl⟩
+      have h0 : inner ℝ (rotationMap t p + v - v) (normalVector t) = p 0 := by
+        rw [add_sub_cancel_right, inner_rotationMap_normalVector]
+      have h1 : inner ℝ (rotationMap t p + v - v) (tangentVector t) = p 1 := by
+        rw [add_sub_cancel_right, inner_rotationMap_tangentVector]
+      simp only [h0, h1]
+      exact (mem_hallway_iff p).1 hp
+    · intro h
+      obtain ⟨p, hp⟩ := (EuclideanGeometry.o.rotation t).surjective (q - v)
+      have hpq : rotationMap t p = q - v := hp
+      have h0 : p 0 = inner ℝ (q - v) (normalVector t) := by
+        rw [← inner_rotationMap_normalVector p t, hpq]
+      have h1 : p 1 = inner ℝ (q - v) (tangentVector t) := by
+        rw [← inner_rotationMap_tangentVector p t, hpq]
+      refine ⟨p, (mem_hallway_iff p).2 ?_, ?_⟩
+      · rw [h0, h1]; exact h
+      · change rotationMap t p + v = q
+        rw [hpq]; abel
+  have hconttan : Continuous (fun t : ℝ ↦ tangentVector (t : Real.Angle)) := by
+    let c : ℝ → (i : Fin 2) → ℝ :=
+      fun t i ↦ Fin.cases (-Real.sin t) (fun _ ↦ Real.cos t) i
+    have hc : Continuous c := by
+      apply continuous_pi
+      intro i
+      fin_cases i
+      · exact Real.continuous_sin.neg
+      · exact Real.continuous_cos
+    have heq : (fun t : ℝ ↦ tangentVector (t : Real.Angle)) =
+        (fun t ↦ WithLp.toLp 2 (c t)) := by
+      funext t
+      ext i
+      fin_cases i <;> rfl
+    rw [heq]
+    exact (PiLp.continuous_toLp (2 : ENNReal) (fun _ : Fin 2 ↦ ℝ)).comp hc
+  have hreg := gerver_direct_path_regularity GerverSofa.PartB.params
+    GerverSofa.PartB.params_mem GerverSofa.PartB.params_equations
+  have hpathcont : Continuous paperGerverPath :=
+    GerverSofa.PartF.Coordinates.continuous_toPlane.comp hreg.1.continuous
+  have hpath0 : paperGerverPath 0 = 0 := by
+    change GerverSofa.PartF.Coordinates.toPlane
+      (GerverSofa.Romik.path GerverSofa.PartB.params 0) = 0
+    rw [hreg.2.1]
+    ext i
+    fin_cases i <;> rfl
+  have hstrip : ∀ q : Point,
+      q ∈ (strips (Real.pi / 2)).1 ∩ (strips (Real.pi / 2)).2.2 ↔ 0 ≤ q 1 ∧ q 1 ≤ 1 := by
+    intro q
+    have h := mem_stripParallelogram_iff (Real.pi / 2) q
+    rw [inner_normalVector_pi_div_two] at h
+    have hset : (strips (Real.pi / 2)).1 ∩ (strips (Real.pi / 2)).2.2 =
+        (stripParallelogram (Real.pi / 2)).1 := rfl
+    rw [hset, h]
+    tauto
+  have hshift : ∀ w z n : Point,
+      (inner ℝ w n ≤ inner ℝ z n + 1 ↔ inner ℝ (w - z) n ≤ 1) := by
+    intro w z n
+    rw [inner_sub_left]
+    constructor <;> intro h <;> linarith
+  ext q
+  constructor
+  · rintro ⟨hs, hint⟩
+    have hhall : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+        (inner ℝ (q - paperGerverPath t) (normalVector (t : Real.Angle)) ≤ 1 ∧
+          inner ℝ (q - paperGerverPath t) (tangentVector (t : Real.Angle)) ≤ 1) ∧
+        (0 ≤ inner ℝ (q - paperGerverPath t) (normalVector (t : Real.Angle)) ∨
+          0 ≤ inner ℝ (q - paperGerverPath t) (tangentVector (t : Real.Angle))) := fun t ht ↦
+      (hmemhall _ _ _).1 (Set.mem_iInter₂.1 hint t ht)
+    refine ⟨⟨((hstrip q).1 hs).1, fun t ht ↦
+      ⟨(hshift _ _ _).2 (hhall t ht).1.1, (hshift _ _ _).2 (hhall t ht).1.2⟩⟩, ?_⟩
+    rintro ⟨-, t, ht, hlt1, hlt2⟩
+    rcases (hhall t (Set.Ioo_subset_Icc_self ht)).2 with h | h
+    · linarith
+    · linarith
+  · rintro ⟨⟨hy, hcap⟩, hniche⟩
+    have hcap' : ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+        inner ℝ (q - paperGerverPath t) (normalVector (t : Real.Angle)) ≤ 1 ∧
+        inner ℝ (q - paperGerverPath t) (tangentVector (t : Real.Angle)) ≤ 1 := fun t ht ↦
+      ⟨(hshift _ _ _).1 (hcap t ht).1, (hshift _ _ _).1 (hcap t ht).2⟩
+    have hopen : ∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+        0 ≤ inner ℝ (q - paperGerverPath t) (normalVector (t : Real.Angle)) ∨
+        0 ≤ inner ℝ (q - paperGerverPath t) (tangentVector (t : Real.Angle)) := by
+      intro t ht
+      by_contra hcon
+      exact hniche ⟨hy, t, ht, not_le.mp fun h ↦ hcon (Or.inl h),
+        not_le.mp fun h ↦ hcon (Or.inr h)⟩
+    have hclosed : IsClosed {t : ℝ |
+        0 ≤ max (inner ℝ (q - paperGerverPath t) (normalVector (t : Real.Angle)))
+          (inner ℝ (q - paperGerverPath t) (tangentVector (t : Real.Angle)))} :=
+      isClosed_le continuous_const
+        (((continuous_const.sub hpathcont).inner continuous_normalVector_real).max
+          ((continuous_const.sub hpathcont).inner hconttan))
+    have hIcc : Set.Icc (0 : ℝ) (Real.pi / 2) ⊆ {t : ℝ |
+        0 ≤ max (inner ℝ (q - paperGerverPath t) (normalVector (t : Real.Angle)))
+          (inner ℝ (q - paperGerverPath t) (tangentVector (t : Real.Angle)))} := by
+      rw [← closure_Ioo (ne_of_lt hTpos)]
+      exact hclosed.closure_subset_iff.mpr fun t ht ↦ le_max_iff.mpr (hopen t ht)
+    have hy1 : q 1 ≤ 1 := by
+      have h := (hcap' 0 (Set.left_mem_Icc.mpr hTpos.le)).2
+      rwa [hpath0, sub_zero, inner_tangentVector_zero] at h
+    refine ⟨(hstrip q).2 ⟨hy, hy1⟩, Set.mem_iInter₂.2 fun t ht ↦ ?_⟩
+    exact (hmemhall _ _ _).2
+      ⟨hcap' t ht, le_max_iff.mp (hIcc ht)⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver / Literal Connected
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem gerver_literal_connected : IsConnected gerverLiteralSofa := by
+  have hu : ∀ (q : Point) (t : ℝ),
+      inner ℝ q (normalVector (t : Real.Angle)) =
+        GerverSofa.dot (GerverSofa.PartF.Coordinates.fromPlane q) (GerverSofa.u t) := by
+    intro q t
+    simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two, GerverSofa.dot,
+      GerverSofa.u, GerverSofa.PartF.Coordinates.fromPlane, mul_comm]
+  have hv : ∀ (q : Point) (t : ℝ),
+      inner ℝ q (tangentVector (t : Real.Angle)) =
+        GerverSofa.dot (GerverSofa.PartF.Coordinates.fromPlane q) (GerverSofa.v t) := by
+    intro q t
+    simp [tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two, GerverSofa.dot,
+      GerverSofa.v, GerverSofa.PartF.Coordinates.fromPlane, mul_comm]
+  have hpath : ∀ t : ℝ,
+      GerverSofa.PartF.Coordinates.fromPlane (paperGerverPath t) =
+        GerverSofa.Romik.path GerverSofa.PartC.params t := fun _ => rfl
+  have hcap : ∀ q : Point,
+      q ∈ gerverOuterCap ↔
+        GerverSofa.PartF.Coordinates.fromPlane q ∈
+          GerverSofa.Romik.K0 GerverSofa.PartC.params := by
+    intro q
+    constructor
+    · rintro ⟨hy, hs⟩
+      refine ⟨hy, fun t ht => ?_⟩
+      obtain ⟨h1, h2⟩ := hs t ht
+      rw [hu, hu, hpath] at h1
+      rw [hv, hv, hpath] at h2
+      exact ⟨h1, h2⟩
+    · rintro ⟨hy, hs⟩
+      refine ⟨hy, fun t ht => ?_⟩
+      obtain ⟨h1, h2⟩ := hs t ht
+      rw [hu, hu, hpath]
+      rw [hv, hv, hpath]
+      exact ⟨h1, h2⟩
+  have hsub : ∀ (q : Point) (t : ℝ),
+      GerverSofa.PartF.Coordinates.fromPlane (q - paperGerverPath t) =
+        ((GerverSofa.PartF.Coordinates.fromPlane q).1 -
+            (GerverSofa.Romik.path GerverSofa.PartC.params t).1,
+          (GerverSofa.PartF.Coordinates.fromPlane q).2 -
+            (GerverSofa.Romik.path GerverSofa.PartC.params t).2) := fun _ _ => rfl
+  have hniche : ∀ q : Point,
+      q ∈ gerverLiteralNiche ↔
+        GerverSofa.PartF.Coordinates.fromPlane q ∈
+          GerverSofa.Romik.niche GerverSofa.PartC.params := by
+    intro q
+    constructor
+    · rintro ⟨hy, t, ht, h1, h2⟩
+      rw [hu, hsub] at h1
+      rw [hv, hsub] at h2
+      exact ⟨hy, t, ht, h1, h2⟩
+    · rintro ⟨hy, t, ht, h1, h2⟩
+      refine ⟨hy, t, ht, ?_, ?_⟩
+      · rw [hu, hsub]; exact h1
+      · rw [hv, hsub]; exact h2
+  have hset : gerverLiteralSofa =
+      GerverSofa.PartF.Coordinates.toPlane '' GerverSofa.PartC.G := by
+    rw [GerverSofa.PartF.Coordinates.image_eq_preimage]
+    ext q
+    exact and_congr (hcap q) (not_congr (hniche q))
+  rw [hset]
+  exact GerverSofa.PartF.Coordinates.isConnected_image
+    GerverSofa.PartC.Stage4.G_connected_direct
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver / Parameter Identification
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open GerverSofa.PartF.ProjectAdapter (selected)
+
+/-- The adapter's selected reverse parameter vector carries the paper's angle `φ` in its
+`phi` field.  This is true by definition — `selected` is built from the very quadruple that
+defines `GerversSofa.φ` — but it is the only bridge from the vendor parameter vector to the
+paper's stage times `gerverStageTimes`, so it is recorded as a named lemma rather than left
+to an invisible unfolding at each use site. -/
+theorem selected_phi : selected.phi = GerversSofa.φ := rfl
+
+/-- The adapter's selected reverse parameter vector carries the paper's angle `θ` in its
+`theta` field; see `selected_phi` for why this `rfl` is worth naming. -/
+theorem selected_theta : selected.theta = GerversSofa.θ := rfl
+
+theorem gerver_parameter_identification :
+    (∀ p : GerverSofa.Romik.Params, p ∈ gerverDirectBox → gerverDirectEquations p →
+      p = gerverParameterDictionary.1 selected ∧
+      p.phi = selected.phi ∧ p.theta = selected.theta ∧
+      0 < selected.phi ∧ selected.phi < selected.theta ∧ selected.theta < Real.pi / 4 ∧
+      (39 : ℝ) / 1000 ≤ selected.phi ∧ selected.phi ≤ (40 : ℝ) / 1000 ∧
+      GerversSofa.ABφθSpec selected.a selected.b selected.phi selected.theta) ∧
+    (∃ p : GerverSofa.Romik.Params, p ∈ gerverDirectBox ∧ gerverDirectEquations p ∧
+      p = gerverParameterDictionary.1 selected) := by
+  -- The reverse parameters of any direct solution in the box satisfy the whole
+  -- reduced specification, whose unique solution is the selected quadruple.
+  have key : ∀ p : GerverSofa.Romik.Params, p ∈ gerverDirectBox → gerverDirectEquations p →
+      gerverParameterDictionary.2 p = selected := by
+    intro p hp heq
+    rw [GerverSofa.PartF.ProjectAdapter.selected_eq_certified]
+    exact GerverSofa.PartF.Parameters.undictionary_eq_reduced_certified hp heq
+  -- Reconstruction then recovers the direct vector from its reverse parameters.
+  have hdict : ∀ p : GerverSofa.Romik.Params, p ∈ gerverDirectBox → gerverDirectEquations p →
+      p = gerverParameterDictionary.1 selected := by
+    intro p hp heq
+    calc p = GerverSofa.PartF.Phases.dictionary (gerverParameterDictionary.2 p) :=
+          (GerverSofa.PartF.Phases.dictionary_undictionary_of_equations heq).symm
+      _ = gerverParameterDictionary.1 selected :=
+          congrArg GerverSofa.PartF.Phases.dictionary (key p hp heq)
+  have hspec : GerversSofa.ABφθSpec selected.a selected.b selected.phi selected.theta :=
+    GerversSofa.ABφθSpec.existsUnique.choose_spec.1
+  refine ⟨fun p hp heq => ?_, GerverSofa.PartB.params, GerverSofa.PartB.params_mem,
+    GerverSofa.PartB.params_equations,
+    hdict _ GerverSofa.PartB.params_mem GerverSofa.PartB.params_equations⟩
+  obtain ⟨hphipos, hphitheta, htheta, -, -, hphilo, hphihi⟩ :=
+    gerver_reverse_physical_domain p hp
+  have hu := key p hp heq
+  rw [hu] at hphipos hphitheta htheta hphilo hphihi
+  exact ⟨hdict p hp heq, congrArg GerverSofa.Reduced.Params.phi hu,
+    congrArg GerverSofa.Reduced.Params.theta hu, hphipos, hphitheta, htheta, hphilo, hphihi,
+    hspec⟩
+
+/-- The two distinguished Gerver angles are interior and complementary. -/
+theorem paperGerverConstants_snd_mem_Ioo :
+    paperGerverConstants.2.1 ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) ∧
+      paperGerverConstants.2.2 ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) ∧
+      paperGerverConstants.2.1 + paperGerverConstants.2.2 = Real.pi / 2 := by
+  have hpos : 0 < GerversSofa.φ := by
+    obtain ⟨hall, q, hqbox, hqeq, -⟩ := gerver_parameter_identification
+    exact (hall q hqbox hqeq).2.2.2.1
+  have hle : GerversSofa.φ ≤ Real.pi / 4 := by
+    have h := GerversSofa.ABφθSpec.existsUnique.choose_spec.1
+    exact le_trans h.2.1 h.2.2.1
+  have hpi := Real.pi_pos
+  have h1 : paperGerverConstants.2.1 = GerversSofa.φ := rfl
+  have h2 : paperGerverConstants.2.2 = Real.pi / 2 - GerversSofa.φ := rfl
+  rw [h1, h2]
+  refine ⟨⟨hpos, by linarith only [hle, hpi]⟩,
+    ⟨by linarith only [hle, hpi], by linarith only [hpos]⟩, by ring⟩
+
+/-- The two distinguished angles are ordered: the certified bound `φ ≤ 1/25` puts `φ` well below
+`π/4 = π/2 - π/4`. -/
+theorem paperGerverConstants_snd_fst_lt_snd_snd :
+    paperGerverConstants.2.1 < paperGerverConstants.2.2 := by
+  obtain ⟨-, -, hsum⟩ := paperGerverConstants_snd_mem_Ioo
+  have hphi : paperGerverConstants.2.1 ≤ (40 : ℝ) / 1000 := by
+    obtain ⟨hall, p, hpbox, hpeq, -⟩ := gerver_parameter_identification
+    exact (hall p hpbox hpeq).2.2.2.2.2.2.2.1
+  have := Real.pi_gt_three
+  linarith
+
+/-! ### The stage times as certified angles
+
+`gerverStageTimes` is defined from the paper angles `GerversSofa.φ` and `GerversSofa.θ`,
+while the certified Part C geometry is indexed by the vendor parameter fields
+`params.phi`, `params.theta` and the derived angles `eta`, `tau`, `T`.  The six lemmas
+below are the dictionary between the two indexings; they are the only place where
+`gerver_parameter_identification` is needed to see a stage time. -/
+
+/-- The zeroth stage time is the start of the rotation interval. -/
+theorem gerverStageTimes_zero : gerverStageTimes 0 = 0 := rfl
+
+/-- The first stage time is the certified first switching angle `params.phi`. -/
+theorem gerverStageTimes_one : gerverStageTimes 1 = GerverSofa.PartC.params.phi := by
+  obtain ⟨-, hphi, -⟩ := gerver_parameter_identification.1 GerverSofa.PartB.params
+    GerverSofa.PartB.params_mem GerverSofa.PartB.params_equations
+  simp [gerverStageTimes, hphi.trans selected_phi]
+
+/-- The second stage time is the certified second switching angle `params.theta`. -/
+theorem gerverStageTimes_two : gerverStageTimes 2 = GerverSofa.PartC.params.theta := by
+  obtain ⟨-, -, htheta, -⟩ := gerver_parameter_identification.1 GerverSofa.PartB.params
+    GerverSofa.PartB.params_mem GerverSofa.PartB.params_equations
+  simp [gerverStageTimes, htheta.trans selected_theta]
+
+/-- The third stage time is the certified reflected angle `eta = T - params.theta`. -/
+theorem gerverStageTimes_three : gerverStageTimes 3 = GerverSofa.PartC.eta := by
+  have h := gerverStageTimes_two
+  simp only [gerverStageTimes, Matrix.cons_val] at h ⊢
+  rw [GerverSofa.PartC.eta, GerverSofa.PartC.T, ← h]
+
+/-- The fourth stage time is the certified reflected angle `tau = T - params.phi`. -/
+theorem gerverStageTimes_four : gerverStageTimes 4 = GerverSofa.PartC.tau := by
+  have h := gerverStageTimes_one
+  simp only [gerverStageTimes, Matrix.cons_val] at h ⊢
+  rw [GerverSofa.PartC.tau, GerverSofa.PartC.T, ← h]
+
+/-- The fifth stage time is the certified end `T` of the rotation interval. -/
+theorem gerverStageTimes_five : gerverStageTimes 5 = GerverSofa.PartC.T := rfl
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver / Contacts
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped ContDiff
+
+namespace MovingSofa
+
+/-- Resolve the Gerver path derivative into its normal and tangent frame components. -/
+def paperGerverVelocityComponents (t : ℝ) : ℝ × ℝ :=
+  (inner ℝ (deriv paperGerverPath t) (normalVector (t : Real.Angle)),
+    inner ℝ (deriv paperGerverPath t) (tangentVector (t : Real.Angle)))
+
+/-- The four standard support-contact points determined by the Gerver path and its derivative. -/
+def paperGerverContacts (t : ℝ) : Fin 4 → Point :=
+  ![paperGerverPath t + (paperGerverVelocityComponents t).1 • tangentVector (t : Real.Angle) +
+      normalVector (t : Real.Angle),
+    paperGerverPath t + (paperGerverVelocityComponents t).1 • tangentVector (t : Real.Angle),
+    paperGerverPath t - (paperGerverVelocityComponents t).2 • normalVector (t : Real.Angle) +
+      tangentVector (t : Real.Angle),
+    paperGerverPath t - (paperGerverVelocityComponents t).2 • normalVector (t : Real.Angle)]
+
+/-- Bundle the path velocity components and the four contact curves. -/
+def paperGerverContactData : (ℝ → ℝ × ℝ) × (ℝ → Fin 4 → Point) :=
+  (paperGerverVelocityComponents, paperGerverContacts)
+
+/-- The explicit body-frame velocity coefficients on a selected Romik branch. -/
+def gerverBranchVelocityComponents (i : Fin 5) (t : ℝ) : ℝ × ℝ :=
+  ![GerverSofa.Romik.alphaBeta1 GerverSofa.PartB.params t,
+    GerverSofa.Romik.alphaBeta2 GerverSofa.PartB.params t,
+    GerverSofa.Romik.alphaBeta3 GerverSofa.PartB.params t,
+    GerverSofa.Romik.alphaBeta4 GerverSofa.PartB.params t,
+    GerverSofa.Romik.alphaBeta5 GerverSofa.PartB.params t] i
+
+/-! ### The common shape of the four contact curves
+
+`paperGerverContacts` is an instance of a generic four-slot shape built from a base curve
+and two scalar coefficient functions.  Continuity and smoothness of the shape are proved
+once here and reused for the ambient curve and for each analytic stage branch. -/
+
+/-- The four contact curves are continuous as soon as the base curve and the two
+coefficient functions are. -/
+private theorem continuous_contactShape (X : ℝ → Point) (A B : ℝ → ℝ) (hX : Continuous X)
+    (hA : Continuous A) (hB : Continuous B) :
+    Continuous fun t : ℝ => (![
+      X t + A t • tangentVector (t : Real.Angle) + normalVector (t : Real.Angle),
+      X t + A t • tangentVector (t : Real.Angle),
+      X t - B t • normalVector (t : Real.Angle) + tangentVector (t : Real.Angle),
+      X t - B t • normalVector (t : Real.Angle)] : Fin 4 → Point) := by
+  have hN : Continuous fun t : ℝ => normalVector (t : Real.Angle) :=
+    contDiff_normalVector.continuous
+  have hT : Continuous fun t : ℝ => tangentVector (t : Real.Angle) :=
+    contDiff_tangentVector.continuous
+  refine continuous_pi fun i => ?_
+  fin_cases i
+  · exact (hX.add (hA.smul hT)).add hN
+  · exact hX.add (hA.smul hT)
+  · exact (hX.sub (hB.smul hN)).add hT
+  · exact hX.sub (hB.smul hN)
+
+/-- The four contact curves are smooth as soon as the base curve and the two coefficient
+functions are. -/
+private theorem contDiff_contactShape (X : ℝ → Point) (A B : ℝ → ℝ)
+    (hX : ContDiff ℝ ∞ X) (hA : ContDiff ℝ ∞ A) (hB : ContDiff ℝ ∞ B) :
+    ContDiff ℝ ∞ fun t : ℝ => (![
+      X t + A t • tangentVector (t : Real.Angle) + normalVector (t : Real.Angle),
+      X t + A t • tangentVector (t : Real.Angle),
+      X t - B t • normalVector (t : Real.Angle) + tangentVector (t : Real.Angle),
+      X t - B t • normalVector (t : Real.Angle)] : Fin 4 → Point) := by
+  rw [contDiff_pi]
+  intro i
+  fin_cases i
+  · exact (hX.add (hA.smul contDiff_tangentVector)).add contDiff_normalVector
+  · exact hX.add (hA.smul contDiff_tangentVector)
+  · exact (hX.sub (hB.smul contDiff_normalVector)).add contDiff_tangentVector
+  · exact hX.sub (hB.smul contDiff_normalVector)
+
+open GerverSofa.Romik GerverSofa.PartF.Coordinates in
+theorem paperGerverContactData_properties :
+    Continuous paperGerverVelocityComponents ∧ Continuous paperGerverContacts ∧
+    (∀ i : Fin 5, ContDiffOn ℝ ∞ paperGerverContacts (gerverStageIntervals i)) ∧
+    (∀ i : Fin 5, ∀ t ∈ gerverStageIntervals i,
+      paperGerverVelocityComponents t = gerverBranchVelocityComponents i t) := by
+  -- Abbreviate the certified direct parameter vector, and order its switching angles.
+  obtain ⟨p, hp⟩ : ∃ p : Params, GerverSofa.PartB.params = p := ⟨_, rfl⟩
+  have hmem : p ∈ gerverDirectBox := hp ▸ GerverSofa.PartB.params_mem
+  have heqs : gerverDirectEquations p := hp ▸ GerverSofa.PartB.params_equations
+  obtain ⟨hC1, -, -⟩ := gerver_direct_path_regularity p hmem heqs
+  obtain ⟨-, hphi, htheta, hphipos, hphitheta, hthetalt, -⟩ :=
+    gerver_parameter_identification.1 p hmem heqs
+  have hphi' : p.phi = GerversSofa.φ := hphi.trans selected_phi
+  have htheta' : p.theta = GerversSofa.θ := htheta.trans selected_theta
+  have h0 : (0 : ℝ) < p.phi := by rw [hphi]; exact hphipos
+  have h1 : p.phi < p.theta := by rw [hphi, htheta]; exact hphitheta
+  have h2 : p.theta < Real.pi / 4 := by rw [htheta]; exact hthetalt
+  have hC1' : ContDiff ℝ 1 (path GerverSofa.PartB.params) := by rw [hp]; exact hC1
+  -- The velocity components read off the body-frame derivative of the direct path.
+  have hvel : ∀ (w : GerverSofa.Point) (t : ℝ), deriv (path p) t = rot t w →
+      paperGerverVelocityComponents t = w := by
+    intro w t hw
+    have hd : deriv paperGerverPath t = toPlane (rot t w) := by
+      rw [deriv_paperGerverPath hC1' t, hp, hw]
+    simp only [paperGerverVelocityComponents, hd, inner_toPlane_rot_normalVector,
+      inner_toPlane_rot_tangentVector]
+  have hvelC : Continuous paperGerverVelocityComponents :=
+    ((continuous_deriv_paperGerverPath hC1').inner contDiff_normalVector.continuous).prodMk
+      ((continuous_deriv_paperGerverPath hC1').inner contDiff_tangentVector.continuous)
+  -- Stage smoothness: the contacts agree there with the smooth analytic branch data.
+  have hstage : ∀ (X W : ℝ → GerverSofa.Point) (s : Set ℝ), ContDiff ℝ ∞ X →
+      ContDiff ℝ ∞ W → (∀ t ∈ s, path p t = X t) →
+      (∀ t ∈ s, deriv (path p) t = rot t (W t)) →
+      ContDiffOn ℝ ∞ paperGerverContacts s := by
+    intro X W s hX hW hXe hWe
+    refine (contDiff_contactShape (fun t => toPlane (X t)) (fun t => (W t).1)
+      (fun t => (W t).2) (contDiff_toPlane hX) hW.fst hW.snd).contDiffOn.congr ?_
+    intro t ht
+    have hx : paperGerverPath t = toPlane (X t) := by
+      change toPlane (path GerverSofa.PartB.params t) = toPlane (X t)
+      rw [hp]
+      exact congrArg toPlane (hXe t ht)
+    simp only [paperGerverContacts, hx, hvel (W t) t (hWe t ht)]
+  -- The five stage intervals, in terms of the direct switching angles.
+  have hI0 : gerverStageIntervals 0 = Set.Icc 0 p.phi := by
+    simp [gerverStageIntervals, gerverStageTimes, hphi']
+  have hI1 : gerverStageIntervals 1 = Set.Icc p.phi p.theta := by
+    simp [gerverStageIntervals, gerverStageTimes, hphi', htheta']
+  have hI2 : gerverStageIntervals 2 = Set.Icc p.theta (Real.pi / 2 - p.theta) := by
+    simp [gerverStageIntervals, gerverStageTimes, htheta']
+  have hI3 : gerverStageIntervals 3 =
+      Set.Icc (Real.pi / 2 - p.theta) (Real.pi / 2 - p.phi) := by
+    simp [gerverStageIntervals, gerverStageTimes, hphi', htheta']
+  have hI4 : gerverStageIntervals 4 = Set.Icc (Real.pi / 2 - p.phi) (Real.pi / 2) := by
+    simp [gerverStageIntervals, gerverStageTimes, hphi']
+  -- The stage coefficient pairs, read off the vector of branch velocities.
+  have hB0 : ∀ t, gerverBranchVelocityComponents 0 t = alphaBeta1 p t := fun t => by
+    simp [gerverBranchVelocityComponents, hp]
+  have hB1 : ∀ t, gerverBranchVelocityComponents 1 t = alphaBeta2 p t := fun t => by
+    simp [gerverBranchVelocityComponents, hp]
+  have hB2 : ∀ t, gerverBranchVelocityComponents 2 t = alphaBeta3 p t := fun t => by
+    simp [gerverBranchVelocityComponents, hp]
+  have hB3 : ∀ t, gerverBranchVelocityComponents 3 t = alphaBeta4 p t := fun t => by
+    simp [gerverBranchVelocityComponents, hp]
+  have hB4 : ∀ t, gerverBranchVelocityComponents 4 t = alphaBeta5 p t := fun t => by
+    simp [gerverBranchVelocityComponents, hp]
+  refine ⟨hvelC, ?_, ?_, ?_⟩
+  · exact continuous_contactShape paperGerverPath
+      (fun t => (paperGerverVelocityComponents t).1)
+      (fun t => (paperGerverVelocityComponents t).2) (continuous_paperGerverPath hC1')
+      hvelC.fst hvelC.snd
+  · intro i
+    fin_cases i
+    · exact hstage (path1 p) (alphaBeta1 p) (gerverStageIntervals 0) (contDiff_path1 p)
+        (contDiff_alphaBeta1 p) (fun t ht => path_eq_path1_of_mem_Icc p (hI0 ▸ ht))
+        (fun t ht => deriv_path_eq_rot_alphaBeta1 hC1 h0 (hI0 ▸ ht))
+    · exact hstage (path2 p) (alphaBeta2 p) (gerverStageIntervals 1) (contDiff_path2 p)
+        (contDiff_alphaBeta2 p) (fun t ht => path_eq_path2_of_mem_Icc heqs (hI1 ▸ ht))
+        (fun t ht => deriv_path_eq_rot_alphaBeta2 hC1 heqs h1 (hI1 ▸ ht))
+    · exact hstage (path3 p) (alphaBeta3 p) (gerverStageIntervals 2) (contDiff_path3 p)
+        (contDiff_alphaBeta3 p) (fun t ht => path_eq_path3_of_mem_Icc heqs h1 (hI2 ▸ ht))
+        (fun t ht => deriv_path_eq_rot_alphaBeta3 hC1 heqs h1 h2 (hI2 ▸ ht))
+    · exact hstage (path4 p) (alphaBeta4 p) (gerverStageIntervals 3) (contDiff_path4 p)
+        (contDiff_alphaBeta4 p) (fun t ht => path_eq_path4_of_mem_Icc heqs h1 h2 (hI3 ▸ ht))
+        (fun t ht => deriv_path_eq_rot_alphaBeta4 hC1 heqs h1 h2 (hI3 ▸ ht))
+    · exact hstage (path5 p) (alphaBeta5 p) (gerverStageIntervals 4) (contDiff_path5 p)
+        (contDiff_alphaBeta5 p) (fun t ht => path_eq_path5_of_mem_Icc heqs h1 h2 (hI4 ▸ ht))
+        (fun t ht => deriv_path_eq_rot_alphaBeta5 hC1 heqs h0 h1 h2 (hI4 ▸ ht))
+  · intro i
+    fin_cases i
+    · exact fun t ht => (hvel _ t (deriv_path_eq_rot_alphaBeta1 hC1 h0
+        (hI0 ▸ (ht : t ∈ gerverStageIntervals 0)))).trans (hB0 t).symm
+    · exact fun t ht => (hvel _ t (deriv_path_eq_rot_alphaBeta2 hC1 heqs h1
+        (hI1 ▸ (ht : t ∈ gerverStageIntervals 1)))).trans (hB1 t).symm
+    · exact fun t ht => (hvel _ t (deriv_path_eq_rot_alphaBeta3 hC1 heqs h1 h2
+        (hI2 ▸ (ht : t ∈ gerverStageIntervals 2)))).trans (hB2 t).symm
+    · exact fun t ht => (hvel _ t (deriv_path_eq_rot_alphaBeta4 hC1 heqs h1 h2
+        (hI3 ▸ (ht : t ∈ gerverStageIntervals 3)))).trans (hB3 t).symm
+    · exact fun t ht => (hvel _ t (deriv_path_eq_rot_alphaBeta5 hC1 heqs h0 h1 h2
+        (hI4 ▸ (ht : t ∈ gerverStageIntervals 4)))).trans (hB4 t).symm
+
+/-! ### Identification with the certified piecewise contact data -/
+
+/-- On the physical rotation interval the paper velocity components agree with the vendor
+piecewise body-frame coefficients. -/
+theorem paperGerverVelocityComponents_eq_alphaBetaAt (t : ℝ)
+    (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    paperGerverVelocityComponents t = GerverSofa.PartC.alphaBetaAt t := by
+  obtain ⟨-, hphi, htheta, -⟩ := gerver_parameter_identification.1 GerverSofa.PartB.params
+    GerverSofa.PartB.params_mem GerverSofa.PartB.params_equations
+  have hphi' : GerverSofa.PartB.params.phi = GerversSofa.φ := hphi.trans selected_phi
+  have htheta' : GerverSofa.PartB.params.theta = GerversSofa.θ := htheta.trans selected_theta
+  have hvel := paperGerverContactData_properties.2.2.2
+  have hI0 : gerverStageIntervals 0 = Set.Icc 0 GerversSofa.φ := by
+    simp [gerverStageIntervals, gerverStageTimes]
+  have hI1 : gerverStageIntervals 1 = Set.Icc GerversSofa.φ GerversSofa.θ := by
+    simp [gerverStageIntervals, gerverStageTimes]
+  have hI2 : gerverStageIntervals 2 = Set.Icc GerversSofa.θ (Real.pi / 2 - GerversSofa.θ) := by
+    simp [gerverStageIntervals, gerverStageTimes]
+  have hI3 : gerverStageIntervals 3 =
+      Set.Icc (Real.pi / 2 - GerversSofa.θ) (Real.pi / 2 - GerversSofa.φ) := by
+    simp [gerverStageIntervals, gerverStageTimes]
+  have hI4 : gerverStageIntervals 4 = Set.Icc (Real.pi / 2 - GerversSofa.φ) (Real.pi / 2) := by
+    simp [gerverStageIntervals, gerverStageTimes]
+  obtain ⟨ht0, ht2⟩ := ht
+  simp only [GerverSofa.PartC.alphaBetaAt, GerverSofa.PartC.eta, GerverSofa.PartC.tau,
+    GerverSofa.PartC.T, hphi', htheta']
+  split_ifs with h1 h2 h3 h4
+  · have := hvel 0 t (by rw [hI0]; exact ⟨ht0, h1⟩)
+    simpa [gerverBranchVelocityComponents] using this
+  · have := hvel 1 t (by rw [hI1]; exact ⟨le_of_not_ge h1, h2⟩)
+    simpa [gerverBranchVelocityComponents] using this
+  · have := hvel 2 t (by rw [hI2]; exact ⟨le_of_not_ge h2, h3⟩)
+    simpa [gerverBranchVelocityComponents] using this
+  · have := hvel 3 t (by rw [hI3]; exact ⟨le_of_not_ge h3, h4⟩)
+    simpa [gerverBranchVelocityComponents] using this
+  · have := hvel 4 t (by rw [hI4]; exact ⟨le_of_not_ge h4, ht2⟩)
+    simpa [gerverBranchVelocityComponents] using this
+
+/-- On the physical rotation interval the four paper contact curves are the coordinate
+transports of the four certified contact curves. -/
+theorem fromPlane_paperGerverContacts (t : ℝ) (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2))
+    (i : Fin 4) :
+    GerverSofa.PartF.Coordinates.fromPlane (paperGerverContacts t i) =
+      ![GerverSofa.PartC.A t, GerverSofa.PartC.B t, GerverSofa.PartC.C t,
+        GerverSofa.PartC.D t] i := by
+  have hab := paperGerverVelocityComponents_eq_alphaBetaAt t ht
+  fin_cases i <;> simp only [paperGerverContacts, hab] <;> rfl
+
+/-- On the physical rotation interval the second paper contact curve is the certified curve
+`B`, read in plane coordinates. -/
+theorem paperGerverContacts_one_eq_toPlane {t : ℝ} (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    paperGerverContacts t 1 =
+      GerverSofa.PartF.Coordinates.toPlane (GerverSofa.PartC.B t) :=
+  (GerverSofa.PartF.Coordinates.toPlane_fromPlane _).symm.trans
+    (congrArg GerverSofa.PartF.Coordinates.toPlane (fromPlane_paperGerverContacts t ht 1))
+
+/-- On the physical rotation interval the fourth paper contact curve is the certified curve
+`D`, read in plane coordinates. -/
+theorem paperGerverContacts_three_eq_toPlane {t : ℝ} (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    paperGerverContacts t 3 =
+      GerverSofa.PartF.Coordinates.toPlane (GerverSofa.PartC.D t) :=
+  (GerverSofa.PartF.Coordinates.toPlane_fromPlane _).symm.trans
+    (congrArg GerverSofa.PartF.Coordinates.toPlane (fromPlane_paperGerverContacts t ht 3))
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The Gerver niche roof
+
+The upper boundary of the paper's literal Gerver niche is a three-piece graph over the
+rotation interval: the fourth contact curve up to the second stage time, the direct path run
+*backwards* through the affine reversal `gerverRoofReverseTime` on the middle stage, and the
+second contact curve from the third stage time on.  `gerverRoofCurve` is that graph as a
+function of an unrestricted real parameter and `gerverNicheRoof` its restriction to the
+rotation interval; `gerverRoofCurve_eq` relates the two.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The affine reversal of the middle roof stage: it maps `gerverStageTimes 2` to
+`gerverStageTimes 4` and `gerverStageTimes 3` to `gerverStageTimes 1`, so it reparametrizes
+the central part of the direct path backwards. -/
+def gerverRoofReverseTime (s : ℝ) : ℝ :=
+  gerverStageTimes 4 - (gerverStageTimes 4 - gerverStageTimes 1) /
+    (gerverStageTimes 3 - gerverStageTimes 2) * (s - gerverStageTimes 2)
+
+/-- The Gerver niche roof as a curve of an unrestricted real parameter.  On the rotation
+interval it agrees with `gerverNicheRoof` (`gerverRoofCurve_eq`); the unrestricted form is
+what the intermediate value theorem and `continuous_if_le` consume. -/
+def gerverRoofCurve (s : ℝ) : Point :=
+  if s ≤ gerverStageTimes 2 then paperGerverContacts s 3
+  else if s ≤ gerverStageTimes 3 then paperGerverPath (gerverRoofReverseTime s)
+  else paperGerverContacts s 1
+
+/-- The three-branch roof of the Gerver niche, using two contact arcs and the reversed path. -/
+def gerverNicheRoof (s : Set.Icc (0 : ℝ) (Real.pi / 2)) : Point :=
+  if s.val ≤ gerverStageTimes 2 then paperGerverContacts s.val 3
+  else if s.val ≤ gerverStageTimes 3 then paperGerverPath (gerverRoofReverseTime s.val)
+  else paperGerverContacts s.val 1
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver / ODEs
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Tangential components of the four contact-curve derivatives on a selected stage. -/
+def gerverStageContactDerivatives (i : Fin 5) (t : ℝ) : Fin 4 → ℝ :=
+  ![inner ℝ (derivWithin (fun s ↦ paperGerverContacts s 0) (gerverStageIntervals i) t)
+      (tangentVector (t : Real.Angle)),
+    inner ℝ (derivWithin (fun s ↦ paperGerverContacts s 1) (gerverStageIntervals i) t)
+      (tangentVector (t : Real.Angle)),
+    inner ℝ (-(derivWithin (fun s ↦ paperGerverContacts s 2) (gerverStageIntervals i) t))
+      (normalVector (t : Real.Angle)),
+    inner ℝ (derivWithin (fun s ↦ paperGerverContacts s 3) (gerverStageIntervals i) t)
+      (normalVector (t : Real.Angle))]
+
+private theorem gerver_contact_derivatives_from_frame (i : Fin 5) (t : ℝ)
+    (ht : t ∈ gerverStageIntervals i)
+    (huniq : UniqueDiffWithinAt ℝ (gerverStageIntervals i) t) :
+    ∀ (X : ℝ → Point) (X' : Point) (A B : ℝ → ℝ) (a' b' : ℝ),
+     HasDerivAt X X' t →
+     inner ℝ X' (normalVector (t : Real.Angle)) = A t →
+     inner ℝ X' (tangentVector (t : Real.Angle)) = B t →
+     HasDerivAt A a' t → HasDerivAt B b' t →
+     (∀ r ∈ gerverStageIntervals i, paperGerverPath r = X r) →
+     (∀ r ∈ gerverStageIntervals i, paperGerverVelocityComponents r = (A r, B r)) →
+     gerverStageContactDerivatives i t 0 = B t + a' + 1 ∧
+     gerverStageContactDerivatives i t 1 = B t + a' ∧
+     gerverStageContactDerivatives i t 2 = b' + 1 - A t ∧
+     gerverStageContactDerivatives i t 3 = A t - b' := by
+  -- Frame orthonormality at the fixed angle `t`.
+  have huu :
+      inner ℝ (normalVector (t : Real.Angle)) (normalVector (t : Real.Angle)) = (1 : ℝ) :=
+    inner_normalVector_self t
+  have hvv :
+      inner ℝ (tangentVector (t : Real.Angle)) (tangentVector (t : Real.Angle)) = (1 : ℝ) :=
+    inner_tangentVector_self t
+  have huv :
+      inner ℝ (normalVector (t : Real.Angle)) (tangentVector (t : Real.Angle)) = (0 : ℝ) :=
+    inner_normalVector_tangentVector t
+  have hvu :
+      inner ℝ (tangentVector (t : Real.Angle)) (normalVector (t : Real.Angle)) = (0 : ℝ) := by
+    rw [real_inner_comm]; exact huv
+  intro X X' A B a' b' hX hXu hXv hA hB hpath hvel
+  have hn : HasDerivAt (fun s : ℝ => normalVector (s : Real.Angle))
+      (tangentVector (t : Real.Angle)) t := hasDerivAt_normalVector t
+  have hg : HasDerivAt (fun s : ℝ => tangentVector (s : Real.Angle))
+      (-normalVector (t : Real.Angle)) t := hasDerivAt_tangentVector t
+  have hAv := hA.fun_smul hg
+  have hBn := hB.fun_smul hn
+  have hc0 : HasDerivWithinAt (fun s => paperGerverContacts s 0)
+      (X' + (A t • (-normalVector (t : Real.Angle)) + a' • tangentVector (t : Real.Angle)) +
+        tangentVector (t : Real.Angle)) (gerverStageIntervals i) t := by
+    refine ((hX.fun_add hAv).fun_add hn).hasDerivWithinAt.congr (fun r hr => ?_) ?_
+    · simp only [paperGerverContacts, Matrix.cons_val_zero, hpath r hr, hvel r hr]
+    · simp only [paperGerverContacts, Matrix.cons_val_zero, hpath t ht, hvel t ht]
+  have hc1 : HasDerivWithinAt (fun s => paperGerverContacts s 1)
+      (X' + (A t • (-normalVector (t : Real.Angle)) + a' • tangentVector (t : Real.Angle)))
+      (gerverStageIntervals i) t := by
+    refine (hX.fun_add hAv).hasDerivWithinAt.congr (fun r hr => ?_) ?_
+    · simp only [paperGerverContacts, hpath r hr, hvel r hr]
+      rfl
+    · simp only [paperGerverContacts, hpath t ht, hvel t ht]
+      rfl
+  have hc2 : HasDerivWithinAt (fun s => paperGerverContacts s 2)
+      (X' - (B t • tangentVector (t : Real.Angle) + b' • normalVector (t : Real.Angle)) +
+        -normalVector (t : Real.Angle)) (gerverStageIntervals i) t := by
+    refine ((hX.fun_sub hBn).fun_add hg).hasDerivWithinAt.congr (fun r hr => ?_) ?_
+    · simp only [paperGerverContacts, hpath r hr, hvel r hr]
+      rfl
+    · simp only [paperGerverContacts, hpath t ht, hvel t ht]
+      rfl
+  have hc3 : HasDerivWithinAt (fun s => paperGerverContacts s 3)
+      (X' - (B t • tangentVector (t : Real.Angle) + b' • normalVector (t : Real.Angle)))
+      (gerverStageIntervals i) t := by
+    refine (hX.fun_sub hBn).hasDerivWithinAt.congr (fun r hr => ?_) ?_
+    · simp only [paperGerverContacts, hpath r hr, hvel r hr]
+      rfl
+    · simp only [paperGerverContacts, hpath t ht, hvel t ht]
+      rfl
+  refine ⟨?_, ?_, ?_, ?_⟩
+  · change inner ℝ (derivWithin (fun s => paperGerverContacts s 0) (gerverStageIntervals i) t)
+      (tangentVector (t : Real.Angle)) = B t + a' + 1
+    rw [hc0.derivWithin huniq]
+    simp only [inner_add_left, real_inner_smul_left, inner_neg_left, hXv, huv, hvv]
+    ring
+  · change inner ℝ (derivWithin (fun s => paperGerverContacts s 1) (gerverStageIntervals i) t)
+      (tangentVector (t : Real.Angle)) = B t + a'
+    rw [hc1.derivWithin huniq]
+    simp only [inner_add_left, real_inner_smul_left, inner_neg_left, hXv, huv, hvv]
+    ring
+  · change inner ℝ (-derivWithin (fun s => paperGerverContacts s 2) (gerverStageIntervals i) t)
+      (normalVector (t : Real.Angle)) = b' + 1 - A t
+    rw [hc2.derivWithin huniq]
+    simp only [inner_neg_left, inner_add_left, inner_sub_left, real_inner_smul_left, hXu, hvu,
+      huu]
+    ring
+  · change inner ℝ (derivWithin (fun s => paperGerverContacts s 3) (gerverStageIntervals i) t)
+      (normalVector (t : Real.Angle)) = A t - b'
+    rw [hc3.derivWithin huniq]
+    simp only [inner_sub_left, inner_add_left, real_inner_smul_left, hXu, hvu, huu]
+    ring
+
+theorem gerver_stageODEs (i : Fin 5) (t : ℝ) (ht : t ∈ gerverStageIntervals i) :
+    (gerverStageContactDerivatives i t 0, gerverStageContactDerivatives i t 2) =
+      ![(0, gerverStageContactDerivatives i t 3),
+        ((paperGerverVelocityComponents t).2,
+          gerverStageContactDerivatives i t 3 - (paperGerverVelocityComponents t).1),
+        ((paperGerverVelocityComponents t).2, -(paperGerverVelocityComponents t).1),
+        (-gerverStageContactDerivatives i t 1 + (paperGerverVelocityComponents t).2,
+          -(paperGerverVelocityComponents t).1),
+        (-gerverStageContactDerivatives i t 1, 0)] i := by
+  open GerverSofa.Romik GerverSofa.PartF.Coordinates in
+  -- The certified direct parameter vector and the ordering of its switching angles.
+  obtain ⟨p, hp⟩ : ∃ p : Params, GerverSofa.PartB.params = p := ⟨_, rfl⟩
+  have hmem : p ∈ gerverDirectBox := hp ▸ GerverSofa.PartB.params_mem
+  have heqs : gerverDirectEquations p := hp ▸ GerverSofa.PartB.params_equations
+  obtain ⟨-, hphieq, hthetaeq, hphipos, hphitheta, hthetalt, -⟩ :=
+    gerver_parameter_identification.1 p hmem heqs
+  have hphi' : p.phi = GerversSofa.φ := hphieq.trans selected_phi
+  have htheta' : p.theta = GerversSofa.θ := hthetaeq.trans selected_theta
+  have h0 : (0 : ℝ) < p.phi := by rw [hphieq]; exact hphipos
+  have h1 : p.phi < p.theta := by rw [hphieq, hthetaeq]; exact hphitheta
+  have h2 : p.theta < Real.pi / 4 := by rw [hthetaeq]; exact hthetalt
+  -- The five closed stage intervals in terms of the direct switching angles.
+  have hI0 : gerverStageIntervals 0 = Set.Icc 0 p.phi := by
+    simp [gerverStageIntervals, gerverStageTimes, hphi']
+  have hI1 : gerverStageIntervals 1 = Set.Icc p.phi p.theta := by
+    simp [gerverStageIntervals, gerverStageTimes, hphi', htheta']
+  have hI2 : gerverStageIntervals 2 = Set.Icc p.theta (Real.pi / 2 - p.theta) := by
+    simp [gerverStageIntervals, gerverStageTimes, htheta']
+  have hI3 : gerverStageIntervals 3 =
+      Set.Icc (Real.pi / 2 - p.theta) (Real.pi / 2 - p.phi) := by
+    simp [gerverStageIntervals, gerverStageTimes, hphi', htheta']
+  have hI4 : gerverStageIntervals 4 = Set.Icc (Real.pi / 2 - p.phi) (Real.pi / 2) := by
+    simp [gerverStageIntervals, gerverStageTimes, hphi']
+  have hval := paperGerverContactData_properties.2.2.2
+  -- Derivatives of the ten branch coefficient functions, in one normal form.
+  have hidt : HasDerivAt (fun s : ℝ => s) 1 t := hasDerivAt_id t
+  have hshape : ∀ (c₀ c₁ c₂ c₃ c₄ : ℝ) (f : ℝ → ℝ),
+      (∀ s : ℝ,
+        f s = c₀ + c₁ * s + c₂ * s * s + c₃ * Real.sin s + c₄ * Real.cos s) →
+      HasDerivAt f (c₁ + 2 * c₂ * t + c₃ * Real.cos t - c₄ * Real.sin t) t := by
+    intro c₀ c₁ c₂ c₃ c₄ f hf
+    have h : HasDerivAt
+        (fun s : ℝ => c₀ + c₁ * s + c₂ * s * s + c₃ * Real.sin s + c₄ * Real.cos s)
+        (c₁ + 2 * c₂ * t + c₃ * Real.cos t - c₄ * Real.sin t) t := by
+      refine ((((hasDerivAt_const t c₀).fun_add (hidt.const_mul c₁)).fun_add
+        ((hidt.const_mul c₂).fun_mul hidt)).fun_add
+        ((Real.hasDerivAt_sin t).const_mul c₃)).fun_add
+        ((Real.hasDerivAt_cos t).const_mul c₄) |>.congr_deriv ?_
+      ring
+    exact h.congr_of_eventuallyEq (Filter.Eventually.of_forall fun s => hf s)
+  -- Unique differentiability of the closed stage interval at `t`.
+  have g0 : (0 : ℝ) < GerversSofa.φ := by rw [← hphi']; exact h0
+  have g1 : GerversSofa.φ < GerversSofa.θ := by rw [← hphi', ← htheta']; exact h1
+  have g2 : GerversSofa.θ < Real.pi / 4 := by rw [← htheta']; exact h2
+  have huniq : UniqueDiffWithinAt ℝ (gerverStageIntervals i) t := by
+    fin_cases i
+    · exact uniqueDiffOn_Icc g0 t ht
+    · exact uniqueDiffOn_Icc g1 t ht
+    · exact uniqueDiffOn_Icc
+        (show GerversSofa.θ < Real.pi / 2 - GerversSofa.θ by linarith) t ht
+    · exact uniqueDiffOn_Icc
+        (show Real.pi / 2 - GerversSofa.θ < Real.pi / 2 - GerversSofa.φ by linarith) t ht
+    · exact uniqueDiffOn_Icc
+        (show Real.pi / 2 - GerversSofa.φ < Real.pi / 2 by linarith) t ht
+  -- The frame computation: the four contact derivatives in terms of `α, β, α', β'`.
+  have hcore := gerver_contact_derivatives_from_frame i t ht huniq
+  fin_cases i
+  · -- Stage 1
+    have hvel : ∀ r ∈ gerverStageIntervals 0,
+        paperGerverVelocityComponents r = alphaBeta1 p r := by
+      intro r hr
+      rw [hval 0 r hr]
+      simp [gerverBranchVelocityComponents, hp]
+    obtain ⟨e0, e1, e2, e3⟩ := hcore (fun s => toPlane (path1 p s))
+      (toPlane (rot t (alphaBeta1 p t))) (fun s => (alphaBeta1 p s).1)
+      (fun s => (alphaBeta1 p s).2) _ _ (hasDerivAt_toPlane (hasDerivAt_path1 p t))
+      (inner_toPlane_rot_normalVector t _) (inner_toPlane_rot_tangentVector t _)
+      (hshape (1 / 2) 0 0 (-2 * p.a1) (2 * p.a2) _ fun s => by dsimp [alphaBeta1]; ring)
+      (hshape (-1) 0 0 (2 * p.a2) (2 * p.a1) _ fun s => by dsimp [alphaBeta1]; ring)
+      (fun r hr => by
+        change toPlane (path GerverSofa.PartB.params r) = toPlane (path1 p r)
+        rw [hp]
+        exact congrArg toPlane (path_eq_path1_of_mem_Icc p (hI0 ▸ hr)))
+      (fun r hr => hvel r hr)
+    rw [Prod.mk.injEq, e0, e2, e3]
+    refine ⟨?_, ?_⟩ <;> dsimp [alphaBeta1] <;> ring
+  · -- Stage 2
+    have hvel : ∀ r ∈ gerverStageIntervals 1,
+        paperGerverVelocityComponents r = alphaBeta2 p r := by
+      intro r hr
+      rw [hval 1 r hr]
+      simp [gerverBranchVelocityComponents, hp]
+    obtain ⟨e0, e1, e2, e3⟩ := hcore (fun s => toPlane (path2 p s))
+      (toPlane (rot t (alphaBeta2 p t))) (fun s => (alphaBeta2 p s).1)
+      (fun s => (alphaBeta2 p s).2) _ _ (hasDerivAt_toPlane (hasDerivAt_path2 p t))
+      (inner_toPlane_rot_normalVector t _) (inner_toPlane_rot_tangentVector t _)
+      (hshape (1 + 2 * p.b1) (-1) 0 0 0 _ fun s => by dsimp [alphaBeta2]; ring)
+      (hshape (p.b2 + 1 / 2) p.b1 (-(1 / 4)) 0 0 _ fun s => by dsimp [alphaBeta2]; ring)
+      (fun r hr => by
+        change toPlane (path GerverSofa.PartB.params r) = toPlane (path2 p r)
+        rw [hp]
+        exact congrArg toPlane (path_eq_path2_of_mem_Icc heqs (hI1 ▸ hr)))
+      (fun r hr => hvel r hr)
+    rw [Prod.mk.injEq, hvel t ht, e0, e2, e3]
+    refine ⟨?_, ?_⟩ <;> dsimp [alphaBeta2] <;> ring
+  · -- Stage 3
+    have hvel : ∀ r ∈ gerverStageIntervals 2,
+        paperGerverVelocityComponents r = alphaBeta3 p r := by
+      intro r hr
+      rw [hval 2 r hr]
+      simp [gerverBranchVelocityComponents, hp]
+    obtain ⟨e0, e1, e2, e3⟩ := hcore (fun s => toPlane (path3 p s))
+      (toPlane (rot t (alphaBeta3 p t))) (fun s => (alphaBeta3 p s).1)
+      (fun s => (alphaBeta3 p s).2) _ _ (hasDerivAt_toPlane (hasDerivAt_path3 p t))
+      (inner_toPlane_rot_normalVector t _) (inner_toPlane_rot_tangentVector t _)
+      (hshape (-1 - p.c2) (-1) 0 0 0 _ fun s => by dsimp [alphaBeta3]; ring)
+      (hshape (1 + p.c1) (-1) 0 0 0 _ fun s => by dsimp [alphaBeta3]; ring)
+      (fun r hr => by
+        change toPlane (path GerverSofa.PartB.params r) = toPlane (path3 p r)
+        rw [hp]
+        exact congrArg toPlane (path_eq_path3_of_mem_Icc heqs h1 (hI2 ▸ hr)))
+      (fun r hr => hvel r hr)
+    rw [Prod.mk.injEq, hvel t ht, e0, e2]
+    refine ⟨?_, ?_⟩ <;> dsimp [alphaBeta3] <;> ring
+  · -- Stage 4
+    have hvel : ∀ r ∈ gerverStageIntervals 3,
+        paperGerverVelocityComponents r = alphaBeta4 p r := by
+      intro r hr
+      rw [hval 3 r hr]
+      simp [gerverBranchVelocityComponents, hp]
+    obtain ⟨e0, e1, e2, e3⟩ := hcore (fun s => toPlane (path4 p s))
+      (toPlane (rot t (alphaBeta4 p t))) (fun s => (alphaBeta4 p s).1)
+      (fun s => (alphaBeta4 p s).2) _ _ (hasDerivAt_toPlane (hasDerivAt_path4 p t))
+      (inner_toPlane_rot_normalVector t _) (inner_toPlane_rot_tangentVector t _)
+      (hshape (-p.d2 - 1 / 2) (-p.d1) (1 / 4) 0 0 _ fun s => by dsimp [alphaBeta4]; ring)
+      (hshape (2 * p.d1 - 1) (-1) 0 0 0 _ fun s => by dsimp [alphaBeta4]; ring)
+      (fun r hr => by
+        change toPlane (path GerverSofa.PartB.params r) = toPlane (path4 p r)
+        rw [hp]
+        exact congrArg toPlane (path_eq_path4_of_mem_Icc heqs h1 h2 (hI3 ▸ hr)))
+      (fun r hr => hvel r hr)
+    rw [Prod.mk.injEq, hvel t ht, e0, e1, e2]
+    refine ⟨?_, ?_⟩ <;> dsimp [alphaBeta4] <;> ring
+  · -- Stage 5
+    have hvel : ∀ r ∈ gerverStageIntervals 4,
+        paperGerverVelocityComponents r = alphaBeta5 p r := by
+      intro r hr
+      rw [hval 4 r hr]
+      simp [gerverBranchVelocityComponents, hp]
+    obtain ⟨e0, e1, e2, e3⟩ := hcore (fun s => toPlane (path5 p s))
+      (toPlane (rot t (alphaBeta5 p t))) (fun s => (alphaBeta5 p s).1)
+      (fun s => (alphaBeta5 p s).2) _ _ (hasDerivAt_toPlane (hasDerivAt_path5 p t))
+      (inner_toPlane_rot_normalVector t _) (inner_toPlane_rot_tangentVector t _)
+      (hshape 1 0 0 (-2 * p.e1) (2 * p.e2) _ fun s => by dsimp [alphaBeta5]; ring)
+      (hshape (-(1 / 2)) 0 0 (2 * p.e2) (2 * p.e1) _ fun s => by dsimp [alphaBeta5]; ring)
+      (fun r hr => by
+        change toPlane (path GerverSofa.PartB.params r) = toPlane (path5 p r)
+        rw [hp]
+        exact congrArg toPlane (path_eq_path5_of_mem_Icc heqs h1 h2 (hI4 ▸ hr)))
+      (fun r hr => hvel r hr)
+    rw [Prod.mk.injEq, e0, e1, e2]
+    refine ⟨?_, ?_⟩ <;> dsimp [alphaBeta5] <;> ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Gerver / Outer Contacts
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem gerver_outer_contact_A (t : ℝ) (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    paperGerverContacts t 0 ∈ gerverOuterCap := by
+  rw [mem_gerverOuterCap_iff, fromPlane_paperGerverContacts t ht 0]
+  exact GerverSofa.PartC.Stage3.supportA_direct t ht
+
+theorem gerver_outer_contact_C (t : ℝ) (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    paperGerverContacts t 2 ∈ gerverOuterCap := by
+  rw [mem_gerverOuterCap_iff, fromPlane_paperGerverContacts t ht 2]
+  exact GerverSofa.PartC.Stage3.supportC_direct t ht
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Regularity of the certified Gerver stage data
+
+The five Gerver stages are nondegenerate: the six stage endpoints increase strictly from `0`
+to `π / 2` (`gerverStageTimes_strictMono`), and `gerverStageIntervals_zero` through
+`gerverStageIntervals_four` name the resulting closed stage intervals.  The certified direct
+path is continuously differentiable on the whole rotation interval
+(`contDiff_paperGerverPath`), while each contact curve is only continuous globally
+(`continuous_paperGerverContact`) and continuously differentiable on a single stage
+(`contDiffOn_paperGerverContact`).  Gluing two consecutive stages presents the two inner
+contact curves on the parameter ranges where they touch the cap as continuous paths of bounded
+variation (`gerverRightContactBV`, `gerverLeftContactBV`).
+
+The second half of the file differentiates the two inner contact curves stagewise.  Writing
+the velocity components of the direct path as `(α, β)`, the contact formulas are
+`B = x + α v` and `D = x - β u`, so the product rule and the frame derivatives
+`hasDerivAt_normalVector`, `hasDerivAt_tangentVector` give
+`B' = (β + α') v` and `D' = (α - β') u`
+(`hasDerivWithinAt_paperGerverContacts_one`, `hasDerivWithinAt_paperGerverContacts_three`).
+Feeding the five analytic stage branches of `paperGerverContactData_properties` into these
+two lemmas yields the signs of the two speeds on the stages where they are needed
+(`hasDerivWithinAt_paperGerverContacts_one_neg_smul`,
+`hasDerivWithinAt_paperGerverContacts_three_pos_smul`); the two nonconstant coefficients are
+signed by the coarse box bounds `gerverDirectBox_b1_lower_bound`,
+`gerverDirectBox_d1_upper_bound` and `gerverStageTimes_two_le_seven_div_ten`.
+
+The last two sections record the consequences used downstream: the frame coordinates
+`s ↦ x s ⋅ u_s` and `s ↦ x s ⋅ v_s` of the direct path are differentiable with derivatives
+`B ⋅ v` and `-D ⋅ u` (`hasDerivAt_inner_paperGerverPath_normalVector`,
+`hasDerivAt_inner_paperGerverPath_tangentVector`), and against a fixed frame direction outside
+the stage the two inner contact curves are strictly monotone on each stage
+(`strictAntiOn_inner_paperGerverContacts_three`, `strictMonoOn_inner_paperGerverContacts_one`).
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The six Gerver stage endpoints increase strictly along the rotation interval. -/
+theorem gerverStageTimes_strictMono : StrictMono gerverStageTimes := by
+  obtain ⟨-, -, -, hpos, hlt, hqt, -⟩ := gerver_parameter_identification.1
+    GerverSofa.PartB.params GerverSofa.PartB.params_mem GerverSofa.PartB.params_equations
+  rw [selected_phi] at hpos
+  rw [selected_phi, selected_theta] at hlt
+  rw [selected_theta] at hqt
+  refine Fin.strictMono_iff_lt_succ.mpr fun i ↦ ?_
+  fin_cases i
+  · change gerverStageTimes 0 < gerverStageTimes 1
+    simp only [gerverStageTimes, Matrix.cons_val_zero, Matrix.cons_val_one]
+    linarith
+  · change gerverStageTimes 1 < gerverStageTimes 2
+    simp only [gerverStageTimes, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val]
+    linarith
+  · change gerverStageTimes 2 < gerverStageTimes 3
+    simp only [gerverStageTimes, Matrix.cons_val]
+    linarith
+  · change gerverStageTimes 3 < gerverStageTimes 4
+    simp only [gerverStageTimes, Matrix.cons_val]
+    linarith
+  · change gerverStageTimes 4 < gerverStageTimes 5
+    simp only [gerverStageTimes, Matrix.cons_val]
+    linarith
+
+/-- The second Gerver stage time is at most `7/10`.  This coarse bound on the certified
+switching angle `θ` is what the stage speed estimates below consume. -/
+theorem gerverStageTimes_two_le_seven_div_ten : gerverStageTimes 2 ≤ (7 : ℝ) / 10 := by
+  have h := GerverSofa.PartB.theta_bounds.2
+  rw [gerverStageTimes_two]
+  norm_num at h ⊢
+  linarith
+
+/-- The first Gerver stage runs between the first two stage times. -/
+theorem gerverStageIntervals_zero :
+    gerverStageIntervals 0 = Set.Icc (gerverStageTimes 0) (gerverStageTimes 1) := rfl
+
+/-- The second Gerver stage runs between the second and third stage times. -/
+theorem gerverStageIntervals_one :
+    gerverStageIntervals 1 = Set.Icc (gerverStageTimes 1) (gerverStageTimes 2) := rfl
+
+/-- The fourth Gerver stage runs between the fourth and fifth stage times. -/
+theorem gerverStageIntervals_three :
+    gerverStageIntervals 3 = Set.Icc (gerverStageTimes 3) (gerverStageTimes 4) := rfl
+
+/-- The fifth Gerver stage runs between the last two stage times. -/
+theorem gerverStageIntervals_four :
+    gerverStageIntervals 4 = Set.Icc (gerverStageTimes 4) (gerverStageTimes 5) := rfl
+
+/-- The certified direct Gerver path is continuously differentiable. -/
+theorem contDiff_paperGerverPath : ContDiff ℝ 1 paperGerverPath := by
+  obtain ⟨hC1, -, -⟩ := gerver_direct_path_regularity GerverSofa.PartB.params
+    GerverSofa.PartB.params_mem GerverSofa.PartB.params_equations
+  obtain ⟨L, hL⟩ : ∃ L : GerverSofa.Point →L[ℝ] Point,
+      ⇑L = GerverSofa.PartF.Coordinates.toPlane :=
+    ⟨{ toLinearMap := GerverSofa.PartF.Coordinates.linearEquiv.toLinearMap
+       cont := GerverSofa.PartF.Coordinates.continuous_toPlane }, rfl⟩
+  have hcomp : paperGerverPath = ⇑L ∘ GerverSofa.Romik.path GerverSofa.PartB.params := by
+    rw [hL]
+    rfl
+  rw [hcomp]
+  exact L.contDiff.comp hC1
+
+/-- Each Gerver contact curve is continuous. -/
+theorem continuous_paperGerverContact (i : Fin 4) :
+    Continuous fun t : ℝ ↦ paperGerverContacts t i :=
+  (continuous_apply i).comp paperGerverContactData_properties.2.1
+
+/-- Each Gerver contact curve is continuously differentiable on each closed stage interval. -/
+theorem contDiffOn_paperGerverContact (i : Fin 4) (j : Fin 5) :
+    ContDiffOn ℝ 1 (fun t : ℝ ↦ paperGerverContacts t i) (gerverStageIntervals j) :=
+  (contDiffOn_pi.mp (paperGerverContactData_properties.2.2.1 j) i).of_le (by norm_num)
+
+/-! ### The two inner contact curves as bounded-variation paths -/
+
+/-- The second Gerver contact curve, on the two stages `[t₃, t₅]` where it is an inner contact
+of the cap, as a continuous path of bounded variation. -/
+def gerverRightContactBV : ContinuousBVPaths (gerverStageTimes 3) (gerverStageTimes 5) :=
+  continuousBVOfContDiffOnIccUnionIcc (fun t ↦ paperGerverContacts t 1)
+    (gerverStageTimes_strictMono (show (3 : Fin 6) < 4 by decide)).le
+    (gerverStageTimes_strictMono (show (4 : Fin 6) < 5 by decide)).le
+    (contDiffOn_paperGerverContact 1 3) (contDiffOn_paperGerverContact 1 4)
+
+/-- The fourth Gerver contact curve, on the two stages `[t₀, t₂]` where it is an inner contact
+of the cap, as a continuous path of bounded variation. -/
+def gerverLeftContactBV : ContinuousBVPaths (gerverStageTimes 0) (gerverStageTimes 2) :=
+  continuousBVOfContDiffOnIccUnionIcc (fun t ↦ paperGerverContacts t 3)
+    (gerverStageTimes_strictMono (show (0 : Fin 6) < 1 by decide)).le
+    (gerverStageTimes_strictMono (show (1 : Fin 6) < 2 by decide)).le
+    (contDiffOn_paperGerverContact 3 0) (contDiffOn_paperGerverContact 3 1)
+
+/-- `gerverRightContactBV` is the second Gerver contact curve. -/
+theorem gerverRightContactBV_apply (t : Set.Icc (gerverStageTimes 3) (gerverStageTimes 5)) :
+    gerverRightContactBV.val t = paperGerverContacts t 1 := rfl
+
+/-- `gerverLeftContactBV` is the fourth Gerver contact curve. -/
+theorem gerverLeftContactBV_apply (t : Set.Icc (gerverStageTimes 0) (gerverStageTimes 2)) :
+    gerverLeftContactBV.val t = paperGerverContacts t 3 := rfl
+
+/-! ### Stage derivatives of the two inner contact curves -/
+
+/-- The velocity components are the frame coordinates of the derivative of the direct path. -/
+theorem deriv_paperGerverPath_eq_smul_add_smul {t a b : ℝ}
+    (h : paperGerverVelocityComponents t = (a, b)) :
+    deriv paperGerverPath t =
+      a • normalVector (t : Real.Angle) + b • tangentVector (t : Real.Angle) := by
+  have hframe := inner_normalVector_smul_add_inner_tangentVector_smul
+    (deriv paperGerverPath t) (t : Real.Angle)
+  rw [paperGerverVelocityComponents, Prod.mk.injEq] at h
+  rw [h.1, h.2] at hframe
+  exact hframe.symm
+
+/-- On a stage where the direct path has velocity components `(α, β)`, the second contact
+curve `B = x + α v` has derivative `(β + α') v`: the frame derivative `v' = -u` cancels the
+normal component `α u` of `x'`, leaving the tangential component `β v` and the derivative of
+the coefficient. -/
+theorem hasDerivWithinAt_paperGerverContacts_one {i : Fin 5} {t c α' : ℝ} {α β : ℝ → ℝ}
+    (ht : t ∈ gerverStageIntervals i) (hα : HasDerivAt α α' t)
+    (hαβ : ∀ s ∈ gerverStageIntervals i, paperGerverVelocityComponents s = (α s, β s))
+    (hc : β t + α' = c) :
+    HasDerivWithinAt (fun s ↦ paperGerverContacts s 1)
+      (c • tangentVector (t : Real.Angle)) (gerverStageIntervals i) t := by
+  have hx : HasDerivAt paperGerverPath (deriv paperGerverPath t) t :=
+    (contDiff_paperGerverPath.differentiable one_ne_zero t).hasDerivAt
+  have hd := deriv_paperGerverPath_eq_smul_add_smul (hαβ t ht)
+  have hmain : HasDerivAt (fun s ↦ paperGerverPath s + α s • tangentVector (s : Real.Angle))
+      (c • tangentVector (t : Real.Angle)) t := by
+    have h := hx.add (hα.smul (hasDerivAt_tangentVector t))
+    rw [hd] at h
+    rw [← hc]
+    convert h using 1
+    module
+  refine hmain.hasDerivWithinAt.congr (fun y hy ↦ ?_) ?_
+  · change paperGerverPath y + (paperGerverVelocityComponents y).1 •
+      tangentVector (y : Real.Angle) = _
+    rw [hαβ y hy]
+  · change paperGerverPath t + (paperGerverVelocityComponents t).1 •
+      tangentVector (t : Real.Angle) = _
+    rw [hαβ t ht]
+
+/-- On a stage where the direct path has velocity components `(α, β)`, the fourth contact
+curve `D = x - β u` has derivative `(α - β') u`; see
+`hasDerivWithinAt_paperGerverContacts_one` for the shape of the computation. -/
+theorem hasDerivWithinAt_paperGerverContacts_three {i : Fin 5} {t c β' : ℝ} {α β : ℝ → ℝ}
+    (ht : t ∈ gerverStageIntervals i) (hβ : HasDerivAt β β' t)
+    (hαβ : ∀ s ∈ gerverStageIntervals i, paperGerverVelocityComponents s = (α s, β s))
+    (hc : α t - β' = c) :
+    HasDerivWithinAt (fun s ↦ paperGerverContacts s 3)
+      (c • normalVector (t : Real.Angle)) (gerverStageIntervals i) t := by
+  have hx : HasDerivAt paperGerverPath (deriv paperGerverPath t) t :=
+    (contDiff_paperGerverPath.differentiable one_ne_zero t).hasDerivAt
+  have hd := deriv_paperGerverPath_eq_smul_add_smul (hαβ t ht)
+  have hmain : HasDerivAt (fun s ↦ paperGerverPath s - β s • normalVector (s : Real.Angle))
+      (c • normalVector (t : Real.Angle)) t := by
+    have h := hx.sub (hβ.smul (hasDerivAt_normalVector t))
+    rw [hd] at h
+    rw [← hc]
+    convert h using 1
+    module
+  refine hmain.hasDerivWithinAt.congr (fun y hy ↦ ?_) ?_
+  · change paperGerverPath y - (paperGerverVelocityComponents y).2 •
+      normalVector (y : Real.Angle) = _
+    rw [hαβ y hy]
+  · change paperGerverPath t - (paperGerverVelocityComponents t).2 •
+      normalVector (t : Real.Angle) = _
+    rw [hαβ t ht]
+
+/-- On each of the last two stages the second contact curve moves strictly backwards along
+the tangent direction: its one-sided derivative is a negative multiple of `v_t`.  On the
+fourth stage the tangential speed is `d₁ - 1 - t/2`, negative because `t ≥ π/2 - θ > 4/5` and
+`d₁ ≤ 33/25`; on the fifth it is the exact constant `-1/2`. -/
+theorem hasDerivWithinAt_paperGerverContacts_one_neg_smul {i : Fin 5} (hi : i = 3 ∨ i = 4)
+    {t : ℝ} (ht : t ∈ gerverStageIntervals i) :
+    ∃ c : ℝ, c < 0 ∧
+      HasDerivWithinAt (fun s ↦ paperGerverContacts s 1)
+        (c • tangentVector (t : Real.Angle)) (gerverStageIntervals i) t := by
+  rcases hi with rfl | rfl
+  · have hbranch : ∀ s ∈ gerverStageIntervals 3, paperGerverVelocityComponents s =
+        ((1 / 4 : ℝ) * s * s - GerverSofa.PartB.params.d1 * s -
+            GerverSofa.PartB.params.d2 - 1 / 2,
+          2 * GerverSofa.PartB.params.d1 - 1 - s) := by
+      intro s hs
+      rw [paperGerverContactData_properties.2.2.2 3 s hs]
+      rfl
+    have hα : HasDerivAt (fun s : ℝ ↦ (1 / 4 : ℝ) * s * s -
+        GerverSofa.PartB.params.d1 * s - GerverSofa.PartB.params.d2 - 1 / 2)
+        (t / 2 - GerverSofa.PartB.params.d1) t := by
+      have h := ((((hasDerivAt_id' t).const_mul (1 / 4 : ℝ)).mul (hasDerivAt_id' t)).sub
+        ((hasDerivAt_id' t).const_mul GerverSofa.PartB.params.d1)).sub_const
+          GerverSofa.PartB.params.d2 |>.sub_const (1 / 2 : ℝ)
+      convert h using 1
+      ring
+    refine ⟨GerverSofa.PartB.params.d1 - 1 - t / 2, ?_,
+      hasDerivWithinAt_paperGerverContacts_one ht hα hbranch (by ring)⟩
+    rw [gerverStageIntervals_three] at ht
+    have hrefl : gerverStageTimes 3 = Real.pi / 2 - gerverStageTimes 2 := rfl
+    linarith [gerverDirectBox_d1_upper_bound GerverSofa.PartB.params_mem,
+      gerverStageTimes_two_le_seven_div_ten, ht.1, Real.pi_gt_three]
+  · have hbranch : ∀ s ∈ gerverStageIntervals 4, paperGerverVelocityComponents s =
+        (1 - 2 * GerverSofa.PartB.params.e1 * Real.sin s +
+            2 * GerverSofa.PartB.params.e2 * Real.cos s,
+          2 * GerverSofa.PartB.params.e1 * Real.cos s +
+            2 * GerverSofa.PartB.params.e2 * Real.sin s - 1 / 2) := by
+      intro s hs
+      rw [paperGerverContactData_properties.2.2.2 4 s hs]
+      rfl
+    have hα : HasDerivAt (fun s : ℝ ↦ 1 - 2 * GerverSofa.PartB.params.e1 * Real.sin s +
+        2 * GerverSofa.PartB.params.e2 * Real.cos s)
+        (-(2 * GerverSofa.PartB.params.e1 * Real.cos t) -
+          2 * GerverSofa.PartB.params.e2 * Real.sin t) t := by
+      have h := ((hasDerivAt_const t (1 : ℝ)).sub
+        ((Real.hasDerivAt_sin t).const_mul (2 * GerverSofa.PartB.params.e1))).add
+        ((Real.hasDerivAt_cos t).const_mul (2 * GerverSofa.PartB.params.e2))
+      convert h using 1
+      ring
+    exact ⟨-(1 / 2), by norm_num,
+      hasDerivWithinAt_paperGerverContacts_one ht hα hbranch (by ring)⟩
+
+/-- On each of the first two stages the fourth contact curve moves strictly forwards along
+the normal direction: its one-sided derivative is a positive multiple of `u_t`.  On the first
+stage the normal speed is the exact constant `1/2`; on the second it is `1 + b₁ - t/2`,
+positive because `t ≤ θ ≤ 7/10` and `b₁ ≥ -53/100`. -/
+theorem hasDerivWithinAt_paperGerverContacts_three_pos_smul {i : Fin 5} (hi : i = 0 ∨ i = 1)
+    {t : ℝ} (ht : t ∈ gerverStageIntervals i) :
+    ∃ c : ℝ, 0 < c ∧
+      HasDerivWithinAt (fun s ↦ paperGerverContacts s 3)
+        (c • normalVector (t : Real.Angle)) (gerverStageIntervals i) t := by
+  rcases hi with rfl | rfl
+  · have hbranch : ∀ s ∈ gerverStageIntervals 0, paperGerverVelocityComponents s =
+        (-2 * GerverSofa.PartB.params.a1 * Real.sin s +
+            2 * GerverSofa.PartB.params.a2 * Real.cos s + 1 / 2,
+          2 * GerverSofa.PartB.params.a1 * Real.cos s +
+            2 * GerverSofa.PartB.params.a2 * Real.sin s - 1) := by
+      intro s hs
+      rw [paperGerverContactData_properties.2.2.2 0 s hs]
+      rfl
+    have hβ : HasDerivAt (fun s : ℝ ↦ 2 * GerverSofa.PartB.params.a1 * Real.cos s +
+        2 * GerverSofa.PartB.params.a2 * Real.sin s - 1)
+        (-(2 * GerverSofa.PartB.params.a1 * Real.sin t) +
+          2 * GerverSofa.PartB.params.a2 * Real.cos t) t := by
+      have h := (((Real.hasDerivAt_cos t).const_mul (2 * GerverSofa.PartB.params.a1)).add
+        ((Real.hasDerivAt_sin t).const_mul (2 * GerverSofa.PartB.params.a2))).sub_const (1 : ℝ)
+      convert h using 1
+      ring
+    exact ⟨1 / 2, by norm_num,
+      hasDerivWithinAt_paperGerverContacts_three ht hβ hbranch (by ring)⟩
+  · have hbranch : ∀ s ∈ gerverStageIntervals 1, paperGerverVelocityComponents s =
+        (1 + 2 * GerverSofa.PartB.params.b1 - s,
+          -(1 / 4 : ℝ) * s * s + GerverSofa.PartB.params.b1 * s +
+            GerverSofa.PartB.params.b2 + 1 / 2) := by
+      intro s hs
+      rw [paperGerverContactData_properties.2.2.2 1 s hs]
+      rfl
+    have hβ : HasDerivAt (fun s : ℝ ↦ -(1 / 4 : ℝ) * s * s +
+        GerverSofa.PartB.params.b1 * s + GerverSofa.PartB.params.b2 + 1 / 2)
+        (-(t / 2) + GerverSofa.PartB.params.b1) t := by
+      have h := ((((hasDerivAt_id' t).const_mul (-(1 / 4) : ℝ)).mul (hasDerivAt_id' t)).add
+        ((hasDerivAt_id' t).const_mul GerverSofa.PartB.params.b1)).add_const
+          GerverSofa.PartB.params.b2 |>.add_const (1 / 2 : ℝ)
+      convert h using 1
+      ring
+    refine ⟨1 + GerverSofa.PartB.params.b1 - t / 2, ?_,
+      hasDerivWithinAt_paperGerverContacts_three ht hβ hbranch (by ring)⟩
+    rw [gerverStageIntervals_one] at ht
+    linarith [gerverDirectBox_b1_lower_bound GerverSofa.PartB.params_mem,
+      gerverStageTimes_two_le_seven_div_ten, ht.2]
+
+/-! ### Frame coordinates of the direct path -/
+
+/-- The derivative of the direct Gerver path in the moving frame. -/
+theorem hasDerivAt_paperGerverPath_frame (t : ℝ) :
+    HasDerivAt paperGerverPath
+      ((paperGerverVelocityComponents t).1 • normalVector (t : Real.Angle) +
+        (paperGerverVelocityComponents t).2 • tangentVector (t : Real.Angle)) t := by
+  have h := (contDiff_paperGerverPath.differentiable one_ne_zero t).hasDerivAt
+  rwa [deriv_paperGerverPath_eq_smul_add_smul rfl] at h
+
+/-- The angular derivative of the normal frame coordinate of the Gerver path is the tangent
+coordinate of the second contact curve: with `B = x + α v` the frame derivative `u' = v`
+contributes `x ⋅ v` and the normal component of `x'` contributes `α`. -/
+theorem hasDerivAt_inner_paperGerverPath_normalVector (t : ℝ) :
+    HasDerivAt (fun s : ℝ ↦ inner ℝ (paperGerverPath s) (normalVector (s : Real.Angle)))
+      (inner ℝ (paperGerverContacts t 1) (tangentVector (t : Real.Angle))) t := by
+  have h := (hasDerivAt_paperGerverPath_frame t).inner ℝ (hasDerivAt_normalVector t)
+  convert h using 1
+  change inner ℝ (paperGerverPath t + (paperGerverVelocityComponents t).1 •
+    tangentVector (t : Real.Angle)) (tangentVector (t : Real.Angle)) = _
+  rw [inner_add_left, inner_add_left, real_inner_smul_left, real_inner_smul_left,
+    real_inner_smul_left, inner_tangentVector_self, inner_normalVector_self,
+    inner_tangentVector_normalVector_real, sub_self, Real.sin_zero]
+  ring
+
+/-- The angular derivative of the tangent frame coordinate of the Gerver path is minus the
+normal coordinate of the fourth contact curve: with `D = x - β u` the frame derivative
+`v' = -u` contributes `-x ⋅ u` and the tangent component of `x'` contributes `β`. -/
+theorem hasDerivAt_inner_paperGerverPath_tangentVector (t : ℝ) :
+    HasDerivAt (fun s : ℝ ↦ inner ℝ (paperGerverPath s) (tangentVector (s : Real.Angle)))
+      (-inner ℝ (paperGerverContacts t 3) (normalVector (t : Real.Angle))) t := by
+  have h := (hasDerivAt_paperGerverPath_frame t).inner ℝ (hasDerivAt_tangentVector t)
+  convert h using 1
+  change -inner ℝ (paperGerverPath t - (paperGerverVelocityComponents t).2 •
+    normalVector (t : Real.Angle)) (normalVector (t : Real.Angle)) = _
+  rw [inner_sub_left, inner_add_left, real_inner_smul_left, real_inner_smul_left,
+    real_inner_smul_left, inner_neg_right, inner_normalVector_self, inner_tangentVector_self,
+    inner_normalVector_tangentVector]
+  ring
+
+/-! ### Stagewise monotonicity of the inner contact curves against a fixed direction -/
+
+/-- Against the tangent direction at a later angle `c`, the fourth Gerver contact curve is
+strictly antitone on each of its first two stages: its stage speed is a positive multiple of
+`u_s`, whose `v_c` coordinate is `sin (s - c) < 0`. -/
+theorem strictAntiOn_inner_paperGerverContacts_three {i : Fin 5} (hi : i = 0 ∨ i = 1) {c : ℝ}
+    (hlt : ∀ s ∈ gerverStageIntervals i, s < c)
+    (hwide : ∀ s ∈ gerverStageIntervals i, c - Real.pi < s) :
+    StrictAntiOn (fun s ↦ inner ℝ (paperGerverContacts s 3) (tangentVector (c : Real.Angle)))
+      (gerverStageIntervals i) := by
+  rw [show gerverStageIntervals i =
+    Set.Icc (gerverStageTimes i.castSucc) (gerverStageTimes i.succ) from rfl] at hlt hwide ⊢
+  have hspeed : ∀ s ∈ interior (Set.Icc (gerverStageTimes i.castSucc)
+      (gerverStageTimes i.succ)), ∃ cs : ℝ, 0 < cs ∧
+      derivWithin (fun r ↦ paperGerverContacts r 3)
+          (interior (Set.Icc (gerverStageTimes i.castSucc) (gerverStageTimes i.succ))) s =
+        cs • normalVector (s : Real.Angle) ∧
+      HasDerivWithinAt (fun r ↦ paperGerverContacts r 3) (cs • normalVector (s : Real.Angle))
+        (interior (Set.Icc (gerverStageTimes i.castSucc) (gerverStageTimes i.succ))) s := by
+    intro s hs
+    obtain ⟨cs, hcs, hder⟩ :=
+      hasDerivWithinAt_paperGerverContacts_three_pos_smul hi (interior_subset hs)
+    exact ⟨cs, hcs, (hder.mono interior_subset).derivWithin
+      (isOpen_interior.uniqueDiffWithinAt hs), hder.mono interior_subset⟩
+  refine strictAntiOn_of_hasDerivWithinAt_neg (convex_Icc _ _)
+    ((continuous_paperGerverContact 3).continuousOn.inner continuousOn_const)
+    (f' := fun s ↦ inner ℝ (derivWithin (fun r ↦ paperGerverContacts r 3)
+      (interior (Set.Icc (gerverStageTimes i.castSucc) (gerverStageTimes i.succ))) s)
+      (tangentVector (c : Real.Angle))) ?_ ?_
+  · intro s hs
+    obtain ⟨cs, -, hu, hder⟩ := hspeed s hs
+    rw [hu]
+    simpa using
+      hder.inner ℝ (hasDerivWithinAt_const s _ (tangentVector (c : Real.Angle)))
+  · intro s hs
+    obtain ⟨cs, hcs, hu, -⟩ := hspeed s hs
+    rw [hu, real_inner_smul_left,
+      real_inner_comm (tangentVector (c : Real.Angle)) (normalVector (s : Real.Angle)),
+      inner_tangentVector_normalVector_real]
+    exact mul_neg_of_pos_of_neg hcs (Real.sin_neg_of_neg_of_neg_pi_lt
+      (by linarith only [hlt s (interior_subset hs)])
+      (by linarith only [hwide s (interior_subset hs)]))
+
+/-- Against the normal direction at an earlier angle `c`, the second Gerver contact curve is
+strictly monotone on each of its last two stages: its stage speed is a negative multiple of
+`v_s`, whose `u_c` coordinate is `sin (c - s) < 0`. -/
+theorem strictMonoOn_inner_paperGerverContacts_one {i : Fin 5} (hi : i = 3 ∨ i = 4) {c : ℝ}
+    (hlt : ∀ s ∈ gerverStageIntervals i, c < s)
+    (hwide : ∀ s ∈ gerverStageIntervals i, s < c + Real.pi) :
+    StrictMonoOn (fun s ↦ inner ℝ (paperGerverContacts s 1) (normalVector (c : Real.Angle)))
+      (gerverStageIntervals i) := by
+  rw [show gerverStageIntervals i =
+    Set.Icc (gerverStageTimes i.castSucc) (gerverStageTimes i.succ) from rfl] at hlt hwide ⊢
+  have hspeed : ∀ s ∈ interior (Set.Icc (gerverStageTimes i.castSucc)
+      (gerverStageTimes i.succ)), ∃ cs : ℝ, cs < 0 ∧
+      derivWithin (fun r ↦ paperGerverContacts r 1)
+          (interior (Set.Icc (gerverStageTimes i.castSucc) (gerverStageTimes i.succ))) s =
+        cs • tangentVector (s : Real.Angle) ∧
+      HasDerivWithinAt (fun r ↦ paperGerverContacts r 1) (cs • tangentVector (s : Real.Angle))
+        (interior (Set.Icc (gerverStageTimes i.castSucc) (gerverStageTimes i.succ))) s := by
+    intro s hs
+    obtain ⟨cs, hcs, hder⟩ :=
+      hasDerivWithinAt_paperGerverContacts_one_neg_smul hi (interior_subset hs)
+    exact ⟨cs, hcs, (hder.mono interior_subset).derivWithin
+      (isOpen_interior.uniqueDiffWithinAt hs), hder.mono interior_subset⟩
+  refine strictMonoOn_of_hasDerivWithinAt_pos (convex_Icc _ _)
+    ((continuous_paperGerverContact 1).continuousOn.inner continuousOn_const)
+    (f' := fun s ↦ inner ℝ (derivWithin (fun r ↦ paperGerverContacts r 1)
+      (interior (Set.Icc (gerverStageTimes i.castSucc) (gerverStageTimes i.succ))) s)
+      (normalVector (c : Real.Angle))) ?_ ?_
+  · intro s hs
+    obtain ⟨cs, -, hu, hder⟩ := hspeed s hs
+    rw [hu]
+    simpa using
+      hder.inner ℝ (hasDerivWithinAt_const s _ (normalVector (c : Real.Angle)))
+  · intro s hs
+    obtain ⟨cs, hcs, hu, -⟩ := hspeed s hs
+    rw [hu, real_inner_smul_left, inner_tangentVector_normalVector_real]
+    exact mul_pos_of_neg_of_neg hcs (Real.sin_neg_of_neg_of_neg_pi_lt
+      (by linarith only [hlt s (interior_subset hs)])
+      (by linarith only [hwide s (interior_subset hs)]))
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Stage endpoints, grid angles and branch selection
+
+The certificate subdivides each of the five analytic stages of Gerver's sofa into
+`NN = 64` equal parts.  This module carries the real-valued mirror of that grid:
+`gerverStageTime` re-indexes the six stage endpoints by a natural number,
+`gerverGridTime m` is the `m`-th grid angle, `gerverContactPoint` is the dictionary of the
+four contact curves and the ambient path, and `GerverAreaCert.stageOf m` is the analytic
+branch that the piecewise definitions select at the `m`-th angle.  The soundness statements
+`endZ_sound` and `ttZ_sound` say that the integer data of the certificate encloses these
+real quantities; the remaining lemmas are the order facts the evaluator needs.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open GerverAreaCert
+
+/-- The six stage endpoints indexed by a natural number, constant past `5`. -/
+def gerverStageTime (s : ℕ) : ℝ := gerverStageTimes ⟨min 5 s, by omega⟩
+
+/-- The real grid angle at index `m`, mirroring `GerverAreaCert.ttZ`. -/
+def gerverGridTime (m : ℕ) : ℝ :=
+  (((NN - m % NN : ℕ) : ℝ) * gerverStageTime (m / NN) +
+    ((m % NN : ℕ) : ℝ) * gerverStageTime (m / NN + 1)) / (NN : ℝ)
+
+/-- The curve selected by a certificate `kind`: `0` the path, `1` `A`, `2` `B`, `3` `C`,
+`4` `D`. -/
+def gerverContactPoint : ℕ → ℝ → Point
+  | 0 => paperGerverPath
+  | 1 => fun t ↦ paperGerverContacts t 0
+  | 2 => fun t ↦ paperGerverContacts t 1
+  | 3 => fun t ↦ paperGerverContacts t 2
+  | _ => fun t ↦ paperGerverContacts t 3
+
+/-- The rotation interval starts at the first stage endpoint `0`. -/
+theorem gerverStageTime_zero : gerverStageTime 0 = 0 := gerverStageTimes_zero
+
+/-- The first stage ends at `φ`. -/
+theorem gerverStageTime_one : gerverStageTime 1 = GerverSofa.PartB.params.phi :=
+  gerverStageTimes_one
+
+/-- The second stage ends at `θ`. -/
+theorem gerverStageTime_two : gerverStageTime 2 = GerverSofa.PartB.params.theta :=
+  gerverStageTimes_two
+
+/-- The third stage ends at `η = π / 2 - θ`. -/
+theorem gerverStageTime_three : gerverStageTime 3 = GerverSofa.PartC.eta :=
+  gerverStageTimes_three
+
+/-- The fourth stage ends at `τ = π / 2 - φ`. -/
+theorem gerverStageTime_four : gerverStageTime 4 = GerverSofa.PartC.tau :=
+  gerverStageTimes_four
+
+/-- The fifth stage ends at `π / 2`. -/
+theorem gerverStageTime_five : gerverStageTime 5 = Real.pi / 2 := gerverStageTimes_five
+
+/-! ### Order properties of the stage endpoints and the grid -/
+
+/-- The five stage endpoints are strictly increasing. -/
+theorem gerverStageTime_lt_succ (s : ℕ) (hs : s < 5) :
+    gerverStageTime s < gerverStageTime (s + 1) := by
+  refine gerverStageTimes_strictMono ?_
+  simp only [Fin.mk_lt_mk]
+  omega
+
+/-- Past index `5` the `ℕ`-indexed stage endpoint is constantly `π / 2`. -/
+private theorem gerverStageTime_of_five_le {s : ℕ} (hs : 5 ≤ s) :
+    gerverStageTime s = Real.pi / 2 := by
+  simp only [gerverStageTime, min_eq_left hs]
+  exact gerverStageTimes_five
+
+/-- The stage-endpoint enclosure at every index, including those past `5`. -/
+private theorem endZ_sound_all (s : ℕ) : SI.Contains (endZ s) (gerverStageTime s) := by
+  obtain ⟨-, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, -, hphi, htheta⟩ :=
+    contains_params
+  rcases s with _ | _ | _ | _ | _ | _ | n
+  · rw [gerverStageTime_zero]
+    exact SI.contains_zero
+  · rw [gerverStageTime_one]
+    exact hphi
+  · rw [gerverStageTime_two]
+    exact htheta
+  · rw [show gerverStageTime 3 = Real.pi / 2 - GerverSofa.PartB.params.theta from
+      gerverStageTime_three]
+    exact SI.contains_sub contains_piHalfZ htheta
+  · rw [show gerverStageTime 4 = Real.pi / 2 - GerverSofa.PartB.params.phi from
+      gerverStageTime_four]
+    exact SI.contains_sub contains_piHalfZ hphi
+  · rw [gerverStageTime_five]
+    exact contains_piHalfZ
+  · rw [gerverStageTime_of_five_le (s := n + 6) (by omega)]
+    exact contains_piHalfZ
+
+/-- The integer stage endpoints enclose the real ones. -/
+theorem endZ_sound (s : ℕ) (hs : s ≤ 5) : SI.Contains (endZ s) (gerverStageTime s) := by
+  interval_cases s <;> exact endZ_sound_all _
+
+/-- The integer grid angles enclose the real ones. -/
+theorem ttZ_sound (m : ℕ) (hm : m ≤ 5 * NN) : SI.Contains (ttZ m) (gerverGridTime m) := by
+  have hNN : (0 : ℤ) < (NN : ℤ) := by norm_num [NN]
+  have hdiv : m / NN ≤ 5 := by
+    rw [show NN = 64 from rfl] at hm ⊢
+    omega
+  have key : SI.Contains
+      (SI.add (SI.imul ((NN - m % NN : ℕ) : ℤ) (endZ (m / NN)))
+        (SI.imul ((m % NN : ℕ) : ℤ) (endZ (m / NN + 1))))
+      (((NN - m % NN : ℕ) : ℝ) * gerverStageTime (m / NN) +
+        ((m % NN : ℕ) : ℝ) * gerverStageTime (m / NN + 1)) := by
+    refine SI.contains_add ?_ ?_
+    · have h := SI.contains_imul ((NN - m % NN : ℕ) : ℤ) (endZ_sound (m / NN) hdiv)
+      rwa [Int.cast_natCast] at h
+    · have h := SI.contains_imul ((m % NN : ℕ) : ℤ) (endZ_sound_all (m / NN + 1))
+      rwa [Int.cast_natCast] at h
+  have hcast : (((NN : ℤ)) : ℝ) = (NN : ℝ) := by push_cast; ring
+  rw [gerverGridTime, ← hcast]
+  exact SI.contains_divn hNN key
+
+/-- The grid step is uniform across stage joins. -/
+theorem gerverGridTime_succ_sub (m : ℕ) :
+    gerverGridTime (m + 1) - gerverGridTime m =
+      (gerverStageTime (m / NN + 1) - gerverStageTime (m / NN)) / (NN : ℝ) := by
+  have hNNR : (0 : ℝ) < (NN : ℝ) := by norm_num [NN]
+  have hr : m % NN < NN := Nat.mod_lt _ (by norm_num [NN])
+  have hsplit : ((m + 1) % NN = m % NN + 1 ∧ (m + 1) / NN = m / NN) ∨
+      (m % NN + 1 = NN ∧ (m + 1) % NN = 0 ∧ (m + 1) / NN = m / NN + 1) := by
+    rw [show NN = 64 from rfl]
+    omega
+  rcases hsplit with ⟨hmod, hdiv⟩ | ⟨hfull, hmod, hdiv⟩
+  · rw [gerverGridTime, gerverGridTime, hmod, hdiv,
+      Nat.cast_sub (by omega : m % NN + 1 ≤ NN), Nat.cast_sub hr.le]
+    push_cast
+    field_simp
+    ring
+  · rw [gerverGridTime, gerverGridTime, hmod, hdiv, Nat.cast_sub hr.le]
+    have hrR : ((m % NN : ℕ) : ℝ) = (NN : ℝ) - 1 := by
+      have : ((m % NN + 1 : ℕ) : ℝ) = ((NN : ℕ) : ℝ) := by rw [hfull]
+      push_cast at this
+      linarith
+    rw [hrR]
+    simp only [Nat.sub_zero]
+    push_cast
+    field_simp
+    ring
+
+/-- The grid angles are strictly increasing. -/
+theorem gerverGridTime_lt_succ (m : ℕ) (hm : m < 5 * NN) :
+    gerverGridTime m < gerverGridTime (m + 1) := by
+  have hNNR : (0 : ℝ) < (NN : ℝ) := by norm_num [NN]
+  have hlt : m / NN < 5 := (Nat.div_lt_iff_lt_mul (by norm_num [NN])).mpr hm
+  have hstep := gerverGridTime_succ_sub m
+  have hstage := gerverStageTime_lt_succ (m / NN) hlt
+  have hpos : 0 < (gerverStageTime (m / NN + 1) - gerverStageTime (m / NN)) / (NN : ℝ) :=
+    div_pos (by linarith) hNNR
+  linarith
+
+/-- Strict monotonicity of the grid angles below the top index, from `gerverGridTime_lt_succ`. -/
+theorem gerverGridTime_lt_of_lt {m n : ℕ} (hmn : m < n) (hn : n ≤ 5 * NN) :
+    gerverGridTime m < gerverGridTime n := by
+  induction n with
+  | zero => omega
+  | succ n ih =>
+    rcases Nat.lt_succ_iff_lt_or_eq.1 hmn with h | h
+    · exact (ih h (by omega)).trans (gerverGridTime_lt_succ n (by omega))
+    · subst h
+      exact gerverGridTime_lt_succ m (by omega)
+
+/-- Weak monotonicity of the grid angles below the top index, from `gerverGridTime_lt_succ`. -/
+theorem gerverGridTime_le_of_le {m n : ℕ} (hmn : m ≤ n) (hn : n ≤ 5 * NN) :
+    gerverGridTime m ≤ gerverGridTime n := by
+  rcases eq_or_lt_of_le hmn with rfl | h
+  · exact le_rfl
+  · exact (gerverGridTime_lt_of_lt h hn).le
+
+/-- The grid angle at a stage boundary is the stage endpoint itself. -/
+theorem gerverGridTime_mul_NN (s : ℕ) : gerverGridTime (s * NN) = gerverStageTime s := by
+  have hNN : 0 < NN := by norm_num [NN]
+  have hne : (NN : ℝ) ≠ 0 := by norm_num [NN]
+  rw [gerverGridTime, Nat.mul_mod_left, Nat.mul_div_cancel _ hNN]
+  simp only [Nat.sub_zero, Nat.cast_zero, zero_mul, add_zero]
+  exact mul_div_cancel_left₀ _ hne
+
+/-- The grid starts at `0`. -/
+theorem gerverGridTime_zero : gerverGridTime 0 = 0 := by
+  have h := gerverGridTime_mul_NN 0
+  rw [zero_mul] at h
+  rw [h, gerverStageTime_zero]
+
+/-- The grid ends at `π / 2`. -/
+theorem gerverGridTime_top : gerverGridTime (5 * NN) = Real.pi / 2 := by
+  rw [gerverGridTime_mul_NN 5, gerverStageTime_five]
+
+/-- Every grid angle lies in the rotation interval. -/
+theorem gerverGridTime_mem_Icc (m : ℕ) (hm : m ≤ 5 * NN) :
+    gerverGridTime m ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := by
+  refine ⟨?_, ?_⟩
+  · rw [← gerverGridTime_zero]
+    exact gerverGridTime_le_of_le (Nat.zero_le m) hm
+  · rw [← gerverGridTime_top]
+    exact gerverGridTime_le_of_le hm le_rfl
+
+/-- The branch index is one of the five stages. -/
+theorem stageOf_mem (m : ℕ) (hm : m ≤ 5 * NN) : 1 ≤ stageOf m ∧ stageOf m ≤ 5 := by
+  have hNN : 0 < NN := by norm_num [NN]
+  have hd : m / NN ≤ 5 := Nat.div_le_of_le_mul (by rw [Nat.mul_comm]; exact hm)
+  rw [stageOf]
+  split_ifs with h
+  · exact ⟨le_max_left _ _, max_le (by omega) hd⟩
+  · have hmlt : m < 5 * NN := by
+      rcases lt_or_eq_of_le hm with h' | h'
+      · exact h'
+      · subst h'; exact absurd (Nat.mul_mod_left 5 NN) h
+    have hd4 : m / NN < 5 := (Nat.div_lt_iff_lt_mul hNN).mpr hmlt
+    exact ⟨Nat.le_add_left 1 _, hd4⟩
+
+/-- The grid angle does not exceed the right endpoint of its branch. -/
+theorem gerverGridTime_le_stageTime (m : ℕ) (hm : m ≤ 5 * NN) :
+    gerverGridTime m ≤ gerverStageTime (stageOf m) := by
+  have hNN : 0 < NN := by norm_num [NN]
+  rw [stageOf]
+  split_ifs with h
+  · have hmm : m / NN * NN = m := Nat.div_mul_cancel (Nat.dvd_of_mod_eq_zero h)
+    have key : gerverGridTime m = gerverStageTime (m / NN) := by
+      have h' := gerverGridTime_mul_NN (m / NN)
+      rwa [hmm] at h'
+    rcases Nat.eq_zero_or_pos (m / NN) with h0 | h0
+    · have hm0 : m = 0 := by rw [← hmm, h0, zero_mul]
+      rw [h0, show max 1 0 = 1 from rfl, hm0, gerverGridTime_zero]
+      have h01 := gerverStageTime_lt_succ 0 (by norm_num)
+      rw [gerverStageTime_zero] at h01
+      exact h01.le
+    · rw [max_eq_right h0, key]
+  · have hmlt : m < 5 * NN := by
+      rcases lt_or_eq_of_le hm with h' | h'
+      · exact h'
+      · subst h'; exact absurd (Nat.mul_mod_left 5 NN) h
+    have hd4 : m / NN < 5 := (Nat.div_lt_iff_lt_mul hNN).mpr hmlt
+    have hlt : m < (m / NN + 1) * NN := (Nat.div_lt_iff_lt_mul hNN).mp (Nat.lt_succ_self _)
+    have hub : (m / NN + 1) * NN ≤ 5 * NN := Nat.mul_le_mul_right NN hd4
+    calc gerverGridTime m ≤ gerverGridTime ((m / NN + 1) * NN) :=
+          gerverGridTime_le_of_le hlt.le hub
+      _ = gerverStageTime (m / NN + 1) := gerverGridTime_mul_NN _
+
+/-- The grid angle strictly exceeds the left endpoint of its branch, except on the
+first branch. -/
+theorem stageTime_lt_gerverGridTime (m : ℕ) (hm : m ≤ 5 * NN) (h2 : 2 ≤ stageOf m) :
+    gerverStageTime (stageOf m - 1) < gerverGridTime m := by
+  have hNN : 0 < NN := by norm_num [NN]
+  have hd5 : m / NN ≤ 5 := Nat.div_le_of_le_mul (by rw [Nat.mul_comm]; exact hm)
+  rw [stageOf] at h2 ⊢
+  split_ifs at h2 ⊢ with h
+  · have hd : 2 ≤ m / NN := by
+      rcases le_max_iff.mp h2 with h' | h'
+      · exact absurd h' (by norm_num)
+      · exact h'
+    have hpos : 0 < m / NN := lt_of_lt_of_le (by norm_num) hd
+    have hmm : m / NN * NN = m := Nat.div_mul_cancel (Nat.dvd_of_mod_eq_zero h)
+    have key : gerverGridTime m = gerverStageTime (m / NN) := by
+      have h' := gerverGridTime_mul_NN (m / NN)
+      rwa [hmm] at h'
+    rw [max_eq_right (Nat.le_of_succ_le hd), key]
+    have h4 : m / NN - 1 < 5 := lt_of_le_of_lt (Nat.sub_le_sub_right hd5 1) (by norm_num)
+    have hlt := gerverStageTime_lt_succ (m / NN - 1) h4
+    rwa [Nat.sub_add_cancel hpos] at hlt
+  · rw [Nat.add_sub_cancel]
+    have hle : m / NN * NN ≤ m := Nat.div_mul_le_self m NN
+    have hne : m / NN * NN ≠ m := by
+      intro he
+      exact h (by rw [← he]; exact Nat.mul_mod_left _ _)
+    have key : gerverGridTime (m / NN * NN) = gerverStageTime (m / NN) := gerverGridTime_mul_NN _
+    rw [← key]
+    exact gerverGridTime_lt_of_lt (lt_of_le_of_ne hle hne) hm
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Properties of the Gerver niche roof
+
+The main results of this file describe the upper boundary of the paper's literal Gerver
+niche: the three graph pieces join up (`gerver_niche_piece_endpoints`), each piece stays
+inside the outer cap (`gerver_niche_roof_membership`), the roof is a strictly monotone
+positive graph (`gerver_niche_roof_strictMono`, `gerver_niche_roof_positive`), and the
+niche is exactly the strict vertical region under that roof (`gerver_niche_vertical_fills`).
+The three pieces are identified branch by branch in `gerverNicheRoof_of_le_two`,
+`gerverNicheRoof_mid` and `gerverNicheRoof_of_ge_three`, each valid on the closed stage.
+
+Alongside them sits the elementary API of the reverse-time reparametrization
+(`continuous_gerverRoofReverseTime`, `gerverRoofReverseTime_two`,
+`gerverRoofReverseTime_three`, `gerverRoofReverseTime_mem_Icc`) and the continuity of the
+roof, in both its real-parameter and its restricted form (`continuous_gerverRoofCurve`,
+`continuous_gerverNicheRoof`).
+
+Every one of them is the certified Part C niche geometry read through the coordinate
+dictionary `GerverSofa.PartF.Coordinates.toPlane`.  The dictionary itself is supplied by
+the public readers of the lower modules — `paperGerverContacts_one_eq_toPlane` and
+`paperGerverContacts_three_eq_toPlane` for the contact curves, `gerverStageTimes_zero`
+through `gerverStageTimes_five` for the stage times, and `mem_gerverLiteralNiche_iff`,
+`toPlane_mem_gerverOuterCap` for the two literal sets.  What remains here, and is kept
+private, is the roof-specific part of the dictionary: the reverse-time reparametrization
+and the identification of `gerverNicheRoof` with the certified upper arc.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+section Roof
+
+open GerverSofa.PartF.Coordinates
+open GerverSofa.PartC (params eta tau)
+
+/-- The roof reverse-time reparametrization is the certified affine reversal of the core
+stage, which sends `params.theta` to `tau` and `eta` to `params.phi`. -/
+private theorem gerverRoofReverseTime_eq (s : ℝ) :
+    gerverRoofReverseTime s = GerverSofa.PartC.Stage4.coreReverseTime s := by
+  rw [gerverRoofReverseTime, gerverStageTimes_four, gerverStageTimes_one,
+    gerverStageTimes_three, gerverStageTimes_two]
+  rfl
+
+/-- The Gerver roof is the certified upper niche arc, read in plane coordinates.  The three
+branches of `gerverNicheRoof` match the three branches of `nicheTopArc` piece for piece,
+with the same cut points `params.theta` and `eta`. -/
+private theorem gerverNicheRoof_eq_toPlane (s : Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    gerverNicheRoof s = toPlane (GerverSofa.PartC.Stage4.nicheTopArc s.val) := by
+  have hsI : (s : ℝ) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := s.2
+  simp only [gerverNicheRoof, GerverSofa.PartC.Stage4.nicheTopArc, gerverStageTimes_two,
+    gerverStageTimes_three, gerverRoofReverseTime_eq]
+  split_ifs with h1 h2
+  · exact paperGerverContacts_three_eq_toPlane hsI
+  · rfl
+  · exact paperGerverContacts_one_eq_toPlane hsI
+
+end Roof
+
+/-- The three graph pieces of the niche roof join up: the second contact curve at `eta`
+meets the ambient path at `params.phi`, the fourth contact curve at `params.theta` meets
+the ambient path at `tau`, and the two outer ends touch the wall. -/
+theorem gerver_niche_piece_endpoints :
+    paperGerverContacts (gerverStageTimes 3) 1 = paperGerverPath (gerverStageTimes 1) ∧
+    paperGerverContacts (gerverStageTimes 2) 3 = paperGerverPath (gerverStageTimes 4) ∧
+    paperGerverContacts (Real.pi / 2) 1 1 = 0 ∧ paperGerverContacts 0 3 1 = 0 := by
+  open GerverSofa.PartF.Coordinates GerverSofa.PartC GerverSofa.PartC.Stage4 in
+  obtain ⟨hB, hD, hBT, hD0⟩ := GerverSofa.PartC.Stage4.niche_piece_endpoints
+  have hpi : (0 : ℝ) < Real.pi / 2 := by positivity
+  have hTeq : GerverSofa.PartC.T = Real.pi / 2 := rfl
+  have hchain : 0 < params.theta ∧ params.theta < eta ∧ eta < tau ∧ tau < Real.pi / 2 :=
+    ⟨GerverSofa.PartC.Stage4.theta_pos, GerverSofa.PartC.Stage4.theta_lt_eta,
+      GerverSofa.PartC.Stage4.eta_lt_tau, hTeq ▸ GerverSofa.PartC.Stage4.tau_lt_T⟩
+  obtain ⟨hθ0, hθη, hητ, hτT⟩ := hchain
+  have hth : params.theta ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨hθ0.le, by linarith⟩
+  have heta : eta ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨by linarith, by linarith⟩
+  have hTm : (Real.pi / 2) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨hpi.le, le_rfl⟩
+  have h0m : (0 : ℝ) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨le_rfl, hpi.le⟩
+  refine ⟨?_, ?_, ?_, ?_⟩
+  · rw [gerverStageTimes_three, gerverStageTimes_one,
+      paperGerverContacts_one_eq_toPlane heta, paperGerverPath_eq_toPlane]
+    exact congrArg toPlane hB
+  · rw [gerverStageTimes_two, gerverStageTimes_four,
+      paperGerverContacts_three_eq_toPlane hth, paperGerverPath_eq_toPlane]
+    exact congrArg toPlane hD
+  · rw [paperGerverContacts_one_eq_toPlane hTm]
+    exact hBT
+  · rw [paperGerverContacts_three_eq_toPlane h0m]
+    exact hD0
+
+/-! ## The reverse-time reparametrization, and continuity of the roof -/
+
+/-- The reverse-time reparametrization is affine, hence continuous. -/
+theorem continuous_gerverRoofReverseTime : Continuous gerverRoofReverseTime := by
+  unfold gerverRoofReverseTime
+  fun_prop
+
+/-- The reverse-time map sends the start of the middle roof stage to the late path time. -/
+theorem gerverRoofReverseTime_two :
+    gerverRoofReverseTime (gerverStageTimes 2) = gerverStageTimes 4 := by
+  rw [gerverRoofReverseTime, sub_self, mul_zero, sub_zero]
+
+/-- The reverse-time map sends the end of the middle roof stage to the early path time. -/
+theorem gerverRoofReverseTime_three :
+    gerverRoofReverseTime (gerverStageTimes 3) = gerverStageTimes 1 := by
+  have h23 : gerverStageTimes 2 < gerverStageTimes 3 := gerverStageTimes_strictMono (by decide)
+  rw [gerverRoofReverseTime, div_mul_cancel₀ _ (sub_ne_zero.mpr h23.ne')]
+  ring
+
+/-- The reverse-time map carries the middle roof interval into the central path interval. -/
+theorem gerverRoofReverseTime_mem_Icc {s : ℝ} (h1 : gerverStageTimes 2 ≤ s)
+    (h2 : s ≤ gerverStageTimes 3) :
+    gerverRoofReverseTime s ∈ Set.Icc (gerverStageTimes 1) (gerverStageTimes 4) := by
+  have h12 : gerverStageTimes 1 < gerverStageTimes 2 := gerverStageTimes_strictMono (by decide)
+  have h23 : gerverStageTimes 2 < gerverStageTimes 3 := gerverStageTimes_strictMono (by decide)
+  have h34 : gerverStageTimes 3 < gerverStageTimes 4 := gerverStageTimes_strictMono (by decide)
+  have hd : (0 : ℝ) < gerverStageTimes 3 - gerverStageTimes 2 := by linarith
+  have hL : 0 < (gerverStageTimes 4 - gerverStageTimes 1) /
+      (gerverStageTimes 3 - gerverStageTimes 2) := div_pos (by linarith) hd
+  have hmul := mul_le_mul_of_nonneg_left (show s - gerverStageTimes 2 ≤
+    gerverStageTimes 3 - gerverStageTimes 2 by linarith) hL.le
+  rw [div_mul_cancel₀ _ hd.ne'] at hmul
+  have hnn : 0 ≤ (gerverStageTimes 4 - gerverStageTimes 1) /
+      (gerverStageTimes 3 - gerverStageTimes 2) * (s - gerverStageTimes 2) :=
+    mul_nonneg hL.le (by linarith)
+  rw [gerverRoofReverseTime]
+  constructor <;> linarith
+
+/-! ## The three graph pieces of the roof -/
+
+/-- Up to the second stage time the niche roof is the fourth contact curve. -/
+theorem gerverNicheRoof_of_le_two {s : ℝ} (hs : s ∈ Set.Icc (0 : ℝ) (Real.pi / 2))
+    (h : s ≤ gerverStageTimes 2) : gerverNicheRoof ⟨s, hs⟩ = paperGerverContacts s 3 :=
+  ite_eq_left h
+
+/-- On the middle stage the niche roof is the reverse-time ambient path.  The identification
+extends to the left endpoint of the stage by the piece-endpoint gluing. -/
+theorem gerverNicheRoof_mid {s : ℝ} (hs : s ∈ Set.Icc (0 : ℝ) (Real.pi / 2))
+    (h1 : gerverStageTimes 2 ≤ s) (h2 : s ≤ gerverStageTimes 3) :
+    gerverNicheRoof ⟨s, hs⟩ = paperGerverPath (gerverRoofReverseTime s) := by
+  rcases eq_or_lt_of_le h1 with heq | hlt
+  · rw [gerverNicheRoof_of_le_two hs heq.ge, ← heq, gerverRoofReverseTime_two]
+    exact gerver_niche_piece_endpoints.2.1
+  · rw [gerverNicheRoof, ite_eq_right (not_le.mpr hlt), ite_eq_left h2]
+
+/-- From the third stage time on the niche roof is the second contact curve.  The
+identification extends to the left endpoint of the stage by the piece-endpoint gluing. -/
+theorem gerverNicheRoof_of_ge_three {s : ℝ} (hs : s ∈ Set.Icc (0 : ℝ) (Real.pi / 2))
+    (h : gerverStageTimes 3 ≤ s) : gerverNicheRoof ⟨s, hs⟩ = paperGerverContacts s 1 := by
+  have h23 : gerverStageTimes 2 < gerverStageTimes 3 := gerverStageTimes_strictMono (by decide)
+  rcases eq_or_lt_of_le h with heq | hlt
+  · rw [gerverNicheRoof_mid hs (by linarith) heq.ge, ← heq, gerverRoofReverseTime_three]
+    exact gerver_niche_piece_endpoints.1.symm
+  · rw [gerverNicheRoof, ite_eq_right (not_le.mpr (by linarith)),
+      ite_eq_right (not_le.mpr hlt)]
+
+/-- On the rotation interval the real-parameter roof curve is the niche roof. -/
+theorem gerverRoofCurve_eq (s : Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    gerverRoofCurve s.val = gerverNicheRoof s := rfl
+
+/-- The roof curve is continuous: its three graph pieces join up at the two cut times, by the
+two endpoint identities of `gerver_niche_piece_endpoints`. -/
+theorem continuous_gerverRoofCurve : Continuous gerverRoofCurve := by
+  have h23 : gerverStageTimes 2 < gerverStageTimes 3 := gerverStageTimes_strictMono (by decide)
+  have hinner : Continuous fun s : ℝ ↦
+      if s ≤ gerverStageTimes 3 then paperGerverPath (gerverRoofReverseTime s)
+      else paperGerverContacts s 1 := by
+    refine continuous_if_le continuous_id continuous_const
+      ((contDiff_paperGerverPath.continuous.comp continuous_gerverRoofReverseTime).continuousOn)
+      ((continuous_paperGerverContact 1).continuousOn) ?_
+    intro s hs
+    rw [hs, gerverRoofReverseTime_three]
+    exact gerver_niche_piece_endpoints.1.symm
+  refine continuous_if_le continuous_id continuous_const
+    ((continuous_paperGerverContact 3).continuousOn) hinner.continuousOn ?_
+  intro s hs
+  rw [ite_eq_left (by rw [hs]; exact h23.le), hs, gerverRoofReverseTime_two]
+  exact gerver_niche_piece_endpoints.2.1
+
+/-- The niche roof is continuous. -/
+theorem continuous_gerverNicheRoof : Continuous gerverNicheRoof :=
+  continuous_gerverRoofCurve.comp continuous_subtype_val
+
+/-- The first coordinate of the niche roof is strictly increasing, so the roof really is a
+graph over the horizontal axis. -/
+theorem gerver_niche_roof_strictMono :
+    StrictMono (fun s : Set.Icc (0 : ℝ) (Real.pi / 2) ↦ gerverNicheRoof s 0) := by
+  intro a b hab
+  have h := GerverSofa.PartC.Stage4.nicheTopArc_fst_strictMono a.2 b.2 hab
+  simp only [gerverNicheRoof_eq_toPlane]
+  exact h
+
+/-- The niche roof has positive height strictly inside the rotation interval. -/
+theorem gerver_niche_roof_positive
+    (s : Set.Icc (0 : ℝ) (Real.pi / 2)) (hs : 0 < s.val ∧ s.val < Real.pi / 2) :
+    0 < gerverNicheRoof s 1 := by
+  rw [gerverNicheRoof_eq_toPlane]
+  exact GerverSofa.PartC.Stage4.nicheTopArc_y_pos ⟨hs.1, hs.2⟩
+
+/-- Each of the three graph pieces of the niche roof stays inside the paper outer cap. -/
+theorem gerver_niche_roof_membership :
+    (∀ t ∈ Set.Icc (gerverStageTimes 0) (gerverStageTimes 2),
+      paperGerverContacts t 3 ∈ gerverOuterCap) ∧
+    (∀ t ∈ Set.Icc (gerverStageTimes 1) (gerverStageTimes 4),
+      paperGerverPath t ∈ gerverOuterCap) ∧
+    (∀ t ∈ Set.Icc (gerverStageTimes 3) (gerverStageTimes 5),
+      paperGerverContacts t 1 ∈ gerverOuterCap) := by
+  open GerverSofa.PartF.Coordinates GerverSofa.PartC GerverSofa.PartC.Stage4 in
+  obtain ⟨hD, hx, hB⟩ := GerverSofa.PartC.Stage4.certified_roof_mem_K
+  have hTeq : GerverSofa.PartC.T = Real.pi / 2 := rfl
+  have hθ0 : 0 < params.theta := GerverSofa.PartC.Stage4.theta_pos
+  have hθη : params.theta < eta := GerverSofa.PartC.Stage4.theta_lt_eta
+  have hητ : eta < tau := GerverSofa.PartC.Stage4.eta_lt_tau
+  have hτT : tau < Real.pi / 2 := hTeq ▸ GerverSofa.PartC.Stage4.tau_lt_T
+  refine ⟨?_, ?_, ?_⟩
+  · intro t ht
+    rw [gerverStageTimes_zero, gerverStageTimes_two] at ht
+    have htI : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := ⟨ht.1, by linarith [ht.2]⟩
+    rw [paperGerverContacts_three_eq_toPlane htI]
+    exact toPlane_mem_gerverOuterCap (hD t ht)
+  · intro t ht
+    rw [gerverStageTimes_one, gerverStageTimes_four] at ht
+    exact toPlane_mem_gerverOuterCap (hx t ht)
+  · intro t ht
+    rw [gerverStageTimes_three, gerverStageTimes_five] at ht
+    have htI : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+      ⟨by linarith [ht.1], hTeq ▸ ht.2⟩
+    rw [paperGerverContacts_one_eq_toPlane htI]
+    exact toPlane_mem_gerverOuterCap (hB t ht)
+
+/-- The strict region between the wall and the graph of `f` over the parameter set `I`. -/
+def strictVerticalFill (f : ℝ → Point) (I : Set ℝ) : Set Point :=
+  {q | ∃ t ∈ I, q 0 = f t 0 ∧ 0 ≤ q 1 ∧ q 1 < f t 1}
+
+section Fills
+
+open GerverSofa.PartF.Coordinates
+open GerverSofa.PartC (params eta tau)
+
+/-- A strict vertical fill is the coordinate preimage of the certified vertical fill under
+the plane dictionary. -/
+private theorem strictVerticalFill_eq_preimage {F : ℝ → Point} {g : ℝ → GerverSofa.Point}
+    {I : Set ℝ} (h : ∀ t ∈ I, F t = toPlane (g t)) :
+    strictVerticalFill F I = fromPlane ⁻¹' GerverSofa.PartC.Stage4.verticalFill g I := by
+  ext q
+  constructor
+  · rintro ⟨t, ht, h0, h1, h2⟩
+    rw [h t ht] at h0 h2
+    exact ⟨t, ht, h0, h1, h2⟩
+  · rintro ⟨t, ht, h0, h1, h2⟩
+    refine ⟨t, ht, ?_, h1, ?_⟩
+    · rw [h t ht]; exact h0
+    · rw [h t ht]; exact h2
+
+end Fills
+
+/-- The paper literal niche is exactly the union of the three strict vertical fills under
+the three graph pieces of the niche roof. -/
+theorem gerver_niche_vertical_fills :
+    gerverLiteralNiche =
+      strictVerticalFill (fun t ↦ paperGerverContacts t 3)
+        (Set.Icc (gerverStageTimes 0) (gerverStageTimes 2)) ∪
+      strictVerticalFill paperGerverPath
+        (Set.Icc (gerverStageTimes 1) (gerverStageTimes 4)) ∪
+      strictVerticalFill (fun t ↦ paperGerverContacts t 1)
+        (Set.Icc (gerverStageTimes 3) (gerverStageTimes 5)) := by
+  open GerverSofa.PartF.Coordinates GerverSofa.PartC GerverSofa.PartC.Stage4 in
+  have hTeq : GerverSofa.PartC.T = Real.pi / 2 := rfl
+  have hθ0 : 0 < params.theta := GerverSofa.PartC.Stage4.theta_pos
+  have hθη : params.theta < eta := GerverSofa.PartC.Stage4.theta_lt_eta
+  have hητ : eta < tau := GerverSofa.PartC.Stage4.eta_lt_tau
+  have hτT : tau < Real.pi / 2 := hTeq ▸ GerverSofa.PartC.Stage4.tau_lt_T
+  have hDeq : ∀ t ∈ Set.Icc (0 : ℝ) params.theta,
+      (fun t ↦ paperGerverContacts t 3) t = toPlane (GerverSofa.PartC.D t) := fun t ht =>
+    paperGerverContacts_three_eq_toPlane ⟨ht.1, by linarith [ht.2]⟩
+  have hpeq : ∀ t ∈ Set.Icc params.phi tau,
+      paperGerverPath t = toPlane (GerverSofa.Romik.path params t) := fun _ _ => rfl
+  have hBeq : ∀ t ∈ Set.Icc eta GerverSofa.PartC.T,
+      (fun t ↦ paperGerverContacts t 1) t = toPlane (GerverSofa.PartC.B t) := fun t ht =>
+    paperGerverContacts_one_eq_toPlane ⟨by linarith [ht.1], hTeq ▸ ht.2⟩
+  rw [gerverStageTimes_zero, gerverStageTimes_two, gerverStageTimes_one, gerverStageTimes_four,
+    gerverStageTimes_three, gerverStageTimes_five, strictVerticalFill_eq_preimage hDeq,
+    strictVerticalFill_eq_preimage hpeq, strictVerticalFill_eq_preimage hBeq,
+    ← Set.preimage_union, ← Set.preimage_union]
+  exact Set.ext fun q =>
+    (mem_gerverLiteralNiche_iff q).trans
+      (by rw [GerverSofa.PartC.Stage4.niche_eq_certifiedNicheRegion]; rfl)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Cap.Tail.Arcs`.
+* `Cap.Tail.Space`.
+* `Cap.CornerMeasure`.
+* `Cap.CornerPaths`.
+* `Cap.Tail.Canonical`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Tail / Arcs
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A planar carrier set equipped with ordered start and end points. -/
+structure DirectedArcData where
+  /-- The set traced by the directed arc. -/
+  carrier : Set Point
+  /-- The starting point of the arc. -/
+  startPoint : Point
+  /-- The ending point of the arc. -/
+  endPoint : Point
+
+/-- The directed convex boundary arcs used for the right and left tail bodies. -/
+def rightLeftTailArcs (B D : ConvexBody Point) : DirectedArcData × DirectedArcData :=
+  (⟨convexBoundaryArc B (Real.pi + paperGerverConstants.2.1) (3 * Real.pi / 2),
+    (edgeVertices B ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle)).1,
+    (edgeVertices B ((3 * Real.pi / 2 : ℝ) : Real.Angle)).2⟩,
+   ⟨convexBoundaryArc D (3 * Real.pi / 2) (3 * Real.pi / 2 + paperGerverConstants.2.2),
+    (edgeVertices D ((3 * Real.pi / 2 : ℝ) : Real.Angle)).1,
+    (edgeVertices D ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle)).2⟩)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Tail / Space
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+open scoped NNReal ENNReal
+
+namespace MovingSofa
+
+/-- The real-angle inner-corner path of a cap. -/
+def capInnerCorner (K : RightAngleCapSpace) (t : ℝ) : Point :=
+  (rotatingHallwayParts (K.1 : Set Point) (t : Real.Angle)).innerCorner
+
+/-- The two surface densities on the upper half-circle, with the top atom excluded. -/
+def HasCapDensities (K : RightAngleCapSpace) (r s : ℝ → ℝ≥0) : Prop :=
+  Measurable r ∧ Measurable s ∧
+    (surfaceAreaMeasure K.1).restrict
+        ((fun t : ℝ ↦ (t : Real.Angle)) '' Set.Ico 0 (Real.pi / 2)) =
+      Measure.map (fun t : ℝ ↦ (t : Real.Angle))
+        ((volume.restrict (Set.Ico 0 (Real.pi / 2))).withDensity (fun t ↦ (r t : ℝ≥0∞))) ∧
+    (surfaceAreaMeasure K.1).restrict
+        ((fun t : ℝ ↦ (t : Real.Angle)) '' Set.Ioc (Real.pi / 2) Real.pi) =
+      Measure.map (fun t : ℝ ↦ ((t + Real.pi / 2 : ℝ) : Real.Angle))
+        ((volume.restrict (Set.Ioc 0 (Real.pi / 2))).withDensity (fun t ↦ (s t : ℝ≥0∞)))
+
+/-- The density, corner regularity and strict interior signs of the injectivity condition. -/
+def SatisfiesInjectivityCondition (K : RightAngleCapSpace) : Prop :=
+  (∃ r s : ℝ → ℝ≥0, HasCapDensities K r s ∧
+    ∀ r' s', HasCapDensities K r' s' →
+      (r =ᵐ[volume.restrict (Set.Icc 0 (Real.pi / 2))] r') ∧
+      (s =ᵐ[volume.restrict (Set.Icc 0 (Real.pi / 2))] s')) ∧
+    ContDiffOn ℝ 1 (capInnerCorner K) (Set.Icc 0 (Real.pi / 2)) ∧
+    ∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+      inner ℝ (derivWithin (capInnerCorner K) (Set.Icc 0 (Real.pi / 2)) t)
+        (normalVector (t : Real.Angle)) < 0 ∧
+      0 < inner ℝ (derivWithin (capInnerCorner K) (Set.Icc 0 (Real.pi / 2)) t)
+        (tangentVector (t : Real.Angle))
+
+/-- Right-angle caps satisfying injectivity and the cap-area threshold. -/
+def SpecialCapSpace :=
+  {K : RightAngleCapSpace // SatisfiesInjectivityCondition K ∧
+    (11 : ℝ) / 5 ≤ ClassicalResults.area (K.1 : Set Point)}
+
+/-- A special cap and two convex tails satisfying the support constraints. -/
+structure CapTailSpace where
+  /-- The special cap forming the central component of the triple. -/
+  cap : SpecialCapSpace
+  /-- The convex body used as the right tail. -/
+  rightBody : ConvexBody Point
+  /-- The convex body used as the left tail. -/
+  leftBody : ConvexBody Point
+  right_subset : (rightBody : Set Point) ⊆ (cap.1.1 : Set Point)
+  left_subset : (leftBody : Set Point) ⊆ (cap.1.1 : Set Point)
+  right_bound : ∀ t ∈ Set.Icc paperGerverConstants.2.1 (Real.pi / 2),
+    supportValue cap.1.1 (t : Real.Angle) +
+      supportValue rightBody ((Real.pi + t : ℝ) : Real.Angle) ≤ 1
+  right_eq : ∀ t ∈ ({paperGerverConstants.2.1, Real.pi / 2} : Set ℝ),
+    supportValue cap.1.1 (t : Real.Angle) +
+      supportValue rightBody ((Real.pi + t : ℝ) : Real.Angle) = 1
+  left_bound : ∀ t ∈ Set.Icc (0 : ℝ) paperGerverConstants.2.2,
+    supportValue cap.1.1 ((Real.pi / 2 + t : ℝ) : Real.Angle) +
+      supportValue leftBody ((3 * Real.pi / 2 + t : ℝ) : Real.Angle) ≤ 1
+  left_eq : ∀ t ∈ ({0, paperGerverConstants.2.2} : Set ℝ),
+    supportValue cap.1.1 ((Real.pi / 2 + t : ℝ) : Real.Angle) +
+      supportValue leftBody ((3 * Real.pi / 2 + t : ℝ) : Real.Angle) = 1
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Corner Measure
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- The normal and tangent components of the inner-corner path derivative. -/
+def capVelocityCoefficients (K : SpecialCapSpace) (t : ℝ) : ℝ × ℝ :=
+  (inner ℝ (derivWithin (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2)) t)
+      (normalVector (t : Real.Angle)),
+    inner ℝ (derivWithin (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2)) t)
+      (tangentVector (t : Real.Angle)))
+
+/-- The density formed by joining the two signed corner-velocity components over `[0, π]`. -/
+def capCornerDensity (K : SpecialCapSpace) (s : ℝ) : ℝ :=
+  if 0 < s ∧ s ≤ Real.pi / 2 then (capVelocityCoefficients K s).2
+  else if Real.pi / 2 < s ∧ s ≤ Real.pi then
+    -(capVelocityCoefficients K (s - Real.pi / 2)).1
+  else 0
+
+/-- The measure on the real angle interval defined by the nonnegative corner density. -/
+def capCornerMeasure (K : SpecialCapSpace) : Measure ℝ :=
+  (volume.restrict (Set.Icc 0 Real.pi)).withDensity
+    (fun s ↦ ENNReal.ofReal (capCornerDensity K s))
+
+/-- Push the corner-density measure to angles modulo a full turn. -/
+def capCornerAngleMeasure (K : SpecialCapSpace) : Measure Real.Angle :=
+  Measure.map (fun s : ℝ ↦ (s : Real.Angle)) (capCornerMeasure K)
+
+/-- Bundle the corner density and its real and angular measures. -/
+def capCornerMeasureData (K : SpecialCapSpace) :
+    (ℝ → ℝ) × Measure ℝ × Measure Real.Angle :=
+  (capCornerDensity K, capCornerMeasure K, capCornerAngleMeasure K)
+
+/-- The corner measure of a special cap reads its density on the angular image of every
+measurable subset of `[0, π]`. -/
+theorem capCornerAngleMeasure_angleImage_eq_setLIntegral (K : SpecialCapSpace) {T : Set ℝ}
+    (hT : MeasurableSet T) (hT' : T ⊆ Set.Icc 0 Real.pi) :
+    capCornerAngleMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' T) =
+      ∫⁻ s in T, ENNReal.ofReal (capCornerDensity K s) := by
+  have hturn : Real.pi ≤ -1 + 2 * Real.pi := by linarith [Real.pi_gt_three]
+  have hsub : Set.Icc (0 : ℝ) Real.pi ⊆ Set.Ioc (-1 : ℝ) Real.pi :=
+    fun x hx ↦ ⟨by linarith [hx.1], hx.2⟩
+  change Measure.map (fun s : ℝ ↦ (s : Real.Angle))
+      ((volume.restrict (Set.Icc 0 Real.pi)).withDensity
+        (fun s ↦ ENNReal.ofReal (capCornerDensity K s)))
+      ((fun s : ℝ ↦ (s : Real.Angle)) '' T) = _
+  exact Real.Angle.map_coe_withDensity_image_eq_setLIntegral hturn hsub hT hT'
+
+/-! ### Regularity and signs of the corner density
+
+The injectivity condition makes the inner corner continuously differentiable on the closed
+quarter turn, so both frame coefficients of its velocity are continuous there; the strict interior
+signs then extend to the two endpoints by continuity.  The corner density is the sum of the two
+branches extended by zero, whence its measurability and its bound. -/
+
+/-- The inner-corner velocity of a special cap is continuous on the cap domain. -/
+theorem continuousOn_derivWithin_capInnerCorner (K : SpecialCapSpace) :
+    ContinuousOn (derivWithin (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2)))
+      (Set.Icc 0 (Real.pi / 2)) :=
+  K.property.1.2.1.continuousOn_derivWithin
+    (uniqueDiffOn_Icc (by positivity : (0 : ℝ) < Real.pi / 2)) le_rfl
+
+/-- Both frame components of the inner-corner velocity are continuous on the cap domain. -/
+theorem continuousOn_capVelocityCoefficients (K : SpecialCapSpace) :
+    ContinuousOn (fun s ↦ (capVelocityCoefficients K s).1) (Set.Icc 0 (Real.pi / 2)) ∧
+      ContinuousOn (fun s ↦ (capVelocityCoefficients K s).2) (Set.Icc 0 (Real.pi / 2)) :=
+  ⟨(continuousOn_derivWithin_capInnerCorner K).inner
+      continuous_normalVector_real.continuousOn,
+    (continuousOn_derivWithin_capInnerCorner K).inner
+      lipschitzWith_tangentVector_real.continuous.continuousOn⟩
+
+/-- The corner density is bounded, being continuous on the two compact halves of the cap domain
+and zero outside. -/
+theorem exists_bound_capCornerDensity (K : SpecialCapSpace) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ s, capCornerDensity K s ≤ C := by
+  obtain ⟨hα, hβ⟩ := continuousOn_capVelocityCoefficients K
+  obtain ⟨C₁, hC₁⟩ := isCompact_Icc.exists_bound_of_continuousOn hα
+  obtain ⟨C₂, hC₂⟩ := isCompact_Icc.exists_bound_of_continuousOn hβ
+  refine ⟨max (max C₁ C₂) 0, le_max_right _ _, fun s ↦ ?_⟩
+  rw [capCornerDensity]
+  split_ifs with h1 h2
+  · exact le_max_of_le_left (le_max_of_le_right
+      ((le_abs_self _).trans (hC₂ s ⟨h1.1.le, h1.2⟩)))
+  · refine le_max_of_le_left (le_max_of_le_left ?_)
+    have h := neg_le_abs (capVelocityCoefficients K (s - Real.pi / 2)).1
+    exact h.trans (hC₁ _ ⟨by linarith [h2.1], by linarith [h2.2]⟩)
+  · exact le_max_right _ _
+
+/-! ### Finiteness and atomlessness of the corner measure -/
+
+instance isFiniteMeasure_capCornerMeasure (K : SpecialCapSpace) :
+    IsFiniteMeasure (capCornerMeasure K) := by
+  obtain ⟨C, -, hC⟩ := exists_bound_capCornerDensity K
+  refine ⟨?_⟩
+  rw [capCornerMeasure, withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ]
+  calc ∫⁻ s, ENNReal.ofReal (capCornerDensity K s) ∂volume.restrict (Set.Icc 0 Real.pi)
+      ≤ ∫⁻ _, ENNReal.ofReal C ∂volume.restrict (Set.Icc 0 Real.pi) :=
+        lintegral_mono fun s ↦ ENNReal.ofReal_le_ofReal (hC s)
+    _ < ⊤ := by
+        rw [lintegral_const, Measure.restrict_apply_univ, Real.volume_Icc]
+        exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top ENNReal.ofReal_lt_top
+
+instance isFiniteMeasure_capCornerAngleMeasure (K : SpecialCapSpace) :
+    IsFiniteMeasure (capCornerAngleMeasure K) := by
+  rw [capCornerAngleMeasure]
+  exact Measure.isFiniteMeasure_map _ _
+
+instance nullSingletonClass_capCornerAngleMeasure (K : SpecialCapSpace) :
+    NullSingletonClass (capCornerAngleMeasure K) := by
+  refine ⟨fun x ↦ ?_⟩
+  rw [capCornerAngleMeasure,
+    Measure.map_apply Real.Angle.continuous_coe.measurable (measurableSet_singleton x),
+    capCornerMeasure]
+  exact ((withDensity_absolutelyContinuous _ _).trans
+    Measure.restrict_le_self.absolutelyContinuous)
+    ((Real.Angle.countable_preimage_coe_singleton x).measure_zero volume)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Corner Paths
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The inner-corner path restricted and bundled as a continuous BV path. -/
+def capCornerBV (K : SpecialCapSpace) {a b : ℝ}
+    (h : Set.Icc a b ⊆ Set.Icc 0 (Real.pi / 2)) : ContinuousBVPaths a b :=
+  continuousBVOfContDiffOn (capInnerCorner K.val) (K.property.1.2.1.mono h)
+
+/-- The corner BV path between the two reflected Gerver switching angles. -/
+def capMiddleBV (K : SpecialCapSpace) :
+    ContinuousBVPaths paperGerverConstants.2.1 paperGerverConstants.2.2 :=
+  capCornerBV K (by
+    have hφ : 0 ≤ GerversSofa.φ := GerversSofa.ABφθSpec.existsUnique.choose_spec.1.1
+    intro t ht
+    change GerversSofa.φ ≤ t ∧ t ≤ Real.pi / 2 - GerversSofa.φ at ht
+    exact ⟨le_trans hφ ht.1, by linarith [ht.2]⟩)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Tail / Canonical
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The closed half-planes above the right and left inner supporting walls. -/
+def innerWallUpperHalfPlanes (K : RightAngleCapSpace) (t : ℝ) : Set Point × Set Point :=
+  (normalHalfPlane (t : Real.Angle) (supportValue K.1 (t : Real.Angle) - 1) true false,
+    normalHalfPlane ((t + Real.pi / 2 : ℝ) : Real.Angle)
+      (supportValue K.1 ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) true false)
+
+/-- The right and left canonical tail sets, before bundling their convex-body proofs. -/
+def canonicalTailSets (K : SpecialCapSpace) : Set Point × Set Point :=
+  ((K.1.1 : Set Point) ∩
+      ⋂ t ∈ Set.Icc paperGerverConstants.2.1 (Real.pi / 2),
+        (innerWallUpperHalfPlanes K.1 t).1,
+    (K.1.1 : Set Point) ∩
+      ⋂ t ∈ Set.Icc (0 : ℝ) paperGerverConstants.2.2,
+        (innerWallUpperHalfPlanes K.1 t).2)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/Development/Geometry/Foundations/Development006.lean b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development006.lean
new file mode 100644
index 0000000000..a2aed17cf5
--- /dev/null
+++ b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development006.lean
@@ -0,0 +1,10860 @@
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+module
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development005
+
+
+
+
+
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development002
+
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development003
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development001
+
+
+
+
+
+public import Mathlib.Analysis.Convex.Exposed
+public import Mathlib.Analysis.Convex.Join
+public import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
+public import Mathlib.MeasureTheory.Constructions.Polish.Basic
+public import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
+public import Mathlib.Topology.Connected.Clopen
+public import Mathlib.Topology.Connected.TotallyDisconnected
+public import Mathlib.Topology.MetricSpace.HausdorffDistance
+public import Mathlib.Topology.Order.IntermediateValue
+/-!
+# Moving sofa: related mathematical developments
+
+* `Motion.Foundations.Development001`.
+* `Bounds.Foundations.Development001`.
+* `Cap.Foundations.Development004`.
+* `Bounds.Foundations.Development002`.
+* `Geometry.Foundations.Development005`.
+* `Analysis.Foundations.Development005`.
+* `Convex.Foundations.Development003`.
+* `Convex.Foundations.Development004`.
+* `Area.Foundations.Development004`.
+* `Optimality`.
+* `Motion.Foundations.Development003`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Motion.Basic`.
+* `Motion.CommonSubset`.
+* `Motion.Compactness`.
+* `Motion.Rotation`.
+* `Motion.AngleLift`.
+* `Motion.RotationAngleCalculation`.
+* `Motion.SupportingHallways`.
+* `Motion.Translation`.
+* `Motion.StandardPosition`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Motion / Basic
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval
+
+namespace MovingSofa
+
+/-- A paper motion permits an initial translation and preserves orientation at every time. -/
+def IsPaperMotion (s : Set Point) (m : I → Point ≃ᵃⁱ[ℝ] Point) : Prop :=
+  IsConnected s ∧ IsClosed s ∧ Continuous m ∧
+    (∃ q : Point, ∀ p, m 0 p = p + q) ∧
+    (∀ t, ∃ a : Real.Angle, ∀ p, m t p = rotationMap a p + m t 0) ∧
+    m 0 '' s ⊆ horizontalHallway ∧
+    (∀ t, m t '' s ⊆ hallway) ∧ m 1 '' s ⊆ verticalHallway
+
+/-- Movability in the paper's translation-invariant convention. -/
+def IsPaperMovingSofa (s : Set Point) : Prop :=
+  ∃ m, IsPaperMotion s m
+
+/-- Clockwise rotation angle of a particular admissible lifted motion witness. -/
+def HasRotationAngle (s : Set Point) (ω : ℝ) : Prop :=
+  ∃ (m : I → Point ≃ᵃⁱ[ℝ] Point), IsPaperMotion s m ∧
+    ∃ α : I → ℝ, Continuous α ∧ α 0 = 0 ∧ α 1 = -ω ∧
+      ∀ t p, m t p = rotationMap (α t : Real.Angle) p + m t 0
+
+/-- Standard position for a compact moving sofa with the specified rotation angle. -/
+def IsStandardPosition (s : Set Point) (ω : ℝ) : Prop :=
+  IsCompact s ∧ HasRotationAngle s ω ∧ 0 < ω ∧ ω ≤ Real.pi / 2 ∧
+    supportValue s (ω : Real.Angle) = 1 ∧
+    supportValue s ((Real.pi / 2 : ℝ) : Real.Angle) = 1
+
+/-- The cap set constructed from all supporting outer quadrants. -/
+def capOfSofa (s : Set Point) (ω : ℝ) : Set Point :=
+  (stripParallelogram ω).1 ∩
+    ⋂ t ∈ Set.Icc 0 ω, (rotatingHallwayParts s (t : Real.Angle)).outerQuadrant
+
+/-- The intersection of supporting hallways used for monotonization. -/
+def monotonization (s : Set Point) (ω : ℝ) : Set Point :=
+  (stripParallelogram ω).1 ∩ ⋂ t ∈ Set.Icc 0 ω, supportingHallway s (t : Real.Angle)
+
+/-- A monotone sofa is the monotonization of a sofa in standard position. -/
+def IsMonotoneSofa (s : Set Point) : Prop :=
+  ∃ (s₀ : Set Point) (ω : ℝ), IsStandardPosition s₀ ω ∧ s = monotonization s₀ ω
+
+/-- The finite-angle outer approximation to a cap. -/
+def angleCap (Θ : AngleSet) (K : CapSpace Θ.angle) : Set Point :=
+  (stripParallelogram Θ.angle).1 ∩
+    ⋂ t ∈ Θ.directions, (rotatingHallwayParts (K.1 : Set Point) (t : Real.Angle)).outerQuadrant
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Motion / Common Subset
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem HasRotationAngle.exists_translated_rotated_hallway {s : Set Point} {ω : ℝ}
+    (hs : HasRotationAngle s ω) {t : ℝ} (ht : t ∈ Set.Icc 0 ω) :
+    ∃ v : Point, s ⊆ (fun p ↦ rotationMap (t : Real.Angle) p + v) '' hallway := by
+  obtain ⟨m, hm, α, hα, hα0, hα1, hmotion⟩ := hs
+  obtain ⟨_, _, _, _, _, _, hhallway, _⟩ := hm
+  obtain ⟨τ, hτ⟩ := mem_range_of_exists_le_of_exists_ge (c := -t) hα
+    ⟨1, by rw [hα1]; linarith [ht.2]⟩ ⟨0, by rw [hα0]; linarith [ht.1]⟩
+  refine ⟨-rotationMap (t : Real.Angle) (m τ 0), ?_⟩
+  intro p hp
+  refine ⟨m τ p, hhallway τ ⟨p, hp, rfl⟩, ?_⟩
+  rw [hmotion τ p, hτ]
+  simp only [rotationMap, Real.Angle.coe_neg, map_add,
+    ← EuclideanGeometry.o.rotation_symm, LinearIsometryEquiv.apply_symm_apply,
+    add_neg_cancel_right]
+
+theorem HasRotationAngle.exists_translated_horizontal_strip {s : Set Point} {ω : ℝ}
+    (hs : HasRotationAngle s ω) :
+    ∃ v : Point, s ⊆ (fun p ↦ p + v) '' (strips ω).1 := by
+  obtain ⟨m, hm, _⟩ := hs
+  obtain ⟨_, _, _, ⟨v, hv⟩, _, hstart, _, _⟩ := hm
+  refine ⟨-v, ?_⟩
+  intro p hp
+  have hmem := hstart ⟨p, hp, rfl⟩
+  obtain ⟨x, y, hxy, heq⟩ := hmem
+  refine ⟨p + v, ?_, by simp⟩
+  change 0 ≤ (p + v) 1 ∧ (p + v) 1 ≤ 1
+  rw [← hv p, ← heq]
+  exact ⟨hxy.2.1, hxy.2.2⟩
+
+theorem HasRotationAngle.exists_translated_vertical_strip {s : Set Point} {ω : ℝ}
+    (hs : HasRotationAngle s ω) :
+    ∃ v : Point, s ⊆ (fun p ↦ p + v) '' (strips ω).2.2 := by
+  obtain ⟨m, hm, α, _, _, hα1, hmotion⟩ := hs
+  obtain ⟨_, _, _, _, _, _, _, hend⟩ := hm
+  refine ⟨-rotationMap (ω : Real.Angle) (m 1 0), ?_⟩
+  intro p hp
+  have hmem := hend ⟨p, hp, rfl⟩
+  obtain ⟨x, y, hxy, heq⟩ := hmem
+  refine ⟨rotationMap (ω : Real.Angle) (m 1 p), ?_, ?_⟩
+  · refine ⟨m 1 p, ?_, rfl⟩
+    change 0 ≤ (m 1 p) 0 ∧ (m 1 p) 0 ≤ 1
+    rw [← heq]
+    exact ⟨hxy.1, hxy.2.1⟩
+  · rw [hmotion 1 p, hα1]
+    simp only [rotationMap, Real.Angle.coe_neg, map_add,
+      ← EuclideanGeometry.o.rotation_symm, LinearIsometryEquiv.apply_symm_apply,
+      add_neg_cancel_right]
+
+private theorem isBounded_of_oblique_bounds (s : Set Point) (a c d A B : ℝ)
+    (hc : 0 < c) (hd : 0 < d)
+    (hy : ∀ p ∈ s, a ≤ p 1 ∧ p 1 ≤ a + 1)
+    (hx : ∀ p ∈ s, c * p 0 + d * p 1 ≤ A ∧ -d * p 0 + c * p 1 ≤ B) :
+    Bornology.IsBounded s := by
+  let l := (c * a - B) / d
+  let u := (A - d * a) / c
+  let M := |l| + |u|
+  let N := |a| + |a + 1|
+  apply isBounded_iff_forall_norm_le.mpr
+  refine ⟨M + N, ?_⟩
+  intro p hp
+  obtain ⟨hy0, hy1⟩ := hy p hp
+  obtain ⟨hx0, hx1⟩ := hx p hp
+  have hl : l ≤ p 0 := by
+    apply (div_le_iff₀ hd).2
+    nlinarith
+  have hu : p 0 ≤ u := by
+    apply (le_div_iff₀ hc).2
+    nlinarith
+  have hM : 0 ≤ M := add_nonneg (abs_nonneg _) (abs_nonneg _)
+  have hN : 0 ≤ N := add_nonneg (abs_nonneg _) (abs_nonneg _)
+  have hxabs : |p 0| ≤ M := by
+    apply abs_le.mpr
+    dsimp [M]
+    constructor <;> linarith [neg_abs_le l, le_abs_self u, abs_nonneg l, abs_nonneg u]
+  have hyabs : |p 1| ≤ N := by
+    apply abs_le.mpr
+    dsimp [N]
+    constructor <;> linarith [neg_abs_le a, le_abs_self (a + 1),
+      abs_nonneg a, abs_nonneg (a + 1)]
+  have hx2 := pow_le_pow_left₀ (abs_nonneg (p 0)) hxabs 2
+  have hy2 := pow_le_pow_left₀ (abs_nonneg (p 1)) hyabs 2
+  have hnorm := EuclideanSpace.norm_sq_eq p
+  simp only [Fin.sum_univ_two, Real.norm_eq_abs, sq_abs] at hnorm hx2 hy2
+  nlinarith [mul_nonneg hM hN, norm_nonneg p]
+
+theorem HasRotationAngle.isCompact {s : Set Point} {ω : ℝ}
+    (hs : HasRotationAngle s ω) (hω : ω ∈ Set.Ioc 0 (Real.pi / 2)) : IsCompact s := by
+  obtain ⟨v, hv⟩ := hs.exists_translated_horizontal_strip
+  have ht : ω / 2 ∈ Set.Icc 0 ω := ⟨by linarith [hω.1], by linarith [hω.1]⟩
+  obtain ⟨w, hw⟩ := hs.exists_translated_rotated_hallway ht
+  have hc : 0 < Real.cos (ω / 2) := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [hω.1, Real.pi_pos], by linarith [hω.2, Real.pi_pos]⟩
+  have hd : 0 < Real.sin (ω / 2) := Real.sin_pos_of_pos_of_lt_pi
+    (by linarith [hω.1]) (by linarith [hω.2, Real.pi_pos])
+  have hb : Bornology.IsBounded s := isBounded_of_oblique_bounds s (v 1)
+      (Real.cos (ω / 2)) (Real.sin (ω / 2))
+      (1 + inner ℝ w (normalVector ((ω / 2 : ℝ) : Real.Angle)))
+      (1 + inner ℝ w (tangentVector ((ω / 2 : ℝ) : Real.Angle))) hc hd
+      (by
+        intro p hp
+        obtain ⟨q, hq, rfl⟩ := hv hp
+        change 0 ≤ q 1 ∧ q 1 ≤ 1 at hq
+        change v 1 ≤ q 1 + v 1 ∧ q 1 + v 1 ≤ v 1 + 1
+        constructor <;> linarith [hq.1, hq.2])
+      (by
+        intro p hp
+        obtain ⟨q, hq, rfl⟩ := hw hp
+        have hq01 : q 0 ≤ 1 ∧ q 1 ≤ 1 := by
+          rcases hq with ⟨a, b, h, rfl⟩ | ⟨a, b, h, rfl⟩
+          · exact ⟨h.1, h.2.2⟩
+          · exact ⟨h.2.1, h.2.2⟩
+        obtain ⟨hq0, hq1⟩ := hq01
+        have hn : inner ℝ (rotationMap ((ω / 2 : ℝ) : Real.Angle) q + w)
+            (normalVector ((ω / 2 : ℝ) : Real.Angle)) ≤
+              1 + inner ℝ w (normalVector ((ω / 2 : ℝ) : Real.Angle)) := by
+          rw [inner_add_left, inner_rotationMap_normalVector]
+          linarith
+        have ht : inner ℝ (rotationMap ((ω / 2 : ℝ) : Real.Angle) q + w)
+            (tangentVector ((ω / 2 : ℝ) : Real.Angle)) ≤
+              1 + inner ℝ w (tangentVector ((ω / 2 : ℝ) : Real.Angle)) := by
+          rw [inner_add_left, inner_rotationMap_tangentVector]
+          linarith
+        simpa [normalVector, tangentVector, frame, PiLp.inner_apply,
+          Fin.sum_univ_two, Real.inner_apply, Real.Angle.cos_coe, Real.Angle.sin_coe,
+          mul_comm] using And.intro hn ht)
+  obtain ⟨m, hm, _⟩ := hs
+  exact Metric.isCompact_iff_isClosed_bounded.mpr ⟨hm.2.1, hb⟩
+
+private theorem width_le_of_forall_mem_Icc {s : Set Point} (hne : s.Nonempty)
+    (f : Point → ℝ) (a b : ℝ) (h : ∀ p ∈ s, a ≤ f p ∧ f p ≤ b) :
+    sSup (f '' s) - sInf (f '' s) ≤ b - a := by
+  have hu : sSup (f '' s) ≤ b := csSup_le (hne.image f) (by
+    rintro _ ⟨p, hp, rfl⟩
+    exact (h p hp).2)
+  have hl : a ≤ sInf (f '' s) := le_csInf (hne.image f) (by
+    rintro _ ⟨p, hp, rfl⟩
+    exact (h p hp).1)
+  linarith
+
+theorem HasRotationAngle.horizontal_width_le {s : Set Point} {ω : ℝ}
+    (hs : HasRotationAngle s ω) :
+    sSup ((fun p : Point ↦ p 1) '' s) - sInf ((fun p : Point ↦ p 1) '' s) ≤ 1 := by
+  obtain ⟨v, hv⟩ := hs.exists_translated_horizontal_strip
+  obtain ⟨m, hm, _⟩ := hs
+  have hb := width_le_of_forall_mem_Icc hm.1.nonempty (fun p ↦ p 1) (v 1) (v 1 + 1)
+    (by
+      intro p hp
+      obtain ⟨q, hq, rfl⟩ := hv hp
+      change 0 ≤ q 1 ∧ q 1 ≤ 1 at hq
+      change v 1 ≤ q 1 + v 1 ∧ q 1 + v 1 ≤ v 1 + 1
+      constructor <;> linarith [hq.1, hq.2])
+  simpa using hb
+
+theorem HasRotationAngle.normal_width_le {s : Set Point} {ω : ℝ}
+    (hs : HasRotationAngle s ω) :
+    sSup ((fun p ↦ inner ℝ p (normalVector (ω : Real.Angle))) '' s) -
+      sInf ((fun p ↦ inner ℝ p (normalVector (ω : Real.Angle))) '' s) ≤ 1 := by
+  obtain ⟨v, hv⟩ := hs.exists_translated_vertical_strip
+  obtain ⟨m, hm, _⟩ := hs
+  have hb := width_le_of_forall_mem_Icc hm.1.nonempty
+    (fun p ↦ inner ℝ p (normalVector (ω : Real.Angle)))
+    (inner ℝ v (normalVector (ω : Real.Angle)))
+    (inner ℝ v (normalVector (ω : Real.Angle)) + 1) (by
+      intro p hp
+      obtain ⟨q, ⟨r, hr, rfl⟩, rfl⟩ := hv hp
+      change 0 ≤ r 0 ∧ r 0 ≤ 1 at hr
+      rw [inner_add_left, inner_rotationMap_normalVector]
+      constructor <;> linarith [hr.1, hr.2])
+  simpa using hb
+
+private theorem subset_Icc_of_width_le_of_csSup_eq {s : Set ℝ}
+    (hbelow : BddBelow s) (habove : BddAbove s)
+    (hwidth : sSup s - sInf s ≤ 1) (hsup : sSup s = 1) : s ⊆ Set.Icc 0 1 := by
+  intro x hx
+  have hl := csInf_le hbelow hx
+  have hu := le_csSup habove hx
+  rw [hsup] at hwidth hu
+  exact ⟨by linarith, hu⟩
+
+theorem IsStandardPosition.subset_strips {s : Set Point} {ω : ℝ}
+    (hs : IsStandardPosition s ω) : s ⊆ (strips ω).1 ∩ (strips ω).2.2 := by
+  obtain ⟨hcompact, hmotion, _, _, hnormal, hvertical⟩ := hs
+  have hcy : IsCompact ((fun p : Point ↦ p 1) '' s) := hcompact.image (by fun_prop)
+  have hcn : IsCompact ((fun p : Point ↦ inner ℝ p (normalVector (ω : Real.Angle))) '' s) :=
+    hcompact.image (by fun_prop)
+  have hsupy : sSup ((fun p : Point ↦ p 1) '' s) = 1 := by
+    simpa [supportValue, normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two] using hvertical
+  have hy := subset_Icc_of_width_le_of_csSup_eq hcy.bddBelow hcy.bddAbove
+    hmotion.horizontal_width_le hsupy
+  have hn := subset_Icc_of_width_le_of_csSup_eq hcn.bddBelow hcn.bddAbove
+    hmotion.normal_width_le hnormal
+  intro p hp
+  refine ⟨hy ⟨p, hp, rfl⟩, ?_⟩
+  obtain ⟨q, rfl⟩ := (EuclideanGeometry.o.rotation (ω : Real.Angle)).surjective p
+  refine ⟨q, ?_, rfl⟩
+  have hb := hn ⟨rotationMap (ω : Real.Angle) q, hp, rfl⟩
+  change 0 ≤ q 0 ∧ q 0 ≤ 1
+  simpa only [Set.mem_Icc, inner_rotationMap_normalVector] using hb
+
+theorem movingSofa_commonSubset (s : Set Point) (ω : ℝ)
+    (hs : HasRotationAngle s ω) (hω : ω ∈ Set.Ioc 0 (Real.pi / 2)) :
+    IsCompact s ∧
+    (∃ v : Point, s ⊆ (fun p ↦ p + v) '' (strips ω).1) ∧
+    (∃ v : Point, s ⊆ (fun p ↦ p + v) '' (strips ω).2.2) ∧
+    (∀ t ∈ Set.Icc 0 ω, ∃ v : Point,
+      s ⊆ (fun p ↦ rotationMap (t : Real.Angle) p + v) '' hallway) ∧
+    (sSup ((fun p : Point ↦ p 1) '' s) - sInf ((fun p : Point ↦ p 1) '' s) ≤ 1) ∧
+    (sSup ((fun p ↦ inner ℝ p (normalVector (ω : Real.Angle))) '' s) -
+      sInf ((fun p ↦ inner ℝ p (normalVector (ω : Real.Angle))) '' s) ≤ 1) ∧
+    (IsStandardPosition s ω → s ⊆ (strips ω).1 ∩ (strips ω).2.2) := by
+  exact ⟨hs.isCompact hω, hs.exists_translated_horizontal_strip,
+    hs.exists_translated_vertical_strip, fun _ ht ↦ hs.exists_translated_rotated_hallway ht,
+    hs.horizontal_width_le, hs.normal_width_le, fun hstd ↦ hstd.subset_strips⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Motion / Compactness
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Set MeasureTheory
+open scoped unitInterval
+
+namespace MovingSofa
+
+private theorem abs_le_of_affine_coordinate {a b c x y z : ℝ}
+    (ha : a ≠ 0) (hy : |y| ≤ 1) (hz : |z| ≤ 1) (heq : z = a * x + b * y + c) :
+    |x| ≤ (1 + |b| + |c|) / |a| := by
+  have hmul : |a| * |x| ≤ 1 + |b| + |c| := by
+    calc
+      |a| * |x| = |z - (b * y + c)| := by rw [← abs_mul]; congr 1; linarith
+      _ ≤ |z| + |b * y + c| := abs_sub _ _
+      _ ≤ 1 + (|b| * |y| + |c|) := by
+        grw [hz, abs_add_le, abs_mul]
+      _ ≤ 1 + |b| + |c| := by nlinarith [abs_nonneg b]
+  exact (le_div_iff₀ (abs_pos.mpr ha)).mpr (by nlinarith)
+
+private theorem norm_le_of_coordinate_bounds {p : Point} {C : ℝ}
+    (hC : 0 ≤ C) (hx : |p 0| ≤ C) (hy : |p 1| ≤ 1) : ‖p‖ ≤ C + 1 := by
+  have hn : ‖p‖ ^ 2 = (p 0) ^ 2 + (p 1) ^ 2 := by
+    simpa [Fin.sum_univ_two] using EuclideanSpace.real_norm_sq_eq p
+  have hx2 := sq_le_sq₀ (abs_nonneg (p 0)) hC |>.mpr hx
+  have hy2 := sq_le_sq₀ (abs_nonneg (p 1)) (by positivity : (0 : ℝ) ≤ 1) |>.mpr hy
+  rw [sq_abs] at hx2 hy2
+  nlinarith [norm_nonneg p]
+
+private theorem affineIsometry_coordinate (e : Point ≃ᵃⁱ[ℝ] Point) (p : Point) (i : Fin 2) :
+    e p i = (e.linearIsometryEquiv !₂[1, 0]) i * p 0 +
+      (e.linearIsometryEquiv !₂[0, 1]) i * p 1 + e 0 i := by
+  have hp : p = p 0 • !₂[1, 0] + p 1 • !₂[0, 1] := by
+    ext j
+    fin_cases j <;> simp
+  have he := e.map_vadd (0 : Point) p
+  change e (p + 0) = e.linearIsometryEquiv p + e 0 at he
+  rw [add_zero] at he
+  rw [he, hp, map_add, map_smul, map_smul]
+  simp [mul_comm]
+
+private theorem isBounded_affine_strip (e : Point ≃ᵃⁱ[ℝ] Point) (i : Fin 2)
+    (hi : (e.linearIsometryEquiv !₂[1, 0]) i ≠ 0) :
+    Bornology.IsBounded {p : Point | |p 1| ≤ 1 ∧ |e p i| ≤ 1} := by
+  let C := (1 + |(e.linearIsometryEquiv !₂[0, 1]) i| + |e 0 i|) /
+    |(e.linearIsometryEquiv !₂[1, 0]) i|
+  have hC : 0 ≤ C := by dsimp [C]; positivity
+  apply isBounded_iff_forall_norm_le.mpr
+  refine ⟨C + 1, fun p hp ↦ ?_⟩
+  apply norm_le_of_coordinate_bounds hC _ hp.1
+  exact abs_le_of_affine_coordinate hi hp.1 hp.2 (affineIsometry_coordinate e p i)
+
+private theorem abs_snd_le_one_of_mem_horizontal {p : Point}
+    (hp : p ∈ horizontalHallway) : |p 1| ≤ 1 := by
+  obtain ⟨x, y, hxy, rfl⟩ := hp
+  simpa using
+    (abs_le.mpr ⟨by linarith [hxy.2.1], hxy.2.2⟩ : |y| ≤ 1)
+
+private theorem abs_fst_le_one_of_mem_vertical {p : Point}
+    (hp : p ∈ verticalHallway) : |p 0| ≤ 1 := by
+  obtain ⟨x, y, hxy, rfl⟩ := hp
+  simpa using
+    (abs_le.mpr ⟨by linarith [hxy.1], hxy.2.1⟩ : |x| ≤ 1)
+
+private theorem isBounded_of_mixed_direction (s : Set Point)
+    (m : I → Point ≃ᵃⁱ[ℝ] Point) (h : IsMovingSofa s m) (t : I)
+    (h0 : ((m t).linearIsometryEquiv !₂[1, 0]) 0 ≠ 0)
+    (h1 : ((m t).linearIsometryEquiv !₂[1, 0]) 1 ≠ 0) : Bornology.IsBounded s := by
+  apply ((isBounded_affine_strip (m t) 0 h0).union
+    (isBounded_affine_strip (m t) 1 h1)).subset
+  intro p hp
+  have hy := abs_snd_le_one_of_mem_horizontal (h.initial hp)
+  rcases h.subset_hallway t ⟨p, hp, rfl⟩ with hhor | hvert
+  · exact Or.inr ⟨hy, abs_snd_le_one_of_mem_horizontal hhor⟩
+  · exact Or.inl ⟨hy, abs_fst_le_one_of_mem_vertical hvert⟩
+
+/-- A set admitting a canonical hallway motion is bounded. -/
+theorem IsMovingSofa.isBounded {s : Set Point}
+    {m : I → Point ≃ᵃⁱ[ℝ] Point} (h : IsMovingSofa s m) : Bornology.IsBounded s := by
+  let d : I → Point := fun t ↦ (m t).linearIsometryEquiv !₂[1, 0]
+  have hd : Continuous d := by
+    have he (t : I) : d t = m t !₂[1, 0] - m t 0 := by
+      have he := (m t).map_vadd (0 : Point) !₂[1, 0]
+      change m t (!₂[1, 0] + 0) = d t + m t 0 at he
+      rw [add_zero] at he
+      exact eq_sub_iff_add_eq.mpr he.symm
+    simp_rw [show d = (fun t ↦ m t !₂[1, 0] - m t 0) from funext he]
+    have hm : Continuous (fun t ↦ (m t).toAffineIsometry.toContinuousAffineMap) :=
+      continuous_induced_dom.comp h.continuous
+    exact (hm.eval_const _).sub (hm.eval_const _)
+  have hnorm (t : I) : (d t 0) ^ 2 + (d t 1) ^ 2 = 1 := by
+    have hn : ‖d t‖ = 1 := by
+      dsimp [d]
+      rw [(m t).linearIsometryEquiv.norm_map]
+      simp [EuclideanSpace.norm_eq, Fin.sum_univ_two]
+    have hs := EuclideanSpace.real_norm_sq_eq (d t)
+    simpa [hn, Fin.sum_univ_two] using hs.symm
+  by_cases hmix : ∃ t, d t 0 ≠ 0 ∧ d t 1 ≠ 0
+  · obtain ⟨t, h0, h1⟩ := hmix
+    exact isBounded_of_mixed_direction s m h t h0 h1
+  have hmem (t : I) : d t 0 ∈ ({-1, 0, 1} : Set ℝ) := by
+    by_cases h0 : d t 0 = 0
+    · simp [h0]
+    have h1 : d t 1 = 0 := by
+      by_contra hn
+      exact hmix ⟨t, h0, hn⟩
+    have hs : (d t 0) ^ 2 = (1 : ℝ) ^ 2 := by simpa [h1] using hnorm t
+    rcases (sq_eq_sq_iff_eq_or_eq_neg.mp hs) with he | he <;> simp [he]
+  have hc : Continuous (fun t ↦ d t 0) :=
+    (PiLp.proj (𝕜 := ℝ) 2 (fun _ : Fin 2 ↦ ℝ) 0).continuous.comp hd
+  have hconst (t : I) : d t 0 = d 0 0 :=
+    isPreconnected_univ.constant_of_mapsTo
+      (((finite_singleton (1 : ℝ)).insert 0).insert (-1)).isDiscrete
+      hc.continuousOn (fun t _ ↦ hmem t) (mem_univ t) (mem_univ 0)
+  have hd0 : d 0 0 = 1 := by simp only [d, h.zero]; rfl
+  have hdir : ((m 1).linearIsometryEquiv !₂[1, 0]) 0 ≠ 0 := by
+    change d 1 0 ≠ 0
+    rw [hconst, hd0]
+    norm_num
+  apply (isBounded_affine_strip (m 1) 0 hdir).subset
+  intro p hp
+  exact ⟨abs_snd_le_one_of_mem_horizontal (h.initial hp),
+    abs_fst_le_one_of_mem_vertical (h.final ⟨p, hp, rfl⟩)⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Motion / Rotation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Set
+open scoped unitInterval
+
+namespace MovingSofa
+
+/-- The linear part of a continuous rigid motion varies continuously on each vector. -/
+theorem continuous_motion_linear_apply (m : I → Point ≃ᵃⁱ[ℝ] Point)
+    (hm : Continuous m) (x : Point) :
+    Continuous (fun t ↦ (m t).linearIsometryEquiv x) := by
+  have hmc : Continuous (fun t ↦ (m t).toAffineIsometry.toContinuousAffineMap) :=
+    continuous_induced_dom.comp hm
+  have he (t : I) : (m t).linearIsometryEquiv x = m t x - m t 0 := by
+    have he := (m t).map_vadd (0 : Point) x
+    change m t (x + 0) = (m t).linearIsometryEquiv x + m t 0 at he
+    rw [add_zero] at he
+    exact eq_sub_iff_add_eq.mpr he.symm
+  simp_rw [he]
+  exact (hmc.eval_const x).sub (hmc.eval_const 0)
+
+/-- The linear parts of an identity-starting continuous rigid motion have positive determinant. -/
+theorem motion_linear_det_pos (m : I → Point ≃ᵃⁱ[ℝ] Point)
+    (hm : Continuous m) (hzero : m 0 = AffineIsometryEquiv.refl ℝ Point) (t : I) :
+    0 < LinearMap.det (m t).linearIsometryEquiv.toLinearEquiv.toLinearMap := by
+  let L : I → Point →L[ℝ] Point := fun t ↦ (m t).linearIsometryEquiv.toContinuousLinearEquiv
+  have hL : Continuous L := continuous_clm_apply.mpr
+    (continuous_motion_linear_apply m hm)
+  have hdet : Continuous (fun t ↦ (L t).det) := ContinuousLinearMap.continuous_det.comp hL
+  have hne (u : I) : (L u).det ≠ 0 := (m u).linearIsometryEquiv.toLinearEquiv.isUnit_det'.ne_zero
+  have hL0 : L 0 = ContinuousLinearMap.id ℝ Point := by
+    ext x
+    simp only [L, hzero]
+    rfl
+  have hzero' : (L 0).det = 1 := by rw [hL0]; simp [ContinuousLinearMap.det]
+  change 0 < (L t).det
+  by_contra hneg
+  have hle : (L t).det ≤ 0 := le_of_not_gt hneg
+  obtain ⟨u, hu⟩ := intermediate_value_univ t 0 hdet ⟨hle, by rw [hzero']; norm_num⟩
+  exact hne u hu
+
+/-- Each placement of an identity-starting continuous rigid motion is a rotation and translation. -/
+theorem exists_motion_rotation (m : I → Point ≃ᵃⁱ[ℝ] Point)
+    (hm : Continuous m) (hzero : m 0 = AffineIsometryEquiv.refl ℝ Point) (t : I) :
+    ∃ θ : Real.Angle, ∀ x, m t x = rotationMap θ x + m t 0 := by
+  obtain ⟨θ, hθ⟩ := EuclideanGeometry.o.exists_linearIsometryEquiv_eq_of_det_pos
+    (motion_linear_det_pos m hm hzero t)
+  refine ⟨θ, fun x ↦ ?_⟩
+  have he := (m t).map_vadd (0 : Point) x
+  change m t (x + 0) = (m t).linearIsometryEquiv x + m t 0 at he
+  simpa only [add_zero, hθ, rotationMap] using he
+
+/-- Translation followed by a varying rotation depends continuously on both parameters. -/
+theorem continuous_vaddConst_trans_rotation :
+    Continuous (fun q : Real.Angle × Point ↦ ((AffineIsometryEquiv.vaddConst ℝ q.2).trans
+        (EuclideanGeometry.o.rotation q.1).toAffineIsometryEquiv)) := by
+  rw [continuous_induced_rng]
+  apply ContinuousAffineMap.continuous_rng
+  · intro p
+    change Continuous (fun q : Real.Angle × Point ↦
+      (EuclideanGeometry.o.rotation q.1) (p + q.2))
+    simp only [Orientation.rotation_apply]
+    exact ((Real.Angle.continuous_cos.comp continuous_fst).smul
+      (continuous_const.add continuous_snd)).add
+      ((Real.Angle.continuous_sin.comp continuous_fst).smul
+        (EuclideanGeometry.o.rightAngleRotation.continuous.comp
+          (continuous_const.add continuous_snd)))
+  · have heq : (fun q : Real.Angle × Point ↦
+        (((AffineIsometryEquiv.vaddConst ℝ q.2).trans
+        (EuclideanGeometry.o.rotation
+          q.1).toAffineIsometryEquiv)).toAffineIsometry.toContinuousAffineMap.contLinear) =
+        (fun q : Real.Angle × Point ↦
+          q.1.cos • ContinuousLinearMap.id ℝ Point +
+          q.1.sin • EuclideanGeometry.o.rightAngleRotation.toContinuousLinearMap) := by
+      funext q
+      apply ContinuousLinearMap.ext
+      intro p
+      change (EuclideanGeometry.o.rotation q.1) p = _
+      exact EuclideanGeometry.o.rotation_apply q.1 p
+    change Continuous (fun q : Real.Angle × Point ↦
+      (((AffineIsometryEquiv.vaddConst ℝ q.2).trans
+        (EuclideanGeometry.o.rotation
+          q.1).toAffineIsometryEquiv)).toAffineIsometry.toContinuousAffineMap.contLinear)
+    rw [heq]
+    exact ((Real.Angle.continuous_cos.comp continuous_fst).smul continuous_const).add
+      ((Real.Angle.continuous_sin.comp continuous_fst).smul continuous_const)
+
+/-- A continuously varying rotation about the origin followed by a continuously varying
+translation is a continuous family of rigid motions. -/
+theorem continuous_rotation_trans_vaddConst {X : Type*} [TopologicalSpace X]
+    {θ : X → Real.Angle} {c : X → Point} (hθ : Continuous θ) (hc : Continuous c) :
+    Continuous (fun x ↦ (EuclideanGeometry.o.rotation (θ x)).toAffineIsometryEquiv.trans
+      (AffineIsometryEquiv.vaddConst ℝ (c x))) := by
+  rw [continuous_induced_rng]
+  apply ContinuousAffineMap.continuous_rng
+  · intro p
+    change Continuous (fun x ↦ (EuclideanGeometry.o.rotation (θ x)) p + c x)
+    simp only [Orientation.rotation_apply]
+    exact (((Real.Angle.continuous_cos.comp hθ).smul continuous_const).add
+      ((Real.Angle.continuous_sin.comp hθ).smul continuous_const)).add hc
+  · have heq : (fun x ↦
+        ((EuclideanGeometry.o.rotation (θ x)).toAffineIsometryEquiv.trans
+          (AffineIsometryEquiv.vaddConst ℝ
+            (c x))).toAffineIsometry.toContinuousAffineMap.contLinear) =
+        (fun x ↦ (θ x).cos • ContinuousLinearMap.id ℝ Point +
+            (θ x).sin • EuclideanGeometry.o.rightAngleRotation.toContinuousLinearMap) := by
+      funext x
+      apply ContinuousLinearMap.ext
+      intro p
+      change (EuclideanGeometry.o.rotation (θ x)) p = _
+      exact EuclideanGeometry.o.rotation_apply (θ x) p
+    change Continuous (fun x ↦
+      ((EuclideanGeometry.o.rotation (θ x)).toAffineIsometryEquiv.trans
+        (AffineIsometryEquiv.vaddConst ℝ
+          (c x))).toAffineIsometry.toContinuousAffineMap.contLinear)
+    rw [heq]
+    exact ((Real.Angle.continuous_cos.comp hθ).smul continuous_const).add
+      ((Real.Angle.continuous_sin.comp hθ).smul continuous_const)
+
+/-- Planar area is invariant under rotation about the origin, with no measurability
+hypothesis on the set. -/
+theorem area_image_rotationMap (α : Real.Angle) (S : Set Point) :
+    ClassicalResults.area (rotationMap α '' S) = ClassicalResults.area S := by
+  have himg : rotationMap α '' S = (EuclideanGeometry.o.rotation α).symm ⁻¹' S := by
+    ext p
+    simp only [rotationMap, Set.mem_image, Set.mem_preimage]
+    constructor
+    · rintro ⟨x, hx, rfl⟩
+      simpa using hx
+    · intro h
+      exact ⟨_, h, (EuclideanGeometry.o.rotation α).apply_symm_apply p⟩
+  change (MeasureTheory.volume (rotationMap α '' S)).toReal =
+    (MeasureTheory.volume S).toReal
+  rw [himg]
+  congr 1
+  exact (LinearIsometryEquiv.measurePreserving (EuclideanGeometry.o.rotation α).symm
+    (E := Point) (F := Point)).measure_preimage_emb
+    ((EuclideanGeometry.o.rotation α).symm.toHomeomorph.measurableEmbedding) S
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Motion / Angle Lift
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval
+
+namespace MovingSofa
+
+/-- An identity-starting continuous rigid motion has a normalized continuous real angle lift. -/
+theorem exists_continuous_motion_angle_lift (m : I → Point ≃ᵃⁱ[ℝ] Point)
+    (hm : Continuous m) (hzero : m 0 = AffineIsometryEquiv.refl ℝ Point) :
+    ∃ (α : I → ℝ) (p : I → Point), Continuous α ∧ Continuous p ∧
+      α 0 = 0 ∧ p 0 = 0 ∧ ∀ t x, m t x = rotationMap (α t : Real.Angle) x + p t := by
+  let e : Point := !₂[1, 0]
+  have he : e ≠ 0 := by
+    intro h
+    have := congrArg (fun p : Point ↦ p 0) h
+    norm_num [e] at this
+  let θ : I → Real.Angle := fun t ↦ EuclideanGeometry.o.oangle e
+    ((m t).linearIsometryEquiv e)
+  have hθ : Continuous θ := by
+    apply continuous_iff_continuousAt.mpr
+    intro t
+    have hne : (m t).linearIsometryEquiv e ≠ 0 := by
+      exact (m t).linearIsometryEquiv.map_ne_zero_iff.mpr he
+    have ha : ContinuousAt (fun z : Point × Point ↦ EuclideanGeometry.o.oangle z.1 z.2)
+        (e, (m t).linearIsometryEquiv e) :=
+      EuclideanGeometry.o.continuousAt_oangle he hne
+    have hp : ContinuousAt (fun u : I ↦ (e, (m u).linearIsometryEquiv e)) t :=
+      continuousAt_const.prodMk (continuous_motion_linear_apply m hm e).continuousAt
+    exact ContinuousAt.comp (f := fun u : I ↦ (e, (m u).linearIsometryEquiv e))
+      (x := t) ha hp
+  have hθ0 : θ 0 = 0 := by
+    have h0 : (m 0).linearIsometryEquiv e = e := by rw [hzero]; rfl
+    simp [θ, h0]
+  have hrot (t : I) : (m t).linearIsometryEquiv = EuclideanGeometry.o.rotation (θ t) := by
+    obtain ⟨u, hu⟩ := EuclideanGeometry.o.exists_linearIsometryEquiv_eq_of_det_pos
+      (motion_linear_det_pos m hm hzero t)
+    have ht : θ t = u := by
+      dsimp [θ]
+      rw [hu, EuclideanGeometry.o.oangle_rotation_self_right he]
+    rw [ht, hu]
+  obtain ⟨α, hα, hα0, hαθ⟩ := Real.Angle.exists_continuous_lift_zero θ hθ hθ0
+  have hp : Continuous (fun t ↦ m t (0 : Point)) := by
+    have hmc : Continuous (fun t ↦ (m t).toAffineIsometry.toContinuousAffineMap) :=
+      continuous_induced_dom.comp hm
+    exact hmc.eval_const 0
+  refine ⟨α, fun t ↦ m t 0, hα, hp, hα0, ?_, ?_⟩
+  · change m 0 0 = 0
+    rw [hzero]
+    rfl
+  · intro t x
+    have hx := (m t).map_vadd (0 : Point) x
+    change m t (x + 0) = (m t).linearIsometryEquiv x + m t 0 at hx
+    simpa only [add_zero, hrot t, hαθ t, rotationMap] using hx
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Motion / Rotation Angle Calculation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The rotation-angle interval from `arccos(5/11)` up to, but excluding, `π/2`. -/
+abbrev RotationCalculationAngle := Set.Ico (Real.arccos (5 / 11 : ℝ)) (Real.pi / 2)
+
+/-- The piecewise lower cutoff on the auxiliary distance used in the rotation estimate. -/
+def rotationCalculationMinimum (ω : RotationCalculationAngle) : ℝ :=
+  if ω.val < Real.arctan (11 / 5 : ℝ) then 5 / 4 else 11 / 10
+
+/-- Auxiliary radii and points determined by an admissible rotation angle and distance. -/
+def rotationCalculationValues (ω : RotationCalculationAngle)
+    (d : Set.Icc (rotationCalculationMinimum ω) (Real.tan ω.val)) :
+    ℝ × ℝ × Point × Point :=
+  let r := 1 - d.val * (Real.cos ω.val / Real.sin ω.val)
+  let g := Real.sqrt (1 - r ^ 2)
+  let o := (stripParallelogram ω.val).2.2
+  (r, g, o - tangentVector 0 + d.val • normalVector 0, o - g • normalVector 0)
+
+theorem rotationCalculation_convex (d : ℝ) (hd : 1 ≤ d) :
+    ConvexOn ℝ (Set.Icc (Real.pi / 4) (Real.pi / 2))
+      (fun t ↦ (1 - d * (Real.cos t / Real.sin t)) ^ 2) ∧
+    ConvexOn ℝ (Set.Icc (Real.pi / 4) (Real.pi / 2))
+      (fun t ↦ Real.cos t ^ 2) := by
+  let D := Set.Icc (Real.pi / 4) (Real.pi / 2)
+  have hsin (t : ℝ) (ht : t ∈ D) : Real.sin t ≠ 0 := by
+    have hpi : 0 < Real.pi := Real.pi_pos
+    have ht0 : 0 < t := lt_of_lt_of_le (by positivity : 0 < Real.pi / 4) ht.1
+    have htpi : t < Real.pi := lt_of_le_of_lt ht.2 (by linarith)
+    exact (Real.sin_pos_of_pos_of_lt_pi ht0 htpi).ne'
+  have hfirst (t : ℝ) (ht : t ∈ D) : HasDerivAt
+      (fun s ↦ (1 - d * (Real.cos s / Real.sin s)) ^ 2)
+      (2 * d * (1 - d * (Real.cos t / Real.sin t)) / Real.sin t ^ 2) t := by
+    have htrig : -(Real.sin t * Real.sin t) - Real.cos t * Real.cos t = -1 := by
+      nlinarith [Real.sin_sq_add_cos_sq t]
+    convert (((hasDerivAt_const t 1).sub
+      ((hasDerivAt_const t d).mul ((Real.hasDerivAt_cos t).div
+        (Real.hasDerivAt_sin t) (hsin t ht)))).pow 2) using 1
+    all_goals simp only [Pi.mul_apply, Pi.sub_apply, Pi.div_apply]
+    all_goals norm_num
+    all_goals simp only [htrig]
+    all_goals field_simp [hsin t ht]
+  have hsecond (t : ℝ) (ht : t ∈ D) : HasDerivAt
+      (fun s ↦ 2 * d * (1 - d * (Real.cos s / Real.sin s)) / Real.sin s ^ 2)
+      (2 * d / Real.sin t ^ 4 *
+        (d + 2 * d * Real.cos t ^ 2 - 2 * Real.sin t * Real.cos t)) t := by
+    have hs := hsin t ht
+    have htrig : -(Real.sin t * Real.sin t) - Real.cos t * Real.cos t = -1 := by
+      nlinarith [Real.sin_sq_add_cos_sq t]
+    convert (((hasDerivAt_const t (2 * d)).mul
+      ((hasDerivAt_const t 1).sub
+        ((hasDerivAt_const t d).mul ((Real.hasDerivAt_cos t).div
+          (Real.hasDerivAt_sin t) hs)))).div
+      ((Real.hasDerivAt_sin t).pow 2) (pow_ne_zero 2 hs)) using 1
+    all_goals simp only [Pi.mul_apply, Pi.sub_apply, Pi.div_apply, Pi.pow_apply]
+    all_goals norm_num
+    all_goals simp only [htrig]
+    all_goals field_simp [hs]
+    all_goals ring
+  constructor
+  · apply convexOn_of_hasDerivWithinAt2_nonneg
+      (f' := fun t ↦ 2 * d * (1 - d * (Real.cos t / Real.sin t)) / Real.sin t ^ 2)
+      (f'' := fun t ↦ 2 * d / Real.sin t ^ 4 *
+        (d + 2 * d * Real.cos t ^ 2 - 2 * Real.sin t * Real.cos t)) (convex_Icc _ _)
+    · fun_prop
+    · intro t ht
+      exact (hfirst t (interior_subset ht)).hasDerivWithinAt
+    · intro t ht
+      exact (hsecond t (interior_subset ht)).hasDerivWithinAt
+    · intro t ht
+      have hc : 2 * Real.sin t * Real.cos t ≤ 1 := by
+        nlinarith [sq_nonneg (Real.sin t - Real.cos t), Real.sin_sq_add_cos_sq t]
+      have : 0 ≤ d + 2 * d * Real.cos t ^ 2 - 2 * Real.sin t * Real.cos t := by
+        nlinarith [sq_nonneg (Real.cos t)]
+      positivity
+  · apply convexOn_of_hasDerivWithinAt2_nonneg
+      (f' := fun t ↦ -2 * Real.sin t * Real.cos t)
+      (f'' := fun t ↦ -2 * Real.cos (2 * t)) (convex_Icc _ _)
+    · fun_prop
+    · intro t ht
+      convert ((Real.hasDerivAt_cos t).mul (Real.hasDerivAt_cos t)).hasDerivWithinAt using 1 <;>
+        first | funext s; simp [Pi.mul_apply]; ring | ring
+    · intro t ht
+      convert (((Real.hasDerivAt_sin t).mul (Real.hasDerivAt_cos t)).const_mul (-2))
+        |>.hasDerivWithinAt using 1 <;>
+          first | funext s; simp [Pi.mul_apply]; ring |
+            rw [Real.cos_two_mul, ← Real.sin_sq_add_cos_sq t]; ring
+    · intro t ht
+      have htD := interior_subset ht
+      change t ∈ Set.Icc (Real.pi / 4) (Real.pi / 2) at htD
+      have h1 : Real.pi / 2 ≤ 2 * t := by
+        calc
+          Real.pi / 2 = 2 * (Real.pi / 4) := by ring
+          _ ≤ 2 * t := mul_le_mul_of_nonneg_left htD.1 (by norm_num)
+      have h2 : 2 * t ≤ Real.pi + Real.pi / 2 := by
+        calc
+          2 * t ≤ 2 * (Real.pi / 2) := mul_le_mul_of_nonneg_left htD.2 (by norm_num)
+          _ ≤ Real.pi + Real.pi / 2 := by linarith [Real.pi_pos]
+      exact mul_nonneg_of_nonpos_of_nonpos (by norm_num)
+        (Real.cos_nonpos_of_pi_div_two_le_of_le h1 h2)
+
+private def rotationBound (d t : ℝ) : ℝ :=
+  (1 - d * (Real.cos t / Real.sin t)) ^ 2 + 4 * Real.cos t ^ 2
+
+private theorem rotationBound_convex (d : ℝ) (hd : 1 ≤ d) :
+    ConvexOn ℝ (Set.Icc (Real.pi / 4) (Real.pi / 2)) (rotationBound d) := by
+  obtain ⟨h₁, h₂⟩ := rotationCalculation_convex d hd
+  exact h₁.add (h₂.smul (by norm_num : (0 : ℝ) ≤ 4))
+
+private theorem rotationBound_arccos :
+    rotationBound (5 / 4) (Real.arccos (5 / 11)) < 1 := by
+  have hs : Real.sin (Real.arccos (5 / 11)) = Real.sqrt 96 / 11 := by
+    rw [Real.sin_arccos]
+    norm_num
+  have hsq : Real.sqrt 96 ^ 2 = 96 := Real.sq_sqrt (by norm_num)
+  have hlo : 97 / 10 < Real.sqrt 96 := by
+    nlinarith [Real.sqrt_nonneg 96]
+  have hhi : Real.sqrt 96 < 49 / 5 := by
+    nlinarith [Real.sqrt_nonneg 96]
+  unfold rotationBound
+  rw [Real.cos_arccos (by norm_num) (by norm_num), hs]
+  have hpos : 0 < Real.sqrt 96 := by positivity
+  field_simp
+  nlinarith
+
+private theorem rotationBound_arctan_left :
+    rotationBound (5 / 4) (Real.arctan (11 / 5)) < 1 := by
+  unfold rotationBound
+  rw [Real.cos_arctan, Real.sin_arctan]
+  norm_num
+  have hs : Real.sqrt (146 : ℝ) ^ 2 = 146 := Real.sq_sqrt (by norm_num)
+  have hp : 0 < Real.sqrt (146 : ℝ) := by positivity
+  field_simp
+  nlinarith
+
+private theorem rotationBound_arctan_right :
+    rotationBound (11 / 10) (Real.arctan (11 / 5)) < 1 := by
+  unfold rotationBound
+  rw [Real.cos_arctan, Real.sin_arctan]
+  norm_num
+  have hs : Real.sqrt (146 : ℝ) ^ 2 = 146 := Real.sq_sqrt (by norm_num)
+  have hp : 0 < Real.sqrt (146 : ℝ) := by positivity
+  field_simp
+  nlinarith
+
+private theorem rotationBound_pi_div_two (d : ℝ) : rotationBound d (Real.pi / 2) = 1 := by
+  simp [rotationBound]
+
+/-- The three landmark angles of the rotation calculation are strictly ordered:
+`π / 4 < arccos (5 / 11) < arctan (11 / 5) < π / 2`. -/
+theorem rotationCalculation_angle_bounds :
+    Real.pi / 4 < Real.arccos (5 / 11 : ℝ) ∧
+    Real.arccos (5 / 11 : ℝ) < Real.arctan (11 / 5 : ℝ) ∧
+    Real.arctan (11 / 5 : ℝ) < Real.pi / 2 := by
+  have ha : Real.arccos (5 / 11 : ℝ) = Real.arctan (Real.sqrt 96 / 5) := by
+    rw [Real.arccos_eq_arctan (by norm_num)]
+    norm_num
+    ring
+  have hs : Real.sqrt 96 ^ 2 = 96 := Real.sq_sqrt (by norm_num)
+  have hpos := Real.sqrt_nonneg 96
+  refine ⟨?_, ?_, Real.arctan_lt_pi_div_two _⟩
+  · rw [ha, ← Real.arctan_one]
+    apply Real.arctan_strictMono
+    nlinarith
+  · rw [ha]
+    apply Real.arctan_strictMono
+    nlinarith
+
+private theorem rotationBound_lt_one (ω : RotationCalculationAngle) :
+    rotationBound (rotationCalculationMinimum ω) ω.val < 1 := by
+  obtain ⟨ha, hab, hb⟩ := rotationCalculation_angle_bounds
+  have hdom₁ : Set.Icc (Real.arccos (5 / 11 : ℝ)) (Real.arctan (11 / 5 : ℝ)) ⊆
+      Set.Icc (Real.pi / 4) (Real.pi / 2) := by
+    intro t ht
+    exact ⟨ha.le.trans ht.1, ht.2.trans hb.le⟩
+  have hdom₂ : Set.Icc (Real.arctan (11 / 5 : ℝ)) (Real.pi / 2) ⊆
+      Set.Icc (Real.pi / 4) (Real.pi / 2) := by
+    intro t ht
+    exact ⟨(ha.le.trans hab.le).trans ht.1, ht.2⟩
+  unfold rotationCalculationMinimum
+  split_ifs with h
+  · exact ConvexOn.lt_on_Ico_of_lt_of_le
+      ((rotationBound_convex (5 / 4) (by norm_num)).subset hdom₁ (convex_Icc _ _))
+      rotationBound_arccos rotationBound_arctan_left.le ω.property.1 h
+  · exact ConvexOn.lt_on_Ico_of_lt_of_le
+      ((rotationBound_convex (11 / 10) (by norm_num)).subset hdom₂ (convex_Icc _ _))
+      rotationBound_arctan_right (rotationBound_pi_div_two _).le (le_of_not_gt h) ω.property.2
+
+private theorem rotationCalculation_sin_bound (ω : RotationCalculationAngle) :
+    1 < rotationCalculationMinimum ω * Real.sin ω.val := by
+  obtain ⟨ha, hab, hb⟩ := rotationCalculation_angle_bounds
+  have hsinmono := Real.strictMonoOn_sin.monotoneOn
+  unfold rotationCalculationMinimum
+  split_ifs with h
+  · have hs := hsinmono
+      (show Real.arccos (5 / 11 : ℝ) ∈ Set.Icc (-(Real.pi / 2)) (Real.pi / 2) by
+        constructor <;> linarith [Real.pi_pos])
+      (show ω.val ∈ Set.Icc (-(Real.pi / 2)) (Real.pi / 2) by
+        constructor <;> linarith [ω.property.1, ω.property.2, Real.pi_pos]) ω.property.1
+    rw [Real.sin_arccos] at hs
+    norm_num at hs
+    have hsqrt := Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 96)
+    have hp := Real.sqrt_nonneg 96
+    have : 44 / 5 < Real.sqrt 96 := by nlinarith
+    nlinarith
+  · have hs := hsinmono
+      (show Real.arctan (11 / 5 : ℝ) ∈ Set.Icc (-(Real.pi / 2)) (Real.pi / 2) by
+        constructor <;> linarith [Real.pi_pos])
+      (show ω.val ∈ Set.Icc (-(Real.pi / 2)) (Real.pi / 2) by
+        constructor <;> linarith [ω.property.1, ω.property.2, Real.pi_pos]) (le_of_not_gt h)
+    rw [Real.sin_arctan] at hs
+    norm_num at hs
+    have hsqrt := Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 146)
+    have hp : 0 < Real.sqrt 146 := by positivity
+    have hupper : Real.sqrt 146 < 121 / 10 := by nlinarith
+    have hbase : 10 / 11 < 11 / Real.sqrt 146 := (lt_div_iff₀ hp).2 (by nlinarith)
+    nlinarith
+
+private theorem rotationCalculation_g_bound (ω : RotationCalculationAngle)
+    (d : Set.Icc (rotationCalculationMinimum ω) (Real.tan ω.val))
+    (hr : 0 ≤ (rotationCalculationValues ω d).1) :
+    2 * Real.cos ω.val < (rotationCalculationValues ω d).2.1 := by
+  obtain ⟨ha, hab, hb⟩ := rotationCalculation_angle_bounds
+  have hw0 : 0 < ω.val := by linarith [ω.property.1, Real.pi_pos]
+  have hsin : 0 < Real.sin ω.val :=
+    Real.sin_pos_of_pos_of_lt_pi hw0 (by linarith [ω.property.2, Real.pi_pos])
+  have hcos : 0 < Real.cos ω.val := Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos],
+    ω.property.2⟩
+  have hcot : 0 ≤ Real.cos ω.val / Real.sin ω.val := le_of_lt (div_pos hcos hsin)
+  have hmin : 0 ≤ rotationCalculationMinimum ω := by
+    unfold rotationCalculationMinimum
+    split_ifs <;> norm_num
+  have hdd := mul_le_mul_of_nonneg_right d.property.1 hcot
+  change 0 ≤ 1 - d.val * (Real.cos ω.val / Real.sin ω.val) at hr
+  have hf := rotationBound_lt_one ω
+  unfold rotationBound at hf
+  have hrle : 1 - d.val * (Real.cos ω.val / Real.sin ω.val) ≤
+      1 - rotationCalculationMinimum ω * (Real.cos ω.val / Real.sin ω.val) := by linarith
+  have hr0 : 0 ≤ 1 - rotationCalculationMinimum ω * (Real.cos ω.val / Real.sin ω.val) := by linarith
+  have hsquares := sq_le_sq₀ hr hr0 |>.2 hrle
+  have hrad : 0 ≤ 1 - (1 - d.val * (Real.cos ω.val / Real.sin ω.val)) ^ 2 := by
+    nlinarith [sq_nonneg (Real.cos ω.val)]
+  have hsqrt := Real.sq_sqrt hrad
+  have hnonneg := Real.sqrt_nonneg (1 - (1 - d.val * (Real.cos ω.val / Real.sin ω.val)) ^ 2)
+  change 2 * Real.cos ω.val < Real.sqrt _
+  nlinarith
+
+theorem rotationCalculation_inequalities (ω : RotationCalculationAngle)
+    (d : Set.Icc (rotationCalculationMinimum ω) (Real.tan ω.val))
+    (hr : 0 ≤ (rotationCalculationValues ω d).1) :
+    1 < inner ℝ
+      ((rotationCalculationValues ω d).2.2.1 -
+        ((stripParallelogram ω.val).2.2 - tangentVector 0))
+      (normalVector ((Real.pi / 2 - ω.val : ℝ) : Real.Angle)) ∧
+    1 < inner ℝ
+      ((rotationCalculationValues ω d).2.2.2 -
+        ((stripParallelogram ω.val).2.2 - normalVector (ω.val : Real.Angle)))
+      (tangentVector ((Real.pi / 2 - ω.val : ℝ) : Real.Angle)) := by
+  obtain ⟨ha, hab, hb⟩ := rotationCalculation_angle_bounds
+  have hw0 : 0 < ω.val := by linarith [ω.property.1, Real.pi_pos]
+  have hsin : 0 < Real.sin ω.val :=
+    Real.sin_pos_of_pos_of_lt_pi hw0 (by linarith [ω.property.2, Real.pi_pos])
+  have hcos : 0 < Real.cos ω.val :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos], ω.property.2⟩
+  have hg := rotationCalculation_g_bound ω d hr
+  have hd := mul_le_mul_of_nonneg_right d.property.1 hsin.le
+  have hds := rotationCalculation_sin_bound ω
+  constructor
+  · simp [rotationCalculationValues, inner_smul_left,
+      normalVector, tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two,
+      Real.cos_pi_div_two_sub, -Real.Angle.coe_sub]
+    linarith
+  · have hvec : (rotationCalculationValues ω d).2.2.2 -
+        ((stripParallelogram ω.val).2.2 - normalVector (ω.val : Real.Angle)) =
+        normalVector (ω.val : Real.Angle) - (rotationCalculationValues ω d).2.1 • normalVector 0
+          := by
+      simp only [rotationCalculationValues]
+      abel
+    rw [hvec]
+    simp only [inner_sub_left, inner_smul_left]
+    simp [normalVector, tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two,
+      Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub, -Real.Angle.coe_sub]
+    have hm := mul_lt_mul_of_pos_right hg hcos
+    nlinarith [Real.sin_sq_add_cos_sq ω.val]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Motion / Supporting Hallways
+-/
+
+@[expose] public section
+
+noncomputable section
+open Set
+open scoped unitInterval
+namespace MovingSofa
+
+/-- The supporting placement regarded as an affine isometry equivalence. -/
+def supportingPlacementEquiv (s : Set Point) (t : Real.Angle) :
+    Point ≃ᵃⁱ[ℝ] Point :=
+  (EuclideanGeometry.o.rotation t).toAffineIsometryEquiv.trans
+    (AffineIsometryEquiv.vaddConst ℝ
+      ((supportValue s t - 1) • normalVector t +
+       (supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) - 1) • tangentVector t))
+
+theorem supportingPlacementEquiv_apply (s : Set Point) (t : Real.Angle) (p : Point) :
+    supportingPlacementEquiv s t p = supportingPlacement s t p := by
+  simp [supportingPlacementEquiv, supportingPlacement, rotationMap, add_assoc]
+
+private def supportCorner (s : Set Point) (t : Real.Angle) : Point :=
+  (supportValue s t - 1) • normalVector t +
+    (supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) - 1) • tangentVector t
+
+private theorem continuous_supportCorner_real (s : Set Point)
+    (hs : s.Nonempty) (hc : IsCompact s) :
+    Continuous (fun t : ℝ ↦ supportCorner s (t : Real.Angle)) := by
+  have hh := (compactSet_support_continuity s s hs hc hs hc).2.2.1
+  have hn := continuous_normalVector_real
+  have hv : Continuous (fun t : ℝ ↦ tangentVector (t : Real.Angle)) := by
+    have h := hn.comp (continuous_id.add (continuous_const (y := Real.pi / 2)))
+    simpa only [Function.comp_def, Pi.add_apply, id_eq, Real.Angle.coe_add,
+      normalVector_add_pi_div_two] using h
+  exact (((hh.comp Real.Angle.continuous_coe).sub continuous_const).smul hn).add
+    (((hh.comp (Real.Angle.continuous_coe.add continuous_const)).sub continuous_const).smul hv)
+
+private theorem supportingPlacementEquiv_symm_apply (s : Set Point) (t : Real.Angle)
+    (p : Point) :
+    (supportingPlacementEquiv s t).symm p =
+      ((AffineIsometryEquiv.vaddConst ℝ (-supportCorner s t)).trans
+        (EuclideanGeometry.o.rotation (-t)).toAffineIsometryEquiv) p := by
+  change (EuclideanGeometry.o.rotation t).symm (p - supportCorner s t) =
+    (EuclideanGeometry.o.rotation (-t)) (p + -supportCorner s t)
+  simp only [Orientation.rotation_symm_apply, Orientation.rotation_apply,
+    Real.Angle.cos_neg, Real.Angle.sin_neg, neg_smul, sub_eq_add_neg]
+
+private theorem continuous_inverse_supportingPlacement (s : Set Point)
+    (hs : s.Nonempty) (hc : IsCompact s) :
+    Continuous (fun t : ℝ ↦ (supportingPlacementEquiv s (t : Real.Angle)).symm) := by
+  have h := continuous_vaddConst_trans_rotation.comp
+    (Real.Angle.continuous_coe.neg.prodMk (continuous_supportCorner_real s hs hc).neg)
+  apply h.congr
+  intro t
+  apply AffineIsometryEquiv.ext
+  intro p
+  exact (supportingPlacementEquiv_symm_apply s (t : Real.Angle) p).symm
+
+private theorem inverse_supportingPlacement_coordinates (s : Set Point)
+    (t : Real.Angle) (p : Point) :
+    ((supportingPlacementEquiv s t).symm p) 0 =
+      inner ℝ p (normalVector t) - supportValue s t + 1 ∧
+    ((supportingPlacementEquiv s t).symm p) 1 =
+      inner ℝ p (tangentVector t) -
+        supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) + 1 := by
+  have hn := inner_supportingPlacement_normalVector s t ((supportingPlacementEquiv s t).symm p)
+  have ht := inner_supportingPlacement_tangentVector s t ((supportingPlacementEquiv s t).symm p)
+  rw [← supportingPlacementEquiv_apply, AffineIsometryEquiv.apply_symm_apply] at hn ht
+  constructor <;> linarith
+
+private theorem inverse_supportingPlacement_mem_hallway (s : Set Point)
+    (t : Real.Angle) {p : Point} (hp : p ∈ supportingHallway s t) :
+    (supportingPlacementEquiv s t).symm p ∈ hallway := by
+  obtain ⟨q, hq, rfl⟩ := hp
+  rw [← supportingPlacementEquiv_apply, AffineIsometryEquiv.symm_apply_apply]
+  exact hq
+
+private theorem hallway_coordinates_le {p : Point} (hp : p ∈ hallway) :
+    p 0 ≤ 1 ∧ p 1 ≤ 1 := by
+  rcases hp with ⟨a, b, h, rfl⟩ | ⟨a, b, h, rfl⟩
+  · exact ⟨h.1, h.2.2⟩
+  · exact ⟨h.2.1, h.2.2⟩
+
+/-- A closed connected subset of the supporting-hallway intersection inherits its
+clockwise hallway motion from the reference compact set. -/
+theorem hasRotationAngle_of_subset_supportingHallways (s S : Set Point) (ω : ℝ)
+    (hsne : s.Nonempty) (hscompact : IsCompact s) (hω : 0 ≤ ω)
+    (hsω : supportValue s (ω : Real.Angle) = 1)
+    (hsπ : supportValue s ((Real.pi / 2 : ℝ) : Real.Angle) = 1)
+    (hSconn : IsConnected S) (hSclosed : IsClosed S)
+    (hSsub : S ⊆ monotonization s ω) : HasRotationAngle S ω := by
+  let m : I → Point ≃ᵃⁱ[ℝ] Point := fun r ↦
+    (supportingPlacementEquiv s ((ω * (r : ℝ) : ℝ) : Real.Angle)).symm
+  have hθ (r : I) : ω * (r : ℝ) ∈ Icc (0 : ℝ) ω := by
+    exact ⟨mul_nonneg hω r.2.1,
+      (mul_le_mul_of_nonneg_left r.2.2 hω).trans_eq (mul_one ω)⟩
+  have hall (r : I) : m r '' S ⊆ hallway := by
+    rintro p ⟨q, hq, rfl⟩
+    apply inverse_supportingPlacement_mem_hallway
+    exact mem_iInter.mp (mem_iInter.mp (hSsub hq).2 (ω * (r : ℝ))) (hθ r)
+  have hrot (r : I) (p : Point) :
+      m r p = rotationMap ((-ω * (r : ℝ) : ℝ) : Real.Angle) p + m r 0 := by
+    dsimp [m]
+    rw [supportingPlacementEquiv_symm_apply, supportingPlacementEquiv_symm_apply]
+    change rotationMap (-((ω * (r : ℝ) : ℝ) : Real.Angle))
+      (p + -supportCorner s _) = _
+    rw [show ((-ω * (r : ℝ) : ℝ) : Real.Angle) =
+      -((ω * (r : ℝ) : ℝ) : Real.Angle) by rw [neg_mul, Real.Angle.coe_neg]]
+    simp [rotationMap, map_add]
+  refine ⟨m, ?_, (fun r ↦ -ω * (r : ℝ)), by fun_prop, by simp, by simp, hrot⟩
+  refine ⟨hSconn,
+    hSclosed,
+    (continuous_inverse_supportingPlacement s hsne hscompact).comp
+      (continuous_const.mul continuous_subtype_val), ?_, ?_, ?_, hall, ?_⟩
+  · refine ⟨-supportCorner s 0, ?_⟩
+    intro p
+    simp [m, supportingPlacementEquiv_symm_apply,
+      Orientation.rotation_zero]
+  · intro r
+    exact ⟨((-ω * (r : ℝ) : ℝ) : Real.Angle), hrot r⟩
+  · rintro p ⟨q, hq, rfl⟩
+    apply mem_horizontalHallway_of_coordinates
+    · exact (hallway_coordinates_le (hall 0 ⟨q, hq, rfl⟩)).1
+    · have hc := (inverse_supportingPlacement_coordinates s 0 q).2
+      have hqP := (mem_stripParallelogram_iff ω q).mp (hSsub hq).1
+      have hy : (m 0 q) 1 = q 1 := by
+        simpa [m, hsπ, tangentVector, frame, PiLp.inner_apply,
+          Fin.sum_univ_two] using hc
+      rw [hy]
+      exact hqP.1
+  · rintro p ⟨q, hq, rfl⟩
+    apply mem_verticalHallway_of_coordinates
+    · have hc := (inverse_supportingPlacement_coordinates s (ω : Real.Angle) q).1
+      have hqP := (mem_stripParallelogram_iff ω q).mp (hSsub hq).1
+      have hx : (m 1 q) 0 = inner ℝ q (normalVector (ω : Real.Angle)) := by
+        simpa [m, hsω] using hc
+      rw [hx]
+      exact hqP.2
+    · exact (hallway_coordinates_le (hall 1 ⟨q, hq, rfl⟩)).2
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Motion / Translation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval
+
+namespace MovingSofa
+
+private def translatedMotion (v : Point) (m : I → Point ≃ᵃⁱ[ℝ] Point) (t : I) :
+    Point ≃ᵃⁱ[ℝ] Point :=
+  (AffineIsometryEquiv.vaddConst ℝ (-v)).trans (m t)
+
+private theorem continuous_translatedMotion (v : Point)
+    (m : I → Point ≃ᵃⁱ[ℝ] Point) (hm : Continuous m) :
+    Continuous (translatedMotion v m) := by
+  rw [continuous_induced_rng]
+  have hmc : Continuous (fun t ↦ (m t).toAffineIsometry.toContinuousAffineMap) :=
+    continuous_induced_dom.comp hm
+  apply (ContinuousAffineMap.continuous_comp_right
+    (AffineIsometryEquiv.vaddConst ℝ (-v)).toAffineIsometry.toContinuousAffineMap).comp
+      hmc |>.congr
+  intro t
+  rfl
+
+/-- Translating a sofa preserves each admitted rotation angle. -/
+theorem hasRotationAngle_image_add (s : Set Point) (v : Point) (ω : ℝ)
+    (hs : HasRotationAngle s ω) :
+    HasRotationAngle ((fun p ↦ p + v) '' s) ω := by
+  obtain ⟨m, hm, α, hα, hα0, hα1, hmotion⟩ := hs
+  refine ⟨translatedMotion v m, ?_, α, hα, hα0, hα1, ?_⟩
+  · obtain ⟨hconn, hclosed, hcont, ⟨q, hq⟩, hrot, hini, hall, hfinal⟩ := hm
+    refine ⟨hconn.image _ (by fun_prop), ?_, continuous_translatedMotion v m hcont,
+      ⟨q - v, ?_⟩, ?_, ?_, ?_, ?_⟩
+    · change IsClosed ((AffineIsometryEquiv.vaddConst ℝ v) '' s)
+      exact (AffineIsometryEquiv.vaddConst ℝ v).toHomeomorph.isClosedMap s hclosed
+    · intro p
+      change m 0 (p - v) = p + (q - v)
+      rw [hq]
+      abel
+    · intro t
+      obtain ⟨a, ha⟩ := hrot t
+      refine ⟨a, fun p ↦ ?_⟩
+      change m t ((AffineIsometryEquiv.vaddConst ℝ (-v)) p) =
+        rotationMap a p + m t ((AffineIsometryEquiv.vaddConst ℝ (-v)) 0)
+      rw [ha ((AffineIsometryEquiv.vaddConst ℝ (-v)) p),
+        ha ((AffineIsometryEquiv.vaddConst ℝ (-v)) 0)]
+      have hvp : (AffineIsometryEquiv.vaddConst ℝ (-v)) p = p - v := by rfl
+      have hv0 : (AffineIsometryEquiv.vaddConst ℝ (-v)) 0 = -v := by simp
+      rw [hvp, hv0]
+      simp only [rotationMap, map_sub, map_neg]
+      abel
+    · rintro _ ⟨_, ⟨y, hy, rfl⟩, rfl⟩
+      exact hini ⟨y, hy, by simp [translatedMotion]⟩
+    · intro t
+      rintro _ ⟨x, ⟨y, hy, rfl⟩, rfl⟩
+      exact hall t ⟨y, hy, by simp [translatedMotion]⟩
+    · rintro _ ⟨x, ⟨y, hy, rfl⟩, rfl⟩
+      exact hfinal ⟨y, hy, by simp [translatedMotion]⟩
+  · intro t p
+    change m t ((AffineIsometryEquiv.vaddConst ℝ (-v)) p) =
+      rotationMap (α t : Real.Angle) p +
+        m t ((AffineIsometryEquiv.vaddConst ℝ (-v)) 0)
+    rw [hmotion t ((AffineIsometryEquiv.vaddConst ℝ (-v)) p),
+      hmotion t ((AffineIsometryEquiv.vaddConst ℝ (-v)) 0)]
+    have hvp : (AffineIsometryEquiv.vaddConst ℝ (-v)) p = p - v := by rfl
+    have hv0 : (AffineIsometryEquiv.vaddConst ℝ (-v)) 0 = -v := by simp
+    rw [hvp, hv0]
+    simp only [rotationMap, map_sub, map_neg]
+    abel
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Motion / Standard Position
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem exists_standardPosition_translation (s : Set Point) (ω : ℝ)
+    (hs : HasRotationAngle s ω) (hω : ω ∈ Set.Ioc 0 (Real.pi / 2)) :
+    (∃ v : Point, IsStandardPosition ((fun p ↦ p + v) '' s) ω) ∧
+    (∀ v w : Point,
+      IsStandardPosition ((fun p ↦ p + v) '' s) ω →
+      IsStandardPosition ((fun p ↦ p + w) '' s) ω →
+      (ω < Real.pi / 2 → v = w) ∧ (ω = Real.pi / 2 → v 1 = w 1)) ∧
+    (∀ v : Point, IsStandardPosition ((fun p ↦ p + v) '' s) ω →
+      ω = Real.pi / 2 → ∀ a : ℝ,
+        IsStandardPosition ((fun p ↦ p + (v + a • normalVector 0)) '' s) ω) ∧
+    (∀ v : Point, IsStandardPosition ((fun p ↦ p + v) '' s) ω →
+      (fun p ↦ p + v) '' s ⊆ (stripParallelogram ω).1) := by
+  have hc : IsCompact s := hs.isCompact hω
+  have hne : s.Nonempty := by
+    obtain ⟨_, hm, _⟩ := hs
+    exact hm.1.nonempty
+  have hadd (v : Point) (t : Real.Angle) :
+      supportValue ((fun p ↦ p + v) '' s) t =
+        supportValue s t + inner ℝ v (normalVector t) :=
+    supportValue_image_add_of_isCompact hc hne v t
+  have hcos (hlt : ω < Real.pi / 2) : Real.cos ω ≠ 0 :=
+    (Real.cos_pos_of_mem_Ioo ⟨by linarith [hω.1, Real.pi_pos], hlt⟩).ne'
+  constructor
+  · rcases lt_or_eq_of_le hω.2 with hlt | rfl
+    · let v : Point := !₂[
+          (1 - supportValue s (ω : Real.Angle) -
+            (1 - supportValue s ((Real.pi / 2 : ℝ) : Real.Angle)) * Real.sin ω) /
+              Real.cos ω,
+          1 - supportValue s ((Real.pi / 2 : ℝ) : Real.Angle)]
+      refine ⟨v, hc.image (by fun_prop), hasRotationAngle_image_add s v ω hs,
+        hω.1, hω.2, ?_, ?_⟩
+      · rw [hadd]
+        simp only [v, normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two, Real.inner_apply,
+          Real.Angle.cos_coe, Real.Angle.sin_coe, Matrix.cons_val_zero,
+          Matrix.cons_val_one]
+        field_simp [hcos hlt]
+        ring
+      · rw [hadd]
+        simp [v, normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+    · let v : Point := !₂[0,
+          1 - supportValue s ((Real.pi / 2 : ℝ) : Real.Angle)]
+      refine ⟨v, hc.image (by fun_prop),
+        hasRotationAngle_image_add s v (Real.pi / 2) hs, hω.1, hω.2, ?_, ?_⟩
+      · rw [hadd]
+        simp [v, normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+      · rw [hadd]
+        simp [v, normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+  · constructor
+    · intro v w hv hw
+      have hvω := hv.2.2.2.2.1
+      have hvπ := hv.2.2.2.2.2
+      have hwω := hw.2.2.2.2.1
+      have hwπ := hw.2.2.2.2.2
+      rw [hadd] at hvω hvπ hwω hwπ
+      constructor
+      · intro hlt
+        have hy : v 1 = w 1 := by
+          simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two] at hvπ hwπ
+          linarith
+        apply PiLp.ext
+        intro i
+        fin_cases i
+        · have hcω := hcos hlt
+          change v 0 = w 0
+          simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply,
+            RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue,
+            Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one] at hvω hwω
+          rw [hy] at hvω
+          apply mul_left_cancel₀ hcω
+          linarith [hvω, hwω]
+        · exact hy
+      · intro _
+        simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two] at hvπ hwπ
+        linarith
+    · constructor
+      · intro v hv hωeq a
+        subst ω
+        have hvπ := hv.2.2.2.2.2
+        rw [hadd] at hvπ
+        refine ⟨hc.image (by fun_prop),
+          hasRotationAngle_image_add s (v + a • normalVector 0) (Real.pi / 2) hs,
+          hω.1, hω.2, ?_, ?_⟩
+        · rw [hadd]
+          simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two] at hvπ ⊢
+          linarith
+        · rw [hadd]
+          simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two] at hvπ ⊢
+          linarith
+      · intro v hv
+        exact hv.subset_strips
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Bounds.WedgeContainment`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Bounds / Wedge Containment
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private theorem smul_mem_of_mem_of_le_of_le {K : ConvexBody Point} {v : Point}
+    {l x r : ℝ} (hl : l • v ∈ (K : Set Point)) (hr : r • v ∈ (K : Set Point))
+    (hlx : l ≤ x) (hxr : x ≤ r) : x • v ∈ (K : Set Point) := by
+  by_cases hlr : l = r
+  · have hx : x = l := by linarith
+    simpa [hx] using hl
+  · have hlrpos : 0 < r - l := sub_pos.mpr (lt_of_le_of_ne (hlx.trans hxr) hlr)
+    have hratio : (x - l) / (r - l) ∈ Set.Icc (0 : ℝ) 1 := by
+      constructor
+      · exact div_nonneg (sub_nonneg.mpr hlx) hlrpos.le
+      · exact (div_le_one hlrpos).2 (by linarith)
+    have hmem := K.convex.lineMap_mem hl hr hratio
+    rw [AffineMap.lineMap_apply_module] at hmem
+    have heq : (1 - (x - l) / (r - l)) • l • v +
+        ((x - l) / (r - l)) • r • v = x • v := by
+      rw [smul_smul, smul_smul, ← add_smul]
+      congr 1
+      field_simp
+      ring
+    rw [heq] at hmem
+    exact hmem
+
+private theorem inner_lineMap (x y u : Point) (s : ℝ) :
+    inner ℝ (AffineMap.lineMap x y s) u =
+      inner ℝ x u + s * (inner ℝ y u - inner ℝ x u) := by
+  rw [AffineMap.lineMap_apply_module, inner_add_left, real_inner_smul_left,
+    real_inner_smul_left]
+  ring
+
+private theorem mem_segment_lineMap_of_one_le (x y : Point) {s : ℝ} (hs : 1 ≤ s) :
+    y ∈ segment ℝ x (AffineMap.lineMap x y s) := by
+  have hspos : 0 < s := zero_lt_one.trans_le hs
+  rw [segment_eq_image_lineMap]
+  refine ⟨1 / s, ⟨by positivity, (div_le_one hspos).2 hs⟩, ?_⟩
+  ext i
+  simp [AffineMap.lineMap_apply_module, one_div, hspos.ne']
+  field_simp
+  ring
+
+private theorem exists_lineMap_fan_boundary (x y u v : Point)
+    (hyu : 0 ≤ inner ℝ y u) (hyv : 0 ≤ inner ℝ y v)
+    (hdu : inner ℝ (y - x) u < 0) (hdv : inner ℝ (y - x) v < 0) :
+    ∃ s : ℝ, 1 ≤ s ∧
+      0 ≤ inner ℝ (AffineMap.lineMap x y s) u ∧
+      0 ≤ inner ℝ (AffineMap.lineMap x y s) v ∧
+      (inner ℝ (AffineMap.lineMap x y s) u = 0 ∨
+        inner ℝ (AffineMap.lineMap x y s) v = 0) := by
+  let du := inner ℝ x u - inner ℝ y u
+  let dv := inner ℝ x v - inner ℝ y v
+  have hdu' : 0 < du := by
+    rw [inner_sub_left] at hdu
+    dsimp [du]
+    linarith
+  have hdv' : 0 < dv := by
+    rw [inner_sub_left] at hdv
+    dsimp [dv]
+    linarith
+  by_cases hratio : inner ℝ x u / du ≤ inner ℝ x v / dv
+  · let s := inner ℝ x u / du
+    have hs : 1 ≤ s := by
+      apply (le_div_iff₀ hdu').2
+      dsimp [du]
+      linarith
+    have hsdu : s * du = inner ℝ x u := by
+      dsimp [s]
+      field_simp
+    have hsu : inner ℝ (AffineMap.lineMap x y s) u = 0 := by
+      rw [inner_lineMap]
+      have hdiff : inner ℝ y u - inner ℝ x u = -du := by simp [du]
+      rw [hdiff, mul_neg, hsdu]
+      ring
+    have hsv : 0 ≤ inner ℝ (AffineMap.lineMap x y s) v := by
+      have hmul : s * dv ≤ inner ℝ x v := (le_div_iff₀ hdv').mp hratio
+      rw [inner_lineMap]
+      dsimp [dv] at hmul
+      linarith
+    exact ⟨s, hs, hsu.ge, hsv, Or.inl hsu⟩
+  · let s := inner ℝ x v / dv
+    have hs : 1 ≤ s := by
+      apply (le_div_iff₀ hdv').2
+      dsimp [dv]
+      linarith
+    have hsdv : s * dv = inner ℝ x v := by
+      dsimp [s]
+      field_simp
+    have hsv : inner ℝ (AffineMap.lineMap x y s) v = 0 := by
+      rw [inner_lineMap]
+      have hdiff : inner ℝ y v - inner ℝ x v = -dv := by simp [dv]
+      rw [hdiff, mul_neg, hsdv]
+      ring
+    have hsu : 0 ≤ inner ℝ (AffineMap.lineMap x y s) u := by
+      have hmul : s * du ≤ inner ℝ x u := (le_div_iff₀ hdu').mp (not_le.mp hratio).le
+      rw [inner_lineMap]
+      dsimp [du] at hmul
+      linarith
+    exact ⟨s, hs, hsu, hsv.ge, Or.inr hsv⟩
+
+private theorem cap_fan_boundary_mem {ω : ℝ} (K : CapSpace ω)
+    (t : ℝ) (ht : t ∈ Set.Ioo 0 ω) (p : Point)
+    (hpy : 0 ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)))
+    (hpω : 0 ≤ inner ℝ p (normalVector (ω : Real.Angle)))
+    (hpboundary : inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 ∨
+    inner ℝ p (normalVector (ω : Real.Angle)) = 0)
+    (hpt : inner ℝ p (normalVector (t : Real.Angle)) <
+      supportValue K.val (t : Real.Angle) - 1)
+    (hptv : inner ℝ p (tangentVector (t : Real.Angle)) <
+      supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) :
+    p ∈ (K.val : Set Point) := by
+  have hsint : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi ht.1
+    (by linarith [ht.2, K.property.2.1, Real.pi_pos])
+  have hcost : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, ht.1], ht.2.trans_le K.property.2.1⟩
+  have hsinδ : 0 < Real.sin (ω - t) := Real.sin_pos_of_pos_of_lt_pi
+    (sub_pos.mpr ht.2) (by linarith [ht.1, K.property.2.1, Real.pi_pos])
+  have hcosδ : 0 < Real.cos (ω - t) := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [sub_pos.mpr ht.2, Real.pi_pos],
+      by linarith [ht.1, K.property.2.1]⟩
+  by_cases hωeq : ω = Real.pi / 2
+  · have hpy0 : p 1 = 0 := by
+      rcases hpboundary with hpy0 | hpω0
+      · simpa [normalVector, frame, PiLp.inner_apply] using hpy0
+      · rw [hωeq] at hpω0
+        simpa [normalVector, frame, PiLp.inner_apply] using hpω0
+    let l := -supportValue K.val (Real.pi : Real.Angle)
+    let r := supportValue K.val (0 : Real.Angle)
+    have hleft : l • normalVector (0 : Real.Angle) ∈ (K.val : Set Point) := by
+      have h := supportValue_pi_smul_normalVector_mem_of_eq K hωeq
+      convert h using 1
+      ext i
+      fin_cases i <;> simp [l, normalVector, frame]
+    have hright : r • normalVector (0 : Real.Angle) ∈ (K.val : Set Point) := by
+      exact supportValue_zero_smul_normalVector_mem K
+    have hbounds := K.supportValue_upper_bounds ht
+    have hsint_lt : Real.sin t < 1 := by
+      nlinarith only [Real.sin_sq_add_cos_sq t, sq_pos_of_pos hcost]
+    have hcost_lt : Real.cos t < 1 := by
+      nlinarith only [Real.sin_sq_add_cos_sq t, sq_pos_of_pos hsint]
+    have hpr : p 0 ≤ r := by
+      have hpt' := hpt
+      simp [normalVector, frame, PiLp.inner_apply, hpy0] at hpt'
+      dsimp [r]
+      nlinarith [hbounds.1]
+    have hangle : ((ω + Real.pi / 2 : ℝ) : Real.Angle) =
+        (Real.pi : Real.Angle) := by
+      congr 1
+      rw [hωeq]
+      ring
+    have hbound₂ := hbounds.2
+    rw [hangle] at hbound₂
+    have hsinωt : Real.sin (ω - t) = Real.cos t := by
+      rw [hωeq, Real.sin_pi_div_two_sub]
+    have hcosωt : Real.cos (ω - t) = Real.sin t := by
+      rw [hωeq, Real.cos_pi_div_two_sub]
+    rw [hsinωt, hcosωt] at hbound₂
+    have hlp : l ≤ p 0 := by
+      have hptv' := hptv
+      simp only [tangentVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply,
+        RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue, Matrix.cons_val_zero,
+        neg_mul, Matrix.cons_val_one, Matrix.cons_val_fin_one, hpy0, mul_zero, add_zero,
+        Real.Angle.coe_add, neg_lt_sub_iff_lt_add] at hptv'
+      dsimp [l]
+      by_contra hp
+      have hmul := mul_lt_mul_of_pos_left (lt_of_not_ge hp) hsint
+      have hsum : Real.sin t * p 0 + supportValue K.val
+          ((t + Real.pi / 2 : ℝ) : Real.Angle) < 1 := by
+        calc
+          _ < Real.sin t * (-supportValue K.val (Real.pi : Real.Angle)) +
+              supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) :=
+            by
+              simpa [add_comm] using add_lt_add_right hmul
+                (supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle))
+          _ ≤ Real.sin t * (-supportValue K.val (Real.pi : Real.Angle)) +
+              (Real.cos t + Real.sin t * supportValue K.val (Real.pi : Real.Angle)) :=
+            by
+              simpa [add_comm] using
+                add_le_add_left hbound₂
+                  (Real.sin t * (-supportValue K.val (Real.pi : Real.Angle)))
+          _ = Real.cos t := by ring
+          _ < 1 := hcost_lt
+      exact (not_lt_of_ge hsum.le) hptv'
+    have hpEq : p = p 0 • normalVector (0 : Real.Angle) := by
+      ext i
+      fin_cases i <;> simp [normalVector, frame, hpy0]
+    rw [hpEq]
+    exact smul_mem_of_mem_of_le_of_le hleft hright hlp hpr
+  · have hωlt : ω < Real.pi / 2 := K.property.2.1.lt_of_ne hωeq
+    have hzero := zero_mem_cap_of_lt K hωlt
+    have hbounds := K.supportValue_upper_bounds ht
+    have hcosω : 0 < Real.cos ω := Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [K.property.1, Real.pi_pos], hωlt⟩
+    rcases hpboundary with hpy0 | hpω0
+    · have hpy0' : p 1 = 0 := by
+        simpa [normalVector, frame, PiLp.inner_apply] using hpy0
+      have hpx0 : 0 ≤ p 0 := by
+        have hpω' := hpω
+        simp [normalVector, frame, PiLp.inner_apply, hpy0'] at hpω'
+        by_contra hneg
+        have := mul_neg_of_pos_of_neg hcosω (lt_of_not_ge hneg)
+        linarith
+      have hsint_lt : Real.sin t < 1 := by
+        nlinarith [Real.sin_sq_add_cos_sq t, sq_pos_of_pos hcost]
+      have hpxr : p 0 ≤ supportValue K.val (0 : Real.Angle) := by
+        have hpt' := hpt
+        simp [normalVector, frame, PiLp.inner_apply, hpy0'] at hpt'
+        nlinarith [hbounds.1]
+      have hpEq : p = p 0 • normalVector (0 : Real.Angle) := by
+        ext i
+        fin_cases i <;> simp [normalVector, frame, hpy0']
+      rw [hpEq]
+      exact K.val.convex.smul_mem_of_nonneg_of_le hzero
+        (supportValue_zero_smul_normalVector_mem K) hpx0 hpxr
+    · let μ := inner ℝ p (tangentVector (ω : Real.Angle))
+      have hpEq : μ • tangentVector (ω : Real.Angle) = p := by
+        have hframe := inner_normalVector_smul_add_inner_tangentVector_smul
+          p (ω : Real.Angle)
+        rw [hpω0, zero_smul, zero_add] at hframe
+        exact hframe
+      have hμ0 : 0 ≤ μ := by
+        have hp1 := congrArg (fun x : Point ↦ x 1) hpEq
+        change μ * Real.cos ω = p 1 at hp1
+        have hpy' : 0 ≤ p 1 := by
+          simpa [normalVector, frame, PiLp.inner_apply] using hpy
+        rw [← hp1] at hpy'
+        by_contra hneg
+        have := mul_neg_of_neg_of_pos (lt_of_not_ge hneg) hcosω
+        linarith
+      have hsinδ_lt : Real.sin (ω - t) < 1 := by
+        nlinarith only [Real.sin_sq_add_cos_sq (ω - t), sq_pos_of_pos hcosδ]
+      have htv : inner ℝ (tangentVector (ω : Real.Angle))
+          (tangentVector (t : Real.Angle)) = Real.cos (ω - t) := by
+        simp [tangentVector, frame, PiLp.inner_apply, Real.cos_sub]
+        ring
+      have hμL : μ ≤ supportValue K.val
+          ((ω + Real.pi / 2 : ℝ) : Real.Angle) := by
+        have hptv' := hptv
+        rw [← hpEq, real_inner_smul_left, htv] at hptv'
+        nlinarith [hbounds.2]
+      rw [← hpEq]
+      exact K.val.convex.smul_mem_of_nonneg_of_le hzero
+        (supportValue_add_pi_div_two_smul_tangentVector_mem_of_lt K hωlt) hμ0 hμL
+
+theorem capWedge_subset_of_innerCorner_mem {ω : ℝ} (K : CapSpace ω)
+    (t : ℝ) (ht : t ∈ Set.Ioo 0 ω)
+    (hz : (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner ∈
+      (K.val : Set Point)) :
+    capWedge K t ⊆ (K.val : Set Point) := by
+  intro q hq
+  rcases hq with ⟨hqFan, hqQuad⟩
+  let z := (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner
+  change z ∈ (K.val : Set Point) at hz
+  change q ∈ supportingPlacement (K.val : Set Point) (t : Real.Angle) ''
+    hallwayParts.innerQuadrant at hqQuad
+  obtain ⟨a, ha, hqa⟩ := hqQuad
+  change a 0 < 0 ∧ a 1 < 0 at ha
+  have hqQuad : inner ℝ q (normalVector (t : Real.Angle)) <
+      supportValue K.val (t : Real.Angle) - 1 ∧
+    inner ℝ q (tangentVector (t : Real.Angle)) <
+      supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+    constructor
+    · rw [← hqa, inner_supportingPlacement_normalVector]
+      linarith [ha.1]
+    · rw [← hqa, inner_supportingPlacement_tangentVector]
+      change a 1 + supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 < _
+      linarith [ha.2]
+  have hzu : inner ℝ z (normalVector (t : Real.Angle)) =
+      supportValue K.val (t : Real.Angle) - 1 := by
+    have h := inner_supportingPlacement_normalVector (K.val : Set Point)
+      (t : Real.Angle) (0 : Point)
+    simpa [z, rotatingHallwayParts, hallwayParts] using h
+  have hzv : inner ℝ z (tangentVector (t : Real.Angle)) =
+      supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+    have h := inner_supportingPlacement_tangentVector (K.val : Set Point)
+      (t : Real.Angle) (0 : Point)
+    simpa [z, rotatingHallwayParts, hallwayParts] using h
+  let d := q - z
+  have hdu : inner ℝ d (normalVector (t : Real.Angle)) < 0 := by
+    change inner ℝ (q - z) (normalVector (t : Real.Angle)) < 0
+    rw [inner_sub_left]
+    linarith [hqQuad.1, hzu]
+  have hdv : inner ℝ d (tangentVector (t : Real.Angle)) < 0 := by
+    change inner ℝ (q - z) (tangentVector (t : Real.Angle)) < 0
+    rw [inner_sub_left]
+    linarith [hqQuad.2, hzv]
+  have hdφ (φ : ℝ) : inner ℝ d (normalVector (φ : Real.Angle)) =
+      inner ℝ d (normalVector (t : Real.Angle)) * Real.cos (φ - t) +
+        inner ℝ d (tangentVector (t : Real.Angle)) * Real.sin (φ - t) := by
+    have hframe := inner_normalVector_smul_add_inner_tangentVector_smul
+      d (t : Real.Angle)
+    nth_rewrite 1 [← hframe]
+    rw [inner_add_left, real_inner_smul_left, real_inner_smul_left]
+    have hcos : Real.cos (t - φ) = Real.cos (φ - t) := by
+      rw [show t - φ = -(φ - t) by ring, Real.cos_neg]
+    rw [inner_normalVector_normalVector, hcos]
+    have hsin : inner ℝ (tangentVector (t : Real.Angle))
+        (normalVector (φ : Real.Angle)) = Real.sin (φ - t) := by
+      simp [normalVector, tangentVector, frame, PiLp.inner_apply, Real.sin_sub]
+      ring
+    rw [hsin]
+  have hsint : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi ht.1
+    (by linarith [ht.2, K.property.2.1, Real.pi_pos])
+  have hcost : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, ht.1], ht.2.trans_le K.property.2.1⟩
+  have hsinδ : 0 < Real.sin (ω - t) := Real.sin_pos_of_pos_of_lt_pi
+    (sub_pos.mpr ht.2) (by linarith [ht.1, K.property.2.1, Real.pi_pos])
+  have hcosδ : 0 < Real.cos (ω - t) := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [sub_pos.mpr ht.2, Real.pi_pos],
+      by linarith [ht.1, K.property.2.1]⟩
+  have hdy : inner ℝ d (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) < 0 := by
+    rw [hdφ, Real.cos_pi_div_two_sub, Real.sin_pi_div_two_sub]
+    exact add_neg (mul_neg_of_neg_of_pos hdu hsint) (mul_neg_of_neg_of_pos hdv hcost)
+  have hdω : inner ℝ d (normalVector (ω : Real.Angle)) < 0 := by
+    rw [hdφ]
+    exact add_neg (mul_neg_of_neg_of_pos hdu hcosδ)
+      (mul_neg_of_neg_of_pos hdv hsinδ)
+  have hqy : 0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) := hqFan.2
+  have hqω : 0 ≤ inner ℝ q (normalVector (ω : Real.Angle)) := hqFan.1
+  obtain ⟨s, hs, hpy, hpω, hpboundary⟩ := exists_lineMap_fan_boundary z q
+    (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) (normalVector (ω : Real.Angle))
+    hqy hqω (by simpa only [d] using hdy) (by simpa only [d] using hdω)
+  let p := AffineMap.lineMap z q s
+  change 0 ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at hpy
+  change 0 ≤ inner ℝ p (normalVector (ω : Real.Angle)) at hpω
+  change inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 ∨
+    inner ℝ p (normalVector (ω : Real.Angle)) = 0 at hpboundary
+  have hspos : 0 < s := zero_lt_one.trans_le hs
+  have hpt : inner ℝ p (normalVector (t : Real.Angle)) <
+      supportValue K.val (t : Real.Angle) - 1 := by
+    rw [inner_lineMap]
+    have hdiff : inner ℝ q (normalVector (t : Real.Angle)) -
+        inner ℝ z (normalVector (t : Real.Angle)) =
+        inner ℝ d (normalVector (t : Real.Angle)) := by
+      change _ = inner ℝ (q - z) (normalVector (t : Real.Angle))
+      rw [inner_sub_left]
+    rw [hdiff, hzu]
+    have := mul_neg_of_pos_of_neg hspos hdu
+    linarith
+  have hptv : inner ℝ p (tangentVector (t : Real.Angle)) <
+      supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+    have hpinner := inner_lineMap z q (tangentVector (t : Real.Angle)) s
+    change inner ℝ p (tangentVector (t : Real.Angle)) = _ at hpinner
+    rw [hpinner]
+    have hdiff : inner ℝ q (tangentVector (t : Real.Angle)) -
+        inner ℝ z (tangentVector (t : Real.Angle)) =
+        inner ℝ d (tangentVector (t : Real.Angle)) := by
+      change _ = inner ℝ (q - z) (tangentVector (t : Real.Angle))
+      rw [inner_sub_left]
+    rw [hdiff, hzv]
+    have := mul_neg_of_pos_of_neg hspos hdv
+    linarith
+  have hpFan : p ∈ capFan ω := ⟨hpω, hpy⟩
+  have hqp : q ∈ segment ℝ z p := mem_segment_lineMap_of_one_le z q hs
+  have hpK := cap_fan_boundary_mem K t ht p hpy hpω hpboundary hpt hptv
+  exact K.val.convex.segment_subset hz hpK hqp
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Cap.Balanced`.
+* `Cap.Clipped.Estimates`.
+* `Cap.Clipped`.
+* `Cap.Densities`.
+* `Cap.UpperBoundary`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Balanced
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter
+open scoped Topology
+
+namespace MovingSofa
+
+/-- The polygonal cap area minus the area of its polygonal niche. -/
+def polygonAreaFunctional (Θ : AngleSet) (K : CapSpace Θ.angle) : ℝ :=
+  ClassicalResults.area (angleCap Θ K) - ClassicalResults.area (polygonNiche Θ K)
+
+/-- A cap containing the distinguished fan point maximizes the polygonal area functional. -/
+def IsMaximumPolygonCap (Θ : AngleSet) (K : PolygonCapSpace Θ) : Prop :=
+  (stripParallelogram Θ.angle).2.2 ∈ (K.val.val : Set Point) ∧
+    ∀ L : PolygonCapSpace Θ, polygonAreaFunctional Θ L.val ≤ polygonAreaFunctional Θ K.val
+
+/-- The cap is a Hausdorff limit of polygonal maxima on increasingly fine dyadic meshes. -/
+def IsBalancedMaximumCap {ω : ℝ} (K : CapSpace ω) : Prop :=
+  ∃ (n : ℕ → ℕ) (hn : ∀ i, 2 ≤ n i), StrictMono n ∧
+    (∀ i, ∃ k : ℕ, n i = 2 ^ k) ∧
+    ∃ P : ∀ i, PolygonCapSpace (uniformAngleSet ω K.property.1 K.property.2.1 (n i) (hn i)),
+      (∀ i, IsMaximumPolygonCap _ (P i)) ∧
+      Tendsto (fun i ↦ Metric.hausdorffDist ((P i).val.val : Set Point) (K.val : Set Point))
+        atTop (𝓝 0)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Clipped / Estimates
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private theorem clippedArea_bound_low {u t : ℝ}
+    (ht : 39 / 20 < t) (ht' : t ≤ 11 / 5) (hu : u < 121 / 50) :
+    u - (t - 5 / 4) ^ 2 / t < 11 / 5 := by
+  have ht0 : 0 < t := by linarith
+  have hsq : 49 / 100 < (t - 5 / 4) ^ 2 := by nlinarith
+  have hquot : 49 / 220 < (t - 5 / 4) ^ 2 / t := by
+    apply (lt_div_iff₀ ht0).2
+    nlinarith
+  linarith
+
+private theorem clippedArea_bound {C S : ℝ} (hC : 0 < C) (hS : 0 < S)
+    (hcircle : S ^ 2 + C ^ 2 = 1) (hC' : C ≤ 5 / 11) :
+    let d : ℝ := if S / C < 11 / 5 then 5 / 4 else 11 / 10
+    1 / C - (S / C - d) ^ 2 * (C / S) < 11 / 5 := by
+  dsimp
+  split_ifs with h
+  · have ht : 0 < S / C := div_pos hS hC
+    have hid : (1 / C) ^ 2 = 1 + (S / C) ^ 2 := by
+      field_simp
+      nlinarith
+    have hu : 11 / 5 ≤ 1 / C := by
+      apply (le_div_iff₀ hC).2
+      linarith
+    have hlo : 39 / 20 < S / C := by nlinarith
+    have hhi : 1 / C < 121 / 50 := by
+      have hu0 : 0 < 1 / C := by positivity
+      nlinarith
+    have hb := clippedArea_bound_low hlo h.le hhi
+    convert hb using 1
+    field_simp
+  · have hs1 : S ≤ 1 := by nlinarith [sq_nonneg C]
+    have hc : C / (1 + S) ≤ C / S / 2 := by
+      have hs' : 0 < 1 + S := by linarith
+      apply (div_le_iff₀ hs').2
+      field_simp
+      nlinarith [mul_nonneg hC.le (sub_nonneg.mpr hs1)]
+    have hid : 1 / C - (S / C - 11 / 10) ^ 2 * (C / S) =
+        C / (1 + S) + 11 / 5 - (121 / 100) * (C / S) := by
+      field_simp
+      nlinarith [hcircle]
+    rw [hid]
+    have hp : 0 < C / S := div_pos hC hS
+    linarith
+
+theorem rotationCalculation_area_estimate (ω : RotationCalculationAngle) :
+    1 / Real.cos ω.val - (Real.tan ω.val - rotationCalculationMinimum ω) ^ 2 *
+      (Real.cos ω.val / Real.sin ω.val) < 11 / 5 := by
+  have hw0 : 0 < ω.val :=
+    (Real.arccos_pos.mpr (by norm_num : (5 / 11 : ℝ) < 1)).trans_le ω.property.1
+  have hC : 0 < Real.cos ω.val :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos], ω.property.2⟩
+  have hS : 0 < Real.sin ω.val :=
+    Real.sin_pos_of_pos_of_lt_pi hw0 (by linarith [ω.property.2, Real.pi_pos])
+  have hC' : Real.cos ω.val ≤ 5 / 11 := by
+    have h := Real.strictAntiOn_cos.antitoneOn
+      (show Real.arccos (5 / 11 : ℝ) ∈ Set.Icc 0 Real.pi from
+        ⟨Real.arccos_nonneg _, Real.arccos_le_pi _⟩)
+      (show ω.val ∈ Set.Icc 0 Real.pi from
+        ⟨hw0.le, by linarith [ω.property.2, Real.pi_pos]⟩) ω.property.1
+    simpa only [Real.cos_arccos (by norm_num : (-1 : ℝ) ≤ 5 / 11)
+      (by norm_num : (5 / 11 : ℝ) ≤ 1)] using h
+  have hs : (Real.sin ω.val / Real.cos ω.val < 11 / 5) ↔
+      ω.val < Real.arctan (11 / 5 : ℝ) := by
+    rw [← Real.tan_eq_sin_div_cos, ← Real.arctan_lt_arctan_iff]
+    rw [Real.arctan_tan (by linarith [Real.pi_pos]) ω.property.2]
+  have hb := clippedArea_bound hC hS (Real.sin_sq_add_cos_sq ω.val) hC'
+  dsimp at hb
+  simp only [hs] at hb
+  simpa only [← Real.tan_eq_sin_div_cos, rotationCalculationMinimum] using hb
+
+theorem rotationCalculationMinimum_pos (ω : RotationCalculationAngle) :
+    0 < rotationCalculationMinimum ω := by
+  unfold rotationCalculationMinimum
+  split_ifs <;> norm_num
+
+theorem rotationCalculationMinimum_lt_tan (ω : RotationCalculationAngle) :
+    rotationCalculationMinimum ω < Real.tan ω.val := by
+  have hw0 : 0 < ω.val :=
+    (Real.arccos_pos.mpr (by norm_num : (5 / 11 : ℝ) < 1)).trans_le ω.property.1
+  have hC : 0 < Real.cos ω.val :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos], ω.property.2⟩
+  have hS : 0 < Real.sin ω.val :=
+    Real.sin_pos_of_pos_of_lt_pi hw0 (by linarith [ω.property.2, Real.pi_pos])
+  have hC' : Real.cos ω.val ≤ 5 / 11 := by
+    have h := Real.cos_le_cos_of_nonneg_of_le_pi (Real.arccos_nonneg (5 / 11))
+      (by linarith [ω.property.2, Real.pi_pos] : ω.val ≤ Real.pi) ω.property.1
+    simpa only [Real.cos_arccos (by norm_num : (-1 : ℝ) ≤ 5 / 11)
+      (by norm_num : (5 / 11 : ℝ) ≤ 1)] using h
+  have hsq : Real.cos ω.val ^ 2 ≤ (5 / 11 : ℝ) ^ 2 :=
+    pow_le_pow_left₀ hC.le hC' 2
+  have ht : (5 / 4 : ℝ) < Real.tan ω.val := by
+    rw [Real.tan_eq_sin_div_cos, lt_div_iff₀ hC]
+    nlinarith [Real.sin_sq_add_cos_sq ω.val]
+  unfold rotationCalculationMinimum
+  split_ifs <;> linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Clipped
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open MeasureTheory Set
+
+/-- Clip the strip parallelogram by the two additional symmetric wall constraints. -/
+def clippedCap (ω d : ℝ) : Set Point :=
+  (stripParallelogram ω).1 ∩
+    normalHalfPlane 0 (d + Real.tan ((Real.pi / 2 - ω) / 2)) false false ∩
+    normalHalfPlane ((ω + Real.pi / 2 : ℝ) : Real.Angle)
+      (d + Real.tan ((Real.pi / 2 - ω) / 2)) false false
+
+theorem mem_clippedCap_iff (ω d : ℝ) (p : Point) :
+    p ∈ clippedCap ω d ↔
+      (0 ≤ p 1 ∧ p 1 ≤ 1) ∧
+      (0 ≤ Real.cos ω * p 0 + Real.sin ω * p 1 ∧
+        Real.cos ω * p 0 + Real.sin ω * p 1 ≤ 1) ∧
+      p 0 ≤ d + Real.tan ((Real.pi / 2 - ω) / 2) ∧
+      -Real.sin ω * p 0 + Real.cos ω * p 1 ≤
+        d + Real.tan ((Real.pi / 2 - ω) / 2) := by
+  simp only [clippedCap, Set.mem_inter_iff, mem_stripParallelogram_iff]
+  simp [normalHalfPlane, normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two,
+    Real.cos_add, Real.sin_add, -Real.Angle.coe_add, and_assoc, mul_comm]
+
+theorem clippedCap_removed_pieces_disjoint {C S c d : ℝ}
+    (hC : 0 < C) (hS : 0 ≤ S) (hd : 0 ≤ d) (hc : c * (1 + S) = C) :
+    Disjoint {p : Point | p 1 ≤ 1 ∧ c + d < p 0}
+      {p : Point | c + d < -S * p 0 + C * p 1} := by
+  rw [Set.disjoint_left]
+  intro p hp hq
+  change p 1 ≤ 1 ∧ c + d < p 0 at hp
+  change c + d < -S * p 0 + C * p 1 at hq
+  have hmul := mul_le_mul_of_nonneg_left hp.1 hC.le
+  have hmul' := mul_le_mul_of_nonneg_left (le_of_lt hp.2) hS
+  have hprod := mul_nonneg hd (by linarith : 0 ≤ 1 + S)
+  nlinarith
+
+private theorem mem_stripParallelogram_coordinates (ω : ℝ) (p : Point) :
+    p ∈ (stripParallelogram ω).1 ↔
+      (0 ≤ p 1 ∧ p 1 ≤ 1) ∧
+      (0 ≤ Real.cos ω * p 0 + Real.sin ω * p 1 ∧
+        Real.cos ω * p 0 + Real.sin ω * p 1 ≤ 1) := by
+  rw [mem_stripParallelogram_iff]
+  simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+
+private theorem capReflection_rotatedCoordinate (ω : ℝ) (p : Point) :
+    Real.cos ω * capReflection ω p 0 +
+        Real.sin ω * capReflection ω p 1 = p 1 := by
+  rw [capReflection_apply_zero, capReflection_apply_one]
+  calc
+    _ = (Real.sin ω ^ 2 + Real.cos ω ^ 2) * p 1 := by ring
+    _ = p 1 := by rw [Real.sin_sq_add_cos_sq]; ring
+
+private def rightRemovedTriangle (ω d : ℝ) : Set Point :=
+  (stripParallelogram ω).1 ∩
+    {p | d + Real.tan ((Real.pi / 2 - ω) / 2) < p 0}
+
+private def leftRemovedTriangle (ω d : ℝ) : Set Point :=
+  (stripParallelogram ω).1 ∩
+    {p | d + Real.tan ((Real.pi / 2 - ω) / 2) <
+      -Real.sin ω * p 0 + Real.cos ω * p 1}
+
+private theorem leftRemovedTriangle_eq_preimage (ω d : ℝ) :
+    leftRemovedTriangle ω d =
+      capReflection ω ⁻¹' rightRemovedTriangle ω d := by
+  ext p
+  simp only [leftRemovedTriangle, rightRemovedTriangle, Set.mem_inter_iff,
+    Set.mem_ofPred_eq, Set.mem_preimage]
+  rw [mem_stripParallelogram_coordinates,
+    mem_stripParallelogram_coordinates,
+    capReflection_rotatedCoordinate, capReflection_apply_zero,
+    capReflection_apply_one]
+  tauto
+
+private theorem volume_stripParallelogram (ω : ℝ) (hC : 0 < Real.cos ω) :
+    volume (stripParallelogram ω).1 = ENNReal.ofReal (1 / Real.cos ω) := by
+  let f : ℝ → ℝ := fun y ↦ (-Real.sin ω * y) / Real.cos ω
+  let g : ℝ → ℝ := fun y ↦ (1 - Real.sin ω * y) / Real.cos ω
+  have hf : Measurable f := by fun_prop
+  have hg : Measurable g := by fun_prop
+  have hfi : IntegrableOn f (Icc (0 : ℝ) 1) :=
+    (intervalIntegrable_iff_integrableOn_Icc_of_le (by norm_num)).mp
+      ((by fun_prop : Continuous f).intervalIntegrable 0 1)
+  have hgi : IntegrableOn g (Icc (0 : ℝ) 1) :=
+    (intervalIntegrable_iff_integrableOn_Icc_of_le (by norm_num)).mp
+      ((by fun_prop : Continuous g).intervalIntegrable 0 1)
+  have hfg : ∀ y ∈ Icc (0 : ℝ) 1, f y ≤ g y := by
+    intro y _
+    dsimp [f, g]
+    rw [div_le_div_iff_of_pos_right hC]
+    linarith
+  have hset : (stripParallelogram ω).1 =
+      {p : Point | p 1 ∈ Icc (0 : ℝ) 1 ∧ p 0 ∈ Icc (f (p 1)) (g (p 1))} := by
+    ext p
+    rw [mem_stripParallelogram_coordinates]
+    simp only [Set.mem_ofPred_eq, Set.mem_Icc]
+    constructor
+    · rintro ⟨hy, hz⟩
+      refine ⟨hy, ?_, ?_⟩
+      · dsimp [f]
+        apply (div_le_iff₀ hC).2
+        linarith [hz.1]
+      · dsimp [g]
+        apply (le_div_iff₀ hC).2
+        linarith [hz.2]
+    · rintro ⟨hy, hx0, hx1⟩
+      refine ⟨hy, ?_, ?_⟩
+      · dsimp [f] at hx0
+        have := (div_le_iff₀ hC).1 hx0
+        linarith
+      · dsimp [g] at hx1
+        have := (le_div_iff₀ hC).1 hx1
+        linarith
+  rw [hset, volume_horizontalIcc hf hg measurableSet_Icc hfi hgi hfg]
+  have hdiff : g - f = fun _ ↦ 1 / Real.cos ω := by
+    funext y
+    dsimp [f, g]
+    field_simp [hC.ne']
+    ring
+  rw [hdiff, setIntegral_const]
+  simp
+
+private theorem volume_rightRemovedTriangle (ω d : ℝ)
+    (hω : ω ∈ Ioo 0 (Real.pi / 2)) (hd0 : 0 ≤ d) (hdt : d ≤ Real.tan ω) :
+    volume (rightRemovedTriangle ω d) =
+      ENNReal.ofReal ((Real.tan ω - d) ^ 2 *
+        (Real.cos ω / Real.sin ω) / 2) := by
+  let C := Real.cos ω
+  let S := Real.sin ω
+  let c := Real.tan ((Real.pi / 2 - ω) / 2)
+  let A := c + d
+  let b := Real.tan ω - d
+  let H := b * C / S
+  let f : ℝ → ℝ := fun _ ↦ A
+  let g : ℝ → ℝ := fun y ↦ (1 - S * y) / C
+  have hC : 0 < C :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos, hω.1], hω.2⟩
+  have hS : 0 < S := Real.sin_pos_of_pos_of_lt_pi hω.1
+    (by linarith [hω.2, Real.pi_pos])
+  have hcircle : S ^ 2 + C ^ 2 = 1 := Real.sin_sq_add_cos_sq ω
+  have hgap : c = C⁻¹ - Real.tan ω :=
+    (parallelogram_gap ω ⟨hω.1.le, hω.2⟩).2.2.2.2
+  have htan : Real.tan ω = S / C := Real.tan_eq_sin_div_cos ω
+  have hcC : c * C = 1 - S := by
+    rw [hgap, htan]
+    field_simp [hC.ne']
+  have hS1 : S < 1 := by nlinarith [sq_pos_of_pos hC]
+  have hc0 : 0 < c := by
+    have hcdiv : c = (1 - S) / C := by
+      apply (eq_div_iff hC.ne').2
+      exact hcC
+    rw [hcdiv]
+    exact div_pos (sub_pos.mpr hS1) hC
+  have hb0 : 0 ≤ b := by dsimp [b]; linarith
+  have hbC : b * C = S - d * C := by
+    dsimp [b]
+    rw [htan]
+    field_simp [hC.ne']
+  have hAC : A * C = 1 - b * C := by
+    dsimp [A]
+    nlinarith [hcC, hbC]
+  have hH0 : 0 ≤ H := by positivity
+  have hH1 : H ≤ 1 := by
+    dsimp [H, b]
+    apply (div_le_iff₀ hS).2
+    dsimp [b] at hbC
+    nlinarith
+  have hf : Measurable f := measurable_const
+  have hg : Measurable g := by fun_prop
+  have hfi : IntegrableOn f (Icc (0 : ℝ) H) :=
+    (intervalIntegrable_iff_integrableOn_Icc_of_le hH0).mp
+      ((by fun_prop : Continuous f).intervalIntegrable 0 H)
+  have hgi : IntegrableOn g (Icc (0 : ℝ) H) :=
+    (intervalIntegrable_iff_integrableOn_Icc_of_le hH0).mp
+      ((by fun_prop : Continuous g).intervalIntegrable 0 H)
+  have hfg : ∀ y ∈ Icc (0 : ℝ) H, f y ≤ g y := by
+    intro y hy
+    dsimp [f, g, A]
+    apply (le_div_iff₀ hC).2
+    dsimp [H, b] at hy
+    dsimp [b] at hbC hAC
+    have hy' := (le_div_iff₀ hS).1 hy.2
+    nlinarith
+  have hset : rightRemovedTriangle ω d =
+      {p : Point | p 1 ∈ Icc (0 : ℝ) H ∧ p 0 ∈ Ioc (f (p 1)) (g (p 1))} := by
+    ext p
+    simp only [rightRemovedTriangle, Set.mem_inter_iff, Set.mem_ofPred_eq,
+      Set.mem_Icc, Set.mem_Ioc]
+    rw [mem_stripParallelogram_coordinates]
+    constructor
+    · rintro ⟨⟨hy, hz⟩, hxA⟩
+      refine ⟨⟨hy.1, ?_⟩, ?_, ?_⟩
+      · dsimp [H, b]
+        have hu : C * A + S * p 1 < 1 := by
+          dsimp [A]
+          nlinarith [mul_lt_mul_of_pos_left hxA hC]
+        apply (le_div_iff₀ hS).2
+        nlinarith [hAC]
+      · simpa [f, A, add_comm]
+      · dsimp [g]
+        apply (le_div_iff₀ hC).2
+        linarith [hz.2]
+    · rintro ⟨hy, hxA, hxU⟩
+      refine ⟨⟨⟨hy.1, hy.2.trans hH1⟩, ?_, ?_⟩, ?_⟩
+      · have hA0 : 0 < A := by dsimp [A]; linarith
+        dsimp [f, A] at hxA
+        nlinarith [mul_pos hC (lt_trans hA0 hxA)]
+      · dsimp [g] at hxU
+        have := (le_div_iff₀ hC).1 hxU
+        linarith
+      · simpa [f, A, add_comm] using hxA
+  rw [hset, volume_horizontalIoc hf hg measurableSet_Icc hfi hgi hfg]
+  have hdiff : g - f = fun y ↦ b - S / C * y := by
+    funext y
+    dsimp [f, g, A, b]
+    rw [hgap, htan]
+    field_simp [hC.ne']
+    ring
+  rw [hdiff, integral_Icc_eq_integral_Ioc,
+    ← intervalIntegral.integral_of_le hH0]
+  have hconst : IntervalIntegrable (fun _ : ℝ ↦ b) volume 0 H :=
+    continuous_const.intervalIntegrable 0 H
+  have hlinear : IntervalIntegrable (fun y : ℝ ↦ S / C * y) volume 0 H :=
+    (continuous_const.mul continuous_id).intervalIntegrable 0 H
+  rw [intervalIntegral.integral_sub hconst hlinear,
+    intervalIntegral.integral_const, intervalIntegral.integral_const_mul,
+    integral_id]
+  simp only [sub_zero, smul_eq_mul]
+  congr 1
+  norm_num
+  change H * b - S / C * (H ^ 2 / 2) = b ^ 2 * (C / S) / 2
+  dsimp [H]
+  field_simp [hC.ne', hS.ne']
+  ring
+
+private theorem measurableSet_stripParallelogram (ω : ℝ) :
+    MeasurableSet (stripParallelogram ω).1 := by
+  rw [show (stripParallelogram ω).1 =
+      {p : Point |
+        (0 ≤ p 1 ∧ p 1 ≤ 1) ∧
+        (0 ≤ Real.cos ω * p 0 + Real.sin ω * p 1 ∧
+          Real.cos ω * p 0 + Real.sin ω * p 1 ≤ 1)} by
+    ext p
+    exact mem_stripParallelogram_coordinates ω p]
+  measurability
+
+private theorem measurableSet_rightRemovedTriangle (ω d : ℝ) :
+    MeasurableSet (rightRemovedTriangle ω d) := by
+  unfold rightRemovedTriangle
+  apply (measurableSet_stripParallelogram ω).inter
+  measurability
+
+private theorem measurableSet_leftRemovedTriangle (ω d : ℝ) :
+    MeasurableSet (leftRemovedTriangle ω d) := by
+  unfold leftRemovedTriangle
+  apply (measurableSet_stripParallelogram ω).inter
+  measurability
+
+private theorem volume_leftRemovedTriangle_eq_right (ω d : ℝ) :
+    volume (leftRemovedTriangle ω d) = volume (rightRemovedTriangle ω d) := by
+  rw [leftRemovedTriangle_eq_preimage]
+  exact (LinearIsometryEquiv.measurePreserving (capReflection ω)).measure_preimage
+    (measurableSet_rightRemovedTriangle ω d).nullMeasurableSet
+
+private theorem stripParallelogram_decomposition (ω d : ℝ) :
+    (stripParallelogram ω).1 =
+      (clippedCap ω d ∪ rightRemovedTriangle ω d) ∪ leftRemovedTriangle ω d := by
+  ext p
+  simp only [Set.mem_union]
+  constructor
+  · intro hp
+    by_cases hx : p 0 ≤ d + Real.tan ((Real.pi / 2 - ω) / 2)
+    · by_cases hz : -Real.sin ω * p 0 + Real.cos ω * p 1 ≤
+          d + Real.tan ((Real.pi / 2 - ω) / 2)
+      · exact Or.inl (Or.inl ((mem_clippedCap_iff ω d p).2
+          ⟨((mem_stripParallelogram_coordinates ω p).1 hp).1,
+            ((mem_stripParallelogram_coordinates ω p).1 hp).2, hx, hz⟩))
+      · exact Or.inr ⟨hp, by
+          change d + Real.tan ((Real.pi / 2 - ω) / 2) <
+            -Real.sin ω * p 0 + Real.cos ω * p 1
+          exact lt_of_not_ge hz⟩
+    · exact Or.inl (Or.inr ⟨hp, by
+        change d + Real.tan ((Real.pi / 2 - ω) / 2) < p 0
+        exact lt_of_not_ge hx⟩)
+  · rintro (hp | hp)
+    · rcases hp with hp | hp
+      · exact (mem_stripParallelogram_coordinates ω p).2
+          ⟨((mem_clippedCap_iff ω d p).1 hp).1,
+            ((mem_clippedCap_iff ω d p).1 hp).2.1⟩
+      · exact hp.1
+    · exact hp.1
+
+private theorem clippedCap_disjoint_rightRemovedTriangle (ω d : ℝ) :
+    Disjoint (clippedCap ω d) (rightRemovedTriangle ω d) := by
+  rw [Set.disjoint_left]
+  intro p hp hq
+  have hp' := ((mem_clippedCap_iff ω d p).1 hp).2.2.1
+  exact (not_lt_of_ge hp') hq.2
+
+private theorem clippedCap_disjoint_leftRemovedTriangle (ω d : ℝ) :
+    Disjoint (clippedCap ω d) (leftRemovedTriangle ω d) := by
+  rw [Set.disjoint_left]
+  intro p hp hq
+  have hp' := ((mem_clippedCap_iff ω d p).1 hp).2.2.2
+  exact (not_lt_of_ge hp') hq.2
+
+private theorem rightRemovedTriangle_disjoint_leftRemovedTriangle
+    {ω d : ℝ} (hC : 0 < Real.cos ω) (hS : 0 ≤ Real.sin ω) (hd : 0 ≤ d)
+    (hc : Real.tan ((Real.pi / 2 - ω) / 2) * (1 + Real.sin ω) = Real.cos ω) :
+    Disjoint (rightRemovedTriangle ω d) (leftRemovedTriangle ω d) := by
+  apply (clippedCap_removed_pieces_disjoint hC hS hd hc).mono
+  · rintro p ⟨hp, hright⟩
+    change d + Real.tan ((Real.pi / 2 - ω) / 2) < p 0 at hright
+    exact ⟨((mem_stripParallelogram_coordinates ω p).1 hp).1.2,
+      by simpa [add_comm] using hright⟩
+  · rintro p ⟨_, hleft⟩
+    change d + Real.tan ((Real.pi / 2 - ω) / 2) <
+      -Real.sin ω * p 0 + Real.cos ω * p 1 at hleft
+    simpa [add_comm] using hleft
+
+theorem clippedCap_area_formula (ω d : ℝ)
+    (hω : ω ∈ Ioo 0 (Real.pi / 2)) (hd0 : 0 ≤ d) (hdt : d ≤ Real.tan ω) :
+    ClassicalResults.area (clippedCap ω d) =
+      1 / Real.cos ω - (Real.tan ω - d) ^ 2 *
+        (Real.cos ω / Real.sin ω) := by
+  let C := Real.cos ω
+  let S := Real.sin ω
+  let c := Real.tan ((Real.pi / 2 - ω) / 2)
+  let T := (Real.tan ω - d) ^ 2 * (C / S) / 2
+  have hC : 0 < C :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos, hω.1], hω.2⟩
+  have hS : 0 < S := Real.sin_pos_of_pos_of_lt_pi hω.1
+    (by linarith [hω.2, Real.pi_pos])
+  have hgap : c = C⁻¹ - Real.tan ω :=
+    (parallelogram_gap ω ⟨hω.1.le, hω.2⟩).2.2.2.2
+  have htan : Real.tan ω = S / C := Real.tan_eq_sin_div_cos ω
+  have hcC : c * C = 1 - S := by
+    rw [hgap, htan]
+    field_simp [hC.ne']
+  have hc : c * (1 + S) = C := by
+    apply (mul_right_cancel₀ hC.ne')
+    nlinarith [hcC, Real.sin_sq_add_cos_sq ω]
+  have hP := volume_stripParallelogram ω hC
+  have hR := volume_rightRemovedTriangle ω d hω hd0 hdt
+  have hL : volume (leftRemovedTriangle ω d) = ENNReal.ofReal T := by
+    rw [volume_leftRemovedTriangle_eq_right, hR]
+  have hR' : volume (rightRemovedTriangle ω d) = ENNReal.ofReal T := by
+    simpa [T, C, S] using hR
+  have hCR := clippedCap_disjoint_rightRemovedTriangle ω d
+  have hCL := clippedCap_disjoint_leftRemovedTriangle ω d
+  have hRL := rightRemovedTriangle_disjoint_leftRemovedTriangle hC hS.le hd0 hc
+  have hdecomp : volume (stripParallelogram ω).1 =
+      (volume (clippedCap ω d) + volume (rightRemovedTriangle ω d)) +
+        volume (leftRemovedTriangle ω d) := by
+    rw [stripParallelogram_decomposition,
+      measure_union (hCL.union_left hRL) (measurableSet_leftRemovedTriangle ω d),
+      measure_union hCR (measurableSet_rightRemovedTriangle ω d)]
+  rw [hP, hR', hL] at hdecomp
+  have hclip_subset : clippedCap ω d ⊆ (stripParallelogram ω).1 := by
+    intro p hp
+    exact (mem_stripParallelogram_coordinates ω p).2
+      ⟨((mem_clippedCap_iff ω d p).1 hp).1,
+        ((mem_clippedCap_iff ω d p).1 hp).2.1⟩
+  have hclip_ne : volume (clippedCap ω d) ≠ ⊤ := by
+    apply ne_of_lt
+    refine lt_of_le_of_lt (measure_mono hclip_subset) ?_
+    rw [hP]
+    exact ENNReal.ofReal_lt_top
+  have hT0 : 0 ≤ T := by dsimp [T]; positivity
+  have hTne : ENNReal.ofReal T ≠ ⊤ := ENNReal.ofReal_ne_top
+  have hsum_ne : volume (clippedCap ω d) + ENNReal.ofReal T ≠ ⊤ :=
+    ENNReal.add_ne_top.mpr ⟨hclip_ne, hTne⟩
+  have hreal := congrArg ENNReal.toReal hdecomp
+  rw [ENNReal.toReal_add hsum_ne hTne,
+    ENNReal.toReal_add hclip_ne hTne,
+    ENNReal.toReal_ofReal (by positivity : 0 ≤ 1 / Real.cos ω),
+    ENNReal.toReal_ofReal hT0] at hreal
+  unfold ClassicalResults.area
+  dsimp [T, C, S] at hreal ⊢
+  linarith
+
+theorem clippedCap_minimum_area (ω : RotationCalculationAngle) :
+    ClassicalResults.area (clippedCap ω.val (rotationCalculationMinimum ω)) < 11 / 5 := by
+  have hω0 : 0 < ω.val :=
+    (Real.arccos_pos.mpr (by norm_num : (5 / 11 : ℝ) < 1)).trans_le ω.property.1
+  calc
+    ClassicalResults.area (clippedCap ω.val (rotationCalculationMinimum ω)) =
+        1 / Real.cos ω.val -
+          (Real.tan ω.val - rotationCalculationMinimum ω) ^ 2 *
+            (Real.cos ω.val / Real.sin ω.val) :=
+      clippedCap_area_formula ω.val (rotationCalculationMinimum ω)
+        ⟨hω0, ω.property.2⟩ (rotationCalculationMinimum_pos ω).le
+        (rotationCalculationMinimum_lt_tan ω).le
+    _ < 11 / 5 := rotationCalculation_area_estimate ω
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Densities
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped NNReal ENNReal
+
+namespace MovingSofa
+
+private theorem coe_injOn_halfTurn :
+    Set.InjOn (fun t : ℝ ↦ (t : Real.Angle)) (Icc 0 Real.pi) := by
+  let : Fact (0 < 2 * Real.pi) := ⟨by positivity⟩
+  intro x hx y hy heq
+  exact (AddCircle.coe_eq_coe_iff_of_mem_Ico
+    (show x ∈ Ico 0 (0 + 2 * Real.pi) from ⟨hx.1, by linarith [hx.2, Real.pi_pos]⟩)
+    (show y ∈ Ico 0 (0 + 2 * Real.pi) from ⟨hy.1, by linarith [hy.2, Real.pi_pos]⟩)).mp heq
+
+/-- A right-angle cap whose surface area measure has angular densities on the two upper quarter
+circles has no atom at a normal direction of the right upper quarter. -/
+theorem HasCapDensities.surfaceAreaMeasure_singleton_eq_zero_of_mem_Ico
+    {K : RightAngleCapSpace} {r s : ℝ → ℝ≥0} (hK : HasCapDensities K r s) {t : ℝ}
+    (ht : t ∈ Ico 0 (Real.pi / 2)) :
+    surfaceAreaMeasure K.1 {(t : Real.Angle)} = 0 := by
+  have he := congrArg (fun μ : Measure Real.Angle ↦ μ {(t : Real.Angle)}) hK.2.2.1
+  have hm : (t : Real.Angle) ∈
+      (fun x : ℝ ↦ (x : Real.Angle)) '' Ico 0 (Real.pi / 2) := ⟨t, ht, rfl⟩
+  rw [Measure.restrict_apply (measurableSet_singleton _),
+    inter_eq_left.mpr (singleton_subset_iff.mpr hm)] at he
+  rw [he]
+  apply Measure.map_restrict_withDensity_singleton volume _ Real.Angle.continuous_coe.measurable
+    _ (coe_injOn_halfTurn.mono ?_) _ ht
+  intro x hx
+  exact ⟨hx.1, by linarith [hx.2, Real.pi_pos]⟩
+
+/-- A right-angle cap whose surface area measure has angular densities on the two upper quarter
+circles has no atom at a normal direction of the left upper quarter. -/
+theorem HasCapDensities.surfaceAreaMeasure_singleton_eq_zero_of_mem_Ioc
+    {K : RightAngleCapSpace} {r s : ℝ → ℝ≥0} (hK : HasCapDensities K r s) {t : ℝ}
+    (ht : t ∈ Ioc 0 (Real.pi / 2)) :
+    surfaceAreaMeasure K.1 {((t + Real.pi / 2 : ℝ) : Real.Angle)} = 0 := by
+  have he := congrArg (fun μ : Measure Real.Angle ↦
+    μ {((t + Real.pi / 2 : ℝ) : Real.Angle)}) hK.2.2.2
+  have hm : ((t + Real.pi / 2 : ℝ) : Real.Angle) ∈
+      (fun x : ℝ ↦ (x : Real.Angle)) '' Ioc (Real.pi / 2) Real.pi :=
+    ⟨t + Real.pi / 2, ⟨by linarith [ht.1], by linarith [ht.2]⟩, rfl⟩
+  rw [Measure.restrict_apply (measurableSet_singleton _),
+    inter_eq_left.mpr (singleton_subset_iff.mpr hm)] at he
+  rw [he]
+  apply Measure.map_restrict_withDensity_singleton volume _
+    (Real.Angle.continuous_coe.comp (continuous_id.add continuous_const)).measurable
+    _ ?_ _ ht
+  intro x hx y hy hxy
+  have heq := coe_injOn_halfTurn
+    (show x + Real.pi / 2 ∈ Icc 0 Real.pi from
+      ⟨by linarith [hx.1, Real.pi_pos], by linarith [hx.2]⟩)
+    (show y + Real.pi / 2 ∈ Icc 0 Real.pi from
+      ⟨by linarith [hy.1, Real.pi_pos], by linarith [hy.2]⟩) hxy
+  linarith
+
+theorem capDensities_contact_eq (K : RightAngleCapSpace)
+    (hK : ∃ r s, HasCapDensities K r s) :
+    (∀ t ∈ Set.Ico (0 : ℝ) (Real.pi / 2),
+      (capVertices K t).1.1 = (capVertices K t).1.2 ∧
+      (tangentArmLengths K t).1.1 = (tangentArmLengths K t).1.2) ∧
+    (∀ t ∈ Set.Ioc (0 : ℝ) (Real.pi / 2),
+      (capVertices K t).2.1 = (capVertices K t).2.2 ∧
+      (tangentArmLengths K t).2.1 = (tangentArmLengths K t).2.2) := by
+  obtain ⟨r, s, hK⟩ := hK
+  constructor
+  · intro t ht
+    have h := (surfaceAreaMeasure_atom_length K.1 (t : Real.Angle)).2.2
+    rw [hK.surfaceAreaMeasure_singleton_eq_zero_of_mem_Ico ht] at h
+    simp only [ENNReal.toReal_zero, zero_smul, add_zero] at h
+    have h' : (capVertices K t).1.1 = (capVertices K t).1.2 := h
+    exact ⟨h', by simp only [tangentArmLengths, h']⟩
+  · intro t ht
+    have h := (surfaceAreaMeasure_atom_length K.1
+      ((t + Real.pi / 2 : ℝ) : Real.Angle)).2.2
+    rw [hK.surfaceAreaMeasure_singleton_eq_zero_of_mem_Ioc ht] at h
+    simp only [ENNReal.toReal_zero, zero_smul, add_zero] at h
+    have h' : (capVertices K t).2.1 = (capVertices K t).2.2 := h
+    exact ⟨h', by simp only [tangentArmLengths, h']⟩
+
+/-- A right-angle cap carrying angular densities has singleton extreme faces at every upper normal
+direction except possibly the vertical one: the densities exclude atoms of the surface area measure
+on the two open quarter circles, so the corresponding faces have zero side length. -/
+theorem capDensities_edgeVertices_eq (K : RightAngleCapSpace)
+    (hK : ∃ r s, HasCapDensities K r s) {t : ℝ} (ht : t ∈ Icc 0 Real.pi)
+    (htop : t ≠ Real.pi / 2) :
+    (edgeVertices K.val (t : Real.Angle)).1 = (edgeVertices K.val (t : Real.Angle)).2 := by
+  rcases lt_or_gt_of_ne htop with h | h
+  · exact ((capDensities_contact_eq K hK).1 t ⟨ht.1, h⟩).1
+  · have hmem : t - Real.pi / 2 ∈ Ioc (0 : ℝ) (Real.pi / 2) := ⟨by linarith, by linarith [ht.2]⟩
+    have h' : (edgeVertices K.val ((t - Real.pi / 2 + Real.pi / 2 : ℝ) : Real.Angle)).1 =
+        (edgeVertices K.val ((t - Real.pi / 2 + Real.pi / 2 : ℝ) : Real.Angle)).2 :=
+      ((capDensities_contact_eq K hK).2 (t - Real.pi / 2) hmem).1
+    rwa [show t - Real.pi / 2 + Real.pi / 2 = t from by ring] at h'
+
+/-- The two cap contact paths and their scalar density functions on the quarter-turn interval. -/
+def nondegenerateCapData (K : RightAngleCapSpace)
+    (_hK : ∃ r s, HasCapDensities K r s) :
+    ((Set.Icc (0 : ℝ) (Real.pi / 2) → Point) ×
+      (Set.Icc (0 : ℝ) (Real.pi / 2) → Point)) ×
+    ((Set.Icc (0 : ℝ) (Real.pi / 2) → ℝ) ×
+      (Set.Icc (0 : ℝ) (Real.pi / 2) → ℝ)) := by
+  classical
+  exact ((fun t ↦ if (t : ℝ) = Real.pi / 2 then (capVertices K t).1.2
+      else (capVertices K t).1.1,
+    fun t ↦ (capVertices K t).2.1),
+    (fun t ↦ if (t : ℝ) = Real.pi / 2 then (tangentArmLengths K t).1.2
+      else (tangentArmLengths K t).1.1,
+    fun t ↦ (tangentArmLengths K t).2.1))
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Upper Boundary
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The union of exposed cap edges over the upper range of normal directions. -/
+def capUpperBoundary {ω : ℝ} (K : CapSpace ω) : Set Point :=
+  ⋃ t ∈ Set.Icc 0 (ω + Real.pi / 2), exposedEdge K.val (t : Real.Angle)
+
+theorem capUpperBoundary_connected {ω : ℝ} (K : CapSpace ω) :
+    IsConnected (capUpperBoundary K) := by
+  let I : Set ℝ := Set.Icc 0 (ω + Real.pi / 2)
+  let G : Set (I × Point) :=
+    {q | q.2 ∈ exposedEdge K.val ((q.1 : ℝ) : Real.Angle)}
+  have hω : 0 ≤ ω + Real.pi / 2 := by
+    have := K.property.1
+    positivity
+  have hI : IsConnected I := isConnected_Icc hω
+  let _ : ConnectedSpace I := Subtype.connectedSpace hI
+  have hnormal : Continuous (fun q : I × Point ↦
+      normalVector (((q.1 : I) : ℝ) : Real.Angle)) :=
+    continuous_normalVector_real.comp (continuous_subtype_val.comp continuous_fst)
+  have hsupport : Continuous (fun q : I × Point ↦
+      supportValue K.val (((q.1 : I) : ℝ) : Real.Angle)) :=
+    (continuous_supportValue_real K.val).comp (continuous_subtype_val.comp continuous_fst)
+  have heq : IsClosed {q : I × Point |
+      inner ℝ q.2 (normalVector (((q.1 : I) : ℝ) : Real.Angle)) =
+        supportValue K.val (((q.1 : I) : ℝ) : Real.Angle)} :=
+    isClosed_eq (continuous_snd.inner hnormal) hsupport
+  have hG : IsCompact G := by
+    rw [show G = Set.univ ×ˢ (K.val : Set Point) ∩
+        {q : I × Point | inner ℝ q.2 (normalVector (((q.1 : I) : ℝ) : Real.Angle)) =
+          supportValue K.val (((q.1 : I) : ℝ) : Real.Angle)} by
+      ext q
+      simp only [G, exposedEdge, supportingLineHalfPlane, normalLine, Set.mem_ofPred_eq,
+        Set.mem_inter_iff, Set.mem_prod, Set.mem_univ, true_and]]
+    exact (isCompact_univ.prod K.val.isCompact).inter_right heq
+  let _ : CompactSpace G := isCompact_iff_compactSpace.mp hG
+  let π : G → I := fun q ↦ q.1.1
+  have hπcont : Continuous π := continuous_fst.comp continuous_subtype_val
+  have hπsurj : Function.Surjective π := by
+    intro t
+    obtain ⟨x, hx⟩ := exposedEdge_nonempty K.val ((t : ℝ) : Real.Angle)
+    exact ⟨⟨(t, x), hx⟩, rfl⟩
+  have hπquot : Topology.IsQuotientMap π :=
+    Topology.IsQuotientMap.of_surjective_continuous hπsurj hπcont
+  have hfiber (t : I) : IsConnected (π ⁻¹' {t}) := by
+    let E := exposedEdge K.val ((t : ℝ) : Real.Angle)
+    let e : E → G := fun x ↦ ⟨(t, x), x.property⟩
+    have hecont : Continuous e := by
+      apply Continuous.subtype_mk
+      exact continuous_const.prodMk continuous_subtype_val
+    have himage : e '' Set.univ = π ⁻¹' {t} := by
+      ext q
+      constructor
+      · rintro ⟨x, -, rfl⟩
+        simp [π, e]
+      · intro hq
+        have hqt : q.1.1 = t := by simpa [π] using hq
+        have hqx : q.1.2 ∈ exposedEdge K.val ((t : ℝ) : Real.Angle) := by
+          have hqG := q.property
+          change q.1.2 ∈ exposedEdge K.val (((q.1.1 : I) : ℝ) : Real.Angle) at hqG
+          simpa [hqt] using hqG
+        let x : E := ⟨q.1.2, hqx⟩
+        refine ⟨x, Set.mem_univ x, ?_⟩
+        apply Subtype.ext
+        apply Prod.ext
+        · exact hqt.symm
+        · rfl
+    rw [← himage]
+    let _ : ConnectedSpace E :=
+      Subtype.connectedSpace (isConnected_exposedEdge K.val ((t : ℝ) : Real.Angle))
+    exact isConnected_univ.image e hecont.continuousOn
+  have hGconn : IsConnected (Set.univ : Set G) := by
+    exact hπquot.isCoinducing.isConnected_preimage_of_isClosed hfiber isClosed_univ
+      isConnected_univ
+  have himage : (fun q : G ↦ q.1.2) '' Set.univ = capUpperBoundary K := by
+    ext x
+    simp only [Set.mem_image, Set.mem_univ, true_and, capUpperBoundary, Set.mem_iUnion]
+    constructor
+    · rintro ⟨q, rfl⟩
+      exact ⟨q.1.1, q.1.1.property, q.property⟩
+    · rintro ⟨t, ht, hx⟩
+      exact ⟨⟨(⟨t, ht⟩, x), hx⟩, rfl⟩
+  rw [← himage]
+  exact hGconn.image _ (continuous_snd.comp continuous_subtype_val).continuousOn
+
+private theorem CapSpace.mem_interior_of_mem_not_upperBoundary {ω : ℝ}
+    (K : CapSpace ω) {p : Point} (hpK : p ∈ (K.val : Set Point))
+    (hpδ : p ∉ ⋃ t ∈ Set.Icc 0 (ω + Real.pi / 2),
+      exposedEdge K.val (t : Real.Angle)) :
+    (⟨p, K.subset_capFan hpK⟩ : capFan ω) ∈
+      interior {q : capFan ω | (q : Point) ∈ (K.val : Set Point)} := by
+  let I := Set.Icc 0 (ω + Real.pi / 2)
+  let gap : ℝ → ℝ := fun t ↦
+    supportValue K.val (t : Real.Angle) - inner ℝ p (normalVector (t : Real.Angle))
+  have hgap_cont : Continuous gap := (continuous_supportValue_real K.val).sub
+    (continuous_const.inner continuous_normalVector_real)
+  have hgap_pos : ∀ t ∈ I, 0 < gap t := by
+    intro t ht
+    have hle := inner_le_supportValue K.val hpK (t : Real.Angle)
+    have hne : inner ℝ p (normalVector (t : Real.Angle)) ≠
+        supportValue K.val (t : Real.Angle) := by
+      intro heq
+      apply hpδ
+      exact Set.mem_iUnion.mpr ⟨t, Set.mem_iUnion.mpr ⟨ht,
+        ⟨hpK, by simpa [supportingLineHalfPlane, normalLine]⟩⟩⟩
+    dsimp [gap]
+    exact sub_pos.mpr (lt_of_le_of_ne hle hne)
+  have hIne : I.Nonempty := ⟨0, by
+    change 0 ∈ Set.Icc 0 (ω + Real.pi / 2)
+    exact ⟨le_rfl, by linarith [K.property.1, Real.pi_pos]⟩⟩
+  have hIcompact : IsCompact I := by
+    dsimp [I]
+    exact isCompact_Icc
+  obtain ⟨m, hm, hmle⟩ := IsCompact.exists_pos_forall_le hIcompact hIne
+    hgap_cont.continuousOn hgap_pos
+  apply mem_interior_iff_mem_nhds.mpr
+  refine Filter.mem_of_superset (Metric.ball_mem_nhds _ (half_pos hm)) ?_
+  intro q hq
+  change (q : Point) ∈ (K.val : Set Point)
+  apply K.mem_of_mem_capFan_of_lt_supportValue q.property
+  intro t ht
+  have hdist : ‖(q : Point) - p‖ < m / 2 := by
+    change dist (q : Point) p < m / 2 at hq
+    simpa [dist_eq_norm] using hq
+  have hnorm := norm_normalVector_real t
+  have hinner : inner ℝ ((q : Point) - p) (normalVector (t : Real.Angle)) < m / 2 := by
+    calc
+      inner ℝ ((q : Point) - p) (normalVector (t : Real.Angle))
+          ≤ ‖(q : Point) - p‖ * ‖normalVector (t : Real.Angle)‖ := real_inner_le_norm _ _
+      _ < m / 2 := by simpa [hnorm] using hdist
+  have := hmle t ht
+  dsimp [gap] at this
+  rw [inner_sub_left] at hinner
+  linarith
+
+private theorem CapSpace.mem_frontier_of_mem_upperBoundary {ω : ℝ}
+    (K : CapSpace ω) {p : Point}
+    (hp : p ∈ ⋃ t ∈ Set.Icc 0 (ω + Real.pi / 2), exposedEdge K.val (t : Real.Angle)) :
+    ∃ z : capFan ω, (z : Point) = p ∧
+      z ∈ frontier {q : capFan ω | (q : Point) ∈ (K.val : Set Point)} := by
+  obtain ⟨t, hp⟩ := Set.mem_iUnion.mp hp
+  obtain ⟨htI, hpedge⟩ := Set.mem_iUnion.mp hp
+  let z : capFan ω := ⟨p, K.subset_capFan hpedge.1⟩
+  have hzK : z ∈ {q : capFan ω | (q : Point) ∈ (K.val : Set Point)} := hpedge.1
+  refine ⟨z, rfl, (mem_frontier_iff_notMem_interior hzK).2 ?_⟩
+  intro hzint
+  obtain ⟨ε, hε, hball⟩ := Metric.isOpen_iff.mp isOpen_interior z hzint
+  let e := ε / 2
+  have he : 0 < e := half_pos hε
+  let q : Point := p + e • normalVector (t : Real.Angle)
+  have hsin : 0 ≤ Real.sin t := Real.sin_nonneg_of_nonneg_of_le_pi htI.1
+    (htI.2.trans (by linarith [K.property.2.1, Real.pi_pos]))
+  have hcos : 0 ≤ Real.cos (t - ω) := Real.cos_nonneg_of_mem_Icc ⟨by
+    linarith [htI.1, K.property.2.1, Real.pi_pos], by linarith [htI.2]⟩
+  have hqfan : q ∈ capFan ω := by
+    constructor
+    · change 0 ≤ inner ℝ q (normalVector (ω : Real.Angle))
+      dsimp [q]
+      rw [inner_add_left, inner_smul_left, inner_normalVector_normalVector]
+      have hpω := (K.subset_capFan hpedge.1).1
+      change 0 ≤ inner ℝ p (normalVector (ω : Real.Angle)) at hpω
+      simp at *
+      nlinarith
+    · change 0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+      dsimp [q]
+      rw [inner_add_left, inner_smul_left, inner_normalVector_normalVector]
+      have hpT := (K.subset_capFan hpedge.1).2
+      change 0 ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at hpT
+      rw [Real.cos_sub]
+      simp
+      nlinarith
+  have hqball : (⟨q, hqfan⟩ : capFan ω) ∈ Metric.ball z ε := by
+    change dist q p < ε
+    simp only [q, dist_eq_norm, add_sub_cancel_left, norm_smul, norm_normalVector_real,
+      mul_one]
+    rw [Real.norm_eq_abs, abs_of_pos he]
+    dsimp [e]
+    linarith
+  have hqK := interior_subset (hball hqball)
+  change q ∈ (K.val : Set Point) at hqK
+  have hqle := inner_le_supportValue K.val hqK (t : Real.Angle)
+  have hpedge_eq := hpedge.2
+  change inner ℝ p (normalVector (t : Real.Angle)) = supportValue K.val (t : Real.Angle)
+    at hpedge_eq
+  dsimp [q] at hqle
+  rw [inner_add_left, inner_smul_left, inner_normalVector_self, hpedge_eq] at hqle
+  simp at hqle
+  nlinarith
+
+theorem capUpperBoundary_relativeBoundary {ω : ℝ} (K : CapSpace ω) :
+    capUpperBoundary K =
+      Subtype.val '' frontier {p : capFan ω | (p : Point) ∈ (K.val : Set Point)} := by
+  ext p
+  constructor
+  · intro hp
+    obtain ⟨z, rfl, hz⟩ := K.mem_frontier_of_mem_upperBoundary hp
+    exact ⟨z, hz, rfl⟩
+  · rintro ⟨z, hzfront, rfl⟩
+    let S : Set (capFan ω) := {p | (p : Point) ∈ (K.val : Set Point)}
+    have hSclosed : IsClosed S := K.val.isClosed.preimage continuous_subtype_val
+    have hzS : z ∈ S := by
+      apply hSclosed.closure_subset
+      exact frontier_subset_closure hzfront
+    by_contra hzupper
+    have hzint := K.mem_interior_of_mem_not_upperBoundary hzS hzupper
+    exact ((mem_frontier_iff_notMem_interior hzS).1 hzfront) hzint
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Bounds.Niche`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Bounds / Niche
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter MeasureTheory
+open scoped Topology
+
+namespace MovingSofa
+
+/-- The infimum of the set’s horizontal coordinates. -/
+def horizontalMin (S : Set Point) : ℝ := sInf ((fun p ↦ p 0) '' S)
+
+/-- The supremum of the set’s horizontal coordinates. -/
+def horizontalMax (S : Set Point) : ℝ := sSup ((fun p ↦ p 0) '' S)
+
+/-- The niche is measurable, finite-area and enclosed by the specified horizontal-span
+rectangle. -/
+def HasNicheRectangleBounds (S N : Set Point) : Prop :=
+  MeasurableSet N ∧
+  N ⊆ {p | horizontalMin S < p 0 ∧ p 0 < horizontalMax S ∧
+    0 ≤ p 1 ∧ p 1 < (horizontalMax S - horizontalMin S) / 2} ∧
+  volume N < ⊤ ∧
+  (horizontalMax S - horizontalMin S = 0 → ClassicalResults.area N = 0)
+
+private theorem niche_coordinate_bounds {c s l r x y : ℝ}
+    (hc : 0 < c) (hs : 0 < s) (hcircle : s ^ 2 + c ^ 2 = 1)
+    (hlr : l ≤ r) (hy : 0 ≤ y)
+    (hx : c * x + s * y < c * r + s - 1)
+    (hz : -s * x + c * y < -s * l + c - 1) :
+    l < x ∧ x < r ∧ y < (r - l) / 2 := by
+  have hc1 : c ≤ 1 := by nlinarith [sq_nonneg s]
+  have hs1 : s ≤ 1 := by nlinarith [sq_nonneg c]
+  have hsc : s * c ≤ 1 / 2 := by nlinarith [sq_nonneg (s - c)]
+  have hsum : 1 ≤ s + c := by nlinarith [mul_pos hs hc]
+  have hxlo : l < x := by nlinarith [mul_nonneg hc.le hy]
+  have hxhi : x < r := by nlinarith [mul_nonneg hs.le hy]
+  have h₁ := mul_lt_mul_of_pos_left hx hs
+  have h₂ := mul_lt_mul_of_pos_left hz hc
+  have hcomb : y < s * c * (r - l) + 1 - s - c := by
+    nlinarith [show s ^ 2 * y + c ^ 2 * y = y by nlinarith [hcircle]]
+  refine ⟨hxlo, hxhi, hcomb.trans_le ?_⟩
+  have := mul_le_mul_of_nonneg_right hsc (sub_nonneg.mpr hlr)
+  linarith
+
+theorem horizontalMin_le (K : ConvexBody Point) {p : Point} (hp : p ∈ K) :
+    horizontalMin K ≤ p 0 := by
+  apply csInf_le ((K.isCompact.image (by fun_prop : Continuous (fun p : Point ↦ p 0))).bddBelow)
+  exact ⟨p, hp, rfl⟩
+
+theorem le_horizontalMax (K : ConvexBody Point) {p : Point} (hp : p ∈ K) :
+    p 0 ≤ horizontalMax K := by
+  apply le_csSup ((K.isCompact.image (by fun_prop : Continuous (fun p : Point ↦ p 0))).bddAbove)
+  exact ⟨p, hp, rfl⟩
+
+theorem horizontalMin_le_horizontalMax (K : ConvexBody Point) :
+    horizontalMin K ≤ horizontalMax K := by
+  obtain ⟨p, hp⟩ := K.nonempty
+  exact (horizontalMin_le K hp).trans (le_horizontalMax K hp)
+
+/-- The support value at the horizontal normal is the horizontal maximum. -/
+theorem supportValue_zero_eq_horizontalMax (S : Set Point) :
+    supportValue S ((0 : ℝ) : Real.Angle) = horizontalMax S := by
+  rw [supportValue, horizontalMax]
+  congr 1
+  refine Set.image_congr fun p _ ↦ ?_
+  rw [inner_normalVector_real, Real.cos_zero, Real.sin_zero, mul_one, mul_zero, add_zero]
+
+/-- The support value at the straight angle negates the horizontal minimum. -/
+theorem supportValue_pi_eq_neg_horizontalMin (S : Set Point) :
+    supportValue S ((Real.pi : ℝ) : Real.Angle) = -horizontalMin S := by
+  have himage : (fun p : Point ↦ -p 0) '' S = -((fun p : Point ↦ p 0) '' S) := by
+    ext x
+    simp [neg_eq_iff_eq_neg]
+  rw [supportValue, horizontalMin, ← Real.sSup_neg, ← himage]
+  congr 1
+  refine Set.image_congr fun p _ ↦ ?_
+  rw [inner_normalVector_real, Real.cos_pi, Real.sin_pi, mul_neg_one, mul_zero, add_zero]
+
+/-- Both horizontal extrema of a compact convex body are attained. -/
+theorem exists_horizontal_extrema (K : ConvexBody Point) :
+    ∃ l r : Point, l ∈ (K : Set Point) ∧ r ∈ (K : Set Point) ∧
+      l 0 = horizontalMin K ∧ r 0 = horizontalMax K := by
+  obtain ⟨l, hlK, hl⟩ := K.isCompact.exists_isMinOn K.nonempty
+    (by fun_prop : Continuous (fun p : Point ↦ p 0)).continuousOn
+  obtain ⟨r, hrK, hr⟩ := K.isCompact.exists_isMaxOn K.nonempty
+    (by fun_prop : Continuous (fun p : Point ↦ p 0)).continuousOn
+  refine ⟨l, r, hlK, hrK, ?_, ?_⟩
+  · apply le_antisymm
+    · apply le_csInf (K.nonempty.image _)
+      rintro _ ⟨p, hp, rfl⟩
+      exact hl hp
+    · exact horizontalMin_le K hlK
+  · apply le_antisymm
+    · exact le_horizontalMax K hrK
+    · apply csSup_le (K.nonempty.image _)
+      rintro _ ⟨p, hp, rfl⟩
+      exact hr hp
+
+theorem CapSpace.mem_horizontalStrip {ω : ℝ} (K : CapSpace ω)
+    {p : Point} (hp : p ∈ K.val) : 0 ≤ p 1 ∧ p 1 ≤ 1 := by
+  have hupper := inner_le_supportValue K.val hp ((Real.pi / 2 : ℝ) : Real.Angle)
+  have hlower := inner_le_supportValue K.val hp ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+  rw [K.property.2.2.2.1] at hupper
+  rw [K.property.2.2.2.2.2.1] at hlower
+  rw [show 3 * Real.pi / 2 = Real.pi + Real.pi / 2 by ring] at hlower
+  simp only [normalVector, frame, Real.Angle.cos_coe, Real.cos_pi_div_two, Real.Angle.sin_coe,
+    Real.sin_pi_div_two, PiLp.inner_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two,
+    Fin.isValue, Matrix.cons_val_zero, zero_mul, Matrix.cons_val_one, Matrix.cons_val_fin_one,
+    one_mul, zero_add, Real.cos_add, Real.cos_pi, mul_zero, Real.sin_pi, mul_one, sub_self,
+    Real.sin_add, neg_neg, neg_mul, Left.neg_nonpos_iff] at hupper hlower
+  exact ⟨by linarith, hupper⟩
+
+/-- A cap has area at most its horizontal width. -/
+theorem CapSpace.area_le_horizontalWidth {ω : ℝ} (K : CapSpace ω) :
+    ClassicalResults.area (K.val : Set Point) ≤ horizontalMax K.val - horizontalMin K.val := by
+  let R : Set Point := {p | p 0 ∈ Set.Icc (horizontalMin K.val) (horizontalMax K.val) ∧
+    p 1 ∈ Set.Icc 0 1}
+  have hsub : (K.val : Set Point) ⊆ R := fun p hp ↦
+    ⟨⟨horizontalMin_le K.val hp, le_horizontalMax K.val hp⟩, K.mem_horizontalStrip hp⟩
+  have hvol : volume R = ENNReal.ofReal (horizontalMax K.val - horizontalMin K.val) := by
+    rw [EuclideanSpace.volume_setOf_apply_mem_Icc]
+    simp
+  have hfinite : volume R ≠ ⊤ := by rw [hvol]; exact ENNReal.ofReal_ne_top
+  calc
+    ClassicalResults.area (K.val : Set Point) ≤ ClassicalResults.area R :=
+      ENNReal.toReal_mono hfinite (measure_mono hsub)
+    _ = horizontalMax K.val - horizontalMin K.val := by
+      change (volume R).toReal = _
+      rw [hvol, ENNReal.toReal_ofReal (sub_nonneg.mpr (horizontalMin_le_horizontalMax K.val))]
+
+/-- Upper support bounds from the horizontal extrema and the unit-height strip. -/
+theorem CapSpace.supportValue_horizontal_bounds {ω t : ℝ} (K : CapSpace ω)
+    (ht : t ∈ Set.Ioo 0 ω) :
+    supportValue K.val (t : Real.Angle) ≤
+      Real.cos t * horizontalMax K.val + Real.sin t ∧
+    supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) ≤
+      -Real.sin t * horizontalMin K.val + Real.cos t := by
+  have hc : 0 ≤ Real.cos t := (Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, ht.1], ht.2.trans_le K.property.2.1⟩).le
+  have hs : 0 ≤ Real.sin t := (Real.sin_pos_of_pos_of_lt_pi ht.1
+    (by linarith [ht.2, K.property.2.1, Real.pi_pos])).le
+  constructor
+  · apply csSup_le (K.val.nonempty.image _)
+    rintro _ ⟨p, hp, rfl⟩
+    have hx := mul_le_mul_of_nonneg_left (le_horizontalMax K.val hp) hc
+    have hy := mul_le_mul_of_nonneg_left (K.mem_horizontalStrip hp).2 hs
+    simp [normalVector, frame, PiLp.inner_apply]
+    nlinarith
+  · apply csSup_le (K.val.nonempty.image _)
+    rintro _ ⟨p, hp, rfl⟩
+    have hx := mul_le_mul_of_nonneg_left (horizontalMin_le K.val hp) hs
+    have hy := mul_le_mul_of_nonneg_left (K.mem_horizontalStrip hp).2 hc
+    simp [normalVector, frame, PiLp.inner_apply, Real.cos_add, Real.sin_add,
+      -Real.Angle.coe_add]
+    nlinarith
+
+/-- Lower support bounds from the horizontal extrema and the nonnegative heights of a cap. -/
+theorem CapSpace.horizontal_le_supportValue {ω t : ℝ} (K : CapSpace ω)
+    (hs : 0 ≤ Real.sin t) (hc : 0 ≤ Real.cos t) :
+    Real.cos t * horizontalMax K.val ≤ supportValue K.val (t : Real.Angle) ∧
+    -Real.sin t * horizontalMin K.val ≤
+      supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) := by
+  have hcompact := K.val.isCompact.image (by fun_prop : Continuous (fun p : Point ↦ p 0))
+  have hne := K.val.nonempty.image (fun p : Point ↦ p 0)
+  constructor
+  · obtain ⟨p, hp, hpmax⟩ := hcompact.sSup_mem hne
+    have hinner := inner_le_supportValue K.val hp (t : Real.Angle)
+    have hy := (K.mem_horizontalStrip hp).1
+    simp [normalVector, frame, PiLp.inner_apply] at hinner
+    change p 0 = horizontalMax K.val at hpmax
+    rw [← hpmax]
+    nlinarith [mul_nonneg hs hy]
+  · obtain ⟨p, hp, hpmin⟩ := hcompact.sInf_mem hne
+    have hinner := inner_le_supportValue K.val hp ((t + Real.pi / 2 : ℝ) : Real.Angle)
+    have hy := (K.mem_horizontalStrip hp).1
+    simp [normalVector, frame, PiLp.inner_apply, Real.cos_add, Real.sin_add,
+      -Real.Angle.coe_add] at hinner
+    change p 0 = horizontalMin K.val at hpmin
+    rw [← hpmin]
+    nlinarith [mul_nonneg hc hy]
+
+private theorem mem_niche_rectangle {ω t : ℝ} (K : CapSpace ω)
+    (ht : t ∈ Set.Ioo 0 ω) {p : Point}
+    (hf : p ∈ capFan ω) (hq : p ∈ innerQuadrant K.val t) :
+    horizontalMin K.val < p 0 ∧ p 0 < horizontalMax K.val ∧
+      0 ≤ p 1 ∧ p 1 < (horizontalMax K.val - horizontalMin K.val) / 2 := by
+  have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, ht.1], ht.2.trans_le K.property.2.1⟩
+  have hs : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi ht.1
+    (by linarith [ht.2, K.property.2.1, Real.pi_pos])
+  have hy : 0 ≤ p 1 := by
+    have := hf.2
+    simpa [normalHalfPlane, normalVector, frame, PiLp.inner_apply] using this
+  obtain ⟨h₁, h₂⟩ := hq
+  change inner ℝ p (normalVector (t : Real.Angle)) < supportValue K.val _ - 1 at h₁
+  change inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) <
+    supportValue K.val _ - 1 at h₂
+  have hbounds := K.supportValue_horizontal_bounds ht
+  have hx : Real.cos t * p 0 + Real.sin t * p 1 <
+      Real.cos t * horizontalMax K.val + Real.sin t - 1 := by
+    have := h₁.trans_le (sub_le_sub_right hbounds.1 1)
+    simpa [normalVector, frame, PiLp.inner_apply, mul_comm] using this
+  have hz : -Real.sin t * p 0 + Real.cos t * p 1 <
+      -Real.sin t * horizontalMin K.val + Real.cos t - 1 := by
+    have := h₂.trans_le (sub_le_sub_right hbounds.2 1)
+    simpa [normalVector, frame, PiLp.inner_apply, Real.cos_add, Real.sin_add,
+      -Real.Angle.coe_add, mul_comm] using this
+  obtain ⟨hl, hr, hh⟩ := niche_coordinate_bounds hc hs (Real.sin_sq_add_cos_sq t)
+    (horizontalMin_le_horizontalMax K.val) hy hx hz
+  exact ⟨hl, hr, hy, hh⟩
+
+theorem capNiche_subset_rectangle {ω : ℝ} (K : CapSpace ω) :
+    capNiche K ⊆ {p | horizontalMin K.val < p 0 ∧ p 0 < horizontalMax K.val ∧
+      0 ≤ p 1 ∧ p 1 < (horizontalMax K.val - horizontalMin K.val) / 2} := by
+  rintro p ⟨hf, hq⟩
+  obtain ⟨t, ht, hq⟩ := Set.mem_iUnion₂.mp hq
+  exact mem_niche_rectangle K ht hf hq
+
+theorem polygonNiche_subset_capNiche (Θ : AngleSet) (K : CapSpace Θ.angle) :
+    polygonNiche Θ K ⊆ capNiche K := by
+  rintro p ⟨hf, hq⟩
+  obtain ⟨t, ht, hq⟩ := Set.mem_iUnion₂.mp hq
+  exact ⟨hf, Set.mem_iUnion₂.mpr ⟨t, Θ.interior t ht, hq⟩⟩
+
+private theorem isBounded_coordinate_rectangle (l r h : ℝ) :
+    Bornology.IsBounded {p : Point | l < p 0 ∧ p 0 < r ∧ 0 ≤ p 1 ∧ p 1 < h} := by
+  apply isBounded_iff_forall_norm_le.mpr
+  refine ⟨|l| + |r| + |h|, ?_⟩
+  rintro p ⟨hl, hr, hy, hh⟩
+  have hx : |p 0| ≤ |l| + |r| := abs_le.mpr ⟨by
+    have := neg_abs_le l
+    linarith [abs_nonneg r], by linarith [le_abs_self r, abs_nonneg l]⟩
+  have hy' : |p 1| ≤ |h| := by rw [abs_of_nonneg hy]; exact hh.le.trans (le_abs_self h)
+  have hx2 := sq_le_sq₀ (abs_nonneg (p 0)) (by positivity : 0 ≤ |l| + |r|) |>.mpr hx
+  have hy2 := sq_le_sq₀ (abs_nonneg (p 1)) (abs_nonneg h) |>.mpr hy'
+  have hn := EuclideanSpace.norm_sq_eq p
+  simp only [Fin.sum_univ_two, Real.norm_eq_abs, sq_abs] at hn hx2 hy2
+  nlinarith [mul_nonneg (by positivity : 0 ≤ |l| + |r|) (abs_nonneg h), norm_nonneg p,
+    abs_nonneg l, abs_nonneg r, abs_nonneg h, sq_abs h]
+
+private theorem isClosed_capFan (ω : ℝ) : IsClosed (capFan ω) := by
+  apply IsClosed.inter
+  · exact isClosed_le continuous_const (by fun_prop)
+  · exact isClosed_le continuous_const (by fun_prop)
+
+private theorem isOpen_innerQuadrant (S : Set Point) (t : ℝ) :
+    IsOpen (innerQuadrant S t) := by
+  apply IsOpen.inter
+  · exact isOpen_lt (by fun_prop) continuous_const
+  · exact isOpen_lt (by fun_prop) continuous_const
+
+theorem measurableSet_capNiche {ω : ℝ} (K : CapSpace ω) : MeasurableSet (capNiche K) :=
+  (isClosed_capFan ω).measurableSet.inter
+    (isOpen_iUnion fun t ↦ isOpen_iUnion fun _ ↦ isOpen_innerQuadrant K.val t).measurableSet
+
+theorem measurableSet_polygonNiche (Θ : AngleSet) (K : CapSpace Θ.angle) :
+    MeasurableSet (polygonNiche Θ K) :=
+  (isClosed_capFan Θ.angle).measurableSet.inter
+    (isOpen_iUnion fun t ↦ isOpen_iUnion fun _ ↦ isOpen_innerQuadrant K.val t).measurableSet
+
+private theorem hasNicheRectangleBounds_of_subset {ω : ℝ} (K : CapSpace ω)
+    {N : Set Point} (hm : MeasurableSet N) (hN : N ⊆ capNiche K) :
+    HasNicheRectangleBounds K.val N := by
+  have hb := hN.trans (capNiche_subset_rectangle K)
+  refine ⟨hm, hb, (isBounded_coordinate_rectangle _ _ _).subset hb |>.measure_lt_top, ?_⟩
+  intro hd
+  have he : N = ∅ := by
+    apply Set.eq_empty_iff_forall_notMem.mpr
+    intro p hp
+    have := hb hp
+    linarith [this.1, this.2.1]
+  simp [he, ClassicalResults.area]
+
+private theorem niche_subset_uniform_rectangle {ω R : ℝ} (K : CapSpace ω)
+    (hR : ∀ p ∈ (K.val : Set Point), |p 0| ≤ R) :
+    capNiche K ⊆ {p | |p 0| ≤ R ∧ 0 ≤ p 1 ∧ p 1 ≤ R} := by
+  have hlo : -R ≤ horizontalMin K.val := by
+    apply le_csInf (K.val.nonempty.image _)
+    rintro _ ⟨p, hp, rfl⟩
+    exact (abs_le.mp (hR p hp)).1
+  have hhi : horizontalMax K.val ≤ R := by
+    apply csSup_le (K.val.nonempty.image _)
+    rintro _ ⟨p, hp, rfl⟩
+    exact (abs_le.mp (hR p hp)).2
+  intro p hp
+  obtain ⟨hl, hr, hy, hh⟩ := capNiche_subset_rectangle K hp
+  exact ⟨abs_le.mpr ⟨hlo.trans hl.le, hr.le.trans hhi⟩, hy, by linarith⟩
+
+theorem niche_uniform_bounds :
+    (∀ (ω : ℝ) (K : CapSpace ω),
+      HasNicheRectangleBounds (K.val : Set Point) (capNiche K)) ∧
+    (∀ (Θ : AngleSet) (K : CapSpace Θ.angle),
+      HasNicheRectangleBounds (K.val : Set Point) (polygonNiche Θ K)) ∧
+    (∀ (ω : ℕ → ℝ) (K : ∀ i, CapSpace (ω i)) (L : ConvexBody Point),
+      Tendsto (fun i ↦ Metric.hausdorffDist ((K i).val : Set Point) (L : Set Point))
+        atTop (𝓝 0) →
+      ∃ R : ℝ, 0 ≤ R ∧
+        (∀ i, capNiche (K i) ⊆ {p | |p 0| ≤ R ∧ 0 ≤ p 1 ∧ p 1 ≤ R}) ∧
+        (∀ i (Θ : AngleSet) (P : CapSpace Θ.angle),
+          (P.val : Set Point) = ((K i).val : Set Point) →
+          polygonNiche Θ P ⊆ {p | |p 0| ≤ R ∧ 0 ≤ p 1 ∧ p 1 ≤ R})) := by
+  refine ⟨fun _ K ↦ hasNicheRectangleBounds_of_subset K (measurableSet_capNiche K) Set.Subset.rfl,
+    fun Θ K ↦ hasNicheRectangleBounds_of_subset K (measurableSet_polygonNiche Θ K)
+      (polygonNiche_subset_capNiche Θ K), ?_⟩
+  intro ω K L hlim
+  obtain ⟨A, hA⟩ := (Metric.isBounded_range_of_tendsto _ hlim).exists_norm_le
+  obtain ⟨B, hB⟩ := L.isCompact.isBounded.exists_norm_le
+  let R := |A| + |B| + 1
+  have hR0 : 0 ≤ R := by dsimp [R]; positivity
+  have hcarriers : ∀ i p, p ∈ ((K i).val : Set Point) → |p 0| ≤ R := by
+    intro i p hp
+    have hdist : Metric.hausdorffDist ((K i).val : Set Point) (L : Set Point) < |A| + 1 := by
+      have := hA _ (Set.mem_range_self i)
+      rw [Real.norm_eq_abs] at this
+      exact lt_of_le_of_lt ((le_abs_self _).trans (this.trans (le_abs_self A))) (by linarith)
+    obtain ⟨q, hq, hpq⟩ := Metric.exists_dist_lt_of_hausdorffDist_lt hp hdist
+      (Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded
+        (K i).val.nonempty L.nonempty (K i).val.isCompact.isBounded L.isCompact.isBounded)
+    have hnorm : ‖p‖ ≤ R := by
+      have ht := norm_add_le (p - q) q
+      rw [sub_add_cancel] at ht
+      rw [dist_eq_norm] at hpq
+      have hb := (hB q hq).trans (le_abs_self B)
+      dsimp [R]
+      linarith
+    have hc : |p 0| ≤ ‖p‖ := by
+      simpa only [Real.norm_eq_abs] using (PiLp.norm_apply_le p 0)
+    exact hc.trans hnorm
+  refine ⟨R, hR0, fun i ↦ niche_subset_uniform_rectangle (K i) (hcarriers i), ?_⟩
+  intro i Θ P hP
+  apply (polygonNiche_subset_capNiche Θ P).trans
+  apply niche_subset_uniform_rectangle P
+  intro p hp
+  exact hcarriers i p (hP ▸ hp)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Geometry.ContactGeometry`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Geometry / Contact Geometry
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter
+open scoped Topology unitInterval
+
+namespace MovingSofa
+
+private theorem inner_tangent_le_fst (K : ConvexBody Point) (t : Real.Angle)
+    {p : Point} (hp : p ∈ exposedEdge K t) :
+    inner ℝ p (tangentVector t) ≤ inner ℝ (edgeVertices K t).1 (tangentVector t) := by
+  rw [inner_edgeVertices_fst_tangent]
+  exact le_csSup ((isCompact_exposedEdge K t).image
+    (continuous_id.inner continuous_const)).bddAbove ⟨p, hp, rfl⟩
+
+private theorem snd_le_inner_tangent (K : ConvexBody Point) (t : Real.Angle)
+    {p : Point} (hp : p ∈ exposedEdge K t) :
+    inner ℝ (edgeVertices K t).2 (tangentVector t) ≤ inner ℝ p (tangentVector t) := by
+  rw [inner_edgeVertices_snd_tangent]
+  exact csInf_le ((isCompact_exposedEdge K t).image
+    (continuous_id.inner continuous_const)).bddBelow ⟨p, hp, rfl⟩
+
+private theorem fst_tangent_le_of_mem_exposedEdge_add (K : ConvexBody Point) (t δ : ℝ)
+    (hδ : δ ∈ Set.Ioo (0 : ℝ) (Real.pi / 2)) {q : Point}
+    (hq : q ∈ exposedEdge K ((t + δ : ℝ) : Real.Angle)) :
+    inner ℝ (edgeVertices K (t : Real.Angle)).1 (tangentVector (t : Real.Angle)) ≤
+      inner ℝ q (tangentVector (t : Real.Angle)) := by
+  have hcomp := inner_le_supportValue K (edgeVertices_fst_mem K (t : Real.Angle)).1
+    ((t + δ : ℝ) : Real.Angle)
+  have hqnormal : inner ℝ q (normalVector ((t + δ : ℝ) : Real.Angle)) =
+      supportValue K ((t + δ : ℝ) : Real.Angle) := hq.2
+  rw [← hqnormal, normalVector_add_real] at hcomp
+  simp only [inner_add_right, inner_smul_right] at hcomp
+  have hpnormal : inner ℝ (edgeVertices K (t : Real.Angle)).1
+      (normalVector (t : Real.Angle)) = supportValue K (t : Real.Angle) :=
+    (edgeVertices_fst_mem K (t : Real.Angle)).2
+  rw [hpnormal] at hcomp
+  have hqle := inner_le_supportValue K hq.1 (t : Real.Angle)
+  have hsin := Real.sin_pos_of_pos_of_lt_pi hδ.1 (hδ.2.trans (by linarith [Real.pi_pos]))
+  have hcos := Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos, hδ.1], hδ.2⟩
+  nlinarith
+
+private theorem tendsto_facePoint_right (K : ConvexBody Point) (t : ℝ) (q : ℝ → Point)
+    (hq : ∀ s : ℝ, q s ∈ exposedEdge K (s : Real.Angle)) :
+    Tendsto q (𝓝[>] t) (𝓝 (edgeVertices K (t : Real.Angle)).1) := by
+  apply K.isCompact.tendsto_nhds_of_unique_mapClusterPt
+    (Filter.Eventually.of_forall fun s ↦ (hq s).1)
+  intro p hp hcluster
+  obtain ⟨s, hqs, hs⟩ := hcluster.exists_seq_tendsto
+  have hst : Tendsto s atTop (𝓝 t) := hs.mono_right nhdsWithin_le_nhds
+  have hnormal := hqs.inner (𝕜 := ℝ) (continuous_normalVector_real.continuousAt.tendsto.comp hst)
+  have hsupport := (continuous_supportValue_real K).continuousAt.tendsto.comp hst
+  have hpnormal : inner ℝ p (normalVector (t : Real.Angle)) =
+      supportValue K (t : Real.Angle) := by
+    apply tendsto_nhds_unique hnormal
+    exact hsupport.congr (fun n ↦ (hq (s n)).2.symm)
+  have hpface : p ∈ exposedEdge K (t : Real.Angle) := ⟨hp, hpnormal⟩
+  have hnear : ∀ᶠ n in atTop, s n ∈ Set.Ioo t (t + Real.pi / 2) :=
+    hs.eventually (Ioo_mem_nhdsGT (by linarith [Real.pi_pos]))
+  have hle : inner ℝ (edgeVertices K (t : Real.Angle)).1 (tangentVector (t : Real.Angle)) ≤
+      inner ℝ p (tangentVector (t : Real.Angle)) := by
+    apply ge_of_tendsto (hqs.inner tendsto_const_nhds)
+    filter_upwards [hnear] with n hn
+    have hn' : s n - t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) := by
+      constructor <;> linarith [hn.1, hn.2]
+    simpa only [add_sub_cancel, Function.comp_apply] using
+      fst_tangent_le_of_mem_exposedEdge_add K t (s n - t) hn'
+        (by simpa only [add_sub_cancel, Function.comp_apply] using hq (s n))
+  have heq := le_antisymm (inner_tangent_le_fst K (t : Real.Angle) hpface) hle
+  rw [← inner_normalVector_smul_add_inner_tangentVector_smul p (t : Real.Angle), hpnormal, heq]
+  rw [← (edgeVertices_fst_mem K (t : Real.Angle)).2]
+  exact inner_normalVector_smul_add_inner_tangentVector_smul _ _
+
+private theorem tangent_le_snd_of_mem_exposedEdge_add (K : ConvexBody Point) (t δ : ℝ)
+    (hδ : δ ∈ Set.Ioo (-(Real.pi / 2)) (0 : ℝ)) {q : Point}
+    (hq : q ∈ exposedEdge K ((t + δ : ℝ) : Real.Angle)) :
+    inner ℝ q (tangentVector (t : Real.Angle)) ≤
+      inner ℝ (edgeVertices K (t : Real.Angle)).2 (tangentVector (t : Real.Angle)) := by
+  have hcomp := inner_le_supportValue K (edgeVertices_snd_mem K (t : Real.Angle)).1
+    ((t + δ : ℝ) : Real.Angle)
+  have hqnormal : inner ℝ q (normalVector ((t + δ : ℝ) : Real.Angle)) =
+      supportValue K ((t + δ : ℝ) : Real.Angle) := hq.2
+  rw [← hqnormal, normalVector_add_real] at hcomp
+  simp only [inner_add_right, inner_smul_right] at hcomp
+  have hpnormal : inner ℝ (edgeVertices K (t : Real.Angle)).2
+      (normalVector (t : Real.Angle)) = supportValue K (t : Real.Angle) :=
+    (edgeVertices_snd_mem K (t : Real.Angle)).2
+  rw [hpnormal] at hcomp
+  have hqle := inner_le_supportValue K hq.1 (t : Real.Angle)
+  have hsin : Real.sin δ < 0 := by
+    have h := Real.sin_pos_of_pos_of_lt_pi (by linarith [hδ.2] : 0 < -δ)
+      (by linarith [Real.pi_pos, hδ.1] : -δ < Real.pi)
+    simpa using h
+  have hcos : 0 < Real.cos δ := Real.cos_pos_of_mem_Ioo
+    ⟨hδ.1, by linarith [Real.pi_pos, hδ.2]⟩
+  nlinarith
+
+private theorem tendsto_facePoint_left (K : ConvexBody Point) (t : ℝ) (q : ℝ → Point)
+    (hq : ∀ s : ℝ, q s ∈ exposedEdge K (s : Real.Angle)) :
+    Tendsto q (𝓝[<] t) (𝓝 (edgeVertices K (t : Real.Angle)).2) := by
+  apply K.isCompact.tendsto_nhds_of_unique_mapClusterPt
+    (Filter.Eventually.of_forall fun s ↦ (hq s).1)
+  intro p hp hcluster
+  obtain ⟨s, hqs, hs⟩ := hcluster.exists_seq_tendsto
+  have hst : Tendsto s atTop (𝓝 t) := hs.mono_right nhdsWithin_le_nhds
+  have hnormal := hqs.inner (𝕜 := ℝ) (continuous_normalVector_real.continuousAt.tendsto.comp hst)
+  have hsupport := (continuous_supportValue_real K).continuousAt.tendsto.comp hst
+  have hpnormal : inner ℝ p (normalVector (t : Real.Angle)) =
+      supportValue K (t : Real.Angle) := by
+    apply tendsto_nhds_unique hnormal
+    exact hsupport.congr (fun n ↦ (hq (s n)).2.symm)
+  have hpface : p ∈ exposedEdge K (t : Real.Angle) := ⟨hp, hpnormal⟩
+  have hnear : ∀ᶠ n in atTop, s n ∈ Set.Ioo (t - Real.pi / 2) t :=
+    hs.eventually (Ioo_mem_nhdsLT (by linarith [Real.pi_pos]))
+  have hle : inner ℝ p (tangentVector (t : Real.Angle)) ≤
+      inner ℝ (edgeVertices K (t : Real.Angle)).2 (tangentVector (t : Real.Angle)) := by
+    apply le_of_tendsto (hqs.inner tendsto_const_nhds)
+    filter_upwards [hnear] with n hn
+    have hn' : s n - t ∈ Set.Ioo (-(Real.pi / 2)) (0 : ℝ) := by
+      constructor <;> linarith [hn.1, hn.2]
+    simpa only [add_sub_cancel, Function.comp_apply] using
+      tangent_le_snd_of_mem_exposedEdge_add K t (s n - t) hn'
+        (by simpa only [add_sub_cancel, Function.comp_apply] using hq (s n))
+  have heq := le_antisymm hle (snd_le_inner_tangent K (t : Real.Angle) hpface)
+  rw [← inner_normalVector_smul_add_inner_tangentVector_smul p (t : Real.Angle), hpnormal, heq]
+  rw [← (edgeVertices_snd_mem K (t : Real.Angle)).2]
+  exact inner_normalVector_smul_add_inner_tangentVector_smul _ _
+
+private theorem intersection_tangent_bounds_right (K : ConvexBody Point) (t δ : ℝ)
+    (hδ : δ ∈ Set.Ioo (0 : ℝ) (Real.pi / 2)) {q : Point}
+    (hq : q ∈ exposedEdge K ((t + δ : ℝ) : Real.Angle)) :
+    inner ℝ (edgeVertices K (t : Real.Angle)).1 (tangentVector (t : Real.Angle)) ≤
+        (supportValue K ((t + δ : ℝ) : Real.Angle) -
+          supportValue K (t : Real.Angle) * Real.cos δ) / Real.sin δ ∧
+      (supportValue K ((t + δ : ℝ) : Real.Angle) -
+          supportValue K (t : Real.Angle) * Real.cos δ) / Real.sin δ ≤
+        inner ℝ q (tangentVector (t : Real.Angle)) := by
+  have hs : 0 < Real.sin δ := Real.sin_pos_of_pos_of_lt_pi hδ.1
+    (by linarith [hδ.2, Real.pi_pos])
+  have hc : 0 ≤ Real.cos δ := (Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [hδ.1, Real.pi_pos], hδ.2⟩).le
+  have hp := inner_le_supportValue K (edgeVertices_fst_mem K (t : Real.Angle)).1
+    ((t + δ : ℝ) : Real.Angle)
+  have hqn : inner ℝ q (normalVector ((t + δ : ℝ) : Real.Angle)) =
+      supportValue K ((t + δ : ℝ) : Real.Angle) := hq.2
+  have hpn : inner ℝ (edgeVertices K (t : Real.Angle)).1
+      (normalVector (t : Real.Angle)) = supportValue K (t : Real.Angle) :=
+    (edgeVertices_fst_mem K (t : Real.Angle)).2
+  rw [normalVector_add_real] at hp hqn
+  simp only [inner_add_right, inner_smul_right] at hp hqn
+  rw [hpn] at hp
+  have hqle := mul_le_mul_of_nonneg_left (inner_le_supportValue K hq.1 (t : Real.Angle)) hc
+  constructor
+  · rw [le_div_iff₀ hs]
+    linarith
+  · rw [div_le_iff₀ hs]
+    linarith
+
+private theorem tendsto_supportingIntersection_right (K : ConvexBody Point) (t : ℝ) :
+    Tendsto (fun s : ℝ ↦ supportingIntersection K (t : Real.Angle) (s : Real.Angle))
+      (𝓝[>] t) (𝓝 (edgeVertices K (t : Real.Angle)).1) := by
+  let r (s : ℝ) := (supportValue K (s : Real.Angle) -
+    supportValue K (t : Real.Angle) * Real.cos (s - t)) / Real.sin (s - t)
+  have hr : Tendsto r (𝓝[>] t)
+      (𝓝 (inner ℝ (edgeVertices K (t : Real.Angle)).1 (tangentVector (t : Real.Angle)))) := by
+    have hq := (tendsto_facePoint_right K t
+      (fun s ↦ (edgeVertices K (s : Real.Angle)).1) (fun s ↦ edgeVertices_fst_mem K _)).inner
+      (𝕜 := ℝ) (tendsto_const_nhds (x := tangentVector (t : Real.Angle)))
+    have hbounds : ∀ᶠ s in 𝓝[>] t,
+        inner ℝ (edgeVertices K (t : Real.Angle)).1 (tangentVector (t : Real.Angle)) ≤ r s ∧
+        r s ≤ inner ℝ (edgeVertices K (s : Real.Angle)).1 (tangentVector (t : Real.Angle)) := by
+      filter_upwards [Ioo_mem_nhdsGT (show t < t + Real.pi / 2 by linarith [Real.pi_pos])]
+        with s hs
+      have hδ : s - t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) := by
+        constructor <;> linarith [hs.1, hs.2]
+      simpa only [add_sub_cancel, r] using intersection_tangent_bounds_right K t (s - t) hδ
+        (by simpa only [add_sub_cancel] using edgeVertices_fst_mem K (s : Real.Angle))
+    exact tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hq
+      (hbounds.mono fun _ h ↦ h.1) (hbounds.mono fun _ h ↦ h.2)
+  have hz := (tendsto_const_nhds
+    (x := supportValue K (t : Real.Angle) • normalVector (t : Real.Angle))).add (hr.smul
+    (tendsto_const_nhds (x := tangentVector (t : Real.Angle))))
+  have hpnormal := (edgeVertices_fst_mem K (t : Real.Angle)).2
+  change inner ℝ (edgeVertices K (t : Real.Angle)).1 (normalVector (t : Real.Angle)) =
+    supportValue K (t : Real.Angle) at hpnormal
+  have heq := inner_normalVector_smul_add_inner_tangentVector_smul (edgeVertices K (t :
+    Real.Angle)).1 (t : Real.Angle)
+  rw [hpnormal] at heq
+  simpa only [supportingIntersection, ← Real.Angle.coe_sub, Real.Angle.cos_coe,
+    Real.Angle.sin_coe, heq, r] using hz
+
+private theorem intersection_tangent_bounds_left (K : ConvexBody Point) (t δ : ℝ)
+    (hδ : δ ∈ Set.Ioo (-(Real.pi / 2)) (0 : ℝ)) {q : Point}
+    (hq : q ∈ exposedEdge K ((t + δ : ℝ) : Real.Angle)) :
+    (supportValue K ((t + δ : ℝ) : Real.Angle) -
+          supportValue K (t : Real.Angle) * Real.cos δ) / Real.sin δ ≤
+      inner ℝ (edgeVertices K (t : Real.Angle)).2 (tangentVector (t : Real.Angle)) ∧
+      inner ℝ q (tangentVector (t : Real.Angle)) ≤
+        (supportValue K ((t + δ : ℝ) : Real.Angle) -
+          supportValue K (t : Real.Angle) * Real.cos δ) / Real.sin δ := by
+  have hs : Real.sin δ < 0 := by
+    have h := Real.sin_pos_of_pos_of_lt_pi (by linarith [hδ.2] : 0 < -δ)
+      (by linarith [hδ.1, Real.pi_pos] : -δ < Real.pi)
+    simpa using h
+  have hc : 0 ≤ Real.cos δ := (Real.cos_pos_of_mem_Ioo
+    ⟨hδ.1, by linarith [hδ.2, Real.pi_pos]⟩).le
+  have hp := inner_le_supportValue K (edgeVertices_snd_mem K (t : Real.Angle)).1
+    ((t + δ : ℝ) : Real.Angle)
+  have hqn : inner ℝ q (normalVector ((t + δ : ℝ) : Real.Angle)) =
+      supportValue K ((t + δ : ℝ) : Real.Angle) := hq.2
+  have hpn : inner ℝ (edgeVertices K (t : Real.Angle)).2
+      (normalVector (t : Real.Angle)) = supportValue K (t : Real.Angle) :=
+    (edgeVertices_snd_mem K (t : Real.Angle)).2
+  rw [normalVector_add_real] at hp hqn
+  simp only [inner_add_right, inner_smul_right] at hp hqn
+  rw [hpn] at hp
+  have hqle := mul_le_mul_of_nonneg_left (inner_le_supportValue K hq.1 (t : Real.Angle)) hc
+  constructor
+  · rw [div_le_iff_of_neg hs]
+    linarith
+  · rw [le_div_iff_of_neg hs]
+    linarith
+
+private theorem tendsto_supportingIntersection_left_aux (K : ConvexBody Point) (t : ℝ) :
+    Tendsto (fun s : ℝ ↦ supportingIntersection K (t : Real.Angle) (s : Real.Angle))
+      (𝓝[<] t) (𝓝 (edgeVertices K (t : Real.Angle)).2) := by
+  let r (s : ℝ) := (supportValue K (s : Real.Angle) -
+    supportValue K (t : Real.Angle) * Real.cos (s - t)) / Real.sin (s - t)
+  have hr : Tendsto r (𝓝[<] t)
+      (𝓝 (inner ℝ (edgeVertices K (t : Real.Angle)).2 (tangentVector (t : Real.Angle)))) := by
+    have hq := (tendsto_facePoint_left K t
+      (fun s ↦ (edgeVertices K (s : Real.Angle)).2) (fun s ↦ edgeVertices_snd_mem K _)).inner
+      (𝕜 := ℝ) (tendsto_const_nhds (x := tangentVector (t : Real.Angle)))
+    have hbounds : ∀ᶠ s in 𝓝[<] t,
+        r s ≤ inner ℝ (edgeVertices K (t : Real.Angle)).2 (tangentVector (t : Real.Angle)) ∧
+        inner ℝ (edgeVertices K (s : Real.Angle)).2 (tangentVector (t : Real.Angle)) ≤ r s := by
+      filter_upwards [Ioo_mem_nhdsLT (show t - Real.pi / 2 < t by linarith [Real.pi_pos])]
+        with s hs
+      have hδ : s - t ∈ Set.Ioo (-(Real.pi / 2)) (0 : ℝ) := by
+        constructor <;> linarith [hs.1, hs.2]
+      simpa only [add_sub_cancel, r] using intersection_tangent_bounds_left K t (s - t) hδ
+        (by simpa only [add_sub_cancel] using edgeVertices_snd_mem K (s : Real.Angle))
+    exact tendsto_of_tendsto_of_tendsto_of_le_of_le' hq tendsto_const_nhds
+      (hbounds.mono fun _ h ↦ h.2) (hbounds.mono fun _ h ↦ h.1)
+  have hz := (tendsto_const_nhds
+    (x := supportValue K (t : Real.Angle) • normalVector (t : Real.Angle))).add (hr.smul
+    (tendsto_const_nhds (x := tangentVector (t : Real.Angle))))
+  have hpnormal := (edgeVertices_snd_mem K (t : Real.Angle)).2
+  change inner ℝ (edgeVertices K (t : Real.Angle)).2 (normalVector (t : Real.Angle)) =
+    supportValue K (t : Real.Angle) at hpnormal
+  have heq := inner_normalVector_smul_add_inner_tangentVector_smul (edgeVertices K (t :
+    Real.Angle)).2 (t : Real.Angle)
+  rw [hpnormal] at heq
+  simpa only [supportingIntersection, ← Real.Angle.coe_sub, Real.Angle.cos_coe,
+    Real.Angle.sin_coe, heq, r] using hz
+
+/-- Both face endpoints and the supporting intersections have the stated one-sided limits. -/
+theorem contact_oneSided_limits (K : ConvexBody Point) (t : ℝ) :
+    Tendsto (fun s : ℝ ↦ (edgeVertices K (s : Real.Angle)).1)
+      (𝓝[>] t) (𝓝 (edgeVertices K (t : Real.Angle)).1) ∧
+    Tendsto (fun s : ℝ ↦ (edgeVertices K (s : Real.Angle)).2)
+      (𝓝[>] t) (𝓝 (edgeVertices K (t : Real.Angle)).1) ∧
+    Tendsto (fun s : ℝ ↦ supportingIntersection K (t : Real.Angle) (s : Real.Angle))
+      (𝓝[>] t) (𝓝 (edgeVertices K (t : Real.Angle)).1) ∧
+    Tendsto (fun s : ℝ ↦ (edgeVertices K (s : Real.Angle)).1)
+      (𝓝[<] t) (𝓝 (edgeVertices K (t : Real.Angle)).2) ∧
+    Tendsto (fun s : ℝ ↦ (edgeVertices K (s : Real.Angle)).2)
+      (𝓝[<] t) (𝓝 (edgeVertices K (t : Real.Angle)).2) ∧
+    Tendsto (fun s : ℝ ↦ supportingIntersection K (s : Real.Angle) (t : Real.Angle))
+      (𝓝[<] t) (𝓝 (edgeVertices K (t : Real.Angle)).2) := by
+  refine ⟨tendsto_facePoint_right K t _ (fun s ↦ edgeVertices_fst_mem K _),
+    tendsto_facePoint_right K t _ (fun s ↦ edgeVertices_snd_mem K _),
+    tendsto_supportingIntersection_right K t,
+    tendsto_facePoint_left K t _ (fun s ↦ edgeVertices_fst_mem K _),
+    tendsto_facePoint_left K t _ (fun s ↦ edgeVertices_snd_mem K _), ?_⟩
+  apply (tendsto_supportingIntersection_left_aux K t).congr'
+  filter_upwards [Ioo_mem_nhdsLT (show t - Real.pi / 2 < t by linarith [Real.pi_pos])]
+    with s hs
+  apply supportingIntersection_comm
+  have hpos : 0 < Real.sin (t - s) := Real.sin_pos_of_pos_of_lt_pi
+    (by linarith [hs.2]) (by linarith [hs.1, Real.pi_pos])
+  rw [show s - t = -(t - s) by ring, Real.sin_neg]
+  exact neg_ne_zero.mpr hpos.ne'
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Analysis.Stieltjes.ConvexBoundary`.
+* `Analysis.SurfaceMeasure.Polygon`.
+* `Analysis.SurfaceMeasure.BoundaryLimit`.
+* `Analysis.SurfaceMeasure.DiscreteBounds`.
+* `Analysis.SurfaceMeasure.Integrals`.
+* `Analysis.SurfaceMeasure.VertexBoundary`.
+* `Analysis.SurfaceMeasure.AngularDensity`.
+* `Analysis.SurfaceMeasure.FrameProducts`.
+* `Analysis.SurfaceMeasure.Boundary`.
+* `Analysis.SurfaceMeasure.Linearity`.
+* `Analysis.SurfaceMeasure.OppositeDensity`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Stieltjes / Convex Boundary
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped Topology
+
+namespace MovingSofa
+
+/-- Each coordinate of the positive vertex is right-continuous and has bounded
+variation on its closed interval domain. -/
+theorem exists_positiveVertex_intervalBV (K : ConvexBody Point) {a b : ℝ} (hab : a ≤ b) :
+    ∃ f : Fin 2 → RightContinuousIntervalBV a b,
+      ∀ i t, (f i).toFun t = (edgeVertices K ((t : ℝ) : Real.Angle)).1 i := by
+  have hv : BoundedVariationOn
+      (fun t : Set.Icc a b ↦ (edgeVertices K ((t : ℝ) : Real.Angle)).1) Set.univ := by
+    exact ne_top_of_le_ne_top (positiveVertex_boundedVariation K a b hab)
+      (eVariationOn.comp_le_of_monotoneOn
+        (fun t : ℝ ↦ (edgeVertices K (t : Real.Angle)).1)
+        (s := Set.Icc a b) (t := Set.univ) (fun t : Set.Icc a b ↦ (t : ℝ))
+        (fun _ _ _ _ h ↦ h) (fun t _ ↦ t.property))
+  have hc (i : Fin 2) : IsIntervalBoundedVariation a b
+      (fun t : Set.Icc a b ↦ (edgeVertices K ((t : ℝ) : Real.Angle)).1 i) := by
+    change BoundedVariationOn ((fun p : Point ↦ p i) ∘
+      fun t : Set.Icc a b ↦ (edgeVertices K ((t : ℝ) : Real.Angle)).1) Set.univ
+    exact (PiLp.proj (𝕜 := ℝ) 2 (fun _ : Fin 2 ↦ ℝ) i).lipschitzWith.comp_boundedVariationOn
+      (g := fun t : Set.Icc a b ↦ (edgeVertices K ((t : ℝ) : Real.Angle)).1) hv
+  have hr (i : Fin 2) (t : Set.Icc a b) : ContinuousWithinAt
+      (fun s : Set.Icc a b ↦ (edgeVertices K ((s : ℝ) : Real.Angle)).1 i)
+      (Set.Ici t) t := by
+    have h : ContinuousWithinAt (fun s : ℝ ↦ (edgeVertices K (s : Real.Angle)).1)
+        (Set.Ici (t : ℝ)) (t : ℝ) :=
+      continuousWithinAt_Ioi_iff_Ici.mp (contact_oneSided_limits K (t : ℝ)).1
+    have hcomp : ContinuousWithinAt
+        (fun s : Set.Icc a b ↦ (edgeVertices K ((s : ℝ) : Real.Angle)).1)
+        (Set.Ici t) t := h.comp
+      (show ContinuousWithinAt (fun s : Set.Icc a b ↦ (s : ℝ)) (Set.Ici t) t from
+        continuous_subtype_val.continuousWithinAt) (fun _ hy ↦ hy)
+    exact (PiLp.proj (𝕜 := ℝ) 2 (fun _ : Fin 2 ↦ ℝ) i).continuous.continuousAt
+      |>.comp_continuousWithinAt hcomp
+  exact ⟨fun i ↦ ⟨_, hc i, hr i⟩, fun _ _ ↦ rfl⟩
+
+/-- Each coordinate of the positive vertex is measurable on the closed parameter interval,
+being the difference of two monotone functions by bounded variation. -/
+theorem measurable_positiveVertex_coordinate (K : ConvexBody Point) {a b : ℝ}
+    (hab : a ≤ b) (i : Fin 2) :
+    Measurable fun t : Set.Icc a b ↦ (edgeVertices K (((t : ℝ) : Real.Angle))).1 i := by
+  obtain ⟨f, hf⟩ := exists_positiveVertex_intervalBV K hab
+  have hfun : (fun t : Set.Icc a b ↦ (edgeVertices K (((t : ℝ) : Real.Angle))).1 i) =
+      (f i).toFun := by
+    funext t
+    exact (hf i t).symm
+  have hbv : LocallyBoundedVariationOn (f i).toFun (Set.univ : Set (Set.Icc a b)) :=
+    (f i).boundedVariation.locallyBoundedVariationOn
+  obtain ⟨p, q, hp, hq, hpq⟩ := hbv.exists_monotoneOn_sub_monotoneOn
+  rw [hfun, hpq]
+  exact (monotoneOn_univ.mp hp).measurable.sub (monotoneOn_univ.mp hq).measurable
+
+/-- Each coordinate of the positive vertex is measurable on the half-open parameter
+interval. -/
+theorem measurable_positiveVertex_coordinate_Ioc (K : ConvexBody Point) {a b : ℝ}
+    (hab : a ≤ b) (i : Fin 2) :
+    Measurable fun t : Set.Ioc a b ↦ (edgeVertices K (((t : ℝ) : Real.Angle))).1 i := by
+  have hpos := measurable_positiveVertex_coordinate K hab i
+  have hmem (t : Set.Ioc a b) : (t : ℝ) ∈ Set.Icc a b := ⟨t.property.1.le, t.property.2⟩
+  have hcont : Continuous fun t : Set.Ioc a b ↦ (⟨(t : ℝ), hmem t⟩ : Set.Icc a b) :=
+    Continuous.subtype_mk continuous_subtype_val hmem
+  have hcomp : Measurable
+      ((fun s : Set.Icc a b ↦ (edgeVertices K (((s : ℝ) : Real.Angle))).1 i) ∘
+        fun t : Set.Ioc a b ↦ (⟨(t : ℝ), hmem t⟩ : Set.Icc a b)) :=
+    hpos.comp hcont.measurable
+  have hfun : ((fun s : Set.Icc a b ↦ (edgeVertices K (((s : ℝ) : Real.Angle))).1 i) ∘
+      fun t : Set.Ioc a b ↦ (⟨(t : ℝ), hmem t⟩ : Set.Icc a b)) =
+      fun t : Set.Ioc a b ↦ (edgeVertices K (((t : ℝ) : Real.Angle))).1 i := by
+    funext t
+    rfl
+  rwa [hfun] at hcomp
+
+/-- Each coordinate of the negative vertex is measurable on the half-open parameter interval,
+being the pointwise left limit of the positive vertex. -/
+theorem measurable_negativeVertex_coordinate (K : ConvexBody Point) {a b : ℝ}
+    (hab : a ≤ b) (i : Fin 2) :
+    Measurable fun t : Set.Ioc a b ↦ (edgeVertices K (((t : ℝ) : Real.Angle))).2 i := by
+  have hpos := measurable_positiveVertex_coordinate K hab i
+  have hmem (n : ℕ) (t : Set.Ioc a b) :
+      max a ((t : ℝ) - 1 / (n + 1 : ℝ)) ∈ Set.Icc a b := by
+    refine ⟨le_max_left _ _, max_le hab ?_⟩
+    have : (0 : ℝ) < 1 / (n + 1 : ℝ) := by positivity
+    linarith [t.property.2]
+  set g : ℕ → Set.Ioc a b → ℝ := fun n t ↦
+    (edgeVertices K ((max a ((t : ℝ) - 1 / (n + 1 : ℝ)) : ℝ) : Real.Angle)).1 i
+  have hgmeas (n : ℕ) : Measurable (g n) := by
+    have hcont : Continuous fun t : Set.Ioc a b ↦ (⟨max a ((t : ℝ) - 1 / (n + 1 : ℝ)),
+        hmem n t⟩ : Set.Icc a b) :=
+      Continuous.subtype_mk
+        (continuous_const.max (continuous_subtype_val.sub continuous_const)) (hmem n)
+    have hcomp : Measurable
+        ((fun s : Set.Icc a b ↦ (edgeVertices K (((s : ℝ) : Real.Angle))).1 i) ∘
+          fun t : Set.Ioc a b ↦ (⟨max a ((t : ℝ) - 1 / (n + 1 : ℝ)),
+            hmem n t⟩ : Set.Icc a b)) :=
+      hpos.comp hcont.measurable
+    have hfun :
+        ((fun s : Set.Icc a b ↦ (edgeVertices K (((s : ℝ) : Real.Angle))).1 i) ∘
+          fun t : Set.Ioc a b ↦ (⟨max a ((t : ℝ) - 1 / (n + 1 : ℝ)),
+            hmem n t⟩ : Set.Icc a b)) = g n := by
+      funext t
+      rfl
+    rwa [hfun] at hcomp
+  refine measurable_of_tendsto_metrizable hgmeas (tendsto_pi_nhds.2 fun t ↦ ?_)
+  have hta : a < (t : ℝ) := t.property.1
+  have hlim := (contact_oneSided_limits K (t : ℝ)).2.2.2.1
+  have hlimi : Filter.Tendsto (fun s : ℝ ↦ (edgeVertices K (s : Real.Angle)).1 i)
+      (𝓝[<] (t : ℝ)) (𝓝 ((edgeVertices K ((t : ℝ) : Real.Angle)).2 i)) :=
+    ((PiLp.proj (𝕜 := ℝ) 2 (fun _ : Fin 2 ↦ ℝ) i).continuous.tendsto _).comp hlim
+  refine hlimi.comp ?_
+  rw [tendsto_nhdsWithin_iff]
+  constructor
+  · have h0 : Filter.Tendsto (fun n : ℕ ↦ (t : ℝ) - 1 / (n + 1 : ℝ))
+        Filter.atTop (𝓝 (t : ℝ)) := by
+      simpa using tendsto_one_div_add_atTop_nhds_zero_nat.const_sub ((t : ℝ))
+    have hconst : Filter.Tendsto (fun _ : ℕ ↦ a) Filter.atTop (𝓝 a) := tendsto_const_nhds
+    have h1 := hconst.max h0
+    rwa [max_eq_right hta.le] at h1
+  · obtain ⟨N, hN⟩ := exists_nat_gt (1 / ((t : ℝ) - a))
+    filter_upwards [Filter.eventually_ge_atTop N] with n hn
+    have hpos : (0 : ℝ) < (t : ℝ) - a := by linarith
+    have hNle : (N : ℝ) ≤ (n : ℝ) := Nat.cast_le.2 hn
+    have hlt : 1 / (n + 1 : ℝ) < (t : ℝ) - a := by
+      have h1 : 1 / ((t : ℝ) - a) < (n + 1 : ℝ) := by linarith
+      rw [div_lt_iff₀ (by positivity : (0 : ℝ) < (n + 1 : ℝ))]
+      rw [div_lt_iff₀ hpos] at h1
+      linarith
+    have hmaxeq : max a ((t : ℝ) - 1 / (n + 1 : ℝ)) = (t : ℝ) - 1 / (n + 1 : ℝ) :=
+      max_eq_right (by linarith)
+    rw [Set.mem_Iio, hmaxeq]
+    have : (0 : ℝ) < 1 / (n + 1 : ℝ) := by positivity
+    linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Polygon
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter MeasureTheory
+open scoped Topology
+
+namespace MovingSofa
+
+private def quarterTurn (p : Point) : Point := !₂[-p 1, p 0]
+
+private theorem quarterTurn_ne_zero {p : Point} (hp : p ≠ 0) : quarterTurn p ≠ 0 := by
+  intro h
+  apply hp
+  ext i
+  fin_cases i
+  · exact congrFun (congrArg WithLp.ofLp h) 1
+  · simpa [quarterTurn] using congrFun (congrArg WithLp.ofLp h) 0
+
+private theorem inner_quarterTurn (p : Point) : inner ℝ p (quarterTurn p) = 0 := by
+  rw [EuclideanSpace.inner_eq_star_dotProduct]
+  simp [quarterTurn, dotProduct, Fin.sum_univ_two]
+  ring
+
+private theorem inner_normalVector_vectorNormalAngle_quarterTurn {p : Point} (hp : p ≠ 0) :
+    inner ℝ p (normalVector (vectorNormalAngle (quarterTurn p))) = 0 := by
+  rw [normalVector_vectorNormalAngle (quarterTurn_ne_zero hp), inner_smul_right,
+    inner_quarterTurn, mul_zero]
+
+/-- A nonzero planar direction has only finitely many perpendicular angular normals. -/
+theorem finite_orthogonalNormal_angles {p : Point} (hp : p ≠ 0) :
+    Set.Finite {t : Real.Angle | inner ℝ p (normalVector t) = 0} := by
+  let t₀ := vectorNormalAngle (quarterTurn p)
+  apply Set.Finite.subset
+    ((Set.finite_singleton (t₀ + (Real.pi : Real.Angle))).insert t₀)
+  intro t ht
+  have h := normalVector_eq_or_eq_add_pi_of_orthogonal hp ht
+    (inner_normalVector_vectorNormalAngle_quarterTurn hp)
+  rcases h with h | h
+  · exact Set.mem_insert_iff.mpr (Or.inl h)
+  · exact Set.mem_insert_iff.mpr (Or.inr (Set.mem_singleton_iff.mpr h))
+
+/-- The possible normals perpendicular to differences of points in a finite set are finite. -/
+theorem finite_pairDifferenceNormal_angles (V : Finset Point) :
+    Set.Finite {t : Real.Angle | ∃ p ∈ V, ∃ q ∈ V,
+      p ≠ q ∧ inner ℝ (p - q) (normalVector t) = 0} := by
+  classical
+  let N : Point → Point → Set Real.Angle := fun p q ↦
+    if h : p = q then ∅ else {t | inner ℝ (p - q) (normalVector t) = 0}
+  have hN (p q : Point) : Set.Finite (N p q) := by
+    by_cases h : p = q
+    · simp [N, h]
+    · simpa [N, h] using finite_orthogonalNormal_angles (sub_ne_zero.mpr h)
+  apply Set.Finite.subset (V.finite_toSet.biUnion fun p _ ↦
+    V.finite_toSet.biUnion fun q _ ↦ hN p q)
+  rintro t ⟨p, hp, q, hq, hpq, ht⟩
+  apply Set.mem_iUnion₂.mpr
+  refine ⟨p, hp, Set.mem_iUnion₂.mpr ⟨q, hq, ?_⟩⟩
+  simp [N, hpq, ht]
+
+/-- The support edge of a convex body is an exposed face. -/
+theorem isExposed_exposedEdge (K : ConvexBody Point) (t : Real.Angle) :
+    IsExposed ℝ (K : Set Point) (exposedEdge K t) := by
+  intro _
+  refine ⟨innerSL ℝ (normalVector t), ?_⟩
+  ext p
+  simp only [Set.mem_ofPred_eq, innerSL_apply_apply, real_inner_comm]
+  constructor
+  · intro hp
+    refine ⟨hp.1, fun q hq ↦ ?_⟩
+    rw [hp.2]
+    exact inner_le_supportValue K hq t
+  · rintro ⟨hp, hmax⟩
+    refine ⟨hp, le_antisymm (inner_le_supportValue K hp t) ?_⟩
+    apply csSup_le (K.nonempty.image _)
+    rintro _ ⟨q, hq, rfl⟩
+    exact hmax q hq
+
+private theorem isExposed_singleton_edgeVertices_fst (K : ConvexBody Point)
+    (t : Real.Angle) :
+    IsExposed ℝ (exposedEdge K t) {(edgeVertices K t).1} := by
+  intro _
+  refine ⟨innerSL ℝ (tangentVector t), ?_⟩
+  ext p
+  simp only [Set.mem_singleton_iff, Set.mem_ofPred_eq, innerSL_apply_apply,
+    real_inner_comm]
+  constructor
+  · rintro rfl
+    refine ⟨edgeVertices_fst_mem K t, fun q hq ↦ ?_⟩
+    rw [inner_edgeVertices_fst_tangent]
+    exact le_csSup ((isCompact_exposedEdge K t).image
+      (continuous_id.inner continuous_const)).bddAbove ⟨q, hq, rfl⟩
+  · rintro ⟨hp, hmax⟩
+    have htangent : inner ℝ p (tangentVector t) =
+        inner ℝ (edgeVertices K t).1 (tangentVector t) := by
+      apply le_antisymm
+      · rw [inner_edgeVertices_fst_tangent]
+        exact le_csSup ((isCompact_exposedEdge K t).image
+          (continuous_id.inner continuous_const)).bddAbove ⟨p, hp, rfl⟩
+      · exact hmax _ (edgeVertices_fst_mem K t)
+    rw [← inner_normalVector_smul_add_inner_tangentVector_smul p t,
+      ← inner_normalVector_smul_add_inner_tangentVector_smul (edgeVertices K t).1 t,
+      hp.2, (edgeVertices_fst_mem K t).2, htangent]
+
+private theorem isExposed_singleton_edgeVertices_snd (K : ConvexBody Point)
+    (t : Real.Angle) :
+    IsExposed ℝ (exposedEdge K t) {(edgeVertices K t).2} := by
+  intro _
+  refine ⟨innerSL ℝ (-tangentVector t), ?_⟩
+  ext p
+  simp only [Set.mem_singleton_iff, Set.mem_ofPred_eq, innerSL_apply_apply,
+    real_inner_comm, inner_neg_left]
+  constructor
+  · rintro rfl
+    refine ⟨edgeVertices_snd_mem K t, fun q hq ↦ neg_le_neg ?_⟩
+    rw [inner_edgeVertices_snd_tangent]
+    exact csInf_le ((isCompact_exposedEdge K t).image
+      (continuous_id.inner continuous_const)).bddBelow ⟨q, hq, rfl⟩
+  · rintro ⟨hp, hmax⟩
+    have htangent : inner ℝ p (tangentVector t) =
+        inner ℝ (edgeVertices K t).2 (tangentVector t) := by
+      apply le_antisymm
+      · exact neg_le_neg_iff.mp (hmax _ (edgeVertices_snd_mem K t))
+      · rw [inner_edgeVertices_snd_tangent]
+        exact csInf_le ((isCompact_exposedEdge K t).image
+          (continuous_id.inner continuous_const)).bddBelow ⟨p, hp, rfl⟩
+    rw [← inner_normalVector_smul_add_inner_tangentVector_smul p t,
+      ← inner_normalVector_smul_add_inner_tangentVector_smul (edgeVertices K t).2 t,
+      hp.2, (edgeVertices_snd_mem K t).2, htangent]
+
+/-- Both endpoints of a polygon's exposed edge belong to its finite generating set. -/
+theorem edgeVertices_mem_of_eq_convexHull (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point)) (t : Real.Angle) :
+    (edgeVertices K t).1 ∈ V ∧ (edgeVertices K t).2 ∈ V := by
+  have hedge := (isExposed_exposedEdge K t).isExtreme
+  constructor
+  · have hext : (edgeVertices K t).1 ∈ Set.extremePoints ℝ (K : Set Point) :=
+      (hedge.trans (isExposed_singleton_edgeVertices_fst K t).isExtreme).mem_extremePoints
+    rw [hKV] at hext
+    exact Finset.mem_coe.mp (extremePoints_convexHull_subset hext)
+  · have hext : (edgeVertices K t).2 ∈ Set.extremePoints ℝ (K : Set Point) :=
+      (hedge.trans (isExposed_singleton_edgeVertices_snd K t).isExtreme).mem_extremePoints
+    rw [hKV] at hext
+    exact Finset.mem_coe.mp (extremePoints_convexHull_subset hext)
+
+/-- Every proper edge normal of a finite convex hull is one of finitely many pair normals. -/
+theorem finite_properEdgeNormal_angles (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point)) :
+    Set.Finite {t : Real.Angle | (edgeVertices K t).1 ≠ (edgeVertices K t).2} := by
+  apply (finite_pairDifferenceNormal_angles V).subset
+  intro t ht
+  refine ⟨(edgeVertices K t).1, (edgeVertices_mem_of_eq_convexHull K V hKV t).1,
+    (edgeVertices K t).2, (edgeVertices_mem_of_eq_convexHull K V hKV t).2, ht, ?_⟩
+  rw [inner_sub_left, (edgeVertices_fst_mem K t).2, (edgeVertices_snd_mem K t).2,
+    sub_self]
+
+private theorem continuousAt_positiveVertex_of_eq_negativeVertex (K : ConvexBody Point)
+    (t : ℝ) (ht : (edgeVertices K (t : Real.Angle)).1 =
+      (edgeVertices K (t : Real.Angle)).2) :
+    ContinuousAt (fun s : ℝ ↦ (edgeVertices K (s : Real.Angle)).1) t := by
+  rw [continuousAt_iff_continuous_left'_right']
+  constructor
+  · change Tendsto _ (𝓝[<] t) _
+    simpa only [ht] using (contact_oneSided_limits K t).2.2.2.1
+  · change Tendsto _ (𝓝[>] t) _
+    exact (contact_oneSided_limits K t).1
+
+/-- Away from a proper-edge normal, the positive vertex of a finite convex hull is locally
+constant. -/
+theorem eventuallyEq_positiveVertex_of_eq_convexHull (K : ConvexBody Point)
+    (V : Finset Point) (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    (t : ℝ) (ht : (edgeVertices K (t : Real.Angle)).1 =
+      (edgeVertices K (t : Real.Angle)).2) :
+    ∀ᶠ s : ℝ in 𝓝 t, (edgeVertices K (s : Real.Angle)).1 =
+      (edgeVertices K (t : Real.Angle)).1 := by
+  classical
+  let v := (edgeVertices K (t : Real.Angle)).1
+  let W : Set Point := (↑(V.erase v) : Set Point)
+  have hvV : v ∈ V := (edgeVertices_mem_of_eq_convexHull K V hKV _).1
+  have hvW : v ∈ Wᶜ := by simp [W, hvV]
+  have hopen : IsOpen Wᶜ := (V.erase v).finite_toSet.isClosed.isOpen_compl
+  have hnhds : Wᶜ ∈ 𝓝 v := hopen.mem_nhds hvW
+  have htend :=
+    (continuousAt_positiveVertex_of_eq_negativeVertex K t ht).eventually hnhds
+  exact htend.mono (by
+    intro s hs
+    have hsV : (edgeVertices K (s : Real.Angle)).1 ∈ V :=
+      (edgeVertices_mem_of_eq_convexHull K V hKV _).1
+    by_contra hne
+    have hnev : (edgeVertices K (s : Real.Angle)).1 ≠ v := by simpa [v] using hne
+    exact hs (by simp [W, hsV, hnev]))
+
+/-- For a two-dimensional finite convex hull, surface measure is supported on its finite set of
+proper-edge normals. -/
+theorem surfaceAreaMeasure_compl_properEdgeNormals_eq_zero
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    (hint : (interior (K : Set Point)).Nonempty) :
+    surfaceAreaMeasure K {t | (edgeVertices K t).1 = (edgeVertices K t).2} = 0 := by
+  let N : Set Real.Angle := {t | (edgeVertices K t).1 ≠ (edgeVertices K t).2}
+  have hNfinite : N.Finite := finite_properEdgeNormal_angles K V hKV
+  have hmeas : MeasurableSet Nᶜ := hNfinite.measurableSet.compl
+  have hface := (surfaceAreaMeasure_face_union K).2.2.2.2 Nᶜ hmeas (Or.inl hint)
+  have hunion : (⋃ t ∈ Nᶜ, exposedEdge K t) ⊆ (V : Set Point) := by
+    intro p hp
+    obtain ⟨t, htN, hpt⟩ := Set.mem_iUnion₂.mp hp
+    have ht : (edgeVertices K t).1 = (edgeVertices K t).2 := by
+      simpa only [N, Set.mem_compl_iff, Set.mem_ofPred_eq, not_not] using htN
+    have hp' : p = (edgeVertices K t).1 := by
+      rw [exposedEdge_eq_segment_edgeVertices, ← ht] at hpt
+      simpa using hpt
+    rw [hp']
+    exact (edgeVertices_mem_of_eq_convexHull K V hKV t).1
+  rw [show {t | (edgeVertices K t).1 = (edgeVertices K t).2} = Nᶜ by
+    ext t
+    simp [N], hface]
+  let _ := Measure.nullSingletonClass_hausdorff Point (by norm_num : (0 : ℝ) < 1)
+  exact measure_mono_null hunion
+    (V.finite_toSet.measure_zero (Measure.hausdorffMeasure 1))
+
+/-- Surface measure of any finite convex hull, including a point or segment, is supported on its
+finite set of proper-edge normals. -/
+theorem surfaceAreaMeasure_compl_properEdgeNormals_eq_zero_of_eq_convexHull
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point)) :
+    surfaceAreaMeasure K {t | (edgeVertices K t).1 = (edgeVertices K t).2} = 0 := by
+  by_cases hsub : (K : Set Point).Subsingleton
+  · rw [surfaceAreaMeasure_eq_zero_of_subsingleton K hsub]
+    simp
+  by_cases hint : (interior (K : Set Point)).Nonempty
+  · exact surfaceAreaMeasure_compl_properEdgeNormals_eq_zero K V hKV hint
+  obtain ⟨d, hd⟩ := exists_segmentPresentation_of_interior_empty K hsub
+    (Set.not_nonempty_iff_eq_empty.mp hint)
+  have hproper (u : Real.Angle)
+      (hu : inner ℝ (d.2.1 - d.1) (normalVector u) = 0) :
+      (edgeVertices K u).1 ≠ (edgeVertices K u).2 := by
+    intro heq
+    have hedge := exposedEdge_eq_segment_of_orthogonal K d hd u hu
+    have hsingle : (K : Set Point).Subsingleton := by
+      rw [← hedge, exposedEdge_eq_segment_edgeVertices, heq]
+      simp
+    exact hsub hsingle
+  have hpi : inner ℝ (d.2.1 - d.1)
+      (normalVector (d.2.2 + (Real.pi : Real.Angle))) = 0 := by
+    have hn : normalVector (d.2.2 + (Real.pi : Real.Angle)) =
+        -normalVector d.2.2 := by
+      induction d.2.2 using Real.Angle.induction_on with
+      | _ t => simpa only [← Real.Angle.coe_add] using normalVector_add_pi t
+    rw [hn, inner_neg_right, hd.2.2, neg_zero]
+  rw [surfaceAreaMeasure_eq_segmentPresentation K d hd]
+  simp [hproper d.2.2 hd.2.2, hproper _ hpi]
+
+/-- Surface measure is carried by the proper edge normals as soon as these are finitely many
+and the degenerate faces lie in a one-dimensional null set. -/
+theorem surfaceAreaMeasure_compl_properEdgeNormals_eq_zero_of_finite_carrier
+    (K : ConvexBody Point)
+    (hN : {t | (edgeVertices K t).1 ≠ (edgeVertices K t).2}.Finite)
+    (V : Set Point) (hV : Measure.hausdorffMeasure 1 V = 0)
+    (hcarrier : (⋃ t ∈ {t | (edgeVertices K t).1 = (edgeVertices K t).2},
+      exposedEdge K t) ⊆ V) :
+    surfaceAreaMeasure K {t | (edgeVertices K t).1 = (edgeVertices K t).2} = 0 := by
+  let N : Set Real.Angle := {t | (edgeVertices K t).1 ≠ (edgeVertices K t).2}
+  have hset : {t | (edgeVertices K t).1 = (edgeVertices K t).2} = Nᶜ := by
+    ext t
+    simp [N]
+  by_cases hsub : (K : Set Point).Subsingleton
+  · rw [surfaceAreaMeasure_eq_zero_of_subsingleton K hsub]
+    simp
+  by_cases hint : (interior (K : Set Point)).Nonempty
+  · rw [hset, (surfaceAreaMeasure_face_union K).2.2.2.2 Nᶜ
+      hN.measurableSet.compl (Or.inl hint)]
+    exact measure_mono_null (by simpa only [hset] using hcarrier) hV
+  obtain ⟨d, hd⟩ := exists_segmentPresentation_of_interior_empty K hsub
+    (Set.not_nonempty_iff_eq_empty.mp hint)
+  have hproper (u : Real.Angle)
+      (hu : inner ℝ (d.2.1 - d.1) (normalVector u) = 0) :
+      (edgeVertices K u).1 ≠ (edgeVertices K u).2 := by
+    intro heq
+    have hedge := exposedEdge_eq_segment_of_orthogonal K d hd u hu
+    have hsingle : (K : Set Point).Subsingleton := by
+      rw [← hedge, exposedEdge_eq_segment_edgeVertices, heq]
+      simp
+    exact hsub hsingle
+  have hpi : inner ℝ (d.2.1 - d.1)
+      (normalVector (d.2.2 + (Real.pi : Real.Angle))) = 0 := by
+    rw [normalVector_add_pi_angle, inner_neg_right, hd.2.2, neg_zero]
+  rw [surfaceAreaMeasure_eq_segmentPresentation K d hd]
+  simp [hproper d.2.2 hd.2.2, hproper _ hpi]
+
+/-- The surface integral of an arbitrary integrand over a finite convex hull is the sum of its
+proper-edge atoms, including the point and segment cases. No regularity of the integrand is
+needed: the measure is carried by a finite set. -/
+theorem integral_surfaceAreaMeasure_eq_sum_properEdgeNormals
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    (φ : Real.Angle → ℝ) (E : Set Real.Angle) (hE : MeasurableSet E) :
+    ∫ t in E, φ t ∂surfaceAreaMeasure K =
+      Finset.sum ((finite_properEdgeNormal_angles K V hKV).inter_of_left E).toFinset
+        (fun t ↦ (surfaceAreaMeasure K).real {t} * φ t) := by
+  classical
+  let N : Set Real.Angle := {t | (edgeVertices K t).1 ≠ (edgeVertices K t).2}
+  have hNfinite : N.Finite := finite_properEdgeNormal_angles K V hKV
+  have hae : ∀ᵐ t ∂surfaceAreaMeasure K, t ∈ N := by
+    rw [ae_iff]
+    simpa only [N, Set.mem_ofPred_eq, not_ne_iff] using
+      surfaceAreaMeasure_compl_properEdgeNormals_eq_zero_of_eq_convexHull K V hKV
+  have hrestrict : (surfaceAreaMeasure K).restrict N = surfaceAreaMeasure K :=
+    Measure.restrict_eq_self_of_ae_mem hae
+  let _ := (surfaceAreaMeasure_face_union K).1
+  have hEN : E ∩ N = ↑(hNfinite.inter_of_left E).toFinset := by
+    ext t
+    simp [N, and_comm]
+  calc
+    ∫ t in E, φ t ∂surfaceAreaMeasure K =
+        ∫ t in E, φ t ∂(surfaceAreaMeasure K).restrict N := by rw [hrestrict]
+    _ = ∫ t in E ∩ N, φ t ∂surfaceAreaMeasure K := by
+      rw [Measure.restrict_restrict hE]
+    _ = _ := by
+      rw [hEN]
+      exact MeasureTheory.setIntegral_finset _
+        (μ := surfaceAreaMeasure K) (f := φ) IntegrableOn.finset
+
+/-- The coordinate tangent integral of a finite convex hull is the sum of its proper-edge atoms,
+including the point and segment cases. -/
+theorem integral_tangentCoordinate_surfaceAreaMeasure_eq_sum_properEdgeNormals
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    (i : Fin 2) (E : Set Real.Angle) (hE : MeasurableSet E) :
+    ∫ t in E, tangentVector t i ∂surfaceAreaMeasure K =
+      Finset.sum ((finite_properEdgeNormal_angles K V hKV).inter_of_left E).toFinset
+        (fun t ↦ (surfaceAreaMeasure K).real {t} * tangentVector t i) :=
+  integral_surfaceAreaMeasure_eq_sum_properEdgeNormals K V hKV
+    (fun t ↦ tangentVector t i) E hE
+
+/-- On a real interval, the coordinate Stieltjes measure of a polygon's positive vertex is
+supported at proper-edge normals (apart from the excluded left endpoint). -/
+theorem intervalStieltjesMeasure_variation_eq_zero_off_properEdgeNormals
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    {a b : ℝ} (hab : a < b) (f : Fin 2 → RightContinuousIntervalBV a b)
+    (hf : ∀ i t, (f i).toFun t =
+      (edgeVertices K (((t : Set.Icc a b) : ℝ) : Real.Angle)).1 i) (i : Fin 2) :
+    (intervalStieltjesMeasure (f i)).variation
+      {t | a < (t : ℝ) ∧ (edgeVertices K (((t : Set.Icc a b) : ℝ) : Real.Angle)).1 =
+        (edgeVertices K (((t : Set.Icc a b) : ℝ) : Real.Angle)).2} = 0 := by
+  let T : Set (Set.Icc a b) :=
+    {t | a < (t : ℝ) ∧ (edgeVertices K ((t : ℝ) : Real.Angle)).1 =
+      (edgeVertices K ((t : ℝ) : Real.Angle)).2}
+  change (f i).boundedVariation.vectorMeasure.variation T = 0
+  apply measure_null_of_locally_null T
+  intro x hx
+  have hconst := eventuallyEq_positiveVertex_of_eq_convexHull K V hKV (x : ℝ) hx.2
+  have hconst' : (f i).toFun =ᶠ[𝓝 x] fun _ ↦ (f i).toFun x := by
+    filter_upwards [continuousAt_subtype_val.eventually hconst] with y hy
+    rw [hf i y, hf i x, hy]
+  obtain ⟨U, hU, hUzero⟩ :=
+    BoundedVariationOn.exists_nhds_variation_vectorMeasure_eq_zero_of_eventuallyEq_const
+    hab (f i).boundedVariation (f i).right_continuous x hx.1 hconst'
+  exact ⟨U, mem_nhdsWithin_of_mem_nhds hU, hUzero⟩
+
+/-- The left limit of a positive-vertex coordinate at a noninitial parameter is the corresponding
+coordinate of the negative vertex. -/
+theorem leftLim_positiveVertex_coordinate
+    (K : ConvexBody Point) {a b : ℝ} (f : Fin 2 → RightContinuousIntervalBV a b)
+    (hf : ∀ i t, (f i).toFun t =
+      (edgeVertices K (((t : Set.Icc a b) : ℝ) : Real.Angle)).1 i)
+    (i : Fin 2) (t : Set.Icc a b) (ht : a < (t : ℝ)) :
+    Function.leftLim (f i).toFun t =
+      (edgeVertices K (((t : Set.Icc a b) : ℝ) : Real.Angle)).2 i := by
+  have hab : a < b := ht.trans_le t.property.2
+  let _ : Fact (a ≤ b) := ⟨hab.le⟩
+  let _ : Nontrivial (Set.Icc a b) :=
+    ⟨⟨⟨a, le_rfl, hab.le⟩, ⟨b, hab.le, le_rfl⟩, fun h ↦
+      hab.ne (congrArg Subtype.val h)⟩⟩
+  have hcoe : Tendsto (fun s : Set.Icc a b ↦ (s : ℝ)) (𝓝[<] t) (𝓝[<] (t : ℝ)) := by
+    rw [nhdsWithin_subtype, Set.image_subtype_val_Icc_Iio,
+      nhdsWithin_Ico_eq_nhdsLT ht]
+    exact tendsto_comap
+  have hcontact := ((contact_oneSided_limits K (t : ℝ)).2.2.2.1).comp hcoe
+  have hcoord := (PiLp.proj (𝕜 := ℝ) 2 (fun _ : Fin 2 ↦ ℝ) i).continuous.continuousAt
+    |>.tendsto.comp hcontact
+  have hfleft := (f i).boundedVariation.tendsto_leftLim t
+  let _ : (𝓝[<] t).NeBot := nhdsLT_neBot_of_exists_lt ⟨⊥, ht⟩
+  apply tendsto_nhds_unique hfleft
+  apply hcoord.congr
+  intro s
+  simpa [Function.comp_apply] using (hf i s).symm
+
+/-- Each noninitial Stieltjes atom of a positive-vertex coordinate is the corresponding
+coordinate of the tangent-weighted surface-measure atom. -/
+theorem intervalStieltjesMeasure_singleton_positiveVertex
+    (K : ConvexBody Point) {a b : ℝ} (f : Fin 2 → RightContinuousIntervalBV a b)
+    (hf : ∀ i t, (f i).toFun t =
+      (edgeVertices K (((t : Set.Icc a b) : ℝ) : Real.Angle)).1 i)
+    (i : Fin 2) (t : Set.Icc a b) (ht : a < (t : ℝ)) :
+    intervalStieltjesMeasure (f i) {t} =
+      (surfaceAreaMeasure K {((t : ℝ) : Real.Angle)}).toReal *
+        tangentVector ((t : ℝ) : Real.Angle) i := by
+  rw [intervalStieltjesMeasure, (f i).boundedVariation.vectorMeasure_singleton,
+    (f i).right_continuous t |>.rightLim_eq,
+    leftLim_positiveVertex_coordinate K f hf i t ht, hf]
+  have h := congrArg (fun p : Point ↦ p i)
+    (surfaceAreaMeasure_atom_length K ((t : ℝ) : Real.Angle)).2.2
+  simpa [PiLp.smul_apply] using congrArg (fun z : ℝ ↦
+    z - (edgeVertices K ((t : ℝ) : Real.Angle)).2 i) h
+
+/-- A proper-edge normal has only finitely many lifts in a half-open interval of length at most one
+full turn. -/
+theorem finite_properEdgeNormal_lifts (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    {a b : ℝ} (hturn : b ≤ a + 2 * Real.pi) :
+    Set.Finite {t : Set.Icc a b | a < (t : ℝ) ∧
+      (edgeVertices K (((t : Set.Icc a b) : ℝ) : Real.Angle)).1 ≠
+        (edgeVertices K (((t : Set.Icc a b) : ℝ) : Real.Angle)).2} := by
+  let S : Set (Set.Icc a b) := {t | a < (t : ℝ) ∧
+    (edgeVertices K ((t : ℝ) : Real.Angle)).1 ≠
+      (edgeVertices K ((t : ℝ) : Real.Angle)).2}
+  let N : Set Real.Angle :=
+    {u | (edgeVertices K u).1 ≠ (edgeVertices K u).2}
+  let c : Set.Icc a b → Real.Angle := fun t ↦ ((t : ℝ) : Real.Angle)
+  let _ : Fact (0 < 2 * Real.pi) := ⟨mul_pos (by norm_num) Real.pi_pos⟩
+  change S.Finite
+  apply Set.Finite.of_finite_image (f := c)
+  · apply (finite_properEdgeNormal_angles K V hKV).subset
+    rintro _ ⟨t, ht, rfl⟩
+    exact ht.2
+  · intro x hx y hy hxy
+    apply Subtype.ext
+    apply (AddCircle.coe_eq_coe_iff_of_mem_Ioc
+      (p := 2 * Real.pi) ⟨hx.1, x.property.2.trans hturn⟩
+      ⟨hy.1, y.property.2.trans hturn⟩).mp
+    change ((x : ℝ) : Real.Angle) = ((y : ℝ) : Real.Angle)
+    simpa [c] using hxy
+
+/-- On a polygon, every measurable noninitial set has Stieltjes mass equal to the finite sum of
+its proper-edge atoms. -/
+theorem intervalStieltjesMeasure_eq_sum_properEdgeNormals
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi)
+    (f : Fin 2 → RightContinuousIntervalBV a b)
+    (hf : ∀ i t, (f i).toFun t =
+      (edgeVertices K (((t : Set.Icc a b) : ℝ) : Real.Angle)).1 i)
+    (i : Fin 2) (E : Set (Set.Icc a b)) (hE : MeasurableSet E)
+    (hEa : ∀ t ∈ E, a < (t : ℝ)) :
+    intervalStieltjesMeasure (f i) E =
+      Finset.sum ((finite_properEdgeNormal_lifts K V hKV hturn).inter_of_left E).toFinset
+        (fun t ↦ (surfaceAreaMeasure K
+          {(((t : Set.Icc a b) : ℝ) : Real.Angle)}).toReal *
+            tangentVector (((t : Set.Icc a b) : ℝ) : Real.Angle) i) := by
+  classical
+  let S : Set (Set.Icc a b) := {t | a < (t : ℝ) ∧
+    (edgeVertices K ((t : ℝ) : Real.Angle)).1 ≠
+      (edgeVertices K ((t : ℝ) : Real.Angle)).2}
+  let T : Set (Set.Icc a b) := {t | a < (t : ℝ) ∧
+    (edgeVertices K ((t : ℝ) : Real.Angle)).1 =
+      (edgeVertices K ((t : ℝ) : Real.Angle)).2}
+  let μ := intervalStieltjesMeasure (f i)
+  change μ E = _
+  have hSfinite : S.Finite := finite_properEdgeNormal_lifts K V hKV hturn
+  have hTzero : μ.variation T = 0 :=
+    intervalStieltjesMeasure_variation_eq_zero_off_properEdgeNormals K V hKV hab f hf i
+  have hdiffsub : E \ S ⊆ T := by
+    intro t ht
+    exact ⟨hEa t ht.1, not_ne_iff.mp (fun hne ↦ ht.2 ⟨hEa t ht.1, hne⟩)⟩
+  have hdiffzero : μ (E \ S) = 0 := by
+    rw [← enorm_eq_zero, ← le_zero_iff]
+    exact (μ.enorm_measure_le_variation (E \ S)).trans
+      (by rw [measure_mono_null hdiffsub hTzero])
+  have hsplit : E = S ∩ E ∪ (E \ S) := by
+    ext t
+    constructor
+    · intro ht
+      by_cases htS : t ∈ S
+      · exact Or.inl ⟨htS, ht⟩
+      · exact Or.inr ⟨ht, htS⟩
+    · rintro (ht | ht)
+      · exact ht.2
+      · exact ht.1
+  have hfinite : S ∩ E =
+      ↑((hSfinite.inter_of_left E).toFinset) := by
+    exact (hSfinite.inter_of_left E).coe_toFinset.symm
+  have hES : MeasurableSet (S ∩ E) := hSfinite.measurableSet.inter hE
+  have hEdiff : MeasurableSet (E \ S) := hE.diff hSfinite.measurableSet
+  have hdisj : Disjoint (S ∩ E) (E \ S) :=
+    Set.disjoint_of_subset_left Set.inter_subset_left Set.disjoint_sdiff_right
+  have hreduce : μ E = μ (S ∩ E) := calc
+    μ E = μ (S ∩ E ∪ (E \ S)) := congrArg μ hsplit
+    _ = μ (S ∩ E) + μ (E \ S) := μ.of_union hdisj hES hEdiff
+    _ = μ (S ∩ E) := by rw [hdiffzero, add_zero]
+  rw [hreduce]
+  conv_lhs => rw [hfinite]
+  let F := (hSfinite.inter_of_left E).toFinset
+  have hU : (↑F : Set (Set.Icc a b)) = ⋃ t ∈ F, {t} := by
+    ext t
+    simp only [Finset.mem_coe, Set.mem_iUnion, Set.mem_singleton_iff]
+    constructor
+    · intro ht
+      exact ⟨t, ht, rfl⟩
+    · rintro ⟨u, hu, _, rfl⟩
+      exact hu
+  rw [hU, μ.of_biUnion_finset (by
+    intro x hx y hy hxy
+    simpa [Set.disjoint_singleton] using hxy) (by simp)]
+  calc
+    Finset.sum F (fun t ↦ μ {t}) =
+        Finset.sum F (fun t ↦
+          (surfaceAreaMeasure K {((t : ℝ) : Real.Angle)}).toReal *
+            tangentVector ((t : ℝ) : Real.Angle) i) := by
+      apply Finset.sum_congr rfl
+      intro t ht
+      have ht' : t ∈ S ∩ E := by simpa [F] using ht
+      exact intervalStieltjesMeasure_singleton_positiveVertex K f hf i t
+        (by simpa [S] using ht'.1.1)
+    _ = _ := by
+      exact congrArg (fun G : Finset (Set.Icc a b) ↦
+        Finset.sum G (fun t ↦
+          (surfaceAreaMeasure K {((t : ℝ) : Real.Angle)}).toReal *
+            tangentVector ((t : ℝ) : Real.Angle) i)) (by
+              apply Finset.ext
+              intro t
+              simp [F, S])
+
+/-- The surface integral of an arbitrary integrand over the angular image of a measurable
+interval set, as a finite sum indexed by the angular lifts carrying a proper edge. The left
+endpoint `a` is excluded from `E` so that each angle has at most one lift in `E`. -/
+theorem integral_surfaceAreaMeasure_image_eq_sum_properEdgeNormal_lifts
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    {a b : ℝ} (hturn : b ≤ a + 2 * Real.pi) (φ : Real.Angle → ℝ)
+    (E : Set (Set.Icc a b)) (hE : MeasurableSet E)
+    (hEa : ∀ t ∈ E, a < (t : ℝ)) :
+    (∫ u in (fun t : Set.Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+      φ u ∂surfaceAreaMeasure K) =
+      Finset.sum ((finite_properEdgeNormal_lifts K V hKV hturn).inter_of_left E).toFinset
+        (fun t ↦ (surfaceAreaMeasure K).real {((t : ℝ) : Real.Angle)} *
+          φ ((t : ℝ) : Real.Angle)) := by
+  classical
+  let c : Set.Icc a b → Real.Angle := fun t ↦ ((t : ℝ) : Real.Angle)
+  let S : Set (Set.Icc a b) := {t | a < (t : ℝ) ∧
+    (edgeVertices K ((t : ℝ) : Real.Angle)).1 ≠
+      (edgeVertices K ((t : ℝ) : Real.Angle)).2}
+  have hinj : Set.InjOn c E := by
+    intro x hx y hy hxy
+    apply Subtype.ext
+    let _ : Fact (0 < 2 * Real.pi) := ⟨mul_pos (by norm_num) Real.pi_pos⟩
+    apply (AddCircle.coe_eq_coe_iff_of_mem_Ioc
+      (p := 2 * Real.pi) ⟨hEa x hx, x.property.2.trans hturn⟩
+      ⟨hEa y hy, y.property.2.trans hturn⟩).mp
+    change c x = c y
+    exact hxy
+  have hAmeas : MeasurableSet (c '' E) :=
+    hE.image_of_continuousOn_injOn
+      ((Real.Angle.continuous_coe : Continuous fun x : ℝ ↦ (x : Real.Angle)).comp
+        continuous_subtype_val).continuousOn hinj
+  rw [integral_surfaceAreaMeasure_eq_sum_properEdgeNormals K V hKV φ (c '' E) hAmeas]
+  let F := ((finite_properEdgeNormal_lifts K V hKV hturn).inter_of_left E).toFinset
+  let G := ((finite_properEdgeNormal_angles K V hKV).inter_of_left (c '' E)).toFinset
+  have hG : G = F.image c := by
+    ext u
+    simp only [G, F, Set.Finite.mem_toFinset, Set.mem_inter_iff, Set.mem_image,
+      Finset.mem_image]
+    constructor
+    · rintro ⟨huN, t, htE, rfl⟩
+      exact ⟨t, ⟨⟨hEa t htE, huN⟩, htE⟩, rfl⟩
+    · rintro ⟨t, ⟨⟨hta, htN⟩, htE⟩, rfl⟩
+      exact ⟨htN, t, htE, rfl⟩
+  change Finset.sum G (fun u ↦ (surfaceAreaMeasure K).real {u} * φ u) = _
+  rw [hG, Finset.sum_image]
+  · intro x hx y hy hxy
+    have hx' : x ∈ S ∩ E := by
+      change (a < (x : ℝ) ∧ (edgeVertices K ((x : ℝ) : Real.Angle)).1 ≠
+        (edgeVertices K ((x : ℝ) : Real.Angle)).2) ∧ x ∈ E
+      simpa [F] using hx
+    have hy' : y ∈ S ∩ E := by
+      change (a < (y : ℝ) ∧ (edgeVertices K ((y : ℝ) : Real.Angle)).1 ≠
+        (edgeVertices K ((y : ℝ) : Real.Angle)).2) ∧ y ∈ E
+      simpa [F] using hy
+    exact hinj hx'.2 hy'.2 hxy
+
+/-- The tangent-coordinate surface integral over the angular image of a measurable interval set is
+the same finite proper-edge sum as the positive-vertex Stieltjes measure. -/
+theorem integral_tangentCoordinate_image_eq_sum_properEdgeNormal_lifts
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    {a b : ℝ} (hturn : b ≤ a + 2 * Real.pi) (i : Fin 2)
+    (E : Set (Set.Icc a b)) (hE : MeasurableSet E)
+    (hEa : ∀ t ∈ E, a < (t : ℝ)) :
+    (∫ u in (fun t : Set.Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+      tangentVector u i ∂surfaceAreaMeasure K) =
+      Finset.sum ((finite_properEdgeNormal_lifts K V hKV hturn).inter_of_left E).toFinset
+        (fun t ↦ (surfaceAreaMeasure K {((t : ℝ) : Real.Angle)}).toReal *
+          tangentVector ((t : ℝ) : Real.Angle) i) :=
+  integral_surfaceAreaMeasure_image_eq_sum_properEdgeNormal_lifts K V hKV hturn
+    (fun u ↦ tangentVector u i) E hE hEa
+
+/-- Polygon case of the positive-vertex Stieltjes/surface-measure identity, including degenerate
+point and segment convex hulls. -/
+theorem positiveVertex_stieltjes_surface_of_eq_convexHull
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    (a b : ℝ) (hab : a < b) (hturn : b ≤ a + 2 * Real.pi) :
+    ∃ f : Fin 2 → RightContinuousIntervalBV a b,
+      (∀ i t, (f i).toFun t = (edgeVertices K ((t : ℝ) : Real.Angle)).1 i) ∧
+      ∀ (i : Fin 2) (E : Set (Set.Icc a b)), MeasurableSet E →
+        (∀ t ∈ E, a < (t : ℝ)) →
+        intervalStieltjesMeasure (f i) E =
+          ∫ u in (fun t : Set.Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+            tangentVector u i ∂surfaceAreaMeasure K := by
+  obtain ⟨f, hf⟩ := exists_positiveVertex_intervalBV K hab.le
+  refine ⟨f, hf, fun i E hE hEa ↦ ?_⟩
+  rw [intervalStieltjesMeasure_eq_sum_properEdgeNormals K V hKV hab hturn f hf i E hE hEa,
+    integral_tangentCoordinate_image_eq_sum_properEdgeNormal_lifts
+      K V hKV hturn i E hE hEa]
+
+/-- The positive vertex increment of a polygon is the tangent-coordinate surface integral over
+the corresponding angular interval. -/
+theorem positiveVertex_sub_eq_integral_of_eq_convexHull
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi) (i : Fin 2) :
+    (edgeVertices K (b : Real.Angle)).1 i -
+        (edgeVertices K (a : Real.Angle)).1 i =
+      ∫ u in (fun t : ℝ ↦ (t : Real.Angle)) '' Set.Ioc a b,
+        tangentVector u i ∂surfaceAreaMeasure K := by
+  obtain ⟨f, hf, hmeasure⟩ :=
+    positiveVertex_stieltjes_surface_of_eq_convexHull K V hKV a b hab hturn
+  let aa : Set.Icc a b := ⟨a, le_rfl, hab.le⟩
+  let bb : Set.Icc a b := ⟨b, hab.le, le_rfl⟩
+  have hangularImage :
+      (fun t : Set.Icc a b ↦ ((t : ℝ) : Real.Angle)) '' Set.Ioc aa bb =
+        (fun t : ℝ ↦ (t : Real.Angle)) '' Set.Ioc a b := by
+    ext u
+    constructor
+    · rintro ⟨t, ht, rfl⟩
+      exact ⟨t, ⟨ht.1, ht.2⟩, rfl⟩
+    · rintro ⟨t, ht, rfl⟩
+      exact ⟨⟨t, ht.1.le, ht.2⟩, ⟨ht.1, ht.2⟩, rfl⟩
+  have hstieltjes := hmeasure i (Set.Ioc aa bb) measurableSet_Ioc
+    (fun t ht ↦ ht.1)
+  rw [intervalStieltjesMeasure_Ioc (f i) aa bb hab.le] at hstieltjes
+  rw [hf i bb, hf i aa, hangularImage] at hstieltjes
+  exact hstieltjes
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Boundary Limit
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter MeasureTheory Set
+open scoped Topology BoundedContinuousFunction
+
+namespace MovingSofa
+
+/-- Face-preserving polygon approximation transfers the boundary increment identity. -/
+private theorem positiveVertex_sub_eq_integral_of_polygon_identity
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi)
+    (hpoly : ∀ (P : ConvexBody Point) (V : Finset Point),
+      (P : Set Point) = convexHull ℝ (V : Set Point) → ∀ i : Fin 2,
+      (edgeVertices P (b : Real.Angle)).1 i - (edgeVertices P (a : Real.Angle)).1 i =
+        ∫ u in (fun t : ℝ ↦ (t : Real.Angle)) '' Ioc a b,
+          tangentVector u i ∂surfaceAreaMeasure P) (i : Fin 2) :
+    (edgeVertices K (b : Real.Angle)).1 i - (edgeVertices K (a : Real.Angle)).1 i =
+      ∫ u in (fun t : ℝ ↦ (t : Real.Angle)) '' Ioc a b,
+        tangentVector u i ∂surfaceAreaMeasure K := by
+  classical
+  obtain ⟨V, P, hP, hdist⟩ := exists_facePreserving_polygonApproximation K
+    {(a : Real.Angle), (b : Real.Angle)}
+  have ha (n : ℕ) : exposedEdge (P n) (a : Real.Angle) = exposedEdge K (a : Real.Angle) :=
+    (hP n).2.2.2 _ (by simp)
+  have hb (n : ℕ) : exposedEdge (P n) (b : Real.Angle) = exposedEdge K (b : Real.Angle) :=
+    (hP n).2.2.2 _ (by simp)
+  have hlim : Tendsto (fun n ↦ Metric.hausdorffDist (P n : Set Point) (K : Set Point))
+      atTop (𝓝 0) := by
+    apply squeeze_zero' (Filter.Eventually.of_forall fun _ ↦ Metric.hausdorffDist_nonneg)
+      (Filter.eventually_atTop.2 ⟨1, fun n hn ↦ hdist n hn⟩)
+    exact tendsto_one_div_atTop_nhds_zero_nat
+  have hcontinuous : Continuous (fun u : Real.Angle ↦ tangentVector u i) := by
+    fin_cases i
+    · exact Real.Angle.continuous_sin.neg
+    · exact Real.Angle.continuous_cos
+  have hint := tendsto_integral_surfaceAreaMeasure_Ioc_of_preserves_faces
+    P K hlim hab hturn ha hb (fun u ↦ tangentVector u i) hcontinuous
+  have hvalue (n : ℕ) :
+      (∫ u in (fun t : ℝ ↦ (t : Real.Angle)) '' Ioc a b,
+        tangentVector u i ∂surfaceAreaMeasure (P n)) =
+      (edgeVertices K (b : Real.Angle)).1 i - (edgeVertices K (a : Real.Angle)).1 i := by
+    rw [← hpoly (P n) (V n) (hP n).2.1 i,
+      edgeVertices_eq_of_exposedEdge_eq (P n) K _ (ha n),
+      edgeVertices_eq_of_exposedEdge_eq (P n) K _ (hb n)]
+  simp only [hvalue] at hint
+  exact tendsto_nhds_unique tendsto_const_nhds hint
+
+/-- The positive vertex increment is the tangent-coordinate surface integral over
+the corresponding half-open angular interval. -/
+theorem positiveVertex_sub_eq_integral
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi)
+    (i : Fin 2) :
+    (edgeVertices K (b : Real.Angle)).1 i - (edgeVertices K (a : Real.Angle)).1 i =
+      ∫ u in (fun t : ℝ ↦ (t : Real.Angle)) '' Ioc a b,
+        tangentVector u i ∂surfaceAreaMeasure K := by
+  exact positiveVertex_sub_eq_integral_of_polygon_identity K hab hturn
+    (fun P V hPV i ↦ positiveVertex_sub_eq_integral_of_eq_convexHull P V hPV hab hturn i) i
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Discrete Bounds
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- The integral of tangent vectors gives the increment of the positive supporting vertex. -/
+theorem integral_tangentVector_surfaceAreaMeasure (K : ConvexBody Point)
+    {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi) :
+    (∫ u in (fun t : ℝ ↦ (t : Real.Angle)) '' Set.Ioc a b,
+      tangentVector u ∂surfaceAreaMeasure K) =
+      (edgeVertices K (b : Real.Angle)).1 - (edgeVertices K (a : Real.Angle)).1 := by
+  let A := (fun t : ℝ ↦ (t : Real.Angle)) '' Set.Ioc a b
+  let _ : IsFiniteMeasure (surfaceAreaMeasure K) := (surfaceAreaMeasure_face_union K).1
+  have hu : Integrable (fun u : Real.Angle ↦ tangentVector u)
+      ((surfaceAreaMeasure K).restrict A) := by
+    rw [← integrableOn_univ]
+    apply ContinuousOn.integrableOn_compact isCompact_univ
+    apply Continuous.continuousOn
+    apply (PiLp.continuous_toLp 2 (fun _ : Fin 2 ↦ ℝ)).comp
+    apply continuous_pi
+    intro i
+    fin_cases i
+    · exact Real.Angle.continuous_sin.neg
+    · exact Real.Angle.continuous_cos
+  ext i
+  have hi := ContinuousLinearMap.integral_comp_comm
+    (PiLp.proj (𝕜 := ℝ) 2 (fun _ : Fin 2 ↦ ℝ) i) hu
+  change (∫ u in A, tangentVector u i ∂surfaceAreaMeasure K) =
+    (∫ u in A, tangentVector u ∂surfaceAreaMeasure K) i at hi
+  rw [← hi]
+  exact (positiveVertex_sub_eq_integral K hab hturn i).symm
+
+/-- The projected boundary integral computes the support value relative to the initial vertex. -/
+theorem integral_inner_tangentVector_eq_support_sub (K : ConvexBody Point)
+    {s : ℝ} (hs : s ∈ Set.Ioo 0 Real.pi) :
+    (∫ u in (fun t : ℝ ↦ (t : Real.Angle)) '' Set.Ioc 0 s,
+      inner ℝ (normalVector (s : Real.Angle)) (tangentVector u) ∂surfaceAreaMeasure K) =
+      supportValue K (s : Real.Angle) -
+        inner ℝ (normalVector (s : Real.Angle)) (edgeVertices K 0).1 := by
+  let A := (fun t : ℝ ↦ (t : Real.Angle)) '' Set.Ioc 0 s
+  let _ : IsFiniteMeasure (surfaceAreaMeasure K) := (surfaceAreaMeasure_face_union K).1
+  have hu : Integrable (fun u : Real.Angle ↦ tangentVector u)
+      ((surfaceAreaMeasure K).restrict A) := by
+    rw [← integrableOn_univ]
+    apply ContinuousOn.integrableOn_compact isCompact_univ
+    apply Continuous.continuousOn
+    apply (PiLp.continuous_toLp 2 (fun _ : Fin 2 ↦ ℝ)).comp
+    apply continuous_pi
+    intro i
+    fin_cases i
+    · exact Real.Angle.continuous_sin.neg
+    · exact Real.Angle.continuous_cos
+  have hi := ContinuousLinearMap.integral_comp_comm
+    (innerSL ℝ (normalVector (s : Real.Angle))) hu
+  change (∫ u in A, inner ℝ (normalVector (s : Real.Angle)) (tangentVector u)
+    ∂surfaceAreaMeasure K) = inner ℝ (normalVector (s : Real.Angle))
+      (∫ u in A, tangentVector u ∂surfaceAreaMeasure K) at hi
+  rw [show (∫ u in A, tangentVector u ∂surfaceAreaMeasure K) =
+      (edgeVertices K (s : Real.Angle)).1 - (edgeVertices K 0).1 from
+        integral_tangentVector_surfaceAreaMeasure K hs.1 (by linarith [hs.2, Real.pi_pos]),
+    inner_sub_right,
+    real_inner_comm (edgeVertices K (s : Real.Angle)).1 (normalVector (s : Real.Angle)),
+    (edgeVertices_fst_mem K (s : Real.Angle)).2] at hi
+  exact hi
+
+/-- Positive atomic sine contributions are bounded by the corresponding support increment. -/
+theorem sum_surfaceAreaMeasure_mul_pos_sin_le (K : ConvexBody Point)
+    (D : Finset ℝ) (hD : ∀ t ∈ D, t ∈ Set.Ioo 0 Real.pi)
+    {s : ℝ} (hs : s ∈ Set.Ioo 0 Real.pi) :
+    ∑ t ∈ D, (surfaceAreaMeasure K {(t : Real.Angle)}).toReal *
+      max (Real.sin (s - t)) 0 ≤ supportValue K (s : Real.Angle) -
+        inner ℝ (normalVector (s : Real.Angle)) (edgeVertices K 0).1 := by
+  classical
+  let F := D.filter (fun t ↦ t ≤ s)
+  let E := (fun t : ℝ ↦ (t : Real.Angle)) '' Set.Ioc 0 s
+  let g : Real.Angle → ℝ :=
+    fun u ↦ inner ℝ (normalVector (s : Real.Angle)) (tangentVector u)
+  have hcoe : Set.InjOn (fun t : ℝ ↦ (t : Real.Angle)) (Set.Ioc 0 s) :=
+    Real.Angle.injOn_coe_Ioc (by linarith [hs.2, Real.pi_pos])
+  have hE : MeasurableSet E := measurableSet_Ioc.image_of_continuousOn_injOn
+    Real.Angle.continuous_coe.continuousOn hcoe
+  have hF (t : ℝ) (ht : t ∈ F) : t ∈ Set.Ioc 0 s :=
+    ⟨(hD t (Finset.mem_filter.mp ht).1).1, (Finset.mem_filter.mp ht).2⟩
+  let _ : IsFiniteMeasure (surfaceAreaMeasure K) := (surfaceAreaMeasure_face_union K).1
+  have hgcont : Continuous g := by
+    apply Continuous.inner continuous_const
+    apply (PiLp.continuous_toLp 2 (fun _ : Fin 2 ↦ ℝ)).comp
+    apply continuous_pi
+    intro i
+    fin_cases i
+    · exact Real.Angle.continuous_sin.neg
+    · exact Real.Angle.continuous_cos
+  have hg : IntegrableOn g E (surfaceAreaMeasure K) :=
+    by
+    have hall : Integrable g (surfaceAreaMeasure K) := by
+      rw [← integrableOn_univ]
+      exact hgcont.continuousOn.integrableOn_compact isCompact_univ
+    exact hall.integrableOn
+  have hgnonneg : ∀ u ∈ E, 0 ≤ g u := by
+    rintro u ⟨t, ht, rfl⟩
+    change 0 ≤ inner ℝ (normalVector (s : Real.Angle)) (tangentVector (t : Real.Angle))
+    rw [real_inner_comm (tangentVector (t : Real.Angle)), inner_tangentVector_normalVector_real]
+    exact Real.sin_nonneg_of_mem_Icc ⟨by linarith [ht.2], by linarith [ht.1, hs.2]⟩
+  have hb := sum_measureReal_mul_le_setIntegral (surfaceAreaMeasure K)
+    (F.image fun t : ℝ ↦ (t : Real.Angle)) hE
+    (by rintro u hu; obtain ⟨t, ht, rfl⟩ := Finset.mem_image.mp hu; exact ⟨t, hF t ht, rfl⟩)
+    hg hgnonneg
+  rw [Finset.sum_image (fun x hx y hy hxy ↦ hcoe (hF x hx) (hF y hy) hxy)] at hb
+  have heval (t : ℝ) : g (t : Real.Angle) = Real.sin (s - t) := by
+    dsimp [g]
+    rw [real_inner_comm (tangentVector (t : Real.Angle)), inner_tangentVector_normalVector_real]
+  simp_rw [heval] at hb
+  change (∑ t ∈ F, (surfaceAreaMeasure K {(t : Real.Angle)}).toReal *
+    Real.sin (s - t)) ≤ _ at hb
+  have hsum : (∑ t ∈ D, (surfaceAreaMeasure K {(t : Real.Angle)}).toReal *
+      max (Real.sin (s - t)) 0) =
+      ∑ t ∈ F, (surfaceAreaMeasure K {(t : Real.Angle)}).toReal * Real.sin (s - t) := by
+    rw [Finset.sum_filter]
+    apply Finset.sum_congr rfl
+    intro t ht
+    by_cases hts : t ≤ s
+    · rw [ite_eq_left hts, max_eq_left (Real.sin_nonneg_of_mem_Icc
+        ⟨by linarith, by linarith [(hD t ht).1, hs.2]⟩)]
+    · have hsin : Real.sin (s - t) ≤ 0 := by
+        have h := Real.sin_nonneg_of_mem_Icc
+          (show t - s ∈ Set.Icc 0 Real.pi from
+            ⟨by linarith, by linarith [(hD t ht).2, hs.1]⟩)
+        rw [show s - t = -(t - s) by ring, Real.sin_neg]
+        linarith
+      rw [ite_eq_right hts, max_eq_right hsin, mul_zero]
+  rw [hsum]
+  exact hb.trans_eq (integral_inner_tangentVector_eq_support_sub K hs)
+
+/-- For upper normals, the negative zero-angle vertex projects below the positive vertex. -/
+theorem inner_negativeVertex_zero_le_positiveVertex (K : ConvexBody Point)
+    {s : ℝ} (hs : s ∈ Set.Icc 0 Real.pi) :
+    inner ℝ (normalVector (s : Real.Angle)) (edgeVertices K 0).2 ≤
+      inner ℝ (normalVector (s : Real.Angle)) (edgeVertices K 0).1 := by
+  have hy : inner ℝ (edgeVertices K 0).2 (tangentVector 0) ≤
+      inner ℝ (edgeVertices K 0).1 (tangentVector 0) := by
+    rw [inner_edgeVertices_snd_tangent]
+    exact csInf_le ((isCompact_exposedEdge K 0).image
+      (continuous_id.inner continuous_const)).bddBelow
+      ⟨(edgeVertices K 0).1, edgeVertices_fst_mem K 0, rfl⟩
+  have hx := (edgeVertices_fst_mem K 0).2.trans (edgeVertices_snd_mem K 0).2.symm
+  simp only [normalVector, frame, Real.Angle.cos_zero, Real.Angle.sin_zero, neg_zero,
+    PiLp.inner_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue,
+    Matrix.cons_val_zero, one_mul, Matrix.cons_val_one, Matrix.cons_val_fin_one, zero_mul,
+    add_zero, tangentVector, zero_add, Real.Angle.cos_coe, Real.Angle.sin_coe, ge_iff_le] at hx hy ⊢
+  have hsin := Real.sin_nonneg_of_mem_Icc hs
+  rw [hx]
+  linarith [mul_le_mul_of_nonneg_right hy hsin]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Integrals
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter MeasureTheory
+open scoped Topology
+
+namespace MovingSofa
+
+/-- A sine convolution of tangent directions is the negative normal projection of their
+Bochner integral. -/
+private theorem integral_sin_sub_eq_neg_inner_integral_tangentVector
+    {α : Type*} [MeasurableSpace α] (μ : Measure α) (s : Set α)
+    (u : α → Real.Angle) (t : Real.Angle)
+    (hu : Integrable (fun x ↦ tangentVector (u x)) (μ.restrict s)) :
+    (∫ x in s, (u x - t).sin ∂μ) =
+      -inner ℝ (normalVector t) (∫ x in s, tangentVector (u x) ∂μ) := by
+  let L : Point →L[ℝ] ℝ := innerSL ℝ (normalVector t)
+  calc
+    (∫ x in s, (u x - t).sin ∂μ) =
+        ∫ x in s, -L (tangentVector (u x)) ∂μ := by
+          apply integral_congr_ae
+          filter_upwards [] with x
+          exact sin_sub_eq_neg_inner_normalVector_tangentVector t (u x)
+    _ = -(∫ x in s, L (tangentVector (u x)) ∂μ) := integral_neg _
+    _ = -L (∫ x in s, tangentVector (u x) ∂μ) := by
+      rw [ContinuousLinearMap.integral_comp_comm L hu]
+    _ = -inner ℝ (normalVector t) (∫ x in s, tangentVector (u x) ∂μ) := rfl
+
+theorem tangentArm_convolution (K : RightAngleCapSpace) (t : ℝ) :
+    (tangentArmLengths K t).2.1 =
+      ∫ u in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioc t (t + Real.pi / 2),
+        (u - (t : Real.Angle)).sin ∂surfaceAreaMeasure K.val := by
+  let A := (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioc t (t + Real.pi / 2)
+  let _ : IsFiniteMeasure (surfaceAreaMeasure K.val) :=
+    (surfaceAreaMeasure_face_union K.val).1
+  have hu : Integrable (fun u : Real.Angle ↦ tangentVector u)
+      ((surfaceAreaMeasure K.val).restrict A) := by
+    rw [← integrableOn_univ]
+    apply ContinuousOn.integrableOn_compact isCompact_univ
+    apply Continuous.continuousOn
+    apply (PiLp.continuous_toLp 2 (fun _ : Fin 2 ↦ ℝ)).comp
+    apply continuous_pi
+    intro i
+    fin_cases i
+    · exact Real.Angle.continuous_sin.neg
+    · exact Real.Angle.continuous_cos
+  have hvec : (∫ u in A, tangentVector u ∂surfaceAreaMeasure K.val) =
+      (edgeVertices K.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).1 -
+        (edgeVertices K.val (t : Real.Angle)).1 := by
+    ext i
+    have hi := ContinuousLinearMap.integral_comp_comm
+      (PiLp.proj (𝕜 := ℝ) 2 (fun _ : Fin 2 ↦ ℝ) i) hu
+    change (∫ u in A, tangentVector u i ∂surfaceAreaMeasure K.val) =
+      (∫ u in A, tangentVector u ∂surfaceAreaMeasure K.val) i at hi
+    rw [← hi]
+    symm
+    simpa [A] using positiveVertex_sub_eq_integral K.val (a := t)
+      (b := t + Real.pi / 2) (by linarith [Real.pi_pos])
+      (by linarith [Real.pi_pos]) i
+  have harm : (tangentArmLengths K t).2.1 =
+      -inner ℝ (normalVector (t : Real.Angle))
+        ((edgeVertices K.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).1 -
+          (edgeVertices K.val (t : Real.Angle)).1) := by
+    obtain ⟨hA, _, hC, _⟩ := capTangentArm_identities K t
+    have hnt : inner ℝ (tangentVector (t : Real.Angle))
+        (normalVector (t : Real.Angle)) = 0 := by
+      rw [real_inner_comm]
+      exact inner_normalVector_tangentVector t
+    have hnn : inner ℝ (normalVector (t : Real.Angle))
+        (normalVector (t : Real.Angle)) = 1 := inner_normalVector_self t
+    rw [hA] at hC
+    have hproj := congrArg (fun p : Point ↦ inner ℝ p (normalVector (t : Real.Angle))) hC
+    simp only [inner_add_left, real_inner_smul_left, hnt, hnn, mul_zero, add_zero,
+      mul_one] at hproj
+    simp only [capVertices] at hproj
+    rw [inner_sub_right]
+    rw [real_inner_comm
+      (edgeVertices K.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).1
+      (normalVector (t : Real.Angle)),
+      real_inner_comm (edgeVertices K.val (t : Real.Angle)).1
+        (normalVector (t : Real.Angle))]
+    linarith
+  have hkernel := integral_sin_sub_eq_neg_inner_integral_tangentVector
+    (surfaceAreaMeasure K.val) A id (t : Real.Angle) hu
+  simp only [id_eq] at hkernel
+  rw [show (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioc t (t + Real.pi / 2) = A from rfl,
+    hkernel, hvec, harm]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Vertex Boundary
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped Topology
+
+namespace MovingSofa
+
+theorem positiveVertex_stieltjes_surface (K : ConvexBody Point) (a b : ℝ)
+    (hab : a < b) (hturn : b ≤ a + 2 * Real.pi) :
+    ∃ f : Fin 2 → RightContinuousIntervalBV a b,
+      (∀ i t, (f i).toFun t = (edgeVertices K ((t : ℝ) : Real.Angle)).1 i) ∧
+      ∀ (i : Fin 2) (E : Set (Set.Icc a b)), MeasurableSet E →
+        (∀ t ∈ E, a < (t : ℝ)) →
+        intervalStieltjesMeasure (f i) E =
+          ∫ u in (fun t : Set.Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+            tangentVector u i ∂surfaceAreaMeasure K := by
+  obtain ⟨f, hf⟩ := exists_positiveVertex_intervalBV K hab.le
+  refine ⟨f, hf, ?_⟩
+  apply intervalStieltjesMeasure_eq_surfaceIntegral_of_increment K hab hturn f hf
+  intro c d hcd i
+  have hcdturn : (d : ℝ) ≤ (c : ℝ) + 2 * Real.pi :=
+    d.property.2.trans (hturn.trans (by linarith [c.property.1]))
+  exact positiveVertex_sub_eq_integral K hcd hcdturn i
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Angular densities of the surface-area measure
+
+On an angular window of at most one turn the positive vertex of a convex body is a function of
+bounded variation whose Stieltjes measure, paired with the moving tangent, is the surface-area
+measure (`sum_intervalStieltjesIntegral_positiveVertex_tangent`).  If on such a window the
+positive vertex happens to be a differentiable curve with derivative `g s • tangentVector s`,
+this identifies the surface-area measure with the Lebesgue density `g`.
+
+This file records that identification (`surfaceAreaMeasure_angleImage_eq_setLIntegral`,
+`surfaceAreaMeasure_angleImage_eq_withDensity_of_hasDerivAt`), and the bookkeeping that glues
+finitely many or countably many such windows together (`measure_angleImage_eq_of_union`,
+`measure_angleImage_eq_of_iUnion`) and turns the resulting set-level identities into the
+`Measure.restrict = Measure.map (Measure.withDensity …)` form used by cap-density statements
+(`measure_restrict_eq_map_withDensity`, `surfaceAreaMeasure_restrict_eq_map_withDensity`,
+`surfaceAreaMeasure_restrict_eq_map_add_withDensity`).
+
+The gluing lemmas are stated for an arbitrary pair of measures on `Real.Angle` and on `ℝ`,
+since they only use additivity and the injectivity of the angular projection on a window of at
+most one turn.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped ENNReal
+
+namespace MovingSofa
+
+/-- Pairing the frame tangent with a multiple of itself, in coordinates. -/
+private theorem sum_mul_tangentVector_sq (c s : ℝ) :
+    tangentVector (s : Real.Angle) 0 * (c * tangentVector (s : Real.Angle) 0) +
+      tangentVector (s : Real.Angle) 1 * (c * tangentVector (s : Real.Angle) 1) = c := by
+  have h : tangentVector (s : Real.Angle) 0 ^ 2 + tangentVector (s : Real.Angle) 1 ^ 2 = 1 := by
+    have h := inner_tangentVector_self s
+    rw [PiLp.inner_apply] at h
+    simpa [Fin.sum_univ_two] using h
+  linear_combination c * h
+
+/-! ### The surface measure of an arc with a differentiable positive vertex -/
+
+/-- Suppose that on the angular window `Ioc a b`, of at most one turn, the positive vertex of `K`
+is traced by a curve `F` with derivative `g s • tangentVector s`, where `g` is continuous and
+nonnegative.  Then the surface-area measure of the angular image of a measurable
+`S ⊆ Ioc a b` is the Lebesgue integral of `g` over `S`. -/
+theorem surfaceAreaMeasure_angleImage_eq_setLIntegral (K : ConvexBody Point) {a b : ℝ}
+    (hab : a < b) (hturn : b ≤ a + 2 * Real.pi) (F : ℝ → Point) (g : ℝ → ℝ)
+    (hF : ∀ s, HasDerivAt F (g s • tangentVector (s : Real.Angle)) s) (hg : Continuous g)
+    (hgnn : ∀ s ∈ Ioc a b, 0 ≤ g s)
+    (hvertex : ∀ s ∈ Icc a b, (edgeVertices K (s : Real.Angle)).1 = F s)
+    {S : Set ℝ} (hS : MeasurableSet S) (hSsub : S ⊆ Ioc a b) :
+    surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+      ∫⁻ s in S, ENNReal.ofReal (g s) := by
+  have hfin : IsFiniteMeasure (surfaceAreaMeasure K) := (surfaceAreaMeasure_face_union K).1
+  have hSIcc : S ⊆ Icc a b := hSsub.trans Ioc_subset_Icc_self
+  obtain ⟨f, hftoFun, hfmeas⟩ := positiveVertex_stieltjes_surface K a b hab hturn
+  have hcontT : ∀ i : Fin 2, Continuous fun s : ℝ ↦ tangentVector (s : Real.Angle) i :=
+    fun i ↦ (continuous_tangentVector_coordinate i).comp Real.Angle.continuous_coe
+  -- the coordinate functions of the differentiable vertex curve and their derivatives
+  set φ : Fin 2 → ℝ → ℝ := fun i s ↦ F s i with hφ
+  set ψ : Fin 2 → ℝ → ℝ := fun i s ↦ g s * tangentVector (s : Real.Angle) i with hψ
+  have hderiv : ∀ (i : Fin 2) (s : ℝ), HasDerivAt (φ i) (ψ i s) s := by
+    intro i s
+    have h := (EuclideanSpace.proj (𝕜 := ℝ) i).hasFDerivAt.comp_hasDerivAt s (hF s)
+    simpa [φ, ψ, Function.comp_def] using h
+  have hcontψ : ∀ i : Fin 2, Continuous (ψ i) := fun i ↦ hg.mul (hcontT i)
+  -- the Stieltjes density of each vertex coordinate
+  have hdens : ∀ i : Fin 2, HasIntervalStieltjesDensity (f i) (ψ i) := by
+    intro i
+    obtain ⟨G, hGfun, hGdens⟩ :=
+      exists_intervalBV_of_hasDerivAt hab.le (φ i) (ψ i) (hderiv i) (hcontψ i)
+    have hfe : f i = G := by
+      refine RightContinuousIntervalBV.toFun_injective (funext fun t ↦ ?_)
+      rw [hftoFun i t, hGfun t, hvertex (t : ℝ) t.2]
+    rw [hfe]
+    exact hGdens
+  -- the sum over the two coordinates of the Stieltjes integrals
+  set E : Set (Icc a b) := {x : Icc a b | (x : ℝ) ∈ S} with hE
+  have hEmeas : MeasurableSet E := hS.preimage measurable_subtype_coe
+  have hEa : ∀ t ∈ E, a < (t : ℝ) := fun t ht ↦ (hSsub ht).1
+  have himage : (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E =
+      (fun s : ℝ ↦ (s : Real.Angle)) '' S := by
+    ext u
+    constructor
+    · rintro ⟨t, ht, rfl⟩
+      exact ⟨(t : ℝ), ht, rfl⟩
+    · rintro ⟨s, hs, rfl⟩
+      exact ⟨⟨s, hSIcc hs⟩, hs, rfl⟩
+  have hsum := sum_intervalStieltjesIntegral_positiveVertex_tangent K hab hturn f hfmeas
+    E hEmeas hEa
+  rw [himage] at hsum
+  -- each Stieltjes integral is an ordinary set integral
+  have hstep : ∀ i : Fin 2,
+      intervalStieltjesIntegral (f i) (fun t ↦ tangentVector ((t : ℝ) : Real.Angle) i) E =
+        ∫ s in S, tangentVector (s : Real.Angle) i * ψ i s := by
+    intro i
+    have hq : Continuous fun t : Icc a b ↦ tangentVector ((t : ℝ) : Real.Angle) i :=
+      (hcontT i).comp continuous_subtype_val
+    rw [intervalStieltjesIntegral_eq_integral_mul_of_density (f i) (hdens i) hq E hEmeas]
+    rw [MeasureTheory.integral_subtype_preimage measurableSet_Icc hS
+      (fun s ↦ tangentVector (s : Real.Angle) i * ψ i s)]
+    rw [Measure.restrict_restrict_of_subset hSIcc]
+  simp only [hstep] at hsum
+  -- integrability of the two summands and of the density
+  have hintegrand : ∀ i : Fin 2,
+      IntegrableOn (fun s : ℝ ↦ tangentVector (s : Real.Angle) i * ψ i s) S volume := by
+    intro i
+    refine (ContinuousOn.integrableOn_compact isCompact_Icc ?_).mono_set hSIcc
+    exact ((hcontT i).mul (hcontψ i)).continuousOn
+  have hgint : IntegrableOn g S volume :=
+    (ContinuousOn.integrableOn_compact isCompact_Icc hg.continuousOn).mono_set hSIcc
+  have hsum2 : (surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S)).toReal =
+      ∫ s in S, g s := by
+    rw [← hsum, Fin.sum_univ_two, ← integral_add (hintegrand 0) (hintegrand 1)]
+    refine setIntegral_congr_fun hS fun s _ ↦ ?_
+    exact sum_mul_tangentVector_sq (g s) s
+  rw [← ENNReal.ofReal_toReal (measure_ne_top (surfaceAreaMeasure K) _), hsum2]
+  rw [MeasureTheory.ofReal_integral_eq_lintegral_ofReal hgint]
+  filter_upwards [ae_restrict_mem hS] with s hs
+  exact hgnn s (hSsub hs)
+
+/-- The same identification as `surfaceAreaMeasure_angleImage_eq_setLIntegral`, phrased as
+agreement with a `Measure.withDensity` for any nonnegative weight `w` that agrees with the
+derivative factor `g` on the window. -/
+theorem surfaceAreaMeasure_angleImage_eq_withDensity_of_hasDerivAt (K : ConvexBody Point)
+    {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi) (F : ℝ → Point) (g w : ℝ → ℝ)
+    (hF : ∀ s, HasDerivAt F (g s • tangentVector (s : Real.Angle)) s) (hg : Continuous g)
+    (hvertex : ∀ s ∈ Icc a b, (edgeVertices K (s : Real.Angle)).1 = F s)
+    (hw : ∀ s ∈ Ioc a b, w s = g s) (hwnn : ∀ s ∈ Ioc a b, 0 ≤ w s)
+    (S : Set ℝ) (hS : MeasurableSet S) (hSsub : S ⊆ Ioc a b) :
+    surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+      volume.withDensity (fun s ↦ ENNReal.ofReal (w s)) S := by
+  have hgnn : ∀ s ∈ Ioc a b, 0 ≤ g s := fun s hs ↦ (hw s hs) ▸ hwnn s hs
+  rw [withDensity_apply _ hS, surfaceAreaMeasure_angleImage_eq_setLIntegral K hab hturn F g hF hg
+    hgnn hvertex hS hSsub]
+  exact setLIntegral_congr_fun hS fun s hs ↦ by rw [hw s (hSsub hs)]
+
+/-! ### Gluing angular density identities -/
+
+/-- Two measures that read one another through the angular projection on each of two disjoint
+measurable subsets of a window of at most one turn do so on their union. -/
+theorem measure_angleImage_eq_of_union {μ : Measure Real.Angle} {ν : Measure ℝ} {c d : ℝ}
+    (hturn : d ≤ c + 2 * Real.pi) {I J : Set ℝ}
+    (hI : I ⊆ Ioc c d) (hJ : J ⊆ Ioc c d)
+    (hImeas : MeasurableSet I) (hJmeas : MeasurableSet J) (hdisj : Disjoint I J)
+    (hA : ∀ S, MeasurableSet S → S ⊆ I → μ ((fun s : ℝ ↦ (s : Real.Angle)) '' S) = ν S)
+    (hB : ∀ S, MeasurableSet S → S ⊆ J → μ ((fun s : ℝ ↦ (s : Real.Angle)) '' S) = ν S)
+    (S : Set ℝ) (hS : MeasurableSet S) (hSsub : S ⊆ I ∪ J) :
+    μ ((fun s : ℝ ↦ (s : Real.Angle)) '' S) = ν S := by
+  have hsplit : S = S ∩ I ∪ S ∩ J := by
+    rw [← Set.inter_union_distrib_left, Set.inter_eq_left.2 hSsub]
+  have hSI : MeasurableSet (S ∩ I) := hS.inter hImeas
+  have hSJ : MeasurableSet (S ∩ J) := hS.inter hJmeas
+  have hdisj' : Disjoint (S ∩ I) (S ∩ J) :=
+    hdisj.mono Set.inter_subset_right Set.inter_subset_right
+  have hinj := Real.Angle.injOn_coe_Ioc hturn
+  have himdisj : Disjoint ((fun s : ℝ ↦ (s : Real.Angle)) '' (S ∩ I))
+      ((fun s : ℝ ↦ (s : Real.Angle)) '' (S ∩ J)) := by
+    rw [Set.disjoint_left]
+    rintro u ⟨x, hx, rfl⟩ ⟨y, hy, hxy⟩
+    have hyx : y = x := hinj (hJ hy.2) (hI hx.2) hxy
+    exact (Set.disjoint_left.1 hdisj' hx) (hyx ▸ hy)
+  have himmeas : MeasurableSet ((fun s : ℝ ↦ (s : Real.Angle)) '' (S ∩ J)) :=
+    Real.Angle.measurableSet_image_of_subset_Ioc hturn hSJ (Set.inter_subset_right.trans hJ)
+  calc μ ((fun s : ℝ ↦ (s : Real.Angle)) '' S)
+      = μ ((fun s : ℝ ↦ (s : Real.Angle)) '' (S ∩ I) ∪
+          (fun s : ℝ ↦ (s : Real.Angle)) '' (S ∩ J)) := by
+        rw [← Set.image_union, ← hsplit]
+    _ = μ ((fun s : ℝ ↦ (s : Real.Angle)) '' (S ∩ I)) +
+          μ ((fun s : ℝ ↦ (s : Real.Angle)) '' (S ∩ J)) := measure_union himdisj himmeas
+    _ = ν (S ∩ I) + ν (S ∩ J) := by
+        rw [hA _ hSI Set.inter_subset_right, hB _ hSJ Set.inter_subset_right]
+    _ = ν S := by rw [← measure_union hdisj' hSJ, ← hsplit]
+
+/-- Two measures that read one another through the angular projection on each member of a
+monotone sequence of measurable sets do so on the union of that sequence. -/
+theorem measure_angleImage_eq_of_iUnion {μ : Measure Real.Angle} {ν : Measure ℝ}
+    {J : ℕ → Set ℝ} (hmono : Monotone J) (hJmeas : ∀ n, MeasurableSet (J n))
+    (hA : ∀ (n : ℕ) (S : Set ℝ), MeasurableSet S → S ⊆ J n →
+      μ ((fun s : ℝ ↦ (s : Real.Angle)) '' S) = ν S)
+    (S : Set ℝ) (hS : MeasurableSet S) (hSsub : S ⊆ ⋃ n, J n) :
+    μ ((fun s : ℝ ↦ (s : Real.Angle)) '' S) = ν S := by
+  have hSeq : S = ⋃ n, S ∩ J n := by
+    rw [← Set.inter_iUnion, Set.inter_eq_left.2 hSsub]
+  have hmono' : Monotone fun n ↦ S ∩ J n := fun m n h ↦
+    Set.inter_subset_inter_right _ (hmono h)
+  have hmonoimg : Monotone fun n ↦ (fun s : ℝ ↦ (s : Real.Angle)) '' (S ∩ J n) :=
+    fun m n h ↦ Set.image_mono (hmono' h)
+  calc μ ((fun s : ℝ ↦ (s : Real.Angle)) '' S)
+      = μ (⋃ n, (fun s : ℝ ↦ (s : Real.Angle)) '' (S ∩ J n)) := by
+        rw [← Set.image_iUnion, ← hSeq]
+    _ = ⨆ n, μ ((fun s : ℝ ↦ (s : Real.Angle)) '' (S ∩ J n)) := hmonoimg.measure_iUnion
+    _ = ⨆ n, ν (S ∩ J n) :=
+        iSup_congr fun n ↦ hA n _ (hS.inter (hJmeas n)) Set.inter_subset_right
+    _ = ν S := by rw [← hmono'.measure_iUnion, ← hSeq]
+
+/-! ### From set-level identities to restricted measures -/
+
+/-- A set-level angular density identity on a measurable parameter set `I` says exactly that the
+measure restricted to the angular image of `I` is the pushforward of the weighted Lebesgue
+measure on `I`. -/
+theorem measure_restrict_eq_map_withDensity {μ : Measure Real.Angle} {I : Set ℝ}
+    {w : ℝ → ℝ≥0∞} (hImeas : MeasurableSet I)
+    (hagree : ∀ S, MeasurableSet S → S ⊆ I →
+      μ ((fun s : ℝ ↦ (s : Real.Angle)) '' S) = volume.withDensity w S) :
+    μ.restrict ((fun s : ℝ ↦ (s : Real.Angle)) '' I) =
+      Measure.map (fun s : ℝ ↦ (s : Real.Angle)) ((volume.restrict I).withDensity w) := by
+  have hmeas : Measurable fun s : ℝ ↦ (s : Real.Angle) := Real.Angle.continuous_coe.measurable
+  ext B hB
+  have hpre : MeasurableSet ((fun s : ℝ ↦ (s : Real.Angle)) ⁻¹' B) := hB.preimage hmeas
+  have hset : B ∩ (fun s : ℝ ↦ (s : Real.Angle)) '' I =
+      (fun s : ℝ ↦ (s : Real.Angle)) '' ((fun s : ℝ ↦ (s : Real.Angle)) ⁻¹' B ∩ I) := by
+    ext u
+    constructor
+    · rintro ⟨hu, s, hs, rfl⟩
+      exact ⟨s, ⟨hu, hs⟩, rfl⟩
+    · rintro ⟨s, ⟨hs1, hs2⟩, rfl⟩
+      exact ⟨hs1, s, hs2, rfl⟩
+  rw [Measure.restrict_apply hB, hset, hagree _ (hpre.inter hImeas) Set.inter_subset_right,
+    Measure.map_apply hmeas hB, withDensity_apply _ hpre,
+    withDensity_apply _ (hpre.inter hImeas), Measure.restrict_restrict hpre]
+
+/-- The surface-area measure instance of `measure_restrict_eq_map_withDensity`. -/
+theorem surfaceAreaMeasure_restrict_eq_map_withDensity {K : ConvexBody Point}
+    {I : Set ℝ} {w : ℝ → ℝ≥0∞} (hImeas : MeasurableSet I)
+    (hagree : ∀ S, MeasurableSet S → S ⊆ I →
+      surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) = volume.withDensity w S) :
+    (surfaceAreaMeasure K).restrict ((fun s : ℝ ↦ (s : Real.Angle)) '' I) =
+      Measure.map (fun s : ℝ ↦ (s : Real.Angle)) ((volume.restrict I).withDensity w) :=
+  measure_restrict_eq_map_withDensity hImeas hagree
+
+/-- The shifted form of `surfaceAreaMeasure_restrict_eq_map_withDensity`: an angular density
+identity on the translated window `Ioc c (c + T)` with the translated weight `u ↦ w (u - c)`
+says that the surface-area measure restricted to that angular arc is the pushforward of the
+`w`-weighted Lebesgue measure on `Ioc 0 T` along `t ↦ ↑(t + c)`. -/
+theorem surfaceAreaMeasure_restrict_eq_map_add_withDensity {K : ConvexBody Point}
+    {c T : ℝ} {w : ℝ → ℝ≥0∞}
+    (hagree : ∀ S, MeasurableSet S → S ⊆ Ioc c (c + T) →
+      surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+        volume.withDensity (fun u ↦ w (u - c)) S) :
+    (surfaceAreaMeasure K).restrict ((fun s : ℝ ↦ (s : Real.Angle)) '' Ioc c (c + T)) =
+      Measure.map (fun t : ℝ ↦ ((t + c : ℝ) : Real.Angle))
+        ((volume.restrict (Ioc 0 T)).withDensity w) := by
+  have hmeas : Measurable fun s : ℝ ↦ (s : Real.Angle) := Real.Angle.continuous_coe.measurable
+  have hmeas' : Measurable fun t : ℝ ↦ ((t + c : ℝ) : Real.Angle) :=
+    Real.Angle.continuous_coe.measurable.comp (measurable_id.add_const c)
+  ext B hB
+  have hpre : MeasurableSet ((fun s : ℝ ↦ (s : Real.Angle)) ⁻¹' B) := hB.preimage hmeas
+  have hpre' : MeasurableSet ((fun t : ℝ ↦ ((t + c : ℝ) : Real.Angle)) ⁻¹' B) :=
+    hB.preimage hmeas'
+  have hset : B ∩ (fun s : ℝ ↦ (s : Real.Angle)) '' Ioc c (c + T) =
+      (fun s : ℝ ↦ (s : Real.Angle)) ''
+        ((fun s : ℝ ↦ (s : Real.Angle)) ⁻¹' B ∩ Ioc c (c + T)) := by
+    ext u
+    constructor
+    · rintro ⟨hu, s, hs, rfl⟩
+      exact ⟨s, ⟨hu, hs⟩, rfl⟩
+    · rintro ⟨s, ⟨hs1, hs2⟩, rfl⟩
+      exact ⟨hs1, s, hs2, rfl⟩
+  have hpreimage : (fun t : ℝ ↦ t + c) ⁻¹'
+      ((fun s : ℝ ↦ (s : Real.Angle)) ⁻¹' B ∩ Ioc c (c + T)) =
+      (fun t : ℝ ↦ ((t + c : ℝ) : Real.Angle)) ⁻¹' B ∩ Ioc 0 T := by
+    ext t
+    simp only [Set.mem_preimage, Set.mem_inter_iff, Set.mem_Ioc]
+    constructor
+    · rintro ⟨h1, h2, h3⟩
+      exact ⟨h1, by linarith, by linarith⟩
+    · rintro ⟨h1, h2, h3⟩
+      exact ⟨h1, by linarith, by linarith⟩
+  rw [Measure.restrict_apply hB, hset,
+    hagree _ (hpre.inter measurableSet_Ioc) Set.inter_subset_right,
+    withDensity_apply _ (hpre.inter measurableSet_Ioc), setLIntegral_comp_sub_right, hpreimage,
+    Measure.map_apply hmeas' hB, withDensity_apply _ hpre',
+    Measure.restrict_restrict hpre']
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Frame Products
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped Topology
+
+namespace MovingSofa
+
+/-- The normal and tangent projections of the positive vertex, with their Stieltjes measures. -/
+theorem exists_positiveVertex_frame_products
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi) :
+    ∃ H P : RightContinuousIntervalBV a b,
+      (∀ t, H.toFun t = inner ℝ (edgeVertices K (((t : ℝ) : Real.Angle))).1
+        (normalVector (((t : ℝ) : Real.Angle)))) ∧
+      (∀ t, P.toFun t = inner ℝ (edgeVertices K (((t : ℝ) : Real.Angle))).1
+        (tangentVector (((t : ℝ) : Real.Angle)))) ∧
+      ∀ E : Set (Icc a b), MeasurableSet E → (∀ t ∈ E, a < (t : ℝ)) →
+        intervalStieltjesMeasure H E =
+          ∫ t in (Subtype.val : Icc a b → ℝ) '' E,
+            inner ℝ (edgeVertices K (t : Real.Angle)).1 (tangentVector (t : Real.Angle)) ∧
+        intervalStieltjesMeasure P E =
+          (surfaceAreaMeasure K
+              ((fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E)).toReal -
+            ∫ t in (Subtype.val : Icc a b → ℝ) '' E,
+              inner ℝ (edgeVertices K (t : Real.Angle)).1 (normalVector (t : Real.Angle)) := by
+  obtain ⟨v, hv, hvm⟩ := positiveVertex_stieltjes_surface K a b hab hturn
+  choose n hn hnd using fun i : Fin 2 ↦ exists_normalVector_coordinate_intervalBV hab.le i
+  choose τ hτ hτd using fun i : Fin 2 ↦ exists_tangentVector_coordinate_intervalBV hab.le i
+  have hnc (i : Fin 2) : Continuous (n i).toFun := by
+    convert ((by
+      fin_cases i
+      · exact Real.Angle.continuous_cos
+      · exact Real.Angle.continuous_sin :
+          Continuous (fun u : Real.Angle ↦ normalVector u i))).comp
+          (Real.Angle.continuous_coe.comp continuous_subtype_val) using 1
+    funext t
+    exact hn i t
+  have hτc (i : Fin 2) : Continuous (τ i).toFun := by
+    convert ((by
+      fin_cases i
+      · exact Real.Angle.continuous_sin.neg
+      · exact Real.Angle.continuous_cos :
+          Continuous (fun u : Real.Angle ↦ tangentVector u i))).comp
+          (Real.Angle.continuous_coe.comp continuous_subtype_val) using 1
+    funext t
+    exact hτ i t
+  obtain ⟨H, hH, hHm⟩ := intervalStieltjes_inner_fin_two v n hnc
+  obtain ⟨P, hP, hPm⟩ := intervalStieltjes_inner_fin_two v τ hτc
+  refine ⟨H, P, ?_, ?_, ?_⟩
+  · intro t
+    rw [hH]
+    simp_rw [hv, hn]
+    rw [PiLp.inner_apply]
+    simp only [Real.inner_apply]
+  · intro t
+    rw [hP]
+    simp_rw [hv, hτ]
+    rw [PiLp.inner_apply]
+    simp only [Real.inner_apply]
+  · intro E hE hEa
+    have hnormal := sum_intervalStieltjesIntegral_positiveVertex_normal
+      K hab hturn v hvm E hE hEa
+    have htangent := sum_intervalStieltjesIntegral_positiveVertex_tangent
+      K hab hturn v hvm E hE hEa
+    have hnfun (i : Fin 2) : (n i).toFun =
+        fun t : Icc a b ↦ normalVector (((t : ℝ) : Real.Angle)) i :=
+      funext (hn i)
+    have hτfun (i : Fin 2) : (τ i).toFun =
+        fun t : Icc a b ↦ tangentVector (((t : ℝ) : Real.Angle)) i :=
+      funext (hτ i)
+    have hnormal' :
+        (∑ i : Fin 2, intervalStieltjesIntegral (v i) (n i).toFun E) = 0 := by
+      simpa only [hnfun] using hnormal
+    have htangent' :
+        (∑ i : Fin 2, intervalStieltjesIntegral (v i) (τ i).toFun E) =
+          (surfaceAreaMeasure K
+            ((fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E)).toReal := by
+      simpa only [hτfun] using htangent
+    have hfinite : IsFiniteMeasure (volume.comap (Subtype.val : Icc a b → ℝ)) :=
+      ⟨by
+        rw [comap_subtype_coe_apply measurableSet_Icc]
+        simp only [image_univ, Subtype.range_val]
+        exact measure_Icc_lt_top⟩
+    let _ := hfinite
+    have hsmoothN (i : Fin 2) :
+        intervalStieltjesIntegral (n i) (v i).toFun E =
+          ∫ t in E, (v i).toFun t * tangentVector (((t : ℝ) : Real.Angle)) i
+            ∂volume.comap (Subtype.val : Icc a b → ℝ) :=
+      intervalStieltjesIntegral_eq_integral_mul_of_density_bv hab.le (n i) (hnd i)
+        (v i).boundedVariation E hE
+    have hsmoothT (i : Fin 2) :
+        intervalStieltjesIntegral (τ i) (v i).toFun E =
+          ∫ t in E, (v i).toFun t * -normalVector (((t : ℝ) : Real.Angle)) i
+            ∂volume.comap (Subtype.val : Icc a b → ℝ) :=
+      intervalStieltjesIntegral_eq_integral_mul_of_density_bv hab.le (τ i) (hτd i)
+        (v i).boundedVariation E hE
+    have hintN (i : Fin 2) : Integrable
+        (fun t : Icc a b ↦ (v i).toFun t * tangentVector (((t : ℝ) : Real.Angle)) i)
+        (volume.comap (Subtype.val : Icc a b → ℝ)) := by
+      apply (v i).boundedVariation.integrable.mul_bdd
+      · exact ((by
+          fin_cases i
+          · exact Real.Angle.continuous_sin.neg
+          · exact Real.Angle.continuous_cos :
+              Continuous (fun u : Real.Angle ↦ tangentVector u i))).comp
+              (Real.Angle.continuous_coe.comp continuous_subtype_val) |>.aestronglyMeasurable
+      · filter_upwards with t
+        fin_cases i
+        · simpa [tangentVector, frame] using Real.abs_sin_le_one (t : ℝ)
+        · simpa [tangentVector, frame] using Real.abs_cos_le_one (t : ℝ)
+    have hintT (i : Fin 2) : Integrable
+        (fun t : Icc a b ↦ (v i).toFun t * -normalVector (((t : ℝ) : Real.Angle)) i)
+        (volume.comap (Subtype.val : Icc a b → ℝ)) := by
+      apply (v i).boundedVariation.integrable.mul_bdd
+      · exact ((by
+          fin_cases i
+          · exact Real.Angle.continuous_cos.neg
+          · exact Real.Angle.continuous_sin.neg :
+              Continuous (fun u : Real.Angle ↦ -normalVector u i))).comp
+              (Real.Angle.continuous_coe.comp continuous_subtype_val) |>.aestronglyMeasurable
+      · filter_upwards with t
+        fin_cases i
+        · simpa [normalVector, frame] using Real.abs_cos_le_one (t : ℝ)
+        · simpa [normalVector, frame] using Real.abs_sin_le_one (t : ℝ)
+    let R := (Subtype.val : Icc a b → ℝ) '' E
+    have hR : MeasurableSet R :=
+      (MeasurableEmbedding.subtype_coe measurableSet_Icc).measurableSet_image' hE
+    have hRsub : R ⊆ Icc a b := by
+      rintro x ⟨t, -, rfl⟩
+      exact t.property
+    have hpre : {t : Icc a b | (t : ℝ) ∈ R} = E :=
+      Set.preimage_image_eq E Subtype.val_injective
+    have hsubN :
+        (∑ i : Fin 2, ∫ t in E,
+          (v i).toFun t * tangentVector (((t : ℝ) : Real.Angle)) i
+            ∂volume.comap (Subtype.val : Icc a b → ℝ)) =
+          ∫ t in R, inner ℝ (edgeVertices K (t : Real.Angle)).1
+            (tangentVector (t : Real.Angle)) := by
+      rw [Fin.sum_univ_two, ← integral_add (hintN 0).integrableOn (hintN 1).integrableOn]
+      have hsubtype := integral_subtype_preimage (μ := volume) (s := Icc a b) (t := R)
+        measurableSet_Icc hR
+        (fun t : ℝ ↦ inner ℝ (edgeVertices K (t : Real.Angle)).1
+          (tangentVector (t : Real.Angle)))
+      rw [hpre, Measure.restrict_restrict_of_subset hRsub] at hsubtype
+      rw [← hsubtype]
+      apply integral_congr_ae
+      filter_upwards with t
+      rw [hv 0 t, hv 1 t]
+      simp [PiLp.inner_apply, Fin.sum_univ_two]
+      ring
+    have hsubT :
+        (∑ i : Fin 2, ∫ t in E,
+          (v i).toFun t * -normalVector (((t : ℝ) : Real.Angle)) i
+            ∂volume.comap (Subtype.val : Icc a b → ℝ)) =
+          -(∫ t in R, inner ℝ (edgeVertices K (t : Real.Angle)).1
+            (normalVector (t : Real.Angle))) := by
+      rw [Fin.sum_univ_two, ← integral_add (hintT 0).integrableOn (hintT 1).integrableOn]
+      have hsubtype := integral_subtype_preimage (μ := volume) (s := Icc a b) (t := R)
+        measurableSet_Icc hR
+        (fun t : ℝ ↦ inner ℝ (edgeVertices K (t : Real.Angle)).1
+          (normalVector (t : Real.Angle)))
+      rw [hpre, Measure.restrict_restrict_of_subset hRsub] at hsubtype
+      rw [← hsubtype, ← integral_neg]
+      apply integral_congr_ae
+      filter_upwards with t
+      rw [hv 0 t, hv 1 t]
+      simp [PiLp.inner_apply, Fin.sum_univ_two]
+      ring
+    constructor
+    · rw [hHm E hE, Finset.sum_add_distrib, hnormal']
+      simp only [zero_add]
+      simpa only [hsmoothN] using hsubN
+    · rw [hPm E hE, Finset.sum_add_distrib, htangent']
+      rw [show (∑ i : Fin 2, intervalStieltjesIntegral (τ i) (v i).toFun E) =
+          -(∫ t in R, inner ℝ (edgeVertices K (t : Real.Angle)).1
+            (normalVector (t : Real.Angle))) by simpa only [hsmoothT] using hsubT]
+      ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Boundary
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped Topology
+
+namespace MovingSofa
+
+theorem positiveArm_stieltjes_surface (K : RightAngleCapSpace) :
+    ∃ f : RightContinuousIntervalBV 0 (Real.pi / 2),
+      (∀ t, f.toFun t = (tangentArmLengths K t).1.1) ∧
+      ∀ E : Set (Set.Icc (0 : ℝ) (Real.pi / 2)), MeasurableSet E →
+        (∀ t ∈ E, 0 < (t : ℝ)) →
+        intervalStieltjesMeasure f E =
+          (∫ t in (fun s : Set.Icc (0 : ℝ) (Real.pi / 2) ↦ (s : ℝ)) '' E,
+            (tangentArmLengths K t).2.1) -
+          (surfaceAreaMeasure K.val
+            ((fun s : Set.Icc (0 : ℝ) (Real.pi / 2) ↦ ((s : ℝ) : Real.Angle)) '' E)).toReal := by
+  have hT : 0 < (Real.pi / 2) := by positivity
+  obtain ⟨HA, PA, hHA, hPA, hmA⟩ := exists_positiveVertex_frame_products K.val hT
+    (by linarith [Real.pi_pos])
+  obtain ⟨HC, PC, hHC, hPC, hmC⟩ := exists_positiveVertex_frame_products K.val
+    (a := 0 + (Real.pi / 2)) (b := (Real.pi / 2) + (Real.pi / 2)) (by linarith)
+    (by linarith [Real.pi_pos])
+  have hHC_support (t : Icc (0 + (Real.pi / 2)) ((Real.pi / 2) + (Real.pi / 2))) :
+      HC.toFun t = supportValue K.val (((t : ℝ) : Real.Angle)) := by
+    rw [hHC, (edgeVertices_fst_mem K.val (((t : ℝ) : Real.Angle))).2]
+  have hHCc : Continuous HC.toFun := by
+    convert (continuous_supportValue_real K.val).comp continuous_subtype_val using 1
+    funext t
+    exact hHC_support t
+  obtain ⟨Hs, hHs, hHsm⟩ := exists_intervalBV_shift_from_zero hT.le HC hHCc
+  obtain ⟨f, hf, hfm⟩ := intervalStieltjes_linear_combination 0 (Real.pi / 2) Hs PA 1 (-1)
+  refine ⟨f, ?_, ?_⟩
+  · intro t
+    rw [hf, hHs, hHC_support, hPA, (tangentArmLengths_positive_frame K t).1]
+    simp only [one_mul, neg_mul, one_mul]
+    ring
+  · intro E hE hE0
+    change Set (Icc 0 (Real.pi / 2)) at E
+    let φ : Icc 0 (Real.pi / 2) →
+        Icc (0 + (Real.pi / 2)) ((Real.pi / 2) + (Real.pi / 2)) := fun t ↦
+      ⟨(t : ℝ) + (Real.pi / 2), by constructor <;> linarith [t.property.1, t.property.2]⟩
+    let R := (Subtype.val : Icc 0 (Real.pi / 2) → ℝ) '' E
+    have hR : MeasurableSet R :=
+      (MeasurableEmbedding.subtype_coe measurableSet_Icc).measurableSet_image' hE
+    have hφE : MeasurableSet (φ '' E) := by
+      let ψ : Icc (0 + (Real.pi / 2)) ((Real.pi / 2) + (Real.pi / 2)) →
+          Icc 0 (Real.pi / 2) := fun t ↦
+        ⟨(t : ℝ) - (Real.pi / 2), by
+          constructor
+          · linarith [t.property.1]
+          · linarith [t.property.2]⟩
+      let e : Icc 0 (Real.pi / 2) ≃ₜ
+          Icc (0 + (Real.pi / 2)) ((Real.pi / 2) + (Real.pi / 2)) :=
+        { toFun := φ
+          invFun := ψ
+          left_inv := fun x ↦ by apply Subtype.ext; simp [φ, ψ]
+          right_inv := fun x ↦ by apply Subtype.ext; simp [φ, ψ]
+          continuous_toFun := (continuous_subtype_val.add_const (Real.pi / 2)).subtype_mk _
+          continuous_invFun := (continuous_subtype_val.sub continuous_const).subtype_mk _ }
+      exact e.measurableEmbedding.measurableSet_image' hE
+    have hφleft : ∀ t ∈ φ '' E, 0 + (Real.pi / 2) < (t : ℝ) := by
+      rintro t ⟨s, hs, rfl⟩
+      dsimp [φ]
+      linarith [hE0 s hs]
+    have hreal : (Subtype.val :
+        Icc (0 + (Real.pi / 2)) ((Real.pi / 2) + (Real.pi / 2)) → ℝ) '' (φ '' E) =
+        (fun x : ℝ ↦ x + (Real.pi / 2)) '' R := by
+      ext x
+      constructor
+      · rintro ⟨-, ⟨s, hs, rfl⟩, rfl⟩
+        exact ⟨s, ⟨s, hs, rfl⟩, rfl⟩
+      · rintro ⟨-, ⟨s, hs, rfl⟩, rfl⟩
+        exact ⟨φ s, ⟨s, hs, rfl⟩, rfl⟩
+    have hHs_measure : intervalStieltjesMeasure Hs E =
+        ∫ t in R, inner ℝ
+          (edgeVertices K.val (((t + (Real.pi / 2) : ℝ) : Real.Angle))).1
+          (tangentVector (((t + (Real.pi / 2) : ℝ) : Real.Angle))) := by
+      rw [hHsm E hE, (hmC (φ '' E) hφE hφleft).1, hreal,
+        integral_image_add_right_eq (Real.pi / 2) R hR]
+    have hPA_measure := (hmA E hE hE0).2
+    have hlin : intervalStieltjesMeasure f E =
+        intervalStieltjesMeasure Hs E - intervalStieltjesMeasure PA E := by
+      have := congrArg (fun m : SignedMeasure (Icc 0 (Real.pi / 2)) ↦ m E) hfm
+      rw [add_apply, _root_.smul_apply, _root_.smul_apply] at this
+      simpa only [smul_eq_mul, one_mul, neg_one_mul, sub_eq_add_neg] using this
+    change intervalStieltjesMeasure f E = _
+    rw [hlin, hHs_measure, hPA_measure]
+    have hAint : IntegrableOn
+        (fun t : ℝ ↦ inner ℝ (edgeVertices K.val (t : Real.Angle)).1
+          (normalVector (t : Real.Angle))) R := by
+      have hAIcc := HA.integrableOn_Icc_of_eq
+        (fun t : ℝ ↦ inner ℝ (edgeVertices K.val (t : Real.Angle)).1
+          (normalVector (t : Real.Angle))) (fun t ↦ (hHA t).symm)
+      apply hAIcc.mono_set
+      rintro x ⟨t, -, rfl⟩
+      exact t.property
+    have hCint : IntegrableOn
+        (fun t : ℝ ↦ inner ℝ
+          (edgeVertices K.val (((t + (Real.pi / 2) : ℝ) : Real.Angle))).1
+          (tangentVector (((t + (Real.pi / 2) : ℝ) : Real.Angle)))) R := by
+      let q : ℝ → ℝ := fun s ↦ inner ℝ (edgeVertices K.val (s : Real.Angle)).1
+        (tangentVector (s : Real.Angle))
+      have hqIcc := PC.integrableOn_Icc_of_eq q (fun t ↦ (hPC t).symm)
+      have hqshift : IntegrableOn q
+          ((fun x : ℝ ↦ x + Real.pi / 2) '' R) := by
+        apply hqIcc.mono_set
+        rintro x ⟨-, ⟨t, -, rfl⟩, rfl⟩
+        constructor <;> linarith [t.property.1, t.property.2]
+      exact (integrableOn_comp_add_right_iff (Real.pi / 2) R hR q).2 hqshift
+    rw [show (∫ t in R, inner ℝ
+          (edgeVertices K.val (((t + Real.pi / 2 : ℝ) : Real.Angle))).1
+          (tangentVector (((t + Real.pi / 2 : ℝ) : Real.Angle)))) -
+        ((surfaceAreaMeasure K.val
+          ((fun t : Icc 0 (Real.pi / 2) ↦ ((t : ℝ) : Real.Angle)) '' E)).toReal -
+          ∫ t in R, inner ℝ (edgeVertices K.val (t : Real.Angle)).1
+            (normalVector (t : Real.Angle))) =
+        ((∫ t in R, inner ℝ
+            (edgeVertices K.val (((t + Real.pi / 2 : ℝ) : Real.Angle))).1
+            (tangentVector (((t + Real.pi / 2 : ℝ) : Real.Angle)))) +
+          ∫ t in R, inner ℝ (edgeVertices K.val (t : Real.Angle)).1
+            (normalVector (t : Real.Angle))) -
+        (surfaceAreaMeasure K.val
+          ((fun t : Icc 0 (Real.pi / 2) ↦ ((t : ℝ) : Real.Angle)) '' E)).toReal by ring]
+    rw [← integral_add hCint hAint]
+    apply congrArg (fun z : ℝ ↦ z -
+      (surfaceAreaMeasure K.val
+        ((fun s : Icc (0 : ℝ) (Real.pi / 2) ↦ ((s : ℝ) : Real.Angle)) '' E)).toReal)
+    apply integral_congr_ae
+    filter_upwards with t
+    rw [(tangentArmLengths_positive_frame K t).2]
+    ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Analysis / Surface Measure / Linearity
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+open scoped unitInterval
+
+namespace MovingSofa
+
+private theorem intervalStieltjesMeasure_congr {a b : ℝ}
+    (f g : RightContinuousIntervalBV a b) (h : ∀ x, f.toFun x = g.toFun x) :
+    intervalStieltjesMeasure f = intervalStieltjesMeasure g := by
+  have hfun : f.toFun = g.toFun := funext h
+  cases f with
+  | mk ff hfb hfr =>
+    cases g with
+    | mk gf hgb hgr =>
+      simp only at hfun ⊢
+      subst gf
+      rfl
+
+/-- Planar surface area measures commute with convex combinations. -/
+theorem surfaceAreaMeasure_convexBodyCombination (t : I)
+    (K L : ConvexBody Point) :
+    surfaceAreaMeasure (convexBodyCombination t K L) =
+      ENNReal.ofReal (1 - (t : ℝ)) • surfaceAreaMeasure K +
+        ENNReal.ofReal (t : ℝ) • surfaceAreaMeasure L := by
+  let r : ℝ := 1 - (t : ℝ)
+  let s : ℝ := t
+  let M := convexBodyCombination t K L
+  have hvertex (u : Real.Angle) :
+      (edgeVertices M u).1 = r • (edgeVertices K u).1 + s • (edgeVertices L u).1 :=
+    (edgeVertices_convexBodyCombination t K L u).1
+  have hab : (0 : ℝ) < 2 * Real.pi := mul_pos (by norm_num) Real.pi_pos
+  have hturn : 2 * Real.pi ≤ (0 : ℝ) + 2 * Real.pi := by simp
+  obtain ⟨fM, hfM, hmM⟩ := positiveVertex_stieltjes_surface M 0 (2 * Real.pi) hab hturn
+  obtain ⟨fK, hfK, hmK⟩ := positiveVertex_stieltjes_surface K 0 (2 * Real.pi) hab hturn
+  obtain ⟨fL, hfL, hmL⟩ := positiveVertex_stieltjes_surface L 0 (2 * Real.pi) hab hturn
+  have hmeasure (i : Fin 2) : intervalStieltjesMeasure (fM i) =
+      r • intervalStieltjesMeasure (fK i) + s • intervalStieltjesMeasure (fL i) := by
+    obtain ⟨f, hfun, hfm⟩ := intervalStieltjes_linear_combination
+      0 (2 * Real.pi) (fK i) (fL i) r s
+    have heq : intervalStieltjesMeasure f = intervalStieltjesMeasure (fM i) := by
+      apply intervalStieltjesMeasure_congr
+      intro x
+      rw [hfun, hfK, hfL, hfM]
+      have hv := congrArg (fun p : Point ↦ p i)
+        (hvertex (((x : Set.Icc (0 : ℝ) (2 * Real.pi)) : ℝ) : Real.Angle))
+      simpa [r, s] using hv.symm
+    rwa [← heq]
+  ext A hA
+  let E : Set (Set.Icc (0 : ℝ) (2 * Real.pi)) :=
+    {x | 0 < (x : ℝ) ∧ (((x : ℝ) : Real.Angle)) ∈ A}
+  have hE : MeasurableSet E := by
+    exact (measurableSet_Ioi.preimage measurable_subtype_coe).inter
+      (hA.preimage (Real.Angle.continuous_coe.comp continuous_subtype_val).measurable)
+  have hEa : ∀ x ∈ E, 0 < (x : ℝ) := fun _ hx ↦ hx.1
+  have himage : (fun x : Set.Icc (0 : ℝ) (2 * Real.pi) ↦
+      (((x : ℝ) : Real.Angle))) '' E = A := by
+    ext u
+    constructor
+    · rintro ⟨x, hx, rfl⟩
+      exact hx.2
+    · intro hu
+      let _ : Fact (0 < 2 * Real.pi) := ⟨hab⟩
+      let x := AddCircle.equivIoc (2 * Real.pi) 0 u
+      have hx : (x : ℝ) ∈ Set.Ioc (0 : ℝ) (2 * Real.pi) := by simpa using x.property
+      let y : Set.Icc (0 : ℝ) (2 * Real.pi) := ⟨x, hx.1.le, hx.2⟩
+      have hxu : (((x : ℝ) : Real.Angle)) = u := AddCircle.coe_equivIoc
+      refine ⟨y, ⟨hx.1, ?_⟩, ?_⟩
+      · simpa [y, hxu] using hu
+      · simpa [y] using hxu
+  have hmassM := sum_intervalStieltjesIntegral_positiveVertex_tangent
+    M hab hturn fM hmM E hE hEa
+  have hmassK := sum_intervalStieltjesIntegral_positiveVertex_tangent
+    K hab hturn fK hmK E hE hEa
+  have hmassL := sum_intervalStieltjesIntegral_positiveVertex_tangent
+    L hab hturn fL hmL E hE hEa
+  rw [himage] at hmassM hmassK hmassL
+  have hstieltjes (i : Fin 2) :
+      intervalStieltjesIntegral (fM i)
+          (fun x ↦ tangentVector ((((x : Set.Icc (0 : ℝ) (2 * Real.pi)) : ℝ) :
+            Real.Angle)) i) E =
+        r * intervalStieltjesIntegral (fK i)
+            (fun x ↦ tangentVector ((((x : Set.Icc (0 : ℝ) (2 * Real.pi)) : ℝ) :
+              Real.Angle)) i) E +
+          s * intervalStieltjesIntegral (fL i)
+            (fun x ↦ tangentVector ((((x : Set.Icc (0 : ℝ) (2 * Real.pi)) : ℝ) :
+              Real.Angle)) i) E := by
+    have hq : Continuous (fun x : Set.Icc (0 : ℝ) (2 * Real.pi) ↦
+        tangentVector (((x : ℝ) : Real.Angle)) i) := by
+      have hi : Continuous (fun u : Real.Angle ↦ tangentVector u i) := by
+        fin_cases i
+        · exact Real.Angle.continuous_sin.neg
+        · exact Real.Angle.continuous_cos
+      exact hi.comp (Real.Angle.continuous_coe.comp continuous_subtype_val)
+    have hrint : (r • VectorMeasure.restrict (intervalStieltjesMeasure (fK i)) E).Integrable
+        (fun x ↦ tangentVector (((x : ℝ) : Real.Angle)) i) :=
+      ((fK i).integrable_of_continuous hq).integrableOn.smul_vectorMeasure r
+    have hsint : (s • VectorMeasure.restrict (intervalStieltjesMeasure (fL i)) E).Integrable
+        (fun x ↦ tangentVector (((x : ℝ) : Real.Angle)) i) :=
+      ((fL i).integrable_of_continuous hq).integrableOn.smul_vectorMeasure s
+    unfold intervalStieltjesIntegral
+    rw [hmeasure i, VectorMeasure.restrict_add, VectorMeasure.restrict_smul,
+      VectorMeasure.restrict_smul, VectorMeasure.integral_add_vectorMeasure hrint hsint,
+      VectorMeasure.integral_smul_vectorMeasure, VectorMeasure.integral_smul_vectorMeasure]
+    rfl
+  have hreal : (surfaceAreaMeasure M A).toReal =
+      r * (surfaceAreaMeasure K A).toReal + s * (surfaceAreaMeasure L A).toReal := by
+    rw [← hmassM, ← hmassK, ← hmassL, Fin.sum_univ_two]
+    rw [hstieltjes 0, hstieltjes 1]
+    simp only [Fin.sum_univ_two]
+    ring
+  let _ : IsFiniteMeasure (surfaceAreaMeasure M) := (surfaceAreaMeasure_face_union M).1
+  let _ : IsFiniteMeasure (surfaceAreaMeasure K) := (surfaceAreaMeasure_face_union K).1
+  let _ : IsFiniteMeasure (surfaceAreaMeasure L) := (surfaceAreaMeasure_face_union L).1
+  have hKtop : surfaceAreaMeasure K A ≠ ⊤ := measure_ne_top _ A
+  have hLtop : surfaceAreaMeasure L A ≠ ⊤ := measure_ne_top _ A
+  have hright :
+      (ENNReal.ofReal (1 - (t : ℝ)) • surfaceAreaMeasure K +
+        ENNReal.ofReal (t : ℝ) • surfaceAreaMeasure L) A ≠ ⊤ := by
+    rw [Measure.add_apply]
+    simp only [Measure.smul_apply]
+    exact ENNReal.add_ne_top.mpr ⟨ENNReal.mul_ne_top ENNReal.ofReal_ne_top hKtop,
+      ENNReal.mul_ne_top ENNReal.ofReal_ne_top hLtop⟩
+  apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ A) hright).mp
+  rw [show convexBodyCombination t K L = M from rfl, hreal]
+  simp only [Measure.add_apply, Measure.smul_apply, smul_eq_mul]
+  rw [ENNReal.toReal_add
+    (ENNReal.mul_ne_top ENNReal.ofReal_ne_top hKtop)
+    (ENNReal.mul_ne_top ENNReal.ofReal_ne_top hLtop),
+    ENNReal.toReal_mul, ENNReal.toReal_mul]
+  rw [ENNReal.toReal_ofReal (sub_nonneg.mpr t.2.2), ENNReal.toReal_ofReal t.2.1]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Angular densities of the opposite surface measure
+
+The opposite surface measure `(oppositeSurfaceData K).1` is the surface-area measure of `K`
+translated by `π`, so it reads the *negative* vertex data of `K` at the angle `s` as the positive
+vertex data at `π + s`.  Transporting `surfaceAreaMeasure_angleImage_eq_setLIntegral` along that
+translation identifies it with a Lebesgue density whenever the positive vertex of `K` at normal
+`π + s` is traced by a differentiable curve `F` with derivative `-(g s) • tangentVector s`
+(`oppositeSurfaceData_angleImage_eq_withDensity`); the extra minus sign is exactly
+`tangentVector_add_pi`.
+
+The two variants `oppositeSurfaceData_angleImage_eq_withDensity_of_openLeft` and
+`…_of_openRight` drop the vertex information at one endpoint of the window by exhausting the
+window from the other side with `measure_angleImage_eq_of_iUnion`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- An Archimedean step used to exhaust an open interval endpoint. -/
+private theorem exists_nat_div_add_two_lt {d e : ℝ} (he : 0 < e) :
+    ∃ n : ℕ, d / (n + 2) < e := by
+  obtain ⟨n, hn⟩ := exists_nat_gt (d / e)
+  refine ⟨n, ?_⟩
+  have hpos : (0 : ℝ) < n + 2 := by positivity
+  rw [div_lt_iff₀ hpos]
+  have h : d / e < n + 2 := by linarith
+  rw [div_lt_iff₀ he] at h
+  linarith
+
+/-- The angular projection reads the opposite surface measure through the `π`-shifted window. -/
+theorem oppositeSurfaceData_angleImage (K : ConvexBody Point) {S : Set ℝ}
+    (hS : MeasurableSet ((fun s : ℝ ↦ (s : Real.Angle)) '' S)) :
+    (oppositeSurfaceData K).1 ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+      surfaceAreaMeasure K
+        ((fun s : ℝ ↦ (s : Real.Angle)) '' ((fun s : ℝ ↦ s + Real.pi) '' S)) := by
+  have hm : Measurable fun t : Real.Angle ↦ t - ((Real.pi : ℝ) : Real.Angle) :=
+    (continuous_id.sub continuous_const).measurable
+  have hpre : (fun t : Real.Angle ↦ t - ((Real.pi : ℝ) : Real.Angle)) ⁻¹'
+      ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+      (fun s : ℝ ↦ (s : Real.Angle)) '' ((fun s : ℝ ↦ s + Real.pi) '' S) := by
+    ext x
+    simp only [Set.mem_preimage, Set.mem_image]
+    constructor
+    · rintro ⟨s, hs, hsx⟩
+      refine ⟨s + Real.pi, ⟨s, hs, rfl⟩, ?_⟩
+      rw [Real.Angle.coe_add, hsx]
+      abel
+    · rintro ⟨r, ⟨s, hs, rfl⟩, rfl⟩
+      refine ⟨s, hs, ?_⟩
+      rw [Real.Angle.coe_add]
+      abel
+  change Measure.map (fun t : Real.Angle ↦ t - ((Real.pi : ℝ) : Real.Angle))
+    (surfaceAreaMeasure K) _ = _
+  rw [Measure.map_apply hm hS, hpre]
+
+/-- Suppose that on the angular window `(π + a, π + b]`, of at most one turn, the positive vertex
+of `K` at normal `π + s` is traced by a curve `F` with derivative `-(g s) • v_s`, where `g` is
+continuous and nonnegative.  Then the opposite surface measure of the angular image of a
+measurable `S ⊆ (a, b]` is the weighted Lebesgue measure of `S` for any weight `f` agreeing with
+`g` on the open window. -/
+theorem oppositeSurfaceData_angleImage_eq_withDensity (K : ConvexBody Point) {a b : ℝ}
+    (hab : a < b) (hturn : b ≤ a + 2 * Real.pi) (F : ℝ → Point) (f g : ℝ → ℝ)
+    (hF : ∀ s, HasDerivAt F (-(g s) • tangentVector (s : Real.Angle)) s) (hg : Continuous g)
+    (hgnn : ∀ s ∈ Set.Ioc a b, 0 ≤ g s) (hfg : ∀ s ∈ Set.Ioo a b, f s = g s)
+    (hvertex : ∀ s ∈ Set.Icc a b, (edgeVertices K ((Real.pi + s : ℝ) : Real.Angle)).1 = F s)
+    {S : Set ℝ} (hS : MeasurableSet S) (hSsub : S ⊆ Set.Ioc a b) :
+    (oppositeSurfaceData K).1 ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+      volume.withDensity (fun s ↦ ENNReal.ofReal (f s)) S := by
+  have hemb : MeasurableEmbedding fun s : ℝ ↦ s + Real.pi :=
+    (MeasurableEquiv.addRight Real.pi).measurableEmbedding
+  have himg : MeasurableSet ((fun s : ℝ ↦ (s : Real.Angle)) '' S) :=
+    Real.Angle.measurableSet_image_of_subset_Ioc hturn hS hSsub
+  have hSmeas : MeasurableSet ((fun s : ℝ ↦ s + Real.pi) '' S) := hemb.measurableSet_image' hS
+  have hSsub' : (fun s : ℝ ↦ s + Real.pi) '' S ⊆ Set.Ioc (a + Real.pi) (b + Real.pi) := by
+    rintro r ⟨s, hs, rfl⟩
+    exact ⟨by linarith [(hSsub hs).1], by linarith [(hSsub hs).2]⟩
+  -- the vertex curve of the shifted window and its derivative
+  have hshift : ∀ r : ℝ, tangentVector ((r - Real.pi : ℝ) : Real.Angle) =
+      -tangentVector (r : Real.Angle) := by
+    intro r
+    have h := tangentVector_add_pi (r - Real.pi)
+    rw [show r - Real.pi + Real.pi = r from by ring] at h
+    rw [h, neg_neg]
+  have hF' : ∀ r : ℝ, HasDerivAt (fun r : ℝ ↦ F (r - Real.pi))
+      (g (r - Real.pi) • tangentVector (r : Real.Angle)) r := by
+    intro r
+    have h := (hF (r - Real.pi)).scomp r ((hasDerivAt_id r).sub_const Real.pi)
+    rw [hshift r] at h
+    refine h.congr_deriv ?_
+    module
+  have hvertex' : ∀ r ∈ Set.Icc (a + Real.pi) (b + Real.pi),
+      (edgeVertices K (r : Real.Angle)).1 = F (r - Real.pi) := by
+    intro r hr
+    have h := hvertex (r - Real.pi) ⟨by linarith [hr.1], by linarith [hr.2]⟩
+    rw [show Real.pi + (r - Real.pi) = r from by ring] at h
+    exact h
+  have hmain := surfaceAreaMeasure_angleImage_eq_setLIntegral K (a := a + Real.pi)
+    (b := b + Real.pi) (by linarith) (by linarith) (fun r ↦ F (r - Real.pi))
+    (fun r ↦ g (r - Real.pi)) hF' (hg.comp (continuous_id.sub continuous_const))
+    (fun r hr ↦ hgnn (r - Real.pi) ⟨by linarith [hr.1], by linarith [hr.2]⟩) hvertex'
+    hSmeas hSsub'
+  -- change variables back to the unshifted window
+  rw [oppositeSurfaceData_angleImage K himg, hmain,
+    setLIntegral_comp_sub_right (fun s ↦ ENNReal.ofReal (g s)) Real.pi,
+    hemb.injective.preimage_image, withDensity_apply _ hS]
+  -- the two densities agree off the right endpoint
+  refine (lintegral_congr_ae ?_).symm
+  have hne : ∀ᵐ s ∂volume.restrict S, s ∈ Set.Ioo a b := by
+    refine ae_restrict_mem_of_countable_diff hS (Set.countable_singleton b) ?_
+    rintro s ⟨hs, hsA⟩
+    have h := hSsub hs
+    rcases eq_or_lt_of_le h.2 with heq | hlt
+    · exact heq
+    · exact absurd (Set.mem_Ioo.2 ⟨h.1, hlt⟩) hsA
+  filter_upwards [hne] with s hs
+  rw [hfg s hs]
+
+/-- The variant of `oppositeSurfaceData_angleImage_eq_withDensity` whose left endpoint carries no
+vertex information: the window is exhausted from the right. -/
+theorem oppositeSurfaceData_angleImage_eq_withDensity_of_openLeft (K : ConvexBody Point)
+    {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi) (F : ℝ → Point)
+    (f g : ℝ → ℝ)
+    (hF : ∀ s, HasDerivAt F (-(g s) • tangentVector (s : Real.Angle)) s) (hg : Continuous g)
+    (hgnn : ∀ s ∈ Set.Ioc a b, 0 ≤ g s) (hfg : ∀ s ∈ Set.Ioo a b, f s = g s)
+    (hvertex : ∀ s ∈ Set.Ioc a b, (edgeVertices K ((Real.pi + s : ℝ) : Real.Angle)).1 = F s)
+    {S : Set ℝ} (hS : MeasurableSet S) (hSsub : S ⊆ Set.Ioc a b) :
+    (oppositeSurfaceData K).1 ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+      volume.withDensity (fun s ↦ ENNReal.ofReal (f s)) S := by
+  have hba : 0 < b - a := by linarith
+  have hpos : ∀ n : ℕ, 0 < (b - a) / (n + 2) := fun n ↦ by positivity
+  have hlt : ∀ n : ℕ, a + (b - a) / (n + 2) < b := by
+    intro n
+    have hn : (0 : ℝ) ≤ n := Nat.cast_nonneg n
+    have h1 : (b - a) / ((n : ℝ) + 2) ≤ (b - a) / 2 := by gcongr; linarith
+    linarith
+  refine measure_angleImage_eq_of_iUnion (μ := (oppositeSurfaceData K).1)
+    (J := fun n : ℕ ↦ Set.Ioc (a + (b - a) / (n + 2)) b)
+    (ν := volume.withDensity (fun s ↦ ENNReal.ofReal (f s))) ?_ (fun n ↦ measurableSet_Ioc) ?_
+    S hS ?_
+  · intro m n hmn
+    refine Set.Ioc_subset_Ioc ?_ le_rfl
+    have hmn' : ((m : ℝ)) ≤ n := Nat.cast_le.2 hmn
+    have hm : (0 : ℝ) ≤ m := Nat.cast_nonneg m
+    have h1 : (b - a) / ((n : ℝ) + 2) ≤ (b - a) / ((m : ℝ) + 2) := by gcongr
+    linarith
+  · intro n T hT hTsub
+    refine oppositeSurfaceData_angleImage_eq_withDensity K (hlt n) (by linarith [hpos n])
+      F f g hF hg (fun s hs ↦ hgnn s ⟨by linarith [hs.1, hpos n], hs.2⟩)
+      (fun s hs ↦ hfg s ⟨by linarith [hs.1, hpos n], hs.2⟩)
+      (fun s hs ↦ hvertex s ⟨by linarith [hs.1, hpos n], hs.2⟩) hT hTsub
+  · intro s hs
+    obtain ⟨n, hn⟩ := exists_nat_div_add_two_lt (d := b - a) (e := s - a)
+      (by linarith [(hSsub hs).1])
+    exact Set.mem_iUnion.2 ⟨n, ⟨by linarith, (hSsub hs).2⟩⟩
+
+/-- The variant of `oppositeSurfaceData_angleImage_eq_withDensity` whose right endpoint carries no
+vertex information: the window is exhausted from the left. -/
+theorem oppositeSurfaceData_angleImage_eq_withDensity_of_openRight (K : ConvexBody Point)
+    {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi) (F : ℝ → Point)
+    (f g : ℝ → ℝ)
+    (hF : ∀ s, HasDerivAt F (-(g s) • tangentVector (s : Real.Angle)) s) (hg : Continuous g)
+    (hgnn : ∀ s ∈ Set.Ioo a b, 0 ≤ g s) (hfg : ∀ s ∈ Set.Ioo a b, f s = g s)
+    (hvertex : ∀ s ∈ Set.Ico a b, (edgeVertices K ((Real.pi + s : ℝ) : Real.Angle)).1 = F s)
+    {S : Set ℝ} (hS : MeasurableSet S) (hSsub : S ⊆ Set.Ioo a b) :
+    (oppositeSurfaceData K).1 ((fun s : ℝ ↦ (s : Real.Angle)) '' S) =
+      volume.withDensity (fun s ↦ ENNReal.ofReal (f s)) S := by
+  have hba : 0 < b - a := by linarith
+  have hpos : ∀ n : ℕ, 0 < (b - a) / (n + 2) := fun n ↦ by positivity
+  have hlt : ∀ n : ℕ, a < b - (b - a) / (n + 2) := by
+    intro n
+    have hn : (0 : ℝ) ≤ n := Nat.cast_nonneg n
+    have h1 : (b - a) / ((n : ℝ) + 2) ≤ (b - a) / 2 := by gcongr; linarith
+    linarith
+  refine measure_angleImage_eq_of_iUnion (μ := (oppositeSurfaceData K).1)
+    (J := fun n : ℕ ↦ Set.Ioc a (b - (b - a) / (n + 2)))
+    (ν := volume.withDensity (fun s ↦ ENNReal.ofReal (f s))) ?_ (fun n ↦ measurableSet_Ioc) ?_
+    S hS ?_
+  · intro m n hmn
+    refine Set.Ioc_subset_Ioc le_rfl ?_
+    have hmn' : ((m : ℝ)) ≤ n := Nat.cast_le.2 hmn
+    have hm : (0 : ℝ) ≤ m := Nat.cast_nonneg m
+    have h1 : (b - a) / ((n : ℝ) + 2) ≤ (b - a) / ((m : ℝ) + 2) := by gcongr
+    linarith
+  · intro n T hT hTsub
+    refine oppositeSurfaceData_angleImage_eq_withDensity K (hlt n) (by linarith [hpos n])
+      F f g hF hg (fun s hs ↦ hgnn s ⟨hs.1, by linarith [hs.2, hpos n]⟩)
+      (fun s hs ↦ hfg s ⟨hs.1, by linarith [hs.2, hpos n]⟩)
+      (fun s hs ↦ hvertex s ⟨hs.1, by linarith [hs.2, hpos n]⟩) hT hTsub
+  · intro s hs
+    obtain ⟨n, hn⟩ := exists_nat_div_add_two_lt (d := b - a) (e := b - s)
+      (by linarith [(hSsub hs).2])
+    exact Set.mem_iUnion.2 ⟨n, ⟨(hSsub hs).1, by linarith⟩⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Convex.SupportArea`.
+* `Convex.Linearity`.
+* `Convex.MixedArea`.
+* `Convex.TangentLinePath`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The area of a convex body as a support-function surface integral
+
+The planar area of a nonempty compact convex set is one half of the integral of its support
+function against its surface area measure. The identity is proved for finite convex hulls by
+fanning the polygon into triangles over an interior base point, and then transported to an
+arbitrary body by polygon approximation and weak convergence of surface measures. The same
+integral, taken with the two bodies decoupled, is convex-bilinear.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval Pointwise Topology
+open MeasureTheory Filter
+
+namespace MovingSofa
+
+private theorem one_lt_finrank_point : 1 < Module.finrank ℝ Point := by
+  simp [Point]
+
+private theorem normalVector_ne_zero (t : Real.Angle) : normalVector t ≠ 0 := by
+  intro h
+  have hone := norm_normalVector t
+  rw [h, norm_zero] at hone
+  exact zero_ne_one hone
+
+/-- The ordered endpoints of an exposed edge differ by its length in the positive tangent
+direction. -/
+theorem edgeVertices_fst_sub_snd_eq_dist_smul_tangentVector
+    (K : ConvexBody Point) (t : Real.Angle) :
+    (edgeVertices K t).1 - (edgeVertices K t).2 =
+      dist (edgeVertices K t).1 (edgeVertices K t).2 • tangentVector t := by
+  let a := (edgeVertices K t).1
+  let b := (edgeVertices K t).2
+  let r := inner ℝ (a - b) (tangentVector t)
+  have hnormal : inner ℝ (a - b) (normalVector t) = 0 := by
+    dsimp only [a, b]
+    rw [inner_sub_left]
+    rw [(edgeVertices_fst_mem K t).2, (edgeVertices_snd_mem K t).2, sub_self]
+  have hdecomp := inner_normalVector_smul_add_inner_tangentVector_smul (a - b) t
+  have hvec : r • tangentVector t = a - b := by
+    simpa only [r, hnormal, zero_smul, zero_add] using hdecomp
+  have hr : 0 ≤ r := by
+    let S : Set ℝ :=
+      (fun p ↦ inner ℝ p (tangentVector t)) '' exposedEdge K t
+    simp only [r, inner_sub_left, a, b, inner_edgeVertices_fst_tangent,
+      inner_edgeVertices_snd_tangent]
+    have hcompact : IsCompact S :=
+      (isCompact_exposedEdge K t).image (continuous_id.inner continuous_const)
+    apply sub_nonneg.mpr
+    simpa only [S] using
+      (csInf_le_csSup (s := S) ((exposedEdge_nonempty K t).image _)
+        (hb := hcompact.bddBelow) (ha := hcompact.bddAbove))
+  have htangent : ‖tangentVector t‖ = 1 := by
+    induction t using Real.Angle.induction_on with
+    | _ t =>
+      rw [← sq_eq_sq₀ (norm_nonneg _) (by norm_num : (0 : ℝ) ≤ 1)]
+      rw [EuclideanSpace.norm_sq_eq]
+      simp [tangentVector, frame, Fin.sum_univ_two]
+  have hnorm := congrArg norm hvec
+  have hre : r = dist a b := by
+    simpa only [norm_smul, htangent, mul_one, Real.norm_eq_abs,
+      abs_of_nonneg hr, dist_eq_norm] using hnorm
+  simpa only [a, b, ← hre] using hvec.symm
+
+/-- The half support integral is convex-bilinear in the two body arguments. -/
+theorem supportIntegral_bilinear :
+    IsConvexBilinear convexBodyCombination convexBodyCombination realCombination
+      (fun K L : ConvexBody Point ↦ (1 / 2 : ℝ) * ∫ t,
+          supportValue K t ∂surfaceAreaMeasure L) := by
+  have hcont (K : ConvexBody Point) : Continuous (fun u : Real.Angle ↦ supportValue K u) :=
+    (compactSet_support_continuity K K K.nonempty K.isCompact K.nonempty K.isCompact).2.2.1
+  have hint (K L : ConvexBody Point) :
+      Integrable (fun u : Real.Angle ↦ supportValue K u) (surfaceAreaMeasure L) := by
+    let _ : IsFiniteMeasure (surfaceAreaMeasure L) := (surfaceAreaMeasure_face_union L).1
+    exact (hcont K).integrable_of_hasCompactSupport
+      (isCompact_univ.of_isClosed_subset isClosed_closure (Set.subset_univ _))
+  constructor
+  · intro K t L M
+    have hmeasure := surfaceAreaMeasure_convexBodyCombination t L M
+    change (1 / 2 : ℝ) * (∫ u, supportValue K u ∂surfaceAreaMeasure
+        (convexBodyCombination t L M)) = _
+    simp only [realCombination]
+    rw [hmeasure, integral_add_measure
+      ((hint K L).smul_measure ENNReal.ofReal_ne_top)
+      ((hint K M).smul_measure ENNReal.ofReal_ne_top)]
+    simp only [integral_smul_measure, ENNReal.toReal_ofReal,
+      sub_nonneg.mpr (show (t : ℝ) ≤ 1 from t.property.2), t.property.1]
+    ring
+  · intro L t K M
+    change (1 / 2 : ℝ) * (∫ u, supportValue (convexBodyCombination t K M) u
+      ∂surfaceAreaMeasure L) = _
+    simp only [realCombination]
+    have hfun : (fun u : Real.Angle ↦ supportValue (convexBodyCombination t K M) u) =
+        fun u ↦ (1 - (t : ℝ)) * supportValue K u + (t : ℝ) * supportValue M u := by
+      funext u
+      exact supportValue_convexBodyCombination t K M u
+    rw [hfun, integral_add ((hint K L).const_mul _) ((hint M L).const_mul _),
+      integral_const_mul, integral_const_mul]
+    ring
+
+/-- The mixed support integral is Hausdorff continuous in the two bodies simultaneously. -/
+theorem tendsto_integral_supportValue_of_hausdorff_pair
+    (P Q : ℕ → ConvexBody Point) (K L : ConvexBody Point)
+    (hPK : Tendsto (fun n ↦ Metric.hausdorffDist (P n : Set Point) (K : Set Point))
+      atTop (𝓝 0))
+    (hQL : Tendsto (fun n ↦ Metric.hausdorffDist (Q n : Set Point) (L : Set Point))
+      atTop (𝓝 0)) :
+    Tendsto (fun n ↦ ∫ u, supportValue (P n) u ∂surfaceAreaMeasure (Q n)) atTop
+      (𝓝 (∫ u, supportValue K u ∂surfaceAreaMeasure L)) := by
+  have hcont (M : ConvexBody Point) : Continuous (supportValue M) :=
+    (compactSet_support_continuity M M M.nonempty M.isCompact M.nonempty M.isCompact).2.2.1
+  have hint (M N : ConvexBody Point) : Integrable (supportValue M) (surfaceAreaMeasure N) := by
+    let _ : IsFiniteMeasure (surfaceAreaMeasure N) := (surfaceAreaMeasure_face_union N).1
+    exact (hcont M).integrable_of_hasCompactSupport
+      (isCompact_univ.of_isClosed_subset isClosed_closure (Set.subset_univ _))
+  have hmass : Tendsto (fun n ↦ (surfaceAreaMeasure (Q n)).real Set.univ) atTop
+      (𝓝 ((surfaceAreaMeasure L).real Set.univ)) := by
+    simpa using surfaceAreaMeasure_weak_continuity Q L hQL (fun _ ↦ (1 : ℝ)) continuous_const
+  have hbound (n : ℕ) :
+      ‖∫ u, supportValue (P n) u - supportValue K u ∂surfaceAreaMeasure (Q n)‖ ≤
+        Metric.hausdorffDist (P n : Set Point) (K : Set Point) *
+          (surfaceAreaMeasure (Q n)).real Set.univ := by
+    let _ : IsFiniteMeasure (surfaceAreaMeasure (Q n)) :=
+      (surfaceAreaMeasure_face_union (Q n)).1
+    apply norm_integral_le_of_norm_le_const
+    filter_upwards [] with u
+    simpa only [Real.norm_eq_abs, vectorSupport, supportValue] using
+      (compactSet_support_continuity (P n) K (P n).nonempty
+        (P n).isCompact K.nonempty K.isCompact).2.1 (normalVector u) (norm_normalVector u)
+  have herror : Tendsto
+      (fun n ↦ ∫ u, supportValue (P n) u - supportValue K u ∂surfaceAreaMeasure (Q n))
+      atTop (𝓝 0) := by
+    apply tendsto_zero_iff_norm_tendsto_zero.mpr
+    exact squeeze_zero (fun _ ↦ norm_nonneg _) hbound (by simpa using hPK.mul hmass)
+  have hfixed := surfaceAreaMeasure_weak_continuity Q L hQL (supportValue K) (hcont K)
+  have hsum := herror.add hfixed
+  simp only [zero_add] at hsum
+  convert hsum using 1
+  funext n
+  rw [integral_sub (hint (P n) (Q n)) (hint K (Q n)), sub_add_cancel]
+
+/-- The support integral of a body against its own surface measure is Hausdorff continuous. -/
+theorem tendsto_integral_supportValue_of_hausdorff
+    (K : ℕ → ConvexBody Point) (L : ConvexBody Point)
+    (hlim : Tendsto (fun n ↦ Metric.hausdorffDist (K n : Set Point) (L : Set Point))
+      atTop (𝓝 0)) :
+    Tendsto (fun n ↦ ∫ u, supportValue (K n) u ∂surfaceAreaMeasure (K n)) atTop
+      (𝓝 (∫ u, supportValue L u ∂surfaceAreaMeasure L)) :=
+  tendsto_integral_supportValue_of_hausdorff_pair K K L L hlim hlim
+
+/-- For a finite convex hull, the support integral is the finite sum over its proper edges. -/
+private theorem integral_supportValue_surfaceAreaMeasure_eq_sum_properEdgeNormals
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point)) :
+    ∫ t, supportValue K t ∂surfaceAreaMeasure K =
+      Finset.sum (finite_properEdgeNormal_angles K V hKV).toFinset
+        (fun t ↦ (surfaceAreaMeasure K).real {t} * supportValue K t) := by
+  classical
+  let N : Set Real.Angle := {t | (edgeVertices K t).1 ≠ (edgeVertices K t).2}
+  have hNfinite : N.Finite := finite_properEdgeNormal_angles K V hKV
+  have hae : ∀ᵐ t ∂surfaceAreaMeasure K, t ∈ N := by
+    rw [ae_iff]
+    simpa only [N, Set.mem_ofPred_eq, not_ne_iff] using
+      surfaceAreaMeasure_compl_properEdgeNormals_eq_zero_of_eq_convexHull K V hKV
+  have hrestrict : (surfaceAreaMeasure K).restrict N = surfaceAreaMeasure K :=
+    Measure.restrict_eq_self_of_ae_mem hae
+  let _ : IsFiniteMeasure (surfaceAreaMeasure K) := (surfaceAreaMeasure_face_union K).1
+  calc
+    ∫ t, supportValue K t ∂surfaceAreaMeasure K =
+        ∫ t in N, supportValue K t ∂surfaceAreaMeasure K := by
+          rw [hrestrict]
+    _ = _ := by
+      rw [show N = ↑hNfinite.toFinset by ext t; simp [N]]
+      exact MeasureTheory.setIntegral_finset _
+        (μ := surfaceAreaMeasure K) (f := supportValue K) IntegrableOn.finset
+
+/-- On a finite convex hull, each coefficient in the support sum is the corresponding edge
+length. -/
+theorem integral_supportValue_surfaceAreaMeasure_eq_sum_edgeLength
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point)) :
+    ∫ t, supportValue K t ∂surfaceAreaMeasure K =
+      Finset.sum (finite_properEdgeNormal_angles K V hKV).toFinset
+        (fun t ↦ dist (edgeVertices K t).1 (edgeVertices K t).2 * supportValue K t) := by
+  rw [integral_supportValue_surfaceAreaMeasure_eq_sum_properEdgeNormals K V hKV]
+  apply Finset.sum_congr rfl
+  intro t _
+  change (surfaceAreaMeasure K {t}).toReal * supportValue K t = _
+  rw [(surfaceAreaMeasure_atom_length K t).2.1, ENNReal.toReal_ofReal dist_nonneg]
+
+/-- The total tangent vector of the surface-area measure vanishes. -/
+theorem integral_tangentVector_surfaceAreaMeasure_eq_zero (K : ConvexBody Point) :
+    ∫ t, tangentVector t ∂surfaceAreaMeasure K = 0 := by
+  have himage :
+      (fun t : ℝ ↦ (t : Real.Angle)) '' Set.Ioc 0 (2 * Real.pi) = Set.univ := by
+    ext u
+    simp only [Set.mem_image, Set.mem_Ioc, Set.mem_univ, iff_true]
+    let _ : Fact (0 < 2 * Real.pi) := ⟨Real.two_pi_pos⟩
+    let s := AddCircle.equivIoc (2 * Real.pi) 0 u
+    have hs : (s : ℝ) ∈ Set.Ioc (0 : ℝ) (2 * Real.pi) := by
+      simpa using s.property
+    exact ⟨s, hs, AddCircle.coe_equivIoc⟩
+  rw [← setIntegral_univ, ← himage,
+    integral_tangentVector_surfaceAreaMeasure K Real.two_pi_pos (by simp)]
+  simp
+
+/-- Every non-generating boundary point of a two-dimensional finite convex hull lies on a
+proper exposed edge. -/
+theorem exists_mem_properExposedEdge_of_mem_frontier
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    (hK : (interior (K : Set Point)).Nonempty) {p : Point}
+    (hp : p ∈ frontier (K : Set Point)) (hpV : p ∉ V) :
+    ∃ t : Real.Angle,
+      (edgeVertices K t).1 ≠ (edgeVertices K t).2 ∧ p ∈ exposedEdge K t := by
+  obtain ⟨t, hpt⟩ := exists_mem_exposedEdge_of_mem_frontier K hK hp
+  refine ⟨t, ?_, hpt⟩
+  intro heq
+  rw [exposedEdge_eq_segment_edgeVertices, heq, segment_same] at hpt
+  apply hpV
+  simpa only [Set.mem_singleton_iff] using
+    hpt.symm ▸ (edgeVertices_mem_of_eq_convexHull K V hKV t).2
+
+/-- The support-area identity holds for a singleton convex body. -/
+theorem convexBody_area_support_integral_of_subsingleton (K : ConvexBody Point)
+    (hK : (K : Set Point).Subsingleton) :
+    ClassicalResults.area (K : Set Point) =
+      (1 / 2 : ℝ) * ∫ t, supportValue K t ∂surfaceAreaMeasure K := by
+  obtain ⟨p, hp⟩ := K.nonempty
+  have hsingleton : (K : Set Point) = {p} := hK.eq_singleton_of_mem hp
+  rw [surfaceAreaMeasure_eq_zero_of_subsingleton K hK, integral_zero_measure, mul_zero]
+  simp [ClassicalResults.area, hsingleton]
+
+/-- The support-area identity holds for a nondegenerate segment presentation. -/
+theorem convexBody_area_support_integral_of_segmentPresentation
+    (K : ConvexBody Point) (d : Point × Point × Real.Angle)
+    (hd : IsSegmentPresentation K d) :
+    ClassicalResults.area (K : Set Point) =
+      (1 / 2 : ℝ) * ∫ t, supportValue K t ∂surfaceAreaMeasure K := by
+  have hline : (K : Set Point) ⊆
+      {p | inner ℝ p (normalVector d.2.2) = inner ℝ d.1 (normalVector d.2.2)} := by
+    intro p hp
+    rw [hd.2.1, segment_eq_image'] at hp
+    obtain ⟨s, _, rfl⟩ := hp
+    change inner ℝ (d.1 + s • (d.2.1 - d.1)) (normalVector d.2.2) = _
+    rw [inner_add_left, inner_smul_left, hd.2.2, mul_zero, add_zero]
+  have hvolume : volume (K : Set Point) = 0 :=
+    measure_mono_null hline
+      (volume.addHaar_setOf_real_inner_eq (normalVector_ne_zero d.2.2) _)
+  have hnormalPi : normalVector (d.2.2 + (Real.pi : Real.Angle)) =
+      -normalVector d.2.2 := by
+    induction d.2.2 using Real.Angle.induction_on with
+    | _ t => simpa only [← Real.Angle.coe_add] using normalVector_add_pi t
+  have hdmem : d.1 ∈ exposedEdge K d.2.2 := by
+    rw [exposedEdge_eq_segment_of_orthogonal K d hd d.2.2 hd.2.2, hd.2.1]
+    exact left_mem_segment ℝ _ _
+  have hpimem : d.1 ∈ exposedEdge K (d.2.2 + (Real.pi : Real.Angle)) := by
+    rw [exposedEdge_eq_segment_of_orthogonal K d hd _ (by
+      rw [hnormalPi, inner_neg_right, hd.2.2, neg_zero]), hd.2.1]
+    exact left_mem_segment ℝ _ _
+  have hsupport : supportValue K d.2.2 +
+      supportValue K (d.2.2 + (Real.pi : Real.Angle)) = 0 := by
+    rw [← hdmem.2, ← hpimem.2, hnormalPi, inner_neg_right, add_neg_cancel]
+  have hint₁ : Integrable (supportValue K) (Measure.dirac d.2.2) :=
+    integrable_dirac (by simp)
+  have hint₂ : Integrable (supportValue K)
+      (Measure.dirac (d.2.2 + (Real.pi : Real.Angle))) := integrable_dirac (by simp)
+  rw [show ClassicalResults.area (K : Set Point) = 0 by
+    simp [ClassicalResults.area, hvolume],
+    surfaceAreaMeasure_eq_segmentPresentation K d hd]
+  rw [integral_smul_measure, integral_add_measure hint₁ hint₂]
+  simp [hsupport]
+
+/-- The support-area identity for finite convex hulls extends to every convex body. -/
+theorem convexBody_area_support_integral_of_finiteHull
+    (hpolygon : ∀ (P : ConvexBody Point) (V : Finset Point), V.Nonempty →
+      (P : Set Point) = convexHull ℝ (V : Set Point) →
+      ClassicalResults.area (P : Set Point) =
+        (1 / 2 : ℝ) * ∫ t, supportValue P t ∂surfaceAreaMeasure P) :
+    ∀ K : ConvexBody Point, ClassicalResults.area (K : Set Point) =
+      (1 / 2 : ℝ) * ∫ t, supportValue K t ∂surfaceAreaMeasure K := by
+  intro K
+  obtain ⟨V, P, hP, hdist⟩ := exists_facePreserving_polygonApproximation K ∅
+  have hlim : Tendsto (fun n ↦ Metric.hausdorffDist (P n : Set Point) (K : Set Point))
+      atTop (𝓝 0) := by
+    apply squeeze_zero' (Filter.Eventually.of_forall fun _ ↦ Metric.hausdorffDist_nonneg)
+      (Filter.eventually_atTop.2 ⟨1, fun n hn ↦ hdist n hn⟩)
+    exact tendsto_one_div_atTop_nhds_zero_nat
+  have harea := convexArea_hausdorff_continuity P K hlim
+  have hintegral := (tendsto_integral_supportValue_of_hausdorff P K hlim).const_mul (1 / 2)
+  have heq (n : ℕ) : ClassicalResults.area (P n : Set Point) =
+      (1 / 2 : ℝ) * ∫ t, supportValue (P n) t ∂surfaceAreaMeasure (P n) :=
+    hpolygon (P n) (V n) (hP n).1 (hP n).2.1
+  exact tendsto_nhds_unique harea
+    (hintegral.congr' (Filter.Eventually.of_forall fun n ↦ (heq n).symm))
+
+/-- The area identity and the explicit support integral's bilinearity give quadraticity. -/
+theorem convexBody_area_support_integral_of_area_identity
+    (harea : ∀ K : ConvexBody Point, ClassicalResults.area (K : Set Point) =
+      (1 / 2 : ℝ) * ∫ t, supportValue K t ∂surfaceAreaMeasure K) :
+    (∀ K : ConvexBody Point, ClassicalResults.area (K : Set Point) =
+      (1 / 2 : ℝ) * ∫ t, supportValue K t ∂surfaceAreaMeasure K) ∧
+    IsConvexBilinear convexBodyCombination convexBodyCombination realCombination
+      (fun K L : ConvexBody Point ↦
+        (1 / 2 : ℝ) * ∫ t, supportValue K t ∂surfaceAreaMeasure L) ∧
+    IsQuadraticFunctional convexBodyCombination
+      (fun K : ConvexBody Point ↦ ClassicalResults.area (K : Set Point)) := by
+  refine ⟨harea, supportIntegral_bilinear, ?_⟩
+  exact ⟨_, supportIntegral_bilinear, harea⟩
+
+/-- An exposed-edge point realizes the support value. -/
+theorem inner_eq_supportValue_of_mem_exposedEdge (K : ConvexBody Point) (t : Real.Angle)
+    {p : Point} (hp : p ∈ exposedEdge K t) :
+    inner ℝ p (normalVector t) = supportValue K t := hp.2
+
+/-- An interior base point lies strictly inside every supporting line. -/
+theorem inner_lt_supportValue_of_mem_interior (K : ConvexBody Point) {o : Point}
+    (ho : o ∈ interior (K : Set Point)) (t : Real.Angle) :
+    inner ℝ o (normalVector t) < supportValue K t := by
+  obtain ⟨ε, hε, hball⟩ := Metric.isOpen_iff.mp isOpen_interior o ho
+  have hnorm : inner ℝ (normalVector t) (normalVector t) = 1 := by
+    rw [real_inner_self_eq_norm_sq, norm_normalVector]
+    norm_num
+  have hpball : o + (ε / 2) • normalVector t ∈ Metric.ball o ε := by
+    simp only [Metric.mem_ball, dist_eq_norm, add_sub_cancel_left, norm_smul,
+      norm_normalVector, mul_one, Real.norm_eq_abs, abs_of_pos (half_pos hε)]
+    linarith
+  have hle := inner_le_supportValue K (interior_subset (hball hpball)) t
+  rw [inner_add_left, real_inner_smul_left, hnorm, mul_one] at hle
+  linarith
+
+/-- The oriented determinant of a triangle with one side in the positive tangent direction. -/
+private theorem planeCrossProduct_add_smul_tangentVector (w : Point) (d : ℝ)
+    (t : Real.Angle) :
+    planeCrossProduct w (w + d • tangentVector t) = d * inner ℝ w (normalVector t) := by
+  induction t using Real.Angle.induction_on with
+  | _ t =>
+    simp [planeCrossProduct, PiLp.add_apply, PiLp.smul_apply, smul_eq_mul,
+      PiLp.inner_apply, Fin.sum_univ_two, normalVector, tangentVector, frame,
+      Real.Angle.cos_coe, Real.Angle.sin_coe]
+    ring
+
+/-- Each exposed-edge summand is the oriented determinant of its triangle over the base
+point. -/
+theorem edgeLength_mul_supportValue_sub_eq_planeCrossProduct
+    (K : ConvexBody Point) (o : Point) (t : Real.Angle) :
+    dist (edgeVertices K t).1 (edgeVertices K t).2 *
+        (supportValue K t - inner ℝ o (normalVector t)) =
+      planeCrossProduct ((edgeVertices K t).2 - o) ((edgeVertices K t).1 - o) := by
+  have hedge := edgeVertices_fst_sub_snd_eq_dist_smul_tangentVector K t
+  have ha : (edgeVertices K t).1 - o =
+      ((edgeVertices K t).2 - o) +
+        dist (edgeVertices K t).1 (edgeVertices K t).2 • tangentVector t := by
+    rw [← hedge]
+    abel
+  rw [ha, planeCrossProduct_add_smul_tangentVector, inner_sub_left,
+    inner_eq_supportValue_of_mem_exposedEdge K t (edgeVertices_snd_mem K t)]
+
+/-- The fan triangle over an exposed edge, based at a chosen interior point. -/
+private def edgeTriangle (K : ConvexBody Point) (o : Point) (t : Real.Angle) : Set Point :=
+  convexHull ℝ ({o, (edgeVertices K t).2, (edgeVertices K t).1} : Set Point)
+
+/-- The finitely many rays from a base point through the generating vertices. -/
+private def vertexRayUnion (o : Point) (V : Finset Point) : Set Point :=
+  ⋃ v : ↥V, (o +ᵥ (ℝ ∙ ((v : Point) - o) : Set Point))
+
+/-- The vertex rays through a base point have planar area zero. -/
+private theorem volume_vertexRayUnion (o : Point) (V : Finset Point) :
+    volume (vertexRayUnion o V) = 0 := by
+  exact volume.addHaar_iUnion_vadd_span_singleton one_lt_finrank_point o _
+
+/-- Away from the finitely many vertex rays, every polygon point lies in a proper-edge fan
+triangle over the base point. -/
+private theorem exists_mem_edgeTriangle_of_notMem_vertexRayUnion
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    {o : Point} (ho : o ∈ interior (K : Set Point)) {x : Point}
+    (hx : x ∈ (K : Set Point)) (hxline : x ∉ vertexRayUnion o V) :
+    ∃ t : Real.Angle,
+      (edgeVertices K t).1 ≠ (edgeVertices K t).2 ∧ x ∈ edgeTriangle K o t := by
+  obtain ⟨p, hpfront, hxp⟩ := K.convex.exists_mem_frontier_mem_segment K.isCompact ho hx
+  have hpV : p ∉ V := by
+    intro hpV
+    apply hxline
+    refine Set.mem_iUnion.2 ⟨⟨p, hpV⟩, ?_⟩
+    rw [segment_eq_image'] at hxp
+    obtain ⟨c, _, hc⟩ := hxp
+    have hc' : o + c • (p - o) = x := hc
+    refine ⟨x - o, ?_, by simp only [vadd_eq_add]; abel⟩
+    exact Submodule.mem_span_singleton.2 ⟨c, by rw [← hc']; abel⟩
+  obtain ⟨t, htproper, hpt⟩ :=
+    exists_mem_properExposedEdge_of_mem_frontier K V hKV ⟨o, ho⟩ hpfront hpV
+  refine ⟨t, htproper, ?_⟩
+  have hpT : p ∈ edgeTriangle K o t := by
+    rw [exposedEdge_eq_segment_edgeVertices] at hpt
+    exact (convex_convexHull ℝ _).segment_subset (subset_convexHull ℝ _ (by simp))
+      (subset_convexHull ℝ _ (by simp)) hpt
+  exact (convex_convexHull ℝ _).segment_subset
+    (subset_convexHull ℝ _ (by simp)) hpT hxp
+
+/-- Two unit normals annihilating a common nonzero vector have vanishing angular sine. -/
+private theorem sin_sub_eq_zero_of_inner_normalVector_eq_zero {u : Point} (hu : u ≠ 0)
+    {s t : Real.Angle} (h1 : inner ℝ u (normalVector s) = 0)
+    (h2 : inner ℝ u (normalVector t) = 0) : (t - s).sin = 0 := by
+  induction s using Real.Angle.induction_on with
+  | _ s =>
+    induction t using Real.Angle.induction_on with
+    | _ t =>
+      have hu' : ¬(u 0 = 0 ∧ u 1 = 0) := by
+        rintro ⟨ha, hb⟩
+        refine hu ?_
+        ext i
+        fin_cases i <;> simpa
+      simp only [PiLp.inner_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two,
+        normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe,
+        Matrix.cons_val_zero, Matrix.cons_val_one] at h1 h2
+      rw [← Real.Angle.coe_sub, Real.Angle.sin_coe, Real.sin_sub]
+      have hd0 : (Real.sin t * Real.cos s - Real.cos t * Real.sin s) * u 0 = 0 := by
+        linear_combination Real.sin t * h1 - Real.sin s * h2
+      have hd1 : (Real.sin t * Real.cos s - Real.cos t * Real.sin s) * u 1 = 0 := by
+        linear_combination (-Real.cos t) * h1 + Real.cos s * h2
+      rcases mul_eq_zero.mp hd0 with hD | hu0
+      · exact hD
+      rcases mul_eq_zero.mp hd1 with hD | hu1
+      · exact hD
+      exact absurd ⟨hu0, hu1⟩ hu'
+
+/-- Distinct normal directions share at most one exposed-edge point. -/
+theorem subsingleton_exposedEdge_inter (K : ConvexBody Point) {o : Point}
+    (ho : o ∈ interior (K : Set Point)) {s t : Real.Angle} (hst : s ≠ t) :
+    (exposedEdge K s ∩ exposedEdge K t).Subsingleton := by
+  intro z hz w hw
+  by_contra hzw
+  have h1 : inner ℝ (z - w) (normalVector s) = 0 := by
+    rw [inner_sub_left, inner_eq_supportValue_of_mem_exposedEdge K s hz.1,
+      inner_eq_supportValue_of_mem_exposedEdge K s hw.1, sub_self]
+  have h2 : inner ℝ (z - w) (normalVector t) = 0 := by
+    rw [inner_sub_left, inner_eq_supportValue_of_mem_exposedEdge K t hz.2,
+      inner_eq_supportValue_of_mem_exposedEdge K t hw.2, sub_self]
+  have hsin := sin_sub_eq_zero_of_inner_normalVector_eq_zero (sub_ne_zero.mpr hzw) h1 h2
+  rcases Real.Angle.sin_eq_zero_iff.mp hsin with hz0 | hpi
+  · exact hst (by linear_combination (norm := abel) -hz0)
+  · have ht : t = s + ((Real.pi : ℝ) : Real.Angle) := by
+      linear_combination (norm := abel) hpi
+    have hopp : normalVector t = -normalVector s := by
+      rw [ht, normalVector_add_pi_angle]
+    have hsum : supportValue K s + supportValue K t = 0 := by
+      rw [← inner_eq_supportValue_of_mem_exposedEdge K s hz.1,
+        ← inner_eq_supportValue_of_mem_exposedEdge K t hz.2, hopp, inner_neg_right,
+        add_neg_cancel]
+    have hs' := inner_lt_supportValue_of_mem_interior K ho s
+    have ht' := inner_lt_supportValue_of_mem_interior K ho t
+    rw [hopp, inner_neg_right] at ht'
+    linarith
+
+/-- A fan triangle is the union of the segments from the base point to its exposed edge. -/
+private theorem edgeTriangle_eq_iUnion_segment (K : ConvexBody Point) (o : Point)
+    (t : Real.Angle) :
+    edgeTriangle K o t = ⋃ z ∈ exposedEdge K t, segment ℝ o z := by
+  rw [edgeTriangle, exposedEdge_eq_segment_edgeVertices,
+    show ({o, (edgeVertices K t).2, (edgeVertices K t).1} : Set Point) =
+      insert o {(edgeVertices K t).2, (edgeVertices K t).1} from rfl,
+    convexHull_insert ⟨(edgeVertices K t).2, by simp⟩, convexHull_pair,
+    convexJoin_singleton_left]
+
+/-- Fan triangles over distinct proper edges meet in a planar null set. -/
+private theorem volume_edgeTriangle_inter_eq_zero (K : ConvexBody Point) {o : Point}
+    (ho : o ∈ interior (K : Set Point)) {s t : Real.Angle} (hst : s ≠ t) :
+    volume (edgeTriangle K o s ∩ edgeTriangle K o t) = 0 := by
+  classical
+  have hsub : ∀ x ∈ edgeTriangle K o s ∩ edgeTriangle K o t,
+      x = o ∨ ∃ z ∈ exposedEdge K s ∩ exposedEdge K t, x ∈ segment ℝ o z := by
+    rintro x ⟨hxs, hxt⟩
+    rw [edgeTriangle_eq_iUnion_segment] at hxs hxt
+    obtain ⟨z, hz, hxz⟩ := Set.mem_iUnion₂.mp hxs
+    obtain ⟨w, hw, hxw⟩ := Set.mem_iUnion₂.mp hxt
+    by_cases hxo : x = o
+    · exact Or.inl hxo
+    have hzw : z = w := K.convex.eq_of_mem_frontier_of_mem_segment K.isClosed ho hxo
+      (exposedEdge_subset_frontier K s hz)
+      (exposedEdge_subset_frontier K t hw) hxz hxw
+    exact Or.inr ⟨z, ⟨hz, hzw ▸ hw⟩, hxz⟩
+  by_cases hne : (exposedEdge K s ∩ exposedEdge K t).Nonempty
+  · obtain ⟨q, hq⟩ := hne
+    refine measure_mono_null (t := (o +ᵥ (ℝ ∙ (q - o) : Set Point))) ?_ ?_
+    · intro x hx
+      rcases hsub x hx with hxo | ⟨z, hzq, hxz⟩
+      · exact ⟨0, Submodule.zero_mem _, by simp only [vadd_eq_add, hxo]; abel⟩
+      · have hzeq : z = q := subsingleton_exposedEdge_inter K ho hst hzq hq
+        subst hzeq
+        rw [segment_eq_image'] at hxz
+        obtain ⟨c, _, hc⟩ := hxz
+        have hc' : o + c • (z - o) = x := hc
+        exact ⟨x - o, Submodule.mem_span_singleton.2 ⟨c, by rw [← hc']; abel⟩,
+          by simp only [vadd_eq_add]; abel⟩
+    · exact volume.addHaar_vadd_span_singleton one_lt_finrank_point o (q - o)
+  · refine measure_mono_null (t := ({o} : Set Point)) ?_ (measure_singleton o)
+    intro x hx
+    rcases hsub x hx with hxo | ⟨z, hz, _⟩
+    · exact hxo
+    · exact absurd ⟨z, hz⟩ hne
+
+/-- The coordinate tangent integral of a surface measure vanishes. -/
+theorem integral_tangentCoordinate_surfaceAreaMeasure_eq_zero (K : ConvexBody Point)
+    (i : Fin 2) : ∫ t, tangentVector t i ∂surfaceAreaMeasure K = 0 := by
+  let _ : IsFiniteMeasure (surfaceAreaMeasure K) := (surfaceAreaMeasure_face_union K).1
+  have hcont : Continuous (fun u : Real.Angle ↦ tangentVector u) := by
+    apply (PiLp.continuous_toLp 2 (fun _ : Fin 2 ↦ ℝ)).comp
+    apply continuous_pi
+    intro j
+    fin_cases j
+    · exact Real.Angle.continuous_sin.neg
+    · exact Real.Angle.continuous_cos
+  have hu : Integrable (fun u : Real.Angle ↦ tangentVector u) (surfaceAreaMeasure K) :=
+    hcont.integrable_of_hasCompactSupport
+      (isCompact_univ.of_isClosed_subset isClosed_closure (Set.subset_univ _))
+  have hi := ContinuousLinearMap.integral_comp_comm
+    (PiLp.proj (𝕜 := ℝ) 2 (fun _ : Fin 2 ↦ ℝ) i) hu
+  change (∫ u, tangentVector u i ∂surfaceAreaMeasure K) =
+    (∫ u, tangentVector u ∂surfaceAreaMeasure K) i at hi
+  rw [hi, integral_tangentVector_surfaceAreaMeasure_eq_zero K]
+  simp
+
+/-- Atomic surface mass is the corresponding edge length. -/
+private theorem surfaceAreaMeasure_real_singleton (K : ConvexBody Point) (t : Real.Angle) :
+    (surfaceAreaMeasure K).real {t} = dist (edgeVertices K t).1 (edgeVertices K t).2 := by
+  change (surfaceAreaMeasure K {t}).toReal = _
+  rw [(surfaceAreaMeasure_atom_length K t).2.1, ENNReal.toReal_ofReal dist_nonneg]
+
+/-- The proper-edge lengths of a polygon weight its tangent coordinates to zero. -/
+theorem sum_edgeLength_mul_tangentCoordinate_eq_zero (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point)) (i : Fin 2) :
+    ∑ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+      dist (edgeVertices K t).1 (edgeVertices K t).2 * tangentVector t i = 0 := by
+  classical
+  have h := integral_tangentCoordinate_surfaceAreaMeasure_eq_sum_properEdgeNormals K V hKV i
+    Set.univ MeasurableSet.univ
+  rw [setIntegral_univ, integral_tangentCoordinate_surfaceAreaMeasure_eq_zero K i] at h
+  rw [show (finite_properEdgeNormal_angles K V hKV).toFinset =
+      ((finite_properEdgeNormal_angles K V hKV).inter_of_left Set.univ).toFinset from by
+    ext u
+    simp]
+  refine Eq.trans (Finset.sum_congr rfl fun u _ ↦ ?_) h.symm
+  rw [surfaceAreaMeasure_real_singleton]
+
+/-- The proper-edge lengths of a polygon weight its normal directions to zero. -/
+theorem sum_edgeLength_mul_inner_normalVector_eq_zero (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point)) (o : Point) :
+    ∑ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+      dist (edgeVertices K t).1 (edgeVertices K t).2 * inner ℝ o (normalVector t) = 0 := by
+  have hkey : ∀ t : Real.Angle, inner ℝ o (normalVector t) =
+      o 0 * tangentVector t 1 - o 1 * tangentVector t 0 := by
+    intro t
+    induction t using Real.Angle.induction_on with
+    | _ t =>
+      simp [PiLp.inner_apply, Fin.sum_univ_two, normalVector, tangentVector, frame,
+        Real.Angle.cos_coe, Real.Angle.sin_coe]
+      ring
+  have h0 := sum_edgeLength_mul_tangentCoordinate_eq_zero K V hKV 0
+  have h1 := sum_edgeLength_mul_tangentCoordinate_eq_zero K V hKV 1
+  calc ∑ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+        dist (edgeVertices K t).1 (edgeVertices K t).2 * inner ℝ o (normalVector t)
+      = o 0 * ∑ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+            dist (edgeVertices K t).1 (edgeVertices K t).2 * tangentVector t 1 -
+          o 1 * ∑ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+            dist (edgeVertices K t).1 (edgeVertices K t).2 * tangentVector t 0 := by
+        rw [Finset.mul_sum, Finset.mul_sum, ← Finset.sum_sub_distrib]
+        refine Finset.sum_congr rfl fun t _ ↦ ?_
+        rw [hkey t]
+        ring
+    _ = 0 := by rw [h0, h1]; ring
+
+/-- The support-area identity for a polygon with interior, by fan triangulation. -/
+theorem convexBody_area_support_integral_of_convexHull_interior
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    (hint : (interior (K : Set Point)).Nonempty) :
+    ClassicalResults.area (K : Set Point) =
+      (1 / 2 : ℝ) * ∫ t, supportValue K t ∂surfaceAreaMeasure K := by
+  classical
+  obtain ⟨o, ho⟩ := hint
+  have hmemF : ∀ t : Real.Angle,
+      t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset ↔
+        (edgeVertices K t).1 ≠ (edgeVertices K t).2 := by
+    intro t
+    simp
+  have hpos : ∀ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+      0 < dist (edgeVertices K t).1 (edgeVertices K t).2 *
+        (supportValue K t - inner ℝ o (normalVector t)) := by
+    intro t htF
+    exact mul_pos (dist_pos.mpr ((hmemF t).1 htF))
+      (sub_pos.mpr (inner_lt_supportValue_of_mem_interior K ho t))
+  have hTvol : ∀ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+      volume (edgeTriangle K o t) = ENNReal.ofReal
+        (dist (edgeVertices K t).1 (edgeVertices K t).2 *
+          (supportValue K t - inner ℝ o (normalVector t)) / 2) := by
+    intro t htF
+    rw [edgeTriangle, EuclideanSpace.volume_convexHull_triple]
+    congr 1
+    rw [show ((edgeVertices K t).2 - o) 0 * ((edgeVertices K t).1 - o) 1 -
+          ((edgeVertices K t).1 - o) 0 * ((edgeVertices K t).2 - o) 1 =
+        planeCrossProduct ((edgeVertices K t).2 - o) ((edgeVertices K t).1 - o) from by
+      simp only [planeCrossProduct]
+      ring, ← edgeLength_mul_supportValue_sub_eq_planeCrossProduct K o t,
+      abs_of_pos (hpos t htF)]
+  have hTsubK : ∀ t : Real.Angle, edgeTriangle K o t ⊆ (K : Set Point) := by
+    intro t
+    rw [edgeTriangle]
+    refine convexHull_min ?_ K.convex
+    intro p hp
+    simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hp
+    rcases hp with rfl | rfl | rfl
+    · exact interior_subset ho
+    · exact (edgeVertices_snd_mem K t).1
+    · exact (edgeVertices_fst_mem K t).1
+  have hvolK : volume (K : Set Point) =
+      ∑ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+        volume (edgeTriangle K o t) := by
+    have hcover : (K : Set Point) ⊆
+        (⋃ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset, edgeTriangle K o t) ∪
+          vertexRayUnion o V := by
+      intro x hx
+      by_cases hxline : x ∈ vertexRayUnion o V
+      · exact Or.inr hxline
+      obtain ⟨t, htproper, hxt⟩ :=
+        exists_mem_edgeTriangle_of_notMem_vertexRayUnion K V hKV ho hx hxline
+      exact Or.inl (Set.mem_biUnion ((hmemF t).2 htproper) hxt)
+    have hle : volume (K : Set Point) ≤
+        volume (⋃ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+          edgeTriangle K o t) := by
+      calc volume (K : Set Point)
+          ≤ volume ((⋃ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+              edgeTriangle K o t) ∪ vertexRayUnion o V) := measure_mono hcover
+        _ ≤ volume (⋃ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+              edgeTriangle K o t) + volume (vertexRayUnion o V) := measure_union_le _ _
+        _ = volume (⋃ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+              edgeTriangle K o t) := by rw [volume_vertexRayUnion, add_zero]
+    have hge : volume (⋃ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+        edgeTriangle K o t) ≤ volume (K : Set Point) :=
+      measure_mono (Set.iUnion₂_subset fun t _ ↦ hTsubK t)
+    rw [le_antisymm hle hge]
+    refine measure_biUnion_finset₀ ?_ ?_
+    · intro s _ t _ hst
+      exact volume_edgeTriangle_inter_eq_zero K ho hst
+    · intro t _
+      exact ((Set.toFinite ({o, (edgeVertices K t).2, (edgeVertices K t).1} :
+        Set Point)).isCompact_convexHull ℝ).isClosed.measurableSet.nullMeasurableSet
+  have hnonneg : ∀ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+      0 ≤ dist (edgeVertices K t).1 (edgeVertices K t).2 *
+        (supportValue K t - inner ℝ o (normalVector t)) / 2 := by
+    intro t htF
+    linarith [hpos t htF]
+  have harea : ClassicalResults.area (K : Set Point) =
+      ∑ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+        dist (edgeVertices K t).1 (edgeVertices K t).2 *
+          (supportValue K t - inner ℝ o (normalVector t)) / 2 := by
+    simp only [ClassicalResults.area]
+    rw [hvolK, Finset.sum_congr rfl hTvol, ← ENNReal.ofReal_sum_of_nonneg hnonneg,
+      ENNReal.toReal_ofReal (Finset.sum_nonneg hnonneg)]
+  have hsplit : ∑ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+        dist (edgeVertices K t).1 (edgeVertices K t).2 *
+          (supportValue K t - inner ℝ o (normalVector t)) / 2 =
+      (1 / 2 : ℝ) * ∑ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+          dist (edgeVertices K t).1 (edgeVertices K t).2 * supportValue K t -
+        (1 / 2 : ℝ) * ∑ t ∈ (finite_properEdgeNormal_angles K V hKV).toFinset,
+          dist (edgeVertices K t).1 (edgeVertices K t).2 * inner ℝ o (normalVector t) := by
+    rw [Finset.mul_sum, Finset.mul_sum, ← Finset.sum_sub_distrib]
+    refine Finset.sum_congr rfl fun t _ ↦ ?_
+    ring
+  rw [harea, hsplit, sum_edgeLength_mul_inner_normalVector_eq_zero K V hKV o, mul_zero,
+    sub_zero, integral_supportValue_surfaceAreaMeasure_eq_sum_edgeLength K V hKV]
+
+/-- The support-area identity for every finite convex hull. -/
+theorem convexBody_area_support_integral_of_convexHull
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point)) :
+    ClassicalResults.area (K : Set Point) =
+      (1 / 2 : ℝ) * ∫ t, supportValue K t ∂surfaceAreaMeasure K := by
+  by_cases hsub : (K : Set Point).Subsingleton
+  · exact convexBody_area_support_integral_of_subsingleton K hsub
+  by_cases hint : (interior (K : Set Point)).Nonempty
+  · exact convexBody_area_support_integral_of_convexHull_interior K V hKV hint
+  obtain ⟨d, hd⟩ := exists_segmentPresentation_of_interior_empty K hsub
+    (Set.not_nonempty_iff_eq_empty.mp hint)
+  exact convexBody_area_support_integral_of_segmentPresentation K d hd
+
+/-! ### Symmetry of the mixed support integral
+
+The mixed integral `∫ h_P dσ_Q` of two finite convex hulls is computed by lifting the circle to
+`(0, 2π]` and indexing by the finite set `F` of lifts carrying a proper edge of `P` or of `Q`.
+The positive-vertex increment formula turns `h_P` into a partial sum along `F`, so the integral
+becomes a lower-triangular double sum `∑_{t ∈ F} ∑_{u ≤ t} α_u β_t sin (t - u)`, whose symmetry
+in `(α, β)` is `Finset.sum_filter_le_add_sum_filter_le_swap` together with the vanishing of the
+first trigonometric moments of the edge lengths. Polygon approximation transfers the identity to
+arbitrary convex bodies. -/
+
+/-- The first tangent coordinate at a real angle lift. -/
+private theorem tangentVector_coe_zero (s : ℝ) :
+    tangentVector ((s : ℝ) : Real.Angle) 0 = -Real.sin s := by
+  simp [tangentVector, frame, Real.Angle.sin_coe]
+
+/-- The second tangent coordinate at a real angle lift. -/
+private theorem tangentVector_coe_one (s : ℝ) :
+    tangentVector ((s : ℝ) : Real.Angle) 1 = Real.cos s := by
+  simp [tangentVector, frame, Real.Angle.cos_coe]
+
+/-- A finite set of angular lifts in `(0, 2π]` covering all proper-edge normals of a polygon.
+Excluding `0` makes the lift of each angle unique, so no edge atom is counted twice. -/
+private def IsProperEdgeLiftIndex (K : ConvexBody Point)
+    (F : Finset (Set.Icc (0 : ℝ) (2 * Real.pi))) : Prop :=
+  (∀ u ∈ F, 0 < (u : ℝ)) ∧
+    ∀ u : Set.Icc (0 : ℝ) (2 * Real.pi), 0 < (u : ℝ) →
+      (edgeVertices K ((u : ℝ) : Real.Angle)).1 ≠
+        (edgeVertices K ((u : ℝ) : Real.Angle)).2 → u ∈ F
+
+/-- The surface integral over an initial angular arc is the finite sum of its edge atoms. -/
+private theorem integral_surfaceAreaMeasure_image_Ioc_eq_sum_lifts
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    (F : Finset (Set.Icc (0 : ℝ) (2 * Real.pi))) (hF : IsProperEdgeLiftIndex K F)
+    (φ : Real.Angle → ℝ) (t : Set.Icc (0 : ℝ) (2 * Real.pi)) :
+    (∫ u in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioc 0 (t : ℝ),
+        φ u ∂surfaceAreaMeasure K) =
+      ∑ u ∈ F.filter (fun u ↦ u ≤ t),
+        dist (edgeVertices K ((u : ℝ) : Real.Angle)).1
+            (edgeVertices K ((u : ℝ) : Real.Angle)).2 * φ ((u : ℝ) : Real.Angle) := by
+  classical
+  have hturn : 2 * Real.pi ≤ 0 + 2 * Real.pi := le_of_eq (by ring)
+  have hpi : (0 : ℝ) ≤ 2 * Real.pi := by positivity
+  let a' : Set.Icc (0 : ℝ) (2 * Real.pi) := ⟨0, le_rfl, hpi⟩
+  have himg : (fun s : Set.Icc (0 : ℝ) (2 * Real.pi) ↦ ((s : ℝ) : Real.Angle)) ''
+      Set.Ioc a' t = (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioc 0 (t : ℝ) := by
+    ext u
+    constructor
+    · rintro ⟨s, hs, rfl⟩
+      exact ⟨(s : ℝ), ⟨hs.1, hs.2⟩, rfl⟩
+    · rintro ⟨s, hs, rfl⟩
+      exact ⟨⟨s, hs.1.le, hs.2.trans t.property.2⟩, ⟨hs.1, hs.2⟩, rfl⟩
+  rw [← himg, integral_surfaceAreaMeasure_image_eq_sum_properEdgeNormal_lifts K V hKV hturn φ
+    (Set.Ioc a' t) measurableSet_Ioc (fun u hu ↦ hu.1)]
+  have hatom : ∀ u : Set.Icc (0 : ℝ) (2 * Real.pi),
+      (surfaceAreaMeasure K).real {((u : ℝ) : Real.Angle)} =
+        dist (edgeVertices K ((u : ℝ) : Real.Angle)).1
+          (edgeVertices K ((u : ℝ) : Real.Angle)).2 := by
+    intro u
+    change (surfaceAreaMeasure K {((u : ℝ) : Real.Angle)}).toReal = _
+    rw [(surfaceAreaMeasure_atom_length K ((u : ℝ) : Real.Angle)).2.1,
+      ENNReal.toReal_ofReal dist_nonneg]
+  simp only [hatom]
+  refine Finset.sum_subset ?_ ?_
+  · intro u hu
+    simp only [Set.Finite.mem_toFinset, Set.mem_inter_iff] at hu
+    exact Finset.mem_filter.2 ⟨hF.2 u hu.1.1 hu.1.2, hu.2.2⟩
+  · intro u hu hnot
+    simp only [Set.Finite.mem_toFinset, Set.mem_inter_iff, not_and] at hnot
+    obtain ⟨huF, hut⟩ := Finset.mem_filter.1 hu
+    have hu0 : 0 < (u : ℝ) := hF.1 u huF
+    have hdeg : ¬((edgeVertices K ((u : ℝ) : Real.Angle)).1 ≠
+        (edgeVertices K ((u : ℝ) : Real.Angle)).2) := fun hne ↦
+      (hnot ⟨hu0, hne⟩) ⟨hu0, hut⟩
+    rw [not_not.mp hdeg, dist_self, zero_mul]
+
+/-- The total surface integral is the finite sum of the edge atoms of a polygon. -/
+private theorem integral_surfaceAreaMeasure_eq_sum_lifts
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    (F : Finset (Set.Icc (0 : ℝ) (2 * Real.pi))) (hF : IsProperEdgeLiftIndex K F)
+    (φ : Real.Angle → ℝ) :
+    (∫ u, φ u ∂surfaceAreaMeasure K) =
+      ∑ u ∈ F,
+        dist (edgeVertices K ((u : ℝ) : Real.Angle)).1
+            (edgeVertices K ((u : ℝ) : Real.Angle)).2 * φ ((u : ℝ) : Real.Angle) := by
+  classical
+  have hpi : (0 : ℝ) ≤ 2 * Real.pi := by positivity
+  let b' : Set.Icc (0 : ℝ) (2 * Real.pi) := ⟨2 * Real.pi, hpi, le_rfl⟩
+  have hfull : (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioc 0 ((b' : ℝ)) = Set.univ := by
+    ext u
+    simp only [Set.mem_image, Set.mem_Ioc, Set.mem_univ, iff_true]
+    let _ : Fact (0 < 2 * Real.pi) := ⟨Real.two_pi_pos⟩
+    refine ⟨(AddCircle.equivIoc (2 * Real.pi) 0 u : ℝ), ?_, AddCircle.coe_equivIoc⟩
+    simpa using (AddCircle.equivIoc (2 * Real.pi) 0 u).property
+  have hfilter : F.filter (fun u ↦ u ≤ b') = F := by
+    refine Finset.filter_true_of_mem fun u _ ↦ ?_
+    exact u.property.2
+  have := integral_surfaceAreaMeasure_image_Ioc_eq_sum_lifts K V hKV F hF φ b'
+  rw [hfull, setIntegral_univ, hfilter] at this
+  exact this
+
+/-- The edge lengths of a polygon weight the cosines and sines of its normal directions to
+zero. -/
+private theorem sum_edgeLength_lift_cos_sin_eq_zero
+    (K : ConvexBody Point) (V : Finset Point)
+    (hKV : (K : Set Point) = convexHull ℝ (V : Set Point))
+    (F : Finset (Set.Icc (0 : ℝ) (2 * Real.pi))) (hF : IsProperEdgeLiftIndex K F) :
+    (∑ u ∈ F, dist (edgeVertices K ((u : ℝ) : Real.Angle)).1
+            (edgeVertices K ((u : ℝ) : Real.Angle)).2 * Real.cos (u : ℝ) = 0) ∧
+      ∑ u ∈ F, dist (edgeVertices K ((u : ℝ) : Real.Angle)).1
+            (edgeVertices K ((u : ℝ) : Real.Angle)).2 * Real.sin (u : ℝ) = 0 := by
+  have hzero : ∀ i : Fin 2, ∑ u ∈ F,
+      dist (edgeVertices K ((u : ℝ) : Real.Angle)).1
+          (edgeVertices K ((u : ℝ) : Real.Angle)).2 *
+        tangentVector ((u : ℝ) : Real.Angle) i = 0 := by
+    intro i
+    rw [← integral_surfaceAreaMeasure_eq_sum_lifts K V hKV F hF (fun v ↦ tangentVector v i)]
+    exact integral_tangentCoordinate_surfaceAreaMeasure_eq_zero K i
+  refine ⟨by simpa only [tangentVector_coe_one] using hzero 1, ?_⟩
+  have h := hzero 0
+  simp only [tangentVector_coe_zero, mul_neg, Finset.sum_neg_distrib] at h
+  linarith
+
+/-- The mixed support integral of two polygons, as a lower-triangular double sum over a common
+index set of angular lifts. -/
+private theorem integral_supportValue_eq_sum_sum_lifts
+    (P Q : ConvexBody Point) (VP VQ : Finset Point)
+    (hP : (P : Set Point) = convexHull ℝ (VP : Set Point))
+    (hQ : (Q : Set Point) = convexHull ℝ (VQ : Set Point))
+    (F : Finset (Set.Icc (0 : ℝ) (2 * Real.pi)))
+    (hidxP : IsProperEdgeLiftIndex P F) (hidxQ : IsProperEdgeLiftIndex Q F) :
+    (∫ t, supportValue P t ∂surfaceAreaMeasure Q) =
+      ∑ t ∈ F, ∑ u ∈ F.filter (fun u ↦ u ≤ t),
+        dist (edgeVertices P ((u : ℝ) : Real.Angle)).1
+              (edgeVertices P ((u : ℝ) : Real.Angle)).2 *
+            dist (edgeVertices Q ((t : ℝ) : Real.Angle)).1
+              (edgeVertices Q ((t : ℝ) : Real.Angle)).2 *
+          Real.sin ((t : ℝ) - (u : ℝ)) := by
+  classical
+  have hcosQ : ∑ u ∈ F, dist (edgeVertices Q ((u : ℝ) : Real.Angle)).1
+      (edgeVertices Q ((u : ℝ) : Real.Angle)).2 * Real.cos (u : ℝ) = 0 :=
+    (sum_edgeLength_lift_cos_sin_eq_zero Q VQ hQ F hidxQ).1
+  have hsinQ : ∑ u ∈ F, dist (edgeVertices Q ((u : ℝ) : Real.Angle)).1
+      (edgeVertices Q ((u : ℝ) : Real.Angle)).2 * Real.sin (u : ℝ) = 0 :=
+    (sum_edgeLength_lift_cos_sin_eq_zero Q VQ hQ F hidxQ).2
+  have hincP : ∀ (i : Fin 2) (t : Set.Icc (0 : ℝ) (2 * Real.pi)), 0 < (t : ℝ) →
+      (edgeVertices P ((t : ℝ) : Real.Angle)).1 i =
+        (edgeVertices P ((0 : ℝ) : Real.Angle)).1 i +
+          ∑ u ∈ F.filter (fun u ↦ u ≤ t),
+            dist (edgeVertices P ((u : ℝ) : Real.Angle)).1
+                (edgeVertices P ((u : ℝ) : Real.Angle)).2 *
+              tangentVector ((u : ℝ) : Real.Angle) i := by
+    intro i t ht
+    have h1 := positiveVertex_sub_eq_integral_of_eq_convexHull P VP hP (a := 0)
+      (b := (t : ℝ)) ht (by simpa using t.property.2) i
+    rw [integral_surfaceAreaMeasure_image_Ioc_eq_sum_lifts P VP hP F hidxP
+      (fun v ↦ tangentVector v i) t] at h1
+    linarith
+  have hsupp : ∀ t : Set.Icc (0 : ℝ) (2 * Real.pi), 0 < (t : ℝ) →
+      supportValue P ((t : ℝ) : Real.Angle) =
+        (edgeVertices P ((0 : ℝ) : Real.Angle)).1 0 * Real.cos (t : ℝ) +
+            (edgeVertices P ((0 : ℝ) : Real.Angle)).1 1 * Real.sin (t : ℝ) +
+          ∑ u ∈ F.filter (fun u ↦ u ≤ t),
+            dist (edgeVertices P ((u : ℝ) : Real.Angle)).1
+                (edgeVertices P ((u : ℝ) : Real.Angle)).2 *
+              Real.sin ((t : ℝ) - (u : ℝ)) := by
+    intro t ht
+    rw [← (edgeVertices_fst_mem P ((t : ℝ) : Real.Angle)).2, inner_normalVector_real,
+      hincP 0 t ht, hincP 1 t ht]
+    have hsplit :
+        (∑ u ∈ F.filter (fun u ↦ u ≤ t),
+              dist (edgeVertices P ((u : ℝ) : Real.Angle)).1
+                  (edgeVertices P ((u : ℝ) : Real.Angle)).2 *
+                tangentVector ((u : ℝ) : Real.Angle) 0) * Real.cos (t : ℝ) +
+            (∑ u ∈ F.filter (fun u ↦ u ≤ t),
+              dist (edgeVertices P ((u : ℝ) : Real.Angle)).1
+                  (edgeVertices P ((u : ℝ) : Real.Angle)).2 *
+                tangentVector ((u : ℝ) : Real.Angle) 1) * Real.sin (t : ℝ) =
+          ∑ u ∈ F.filter (fun u ↦ u ≤ t),
+            dist (edgeVertices P ((u : ℝ) : Real.Angle)).1
+                (edgeVertices P ((u : ℝ) : Real.Angle)).2 *
+              Real.sin ((t : ℝ) - (u : ℝ)) := by
+      rw [Finset.sum_mul, Finset.sum_mul, ← Finset.sum_add_distrib]
+      refine Finset.sum_congr rfl fun u _ ↦ ?_
+      rw [tangentVector_coe_zero, tangentVector_coe_one, Real.sin_sub]
+      ring
+    linarith
+  rw [integral_surfaceAreaMeasure_eq_sum_lifts Q VQ hQ F hidxQ (fun v ↦ supportValue P v)]
+  have step1 : ∑ t ∈ F, dist (edgeVertices Q ((t : ℝ) : Real.Angle)).1
+          (edgeVertices Q ((t : ℝ) : Real.Angle)).2 * supportValue P ((t : ℝ) : Real.Angle) =
+      ∑ t ∈ F, ((edgeVertices P ((0 : ℝ) : Real.Angle)).1 0 *
+            (dist (edgeVertices Q ((t : ℝ) : Real.Angle)).1
+              (edgeVertices Q ((t : ℝ) : Real.Angle)).2 * Real.cos (t : ℝ)) +
+          (edgeVertices P ((0 : ℝ) : Real.Angle)).1 1 *
+            (dist (edgeVertices Q ((t : ℝ) : Real.Angle)).1
+              (edgeVertices Q ((t : ℝ) : Real.Angle)).2 * Real.sin (t : ℝ)) +
+          ∑ u ∈ F.filter (fun u ↦ u ≤ t),
+            dist (edgeVertices P ((u : ℝ) : Real.Angle)).1
+                  (edgeVertices P ((u : ℝ) : Real.Angle)).2 *
+                dist (edgeVertices Q ((t : ℝ) : Real.Angle)).1
+                  (edgeVertices Q ((t : ℝ) : Real.Angle)).2 *
+              Real.sin ((t : ℝ) - (u : ℝ))) := by
+    refine Finset.sum_congr rfl fun t ht ↦ ?_
+    have hms : dist (edgeVertices Q ((t : ℝ) : Real.Angle)).1
+          (edgeVertices Q ((t : ℝ) : Real.Angle)).2 *
+        (∑ u ∈ F.filter (fun u ↦ u ≤ t), dist (edgeVertices P ((u : ℝ) : Real.Angle)).1
+          (edgeVertices P ((u : ℝ) : Real.Angle)).2 * Real.sin ((t : ℝ) - (u : ℝ))) =
+          ∑ u ∈ F.filter (fun u ↦ u ≤ t),
+            dist (edgeVertices P ((u : ℝ) : Real.Angle)).1
+                  (edgeVertices P ((u : ℝ) : Real.Angle)).2 *
+                dist (edgeVertices Q ((t : ℝ) : Real.Angle)).1
+                  (edgeVertices Q ((t : ℝ) : Real.Angle)).2 *
+              Real.sin ((t : ℝ) - (u : ℝ)) := by
+      rw [Finset.mul_sum]
+      exact Finset.sum_congr rfl fun u _ ↦ by ring
+    rw [hsupp t (hidxQ.1 t ht)]
+    linear_combination hms
+  have step2 : ∑ t ∈ F, ((edgeVertices P ((0 : ℝ) : Real.Angle)).1 0 *
+          (dist (edgeVertices Q ((t : ℝ) : Real.Angle)).1
+            (edgeVertices Q ((t : ℝ) : Real.Angle)).2 * Real.cos (t : ℝ)) +
+        (edgeVertices P ((0 : ℝ) : Real.Angle)).1 1 *
+          (dist (edgeVertices Q ((t : ℝ) : Real.Angle)).1
+            (edgeVertices Q ((t : ℝ) : Real.Angle)).2 * Real.sin (t : ℝ)) +
+        ∑ u ∈ F.filter (fun u ↦ u ≤ t),
+          dist (edgeVertices P ((u : ℝ) : Real.Angle)).1
+                (edgeVertices P ((u : ℝ) : Real.Angle)).2 *
+              dist (edgeVertices Q ((t : ℝ) : Real.Angle)).1
+                (edgeVertices Q ((t : ℝ) : Real.Angle)).2 *
+            Real.sin ((t : ℝ) - (u : ℝ))) =
+      (edgeVertices P ((0 : ℝ) : Real.Angle)).1 0 *
+          (∑ t ∈ F, dist (edgeVertices Q ((t : ℝ) : Real.Angle)).1
+            (edgeVertices Q ((t : ℝ) : Real.Angle)).2 * Real.cos (t : ℝ)) +
+        (edgeVertices P ((0 : ℝ) : Real.Angle)).1 1 *
+          (∑ t ∈ F, dist (edgeVertices Q ((t : ℝ) : Real.Angle)).1
+            (edgeVertices Q ((t : ℝ) : Real.Angle)).2 * Real.sin (t : ℝ)) +
+        ∑ t ∈ F, ∑ u ∈ F.filter (fun u ↦ u ≤ t),
+          dist (edgeVertices P ((u : ℝ) : Real.Angle)).1
+                (edgeVertices P ((u : ℝ) : Real.Angle)).2 *
+              dist (edgeVertices Q ((t : ℝ) : Real.Angle)).1
+                (edgeVertices Q ((t : ℝ) : Real.Angle)).2 *
+            Real.sin ((t : ℝ) - (u : ℝ)) := by
+    rw [Finset.mul_sum, Finset.mul_sum, ← Finset.sum_add_distrib, ← Finset.sum_add_distrib]
+  rw [step1, step2, hcosQ, hsinQ]
+  ring
+
+/-- The lower-triangular sine double sum is symmetric under swapping the two weight families,
+provided the second family has vanishing first trigonometric moments. -/
+private theorem sum_filter_le_sin_sub_comm
+    {ι : Type*} [LinearOrder ι] (F : Finset ι) (θ : ι → ℝ) (α β : ι → ℝ)
+    (hβ0 : ∑ u ∈ F, β u * Real.cos (θ u) = 0)
+    (hβ1 : ∑ u ∈ F, β u * Real.sin (θ u) = 0) :
+    (∑ t ∈ F, ∑ u ∈ F.filter (fun u ↦ u ≤ t), α u * β t * Real.sin (θ t - θ u)) =
+      ∑ t ∈ F, ∑ u ∈ F.filter (fun u ↦ u ≤ t), β u * α t * Real.sin (θ t - θ u) := by
+  set f : ι → ι → ℝ := fun x y ↦ α x * β y * Real.sin (θ y - θ x) with hf
+  have hdiag : ∀ x, f x x = 0 := by
+    intro x
+    simp [hf]
+  have hfull : (∑ x ∈ F, ∑ y ∈ F, f x y) = 0 := by
+    refine Finset.sum_eq_zero fun x _ ↦ ?_
+    have hinner : (∑ y ∈ F, f x y) =
+        (α x * Real.cos (θ x)) * (∑ y ∈ F, β y * Real.sin (θ y)) -
+          (α x * Real.sin (θ x)) * (∑ y ∈ F, β y * Real.cos (θ y)) := by
+      rw [Finset.mul_sum, Finset.mul_sum, ← Finset.sum_sub_distrib]
+      refine Finset.sum_congr rfl fun y _ ↦ ?_
+      simp only [hf, Real.sin_sub]
+      ring
+    rw [hinner, hβ0, hβ1]
+    ring
+  have htrans : (∑ t ∈ F, ∑ u ∈ F.filter (fun u ↦ u ≤ t), f u t) +
+      ∑ t ∈ F, ∑ u ∈ F.filter (fun u ↦ u ≤ t), f t u = 0 := by
+    rw [Finset.sum_filter_le_add_sum_filter_le_swap F f hdiag, hfull]
+  have hright : (∑ t ∈ F, ∑ u ∈ F.filter (fun u ↦ u ≤ t), β u * α t * Real.sin (θ t - θ u)) =
+      -∑ t ∈ F, ∑ u ∈ F.filter (fun u ↦ u ≤ t), f t u := by
+    rw [← Finset.sum_neg_distrib]
+    refine Finset.sum_congr rfl fun t _ ↦ ?_
+    rw [← Finset.sum_neg_distrib]
+    refine Finset.sum_congr rfl fun u _ ↦ ?_
+    simp only [hf]
+    rw [show θ u - θ t = -(θ t - θ u) by ring, Real.sin_neg]
+    ring
+  rw [hright]
+  linarith [htrans]
+
+/-- Symmetry of the mixed support integral for finite convex hulls, including the degenerate
+point and segment cases. -/
+theorem supportIntegral_symm_of_convexHull (P Q : ConvexBody Point) (VP VQ : Finset Point)
+    (hP : (P : Set Point) = convexHull ℝ (VP : Set Point))
+    (hQ : (Q : Set Point) = convexHull ℝ (VQ : Set Point)) :
+    (∫ t, supportValue P t ∂surfaceAreaMeasure Q) =
+      ∫ t, supportValue Q t ∂surfaceAreaMeasure P := by
+  classical
+  have hturn : 2 * Real.pi ≤ 0 + 2 * Real.pi := le_of_eq (by ring)
+  have hSP := finite_properEdgeNormal_lifts P VP hP (a := 0) (b := 2 * Real.pi) hturn
+  have hSQ := finite_properEdgeNormal_lifts Q VQ hQ (a := 0) (b := 2 * Real.pi) hturn
+  set F : Finset (Set.Icc (0 : ℝ) (2 * Real.pi)) := hSP.toFinset ∪ hSQ.toFinset with hFdef
+  have hidxP : IsProperEdgeLiftIndex P F := by
+    refine ⟨fun u hu ↦ ?_, fun u h0 hne ↦ ?_⟩
+    · rcases Finset.mem_union.mp hu with h | h
+      · exact (hSP.mem_toFinset.mp h).1
+      · exact (hSQ.mem_toFinset.mp h).1
+    · exact Finset.mem_union_left _ (hSP.mem_toFinset.mpr ⟨h0, hne⟩)
+  have hidxQ : IsProperEdgeLiftIndex Q F := by
+    refine ⟨fun u hu ↦ ?_, fun u h0 hne ↦ ?_⟩
+    · rcases Finset.mem_union.mp hu with h | h
+      · exact (hSP.mem_toFinset.mp h).1
+      · exact (hSQ.mem_toFinset.mp h).1
+    · exact Finset.mem_union_right _ (hSQ.mem_toFinset.mpr ⟨h0, hne⟩)
+  rw [integral_supportValue_eq_sum_sum_lifts P Q VP VQ hP hQ F hidxP hidxQ,
+    integral_supportValue_eq_sum_sum_lifts Q P VQ VP hQ hP F hidxQ hidxP]
+  exact sum_filter_le_sin_sub_comm F (fun u ↦ (u : ℝ))
+    (fun u ↦ dist (edgeVertices P ((u : ℝ) : Real.Angle)).1
+      (edgeVertices P ((u : ℝ) : Real.Angle)).2)
+    (fun u ↦ dist (edgeVertices Q ((u : ℝ) : Real.Angle)).1
+      (edgeVertices Q ((u : ℝ) : Real.Angle)).2)
+    (sum_edgeLength_lift_cos_sin_eq_zero Q VQ hQ F hidxQ).1
+    (sum_edgeLength_lift_cos_sin_eq_zero Q VQ hQ F hidxQ).2
+
+/-- Symmetry of the mixed support integral for arbitrary planar convex bodies:
+`∫ h_K dσ_L = ∫ h_L dσ_K`. -/
+theorem supportIntegral_symm (K L : ConvexBody Point) :
+    (∫ t, supportValue K t ∂surfaceAreaMeasure L) =
+      ∫ t, supportValue L t ∂surfaceAreaMeasure K := by
+  obtain ⟨VP, P, hPspec, hPdist⟩ := exists_facePreserving_polygonApproximation K ∅
+  obtain ⟨VQ, Q, hQspec, hQdist⟩ := exists_facePreserving_polygonApproximation L ∅
+  have hlimP : Tendsto (fun n ↦ Metric.hausdorffDist (P n : Set Point) (K : Set Point))
+      atTop (𝓝 0) := by
+    apply squeeze_zero' (Filter.Eventually.of_forall fun _ ↦ Metric.hausdorffDist_nonneg)
+      (Filter.eventually_atTop.2 ⟨1, fun n hn ↦ hPdist n hn⟩)
+    exact tendsto_one_div_atTop_nhds_zero_nat
+  have hlimQ : Tendsto (fun n ↦ Metric.hausdorffDist (Q n : Set Point) (L : Set Point))
+      atTop (𝓝 0) := by
+    apply squeeze_zero' (Filter.Eventually.of_forall fun _ ↦ Metric.hausdorffDist_nonneg)
+      (Filter.eventually_atTop.2 ⟨1, fun n hn ↦ hQdist n hn⟩)
+    exact tendsto_one_div_atTop_nhds_zero_nat
+  refine tendsto_nhds_unique
+    (tendsto_integral_supportValue_of_hausdorff_pair P Q K L hlimP hlimQ)
+    (Filter.Tendsto.congr (fun n ↦ ?_)
+      (tendsto_integral_supportValue_of_hausdorff_pair Q P L K hlimQ hlimP))
+  exact supportIntegral_symm_of_convexHull (Q n) (P n) (VQ n) (VP n)
+    (hQspec n).2.1 (hPspec n).2.1
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Linearity
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open scoped unitInterval
+
+namespace MovingSofa
+
+theorem convexBody_maps_linear (t : I) (K L : ConvexBody Point) :
+    (∀ a : Real.Angle,
+      supportValue (convexBodyCombination t K L) a =
+        (1 - (t : ℝ)) * supportValue K a + (t : ℝ) * supportValue L a) ∧
+    (∀ a : Real.Angle,
+      (edgeVertices (convexBodyCombination t K L) a).1 =
+        (1 - (t : ℝ)) • (edgeVertices K a).1 + (t : ℝ) • (edgeVertices L a).1 ∧
+      (edgeVertices (convexBodyCombination t K L) a).2 =
+        (1 - (t : ℝ)) • (edgeVertices K a).2 + (t : ℝ) • (edgeVertices L a).2) ∧
+    (∀ a b : ℝ, a < b → b < a + Real.pi →
+      supportingIntersection (convexBodyCombination t K L) a b =
+        (1 - (t : ℝ)) • supportingIntersection K a b +
+          (t : ℝ) • supportingIntersection L a b) ∧
+    surfaceAreaMeasure (convexBodyCombination t K L) =
+      ENNReal.ofReal (1 - (t : ℝ)) • surfaceAreaMeasure K +
+        ENNReal.ofReal (t : ℝ) • surfaceAreaMeasure L := by
+  refine ⟨supportValue_convexBodyCombination t K L, ?_, ?_,
+    surfaceAreaMeasure_convexBodyCombination t K L⟩
+  · exact edgeVertices_convexBodyCombination t K L
+  · intro a b _hab _hba
+    simp only [supportingIntersection, supportValue_convexBodyCombination]
+    ext i
+    simp only [PiLp.add_apply, PiLp.smul_apply, smul_eq_mul]
+    ring
+
+theorem convexBody_area_support_integral :
+    (∀ K : ConvexBody Point, ClassicalResults.area (K : Set Point) =
+      (1 / 2 : ℝ) * ∫ a : Real.Angle, supportValue K a ∂surfaceAreaMeasure K) ∧
+    IsConvexBilinear convexBodyCombination convexBodyCombination realCombination
+      (fun K L : ConvexBody Point ↦
+        (1 / 2 : ℝ) * ∫ a : Real.Angle, supportValue K a ∂surfaceAreaMeasure L) ∧
+    IsQuadraticFunctional convexBodyCombination
+      (fun K : ConvexBody Point ↦ ClassicalResults.area (K : Set Point)) :=
+  convexBody_area_support_integral_of_area_identity
+    (convexBody_area_support_integral_of_finiteHull
+      (fun P W _hW hPW ↦ convexBody_area_support_integral_of_convexHull P W hPW))
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# The mixed area of two planar convex bodies
+
+The mixed support integral `∫ h_K dσ_L` is symmetric in the two bodies, and the area of the
+Minkowski segment `λ ↦ |(1 - λ) K + λ L|` is therefore the quadratic
+`(1 - λ)² |K| + λ (1 - λ) ∫ h_L dσ_K + λ² |L|`, whose right derivative at `λ = 0` is
+`∫ (h_L - h_K) dσ_K`.
+
+The symmetry itself is `MovingSofa.supportIntegral_symm` in `MovingSofa.Convex.SupportArea`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- Symmetry of the mixed support integral, together with the right derivative at `0` of the
+area along the Minkowski segment from `K` to `L`. -/
+theorem supportMeasure_mixedArea_symmetry (K L : ConvexBody Point) :
+    (∫ t : Real.Angle, supportValue K t ∂surfaceAreaMeasure L) =
+      (∫ t : Real.Angle, supportValue L t ∂surfaceAreaMeasure K) ∧
+    HasDerivWithinAt
+      (segmentFunctional convexBodyCombination
+        (fun M : ConvexBody Point ↦ ClassicalResults.area (M : Set Point)) K L)
+      (∫ t : Real.Angle, supportValue L t - supportValue K t ∂surfaceAreaMeasure K)
+      (Set.Icc (0 : ℝ) 1) 0 := by
+  classical
+  have hsymm := supportIntegral_symm K L
+  refine ⟨hsymm, ?_⟩
+  obtain ⟨harea, hbil, -⟩ := convexBody_area_support_integral
+  have hcont (M : ConvexBody Point) : Continuous (supportValue M) :=
+    (compactSet_support_continuity M M M.nonempty M.isCompact M.nonempty M.isCompact).2.2.1
+  have hint : ∀ M N : ConvexBody Point, Integrable (supportValue M) (surfaceAreaMeasure N) := by
+    intro M N
+    let _ : IsFiniteMeasure (surfaceAreaMeasure N) := (surfaceAreaMeasure_face_union N).1
+    exact (hcont M).integrable_of_hasCompactSupport
+      (isCompact_univ.of_isClosed_subset isClosed_closure (Set.subset_univ _))
+  rw [integral_sub (hint L K) (hint K K)]
+  have hfun : ∀ lam ∈ Set.Icc (0 : ℝ) 1,
+      segmentFunctional convexBodyCombination
+          (fun M : ConvexBody Point ↦ ClassicalResults.area (M : Set Point)) K L lam =
+        1 / 2 * (∫ t, supportValue K t ∂surfaceAreaMeasure K) +
+          ((∫ t, supportValue L t ∂surfaceAreaMeasure K) -
+            ∫ t, supportValue K t ∂surfaceAreaMeasure K) * lam +
+          (1 / 2 * (∫ t, supportValue K t ∂surfaceAreaMeasure K) -
+            (∫ t, supportValue L t ∂surfaceAreaMeasure K) +
+            1 / 2 * ∫ t, supportValue L t ∂surfaceAreaMeasure L) * lam ^ 2 := by
+    intro lam hlam
+    have hseg : segmentFunctional convexBodyCombination
+        (fun M : ConvexBody Point ↦ ClassicalResults.area (M : Set Point)) K L lam =
+        ClassicalResults.area
+          ((convexBodyCombination ⟨lam, hlam⟩ K L : ConvexBody Point) : Set Point) :=
+      dite_eq_left hlam
+    have h2 := hbil.2 (convexBodyCombination ⟨lam, hlam⟩ K L) ⟨lam, hlam⟩ K L
+    have h3 := hbil.1 K ⟨lam, hlam⟩ K L
+    have h4 := hbil.1 L ⟨lam, hlam⟩ K L
+    simp only [realCombination] at h2 h3 h4
+    rw [hseg, harea (convexBodyCombination ⟨lam, hlam⟩ K L), h2, h3, h4, hsymm]
+    ring
+  have hpoly : ∀ A B C : ℝ, HasDerivAt (fun lam : ℝ ↦ A + B * lam + C * lam ^ 2) B 0 := by
+    intro A B C
+    have h : HasDerivAt (fun lam : ℝ ↦ A + B * lam + C * lam ^ 2)
+        (0 + B * 1 + C * (2 * 0 ^ 1)) 0 :=
+      HasDerivAt.add (HasDerivAt.add (hasDerivAt_const _ _)
+        ((hasDerivAt_id _).const_mul _)) ((hasDerivAt_pow _ _).const_mul _)
+    simpa using h
+  exact HasDerivWithinAt.congr (hpoly _ _ _).hasDerivWithinAt hfun
+    (hfun 0 ⟨le_rfl, zero_le_one⟩)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Tangent Line Path
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set Filter
+open scoped Topology
+
+namespace MovingSofa
+
+/-- Trace intersections with a fixed supporting line, ending at its exposed-edge endpoint. -/
+def tangentLinePath (K : ConvexBody Point) (t : ℝ) (s : Set.Ioc (t - Real.pi) t) : Point :=
+  if s.val < t then supportingIntersection K (s.val : Real.Angle) (t : Real.Angle)
+  else (edgeVertices K (t : Real.Angle)).2
+
+/-- Restrict the tangent-line path to a closed interval of supporting directions. -/
+def tangentLineRestriction (K : ConvexBody Point) (t a b : ℝ)
+    (ha : a ∈ Set.Ioc (t - Real.pi) t) (hb : b ∈ Set.Ioc (t - Real.pi) t)
+    (s : Set.Icc a b) : Point :=
+  tangentLinePath K t ⟨s.val, lt_of_lt_of_le ha.1 s.property.1,
+    le_trans s.property.2 hb.2⟩
+
+private def tangentLineCoordinate (K : ConvexBody Point) (t s : ℝ) : ℝ :=
+  (supportValue K (s : Real.Angle) - supportValue K (t : Real.Angle) * Real.cos (s - t)) /
+    Real.sin (s - t)
+
+private theorem tangentLineCoordinate_le (K : ConvexBody Point) {t s : ℝ}
+    (hs : s ∈ Ioo (t - Real.pi) t) {p : Point} (hp : p ∈ K) :
+    tangentLineCoordinate K t s ≤ inner ℝ p (tangentVector (t : Real.Angle)) +
+      (inner ℝ p (normalVector (t : Real.Angle)) - supportValue K (t : Real.Angle)) *
+        (Real.cos (s - t) / Real.sin (s - t)) := by
+  have hsin : Real.sin (s - t) < 0 :=
+    Real.sin_neg_of_neg_of_neg_pi_lt (by linarith [hs.2]) (by linarith [hs.1])
+  have hbound := inner_le_supportValue K hp (s : Real.Angle)
+  have hn : normalVector (s : Real.Angle) =
+      Real.cos (s - t) • normalVector (t : Real.Angle) +
+        Real.sin (s - t) • tangentVector (t : Real.Angle) := by
+    simpa only [add_sub_cancel] using normalVector_add_real t (s - t)
+  rw [hn, inner_add_right, inner_smul_right, inner_smul_right] at hbound
+  dsimp [tangentLineCoordinate]
+  apply (div_le_iff_of_neg hsin).mpr
+  have hc : (inner ℝ p (tangentVector (t : Real.Angle)) +
+      (inner ℝ p (normalVector (t : Real.Angle)) - supportValue K (t : Real.Angle)) *
+        (Real.cos (s - t) / Real.sin (s - t))) * Real.sin (s - t) =
+      inner ℝ p (tangentVector (t : Real.Angle)) * Real.sin (s - t) +
+      (inner ℝ p (normalVector (t : Real.Angle)) - supportValue K (t : Real.Angle)) *
+        Real.cos (s - t) := by
+    field_simp [hsin.ne]
+  rw [hc]
+  nlinarith
+
+private theorem tangentLineCoordinate_eq_of_mem_support (K : ConvexBody Point) {t s : ℝ}
+    (hs : s ∈ Ioo (t - Real.pi) t) {p : Point}
+    (hp : inner ℝ p (normalVector (s : Real.Angle)) = supportValue K (s : Real.Angle)) :
+    tangentLineCoordinate K t s = inner ℝ p (tangentVector (t : Real.Angle)) +
+      (inner ℝ p (normalVector (t : Real.Angle)) - supportValue K (t : Real.Angle)) *
+        (Real.cos (s - t) / Real.sin (s - t)) := by
+  have hsin : Real.sin (s - t) ≠ 0 :=
+    (Real.sin_neg_of_neg_of_neg_pi_lt (by linarith [hs.2]) (by linarith [hs.1])).ne
+  have hn : normalVector (s : Real.Angle) =
+      Real.cos (s - t) • normalVector (t : Real.Angle) +
+        Real.sin (s - t) • tangentVector (t : Real.Angle) := by
+    simpa only [add_sub_cancel] using normalVector_add_real t (s - t)
+  dsimp [tangentLineCoordinate]
+  rw [← hp, hn, inner_add_right, inner_smul_right, inner_smul_right]
+  field_simp
+  ring
+
+private theorem monotoneOn_tangentLineCoordinate (K : ConvexBody Point) (t : ℝ) :
+    MonotoneOn (tangentLineCoordinate K t) (Ioo (t - Real.pi) t) := by
+  intro x hx y hy hxy
+  obtain ⟨p, hp, hmax⟩ := exists_mem_inner_eq_supportValue K (y : Real.Angle)
+  rw [tangentLineCoordinate_eq_of_mem_support K hy hmax]
+  apply (tangentLineCoordinate_le K hx hp).trans
+  apply add_le_add_right
+  exact mul_le_mul_of_nonpos_left
+    (Real.antitoneOn_cos_div_sin_Ioo_neg_pi_zero ⟨by linarith [hx.1], by linarith [hx.2]⟩
+      ⟨by linarith [hy.1], by linarith [hy.2]⟩ (by linarith))
+    (sub_nonpos.mpr (inner_le_supportValue K hp (t : Real.Angle)))
+
+private theorem supportingIntersection_eq_tangentLineCoordinate (K : ConvexBody Point)
+    {t s : ℝ} (hs : s ∈ Ioo (t - Real.pi) t) :
+    supportingIntersection K (s : Real.Angle) (t : Real.Angle) =
+      supportValue K (t : Real.Angle) • normalVector (t : Real.Angle) +
+        tangentLineCoordinate K t s • tangentVector (t : Real.Angle) := by
+  rw [supportingIntersection_comm K s t
+    (Real.sin_pos_of_pos_of_lt_pi (by linarith [hs.2]) (by linarith [hs.1])).ne']
+  simp only [supportingIntersection, tangentLineCoordinate, ← Real.Angle.coe_sub,
+    Real.Angle.cos_coe, Real.Angle.sin_coe]
+
+private theorem continuousAt_tangentLineCoordinate (K : ConvexBody Point) {t s : ℝ}
+    (hs : s ∈ Ioo (t - Real.pi) t) : ContinuousAt (tangentLineCoordinate K t) s := by
+  have hsin : Real.sin (s - t) ≠ 0 :=
+    (Real.sin_neg_of_neg_of_neg_pi_lt (by linarith [hs.2]) (by linarith [hs.1])).ne
+  exact ((continuous_supportValue_real K).continuousAt.sub
+    (continuousAt_const.mul (Real.continuous_cos.continuousAt.comp
+      (continuousAt_id.sub continuousAt_const)))).div
+    (Real.continuous_sin.continuousAt.comp (continuousAt_id.sub continuousAt_const)) hsin
+
+private theorem tendsto_tangentLineCoordinate_left (K : ConvexBody Point) (t : ℝ) :
+    Tendsto (tangentLineCoordinate K t) (𝓝[<] t)
+      (𝓝 (inner ℝ (edgeVertices K (t : Real.Angle)).2 (tangentVector (t : Real.Angle)))) := by
+  have h := (contact_oneSided_limits K t).2.2.2.2.2.inner
+    (𝕜 := ℝ) (tendsto_const_nhds (x := tangentVector (t : Real.Angle)))
+  apply h.congr'
+  filter_upwards [Ioo_mem_nhdsLT (by linarith [Real.pi_pos] : t - Real.pi < t)] with s hs
+  rw [supportingIntersection_eq_tangentLineCoordinate K hs]
+  simp only [inner_add_left, real_inner_smul_left, inner_normalVector_tangentVector,
+    inner_tangentVector_self, mul_zero, mul_one, zero_add]
+
+private def closedTangentLineCoordinate (K : ConvexBody Point) (t s : ℝ) : ℝ :=
+  if s = t then inner ℝ (edgeVertices K (t : Real.Angle)).2 (tangentVector (t : Real.Angle))
+  else tangentLineCoordinate K t s
+
+private theorem continuousOn_closedTangentLineCoordinate (K : ConvexBody Point) {t a : ℝ}
+    (ha : a ∈ Ioo (t - Real.pi) t) :
+    ContinuousOn (closedTangentLineCoordinate K t) (Icc a t) := by
+  exact continuousOn_replace_right_endpoint ha.2 _ _
+    (fun s hs ↦
+      (continuousAt_tangentLineCoordinate K ⟨ha.1.trans_le hs.1, hs.2⟩).tendsto.mono_left
+      nhdsWithin_le_nhds)
+    (fun s hs ↦
+      (continuousAt_tangentLineCoordinate K ⟨ha.1.trans hs.1, hs.2⟩).tendsto.mono_left
+      nhdsWithin_le_nhds)
+    (tendsto_tangentLineCoordinate_left K t)
+
+private theorem monotoneOn_closedTangentLineCoordinate (K : ConvexBody Point) (t : ℝ) :
+    MonotoneOn (closedTangentLineCoordinate K t) (Ioc (t - Real.pi) t) := by
+  intro x hx y hy hxy
+  by_cases hyEq : y = t
+  · subst y
+    by_cases hxEq : x = t
+    · subst x
+      rfl
+    · have hxt : x < t := lt_of_le_of_ne hx.2 hxEq
+      change (if x = t then _ else _) ≤ (if t = t then _ else _)
+      simp only [ite_eq_right hxEq, ite_true]
+      apply ge_of_tendsto (tendsto_tangentLineCoordinate_left K t)
+      filter_upwards [Ioo_mem_nhdsLT hxt] with u hu
+      exact monotoneOn_tangentLineCoordinate K t ⟨hx.1, hxt⟩
+        ⟨hx.1.trans hu.1, hu.2⟩ hu.1.le
+  · have hyt : y < t := lt_of_le_of_ne hy.2 hyEq
+    have hxt : x < t := hxy.trans_lt hyt
+    simp only [closedTangentLineCoordinate, ite_eq_right hxt.ne, ite_eq_right hyEq]
+    exact monotoneOn_tangentLineCoordinate K t ⟨hx.1, hxt⟩ ⟨hy.1, hyt⟩ hxy
+
+private theorem tangentLinePath_eq_coordinate (K : ConvexBody Point) (t : ℝ)
+    (s : Ioc (t - Real.pi) t) :
+    tangentLinePath K t s = supportValue K (t : Real.Angle) • normalVector (t : Real.Angle) +
+      closedTangentLineCoordinate K t s • tangentVector (t : Real.Angle) := by
+  dsimp [tangentLinePath]
+  split_ifs with hs
+  · rw [supportingIntersection_eq_tangentLineCoordinate K ⟨s.property.1, hs⟩]
+    simp only [closedTangentLineCoordinate, ite_eq_right hs.ne]
+  · have heq : (s : ℝ) = t := le_antisymm s.property.2 (le_of_not_gt hs)
+    simp only [closedTangentLineCoordinate, heq]
+    have hp := (edgeVertices_snd_mem K (t : Real.Angle)).2
+    change inner ℝ (edgeVertices K (t : Real.Angle)).2 (normalVector (t : Real.Angle)) =
+      supportValue K (t : Real.Angle) at hp
+    rw [← hp]
+    exact (inner_normalVector_smul_add_inner_tangentVector_smul _ _).symm
+
+/-- Every value of a tangent-line path lies on the supporting line at its own path parameter. -/
+theorem tangentLinePath_mem_supportingLine (K : ConvexBody Point) (t : ℝ)
+    (s : Set.Ioc (t - Real.pi) t) :
+    tangentLinePath K t s ∈
+      (supportingLineHalfPlane (K : Set Point) ((s : ℝ) : Real.Angle)).1 := by
+  unfold tangentLinePath
+  split_ifs with hs
+  · exact supportingIntersection_inner_left K (s : ℝ) t
+  · have hst : (s : ℝ) = t := le_antisymm s.property.2 (le_of_not_gt hs)
+    rw [hst]
+    exact (edgeVertices_snd_mem K (t : Real.Angle)).2
+
+theorem tangentLinePath_segment_area (K : ConvexBody Point) (t a b : ℝ)
+    (ha : a ∈ Set.Ioc (t - Real.pi) t) (hb : b ∈ Set.Ioc (t - Real.pi) t)
+    (hab : a ≤ b) :
+    ∃ γ : ContinuousBVPaths a b,
+      γ.val = tangentLineRestriction K t a b ha hb ∧
+      Set.range γ.val = segment ℝ (tangentLinePath K t ⟨a, ha⟩)
+        (tangentLinePath K t ⟨b, hb⟩) ∧
+      Set.range γ.val ⊆ (supportingLineHalfPlane (K : Set Point) (t : Real.Angle)).1 ∧
+      curveAreaFunctional γ =
+        segmentArea (tangentLinePath K t ⟨a, ha⟩) (tangentLinePath K t ⟨b, hb⟩) := by
+  let Q : Set.Icc a b → ℝ := fun s ↦ closedTangentLineCoordinate K t s
+  have hQmono : Monotone Q := by
+    intro x y hxy
+    exact monotoneOn_closedTangentLineCoordinate K t
+      ⟨ha.1.trans_le x.property.1, x.property.2.trans hb.2⟩
+      ⟨ha.1.trans_le y.property.1, y.property.2.trans hb.2⟩ hxy
+  have hQcont : Continuous Q := by
+    change Continuous (Set.domRestrict (Set.Icc a b) (closedTangentLineCoordinate K t))
+    by_cases hat : a < t
+    · exact continuousOn_iff_continuous_domRestrict.mp
+        ((continuousOn_closedTangentLineCoordinate K ⟨ha.1, hat⟩).mono
+          (Icc_subset_Icc_right hb.2))
+    · have heq : a = b := le_antisymm hab (hb.2.trans (le_of_not_gt hat))
+      have hc : ContinuousOn (closedTangentLineCoordinate K t) ({a} : Set ℝ) :=
+        (show ({a} : Set ℝ).Subsingleton from Set.subsingleton_singleton).continuousOn _
+      exact continuousOn_iff_continuous_domRestrict.mp (hc.mono (by simp [heq]))
+  have hQbv : BoundedVariationOn Q Set.univ := by
+    apply MonotoneOn.boundedVariationOn (f := Q) (s := Set.univ)
+      (fun _ _ _ _ hxy ↦ hQmono hxy)
+      (C := |Q ⟨a, le_rfl, hab⟩| + |Q ⟨b, hab, le_rfl⟩|)
+    intro s _
+    have hlo := hQmono (show (⟨a, le_rfl, hab⟩ : Set.Icc a b) ≤ s from s.property.1)
+    have hhi := hQmono (show s ≤ (⟨b, hab, le_rfl⟩ : Set.Icc a b) from s.property.2)
+    exact abs_le.mpr ⟨by
+      linarith [neg_abs_le (Q ⟨a, le_rfl, hab⟩), abs_nonneg (Q ⟨b, hab, le_rfl⟩)], by
+      linarith [le_abs_self (Q ⟨b, hab, le_rfl⟩), abs_nonneg (Q ⟨a, le_rfl, hab⟩)]⟩
+  let h := supportValue K (t : Real.Angle)
+  let γ : ContinuousBVPaths a b := {
+    val := fun s ↦ h • normalVector (t : Real.Angle) + Q s • tangentVector (t : Real.Angle)
+    property := ⟨continuous_const.add (hQcont.smul continuous_const), fun i ↦ by
+      let C : NNReal := ⟨|tangentVector (t : Real.Angle) i|, abs_nonneg _⟩
+      have hLip : LipschitzWith C (fun r : ℝ ↦
+          h * normalVector (t : Real.Angle) i + r * tangentVector (t : Real.Angle) i) := by
+        apply LipschitzWith.of_dist_le_mul
+        intro x y
+        simp only [Real.dist_eq]
+        rw [show (h * normalVector (t : Real.Angle) i + x * tangentVector (t : Real.Angle) i) -
+          (h * normalVector (t : Real.Angle) i + y * tangentVector (t : Real.Angle) i) =
+            (x - y) * tangentVector (t : Real.Angle) i by ring, abs_mul, mul_comm]
+        rfl
+      have hcomp := hLip.comp_boundedVariationOn hQbv
+      change BoundedVariationOn (fun t_1 ↦
+        h * normalVector (t : Real.Angle) i + Q t_1 * tangentVector (t : Real.Angle) i) Set.univ
+      exact hcomp⟩ }
+  refine ⟨γ, ?_, ?_, ?_, ?_⟩
+  · funext s
+    unfold tangentLineRestriction
+    exact (tangentLinePath_eq_coordinate K t
+      ⟨s, lt_of_lt_of_le ha.1 s.property.1, s.property.2.trans hb.2⟩).symm
+  · rw [show γ.val = fun s ↦ h • normalVector (t : Real.Angle) +
+        Q s • tangentVector (t : Real.Angle) from rfl]
+    let A := Q ⟨a, le_rfl, hab⟩
+    let B := Q ⟨b, hab, le_rfl⟩
+    have hAB : A ≤ B := hQmono hab
+    have hQrange : Set.range Q = Set.Icc (Q ⟨a, le_rfl, hab⟩) (Q ⟨b, hab, le_rfl⟩) := by
+      apply Set.Subset.antisymm
+      · rintro _ ⟨s, rfl⟩
+        exact ⟨hQmono s.property.1, hQmono s.property.2⟩
+      · intro y hy
+        let _ : PreconnectedSpace (Set.Icc a b) :=
+          Subtype.preconnectedSpace isPreconnected_Icc
+        exact intermediate_value_univ ⟨a, le_rfl, hab⟩ ⟨b, hab, le_rfl⟩ hQcont hy
+    have haeq := tangentLinePath_eq_coordinate K t ⟨a, ha⟩
+    have hbeq := tangentLinePath_eq_coordinate K t ⟨b, hb⟩
+    have hAa : closedTangentLineCoordinate K t a = A := rfl
+    have hBb : closedTangentLineCoordinate K t b = B := rfl
+    change Set.range (fun s ↦ h • normalVector (t : Real.Angle) +
+      Q s • tangentVector (t : Real.Angle)) = _
+    rw [haeq, hbeq]
+    rw [hAa, hBb]
+    rw [segment_eq_image_lineMap]
+    ext p
+    constructor
+    · rintro ⟨s, rfl⟩
+      have hs : Q s ∈ Set.Icc A B := by
+        exact ⟨hQmono s.property.1, hQmono s.property.2⟩
+      by_cases hABeq : A = B
+      · refine ⟨0, by simp, ?_⟩
+        have : Q s = A := le_antisymm (hs.2.trans_eq hABeq.symm) hs.1
+        ext i
+        simp [h, hABeq, this]
+      · let u := (Q s - A) / (B - A)
+        have hpos : 0 < B - A := sub_pos.mpr (lt_of_le_of_ne hAB hABeq)
+        refine ⟨u, ⟨div_nonneg (sub_nonneg.mpr hs.1) hpos.le,
+          (div_le_one hpos).2 (by linarith [hs.2])⟩, ?_⟩
+        ext i
+        simp only [AffineMap.lineMap_apply_module, PiLp.add_apply, PiLp.smul_apply,
+          smul_eq_mul]
+        dsimp [u]
+        field_simp [hpos.ne']
+        ring
+    · rintro ⟨u, hu, rfl⟩
+      let r := A + u * (B - A)
+      have hr : r ∈ Set.Icc A B := by
+        dsimp [r]
+        constructor
+        · exact le_add_of_nonneg_right (mul_nonneg hu.1 (sub_nonneg.mpr hAB))
+        · nlinarith [mul_le_mul_of_nonneg_right hu.2 (sub_nonneg.mpr hAB)]
+      have hrange : r ∈ Set.range Q := by simpa [A, B, hQrange] using hr
+      obtain ⟨s, hs⟩ := hrange
+      refine ⟨s, ?_⟩
+      ext i
+      simp only [AffineMap.lineMap_apply_module, PiLp.add_apply, PiLp.smul_apply,
+        smul_eq_mul]
+      rw [hs]
+      ring
+  · intro p hp
+    obtain ⟨s, rfl⟩ := hp
+    change inner ℝ (γ.val s) (normalVector (t : Real.Angle)) = supportValue K _
+    rw [inner_add_left, real_inner_smul_left, real_inner_smul_left,
+      inner_normalVector_self, real_inner_comm, inner_normalVector_tangentVector]
+    simp [h]
+  · let Qbv : RightContinuousIntervalBV a b := {
+      toFun := Q
+      boundedVariation := hQbv
+      right_continuous := fun s ↦ hQcont.continuousAt.continuousWithinAt }
+    have hcoord (i : Fin 2) (s : Set.Icc a b) :
+        (continuousBVCoordinate γ i).toFun s =
+          h * normalVector (t : Real.Angle) i +
+            tangentVector (t : Real.Angle) i * Qbv.toFun s := by
+      change (h • normalVector (t : Real.Angle) + Q s • tangentVector (t : Real.Angle)) i = _
+      simp [Qbv, mul_comm]
+    unfold curveAreaFunctional
+    change (intervalStieltjesIntegral (continuousBVCoordinate γ 1)
+        (continuousBVCoordinate γ 0).toFun Set.univ -
+      intervalStieltjesIntegral (continuousBVCoordinate γ 0)
+        (continuousBVCoordinate γ 1).toFun Set.univ) / 2 = _
+    rw [intervalStieltjesIntegral_affine_driver_cross hab Qbv
+      (continuousBVCoordinate γ 0) (continuousBVCoordinate γ 1) hQcont
+      (h * normalVector (t : Real.Angle) 0) (tangentVector (t : Real.Angle) 0)
+      (h * normalVector (t : Real.Angle) 1) (tangentVector (t : Real.Angle) 1)
+      (hcoord 0) (hcoord 1)]
+    have haeq := tangentLinePath_eq_coordinate K t ⟨a, ha⟩
+    have hbeq := tangentLinePath_eq_coordinate K t ⟨b, hb⟩
+    rw [haeq, hbeq]
+    simp only [Qbv]
+    simp [segmentArea, planeCrossProduct, normalVector, tangentVector, frame, h]
+    simp only [Q]
+    ring
+
+theorem tangentLinePath_convexLinear (t a b : ℝ)
+    (ha : a ∈ Set.Ioc (t - Real.pi) t) (hb : b ∈ Set.Ioc (t - Real.pi) t)
+    (hab : a ≤ b) :
+    ∃ F : ConvexBody Point → ContinuousBVPaths a b,
+      (∀ K, (F K).val = tangentLineRestriction K t a b ha hb) ∧
+      IsConvexLinear convexBodyCombination
+        (fun r x y ↦ (1 - (r : ℝ)) • x + (r : ℝ) • y) F := by
+  let F : ConvexBody Point → ContinuousBVPaths a b := fun K ↦
+    (tangentLinePath_segment_area K t a b ha hb hab).choose
+  have hF (K : ConvexBody Point) :
+      (F K).val = tangentLineRestriction K t a b ha hb :=
+    (tangentLinePath_segment_area K t a b ha hb hab).choose_spec.1
+  refine ⟨F, hF, ?_⟩
+  intro r K L
+  apply Subtype.ext
+  funext s
+  change (F (convexBodyCombination r K L)).val s =
+    ((1 - (r : ℝ)) • (F K).val + (r : ℝ) • (F L).val) s
+  rw [hF, hF, hF]
+  unfold tangentLineRestriction tangentLinePath
+  by_cases hs : (s : ℝ) < t
+  · simp only [hs, ↓reduceIte, Pi.add_apply, Pi.smul_apply]
+    exact (convexBody_maps_linear r K L).2.2.1 (s : ℝ) t hs
+      (by linarith [ha.1, s.property.1])
+  · have hst : (s : ℝ) = t := by
+      apply le_antisymm
+      · exact le_trans s.property.2 hb.2
+      · exact le_of_not_gt hs
+    simp only [hs, ↓reduceIte, Pi.add_apply, Pi.smul_apply]
+    simpa [hst] using (convexBody_maps_linear r K L).2.1 (t : Real.Angle) |>.2
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Convex.ArcCutBoundary`.
+* `Convex.ArcArea`.
+* `Convex.ArcBilinear`.
+* `Convex.ArcJordan`.
+* `Convex.ArcRegionArea`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Arc Cut Boundary
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open Set
+
+private theorem normalVector_injective_cut : Function.Injective normalVector := by
+  intro a b hab
+  induction a using Real.Angle.induction_on with
+  | _ a =>
+    induction b using Real.Angle.induction_on with
+    | _ b =>
+      apply Real.Angle.cos_sin_inj
+      · exact congrFun (congrArg WithLp.ofLp hab) 0
+      · exact congrFun (congrArg WithLp.ofLp hab) 1
+
+-- Duplicate of the current private UpperGraph helper; promote with the separation helper.
+private theorem exteriorNormal_eq_of_orthogonal_of_interior_nonempty_cut
+    (K : ConvexBody Point) (hK : (interior (K : Set Point)).Nonempty)
+    {p v : Point} (hv : v ≠ 0) {a b : Real.Angle}
+    (ha : IsExteriorNormal K p a) (hb : IsExteriorNormal K p b)
+    (hva : inner ℝ v (normalVector a) = 0)
+    (hvb : inner ℝ v (normalVector b) = 0) : a = b := by
+  let orientation : Orientation ℝ Point (Fin 2) :=
+    (EuclideanSpace.basisFun (Fin 2) ℝ).toBasis.orientation
+  rcases EuclideanGeometry.eq_or_eq_neg_of_unit_orthogonal orientation hv
+      (norm_normalVector a) (norm_normalVector b) hva hvb with hab | hab
+  · exact normalVector_injective_cut hab
+  · exfalso
+    obtain ⟨z, hz⟩ := hK
+    obtain ⟨ε, hε, hball⟩ := Metric.isOpen_iff.mp isOpen_interior z hz
+    let q := z + (ε / 2) • normalVector a
+    have hq : q ∈ K := interior_subset (hball (by
+      rw [Metric.mem_ball, dist_eq_norm]
+      rw [show q - z = (ε / 2) • normalVector a by simp [q], norm_smul,
+        norm_normalVector, Real.norm_eq_abs, abs_of_pos (div_pos hε (by norm_num))]
+      norm_num
+      linarith))
+    have haz := ha z (interior_subset hz)
+    have hbz := hb z (interior_subset hz)
+    have hba : normalVector b = -normalVector a := by rw [hab]; simp
+    rw [hba, inner_neg_right] at hbz
+    have heq : inner ℝ (z - p) (normalVector a) = 0 := by linarith
+    have haq := ha q hq
+    have hqp : q - p = (z - p) + (ε / 2) • normalVector a := by
+      dsimp only [q]
+      module
+    rw [hqp, inner_add_left, inner_smul_left, real_inner_self_eq_norm_sq,
+      norm_normalVector a, heq, zero_add] at haq
+    have : ε / 2 ≤ 0 := by simpa using haq
+    linarith
+
+private theorem isExteriorNormal_of_mem_exposedEdge
+    (K : ConvexBody Point) {p : Point} {a : Real.Angle}
+    (hp : p ∈ exposedEdge K a) : IsExteriorNormal K p a := by
+  intro q hq
+  have hqle := inner_le_supportValue K hq a
+  have hpEq := hp.2
+  change inner ℝ p (normalVector a) = supportValue K a at hpEq
+  rw [inner_sub_left, hpEq]
+  linarith
+
+/-- The frontier of a cut body is the retained convex boundary arc together with its chord. -/
+theorem frontier_eq_convexBoundaryArc_union_segment_of_cut
+    (K K' : ConvexBody Point) (a b t : ℝ) (P Q : Point)
+    (hat : a < t) (htb : t < b) (hba : b < a + Real.pi)
+    (hP : P = (edgeVertices K (a : Real.Angle)).1)
+    (hQ : Q = (edgeVertices K (b : Real.Angle)).2)
+    (hleft : ∀ s ∈ Set.Ioc (t - Real.pi) a, exposedEdge K' s = {P})
+    (hmiddle : ∀ s ∈ Set.Ioo a b, exposedEdge K' s = exposedEdge K s)
+    (hright : ∀ s ∈ Set.Ico b (t + Real.pi), exposedEdge K' s = {Q})
+    (hterminal : exposedEdge K' (t + Real.pi) = segment ℝ Q P)
+    (hInt : (interior (K' : Set Point)).Nonempty) :
+    frontier (K' : Set Point) = convexBoundaryArc K a b ∪ segment ℝ P Q := by
+  apply Set.Subset.antisymm
+  · intro x hx
+    obtain ⟨u, hxu⟩ := exists_mem_exposedEdge_of_mem_frontier K' hInt hx
+    let _ : Fact (0 < 2 * Real.pi) := ⟨by positivity⟩
+    let s := AddCircle.equivIoc (2 * Real.pi) (t - Real.pi) u
+    have hs : (s : ℝ) ∈ Set.Ioc (t - Real.pi) (t + Real.pi) := by
+      have := s.property
+      convert this using 1
+      ring_nf
+    have hsu : (((s : ℝ) : Real.Angle)) = u := AddCircle.coe_equivIoc
+    have hxs : x ∈ exposedEdge K' (s : ℝ) := by simpa [hsu] using hxu
+    rcases le_or_gt (s : ℝ) a with hsa | has
+    · left
+      left
+      left
+      have hface := hleft (s : ℝ) ⟨hs.1, hsa⟩
+      rw [hface] at hxs
+      simpa [hP] using hxs
+    · rcases lt_or_ge (s : ℝ) b with hsb | hbs
+      · left
+        left
+        right
+        refine Set.mem_iUnion_of_mem (s : ℝ) ?_
+        refine Set.mem_iUnion_of_mem ⟨has, hsb⟩ ?_
+        · rw [← hmiddle (s : ℝ) ⟨has, hsb⟩]
+          exact hxs
+      · rcases lt_or_eq_of_le hs.2 with hst | hst
+        · left
+          right
+          have hface := hright (s : ℝ) ⟨hbs, hst⟩
+          rw [hface] at hxs
+          simpa [hQ] using hxs
+        · right
+          rw [hst] at hxs
+          change x ∈ exposedEdge K'
+            ((t : Real.Angle) + (Real.pi : Real.Angle)) at hxs
+          rw [hterminal] at hxs
+          simpa [segment_symm] using hxs
+  · rintro x (hx | hx)
+    · rcases hx with hx | hx
+      · rcases hx with hx | hx
+        · have hPa : P ∈ exposedEdge K' (a : ℝ) := by
+            rw [hleft a ⟨by linarith [Real.pi_pos], le_rfl⟩]
+            simp
+          have hxP : x = P := by simpa [hP] using hx
+          rw [hxP]
+          exact exposedEdge_subset_frontier K'
+            (a : Real.Angle) hPa
+        · rcases Set.mem_iUnion.mp hx with ⟨s, hx⟩
+          rcases Set.mem_iUnion.mp hx with ⟨hs, hx⟩
+          rw [← hmiddle s hs] at hx
+          exact exposedEdge_subset_frontier K' (s : Real.Angle) hx
+      · have hQb : Q ∈ exposedEdge K' (b : ℝ) := by
+          rw [hright b ⟨le_rfl, by linarith [Real.pi_pos]⟩]
+          simp
+        have hxQ : x = Q := by simpa [hQ] using hx
+        rw [hxQ]
+        exact exposedEdge_subset_frontier K'
+          (b : Real.Angle) hQb
+    · rw [segment_symm, ← hterminal] at hx
+      exact exposedEdge_subset_frontier K'
+        ((t + Real.pi : ℝ) : Real.Angle) hx
+
+/-- A retained convex boundary arc meets its cutting chord only at the endpoints. -/
+theorem convexBoundaryArc_inter_segment_eq_endpoints_of_cut
+    (K K' : ConvexBody Point) (a b t c : ℝ) (P Q : Point)
+    (hat : a < t) (htb : t < b) (hba : b < a + Real.pi)
+    (hPQ : P ≠ Q)
+    (hPt : inner ℝ P (normalVector (t : Real.Angle)) = c)
+    (hQt : inner ℝ Q (normalVector (t : Real.Angle)) = c)
+    (hP : P = (edgeVertices K (a : Real.Angle)).1)
+    (hQ : Q = (edgeVertices K (b : Real.Angle)).2)
+    (hleft : ∀ s ∈ Set.Ioc (t - Real.pi) a, exposedEdge K' s = {P})
+    (hmiddle : ∀ s ∈ Set.Ioo a b, exposedEdge K' s = exposedEdge K s)
+    (hright : ∀ s ∈ Set.Ico b (t + Real.pi), exposedEdge K' s = {Q})
+    (hterminal : exposedEdge K' (t + Real.pi) = segment ℝ Q P)
+    (hInt : (interior (K' : Set Point)).Nonempty) :
+    convexBoundaryArc K a b ∩ segment ℝ P Q = {P, Q} := by
+  have hPa : P ∈ exposedEdge K' (a : ℝ) := by
+    rw [hleft a ⟨by linarith [Real.pi_pos], le_rfl⟩]
+    simp
+  have hQb : Q ∈ exposedEdge K' (b : ℝ) := by
+    rw [hright b ⟨le_rfl, by linarith [Real.pi_pos]⟩]
+    simp
+  have hPterm : P ∈ exposedEdge K' (t + Real.pi) := by
+    rw [hterminal]
+    exact right_mem_segment ℝ Q P
+  have hQterm : Q ∈ exposedEdge K' (t + Real.pi) := by
+    rw [hterminal]
+    exact left_mem_segment ℝ Q P
+  apply Set.Subset.antisymm
+  · rintro x ⟨hxarc, hxseg⟩
+    rcases hxarc with hxarc | hxQ
+    · rcases hxarc with hxP | hxmid
+      · left
+        simpa [hP] using hxP
+      · rcases Set.mem_iUnion.mp hxmid with ⟨s, hxmid⟩
+        rcases Set.mem_iUnion.mp hxmid with ⟨hs, hxsK⟩
+        have hxs : x ∈ exposedEdge K' (s : ℝ) := by
+          rw [hmiddle s hs]
+          exact hxsK
+        rw [segment_eq_image] at hxseg
+        obtain ⟨r, hr, hxr⟩ := hxseg
+        by_cases hr0 : r = 0
+        · left
+          subst r
+          simpa using hxr.symm
+        by_cases hr1 : r = 1
+        · right
+          subst r
+          simpa using hxr.symm
+        have hrpos : 0 < r := lt_of_le_of_ne hr.1 (Ne.symm hr0)
+        have hrlt : r < 1 := lt_of_le_of_ne hr.2 hr1
+        have hPK : P ∈ K' := hPa.1
+        have hQK : Q ∈ K' := hQb.1
+        have hPLe := inner_le_supportValue K' hPK (s : Real.Angle)
+        have hQLe := inner_le_supportValue K' hQK (s : Real.Angle)
+        have hxEq := hxs.2
+        change inner ℝ x (normalVector (s : Real.Angle)) =
+          supportValue K' (s : Real.Angle) at hxEq
+        rw [← hxr, inner_add_left, inner_smul_left, inner_smul_left] at hxEq
+        simp only [map_sub, map_one, RCLike.conj_to_real] at hxEq
+        have hPEq : inner ℝ P (normalVector (s : Real.Angle)) =
+            supportValue K' (s : Real.Angle) := by
+          nlinarith
+        have hQEq : inner ℝ Q (normalVector (s : Real.Angle)) =
+            supportValue K' (s : Real.Angle) := by
+          nlinarith
+        have hPs : P ∈ exposedEdge K' (s : ℝ) := ⟨hPK, hPEq⟩
+        have hQs : Q ∈ exposedEdge K' (s : ℝ) := ⟨hQK, hQEq⟩
+        have hos : inner ℝ (Q - P) (normalVector (s : Real.Angle)) = 0 := by
+          rw [inner_sub_left, hPEq, hQEq]
+          ring
+        have hot : inner ℝ (Q - P) (normalVector (t + Real.pi : Real.Angle)) = 0 := by
+          change inner ℝ (Q - P)
+            (normalVector (((t + Real.pi : ℝ) : Real.Angle))) = 0
+          rw [normalVector_add_pi, inner_neg_right, inner_sub_left, hPt, hQt]
+          ring
+        have hQP : Q - P ≠ 0 := sub_ne_zero.mpr (Ne.symm hPQ)
+        have hang : (s : Real.Angle) = ((t + Real.pi : ℝ) : Real.Angle) :=
+          exteriorNormal_eq_of_orthogonal_of_interior_nonempty_cut K' hInt hQP
+            (isExteriorNormal_of_mem_exposedEdge K' hPs)
+            (isExteriorNormal_of_mem_exposedEdge K' hPterm) hos hot
+        have hnv := congrArg normalVector hang
+        have hcos : Real.cos (s - (t + Real.pi)) = 1 := by
+          rw [← inner_normalVector_normalVector s (t + Real.pi), hnv,
+            inner_normalVector_self]
+        have htupper : t < a + Real.pi := lt_trans htb hba
+        have hlower : -(2 * Real.pi) < s - (t + Real.pi) := by
+          nlinarith [hs.1, htupper]
+        have hupper : s - (t + Real.pi) < 2 * Real.pi := by
+          nlinarith [hs.2, hat, Real.pi_pos]
+        have := (Real.cos_eq_one_iff_of_lt_of_lt hlower hupper).mp hcos
+        have hbtpi : b < t + Real.pi := by nlinarith [hba, hat]
+        linarith [hs.2, hbtpi]
+    · right
+      simpa [hQ] using hxQ
+  · intro x hx
+    rcases hx with hx | hx
+    · subst x
+      constructor
+      · left
+        left
+        simp [hP]
+      · exact left_mem_segment ℝ P Q
+    · subst x
+      constructor
+      · right
+        simp [hQ]
+      · exact right_mem_segment ℝ P Q
+
+/-- A convex body admits a counterclockwise BV frontier parametrization based off a fixed face. -/
+theorem exists_closedBVJordan_frontier_base_not_mem_chord
+    (K : ConvexBody Point) (t : ℝ) (P Q : Point)
+    (hInt : (interior (K : Set Point)).Nonempty)
+    (hterminal : exposedEdge K (t + Real.pi) = segment ℝ Q P) :
+    ∃ (a b : ℝ) (x : ContinuousBVPaths a b),
+      ∃ hab : a < b, IsOrientedJordanParametrization hab.le (frontier (K : Set Point))
+        true x.val ∧ x.val ⟨a, le_rfl, hab.le⟩ ∉ segment ℝ P Q := by
+  obtain ⟨R, hRt⟩ := exposedEdge_nonempty K (t : Real.Angle)
+  have hRnot : R ∉ segment ℝ P Q := by
+    intro hRseg
+    have hRopp : R ∈ exposedEdge K (t + Real.pi) := by
+      rw [hterminal]
+      simpa [segment_symm] using hRseg
+    have hv : tangentVector (t : Real.Angle) ≠ 0 := by
+      intro hzero
+      have := inner_tangentVector_self t
+      rw [hzero, inner_zero_left] at this
+      norm_num at this
+    have hot : inner ℝ (tangentVector (t : Real.Angle))
+        (normalVector (t : Real.Angle)) = 0 := by
+      rw [real_inner_comm]
+      exact inner_normalVector_tangentVector t
+    have hopp : inner ℝ (tangentVector (t : Real.Angle))
+        (normalVector (t + Real.pi : Real.Angle)) = 0 := by
+      change inner ℝ (tangentVector (t : Real.Angle))
+        (normalVector (((t + Real.pi : ℝ) : Real.Angle))) = 0
+      rw [normalVector_add_pi, inner_neg_right, hot, neg_zero]
+    have hang : (t : Real.Angle) = ((t + Real.pi : ℝ) : Real.Angle) :=
+      exteriorNormal_eq_of_orthogonal_of_interior_nonempty_cut K hInt hv
+        (isExteriorNormal_of_mem_exposedEdge K hRt)
+        (isExteriorNormal_of_mem_exposedEdge K hRopp) hot hopp
+    have hnv := congrArg normalVector hang
+    change normalVector (t : Real.Angle) =
+      normalVector (((t + Real.pi : ℝ) : Real.Angle)) at hnv
+    rw [normalVector_add_pi] at hnv
+    have hz : normalVector (t : Real.Angle) = 0 := by
+      ext i
+      have hi := congrFun (congrArg WithLp.ofLp hnv) i
+      simp only [PiLp.neg_apply] at hi
+      have : (normalVector (t : Real.Angle)) i = 0 := by linarith
+      exact this
+    have := norm_normalVector (t : Real.Angle)
+    rw [hz, norm_zero] at this
+    norm_num at this
+  obtain ⟨o, ho⟩ := hInt
+  obtain ⟨ρ, e, C, γ, hradial, he, hLip, hγ, hγJordan, hinterior⟩ :=
+    convexBody_radial_boundary K o ho
+  have hRrange : R ∈ Set.range γ.val := by
+    rw [hγJordan.2.2.2.1]
+    exact exposedEdge_subset_frontier K (t : Real.Angle) hRt
+  obtain ⟨s, hstop, hsR⟩ := exists_param_lt_top_of_mem_range (by positivity) γ.val
+    hγJordan.2.2.2.2.1 hRrange
+  by_cases hs0 : (s : ℝ) = 0
+  · refine ⟨0, 2 * Real.pi, γ, mul_pos (by norm_num) Real.pi_pos, hγJordan, ?_⟩
+    have hs : s = ⟨0, le_rfl, (mul_pos (by norm_num) Real.pi_pos).le⟩ :=
+      Subtype.ext hs0
+    rw [← hsR, hs] at hRnot
+    exact hRnot
+  · have hspos : 0 < (s : ℝ) := lt_of_le_of_ne s.property.1 (Ne.symm hs0)
+    obtain ⟨r, hrJordan, hr⟩ :=
+      exists_oriented_cyclic_rotation_eq_concat (by positivity) γ hγJordan s hspos hstop
+    refine ⟨0, 2, r, by norm_num, hrJordan, ?_⟩
+    simpa [hr, Function.concatUnitIntervals, hsR] using hRnot
+
+end MovingSofa
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The nonterminal boundary of a convex cut body realizes the corresponding convex boundary arc. -/
+theorem exists_rectifiableOrientedArc_convexBoundaryArc_of_cut
+    (K K' : ConvexBody Point) (a b t c : ℝ) (P Q : Point)
+    (hat : a < t) (htb : t < b) (hba : b < a + Real.pi)
+    (hPQ : P ≠ Q)
+    (hPt : inner ℝ P (normalVector (t : Real.Angle)) = c)
+    (hQt : inner ℝ Q (normalVector (t : Real.Angle)) = c)
+    (hP : P = (edgeVertices K (a : Real.Angle)).1)
+    (hQ : Q = (edgeVertices K (b : Real.Angle)).2)
+    (hleft : ∀ s ∈ Set.Ioc (t - Real.pi) a, exposedEdge K' s = {P})
+    (hmiddle : ∀ s ∈ Set.Ioo a b, exposedEdge K' s = exposedEdge K s)
+    (hright : ∀ s ∈ Set.Ico b (t + Real.pi), exposedEdge K' s = {Q})
+    (hterminal : exposedEdge K' (t + Real.pi) = segment ℝ Q P)
+    (hInt : (interior (K' : Set Point)).Nonempty) :
+    ∃ A : RectifiableOrientedArc,
+      A.val.carrier = convexBoundaryArc K a b ∧
+        A.val.startPoint = P ∧ A.val.endPoint = Q := by
+  have hfrontier := frontier_eq_convexBoundaryArc_union_segment_of_cut
+    K K' a b t P Q hat htb hba hP hQ hleft hmiddle hright hterminal hInt
+  have hinter := convexBoundaryArc_inter_segment_eq_endpoints_of_cut
+    K K' a b t c P Q hat htb hba hPQ hPt hQt hP hQ hleft hmiddle hright
+      hterminal hInt
+  obtain ⟨α, β, x, hαβ, hx, hbase⟩ :=
+    exists_closedBVJordan_frontier_base_not_mem_chord K' t P Q hInt hterminal
+  exact exists_rectifiableOrientedArc_of_closedJordan_cut hαβ hx P Q hPQ hbase
+    hfrontier hinter
+
+private theorem isExposed_exposedEdge_jordan (K : ConvexBody Point) (t : Real.Angle) :
+    IsExposed ℝ (K : Set Point) (exposedEdge K t) :=
+  isExposed_exposedEdge K t
+
+/-- A singleton exposed face of a segment is one of its endpoints. -/
+theorem endpoint_of_exposedEdge_eq_singleton_of_eq_segment
+    (K : ConvexBody Point) (x y P : Point) (s : Real.Angle)
+    (hK : (K : Set Point) = segment ℝ x y)
+    (hface : exposedEdge K s = {P}) : x = P ∨ y = P := by
+  have hPextreme : P ∈ Set.extremePoints ℝ (K : Set Point) := by
+    have hexposed : IsExposed ℝ (K : Set Point) {P} := by
+      rw [← hface]
+      exact isExposed_exposedEdge_jordan K s
+    exact hexposed.isExtreme.mem_extremePoints
+  rw [mem_extremePoints_iff_forall_segment] at hPextreme
+  have hxK : x ∈ (K : Set Point) := by rw [hK]; exact left_mem_segment ℝ x y
+  have hyK : y ∈ (K : Set Point) := by rw [hK]; exact right_mem_segment ℝ x y
+  have hPseg : P ∈ segment ℝ x y := by rw [← hK]; exact hPextreme.1
+  exact hPextreme.2 x hxK y hyK hPseg
+
+/-- A cut body with empty interior has its selected boundary arc equal to the endpoint segment. -/
+theorem convexBoundaryArc_eq_segment_of_cut_interior_empty
+    (K K' : ConvexBody Point) (a b t c : ℝ) (P Q : Point)
+    (hat : a < t) (htb : t < b) (hba : b < a + Real.pi)
+    (hPQ : P ≠ Q)
+    (hPt : inner ℝ P (normalVector (t : Real.Angle)) = c)
+    (hQt : inner ℝ Q (normalVector (t : Real.Angle)) = c)
+    (hP : P = (edgeVertices K (a : Real.Angle)).1)
+    (hQ : Q = (edgeVertices K (b : Real.Angle)).2)
+    (hleft : ∀ s ∈ Set.Ioc (t - Real.pi) a, exposedEdge K' s = {P})
+    (hmiddle : ∀ s ∈ Set.Ioo a b, exposedEdge K' s = exposedEdge K s)
+    (hright : ∀ s ∈ Set.Ico b (t + Real.pi), exposedEdge K' s = {Q})
+    (hInt : interior (K' : Set Point) = ∅) :
+    convexBoundaryArc K a b = segment ℝ P Q := by
+  have hPmem : P ∈ (K' : Set Point) := by
+    have ha : a ∈ Set.Ioc (t - Real.pi) a := ⟨by linarith, le_rfl⟩
+    have : P ∈ exposedEdge K' (a : Real.Angle) := by rw [hleft a ha]; simp
+    exact this.1
+  have hQmem : Q ∈ (K' : Set Point) := by
+    have hb : b ∈ Set.Ico b (t + Real.pi) := ⟨le_rfl, by linarith⟩
+    have : Q ∈ exposedEdge K' (b : Real.Angle) := by rw [hright b hb]; simp
+    exact this.1
+  have hnsub : ¬(K' : Set Point).Subsingleton := by
+    intro hs
+    exact hPQ (hs hPmem hQmem)
+  obtain ⟨x, y, hxy, hK'⟩ := K'.exists_eq_segment_of_interior_empty hnsub hInt
+  have hxP : x = P ∨ y = P := endpoint_of_exposedEdge_eq_singleton_of_eq_segment
+    K' x y P (a : Real.Angle) hK' (hleft a ⟨by linarith, le_rfl⟩)
+  have hxQ : x = Q ∨ y = Q := endpoint_of_exposedEdge_eq_singleton_of_eq_segment
+    K' x y Q (b : Real.Angle) hK' (hright b ⟨le_rfl, by linarith⟩)
+  have hK'PQ : (K' : Set Point) = segment ℝ P Q := by
+    rcases hxP with rfl | rfl <;> rcases hxQ with hxQ | hxQ
+    · exact (hPQ hxQ).elim
+    · simpa [hxQ] using hK'
+    · simpa [hxQ, segment_symm ℝ] using hK'
+    · exact (hPQ hxQ).elim
+  have horth : inner ℝ (Q - P) (normalVector (t : Real.Angle)) = 0 := by
+    rw [inner_sub_left, hQt, hPt, sub_self]
+  let d : Point × Point × Real.Angle := (P, Q, (t : Real.Angle))
+  have hd : IsSegmentPresentation K' d := ⟨hPQ, hK'PQ, horth⟩
+  have hface : exposedEdge K (t : Real.Angle) = (K' : Set Point) := by
+    rw [← hmiddle t ⟨hat, htb⟩]
+    exact exposedEdge_eq_segment_of_orthogonal K' d hd (t : Real.Angle) horth
+  apply Set.Subset.antisymm
+  · rw [convexBoundaryArc]
+    refine Set.union_subset (Set.union_subset (by
+      intro z hz
+      simp only [Set.mem_singleton_iff] at hz
+      subst z
+      rw [hP]
+      exact left_mem_segment ℝ _ _) ?_) (by
+      intro z hz
+      simp only [Set.mem_singleton_iff] at hz
+      subst z
+      rw [hQ]
+      exact right_mem_segment ℝ _ _)
+    refine Set.iUnion₂_subset fun s hs z hz ↦ ?_
+    rw [← hK'PQ]
+    have hz' : z ∈ exposedEdge K' (s : Real.Angle) := by
+      rw [hmiddle s hs]
+      exact hz
+    exact hz'.1
+  · intro z hz
+    have hzK' : z ∈ (K' : Set Point) := by rw [hK'PQ]; exact hz
+    have hzface : z ∈ exposedEdge K (t : Real.Angle) := by rw [hface]; exact hzK'
+    rw [convexBoundaryArc]
+    apply Set.mem_union_left
+    apply Set.mem_union_right
+    apply Set.mem_iUnion.mpr
+    refine ⟨t, Set.mem_iUnion.mpr ⟨⟨hat, htb⟩, hzface⟩⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Arc Area
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+/-- An oriented rectifiable arc agrees with the prescribed convex boundary arc and its
+endpoints. -/
+def RealizesConvexArc (K : ConvexBody Point) (a b : ℝ)
+    (Γ : RectifiableOrientedArc) : Prop :=
+  Γ.val.carrier = convexBoundaryArc K a b ∧
+    Γ.val.startPoint = (edgeVertices K (a : Real.Angle)).1 ∧
+    Γ.val.endPoint = (edgeVertices K (b : Real.Angle)).2
+
+/-- The signed area of a realization of the convex boundary arc, or zero if none exists. -/
+def convexArcArea (K : ConvexBody Point) (a b : ℝ) : ℝ := by
+  classical
+  exact if h : ∃ Γ, RealizesConvexArc K a b Γ then jordanArcArea h.choose else 0
+
+/-- Every realization of a convex boundary arc computes that arc's signed area. -/
+theorem convexArcArea_eq_jordanArcArea_of_realizes {K : ConvexBody Point} {a b : ℝ}
+    {Γ : RectifiableOrientedArc} (hΓ : RealizesConvexArc K a b Γ) :
+    convexArcArea K a b = jordanArcArea Γ := by
+  rw [convexArcArea, dite_eq_left ⟨Γ, hΓ⟩]
+  let Δ := Classical.choose (show ∃ Γ, RealizesConvexArc K a b Γ from ⟨Γ, hΓ⟩)
+  have hΔ : RealizesConvexArc K a b Δ :=
+    Classical.choose_spec (show ∃ Γ, RealizesConvexArc K a b Γ from ⟨Γ, hΓ⟩)
+  change curveAreaFunctional (Classical.choice Δ.property).path =
+    curveAreaFunctional (Classical.choice Γ.property).path
+  exact (curveArea_reparametrization.2.1 Δ Γ
+    (Classical.choice Δ.property) (Classical.choice Γ.property)
+    (hΔ.1.trans hΓ.1.symm)).1
+      (hΔ.2.1.trans hΓ.2.1.symm) (hΔ.2.2.trans hΓ.2.2.symm)
+
+/-- Any bounded-variation parametrization of a convex boundary arc computes that arc's signed
+area. -/
+theorem convexArcArea_eq_curveAreaFunctional_of_realizes {K : ConvexBody Point} {a b : ℝ}
+    {Γ : OrientedJordanArc} (p : ArcBVParametrization Γ)
+    (hΓ : RealizesConvexArc K a b ⟨Γ, ⟨p⟩⟩) :
+    convexArcArea K a b = curveAreaFunctional p.path := by
+  rw [convexArcArea_eq_jordanArcArea_of_realizes hΓ]
+  exact (curveArea_arc_same_carrier Γ Γ
+    (Classical.choice (⟨Γ, ⟨p⟩⟩ : RectifiableOrientedArc).property) p rfl).1 rfl rfl
+
+/-- A continuous bounded-variation path on `[a, b]` that traces a convex boundary arc
+injectively, from the arc's first vertex to its last, computes that arc's signed area. -/
+theorem convexArcArea_eq_curveAreaFunctional_of_injOn {K : ConvexBody Point} {α β a b : ℝ}
+    {f : ℝ → Point} (x : ContinuousBVPaths a b) (hab : a < b)
+    (hx : ∀ t : Set.Icc a b, x.val t = f t) (hinj : Set.InjOn f (Set.Icc a b))
+    (himage : f '' Set.Icc a b = convexBoundaryArc K α β)
+    (hstart : f a = (edgeVertices K (α : Real.Angle)).1)
+    (hend : f b = (edgeVertices K (β : Real.Angle)).2) :
+    convexArcArea K α β = curveAreaFunctional x := by
+  have hxinj : Function.Injective x.val := fun s t hst ↦
+    Subtype.ext (hinj s.2 t.2 (by rw [← hx s, ← hx t, hst]))
+  have hrange : Set.range x.val = convexBoundaryArc K α β := by
+    rw [← himage, Set.image_eq_range]
+    exact congrArg Set.range (funext hx)
+  have hstart' : x.val ⟨a, le_rfl, hab.le⟩ = (edgeVertices K (α : Real.Angle)).1 := by
+    rw [hx, hstart]
+  have hend' : x.val ⟨b, hab.le, le_rfl⟩ = (edgeVertices K (β : Real.Angle)).2 := by
+    rw [hx, hend]
+  let Γ : OrientedJordanArc :=
+    { carrier := convexBoundaryArc K α β
+      startPoint := (edgeVertices K (α : Real.Angle)).1
+      endPoint := (edgeVertices K (β : Real.Angle)).2
+      parametrizable :=
+        ⟨a, b, hab.le, x.val, x.property.1, hxinj, hrange, hstart', hend'⟩ }
+  exact convexArcArea_eq_curveAreaFunctional_of_realizes (Γ := Γ)
+    ⟨a, b, hab.le, x, hxinj, hrange, hstart', hend'⟩ ⟨rfl, rfl, rfl⟩
+
+private theorem exists_degenerate_convexArc (K : ConvexBody Point) (a b : ℝ)
+    (hab : a < b) (hba : b < a + Real.pi)
+    (heq : (edgeVertices K (a : Real.Angle)).1 =
+      (edgeVertices K (b : Real.Angle)).2) :
+    ∃ Γ : RectifiableOrientedArc,
+      RealizesConvexArc K a b Γ ∧ convexArcArea K a b = jordanArcArea Γ ∧
+      (Γ.val.startPoint = Γ.val.endPoint → Γ.val.carrier = {Γ.val.startPoint}) := by
+  let P := (edgeVertices K (a : Real.Angle)).1
+  have hcut := (convexBoundaryArc_cut K a b hab hba P P (supportingIntersection K a b)
+    rfl heq rfl).1 rfl
+  obtain ⟨Γ, hcarrier, hstart, hend, _⟩ := (segmentArea_jordan_and_frame P P).1
+  have hreal : RealizesConvexArc K a b Γ := by
+    refine ⟨hcarrier.trans (segment_same ℝ P) |>.trans hcut.2.symm,
+      hstart, hend.trans heq⟩
+  exact ⟨Γ, hreal, convexArcArea_eq_jordanArcArea_of_realizes hreal, fun _ ↦ by
+    rw [hcarrier, segment_same, hstart]⟩
+
+private theorem surfaceAreaMeasure_openArc_eq_zero_of_convexBoundaryArc_eq_singleton
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    {P : Point} (hArc : convexBoundaryArc K a b = {P}) :
+    surfaceAreaMeasure K ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b) = 0 := by
+  let E := (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b
+  have hmeas : MeasurableSet E := (Real.Angle.isOpen_image_Ioo a b).measurableSet
+  have hsubset : E ⊆ (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Icc a b :=
+    Set.image_mono Set.Ioo_subset_Icc_self
+  rw [(surfaceAreaMeasure_face_union K).2.2.2.2 E hmeas
+    (Or.inr ⟨a, b, hab.le, hba, hsubset⟩)]
+  have hunion_sub : (⋃ t ∈ E, exposedEdge K t) ⊆ {P} := by
+    intro x hx
+    simp only [Set.mem_iUnion] at hx
+    obtain ⟨t, ht⟩ := hx
+    obtain ⟨htE, hxt⟩ := ht
+    obtain ⟨r, hr, rfl⟩ := htE
+    have hxArc : x ∈ convexBoundaryArc K a b := by
+      change x ∈ ({(edgeVertices K (a : Real.Angle)).1} ∪
+        (⋃ t ∈ Set.Ioo a b, exposedEdge K (t : Real.Angle))) ∪
+        {(edgeVertices K (b : Real.Angle)).2}
+      apply Set.mem_union_left
+      apply Set.mem_union_right
+      exact Set.mem_iUnion.2 ⟨r, Set.mem_iUnion.2 ⟨hr, hxt⟩⟩
+    rw [hArc] at hxArc
+    exact hxArc
+  have hunion_nonempty : (⋃ t ∈ E, exposedEdge K t).Nonempty := by
+    let r := (a + b) / 2
+    have hr : r ∈ Set.Ioo a b := by dsimp [r]; constructor <;> linarith
+    obtain ⟨x, hx⟩ := exposedEdge_nonempty K (r : Real.Angle)
+    exact ⟨x, Set.mem_iUnion.2 ⟨(r : Real.Angle), Set.mem_iUnion.2
+      ⟨⟨r, hr, rfl⟩, hx⟩⟩⟩
+  have hunion : (⋃ t ∈ E, exposedEdge K t) = {P} :=
+    Set.Nonempty.subset_singleton_iff hunion_nonempty |>.mp hunion_sub
+  rw [hunion]
+  let _ := MeasureTheory.Measure.nullSingletonClass_hausdorff Point
+    (by norm_num : (0 : ℝ) < 1)
+  exact measure_singleton P
+
+private theorem convexArcArea_eq_integral_of_endpoints_eq (K : ConvexBody Point)
+    {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (heq : (edgeVertices K (a : Real.Angle)).1 =
+      (edgeVertices K (b : Real.Angle)).2) :
+    convexArcArea K a b =
+      (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        supportValue K t ∂surfaceAreaMeasure K) / 2 := by
+  let P := (edgeVertices K (a : Real.Angle)).1
+  have hcut := (convexBoundaryArc_cut K a b hab hba P P (supportingIntersection K a b)
+    rfl heq rfl).1 rfl
+  obtain ⟨Γ, hcarrier, hstart, hend, hΓarea⟩ := (segmentArea_jordan_and_frame P P).1
+  have hreal : RealizesConvexArc K a b Γ := by
+    refine ⟨hcarrier.trans (segment_same ℝ P) |>.trans hcut.2.symm,
+      hstart, hend.trans heq⟩
+  rw [convexArcArea_eq_jordanArcArea_of_realizes hreal, hΓarea]
+  have hμ := surfaceAreaMeasure_openArc_eq_zero_of_convexBoundaryArc_eq_singleton
+    K hab hba hcut.2
+  rw [MeasureTheory.setIntegral_measure_zero _ hμ]
+  simp [segmentArea, planeCrossProduct]
+  ring
+
+private theorem exists_pos_smul_tangentVector_of_cut
+    (K : ConvexBody Point) {a t : ℝ} (hat : a < t) (hta : t < a + Real.pi)
+    {P Q : Point} (hP : P = (edgeVertices K (a : Real.Angle)).1)
+    (hQK : Q ∈ K) (hne : P ≠ Q)
+    (hnormal : inner ℝ P (normalVector (t : Real.Angle)) =
+      inner ℝ Q (normalVector (t : Real.Angle))) :
+    ∃ d : ℝ, 0 < d ∧ Q - P = d • tangentVector (t : Real.Angle) := by
+  let d := inner ℝ (Q - P) (tangentVector (t : Real.Angle))
+  have hnormal0 : inner ℝ (Q - P) (normalVector (t : Real.Angle)) = 0 := by
+    rw [inner_sub_left, hnormal, sub_self]
+  have hdecomp : Q - P = d • tangentVector (t : Real.Angle) := by
+    rw [← inner_normalVector_smul_add_inner_tangentVector_smul
+      (Q - P) (t : Real.Angle), hnormal0, zero_smul, zero_add]
+  have hPa : inner ℝ P (normalVector (a : Real.Angle)) = supportValue K a := by
+    rw [hP]
+    exact (edgeVertices_fst_mem K (a : Real.Angle)).2
+  have hQa : inner ℝ Q (normalVector (a : Real.Angle)) ≤ supportValue K a :=
+    inner_le_supportValue K hQK (a : Real.Angle)
+  have hinner : inner ℝ (Q - P) (normalVector (a : Real.Angle)) ≤ 0 := by
+    rw [inner_sub_left, hPa]
+    linarith
+  have hsin : 0 < Real.sin (t - a) := Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hat) (by linarith)
+  have htana : inner ℝ (tangentVector (t : Real.Angle))
+      (normalVector (a : Real.Angle)) = -Real.sin (t - a) := by
+    rw [real_inner_comm]
+    have h := sin_sub_eq_neg_inner_normalVector_tangentVector
+      (a : Real.Angle) (t : Real.Angle)
+    have h' : (((t - a : ℝ) : Real.Angle)).sin =
+        -inner ℝ (normalVector (a : Real.Angle)) (tangentVector (t : Real.Angle)) := by
+      simpa only [Real.Angle.coe_sub] using h
+    rw [Real.Angle.sin_coe] at h'
+    linarith
+  have hdnonneg : 0 ≤ d := by
+    rw [hdecomp, real_inner_smul_left, htana] at hinner
+    nlinarith
+  have hdne : d ≠ 0 := by
+    intro hd
+    apply hne
+    have : Q - P = 0 := by rw [hdecomp, hd, zero_smul]
+    exact (sub_eq_zero.mp this).symm
+  exact ⟨d, lt_of_le_of_ne hdnonneg (Ne.symm hdne), hdecomp⟩
+
+/-- The half support integral against a surface measure is convex-bilinear in the pair of
+bodies, by convex-linearity of support functions and surface measures. -/
+theorem convexArcIntegral_bilinear (a b : ℝ) :
+    IsConvexBilinear convexBodyCombination convexBodyCombination realCombination
+      (fun K L : ConvexBody Point ↦ (1 / 2 : ℝ) *
+        ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+          supportValue K t ∂surfaceAreaMeasure L) := by
+  let E := (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b
+  have hcont (K : ConvexBody Point) : Continuous (fun u : Real.Angle ↦ supportValue K u) :=
+    (compactSet_support_continuity K K K.nonempty K.isCompact K.nonempty K.isCompact).2.2.1
+  have hint (K L : ConvexBody Point) :
+      Integrable (fun u : Real.Angle ↦ supportValue K u) (surfaceAreaMeasure L) := by
+    let _ : IsFiniteMeasure (surfaceAreaMeasure L) := (surfaceAreaMeasure_face_union L).1
+    exact (hcont K).integrable_of_hasCompactSupport
+      (isCompact_univ.of_isClosed_subset isClosed_closure (Set.subset_univ _))
+  constructor
+  · intro K t L M
+    have hmeasure := (convexBody_maps_linear t L M).2.2.2
+    change (1 / 2 : ℝ) * (∫ u in E, supportValue K u ∂surfaceAreaMeasure
+        (convexBodyCombination t L M)) = _
+    simp only [realCombination]
+    rw [hmeasure, Measure.restrict_add, Measure.restrict_smul, Measure.restrict_smul,
+      integral_add_measure ((hint K L).restrict.smul_measure ENNReal.ofReal_ne_top)
+        ((hint K M).restrict.smul_measure ENNReal.ofReal_ne_top)]
+    simp only [integral_smul_measure, ENNReal.toReal_ofReal,
+      sub_nonneg.mpr (show (t : ℝ) ≤ 1 from t.property.2), t.property.1]
+    ring
+  · intro L t K M
+    change (1 / 2 : ℝ) * (∫ u in E, supportValue (convexBodyCombination t K M) u
+      ∂surfaceAreaMeasure L) = _
+    simp only [realCombination]
+    have hfun : (fun u : Real.Angle ↦ supportValue (convexBodyCombination t K M) u) =
+        fun u ↦ (1 - (t : ℝ)) * supportValue K u + (t : ℝ) * supportValue M u := by
+      funext u
+      exact (convexBody_maps_linear t K M).1 u
+    rw [hfun, integral_add ((hint K L).restrict.const_mul _) ((hint M L).restrict.const_mul _),
+      integral_const_mul, integral_const_mul]
+    ring
+
+private theorem convexArcArea_quadratic_of_integral_eq (a b : ℝ)
+    (harea : ∀ K : ConvexBody Point, convexArcArea K a b =
+      (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        supportValue K t ∂surfaceAreaMeasure K) / 2) :
+    IsQuadraticFunctional convexBodyCombination (fun K ↦ convexArcArea K a b) := by
+  let B := fun K L : ConvexBody Point ↦ (1 / 2 : ℝ) *
+    ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+      supportValue K t ∂surfaceAreaMeasure L
+  refine ⟨B, convexArcIntegral_bilinear a b, ?_⟩
+  intro K
+  change convexArcArea K a b = B K K
+  rw [harea K]
+  simp only [B]
+  ring
+
+private theorem convexArc_area_of_realization_and_integral
+    (a b : ℝ)
+    (hgeom : ∀ K : ConvexBody Point,
+      ∃ Γ : RectifiableOrientedArc,
+        RealizesConvexArc K a b Γ ∧
+        (Γ.val.startPoint = Γ.val.endPoint → Γ.val.carrier = {Γ.val.startPoint}))
+    (harea : ∀ K : ConvexBody Point, convexArcArea K a b =
+      (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        supportValue K t ∂surfaceAreaMeasure K) / 2) :
+    (∀ K : ConvexBody Point, ∃ Γ : RectifiableOrientedArc,
+      RealizesConvexArc K a b Γ ∧ convexArcArea K a b = jordanArcArea Γ ∧
+      (Γ.val.startPoint = Γ.val.endPoint → Γ.val.carrier = {Γ.val.startPoint})) ∧
+    (∀ K : ConvexBody Point, convexArcArea K a b =
+      (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        supportValue K t ∂surfaceAreaMeasure K) / 2) ∧
+    IsQuadraticFunctional convexBodyCombination (fun K ↦ convexArcArea K a b) := by
+  refine ⟨?_, harea, convexArcArea_quadratic_of_integral_eq a b harea⟩
+  intro K
+  obtain ⟨Γ, hreal, hsingleton⟩ := hgeom K
+  exact ⟨Γ, hreal, convexArcArea_eq_jordanArcArea_of_realizes hreal, hsingleton⟩
+
+private theorem exists_convexArc_realization (a b : ℝ) (hab : a < b)
+    (hba : b < a + Real.pi) (K : ConvexBody Point) :
+    ∃ Γ : RectifiableOrientedArc,
+      RealizesConvexArc K a b Γ ∧
+      (Γ.val.startPoint = Γ.val.endPoint → Γ.val.carrier = {Γ.val.startPoint}) := by
+  by_cases heq : (edgeVertices K (a : Real.Angle)).1 =
+      (edgeVertices K (b : Real.Angle)).2
+  · obtain ⟨Γ, hreal, -, hsingleton⟩ := exists_degenerate_convexArc K a b hab hba heq
+    exact ⟨Γ, hreal, hsingleton⟩
+  · let P := (edgeVertices K (a : Real.Angle)).1
+    let Q := (edgeVertices K (b : Real.Angle)).2
+    let O := supportingIntersection K a b
+    obtain ⟨-, hcut⟩ := convexBoundaryArc_cut K a b hab hba P Q O rfl rfl rfl
+    obtain ⟨hncol, t, c, K', hat, htb, hPt, hQt, hcO, hK', hleft, hmiddle,
+      hright, hterminal⟩ := hcut heq
+    by_cases hInt : (interior (K' : Set Point)).Nonempty
+    · obtain ⟨Γ, hcarrier, hstart, hend⟩ :=
+        exists_rectifiableOrientedArc_convexBoundaryArc_of_cut K K' a b t c P Q
+          hat htb hba heq hPt hQt rfl rfl hleft hmiddle hright hterminal hInt
+      have hreal : RealizesConvexArc K a b Γ := ⟨hcarrier, hstart, hend⟩
+      refine ⟨Γ, hreal, ?_⟩
+      intro hendpoints
+      exact (heq (hstart.symm.trans (hendpoints.trans hend))) |>.elim
+    · have hInt' : interior (K' : Set Point) = ∅ := Set.not_nonempty_iff_eq_empty.mp hInt
+      have harc := convexBoundaryArc_eq_segment_of_cut_interior_empty K K' a b t c P Q
+        hat htb hba heq hPt hQt rfl rfl hleft hmiddle hright hInt'
+      obtain ⟨Γ, hcarrier, hstart, hend, -⟩ := (segmentArea_jordan_and_frame P Q).1
+      have hreal : RealizesConvexArc K a b Γ :=
+        ⟨hcarrier.trans harc.symm, hstart, hend⟩
+      refine ⟨Γ, hreal, ?_⟩
+      intro hendpoints
+      exact (heq (hstart.symm.trans (hendpoints.trans hend))) |>.elim
+
+private theorem surfaceAreaMeasure_restrict_openArc_eq_of_exposedEdge_eq
+    (K L : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hfaces : ∀ s ∈ Set.Ioo a b,
+      exposedEdge K (s : Real.Angle) = exposedEdge L (s : Real.Angle)) :
+    (surfaceAreaMeasure K).restrict
+        ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b) =
+      (surfaceAreaMeasure L).restrict
+        ((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b) := by
+  let E := (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b
+  have hE : MeasurableSet E := Real.Angle.isOpen_image_Ioo a b |>.measurableSet
+  apply Measure.ext
+  intro S hS
+  rw [Measure.restrict_apply hS, Measure.restrict_apply hS]
+  have hSE : MeasurableSet (S ∩ E) := hS.inter hE
+  have hdomain : S ∩ E ⊆ (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Icc a b :=
+    Set.inter_subset_right.trans (Set.image_mono Set.Ioo_subset_Icc_self)
+  rw [(surfaceAreaMeasure_face_union K).2.2.2.2 (S ∩ E) hSE
+      (Or.inr ⟨a, b, hab.le, hba, hdomain⟩),
+    (surfaceAreaMeasure_face_union L).2.2.2.2 (S ∩ E) hSE
+      (Or.inr ⟨a, b, hab.le, hba, hdomain⟩)]
+  congr 2
+  ext p
+  simp only [Set.mem_iUnion, Set.mem_inter_iff, E]
+  constructor
+  · rintro ⟨⟨hpS, s, hs, hsp⟩, hx⟩
+    subst p
+    exact ⟨⟨hpS, ⟨s, hs, rfl⟩⟩, hfaces s hs ▸ hx⟩
+  · rintro ⟨⟨hpS, s, hs, hsp⟩, hx⟩
+    subst p
+    exact ⟨⟨hpS, ⟨s, hs, rfl⟩⟩, hfaces s hs |>.symm ▸ hx⟩
+
+private theorem surfaceAreaMeasure_compl_openArc_union_terminal_eq_zero_of_cut
+    (K : ConvexBody Point) {a b t : ℝ} {P Q : Point}
+    (hleft : ∀ s ∈ Set.Ioc (t - Real.pi) a, exposedEdge K s = {P})
+    (hright : ∀ s ∈ Set.Ico b (t + Real.pi), exposedEdge K s = {Q})
+    (hInt : (interior (K : Set Point)).Nonempty) :
+    surfaceAreaMeasure K
+      (((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b ∪
+        {((t + Real.pi : ℝ) : Real.Angle)})ᶜ) = 0 := by
+  let E := (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b
+  let θ : Real.Angle := (t + Real.pi : ℝ)
+  let D := (E ∪ {θ})ᶜ
+  have hD : MeasurableSet D :=
+    ((Real.Angle.isOpen_image_Ioo a b).measurableSet.union
+      (measurableSet_singleton θ)).compl
+  have hface := (surfaceAreaMeasure_face_union K).2.2.2.2 D hD (Or.inl hInt)
+  have hunion : (⋃ u ∈ D, exposedEdge K u) ⊆ {P, Q} := by
+    intro x hx
+    obtain ⟨u, huD, hxu⟩ := Set.mem_iUnion₂.mp hx
+    let _ : Fact (0 < 2 * Real.pi) := ⟨by positivity⟩
+    let s := AddCircle.equivIoc (2 * Real.pi) (t - Real.pi) u
+    have hs : (s : ℝ) ∈ Set.Ioc (t - Real.pi) (t + Real.pi) := by
+      have hs' := s.property
+      convert hs' using 1
+      ring_nf
+    have hsu : (((s : ℝ) : Real.Angle)) = u := AddCircle.coe_equivIoc
+    have hxs : x ∈ exposedEdge K (s : ℝ) := by simpa [hsu] using hxu
+    rcases le_or_gt (s : ℝ) a with hsa | has
+    · rw [hleft (s : ℝ) ⟨hs.1, hsa⟩] at hxs
+      exact Or.inl (by simpa using hxs)
+    · rcases lt_or_ge (s : ℝ) b with hsb | hbs
+      · exfalso
+        apply huD
+        apply Set.mem_union_left
+        exact ⟨s, ⟨has, hsb⟩, hsu⟩
+      · rcases lt_or_eq_of_le hs.2 with hst | hst
+        · rw [hright (s : ℝ) ⟨hbs, hst⟩] at hxs
+          exact Or.inr (by simpa using hxs)
+        · exfalso
+          apply huD
+          apply Set.mem_union_right
+          simp only [Set.mem_singleton_iff, θ]
+          rw [← hsu, hst]
+  rw [show (((fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b ∪
+      {((t + Real.pi : ℝ) : Real.Angle)})ᶜ) = D by rfl, hface]
+  let _ := Measure.nullSingletonClass_hausdorff Point (by norm_num : (0 : ℝ) < 1)
+  exact measure_mono_null hunion
+    ((Set.toFinite {P, Q}).measure_zero (Measure.hausdorffMeasure 1))
+
+private theorem integral_eq_openArc_add_terminal_of_cut
+    (K : ConvexBody Point) {a b t : ℝ} {P Q : Point}
+    (hat : a < t) (htb : t < b) (hba : b < a + Real.pi)
+    (hleft : ∀ s ∈ Set.Ioc (t - Real.pi) a, exposedEdge K s = {P})
+    (hright : ∀ s ∈ Set.Ico b (t + Real.pi), exposedEdge K s = {Q})
+    (hInt : (interior (K : Set Point)).Nonempty) :
+    (∫ u, supportValue K u ∂surfaceAreaMeasure K) =
+      (∫ u in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        supportValue K u ∂surfaceAreaMeasure K) +
+      (surfaceAreaMeasure K).real {((t + Real.pi : ℝ) : Real.Angle)} *
+        supportValue K ((t + Real.pi : ℝ) : Real.Angle) := by
+  let E := (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b
+  let θ : Real.Angle := (t + Real.pi : ℝ)
+  let C := E ∪ {θ}
+  let _ : IsFiniteMeasure (surfaceAreaMeasure K) := (surfaceAreaMeasure_face_union K).1
+  have hcont : Continuous (fun u : Real.Angle ↦ supportValue K u) :=
+    (compactSet_support_continuity K K K.nonempty K.isCompact K.nonempty K.isCompact).2.2.1
+  have hint : Integrable (fun u : Real.Angle ↦ supportValue K u) (surfaceAreaMeasure K) :=
+    hcont.integrable_of_hasCompactSupport
+      (isCompact_univ.of_isClosed_subset isClosed_closure (Set.subset_univ _))
+  have hC : MeasurableSet C :=
+    (Real.Angle.isOpen_image_Ioo a b).measurableSet.union (measurableSet_singleton θ)
+  have hzero : surfaceAreaMeasure K Cᶜ = 0 := by
+    exact surfaceAreaMeasure_compl_openArc_union_terminal_eq_zero_of_cut
+      K hleft hright hInt
+  have hdisj : Disjoint E {θ} := by
+    rw [Set.disjoint_singleton_right]
+    rintro ⟨s, hs, heq⟩
+    have hlow : t - Real.pi < a := by linarith
+    have hupp : b < t + Real.pi := by linarith
+    have hsrange : s ∈ Set.Ioc (t - Real.pi) (t + Real.pi) :=
+      ⟨hlow.trans hs.1, (hs.2.trans hupp).le⟩
+    have htrange : t + Real.pi ∈ Set.Ioc (t - Real.pi) (t + Real.pi) :=
+      ⟨by linarith [Real.pi_pos], le_rfl⟩
+    have hinj := Real.Angle.injOn_coe_Ioc (by linarith [Real.pi_pos]) hsrange htrange heq
+    rw [hinj] at hs
+    exact (not_lt_of_ge hupp.le hs.2)
+  have hsplit := MeasureTheory.setIntegral_union hdisj (measurableSet_singleton θ)
+    hint.integrableOn hint.integrableOn
+  have hall := MeasureTheory.integral_add_compl hC hint
+  rw [MeasureTheory.setIntegral_measure_zero _ hzero, add_zero] at hall
+  rw [← hall, hsplit, MeasureTheory.integral_singleton]
+  rfl
+
+private theorem terminal_surface_term_eq_segmentArea_of_cut
+    (K : ConvexBody Point) {t c d : ℝ} {P Q : Point}
+    (hPt : inner ℝ P (normalVector (t : Real.Angle)) = c)
+    (hQt : inner ℝ Q (normalVector (t : Real.Angle)) = c)
+    (hterminal : exposedEdge K (t + Real.pi) = segment ℝ Q P)
+    (hd : 0 < d) (hdir : Q - P = d • tangentVector (t : Real.Angle)) :
+    (surfaceAreaMeasure K).real {((t + Real.pi : ℝ) : Real.Angle)} *
+        supportValue K ((t + Real.pi : ℝ) : Real.Angle) / 2 =
+      segmentArea Q P := by
+  let θ : Real.Angle := (t + Real.pi : ℝ)
+  have hterminal' : exposedEdge K θ = segment ℝ Q P := by
+    simpa only [θ, Real.Angle.coe_add] using hterminal
+  have hmass : (surfaceAreaMeasure K).real {θ} = d := by
+    have htangent_norm : ‖tangentVector (t : Real.Angle)‖ = 1 := by
+      have hsq : ‖tangentVector (t : Real.Angle)‖ ^ 2 = 1 := by
+        rw [← real_inner_self_eq_norm_sq, inner_tangentVector_self]
+      nlinarith [norm_nonneg (tangentVector (t : Real.Angle))]
+    rw [Measure.real, (surfaceAreaMeasure_atom_length K θ).1, hterminal',
+      MeasureTheory.hausdorffMeasure_segment, edist_dist]
+    simp only [dist_eq_norm, sub_eq_add_neg]
+    rw [show Q + -P = Q - P by rfl, hdir, norm_smul, htangent_norm,
+      mul_one, Real.norm_eq_abs, abs_of_pos hd]
+    exact ENNReal.toReal_ofReal hd.le
+  have hsupp : supportValue K θ = -c := by
+    have hPterm : P ∈ exposedEdge K θ := by
+      rw [hterminal']
+      exact right_mem_segment ℝ Q P
+    have h := hPterm.2
+    change inner ℝ P (normalVector θ) = supportValue K θ at h
+    change inner ℝ P (normalVector (((t + Real.pi : ℝ) : Real.Angle))) = _ at h
+    rw [normalVector_add_pi, inner_neg_right, hPt] at h
+    linarith
+  have hQline : Q ∈ normalLine θ (-c) := by
+    change inner ℝ Q (normalVector θ) = -c
+    change inner ℝ Q (normalVector (((t + Real.pi : ℝ) : Real.Angle))) = -c
+    rw [normalVector_add_pi, inner_neg_right, hQt]
+  have hPline : P ∈ normalLine θ (-c) := by
+    change inner ℝ P (normalVector θ) = -c
+    change inner ℝ P (normalVector (((t + Real.pi : ℝ) : Real.Angle))) = -c
+    rw [normalVector_add_pi, inner_neg_right, hPt]
+  have hdir' : P - Q = d • tangentVector θ := by
+    have htangent : tangentVector θ = -tangentVector (t : Real.Angle) := by
+      ext i
+      fin_cases i <;> simp [θ, tangentVector, frame]
+    calc
+      P - Q = -(Q - P) := by module
+      _ = -(d • tangentVector (t : Real.Angle)) := congrArg Neg.neg hdir
+      _ = d • tangentVector θ := by rw [htangent]; module
+  rw [hmass, hsupp]
+  rw [mul_comm]
+  exact ((segmentArea_jordan_and_frame Q P).2 θ (-c) d hQline hPline hdir').symm
+
+private theorem supportValue_eq_of_exposedEdge_eq (K L : ConvexBody Point)
+    (t : Real.Angle) (hface : exposedEdge K t = exposedEdge L t) :
+    supportValue K t = supportValue L t := by
+  obtain ⟨p, hp⟩ := exposedEdge_nonempty K t
+  have hp' : p ∈ exposedEdge L t := hface ▸ hp
+  exact hp.2.symm.trans hp'.2
+
+private theorem integral_openArc_eq_of_exposedEdge_eq
+    (K L : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hfaces : ∀ s ∈ Set.Ioo a b,
+      exposedEdge K (s : Real.Angle) = exposedEdge L (s : Real.Angle)) :
+    (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        supportValue K t ∂surfaceAreaMeasure K) =
+      ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        supportValue L t ∂surfaceAreaMeasure L := by
+  let E := (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b
+  have hμ := surfaceAreaMeasure_restrict_openArc_eq_of_exposedEdge_eq
+    K L hab hba hfaces
+  change (∫ t, supportValue K t ∂(surfaceAreaMeasure K).restrict E) =
+    ∫ t, supportValue L t ∂(surfaceAreaMeasure L).restrict E
+  rw [hμ]
+  apply MeasureTheory.integral_congr_ae
+  have hE : MeasurableSet E := Real.Angle.isOpen_image_Ioo a b |>.measurableSet
+  filter_upwards [ae_restrict_mem hE] with t ht
+  obtain ⟨s, hs, rfl⟩ := ht
+  exact supportValue_eq_of_exposedEdge_eq K L _ (hfaces s hs)
+
+private theorem jordanArcArea_eq_integral_of_cut_interior_nonempty
+    (K : ConvexBody Point) {a b t c d α β : ℝ} {P Q : Point}
+    {x : ContinuousBVPaths α β} {A : RectifiableOrientedArc}
+    (hαβ : α < β)
+    (hx : IsOrientedJordanParametrization hαβ.le (frontier (K : Set Point)) true x.val)
+    (hAarea : curveAreaFunctional x = jordanArcArea A + segmentArea Q P)
+    (hat : a < t) (htb : t < b) (hba : b < a + Real.pi)
+    (hPt : inner ℝ P (normalVector (t : Real.Angle)) = c)
+    (hQt : inner ℝ Q (normalVector (t : Real.Angle)) = c)
+    (hleft : ∀ s ∈ Set.Ioc (t - Real.pi) a, exposedEdge K s = {P})
+    (hright : ∀ s ∈ Set.Ico b (t + Real.pi), exposedEdge K s = {Q})
+    (hterminal : exposedEdge K (t + Real.pi) = segment ℝ Q P)
+    (hInt : (interior (K : Set Point)).Nonempty)
+    (hd : 0 < d) (hdir : Q - P = d • tangentVector (t : Real.Angle)) :
+    jordanArcArea A =
+      (∫ u in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        supportValue K u ∂surfaceAreaMeasure K) / 2 := by
+  have hxarea := curveArea_eq_jordanInterior_area α β hαβ.le
+    (frontier (K : Set Point)) x hx
+  rw [jordanInterior_frontier_eq_interior K.convex K.isCompact.isClosed
+    K.isCompact.isBounded K.nonempty] at hxarea
+  have hinterarea : ClassicalResults.area (interior (K : Set Point)) =
+      ClassicalResults.area (K : Set Point) := by
+    simp only [ClassicalResults.area]
+    rw [measure_interior_of_null_frontier (K.convex.addHaar_frontier volume)]
+  rw [hinterarea, (convexBody_area_support_integral.1 K)] at hxarea
+  have hsplit := integral_eq_openArc_add_terminal_of_cut K hat htb hba hleft hright hInt
+  have hterminalArea := terminal_surface_term_eq_segmentArea_of_cut K hPt hQt hterminal
+    hd hdir
+  rw [hAarea, hsplit] at hxarea
+  linarith
+
+private theorem convexArcArea_eq_integral_of_cut_interior_nonempty
+    (K K' : ConvexBody Point) {a b t c d α β : ℝ} {P Q : Point}
+    {x : ContinuousBVPaths α β} {A : RectifiableOrientedArc}
+    (hαβ : α < β)
+    (hx : IsOrientedJordanParametrization hαβ.le (frontier (K' : Set Point)) true x.val)
+    (hAcarrier : A.val.carrier = convexBoundaryArc K a b)
+    (hAstart : A.val.startPoint = P) (hAend : A.val.endPoint = Q)
+    (hAarea : curveAreaFunctional x = jordanArcArea A + segmentArea Q P)
+    (hP : P = (edgeVertices K (a : Real.Angle)).1)
+    (hQ : Q = (edgeVertices K (b : Real.Angle)).2)
+    (hat : a < t) (htb : t < b) (hba : b < a + Real.pi)
+    (hPt : inner ℝ P (normalVector (t : Real.Angle)) = c)
+    (hQt : inner ℝ Q (normalVector (t : Real.Angle)) = c)
+    (hmiddle : ∀ s ∈ Set.Ioo a b, exposedEdge K' s = exposedEdge K s)
+    (hleft : ∀ s ∈ Set.Ioc (t - Real.pi) a, exposedEdge K' s = {P})
+    (hright : ∀ s ∈ Set.Ico b (t + Real.pi), exposedEdge K' s = {Q})
+    (hterminal : exposedEdge K' (t + Real.pi) = segment ℝ Q P)
+    (hInt : (interior (K' : Set Point)).Nonempty)
+    (hd : 0 < d) (hdir : Q - P = d • tangentVector (t : Real.Angle)) :
+    convexArcArea K a b =
+      (∫ u in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        supportValue K u ∂surfaceAreaMeasure K) / 2 := by
+  have hreal : RealizesConvexArc K a b A := ⟨hAcarrier, hAstart.trans hP, hAend.trans hQ⟩
+  rw [convexArcArea_eq_jordanArcArea_of_realizes hreal]
+  rw [jordanArcArea_eq_integral_of_cut_interior_nonempty K' hαβ hx hAarea
+    hat htb hba hPt hQt hleft hright hterminal hInt hd hdir]
+  exact congrArg (fun z : ℝ ↦ z / 2)
+    (integral_openArc_eq_of_exposedEdge_eq K' K (hat.trans htb) hba hmiddle)
+
+private theorem integral_openArc_eq_segmentArea_of_segment
+    (K : ConvexBody Point) {a b t c d : ℝ} {P Q : Point}
+    (hat : a < t) (htb : t < b) (hba : b < a + Real.pi)
+    (hK : (K : Set Point) = segment ℝ P Q) (hPQ : P ≠ Q)
+    (hPt : inner ℝ P (normalVector (t : Real.Angle)) = c)
+    (hQt : inner ℝ Q (normalVector (t : Real.Angle)) = c)
+    (hd : 0 < d) (hdir : Q - P = d • tangentVector (t : Real.Angle)) :
+    (∫ u in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+      supportValue K u ∂surfaceAreaMeasure K) / 2 = segmentArea P Q := by
+  let E := (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b
+  let θ : Real.Angle := (t : ℝ)
+  have horth : inner ℝ (Q - P) (normalVector θ) = 0 := by
+    change inner ℝ (Q - P) (normalVector (t : Real.Angle)) = 0
+    rw [inner_sub_left, hQt, hPt, sub_self]
+  have hpres : IsSegmentPresentation K (P, Q, θ) := ⟨hPQ, hK, horth⟩
+  have htangent_norm : ‖tangentVector (t : Real.Angle)‖ = 1 := by
+    have hsq : ‖tangentVector (t : Real.Angle)‖ ^ 2 = 1 := by
+      rw [← real_inner_self_eq_norm_sq, inner_tangentVector_self]
+    nlinarith [norm_nonneg (tangentVector (t : Real.Angle))]
+  have hdist : dist P Q = d := by
+    rw [dist_eq_norm, show P - Q = -(Q - P) by module, norm_neg, hdir,
+      norm_smul, htangent_norm, mul_one, Real.norm_eq_abs, abs_of_pos hd]
+  have hE : MeasurableSet E := Real.Angle.isOpen_image_Ioo a b |>.measurableSet
+  have htE : θ ∈ E := ⟨t, ⟨hat, htb⟩, rfl⟩
+  have hopp : θ + (Real.pi : Real.Angle) ∉ E := by
+    rintro ⟨s, hs, heq⟩
+    have hsupper : s ≤ a + 2 * Real.pi := by
+      have : s < a + Real.pi := hs.2.trans hba
+      linarith [Real.pi_pos]
+    have htupper : t + Real.pi ≤ a + 2 * Real.pi := by
+      linarith [htb, hba, Real.pi_pos]
+    have hsrange : s ∈ Set.Ioc a (a + 2 * Real.pi) := ⟨hs.1, hsupper⟩
+    have htrange : t + Real.pi ∈ Set.Ioc a (a + 2 * Real.pi) :=
+      ⟨by linarith [hat, Real.pi_pos], htupper⟩
+    have hinj := Real.Angle.injOn_coe_Ioc (by linarith [Real.pi_pos])
+      hsrange htrange
+    have hst : s = t + Real.pi := hinj (by
+      change (s : Real.Angle) = ((t + Real.pi : ℝ) : Real.Angle)
+      simpa only [θ, Real.Angle.coe_add] using heq)
+    linarith [hs.2, hba]
+  have hmeasure : (surfaceAreaMeasure K).restrict E =
+      ENNReal.ofReal d • Measure.dirac θ := by
+    classical
+    rw [surfaceAreaMeasure_eq_segmentPresentation K (P, Q, θ) hpres, hdist,
+      Measure.restrict_smul, Measure.restrict_add]
+    simp [restrict_dirac, htE, hopp]
+  change (∫ u, supportValue K u ∂(surfaceAreaMeasure K).restrict E) / 2 = _
+  rw [hmeasure, MeasureTheory.integral_smul_measure, MeasureTheory.integral_dirac,
+    ENNReal.toReal_ofReal hd.le]
+  have hPline : P ∈ normalLine θ c := hPt
+  have hQline : Q ∈ normalLine θ c := hQt
+  have hsupp : supportValue K θ = c := by
+    have hpK : P ∈ K := by
+      change P ∈ (K : Set Point)
+      rw [hK]
+      exact left_mem_segment ℝ P Q
+    have hp : P ∈ exposedEdge K θ := by
+      rw [exposedEdge_eq_segment_of_orthogonal K (P, Q, θ) hpres θ horth]
+      exact hpK
+    have hpEq := hp.2
+    change inner ℝ P (normalVector θ) = supportValue K θ at hpEq
+    exact hpEq.symm.trans hPt
+  rw [hsupp]
+  simp only [smul_eq_mul]
+  convert ((segmentArea_jordan_and_frame P Q).2 θ c d hPline hQline hdir).symm using 1
+  ring
+
+private theorem convexArcArea_eq_integral_of_cut_interior_empty
+    (K K' : ConvexBody Point) {a b t c d : ℝ} {P Q : Point}
+    (hat : a < t) (htb : t < b) (hba : b < a + Real.pi)
+    (hPQ : P ≠ Q)
+    (hPt : inner ℝ P (normalVector (t : Real.Angle)) = c)
+    (hQt : inner ℝ Q (normalVector (t : Real.Angle)) = c)
+    (hP : P = (edgeVertices K (a : Real.Angle)).1)
+    (hQ : Q = (edgeVertices K (b : Real.Angle)).2)
+    (hleft : ∀ s ∈ Set.Ioc (t - Real.pi) a, exposedEdge K' s = {P})
+    (hmiddle : ∀ s ∈ Set.Ioo a b, exposedEdge K' s = exposedEdge K s)
+    (hright : ∀ s ∈ Set.Ico b (t + Real.pi), exposedEdge K' s = {Q})
+    (hInt : interior (K' : Set Point) = ∅)
+    (hd : 0 < d) (hdir : Q - P = d • tangentVector (t : Real.Angle)) :
+    convexArcArea K a b =
+      (∫ u in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        supportValue K u ∂surfaceAreaMeasure K) / 2 := by
+  have hPmem : P ∈ (K' : Set Point) := by
+    have : P ∈ exposedEdge K' (a : Real.Angle) := by
+      rw [hleft a ⟨by linarith [Real.pi_pos], le_rfl⟩]
+      simp
+    exact this.1
+  have hQmem : Q ∈ (K' : Set Point) := by
+    have : Q ∈ exposedEdge K' (b : Real.Angle) := by
+      rw [hright b ⟨le_rfl, by linarith [Real.pi_pos]⟩]
+      simp
+    exact this.1
+  have hnsub : ¬(K' : Set Point).Subsingleton := by
+    intro hs
+    exact hPQ (hs hPmem hQmem)
+  obtain ⟨x, y, hxy, hK'⟩ := K'.exists_eq_segment_of_interior_empty hnsub hInt
+  have hxP : x = P ∨ y = P := endpoint_of_exposedEdge_eq_singleton_of_eq_segment
+    K' x y P (a : Real.Angle) hK' (hleft a ⟨by linarith [Real.pi_pos], le_rfl⟩)
+  have hxQ : x = Q ∨ y = Q := endpoint_of_exposedEdge_eq_singleton_of_eq_segment
+    K' x y Q (b : Real.Angle) hK' (hright b ⟨le_rfl, by linarith [Real.pi_pos]⟩)
+  have hK'PQ : (K' : Set Point) = segment ℝ P Q := by
+    rcases hxP with rfl | rfl <;> rcases hxQ with hxQ | hxQ
+    · exact (hPQ hxQ).elim
+    · simpa [hxQ] using hK'
+    · simpa [hxQ, segment_symm ℝ] using hK'
+    · exact (hPQ hxQ).elim
+  have harc := convexBoundaryArc_eq_segment_of_cut_interior_empty K K' a b t c P Q
+    hat htb hba hPQ hPt hQt hP hQ hleft hmiddle hright hInt
+  obtain ⟨A, hAcarrier, hAstart, hAend, hAarea⟩ :=
+    (segmentArea_jordan_and_frame P Q).1
+  have hreal : RealizesConvexArc K a b A :=
+    ⟨hAcarrier.trans harc.symm, hAstart.trans hP, hAend.trans hQ⟩
+  rw [convexArcArea_eq_jordanArcArea_of_realizes hreal]
+  rw [hAarea]
+  rw [← integral_openArc_eq_segmentArea_of_segment K' hat htb hba hK'PQ hPQ
+    hPt hQt hd hdir]
+  exact congrArg (fun z : ℝ ↦ z / 2)
+    (integral_openArc_eq_of_exposedEdge_eq K' K (hat.trans htb) hba hmiddle)
+
+private theorem convexArcArea_eq_integral_of_endpoints_ne
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hne : (edgeVertices K (a : Real.Angle)).1 ≠
+      (edgeVertices K (b : Real.Angle)).2) :
+    convexArcArea K a b =
+      (∫ u in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        supportValue K u ∂surfaceAreaMeasure K) / 2 := by
+  let P := (edgeVertices K (a : Real.Angle)).1
+  let Q := (edgeVertices K (b : Real.Angle)).2
+  let O := supportingIntersection K a b
+  obtain ⟨-, hcut⟩ := convexBoundaryArc_cut K a b hab hba P Q O rfl rfl rfl
+  obtain ⟨hncol, t, c, K', hat, htb, hPt, hQt, hcO, hK', hleft, hmiddle,
+    hright, hterminal⟩ := hcut hne
+  have hnormal : inner ℝ P (normalVector (t : Real.Angle)) =
+      inner ℝ Q (normalVector (t : Real.Angle)) := hPt.trans hQt.symm
+  obtain ⟨d, hd, hdir⟩ := exists_pos_smul_tangentVector_of_cut K hat
+    (htb.trans hba) rfl (edgeVertices_snd_mem K (b : Real.Angle)).1 hne hnormal
+  by_cases hInt : (interior (K' : Set Point)).Nonempty
+  · have hfrontier := frontier_eq_convexBoundaryArc_union_segment_of_cut
+      K K' a b t P Q hat htb hba rfl rfl hleft hmiddle hright hterminal hInt
+    have hinter := convexBoundaryArc_inter_segment_eq_endpoints_of_cut
+      K K' a b t c P Q hat htb hba hne hPt hQt rfl rfl hleft hmiddle hright
+        hterminal hInt
+    obtain ⟨α, β, x, hαβ, hx, hbase⟩ :=
+      exists_closedBVJordan_frontier_base_not_mem_chord K' t P Q hInt hterminal
+    obtain ⟨A, hAcarrier, hAstart, hAend, hAarea⟩ :=
+      exists_rectifiableOrientedArc_of_cut_with_area K' hαβ hx hne hbase
+        hfrontier hinter hPt hQt hterminal hd hdir
+    exact convexArcArea_eq_integral_of_cut_interior_nonempty K K' hαβ hx
+      hAcarrier hAstart hAend hAarea rfl rfl hat htb hba hPt hQt hmiddle hleft hright
+        hterminal hInt hd hdir
+  · exact convexArcArea_eq_integral_of_cut_interior_empty K K' hat htb hba hne
+      hPt hQt rfl rfl hleft hmiddle hright (Set.not_nonempty_iff_eq_empty.mp hInt)
+      hd hdir
+
+theorem convexArc_area (a b : ℝ) (hab : a < b) (hba : b < a + Real.pi) :
+    (∀ K : ConvexBody Point, ∃ Γ : RectifiableOrientedArc,
+      RealizesConvexArc K a b Γ ∧ convexArcArea K a b = jordanArcArea Γ ∧
+      (Γ.val.startPoint = Γ.val.endPoint → Γ.val.carrier = {Γ.val.startPoint})) ∧
+    (∀ K : ConvexBody Point, convexArcArea K a b =
+      (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        supportValue K t ∂surfaceAreaMeasure K) / 2) ∧
+    IsQuadraticFunctional convexBodyCombination (fun K ↦ convexArcArea K a b) := by
+  apply convexArc_area_of_realization_and_integral a b
+    (exists_convexArc_realization a b hab hba)
+  intro K
+  by_cases heq : (edgeVertices K (a : Real.Angle)).1 =
+      (edgeVertices K (b : Real.Angle)).2
+  · exact convexArcArea_eq_integral_of_endpoints_eq K hab hba heq
+  · exact convexArcArea_eq_integral_of_endpoints_ne K hab hba heq
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Arc Bilinear
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+
+namespace MovingSofa
+
+/-- Half the coordinate cross Stieltjes integral over the open parameter interval. -/
+def openIntervalCrossIntegral {a b : ℝ} (f : Fin 2 → RightContinuousIntervalBV a b)
+    (g : Set.Icc a b → Point) : ℝ :=
+  (intervalStieltjesIntegral (f 1) (fun t ↦ g t 0) {t | a < (t : ℝ) ∧ (t : ℝ) < b} -
+    intervalStieltjesIntegral (f 0) (fun t ↦ g t 1) {t | a < (t : ℝ) ∧ (t : ℝ) < b}) / 2
+
+/-- The half-support integral is the vertex cross Stieltjes integral, for any bounded
+measurable selection from the first body's exposed edges. -/
+private theorem halfSupportIntegral_eq_openIntervalCrossIntegral
+    (K L : ConvexBody Point) {a b : ℝ} (hab : a < b) (hturn : b ≤ a + 2 * Real.pi)
+    (F : Fin 2 → RightContinuousIntervalBV a b)
+    (hF : ∀ (i : Fin 2) (E : Set (Icc a b)), MeasurableSet E →
+      (∀ t ∈ E, a < (t : ℝ)) →
+      intervalStieltjesMeasure (F i) E =
+        ∫ u in (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) '' E,
+          tangentVector u i ∂surfaceAreaMeasure L)
+    (W : Real.Angle → Point) (C : ℝ)
+    (hWm : ∀ i : Fin 2, Measurable fun t : Ioc a b ↦ W (((t : ℝ) : Real.Angle)) i)
+    (hWb : ∀ (u : Real.Angle) (i : Fin 2), ‖W u i‖ ≤ C)
+    (hWsupp : ∀ u : Real.Angle, planeCrossProduct (W u) (tangentVector u) = supportValue K u) :
+    ((1 / 2 : ℝ) * ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+      supportValue K t ∂surfaceAreaMeasure L) =
+      openIntervalCrossIntegral F (fun t ↦ W (((t : ℝ) : Real.Angle))) := by
+  have hWcoord : ∀ u : Real.Angle,
+      W u 0 * tangentVector u 1 - W u 1 * tangentVector u 0 = supportValue K u := hWsupp
+  have hE : MeasurableSet {t : Icc a b | a < (t : ℝ) ∧ (t : ℝ) < b} :=
+    measurableSet_Ioo.preimage measurable_subtype_coe
+  have hEa : ∀ t ∈ {t : Icc a b | a < (t : ℝ) ∧ (t : ℝ) < b}, a < (t : ℝ) :=
+    fun _ ht ↦ ht.1
+  have himg : (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) ''
+      {t : Icc a b | a < (t : ℝ) ∧ (t : ℝ) < b} =
+      (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b := by
+    ext u
+    constructor
+    · rintro ⟨t, ht, rfl⟩
+      exact ⟨(t : ℝ), ht, rfl⟩
+    · rintro ⟨s, hs, rfl⟩
+      exact ⟨⟨s, hs.1.le, hs.2.le⟩, hs, rfl⟩
+  have hsub : (fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) ''
+      {t : Icc a b | a < (t : ℝ) ∧ (t : ℝ) < b} ⊆
+      Set.range fun t : Ioc a b ↦ ((t : ℝ) : Real.Angle) := by
+    rintro u ⟨t, ht, rfl⟩
+    exact ⟨⟨(t : ℝ), ht.1, ht.2.le⟩, rfl⟩
+  have hSmeas : MeasurableSet ((fun t : Icc a b ↦ ((t : ℝ) : Real.Angle)) ''
+      {t : Icc a b | a < (t : ℝ) ∧ (t : ℝ) < b}) := by
+    rw [himg]
+    exact (Real.Angle.isOpen_image_Ioo a b).measurableSet
+  have h1 := intervalStieltjesIntegral_positiveVertex_coordinate_of_measurable L hab hturn F hF
+    1 (fun u ↦ W u 0) C (hWm 0) (fun u ↦ hWb u 0) _ hE hEa
+  have h0 := intervalStieltjesIntegral_positiveVertex_coordinate_of_measurable L hab hturn F hF
+    0 (fun u ↦ W u 1) C (hWm 1) (fun u ↦ hWb u 1) _ hE hEa
+  have hi1 := integrableOn_mul_tangentVector_of_bounded L hturn 1 (fun u ↦ W u 0) C (hWm 0)
+    (fun u ↦ hWb u 0) _ hSmeas hsub
+  have hi0 := integrableOn_mul_tangentVector_of_bounded L hturn 0 (fun u ↦ W u 1) C (hWm 1)
+    (fun u ↦ hWb u 1) _ hSmeas hsub
+  rw [openIntervalCrossIntegral, h1, h0, ← integral_sub hi1 hi0,
+    setIntegral_congr_fun hSmeas (fun u _ ↦ hWcoord u), himg]
+  ring
+
+theorem convexArc_bilinear_computation (a b : ℝ) (hab : a < b) (hba : b < a + Real.pi) :
+    IsConvexBilinear convexBodyCombination convexBodyCombination realCombination
+      (fun K L : ConvexBody Point ↦ (1 / 2 : ℝ) *
+        ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+          supportValue K t ∂surfaceAreaMeasure L) ∧
+    ∃ F : ConvexBody Point → Fin 2 → RightContinuousIntervalBV a b,
+      (∀ K i t, (F K i).toFun t = (edgeVertices K ((t : ℝ) : Real.Angle)).1 i) ∧
+      (∀ K L : ConvexBody Point,
+        ((1 / 2 : ℝ) * ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+          supportValue K t ∂surfaceAreaMeasure L) =
+          openIntervalCrossIntegral (F L) (fun t ↦ (edgeVertices K ((t : ℝ) : Real.Angle)).1) ∧
+        ((1 / 2 : ℝ) * ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+          supportValue K t ∂surfaceAreaMeasure L) =
+          openIntervalCrossIntegral (F L) (fun t ↦ (edgeVertices K ((t : ℝ) : Real.Angle)).2)) ∧
+      (∀ K : ConvexBody Point, convexArcArea K a b =
+        openIntervalCrossIntegral (F K) (fun t ↦ (edgeVertices K ((t : ℝ) : Real.Angle)).1)) := by
+  have hturn : b ≤ a + 2 * Real.pi := by linarith [Real.pi_pos]
+  refine ⟨convexArcIntegral_bilinear a b, ?_⟩
+  choose F hFtoFun hFmeasure using fun L : ConvexBody Point ↦
+    positiveVertex_stieltjes_surface L a b hab hturn
+  have hcross1 (K L : ConvexBody Point) :
+      ((1 / 2 : ℝ) * ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        supportValue K t ∂surfaceAreaMeasure L) =
+        openIntervalCrossIntegral (F L)
+          (fun t ↦ (edgeVertices K ((t : ℝ) : Real.Angle)).1) := by
+    obtain ⟨R, hR⟩ := K.isCompact.isBounded.subset_closedBall (0 : Point)
+    refine halfSupportIntegral_eq_openIntervalCrossIntegral K L hab hturn (F L) (hFmeasure L)
+      (fun u ↦ (edgeVertices K u).1) R
+      (fun i ↦ measurable_positiveVertex_coordinate_Ioc K hab.le i)
+      (fun u i ↦ ?_) (fun u ↦ ?_)
+    · have hmem := hR (edgeVertices_fst_mem K u).1
+      rw [Metric.mem_closedBall, dist_zero_right] at hmem
+      exact le_trans (by simpa [Real.norm_eq_abs] using Point.abs_apply_le_norm _ i) hmem
+    · rw [planeCrossProduct_tangentVector, (edgeVertices_fst_mem K u).2]
+  have hcross2 (K L : ConvexBody Point) :
+      ((1 / 2 : ℝ) * ∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        supportValue K t ∂surfaceAreaMeasure L) =
+        openIntervalCrossIntegral (F L)
+          (fun t ↦ (edgeVertices K ((t : ℝ) : Real.Angle)).2) := by
+    obtain ⟨R, hR⟩ := K.isCompact.isBounded.subset_closedBall (0 : Point)
+    refine halfSupportIntegral_eq_openIntervalCrossIntegral K L hab hturn (F L) (hFmeasure L)
+      (fun u ↦ (edgeVertices K u).2) R
+      (fun i ↦ measurable_negativeVertex_coordinate K hab.le i) (fun u i ↦ ?_) (fun u ↦ ?_)
+    · have hmem := hR (edgeVertices_snd_mem K u).1
+      rw [Metric.mem_closedBall, dist_zero_right] at hmem
+      exact le_trans (by simpa [Real.norm_eq_abs] using Point.abs_apply_le_norm _ i) hmem
+    · rw [planeCrossProduct_tangentVector, (edgeVertices_snd_mem K u).2]
+  refine ⟨F, hFtoFun, fun K L ↦ ⟨hcross1 K L, hcross2 K L⟩, fun K ↦ ?_⟩
+  have harea := (convexArc_area a b hab hba).2.1 K
+  have hK := hcross1 K K
+  rw [harea]
+  linarith
+
+/-- Antisymmetry of the vertex cross Stieltjes integral over the open interval, up to the two
+endpoint corrections: the negative vertices at `b` and the positive vertices at `a`. -/
+theorem openIntervalCrossIntegral_antisymm {a b : ℝ} (hab : a < b)
+    (K L : ConvexBody Point) (FK FL : Fin 2 → RightContinuousIntervalBV a b)
+    (hFK : ∀ i t, (FK i).toFun t =
+      (edgeVertices K (((t : Set.Icc a b) : ℝ) : Real.Angle)).1 i)
+    (hFL : ∀ i t, (FL i).toFun t =
+      (edgeVertices L (((t : Set.Icc a b) : ℝ) : Real.Angle)).1 i) :
+    openIntervalCrossIntegral FL (fun t ↦ (edgeVertices K ((t : ℝ) : Real.Angle)).2) -
+        openIntervalCrossIntegral FK (fun t ↦ (edgeVertices L ((t : ℝ) : Real.Angle)).1) =
+      segmentArea (edgeVertices K (b : Real.Angle)).2 (edgeVertices L (b : Real.Angle)).2 -
+        segmentArea (edgeVertices K (a : Real.Angle)).1 (edgeVertices L (a : Real.Angle)).1 := by
+  have key (i j : Fin 2) :
+      intervalStieltjesIntegral (FL j)
+          (fun t ↦ (edgeVertices K ((t : ℝ) : Real.Angle)).2 i)
+          {t : Set.Icc a b | a < (t : ℝ) ∧ (t : ℝ) < b} +
+        intervalStieltjesIntegral (FK i)
+          (fun t ↦ (edgeVertices L ((t : ℝ) : Real.Angle)).1 j)
+          {t : Set.Icc a b | a < (t : ℝ) ∧ (t : ℝ) < b} =
+      (edgeVertices K (b : Real.Angle)).2 i * (edgeVertices L (b : Real.Angle)).2 j -
+        (edgeVertices K (a : Real.Angle)).1 i * (edgeVertices L (a : Real.Angle)).1 j := by
+    have h := intervalStieltjes_integration_by_parts_Ioo a b hab (FK i) (FL j)
+    have h1 : intervalStieltjesIntegral (FL j) (Function.leftLim (FK i).toFun)
+          {t : Set.Icc a b | a < (t : ℝ) ∧ (t : ℝ) < b} =
+        intervalStieltjesIntegral (FL j)
+          (fun t ↦ (edgeVertices K ((t : ℝ) : Real.Angle)).2 i)
+          {t : Set.Icc a b | a < (t : ℝ) ∧ (t : ℝ) < b} :=
+      VectorMeasure.setIntegral_congr_fun
+        (fun t ht ↦ leftLim_positiveVertex_coordinate K FK hFK i t ht.1)
+    have h2 : (FL j).toFun =
+        fun t : Set.Icc a b ↦ (edgeVertices L ((t : ℝ) : Real.Angle)).1 j :=
+      funext (hFL j)
+    rw [h1, leftLim_positiveVertex_coordinate K FK hFK i ⟨b, hab.le, le_rfl⟩ hab,
+      leftLim_positiveVertex_coordinate L FL hFL j ⟨b, hab.le, le_rfl⟩ hab,
+      hFK i ⟨a, le_rfl, hab.le⟩, hFL j ⟨a, le_rfl, hab.le⟩, h2] at h
+    exact h
+  have k01 := key 0 1
+  have k10 := key 1 0
+  simp only [openIntervalCrossIntegral, segmentArea, planeCrossProduct]
+  linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Convex / Arc Jordan
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open Set
+
+/-- A rectifiable path traverses a segment with monotone surjective reparametrizations. -/
+def IsSegmentTraversal (γ : RectifiablePathData) (P Q : Point) : Prop :=
+  ∃ (φ : Set.Icc (0 : ℝ) 1 → Set.Icc γ.a γ.b)
+    (τ : Set.Icc (0 : ℝ) 1 → Set.Icc (0 : ℝ) 1),
+    Continuous φ ∧ Monotone φ ∧ Function.Surjective φ ∧
+    Continuous τ ∧ Monotone τ ∧ Function.Surjective τ ∧
+    ∀ s, γ.path.val (φ s) = (1 - (τ s : ℝ)) • P + (τ s : ℝ) • Q
+
+/-- A rectifiable path traverses an oriented Jordan arc in reverse. -/
+def IsReverseArcTraversal (γ : RectifiablePathData) {A : OrientedJordanArc}
+    (p : ArcBVParametrization A) : Prop :=
+  ∃ (φ : Set.Icc (0 : ℝ) 1 → Set.Icc γ.a γ.b)
+    (ψ : Set.Icc (0 : ℝ) 1 → Set.Icc p.a p.b),
+    Continuous φ ∧ Monotone φ ∧ Function.Surjective φ ∧
+    Continuous ψ ∧ Monotone ψ ∧ Function.Surjective ψ ∧
+    ∀ s, γ.path.val (φ s) = p.path.val
+      ⟨p.a + p.b - (ψ s : ℝ), by
+        constructor <;> linarith [(ψ s).property.1, (ψ s).property.2]⟩
+
+/-- A path traversing an oriented segment has the segment's signed area. -/
+theorem IsSegmentTraversal.curveAreaFunctional_eq {γ : RectifiablePathData} {P Q : Point}
+    (h : IsSegmentTraversal γ P Q) : curveAreaFunctional γ.path = segmentArea P Q := by
+  obtain ⟨φ, τ, hφc, hφm, hφs, hτc, hτm, hτs, heq⟩ := h
+  obtain ⟨y, hy, hymono, -⟩ := curveArea_comp_monotone_or_antitone_surjective
+    γ.ordered zero_le_one γ.path φ hφc hφs (Or.inl hφm)
+  obtain ⟨y', hy', hy'mono, -⟩ := curveArea_comp_monotone_or_antitone_surjective
+    zero_le_one zero_le_one (lineSegmentBVPath P Q) τ hτc hτs (Or.inl hτm)
+  have hyy : y = y' := by
+    refine Subtype.ext ?_
+    rw [hy, hy']
+    funext s
+    rw [Function.comp_apply, Function.comp_apply, heq s, lineSegmentBVPath_apply]
+  rw [← hymono hφm, hyy, hy'mono hτm, curveAreaFunctional_lineSegmentBVPath]
+
+/-- A path traversing an oriented Jordan arc backwards has the opposite signed area. -/
+theorem IsReverseArcTraversal.curveAreaFunctional_eq {γ : RectifiablePathData}
+    {A : OrientedJordanArc} {p : ArcBVParametrization A} (h : IsReverseArcTraversal γ p) :
+    curveAreaFunctional γ.path = -curveAreaFunctional p.path := by
+  obtain ⟨φ, ψ, hφc, hφm, hφs, hψc, hψm, hψs, heq⟩ := h
+  set ρ : Set.Icc (0 : ℝ) 1 → Set.Icc p.a p.b := fun s ↦
+    ⟨p.a + p.b - (ψ s : ℝ), by
+      constructor <;> linarith [(ψ s).property.1, (ψ s).property.2]⟩ with hρdef
+  have hcomp : ∀ s, γ.path.val (φ s) = p.path.val (ρ s) := heq
+  have hρc : Continuous ρ :=
+    (continuous_const.sub (continuous_subtype_val.comp hψc)).subtype_mk _
+  have hρa : Antitone ρ := fun s t hst ↦
+    Subtype.coe_le_coe.mp (by
+      simp only [hρdef]
+      linarith [Subtype.coe_le_coe.mpr (hψm hst)])
+  have hρs : Function.Surjective ρ := by
+    intro u
+    obtain ⟨s, hs⟩ := hψs ⟨p.a + p.b - (u : ℝ), by
+      constructor <;> linarith [u.property.1, u.property.2]⟩
+    refine ⟨s, Subtype.ext ?_⟩
+    have hval : ((ψ s : ℝ)) = p.a + p.b - (u : ℝ) := congrArg Subtype.val hs
+    simp only [hρdef, hval]
+    ring
+  obtain ⟨y, hy, hymono, -⟩ := curveArea_comp_monotone_or_antitone_surjective
+    γ.ordered zero_le_one γ.path φ hφc hφs (Or.inl hφm)
+  obtain ⟨y', hy', -, hy'anti⟩ := curveArea_comp_monotone_or_antitone_surjective
+    p.ordered zero_le_one p.path ρ hρc hρs (Or.inr hρa)
+  have hyy : y = y' := by
+    refine Subtype.ext ?_
+    rw [hy, hy']
+    funext s
+    exact hcomp s
+  rw [← hymono hφm, hyy, hy'anti hρa]
+
+private def reverseArcPath {A : OrientedJordanArc} (p : ArcBVParametrization A) :
+    ContinuousBVPaths p.a p.b where
+  val := p.path.val ∘ Set.Icc.reverse p.ordered
+  property := by
+    constructor
+    · exact p.path.property.1.comp (Set.Icc.continuous_reverse p.ordered)
+    · intro i
+      exact BoundedVariationOn.comp_antitone_surjective_Icc p.ordered
+        (p.path.property.2 i) (Set.Icc.antitone_reverse p.ordered)
+        (Set.Icc.surjective_reverse p.ordered)
+
+private lemma reverseArcPath_start {A : OrientedJordanArc}
+    (p : ArcBVParametrization A) :
+    (reverseArcPath p).val ⟨p.a, le_rfl, p.ordered⟩ = A.endPoint := by
+  simpa [reverseArcPath, Set.Icc.reverse] using p.end_eq
+
+private lemma isSegmentTraversal_lineSegmentBVPath (P Q : Point) :
+    IsSegmentTraversal
+      { a := 0, b := 1, ordered := by norm_num, path := lineSegmentBVPath P Q } P Q := by
+  refine ⟨id, id, continuous_id, monotone_id, Function.surjective_id,
+    continuous_id, monotone_id, Function.surjective_id, ?_⟩
+  intro s
+  simp [lineSegmentBVPath, Path.segment_apply, AffineMap.lineMap_apply_module']
+  module
+
+private theorem boundedVariation_concatUnitIntervals_coordinate_jordan
+    (p q : ContinuousBVPaths 0 1)
+    (hjoin : p.val ⟨1, by norm_num⟩ = q.val ⟨0, by norm_num⟩) (i : Fin 2) :
+    BoundedVariationOn (fun t ↦ Function.concatUnitIntervals p.val q.val t i) Set.univ :=
+  boundedVariation_concatUnitIntervals_coordinate p q hjoin i
+
+private def concatUnitPaths_jordan (p q : ContinuousBVPaths 0 1)
+    (hjoin : p.val ⟨1, by norm_num⟩ = q.val ⟨0, by norm_num⟩) :
+    ContinuousBVPaths 0 2 :=
+  ⟨Function.concatUnitIntervals p.val q.val,
+    Function.continuous_concatUnitIntervals p.property.1 q.property.1 hjoin,
+    boundedVariation_concatUnitIntervals_coordinate_jordan p q hjoin⟩
+
+private lemma concatUnitPaths_jordan_end (p q : ContinuousBVPaths 0 1)
+    (hjoin : p.val ⟨1, by norm_num⟩ = q.val ⟨0, by norm_num⟩) :
+    (concatUnitPaths_jordan p q hjoin).val ⟨2, by norm_num⟩ =
+      q.val ⟨1, by norm_num⟩ := by
+  change Function.concatUnitIntervals p.val q.val ⟨2, by norm_num⟩ = _
+  unfold Function.concatUnitIntervals
+  simp only
+  rw [ite_eq_right (by norm_num : ¬(2 : ℝ) ≤ 1)]
+  apply congrArg q.val
+  apply Subtype.ext
+  norm_num
+
+private def unitParam_jordan (a b : ℝ) (hab : a ≤ b) :
+    Set.Icc (0 : ℝ) 1 → Set.Icc a b :=
+  Set.Icc.convexComb ⟨a, le_rfl, hab⟩ ⟨b, hab, le_rfl⟩
+
+private lemma continuous_unitParam_jordan (a b : ℝ) (hab : a ≤ b) :
+    Continuous (unitParam_jordan a b hab) :=
+  Set.Icc.continuous_convexComb _ _
+
+private lemma monotone_unitParam_jordan (a b : ℝ) (hab : a ≤ b) :
+    Monotone (unitParam_jordan a b hab) := by
+  intro s t hst
+  apply Subtype.coe_le_coe.mp
+  simp only [unitParam_jordan, Set.Icc.coe_convexComb]
+  nlinarith [show (s : ℝ) ≤ t from hst]
+
+private lemma surjective_unitParam_jordan (a b : ℝ) (hab : a ≤ b) :
+    Function.Surjective (unitParam_jordan a b hab) :=
+  surjective_convexComb_endpoints a b hab
+
+private theorem exists_reverseArcUnitPath {A : OrientedJordanArc}
+    (p : ArcBVParametrization A) :
+    ∃ q : ContinuousBVPaths 0 1,
+      q.val = (reverseArcPath p).val ∘ unitParam_jordan p.a p.b p.ordered ∧
+      IsReverseArcTraversal
+        { a := 0, b := 1, ordered := by norm_num, path := q } p := by
+  obtain ⟨q, hq⟩ := continuousBVPaths_comp_monotone_surjective p.ordered
+    (reverseArcPath p) (unitParam_jordan p.a p.b p.ordered)
+    (continuous_unitParam_jordan _ _ _) (monotone_unitParam_jordan _ _ _)
+    (surjective_unitParam_jordan _ _ _)
+  refine ⟨q, hq, ?_⟩
+  let φ : Set.Icc (0 : ℝ) 1 → Set.Icc (0 : ℝ) 1 := id
+  let ψ := unitParam_jordan p.a p.b p.ordered
+  refine ⟨φ, ψ, continuous_id, monotone_id, Function.surjective_id,
+    continuous_unitParam_jordan _ _ _, monotone_unitParam_jordan _ _ _,
+    surjective_unitParam_jordan _ _ _, ?_⟩
+  intro s
+  rw [hq]
+  change p.path.val (Set.Icc.reverse p.ordered (ψ s)) = p.path.val _
+  simp only [ψ, unitParam_jordan, Set.Icc.reverse, Set.Icc.coe_convexComb]
+
+private def doubleParam_jordan : Set.Icc (0 : ℝ) 1 → Set.Icc (0 : ℝ) 2 :=
+  fun t ↦ ⟨2 * (t : ℝ), by constructor <;> nlinarith [t.property.1, t.property.2]⟩
+
+private lemma continuous_doubleParam_jordan : Continuous doubleParam_jordan :=
+  Continuous.subtype_mk (continuous_const.mul continuous_subtype_val) _
+
+private lemma monotone_doubleParam_jordan : Monotone doubleParam_jordan := by
+  intro s t hst
+  exact Subtype.coe_le_coe.mp (mul_le_mul_of_nonneg_left hst (by norm_num))
+
+private lemma surjective_doubleParam_jordan : Function.Surjective doubleParam_jordan := by
+  intro t
+  refine ⟨⟨(t : ℝ) / 2, by constructor <;> nlinarith [t.property.1, t.property.2]⟩, ?_⟩
+  apply Subtype.ext
+  change 2 * ((t : ℝ) / 2) = t
+  ring
+
+private def reparamTwoToUnit_jordan (q : ContinuousBVPaths 0 2) :
+    ContinuousBVPaths 0 1 :=
+  ⟨q.val ∘ doubleParam_jordan,
+    q.property.1.comp continuous_doubleParam_jordan,
+    fun i ↦ BoundedVariationOn.comp_monotone_surjective_Icc (by norm_num)
+      (q.property.2 i) monotone_doubleParam_jordan surjective_doubleParam_jordan⟩
+
+private lemma reparamTwoToUnit_one (q : ContinuousBVPaths 0 2) :
+    (reparamTwoToUnit_jordan q).val ⟨1, by norm_num⟩ = q.val ⟨2, by norm_num⟩ := by
+  change q.val (doubleParam_jordan ⟨1, by norm_num⟩) = q.val ⟨2, by norm_num⟩
+  rw [show doubleParam_jordan ⟨1, by norm_num⟩ = ⟨2, by norm_num⟩ by
+    apply Subtype.ext
+    norm_num [doubleParam_jordan]]
+
+private def concatThreeUnitPaths_jordan (p₀ p₁ p₂ : ContinuousBVPaths 0 1)
+    (h01 : p₀.val ⟨1, by norm_num⟩ = p₁.val ⟨0, by norm_num⟩)
+    (h12 : p₁.val ⟨1, by norm_num⟩ = p₂.val ⟨0, by norm_num⟩) :
+    ContinuousBVPaths 0 2 :=
+  let q := concatUnitPaths_jordan p₀ p₁ h01
+  concatUnitPaths_jordan (reparamTwoToUnit_jordan q) p₂ (by
+    rw [reparamTwoToUnit_one]
+    exact (concatUnitPaths_jordan_end p₀ p₁ h01).trans h12)
+
+private lemma concatThreeUnitPaths_first_jordan
+    (p₀ p₁ p₂ : ContinuousBVPaths 0 1)
+    (h01 : p₀.val ⟨1, by norm_num⟩ = p₁.val ⟨0, by norm_num⟩)
+    (h12 : p₁.val ⟨1, by norm_num⟩ = p₂.val ⟨0, by norm_num⟩)
+    (t : Set.Icc (0 : ℝ) 1) :
+    (concatThreeUnitPaths_jordan p₀ p₁ p₂ h01 h12).val
+      ⟨(t : ℝ) / 2, by constructor <;> nlinarith [t.property.1, t.property.2]⟩ =
+        p₀.val t := by
+  simp only [concatThreeUnitPaths_jordan, concatUnitPaths_jordan,
+    reparamTwoToUnit_jordan]
+  unfold Function.concatUnitIntervals
+  rw [ite_eq_left (by nlinarith [t.property.1, t.property.2] : (t : ℝ) / 2 ≤ 1)]
+  have houter : Set.projIcc 0 1 (by norm_num) ((t : ℝ) / 2) =
+      ⟨(t : ℝ) / 2, by constructor <;> nlinarith [t.property.1, t.property.2]⟩ := by
+    exact Set.projIcc_of_mem (by norm_num) (by
+      constructor <;> nlinarith [t.property.1, t.property.2])
+  rw [houter]
+  change Function.concatUnitIntervals p₀.val p₁.val
+    (doubleParam_jordan ⟨(t : ℝ) / 2, by
+      constructor <;> nlinarith [t.property.1, t.property.2]⟩) = p₀.val t
+  rw [show doubleParam_jordan ⟨(t : ℝ) / 2, by
+      constructor <;> nlinarith [t.property.1, t.property.2]⟩ =
+      ⟨(t : ℝ), ⟨t.property.1, t.property.2.trans (by norm_num)⟩⟩ by
+    apply Subtype.ext
+    simp [doubleParam_jordan]
+    ring]
+  unfold Function.concatUnitIntervals
+  rw [ite_eq_left t.property.2]
+  apply congrArg p₀.val
+  exact Set.projIcc_of_mem (by norm_num) t.property
+
+private lemma concatThreeUnitPaths_middle_jordan
+    (p₀ p₁ p₂ : ContinuousBVPaths 0 1)
+    (h01 : p₀.val ⟨1, by norm_num⟩ = p₁.val ⟨0, by norm_num⟩)
+    (h12 : p₁.val ⟨1, by norm_num⟩ = p₂.val ⟨0, by norm_num⟩)
+    (t : Set.Icc (0 : ℝ) 1) :
+    (concatThreeUnitPaths_jordan p₀ p₁ p₂ h01 h12).val
+      ⟨(1 + (t : ℝ)) / 2, by
+        constructor <;> nlinarith [t.property.1, t.property.2]⟩ = p₁.val t := by
+  simp only [concatThreeUnitPaths_jordan, concatUnitPaths_jordan,
+    reparamTwoToUnit_jordan]
+  unfold Function.concatUnitIntervals
+  rw [ite_eq_left (by nlinarith [t.property.2] : (1 + (t : ℝ)) / 2 ≤ 1)]
+  have houter : Set.projIcc 0 1 (by norm_num) ((1 + (t : ℝ)) / 2) =
+      ⟨(1 + (t : ℝ)) / 2, by
+        constructor <;> nlinarith [t.property.1, t.property.2]⟩ := by
+    exact Set.projIcc_of_mem (by norm_num) (by
+      constructor <;> nlinarith [t.property.1, t.property.2])
+  rw [houter]
+  change Function.concatUnitIntervals p₀.val p₁.val
+    (doubleParam_jordan ⟨(1 + (t : ℝ)) / 2, by
+      constructor <;> nlinarith [t.property.1, t.property.2]⟩) = p₁.val t
+  rw [show doubleParam_jordan ⟨(1 + (t : ℝ)) / 2, by
+      constructor <;> nlinarith [t.property.1, t.property.2]⟩ =
+      ⟨1 + (t : ℝ), by
+        constructor <;> nlinarith [t.property.1, t.property.2]⟩ by
+    apply Subtype.ext
+    simp [doubleParam_jordan]
+    ring]
+  unfold Function.concatUnitIntervals
+  by_cases ht : (t : ℝ) = 0
+  · have ht' : t = ⟨0, by norm_num⟩ := Subtype.ext ht
+    subst t
+    simpa using h01
+  · have htpos : 0 < (t : ℝ) := lt_of_le_of_ne t.property.1 (Ne.symm ht)
+    rw [ite_eq_right (by linarith : ¬(1 + (t : ℝ) ≤ 1))]
+    apply congrArg p₁.val
+    apply Subtype.ext
+    simp only [Set.coe_projIcc]
+    rw [show 1 + (t : ℝ) - 1 = t by ring, min_eq_right t.property.2,
+      max_eq_right t.property.1]
+
+private lemma concatThreeUnitPaths_last_jordan
+    (p₀ p₁ p₂ : ContinuousBVPaths 0 1)
+    (h01 : p₀.val ⟨1, by norm_num⟩ = p₁.val ⟨0, by norm_num⟩)
+    (h12 : p₁.val ⟨1, by norm_num⟩ = p₂.val ⟨0, by norm_num⟩)
+    (t : Set.Icc (0 : ℝ) 1) :
+    (concatThreeUnitPaths_jordan p₀ p₁ p₂ h01 h12).val
+      ⟨1 + (t : ℝ), by
+        constructor <;> nlinarith [t.property.1, t.property.2]⟩ = p₂.val t := by
+  by_cases ht : (t : ℝ) = 0
+  · have ht' : t = ⟨0, by norm_num⟩ := Subtype.ext ht
+    rw [ht']
+    rw [show (⟨1 + ((⟨0, by norm_num⟩ : Set.Icc (0 : ℝ) 1) : ℝ), by
+      norm_num⟩ : Set.Icc (0 : ℝ) 2) = ⟨1, by norm_num⟩ by
+      apply Subtype.ext
+      norm_num]
+    change (concatThreeUnitPaths_jordan p₀ p₁ p₂ h01 h12).val
+      ⟨1, by norm_num⟩ = p₂.val ⟨0, by norm_num⟩
+    unfold concatThreeUnitPaths_jordan
+    change Function.concatUnitIntervals _ p₂.val ⟨1, by norm_num⟩ = _
+    unfold Function.concatUnitIntervals
+    rw [ite_eq_left (by norm_num : (1 : ℝ) ≤ 1)]
+    calc
+      _ = (reparamTwoToUnit_jordan
+          (concatUnitPaths_jordan p₀ p₁ h01)).val ⟨1, by norm_num⟩ := by
+        apply congrArg _
+        apply Subtype.ext
+        norm_num [Set.coe_projIcc]
+      _ = _ := by
+        rw [reparamTwoToUnit_one, concatUnitPaths_jordan_end]
+        exact h12
+  · have htpos : 0 < (t : ℝ) := lt_of_le_of_ne t.property.1 (Ne.symm ht)
+    simp only [concatThreeUnitPaths_jordan, concatUnitPaths_jordan]
+    unfold Function.concatUnitIntervals
+    rw [ite_eq_right (by linarith : ¬(1 + (t : ℝ) ≤ 1))]
+    apply congrArg p₂.val
+    apply Subtype.ext
+    simp only [Set.coe_projIcc]
+    rw [show 1 + (t : ℝ) - 1 = t by ring, min_eq_right t.property.2,
+      max_eq_right t.property.1]
+
+private theorem isPathConcatenation_concatThreeUnitPaths_jordan
+    (p₀ p₁ p₂ : ContinuousBVPaths 0 1)
+    (h01 : p₀.val ⟨1, by norm_num⟩ = p₁.val ⟨0, by norm_num⟩)
+    (h12 : p₁.val ⟨1, by norm_num⟩ = p₂.val ⟨0, by norm_num⟩) :
+    IsPathConcatenation
+      { a := 0, b := 2, ordered := by norm_num,
+        path := concatThreeUnitPaths_jordan p₀ p₁ p₂ h01 h12 }
+      ![{ a := 0, b := 1, ordered := by norm_num, path := p₀ },
+        { a := 0, b := 1, ordered := by norm_num, path := p₁ },
+        { a := 0, b := 1, ordered := by norm_num, path := p₂ }] := by
+  let cuts : Fin 4 → Set.Icc (0 : ℝ) 2 :=
+    ![⟨0, by norm_num⟩, ⟨1 / 2, by norm_num⟩,
+      ⟨1, by norm_num⟩, ⟨2, by norm_num⟩]
+  refine ⟨by norm_num, cuts, ?_, rfl, rfl, ?_⟩
+  · intro i j hij
+    fin_cases i <;> fin_cases j <;>
+      norm_num [cuts, Matrix.cons_val_zero, Matrix.cons_val_one] at hij <;>
+      norm_num [cuts, Matrix.cons_val_zero, Matrix.cons_val_one]
+  intro i
+  fin_cases i
+  · let φ : Set.Icc (0 : ℝ) 1 → Set.Icc (cuts (Fin.castSucc 0) : ℝ)
+        (cuts (Fin.succ 0) : ℝ) := fun t ↦ ⟨(t : ℝ) / 2, by
+          change (0 : ℝ) ≤ (t : ℝ) / 2 ∧ (t : ℝ) / 2 ≤ 1 / 2
+          constructor <;> nlinarith [t.property.1, t.property.2]⟩
+    refine ⟨φ, id, Continuous.subtype_mk (continuous_subtype_val.div_const 2) _,
+      ?_, ?_, continuous_id, monotone_id, Function.surjective_id, ?_⟩
+    · intro s t hst
+      exact Subtype.coe_le_coe.mp (div_le_div_of_nonneg_right hst (by norm_num))
+    · intro t
+      have ht : (0 : ℝ) ≤ t ∧ (t : ℝ) ≤ 1 / 2 := by
+        simpa [cuts, Matrix.cons_val_zero, Matrix.cons_val_one] using t.property
+      refine ⟨⟨2 * (t : ℝ), by
+        change (0 : ℝ) ≤ 2 * (t : ℝ) ∧ 2 * (t : ℝ) ≤ 1
+        constructor <;> nlinarith [ht.1, ht.2]⟩, ?_⟩
+      apply Subtype.ext
+      change 2 * (t : ℝ) / 2 = t
+      ring
+    · intro t
+      exact concatThreeUnitPaths_first_jordan p₀ p₁ p₂ h01 h12 t
+  · let φ : Set.Icc (0 : ℝ) 1 → Set.Icc (cuts (Fin.castSucc 1) : ℝ)
+        (cuts (Fin.succ 1) : ℝ) := fun t ↦ ⟨(1 + (t : ℝ)) / 2, by
+          change (1 / 2 : ℝ) ≤ (1 + (t : ℝ)) / 2 ∧ (1 + (t : ℝ)) / 2 ≤ 1
+          constructor <;> nlinarith [t.property.1, t.property.2]⟩
+    refine ⟨φ, id, Continuous.subtype_mk
+      ((continuous_const.add continuous_subtype_val).div_const 2) _, ?_, ?_,
+      continuous_id, monotone_id, Function.surjective_id, ?_⟩
+    · intro s t hst
+      apply Subtype.coe_le_coe.mp
+      have hst' : (s : ℝ) ≤ t := hst
+      change (1 + (s : ℝ)) / 2 ≤ (1 + (t : ℝ)) / 2
+      linarith
+    · intro t
+      have ht : (1 / 2 : ℝ) ≤ t ∧ (t : ℝ) ≤ 1 := by
+        simpa [cuts, Matrix.cons_val_zero, Matrix.cons_val_one] using t.property
+      refine ⟨⟨2 * (t : ℝ) - 1, by
+        change (0 : ℝ) ≤ 2 * (t : ℝ) - 1 ∧ 2 * (t : ℝ) - 1 ≤ 1
+        constructor <;> nlinarith [ht.1, ht.2]⟩, ?_⟩
+      apply Subtype.ext
+      change (1 + (2 * (t : ℝ) - 1)) / 2 = t
+      ring
+    · intro t
+      exact concatThreeUnitPaths_middle_jordan p₀ p₁ p₂ h01 h12 t
+  · let φ : Set.Icc (0 : ℝ) 1 → Set.Icc (cuts (Fin.castSucc 2) : ℝ)
+        (cuts (Fin.succ 2) : ℝ) := fun t ↦ ⟨1 + (t : ℝ), by
+          change (1 : ℝ) ≤ 1 + (t : ℝ) ∧ 1 + (t : ℝ) ≤ 2
+          constructor <;> nlinarith [t.property.1, t.property.2]⟩
+    refine ⟨φ, id, Continuous.subtype_mk
+      (continuous_const.add continuous_subtype_val) _, ?_, ?_,
+      continuous_id, monotone_id, Function.surjective_id, ?_⟩
+    · intro s t hst
+      apply Subtype.coe_le_coe.mp
+      have hst' : (s : ℝ) ≤ t := hst
+      change 1 + (s : ℝ) ≤ 1 + (t : ℝ)
+      linarith
+    · intro t
+      have ht : (1 : ℝ) ≤ t ∧ (t : ℝ) ≤ 2 := by
+        simpa [cuts, Matrix.cons_val_zero, Matrix.cons_val_one] using t.property
+      refine ⟨⟨(t : ℝ) - 1, by
+        change (0 : ℝ) ≤ (t : ℝ) - 1 ∧ (t : ℝ) - 1 ≤ 1
+        constructor <;> nlinarith [ht.1, ht.2]⟩, ?_⟩
+      apply Subtype.ext
+      change 1 + ((t : ℝ) - 1) = t
+      ring
+    · intro t
+      exact concatThreeUnitPaths_last_jordan p₀ p₁ p₂ h01 h12 t
+
+private lemma range_comp_surjective {α β γ : Type*} (f : β → γ) (g : α → β)
+    (hg : Function.Surjective g) : Set.range (f ∘ g) = Set.range f := by
+  apply Set.Subset.antisymm
+  · exact Set.range_comp_subset_range _ _
+  · rintro y ⟨x, rfl⟩
+    obtain ⟨z, rfl⟩ := hg x
+    exact ⟨z, rfl⟩
+
+private lemma range_concatThreeUnitPaths_jordan
+    (p₀ p₁ p₂ : ContinuousBVPaths 0 1)
+    (h01 : p₀.val ⟨1, by norm_num⟩ = p₁.val ⟨0, by norm_num⟩)
+    (h12 : p₁.val ⟨1, by norm_num⟩ = p₂.val ⟨0, by norm_num⟩) :
+    Set.range (concatThreeUnitPaths_jordan p₀ p₁ p₂ h01 h12).val =
+      Set.range p₀.val ∪ Set.range p₁.val ∪ Set.range p₂.val := by
+  let q := concatUnitPaths_jordan p₀ p₁ h01
+  have hqend : q.val ⟨2, by norm_num⟩ = p₂.val ⟨0, by norm_num⟩ :=
+    (concatUnitPaths_jordan_end p₀ p₁ h01).trans h12
+  have houter : (reparamTwoToUnit_jordan q).val ⟨1, by norm_num⟩ =
+      p₂.val ⟨0, by norm_num⟩ := (reparamTwoToUnit_one q).trans hqend
+  change Set.range (Function.concatUnitIntervals
+    (reparamTwoToUnit_jordan q).val p₂.val) = _
+  rw [Function.range_concatUnitIntervals _ _ houter]
+  have hrepr : Set.range (reparamTwoToUnit_jordan q).val = Set.range q.val := by
+    exact range_comp_surjective q.val doubleParam_jordan surjective_doubleParam_jordan
+  rw [hrepr]
+  change Set.range (Function.concatUnitIntervals p₀.val p₁.val) ∪ Set.range p₂.val = _
+  rw [Function.range_concatUnitIntervals _ _ h01]
+
+private lemma range_reverseArcUnitPath {A : OrientedJordanArc}
+    (p : ArcBVParametrization A) (q : ContinuousBVPaths 0 1)
+    (hq : q.val = (reverseArcPath p).val ∘ unitParam_jordan p.a p.b p.ordered) :
+    Set.range q.val = A.carrier := by
+  rw [hq, range_comp_surjective _ _ (surjective_unitParam_jordan _ _ _)]
+  change Set.range (p.path.val ∘ Set.Icc.reverse p.ordered) = _
+  rw [range_comp_surjective _ _ (Set.Icc.surjective_reverse p.ordered), p.range_eq]
+
+private theorem segment_fst_supportingIntersection_inter_body
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hne : (edgeVertices K (a : Real.Angle)).1 ≠
+      (edgeVertices K (b : Real.Angle)).2) :
+    segment ℝ (edgeVertices K (a : Real.Angle)).1 (supportingIntersection K a b) ∩
+        (K : Set Point) = {(edgeVertices K (a : Real.Angle)).1} := by
+  let P := (edgeVertices K (a : Real.Angle)).1
+  let O := supportingIntersection K (a : Real.Angle) (b : Real.Angle)
+  obtain ⟨d, hd, hO⟩ := supportingIntersection_eq_fst_add_pos_tangent K hab hba hne
+  change O = P + d • tangentVector (a : Real.Angle) at hO
+  apply Set.Subset.antisymm
+  · rintro z ⟨hzseg, hzK⟩
+    rw [segment_eq_image] at hzseg
+    obtain ⟨r, hr, rfl⟩ := hzseg
+    have hPn := (edgeVertices_fst_mem K (a : Real.Angle)).2
+    have hOn := supportingIntersection_inner_left K a b
+    have hzline : inner ℝ ((1 - r) • P + r • O) (normalVector (a : Real.Angle)) =
+        supportValue K a := by
+      rw [inner_add_left, real_inner_smul_left, real_inner_smul_left]
+      change (1 - r) * inner ℝ P (normalVector (a : Real.Angle)) +
+        r * inner ℝ O (normalVector (a : Real.Angle)) = _
+      rw [show inner ℝ P (normalVector (a : Real.Angle)) = supportValue K a from hPn,
+        show inner ℝ O (normalVector (a : Real.Angle)) = supportValue K a from hOn]
+      ring
+    have hzedge : (1 - r) • P + r • O ∈ exposedEdge K (a : Real.Angle) :=
+      ⟨hzK, hzline⟩
+    have hzle : inner ℝ ((1 - r) • P + r • O) (tangentVector (a : Real.Angle)) ≤
+        inner ℝ P (tangentVector (a : Real.Angle)) := by
+      rw [inner_edgeVertices_fst_tangent]
+      exact le_csSup ((isCompact_exposedEdge K _).image
+        (continuous_id.inner continuous_const)).bddAbove ⟨_, hzedge, rfl⟩
+    rw [hO, inner_add_left, real_inner_smul_left, real_inner_smul_left,
+      inner_add_left, real_inner_smul_left, inner_tangentVector_self] at hzle
+    have hr0 : r = 0 := by nlinarith [hr.1]
+    simp [hr0]
+  · intro z hz
+    have hzP : z = P := by simpa [P] using hz
+    subst z
+    exact ⟨left_mem_segment ℝ P O, (edgeVertices_fst_mem K _).1⟩
+
+private theorem segment_supportingIntersection_snd_inter_body
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hne : (edgeVertices K (a : Real.Angle)).1 ≠
+      (edgeVertices K (b : Real.Angle)).2) :
+    segment ℝ (supportingIntersection K a b) (edgeVertices K (b : Real.Angle)).2 ∩
+        (K : Set Point) = {(edgeVertices K (b : Real.Angle)).2} := by
+  let Q := (edgeVertices K (b : Real.Angle)).2
+  let O := supportingIntersection K (a : Real.Angle) (b : Real.Angle)
+  obtain ⟨d, hd, hO⟩ := supportingIntersection_eq_snd_sub_pos_tangent K hab hba hne
+  change O = Q - d • tangentVector (b : Real.Angle) at hO
+  apply Set.Subset.antisymm
+  · rintro z ⟨hzseg, hzK⟩
+    rw [segment_eq_image] at hzseg
+    obtain ⟨r, hr, rfl⟩ := hzseg
+    have hQn := (edgeVertices_snd_mem K (b : Real.Angle)).2
+    have hOn := supportingIntersection_inner_right K a b
+      (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hab) (by linarith)).ne'
+    have hzline : inner ℝ ((1 - r) • O + r • Q) (normalVector (b : Real.Angle)) =
+        supportValue K b := by
+      rw [inner_add_left, real_inner_smul_left, real_inner_smul_left]
+      change (1 - r) * inner ℝ O (normalVector (b : Real.Angle)) +
+        r * inner ℝ Q (normalVector (b : Real.Angle)) = _
+      rw [show inner ℝ O (normalVector (b : Real.Angle)) = supportValue K b from hOn,
+        show inner ℝ Q (normalVector (b : Real.Angle)) = supportValue K b from hQn]
+      ring
+    have hzedge : (1 - r) • O + r • Q ∈ exposedEdge K (b : Real.Angle) :=
+      ⟨hzK, hzline⟩
+    have hzge : inner ℝ Q (tangentVector (b : Real.Angle)) ≤
+        inner ℝ ((1 - r) • O + r • Q) (tangentVector (b : Real.Angle)) := by
+      rw [inner_edgeVertices_snd_tangent]
+      exact csInf_le ((isCompact_exposedEdge K _).image
+        (continuous_id.inner continuous_const)).bddBelow ⟨_, hzedge, rfl⟩
+    rw [hO, inner_add_left, real_inner_smul_left, real_inner_smul_left,
+      inner_sub_left, real_inner_smul_left, inner_tangentVector_self] at hzge
+    have hr1 : r = 1 := by nlinarith [hr.2]
+    simp [hr1]
+  · intro z hz
+    have hzQ : z = Q := by simpa [Q] using hz
+    subst z
+    exact ⟨right_mem_segment ℝ O Q, (edgeVertices_snd_mem K _).1⟩
+
+/-- A convex boundary arc lies in its convex body. -/
+theorem convexBoundaryArc_subset_body (K : ConvexBody Point) (a b : ℝ) :
+    convexBoundaryArc K a b ⊆ (K : Set Point) := by
+  rintro z (hz | hz)
+  · rcases hz with hz | hz
+    · simpa using hz ▸ (edgeVertices_fst_mem K (a : Real.Angle)).1
+    · rcases Set.mem_iUnion.mp hz with ⟨t, hz⟩
+      rcases Set.mem_iUnion.mp hz with ⟨_, hz⟩
+      exact hz.1
+  · simpa using hz ▸ (edgeVertices_snd_mem K (b : Real.Angle)).1
+
+private theorem segment_fst_inter_convexBoundaryArc
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hne : (edgeVertices K (a : Real.Angle)).1 ≠
+      (edgeVertices K (b : Real.Angle)).2) :
+    segment ℝ (edgeVertices K (a : Real.Angle)).1 (supportingIntersection K a b) ∩
+        convexBoundaryArc K a b = {(edgeVertices K (a : Real.Angle)).1} := by
+  apply Set.Subset.antisymm
+  · intro z hz
+    have hz' : z ∈ segment ℝ (edgeVertices K (a : Real.Angle)).1
+        (supportingIntersection K a b) ∩ (K : Set Point) :=
+      ⟨hz.1, convexBoundaryArc_subset_body K a b hz.2⟩
+    simpa [segment_fst_supportingIntersection_inter_body K hab hba hne] using hz'
+  · intro z hz
+    have hzP : z = (edgeVertices K (a : Real.Angle)).1 := by simpa using hz
+    subst z
+    exact ⟨left_mem_segment _ _ _, Or.inl (Or.inl rfl)⟩
+
+private theorem segment_snd_inter_convexBoundaryArc
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hne : (edgeVertices K (a : Real.Angle)).1 ≠
+      (edgeVertices K (b : Real.Angle)).2) :
+    segment ℝ (supportingIntersection K a b) (edgeVertices K (b : Real.Angle)).2 ∩
+        convexBoundaryArc K a b = {(edgeVertices K (b : Real.Angle)).2} := by
+  apply Set.Subset.antisymm
+  · intro z hz
+    have hz' : z ∈ segment ℝ (supportingIntersection K a b)
+        (edgeVertices K (b : Real.Angle)).2 ∩ (K : Set Point) :=
+      ⟨hz.1, convexBoundaryArc_subset_body K a b hz.2⟩
+    simpa [segment_supportingIntersection_snd_inter_body K hab hba hne] using hz'
+  · intro z hz
+    have hzQ : z = (edgeVertices K (b : Real.Angle)).2 := by simpa using hz
+    subst z
+    exact ⟨right_mem_segment _ _ _, Or.inr rfl⟩
+
+private theorem supporting_segments_inter
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi) :
+    segment ℝ (edgeVertices K (a : Real.Angle)).1 (supportingIntersection K a b) ∩
+      segment ℝ (supportingIntersection K a b) (edgeVertices K (b : Real.Angle)).2 =
+        {supportingIntersection K a b} := by
+  let P := (edgeVertices K (a : Real.Angle)).1
+  let Q := (edgeVertices K (b : Real.Angle)).2
+  let O := supportingIntersection K (a : Real.Angle) (b : Real.Angle)
+  have hsin : 0 < Real.sin (b - a) :=
+    Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hab) (by linarith)
+  have hPa : inner ℝ P (normalVector (a : Real.Angle)) = supportValue K a :=
+    (edgeVertices_fst_mem K _).2
+  have hOa : inner ℝ O (normalVector (a : Real.Angle)) = supportValue K a :=
+    supportingIntersection_inner_left K a b
+  have hOb : inner ℝ O (normalVector (b : Real.Angle)) = supportValue K b :=
+    supportingIntersection_inner_right K a b hsin.ne'
+  have hQb : inner ℝ Q (normalVector (b : Real.Angle)) = supportValue K b :=
+    (edgeVertices_snd_mem K _).2
+  have inner_eq_of_segment {X Y z v : Point} {c : ℝ}
+      (hX : inner ℝ X v = c) (hY : inner ℝ Y v = c)
+      (hz : z ∈ segment ℝ X Y) : inner ℝ z v = c := by
+    rw [segment_eq_image] at hz
+    obtain ⟨r, hr, rfl⟩ := hz
+    rw [inner_add_left, real_inner_smul_left, real_inner_smul_left, hX, hY]
+    ring
+  apply Set.Subset.antisymm
+  · rintro z ⟨hz₁, hz₂⟩
+    have hza : inner ℝ z (normalVector (a : Real.Angle)) = supportValue K a :=
+      inner_eq_of_segment hPa hOa hz₁
+    have hzb : inner ℝ z (normalVector (b : Real.Angle)) = supportValue K b :=
+      inner_eq_of_segment hOb hQb hz₂
+    have hna : inner ℝ (z - O) (normalVector (a : Real.Angle)) = 0 := by
+      rw [inner_sub_left, hza, hOa, sub_self]
+    have hnb : inner ℝ (z - O) (normalVector (b : Real.Angle)) = 0 := by
+      rw [inner_sub_left, hzb, hOb, sub_self]
+    have hnexp : normalVector (b : Real.Angle) =
+        Real.cos (b - a) • normalVector (a : Real.Angle) +
+          Real.sin (b - a) • tangentVector (a : Real.Angle) := by
+      simpa only [add_sub_cancel] using normalVector_add_real a (b - a)
+    have hta : inner ℝ (z - O) (tangentVector (a : Real.Angle)) = 0 := by
+      rw [hnexp, inner_add_right, inner_smul_right, inner_smul_right, hna,
+        mul_zero, zero_add] at hnb
+      exact (mul_eq_zero.mp hnb).resolve_left hsin.ne'
+    have hzo : z = O := by
+      rw [show z = O + (z - O) by abel, ← inner_normalVector_smul_add_inner_tangentVector_smul
+        (z - O) (a : Real.Angle), hna, hta, zero_smul, zero_smul, add_zero]
+      simp
+    simp [O, hzo]
+  · intro z hz
+    have hzO : z = O := by simpa [O] using hz
+    subst z
+    exact ⟨right_mem_segment _ _ _, left_mem_segment _ _ _⟩
+
+private def brokenSupportPath (K : ConvexBody Point) (a b : ℝ) :
+    Path (edgeVertices K (a : Real.Angle)).1 (edgeVertices K (b : Real.Angle)).2 :=
+  (Path.segment (edgeVertices K (a : Real.Angle)).1 (supportingIntersection K a b)).trans
+    (Path.segment (supportingIntersection K a b) (edgeVertices K (b : Real.Angle)).2)
+
+private theorem path_segment_injective {P Q : Point} (hPQ : P ≠ Q) :
+    Function.Injective (Path.segment P Q) := by
+  intro s t hst
+  apply Subtype.ext
+  exact AffineMap.lineMap_injective ℝ hPQ (by
+    simpa only [Path.segment_apply] using hst)
+
+private theorem brokenSupportPath_injective
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hne : (edgeVertices K (a : Real.Angle)).1 ≠
+      (edgeVertices K (b : Real.Angle)).2) :
+    Function.Injective (brokenSupportPath K a b) := by
+  let P := (edgeVertices K (a : Real.Angle)).1
+  let Q := (edgeVertices K (b : Real.Angle)).2
+  let O := supportingIntersection K (a : Real.Angle) (b : Real.Angle)
+  obtain ⟨d₀, hd₀, hPO⟩ := supportingIntersection_eq_fst_add_pos_tangent K hab hba hne
+  obtain ⟨d₁, hd₁, hQO⟩ := supportingIntersection_eq_snd_sub_pos_tangent K hab hba hne
+  change O = P + d₀ • tangentVector (a : Real.Angle) at hPO
+  change O = Q - d₁ • tangentVector (b : Real.Angle) at hQO
+  have htne (u : Real.Angle) : tangentVector u ≠ 0 := by
+    intro hu
+    have hone : inner ℝ (tangentVector u) (tangentVector u) = 1 := by
+      rw [← u.coe_toReal]
+      exact inner_tangentVector_self u.toReal
+    rw [hu] at hone
+    simp at hone
+  have hPOne : P ≠ O := by
+    intro h
+    have hz : d₀ • tangentVector (a : Real.Angle) = 0 := by
+      calc
+        d₀ • tangentVector (a : Real.Angle) = O - P := by rw [hPO]; abel
+        _ = 0 := by rw [← h, sub_self]
+    exact hd₀.ne' ((smul_eq_zero.mp hz).resolve_right (htne _))
+  have hOQne : O ≠ Q := by
+    intro h
+    have hz : d₁ • tangentVector (b : Real.Angle) = 0 := by
+      calc
+        d₁ • tangentVector (b : Real.Angle) = Q - O := by rw [hQO]; abel
+        _ = 0 := by rw [h, sub_self]
+    exact hd₁.ne' ((smul_eq_zero.mp hz).resolve_right (htne _))
+  intro s t hst
+  change ((Path.segment P O).trans (Path.segment O Q)) s =
+    ((Path.segment P O).trans (Path.segment O Q)) t at hst
+  simp only [Path.trans_apply] at hst
+  split_ifs at hst with hs ht ht
+  · have heq := path_segment_injective hPOne hst
+    apply Subtype.ext
+    have := congrArg Subtype.val heq
+    norm_num at this ⊢
+    linarith
+  · have hmem : (Path.segment P O) ⟨2 * (s : ℝ), by constructor <;> linarith
+        [s.property.1, s.property.2]⟩ ∈ segment ℝ P O ∩ segment ℝ O Q := by
+      constructor
+      · rw [← Path.range_segment]
+        exact Set.mem_range_self _
+      · rw [hst, ← Path.range_segment]
+        exact Set.mem_range_self _
+    have hO := Set.ext_iff.mp (supporting_segments_inter K hab hba) _ |>.mp hmem
+    have hs1 : (2 : ℝ) * s = 1 := by
+      have heq := path_segment_injective hPOne (hO.trans (Path.target _).symm)
+      exact congrArg Subtype.val heq
+    have ht0 : 2 * (t : ℝ) - 1 = 0 := by
+      have heq := path_segment_injective hOQne (hst.symm.trans (hO.trans (Path.source _).symm))
+      exact congrArg Subtype.val heq
+    apply Subtype.ext
+    linarith
+  · have hmem : (Path.segment P O) ⟨2 * (t : ℝ), by constructor <;> linarith
+        [t.property.1, t.property.2]⟩ ∈ segment ℝ P O ∩ segment ℝ O Q := by
+      constructor
+      · rw [← Path.range_segment]
+        exact Set.mem_range_self _
+      · rw [← hst, ← Path.range_segment]
+        exact Set.mem_range_self _
+    have hO := Set.ext_iff.mp (supporting_segments_inter K hab hba) _ |>.mp hmem
+    have ht1 : (2 : ℝ) * t = 1 := by
+      have heq := path_segment_injective hPOne (hO.trans (Path.target _).symm)
+      exact congrArg Subtype.val heq
+    have hs0 : 2 * (s : ℝ) - 1 = 0 := by
+      have heq := path_segment_injective hOQne (hst.trans (hO.trans (Path.source _).symm))
+      exact congrArg Subtype.val heq
+    apply Subtype.ext
+    linarith
+  · have heq := path_segment_injective hOQne hst
+    apply Subtype.ext
+    have := congrArg Subtype.val heq
+    norm_num at this ⊢
+    linarith
+
+private theorem brokenSupportPath_range
+    (K : ConvexBody Point) (a b : ℝ) :
+    Set.range (brokenSupportPath K a b) =
+      segment ℝ (edgeVertices K (a : Real.Angle)).1 (supportingIntersection K a b) ∪
+      segment ℝ (supportingIntersection K a b) (edgeVertices K (b : Real.Angle)).2 := by
+  simp [brokenSupportPath, Path.trans_range, Path.range_segment]
+
+private theorem brokenSupportPath_range_inter_convexBoundaryArc
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hne : (edgeVertices K (a : Real.Angle)).1 ≠
+      (edgeVertices K (b : Real.Angle)).2) :
+    Set.range (brokenSupportPath K a b) ∩ convexBoundaryArc K a b =
+      {(edgeVertices K (a : Real.Angle)).1, (edgeVertices K (b : Real.Angle)).2} := by
+  rw [brokenSupportPath_range, Set.union_inter_distrib_right,
+    segment_fst_inter_convexBoundaryArc K hab hba hne,
+    segment_snd_inter_convexBoundaryArc K hab hba hne]
+  ext z
+  simp [or_comm]
+
+private def arcUnitPath {A : OrientedJordanArc} (p : ArcBVParametrization A) :
+    Path A.startPoint A.endPoint where
+  toFun := p.path.val ∘ unitParam_jordan p.a p.b p.ordered
+  source' := by
+    change p.path.val (unitParam_jordan p.a p.b p.ordered 0) = A.startPoint
+    rw [show unitParam_jordan p.a p.b p.ordered 0 = ⟨p.a, le_rfl, p.ordered⟩ by
+      exact Set.Icc.convexComb_zero _ _]
+    exact p.start_eq
+  target' := by
+    change p.path.val (unitParam_jordan p.a p.b p.ordered 1) = A.endPoint
+    rw [show unitParam_jordan p.a p.b p.ordered 1 = ⟨p.b, p.ordered, le_rfl⟩ by
+      exact Set.Icc.convexComb_one _ _]
+    exact p.end_eq
+  continuous_toFun := p.path.property.1.comp (continuous_unitParam_jordan _ _ _)
+
+private theorem arcUnitPath_injective {A : OrientedJordanArc}
+    (p : ArcBVParametrization A) (hne : A.startPoint ≠ A.endPoint) :
+    Function.Injective (arcUnitPath p) := by
+  intro s t hst
+  have hpab : p.a < p.b := lt_of_le_of_ne p.ordered fun hab ↦ by
+    have heq : (⟨p.a, le_rfl, p.ordered⟩ : Set.Icc p.a p.b) =
+        ⟨p.b, p.ordered, le_rfl⟩ := Subtype.ext hab
+    exact hne (p.start_eq.symm.trans ((congrArg p.path.val heq).trans p.end_eq))
+  have huv := p.injective (by simpa [arcUnitPath, Function.comp_apply] using hst)
+  exact (strictMono_convexComb_of_lt _ _ hpab).injective huv
+
+private theorem arcUnitPath_range {A : OrientedJordanArc}
+    (p : ArcBVParametrization A) : Set.range (arcUnitPath p) = A.carrier := by
+  change Set.range (p.path.val ∘ unitParam_jordan p.a p.b p.ordered) = _
+  rw [range_comp_surjective _ _ (surjective_unitParam_jordan _ _ _), p.range_eq]
+
+private theorem isJordanCurve_of_tau {Γ : Set Point} (hΓ : TauCeti.IsJordanCurve Γ) :
+    IsJordanCurve Γ := by
+  obtain ⟨e⟩ := hΓ
+  refine ⟨fun z ↦ (e.symm z : Point), continuous_subtype_val.comp e.symm.continuous,
+    fun z w h ↦ e.symm.injective (Subtype.ext h), ?_⟩
+  apply Set.Subset.antisymm
+  · rintro z ⟨u, rfl⟩
+    exact (e.symm u).property
+  · intro z hz
+    exact ⟨e ⟨z, hz⟩, congrArg Subtype.val (e.symm_apply_apply ⟨z, hz⟩)⟩
+
+private theorem convexArc_broken_isJordanCurve
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hne : (edgeVertices K (a : Real.Angle)).1 ≠
+      (edgeVertices K (b : Real.Angle)).2)
+    (A : RectifiableOrientedArc) (p : ArcBVParametrization A.val)
+    (hA : RealizesConvexArc K a b A) :
+    IsJordanCurve (Set.range (brokenSupportPath K a b) ∪ A.val.carrier) := by
+  have hmeet : Set.range (brokenSupportPath K a b) ∩ Set.range (arcUnitPath p) =
+      {(edgeVertices K (a : Real.Angle)).1, (edgeVertices K (b : Real.Angle)).2} := by
+    rw [arcUnitPath_range, hA.1]
+    exact brokenSupportPath_range_inter_convexBoundaryArc K hab hba hne
+  have hstart : A.val.startPoint = (edgeVertices K (a : Real.Angle)).1 := hA.2.1
+  have hend : A.val.endPoint = (edgeVertices K (b : Real.Angle)).2 := hA.2.2
+  have hstartend : A.val.startPoint ≠ A.val.endPoint := by simpa [hstart, hend] using hne
+  let δ : Path (edgeVertices K (a : Real.Angle)).1
+      (edgeVertices K (b : Real.Angle)).2 :=
+    (arcUnitPath p).cast hstart.symm hend.symm
+  have hδinj : Function.Injective δ := by
+    simpa only [δ, Path.cast_coe] using arcUnitPath_injective p hstartend
+  have hδrange : Set.range δ = A.val.carrier := by
+    simpa only [δ, Path.cast_coe] using arcUnitPath_range p
+  have htau := TauCeti.isJordanCurve_range_union_range_of_inter_eq_pair
+    (brokenSupportPath_injective K hab hba hne) hδinj (by
+      rw [hδrange]
+      simpa [hstart, hend, arcUnitPath_range] using hmeet)
+  rw [hδrange] at htau
+  exact isJordanCurve_of_tau htau
+
+private theorem jordanInterior_subset_closedConvexHull {Γ : Set Point}
+    (hΓ : Γ.Nonempty) : jordanInterior Γ ⊆ closedConvexHull ℝ Γ := by
+  intro p hp
+  exact TauCeti.filledHull_subset_closedConvexHull hΓ hp.2
+
+/-- The region enclosed by a loop inside a closed convex set stays inside that set. -/
+theorem jordanInterior_subset_of_subset_closed_convex
+    {Γ C : Set Point} (hΓ : Γ.Nonempty) (hsub : Γ ⊆ C)
+    (hconv : Convex ℝ C) (hclosed : IsClosed C) : jordanInterior Γ ⊆ C := by
+  exact (jordanInterior_subset_closedConvexHull hΓ).trans
+    (closedConvexHull_min hsub hconv hclosed)
+
+private theorem isOpen_jordanInterior_of_isJordanCurve {Γ : Set Point}
+    (hΓ : IsJordanCurve Γ) : IsOpen (jordanInterior Γ) := by
+  obtain ⟨U, V, hUopen, hVopen, hUconn, hVconn, hUbounded, hVunbounded, hdis,
+    hcover, hfrontU, hfrontV, hcompU, hcompV⟩ := jordan_separation hΓ
+  have heq : jordanInterior Γ = U := by
+    ext p
+    constructor
+    · intro hp
+      have hpUV : p ∈ U ∪ V := hcover.symm.subset hp.1
+      rcases hpUV with hpU | hpV
+      · exact hpU
+      · exfalso
+        exact hVunbounded (by simpa only [hcompV p hpV] using hp.2)
+    · intro hpU
+      constructor
+      · intro hpΓ
+        have hpcompl : p ∈ Γᶜ := hcover ▸ Or.inl hpU
+        exact hpcompl hpΓ
+      · simpa only [hcompU p hpU] using hUbounded
+  rw [heq]
+  exact hUopen
+
+private theorem jordanInterior_subset_interior_of_subset_closed_convex
+    {Γ C : Set Point} (hJordan : IsJordanCurve Γ) (hΓ : Γ.Nonempty)
+    (hsub : Γ ⊆ C) (hconv : Convex ℝ C) (hclosed : IsClosed C) :
+    jordanInterior Γ ⊆ interior C := by
+  intro p hp
+  apply mem_interior_iff_mem_nhds.mpr
+  exact Filter.mem_of_superset
+    ((isOpen_jordanInterior_of_isJordanCurve hJordan).mem_nhds hp)
+    (jordanInterior_subset_of_subset_closed_convex hΓ hsub hconv hclosed)
+
+private theorem injOn_concatUnitPaths_jordan
+    (p q : ContinuousBVPaths 0 1)
+    (hjoin : p.val ⟨1, by norm_num⟩ = q.val ⟨0, by norm_num⟩)
+    (hclose : q.val ⟨1, by norm_num⟩ = p.val ⟨0, by norm_num⟩)
+    (hp : Function.Injective p.val) (hq : Function.Injective q.val)
+    (hmeet : Set.range p.val ∩ Set.range q.val =
+      {p.val ⟨0, by norm_num⟩, p.val ⟨1, by norm_num⟩}) :
+    Set.InjOn (concatUnitPaths_jordan p q hjoin).val {t | (t : ℝ) < 2} := by
+  intro s hs t ht hst
+  change (s : ℝ) < 2 at hs
+  change (t : ℝ) < 2 at ht
+  change Function.concatUnitIntervals p.val q.val s =
+    Function.concatUnitIntervals p.val q.val t at hst
+  unfold Function.concatUnitIntervals at hst
+  split_ifs at hst with hs₁ ht₁ ht₁
+  · have heq := congrArg Subtype.val (hp hst)
+    apply Subtype.ext
+    simp only [Set.coe_projIcc] at heq
+    rw [min_eq_right hs₁, max_eq_right s.property.1,
+      min_eq_right ht₁, max_eq_right t.property.1] at heq
+    exact heq
+  · have hmem : p.val (Set.projIcc 0 1 (by norm_num) (s : ℝ)) ∈
+        Set.range p.val ∩ Set.range q.val :=
+      ⟨Set.mem_range_self _, hst ▸ Set.mem_range_self _⟩
+    rw [hmeet] at hmem
+    rcases hmem with hP | hQ
+    · have hs0 := congrArg Subtype.val (hp hP)
+      have ht1 := congrArg Subtype.val (hq (hst.symm.trans (hP.trans hclose.symm)))
+      simp only [Set.coe_projIcc] at hs0 ht1
+      rw [min_eq_right hs₁, max_eq_right s.property.1] at hs0
+      rw [min_eq_right (by linarith [t.property.2]),
+        max_eq_right (by linarith)] at ht1
+      exfalso
+      linarith
+    · have hs1 := congrArg Subtype.val (hp hQ)
+      have ht0 := congrArg Subtype.val (hq (hst.symm.trans (hQ.trans hjoin)))
+      simp only [Set.coe_projIcc] at hs1 ht0
+      rw [min_eq_right hs₁, max_eq_right s.property.1] at hs1
+      rw [min_eq_right (by linarith [t.property.2]),
+        max_eq_right (by linarith)] at ht0
+      apply Subtype.ext
+      linarith
+  · have hmem : p.val (Set.projIcc 0 1 (by norm_num) (t : ℝ)) ∈
+        Set.range p.val ∩ Set.range q.val :=
+      ⟨Set.mem_range_self _, hst.symm ▸ Set.mem_range_self _⟩
+    rw [hmeet] at hmem
+    rcases hmem with hP | hQ
+    · have ht0 := congrArg Subtype.val (hp hP)
+      have hs1 := congrArg Subtype.val (hq (hst.trans (hP.trans hclose.symm)))
+      simp only [Set.coe_projIcc] at ht0 hs1
+      rw [min_eq_right ht₁, max_eq_right t.property.1] at ht0
+      rw [min_eq_right (by linarith [s.property.2]),
+        max_eq_right (by linarith)] at hs1
+      exfalso
+      linarith
+    · have ht1 := congrArg Subtype.val (hp hQ)
+      have hs0 := congrArg Subtype.val (hq (hst.trans (hQ.trans hjoin)))
+      simp only [Set.coe_projIcc] at ht1 hs0
+      rw [min_eq_right ht₁, max_eq_right t.property.1] at ht1
+      rw [min_eq_right (by linarith [s.property.2]),
+        max_eq_right (by linarith)] at hs0
+      apply Subtype.ext
+      linarith
+  · have heq := congrArg Subtype.val (hq hst)
+    apply Subtype.ext
+    simp only [Set.coe_projIcc] at heq
+    rw [min_eq_right (by linarith [s.property.2]), max_eq_right (by linarith),
+      min_eq_right (by linarith [t.property.2]), max_eq_right (by linarith)] at heq
+    linarith
+
+private theorem reparam_support_segments_eq_brokenSupportPath
+    (K : ConvexBody Point) (a b : ℝ)
+    (h01 : (lineSegmentBVPath (edgeVertices K a).1 (supportingIntersection K a b)).val
+      ⟨1, by norm_num⟩ =
+      (lineSegmentBVPath (supportingIntersection K a b) (edgeVertices K b).2).val
+        ⟨0, by norm_num⟩) :
+    (reparamTwoToUnit_jordan (concatUnitPaths_jordan
+      (lineSegmentBVPath (edgeVertices K a).1 (supportingIntersection K a b))
+      (lineSegmentBVPath (supportingIntersection K a b) (edgeVertices K b).2)
+      h01)).val = brokenSupportPath K a b := by
+  funext t
+  simp only [reparamTwoToUnit_jordan, Function.comp_apply, doubleParam_jordan,
+    concatUnitPaths_jordan, lineSegmentBVPath, brokenSupportPath]
+  by_cases ht : (t : ℝ) ≤ 1 / 2
+  · simp only [Path.trans_apply, dite_eq_left ht]
+    unfold Function.concatUnitIntervals
+    rw [ite_eq_left (by linarith)]
+    congr 1
+    apply Subtype.ext
+    simp only [Set.coe_projIcc]
+    rw [min_eq_right (by linarith [t.property.2]),
+      max_eq_right (by linarith [t.property.1])]
+  · simp only [Path.trans_apply, dite_eq_right ht]
+    unfold Function.concatUnitIntervals
+    rw [ite_eq_right (by linarith)]
+    congr 1
+    apply Subtype.ext
+    simp only [Set.coe_projIcc]
+    rw [min_eq_right (by linarith [t.property.2]),
+      max_eq_right (by linarith [t.property.1])]
+
+private theorem reverseArcUnitPath_injective {A : OrientedJordanArc}
+    (p : ArcBVParametrization A) (q : ContinuousBVPaths 0 1)
+    (hq : q.val = (reverseArcPath p).val ∘ unitParam_jordan p.a p.b p.ordered)
+    (hne : A.startPoint ≠ A.endPoint) : Function.Injective q.val := by
+  have hpab : p.a < p.b := lt_of_le_of_ne p.ordered fun hab ↦ by
+    have heq : (⟨p.a, le_rfl, p.ordered⟩ : Set.Icc p.a p.b) =
+        ⟨p.b, p.ordered, le_rfl⟩ := Subtype.ext hab
+    exact hne (p.start_eq.symm.trans ((congrArg p.path.val heq).trans p.end_eq))
+  intro s t hst
+  rw [hq] at hst
+  have hrev := p.injective hst
+  apply Subtype.ext
+  have hval := congrArg Subtype.val hrev
+  simp only [Set.Icc.reverse] at hval
+  have huv : (unitParam_jordan p.a p.b p.ordered s : ℝ) =
+      unitParam_jordan p.a p.b p.ordered t := by linarith
+  exact congrArg Subtype.val
+    ((strictMono_convexComb_of_lt _ _ hpab).injective (Subtype.ext huv))
+
+private theorem brokenSupportPath_union_convexArc_subset_endpointHalfPlanes
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (A : RectifiableOrientedArc) (hA : RealizesConvexArc K a b A) :
+    Set.range (brokenSupportPath K a b) ∪ A.val.carrier ⊆
+      (supportingLineHalfPlane K (a : Real.Angle)).2 ∩
+        (supportingLineHalfPlane K (b : Real.Angle)).2 := by
+  have hsin : 0 < Real.sin (b - a) :=
+    Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hab) (by linarith)
+  have hK (t : Real.Angle) : (K : Set Point) ⊆ (supportingLineHalfPlane K t).2 := by
+    intro p hp
+    exact inner_le_supportValue K hp t
+  have hOa : supportingIntersection K a b ∈ (supportingLineHalfPlane K a).2 := by
+    exact le_of_eq (supportingIntersection_inner_left K a b)
+  have hOb : supportingIntersection K a b ∈ (supportingLineHalfPlane K b).2 := by
+    exact le_of_eq (supportingIntersection_inner_right K a b hsin.ne')
+  have hconv (t : Real.Angle) : Convex ℝ (supportingLineHalfPlane K t).2 := by
+    intro x hx y hy u v hu hv huv
+    change inner ℝ x (normalVector t) ≤ supportValue K t at hx
+    change inner ℝ y (normalVector t) ≤ supportValue K t at hy
+    change inner ℝ (u • x + v • y) (normalVector t) ≤ supportValue K t
+    rw [inner_add_left, inner_smul_left, inner_smul_left]
+    simp only [RCLike.conj_to_real]
+    calc
+      u * inner ℝ x (normalVector t) + v * inner ℝ y (normalVector t) ≤
+          u * supportValue K t + v * supportValue K t :=
+        add_le_add (mul_le_mul_of_nonneg_left hx hu) (mul_le_mul_of_nonneg_left hy hv)
+      _ = supportValue K t := by rw [← add_mul, huv, one_mul]
+  intro p hp
+  rcases hp with hp | hp
+  · rw [brokenSupportPath_range] at hp
+    rcases hp with hp | hp
+    · constructor
+      · exact (hconv a).segment_subset
+          (hK a (edgeVertices_fst_mem K (a : Real.Angle)).1) hOa hp
+      · exact (hconv b).segment_subset
+          (hK b (edgeVertices_fst_mem K (a : Real.Angle)).1) hOb hp
+    · constructor
+      · exact (hconv a).segment_subset hOa
+          (hK a (edgeVertices_snd_mem K (b : Real.Angle)).1) hp
+      · exact (hconv b).segment_subset hOb
+          (hK b (edgeVertices_snd_mem K (b : Real.Angle)).1) hp
+  · have hpK : p ∈ K := convexBoundaryArc_subset_body K a b (hA.1 ▸ hp)
+    exact ⟨hK a hpK, hK b hpK⟩
+
+private theorem convexBoundaryArc_exists_support_eq
+    (K : ConvexBody Point) {a b : ℝ} (hab : a ≤ b) {p : Point}
+    (hp : p ∈ convexBoundaryArc K a b) :
+    ∃ t ∈ Set.Icc a b,
+      inner ℝ p (normalVector (t : Real.Angle)) = supportValue K (t : Real.Angle) := by
+  rcases hp with (hp | hp) | hp
+  · subst p
+    exact ⟨a, ⟨le_rfl, hab⟩, (edgeVertices_fst_mem K (a : Real.Angle)).2⟩
+  · simp only [Set.mem_iUnion] at hp
+    obtain ⟨t, htab, hp⟩ := hp
+    exact ⟨t, ⟨htab.1.le, htab.2.le⟩, hp.2⟩
+  · subst p
+    exact ⟨b, ⟨hab, le_rfl⟩, (edgeVertices_snd_mem K (b : Real.Angle)).2⟩
+
+private theorem inner_midpoint_normal_pos {a b t : ℝ}
+    (_hab : a < b) (hba : b < a + Real.pi) (ht : t ∈ Set.Icc a b) :
+    0 < inner ℝ (normalVector (((a + b) / 2 : ℝ) : Real.Angle))
+      (normalVector (t : Real.Angle)) := by
+  rw [inner_normalVector_normalVector]
+  apply Real.cos_pos_of_mem_Ioo
+  constructor <;> linarith [ht.1, ht.2]
+
+private theorem sub_pos_midpoint_normal_inner_lt_support
+    (K : ConvexBody Point) {a b r : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    {p : Point}
+    (hp : p ∈ ⋂ t ∈ Set.Icc a b,
+      (supportingLineHalfPlane K (t : Real.Angle)).2) (hr : 0 < r)
+    (t : ℝ) (ht : t ∈ Set.Icc a b) :
+    inner ℝ (p - r • normalVector (((a + b) / 2 : ℝ) : Real.Angle))
+        (normalVector (t : Real.Angle)) < supportValue K (t : Real.Angle) := by
+  have hple : inner ℝ p (normalVector (t : Real.Angle)) ≤
+      supportValue K (t : Real.Angle) := by
+    exact Set.mem_iInter₂.mp hp t ht
+  rw [inner_sub_left, real_inner_smul_left]
+  have hdot := inner_midpoint_normal_pos hab hba ht
+  nlinarith
+
+private theorem brokenSupportPath_mem_support_eq
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi) {p : Point}
+    (hp : p ∈ Set.range (brokenSupportPath K a b)) :
+    inner ℝ p (normalVector (a : Real.Angle)) = supportValue K a ∨
+      inner ℝ p (normalVector (b : Real.Angle)) = supportValue K b := by
+  have hsin : Real.sin (b - a) ≠ 0 :=
+    (Real.sin_pos_of_pos_of_lt_pi (sub_pos.mpr hab) (by linarith)).ne'
+  rw [brokenSupportPath_range] at hp
+  rcases hp with hp | hp
+  · left
+    rw [segment_eq_image] at hp
+    obtain ⟨r, hr, rfl⟩ := hp
+    rw [inner_add_left, real_inner_smul_left, real_inner_smul_left,
+      (edgeVertices_fst_mem K (a : Real.Angle)).2,
+      supportingIntersection_inner_left]
+    ring
+  · right
+    rw [segment_eq_image] at hp
+    obtain ⟨r, hr, rfl⟩ := hp
+    rw [inner_add_left, real_inner_smul_left, real_inner_smul_left,
+      supportingIntersection_inner_right K a b hsin,
+      (edgeVertices_snd_mem K (b : Real.Angle)).2]
+    ring
+
+private theorem sub_pos_midpoint_normal_not_mem_boundary_loop
+    (K : ConvexBody Point) {a b r : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (A : RectifiableOrientedArc) (hA : RealizesConvexArc K a b A) {p : Point}
+    (hp : p ∈ ⋂ t ∈ Set.Icc a b,
+      (supportingLineHalfPlane K (t : Real.Angle)).2) (hr : 0 < r) :
+    p - r • normalVector (((a + b) / 2 : ℝ) : Real.Angle) ∉
+      Set.range (brokenSupportPath K a b) ∪ A.val.carrier := by
+  intro hq
+  rcases hq with hq | hq
+  · rcases brokenSupportPath_mem_support_eq K hab hba hq with hqa | hqb
+    · exact (ne_of_lt (sub_pos_midpoint_normal_inner_lt_support K hab hba hp hr a
+        ⟨le_rfl, hab.le⟩)) hqa
+    · exact (ne_of_lt (sub_pos_midpoint_normal_inner_lt_support K hab hba hp hr b
+        ⟨hab.le, le_rfl⟩)) hqb
+  · obtain ⟨t, ht, hqt⟩ := convexBoundaryArc_exists_support_eq K hab.le (hA.1 ▸ hq)
+    exact (ne_of_lt (sub_pos_midpoint_normal_inner_lt_support K hab hba hp hr t ht)) hqt
+
+private theorem boundaryLoop_component_unbounded_of_mem_supportIntersection
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (A : RectifiableOrientedArc) (hA : RealizesConvexArc K a b A) {p : Point}
+    (hpX : p ∈ ⋂ t ∈ Set.Icc a b,
+      (supportingLineHalfPlane K (t : Real.Angle)).2)
+    (hpΓ : p ∉ Set.range (brokenSupportPath K a b) ∪ A.val.carrier) :
+    ¬ Bornology.IsBounded (connectedComponentIn
+      (Set.range (brokenSupportPath K a b) ∪ A.val.carrier)ᶜ p) := by
+  let n := normalVector (((a + b) / 2 : ℝ) : Real.Angle)
+  let f : ℝ → Point := fun r ↦ p - r • n
+  have hf : Continuous f := continuous_const.sub (continuous_id.smul continuous_const)
+  have hpre : IsPreconnected (f '' Set.Ici 0) :=
+    isPreconnected_Ici.image f hf.continuousOn
+  have hsub : f '' Set.Ici 0 ⊆
+      (Set.range (brokenSupportPath K a b) ∪ A.val.carrier)ᶜ := by
+    rintro q ⟨r, hr, rfl⟩
+    by_cases hr0 : r = 0
+    · simpa [f, hr0] using hpΓ
+    · exact sub_pos_midpoint_normal_not_mem_boundary_loop K hab hba A hA hpX
+        (lt_of_le_of_ne hr (Ne.symm hr0))
+  have hpmem : p ∈ f '' Set.Ici 0 := ⟨0, by simp, by simp [f]⟩
+  have hcomp := hpre.subset_connectedComponentIn hpmem hsub
+  intro hb
+  obtain ⟨C, hC⟩ := hb.exists_norm_le
+  have hn : ‖n‖ = 1 := norm_normalVector_real ((a + b) / 2)
+  let r := max 0 (C + ‖p‖ + 1)
+  have hr : 0 ≤ r := le_max_left _ _
+  have hq := hC (f r) (hcomp ⟨r, hr, rfl⟩)
+  have hdiff : ‖r • n‖ ≤ C + ‖p‖ := by
+    calc
+      ‖r • n‖ = ‖p - f r‖ := by simp [f]
+      _ ≤ ‖p‖ + ‖f r‖ := norm_sub_le _ _
+      _ ≤ C + ‖p‖ := by linarith
+  rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg hr, hn, mul_one] at hdiff
+  have hlower : C + ‖p‖ + 1 ≤ r := le_max_right _ _
+  linarith
+
+private theorem jordanInterior_boundaryLoop_disjoint_supportIntersection
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (A : RectifiableOrientedArc) (hA : RealizesConvexArc K a b A) :
+    Disjoint (jordanInterior (Set.range (brokenSupportPath K a b) ∪ A.val.carrier))
+      (⋂ t ∈ Set.Icc a b, (supportingLineHalfPlane K (t : Real.Angle)).2) := by
+  rw [Set.disjoint_left]
+  intro p hpI hpX
+  exact boundaryLoop_component_unbounded_of_mem_supportIntersection
+    K hab hba A hA hpX hpI.1 hpI.2
+
+private theorem isClosed_supportingLineHalfPlane_lower
+    (K : ConvexBody Point) (t : Real.Angle) :
+    IsClosed (supportingLineHalfPlane K t).2 := by
+  exact isClosed_le (continuous_id.inner continuous_const) continuous_const
+
+private theorem convex_supportingLineHalfPlane_lower
+    (K : ConvexBody Point) (t : Real.Angle) :
+    Convex ℝ (supportingLineHalfPlane K t).2 := by
+  intro x hx y hy u v hu hv huv
+  change inner ℝ x (normalVector t) ≤ supportValue K t at hx
+  change inner ℝ y (normalVector t) ≤ supportValue K t at hy
+  change inner ℝ (u • x + v • y) (normalVector t) ≤ supportValue K t
+  rw [inner_add_left, inner_smul_left, inner_smul_left]
+  simp only [RCLike.conj_to_real]
+  calc
+    u * inner ℝ x (normalVector t) + v * inner ℝ y (normalVector t) ≤
+        u * supportValue K t + v * supportValue K t :=
+      add_le_add (mul_le_mul_of_nonneg_left hx hu) (mul_le_mul_of_nonneg_left hy hv)
+    _ = supportValue K t := by rw [← add_mul, huv, one_mul]
+
+private theorem jordanInterior_boundaryLoop_subset_endpointHalfPlanes
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hne : (edgeVertices K (a : Real.Angle)).1 ≠
+      (edgeVertices K (b : Real.Angle)).2)
+    (A : RectifiableOrientedArc) (p : ArcBVParametrization A.val)
+    (hA : RealizesConvexArc K a b A) :
+    jordanInterior (Set.range (brokenSupportPath K a b) ∪ A.val.carrier) ⊆
+      interior ((supportingLineHalfPlane K a).2 ∩
+        (supportingLineHalfPlane K b).2) := by
+  apply jordanInterior_subset_interior_of_subset_closed_convex
+    (convexArc_broken_isJordanCurve K hab hba hne A p hA)
+  · exact (Set.range_nonempty _).inl
+  · exact brokenSupportPath_union_convexArc_subset_endpointHalfPlanes K hab hba A hA
+  · exact (convex_supportingLineHalfPlane_lower K a).inter
+      (convex_supportingLineHalfPlane_lower K b)
+  · exact (isClosed_supportingLineHalfPlane_lower K a).inter
+      (isClosed_supportingLineHalfPlane_lower K b)
+
+private theorem concatThree_first_image_jordan
+    (p₀ p₁ p₂ : ContinuousBVPaths 0 1)
+    (h01 : p₀.val ⟨1, by norm_num⟩ = p₁.val ⟨0, by norm_num⟩)
+    (h12 : p₁.val ⟨1, by norm_num⟩ = p₂.val ⟨0, by norm_num⟩) :
+    (concatThreeUnitPaths_jordan p₀ p₁ p₂ h01 h12).val ''
+        Set.Icc (⟨0, by norm_num⟩ : Set.Icc (0 : ℝ) 2) ⟨1 / 2, by norm_num⟩ =
+      Set.range p₀.val := by
+  apply Set.Subset.antisymm
+  · rintro z ⟨t, ht, rfl⟩
+    have htt : (t : ℝ) ≤ 1 / 2 := by exact_mod_cast ht.2
+    let u : Set.Icc (0 : ℝ) 1 := ⟨2 * (t : ℝ), by
+      constructor <;> nlinarith [t.property.1, htt]⟩
+    refine ⟨u, ?_⟩
+    have heq : (⟨(u : ℝ) / 2, by
+        constructor <;> nlinarith [u.property.1, u.property.2]⟩ : Set.Icc (0 : ℝ) 2) = t := by
+      apply Subtype.ext
+      dsimp [u]
+      ring
+    rw [← heq]
+    exact (concatThreeUnitPaths_first_jordan p₀ p₁ p₂ h01 h12 u).symm
+  · rintro z ⟨u, rfl⟩
+    let t : Set.Icc (0 : ℝ) 2 := ⟨(u : ℝ) / 2, by
+      constructor <;> nlinarith [u.property.1, u.property.2]⟩
+    refine ⟨t, ⟨by exact Subtype.coe_le_coe.mp t.property.1,
+      by change (t : ℝ) ≤ 1 / 2; dsimp [t]; linarith [u.property.2]⟩, ?_⟩
+    exact concatThreeUnitPaths_first_jordan p₀ p₁ p₂ h01 h12 u
+
+private theorem concatThree_convexArc_oriented
+    (K : ConvexBody Point) {a b : ℝ} (hab : a < b) (hba : b < a + Real.pi)
+    (hne : (edgeVertices K (a : Real.Angle)).1 ≠
+      (edgeVertices K (b : Real.Angle)).2)
+    (A : RectifiableOrientedArc) (p : ArcBVParametrization A.val)
+    (hA : RealizesConvexArc K a b A)
+    (p₂ : ContinuousBVPaths 0 1)
+    (hp₂ : p₂.val = (reverseArcPath p).val ∘ unitParam_jordan p.a p.b p.ordered)
+    (h01 : (lineSegmentBVPath (edgeVertices K a).1 (supportingIntersection K a b)).val
+      ⟨1, by norm_num⟩ =
+      (lineSegmentBVPath (supportingIntersection K a b) (edgeVertices K b).2).val
+        ⟨0, by norm_num⟩)
+    (h12 : (lineSegmentBVPath (supportingIntersection K a b) (edgeVertices K b).2).val
+      ⟨1, by norm_num⟩ = p₂.val ⟨0, by norm_num⟩) :
+    IsOrientedJordanParametrization (by norm_num : (0 : ℝ) ≤ 2)
+      (Set.range (concatThreeUnitPaths_jordan
+        (lineSegmentBVPath (edgeVertices K a).1 (supportingIntersection K a b))
+        (lineSegmentBVPath (supportingIntersection K a b) (edgeVertices K b).2)
+        p₂ h01 h12).val) true
+      (concatThreeUnitPaths_jordan
+        (lineSegmentBVPath (edgeVertices K a).1 (supportingIntersection K a b))
+        (lineSegmentBVPath (supportingIntersection K a b) (edgeVertices K b).2)
+        p₂ h01 h12).val := by
+  let p₀ := lineSegmentBVPath (edgeVertices K a).1 (supportingIntersection K a b)
+  let p₁ := lineSegmentBVPath (supportingIntersection K a b) (edgeVertices K b).2
+  let q := concatUnitPaths_jordan p₀ p₁ h01
+  let q' := reparamTwoToUnit_jordan q
+  let γ := concatThreeUnitPaths_jordan p₀ p₁ p₂ h01 h12
+  have hq' : q'.val = brokenSupportPath K a b :=
+    reparam_support_segments_eq_brokenSupportPath K a b h01
+  have hpq : Function.Injective q'.val := hq' ▸ brokenSupportPath_injective K hab hba hne
+  have hPneQ : A.val.startPoint ≠ A.val.endPoint := by simpa [hA.2.1, hA.2.2] using hne
+  have hp₂inj := reverseArcUnitPath_injective p p₂ hp₂ hPneQ
+  have hqrange : Set.range q'.val = Set.range (brokenSupportPath K a b) := by rw [hq']
+  have hp₂range : Set.range p₂.val = A.val.carrier := range_reverseArcUnitPath p p₂ hp₂
+  have hmeet : Set.range q'.val ∩ Set.range p₂.val =
+      {q'.val ⟨0, by norm_num⟩, q'.val ⟨1, by norm_num⟩} := by
+    rw [hqrange, hp₂range, hA.1,
+      brokenSupportPath_range_inter_convexBoundaryArc K hab hba hne, hq']
+    simp [brokenSupportPath]
+  have hclose : p₂.val ⟨1, by norm_num⟩ = q'.val ⟨0, by norm_num⟩ := by
+    rw [hp₂]
+    simp [q', hq', brokenSupportPath, reverseArcPath, unitParam_jordan, Set.Icc.reverse,
+      hA.2.1, p.start_eq]
+  have hjoin : q'.val ⟨1, by norm_num⟩ = p₂.val ⟨0, by norm_num⟩ := by
+    rw [show q'.val ⟨1, by norm_num⟩ = q.val ⟨2, by norm_num⟩ from
+      reparamTwoToUnit_one q]
+    exact (concatUnitPaths_jordan_end p₀ p₁ h01).trans h12
+  have hinj := injOn_concatUnitPaths_jordan q' p₂ hjoin
+    hclose hpq hp₂inj hmeet
+  have hrange : Set.range γ.val = Set.range (brokenSupportPath K a b) ∪ A.val.carrier := by
+    rw [show γ = concatThreeUnitPaths_jordan p₀ p₁ p₂ h01 h12 by rfl]
+    rw [range_concatThreeUnitPaths_jordan, hp₂range]
+    have hq'q : Set.range q'.val = Set.range q.val :=
+      range_comp_surjective q.val doubleParam_jordan surjective_doubleParam_jordan
+    rw [← Function.range_concatUnitIntervals p₀.val p₁.val h01]
+    change Set.range q.val ∪ A.val.carrier = _
+    rw [← hq'q, hqrange]
+  have hJordan := convexArc_broken_isJordanCurve K hab hba hne A p hA
+  obtain ⟨d, hd, hdir⟩ := supportingIntersection_eq_fst_add_pos_tangent K hab hba hne
+  have hclosedγ : γ.val ⟨0, by norm_num⟩ = γ.val ⟨2, by norm_num⟩ := by
+    simpa [γ, concatThreeUnitPaths_jordan, concatUnitPaths_jordan, q', q] using hclose.symm
+  have hinjγ : Set.InjOn γ.val {t | (t : ℝ) < 2} := by
+    simpa [γ, concatThreeUnitPaths_jordan, q', q] using hinj
+  have hhalfγ (t : Set.Icc (0 : ℝ) 2) :
+      γ.val t ∈ normalHalfPlane (a : Real.Angle) (supportValue K a) false false :=
+    (brokenSupportPath_union_convexArc_subset_endpointHalfPlanes K hab hba A hA
+      (hrange ▸ Set.mem_range_self t)).1
+  let s : Set.Icc (0 : ℝ) 2 := ⟨0, by norm_num⟩
+  let t : Set.Icc (0 : ℝ) 2 := ⟨1 / 2, by norm_num⟩
+  have hs : γ.val s = (edgeVertices K (a : Real.Angle)).1 := by
+    simpa [s, γ, p₀, lineSegmentBVPath] using
+      concatThreeUnitPaths_first_jordan p₀ p₁ p₂ h01 h12 ⟨0, by norm_num⟩
+  have ht : γ.val t = supportingIntersection K a b := by
+    simpa [t, γ, p₀, lineSegmentBVPath] using
+      concatThreeUnitPaths_first_jordan p₀ p₁ p₂ h01 h12 ⟨1, by norm_num⟩
+  exact jordan_counterclockwise_of_supporting_segment 0 2 (by norm_num) γ.val
+    γ.property.1 (hrange ▸ hJordan) hclosedγ hinjγ
+    (a : Real.Angle) (supportValue K a) hhalfγ s t (by
+      change (0 : ℝ) < 1 / 2
+      norm_num)
+    (by rw [hs]; exact (edgeVertices_fst_mem K (a : Real.Angle)).2)
+    d hd (by rw [hs, ht]; exact hdir)
+    (by
+      rw [hs, ht]
+      rw [show γ.val '' Set.Icc s t = Set.range p₀.val by
+        simpa [s, t, γ] using concatThree_first_image_jordan p₀ p₁ p₂ h01 h12]
+      exact Path.range_segment _ _)
+
+private theorem exists_rectifiableOrientedArc_segment_jordan (P Q : Point) (hPQ : P ≠ Q) :
+    ∃ A : RectifiableOrientedArc,
+      A.val.carrier = segment ℝ P Q ∧ A.val.startPoint = P ∧ A.val.endPoint = Q := by
+  let Γ : OrientedJordanArc :=
+    { carrier := segment ℝ P Q
+      startPoint := P
+      endPoint := Q
+      parametrizable := by
+        refine ⟨0, 1, by norm_num, Path.segment P Q,
+          (Path.segment P Q).continuous, Path.segment_injective_of_ne hPQ,
+          Path.range_segment P Q, ?_, ?_⟩ <;> simp }
+  let z : ArcBVParametrization Γ :=
+    { a := 0
+      b := 1
+      ordered := by norm_num
+      path := lineSegmentBVPath P Q
+      injective := Path.segment_injective_of_ne hPQ
+      range_eq := Path.range_segment P Q
+      start_eq := by simp [lineSegmentBVPath, Γ]
+      end_eq := by simp [lineSegmentBVPath, Γ] }
+  let A : RectifiableOrientedArc := ⟨Γ, ⟨z⟩⟩
+  exact ⟨A, rfl, rfl, rfl⟩
+
+private theorem exists_rectifiableOrientedArc_convexBoundaryArc_jordan
+    (K : ConvexBody Point) (a b : ℝ) (hab : a < b) (hba : b < a + Real.pi)
+    (hne : (edgeVertices K a).1 ≠ (edgeVertices K b).2) :
+    ∃ A : RectifiableOrientedArc, RealizesConvexArc K a b A := by
+  let P := (edgeVertices K (a : Real.Angle)).1
+  let Q := (edgeVertices K (b : Real.Angle)).2
+  let O := supportingIntersection K a b
+  obtain ⟨-, hcut⟩ := convexBoundaryArc_cut K a b hab hba P Q O rfl rfl rfl
+  obtain ⟨hncol, t, c, K', hat, htb, hPt, hQt, hcO, hK', hleft, hmiddle,
+      hright, hterminal⟩ := hcut hne
+  by_cases hInt : (interior (K' : Set Point)).Nonempty
+  · obtain ⟨A, hcarrier, hstart, hend⟩ :=
+      exists_rectifiableOrientedArc_convexBoundaryArc_of_cut K K' a b t c P Q
+        hat htb hba hne hPt hQt rfl rfl hleft hmiddle hright hterminal hInt
+    exact ⟨A, hcarrier, hstart, hend⟩
+  · have hInt' : interior (K' : Set Point) = ∅ := Set.not_nonempty_iff_eq_empty.mp hInt
+    have harc := convexBoundaryArc_eq_segment_of_cut_interior_empty K K' a b t c P Q
+      hat htb hba hne hPt hQt rfl rfl hleft hmiddle hright hInt'
+    obtain ⟨A, hcarrier, hstart, hend⟩ :=
+      exists_rectifiableOrientedArc_segment_jordan P Q hne
+    exact ⟨A, hcarrier.trans harc.symm, hstart, hend⟩
+
+/-- The supporting segments and reversed convex boundary arc form a counterclockwise Jordan
+curve. -/
+theorem convexBoundaryArc_jordan (K : ConvexBody Point) (a b : ℝ)
+    (hab : a < b) (hba : b < a + Real.pi)
+    (hne : (edgeVertices K a).1 ≠ (edgeVertices K b).2) :
+    ∃ (A : RectifiableOrientedArc) (p : ArcBVParametrization A.val)
+      (γ : RectifiablePathData) (pieces : Fin 3 → RectifiablePathData),
+      RealizesConvexArc K a b A ∧
+      IsSegmentTraversal (pieces 0) (edgeVertices K a).1 (supportingIntersection K a b) ∧
+      IsSegmentTraversal (pieces 1) (supportingIntersection K a b) (edgeVertices K b).2 ∧
+      IsReverseArcTraversal (pieces 2) p ∧ IsPathConcatenation γ pieces ∧
+      IsOrientedJordanParametrization γ.ordered (Set.range γ.path.val) true γ.path.val ∧
+      jordanInterior (Set.range γ.path.val) ⊆
+        interior ((supportingLineHalfPlane K a).2 ∩ (supportingLineHalfPlane K b).2) ∧
+      Disjoint (jordanInterior (Set.range γ.path.val))
+        (⋂ t ∈ Set.Icc a b, (supportingLineHalfPlane K (t : Real.Angle)).2) := by
+  obtain ⟨A, hA⟩ :=
+    exists_rectifiableOrientedArc_convexBoundaryArc_jordan K a b hab hba hne
+  let p : ArcBVParametrization A.val := Classical.choice A.property
+  let P := (edgeVertices K (a : Real.Angle)).1
+  let Q := (edgeVertices K (b : Real.Angle)).2
+  let O := supportingIntersection K a b
+  let p₀ := lineSegmentBVPath P O
+  let p₁ := lineSegmentBVPath O Q
+  obtain ⟨p₂, hp₂, hp₂rev⟩ := exists_reverseArcUnitPath p
+  have h01 : p₀.val ⟨1, by norm_num⟩ = p₁.val ⟨0, by norm_num⟩ := by
+    simp [p₀, p₁, lineSegmentBVPath]
+  have h12 : p₁.val ⟨1, by norm_num⟩ = p₂.val ⟨0, by norm_num⟩ := by
+    rw [hp₂]
+    change p₁.val ⟨1, by norm_num⟩ =
+      (reverseArcPath p).val (unitParam_jordan p.a p.b p.ordered ⟨0, by norm_num⟩)
+    rw [show unitParam_jordan p.a p.b p.ordered ⟨0, by norm_num⟩ =
+      ⟨p.a, le_rfl, p.ordered⟩ by exact Set.Icc.convexComb_zero _ _]
+    rw [reverseArcPath_start]
+    simpa [p₁, Q, lineSegmentBVPath, Path.segment_apply,
+      AffineMap.lineMap_apply_module'] using hA.2.2.symm
+  let γpath := concatThreeUnitPaths_jordan p₀ p₁ p₂ h01 h12
+  let γ : RectifiablePathData :=
+    { a := 0, b := 2, ordered := by norm_num, path := γpath }
+  let pieces : Fin 3 → RectifiablePathData :=
+    ![{ a := 0, b := 1, ordered := by norm_num, path := p₀ },
+      { a := 0, b := 1, ordered := by norm_num, path := p₁ },
+      { a := 0, b := 1, ordered := by norm_num, path := p₂ }]
+  have horient := concatThree_convexArc_oriented K hab hba hne A p hA p₂ hp₂ h01 h12
+  have hrange : Set.range γ.path.val =
+      Set.range (brokenSupportPath K a b) ∪ A.val.carrier := by
+    change Set.range (concatThreeUnitPaths_jordan p₀ p₁ p₂ h01 h12).val = _
+    rw [range_concatThreeUnitPaths_jordan, range_reverseArcUnitPath p p₂ hp₂]
+    rw [← Function.range_concatUnitIntervals p₀.val p₁.val h01]
+    let q := concatUnitPaths_jordan p₀ p₁ h01
+    have hrepr : Set.range (reparamTwoToUnit_jordan q).val = Set.range q.val :=
+      range_comp_surjective q.val doubleParam_jordan surjective_doubleParam_jordan
+    change Set.range q.val ∪ A.val.carrier = _
+    rw [← hrepr, show (reparamTwoToUnit_jordan q).val = brokenSupportPath K a b by
+      simpa [q, p₀, p₁, P, Q, O] using
+        reparam_support_segments_eq_brokenSupportPath K a b h01]
+  refine ⟨A, p, γ, pieces, hA, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
+  · simpa [pieces, P, O, Matrix.cons_val_zero] using
+      isSegmentTraversal_lineSegmentBVPath P O
+  · simpa [pieces, Q, O, Matrix.cons_val_zero, Matrix.cons_val_one] using
+      isSegmentTraversal_lineSegmentBVPath O Q
+  · simpa [pieces, Matrix.cons_val_zero, Matrix.cons_val_one] using hp₂rev
+  · simpa [γ, γpath, pieces] using
+      isPathConcatenation_concatThreeUnitPaths_jordan p₀ p₁ p₂ h01 h12
+  · simpa [γ, γpath, p₀, p₁] using horient
+  · rw [hrange]
+    exact jordanInterior_boundaryLoop_subset_endpointHalfPlanes K hab hba hne A p hA
+  · rw [hrange]
+    exact jordanInterior_boundaryLoop_disjoint_supportIntersection K hab hba A hA
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Area of the region between a convex arc and its supporting tangents
+
+The loop built from a convex boundary arc and the two tangent segments meeting at the
+intersection of the arc's endpoint supporting lines bounds the "Mamikon region" cut off by
+those tangents.  The single result here bounds that region's signed area by the measure of any
+set that receives it.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The region between a convex boundary arc and its two supporting tangent segments has area
+at most that of any finite-measure set that receives every point of a closed convex carrier of
+the body which lies inside both endpoint supporting half-planes but outside the body. -/
+theorem convexArc_tangentRegion_area_le (B : ConvexBody Point) (a b : ℝ)
+    (hab : a < b) (hba : b < a + Real.pi)
+    {C : Set Point} (hconv : Convex ℝ C) (hclosed : IsClosed C)
+    (hBC : (B : Set Point) ⊆ C)
+    (hOC : supportingIntersection B (a : Real.Angle) (b : Real.Angle) ∈ C)
+    {E : Set Point} (hE : MeasureTheory.volume E ≠ ⊤)
+    (hsubE : ∀ q ∈ interior ((supportingLineHalfPlane B (a : Real.Angle)).2 ∩
+        (supportingLineHalfPlane B (b : Real.Angle)).2),
+      q ∈ C → q ∉ (B : Set Point) → q ∈ E) :
+    segmentArea (edgeVertices B (a : Real.Angle)).1
+        (supportingIntersection B (a : Real.Angle) (b : Real.Angle)) +
+      segmentArea (supportingIntersection B (a : Real.Angle) (b : Real.Angle))
+        (edgeVertices B (b : Real.Angle)).2 -
+      convexArcArea B a b ≤ ClassicalResults.area E := by
+  have harea0 : (0 : ℝ) ≤ ClassicalResults.area E := ENNReal.toReal_nonneg
+  by_cases hPQ : (edgeVertices B (a : Real.Angle)).1 = (edgeVertices B (b : Real.Angle)).2
+  · -- coincident endpoints: the cut lemma collapses the arc and the intersection to one point
+    obtain ⟨hO, hcar⟩ := (convexBoundaryArc_cut B a b hab hba _ _ _ rfl rfl rfl).1 hPQ
+    obtain ⟨Δ, hΔcar, hΔstart, hΔend, hΔarea⟩ :=
+      (segmentArea_jordan_and_frame (edgeVertices B (a : Real.Angle)).1
+        (edgeVertices B (a : Real.Angle)).1).1
+    have hreal : RealizesConvexArc B a b Δ :=
+      ⟨hΔcar.trans ((segment_same ℝ _).trans hcar.symm), hΔstart, hΔend.trans hPQ⟩
+    have hzero : segmentArea (edgeVertices B (a : Real.Angle)).1
+        (edgeVertices B (a : Real.Angle)).1 = 0 := by
+      simp only [segmentArea, planeCrossProduct]
+      ring
+    rw [convexArcArea_eq_jordanArcArea_of_realizes hreal, hΔarea, hO, ← hPQ, hzero]
+    simpa using harea0
+  · obtain ⟨A, p, γ, pieces, hA, hs0, hs1, hrev, hconcat, horient, hint, hdisj⟩ :=
+      convexBoundaryArc_jordan B a b hab hba hPQ
+    have hΓne : (Set.range γ.path.val).Nonempty :=
+      ⟨_, ⟨⟨γ.a, le_rfl, γ.ordered⟩, rfl⟩⟩
+    have hpiece0 : Set.range (pieces 0).path.val ⊆ C := by
+      obtain ⟨φ, τ, -, -, hφs, -, -, -, heq⟩ := hs0
+      rintro _ ⟨u, rfl⟩
+      obtain ⟨s, hs⟩ := hφs u
+      rw [← hs, heq s]
+      exact hconv (hBC (edgeVertices_fst_mem B _).1) hOC
+        (by linarith [(τ s).property.2]) (τ s).property.1 (by ring)
+    have hpiece1 : Set.range (pieces 1).path.val ⊆ C := by
+      obtain ⟨φ, τ, -, -, hφs, -, -, -, heq⟩ := hs1
+      rintro _ ⟨u, rfl⟩
+      obtain ⟨s, hs⟩ := hφs u
+      rw [← hs, heq s]
+      exact hconv hOC (hBC (edgeVertices_snd_mem B _).1)
+        (by linarith [(τ s).property.2]) (τ s).property.1 (by ring)
+    have hpiece2 : Set.range (pieces 2).path.val ⊆ C := by
+      obtain ⟨φ, ψ, -, -, hφs, -, -, -, heq⟩ := hrev
+      rintro _ ⟨u, rfl⟩
+      obtain ⟨s, hs⟩ := hφs u
+      rw [← hs, heq s]
+      have hmem : ∀ v : Set.Icc p.a p.b, p.path.val v ∈ (B : Set Point) := by
+        intro v
+        have h : p.path.val v ∈ Set.range p.path.val := ⟨v, rfl⟩
+        rw [p.range_eq, hA.1] at h
+        exact convexBoundaryArc_subset_body B a b h
+      exact hBC (hmem _)
+    -- the whole loop lies in `C`, hence so does the region it encloses
+    have hrangeC : Set.range γ.path.val ⊆ C := by
+      refine hconcat.range_subset_iUnion.trans (Set.iUnion_subset ?_)
+      intro i
+      fin_cases i
+      · exact hpiece0
+      · exact hpiece1
+      · exact hpiece2
+    have hRC : jordanInterior (Set.range γ.path.val) ⊆ C :=
+      jordanInterior_subset_of_subset_closed_convex hΓne hrangeC hconv hclosed
+    -- `B` sits inside all of its supporting half-planes, so the region avoids `B`
+    have hRnotB : ∀ q ∈ jordanInterior (Set.range γ.path.val), q ∉ (B : Set Point) := by
+      intro q hq hqB
+      refine Set.disjoint_left.mp hdisj hq (Set.mem_iInter₂.mpr fun t _ ↦ ?_)
+      change inner ℝ q (normalVector (t : Real.Angle)) ≤ supportValue (B : Set Point) _
+      exact inner_le_supportValue B hqB _
+    have hRE : jordanInterior (Set.range γ.path.val) ⊆ E := fun q hq =>
+      hsubE q (hint hq) (hRC hq) (hRnotB q hq)
+    -- signed area of the counterclockwise loop is the area it encloses, and path additivity
+    -- splits it into the two segments and the reversed arc
+    have hloop : curveAreaFunctional γ.path =
+        ClassicalResults.area (jordanInterior (Set.range γ.path.val)) :=
+      curveArea_eq_jordanInterior_area γ.a γ.b γ.ordered _ γ.path horient
+    have hparc : curveAreaFunctional p.path = convexArcArea B a b := by
+      rw [convexArcArea_eq_jordanArcArea_of_realizes hA]
+      change _ = curveAreaFunctional (Classical.choice A.property).path
+      exact (curveArea_reparametrization.2.1 A.val A.val p
+        (Classical.choice A.property) rfl).1 rfl rfl
+    have hval : segmentArea (edgeVertices B (a : Real.Angle)).1
+          (supportingIntersection B (a : Real.Angle) (b : Real.Angle)) +
+        segmentArea (supportingIntersection B (a : Real.Angle) (b : Real.Angle))
+          (edgeVertices B (b : Real.Angle)).2 - convexArcArea B a b =
+        curveAreaFunctional γ.path := by
+      rw [curveArea_concatenation γ pieces hconcat, Fin.sum_univ_three,
+        hs0.curveAreaFunctional_eq, hs1.curveAreaFunctional_eq,
+        hrev.curveAreaFunctional_eq, hparc]
+      ring
+    rw [hval, hloop]
+    have hle : MeasureTheory.volume (jordanInterior (Set.range γ.path.val)) ≤
+        MeasureTheory.volume E := MeasureTheory.measure_mono hRE
+    exact (ENNReal.toReal_le_toReal (ne_top_of_le_ne_top hE hle) hE).mpr hle
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Area.Mamikon.Basic`.
+* `Area.Variation`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Area / Mamikon / Basic
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The signed area between the convex boundary arc and a path on its supporting lines. -/
+def mamikonFunctional (K : ConvexBody Point) (a b : ℝ)
+    (hab : a < b) (_hba : b < a + Real.pi) (z : ContinuousBVPaths a b)
+    (_hz : ∀ t : Set.Icc a b,
+      z.val t ∈ (supportingLineHalfPlane K (t.val : Real.Angle)).1) : ℝ :=
+  segmentArea (edgeVertices K (a : Real.Angle)).1
+      (z.val ⟨a, le_rfl, hab.le⟩) +
+    curveAreaFunctional z +
+    segmentArea (z.val ⟨b, hab.le, le_rfl⟩)
+      (edgeVertices K (b : Real.Angle)).2 -
+    convexArcArea K a b
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Area / Variation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory Set
+open scoped unitInterval
+
+namespace MovingSofa
+
+/-- Coordinatewise convex combination of two pairs of planar points. -/
+def pointPairCombination (t : I) (x y : Point × Point) : Point × Point :=
+  ((1 - (t : ℝ)) • x.1 + (t : ℝ) • y.1,
+    (1 - (t : ℝ)) • x.2 + (t : ℝ) • y.2)
+
+theorem segmentArea_variation :
+    IsQuadraticFunctional pointPairCombination (fun x ↦ segmentArea x.1 x.2) ∧
+    ∀ p q p' q' : Point,
+      convexDirectionalDerivative pointPairCombination (fun x ↦ segmentArea x.1 x.2)
+        (p, q) (p', q') =
+      (planeCrossProduct (p' + q') (q - p) - 2 * planeCrossProduct p q) / 2 +
+        (segmentArea q q' - segmentArea p p') := by
+  let h : (Point × Point) → (Point × Point) → ℝ := fun x y ↦
+    (planeCrossProduct x.1 y.2 + planeCrossProduct y.1 x.2) / 4
+  have hh : IsConvexBilinear pointPairCombination pointPairCombination realCombination h := by
+    constructor <;> intro x t y z <;>
+      simp [h, pointPairCombination, realCombination, planeCrossProduct] <;> ring
+  have hf (x : Point × Point) : segmentArea x.1 x.2 = h x x := by
+    simp only [h, segmentArea]
+    ring
+  refine ⟨⟨h, hh, hf⟩, ?_⟩
+  intro p q p' q'
+  rw [(quadratic_directional_derivative pointPairCombination _ h hh hf).2.1]
+  simp [h, segmentArea, planeCrossProduct]
+  ring
+
+/-- The pointwise convex combination of two continuous BV paths. -/
+def bvPathCombination {a b : ℝ} (t : I) (x y : ContinuousBVPaths a b) :
+    ContinuousBVPaths a b :=
+  (1 - (t : ℝ)) • x + (t : ℝ) • y
+
+private theorem coordinate_measure_smul_add_smul {a b : ℝ} (r s : ℝ)
+    (x y : ContinuousBVPaths a b) (i : Fin 2) :
+    intervalStieltjesMeasure (continuousBVCoordinate (r • x + s • y) i) =
+      r • intervalStieltjesMeasure (continuousBVCoordinate x i) +
+        s • intervalStieltjesMeasure (continuousBVCoordinate y i) := by
+  obtain ⟨F, hF, hμ⟩ := intervalStieltjes_linear_combination a b
+    (continuousBVCoordinate x i) (continuousBVCoordinate y i) r s
+  rw [← hμ]
+  refine congrArg _ (RightContinuousIntervalBV.toFun_injective (funext fun u ↦ ?_))
+  simpa [continuousBVCoordinate] using (hF u).symm
+
+private def coordinateIntegral {a b : ℝ} (x y : ContinuousBVPaths a b) (i j : Fin 2) : ℝ :=
+  intervalStieltjesIntegral (continuousBVCoordinate y j) (fun t ↦ x.val t i) univ
+
+private theorem coordinateIntegral_smul_add_smul_right {a b : ℝ} (r s : ℝ)
+    (x y z : ContinuousBVPaths a b) (i j : Fin 2) :
+    coordinateIntegral x (r • y + s • z) i j =
+      r * coordinateIntegral x y i j + s * coordinateIntegral x z i j := by
+  have hi (w : ContinuousBVPaths a b) :
+      (intervalStieltjesMeasure (continuousBVCoordinate w j)).Integrable (fun s ↦ x.val s i) :=
+    RightContinuousIntervalBV.integrable_of_continuous _
+      ((PiLp.continuous_apply 2 _ i).comp x.property.1)
+  unfold coordinateIntegral intervalStieltjesIntegral
+  simp only [VectorMeasure.restrict_univ]
+  rw [coordinate_measure_smul_add_smul, VectorMeasure.integral_add_vectorMeasure
+    ((hi y).smul_vectorMeasure _) ((hi z).smul_vectorMeasure _)]
+  simp only [VectorMeasure.integral_smul_vectorMeasure, smul_eq_mul]
+
+private theorem coordinateIntegral_smul_add_smul_left {a b : ℝ} (r s : ℝ)
+    (x y z : ContinuousBVPaths a b) (i j : Fin 2) :
+    coordinateIntegral (r • x + s • y) z i j =
+      r * coordinateIntegral x z i j + s * coordinateIntegral y z i j := by
+  have hi (w : ContinuousBVPaths a b) :
+      (intervalStieltjesMeasure (continuousBVCoordinate z j)).Integrable (fun s ↦ w.val s i) :=
+    RightContinuousIntervalBV.integrable_of_continuous _
+      ((PiLp.continuous_apply 2 _ i).comp w.property.1)
+  unfold coordinateIntegral intervalStieltjesIntegral
+  simp only [VectorMeasure.restrict_univ]
+  change (∫ᵛ u, r • x.val u i + s • y.val u i
+    ∂[ContinuousLinearMap.mul ℝ ℝ; intervalStieltjesMeasure (continuousBVCoordinate z j)]) = _
+  have hadd := VectorMeasure.integral_fun_add (B := ContinuousLinearMap.mul ℝ ℝ)
+    ((hi x).smul r) ((hi y).smul s)
+  simp only [Pi.smul_apply, VectorMeasure.integral_fun_smul] at hadd
+  simpa only [smul_eq_mul] using hadd
+
+private theorem coordinateIntegral_combination_right {a b : ℝ}
+    (t : I) (x y z : ContinuousBVPaths a b) (i j : Fin 2) :
+    coordinateIntegral x (bvPathCombination t y z) i j =
+      (1 - (t : ℝ)) * coordinateIntegral x y i j + (t : ℝ) * coordinateIntegral x z i j :=
+  coordinateIntegral_smul_add_smul_right _ _ x y z i j
+
+private theorem coordinateIntegral_combination_left {a b : ℝ}
+    (t : I) (x y z : ContinuousBVPaths a b) (i j : Fin 2) :
+    coordinateIntegral (bvPathCombination t x y) z i j =
+      (1 - (t : ℝ)) * coordinateIntegral x z i j + (t : ℝ) * coordinateIntegral y z i j :=
+  coordinateIntegral_smul_add_smul_left _ _ x y z i j
+
+private theorem coordinateIntegral_difference {a b : ℝ} (x y : ContinuousBVPaths a b)
+    (i j : Fin 2) :
+    intervalStieltjesIntegral (continuousBVCoordinate x j) (fun t ↦ y.val t i - x.val t i) univ =
+      coordinateIntegral y x i j - coordinateIntegral x x i j := by
+  unfold coordinateIntegral intervalStieltjesIntegral
+  simp only [VectorMeasure.restrict_univ]
+  exact VectorMeasure.integral_fun_sub
+    (RightContinuousIntervalBV.integrable_of_continuous _
+      ((PiLp.continuous_apply 2 _ i).comp y.property.1))
+    (RightContinuousIntervalBV.integrable_of_continuous _
+      ((PiLp.continuous_apply 2 _ i).comp x.property.1))
+
+private theorem curveArea_quadratic_derivative (a b : ℝ) :
+    IsQuadraticFunctional bvPathCombination (@curveAreaFunctional a b) ∧
+    ∀ x y : ContinuousBVPaths a b,
+      convexDirectionalDerivative bvPathCombination curveAreaFunctional x y =
+        (coordinateIntegral x y 0 1 - coordinateIntegral x y 1 0 +
+          (coordinateIntegral y x 0 1 - coordinateIntegral y x 1 0) -
+          2 * (coordinateIntegral x x 0 1 - coordinateIntegral x x 1 0)) / 2 := by
+  let h : ContinuousBVPaths a b → ContinuousBVPaths a b → ℝ := fun x y ↦
+    (coordinateIntegral x y 0 1 - coordinateIntegral x y 1 0) / 2
+  have hh : IsConvexBilinear bvPathCombination bvPathCombination realCombination h := by
+    constructor
+    · intro x t y z
+      dsimp only [h]
+      rw [coordinateIntegral_combination_right, coordinateIntegral_combination_right]
+      simp only [realCombination]
+      ring
+    · intro x t y z
+      dsimp only [h]
+      rw [coordinateIntegral_combination_left, coordinateIntegral_combination_left]
+      simp only [realCombination]
+      ring
+  have hf (x : ContinuousBVPaths a b) : curveAreaFunctional x = h x x := rfl
+  refine ⟨⟨h, hh, hf⟩, ?_⟩
+  intro x y
+  rw [(quadratic_directional_derivative bvPathCombination _ h hh hf).2.1]
+  dsimp only [h]
+  ring
+
+theorem curveArea_variation (a b : ℝ) (hab : a ≤ b) :
+    IsQuadraticFunctional bvPathCombination (@curveAreaFunctional a b) ∧
+    ∀ x y : ContinuousBVPaths a b,
+      convexDirectionalDerivative bvPathCombination curveAreaFunctional x y =
+      intervalStieltjesIntegral (continuousBVCoordinate x 1)
+          (fun t ↦ y.val t 0 - x.val t 0) Set.univ -
+        intervalStieltjesIntegral (continuousBVCoordinate x 0)
+          (fun t ↦ y.val t 1 - x.val t 1) Set.univ +
+        (segmentArea (x.val ⟨b, hab, le_rfl⟩) (y.val ⟨b, hab, le_rfl⟩) -
+          segmentArea (x.val ⟨a, le_rfl, hab⟩) (y.val ⟨a, le_rfl, hab⟩)) := by
+  obtain ⟨hq, hd⟩ := curveArea_quadratic_derivative a b
+  refine ⟨hq, ?_⟩
+  intro x y
+  have hc (w : ContinuousBVPaths a b) (i : Fin 2) :
+      Continuous (continuousBVCoordinate w i).toFun :=
+    (PiLp.continuous_apply 2 _ i).comp w.property.1
+  have h01 := intervalStieltjes_integration_by_parts_of_continuous a b hab
+    (continuousBVCoordinate x 0) (continuousBVCoordinate y 1) (hc x 0) (hc y 1)
+  have h10 := intervalStieltjes_integration_by_parts_of_continuous a b hab
+    (continuousBVCoordinate x 1) (continuousBVCoordinate y 0) (hc x 1) (hc y 0)
+  change coordinateIntegral y x 1 0 + coordinateIntegral x y 0 1 = _ at h01
+  change coordinateIntegral y x 0 1 + coordinateIntegral x y 1 0 = _ at h10
+  rw [hd, coordinateIntegral_difference, coordinateIntegral_difference]
+  simp only [continuousBVCoordinate] at h01 h10
+  simp only [segmentArea, planeCrossProduct]
+  linarith
+
+private theorem curveAreaFunctional_eq_coordinateIntegral {a b : ℝ}
+    (x : ContinuousBVPaths a b) :
+    curveAreaFunctional x = (coordinateIntegral x x 0 1 - coordinateIntegral x x 1 0) / 2 := rfl
+
+/-- Translating a continuous BV path by another one adds the translating path's own signed area
+and the two mixed cross integrals. -/
+private theorem curveAreaFunctional_add {a b : ℝ}
+    (x c : ContinuousBVPaths a b) :
+    curveAreaFunctional (x + c) = curveAreaFunctional x + curveAreaFunctional c +
+      (coordinateIntegral x c 0 1 - coordinateIntegral x c 1 0 +
+        (coordinateIntegral c x 0 1 - coordinateIntegral c x 1 0)) / 2 := by
+  have hone : x + c = (1 : ℝ) • x + (1 : ℝ) • c := by rw [one_smul, one_smul]
+  simp only [curveAreaFunctional_eq_coordinateIntegral, hone,
+    coordinateIntegral_smul_add_smul_left, coordinateIntegral_smul_add_smul_right]
+  ring
+
+/-- Translating a continuous BV path by a fixed one changes its signed area by a convex-linear
+functional of the path: the two mixed Stieltjes cross integrals are separately linear and the
+translating path's own area is constant. -/
+theorem curveArea_translation_convexLinear {a b : ℝ}
+    (c : ContinuousBVPaths a b) :
+    IsConvexLinear bvPathCombination realCombination
+      (fun x ↦ curveAreaFunctional (x + c) - curveAreaFunctional x) := by
+  intro t x y
+  change curveAreaFunctional ((1 - (t : ℝ)) • x + (t : ℝ) • y + c) -
+      curveAreaFunctional ((1 - (t : ℝ)) • x + (t : ℝ) • y) =
+    realCombination t (curveAreaFunctional (x + c) - curveAreaFunctional x)
+      (curveAreaFunctional (y + c) - curveAreaFunctional y)
+  rw [curveAreaFunctional_add ((1 - (t : ℝ)) • x + (t : ℝ) • y) c,
+    curveAreaFunctional_add x c, curveAreaFunctional_add y c, realCombination]
+  simp only [coordinateIntegral_smul_add_smul_left, coordinateIntegral_smul_add_smul_right]
+  ring
+
+theorem convexArcArea_variation (a b : ℝ) (hab : a < b) (hba : b < a + Real.pi) :
+    IsQuadraticFunctional convexBodyCombination (fun K ↦ convexArcArea K a b) ∧
+    ∀ K L : ConvexBody Point,
+      convexDirectionalDerivative convexBodyCombination (fun M ↦ convexArcArea M a b) K L =
+      (∫ t in (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b,
+        (supportValue L t - supportValue K t) ∂surfaceAreaMeasure K) +
+        (segmentArea (edgeVertices K (b : Real.Angle)).2 (edgeVertices L (b : Real.Angle)).2 -
+          segmentArea (edgeVertices K (a : Real.Angle)).1 (edgeVertices L (a : Real.Angle)).1) := by
+  obtain ⟨hbil0, F, hFtoFun, hFcross, hFarea⟩ := convexArc_bilinear_computation a b hab hba
+  let E := (fun s : ℝ ↦ (s : Real.Angle)) '' Set.Ioo a b
+  let B : ConvexBody Point → ConvexBody Point → ℝ := fun K L ↦
+    (1 / 2 : ℝ) * ∫ t in E, supportValue K t ∂surfaceAreaMeasure L
+  have hbil : IsConvexBilinear convexBodyCombination convexBodyCombination realCombination B :=
+    hbil0
+  have hfB : ∀ K : ConvexBody Point, convexArcArea K a b = B K K := fun K ↦
+    (hFarea K).trans ((hFcross K K).1).symm
+  refine ⟨⟨B, hbil, hfB⟩, ?_⟩
+  intro K L
+  have hcont (M : ConvexBody Point) : Continuous fun u : Real.Angle ↦ supportValue M u :=
+    (compactSet_support_continuity M M M.nonempty M.isCompact M.nonempty M.isCompact).2.2.1
+  have hint (M N : ConvexBody Point) :
+      Integrable (fun u : Real.Angle ↦ supportValue M u) (surfaceAreaMeasure N) := by
+    let _ : IsFiniteMeasure (surfaceAreaMeasure N) := (surfaceAreaMeasure_face_union N).1
+    exact (hcont M).integrable_of_hasCompactSupport
+      (isCompact_univ.of_isClosed_subset isClosed_closure (Set.subset_univ _))
+  have hsub : (∫ t in E, (supportValue L t - supportValue K t) ∂surfaceAreaMeasure K) =
+      2 * B L K - 2 * B K K := by
+    rw [integral_sub (hint L K).restrict (hint K K).restrict]
+    simp only [B, E]
+    ring
+  have hanti : B K L - B L K =
+      segmentArea (edgeVertices K (b : Real.Angle)).2 (edgeVertices L (b : Real.Angle)).2 -
+        segmentArea (edgeVertices K (a : Real.Angle)).1 (edgeVertices L (a : Real.Angle)).1 := by
+    have h1 : B K L = openIntervalCrossIntegral (F L)
+        (fun t : Set.Icc a b ↦ (edgeVertices K ((t : ℝ) : Real.Angle)).2) := (hFcross K L).2
+    have h2 : B L K = openIntervalCrossIntegral (F K)
+        (fun t : Set.Icc a b ↦ (edgeVertices L ((t : ℝ) : Real.Angle)).1) := (hFcross L K).1
+    rw [h1, h2]
+    exact openIntervalCrossIntegral_antisymm hab K L (F K) (F L) (hFtoFun K) (hFtoFun L)
+  rw [(quadratic_directional_derivative convexBodyCombination
+    (fun M ↦ convexArcArea M a b) B hbil hfB).2.1 K L, hsub]
+  linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Optimality from the area upper bound
+
+`AreaUpperBound` says that no moving sofa has larger area than Gerver's sofa. Since Gerver's sofa
+is itself a moving sofa, the bound gives `sofaConstant = volume gerversSofa`.
+-/
+
+@[expose] public section
+
+namespace MovingSofa
+
+open MeasureTheory
+open scoped EuclideanGeometry unitInterval
+
+/-- Every moving sofa has area at most that of Gerver's sofa. This is the upper bound proved in
+Baek's paper; together with the fact that Gerver's sofa is a moving sofa it gives optimality. -/
+def AreaUpperBound : Prop :=
+  ∀ (s : Set ℝ²) (m : I → E(2)), IsMovingSofa s m → volume s ≤ volume gerversSofa
+
+/-- The area upper bound implies that Gerver's sofa attains the sofa constant. -/
+theorem optimality_of_areaUpperBound (h : AreaUpperBound) :
+    sofaConstant = volume gerversSofa := by
+  apply le_antisymm _ volume_gerversSofa_le_sofaConstant
+  unfold sofaConstant
+  refine iSup_le fun s ↦ iSup_le fun hs ↦ ?_
+  obtain ⟨m, hm⟩ := hs
+  exact h s m hm
+
+end MovingSofa
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Motion.CanonicalBridge`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Motion / Canonical Bridge
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open MeasureTheory
+open scoped unitInterval
+
+private def normalizeMotion (m : I → Point ≃ᵃⁱ[ℝ] Point) (t : I) :
+    Point ≃ᵃⁱ[ℝ] Point :=
+  (m 0).symm.trans (m t)
+
+private theorem continuous_normalizeMotion (m : I → Point ≃ᵃⁱ[ℝ] Point)
+    (hm : Continuous m) : Continuous (normalizeMotion m) := by
+  rw [continuous_induced_rng]
+  have hmc : Continuous (fun t ↦ (m t).toAffineIsometry.toContinuousAffineMap) :=
+    continuous_induced_dom.comp hm
+  apply (ContinuousAffineMap.continuous_comp_right
+    (m 0).symm.toAffineIsometry.toContinuousAffineMap).comp hmc |>.congr
+  intro t
+  rfl
+
+/-- A canonical hallway motion is also a paper motion of the same set. -/
+theorem IsMovingSofa.isPaperMotion {s : Set Point} {m : I → Point ≃ᵃⁱ[ℝ] Point}
+    (h : IsMovingSofa s m) : IsPaperMotion s m := by
+  refine ⟨h.isConnected, h.isClosed, h.continuous, ⟨0, ?_⟩,
+    exists_motion_rotation m h.continuous h.zero, ?_, h.subset_hallway, h.final⟩
+  · intro p
+    simp [h.zero]
+  · simpa [h.zero] using h.initial
+
+theorem canonical_paper_motion_bridge :
+    (∀ s : Set Point, IsPaperMovingSofa s →
+      ∃ (q : Point) (m : I → Point ≃ᵃⁱ[ℝ] Point),
+        IsMovingSofa ((fun p ↦ p + q) '' s) m ∧
+        volume ((fun p ↦ p + q) '' s) = volume s) ∧
+    (∀ (s : Set Point) (m : I → Point ≃ᵃⁱ[ℝ] Point),
+      IsMovingSofa s m → IsPaperMovingSofa s) := by
+  constructor
+  · rintro s ⟨m, hsconn, hsclosed, hmcont, ⟨q, hq⟩, -, hini, hall, hfinal⟩
+    refine ⟨q, normalizeMotion m, ?_, ?_⟩
+    · refine
+        { isConnected := hsconn.image _ (by fun_prop)
+          isClosed := ?_
+          continuous := continuous_normalizeMotion m hmcont
+          zero := ?_
+          initial := ?_
+          subset_hallway := ?_
+          final := ?_ }
+      · change IsClosed ((AffineIsometryEquiv.vaddConst ℝ q) '' s)
+        exact (AffineIsometryEquiv.vaddConst ℝ q).toHomeomorph.isClosedMap s hsclosed
+      · ext x
+        simp [normalizeMotion]
+      · rintro _ ⟨x, hx, rfl⟩
+        change x + q ∈ horizontalHallway
+        rw [← hq x]
+        exact hini ⟨x, hx, rfl⟩
+      · intro t
+        rintro _ ⟨_, ⟨x, hx, rfl⟩, rfl⟩
+        change normalizeMotion m t (x + q) ∈ hallway
+        rw [← hq x]
+        exact hall t ⟨x, hx, by simp [normalizeMotion]⟩
+      · rintro _ ⟨_, ⟨x, hx, rfl⟩, rfl⟩
+        change normalizeMotion m 1 (x + q) ∈ verticalHallway
+        rw [← hq x]
+        exact hfinal ⟨x, hx, by simp [normalizeMotion]⟩
+    · have himage : (fun p ↦ p + q) '' s = (fun p ↦ p - q) ⁻¹' s := by
+        ext p
+        constructor
+        · rintro ⟨x, hx, rfl⟩
+          simpa using hx
+        · intro hp
+          exact ⟨p - q, hp, by simp⟩
+      rw [himage, ← Measure.map_apply_of_aemeasurable (μ := volume)
+        (f := fun p : Point ↦ p - q) (by fun_prop) hsclosed.measurableSet]
+      have hmap : Measure.map (fun p : Point ↦ p - q) volume = volume := by
+        simpa only [sub_eq_add_neg] using map_add_right_eq_self volume (-q)
+      rw [hmap]
+  · exact fun s m h ↦ ⟨m, h.isPaperMotion⟩
+
+theorem canonical_motion_compactness (s : Set Point) (m : I → Point ≃ᵃⁱ[ℝ] Point)
+    (h : IsMovingSofa s m) :
+    IsCompact s ∧ MeasurableSet s ∧ volume s < ⊤ := by
+  have hc : IsCompact s := Metric.isCompact_iff_isClosed_bounded.mpr
+    ⟨h.isClosed, h.isBounded⟩
+  exact ⟨hc, hc.measurableSet, hc.measure_lt_top⟩
+
+theorem real_area_le_iff_volume_le (s t : Set Point)
+    (hfinS : volume s < ⊤) (hfinT : volume t < ⊤) :
+    ClassicalResults.area s ≤ ClassicalResults.area t ↔ volume s ≤ volume t := by
+  exact ENNReal.toReal_le_toReal hfinS.ne hfinT.ne
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/Development/Geometry/Foundations/Development007.lean b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development007.lean
new file mode 100644
index 0000000000..1febd02c61
--- /dev/null
+++ b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development007.lean
@@ -0,0 +1,10967 @@
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+module
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development006
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development005
+
+
+
+
+
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development002
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development001
+
+
+
+public import Mathlib.MeasureTheory.Integral.Prod
+public import Mathlib.MeasureTheory.Measure.Hausdorff
+public import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
+public import Mathlib.MeasureTheory.Measure.Real
+public import Mathlib.Topology.ContinuousMap.Basic
+public import Mathlib.Topology.Order.ProjIcc
+/-!
+# Moving sofa: related mathematical developments
+
+* `Polygon.Foundations.Development002`.
+* `Bounds.Foundations.Development003`.
+* `Cap.Foundations.Development005`.
+* `Sofa.Foundations.Development001`.
+* `Cap.Foundations.Development006`.
+* `Bounds.Foundations.Development004`.
+* `Cap.Foundations.Development007`.
+* `Area.Foundations.Development003`.
+* `Bounds.Foundations.Development005`.
+* `Bounds.Foundations.Development006`.
+* `Motion.Foundations.Development002`.
+* `Sofa.Foundations.Development002`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Polygon.Approximation`.
+* `Polygon.CapWidthBound`.
+* `Polygon.DiscreteCapData`.
+* `Polygon.Height.Properties`.
+* `Polygon.Nef.Slices`.
+* `Polygon.Nef.Variation.LocalSlices`.
+* `Polygon.Nef.Variation.Area`.
+* `Polygon.Nef.Variation.Boundary`.
+* `Polygon.Nef.Variation`.
+* `Polygon.Nef.SignedVariation`.
+* `Polygon.Height.WallVariation`.
+* `Polygon.RightAngleGrid`.
+* `Polygon.Translation`.
+* `Polygon.Height.Reconstruction`.
+* `Polygon.Height.PositiveIncrement.Contacts`.
+* `Polygon.Height.PositiveIncrement`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Approximation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Membership in a polygon cap is given by the strip and selected support inequalities. -/
+theorem mem_angleCap_iff (Θ : AngleSet) (K : CapSpace Θ.angle) (p : Point) :
+    p ∈ angleCap Θ K ↔
+      ((0 ≤ p 1 ∧ p 1 ≤ 1) ∧
+        (0 ≤ inner ℝ p (normalVector (Θ.angle : Real.Angle)) ∧
+          inner ℝ p (normalVector (Θ.angle : Real.Angle)) ≤ 1)) ∧
+      ∀ t ∈ Θ.directions,
+        inner ℝ p (normalVector (t : Real.Angle)) ≤ supportValue K.val (t : Real.Angle) ∧
+        inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) ≤
+          supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) := by
+  simp only [angleCap, Set.mem_inter_iff, Set.mem_iInter, mem_stripParallelogram_iff]
+  apply and_congr_right
+  intro _
+  apply forall_congr'
+  intro t
+  apply forall_congr'
+  intro _
+  rw [(rotatingHallwayParts_formulas (K.val : Set Point) (t : Real.Angle)).2.2.2.2.2.2.2.1]
+  rfl
+
+/-- The polygon-cap approximation contains the original cap. -/
+theorem subset_angleCap (Θ : AngleSet) (K : CapSpace Θ.angle) :
+    (K.val : Set Point) ⊆ angleCap Θ K := by
+  intro p hp
+  apply (mem_angleCap_iff Θ K p).mpr
+  refine ⟨(mem_stripParallelogram_iff _ _).mp (K.subset_stripParallelogram hp), ?_⟩
+  intro t _
+  exact ⟨inner_le_supportValue K.val hp _, inner_le_supportValue K.val hp _⟩
+
+/-- A polygon cap is the intersection of its selected support half-planes. -/
+theorem angleCap_eq_iInter_supportValue (Θ : AngleSet) (K : CapSpace Θ.angle) :
+    angleCap Θ K = ⋂ t ∈
+      ((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain Θ) ∪ capLowerNormals Θ.angle,
+        normalHalfPlane t (supportValue K.val t) false false := by
+  ext p
+  simp only [Set.mem_iInter, mem_angleCap_iff]
+  constructor
+  · rintro ⟨⟨hy, hw⟩, ht⟩ u hu
+    change inner ℝ p (normalVector u) ≤ supportValue K.val u
+    rcases hu with ⟨u, hu, rfl⟩ | hu
+    · rcases hu with (hu | ⟨t, ht', rfl⟩) | hu
+      · exact (ht u hu).1
+      · exact (ht t ht').2
+      · rcases hu with rfl | hu
+        · rw [K.property.2.2.1]
+          exact hw.2
+        · have heq : u = Real.pi / 2 := hu
+          subst u
+          rw [K.property.2.2.2.1]
+          simpa [normalVector, frame, PiLp.inner_apply] using hy.2
+    · rcases hu with rfl | hu
+      · rw [K.property.2.2.2.2.1, normalVector_add_pi, inner_neg_right]
+        exact neg_nonpos.mpr hw.1
+      · have heq : u = ((3 * Real.pi / 2 : ℝ) : Real.Angle) := hu
+        subst u
+        rw [K.property.2.2.2.2.2.1, inner_normalVector_three_pi_div_two]
+        exact neg_nonpos.mpr hy.1
+  · intro hp
+    have ht (t : ℝ) (ht : t ∈ Θ.directions) :=
+      hp (t : Real.Angle) (Or.inl ⟨t, Or.inl (Or.inl ht), rfl⟩)
+    have ht' (t : ℝ) (ht : t ∈ Θ.directions) :=
+      hp ((t + Real.pi / 2 : ℝ) : Real.Angle)
+        (Or.inl ⟨t + Real.pi / 2, Or.inl (Or.inr ⟨t, ht, rfl⟩), rfl⟩)
+    have hw := hp (Θ.angle : Real.Angle)
+      (Or.inl ⟨Θ.angle, Or.inr (Or.inl rfl), rfl⟩)
+    have hy := hp ((Real.pi / 2 : ℝ) : Real.Angle)
+      (Or.inl ⟨Real.pi / 2, Or.inr (Or.inr rfl), rfl⟩)
+    have hl := hp ((Θ.angle + Real.pi : ℝ) : Real.Angle) (Or.inr (Or.inl rfl))
+    have hb := hp ((3 * Real.pi / 2 : ℝ) : Real.Angle) (Or.inr (Or.inr rfl))
+    change inner ℝ p (normalVector _) ≤ supportValue K.val _ at hw hy hl hb
+    rw [K.property.2.2.1] at hw
+    rw [K.property.2.2.2.1] at hy
+    rw [K.property.2.2.2.2.1, normalVector_add_pi, inner_neg_right] at hl
+    rw [K.property.2.2.2.2.2.1, inner_normalVector_three_pi_div_two] at hb
+    have hy' : p 1 ≤ 1 := by simpa [normalVector, frame, PiLp.inner_apply] using hy
+    exact ⟨⟨⟨by linarith, hy'⟩, by linarith, hw⟩, fun t h ↦ ⟨ht t h, ht' t h⟩⟩
+
+/-- Polygon-cap approximations are closed. -/
+theorem isClosed_angleCap (Θ : AngleSet) (K : CapSpace Θ.angle) :
+    IsClosed (angleCap Θ K) := by
+  rw [angleCap_eq_iInter_supportValue]
+  apply isClosed_iInter
+  intro t
+  apply isClosed_iInter
+  intro _
+  exact isClosed_le (by fun_prop) continuous_const
+
+/-- Polygon-cap approximations are convex. -/
+theorem convex_angleCap (Θ : AngleSet) (K : CapSpace Θ.angle) :
+    Convex ℝ (angleCap Θ K) := by
+  rw [angleCap_eq_iInter_supportValue]
+  apply convex_iInter
+  intro t
+  apply convex_iInter
+  intro _
+  apply convex_halfSpace_le
+  exact ⟨fun x y ↦ inner_add_left x y _, fun a x ↦ by simp [real_inner_smul_left]⟩
+
+/-- Polygon-cap approximation preserves support at every selected normal. -/
+theorem supportValue_angleCap (Θ : AngleSet) (K : CapSpace Θ.angle)
+    {t : Real.Angle}
+    (ht : t ∈ ((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain Θ) ∪ capLowerNormals Θ.angle) :
+    supportValue (angleCap Θ K) t = supportValue K.val t := by
+  have hbound : ∀ p ∈ angleCap Θ K, inner ℝ p (normalVector t) ≤ supportValue K.val t := by
+    intro p hp
+    rw [angleCap_eq_iInter_supportValue] at hp
+    exact Set.mem_iInter.mp (Set.mem_iInter.mp hp t) ht
+  have hbd : BddAbove ((fun p ↦ inner ℝ p (normalVector t)) '' angleCap Θ K) := by
+    refine ⟨supportValue K.val t, ?_⟩
+    rintro _ ⟨p, hp, rfl⟩
+    exact hbound p hp
+  apply le_antisymm
+  · apply csSup_le ((K.val.nonempty.mono (subset_angleCap Θ K)).image _)
+    rintro _ ⟨p, hp, rfl⟩
+    exact hbound p hp
+  · apply csSup_le (K.val.nonempty.image _)
+    rintro _ ⟨p, hp, rfl⟩
+    exact le_csSup hbd ⟨p, subset_angleCap Θ K hp, rfl⟩
+
+/-- Polygon-cap approximation fixes caps with the prescribed normals. -/
+theorem angleCap_eq_self (Θ : AngleSet) (P : PolygonCapSpace Θ) :
+    angleCap Θ P.val = (P.val.val : Set Point) := by
+  rw [angleCap_eq_iInter_supportValue, ← P.property.eq_iInter_supportValue]
+
+/-- An interior selected direction bounds the polygon cap, including at right angle. -/
+theorem isBounded_angleCap (Θ : AngleSet) (K : CapSpace Θ.angle) :
+    Bornology.IsBounded (angleCap Θ K) := by
+  obtain ⟨t, ht⟩ := Θ.nonempty
+  have hti := Θ.interior t ht
+  have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, hti.1], hti.2.trans_le Θ.angle_le⟩
+  have hs : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi hti.1
+    (by linarith [hti.2, Θ.angle_le, Real.pi_pos])
+  let l := -supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) / Real.sin t
+  let r := supportValue K.val (t : Real.Angle) / Real.cos t
+  let M := |l| + |r|
+  have hM : 0 ≤ M := by dsimp [M]; positivity
+  apply isBounded_iff_forall_norm_le.mpr
+  refine ⟨M + 1, ?_⟩
+  intro p hp
+  obtain ⟨⟨hy, _⟩, hnormals⟩ := (mem_angleCap_iff Θ K p).mp hp
+  obtain ⟨ha, hb⟩ := hnormals t ht
+  simp [normalVector, frame, PiLp.inner_apply, Real.cos_add, Real.sin_add,
+    -Real.Angle.coe_add] at ha hb
+  have hl : l ≤ p 0 := by
+    apply (div_le_iff₀ hs).mpr
+    nlinarith [mul_nonneg hc.le hy.1]
+  have hr : p 0 ≤ r := by
+    apply (le_div_iff₀ hc).mpr
+    nlinarith [mul_nonneg hs.le hy.1]
+  have hx : |p 0| ≤ M := by
+    apply abs_le.mpr
+    dsimp [M]
+    constructor <;> linarith [neg_abs_le l, le_abs_self r, abs_nonneg l, abs_nonneg r]
+  have hx2 := (sq_le_sq₀ (abs_nonneg (p 0)) hM).mpr hx
+  have hy2 : (p 1) ^ 2 ≤ 1 := by nlinarith [hy.1, hy.2]
+  have hn := EuclideanSpace.norm_sq_eq p
+  simp only [Fin.sum_univ_two, Real.norm_eq_abs, sq_abs] at hn hx2
+  nlinarith [norm_nonneg p]
+
+/-- Polygon-cap approximations are compact. -/
+theorem isCompact_angleCap (Θ : AngleSet) (K : CapSpace Θ.angle) :
+    IsCompact (angleCap Θ K) :=
+  Metric.isCompact_iff_isClosed_bounded.mpr ⟨isClosed_angleCap Θ K, isBounded_angleCap Θ K⟩
+
+theorem angleCap_properties (Θ : AngleSet) (K : CapSpace Θ.angle) :
+    (∃ P : PolygonCapSpace Θ, (P.val.val : Set Point) = angleCap Θ K) ∧
+    (K.val : Set Point) ⊆ angleCap Θ K ∧
+    (∀ t ∈ ((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain Θ) ∪ capLowerNormals Θ.angle,
+      supportValue (angleCap Θ K) t = supportValue (K.val : Set Point) t) ∧
+    (∀ P : PolygonCapSpace Θ, angleCap Θ P.val = (P.val.val : Set Point)) := by
+  refine ⟨?_, subset_angleCap Θ K, fun _ ht ↦ supportValue_angleCap Θ K ht,
+    angleCap_eq_self Θ⟩
+  let L : ConvexBody Point := ⟨angleCap Θ K, convex_angleCap Θ K,
+    isCompact_angleCap Θ K, K.val.nonempty.mono (subset_angleCap Θ K)⟩
+  let N := ((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain Θ) ∪ capLowerNormals Θ.angle
+  have hrepr : HasHalfPlaneRepresentation L N := by
+    refine ⟨(fun t ↦ (t, supportValue K.val t)) '' N, ?_, ?_⟩
+    · rintro _ ⟨t, ht, rfl⟩
+      exact ht
+    · simp only [Set.biInter_image]
+      exact angleCap_eq_iInter_supportValue Θ K
+  have hdomain : angleDomain Θ ⊆ capUpperAngles Θ.angle := by
+    rintro t ((ht | ⟨s, hs, rfl⟩) | ht)
+    · exact Or.inl ⟨(Θ.interior t ht).1.le, (Θ.interior t ht).2.le⟩
+    · right
+      constructor <;> linarith [(Θ.interior s hs).1, (Θ.interior s hs).2]
+    · rcases ht with rfl | rfl
+      · exact Or.inl ⟨Θ.angle_pos.le, le_rfl⟩
+      · exact Or.inr ⟨le_rfl, le_add_of_nonneg_left Θ.angle_pos.le⟩
+  have hrepr' : HasHalfPlaneRepresentation L
+      (((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles Θ.angle) ∪ capLowerNormals Θ.angle) := by
+    obtain ⟨C, hC, hLC⟩ := hrepr
+    refine ⟨C, ?_, hLC⟩
+    intro c hc
+    rcases hC c hc with ⟨t, ht, heq⟩ | ht
+    · exact Or.inl ⟨t, hdomain ht, heq⟩
+    · exact Or.inr ht
+  have hsupp (t : Real.Angle) (ht : t ∈ N) : supportValue L t = supportValue K.val t :=
+    supportValue_angleCap Θ K ht
+  have hcap : IsCap Θ.angle L := by
+    refine ⟨Θ.angle_pos, Θ.angle_le, ?_, ?_, ?_, ?_, hrepr'⟩
+    · rw [hsupp _ (Or.inl ⟨Θ.angle, Or.inr (Or.inl rfl), rfl⟩)]
+      exact K.property.2.2.1
+    · rw [hsupp _ (Or.inl ⟨Real.pi / 2, Or.inr (Or.inr rfl), rfl⟩)]
+      exact K.property.2.2.2.1
+    · rw [hsupp _ (Or.inr (Or.inl rfl))]
+      exact K.property.2.2.2.2.1
+    · rw [hsupp _ (Or.inr (Or.inr rfl))]
+      exact K.property.2.2.2.2.2.1
+  exact ⟨⟨⟨L, hcap⟩, hrepr⟩, rfl⟩
+
+theorem polygonNiche_angleCap (Θ : AngleSet) (K P : CapSpace Θ.angle)
+    (hP : (P.val : Set Point) = angleCap Θ K) :
+    polygonNiche Θ K = polygonNiche Θ P ∧ polygonNiche Θ K ⊆ capNiche K := by
+  refine ⟨?_, polygonNiche_subset_capNiche Θ K⟩
+  have hs := (angleCap_properties Θ K).2.2.1
+  have hq (t : ℝ) (ht : t ∈ Θ.directions) :
+      innerQuadrant (K.val : Set Point) t = innerQuadrant (P.val : Set Point) t := by
+    unfold innerQuadrant
+    rw [hP, hs (t : Real.Angle) (Or.inl ⟨t, Or.inl (Or.inl ht), rfl⟩),
+      hs ((t + Real.pi / 2 : ℝ) : Real.Angle)
+        (Or.inl ⟨t + Real.pi / 2, Or.inl (Or.inr ⟨t, ht, rfl⟩), rfl⟩)]
+  unfold polygonNiche
+  congr 1
+  exact Set.iUnion_congr fun t ↦ Set.iUnion_congr fun ht ↦ hq t ht
+
+theorem polygonArea_upperBound (Θ : AngleSet) :
+    (∀ K : PolygonCapSpace Θ, polygonAreaFunctional Θ K.val =
+      ClassicalResults.area (K.val.val : Set Point) -
+        ClassicalResults.area (polygonNiche Θ K.val)) ∧
+    (∀ K : CapSpace Θ.angle, capAreaFunctional K ≤ polygonAreaFunctional Θ K) := by
+  constructor
+  · intro K
+    unfold polygonAreaFunctional
+    rw [(angleCap_properties Θ K.val).2.2.2 K]
+  · intro K
+    obtain ⟨P, hP⟩ := (angleCap_properties Θ K).1
+    have hcap : ClassicalResults.area (K.val : Set Point) ≤
+        ClassicalResults.area (angleCap Θ K) := by
+      apply ENNReal.toReal_mono
+      · rw [← hP]
+        exact P.val.val.isCompact.measure_ne_top
+      · exact MeasureTheory.measure_mono (angleCap_properties Θ K).2.1
+    have hniche : ClassicalResults.area (polygonNiche Θ K) ≤
+        ClassicalResults.area (capNiche K) := by
+      apply ENNReal.toReal_mono (niche_uniform_bounds.1 Θ.angle K).2.2.1.ne
+      exact MeasureTheory.measure_mono (polygonNiche_angleCap Θ K P.val hP).2
+    exact sub_le_sub hcap hniche
+
+/-- A maximum polygon cap dominates every cap under the polygon area functional. -/
+theorem polygonAreaFunctional_le_maximum (Θ : AngleSet) (P : PolygonCapSpace Θ)
+    (hP : IsMaximumPolygonCap Θ P) (K : CapSpace Θ.angle) :
+    polygonAreaFunctional Θ K ≤ polygonAreaFunctional Θ P.val := by
+  obtain ⟨Q, hQ⟩ := (angleCap_properties Θ K).1
+  have hn := (polygonNiche_angleCap Θ K Q.val hQ).1
+  have harea : polygonAreaFunctional Θ Q.val = polygonAreaFunctional Θ K := by
+    unfold polygonAreaFunctional
+    rw [(angleCap_properties Θ Q.val).2.2.2 Q, hQ, hn]
+  rw [← harea]
+  exact hP.2 Q
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Cap Width Bound
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open MeasureTheory
+open scoped Pointwise
+
+private theorem directionalWidth_zero (K : ConvexBody Point) :
+    directionalWidth K 0 = horizontalMax K - horizontalMin K := by
+  simp only [directionalWidth, zero_add]
+  rw [← Real.Angle.coe_zero, supportValue_zero_eq_horizontalMax,
+    supportValue_pi_eq_neg_horizontalMin, ← sub_eq_add_neg]
+
+private theorem volume_coordinate_open_rectangle (l r b t : ℝ) :
+    volume {p : Point | p 0 ∈ Set.Ioo l r ∧ p 1 ∈ Set.Ioo b t} =
+      ENNReal.ofReal (r - l) * ENNReal.ofReal (t - b) := by
+  have h := EuclideanSpace.volume_preserving_finTwoCoordinates.measure_preimage
+    ((measurableSet_Ioo.prod measurableSet_Ioo).nullMeasurableSet :
+      NullMeasurableSet (Set.Ioo l r ×ˢ Set.Ioo b t) (volume : Measure (ℝ × ℝ)))
+  simpa [Measure.volume_eq_prod, Measure.prod_prod, Real.volume_Ioo, Set.preimage, Set.prod] using h
+
+private theorem cap_horizontalWidth_le_of_cos_pos {ω : ℝ} (K : CapSpace ω)
+    (hc : 0 < Real.cos ω) :
+    directionalWidth (K.val : Set Point) 0 ≤ (1 + Real.sin ω) / Real.cos ω := by
+  have hs : 0 ≤ Real.sin ω := Real.sin_nonneg_of_nonneg_of_le_pi K.property.1.le
+    (K.property.2.1.trans (by linarith [Real.pi_pos]))
+  have hx (p : Point) (hp : p ∈ (K.val : Set Point)) :
+      -Real.sin ω / Real.cos ω ≤ p 0 ∧ p 0 ≤ 1 / Real.cos ω := by
+    obtain ⟨hy, hv⟩ := (mem_stripParallelogram_iff ω p).mp (K.subset_stripParallelogram hp)
+    simp [normalVector, frame, PiLp.inner_apply] at hv
+    constructor
+    · apply (div_le_iff₀ hc).mpr
+      nlinarith [mul_nonneg hs (sub_nonneg.mpr hy.2)]
+    · apply (le_div_iff₀ hc).mpr
+      nlinarith [mul_nonneg hs hy.1]
+  have hmax : horizontalMax K.val ≤ 1 / Real.cos ω :=
+    csSup_le (K.val.nonempty.image _) (by rintro _ ⟨p, hp, rfl⟩; exact (hx p hp).2)
+  have hmin : -Real.sin ω / Real.cos ω ≤ horizontalMin K.val :=
+    le_csInf (K.val.nonempty.image _) (by rintro _ ⟨p, hp, rfl⟩; exact (hx p hp).1)
+  rw [directionalWidth_zero]
+  calc
+    horizontalMax K.val - horizontalMin K.val ≤
+        1 / Real.cos ω - -Real.sin ω / Real.cos ω := sub_le_sub hmax hmin
+    _ = (1 + Real.sin ω) / Real.cos ω := by ring
+
+private theorem wedge_rectangle_inequalities {s c a b x y : ℝ}
+    (hs : 0 < s) (hc : 0 < c) (hs1 : s ≤ 1) (hc1 : c ≤ 1)
+    (hL : 0 < a / c + b / s)
+    (hx : x ∈ Set.Ioo (-b / s + (a / c + b / s) / 4)
+      (a / c - (a / c + b / s) / 4))
+    (hy : y ∈ Set.Ioo 0 ((a / c + b / s) * s * c / 4)) :
+    0 ≤ y ∧ c * x + s * y < a ∧ -s * x + c * y < b := by
+  let L := a / c + b / s
+  have hLc : 0 ≤ L * c / 4 := by dsimp [L]; positivity
+  have hLs : 0 ≤ L * s / 4 := by dsimp [L]; positivity
+  have hyc : y < L * c / 4 := by
+    apply hy.2.trans_le
+    calc
+      L * s * c / 4 = s * (L * c / 4) := by ring
+      _ ≤ 1 * (L * c / 4) := mul_le_mul_of_nonneg_right hs1 hLc
+      _ = L * c / 4 := one_mul _
+  have hys : y < L * s / 4 := by
+    apply hy.2.trans_le
+    calc
+      L * s * c / 4 = c * (L * s / 4) := by ring
+      _ ≤ 1 * (L * s / 4) := mul_le_mul_of_nonneg_right hc1 hLs
+      _ = L * s / 4 := one_mul _
+  have hsy : s * y ≤ y := by nlinarith [mul_nonneg (sub_nonneg.mpr hs1) hy.1.le]
+  have hcy : c * y ≤ y := by nlinarith [mul_nonneg (sub_nonneg.mpr hc1) hy.1.le]
+  have hright : (x + L / 4) * c < a :=
+    (lt_div_iff₀ hc).mp (by dsimp [L]; linarith [hx.2])
+  have hleft : -b < (x - L / 4) * s :=
+    (div_lt_iff₀ hs).mp (by dsimp [L]; linarith [hx.1])
+  exact ⟨hy.1.le, by nlinarith, by nlinarith⟩
+
+private theorem polygonNiche_area_ge_rectangle (Θ : AngleSet)
+    (hΘ : Θ.angle = Real.pi / 2) {t : ℝ} (ht : t ∈ Θ.directions)
+    (K : CapSpace Θ.angle) (hs : 0 < Real.sin t) (hc : 0 < Real.cos t)
+    (hL : 0 < (supportValue K.val (t : Real.Angle) - 1) / Real.cos t +
+      (supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) / Real.sin t) :
+    ((supportValue K.val (t : Real.Angle) - 1) / Real.cos t +
+      (supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) / Real.sin t) ^ 2 *
+      Real.sin t * Real.cos t / 8 ≤ ClassicalResults.area (polygonNiche Θ K) := by
+  let a := supportValue K.val (t : Real.Angle) - 1
+  let b := supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1
+  let L := a / Real.cos t + b / Real.sin t
+  let R : Set Point := {p | p 0 ∈ Set.Ioo (-b / Real.sin t + L / 4)
+    (a / Real.cos t - L / 4) ∧ p 1 ∈ Set.Ioo 0 (L * Real.sin t * Real.cos t / 4)}
+  have hsub : R ⊆ polygonNiche Θ K := by
+    intro p hp
+    have hineq := wedge_rectangle_inequalities hs hc (Real.sin_le_one t) (Real.cos_le_one t)
+      hL hp.1 hp.2
+    refine ⟨?_, Set.mem_iUnion₂.mpr ⟨t, ht, ?_⟩⟩
+    · rw [hΘ]
+      simpa [capFan, normalHalfPlane, normalVector, frame, PiLp.inner_apply] using hineq.1
+    · simpa [innerQuadrant, normalHalfPlane, normalVector, frame, PiLp.inner_apply,
+        Real.sin_add, Real.cos_add, -Real.Angle.coe_add, a, b] using hineq.2
+  have hwidth : a / Real.cos t - L / 4 - (-b / Real.sin t + L / 4) = L / 2 := by
+    dsimp [L]
+    ring
+  have harea : ClassicalResults.area R = L ^ 2 * Real.sin t * Real.cos t / 8 := by
+    change (volume R).toReal = _
+    rw [volume_coordinate_open_rectangle, hwidth, sub_zero, ENNReal.toReal_mul,
+      ENNReal.toReal_ofReal (show 0 ≤ L / 2 by dsimp [L, a, b]; positivity),
+      ENNReal.toReal_ofReal (show 0 ≤ L * Real.sin t * Real.cos t / 4 by
+        dsimp [L, a, b]; positivity)]
+    ring
+  rw [← harea]
+  exact ENNReal.toReal_mono (niche_uniform_bounds.2.1 Θ K).2.2.1.ne (measure_mono hsub)
+
+theorem polygonCap_width_bound (ω t : ℝ) (hω : 0 < ω)
+    (hω' : ω ≤ Real.pi / 2) (ht : t ∈ Set.Ioo 0 ω) :
+    ∃ c : ℝ, 0 < c ∧ ∀ (Θ : AngleSet), Θ.angle = ω → t ∈ Θ.directions →
+      ∀ K : PolygonCapSpace Θ, 0 ≤ polygonAreaFunctional Θ K.val →
+        directionalWidth (K.val.val : Set Point) (0 : Real.Angle) ≤ c := by
+  rcases lt_or_eq_of_le hω' with hlt | rfl
+  · have hc : 0 < Real.cos ω := Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [Real.pi_pos], hlt⟩
+    have hs : 0 ≤ Real.sin ω := Real.sin_nonneg_of_nonneg_of_le_pi hω.le
+      (by linarith [Real.pi_pos])
+    refine ⟨(1 + Real.sin ω) / Real.cos ω, div_pos (by linarith) hc, ?_⟩
+    intro Θ hΘ _ K _
+    have hcΘ : 0 < Real.cos Θ.angle := by simpa only [hΘ] using hc
+    simpa only [hΘ] using cap_horizontalWidth_le_of_cos_pos K.val hcΘ
+  · have hs : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi ht.1
+      (by linarith [ht.2, Real.pi_pos])
+    have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [ht.1, Real.pi_pos], ht.2⟩
+    let D := 1 / Real.cos t + 1 / Real.sin t
+    let C := max (2 * D) (32 / (Real.sin t * Real.cos t))
+    have hC : 0 < C := lt_of_lt_of_le (div_pos (by norm_num) (mul_pos hs hc)) (le_max_right _ _)
+    refine ⟨C, hC, ?_⟩
+    intro Θ hΘ htΘ K hnonneg
+    by_contra hwidth
+    have hdC : C < directionalWidth (K.val.val : Set Point) 0 := lt_of_not_ge hwidth
+    rw [directionalWidth_zero] at hdC
+    let d := horizontalMax K.val.val - horizontalMin K.val.val
+    have hd0 : 0 < d := hC.trans hdC
+    let a := supportValue K.val.val (t : Real.Angle) - 1
+    let b := supportValue K.val.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1
+    let L := a / Real.cos t + b / Real.sin t
+    have hsupport := CapSpace.horizontal_le_supportValue K.val hs.le hc.le
+    simp only [Real.Angle.coe_add] at hsupport
+    have ha : horizontalMax K.val.val - 1 / Real.cos t ≤ a / Real.cos t := by
+      apply (le_div_iff₀ hc).mpr
+      rw [sub_mul, div_mul_cancel₀ _ hc.ne']
+      dsimp [a]
+      linarith [hsupport.1]
+    have hb : -horizontalMin K.val.val - 1 / Real.sin t ≤ b / Real.sin t := by
+      apply (le_div_iff₀ hs).mpr
+      rw [sub_mul, div_mul_cancel₀ _ hs.ne']
+      dsimp [b]
+      linarith [hsupport.2]
+    have hLd : d - D ≤ L := by dsimp [L, d, D]; linarith
+    have hdD : 2 * D < d := (le_max_left _ _).trans_lt hdC
+    have hdlarge : 32 / (Real.sin t * Real.cos t) < d := (le_max_right _ _).trans_lt hdC
+    have hhalf : d / 2 < L := by linarith
+    have hL : 0 < L := (half_pos hd0).trans hhalf
+    have hlower := polygonNiche_area_ge_rectangle Θ hΘ htΘ K.val hs hc hL
+    have hsq : (d / 2) ^ 2 < L ^ 2 := (sq_lt_sq₀ (half_pos hd0).le hL.le).mpr hhalf
+    have hsq' := mul_lt_mul_of_pos_right hsq (mul_pos hs hc)
+    have hdlarge' : 32 < d * (Real.sin t * Real.cos t) :=
+      (div_lt_iff₀ (mul_pos hs hc)).mp hdlarge
+    have hquad : d < L ^ 2 * Real.sin t * Real.cos t / 8 := by
+      nlinarith [mul_pos hd0 (sub_pos.mpr hdlarge')]
+    have harea := K.val.area_le_horizontalWidth
+    have hidentity := (polygonArea_upperBound Θ).1 K
+    change L ^ 2 * Real.sin t * Real.cos t / 8 ≤ _ at hlower
+    change ClassicalResults.area (K.val.val : Set Point) ≤ d at harea
+    rw [hidentity] at hnonneg
+    linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Discrete Cap Data
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The uniformly spaced interior directions for a right-angle polygonal approximation. -/
+def rightAngleSet (n : ℕ) (hn : 2 ≤ n) : AngleSet :=
+  uniformAngleSet (Real.pi / 2) (by positivity) le_rfl n hn
+
+/-- The angular mesh size `π/(2n)`. -/
+def polygonStepSize (n : ℕ) : ℝ := (Real.pi / 2) / n
+
+/-- The cap maximizes the polygonal functional on a uniform mesh with a power-of-two step
+count. -/
+def IsMaximumPolygonCapSteps (n : ℕ) (K : RightAngleCapSpace) : Prop :=
+  ∃ hn : 2 ≤ n, (∃ k : ℕ, n = 2 ^ k) ∧
+    ∃ P : PolygonCapSpace (rightAngleSet n hn),
+      P.val = K ∧ IsMaximumPolygonCap (rightAngleSet n hn) P
+
+/-- The two piecewise scalar functions used in the arm-length inequalities. -/
+def magicFunctions : (NNReal → ℝ) × (NNReal → ℝ) :=
+  (fun x ↦ max |(x : ℝ) - 1| ((|(x : ℝ) - 1| + 1) / 2),
+   fun x ↦ (x : ℝ) - max |(x : ℝ) - 1| ((|(x : ℝ) - 1| + 1) / 2))
+
+/-- The magic function `m₀` is nondecreasing: it is `3 * x / 2 - 1` on `[0, 1]`, `x / 2` on
+`[1, 2]`, and constant equal to `1` afterwards. -/
+theorem magicFunctions_snd_monotone : Monotone magicFunctions.2 := by
+  intro a b hab
+  have hab' : (a : ℝ) ≤ (b : ℝ) := hab
+  simp only [magicFunctions, max_def]
+  rcases abs_cases ((a : ℝ) - 1) with ⟨ha1, ha2⟩ | ⟨ha1, ha2⟩ <;>
+    rcases abs_cases ((b : ℝ) - 1) with ⟨hb1, hb2⟩ | ⟨hb1, hb2⟩ <;>
+      rw [ha1, hb1] <;> split_ifs <;> linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Height / Properties
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private theorem polygonHeightValue_supportValue {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    {t : ℝ} (ht : t ∈ angleDomain Θ) :
+    polygonHeightValue (Θ := Θ)
+      (fun s ↦ supportValue (K.val.val : Set Point) (s.val : Real.Angle)) t =
+      supportValue K.val.val (t : Real.Angle) := by
+  simp [polygonHeightValue, ht]
+
+private theorem polygonHeightFan_supportValue {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    polygonHeightFan (Θ := Θ)
+      (fun s ↦ supportValue (K.val.val : Set Point) (s.val : Real.Angle)) =
+      capFan Θ.angle := by
+  have hω : Θ.angle ∈ angleDomain Θ := by simp [angleDomain]
+  have hT : Real.pi / 2 ∈ angleDomain Θ := by simp [angleDomain]
+  ext q
+  simp only [polygonHeightFan, Set.mem_iInter, Set.mem_insert_iff, Set.mem_singleton_iff,
+    forall_eq_or_imp, forall_eq]
+  rw [polygonHeightValue_supportValue K hω, polygonHeightValue_supportValue K hT,
+    K.val.property.2.2.1, K.val.property.2.2.2.1]
+  simp [capFan]
+
+private theorem polygonHeightNiche_supportValue {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    polygonHeightNiche (Θ := Θ)
+      (fun s ↦ supportValue (K.val.val : Set Point) (s.val : Real.Angle)) =
+      polygonNiche Θ K.val := by
+  unfold polygonHeightNiche polygonNiche
+  rw [polygonHeightFan_supportValue K]
+  congr 1
+  apply Set.iUnion_congr
+  intro t
+  apply Set.iUnion_congr
+  intro ht
+  have htD : t ∈ angleDomain Θ := by simp [angleDomain, ht]
+  have htTD : t + Real.pi / 2 ∈ angleDomain Θ := by
+    exact Or.inl (Or.inr ⟨t, ht, rfl⟩)
+  rw [polygonHeightValue_supportValue K htD, polygonHeightValue_supportValue K htTD]
+  rfl
+
+private theorem polygonHeightParallelogram_supportValue {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    polygonHeightParallelogram (Θ := Θ)
+      (fun s ↦ supportValue (K.val.val : Set Point) (s.val : Real.Angle)) =
+      (stripParallelogram Θ.angle).1 := by
+  have hω : Θ.angle ∈ angleDomain Θ := by simp [angleDomain]
+  have hT : Real.pi / 2 ∈ angleDomain Θ := by simp [angleDomain]
+  ext p
+  simp only [polygonHeightParallelogram, Set.mem_iInter, Set.mem_insert_iff,
+    Set.mem_singleton_iff, forall_eq_or_imp, forall_eq]
+  rw [polygonHeightValue_supportValue K hω, polygonHeightValue_supportValue K hT,
+    K.val.property.2.2.1, K.val.property.2.2.2.1, mem_stripParallelogram_iff]
+  simp [normalHalfPlane, normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+  tauto
+
+private theorem polygonHeightCap_supportValue {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    polygonHeightCap (Θ := Θ)
+      (fun s ↦ supportValue (K.val.val : Set Point) (s.val : Real.Angle)) =
+      angleCap Θ K.val := by
+  unfold polygonHeightCap
+  rw [polygonHeightParallelogram_supportValue K]
+  ext p
+  rw [mem_angleCap_iff]
+  simp only [Set.mem_inter_iff, mem_stripParallelogram_iff, Set.mem_iInter]
+  apply and_congr_right
+  intro _
+  constructor
+  · intro h t ht
+    have htD : t ∈ angleDomain Θ := Or.inl (Or.inl ht)
+    have htTD : t + Real.pi / 2 ∈ angleDomain Θ := Or.inl (Or.inr ⟨t, ht, rfl⟩)
+    have h₁ := h t (Or.inl ht)
+    have h₂ := h (t + Real.pi / 2) (Or.inr ⟨t, ht, rfl⟩)
+    rw [polygonHeightValue_supportValue K htD] at h₁
+    rw [polygonHeightValue_supportValue K htTD] at h₂
+    exact ⟨h₁, h₂⟩
+  · intro h t ht
+    have htD : t ∈ angleDomain Θ := Or.inl ht
+    rw [polygonHeightValue_supportValue K htD]
+    rcases ht with ht | ⟨s, hs, rfl⟩
+    · exact (h t ht).1
+    · exact (h s hs).2
+
+private theorem mem_polygonHeightCap_supportValue_image_add {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (q p : Point) :
+    p + q ∈ polygonHeightCap (Θ := Θ)
+      (fun t ↦ supportValue ((fun x ↦ x + q) '' (K.val.val : Set Point))
+        (t.val : Real.Angle)) ↔
+    p ∈ polygonHeightCap (Θ := Θ)
+      (fun t ↦ supportValue (K.val.val : Set Point) (t.val : Real.Angle)) := by
+  have hval (t : ℝ) (ht : t ∈ angleDomain Θ) :
+      polygonHeightValue (Θ := Θ)
+        (fun t ↦ supportValue ((fun x ↦ x + q) '' (K.val.val : Set Point))
+          (t.val : Real.Angle)) t =
+      polygonHeightValue (Θ := Θ)
+        (fun t ↦ supportValue (K.val.val : Set Point) (t.val : Real.Angle)) t +
+          inner ℝ q (normalVector (t : Real.Angle)) := by
+    simp only [polygonHeightValue, dite_eq_left ht]
+    exact supportValue_image_add K.val.val q (t : Real.Angle)
+  simp only [polygonHeightCap, polygonHeightParallelogram, Set.mem_inter_iff,
+    Set.mem_iInter]
+  apply and_congr
+  · apply forall_congr'
+    intro t
+    apply forall_congr'
+    intro ht
+    rw [hval t (Or.inr ht)]
+    simp [normalHalfPlane, inner_add_left, add_sub_right_comm]
+  · apply forall_congr'
+    intro t
+    apply forall_congr'
+    intro ht
+    rw [hval t (Or.inl ht)]
+    simp [normalHalfPlane, inner_add_left]
+
+private theorem polygonHeightNiche_supportValue_image_add {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (q : Point) :
+    polygonHeightNiche (Θ := Θ)
+      (fun t ↦ supportValue ((fun x ↦ x + q) '' (K.val.val : Set Point))
+        (t.val : Real.Angle)) =
+      (fun p ↦ p + q) '' polygonNiche Θ K.val := by
+  have hval (t : ℝ) (ht : t ∈ angleDomain Θ) :
+      polygonHeightValue (Θ := Θ)
+        (fun t ↦ supportValue ((fun x ↦ x + q) '' (K.val.val : Set Point))
+          (t.val : Real.Angle)) t =
+      polygonHeightValue (Θ := Θ)
+        (fun t ↦ supportValue (K.val.val : Set Point) (t.val : Real.Angle)) t +
+          inner ℝ q (normalVector (t : Real.Angle)) := by
+    simp only [polygonHeightValue, dite_eq_left ht]
+    exact supportValue_image_add K.val.val q (t : Real.Angle)
+  ext x
+  obtain ⟨p, rfl⟩ : ∃ p : Point, x = p + q := ⟨x - q, by simp⟩
+  rw [← polygonHeightNiche_supportValue K]
+  simp only [Set.mem_image, add_left_inj, exists_eq_right]
+  simp only [polygonHeightNiche, polygonHeightFan, Set.mem_inter_iff,
+    Set.mem_iInter, Set.mem_iUnion]
+  apply and_congr
+  · apply forall_congr'
+    intro t
+    apply forall_congr'
+    intro ht
+    rw [hval t (Or.inr ht)]
+    simp [normalHalfPlane, inner_add_left, add_sub_right_comm]
+  · apply exists_congr
+    intro t
+    apply exists_congr
+    intro ht
+    rw [hval t (Or.inl (Or.inl ht)),
+      hval (t + Real.pi / 2) (Or.inl (Or.inr ⟨t, ht, rfl⟩))]
+    simp [normalHalfPlane, inner_add_left, add_sub_right_comm]
+
+theorem polygonHeightCap_of_translate {Θ : AngleSet} (K : PolygonCapTranslateSpace Θ) :
+    polygonHeightCap (polygonTranslateHeight K) = K.val := by
+  obtain ⟨P, q, hK⟩ := K.property
+  unfold polygonTranslateHeight
+  rw [hK]
+  ext x
+  obtain ⟨p, rfl⟩ : ∃ p : Point, x = p + q := ⟨x - q, by simp⟩
+  rw [mem_polygonHeightCap_supportValue_image_add, polygonHeightCap_supportValue, angleCap_eq_self]
+  simp
+
+theorem polygonTranslateHeight_injective (Θ : AngleSet) :
+    Function.Injective (polygonTranslateHeight (Θ := Θ)) := by
+  intro K L h
+  apply Subtype.ext
+  rw [← polygonHeightCap_of_translate K, ← polygonHeightCap_of_translate L, h]
+
+theorem polygonHeightNiche_of_cap {Θ : AngleSet} (K : PolygonCapSpace Θ) :
+    polygonHeightNiche (Θ := Θ) (fun t ↦ supportValue (K.val.val : Set Point) (t.val :
+      Real.Angle)) =
+      polygonNiche Θ K.val ∧
+    polygonHeightArea (Θ := Θ) (fun t ↦ supportValue (K.val.val : Set Point) (t.val : Real.Angle)) =
+      polygonAreaFunctional Θ K.val := by
+  refine ⟨polygonHeightNiche_supportValue K, ?_⟩
+  unfold polygonHeightArea polygonAreaFunctional
+  rw [polygonHeightCap_supportValue K, polygonHeightNiche_supportValue K]
+
+theorem polygonTranslateExtensions_eq {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (q : Point) (K' : PolygonCapTranslateSpace Θ)
+    (hK : K'.val = (fun p ↦ p + q) '' (K.val.val : Set Point)) :
+    (polygonTranslateExtensions K').1 = (fun p ↦ p + q) '' polygonNiche Θ K.val ∧
+    (polygonTranslateExtensions K').2 = polygonAreaFunctional Θ K.val := by
+  have hn : polygonHeightNiche (polygonTranslateHeight K') =
+      (fun p ↦ p + q) '' polygonNiche Θ K.val := by
+    unfold polygonTranslateHeight
+    rw [hK]
+    exact polygonHeightNiche_supportValue_image_add K q
+  refine ⟨hn, ?_⟩
+  change ClassicalResults.area (polygonHeightCap (polygonTranslateHeight K')) -
+    ClassicalResults.area (polygonHeightNiche (polygonTranslateHeight K')) = _
+  rw [polygonHeightCap_of_translate K', hn, hK, ClassicalResults.area_image_add,
+    ClassicalResults.area_image_add]
+  unfold polygonAreaFunctional
+  rw [angleCap_eq_self]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Nef / Slices
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa.Nef
+
+open Filter Topology
+
+/-- Convert normal and tangent coordinates into a point in the frame at angle `a`. -/
+def framePoint (a : Real.Angle) (x y : ℝ) : Point :=
+  rotationMap a !₂[x, y]
+
+lemma framePoint_eq (a : Real.Angle) (x y : ℝ) :
+    framePoint a x y = x • normalVector a + y • tangentVector a := by
+  ext j
+  fin_cases j <;>
+    simp [framePoint, rotationMap, Orientation.rotation_apply,
+      rightAngleRotation_apply, normalVector, tangentVector, frame]
+  <;> ring
+
+@[simp] lemma inner_framePoint_normalVector (a : Real.Angle) (x y : ℝ) :
+    inner ℝ (framePoint a x y) (normalVector a) = x := by
+  rw [framePoint, inner_rotationMap_normalVector]
+  rfl
+
+@[simp] lemma inner_framePoint_tangentVector (a : Real.Angle) (x y : ℝ) :
+    inner ℝ (framePoint a x y) (tangentVector a) = y := by
+  rw [framePoint, inner_rotationMap_tangentVector]
+  rfl
+
+@[simp] lemma norm_tangentVector_angle (a : Real.Angle) : ‖tangentVector a‖ = 1 := by
+  rw [← sq_eq_sq₀ (norm_nonneg _) (by norm_num : (0 : ℝ) ≤ 1)]
+  rw [EuclideanSpace.norm_sq_eq]
+  simp [tangentVector, frame, Fin.sum_univ_two]
+  nlinarith [Real.Angle.cos_sq_add_sin_sq a]
+
+/-- The normal-normal coefficient for changing between two oriented frames. -/
+def frameNormalCoeff (a b : Real.Angle) : ℝ :=
+  inner ℝ (normalVector a) (normalVector b)
+
+/-- The tangent-normal coefficient for changing between two oriented frames. -/
+def frameTangentCoeff (a b : Real.Angle) : ℝ :=
+  inner ℝ (tangentVector a) (normalVector b)
+
+lemma inner_framePoint_normalVector_eq (a b : Real.Angle) (x y : ℝ) :
+    inner ℝ (framePoint a x y) (normalVector b) =
+      frameNormalCoeff a b * x + frameTangentCoeff a b * y := by
+  rw [framePoint_eq, inner_add_left, real_inner_smul_left,
+    real_inner_smul_left]
+  simp only [frameNormalCoeff, frameTangentCoeff]
+  ring
+
+/-- Select the half-plane side required by a Boolean cell’s membership pattern. -/
+def cellUpper {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (P : Fin n → Bool) (j : Fin n) : Bool :=
+  if P j then (H j).upper else !(H j).upper
+
+/-- Solve the wall equation for the tangent coordinate at a fixed normal coordinate. -/
+def frameBoundaryValue (a : Real.Angle) (H : PlanarHalfPlaneData) (x : ℝ) : ℝ :=
+  (H.height - frameNormalCoeff a H.angle * x) / frameTangentCoeff a H.angle
+
+lemma frameBoundaryValue_sub (a : Real.Angle) (H : PlanarHalfPlaneData)
+    (x z : ℝ) :
+    frameBoundaryValue a H x - frameBoundaryValue a H z =
+      -(frameNormalCoeff a H.angle / frameTangentCoeff a H.angle) * (x - z) := by
+  simp only [frameBoundaryValue]
+  ring
+
+lemma abs_frameBoundaryValue_sub (a : Real.Angle) (H : PlanarHalfPlaneData)
+    (x z : ℝ) :
+    |frameBoundaryValue a H x - frameBoundaryValue a H z| =
+      |frameNormalCoeff a H.angle / frameTangentCoeff a H.angle| * |x - z| := by
+  rw [frameBoundaryValue_sub, abs_mul, abs_neg]
+
+/-- The wall bounds a Boolean cell’s slice from above in the selected frame. -/
+def isUpperEndpoint {n : ℕ} (a : Real.Angle) (H : Fin n → PlanarHalfPlaneData)
+    (P : Fin n → Bool) (j : Fin n) : Prop :=
+  (cellUpper H P j = false ∧ 0 < frameTangentCoeff a (H j).angle) ∨
+    (cellUpper H P j = true ∧ frameTangentCoeff a (H j).angle < 0)
+
+/-- The wall bounds a Boolean cell’s slice from below in the selected frame. -/
+def isLowerEndpoint {n : ℕ} (a : Real.Angle) (H : Fin n → PlanarHalfPlaneData)
+    (P : Fin n → Bool) (j : Fin n) : Prop :=
+  (cellUpper H P j = false ∧ frameTangentCoeff a (H j).angle < 0) ∨
+    (cellUpper H P j = true ∧ 0 < frameTangentCoeff a (H j).angle)
+
+/-- The affine wall functions imposing upper bounds on the slice. -/
+def upperBoundaryFunctions {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) : List (ℝ → ℝ) := by
+  classical
+  exact ((Finset.univ.filter (isUpperEndpoint a H P)).toList.map fun j x ↦
+    frameBoundaryValue a (H j) x)
+
+/-- The affine wall functions imposing lower bounds on the slice. -/
+def lowerBoundaryFunctions {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) : List (ℝ → ℝ) := by
+  classical
+  exact ((Finset.univ.filter (isLowerEndpoint a H P)).toList.map fun j x ↦
+    frameBoundaryValue a (H j) x)
+
+/-- The minimum of all upper wall functions, capped by the truncation height `R`. -/
+def upperEnvelope {n : ℕ} (a : Real.Angle) (H : Fin n → PlanarHalfPlaneData)
+    (P : Fin n → Bool) (R x : ℝ) : ℝ :=
+  (upperBoundaryFunctions a H P).foldr (fun f r ↦ min (f x) r) R
+
+/-- The maximum of all lower wall functions, bounded below by `-R`. -/
+def lowerEnvelope {n : ℕ} (a : Real.Angle) (H : Fin n → PlanarHalfPlaneData)
+    (P : Fin n → Bool) (R x : ℝ) : ℝ :=
+  (lowerBoundaryFunctions a H P).foldr (fun f r ↦ max (f x) r) (-R)
+
+/-- The sum of absolute wall slopes, bounding the variation of slice envelopes. -/
+def slopeBound {n : ℕ} (a : Real.Angle) (H : Fin n → PlanarHalfPlaneData) : ℝ :=
+  ∑ j, |frameNormalCoeff a (H j).angle / frameTangentCoeff a (H j).angle|
+
+lemma slopeBound_nonneg {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) : 0 ≤ slopeBound a H := by
+  exact Finset.sum_nonneg fun _ _ ↦ abs_nonneg _
+
+lemma abs_upperEnvelope_sub_le {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) (R x z : ℝ) :
+    |upperEnvelope a H P R x - upperEnvelope a H P R z| ≤
+      slopeBound a H * |x - z| := by
+  apply List.abs_foldr_min_apply_sub_le
+  · exact slopeBound_nonneg a H
+  · intro f hf
+    simp only [upperBoundaryFunctions, List.mem_map, Finset.mem_toList,
+      Finset.mem_filter, Finset.mem_univ, true_and] at hf
+    obtain ⟨j, _, rfl⟩ := hf
+    rw [abs_frameBoundaryValue_sub]
+    apply mul_le_mul_of_nonneg_right _ (abs_nonneg _)
+    simpa only [slopeBound] using
+      (Finset.single_le_sum (s := Finset.univ)
+        (f := fun k : Fin n ↦
+          |frameNormalCoeff a (H k).angle / frameTangentCoeff a (H k).angle|)
+        (fun _ _ ↦ abs_nonneg _) (Finset.mem_univ j))
+
+lemma abs_lowerEnvelope_sub_le {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) (R x z : ℝ) :
+    |lowerEnvelope a H P R x - lowerEnvelope a H P R z| ≤
+      slopeBound a H * |x - z| := by
+  apply List.abs_foldr_max_apply_sub_le
+  · exact slopeBound_nonneg a H
+  · intro f hf
+    simp only [lowerBoundaryFunctions, List.mem_map, Finset.mem_toList,
+      Finset.mem_filter, Finset.mem_univ, true_and] at hf
+    obtain ⟨j, _, rfl⟩ := hf
+    rw [abs_frameBoundaryValue_sub]
+    apply mul_le_mul_of_nonneg_right _ (abs_nonneg _)
+    simpa only [slopeBound] using
+      (Finset.single_le_sum (s := Finset.univ)
+        (f := fun k : Fin n ↦
+          |frameNormalCoeff a (H k).angle / frameTangentCoeff a (H k).angle|)
+        (fun _ _ ↦ abs_nonneg _) (Finset.mem_univ j))
+
+/-- The nonnegative difference between the truncated upper and lower slice envelopes. -/
+def cellSliceLength {n : ℕ} (a : Real.Angle) (H : Fin n → PlanarHalfPlaneData)
+    (P : Fin n → Bool) (R x : ℝ) : ℝ :=
+  max (upperEnvelope a H P R x - lowerEnvelope a H P R x) 0
+
+lemma abs_cellSliceLength_sub_le {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) (R x z : ℝ) :
+    |cellSliceLength a H P R x - cellSliceLength a H P R z| ≤
+      (2 * slopeBound a H) * |x - z| := by
+  unfold cellSliceLength
+  refine (abs_max_sub_max_le_max _ _ _ _).trans (max_le ?_ ?_)
+  · calc
+      |(upperEnvelope a H P R x - lowerEnvelope a H P R x) -
+          (upperEnvelope a H P R z - lowerEnvelope a H P R z)| ≤
+          |upperEnvelope a H P R x - upperEnvelope a H P R z| +
+            |lowerEnvelope a H P R x - lowerEnvelope a H P R z| := by
+              rw [sub_sub_sub_comm]
+              exact abs_sub _ _
+      _ ≤ slopeBound a H * |x - z| + slopeBound a H * |x - z| :=
+        add_le_add (abs_upperEnvelope_sub_le a H P R x z)
+          (abs_lowerEnvelope_sub_le a H P R x z)
+      _ = (2 * slopeBound a H) * |x - z| := by ring
+  · simp only [sub_self, abs_zero]
+    exact mul_nonneg (mul_nonneg (by norm_num) (slopeBound_nonneg a H))
+      (abs_nonneg _)
+
+lemma continuous_cellSliceLength {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) (R : ℝ) :
+    Continuous (cellSliceLength a H P R) := by
+  let K : NNReal :=
+    ⟨2 * slopeBound a H, mul_nonneg (by norm_num) (slopeBound_nonneg a H)⟩
+  apply (LipschitzWith.of_dist_le_mul (K := K) fun x z ↦ ?_).continuous
+  change |cellSliceLength a H P R x - cellSliceLength a H P R z| ≤
+    (2 * slopeBound a H) * |x - z|
+  exact abs_cellSliceLength_sub_le a H P R x z
+
+/-- The closed Boolean cell’s slice at a fixed normal coordinate, truncated to `[-R, R]`. -/
+def closedCellSlice {n : ℕ} (a : Real.Angle) (H : Fin n → PlanarHalfPlaneData)
+    (P : Fin n → Bool) (R x : ℝ) : Set ℝ :=
+  {y | framePoint a x y ∈ closedBooleanCell H P} ∩ Set.Icc (-R) R
+
+/-- Every wall parallel to the slice is satisfied at the selected normal coordinate. -/
+def parallelCellFeasible {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) (x : ℝ) : Prop :=
+  ∀ j, frameTangentCoeff a (H j).angle = 0 →
+    framePoint a x 0 ∈ normalHalfPlane (H j).angle (H j).height
+      (cellUpper H P j) false
+
+lemma mem_closedCellSlice_iff {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) (R x y : ℝ)
+    (hparallel : parallelCellFeasible a H P x) :
+    y ∈ closedCellSlice a H P R x ↔
+      lowerEnvelope a H P R x ≤ y ∧ y ≤ upperEnvelope a H P R x := by
+  classical
+  rw [closedCellSlice, Set.mem_inter_iff, Set.mem_ofPred_eq, Set.mem_Icc,
+    lowerEnvelope, upperEnvelope, List.foldr_max_apply_le_iff, List.le_foldr_min_apply_iff]
+  constructor
+  · rintro ⟨hycell, hyR⟩
+    simp only [closedBooleanCell, Set.mem_iInter] at hycell
+    refine ⟨⟨hyR.1, ?_⟩, hyR.2, ?_⟩
+    · intro f hf
+      simp only [lowerBoundaryFunctions, List.mem_map, Finset.mem_toList,
+        Finset.mem_filter, Finset.mem_univ, true_and] at hf
+      obtain ⟨j, hj, rfl⟩ := hf
+      have hjcell := hycell j
+      change framePoint a x y ∈ normalHalfPlane (H j).angle (H j).height
+        (cellUpper H P j) false at hjcell
+      rcases hj with ⟨hu, hb⟩ | ⟨hu, hb⟩
+      · simp only [normalHalfPlane, hu, Bool.false_eq_true, ↓reduceIte,
+          Set.mem_ofPred_eq] at hjcell
+        rw [inner_framePoint_normalVector_eq] at hjcell
+        simp only [frameBoundaryValue]
+        apply (div_le_iff_of_neg hb).2
+        nlinarith
+      · simp only [normalHalfPlane, hu, Bool.false_eq_true, ↓reduceIte,
+          Set.mem_ofPred_eq] at hjcell
+        rw [inner_framePoint_normalVector_eq] at hjcell
+        simp only [frameBoundaryValue]
+        apply (div_le_iff₀ hb).2
+        nlinarith
+    · intro f hf
+      simp only [upperBoundaryFunctions, List.mem_map, Finset.mem_toList,
+        Finset.mem_filter, Finset.mem_univ, true_and] at hf
+      obtain ⟨j, hj, rfl⟩ := hf
+      have hjcell := hycell j
+      change framePoint a x y ∈ normalHalfPlane (H j).angle (H j).height
+        (cellUpper H P j) false at hjcell
+      rcases hj with ⟨hu, hb⟩ | ⟨hu, hb⟩
+      · simp only [normalHalfPlane, hu, Bool.false_eq_true, ↓reduceIte,
+          Set.mem_ofPred_eq] at hjcell
+        rw [inner_framePoint_normalVector_eq] at hjcell
+        simp only [frameBoundaryValue]
+        apply (le_div_iff₀ hb).2
+        nlinarith
+      · simp only [normalHalfPlane, hu, Bool.false_eq_true, ↓reduceIte,
+          Set.mem_ofPred_eq] at hjcell
+        rw [inner_framePoint_normalVector_eq] at hjcell
+        simp only [frameBoundaryValue]
+        apply (le_div_iff_of_neg hb).2
+        nlinarith
+  · rintro ⟨⟨hyRneg, hylower⟩, hyR, hyupper⟩
+    refine ⟨?_, hyRneg, hyR⟩
+    simp only [closedBooleanCell, Set.mem_iInter]
+    intro j
+    change framePoint a x y ∈ normalHalfPlane (H j).angle (H j).height
+      (cellUpper H P j) false
+    by_cases hb0 : frameTangentCoeff a (H j).angle = 0
+    · have hj := hparallel j hb0
+      cases hu : cellUpper H P j <;>
+        simp only [normalHalfPlane, hu, Bool.false_eq_true, ↓reduceIte,
+          Set.mem_ofPred_eq] at hj ⊢ <;>
+        rw [inner_framePoint_normalVector_eq] at hj ⊢ <;>
+        simp only [hb0, mul_zero, zero_mul, add_zero] at hj ⊢ <;>
+        exact hj
+    · rcases lt_or_gt_of_ne hb0 with hb | hb
+      · by_cases hu : cellUpper H P j = false
+        · have hjindex : isLowerEndpoint a H P j := Or.inl ⟨hu, hb⟩
+          have hjfun : (fun x ↦ frameBoundaryValue a (H j) x) ∈
+              lowerBoundaryFunctions a H P := by
+            simp only [lowerBoundaryFunctions, List.mem_map, Finset.mem_toList,
+              Finset.mem_filter, Finset.mem_univ, true_and]
+            exact ⟨j, hjindex, rfl⟩
+          have hjbound := hylower _ hjfun
+          simp only [normalHalfPlane, hu, Bool.false_eq_true, ↓reduceIte,
+            Set.mem_ofPred_eq, inner_framePoint_normalVector_eq,
+            frameBoundaryValue] at hjbound ⊢
+          have := (div_le_iff_of_neg hb).mp hjbound
+          nlinarith
+        · have hu' : cellUpper H P j = true := Bool.eq_true_of_not_eq_false hu
+          have hjindex : isUpperEndpoint a H P j := Or.inr ⟨hu', hb⟩
+          have hjfun : (fun x ↦ frameBoundaryValue a (H j) x) ∈
+              upperBoundaryFunctions a H P := by
+            simp only [upperBoundaryFunctions, List.mem_map, Finset.mem_toList,
+              Finset.mem_filter, Finset.mem_univ, true_and]
+            exact ⟨j, hjindex, rfl⟩
+          have hjbound := hyupper _ hjfun
+          simp only [normalHalfPlane, hu', Bool.false_eq_true, ↓reduceIte, Set.mem_ofPred_eq,
+            inner_framePoint_normalVector_eq, frameBoundaryValue,
+            ] at hjbound ⊢
+          have := (le_div_iff_of_neg hb).mp hjbound
+          nlinarith
+      · by_cases hu : cellUpper H P j = false
+        · have hjindex : isUpperEndpoint a H P j := Or.inl ⟨hu, hb⟩
+          have hjfun : (fun x ↦ frameBoundaryValue a (H j) x) ∈
+              upperBoundaryFunctions a H P := by
+            simp only [upperBoundaryFunctions, List.mem_map, Finset.mem_toList,
+              Finset.mem_filter, Finset.mem_univ, true_and]
+            exact ⟨j, hjindex, rfl⟩
+          have hjbound := hyupper _ hjfun
+          simp only [normalHalfPlane, hu, Bool.false_eq_true, ↓reduceIte,
+            Set.mem_ofPred_eq, inner_framePoint_normalVector_eq,
+            frameBoundaryValue] at hjbound ⊢
+          have := (le_div_iff₀ hb).mp hjbound
+          nlinarith
+        · have hu' : cellUpper H P j = true := Bool.eq_true_of_not_eq_false hu
+          have hjindex : isLowerEndpoint a H P j := Or.inr ⟨hu', hb⟩
+          have hjfun : (fun x ↦ frameBoundaryValue a (H j) x) ∈
+              lowerBoundaryFunctions a H P := by
+            simp only [lowerBoundaryFunctions, List.mem_map, Finset.mem_toList,
+              Finset.mem_filter, Finset.mem_univ, true_and]
+            exact ⟨j, hjindex, rfl⟩
+          have hjbound := hylower _ hjfun
+          simp only [normalHalfPlane, hu', Bool.false_eq_true, ↓reduceIte, Set.mem_ofPred_eq,
+            inner_framePoint_normalVector_eq, frameBoundaryValue,
+            ] at hjbound ⊢
+          have := (div_le_iff₀ hb).mp hjbound
+          nlinarith
+
+lemma volume_closedCellSlice_toReal {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) (R x : ℝ)
+    (hparallel : parallelCellFeasible a H P x) :
+    (MeasureTheory.volume (closedCellSlice a H P R x)).toReal =
+      cellSliceLength a H P R x := by
+  have hset : closedCellSlice a H P R x =
+      Set.Icc (lowerEnvelope a H P R x) (upperEnvelope a H P R x) := by
+    ext y
+    simpa [Set.mem_Icc] using mem_closedCellSlice_iff a H P R x y hparallel
+  rw [hset, Real.volume_Icc]
+  unfold cellSliceLength
+  by_cases h : 0 ≤ upperEnvelope a H P R x - lowerEnvelope a H P R x
+  · rw [ENNReal.toReal_ofReal h, max_eq_left h]
+  · have h' : upperEnvelope a H P R x - lowerEnvelope a H P R x ≤ 0 := le_of_not_ge h
+    rw [ENNReal.ofReal_of_nonpos h', max_eq_right h']
+    rfl
+
+/-- The actual Boolean membership cell’s slice, truncated to `[-R, R]`. -/
+def booleanCellSlice {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) (R x : ℝ) : Set ℝ :=
+  {y | framePoint a x y ∈ booleanCell (fun j ↦ (H j).carrier) P} ∩ Set.Icc (-R) R
+
+/-- No wall parallel to the slice has its boundary at the selected normal coordinate. -/
+def noParallelBoundaryAt {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (x : ℝ) : Prop :=
+  ∀ j, frameTangentCoeff a (H j).angle = 0 →
+    frameNormalCoeff a (H j).angle * x ≠ (H j).height
+
+/-- The tangent coordinates at which the slice meets a wall boundary. -/
+def cellSliceBoundaryExceptions {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (x : ℝ) : Set ℝ :=
+  ⋃ j, {y | inner ℝ (framePoint a x y) (normalVector (H j).angle) = (H j).height}
+
+lemma finite_cellSliceBoundaryExceptions {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (x : ℝ) (hx : noParallelBoundaryAt a H x) :
+    (cellSliceBoundaryExceptions a H x).Finite := by
+  classical
+  unfold cellSliceBoundaryExceptions
+  apply Set.Finite.iUnion Set.finite_univ
+  · intro j _
+    apply Set.Subsingleton.finite
+    intro y hy z hz
+    simp only [Set.mem_ofPred_eq, inner_framePoint_normalVector_eq] at hy hz
+    by_cases hb : frameTangentCoeff a (H j).angle = 0
+    · exact (hx j hb (by simpa [hb] using hy)).elim
+    · rcases lt_or_gt_of_ne hb with hb | hb <;> nlinarith
+  · simp
+
+lemma volume_booleanCellSlice_eq_closedCellSlice {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) (R x : ℝ)
+    (hx : noParallelBoundaryAt a H x) :
+    MeasureTheory.volume (booleanCellSlice a H P R x) =
+      MeasureTheory.volume (closedCellSlice a H P R x) := by
+  apply MeasureTheory.measure_congr
+  rw [MeasureTheory.ae_eq_set]
+  have hzero :=
+    (finite_cellSliceBoundaryExceptions a H x hx).measure_zero MeasureTheory.volume
+  constructor <;> apply MeasureTheory.measure_mono_null _ hzero
+  · intro y hy
+    by_contra hyexception
+    have hyne : ∀ j, inner ℝ (framePoint a x y) (normalVector (H j).angle) ≠
+        (H j).height := by
+      intro j hj
+      apply hyexception
+      exact Set.mem_iUnion.mpr ⟨j, hj⟩
+    have heq := mem_booleanCell_iff_mem_closedBooleanCell_of_ne H P _ hyne
+    rcases hy.1 with ⟨hycell, hyR⟩
+    exact hy.2 ⟨heq.mp hycell, hyR⟩
+  · intro y hy
+    by_contra hyexception
+    have hyne : ∀ j, inner ℝ (framePoint a x y) (normalVector (H j).angle) ≠
+        (H j).height := by
+      intro j hj
+      apply hyexception
+      exact Set.mem_iUnion.mpr ⟨j, hj⟩
+    have heq := mem_booleanCell_iff_mem_closedBooleanCell_of_ne H P _ hyne
+    rcases hy.1 with ⟨hycell, hyR⟩
+    exact hy.2 ⟨heq.mpr hycell, hyR⟩
+
+lemma volume_booleanCellSlice_toReal {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) (R x : ℝ)
+    (hparallel : parallelCellFeasible a H P x)
+    (hx : noParallelBoundaryAt a H x) :
+    (MeasureTheory.volume (booleanCellSlice a H P R x)).toReal =
+      cellSliceLength a H P R x := by
+  rw [volume_booleanCellSlice_eq_closedCellSlice a H P R x hx]
+  exact volume_closedCellSlice_toReal a H P R x hparallel
+
+lemma normalVector_eq_frameCombination (a b : Real.Angle) :
+    normalVector b = frameNormalCoeff a b • normalVector a +
+      frameTangentCoeff a b • tangentVector a := by
+  symm
+  simpa only [frameNormalCoeff, frameTangentCoeff, real_inner_comm] using
+    inner_normalVector_smul_add_inner_tangentVector_smul (normalVector b) a
+
+lemma normalLine_eq_of_frameTangentCoeff_eq_zero {a b : Real.Angle} {h k : ℝ}
+    (hb : frameTangentCoeff a b = 0) (hk : frameNormalCoeff a b * h = k) :
+    normalLine b k = normalLine a h := by
+  have hvec : normalVector b = frameNormalCoeff a b • normalVector a := by
+    rw [normalVector_eq_frameCombination a b, hb, zero_smul, add_zero]
+  have hcoeff : frameNormalCoeff a b ≠ 0 := by
+    intro hc
+    have : normalVector b = 0 := by simp [hvec, hc]
+    have hnorm : ‖normalVector b‖ = 1 := by
+      simpa only [b.coe_toReal] using norm_normalVector_real b.toReal
+    rw [this, norm_zero] at hnorm
+    norm_num at hnorm
+  ext p
+  simp only [normalLine, Set.mem_ofPred_eq, hvec, inner_smul_right]
+  rw [← hk]
+  constructor <;> intro hp
+  · exact (mul_left_cancel₀ hcoeff hp)
+  · exact congrArg (frameNormalCoeff a b * ·) hp
+
+@[simp] lemma frameNormalCoeff_self (a : Real.Angle) : frameNormalCoeff a a = 1 := by
+  simpa only [frameNormalCoeff, a.coe_toReal] using inner_normalVector_self a.toReal
+
+lemma measurableSet_booleanCellSlice {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) (R x : ℝ) :
+    MeasurableSet (booleanCellSlice a H P R x) := by
+  apply MeasurableSet.inter _ measurableSet_Icc
+  have hcont : Continuous (fun y ↦ framePoint a x y) := by
+    simp_rw [framePoint_eq]
+    fun_prop
+  exact (measurableSet_booleanCell H P).preimage hcont.measurable
+
+/-- The envelope slice length when parallel walls are feasible, and zero otherwise. -/
+def actualCellSliceLength {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) (R x : ℝ) : ℝ := by
+  classical
+  exact if parallelCellFeasible a H P x then cellSliceLength a H P R x else 0
+
+lemma closedCellSlice_eq_empty_of_not_parallelCellFeasible {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) (R x : ℝ)
+    (hparallel : ¬parallelCellFeasible a H P x) :
+    closedCellSlice a H P R x = ∅ := by
+  classical
+  unfold parallelCellFeasible at hparallel
+  push Not at hparallel
+  obtain ⟨j, hb, hj⟩ := hparallel
+  apply Set.eq_empty_iff_forall_notMem.mpr
+  intro y hy
+  have hycell := hy.1
+  simp only [closedBooleanCell, Set.mem_iInter] at hycell
+  have hjy := hycell j
+  change framePoint a x y ∈ normalHalfPlane (H j).angle (H j).height
+    (cellUpper H P j) false at hjy
+  cases hu : cellUpper H P j <;>
+    simp only [normalHalfPlane, hu, Bool.false_eq_true, ↓reduceIte,
+      Set.mem_ofPred_eq, inner_framePoint_normalVector_eq, hb, zero_mul, add_zero]
+      at hj hjy
+  · exact hj hjy
+  · exact hj hjy
+
+lemma volume_booleanCellSlice_toReal_eq_actual {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (P : Fin n → Bool) (R x : ℝ)
+    (_hx : noParallelBoundaryAt a H x) :
+    (MeasureTheory.volume (booleanCellSlice a H P R x)).toReal =
+      actualCellSliceLength a H P R x := by
+  by_cases hp : parallelCellFeasible a H P x
+  · simp only [actualCellSliceLength, hp, ↓reduceIte]
+    exact volume_booleanCellSlice_toReal a H P R x hp _hx
+  · simp only [actualCellSliceLength, hp, ↓reduceIte]
+    rw [volume_booleanCellSlice_eq_closedCellSlice a H P R x _hx,
+      closedCellSlice_eq_empty_of_not_parallelCellFeasible a H P R x hp,
+      MeasureTheory.measure_empty]
+    rfl
+
+/-- The measurable equivalence between the Euclidean plane and a pair of real coordinates. -/
+def frameCoordinates : Point ≃ᵐ ℝ × ℝ :=
+  (MeasurableEquiv.toLp 2 (Fin 2 → ℝ)).symm.trans MeasurableEquiv.finTwoArrow
+
+lemma frameCoordinates_measurePreserving :
+    MeasureTheory.MeasurePreserving frameCoordinates MeasureTheory.volume
+      ((MeasureTheory.volume : MeasureTheory.Measure ℝ).prod MeasureTheory.volume) := by
+  rw [← MeasureTheory.Measure.volume_eq_prod]
+  convert EuclideanSpace.volume_preserving_finTwoCoordinates using 1
+  funext p
+  rfl
+
+lemma framePoint_measurePreserving (a : Real.Angle) :
+    MeasureTheory.MeasurePreserving (fun p : ℝ × ℝ ↦ framePoint a p.1 p.2)
+      ((MeasureTheory.volume : MeasureTheory.Measure ℝ).prod MeasureTheory.volume)
+      MeasureTheory.volume := by
+  have hcoordinates : MeasureTheory.MeasurePreserving frameCoordinates.symm
+      ((MeasureTheory.volume : MeasureTheory.Measure ℝ).prod MeasureTheory.volume)
+      MeasureTheory.volume :=
+    frameCoordinates_measurePreserving.symm frameCoordinates
+  have hrotation : MeasureTheory.MeasurePreserving
+      (EuclideanGeometry.o.rotation a : Point → Point) :=
+    LinearIsometryEquiv.measurePreserving _
+  convert hrotation.comp hcoordinates using 1
+  funext p
+  rfl
+
+lemma volume_toReal_eq_integral_frameSlice (a : Real.Angle) {S : Set Point}
+    (hS : MeasurableSet S) (hSfinite : MeasureTheory.volume S ≠ ⊤) :
+    (MeasureTheory.volume S).toReal =
+      ∫ x : ℝ, (MeasureTheory.volume {y : ℝ | framePoint a x y ∈ S}).toReal := by
+  let f : ℝ × ℝ → Point := fun p ↦ framePoint a p.1 p.2
+  have hf := framePoint_measurePreserving a
+  have hpre : MeasurableSet (f ⁻¹' S) := hS.preimage hf.measurable
+  have hmeasure : MeasureTheory.volume (f ⁻¹' S) = MeasureTheory.volume S :=
+    hf.measure_preimage hS.nullMeasurableSet
+  have hprod : ((MeasureTheory.volume : MeasureTheory.Measure ℝ).prod
+      MeasureTheory.volume) (f ⁻¹' S) =
+      ∫⁻ x : ℝ, MeasureTheory.volume (Prod.mk x ⁻¹' (f ⁻¹' S)) :=
+    MeasureTheory.Measure.prod_apply hpre
+  have hsectionMeas : Measurable
+      (fun x : ℝ ↦ MeasureTheory.volume (Prod.mk x ⁻¹' (f ⁻¹' S))) :=
+    measurable_measure_prodMk_left hpre
+  have hsectionFinite : ∀ᵐ x : ℝ ∂MeasureTheory.volume,
+      MeasureTheory.volume (Prod.mk x ⁻¹' (f ⁻¹' S)) < ⊤ :=
+    MeasureTheory.Measure.ae_measure_lt_top hpre (by
+      rw [← MeasureTheory.Measure.volume_eq_prod, hmeasure]
+      exact hSfinite)
+  rw [← hmeasure]
+  calc
+    (MeasureTheory.volume (f ⁻¹' S)).toReal =
+        (((MeasureTheory.volume : MeasureTheory.Measure ℝ).prod
+          MeasureTheory.volume) (f ⁻¹' S)).toReal := by
+      rw [← MeasureTheory.Measure.volume_eq_prod]
+    _ = (∫⁻ x : ℝ,
+        MeasureTheory.volume (Prod.mk x ⁻¹' (f ⁻¹' S))).toReal :=
+      congrArg ENNReal.toReal hprod
+    _ = ∫ x : ℝ,
+        (MeasureTheory.volume (Prod.mk x ⁻¹' (f ⁻¹' S))).toReal :=
+      (MeasureTheory.integral_toReal hsectionMeas.aemeasurable hsectionFinite).symm
+    _ = ∫ x : ℝ,
+        (MeasureTheory.volume {y : ℝ | framePoint a x y ∈ S}).toReal := by
+      rfl
+
+end MovingSofa.Nef
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Nef / Variation / Local Slices
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa.Nef
+
+open Filter Topology
+
+/-- Replace one wall by a closed upper-bound wall at auxiliary height `M`. -/
+def auxiliaryHalfPlaneFamily {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (i : Fin n) (M : ℝ) : Fin n → PlanarHalfPlaneData :=
+  Function.update H i
+    { angle := (H i).angle, height := M, upper := false, strict := false }
+
+@[simp] lemma auxiliaryHalfPlaneFamily_self {n : ℕ}
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n) (M : ℝ) :
+    auxiliaryHalfPlaneFamily H i M i =
+      { angle := (H i).angle, height := M, upper := false, strict := false } := by
+  simp [auxiliaryHalfPlaneFamily]
+
+lemma auxiliaryHalfPlaneFamily_of_ne {n : ℕ}
+    (H : Fin n → PlanarHalfPlaneData) (i j : Fin n) (M : ℝ) (hji : j ≠ i) :
+    auxiliaryHalfPlaneFamily H i M j = H j := by
+  simp [auxiliaryHalfPlaneFamily, Function.update_of_ne hji]
+
+lemma noParallelBoundaryAt_auxiliary {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (i : Fin n) (hLines : Function.Injective (fun j ↦ (H j).boundaryLine))
+    (R ε₀ : ℝ) (hR : 0 < R) (hε₀ : 0 < ε₀) :
+    noParallelBoundaryAt (H i).angle
+      (auxiliaryHalfPlaneFamily H i (|(H i).height| + ε₀ + R + 1)) (H i).height := by
+  intro j hb
+  by_cases hji : j = i
+  · subst j
+    simp only [auxiliaryHalfPlaneFamily_self, frameNormalCoeff_self]
+    have hh : (H i).height < |(H i).height| + ε₀ + R + 1 := by
+      linarith [le_abs_self (H i).height]
+    simpa only [one_mul] using ne_of_lt hh
+  · rw [auxiliaryHalfPlaneFamily_of_ne H i j _ hji] at hb ⊢
+    intro heq
+    have hline : (H j).boundaryLine = (H i).boundaryLine := by
+      exact normalLine_eq_of_frameTangentCoeff_eq_zero hb heq
+    exact hji (hLines hline)
+
+lemma eventually_parallel_inequalities_iff {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (h : ℝ) (hh : noParallelBoundaryAt a H h) :
+    ∀ᶠ x in 𝓝 h, ∀ j, frameTangentCoeff a (H j).angle = 0 →
+      (frameNormalCoeff a (H j).angle * x ≤ (H j).height ↔
+          frameNormalCoeff a (H j).angle * h ≤ (H j).height) ∧
+        ((H j).height ≤ frameNormalCoeff a (H j).angle * x ↔
+          (H j).height ≤ frameNormalCoeff a (H j).angle * h) := by
+  suffices ∀ᶠ x in 𝓝 h, ∀ j ∈ (Set.univ : Set (Fin n)),
+      frameTangentCoeff a (H j).angle = 0 →
+        (frameNormalCoeff a (H j).angle * x ≤ (H j).height ↔
+            frameNormalCoeff a (H j).angle * h ≤ (H j).height) ∧
+          ((H j).height ≤ frameNormalCoeff a (H j).angle * x ↔
+            (H j).height ≤ frameNormalCoeff a (H j).angle * h) by
+    exact this.mono fun x hx j ↦ hx j (Set.mem_univ j)
+  apply (Filter.eventually_all_finite Set.finite_univ).2
+  intro j _
+  by_cases hb : frameTangentCoeff a (H j).angle = 0
+  · have hne := hh j hb
+    rcases lt_or_gt_of_ne hne with hlt | hgt
+    · have hev : ∀ᶠ x in 𝓝 h,
+          frameNormalCoeff a (H j).angle * x < (H j).height :=
+        (continuousAt_const.mul continuousAt_id).eventually_lt continuousAt_const hlt
+      filter_upwards [hev] with x hx
+      intro _
+      constructor <;> constructor <;> intro <;> linarith
+    · have hev : ∀ᶠ x in 𝓝 h,
+          (H j).height < frameNormalCoeff a (H j).angle * x :=
+        continuousAt_const.eventually_lt (continuousAt_const.mul continuousAt_id) hgt
+      filter_upwards [hev] with x hx
+      intro _
+      constructor <;> constructor <;> intro <;> linarith
+  · exact Filter.Eventually.of_forall fun _ hb' ↦ (hb hb').elim
+
+lemma eventually_parallelCellFeasible_iff {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (h : ℝ) (hh : noParallelBoundaryAt a H h) :
+    ∀ᶠ x in 𝓝 h, ∀ P : Fin n → Bool,
+      parallelCellFeasible a H P x ↔ parallelCellFeasible a H P h := by
+  filter_upwards [eventually_parallel_inequalities_iff a H h hh] with x hx
+  intro P
+  constructor <;> intro hp j hb
+  · have hj := hp j hb
+    have hiff := hx j hb
+    cases hu : cellUpper H P j <;>
+      simp only [normalHalfPlane, hu, Bool.false_eq_true, ↓reduceIte,
+        Set.mem_ofPred_eq, inner_framePoint_normalVector_eq, hb, mul_zero, add_zero]
+        at hj ⊢
+    · exact hiff.1.mp hj
+    · exact hiff.2.mp hj
+  · have hj := hp j hb
+    have hiff := hx j hb
+    cases hu : cellUpper H P j <;>
+      simp only [normalHalfPlane, hu, Bool.false_eq_true, ↓reduceIte,
+        Set.mem_ofPred_eq, inner_framePoint_normalVector_eq, hb, mul_zero, add_zero]
+        at hj ⊢
+    · exact hiff.1.mpr hj
+    · exact hiff.2.mpr hj
+
+lemma exists_parallel_stability_radius {n : ℕ} (a : Real.Angle)
+    (H : Fin n → PlanarHalfPlaneData) (h : ℝ) (hh : noParallelBoundaryAt a H h) :
+    ∃ ε > 0, ∀ x, |x - h| ≤ ε →
+      noParallelBoundaryAt a H x ∧
+        ∀ P : Fin n → Bool,
+          (parallelCellFeasible a H P x ↔ parallelCellFeasible a H P h) := by
+  have hev : ∀ᶠ x in 𝓝 h,
+      (∀ j, frameTangentCoeff a (H j).angle = 0 →
+        (frameNormalCoeff a (H j).angle * x ≤ (H j).height ↔
+            frameNormalCoeff a (H j).angle * h ≤ (H j).height) ∧
+          ((H j).height ≤ frameNormalCoeff a (H j).angle * x ↔
+            (H j).height ≤ frameNormalCoeff a (H j).angle * h)) ∧
+        ∀ P : Fin n → Bool,
+          (parallelCellFeasible a H P x ↔ parallelCellFeasible a H P h) :=
+    (eventually_parallel_inequalities_iff a H h hh).and
+      (eventually_parallelCellFeasible_iff a H h hh)
+  rcases (Metric.mem_nhds_iff.mp hev) with ⟨r, hr, hball⟩
+  refine ⟨r / 2, by positivity, ?_⟩
+  intro x hx
+  have hxr : x ∈ Metric.ball h r := by
+    rw [Metric.mem_ball, Real.dist_eq]
+    calc
+      |x - h| ≤ r / 2 := hx
+      _ < r := by linarith
+  have hstable := hball hxr
+  refine ⟨?_, hstable.2⟩
+  intro j hb heq
+  have hj := hstable.1 j hb
+  apply hh j hb
+  apply le_antisymm
+  · exact hj.1.mp (by rw [heq])
+  · exact hj.2.mp (by rw [heq])
+
+lemma setMembershipPattern_auxiliary {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (i : Fin n) (M : ℝ) (p : Point)
+    (hp : p ∈ (auxiliaryHalfPlaneFamily H i M i).carrier) :
+    setMembershipPattern (fun j ↦ (auxiliaryHalfPlaneFamily H i M j).carrier) p =
+      Function.update (setMembershipPattern (fun j ↦ (H j).carrier) p) i true := by
+  classical
+  funext j
+  by_cases hji : j = i
+  · subst j
+    rw [auxiliaryHalfPlaneFamily_self] at hp
+    simp [setMembershipPattern, hp]
+  · simp [setMembershipPattern, auxiliaryHalfPlaneFamily_of_ne H i j M hji,
+      Function.update_of_ne hji]
+
+lemma mem_activeBooleanRegion_iff_mem_booleanCell_auxiliary {n : ℕ}
+    (E : BooleanFunction n) (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (M : ℝ) (p : Point) (hp : p ∈ (auxiliaryHalfPlaneFamily H i M i).carrier) :
+    p ∈ activeBooleanRegion E H i ↔
+      ∃ P ∈ activeBooleanPatterns E i,
+        p ∈ booleanCell (fun j ↦ (auxiliaryHalfPlaneFamily H i M j).carrier) P := by
+  classical
+  rw [mem_activeBooleanRegion_iff]
+  constructor
+  · intro hactive
+    unfold IsActiveBooleanPattern at hactive
+    let P := setMembershipPattern
+      (fun j ↦ (auxiliaryHalfPlaneFamily H i M j).carrier) p
+    have hP : P = Function.update
+        (setMembershipPattern (fun j ↦ (H j).carrier) p) i true :=
+      setMembershipPattern_auxiliary H i M p hp
+    refine ⟨P, ?_, (mem_booleanCell_iff _ _ _).mpr rfl⟩
+    simp only [activeBooleanPatterns, Finset.mem_filter, Finset.mem_univ, true_and]
+    unfold IsActiveBooleanPattern
+    rw [hP]
+    simpa only [Function.update_idem] using hactive
+  · rintro ⟨P, hP, hcell⟩
+    simp only [activeBooleanPatterns, Finset.mem_filter, Finset.mem_univ,
+      true_and] at hP
+    have hpattern := (mem_booleanCell_iff _ P p).mp hcell
+    have haux := setMembershipPattern_auxiliary H i M p hp
+    unfold IsActiveBooleanPattern at hP ⊢
+    rw [hpattern] at haux
+    rw [haux] at hP
+    simpa only [Function.update_idem] using hP
+
+/-- The slice of the region where the selected Boolean wall is active, truncated to `[-R, R]`. -/
+def activeRegionSlice {n : ℕ} (a : Real.Angle) (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n) (R x : ℝ) : Set ℝ :=
+  {y | framePoint a x y ∈ activeBooleanRegion E H i} ∩ Set.Icc (-R) R
+
+lemma frameSlice_perturbSdiff_eq {n : ℕ} {E : BooleanFunction n}
+    (hE : IsMonotoneBooleanFunction E) (H : Fin n → PlanarHalfPlaneData)
+    (i : Fin n) (hSide : (H i).upper = false) (R ε₀ δ ε x : ℝ)
+    (hδ : |δ| ≤ ε₀)
+    (hBound : ∀ z : ℝ, |z| ≤ ε₀ →
+      perturbNefHeight E H i z ⊆ Metric.closedBall 0 R) :
+    (x ∈ heightInterval (H i).strict (H i).height δ ε →
+      {y : ℝ | framePoint (H i).angle x y ∈
+          perturbNefHeight E H i δ \ perturbNefHeight E H i ε} =
+        activeRegionSlice (H i).angle E H i R x) ∧
+    (x ∉ heightInterval (H i).strict (H i).height δ ε →
+      {y : ℝ | framePoint (H i).angle x y ∈
+          perturbNefHeight E H i δ \ perturbNefHeight E H i ε} = ∅) := by
+  constructor
+  · intro hx
+    rw [perturbNefHeight_sdiff_eq_heightInterval hE H i hSide]
+    ext y
+    simp only [Set.mem_ofPred_eq, Set.mem_inter_iff,
+      inner_framePoint_normalVector, hx, true_and, activeRegionSlice]
+    constructor
+    · intro hy
+      refine ⟨hy, ?_⟩
+      have hdiff : framePoint (H i).angle x y ∈
+          perturbNefHeight E H i δ \ perturbNefHeight E H i ε := by
+        rw [perturbNefHeight_sdiff_eq_heightInterval hE H i hSide]
+        exact ⟨by simpa using hx, hy⟩
+      have hball := hBound δ hδ hdiff.1
+      have hynorm := abs_real_inner_le_norm (framePoint (H i).angle x y)
+        (tangentVector (H i).angle)
+      rw [inner_framePoint_tangentVector, norm_tangentVector_angle, mul_one] at hynorm
+      have hpR : ‖framePoint (H i).angle x y‖ ≤ R := by
+        simpa [Metric.mem_closedBall, dist_zero_left] using hball
+      exact abs_le.mp (hynorm.trans hpR)
+    · exact fun hy ↦ hy.1
+  · intro hx
+    rw [perturbNefHeight_sdiff_eq_heightInterval hE H i hSide]
+    ext y
+    simp [hx]
+
+lemma activeRegionSlice_eq_biUnion_booleanCellSlice_auxiliary {n : ℕ}
+    (E : BooleanFunction n) (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (M R x : ℝ) (hxM : x ≤ M) :
+    activeRegionSlice (H i).angle E H i R x =
+      ⋃ P ∈ activeBooleanPatterns E i,
+        booleanCellSlice (H i).angle (auxiliaryHalfPlaneFamily H i M) P R x := by
+  classical
+  ext y
+  have haux : framePoint (H i).angle x y ∈
+      (auxiliaryHalfPlaneFamily H i M i).carrier := by
+    simp [PlanarHalfPlaneData.carrier, normalHalfPlane, hxM]
+  rw [activeRegionSlice, Set.mem_inter_iff, Set.mem_ofPred_eq,
+    mem_activeBooleanRegion_iff_mem_booleanCell_auxiliary E H i M _ haux]
+  simp only [Set.mem_iUnion, booleanCellSlice, Set.mem_inter_iff, Set.mem_ofPred_eq]
+  constructor
+  · rintro ⟨⟨P, hP, hyP⟩, hyR⟩
+    exact ⟨P, ⟨hP, hyP, hyR⟩⟩
+  · rintro ⟨P, hP, hyP, hyR⟩
+    exact ⟨⟨P, hP, hyP⟩, hyR⟩
+
+lemma measure_activeRegionSlice_eq_sum {n : ℕ}
+    (E : BooleanFunction n) (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (M R x : ℝ) (hxM : x ≤ M) :
+    MeasureTheory.volume (activeRegionSlice (H i).angle E H i R x) =
+      ∑ P ∈ activeBooleanPatterns E i,
+        MeasureTheory.volume
+          (booleanCellSlice (H i).angle (auxiliaryHalfPlaneFamily H i M) P R x) := by
+  rw [activeRegionSlice_eq_biUnion_booleanCellSlice_auxiliary E H i M R x hxM]
+  apply MeasureTheory.measure_biUnion_finset
+  · intro P hP Q hQ hPQ
+    change Disjoint
+      (booleanCellSlice (H i).angle (auxiliaryHalfPlaneFamily H i M) P R x)
+      (booleanCellSlice (H i).angle (auxiliaryHalfPlaneFamily H i M) Q R x)
+    rw [Set.disjoint_left]
+    intro y hyP hyQ
+    apply hPQ
+    have hcellP := hyP.1
+    have hcellQ := hyQ.1
+    have hp := (mem_booleanCell_iff _ P _).mp hcellP
+    have hq := (mem_booleanCell_iff _ Q _).mp hcellQ
+    exact hp.symm.trans hq
+  · intro P _
+    exact measurableSet_booleanCellSlice _ _ _ _ _
+
+/-- Sum the feasible slice lengths over patterns where the selected wall is active. -/
+def activeSliceLength {n : ℕ} (a : Real.Angle) (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n) (R x : ℝ) : ℝ :=
+  ∑ P ∈ activeBooleanPatterns E i, actualCellSliceLength a H P R x
+
+/-- Sum slice lengths while freezing all parallel-wall feasibility decisions at the reference
+height. -/
+def frozenActiveSliceLength {n : ℕ} (a : Real.Angle) (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n) (R h x : ℝ) : ℝ := by
+  classical
+  exact ∑ P ∈ activeBooleanPatterns E i,
+    if parallelCellFeasible a H P h then cellSliceLength a H P R x else 0
+
+lemma continuous_frozenActiveSliceLength {n : ℕ} (a : Real.Angle)
+    (E : BooleanFunction n) (H : Fin n → PlanarHalfPlaneData)
+    (i : Fin n) (R h : ℝ) :
+    Continuous (frozenActiveSliceLength a E H i R h) := by
+  classical
+  unfold frozenActiveSliceLength
+  apply continuous_finsetSum
+  intro P hP
+  by_cases hp : parallelCellFeasible a H P h
+  · simpa [hp] using continuous_cellSliceLength a H P R
+  · simpa [hp] using (continuous_const : Continuous (fun _ : ℝ ↦ (0 : ℝ)))
+
+lemma activeSliceLength_eq_frozen_of_stable {n : ℕ} (a : Real.Angle)
+    (E : BooleanFunction n) (H : Fin n → PlanarHalfPlaneData)
+    (i : Fin n) (R h x : ℝ)
+    (hstable : ∀ P : Fin n → Bool,
+      parallelCellFeasible a H P x ↔ parallelCellFeasible a H P h) :
+    activeSliceLength a E H i R x = frozenActiveSliceLength a E H i R h x := by
+  classical
+  unfold activeSliceLength frozenActiveSliceLength
+  apply Finset.sum_congr rfl
+  intro P hP
+  by_cases hp : parallelCellFeasible a H P h
+  · have hpx := (hstable P).mpr hp
+    simp [actualCellSliceLength, hp, hpx]
+  · have hpx : ¬parallelCellFeasible a H P x := fun hx ↦ hp ((hstable P).mp hx)
+    simp [actualCellSliceLength, hp, hpx]
+
+lemma abs_frozenActiveSliceLength_sub_le {n : ℕ} (a : Real.Angle)
+    (E : BooleanFunction n) (H : Fin n → PlanarHalfPlaneData)
+    (i : Fin n) (R h x z : ℝ) :
+    |frozenActiveSliceLength a E H i R h x -
+        frozenActiveSliceLength a E H i R h z| ≤
+      ((activeBooleanPatterns E i).card : ℝ) * (2 * slopeBound a H) * |x - z| := by
+  classical
+  unfold frozenActiveSliceLength
+  rw [← Finset.sum_sub_distrib]
+  calc
+    |∑ P ∈ activeBooleanPatterns E i,
+        ((if parallelCellFeasible a H P h then cellSliceLength a H P R x else 0) -
+          if parallelCellFeasible a H P h then cellSliceLength a H P R z else 0)| ≤
+        ∑ P ∈ activeBooleanPatterns E i,
+          |(if parallelCellFeasible a H P h then cellSliceLength a H P R x else 0) -
+            if parallelCellFeasible a H P h then cellSliceLength a H P R z else 0| :=
+      Finset.abs_sum_le_sum_abs _ _
+    _ ≤ ∑ _P ∈ activeBooleanPatterns E i,
+        (2 * slopeBound a H) * |x - z| := by
+      apply Finset.sum_le_sum
+      intro P hP
+      by_cases hp : parallelCellFeasible a H P h
+      · simpa [hp] using abs_cellSliceLength_sub_le a H P R x z
+      · simp only [hp, ↓reduceIte, sub_self, abs_zero]
+        exact mul_nonneg (mul_nonneg (by norm_num) (slopeBound_nonneg a H))
+          (abs_nonneg _)
+    _ = ((activeBooleanPatterns E i).card : ℝ) * (2 * slopeBound a H) *
+        |x - z| := by
+      simp
+      ring
+
+lemma volume_activeRegionSlice_toReal_eq_activeSliceLength {n : ℕ}
+    (E : BooleanFunction n) (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (M R x : ℝ) (hxM : x ≤ M)
+    (hx : noParallelBoundaryAt (H i).angle (auxiliaryHalfPlaneFamily H i M) x) :
+    (MeasureTheory.volume (activeRegionSlice (H i).angle E H i R x)).toReal =
+      activeSliceLength (H i).angle E (auxiliaryHalfPlaneFamily H i M) i R x := by
+  rw [measure_activeRegionSlice_eq_sum E H i M R x hxM]
+  unfold activeSliceLength
+  rw [ENNReal.toReal_sum]
+  · apply Finset.sum_congr rfl
+    intro P hP
+    exact volume_booleanCellSlice_toReal_eq_actual _ _ _ _ _ hx
+  · intro P hP
+    apply ne_of_lt
+    calc
+      MeasureTheory.volume
+          (booleanCellSlice (H i).angle (auxiliaryHalfPlaneFamily H i M) P R x) ≤
+          MeasureTheory.volume (Set.Icc (-R) R) :=
+        MeasureTheory.measure_mono Set.inter_subset_right
+      _ = ENNReal.ofReal (R - -R) := Real.volume_Icc
+      _ < ⊤ := ENNReal.ofReal_lt_top
+
+end MovingSofa.Nef
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Nef / Variation / Area
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa.Nef
+
+open Filter Topology
+
+lemma volume_perturbSdiff_toReal_eq_integral_slice {n : ℕ}
+    {E : BooleanFunction n} (hE : IsMonotoneBooleanFunction E)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (hSide : (H i).upper = false) (R ε₀ δ ε : ℝ) (hδ : |δ| ≤ ε₀)
+    (hBound : ∀ z : ℝ, |z| ≤ ε₀ →
+      perturbNefHeight E H i z ⊆ Metric.closedBall 0 R) :
+    (MeasureTheory.volume
+      (perturbNefHeight E H i δ \ perturbNefHeight E H i ε)).toReal =
+      ∫ x in heightInterval (H i).strict (H i).height δ ε,
+        (MeasureTheory.volume
+          (activeRegionSlice (H i).angle E H i R x)).toReal := by
+  let S := perturbNefHeight E H i δ \ perturbNefHeight E H i ε
+  have hS : MeasurableSet S :=
+    MeasurableSet.diff (measurableSet_perturbNefHeight E H i δ)
+      (measurableSet_perturbNefHeight E H i ε)
+  have hSfinite : MeasureTheory.volume S ≠ ⊤ := by
+    apply ne_of_lt
+    calc
+      MeasureTheory.volume S ≤ MeasureTheory.volume (perturbNefHeight E H i δ) :=
+        MeasureTheory.measure_mono Set.sdiff_subset
+      _ ≤ MeasureTheory.volume (Metric.closedBall (0 : Point) R) :=
+        MeasureTheory.measure_mono (hBound δ hδ)
+      _ < ⊤ := MeasureTheory.measure_closedBall_lt_top
+  rw [show (MeasureTheory.volume S).toReal =
+      ∫ x : ℝ, (MeasureTheory.volume
+        {y : ℝ | framePoint (H i).angle x y ∈ S}).toReal by
+    exact volume_toReal_eq_integral_frameSlice (H i).angle hS hSfinite]
+  rw [← MeasureTheory.integral_indicator
+    (measurableSet_heightInterval (H i).strict (H i).height δ ε)]
+  apply MeasureTheory.integral_congr_ae
+  filter_upwards [] with x
+  by_cases hx : x ∈ heightInterval (H i).strict (H i).height δ ε
+  · rw [(frameSlice_perturbSdiff_eq hE H i hSide R ε₀ δ ε x hδ hBound).1 hx]
+    simp [hx]
+  · rw [(frameSlice_perturbSdiff_eq hE H i hSide R ε₀ δ ε x hδ hBound).2 hx]
+    simp [hx]
+
+lemma area_perturb_sub_eq_integral_activeSliceLength {n : ℕ}
+    {E : BooleanFunction n} (hE : IsMonotoneBooleanFunction E)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (hSide : (H i).upper = false) (R ε₀ M δ ε : ℝ)
+    (hεδ : ε ≤ δ) (hδ : |δ| ≤ ε₀) (hε : |ε| ≤ ε₀)
+    (hBound : ∀ z : ℝ, |z| ≤ ε₀ →
+      perturbNefHeight E H i z ⊆ Metric.closedBall 0 R)
+    (hxM : ∀ x ∈ heightInterval (H i).strict (H i).height δ ε, x ≤ M)
+    (hxregular : ∀ x ∈ heightInterval (H i).strict (H i).height δ ε,
+      noParallelBoundaryAt (H i).angle (auxiliaryHalfPlaneFamily H i M) x) :
+    ClassicalResults.area (perturbNefHeight E H i δ) -
+        ClassicalResults.area (perturbNefHeight E H i ε) =
+      ∫ x in heightInterval (H i).strict (H i).height δ ε,
+        activeSliceLength (H i).angle E (auxiliaryHalfPlaneFamily H i M) i R x := by
+  rw [area_perturb_sub_eq_volume_sdiff hE H i hSide R ε₀ δ ε hεδ hδ hε hBound,
+    volume_perturbSdiff_toReal_eq_integral_slice hE H i hSide R ε₀ δ ε hδ hBound]
+  apply MeasureTheory.setIntegral_congr_fun
+    (measurableSet_heightInterval (H i).strict (H i).height δ ε)
+  intro x hx
+  exact volume_activeRegionSlice_toReal_eq_activeSliceLength E H i M R x
+    (hxM x hx) (hxregular x hx)
+
+lemma area_perturb_sub_eq_intervalIntegral_activeSliceLength {n : ℕ}
+    {E : BooleanFunction n} (hE : IsMonotoneBooleanFunction E)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (hSide : (H i).upper = false) (R ε₀ M ρ : ℝ)
+    (hε₀ : 0 < ε₀) (hρ : 0 < ρ)
+    (hM : (H i).height + ε₀ ≤ M)
+    (hBound : ∀ z : ℝ, |z| ≤ ε₀ →
+      perturbNefHeight E H i z ⊆ Metric.closedBall 0 R)
+    (hregular : ∀ x, |x - (H i).height| ≤ ρ →
+      noParallelBoundaryAt (H i).angle (auxiliaryHalfPlaneFamily H i M) x) :
+    ∀ δ : ℝ, |δ| ≤ min ε₀ ρ →
+      ClassicalResults.area (perturbNefHeight E H i δ) -
+          ClassicalResults.area (perturbNefHeight E H i 0) =
+        ∫ x in (H i).height..(H i).height + δ,
+          activeSliceLength (H i).angle E
+            (auxiliaryHalfPlaneFamily H i M) i R x := by
+  intro δ hδ
+  have hδε₀ : |δ| ≤ ε₀ := hδ.trans (min_le_left _ _)
+  have hδρ : |δ| ≤ ρ := hδ.trans (min_le_right _ _)
+  have hzero : |(0 : ℝ)| ≤ ε₀ := by simpa using hε₀.le
+  by_cases hδ0 : 0 ≤ δ
+  · have hformula := area_perturb_sub_eq_integral_activeSliceLength
+      hE H i hSide R ε₀ M δ 0 hδ0 hδε₀ hzero hBound
+      (fun x hx ↦ by
+        have hxb := mem_heightInterval_bounds hx
+        linarith [le_abs_self δ])
+      (fun x hx ↦ by
+        apply hregular
+        have hxb := mem_heightInterval_bounds hx
+        rw [abs_le]
+        constructor <;> linarith [le_abs_self δ, neg_le_abs δ])
+    rw [integral_heightInterval_eq_intervalIntegral _ _ _ _ _ hδ0] at hformula
+    simpa using hformula
+  · have hδle : δ ≤ 0 := le_of_not_ge hδ0
+    have hformula := area_perturb_sub_eq_integral_activeSliceLength
+      hE H i hSide R ε₀ M 0 δ hδle hzero hδε₀ hBound
+      (fun x hx ↦ by
+        have hxb := mem_heightInterval_bounds hx
+        linarith)
+      (fun x hx ↦ by
+        apply hregular
+        have hxb := mem_heightInterval_bounds hx
+        rw [abs_le]
+        constructor <;> linarith [le_abs_self δ, neg_le_abs δ])
+    rw [integral_heightInterval_eq_intervalIntegral _ _ _ _ _ hδle] at hformula
+    simp only [add_zero] at hformula
+    rw [intervalIntegral.integral_symm] at hformula
+    linarith
+
+lemma area_perturb_remainder_le {n : ℕ} {E : BooleanFunction n}
+    (hE : IsMonotoneBooleanFunction E) (H : Fin n → PlanarHalfPlaneData)
+    (i : Fin n) (hSide : (H i).upper = false) (R ε₀ M ρ : ℝ)
+    (hε₀ : 0 < ε₀) (hρ : 0 < ρ) (hM : (H i).height + ε₀ ≤ M)
+    (hBound : ∀ z : ℝ, |z| ≤ ε₀ →
+      perturbNefHeight E H i z ⊆ Metric.closedBall 0 R)
+    (hstable : ∀ x, |x - (H i).height| ≤ ρ →
+      noParallelBoundaryAt (H i).angle (auxiliaryHalfPlaneFamily H i M) x ∧
+        ∀ P : Fin n → Bool,
+          (parallelCellFeasible (H i).angle (auxiliaryHalfPlaneFamily H i M) P x ↔
+            parallelCellFeasible (H i).angle (auxiliaryHalfPlaneFamily H i M) P
+              (H i).height)) :
+    ∀ δ : ℝ, |δ| ≤ min ε₀ ρ →
+      |ClassicalResults.area (perturbNefHeight E H i δ) -
+          ClassicalResults.area (perturbNefHeight E H i 0) -
+          activeSliceLength (H i).angle E (auxiliaryHalfPlaneFamily H i M) i R
+            (H i).height * δ| ≤
+        (((activeBooleanPatterns E i).card : ℝ) *
+          (2 * slopeBound (H i).angle (auxiliaryHalfPlaneFamily H i M))) * δ ^ 2 := by
+  intro δ hδ
+  let a := (H i).angle
+  let h := (H i).height
+  let G := frozenActiveSliceLength a E (auxiliaryHalfPlaneFamily H i M) i R h
+  let L := ((activeBooleanPatterns E i).card : ℝ) *
+    (2 * slopeBound a (auxiliaryHalfPlaneFamily H i M))
+  have hδρ : |δ| ≤ ρ := hδ.trans (min_le_right _ _)
+  have harea := area_perturb_sub_eq_intervalIntegral_activeSliceLength
+    hE H i hSide R ε₀ M ρ hε₀ hρ hM hBound (fun x hx ↦ (hstable x hx).1) δ hδ
+  have hGh : activeSliceLength a E (auxiliaryHalfPlaneFamily H i M) i R h = G h := by
+    apply activeSliceLength_eq_frozen_of_stable
+    intro P
+    rfl
+  have hFG : (∫ x in h..h + δ,
+      activeSliceLength a E (auxiliaryHalfPlaneFamily H i M) i R x) =
+      ∫ x in h..h + δ, G x := by
+    apply intervalIntegral.integral_congr
+    intro x hx
+    apply activeSliceLength_eq_frozen_of_stable
+    apply (hstable x ?_).2
+    rw [abs_le]
+    rcases Set.mem_uIcc.mp hx with hx | hx <;>
+      constructor <;> linarith [le_abs_self δ, neg_le_abs δ]
+  rw [show (H i).angle = a by rfl, show (H i).height = h by rfl] at harea ⊢
+  rw [hFG] at harea
+  rw [hGh]
+  have hGint : IntervalIntegrable G MeasureTheory.volume h (h + δ) :=
+    (continuous_frozenActiveSliceLength a E
+      (auxiliaryHalfPlaneFamily H i M) i R h).intervalIntegrable h (h + δ)
+  have hcint : IntervalIntegrable (fun _ : ℝ ↦ G h) MeasureTheory.volume h (h + δ) :=
+    continuous_const.intervalIntegrable h (h + δ)
+  have hremainder :
+      ClassicalResults.area (perturbNefHeight E H i δ) -
+          ClassicalResults.area (perturbNefHeight E H i 0) - G h * δ =
+        ∫ x in h..h + δ, (G x - G h) := by
+    rw [intervalIntegral.integral_sub hGint hcint,
+      intervalIntegral.integral_const]
+    simp only [smul_eq_mul]
+    rw [← harea]
+    ring
+  have hL : 0 ≤ L :=
+    mul_nonneg (Nat.cast_nonneg _)
+      (mul_nonneg (by norm_num) (slopeBound_nonneg _ _))
+  rw [hremainder, ← Real.norm_eq_abs]
+  calc
+    ‖∫ x in h..h + δ, (G x - G h)‖ ≤ (L * |δ|) * |(h + δ) - h| := by
+      apply intervalIntegral.norm_integral_le_of_norm_le_const
+      intro x hx
+      rw [Real.norm_eq_abs]
+      calc
+        |G x - G h| ≤ L * |x - h| := by
+          exact abs_frozenActiveSliceLength_sub_le a E
+            (auxiliaryHalfPlaneFamily H i M) i R h x h
+        _ ≤ L * |δ| := by
+          apply mul_le_mul_of_nonneg_left _ hL
+          rw [abs_le]
+          rcases Set.mem_uIoc.mp hx with hx | hx <;>
+            constructor <;> linarith [le_abs_self δ, neg_le_abs δ]
+    _ = L * δ ^ 2 := by
+      rw [show (h + δ) - h = δ by ring]
+      calc
+        L * |δ| * |δ| = L * |δ| ^ 2 := by ring
+        _ = L * δ ^ 2 := by rw [sq_abs]
+    _ = (((activeBooleanPatterns E i).card : ℝ) *
+          (2 * slopeBound a (auxiliaryHalfPlaneFamily H i M))) * δ ^ 2 := rfl
+
+end MovingSofa.Nef
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Nef / Variation / Boundary
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa.Nef
+
+open Filter Topology
+
+lemma tendsto_sub_one_div_normalVector (p : Point) (a : Real.Angle) :
+    Filter.Tendsto
+      (fun k : ℕ ↦ p - (1 / ((k + 1 : ℕ) : ℝ)) • normalVector a)
+      Filter.atTop (𝓝 p) := by
+  simpa using tendsto_const_nhds.sub
+    ((tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).smul_const
+      (normalVector a))
+
+lemma tendsto_add_one_div_normalVector (p : Point) (a : Real.Angle) :
+    Filter.Tendsto
+      (fun k : ℕ ↦ p + (1 / ((k + 1 : ℕ) : ℝ)) • normalVector a)
+      Filter.atTop (𝓝 p) := by
+  simpa using tendsto_const_nhds.add
+    ((tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).smul_const
+      (normalVector a))
+
+lemma sub_smul_normalVector_mem_carrier (H : PlanarHalfPlaneData)
+    (hupper : H.upper = false) {p : Point} (hp : p ∈ H.boundaryLine)
+    {t : ℝ} (ht : 0 < t) :
+    p - t • normalVector H.angle ∈ H.carrier := by
+  change inner ℝ p (normalVector H.angle) = H.height at hp
+  rw [PlanarHalfPlaneData.carrier, normalHalfPlane, hupper]
+  simp only [Bool.false_eq_true, ↓reduceIte, Set.mem_ofPred_eq]
+  rw [inner_sub_left, real_inner_smul_left, hp]
+  rw [show inner ℝ (normalVector H.angle) (normalVector H.angle) = 1 by
+    simpa using inner_normalVector_self H.angle.toReal]
+  cases H.strict <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> linarith
+
+lemma add_smul_normalVector_not_mem_carrier (H : PlanarHalfPlaneData)
+    (hupper : H.upper = false) {p : Point} (hp : p ∈ H.boundaryLine)
+    {t : ℝ} (ht : 0 < t) :
+    p + t • normalVector H.angle ∉ H.carrier := by
+  change inner ℝ p (normalVector H.angle) = H.height at hp
+  rw [PlanarHalfPlaneData.carrier, normalHalfPlane, hupper]
+  simp only [Bool.false_eq_true, ↓reduceIte, Set.mem_ofPred_eq]
+  rw [inner_add_left, real_inner_smul_left, hp]
+  rw [show inner ℝ (normalVector H.angle) (normalVector H.angle) = 1 by
+    simpa using inner_normalVector_self H.angle.toReal]
+  cases H.strict <;> simp only [Bool.false_eq_true, ↓reduceIte] <;> linarith
+
+lemma mem_frontier_booleanSet_of_active {n : ℕ} (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (hupper : (H i).upper = false) (p : Point)
+    (hline : p ∈ (H i).boundaryLine)
+    (hother : ∀ j, j ≠ i → p ∉ (H j).boundaryLine)
+    (hactive : IsActiveBooleanPattern E i
+      (setMembershipPattern (fun j ↦ (H j).carrier) p)) :
+    p ∈ frontier (booleanSet E (fun j ↦ (H j).carrier)) := by
+  let P := setMembershipPattern (fun j ↦ (H j).carrier) p
+  let qminus : ℕ → Point := fun k ↦
+    p - (1 / ((k + 1 : ℕ) : ℝ)) • normalVector (H i).angle
+  let qplus : ℕ → Point := fun k ↦
+    p + (1 / ((k + 1 : ℕ) : ℝ)) • normalVector (H i).angle
+  have hminus_lim : Filter.Tendsto qminus Filter.atTop (𝓝 p) :=
+    tendsto_sub_one_div_normalVector p (H i).angle
+  have hplus_lim : Filter.Tendsto qplus Filter.atTop (𝓝 p) :=
+    tendsto_add_one_div_normalVector p (H i).angle
+  have hlocal := eventually_setMembershipPattern_eq_of_ne H i p hother
+  have hminus_local : ∀ᶠ k in Filter.atTop, ∀ j, j ≠ i →
+      setMembershipPattern (fun r ↦ (H r).carrier) (qminus k) j = P j :=
+    hminus_lim.eventually hlocal
+  have hplus_local : ∀ᶠ k in Filter.atTop, ∀ j, j ≠ i →
+      setMembershipPattern (fun r ↦ (H r).carrier) (qplus k) j = P j :=
+    hplus_lim.eventually hlocal
+  have hminus_mem : ∀ᶠ k in Filter.atTop,
+      qminus k ∈ booleanSet E (fun j ↦ (H j).carrier) := by
+    filter_upwards [hminus_local] with k hk
+    have hki : qminus k ∈ (H i).carrier :=
+      sub_smul_normalVector_mem_carrier (H i) hupper hline (by positivity)
+    have hpattern : setMembershipPattern (fun r ↦ (H r).carrier) (qminus k) =
+        Function.update P i true := by
+      funext j
+      by_cases hji : j = i
+      · subst j
+        simp [setMembershipPattern, hki]
+      · simpa [Function.update_of_ne hji] using hk j hji
+    change E (setMembershipPattern (fun r ↦ (H r).carrier) (qminus k)) = true
+    rw [hpattern]
+    exact hactive.2
+  have hplus_mem : ∀ᶠ k in Filter.atTop,
+      qplus k ∈ (booleanSet E (fun j ↦ (H j).carrier))ᶜ := by
+    filter_upwards [hplus_local] with k hk
+    have hki : qplus k ∉ (H i).carrier :=
+      add_smul_normalVector_not_mem_carrier (H i) hupper hline (by positivity)
+    have hpattern : setMembershipPattern (fun r ↦ (H r).carrier) (qplus k) =
+        Function.update P i false := by
+      funext j
+      by_cases hji : j = i
+      · subst j
+        simp [setMembershipPattern, hki]
+      · simpa [Function.update_of_ne hji] using hk j hji
+    change E (setMembershipPattern (fun r ↦ (H r).carrier) (qplus k)) ≠ true
+    rw [hpattern, hactive.1]
+    decide
+  rw [frontier_eq_closure_inter_closure]
+  exact ⟨mem_closure_of_tendsto hminus_lim hminus_mem,
+    mem_closure_of_tendsto hplus_lim hplus_mem⟩
+
+lemma not_mem_frontier_booleanSet_of_not_active {n : ℕ} {E : BooleanFunction n}
+    (hE : IsMonotoneBooleanFunction E) (H : Fin n → PlanarHalfPlaneData)
+    (i : Fin n) (p : Point)
+    (hother : ∀ j, j ≠ i → p ∉ (H j).boundaryLine)
+    (hinactive : ¬IsActiveBooleanPattern E i
+      (setMembershipPattern (fun j ↦ (H j).carrier) p)) :
+    p ∉ frontier (booleanSet E (fun j ↦ (H j).carrier)) := by
+  let P := setMembershipPattern (fun j ↦ (H j).carrier) p
+  let X := booleanSet E (fun j ↦ (H j).carrier)
+  have heq : E (Function.update P i false) = E (Function.update P i true) :=
+    (not_isActiveBooleanPattern_iff hE i P).mp hinactive
+  have hlocal := eventually_setMembershipPattern_eq_of_ne H i p hother
+  have hevent : ∀ᶠ q in 𝓝 p,
+      (q ∈ X ↔ E (Function.update P i false) = true) := by
+    filter_upwards [hlocal] with q hq
+    have hpattern : setMembershipPattern (fun r ↦ (H r).carrier) q =
+        Function.update P i
+          (setMembershipPattern (fun r ↦ (H r).carrier) q i) := by
+      funext j
+      by_cases hji : j = i
+      · subst j
+        simp
+      · simpa [Function.update_of_ne hji] using hq j hji
+    change (E (setMembershipPattern (fun r ↦ (H r).carrier) q) = true ↔ _)
+    rw [hpattern]
+    cases hqi : setMembershipPattern (fun r ↦ (H r).carrier) q i
+    · rfl
+    · simp [heq]
+  cases hvalue : E (Function.update P i false)
+  · have hcompl : Xᶜ ∈ 𝓝 p := by
+      filter_upwards [hevent] with q hq
+      simpa [hvalue] using hq
+    have hinter : p ∈ interior Xᶜ := mem_interior_iff_mem_nhds.mpr hcompl
+    have hpcompl : p ∈ Xᶜ := interior_subset hinter
+    have hnot : p ∉ frontier Xᶜ :=
+      (mem_interior_iff_notMem_frontier hpcompl).mp hinter
+    simpa [X] using hnot
+  · have hset : X ∈ 𝓝 p := by
+      filter_upwards [hevent] with q hq
+      simpa [hvalue] using hq
+    have hinter : p ∈ interior X := mem_interior_iff_mem_nhds.mpr hset
+    have hp : p ∈ X := interior_subset hinter
+    exact (mem_interior_iff_notMem_frontier hp).mp hinter
+
+lemma mem_frontier_booleanSet_iff_mem_activeBooleanRegion {n : ℕ}
+    {E : BooleanFunction n} (hE : IsMonotoneBooleanFunction E)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (hupper : (H i).upper = false) (p : Point)
+    (hline : p ∈ (H i).boundaryLine)
+    (hother : ∀ j, j ≠ i → p ∉ (H j).boundaryLine) :
+    p ∈ frontier (booleanSet E (fun j ↦ (H j).carrier)) ↔
+      p ∈ activeBooleanRegion E H i := by
+  rw [mem_activeBooleanRegion_iff]
+  constructor
+  · intro hfrontier
+    by_contra hinactive
+    exact (not_mem_frontier_booleanSet_of_not_active hE H i p hother hinactive)
+      hfrontier
+  · intro hactive
+    exact mem_frontier_booleanSet_of_active E H i hupper p hline hother hactive
+
+/-- The truncated slice of the Boolean set’s frontier along a selected wall boundary. -/
+def frontierLineSlice {n : ℕ} (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n) (R : ℝ) : Set ℝ :=
+  {y | framePoint (H i).angle (H i).height y ∈
+    frontier (booleanSet E (fun j ↦ (H j).carrier))} ∩ Set.Icc (-R) R
+
+lemma volume_frontierLineSlice_eq_activeRegionSlice {n : ℕ}
+    {E : BooleanFunction n} (hE : IsMonotoneBooleanFunction E)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (hupper : (H i).upper = false) (M R : ℝ)
+    (hno : noParallelBoundaryAt (H i).angle
+      (auxiliaryHalfPlaneFamily H i M) (H i).height) :
+    MeasureTheory.volume (frontierLineSlice E H i R) =
+      MeasureTheory.volume (activeRegionSlice (H i).angle E H i R (H i).height) := by
+  apply MeasureTheory.measure_congr
+  rw [MeasureTheory.ae_eq_set]
+  have hzero := (finite_cellSliceBoundaryExceptions (H i).angle
+    (auxiliaryHalfPlaneFamily H i M) (H i).height hno).measure_zero
+      MeasureTheory.volume
+  have heq : ∀ y,
+      y ∉ cellSliceBoundaryExceptions (H i).angle
+          (auxiliaryHalfPlaneFamily H i M) (H i).height →
+        (y ∈ frontierLineSlice E H i R ↔
+          y ∈ activeRegionSlice (H i).angle E H i R (H i).height) := by
+    intro y hy
+    let p := framePoint (H i).angle (H i).height y
+    have hline : p ∈ (H i).boundaryLine := by
+      change inner ℝ p (normalVector (H i).angle) = (H i).height
+      exact inner_framePoint_normalVector _ _ _
+    have hother : ∀ j, j ≠ i → p ∉ (H j).boundaryLine := by
+      intro j hji hj
+      apply hy
+      apply Set.mem_iUnion.mpr
+      refine ⟨j, ?_⟩
+      change inner ℝ p (normalVector (H j).angle) = (H j).height at hj
+      simpa [p, auxiliaryHalfPlaneFamily_of_ne H i j M hji] using hj
+    have hfrontier := mem_frontier_booleanSet_iff_mem_activeBooleanRegion
+      hE H i hupper p hline hother
+    constructor
+    · rintro ⟨hyfrontier, hyR⟩
+      exact ⟨hfrontier.mp hyfrontier, hyR⟩
+    · rintro ⟨hyactive, hyR⟩
+      exact ⟨hfrontier.mpr hyactive, hyR⟩
+  constructor <;> apply MeasureTheory.measure_mono_null _ hzero
+  · intro y hy
+    by_contra hyexception
+    exact hy.2 ((heq y hyexception).mp hy.1)
+  · intro y hy
+    by_contra hyexception
+    exact hy.2 ((heq y hyexception).mpr hy.1)
+
+lemma lineMap_framePoint (a : Real.Angle) (h y : ℝ) :
+    AffineMap.lineMap (framePoint a h 0) (framePoint a h 1) y =
+      framePoint a h y := by
+  simp only [AffineMap.lineMap_apply_module', framePoint_eq, zero_smul,
+    add_zero, one_smul]
+  module
+
+lemma dist_framePoint_zero_one (a : Real.Angle) (h : ℝ) :
+    dist (framePoint a h 0) (framePoint a h 1) = 1 := by
+  rw [dist_eq_norm]
+  simp only [framePoint_eq, zero_smul, add_zero, one_smul]
+  rw [show h • normalVector a - (h • normalVector a + tangentVector a) =
+      -tangentVector a by module]
+  simp [norm_tangentVector_angle]
+
+lemma frontier_booleanSet_inter_boundaryLine_eq_image_frontierLineSlice {n : ℕ}
+    (E : BooleanFunction n) (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (R : ℝ)
+    (hBound : booleanSet E (fun j ↦ (H j).carrier) ⊆ Metric.closedBall 0 R) :
+    frontier (booleanSet E (fun j ↦ (H j).carrier)) ∩ (H i).boundaryLine =
+      AffineMap.lineMap
+        (framePoint (H i).angle (H i).height 0)
+        (framePoint (H i).angle (H i).height 1) ''
+          frontierLineSlice E H i R := by
+  ext p
+  constructor
+  · rintro ⟨hpfrontier, hpline⟩
+    let y := inner ℝ p (tangentVector (H i).angle)
+    have hpnormal : inner ℝ p (normalVector (H i).angle) = (H i).height := hpline
+    have hparam : framePoint (H i).angle (H i).height y = p := by
+      rw [framePoint_eq, ← hpnormal]
+      exact inner_normalVector_smul_add_inner_tangentVector_smul p (H i).angle
+    have hpball : p ∈ Metric.closedBall (0 : Point) R := by
+      have hpclosure : p ∈ closure (booleanSet E (fun j ↦ (H j).carrier)) :=
+        frontier_subset_closure hpfrontier
+      exact (closure_minimal hBound Metric.isClosed_closedBall) hpclosure
+    have hynorm := abs_real_inner_le_norm p (tangentVector (H i).angle)
+    rw [norm_tangentVector_angle, mul_one] at hynorm
+    have hpnorm : ‖p‖ ≤ R := by
+      simpa [Metric.mem_closedBall, dist_zero_left] using hpball
+    have hyR : y ∈ Set.Icc (-R) R := abs_le.mp (hynorm.trans hpnorm)
+    refine ⟨y, ⟨?_, ?_⟩⟩
+    · exact ⟨by simpa [frontierLineSlice, hparam] using hpfrontier, hyR⟩
+    · rw [lineMap_framePoint, hparam]
+  · rintro ⟨y, hy, rfl⟩
+    rw [lineMap_framePoint]
+    refine ⟨hy.1, ?_⟩
+    change inner ℝ (framePoint (H i).angle (H i).height y)
+      (normalVector (H i).angle) = (H i).height
+    exact inner_framePoint_normalVector _ _ _
+
+lemma hausdorffMeasure_frontier_inter_boundaryLine_eq_volume_slice {n : ℕ}
+    (E : BooleanFunction n) (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (R : ℝ)
+    (hBound : booleanSet E (fun j ↦ (H j).carrier) ⊆ Metric.closedBall 0 R) :
+    MeasureTheory.Measure.hausdorffMeasure 1
+        (frontier (booleanSet E (fun j ↦ (H j).carrier)) ∩ (H i).boundaryLine) =
+      MeasureTheory.volume (frontierLineSlice E H i R) := by
+  rw [frontier_booleanSet_inter_boundaryLine_eq_image_frontierLineSlice
+    E H i R hBound, MeasureTheory.hausdorffMeasure_lineMap_image,
+    MeasureTheory.hausdorffMeasure_real]
+  have hdist := dist_framePoint_zero_one (H i).angle (H i).height
+  have hnndist : nndist
+      (framePoint (H i).angle (H i).height 0)
+      (framePoint (H i).angle (H i).height 1) = 1 := by
+    apply NNReal.eq
+    simpa using hdist
+  rw [hnndist, one_smul]
+
+lemma hausdorffMeasure_frontier_toReal_eq_activeSliceLength {n : ℕ}
+    {E : BooleanFunction n} (hE : IsMonotoneBooleanFunction E)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (hupper : (H i).upper = false) (M R : ℝ)
+    (hM : (H i).height ≤ M)
+    (hBound : booleanSet E (fun j ↦ (H j).carrier) ⊆ Metric.closedBall 0 R)
+    (hno : noParallelBoundaryAt (H i).angle
+      (auxiliaryHalfPlaneFamily H i M) (H i).height) :
+    (MeasureTheory.Measure.hausdorffMeasure 1
+      (frontier (booleanSet E (fun j ↦ (H j).carrier)) ∩
+        (H i).boundaryLine)).toReal =
+      activeSliceLength (H i).angle E
+        (auxiliaryHalfPlaneFamily H i M) i R (H i).height := by
+  rw [hausdorffMeasure_frontier_inter_boundaryLine_eq_volume_slice E H i R hBound,
+    volume_frontierLineSlice_eq_activeRegionSlice hE H i hupper M R hno]
+  exact volume_activeRegionSlice_toReal_eq_activeSliceLength E H i M R
+    (H i).height hM hno
+
+end MovingSofa.Nef
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Nef / Variation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+open Filter Topology
+open Nef
+
+theorem simpleNefPolygon_area_variation {n : ℕ} (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (hE : IsMonotoneBooleanFunction E)
+    (hLines : Function.Injective (fun j ↦ (H j).boundaryLine))
+    (hSide : (H i).upper = false) (R ε₀ : ℝ) (hR : 0 < R) (hε₀ : 0 < ε₀)
+    (hBound : ∀ δ : ℝ, |δ| ≤ ε₀ →
+      perturbNefHeight E H i δ ⊆ Metric.closedBall 0 R) :
+    ∃ C ε : ℝ, 0 ≤ C ∧ 0 < ε ∧ ε ≤ ε₀ ∧
+      ∀ δ : ℝ, |δ| ≤ ε →
+        |ClassicalResults.area (perturbNefHeight E H i δ) -
+          ClassicalResults.area (perturbNefHeight E H i 0) -
+          (MeasureTheory.Measure.hausdorffMeasure 1
+            (frontier (perturbNefHeight E H i 0) ∩ (H i).boundaryLine)).toReal * δ| ≤
+          C * δ ^ 2 := by
+  let M := |(H i).height| + ε₀ + R + 1
+  have hno : noParallelBoundaryAt (H i).angle
+      (auxiliaryHalfPlaneFamily H i M) (H i).height := by
+    exact noParallelBoundaryAt_auxiliary H i hLines R ε₀ hR hε₀
+  rcases exists_parallel_stability_radius (H i).angle
+      (auxiliaryHalfPlaneFamily H i M) (H i).height hno with
+    ⟨ρ, hρ, hstable⟩
+  let C := ((activeBooleanPatterns E i).card : ℝ) *
+    (2 * slopeBound (H i).angle (auxiliaryHalfPlaneFamily H i M))
+  let ε := min ε₀ ρ
+  refine ⟨C, ε, ?_, ?_, ?_, ?_⟩
+  · exact mul_nonneg (Nat.cast_nonneg _)
+      (mul_nonneg (by norm_num) (slopeBound_nonneg _ _))
+  · exact lt_min hε₀ hρ
+  · exact min_le_left _ _
+  · intro δ hδ
+    have hM : (H i).height + ε₀ ≤ M := by
+      dsimp [M]
+      linarith [le_abs_self (H i).height]
+    have hzero : |(0 : ℝ)| ≤ ε₀ := by simpa using hε₀.le
+    have hBoundZero := hBound 0 hzero
+    rw [perturbNefHeight_zero] at hBoundZero
+    have hcoefficient :
+        (MeasureTheory.Measure.hausdorffMeasure 1
+          (frontier (perturbNefHeight E H i 0) ∩ (H i).boundaryLine)).toReal =
+        activeSliceLength (H i).angle E
+          (auxiliaryHalfPlaneFamily H i M) i R (H i).height := by
+      rw [perturbNefHeight_zero]
+      exact hausdorffMeasure_frontier_toReal_eq_activeSliceLength
+        hE H i hSide M R (le_trans (le_add_of_nonneg_right hε₀.le) hM)
+          hBoundZero hno
+    have hremainder := area_perturb_remainder_le hE H i hSide R ε₀ M ρ
+      hε₀ hρ hM hBound hstable δ hδ
+    rw [hcoefficient]
+    exact hremainder
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Nef / Signed Variation
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace MovingSofa
+
+private def PlanarHalfPlaneData.reverseOrientation
+    (H : PlanarHalfPlaneData) : PlanarHalfPlaneData :=
+  ⟨H.angle + ((Real.pi : ℝ) : Real.Angle), -H.height, !H.upper, H.strict⟩
+
+private theorem PlanarHalfPlaneData.carrier_reverseOrientation (H : PlanarHalfPlaneData) :
+    H.reverseOrientation.carrier = H.carrier := by
+  ext p
+  simp only [reverseOrientation, carrier, normalHalfPlane, Set.mem_ofPred_eq,
+    normalVector_add_pi_angle, inner_neg_right]
+  cases H.upper <;> cases H.strict <;> simp
+
+private theorem PlanarHalfPlaneData.boundaryLine_reverseOrientation (H : PlanarHalfPlaneData) :
+    H.reverseOrientation.boundaryLine = H.boundaryLine := by
+  ext p
+  simp [reverseOrientation, boundaryLine, normalLine, normalVector_add_pi_angle]
+
+private theorem PlanarHalfPlaneData.carrier_reverseOrientation_add_height
+    (H : PlanarHalfPlaneData) (δ : ℝ) :
+    ({H.reverseOrientation with height := H.reverseOrientation.height + δ}).carrier =
+      ({H with height := H.height + -δ}).carrier := by
+  have heq : {H.reverseOrientation with height := H.reverseOrientation.height + δ} =
+      ({H with height := H.height + -δ}).reverseOrientation := by
+    cases H
+    simp [reverseOrientation, add_comm]
+  rw [heq, carrier_reverseOrientation]
+
+private theorem perturbNefHeight_reverseOrientation {n : ℕ} (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n) (δ : ℝ) :
+    perturbNefHeight E (fun j ↦ (H j).reverseOrientation) i δ =
+      perturbNefHeight E H i (-δ) := by
+  unfold perturbNefHeight
+  congr 1
+  funext j
+  split_ifs
+  · exact PlanarHalfPlaneData.carrier_reverseOrientation_add_height _ _
+  · exact PlanarHalfPlaneData.carrier_reverseOrientation _
+
+/-- Quadratic area variation when the moved half-plane is a lower constraint. -/
+theorem simpleNefPolygon_area_variation_lower {n : ℕ} (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (hE : IsMonotoneBooleanFunction E)
+    (hLines : Function.Injective (fun j ↦ (H j).boundaryLine))
+    (hSide : (H i).upper = true) (R ε₀ : ℝ) (hR : 0 < R) (hε₀ : 0 < ε₀)
+    (hBound : ∀ δ : ℝ, |δ| ≤ ε₀ →
+      perturbNefHeight E H i δ ⊆ Metric.closedBall 0 R) :
+    ∃ C ε : ℝ, 0 ≤ C ∧ 0 < ε ∧ ε ≤ ε₀ ∧
+      ∀ δ : ℝ, |δ| ≤ ε →
+        |ClassicalResults.area (perturbNefHeight E H i δ) -
+          ClassicalResults.area (perturbNefHeight E H i 0) +
+          (MeasureTheory.Measure.hausdorffMeasure 1
+            (frontier (perturbNefHeight E H i 0) ∩ (H i).boundaryLine)).toReal * δ| ≤
+          C * δ ^ 2 := by
+  have hLines' : Function.Injective (fun j ↦ (H j).reverseOrientation.boundaryLine) := by
+    simpa only [PlanarHalfPlaneData.boundaryLine_reverseOrientation] using hLines
+  have hSide' : (H i).reverseOrientation.upper = false := by
+    simp [PlanarHalfPlaneData.reverseOrientation, hSide]
+  obtain ⟨C, ε, hC, hε, hε₀', h⟩ := simpleNefPolygon_area_variation E
+    (fun j ↦ (H j).reverseOrientation) i hE hLines' hSide' R ε₀ hR hε₀ (by
+      intro δ hδ
+      rw [perturbNefHeight_reverseOrientation]
+      exact hBound (-δ) (by simpa only [abs_neg] using hδ))
+  refine ⟨C, ε, hC, hε, hε₀', fun δ hδ ↦ ?_⟩
+  have hb := h (-δ) (by simpa only [abs_neg] using hδ)
+  simpa only [perturbNefHeight_reverseOrientation, neg_neg, neg_zero,
+    PlanarHalfPlaneData.boundaryLine_reverseOrientation, mul_neg, sub_neg_eq_add,
+    neg_sq] using hb
+/-- Signed quadratic area variation for either orientation of a simple Nef wall. -/
+theorem simpleNefPolygon_area_variation_signed {n : ℕ} (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (i : Fin n)
+    (hE : IsMonotoneBooleanFunction E)
+    (hLines : Function.Injective (fun j ↦ (H j).boundaryLine))
+    (R ε₀ : ℝ) (hR : 0 < R) (hε₀ : 0 < ε₀)
+    (hBound : ∀ δ : ℝ, |δ| ≤ ε₀ →
+      perturbNefHeight E H i δ ⊆ Metric.closedBall 0 R) :
+    ∃ C ε : ℝ, 0 ≤ C ∧ 0 < ε ∧ ε ≤ ε₀ ∧
+      ∀ δ : ℝ, |δ| ≤ ε →
+        |ClassicalResults.area (perturbNefHeight E H i δ) -
+          ClassicalResults.area (perturbNefHeight E H i 0) -
+          (if (H i).upper then -1 else 1) *
+            (MeasureTheory.Measure.hausdorffMeasure 1
+              (frontier (perturbNefHeight E H i 0) ∩ (H i).boundaryLine)).toReal * δ| ≤
+          C * δ ^ 2 := by
+  cases hs : (H i).upper
+  · simpa only [hs, Bool.false_eq_true, ite_false, one_mul] using
+      simpleNefPolygon_area_variation E H i hE hLines hs R ε₀ hR hε₀ hBound
+  · simpa only [hs, ite_true, neg_one_mul, neg_mul, one_mul, sub_neg_eq_add] using
+      simpleNefPolygon_area_variation_lower E H i hE hLines hs R ε₀ hR hε₀ hBound
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Height / Wall Variation
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace MovingSofa
+
+private theorem endpoint_wall_changes_disjoint (a : Real.Angle) (h ε : ℝ)
+    (hε : 0 ≤ ε) (hε' : ε ≤ 1) (p : Point) :
+    (p ∈ normalHalfPlane a (h + ε) false false ↔
+      p ∈ normalHalfPlane a h false false) ∨
+    (p ∈ normalHalfPlane a (h - 1 + ε) true false ↔
+      p ∈ normalHalfPlane a (h - 1) true false) := by
+  change (inner ℝ p (normalVector a) ≤ h + ε ↔ inner ℝ p (normalVector a) ≤ h) ∨
+    (h - 1 + ε ≤ inner ℝ p (normalVector a) ↔ h - 1 ≤ inner ℝ p (normalVector a))
+  by_cases hp : inner ℝ p (normalVector a) ≤ h
+  · left
+    constructor <;> intro _ <;> linarith
+  · right
+    constructor <;> intro _ <;> linarith
+
+private theorem area_add_of_indicator_add_eq (A B C D : Set Point)
+    (hA : MeasurableSet A) (hB : MeasurableSet B)
+    (hC : MeasurableSet C) (hD : MeasurableSet D)
+    (hAfin : MeasureTheory.volume A ≠ ⊤) (hBfin : MeasureTheory.volume B ≠ ⊤)
+    (hCfin : MeasureTheory.volume C ≠ ⊤) (hDfin : MeasureTheory.volume D ≠ ⊤)
+    (h : ∀ p, A.indicator (fun _ ↦ (1 : ℝ)) p + B.indicator (fun _ ↦ (1 : ℝ)) p =
+      C.indicator (fun _ ↦ (1 : ℝ)) p + D.indicator (fun _ ↦ (1 : ℝ)) p) :
+    ClassicalResults.area A + ClassicalResults.area B =
+      ClassicalResults.area C + ClassicalResults.area D := by
+  have hint (S : Set Point) (hm : MeasurableSet S) (hf : MeasureTheory.volume S ≠ ⊤) :
+      MeasureTheory.Integrable (S.indicator (fun _ ↦ (1 : ℝ))) :=
+    (MeasureTheory.integrableOn_const hf).integrable_indicator hm
+  have heq := congrArg (fun f : Point → ℝ ↦ ∫ p, f p) (funext h)
+  rw [MeasureTheory.integral_add (hint A hA hAfin) (hint B hB hBfin),
+    MeasureTheory.integral_add (hint C hC hCfin) (hint D hD hDfin)] at heq
+  have harea (S : Set Point) (hm : MeasurableSet S) :
+      (∫ p, S.indicator (fun _ ↦ (1 : ℝ)) p) = ClassicalResults.area S :=
+    MeasureTheory.integral_indicator_one hm
+  simpa only [harea A hA, harea B hB, harea C hC, harea D hD] using heq
+private def BooleanFunction.all (n : ℕ) : BooleanFunction n :=
+  fun P ↦ decide (∀ i, P i = true)
+
+private theorem BooleanFunction.all_monotone (n : ℕ) :
+    IsMonotoneBooleanFunction (BooleanFunction.all n) := by
+  intro P Q hPQ hP
+  simp only [BooleanFunction.all, decide_eq_true_eq] at hP ⊢
+  exact fun i ↦ hPQ i (hP i)
+
+private theorem booleanSet_all {n : ℕ} (H : Fin n → Set Point) :
+    booleanSet (BooleanFunction.all n) H = ⋂ i, H i := by
+  classical
+  ext p
+  simp [booleanSet, BooleanFunction.all]
+
+private theorem independentWallCap_eq_booleanSet {Θ : AngleSet}
+    (h upper lower : PolygonHeightSpace Θ) {n : ℕ}
+    (H : Fin n → PlanarHalfPlaneData) (hH : Set.range H = polygonCapWalls h)
+    (F : PlanarHalfPlaneData → Set Point)
+    (hupper : ∀ t ∈ angleDomain Θ,
+      F ⟨(t : Real.Angle), polygonHeightValue h t, false, false⟩ =
+        normalHalfPlane (t : Real.Angle) (polygonHeightValue upper t) false false)
+    (hlower : ∀ t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+      F ⟨(t : Real.Angle), polygonHeightValue h t - 1, true, false⟩ =
+        normalHalfPlane (t : Real.Angle) (polygonHeightValue lower t) true false) :
+    independentWallCap upper lower =
+      booleanSet (BooleanFunction.all n) (fun i ↦ F (H i)) := by
+  rw [booleanSet_all]
+  ext p
+  have hrange : (∀ i, p ∈ F (H i)) ↔ ∀ W ∈ polygonCapWalls h, p ∈ F W := by
+    rw [← hH]
+    simp
+  simp only [Set.mem_iInter]
+  rw [hrange]
+  simp only [independentWallCap, Set.mem_inter_iff, Set.mem_iInter]
+  constructor
+  · rintro ⟨hend, hint⟩ W hW
+    rcases hW with ⟨t, ht, rfl⟩ | ⟨t, ht, rfl⟩
+    · rw [hupper t ht]
+      rcases ht with ht | ht
+      · exact hint t ht
+      · exact (hend t ht).1
+    · rw [hlower t ht]
+      exact (hend t ht).2
+  · intro hall
+    constructor
+    · intro t ht
+      constructor
+      · rw [← hupper t (Or.inr ht)]
+        exact hall _ (Or.inl ⟨t, Or.inr ht, rfl⟩)
+      · rw [← hlower t ht]
+        exact hall _ (Or.inr ⟨t, ht, rfl⟩)
+    · intro t ht
+      rw [← hupper t (Or.inl ht)]
+      exact hall _ (Or.inl ⟨t, Or.inl ht, rfl⟩)
+
+private def BooleanFunction.niche (Θ : AngleSet) {n : ℕ}
+    (endpoint : {t : ℝ // t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)} → Fin n)
+    (left right : {t : ℝ // t ∈ Θ.directions} → Fin n) : BooleanFunction n := by
+  classical
+  exact fun P ↦ decide ((∀ t, P (endpoint t) = true) ∧
+    ∃ t, P (left t) = true ∧ P (right t) = true)
+
+private theorem BooleanFunction.niche_monotone (Θ : AngleSet) {n : ℕ}
+    (endpoint : {t : ℝ // t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)} → Fin n)
+    (left right : {t : ℝ // t ∈ Θ.directions} → Fin n) :
+    IsMonotoneBooleanFunction (BooleanFunction.niche Θ endpoint left right) := by
+  classical
+  intro P Q hPQ hP
+  simp only [BooleanFunction.niche, decide_eq_true_eq] at hP ⊢
+  obtain ⟨he, t, hl, hr⟩ := hP
+  exact ⟨fun s ↦ hPQ _ (he s), t, hPQ _ hl, hPQ _ hr⟩
+
+private theorem independentWallNiche_eq_booleanSet {Θ : AngleSet}
+    (lower : PolygonHeightSpace Θ) {n : ℕ} (H : Fin n → Set Point)
+    (endpoint : {t : ℝ // t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)} → Fin n)
+    (left right : {t : ℝ // t ∈ Θ.directions} → Fin n)
+    (he : ∀ t, H (endpoint t) =
+      normalHalfPlane (t.val : Real.Angle) (polygonHeightValue lower t.val) true false)
+    (hl : ∀ t, H (left t) =
+      normalHalfPlane (t.val : Real.Angle) (polygonHeightValue lower t.val) false true)
+    (hr : ∀ t, H (right t) =
+      normalHalfPlane ((t.val + Real.pi / 2 : ℝ) : Real.Angle)
+        (polygonHeightValue lower (t.val + Real.pi / 2)) false true) :
+    independentWallNiche lower =
+      booleanSet (BooleanFunction.niche Θ endpoint left right) H := by
+  classical
+  ext p
+  simp only [independentWallNiche, Set.mem_inter_iff, Set.mem_iInter,
+    Set.mem_iUnion, booleanSet, Set.mem_ofPred_eq, BooleanFunction.niche,
+    decide_eq_true_eq, he, hl, hr]
+  constructor
+  · rintro ⟨he, t, ht, hl, hr⟩
+    exact ⟨fun s ↦ he s.val s.property, ⟨t, ht⟩, hl, hr⟩
+  · rintro ⟨he, t, hl, hr⟩
+    exact ⟨fun s hs ↦ he ⟨s, hs⟩, t.val, t.property, hl, hr⟩
+
+private theorem exists_polygonNiche_wall_indices {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (hH : Set.range H = polygonNicheWalls h) :
+    ∃ (endpoint : {t : ℝ // t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)} → Fin n)
+      (left right : {t : ℝ // t ∈ Θ.directions} → Fin n),
+      (∀ t, H (endpoint t) =
+        ⟨(t.val : Real.Angle), polygonHeightValue h t.val - 1, true, false⟩) ∧
+      (∀ t, H (left t) =
+        ⟨(t.val : Real.Angle), polygonHeightValue h t.val - 1, false, true⟩) ∧
+      (∀ t, H (right t) =
+        ⟨((t.val + Real.pi / 2 : ℝ) : Real.Angle),
+          polygonHeightValue h (t.val + Real.pi / 2) - 1, false, true⟩) := by
+  classical
+  have hpre (W : PlanarHalfPlaneData) (hW : W ∈ polygonNicheWalls h) :
+      ∃ i, H i = W := by
+    change W ∈ Set.range H
+    rwa [hH]
+  have he (t : {t : ℝ // t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)}) :
+      ∃ i, H i = ⟨(t.val : Real.Angle), polygonHeightValue h t.val - 1, true, false⟩ :=
+    hpre _ (Or.inr ⟨t.val, t.property, rfl⟩)
+  have hl (t : {t : ℝ // t ∈ Θ.directions}) :
+      ∃ i, H i = ⟨(t.val : Real.Angle), polygonHeightValue h t.val - 1, false, true⟩ :=
+    hpre _ (Or.inl ⟨t.val, Or.inl t.property, rfl⟩)
+  have hr (t : {t : ℝ // t ∈ Θ.directions}) :
+      ∃ i, H i = ⟨((t.val + Real.pi / 2 : ℝ) : Real.Angle),
+        polygonHeightValue h (t.val + Real.pi / 2) - 1, false, true⟩ :=
+    hpre _ (Or.inl ⟨t.val + Real.pi / 2, Or.inr ⟨t.val, t.property, rfl⟩, rfl⟩)
+  choose endpoint he using he
+  choose left hl using hl
+  choose right hr using hr
+  exact ⟨endpoint, left, right, he, hl, hr⟩
+
+private def PlanarHalfPlaneData.moveWall (W : PlanarHalfPlaneData) (δ : ℝ)
+    (V : PlanarHalfPlaneData) : PlanarHalfPlaneData := by
+  classical
+  exact if V = W then {V with height := V.height + δ} else V
+
+private theorem perturbNefHeight_eq_moveWall {n : ℕ} (E : BooleanFunction n)
+    (H : Fin n → PlanarHalfPlaneData) (hH : Function.Injective H) (i : Fin n) (δ : ℝ) :
+    perturbNefHeight E H i δ =
+      booleanSet E (fun j ↦ ((H i).moveWall δ (H j)).carrier) := by
+  classical
+  unfold perturbNefHeight
+  congr 1
+  funext j
+  simp only [PlanarHalfPlaneData.moveWall, hH.eq_iff]
+
+/-- Angles in a polygon angle domain have distinct classes modulo a full turn. -/
+theorem angleDomain_coe_injective (Θ : AngleSet) :
+    Function.Injective (fun t : angleDomain Θ ↦ (t.val : Real.Angle)) := by
+  intro s t hst
+  apply Subtype.ext
+  have hs := angleDomain_subset_Ioo Θ s.property
+  have ht := angleDomain_subset_Ioo Θ t.property
+  exact ((normalLine_eq_iff_of_mem_Ioo (c := 0) (d := 0) hs ht).mp
+    (congrArg (fun a ↦ normalLine a 0) hst)).1
+
+private theorem polygonHeightValue_update {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    (t : angleDomain Θ) (δ : ℝ) (s : ℝ) (hs : s ∈ angleDomain Θ) :
+    polygonHeightValue (Function.update h t (h t + δ)) s =
+      polygonHeightValue h s + if s = t.val then δ else 0 := by
+  classical
+  by_cases hst : s = t.val
+  · subst s
+    simp [polygonHeightValue, Function.update]
+  · have hne : (⟨s, hs⟩ : angleDomain Θ) ≠ t := by
+      intro heq
+      exact hst (congrArg Subtype.val heq)
+    simp [polygonHeightValue, Function.update, hs, hst, hne]
+
+private theorem perturbNefHeight_cap_upper {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (hH : Set.range H = polygonCapWalls h) (hHinj : Function.Injective H)
+    (i : Fin n) (t : angleDomain Θ)
+    (hi : H i = ⟨(t.val : Real.Angle), h t, false, false⟩) (δ : ℝ) :
+    perturbNefHeight (BooleanFunction.all n) H i δ =
+      independentWallCap (Function.update h t (h t + δ)) (fun s ↦ h s - 1) := by
+  classical
+  rw [perturbNefHeight_eq_moveWall _ _ hHinj]
+  symm
+  apply independentWallCap_eq_booleanSet h _ _ H hH
+    (fun W ↦ ((H i).moveWall δ W).carrier)
+  · intro s hs
+    have heq : (⟨(s : Real.Angle), polygonHeightValue h s, false, false⟩ :
+        PlanarHalfPlaneData) = H i ↔ s = t.val := by
+      rw [hi]
+      constructor
+      · intro heq
+        have ha := congrArg PlanarHalfPlaneData.angle heq
+        exact congrArg Subtype.val (angleDomain_coe_injective Θ
+          (a₁ := ⟨s, hs⟩) (a₂ := t) ha)
+      · intro hst
+        subst s
+        simp [polygonHeightValue]
+    simp only [PlanarHalfPlaneData.moveWall, heq, polygonHeightValue_update h t δ s hs]
+    split_ifs with hst
+    · rfl
+    · simp only [add_zero]
+      rfl
+  · intro s hs
+    have hsD : s ∈ angleDomain Θ := Or.inr hs
+    have hne : (⟨(s : Real.Angle), polygonHeightValue h s - 1, true, false⟩ :
+        PlanarHalfPlaneData) ≠ H i := by
+      rw [hi]
+      intro heq
+      have hb := congrArg PlanarHalfPlaneData.upper heq
+      cases hb
+    simp only [PlanarHalfPlaneData.moveWall, hne, ite_false]
+    simp [PlanarHalfPlaneData.carrier, polygonHeightValue, hsD]
+
+private theorem perturbNefHeight_cap_lower {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (hH : Set.range H = polygonCapWalls h) (hHinj : Function.Injective H)
+    (i : Fin n) (t : angleDomain Θ)
+    (hi : H i = ⟨(t.val : Real.Angle), h t - 1, true, false⟩) (δ : ℝ) :
+    perturbNefHeight (BooleanFunction.all n) H i δ =
+      independentWallCap h (Function.update (fun s ↦ h s - 1) t (h t - 1 + δ)) := by
+  classical
+  rw [perturbNefHeight_eq_moveWall _ _ hHinj]
+  symm
+  apply independentWallCap_eq_booleanSet h _ _ H hH
+    (fun W ↦ ((H i).moveWall δ W).carrier)
+  · intro s hs
+    have hne : (⟨(s : Real.Angle), polygonHeightValue h s, false, false⟩ :
+        PlanarHalfPlaneData) ≠ H i := by
+      rw [hi]
+      intro heq
+      have hb := congrArg PlanarHalfPlaneData.upper heq
+      cases hb
+    simp only [PlanarHalfPlaneData.moveWall, hne, ite_false]
+    rfl
+  · intro s hs
+    have hsD : s ∈ angleDomain Θ := Or.inr hs
+    have heq : (⟨(s : Real.Angle), polygonHeightValue h s - 1, true, false⟩ :
+        PlanarHalfPlaneData) = H i ↔ s = t.val := by
+      rw [hi]
+      constructor
+      · intro heq
+        have ha := congrArg PlanarHalfPlaneData.angle heq
+        exact congrArg Subtype.val (angleDomain_coe_injective Θ
+          (a₁ := ⟨s, hsD⟩) (a₂ := t) ha)
+      · intro hst
+        subst s
+        simp [polygonHeightValue]
+    simp only [PlanarHalfPlaneData.moveWall, heq,
+      polygonHeightValue_update (fun s ↦ h s - 1) t δ s hsD]
+    split_ifs <;> simp [PlanarHalfPlaneData.carrier, polygonHeightValue, hsD]
+
+private theorem polygonCap_singleWall_uniform_bounds {Θ : AngleSet} (h : PolygonHeightSpace Θ) :
+    ∃ R ε₀ : ℝ, 0 < R ∧ 0 < ε₀ ∧
+      ∀ (n : ℕ) (H : Fin n → PlanarHalfPlaneData),
+        Set.range H = polygonCapWalls h → Function.Injective H →
+        ∀ (i : Fin n) (t : angleDomain Θ),
+          (H i = ⟨(t.val : Real.Angle), h t, false, false⟩ ∨
+            H i = ⟨(t.val : Real.Angle), h t - 1, true, false⟩) →
+          ∀ δ : ℝ, |δ| ≤ ε₀ →
+            perturbNefHeight (BooleanFunction.all n) H i δ ⊆ Metric.closedBall 0 R := by
+  classical
+  obtain ⟨R, ε₀, hR, hε₀, hb⟩ := polygonPerturbation_uniform_bounds Θ h
+  refine ⟨R, ε₀, hR, hε₀, ?_⟩
+  intro n H hH hHinj i t hi δ hδ
+  have hupdate (f : PolygonHeightSpace Θ) :
+      ∀ s, |Function.update f t (f t + δ) s - f s| ≤ ε₀ := by
+    intro s
+    by_cases hst : s = t
+    · subst s
+      simpa [Function.update] using hδ
+    · simp [Function.update, hst, le_of_lt hε₀]
+  rcases hi with hi | hi
+  · rw [perturbNefHeight_cap_upper h H hH hHinj i t hi δ]
+    exact (hb _ _ (hupdate h) (by intro s; simpa using le_of_lt hε₀)).1
+  · rw [perturbNefHeight_cap_lower h H hH hHinj i t hi δ]
+    exact (hb _ _ (by intro s; simpa using le_of_lt hε₀)
+      (hupdate (fun s ↦ h s - 1))).1
+
+private theorem perturbNefHeight_niche {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (hangle : Function.Injective (fun j ↦ (H j).angle))
+    (endpoint : {t : ℝ // t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)} → Fin n)
+    (left right : {t : ℝ // t ∈ Θ.directions} → Fin n)
+    (he : ∀ t, H (endpoint t) =
+      ⟨(t.val : Real.Angle), polygonHeightValue h t.val - 1, true, false⟩)
+    (hl : ∀ t, H (left t) =
+      ⟨(t.val : Real.Angle), polygonHeightValue h t.val - 1, false, true⟩)
+    (hr : ∀ t, H (right t) =
+      ⟨((t.val + Real.pi / 2 : ℝ) : Real.Angle),
+        polygonHeightValue h (t.val + Real.pi / 2) - 1, false, true⟩)
+    (i : Fin n) (t : angleDomain Θ) (hi : (H i).angle = (t.val : Real.Angle)) (δ : ℝ) :
+    perturbNefHeight (BooleanFunction.niche Θ endpoint left right) H i δ =
+      independentWallNiche (Function.update (fun s ↦ h s - 1) t (h t - 1 + δ)) := by
+  classical
+  have hwall (j : Fin n) (s : ℝ) (hs : s ∈ angleDomain Θ) (upper strict : Bool)
+      (hj : H j = ⟨(s : Real.Angle), polygonHeightValue h s - 1, upper, strict⟩) :
+      (if j = i then {H j with height := (H j).height + δ} else H j).carrier =
+        normalHalfPlane (s : Real.Angle)
+          (polygonHeightValue (Function.update (fun s ↦ h s - 1) t (h t - 1 + δ)) s)
+          upper strict := by
+    have hji : j = i ↔ s = t.val := by
+      constructor
+      · intro hji
+        have ha : (s : Real.Angle) = (t.val : Real.Angle) := by
+          rw [← hi, ← hji, hj]
+        exact congrArg Subtype.val (angleDomain_coe_injective Θ
+          (a₁ := ⟨s, hs⟩) (a₂ := t) ha)
+      · intro hst
+        apply hangle
+        simp only [hj, hi, hst]
+    simp only [hji, hj, polygonHeightValue_update (fun s ↦ h s - 1) t δ s hs]
+    split_ifs <;> simp [PlanarHalfPlaneData.carrier, polygonHeightValue, hs]
+  unfold perturbNefHeight
+  symm
+  apply independentWallNiche_eq_booleanSet _ _ endpoint left right
+  · intro s
+    exact hwall _ s.val (Or.inr s.property) true false (he s)
+  · intro s
+    exact hwall _ s.val (Or.inl (Or.inl s.property)) false true (hl s)
+  · intro s
+    exact hwall _ (s.val + Real.pi / 2)
+      (Or.inl (Or.inr ⟨s.val, s.property, rfl⟩)) false true (hr s)
+
+private theorem exists_angle_of_mem_polygonNicheWalls {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    (W : PlanarHalfPlaneData) (hW : W ∈ polygonNicheWalls h) :
+    ∃ s : angleDomain Θ, W.angle = (s.val : Real.Angle) ∧ W.height = h s - 1 := by
+  rcases hW with ⟨s, hs, rfl⟩ | ⟨s, hs, rfl⟩
+  · refine ⟨⟨s, Or.inl hs⟩, rfl, ?_⟩
+    simp [polygonHeightValue, show s ∈ angleDomain Θ from Or.inl hs]
+  · refine ⟨⟨s, Or.inr hs⟩, rfl, ?_⟩
+    simp [polygonHeightValue, show s ∈ angleDomain Θ from Or.inr hs]
+
+private theorem polygonNicheWalls_angle_injective {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (hH : Set.range H = polygonNicheWalls h)
+    (hLines : Function.Injective (fun j ↦ (H j).boundaryLine)) :
+    Function.Injective (fun j ↦ (H j).angle) := by
+  intro i j hij
+  obtain ⟨s, hs, hsheight⟩ := exists_angle_of_mem_polygonNicheWalls h
+    (H i) (hH ▸ Set.mem_range_self i)
+  obtain ⟨t, ht, htheight⟩ := exists_angle_of_mem_polygonNicheWalls h
+    (H j) (hH ▸ Set.mem_range_self j)
+  have hst : s = t := angleDomain_coe_injective Θ (hs.symm.trans (hij.trans ht))
+  apply hLines
+  simp only [PlanarHalfPlaneData.boundaryLine, hij, hsheight, htheight, hst]
+
+private theorem exists_polygonNiche_perturbation_formula {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (hH : Set.range H = polygonNicheWalls h)
+    (hLines : Function.Injective (fun j ↦ (H j).boundaryLine)) :
+    ∃ E : BooleanFunction n, IsMonotoneBooleanFunction E ∧
+      ∀ i, ∃ t : angleDomain Θ, (H i).angle = (t.val : Real.Angle) ∧
+        ∀ δ : ℝ, perturbNefHeight E H i δ =
+          independentWallNiche (Function.update (fun s ↦ h s - 1) t (h t - 1 + δ)) := by
+  obtain ⟨endpoint, left, right, he, hl, hr⟩ := exists_polygonNiche_wall_indices h H hH
+  refine ⟨BooleanFunction.niche Θ endpoint left right,
+    BooleanFunction.niche_monotone Θ endpoint left right, ?_⟩
+  intro i
+  obtain ⟨t, ht, _⟩ := exists_angle_of_mem_polygonNicheWalls h
+    (H i) (hH ▸ Set.mem_range_self i)
+  refine ⟨t, ht, fun δ ↦ ?_⟩
+  exact perturbNefHeight_niche h H (polygonNicheWalls_angle_injective h H hH hLines)
+    endpoint left right he hl hr i t ht δ
+
+/-- The niche with every lower wall shifted by one is the height-defined niche. -/
+theorem independentWallNiche_sub_one {Θ : AngleSet} (h : PolygonHeightSpace Θ) :
+    independentWallNiche (fun s ↦ h s - 1) = polygonHeightNiche h := by
+  have hv (s : ℝ) (hs : s ∈ angleDomain Θ) :
+      polygonHeightValue (fun s ↦ h s - 1) s = polygonHeightValue h s - 1 := by
+    simp [polygonHeightValue, hs]
+  unfold independentWallNiche polygonHeightNiche polygonHeightFan
+  congr 1
+  · apply Set.iInter_congr
+    intro s
+    apply Set.iInter_congr
+    intro hs
+    rw [hv s (Or.inr hs)]
+  · apply Set.iUnion_congr
+    intro s
+    apply Set.iUnion_congr
+    intro hs
+    rw [hv s (Or.inl (Or.inl hs)),
+      hv (s + Real.pi / 2) (Or.inl (Or.inr ⟨s, hs, rfl⟩))]
+
+private theorem exists_polygonNiche_stable_presentation {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (hH : Set.range H = polygonNicheWalls h)
+    (hLines : Function.Injective (fun j ↦ (H j).boundaryLine)) :
+    ∃ (E : BooleanFunction n) (R ε₀ : ℝ),
+      IsMonotoneBooleanFunction E ∧ 0 < R ∧ 0 < ε₀ ∧
+      polygonHeightNiche h = booleanSet E (fun j ↦ (H j).carrier) ∧
+      (∀ i, ∃ t : angleDomain Θ, (H i).angle = (t.val : Real.Angle) ∧
+        ∀ δ : ℝ, perturbNefHeight E H i δ =
+          independentWallNiche (Function.update (fun s ↦ h s - 1) t (h t - 1 + δ))) ∧
+      ∀ i δ, |δ| ≤ ε₀ → perturbNefHeight E H i δ ⊆ Metric.closedBall 0 R := by
+  classical
+  obtain ⟨E, hE, htransport⟩ := exists_polygonNiche_perturbation_formula h H hH hLines
+  obtain ⟨R, ε₀, hR, hε₀, hb⟩ := polygonPerturbation_uniform_bounds Θ h
+  refine ⟨E, R, ε₀, hE, hR, hε₀, ?_, htransport, ?_⟩
+  · have hW : (⟨(Θ.angle : Real.Angle), polygonHeightValue h Θ.angle - 1,
+        true, false⟩ : PlanarHalfPlaneData) ∈ Set.range H := by
+      rw [hH]
+      exact Or.inr ⟨Θ.angle, by simp, rfl⟩
+    obtain ⟨i, _⟩ := hW
+    obtain ⟨t, _, ht⟩ := htransport i
+    have hz := ht 0
+    simpa [perturbNefHeight, independentWallNiche_sub_one] using hz.symm
+  · intro i δ hδ
+    obtain ⟨t, _, ht⟩ := htransport i
+    rw [ht δ]
+    apply (hb h _ (by intro s; simpa using le_of_lt hε₀) ?_).2
+    intro s
+    by_cases hst : s = t
+    · subst s
+      simpa [Function.update] using hδ
+    · simp [Function.update, hst, le_of_lt hε₀]
+
+end MovingSofa
+
+namespace MovingSofa
+
+/-- Quadratic area variation for any wall in a polygon-height niche presentation. -/
+theorem polygonNiche_wall_area_variation {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (hH : Set.range H = polygonNicheWalls h)
+    (hLines : Function.Injective (fun j ↦ (H j).boundaryLine)) (i : Fin n) :
+    ∃ (t : angleDomain Θ) (C η : ℝ), (H i).angle = (t.val : Real.Angle) ∧
+      0 ≤ C ∧ 0 < η ∧ ∀ δ : ℝ, |δ| ≤ η →
+        |ClassicalResults.area
+            (independentWallNiche (Function.update (fun s ↦ h s - 1) t (h t - 1 + δ))) -
+          ClassicalResults.area (polygonHeightNiche h) -
+          (if (H i).upper then -1 else 1) *
+            (MeasureTheory.Measure.hausdorffMeasure 1
+              (frontier (polygonHeightNiche h) ∩ (H i).boundaryLine)).toReal * δ| ≤
+          C * δ ^ 2 := by
+  classical
+  obtain ⟨E, R, ε₀, hE, hR, hε₀, hbase, htransport, hbound⟩ :=
+    exists_polygonNiche_stable_presentation h H hH hLines
+  obtain ⟨t, ht, hmove⟩ := htransport i
+  have hz : perturbNefHeight E H i 0 = polygonHeightNiche h := by
+    simpa [perturbNefHeight] using hbase.symm
+  obtain ⟨C, η, hC, hη, _, hvar⟩ := simpleNefPolygon_area_variation_signed
+    E H i hE hLines R ε₀ hR hε₀ (hbound i)
+  refine ⟨t, C, η, ht, hC, hη, ?_⟩
+  intro δ hδ
+  have hv := hvar δ hδ
+  rw [hz, hmove δ] at hv
+  exact hv
+
+end MovingSofa
+
+namespace MovingSofa
+
+/-- The cap with every lower endpoint wall shifted by one is the height-defined cap. -/
+theorem independentWallCap_sub_one {Θ : AngleSet} (h : PolygonHeightSpace Θ) :
+    independentWallCap h (fun s ↦ h s - 1) = polygonHeightCap h := by
+  unfold independentWallCap polygonHeightCap polygonHeightParallelogram
+  congr 1
+  apply Set.iInter_congr
+  intro s
+  apply Set.iInter_congr
+  intro hs
+  have hsD : s ∈ angleDomain Θ := Or.inr hs
+  simp [polygonHeightValue, hsD]
+
+/-- Quadratic area variation for one upper wall of a polygon-height cap. -/
+theorem polygonCap_upper_wall_area_variation {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (hH : Set.range H = polygonCapWalls h)
+    (hLines : Function.Injective (fun j ↦ (H j).boundaryLine))
+    (i : Fin n) (t : angleDomain Θ)
+    (hi : H i = ⟨(t.val : Real.Angle), h t, false, false⟩) :
+    ∃ C η : ℝ, 0 ≤ C ∧ 0 < η ∧ ∀ δ : ℝ, |δ| ≤ η →
+      |ClassicalResults.area
+          (independentWallCap (Function.update h t (h t + δ)) (fun s ↦ h s - 1)) -
+        ClassicalResults.area (polygonHeightCap h) -
+        (MeasureTheory.Measure.hausdorffMeasure 1
+          (frontier (polygonHeightCap h) ∩ (H i).boundaryLine)).toReal * δ| ≤
+        C * δ ^ 2 := by
+  classical
+  have hHinj : Function.Injective H := by
+    intro j k hjk
+    exact hLines (congrArg PlanarHalfPlaneData.boundaryLine hjk)
+  obtain ⟨R, ε₀, hR, hε₀, hbound⟩ := polygonCap_singleWall_uniform_bounds h
+  have hmove := perturbNefHeight_cap_upper h H hH hHinj i t hi
+  have hz : perturbNefHeight (BooleanFunction.all n) H i 0 = polygonHeightCap h := by
+    rw [hmove 0]
+    have hu : Function.update h t (h t + 0) = h := by
+      funext s
+      simp only [Function.update, add_zero, eq_rec_constant, dite_eq_ite, ite_eq_right_iff]
+      rintro rfl
+      rfl
+    rw [hu, independentWallCap_sub_one]
+  obtain ⟨C, η, hC, hη, _, hvar⟩ := simpleNefPolygon_area_variation
+    (BooleanFunction.all n) H i (BooleanFunction.all_monotone n) hLines
+    (by rw [hi]) R ε₀ hR hε₀ (hbound n H hH hHinj i t (Or.inl hi))
+  refine ⟨C, η, hC, hη, ?_⟩
+  intro δ hδ
+  have hv := hvar δ hδ
+  rw [hz, hmove δ] at hv
+  exact hv
+
+end MovingSofa
+
+namespace MovingSofa
+
+/-- Quadratic area variation for one lower endpoint wall of a polygon-height cap. -/
+theorem polygonCap_lower_wall_area_variation {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    {n : ℕ} (H : Fin n → PlanarHalfPlaneData)
+    (hH : Set.range H = polygonCapWalls h)
+    (hLines : Function.Injective (fun j ↦ (H j).boundaryLine))
+    (i : Fin n) (t : angleDomain Θ)
+    (hi : H i = ⟨(t.val : Real.Angle), h t - 1, true, false⟩) :
+    ∃ C η : ℝ, 0 ≤ C ∧ 0 < η ∧ ∀ δ : ℝ, |δ| ≤ η →
+      |ClassicalResults.area
+          (independentWallCap h (Function.update (fun s ↦ h s - 1) t (h t - 1 + δ))) -
+        ClassicalResults.area (polygonHeightCap h) +
+        (MeasureTheory.Measure.hausdorffMeasure 1
+          (frontier (polygonHeightCap h) ∩ (H i).boundaryLine)).toReal * δ| ≤
+        C * δ ^ 2 := by
+  classical
+  have hHinj : Function.Injective H := by
+    intro j k hjk
+    exact hLines (congrArg PlanarHalfPlaneData.boundaryLine hjk)
+  obtain ⟨R, ε₀, hR, hε₀, hbound⟩ := polygonCap_singleWall_uniform_bounds h
+  have hmove := perturbNefHeight_cap_lower h H hH hHinj i t hi
+  have hz : perturbNefHeight (BooleanFunction.all n) H i 0 = polygonHeightCap h := by
+    simpa [independentWallCap_sub_one] using hmove 0
+  obtain ⟨C, η, hC, hη, _, hvar⟩ := simpleNefPolygon_area_variation_lower
+    (BooleanFunction.all n) H i (BooleanFunction.all_monotone n) hLines
+    (by rw [hi]) R ε₀ hR hε₀ (hbound n H hH hHinj i t (Or.inr hi))
+  refine ⟨C, η, hC, hη, ?_⟩
+  intro δ hδ
+  have hv := hvar δ hδ
+  rw [hz, hmove δ] at hv
+  exact hv
+
+end MovingSofa
+
+namespace MovingSofa
+
+private theorem mem_independentWallCap_congr {Θ : AngleSet}
+    (upper upper' lower lower' : PolygonHeightSpace Θ) (p : Point)
+    (hu : ∀ s : angleDomain Θ,
+      (inner ℝ p (normalVector (s.val : Real.Angle)) ≤ upper s ↔
+        inner ℝ p (normalVector (s.val : Real.Angle)) ≤ upper' s))
+    (hl : ∀ s : angleDomain Θ,
+      (lower s ≤ inner ℝ p (normalVector (s.val : Real.Angle)) ↔
+        lower' s ≤ inner ℝ p (normalVector (s.val : Real.Angle)))) :
+    p ∈ independentWallCap upper lower ↔ p ∈ independentWallCap upper' lower' := by
+  have hu' (s : ℝ) (hs : s ∈ angleDomain Θ) :
+      p ∈ normalHalfPlane (s : Real.Angle) (polygonHeightValue upper s) false false ↔
+        p ∈ normalHalfPlane (s : Real.Angle) (polygonHeightValue upper' s) false false := by
+    simpa [normalHalfPlane, polygonHeightValue, hs] using hu ⟨s, hs⟩
+  have hl' (s : ℝ) (hs : s ∈ angleDomain Θ) :
+      p ∈ normalHalfPlane (s : Real.Angle) (polygonHeightValue lower s) true false ↔
+        p ∈ normalHalfPlane (s : Real.Angle) (polygonHeightValue lower' s) true false := by
+    simpa [normalHalfPlane, polygonHeightValue, hs] using hl ⟨s, hs⟩
+  simp only [independentWallCap, Set.mem_inter_iff, Set.mem_iInter]
+  constructor
+  · rintro ⟨he, hi⟩
+    exact ⟨fun s hs ↦ ⟨(hu' s (Or.inr hs)).mp (he s hs).1,
+      (hl' s (Or.inr hs)).mp (he s hs).2⟩,
+      fun s hs ↦ (hu' s (Or.inl hs)).mp (hi s hs)⟩
+  · rintro ⟨he, hi⟩
+    exact ⟨fun s hs ↦ ⟨(hu' s (Or.inr hs)).mpr (he s hs).1,
+      (hl' s (Or.inr hs)).mpr (he s hs).2⟩,
+      fun s hs ↦ (hu' s (Or.inl hs)).mpr (hi s hs)⟩
+
+private theorem independentWallCap_indicator_update_add {Θ : AngleSet}
+    (h : PolygonHeightSpace Θ) (t : angleDomain Θ) (ε : ℝ)
+    (hε : 0 ≤ ε) (hε' : ε ≤ 1) (p : Point) :
+    (independentWallCap (Function.update h t (h t + ε))
+        (Function.update (fun s ↦ h s - 1) t (h t - 1 + ε))).indicator
+        (fun _ ↦ (1 : ℝ)) p +
+      (independentWallCap h (fun s ↦ h s - 1)).indicator (fun _ ↦ (1 : ℝ)) p =
+      (independentWallCap (Function.update h t (h t + ε))
+        (fun s ↦ h s - 1)).indicator (fun _ ↦ (1 : ℝ)) p +
+      (independentWallCap h
+        (Function.update (fun s ↦ h s - 1) t (h t - 1 + ε))).indicator
+        (fun _ ↦ (1 : ℝ)) p := by
+  classical
+  have hsplit := endpoint_wall_changes_disjoint (t.val : Real.Angle) (h t) ε hε hε' p
+  have hI (A B : Set Point) (hab : p ∈ A ↔ p ∈ B) :
+      A.indicator (fun _ ↦ (1 : ℝ)) p = B.indicator (fun _ ↦ (1 : ℝ)) p := by
+    by_cases ha : p ∈ A
+    · rw [Set.indicator_of_mem ha, Set.indicator_of_mem (hab.mp ha)]
+    · rw [Set.indicator_of_notMem ha, Set.indicator_of_notMem (fun hb ↦ ha (hab.mpr hb))]
+  change (_ ↔ _) ∨ (_ ↔ _) at hsplit
+  rcases hsplit with hu | hl
+  · have hu' (s : angleDomain Θ) :
+        inner ℝ p (normalVector (s.val : Real.Angle)) ≤ Function.update h t (h t + ε) s ↔
+          inner ℝ p (normalVector (s.val : Real.Angle)) ≤ h s := by
+      by_cases hst : s = t
+      · subst s
+        simpa [Function.update, normalHalfPlane] using hu
+      · simp [Function.update, hst]
+    have heq (lower : PolygonHeightSpace Θ) :=
+      mem_independentWallCap_congr (Function.update h t (h t + ε)) h lower lower p
+        hu' (fun _ ↦ Iff.rfl)
+    exact (congrArg₂ (fun a b : ℝ ↦ a + b)
+      (hI _ _ (heq (Function.update (fun s ↦ h s - 1) t (h t - 1 + ε)))) rfl).trans
+      ((add_comm _ _).trans (congrArg₂ (fun a b : ℝ ↦ a + b)
+        (hI _ _ (heq (fun s ↦ h s - 1))).symm rfl))
+  · have hl' (s : angleDomain Θ) :
+        Function.update (fun s ↦ h s - 1) t (h t - 1 + ε) s ≤
+            inner ℝ p (normalVector (s.val : Real.Angle)) ↔
+          h s - 1 ≤ inner ℝ p (normalVector (s.val : Real.Angle)) := by
+      by_cases hst : s = t
+      · subst s
+        simpa [Function.update, normalHalfPlane] using hl
+      · simp [Function.update, hst]
+    have heq (upper : PolygonHeightSpace Θ) :=
+      mem_independentWallCap_congr upper upper
+        (Function.update (fun s ↦ h s - 1) t (h t - 1 + ε)) (fun s ↦ h s - 1) p
+        (fun _ ↦ Iff.rfl) hl'
+    exact congrArg₂ (fun a b : ℝ ↦ a + b)
+      (hI _ _ (heq (Function.update h t (h t + ε)))) (hI _ _ (heq h)).symm
+
+end MovingSofa
+
+namespace MovingSofa
+
+private theorem isClosed_independentWallCap {Θ : AngleSet} (upper lower : PolygonHeightSpace Θ) :
+    IsClosed (independentWallCap upper lower) := by
+  have hu (a : Real.Angle) (c : ℝ) : IsClosed (normalHalfPlane a c false false) :=
+    isClosed_le (by fun_prop) continuous_const
+  have hl (a : Real.Angle) (c : ℝ) : IsClosed (normalHalfPlane a c true false) :=
+    isClosed_le continuous_const (by fun_prop)
+  unfold independentWallCap
+  exact (isClosed_iInter fun s ↦ isClosed_iInter fun _ ↦ (hu _ _).inter (hl _ _)).inter
+    (isClosed_iInter fun s ↦ isClosed_iInter fun _ ↦ hu _ _)
+
+/-- Simultaneous endpoint-wall variation is the sum of the two independent variations. -/
+theorem independentWallCap_area_update_add {Θ : AngleSet}
+    (h : PolygonHeightSpace Θ) (t : angleDomain Θ) (ε R : ℝ)
+    (hε : 0 ≤ ε) (hε' : ε ≤ 1)
+    (hbound : ∀ (upper lower : PolygonHeightSpace Θ),
+      (upper = h ∨ upper = Function.update h t (h t + ε)) →
+      (lower = (fun s ↦ h s - 1) ∨
+        lower = Function.update (fun s ↦ h s - 1) t (h t - 1 + ε)) →
+      independentWallCap upper lower ⊆ Metric.closedBall 0 R) :
+    ClassicalResults.area (independentWallCap (Function.update h t (h t + ε))
+        (Function.update (fun s ↦ h s - 1) t (h t - 1 + ε))) +
+      ClassicalResults.area (independentWallCap h (fun s ↦ h s - 1)) =
+      ClassicalResults.area (independentWallCap (Function.update h t (h t + ε))
+        (fun s ↦ h s - 1)) +
+      ClassicalResults.area (independentWallCap h
+        (Function.update (fun s ↦ h s - 1) t (h t - 1 + ε))) := by
+  have hf (upper lower : PolygonHeightSpace Θ)
+      (hu : upper = h ∨ upper = Function.update h t (h t + ε))
+      (hl : lower = (fun s ↦ h s - 1) ∨
+        lower = Function.update (fun s ↦ h s - 1) t (h t - 1 + ε)) :
+      MeasureTheory.volume (independentWallCap upper lower) ≠ ⊤ :=
+    ne_of_lt (lt_of_le_of_lt (MeasureTheory.measure_mono (hbound upper lower hu hl))
+      MeasureTheory.measure_closedBall_lt_top)
+  apply area_add_of_indicator_add_eq
+  · exact (isClosed_independentWallCap _ _).measurableSet
+  · exact (isClosed_independentWallCap _ _).measurableSet
+  · exact (isClosed_independentWallCap _ _).measurableSet
+  · exact (isClosed_independentWallCap _ _).measurableSet
+  · exact hf _ _ (Or.inr rfl) (Or.inr rfl)
+  · exact hf _ _ (Or.inl rfl) (Or.inl rfl)
+  · exact hf _ _ (Or.inr rfl) (Or.inl rfl)
+  · exact hf _ _ (Or.inl rfl) (Or.inr rfl)
+  · exact independentWallCap_indicator_update_add h t ε hε hε'
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Right Angle Grid
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Set MeasureTheory
+
+namespace MovingSofa
+
+/-- Directions in the right-angle grid are positive integer multiples of its step size. -/
+theorem mem_rightAngleSet_directions_iff (n : ℕ) (hn : 2 ≤ n) (t : ℝ) :
+    t ∈ (rightAngleSet n hn).directions ↔
+      ∃ i : ℕ, i ∈ Finset.Ioo 0 n ∧ t = i * polygonStepSize n := by
+  simp only [rightAngleSet, uniformAngleSet, Finset.mem_image]
+  constructor
+  · rintro ⟨i, hi, rfl⟩
+    refine ⟨i, hi, ?_⟩
+    simp only [polygonStepSize]
+    ring
+  · rintro ⟨i, hi, rfl⟩
+    refine ⟨i, hi, ?_⟩
+    simp only [polygonStepSize]
+    ring
+
+/-- No grid direction lies strictly between the predecessor of a grid point and the point. -/
+theorem rightAngleSet_direction_le_pred_or_ge (n : ℕ) (hn : 2 ≤ n) {r t : ℝ}
+    (hr : r ∈ (rightAngleSet n hn).directions)
+    (ht : t ∈ (rightAngleSet n hn).directions) :
+    r ≤ t - polygonStepSize n ∨ t ≤ r := by
+  obtain ⟨j, hj, rfl⟩ := (mem_rightAngleSet_directions_iff n hn r).mp hr
+  obtain ⟨i, hi, rfl⟩ := (mem_rightAngleSet_directions_iff n hn t).mp ht
+  have hnpos : (0 : ℝ) < n := by positivity
+  have hδ : 0 < polygonStepSize n := by simp only [polygonStepSize]; positivity
+  by_cases hji : j < i
+  · left
+    have hnat : j + 1 ≤ i := hji
+    have hcast : (j : ℝ) + 1 ≤ (i : ℝ) := by exact_mod_cast hnat
+    nlinarith
+  · right
+    have hij : i ≤ j := Nat.le_of_not_gt hji
+    exact mul_le_mul_of_nonneg_right (by exact_mod_cast hij) hδ.le
+
+/-- No grid direction lies strictly between a grid point and its successor. -/
+theorem rightAngleSet_direction_le_or_succ_le (n : ℕ) (hn : 2 ≤ n) {r t : ℝ}
+    (hr : r ∈ (rightAngleSet n hn).directions)
+    (ht : t ∈ (rightAngleSet n hn).directions) :
+    r ≤ t ∨ t + polygonStepSize n ≤ r := by
+  obtain ⟨j, hj, rfl⟩ := (mem_rightAngleSet_directions_iff n hn r).mp hr
+  obtain ⟨i, hi, rfl⟩ := (mem_rightAngleSet_directions_iff n hn t).mp ht
+  have hδ : 0 < polygonStepSize n := by simp only [polygonStepSize]; positivity
+  by_cases hji : j ≤ i
+  · left
+    exact mul_le_mul_of_nonneg_right (by exact_mod_cast hji) hδ.le
+  · right
+    have hcast : (i : ℝ) + 1 ≤ (j : ℝ) := by
+      exact_mod_cast (Nat.add_one_le_iff.mpr (Nat.lt_of_not_ge hji))
+    nlinarith
+
+/-- Every interior grid direction stays at least one step from both endpoints. -/
+theorem rightAngleSet_direction_bounds (n : ℕ) (hn : 2 ≤ n) {t : ℝ}
+    (ht : t ∈ (rightAngleSet n hn).directions) :
+    polygonStepSize n ≤ t ∧ t ≤ Real.pi / 2 - polygonStepSize n := by
+  obtain ⟨i, hi, rfl⟩ := (mem_rightAngleSet_directions_iff n hn t).mp ht
+  obtain ⟨hi0, hin⟩ := Finset.mem_Ioo.mp hi
+  have hnpos : (0 : ℝ) < n := by positivity
+  have hδ : 0 < polygonStepSize n := by simp only [polygonStepSize]; positivity
+  have hi1 : (1 : ℝ) ≤ i := by exact_mod_cast hi0
+  have hin1 : (i : ℝ) + 1 ≤ n := by exact_mod_cast hin
+  constructor
+  · simpa only [one_mul] using mul_le_mul_of_nonneg_right hi1 hδ.le
+  · have hstep : (n : ℝ) * polygonStepSize n = Real.pi / 2 := by
+      simp only [polygonStepSize]
+      field_simp
+    nlinarith
+
+/-- The two lower normals of a right-angle polygon cap coincide at `3π/2`. -/
+theorem rightAngle_polygon_normals_eq (n : ℕ) (hn : 2 ≤ n) :
+    ((fun r : ℝ ↦ (r : Real.Angle)) '' angleDomain (rightAngleSet n hn)) ∪
+        capLowerNormals (rightAngleSet n hn).angle =
+      (fun r : ℝ ↦ (r : Real.Angle)) ''
+        (angleDomain (rightAngleSet n hn) ∪ {3 * Real.pi / 2}) := by
+  ext a
+  simp only [Set.mem_union, Set.mem_image, Set.mem_singleton_iff]
+  constructor
+  · rintro (⟨r, hr, rfl⟩ | ha)
+    · exact ⟨r, Or.inl hr, rfl⟩
+    · simp only [capLowerNormals, rightAngleSet, uniformAngleSet,
+        Set.mem_insert_iff] at ha
+      rcases ha with ha | ha
+      · subst a
+        refine ⟨3 * Real.pi / 2, Or.inr rfl, ?_⟩
+        congr 1
+        ring
+      · subst a
+        exact ⟨3 * Real.pi / 2, Or.inr rfl, rfl⟩
+  · rintro ⟨r, hr | rfl, rfl⟩
+    · exact Or.inl ⟨r, hr, rfl⟩
+    · right
+      simp [capLowerNormals, rightAngleSet, uniformAngleSet]
+
+/-- The predecessor of a right-angle grid direction is zero or again a grid direction. -/
+theorem rightAngleSet_sub_step (n : ℕ) (hn : 2 ≤ n) {t : ℝ}
+    (ht : t ∈ (rightAngleSet n hn).directions) :
+    t - polygonStepSize n = 0 ∨
+      t - polygonStepSize n ∈ (rightAngleSet n hn).directions := by
+  obtain ⟨i, hi, rfl⟩ := (mem_rightAngleSet_directions_iff n hn t).mp ht
+  obtain ⟨hi0, hin⟩ := Finset.mem_Ioo.mp hi
+  by_cases hi1 : i = 1
+  · left
+    subst hi1
+    push_cast
+    ring
+  · right
+    refine (mem_rightAngleSet_directions_iff n hn _).mpr ⟨i - 1, Finset.mem_Ioo.mpr ⟨?_, ?_⟩, ?_⟩
+    · omega
+    · omega
+    · rw [Nat.cast_sub (by omega)]
+      push_cast
+      ring
+
+/-- The successor of a right-angle grid direction is the terminal angle or a grid direction. -/
+theorem rightAngleSet_add_step (n : ℕ) (hn : 2 ≤ n) {t : ℝ}
+    (ht : t ∈ (rightAngleSet n hn).directions) :
+    t + polygonStepSize n = Real.pi / 2 ∨
+      t + polygonStepSize n ∈ (rightAngleSet n hn).directions := by
+  obtain ⟨i, hi, rfl⟩ := (mem_rightAngleSet_directions_iff n hn t).mp ht
+  obtain ⟨hi0, hin⟩ := Finset.mem_Ioo.mp hi
+  have hn0 : (n : ℝ) ≠ 0 := by positivity
+  by_cases hi1 : i + 1 = n
+  · left
+    have : ((i : ℝ) + 1) = (n : ℝ) := by exact_mod_cast congrArg (fun m : ℕ ↦ (m : ℝ)) hi1
+    simp only [polygonStepSize]
+    field_simp
+    linarith [this]
+  · right
+    refine (mem_rightAngleSet_directions_iff n hn _).mpr ⟨i + 1, Finset.mem_Ioo.mpr ⟨?_, ?_⟩, ?_⟩
+    · omega
+    · omega
+    · push_cast
+      ring
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Translation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem polygonCapTranslate_iff (Θ : AngleSet) (K : ConvexBody Point) :
+    (∃ K' : PolygonCapTranslateSpace Θ, K'.val = (K : Set Point)) ↔
+      (supportValue K (Θ.angle : Real.Angle) +
+        supportValue K ((Θ.angle + Real.pi : ℝ) : Real.Angle) = 1) ∧
+      (supportValue K ((Real.pi / 2 : ℝ) : Real.Angle) +
+        supportValue K ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 1) ∧
+      ∃ constraints : Set (Real.Angle × ℝ), constraints.Finite ∧
+        (∀ c ∈ constraints, c.1 ∈
+          ((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain Θ) ∪ capLowerNormals Θ.angle) ∧
+        (K : Set Point) = ⋂ c ∈ constraints, normalHalfPlane c.1 c.2 false false := by
+  classical
+  let N : Set Real.Angle :=
+    ((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain Θ) ∪ capLowerNormals Θ.angle
+  have hNfinite : N.Finite := by
+    apply Set.Finite.union
+    · apply Set.Finite.image
+      simp only [angleDomain]
+      exact ((Θ.directions.finite_toSet.union
+        (Θ.directions.finite_toSet.image (fun t : ℝ ↦ t + Real.pi / 2))).union
+          ((Set.finite_singleton (Real.pi / 2)).insert Θ.angle))
+    · simp [capLowerNormals]
+  constructor
+  · rintro ⟨K', hK'eq⟩
+    obtain ⟨P, q, hPq⟩ := K'.property
+    have hKeq : (K : Set Point) = (fun p ↦ p + q) '' (P.val.val : Set Point) := by
+      rw [← hK'eq, hPq]
+    have hsupp (t : Real.Angle) :
+        supportValue K t = supportValue P.val.val t + inner ℝ q (normalVector t) := by
+      rw [show (K : Set Point) = (ConvexBody.translate P.val.val q : Set Point) from hKeq]
+      exact supportValue_image_add P.val.val q t
+    have hopen := P.val.property
+    refine ⟨?_, ?_, ?_⟩
+    · rw [hsupp, hsupp, hopen.2.2.1, hopen.2.2.2.2.1, normalVector_add_pi]
+      simp only [inner_neg_right]
+      ring
+    · have hang : (((Real.pi / 2 + Real.pi : ℝ) : Real.Angle)) =
+          ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+        congr 1
+        ring
+      rw [hsupp]
+      rw [← hang]
+      rw [hsupp, hopen.2.2.2.1]
+      rw [show supportValue P.val.val (((Real.pi / 2 + Real.pi : ℝ) : Real.Angle)) = 0 by
+        rw [hang]; exact hopen.2.2.2.2.2.1]
+      rw [normalVector_add_pi]
+      simp only [inner_neg_right]
+      ring
+    · have hrepr : HasHalfPlaneRepresentation K N := by
+        rw [show (K : Set Point) = (ConvexBody.translate P.val.val q : Set Point) from hKeq]
+        exact P.property.translate q
+      exact hrepr.finite_constraints hNfinite
+  · rintro ⟨hwidthω, hwidthT, C, hCfinite, hCN, hKC⟩
+    let v : Point := if h : Θ.angle = Real.pi / 2 then
+      !₂[0, 1 - supportValue K ((Real.pi / 2 : ℝ) : Real.Angle)]
+    else
+      !₂[(1 - supportValue K (Θ.angle : Real.Angle) -
+          (1 - supportValue K ((Real.pi / 2 : ℝ) : Real.Angle)) * Real.sin Θ.angle) /
+          Real.cos Θ.angle,
+        1 - supportValue K ((Real.pi / 2 : ℝ) : Real.Angle)]
+    have hvT : inner ℝ v (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) =
+        1 - supportValue K ((Real.pi / 2 : ℝ) : Real.Angle) := by
+      by_cases h : Θ.angle = Real.pi / 2
+      · simp [v, h, normalVector, frame, PiLp.inner_apply]
+      · simp [v, h, normalVector, frame, PiLp.inner_apply]
+    have hvω : inner ℝ v (normalVector (Θ.angle : Real.Angle)) =
+        1 - supportValue K (Θ.angle : Real.Angle) := by
+      by_cases h : Θ.angle = Real.pi / 2
+      · simpa [h] using hvT
+      · have hlt : Θ.angle < Real.pi / 2 := lt_of_le_of_ne Θ.angle_le h
+        have hcos : Real.cos Θ.angle ≠ 0 := (Real.cos_pos_of_mem_Ioo
+          ⟨lt_trans (neg_neg_of_pos (by positivity : 0 < Real.pi / 2)) Θ.angle_pos,
+            hlt⟩).ne'
+        simp [v, h, normalVector, frame, PiLp.inner_apply]
+        field_simp [hcos]
+        ring
+    let L := ConvexBody.translate K v
+    have hsupp (t : Real.Angle) :
+        supportValue L t = supportValue K t + inner ℝ v (normalVector t) :=
+      supportValue_image_add K v t
+    have htopω : supportValue L (Θ.angle : Real.Angle) = 1 := by
+      rw [hsupp, hvω]
+      ring
+    have htopT : supportValue L ((Real.pi / 2 : ℝ) : Real.Angle) = 1 := by
+      rw [hsupp, hvT]
+      ring
+    have hbotω : supportValue L ((Θ.angle + Real.pi : ℝ) : Real.Angle) = 0 := by
+      rw [hsupp, normalVector_add_pi, inner_neg_right]
+      rw [hvω]
+      linarith
+    have hang : (((Real.pi / 2 + Real.pi : ℝ) : Real.Angle)) =
+        ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+      congr 1
+      ring
+    have hbotT : supportValue L ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := by
+      rw [← hang, hsupp, normalVector_add_pi, inner_neg_right]
+      rw [hang, hvT]
+      linarith
+    have hKrepr : HasHalfPlaneRepresentation K N := ⟨C, hCN, hKC⟩
+    have hLrepr : HasHalfPlaneRepresentation L N := hKrepr.translate v
+    have hangleDomain : angleDomain Θ ⊆ capUpperAngles Θ.angle := by
+      rintro t ((ht | ⟨s, hs, rfl⟩) | ht)
+      · left
+        exact ⟨(Θ.interior t ht).1.le, (Θ.interior t ht).2.le⟩
+      · right
+        constructor <;> linarith [(Θ.interior s hs).1, (Θ.interior s hs).2]
+      · rcases ht with (rfl | rfl)
+        · exact Or.inl ⟨Θ.angle_pos.le, le_rfl⟩
+        · exact Or.inr ⟨le_rfl, le_add_of_nonneg_left Θ.angle_pos.le⟩
+    have hNsubset : N ⊆
+        ((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles Θ.angle) ∪
+          capLowerNormals Θ.angle := by
+      rintro t (ht | ht)
+      · obtain ⟨s, hs, rfl⟩ := ht
+        exact Or.inl ⟨s, hangleDomain hs, rfl⟩
+      · exact Or.inr ht
+    have hLcapRepr : HasHalfPlaneRepresentation L
+        (((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles Θ.angle) ∪
+          capLowerNormals Θ.angle) := by
+      obtain ⟨D, hDN, hLD⟩ := hLrepr
+      exact ⟨D, fun c hc ↦ hNsubset (hDN c hc), hLD⟩
+    have hLcap : IsCap Θ.angle L :=
+      ⟨Θ.angle_pos, Θ.angle_le, htopω, htopT, hbotω, hbotT, hLcapRepr⟩
+    let P : PolygonCapSpace Θ := ⟨⟨L, hLcap⟩, hLrepr⟩
+    refine ⟨⟨(K : Set Point), ?_⟩, rfl⟩
+    refine ⟨P, -v, ?_⟩
+    ext x
+    constructor
+    · intro hx
+      exact ⟨x + v, ⟨x, hx, rfl⟩, by simp⟩
+    · rintro ⟨_, ⟨y, hy, rfl⟩, rfl⟩
+      simpa using hy
+
+theorem polygonHeightArea_le_translateArea {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    (K : PolygonCapTranslateSpace Θ) (hK : polygonHeightCap h = K.val) :
+    polygonHeightArea h ≤ (polygonTranslateExtensions K).2 := by
+  obtain ⟨P, q, hPq⟩ := K.property
+  let C := ConvexBody.translate P.val.val q
+  have hC : (C : Set Point) = K.val := hPq.symm
+  have hBody : polygonHeightCap h = (C : Set Point) := hK.trans hC.symm
+  have hw := (polygonCapTranslate_iff Θ C).1 ⟨K, hC.symm⟩
+  let g := polygonTranslateHeight K
+  have hle : ∀ t, g t ≤ h t := by
+    intro t
+    change supportValue K.val (t.val : Real.Angle) ≤ h t
+    rw [← hC]
+    simpa only [polygonHeightValue, dite_eq_left t.property] using
+      supportValue_le_polygonHeightValue h C hBody t.property
+  have heq : ∀ t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+      polygonHeightValue g t = polygonHeightValue h t := by
+    intro t ht
+    have htD : t ∈ angleDomain Θ := Or.inr ht
+    have hg : polygonHeightValue g t = supportValue C (t : Real.Angle) := by
+      simp only [g, polygonTranslateHeight, polygonHeightValue, dite_eq_left htD]
+      rw [hC]
+    rw [hg]
+    apply supportValue_eq_polygonHeightValue_of_width_one h C hBody ht
+    rcases ht with rfl | ht
+    · exact hw.1
+    · have ht : t = Real.pi / 2 := ht
+      subst t
+      simpa only [show Real.pi / 2 + Real.pi = 3 * Real.pi / 2 by ring] using hw.2.1
+  have hsub := polygonHeightNiche_mono_of_eq_endpoints hle heq
+  have harea : ClassicalResults.area (polygonHeightNiche g) ≤
+      ClassicalResults.area (polygonHeightNiche h) := by
+    apply ENNReal.toReal_mono (isBounded_polygonHeightNiche h).measure_lt_top.ne
+    exact MeasureTheory.measure_mono hsub
+  change ClassicalResults.area (polygonHeightCap h) -
+      ClassicalResults.area (polygonHeightNiche h) ≤
+    ClassicalResults.area (polygonHeightCap g) - ClassicalResults.area (polygonHeightNiche g)
+  rw [hK, polygonHeightCap_of_translate K]
+  exact sub_le_sub_left harea _
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Height / Reconstruction
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private theorem isClosed_polygonHeightCap {Θ : AngleSet} (h : PolygonHeightSpace Θ) :
+    IsClosed (polygonHeightCap h) := by
+  unfold polygonHeightCap polygonHeightParallelogram
+  apply IsClosed.inter
+  · exact isClosed_iInter fun t ↦ isClosed_iInter fun _ ↦
+      (isClosed_normalHalfPlane _ _ false).inter (isClosed_normalHalfPlane _ _ true)
+  · exact isClosed_iInter fun t ↦ isClosed_iInter fun _ ↦
+      isClosed_normalHalfPlane _ _ false
+
+private theorem convex_polygonHeightCap {Θ : AngleSet} (h : PolygonHeightSpace Θ) :
+    Convex ℝ (polygonHeightCap h) := by
+  unfold polygonHeightCap polygonHeightParallelogram
+  apply Convex.inter
+  · exact convex_iInter fun t ↦ convex_iInter fun _ ↦
+      (convex_normalHalfPlane _ _ false).inter (convex_normalHalfPlane _ _ true)
+  · exact convex_iInter fun t ↦ convex_iInter fun _ ↦
+      convex_normalHalfPlane _ _ false
+
+private theorem normalHalfPlane_upper_eq_lower_add_pi (t c : ℝ) :
+    normalHalfPlane (t : Real.Angle) c true false =
+      normalHalfPlane ((t + Real.pi : ℝ) : Real.Angle) (-c) false false := by
+  ext p
+  change c ≤ inner ℝ p (normalVector (t : Real.Angle)) ↔
+    inner ℝ p (normalVector ((t + Real.pi : ℝ) : Real.Angle)) ≤ -c
+  rw [normalVector_add_pi, inner_neg_right]
+  constructor <;> intro h <;> linarith
+
+private theorem polygonHeightCap_halfPlaneRepresentation {Θ : AngleSet}
+    (h : PolygonHeightSpace Θ) :
+    HasHalfPlaneRepresentation (polygonHeightCap h)
+      (((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain Θ) ∪ capLowerNormals Θ.angle) := by
+  let C : Set (Real.Angle × ℝ) :=
+    ((fun t : ℝ ↦ ((t : Real.Angle), polygonHeightValue h t)) '' angleDomain Θ) ∪
+      ((fun t : ℝ ↦ (((t + Real.pi : ℝ) : Real.Angle),
+        -(polygonHeightValue h t - 1))) '' ({Θ.angle, Real.pi / 2} : Set ℝ))
+  refine ⟨C, ?_, ?_⟩
+  · rintro c (⟨t, ht, rfl⟩ | ⟨t, ht, rfl⟩)
+    · exact Or.inl ⟨t, ht, rfl⟩
+    · right
+      rcases ht with rfl | rfl
+      · exact Or.inl rfl
+      · change (((Real.pi / 2 + Real.pi : ℝ) : Real.Angle)) ∈ capLowerNormals Θ.angle
+        rw [show Real.pi / 2 + Real.pi = 3 * Real.pi / 2 by ring]
+        simp [capLowerNormals]
+  · ext p
+    simp only [polygonHeightCap, polygonHeightParallelogram, Set.mem_inter_iff,
+      Set.mem_iInter, C, Set.mem_union, Set.mem_image]
+    constructor
+    · rintro ⟨hendpoint, hinterior⟩ c (⟨t, ht, rfl⟩ | ⟨t, ht, rfl⟩)
+      · rcases ht with (ht | ht) | ht
+        · exact (hinterior t (Or.inl ht))
+        · obtain ⟨s, hs, rfl⟩ := ht
+          exact hinterior (s + Real.pi / 2) (Or.inr ⟨s, hs, rfl⟩)
+        · exact (hendpoint t ht).1
+      · rw [← normalHalfPlane_upper_eq_lower_add_pi]
+        exact (hendpoint t ht).2
+    · intro hall
+      constructor
+      · intro t ht
+        constructor
+        · exact hall ((t : Real.Angle), polygonHeightValue h t)
+            (Or.inl ⟨t, Or.inr ht, rfl⟩)
+        · rw [normalHalfPlane_upper_eq_lower_add_pi]
+          exact hall (((t + Real.pi : ℝ) : Real.Angle), -(polygonHeightValue h t - 1))
+            (Or.inr ⟨t, ht, rfl⟩)
+      · intro t ht
+        exact hall ((t : Real.Angle), polygonHeightValue h t)
+          (Or.inl ⟨t, Or.inl ht, rfl⟩)
+
+private theorem polygonHeightCap_upper_bound {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    {p : Point} (hp : p ∈ polygonHeightCap h) {t : ℝ} (ht : t ∈ angleDomain Θ) :
+    inner ℝ p (normalVector (t : Real.Angle)) ≤ polygonHeightValue h t := by
+  exact polygonHeightCap_subset_normalHalfPlane h ht hp
+
+private theorem polygonHeightCap_lower_bound {Θ : AngleSet} (h : PolygonHeightSpace Θ)
+    {p : Point} (hp : p ∈ polygonHeightCap h) {t : ℝ}
+    (ht : t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    polygonHeightValue h t - 1 ≤ inner ℝ p (normalVector (t : Real.Angle)) := by
+  change p ∈ polygonHeightParallelogram h ∩ _ at hp
+  exact (Set.mem_iInter.mp (Set.mem_iInter.mp hp.1 t) ht).2
+
+/-- Attainment of both endpoint strip bounds reconstructs a translated polygon cap. -/
+theorem exists_polygonCapTranslate_eq_polygonHeightCap {Θ : AngleSet}
+    (h : PolygonHeightSpace Θ)
+    (hbounded : Bornology.IsBounded (polygonHeightCap h))
+    (hupper : ∀ t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+      ∃ p ∈ polygonHeightCap h,
+        inner ℝ p (normalVector (t : Real.Angle)) = polygonHeightValue h t)
+    (hlower : ∀ t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+      ∃ p ∈ polygonHeightCap h,
+        inner ℝ p (normalVector (t : Real.Angle)) = polygonHeightValue h t - 1) :
+    ∃ K' : PolygonCapTranslateSpace Θ, K'.val = polygonHeightCap h := by
+  obtain ⟨p, hp, _⟩ := hupper (Real.pi / 2) (by simp)
+  let L : ConvexBody Point := {
+    carrier := polygonHeightCap h
+    convex' := convex_polygonHeightCap h
+    isCompact' := Metric.isCompact_iff_isClosed_bounded.mpr
+      ⟨isClosed_polygonHeightCap h, hbounded⟩
+    nonempty' := ⟨p, hp⟩ }
+  have hsuppUpper (t : ℝ) (ht : t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+      supportValue L (t : Real.Angle) = polygonHeightValue h t := by
+    obtain ⟨q, hq, hqeq⟩ := hupper t ht
+    apply le_antisymm
+    · apply supportValue_le_of_subset_normalHalfPlane
+      intro x hx
+      exact polygonHeightCap_upper_bound h hx (Or.inr ht)
+    · simpa only [hqeq] using inner_le_supportValue L hq (t : Real.Angle)
+  have hsuppLower (t : ℝ) (ht : t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+      supportValue L ((t + Real.pi : ℝ) : Real.Angle) =
+        -(polygonHeightValue h t - 1) := by
+    obtain ⟨q, hq, hqeq⟩ := hlower t ht
+    apply le_antisymm
+    · apply supportValue_le_of_subset_normalHalfPlane
+      intro x hx
+      change inner ℝ x (normalVector ((t + Real.pi : ℝ) : Real.Angle)) ≤
+        -(polygonHeightValue h t - 1)
+      rw [normalVector_add_pi, inner_neg_right]
+      exact neg_le_neg (polygonHeightCap_lower_bound h hx ht)
+    · have hle := inner_le_supportValue L hq ((t + Real.pi : ℝ) : Real.Angle)
+      rw [normalVector_add_pi, inner_neg_right, hqeq] at hle
+      exact hle
+  have hwidthω : supportValue L (Θ.angle : Real.Angle) +
+      supportValue L ((Θ.angle + Real.pi : ℝ) : Real.Angle) = 1 := by
+    rw [hsuppUpper Θ.angle (by simp), hsuppLower Θ.angle (by simp)]
+    ring
+  have hangle : (((Real.pi / 2 + Real.pi : ℝ) : Real.Angle)) =
+      ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+    congr 1
+    ring
+  have hwidthT : supportValue L ((Real.pi / 2 : ℝ) : Real.Angle) +
+      supportValue L ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 1 := by
+    rw [← hangle, hsuppUpper (Real.pi / 2) (by simp),
+      hsuppLower (Real.pi / 2) (by simp)]
+    ring
+  have hdomain : (angleDomain Θ).Finite := by
+    unfold angleDomain
+    exact (Θ.directions.finite_toSet.union
+      (Θ.directions.finite_toSet.image (fun t : ℝ ↦ t + Real.pi / 2))).union
+        (Set.finite_singleton (Real.pi / 2) |>.insert Θ.angle)
+  have hlowerNormals : (capLowerNormals Θ.angle).Finite := by
+    simp [capLowerNormals]
+  have hrepr : HasHalfPlaneRepresentation (L : Set Point)
+      (((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain Θ) ∪ capLowerNormals Θ.angle) := by
+    exact polygonHeightCap_halfPlaneRepresentation h
+  obtain ⟨C, hCfinite, hCN, hLC⟩ :=
+    hrepr.finite_constraints
+      (hdomain.image _ |>.union hlowerNormals)
+  exact (polygonCapTranslate_iff Θ L).mpr
+    ⟨hwidthω, hwidthT, C, hCfinite, hCN, hLC⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Height / Positive Increment / Contacts
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private theorem polygonHeightValue_raisedPolygonSupport {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (t : angleDomain Θ) (ε : ℝ) {s : ℝ}
+    (hs : s ∈ angleDomain Θ) :
+    polygonHeightValue (raisedPolygonSupport K t ε) s =
+      supportValue K.val.val (s : Real.Angle) +
+        if (⟨s, hs⟩ : angleDomain Θ) = t then ε else 0 := by
+  simp [polygonHeightValue, raisedPolygonSupport, hs]
+
+private theorem mem_polygonHeightCap_raised_of_mem_of_not_endpoint {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (t : angleDomain Θ) {ε : ℝ} (hε : 0 ≤ ε)
+    (ht : t.val ∉ ({Θ.angle, Real.pi / 2} : Set ℝ)) {p : Point}
+    (hp : p ∈ (K.val.val : Set Point)) :
+    p ∈ polygonHeightCap (raisedPolygonSupport K t ε) := by
+  have hupper (s : ℝ) (hs : s ∈ angleDomain Θ) :
+      inner ℝ p (normalVector (s : Real.Angle)) ≤
+        polygonHeightValue (raisedPolygonSupport K t ε) s := by
+    rw [polygonHeightValue_raisedPolygonSupport K t ε hs]
+    exact (inner_le_supportValue K.val.val hp _).trans
+      (le_add_of_nonneg_right (by split_ifs <;> positivity))
+  unfold polygonHeightCap polygonHeightParallelogram
+  simp only [Set.mem_inter_iff, Set.mem_iInter]
+  constructor
+  · intro s hs
+    constructor
+    · exact hupper s (Or.inr hs)
+    · change polygonHeightValue (raisedPolygonSupport K t ε) s - 1 ≤
+        inner ℝ p (normalVector (s : Real.Angle))
+      rw [polygonHeightValue_raisedPolygonSupport K t ε (Or.inr hs)]
+      have hne : (⟨s, Or.inr hs⟩ : angleDomain Θ) ≠ t := by
+        intro heq
+        apply ht
+        have hval : s = t.val := congrArg Subtype.val heq
+        simpa [← hval] using hs
+      rw [ite_eq_right hne]
+      simp only [add_zero]
+      rcases hs with rfl | rfl
+      · have hlower := inner_le_supportValue K.val.val hp
+          ((Θ.angle + Real.pi : ℝ) : Real.Angle)
+        rw [K.val.property.2.2.2.2.1, normalVector_add_pi, inner_neg_right] at hlower
+        rw [K.val.property.2.2.1]
+        linarith
+      · have hlower := inner_le_supportValue K.val.val hp
+          ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+        rw [K.val.property.2.2.2.2.2.1, inner_normalVector_three_pi_div_two] at hlower
+        rw [K.val.property.2.2.2.1]
+        simpa [normalVector, frame, PiLp.inner_apply] using hlower
+  · intro s hs
+    exact hupper s (Or.inl hs)
+
+private theorem raisedPolygonSupport_endpoint_contacts_of_not_endpoint {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (t : angleDomain Θ) {ε : ℝ} (hε : 0 ≤ ε)
+    (ht : t.val ∉ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    (∀ s ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+      ∃ p ∈ polygonHeightCap (raisedPolygonSupport K t ε),
+        inner ℝ p (normalVector (s : Real.Angle)) =
+          polygonHeightValue (raisedPolygonSupport K t ε) s) ∧
+    (∀ s ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+      ∃ p ∈ polygonHeightCap (raisedPolygonSupport K t ε),
+        inner ℝ p (normalVector (s : Real.Angle)) =
+          polygonHeightValue (raisedPolygonSupport K t ε) s - 1) := by
+  have hne (s : ℝ) (hs : s ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+      (⟨s, Or.inr hs⟩ : angleDomain Θ) ≠ t := by
+    intro heq
+    apply ht
+    have hval : s = t.val := congrArg Subtype.val heq
+    simpa [← hval] using hs
+  have hvalue (s : ℝ) (hs : s ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+      polygonHeightValue (raisedPolygonSupport K t ε) s =
+        supportValue K.val.val (s : Real.Angle) := by
+    rw [polygonHeightValue_raisedPolygonSupport K t ε (Or.inr hs), ite_eq_right (hne s hs),
+      add_zero]
+  constructor
+  · intro s hs
+    obtain ⟨p, hp, hpinner⟩ := exists_mem_inner_eq_supportValue K.val.val
+      (s : Real.Angle)
+    exact ⟨p, mem_polygonHeightCap_raised_of_mem_of_not_endpoint K t hε ht hp,
+      hpinner.trans (hvalue s hs).symm⟩
+  · intro s hs
+    rcases hs with rfl | rfl
+    · obtain ⟨p, hp, hpinner⟩ := exists_mem_inner_eq_supportValue K.val.val
+        ((Θ.angle + Real.pi : ℝ) : Real.Angle)
+      refine ⟨p, mem_polygonHeightCap_raised_of_mem_of_not_endpoint K t hε ht hp, ?_⟩
+      rw [normalVector_add_pi, inner_neg_right,
+        K.val.property.2.2.2.2.1] at hpinner
+      rw [hvalue Θ.angle (by simp), K.val.property.2.2.1]
+      linarith
+    · obtain ⟨p, hp, hpinner⟩ := exists_mem_inner_eq_supportValue K.val.val
+        ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+      refine ⟨p, mem_polygonHeightCap_raised_of_mem_of_not_endpoint K t hε ht hp, ?_⟩
+      rw [K.val.property.2.2.2.2.2.1, inner_normalVector_three_pi_div_two] at hpinner
+      rw [hvalue (Real.pi / 2) (by simp), K.val.property.2.2.2.1]
+      simpa [normalVector, frame, PiLp.inner_apply] using hpinner
+
+private theorem independentWallCap_self_sub_one_eq {Θ : AngleSet}
+    (h : PolygonHeightSpace Θ) :
+    independentWallCap h (fun t ↦ h t - 1) = polygonHeightCap h := by
+  ext p
+  simp only [independentWallCap, polygonHeightCap, polygonHeightParallelogram,
+    Set.mem_inter_iff, Set.mem_iInter]
+  apply and_congr
+  · apply forall_congr'
+    intro t
+    apply forall_congr'
+    intro ht
+    have htD : t ∈ angleDomain Θ := Or.inr ht
+    simp only [polygonHeightValue, dite_eq_left htD]
+  · rfl
+
+private theorem polygonHeightCap_raised_bounded {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (t : angleDomain Θ) :
+    ∃ ε₀ : ℝ, 0 < ε₀ ∧ ∀ ε : ℝ, |ε| ≤ ε₀ →
+      Bornology.IsBounded (polygonHeightCap (raisedPolygonSupport K t ε)) := by
+  let h : PolygonHeightSpace Θ :=
+    fun s ↦ supportValue K.val.val (s.val : Real.Angle)
+  obtain ⟨R, ε₀, hR, hε₀, hbound⟩ := polygonPerturbation_uniform_bounds Θ h
+  refine ⟨ε₀, hε₀, fun ε hε ↦ ?_⟩
+  let hε' := raisedPolygonSupport K t ε
+  let lower : PolygonHeightSpace Θ := fun s ↦ hε' s - 1
+  have hu (s : angleDomain Θ) : |hε' s - h s| ≤ ε₀ := by
+    change |(supportValue K.val.val (s.val : Real.Angle) + if s = t then ε else 0) -
+      supportValue K.val.val (s.val : Real.Angle)| ≤ ε₀
+    split_ifs
+    · simpa using hε
+    · simp [hε₀.le]
+  have hl (s : angleDomain Θ) : |lower s - (h s - 1)| ≤ ε₀ := by
+    simpa only [lower, sub_sub_sub_cancel_right] using hu s
+  have hsub := (hbound hε' lower hu hl).1
+  rw [independentWallCap_self_sub_one_eq] at hsub
+  exact Metric.isBounded_closedBall.subset hsub
+
+/-- A sufficiently small interior height increase yields a translated polygon cap. -/
+theorem polygonCap_positive_height_increment_of_not_endpoint {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (t : angleDomain Θ)
+    (ht : t.val ∉ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    ∃ ε₀ : ℝ, 0 < ε₀ ∧ ∀ ε : ℝ, 0 < ε → ε < ε₀ →
+      ∃ K' : PolygonCapTranslateSpace Θ,
+        K'.val = polygonHeightCap (raisedPolygonSupport K t ε) := by
+  obtain ⟨ε₀, hε₀, hbounded⟩ := polygonHeightCap_raised_bounded K t
+  refine ⟨ε₀, hε₀, fun ε hε hεlt ↦ ?_⟩
+  have hcontacts := raisedPolygonSupport_endpoint_contacts_of_not_endpoint
+    K t hε.le ht
+  exact exists_polygonCapTranslate_eq_polygonHeightCap _
+    (hbounded ε (abs_le.mpr ⟨by linarith, hεlt.le⟩)) hcontacts.1 hcontacts.2
+
+private theorem midpoint_mem_strict_support_of_face {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) {a b : Point} {t c : ℝ} (ht : t ∈ angleDomain Θ)
+    (ha : a ∈ (K.val.val : Set Point)) (hb : b ∈ (K.val.val : Set Point))
+    (hat : inner ℝ a (normalVector (t : Real.Angle)) = c)
+    (hbt : inner ℝ b (normalVector (t : Real.Angle)) = c)
+    (hab : a ≠ b) :
+    let m := (2 : ℝ)⁻¹ • (a + b)
+    m ∈ (K.val.val : Set Point) ∧
+      inner ℝ m (normalVector (t : Real.Angle)) = c ∧
+      ∀ s ∈ angleDomain Θ, s ≠ t →
+        inner ℝ m (normalVector (s : Real.Angle)) <
+          supportValue K.val.val (s : Real.Angle) := by
+  dsimp
+  have hm : (2 : ℝ)⁻¹ • (a + b) ∈ (K.val.val : Set Point) := by
+    rw [smul_add]
+    exact K.val.val.convex ha hb (by norm_num) (by norm_num) (by norm_num)
+  have hmt : inner ℝ ((2 : ℝ)⁻¹ • (a + b)) (normalVector (t : Real.Angle)) =
+      c := by
+    rw [inner_smul_left, inner_add_left]
+    simp only [RCLike.conj_to_real, hat, hbt]
+    ring
+  refine ⟨hm, hmt, ?_⟩
+  intro s hs hst
+  have hle := inner_le_supportValue K.val.val hm (s : Real.Angle)
+  apply lt_of_le_of_ne hle
+  intro heq
+  have hae := inner_le_supportValue K.val.val ha (s : Real.Angle)
+  have hbe := inner_le_supportValue K.val.val hb (s : Real.Angle)
+  have hmavg : inner ℝ ((2 : ℝ)⁻¹ • (a + b)) (normalVector (s : Real.Angle)) =
+      (2 : ℝ)⁻¹ * (inner ℝ a (normalVector (s : Real.Angle)) +
+        inner ℝ b (normalVector (s : Real.Angle))) := by
+    rw [inner_smul_left, inner_add_left]
+    simp only [RCLike.conj_to_real]
+  have haeq : inner ℝ a (normalVector (s : Real.Angle)) =
+      supportValue K.val.val (s : Real.Angle) := by
+    rw [hmavg] at heq
+    nlinarith
+  have hbeq : inner ℝ b (normalVector (s : Real.Angle)) =
+      supportValue K.val.val (s : Real.Angle) := by
+    rw [hmavg] at heq
+    nlinarith
+  have hortht : inner ℝ (b - a) (normalVector (t : Real.Angle)) = 0 := by
+    rw [inner_sub_left, hbt, hat, sub_self]
+  have horths : inner ℝ (b - a) (normalVector (s : Real.Angle)) = 0 := by
+    rw [inner_sub_left, hbeq, haeq, sub_self]
+  exact hst (eq_of_inner_sub_normalVector_eq_zero_of_ne
+    (angleDomain_subset_Ioo Θ hs) (angleDomain_subset_Ioo Θ ht) hab horths hortht)
+
+private theorem exists_pos_uniform_strict_support_gap {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (m : Point) (t : angleDomain Θ)
+    (hstrict : ∀ s ∈ angleDomain Θ, s ≠ t.val →
+      inner ℝ m (normalVector (s : Real.Angle)) <
+        supportValue K.val.val (s : Real.Angle)) :
+    ∃ δ : ℝ, 0 < δ ∧ ∀ s ∈ angleDomain Θ, s ≠ t.val →
+      δ ≤ supportValue K.val.val (s : Real.Angle) -
+        inner ℝ m (normalVector (s : Real.Angle)) := by
+  let gap : ℝ → ℝ := fun s ↦ if s = t.val then 1 else
+    supportValue K.val.val (s : Real.Angle) - inner ℝ m (normalVector (s : Real.Angle))
+  have hdomain : (angleDomain Θ).Finite := by
+    unfold angleDomain
+    exact (Θ.directions.finite_toSet.union
+      (Θ.directions.finite_toSet.image (fun s : ℝ ↦ s + Real.pi / 2))).union
+        (Set.finite_singleton (Real.pi / 2) |>.insert Θ.angle)
+  have hnonempty : (angleDomain Θ).Nonempty := ⟨Θ.angle, by simp [angleDomain]⟩
+  obtain ⟨δ, hδ, hδle⟩ := hdomain.isCompact.exists_pos_forall_le hnonempty
+    (hdomain.continuousOn gap) (fun s hs ↦ by
+      dsimp only [gap]
+      split_ifs with heq
+      · positivity
+      · linarith [hstrict s hs heq])
+  refine ⟨δ, hδ, ?_⟩
+  intro s hs hne
+  simpa [gap, hne] using hδle s hs
+
+private theorem cos_sub_nonneg_of_mem_endpoints {Θ : AngleSet} {s t : ℝ}
+    (hs : s ∈ ({Θ.angle, Real.pi / 2} : Set ℝ))
+    (ht : t ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) : 0 ≤ Real.cos (s - t) := by
+  apply Real.cos_nonneg_of_mem_Icc
+  rcases hs with rfl | rfl <;> rcases ht with rfl | rfl
+  all_goals constructor <;> linarith [Θ.angle_pos, Θ.angle_le, Real.pi_pos]
+
+private theorem add_smul_normal_mem_raisedPolygonSupport {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (t : angleDomain Θ)
+    (htendpoint : t.val ∈ ({Θ.angle, Real.pi / 2} : Set ℝ))
+    (m : Point) (hm : m ∈ (K.val.val : Set Point))
+    (hmLower : supportValue K.val.val (t.val : Real.Angle) - 1 ≤
+      inner ℝ m (normalVector (t.val : Real.Angle)))
+    (hmUpper : inner ℝ m (normalVector (t.val : Real.Angle)) ≤
+      supportValue K.val.val (t.val : Real.Angle))
+    {δ ε : ℝ} (hδ : ∀ s ∈ angleDomain Θ, s ≠ t.val →
+      δ ≤ supportValue K.val.val (s : Real.Angle) -
+        inner ℝ m (normalVector (s : Real.Angle)))
+    (hε : 0 < ε) (hεδ : ε < δ) :
+    m + ε • normalVector (t.val : Real.Angle) ∈
+      polygonHeightCap (raisedPolygonSupport K t ε) := by
+  unfold polygonHeightCap polygonHeightParallelogram
+  simp only [Set.mem_inter_iff, Set.mem_iInter]
+  constructor
+  · intro r hr
+    have hrD : r ∈ angleDomain Θ := Or.inr hr
+    have hval := polygonHeightValue_raisedPolygonSupport K t ε hrD
+    constructor
+    · change inner ℝ (m + ε • normalVector (t.val : Real.Angle))
+          (normalVector (r : Real.Angle)) ≤
+        polygonHeightValue (raisedPolygonSupport K t ε) r
+      rw [hval, inner_add_left, inner_smul_left]
+      simp only [RCLike.conj_to_real]
+      by_cases hrt : r = t.val
+      · have hsub : (⟨r, hrD⟩ : angleDomain Θ) = t := Subtype.ext hrt
+        rw [ite_eq_left hsub, hrt, inner_normalVector_self]
+        linarith
+      · have hsub : (⟨r, hrD⟩ : angleDomain Θ) ≠ t := by
+          intro heq
+          exact hrt (congrArg Subtype.val heq)
+        rw [ite_eq_right hsub]
+        have hcos := Real.cos_le_one (t.val - r)
+        rw [inner_normalVector_normalVector]
+        have hgap := hδ r hrD hrt
+        nlinarith
+    · change polygonHeightValue (raisedPolygonSupport K t ε) r - 1 ≤
+        inner ℝ (m + ε • normalVector (t.val : Real.Angle))
+          (normalVector (r : Real.Angle))
+      rw [hval, inner_add_left, inner_smul_left]
+      simp only [RCLike.conj_to_real]
+      by_cases hrt : r = t.val
+      · have hsub : (⟨r, hrD⟩ : angleDomain Θ) = t := Subtype.ext hrt
+        rw [ite_eq_left hsub, hrt, inner_normalVector_self]
+        nlinarith
+      · have hsub : (⟨r, hrD⟩ : angleDomain Θ) ≠ t := by
+          intro heq
+          exact hrt (congrArg Subtype.val heq)
+        rw [ite_eq_right hsub, add_zero]
+        have hbase : supportValue K.val.val (r : Real.Angle) - 1 ≤
+            inner ℝ m (normalVector (r : Real.Angle)) := by
+          rcases hr with rfl | rfl
+          · have hl := inner_le_supportValue K.val.val hm
+              ((Θ.angle + Real.pi : ℝ) : Real.Angle)
+            rw [K.val.property.2.2.2.2.1, normalVector_add_pi, inner_neg_right] at hl
+            rw [K.val.property.2.2.1]
+            linarith
+          · have hl := inner_le_supportValue K.val.val hm
+              ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+            rw [K.val.property.2.2.2.2.2.1, inner_normalVector_three_pi_div_two] at hl
+            rw [K.val.property.2.2.2.1]
+            simpa [normalVector, frame, PiLp.inner_apply] using hl
+        have hcos := cos_sub_nonneg_of_mem_endpoints htendpoint hr
+        rw [inner_normalVector_normalVector]
+        exact hbase.trans (le_add_of_nonneg_right (mul_nonneg hε.le hcos))
+  · intro r hr
+    have hrD : r ∈ angleDomain Θ := Or.inl hr
+    have hval := polygonHeightValue_raisedPolygonSupport K t ε hrD
+    change inner ℝ (m + ε • normalVector (t.val : Real.Angle))
+        (normalVector (r : Real.Angle)) ≤
+      polygonHeightValue (raisedPolygonSupport K t ε) r
+    rw [hval, inner_add_left, inner_smul_left]
+    simp only [RCLike.conj_to_real]
+    by_cases hrt : r = t.val
+    · have hsub : (⟨r, hrD⟩ : angleDomain Θ) = t := Subtype.ext hrt
+      rw [ite_eq_left hsub, hrt, inner_normalVector_self]
+      linarith
+    · have hsub : (⟨r, hrD⟩ : angleDomain Θ) ≠ t := by
+        intro heq
+        exact hrt (congrArg Subtype.val heq)
+      rw [ite_eq_right hsub]
+      have hcos := Real.cos_le_one (t.val - r)
+      rw [inner_normalVector_normalVector]
+      have hgap := hδ r hrD hrt
+      nlinarith
+
+private theorem mem_raisedPolygonSupport_of_endpoint_of_le_inner {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (t : angleDomain Θ)
+    (ht : t.val ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) {p : Point}
+    (hp : p ∈ (K.val.val : Set Point)) {ε : ℝ} (hε : 0 ≤ ε)
+    (hinner : ε ≤ inner ℝ p (normalVector (t.val : Real.Angle))) :
+    p ∈ polygonHeightCap (raisedPolygonSupport K t ε) := by
+  unfold polygonHeightCap polygonHeightParallelogram
+  simp only [Set.mem_inter_iff, Set.mem_iInter]
+  constructor
+  · intro r hr
+    have hrD : r ∈ angleDomain Θ := Or.inr hr
+    have hvalue := polygonHeightValue_raisedPolygonSupport K t ε hrD
+    constructor
+    · change inner ℝ p (normalVector (r : Real.Angle)) ≤
+          polygonHeightValue (raisedPolygonSupport K t ε) r
+      rw [hvalue]
+      exact (inner_le_supportValue K.val.val hp _).trans
+        (le_add_of_nonneg_right (by split_ifs <;> positivity))
+    · change polygonHeightValue (raisedPolygonSupport K t ε) r - 1 ≤
+          inner ℝ p (normalVector (r : Real.Angle))
+      rw [hvalue]
+      by_cases hrt : r = t.val
+      · have hsub : (⟨r, hrD⟩ : angleDomain Θ) = t := Subtype.ext hrt
+        rw [ite_eq_left hsub, hrt]
+        have hsupport : supportValue K.val.val (t.val : Real.Angle) = 1 := by
+          rcases ht with ht | ht
+          · simpa [ht] using K.val.property.2.2.1
+          · have ht' : t.val = Real.pi / 2 := by simpa using ht
+            simpa [ht'] using K.val.property.2.2.2.1
+        rw [hsupport]
+        simpa using hinner
+      · have hsub : (⟨r, hrD⟩ : angleDomain Θ) ≠ t := by
+          intro heq
+          exact hrt (congrArg Subtype.val heq)
+        rw [ite_eq_right hsub, add_zero]
+        rcases hr with rfl | rfl
+        · have hlower := inner_le_supportValue K.val.val hp
+              ((Θ.angle + Real.pi : ℝ) : Real.Angle)
+          rw [K.val.property.2.2.2.2.1, normalVector_add_pi, inner_neg_right] at hlower
+          rw [K.val.property.2.2.1]
+          linarith
+        · have hlower := inner_le_supportValue K.val.val hp
+              ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+          rw [K.val.property.2.2.2.2.2.1, inner_normalVector_three_pi_div_two] at hlower
+          rw [K.val.property.2.2.2.1]
+          simpa [normalVector, frame, PiLp.inner_apply] using hlower
+  · intro r hr
+    have hrD : r ∈ angleDomain Θ := Or.inl hr
+    rw [polygonHeightValue_raisedPolygonSupport K t ε hrD]
+    exact (inner_le_supportValue K.val.val hp _).trans
+      (le_add_of_nonneg_right (by split_ifs <;> positivity))
+
+private theorem moved_upper_lower_contacts {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (t : angleDomain Θ) (ht : t.val ∈ ({Θ.angle, Real.pi / 2} : Set ℝ))
+    (u l : Point) (hu : u ∈ (K.val.val : Set Point))
+    (huval : inner ℝ u (normalVector (t.val : Real.Angle)) =
+      supportValue K.val.val (t.val : Real.Angle))
+    (hl : l ∈ (K.val.val : Set Point))
+    (hlval : inner ℝ l (normalVector (t.val : Real.Angle)) =
+      supportValue K.val.val (t.val : Real.Angle) - 1)
+    {δu δl ε : ℝ}
+    (hδu : ∀ s ∈ angleDomain Θ, s ≠ t.val →
+      δu ≤ supportValue K.val.val (s : Real.Angle) -
+        inner ℝ u (normalVector (s : Real.Angle)))
+    (hδl : ∀ s ∈ angleDomain Θ, s ≠ t.val →
+      δl ≤ supportValue K.val.val (s : Real.Angle) -
+        inner ℝ l (normalVector (s : Real.Angle)))
+    (hε : 0 < ε) (hεu : ε < δu) (hεl : ε < δl) :
+    (∃ p ∈ polygonHeightCap (raisedPolygonSupport K t ε),
+      inner ℝ p (normalVector (t.val : Real.Angle)) =
+        polygonHeightValue (raisedPolygonSupport K t ε) t.val) ∧
+    (∃ p ∈ polygonHeightCap (raisedPolygonSupport K t ε),
+      inner ℝ p (normalVector (t.val : Real.Angle)) =
+        polygonHeightValue (raisedPolygonSupport K t ε) t.val - 1) := by
+  have huLower : supportValue K.val.val (t.val : Real.Angle) - 1 ≤
+      inner ℝ u (normalVector (t.val : Real.Angle)) := by linarith
+  have hlUpper : inner ℝ l (normalVector (t.val : Real.Angle)) ≤
+      supportValue K.val.val (t.val : Real.Angle) := by linarith
+  have hu' := add_smul_normal_mem_raisedPolygonSupport K t ht u hu huLower huval.le
+    hδu hε hεu
+  have hl' := add_smul_normal_mem_raisedPolygonSupport K t ht l hl hlval.ge hlUpper
+    hδl hε hεl
+  have hvalue : polygonHeightValue (raisedPolygonSupport K t ε) t.val =
+      supportValue K.val.val (t.val : Real.Angle) + ε := by
+    rw [polygonHeightValue_raisedPolygonSupport K t ε t.property]
+    simp
+  constructor
+  · refine ⟨u + ε • normalVector (t.val : Real.Angle), hu', ?_⟩
+    rw [inner_add_left, inner_smul_left, huval, inner_normalVector_self, hvalue]
+    simp
+  · refine ⟨l + ε • normalVector (t.val : Real.Angle), hl', ?_⟩
+    rw [inner_add_left, inner_smul_left, hlval, inner_normalVector_self, hvalue]
+    simp
+    ring
+
+private theorem exists_upperFace_midpoint_strict {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (t : angleDomain Θ)
+    (ht : 0 < surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}) :
+    ∃ m : Point,
+      m ∈ (K.val.val : Set Point) ∧
+      inner ℝ m (normalVector (t.val : Real.Angle)) =
+        supportValue K.val.val (t.val : Real.Angle) ∧
+      ∀ s ∈ angleDomain Θ, s ≠ t.val →
+        inner ℝ m (normalVector (s : Real.Angle)) <
+          supportValue K.val.val (s : Real.Angle) := by
+  let a := (edgeVertices K.val.val (t.val : Real.Angle)).1
+  let b := (edgeVertices K.val.val (t.val : Real.Angle)).2
+  have haedge := edgeVertices_fst_mem K.val.val (t.val : Real.Angle)
+  have hbedge := edgeVertices_snd_mem K.val.val (t.val : Real.Angle)
+  have hab : a ≠ b := by
+    have hdist : 0 < dist a b := by
+      rw [(surfaceAreaMeasure_atom_length K.val.val (t.val : Real.Angle)).2.1] at ht
+      exact ENNReal.ofReal_pos.mp ht
+    exact dist_ne_zero.mp hdist.ne'
+  let m := (2 : ℝ)⁻¹ • (a + b)
+  have hm := midpoint_mem_strict_support_of_face K t.property
+    haedge.1 hbedge.1 haedge.2 hbedge.2 hab
+  exact ⟨m, hm⟩
+
+private theorem stripParallelogram_top_inner_angle (Θ : AngleSet)
+    (hΘ : Θ.angle < Real.pi / 2) :
+    inner ℝ (stripParallelogram Θ.angle).2.2 (normalVector (Θ.angle : Real.Angle)) = 1 := by
+  have hc : 0 < Real.cos Θ.angle := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Θ.angle_pos, Real.pi_pos], hΘ⟩
+  have hgap : Real.tan (Real.pi / 4 - Θ.angle / 2) =
+      (Real.cos Θ.angle)⁻¹ - Real.tan Θ.angle := by
+    simpa [show Real.pi / 4 - Θ.angle / 2 =
+        (Real.pi / 2 - Θ.angle) / 2 by ring]
+      using Real.tan_pi_div_two_sub_div_two Θ.angle ⟨Θ.angle_pos.le, hΘ⟩
+  have htan := Real.tan_eq_sin_div_cos Θ.angle
+  simp [stripParallelogram, normalVector, frame, PiLp.inner_apply, hgap, htan]
+  field_simp [hc.ne']
+  ring
+
+private theorem stripParallelogram_top_sub_vertical_mem_of_angle_lt {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hΘ : Θ.angle < Real.pi / 2) :
+    (stripParallelogram Θ.angle).2.2 - tangentVector 0 ∈ (K.val.val : Set Point) := by
+  let o := (stripParallelogram Θ.angle).2.2
+  let a := o - tangentVector 0
+  have ho := stripParallelogram_top_mem_of_angle_lt Θ hΘ K
+  have haFan : a ∈ capFan Θ.angle := by
+    constructor
+    · change 0 ≤ inner ℝ a (normalVector (Θ.angle : Real.Angle))
+      dsimp [a]
+      have hoω : inner ℝ o (normalVector (Θ.angle : Real.Angle)) = 1 :=
+        stripParallelogram_top_inner_angle Θ hΘ
+      rw [inner_sub_left, hoω]
+      simp only [tangentVector, frame, Real.Angle.cos_zero, Real.Angle.sin_zero, neg_zero,
+        normalVector, Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply,
+        RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue, Matrix.cons_val_zero,
+        mul_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one, mul_one, zero_add, sub_nonneg,
+        ge_iff_le]
+      exact Real.sin_le_one Θ.angle
+    · change 0 ≤ inner ℝ a (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+      simp [a, o, stripParallelogram, tangentVector, normalVector, frame,
+        PiLp.inner_apply]
+  apply K.val.mem_of_mem_capFan_of_le_supportValue haFan
+  intro s hs
+  have ho_le := inner_le_supportValue K.val.val ho (s : Real.Angle)
+  have hale : inner ℝ a (normalVector (s : Real.Angle)) ≤
+      inner ℝ o (normalVector (s : Real.Angle)) := by
+    rw [show a = o - tangentVector 0 by rfl, inner_sub_left]
+    apply sub_le_self
+    have hsI : s ∈ Set.Icc 0 Real.pi := by
+      rcases hs with hs | hs
+      · exact ⟨hs.1, hs.2.trans (Θ.angle_le.trans (by linarith [Real.pi_pos]))⟩
+      · exact ⟨(by positivity : 0 ≤ Real.pi / 2).trans hs.1,
+          hs.2.trans (by linarith [Θ.angle_le, Real.pi_pos])⟩
+    simpa [tangentVector, normalVector, frame, PiLp.inner_apply] using
+      Real.sin_nonneg_of_mem_Icc hsI
+  exact hale.trans ho_le
+
+private theorem stripParallelogram_top_sub_normal_mem_of_angle_lt {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hΘ : Θ.angle < Real.pi / 2) :
+    (stripParallelogram Θ.angle).2.2 - normalVector (Θ.angle : Real.Angle) ∈
+      (K.val.val : Set Point) := by
+  let o := (stripParallelogram Θ.angle).2.2
+  let b := o - normalVector (Θ.angle : Real.Angle)
+  have ho := stripParallelogram_top_mem_of_angle_lt Θ hΘ K
+  have hbFan : b ∈ capFan Θ.angle := by
+    constructor
+    · change 0 ≤ inner ℝ b (normalVector (Θ.angle : Real.Angle))
+      dsimp [b]
+      have hoω : inner ℝ o (normalVector (Θ.angle : Real.Angle)) = 1 :=
+        stripParallelogram_top_inner_angle Θ hΘ
+      rw [inner_sub_left, hoω, inner_normalVector_self, sub_self]
+    · change 0 ≤ inner ℝ b (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+      dsimp [b]
+      have hoT : inner ℝ o (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 1 := by
+        simp [o, stripParallelogram, normalVector, frame, PiLp.inner_apply]
+      rw [inner_sub_left, hoT]
+      simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, Real.cos_pi_div_two,
+        Real.sin_pi_div_two, PiLp.inner_apply, RCLike.inner_apply, conj_trivial,
+        Fin.sum_univ_two, Fin.isValue, Matrix.cons_val_zero, zero_mul, Matrix.cons_val_one,
+        Matrix.cons_val_fin_one, one_mul, zero_add, sub_nonneg, ge_iff_le]
+      exact Real.sin_le_one Θ.angle
+  apply K.val.mem_of_mem_capFan_of_le_supportValue hbFan
+  intro s hs
+  have ho_le := inner_le_supportValue K.val.val ho (s : Real.Angle)
+  have hble : inner ℝ b (normalVector (s : Real.Angle)) ≤
+      inner ℝ o (normalVector (s : Real.Angle)) := by
+    rw [show b = o - normalVector (Θ.angle : Real.Angle) by rfl, inner_sub_left]
+    apply sub_le_self
+    rw [inner_normalVector_normalVector]
+    apply Real.cos_nonneg_of_mem_Icc
+    rcases hs with hs | hs
+    · constructor <;> linarith [hs.1, hs.2, Θ.angle_pos, Θ.angle_le, Real.pi_pos]
+    · constructor <;> linarith [hs.1, hs.2, Θ.angle_pos, Θ.angle_le, Real.pi_pos]
+  exact hble.trans ho_le
+
+private theorem exists_lowerFace_midpoint_strict_of_angle_lt {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hΘ : Θ.angle < Real.pi / 2)
+    (t : angleDomain Θ) (ht : t.val ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)) :
+    ∃ m : Point,
+      m ∈ (K.val.val : Set Point) ∧
+      inner ℝ m (normalVector (t.val : Real.Angle)) =
+        supportValue K.val.val (t.val : Real.Angle) - 1 ∧
+      ∀ s ∈ angleDomain Θ, s ≠ t.val →
+        inner ℝ m (normalVector (s : Real.Angle)) <
+          supportValue K.val.val (s : Real.Angle) := by
+  have hzero := zero_mem_cap_of_lt K.val hΘ
+  rcases ht with ht | ht
+  · have htval : t.val = Θ.angle := ht
+    let b := (stripParallelogram Θ.angle).2.2 - normalVector (Θ.angle : Real.Angle)
+    have hb := stripParallelogram_top_sub_normal_mem_of_angle_lt K hΘ
+    have hbnormal : inner ℝ b (normalVector (Θ.angle : Real.Angle)) = 0 := by
+      dsimp [b]
+      rw [inner_sub_left, stripParallelogram_top_inner_angle Θ hΘ,
+        inner_normalVector_self, sub_self]
+    have hbne : (0 : Point) ≠ b := by
+      intro heq
+      have hgap := (parallelogram_gap Θ.angle ⟨Θ.angle_pos.le, hΘ⟩).2.1
+      rw [← show b = (stripParallelogram Θ.angle).2.2 -
+        normalVector (Θ.angle : Real.Angle) by rfl, ← heq] at hgap
+      have hgappos : 0 < Real.tan ((Real.pi / 2 - Θ.angle) / 2) :=
+        Real.tan_pos_of_pos_of_lt_pi_div_two (by linarith)
+          (by linarith [Θ.angle_pos, Real.pi_pos])
+      have hcoord := congrArg (fun p : Point ↦ p 1) hgap
+      have hcos : 0 < Real.cos Θ.angle := Real.cos_pos_of_mem_Ioo
+        ⟨by linarith [Θ.angle_pos, Real.pi_pos], hΘ⟩
+      simp only [Fin.isValue, PiLp.zero_apply, tangentVector, frame, Real.Angle.cos_coe,
+        Real.Angle.sin_coe, PiLp.smul_apply, Matrix.cons_val_one,
+        Matrix.cons_val_fin_one, smul_eq_mul, zero_eq_mul] at hcoord
+      rcases hcoord with hgapzero | hcoszero
+      · exact hgappos.ne' hgapzero
+      · exact hcos.ne' hcoszero
+    have hm := midpoint_mem_strict_support_of_face (c := 0) K t.property hzero hb
+      (by simp) (by simpa [htval, K.val.property.2.2.1] using hbnormal) hbne
+    refine ⟨(2 : ℝ)⁻¹ • ((0 : Point) + b), hm.1, ?_, ?_⟩
+    · rw [hm.2.1, htval, K.val.property.2.2.1]
+      ring
+    · simpa [htval] using hm.2.2
+  · have htval : t.val = Real.pi / 2 := ht
+    let a := (stripParallelogram Θ.angle).2.2 - tangentVector 0
+    have ha := stripParallelogram_top_sub_vertical_mem_of_angle_lt K hΘ
+    have hanormal : inner ℝ a (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := by
+      simp [a, stripParallelogram, tangentVector, normalVector, frame, PiLp.inner_apply]
+    have hane : (0 : Point) ≠ a := by
+      intro heq
+      have hgap := (parallelogram_gap Θ.angle ⟨Θ.angle_pos.le, hΘ⟩).1
+      rw [← show a = (stripParallelogram Θ.angle).2.2 - tangentVector 0 by rfl,
+        ← heq] at hgap
+      have hgappos : 0 < Real.tan ((Real.pi / 2 - Θ.angle) / 2) :=
+        Real.tan_pos_of_pos_of_lt_pi_div_two (by linarith)
+          (by linarith [Θ.angle_pos, Real.pi_pos])
+      have hcoord := congrArg (fun p : Point ↦ p 0) hgap
+      simp [normalVector, frame] at hcoord
+      linarith
+    have hm := midpoint_mem_strict_support_of_face (c := 0) K t.property hzero ha
+      (by simp) (by simpa [htval, K.val.property.2.2.2.1] using hanormal) hane
+    refine ⟨(2 : ℝ)⁻¹ • ((0 : Point) + a), hm.1, ?_, ?_⟩
+    · rw [hm.2.1, htval, K.val.property.2.2.2.1]
+      ring
+    · simpa [htval] using hm.2.2
+
+private theorem sub_normalVector_pi_div_two_mem_of_mem_of_angle_eq {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hΘ : Θ.angle = Real.pi / 2) {p : Point}
+    (hp : p ∈ (K.val.val : Set Point))
+    (hpT : inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 1) :
+    p - normalVector ((Real.pi / 2 : ℝ) : Real.Angle) ∈ (K.val.val : Set Point) := by
+  apply K.val.mem_of_mem_capFan_of_le_supportValue
+  · constructor
+    · change 0 ≤ inner ℝ (p - normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+          (normalVector (Θ.angle : Real.Angle))
+      rw [hΘ, inner_sub_left, hpT, inner_normalVector_self, sub_self]
+    · change 0 ≤ inner ℝ (p - normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+          (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+      rw [inner_sub_left, hpT, inner_normalVector_self, sub_self]
+  · intro s hs
+    have hsI : s ∈ Set.Icc 0 Real.pi := by
+      rcases hs with hs | hs
+      · exact ⟨hs.1, by rw [hΘ] at hs; linarith [hs.2, Real.pi_pos]⟩
+      · exact ⟨by linarith [hs.1, Real.pi_pos], by rw [hΘ] at hs; linarith [hs.2]⟩
+    have hsin : 0 ≤ Real.sin s := Real.sin_nonneg_of_mem_Icc hsI
+    have hle : inner ℝ (p - normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+        (normalVector (s : Real.Angle)) ≤ inner ℝ p (normalVector (s : Real.Angle)) := by
+      rw [inner_sub_left]
+      apply sub_le_self
+      rw [inner_normalVector_normalVector]
+      simpa [Real.cos_pi_div_two_sub] using hsin
+    exact hle.trans (inner_le_supportValue K.val.val hp (s : Real.Angle))
+
+private theorem exists_lowerFace_midpoint_strict_of_angle_eq {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hΘ : Θ.angle = Real.pi / 2)
+    (t : angleDomain Θ) (ht : t.val ∈ ({Θ.angle, Real.pi / 2} : Set ℝ))
+    (hmass : 0 < surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}) :
+    ∃ m : Point,
+      m ∈ (K.val.val : Set Point) ∧
+      inner ℝ m (normalVector (t.val : Real.Angle)) =
+        supportValue K.val.val (t.val : Real.Angle) - 1 ∧
+      ∀ s ∈ angleDomain Θ, s ≠ t.val →
+        inner ℝ m (normalVector (s : Real.Angle)) <
+          supportValue K.val.val (s : Real.Angle) := by
+  have htval : t.val = Real.pi / 2 := by
+    rcases ht with ht | ht
+    · simpa [hΘ] using ht
+    · exact ht
+  let a := (edgeVertices K.val.val (t.val : Real.Angle)).1
+  let b := (edgeVertices K.val.val (t.val : Real.Angle)).2
+  let a' := a - normalVector ((Real.pi / 2 : ℝ) : Real.Angle)
+  let b' := b - normalVector ((Real.pi / 2 : ℝ) : Real.Angle)
+  have haedge := edgeVertices_fst_mem K.val.val (t.val : Real.Angle)
+  have hbedge := edgeVertices_snd_mem K.val.val (t.val : Real.Angle)
+  have hab : a ≠ b := by
+    have hdist : 0 < dist a b := by
+      rw [(surfaceAreaMeasure_atom_length K.val.val (t.val : Real.Angle)).2.1] at hmass
+      exact ENNReal.ofReal_pos.mp hmass
+    exact dist_ne_zero.mp hdist.ne'
+  have haT : inner ℝ a (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 1 := by
+    have haSupport : inner ℝ a (normalVector (t.val : Real.Angle)) =
+        supportValue K.val.val (t.val : Real.Angle) := haedge.2
+    simpa [htval, K.val.property.2.2.2.1] using haSupport
+  have hbT : inner ℝ b (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 1 := by
+    have hbSupport : inner ℝ b (normalVector (t.val : Real.Angle)) =
+        supportValue K.val.val (t.val : Real.Angle) := hbedge.2
+    simpa [htval, K.val.property.2.2.2.1] using hbSupport
+  have ha' : a' ∈ (K.val.val : Set Point) :=
+    sub_normalVector_pi_div_two_mem_of_mem_of_angle_eq K hΘ haedge.1 haT
+  have hb' : b' ∈ (K.val.val : Set Point) :=
+    sub_normalVector_pi_div_two_mem_of_mem_of_angle_eq K hΘ hbedge.1 hbT
+  have ha'normal : inner ℝ a' (normalVector (t.val : Real.Angle)) = 0 := by
+    rw [htval]
+    dsimp [a']
+    rw [inner_sub_left, haT, inner_normalVector_self, sub_self]
+  have hb'normal : inner ℝ b' (normalVector (t.val : Real.Angle)) = 0 := by
+    rw [htval]
+    dsimp [b']
+    rw [inner_sub_left, hbT, inner_normalVector_self, sub_self]
+  have hab' : a' ≠ b' := by
+    intro heq
+    apply hab
+    dsimp [a', b'] at heq
+    exact sub_left_injective heq
+  have hm := midpoint_mem_strict_support_of_face (c := 0) K t.property ha' hb'
+    ha'normal hb'normal hab'
+  refine ⟨(2 : ℝ)⁻¹ • (a' + b'), hm.1, ?_, hm.2.2⟩
+  rw [hm.2.1, htval, K.val.property.2.2.2.1]
+  ring
+
+private theorem raisedPolygonSupport_endpoint_contacts_of_angle_lt {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hΘ : Θ.angle < Real.pi / 2)
+    (t : angleDomain Θ) (ht : t.val ∈ ({Θ.angle, Real.pi / 2} : Set ℝ))
+    (hmass : 0 < surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}) :
+    ∃ ε₀ : ℝ, 0 < ε₀ ∧ ∀ ε : ℝ, 0 < ε → ε < ε₀ →
+      ((∀ s ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+        ∃ p ∈ polygonHeightCap (raisedPolygonSupport K t ε),
+          inner ℝ p (normalVector (s : Real.Angle)) =
+            polygonHeightValue (raisedPolygonSupport K t ε) s) ∧
+      (∀ s ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+        ∃ p ∈ polygonHeightCap (raisedPolygonSupport K t ε),
+          inner ℝ p (normalVector (s : Real.Angle)) =
+            polygonHeightValue (raisedPolygonSupport K t ε) s - 1)) := by
+  obtain ⟨u, hu, huval, hustrict⟩ := exists_upperFace_midpoint_strict K t hmass
+  obtain ⟨l, hl, hlval, hlstrict⟩ :=
+    exists_lowerFace_midpoint_strict_of_angle_lt K hΘ t ht
+  obtain ⟨δu, hδupos, hδu⟩ := exists_pos_uniform_strict_support_gap K u t hustrict
+  obtain ⟨δl, hδlpos, hδl⟩ := exists_pos_uniform_strict_support_gap K l t hlstrict
+  let o := (stripParallelogram Θ.angle).2.2
+  let a := o - tangentVector 0
+  let b := o - normalVector (Θ.angle : Real.Angle)
+  have ho : o ∈ (K.val.val : Set Point) :=
+    stripParallelogram_top_mem_of_angle_lt Θ hΘ K
+  have ha : a ∈ (K.val.val : Set Point) :=
+    stripParallelogram_top_sub_vertical_mem_of_angle_lt K hΘ
+  have hb : b ∈ (K.val.val : Set Point) :=
+    stripParallelogram_top_sub_normal_mem_of_angle_lt K hΘ
+  have hoω : inner ℝ o (normalVector (Θ.angle : Real.Angle)) = 1 :=
+    stripParallelogram_top_inner_angle Θ hΘ
+  have hoT : inner ℝ o (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 1 := by
+    simp [o, stripParallelogram, normalVector, frame, PiLp.inner_apply]
+  have haT : inner ℝ a (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := by
+    simp [a, o, stripParallelogram, tangentVector, normalVector, frame,
+      PiLp.inner_apply]
+  have hbω : inner ℝ b (normalVector (Θ.angle : Real.Angle)) = 0 := by
+    dsimp [b]
+    rw [inner_sub_left, hoω, inner_normalVector_self, sub_self]
+  let gap := 1 - Real.sin Θ.angle
+  have hcos : 0 < Real.cos Θ.angle := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Θ.angle_pos, Real.pi_pos], hΘ⟩
+  have hgappos : 0 < gap := by
+    have hcosSq : 0 < Real.cos Θ.angle ^ 2 := sq_pos_of_pos hcos
+    have htrig := Real.sin_sq_add_cos_sq Θ.angle
+    have hsinle := Real.sin_le_one Θ.angle
+    dsimp [gap]
+    nlinarith
+  have haω : inner ℝ a (normalVector (Θ.angle : Real.Angle)) = gap := by
+    dsimp [a, gap]
+    rw [inner_sub_left, hoω]
+    simp [tangentVector, normalVector, frame, PiLp.inner_apply]
+  have hbT : inner ℝ b (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = gap := by
+    dsimp [b, gap]
+    rw [inner_sub_left, hoT]
+    simp [normalVector, frame, PiLp.inner_apply]
+  refine ⟨min δu (min δl (min 1 gap)), by positivity, ?_⟩
+  intro ε hε hεlt
+  have hεu : ε < δu := hεlt.trans_le (min_le_left _ _)
+  have hεl : ε < δl := hεlt.trans_le ((min_le_right _ _).trans (min_le_left _ _))
+  have hε1 : ε < 1 :=
+    hεlt.trans_le ((min_le_right _ _).trans ((min_le_right _ _).trans (min_le_left _ _)))
+  have hεgap : ε < gap :=
+    hεlt.trans_le ((min_le_right _ _).trans ((min_le_right _ _).trans (min_le_right _ _)))
+  have hmoved := moved_upper_lower_contacts K t ht u l hu huval hl hlval
+    hδu hδl hε hεu hεl
+  rcases ht with ht | ht
+  · have htval : t.val = Θ.angle := ht
+    have hoMem : o ∈ polygonHeightCap (raisedPolygonSupport K t ε) := by
+      apply mem_raisedPolygonSupport_of_endpoint_of_le_inner K t (Or.inl htval) ho hε.le
+      rw [htval, hoω]
+      exact hε1.le
+    have haMem : a ∈ polygonHeightCap (raisedPolygonSupport K t ε) := by
+      apply mem_raisedPolygonSupport_of_endpoint_of_le_inner K t (Or.inl htval) ha hε.le
+      rw [htval, haω]
+      exact hεgap.le
+    have hother : polygonHeightValue (raisedPolygonSupport K t ε) (Real.pi / 2) = 1 := by
+      rw [polygonHeightValue_raisedPolygonSupport K t ε (Or.inr (by simp))]
+      have hne : (⟨Real.pi / 2, Or.inr (by simp)⟩ : angleDomain Θ) ≠ t := by
+        intro heq
+        have := congrArg Subtype.val heq
+        linarith [hΘ]
+      rw [ite_eq_right hne, add_zero, K.val.property.2.2.2.1]
+    constructor
+    · intro s hs
+      rcases hs with hs | hs
+      · have hsval : s = Θ.angle := hs
+        subst s
+        simpa [htval] using hmoved.1
+      · have hsval : s = Real.pi / 2 := by simpa using hs
+        subst s
+        exact ⟨o, hoMem, hoT.trans hother.symm⟩
+    · intro s hs
+      rcases hs with hs | hs
+      · have hsval : s = Θ.angle := hs
+        subst s
+        simpa [htval] using hmoved.2
+      · have hsval : s = Real.pi / 2 := by simpa using hs
+        subst s
+        refine ⟨a, haMem, ?_⟩
+        rw [haT, hother]
+        ring
+  · have htval : t.val = Real.pi / 2 := by simpa using ht
+    have hoMem : o ∈ polygonHeightCap (raisedPolygonSupport K t ε) := by
+      apply mem_raisedPolygonSupport_of_endpoint_of_le_inner K t (Or.inr ht) ho hε.le
+      rw [htval, hoT]
+      exact hε1.le
+    have hbMem : b ∈ polygonHeightCap (raisedPolygonSupport K t ε) := by
+      apply mem_raisedPolygonSupport_of_endpoint_of_le_inner K t (Or.inr ht) hb hε.le
+      rw [htval, hbT]
+      exact hεgap.le
+    have hother : polygonHeightValue (raisedPolygonSupport K t ε) Θ.angle = 1 := by
+      rw [polygonHeightValue_raisedPolygonSupport K t ε (Or.inr (by simp))]
+      have hne : (⟨Θ.angle, Or.inr (by simp)⟩ : angleDomain Θ) ≠ t := by
+        intro heq
+        have := congrArg Subtype.val heq
+        linarith [hΘ]
+      rw [ite_eq_right hne, add_zero, K.val.property.2.2.1]
+    constructor
+    · intro s hs
+      rcases hs with hs | hs
+      · have hsval : s = Θ.angle := hs
+        subst s
+        exact ⟨o, hoMem, hoω.trans hother.symm⟩
+      · have hsval : s = Real.pi / 2 := by simpa using hs
+        subst s
+        simpa [htval] using hmoved.1
+    · intro s hs
+      rcases hs with hs | hs
+      · have hsval : s = Θ.angle := hs
+        subst s
+        refine ⟨b, hbMem, ?_⟩
+        rw [hbω, hother]
+        ring
+      · have hsval : s = Real.pi / 2 := by simpa using hs
+        subst s
+        simpa [htval] using hmoved.2
+
+private theorem raisedPolygonSupport_endpoint_contacts_of_angle_eq {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (hΘ : Θ.angle = Real.pi / 2)
+    (t : angleDomain Θ) (ht : t.val ∈ ({Θ.angle, Real.pi / 2} : Set ℝ))
+    (hmass : 0 < surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}) :
+    ∃ ε₀ : ℝ, 0 < ε₀ ∧ ∀ ε : ℝ, 0 < ε → ε < ε₀ →
+      ((∀ s ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+        ∃ p ∈ polygonHeightCap (raisedPolygonSupport K t ε),
+          inner ℝ p (normalVector (s : Real.Angle)) =
+            polygonHeightValue (raisedPolygonSupport K t ε) s) ∧
+      (∀ s ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+        ∃ p ∈ polygonHeightCap (raisedPolygonSupport K t ε),
+          inner ℝ p (normalVector (s : Real.Angle)) =
+            polygonHeightValue (raisedPolygonSupport K t ε) s - 1)) := by
+  obtain ⟨u, hu, huval, hustrict⟩ := exists_upperFace_midpoint_strict K t hmass
+  obtain ⟨l, hl, hlval, hlstrict⟩ :=
+    exists_lowerFace_midpoint_strict_of_angle_eq K hΘ t ht hmass
+  obtain ⟨δu, hδupos, hδu⟩ := exists_pos_uniform_strict_support_gap K u t hustrict
+  obtain ⟨δl, hδlpos, hδl⟩ := exists_pos_uniform_strict_support_gap K l t hlstrict
+  have htval : t.val = Real.pi / 2 := by
+    rcases ht with ht | ht
+    · simpa [hΘ] using ht
+    · simpa using ht
+  refine ⟨min δu δl, by positivity, ?_⟩
+  intro ε hε hεlt
+  have hmoved := moved_upper_lower_contacts K t ht u l hu huval hl hlval
+    hδu hδl hε (hεlt.trans_le (min_le_left _ _))
+      (hεlt.trans_le (min_le_right _ _))
+  constructor
+  · intro s hs
+    have hsval : s = Real.pi / 2 := by
+      rcases hs with hs | hs
+      · simpa [hΘ] using hs
+      · simpa using hs
+    subst s
+    simpa [htval] using hmoved.1
+  · intro s hs
+    have hsval : s = Real.pi / 2 := by
+      rcases hs with hs | hs
+      · simpa [hΘ] using hs
+      · simpa using hs
+    subst s
+    simpa [htval] using hmoved.2
+
+/-- A sufficiently small endpoint height increase preserves both unit widths. -/
+theorem polygonCap_positive_height_increment_of_endpoint {Θ : AngleSet}
+    (K : PolygonCapSpace Θ) (t : angleDomain Θ)
+    (ht : t.val ∈ ({Θ.angle, Real.pi / 2} : Set ℝ))
+    (hmass : 0 < surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}) :
+    ∃ ε₀ : ℝ, 0 < ε₀ ∧ ∀ ε : ℝ, 0 < ε → ε < ε₀ →
+      ∃ K' : PolygonCapTranslateSpace Θ,
+        K'.val = polygonHeightCap (raisedPolygonSupport K t ε) := by
+  obtain ⟨εb, hεb, hbounded⟩ := polygonHeightCap_raised_bounded K t
+  obtain ⟨εc, hεc, hcontacts⟩ : ∃ εc : ℝ, 0 < εc ∧ ∀ ε : ℝ, 0 < ε → ε < εc →
+      ((∀ s ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+        ∃ p ∈ polygonHeightCap (raisedPolygonSupport K t ε),
+          inner ℝ p (normalVector (s : Real.Angle)) =
+            polygonHeightValue (raisedPolygonSupport K t ε) s) ∧
+      (∀ s ∈ ({Θ.angle, Real.pi / 2} : Set ℝ),
+        ∃ p ∈ polygonHeightCap (raisedPolygonSupport K t ε),
+          inner ℝ p (normalVector (s : Real.Angle)) =
+            polygonHeightValue (raisedPolygonSupport K t ε) s - 1)) := by
+    rcases lt_or_eq_of_le Θ.angle_le with hΘ | hΘ
+    · exact raisedPolygonSupport_endpoint_contacts_of_angle_lt K hΘ t ht hmass
+    · exact raisedPolygonSupport_endpoint_contacts_of_angle_eq K hΘ t ht hmass
+  refine ⟨min εb εc, by positivity, ?_⟩
+  intro ε hε hεlt
+  have hb : ε < εb := hεlt.trans_le (min_le_left _ _)
+  have hc : ε < εc := hεlt.trans_le (min_le_right _ _)
+  have hcontact := hcontacts ε hε hc
+  exact exists_polygonCapTranslate_eq_polygonHeightCap _
+    (hbounded ε (abs_le.mpr ⟨by linarith, hb.le⟩)) hcontact.1 hcontact.2
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Polygon / Height / Positive Increment
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem polygonCap_positive_height_increment {Θ : AngleSet} (K : PolygonCapSpace Θ)
+    (t : angleDomain Θ)
+    (ht : 0 < surfaceAreaMeasure K.val.val {(t.val : Real.Angle)}) :
+    ∃ ε₀ : ℝ, 0 < ε₀ ∧ ∀ ε : ℝ, 0 < ε → ε < ε₀ →
+      ∃ K' : PolygonCapTranslateSpace Θ,
+        K'.val = polygonHeightCap (raisedPolygonSupport K t ε) := by
+  by_cases htendpoint : t.val ∈ ({Θ.angle, Real.pi / 2} : Set ℝ)
+  · exact polygonCap_positive_height_increment_of_endpoint K t htendpoint ht
+  · exact polygonCap_positive_height_increment_of_not_endpoint K t htendpoint
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Bounds.LegComputation`.
+* `Bounds.Lower.Profile`.
+* `Bounds.NicheLimits`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Bounds / Leg Computation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Set MeasureTheory
+
+namespace MovingSofa
+
+private theorem rightAngle_allowedReal_gap_left (n : ℕ) (hn : 2 ≤ n) {t r : ℝ}
+    (ht : t ∈ (rightAngleSet n hn).directions)
+    (hr : r ∈ angleDomain (rightAngleSet n hn) ∪ {3 * Real.pi / 2}) :
+    r ∈ Set.Icc (t - polygonStepSize n + Real.pi / 2 - Real.pi)
+        (t - polygonStepSize n + Real.pi / 2) ∨
+    r ∈ Set.Icc (t + Real.pi / 2) (t + Real.pi / 2 + Real.pi) := by
+  have hδpos : 0 < polygonStepSize n := by simp only [polygonStepSize]; positivity
+  obtain ⟨hδt, htδ⟩ := rightAngleSet_direction_bounds n hn ht
+  have htI := (rightAngleSet n hn).interior t ht
+  change t ∈ Set.Ioo 0 (Real.pi / 2) at htI
+  obtain ⟨ht0, htT⟩ := htI
+  rcases hr with hr | rfl
+  · change r ∈ ((rightAngleSet n hn).directions : Set ℝ) ∪
+        ((fun q : ℝ ↦ q + Real.pi / 2) ''
+          ((rightAngleSet n hn).directions : Set ℝ)) ∪
+        {(rightAngleSet n hn).angle, Real.pi / 2} at hr
+    rcases hr with (hr | ⟨q, hq, rfl⟩) | hr
+    · have hrI := (rightAngleSet n hn).interior r hr
+      change r ∈ Set.Ioo 0 (Real.pi / 2) at hrI
+      obtain ⟨hr0, hrT⟩ := hrI
+      left
+      constructor <;> linarith [Real.pi_pos]
+    · have hqI := (rightAngleSet n hn).interior q hq
+      change q ∈ Set.Ioo 0 (Real.pi / 2) at hqI
+      obtain ⟨hq0, hqT⟩ := hqI
+      simp only
+      rcases rightAngleSet_direction_le_pred_or_ge n hn hq ht with hqle | hqge
+      · left
+        constructor
+        · linarith [Real.pi_pos]
+        · linarith
+      · right
+        constructor <;> linarith [Real.pi_pos]
+    · simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hr
+      rcases hr with rfl | rfl <;> left
+      all_goals
+        change _ ≤ Real.pi / 2 ∧ Real.pi / 2 ≤ _
+        constructor <;> linarith [Real.pi_pos]
+  · right
+    constructor <;> linarith [Real.pi_pos]
+
+private theorem rightAngle_allowedReal_gap_right (n : ℕ) (hn : 2 ≤ n) {t r : ℝ}
+    (ht : t ∈ (rightAngleSet n hn).directions)
+    (hr : r ∈ angleDomain (rightAngleSet n hn) ∪ {3 * Real.pi / 2}) :
+    r ∈ Set.Icc (t + Real.pi / 2 - Real.pi) (t + Real.pi / 2) ∨
+      r ∈ Set.Icc (t + polygonStepSize n + Real.pi / 2)
+        (t + polygonStepSize n + Real.pi / 2 + Real.pi) := by
+  have hδpos : 0 < polygonStepSize n := by simp only [polygonStepSize]; positivity
+  obtain ⟨hδt, htδ⟩ := rightAngleSet_direction_bounds n hn ht
+  have htI := (rightAngleSet n hn).interior t ht
+  change t ∈ Set.Ioo 0 (Real.pi / 2) at htI
+  obtain ⟨ht0, htT⟩ := htI
+  rcases hr with hr | rfl
+  · change r ∈ ((rightAngleSet n hn).directions : Set ℝ) ∪
+        ((fun q : ℝ ↦ q + Real.pi / 2) ''
+          ((rightAngleSet n hn).directions : Set ℝ)) ∪
+        {(rightAngleSet n hn).angle, Real.pi / 2} at hr
+    rcases hr with (hr | ⟨q, hq, rfl⟩) | hr
+    · have hrI := (rightAngleSet n hn).interior r hr
+      change r ∈ Set.Ioo 0 (Real.pi / 2) at hrI
+      obtain ⟨hr0, hrT⟩ := hrI
+      left
+      constructor <;> linarith [Real.pi_pos]
+    · have hqI := (rightAngleSet n hn).interior q hq
+      change q ∈ Set.Ioo 0 (Real.pi / 2) at hqI
+      obtain ⟨hq0, hqT⟩ := hqI
+      simp only
+      rcases rightAngleSet_direction_le_or_succ_le n hn hq ht with hqle | hqge
+      · left
+        constructor <;> linarith [Real.pi_pos]
+      · right
+        constructor <;> linarith [Real.pi_pos]
+    · simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hr
+      rcases hr with rfl | rfl <;> left
+      all_goals
+        change _ ≤ Real.pi / 2 ∧ Real.pi / 2 ≤ _
+        constructor <;> linarith [Real.pi_pos]
+  · right
+    constructor <;> linarith [Real.pi_pos]
+
+/-- The preceding shifted grid support is attained at the negative left-wall contact. -/
+private theorem rightAngle_supportValue_prev_shift (n : ℕ) (hn : 2 ≤ n)
+    (K : PolygonCapSpace (rightAngleSet n hn)) {t : ℝ}
+    (ht : t ∈ (rightAngleSet n hn).directions) :
+    supportValue K.val.val
+        ((t - polygonStepSize n + Real.pi / 2 : ℝ) : Real.Angle) =
+      inner ℝ (capVertices K.val t).2.2
+        (normalVector ((t - polygonStepSize n + Real.pi / 2 : ℝ) : Real.Angle)) := by
+  let a := t - polygonStepSize n + Real.pi / 2
+  let b := t + Real.pi / 2
+  let R := angleDomain (rightAngleSet n hn) ∪ {3 * Real.pi / 2}
+  have hδpos : 0 < polygonStepSize n := by simp only [polygonStepSize]; positivity
+  have hδpi : polygonStepSize n < Real.pi := by
+    have hnpos : (0 : ℝ) < n := by positivity
+    have hn2 : (2 : ℝ) ≤ n := by exact_mod_cast hn
+    have hδT : polygonStepSize n < Real.pi / 2 := by
+      simp only [polygonStepSize]
+      apply (div_lt_iff₀ hnpos).2
+      nlinarith [Real.pi_pos]
+    linarith [Real.pi_pos]
+  have hrepr : HasHalfPlaneRepresentation K.val.val
+      ((fun r : ℝ ↦ (r : Real.Angle)) '' R) := by
+    rw [← rightAngle_polygon_normals_eq n hn]
+    exact K.property
+  have hp : supportingIntersection K.val.val (a : Real.Angle) (b : Real.Angle) ∈ K.val.val := by
+    apply hrepr.supportingIntersection_mem_of_gap K.val.val R
+    · dsimp [a, b]
+      linarith
+    · dsimp [a, b]
+      linarith
+    · intro r hr
+      simpa only [a, b] using rightAngle_allowedReal_gap_left n hn ht hr
+  have heq := supportingIntersection_eq_edgeVertices_snd_of_mem K.val.val
+    (a := a) (b := b) (by dsimp [a, b]; linarith)
+    (by dsimp [a, b]; linarith) hp
+  have hinter := supportingIntersection_inner_left K.val.val a b
+  change inner ℝ (supportingIntersection K.val.val (a : Real.Angle) (b : Real.Angle))
+      (normalVector (a : Real.Angle)) = supportValue K.val.val (a : Real.Angle) at hinter
+  change supportValue K.val.val (a : Real.Angle) =
+    inner ℝ (edgeVertices K.val.val (b : Real.Angle)).2 (normalVector (a : Real.Angle))
+  rw [← heq, hinter]
+
+/-- The next shifted support is attained at the positive contact, including the final grid point. -/
+private theorem rightAngle_supportValue_next_shift (n : ℕ) (hn : 2 ≤ n)
+    (K : PolygonCapSpace (rightAngleSet n hn)) {t : ℝ}
+    (ht : t ∈ (rightAngleSet n hn).directions) :
+    supportValue K.val.val
+        ((t + polygonStepSize n + Real.pi / 2 : ℝ) : Real.Angle) =
+      inner ℝ (capVertices K.val t).2.1
+        (normalVector ((t + polygonStepSize n + Real.pi / 2 : ℝ) : Real.Angle)) := by
+  let a := t + Real.pi / 2
+  let b := t + polygonStepSize n + Real.pi / 2
+  let R := angleDomain (rightAngleSet n hn) ∪ {3 * Real.pi / 2}
+  have hδpos : 0 < polygonStepSize n := by simp only [polygonStepSize]; positivity
+  have hδpi : polygonStepSize n < Real.pi := by
+    have hnpos : (0 : ℝ) < n := by positivity
+    have hn2 : (2 : ℝ) ≤ n := by exact_mod_cast hn
+    have hδT : polygonStepSize n < Real.pi / 2 := by
+      simp only [polygonStepSize]
+      apply (div_lt_iff₀ hnpos).2
+      nlinarith [Real.pi_pos]
+    linarith [Real.pi_pos]
+  have hrepr : HasHalfPlaneRepresentation K.val.val
+      ((fun r : ℝ ↦ (r : Real.Angle)) '' R) := by
+    rw [← rightAngle_polygon_normals_eq n hn]
+    exact K.property
+  have hp : supportingIntersection K.val.val (a : Real.Angle) (b : Real.Angle) ∈ K.val.val := by
+    apply hrepr.supportingIntersection_mem_of_gap K.val.val R
+    · dsimp [a, b]
+      linarith
+    · dsimp [a, b]
+      linarith
+    · intro r hr
+      simpa only [a, b] using rightAngle_allowedReal_gap_right n hn ht hr
+  have heq := supportingIntersection_eq_edgeVertices_fst_of_mem K.val.val
+    (a := a) (b := b) (by dsimp [a, b]; linarith)
+    (by dsimp [a, b]; linarith) hp
+  have hinter := supportingIntersection_inner_right K.val.val a b
+    (show Real.sin (b - a) ≠ 0 by
+      apply ne_of_gt
+      apply Real.sin_pos_of_pos_of_lt_pi <;> dsimp [a, b] <;> linarith)
+  change inner ℝ (supportingIntersection K.val.val (a : Real.Angle) (b : Real.Angle))
+      (normalVector (b : Real.Angle)) = supportValue K.val.val (b : Real.Angle) at hinter
+  change supportValue K.val.val (b : Real.Angle) =
+    inner ℝ (edgeVertices K.val.val (a : Real.Angle)).1 (normalVector (b : Real.Angle))
+  rw [← heq, hinter]
+
+private theorem one_sub_cos_div_cos_eq_tan_mul_tan_half {δ : ℝ}
+    (hδ0 : 0 < δ) (hδT : δ < Real.pi / 2) :
+    (1 - Real.cos δ) / Real.cos δ = Real.tan δ * Real.tan (δ / 2) := by
+  have hc : Real.cos δ ≠ 0 := ne_of_gt
+    (Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos], hδT⟩)
+  have hch : Real.cos (δ / 2) ≠ 0 := ne_of_gt
+    (Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos], by linarith⟩)
+  rw [Real.tan_eq_sin_div_cos, Real.tan_eq_sin_div_cos]
+  field_simp [hc, hch]
+  have htwo : δ = 2 * (δ / 2) := by ring
+  have hcos : Real.cos δ = 1 - 2 * Real.sin (δ / 2) ^ 2 :=
+    (congrArg Real.cos htwo).trans (Real.cos_two_mul_eq_one_sub _)
+  have hsin : Real.sin δ = 2 * Real.sin (δ / 2) * Real.cos (δ / 2) :=
+    (congrArg Real.sin htwo).trans (Real.sin_two_mul _)
+  rw [hcos, hsin]
+  ring
+
+private theorem polygonStepSize_mem_Ioo (n : ℕ) (hn : 2 ≤ n) :
+    polygonStepSize n ∈ Set.Ioo 0 (Real.pi / 2) := by
+  have hnpos : (0 : ℝ) < n := by positivity
+  have hn2 : (2 : ℝ) ≤ n := by exact_mod_cast hn
+  constructor
+  · simp only [polygonStepSize]
+    positivity
+  · simp only [polygonStepSize]
+    apply (div_lt_iff₀ hnpos).2
+    nlinarith [Real.pi_pos]
+
+private theorem rightAngle_bRay_prev_leg_length (n : ℕ) (hn : 2 ≤ n)
+    (K : PolygonCapSpace (rightAngleSet n hn)) {t : ℝ}
+    (ht : t ∈ (rightAngleSet n hn).directions) :
+    Measure.hausdorffMeasure 1
+        ((rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).bRay ∩
+          (innerWallUpperHalfPlanes K.val (t - polygonStepSize n)).2) =
+      ENNReal.ofReal (Real.tan (polygonStepSize n) *
+        max 0 ((tangentArmLengths K.val t).2.2 - 1 +
+          Real.tan (polygonStepSize n / 2))) := by
+  let δ := polygonStepSize n
+  let x := (rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).innerCorner
+  let y := (rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).outerCorner
+  let c := (capVertices K.val t).2.2
+  let g := (tangentArmLengths K.val t).2.2
+  let u := t - δ + Real.pi / 2
+  have hδI : δ ∈ Set.Ioo 0 (Real.pi / 2) := polygonStepSize_mem_Ioo n hn
+  have hcδ : 0 < Real.cos δ := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [hδI.1, Real.pi_pos], hδI.2⟩
+  have hnormal : normalVector (u : Real.Angle) =
+      Real.sin δ • normalVector (t : Real.Angle) +
+        Real.cos δ • tangentVector (t : Real.Angle) := by
+    have h := normalVector_add_real t (Real.pi / 2 - δ)
+    rw [Real.cos_pi_div_two_sub, Real.sin_pi_div_two_sub] at h
+    have hu : u = t + (Real.pi / 2 - δ) := by dsimp [u]; ring
+    rw [hu]
+    exact h
+  have htrans : 0 < inner ℝ (tangentVector (t : Real.Angle))
+      (normalVector (u : Real.Angle)) := by
+    rw [hnormal, inner_add_right, inner_smul_right, inner_smul_right,
+      real_inner_comm, inner_normalVector_tangentVector, inner_tangentVector_self]
+    simpa using hcδ
+  rw [show (innerWallUpperHalfPlanes K.val (t - polygonStepSize n)).2 =
+      normalHalfPlane (u : Real.Angle)
+        (supportValue K.val.val (u : Real.Angle) - 1) true false by
+    simp only [innerWallUpperHalfPlanes, u, δ]
+    ]
+  rw [hausdorffMeasure_bRay_inter_normalHalfPlane _ _ _ _ htrans]
+  have hx : x = y - normalVector (t : Real.Angle) - tangentVector (t : Real.Angle) := by
+    have hf := rotatingHallwayParts_formulas (K.val.val : Set Point) (t : Real.Angle)
+    rw [show x = (rotatingHallwayParts (K.val.val : Set Point)
+      (t : Real.Angle)).innerCorner by rfl, hf.2.1]
+    rw [show y = (rotatingHallwayParts (K.val.val : Set Point)
+      (t : Real.Angle)).outerCorner by rfl, hf.2.2.1]
+    module
+  have hy : y = c + g • normalVector (t : Real.Angle) := by
+    change (rotatingHallwayParts (K.val.val : Set Point)
+      (t : Real.Angle)).outerCorner =
+        (capVertices K.val t).2.2 +
+          (tangentArmLengths K.val t).2.2 • normalVector (t : Real.Angle)
+    exact (capTangentArm_identities K.val t).2.2.2
+  have hsupport : supportValue K.val.val (u : Real.Angle) =
+      inner ℝ c (normalVector (u : Real.Angle)) := by
+    simpa only [u, δ, c] using rightAngle_supportValue_prev_shift n hn K ht
+  have hd : inner ℝ (tangentVector (t : Real.Angle))
+      (normalVector (u : Real.Angle)) = Real.cos δ := by
+    rw [hnormal, inner_add_right, inner_smul_right, inner_smul_right,
+      real_inner_comm, inner_normalVector_tangentVector, inner_tangentVector_self]
+    ring
+  have hnum : inner ℝ x (normalVector (u : Real.Angle)) -
+      (supportValue K.val.val (u : Real.Angle) - 1) =
+      (g - 1) * Real.sin δ + 1 - Real.cos δ := by
+    rw [hx, hy, hsupport, hnormal]
+    simp only [inner_sub_left, inner_add_left,
+      inner_add_right, inner_smul_right, inner_normalVector_self,
+      inner_normalVector_tangentVector, real_inner_comm, inner_tangentVector_self]
+    ring
+  have hratio :
+      (inner ℝ x (normalVector (u : Real.Angle)) -
+          (supportValue K.val.val (u : Real.Angle) - 1)) /
+          inner ℝ (tangentVector (t : Real.Angle)) (normalVector (u : Real.Angle)) =
+        Real.tan δ * (g - 1 + Real.tan (δ / 2)) := by
+    rw [hnum, hd]
+    calc
+      ((g - 1) * Real.sin δ + 1 - Real.cos δ) / Real.cos δ =
+          Real.tan δ * (g - 1) + (1 - Real.cos δ) / Real.cos δ := by
+            rw [Real.tan_eq_sin_div_cos]
+            field_simp [hcδ.ne']
+            ring
+      _ = Real.tan δ * (g - 1) + Real.tan δ * Real.tan (δ / 2) := by
+        rw [one_sub_cos_div_cos_eq_tan_mul_tan_half hδI.1 hδI.2]
+      _ = Real.tan δ * (g - 1 + Real.tan (δ / 2)) := by ring
+  change ENNReal.ofReal (max 0 _) = ENNReal.ofReal (_ * max 0 _)
+  rw [hratio]
+  congr 1
+  rw [mul_max_of_nonneg _ _ (Real.tan_pos_of_pos_of_lt_pi_div_two hδI.1 hδI.2).le,
+    mul_zero]
+
+private theorem rightAngle_bRay_next_leg_length (n : ℕ) (hn : 2 ≤ n)
+    (K : PolygonCapSpace (rightAngleSet n hn)) {t : ℝ}
+    (ht : t ∈ (rightAngleSet n hn).directions) :
+    Measure.hausdorffMeasure 1
+        ((rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).bRay ∩
+          (innerWallUpperHalfPlanes K.val (t + polygonStepSize n)).2) =
+      ENNReal.ofReal (Real.tan (polygonStepSize n) *
+        max 0 (1 - (tangentArmLengths K.val t).2.1 +
+          Real.tan (polygonStepSize n / 2))) := by
+  let δ := polygonStepSize n
+  let x := (rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).innerCorner
+  let y := (rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).outerCorner
+  let c := (capVertices K.val t).2.1
+  let g := (tangentArmLengths K.val t).2.1
+  let u := t + δ + Real.pi / 2
+  have hδI : δ ∈ Set.Ioo 0 (Real.pi / 2) := polygonStepSize_mem_Ioo n hn
+  have hcδ : 0 < Real.cos δ := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [hδI.1, Real.pi_pos], hδI.2⟩
+  have hnormal : normalVector (u : Real.Angle) =
+      (-Real.sin δ) • normalVector (t : Real.Angle) +
+        Real.cos δ • tangentVector (t : Real.Angle) := by
+    have h := normalVector_add_real t (Real.pi / 2 + δ)
+    simp only [Real.cos_add, Real.sin_add, Real.cos_pi_div_two,
+      Real.sin_pi_div_two, zero_mul, one_mul, zero_sub, add_zero] at h
+    have hu : u = t + (Real.pi / 2 + δ) := by dsimp [u]; ring
+    rw [hu]
+    exact h
+  have htrans : 0 < inner ℝ (tangentVector (t : Real.Angle))
+      (normalVector (u : Real.Angle)) := by
+    rw [hnormal, inner_add_right, inner_smul_right, inner_smul_right,
+      real_inner_comm, inner_normalVector_tangentVector, inner_tangentVector_self]
+    simpa using hcδ
+  rw [show (innerWallUpperHalfPlanes K.val (t + polygonStepSize n)).2 =
+      normalHalfPlane (u : Real.Angle)
+        (supportValue K.val.val (u : Real.Angle) - 1) true false by
+    simp only [innerWallUpperHalfPlanes, u, δ]
+    ]
+  rw [hausdorffMeasure_bRay_inter_normalHalfPlane _ _ _ _ htrans]
+  have hx : x = y - normalVector (t : Real.Angle) - tangentVector (t : Real.Angle) := by
+    have hf := rotatingHallwayParts_formulas (K.val.val : Set Point) (t : Real.Angle)
+    rw [show x = (rotatingHallwayParts (K.val.val : Set Point)
+      (t : Real.Angle)).innerCorner by rfl, hf.2.1]
+    rw [show y = (rotatingHallwayParts (K.val.val : Set Point)
+      (t : Real.Angle)).outerCorner by rfl, hf.2.2.1]
+    module
+  have hy : y = c + g • normalVector (t : Real.Angle) := by
+    change (rotatingHallwayParts (K.val.val : Set Point)
+      (t : Real.Angle)).outerCorner =
+        (capVertices K.val t).2.1 +
+          (tangentArmLengths K.val t).2.1 • normalVector (t : Real.Angle)
+    exact (capTangentArm_identities K.val t).2.2.1
+  have hsupport : supportValue K.val.val (u : Real.Angle) =
+      inner ℝ c (normalVector (u : Real.Angle)) := by
+    simpa only [u, δ, c] using rightAngle_supportValue_next_shift n hn K ht
+  have hd : inner ℝ (tangentVector (t : Real.Angle))
+      (normalVector (u : Real.Angle)) = Real.cos δ := by
+    rw [hnormal, inner_add_right, inner_smul_right, inner_smul_right,
+      real_inner_comm, inner_normalVector_tangentVector, inner_tangentVector_self]
+    ring
+  have hnum : inner ℝ x (normalVector (u : Real.Angle)) -
+      (supportValue K.val.val (u : Real.Angle) - 1) =
+      (1 - g) * Real.sin δ + 1 - Real.cos δ := by
+    rw [hx, hy, hsupport, hnormal]
+    simp only [inner_sub_left, inner_add_left, inner_add_right, inner_smul_right,
+      inner_normalVector_self, inner_normalVector_tangentVector, real_inner_comm,
+      inner_tangentVector_self]
+    ring
+  have hratio :
+      (inner ℝ x (normalVector (u : Real.Angle)) -
+          (supportValue K.val.val (u : Real.Angle) - 1)) /
+          inner ℝ (tangentVector (t : Real.Angle)) (normalVector (u : Real.Angle)) =
+        Real.tan δ * (1 - g + Real.tan (δ / 2)) := by
+    rw [hnum, hd]
+    calc
+      ((1 - g) * Real.sin δ + 1 - Real.cos δ) / Real.cos δ =
+          Real.tan δ * (1 - g) + (1 - Real.cos δ) / Real.cos δ := by
+            rw [Real.tan_eq_sin_div_cos]
+            field_simp [hcδ.ne']
+            ring
+      _ = Real.tan δ * (1 - g) + Real.tan δ * Real.tan (δ / 2) := by
+        rw [one_sub_cos_div_cos_eq_tan_mul_tan_half hδI.1 hδI.2]
+      _ = Real.tan δ * (1 - g + Real.tan (δ / 2)) := by ring
+  change ENNReal.ofReal (max 0 _) = ENNReal.ofReal (_ * max 0 _)
+  rw [hratio]
+  congr 1
+  rw [mul_max_of_nonneg _ _ (Real.tan_pos_of_pos_of_lt_pi_div_two hδI.1 hδI.2).le,
+    mul_zero]
+
+theorem maximumPolygonCap_leg_lengths (n : ℕ) (hn : 2 ≤ n)
+    (K : PolygonCapSpace (rightAngleSet n hn))
+    (t : ℝ) (ht : t ∈ (rightAngleSet n hn).directions) :
+    MeasureTheory.Measure.hausdorffMeasure 1
+      ((rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).bRay ∩
+        (innerWallUpperHalfPlanes K.val (t - polygonStepSize n)).2) =
+      ENNReal.ofReal (Real.tan (polygonStepSize n) *
+        max 0 ((tangentArmLengths K.val t).2.2 - 1 +
+          Real.tan (polygonStepSize n / 2))) ∧
+    MeasureTheory.Measure.hausdorffMeasure 1
+      ((rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).bRay ∩
+        (innerWallUpperHalfPlanes K.val (t + polygonStepSize n)).2) =
+      ENNReal.ofReal (Real.tan (polygonStepSize n) *
+        max 0 (1 - (tangentArmLengths K.val t).2.1 +
+          Real.tan (polygonStepSize n / 2))) := by
+  exact ⟨rightAngle_bRay_prev_leg_length n hn K ht,
+    rightAngle_bRay_next_leg_length n hn K ht⟩
+
+/-- The face line of a convex body meets the two adjacent inner-wall half-planes in a set whose
+length is at most `2 tan(δ/2)` less the face length. -/
+theorem hausdorffMeasure_faceLine_inter_innerWalls_le (K : ConvexBody Point)
+    (t δ : ℝ) (hδ0 : 0 < δ) (hδ : δ < Real.pi) :
+    Measure.hausdorffMeasure 1
+        ({p : Point | inner ℝ p (normalVector (t : Real.Angle)) =
+            supportValue (K : Set Point) (t : Real.Angle) - 1} ∩
+          (normalHalfPlane ((t - δ : ℝ) : Real.Angle)
+              (supportValue (K : Set Point) ((t - δ : ℝ) : Real.Angle) - 1) true false ∩
+            normalHalfPlane ((t + δ : ℝ) : Real.Angle)
+              (supportValue (K : Set Point) ((t + δ : ℝ) : Real.Angle) - 1) true false)) ≤
+      ENNReal.ofReal (max 0 (2 * Real.tan (δ / 2) -
+        (surfaceAreaMeasure K {(t : Real.Angle)}).toReal)) := by
+  have hpi := Real.pi_pos
+  have hsin : 0 < Real.sin δ := Real.sin_pos_of_pos_of_lt_pi hδ0 hδ
+  have hhalfcos : 0 < Real.cos (δ / 2) := Real.cos_pos_of_mem_Ioo ⟨by linarith, by linarith⟩
+  have hhalfsin : 0 < Real.sin (δ / 2) := Real.sin_pos_of_pos_of_lt_pi (by linarith) (by linarith)
+  have htanhalf : (1 - Real.cos δ) / Real.sin δ = Real.tan (δ / 2) := by
+    have h1 : Real.sin δ = 2 * Real.sin (δ / 2) * Real.cos (δ / 2) := by
+      have h := Real.sin_two_mul (δ / 2)
+      rwa [show 2 * (δ / 2) = δ by ring] at h
+    have h2 : Real.cos δ = 2 * Real.cos (δ / 2) ^ 2 - 1 := by
+      have h := Real.cos_two_mul (δ / 2)
+      rwa [show 2 * (δ / 2) = δ by ring] at h
+    rw [Real.tan_eq_sin_div_cos, h1, h2, div_eq_div_iff (by positivity) hhalfcos.ne']
+    nlinarith [Real.sin_sq_add_cos_sq (δ / 2)]
+  have hneg : t - (t + δ) = -δ := by ring
+  have hpos : t - (t - δ) = δ := by ring
+  have hAn : inner ℝ (edgeVertices K (t : Real.Angle)).1 (normalVector (t : Real.Angle)) =
+      supportValue (K : Set Point) (t : Real.Angle) := (edgeVertices_fst_mem K _).2
+  have hBn : inner ℝ (edgeVertices K (t : Real.Angle)).2 (normalVector (t : Real.Angle)) =
+      supportValue (K : Set Point) (t : Real.Angle) := (edgeVertices_snd_mem K _).2
+  have hApb := inner_le_supportValue K (edgeVertices_fst_mem K (t : Real.Angle)).1
+    ((t + δ : ℝ) : Real.Angle)
+  have hBmb := inner_le_supportValue K (edgeVertices_snd_mem K (t : Real.Angle)).1
+    ((t - δ : ℝ) : Real.Angle)
+  rw [inner_normalVector_eq_frame_rotate _ t (t + δ), hneg, Real.cos_neg, Real.sin_neg,
+    hAn] at hApb
+  rw [inner_normalVector_eq_frame_rotate _ t (t - δ), hpos, hBn] at hBmb
+  have hAB := (surfaceAreaMeasure_atom_length K (t : Real.Angle)).2.2
+  have hgap : inner ℝ (edgeVertices K (t : Real.Angle)).1 (tangentVector (t : Real.Angle)) =
+      inner ℝ (edgeVertices K (t : Real.Angle)).2 (tangentVector (t : Real.Angle)) +
+        (surfaceAreaMeasure K {(t : Real.Angle)}).toReal := by
+    rw [hAB, inner_add_left, real_inner_smul_left, inner_tangentVector_self, mul_one]
+  refine le_trans (hausdorffMeasure_le_of_frame_bounds (t := t)
+    (e := supportValue (K : Set Point) (t : Real.Angle) - 1)
+    (lo := inner ℝ (edgeVertices K (t : Real.Angle)).1 (tangentVector (t : Real.Angle)) -
+      Real.tan (δ / 2))
+    (hi := inner ℝ (edgeVertices K (t : Real.Angle)).2 (tangentVector (t : Real.Angle)) +
+      Real.tan (δ / 2)) ?_) (le_of_eq ?_)
+  · rintro p ⟨hline, hprev, hnext⟩
+    have hpn : inner ℝ p (normalVector (t : Real.Angle)) =
+        supportValue (K : Set Point) (t : Real.Angle) - 1 := hline
+    change supportValue (K : Set Point) ((t - δ : ℝ) : Real.Angle) - 1 ≤
+      inner ℝ p (normalVector ((t - δ : ℝ) : Real.Angle)) at hprev
+    change supportValue (K : Set Point) ((t + δ : ℝ) : Real.Angle) - 1 ≤
+      inner ℝ p (normalVector ((t + δ : ℝ) : Real.Angle)) at hnext
+    rw [inner_normalVector_eq_frame_rotate p t (t - δ), hpos, hpn] at hprev
+    rw [inner_normalVector_eq_frame_rotate p t (t + δ), hneg, Real.cos_neg, Real.sin_neg,
+      hpn] at hnext
+    have hkeyU : (inner ℝ p (tangentVector (t : Real.Angle)) -
+        inner ℝ (edgeVertices K (t : Real.Angle)).2 (tangentVector (t : Real.Angle))) *
+          Real.sin δ ≤ 1 - Real.cos δ := by nlinarith [hprev, hBmb]
+    have hkeyL : (inner ℝ (edgeVertices K (t : Real.Angle)).1 (tangentVector (t : Real.Angle)) -
+        inner ℝ p (tangentVector (t : Real.Angle))) * Real.sin δ ≤ 1 - Real.cos δ := by
+      nlinarith [hnext, hApb]
+    have hU := (le_div_iff₀ hsin).mpr hkeyU
+    have hL := (le_div_iff₀ hsin).mpr hkeyL
+    rw [htanhalf] at hU hL
+    exact ⟨hpn, by linarith, by linarith⟩
+  · have hlen : inner ℝ (edgeVertices K (t : Real.Angle)).2 (tangentVector (t : Real.Angle)) +
+        Real.tan (δ / 2) -
+        (inner ℝ (edgeVertices K (t : Real.Angle)).1 (tangentVector (t : Real.Angle)) -
+          Real.tan (δ / 2)) =
+        2 * Real.tan (δ / 2) - (surfaceAreaMeasure K {(t : Real.Angle)}).toReal := by
+      rw [hgap]
+      ring
+    rw [hlen]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Bounds / Lower / Profile
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Integrate the reflected arm profile through the second magic function and add one. -/
+def armIntegralOperator (f : C(Set.Icc (0 : ℝ) (Real.pi / 2), NNReal)) :
+    C(Set.Icc (0 : ℝ) (Real.pi / 2), ℝ) where
+  toFun x := 1 + ∫ u in (0 : ℝ)..(x : ℝ),
+    magicFunctions.2 (f (Set.projIcc 0 (Real.pi / 2) (by positivity) (Real.pi / 2 - u)))
+  continuous_toFun := by
+    have hf : Continuous (fun u : ℝ ↦
+        magicFunctions.2 (f (Set.projIcc 0 (Real.pi / 2) (by positivity)
+          (Real.pi / 2 - u)))) := by
+      unfold magicFunctions
+      fun_prop
+    exact continuous_const.add
+      ((intervalIntegral.differentiable_integral_of_continuous hf).continuous.comp
+        continuous_subtype_val)
+
+/-- Iteratively improve the nonnegative arm lower bound by taking the maximum with its
+integral update. -/
+def armLowerBoundSequence : ℕ → C(Set.Icc (0 : ℝ) (Real.pi / 2), NNReal)
+  | 0 => ⟨fun _ ↦ 0, continuous_const⟩
+  | n + 1 =>
+      ⟨fun x ↦ max (armLowerBoundSequence n x)
+          (Real.toNNReal (armIntegralOperator (armLowerBoundSequence n) x)),
+        (armLowerBoundSequence n).continuous.max
+          (continuous_real_toNNReal.comp (armIntegralOperator (armLowerBoundSequence
+            n)).continuous)⟩
+
+/-- The continuous profile `max (1 - x) c` on the quarter-turn interval. -/
+def lowerBoundProfile (c : Set.Icc (0 : ℝ) 1) :
+    C(Set.Icc (0 : ℝ) (Real.pi / 2), ℝ) where
+  toFun x := max (1 - (x : ℝ)) (c : ℝ)
+  continuous_toFun := (continuous_const.sub continuous_subtype_val).max continuous_const
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Bounds / Niche Limits
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter MeasureTheory
+open scoped Topology
+
+namespace MovingSofa
+
+/-- Support values converge under Hausdorff convergence of caps. -/
+theorem tendsto_supportValue_of_hausdorff {ω : ℝ}
+    (K : ℕ → CapSpace ω) (L : CapSpace ω)
+    (hlim : Tendsto (fun i ↦ Metric.hausdorffDist ((K i).val : Set Point)
+      (L.val : Set Point)) atTop (𝓝 0)) (t : ℝ) :
+    Tendsto (fun i ↦ supportValue (K i).val (t : Real.Angle)) atTop
+      (𝓝 (supportValue L.val (t : Real.Angle))) := by
+  apply tendsto_iff_dist_tendsto_zero.mpr
+  apply squeeze_zero (fun _ ↦ dist_nonneg) _ hlim
+  intro i
+  simpa only [Real.dist_eq, vectorSupport, supportValue] using
+    (compactSet_support_continuity (K i).val L.val (K i).val.nonempty
+      (K i).val.isCompact L.val.nonempty L.val.isCompact).2.1
+        (normalVector (t : Real.Angle)) (norm_normalVector_real t)
+
+private theorem volume_normalLine_eq_zero (t c : ℝ) :
+    volume {p : Point | inner ℝ p (normalVector (t : Real.Angle)) = c} = 0 := by
+  let u := normalVector (t : Real.Angle)
+  let f : Point →ᵃ[ℝ] ℝ := (innerSL ℝ u).toLinearMap.toAffineMap
+  let A := (AffineSubspace.mk' c (⊥ : Submodule ℝ ℝ)).comap f
+  have hA : (A : Set Point) = {p : Point | inner ℝ p u = c} := by
+    ext p
+    simp [A, f, AffineSubspace.mem_mk', real_inner_comm, sub_eq_zero]
+  rw [← hA]
+  apply Measure.addHaar_affineSubspace
+  intro htop
+  have hp : (c + 1) • u ∈ A := by rw [htop]; trivial
+  rw [← SetLike.mem_coe, hA] at hp
+  have hu : inner ℝ u u = 1 := by
+    rw [real_inner_self_eq_norm_sq, norm_normalVector_real]
+    norm_num
+  change inner ℝ ((c + 1) • u) u = c at hp
+  rw [real_inner_smul_left, hu, mul_one] at hp
+  linarith
+
+private theorem eventually_lt_iff_of_ne {f : ℕ → ℝ} {a x : ℝ}
+    (hf : Tendsto f atTop (𝓝 a)) (hxa : x ≠ a) :
+    ∀ᶠ i in atTop, (x < f i ↔ x < a) := by
+  rcases lt_or_gt_of_ne hxa with h | h
+  · filter_upwards [hf.eventually_const_lt h] with i hi
+    exact iff_of_true hi h
+  · filter_upwards [hf.eventually_lt_const h] with i hi
+    exact iff_of_false (not_lt_of_ge hi.le) (not_lt_of_ge h.le)
+
+private theorem ae_eventually_mem_polygonNiche_iff (Θ : AngleSet)
+    (K : ℕ → CapSpace Θ.angle) (L : CapSpace Θ.angle)
+    (hlim : Tendsto (fun i ↦ Metric.hausdorffDist ((K i).val : Set Point)
+      (L.val : Set Point)) atTop (𝓝 0)) :
+    ∀ᵐ p ∂volume, ∀ᶠ i in atTop, (p ∈ polygonNiche Θ (K i) ↔ p ∈ polygonNiche Θ L) := by
+  have hae (t : ℝ) : ∀ᵐ p ∂volume,
+      inner ℝ p (normalVector (t : Real.Angle)) ≠ supportValue L.val (t : Real.Angle) - 1 := by
+    apply ae_iff.mpr
+    simpa only [not_not] using volume_normalLine_eq_zero t
+      (supportValue L.val (t : Real.Angle) - 1)
+  have hall : ∀ᵐ p ∂volume, ∀ t ∈ Θ.directions,
+      inner ℝ p (normalVector (t : Real.Angle)) ≠ supportValue L.val (t : Real.Angle) - 1 ∧
+      inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) ≠
+        supportValue L.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 :=
+    Θ.directions.eventually_all.mpr fun t _ ↦ (hae t).and (hae (t + Real.pi / 2))
+  filter_upwards [hall] with p hp
+  have hq (t : ℝ) (ht : t ∈ Θ.directions) : ∀ᶠ i in atTop,
+      (p ∈ innerQuadrant (K i).val t ↔ p ∈ innerQuadrant L.val t) := by
+    have h₁ := eventually_lt_iff_of_ne
+      ((tendsto_supportValue_of_hausdorff K L hlim t).sub_const 1) (hp t ht).1
+    have h₂ := eventually_lt_iff_of_ne
+      ((tendsto_supportValue_of_hausdorff K L hlim (t + Real.pi / 2)).sub_const 1)
+      (hp t ht).2
+    filter_upwards [h₁, h₂] with i h₁ h₂
+    exact and_congr h₁ h₂
+  filter_upwards [Θ.directions.eventually_all.mpr hq] with i hi
+  simp only [polygonNiche, Set.mem_inter_iff, Set.mem_iUnion]
+  constructor
+  · rintro ⟨hf, t, ht, hqt⟩
+    exact ⟨hf, t, ht, (hi t ht).mp hqt⟩
+  · rintro ⟨hf, t, ht, hqt⟩
+    exact ⟨hf, t, ht, (hi t ht).mpr hqt⟩
+
+private theorem eventually_le_iff_of_ne {f : ℕ → ℝ} {a x : ℝ}
+    (hf : Tendsto f atTop (𝓝 a)) (hxa : x ≠ a) :
+    ∀ᶠ i in atTop, (x ≤ f i ↔ x ≤ a) := by
+  rcases lt_or_gt_of_ne hxa with h | h
+  · filter_upwards [hf.eventually_const_lt h] with i hi
+    exact iff_of_true hi.le h.le
+  · filter_upwards [hf.eventually_lt_const h] with i hi
+    exact iff_of_false (not_le_of_gt hi) (not_le_of_gt h)
+
+private theorem ae_eventually_mem_angleCap_iff (Θ : AngleSet)
+    (K : ℕ → CapSpace Θ.angle) (L : CapSpace Θ.angle)
+    (hlim : Tendsto (fun i ↦ Metric.hausdorffDist ((K i).val : Set Point)
+      (L.val : Set Point)) atTop (𝓝 0)) :
+    ∀ᵐ p ∂volume, ∀ᶠ i in atTop, (p ∈ angleCap Θ (K i) ↔ p ∈ angleCap Θ L) := by
+  have hae (t : ℝ) : ∀ᵐ p ∂volume,
+      inner ℝ p (normalVector (t : Real.Angle)) ≠ supportValue L.val (t : Real.Angle) := by
+    apply ae_iff.mpr
+    simpa only [not_not] using volume_normalLine_eq_zero t
+      (supportValue L.val (t : Real.Angle))
+  have hall : ∀ᵐ p ∂volume, ∀ t ∈ Θ.directions,
+      inner ℝ p (normalVector (t : Real.Angle)) ≠ supportValue L.val (t : Real.Angle) ∧
+      inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) ≠
+        supportValue L.val ((t + Real.pi / 2 : ℝ) : Real.Angle) :=
+    Θ.directions.eventually_all.mpr fun t _ ↦ (hae t).and (hae (t + Real.pi / 2))
+  filter_upwards [hall] with p hp
+  have hq (t : ℝ) (ht : t ∈ Θ.directions) : ∀ᶠ i in atTop,
+      (inner ℝ p (normalVector (t : Real.Angle)) ≤ supportValue (K i).val (t : Real.Angle) ↔
+        inner ℝ p (normalVector (t : Real.Angle)) ≤ supportValue L.val (t : Real.Angle)) ∧
+      (inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) ≤
+          supportValue (K i).val ((t + Real.pi / 2 : ℝ) : Real.Angle) ↔
+        inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) ≤
+          supportValue L.val ((t + Real.pi / 2 : ℝ) : Real.Angle)) :=
+    (eventually_le_iff_of_ne (tendsto_supportValue_of_hausdorff K L hlim t) (hp t ht).1).and
+      (eventually_le_iff_of_ne (tendsto_supportValue_of_hausdorff K L hlim
+        (t + Real.pi / 2)) (hp t ht).2)
+  filter_upwards [Θ.directions.eventually_all.mpr hq] with i hi
+  simp only [mem_angleCap_iff]
+  apply and_congr_right
+  intro _
+  exact forall_congr' fun t ↦ forall_congr' fun ht ↦ and_congr (hi t ht).1 (hi t ht).2
+
+private theorem exists_angleCap_norm_bound (Θ : AngleSet)
+    (K : ℕ → CapSpace Θ.angle) (L : CapSpace Θ.angle)
+    (hlim : Tendsto (fun i ↦ Metric.hausdorffDist ((K i).val : Set Point)
+      (L.val : Set Point)) atTop (𝓝 0)) :
+    ∃ R : ℝ, ∀ i p, p ∈ angleCap Θ (K i) → ‖p‖ ≤ R := by
+  obtain ⟨t, ht⟩ := Θ.nonempty
+  have hti := Θ.interior t ht
+  have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, hti.1], hti.2.trans_le Θ.angle_le⟩
+  have hs : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi hti.1
+    (by linarith [hti.2, Θ.angle_le, Real.pi_pos])
+  obtain ⟨A, hA⟩ := (Metric.isBounded_range_of_tendsto _
+    (tendsto_supportValue_of_hausdorff K L hlim t)).exists_norm_le
+  obtain ⟨B, hB⟩ := (Metric.isBounded_range_of_tendsto _
+    (tendsto_supportValue_of_hausdorff K L hlim (t + Real.pi / 2))).exists_norm_le
+  let l := -|B| / Real.sin t
+  let r := |A| / Real.cos t
+  let M := |l| + |r|
+  have hM : 0 ≤ M := by dsimp [M]; positivity
+  refine ⟨M + 1, fun i p hp ↦ ?_⟩
+  have haBound : supportValue (K i).val (t : Real.Angle) ≤ |A| :=
+    (le_abs_self _).trans ((hA _ (Set.mem_range_self i)).trans (le_abs_self A))
+  have hbBound : supportValue (K i).val ((t + Real.pi / 2 : ℝ) : Real.Angle) ≤ |B| :=
+    (le_abs_self _).trans ((hB _ (Set.mem_range_self i)).trans (le_abs_self B))
+  obtain ⟨⟨hy, _⟩, hnormals⟩ := (mem_angleCap_iff Θ (K i) p).mp hp
+  obtain ⟨ha, hb⟩ := hnormals t ht
+  simp [normalVector, frame, PiLp.inner_apply, Real.cos_add, Real.sin_add,
+    -Real.Angle.coe_add] at ha hb
+  have hl : l ≤ p 0 := by
+    apply (div_le_iff₀ hs).mpr
+    nlinarith [mul_nonneg hc.le hy.1]
+  have hr : p 0 ≤ r := by
+    apply (le_div_iff₀ hc).mpr
+    nlinarith [mul_nonneg hs.le hy.1]
+  have hx : |p 0| ≤ M := by
+    apply abs_le.mpr
+    dsimp [M]
+    constructor <;> linarith [neg_abs_le l, le_abs_self r, abs_nonneg l, abs_nonneg r]
+  have hx2 := (sq_le_sq₀ (abs_nonneg (p 0)) hM).mpr hx
+  have hy2 : (p 1) ^ 2 ≤ 1 := by nlinarith [hy.1, hy.2]
+  have hn := EuclideanSpace.norm_sq_eq p
+  simp only [Fin.sum_univ_two, Real.norm_eq_abs, sq_abs] at hn hx2
+  nlinarith [norm_nonneg p]
+
+private theorem tendsto_polygonNiche_area (Θ : AngleSet)
+    (K : ℕ → CapSpace Θ.angle) (L : CapSpace Θ.angle)
+    (hlim : Tendsto (fun i ↦ Metric.hausdorffDist ((K i).val : Set Point)
+      (L.val : Set Point)) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ ClassicalResults.area (polygonNiche Θ (K i))) atTop
+      (𝓝 (ClassicalResults.area (polygonNiche Θ L))) := by
+  obtain ⟨R, hR, _, hpoly⟩ := niche_uniform_bounds.2.2 (fun _ ↦ Θ.angle) K L.val hlim
+  let C := Metric.closedBall (0 : Point) (2 * R)
+  have hcompact : IsCompact C := isCompact_closedBall _ _
+  have hsubset (i : ℕ) : polygonNiche Θ (K i) ⊆ C := by
+    intro p hp
+    obtain ⟨hx, hy, hh⟩ := hpoly i Θ (K i) rfl hp
+    change dist p 0 ≤ 2 * R
+    rw [dist_zero_right]
+    have hx2 := (sq_le_sq₀ (abs_nonneg (p 0)) hR).mpr hx
+    have hy2 := (sq_le_sq₀ hy hR).mpr hh
+    have hn := EuclideanSpace.norm_sq_eq p
+    simp only [Fin.sum_univ_two, Real.norm_eq_abs, sq_abs] at hn hx2
+    nlinarith [norm_nonneg p, sq_nonneg R]
+  have hCint : Integrable (C.indicator fun _ ↦ (1 : ℝ)) :=
+    (integrableOn_const hcompact.measure_lt_top.ne).integrable_indicator hcompact.measurableSet
+  have hpointwise : ∀ᵐ p ∂volume,
+      Tendsto (fun i ↦ (polygonNiche Θ (K i)).indicator (fun _ ↦ (1 : ℝ)) p)
+        atTop (𝓝 ((polygonNiche Θ L).indicator (fun _ ↦ (1 : ℝ)) p)) := by
+    filter_upwards [ae_eventually_mem_polygonNiche_iff Θ K L hlim] with p hp
+    apply Tendsto.congr' _ tendsto_const_nhds
+    filter_upwards [hp] with i hi
+    by_cases h : p ∈ polygonNiche Θ L
+    · simp [h, hi.mpr h]
+    · have hn : p ∉ polygonNiche Θ (K i) := fun hk ↦ h (hi.mp hk)
+      simp [h, hn]
+  have hconv := tendsto_integral_filter_of_dominated_convergence (μ := volume)
+    (F := fun i ↦ (polygonNiche Θ (K i)).indicator fun _ ↦ (1 : ℝ))
+    (f := (polygonNiche Θ L).indicator fun _ ↦ (1 : ℝ))
+    (C.indicator fun _ ↦ (1 : ℝ))
+    (Eventually.of_forall fun i ↦
+      (measurable_const.indicator (measurableSet_polygonNiche Θ (K i))).aestronglyMeasurable)
+    (Eventually.of_forall fun i ↦ Eventually.of_forall fun p ↦ by
+      by_cases hp : p ∈ polygonNiche Θ (K i)
+      · simp [hp, hsubset i hp]
+      · by_cases hpc : p ∈ C <;> simp [hp, hpc]) hCint hpointwise
+  have harea (P : CapSpace Θ.angle) :
+      (∫ p, (polygonNiche Θ P).indicator (fun _ ↦ (1 : ℝ)) p) =
+        ClassicalResults.area (polygonNiche Θ P) := by
+    exact integral_indicator_one (measurableSet_polygonNiche Θ P)
+  simpa only [harea] using hconv
+
+private theorem tendsto_angleCap_area (Θ : AngleSet)
+    (K : ℕ → CapSpace Θ.angle) (L : CapSpace Θ.angle)
+    (hlim : Tendsto (fun i ↦ Metric.hausdorffDist ((K i).val : Set Point)
+      (L.val : Set Point)) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ ClassicalResults.area (angleCap Θ (K i))) atTop
+      (𝓝 (ClassicalResults.area (angleCap Θ L))) := by
+  obtain ⟨R, hR⟩ := exists_angleCap_norm_bound Θ K L hlim
+  let C := Metric.closedBall (0 : Point) R
+  have hcompact : IsCompact C := isCompact_closedBall _ _
+  have hsubset (i : ℕ) : angleCap Θ (K i) ⊆ C := by
+    intro p hp
+    simpa only [C, Metric.mem_closedBall, dist_zero_right] using hR i p hp
+  have hCint : Integrable (C.indicator fun _ ↦ (1 : ℝ)) :=
+    (integrableOn_const hcompact.measure_lt_top.ne).integrable_indicator hcompact.measurableSet
+  have hpointwise : ∀ᵐ p ∂volume,
+      Tendsto (fun i ↦ (angleCap Θ (K i)).indicator (fun _ ↦ (1 : ℝ)) p)
+        atTop (𝓝 ((angleCap Θ L).indicator (fun _ ↦ (1 : ℝ)) p)) := by
+    filter_upwards [ae_eventually_mem_angleCap_iff Θ K L hlim] with p hp
+    apply Tendsto.congr' _ tendsto_const_nhds
+    filter_upwards [hp] with i hi
+    by_cases h : p ∈ angleCap Θ L
+    · simp [h, hi.mpr h]
+    · have hn : p ∉ angleCap Θ (K i) := fun hk ↦ h (hi.mp hk)
+      simp [h, hn]
+  have hconv := tendsto_integral_filter_of_dominated_convergence (μ := volume)
+    (F := fun i ↦ (angleCap Θ (K i)).indicator fun _ ↦ (1 : ℝ))
+    (f := (angleCap Θ L).indicator fun _ ↦ (1 : ℝ))
+    (C.indicator fun _ ↦ (1 : ℝ))
+    (Eventually.of_forall fun i ↦
+      (measurable_const.indicator (isClosed_angleCap Θ (K i)).measurableSet).aestronglyMeasurable)
+    (Eventually.of_forall fun i ↦ Eventually.of_forall fun p ↦ by
+      by_cases hp : p ∈ angleCap Θ (K i)
+      · simp [hp, hsubset i hp]
+      · by_cases hpc : p ∈ C <;> simp [hp, hpc]) hCint hpointwise
+  have harea (P : CapSpace Θ.angle) :
+      (∫ p, (angleCap Θ P).indicator (fun _ ↦ (1 : ℝ)) p) =
+        ClassicalResults.area (angleCap Θ P) := by
+    exact integral_indicator_one (isClosed_angleCap Θ P).measurableSet
+  simpa only [harea] using hconv
+
+theorem polygonArea_continuity (Θ : AngleSet)
+    (K : ℕ → CapSpace Θ.angle) (L : CapSpace Θ.angle)
+    (hlim : Tendsto (fun i ↦ Metric.hausdorffDist ((K i).val : Set Point)
+      (L.val : Set Point)) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ ClassicalResults.area (polygonNiche Θ (K i)))
+      atTop (𝓝 (ClassicalResults.area (polygonNiche Θ L))) ∧
+    Tendsto (fun i ↦ polygonAreaFunctional Θ (K i))
+      atTop (𝓝 (polygonAreaFunctional Θ L)) := by
+  exact ⟨tendsto_polygonNiche_area Θ K L hlim,
+    (tendsto_angleCap_area Θ K L hlim).sub (tendsto_polygonNiche_area Θ K L hlim)⟩
+
+private theorem polygonNiche_mono_directions (Θ Ψ : AngleSet)
+    (K : CapSpace Θ.angle) (L : CapSpace Ψ.angle)
+    (hangle : Θ.angle = Ψ.angle) (hcarrier : (K.val : Set Point) = (L.val : Set Point))
+    (hsub : Θ.directions ⊆ Ψ.directions) :
+    polygonNiche Θ K ⊆ polygonNiche Ψ L := by
+  rintro p ⟨hp, hq⟩
+  obtain ⟨t, ht, hpt⟩ := Set.mem_iUnion₂.mp hq
+  refine ⟨?_, Set.mem_iUnion₂.mpr ⟨t, hsub ht, ?_⟩⟩
+  · simpa only [hangle] using hp
+  · simpa only [hcarrier] using hpt
+
+/-- Every niche point belongs to all sufficiently fine uniform polygon niches. -/
+theorem eventually_mem_polygonNiche_of_mem_capNiche (ω : ℝ) (hω : 0 < ω)
+    (hω' : ω ≤ Real.pi / 2) (n : ℕ → ℕ) (hn : ∀ i, 2 ≤ n i)
+    (hmono : StrictMono n) (K : CapSpace ω) {p : Point} (hp : p ∈ capNiche K) :
+    ∀ᶠ i in atTop, p ∈ polygonNiche (uniformAngleSet ω hω hω' (n i) (hn i)) K := by
+  obtain ⟨hfan, hq⟩ := hp
+  obtain ⟨t, ht, hpt⟩ := Set.mem_iUnion₂.mp hq
+  have hopen : IsOpen {s : ℝ | p ∈ innerQuadrant K.val s} := by
+    apply IsOpen.inter
+    · exact isOpen_lt (continuous_const.inner continuous_normalVector_real)
+        ((continuous_supportValue_real K.val).sub continuous_const)
+    · exact isOpen_lt
+        (continuous_const.inner (continuous_normalVector_real.comp
+          (continuous_id.add continuous_const)))
+        (((continuous_supportValue_real K.val).comp
+          (continuous_id.add continuous_const)).sub continuous_const)
+  obtain ⟨a, b, ⟨hat, htb⟩, hab⟩ := mem_nhds_iff_exists_Ioo_subset.mp (hopen.mem_nhds hpt)
+  have hat' : max 0 a < t := max_lt ht.1 hat
+  have htb' : t < min ω b := lt_min ht.2 htb
+  filter_upwards [eventually_exists_uniformAngleSet_mem_Ioo ω hω hω' n hn hmono
+    (le_max_left 0 a) (hat'.trans htb') (min_le_left ω b)] with i hi
+  obtain ⟨s, hs, hsa, hsb⟩ := hi
+  refine ⟨hfan, Set.mem_iUnion₂.mpr ⟨s, hs, hab ?_⟩⟩
+  exact ⟨(le_max_right 0 a).trans_lt hsa, hsb.trans_le (min_le_right ω b)⟩
+
+private theorem iUnion_uniform_polygonNiche (ω : ℝ) (hω : 0 < ω)
+    (hω' : ω ≤ Real.pi / 2) (n : ℕ → ℕ) (hn : ∀ i, 2 ≤ n i)
+    (hmono : StrictMono n) (K : CapSpace ω) :
+    (⋃ i, polygonNiche (uniformAngleSet ω hω hω' (n i) (hn i)) K) = capNiche K := by
+  apply Set.Subset.antisymm
+  · exact Set.iUnion_subset fun i ↦
+      polygonNiche_subset_capNiche (uniformAngleSet ω hω hω' (n i) (hn i)) K
+  · intro p hp
+    obtain ⟨i, hi⟩ := (eventually_mem_polygonNiche_of_mem_capNiche ω hω hω' n hn hmono K hp).exists
+    exact Set.mem_iUnion.mpr ⟨i, hi⟩
+
+private theorem tendsto_uniform_polygonNiche_area (ω : ℝ) (hω : 0 < ω)
+    (hω' : ω ≤ Real.pi / 2) (n : ℕ → ℕ) (hn : ∀ i, 2 ≤ n i)
+    (hmono : StrictMono n) (hdyadic : ∀ i, ∃ k : ℕ, n i = 2 ^ k) (K : CapSpace ω) :
+    Tendsto (fun i ↦ ClassicalResults.area
+      (polygonNiche (uniformAngleSet ω hω hω' (n i) (hn i)) K)) atTop
+      (𝓝 (ClassicalResults.area (capNiche K))) := by
+  have hmon : Monotone (fun i ↦ polygonNiche (uniformAngleSet ω hω hω' (n i) (hn i)) K) := by
+    intro i j hij
+    exact polygonNiche_mono_directions
+      (uniformAngleSet ω hω hω' (n i) (hn i))
+      (uniformAngleSet ω hω hω' (n j) (hn j)) K K rfl rfl
+      (uniformAngleSet_directions_mono_of_dyadic ω hω hω' n hn hmono.monotone hdyadic hij)
+  have hlim := tendsto_measure_iUnion_atTop (μ := volume) hmon
+  rw [iUnion_uniform_polygonNiche ω hω hω' n hn hmono K] at hlim
+  exact (ENNReal.tendsto_toReal (niche_uniform_bounds.1 ω K).2.2.1.ne).comp hlim
+
+theorem maximizingPolygon_nicheArea_limit (ω : ℝ) (hω : 0 < ω)
+    (hω' : ω ≤ Real.pi / 2) (n : ℕ → ℕ) (hn : ∀ i, 2 ≤ n i)
+    (hmono : StrictMono n) (hdyadic : ∀ i, ∃ k : ℕ, n i = 2 ^ k)
+    (P : ∀ i, PolygonCapSpace (uniformAngleSet ω hω hω' (n i) (hn i)))
+    (hmax : ∀ i, IsMaximumPolygonCap _ (P i)) (K : CapSpace ω)
+    (hlim : Tendsto (fun i ↦ Metric.hausdorffDist ((P i).val.val : Set Point)
+      (K.val : Set Point)) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ ClassicalResults.area
+      (polygonNiche (uniformAngleSet ω hω hω' (n i) (hn i)) (P i).val))
+      atTop (𝓝 (ClassicalResults.area (capNiche K))) := by
+  let Θ (i : ℕ) := uniformAngleSet ω hω hω' (n i) (hn i)
+  have hupper (i : ℕ) :
+      ClassicalResults.area (polygonNiche (Θ i) (P i).val) ≤
+        ClassicalResults.area ((P i).val.val : Set Point) - capAreaFunctional K := by
+    have hmax' := polygonAreaFunctional_le_maximum (Θ i) (P i) (hmax i) K
+    have hbound := (polygonArea_upperBound (Θ i)).2 K
+    have harea := (polygonArea_upperBound (Θ i)).1 (P i)
+    have h := hbound.trans hmax'
+    rw [harea] at h
+    exact le_sub_comm.mp h
+  have hupperlim : Tendsto
+      (fun i ↦ ClassicalResults.area ((P i).val.val : Set Point) - capAreaFunctional K)
+      atTop (𝓝 (ClassicalResults.area (capNiche K))) := by
+    convert (convexArea_hausdorff_continuity (fun i ↦ (P i).val.val) K.val hlim).sub_const
+      (capAreaFunctional K) using 1
+    simp only [capAreaFunctional, sub_sub_cancel]
+  apply tendsto_order.mpr
+  constructor
+  · intro a ha
+    have hex := (tendsto_uniform_polygonNiche_area ω hω hω' n hn hmono hdyadic
+      K).eventually_const_lt ha
+    obtain ⟨m, hm⟩ := hex.exists
+    have hfixed := (polygonArea_continuity (Θ m) (fun i ↦ (P i).val) K hlim).1
+    filter_upwards [hfixed.eventually_const_lt hm, eventually_ge_atTop m] with i hi hmi
+    have hsub : polygonNiche (Θ m) (P i).val ⊆ polygonNiche (Θ i) (P i).val :=
+      polygonNiche_mono_directions (Θ m) (Θ i) (P i).val (P i).val rfl rfl
+        (uniformAngleSet_directions_mono_of_dyadic ω hω hω' n hn hmono.monotone hdyadic hmi)
+    have hle : ClassicalResults.area (polygonNiche (Θ m) (P i).val) ≤
+        ClassicalResults.area (polygonNiche (Θ i) (P i).val) :=
+      ENNReal.toReal_mono (niche_uniform_bounds.2.1 (Θ i) (P i).val).2.2.1.ne (measure_mono hsub)
+    exact hi.trans_le hle
+  · intro b hb
+    filter_upwards [hupperlim.eventually_lt_const hb] with i hi
+    exact (hupper i).trans_lt hi
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Cap.NicheLimit`.
+* `Cap.Reflection`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Niche Limit
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter
+open scoped Topology
+
+namespace MovingSofa
+
+/-- Containment of uniformly approximating polygon niches persists in the cap limit. -/
+theorem capNiche_subset_of_uniform_polygonNiche_subset {ω : ℝ} (K : CapSpace ω)
+    (n : ℕ → ℕ) (hn : ∀ i, 2 ≤ n i) (hmono : StrictMono n)
+    (hdyadic : ∀ i, ∃ k : ℕ, n i = 2 ^ k)
+    (P : ∀ i, PolygonCapSpace
+      (uniformAngleSet ω K.property.1 K.property.2.1 (n i) (hn i)))
+    (hcontain : ∀ i, polygonNiche
+      (uniformAngleSet ω K.property.1 K.property.2.1 (n i) (hn i)) (P i).val ⊆
+        ((P i).val.val : Set Point))
+    (hlim : Tendsto (fun i ↦ Metric.hausdorffDist ((P i).val.val : Set Point)
+      (K.val : Set Point)) atTop (𝓝 0)) : capNiche K ⊆ (K.val : Set Point) := by
+  intro p hp
+  obtain ⟨m, hm⟩ := (eventually_mem_polygonNiche_of_mem_capNiche ω K.property.1
+    K.property.2.1 n hn hmono K hp).exists
+  obtain ⟨hfan, hquad⟩ := hm
+  obtain ⟨t, ht, hpt⟩ := Set.mem_iUnion₂.mp hquad
+  change inner ℝ p (normalVector (t : Real.Angle)) < supportValue K.val (t : Real.Angle) - 1 ∧
+    inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) <
+      supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 at hpt
+  have htend := tendsto_supportValue_of_hausdorff (fun i ↦ (P i).val) K hlim t
+  have htend' := tendsto_supportValue_of_hausdorff (fun i ↦ (P i).val) K hlim
+    (t + Real.pi / 2)
+  apply ConvexBody.mem_of_tendsto_hausdorffDist p (fun i ↦ (P i).val.val) K.val _ hlim
+  filter_upwards [(htend.sub_const 1).eventually_const_lt hpt.1,
+    (htend'.sub_const 1).eventually_const_lt hpt.2, eventually_ge_atTop m] with i hi hi' hmi
+  apply hcontain i
+  refine ⟨hfan, Set.mem_iUnion₂.mpr ⟨t, ?_, ?_⟩⟩
+  · exact uniformAngleSet_directions_mono_of_dyadic ω K.property.1 K.property.2.1
+      n hn hmono.monotone hdyadic hmi ht
+  · exact ⟨hi, hi'⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Reflection
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Reflect about the line through the origin and the distinguished parallelogram point. -/
+def mirrorReflection (ω : ℝ) (p : Point) : Point :=
+  (2 * inner ℝ p (stripParallelogram ω).2.2 /
+    inner ℝ (stripParallelogram ω).2.2 (stripParallelogram ω).2.2) •
+      (stripParallelogram ω).2.2 - p
+
+/-- Reflect every interior wall direction about half the terminal angle. -/
+def reflectedAngleSet (Θ : AngleSet) : AngleSet where
+  angle := Θ.angle
+  angle_pos := Θ.angle_pos
+  angle_le := Θ.angle_le
+  directions := Θ.directions.image (fun t ↦ Θ.angle - t)
+  nonempty := Θ.nonempty.image _
+  interior := by
+    intro t ht
+    obtain ⟨s, hs, rfl⟩ := Finset.mem_image.mp ht
+    obtain ⟨hs0, hsω⟩ := Θ.interior s hs
+    constructor <;> linarith
+
+private theorem mirrorReflection_eq_capReflection_m152940c (ω : ℝ)
+    (hω0 : 0 < ω) (hωle : ω ≤ Real.pi / 2) :
+    mirrorReflection ω = capReflection ω := by
+  change stripTopReflection ω = capReflection ω
+  exact stripTopReflection_eq_capReflection ω hω0 hωle
+
+private theorem reflectedAngle_mem_polygonNormals (Θ : AngleSet) (a : Real.Angle)
+    (ha : a ∈ ((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain Θ) ∪
+      capLowerNormals Θ.angle) :
+    reflectedAngle Θ.angle a ∈
+      ((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain (reflectedAngleSet Θ)) ∪
+        capLowerNormals Θ.angle := by
+  rcases ha with ha | ha
+  · obtain ⟨t, ht, rfl⟩ := ha
+    left
+    rw [reflectedAngle_coe]
+    refine ⟨Θ.angle + Real.pi / 2 - t, ?_, rfl⟩
+    simp only [angleDomain, Set.mem_union, Set.mem_image, Set.mem_insert_iff,
+      Set.mem_singleton_iff, Finset.mem_coe] at ht ⊢
+    rcases ht with (ht | ht) | ht
+    · left
+      right
+      refine ⟨Θ.angle - t, Finset.mem_image.mpr ⟨t, ht, rfl⟩, by ring⟩
+    · obtain ⟨s, hs, rfl⟩ := ht
+      left
+      left
+      exact Finset.mem_image.mpr ⟨s, hs, by ring⟩
+    · rcases ht with rfl | rfl
+      · exact Or.inr (Or.inr (by simp))
+      · apply Or.inr
+        apply Or.inl
+        change Θ.angle + Real.pi / 2 - Real.pi / 2 = Θ.angle
+        ring
+  · right
+    simp only [capLowerNormals, Set.mem_insert_iff, Set.mem_singleton_iff] at ha ⊢
+    rcases ha with rfl | rfl
+    · right
+      exact reflectedAngle_at_omega_add_pi Θ.angle
+    · left
+      exact reflectedAngle_at_three_pi_div_two Θ.angle
+
+private theorem reflectedBody_polygonRepresentation (Θ : AngleSet)
+    (K : PolygonCapSpace Θ) :
+    HasHalfPlaneRepresentation (reflectedBody Θ.angle K.val.val)
+      (((fun t : ℝ ↦ (t : Real.Angle)) '' angleDomain (reflectedAngleSet Θ)) ∪
+        capLowerNormals Θ.angle) := by
+  exact K.property.reflectedBody fun a ha ↦ reflectedAngle_mem_polygonNormals Θ a ha
+
+private def reflectedPolygonCap (Θ : AngleSet) (K : PolygonCapSpace Θ) :
+    PolygonCapSpace (reflectedAngleSet Θ) :=
+  ⟨⟨reflectedBody Θ.angle K.val.val, reflectedBody_isCap K.val⟩,
+    reflectedBody_polygonRepresentation Θ K⟩
+
+private theorem polygonNiche_reflection (Θ : AngleSet) (K : PolygonCapSpace Θ) :
+    polygonNiche (reflectedAngleSet Θ) (reflectedPolygonCap Θ K).val =
+      capReflection Θ.angle '' polygonNiche Θ K.val := by
+  unfold polygonNiche
+  rw [Set.image_inter (capReflection Θ.angle).injective,
+    capReflection_image_capFan]
+  congr 1
+  ext p
+  constructor
+  · intro hp
+    obtain ⟨t, ht, hp⟩ := Set.mem_iUnion₂.mp hp
+    obtain ⟨s, hs, rfl⟩ := Finset.mem_image.mp ht
+    change p ∈ innerQuadrant (reflectedBody Θ.angle K.val.val) (Θ.angle - s) at hp
+    rw [innerQuadrant_reflection] at hp
+    obtain ⟨q, hq, rfl⟩ := hp
+    refine ⟨q, Set.mem_iUnion₂.mpr ⟨s, hs, ?_⟩, rfl⟩
+    simpa only [sub_sub_cancel] using hq
+  · rintro ⟨q, hq, rfl⟩
+    obtain ⟨s, hs, hq⟩ := Set.mem_iUnion₂.mp hq
+    apply Set.mem_iUnion₂.mpr
+    refine ⟨Θ.angle - s, Finset.mem_image.mpr ⟨s, hs, rfl⟩, ?_⟩
+    change capReflection Θ.angle q ∈
+      innerQuadrant (reflectedBody Θ.angle K.val.val) (Θ.angle - s)
+    rw [innerQuadrant_reflection]
+    exact ⟨q, by simpa only [sub_sub_cancel], rfl⟩
+
+private theorem polygonAreaFunctional_reflectedPolygonCap
+    (Θ : AngleSet) (K : PolygonCapSpace Θ) :
+    polygonAreaFunctional (reflectedAngleSet Θ) (reflectedPolygonCap Θ K).val =
+      polygonAreaFunctional Θ K.val := by
+  unfold polygonAreaFunctional
+  rw [angleCap_eq_self, angleCap_eq_self, polygonNiche_reflection]
+  change ClassicalResults.area (capReflection Θ.angle '' (K.val.val : Set Point)) -
+      ClassicalResults.area (capReflection Θ.angle '' polygonNiche Θ K.val) = _
+  rw [area_image_capReflection Θ.angle (K.val.val : Set Point)
+      K.val.val.isCompact.measurableSet,
+    area_image_capReflection Θ.angle (polygonNiche Θ K.val)
+      (measurableSet_polygonNiche Θ K.val)]
+
+private def reflectedPolygonCapBack (Θ : AngleSet)
+    (L : PolygonCapSpace (reflectedAngleSet Θ)) : PolygonCapSpace Θ :=
+  ⟨⟨reflectedBody Θ.angle L.val.val, reflectedBody_isCap L.val⟩, by
+    simpa [reflectedAngleSet, Finset.image_image] using
+      reflectedBody_polygonRepresentation (reflectedAngleSet Θ) L⟩
+
+private theorem polygonNiche_reflection_back (Θ : AngleSet)
+    (L : PolygonCapSpace (reflectedAngleSet Θ)) :
+    polygonNiche Θ (reflectedPolygonCapBack Θ L).val =
+      capReflection Θ.angle '' polygonNiche (reflectedAngleSet Θ) L.val := by
+  unfold polygonNiche
+  change capFan Θ.angle ∩
+      (⋃ t ∈ Θ.directions, innerQuadrant (reflectedBody Θ.angle L.val.val) t) =
+    capReflection Θ.angle ''
+      (capFan Θ.angle ∩
+        ⋃ t ∈ (reflectedAngleSet Θ).directions, innerQuadrant L.val.val t)
+  rw [Set.image_inter (capReflection Θ.angle).injective,
+    capReflection_image_capFan]
+  congr 1
+  ext p
+  constructor
+  · intro hp
+    obtain ⟨t, ht, hp⟩ := Set.mem_iUnion₂.mp hp
+    change p ∈ innerQuadrant (reflectedBody Θ.angle L.val.val) t at hp
+    rw [innerQuadrant_reflection] at hp
+    obtain ⟨q, hq, rfl⟩ := hp
+    refine ⟨q, Set.mem_iUnion₂.mpr ⟨Θ.angle - t, ?_, hq⟩, rfl⟩
+    exact Finset.mem_image.mpr ⟨t, ht, rfl⟩
+  · rintro ⟨q, hq, rfl⟩
+    obtain ⟨s, hs, hq⟩ := Set.mem_iUnion₂.mp hq
+    obtain ⟨t, ht, rfl⟩ := Finset.mem_image.mp hs
+    apply Set.mem_iUnion₂.mpr
+    refine ⟨t, ht, ?_⟩
+    rw [innerQuadrant_reflection]
+    exact ⟨q, hq, rfl⟩
+
+private theorem polygonAreaFunctional_reflectedPolygonCapBack (Θ : AngleSet)
+    (L : PolygonCapSpace (reflectedAngleSet Θ)) :
+    polygonAreaFunctional Θ (reflectedPolygonCapBack Θ L).val =
+      polygonAreaFunctional (reflectedAngleSet Θ) L.val := by
+  unfold polygonAreaFunctional
+  rw [angleCap_eq_self, angleCap_eq_self, polygonNiche_reflection_back]
+  change ClassicalResults.area (capReflection Θ.angle '' (L.val.val : Set Point)) -
+      ClassicalResults.area
+        (capReflection Θ.angle '' polygonNiche (reflectedAngleSet Θ) L.val) = _
+  rw [area_image_capReflection Θ.angle (L.val.val : Set Point)
+      L.val.val.isCompact.measurableSet,
+    area_image_capReflection Θ.angle (polygonNiche (reflectedAngleSet Θ) L.val)
+      (measurableSet_polygonNiche (reflectedAngleSet Θ) L.val)]
+
+private theorem mirrorReflection_stripTop (ω : ℝ) :
+    mirrorReflection ω (stripParallelogram ω).2.2 =
+      (stripParallelogram ω).2.2 := by
+  change stripTopReflection ω (stripParallelogram ω).2.2 = _
+  exact stripTopReflection_stripTop ω
+
+theorem maximumPolygonCap_mirror (Θ : AngleSet) (K : PolygonCapSpace Θ)
+    (hK : IsMaximumPolygonCap Θ K) :
+    ∃ P : PolygonCapSpace (reflectedAngleSet Θ),
+      (P.val.val : Set Point) = mirrorReflection Θ.angle '' (K.val.val : Set Point) ∧
+      IsMaximumPolygonCap (reflectedAngleSet Θ) P := by
+  let P := reflectedPolygonCap Θ K
+  refine ⟨P, ?_, ?_⟩
+  · change capReflection Θ.angle '' (K.val.val : Set Point) =
+      mirrorReflection Θ.angle '' (K.val.val : Set Point)
+    rw [mirrorReflection_eq_capReflection_m152940c Θ.angle Θ.angle_pos Θ.angle_le]
+  · constructor
+    · change (stripParallelogram Θ.angle).2.2 ∈
+        capReflection Θ.angle '' (K.val.val : Set Point)
+      refine ⟨(stripParallelogram Θ.angle).2.2, hK.1, ?_⟩
+      rw [← mirrorReflection_eq_capReflection_m152940c Θ.angle Θ.angle_pos Θ.angle_le,
+        mirrorReflection_stripTop]
+    · intro L
+      calc
+        polygonAreaFunctional (reflectedAngleSet Θ) L.val =
+            polygonAreaFunctional Θ (reflectedPolygonCapBack Θ L).val :=
+          (polygonAreaFunctional_reflectedPolygonCapBack Θ L).symm
+        _ ≤ polygonAreaFunctional Θ K.val := hK.2 (reflectedPolygonCapBack Θ L)
+        _ = polygonAreaFunctional (reflectedAngleSet Θ) P.val :=
+          (polygonAreaFunctional_reflectedPolygonCap Θ K).symm
+
+private def castPolygonCap {Θ Φ : AngleSet} (h : Θ = Φ)
+    (K : PolygonCapSpace Θ) : PolygonCapSpace Φ :=
+  cast (congrArg PolygonCapSpace h) K
+
+@[simp] private theorem castPolygonCap_coe {Θ Φ : AngleSet} (h : Θ = Φ)
+    (K : PolygonCapSpace Θ) :
+    ((castPolygonCap h K).val.val : Set Point) = (K.val.val : Set Point) := by
+  subst h
+  rfl
+
+private theorem isMaximumPolygonCap_castPolygonCap {Θ Φ : AngleSet} (h : Θ = Φ)
+    {K : PolygonCapSpace Θ} (hK : IsMaximumPolygonCap Θ K) :
+    IsMaximumPolygonCap Φ (castPolygonCap h K) := by
+  subst h
+  exact hK
+
+private theorem reflectedAngleSet_uniformAngleSet (ω : ℝ) (hω : 0 < ω)
+    (hω' : ω ≤ Real.pi / 2) (n : ℕ) (hn : 2 ≤ n) :
+    reflectedAngleSet (uniformAngleSet ω hω hω' n hn) =
+      uniformAngleSet ω hω hω' n hn := by
+  unfold reflectedAngleSet uniformAngleSet
+  congr 1
+  ext t
+  constructor
+  · intro ht
+    obtain ⟨s, hs, rfl⟩ := Finset.mem_image.mp ht
+    obtain ⟨j, hj, rfl⟩ := Finset.mem_image.mp hs
+    obtain ⟨hj0, hjn⟩ := Finset.mem_Ioo.mp hj
+    apply Finset.mem_image.mpr
+    refine ⟨n - j, Finset.mem_Ioo.mpr ⟨Nat.sub_pos_of_lt hjn, ?_⟩, ?_⟩
+    · omega
+    have hn0 : (n : ℝ) ≠ 0 := by positivity
+    rw [Nat.cast_sub hjn.le]
+    field_simp
+  · intro ht
+    obtain ⟨j, hj, rfl⟩ := Finset.mem_image.mp ht
+    obtain ⟨hj0, hjn⟩ := Finset.mem_Ioo.mp hj
+    apply Finset.mem_image.mpr
+    refine ⟨ω - (j : ℝ) / n * ω, ?_, by ring⟩
+    apply Finset.mem_image.mpr
+    refine ⟨n - j, Finset.mem_Ioo.mpr ⟨Nat.sub_pos_of_lt hjn, by omega⟩, ?_⟩
+    have hn0 : (n : ℝ) ≠ 0 := by positivity
+    rw [Nat.cast_sub hjn.le]
+    field_simp
+
+theorem balancedMaximumCap_mirror {ω : ℝ} (K : CapSpace ω)
+    (hK : IsBalancedMaximumCap K) :
+    ∃ P : CapSpace ω,
+      (P.val : Set Point) = mirrorReflection ω '' (K.val : Set Point) ∧
+      IsBalancedMaximumCap P := by
+  obtain ⟨n, hn, hmono, hdyadic, Q, hQmax, hQlim⟩ := hK
+  let P : CapSpace ω := ⟨reflectedBody ω K.val, reflectedBody_isCap K⟩
+  have hM : mirrorReflection ω = capReflection ω :=
+    mirrorReflection_eq_capReflection_m152940c ω K.property.1 K.property.2.1
+  have hmirror : (P.val : Set Point) = mirrorReflection ω '' (K.val : Set Point) := by
+    change capReflection ω '' (K.val : Set Point) = mirrorReflection ω '' (K.val : Set Point)
+    rw [hM]
+  refine ⟨P, hmirror, n, hn, hmono, hdyadic, ?_⟩
+  have hR (i : ℕ) : ∃ R : PolygonCapSpace
+      (uniformAngleSet ω P.property.1 P.property.2.1 (n i) (hn i)),
+      (R.val.val : Set Point) = mirrorReflection ω '' (Q i).val.val ∧
+        IsMaximumPolygonCap _ R := by
+    have heq := reflectedAngleSet_uniformAngleSet ω K.property.1 K.property.2.1
+      (n i) (hn i)
+    let W := maximumPolygonCap_mirror _ (Q i) (hQmax i)
+    let R := castPolygonCap heq W.choose
+    refine ⟨R, ?_, ?_⟩
+    · rw [castPolygonCap_coe]
+      simpa [W, uniformAngleSet] using W.choose_spec.1
+    · exact isMaximumPolygonCap_castPolygonCap heq W.choose_spec.2
+  let R : ∀ i, PolygonCapSpace
+      (uniformAngleSet ω P.property.1 P.property.2.1 (n i) (hn i)) :=
+    fun i ↦ (hR i).choose
+  refine ⟨R, ?_, ?_⟩
+  · exact fun i ↦ (hR i).choose_spec.2
+  · apply hQlim.congr'
+    filter_upwards [] with i
+    rw [(hR i).choose_spec.1, hmirror, hM]
+    exact (Metric.hausdorffDist_image (capReflection ω).isometry).symm
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Sofa.Support`.
+* `Sofa.Cap`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Sofa / Support
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem standardPosition_subset_monotonization (s : Set Point) (ω : ℝ)
+    (hs : IsStandardPosition s ω) : s ⊆ monotonization s ω := by
+  have hω : ω ∈ Set.Ioc 0 (Real.pi / 2) := ⟨hs.2.2.1, hs.2.2.2.1⟩
+  have hcommon := movingSofa_commonSubset s ω hs.2.1 hω
+  have hs_ne : s.Nonempty := by
+    obtain ⟨_, hm, _⟩ := hs.2.1
+    exact hm.1.nonempty
+  have hshape : s ⊆ (stripParallelogram ω).1 := by
+    have htranslated :=
+      (exists_standardPosition_translation s ω hs.2.1 hω).2.2.2 (0 : Point)
+    have hstandard :
+        IsStandardPosition ((fun p : Point ↦ p + (0 : Point)) '' s) ω := by
+      simpa using hs
+    simpa using htranslated hstandard
+  rw [monotonization]
+  intro p hp
+  refine ⟨hshape hp, Set.mem_iInter.2 fun t ↦ Set.mem_iInter.2 fun ht ↦ ?_⟩
+  exact subset_supportingHallway s (t : Real.Angle) hs_ne hcommon.1
+    (hcommon.2.2.2.1 t ht) hp
+
+theorem standardPosition_subset_cap (s : Set Point) (ω : ℝ)
+    (hs : IsStandardPosition s ω) :
+    s ⊆ monotonization s ω ∧ monotonization s ω ⊆ capOfSofa s ω := by
+  refine ⟨standardPosition_subset_monotonization s ω hs, ?_⟩
+  rw [monotonization, capOfSofa]
+  rintro p ⟨hpP, hpL⟩
+  refine ⟨hpP, Set.mem_iInter.2 fun t ↦ Set.mem_iInter.2 fun ht ↦ ?_⟩
+  apply supportingHallway_subset_outerQuadrant s (t : Real.Angle)
+  exact Set.mem_iInter.1 (Set.mem_iInter.1 hpL t) ht
+
+private theorem inner_le_supportValue_of_mem_capOfSofa {s : Set Point} {ω t : ℝ}
+    (ht : t ∈ capUpperAngles ω) {p : Point} (hp : p ∈ capOfSofa s ω) :
+    inner ℝ p (normalVector (t : Real.Angle)) ≤ supportValue s (t : Real.Angle) := by
+  rcases ht with ht | ht
+  · have hq := Set.mem_iInter.1 (Set.mem_iInter.1 hp.2 t) ht
+    obtain ⟨q, hq, rfl⟩ := hq
+    rw [inner_supportingPlacement_normalVector]
+    change q 0 ≤ 1 ∧ q 1 ≤ 1 at hq
+    linarith [hq.1]
+  · have ht' : t - Real.pi / 2 ∈ Set.Icc 0 ω := by
+      constructor <;> linarith [ht.1, ht.2]
+    have hq := Set.mem_iInter.1 (Set.mem_iInter.1 hp.2 (t - Real.pi / 2)) ht'
+    obtain ⟨q, hq, rfl⟩ := hq
+    have ha : (t : Real.Angle) = ((t - Real.pi / 2 : ℝ) : Real.Angle) +
+        ((Real.pi / 2 : ℝ) : Real.Angle) := by
+      rw [← Real.Angle.coe_add, sub_add_cancel]
+    rw [ha, normalVector_add_pi_div_two, inner_supportingPlacement_tangentVector]
+    change q 0 ≤ 1 ∧ q 1 ≤ 1 at hq
+    linarith [hq.2]
+
+theorem standardPosition_support_eq (s : Set Point) (ω : ℝ)
+    (hs : IsStandardPosition s ω) :
+    (∀ t ∈ capUpperAngles ω,
+      supportValue (monotonization s ω) (t : Real.Angle) = supportValue s (t : Real.Angle) ∧
+      supportValue (capOfSofa s ω) (t : Real.Angle) = supportValue s (t : Real.Angle)) ∧
+    (∀ t ∈ Set.Icc 0 ω,
+      supportingHallway (monotonization s ω) (t : Real.Angle) =
+        supportingHallway s (t : Real.Angle) ∧
+      supportingHallway (capOfSofa s ω) (t : Real.Angle) =
+        supportingHallway s (t : Real.Angle)) := by
+  have hs_ne : s.Nonempty := by
+    by_contra h
+    have he : s = ∅ := Set.not_nonempty_iff_eq_empty.mp h
+    have hnorm := hs.2.2.2.2.1
+    simp [he, supportValue] at hnorm
+  obtain ⟨hsm, hmc⟩ := standardPosition_subset_cap s ω hs
+  have hsupport (t : ℝ) (ht : t ∈ capUpperAngles ω) :
+      supportValue (monotonization s ω) (t : Real.Angle) = supportValue s (t : Real.Angle) ∧
+      supportValue (capOfSofa s ω) (t : Real.Angle) = supportValue s (t : Real.Angle) := by
+    have hc (p : Point) (hp : p ∈ capOfSofa s ω) :=
+      inner_le_supportValue_of_mem_capOfSofa ht hp
+    exact ⟨supportValue_eq_of_subset_of_inner_le hs_ne hsm _ (fun p hp ↦ hc p (hmc hp)),
+      supportValue_eq_of_subset_of_inner_le hs_ne (hsm.trans hmc) _ hc⟩
+  refine ⟨hsupport, ?_⟩
+  intro t ht
+  have ht' : t + Real.pi / 2 ∈ capUpperAngles ω :=
+    Or.inr ⟨by linarith [ht.1], by linarith [ht.2]⟩
+  have h₀ := hsupport t (Or.inl ht)
+  have h₁ := hsupport (t + Real.pi / 2) ht'
+  simp only [Real.Angle.coe_add] at h₁
+  constructor
+  · unfold supportingHallway
+    congr 1
+    funext p
+    simp only [supportingPlacement, h₀.1, h₁.1]
+  · unfold supportingHallway
+    congr 1
+    funext p
+    simp only [supportingPlacement, h₀.2, h₁.2]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Sofa / Cap
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private theorem capOfSofa_halfPlaneRepresentation (s : Set Point) (ω : ℝ) (hω : 0 ≤ ω) :
+    HasHalfPlaneRepresentation (capOfSofa s ω)
+      (((fun t : ℝ ↦ (t : Real.Angle)) '' capUpperAngles ω) ∪ capLowerNormals ω) := by
+  let U : Set (Real.Angle × ℝ) :=
+    (fun t : ℝ ↦ ((t : Real.Angle), supportValue s (t : Real.Angle))) '' capUpperAngles ω
+  let B : Set (Real.Angle × ℝ) :=
+    {(((ω : ℝ) : Real.Angle), 1),
+      (((Real.pi / 2 : ℝ) : Real.Angle), 1),
+      (((ω + Real.pi : ℝ) : Real.Angle), 0),
+      (((3 * Real.pi / 2 : ℝ) : Real.Angle), 0)}
+  refine ⟨U ∪ B, ?_, ?_⟩
+  · rintro c (hc | hc)
+    · obtain ⟨t, ht, rfl⟩ := hc
+      exact Or.inl ⟨t, ht, rfl⟩
+    · rcases hc with rfl | rfl | rfl | rfl
+      · exact Or.inl ⟨ω, Or.inl ⟨hω, le_rfl⟩, rfl⟩
+      · exact Or.inl ⟨Real.pi / 2, Or.inr ⟨le_rfl, le_add_of_nonneg_left hω⟩, rfl⟩
+      · exact Or.inr (Or.inl rfl)
+      · exact Or.inr (Or.inr rfl)
+  · ext p
+    simp only [Set.mem_iInter]
+    constructor
+    · intro hp c hc
+      rcases hc with hc | hc
+      · obtain ⟨φ, hφ, rfl⟩ := hc
+        change inner ℝ p (normalVector (φ : Real.Angle)) ≤ supportValue s _
+        rcases hφ with hφ | hφ
+        · have hout := Set.mem_iInter.1 (Set.mem_iInter.1 hp.2 φ) hφ
+          rw [(rotatingHallwayParts_formulas s (φ : Real.Angle)).2.2.2.2.2.2.2.1]
+            at hout
+          exact hout.1
+        · let t := φ - Real.pi / 2
+          have ht : t ∈ Set.Icc 0 ω := by
+            constructor <;> dsimp [t] <;> linarith [hφ.1, hφ.2]
+          have hout := Set.mem_iInter.1 (Set.mem_iInter.1 hp.2 t) ht
+          rw [(rotatingHallwayParts_formulas s (t : Real.Angle)).2.2.2.2.2.2.2.1]
+            at hout
+          have hang : ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle)) =
+              (φ : Real.Angle) := by
+            rw [← Real.Angle.coe_add]
+            congr 1
+            dsimp [t]
+            ring
+          have hout' := hout.2
+          change inner ℝ p (normalVector ((t : Real.Angle) +
+            ((Real.pi / 2 : ℝ) : Real.Angle))) ≤ _ at hout'
+          simpa [hang] using hout'
+      · rcases hc with rfl | rfl | rfl | rfl
+        · exact ((mem_stripParallelogram_iff ω p).1 hp.1).2.2
+        · simpa [normalHalfPlane, normalVector, frame, PiLp.inner_apply,
+            Fin.sum_univ_two] using ((mem_stripParallelogram_iff ω p).1 hp.1).1.2
+        · change inner ℝ p (normalVector ((ω + Real.pi : ℝ) : Real.Angle)) ≤ 0
+          rw [normalVector_add_pi, inner_neg_right]
+          exact neg_nonpos.mpr ((mem_stripParallelogram_iff ω p).1 hp.1).2.1
+        · simpa [normalHalfPlane, inner_normalVector_three_pi_div_two] using
+            neg_nonpos.mpr ((mem_stripParallelogram_iff ω p).1 hp.1).1.1
+    · intro hp
+      have hupper (φ : ℝ) (hφ : φ ∈ capUpperAngles ω) :
+          inner ℝ p (normalVector (φ : Real.Angle)) ≤ supportValue s (φ : Real.Angle) :=
+        hp ((φ : Real.Angle), supportValue s (φ : Real.Angle))
+          (Or.inl ⟨φ, hφ, rfl⟩)
+      have hω' := hp (((ω : ℝ) : Real.Angle), 1) (Or.inr (Or.inl rfl))
+      have hπ := hp (((Real.pi / 2 : ℝ) : Real.Angle), 1)
+        (Or.inr (Or.inr (Or.inl rfl)))
+      have hlowerω := hp (((ω + Real.pi : ℝ) : Real.Angle), 0)
+        (Or.inr (Or.inr (Or.inr (Or.inl rfl))))
+      have hlowerπ := hp (((3 * Real.pi / 2 : ℝ) : Real.Angle), 0)
+        (Or.inr (Or.inr (Or.inr (Or.inr rfl))))
+      refine ⟨(mem_stripParallelogram_iff ω p).2 ?_, ?_⟩
+      · change inner ℝ p (normalVector (ω : Real.Angle)) ≤ 1 at hω'
+        change inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) ≤ 1 at hπ
+        change inner ℝ p (normalVector ((ω + Real.pi : ℝ) : Real.Angle)) ≤ 0 at hlowerω
+        change inner ℝ p (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) ≤ 0 at hlowerπ
+        rw [normalVector_add_pi, inner_neg_right] at hlowerω
+        rw [inner_normalVector_three_pi_div_two] at hlowerπ
+        have hπ' : p 1 ≤ 1 := by
+          simpa [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two] using hπ
+        exact ⟨⟨by linarith, hπ'⟩, by linarith, hω'⟩
+      · refine Set.mem_iInter.2 fun t ↦ Set.mem_iInter.2 fun ht ↦ ?_
+        rw [(rotatingHallwayParts_formulas s (t : Real.Angle)).2.2.2.2.2.2.2.1]
+        refine ⟨hupper t (Or.inl ht), ?_⟩
+        have hshift : t + Real.pi / 2 ∈ capUpperAngles ω :=
+          Or.inr ⟨by linarith [ht.1], by linarith [ht.2]⟩
+        change inner ℝ p (normalVector ((t : Real.Angle) +
+          ((Real.pi / 2 : ℝ) : Real.Angle))) ≤ _
+        simpa only [Real.Angle.coe_add] using hupper (t + Real.pi / 2) hshift
+
+private theorem isBounded_capOfSofa (s : Set Point) (ω : ℝ)
+    (hω : 0 < ω) (hωπ : ω ≤ Real.pi / 2) : Bornology.IsBounded (capOfSofa s ω) := by
+  let t := ω / 2
+  let A := supportValue s (t : Real.Angle)
+  let B := supportValue s ((t + Real.pi / 2 : ℝ) : Real.Angle)
+  have ht : t ∈ Set.Icc 0 ω := by
+    constructor <;> dsimp [t] <;> linarith
+  have htI : t ∈ Set.Ioo (-(Real.pi / 2)) (Real.pi / 2) := by
+    constructor <;> dsimp [t] <;> linarith [Real.pi_pos]
+  have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo htI
+  have hsine : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi (by dsimp [t]; linarith)
+    (by dsimp [t]; linarith [Real.pi_pos])
+  refine (isBounded_iff_forall_norm_le).2 ⟨|A / Real.cos t| +
+    |B / Real.sin t| + 1, ?_⟩
+  intro p hp
+  have hstrip := (mem_stripParallelogram_iff ω p).1 hp.1
+  have hout := Set.mem_iInter.1 (Set.mem_iInter.1 hp.2 t) ht
+  rw [(rotatingHallwayParts_formulas s (t : Real.Angle)).2.2.2.2.2.2.2.1]
+    at hout
+  have h₀ := hout.1
+  have h₁ := hout.2
+  change inner ℝ p (normalVector (t : Real.Angle)) ≤ A at h₀
+  change inner ℝ p (normalVector ((t : Real.Angle) +
+    ((Real.pi / 2 : ℝ) : Real.Angle))) ≤ B at h₁
+  rw [normalVector_add_pi_div_two] at h₁
+  simp only [normalVector, tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two,
+    Real.inner_apply, Real.Angle.cos_coe, Real.Angle.sin_coe, Matrix.cons_val_zero,
+    Matrix.cons_val_one] at h₀ h₁
+  have hxhi : p 0 ≤ A / Real.cos t := (le_div_iff₀ hc).2 (by
+    have := mul_nonneg hstrip.1.1 hsine.le
+    nlinarith)
+  have hxlo : -(B / Real.sin t) ≤ p 0 := by
+    rw [← neg_div]
+    apply (div_le_iff₀ hsine).2
+    have := mul_nonneg hstrip.1.1 hc.le
+    nlinarith
+  have hxabs : |p 0| ≤ |A / Real.cos t| + |B / Real.sin t| := by
+    rw [abs_le]
+    constructor
+    · calc
+        -(|A / Real.cos t| + |B / Real.sin t|) ≤ -(B / Real.sin t) := by
+          have hA := abs_nonneg (A / Real.cos t)
+          have hB := le_abs_self (B / Real.sin t)
+          linarith
+        _ ≤ p 0 := hxlo
+    · calc
+        p 0 ≤ A / Real.cos t := hxhi
+        _ ≤ |A / Real.cos t| + |B / Real.sin t| := by
+          have hA := le_abs_self (A / Real.cos t)
+          have hB := abs_nonneg (B / Real.sin t)
+          linarith
+  have hyabs : |p 1| ≤ 1 := by
+    rw [abs_le]
+    exact ⟨by linarith [hstrip.1.1], hstrip.1.2⟩
+  have hpdecomp : p = p 0 • (!₂[1, 0] : Point) + p 1 • !₂[0, 1] := by
+    ext i
+    fin_cases i <;> simp
+  have he₀ : ‖(!₂[1, 0] : Point)‖ = 1 := by
+    have h : (!₂[1, 0] : Point) = normalVector (0 : Real.Angle) := by
+      ext i
+      fin_cases i <;> simp [normalVector, frame]
+    rw [h]
+    exact norm_normalVector_real 0
+  have he₁ : ‖(!₂[0, 1] : Point)‖ = 1 := by
+    have h : (!₂[0, 1] : Point) = normalVector ((Real.pi / 2 : ℝ) : Real.Angle) := by
+      ext i
+      fin_cases i <;> simp [normalVector, frame]
+    rw [h]
+    exact norm_normalVector_real (Real.pi / 2)
+  rw [hpdecomp]
+  calc
+    ‖p 0 • (!₂[1, 0] : Point) + p 1 • !₂[0, 1]‖ ≤
+        ‖p 0 • (!₂[1, 0] : Point)‖ + ‖p 1 • (!₂[0, 1] : Point)‖ := norm_add_le _ _
+    _ = |p 0| + |p 1| := by simp [norm_smul, he₀, he₁]
+    _ ≤ |A / Real.cos t| + |B / Real.sin t| + 1 := add_le_add hxabs hyabs
+
+private theorem mem_capOfSofa_of_mem_strip_of_upper {s : Set Point} {ω : ℝ} {p : Point}
+    (hp : p ∈ (stripParallelogram ω).1)
+    (hupper : ∀ t ∈ capUpperAngles ω,
+      inner ℝ p (normalVector (t : Real.Angle)) ≤ supportValue s (t : Real.Angle)) :
+    p ∈ capOfSofa s ω := by
+  refine ⟨hp, Set.mem_iInter.2 fun t ↦ Set.mem_iInter.2 fun ht ↦ ?_⟩
+  rw [(rotatingHallwayParts_formulas s (t : Real.Angle)).2.2.2.2.2.2.2.1]
+  refine ⟨hupper t (Or.inl ht), ?_⟩
+  have hshift : t + Real.pi / 2 ∈ capUpperAngles ω :=
+    Or.inr ⟨by linarith [ht.1], by linarith [ht.2]⟩
+  change inner ℝ p (normalVector ((t : Real.Angle) +
+    ((Real.pi / 2 : ℝ) : Real.Angle))) ≤ _
+  simpa only [Real.Angle.coe_add] using hupper (t + Real.pi / 2) hshift
+
+private theorem supportValue_eq_zero_of_nonpos_of_attains_zero (K : ConvexBody Point)
+    (a : Real.Angle) (hupper : ∀ p ∈ K, inner ℝ p (normalVector a) ≤ 0)
+    {q : Point} (hq : q ∈ K) (hqzero : inner ℝ q (normalVector a) = 0) :
+    supportValue K a = 0 := by
+  apply le_antisymm
+  · exact supportValue_le_of_subset_normalHalfPlane K a 0 hupper
+  · simpa only [hqzero] using inner_le_supportValue K hq a
+
+private theorem standardPosition_right_cap_lower_support (s : Set Point)
+    (hs : IsStandardPosition s (Real.pi / 2)) (K : ConvexBody Point)
+    (hK : (K : Set Point) = capOfSofa s (Real.pi / 2))
+    (hs_ne : s.Nonempty) :
+    supportValue K ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := by
+  have hpi := Real.pi_pos
+  obtain ⟨p, hp, hpeq⟩ := (hs.1.image
+    (continuous_inner.comp (continuous_id.prodMk continuous_const))).sSup_mem
+      (hs_ne.image
+        (fun p ↦ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))))
+  have hpT : inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 1 := by
+    change inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) =
+      supportValue s _ at hpeq
+    simpa only [hs.2.2.2.2.2] using hpeq
+  let q := p - tangentVector (0 : Real.Angle)
+  have hqy : q 1 = 0 := by
+    have hpcoord : p 1 = 1 := by
+      simpa [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two] using hpT
+    simp [q, tangentVector, frame, hpcoord]
+  have hqP : q ∈ (stripParallelogram (Real.pi / 2)).1 := by
+    rw [mem_stripParallelogram_iff]
+    have hqn : inner ℝ q
+        (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := by
+      simpa [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two] using hqy
+    exact ⟨⟨by rw [hqy], by rw [hqy]; norm_num⟩,
+      by rw [hqn], by rw [hqn]; norm_num⟩
+  have hq : q ∈ capOfSofa s (Real.pi / 2) := by
+    apply mem_capOfSofa_of_mem_strip_of_upper hqP
+    intro u hu
+    have hsu : 0 ≤ Real.sin u := Real.sin_nonneg_of_nonneg_of_le_pi
+      (by rcases hu with hu | hu <;> linarith [hu.1])
+      (by rcases hu with hu | hu <;> linarith [hu.2, Real.pi_pos])
+    calc
+      inner ℝ q (normalVector (u : Real.Angle)) =
+          inner ℝ p (normalVector (u : Real.Angle)) - Real.sin u := by
+            simp [q, tangentVector, normalVector, frame, PiLp.inner_apply,
+              Fin.sum_univ_two]
+            ring
+      _ ≤ inner ℝ p (normalVector (u : Real.Angle)) := sub_le_self _ hsu
+      _ ≤ supportValue s (u : Real.Angle) :=
+        inner_le_supportValue_of_isCompact hs.1 hp _
+  apply supportValue_eq_zero_of_nonpos_of_attains_zero K
+  · intro z hz
+    change z ∈ (K : Set Point) at hz
+    rw [hK] at hz
+    have hz0 := ((mem_stripParallelogram_iff (Real.pi / 2) z).1 hz.1).1.1
+    simpa [inner_normalVector_three_pi_div_two] using neg_nonpos.mpr hz0
+  · exact hK.symm ▸ hq
+  · simpa [inner_normalVector_three_pi_div_two] using congrArg Neg.neg hqy
+
+theorem standardPosition_cap (s : Set Point) (ω : ℝ)
+    (hs : IsStandardPosition s ω) :
+    ∃ K : CapSpace ω, (K.val : Set Point) = capOfSofa s ω := by
+  have hs_ne : s.Nonempty := by
+    obtain ⟨_, hm, _⟩ := hs.2.1
+    exact hm.1.nonempty
+  have hω0 : 0 ≤ ω := hs.2.2.1.le
+  obtain ⟨D, hDN, hC⟩ := capOfSofa_halfPlaneRepresentation s ω hω0
+  have hclosed : IsClosed (capOfSofa s ω) := by
+    rw [hC]
+    exact isClosed_iInter fun c ↦ isClosed_iInter fun _ ↦
+      isClosed_normalHalfPlane c.1 c.2 false
+  have hconvex : Convex ℝ (capOfSofa s ω) := by
+    rw [hC]
+    exact convex_iInter fun c ↦ convex_iInter fun _ ↦
+      convex_normalHalfPlane c.1 c.2 false
+  have hscap : s ⊆ capOfSofa s ω :=
+    let h := standardPosition_subset_cap s ω hs
+    h.1.trans h.2
+  have hne : (capOfSofa s ω).Nonempty := hs_ne.mono hscap
+  let K : ConvexBody Point := {
+    carrier := capOfSofa s ω
+    convex' := hconvex
+    isCompact' := Metric.isCompact_iff_isClosed_bounded.2
+      ⟨hclosed, isBounded_capOfSofa s ω hs.2.2.1 hs.2.2.2.1⟩
+    nonempty' := hne }
+  have hsupport := (standardPosition_support_eq s ω hs).1
+  have hKω : supportValue K (ω : Real.Angle) = 1 := by
+    change supportValue (capOfSofa s ω) (ω : Real.Angle) = 1
+    rw [(hsupport ω (Or.inl ⟨hω0, le_rfl⟩)).2, hs.2.2.2.2.1]
+  have hKπ : supportValue K ((Real.pi / 2 : ℝ) : Real.Angle) = 1 := by
+    change supportValue (capOfSofa s ω) ((Real.pi / 2 : ℝ) : Real.Angle) = 1
+    rw [(hsupport (Real.pi / 2) (Or.inr ⟨le_rfl,
+      le_add_of_nonneg_left hω0⟩)).2, hs.2.2.2.2.2]
+  by_cases hright : ω = Real.pi / 2
+  · subst ω
+    have hbottom := standardPosition_right_cap_lower_support s hs K rfl hs_ne
+    have hsame : (((Real.pi / 2 + Real.pi : ℝ) : Real.Angle)) =
+        ((3 * Real.pi / 2 : ℝ) : Real.Angle) := by
+      congr 1
+      ring
+    refine ⟨⟨K, hs.2.2.1, hs.2.2.2.1, hKω, hKπ, ?_, hbottom,
+      ⟨D, hDN, ?_⟩⟩, rfl⟩
+    · simpa [hsame] using hbottom
+    · exact hC
+  · have hlt : ω < Real.pi / 2 := lt_of_le_of_ne hs.2.2.2.1 hright
+    let o := (stripParallelogram ω).2.2
+    have hc : 0 < Real.cos ω := Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [hs.2.2.1, Real.pi_pos], hlt⟩
+    obtain ⟨pω, hpω, hpωeq⟩ := (hs.1.image
+      (continuous_inner.comp (continuous_id.prodMk continuous_const))).sSup_mem
+        (hs_ne.image (fun p ↦ inner ℝ p (normalVector (ω : Real.Angle))))
+    obtain ⟨pT, hpT, hpTeq⟩ := (hs.1.image
+      (continuous_inner.comp (continuous_id.prodMk continuous_const))).sSup_mem
+        (hs_ne.image
+          (fun p ↦ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))))
+    have hpω' : inner ℝ pω (normalVector (ω : Real.Angle)) = 1 := by
+      change inner ℝ pω (normalVector (ω : Real.Angle)) = supportValue s _ at hpωeq
+      simpa only [hs.2.2.2.2.1] using hpωeq
+    have hpT' : inner ℝ pT (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 1 := by
+      change inner ℝ pT (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) =
+        supportValue s _ at hpTeq
+      simpa only [hs.2.2.2.2.2] using hpTeq
+    have hpωT : inner ℝ pω (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) ≤ 1 := by
+      simpa [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two] using
+        ((mem_stripParallelogram_iff ω pω).1 (hscap hpω).1).1.2
+    have hpTω : inner ℝ pT (normalVector (ω : Real.Angle)) ≤ 1 :=
+      ((mem_stripParallelogram_iff ω pT).1
+        (hscap hpT).1).2.2
+    have hoω : inner ℝ o (normalVector (ω : Real.Angle)) = 1 := by
+      have hgap : Real.tan (Real.pi / 4 - ω / 2) =
+          (Real.cos ω)⁻¹ - Real.tan ω := by
+        simpa [show Real.pi / 4 - ω / 2 = (Real.pi / 2 - ω) / 2 by ring]
+          using Real.tan_pi_div_two_sub_div_two ω ⟨hω0, hlt⟩
+      have htan := Real.tan_eq_sin_div_cos ω
+      simp [o, stripParallelogram, normalVector, frame, PiLp.inner_apply, hgap, htan]
+      field_simp [hc.ne']
+      ring
+    have hoT : inner ℝ o (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 1 := by
+      simp [o, stripParallelogram, normalVector, frame, PiLp.inner_apply]
+    have hou (u : ℝ) (hu : u ∈ capUpperAngles ω) :
+        inner ℝ o (normalVector (u : Real.Angle)) ≤ supportValue s (u : Real.Angle) := by
+      rcases hu with hu | hu
+      · let c := Real.cos u / Real.cos ω
+        let d := Real.sin (ω - u) / Real.cos ω
+        have hd0 : 0 ≤ d := div_nonneg
+          (Real.sin_nonneg_of_nonneg_of_le_pi (sub_nonneg.mpr hu.2)
+            (by linarith [hu.1, hs.2.2.2.1, Real.pi_pos])) hc.le
+        have hdecomp : normalVector (u : Real.Angle) =
+            c • normalVector (ω : Real.Angle) -
+              d • normalVector ((Real.pi / 2 : ℝ) : Real.Angle) := by
+          ext i
+          fin_cases i
+          · simp [c, d, normalVector, frame, Real.sin_sub]
+            field_simp [hc.ne']
+          · simp [c, d, normalVector, frame, Real.sin_sub]
+            field_simp [hc.ne']
+            ring
+        rw [hdecomp, inner_sub_right, inner_smul_right, inner_smul_right, hoω, hoT]
+        apply (inner_le_supportValue_of_isCompact hs.1 hpω (u : Real.Angle)).trans'
+        rw [hdecomp, inner_sub_right, inner_smul_right, inner_smul_right, hpω']
+        simpa only [mul_one] using
+          sub_le_sub_left (mul_le_mul_of_nonneg_left hpωT hd0) c
+      · let c := Real.sin (u - ω) / Real.cos ω
+        let d := -Real.cos u / Real.cos ω
+        have hd0 : 0 ≤ d := div_nonneg
+          (neg_nonneg.mpr (Real.cos_nonpos_of_pi_div_two_le_of_le hu.1
+            (by linarith [hu.2, hs.2.2.2.1, Real.pi_pos]))) hc.le
+        have hdecomp : normalVector (u : Real.Angle) =
+            c • normalVector ((Real.pi / 2 : ℝ) : Real.Angle) -
+              d • normalVector (ω : Real.Angle) := by
+          ext i
+          fin_cases i
+          · simp [c, d, normalVector, frame, Real.sin_sub]
+            field_simp [hc.ne']
+          · simp [c, d, normalVector, frame, Real.sin_sub]
+            field_simp [hc.ne']
+            ring
+        rw [hdecomp, inner_sub_right, inner_smul_right, inner_smul_right, hoT, hoω]
+        apply (inner_le_supportValue_of_isCompact hs.1 hpT (u : Real.Angle)).trans'
+        rw [hdecomp, inner_sub_right, inner_smul_right, inner_smul_right, hpT']
+        simpa only [mul_one] using
+          sub_le_sub_left (mul_le_mul_of_nonneg_left hpTω hd0) c
+    let q₀ := o - tangentVector (0 : Real.Angle)
+    have hsinω0 : 0 ≤ Real.sin ω := Real.sin_nonneg_of_nonneg_of_le_pi hω0
+      (by linarith [hs.2.2.2.1, Real.pi_pos])
+    have hsinω1 : Real.sin ω ≤ 1 := Real.sin_le_one ω
+    have hq₀P : q₀ ∈ (stripParallelogram ω).1 := by
+      rw [mem_stripParallelogram_iff]
+      have hq₀y : q₀ 1 = 0 := by
+        simp [q₀, o, stripParallelogram, tangentVector, frame]
+      have hq₀n : inner ℝ q₀ (normalVector (ω : Real.Angle)) = 1 - Real.sin ω := by
+        rw [inner_sub_left, hoω]
+        simp [tangentVector, normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+      exact ⟨⟨by rw [hq₀y], by rw [hq₀y]; norm_num⟩,
+        by rw [hq₀n]; exact sub_nonneg.mpr hsinω1,
+        by rw [hq₀n]; linarith⟩
+    have hq₀ : q₀ ∈ capOfSofa s ω := by
+      apply mem_capOfSofa_of_mem_strip_of_upper hq₀P
+      intro u hu
+      have hsu : 0 ≤ Real.sin u := Real.sin_nonneg_of_nonneg_of_le_pi
+        (by rcases hu with hu | hu <;> linarith [hu.1])
+        (by rcases hu with hu | hu <;> linarith [hu.2, hs.2.2.2.1])
+      calc
+        inner ℝ q₀ (normalVector (u : Real.Angle)) =
+            inner ℝ o (normalVector (u : Real.Angle)) - Real.sin u := by
+              simp [q₀, tangentVector, normalVector, frame, PiLp.inner_apply,
+                Fin.sum_univ_two]
+              ring
+        _ ≤ inner ℝ o (normalVector (u : Real.Angle)) := sub_le_self _ hsu
+        _ ≤ supportValue s (u : Real.Angle) := hou u hu
+    let qω := o - normalVector (ω : Real.Angle)
+    have hqωP : qω ∈ (stripParallelogram ω).1 := by
+      rw [mem_stripParallelogram_iff]
+      have hqωn : inner ℝ qω (normalVector (ω : Real.Angle)) = 0 := by
+        rw [inner_sub_left, hoω, inner_normalVector_self]
+        norm_num
+      have hqωy : qω 1 = 1 - Real.sin ω := by
+        simp [qω, o, stripParallelogram, normalVector, frame]
+      exact ⟨⟨by rw [hqωy]; linarith, by rw [hqωy]; linarith⟩,
+        by rw [hqωn], by rw [hqωn]; norm_num⟩
+    have hqω : qω ∈ capOfSofa s ω := by
+      apply mem_capOfSofa_of_mem_strip_of_upper hqωP
+      intro u hu
+      have hcos : 0 ≤ Real.cos (ω - u) := Real.cos_nonneg_of_mem_Icc
+        ⟨by rcases hu with hu | hu <;> linarith [hu.2, hs.2.2.2.1],
+          by rcases hu with hu | hu <;> linarith [hu.1, hs.2.2.1]⟩
+      calc
+        inner ℝ qω (normalVector (u : Real.Angle)) =
+            inner ℝ o (normalVector (u : Real.Angle)) - Real.cos (ω - u) := by
+              simp [qω, inner_sub_left, inner_normalVector_normalVector]
+        _ ≤ inner ℝ o (normalVector (u : Real.Angle)) := sub_le_self _ hcos
+        _ ≤ supportValue s (u : Real.Angle) := hou u hu
+    have hlowerT : supportValue K ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := by
+      apply supportValue_eq_zero_of_nonpos_of_attains_zero K
+      · intro p hp
+        change p ∈ capOfSofa s ω at hp
+        have := ((mem_stripParallelogram_iff ω p).1 hp.1).1.1
+        simpa [inner_normalVector_three_pi_div_two] using neg_nonpos.mpr this
+      · exact hq₀
+      · simp [q₀, o, stripParallelogram, inner_normalVector_three_pi_div_two,
+          tangentVector, frame]
+    have hlowerω : supportValue K ((ω + Real.pi : ℝ) : Real.Angle) = 0 := by
+      apply supportValue_eq_zero_of_nonpos_of_attains_zero K
+      · intro p hp
+        change p ∈ capOfSofa s ω at hp
+        rw [normalVector_add_pi, inner_neg_right]
+        exact neg_nonpos.mpr ((mem_stripParallelogram_iff ω p).1 hp.1).2.1
+      · exact hqω
+      · rw [normalVector_add_pi, inner_neg_right]
+        have hqωn : inner ℝ qω (normalVector (ω : Real.Angle)) = 0 := by
+          rw [inner_sub_left, hoω, inner_normalVector_self]
+          norm_num
+        rw [hqωn, neg_zero]
+    exact ⟨⟨K, hs.2.2.1, hs.2.2.2.1, hKω, hKπ, hlowerω, hlowerT,
+      ⟨D, hDN, hC⟩⟩, rfl⟩
+
+theorem standardPosition_monotonization (s : Set Point) (ω : ℝ)
+    (hs : IsStandardPosition s ω) (K : CapSpace ω)
+    (hK : (K.val : Set Point) = capOfSofa s ω) :
+    monotonization s ω = (K.val : Set Point) \ capNiche K := by
+  have hsupport := (standardPosition_support_eq s ω hs).1
+  have hparts (t : ℝ) (ht : t ∈ Set.Icc 0 ω) :
+      supportingHallway s (t : Real.Angle) =
+        (rotatingHallwayParts s (t : Real.Angle)).outerQuadrant \
+          innerQuadrant (K.val : Set Point) t := by
+    obtain ⟨h, _⟩ := rotatingHallwayParts_formulas s (t : Real.Angle)
+    have h₀ := (hsupport t (Or.inl ht)).2
+    have ht' : t + Real.pi / 2 ∈ capUpperAngles ω :=
+      Or.inr ⟨by linarith [ht.1], by linarith [ht.2]⟩
+    have h₁ := (hsupport (t + Real.pi / 2) ht').2
+    simp only [Real.Angle.coe_add] at h₁
+    rw [h, rotatingHallwayParts_innerQuadrant]
+    simp only [innerQuadrant, hK, h₀, h₁, Real.Angle.coe_add]
+  ext p
+  constructor
+  · intro hp
+    have hpH (t : ℝ) (ht : t ∈ Set.Icc 0 ω) :
+        p ∈ (rotatingHallwayParts s (t : Real.Angle)).outerQuadrant \
+          innerQuadrant (K.val : Set Point) t := by
+      rw [← hparts t ht]
+      exact Set.mem_iInter.1 (Set.mem_iInter.1 hp.2 t) ht
+    have hpK : p ∈ (K.val : Set Point) := by
+      rw [hK]
+      exact ⟨hp.1, Set.mem_iInter.2 fun t ↦ Set.mem_iInter.2 fun ht ↦ (hpH t ht).1⟩
+    refine ⟨hpK, ?_⟩
+    rintro ⟨_, hN⟩
+    obtain ⟨t, ht, hq⟩ := Set.mem_iUnion₂.mp hN
+    exact (hpH t ⟨ht.1.le, ht.2.le⟩).2 hq
+  · rintro ⟨hpK, hpN⟩
+    have hpC : p ∈ capOfSofa s ω := hK ▸ hpK
+    have hpF := K.subset_capFan hpK
+    refine ⟨hpC.1, Set.mem_iInter.2 fun t ↦ Set.mem_iInter.2 fun ht ↦ ?_⟩
+    rw [hparts t ht]
+    refine ⟨Set.mem_iInter.1 (Set.mem_iInter.1 hpC.2 t) ht, ?_⟩
+    intro hq
+    have ht0 : t ≠ 0 := by
+      rintro rfl
+      have h := hq.2
+      change inner ℝ p (normalVector ((0 + Real.pi / 2 : ℝ) : Real.Angle)) <
+        supportValue K.val ((0 + Real.pi / 2 : ℝ) : Real.Angle) - 1 at h
+      rw [zero_add, K.property.2.2.2.1] at h
+      have hf := hpF.2
+      change 0 ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at hf
+      linarith
+    have htω : t ≠ ω := by
+      intro he
+      subst t
+      have h := hq.1
+      change inner ℝ p (normalVector (ω : Real.Angle)) <
+        supportValue K.val (ω : Real.Angle) - 1 at h
+      rw [K.property.2.2.1] at h
+      have hf := hpF.1
+      change 0 ≤ inner ℝ p (normalVector (ω : Real.Angle)) at hf
+      linarith
+    exact hpN ⟨hpF, Set.mem_iUnion₂.mpr
+      ⟨t, ⟨lt_of_le_of_ne ht.1 (Ne.symm ht0), lt_of_le_of_ne ht.2 htω⟩, hq⟩⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Cap.Connectedness`.
+* `Cap.MirrorFeatures`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Connectedness
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private theorem isOpen_capNiche_subtype {ω : ℝ} (K : CapSpace ω) :
+    IsOpen {p : capFan ω | (p : Point) ∈ capNiche K} := by
+  let U : Set Point := ⋃ t ∈ Set.Ioo 0 ω, innerQuadrant (K.val : Set Point) t
+  have hU : IsOpen U := isOpen_iUnion fun t ↦ isOpen_iUnion fun _ ↦ by
+    apply IsOpen.inter
+    · exact isOpen_lt (by fun_prop) continuous_const
+    · exact isOpen_lt (by fun_prop) continuous_const
+  have heq : {p : capFan ω | (p : Point) ∈ capNiche K} =
+      Subtype.val ⁻¹' U := by
+    ext p
+    change ((p : Point) ∈ capNiche K) ↔ (p : Point) ∈ U
+    simp only [capNiche, Set.mem_inter_iff, p.property, true_and, U]
+  rw [heq]
+  exact hU.preimage continuous_subtype_val
+
+private theorem capNiche_disjoint_upperBoundary_of_subset {ω : ℝ} (K : CapSpace ω)
+    (hsub : capNiche K ⊆ (K.val : Set Point)) :
+    Disjoint (capNiche K) (capUpperBoundary K) := by
+  rw [Set.disjoint_left]
+  intro p hpN hpδ
+  let z : capFan ω := ⟨p, hpN.1⟩
+  have hzopen : z ∈ {q : capFan ω | (q : Point) ∈ capNiche K} := hpN
+  have hzint : z ∈ interior {q : capFan ω | (q : Point) ∈ (K.val : Set Point)} := by
+    apply mem_interior_iff_mem_nhds.mpr
+    exact Filter.mem_of_superset (isOpen_capNiche_subtype K |>.mem_nhds hzopen)
+      (fun q hq ↦ hsub hq)
+  have hzfront : z ∈ frontier {q : capFan ω | (q : Point) ∈ (K.val : Set Point)} := by
+    rw [capUpperBoundary_relativeBoundary] at hpδ
+    obtain ⟨q, hq, hqp⟩ := hpδ
+    have : q = z := Subtype.ext hqp
+    simpa [this] using hq
+  have hzK : z ∈ {q : capFan ω | (q : Point) ∈ (K.val : Set Point)} := hsub hpN
+  exact (mem_frontier_iff_notMem_interior hzK).1 hzfront hzint
+
+private theorem mem_interior_capFan_of_pos {ω : ℝ} {p : Point}
+    (hω : 0 < inner ℝ p (normalVector (ω : Real.Angle)))
+    (hπ : 0 < inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))) :
+    p ∈ interior (capFan ω) := by
+  let U := normalHalfPlane (ω : Real.Angle) 0 true true ∩
+    normalHalfPlane ((Real.pi / 2 : ℝ) : Real.Angle) 0 true true
+  have hU : IsOpen U := (isOpen_lt continuous_const (by fun_prop)).inter
+    (isOpen_lt continuous_const (by fun_prop))
+  apply mem_interior_iff_mem_nhds.mpr
+  refine Filter.mem_of_superset (hU.mem_nhds ?_) ?_
+  · exact ⟨hω, hπ⟩
+  · rintro q ⟨hqω, hqπ⟩
+    change 0 < inner ℝ q (normalVector (ω : Real.Angle)) at hqω
+    change 0 < inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at hqπ
+    change 0 ≤ inner ℝ q (normalVector (ω : Real.Angle)) ∧
+      0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+    exact ⟨hqω.le, hqπ.le⟩
+
+private theorem innerCorner_mem_interior_capFan_of_mem_capWedge {ω t : ℝ}
+    (K : CapSpace ω) (ht : t ∈ Set.Ioo 0 ω) {q : Point} (hq : q ∈ capWedge K t) :
+    (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner ∈
+      interior (capFan ω) := by
+  let z := (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner
+  rcases hq with ⟨hqFan, hqQuad⟩
+  have hzu : inner ℝ z (normalVector (t : Real.Angle)) =
+      supportValue K.val (t : Real.Angle) - 1 := by
+    simpa [z, rotatingHallwayParts, hallwayParts] using
+      inner_supportingPlacement_normalVector (K.val : Set Point) (t : Real.Angle) (0 : Point)
+  have hzv : inner ℝ z (tangentVector (t : Real.Angle)) =
+      supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+    simpa [z, rotatingHallwayParts, hallwayParts] using
+      inner_supportingPlacement_tangentVector (K.val : Set Point) (t : Real.Angle) (0 : Point)
+  change q ∈ supportingPlacement (K.val : Set Point) (t : Real.Angle) ''
+    hallwayParts.innerQuadrant at hqQuad
+  obtain ⟨a, ha, hqa⟩ := hqQuad
+  change a 0 < 0 ∧ a 1 < 0 at ha
+  have hqQuad : inner ℝ q (normalVector (t : Real.Angle)) <
+      supportValue K.val (t : Real.Angle) - 1 ∧
+    inner ℝ q (tangentVector (t : Real.Angle)) <
+      supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+    constructor
+    · rw [← hqa, inner_supportingPlacement_normalVector]
+      linarith [ha.1]
+    · rw [← hqa, inner_supportingPlacement_tangentVector]
+      change a 1 + supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 < _
+      linarith [ha.2]
+  let d := q - z
+  have hdu : inner ℝ d (normalVector (t : Real.Angle)) < 0 := by
+    rw [inner_sub_left]
+    linarith [hqQuad.1, hzu]
+  have hdv : inner ℝ d (tangentVector (t : Real.Angle)) < 0 := by
+    rw [inner_sub_left]
+    linarith [hqQuad.2, hzv]
+  have hdφ (φ : ℝ) : inner ℝ d (normalVector (φ : Real.Angle)) =
+      inner ℝ d (normalVector (t : Real.Angle)) * Real.cos (φ - t) +
+        inner ℝ d (tangentVector (t : Real.Angle)) * Real.sin (φ - t) := by
+    nth_rewrite 1 [← inner_normalVector_smul_add_inner_tangentVector_smul d (t : Real.Angle)]
+    rw [inner_add_left, real_inner_smul_left, real_inner_smul_left,
+      inner_normalVector_normalVector]
+    have hc : Real.cos (t - φ) = Real.cos (φ - t) := by
+      rw [show t - φ = -(φ - t) by ring, Real.cos_neg]
+    rw [hc]
+    have hs : inner ℝ (tangentVector (t : Real.Angle))
+        (normalVector (φ : Real.Angle)) = Real.sin (φ - t) := by
+      simp [normalVector, tangentVector, frame, PiLp.inner_apply, Real.sin_sub]
+      ring
+    rw [hs]
+  have hsint : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi ht.1
+    (by linarith [ht.2, K.property.2.1, Real.pi_pos])
+  have hcost : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, ht.1], ht.2.trans_le K.property.2.1⟩
+  have hsinδ : 0 < Real.sin (ω - t) := Real.sin_pos_of_pos_of_lt_pi
+    (sub_pos.mpr ht.2) (by linarith [ht.1, K.property.2.1, Real.pi_pos])
+  have hcosδ : 0 < Real.cos (ω - t) := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [sub_pos.mpr ht.2, Real.pi_pos], by linarith [ht.1, K.property.2.1]⟩
+  apply mem_interior_capFan_of_pos
+  · change 0 < inner ℝ z (normalVector (ω : Real.Angle))
+    have hdω : inner ℝ d (normalVector (ω : Real.Angle)) < 0 := by
+      rw [hdφ]
+      exact add_neg (mul_neg_of_neg_of_pos hdu hcosδ)
+        (mul_neg_of_neg_of_pos hdv hsinδ)
+    have hqω : 0 ≤ inner ℝ q (normalVector (ω : Real.Angle)) := hqFan.1
+    change inner ℝ (q - z) (normalVector (ω : Real.Angle)) < 0 at hdω
+    rw [inner_sub_left] at hdω
+    linarith
+  · change 0 < inner ℝ z (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+    have hdπ : inner ℝ d (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) < 0 := by
+      rw [hdφ, Real.cos_pi_div_two_sub, Real.sin_pi_div_two_sub]
+      exact add_neg (mul_neg_of_neg_of_pos hdu hsint)
+        (mul_neg_of_neg_of_pos hdv hcost)
+    have hqπ : 0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) := hqFan.2
+    change inner ℝ (q - z) (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) < 0 at hdπ
+    rw [inner_sub_left] at hdπ
+    linarith
+
+private theorem exists_mem_capNiche_not_mem_of_innerCorner {ω t : ℝ}
+    (K : CapSpace ω) (ht : t ∈ Set.Ioo 0 ω)
+    (hzFan : (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner ∈
+      interior (capFan ω))
+    (hzK : (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner ∉
+      (K.val : Set Point)) :
+    ∃ q, q ∈ capNiche K ∧ q ∉ (K.val : Set Point) := by
+  let z := (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner
+  obtain ⟨ε, hε, hεball⟩ := Metric.isOpen_iff.mp isOpen_interior z hzFan
+  obtain ⟨δ, hδ, hδball⟩ := Metric.isOpen_iff.mp K.val.isClosed.isOpen_compl z hzK
+  let e := min ε δ / 2
+  have he : 0 < e := half_pos (lt_min hε hδ)
+  let q := z - e • normalVector ((Real.pi / 2 : ℝ) : Real.Angle)
+  have hdist : dist q z = e := by
+    simp [q, norm_smul, norm_normalVector_real, abs_of_pos he]
+  have hqFan : q ∈ capFan ω := interior_subset (hεball (by
+    change dist q z < ε
+    rw [hdist]
+    dsimp [e]
+    linarith [min_le_left ε δ]))
+  have hqK : q ∉ (K.val : Set Point) := hδball (by
+    change dist q z < δ
+    rw [hdist]
+    dsimp [e]
+    linarith [min_le_right ε δ])
+  have hzu : inner ℝ z (normalVector (t : Real.Angle)) =
+      supportValue K.val (t : Real.Angle) - 1 := by
+    simpa [z, rotatingHallwayParts, hallwayParts] using
+      inner_supportingPlacement_normalVector (K.val : Set Point) (t : Real.Angle) (0 : Point)
+  have hzv : inner ℝ z (tangentVector (t : Real.Angle)) =
+      supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+    simpa [z, rotatingHallwayParts, hallwayParts] using
+      inner_supportingPlacement_tangentVector (K.val : Set Point) (t : Real.Angle) (0 : Point)
+  have hs : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi ht.1
+    (by linarith [ht.2, K.property.2.1, Real.pi_pos])
+  have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, ht.1], ht.2.trans_le K.property.2.1⟩
+  have hquad : q ∈ innerQuadrant (K.val : Set Point) t := by
+    constructor
+    · change inner ℝ q (normalVector (t : Real.Angle)) < _
+      dsimp [q]
+      rw [inner_sub_left, real_inner_smul_left, hzu]
+      simp [normalVector, frame, PiLp.inner_apply]
+      nlinarith
+    · change inner ℝ q (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) < _
+      rw [show ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+        (t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) by rfl,
+        normalVector_add_pi_div_two]
+      dsimp [q]
+      rw [inner_sub_left, real_inner_smul_left, hzv]
+      simp [normalVector, tangentVector, frame, PiLp.inner_apply]
+      nlinarith
+  exact ⟨q, ⟨hqFan, Set.mem_iUnion₂.mpr ⟨t, ht, hquad⟩⟩, hqK⟩
+
+private theorem verticalLine_disjoint_cap_diff_niche_of_innerCorner {ω t : ℝ}
+    (K : CapSpace ω) (ht : t ∈ Set.Ioo 0 ω)
+    (hzFan : (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner ∈
+      interior (capFan ω))
+    (hzK : (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner ∉
+      (K.val : Set Point)) {q : Point}
+    (hq : q ∈ (K.val : Set Point) \ capNiche K) :
+    q 0 ≠ (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner 0 := by
+  let z := (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner
+  intro hx
+  change q 0 = z 0 at hx
+  by_cases hy : z 1 ≤ q 1
+  · have hnot : ¬ ∀ u ∈ capUpperAngles ω,
+        inner ℝ z (normalVector (u : Real.Angle)) ≤ supportValue K.val (u : Real.Angle) := by
+      intro hall
+      exact hzK (K.mem_of_mem_capFan_of_le_supportValue (interior_subset hzFan) hall)
+    push Not at hnot
+    obtain ⟨u, hu, hzu⟩ := hnot
+    have huI : u ∈ Set.Icc 0 (ω + Real.pi / 2) := by
+      rcases hu with hu | hu
+      · exact ⟨hu.1, hu.2.trans (le_add_of_nonneg_right (by positivity))⟩
+      · exact ⟨(by positivity : 0 ≤ Real.pi / 2).trans hu.1, hu.2⟩
+    have hsin : 0 ≤ Real.sin u := Real.sin_nonneg_of_nonneg_of_le_pi huI.1
+      (huI.2.trans (by linarith [K.property.2.1, Real.pi_pos]))
+    have hinner : inner ℝ z (normalVector (u : Real.Angle)) ≤
+        inner ℝ q (normalVector (u : Real.Angle)) := by
+      simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe, PiLp.inner_apply,
+        RCLike.inner_apply, conj_trivial, Fin.sum_univ_two, Fin.isValue, Matrix.cons_val_zero,
+        Matrix.cons_val_one, Matrix.cons_val_fin_one]
+      rw [hx]
+      nlinarith [mul_le_mul_of_nonneg_left hy hsin]
+    have hqle := inner_le_supportValue K.val hq.1 (u : Real.Angle)
+    linarith
+  · have hzu : inner ℝ z (normalVector (t : Real.Angle)) =
+        supportValue K.val (t : Real.Angle) - 1 := by
+      simpa [z, rotatingHallwayParts, hallwayParts] using
+        inner_supportingPlacement_normalVector (K.val : Set Point) (t : Real.Angle) (0 : Point)
+    have hzv : inner ℝ z (tangentVector (t : Real.Angle)) =
+        supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+      simpa [z, rotatingHallwayParts, hallwayParts] using
+        inner_supportingPlacement_tangentVector (K.val : Set Point) (t : Real.Angle) (0 : Point)
+    have hs : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi ht.1
+      (by linarith [ht.2, K.property.2.1, Real.pi_pos])
+    have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [Real.pi_pos, ht.1], ht.2.trans_le K.property.2.1⟩
+    apply hq.2
+    refine ⟨K.subset_capFan hq.1, Set.mem_iUnion₂.mpr ⟨t, ht, ?_⟩⟩
+    constructor
+    · change inner ℝ q (normalVector (t : Real.Angle)) < _
+      simp [normalVector, frame, PiLp.inner_apply] at hzu ⊢
+      nlinarith
+    · change inner ℝ q (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) < _
+      rw [show ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+        (t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) by rfl,
+        normalVector_add_pi_div_two]
+      simp [tangentVector, frame, PiLp.inner_apply] at hzv ⊢
+      nlinarith
+
+private theorem exists_upperBoundary_segment_subset_diff {ω : ℝ} (K : CapSpace ω)
+    {p : Point} (hp : p ∈ (K.val : Set Point) \ capNiche K) :
+    ∃ q ∈ capUpperBoundary K, segment ℝ p q ⊆ (K.val : Set Point) \ capNiche K := by
+  let F : Set Point := (K.val : Set Point) ∩ {q | q 0 = p 0}
+  have hFc : IsCompact F := K.val.isCompact.inter_right
+    (isClosed_eq (by fun_prop) continuous_const)
+  have hpF : p ∈ F := ⟨hp.1, rfl⟩
+  obtain ⟨q, hqF, hqmax⟩ := hFc.exists_isMaxOn ⟨p, hpF⟩
+    (by fun_prop : Continuous (fun q : Point ↦ q 1)).continuousOn
+  have hqK : q ∈ (K.val : Set Point) := hqF.1
+  have hqδ : q ∈ capUpperBoundary K := by
+    rw [capUpperBoundary_relativeBoundary]
+    let z : capFan ω := ⟨q, K.subset_capFan hqK⟩
+    have hzK : z ∈ {w : capFan ω | (w : Point) ∈ (K.val : Set Point)} := hqK
+    refine ⟨z, (mem_frontier_iff_notMem_interior hzK).2 ?_, rfl⟩
+    intro hzint
+    obtain ⟨ε, hε, hball⟩ := Metric.isOpen_iff.mp isOpen_interior z hzint
+    let r : Point := q + (ε / 2) • normalVector ((Real.pi / 2 : ℝ) : Real.Angle)
+    have hrFan : r ∈ capFan ω := by
+      constructor
+      · change 0 ≤ inner ℝ r (normalVector (ω : Real.Angle))
+        dsimp [r]
+        rw [inner_add_left, real_inner_smul_left, inner_normalVector_normalVector]
+        have hsin : 0 ≤ Real.sin ω := Real.sin_nonneg_of_nonneg_of_le_pi K.property.1.le
+          (K.property.2.1.trans (by linarith [Real.pi_pos]))
+        have := (K.subset_capFan hqK).1
+        change 0 ≤ inner ℝ q (normalVector (ω : Real.Angle)) at this
+        rw [Real.cos_pi_div_two_sub]
+        positivity
+      · change 0 ≤ inner ℝ r (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+        dsimp [r]
+        rw [inner_add_left, real_inner_smul_left, inner_normalVector_self]
+        have := (K.subset_capFan hqK).2
+        change 0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at this
+        linarith
+    have hrball : (⟨r, hrFan⟩ : capFan ω) ∈ Metric.ball z ε := by
+      change dist r q < ε
+      simp [r, norm_smul, norm_normalVector_real, abs_of_pos hε]
+      linarith
+    have hrK := interior_subset (hball hrball)
+    have hrF : r ∈ F := ⟨hrK, by simpa [r, normalVector, frame] using hqF.2⟩
+    have := hqmax hrF
+    simp [r, normalVector, frame] at this
+    linarith
+  refine ⟨q, hqδ, ?_⟩
+  intro x hx
+  have hxK := K.val.convex.segment_subset hp.1 hqK hx
+  refine ⟨hxK, ?_⟩
+  intro hxN
+  have hxFan := hxN.1
+  obtain ⟨u, hu, hxQuad⟩ := Set.mem_iUnion₂.mp hxN.2
+  have hxp0 : x 0 = p 0 := by
+    rw [segment_eq_image_lineMap] at hx
+    obtain ⟨s, hs, rfl⟩ := hx
+    simp only [Fin.isValue, AffineMap.lineMap_apply_module, PiLp.add_apply, PiLp.smul_apply,
+      smul_eq_mul]
+    rw [hqF.2]
+    ring
+  have hpx1 : p 1 ≤ x 1 := by
+    rw [segment_eq_image_lineMap] at hx
+    obtain ⟨s, hs, rfl⟩ := hx
+    have hpq := hqmax hpF
+    simp [AffineMap.lineMap_apply_module]
+    nlinarith [mul_nonneg hs.1 (sub_nonneg.mpr hpq)]
+  apply hp.2
+  refine ⟨K.subset_capFan hp.1, Set.mem_iUnion₂.mpr ⟨u, hu, ?_⟩⟩
+  rcases hxQuad with ⟨hxU, hxV⟩
+  constructor
+  · change inner ℝ p (normalVector (u : Real.Angle)) < _
+    change inner ℝ x (normalVector (u : Real.Angle)) < _ at hxU
+    have hs : 0 < Real.sin u := Real.sin_pos_of_pos_of_lt_pi hu.1
+      (by linarith [hu.2, K.property.2.1, Real.pi_pos])
+    simp only [normalVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe,
+      PiLp.inner_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two,
+      Fin.isValue, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one,
+      gt_iff_lt] at hxU ⊢
+    rw [hxp0] at hxU
+    nlinarith [mul_le_mul_of_nonneg_left hpx1 hs.le]
+  · change inner ℝ p (normalVector ((u + Real.pi / 2 : ℝ) : Real.Angle)) < _
+    change inner ℝ x (normalVector ((u + Real.pi / 2 : ℝ) : Real.Angle)) < _ at hxV
+    have hc : 0 < Real.cos u := Real.cos_pos_of_mem_Ioo
+      ⟨by linarith [Real.pi_pos, hu.1], hu.2.trans_le K.property.2.1⟩
+    rw [show ((u + Real.pi / 2 : ℝ) : Real.Angle) =
+      (u : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) by rfl,
+      normalVector_add_pi_div_two] at hxV ⊢
+    simp only [tangentVector, frame, Real.Angle.cos_coe, Real.Angle.sin_coe,
+      PiLp.inner_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_two,
+      Fin.isValue, Matrix.cons_val_zero, neg_mul, Matrix.cons_val_one,
+      Matrix.cons_val_fin_one, neg_add_lt_iff_lt_add, gt_iff_lt] at hxV ⊢
+    rw [hxp0] at hxV
+    nlinarith [mul_le_mul_of_nonneg_left hpx1 hc.le]
+
+theorem cap_niche_connected_iff {ω : ℝ} (K : CapSpace ω) :
+    (capNiche K ⊆ (K.val : Set Point) ↔
+      capNiche K ⊆ (K.val : Set Point) \ capUpperBoundary K) ∧
+    (capNiche K ⊆ (K.val : Set Point) \ capUpperBoundary K ↔
+      ∀ t ∈ Set.Ioo 0 ω,
+        (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner ∉
+          interior (capFan ω) ∨
+        (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner ∈
+          (K.val : Set Point)) ∧
+    ((∀ t ∈ Set.Ioo 0 ω,
+        (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner ∉
+          interior (capFan ω) ∨
+        (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner ∈
+          (K.val : Set Point)) ↔
+      IsConnected ((K.val : Set Point) \ capNiche K)) := by
+  refine ⟨?_, ?_, ?_⟩
+  · constructor
+    · intro hsub p hp
+      refine ⟨hsub hp, ?_⟩
+      exact fun hpδ ↦
+        Set.disjoint_left.mp (capNiche_disjoint_upperBoundary_of_subset K hsub) hp hpδ
+    · intro hsub p hp
+      exact (hsub hp).1
+  · constructor
+    · intro hsub t ht
+      by_cases hzFan : (rotatingHallwayParts (K.val : Set Point)
+          (t : Real.Angle)).innerCorner ∈ interior (capFan ω)
+      · right
+        by_contra hzK
+        obtain ⟨q, hqN, hqK⟩ :=
+          exists_mem_capNiche_not_mem_of_innerCorner K ht hzFan hzK
+        exact hqK (hsub hqN).1
+      · exact Or.inl hzFan
+    · intro hcorner p hp
+      have hpFan := hp.1
+      obtain ⟨t, ht, hpQuad⟩ := Set.mem_iUnion₂.mp hp.2
+      have hpWedge : p ∈ capWedge K t := ⟨hpFan, by
+        change p ∈ innerQuadrant (K.val : Set Point) t at hpQuad
+        have hf := rotatingHallwayParts_formulas (K.val : Set Point) (t : Real.Angle)
+        rw [hf.2.2.2.2.2.2.2.2]
+        exact hpQuad⟩
+      rcases hcorner t ht with hzout | hzK
+      · exact False.elim (hzout (innerCorner_mem_interior_capFan_of_mem_capWedge K ht hpWedge))
+      · have hpK := capWedge_subset_of_innerCorner_mem K t ht hzK hpWedge
+        refine ⟨hpK, ?_⟩
+        exact fun hpδ ↦ Set.disjoint_left.mp
+          (capNiche_disjoint_upperBoundary_of_subset K fun q hq ↦ by
+            have hqFan := hq.1
+            obtain ⟨u, hu, hqQuad⟩ := Set.mem_iUnion₂.mp hq.2
+            have hqWedge : q ∈ capWedge K u := ⟨hqFan, by
+              have hf := rotatingHallwayParts_formulas (K.val : Set Point) (u : Real.Angle)
+              rw [hf.2.2.2.2.2.2.2.2]
+              exact hqQuad⟩
+            rcases hcorner u hu with hzout | hzmem
+            · exact False.elim
+                (hzout (innerCorner_mem_interior_capFan_of_mem_capWedge K hu hqWedge))
+            · exact capWedge_subset_of_innerCorner_mem K u hu hzmem hqWedge) hp hpδ
+  · constructor
+    · intro hcorner
+      have hsub : capNiche K ⊆ (K.val : Set Point) := by
+        intro p hp
+        have hpFan := hp.1
+        obtain ⟨t, ht, hpQuad⟩ := Set.mem_iUnion₂.mp hp.2
+        have hpWedge : p ∈ capWedge K t := ⟨hpFan, by
+          have hf := rotatingHallwayParts_formulas (K.val : Set Point) (t : Real.Angle)
+          rw [hf.2.2.2.2.2.2.2.2]
+          exact hpQuad⟩
+        rcases hcorner t ht with hzout | hzmem
+        · exact False.elim
+            (hzout (innerCorner_mem_interior_capFan_of_mem_capWedge K ht hpWedge))
+        · exact capWedge_subset_of_innerCorner_mem K t ht hzmem hpWedge
+      have hδS : capUpperBoundary K ⊆ (K.val : Set Point) \ capNiche K := by
+        intro q hq
+        have hqK : q ∈ (K.val : Set Point) := by
+          obtain ⟨t, ht, hqedge⟩ := Set.mem_iUnion₂.mp hq
+          exact hqedge.1
+        exact ⟨hqK, fun hqN ↦
+          Set.disjoint_left.mp (capNiche_disjoint_upperBoundary_of_subset K hsub) hqN hq⟩
+      obtain ⟨l, r, hlK, -, hl, -⟩ := exists_horizontal_extrema K.val
+      have hlS : l ∈ (K.val : Set Point) \ capNiche K := ⟨hlK, fun hlN ↦ by
+        have := (capNiche_subset_rectangle K hlN).1
+        linarith⟩
+      obtain ⟨q₀, hq₀δ, hseg₀⟩ := exists_upperBoundary_segment_subset_diff K hlS
+      let S : Set Point := (K.val : Set Point) \ capNiche K
+      let f : S → Point := fun p ↦ Classical.choose
+        (exists_upperBoundary_segment_subset_diff K p.property)
+      have hfδ (p : S) : f p ∈ capUpperBoundary K :=
+        (Classical.choose_spec (exists_upperBoundary_segment_subset_diff K p.property)).1
+      have hfseg (p : S) : segment ℝ (p : Point) (f p) ⊆ S :=
+        (Classical.choose_spec (exists_upperBoundary_segment_subset_diff K p.property)).2
+      let A : S → Set Point := fun p ↦ capUpperBoundary K ∪ segment ℝ (p : Point) (f p)
+      have hA (p : S) : IsPreconnected (A p) := by
+        apply IsPreconnected.union' ⟨f p, hfδ p, right_mem_segment ℝ _ _⟩
+        · exact (capUpperBoundary_connected K).isPreconnected
+        · exact (convex_segment (𝕜 := ℝ) (p : Point) (f p)).isPreconnected
+      have hcommon : (⋂ p, A p).Nonempty := by
+        refine ⟨q₀, Set.mem_iInter.mpr fun p ↦ ?_⟩
+        exact Or.inl hq₀δ
+      have hunion : (⋃ p, A p) = S := by
+        apply Set.Subset.antisymm
+        · intro x hx
+          obtain ⟨p, hxp⟩ := Set.mem_iUnion.mp hx
+          rcases hxp with hxδ | hxseg
+          · exact hδS hxδ
+          · exact hfseg p hxseg
+        · intro x hx
+          let p : S := ⟨x, hx⟩
+          exact Set.mem_iUnion.mpr ⟨p, Or.inr (left_mem_segment ℝ _ _)⟩
+      refine ⟨⟨l, hlS⟩, ?_⟩
+      change IsPreconnected S
+      rw [← hunion]
+      exact isPreconnected_iUnion hcommon hA
+    · intro hconn t ht
+      by_cases hzFan : (rotatingHallwayParts (K.val : Set Point)
+          (t : Real.Angle)).innerCorner ∈ interior (capFan ω)
+      · right
+        by_contra hzK
+        let z := (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner
+        obtain ⟨l, r, hlK, hrK, hl, hr⟩ := exists_horizontal_extrema K.val
+        have hlS : l ∈ (K.val : Set Point) \ capNiche K := ⟨hlK, fun hlN ↦ by
+          have := (capNiche_subset_rectangle K hlN).1
+          linarith⟩
+        have hrS : r ∈ (K.val : Set Point) \ capNiche K := ⟨hrK, fun hrN ↦ by
+          have := (capNiche_subset_rectangle K hrN).2.1
+          linarith⟩
+        have hzy : 0 < z 1 := by
+          by_contra hn
+          obtain ⟨ε, hε, hball⟩ := Metric.isOpen_iff.mp isOpen_interior z hzFan
+          let q := z - (ε / 2) • normalVector ((Real.pi / 2 : ℝ) : Real.Angle)
+          have hq := interior_subset (hball (by
+            change dist q z < ε
+            simp [q, norm_smul, norm_normalVector_real, abs_of_pos hε]
+            linarith))
+          have := hq.2
+          change 0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at this
+          simp [q, normalVector, frame, PiLp.inner_apply] at this
+          linarith
+        have hzu : inner ℝ z (normalVector (t : Real.Angle)) =
+            supportValue K.val (t : Real.Angle) - 1 := by
+          simpa [z, rotatingHallwayParts, hallwayParts] using
+            inner_supportingPlacement_normalVector (K.val : Set Point)
+              (t : Real.Angle) (0 : Point)
+        have hzv : inner ℝ z (tangentVector (t : Real.Angle)) =
+            supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+          simpa [z, rotatingHallwayParts, hallwayParts] using
+            inner_supportingPlacement_tangentVector (K.val : Set Point)
+              (t : Real.Angle) (0 : Point)
+        have hb := K.supportValue_horizontal_bounds ht
+        have hb₁ := hb.1
+        have hb₂ := hb.2
+        change supportValue K.val ((t : Real.Angle) +
+          ((Real.pi / 2 : ℝ) : Real.Angle)) ≤
+            -Real.sin t * horizontalMin K.val + Real.cos t at hb₂
+        have hs : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi ht.1
+          (by linarith [ht.2, K.property.2.1, Real.pi_pos])
+        have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+          ⟨by linarith [Real.pi_pos, ht.1], ht.2.trans_le K.property.2.1⟩
+        have hs1 : Real.sin t ≤ 1 := Real.sin_le_one t
+        have hc1 : Real.cos t ≤ 1 := Real.cos_le_one t
+        have hzlr : l 0 < z 0 ∧ z 0 < r 0 := by
+          change inner ℝ z (tangentVector (t : Real.Angle)) =
+            supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 at hzv
+          simp [normalVector, tangentVector, frame, PiLp.inner_apply] at hzu hzv
+          rw [hl, hr]
+          constructor
+          · nlinarith [hb₂, mul_pos hc hzy, sub_nonneg.mpr hc1]
+          · nlinarith [hb₁, mul_pos hs hzy, sub_nonneg.mpr hs1]
+        obtain ⟨q, hqS, hqx⟩ := hconn.isPreconnected.intermediate_value hlS hrS
+          (by fun_prop : ContinuousOn (fun p : Point ↦ p 0) _)
+          ⟨hzlr.1.le, hzlr.2.le⟩
+        exact verticalLine_disjoint_cap_diff_niche_of_innerCorner K ht hzFan hzK hqS hqx
+      · exact Or.inl hzFan
+
+private theorem exists_line_meeting_of_support_bounds {s : Set Point} {ω : ℝ}
+    (hs : IsCompact s) (hc : IsConnected s) (hω : ω ≤ Real.pi / 2) (p : Point)
+    (hp0 : inner ℝ p (normalVector (0 : Real.Angle)) ≤ supportValue s 0)
+    (hpω : inner ℝ p (tangentVector (ω : Real.Angle)) ≤
+      supportValue s ((ω + Real.pi / 2 : ℝ) : Real.Angle)) :
+    ∃ q ∈ s, ∃ θ ∈ Set.Icc ω (Real.pi / 2),
+      inner ℝ (q - p) (tangentVector (θ : Real.Angle)) = 0 := by
+  obtain ⟨q₀, hq₀, he₀, _⟩ := hs.exists_sSup_image_eq_and_ge
+    (f := fun q : Point ↦ inner ℝ q (normalVector (0 : Real.Angle))) hc.nonempty
+    (continuous_id.inner continuous_const).continuousOn
+  obtain ⟨qω, hqω, heω, _⟩ := hs.exists_sSup_image_eq_and_ge
+    (f := fun q : Point ↦ inner ℝ q (tangentVector (ω : Real.Angle))) hc.nonempty
+    (continuous_id.inner continuous_const).continuousOn
+  have hn : normalVector ((ω + Real.pi / 2 : ℝ) : Real.Angle) =
+      tangentVector (ω : Real.Angle) := by
+    rw [Real.Angle.coe_add, normalVector_add_pi_div_two]
+  have hzero : tangentVector ((Real.pi / 2 : ℝ) : Real.Angle) =
+      -normalVector (0 : Real.Angle) := by
+    simpa using tangentVector_add_pi_div_two 0
+  have hpω' : inner ℝ p (tangentVector (ω : Real.Angle)) ≤
+      inner ℝ qω (tangentVector (ω : Real.Angle)) := by
+    simpa only [supportValue, hn, heω] using hpω
+  have hp0' : inner ℝ p (normalVector (0 : Real.Angle)) ≤
+      inner ℝ q₀ (normalVector (0 : Real.Angle)) := by
+    simpa only [supportValue, he₀] using hp0
+  let f : Point × ℝ → ℝ := fun z ↦ inner ℝ (z.1 - p) (tangentVector (z.2 : Real.Angle))
+  have hv : Continuous (fun θ : ℝ ↦ tangentVector (θ : Real.Angle)) := by
+    have h := continuous_normalVector_real.comp (continuous_id.add continuous_const
+      : Continuous (fun θ : ℝ ↦ θ + Real.pi / 2))
+    simpa only [Function.comp_def, Pi.add_apply, id_eq, Real.Angle.coe_add,
+      normalVector_add_pi_div_two] using h
+  have hf : Continuous f := (continuous_fst.sub continuous_const).inner
+    (hv.comp continuous_snd)
+  have hl : f (q₀, Real.pi / 2) ≤ 0 := by
+    change inner ℝ (q₀ - p) (tangentVector ((Real.pi / 2 : ℝ) : Real.Angle)) ≤ 0
+    rw [hzero, inner_neg_right, inner_sub_left]
+    linarith
+  have hr : 0 ≤ f (qω, ω) := by
+    change 0 ≤ inner ℝ (qω - p) (tangentVector (ω : Real.Angle))
+    rw [inner_sub_left]
+    linarith
+  obtain ⟨⟨q, θ⟩, hmem, he⟩ := (hc.isPreconnected.prod isPreconnected_Icc).intermediate_value
+    (f := f) ⟨hq₀, hω, le_rfl⟩ ⟨hqω, le_rfl, hω⟩ hf.continuousOn ⟨hl, hr⟩
+  exact ⟨q, hmem.1, θ, hmem.2, he⟩
+
+private theorem mem_supportingHallway_iff_coordinates (s : Set Point) (t : Real.Angle)
+    (p : Point) :
+    p ∈ supportingHallway s t ↔
+      inner ℝ p (normalVector t) ≤ supportValue s t ∧
+      inner ℝ p (tangentVector t) ≤ supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) ∧
+      (supportValue s t - 1 ≤ inner ℝ p (normalVector t) ∨
+        supportValue s (t + ((Real.pi / 2 : ℝ) : Real.Angle)) - 1 ≤
+          inner ℝ p (tangentVector t)) := by
+  obtain ⟨h, _, _, _, _, _, _, ho, hi⟩ := rotatingHallwayParts_formulas s t
+  rw [h, ho, hi]
+  simp only [Set.mem_sdiff, Set.mem_inter_iff, normalHalfPlane, Set.mem_ofPred_eq,
+    Bool.false_eq_true, ↓reduceIte, normalVector_add_pi_div_two]
+  simp only [not_and_or, not_lt, and_assoc]
+
+private theorem segment_subset_supportingHallway_of_projections_le {s : Set Point}
+    {t : Real.Angle} {p q : Point} (hp : p ∈ supportingHallway s t)
+    (hq : q ∈ supportingHallway s t)
+    (hn : inner ℝ p (normalVector t) ≤ inner ℝ q (normalVector t))
+    (hv : inner ℝ p (tangentVector t) ≤ inner ℝ q (tangentVector t)) :
+    segment ℝ p q ⊆ supportingHallway s t := by
+  rw [mem_supportingHallway_iff_coordinates] at hp hq
+  rintro z ⟨a, b, ha, hb, hab, rfl⟩
+  rw [mem_supportingHallway_iff_coordinates]
+  simp only [inner_add_left, real_inner_smul_left]
+  have weighted_le (x y m : ℝ) (hx : x ≤ m) (hy : y ≤ m) : a * x + b * y ≤ m := by
+    calc
+      a * x + b * y ≤ a * m + b * m :=
+        add_le_add (mul_le_mul_of_nonneg_left hx ha) (mul_le_mul_of_nonneg_left hy hb)
+      _ = m := by rw [← add_mul, hab, one_mul]
+  have le_weighted (x y m : ℝ) (hx : m ≤ x) (hy : m ≤ y) : m ≤ a * x + b * y := by
+    calc
+      m = a * m + b * m := by rw [← add_mul, hab, one_mul]
+      _ ≤ a * x + b * y :=
+        add_le_add (mul_le_mul_of_nonneg_left hx ha) (mul_le_mul_of_nonneg_left hy hb)
+  refine ⟨weighted_le _ _ _ hp.1 hq.1, weighted_le _ _ _ hp.2.1 hq.2.1, ?_⟩
+  rcases hp.2.2 with hn' | hv'
+  · exact Or.inl (le_weighted _ _ _ hn' (hn'.trans hn))
+  · exact Or.inr (le_weighted _ _ _ hv' (hv'.trans hv))
+
+private theorem convex_stripParallelogram (ω : ℝ) : Convex ℝ (stripParallelogram ω).1 := by
+  intro p hp q hq a b ha hb hab
+  rw [mem_stripParallelogram_iff] at hp hq ⊢
+  simp only [PiLp.add_apply, PiLp.smul_apply, smul_eq_mul, inner_add_left,
+    real_inner_smul_left]
+  exact ⟨convex_Icc (0 : ℝ) 1 hp.1 hq.1 ha hb hab, convex_Icc (0 : ℝ) 1 hp.2 hq.2 ha hb hab⟩
+
+private theorem segment_subset_supportingHallway_of_tangent_eq_zero {s : Set Point}
+    {t θ : ℝ} {p q : Point} (hp : p ∈ supportingHallway s (t : Real.Angle))
+    (hq : q ∈ supportingHallway s (t : Real.Angle))
+    (hθ : θ - t ∈ Set.Icc 0 (Real.pi / 2))
+    (hline : inner ℝ (q - p) (tangentVector (θ : Real.Angle)) = 0) :
+    segment ℝ p q ⊆ supportingHallway s (t : Real.Angle) := by
+  let c := inner ℝ (q - p) (normalVector (θ : Real.Angle))
+  have hdiff : q - p = c • normalVector (θ : Real.Angle) := by
+    have h := inner_normalVector_smul_add_inner_tangentVector_smul (q - p) (θ : Real.Angle)
+    simpa only [hline, zero_smul, add_zero] using h.symm
+  have hcos : 0 ≤ Real.cos (θ - t) :=
+    Real.cos_nonneg_of_mem_Icc ⟨by linarith [Real.pi_pos, hθ.1], hθ.2⟩
+  have hsin : 0 ≤ Real.sin (θ - t) :=
+    Real.sin_nonneg_of_nonneg_of_le_pi hθ.1 (by linarith [Real.pi_pos, hθ.2])
+  have hn : inner ℝ q (normalVector (t : Real.Angle)) -
+      inner ℝ p (normalVector (t : Real.Angle)) = c * Real.cos (θ - t) := by
+    rw [← inner_sub_left, hdiff, real_inner_smul_left, inner_normalVector_normalVector]
+  have hv : inner ℝ q (tangentVector (t : Real.Angle)) -
+      inner ℝ p (tangentVector (t : Real.Angle)) = c * Real.sin (θ - t) := by
+    rw [← inner_sub_left, hdiff, real_inner_smul_left]
+    congr 1
+    simp [normalVector, tangentVector, frame, PiLp.inner_apply, Fin.sum_univ_two,
+      Real.sin_sub]
+    ring
+  by_cases hc : 0 ≤ c
+  · apply segment_subset_supportingHallway_of_projections_le hp hq
+    · linarith [mul_nonneg hc hcos]
+    · linarith [mul_nonneg hc hsin]
+  · rw [segment_symm]
+    apply segment_subset_supportingHallway_of_projections_le hq hp
+    · linarith [mul_nonpos_of_nonpos_of_nonneg (le_of_not_ge hc) hcos]
+    · linarith [mul_nonpos_of_nonpos_of_nonneg (le_of_not_ge hc) hsin]
+
+theorem standardPosition_monotonization_connected (s : Set Point) (ω : ℝ)
+    (hs : IsStandardPosition s ω) : IsConnected (monotonization s ω) := by
+  have hc : IsConnected s := by
+    obtain ⟨m, hm, _⟩ := hs.2.1
+    exact hm.1
+  have hsub := (standardPosition_subset_cap s ω hs).1
+  obtain ⟨q₀, hq₀⟩ := hc.nonempty
+  refine ⟨hc.nonempty.mono hsub, isPreconnected_of_forall q₀ ?_⟩
+  intro p hp
+  have hpH (t : ℝ) (ht : t ∈ Set.Icc 0 ω) :
+      p ∈ supportingHallway s (t : Real.Angle) :=
+    Set.mem_iInter.1 (Set.mem_iInter.1 hp.2 t) ht
+  have hp0 := (mem_supportingHallway_iff_coordinates s 0 p).mp
+    (hpH 0 ⟨le_rfl, hs.2.2.1.le⟩)
+  have hpω := (mem_supportingHallway_iff_coordinates s (ω : Real.Angle) p).mp
+    (hpH ω ⟨hs.2.2.1.le, le_rfl⟩)
+  obtain ⟨q, hqs, θ, hθ, hline⟩ := exists_line_meeting_of_support_bounds hs.1 hc
+    hs.2.2.2.1 p hp0.1 (by simpa only [Real.Angle.coe_add] using hpω.2.1)
+  have hq := hsub hqs
+  have hseg : segment ℝ p q ⊆ monotonization s ω := by
+    intro z hz
+    refine ⟨(convex_stripParallelogram ω).segment_subset hp.1 hq.1 hz,
+      Set.mem_iInter.2 fun t ↦ Set.mem_iInter.2 fun ht ↦ ?_⟩
+    apply segment_subset_supportingHallway_of_tangent_eq_zero (hpH t ht)
+      (Set.mem_iInter.1 (Set.mem_iInter.1 hq.2 t) ht) _ hline hz
+    exact ⟨by linarith [hθ.1, ht.2], by linarith [hθ.2, ht.1]⟩
+  refine ⟨s ∪ segment ℝ p q, Set.union_subset hsub hseg, Or.inl hq₀,
+    Or.inr (left_mem_segment ℝ p q), ?_⟩
+  exact IsPreconnected.union q hqs (right_mem_segment ℝ p q) hc.isPreconnected
+    (convex_segment p q).isPreconnected
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Mirror Features
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open MeasureTheory
+
+namespace MovingSofa
+
+private theorem mirrorReflection_eq_capReflection_m6ef9701 (ω : ℝ)
+    (hω0 : 0 < ω) (hωle : ω ≤ Real.pi / 2) :
+    mirrorReflection ω = capReflection ω := by
+  change stripTopReflection ω = capReflection ω
+  exact stripTopReflection_eq_capReflection ω hω0 hωle
+
+private theorem capReflection_rotationMap (ω s : ℝ) (p : Point) :
+    capReflection ω (rotationMap (s : Real.Angle) p) =
+      rotationMap ((ω - s : ℝ) : Real.Angle) (coordinateSwap p) := by
+  have hang : (ω : Real.Angle) - (s : Real.Angle) =
+      ((ω - s : ℝ) : Real.Angle) := by
+    rw [← Real.Angle.coe_sub]
+  ext i
+  fin_cases i
+  · change capReflection ω (rotationMap (s : Real.Angle) p) 0 =
+      rotationMap ((ω - s : ℝ) : Real.Angle) (coordinateSwap p) 0
+    rw [capReflection_apply_zero]
+    rw [← hang]
+    simp only [Fin.isValue, rotationMap, Orientation.rotation_apply, Real.Angle.cos_coe,
+      Real.Angle.sin_coe, rightAngleRotation_apply, PiLp.add_apply, PiLp.smul_apply,
+      smul_eq_mul, Matrix.cons_val_zero, mul_neg, neg_mul, Matrix.cons_val_one,
+      Matrix.cons_val_fin_one, coordinateSwap, LinearIsometryEquiv.piLpCongrLeft_apply,
+      Equiv.piCongrLeft'_apply, Equiv.symm_swap, Equiv.swap_apply_right, Equiv.swap_apply_left]
+    rw [hang, Real.Angle.sin_coe, Real.Angle.cos_coe, Real.sin_sub, Real.cos_sub]
+    ring
+  · change capReflection ω (rotationMap (s : Real.Angle) p) 1 =
+      rotationMap ((ω - s : ℝ) : Real.Angle) (coordinateSwap p) 1
+    rw [capReflection_apply_one]
+    rw [← hang]
+    simp only [Fin.isValue, rotationMap, Orientation.rotation_apply, Real.Angle.cos_coe,
+      Real.Angle.sin_coe, rightAngleRotation_apply, PiLp.add_apply, PiLp.smul_apply,
+      smul_eq_mul, Matrix.cons_val_zero, mul_neg, Matrix.cons_val_one, Matrix.cons_val_fin_one,
+      coordinateSwap, LinearIsometryEquiv.piLpCongrLeft_apply, Equiv.piCongrLeft'_apply,
+      Equiv.symm_swap, Equiv.swap_apply_right, Equiv.swap_apply_left]
+    rw [hang, Real.Angle.sin_coe, Real.Angle.cos_coe, Real.sin_sub, Real.cos_sub]
+    ring
+
+private theorem capReflection_normal_at_complement (ω t : ℝ) :
+    capReflection ω (normalVector ((ω - t : ℝ) : Real.Angle)) =
+      tangentVector (t : Real.Angle) := by
+  rw [capReflection_normalVector]
+  have hang : ω + Real.pi / 2 - (ω - t) = t + Real.pi / 2 := by ring
+  rw [hang]
+  ext i
+  fin_cases i <;> simp [normalVector, tangentVector, frame,
+    Real.Angle.sin_add_pi_div_two, Real.Angle.cos_add_pi_div_two]
+
+private theorem capReflection_tangent_at_complement (ω t : ℝ) :
+    capReflection ω (tangentVector ((ω - t : ℝ) : Real.Angle)) =
+      normalVector (t : Real.Angle) := by
+  rw [capReflection_tangentVector]
+  have hang : ω + Real.pi / 2 - (ω - t) = t + Real.pi / 2 := by ring
+  rw [hang]
+  ext i
+  fin_cases i <;> simp [normalVector, tangentVector, frame,
+    Real.Angle.sin_add_pi_div_two, Real.Angle.cos_add_pi_div_two]
+
+private theorem supportValue_reflected_at (ω t : ℝ) (K : ConvexBody Point) :
+    supportValue (reflectedBody ω K) (t : Real.Angle) =
+      supportValue K ((ω - t + Real.pi / 2 : ℝ) : Real.Angle) := by
+  rw [supportValue_reflectedBody, reflectedAngle_coe]
+  congr 2
+  ring
+
+private theorem supportValue_reflected_at_add_pi_div_two (ω t : ℝ)
+    (K : ConvexBody Point) :
+    supportValue (reflectedBody ω K) ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+      supportValue K ((ω - t : ℝ) : Real.Angle) := by
+  rw [supportValue_reflectedBody, reflectedAngle_coe]
+  congr 2
+  ring
+
+private theorem supportingPlacement_reflection (ω t : ℝ)
+    (K : ConvexBody Point) (q : Point) :
+    supportingPlacement (reflectedBody ω K) (t : Real.Angle) (coordinateSwap q) =
+      capReflection ω (supportingPlacement K ((ω - t : ℝ) : Real.Angle) q) := by
+  unfold supportingPlacement
+  rw [map_add, map_add, map_smul, map_smul,
+    capReflection_rotationMap]
+  have hrot : ω - (ω - t) = t := by ring
+  have htadd : (t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) =
+      ((t + Real.pi / 2 : ℝ) : Real.Angle) := by
+    rw [← Real.Angle.coe_add]
+  have hsadd : ((ω - t : ℝ) : Real.Angle) +
+      ((Real.pi / 2 : ℝ) : Real.Angle) =
+      ((ω - t + Real.pi / 2 : ℝ) : Real.Angle) := by
+    rw [← Real.Angle.coe_add]
+  rw [hrot, supportValue_reflected_at,
+    htadd, supportValue_reflected_at_add_pi_div_two, hsadd,
+    capReflection_normal_at_complement,
+    capReflection_tangent_at_complement]
+  module
+
+private theorem coordinateSwap_involutive (p : Point) :
+    coordinateSwap (coordinateSwap p) = p := by
+  ext i
+  fin_cases i <;> simp [coordinateSwap]
+
+private theorem coordinateSwap_image_hallway :
+    coordinateSwap '' hallway = hallway := by
+  ext p
+  constructor
+  · rintro ⟨q, hq, rfl⟩
+    rcases hq with ⟨a, b, h, rfl⟩ | ⟨a, b, h, rfl⟩
+    · exact Or.inr ⟨b, a, ⟨h.2.1, h.2.2, h.1⟩, by
+        ext i
+        fin_cases i <;> simp [coordinateSwap]⟩
+    · exact Or.inl ⟨b, a, ⟨h.2.2, h.1, h.2.1⟩, by
+        ext i
+        fin_cases i <;> simp [coordinateSwap]⟩
+  · intro hp
+    refine ⟨coordinateSwap p, ?_, coordinateSwap_involutive p⟩
+    rcases hp with ⟨a, b, h, rfl⟩ | ⟨a, b, h, rfl⟩
+    · exact Or.inr ⟨b, a, ⟨h.2.1, h.2.2, h.1⟩, by
+        ext i
+        fin_cases i <;> simp [coordinateSwap]⟩
+    · exact Or.inl ⟨b, a, ⟨h.2.2, h.1, h.2.1⟩, by
+        ext i
+        fin_cases i <;> simp [coordinateSwap]⟩
+
+private theorem coordinateSwap_image_eq_preimage (s : Set Point) :
+    coordinateSwap '' s = coordinateSwap ⁻¹' s := by
+  ext p
+  constructor
+  · rintro ⟨q, hq, rfl⟩
+    change coordinateSwap (coordinateSwap q) ∈ s
+    rwa [coordinateSwap_involutive]
+  · intro hp
+    exact ⟨coordinateSwap p, hp, coordinateSwap_involutive p⟩
+
+private theorem coordinateSwap_image_outerQuadrant :
+    coordinateSwap '' hallwayParts.outerQuadrant = hallwayParts.outerQuadrant := by
+  rw [coordinateSwap_image_eq_preimage]
+  ext p
+  simp [hallwayParts, coordinateSwap, and_comm]
+
+private theorem coordinateSwap_image_innerQuadrant :
+    coordinateSwap '' hallwayParts.innerQuadrant = hallwayParts.innerQuadrant := by
+  rw [coordinateSwap_image_eq_preimage]
+  ext p
+  simp [hallwayParts, coordinateSwap, and_comm]
+
+private theorem coordinateSwap_image_a :
+    coordinateSwap '' hallwayParts.a = hallwayParts.c := by
+  rw [coordinateSwap_image_eq_preimage]
+  ext p
+  simp [hallwayParts, coordinateSwap]
+
+private theorem coordinateSwap_image_c :
+    coordinateSwap '' hallwayParts.c = hallwayParts.a := by
+  rw [coordinateSwap_image_eq_preimage]
+  ext p
+  simp [hallwayParts, coordinateSwap]
+
+private theorem coordinateSwap_image_b :
+    coordinateSwap '' hallwayParts.b = hallwayParts.d := by
+  rw [coordinateSwap_image_eq_preimage]
+  ext p
+  simp [hallwayParts, coordinateSwap]
+
+private theorem coordinateSwap_image_d :
+    coordinateSwap '' hallwayParts.d = hallwayParts.b := by
+  rw [coordinateSwap_image_eq_preimage]
+  ext p
+  simp [hallwayParts, coordinateSwap]
+
+private theorem coordinateSwap_image_bRay :
+    coordinateSwap '' hallwayParts.bRay = hallwayParts.dRay := by
+  rw [coordinateSwap_image_eq_preimage]
+  ext p
+  simp [hallwayParts, coordinateSwap, and_comm]
+
+private theorem coordinateSwap_image_dRay :
+    coordinateSwap '' hallwayParts.dRay = hallwayParts.bRay := by
+  rw [coordinateSwap_image_eq_preimage]
+  ext p
+  simp [hallwayParts, coordinateSwap, and_comm]
+
+private theorem supportingPlacement_image_reflection (ω t : ℝ)
+    (K : ConvexBody Point) (s : Set Point) :
+    supportingPlacement (reflectedBody ω K) (t : Real.Angle) ''
+        (coordinateSwap '' s) =
+      capReflection ω ''
+        (supportingPlacement K ((ω - t : ℝ) : Real.Angle) '' s) := by
+  rw [Set.image_image, Set.image_image]
+  congr 1
+  funext q
+  exact supportingPlacement_reflection ω t K q
+
+private theorem supportingHallway_reflection (ω t : ℝ)
+    (K : ConvexBody Point) :
+    supportingHallway (reflectedBody ω K) (t : Real.Angle) =
+      capReflection ω '' supportingHallway K ((ω - t : ℝ) : Real.Angle) := by
+  unfold supportingHallway
+  nth_rewrite 1 [← coordinateSwap_image_hallway]
+  exact supportingPlacement_image_reflection ω t K hallway
+
+private theorem rotatingHallwayParts_reflection (ω t : ℝ)
+    (K : ConvexBody Point) :
+    (rotatingHallwayParts (reflectedBody ω K) (t : Real.Angle)).innerCorner =
+        capReflection ω
+          (rotatingHallwayParts K ((ω - t : ℝ) : Real.Angle)).innerCorner ∧
+    (rotatingHallwayParts (reflectedBody ω K) (t : Real.Angle)).outerCorner =
+        capReflection ω
+          (rotatingHallwayParts K ((ω - t : ℝ) : Real.Angle)).outerCorner ∧
+    (rotatingHallwayParts (reflectedBody ω K) (t : Real.Angle)).outerQuadrant =
+        capReflection ω ''
+          (rotatingHallwayParts K ((ω - t : ℝ) : Real.Angle)).outerQuadrant ∧
+    (rotatingHallwayParts (reflectedBody ω K) (t : Real.Angle)).innerQuadrant =
+        capReflection ω ''
+          (rotatingHallwayParts K ((ω - t : ℝ) : Real.Angle)).innerQuadrant ∧
+    (rotatingHallwayParts (reflectedBody ω K) (t : Real.Angle)).a =
+        capReflection ω ''
+          (rotatingHallwayParts K ((ω - t : ℝ) : Real.Angle)).c ∧
+    (rotatingHallwayParts (reflectedBody ω K) (t : Real.Angle)).c =
+        capReflection ω ''
+          (rotatingHallwayParts K ((ω - t : ℝ) : Real.Angle)).a ∧
+    (rotatingHallwayParts (reflectedBody ω K) (t : Real.Angle)).b =
+        capReflection ω ''
+          (rotatingHallwayParts K ((ω - t : ℝ) : Real.Angle)).d ∧
+    (rotatingHallwayParts (reflectedBody ω K) (t : Real.Angle)).d =
+        capReflection ω ''
+          (rotatingHallwayParts K ((ω - t : ℝ) : Real.Angle)).b ∧
+    (rotatingHallwayParts (reflectedBody ω K) (t : Real.Angle)).bRay =
+        capReflection ω ''
+          (rotatingHallwayParts K ((ω - t : ℝ) : Real.Angle)).dRay ∧
+    (rotatingHallwayParts (reflectedBody ω K) (t : Real.Angle)).dRay =
+        capReflection ω ''
+          (rotatingHallwayParts K ((ω - t : ℝ) : Real.Angle)).bRay := by
+  have hzero : coordinateSwap (0 : Point) = 0 := by
+    ext i
+    fin_cases i <;> simp [coordinateSwap]
+  have hone : coordinateSwap (!₂[1, 1] : Point) = !₂[1, 1] := by
+    ext i
+    fin_cases i <;> simp [coordinateSwap]
+  unfold rotatingHallwayParts
+  dsimp only
+  refine ⟨?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
+  · simp only [hallwayParts]
+    rw [← hzero]
+    exact supportingPlacement_reflection ω t K 0
+  · simp only [hallwayParts]
+    nth_rewrite 1 [← hone]
+    exact supportingPlacement_reflection ω t K !₂[1, 1]
+  · nth_rewrite 1 [← coordinateSwap_image_outerQuadrant]
+    exact supportingPlacement_image_reflection ω t K hallwayParts.outerQuadrant
+  · nth_rewrite 1 [← coordinateSwap_image_innerQuadrant]
+    exact supportingPlacement_image_reflection ω t K hallwayParts.innerQuadrant
+  · nth_rewrite 1 [← coordinateSwap_image_c]
+    exact supportingPlacement_image_reflection ω t K hallwayParts.c
+  · nth_rewrite 1 [← coordinateSwap_image_a]
+    exact supportingPlacement_image_reflection ω t K hallwayParts.a
+  · nth_rewrite 1 [← coordinateSwap_image_d]
+    exact supportingPlacement_image_reflection ω t K hallwayParts.d
+  · nth_rewrite 1 [← coordinateSwap_image_b]
+    exact supportingPlacement_image_reflection ω t K hallwayParts.b
+  · nth_rewrite 1 [← coordinateSwap_image_dRay]
+    exact supportingPlacement_image_reflection ω t K hallwayParts.dRay
+  · nth_rewrite 1 [← coordinateSwap_image_bRay]
+    exact supportingPlacement_image_reflection ω t K hallwayParts.bRay
+
+private theorem exposedEdge_reflection (ω : ℝ) (K : ConvexBody Point)
+    (a : Real.Angle) :
+    exposedEdge (reflectedBody ω K) a =
+      capReflection ω '' exposedEdge K (reflectedAngle ω a) := by
+  ext p
+  constructor
+  · rintro ⟨⟨q, hq, rfl⟩, hp⟩
+    refine ⟨q, ⟨hq, ?_⟩, rfl⟩
+    change inner ℝ (capReflection ω q) (normalVector a) =
+      supportValue (reflectedBody ω K) a at hp
+    change inner ℝ q (normalVector (reflectedAngle ω a)) =
+      supportValue K (reflectedAngle ω a)
+    rw [← inner_capReflection_normalVector ω q a,
+      ← supportValue_reflectedBody ω K a]
+    exact hp
+  · rintro ⟨q, ⟨hq, hp⟩, rfl⟩
+    refine ⟨⟨q, hq, rfl⟩, ?_⟩
+    change inner ℝ (capReflection ω q) (normalVector a) =
+      supportValue (reflectedBody ω K) a
+    rw [inner_capReflection_normalVector, supportValue_reflectedBody]
+    exact hp
+
+private theorem exposedEdgeHeights_reflection (ω : ℝ) (K : ConvexBody Point)
+    (a : Real.Angle) :
+    (fun p ↦ inner ℝ p (tangentVector a)) '' exposedEdge (reflectedBody ω K) a =
+      -((fun p ↦ inner ℝ p (tangentVector (reflectedAngle ω a))) ''
+        exposedEdge K (reflectedAngle ω a)) := by
+  rw [exposedEdge_reflection]
+  ext x
+  simp only [Set.mem_image, Set.mem_neg]
+  constructor
+  · rintro ⟨_, ⟨q, hq, rfl⟩, rfl⟩
+    refine ⟨q, hq, ?_⟩
+    rw [inner_capReflection_tangentVector, neg_neg]
+  · rintro ⟨q, hq, hq_inner⟩
+    refine ⟨capReflection ω q, ⟨q, hq, rfl⟩, ?_⟩
+    rw [inner_capReflection_tangentVector, hq_inner, neg_neg]
+
+private theorem edgeVertices_reflection (ω : ℝ) (K : ConvexBody Point)
+    (a : Real.Angle) :
+    edgeVertices (reflectedBody ω K) a =
+      (capReflection ω (edgeVertices K (reflectedAngle ω a)).2,
+        capReflection ω (edgeVertices K (reflectedAngle ω a)).1) := by
+  unfold edgeVertices
+  dsimp only
+  rw [supportValue_reflectedBody, exposedEdgeHeights_reflection,
+    Real.sSup_neg, Real.sInf_neg]
+  apply Prod.ext <;> dsimp only
+  · rw [map_add, map_smul, map_smul, capReflection_normalVector_angle,
+      capReflection_tangentVector_angle, reflectedAngle_involutive]
+    module
+  · rw [map_add, map_smul, map_smul, capReflection_normalVector_angle,
+      capReflection_tangentVector_angle, reflectedAngle_involutive]
+    module
+
+private theorem capVertices_reflection {ω : ℝ} (K : CapSpace ω) (t : ℝ) :
+    let P : CapSpace ω := ⟨reflectedBody ω K.val, reflectedBody_isCap K⟩
+    capVertices P t =
+      ((capReflection ω (capVertices K (ω - t)).2.2,
+        capReflection ω (capVertices K (ω - t)).2.1),
+       (capReflection ω (capVertices K (ω - t)).1.2,
+        capReflection ω (capVertices K (ω - t)).1.1)) := by
+  dsimp only
+  unfold capVertices
+  rw [edgeVertices_reflection, edgeVertices_reflection]
+  dsimp only
+  rw [reflectedAngle_coe, reflectedAngle_coe]
+  congr 1 <;> ring_nf
+
+private theorem capReflection_normal_zero (ω : ℝ) :
+    capReflection ω (normalVector 0) = tangentVector (ω : Real.Angle) := by
+  simpa using capReflection_normal_at_complement ω ω
+
+private theorem capReflection_tangent_omega (ω : ℝ) :
+    capReflection ω (tangentVector (ω : Real.Angle)) = normalVector 0 := by
+  simpa using capReflection_tangent_at_complement ω 0
+
+private theorem wedgeEndpoints_reflection {ω : ℝ} (K : CapSpace ω) (t : ℝ) :
+    let P : CapSpace ω := ⟨reflectedBody ω K.val, reflectedBody_isCap K⟩
+    wedgeEndpoints P t =
+      (capReflection ω (wedgeEndpoints K (ω - t)).2,
+        capReflection ω (wedgeEndpoints K (ω - t)).1) := by
+  dsimp only
+  unfold wedgeEndpoints
+  rw [supportValue_reflected_at, supportValue_reflected_at_add_pi_div_two]
+  apply Prod.ext <;> dsimp only
+  · rw [map_smul, capReflection_tangent_omega]
+    congr 2; ring_nf
+  · rw [map_smul, capReflection_normal_zero]
+
+private theorem wedgeGaps_reflection {ω : ℝ} (K : CapSpace ω) (t : ℝ) :
+    let P : CapSpace ω := ⟨reflectedBody ω K.val, reflectedBody_isCap K⟩
+    wedgeGaps P t = ((wedgeGaps K (ω - t)).2, (wedgeGaps K (ω - t)).1) := by
+  dsimp only
+  unfold wedgeGaps
+  rw [capVertices_reflection K 0, capVertices_reflection K ω,
+    wedgeEndpoints_reflection K t]
+  dsimp only
+  apply Prod.ext <;> dsimp only
+  · rw [← map_sub, ← capReflection_tangent_omega,
+      (capReflection ω).inner_map_map]
+    simp
+  · rw [← map_sub, ← capReflection_normal_zero,
+      (capReflection ω).inner_map_map]
+    simp
+
+private theorem capWedge_reflection {ω : ℝ} (K : CapSpace ω) (t : ℝ) :
+    let P : CapSpace ω := ⟨reflectedBody ω K.val, reflectedBody_isCap K⟩
+    capWedge P t = capReflection ω '' capWedge K (ω - t) := by
+  dsimp only
+  unfold capWedge
+  rw [Set.image_inter (capReflection ω).injective,
+    capReflection_image_capFan,
+    (rotatingHallwayParts_reflection ω t K.val).2.2.2.1]
+
+private theorem iUnion_innerQuadrant_reflection {ω : ℝ} (K : CapSpace ω) :
+    (⋃ t ∈ Set.Ioo 0 ω, innerQuadrant (reflectedBody ω K.val) t) =
+      capReflection ω '' (⋃ s ∈ Set.Ioo 0 ω, innerQuadrant K.val s) := by
+  ext p
+  constructor
+  · intro hp
+    obtain ⟨t, ht, hp⟩ := Set.mem_iUnion₂.mp hp
+    rcases ht with ⟨ht0, htω⟩
+    rw [innerQuadrant_reflection] at hp
+    obtain ⟨q, hq, rfl⟩ := hp
+    refine ⟨q, Set.mem_iUnion₂.mpr ⟨ω - t, ⟨by linarith, by linarith⟩, hq⟩, rfl⟩
+  · rintro ⟨q, hq, rfl⟩
+    obtain ⟨s, hs, hq⟩ := Set.mem_iUnion₂.mp hq
+    rcases hs with ⟨hs0, hsω⟩
+    apply Set.mem_iUnion₂.mpr
+    refine ⟨ω - s, ⟨by linarith, by linarith⟩, ?_⟩
+    rw [innerQuadrant_reflection]
+    exact ⟨q, by simpa, rfl⟩
+
+private theorem capNiche_reflection {ω : ℝ} (K : CapSpace ω) :
+    let P : CapSpace ω := ⟨reflectedBody ω K.val, reflectedBody_isCap K⟩
+    capNiche P = capReflection ω '' capNiche K := by
+  dsimp only
+  unfold capNiche
+  rw [Set.image_inter (capReflection ω).injective,
+    capReflection_image_capFan, ← iUnion_innerQuadrant_reflection]
+
+private theorem iUnion_exposedEdge_reflection {ω : ℝ} (K : CapSpace ω) :
+    (⋃ t ∈ Set.Icc 0 (ω + Real.pi / 2),
+        exposedEdge (reflectedBody ω K.val) (t : Real.Angle)) =
+      capReflection ω ''
+        (⋃ s ∈ Set.Icc 0 (ω + Real.pi / 2),
+          exposedEdge K.val (s : Real.Angle)) := by
+  ext p
+  constructor
+  · intro hp
+    obtain ⟨t, ht, hp⟩ := Set.mem_iUnion₂.mp hp
+    rcases ht with ⟨ht0, htω⟩
+    rw [exposedEdge_reflection] at hp
+    obtain ⟨q, hq, rfl⟩ := hp
+    refine ⟨q, Set.mem_iUnion₂.mpr ⟨ω + Real.pi / 2 - t,
+      ⟨by linarith, by linarith⟩, ?_⟩, rfl⟩
+    simpa only [reflectedAngle_coe] using hq
+  · rintro ⟨q, hq, rfl⟩
+    obtain ⟨s, hs, hq⟩ := Set.mem_iUnion₂.mp hq
+    rcases hs with ⟨hs0, hsω⟩
+    apply Set.mem_iUnion₂.mpr
+    refine ⟨ω + Real.pi / 2 - s, ⟨by linarith, by linarith⟩, ?_⟩
+    rw [exposedEdge_reflection]
+    refine ⟨q, ?_, rfl⟩
+    rw [reflectedAngle_coe]
+    convert hq using 1; ring_nf
+
+private theorem capUpperBoundary_reflection {ω : ℝ} (K : CapSpace ω) :
+    let P : CapSpace ω := ⟨reflectedBody ω K.val, reflectedBody_isCap K⟩
+    capUpperBoundary P = capReflection ω '' capUpperBoundary K := by
+  dsimp only
+  unfold capUpperBoundary
+  exact iUnion_exposedEdge_reflection K
+
+private def reflectedAngleEquiv (ω : ℝ) : Real.Angle ≃ᵐ Real.Angle :=
+  MeasurableEquiv.ofInvolutive (reflectedAngle ω)
+    (reflectedAngle_involutive ω) (by unfold reflectedAngle; fun_prop)
+
+private theorem reflectedAngle_add_pi (ω : ℝ) (a : Real.Angle) :
+    reflectedAngle ω (a + (Real.pi : Real.Angle)) =
+      reflectedAngle ω a + (Real.pi : Real.Angle) := by
+  have hpi : -(Real.pi : Real.Angle) = (Real.pi : Real.Angle) :=
+    neg_eq_iff_add_eq_zero.mpr Real.Angle.coe_pi_add_coe_pi
+  unfold reflectedAngle
+  rw [sub_eq_add_neg, neg_add_rev, hpi]
+  abel
+
+private theorem isSegmentPresentation_reflectedBody (ω : ℝ)
+    (K : ConvexBody Point) (d : Point × Point × Real.Angle)
+    (hd : IsSegmentPresentation K d) :
+    IsSegmentPresentation (reflectedBody ω K)
+      (capReflection ω d.1, capReflection ω d.2.1, reflectedAngle ω d.2.2) := by
+  refine ⟨fun h ↦ hd.1 ((capReflection ω).injective h), ?_, ?_⟩
+  · change capReflection ω '' (K : Set Point) =
+      segment ℝ (capReflection ω d.1) (capReflection ω d.2.1)
+    rw [hd.2.1]
+    exact image_segment ℝ
+      (capReflection ω).toLinearEquiv.toLinearMap.toAffineMap d.1 d.2.1
+  · rw [← map_sub, ← capReflection_normalVector_angle,
+      (capReflection ω).inner_map_map]
+    exact hd.2.2
+
+private theorem reflectedBody_reflectedBody (ω : ℝ) (K : ConvexBody Point) :
+    reflectedBody ω (reflectedBody ω K) = K := by
+  apply ConvexBody.ext
+  ext p
+  constructor
+  · rintro ⟨q, ⟨r, hr, rfl⟩, rfl⟩
+    simpa only [capReflection_involutive] using hr
+  · intro hp
+    exact ⟨capReflection ω p, ⟨p, hp, rfl⟩, capReflection_involutive ω p⟩
+
+private theorem subsingleton_reflectedBody_iff (ω : ℝ) (K : ConvexBody Point) :
+    ((reflectedBody ω K : ConvexBody Point) : Set Point).Subsingleton ↔
+      (K : Set Point).Subsingleton := by
+  constructor
+  · intro hP p hp q hq
+    apply (capReflection ω).injective
+    exact hP ⟨p, hp, rfl⟩ ⟨q, hq, rfl⟩
+  · intro hK _
+    rintro ⟨p, hp, rfl⟩ _ ⟨q, hq, rfl⟩
+    rw [hK hp hq]
+
+private theorem exists_segmentPresentation_reflectedBody_iff (ω : ℝ)
+    (K : ConvexBody Point) :
+    (∃ d, IsSegmentPresentation (reflectedBody ω K) d) ↔
+      ∃ d, IsSegmentPresentation K d := by
+  constructor
+  · rintro ⟨d, hd⟩
+    have h := isSegmentPresentation_reflectedBody ω (reflectedBody ω K) d hd
+    rw [reflectedBody_reflectedBody] at h
+    exact ⟨_, h⟩
+  · rintro ⟨d, hd⟩
+    exact ⟨_, isSegmentPresentation_reflectedBody ω K d hd⟩
+
+private theorem surfaceAreaMeasure_reflectedBody_of_subsingleton (ω : ℝ)
+    (K : ConvexBody Point) (hK : (K : Set Point).Subsingleton) :
+    surfaceAreaMeasure (reflectedBody ω K) =
+      Measure.map (reflectedAngleEquiv ω) (surfaceAreaMeasure K) := by
+  have hP : ((reflectedBody ω K : ConvexBody Point) : Set Point).Subsingleton := by
+    rintro _ ⟨p, hp, rfl⟩ _ ⟨q, hq, rfl⟩
+    rw [hK hp hq]
+  rw [surfaceAreaMeasure_eq_zero_of_subsingleton (reflectedBody ω K) hP,
+    surfaceAreaMeasure_eq_zero_of_subsingleton K hK, Measure.map_zero]
+
+private theorem surfaceAreaMeasure_reflectedBody_of_segmentPresentation (ω : ℝ)
+    (K : ConvexBody Point) (d : Point × Point × Real.Angle)
+    (hd : IsSegmentPresentation K d) :
+    surfaceAreaMeasure (reflectedBody ω K) =
+      Measure.map (reflectedAngleEquiv ω) (surfaceAreaMeasure K) := by
+  rw [surfaceAreaMeasure_eq_segmentPresentation (reflectedBody ω K)
+      (capReflection ω d.1, capReflection ω d.2.1, reflectedAngle ω d.2.2)
+      (isSegmentPresentation_reflectedBody ω K d hd),
+    surfaceAreaMeasure_eq_segmentPresentation K d hd,
+    MeasureTheory.Measure.map_smul _ (reflectedAngleEquiv ω).measurable.aemeasurable,
+    MeasureTheory.Measure.map_add _ _ (reflectedAngleEquiv ω).measurable,
+    MeasureTheory.Measure.map_dirac, MeasureTheory.Measure.map_dirac,
+    (capReflection ω).dist_map]
+  change ENNReal.ofReal (dist d.1 d.2.1) •
+      (Measure.dirac (reflectedAngle ω d.2.2) +
+        Measure.dirac (reflectedAngle ω d.2.2 + (Real.pi : Real.Angle))) =
+    ENNReal.ofReal (dist d.1 d.2.1) •
+      (Measure.dirac (reflectedAngle ω d.2.2) +
+        Measure.dirac (reflectedAngle ω (d.2.2 + (Real.pi : Real.Angle))))
+  rw [reflectedAngle_add_pi]
+
+private theorem isExteriorNormal_reflectedBody_iff (ω : ℝ)
+    (K : ConvexBody Point) (p : Point) (a : Real.Angle) :
+    IsExteriorNormal (reflectedBody ω K) (capReflection ω p) (reflectedAngle ω a) ↔
+      IsExteriorNormal K p a := by
+  constructor
+  · intro h q hq
+    have h' := h (capReflection ω q) ⟨q, hq, rfl⟩
+    rw [← map_sub, ← capReflection_normalVector_angle,
+      (capReflection ω).inner_map_map] at h'
+    exact h'
+  · intro h _
+    rintro ⟨q, hq, rfl⟩
+    rw [← map_sub, ← capReflection_normalVector_angle,
+      (capReflection ω).inner_map_map]
+    exact h q hq
+
+private theorem mem_regularBoundary_reflectedBody_iff (ω : ℝ)
+    (K : ConvexBody Point) (p : Point) :
+    capReflection ω p ∈ regularBoundary (reflectedBody ω K) ↔
+      p ∈ regularBoundary K := by
+  have hfront : capReflection ω p ∈ frontier ((reflectedBody ω K : ConvexBody Point) : Set Point) ↔
+      p ∈ frontier (K : Set Point) := by
+    change capReflection ω p ∈ frontier (capReflection ω '' (K : Set Point)) ↔ _
+    have himage : capReflection ω '' frontier (K : Set Point) =
+        frontier (capReflection ω '' (K : Set Point)) :=
+      (capReflection ω).toHomeomorph.image_frontier (K : Set Point)
+    rw [← himage]
+    constructor
+    · rintro ⟨q, hq, hpq⟩
+      have hqp : q = p := (capReflection ω).injective hpq
+      simpa only [hqp] using hq
+    · exact fun hp ↦ ⟨p, hp, rfl⟩
+  have hunique :
+      (∃! b, IsExteriorNormal (reflectedBody ω K) (capReflection ω p) b) ↔
+        ∃! a, IsExteriorNormal K p a := by
+    constructor
+    · rintro ⟨b, hb, hub⟩
+      have hbK : IsExteriorNormal K p (reflectedAngle ω b) := by
+        apply (isExteriorNormal_reflectedBody_iff ω K p (reflectedAngle ω b)).mp
+        simpa only [reflectedAngle_involutive] using hb
+      refine ⟨reflectedAngle ω b, hbK, ?_⟩
+      intro c hc
+      have hcL := (isExteriorNormal_reflectedBody_iff ω K p c).mpr hc
+      have hcb : reflectedAngle ω c = b := hub _ hcL
+      calc
+        c = reflectedAngle ω (reflectedAngle ω c) :=
+          (reflectedAngle_involutive ω c).symm
+        _ = reflectedAngle ω b := congrArg (reflectedAngle ω) hcb
+    · rintro ⟨a, ha, hua⟩
+      refine ⟨reflectedAngle ω a,
+        (isExteriorNormal_reflectedBody_iff ω K p a).mpr ha, ?_⟩
+      intro b hb
+      have hbK : IsExteriorNormal K p (reflectedAngle ω b) := by
+        apply (isExteriorNormal_reflectedBody_iff ω K p (reflectedAngle ω b)).mp
+        simpa only [reflectedAngle_involutive] using hb
+      have hba : reflectedAngle ω b = a := hua _ hbK
+      calc
+        b = reflectedAngle ω (reflectedAngle ω b) :=
+          (reflectedAngle_involutive ω b).symm
+        _ = reflectedAngle ω a := congrArg (reflectedAngle ω) hba
+  simp only [regularBoundary, Set.mem_ofPred_eq]
+  exact and_congr hfront hunique
+
+private theorem exteriorNormalAngle_reflectedBody (ω : ℝ)
+    (K : ConvexBody Point) (p : Point) (hp : p ∈ regularBoundary K) :
+    exteriorNormalAngle (reflectedBody ω K) (capReflection ω p) =
+      reflectedAngle ω (exteriorNormalAngle K p) := by
+  have hP := (mem_regularBoundary_reflectedBody_iff ω K p).mpr hp
+  have huK : ∃! a, IsExteriorNormal K p a := hp.2
+  have huP : ∃! a,
+      IsExteriorNormal (reflectedBody ω K) (capReflection ω p) a := hP.2
+  have hnK : IsExteriorNormal K p (exteriorNormalAngle K p) := by
+    unfold exteriorNormalAngle
+    rw [dite_eq_left huK]
+    generalize_proofs h
+    exact h.choose_spec
+  have hnP : IsExteriorNormal (reflectedBody ω K) (capReflection ω p)
+      (exteriorNormalAngle (reflectedBody ω K) (capReflection ω p)) := by
+    unfold exteriorNormalAngle
+    rw [dite_eq_left huP]
+    generalize_proofs _ h
+    exact h.choose_spec
+  exact huP.unique hnP ((isExteriorNormal_reflectedBody_iff ω K p _).mpr hnK)
+
+private theorem regularBoundary_reflectedBody (ω : ℝ) (K : ConvexBody Point) :
+    regularBoundary (reflectedBody ω K) =
+      capReflection ω '' regularBoundary K := by
+  ext q
+  constructor
+  · intro hq
+    refine ⟨capReflection ω q, ?_, capReflection_involutive ω q⟩
+    apply (mem_regularBoundary_reflectedBody_iff ω K (capReflection ω q)).mp
+    simpa only [capReflection_involutive] using hq
+  · rintro ⟨p, hp, rfl⟩
+    exact (mem_regularBoundary_reflectedBody_iff ω K p).mpr hp
+
+private theorem map_hausdorffMeasure_restrict_regularBoundary (ω : ℝ)
+    (K : ConvexBody Point) :
+    Measure.map (capReflection ω)
+        ((Measure.hausdorffMeasure 1).restrict (regularBoundary K)) =
+      (Measure.hausdorffMeasure 1).restrict
+        (regularBoundary (reflectedBody ω K)) := by
+  rw [regularBoundary_reflectedBody]
+  let e := (capReflection ω).toHomeomorph.toMeasurableEquiv
+  have hrestrict := e.restrict_map (Measure.hausdorffMeasure 1)
+    (capReflection ω '' regularBoundary K)
+  change (Measure.map (capReflection ω) (Measure.hausdorffMeasure 1)).restrict
+      (capReflection ω '' regularBoundary K) =
+    Measure.map (capReflection ω)
+      ((Measure.hausdorffMeasure 1).restrict
+        (capReflection ω ⁻¹' (capReflection ω '' regularBoundary K))) at hrestrict
+  have hmap : Measure.map (capReflection ω) (Measure.hausdorffMeasure 1) =
+      Measure.hausdorffMeasure 1 :=
+    (capReflection ω).toIsometryEquiv.map_hausdorffMeasure 1
+  rw [hmap] at hrestrict
+  have hpre : capReflection ω ⁻¹' (capReflection ω '' regularBoundary K) =
+      regularBoundary K := by
+    exact Set.preimage_image_eq _ (capReflection ω).injective
+  rw [hpre] at hrestrict
+  exact hrestrict.symm
+
+private theorem reflectedAngle_preimage_eq_image (ω : ℝ)
+    (E : Set Real.Angle) :
+    reflectedAngle ω ⁻¹' E = reflectedAngle ω '' E := by
+  ext a
+  constructor
+  · intro ha
+    exact ⟨reflectedAngle ω a, ha, reflectedAngle_involutive ω a⟩
+  · rintro ⟨b, hb, rfl⟩
+    simpa only [Set.mem_preimage, reflectedAngle_involutive] using hb
+
+private theorem surfaceAreaMeasure_reflectedBody (ω : ℝ)
+    (K : ConvexBody Point) :
+    surfaceAreaMeasure (reflectedBody ω K) =
+      Measure.map (reflectedAngleEquiv ω) (surfaceAreaMeasure K) := by
+  by_cases hsub : (K : Set Point).Subsingleton
+  · exact surfaceAreaMeasure_reflectedBody_of_subsingleton ω K hsub
+  by_cases hseg : ∃ d, IsSegmentPresentation K d
+  · obtain ⟨d, hd⟩ := hseg
+    exact surfaceAreaMeasure_reflectedBody_of_segmentPresentation ω K d hd
+  have hsubP : ¬ ((reflectedBody ω K : ConvexBody Point) : Set Point).Subsingleton :=
+    fun h ↦ hsub ((subsingleton_reflectedBody_iff ω K).mp h)
+  have hsegP : ¬ ∃ d, IsSegmentPresentation (reflectedBody ω K) d :=
+    fun h ↦ hseg ((exists_segmentPresentation_reflectedBody_iff ω K).mp h)
+  let μK := (Measure.hausdorffMeasure 1).restrict (regularBoundary K)
+  let μP := (Measure.hausdorffMeasure 1).restrict
+    (regularBoundary (reflectedBody ω K))
+  have hμ : Measure.map (capReflection ω) μK = μP :=
+    map_hausdorffMeasure_restrict_regularBoundary ω K
+  have haeK : AEMeasurable (exteriorNormalAngle K) μK :=
+    aemeasurable_exteriorNormalAngle_restrict_regularBoundary K _
+  have haeP : AEMeasurable (exteriorNormalAngle (reflectedBody ω K)) μP :=
+    aemeasurable_exteriorNormalAngle_restrict_regularBoundary (reflectedBody ω K) _
+  simp only [surfaceAreaMeasure, hsub, hseg, hsubP, hsegP, ↓reduceIte]
+  change Measure.map (exteriorNormalAngle (reflectedBody ω K)) μP =
+    Measure.map (reflectedAngleEquiv ω)
+      (Measure.map (exteriorNormalAngle K) μK)
+  rw [← hμ]
+  have haeP' : AEMeasurable (exteriorNormalAngle (reflectedBody ω K))
+      (Measure.map (capReflection ω) μK) := by
+    rwa [hμ]
+  rw [AEMeasurable.map_map_of_aemeasurable haeP'
+      (capReflection ω).continuous.measurable.aemeasurable,
+    AEMeasurable.map_map_of_aemeasurable
+      (reflectedAngleEquiv ω).measurable.aemeasurable haeK]
+  apply Measure.map_congr
+  filter_upwards [ae_restrict_mem (measurableSet_regularBoundary K)] with p hp
+  exact exteriorNormalAngle_reflectedBody ω K p hp
+
+theorem cap_mirror_features {ω : ℝ} (K : CapSpace ω) :
+    ∃ P : CapSpace ω,
+      (P.val : Set Point) = mirrorReflection ω '' (K.val : Set Point) ∧
+      (∀ t ∈ Set.Icc 0 ω,
+        supportingHallway (P.val : Set Point) (t : Real.Angle) =
+        mirrorReflection ω '' supportingHallway (K.val : Set Point) ((ω - t : ℝ) : Real.Angle) ∧
+        (rotatingHallwayParts (P.val : Set Point) (t : Real.Angle)).innerCorner = mirrorReflection
+          ω (rotatingHallwayParts (K.val : Set Point) ((ω - t : ℝ) : Real.Angle)).innerCorner ∧
+        (rotatingHallwayParts (P.val : Set Point) (t : Real.Angle)).outerCorner = mirrorReflection
+          ω (rotatingHallwayParts (K.val : Set Point) ((ω - t : ℝ) : Real.Angle)).outerCorner ∧
+        (rotatingHallwayParts (P.val : Set Point) (t : Real.Angle)).outerQuadrant =
+          mirrorReflection ω '' (rotatingHallwayParts (K.val : Set Point) ((ω - t : ℝ) :
+          Real.Angle)).outerQuadrant ∧
+        (rotatingHallwayParts (P.val : Set Point) (t : Real.Angle)).innerQuadrant =
+          mirrorReflection ω '' (rotatingHallwayParts (K.val : Set Point) ((ω - t : ℝ) :
+          Real.Angle)).innerQuadrant ∧
+        (rotatingHallwayParts (P.val : Set Point) (t : Real.Angle)).a = mirrorReflection ω ''
+          (rotatingHallwayParts (K.val : Set Point) ((ω - t : ℝ) : Real.Angle)).c ∧
+        (rotatingHallwayParts (P.val : Set Point) (t : Real.Angle)).c = mirrorReflection ω ''
+          (rotatingHallwayParts (K.val : Set Point) ((ω - t : ℝ) : Real.Angle)).a ∧
+        (rotatingHallwayParts (P.val : Set Point) (t : Real.Angle)).b = mirrorReflection ω ''
+          (rotatingHallwayParts (K.val : Set Point) ((ω - t : ℝ) : Real.Angle)).d ∧
+        (rotatingHallwayParts (P.val : Set Point) (t : Real.Angle)).d = mirrorReflection ω ''
+          (rotatingHallwayParts (K.val : Set Point) ((ω - t : ℝ) : Real.Angle)).b ∧
+        (rotatingHallwayParts (P.val : Set Point) (t : Real.Angle)).bRay = mirrorReflection ω ''
+          (rotatingHallwayParts (K.val : Set Point) ((ω - t : ℝ) : Real.Angle)).dRay ∧
+        (rotatingHallwayParts (P.val : Set Point) (t : Real.Angle)).dRay = mirrorReflection ω ''
+          (rotatingHallwayParts (K.val : Set Point) ((ω - t : ℝ) : Real.Angle)).bRay ∧
+        (capVertices P t).1.1 = mirrorReflection ω (capVertices K (ω - t)).2.2 ∧
+        (capVertices P t).1.2 = mirrorReflection ω (capVertices K (ω - t)).2.1 ∧
+        (capVertices P t).2.1 = mirrorReflection ω (capVertices K (ω - t)).1.2 ∧
+        (capVertices P t).2.2 = mirrorReflection ω (capVertices K (ω - t)).1.1) ∧
+      (∀ t ∈ Set.Ioo 0 ω,
+        (wedgeEndpoints P t).1 = mirrorReflection ω (wedgeEndpoints K (ω - t)).2 ∧
+        (wedgeEndpoints P t).2 = mirrorReflection ω (wedgeEndpoints K (ω - t)).1 ∧
+        (wedgeGaps P t).1 = (wedgeGaps K (ω - t)).2 ∧
+        (wedgeGaps P t).2 = (wedgeGaps K (ω - t)).1 ∧
+        capWedge P t = mirrorReflection ω '' capWedge K (ω - t)) ∧
+      capUpperBoundary P = mirrorReflection ω '' capUpperBoundary K ∧
+      capNiche P = mirrorReflection ω '' capNiche K ∧
+      (∀ E : Set Real.Angle, MeasurableSet E →
+        surfaceAreaMeasure P.val E = surfaceAreaMeasure K.val
+          ((fun a : Real.Angle ↦ ((ω + Real.pi / 2 : ℝ) : Real.Angle) - a) '' E)) := by
+  let P : CapSpace ω := ⟨reflectedBody ω K.val, reflectedBody_isCap K⟩
+  have hM : mirrorReflection ω = capReflection ω :=
+    mirrorReflection_eq_capReflection_m6ef9701 ω K.property.1 K.property.2.1
+  refine ⟨P, ?_, ?_, ?_, ?_, ?_, ?_⟩
+  · change capReflection ω '' (K.val : Set Point) =
+      mirrorReflection ω '' (K.val : Set Point)
+    rw [hM]
+  · intro t _
+    rw [hM]
+    have hh := supportingHallway_reflection ω t K.val
+    have hp := rotatingHallwayParts_reflection ω t K.val
+    have hv := capVertices_reflection K t
+    rcases hp with ⟨hinner, houter, hQouter, hQinner, ha, hc, hb, hd, hbRay, hdRay⟩
+    refine ⟨hh, hinner, houter, hQouter, hQinner, ha, hc, hb, hd, hbRay, hdRay,
+      ?_, ?_, ?_, ?_⟩
+    · simpa only [P] using congrArg (fun v ↦ v.1.1) hv
+    · simpa only [P] using congrArg (fun v ↦ v.1.2) hv
+    · simpa only [P] using congrArg (fun v ↦ v.2.1) hv
+    · simpa only [P] using congrArg (fun v ↦ v.2.2) hv
+  · intro t _
+    rw [hM]
+    have he := wedgeEndpoints_reflection K t
+    have hg := wedgeGaps_reflection K t
+    refine ⟨?_, ?_, ?_, ?_, capWedge_reflection K t⟩
+    · simpa only [P] using congrArg Prod.fst he
+    · simpa only [P] using congrArg Prod.snd he
+    · simpa only [P] using congrArg Prod.fst hg
+    · simpa only [P] using congrArg Prod.snd hg
+  · rw [hM]
+    exact capUpperBoundary_reflection K
+  · rw [hM]
+    exact capNiche_reflection K
+  · intro E hE
+    change surfaceAreaMeasure (reflectedBody ω K.val) E = _
+    rw [surfaceAreaMeasure_reflectedBody,
+      (reflectedAngleEquiv ω).map_apply]
+    change surfaceAreaMeasure K.val (reflectedAngle ω ⁻¹' E) = _
+    rw [reflectedAngle_preimage_eq_image]
+    rfl
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Bounds.Arm.Geometry`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Bounds / Arm / Geometry
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter
+open scoped Topology
+
+namespace MovingSofa
+
+private theorem mirrorReflection_rightAngle_apply (p : Point) :
+    mirrorReflection (Real.pi / 2) p = !₂[-p 0, p 1] := by
+  have harg : Real.pi / 4 - (Real.pi / 2) / 2 = 0 := by ring
+  have hnorm : ‖(!₂[0, 1] : Point)‖ ^ 2 = 1 := by
+    simpa [Fin.sum_univ_two] using EuclideanSpace.norm_sq_eq (!₂[0, 1] : Point)
+  ext i
+  fin_cases i <;>
+    simp [mirrorReflection, stripParallelogram, harg, PiLp.inner_apply,
+      Fin.sum_univ_two, hnorm]
+  ring
+
+private theorem inner_mirror_sub_tangent (p q : Point) (t : ℝ) :
+    inner ℝ (mirrorReflection (Real.pi / 2) p - mirrorReflection (Real.pi / 2) q)
+      (tangentVector (t : Real.Angle)) =
+    inner ℝ (p - q) (normalVector ((Real.pi / 2 - t : ℝ) : Real.Angle)) := by
+  rw [mirrorReflection_rightAngle_apply, mirrorReflection_rightAngle_apply]
+  simp [tangentVector, normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two,
+    Real.cos_pi_div_two_sub, Real.sin_pi_div_two_sub, -Real.Angle.coe_sub]
+  ring
+
+private theorem inner_mirror_sub_normal (p q : Point) (t : ℝ) :
+    inner ℝ (mirrorReflection (Real.pi / 2) p - mirrorReflection (Real.pi / 2) q)
+      (normalVector (t : Real.Angle)) =
+    inner ℝ (p - q) (tangentVector ((Real.pi / 2 - t : ℝ) : Real.Angle)) := by
+  rw [mirrorReflection_rightAngle_apply, mirrorReflection_rightAngle_apply]
+  simp [tangentVector, normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two,
+    Real.cos_pi_div_two_sub, Real.sin_pi_div_two_sub, -Real.Angle.coe_sub]
+  ring
+
+theorem tangentArms_mirror (K P : RightAngleCapSpace)
+    (hP : (P.val : Set Point) = mirrorReflection (Real.pi / 2) '' (K.val : Set Point))
+    (t : ℝ) (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    (tangentArmLengths P t).1.1 = (tangentArmLengths K (Real.pi / 2 - t)).2.2 ∧
+    (tangentArmLengths P t).1.2 = (tangentArmLengths K (Real.pi / 2 - t)).2.1 ∧
+    (tangentArmLengths P t).2.1 = (tangentArmLengths K (Real.pi / 2 - t)).1.2 ∧
+    (tangentArmLengths P t).2.2 = (tangentArmLengths K (Real.pi / 2 - t)).1.1 := by
+  obtain ⟨Q, hQ, hfeatures, _⟩ := cap_mirror_features K
+  have hPQ : P = Q := by
+    apply Subtype.ext
+    apply ConvexBody.ext
+    exact hP.trans hQ.symm
+  subst P
+  obtain ⟨_, _, hy, _, _, _, _, _, _, _, _, hA₁, hA₂, hC₁, hC₂⟩ := hfeatures t ht
+  dsimp only [tangentArmLengths]
+  rw [hy, hA₁, hA₂, hC₁, hC₂]
+  exact ⟨inner_mirror_sub_tangent _ _ _, inner_mirror_sub_tangent _ _ _,
+    inner_mirror_sub_normal _ _ _, inner_mirror_sub_normal _ _ _⟩
+
+private theorem tendsto_tan_sub_div (t : ℝ) :
+    Tendsto (fun s : ℝ ↦ Real.tan (s - t) / (s - t)) (𝓝[≠] t) (𝓝 1) := by
+  have hd : HasDerivAt (fun s : ℝ ↦ Real.tan (s - t)) 1 t := by
+    simpa [Function.comp_def] using
+      (Real.hasDerivAt_tan (x := t - t) (by simp)).comp (h := fun s : ℝ ↦ s - t) t
+        ((hasDerivAt_id t).sub_const t)
+  convert hd.tendsto_slope using 1
+  funext s
+  simp [slope_def_field]
+
+private theorem tendsto_normal_slope {F : Filter ℝ} {t : ℝ}
+    (hF : F ≤ 𝓝[≠] t) (a : ℝ) (y q : ℝ → Point) (p : Point)
+    (hy : Tendsto y F (𝓝 (y t))) (hq : Tendsto q F (𝓝 p))
+    (hfixed : ∀ᶠ s in F, inner ℝ (q s) (normalVector ((t + a : ℝ) : Real.Angle)) =
+      inner ℝ (y t) (normalVector ((t + a : ℝ) : Real.Angle)))
+    (hmoving : ∀ᶠ s in F, inner ℝ (q s) (normalVector ((s + a : ℝ) : Real.Angle)) =
+      inner ℝ (y s) (normalVector ((s + a : ℝ) : Real.Angle))) :
+    Tendsto (fun s : ℝ ↦ inner ℝ ((s - t)⁻¹ • (y s - y t))
+      (normalVector ((t + a : ℝ) : Real.Angle))) F
+      (𝓝 (inner ℝ (p - y t) (tangentVector ((t + a : ℝ) : Real.Angle)))) := by
+  have htan := (tendsto_tan_sub_div t).mono_left hF
+  have hpair := htan.mul ((hq.sub hy).inner (𝕜 := ℝ)
+    (tendsto_const_nhds (x := tangentVector ((t + a : ℝ) : Real.Angle))))
+  simp only [one_mul] at hpair
+  apply hpair.congr'
+  have hnear : ∀ᶠ s in F, s - t ∈ Set.Ioo (-(Real.pi / 2)) (Real.pi / 2) := by
+    have hc : Tendsto (fun s : ℝ ↦ s - t) F (𝓝 0) := by
+      simpa using (tendsto_id.mono_right (hF.trans nhdsWithin_le_nhds)).sub_const t
+    exact hc.eventually (Ioo_mem_nhds (by linarith [Real.pi_pos]) (by linarith [Real.pi_pos]))
+  filter_upwards [hfixed, hmoving, hnear] with s hf hm hs
+  have hn : normalVector ((s + a : ℝ) : Real.Angle) =
+      Real.cos (s - t) • normalVector ((t + a : ℝ) : Real.Angle) +
+      Real.sin (s - t) • tangentVector ((t + a : ℝ) : Real.Angle) := by
+    convert normalVector_add_real (t + a) (s - t) using 1
+    congr 2
+    ring
+  rw [hn] at hm
+  simp only [inner_add_right, inner_smul_right] at hm
+  rw [hf] at hm
+  have hc : Real.cos (s - t) ≠ 0 := (Real.cos_pos_of_mem_Ioo hs).ne'
+  simp only [real_inner_smul_left, inner_sub_left, Real.tan_eq_sin_div_cos]
+  have heq : Real.sin (s - t) *
+      (inner ℝ (q s) (tangentVector ((t + a : ℝ) : Real.Angle)) -
+        inner ℝ (y s) (tangentVector ((t + a : ℝ) : Real.Angle))) =
+      Real.cos (s - t) * (inner ℝ (y s) (normalVector ((t + a : ℝ) : Real.Angle)) -
+        inner ℝ (y t) (normalVector ((t + a : ℝ) : Real.Angle))) := by linarith
+  field_simp
+  rw [heq]
+
+private theorem outerCorner_inner_normal (K : ConvexBody Point) (t : ℝ) :
+    inner ℝ (rotatingHallwayParts (K : Set Point) (t : Real.Angle)).outerCorner
+      (normalVector (t : Real.Angle)) = supportValue K (t : Real.Angle) := by
+  change inner ℝ (supportingPlacement (K : Set Point) (t : Real.Angle) (!₂[1, 1] : Point))
+    (normalVector (t : Real.Angle)) = _
+  rw [inner_supportingPlacement_normalVector]
+  simp
+
+private theorem outerCorner_inner_tangent (K : ConvexBody Point) (t : ℝ) :
+    inner ℝ (rotatingHallwayParts (K : Set Point) (t : Real.Angle)).outerCorner
+      (tangentVector (t : Real.Angle)) = supportValue K ((t + Real.pi / 2 : ℝ) : Real.Angle) := by
+  change inner ℝ (supportingPlacement (K : Set Point) (t : Real.Angle) (!₂[1, 1] : Point))
+    (tangentVector (t : Real.Angle)) = _
+  rw [inner_supportingPlacement_tangentVector]
+  simp
+
+private theorem eventually_sin_sub_ne_zero {F : Filter ℝ} {t : ℝ}
+    (hF : F ≤ 𝓝[≠] t) : ∀ᶠ s in F, Real.sin (s - t) ≠ 0 := by
+  have hn : Tendsto (fun s : ℝ ↦ s - t) F (𝓝 0) := by
+    simpa using (tendsto_id.mono_right (hF.trans nhdsWithin_le_nhds)).sub_const t
+  have hnear := hn.eventually (Ioo_mem_nhds (neg_neg_of_pos Real.pi_pos) Real.pi_pos)
+  have hne : ∀ᶠ s in F, s ≠ t := hF self_mem_nhdsWithin
+  filter_upwards [hnear, hne] with s hs hst
+  exact mt (Real.sin_eq_zero_iff_of_lt_of_lt hs.1 hs.2).mp (sub_ne_zero.mpr hst)
+
+private theorem tendsto_outerCorner_slope {F : Filter ℝ} {t : ℝ}
+    (K : ConvexBody Point) (hF : F ≤ 𝓝[≠] t) (p q : Point)
+    (hp : Tendsto (fun s : ℝ ↦ supportingIntersection K (t : Real.Angle) (s : Real.Angle))
+      F (𝓝 p))
+    (hq : Tendsto (fun s : ℝ ↦ supportingIntersection K
+      ((t + Real.pi / 2 : ℝ) : Real.Angle) ((s + Real.pi / 2 : ℝ) : Real.Angle)) F (𝓝 q)) :
+    Tendsto (fun s : ℝ ↦ (s - t)⁻¹ •
+      ((rotatingHallwayParts (K : Set Point) (s : Real.Angle)).outerCorner -
+        (rotatingHallwayParts (K : Set Point) (t : Real.Angle)).outerCorner)) F
+      (𝓝 (inner ℝ (p - (rotatingHallwayParts (K : Set Point) (t : Real.Angle)).outerCorner)
+          (tangentVector (t : Real.Angle)) • normalVector (t : Real.Angle) +
+        inner ℝ (q - (rotatingHallwayParts (K : Set Point) (t : Real.Angle)).outerCorner)
+          (-normalVector (t : Real.Angle)) • tangentVector (t : Real.Angle))) := by
+  let y (s : ℝ) := (rotatingHallwayParts (K : Set Point) (s : Real.Angle)).outerCorner
+  have hy : Tendsto y F (𝓝 (y t)) :=
+    (continuous_outerCorner K).continuousAt.tendsto.mono_left (hF.trans nhdsWithin_le_nhds)
+  have hA := tendsto_normal_slope hF 0 y
+    (fun s ↦ supportingIntersection K (t : Real.Angle) (s : Real.Angle)) p hy hp
+    (Eventually.of_forall fun s ↦ by
+      simp only [add_zero, supportingIntersection_inner_left, y, outerCorner_inner_normal])
+    (by
+      filter_upwards [eventually_sin_sub_ne_zero hF] with s hs
+      simp only [add_zero, y, outerCorner_inner_normal]
+      exact supportingIntersection_inner_right K t s hs)
+  simp only [add_zero] at hA
+  have hC := tendsto_normal_slope hF (Real.pi / 2) y
+    (fun s ↦ supportingIntersection K ((t + Real.pi / 2 : ℝ) : Real.Angle)
+      ((s + Real.pi / 2 : ℝ) : Real.Angle)) q hy hq
+    (Eventually.of_forall fun s ↦ by
+      rw [supportingIntersection_inner_left]
+      simp only [Real.Angle.coe_add, normalVector_add_pi_div_two, y, outerCorner_inner_tangent])
+    (by
+      filter_upwards [eventually_sin_sub_ne_zero hF] with s hs
+      rw [supportingIntersection_inner_right K (t + Real.pi / 2) (s + Real.pi / 2)
+        (by simpa only [add_sub_add_right_eq_sub] using hs)]
+      simp only [Real.Angle.coe_add, normalVector_add_pi_div_two, y, outerCorner_inner_tangent])
+  simp only [Real.Angle.coe_add, normalVector_add_pi_div_two] at hC
+  have hCt : tangentVector ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle)) =
+      -normalVector (t : Real.Angle) := tangentVector_add_pi_div_two t
+  rw [hCt] at hC
+  have hsum := (hA.smul (tendsto_const_nhds (x := normalVector (t : Real.Angle)))).add
+    (hC.smul (tendsto_const_nhds (x := tangentVector (t : Real.Angle))))
+  simpa only [inner_normalVector_smul_add_inner_tangentVector_smul] using hsum
+
+private theorem hasDerivWithinAt_outerCorner_right (K : RightAngleCapSpace) (t : ℝ) :
+    HasDerivWithinAt
+      (fun s : ℝ ↦ (rotatingHallwayParts (K.val : Set Point) (s : Real.Angle)).outerCorner)
+      (-(tangentArmLengths K t).1.1 • normalVector (t : Real.Angle) +
+        (tangentArmLengths K t).2.1 • tangentVector (t : Real.Angle)) (Set.Ici t) t := by
+  have hshift : Tendsto (fun s : ℝ ↦ s + Real.pi / 2) (𝓝[>] t) (𝓝[>] (t + Real.pi / 2)) := by
+    apply tendsto_nhdsWithin_iff.mpr
+    refine ⟨((continuous_id.add_const _).continuousAt.tendsto.mono_left nhdsWithin_le_nhds), ?_⟩
+    filter_upwards [self_mem_nhdsWithin] with s hs
+    change t + Real.pi / 2 < s + Real.pi / 2
+    linarith [show t < s from hs]
+  have h := tendsto_outerCorner_slope K.val (nhdsGT_le_nhdsNE t)
+    (edgeVertices K.val (t : Real.Angle)).1
+    (edgeVertices K.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).1
+    (contact_oneSided_limits K.val t).2.2.1
+    (((contact_oneSided_limits K.val (t + Real.pi / 2)).2.2.1).comp hshift)
+  rw [hasDerivWithinAt_iff_tendsto_slope, Set.Ici_sdiff_left]
+  have heq :
+      inner ℝ ((edgeVertices K.val (t : Real.Angle)).1 -
+        (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).outerCorner)
+          (tangentVector (t : Real.Angle)) • normalVector (t : Real.Angle) +
+        inner ℝ ((edgeVertices K.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).1 -
+          (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).outerCorner)
+          (-normalVector (t : Real.Angle)) • tangentVector (t : Real.Angle) =
+      -(tangentArmLengths K t).1.1 • normalVector (t : Real.Angle) +
+        (tangentArmLengths K t).2.1 • tangentVector (t : Real.Angle) := by
+    simp only [tangentArmLengths, capVertices, inner_sub_left, inner_neg_right]
+    module
+  rw [heq] at h
+  convert h using 1
+  funext s
+  exact slope_def_module _ _ _
+
+private theorem tendsto_supportingIntersection_left_ordered (K : ConvexBody Point) (t : ℝ) :
+    Tendsto (fun s : ℝ ↦ supportingIntersection K (t : Real.Angle) (s : Real.Angle))
+      (𝓝[<] t) (𝓝 (edgeVertices K (t : Real.Angle)).2) := by
+  apply (contact_oneSided_limits K t).2.2.2.2.2.congr'
+  filter_upwards [eventually_sin_sub_ne_zero (nhdsLT_le_nhdsNE t)] with s hs
+  exact (supportingIntersection_comm K t s hs).symm
+
+private theorem hasDerivWithinAt_outerCorner_left (K : RightAngleCapSpace) (t : ℝ) :
+    HasDerivWithinAt
+      (fun s : ℝ ↦ (rotatingHallwayParts (K.val : Set Point) (s : Real.Angle)).outerCorner)
+      (-(tangentArmLengths K t).1.2 • normalVector (t : Real.Angle) +
+        (tangentArmLengths K t).2.2 • tangentVector (t : Real.Angle)) (Set.Iic t) t := by
+  have hshift : Tendsto (fun s : ℝ ↦ s + Real.pi / 2) (𝓝[<] t) (𝓝[<] (t + Real.pi / 2)) := by
+    apply tendsto_nhdsWithin_iff.mpr
+    refine ⟨((continuous_id.add_const _).continuousAt.tendsto.mono_left nhdsWithin_le_nhds), ?_⟩
+    filter_upwards [self_mem_nhdsWithin] with s hs
+    change s + Real.pi / 2 < t + Real.pi / 2
+    linarith [show s < t from hs]
+  have h := tendsto_outerCorner_slope K.val (nhdsLT_le_nhdsNE t)
+    (edgeVertices K.val (t : Real.Angle)).2
+    (edgeVertices K.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).2
+    (tendsto_supportingIntersection_left_ordered K.val t)
+    ((tendsto_supportingIntersection_left_ordered K.val (t + Real.pi / 2)).comp hshift)
+  rw [hasDerivWithinAt_iff_tendsto_slope, Set.Iic_sdiff_right]
+  have heq :
+      inner ℝ ((edgeVertices K.val (t : Real.Angle)).2 -
+        (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).outerCorner)
+          (tangentVector (t : Real.Angle)) • normalVector (t : Real.Angle) +
+        inner ℝ ((edgeVertices K.val ((t + Real.pi / 2 : ℝ) : Real.Angle)).2 -
+          (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).outerCorner)
+          (-normalVector (t : Real.Angle)) • tangentVector (t : Real.Angle) =
+      -(tangentArmLengths K t).1.2 • normalVector (t : Real.Angle) +
+        (tangentArmLengths K t).2.2 • tangentVector (t : Real.Angle) := by
+    simp only [tangentArmLengths, capVertices, inner_sub_left, inner_neg_right]
+    module
+  rw [heq] at h
+  convert h using 1
+  funext s
+  exact slope_def_module _ _ _
+
+theorem capCorners_oneSided_derivatives (K : RightAngleCapSpace) :
+    (∀ t ∈ Set.Ico (0 : ℝ) (Real.pi / 2),
+      HasDerivWithinAt
+        (fun s : ℝ ↦ (rotatingHallwayParts (K.val : Set Point) (s : Real.Angle)).outerCorner)
+        (-(tangentArmLengths K t).1.1 • normalVector (t : Real.Angle) +
+          (tangentArmLengths K t).2.1 • tangentVector (t : Real.Angle)) (Set.Ici t) t ∧
+      HasDerivWithinAt
+        (fun s : ℝ ↦ (rotatingHallwayParts (K.val : Set Point) (s : Real.Angle)).innerCorner)
+        (-((tangentArmLengths K t).1.1 - 1) • normalVector (t : Real.Angle) +
+          ((tangentArmLengths K t).2.1 - 1) • tangentVector (t : Real.Angle)) (Set.Ici t) t) ∧
+    (∀ t ∈ Set.Ioc (0 : ℝ) (Real.pi / 2),
+      HasDerivWithinAt
+        (fun s : ℝ ↦ (rotatingHallwayParts (K.val : Set Point) (s : Real.Angle)).outerCorner)
+        (-(tangentArmLengths K t).1.2 • normalVector (t : Real.Angle) +
+          (tangentArmLengths K t).2.2 • tangentVector (t : Real.Angle)) (Set.Iic t) t ∧
+      HasDerivWithinAt
+        (fun s : ℝ ↦ (rotatingHallwayParts (K.val : Set Point) (s : Real.Angle)).innerCorner)
+        (-((tangentArmLengths K t).1.2 - 1) • normalVector (t : Real.Angle) +
+          ((tangentArmLengths K t).2.2 - 1) • tangentVector (t : Real.Angle)) (Set.Iic t) t) := by
+  constructor
+  · intro t _
+    have hout := hasDerivWithinAt_outerCorner_right K t
+    refine ⟨hout, ?_⟩
+    convert hout.sub ((hasDerivAt_normalVector t).hasDerivWithinAt.add
+      (hasDerivAt_tangentVector t).hasDerivWithinAt) using 1
+    · funext s
+      exact eq_sub_of_add_eq (outerCorner_eq_innerCorner_add K.val s).symm
+    · module
+  · intro t _
+    have hout := hasDerivWithinAt_outerCorner_left K t
+    refine ⟨hout, ?_⟩
+    convert hout.sub ((hasDerivAt_normalVector t).hasDerivWithinAt.add
+      (hasDerivAt_tangentVector t).hasDerivWithinAt) using 1
+    · funext s
+      exact eq_sub_of_add_eq (outerCorner_eq_innerCorner_add K.val s).symm
+    · module
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Cap.Regularity`.
+* `Cap.Tail.Contacts`.
+* `Cap.Tail.Bodies`.
+* `Cap.Tail.Extension`.
+* `Cap.Tail.Separation`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Regularity
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Set Filter
+open scoped Topology
+
+namespace MovingSofa
+
+private theorem continuous_nondegenerateCapData_right (K : RightAngleCapSpace)
+    (hD : ∃ r s, HasCapDensities K r s) :
+    Continuous (nondegenerateCapData K hD).1.1 := by
+  have h := continuousOn_replace_right_endpoint (by positivity : (0 : ℝ) < Real.pi / 2)
+    (fun t : ℝ ↦ (capVertices K t).1.1) ((capVertices K (Real.pi / 2)).1.2)
+    (fun t ht ↦ (contact_oneSided_limits K.1 t).1)
+    (fun t ht ↦ ?_) (contact_oneSided_limits K.1 (Real.pi / 2)).2.2.2.1
+  · apply (continuousOn_iff_continuous_domRestrict.mp h).congr
+    intro t
+    dsimp [Set.domRestrict, nondegenerateCapData]
+    split_ifs with heq
+    · rw [heq]
+    · rfl
+  · have heq := (capDensities_contact_eq K hD).1 t ⟨ht.1.le, ht.2⟩ |>.1
+    change (edgeVertices K.1 (t : Real.Angle)).1 =
+      (edgeVertices K.1 (t : Real.Angle)).2 at heq
+    simpa only [← heq, capVertices] using (contact_oneSided_limits K.1 t).2.2.2.1
+
+private theorem continuous_nondegenerateCapData_left (K : RightAngleCapSpace)
+    (hD : ∃ r s, HasCapDensities K r s) :
+    Continuous (nondegenerateCapData K hD).1.2 := by
+  let f : ℝ → Point := fun t ↦ (capVertices K t).2.1
+  have hr (t : ℝ) : Tendsto f (𝓝[>] t) (𝓝 (f t)) := by
+    apply (contact_oneSided_limits K.1 (t + Real.pi / 2)).1.comp
+    apply tendsto_nhdsWithin_iff.mpr
+    constructor
+    · exact (tendsto_id.mono_left nhdsWithin_le_nhds).add_const _
+    · filter_upwards [self_mem_nhdsWithin] with u hu
+      change t + Real.pi / 2 < u + Real.pi / 2
+      simpa only [add_comm] using add_lt_add_right (Set.mem_Ioi.mp hu) (Real.pi / 2)
+  have hl (t : ℝ) (ht : t ∈ Ioc 0 (Real.pi / 2)) :
+      Tendsto f (𝓝[<] t) (𝓝 (f t)) := by
+    have h := (contact_oneSided_limits K.1 (t + Real.pi / 2)).2.2.2.1
+    have heq := (capDensities_contact_eq K hD).2 t ht |>.1
+    change (edgeVertices K.1 ((t + Real.pi / 2 : ℝ) : Real.Angle)).1 =
+      (edgeVertices K.1 ((t + Real.pi / 2 : ℝ) : Real.Angle)).2 at heq
+    rw [← heq] at h
+    apply h.comp
+    apply tendsto_nhdsWithin_iff.mpr
+    constructor
+    · exact (tendsto_id.mono_left nhdsWithin_le_nhds).add_const _
+    · filter_upwards [self_mem_nhdsWithin] with u hu
+      change u + Real.pi / 2 < t + Real.pi / 2
+      simpa only [add_comm] using add_lt_add_right (Set.mem_Iio.mp hu) (Real.pi / 2)
+  have h := continuousOn_replace_right_endpoint (by positivity : (0 : ℝ) < Real.pi / 2)
+    f (f (Real.pi / 2)) (fun t _ ↦ hr t) (fun t ht ↦ hl t ⟨ht.1, ht.2.le⟩)
+    (hl (Real.pi / 2) ⟨by positivity, le_rfl⟩)
+  have hc : ContinuousOn f (Icc 0 (Real.pi / 2)) := h.congr fun t ht ↦ by
+    by_cases heq : t = Real.pi / 2 <;> simp [heq]
+  change Continuous ((Icc 0 (Real.pi / 2)).domRestrict f)
+  exact continuousOn_iff_continuous_domRestrict.mp hc
+
+private theorem continuous_nondegenerateCapData_arms (K : RightAngleCapSpace)
+    (hD : ∃ r s, HasCapDensities K r s) :
+    Continuous (nondegenerateCapData K hD).2.1 ∧
+    Continuous (nondegenerateCapData K hD).2.2 := by
+  have hy := (continuous_outerCorner K.1).comp
+    (continuous_subtype_val : Continuous (fun t : Icc (0 : ℝ) (Real.pi / 2) ↦ (t : ℝ)))
+  have hn := continuous_normalVector_real.comp
+    (continuous_subtype_val : Continuous (fun t : Icc (0 : ℝ) (Real.pi / 2) ↦ (t : ℝ)))
+  have hv := (continuous_iff_continuousAt.mpr fun t ↦
+    (hasDerivAt_tangentVector t).continuousAt).comp
+      (continuous_subtype_val : Continuous (fun t : Icc (0 : ℝ) (Real.pi / 2) ↦ (t : ℝ)))
+  constructor
+  · apply ((hy.sub (continuous_nondegenerateCapData_right K hD)).inner hv).congr
+    intro t
+    dsimp [nondegenerateCapData, tangentArmLengths]
+    split_ifs <;> rfl
+  · exact (hy.sub (continuous_nondegenerateCapData_left K hD)).inner hn
+
+private theorem nondegenerateCap_hasDerivWithinAt (K : RightAngleCapSpace)
+    (hD : ∃ r s, HasCapDensities K r s) (t : Icc (0 : ℝ) (Real.pi / 2)) :
+    HasDerivWithinAt (capInnerCorner K)
+      (-((nondegenerateCapData K hD).2.1 t - 1) • normalVector (t : Real.Angle) +
+        ((nondegenerateCapData K hD).2.2 t - 1) • tangentVector (t : Real.Angle))
+      (Icc 0 (Real.pi / 2)) (t : ℝ) ∧
+    HasDerivWithinAt
+      (fun s : ℝ ↦ (rotatingHallwayParts (K.val : Set Point) (s : Real.Angle)).outerCorner)
+      (-(nondegenerateCapData K hD).2.1 t • normalVector (t : Real.Angle) +
+        (nondegenerateCapData K hD).2.2 t • tangentVector (t : Real.Angle))
+      (Icc 0 (Real.pi / 2)) (t : ℝ) := by
+  have hr (ht : (t : ℝ) < Real.pi / 2) :=
+    (capCorners_oneSided_derivatives K).1 t ⟨t.property.1, ht⟩
+  have hl (ht : 0 < (t : ℝ)) :=
+    (capCorners_oneSided_derivatives K).2 t ⟨ht, t.property.2⟩
+  have hfplus (ht : (t : ℝ) < Real.pi / 2) :
+      (nondegenerateCapData K hD).2.1 t = (tangentArmLengths K t).1.1 := by
+    simp only [nondegenerateCapData, ite_eq_right ht.ne]
+  have hfminus (ht : 0 < (t : ℝ)) :
+      (nondegenerateCapData K hD).2.1 t = (tangentArmLengths K t).1.2 := by
+    dsimp [nondegenerateCapData]
+    split_ifs with heq
+    · rfl
+    · exact ((capDensities_contact_eq K hD).1 t
+        ⟨t.property.1, lt_of_le_of_ne t.property.2 heq⟩).2
+  have hgminus (ht : 0 < (t : ℝ)) :
+      (nondegenerateCapData K hD).2.2 t = (tangentArmLengths K t).2.2 :=
+    ((capDensities_contact_eq K hD).2 t ⟨ht, t.property.2⟩).2
+  constructor
+  · apply hasDerivWithinAt_Icc_of_oneSided (by positivity) t.property
+    · intro ht
+      rw [hfplus ht]
+      exact (hr ht).2
+    · intro ht
+      rw [hfminus ht, hgminus ht]
+      exact (hl ht).2
+  · apply hasDerivWithinAt_Icc_of_oneSided (by positivity) t.property
+    · intro ht
+      rw [hfplus ht]
+      exact (hr ht).1
+    · intro ht
+      rw [hfminus ht, hgminus ht]
+      exact (hl ht).1
+
+theorem nondegenerateCap_continuity (K : RightAngleCapSpace)
+    (hD : ∃ r s, HasCapDensities K r s) :
+    Continuous (nondegenerateCapData K hD).1.1 ∧
+    Continuous (nondegenerateCapData K hD).1.2 ∧
+    Continuous (nondegenerateCapData K hD).2.1 ∧
+    Continuous (nondegenerateCapData K hD).2.2 ∧
+    ContDiffOn ℝ 1 (capInnerCorner K) (Set.Icc 0 (Real.pi / 2)) ∧
+    ContDiffOn ℝ 1
+      (fun s : ℝ ↦ (rotatingHallwayParts (K.val : Set Point) (s : Real.Angle)).outerCorner)
+      (Set.Icc 0 (Real.pi / 2)) ∧
+    ∀ t : Set.Icc (0 : ℝ) (Real.pi / 2),
+      HasDerivWithinAt (capInnerCorner K)
+        (-((nondegenerateCapData K hD).2.1 t - 1) • normalVector (t : Real.Angle) +
+          ((nondegenerateCapData K hD).2.2 t - 1) • tangentVector (t : Real.Angle))
+        (Set.Icc 0 (Real.pi / 2)) (t : ℝ) ∧
+      HasDerivWithinAt
+        (fun s : ℝ ↦ (rotatingHallwayParts (K.val : Set Point) (s : Real.Angle)).outerCorner)
+        (-(nondegenerateCapData K hD).2.1 t • normalVector (t : Real.Angle) +
+          (nondegenerateCapData K hD).2.2 t • tangentVector (t : Real.Angle))
+        (Set.Icc 0 (Real.pi / 2)) (t : ℝ) := by
+  have hA := continuous_nondegenerateCapData_right K hD
+  have hC := continuous_nondegenerateCapData_left K hD
+  obtain ⟨hf, hg⟩ := continuous_nondegenerateCapData_arms K hD
+  have hn := continuous_normalVector_real.comp
+    (continuous_subtype_val : Continuous (fun t : Icc (0 : ℝ) (Real.pi / 2) ↦ (t : ℝ)))
+  have hv := (continuous_iff_continuousAt.mpr fun t ↦
+    (hasDerivAt_tangentVector t).continuousAt).comp
+      (continuous_subtype_val : Continuous (fun t : Icc (0 : ℝ) (Real.pi / 2) ↦ (t : ℝ)))
+  refine ⟨hA, hC, hf, hg, ?_, ?_, nondegenerateCap_hasDerivWithinAt K hD⟩
+  · exact contDiffOn_one_of_continuous_derivative
+      (uniqueDiffOn_Icc (by positivity)) _ _
+      (((hf.sub continuous_const).neg.smul hn).add ((hg.sub continuous_const).smul hv))
+      (fun t ↦ (nondegenerateCap_hasDerivWithinAt K hD t).1)
+  · exact contDiffOn_one_of_continuous_derivative
+      (uniqueDiffOn_Icc (by positivity)) _ _
+      ((hf.neg.smul hn).add (hg.smul hv))
+      (fun t ↦ (nondegenerateCap_hasDerivWithinAt K hD t).2)
+
+/-- The corner coordinates of a right-angle cap in its moving frame. -/
+theorem inner_capInnerCorner (K : RightAngleCapSpace) (t : ℝ) :
+    inner ℝ (capInnerCorner K t) (normalVector (t : Real.Angle)) =
+        supportValue (K.1 : Set Point) (t : Real.Angle) - 1 ∧
+      inner ℝ (capInnerCorner K t) (tangentVector (t : Real.Angle)) =
+        supportValue (K.1 : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+  constructor
+  · have h := inner_supportingPlacement_normalVector (K.1 : Set Point) (t : Real.Angle) 0
+    simpa [capInnerCorner, rotatingHallwayParts, hallwayParts] using h
+  · have h := inner_supportingPlacement_tangentVector (K.1 : Set Point) (t : Real.Angle) 0
+    rw [← Real.Angle.coe_add] at h
+    simpa [capInnerCorner, rotatingHallwayParts, hallwayParts] using h
+
+/-- The injectivity condition supplies the support derivatives with their strict signs. -/
+theorem capSupport_hasDerivAt (K : RightAngleCapSpace)
+    (hinj : SatisfiesInjectivityCondition K) :
+    ∃ dh dj : ℝ → ℝ,
+      (∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+        HasDerivAt (fun u : ℝ ↦ supportValue (K.1 : Set Point) (u : Real.Angle)) (dh t) t) ∧
+      (∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+        HasDerivAt (fun u : ℝ ↦ supportValue (K.1 : Set Point) (u : Real.Angle)) (dj t)
+          (t + Real.pi / 2)) ∧
+      (∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+        dh t - supportValue (K.1 : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) + 1 < 0) ∧
+      (∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+        0 < dj t + supportValue (K.1 : Set Point) (t : Real.Angle) - 1) := by
+  obtain ⟨-, hC1, hsign⟩ := hinj
+  have hxn : ∀ t : ℝ, inner ℝ (capInnerCorner K t) (normalVector (t : Real.Angle)) =
+      supportValue (K.1 : Set Point) (t : Real.Angle) - 1 := fun t => (inner_capInnerCorner K t).1
+  have hxt : ∀ t : ℝ, inner ℝ (capInnerCorner K t) (tangentVector (t : Real.Angle)) =
+      supportValue (K.1 : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 :=
+    fun t => (inner_capInnerCorner K t).2
+  have hdiff : ∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+      HasDerivAt (capInnerCorner K) (deriv (capInnerCorner K) t) t := by
+    intro t ht
+    have hmem : Set.Icc (0 : ℝ) (Real.pi / 2) ∈ nhds t := Icc_mem_nhds ht.1 ht.2
+    exact (((hC1.differentiableOn one_ne_zero) t (Set.Ioo_subset_Icc_self ht)).differentiableAt
+      hmem).hasDerivAt
+  have hderivEq : ∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+      derivWithin (capInnerCorner K) (Set.Icc 0 (Real.pi / 2)) t = deriv (capInnerCorner K) t :=
+    fun t ht => derivWithin_of_mem_nhds (Icc_mem_nhds ht.1 ht.2)
+  have hfunn : (fun u : ℝ ↦ supportValue (K.1 : Set Point) (u : Real.Angle)) =
+      fun u : ℝ ↦ inner ℝ (capInnerCorner K u) (normalVector (u : Real.Angle)) + 1 := by
+    funext u; rw [hxn u]; ring
+  have hfunt :
+      (fun u : ℝ ↦ supportValue (K.1 : Set Point) ((u + Real.pi / 2 : ℝ) : Real.Angle)) =
+      fun u : ℝ ↦ inner ℝ (capInnerCorner K u) (tangentVector (u : Real.Angle)) + 1 := by
+    funext u; rw [hxt u]; ring
+  refine ⟨fun t ↦ inner ℝ (capInnerCorner K t) (tangentVector (t : Real.Angle)) +
+      inner ℝ (deriv (capInnerCorner K) t) (normalVector (t : Real.Angle)),
+    fun t ↦ -inner ℝ (capInnerCorner K t) (normalVector (t : Real.Angle)) +
+      inner ℝ (deriv (capInnerCorner K) t) (tangentVector (t : Real.Angle)), ?_, ?_, ?_, ?_⟩
+  · intro t ht
+    rw [hfunn]
+    exact ((hdiff t ht).inner ℝ (hasDerivAt_normalVector t)).add_const 1
+  · intro t ht
+    refine (hasDerivAt_comp_add_const_iff (f := fun u : ℝ ↦
+      supportValue (K.1 : Set Point) (u : Real.Angle)) t (Real.pi / 2)).mp ?_
+    rw [hfunt]
+    have hd := ((hdiff t ht).inner ℝ (hasDerivAt_tangentVector t)).add_const 1
+    rw [inner_neg_right] at hd
+    exact hd
+  · intro t ht
+    have h1 := (hsign t ht).1
+    rw [hderivEq t ht] at h1
+    dsimp only
+    rw [hxt t]
+    linarith only [h1]
+  · intro t ht
+    have h2 := (hsign t ht).2
+    rw [hderivEq t ht] at h2
+    dsimp only
+    rw [hxn t]
+    linarith only [h2]
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Endpoint contacts for the canonical cap tails
+
+The right and left canonical tails of a special cap are cut out of the cap by the upper
+half-planes of its inner supporting walls. This file supplies the contact points that make the
+tail support values meet their endpoint bounds.
+
+The mathematical content is the frame-free lemma
+`exists_mem_inner_eq_sub_one_of_deriv_signs`: let `s` be a nonempty compact convex planar set
+lying above the horizontal axis with `h_s(pi/2) = 1`, whose support function `h` is
+differentiable on the two quarter-turn families with the strict corner velocity signs
+`h'(t) - h(t + pi/2) + 1 < 0` and `h'(t + pi/2) + h(t) - 1 > 0` for `t` in `(0, pi/2)`. Then
+every interior cut line `{q | inner q (normalVector t) = h(t) - 1}` meets `s` in a point that
+stays above the whole cut family on `[t, pi/2]`.
+
+The proof selects the transverse maximum `p` of the first cut section and studies the
+deficiency `g(t) = h(t) - inner p (normalVector t)`. Wherever `g(t) = 1`, the chord joining the
+two frame contacts has nonpositive signed area (`planeCrossProduct_nonpos_of_isMaxOn`), which
+forces `g` to decrease strictly at the cut angle; at a first return of `g` to level one the
+same determinant is strictly positive, a contradiction.
+
+The left tail follows from the same lemma applied to `reflectedBody (pi/2) K`, whose support
+derivatives are the negated right-hand ones read backwards; see `exists_left_tail_contact`.
+
+This argument replaces the step in the paper's proof of `lem:right-left-body` which infers that the
+upper boundary avoids the niche; that inference fails for the stated class of caps (P46 in
+`NOTES.md`).
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- Lowering a right-angle cap point vertically to the base line keeps it inside the cap. -/
+theorem basePoint_mem_of_rightAngleCap (K : RightAngleCapSpace) {q : Point}
+    (hq : q ∈ (K.1 : Set Point)) : (!₂[q 0, 0] : Point) ∈ (K.1 : Set Point) := by
+  have hq0 : (!₂[q 0, 0] : Point) 0 = q 0 := rfl
+  have hq1 : (!₂[q 0, 0] : Point) 1 = 0 := rfl
+  have hqy : 0 ≤ q 1 := by
+    have h := K.inner_normalVector_pi_div_two_nonneg hq
+    rwa [inner_normalVector_pi_div_two] at h
+  have hzero : inner ℝ (!₂[q 0, 0] : Point)
+      (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := by
+    rw [inner_normalVector_pi_div_two, hq1]
+  refine K.mem_of_mem_capFan_of_le_supportValue ⟨le_of_eq hzero.symm, le_of_eq hzero.symm⟩ ?_
+  intro t ht
+  have htI : 0 ≤ t ∧ t ≤ Real.pi := by
+    rcases ht with h | h
+    · exact ⟨h.1, by linarith only [h.2, Real.pi_pos]⟩
+    · exact ⟨by linarith only [h.1, Real.pi_pos], by linarith only [h.2]⟩
+  have hsin : 0 ≤ Real.sin t := Real.sin_nonneg_of_nonneg_of_le_pi htI.1 htI.2
+  have hqt := inner_le_supportValue_of_isCompact K.1.isCompact hq (t : Real.Angle)
+  rw [inner_normalVector_real] at hqt
+  rw [inner_normalVector_real, hq0, hq1]
+  nlinarith only [hqt, hsin, hqy]
+
+/-- The two frame contacts at an interior angle, with their derivative coordinates. -/
+theorem exists_frame_contacts {s : Set Point} (hcomp : IsCompact s) (hne : s.Nonempty)
+    {t dh dj : ℝ}
+    (hdh : HasDerivAt (fun u : ℝ ↦ supportValue s (u : Real.Angle)) dh t)
+    (hdj : HasDerivAt (fun u : ℝ ↦ supportValue s (u : Real.Angle)) dj (t + Real.pi / 2)) :
+    ∃ A ∈ s, ∃ C ∈ s,
+      inner ℝ A (normalVector (t : Real.Angle)) = supportValue s (t : Real.Angle) ∧
+      inner ℝ A (tangentVector (t : Real.Angle)) = dh ∧
+      inner ℝ C (normalVector (t : Real.Angle)) = -dj ∧
+      inner ℝ C (tangentVector (t : Real.Angle)) =
+        supportValue s ((t + Real.pi / 2 : ℝ) : Real.Angle) := by
+  obtain ⟨A, hA, hA1, hA2⟩ := exists_contact_of_hasDerivAt hcomp hne hdh
+  obtain ⟨C, hC, hC1, hC2⟩ := exists_contact_of_hasDerivAt hcomp hne hdj
+  have hnv : normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+      tangentVector (t : Real.Angle) := by
+    rw [Real.Angle.coe_add]
+    exact normalVector_add_pi_div_two (t : Real.Angle)
+  have htv : tangentVector ((t + Real.pi / 2 : ℝ) : Real.Angle) =
+      -normalVector (t : Real.Angle) := tangentVector_add_pi_div_two t
+  rw [hnv] at hC1
+  rw [htv, inner_neg_right] at hC2
+  exact ⟨A, hA, C, hC, hA1, hA2, by linarith, hC1⟩
+
+/-- A transverse chord through a maximizing section point has nonpositive signed area. -/
+theorem planeCrossProduct_nonpos_of_isMaxOn {s : Set Point} (hconv : Convex ℝ s)
+    {r c₀ : ℝ} {p : Point}
+    (hpline : inner ℝ p (normalVector (r : Real.Angle)) = c₀)
+    (hmax : ∀ q ∈ s, inner ℝ q (normalVector (r : Real.Angle)) = c₀ →
+      inner ℝ q (tangentVector (r : Real.Angle)) ≤ inner ℝ p (tangentVector (r : Real.Angle)))
+    {A C : Point} (hA : A ∈ s) (hC : C ∈ s)
+    (hApos : 0 < inner ℝ (A - p) (normalVector (r : Real.Angle)))
+    (hCneg : inner ℝ (C - p) (normalVector (r : Real.Angle)) < 0) :
+    planeCrossProduct (A - p) (C - p) ≤ 0 := by
+  set u := normalVector (r : Real.Angle) with hu
+  set v := tangentVector (r : Real.Angle) with hv
+  set mu : ℝ := inner ℝ (A - p) u with hmu
+  set la : ℝ := -inner ℝ (C - p) u with hla
+  have hlapos : 0 < la := by simp only [hla]; linarith
+  have hsum : 0 < la + mu := by linarith
+  set w : ℝ := la / (la + mu) with hw
+  have hw0 : 0 ≤ w := div_nonneg hlapos.le hsum.le
+  have hw1 : 0 ≤ 1 - w := by
+    rw [hw, sub_nonneg, div_le_one hsum]
+    linarith
+  set q : Point := w • A + (1 - w) • C with hq
+  have hqs : q ∈ s := hconv hA hC hw0 hw1 (by ring)
+  have hwmu : w * mu + (1 - w) * (-la) = 0 := by
+    rw [hw]; field_simp; ring
+  have hqu : inner ℝ q u = c₀ := by
+    have h1 : inner ℝ q u - inner ℝ p u = w * mu + (1 - w) * (-la) := by
+      simp only [hq, hmu, hla, inner_add_left, real_inner_smul_left, inner_sub_left]
+      ring
+    rw [hwmu] at h1
+    rw [hpline] at h1
+    linarith
+  have hqv := hmax q hqs hqu
+  have h2 : inner ℝ q v - inner ℝ p v =
+      w * inner ℝ (A - p) v + (1 - w) * inner ℝ (C - p) v := by
+    simp only [hq, inner_add_left, real_inner_smul_left, inner_sub_left]
+    ring
+  have hnum : w * inner ℝ (A - p) v + (1 - w) * inner ℝ (C - p) v ≤ 0 := by
+    rw [← h2]; linarith
+  have hcross : planeCrossProduct (A - p) (C - p) =
+      mu * inner ℝ (C - p) v - inner ℝ (A - p) v * (-la) := by
+    rw [planeCrossProduct_eq_inner_frame (A - p) (C - p) r, ← hu, ← hv, ← hmu]
+    simp only [hla]
+    ring
+  rw [hcross]
+  have hwid : (1 - w) * (la + mu) = mu := by rw [hw]; field_simp; ring
+  have hwid2 : w * (la + mu) = la := by rw [hw]; field_simp
+  have hexp : (w * inner ℝ (A - p) v + (1 - w) * inner ℝ (C - p) v) * (la + mu) =
+      la * inner ℝ (A - p) v + mu * inner ℝ (C - p) v := by
+    calc (w * inner ℝ (A - p) v + (1 - w) * inner ℝ (C - p) v) * (la + mu)
+        = (w * (la + mu)) * inner ℝ (A - p) v +
+          ((1 - w) * (la + mu)) * inner ℝ (C - p) v := by ring
+      _ = la * inner ℝ (A - p) v + mu * inner ℝ (C - p) v := by rw [hwid, hwid2]
+  have hmul := mul_nonpos_of_nonpos_of_nonneg hnum hsum.le
+  rw [hexp] at hmul
+  linarith
+
+/-- The three chord signs at a first return time, in the frame coordinates at that time. -/
+private theorem chord_signs {r t d al be : ℝ} (hr : 0 < r) (hrt : r < t)
+    (ht : t < Real.pi / 2) (hd : 0 ≤ d) (hal : al < 0) (hbe : 0 < be)
+    (hstrip : Real.sin t + Real.cos t * d ≤ 1) :
+    0 < Real.cos (t - r) - Real.sin (t - r) * d ∧
+      Real.cos (t - r) * (-be) - Real.sin (t - r) * (1 - al + d) < 0 ∧
+      0 < 1 - al + d + be * d := by
+  have hpi := Real.pi_pos
+  have hδpos : 0 < t - r := by linarith only [hrt]
+  have hδlt : t - r < Real.pi / 2 := by linarith only [hr, ht]
+  have hsinδ : 0 < Real.sin (t - r) :=
+    Real.sin_pos_of_pos_of_lt_pi hδpos (by linarith only [hpi, hδlt])
+  have hcosδ : 0 < Real.cos (t - r) :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith only [hpi, hδpos], hδlt⟩
+  have hcost : 0 < Real.cos t :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith only [hpi, hr, hrt], ht⟩
+  have hcosr : Real.cos (t - r) * Real.cos t + Real.sin (t - r) * Real.sin t = Real.cos r := by
+    rw [← Real.cos_sub, show t - r - t = -r by ring, Real.cos_neg]
+  have hcosrgt : Real.sin (t - r) < Real.cos r := by
+    rw [← Real.sin_pi_div_two_sub]
+    exact Real.sin_lt_sin_of_lt_of_le_pi_div_two (by linarith only [hpi, hδpos])
+      (by linarith only [hr]) (by linarith only [ht])
+  refine ⟨?_, ?_, by nlinarith only [hal, hbe, hd]⟩
+  · have hmul := mul_le_mul_of_nonneg_left hstrip hsinδ.le
+    have hstep : Real.cos r - Real.sin (t - r) ≤
+        Real.cos t * (Real.cos (t - r) - Real.sin (t - r) * d) := by
+      nlinarith only [hmul, hcosr]
+    by_contra hc
+    push Not at hc
+    nlinarith only [hstep, hc, hcost, hcosrgt]
+  · nlinarith only [hcosδ, hsinδ, hbe, hal, hd]
+
+/-- Contradiction sign lemma at the cut angle: a nonpositive determinant forces a negative slope. -/
+private theorem neg_of_det_nonpos {d al be : ℝ} (hal : al < 0) (hbe : 0 < be)
+    (hdet : 1 - al + d + be * d ≤ 0) : d < 0 := by
+  by_contra h
+  push Not at h
+  nlinarith only [hal, hbe, hdet, h]
+
+/-- Endpoint contact for the upper cut family of a strip body with strict corner velocities. -/
+theorem exists_mem_inner_eq_sub_one_of_deriv_signs
+    {s : Set Point} (hcomp : IsCompact s) (hconv : Convex ℝ s) (hne : s.Nonempty)
+    (dh dj : ℝ → ℝ)
+    (hdh : ∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+      HasDerivAt (fun u : ℝ ↦ supportValue s (u : Real.Angle)) (dh t) t)
+    (hdj : ∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+      HasDerivAt (fun u : ℝ ↦ supportValue s (u : Real.Angle)) (dj t) (t + Real.pi / 2))
+    (halpha : ∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+      dh t - supportValue s ((t + Real.pi / 2 : ℝ) : Real.Angle) + 1 < 0)
+    (hbeta : ∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+      0 < dj t + supportValue s (t : Real.Angle) - 1)
+    (htop : supportValue s ((Real.pi / 2 : ℝ) : Real.Angle) = 1)
+    (hlow : ∀ q ∈ s, 0 ≤ inner ℝ q (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)))
+    {r : ℝ} (hr : r ∈ Set.Ioo (0 : ℝ) (Real.pi / 2)) :
+    ∃ p ∈ s, inner ℝ p (normalVector (r : Real.Angle)) =
+        supportValue s (r : Real.Angle) - 1 ∧
+      ∀ t ∈ Set.Icc r (Real.pi / 2),
+        supportValue s (t : Real.Angle) - 1 ≤ inner ℝ p (normalVector (t : Real.Angle)) := by
+  -- the cut line at `r` meets the body, between the two frame contacts
+  obtain ⟨A₀, hA₀, C₀, hC₀, hA₀n, hA₀t, hC₀n, hC₀t⟩ :=
+    exists_frame_contacts hcomp hne (hdh r hr) (hdj r hr)
+  have hC₀le : inner ℝ C₀ (normalVector (r : Real.Angle)) ≤
+      supportValue s (r : Real.Angle) - 1 := by
+    rw [hC₀n]; linarith only [hbeta r hr]
+  have hA₀ge : supportValue s (r : Real.Angle) - 1 ≤
+      inner ℝ A₀ (normalVector (r : Real.Angle)) := by rw [hA₀n]; linarith only []
+  obtain ⟨p₀, hp₀, hp₀line⟩ :=
+    exists_mem_inner_eq_of_convex hconv hA₀ hC₀ hC₀le hA₀ge
+  -- the transverse maximizer on that cut section
+  have hcontinner : Continuous (fun q : Point ↦ inner ℝ q (normalVector (r : Real.Angle))) :=
+    continuous_inner.comp (continuous_id.prodMk continuous_const)
+  have hScomp : IsCompact (s ∩ {q : Point |
+      inner ℝ q (normalVector (r : Real.Angle)) = supportValue s (r : Real.Angle) - 1}) :=
+    hcomp.inter_right (isClosed_eq hcontinner continuous_const)
+  have hconttang : Continuous (fun q : Point ↦ inner ℝ q (tangentVector (r : Real.Angle))) :=
+    continuous_inner.comp (continuous_id.prodMk continuous_const)
+  obtain ⟨p, hpS, hpmax⟩ :=
+    hScomp.exists_isMaxOn ⟨p₀, hp₀, hp₀line⟩ hconttang.continuousOn
+  have hps : p ∈ s := hpS.1
+  have hpline : inner ℝ p (normalVector (r : Real.Angle)) =
+    supportValue s (r : Real.Angle) - 1 := hpS.2
+  have hmaxq : ∀ q ∈ s, inner ℝ q (normalVector (r : Real.Angle)) =
+      supportValue s (r : Real.Angle) - 1 →
+      inner ℝ q (tangentVector (r : Real.Angle)) ≤
+        inner ℝ p (tangentVector (r : Real.Angle)) := fun q hq hq2 => hpmax ⟨hq, hq2⟩
+  -- the deficiency function
+  set g : ℝ → ℝ := fun u ↦ supportValue s (u : Real.Angle) -
+    inner ℝ p (normalVector (u : Real.Angle)) with hgdef
+  have hgval : ∀ u : ℝ, g u = supportValue s (u : Real.Angle) -
+      inner ℝ p (normalVector (u : Real.Angle)) := fun _ => rfl
+  have hgr : g r = 1 := by rw [hgval, hpline]; ring
+  have hgderiv : ∀ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+      HasDerivAt g (dh t - inner ℝ p (tangentVector (t : Real.Angle))) t :=
+    fun t ht => (hdh t ht).sub (hasDerivAt_inner_normalVector p t)
+  have hgT : g (Real.pi / 2) ≤ 1 := by
+    rw [hgval, htop]
+    linarith only [hlow p hps]
+  -- coordinates of the two contact chords relative to the maximizer
+  have hcoords : ∀ t : ℝ, g t = 1 → ∀ A C : Point,
+      inner ℝ A (normalVector (t : Real.Angle)) = supportValue s (t : Real.Angle) →
+      inner ℝ A (tangentVector (t : Real.Angle)) = dh t →
+      inner ℝ C (normalVector (t : Real.Angle)) = -dj t →
+      inner ℝ C (tangentVector (t : Real.Angle)) =
+        supportValue s ((t + Real.pi / 2 : ℝ) : Real.Angle) →
+      inner ℝ (A - p) (normalVector (t : Real.Angle)) = 1 ∧
+      inner ℝ (A - p) (tangentVector (t : Real.Angle)) =
+        dh t - inner ℝ p (tangentVector (t : Real.Angle)) ∧
+      inner ℝ (C - p) (normalVector (t : Real.Angle)) =
+        -dj t - (supportValue s (t : Real.Angle) - 1) ∧
+      inner ℝ (C - p) (tangentVector (t : Real.Angle)) =
+        supportValue s ((t + Real.pi / 2 : ℝ) : Real.Angle) -
+          inner ℝ p (tangentVector (t : Real.Angle)) := by
+    intro t hgt A C e1 e2 e3 e4
+    have hpu : inner ℝ p (normalVector (t : Real.Angle)) =
+        supportValue s (t : Real.Angle) - 1 := by
+      have h := hgval t
+      rw [hgt] at h
+      linarith only [h]
+    refine ⟨?_, ?_, ?_, ?_⟩
+    · simp only [inner_sub_left, e1, hpu]; ring
+    · simp only [inner_sub_left, e2]
+    · simp only [inner_sub_left, e3, hpu]
+    · simp only [inner_sub_left, e4]
+  -- the deficiency decreases strictly at the cut angle
+  have hdgr : dh r - inner ℝ p (tangentVector (r : Real.Angle)) < 0 := by
+    obtain ⟨k1, k2, k3, k4⟩ := hcoords r hgr A₀ C₀ hA₀n hA₀t hC₀n hC₀t
+    have hApos : 0 < inner ℝ (A₀ - p) (normalVector (r : Real.Angle)) := by
+      rw [k1]; norm_num
+    have hCneg : inner ℝ (C₀ - p) (normalVector (r : Real.Angle)) < 0 := by
+      rw [k3]; linarith only [hbeta r hr]
+    have hcr := planeCrossProduct_nonpos_of_isMaxOn hconv hpline hmaxq hA₀ hC₀ hApos hCneg
+    rw [planeCrossProduct_eq_inner_frame (A₀ - p) (C₀ - p) r, k1, k2, k3, k4] at hcr
+    exact neg_of_det_nonpos (halpha r hr) (hbeta r hr) (by linarith only [hcr])
+  -- the deficiency never exceeds one on the cut range
+  have hmain : ∀ t ∈ Set.Icc r (Real.pi / 2), g t ≤ 1 := by
+    by_contra hcon
+    push Not at hcon
+    obtain ⟨t₁, ht₁mem, ht₁gt⟩ := hcon
+    have ht₁r : r < t₁ :=
+      lt_of_le_of_ne ht₁mem.1 (by rintro rfl; rw [hgr] at ht₁gt; exact lt_irrefl 1 ht₁gt)
+    have ht₁T : t₁ < Real.pi / 2 :=
+      lt_of_le_of_ne ht₁mem.2 (by rintro rfl; exact absurd hgT (not_le.mpr ht₁gt))
+    -- a time just to the right of the cut where the deficiency is below one
+    have hev : ∀ᶠ y in nhdsWithin r (Set.Ioi r), slope g r y < 0 := by
+      refine (((hasDerivAt_iff_tendsto_slope.mp (hgderiv r hr)).mono_left
+        (nhdsWithin_mono r fun x hx => ne_of_gt hx)).eventually_lt_const hdgr)
+    obtain ⟨r', hr'slope, hr'mem⟩ :=
+      (hev.and (Filter.eventually_of_mem (Ioo_mem_nhdsGT ht₁r) fun y hy => hy)).exists
+    have hrr' : r < r' := hr'mem.1
+    have hr't₁ : r' < t₁ := hr'mem.2
+    have hgr'lt : g r' < 1 := by
+      rw [slope_def_field] at hr'slope
+      have hnum : g r' - g r < 0 := by
+        by_contra hcc
+        push Not at hcc
+        have h0 : 0 ≤ (g r' - g r) / (r' - r) := div_nonneg hcc (by linarith only [hrr'])
+        linarith only [h0, hr'slope]
+      linarith only [hnum, hgr]
+    -- the first return to level one
+    have hgcontOn : ContinuousOn g (Set.Icc r' t₁) := fun u hu =>
+      ((hgderiv u ⟨by linarith only [hu.1, hrr', hr.1],
+        by linarith only [hu.2, ht₁T]⟩).continuousAt).continuousWithinAt
+    obtain ⟨t₂, ht₂mem', hgt₂, hleft⟩ :=
+      exists_first_return hr't₁ hgcontOn hgr'lt ht₁gt.le
+    have ht₂gt : r' < t₂ := ht₂mem'.1
+    have ht₂mem : t₂ ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) :=
+      ⟨by linarith only [hr.1, hrr', ht₂gt], lt_of_le_of_lt ht₂mem'.2 ht₁T⟩
+    have hdgt₂ : 0 ≤ dh t₂ - inner ℝ p (tangentVector (t₂ : Real.Angle)) :=
+      nonneg_of_first_return ht₂gt (hgderiv t₂ ht₂mem) hgt₂ hleft
+    -- the transverse signs at the first return
+    obtain ⟨A, hA, C, hC, e1, e2, e3, e4⟩ :=
+      exists_frame_contacts hcomp hne (hdh t₂ ht₂mem) (hdj t₂ ht₂mem)
+    obtain ⟨k1, k2, k3, k4⟩ := hcoords t₂ hgt₂ A C e1 e2 e3 e4
+    have hframeR : normalVector (r : Real.Angle) =
+        Real.cos (t₂ - r) • normalVector (t₂ : Real.Angle) -
+          Real.sin (t₂ - r) • tangentVector (t₂ : Real.Angle) := by
+      have hrew : ((r : ℝ) : Real.Angle) = ((t₂ + -(t₂ - r) : ℝ) : Real.Angle) := by
+        congr 1; ring
+      rw [hrew, normalVector_add_real t₂ (-(t₂ - r)), Real.cos_neg, Real.sin_neg]
+      module
+    have hframeT : normalVector ((Real.pi / 2 : ℝ) : Real.Angle) =
+        Real.sin t₂ • normalVector (t₂ : Real.Angle) +
+          Real.cos t₂ • tangentVector (t₂ : Real.Angle) := by
+      have hrew : ((Real.pi / 2 : ℝ) : Real.Angle) =
+          ((t₂ + (Real.pi / 2 - t₂) : ℝ) : Real.Angle) := by norm_num
+      rw [hrew, normalVector_add_real t₂ (Real.pi / 2 - t₂), Real.cos_pi_div_two_sub,
+        Real.sin_pi_div_two_sub]
+    have hAT : inner ℝ (A - p) (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) =
+        Real.sin t₂ * 1 + Real.cos t₂ *
+          (dh t₂ - inner ℝ p (tangentVector (t₂ : Real.Angle))) := by
+      rw [hframeT, inner_add_right, real_inner_smul_right, real_inner_smul_right, k1, k2]
+    have hstrip : Real.sin t₂ + Real.cos t₂ *
+        (dh t₂ - inner ℝ p (tangentVector (t₂ : Real.Angle))) ≤ 1 := by
+      have h1 := inner_le_supportValue_of_isCompact hcomp hA ((Real.pi / 2 : ℝ) : Real.Angle)
+      rw [htop] at h1
+      have h2 : inner ℝ (A - p) (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) ≤ 1 := by
+        rw [inner_sub_left]
+        linarith only [h1, hlow p hps]
+      rw [hAT] at h2
+      linarith only [h2]
+    obtain ⟨hs1, hs2, hs3⟩ := chord_signs hr.1 (by linarith only [hrr', ht₂gt])
+      ht₂mem.2 hdgt₂ (halpha t₂ ht₂mem) (hbeta t₂ ht₂mem) hstrip
+    have hAr : inner ℝ (A - p) (normalVector (r : Real.Angle)) =
+        Real.cos (t₂ - r) * 1 - Real.sin (t₂ - r) *
+          (dh t₂ - inner ℝ p (tangentVector (t₂ : Real.Angle))) := by
+      rw [hframeR, inner_sub_right, real_inner_smul_right, real_inner_smul_right, k1, k2]
+    have hCr : inner ℝ (C - p) (normalVector (r : Real.Angle)) =
+        Real.cos (t₂ - r) * (-dj t₂ - (supportValue s (t₂ : Real.Angle) - 1)) -
+          Real.sin (t₂ - r) * (supportValue s ((t₂ + Real.pi / 2 : ℝ) : Real.Angle) -
+            inner ℝ p (tangentVector (t₂ : Real.Angle))) := by
+      rw [hframeR, inner_sub_right, real_inner_smul_right, real_inner_smul_right, k3, k4]
+    have hApos : 0 < inner ℝ (A - p) (normalVector (r : Real.Angle)) := by
+      rw [hAr]; linarith only [hs1]
+    have hCneg : inner ℝ (C - p) (normalVector (r : Real.Angle)) < 0 := by
+      rw [hCr]; linarith only [hs2]
+    have hcr := planeCrossProduct_nonpos_of_isMaxOn hconv hpline hmaxq hA hC hApos hCneg
+    rw [planeCrossProduct_eq_inner_frame (A - p) (C - p) t₂, k1, k2, k3, k4] at hcr
+    linarith only [hcr, hs3]
+  refine ⟨p, hps, hpline, fun t ht => ?_⟩
+  have hle := hmain t ht
+  rw [hgval] at hle
+  linarith only [hle]
+
+/-- The lowered horizontal extremes of a right-angle cap lie under all right and left cuts. -/
+theorem exists_cap_base_points (K : RightAngleCapSpace) :
+    ∃ pR ∈ (K.1 : Set Point), ∃ pL ∈ (K.1 : Set Point),
+      inner ℝ pR (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 ∧
+      inner ℝ pL (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 ∧
+      (∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+        supportValue (K.1 : Set Point) (t : Real.Angle) - 1 ≤
+          inner ℝ pR (normalVector (t : Real.Angle))) ∧
+      (∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+        supportValue (K.1 : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 ≤
+          inner ℝ pL (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle))) := by
+  have hheight : ∀ q ∈ (K.1 : Set Point), 0 ≤ q 1 ∧ q 1 ≤ 1 := by
+    intro q hq
+    have hlo := K.inner_normalVector_pi_div_two_nonneg hq
+    have hhi := inner_le_supportValue_of_isCompact K.1.isCompact hq
+      ((Real.pi / 2 : ℝ) : Real.Angle)
+    rw [K.property.2.2.2.1] at hhi
+    rw [inner_normalVector_pi_div_two] at hlo hhi
+    exact ⟨hlo, hhi⟩
+  obtain ⟨A, hA, hAeq⟩ := exists_mem_inner_eq_supportValue K.1 ((0 : ℝ) : Real.Angle)
+  obtain ⟨C, hC, hCeq⟩ := exists_mem_inner_eq_supportValue K.1 ((Real.pi : ℝ) : Real.Angle)
+  rw [inner_normalVector_real, Real.cos_zero, Real.sin_zero] at hAeq
+  rw [inner_normalVector_real, Real.cos_pi, Real.sin_pi] at hCeq
+  have hAmax : ∀ q ∈ (K.1 : Set Point), q 0 ≤ A 0 := by
+    intro q hq
+    have h := inner_le_supportValue_of_isCompact K.1.isCompact hq ((0 : ℝ) : Real.Angle)
+    rw [inner_normalVector_real, Real.cos_zero, Real.sin_zero] at h
+    linarith only [h, hAeq]
+  have hCmin : ∀ q ∈ (K.1 : Set Point), C 0 ≤ q 0 := by
+    intro q hq
+    have h := inner_le_supportValue_of_isCompact K.1.isCompact hq ((Real.pi : ℝ) : Real.Angle)
+    rw [inner_normalVector_real, Real.cos_pi, Real.sin_pi] at h
+    linarith only [h, hCeq]
+  refine ⟨!₂[A 0, 0], basePoint_mem_of_rightAngleCap K hA, !₂[C 0, 0],
+    basePoint_mem_of_rightAngleCap K hC, ?_, ?_, ?_, ?_⟩
+  · rw [inner_normalVector_pi_div_two]; rfl
+  · rw [inner_normalVector_pi_div_two]; rfl
+  · intro t ht
+    have hcos : 0 ≤ Real.cos t :=
+      Real.cos_nonneg_of_mem_Icc ⟨by linarith only [ht.1, Real.pi_pos], ht.2⟩
+    have hsin : 0 ≤ Real.sin t :=
+      Real.sin_nonneg_of_nonneg_of_le_pi ht.1 (by linarith only [ht.2, Real.pi_pos])
+    have hsin1 : Real.sin t ≤ 1 := Real.sin_le_one t
+    have hbound : supportValue (K.1 : Set Point) (t : Real.Angle) ≤ A 0 * Real.cos t + 1 := by
+      refine csSup_le (K.1.nonempty.image _) ?_
+      rintro _ ⟨q, hq, rfl⟩
+      dsimp only
+      rw [inner_normalVector_real]
+      nlinarith only [hAmax q hq, (hheight q hq).2, hcos, hsin, hsin1]
+    rw [inner_normalVector_real]
+    change supportValue (K.1 : Set Point) (t : Real.Angle) - 1 ≤ A 0 * Real.cos t + 0 * Real.sin t
+    linarith only [hbound]
+  · intro t ht
+    have hcos : 0 ≤ Real.cos t :=
+      Real.cos_nonneg_of_mem_Icc ⟨by linarith only [ht.1, Real.pi_pos], ht.2⟩
+    have hcos1 : Real.cos t ≤ 1 := Real.cos_le_one t
+    have hsin : 0 ≤ Real.sin t :=
+      Real.sin_nonneg_of_nonneg_of_le_pi ht.1 (by linarith only [ht.2, Real.pi_pos])
+    have hbound : supportValue (K.1 : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) ≤
+        -(C 0) * Real.sin t + 1 := by
+      refine csSup_le (K.1.nonempty.image _) ?_
+      rintro _ ⟨q, hq, rfl⟩
+      dsimp only
+      rw [inner_normalVector_real, Real.cos_add_pi_div_two, Real.sin_add_pi_div_two]
+      nlinarith only [hCmin q hq, (hheight q hq).2, hcos, hcos1, hsin]
+    rw [inner_normalVector_real, Real.cos_add_pi_div_two, Real.sin_add_pi_div_two]
+    change supportValue (K.1 : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 ≤
+      C 0 * -Real.sin t + 0 * Real.cos t
+    linarith only [hbound]
+
+/-- The right cut line meets the cap, below the whole right cut family. -/
+theorem exists_right_tail_contact (K : RightAngleCapSpace)
+    (hinj : SatisfiesInjectivityCondition K) {r : ℝ}
+    (hr : r ∈ Set.Ioo (0 : ℝ) (Real.pi / 2)) :
+    ∃ p ∈ (K.1 : Set Point),
+      inner ℝ p (normalVector (r : Real.Angle)) =
+        supportValue (K.1 : Set Point) (r : Real.Angle) - 1 ∧
+      ∀ t ∈ Set.Icc r (Real.pi / 2),
+        supportValue (K.1 : Set Point) (t : Real.Angle) - 1 ≤
+          inner ℝ p (normalVector (t : Real.Angle)) := by
+  obtain ⟨dh, dj, h1, h2, h3, h4⟩ := capSupport_hasDerivAt K hinj
+  exact exists_mem_inner_eq_sub_one_of_deriv_signs K.1.isCompact K.1.convex K.1.nonempty
+    dh dj h1 h2 h3 h4 K.property.2.2.2.1
+    (fun q hq => K.inner_normalVector_pi_div_two_nonneg hq) hr
+
+/-- The left cut line meets the cap, below the whole left cut family. -/
+theorem exists_left_tail_contact (K : RightAngleCapSpace)
+    (hinj : SatisfiesInjectivityCondition K) {l : ℝ}
+    (hl : l ∈ Set.Ioo (0 : ℝ) (Real.pi / 2)) :
+    ∃ p ∈ (K.1 : Set Point),
+      inner ℝ p (normalVector ((l + Real.pi / 2 : ℝ) : Real.Angle)) =
+        supportValue (K.1 : Set Point) ((l + Real.pi / 2 : ℝ) : Real.Angle) - 1 ∧
+      ∀ t ∈ Set.Icc (0 : ℝ) l,
+        supportValue (K.1 : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 ≤
+          inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) := by
+  obtain ⟨dh, dj, h1, h2, h3, h4⟩ := capSupport_hasDerivAt K hinj
+  have hcast : ∀ a b : ℝ, a = b → ((a : ℝ) : Real.Angle) = ((b : ℝ) : Real.Angle) :=
+    fun a b h => by rw [h]
+  have hsv : ∀ u : ℝ,
+      supportValue (reflectedBody (Real.pi / 2) K.1 : Set Point) (u : Real.Angle) =
+        supportValue (K.1 : Set Point) ((Real.pi - u : ℝ) : Real.Angle) :=
+    supportValue_reflectedBody_pi_div_two K.1
+  have hfunS : (fun w : ℝ ↦ supportValue (reflectedBody (Real.pi / 2) K.1 : Set Point)
+      (w : Real.Angle)) =
+      fun w : ℝ ↦ supportValue (K.1 : Set Point) ((Real.pi - w : ℝ) : Real.Angle) :=
+    funext hsv
+  have hrefl : ∀ u : ℝ, u ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) →
+      Real.pi / 2 - u ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) :=
+    fun u hu => ⟨by linarith only [hu.2], by linarith only [hu.1]⟩
+  have hsvL : ∀ u : ℝ,
+      supportValue (reflectedBody (Real.pi / 2) K.1 : Set Point)
+          ((u + Real.pi / 2 : ℝ) : Real.Angle) =
+        supportValue (K.1 : Set Point) ((Real.pi / 2 - u : ℝ) : Real.Angle) := by
+    intro u
+    rw [hsv (u + Real.pi / 2),
+      hcast _ _ (show Real.pi - (u + Real.pi / 2) = Real.pi / 2 - u by ring)]
+  have hsvR : ∀ u : ℝ,
+      supportValue (reflectedBody (Real.pi / 2) K.1 : Set Point) (u : Real.Angle) =
+        supportValue (K.1 : Set Point) ((Real.pi / 2 - u + Real.pi / 2 : ℝ) : Real.Angle) := by
+    intro u
+    rw [hsv u, hcast _ _ (show Real.pi - u = Real.pi / 2 - u + Real.pi / 2 by ring)]
+  obtain ⟨q0, hq0s, hq0line, hq0bound⟩ :=
+    exists_mem_inner_eq_sub_one_of_deriv_signs
+      (reflectedBody (Real.pi / 2) K.1).isCompact (reflectedBody (Real.pi / 2) K.1).convex
+      (reflectedBody (Real.pi / 2) K.1).nonempty
+      (fun u ↦ -dj (Real.pi / 2 - u)) (fun u ↦ -dh (Real.pi / 2 - u))
+      (by
+        intro u hu
+        have hF : HasDerivAt (fun w : ℝ ↦ supportValue (K.1 : Set Point) (w : Real.Angle))
+            (dj (Real.pi / 2 - u)) (Real.pi - u) := by
+          rw [show Real.pi - u = Real.pi / 2 - u + Real.pi / 2 by ring]
+          exact h2 _ (hrefl u hu)
+        rw [hfunS]
+        exact HasDerivAt.comp_const_sub Real.pi u hF)
+      (by
+        intro u hu
+        have hF : HasDerivAt (fun w : ℝ ↦ supportValue (K.1 : Set Point) (w : Real.Angle))
+            (dh (Real.pi / 2 - u)) (Real.pi - (u + Real.pi / 2)) := by
+          rw [show Real.pi - (u + Real.pi / 2) = Real.pi / 2 - u by ring]
+          exact h1 _ (hrefl u hu)
+        rw [hfunS]
+        exact HasDerivAt.comp_const_sub Real.pi (u + Real.pi / 2) hF)
+      (by
+        intro u hu
+        rw [hsvL u]
+        linarith only [h4 _ (hrefl u hu)])
+      (by
+        intro u hu
+        rw [hsvR u]
+        linarith only [h3 _ (hrefl u hu)])
+      (by
+        rw [hsv (Real.pi / 2), hcast _ _ (show Real.pi - Real.pi / 2 = Real.pi / 2 by ring)]
+        exact K.property.2.2.2.1)
+      (by
+        rintro q ⟨v, hv, rfl⟩
+        rw [inner_capReflection_pi_div_two,
+          hcast _ _ (show Real.pi - Real.pi / 2 = Real.pi / 2 by ring)]
+        exact K.inner_normalVector_pi_div_two_nonneg hv)
+      (show Real.pi / 2 - l ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) from hrefl l hl)
+  obtain ⟨p, hpK, rfl⟩ : ∃ p ∈ (K.1 : Set Point), capReflection (Real.pi / 2) p = q0 := hq0s
+  refine ⟨p, hpK, ?_, ?_⟩
+  · rw [inner_capReflection_pi_div_two,
+      hcast _ _ (show Real.pi - (Real.pi / 2 - l) = l + Real.pi / 2 by ring)] at hq0line
+    rw [hsv (Real.pi / 2 - l),
+      hcast _ _ (show Real.pi - (Real.pi / 2 - l) = l + Real.pi / 2 by ring)] at hq0line
+    exact hq0line
+  · intro t ht
+    have hw : Real.pi / 2 - t ∈ Set.Icc (Real.pi / 2 - l) (Real.pi / 2) :=
+      ⟨by linarith only [ht.2], by linarith only [ht.1]⟩
+    have hb := hq0bound (Real.pi / 2 - t) hw
+    rw [inner_capReflection_pi_div_two,
+      hcast _ _ (show Real.pi - (Real.pi / 2 - t) = t + Real.pi / 2 by ring)] at hb
+    rw [hsv (Real.pi / 2 - t),
+      hcast _ _ (show Real.pi - (Real.pi / 2 - t) = t + Real.pi / 2 by ring)] at hb
+    exact hb
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Tail / Bodies
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The distinguished inner wall, upper half-plane, fan point and corner on one side. -/
+structure DistinguishedCapSide where
+  /-- The distinguished inner supporting wall. -/
+  wall : Set Point
+  /-- The upper half-plane selected at the distinguished wall. -/
+  upperHalfPlane : Set Point
+  /-- The corresponding endpoint of the cap fan. -/
+  fanPoint : Point
+  /-- The inner hallway corner at the distinguished angle. -/
+  corner : Point
+
+/-- The right and left cap geometry at the two distinguished Gerver angles. -/
+def distinguishedCapSides (K : RightAngleCapSpace) : DistinguishedCapSide × DistinguishedCapSide :=
+  let r := paperGerverConstants.2.1
+  let l := paperGerverConstants.2.2
+  (⟨(rotatingHallwayParts (K.1 : Set Point) (r : Real.Angle)).b,
+      (innerWallUpperHalfPlanes K r).1, (wedgeEndpoints K r).1, capInnerCorner K r⟩,
+    ⟨(rotatingHallwayParts (K.1 : Set Point) (l : Real.Angle)).d,
+      (innerWallUpperHalfPlanes K l).2, (wedgeEndpoints K l).2, capInnerCorner K l⟩)
+
+/-- The right distinguished fan point is the right wedge endpoint at the right Gerver angle. -/
+theorem distinguishedCapSides_fst_fanPoint (K : RightAngleCapSpace) :
+    (distinguishedCapSides K).1.fanPoint = (wedgeEndpoints K paperGerverConstants.2.1).1 := rfl
+
+/-- The right distinguished corner is the inner corner at the right Gerver angle. -/
+theorem distinguishedCapSides_fst_corner (K : RightAngleCapSpace) :
+    (distinguishedCapSides K).1.corner = capInnerCorner K paperGerverConstants.2.1 := rfl
+
+/-- The left distinguished fan point is the left wedge endpoint at the left Gerver angle. -/
+theorem distinguishedCapSides_snd_fanPoint (K : RightAngleCapSpace) :
+    (distinguishedCapSides K).2.fanPoint = (wedgeEndpoints K paperGerverConstants.2.2).2 := rfl
+
+/-- The left distinguished corner is the inner corner at the left Gerver angle. -/
+theorem distinguishedCapSides_snd_corner (K : RightAngleCapSpace) :
+    (distinguishedCapSides K).2.corner = capInnerCorner K paperGerverConstants.2.2 := rfl
+
+/-- Canonical tails are convex bodies and satisfy all support and endpoint-line identities. -/
+theorem canonicalTailSets_properties (K : SpecialCapSpace) :
+    (canonicalTailSets K).1.Nonempty ∧ IsCompact (canonicalTailSets K).1 ∧ Convex ℝ
+      (canonicalTailSets K).1 ∧ (canonicalTailSets K).1 ⊆ (K.1.1 : Set Point) ∧
+    (canonicalTailSets K).2.Nonempty ∧ IsCompact (canonicalTailSets K).2 ∧ Convex ℝ
+      (canonicalTailSets K).2 ∧ (canonicalTailSets K).2 ⊆ (K.1.1 : Set Point) ∧
+    (∀ t ∈ Set.Icc paperGerverConstants.2.1 (Real.pi / 2),
+      supportValue K.1.1 (t : Real.Angle) +
+        supportValue (canonicalTailSets K).1 ((Real.pi + t : ℝ) : Real.Angle) ≤ 1) ∧
+    (∀ t ∈ ({paperGerverConstants.2.1, Real.pi / 2} : Set ℝ),
+      supportValue K.1.1 (t : Real.Angle) +
+        supportValue (canonicalTailSets K).1 ((Real.pi + t : ℝ) : Real.Angle) = 1) ∧
+    (supportingLineHalfPlane (canonicalTailSets K).1 ((3 * Real.pi / 2 : ℝ) : Real.Angle)).1 =
+      normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0 ∧
+    (supportingLineHalfPlane (canonicalTailSets K).1 ((Real.pi + paperGerverConstants.2.1 : ℝ) :
+      Real.Angle)).1 =
+      (distinguishedCapSides K.1).1.wall ∧
+    (∀ t ∈ Set.Icc (0 : ℝ) paperGerverConstants.2.2,
+      supportValue K.1.1 ((Real.pi / 2 + t : ℝ) : Real.Angle) +
+        supportValue (canonicalTailSets K).2 ((3 * Real.pi / 2 + t : ℝ) : Real.Angle) ≤ 1) ∧
+    (∀ t ∈ ({0, paperGerverConstants.2.2} : Set ℝ),
+      supportValue K.1.1 ((Real.pi / 2 + t : ℝ) : Real.Angle) +
+        supportValue (canonicalTailSets K).2 ((3 * Real.pi / 2 + t : ℝ) : Real.Angle) = 1) ∧
+    (supportingLineHalfPlane (canonicalTailSets K).2 ((3 * Real.pi / 2 : ℝ) : Real.Angle)).1 =
+      normalLine ((Real.pi / 2 : ℝ) : Real.Angle) 0 ∧
+    (supportingLineHalfPlane (canonicalTailSets K).2 ((3 * Real.pi / 2 + paperGerverConstants.2.2
+      : ℝ) : Real.Angle)).1 =
+      (distinguishedCapSides K.1).2.wall := by
+  have hcast : ∀ a b : ℝ, a = b → ((a : ℝ) : Real.Angle) = ((b : ℝ) : Real.Angle) :=
+    fun a b h => by rw [h]
+  obtain ⟨hr, hl, -⟩ := paperGerverConstants_snd_mem_Ioo
+  have htop : supportValue (K.1.1 : Set Point) ((Real.pi / 2 : ℝ) : Real.Angle) = 1 :=
+    K.1.property.2.2.2.1
+  -- membership characterizations of the two tails
+  have hBmem : ∀ p : Point, p ∈ (canonicalTailSets K).1 ↔
+      (p ∈ (K.1.1 : Set Point) ∧ ∀ t ∈ Set.Icc paperGerverConstants.2.1 (Real.pi / 2),
+        supportValue (K.1.1 : Set Point) (t : Real.Angle) - 1 ≤
+          inner ℝ p (normalVector (t : Real.Angle))) := by
+    intro p
+    simp [canonicalTailSets, innerWallUpperHalfPlanes, normalHalfPlane]
+  have hDmem : ∀ p : Point, p ∈ (canonicalTailSets K).2 ↔
+      (p ∈ (K.1.1 : Set Point) ∧ ∀ t ∈ Set.Icc (0 : ℝ) paperGerverConstants.2.2,
+        supportValue (K.1.1 : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 ≤
+          inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle))) := by
+    intro p
+    simp [canonicalTailSets, innerWallUpperHalfPlanes, normalHalfPlane]
+  -- domain properties
+  obtain ⟨pR, hpRK, pL, hpLK, hpRy, hpLy, hpRcut, hpLcut⟩ := exists_cap_base_points K.1
+  have hpRB : pR ∈ (canonicalTailSets K).1 :=
+    (hBmem pR).mpr ⟨hpRK, fun t ht => hpRcut t ⟨le_trans hr.1.le ht.1, ht.2⟩⟩
+  have hpLD : pL ∈ (canonicalTailSets K).2 :=
+    (hDmem pL).mpr ⟨hpLK, fun t ht => hpLcut t ⟨ht.1, le_trans ht.2 hl.2.le⟩⟩
+  have hBne : (canonicalTailSets K).1.Nonempty := ⟨pR, hpRB⟩
+  have hDne : (canonicalTailSets K).2.Nonempty := ⟨pL, hpLD⟩
+  have hBsub : (canonicalTailSets K).1 ⊆ (K.1.1 : Set Point) := fun p hp => ((hBmem p).mp hp).1
+  have hDsub : (canonicalTailSets K).2 ⊆ (K.1.1 : Set Point) := fun p hp => ((hDmem p).mp hp).1
+  have hBcomp : IsCompact (canonicalTailSets K).1 :=
+    K.1.1.isCompact.inter_right (isClosed_biInter fun t _ => isClosed_normalHalfPlane _ _ true)
+  have hDcomp : IsCompact (canonicalTailSets K).2 :=
+    K.1.1.isCompact.inter_right (isClosed_biInter fun t _ => isClosed_normalHalfPlane _ _ true)
+  have hBconv : Convex ℝ (canonicalTailSets K).1 :=
+    K.1.1.convex.inter (convex_iInter fun t => convex_iInter fun _ =>
+      convex_normalHalfPlane _ _ true)
+  have hDconv : Convex ℝ (canonicalTailSets K).2 :=
+    K.1.1.convex.inter (convex_iInter fun t => convex_iInter fun _ =>
+      convex_normalHalfPlane _ _ true)
+  -- the two support inequalities
+  have hBbound : ∀ t ∈ Set.Icc paperGerverConstants.2.1 (Real.pi / 2),
+      supportValue (canonicalTailSets K).1 ((Real.pi + t : ℝ) : Real.Angle) ≤
+        1 - supportValue (K.1.1 : Set Point) (t : Real.Angle) := by
+    intro t ht
+    have h := supportValue_le_of_cut hBne (a := t) (b := Real.pi + t)
+      (hcast _ _ (by ring)) (fun p hp => ((hBmem p).mp hp).2 t ht)
+    linarith only [h]
+  have hDbound : ∀ t ∈ Set.Icc (0 : ℝ) paperGerverConstants.2.2,
+      supportValue (canonicalTailSets K).2 ((3 * Real.pi / 2 + t : ℝ) : Real.Angle) ≤
+        1 - supportValue (K.1.1 : Set Point) ((Real.pi / 2 + t : ℝ) : Real.Angle) := by
+    intro t ht
+    have hc : ∀ p ∈ (canonicalTailSets K).2,
+        supportValue (K.1.1 : Set Point) ((Real.pi / 2 + t : ℝ) : Real.Angle) - 1 ≤
+          inner ℝ p (normalVector ((Real.pi / 2 + t : ℝ) : Real.Angle)) := by
+      intro p hp
+      have h := ((hDmem p).mp hp).2 t ht
+      rwa [hcast _ _ (show t + Real.pi / 2 = Real.pi / 2 + t by ring)] at h
+    have h := supportValue_le_of_cut hDne (a := Real.pi / 2 + t) (b := 3 * Real.pi / 2 + t)
+      (hcast _ _ (by ring)) hc
+    linarith only [h]
+  -- the four endpoint equalities
+  have hBzero :
+      supportValue (canonicalTailSets K).1 ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := by
+    refine le_antisymm ?_ ?_
+    · have h := hBbound (Real.pi / 2) ⟨hr.2.le, le_refl _⟩
+      rw [hcast _ _ (show Real.pi + Real.pi / 2 = 3 * Real.pi / 2 by ring), htop] at h
+      linarith only [h]
+    · have h := le_supportValue_of_cut hBcomp (a := Real.pi / 2) (b := 3 * Real.pi / 2)
+        (hcast _ _ (by ring)) hpRB hpRy
+      linarith only [h]
+  have hDzero :
+      supportValue (canonicalTailSets K).2 ((3 * Real.pi / 2 : ℝ) : Real.Angle) = 0 := by
+    refine le_antisymm ?_ ?_
+    · have h := hDbound 0 ⟨le_refl _, hl.1.le⟩
+      rw [hcast _ _ (show 3 * Real.pi / 2 + (0 : ℝ) = 3 * Real.pi / 2 by ring),
+        hcast _ _ (show Real.pi / 2 + (0 : ℝ) = Real.pi / 2 by ring), htop] at h
+      linarith only [h]
+    · have h := le_supportValue_of_cut hDcomp (a := Real.pi / 2) (b := 3 * Real.pi / 2)
+        (hcast _ _ (by ring)) hpLD hpLy
+      linarith only [h]
+  obtain ⟨pr, hprK, hprline, hprcut⟩ := exists_right_tail_contact K.1 K.property.1 hr
+  obtain ⟨pl, hplK, hplline, hplcut⟩ := exists_left_tail_contact K.1 K.property.1 hl
+  have hprB : pr ∈ (canonicalTailSets K).1 := (hBmem pr).mpr ⟨hprK, hprcut⟩
+  have hplD : pl ∈ (canonicalTailSets K).2 := (hDmem pl).mpr ⟨hplK, hplcut⟩
+  have hBeqR : supportValue (canonicalTailSets K).1
+      ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle) =
+      1 - supportValue (K.1.1 : Set Point) ((paperGerverConstants.2.1 : ℝ) : Real.Angle) := by
+    refine le_antisymm (hBbound _ ⟨le_refl _, hr.2.le⟩) ?_
+    have h := le_supportValue_of_cut hBcomp (a := paperGerverConstants.2.1)
+      (b := Real.pi + paperGerverConstants.2.1) (hcast _ _ (by ring)) hprB hprline
+    linarith only [h]
+  have hDeqL : supportValue (canonicalTailSets K).2
+      ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle) =
+      1 - supportValue (K.1.1 : Set Point)
+        ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle) := by
+    refine le_antisymm (hDbound _ ⟨hl.1.le, le_refl _⟩) ?_
+    rw [hcast _ _ (show paperGerverConstants.2.2 + Real.pi / 2 =
+      Real.pi / 2 + paperGerverConstants.2.2 by ring)] at hplline
+    have h := le_supportValue_of_cut hDcomp (a := Real.pi / 2 + paperGerverConstants.2.2)
+      (b := 3 * Real.pi / 2 + paperGerverConstants.2.2) (hcast _ _ (by ring)) hplD hplline
+    linarith only [h]
+  -- the distinguished walls
+  have hwallR : (distinguishedCapSides K.1).1.wall =
+      normalLine ((paperGerverConstants.2.1 : ℝ) : Real.Angle)
+        (supportValue (K.1.1 : Set Point) ((paperGerverConstants.2.1 : ℝ) : Real.Angle) - 1) :=
+    (rotatingHallwayParts_formulas (K.1.1 : Set Point)
+      ((paperGerverConstants.2.1 : ℝ) : Real.Angle)).2.2.2.2.1
+  have hwallL : (distinguishedCapSides K.1).2.wall =
+      normalLine
+        (((paperGerverConstants.2.2 : ℝ) : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle))
+        (supportValue (K.1.1 : Set Point)
+          (((paperGerverConstants.2.2 : ℝ) : Real.Angle) +
+            ((Real.pi / 2 : ℝ) : Real.Angle)) - 1) :=
+    (rotatingHallwayParts_formulas (K.1.1 : Set Point)
+      ((paperGerverConstants.2.2 : ℝ) : Real.Angle)).2.2.2.2.2.2.1
+  have hangL :
+      ((paperGerverConstants.2.2 : ℝ) : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) =
+      ((Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle) := by
+    rw [← Real.Angle.coe_add]
+    exact hcast _ _ (by ring)
+  refine ⟨hBne, hBcomp, hBconv, hBsub, hDne, hDcomp, hDconv, hDsub,
+    ?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩
+  · intro t ht
+    linarith only [hBbound t ht]
+  · intro t ht
+    simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at ht
+    rcases ht with rfl | rfl
+    · linarith only [hBeqR]
+    · rw [hcast _ _ (show Real.pi + Real.pi / 2 = 3 * Real.pi / 2 by ring), hBzero, htop]
+      ring
+  · change normalLine ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+      (supportValue (canonicalTailSets K).1 ((3 * Real.pi / 2 : ℝ) : Real.Angle)) = _
+    rw [hBzero, normalLine_eq_of_cut (a := Real.pi / 2) (b := 3 * Real.pi / 2)
+      (hcast _ _ (by ring)), neg_zero]
+  · change normalLine ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle)
+      (supportValue (canonicalTailSets K).1
+        ((Real.pi + paperGerverConstants.2.1 : ℝ) : Real.Angle)) = _
+    rw [hBeqR, normalLine_eq_of_cut (a := paperGerverConstants.2.1)
+      (b := Real.pi + paperGerverConstants.2.1) (hcast _ _ (by ring)), hwallR]
+    congr 1
+    ring
+  · intro t ht
+    linarith only [hDbound t ht]
+  · intro t ht
+    simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at ht
+    rcases ht with rfl | rfl
+    · rw [hcast _ _ (show 3 * Real.pi / 2 + (0 : ℝ) = 3 * Real.pi / 2 by ring),
+        hcast _ _ (show Real.pi / 2 + (0 : ℝ) = Real.pi / 2 by ring), hDzero, htop]
+      ring
+    · linarith only [hDeqL]
+  · change normalLine ((3 * Real.pi / 2 : ℝ) : Real.Angle)
+      (supportValue (canonicalTailSets K).2 ((3 * Real.pi / 2 : ℝ) : Real.Angle)) = _
+    rw [hDzero, normalLine_eq_of_cut (a := Real.pi / 2) (b := 3 * Real.pi / 2)
+      (hcast _ _ (by ring)), neg_zero]
+  · change normalLine ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle)
+      (supportValue (canonicalTailSets K).2
+        ((3 * Real.pi / 2 + paperGerverConstants.2.2 : ℝ) : Real.Angle)) = _
+    rw [hDeqL, normalLine_eq_of_cut (a := Real.pi / 2 + paperGerverConstants.2.2)
+      (b := 3 * Real.pi / 2 + paperGerverConstants.2.2) (hcast _ _ (by ring)), hwallL, hangL]
+    congr 1
+    ring
+
+/-- The inner corner's normal support coordinate is the cap's support value less one. -/
+theorem inner_capInnerCorner_normalVector (K : RightAngleCapSpace) (t : ℝ) :
+    inner ℝ (capInnerCorner K t) (normalVector (t : Real.Angle)) =
+      supportValue (K.val : Set Point) (t : Real.Angle) - 1 := by
+  have hform := (rotatingHallwayParts_formulas (K.val : Set Point) (t : Real.Angle)).2.1
+  change inner ℝ (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner
+    (normalVector (t : Real.Angle)) = _
+  rw [hform, inner_add_left, real_inner_smul_left, real_inner_smul_left,
+    inner_normalVector_self, inner_tangentVector_normalVector_real]
+  simp
+
+/-- The inner corner's tangent support coordinate is the quarter-turned support value less one. -/
+theorem inner_capInnerCorner_tangentVector (K : RightAngleCapSpace) (t : ℝ) :
+    inner ℝ (capInnerCorner K t) (tangentVector (t : Real.Angle)) =
+      supportValue (K.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+  have hform := (rotatingHallwayParts_formulas (K.val : Set Point) (t : Real.Angle)).2.1
+  change inner ℝ (rotatingHallwayParts (K.val : Set Point) (t : Real.Angle)).innerCorner
+    (tangentVector (t : Real.Angle)) = _
+  rw [hform, inner_add_left, real_inner_smul_left, real_inner_smul_left,
+    inner_tangentVector_tangentVector, inner_normalVector_tangentVector,
+    ← Real.Angle.coe_add]
+  simp
+
+/-- A point whose displacement from the inner corner has negative coordinates in the rotating
+frame at time `t` lies in the open inward quadrant at that time. -/
+theorem mem_innerQuadrant_of_frame_coordinates_neg (K : RightAngleCapSpace) (t : ℝ) (q : Point)
+    (h1 : (q 0 - capInnerCorner K t 0) * Real.cos t +
+      (q 1 - capInnerCorner K t 1) * Real.sin t < 0)
+    (h2 : -((q 0 - capInnerCorner K t 0) * Real.sin t) +
+      (q 1 - capInnerCorner K t 1) * Real.cos t < 0) :
+    q ∈ innerQuadrant (K.val : Set Point) t := by
+  have e1 : inner ℝ q (normalVector (t : Real.Angle)) <
+      supportValue (K.val : Set Point) (t : Real.Angle) - 1 := by
+    rw [← inner_capInnerCorner_normalVector K t, inner_normalVector_real,
+      inner_normalVector_real]
+    linarith only [h1]
+  have e2 : inner ℝ q (tangentVector (t : Real.Angle)) <
+      supportValue (K.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 := by
+    rw [← inner_capInnerCorner_tangentVector K t, inner_tangentVector_real,
+      inner_tangentVector_real]
+    linarith only [h2]
+  refine ⟨e1, ?_⟩
+  change inner ℝ q (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) < _
+  rw [Real.Angle.coe_add, normalVector_add_pi_div_two]
+  exact e2
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Tail / Extension
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The canonical tail sets give an admissible cap-tail triple with exactly these carriers. -/
+theorem exists_canonicalCapTail (K : SpecialCapSpace) :
+    ∃ T : CapTailSpace, T.cap = K ∧
+      (T.rightBody : Set Point) = (canonicalTailSets K).1 ∧
+      (T.leftBody : Set Point) = (canonicalTailSets K).2 := by
+  obtain ⟨hBne, hBcp, hBcv, hBsub, hDne, hDcp, hDcv, hDsub, hBbd, hBeq, -, -,
+    hDbd, hDeq, -, -⟩ := canonicalTailSets_properties K
+  exact ⟨{ cap := K
+           rightBody := ⟨(canonicalTailSets K).1, hBcv, hBcp, hBne⟩
+           leftBody := ⟨(canonicalTailSets K).2, hDcv, hDcp, hDne⟩
+           right_subset := hBsub
+           left_subset := hDsub
+           right_bound := hBbd
+           right_eq := hBeq
+           left_bound := hDbd
+           left_eq := hDeq }, rfl, rfl, rfl⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Tail / Separation
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- A point above both distinguished inner walls of a special cap lies above mid-height. -/
+private theorem lt_coordinate_of_mem_distinguishedCapSides_upperHalfPlanes (K : SpecialCapSpace)
+    {q : Point} (h₁ : q ∈ (distinguishedCapSides K.val).1.upperHalfPlane)
+    (h₂ : q ∈ (distinguishedCapSides K.val).2.upperHalfPlane) :
+    (horizontalMax K.val.val - horizontalMin K.val.val) / 2 < q 1 := by
+  obtain ⟨hrIoo, hlIoo, hrl⟩ := paperGerverConstants_snd_mem_Ioo
+  have hphi : paperGerverConstants.2.1 ≤ (40 : ℝ) / 1000 := by
+    obtain ⟨hall, p, hpbox, hpeq, -⟩ := gerver_parameter_identification
+    exact (hall p hpbox hpeq).2.2.2.2.2.2.2.1
+  have hcosr : (1249 : ℝ) / 1250 ≤ Real.cos paperGerverConstants.2.1 := by
+    have h := Real.one_sub_sq_div_two_le_cos (x := paperGerverConstants.2.1)
+    nlinarith only [h, hphi, hrIoo.1]
+  have hsinle : Real.sin paperGerverConstants.2.1 ≤ paperGerverConstants.2.1 :=
+    Real.sin_le hrIoo.1.le
+  have hcpos : (0 : ℝ) < Real.cos paperGerverConstants.2.1 := by linarith
+  have hsinr : (0 : ℝ) ≤ Real.sin paperGerverConstants.2.1 :=
+    (Real.sin_pos_of_pos_of_lt_pi hrIoo.1 (by linarith [hrIoo.2, Real.pi_pos])).le
+  have hsinl : (0 : ℝ) ≤ Real.sin paperGerverConstants.2.2 :=
+    (Real.sin_pos_of_pos_of_lt_pi hlIoo.1 (by linarith [hlIoo.2, Real.pi_pos])).le
+  have hcosl : (0 : ℝ) ≤ Real.cos paperGerverConstants.2.2 :=
+    (Real.cos_pos_of_mem_Ioo ⟨by linarith [hlIoo.1, Real.pi_pos], hlIoo.2⟩).le
+  have hleq : paperGerverConstants.2.2 = Real.pi / 2 - paperGerverConstants.2.1 := by linarith
+  have hsinleq : Real.sin paperGerverConstants.2.2 = Real.cos paperGerverConstants.2.1 := by
+    rw [hleq, Real.sin_pi_div_two_sub]
+  have hcosleq : Real.cos paperGerverConstants.2.2 = Real.sin paperGerverConstants.2.1 := by
+    rw [hleq, Real.cos_pi_div_two_sub]
+  -- the cap is wide, by rectangle area monotonicity and the special-cap area threshold
+  have hwidth : (11 : ℝ) / 5 ≤
+      supportValue (K.val.val : Set Point) ((0 : ℝ) : Real.Angle) +
+        supportValue (K.val.val : Set Point) ((Real.pi : ℝ) : Real.Angle) := by
+    have h := K.property.2.trans K.val.area_le_horizontalWidth
+    rwa [supportValue_zero_eq_horizontalMax, supportValue_pi_eq_neg_horizontalMin,
+      ← sub_eq_add_neg]
+  -- support lower bounds at the two distinguished angles
+  have hlowR : Real.cos paperGerverConstants.2.1 *
+      supportValue (K.val.val : Set Point) ((0 : ℝ) : Real.Angle) ≤
+      supportValue (K.val.val : Set Point)
+        ((paperGerverConstants.2.1 : ℝ) : Real.Angle) := by
+    have h := (CapSpace.horizontal_le_supportValue K.val hsinr hcpos.le).1
+    rwa [← supportValue_zero_eq_horizontalMax] at h
+  have hlowL : Real.cos paperGerverConstants.2.1 *
+      supportValue (K.val.val : Set Point) ((Real.pi : ℝ) : Real.Angle) ≤
+      supportValue (K.val.val : Set Point)
+        ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) := by
+    have h := (CapSpace.horizontal_le_supportValue K.val hsinl hcosl).2
+    rw [supportValue_pi_eq_neg_horizontalMin, ← hsinleq, mul_neg, ← neg_mul]
+    exact h
+  -- the two defining support inequalities of the closed tail half-planes
+  have h₁' : supportValue (K.val.val : Set Point)
+      ((paperGerverConstants.2.1 : ℝ) : Real.Angle) - 1 ≤
+      inner ℝ q (normalVector ((paperGerverConstants.2.1 : ℝ) : Real.Angle)) := h₁
+  have h₂' : supportValue (K.val.val : Set Point)
+      ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) - 1 ≤
+      inner ℝ q
+        (normalVector ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle)) := h₂
+  rw [inner_normalVector_real] at h₁'
+  rw [inner_normalVector_real, Real.cos_add, Real.sin_add, Real.cos_pi_div_two,
+    Real.sin_pi_div_two, hsinleq, hcosleq] at h₂'
+  -- the strict product bound of the informal proof
+  rw [← supportValue_zero_eq_horizontalMax, sub_eq_add_neg,
+    ← supportValue_pi_eq_neg_horizontalMin]
+  by_contra hcon
+  rw [not_lt] at hcon
+  have hcs : (1199 : ℝ) / 1250 ≤
+      Real.cos paperGerverConstants.2.1 - Real.sin paperGerverConstants.2.1 := by linarith
+  have hprod : (2 : ℝ) < (Real.cos paperGerverConstants.2.1 -
+      Real.sin paperGerverConstants.2.1) *
+      (supportValue (K.val.val : Set Point) ((0 : ℝ) : Real.Angle) +
+        supportValue (K.val.val : Set Point) ((Real.pi : ℝ) : Real.Angle)) := by
+    have h := mul_le_mul hcs hwidth (by norm_num) (by linarith)
+    linarith
+  have hmul : q 1 * Real.sin paperGerverConstants.2.1 ≤
+      (supportValue (K.val.val : Set Point) ((0 : ℝ) : Real.Angle) +
+        supportValue (K.val.val : Set Point) ((Real.pi : ℝ) : Real.Angle)) / 2 *
+        Real.sin paperGerverConstants.2.1 :=
+    mul_le_mul_of_nonneg_right hcon hsinr
+  nlinarith only [h₁', h₂', hlowR, hlowL, hmul, hprod]
+
+theorem cap_and_niche_tail_separation (K : SpecialCapSpace) :
+    ((K.val.val : Set Point) ∩ (distinguishedCapSides K.val).1.upperHalfPlane) ∩
+        (distinguishedCapSides K.val).2.upperHalfPlane = ∅ ∧
+    (capNiche K.val ∩ (distinguishedCapSides K.val).1.upperHalfPlane) ∩
+        (distinguishedCapSides K.val).2.upperHalfPlane = ∅ := by
+  have hwidth : (11 : ℝ) / 5 ≤ horizontalMax K.val.val - horizontalMin K.val.val :=
+    K.property.2.trans K.val.area_le_horizontalWidth
+  constructor
+  · refine Set.eq_empty_iff_forall_notMem.mpr ?_
+    rintro q ⟨⟨hqK, h₁⟩, h₂⟩
+    have hmid := lt_coordinate_of_mem_distinguishedCapSides_upperHalfPlanes K h₁ h₂
+    have hq := (K.val.mem_horizontalStrip hqK).2
+    linarith
+  · refine Set.eq_empty_iff_forall_notMem.mpr ?_
+    rintro q ⟨⟨hqN, h₁⟩, h₂⟩
+    have hmid := lt_coordinate_of_mem_distinguishedCapSides_upperHalfPlanes K h₁ h₂
+    have hq := (capNiche_subset_rectangle K.val hqN).2.2.2
+    linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Area.NicheDecomposition`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Area / Niche Decomposition
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem niche_three_regions_area (K : SpecialCapSpace) :
+    ∀ E₀ Eᵣ Eₗ : Set Point,
+      E₀ = capNiche K.val \ ((distinguishedCapSides K.val).1.upperHalfPlane ∪
+        (distinguishedCapSides K.val).2.upperHalfPlane) →
+      Eᵣ = capNiche K.val ∩ (distinguishedCapSides K.val).1.upperHalfPlane →
+      Eₗ = capNiche K.val ∩ (distinguishedCapSides K.val).2.upperHalfPlane →
+      MeasurableSet E₀ ∧ MeasurableSet Eᵣ ∧ MeasurableSet Eₗ ∧
+      MeasureTheory.volume E₀ < ⊤ ∧ MeasureTheory.volume Eᵣ < ⊤ ∧
+      MeasureTheory.volume Eₗ < ⊤ ∧
+      Disjoint E₀ Eᵣ ∧ Disjoint E₀ Eₗ ∧ Disjoint Eᵣ Eₗ ∧
+      E₀ ∪ Eᵣ ∪ Eₗ = capNiche K.val ∧
+      ClassicalResults.area (capNiche K.val) =
+        ClassicalResults.area E₀ + ClassicalResults.area Eᵣ + ClassicalResults.area Eₗ := by
+  intro E₀ Eᵣ Eₗ h₀ hᵣ hₗ
+  set N := capNiche K.val with hN
+  set HR := (distinguishedCapSides K.val).1.upperHalfPlane with hHR
+  set HL := (distinguishedCapSides K.val).2.upperHalfPlane with hHL
+  subst h₀ hᵣ hₗ
+  -- Borel measurability: the niche is Borel and both tail half-planes are closed.
+  have hNmeas : MeasurableSet N := measurableSet_capNiche K.val
+  have hRmeas : MeasurableSet HR := (isClosed_normalHalfPlane _ _ true).measurableSet
+  have hLmeas : MeasurableSet HL := (isClosed_normalHalfPlane _ _ true).measurableSet
+  have hm₀ : MeasurableSet (N \ (HR ∪ HL)) := hNmeas.diff (hRmeas.union hLmeas)
+  have hmᵣ : MeasurableSet (N ∩ HR) := hNmeas.inter hRmeas
+  have hmₗ : MeasurableSet (N ∩ HL) := hNmeas.inter hLmeas
+  -- Finite area: the uniform niche bounds confine the niche to a bounded rectangle.
+  have hNfin : MeasureTheory.volume N < ⊤ := (niche_uniform_bounds.1 _ K.val).2.2.1
+  have hf₀ : MeasureTheory.volume (N \ (HR ∪ HL)) < ⊤ :=
+    lt_of_le_of_lt (MeasureTheory.measure_mono Set.sdiff_subset) hNfin
+  have hfᵣ : MeasureTheory.volume (N ∩ HR) < ⊤ :=
+    lt_of_le_of_lt (MeasureTheory.measure_mono Set.inter_subset_left) hNfin
+  have hfₗ : MeasureTheory.volume (N ∩ HL) < ⊤ :=
+    lt_of_le_of_lt (MeasureTheory.measure_mono Set.inter_subset_left) hNfin
+  -- Disjointness: the outer region avoids both tails by definition, and the two tails
+  -- meet the niche in disjoint sets by the strengthened separation theorem.
+  have hd₀ᵣ : Disjoint (N \ (HR ∪ HL)) (N ∩ HR) :=
+    Set.disjoint_left.mpr fun _ hx hy ↦ hx.2 (Or.inl hy.2)
+  have hd₀ₗ : Disjoint (N \ (HR ∪ HL)) (N ∩ HL) :=
+    Set.disjoint_left.mpr fun _ hx hy ↦ hx.2 (Or.inr hy.2)
+  have hdᵣₗ : Disjoint (N ∩ HR) (N ∩ HL) :=
+    Set.disjoint_left.mpr fun x hx hy ↦
+      Set.eq_empty_iff_forall_notMem.mp (cap_and_niche_tail_separation K).2 x ⟨hx, hy.2⟩
+  -- The three regions exhaust the niche by cases on half-plane membership.
+  have hunion : (N \ (HR ∪ HL)) ∪ (N ∩ HR) ∪ (N ∩ HL) = N := by
+    rw [Set.union_assoc, ← Set.inter_union_distrib_left, Set.sdiff_union_inter]
+  refine ⟨hm₀, hmᵣ, hmₗ, hf₀, hfᵣ, hfₗ, hd₀ᵣ, hd₀ₗ, hdᵣₗ, hunion, ?_⟩
+  -- Additivity of the extended measures, transferred to real areas by finiteness.
+  have hvol : MeasureTheory.volume N = MeasureTheory.volume (N \ (HR ∪ HL)) +
+      MeasureTheory.volume (N ∩ HR) + MeasureTheory.volume (N ∩ HL) := by
+    rw [← MeasureTheory.measure_union hd₀ᵣ hmᵣ,
+      ← MeasureTheory.measure_union (hd₀ₗ.union_left hdᵣₗ) hmₗ, hunion]
+  simp only [ClassicalResults.area]
+  rw [hvol, ENNReal.toReal_add (ENNReal.add_ne_top.mpr ⟨hf₀.ne, hfᵣ.ne⟩) hfₗ.ne,
+    ENNReal.toReal_add hf₀.ne hfᵣ.ne]
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Bounds.MonotonicityIntervals`.
+* `Bounds.WedgeEndpoints`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-! # Monotonicity intervals for the distinguished cap sides
+
+`cap_tail_monotonicity_intervals` records that on the right interval
+`(φ, π/2]` the inner corner has left the distinguished right half-plane, and
+that inside that half-plane the inner quadrant is cut out by the single inner
+wall `b(t)`; symmetrically on `[0, π/2 - φ)` for the left side.  Intersecting
+with the fan gives the two wedge forms.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+private theorem capInnerCorner_right_projection_neg (K : SpecialCapSpace) :
+    ∀ t ∈ Set.Ioc GerversSofa.φ (Real.pi / 2),
+     inner ℝ (capInnerCorner K.val t - capInnerCorner K.val GerversSofa.φ)
+       (normalVector ((GerversSofa.φ : ℝ) : Real.Angle)) < 0 := by
+  have hpi := Real.pi_pos
+  have hφ0 : (0 : ℝ) ≤ GerversSofa.φ := GerversSofa.ABφθSpec.existsUnique.choose_spec.1.1
+  have hφ4 : GerversSofa.φ ≤ Real.pi / 4 :=
+    le_trans GerversSofa.ABφθSpec.existsUnique.choose_spec.1.2.1
+      GerversSofa.ABφθSpec.existsUnique.choose_spec.1.2.2.1
+  -- the injectivity condition: a continuously differentiable corner with strict interior signs
+  obtain ⟨-, hcd, hsign⟩ := K.2.1
+  set D : ℝ → Point := derivWithin (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2)) with hDdef
+  have hdw : ∀ s ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      HasDerivWithinAt (capInnerCorner K.val) (D s) (Set.Icc (0 : ℝ) (Real.pi / 2)) s :=
+    fun s hs => (hcd.differentiableOn_one s hs).hasDerivWithinAt
+  have hcont : ContinuousOn (capInnerCorner K.val) (Set.Icc (0 : ℝ) (Real.pi / 2)) :=
+    hcd.continuousOn
+  have hsub : Set.Icc GerversSofa.φ (Real.pi / 2) ⊆ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+    Set.Icc_subset_Icc hφ0 le_rfl
+  have hanti : StrictAntiOn
+      (fun s : ℝ ↦ inner ℝ (capInnerCorner K.val s)
+        (normalVector ((GerversSofa.φ : ℝ) : Real.Angle)))
+      (Set.Icc GerversSofa.φ (Real.pi / 2)) := by
+    refine strictAntiOn_of_hasDerivWithinAt_neg (convex_Icc _ _)
+      ((hcont.mono hsub).inner continuousOn_const)
+      (f' := fun s ↦ inner ℝ (D s) (normalVector ((GerversSofa.φ : ℝ) : Real.Angle)))
+      ?_ ?_
+    · intro s hs
+      rw [interior_Icc] at hs ⊢
+      have hsub' : Set.Ioo GerversSofa.φ (Real.pi / 2) ⊆ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+        fun z hz => ⟨hφ0.trans hz.1.le, hz.2.le⟩
+      simpa using
+        ((hdw s (hsub' hs)).mono hsub').inner ℝ
+          (hasDerivWithinAt_const s _
+            (normalVector ((GerversSofa.φ : ℝ) : Real.Angle)))
+    · intro s hs
+      rw [interior_Icc] at hs
+      obtain ⟨hs1, hs2⟩ := hs
+      obtain ⟨hα, hβ⟩ := hsign s ⟨hφ0.trans_lt hs1, hs2⟩
+      rw [inner_normalVector_eq_frame_rotate (D s) s GerversSofa.φ]
+      have hcos : 0 < Real.cos (s - GerversSofa.φ) :=
+        Real.cos_pos_of_mem_Ioo ⟨by linarith, by linarith⟩
+      have hsin : 0 < Real.sin (s - GerversSofa.φ) :=
+        Real.sin_pos_of_pos_of_lt_pi (by linarith) (by linarith)
+      nlinarith
+  intro t ht
+  have h := hanti ⟨le_rfl, le_trans hφ4 (by linarith)⟩ ⟨ht.1.le, ht.2⟩ ht.1
+  rw [inner_sub_left]
+  simpa using h
+
+private theorem capInnerCorner_left_projection_neg (K : SpecialCapSpace) :
+    ∀ t ∈ Set.Ico (0 : ℝ) (Real.pi / 2 - GerversSofa.φ),
+     inner ℝ (capInnerCorner K.val t -
+         capInnerCorner K.val (Real.pi / 2 - GerversSofa.φ))
+       (tangentVector ((Real.pi / 2 - GerversSofa.φ : ℝ) : Real.Angle)) < 0 := by
+  have hpi := Real.pi_pos
+  have hφ0 : (0 : ℝ) ≤ GerversSofa.φ := GerversSofa.ABφθSpec.existsUnique.choose_spec.1.1
+  have hφ4 : GerversSofa.φ ≤ Real.pi / 4 :=
+    le_trans GerversSofa.ABφθSpec.existsUnique.choose_spec.1.2.1
+      GerversSofa.ABφθSpec.existsUnique.choose_spec.1.2.2.1
+  -- the injectivity condition: a continuously differentiable corner with strict interior signs
+  obtain ⟨-, hcd, hsign⟩ := K.2.1
+  set D : ℝ → Point := derivWithin (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2)) with hDdef
+  have hdw : ∀ s ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      HasDerivWithinAt (capInnerCorner K.val) (D s) (Set.Icc (0 : ℝ) (Real.pi / 2)) s :=
+    fun s hs => (hcd.differentiableOn_one s hs).hasDerivWithinAt
+  have hcont : ContinuousOn (capInnerCorner K.val) (Set.Icc (0 : ℝ) (Real.pi / 2)) :=
+    hcd.continuousOn
+  have hsub : Set.Icc (0 : ℝ) (Real.pi / 2 - GerversSofa.φ) ⊆
+      Set.Icc (0 : ℝ) (Real.pi / 2) := Set.Icc_subset_Icc le_rfl (by linarith)
+  have hmono : StrictMonoOn
+      (fun s : ℝ ↦ inner ℝ (capInnerCorner K.val s)
+        (tangentVector ((Real.pi / 2 - GerversSofa.φ : ℝ) : Real.Angle)))
+      (Set.Icc (0 : ℝ) (Real.pi / 2 - GerversSofa.φ)) := by
+    refine strictMonoOn_of_hasDerivWithinAt_pos (convex_Icc _ _)
+      ((hcont.mono hsub).inner continuousOn_const)
+      (f' := fun s ↦ inner ℝ (D s)
+        (tangentVector ((Real.pi / 2 - GerversSofa.φ : ℝ) : Real.Angle)))
+      ?_ ?_
+    · intro s hs
+      rw [interior_Icc] at hs ⊢
+      have hsub' : Set.Ioo (0 : ℝ) (Real.pi / 2 - GerversSofa.φ) ⊆
+          Set.Icc (0 : ℝ) (Real.pi / 2) := fun z hz => ⟨hz.1.le, by linarith [hz.2]⟩
+      simpa using
+        ((hdw s (hsub' hs)).mono hsub').inner ℝ
+          (hasDerivWithinAt_const s _
+            (tangentVector ((Real.pi / 2 - GerversSofa.φ : ℝ) : Real.Angle)))
+    · intro s hs
+      rw [interior_Icc] at hs
+      obtain ⟨hs1, hs2⟩ := hs
+      obtain ⟨hα, hβ⟩ := hsign s ⟨hs1, by linarith⟩
+      rw [inner_tangentVector_eq_frame_rotate (D s) s (Real.pi / 2 - GerversSofa.φ)]
+      have hcos : 0 < Real.cos (s - (Real.pi / 2 - GerversSofa.φ)) :=
+        Real.cos_pos_of_mem_Ioo ⟨by linarith, by linarith⟩
+      have hsin : Real.sin (s - (Real.pi / 2 - GerversSofa.φ)) < 0 :=
+        Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) (by linarith)
+      nlinarith
+  intro t ht
+  have h := hmono ⟨ht.1, ht.2.le⟩ ⟨by linarith, le_rfl⟩ ht.2
+  rw [inner_sub_left]
+  simpa using h
+
+theorem cap_tail_monotonicity_intervals (K : SpecialCapSpace) :
+    (∀ t ∈ Set.Ioc paperGerverConstants.2.1 (Real.pi / 2),
+      capInnerCorner K.val t ∉ (distinguishedCapSides K.val).1.upperHalfPlane ∧
+      (distinguishedCapSides K.val).1.upperHalfPlane ∩ innerQuadrant K.val.val t =
+        (distinguishedCapSides K.val).1.upperHalfPlane \
+          (innerWallUpperHalfPlanes K.val t).1) ∧
+    (∀ t ∈ Set.Ico (0 : ℝ) paperGerverConstants.2.2,
+      capInnerCorner K.val t ∉ (distinguishedCapSides K.val).2.upperHalfPlane ∧
+      (distinguishedCapSides K.val).2.upperHalfPlane ∩ innerQuadrant K.val.val t =
+        (distinguishedCapSides K.val).2.upperHalfPlane \
+          (innerWallUpperHalfPlanes K.val t).2) ∧
+    (∀ t ∈ Set.Ioc paperGerverConstants.2.1 (Real.pi / 2),
+      t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) →
+      (distinguishedCapSides K.val).1.upperHalfPlane ∩ capWedge K.val t =
+        ((distinguishedCapSides K.val).1.upperHalfPlane ∩ {p : Point | 0 ≤ p 1}) \
+          (innerWallUpperHalfPlanes K.val t).1) ∧
+    (∀ t ∈ Set.Ico (0 : ℝ) paperGerverConstants.2.2,
+      t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2) →
+      (distinguishedCapSides K.val).2.upperHalfPlane ∩ capWedge K.val t =
+        ((distinguishedCapSides K.val).2.upperHalfPlane ∩ {p : Point | 0 ≤ p 1}) \
+          (innerWallUpperHalfPlanes K.val t).2) := by
+  have hpi := Real.pi_pos
+  have hφ0 : (0 : ℝ) ≤ GerversSofa.φ := GerversSofa.ABφθSpec.existsUnique.choose_spec.1.1
+  have hφ4 : GerversSofa.φ ≤ Real.pi / 4 :=
+    le_trans GerversSofa.ABφθSpec.existsUnique.choose_spec.1.2.1
+      GerversSofa.ABφθSpec.existsUnique.choose_spec.1.2.2.1
+  -- frame coordinates of the inner corner path
+  have hxu := inner_capInnerCorner_normalVector K.val
+  have hxv := inner_capInnerCorner_tangentVector K.val
+  -- membership descriptions relative to the moving corner
+  have hHb : ∀ (t : ℝ) (p : Point), p ∈ (innerWallUpperHalfPlanes K.val t).1 ↔
+      0 ≤ inner ℝ (p - capInnerCorner K.val t) (normalVector (t : Real.Angle)) := by
+    intro t p
+    rw [inner_sub_left, hxu]
+    change (supportValue (K.val.val : Set Point) (t : Real.Angle) - 1 ≤
+      inner ℝ p (normalVector (t : Real.Angle))) ↔ _
+    constructor <;> intro hp <;> linarith
+  have hHd : ∀ (t : ℝ) (p : Point), p ∈ (innerWallUpperHalfPlanes K.val t).2 ↔
+      0 ≤ inner ℝ (p - capInnerCorner K.val t) (tangentVector (t : Real.Angle)) := by
+    intro t p
+    rw [inner_sub_left, hxv]
+    change (supportValue (K.val.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1 ≤
+      inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle))) ↔ _
+    rw [Real.Angle.coe_add, normalVector_add_pi_div_two]
+    constructor <;> intro hp <;> linarith
+  have hQ : ∀ (t : ℝ) (p : Point), p ∈ innerQuadrant (K.val.val : Set Point) t ↔
+      inner ℝ (p - capInnerCorner K.val t) (normalVector (t : Real.Angle)) < 0 ∧
+        inner ℝ (p - capInnerCorner K.val t) (tangentVector (t : Real.Angle)) < 0 := by
+    intro t p
+    rw [inner_sub_left, inner_sub_left, hxu, hxv]
+    change (inner ℝ p (normalVector (t : Real.Angle)) <
+        supportValue (K.val.val : Set Point) (t : Real.Angle) - 1 ∧
+      inner ℝ p (normalVector ((t + Real.pi / 2 : ℝ) : Real.Angle)) <
+        supportValue (K.val.val : Set Point) ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) ↔ _
+    rw [Real.Angle.coe_add, normalVector_add_pi_div_two]
+    constructor <;> rintro ⟨h1, h2⟩ <;> exact ⟨by linarith, by linarith⟩
+  have hfan : capFan (Real.pi / 2) = {p : Point | 0 ≤ p 1} := by
+    ext p
+    have hval : inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = p 1 := by
+      simp [normalVector, frame, PiLp.inner_apply, Fin.sum_univ_two]
+    constructor
+    · intro hp
+      have := hp.2
+      change 0 ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at this
+      rw [hval] at this
+      exact this
+    · intro hp
+      have hp' : (0 : ℝ) ≤ p 1 := hp
+      exact ⟨by change 0 ≤ inner ℝ p _; rw [hval]; exact hp',
+        by change 0 ≤ inner ℝ p _; rw [hval]; exact hp'⟩
+  have hwedge : ∀ t : ℝ, capWedge K.val t =
+      {p : Point | 0 ≤ p 1} ∩ innerQuadrant (K.val.val : Set Point) t := by
+    intro t
+    have hiq : (rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).innerQuadrant =
+        innerQuadrant (K.val.val : Set Point) t := by
+      rw [(rotatingHallwayParts_formulas (K.val.val : Set Point)
+        (t : Real.Angle)).2.2.2.2.2.2.2.2, innerQuadrant, ← Real.Angle.coe_add]
+    change capFan (Real.pi / 2) ∩
+      (rotatingHallwayParts (K.val.val : Set Point) (t : Real.Angle)).innerQuadrant = _
+    rw [hfan, hiq]
+  -- monotone comparison of the corner path against the two distinguished frames
+  have hkeyR := capInnerCorner_right_projection_neg K
+  have hkeyL := capInnerCorner_left_projection_neg K
+  -- the right-hand quadrant identity, at all quadrant times
+  have partR : ∀ t ∈ Set.Ioc GerversSofa.φ (Real.pi / 2),
+      capInnerCorner K.val t ∉ (distinguishedCapSides K.val).1.upperHalfPlane ∧
+      (distinguishedCapSides K.val).1.upperHalfPlane ∩ innerQuadrant K.val.val t =
+        (distinguishedCapSides K.val).1.upperHalfPlane \
+          (innerWallUpperHalfPlanes K.val t).1 := by
+    intro t ht
+    have hR := hkeyR t ht
+    have hside : (distinguishedCapSides K.val).1.upperHalfPlane =
+        (innerWallUpperHalfPlanes K.val GerversSofa.φ).1 := rfl
+    have hcos : 0 ≤ Real.cos (t - GerversSofa.φ) :=
+      Real.cos_nonneg_of_mem_Icc ⟨by linarith [ht.1], by linarith [ht.2]⟩
+    have hsin : 0 ≤ Real.sin (t - GerversSofa.φ) :=
+      Real.sin_nonneg_of_nonneg_of_le_pi (by linarith [ht.1]) (by linarith [ht.2])
+    rw [hside]
+    constructor
+    · intro hmem
+      exact absurd ((hHb _ _).mp hmem) (not_le.mpr hR)
+    refine Set.Subset.antisymm ?_ ?_
+    · rintro p ⟨hA, hQp⟩
+      refine ⟨hA, ?_⟩
+      rw [hHb t p]
+      exact not_le.mpr ((hQ t p).mp hQp).1
+    · rintro p ⟨hA, hH⟩
+      refine ⟨hA, ?_⟩
+      rw [hQ t p]
+      have ha : inner ℝ (p - capInnerCorner K.val t) (normalVector (t : Real.Angle)) < 0 :=
+        not_le.mp fun h => hH ((hHb t p).mpr h)
+      refine ⟨ha, ?_⟩
+      by_contra hb
+      rw [not_lt] at hb
+      have hAmem : 0 ≤ inner ℝ (p - capInnerCorner K.val GerversSofa.φ)
+          (normalVector ((GerversSofa.φ : ℝ) : Real.Angle)) := (hHb _ p).mp hA
+      have hsplit : inner ℝ (p - capInnerCorner K.val GerversSofa.φ)
+            (normalVector ((GerversSofa.φ : ℝ) : Real.Angle)) =
+          inner ℝ (p - capInnerCorner K.val t)
+              (normalVector ((GerversSofa.φ : ℝ) : Real.Angle)) +
+            inner ℝ (capInnerCorner K.val t - capInnerCorner K.val GerversSofa.φ)
+              (normalVector ((GerversSofa.φ : ℝ) : Real.Angle)) := by
+        rw [← inner_add_left]
+        congr 1
+        abel
+      have hle : inner ℝ (p - capInnerCorner K.val t)
+          (normalVector ((GerversSofa.φ : ℝ) : Real.Angle)) ≤ 0 := by
+        rw [inner_normalVector_eq_frame_rotate (p - capInnerCorner K.val t) t GerversSofa.φ]
+        linarith [mul_nonneg hb hsin, mul_nonneg (neg_nonneg.mpr ha.le) hcos]
+      linarith
+  -- the left-hand quadrant identity, at all quadrant times
+  have partL : ∀ t ∈ Set.Ico (0 : ℝ) (Real.pi / 2 - GerversSofa.φ),
+      capInnerCorner K.val t ∉ (distinguishedCapSides K.val).2.upperHalfPlane ∧
+      (distinguishedCapSides K.val).2.upperHalfPlane ∩ innerQuadrant K.val.val t =
+        (distinguishedCapSides K.val).2.upperHalfPlane \
+          (innerWallUpperHalfPlanes K.val t).2 := by
+    intro t ht
+    have hL := hkeyL t ht
+    have hside : (distinguishedCapSides K.val).2.upperHalfPlane =
+        (innerWallUpperHalfPlanes K.val (Real.pi / 2 - GerversSofa.φ)).2 := rfl
+    have hcos : 0 ≤ Real.cos (t - (Real.pi / 2 - GerversSofa.φ)) :=
+      Real.cos_nonneg_of_mem_Icc ⟨by linarith [ht.1], by linarith [ht.2]⟩
+    have hsin : Real.sin (t - (Real.pi / 2 - GerversSofa.φ)) ≤ 0 := by
+      have hpos : 0 ≤ Real.sin (Real.pi / 2 - GerversSofa.φ - t) :=
+        Real.sin_nonneg_of_nonneg_of_le_pi (by linarith [ht.2]) (by linarith [ht.1])
+      rw [show t - (Real.pi / 2 - GerversSofa.φ) =
+        -(Real.pi / 2 - GerversSofa.φ - t) by ring, Real.sin_neg]
+      linarith
+    rw [hside]
+    constructor
+    · intro hmem
+      exact absurd ((hHd _ _).mp hmem) (not_le.mpr hL)
+    refine Set.Subset.antisymm ?_ ?_
+    · rintro p ⟨hA, hQp⟩
+      refine ⟨hA, ?_⟩
+      rw [hHd t p]
+      exact not_le.mpr ((hQ t p).mp hQp).2
+    · rintro p ⟨hA, hH⟩
+      refine ⟨hA, ?_⟩
+      rw [hQ t p]
+      have hb : inner ℝ (p - capInnerCorner K.val t) (tangentVector (t : Real.Angle)) < 0 :=
+        not_le.mp fun h => hH ((hHd t p).mpr h)
+      refine ⟨?_, hb⟩
+      by_contra ha
+      rw [not_lt] at ha
+      have hAmem : 0 ≤ inner ℝ
+          (p - capInnerCorner K.val (Real.pi / 2 - GerversSofa.φ))
+          (tangentVector ((Real.pi / 2 - GerversSofa.φ : ℝ) : Real.Angle)) := (hHd _ p).mp hA
+      have hsplit : inner ℝ (p - capInnerCorner K.val (Real.pi / 2 - GerversSofa.φ))
+            (tangentVector ((Real.pi / 2 - GerversSofa.φ : ℝ) : Real.Angle)) =
+          inner ℝ (p - capInnerCorner K.val t)
+              (tangentVector ((Real.pi / 2 - GerversSofa.φ : ℝ) : Real.Angle)) +
+            inner ℝ (capInnerCorner K.val t -
+                capInnerCorner K.val (Real.pi / 2 - GerversSofa.φ))
+              (tangentVector ((Real.pi / 2 - GerversSofa.φ : ℝ) : Real.Angle)) := by
+        rw [← inner_add_left]
+        congr 1
+        abel
+      have hle : inner ℝ (p - capInnerCorner K.val t)
+          (tangentVector ((Real.pi / 2 - GerversSofa.φ : ℝ) : Real.Angle)) ≤ 0 := by
+        rw [inner_tangentVector_eq_frame_rotate (p - capInnerCorner K.val t) t
+          (Real.pi / 2 - GerversSofa.φ)]
+        linarith [mul_nonneg ha (neg_nonneg.mpr hsin),
+          mul_nonneg (neg_nonneg.mpr hb.le) hcos]
+      linarith
+  -- the wedge identities are the quadrant identities intersected with the fan
+  refine ⟨partR, partL, ?_, ?_⟩
+  · intro t ht _
+    rw [hwedge t, Set.inter_left_comm, (partR t ht).2]
+    ext p
+    simp only [Set.mem_inter_iff, Set.mem_sdiff, Set.mem_ofPred_eq]
+    tauto
+  · intro t ht _
+    rw [hwedge t, Set.inter_left_comm, (partL t ht).2]
+    ext p
+    simp only [Set.mem_inter_iff, Set.mem_sdiff, Set.mem_ofPred_eq]
+    tauto
+
+/-- On any subinterval of `[0, π/2]` the inner corner of a special cap has strictly decreasing
+horizontal coordinate: the injectivity condition makes its horizontal derivative negative. -/
+theorem strictAntiOn_capInnerCorner_fst (K : SpecialCapSpace) {a b : ℝ}
+    (ha : 0 ≤ a) (hb : b ≤ Real.pi / 2) :
+    StrictAntiOn (fun t ↦ capInnerCorner K.val t 0) (Set.Icc a b) := by
+  obtain ⟨-, hcd, hsign⟩ := K.property.1
+  set Dv : ℝ → Point := derivWithin (capInnerCorner K.val) (Set.Icc 0 (Real.pi / 2))
+  have hIccsub : Set.Icc a b ⊆ Set.Icc (0 : ℝ) (Real.pi / 2) := fun t ht ↦
+    ⟨le_trans ha ht.1, le_trans ht.2 hb⟩
+  have hIoosub : Set.Ioo a b ⊆ Set.Icc (0 : ℝ) (Real.pi / 2) := fun t ht ↦
+    ⟨le_trans ha ht.1.le, le_trans ht.2.le hb⟩
+  have hdw : ∀ s ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      HasDerivWithinAt (capInnerCorner K.val) (Dv s) (Set.Icc (0 : ℝ) (Real.pi / 2)) s :=
+    fun s hs ↦ (hcd.differentiableOn_one s hs).hasDerivWithinAt
+  have hcont : ContinuousOn (capInnerCorner K.val) (Set.Icc (0 : ℝ) (Real.pi / 2)) :=
+    hcd.continuousOn
+  have hinner0 : ∀ p : Point, inner ℝ p (normalVector ((0 : ℝ) : Real.Angle)) = p 0 := by
+    intro p
+    rw [inner_normalVector_real]
+    simp
+  have hmono0 : StrictAntiOn
+      (fun t ↦ inner ℝ (capInnerCorner K.val t) (normalVector ((0 : ℝ) : Real.Angle)))
+      (Set.Icc a b) := by
+    refine strictAntiOn_of_hasDerivWithinAt_neg (convex_Icc _ _)
+      ((hcont.mono hIccsub).inner continuousOn_const)
+      (f' := fun s ↦ inner ℝ (Dv s) (normalVector ((0 : ℝ) : Real.Angle))) ?_ ?_
+    · intro s hs
+      rw [interior_Icc] at hs ⊢
+      simpa using ((hdw s (hIoosub hs)).mono hIoosub).inner ℝ
+        (hasDerivWithinAt_const s _ (normalVector ((0 : ℝ) : Real.Angle)))
+    · intro s hs
+      rw [interior_Icc] at hs
+      obtain ⟨hα, hβ⟩ := hsign s ⟨lt_of_le_of_lt ha hs.1, lt_of_lt_of_le hs.2 hb⟩
+      rw [inner_normalVector_eq_frame_rotate (Dv s) s 0]
+      have hcos : 0 < Real.cos (s - 0) := by
+        rw [sub_zero]
+        exact Real.cos_pos_of_mem_Ioo ⟨by linarith only [Real.pi_pos, hs.1, ha],
+          by linarith only [hs.2, hb]⟩
+      have hsin : 0 < Real.sin (s - 0) := by
+        rw [sub_zero]
+        exact Real.sin_pos_of_pos_of_lt_pi (lt_of_le_of_lt ha hs.1)
+          (by linarith only [hs.2, hb, Real.pi_pos])
+      have h1 : inner ℝ (Dv s) (normalVector (s : Real.Angle)) * Real.cos (s - 0) < 0 :=
+        mul_neg_of_neg_of_pos hα hcos
+      have h2 : 0 < inner ℝ (Dv s) (tangentVector (s : Real.Angle)) * Real.sin (s - 0) :=
+        mul_pos hβ hsin
+      linarith only [h1, h2]
+  intro u hu w hw huw
+  simpa only [hinner0] using hmono0 hu hw huw
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Bounds / Wedge Endpoints
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem wedgeGaps_positive_lower_bound {ω : ℝ} (K : CapSpace ω)
+    (t : ℝ) (ht : t ∈ Set.Ioo 0 ω) :
+    (1 - Real.sin t) / Real.cos t ≤ (wedgeGaps K t).1 ∧
+    0 < (1 - Real.sin t) / Real.cos t ∧
+    (1 - Real.sin (ω - t)) / Real.cos (ω - t) ≤ (wedgeGaps K t).2 ∧
+    0 < (1 - Real.sin (ω - t)) / Real.cos (ω - t) := by
+  have hcost : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, ht.1], ht.2.trans_le K.property.2.1⟩
+  have hsint_lt : Real.sin t < 1 := by
+    nlinarith only [Real.sin_sq_add_cos_sq t, sq_pos_of_pos hcost]
+  have hcosδ : 0 < Real.cos (ω - t) := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [sub_pos.mpr ht.2, Real.pi_pos],
+      by linarith [ht.1, K.property.2.1]⟩
+  have hsinδ_lt : Real.sin (ω - t) < 1 := by
+    nlinarith only [Real.sin_sq_add_cos_sq (ω - t), sq_pos_of_pos hcosδ]
+  have hbounds := K.supportValue_upper_bounds ht
+  have hC := (edgeVertices_fst_mem K.val
+    ((ω + Real.pi / 2 : ℝ) : Real.Angle)).2
+  change inner ℝ (capVertices K ω).2.1
+      (normalVector ((ω + Real.pi / 2 : ℝ) : Real.Angle)) =
+    supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) at hC
+  simp only [normalVector, frame, Real.Angle.coe_add, Real.Angle.cos_add_pi_div_two,
+    Real.Angle.sin_coe, Real.Angle.sin_add_pi_div_two, Real.Angle.cos_coe] at hC
+  change inner ℝ (capVertices K ω).2.1 (tangentVector (ω : Real.Angle)) =
+    supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) at hC
+  have hgap₁ := wedgeGaps_fst_eq_supportValue K t
+  have hgap₂ : (wedgeGaps K t).2 =
+      supportValue K.val ((ω + Real.pi / 2 : ℝ) : Real.Angle) -
+        (supportValue K.val ((t + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+          Real.cos (ω - t) := by
+    simp only [wedgeGaps, wedgeEndpoints, inner_sub_left,
+      real_inner_smul_left, hC]
+    rw [inner_tangentVector_self]
+    ring
+  rw [hgap₁, hgap₂]
+  refine ⟨?_, div_pos (sub_pos.mpr hsint_lt) hcost, ?_,
+    div_pos (sub_pos.mpr hsinδ_lt) hcosδ⟩
+  · apply (div_le_iff₀ hcost).2
+    rw [sub_mul, div_mul_cancel₀ _ hcost.ne']
+    nlinarith [hbounds.1]
+  · apply (div_le_iff₀ hcosδ).2
+    rw [sub_mul, div_mul_cancel₀ _ hcosδ.ne']
+    nlinarith [hbounds.2]
+
+/-- Base points between the horizontal extrema of a right-angle cap lie in the cap. -/
+private theorem smul_normalVector_zero_mem_of_rightAngleCap (K : RightAngleCapSpace) {x : ℝ}
+    (hlo : -supportValue (K.val : Set Point) ((Real.pi : ℝ) : Real.Angle) ≤ x)
+    (hhi : x ≤ supportValue (K.val : Set Point) ((0 : ℝ) : Real.Angle)) :
+    x • normalVector (0 : Real.Angle) ∈ (K.val : Set Point) := by
+  obtain ⟨A, hA, hAeq⟩ := exists_mem_inner_eq_supportValue K.val ((0 : ℝ) : Real.Angle)
+  obtain ⟨C, hC, hCeq⟩ := exists_mem_inner_eq_supportValue K.val ((Real.pi : ℝ) : Real.Angle)
+  rw [inner_normalVector_zero] at hAeq
+  rw [inner_normalVector_real, Real.cos_pi, Real.sin_pi] at hCeq
+  have hzero : inner ℝ (x • normalVector (0 : Real.Angle))
+      (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) = 0 := by
+    rw [inner_smul_normalVector_zero, Real.cos_pi_div_two, mul_zero]
+  refine K.mem_of_mem_capFan_of_le_supportValue ⟨le_of_eq hzero.symm, le_of_eq hzero.symm⟩ ?_
+  intro t ht
+  have htI : 0 ≤ t ∧ t ≤ Real.pi := by
+    rcases ht with h | h
+    · exact ⟨h.1, by linarith only [h.2, Real.pi_pos]⟩
+    · exact ⟨by linarith only [h.1, Real.pi_pos], by linarith only [h.2]⟩
+  have hsin : 0 ≤ Real.sin t := Real.sin_nonneg_of_nonneg_of_le_pi htI.1 htI.2
+  rw [inner_smul_normalVector_zero]
+  rcases le_or_gt 0 (Real.cos t) with hcos | hcos
+  · have h := inner_le_supportValue K.val hA (t : Real.Angle)
+    rw [inner_normalVector_real] at h
+    have hy := (K.mem_horizontalStrip hA).1
+    nlinarith only [h, hAeq, hhi, hcos, mul_nonneg hsin hy]
+  · have h := inner_le_supportValue K.val hC (t : Real.Angle)
+    rw [inner_normalVector_real] at h
+    have hy := (K.mem_horizontalStrip hC).1
+    nlinarith only [h, hCeq, hlo, hcos, mul_nonneg hsin hy]
+
+/-- The bottom right cap vertex is the horizontal maximum on the base line. -/
+private theorem capVertices_zero_snd_eq (K : RightAngleCapSpace) :
+    (capVertices K 0).1.2 =
+      supportValue (K.val : Set Point) ((0 : ℝ) : Real.Angle) •
+        normalVector (0 : Real.Angle) := by
+  set R := supportValue (K.val : Set Point) ((0 : ℝ) : Real.Angle) with hRdef
+  have hS : -supportValue (K.val : Set Point) ((Real.pi : ℝ) : Real.Angle) ≤ R := by
+    obtain ⟨p, hp⟩ := K.val.nonempty
+    have h0 := inner_le_supportValue K.val hp ((0 : ℝ) : Real.Angle)
+    have hpi := inner_le_supportValue K.val hp ((Real.pi : ℝ) : Real.Angle)
+    rw [inner_normalVector_zero] at h0
+    rw [inner_normalVector_real, Real.cos_pi, Real.sin_pi] at hpi
+    rw [hRdef]
+    linarith
+  have hmem : R • normalVector (0 : Real.Angle) ∈ (K.val : Set Point) :=
+    smul_normalVector_zero_mem_of_rightAngleCap K hS le_rfl
+  have hedge : R • normalVector (0 : Real.Angle) ∈
+      exposedEdge K.val ((0 : ℝ) : Real.Angle) := by
+    refine ⟨hmem, ?_⟩
+    change inner ℝ (R • normalVector (0 : Real.Angle))
+      (normalVector ((0 : ℝ) : Real.Angle)) = _
+    rw [inner_smul_normalVector_zero, Real.cos_zero, mul_one]
+  change (edgeVertices K.val ((0 : ℝ) : Real.Angle)).2 = _
+  refine edgeVertices_snd_eq_of_tangent_isLeast K.val ((0 : ℝ) : Real.Angle) hedge ?_
+  intro q hq
+  have hq1 : 0 ≤ q 1 := (K.mem_horizontalStrip hq.1).1
+  have hp1 : inner ℝ (R • normalVector (0 : Real.Angle))
+      (tangentVector ((0 : ℝ) : Real.Angle)) = 0 := by
+    rw [real_inner_smul_left, show (0 : Real.Angle) = ((0 : ℝ) : Real.Angle) from rfl,
+      inner_normalVector_tangentVector, mul_zero]
+  rw [hp1, inner_tangentVector_zero]
+  exact hq1
+
+/-- The bottom left cap vertex is the horizontal minimum on the base line. -/
+private theorem capVertices_pi_div_two_fst_eq (K : RightAngleCapSpace) :
+    (capVertices K (Real.pi / 2)).2.1 =
+      (-supportValue (K.val : Set Point) ((Real.pi : ℝ) : Real.Angle)) •
+        normalVector (0 : Real.Angle) := by
+  set S := supportValue (K.val : Set Point) ((Real.pi : ℝ) : Real.Angle) with hSdef
+  have hR : -S ≤ supportValue (K.val : Set Point) ((0 : ℝ) : Real.Angle) := by
+    obtain ⟨p, hp⟩ := K.val.nonempty
+    have h0 := inner_le_supportValue K.val hp ((0 : ℝ) : Real.Angle)
+    have hpi := inner_le_supportValue K.val hp ((Real.pi : ℝ) : Real.Angle)
+    rw [inner_normalVector_zero] at h0
+    rw [inner_normalVector_real, Real.cos_pi, Real.sin_pi] at hpi
+    rw [hSdef]
+    linarith
+  have hmem : (-S) • normalVector (0 : Real.Angle) ∈ (K.val : Set Point) :=
+    smul_normalVector_zero_mem_of_rightAngleCap K le_rfl hR
+  have hang : ((Real.pi / 2 + Real.pi / 2 : ℝ) : Real.Angle) =
+      ((Real.pi : ℝ) : Real.Angle) := by
+    congr 1
+    ring
+  have hedge : (-S) • normalVector (0 : Real.Angle) ∈
+      exposedEdge K.val ((Real.pi : ℝ) : Real.Angle) := by
+    refine ⟨hmem, ?_⟩
+    change inner ℝ ((-S) • normalVector (0 : Real.Angle))
+      (normalVector ((Real.pi : ℝ) : Real.Angle)) = _
+    rw [inner_smul_normalVector_zero, Real.cos_pi]
+    ring
+  change (edgeVertices K.val ((Real.pi / 2 + Real.pi / 2 : ℝ) : Real.Angle)).1 = _
+  rw [hang]
+  refine edgeVertices_fst_eq_of_tangent_isGreatest K.val ((Real.pi : ℝ) : Real.Angle) hedge ?_
+  intro q hq
+  have hq1 : 0 ≤ q 1 := (K.mem_horizontalStrip hq.1).1
+  have hp1 : inner ℝ ((-S) • normalVector (0 : Real.Angle))
+      (tangentVector ((Real.pi : ℝ) : Real.Angle)) = 0 := by
+    rw [real_inner_smul_left, inner_tangentVector_pi]
+    simp [normalVector, frame]
+  rw [hp1, inner_tangentVector_pi]
+  linarith
+
+/-- Horizontal coordinates strictly inside the cap width give interior bottom-edge points. -/
+private theorem smul_normalVector_zero_mem_bottomEdge (K : RightAngleCapSpace) {x : ℝ}
+    (hlo : -supportValue (K.val : Set Point) ((Real.pi : ℝ) : Real.Angle) < x)
+    (hhi : x < supportValue (K.val : Set Point) ((0 : ℝ) : Real.Angle)) :
+    x • normalVector (0 : Real.Angle) ∈
+      exposedEdge K.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) \
+        {(capVertices K 0).1.2, (capVertices K (Real.pi / 2)).2.1} := by
+  have hinj : ∀ a b : ℝ, a • normalVector (0 : Real.Angle) =
+      b • normalVector (0 : Real.Angle) → a = b := by
+    intro a b h
+    have h' := congrArg (fun p : Point ↦ inner ℝ p (normalVector ((0 : ℝ) : Real.Angle))) h
+    simpa only [inner_smul_normalVector_zero, Real.cos_zero, mul_one] using h'
+  have hcos : Real.cos (3 * Real.pi / 2) = 0 := by
+    rw [show (3 * Real.pi / 2 : ℝ) = Real.pi + Real.pi / 2 by ring, Real.cos_add,
+      Real.cos_pi_div_two, Real.sin_pi_div_two, Real.cos_pi, Real.sin_pi]
+    ring
+  refine ⟨⟨smul_normalVector_zero_mem_of_rightAngleCap K hlo.le hhi.le, ?_⟩, ?_⟩
+  · change inner ℝ (x • normalVector (0 : Real.Angle))
+      (normalVector ((3 * Real.pi / 2 : ℝ) : Real.Angle)) = _
+    rw [inner_smul_normalVector_zero, hcos, mul_zero, K.property.2.2.2.2.2.1]
+  · rintro (h | h)
+    · rw [capVertices_zero_snd_eq K] at h
+      exact absurd (hinj _ _ h) (ne_of_lt hhi)
+    · rw [capVertices_pi_div_two_fst_eq K] at h
+      exact absurd (hinj _ _ h) (ne_of_gt hlo)
+
+theorem specialCap_wedgeEndpoints_in_bottomEdge (K : SpecialCapSpace) :
+    (distinguishedCapSides K.val).1.fanPoint ∈
+      exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) \
+        {(capVertices K.val 0).1.2, (capVertices K.val (Real.pi / 2)).2.1} ∧
+    (distinguishedCapSides K.val).2.fanPoint ∈
+      exposedEdge K.val.val ((3 * Real.pi / 2 : ℝ) : Real.Angle) \
+        {(capVertices K.val 0).1.2, (capVertices K.val (Real.pi / 2)).2.1} := by
+  obtain ⟨hrIoo, hlIoo, hrl⟩ := paperGerverConstants_snd_mem_Ioo
+  -- the certified parameter bound `0 < varphi ≤ 1/25` and its trigonometric consequences
+  have hphi : paperGerverConstants.2.1 ≤ (40 : ℝ) / 1000 := by
+    obtain ⟨hall, q, hqbox, hqeq, -⟩ := gerver_parameter_identification
+    exact (hall q hqbox hqeq).2.2.2.2.2.2.2.1
+  have hcosr : (1249 : ℝ) / 1250 ≤ Real.cos paperGerverConstants.2.1 := by
+    have h := Real.one_sub_sq_div_two_le_cos (x := paperGerverConstants.2.1)
+    nlinarith only [h, hphi, hrIoo.1]
+  have hcpos : (0 : ℝ) < Real.cos paperGerverConstants.2.1 := by linarith
+  have hsinr : (0 : ℝ) ≤ Real.sin paperGerverConstants.2.1 :=
+    (Real.sin_pos_of_pos_of_lt_pi hrIoo.1 (by linarith [hrIoo.2, Real.pi_pos])).le
+  have hsinl : (0 : ℝ) ≤ Real.sin paperGerverConstants.2.2 :=
+    (Real.sin_pos_of_pos_of_lt_pi hlIoo.1 (by linarith [hlIoo.2, Real.pi_pos])).le
+  have hcosl : (0 : ℝ) ≤ Real.cos paperGerverConstants.2.2 :=
+    (Real.cos_pos_of_mem_Ioo ⟨by linarith [hlIoo.1, Real.pi_pos], hlIoo.2⟩).le
+  have hcosdiff : Real.pi / 2 - paperGerverConstants.2.2 = paperGerverConstants.2.1 := by
+    linarith
+  have hsinleq : Real.sin paperGerverConstants.2.2 = Real.cos paperGerverConstants.2.1 := by
+    rw [show paperGerverConstants.2.2 = Real.pi / 2 - paperGerverConstants.2.1 by linarith,
+      Real.sin_pi_div_two_sub]
+  have hinvc : (1 : ℝ) / Real.cos paperGerverConstants.2.1 ≤ 1250 / 1249 := by
+    rw [div_le_div_iff₀ hcpos (by norm_num)]
+    linarith
+  -- the cap is wide, by rectangle area monotonicity and the special-cap area threshold
+  have hwidth : (11 : ℝ) / 5 ≤
+      supportValue (K.val.val : Set Point) ((0 : ℝ) : Real.Angle) +
+        supportValue (K.val.val : Set Point) ((Real.pi : ℝ) : Real.Angle) := by
+    have h := K.property.2.trans K.val.area_le_horizontalWidth
+    rwa [supportValue_zero_eq_horizontalMax, supportValue_pi_eq_neg_horizontalMin,
+      ← sub_eq_add_neg]
+  -- support lower bounds at the two distinguished angles
+  have hlowR : Real.cos paperGerverConstants.2.1 *
+      supportValue (K.val.val : Set Point) ((0 : ℝ) : Real.Angle) ≤
+      supportValue (K.val.val : Set Point)
+        ((paperGerverConstants.2.1 : ℝ) : Real.Angle) := by
+    have h := (CapSpace.horizontal_le_supportValue K.val hsinr hcpos.le).1
+    rwa [← supportValue_zero_eq_horizontalMax] at h
+  have hlowL : Real.cos paperGerverConstants.2.1 *
+      supportValue (K.val.val : Set Point) ((Real.pi : ℝ) : Real.Angle) ≤
+      supportValue (K.val.val : Set Point)
+        ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) := by
+    have h := (CapSpace.horizontal_le_supportValue K.val hsinl hcosl).2
+    rw [supportValue_pi_eq_neg_horizontalMin, ← hsinleq, mul_neg, ← neg_mul]
+    exact h
+  -- the two positive wedge gaps
+  have hgap₁ := wedgeGaps_positive_lower_bound K.val paperGerverConstants.2.1 hrIoo
+  have hgap₂ := wedgeGaps_positive_lower_bound K.val paperGerverConstants.2.2 hlIoo
+  rw [wedgeGaps_fst_eq_supportValue] at hgap₁
+  rw [wedgeGaps_snd_eq_supportValue,
+    show ((Real.pi / 2 + Real.pi / 2 : ℝ) : Real.Angle) = ((Real.pi : ℝ) : Real.Angle) by
+      congr 1; ring, hcosdiff] at hgap₂
+  have hgap₁' : (1 - Real.sin paperGerverConstants.2.1) / Real.cos paperGerverConstants.2.1 ≤
+      supportValue (K.val.val : Set Point) ((0 : ℝ) : Real.Angle) -
+        (supportValue (K.val.val : Set Point)
+          ((paperGerverConstants.2.1 : ℝ) : Real.Angle) - 1) /
+          Real.cos paperGerverConstants.2.1 := hgap₁.1
+  have hgap₂' : (1 - Real.sin paperGerverConstants.2.1) / Real.cos paperGerverConstants.2.1 ≤
+      supportValue (K.val.val : Set Point) ((Real.pi : ℝ) : Real.Angle) -
+        (supportValue (K.val.val : Set Point)
+          ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+          Real.cos paperGerverConstants.2.1 := hgap₂.2.2.1
+  -- the reciprocal-cosine lower bounds for the two inner-wall coordinates
+  have hRc : supportValue (K.val.val : Set Point) ((0 : ℝ) : Real.Angle) ≤
+      supportValue (K.val.val : Set Point) ((paperGerverConstants.2.1 : ℝ) : Real.Angle) /
+        Real.cos paperGerverConstants.2.1 := by
+    rw [le_div_iff₀ hcpos]
+    linarith
+  have hSc : supportValue (K.val.val : Set Point) ((Real.pi : ℝ) : Real.Angle) ≤
+      supportValue (K.val.val : Set Point)
+        ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) /
+        Real.cos paperGerverConstants.2.1 := by
+    rw [le_div_iff₀ hcpos]
+    linarith
+  have hRc' : supportValue (K.val.val : Set Point) ((0 : ℝ) : Real.Angle) -
+      1 / Real.cos paperGerverConstants.2.1 ≤
+      (supportValue (K.val.val : Set Point)
+        ((paperGerverConstants.2.1 : ℝ) : Real.Angle) - 1) /
+        Real.cos paperGerverConstants.2.1 := by
+    rw [sub_div]
+    linarith
+  have hSc' : supportValue (K.val.val : Set Point) ((Real.pi : ℝ) : Real.Angle) -
+      1 / Real.cos paperGerverConstants.2.1 ≤
+      (supportValue (K.val.val : Set Point)
+        ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+        Real.cos paperGerverConstants.2.1 := by
+    rw [sub_div]
+    linarith
+  -- the two fan points as multiples of the horizontal normal
+  have htv : tangentVector ((Real.pi / 2 : ℝ) : Real.Angle) = -normalVector (0 : Real.Angle) := by
+    have h := tangentVector_add_pi_div_two 0
+    rwa [zero_add, Real.Angle.coe_zero] at h
+  constructor
+  · change ((supportValue (K.val.val : Set Point)
+        ((paperGerverConstants.2.1 : ℝ) : Real.Angle) - 1) /
+        Real.cos paperGerverConstants.2.1) • normalVector (0 : Real.Angle) ∈ _
+    refine smul_normalVector_zero_mem_bottomEdge K.val ?_ ?_
+    · linarith
+    · have hpos : 0 < (1 - Real.sin paperGerverConstants.2.1) /
+          Real.cos paperGerverConstants.2.1 := hgap₁.2.1
+      linarith
+  · change ((supportValue (K.val.val : Set Point)
+        ((paperGerverConstants.2.2 + Real.pi / 2 : ℝ) : Real.Angle) - 1) /
+        Real.cos (Real.pi / 2 - paperGerverConstants.2.2)) •
+        tangentVector ((Real.pi / 2 : ℝ) : Real.Angle) ∈ _
+    rw [hcosdiff, htv, smul_neg, ← neg_smul]
+    refine smul_normalVector_zero_mem_bottomEdge K.val ?_ ?_
+    · have hpos : 0 < (1 - Real.sin paperGerverConstants.2.1) /
+          Real.cos paperGerverConstants.2.1 := hgap₂.2.2.2
+      linarith
+    · linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Bounds.Upper.Q`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Bounds / Upper / Q
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+/-- The cap-tail upper-bound functional Q assembled from cap area, tail arcs and endpoint
+segments. -/
+def upperBoundQ (T : CapTailSpace) : ℝ :=
+  ClassicalResults.area (T.cap.val.val : Set Point) +
+    convexArcArea T.leftBody (3 * Real.pi / 2)
+      (3 * Real.pi / 2 + paperGerverConstants.2.2) +
+    segmentArea (rightLeftTailArcs T.rightBody T.leftBody).2.endPoint
+      (distinguishedCapSides T.cap.val).2.corner -
+    curveAreaFunctional (capMiddleBV T.cap) +
+    segmentArea (distinguishedCapSides T.cap.val).1.corner
+      (rightLeftTailArcs T.rightBody T.leftBody).1.startPoint +
+    convexArcArea T.rightBody (Real.pi + paperGerverConstants.2.1) (3 * Real.pi / 2)
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Motion.Monotonization`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Motion / Monotonization
+-/
+
+@[expose] public section
+
+noncomputable section
+open Set
+open scoped unitInterval
+namespace MovingSofa
+
+private theorem isClosed_supportingHallway (s : Set Point) (t : Real.Angle) :
+    IsClosed (supportingHallway s t) := by
+  have hclosed : IsClosed hallway := by
+    rw [hallway_eq_outerQuadrant_sdiff_innerQuadrant]
+    apply IsClosed.sdiff
+    · change IsClosed {p : Point | p 0 ≤ 1 ∧ p 1 ≤ 1}
+      exact (isClosed_le (by fun_prop : Continuous (fun p : Point ↦ p 0)) continuous_const).inter
+        (isClosed_le (by fun_prop : Continuous (fun p : Point ↦ p 1)) continuous_const)
+    · change IsOpen {p : Point | p 0 < 0 ∧ p 1 < 0}
+      exact (isOpen_lt (by fun_prop : Continuous (fun p : Point ↦ p 0)) continuous_const).inter
+        (isOpen_lt (by fun_prop : Continuous (fun p : Point ↦ p 1)) continuous_const)
+  have heq : supportingPlacement s t = supportingPlacementEquiv s t := by
+    funext p
+    exact (supportingPlacementEquiv_apply s t p).symm
+  rw [supportingHallway, heq]
+  exact (supportingPlacementEquiv s t).toHomeomorph.isClosedMap _ hclosed
+
+private theorem isClosed_monotonization (s : Set Point) (ω : ℝ) :
+    IsClosed (monotonization s ω) := by
+  have hP : (stripParallelogram ω).1 =
+      {p : Point | p 1 ∈ Icc (0 : ℝ) 1 ∧
+        inner ℝ p (normalVector (ω : Real.Angle)) ∈ Icc (0 : ℝ) 1} := by
+    ext p
+    exact mem_stripParallelogram_iff ω p
+  have hclosedP : IsClosed (stripParallelogram ω).1 := by
+    rw [hP]
+    exact (isClosed_Icc.preimage (by fun_prop : Continuous (fun p : Point ↦ p 1))).inter
+      (isClosed_Icc.preimage (continuous_id.inner continuous_const))
+  exact hclosedP.inter (isClosed_iInter fun t ↦
+    isClosed_iInter fun _ ↦ isClosed_supportingHallway s (t : Real.Angle))
+
+private theorem monotonization_subset_capOfSofa (s : Set Point) (ω : ℝ) :
+    monotonization s ω ⊆ capOfSofa s ω := by
+  intro p hp
+  change p ∈ (stripParallelogram ω).1 ∩
+    ⋂ t ∈ Icc 0 ω, supportingHallway s (t : Real.Angle) at hp
+  refine ⟨hp.1, ?_⟩
+  simp only [mem_inter_iff, mem_iInter] at hp ⊢
+  intro t ht
+  exact supportingHallway_subset_outerQuadrant s (t : Real.Angle) (hp.2 t ht)
+
+private theorem isCompact_monotonization (s : Set Point) (ω : ℝ)
+    (hs : IsStandardPosition s ω) : IsCompact (monotonization s ω) := by
+  obtain ⟨K, hK⟩ := standardPosition_cap s ω hs
+  have hc : IsCompact (capOfSofa s ω) := hK ▸ K.val.isCompact
+  exact hc.of_isClosed_subset (isClosed_monotonization s ω)
+    (monotonization_subset_capOfSofa s ω)
+
+private theorem hasRotationAngle_monotonization (s : Set Point) (ω : ℝ)
+    (hs : IsStandardPosition s ω) : HasRotationAngle (monotonization s ω) ω := by
+  have hsne : s.Nonempty := by
+    obtain ⟨m, hm, _⟩ := hs.2.1
+    exact hm.1.nonempty
+  exact hasRotationAngle_of_subset_supportingHallways s (monotonization s ω) ω
+    hsne hs.1 hs.2.2.1.le hs.2.2.2.2.1 hs.2.2.2.2.2
+    (standardPosition_monotonization_connected s ω hs)
+    (isClosed_monotonization s ω) Set.Subset.rfl
+
+theorem standardPosition_monotonization_standard (s : Set Point) (ω : ℝ)
+    (hs : IsStandardPosition s ω) :
+    IsStandardPosition (monotonization s ω) ω ∧ s ⊆ monotonization s ω := by
+  refine ⟨⟨isCompact_monotonization s ω hs, hasRotationAngle_monotonization s ω hs,
+    hs.2.2.1, hs.2.2.2.1, ?_, ?_⟩, standardPosition_subset_monotonization s ω hs⟩
+  · exact ((standardPosition_support_eq s ω hs).1 ω
+      (Or.inl ⟨hs.2.2.1.le, le_rfl⟩)).1.trans hs.2.2.2.2.1
+  · exact ((standardPosition_support_eq s ω hs).1 (Real.pi / 2)
+      (Or.inr ⟨le_rfl, by linarith [hs.2.2.1]⟩)).1.trans hs.2.2.2.2.2
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Sofa.Area`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Sofa / Area
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace MovingSofa
+
+theorem monotoneSofa_structure (s : Set Point) (ω : ℝ)
+    (hs : ∃ s₀ : Set Point, IsStandardPosition s₀ ω ∧ s = monotonization s₀ ω)
+    (K : CapSpace ω) (hK : (K.val : Set Point) = capOfSofa s ω) :
+    s = (K.val : Set Point) \ capNiche K := by
+  obtain ⟨s₀, hs₀, rfl⟩ := hs
+  have heq : capOfSofa (monotonization s₀ ω) ω = capOfSofa s₀ ω := by
+    have hparts (t : ℝ) (ht : t ∈ Set.Icc 0 ω) :
+        (rotatingHallwayParts (monotonization s₀ ω) (t : Real.Angle)).outerQuadrant =
+          (rotatingHallwayParts s₀ (t : Real.Angle)).outerQuadrant := by
+      have h₀ := ((standardPosition_support_eq s₀ ω hs₀).1 t (Or.inl ht)).1
+      have h₁ := ((standardPosition_support_eq s₀ ω hs₀).1 (t + Real.pi / 2)
+        (Or.inr ⟨by linarith [ht.1], by linarith [ht.2]⟩)).1
+      simp only [Real.Angle.coe_add] at h₁
+      change supportingPlacement (monotonization s₀ ω) (t : Real.Angle) ''
+        hallwayParts.outerQuadrant = supportingPlacement s₀ (t : Real.Angle) ''
+          hallwayParts.outerQuadrant
+      have he : supportingPlacement (monotonization s₀ ω) (t : Real.Angle) =
+          supportingPlacement s₀ (t : Real.Angle) := by
+        funext p
+        simp only [supportingPlacement, h₀, h₁]
+      rw [he]
+    unfold capOfSofa
+    congr 1
+    apply Set.iInter_congr
+    intro t
+    apply Set.iInter_congr
+    exact hparts t
+  exact standardPosition_monotonization s₀ ω hs₀ K (hK.trans heq)
+
+theorem capAreaFunctional_eq_sofaArea (s : Set Point) (ω : ℝ)
+    (hs : ∃ s₀ : Set Point, IsStandardPosition s₀ ω ∧ s = monotonization s₀ ω)
+    (K : CapSpace ω) (hK : (K.val : Set Point) = capOfSofa s ω) :
+    capAreaFunctional K = ClassicalResults.area s := by
+  have heq := monotoneSofa_structure s ω hs K hK
+  have hconn : IsConnected s := by
+    obtain ⟨s₀, hs₀, rfl⟩ := hs
+    exact standardPosition_monotonization_connected s₀ ω hs₀
+  have hsub : capNiche K ⊆ (K.val : Set Point) :=
+    (cap_niche_connected_iff K).1.mpr
+      ((cap_niche_connected_iff K).2.1.mpr
+        ((cap_niche_connected_iff K).2.2.mpr (heq ▸ hconn)))
+  rw [capAreaFunctional, heq]
+  change MeasureTheory.volume.real (K.val : Set Point) -
+    MeasureTheory.volume.real (capNiche K) =
+      MeasureTheory.volume.real ((K.val : Set Point) \ capNiche K)
+  have h := MeasureTheory.measureReal_inter_add_sdiff
+    (μ := MeasureTheory.volume) (s := (K.val : Set Point))
+    (measurableSet_capNiche K) K.val.isCompact.measure_lt_top.ne
+  rw [Set.inter_eq_right.mpr hsub] at h
+  linarith
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/Development/Geometry/Foundations/Development008.lean b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development008.lean
new file mode 100644
index 0000000000..c4cd73f620
--- /dev/null
+++ b/LeanPool/MovingSofa/Development/Geometry/Foundations/Development008.lean
@@ -0,0 +1,347 @@
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+module
+
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development006
+public import LeanPool.MovingSofa.Development.Geometry.Foundations.Development007
+/-!
+# Moving sofa: related mathematical developments
+
+* `Cap.Foundations.Development008`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+/-!
+# Moving sofa: related mathematical developments
+
+* `Cap.Monotone`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2026 Dean Cureton. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton
+-/
+/-!
+# Cap / Monotone
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace MovingSofa
+
+/-- Removing the niche from a cap leaves a compact set. -/
+lemma isCompact_sdiff_capNiche {ω : ℝ} (K : CapSpace ω) :
+    IsCompact ((K.val : Set Point) \ capNiche K) := by
+  have hopen : IsOpen (⋃ t ∈ Set.Ioo 0 ω, innerQuadrant (K.val : Set Point) t) := by
+    apply isOpen_iUnion
+    intro t
+    apply isOpen_iUnion
+    intro _
+    exact (isOpen_lt (by fun_prop) continuous_const).inter
+      (isOpen_lt (by fun_prop) continuous_const)
+  have heq : (K.val : Set Point) \ capNiche K =
+      (K.val : Set Point) \ ⋃ t ∈ Set.Ioo 0 ω, innerQuadrant (K.val : Set Point) t := by
+    ext p
+    simp only [Set.mem_sdiff, capNiche, Set.mem_inter_iff]
+    constructor
+    · rintro ⟨hp, hn⟩
+      exact ⟨hp, fun h ↦ hn ⟨K.subset_capFan hp, h⟩⟩
+    · rintro ⟨hp, hn⟩
+      exact ⟨hp, fun h ↦ hn h.2⟩
+  rw [heq]
+  exact K.val.isCompact.diff hopen
+
+private theorem supportValue_sdiff_capNiche_eq {ω t : ℝ} (K : CapSpace ω)
+    (hN : capNiche K ⊆ (K.val : Set Point))
+    (ht : t ∈ Set.Icc 0 (ω + Real.pi / 2)) :
+    supportValue ((K.val : Set Point) \ capNiche K) (t : Real.Angle) =
+      supportValue K.val (t : Real.Angle) := by
+  let S : Set Point := (K.val : Set Point) \ capNiche K
+  have hNupper : capNiche K ⊆ (K.val : Set Point) \ capUpperBoundary K :=
+    (cap_niche_connected_iff K).1.mp hN
+  have hScompact : IsCompact S := isCompact_sdiff_capNiche K
+  have hSsub : S ⊆ (K.val : Set Point) := Set.sdiff_subset
+  have hSnonempty : S.Nonempty := by
+    have hcorners := (cap_niche_connected_iff K).2.1.mp hNupper
+    exact ((cap_niche_connected_iff K).2.2.mp hcorners).nonempty
+  obtain ⟨p, hp⟩ := exposedEdge_nonempty K.val (t : Real.Angle)
+  have hpupper : p ∈ capUpperBoundary K := by
+    rw [capUpperBoundary]
+    exact Set.mem_iUnion₂.mpr ⟨t, ht, hp⟩
+  have hpS : p ∈ S := by
+    refine ⟨hp.1, fun hpN ↦ ?_⟩
+    exact (hNupper hpN).2 hpupper
+  symm
+  apply supportValue_eq_of_subset_of_inner_le hSnonempty hSsub
+  intro q hq
+  calc
+    inner ℝ q (normalVector (t : Real.Angle)) ≤ supportValue K.val (t : Real.Angle) :=
+      inner_le_supportValue K.val hq _
+    _ = inner ℝ p (normalVector (t : Real.Angle)) := hp.2.symm
+    _ ≤ supportValue S (t : Real.Angle) :=
+      inner_le_supportValue_of_isCompact hScompact hpS _
+
+private theorem mem_supportingHallway_of_mem_sdiff_capNiche {ω t : ℝ}
+    (K : CapSpace ω) (ht : t ∈ Set.Icc 0 ω) {p : Point}
+    (hp : p ∈ (K.val : Set Point) \ capNiche K) :
+    p ∈ supportingHallway (K.val : Set Point) (t : Real.Angle) := by
+  obtain ⟨hhall, _, _, _, _, _, _, houter, hinner⟩ :=
+    rotatingHallwayParts_formulas (K.val : Set Point) (t : Real.Angle)
+  rw [hhall]
+  refine ⟨?_, ?_⟩
+  · rw [houter]
+    exact ⟨inner_le_supportValue K.val hp.1 _, inner_le_supportValue K.val hp.1 _⟩
+  · intro hq
+    rw [hinner] at hq
+    change p ∈ innerQuadrant (K.val : Set Point) t at hq
+    by_cases ht0 : t = 0
+    · subst t
+      have h := hq.2
+      change inner ℝ p (normalVector ((0 + Real.pi / 2 : ℝ) : Real.Angle)) <
+        supportValue K.val ((0 + Real.pi / 2 : ℝ) : Real.Angle) - 1 at h
+      rw [zero_add, K.property.2.2.2.1] at h
+      have hf := (K.subset_capFan hp.1).2
+      change 0 ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle)) at hf
+      linarith
+    · by_cases htω : t = ω
+      · subst t
+        have h := hq.1
+        change inner ℝ p (normalVector (ω : Real.Angle)) <
+          supportValue K.val (ω : Real.Angle) - 1 at h
+        rw [K.property.2.2.1] at h
+        have hf := (K.subset_capFan hp.1).1
+        change 0 ≤ inner ℝ p (normalVector (ω : Real.Angle)) at hf
+        linarith
+      · apply hp.2
+        exact ⟨K.subset_capFan hp.1, Set.mem_iUnion₂.mpr
+          ⟨t, ⟨lt_of_le_of_ne ht.1 (Ne.symm ht0), lt_of_le_of_ne ht.2 htω⟩, hq⟩⟩
+
+private theorem cap_sdiff_niche_eq_monotonization_self {ω : ℝ} (K : CapSpace ω) :
+    (K.val : Set Point) \ capNiche K = monotonization (K.val : Set Point) ω := by
+  ext p
+  constructor
+  · intro hp
+    refine ⟨K.subset_stripParallelogram hp.1,
+      Set.mem_iInter.2 fun t ↦ Set.mem_iInter.2 fun ht ↦ ?_⟩
+    exact mem_supportingHallway_of_mem_sdiff_capNiche K ht hp
+  · intro hp
+    change p ∈ (stripParallelogram ω).1 ∩
+      ⋂ t ∈ Set.Icc 0 ω, supportingHallway (K.val : Set Point) (t : Real.Angle) at hp
+    have hpstrip := (mem_stripParallelogram_iff ω p).mp hp.1
+    have hpFan : p ∈ capFan ω := by
+      change 0 ≤ inner ℝ p (normalVector (ω : Real.Angle)) ∧
+        0 ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+      refine ⟨hpstrip.2.1, ?_⟩
+      simpa [normalVector, frame, PiLp.inner_apply] using hpstrip.1.1
+    have hpK : p ∈ (K.val : Set Point) := by
+      apply K.mem_of_mem_capFan_of_le_supportValue hpFan
+      intro u hu
+      rcases hu with hu | hu
+      · have hhall := Set.mem_iInter.1 (Set.mem_iInter.1 hp.2 u) hu
+        obtain ⟨hhallEq, _, _, _, _, _, _, houter, _⟩ :=
+          rotatingHallwayParts_formulas (K.val : Set Point) (u : Real.Angle)
+        rw [hhallEq, houter] at hhall
+        exact hhall.1.1
+      · let t := u - Real.pi / 2
+        have ht : t ∈ Set.Icc 0 ω := by
+          dsimp [t]
+          constructor <;> linarith [hu.1, hu.2]
+        have hhall := Set.mem_iInter.1 (Set.mem_iInter.1 hp.2 t) ht
+        obtain ⟨hhallEq, _, _, _, _, _, _, houter, _⟩ :=
+          rotatingHallwayParts_formulas (K.val : Set Point) (t : Real.Angle)
+        rw [hhallEq, houter] at hhall
+        have heq : (t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) =
+            (u : Real.Angle) := by
+          rw [← Real.Angle.coe_add]
+          congr 1
+          dsimp [t]
+          ring
+        have hq := hhall.1.2
+        change inner ℝ p
+          (normalVector ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle))) ≤
+            supportValue K.val
+              ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle)) at hq
+        rw [heq] at hq
+        exact hq
+    refine ⟨hpK, ?_⟩
+    rintro ⟨_, hN⟩
+    obtain ⟨t, ht, hquad⟩ := Set.mem_iUnion₂.mp hN
+    have hhall := Set.mem_iInter.1 (Set.mem_iInter.1 hp.2 t) ⟨ht.1.le, ht.2.le⟩
+    obtain ⟨hhallEq, _, _, _, _, _, _, _, hinner⟩ :=
+      rotatingHallwayParts_formulas (K.val : Set Point) (t : Real.Angle)
+    rw [hhallEq] at hhall
+    apply hhall.2
+    rw [hinner]
+    exact hquad
+
+private theorem cap_eq_capOfSofa_sdiff_capNiche {ω : ℝ} (K : CapSpace ω)
+    (hN : capNiche K ⊆ (K.val : Set Point)) :
+    (K.val : Set Point) = capOfSofa ((K.val : Set Point) \ capNiche K) ω := by
+  let S : Set Point := (K.val : Set Point) \ capNiche K
+  have hsupp (t : ℝ) (ht : t ∈ Set.Icc 0 (ω + Real.pi / 2)) :
+      supportValue S (t : Real.Angle) = supportValue K.val (t : Real.Angle) :=
+    supportValue_sdiff_capNiche_eq K hN ht
+  ext p
+  constructor
+  · intro hp
+    refine ⟨K.subset_stripParallelogram hp,
+      Set.mem_iInter.2 fun t ↦ Set.mem_iInter.2 fun ht ↦ ?_⟩
+    obtain ⟨_, _, _, _, _, _, _, houter, _⟩ :=
+      rotatingHallwayParts_formulas S (t : Real.Angle)
+    rw [houter]
+    change inner ℝ p (normalVector (t : Real.Angle)) ≤ supportValue S (t : Real.Angle) ∧
+      inner ℝ p
+          (normalVector ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle))) ≤
+        supportValue S ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle))
+    have h₀ := hsupp t ⟨ht.1, ht.2.trans (le_add_of_nonneg_right (by positivity))⟩
+    have h₁ := hsupp (t + Real.pi / 2)
+      ⟨by linarith [ht.1, Real.pi_pos], by linarith [ht.2]⟩
+    simp only [Real.Angle.coe_add] at h₁
+    rw [h₀, h₁]
+    exact ⟨inner_le_supportValue K.val hp _, inner_le_supportValue K.val hp _⟩
+  · intro hp
+    have hpstrip := (mem_stripParallelogram_iff ω p).mp hp.1
+    have hpFan : p ∈ capFan ω := by
+      change 0 ≤ inner ℝ p (normalVector (ω : Real.Angle)) ∧
+        0 ≤ inner ℝ p (normalVector ((Real.pi / 2 : ℝ) : Real.Angle))
+      refine ⟨hpstrip.2.1, ?_⟩
+      simpa [normalVector, frame, PiLp.inner_apply] using hpstrip.1.1
+    apply K.mem_of_mem_capFan_of_le_supportValue hpFan
+    intro u hu
+    rcases hu with hu | hu
+    · have hq := Set.mem_iInter.1 (Set.mem_iInter.1 hp.2 u) hu
+      obtain ⟨_, _, _, _, _, _, _, houter, _⟩ :=
+        rotatingHallwayParts_formulas S (u : Real.Angle)
+      rw [houter] at hq
+      have heq := hsupp u
+        ⟨hu.1, hu.2.trans (le_add_of_nonneg_right (by positivity))⟩
+      have hq' := hq.1
+      change inner ℝ p (normalVector (u : Real.Angle)) ≤
+        supportValue S (u : Real.Angle) at hq'
+      rw [heq] at hq'
+      exact hq'
+    · let t := u - Real.pi / 2
+      have ht : t ∈ Set.Icc 0 ω := by
+        dsimp [t]
+        constructor <;> linarith [hu.1, hu.2]
+      have hq := Set.mem_iInter.1 (Set.mem_iInter.1 hp.2 t) ht
+      obtain ⟨_, _, _, _, _, _, _, houter, _⟩ :=
+        rotatingHallwayParts_formulas S (t : Real.Angle)
+      rw [houter] at hq
+      have hang : (t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle) =
+          (u : Real.Angle) := by
+        rw [← Real.Angle.coe_add]
+        congr 1
+        dsimp [t]
+        ring
+      have heq := hsupp u
+        ⟨(by positivity : 0 ≤ Real.pi / 2).trans hu.1,
+          hu.2⟩
+      have hq' := hq.2
+      change inner ℝ p
+        (normalVector ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle))) ≤
+          supportValue S
+            ((t : Real.Angle) + ((Real.pi / 2 : ℝ) : Real.Angle)) at hq'
+      rw [hang, heq] at hq'
+      exact hq'
+
+theorem cap_isMonotoneCap_iff_niche_subset {ω : ℝ} (K : CapSpace ω) :
+    (∃ s₀ : Set Point, IsStandardPosition s₀ ω ∧
+      (K.val : Set Point) = capOfSofa (monotonization s₀ ω) ω) ↔
+      capNiche K ⊆ (K.val : Set Point) := by
+  constructor
+  · rintro ⟨s₀, hs₀, hK⟩
+    let S := monotonization s₀ ω
+    have hstructure : S = (K.val : Set Point) \ capNiche K :=
+      monotoneSofa_structure S ω ⟨s₀, hs₀, rfl⟩ K hK
+    have hconn : IsConnected S := standardPosition_monotonization_connected s₀ ω hs₀
+    exact (cap_niche_connected_iff K).1.mpr
+      ((cap_niche_connected_iff K).2.1.mpr
+        ((cap_niche_connected_iff K).2.2.mpr (hstructure ▸ hconn)))
+  · intro hN
+    let S : Set Point := (K.val : Set Point) \ capNiche K
+    have hNupper : capNiche K ⊆ (K.val : Set Point) \ capUpperBoundary K :=
+      (cap_niche_connected_iff K).1.mp hN
+    have hcorners := (cap_niche_connected_iff K).2.1.mp hNupper
+    have hSconn : IsConnected S := (cap_niche_connected_iff K).2.2.mp hcorners
+    have hScompact : IsCompact S := isCompact_sdiff_capNiche K
+    have hSsub : S ⊆ monotonization (K.val : Set Point) ω := by
+      rw [← cap_sdiff_niche_eq_monotonization_self K]
+    have hrot : HasRotationAngle S ω :=
+      hasRotationAngle_of_subset_supportingHallways (K.val : Set Point) S ω
+        K.val.nonempty K.val.isCompact K.property.1.le K.property.2.2.1
+        K.property.2.2.2.1 hSconn hScompact.isClosed hSsub
+    have hsuppω : supportValue S (ω : Real.Angle) = 1 := by
+      rw [supportValue_sdiff_capNiche_eq K hN
+        ⟨K.property.1.le, le_add_of_nonneg_right (by positivity)⟩,
+        K.property.2.2.1]
+    have hsuppπ : supportValue S ((Real.pi / 2 : ℝ) : Real.Angle) = 1 := by
+      rw [supportValue_sdiff_capNiche_eq K hN
+        ⟨by positivity, by linarith [K.property.1]⟩,
+        K.property.2.2.2.1]
+    have hstandard : IsStandardPosition S ω :=
+      ⟨hScompact, hrot, K.property.1, K.property.2.1, hsuppω, hsuppπ⟩
+    have hcap : (K.val : Set Point) = capOfSofa S ω :=
+      cap_eq_capOfSofa_sdiff_capNiche K hN
+    have hhall (t : ℝ) (ht : t ∈ Set.Icc 0 ω) :
+        supportingHallway S (t : Real.Angle) =
+          supportingHallway (K.val : Set Point) (t : Real.Angle) := by
+      unfold supportingHallway
+      congr 1
+      funext p
+      simp only [supportingPlacement]
+      rw [supportValue_sdiff_capNiche_eq K hN
+          ⟨ht.1, ht.2.trans (le_add_of_nonneg_right (by positivity))⟩]
+      have hshift : t + Real.pi / 2 ∈ Set.Icc 0 (ω + Real.pi / 2) :=
+        ⟨by linarith [ht.1, Real.pi_pos], by linarith [ht.2]⟩
+      have h₁ := supportValue_sdiff_capNiche_eq K hN hshift
+      simp only [Real.Angle.coe_add] at h₁
+      rw [h₁]
+    have hmon : monotonization S ω = S := by
+      calc
+        monotonization S ω = monotonization (K.val : Set Point) ω := by
+          unfold monotonization
+          congr 1
+          apply Set.iInter_congr
+          intro t
+          apply Set.iInter_congr
+          exact hhall t
+        _ = S := (cap_sdiff_niche_eq_monotonization_self K).symm
+    exact ⟨S, hstandard, by rw [hmon]; exact hcap⟩
+
+end MovingSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/Development/IntervalArithmetic/Foundations/Development001.lean b/LeanPool/MovingSofa/Development/IntervalArithmetic/Foundations/Development001.lean
new file mode 100644
index 0000000000..7b3e22770f
--- /dev/null
+++ b/LeanPool/MovingSofa/Development/IntervalArithmetic/Foundations/Development001.lean
@@ -0,0 +1,13358 @@
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+module
+
+public import Mathlib.Algebra.Order.Field.Basic
+public import Mathlib.Algebra.Order.Floor.Defs
+public import Mathlib.Algebra.Order.Floor.Ring
+public import Mathlib.Analysis.Analytic.Order
+public import Mathlib.Analysis.Calculus.ContDiff.Defs
+public import Mathlib.Analysis.Calculus.Deriv.Add
+public import Mathlib.Analysis.Calculus.Deriv.Comp
+public import Mathlib.Analysis.Calculus.Deriv.Inv
+public import Mathlib.Analysis.Calculus.Deriv.Mul
+public import Mathlib.Analysis.Calculus.Deriv.Slope
+public import Mathlib.Analysis.Calculus.MeanValue
+public import Mathlib.Analysis.Calculus.ParametricIntegral
+public import Mathlib.Analysis.Calculus.Taylor
+public import Mathlib.Analysis.Complex.ExponentialBounds
+public import Mathlib.Analysis.Real.Pi.Bounds
+public import Mathlib.Analysis.SpecialFunctions.Arsinh
+public import Mathlib.Analysis.SpecialFunctions.ExpDeriv
+public import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
+public import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
+public import Mathlib.Analysis.SpecialFunctions.Log.Basic
+public import Mathlib.Analysis.SpecialFunctions.Log.Deriv
+public import Mathlib.Analysis.SpecialFunctions.Pow.Real
+public import Mathlib.Analysis.SpecialFunctions.Sqrt
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc
+public import Mathlib.Analysis.SpecificLimits.Normed
+public import Mathlib.Basic.Real.Basic
+public import Mathlib.Data.List.Range
+public import Mathlib.Data.Nat.Log
+public import Mathlib.Data.Rat.Cast.Order
+public import Mathlib.Data.Rat.Defs
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
+public import Mathlib.NumberTheory.Harmonic.EulerMascheroni
+public import Mathlib.Tactic.Linarith
+public import Mathlib.Tactic.NormNum
+public import Mathlib.Topology.Order.Basic
+
+/-!
+# Moving sofa: related mathematical developments
+
+* `LeanCert.Contrib.Sinc`.
+* `LeanCert.Core.DerivativeIntervals`.
+* `LeanCert.Core.Dyadic`.
+* `LeanCert.Core.Expr`.
+* `LeanCert.Core.Interval`.
+* `LeanCert.Core.IntervalRat.Basic`.
+* `LeanCert.Core.IntervalRat.LogReduction`.
+* `LeanCert.Core.IntervalRat.Transcendental`.
+* `LeanCert.Core.Support`.
+* `LeanCert.Core.Taylor`.
+* `LeanCert.Core.IntervalRat.Taylor`.
+* `LeanCert.Core.IntervalReal`.
+* `LeanCert.Core.IntervalDyadic`.
+* `LeanCert.Core.IntervalRealEndpoints`.
+* `LeanCert.Core.TrigReduction`.
+* `LeanCert.Core.IntervalRat.TrigReduced`.
+* `LeanCert.Engine.Eval.Core`.
+* `LeanCert.Engine.Eval.Result`.
+* `LeanCert.Engine.Eval.Extended`.
+* `LeanCert.Engine.Bounds.Lemmas`.
+* `LeanCert.Engine.IntervalEval`.
+* `LeanCert.Engine.AD.Basic`.
+* `LeanCert.Engine.AD.Transcendental`.
+* `LeanCert.Engine.AD.Eval`.
+* `LeanCert.Engine.AD.Correctness`.
+* `LeanCert.Engine.AD.Computable`.
+* `LeanCert.Engine.AD.DomainChecked`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2025 LeanCert Authors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Differentiability of sinc and dslope
+
+This file proves that the sinc function is differentiable everywhere, including at 0.
+
+## Main results
+
+* `Real.differentiableAt_sinc`: sinc is differentiable at every point
+* `Real.hasDerivAt_sinc_zero`: sinc has derivative 0 at x = 0
+
+## Mathematical background
+
+The sinc function is defined as:
+- `sinc x = sin x / x` for `x ≠ 0`
+- `sinc 0 = 1`
+
+Equivalently, `sinc = dslope sin 0`.
+
+The derivative of sinc at 0 can be computed using Taylor expansion:
+- `sin x = x - x³/6 + x⁵/120 - ...`
+- `sinc x = 1 - x²/6 + x⁴/120 - ...`
+- `sinc'(0) = 0`
+-/
+
+@[expose] public section
+
+open Filter Topology
+open scoped Topology
+
+namespace Real
+
+variable {x : ℝ}
+
+/-- Bound on |sin x - x| in terms of |x|³.
+For x > 0: sin x < x and x - x³/4 < sin x, so 0 < x - sin x < x³/4.
+By symmetry (sin(-x) = -sin(x)), |sin x - x| ≤ |x|³/4 for all x with |x| ≤ 1. -/
+theorem abs_sin_sub_self_le (x : ℝ) (_hx : |x| ≤ 1) : |sin x - x| ≤ |x| ^ 3 / 4 := by
+  rcases le_or_gt x 0 with hx_neg | hx_pos
+  · -- Case x ≤ 0
+    have hx' : 0 ≤ -x := neg_nonneg.mpr hx_neg
+    rcases eq_or_lt_of_le hx' with hx_zero | hx_pos'
+    · -- x = 0
+      have : x = 0 := neg_eq_zero.mp hx_zero.symm
+      simp [this]
+    · -- x < 0, i.e., 0 < -x ≤ 1
+      have h1 : sin (-x) < -x := sin_lt hx_pos'
+      have h2six : -x - (-x) ^ 3 / 6 < sin (-x) := sin_gt_sub_cube hx_pos'
+      have h2 : -x - (-x) ^ 3 / 4 < sin (-x) := by
+        have hcubed : 0 < (-x) ^ 3 := pow_pos hx_pos' _
+        linarith
+      -- sin x = -sin(-x), so sin x - x = -sin(-x) - x
+      -- From h1: sin(-x) < -x, so sin(-x) + x < 0
+      -- Therefore -sin(-x) - x = -(sin(-x) + x) > 0
+      rw [abs_of_nonpos hx_neg]
+      have hsinx : sin x = -sin (-x) := by simp [sin_neg]
+      rw [hsinx]
+      have hsum_pos : 0 < -sin (-x) - x := by linarith
+      rw [abs_of_pos hsum_pos]
+      -- Goal: -sin(-x) - x ≤ (-x) ^ 3 / 4
+      -- From h2: sin(-x) > -x - (-x)^3/4, so -sin(-x) < x + (-x)^3/4
+      -- Therefore -sin(-x) - x < (-x)^3/4
+      linarith
+  · -- Case x > 0
+    have h1 : sin x < x := sin_lt hx_pos
+    have h2six : x - x ^ 3 / 6 < sin x := sin_gt_sub_cube hx_pos
+    have h2 : x - x ^ 3 / 4 < sin x := by
+      have hcubed : 0 < x ^ 3 := pow_pos hx_pos _
+      linarith
+    rw [abs_of_pos hx_pos]
+    have hsub_pos : 0 < x - sin x := by linarith
+    rw [abs_of_neg (by linarith : sin x - x < 0)]
+    linarith
+
+/-- The derivative of sinc at 0 is 0.
+
+The proof uses the squeeze theorem with the bound `|sinc x - 1| ≤ |x|² / 4`,
+which follows from sin bounds in Mathlib. -/
+theorem hasDerivAt_sinc_zero : HasDerivAt sinc 0 0 := by
+  rw [hasDerivAt_iff_tendsto_slope]
+  -- Need to show: Tendsto (slope sinc 0) (𝓝[≠] 0) (𝓝 0)
+  rw [Metric.tendsto_nhdsWithin_nhds]
+  intro ε hε
+  use min 1 (ε * 4)
+  constructor
+  · positivity
+  intro y hy_ne hy_dist
+  simp only [slope, vsub_eq_sub, sinc_zero, smul_eq_mul, dist_eq_norm, sub_zero] at *
+  by_cases hy : y = 0
+  · exact absurd hy hy_ne
+  rw [Real.norm_eq_abs] at hy_dist ⊢
+  rw [abs_mul, abs_inv]
+  rw [sinc_of_ne_zero hy, div_sub_one hy, abs_div]
+  -- Goal: |y|⁻¹ * (|sin y - y| / |y|) < ε
+  -- i.e., |sin y - y| / |y|² < ε
+  have hy_le_1 : |y| ≤ 1 := (lt_min_iff.mp hy_dist).1.le
+  have hy_abs_pos : 0 < |y| := abs_pos.mpr hy
+  have hy_sq_pos : 0 < |y| ^ 2 := sq_pos_of_pos hy_abs_pos
+  -- |sin y - y| ≤ |y|³/4
+  have hbound : |sin y - y| ≤ |y| ^ 3 / 4 := abs_sin_sub_self_le y hy_le_1
+  -- |y|⁻¹ * (|sin y - y| / |y|) = |sin y - y| / |y|² ≤ |y|³/4 / |y|² = |y|/4
+  calc |y|⁻¹ * (|sin y - y| / |y|)
+      = |sin y - y| / |y| ^ 2 := by rw [inv_mul_eq_div, div_div, sq]
+    _ ≤ (|y| ^ 3 / 4) / |y| ^ 2 := by
+        apply div_le_div_of_nonneg_right hbound
+        exact hy_sq_pos.le
+    _ = |y| / 4 := by
+        have hy_ne : |y| ≠ 0 := ne_of_gt hy_abs_pos
+        have hsq : y ^ 2 = |y| ^ 2 := (sq_abs y).symm
+        field_simp [hy_ne]
+    _ < ε := by
+        have h_eps : |y| < ε * 4 := (lt_min_iff.mp hy_dist).2
+        linarith
+
+theorem differentiableAt_sinc_zero : DifferentiableAt ℝ sinc 0 :=
+  hasDerivAt_sinc_zero.differentiableAt
+
+/-- sinc is differentiable everywhere. -/
+theorem differentiableAt_sinc (x : ℝ) : DifferentiableAt ℝ sinc x := by
+  by_cases hx : x = 0
+  · exact hx ▸ differentiableAt_sinc_zero
+  · rw [sinc_eq_dslope, differentiableAt_dslope_of_ne hx]
+    exact differentiable_sin.differentiableAt
+
+/-- sinc is a differentiable function. -/
+theorem differentiable_sinc : Differentiable ℝ sinc := fun x => differentiableAt_sinc x
+
+/-- Integral representation of sinc:
+    `sinc x = ∫ t in 0..1, cos (x * t)`.
+    This is the standard average-cosine form used for derivative bounds. -/
+theorem sinc_eq_integral_cos (x : ℝ) :
+    sinc x = ∫ t in (0 : ℝ)..1, cos (x * t) := by
+  by_cases hx : x = 0
+  · subst hx
+    simp [sinc_zero]
+  · have hcomp :
+      ∫ t in (0 : ℝ)..1, cos (x * t) = x⁻¹ * ∫ u in (0 : ℝ)..x, cos u := by
+      simpa [mul_comm] using
+        (intervalIntegral.integral_comp_mul_right
+          (f := fun u : ℝ => cos u) (a := (0 : ℝ)) (b := 1) (c := x) hx)
+    rw [hcomp, integral_cos, sin_zero, sub_zero, inv_mul_eq_div, sinc_of_ne_zero hx]
+
+/-- The derivative of sinc at a nonzero point. -/
+theorem hasDerivAt_sinc_of_ne_zero (hx : x ≠ 0) :
+    HasDerivAt sinc ((x * cos x - sin x) / x ^ 2) x := by
+  have h1 : HasDerivAt sin (cos x) x := hasDerivAt_sin x
+  have h2 : HasDerivAt (fun y => y) 1 x := hasDerivAt_id x
+  have h3 : HasDerivAt (fun y => sin y / y) ((cos x * x - sin x * 1) / x ^ 2) x :=
+    h1.div h2 hx
+  simp only [mul_one] at h3
+  have h4 : HasDerivAt (fun y => sin y / y) ((x * cos x - sin x) / x ^ 2) x := by
+    convert h3 using 1; ring
+  -- sinc agrees with sin/id on a neighborhood of x (since x ≠ 0)
+  have heq : sinc =ᶠ[𝓝 x] (fun y => sin y / y) := by
+    filter_upwards [eventually_ne_nhds hx] with y hy
+    exact sinc_of_ne_zero hy
+  exact h4.congr_of_eventuallyEq heq
+
+/-- The derivative of sinc. For x = 0, deriv sinc 0 = 0. For x ≠ 0, deriv sinc x = (x cos x - sin x)
+/ x². -/
+theorem deriv_sinc : deriv sinc x = if x = 0 then 0 else (x * cos x - sin x) / x ^ 2 := by
+  split_ifs with hx
+  · exact hx ▸ hasDerivAt_sinc_zero.deriv
+  · exact (hasDerivAt_sinc_of_ne_zero hx).deriv
+
+@[simp]
+theorem deriv_sinc_zero : deriv sinc 0 = 0 := hasDerivAt_sinc_zero.deriv
+
+/-
+The derivative of sinc is bounded: |sinc'(x)| ≤ 1 for all x.
+
+The key insight is that for x ≠ 0:
+- sinc'(x) = (x cos x - sin x) / x²
+- Let g(x) = x cos x - sin x. Then g(0) = 0 and g'(x) = -x sin x.
+- |g(x)| = |∫₀ˣ -t sin t dt| ≤ ∫₀ˣ |t| dt = x²/2
+- Therefore |sinc'(x)| = |g(x)|/x² ≤ 1/2 < 1
+
+The proof using integration is mathematically straightforward but requires
+formalizing the integral representation. Instead we use a direct monotonicity argument.
+-/
+
+/-- Bound: |x cos x - sin x| ≤ x² for all x.
+    Proof uses monotonicity of auxiliary functions on [0, ∞). -/
+theorem abs_x_mul_cos_sub_sin_le_sq (x : ℝ) : |x * cos x - sin x| ≤ x ^ 2 := by
+  suffices ∀ x ≥ 0, |x * cos x - sin x| ≤ x ^ 2 by
+    obtain h | h := le_total 0 x
+    · exact this x h
+    · rw [← abs_neg, ← neg_sq x]
+      convert this (-x) (neg_nonneg.mpr h) using 2
+      ring_nf
+      simp only [cos_neg, sin_neg]
+      ring
+  intro x hx
+  rw [abs_le]
+  constructor
+  · -- -x^2 ≤ x cos x - sin x, equivalently 0 ≤ x cos x - sin x + x^2
+    -- Define g(t) = t * cos t - sin t + t^2, show g is monotone on [0,∞) with g(0) = 0
+    let g : ℝ → ℝ := fun t => t * cos t - sin t + t^2
+    have hg_diff : ∀ t, HasDerivAt g (t * (2 - sin t)) t := fun t => by
+      have h1 : HasDerivAt (fun t => t * cos t) (1 * cos t + t * (-sin t)) t :=
+        hasDerivAt_id t |>.mul (hasDerivAt_cos t)
+      have h2 : HasDerivAt (fun t => t * cos t - sin t) (1 * cos t + t * (-sin t) - cos t) t :=
+        h1.sub (hasDerivAt_sin t)
+      have hpow : HasDerivAt (fun t => t^2) (2 * t) t := by
+        have := hasDerivAt_pow 2 t
+        simp only [Nat.cast_ofNat, Nat.add_one_sub_one, pow_one] at this
+        exact this
+      have h3 : HasDerivAt (fun t => t * cos t - sin t + t^2)
+          (1 * cos t + t * (-sin t) - cos t + 2 * t) t :=
+        h2.add hpow
+      convert h3 using 1
+      ring
+    have hg_cont : Continuous g :=
+      (continuous_id.mul continuous_cos).sub continuous_sin |>.add (continuous_pow 2)
+    have hg_diffble : Differentiable ℝ g := fun t => (hg_diff t).differentiableAt
+    have hg' : ∀ t, deriv g t = t * (2 - sin t) := fun t => (hg_diff t).deriv
+    have hg_nonneg : ∀ t ∈ interior (Set.Ici (0:ℝ)), 0 ≤ deriv g t := by
+      intro t ht
+      rw [interior_Ici] at ht
+      rw [hg' t]
+      apply mul_nonneg (le_of_lt (Set.mem_Ioi.mp ht))
+      linarith [sin_le_one t]
+    have g_nonneg : 0 ≤ g x := by
+      have hmono := monotoneOn_of_deriv_nonneg (convex_Ici 0) hg_cont.continuousOn
+        hg_diffble.differentiableOn hg_nonneg
+      have hg0 : g 0 = 0 := by simp [g]
+      calc 0 = g 0 := hg0.symm
+        _ ≤ g x := hmono (by simp : (0:ℝ) ∈ Set.Ici 0) (Set.mem_Ici.mpr hx) hx
+    linarith [g_nonneg]
+  · -- x cos x - sin x ≤ x^2, equivalently 0 ≤ x^2 - (x cos x - sin x)
+    -- Define g(t) = t^2 - (t * cos t - sin t), show g is monotone on [0,∞) with g(0) = 0
+    let g : ℝ → ℝ := fun t => t^2 - (t * cos t - sin t)
+    have hg_diff : ∀ t, HasDerivAt g (t * (2 + sin t)) t := fun t => by
+      have h1 : HasDerivAt (fun t => t * cos t) (1 * cos t + t * (-sin t)) t :=
+        hasDerivAt_id t |>.mul (hasDerivAt_cos t)
+      have h2 : HasDerivAt (fun t => t * cos t - sin t) (1 * cos t + t * (-sin t) - cos t) t :=
+        h1.sub (hasDerivAt_sin t)
+      have hpow : HasDerivAt (fun t => t^2) (2 * t) t := by
+        have := hasDerivAt_pow 2 t
+        simp only [Nat.cast_ofNat, Nat.add_one_sub_one, pow_one] at this
+        exact this
+      have h3 : HasDerivAt (fun t => t^2 - (t * cos t - sin t))
+          (2 * t - (1 * cos t + t * (-sin t) - cos t)) t :=
+        hpow.sub h2
+      convert h3 using 1
+      ring
+    have hg_cont : Continuous g :=
+      (continuous_pow 2).sub ((continuous_id.mul continuous_cos).sub continuous_sin)
+    have hg_diffble : Differentiable ℝ g := fun t => (hg_diff t).differentiableAt
+    have hg' : ∀ t, deriv g t = t * (2 + sin t) := fun t => (hg_diff t).deriv
+    have hg_nonneg : ∀ t ∈ interior (Set.Ici (0:ℝ)), 0 ≤ deriv g t := by
+      intro t ht
+      rw [interior_Ici] at ht
+      rw [hg' t]
+      apply mul_nonneg (le_of_lt (Set.mem_Ioi.mp ht))
+      linarith [neg_one_le_sin t]
+    have g_nonneg : 0 ≤ g x := by
+      have hmono := monotoneOn_of_deriv_nonneg (convex_Ici 0) hg_cont.continuousOn
+        hg_diffble.differentiableOn hg_nonneg
+      have hg0 : g 0 = 0 := by simp [g]
+      calc 0 = g 0 := hg0.symm
+        _ ≤ g x := hmono (by simp : (0:ℝ) ∈ Set.Ici 0) (Set.mem_Ici.mpr hx) hx
+    linarith [g_nonneg]
+
+theorem abs_deriv_sinc_le_one (x : ℝ) : |deriv sinc x| ≤ 1 := by
+  by_cases hx : x = 0
+  · simp [hx]
+  · rw [(hasDerivAt_sinc_of_ne_zero hx).deriv]
+    rw [abs_div]
+    have hx_sq : x ^ 2 > 0 := sq_pos_of_ne_zero hx
+    have habs_sq : |x ^ 2| = x ^ 2 := abs_of_pos hx_sq
+    rw [habs_sq, div_le_one hx_sq]
+    exact abs_x_mul_cos_sub_sin_le_sq x
+
+/-- deriv sinc x is in [-1, 1] for all x -/
+theorem deriv_sinc_mem_Icc (x : ℝ) : deriv sinc x ∈ Set.Icc (-1) 1 := by
+  rw [Set.mem_Icc]
+  have h := abs_deriv_sinc_le_one x
+  rw [abs_le] at h
+  exact h
+
+/-!
+## Integral representation and smoothness of sinc
+
+The sinc function is smooth (C^∞). The key insight is the integral representation:
+  sinc(x) = ∫ t in 0..1, cos(t * x) dt
+
+This works because:
+- For x ≠ 0: ∫₀¹ cos(tx) dt = [sin(tx)/x]₀¹ = sin(x)/x = sinc(x)
+- For x = 0: ∫₀¹ cos(0) dt = ∫₀¹ 1 dt = 1 = sinc(0)
+
+Smoothness follows from differentiation under the integral sign (Leibniz rule):
+since cos(t*x) is C^∞ in x and the domain [0,1] is compact, the integral is C^∞.
+-/
+
+open MeasureTheory intervalIntegral in
+/-- The integral representation of sinc: sinc(x) = ∫ t in 0..1, cos(t * x) -/
+theorem sinc_eq_integral (x : ℝ) : sinc x = ∫ t in (0 : ℝ)..1, cos (t * x) := by
+  rcases eq_or_ne x 0 with rfl | hx
+  · -- Case x = 0: both sides equal 1
+    simp only [sinc_zero, mul_zero, cos_zero]
+    rw [intervalIntegral.integral_const, sub_zero, smul_eq_mul, mul_one]
+  · -- Case x ≠ 0: use fundamental theorem of calculus
+    have hcont : Continuous (fun t => cos (t * x)) := continuous_cos.comp (continuous_mul_const x)
+    have hderiv : ∀ t, HasDerivAt (fun u => sin (u * x) / x) (cos (t * x)) t := by
+      intro t
+      have : HasDerivAt (fun u => sin (u * x)) (x * cos (t * x)) t := by
+        have := Real.hasDerivAt_sin (t * x)
+        convert! HasDerivAt.comp t this (hasDerivAt_mul_const x) using 1
+        ring
+      convert! this.div_const x using 1
+      field_simp
+    -- The antiderivative is sin(tx)/x, which is continuous
+    have hcont_anti : ContinuousOn (fun t => sin (t * x) / x) (Set.Icc 0 1) :=
+      (continuous_sin.comp (continuous_mul_const x)).continuousOn.div_const x
+    -- cos(tx) is interval integrable since it's continuous
+    have hint : IntervalIntegrable (fun t => cos (t * x)) volume 0 1 :=
+      hcont.intervalIntegrable 0 1
+    rw [integral_eq_sub_of_hasDerivAt_of_le (by norm_num : (0 : ℝ) ≤ 1)
+        hcont_anti (fun t _ => hderiv t) hint]
+    simp only [one_mul, zero_mul, sin_zero, zero_div, sub_zero]
+    rw [sinc_of_ne_zero hx]
+
+/-- The derivative of sinc equals dslope (cos - sinc) at 0.
+
+For x ≠ 0:
+  dslope (cos - sinc) 0 x = (cos x - sinc x) / x = (x cos x - sin x) / x² = deriv sinc x
+
+For x = 0:
+  dslope (cos - sinc) 0 0 = deriv (cos - sinc) 0 = -sin 0 - deriv sinc 0 = 0 = deriv sinc 0
+-/
+theorem deriv_sinc_eq_dslope : deriv sinc = dslope (cos - sinc) 0 := by
+  ext x
+  by_cases hx : x = 0
+  · -- At x = 0
+    simp only [hx, dslope_same, deriv_sinc_zero]
+    simp only [deriv_sub, differentiableAt_cos, differentiableAt_sinc_zero, deriv_cos, sin_zero,
+      deriv_sinc_zero, sub_zero, neg_zero]
+  · -- At x ≠ 0
+    rw [dslope_of_ne _ hx, slope, vsub_eq_sub]
+    simp only [Pi.sub_apply, cos_zero, sinc_zero, sub_self, sub_zero]
+    rw [(hasDerivAt_sinc_of_ne_zero hx).deriv]
+    rw [sinc_of_ne_zero hx, smul_eq_mul]
+    field_simp
+
+/-- sinc is smooth at every nonzero point. -/
+theorem contDiffAt_sinc_of_ne_zero {x : ℝ} (hx : x ≠ 0) : ContDiffAt ℝ ⊤ sinc x := by
+  have heq : sinc =ᶠ[𝓝 x] fun y => sin y / y := by
+    filter_upwards [eventually_ne_nhds hx] with y hy
+    exact sinc_of_ne_zero hy
+  exact (contDiff_sin.contDiffAt.div contDiff_id.contDiffAt hx).congr_of_eventuallyEq heq
+
+/-- sinc is analytic at 0.
+
+The proof uses the order theory of analytic functions. Since sin is analytic at 0
+with order ≥ 1 (because sin(0) = 0), there exists an analytic function g such that
+sin(z) = z • g(z) near 0. This g must equal sinc away from 0, and by continuity
+of both functions at 0, g = sinc everywhere near 0. Therefore sinc is analytic at 0. -/
+theorem analyticAt_sinc_zero : AnalyticAt ℝ sinc 0 := by
+  -- sin has order ≥ 1 at 0 because sin(0) = 0 and sin is analytic
+  have hsin_an : AnalyticAt ℝ sin 0 := analyticAt_sin
+  -- sin(0) = 0 implies order ≠ 0, hence order ≥ 1
+  have horder_ne_zero : analyticOrderAt sin (0 : ℝ) ≠ 0 :=
+    hsin_an.analyticOrderAt_ne_zero.mpr sin_zero
+  -- From the order, we get an analytic g with sin(z) = z * g(z) near 0
+  have horder : (1 : ℕ) ≤ analyticOrderAt sin (0 : ℝ) := Order.one_le_iff_ne_zero.mpr horder_ne_zero
+  rw [natCast_le_analyticOrderAt hsin_an] at horder
+  simp only [pow_one, sub_zero] at horder
+  obtain ⟨g, hg_an, hg_eq⟩ := horder
+  -- g equals sinc away from 0: from sin z = z • g z, we get g z = sin z / z = sinc z
+  have hg_eq_sinc : g =ᶠ[𝓝[≠] 0] sinc := by
+    filter_upwards [hg_eq.filter_mono nhdsWithin_le_nhds,
+                    self_mem_nhdsWithin] with z hsin_eq hz
+    simp only [smul_eq_mul] at hsin_eq
+    have hz' : z ≠ 0 := Set.mem_compl_singleton_iff.mp hz
+    rw [sinc_of_ne_zero hz']
+    field_simp [hz']
+    linarith [hsin_eq]
+  -- Since g is continuous at 0 and sinc is continuous at 0,
+  -- and they agree on the punctured neighborhood, they agree at 0
+  have hg_zero : g 0 = sinc 0 := by
+    have hg_cont : ContinuousAt g 0 := hg_an.continuousAt
+    have hsinc_cont : ContinuousAt sinc 0 := continuous_sinc.continuousAt
+    -- g → g(0) as x → 0, and sinc(x) → sinc(0) as x → 0
+    -- Since g(x) = sinc(x) for x ≠ 0 near 0, limits must be equal
+    have h : Tendsto g (𝓝[≠] 0) (𝓝 (sinc 0)) := by
+      have := hsinc_cont.tendsto.mono_left (nhdsWithin_le_nhds (s := {0}ᶜ))
+      exact this.congr' hg_eq_sinc.symm
+    have h2 : Tendsto g (𝓝[≠] 0) (𝓝 (g 0)) :=
+      hg_cont.tendsto.mono_left (nhdsWithin_le_nhds (s := {0}ᶜ))
+    exact tendsto_nhds_unique h2 h
+  -- Now sinc = g near 0, so sinc is analytic at 0
+  exact hg_an.congr (hg_eq.mono fun z hsin_eq => by
+    simp only [smul_eq_mul] at hsin_eq
+    by_cases hz : z = 0
+    · simp [hz, hg_zero]
+    · rw [sinc_of_ne_zero hz]
+      field_simp [hz]
+      linarith [hsin_eq])
+
+/-- sinc is analytic at every point. -/
+theorem analyticAt_sinc (x : ℝ) : AnalyticAt ℝ sinc x := by
+  by_cases hx : x = 0
+  · exact hx ▸ analyticAt_sinc_zero
+  · exact contDiffAt_sinc_of_ne_zero hx |>.analyticAt
+
+/-- sinc is smooth (infinitely differentiable).
+
+The proof uses that sinc is analytic everywhere (analyticAt_sinc),
+and analytic functions are smooth. -/
+theorem contDiff_sinc : ContDiff ℝ ⊤ sinc :=
+  AnalyticOnNhd.contDiff (fun x _ => analyticAt_sinc x)
+
+/-- sinc is an analytic function. -/
+theorem analyticOnNhd_sinc : AnalyticOnNhd ℝ sinc Set.univ := fun x _ => analyticAt_sinc x
+
+end Real
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Derivative Interval Library
+
+This file provides a systematic collection of derivative interval facts for basic functions.
+These lemmas establish that derivatives of common functions lie within specified intervals,
+which feeds into:
+- Monotonicity analysis
+- Lipschitz/contractivity bounds
+- Newton method analysis
+- Global optimization pruning
+
+## Main theorems
+
+### Exp (derivative = exp, always positive)
+* `exp_deriv_pos` - exp' x > 0 for all x
+* `exp_deriv_interval` - exp' x ∈ [exp a, exp b] for x ∈ [a, b]
+
+### Log (derivative = 1/x on (0, ∞))
+* `log_deriv_interval` - log' x ∈ [1/b, 1/a] for x ∈ [a, b] with 0 < a
+
+### Arctan (derivative = 1/(1+x²) ∈ (0, 1])
+* `arctan_deriv_interval` - arctan' x ∈ (0, 1] for all x
+* `arctan_deriv_mem_Icc` - arctan' x ∈ [0, 1] for all x
+
+### Arsinh (derivative = 1/√(1+x²) ∈ (0, 1])
+* `arsinh_deriv_interval` - arsinh' x ∈ (0, 1] for all x
+* `arsinh_deriv_mem_Icc` - arsinh' x ∈ [0, 1] for all x
+
+### Sin/Cos (derivatives bounded by 1)
+* `sin_deriv_interval` - |sin' x| = |cos x| ≤ 1
+* `cos_deriv_interval` - |cos' x| = |sin x| ≤ 1
+
+### Sinh/Cosh (derivative relationships)
+* `sinh_deriv_eq_cosh` - sinh' = cosh
+* `cosh_deriv_eq_sinh` - cosh' = sinh
+* `cosh_deriv_interval` - cosh' x = sinh x, monotonicity facts
+
+## Design notes
+
+These lemmas are independent of any specific algorithm; they're just facts about functions.
+They're designed to be composable with the chain rule for AD correctness proofs.
+-/
+
+/-! ### Exp derivative intervals -/
+
+@[expose] public section
+
+namespace LeanCert.Core.DerivativeIntervals
+
+open Real Set
+
+/-- The derivative of exp is positive everywhere -/
+theorem exp_deriv_pos (x : ℝ) : 0 < deriv exp x := by
+  rw [Real.deriv_exp]
+  exact Real.exp_pos x
+
+/-- exp is strictly increasing (follows from positive derivative) -/
+theorem exp_strictMono : StrictMono exp := Real.exp_strictMono
+
+/-- For x ∈ [a, b], we have exp' x = exp x ∈ [exp a, exp b] -/
+theorem exp_deriv_interval {a b x : ℝ} (hx : x ∈ Icc a b) :
+    deriv exp x ∈ Icc (exp a) (exp b) := by
+  rw [Real.deriv_exp]
+  exact ⟨exp_le_exp.mpr hx.1, exp_le_exp.mpr hx.2⟩
+
+/-! ### Log derivative intervals -/
+
+/-- The derivative of log at x > 0 is x⁻¹ -/
+theorem log_deriv_eq_inv {x : ℝ} (_hx : 0 < x) : deriv log x = x⁻¹ :=
+  Real.deriv_log x
+
+/-- For x ∈ [a, b] with 0 < a ≤ b, we have log' x = 1/x ∈ [1/b, 1/a] -/
+theorem log_deriv_interval {a b x : ℝ} (ha : 0 < a) (hx : x ∈ Icc a b) :
+    deriv log x ∈ Icc (b⁻¹) (a⁻¹) := by
+  have hx_pos : 0 < x := lt_of_lt_of_le ha hx.1
+  rw [log_deriv_eq_inv hx_pos]
+  constructor
+  · exact inv_anti₀ hx_pos hx.2
+  · have ha_le_x : a ≤ x := hx.1
+    exact inv_anti₀ ha ha_le_x
+
+/-- log is strictly increasing on (0, ∞) -/
+theorem log_strictMonoOn : StrictMonoOn log (Ioi 0) := Real.strictMonoOn_log
+
+/-! ### Arctan derivative intervals -/
+
+/-- The derivative of arctan at x is 1/(1+x²) -/
+theorem arctan_deriv_eq (x : ℝ) : deriv arctan x = (1 + x ^ 2)⁻¹ :=
+  (Real.hasDerivAt_arctan' x).deriv
+
+/-- 1/(1+x²) is always positive -/
+theorem arctan_deriv_pos (x : ℝ) : 0 < deriv arctan x := by
+  rw [arctan_deriv_eq]
+  exact inv_pos.mpr (by nlinarith [sq_nonneg x])
+
+/-- 1/(1+x²) ≤ 1 for all x -/
+theorem arctan_deriv_le_one (x : ℝ) : deriv arctan x ≤ 1 := by
+  rw [arctan_deriv_eq]
+  have h : 1 ≤ 1 + x ^ 2 := by nlinarith [sq_nonneg x]
+  exact inv_le_one_of_one_le₀ h
+
+/-- arctan' x ∈ (0, 1] for all x -/
+theorem arctan_deriv_interval (x : ℝ) : deriv arctan x ∈ Ioc 0 1 :=
+  ⟨arctan_deriv_pos x, arctan_deriv_le_one x⟩
+
+/-- arctan' x ∈ [0, 1] for all x (closed interval version for interval arithmetic) -/
+theorem arctan_deriv_mem_Icc (x : ℝ) : deriv arctan x ∈ Icc 0 1 :=
+  ⟨le_of_lt (arctan_deriv_pos x), arctan_deriv_le_one x⟩
+
+/-- arctan is strictly increasing (follows from positive derivative) -/
+theorem arctan_strictMono : StrictMono arctan := Real.arctan_strictMono
+
+/-! ### Arsinh derivative intervals -/
+
+/-- The derivative of arsinh at x is (√(1+x²))⁻¹ -/
+theorem arsinh_deriv_eq (x : ℝ) : deriv arsinh x = (sqrt (1 + x ^ 2))⁻¹ := by
+  have h := Real.hasDerivAt_arsinh x
+  rw [h.deriv]
+
+/-- √(1+x²) ≥ 1 for all x -/
+theorem sqrt_one_add_sq_ge_one (x : ℝ) : 1 ≤ sqrt (1 + x ^ 2) := by
+  have h : 1 ≤ 1 + x ^ 2 := by nlinarith [sq_nonneg x]
+  calc sqrt (1 + x ^ 2) ≥ sqrt 1 := sqrt_le_sqrt h
+    _ = 1 := sqrt_one
+
+/-- √(1+x²) > 0 for all x -/
+theorem sqrt_one_add_sq_pos (x : ℝ) : 0 < sqrt (1 + x ^ 2) := by
+  apply sqrt_pos.mpr
+  nlinarith [sq_nonneg x]
+
+/-- arsinh' x > 0 for all x -/
+theorem arsinh_deriv_pos (x : ℝ) : 0 < deriv arsinh x := by
+  rw [arsinh_deriv_eq]
+  exact inv_pos.mpr (sqrt_one_add_sq_pos x)
+
+/-- arsinh' x ≤ 1 for all x -/
+theorem arsinh_deriv_le_one (x : ℝ) : deriv arsinh x ≤ 1 := by
+  rw [arsinh_deriv_eq]
+  exact inv_le_one_of_one_le₀ (sqrt_one_add_sq_ge_one x)
+
+/-- arsinh' x ∈ (0, 1] for all x -/
+theorem arsinh_deriv_interval (x : ℝ) : deriv arsinh x ∈ Ioc 0 1 :=
+  ⟨arsinh_deriv_pos x, arsinh_deriv_le_one x⟩
+
+/-- arsinh' x ∈ [0, 1] for all x (closed interval version) -/
+theorem arsinh_deriv_mem_Icc (x : ℝ) : deriv arsinh x ∈ Icc 0 1 :=
+  ⟨le_of_lt (arsinh_deriv_pos x), arsinh_deriv_le_one x⟩
+
+/-- arsinh is strictly increasing -/
+theorem arsinh_strictMono : StrictMono arsinh := Real.arsinh_strictMono
+
+/-! ### Sin derivative intervals -/
+
+/-- The derivative of sin is cos -/
+theorem sin_deriv_eq (x : ℝ) : deriv sin x = cos x := by
+  have h := Real.deriv_sin
+  exact congrFun h x
+
+/-- |sin' x| = |cos x| ≤ 1 -/
+theorem sin_deriv_abs_le_one (x : ℝ) : |deriv sin x| ≤ 1 := by
+  rw [sin_deriv_eq]
+  exact abs_cos_le_one x
+
+/-- sin' x ∈ [-1, 1] for all x -/
+theorem sin_deriv_mem_Icc (x : ℝ) : deriv sin x ∈ Icc (-1) 1 := by
+  rw [sin_deriv_eq]
+  exact ⟨neg_one_le_cos x, cos_le_one x⟩
+
+/-! ### Cos derivative intervals -/
+
+/-- The derivative of cos is -sin -/
+theorem cos_deriv_eq (x : ℝ) : deriv cos x = -sin x := by
+  have h := Real.deriv_cos'
+  exact congrFun h x
+
+/-- |cos' x| = |sin x| ≤ 1 -/
+theorem cos_deriv_abs_le_one (x : ℝ) : |deriv cos x| ≤ 1 := by
+  rw [cos_deriv_eq, abs_neg]
+  exact abs_sin_le_one x
+
+/-- cos' x ∈ [-1, 1] for all x -/
+theorem cos_deriv_mem_Icc (x : ℝ) : deriv cos x ∈ Icc (-1) 1 := by
+  rw [cos_deriv_eq]
+  constructor
+  · simp only [neg_le_neg_iff]; exact sin_le_one x
+  · have h := neg_one_le_sin x; linarith
+
+/-! ### Sinh derivative intervals -/
+
+/-- The derivative of sinh is cosh -/
+theorem sinh_deriv_eq (x : ℝ) : deriv sinh x = cosh x := by
+  have h := Real.deriv_sinh
+  exact congrFun h x
+
+/-- sinh' x = cosh x ≥ 1 for all x -/
+theorem sinh_deriv_ge_one (x : ℝ) : 1 ≤ deriv sinh x := by
+  rw [sinh_deriv_eq]
+  exact Real.one_le_cosh x
+
+/-- sinh' x > 0 for all x -/
+theorem sinh_deriv_pos (x : ℝ) : 0 < deriv sinh x := by
+  have h := sinh_deriv_ge_one x
+  linarith
+
+/-- sinh is strictly increasing -/
+theorem sinh_strictMono : StrictMono sinh := Real.sinh_strictMono
+
+/-! ### Cosh derivative intervals -/
+
+/-- The derivative of cosh is sinh -/
+theorem cosh_deriv_eq (x : ℝ) : deriv cosh x = sinh x := by
+  have h := Real.deriv_cosh
+  exact congrFun h x
+
+/-- cosh' 0 = sinh 0 = 0 -/
+theorem cosh_deriv_zero : deriv cosh 0 = 0 := by
+  rw [cosh_deriv_eq, Real.sinh_zero]
+
+/-- For x > 0, cosh' x = sinh x > 0 -/
+theorem cosh_deriv_pos_of_pos {x : ℝ} (hx : 0 < x) : 0 < deriv cosh x := by
+  rw [cosh_deriv_eq]
+  exact Real.sinh_pos_iff.mpr hx
+
+/-- For x < 0, cosh' x = sinh x < 0 -/
+theorem cosh_deriv_neg_of_neg {x : ℝ} (hx : x < 0) : deriv cosh x < 0 := by
+  rw [cosh_deriv_eq]
+  exact Real.sinh_neg_iff.mpr hx
+
+/-- cosh is strictly decreasing on (-∞, 0] -/
+theorem cosh_strictAntiOn_nonpos : StrictAntiOn cosh (Iic 0) := by
+  intro x hx y hy hxy
+  simp only [mem_Iic] at hx hy
+  -- Use that cosh is even and decreasing on negative reals
+  calc cosh y = cosh (-(-y)) := by ring_nf
+    _ = cosh (-y) := by rw [Real.cosh_neg]
+    _ < cosh (-x) := by
+        apply Real.cosh_strictMonoOn
+        · simp only [mem_Ici]; linarith
+        · simp only [mem_Ici]; linarith
+        · linarith
+    _ = cosh x := by rw [Real.cosh_neg]
+
+/-- cosh is strictly increasing on [0, ∞) -/
+theorem cosh_strictMonoOn_nonneg : StrictMonoOn cosh (Ici 0) :=
+  Real.cosh_strictMonoOn
+
+end LeanCert.Core.DerivativeIntervals
+
+/-! ## One-Sided Taylor Bounds
+
+For convex functions, the Taylor polynomial provides a one-sided bound.
+For exp on [0, ∞), the n-th Taylor polynomial is a lower bound.
+For concave functions, similar upper bounds hold.
+
+These bounds are tighter than symmetric remainder bounds at endpoints,
+which is important for verified numerics.
+-/
+
+namespace LeanCert.Core.OneSidedTaylor
+
+open Real Set
+
+/-! ### Exp lower bounds (convexity gives Taylor poly ≤ function) -/
+
+/-- 1 ≤ exp x for x ≥ 0 (Taylor degree 0) -/
+theorem one_le_exp_of_nonneg (x : ℝ) (hx : 0 ≤ x) : 1 ≤ exp x :=
+  Real.one_le_exp hx
+
+/-- 1 + x ≤ exp x for all x (Taylor degree 1) -/
+theorem one_add_le_exp (x : ℝ) : 1 + x ≤ exp x := by
+  have h := add_one_le_exp x
+  linarith
+
+/-- x + 1 ≤ exp x for all x (alternate form) -/
+theorem add_one_le_exp' (x : ℝ) : x + 1 ≤ exp x :=
+  add_one_le_exp x
+
+/-! ### Monotonicity from derivative bounds
+
+These wrap the Mathlib lemmas for convenience with our naming. -/
+
+/-- If f' ≥ 0 on [a, b], then f is monotone increasing on [a, b] -/
+theorem monotoneOn_of_deriv_nonneg {f : ℝ → ℝ} {a b : ℝ}
+    (hf : ContinuousOn f (Icc a b))
+    (hf' : DifferentiableOn ℝ f (Ioo a b))
+    (hderiv : ∀ x ∈ Ioo a b, 0 ≤ deriv f x) :
+    MonotoneOn f (Icc a b) := by
+  refine _root_.monotoneOn_of_deriv_nonneg (convex_Icc a b) hf ?_ ?_
+  · rw [interior_Icc]; exact hf'
+  · intro x hx; rw [interior_Icc] at hx; exact hderiv x hx
+
+/-- If f' > 0 on (a, b), then f is strictly monotone on [a, b] -/
+theorem strictMonoOn_of_deriv_pos {f : ℝ → ℝ} {a b : ℝ} (_hab : a < b)
+    (hf : ContinuousOn f (Icc a b))
+    (_hf' : DifferentiableOn ℝ f (Ioo a b))
+    (hderiv : ∀ x ∈ Ioo a b, 0 < deriv f x) :
+    StrictMonoOn f (Icc a b) := by
+  refine _root_.strictMonoOn_of_deriv_pos (convex_Icc a b) hf ?_
+  intro x hx; rw [interior_Icc] at hx; exact hderiv x hx
+
+/-- If f' ≤ 0 on [a, b], then f is antitone (monotone decreasing) on [a, b] -/
+theorem antitoneOn_of_deriv_nonpos {f : ℝ → ℝ} {a b : ℝ}
+    (hf : ContinuousOn f (Icc a b))
+    (hf' : DifferentiableOn ℝ f (Ioo a b))
+    (hderiv : ∀ x ∈ Ioo a b, deriv f x ≤ 0) :
+    AntitoneOn f (Icc a b) := by
+  refine _root_.antitoneOn_of_deriv_nonpos (convex_Icc a b) hf ?_ ?_
+  · rw [interior_Icc]; exact hf'
+  · intro x hx; rw [interior_Icc] at hx; exact hderiv x hx
+
+/-- If f' < 0 on (a, b), then f is strictly antitone on [a, b] -/
+theorem strictAntiOn_of_deriv_neg {f : ℝ → ℝ} {a b : ℝ} (_hab : a < b)
+    (hf : ContinuousOn f (Icc a b))
+    (_hf' : DifferentiableOn ℝ f (Ioo a b))
+    (hderiv : ∀ x ∈ Ioo a b, deriv f x < 0) :
+    StrictAntiOn f (Icc a b) := by
+  refine _root_.strictAntiOn_of_deriv_neg (convex_Icc a b) hf ?_
+  intro x hx; rw [interior_Icc] at hx; exact hderiv x hx
+
+end LeanCert.Core.OneSidedTaylor
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2025 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Dyadic Rationals (n * 2^e)
+
+This file defines Dyadic rationals, which are the backbone of high-performance
+verified numerics. Unlike arbitrary `Rat`, Dyadics do not require GCD
+normalization, making them significantly faster for kernel evaluation.
+
+## Main definitions
+
+* `Dyadic` - A dyadic rational number: `mantissa * 2^exponent`
+* `Dyadic.toRat` - Convert to standard rational
+* `Dyadic.add`, `Dyadic.mul`, `Dyadic.neg` - Arithmetic operations
+* `Dyadic.shiftDown`, `Dyadic.shiftUp` - Directed rounding for interval bounds
+
+## Design notes
+
+Dyadics use power-of-2 multiplications instead of GCD-based normalization.
+This makes them orders of magnitude faster for kernel evaluation via
+`native_decide`, since 2^n can be computed by simple bit operations.
+
+For interval arithmetic, we use directed rounding:
+- Lower bounds use `shiftDown` (round toward -∞)
+- Upper bounds use `shiftUp` (round toward +∞)
+
+This ensures mathematical soundness even when truncating precision.
+-/
+
+@[expose] public section
+
+namespace LeanCert.Core
+
+/-- A Dyadic rational number: `mantissa * 2^exponent`
+
+This representation allows fast arithmetic without GCD normalization. -/
+structure Dyadic where
+  /-- The significand/mantissa -/
+  mantissa : Int
+  /-- The exponent (power of 2) -/
+  exponent : Int
+  deriving Repr, DecidableEq, Inhabited
+
+namespace Dyadic
+
+/-! ### Helper: Power of 2 -/
+
+/-- Compute 2^n for natural n -/
+def pow2Nat (n : Nat) : Nat := Nat.shiftLeft 1 n
+
+/-- Shift an integer left by n bits (multiply by 2^n) -/
+def shiftLeftInt (m : Int) (n : Nat) : Int :=
+  m * (pow2Nat n : Int)
+
+/-- Shift an integer right by n bits (divide by 2^n, floor toward -∞) -/
+def shiftRightInt (m : Int) (n : Nat) : Int :=
+  m / (pow2Nat n : Int)
+
+/-- Check if any of the low n bits are set -/
+def hasLowBits (m : Int) (n : Nat) : Bool :=
+  m % (pow2Nat n : Int) ≠ 0
+
+/-! ### Conversion to Rat -/
+
+/-- Convert Dyadic to standard Rat.
+
+For non-negative exponents: `mantissa * 2^exponent`
+For negative exponents: `mantissa / 2^(-exponent)` -/
+def toRat (d : Dyadic) : ℚ :=
+  if d.exponent ≥ 0 then
+    d.mantissa * (pow2Nat d.exponent.toNat : ℤ)
+  else
+    (d.mantissa : ℚ) / (pow2Nat (-d.exponent).toNat : ℕ)
+
+instance : Coe Dyadic ℚ where coe := toRat
+
+/-- Equality of represented values, ignoring noncanonical mantissa/exponent pairs. -/
+def ValueEq (d₁ d₂ : Dyadic) : Prop := d₁.toRat = d₂.toRat
+
+instance (d₁ d₂ : Dyadic) : Decidable (d₁.ValueEq d₂) :=
+  inferInstanceAs (Decidable (d₁.toRat = d₂.toRat))
+
+instance : Setoid Dyadic where
+  r := ValueEq
+  iseqv := {
+    refl := fun _ => rfl
+    symm := fun h => h.symm
+    trans := fun h₁ h₂ => h₁.trans h₂
+  }
+
+@[simp] theorem valueEq_iff (d₁ d₂ : Dyadic) :
+    d₁.ValueEq d₂ ↔ d₁.toRat = d₂.toRat := Iff.rfl
+
+/-- A lawful identity key for caches, sets, serialization, and deduplication.
+
+Arithmetic keeps the fast, noncanonical `Dyadic` representation. Converting at
+an identity boundary collapses all representations of the same rational value.
+-/
+structure CanonicalKey where
+  /-- The exact rational value used to compare normalized dyadic numbers. -/
+  value : ℚ
+  deriving Repr, DecidableEq, BEq, Hashable
+
+/-- Convert a dyadic value to its canonical identity key. -/
+def canonicalKey (d : Dyadic) : CanonicalKey := ⟨d.toRat⟩
+
+@[simp] theorem canonicalKey_value (d : Dyadic) : d.canonicalKey.value = d.toRat := rfl
+
+/-- Canonical keys agree exactly when represented dyadic values agree. -/
+theorem canonicalKey_eq_iff (d₁ d₂ : Dyadic) :
+    d₁.canonicalKey = d₂.canonicalKey ↔ d₁.ValueEq d₂ := by
+  simp only [canonicalKey, ValueEq, CanonicalKey.mk.injEq]
+
+/-- Cast a Dyadic to ℝ by going through ℚ -/
+def toReal (d : Dyadic) : ℝ := Rat.cast (toRat d)
+
+instance : Coe Dyadic ℝ where coe := toReal
+
+/-- Equality of rational denotations gives semantic dyadic equality. -/
+theorem valueEq_of_toRat_eq {d₁ d₂ : Dyadic}
+    (h : d₁.toRat = d₂.toRat) : d₁.ValueEq d₂ := h
+
+/-! ### Construction -/
+
+/-- Create a dyadic from an integer (exponent = 0) -/
+def ofInt (i : Int) : Dyadic := ⟨i, 0⟩
+
+/-- Create a dyadic for 2^n -/
+def pow2 (n : Int) : Dyadic := ⟨1, n⟩
+
+/-- Zero as a Dyadic -/
+def zero : Dyadic := ⟨0, 0⟩
+
+/-- One as a Dyadic -/
+def one : Dyadic := ⟨1, 0⟩
+
+instance : Zero Dyadic where zero := Dyadic.zero
+instance : One Dyadic where one := Dyadic.one
+
+theorem toRat_ofInt (i : Int) : (ofInt i).toRat = i := by
+  simp only [ofInt, toRat, pow2Nat, Int.toNat_zero, le_refl, ↓reduceIte]
+  have h : Nat.shiftLeft 1 0 = 1 := rfl
+  simp only [h, Nat.cast_one, Int.cast_one, mul_one]
+
+@[simp]
+theorem toRat_zero : (0 : Dyadic).toRat = 0 := by
+  have h := toRat_ofInt 0; rw [Int.cast_zero] at h; exact h
+
+@[simp]
+theorem toRat_one : (1 : Dyadic).toRat = 1 := by
+  have h := toRat_ofInt 1; rw [Int.cast_one] at h; exact h
+
+/-! ### Arithmetic Operations -/
+
+/-- Negation of a Dyadic -/
+def neg (d : Dyadic) : Dyadic := ⟨-d.mantissa, d.exponent⟩
+
+instance : Neg Dyadic where neg := Dyadic.neg
+
+/-- Addition of Dyadics. Aligns exponents by shifting the mantissa with
+larger exponent to match the smaller one. -/
+def add (d₁ d₂ : Dyadic) : Dyadic :=
+  if d₁.exponent ≤ d₂.exponent then
+    let shift := (d₂.exponent - d₁.exponent).toNat
+    ⟨d₁.mantissa + shiftLeftInt d₂.mantissa shift, d₁.exponent⟩
+  else
+    let shift := (d₁.exponent - d₂.exponent).toNat
+    ⟨shiftLeftInt d₁.mantissa shift + d₂.mantissa, d₂.exponent⟩
+
+instance : Add Dyadic where add := Dyadic.add
+
+/-- Subtraction of Dyadics -/
+def sub (d₁ d₂ : Dyadic) : Dyadic := add d₁ (neg d₂)
+
+instance : Sub Dyadic where sub := Dyadic.sub
+
+/-- Multiplication of Dyadics. Simply multiplies mantissas and adds exponents. -/
+def mul (d₁ d₂ : Dyadic) : Dyadic :=
+  ⟨d₁.mantissa * d₂.mantissa, d₁.exponent + d₂.exponent⟩
+
+instance : Mul Dyadic where mul := Dyadic.mul
+
+/-- Absolute value of a Dyadic -/
+def abs (d : Dyadic) : Dyadic := ⟨Int.natAbs d.mantissa, d.exponent⟩
+
+/-- Scale by a power of 2 (efficient: just adjusts exponent) -/
+def scale2 (d : Dyadic) (n : Int) : Dyadic := ⟨d.mantissa, d.exponent + n⟩
+
+/-! ### Directed Rounding for Interval Arithmetic -/
+
+/-- Shift a dyadic to a new (larger) exponent with "Down" rounding (for lower bounds).
+
+If `newExp > d.exponent`, we lose precision by right-shifting the mantissa.
+Bits are discarded (floor division toward -∞). -/
+def shiftDown (d : Dyadic) (newExp : Int) : Dyadic :=
+  if newExp ≤ d.exponent then
+    d -- already at or below target precision
+  else
+    let diff := (newExp - d.exponent).toNat
+    -- Integer division is floor division toward -∞ for positive divisor
+    ⟨shiftRightInt d.mantissa diff, newExp⟩
+
+/-- Shift a dyadic to a new (larger) exponent with "Up" rounding (for upper bounds).
+
+If `newExp > d.exponent`, we lose precision by right-shifting the mantissa.
+If any bits are lost, we add 1 to ensure upward rounding (ceiling toward +∞). -/
+def shiftUp (d : Dyadic) (newExp : Int) : Dyadic :=
+  if newExp ≤ d.exponent then
+    d -- already at or below target precision
+  else
+    let diff := (newExp - d.exponent).toNat
+    if hasLowBits d.mantissa diff then
+      -- Lost bits: round up
+      ⟨shiftRightInt d.mantissa diff + 1, newExp⟩
+    else
+      -- No lost bits: exact
+      ⟨shiftRightInt d.mantissa diff, newExp⟩
+
+/-! ### Normalization (Mantissa Control) -/
+
+/-- Get the bit length of a natural number -/
+def natBitLength (n : Nat) : Nat :=
+  if n = 0 then 0 else Nat.log2 n + 1
+
+/-- Get the bit length of the absolute value of an integer -/
+def bitLength (m : Int) : Nat := natBitLength m.natAbs
+
+/-- Normalize a Dyadic to keep the mantissa within a reasonable bit-limit.
+
+This prevents mantissas from growing without bound during repeated multiplications.
+Similar to how hardware floats work, but with directed rounding.
+
+- `maxBits`: Maximum allowed bits for mantissa (default 256)
+- `roundUp`: If true, round toward +∞; if false, round toward -∞ -/
+def normalize (d : Dyadic) (maxBits : Nat := 256) (roundUp : Bool := false) : Dyadic :=
+  let bits := bitLength d.mantissa
+  if bits ≤ maxBits then d
+  else
+    let shift := bits - maxBits
+    let newExp := d.exponent + shift
+    if roundUp then shiftUp d newExp
+    else shiftDown d newExp
+
+/-- Normalize for lower bounds (round down) -/
+def normalizeDown (d : Dyadic) (maxBits : Nat := 256) : Dyadic :=
+  normalize d maxBits false
+
+/-- Normalize for upper bounds (round up) -/
+def normalizeUp (d : Dyadic) (maxBits : Nat := 256) : Dyadic :=
+  normalize d maxBits true
+
+/-! ### Comparison -/
+
+/-- Compare two Dyadics (decidable) -/
+def compare (d₁ d₂ : Dyadic) : Ordering :=
+  -- Align to smaller exponent for comparison
+  if d₁.exponent ≤ d₂.exponent then
+    let shift := (d₂.exponent - d₁.exponent).toNat
+    Ord.compare d₁.mantissa (shiftLeftInt d₂.mantissa shift)
+  else
+    let shift := (d₁.exponent - d₂.exponent).toNat
+    Ord.compare (shiftLeftInt d₁.mantissa shift) d₂.mantissa
+
+instance : Ord Dyadic where compare := Dyadic.compare
+
+/-- Less-than-or-equal for Dyadics -/
+def le (d₁ d₂ : Dyadic) : Bool := compare d₁ d₂ != .gt
+
+/-- Less-than for Dyadics -/
+def lt (d₁ d₂ : Dyadic) : Bool := compare d₁ d₂ == .lt
+
+instance : LE Dyadic where le d₁ d₂ := le d₁ d₂
+instance : LT Dyadic where lt d₁ d₂ := lt d₁ d₂
+
+instance : DecidableRel (α := Dyadic) (· ≤ ·) := fun d₁ d₂ =>
+  if h : le d₁ d₂ then isTrue h else isFalse h
+
+instance : DecidableRel (α := Dyadic) (· < ·) := fun d₁ d₂ =>
+  if h : lt d₁ d₂ then isTrue h else isFalse h
+
+/-- Minimum of two Dyadics -/
+def min (d₁ d₂ : Dyadic) : Dyadic := if le d₁ d₂ then d₁ else d₂
+
+/-- Maximum of two Dyadics -/
+def max (d₁ d₂ : Dyadic) : Dyadic := if le d₁ d₂ then d₂ else d₁
+
+/-- Minimum of four Dyadics -/
+def min4 (a b c d : Dyadic) : Dyadic := min (min a b) (min c d)
+
+/-- Maximum of four Dyadics -/
+def max4 (a b c d : Dyadic) : Dyadic := max (max a b) (max c d)
+
+/-! ### Correctness Theorems -/
+
+/-! #### Helper lemmas for shift proofs -/
+
+/-- shiftRightInt is floor division by 2^n -/
+private theorem shiftRightInt_eq_div (m : Int) (n : Nat) :
+    shiftRightInt m n = m / (pow2Nat n : Int) := by
+  simp only [shiftRightInt, pow2Nat]
+
+/-- shiftLeftInt is multiplication by 2^n -/
+private theorem shiftLeftInt_eq_mul (m : Int) (n : Nat) :
+    shiftLeftInt m n = m * (pow2Nat n : Int) := by
+  simp only [shiftLeftInt, pow2Nat]
+
+/-- pow2Nat n is always positive.
+    Proof: 2^n ≥ 1 for all n. -/
+private theorem pow2Nat_pos (n : Nat) : 0 < pow2Nat n := by
+  unfold pow2Nat
+  -- Use the fact that shiftLeft 1 n = 2^n
+  conv => lhs; rw [show (0 : ℕ) = 0 from rfl]
+  conv => rhs; rw [show Nat.shiftLeft 1 n = 1 * 2^n from Nat.shiftLeft_eq 1 n]
+  simp only [one_mul]
+  exact Nat.one_le_two_pow
+
+/-- pow2Nat as Int is positive -/
+private theorem pow2Nat_int_pos (n : Nat) : (0 : Int) < (pow2Nat n : Int) := by
+  exact Int.natCast_pos.mpr (pow2Nat_pos n)
+
+/-- Floor division property: (m / d) * d ≤ m for positive d.
+    Follows from m % d ≥ 0 and m = m % d + d * (m / d). -/
+private theorem int_ediv_mul_le (m : Int) (d : Int) (hd : 0 < d) :
+    (m / d) * d ≤ m := Int.ediv_mul_le m (ne_of_gt hd)
+
+/-- Ceiling division property: m ≤ (m / d + 1) * d when m % d ≠ 0 -/
+private theorem int_ediv_add_one_mul_ge (m : Int) (d : Int) (hd : 0 < d) (_hrem : m % d ≠ 0) :
+    m ≤ (m / d + 1) * d := by
+  -- From Int.emod_add_mul_ediv: m % d + d * (m / d) = m
+  have h := Int.emod_add_mul_ediv m d
+  -- m % d < d since d > 0
+  have hlt := Int.emod_lt_of_pos m hd
+  -- d + d * (m / d) = (m / d + 1) * d
+  have heq : d + d * (m / d) = (m / d + 1) * d := by
+    rw [add_mul, one_mul, mul_comm d (m / d), add_comm]
+  -- m = m % d + d * (m / d) < d + d * (m / d) = (m / d + 1) * d
+  have hcalc : m < (m / d + 1) * d :=
+    calc m = m % d + d * (m / d) := h.symm
+      _ < d + d * (m / d) := by gcongr 1
+      _ = (m / d + 1) * d := heq
+  exact le_of_lt hcalc
+
+/-- Exact division: m = (m / d) * d when m % d = 0 -/
+private theorem int_ediv_mul_eq (m : Int) (d : Int) (_hd : 0 < d) (hrem : m % d = 0) :
+    (m / d) * d = m := by
+  -- From Int.emod_add_mul_ediv: m % d + d * (m / d) = m
+  have h := Int.emod_add_mul_ediv m d
+  -- Since m % d = 0, we have d * (m / d) = m
+  simp only [hrem, zero_add] at h
+  -- d * (m / d) = m, so (m / d) * d = m
+  calc (m / d) * d = d * (m / d) := mul_comm _ _
+    _ = m := h
+
+theorem toRat_neg (d : Dyadic) : (neg d).toRat = -(d.toRat) := by
+  by_cases h : d.exponent ≥ 0
+  · simp [neg, toRat, h]
+  · simp only [toRat, neg, ge_iff_le, h, ↓reduceIte, Int.cast_neg]
+    rw [neg_div]
+
+/-! ### Arithmetic Homomorphisms -/
+
+/-- Helper: pow2Nat n = 2^n -/
+private theorem pow2Nat_eq_pow (n : Nat) : pow2Nat n = 2 ^ n := by
+  unfold pow2Nat
+  rw [show Nat.shiftLeft 1 n = 1 * 2^n from Nat.shiftLeft_eq 1 n]
+  simp only [one_mul]
+
+/-- Unification lemma: toRat d = d.mantissa * 2^d.exponent -/
+theorem toRat_eq (d : Dyadic) : d.toRat = d.mantissa * (2 : ℚ) ^ d.exponent := by
+  unfold toRat
+  split_ifs with h
+  · -- Case: exponent ≥ 0
+    have he : d.exponent = (d.exponent.toNat : ℤ) := (Int.toNat_of_nonneg h).symm
+    rw [he, zpow_natCast, pow2Nat_eq_pow]
+    -- Simplify (↑n : ℤ).toNat = n
+    simp only [Nat.cast_pow, Nat.cast_ofNat, Int.cast_pow, Int.cast_ofNat,
+               Int.toNat_natCast]
+  · -- Case: exponent < 0
+    push Not at h
+    have hpos : 0 ≤ -d.exponent := Int.neg_nonneg.mpr (le_of_lt h)
+    -- (-d.exponent).toNat = the natural number n such that d.exponent = -n
+    set n := (-d.exponent).toNat with hn_def
+    have hne : d.exponent = -(n : ℤ) := by
+      rw [Int.toNat_of_nonneg hpos]; omega
+    rw [hne, zpow_neg, zpow_natCast, pow2Nat_eq_pow]
+    -- Goal: mantissa / 2^n = mantissa * (2^n)⁻¹
+    simp only [Nat.cast_pow, Nat.cast_ofNat, div_eq_mul_inv]
+
+theorem toRat_add (d₁ d₂ : Dyadic) : (add d₁ d₂).toRat = d₁.toRat + d₂.toRat := by
+  -- Rewrite using toRat_eq to work with mantissa * 2^exponent form
+  rw [toRat_eq d₁, toRat_eq d₂]
+  unfold add
+  split_ifs with h
+  · -- Case: d₁.exponent ≤ d₂.exponent
+    rw [toRat_eq]
+    simp only [shiftLeftInt, pow2Nat_eq_pow]
+    -- Goal: (m₁ + m₂ * 2^shift) * 2^e₁ = m₁ * 2^e₁ + m₂ * 2^e₂
+    set shift := (d₂.exponent - d₁.exponent).toNat with hshift_def
+    have hshift : (shift : ℤ) = d₂.exponent - d₁.exponent := by
+      rw [hshift_def, Int.toNat_of_nonneg]; omega
+    -- 2^shift * 2^e₁ = 2^e₂
+    have hexp : (2 : ℚ) ^ shift * 2 ^ d₁.exponent = 2 ^ d₂.exponent := by
+      have h2 : (2 : ℚ) ^ shift = (2 : ℚ) ^ (shift : ℤ) := by rw [zpow_natCast]
+      rw [h2, ← zpow_add₀ (two_ne_zero), hshift]
+      congr 1; omega
+    -- Show ↑(2^shift : ℕ) = (2 : ℚ)^shift
+    have hcast : (↑(2 ^ shift : ℕ) : ℚ) = (2 : ℚ) ^ shift := by
+      rw [Nat.cast_pow, Nat.cast_ofNat]
+    -- Push casts inside and simplify
+    simp only [Int.cast_add, Int.cast_mul, Int.cast_natCast, hcast, add_mul, mul_assoc, hexp]
+  · -- Case: d₁.exponent > d₂.exponent
+    rw [toRat_eq]
+    simp only [shiftLeftInt, pow2Nat_eq_pow]
+    set shift := (d₁.exponent - d₂.exponent).toNat with hshift_def
+    have hshift : (shift : ℤ) = d₁.exponent - d₂.exponent := by
+      rw [hshift_def, Int.toNat_of_nonneg]; omega
+    have hexp : (2 : ℚ) ^ shift * 2 ^ d₂.exponent = 2 ^ d₁.exponent := by
+      have h2 : (2 : ℚ) ^ shift = (2 : ℚ) ^ (shift : ℤ) := by rw [zpow_natCast]
+      rw [h2, ← zpow_add₀ (two_ne_zero), hshift]
+      congr 1; omega
+    have hcast : (↑(2 ^ shift : ℕ) : ℚ) = (2 : ℚ) ^ shift := by
+      rw [Nat.cast_pow, Nat.cast_ofNat]
+    simp only [Int.cast_add, Int.cast_mul, Int.cast_natCast, hcast, add_mul, mul_assoc, hexp,
+      add_comm]
+
+theorem toRat_mul (d₁ d₂ : Dyadic) : (mul d₁ d₂).toRat = d₁.toRat * d₂.toRat := by
+  -- Key: (m1*m2) * 2^(e1+e2) = (m1*2^e1) * (m2*2^e2)
+  rw [toRat_eq d₁, toRat_eq d₂, toRat_eq]
+  unfold mul
+  simp only [Int.cast_mul]
+  -- Goal: (m₁ * m₂) * 2^(e₁ + e₂) = (m₁ * 2^e₁) * (m₂ * 2^e₂)
+  -- Use zpow_add₀: 2^(e₁ + e₂) = 2^e₁ * 2^e₂
+  rw [zpow_add₀ (two_ne_zero)]
+  -- Goal: (a * b) * (c * d) = (a * c) * (b * d)
+  -- Use mul_mul_mul_comm: a * b * (c * d) = a * c * (b * d)
+  rw [mul_mul_mul_comm]
+
+/-! ### Rounding Properties -/
+
+/-- shiftDown produces a value ≤ the original -/
+theorem toRat_shiftDown_le (d : Dyadic) (newExp : Int) :
+    (shiftDown d newExp).toRat ≤ d.toRat := by
+  simp only [shiftDown]
+  split_ifs with h
+  · exact le_refl _
+  · -- newExp > d.exponent
+    push Not at h
+    set diff := (newExp - d.exponent).toNat with hdiff_def
+    have hdiff : (diff : ℤ) = newExp - d.exponent := by
+      rw [hdiff_def, Int.toNat_of_nonneg]; omega
+    have hnewExp : newExp = d.exponent + diff := by omega
+    rw [toRat_eq, toRat_eq]
+    simp only [shiftRightInt, pow2Nat_eq_pow]
+    -- Use int_ediv_mul_le: (m / b) * b ≤ m
+    have hpow_int_pos : (0 : ℤ) < (2 ^ diff : ℕ) := Int.natCast_pos.mpr (Nat.one_le_two_pow)
+    have hdiv : d.mantissa / (2 ^ diff : ℕ) * (2 ^ diff : ℕ) ≤ d.mantissa :=
+      int_ediv_mul_le d.mantissa (2 ^ diff : ℕ) hpow_int_pos
+    -- Key: 2^newExp = 2^d.exponent * 2^diff
+    have hpow : (2 : ℚ) ^ newExp = 2 ^ d.exponent * (2 : ℚ) ^ diff := by
+      rw [hnewExp, zpow_add₀ (two_ne_zero), zpow_natCast]
+    rw [hpow]
+    -- Goal: (m / 2^diff) * (2^d.exp * 2^diff) ≤ m * 2^d.exp
+    have hpos : (0 : ℚ) < 2 ^ d.exponent := zpow_pos (by norm_num : (0 : ℚ) < 2) _
+    have hpow2_pos : (0 : ℚ) < (2 : ℚ) ^ diff := pow_pos (by norm_num : (0 : ℚ) < 2) _
+    -- Convert hdiv to ℚ: (m / 2^diff) * 2^diff ≤ m
+    have hq : (↑(d.mantissa / (2 ^ diff : ℕ)) : ℚ) * ((2 : ℚ) ^ diff) ≤ d.mantissa := by
+      have heq : ((2 : ℚ) ^ diff) = ((2 ^ diff : ℕ) : ℚ) := by simp
+      rw [heq]
+      -- Goal: (m/b : ℤ) * (b : ℕ) ≤ m in ℚ
+      -- hdiv: (m/b) * b ≤ m in ℤ
+      calc (↑(d.mantissa / (2 ^ diff : ℕ)) : ℚ) * ↑(2 ^ diff : ℕ)
+          = ↑(d.mantissa / (2 ^ diff : ℕ) * (2 ^ diff : ℕ)) := by simp only [Int.cast_mul,
+            Int.cast_natCast]
+        _ ≤ ↑d.mantissa := Int.cast_le.mpr hdiv
+    -- Rearrange: (m/b) * (c * b) = (m/b) * b * c and apply hq
+    have hrearr : (↑(d.mantissa / (2 ^ diff : ℕ)) : ℚ) * (2 ^ d.exponent * (2 : ℚ) ^ diff)
+                = (↑(d.mantissa / (2 ^ diff : ℕ))) * ((2 : ℚ) ^ diff) * 2 ^ d.exponent := by
+      rw [mul_comm (2 ^ d.exponent) ((2 : ℚ) ^ diff), mul_assoc]
+    rw [hrearr]
+    exact mul_le_mul_of_nonneg_right hq (le_of_lt hpos)
+
+/-- shiftUp produces a value ≥ the original -/
+theorem toRat_shiftUp_ge (d : Dyadic) (newExp : Int) :
+    d.toRat ≤ (shiftUp d newExp).toRat := by
+  simp only [shiftUp]
+  split_ifs with h hlowbits
+  · exact le_refl _
+  · -- Case: Lost bits, added 1
+    push Not at h
+    set diff := (newExp - d.exponent).toNat with hdiff_def
+    have hdiff : (diff : ℤ) = newExp - d.exponent := by
+      rw [hdiff_def, Int.toNat_of_nonneg]; omega
+    have hnewExp : newExp = d.exponent + diff := by omega
+    rw [toRat_eq, toRat_eq]
+    simp only [shiftRightInt, pow2Nat_eq_pow, hasLowBits] at hlowbits ⊢
+    -- Use int_ediv_add_one_mul_ge: m ≤ (m/b + 1) * b when m % b ≠ 0
+    have hpow_int_pos : (0 : ℤ) < (2 ^ diff : ℕ) := Int.natCast_pos.mpr (Nat.one_le_two_pow)
+    have hrem : d.mantissa % (2 ^ diff : ℕ) ≠ 0 := by
+      simpa only [ne_eq, decide_eq_true_eq] using hlowbits
+    have hceil : d.mantissa ≤ (d.mantissa / (2 ^ diff : ℕ) + 1) * (2 ^ diff : ℕ) :=
+      int_ediv_add_one_mul_ge d.mantissa (2 ^ diff : ℕ) hpow_int_pos hrem
+    -- Key: 2^newExp = 2^d.exponent * 2^diff
+    have hpow : (2 : ℚ) ^ newExp = 2 ^ d.exponent * (2 : ℚ) ^ diff := by
+      rw [hnewExp, zpow_add₀ (two_ne_zero), zpow_natCast]
+    rw [hpow]
+    have hpos : (0 : ℚ) < 2 ^ d.exponent := zpow_pos (by norm_num : (0 : ℚ) < 2) _
+    -- Convert hceil to ℚ: m ≤ (m/b + 1) * b
+    have hq : (d.mantissa : ℚ) ≤ (↑(d.mantissa / (2 ^ diff : ℕ) + 1)) * ((2 : ℚ) ^ diff) := by
+      have heq : ((2 : ℚ) ^ diff) = ((2 ^ diff : ℕ) : ℚ) := by simp
+      rw [heq]
+      calc (d.mantissa : ℚ) ≤ ↑((d.mantissa / (2 ^ diff : ℕ) + 1) * (2 ^ diff : ℕ)) :=
+        Int.cast_le.mpr hceil
+        _ = ↑(d.mantissa / (2 ^ diff : ℕ) + 1) * ↑(2 ^ diff : ℕ) := by simp only [Int.cast_mul,
+          Int.cast_natCast]
+    -- Rearrange and apply hq
+    have hrearr : (↑(d.mantissa / (2 ^ diff : ℕ) + 1) : ℚ) * (2 ^ d.exponent * (2 : ℚ) ^ diff)
+                = (↑(d.mantissa / (2 ^ diff : ℕ) + 1)) * ((2 : ℚ) ^ diff) * 2 ^ d.exponent := by
+      rw [mul_comm (2 ^ d.exponent) ((2 : ℚ) ^ diff), mul_assoc]
+    rw [hrearr]
+    calc (d.mantissa : ℚ) * 2 ^ d.exponent
+        ≤ (↑(d.mantissa / (2 ^ diff : ℕ) + 1) * ((2 : ℚ) ^ diff)) * 2 ^ d.exponent :=
+            mul_le_mul_of_nonneg_right hq (le_of_lt hpos)
+      _ = ↑(d.mantissa / (2 ^ diff : ℕ) + 1) * (2 : ℚ) ^ diff * 2 ^ d.exponent := by rfl
+  · -- Case: No lost bits, exact (equality)
+    push Not at h
+    set diff := (newExp - d.exponent).toNat with hdiff_def
+    have hdiff : (diff : ℤ) = newExp - d.exponent := by
+      rw [hdiff_def, Int.toNat_of_nonneg]; omega
+    have hnewExp : newExp = d.exponent + diff := by omega
+    rw [toRat_eq, toRat_eq]
+    simp only [shiftRightInt, pow2Nat_eq_pow, hasLowBits] at hlowbits ⊢
+    -- Use int_ediv_mul_eq: (m/b) * b = m when m % b = 0
+    have hpow_int_pos : (0 : ℤ) < (2 ^ diff : ℕ) := Int.natCast_pos.mpr (Nat.one_le_two_pow)
+    have hrem : d.mantissa % (2 ^ diff : ℕ) = 0 := by
+      simp only [ne_eq, decide_eq_true_eq, not_not] at hlowbits; exact hlowbits
+    have hexact : (d.mantissa / (2 ^ diff : ℕ)) * (2 ^ diff : ℕ) = d.mantissa :=
+      int_ediv_mul_eq d.mantissa (2 ^ diff : ℕ) hpow_int_pos hrem
+    -- Key: 2^newExp = 2^d.exponent * 2^diff
+    have hpow : (2 : ℚ) ^ newExp = 2 ^ d.exponent * (2 : ℚ) ^ diff := by
+      rw [hnewExp, zpow_add₀ (two_ne_zero), zpow_natCast]
+    rw [hpow]
+    -- Goal: m * 2^exp ≤ (m/b) * (2^exp * 2^diff)
+    -- Since (m/b) * b = m, this is equality
+    have heq : ((2 : ℚ) ^ diff) = ((2 ^ diff : ℕ) : ℚ) := by simp
+    calc (d.mantissa : ℚ) * 2 ^ d.exponent
+        = ↑((d.mantissa / (2 ^ diff : ℕ)) * (2 ^ diff : ℕ)) * 2 ^ d.exponent := by
+            simp only [hexact]
+      _ = ↑(d.mantissa / (2 ^ diff : ℕ)) * ↑(2 ^ diff : ℕ) * 2 ^ d.exponent := by
+            simp only [Int.cast_mul, Int.cast_natCast]
+      _ = ↑(d.mantissa / (2 ^ diff : ℕ)) * ((2 : ℚ) ^ diff * 2 ^ d.exponent) := by
+            rw [← heq, mul_assoc]
+      _ = ↑(d.mantissa / (2 ^ diff : ℕ)) * (2 ^ d.exponent * (2 : ℚ) ^ diff) := by
+            rw [mul_comm ((2 : ℚ) ^ diff) (2 ^ d.exponent)]
+      _ ≤ ↑(d.mantissa / (2 ^ diff : ℕ)) * (2 ^ d.exponent * (2 : ℚ) ^ diff) := le_refl _
+
+/-! ### Comparison Helper Lemmas -/
+
+/-- Helper: comparing integers reflects comparing rationals when scaled by positive factor -/
+private theorem int_lt_iff_rat_mul_lt (a b : ℤ) (c : ℚ) (hc : 0 < c) :
+    a < b ↔ (a : ℚ) * c < (b : ℚ) * c :=
+    ⟨fun h ↦ mul_lt_mul_of_pos_right (Int.cast_lt.mpr h) hc, fun h ↦ Int.cast_lt.mp
+      ((Rat.mul_lt_mul_right hc).mp h)⟩
+
+private theorem int_eq_iff_rat_mul_eq (a b : ℤ) (c : ℚ) (hc : c ≠ 0) :
+    a = b ↔ (a : ℚ) * c = (b : ℚ) * c := by
+  constructor
+  · intro h; rw [h]
+  · intro h
+    have hcancel := (mul_eq_mul_right_iff (c := c)).mp h
+    cases hcancel with
+    | inl heq => exact Int.cast_inj.mp heq
+    | inr hzero => exact absurd hzero hc
+
+private theorem int_gt_iff_rat_mul_gt (a b : ℤ) (c : ℚ) (hc : 0 < c) :
+    a > b ↔ (a : ℚ) * c > (b : ℚ) * c := int_lt_iff_rat_mul_lt b a c hc
+
+/-- Key lemma: shiftLeftInt relates to multiplication by 2^shift in ℚ -/
+private theorem shiftLeftInt_toRat (m : ℤ) (n : ℕ) :
+    (shiftLeftInt m n : ℚ) = (m : ℚ) * (2 : ℚ) ^ n := by
+  simp only [shiftLeftInt, pow2Nat_eq_pow, Int.cast_mul, Int.cast_natCast]
+  rw [Nat.cast_pow, Nat.cast_ofNat]
+
+/-- Ord.compare on Int correctly reflects <, =, > -/
+private theorem ord_compare_int_lt (a b : ℤ) : Ord.compare a b = .lt ↔ a < b := by
+  constructor
+  · intro h
+    by_contra! hge
+    have hcases := lt_or_eq_of_le hge
+    cases hcases with
+    | inl hgt =>
+      simp only [Ord.compare, compareOfLessAndEq, not_lt.mpr (le_of_lt hgt), ne_of_gt hgt,
+        ↓reduceIte] at h
+      exact absurd h (by decide)
+    | inr heq =>
+      simp only [Ord.compare, compareOfLessAndEq, heq, lt_irrefl, ↓reduceIte] at h
+      exact absurd h (by decide)
+  · intro h
+    simp only [Ord.compare, compareOfLessAndEq, h, ↓reduceIte]
+
+private theorem ord_compare_int_eq (a b : ℤ) : Ord.compare a b = .eq ↔ a = b := by
+  constructor
+  · intro h
+    by_contra hne
+    have hcases := lt_or_gt_of_ne hne
+    cases hcases with
+    | inl hlt =>
+      simp only [Ord.compare, compareOfLessAndEq, hlt, ↓reduceIte] at h
+      exact absurd h (by decide)
+    | inr hgt =>
+      simp only [Ord.compare, compareOfLessAndEq, not_lt.mpr (le_of_lt hgt), ne_of_gt hgt,
+        ↓reduceIte] at h
+      exact absurd h (by decide)
+  · intro h
+    simp only [Ord.compare, compareOfLessAndEq, h, lt_irrefl, ↓reduceIte]
+
+private theorem ord_compare_int_gt (a b : ℤ) : Ord.compare a b = .gt ↔ a > b := by
+  constructor
+  · intro h
+    by_contra! hle
+    have hcases := lt_or_eq_of_le hle
+    cases hcases with
+    | inl hlt =>
+      simp only [Ord.compare, compareOfLessAndEq, hlt, ↓reduceIte] at h
+      exact absurd h (by decide)
+    | inr heq =>
+      simp only [Ord.compare, compareOfLessAndEq, heq, lt_irrefl, ↓reduceIte] at h
+      exact absurd h (by decide)
+  · intro h
+    simp only [Ord.compare, compareOfLessAndEq, not_lt.mpr (le_of_lt h), ne_of_gt h, ↓reduceIte]
+
+/-- Ord.compare not gt means ≤ -/
+private theorem ord_compare_ne_gt_iff (a b : ℤ) : (Ord.compare a b != .gt) = true ↔ a ≤ b := by
+  rw [bne_iff_ne, ne_eq]
+  constructor
+  · intro h
+    by_contra! hgt
+    have := ord_compare_int_gt a b |>.mpr hgt
+    exact h this
+  · intro hle hgt
+    have := ord_compare_int_gt a b |>.mp hgt
+    exact not_lt.mpr hle this
+
+/-! ### Comparison Theorems -/
+
+/-- Helper: Convert aligned integer comparison to toRat comparison (case e₁ ≤ e₂) -/
+private theorem aligned_lt_iff_toRat_lt_case1 (d₁ d₂ : Dyadic) (h : d₁.exponent ≤ d₂.exponent) :
+    d₁.mantissa < shiftLeftInt d₂.mantissa (d₂.exponent - d₁.exponent).toNat ↔ d₁.toRat < d₂.toRat
+      := by
+  set shift := (d₂.exponent - d₁.exponent).toNat with hshift_def
+  have hshift : (shift : ℤ) = d₂.exponent - d₁.exponent := by
+    rw [hshift_def, Int.toNat_of_nonneg]; omega
+  rw [toRat_eq, toRat_eq]
+  -- Key: 2^e₂ = 2^shift * 2^e₁
+  have hpow : (2 : ℚ) ^ d₂.exponent = (2 : ℚ) ^ (shift : ℤ) * 2 ^ d₁.exponent := by
+    rw [← zpow_add₀ (two_ne_zero : (2 : ℚ) ≠ 0), hshift]; congr 1; omega
+  have hpos : (0 : ℚ) < 2 ^ d₁.exponent := zpow_pos (by norm_num : (0 : ℚ) < 2) _
+  -- Rewrite RHS: m₂ * 2^e₂ = m₂ * (2^shift * 2^e₁) = (m₂ * 2^shift) * 2^e₁
+  have hrhs : (d₂.mantissa : ℚ) * 2 ^ d₂.exponent = d₂.mantissa * (2 : ℚ) ^ shift * 2 ^
+    d₁.exponent := by
+    rw [hpow, zpow_natCast, mul_assoc]
+  rw [hrhs]
+  -- Now: m₁ < shiftLeftInt m₂ shift ↔ m₁ * 2^e₁ < (m₂ * 2^shift) * 2^e₁
+  rw [int_lt_iff_rat_mul_lt _ _ _ hpos, shiftLeftInt_toRat]
+
+private theorem aligned_eq_iff_toRat_eq_case1 (d₁ d₂ : Dyadic) (h : d₁.exponent ≤ d₂.exponent) :
+    d₁.mantissa = shiftLeftInt d₂.mantissa (d₂.exponent - d₁.exponent).toNat ↔ d₁.toRat = d₂.toRat
+      := by
+  set shift := (d₂.exponent - d₁.exponent).toNat with hshift_def
+  have hshift : (shift : ℤ) = d₂.exponent - d₁.exponent := by
+    rw [hshift_def, Int.toNat_of_nonneg]; omega
+  rw [toRat_eq, toRat_eq]
+  have hpow : (2 : ℚ) ^ d₂.exponent = (2 : ℚ) ^ (shift : ℤ) * 2 ^ d₁.exponent := by
+    rw [← zpow_add₀ (two_ne_zero : (2 : ℚ) ≠ 0), hshift]; congr 1; omega
+  have hne : (2 : ℚ) ^ d₁.exponent ≠ 0 := zpow_ne_zero _ (two_ne_zero)
+  have hrhs : (d₂.mantissa : ℚ) * 2 ^ d₂.exponent = d₂.mantissa * (2 : ℚ) ^ shift * 2 ^
+    d₁.exponent := by
+    rw [hpow, zpow_natCast, mul_assoc]
+  rw [hrhs, int_eq_iff_rat_mul_eq _ _ _ hne, shiftLeftInt_toRat]
+
+private theorem aligned_gt_iff_toRat_gt_case1 (d₁ d₂ : Dyadic) (h : d₁.exponent ≤ d₂.exponent) :
+    d₁.mantissa > shiftLeftInt d₂.mantissa (d₂.exponent - d₁.exponent).toNat ↔ d₁.toRat > d₂.toRat
+      := by
+  set shift := (d₂.exponent - d₁.exponent).toNat with hshift_def
+  have hshift : (shift : ℤ) = d₂.exponent - d₁.exponent := by
+    rw [hshift_def, Int.toNat_of_nonneg]; omega
+  rw [toRat_eq, toRat_eq]
+  have hpow : (2 : ℚ) ^ d₂.exponent = (2 : ℚ) ^ (shift : ℤ) * 2 ^ d₁.exponent := by
+    rw [← zpow_add₀ (two_ne_zero : (2 : ℚ) ≠ 0), hshift]; congr 1; omega
+  have hpos : (0 : ℚ) < 2 ^ d₁.exponent := zpow_pos (by norm_num : (0 : ℚ) < 2) _
+  have hrhs : (d₂.mantissa : ℚ) * 2 ^ d₂.exponent = d₂.mantissa * (2 : ℚ) ^ shift * 2 ^
+    d₁.exponent := by
+    rw [hpow, zpow_natCast, mul_assoc]
+  rw [hrhs, int_gt_iff_rat_mul_gt _ _ _ hpos, shiftLeftInt_toRat]
+
+/-- Helper: Convert aligned integer comparison to toRat comparison (case e₁ > e₂) -/
+private theorem aligned_lt_iff_toRat_lt_case2 (d₁ d₂ : Dyadic) (h : d₁.exponent > d₂.exponent) :
+    shiftLeftInt d₁.mantissa (d₁.exponent - d₂.exponent).toNat < d₂.mantissa ↔ d₁.toRat < d₂.toRat
+      := by
+  set shift := (d₁.exponent - d₂.exponent).toNat with hshift_def
+  have hshift : (shift : ℤ) = d₁.exponent - d₂.exponent := by
+    rw [hshift_def, Int.toNat_of_nonneg]; omega
+  rw [toRat_eq, toRat_eq]
+  have hpow : (2 : ℚ) ^ d₁.exponent = (2 : ℚ) ^ (shift : ℤ) * 2 ^ d₂.exponent := by
+    rw [← zpow_add₀ (two_ne_zero : (2 : ℚ) ≠ 0), hshift]; congr 1; omega
+  have hpos : (0 : ℚ) < 2 ^ d₂.exponent := zpow_pos (by norm_num : (0 : ℚ) < 2) _
+  -- Rewrite LHS: m₁ * 2^e₁ = m₁ * (2^shift * 2^e₂) = (m₁ * 2^shift) * 2^e₂
+  have hlhs : (d₁.mantissa : ℚ) * 2 ^ d₁.exponent = d₁.mantissa * (2 : ℚ) ^ shift * 2 ^
+    d₂.exponent := by
+    rw [hpow, zpow_natCast, mul_assoc]
+  rw [hlhs, int_lt_iff_rat_mul_lt _ _ _ hpos, shiftLeftInt_toRat]
+
+private theorem aligned_eq_iff_toRat_eq_case2 (d₁ d₂ : Dyadic) (h : d₁.exponent > d₂.exponent) :
+    shiftLeftInt d₁.mantissa (d₁.exponent - d₂.exponent).toNat = d₂.mantissa ↔ d₁.toRat = d₂.toRat
+      := by
+  set shift := (d₁.exponent - d₂.exponent).toNat with hshift_def
+  have hshift : (shift : ℤ) = d₁.exponent - d₂.exponent := by
+    rw [hshift_def, Int.toNat_of_nonneg]; omega
+  rw [toRat_eq, toRat_eq]
+  have hpow : (2 : ℚ) ^ d₁.exponent = (2 : ℚ) ^ (shift : ℤ) * 2 ^ d₂.exponent := by
+    rw [← zpow_add₀ (two_ne_zero : (2 : ℚ) ≠ 0), hshift]; congr 1; omega
+  have hne : (2 : ℚ) ^ d₂.exponent ≠ 0 := zpow_ne_zero _ (two_ne_zero)
+  have hlhs : (d₁.mantissa : ℚ) * 2 ^ d₁.exponent = d₁.mantissa * (2 : ℚ) ^ shift * 2 ^
+    d₂.exponent := by
+    rw [hpow, zpow_natCast, mul_assoc]
+  rw [hlhs, int_eq_iff_rat_mul_eq _ _ _ hne, shiftLeftInt_toRat]
+
+private theorem aligned_gt_iff_toRat_gt_case2 (d₁ d₂ : Dyadic) (h : d₁.exponent > d₂.exponent) :
+    shiftLeftInt d₁.mantissa (d₁.exponent - d₂.exponent).toNat > d₂.mantissa ↔ d₁.toRat > d₂.toRat
+      := by
+  set shift := (d₁.exponent - d₂.exponent).toNat with hshift_def
+  have hshift : (shift : ℤ) = d₁.exponent - d₂.exponent := by
+    rw [hshift_def, Int.toNat_of_nonneg]; omega
+  rw [toRat_eq, toRat_eq]
+  have hpow : (2 : ℚ) ^ d₁.exponent = (2 : ℚ) ^ (shift : ℤ) * 2 ^ d₂.exponent := by
+    rw [← zpow_add₀ (two_ne_zero : (2 : ℚ) ≠ 0), hshift]; congr 1; omega
+  have hpos : (0 : ℚ) < 2 ^ d₂.exponent := zpow_pos (by norm_num : (0 : ℚ) < 2) _
+  have hlhs : (d₁.mantissa : ℚ) * 2 ^ d₁.exponent = d₁.mantissa * (2 : ℚ) ^ shift * 2 ^
+    d₂.exponent := by
+    rw [hpow, zpow_natCast, mul_assoc]
+  rw [hlhs, int_gt_iff_rat_mul_gt _ _ _ hpos, shiftLeftInt_toRat]
+
+/-- le is correct with respect to toRat ordering -/
+theorem le_iff_toRat_le (d₁ d₂ : Dyadic) :
+    le d₁ d₂ = true ↔ d₁.toRat ≤ d₂.toRat := by
+  unfold le compare
+  split_ifs with h
+  · -- Case: d₁.exponent ≤ d₂.exponent
+    set shift := (d₂.exponent - d₁.exponent).toNat
+    rw [ord_compare_ne_gt_iff]
+    constructor
+    · intro hle
+      by_cases! hlt : d₁.mantissa < shiftLeftInt d₂.mantissa shift
+      · exact le_of_lt ((aligned_lt_iff_toRat_lt_case1 d₁ d₂ h).mp hlt)
+      · have heq_or_gt := lt_or_eq_of_le hlt
+        cases heq_or_gt with
+        | inl hgt =>
+          exfalso; exact not_lt.mpr hle hgt
+        | inr heq =>
+          exact le_of_eq ((aligned_eq_iff_toRat_eq_case1 d₁ d₂ h).mp heq.symm)
+    · intro hle
+      by_contra! hgt
+      have hgt' := (aligned_gt_iff_toRat_gt_case1 d₁ d₂ h).mp hgt
+      exact not_lt.mpr hle hgt'
+  · -- Case: d₁.exponent > d₂.exponent
+    push Not at h
+    set shift := (d₁.exponent - d₂.exponent).toNat
+    rw [ord_compare_ne_gt_iff]
+    constructor
+    · intro hle
+      by_cases! hlt : shiftLeftInt d₁.mantissa shift < d₂.mantissa
+      · exact le_of_lt ((aligned_lt_iff_toRat_lt_case2 d₁ d₂ h).mp hlt)
+      · have heq_or_gt := lt_or_eq_of_le hlt
+        cases heq_or_gt with
+        | inl hgt =>
+          exfalso; exact not_lt.mpr hle hgt
+        | inr heq =>
+          exact le_of_eq ((aligned_eq_iff_toRat_eq_case2 d₁ d₂ h).mp heq.symm)
+    · intro hle
+      by_contra! hgt
+      have hgt' := (aligned_gt_iff_toRat_gt_case2 d₁ d₂ h).mp hgt
+      exact not_lt.mpr hle hgt'
+
+/-- compare reflects toRat ordering: lt case -/
+theorem compare_lt_iff (d₁ d₂ : Dyadic) :
+    compare d₁ d₂ = .lt ↔ d₁.toRat < d₂.toRat := by
+  unfold compare
+  split_ifs with h
+  · rw [ord_compare_int_lt]; exact aligned_lt_iff_toRat_lt_case1 d₁ d₂ h
+  · push Not at h; rw [ord_compare_int_lt]; exact aligned_lt_iff_toRat_lt_case2 d₁ d₂ h
+
+/-- compare reflects toRat ordering: gt case -/
+theorem compare_gt_iff (d₁ d₂ : Dyadic) :
+    compare d₁ d₂ = .gt ↔ d₁.toRat > d₂.toRat := by
+  unfold compare
+  split_ifs with h
+  · rw [ord_compare_int_gt]; exact aligned_gt_iff_toRat_gt_case1 d₁ d₂ h
+  · push Not at h; rw [ord_compare_int_gt]; exact aligned_gt_iff_toRat_gt_case2 d₁ d₂ h
+
+/-- compare reflects toRat ordering: eq case -/
+theorem compare_eq_iff (d₁ d₂ : Dyadic) :
+    compare d₁ d₂ = .eq ↔ d₁.toRat = d₂.toRat := by
+  unfold compare
+  split_ifs with h
+  · rw [ord_compare_int_eq]; exact aligned_eq_iff_toRat_eq_case1 d₁ d₂ h
+  · push Not at h; rw [ord_compare_int_eq]; exact aligned_eq_iff_toRat_eq_case2 d₁ d₂ h
+
+/-! ### Min/Max Lemmas -/
+
+/-- min produces value ≤ first argument -/
+theorem min_toRat_le_left (d₁ d₂ : Dyadic) : (min d₁ d₂).toRat ≤ d₁.toRat := by
+  unfold min
+  split_ifs with h
+  · exact le_refl _
+  · exact le_of_not_ge (fun hle => h (le_iff_toRat_le d₁ d₂ |>.mpr hle))
+
+/-- min produces value ≤ second argument -/
+theorem min_toRat_le_right (d₁ d₂ : Dyadic) : (min d₁ d₂).toRat ≤ d₂.toRat := by
+  unfold min
+  split_ifs with h
+  · exact le_iff_toRat_le d₁ d₂ |>.mp h
+  · exact le_refl _
+
+/-- first argument ≤ max -/
+theorem le_max_toRat_left (d₁ d₂ : Dyadic) : d₁.toRat ≤ (max d₁ d₂).toRat := by
+  unfold max
+  split_ifs with h
+  · exact le_iff_toRat_le d₁ d₂ |>.mp h
+  · exact le_refl _
+
+/-- second argument ≤ max -/
+theorem le_max_toRat_right (d₁ d₂ : Dyadic) : d₂.toRat ≤ (max d₁ d₂).toRat := by
+  unfold max
+  split_ifs with h
+  · exact le_refl _
+  · exact le_of_not_ge (fun hle => h (le_iff_toRat_le d₁ d₂ |>.mpr hle))
+
+/-- Dyadic.min commutes with toRat -/
+theorem min_toRat (d₁ d₂ : Dyadic) : (min d₁ d₂).toRat = Min.min d₁.toRat d₂.toRat := by
+  unfold min
+  split_ifs with h
+  · -- h : le d₁ d₂
+    have hle : d₁.toRat ≤ d₂.toRat := le_iff_toRat_le d₁ d₂ |>.mp h
+    exact (min_eq_left hle).symm
+  · -- ¬ le d₁ d₂, so d₂.toRat < d₁.toRat
+    have hgt : d₂.toRat < d₁.toRat := by
+      have := le_iff_toRat_le d₁ d₂
+      exact lt_of_not_ge (fun hle => h (this.mpr hle))
+    exact (min_eq_right (le_of_lt hgt)).symm
+
+/-- Dyadic.max commutes with toRat -/
+theorem max_toRat (d₁ d₂ : Dyadic) : (max d₁ d₂).toRat = Max.max d₁.toRat d₂.toRat := by
+  unfold max
+  split_ifs with h
+  · -- h : le d₁ d₂
+    have hle : d₁.toRat ≤ d₂.toRat := le_iff_toRat_le d₁ d₂ |>.mp h
+    exact (max_eq_right hle).symm
+  · -- ¬ le d₁ d₂, so d₂.toRat < d₁.toRat
+    have hgt : d₂.toRat < d₁.toRat := by
+      have := le_iff_toRat_le d₁ d₂
+      exact lt_of_not_ge (fun hle => h (this.mpr hle))
+    exact (max_eq_left (le_of_lt hgt)).symm
+
+/-- min4 ≤ max4 -/
+theorem min4_le_max4 (a b c d : Dyadic) : (min4 a b c d).toRat ≤ (max4 a b c d).toRat := by
+  unfold min4 max4
+  calc (min (min a b) (min c d)).toRat
+      ≤ (min a b).toRat := min_toRat_le_left _ _
+    _ ≤ a.toRat := min_toRat_le_left _ _
+    _ ≤ (max a b).toRat := le_max_toRat_left _ _
+    _ ≤ (max (max a b) (max c d)).toRat := le_max_toRat_left _ _
+
+/-! ### Normalize Lemmas -/
+
+/-- normalizeDown produces value ≤ original -/
+theorem toRat_normalizeDown_le (d : Dyadic) (maxBits : Nat) :
+    (d.normalizeDown maxBits).toRat ≤ d.toRat := by
+  unfold normalizeDown normalize
+  simp only [Bool.false_eq_true, ↓reduceIte]
+  split_ifs with h
+  · exact le_refl _
+  · exact toRat_shiftDown_le d _
+
+/-- normalizeUp produces value ≥ original -/
+theorem toRat_normalizeUp_ge (d : Dyadic) (maxBits : Nat) :
+    d.toRat ≤ (d.normalizeUp maxBits).toRat := by
+  unfold normalizeUp normalize
+  simp only [↓reduceIte]
+  split_ifs with h
+  · exact le_refl _
+  · exact toRat_shiftUp_ge d _
+
+/-! ### Scale2 Lemmas -/
+
+/-- scale2 multiplies by 2^n -/
+theorem toRat_scale2 (d : Dyadic) (n : Int) :
+    (d.scale2 n).toRat = d.toRat * (2 : ℚ) ^ n := by
+  unfold scale2
+  rw [toRat_eq, toRat_eq]
+  simp only []
+  rw [mul_assoc, ← zpow_add₀ (two_ne_zero : (2 : ℚ) ≠ 0)]
+
+/-- scale2 preserves order -/
+theorem toRat_scale2_le_scale2 (d₁ d₂ : Dyadic) (n : Int) (h : d₁.toRat ≤ d₂.toRat) :
+    (d₁.scale2 n).toRat ≤ (d₂.scale2 n).toRat := by
+  rw [toRat_scale2, toRat_scale2]
+  exact mul_le_mul_of_nonneg_right h (zpow_nonneg (by norm_num : (0 : ℚ) ≤ 2) n)
+
+/-! ### Square Root Operations -/
+
+/-- Integer square root of a natural number.
+    Satisfies: `(intSqrtNat n)^2 ≤ n < (intSqrtNat n + 1)^2` -/
+def intSqrtNat (n : Nat) : Nat := Nat.sqrt n
+
+/-- Integer square root of a non-negative integer.
+    Returns 0 for negative inputs. -/
+def intSqrt (n : Int) : Int :=
+  if n < 0 then 0 else Int.ofNat (intSqrtNat n.toNat)
+
+/-- Key property of Nat.sqrt: `Nat.sqrt n ^ 2 ≤ n` -/
+private theorem nat_sqrt_sq_le (n : Nat) : (Nat.sqrt n) ^ 2 ≤ n := Nat.sqrt_le' n
+
+/-- Key property of Nat.sqrt: `n < (Nat.sqrt n + 1) ^ 2` -/
+private theorem nat_lt_succ_sqrt_sq (n : Nat) : n < (Nat.sqrt n + 1) ^ 2 := by
+  have h := Nat.lt_succ_sqrt n
+  simp only [Nat.succ_eq_add_one] at h
+  rw [sq]
+  exact h
+
+/-- intSqrt n ^ 2 ≤ n for n ≥ 0 -/
+theorem intSqrt_sq_le {n : Int} (hn : 0 ≤ n) : (intSqrt n) ^ 2 ≤ n := by
+  simp only [intSqrt, not_lt.mpr hn, ↓reduceIte, intSqrtNat]
+  have h := nat_sqrt_sq_le n.toNat
+  calc ((Nat.sqrt n.toNat : ℤ) ^ 2 : ℤ)
+      = ((Nat.sqrt n.toNat) ^ 2 : ℕ) := by norm_cast
+    _ ≤ (n.toNat : ℤ) := Int.ofNat_le.mpr h
+    _ = n := Int.toNat_of_nonneg hn
+
+/-- n < (intSqrt n + 1) ^ 2 for n ≥ 0 -/
+theorem int_lt_succ_sqrt_sq {n : Int} (hn : 0 ≤ n) : n < (intSqrt n + 1) ^ 2 := by
+  simp only [intSqrt, not_lt.mpr hn, ↓reduceIte, intSqrtNat]
+  have h := nat_lt_succ_sqrt_sq n.toNat
+  calc n = (n.toNat : ℤ) := (Int.toNat_of_nonneg hn).symm
+    _ < ((Nat.sqrt n.toNat + 1) ^ 2 : ℕ) := Int.ofNat_lt.mpr h
+    _ = ((Nat.sqrt n.toNat : ℤ) + 1) ^ 2 := by norm_cast
+
+/-- intSqrt is nonnegative for nonnegative inputs -/
+theorem intSqrt_nonneg {n : Int} (hn : 0 ≤ n) : 0 ≤ intSqrt n := by
+  simp only [intSqrt, not_lt.mpr hn, ↓reduceIte, intSqrtNat]
+  exact Int.natCast_nonneg _
+
+/-- Compute sqrt(d) with a target exponent `prec`.
+    Returns a Dyadic with exponent `prec` such that result ≤ sqrt(d).
+
+    For d = m * 2^e, we compute:
+    - shift = e - 2*prec (to align for sqrt)
+    - m' = m * 2^shift (or m / 2^(-shift) if shift < 0)
+    - result = floor(sqrt(m')) * 2^prec
+
+    This ensures: result.toRat ≤ sqrt(d.toRat) -/
+def sqrtDown (d : Dyadic) (prec : Int) : Dyadic :=
+  if d.mantissa < 0 then zero
+  else
+    let shift := d.exponent - 2 * prec
+    let m := if shift ≥ 0 then
+      d.mantissa * (pow2Nat shift.toNat : Int)
+    else
+      d.mantissa / (pow2Nat (-shift).toNat : Int)
+    ⟨intSqrt m, prec⟩
+
+/-- Compute sqrt(d) rounded up with target exponent `prec`.
+    Returns a Dyadic with exponent `prec` such that result ≥ sqrt(d).
+
+    Uses the same computation as sqrtDown, but if the result is not
+    a perfect square, adds 1 to round up. -/
+def sqrtUp (d : Dyadic) (prec : Int) : Dyadic :=
+  if d.mantissa < 0 then zero
+  else
+    let shift := d.exponent - 2 * prec
+    let m := if shift ≥ 0 then
+      d.mantissa * (pow2Nat shift.toNat : Int)
+    else
+      d.mantissa / (pow2Nat (-shift).toNat : Int)
+    let s := intSqrt m
+    -- If m is a perfect square of s, return s; otherwise s+1
+    if s * s = m then ⟨s, prec⟩ else ⟨s + 1, prec⟩
+
+/-! #### Sqrt Correctness Theorems -/
+
+/-- Helper: For non-negative m and shift ≥ 0, the scaled mantissa is non-negative -/
+private theorem sqrt_scaled_shift_nonneg {m : Int} {shift : Int}
+    (hm : 0 ≤ m) (_hshift : 0 ≤ shift) :
+    0 ≤ m * (pow2Nat shift.toNat : Int) := by
+  apply mul_nonneg hm
+  exact Int.natCast_nonneg _
+
+/-- sqrtDown is non-negative for non-negative inputs -/
+theorem sqrtDown_nonneg (d : Dyadic) (prec : Int) (hd : 0 ≤ d.mantissa) :
+    0 ≤ (sqrtDown d prec).mantissa := by
+  simp only [sqrtDown, not_lt.mpr hd, ↓reduceIte]
+  set shift := d.exponent - 2 * prec
+  have hm_nonneg : 0 ≤ (if shift ≥ 0 then
+      d.mantissa * (pow2Nat shift.toNat : Int)
+    else
+      d.mantissa / (pow2Nat (-shift).toNat : Int)) := by
+    by_cases! hsh : shift ≥ 0
+    · simp only [hsh, ↓reduceIte]
+      exact sqrt_scaled_shift_nonneg hd hsh
+    · simp only [not_le.mpr hsh, ↓reduceIte]
+      exact Int.ediv_nonneg hd (Int.natCast_nonneg _)
+  exact intSqrt_nonneg hm_nonneg
+
+/-- sqrtUp is non-negative for non-negative inputs -/
+theorem sqrtUp_nonneg (d : Dyadic) (prec : Int) (hd : 0 ≤ d.mantissa) :
+    0 ≤ (sqrtUp d prec).mantissa := by
+  simp only [sqrtUp, not_lt.mpr hd, ↓reduceIte]
+  set shift := d.exponent - 2 * prec
+  have hm_nonneg : 0 ≤ (if shift ≥ 0 then
+      d.mantissa * (pow2Nat shift.toNat : Int)
+    else
+      d.mantissa / (pow2Nat (-shift).toNat : Int)) := by
+    by_cases! hsh : shift ≥ 0
+    · simp only [hsh, ↓reduceIte]
+      exact sqrt_scaled_shift_nonneg hd hsh
+    · simp only [not_le.mpr hsh, ↓reduceIte]
+      exact Int.ediv_nonneg hd (Int.natCast_nonneg _)
+  set m := (if shift ≥ 0 then
+      d.mantissa * (pow2Nat shift.toNat : Int)
+    else
+      d.mantissa / (pow2Nat (-shift).toNat : Int))
+  set s := intSqrt m
+  by_cases hperfect : s * s = m
+  · simp only [hperfect, ↓reduceIte]
+    exact intSqrt_nonneg hm_nonneg
+  · simp only [hperfect, ↓reduceIte]
+    calc 0 ≤ s := intSqrt_nonneg hm_nonneg
+      _ ≤ s + 1 := by omega
+
+/-- sqrtDown.toRat ≤ sqrtUp.toRat for non-negative inputs -/
+theorem sqrtDown_le_sqrtUp (d : Dyadic) (prec : Int) (hd : 0 ≤ d.mantissa) :
+    (sqrtDown d prec).toRat ≤ (sqrtUp d prec).toRat := by
+  simp only [sqrtDown, sqrtUp, not_lt.mpr hd, ↓reduceIte]
+  set shift := d.exponent - 2 * prec
+  have hm_nonneg : 0 ≤ (if shift ≥ 0 then
+      d.mantissa * (pow2Nat shift.toNat : Int)
+    else
+      d.mantissa / (pow2Nat (-shift).toNat : Int)) := by
+    by_cases! hsh : shift ≥ 0
+    · simp only [hsh, ↓reduceIte]
+      exact sqrt_scaled_shift_nonneg hd hsh
+    · simp only [not_le.mpr hsh, ↓reduceIte]
+      exact Int.ediv_nonneg hd (Int.natCast_nonneg _)
+  set m := (if shift ≥ 0 then
+      d.mantissa * (pow2Nat shift.toNat : Int)
+    else
+      d.mantissa / (pow2Nat (-shift).toNat : Int))
+  set s := intSqrt m
+  by_cases hperfect : s * s = m
+  · -- Perfect square: both return s
+    simp only [hperfect, ↓reduceIte]
+    exact le_refl _
+  · -- Non-perfect: sqrtDown returns s, sqrtUp returns s+1
+    simp only [hperfect, ↓reduceIte]
+    rw [toRat_eq, toRat_eq]
+    apply mul_le_mul_of_nonneg_right _ (le_of_lt (zpow_pos (by norm_num : (0 : ℚ) < 2) _))
+    simp only [Int.cast_add, Int.cast_one]
+    exact le_add_of_nonneg_right (by norm_num : (0 : ℚ) ≤ 1)
+
+end Dyadic
+end LeanCert.Core
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Unified Expression AST
+
+This file defines the unified AST for real expressions (`Expr`) and its
+evaluation semantics. All numerical algorithms in LeanCert operate on this
+single expression type.
+
+## Main definitions
+
+* `LeanCert.Core.Expr` - The expression AST supporting algebraic and transcendental operations
+* `LeanCert.Core.Expr.eval` - Evaluation of expressions given a variable assignment
+
+## Design notes
+
+The expression type uses natural number indices for variables. This simplifies
+the interval evaluation and automatic differentiation implementations.
+-/
+
+/-! ## Auxiliary definition for atanh
+
+Since Mathlib doesn't provide `Real.atanh`, we define it here using the
+standard formula: `atanh x = (1/2) * log((1+x)/(1-x))` for `|x| < 1`.
+-/
+
+@[expose] public section
+
+namespace LeanCert.Core
+
+/-- The inverse hyperbolic tangent function.
+    Defined as `atanh x = (1/2) * log((1+x)/(1-x))` for `|x| < 1`. -/
+noncomputable def Real.atanh (x : ℝ) : ℝ :=
+  (1 / 2) * Real.log ((1 + x) / (1 - x))
+
+/-- For |x| < 1, the argument (1+x)/(1-x) is positive. -/
+theorem Real.atanh_arg_pos {x : ℝ} (hx : |x| < 1) : 0 < (1 + x) / (1 - x) := by
+  have h1 : -1 < x := by linarith [abs_lt.mp hx]
+  have h2 : x < 1 := (abs_lt.mp hx).2
+  have hnum : 0 < 1 + x := by linarith
+  have hdenom : 0 < 1 - x := by linarith
+  exact div_pos hnum hdenom
+
+/-- atanh(0) = 0 -/
+@[simp]
+theorem Real.atanh_zero : Real.atanh 0 = 0 := by
+  simp only [Real.atanh, add_zero, sub_zero, div_one, Real.log_one, mul_zero]
+
+/-- atanh(-x) = -atanh(x) for |x| < 1 -/
+theorem Real.atanh_neg {x : ℝ} (hx : |x| < 1) : Real.atanh (-x) = -Real.atanh x := by
+  simp only [Real.atanh]
+  have hlo : -1 < x := (abs_lt.mp hx).1
+  have hhi : x < 1 := (abs_lt.mp hx).2
+  have hpos1 : 0 < 1 + x := by linarith
+  have hpos2 : 0 < 1 - x := by linarith
+  have h1 : 1 + -x = 1 - x := by ring
+  have h2 : 1 - -x = 1 + x := by ring
+  rw [h1, h2]
+  rw [Real.log_div (ne_of_gt hpos2) (ne_of_gt hpos1)]
+  rw [Real.log_div (ne_of_gt hpos1) (ne_of_gt hpos2)]
+  ring
+
+/-- atanh is strictly monotone on (-1, 1) -/
+theorem Real.atanh_strictMonoOn : StrictMonoOn Real.atanh (Set.Ioo (-1) 1) := by
+  intro x hx y hy hxy
+  simp only [Real.atanh]
+  have hx1 : 0 < 1 + x := by linarith [hx.1]
+  have hx2 : 0 < 1 - x := by linarith [hx.2]
+  have hy1 : 0 < 1 + y := by linarith [hy.1]
+  have hy2 : 0 < 1 - y := by linarith [hy.2]
+  have hargx : 0 < (1 + x) / (1 - x) := div_pos hx1 hx2
+  have hargy : 0 < (1 + y) / (1 - y) := div_pos hy1 hy2
+  gcongr 1
+  rw [Real.log_lt_log_iff hargx hargy]
+  -- Need (1+x)/(1-x) < (1+y)/(1-y) when x < y
+  rw [div_lt_div_iff₀ hx2 hy2]
+  ring_nf
+  linarith
+
+/-- atanh is monotone on (-1, 1): if x ≤ y then atanh x ≤ atanh y -/
+theorem Real.atanh_mono {x y : ℝ} (hx : |x| < 1) (hy : |y| < 1) (hxy : x ≤ y) :
+    Real.atanh x ≤ Real.atanh y := by
+  rcases hxy.lt_or_eq with hlt | heq
+  · have hx' : x ∈ Set.Ioo (-1 : ℝ) 1 := ⟨(abs_lt.mp hx).1, (abs_lt.mp hx).2⟩
+    have hy' : y ∈ Set.Ioo (-1 : ℝ) 1 := ⟨(abs_lt.mp hy).1, (abs_lt.mp hy).2⟩
+    exact le_of_lt (Real.atanh_strictMonoOn hx' hy' hlt)
+  · rw [heq]
+
+/-- The error function: erf(x) = (2/√π) ∫₀ˣ exp(-t²) dt.
+    Essential for statistical and financial modeling (normal distribution CDF).
+    Uses interval integral notation (∫ t in 0..x) which handles negative x correctly. -/
+noncomputable def Real.erf (x : ℝ) : ℝ :=
+  (2 / Real.sqrt Real.pi) * ∫ t in (0:ℝ)..x, Real.exp (-(t^2))
+
+/-- The Gaussian function exp(-t²) is always positive. -/
+private theorem exp_neg_sq_pos (t : ℝ) : 0 < Real.exp (-(t^2)) :=
+  Real.exp_pos _
+
+/-- The Gaussian function exp(-t²) is integrable on any compact interval. -/
+private theorem intervalIntegrable_exp_neg_sq (a b : ℝ) :
+    IntervalIntegrable (fun t => Real.exp (-(t^2))) MeasureTheory.volume a b := by
+  apply Continuous.intervalIntegrable
+  exact Real.continuous_exp.comp (continuous_neg.comp (continuous_pow 2))
+
+/-- Helper: The interval integral of a positive function over [0, x] is bounded by the
+    half-Gaussian integral when x ≥ 0. -/
+private theorem integral_exp_neg_sq_le_half_gaussian (x : ℝ) (hx : 0 ≤ x) :
+    ∫ t in (0:ℝ)..x, Real.exp (-(t^2)) ≤ Real.sqrt Real.pi / 2 := by
+  have hgauss : ∫ t in Set.Ioi 0, Real.exp (-(1:ℝ) * t^2) = Real.sqrt (Real.pi / 1) / 2 :=
+    integral_gaussian_Ioi 1
+  simp only [div_one] at hgauss
+  -- The interval integral equals the set integral on Ioc when x ≥ 0
+  rw [intervalIntegral.integral_of_le hx]
+  -- Need: ∫ t in Ioc 0 x, ... ≤ ∫ t in Ioi 0, ...
+  -- We have Ioc 0 x ⊆ Ioi 0 for x ≥ 0
+  have hint : MeasureTheory.Integrable (fun t => Real.exp (-(1:ℝ) * t^2)) MeasureTheory.volume :=
+    integrable_exp_neg_mul_sq (by norm_num : (0:ℝ) < 1)
+  have hint_Ioi : MeasureTheory.Integrable (fun t => Real.exp (-(t^2)))
+    (MeasureTheory.volume.restrict (Set.Ioi 0)) := by
+    have heq : (fun t : ℝ => Real.exp (-(t^2))) = (fun t => Real.exp (-(1:ℝ) * t^2)) := by
+      funext t; ring_nf
+    rw [heq]
+    exact hint.restrict
+  calc ∫ t in Set.Ioc 0 x, Real.exp (-(t^2))
+      ≤ ∫ t in Set.Ioi 0, Real.exp (-(t^2)) := by
+        apply MeasureTheory.setIntegral_mono_set hint_Ioi
+        · -- Nonnegativity
+          filter_upwards with t using le_of_lt (exp_neg_sq_pos t)
+        · -- Set inclusion: Ioc 0 x ⊆ Ioi 0 (need to convert to ae form)
+          exact Filter.Eventually.of_forall (fun t ht => Set.Ioc_subset_Ioi_self ht)
+      _ = Real.sqrt Real.pi / 2 := by
+        have heq : (fun t : ℝ => Real.exp (-(t^2))) = (fun t => Real.exp (-(1:ℝ) * t^2)) := by
+          funext t; ring_nf
+        rw [heq, hgauss]
+
+/-- Helper: The interval integral of a positive function over [0, x] is nonneg when x ≥ 0. -/
+private theorem integral_exp_neg_sq_nonneg (x : ℝ) (hx : 0 ≤ x) :
+    0 ≤ ∫ t in (0:ℝ)..x, Real.exp (-(t^2)) := by
+  rw [intervalIntegral.integral_of_le hx]
+  apply MeasureTheory.setIntegral_nonneg measurableSet_Ioc
+  intro t _
+  exact le_of_lt (exp_neg_sq_pos t)
+
+/-- The error function is bounded above by 1. -/
+theorem Real.erf_le_one (x : ℝ) : Real.erf x ≤ 1 := by
+  unfold Real.erf
+  by_cases! hx : 0 ≤ x
+  · -- Case x ≥ 0: Use that integral is bounded by √π/2
+    have h1 : Real.sqrt Real.pi > 0 := Real.sqrt_pos.mpr Real.pi_pos
+    have h2 : ∫ t in (0:ℝ)..x, Real.exp (-(t^2)) ≤ Real.sqrt Real.pi / 2 :=
+      integral_exp_neg_sq_le_half_gaussian x hx
+    calc (2 / Real.sqrt Real.pi) * ∫ t in (0:ℝ)..x, Real.exp (-(t^2))
+        ≤ (2 / Real.sqrt Real.pi) * (Real.sqrt Real.pi / 2) := by
+          apply mul_le_mul_of_nonneg_left h2
+          exact div_nonneg (by norm_num) (le_of_lt h1)
+        _ = 1 := by field_simp
+  · -- Case x < 0: Integral is negative (∫ 0..x = -∫ x..0), so erf(x) ≤ 0 ≤ 1
+    have h1 : Real.sqrt Real.pi > 0 := Real.sqrt_pos.mpr Real.pi_pos
+    have hxle : x ≤ 0 := le_of_lt hx
+    -- For x < 0: ∫ 0..x = -∫ x..0, and ∫ x..0 ≥ 0 since we integrate a positive function
+    have h2 : ∫ t in (0:ℝ)..x, Real.exp (-(t^2)) ≤ 0 := by
+      rw [intervalIntegral.integral_symm]
+      apply neg_nonpos.mpr
+      -- ∫ x..0 for x < 0 means integrating from x to 0, i.e., over [x, 0]
+      rw [intervalIntegral.integral_of_le hxle]
+      apply MeasureTheory.setIntegral_nonneg measurableSet_Ioc
+      intro t _
+      exact le_of_lt (exp_neg_sq_pos t)
+    calc (2 / Real.sqrt Real.pi) * ∫ t in (0:ℝ)..x, Real.exp (-(t^2))
+        ≤ (2 / Real.sqrt Real.pi) * 0 := by
+          apply mul_le_mul_of_nonneg_left h2
+          exact div_nonneg (by norm_num) (le_of_lt h1)
+        _ = 0 := by ring
+        _ ≤ 1 := by norm_num
+
+/-- Helper: The integral ∫ x..0 exp(-t²) for x ≤ 0 is bounded by √π/2. -/
+private theorem integral_exp_neg_sq_le_half_gaussian' (x : ℝ) (hx : x ≤ 0) :
+    ∫ t in x..(0:ℝ), Real.exp (-(t^2)) ≤ Real.sqrt Real.pi / 2 := by
+  -- Use substitution: the function is even, so ∫ x..0 = ∫ 0..-x
+  have heq : ∀ t : ℝ, Real.exp (-(t^2)) = Real.exp (-((-t)^2)) := fun t => by ring_nf
+  have hsub : ∫ t in x..(0:ℝ), Real.exp (-(t^2)) = ∫ t in (0:ℝ)..(-x), Real.exp (-(t^2)) := by
+    calc ∫ t in x..(0:ℝ), Real.exp (-(t^2))
+        = ∫ t in x..(0:ℝ), Real.exp (-((-t)^2)) := by simp only [neg_sq]
+      _ = ∫ t in (-0:ℝ)..(-x), Real.exp (-(t^2)) := by
+          rw [intervalIntegral.integral_comp_neg (fun t => Real.exp (-(t^2)))]
+      _ = ∫ t in (0:ℝ)..(-x), Real.exp (-(t^2)) := by simp only [neg_zero]
+  rw [hsub]
+  exact integral_exp_neg_sq_le_half_gaussian (-x) (by linarith)
+
+/-- The error function is bounded below by -1. -/
+theorem Real.neg_one_le_erf (x : ℝ) : -1 ≤ Real.erf x := by
+  unfold Real.erf
+  by_cases! hx : 0 ≤ x
+  · -- Case x ≥ 0: Integral is nonneg, so erf(x) ≥ 0 > -1
+    have h1 : Real.sqrt Real.pi > 0 := Real.sqrt_pos.mpr Real.pi_pos
+    have h2 : 0 ≤ ∫ t in (0:ℝ)..x, Real.exp (-(t^2)) := integral_exp_neg_sq_nonneg x hx
+    have h3 : 0 ≤ (2 / Real.sqrt Real.pi) * ∫ t in (0:ℝ)..x, Real.exp (-(t^2)) := by
+      apply mul_nonneg
+      · exact div_nonneg (by norm_num) (le_of_lt h1)
+      · exact h2
+    linarith
+  · -- Case x < 0: ∫ 0..x = -∫ x..0, and ∫ x..0 ≤ √π/2
+    have h1 : Real.sqrt Real.pi > 0 := Real.sqrt_pos.mpr Real.pi_pos
+    have hxle : x ≤ 0 := le_of_lt hx
+    -- ∫ 0..x = -∫ x..0
+    rw [intervalIntegral.integral_symm]
+    have h2 : ∫ t in x..(0:ℝ), Real.exp (-(t^2)) ≤ Real.sqrt Real.pi / 2 :=
+      integral_exp_neg_sq_le_half_gaussian' x hxle
+    -- Need: -1 ≤ (2/√π) * (-∫ x..0 f)
+    -- i.e., -1 ≤ -(2/√π) * ∫ x..0 f
+    -- i.e., (2/√π) * ∫ x..0 f ≤ 1
+    -- From h2: ∫ x..0 f ≤ √π/2, so (2/√π) * ∫ x..0 f ≤ (2/√π) * (√π/2) = 1
+    have h3 : (2 / Real.sqrt Real.pi) * ∫ t in x..(0:ℝ), Real.exp (-(t^2)) ≤ 1 := by
+      calc (2 / Real.sqrt Real.pi) * ∫ t in x..(0:ℝ), Real.exp (-(t^2))
+          ≤ (2 / Real.sqrt Real.pi) * (Real.sqrt Real.pi / 2) := by
+            apply mul_le_mul_of_nonneg_left h2
+            exact div_nonneg (by norm_num) (le_of_lt h1)
+          _ = 1 := by field_simp
+    linarith
+
+/-- The error function always lies in `[-1, 1]`. -/
+theorem Real.erf_mem_Icc (x : ℝ) : Real.erf x ∈ Set.Icc (-1 : ℝ) 1 :=
+  ⟨Real.neg_one_le_erf x, Real.erf_le_one x⟩
+
+/-- The normalized sinc function always lies in `[-1, 1]`. -/
+theorem Real.sinc_mem_Icc (x : ℝ) : Real.sinc x ∈ Set.Icc (-1 : ℝ) 1 :=
+  ⟨Real.neg_one_le_sinc x, Real.sinc_le_one x⟩
+
+/-- Named mathematical constants with known interval bounds.
+    Adding a new constant (e.g., Catalan's) only requires extending this enum
+    and its lookup tables — zero evaluator files need updating. -/
+inductive MathConst where
+  | pi
+  | eulerMascheroni
+  deriving Repr, DecidableEq, Inhabited
+
+/-- The real value of a named mathematical constant. -/
+noncomputable def MathConst.toReal : MathConst → ℝ
+  | .pi => Real.pi
+  | .eulerMascheroni => Real.eulerMascheroniConstant
+
+/-- Float approximation for heuristic evaluation (unverified). -/
+def MathConst.toFloat : MathConst → Float
+  | .pi => 3.141592653589793
+  | .eulerMascheroni => 0.5772156649015329
+
+/-- Rational approximation for display/debugging (unverified). -/
+def MathConst.toRatApprox : MathConst → ℚ
+  | .pi => 157 / 50
+  | .eulerMascheroni => 577 / 1000
+
+/-- Unified AST for real-valued expressions. -/
+inductive Expr where
+  /-- Rational constant -/
+  | const (q : ℚ)
+  /-- Variable with de Bruijn-style index -/
+  | var (idx : Nat)
+  /-- Addition -/
+  | add (e₁ e₂ : Expr)
+  /-- Multiplication -/
+  | mul (e₁ e₂ : Expr)
+  /-- Negation -/
+  | neg (e : Expr)
+  /-- Multiplicative inverse (partial: undefined at 0) -/
+  | inv (e : Expr)
+  /-- Exponential function -/
+  | exp (e : Expr)
+  /-- Sine function -/
+  | sin (e : Expr)
+  /-- Cosine function -/
+  | cos (e : Expr)
+  /-- Natural logarithm (partial: undefined for x ≤ 0) -/
+  | log (e : Expr)
+  /-- Arctangent function -/
+  | atan (e : Expr)
+  /-- Inverse hyperbolic sine (arsinh) -/
+  | arsinh (e : Expr)
+  /-- Inverse hyperbolic tangent (partial: undefined for |x| ≥ 1) -/
+  | atanh (e : Expr)
+  /-- Sinc function: sinc(x) = sin(x)/x for x ≠ 0, sinc(0) = 1 -/
+  | sinc (e : Expr)
+  /-- Error function: erf(x) = (2/√π) ∫₀ˣ exp(-t²) dt -/
+  | erf (e : Expr)
+  /-- Hyperbolic sine: sinh(x) = (exp(x) - exp(-x)) / 2 -/
+  | sinh (e : Expr)
+  /-- Hyperbolic cosine: cosh(x) = (exp(x) + exp(-x)) / 2 -/
+  | cosh (e : Expr)
+  /-- Hyperbolic tangent: tanh(x) = sinh(x) / cosh(x) ∈ (-1, 1) -/
+  | tanh (e : Expr)
+  /-- Square root (partial: undefined for x < 0) -/
+  | sqrt (e : Expr)
+  /-- A named mathematical constant (π, γ, …) looked up from a table. -/
+  | namedConst (c : MathConst)
+  deriving Repr, DecidableEq, Inhabited
+
+namespace Expr
+
+/-- Subtraction as a derived operation -/
+def sub (e₁ e₂ : Expr) : Expr := add e₁ (neg e₂)
+
+/-- Division as a derived operation -/
+def div (e₁ e₂ : Expr) : Expr := mul e₁ (inv e₂)
+
+/-- Integer power (non-negative exponent) -/
+def pow (e : Expr) : Nat → Expr
+  | 0 => const 1
+  | n + 1 => mul e (pow e n)
+
+/-- Absolute value as a derived operation: |x| = sqrt(x²)
+    This gives correct results for all real x (except at 0 for interval purposes) -/
+def abs (e : Expr) : Expr := sqrt (mul e e)
+
+/-- Evaluate an expression given a variable assignment ρ : Nat → ℝ -/
+noncomputable def eval (ρ : Nat → ℝ) : Expr → ℝ
+  | const q => (q : ℝ)
+  | var idx => ρ idx
+  | add e₁ e₂ => eval ρ e₁ + eval ρ e₂
+  | mul e₁ e₂ => eval ρ e₁ * eval ρ e₂
+  | neg e => -(eval ρ e)
+  | inv e => (eval ρ e)⁻¹
+  | exp e => Real.exp (eval ρ e)
+  | sin e => Real.sin (eval ρ e)
+  | cos e => Real.cos (eval ρ e)
+  | log e => Real.log (eval ρ e)
+  | atan e => Real.arctan (eval ρ e)
+  | arsinh e => Real.arsinh (eval ρ e)
+  | atanh e => Real.atanh (eval ρ e)
+  | sinc e => Real.sinc (eval ρ e)
+  | erf e => Real.erf (eval ρ e)
+  | sinh e => Real.sinh (eval ρ e)
+  | cosh e => Real.cosh (eval ρ e)
+  | tanh e => Real.tanh (eval ρ e)
+  | sqrt e => Real.sqrt (eval ρ e)
+  | namedConst c => c.toReal
+
+/-- Update variable assignment at a specific index -/
+def updateVar (ρ : Nat → ℝ) (idx : Nat) (x : ℝ) : Nat → ℝ :=
+  fun i => if i = idx then x else ρ i
+
+/-- Notation for replacing one coordinate of a variable environment. -/
+notation ρ "[" idx " ↦ " x "]" => updateVar ρ idx x
+
+@[simp]
+theorem updateVar_same (ρ : Nat → ℝ) (idx : Nat) (x : ℝ) :
+    updateVar ρ idx x idx = x := by simp [updateVar]
+
+@[simp]
+theorem updateVar_other (ρ : Nat → ℝ) (idx i : Nat) (x : ℝ) (h : i ≠ idx) :
+    updateVar ρ idx x i = ρ i := by simp [updateVar, h]
+
+theorem updateVar_self (ρ : Nat → ℝ) (idx : Nat) :
+    updateVar ρ idx (ρ idx) = ρ := by
+  funext i
+  simp only [updateVar]
+  split_ifs with h
+  · rw [h]
+  · rfl
+
+/-- Evaluate `e` as a scalar function of variable `idx`, with all other
+    variables fixed by `ρ`. This represents the map t ↦ eval ρ[idx ↦ t] e. -/
+noncomputable abbrev evalAlong (e : Expr) (ρ : Nat → ℝ) (idx : Nat) : ℝ → ℝ :=
+  fun t => eval (updateVar ρ idx t) e
+
+/-- The set of free variable indices in an expression -/
+def freeVars : Expr → Finset Nat
+  | const _ => ∅
+  | var idx => {idx}
+  | add e₁ e₂ => freeVars e₁ ∪ freeVars e₂
+  | mul e₁ e₂ => freeVars e₁ ∪ freeVars e₂
+  | neg e => freeVars e
+  | inv e => freeVars e
+  | exp e => freeVars e
+  | sin e => freeVars e
+  | cos e => freeVars e
+  | log e => freeVars e
+  | atan e => freeVars e
+  | arsinh e => freeVars e
+  | atanh e => freeVars e
+  | sinc e => freeVars e
+  | erf e => freeVars e
+  | sinh e => freeVars e
+  | cosh e => freeVars e
+  | tanh e => freeVars e
+  | sqrt e => freeVars e
+  | namedConst _ => ∅
+
+/-- An expression is closed if it has no free variables -/
+def isClosed (e : Expr) : Prop := freeVars e = ∅
+
+-- Basic lemmas about eval
+
+@[simp]
+theorem eval_const (ρ : Nat → ℝ) (q : ℚ) : eval ρ (const q) = q := rfl
+
+@[simp]
+theorem eval_var (ρ : Nat → ℝ) (idx : Nat) : eval ρ (var idx) = ρ idx := rfl
+
+@[simp]
+theorem eval_add (ρ : Nat → ℝ) (e₁ e₂ : Expr) :
+    eval ρ (add e₁ e₂) = eval ρ e₁ + eval ρ e₂ := rfl
+
+@[simp]
+theorem eval_mul (ρ : Nat → ℝ) (e₁ e₂ : Expr) :
+    eval ρ (mul e₁ e₂) = eval ρ e₁ * eval ρ e₂ := rfl
+
+@[simp]
+theorem eval_neg (ρ : Nat → ℝ) (e : Expr) : eval ρ (neg e) = -(eval ρ e) := rfl
+
+@[simp]
+theorem eval_inv (ρ : Nat → ℝ) (e : Expr) : eval ρ (inv e) = (eval ρ e)⁻¹ := rfl
+
+@[simp]
+theorem eval_exp (ρ : Nat → ℝ) (e : Expr) : eval ρ (exp e) = Real.exp (eval ρ e) := rfl
+
+@[simp]
+theorem eval_sin (ρ : Nat → ℝ) (e : Expr) : eval ρ (sin e) = Real.sin (eval ρ e) := rfl
+
+@[simp]
+theorem eval_cos (ρ : Nat → ℝ) (e : Expr) : eval ρ (cos e) = Real.cos (eval ρ e) := rfl
+
+@[simp]
+theorem eval_log (ρ : Nat → ℝ) (e : Expr) : eval ρ (log e) = Real.log (eval ρ e) := rfl
+
+@[simp]
+theorem eval_atan (ρ : Nat → ℝ) (e : Expr) : eval ρ (atan e) = Real.arctan (eval ρ e) := rfl
+
+@[simp]
+theorem eval_arsinh (ρ : Nat → ℝ) (e : Expr) : eval ρ (arsinh e) = Real.arsinh (eval ρ e) := rfl
+
+@[simp]
+theorem eval_atanh (ρ : Nat → ℝ) (e : Expr) : eval ρ (atanh e) = Real.atanh (eval ρ e) := rfl
+
+@[simp]
+theorem eval_sinc (ρ : Nat → ℝ) (e : Expr) : eval ρ (sinc e) = Real.sinc (eval ρ e) := rfl
+
+@[simp]
+theorem eval_erf (ρ : Nat → ℝ) (e : Expr) : eval ρ (erf e) = Real.erf (eval ρ e) := rfl
+
+@[simp]
+theorem eval_sinh (ρ : Nat → ℝ) (e : Expr) : eval ρ (sinh e) = Real.sinh (eval ρ e) := rfl
+
+@[simp]
+theorem eval_cosh (ρ : Nat → ℝ) (e : Expr) : eval ρ (cosh e) = Real.cosh (eval ρ e) := rfl
+
+@[simp]
+theorem eval_tanh (ρ : Nat → ℝ) (e : Expr) : eval ρ (tanh e) = Real.tanh (eval ρ e) := rfl
+
+@[simp]
+theorem eval_sqrt (ρ : Nat → ℝ) (e : Expr) : eval ρ (sqrt e) = Real.sqrt (eval ρ e) := rfl
+
+@[simp]
+theorem eval_namedConst (ρ : Nat → ℝ) (c : MathConst) :
+    eval ρ (namedConst c) = c.toReal := rfl
+
+theorem eval_eulerMascheroni (ρ : Nat → ℝ) :
+    eval ρ (namedConst .eulerMascheroni) = Real.eulerMascheroniConstant := rfl
+
+@[simp]
+theorem eval_sub (ρ : Nat → ℝ) (e₁ e₂ : Expr) :
+    eval ρ (sub e₁ e₂) = eval ρ e₁ - eval ρ e₂ := by simp [sub, sub_eq_add_neg]
+
+@[simp]
+theorem eval_div (ρ : Nat → ℝ) (e₁ e₂ : Expr) :
+    eval ρ (div e₁ e₂) = eval ρ e₁ / eval ρ e₂ := by simp [div, div_eq_mul_inv]
+
+@[simp]
+theorem eval_pow (ρ : Nat → ℝ) (e : Expr) (n : ℕ) :
+    eval ρ (pow e n) = (eval ρ e) ^ n := by
+  induction n with
+  | zero => simp [pow]
+  | succ n ih => simp only [pow, eval_mul, ih, pow_succ']
+
+/-- Evaluation of abs for any argument: |x| = sqrt(x²) -/
+theorem eval_abs (ρ : Nat → ℝ) (e : Expr) :
+    eval ρ (abs e) = |eval ρ e| := by
+  simp only [abs, eval_sqrt, eval_mul, ← sq, Real.sqrt_sq_eq_abs]
+
+/-- Evaluation of sqrt(x * x) = |x| (unfolded form of `abs`). -/
+theorem eval_sqrt_mul_self_eq_abs (ρ : Nat → ℝ) (e : Expr) :
+    eval ρ (sqrt (mul e e)) = |eval ρ e| := eval_abs ρ e
+
+/-- `√(x * x) = |x|` — LeanCert-namespaced alias of `Real.sqrt_mul_self_eq_abs`. -/
+theorem sqrt_mul_self_eq_abs (x : ℝ) : Real.sqrt (x * x) = |x| :=
+  Real.sqrt_mul_self_eq_abs x
+
+/-- For positive x, sqrt(x²) = x -/
+theorem eval_sqrt_sq_of_pos (ρ : Nat → ℝ) (e : Expr) (hpos : 0 < eval ρ e) :
+    eval ρ (sqrt (mul e e)) = eval ρ e := by
+  simp only [eval_sqrt, eval_mul, ← sq, Real.sqrt_sq (le_of_lt hpos)]
+
+/-- Abs correctly computes absolute value for positive inputs -/
+theorem eval_abs_of_pos (ρ : Nat → ℝ) (e : Expr) (hpos : 0 < eval ρ e) :
+    eval ρ (abs e) = eval ρ e := by
+  rw [eval_abs, abs_of_pos hpos]
+
+/-- Abs correctly computes absolute value for negative inputs -/
+theorem eval_abs_of_neg (ρ : Nat → ℝ) (e : Expr) (hneg : eval ρ e < 0) :
+    eval ρ (abs e) = -(eval ρ e) := by
+  rw [eval_abs, abs_of_neg hneg]
+
+-- Lemmas about evalAlong
+
+theorem evalAlong_eq (e : Expr) (ρ : Nat → ℝ) (idx : Nat) (t : ℝ) :
+    evalAlong e ρ idx t = eval (ρ[idx ↦ t]) e := rfl
+
+theorem evalAlong_at_ρ (e : Expr) (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong e ρ idx (ρ idx) = eval ρ e := by
+  simp only [evalAlong, updateVar_self]
+
+/-- evalAlong for a constant is constant -/
+theorem evalAlong_const' (ρ : Nat → ℝ) (idx : Nat) (q : ℚ) :
+    evalAlong (const q) ρ idx = fun _ => (q : ℝ) := rfl
+
+/-- evalAlong for the active variable is the identity -/
+theorem evalAlong_var_active (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong (var idx) ρ idx = id := by
+  funext t
+  simp only [evalAlong, eval_var, updateVar_same, id_eq]
+
+/-- evalAlong for a passive variable is constant -/
+theorem evalAlong_var_passive (ρ : Nat → ℝ) (idx i : Nat) (h : i ≠ idx) :
+    evalAlong (var i) ρ idx = fun _ => ρ i := by
+  funext t
+  simp only [evalAlong, eval_var, updateVar_other _ _ _ _ h]
+
+/-- evalAlong for addition -/
+theorem evalAlong_add (e₁ e₂ : Expr) (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong (add e₁ e₂) ρ idx = fun t => evalAlong e₁ ρ idx t + evalAlong e₂ ρ idx t := by
+  funext t
+  simp only [evalAlong, eval_add]
+
+/-- evalAlong for addition (Pi form for compatibility with deriv_add) -/
+theorem evalAlong_add_pi (e₁ e₂ : Expr) (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong (add e₁ e₂) ρ idx = evalAlong e₁ ρ idx + evalAlong e₂ ρ idx := rfl
+
+/-- evalAlong for multiplication -/
+theorem evalAlong_mul (e₁ e₂ : Expr) (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong (mul e₁ e₂) ρ idx = fun t => evalAlong e₁ ρ idx t * evalAlong e₂ ρ idx t := by
+  funext t
+  simp only [evalAlong, eval_mul]
+
+/-- evalAlong for multiplication (Pi form for compatibility with deriv_mul) -/
+theorem evalAlong_mul_pi (e₁ e₂ : Expr) (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong (mul e₁ e₂) ρ idx = evalAlong e₁ ρ idx * evalAlong e₂ ρ idx := rfl
+
+/-- evalAlong for negation -/
+theorem evalAlong_neg (e : Expr) (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong (neg e) ρ idx = fun t => -(evalAlong e ρ idx t) := by
+  funext t
+  simp only [evalAlong, eval_neg]
+
+/-- evalAlong for negation (Pi form for compatibility with deriv.neg) -/
+theorem evalAlong_neg_pi (e : Expr) (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong (neg e) ρ idx = -evalAlong e ρ idx := rfl
+
+/-- evalAlong for sin -/
+theorem evalAlong_sin (e : Expr) (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong (sin e) ρ idx = fun t => Real.sin (evalAlong e ρ idx t) := by
+  funext t
+  simp only [evalAlong, eval_sin]
+
+/-- evalAlong for cos -/
+theorem evalAlong_cos (e : Expr) (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong (cos e) ρ idx = fun t => Real.cos (evalAlong e ρ idx t) := by
+  funext t
+  simp only [evalAlong, eval_cos]
+
+/-- evalAlong for exp -/
+theorem evalAlong_exp (e : Expr) (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong (exp e) ρ idx = fun t => Real.exp (evalAlong e ρ idx t) := by
+  funext t
+  simp only [evalAlong, eval_exp]
+
+/-- evalAlong for inv -/
+theorem evalAlong_inv (e : Expr) (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong (inv e) ρ idx = fun t => (evalAlong e ρ idx t)⁻¹ := by
+  funext t
+  simp only [evalAlong, eval_inv]
+
+/-- evalAlong for log -/
+theorem evalAlong_log (e : Expr) (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong (log e) ρ idx = fun t => Real.log (evalAlong e ρ idx t) := by
+  funext t
+  simp only [evalAlong, eval_log]
+
+/-- evalAlong for atan -/
+theorem evalAlong_atan (e : Expr) (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong (atan e) ρ idx = fun t => Real.arctan (evalAlong e ρ idx t) := by
+  funext t
+  simp only [evalAlong, eval_atan]
+
+/-- evalAlong for arsinh -/
+theorem evalAlong_arsinh (e : Expr) (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong (arsinh e) ρ idx = fun t => Real.arsinh (evalAlong e ρ idx t) := by
+  funext t
+  simp only [evalAlong, eval_arsinh]
+
+/-- evalAlong for atanh -/
+theorem evalAlong_atanh (e : Expr) (ρ : Nat → ℝ) (idx : Nat) :
+    evalAlong (atanh e) ρ idx = fun t => Real.atanh (evalAlong e ρ idx t) := by
+  funext t
+  simp only [evalAlong, eval_atanh]
+
+/-! ## Single-variable expressions
+
+For 1D optimization and root finding, we often work with expressions that only use variable 0.
+These lemmas establish that such expressions can be evaluated equivalently with different
+environment representations.
+-/
+
+/-- Check if an expression only uses variable 0 (computable) -/
+def usesOnlyVar0 : Expr → Bool
+  | const _ => true
+  | var n => n == 0
+  | add e₁ e₂ => e₁.usesOnlyVar0 && e₂.usesOnlyVar0
+  | mul e₁ e₂ => e₁.usesOnlyVar0 && e₂.usesOnlyVar0
+  | neg e => e.usesOnlyVar0
+  | inv e => e.usesOnlyVar0
+  | exp e => e.usesOnlyVar0
+  | sin e => e.usesOnlyVar0
+  | cos e => e.usesOnlyVar0
+  | log e => e.usesOnlyVar0
+  | atan e => e.usesOnlyVar0
+  | arsinh e => e.usesOnlyVar0
+  | atanh e => e.usesOnlyVar0
+  | sinc e => e.usesOnlyVar0
+  | erf e => e.usesOnlyVar0
+  | sinh e => e.usesOnlyVar0
+  | cosh e => e.usesOnlyVar0
+  | tanh e => e.usesOnlyVar0
+  | sqrt e => e.usesOnlyVar0
+  | namedConst _ => true
+
+/-- If two environments agree on variable 0, then a usesOnlyVar0 expression evaluates the same -/
+theorem eval_usesOnlyVar0_eq (e : Expr) (he : e.usesOnlyVar0 = true)
+    (ρ₁ ρ₂ : Nat → ℝ) (h0 : ρ₁ 0 = ρ₂ 0) :
+    eval ρ₁ e = eval ρ₂ e := by
+  induction e with
+  | const q => rfl
+  | var n =>
+    simp only [usesOnlyVar0, beq_iff_eq] at he
+    simp only [eval_var, he, h0]
+  | add e₁ e₂ ih1 ih2 =>
+    simp only [usesOnlyVar0, Bool.and_eq_true] at he
+    simp only [eval_add, ih1 he.1, ih2 he.2]
+  | mul e₁ e₂ ih1 ih2 =>
+    simp only [usesOnlyVar0, Bool.and_eq_true] at he
+    simp only [eval_mul, ih1 he.1, ih2 he.2]
+  | neg e ih =>
+    simp only [usesOnlyVar0] at he
+    simp only [eval_neg, ih he]
+  | inv e ih =>
+    simp only [usesOnlyVar0] at he
+    simp only [eval_inv, ih he]
+  | exp e ih =>
+    simp only [usesOnlyVar0] at he
+    simp only [eval_exp, ih he]
+  | sin e ih =>
+    simp only [usesOnlyVar0] at he
+    simp only [eval_sin, ih he]
+  | cos e ih =>
+    simp only [usesOnlyVar0] at he
+    simp only [eval_cos, ih he]
+  | log e ih =>
+    simp only [usesOnlyVar0] at he
+    simp only [eval_log, ih he]
+  | atan e ih =>
+    simp only [usesOnlyVar0] at he
+    simp only [eval_atan, ih he]
+  | arsinh e ih =>
+    simp only [usesOnlyVar0] at he
+    simp only [eval_arsinh, ih he]
+  | atanh e ih =>
+    simp only [usesOnlyVar0] at he
+    simp only [eval_atanh, ih he]
+  | sinc e ih =>
+    simp only [usesOnlyVar0] at he
+    simp only [eval_sinc, ih he]
+  | erf e ih =>
+    simp only [usesOnlyVar0] at he
+    simp only [eval_erf, ih he]
+  | sinh e ih =>
+    simp only [usesOnlyVar0] at he
+    simp only [eval_sinh, ih he]
+  | cosh e ih =>
+    simp only [usesOnlyVar0] at he
+    simp only [eval_cosh, ih he]
+  | tanh e ih =>
+    simp only [usesOnlyVar0] at he
+    simp only [eval_tanh, ih he]
+  | sqrt e ih =>
+    simp only [usesOnlyVar0] at he
+    simp only [eval_sqrt, ih he]
+  | namedConst _ => rfl
+
+/-- For single-variable expressions, `fun n => if n = 0 then x else 0` and `fun _ => x`
+    give the same evaluation result. -/
+theorem eval_1d_equiv (e : Expr) (he : e.usesOnlyVar0 = true) (x : ℝ) :
+    eval (fun n => if n = 0 then x else 0) e = eval (fun _ => x) e := by
+  apply eval_usesOnlyVar0_eq e he
+  simp
+
+/-- Alternative: evaluation with Box-style environment equals 1D evaluation -/
+theorem eval_box1d_eq_eval1d (e : Expr) (he : e.usesOnlyVar0 = true) (x : ℝ) :
+    eval (fun n => if n = 0 then x else 0) e = eval (fun _ => x) e :=
+  eval_1d_equiv e he x
+
+end Expr
+
+end LeanCert.Core
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Abstract Interval Interface
+
+This file defines the abstract notion of intervals as subsets of ℝ,
+with semantic properties. This provides the mathematical foundation
+that `IntervalReal` will implement computationally.
+
+## Main definitions
+
+* Lemmas about `Set.Icc` (closed intervals) relevant to interval arithmetic
+* Convexity and integrability properties
+
+## Design notes
+
+We primarily use `Set.Icc a b` from Mathlib as our semantic interval type.
+This file collects lemmas and abstractions useful for verified numerics.
+-/
+
+/-! ### Interval membership and operations -/
+
+@[expose] public section
+
+namespace LeanCert.Core
+
+/-- A real number is in the interval [a, b] -/
+abbrev memIcc (x a b : ℝ) : Prop := a ≤ x ∧ x ≤ b
+
+theorem mem_Icc_iff (x a b : ℝ) : memIcc x a b ↔ x ∈ Set.Icc a b := Iff.rfl
+
+/-! ### Interval arithmetic semantics
+
+These lemmas establish the semantic foundation for interval arithmetic:
+if x ∈ [a, b] and y ∈ [c, d], then x ⊕ y ∈ [a ⊕ c, b ⊕ d] (for appropriate ⊕).
+-/
+
+/-- Addition preserves interval membership -/
+theorem add_mem_Icc {x y a b c d : ℝ} (hx : x ∈ Set.Icc a b) (hy : y ∈ Set.Icc c d) :
+    x + y ∈ Set.Icc (a + c) (b + d) := by
+  simp only [Set.mem_Icc] at *
+  constructor <;> linarith
+
+/-- Negation reverses interval bounds -/
+theorem neg_mem_Icc {x a b : ℝ} (hx : x ∈ Set.Icc a b) :
+    -x ∈ Set.Icc (-b) (-a) := by
+  simp only [Set.mem_Icc] at *
+  constructor <;> linarith
+
+/-- Subtraction on intervals -/
+theorem sub_mem_Icc {x y a b c d : ℝ} (hx : x ∈ Set.Icc a b) (hy : y ∈ Set.Icc c d) :
+    x - y ∈ Set.Icc (a - d) (b - c) := by
+  simp only [Set.mem_Icc] at *
+  constructor <;> linarith
+
+/-! ### Convexity -/
+
+/-- Closed intervals are convex -/
+theorem Icc_convex (a b : ℝ) : Convex ℝ (Set.Icc a b) := convex_Icc a b
+
+/-! ### Compactness -/
+
+/-- Nonempty closed bounded intervals are compact -/
+theorem Icc_compact (a b : ℝ) : IsCompact (Set.Icc a b) := isCompact_Icc
+
+/-! ### Continuous functions on intervals -/
+
+/-- A continuous function on a compact interval attains its bounds -/
+theorem continuous_Icc_bounds {f : ℝ → ℝ} {a b : ℝ} (hab : a ≤ b)
+    (hf : Continuous f) :
+    ∃ (lo hi : ℝ), (∀ x ∈ Set.Icc a b, lo ≤ f x ∧ f x ≤ hi) ∧
+      (∃ x ∈ Set.Icc a b, f x = lo) ∧ (∃ x ∈ Set.Icc a b, f x = hi) := by
+  have hne : (Set.Icc a b).Nonempty := Set.nonempty_Icc.mpr hab
+  have hcpt : IsCompact (Set.Icc a b) := isCompact_Icc
+  obtain ⟨xmin, hxmin_mem, hxmin⟩ := hcpt.exists_isMinOn hne hf.continuousOn
+  obtain ⟨xmax, hxmax_mem, hxmax⟩ := hcpt.exists_isMaxOn hne hf.continuousOn
+  exact ⟨f xmin, f xmax,
+    fun x hx => ⟨hxmin hx, hxmax hx⟩,
+    ⟨xmin, hxmin_mem, rfl⟩,
+    ⟨xmax, hxmax_mem, rfl⟩⟩
+
+end LeanCert.Core
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Rational Endpoint Intervals - Core Definitions
+
+This file defines `IntervalRat`, a concrete interval type with rational endpoints
+suitable for computation. We prove the Fundamental Theorem of Interval Arithmetic
+(FTIA) for each operation.
+
+## Main definitions
+
+* `LeanCert.Core.IntervalRat` - Intervals with rational endpoints
+* `LeanCert.Core.IntervalRat.toSet` - Semantic interpretation as a subset of ℝ
+* Operations: `add`, `neg`, `sub`, `mul`, `inv`, `div`
+
+## Main theorems
+
+* `mem_add` - FTIA for addition
+* `mem_neg` - FTIA for negation
+* `mem_sub` - FTIA for subtraction
+* `mem_mul` - FTIA for multiplication
+
+## Design notes
+
+All operations maintain the invariant `lo ≤ hi`. Domain restrictions for partial
+operations (like `inv`) are encoded via separate types or explicit hypotheses.
+-/
+
+@[expose] public section
+
+namespace LeanCert.Core
+
+/-- An interval with rational endpoints -/
+structure IntervalRat where
+  /-- Lower endpoint of the interval. -/
+  lo : ℚ
+  /-- Upper endpoint of the interval. -/
+  hi : ℚ
+  le : lo ≤ hi
+  deriving Repr, DecidableEq
+
+/-- Default interval [0, 0] for unsupported expression branches -/
+instance : Inhabited IntervalRat where
+  default := ⟨0, 0, le_refl 0⟩
+
+namespace IntervalRat
+
+/-- The set of reals contained in this interval -/
+def toSet (I : IntervalRat) : Set ℝ := Set.Icc (I.lo : ℝ) (I.hi : ℝ)
+
+/-- Membership in an interval -/
+instance : Membership ℝ IntervalRat where
+  mem I x := (I.lo : ℝ) ≤ x ∧ x ≤ (I.hi : ℝ)
+
+@[simp]
+theorem mem_def (x : ℝ) (I : IntervalRat) : x ∈ I ↔ (I.lo : ℝ) ≤ x ∧ x ≤ (I.hi : ℝ) :=
+  Iff.rfl
+
+/-- Membership in IntervalRat is the same as membership in Set.Icc -/
+theorem mem_iff_mem_Icc (x : ℝ) (I : IntervalRat) : x ∈ I ↔ x ∈ Set.Icc (I.lo : ℝ) (I.hi : ℝ) := by
+  simp only [mem_def, Set.mem_Icc]
+
+/-- Universal quantifier over IntervalRat equals universal over Set.Icc -/
+theorem forall_mem_iff_forall_Icc {P : ℝ → Prop} (I : IntervalRat) :
+    (∀ x ∈ I, P x) ↔ (∀ x ∈ Set.Icc (I.lo : ℝ) (I.hi : ℝ), P x) := by
+  constructor <;> intro h x hx <;> apply h <;> simp only [mem_iff_mem_Icc] at hx ⊢ <;> exact hx
+
+/-- Existence in IntervalRat equals existence in Set.Icc -/
+theorem exists_mem_iff_exists_Icc {P : ℝ → Prop} (I : IntervalRat) :
+    (∃ x ∈ I, P x) ↔ (∃ x ∈ Set.Icc (I.lo : ℝ) (I.hi : ℝ), P x) := by
+  constructor <;> intro ⟨x, hx, hp⟩ <;> exact ⟨x, by simp only [mem_iff_mem_Icc] at hx ⊢; exact
+    hx, hp⟩
+
+/-- Create an interval from a single rational -/
+def singleton (q : ℚ) : IntervalRat := ⟨q, q, le_refl q⟩
+
+theorem mem_singleton (q : ℚ) : (q : ℝ) ∈ singleton q := by
+  simp only [mem_def, singleton, le_refl, and_self]
+
+/-- The width of an interval -/
+def width (I : IntervalRat) : ℚ := I.hi - I.lo
+
+theorem width_nonneg (I : IntervalRat) : 0 ≤ I.width := by
+  simp only [width, sub_nonneg]
+  exact I.le
+
+/-- Midpoint of an interval -/
+def midpoint (I : IntervalRat) : ℚ := (I.lo + I.hi) / 2
+
+/-- The midpoint of an interval is contained in the interval -/
+theorem midpoint_mem (I : IntervalRat) : (I.midpoint : ℝ) ∈ I := by
+  simp only [mem_def, midpoint]
+  have hle : (I.lo : ℝ) ≤ I.hi := by exact_mod_cast I.le
+  constructor
+  · -- lo ≤ (lo + hi) / 2
+    have : (((I.lo + I.hi) / 2 : ℚ) : ℝ) = ((I.lo : ℝ) + I.hi) / 2 := by
+      simp only [Rat.cast_div, Rat.cast_add, Rat.cast_ofNat]
+    rw [this]
+    linarith
+  · -- (lo + hi) / 2 ≤ hi
+    have : (((I.lo + I.hi) / 2 : ℚ) : ℝ) = ((I.lo : ℝ) + I.hi) / 2 := by
+      simp only [Rat.cast_div, Rat.cast_add, Rat.cast_ofNat]
+    rw [this]
+    linarith
+
+/-! ### Interval addition -/
+
+/-- Add two intervals -/
+def add (I J : IntervalRat) : IntervalRat where
+  lo := I.lo + J.lo
+  hi := I.hi + J.hi
+  le := by linarith [I.le, J.le]
+
+/-- FTIA for addition -/
+theorem mem_add {x y : ℝ} {I J : IntervalRat} (hx : x ∈ I) (hy : y ∈ J) :
+    x + y ∈ add I J := by
+  simp only [mem_def] at *
+  simp only [add, Rat.cast_add]
+  constructor <;> linarith
+
+/-! ### Interval negation -/
+
+/-- Negate an interval -/
+def neg (I : IntervalRat) : IntervalRat where
+  lo := -I.hi
+  hi := -I.lo
+  le := by linarith [I.le]
+
+/-- FTIA for negation -/
+theorem mem_neg {x : ℝ} {I : IntervalRat} (hx : x ∈ I) : -x ∈ neg I := by
+  simp only [mem_def] at *
+  simp only [neg, Rat.cast_neg]
+  constructor <;> linarith
+
+/-! ### Interval subtraction -/
+
+/-- Subtract two intervals -/
+def sub (I J : IntervalRat) : IntervalRat := add I (neg J)
+
+/-- FTIA for subtraction -/
+theorem mem_sub {x y : ℝ} {I J : IntervalRat} (hx : x ∈ I) (hy : y ∈ J) :
+    x - y ∈ sub I J := by
+  simp only [sub_eq_add_neg]
+  exact mem_add hx (mem_neg hy)
+
+/-! ### Interval multiplication -/
+
+/-- Helper: minimum of four rationals -/
+def min4 (a b c d : ℚ) : ℚ := min (min a b) (min c d)
+
+/-- Helper: maximum of four rationals -/
+def max4 (a b c d : ℚ) : ℚ := max (max a b) (max c d)
+
+theorem min4_le_all (a b c d : ℚ) :
+    min4 a b c d ≤ a ∧ min4 a b c d ≤ b ∧ min4 a b c d ≤ c ∧ min4 a b c d ≤ d := by
+  simp only [min4]
+  refine ⟨?_, ?_, ?_, ?_⟩
+  · exact le_trans (min_le_left _ _) (min_le_left _ _)
+  · exact le_trans (min_le_left _ _) (min_le_right _ _)
+  · exact le_trans (min_le_right _ _) (min_le_left _ _)
+  · exact le_trans (min_le_right _ _) (min_le_right _ _)
+
+theorem all_le_max4 (a b c d : ℚ) :
+    a ≤ max4 a b c d ∧ b ≤ max4 a b c d ∧ c ≤ max4 a b c d ∧ d ≤ max4 a b c d := by
+  simp only [max4]
+  refine ⟨?_, ?_, ?_, ?_⟩
+  · exact le_trans (le_max_left _ _) (le_max_left _ _)
+  · exact le_trans (le_max_right _ _) (le_max_left _ _)
+  · exact le_trans (le_max_left _ _) (le_max_right _ _)
+  · exact le_trans (le_max_right _ _) (le_max_right _ _)
+
+theorem le_min4_iff (x a b c d : ℚ) :
+    x ≤ min4 a b c d ↔ x ≤ a ∧ x ≤ b ∧ x ≤ c ∧ x ≤ d := by
+  simp only [min4, le_min_iff, and_assoc]
+
+theorem max4_le_iff (x a b c d : ℚ) :
+    max4 a b c d ≤ x ↔ a ≤ x ∧ b ≤ x ∧ c ≤ x ∧ d ≤ x := by
+  simp only [max4, max_le_iff, and_assoc]
+
+/-- Multiply two intervals -/
+def mul (I J : IntervalRat) : IntervalRat where
+  lo := min4 (I.lo * J.lo) (I.lo * J.hi) (I.hi * J.lo) (I.hi * J.hi)
+  hi := max4 (I.lo * J.lo) (I.lo * J.hi) (I.hi * J.lo) (I.hi * J.hi)
+  le := by
+    simp only [min4, max4]
+    exact le_trans (min_le_of_left_le (min_le_left _ _))
+                   (le_max_of_le_left (le_max_left _ _))
+
+/-- Fast interval multiplication using sign-based case splitting.
+    Reduces from 4 multiplications + 12 comparisons to 2 multiplications
+    in the common case (both intervals positive or both negative).
+    Falls back to the full 4-way product for mixed-sign intervals. -/
+def mulFast (I J : IntervalRat) : IntervalRat :=
+  if hIlo : I.lo ≥ 0 then
+    if hJlo : J.lo ≥ 0 then
+      ⟨I.lo * J.lo, I.hi * J.hi, by nlinarith [I.le, J.le]⟩
+    else if hJhi : J.hi ≤ 0 then
+      ⟨I.hi * J.lo, I.lo * J.hi, by nlinarith [I.le, J.le]⟩
+    else
+      ⟨I.hi * J.lo, I.hi * J.hi, by nlinarith [I.le, J.le]⟩
+  else if hIhi : I.hi ≤ 0 then
+    if hJlo : J.lo ≥ 0 then
+      ⟨I.lo * J.hi, I.hi * J.lo, by nlinarith [I.le, J.le]⟩
+    else if hJhi : J.hi ≤ 0 then
+      ⟨I.hi * J.hi, I.lo * J.lo, by nlinarith [I.le, J.le]⟩
+    else
+      ⟨I.lo * J.hi, I.lo * J.lo, by nlinarith [I.le, J.le]⟩
+  else
+    if hJlo : J.lo ≥ 0 then
+      ⟨I.lo * J.hi, I.hi * J.hi, by nlinarith [I.le, J.le]⟩
+    else if hJhi : J.hi ≤ 0 then
+      ⟨I.hi * J.lo, I.lo * J.lo, by nlinarith [I.le, J.le]⟩
+    else
+      mul I J
+
+/- We intentionally do not attach `mulFast` as a native replacement here.
+   Native certificate checking should execute the same definition that the
+   kernel proofs reason about. Keep `mulFast` available only as an internal
+   candidate for future explicitly-audited runtime backends. -/
+
+/-- Helper: for x ∈ [a₁, a₂], x*y lies between endpoint products.
+    When y ≥ 0: a₁*y ≤ x*y ≤ a₂*y
+    When y ≤ 0: a₂*y ≤ x*y ≤ a₁*y -/
+private theorem mul_mem_endpoints_left {x a₁ a₂ y : ℝ}
+    (ha : a₁ ≤ x ∧ x ≤ a₂) :
+    min (a₁ * y) (a₂ * y) ≤ x * y ∧ x * y ≤ max (a₁ * y) (a₂ * y) := by
+  by_cases! hy : 0 ≤ y
+  · -- y ≥ 0: multiplication preserves order, so a₁*y ≤ x*y ≤ a₂*y
+    have h1 : a₁ * y ≤ x * y := mul_le_mul_of_nonneg_right ha.1 hy
+    have h2 : x * y ≤ a₂ * y := mul_le_mul_of_nonneg_right ha.2 hy
+    constructor
+    · exact le_trans (min_le_left _ _) h1
+    · exact le_trans h2 (le_max_right _ _)
+  · -- y < 0: multiplication reverses order, so a₂*y ≤ x*y ≤ a₁*y
+    have hy' := le_of_lt hy
+    have h1 : a₂ * y ≤ x * y := mul_le_mul_of_nonpos_right ha.2 hy'
+    have h2 : x * y ≤ a₁ * y := mul_le_mul_of_nonpos_right ha.1 hy'
+    constructor
+    · exact le_trans (min_le_right _ _) h1
+    · exact le_trans h2 (le_max_left _ _)
+
+/-- Helper: for y ∈ [b₁, b₂], x*y lies between endpoint products.
+    When x ≥ 0: x*b₁ ≤ x*y ≤ x*b₂
+    When x ≤ 0: x*b₂ ≤ x*y ≤ x*b₁ -/
+private theorem mul_mem_endpoints_right {y b₁ b₂ x : ℝ}
+    (hb : b₁ ≤ y ∧ y ≤ b₂) :
+    min (x * b₁) (x * b₂) ≤ x * y ∧ x * y ≤ max (x * b₁) (x * b₂) := by
+  by_cases! hx : 0 ≤ x
+  · -- x ≥ 0: multiplication preserves order, so x*b₁ ≤ x*y ≤ x*b₂
+    have h1 : x * b₁ ≤ x * y := mul_le_mul_of_nonneg_left hb.1 hx
+    have h2 : x * y ≤ x * b₂ := mul_le_mul_of_nonneg_left hb.2 hx
+    constructor
+    · exact le_trans (min_le_left _ _) h1
+    · exact le_trans h2 (le_max_right _ _)
+  · -- x < 0: multiplication reverses order, so x*b₂ ≤ x*y ≤ x*b₁
+    have hx' := le_of_lt hx
+    have h1 : x * b₂ ≤ x * y := mul_le_mul_of_nonpos_left hb.2 hx'
+    have h2 : x * y ≤ x * b₁ := mul_le_mul_of_nonpos_left hb.1 hx'
+    constructor
+    · exact le_trans (min_le_right _ _) h1
+    · exact le_trans h2 (le_max_left _ _)
+
+/-- Lower bound: xy ≥ min of corner products -/
+private theorem mul_lower_bound {x y a₁ a₂ b₁ b₂ : ℝ}
+    (ha : a₁ ≤ x ∧ x ≤ a₂) (hb : b₁ ≤ y ∧ y ≤ b₂) :
+    min (min (a₁ * b₁) (a₁ * b₂)) (min (a₂ * b₁) (a₂ * b₂)) ≤ x * y := by
+  -- First, x*y ≥ min(a₁*y, a₂*y) from mul_mem_endpoints_left
+  have h1 := (mul_mem_endpoints_left (y := y) ha).1
+  -- Now we need: min of corners ≤ min(a₁*y, a₂*y)
+  -- Case split on whether a₁*y ≤ a₂*y
+  by_cases! hcmp : a₁ * y ≤ a₂ * y
+  · -- min(a₁*y, a₂*y) = a₁*y
+    rw [min_eq_left hcmp] at h1
+    -- Need: min corners ≤ a₁*y
+    -- a₁*y is between a₁*b₁ and a₁*b₂, so min(a₁*b₁, a₁*b₂) ≤ a₁*y
+    have h2 := (mul_mem_endpoints_right hb (x := a₁)).1
+    calc min (min (a₁ * b₁) (a₁ * b₂)) (min (a₂ * b₁) (a₂ * b₂))
+        ≤ min (a₁ * b₁) (a₁ * b₂) := min_le_left _ _
+      _ ≤ a₁ * y := h2
+      _ ≤ x * y := h1
+  · -- min(a₁*y, a₂*y) = a₂*y
+    rw [min_eq_right (le_of_lt hcmp)] at h1
+    have h2 := (mul_mem_endpoints_right hb (x := a₂)).1
+    calc min (min (a₁ * b₁) (a₁ * b₂)) (min (a₂ * b₁) (a₂ * b₂))
+        ≤ min (a₂ * b₁) (a₂ * b₂) := min_le_right _ _
+      _ ≤ a₂ * y := h2
+      _ ≤ x * y := h1
+
+/-- Upper bound: xy ≤ max of corner products -/
+private theorem mul_upper_bound {x y a₁ a₂ b₁ b₂ : ℝ}
+    (ha : a₁ ≤ x ∧ x ≤ a₂) (hb : b₁ ≤ y ∧ y ≤ b₂) :
+    x * y ≤ max (max (a₁ * b₁) (a₁ * b₂)) (max (a₂ * b₁) (a₂ * b₂)) := by
+  have h1 := (mul_mem_endpoints_left (y := y) ha).2
+  by_cases! hcmp : a₁ * y ≤ a₂ * y
+  · rw [max_eq_right hcmp] at h1
+    have h2 := (mul_mem_endpoints_right hb (x := a₂)).2
+    calc x * y
+        ≤ a₂ * y := h1
+      _ ≤ max (a₂ * b₁) (a₂ * b₂) := h2
+      _ ≤ max (max (a₁ * b₁) (a₁ * b₂)) (max (a₂ * b₁) (a₂ * b₂)) := le_max_right _ _
+  · rw [max_eq_left (le_of_lt hcmp)] at h1
+    have h2 := (mul_mem_endpoints_right hb (x := a₁)).2
+    calc x * y
+        ≤ a₁ * y := h1
+      _ ≤ max (a₁ * b₁) (a₁ * b₂) := h2
+      _ ≤ max (max (a₁ * b₁) (a₁ * b₂)) (max (a₂ * b₁) (a₂ * b₂)) := le_max_left _ _
+
+/-- Key lemma: product lies in convex hull of corner products.
+    For x ∈ [a₁, a₂] and y ∈ [b₁, b₂], we have
+    min{a₁b₁, a₁b₂, a₂b₁, a₂b₂} ≤ xy ≤ max{a₁b₁, a₁b₂, a₂b₁, a₂b₂}
+
+    This is a standard result from interval arithmetic: extrema of bilinear
+    functions on rectangles occur at corners. -/
+private theorem mul_mem_corners {x y a₁ a₂ b₁ b₂ : ℝ}
+    (ha : a₁ ≤ x ∧ x ≤ a₂) (hb : b₁ ≤ y ∧ y ≤ b₂) :
+    min (min (a₁ * b₁) (a₁ * b₂)) (min (a₂ * b₁) (a₂ * b₂)) ≤ x * y ∧
+    x * y ≤ max (max (a₁ * b₁) (a₁ * b₂)) (max (a₂ * b₁) (a₂ * b₂)) :=
+  ⟨mul_lower_bound ha hb, mul_upper_bound ha hb⟩
+
+/-- FTIA for multiplication -/
+theorem mem_mul {x y : ℝ} {I J : IntervalRat} (hx : x ∈ I) (hy : y ∈ J) :
+    x * y ∈ mul I J := by
+  simp only [mem_def] at *
+  simp only [mul, min4, max4, Rat.cast_mul, Rat.cast_min, Rat.cast_max]
+  exact mul_mem_corners hx hy
+
+/-- Helper: show a = min4 a b c d when a ≤ b, a ≤ c, a ≤ d -/
+theorem eq_min4_of_le {a b c d : ℚ} (h1 : a ≤ b) (h2 : a ≤ c) (h3 : a ≤ d) :
+    a = min4 a b c d := le_antisymm ((le_min4_iff _ _ _ _ _).mpr ⟨le_refl _, h1, h2, h3⟩)
+      (min4_le_all _ _ _ _).1
+
+theorem eq_min4_of_le2 {a b c d : ℚ} (h1 : b ≤ a) (h2 : b ≤ c) (h3 : b ≤ d) :
+    b = min4 a b c d := le_antisymm ((le_min4_iff _ _ _ _ _).mpr ⟨h1, le_refl _, h2, h3⟩)
+      (min4_le_all _ _ _ _).2.1
+
+theorem eq_min4_of_le3 {a b c d : ℚ} (h1 : c ≤ a) (h2 : c ≤ b) (h3 : c ≤ d) :
+    c = min4 a b c d := le_antisymm ((le_min4_iff _ _ _ _ _).mpr ⟨h1, h2, le_refl _, h3⟩)
+      (min4_le_all _ _ _ _).2.2.1
+
+theorem eq_min4_of_le4 {a b c d : ℚ} (h1 : d ≤ a) (h2 : d ≤ b) (h3 : d ≤ c) :
+    d = min4 a b c d := le_antisymm ((le_min4_iff _ _ _ _ _).mpr ⟨h1, h2, h3, le_refl _⟩)
+      (min4_le_all _ _ _ _).2.2.2
+
+theorem eq_max4_of_ge {a b c d : ℚ} (h1 : b ≤ a) (h2 : c ≤ a) (h3 : d ≤ a) :
+    a = max4 a b c d := le_antisymm (all_le_max4 _ _ _ _).1 ((max4_le_iff _ _ _ _ _).mpr ⟨le_refl
+      _, h1, h2, h3⟩)
+
+theorem eq_max4_of_ge2 {a b c d : ℚ} (h1 : a ≤ b) (h2 : c ≤ b) (h3 : d ≤ b) :
+    b = max4 a b c d := le_antisymm (all_le_max4 _ _ _ _).2.1 ((max4_le_iff _ _ _ _ _).mpr ⟨h1,
+      le_refl _, h2, h3⟩)
+
+theorem eq_max4_of_ge3 {a b c d : ℚ} (h1 : a ≤ c) (h2 : b ≤ c) (h3 : d ≤ c) :
+    c = max4 a b c d := le_antisymm (all_le_max4 _ _ _ _).2.2.1 ((max4_le_iff _ _ _ _ _).mpr ⟨h1,
+      h2, le_refl _, h3⟩)
+
+theorem eq_max4_of_ge4 {a b c d : ℚ} (h1 : a ≤ d) (h2 : b ≤ d) (h3 : c ≤ d) :
+    d = max4 a b c d := le_antisymm (all_le_max4 _ _ _ _).2.2.2 ((max4_le_iff _ _ _ _ _).mpr ⟨h1,
+      h2, h3, le_refl _⟩)
+
+/-- mulFast endpoints match mul endpoints: lo -/
+private theorem mulFast_lo (I J : IntervalRat) : (mulFast I J).lo = (mul I J).lo := by
+  simp only [mulFast, mul]
+  split
+  · split
+    · exact eq_min4_of_le (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+        (by nlinarith [I.le, J.le])
+    · split
+      · exact eq_min4_of_le3 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+      · exact eq_min4_of_le3 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+  · split
+    · split
+      · exact eq_min4_of_le2 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+      · split
+        · exact eq_min4_of_le4 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+            (by nlinarith [I.le, J.le])
+        · exact eq_min4_of_le2 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+            (by nlinarith [I.le, J.le])
+    · split
+      · exact eq_min4_of_le2 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+      · split
+        · exact eq_min4_of_le3 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+            (by nlinarith [I.le, J.le])
+        · rfl
+
+/-- mulFast endpoints match mul endpoints: hi -/
+private theorem mulFast_hi (I J : IntervalRat) : (mulFast I J).hi = (mul I J).hi := by
+  simp only [mulFast, mul]
+  split
+  · split
+    · exact eq_max4_of_ge4 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+        (by nlinarith [I.le, J.le])
+    · split
+      · exact eq_max4_of_ge2 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+      · exact eq_max4_of_ge4 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+  · split
+    · split
+      · exact eq_max4_of_ge3 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+      · split
+        · exact eq_max4_of_ge (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+            (by nlinarith [I.le, J.le])
+        · exact eq_max4_of_ge (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+            (by nlinarith [I.le, J.le])
+    · split
+      · exact eq_max4_of_ge4 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+      · split
+        · exact eq_max4_of_ge (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+            (by nlinarith [I.le, J.le])
+        · rfl
+
+/-- `mulFast` preserves the containment property of `mul`.
+    This is retained as documentation and a future audited optimization hook;
+    production certificate checking currently uses `mul` directly. -/
+theorem mem_mulFast {x y : ℝ} {I J : IntervalRat} (hx : x ∈ I) (hy : y ∈ J) :
+    x * y ∈ mulFast I J := by
+  simp only [mem_def]
+  rw [mulFast_lo, mulFast_hi]
+  exact mem_mul hx hy
+
+/-! ### Interval containing zero check -/
+
+/-- Check if an interval contains zero -/
+def containsZero (I : IntervalRat) : Prop := I.lo ≤ 0 ∧ 0 ≤ I.hi
+
+/-- Decidable containsZero -/
+instance (I : IntervalRat) : Decidable (containsZero I) :=
+  inferInstanceAs (Decidable (I.lo ≤ 0 ∧ 0 ≤ I.hi))
+
+/-- An interval that is guaranteed not to contain zero -/
+structure IntervalRatNonzero extends IntervalRat where
+  nonzero : ¬containsZero toIntervalRat
+
+/-! ### Interval inversion (for nonzero intervals) -/
+
+/-- Invert an interval that doesn't contain zero -/
+def invNonzero (I : IntervalRatNonzero) : IntervalRat :=
+  if h : 0 < I.lo then
+    -- Positive interval: [1/hi, 1/lo]
+    { lo := I.hi⁻¹
+      hi := I.lo⁻¹
+      le := by
+        have hlo : (0 : ℚ) < I.lo := h
+        have hhi : (0 : ℚ) < I.hi := lt_of_lt_of_le hlo I.le
+        exact inv_anti₀ hlo I.le }
+  else
+    -- Negative interval: [1/hi, 1/lo] (both negative)
+    { lo := I.hi⁻¹
+      hi := I.lo⁻¹
+      le := by
+        have hlo_le : I.lo ≤ 0 := le_of_not_gt h
+        have hhi_neg : I.hi < 0 := by
+          have hnz := I.nonzero
+          simp only [containsZero, not_and, not_le] at hnz
+          exact hnz hlo_le
+        have hlo_neg : I.lo < 0 := lt_of_le_of_lt I.le hhi_neg
+        exact (inv_le_inv_of_neg hhi_neg hlo_neg).mpr I.le }
+
+/-! ### FTIA for inversion -/
+
+theorem mem_invNonzero {x : ℝ} {I : IntervalRatNonzero} (hx : x ∈ I.toIntervalRat) (_hxne : x ≠ 0) :
+    x⁻¹ ∈ invNonzero I := by
+  simp only [mem_def] at *
+  simp only [invNonzero]
+  split_ifs with h
+  · -- Positive case: 0 < I.lo
+    have hlo_pos : (0 : ℝ) < I.lo := by exact_mod_cast h
+    have hx_pos : 0 < x := lt_of_lt_of_le hlo_pos hx.1
+    have hhi_pos : (0 : ℝ) < I.hi := lt_of_lt_of_le hlo_pos (by exact_mod_cast I.le)
+    simp only [Rat.cast_inv]
+    constructor
+    · exact inv_anti₀ hx_pos hx.2
+    · exact inv_anti₀ (by exact_mod_cast h) hx.1
+  · -- Negative case
+    have hlo_le : I.lo ≤ 0 := le_of_not_gt h
+    have hhi_neg : I.hi < 0 := by
+      have hnz := I.nonzero
+      simp only [containsZero, not_and, not_le] at hnz
+      exact hnz hlo_le
+    have hlo_neg : I.lo < 0 := lt_of_le_of_lt I.le hhi_neg
+    have hx_neg : x < 0 := lt_of_le_of_lt hx.2 (by exact_mod_cast hhi_neg)
+    have hhi_neg_r : (I.hi : ℝ) < 0 := by exact_mod_cast hhi_neg
+    have hlo_neg_r : (I.lo : ℝ) < 0 := by exact_mod_cast hlo_neg
+    simp only [Rat.cast_inv]
+    constructor
+    · exact (inv_le_inv_of_neg hhi_neg_r hx_neg).mpr hx.2
+    · exact (inv_le_inv_of_neg hx_neg hlo_neg_r).mpr hx.1
+
+/-! ### Scalar operations -/
+
+/-- Scale an interval by a rational -/
+def scale (q : ℚ) (I : IntervalRat) : IntervalRat :=
+  if hq : 0 ≤ q then
+    { lo := q * I.lo
+      hi := q * I.hi
+      le := mul_le_mul_of_nonneg_left I.le hq }
+  else
+    { lo := q * I.hi
+      hi := q * I.lo
+      le := mul_le_mul_of_nonpos_left I.le (le_of_lt (not_le.mp hq)) }
+
+/-- FTIA for scaling -/
+theorem mem_scale {x : ℝ} {I : IntervalRat} (q : ℚ) (hx : x ∈ I) :
+    (q : ℝ) * x ∈ scale q I := by
+  simp only [mem_def, scale] at *
+  split_ifs with hq
+  · simp only [Rat.cast_mul]
+    constructor
+    · exact mul_le_mul_of_nonneg_left hx.1 (Rat.cast_nonneg.mpr hq)
+    · exact mul_le_mul_of_nonneg_left hx.2 (Rat.cast_nonneg.mpr hq)
+  · simp only [Rat.cast_mul]
+    have hq' : (q : ℝ) ≤ 0 := Rat.cast_nonpos.mpr (le_of_lt (not_le.mp hq))
+    constructor
+    · exact mul_le_mul_of_nonpos_left hx.2 hq'
+    · exact mul_le_mul_of_nonpos_left hx.1 hq'
+
+/-! ### Interval splitting -/
+
+/-- Split an interval at its midpoint -/
+def bisect (I : IntervalRat) : IntervalRat × IntervalRat :=
+  let m := I.midpoint
+  (⟨I.lo, m, by change I.lo ≤ (I.lo + I.hi) / 2; linarith [I.le]⟩,
+   ⟨m, I.hi, by change (I.lo + I.hi) / 2 ≤ I.hi; linarith [I.le]⟩)
+
+theorem mem_bisect_left {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (hm : x ≤ I.midpoint) :
+    x ∈ (bisect I).1 := by
+  simp only [mem_def, bisect] at *
+  exact ⟨hx.1, hm⟩
+
+theorem mem_bisect_right {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (hm : I.midpoint ≤ x) :
+    x ∈ (bisect I).2 := by
+  simp only [mem_def, bisect] at *
+  exact ⟨hm, hx.2⟩
+
+/-- Distance from midpoint to lo is half the width -/
+theorem midpoint_sub_lo (I : IntervalRat) :
+    (I.midpoint : ℝ) - I.lo = (I.hi - I.lo) / 2 := by
+  simp only [midpoint]
+  simp only [Rat.cast_div, Rat.cast_add, Rat.cast_ofNat]
+  ring
+
+/-- Distance from hi to midpoint is half the width -/
+theorem hi_sub_midpoint (I : IntervalRat) :
+    (I.hi : ℝ) - I.midpoint = (I.hi - I.lo) / 2 := by
+  simp only [midpoint]
+  simp only [Rat.cast_div, Rat.cast_add, Rat.cast_ofNat]
+  ring
+
+/-- Midpoint is at least lo -/
+theorem midpoint_ge_lo (I : IntervalRat) : I.lo ≤ I.midpoint := by
+  simp only [midpoint]
+  have h := I.le
+  linarith
+
+/-- Midpoint is at most hi -/
+theorem midpoint_le_hi (I : IntervalRat) : I.midpoint ≤ I.hi := by
+  simp only [midpoint]
+  have h := I.le
+  linarith
+
+/-- Midpoint is at least lo (real version) -/
+theorem midpoint_ge_lo_real (I : IntervalRat) : (I.lo : ℝ) ≤ I.midpoint := by
+  have := midpoint_ge_lo I
+  exact_mod_cast this
+
+/-- Midpoint is at most hi (real version) -/
+theorem midpoint_le_hi_real (I : IntervalRat) : (I.midpoint : ℝ) ≤ I.hi := by
+  have := midpoint_le_hi I
+  exact_mod_cast this
+
+/-- Left bisection is a subset of the original interval -/
+theorem mem_of_mem_bisect_left {x : ℝ} {I : IntervalRat} (hx : x ∈ (bisect I).1) : x ∈ I := by
+  simp only [mem_def, bisect] at *
+  constructor
+  · exact hx.1
+  · exact le_trans hx.2 (Rat.cast_le.mpr (midpoint_le_hi I))
+
+/-- Right bisection is a subset of the original interval -/
+theorem mem_of_mem_bisect_right {x : ℝ} {I : IntervalRat} (hx : x ∈ (bisect I).2) : x ∈ I := by
+  simp only [mem_def, bisect] at *
+  constructor
+  · exact le_trans (Rat.cast_le.mpr (midpoint_ge_lo I)) hx.1
+  · exact hx.2
+
+/-- Any point in an interval is in one of its bisected halves -/
+theorem mem_bisect_or {x : ℝ} {I : IntervalRat} (hx : x ∈ I) :
+    x ∈ (bisect I).1 ∨ x ∈ (bisect I).2 := by
+  by_cases hm : x ≤ I.midpoint
+  · left; exact mem_bisect_left hx hm
+  · right; exact mem_bisect_right hx (le_of_lt (not_le.mp hm))
+
+/-- The default interval [0,0] contains only 0 -/
+theorem mem_default (x : ℝ) : x ∈ (default : IntervalRat) ↔ x = 0 := by
+  simp only [mem_def, Inhabited.default, Rat.cast_zero]
+  constructor
+  · intro ⟨h1, h2⟩; linarith
+  · intro h; subst h; exact ⟨le_refl _, le_refl _⟩
+
+/-! ### Interval intersection -/
+
+/-- Intersect two intervals. Returns none if they don't intersect. -/
+def intersect (I J : IntervalRat) : Option IntervalRat :=
+  let lo := max I.lo J.lo
+  let hi := min I.hi J.hi
+  if h : lo ≤ hi then
+    some ⟨lo, hi, h⟩
+  else
+    none
+
+/-- If intersection succeeds, the result contains any point in both intervals -/
+theorem mem_intersect {x : ℝ} {I J : IntervalRat} (hI : x ∈ I) (hJ : x ∈ J) :
+    ∃ K, intersect I J = some K ∧ x ∈ K := by
+  simp only [mem_def] at hI hJ
+  -- Work with the max/min at rational level
+  have hle : max I.lo J.lo ≤ min I.hi J.hi := by
+    have hmax_le : (I.lo : ℝ) ⊔ (J.lo : ℝ) ≤ x := sup_le hI.1 hJ.1
+    have hle_min : x ≤ (I.hi : ℝ) ⊓ (J.hi : ℝ) := le_inf hI.2 hJ.2
+    have hR : (I.lo : ℝ) ⊔ (J.lo : ℝ) ≤ (I.hi : ℝ) ⊓ (J.hi : ℝ) := le_trans hmax_le hle_min
+    -- Convert back: on ℚ, max = sup and min = inf
+    calc max I.lo J.lo = I.lo ⊔ J.lo := rfl
+      _ ≤ I.hi ⊓ J.hi := by exact_mod_cast hR
+      _ = min I.hi J.hi := rfl
+  simp only [intersect, hle, ↓reduceDIte]
+  refine ⟨⟨max I.lo J.lo, min I.hi J.hi, hle⟩, rfl, ?_⟩
+  simp only [mem_def]
+  constructor
+  · -- Show (max I.lo J.lo : ℝ) ≤ x
+    have h1 : max I.lo J.lo ≤ I.lo ∨ max I.lo J.lo = I.lo ∨ max I.lo J.lo = J.lo := by
+      simp only [max_def]; split_ifs <;> simp
+    cases le_or_gt I.lo J.lo with
+    | inl h => simp only [max_eq_right h]; exact hJ.1
+    | inr h => simp only [max_eq_left (le_of_lt h)]; exact hI.1
+  · -- Show x ≤ (min I.hi J.hi : ℝ)
+    cases le_or_gt I.hi J.hi with
+    | inl h => simp only [min_eq_left h]; exact hI.2
+    | inr h => simp only [min_eq_right (le_of_lt h)]; exact hJ.2
+
+/-- If intersection returns some K, then K ⊆ I -/
+theorem intersect_subset_left {I J K : IntervalRat} (h : intersect I J = some K) :
+    K.lo ≥ I.lo ∧ K.hi ≤ I.hi := by
+  simp only [intersect] at h
+  by_cases hle : max I.lo J.lo ≤ min I.hi J.hi
+  · simp only [hle, ↓reduceDIte, Option.some.injEq] at h
+    subst h
+    exact ⟨le_sup_left, inf_le_left⟩
+  · simp only [hle, ↓reduceDIte, reduceCtorEq] at h
+
+/-- If intersection returns some K, then K ⊆ J -/
+theorem intersect_subset_right {I J K : IntervalRat} (h : intersect I J = some K) :
+    K.lo ≥ J.lo ∧ K.hi ≤ J.hi := by
+  simp only [intersect] at h
+  by_cases hle : max I.lo J.lo ≤ min I.hi J.hi
+  · simp only [hle, ↓reduceDIte, Option.some.injEq] at h
+    subst h
+    exact ⟨le_sup_right, inf_le_right⟩
+  · simp only [hle, ↓reduceDIte, reduceCtorEq] at h
+
+end IntervalRat
+
+end LeanCert.Core
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+
+/-!
+# Verified Argument Reduction for Logarithm
+
+This file provides argument reduction for computing log(q) using the identity:
+  log(q) = log(m) + k * log(2)
+where m = q * 2^(-k) is in a "good" range [1/2, 2] for Taylor series convergence.
+
+## Main definitions
+
+* `reductionExponent` - Find k such that q * 2^(-k) is in [1/2, 2]
+* `reduceMantissa` - The reduced mantissa m = q * 2^(-k)
+
+## Main theorems
+
+* `reconstruction_eq` - Algebraic identity: q = m * 2^k
+* `reduced_bounds` - Bounds on m: 1 / 2 ≤ m ≤ 2 for q > 0
+
+## Design notes
+
+This reduction allows us to use the rapidly converging atanh-based series:
+  log(m) = 2 * atanh((m-1)/(m+1))
+For m ∈ [1/2, 2], we have (m-1)/(m+1) ∈ [-1/3, 1/3], where atanh converges very fast.
+-/
+
+/-! ### Argument Reduction -/
+
+@[expose] public section
+
+namespace LeanCert.Core.LogReduction
+
+/-- Find k such that q * 2^(-k) is approximately in [1/2, 2].
+    Implementation: k = log2(num) - log2(den) approximately. -/
+def reductionExponent (q : ℚ) : ℤ :=
+  if q ≤ 0 then 0
+  else
+    let n_log := q.num.natAbs.log2
+    let d_log := q.den.log2
+    (n_log : ℤ) - (d_log : ℤ)
+
+/-- The reduced mantissa m = q * 2^(-k). -/
+def reduceMantissa (q : ℚ) : ℚ :=
+  if q ≤ 0 then 1  -- Default for non-positive inputs
+  else
+    let k := reductionExponent q
+    q * (2 : ℚ) ^ (-k)
+
+/-- The main algebraic theorem: q = m * 2^k for positive q -/
+theorem reconstruction_eq {q : ℚ} (hq : 0 < q) :
+    let k := reductionExponent q
+    let m := reduceMantissa q
+    q = m * (2 : ℚ) ^ k := by
+  simp only [reductionExponent, reduceMantissa, not_le.mpr hq, ↓reduceIte]
+  -- Goal: q = (q * 2^(-k)) * 2^k = q * (2^(-k) * 2^k) = q * 2^0 = q
+  rw [mul_assoc, ← zpow_add₀ (by norm_num : (2 : ℚ) ≠ 0)]
+  simp only [neg_add_cancel, zpow_zero, mul_one]
+
+/-- The reduced mantissa is bounded: 1/4 ≤ m ≤ 4 for q > 0.
+    (We use slightly weaker bounds than [1/2, 2] for simpler proofs,
+     but the series still converges rapidly.) -/
+theorem reduced_bounds_weak {q : ℚ} (hq : 0 < q) :
+    let m := reduceMantissa q
+    1 / 4 ≤ m ∧ m ≤ 4 := by
+  simp only [reduceMantissa, reductionExponent, not_le.mpr hq, ↓reduceIte]
+  set n := q.num.natAbs.log2 with hn_def
+  set d := q.den.log2 with hd_def
+  have hnum_pos : q.num > 0 := Rat.num_pos.mpr hq
+  have hnum_ne : q.num.natAbs ≠ 0 := Int.natAbs_ne_zero.mpr (ne_of_gt hnum_pos)
+  have hden_pos := q.den_pos
+  have hden_ne : q.den ≠ 0 := ne_of_gt hden_pos
+  have h_num_lo : (2 : ℕ) ^ n ≤ q.num.natAbs := Nat.log2_self_le hnum_ne
+  have h_num_hi : q.num.natAbs < 2 ^ (n + 1) := (Nat.log2_lt hnum_ne).mp (Nat.lt_succ_self _)
+  have h_den_lo : (2 : ℕ) ^ d ≤ q.den := Nat.log2_self_le hden_ne
+  have h_den_hi : q.den < 2 ^ (d + 1) := (Nat.log2_lt hden_ne).mp (Nat.lt_succ_self _)
+  have h2_ne : (2 : ℚ) ≠ 0 := by norm_num
+  have hden_pos_q : (0 : ℚ) < q.den := by exact_mod_cast hden_pos
+  have hnum_pos_q : (0 : ℚ) < q.num.natAbs := by exact_mod_cast Nat.pos_of_ne_zero hnum_ne
+  have hq_eq : q = q.num / q.den := (Rat.num_div_den q).symm
+  have hnum_eq : (q.num : ℚ) = q.num.natAbs := by simp only [Nat.cast_natAbs, abs_of_pos hnum_pos]
+  have hexp : (-(↑n - ↑d) : ℤ) = (d : ℤ) - (n : ℤ) := by ring
+  rw [hexp]
+  have hpow_diff : (2 : ℚ) ^ ((d : ℤ) - (n : ℤ)) = 2 ^ d / 2 ^ n := by
+    rw [zpow_sub₀ h2_ne, zpow_natCast, zpow_natCast]
+  rw [hpow_diff, hq_eq, hnum_eq]
+  have h_num_lo_q : (2 : ℚ) ^ n ≤ q.num.natAbs := by exact_mod_cast h_num_lo
+  have h_num_hi_q : (q.num.natAbs : ℚ) < 2 ^ (n + 1) := by exact_mod_cast h_num_hi
+  have h_den_lo_q : (2 : ℚ) ^ d ≤ q.den := by exact_mod_cast h_den_lo
+  have h_den_hi_q : (q.den : ℚ) < 2 ^ (d + 1) := by exact_mod_cast h_den_hi
+  constructor
+  · -- Lower bound: 1/4 ≤ m
+    have h_simp : (q.num.natAbs : ℚ) / q.den * (2 ^ d / 2 ^ n) =
+                  (q.num.natAbs * 2 ^ d) / (q.den * 2 ^ n) := by field_simp
+    rw [h_simp, one_div, le_div_iff₀ (by positivity : (0 : ℚ) < q.den * 2 ^ n)]
+    have key : (q.den : ℚ) * 2 ^ n / 4 < q.num.natAbs * 2 ^ d := by
+      have h1 : (q.den : ℚ) * 2 ^ n / 4 < 2 ^ (d + 1) * 2 ^ n / 4 := by
+        apply div_lt_div_of_pos_right _ (by norm_num : (0 : ℚ) < 4)
+        exact mul_lt_mul_of_pos_right h_den_hi_q (by positivity)
+      have h2 : (2 : ℚ) ^ (d + 1) * 2 ^ n / 4 = 2 ^ d * 2 ^ n / 2 := by rw [pow_succ]; ring
+      have h3 : (2 : ℚ) ^ d * 2 ^ n / 2 ≤ 2 ^ n * 2 ^ d := by
+        rw [mul_comm (2 ^ d : ℚ) (2 ^ n)]; apply div_le_self (by positivity) (by norm_num)
+      have h4 : (2 : ℚ) ^ n * 2 ^ d ≤ q.num.natAbs * 2 ^ d := by
+        apply mul_le_mul_of_nonneg_right h_num_lo_q (by positivity)
+      linarith
+    linarith
+  · -- Upper bound: m ≤ 4
+    have h_simp : (q.num.natAbs : ℚ) / q.den * (2 ^ d / 2 ^ n) =
+                  (q.num.natAbs * 2 ^ d) / (q.den * 2 ^ n) := by field_simp
+    rw [h_simp, div_le_iff₀ (by positivity : (0 : ℚ) < q.den * 2 ^ n)]
+    have h1 : (q.num.natAbs : ℚ) * 2 ^ d < 2 ^ (n + 1) * 2 ^ d := by
+      apply mul_lt_mul_of_pos_right h_num_hi_q (by positivity)
+    have h2 : (2 : ℚ) ^ (n + 1) * 2 ^ d = 2 * (2 ^ n * 2 ^ d) := by rw [pow_succ]; ring
+    have h3 : (2 : ℚ) * (2 ^ n * 2 ^ d) ≤ 4 * (2 ^ n * 2 ^ d) := by
+      have hp : (0 : ℚ) ≤ 2 ^ n * 2 ^ d := by positivity
+      linarith
+    have h4 : (4 : ℚ) * (2 ^ n * 2 ^ d) ≤ 4 * (q.den * 2 ^ n) := by
+      have : (2 : ℚ) ^ n * 2 ^ d ≤ q.den * 2 ^ n := by
+        calc (2 : ℚ) ^ n * 2 ^ d = 2 ^ d * 2 ^ n := by ring
+          _ ≤ q.den * 2 ^ n := by apply mul_le_mul_of_nonneg_right h_den_lo_q (by positivity)
+      linarith
+    calc (q.num.natAbs : ℚ) * 2 ^ d ≤ 2 ^ (n + 1) * 2 ^ d := le_of_lt h1
+      _ = 2 * (2 ^ n * 2 ^ d) := h2
+      _ ≤ 4 * (2 ^ n * 2 ^ d) := h3
+      _ ≤ 4 * (q.den * 2 ^ n) := h4
+
+/-- Tighter bounds: 1 / 2 ≤ m ≤ 2 for most q > 0 -/
+theorem reduced_bounds {q : ℚ} (hq : 0 < q) :
+    let m := reduceMantissa q
+    1 / 2 ≤ m ∧ m ≤ 2 := by
+  simp only [reduceMantissa, reductionExponent, not_le.mpr hq, ↓reduceIte]
+  set n := q.num.natAbs.log2 with hn_def
+  set d := q.den.log2 with hd_def
+  have hnum_pos : q.num > 0 := Rat.num_pos.mpr hq
+  have hnum_ne : q.num.natAbs ≠ 0 := Int.natAbs_ne_zero.mpr (ne_of_gt hnum_pos)
+  have hden_pos := q.den_pos
+  have hden_ne : q.den ≠ 0 := ne_of_gt hden_pos
+  have h_num_lo : (2 : ℕ) ^ n ≤ q.num.natAbs := Nat.log2_self_le hnum_ne
+  have h_num_hi : q.num.natAbs < 2 ^ (n + 1) := (Nat.log2_lt hnum_ne).mp (Nat.lt_succ_self _)
+  have h_den_lo : (2 : ℕ) ^ d ≤ q.den := Nat.log2_self_le hden_ne
+  have h_den_hi : q.den < 2 ^ (d + 1) := (Nat.log2_lt hden_ne).mp (Nat.lt_succ_self _)
+  have h2_ne : (2 : ℚ) ≠ 0 := by norm_num
+  have hden_pos_q : (0 : ℚ) < q.den := by exact_mod_cast hden_pos
+  have hnum_pos_q : (0 : ℚ) < q.num.natAbs := by exact_mod_cast Nat.pos_of_ne_zero hnum_ne
+  have hq_eq : q = q.num / q.den := (Rat.num_div_den q).symm
+  have hnum_eq : (q.num : ℚ) = q.num.natAbs := by simp only [Nat.cast_natAbs, abs_of_pos hnum_pos]
+  have hexp : (-(↑n - ↑d) : ℤ) = (d : ℤ) - (n : ℤ) := by ring
+  rw [hexp]
+  have hpow_diff : (2 : ℚ) ^ ((d : ℤ) - (n : ℤ)) = 2 ^ d / 2 ^ n := by
+    rw [zpow_sub₀ h2_ne, zpow_natCast, zpow_natCast]
+  rw [hpow_diff, hq_eq, hnum_eq]
+  have h_num_lo_q : (2 : ℚ) ^ n ≤ q.num.natAbs := by exact_mod_cast h_num_lo
+  have h_num_hi_q : (q.num.natAbs : ℚ) < 2 ^ (n + 1) := by exact_mod_cast h_num_hi
+  have h_den_lo_q : (2 : ℚ) ^ d ≤ q.den := by exact_mod_cast h_den_lo
+  have h_den_hi_q : (q.den : ℚ) < 2 ^ (d + 1) := by exact_mod_cast h_den_hi
+  constructor
+  · -- Lower bound: 1 / 2 ≤ m
+    have h_simp : (q.num.natAbs : ℚ) / q.den * (2 ^ d / 2 ^ n) =
+                  (q.num.natAbs * 2 ^ d) / (q.den * 2 ^ n) := by field_simp
+    rw [h_simp, one_div, le_div_iff₀ (by positivity : (0 : ℚ) < q.den * 2 ^ n)]
+    -- Goal: (1/2) * (den * 2^n) ≤ natAbs * 2^d, i.e., den * 2^n / 2 ≤ natAbs * 2^d
+    have key : (q.den : ℚ) * 2 ^ n / 2 < q.num.natAbs * 2 ^ d := by
+      have h1 : (q.den : ℚ) * 2 ^ n / 2 < 2 ^ (d + 1) * 2 ^ n / 2 := by
+        apply div_lt_div_of_pos_right _ (by norm_num : (0 : ℚ) < 2)
+        exact mul_lt_mul_of_pos_right h_den_hi_q (by positivity)
+      have h2 : (2 : ℚ) ^ (d + 1) * 2 ^ n / 2 = 2 ^ d * 2 ^ n := by rw [pow_succ]; ring
+      have h3 : (2 : ℚ) ^ d * 2 ^ n = 2 ^ n * 2 ^ d := by ring
+      have h4 : (2 : ℚ) ^ n * 2 ^ d ≤ q.num.natAbs * 2 ^ d := by
+        apply mul_le_mul_of_nonneg_right h_num_lo_q (by positivity)
+      linarith
+    linarith
+  · -- Upper bound: m ≤ 2
+    have h_simp : (q.num.natAbs : ℚ) / q.den * (2 ^ d / 2 ^ n) =
+                  (q.num.natAbs * 2 ^ d) / (q.den * 2 ^ n) := by field_simp
+    rw [h_simp, div_le_iff₀ (by positivity : (0 : ℚ) < q.den * 2 ^ n)]
+    -- Goal: natAbs * 2^d ≤ 2 * (den * 2^n)
+    have h1 : (q.num.natAbs : ℚ) * 2 ^ d < 2 ^ (n + 1) * 2 ^ d := by
+      apply mul_lt_mul_of_pos_right h_num_hi_q (by positivity)
+    have h2 : (2 : ℚ) ^ (n + 1) * 2 ^ d = 2 * (2 ^ n * 2 ^ d) := by rw [pow_succ]; ring
+    have h3 : (2 : ℚ) * (2 ^ n * 2 ^ d) ≤ 2 * (q.den * 2 ^ n) := by
+      have : (2 : ℚ) ^ n * 2 ^ d ≤ q.den * 2 ^ n := by
+        calc (2 : ℚ) ^ n * 2 ^ d = 2 ^ d * 2 ^ n := by ring
+          _ ≤ q.den * 2 ^ n := by apply mul_le_mul_of_nonneg_right h_den_lo_q (by positivity)
+      linarith
+    calc (q.num.natAbs : ℚ) * 2 ^ d ≤ 2 ^ (n + 1) * 2 ^ d := le_of_lt h1
+      _ = 2 * (2 ^ n * 2 ^ d) := h2
+      _ ≤ 2 * (q.den * 2 ^ n) := h3
+
+/-- The reduced mantissa is positive for positive input -/
+theorem reduced_pos {q : ℚ} (hq : 0 < q) : 0 < reduceMantissa q := by
+  simp only [reduceMantissa, not_le.mpr hq, ↓reduceIte]
+  apply mul_pos hq
+  apply zpow_pos
+  norm_num
+
+/-! ### Connection to Real.log -/
+
+/-- Key algebraic identity for Real.log: log(q) = log(m) + k * log(2) -/
+theorem log_reduction {q : ℚ} (hq : 0 < q) :
+    let k := reductionExponent q
+    let m := reduceMantissa q
+    Real.log q = Real.log m + k * Real.log 2 := by
+  intro k m  -- Introduce let-bound variables
+  have hm_pos : (0 : ℝ) < m := by exact_mod_cast reduced_pos hq
+  have h_recon := reconstruction_eq hq
+  have hq_eq : (q : ℝ) = (m : ℝ) * (2 : ℝ) ^ k := by
+    conv_lhs => rw [h_recon]
+    push_cast
+    rfl
+  rw [hq_eq]
+  have h2_pos : (0 : ℝ) < 2 ^ k := zpow_pos (by norm_num : (0 : ℝ) < 2) k
+  rw [Real.log_mul (ne_of_gt hm_pos) (ne_of_gt h2_pos), Real.log_zpow]
+
+/-- The transformation y = (m-1)/(m+1) maps m ∈ [1/2, 2] to y ∈ [-1/3, 1/3] -/
+theorem atanh_arg_bounds {m : ℚ} (hlo : 1 / 2 ≤ m) (hhi : m ≤ 2) :
+    let y := (m - 1) / (m + 1)
+    (-1)/3 ≤ y ∧ y ≤ 1/3 := by
+  intro y  -- Introduce let-bound variable
+  have hden_pos : 0 < m + 1 := by linarith
+  have hden_nonneg : 0 ≤ m + 1 := le_of_lt hden_pos
+  constructor
+  · -- Lower bound: (m-1)/(m+1) ≥ -1/3 when m ≥ 1/2
+    -- Cross multiply: -1/3 ≤ (m-1)/(m+1) ↔ -(m+1) ≤ 3(m-1) ↔ 2 ≤ 4m
+    have h : (-1 : ℚ) / 3 * (m + 1) ≤ m - 1 := by linarith
+    have key := div_le_div_of_nonneg_right h hden_nonneg
+    simp only [mul_div_assoc, div_self (ne_of_gt hden_pos), mul_one] at key
+    exact key
+  · -- Upper bound: (m-1)/(m+1) ≤ 1/3 when m ≤ 2
+    -- Cross multiply: (m-1)/(m+1) ≤ 1/3 ↔ 3(m-1) ≤ m+1 ↔ 2m ≤ 4
+    have h : m - 1 ≤ (1 : ℚ) / 3 * (m + 1) := by linarith
+    have key := div_le_div_of_nonneg_right h hden_nonneg
+    simp only [mul_div_assoc, div_self (ne_of_gt hden_pos), mul_one] at key
+    exact key
+
+/-- log(m) = 2 * atanh((m-1)/(m+1)) for m > 0 with m ≠ 1 -/
+theorem log_via_atanh {m : ℚ} (hm_pos : 0 < m) :
+    Real.log m = 2 * Real.atanh ((m - 1) / (m + 1)) := by
+  have hm_pos' : (0 : ℝ) < m := by exact_mod_cast hm_pos
+  have hsum_pos : (0 : ℝ) < m + 1 := by linarith
+  have hdiff_ne : (m : ℝ) + 1 ≠ 0 := ne_of_gt hsum_pos
+  -- atanh(y) = (1/2) * log((1+y)/(1-y))
+  -- Setting y = (m-1)/(m+1):
+  -- 1 + y = 1 + (m-1)/(m+1) = ((m+1) + (m-1))/(m+1) = 2m/(m+1)
+  -- 1 - y = 1 - (m-1)/(m+1) = ((m+1) - (m-1))/(m+1) = 2/(m+1)
+  -- (1+y)/(1-y) = (2m/(m+1)) / (2/(m+1)) = m
+  rw [Real.atanh]
+  have h1 : 1 + ((m : ℝ) - 1) / (m + 1) = 2 * m / (m + 1) := by field_simp; ring
+  have h2 : 1 - ((m : ℝ) - 1) / (m + 1) = 2 / (m + 1) := by field_simp; ring
+  rw [h1, h2]
+  have h3 : 2 * (m : ℝ) / (m + 1) / (2 / (m + 1)) = m := by field_simp
+  rw [h3]
+  ring
+
+end LeanCert.Core.LogReduction
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Rational Endpoint Intervals - Transcendental Functions
+
+This file provides noncomputable interval bounds for transcendental functions
+using floor/ceiling to obtain rational endpoints.
+
+## Main definitions
+
+* `ofRealEndpoints` - Create rational interval from real bounds using floor/ceil
+* `expInterval` - Interval bound for exponential
+* `logInterval` - Interval bound for logarithm (positive intervals)
+* `atanhIntervalComputed` - Interval bound for atanh (intervals in (-1, 1))
+* `sqrtInterval` - Interval bound for square root
+
+## Main theorems
+
+* `mem_expInterval` - FTIA for exp
+* `mem_logInterval` - FTIA for log
+* `mem_atanhIntervalComputed` - FTIA for atanh
+* `mem_sqrtInterval` - FTIA for sqrt
+
+## Design notes
+
+All definitions in this file are noncomputable as they use `Real.exp`, `Real.log`,
+etc. For computable versions, see `IntervalRat.Taylor`.
+-/
+
+/-! ### Rational enclosure of real intervals -/
+
+@[expose] public section
+
+namespace LeanCert.Core
+
+namespace IntervalRat
+
+/-- Coarse rational enclosure of a real interval using floor/ceil.
+    Given a real interval [lo, hi], returns a rational interval [⌊lo⌋, ⌈hi⌉]
+    that is guaranteed to contain all points in the original interval. -/
+noncomputable def ofRealEndpoints (lo hi : ℝ) (hle : lo ≤ hi) : IntervalRat where
+  lo := ⌊lo⌋
+  hi := ⌈hi⌉
+  le := by
+    have h1 : (⌊lo⌋ : ℝ) ≤ lo := Int.floor_le lo
+    have h2 : hi ≤ (⌈hi⌉ : ℝ) := Int.le_ceil hi
+    have h3 : (⌊lo⌋ : ℝ) ≤ (⌈hi⌉ : ℝ) := le_trans h1 (le_trans hle h2)
+    exact_mod_cast h3
+
+/-- Any point in [lo, hi] is in the rational enclosure [⌊lo⌋, ⌈hi⌉] -/
+theorem mem_ofRealEndpoints {x lo hi : ℝ} (hle : lo ≤ hi) (hx : lo ≤ x ∧ x ≤ hi) :
+    x ∈ ofRealEndpoints lo hi hle := by
+  simp only [mem_def, ofRealEndpoints]
+  constructor
+  · have h := Int.floor_le lo
+    exact le_trans h hx.1
+  · have h := Int.le_ceil hi
+    exact le_trans hx.2 h
+
+/-! ### Exponential interval -/
+
+/-- Interval bound for exp on rational intervals.
+    Since exp is strictly increasing, exp([a,b]) ⊆ [⌊exp(a)⌋, ⌈exp(b)⌉].
+    This uses Real.exp and floor/ceil to get rational bounds. -/
+noncomputable def expInterval (I : IntervalRat) : IntervalRat :=
+  ofRealEndpoints (Real.exp I.lo) (Real.exp I.hi)
+    (Real.exp_le_exp.mpr (by exact_mod_cast I.le))
+
+/-- FTIA for exp: if x ∈ I, then exp(x) ∈ expInterval(I) -/
+theorem mem_expInterval {x : ℝ} {I : IntervalRat} (hx : x ∈ I) :
+    Real.exp x ∈ expInterval I := by
+  simp only [expInterval]
+  apply mem_ofRealEndpoints
+  simp only [mem_def] at hx
+  constructor
+  · exact Real.exp_le_exp.mpr hx.1
+  · exact Real.exp_le_exp.mpr hx.2
+
+/-! ### Positive interval check -/
+
+/-- Check if an interval is strictly positive (lo > 0) -/
+def isPositive (I : IntervalRat) : Prop := 0 < I.lo
+
+/-- Decidable isPositive -/
+instance (I : IntervalRat) : Decidable (isPositive I) :=
+  inferInstanceAs (Decidable (0 < I.lo))
+
+/-- An interval that is guaranteed to be strictly positive -/
+structure IntervalRatPos extends IntervalRat where
+  lo_pos : 0 < toIntervalRat.lo
+
+/-! ### Logarithm interval (for positive intervals) -/
+
+/-- Interval bound for log on positive rational intervals.
+    Since log is strictly increasing on (0, ∞), log([a,b]) ⊆ [⌊log(a)⌋, ⌈log(b)⌉] for a > 0.
+    This uses Real.log and floor/ceil to get rational bounds. -/
+noncomputable def logInterval (I : IntervalRatPos) : IntervalRat :=
+  ofRealEndpoints (Real.log I.lo) (Real.log I.hi)
+    (Real.log_le_log (by exact_mod_cast I.lo_pos) (by exact_mod_cast I.le))
+
+/-- FTIA for log: if x ∈ I with lo > 0, then log(x) ∈ logInterval(I) -/
+theorem mem_logInterval {x : ℝ} {I : IntervalRatPos} (hx : x ∈ I.toIntervalRat) :
+    Real.log x ∈ logInterval I := by
+  simp only [logInterval]
+  apply mem_ofRealEndpoints
+  simp only [mem_def] at hx
+  have hlo_pos : (0 : ℝ) < I.lo := by exact_mod_cast I.lo_pos
+  have hx_pos : 0 < x := lt_of_lt_of_le hlo_pos hx.1
+  constructor
+  · exact Real.log_le_log hlo_pos hx.1
+  · exact Real.log_le_log hx_pos hx.2
+
+/-! ### Atanh interval (for intervals in (-1, 1)) -/
+
+/-- An interval strictly contained in (-1, 1), suitable for atanh -/
+structure IntervalRatInUnitBall where
+  /-- Lower endpoint of the interval. -/
+  lo : ℚ
+  /-- Upper endpoint of the interval. -/
+  hi : ℚ
+  le : lo ≤ hi
+  lo_gt : -1 < lo
+  hi_lt : hi < 1
+
+/-- Convert to standard interval -/
+def IntervalRatInUnitBall.toIntervalRat (I : IntervalRatInUnitBall) : IntervalRat :=
+  ⟨I.lo, I.hi, I.le⟩
+
+/-- Membership in the underlying interval -/
+instance : Membership ℝ IntervalRatInUnitBall where
+  mem I x := (I.lo : ℝ) ≤ x ∧ x ≤ I.hi
+
+theorem IntervalRatInUnitBall.mem_def {x : ℝ} {I : IntervalRatInUnitBall} :
+    x ∈ I ↔ (I.lo : ℝ) ≤ x ∧ x ≤ I.hi := Iff.rfl
+
+/-- Interval bound for atanh on intervals strictly inside (-1, 1).
+    Since atanh is strictly increasing on (-1, 1), atanh([a,b]) ⊆ [⌊atanh(a)⌋, ⌈atanh(b)⌉]. -/
+noncomputable def atanhIntervalComputed (I : IntervalRatInUnitBall) : IntervalRat :=
+  have hlo : (-1 : ℝ) < I.lo := by exact_mod_cast I.lo_gt
+  have hhi : (I.hi : ℝ) < 1 := by exact_mod_cast I.hi_lt
+  have hle : (I.lo : ℝ) ≤ I.hi := by exact_mod_cast I.le
+  ofRealEndpoints (Real.atanh I.lo) (Real.atanh I.hi)
+    (Real.atanh_mono
+      (by rw [abs_lt]; constructor <;> linarith)
+      (by rw [abs_lt]; constructor <;> linarith)
+      hle)
+
+/-- FTIA for atanh: if x ∈ I and I ⊂ (-1, 1), then atanh(x) ∈ atanhIntervalComputed(I) -/
+theorem mem_atanhIntervalComputed {x : ℝ} {I : IntervalRatInUnitBall} (hx : x ∈ I) :
+    Real.atanh x ∈ atanhIntervalComputed I := by
+  simp only [atanhIntervalComputed]
+  apply mem_ofRealEndpoints
+  have hlo : (-1 : ℝ) < I.lo := by exact_mod_cast I.lo_gt
+  have hhi : (I.hi : ℝ) < 1 := by exact_mod_cast I.hi_lt
+  rw [IntervalRatInUnitBall.mem_def] at hx
+  have hx_lo : -1 < x := by linarith [hx.1]
+  have hx_hi : x < 1 := by linarith [hx.2]
+  have habs_lo : |(I.lo : ℝ)| < 1 := by rw [abs_lt]; constructor <;> linarith
+  have habs_hi : |(I.hi : ℝ)| < 1 := by rw [abs_lt]; constructor <;> linarith
+  have habs_x : |x| < 1 := by rw [abs_lt]; constructor <;> linarith
+  constructor
+  · exact Real.atanh_mono habs_lo habs_x hx.1
+  · exact Real.atanh_mono habs_x habs_hi hx.2
+
+/-! ### Square Root Interval -/
+
+/-- Integer square root (floor of sqrt).
+    Satisfies: `(intSqrtNat n)^2 ≤ n < (intSqrtNat n + 1)^2` -/
+def intSqrtNat (n : Nat) : Nat := Nat.sqrt n
+
+/-- Key property of Nat.sqrt: `Nat.sqrt n ^ 2 ≤ n` -/
+private theorem nat_sqrt_sq_le (n : Nat) : (Nat.sqrt n) ^ 2 ≤ n := Nat.sqrt_le' n
+
+/-- Key property of Nat.sqrt: `n < (Nat.sqrt n + 1) ^ 2` -/
+private theorem nat_lt_succ_sqrt_sq (n : Nat) : n < (Nat.sqrt n + 1) ^ 2 := by
+  have h := Nat.lt_succ_sqrt n
+  simp only [Nat.succ_eq_add_one] at h
+  rw [sq]
+  exact h
+
+/-- Rational lower bound for sqrt.
+    For q ≥ 0 with q = num/den, we compute floor(sqrt(num * den)) / den.
+    This gives: sqrtRatLower q ≤ sqrt(q).
+
+    The idea: sqrt(num/den) = sqrt(num*den)/den when properly scaled. -/
+def sqrtRatLower (q : ℚ) : ℚ :=
+  if q ≤ 0 then 0
+  else
+    let num := q.num.natAbs
+    let den := q.den
+    -- Compute floor(sqrt(num * den)) / den
+    let s := intSqrtNat (num * den)
+    (s : ℚ) / den
+
+/-- Rational upper bound for sqrt.
+    For q ≥ 0, we compute ceil(sqrt(num * den)) / den.
+    This gives: sqrt(q) ≤ sqrtRatUpper q. -/
+def sqrtRatUpper (q : ℚ) : ℚ :=
+  if q ≤ 0 then 0
+  else
+    let num := q.num.natAbs
+    let den := q.den
+    let prod := num * den
+    let s := intSqrtNat prod
+    -- If s^2 = prod, return s/den; else (s+1)/den
+    if s * s = prod then (s : ℚ) / den
+    else ((s + 1) : ℚ) / den
+
+/-- Scaling exponent for precise sqrt bounds (2^scaleBits = scaling factor for denominator) -/
+def sqrtScaleBits : Nat := 20  -- gives about 6 decimal digits precision
+
+/-- Rational lower bound for sqrt with high precision.
+    For q ≥ 0, we scale by 4^k, compute integer sqrt, and scale back by 2^k.
+    This gives: sqrtRatLowerPrec q ≤ sqrt(q) with precision ~2^(-k). -/
+def sqrtRatLowerPrec (q : ℚ) (scaleBits : Nat := sqrtScaleBits) : ℚ :=
+  if q ≤ 0 then 0
+  else
+    let num := q.num.natAbs
+    let den := q.den
+    -- Scale up: multiply num by 4^scaleBits = 2^(2*scaleBits)
+    let scaledNum := num * (2 ^ (2 * scaleBits))
+    -- Compute floor(sqrt(scaledNum * den))
+    let s := intSqrtNat (scaledNum * den)
+    -- Scale back: divide by den * 2^scaleBits
+    (s : ℚ) / (den * 2 ^ scaleBits)
+
+/-- Rational upper bound for sqrt with high precision.
+    For q ≥ 0, we scale by 4^k, compute ceil of integer sqrt, and scale back by 2^k.
+    This gives: sqrt(q) ≤ sqrtRatUpperPrec q with precision ~2^(-k). -/
+def sqrtRatUpperPrec (q : ℚ) (scaleBits : Nat := sqrtScaleBits) : ℚ :=
+  if q ≤ 0 then 0
+  else
+    let num := q.num.natAbs
+    let den := q.den
+    let scaledNum := num * (2 ^ (2 * scaleBits))
+    let prod := scaledNum * den
+    let s := intSqrtNat prod
+    -- If s^2 = prod (perfect square), use s; else use s+1 (ceiling)
+    let result := if s * s = prod then s else s + 1
+    (result : ℚ) / (den * 2 ^ scaleBits)
+
+/-- Helper: For non-negative q, q.num = q.num.natAbs -/
+private theorem rat_num_of_nonneg {q : ℚ} (hq : 0 ≤ q) : (q.num : ℤ) = q.num.natAbs := by
+  exact (Int.natAbs_of_nonneg (Rat.num_nonneg.mpr hq)).symm
+
+/-- Helper: For positive q, num.natAbs > 0 -/
+private theorem rat_num_natAbs_pos {q : ℚ} (hq : 0 < q) : 0 < q.num.natAbs := by
+  have h := Rat.num_pos.mpr hq
+  exact Int.natAbs_pos.mpr (ne_of_gt h)
+
+/-- Soundness of sqrtRatLower: sqrtRatLower q ≤ Real.sqrt q for q ≥ 0 -/
+theorem sqrtRatLower_le_sqrt {q : ℚ} (hq : 0 ≤ q) : (sqrtRatLower q : ℝ) ≤ Real.sqrt q := by
+  simp only [sqrtRatLower]
+  by_cases! hq0 : q ≤ 0
+  · -- q = 0
+    have heq : q = 0 := le_antisymm hq0 hq
+    simp only [heq, le_refl, ↓reduceIte, Rat.cast_zero, Real.sqrt_zero]
+  · -- q > 0
+    simp only [not_le.mpr hq0, ↓reduceIte]
+    -- Key insight: sqrtRatLower returns s/den where s = floor(sqrt(num*den))
+    -- We want to show s/den ≤ sqrt(q) = sqrt(num/den)
+    -- This follows because s^2 ≤ num*den, so s ≤ sqrt(num*den), hence s/den ≤ sqrt(num*den)/den =
+    -- sqrt(num/den)
+    have hden_pos : (0 : ℝ) < q.den := by exact_mod_cast q.den_pos
+    have hden_nn : (0 : ℝ) ≤ q.den := le_of_lt hden_pos
+    have hsqrt_den_pos : 0 < Real.sqrt q.den := Real.sqrt_pos.mpr hden_pos
+    have hnum_nn : 0 ≤ q.num := Rat.num_nonneg.mpr hq
+    have hnum_real_nn : (0 : ℝ) ≤ q.num.natAbs := Nat.cast_nonneg _
+    -- s^2 ≤ num * den
+    have hsq : (intSqrtNat (q.num.natAbs * q.den))^2 ≤ q.num.natAbs * q.den := nat_sqrt_sq_le _
+    -- s ≤ sqrt(num * den)
+    have hs_le : (intSqrtNat (q.num.natAbs * q.den) : ℝ) ≤ Real.sqrt (q.num.natAbs * q.den) := by
+      have h1 : ((intSqrtNat (q.num.natAbs * q.den) : ℕ) : ℝ) ^ 2 ≤ (q.num.natAbs * q.den : ℕ) := by
+        exact_mod_cast hsq
+      have h2 := Real.sqrt_sq (Nat.cast_nonneg (intSqrtNat (q.num.natAbs * q.den)))
+      calc (intSqrtNat (q.num.natAbs * q.den) : ℝ)
+          = Real.sqrt ((intSqrtNat (q.num.natAbs * q.den) : ℝ) ^ 2) := h2.symm
+        _ ≤ Real.sqrt ((q.num.natAbs * q.den : ℕ) : ℝ) := Real.sqrt_le_sqrt h1
+        _ = Real.sqrt ((q.num.natAbs : ℝ) * q.den) := by rw [Nat.cast_mul]
+    -- q = num/den and sqrt(q) = sqrt(num)/sqrt(den) = sqrt(num*den)/den (after scaling)
+    have hq_eq : (q : ℝ) = (q.num.natAbs : ℝ) / q.den := by
+      have habs : (q.num : ℤ) = q.num.natAbs := (Int.natAbs_of_nonneg hnum_nn).symm
+      rw [Rat.cast_def, habs, Int.cast_natCast]
+      simp only [Int.natAbs_natCast]
+    simp only [Rat.cast_div, Rat.cast_natCast]
+    rw [hq_eq, Real.sqrt_div hnum_real_nn, div_le_div_iff₀ hden_pos hsqrt_den_pos]
+    -- Goal: s * sqrt(den) ≤ sqrt(num) * den
+    calc (intSqrtNat (q.num.natAbs * q.den) : ℝ) * Real.sqrt q.den
+        ≤ Real.sqrt (q.num.natAbs * q.den) * Real.sqrt q.den := by
+            apply mul_le_mul_of_nonneg_right hs_le (le_of_lt hsqrt_den_pos)
+      _ = Real.sqrt (q.num.natAbs) * Real.sqrt q.den * Real.sqrt q.den := by
+            rw [Real.sqrt_mul hnum_real_nn]
+      _ = Real.sqrt (q.num.natAbs) * (Real.sqrt q.den * Real.sqrt q.den) := by ring
+      _ = Real.sqrt (q.num.natAbs) * q.den := by rw [Real.mul_self_sqrt hden_nn]
+
+/-- Soundness of sqrtRatUpper: Real.sqrt q ≤ sqrtRatUpper q for q ≥ 0 -/
+theorem sqrt_le_sqrtRatUpper {q : ℚ} (hq : 0 ≤ q) : Real.sqrt q ≤ (sqrtRatUpper q : ℝ) := by
+  simp only [sqrtRatUpper]
+  by_cases! hq0 : q ≤ 0
+  · -- q = 0
+    have heq : q = 0 := le_antisymm hq0 hq
+    simp only [heq, le_refl, ↓reduceIte, Rat.cast_zero, Real.sqrt_zero]
+  · -- q > 0
+    simp only [not_le.mpr hq0, ↓reduceIte]
+    have hden_pos : (0 : ℝ) < q.den := by exact_mod_cast q.den_pos
+    have hden_nn : (0 : ℝ) ≤ q.den := le_of_lt hden_pos
+    have hnum_nn : 0 ≤ q.num := Rat.num_nonneg.mpr hq
+    have hnum_real_nn : (0 : ℝ) ≤ q.num.natAbs := Nat.cast_nonneg _
+    have hsqrt_den_pos : 0 < Real.sqrt q.den := Real.sqrt_pos.mpr hden_pos
+    have hq_eq : (q : ℝ) = (q.num.natAbs : ℝ) / q.den := by
+      have habs : (q.num : ℤ) = q.num.natAbs := (Int.natAbs_of_nonneg hnum_nn).symm
+      rw [Rat.cast_def, habs, Int.cast_natCast]
+      simp only [Int.natAbs_natCast]
+    set prod := q.num.natAbs * q.den
+    set s := intSqrtNat prod
+    rw [hq_eq, Real.sqrt_div hnum_real_nn]
+    by_cases hperfect : s * s = prod
+    · -- Perfect square case: s^2 = num * den, so sqrt(num*den) = s
+      simp only [hperfect, ↓reduceIte, Rat.cast_div, Rat.cast_natCast]
+      rw [div_le_div_iff₀ hsqrt_den_pos hden_pos]
+      -- sqrt(num*den) = s since s^2 = num*den
+      have heq_sqrt : Real.sqrt ((q.num.natAbs : ℝ) * q.den) = s := by
+        have h : (s : ℝ) ^ 2 = (q.num.natAbs : ℝ) * q.den := by
+          calc (s : ℝ) ^ 2 = ((s * s : ℕ) : ℝ) := by push_cast; ring
+            _ = (prod : ℝ) := by exact_mod_cast hperfect
+            _ = (q.num.natAbs : ℝ) * q.den := by simp only [prod, Nat.cast_mul]
+        rw [← h, Real.sqrt_sq (Nat.cast_nonneg s)]
+      calc Real.sqrt q.num.natAbs * q.den
+          = Real.sqrt q.num.natAbs * (Real.sqrt q.den * Real.sqrt q.den) := by
+              rw [Real.mul_self_sqrt hden_nn]
+        _ = (Real.sqrt q.num.natAbs * Real.sqrt q.den) * Real.sqrt q.den := by ring
+        _ = Real.sqrt ((q.num.natAbs : ℝ) * q.den) * Real.sqrt q.den := by
+              rw [Real.sqrt_mul hnum_real_nn]
+        _ = (s : ℝ) * Real.sqrt q.den := by rw [heq_sqrt]
+        _ ≤ (s : ℝ) * Real.sqrt q.den := le_refl _
+    · -- Non-perfect square case: prod < (s+1)^2
+      simp only [hperfect, ↓reduceIte, Rat.cast_div, Rat.cast_natCast]
+      rw [div_le_div_iff₀ hsqrt_den_pos hden_pos]
+      have hlt : prod < (s + 1) ^ 2 := nat_lt_succ_sqrt_sq prod
+      have hsqrt_le : Real.sqrt ((q.num.natAbs : ℝ) * q.den) ≤ (s + 1 : ℕ) := by
+        have h1 : (q.num.natAbs : ℝ) * q.den < ((s + 1) ^ 2 : ℕ) := by
+          calc (q.num.natAbs : ℝ) * q.den = (prod : ℕ) := by simp only [prod, Nat.cast_mul]
+            _ < ((s + 1) ^ 2 : ℕ) := by exact_mod_cast hlt
+        have h2 : (0 : ℝ) ≤ (s + 1 : ℕ) := Nat.cast_nonneg _
+        have h3 : Real.sqrt ((q.num.natAbs : ℝ) * q.den) < Real.sqrt (((s + 1) ^ 2 : ℕ) : ℝ) :=
+          Real.sqrt_lt_sqrt (mul_nonneg hnum_real_nn hden_nn) h1
+        have h4 : Real.sqrt (((s + 1) ^ 2 : ℕ) : ℝ) = (s + 1 : ℕ) := by
+          have : (((s + 1) ^ 2 : ℕ) : ℝ) = ((s + 1 : ℕ) : ℝ) ^ 2 := by push_cast; ring
+          rw [this, Real.sqrt_sq h2]
+        exact le_of_lt (h3.trans_eq h4)
+      have hgoal : Real.sqrt q.num.natAbs * q.den ≤ ((s + 1 : ℕ) : ℝ) * Real.sqrt q.den := by
+        calc Real.sqrt q.num.natAbs * q.den
+            = Real.sqrt q.num.natAbs * (Real.sqrt q.den * Real.sqrt q.den) := by
+                rw [Real.mul_self_sqrt hden_nn]
+          _ = (Real.sqrt q.num.natAbs * Real.sqrt q.den) * Real.sqrt q.den := by ring
+          _ = Real.sqrt ((q.num.natAbs : ℝ) * q.den) * Real.sqrt q.den := by
+                rw [Real.sqrt_mul hnum_real_nn]
+          _ ≤ ((s + 1 : ℕ) : ℝ) * Real.sqrt q.den := by
+              exact mul_le_mul_of_nonneg_right hsqrt_le (le_of_lt hsqrt_den_pos)
+      convert! hgoal using 2
+      push_cast
+      ring
+
+/-- Square root interval with conservative bounds.
+    For a non-negative interval [lo, hi], sqrt is monotone so:
+    sqrt([lo, hi]) ⊆ [0, max(hi, 1)]
+
+    The lower bound is 0 (always sound for sqrt).
+    The upper bound uses max(hi, 1) which satisfies sqrt(x) ≤ max(x, 1) for x ≥ 0. -/
+def sqrtInterval (I : IntervalRat) : IntervalRat :=
+  ⟨0, max I.hi 1, le_sup_of_le_right rfl⟩
+
+/-- Improved square root interval with tight lower bounds.
+    For a non-negative interval [lo, hi] with lo ≥ 0:
+    - Lower bound: sqrtRatLower(lo)
+    - Upper bound: sqrtRatUpper(hi)
+
+    For intervals crossing zero, we use 0 as lower bound. -/
+def sqrtIntervalTight (I : IntervalRat) : IntervalRat :=
+  if h : 0 ≤ I.lo then
+    ⟨sqrtRatLower I.lo, max (sqrtRatUpper I.hi) 1,
+     by
+       simp only [le_max_iff]
+       left
+       have hlo_le_hi : I.lo ≤ I.hi := I.le
+       -- sqrtRatLower I.lo ≤ sqrtRatUpper I.hi
+       have h1 : (sqrtRatLower I.lo : ℝ) ≤ Real.sqrt I.lo := sqrtRatLower_le_sqrt h
+       have h2 : Real.sqrt I.hi ≤ (sqrtRatUpper I.hi : ℝ) := sqrt_le_sqrtRatUpper (le_trans h
+         hlo_le_hi)
+       have h3 : Real.sqrt I.lo ≤ Real.sqrt I.hi := Real.sqrt_le_sqrt (by exact_mod_cast hlo_le_hi)
+       have h4 : (sqrtRatLower I.lo : ℝ) ≤ (sqrtRatUpper I.hi : ℝ) := le_trans h1 (le_trans h3 h2)
+       exact_mod_cast h4⟩
+  else
+    ⟨0, max (sqrtRatUpper I.hi) 1, le_sup_of_le_right rfl⟩
+
+/-- Soundness of sqrtRatLowerPrec: sqrtRatLowerPrec q k ≤ Real.sqrt q for q ≥ 0 -/
+theorem sqrtRatLowerPrec_le_sqrt {q : ℚ} (hq : 0 ≤ q) (k : Nat) :
+    (sqrtRatLowerPrec q k : ℝ) ≤ Real.sqrt q := by
+  simp only [sqrtRatLowerPrec]
+  by_cases! hq0 : q ≤ 0
+  · have heq : q = 0 := le_antisymm hq0 hq
+    simp only [heq, le_refl, ↓reduceIte, Rat.cast_zero, Real.sqrt_zero]
+  · simp only [not_le.mpr hq0, ↓reduceIte]
+    -- Setup: let num = q.num.natAbs, den = q.den, scaledNum = num * 4^k
+    -- We compute s = floor(sqrt(scaledNum * den)) and return s / (den * 2^k)
+    -- Goal: s / (den * 2^k) ≤ sqrt(q) = sqrt(num/den)
+    have hden_pos : (0 : ℝ) < q.den := by exact_mod_cast q.den_pos
+    have hden_nn : (0 : ℝ) ≤ q.den := le_of_lt hden_pos
+    have hnum_nn : 0 ≤ q.num := Rat.num_nonneg.mpr hq
+    have hnum_real_nn : (0 : ℝ) ≤ q.num.natAbs := Nat.cast_nonneg _
+    have h2k_pos : (0 : ℝ) < 2 ^ k := pow_pos (by norm_num : (0 : ℝ) < 2) k
+    have h2k_nn : (0 : ℝ) ≤ 2 ^ k := le_of_lt h2k_pos
+    have hden2k_pos : (0 : ℝ) < q.den * 2 ^ k := mul_pos hden_pos h2k_pos
+    set num := q.num.natAbs
+    set den := q.den
+    set scaledNum := num * 2 ^ (2 * k)
+    set prod := scaledNum * den
+    set s := intSqrtNat prod
+    -- s^2 ≤ prod, so s ≤ sqrt(prod)
+    have hsq : s ^ 2 ≤ prod := nat_sqrt_sq_le prod
+    have hs_le_sqrt : (s : ℝ) ≤ Real.sqrt prod := by
+      have h1 : (s : ℝ) ^ 2 ≤ prod := by exact_mod_cast hsq
+      have h2 := Real.sqrt_sq (Nat.cast_nonneg s)
+      calc (s : ℝ) = Real.sqrt ((s : ℝ) ^ 2) := h2.symm
+        _ ≤ Real.sqrt prod := Real.sqrt_le_sqrt h1
+    -- sqrt(prod) = sqrt(num * 4^k * den) = sqrt(num * den) * 2^k
+    have h4k_eq : (2 : ℝ) ^ (2 * k) = ((2 : ℝ) ^ k) ^ 2 := by ring
+    have hsqrt_prod : Real.sqrt (prod : ℕ) = Real.sqrt (num * den) * 2 ^ k := by
+      calc Real.sqrt (prod : ℕ) = Real.sqrt ((scaledNum * den : ℕ) : ℝ) := by simp only [prod]
+        _ = Real.sqrt ((num * 2 ^ (2 * k) * den : ℕ) : ℝ) := by simp only [scaledNum]
+        _ = Real.sqrt ((num : ℝ) * 2 ^ (2 * k) * den) := by push_cast; ring_nf
+        _ = Real.sqrt ((num : ℝ) * den * (2 ^ k) ^ 2) := by rw [h4k_eq]; ring_nf
+        _ = Real.sqrt ((num : ℝ) * den) * Real.sqrt ((2 ^ k) ^ 2) := by
+            rw [Real.sqrt_mul (mul_nonneg hnum_real_nn hden_nn)]
+        _ = Real.sqrt ((num : ℝ) * den) * 2 ^ k := by rw [Real.sqrt_sq h2k_nn]
+    -- q = num/den, so sqrt(q) = sqrt(num)/sqrt(den) = sqrt(num*den)/den
+    have hq_eq : (q : ℝ) = (num : ℝ) / den := by
+      have habs : (q.num : ℤ) = num := (Int.natAbs_of_nonneg hnum_nn).symm
+      rw [Rat.cast_def, habs, Int.cast_natCast]
+    -- Final calculation
+    simp only [Rat.cast_div, Rat.cast_natCast, Rat.cast_mul, Rat.cast_pow, Rat.cast_ofNat]
+    rw [hq_eq, Real.sqrt_div hnum_real_nn]
+    have hsqrt_den_pos : 0 < Real.sqrt den := Real.sqrt_pos.mpr hden_pos
+    -- Key helper: sqrt(num * den) / den = sqrt(num) / sqrt(den)
+    have hsqrt_conv : Real.sqrt (num * den) / den = Real.sqrt num / Real.sqrt den := by
+      have hden_sqrt_mul : Real.sqrt den * Real.sqrt den = den := Real.mul_self_sqrt hden_nn
+      calc Real.sqrt (num * den) / den
+          = Real.sqrt num * Real.sqrt den / den := by rw [Real.sqrt_mul hnum_real_nn]
+        _ = Real.sqrt num * Real.sqrt den / (Real.sqrt den * Real.sqrt den) := by rw [hden_sqrt_mul]
+        _ = Real.sqrt num * (Real.sqrt den / (Real.sqrt den * Real.sqrt den)) := by rw
+          [mul_div_assoc]
+        _ = Real.sqrt num * (Real.sqrt den / Real.sqrt den / Real.sqrt den) := by rw
+          [div_mul_eq_div_div]
+        _ = Real.sqrt num * (1 / Real.sqrt den) := by rw [div_self (ne_of_gt hsqrt_den_pos)]
+        _ = Real.sqrt num / Real.sqrt den := by rw [mul_one_div]
+    calc (s : ℝ) / (den * 2 ^ k)
+        ≤ Real.sqrt prod / (den * 2 ^ k) :=
+          div_le_div_of_nonneg_right hs_le_sqrt (le_of_lt hden2k_pos)
+      _ = (Real.sqrt (num * den) * 2 ^ k) / (den * 2 ^ k) := by rw [hsqrt_prod]
+      _ = Real.sqrt (num * den) / den := by rw [mul_div_mul_right _ _ (ne_of_gt h2k_pos)]
+      _ = Real.sqrt num / Real.sqrt den := hsqrt_conv
+
+/-- Soundness of sqrtRatUpperPrec: Real.sqrt q ≤ sqrtRatUpperPrec q k for q ≥ 0 -/
+theorem sqrt_le_sqrtRatUpperPrec {q : ℚ} (hq : 0 ≤ q) (k : Nat) :
+    Real.sqrt q ≤ (sqrtRatUpperPrec q k : ℝ) := by
+  simp only [sqrtRatUpperPrec]
+  by_cases! hq0 : q ≤ 0
+  · have heq : q = 0 := le_antisymm hq0 hq
+    simp only [heq, le_refl, ↓reduceIte, Rat.cast_zero, Real.sqrt_zero]
+  · simp only [not_le.mpr hq0, ↓reduceIte]
+    have hden_pos : (0 : ℝ) < q.den := by exact_mod_cast q.den_pos
+    have hden_nn : (0 : ℝ) ≤ q.den := le_of_lt hden_pos
+    have hnum_nn : 0 ≤ q.num := Rat.num_nonneg.mpr hq
+    have hnum_real_nn : (0 : ℝ) ≤ q.num.natAbs := Nat.cast_nonneg _
+    have h2k_pos : (0 : ℝ) < 2 ^ k := pow_pos (by norm_num : (0 : ℝ) < 2) k
+    have h2k_nn : (0 : ℝ) ≤ 2 ^ k := le_of_lt h2k_pos
+    have hden2k_pos : (0 : ℝ) < q.den * 2 ^ k := mul_pos hden_pos h2k_pos
+    set num := q.num.natAbs
+    set den := q.den
+    set scaledNum := num * 2 ^ (2 * k)
+    set prod := scaledNum * den
+    set s := intSqrtNat prod
+    -- sqrt(prod) ≤ result (either s or s+1)
+    have h4k_eq : (2 : ℝ) ^ (2 * k) = ((2 : ℝ) ^ k) ^ 2 := by ring
+    have hsqrt_prod : Real.sqrt (prod : ℕ) = Real.sqrt (num * den) * 2 ^ k := by
+      calc Real.sqrt (prod : ℕ) = Real.sqrt ((scaledNum * den : ℕ) : ℝ) := by simp only [prod]
+        _ = Real.sqrt ((num * 2 ^ (2 * k) * den : ℕ) : ℝ) := by simp only [scaledNum]
+        _ = Real.sqrt ((num : ℝ) * 2 ^ (2 * k) * den) := by push_cast; ring_nf
+        _ = Real.sqrt ((num : ℝ) * den * (2 ^ k) ^ 2) := by rw [h4k_eq]; ring_nf
+        _ = Real.sqrt ((num : ℝ) * den) * Real.sqrt ((2 ^ k) ^ 2) := by
+            rw [Real.sqrt_mul (mul_nonneg hnum_real_nn hden_nn)]
+        _ = Real.sqrt ((num : ℝ) * den) * 2 ^ k := by rw [Real.sqrt_sq h2k_nn]
+    have hq_eq : (q : ℝ) = (num : ℝ) / den := by
+      have habs : (q.num : ℤ) = num := (Int.natAbs_of_nonneg hnum_nn).symm
+      rw [Rat.cast_def, habs, Int.cast_natCast]
+    have hsqrt_den_pos : 0 < Real.sqrt den := Real.sqrt_pos.mpr hden_pos
+    -- Key helper: sqrt(num * den) / den = sqrt(num) / sqrt(den)
+    have hsqrt_conv : Real.sqrt (num * den) / den = Real.sqrt num / Real.sqrt den := by
+      have hden_sqrt_mul : Real.sqrt den * Real.sqrt den = den := Real.mul_self_sqrt hden_nn
+      calc Real.sqrt (num * den) / den
+          = Real.sqrt num * Real.sqrt den / den := by rw [Real.sqrt_mul hnum_real_nn]
+        _ = Real.sqrt num * Real.sqrt den / (Real.sqrt den * Real.sqrt den) := by rw [hden_sqrt_mul]
+        _ = Real.sqrt num * (Real.sqrt den / (Real.sqrt den * Real.sqrt den)) := by rw
+          [mul_div_assoc]
+        _ = Real.sqrt num * (Real.sqrt den / Real.sqrt den / Real.sqrt den) := by rw
+          [div_mul_eq_div_div]
+        _ = Real.sqrt num * (1 / Real.sqrt den) := by rw [div_self (ne_of_gt hsqrt_den_pos)]
+        _ = Real.sqrt num / Real.sqrt den := by rw [mul_one_div]
+    by_cases hperfect : s * s = prod
+    · -- Perfect square: result = s, and sqrt(prod) = s
+      simp only [hperfect, ↓reduceIte, Rat.cast_div, Rat.cast_natCast, Rat.cast_mul,
+                 Rat.cast_pow, Rat.cast_ofNat]
+      have hsqrt_eq_s : Real.sqrt (prod : ℕ) = s := by
+        have h : (s : ℝ) ^ 2 = prod := by
+          calc (s : ℝ) ^ 2 = ((s * s : ℕ) : ℝ) := by push_cast; ring
+            _ = prod := by exact_mod_cast hperfect
+        rw [← h, Real.sqrt_sq (Nat.cast_nonneg s)]
+      rw [hq_eq, Real.sqrt_div hnum_real_nn]
+      calc Real.sqrt num / Real.sqrt den
+          = Real.sqrt (num * den) / den := hsqrt_conv.symm
+        _ = (Real.sqrt (num * den) * 2 ^ k) / (den * 2 ^ k) := by
+            rw [mul_div_mul_right _ _ (ne_of_gt h2k_pos)]
+        _ = Real.sqrt prod / (den * 2 ^ k) := by rw [hsqrt_prod]
+        _ ≤ (s : ℝ) / (den * 2 ^ k) := by rw [hsqrt_eq_s]
+    · -- Non-perfect square: result = s + 1, and sqrt(prod) < s + 1
+      simp only [hperfect, ↓reduceIte, Rat.cast_div, Rat.cast_natCast, Rat.cast_mul,
+                 Rat.cast_pow, Rat.cast_ofNat]
+      have hlt : prod < (s + 1) ^ 2 := nat_lt_succ_sqrt_sq prod
+      have hsqrt_lt : Real.sqrt (prod : ℕ) < (s + 1 : ℕ) := by
+        have h1 : (prod : ℝ) < ((s + 1) ^ 2 : ℕ) := by exact_mod_cast hlt
+        have h2 : (0 : ℝ) ≤ (s + 1 : ℕ) := Nat.cast_nonneg _
+        have h3 : Real.sqrt (prod : ℕ) < Real.sqrt (((s + 1) ^ 2 : ℕ) : ℝ) :=
+          Real.sqrt_lt_sqrt (Nat.cast_nonneg _) h1
+        have h4 : Real.sqrt (((s + 1) ^ 2 : ℕ) : ℝ) = (s + 1 : ℕ) := by
+          have : (((s + 1) ^ 2 : ℕ) : ℝ) = ((s + 1 : ℕ) : ℝ) ^ 2 := by push_cast; ring
+          rw [this, Real.sqrt_sq h2]
+        exact h3.trans_eq h4
+      rw [hq_eq, Real.sqrt_div hnum_real_nn]
+      calc Real.sqrt num / Real.sqrt den
+          = Real.sqrt (num * den) / den := hsqrt_conv.symm
+        _ = (Real.sqrt (num * den) * 2 ^ k) / (den * 2 ^ k) := by
+            rw [mul_div_mul_right _ _ (ne_of_gt h2k_pos)]
+        _ = Real.sqrt prod / (den * 2 ^ k) := by rw [hsqrt_prod]
+        _ ≤ (s + 1 : ℕ) / (den * 2 ^ k) :=
+            div_le_div_of_nonneg_right (le_of_lt hsqrt_lt) (le_of_lt hden2k_pos)
+
+/-- High-precision square root interval.
+    For a non-negative interval [lo, hi] with lo ≥ 0:
+    - Lower bound: sqrtRatLowerPrec(lo)
+    - Upper bound: sqrtRatUpperPrec(hi)
+
+    Uses scaling to achieve ~6 decimal digits of precision. -/
+def sqrtIntervalTightPrec (I : IntervalRat) : IntervalRat :=
+  if h : 0 ≤ I.lo then
+    ⟨sqrtRatLowerPrec I.lo, max (sqrtRatUpperPrec I.hi) 1,
+     by
+       simp only [le_max_iff]
+       left
+       have hlo_le_hi : I.lo ≤ I.hi := I.le
+       have h1 : (sqrtRatLowerPrec I.lo : ℝ) ≤ Real.sqrt I.lo := sqrtRatLowerPrec_le_sqrt h
+         sqrtScaleBits
+       have h2 : Real.sqrt I.hi ≤ (sqrtRatUpperPrec I.hi : ℝ) :=
+         sqrt_le_sqrtRatUpperPrec (le_trans h hlo_le_hi) sqrtScaleBits
+       have h3 : Real.sqrt I.lo ≤ Real.sqrt I.hi := Real.sqrt_le_sqrt (by exact_mod_cast hlo_le_hi)
+       have h4 : (sqrtRatLowerPrec I.lo : ℝ) ≤ (sqrtRatUpperPrec I.hi : ℝ) := le_trans h1
+         (le_trans h3 h2)
+       exact_mod_cast h4⟩
+  else
+    ⟨0, max (sqrtRatUpperPrec I.hi) 1, le_sup_of_le_right rfl⟩
+
+/-- Membership theorem for sqrtIntervalTightPrec -/
+theorem mem_sqrtIntervalTightPrec' {x : ℝ} {I : IntervalRat} (hx : x ∈ I) :
+    Real.sqrt x ∈ sqrtIntervalTightPrec I := by
+  simp only [sqrtIntervalTightPrec, mem_def]
+  split_ifs with h
+  · -- lo ≥ 0: use tight bounds
+    constructor
+    · calc (sqrtRatLowerPrec I.lo : ℝ) ≤ Real.sqrt I.lo := sqrtRatLowerPrec_le_sqrt h sqrtScaleBits
+        _ ≤ Real.sqrt x := Real.sqrt_le_sqrt hx.1
+    · simp only [Rat.cast_max, Rat.cast_one]
+      calc Real.sqrt x ≤ Real.sqrt I.hi := Real.sqrt_le_sqrt hx.2
+        _ ≤ (sqrtRatUpperPrec I.hi : ℝ) := sqrt_le_sqrtRatUpperPrec (le_trans h I.le) sqrtScaleBits
+        _ ≤ max (sqrtRatUpperPrec I.hi : ℝ) 1 := le_max_left _ _
+  · -- lo < 0: use 0 as lower bound
+    push Not at h
+    simp only [Rat.cast_zero, Rat.cast_max, Rat.cast_one]
+    constructor
+    · exact Real.sqrt_nonneg x
+    · -- sqrt(x) ≤ max(sqrtRatUpperPrec I.hi, 1)
+      by_cases! hhi_neg : I.hi < 0
+      · -- If hi < 0, then x ≤ I.hi < 0, but sqrt(x) ≥ 0, so sqrt(x) ≤ 1 ≤ max(_, 1)
+        have hx_neg : x < 0 := lt_of_le_of_lt hx.2 (by exact_mod_cast hhi_neg)
+        have hsqrt_zero : Real.sqrt x = 0 := Real.sqrt_eq_zero'.mpr (le_of_lt hx_neg)
+        rw [hsqrt_zero]
+        calc (0 : ℝ) ≤ 1 := by norm_num
+          _ ≤ max (sqrtRatUpperPrec I.hi : ℝ) 1 := le_max_right _ _
+      · calc Real.sqrt x ≤ Real.sqrt I.hi := Real.sqrt_le_sqrt hx.2
+          _ ≤ (sqrtRatUpperPrec I.hi : ℝ) := sqrt_le_sqrtRatUpperPrec hhi_neg sqrtScaleBits
+          _ ≤ max (sqrtRatUpperPrec I.hi : ℝ) 1 := le_max_left _ _
+
+/-- Helper: sqrt(x) ≤ max(x, 1) for x ≥ 0 -/
+private theorem sqrt_le_max_one {x : ℝ} (hx : 0 ≤ x) : Real.sqrt x ≤ max x 1 := by
+  rcases le_or_gt x 1 with hle | hgt
+  · -- x ≤ 1: sqrt(x) ≤ 1 ≤ max(x, 1)
+    calc Real.sqrt x ≤ Real.sqrt 1 := Real.sqrt_le_sqrt hle
+      _ = 1 := Real.sqrt_one
+      _ ≤ max x 1 := le_max_right x 1
+  · -- x > 1: sqrt(x) < x ≤ max(x, 1)
+    have hx_pos : 0 < x := lt_trans zero_lt_one hgt
+    have hsqrt_pos : 0 < Real.sqrt x := Real.sqrt_pos.mpr hx_pos
+    have hsqrt_gt_one : 1 < Real.sqrt x := by
+      rw [← Real.sqrt_one]
+      exact Real.sqrt_lt_sqrt (by norm_num) hgt
+    have hsqrt_lt : Real.sqrt x < x := by
+      have h1 : Real.sqrt x * Real.sqrt x = x := Real.mul_self_sqrt hx
+      have h2 : Real.sqrt x * 1 < Real.sqrt x * Real.sqrt x :=
+        mul_lt_mul_of_pos_left hsqrt_gt_one hsqrt_pos
+      simp only [mul_one] at h2
+      linarith
+    calc Real.sqrt x ≤ x := le_of_lt hsqrt_lt
+      _ ≤ max x 1 := le_max_left x 1
+
+/-- Soundness of sqrt interval: if x ∈ I and x ≥ 0, then sqrt(x) ∈ sqrtInterval I -/
+theorem mem_sqrtInterval {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (hx_nn : 0 ≤ x) :
+    Real.sqrt x ∈ sqrtInterval I := by
+  simp only [mem_def, sqrtInterval, Rat.cast_zero, Rat.cast_max, Rat.cast_one]
+  constructor
+  · exact Real.sqrt_nonneg x
+  · calc Real.sqrt x ≤ max x 1 := sqrt_le_max_one hx_nn
+      _ ≤ max (I.hi : ℝ) 1 := max_le_max_right 1 hx.2
+
+/-- General soundness of sqrt interval: works for any x ∈ I (including negative).
+    When x < 0, Real.sqrt x = 0, which is always in [0, max(hi, 1)]. -/
+theorem mem_sqrtInterval' {x : ℝ} {I : IntervalRat} (hx : x ∈ I) :
+    Real.sqrt x ∈ sqrtInterval I := by
+  rcases le_or_gt 0 x with hnn | hneg
+  · -- x ≥ 0: use the standard soundness
+    exact mem_sqrtInterval hx hnn
+  · -- x < 0: sqrt(x) = 0, and 0 ∈ [0, max(hi, 1)]
+    simp only [mem_def, sqrtInterval, Rat.cast_zero, Rat.cast_max, Rat.cast_one]
+    have hsqrt_zero : Real.sqrt x = 0 := Real.sqrt_eq_zero'.mpr (le_of_lt hneg)
+    rw [hsqrt_zero]
+    constructor
+    · exact le_refl 0
+    · calc (0 : ℝ) ≤ 1 := by norm_num
+        _ ≤ max (I.hi : ℝ) 1 := le_max_right _ _
+
+/-- Helper: upper bound for tight sqrt interval is valid -/
+private theorem sqrt_le_sqrtRatUpper_max {x : ℝ} {q : ℚ} (hx : 0 ≤ x) (hxq : x ≤ q) :
+    Real.sqrt x ≤ max (sqrtRatUpper q : ℝ) 1 := by
+  by_cases! hq0 : q ≤ 0
+  · -- q ≤ 0 means x ≤ 0, combined with hx gives x = 0
+    have hx0 : x = 0 := le_antisymm (le_trans hxq (by exact_mod_cast hq0)) hx
+    rw [hx0, Real.sqrt_zero]
+    exact le_trans (by norm_num : (0 : ℝ) ≤ 1) (le_max_right (sqrtRatUpper q : ℝ) (1 : ℝ))
+  · -- q > 0
+    calc Real.sqrt x
+        ≤ Real.sqrt q := Real.sqrt_le_sqrt hxq
+      _ ≤ sqrtRatUpper q := sqrt_le_sqrtRatUpper (le_of_lt hq0)
+      _ ≤ max (sqrtRatUpper q : ℝ) 1 := le_max_left (sqrtRatUpper q : ℝ) (1 : ℝ)
+
+/-- Soundness of tight sqrt interval: if x ∈ I and x ≥ 0, then sqrt(x) ∈ sqrtIntervalTight I -/
+theorem mem_sqrtIntervalTight {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (hx_nn : 0 ≤ x) :
+    Real.sqrt x ∈ sqrtIntervalTight I := by
+  simp only [sqrtIntervalTight]
+  split_ifs with hlo
+  · -- Case: I.lo ≥ 0 (positive interval)
+    simp only [mem_def, Rat.cast_max, Rat.cast_one]
+    constructor
+    · -- Lower bound: sqrtRatLower I.lo ≤ sqrt(x)
+      have h1 : (sqrtRatLower I.lo : ℝ) ≤ Real.sqrt I.lo := sqrtRatLower_le_sqrt hlo
+      have h2 : Real.sqrt (I.lo : ℝ) ≤ Real.sqrt x := Real.sqrt_le_sqrt hx.1
+      exact le_trans h1 h2
+    · -- Upper bound: sqrt(x) ≤ max (sqrtRatUpper I.hi) 1
+      exact sqrt_le_sqrtRatUpper_max hx_nn hx.2
+  · -- Case: I.lo < 0 (interval crosses zero)
+    simp only [mem_def, Rat.cast_zero, Rat.cast_max, Rat.cast_one]
+    constructor
+    · exact Real.sqrt_nonneg x
+    · exact sqrt_le_sqrtRatUpper_max hx_nn hx.2
+
+/-- General soundness of tight sqrt interval: works for any x ∈ I (including negative).
+    When x < 0, Real.sqrt x = 0, which is always in the result interval. -/
+theorem mem_sqrtIntervalTight' {x : ℝ} {I : IntervalRat} (hx : x ∈ I) :
+    Real.sqrt x ∈ sqrtIntervalTight I := by
+  rcases le_or_gt 0 x with hnn | hneg
+  · exact mem_sqrtIntervalTight hx hnn
+  · -- x < 0: sqrt(x) = 0
+    have hsqrt_zero : Real.sqrt x = 0 := Real.sqrt_eq_zero'.mpr (le_of_lt hneg)
+    simp only [sqrtIntervalTight]
+    split_ifs with hlo
+    · -- I.lo ≥ 0, but x ∈ I and x < 0, contradiction
+      have h : (I.lo : ℝ) ≤ x := hx.1
+      have hlo_real : (0 : ℝ) ≤ I.lo := by exact_mod_cast hlo
+      have hx_nn : 0 ≤ x := le_trans hlo_real h
+      exact absurd hx_nn (not_le.mpr hneg)
+    · -- I.lo < 0
+      simp only [mem_def, Rat.cast_zero, Rat.cast_max, Rat.cast_one]
+      rw [hsqrt_zero]
+      constructor
+      · exact le_refl 0
+      · exact le_trans (by norm_num : (0 : ℝ) ≤ 1) (le_max_right (sqrtRatUpper I.hi : ℝ) (1 : ℝ))
+
+end IntervalRat
+
+end LeanCert.Core
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Expression Support Predicates
+
+This file defines predicates indicating which expressions are supported by
+different interval evaluation strategies.
+
+## Main definitions
+
+* `ExprSupportedCore` - Predicate for expressions in the computable subset
+  (const, var, add, mul, neg, sin, cos, exp, log, sqrt, sinh, cosh, tanh, erf, pi)
+
+* `ADSupported` - Predicate for the noncomputable AD subset
+  (const, var, add, mul, neg, sin, cos, exp)
+
+## Design notes
+
+`ADSupported` is the differentiable fragment used by automatic
+differentiation. It is contained in `ExprSupportedCore` via
+`ADSupported.toCore`. Checked evaluators accept arbitrary expressions and
+encode domain failure in their result type, so they need no syntactic support
+predicate.
+
+The core subset is kept computable so that tactics can use `native_decide`
+for interval bound checking. The extended subset uses `Real.exp` with
+floor/ceil bounds, which requires noncomputability.
+-/
+
+/-! ### Core supported expression subset (computable) -/
+
+@[expose] public section
+
+namespace LeanCert.Core
+
+open LeanCert.Core
+
+/-- Predicate indicating an expression is in the computable core subset.
+    Supports: const, var, add, mul, neg, sin, cos, exp, log, sqrt, sinh, cosh, tanh, erf, pi
+
+    Note: log requires positive domain for correctness. The correctness theorem
+    `evalIntervalCore_correct` has an additional hypothesis `evalDomainValid`
+    that ensures log arguments evaluate to positive intervals. -/
+inductive ExprSupportedCore : Expr → Prop where
+  | const (q : ℚ) : ExprSupportedCore (Expr.const q)
+  | var (idx : Nat) : ExprSupportedCore (Expr.var idx)
+  | add {e₁ e₂ : Expr} : ExprSupportedCore e₁ → ExprSupportedCore e₂ →
+      ExprSupportedCore (Expr.add e₁ e₂)
+  | mul {e₁ e₂ : Expr} : ExprSupportedCore e₁ → ExprSupportedCore e₂ →
+      ExprSupportedCore (Expr.mul e₁ e₂)
+  | neg {e : Expr} : ExprSupportedCore e → ExprSupportedCore (Expr.neg e)
+  | sin {e : Expr} : ExprSupportedCore e → ExprSupportedCore (Expr.sin e)
+  | cos {e : Expr} : ExprSupportedCore e → ExprSupportedCore (Expr.cos e)
+  | exp {e : Expr} : ExprSupportedCore e → ExprSupportedCore (Expr.exp e)
+  | log {e : Expr} : ExprSupportedCore e → ExprSupportedCore (Expr.log e)
+  | sqrt {e : Expr} : ExprSupportedCore e → ExprSupportedCore (Expr.sqrt e)
+  | sinh {e : Expr} : ExprSupportedCore e → ExprSupportedCore (Expr.sinh e)
+  | cosh {e : Expr} : ExprSupportedCore e → ExprSupportedCore (Expr.cosh e)
+  | tanh {e : Expr} : ExprSupportedCore e → ExprSupportedCore (Expr.tanh e)
+  | erf {e : Expr} : ExprSupportedCore e → ExprSupportedCore (Expr.erf e)
+  | namedConst (c : MathConst) : ExprSupportedCore (Expr.namedConst c)
+
+/-! ### Extended supported expression subset (with exp) -/
+
+/-- Predicate indicating an expression is in the fully-verified subset for AD.
+    Supports: const, var, add, mul, neg, sin, cos, exp
+    Does NOT support:
+    - sqrt (not differentiable at 0 - use ExprSupportedCore for interval evaluation only)
+    - inv (requires nonzero interval checks)
+    - log (requires positive interval checks)
+    - atan/arsinh/atanh (derivative proofs incomplete in the total AD evaluator) -/
+inductive ADSupported : Expr → Prop where
+  | const (q : ℚ) : ADSupported (Expr.const q)
+  | var (idx : Nat) : ADSupported (Expr.var idx)
+  | add {e₁ e₂ : Expr} : ADSupported e₁ → ADSupported e₂ → ADSupported (Expr.add e₁ e₂)
+  | mul {e₁ e₂ : Expr} : ADSupported e₁ → ADSupported e₂ → ADSupported (Expr.mul e₁ e₂)
+  | neg {e : Expr} : ADSupported e → ADSupported (Expr.neg e)
+  | sin {e : Expr} : ADSupported e → ADSupported (Expr.sin e)
+  | cos {e : Expr} : ADSupported e → ADSupported (Expr.cos e)
+  | exp {e : Expr} : ADSupported e → ADSupported (Expr.exp e)
+
+/-- ADSupported expressions are also in ExprSupportedCore -/
+theorem ADSupported.toCore {e : Expr} (h : ADSupported e) : ExprSupportedCore e := by
+  induction h with
+  | const q => exact ExprSupportedCore.const q
+  | var idx => exact ExprSupportedCore.var idx
+  | add _ _ ih₁ ih₂ => exact ExprSupportedCore.add ih₁ ih₂
+  | mul _ _ ih₁ ih₂ => exact ExprSupportedCore.mul ih₁ ih₂
+  | neg _ ih => exact ExprSupportedCore.neg ih
+  | sin _ ih => exact ExprSupportedCore.sin ih
+  | cos _ ih => exact ExprSupportedCore.cos ih
+  | exp _ ih => exact ExprSupportedCore.exp ih
+
+/-- Computable recognition of the differentiable `ADSupported` subset used
+by Newton/AD backends. -/
+def Expr.checkADSupported : Expr → Bool
+  | .const _ | .var _ => true
+  | .add e₁ e₂ | .mul e₁ e₂ => e₁.checkADSupported && e₂.checkADSupported
+  | .neg e | .exp e | .sin e | .cos e => e.checkADSupported
+  | _ => false
+
+/-- The executable support check recognizes exactly the differentiable
+`ADSupported` fragment used by the checked AD/monotonicity backend. -/
+theorem Expr.checkADSupported_eq_true_iff (e : Expr) :
+    e.checkADSupported = true ↔ ADSupported e := by
+  induction e with
+  | const q => simp [Expr.checkADSupported, ADSupported.const]
+  | var i => simp [Expr.checkADSupported, ADSupported.var]
+  | add e₁ e₂ ih₁ ih₂ =>
+      simp only [Expr.checkADSupported, Bool.and_eq_true, ih₁, ih₂]
+      constructor
+      · rintro ⟨h₁, h₂⟩
+        exact .add h₁ h₂
+      · intro h
+        cases h
+        exact ⟨by assumption, by assumption⟩
+  | mul e₁ e₂ ih₁ ih₂ =>
+      simp only [Expr.checkADSupported, Bool.and_eq_true, ih₁, ih₂]
+      constructor
+      · rintro ⟨h₁, h₂⟩
+        exact .mul h₁ h₂
+      · intro h
+        cases h
+        exact ⟨by assumption, by assumption⟩
+  | neg e ih | exp e ih | sin e ih | cos e ih =>
+      simp only [Expr.checkADSupported, ih]
+      constructor
+      · intro h
+        first | exact .neg h | exact .exp h | exact .sin h | exact .cos h
+      · intro h
+        cases h
+        assumption
+  | inv e ih => exact ⟨by simp [Expr.checkADSupported], fun h => by cases h⟩
+  | log e ih => exact ⟨by simp [Expr.checkADSupported], fun h => by cases h⟩
+  | atan e ih => exact ⟨by simp [Expr.checkADSupported], fun h => by cases h⟩
+  | arsinh e ih => exact ⟨by simp [Expr.checkADSupported], fun h => by cases h⟩
+  | atanh e ih => exact ⟨by simp [Expr.checkADSupported], fun h => by cases h⟩
+  | sinc e ih => exact ⟨by simp [Expr.checkADSupported], fun h => by cases h⟩
+  | erf e ih => exact ⟨by simp [Expr.checkADSupported], fun h => by cases h⟩
+  | sinh e ih => exact ⟨by simp [Expr.checkADSupported], fun h => by cases h⟩
+  | cosh e ih => exact ⟨by simp [Expr.checkADSupported], fun h => by cases h⟩
+  | tanh e ih => exact ⟨by simp [Expr.checkADSupported], fun h => by cases h⟩
+  | sqrt e ih => exact ⟨by simp [Expr.checkADSupported], fun h => by cases h⟩
+  | namedConst c =>
+      constructor
+      · simp [Expr.checkADSupported]
+      · intro h
+        cases h
+
+/-- Computable recognition of `ExprSupportedCore`. -/
+def Expr.checkSupportedCore : Expr → Bool
+  | .const _ | .var _ | .namedConst _ => true
+  | .add e₁ e₂ | .mul e₁ e₂ => e₁.checkSupportedCore && e₂.checkSupportedCore
+  | .neg e | .exp e | .sin e | .cos e | .log e | .sqrt e |
+      .sinh e | .cosh e | .tanh e | .erf e => e.checkSupportedCore
+  | _ => false
+
+/-- A successful computable core-support check produces the corresponding
+proof object required by the tight Rational evaluator theorem. -/
+theorem Expr.checkSupportedCore_correct {e : Expr}
+    (h : e.checkSupportedCore = true) : ExprSupportedCore e := by
+  induction e with
+  | const q => exact .const q
+  | var idx => exact .var idx
+  | add left right ihLeft ihRight =>
+      simp only [Expr.checkSupportedCore, Bool.and_eq_true] at h
+      exact .add (ihLeft h.1) (ihRight h.2)
+  | mul left right ihLeft ihRight =>
+      simp only [Expr.checkSupportedCore, Bool.and_eq_true] at h
+      exact .mul (ihLeft h.1) (ihRight h.2)
+  | neg e ih => exact .neg (ih h)
+  | exp e ih => exact .exp (ih h)
+  | sin e ih => exact .sin (ih h)
+  | cos e ih => exact .cos (ih h)
+  | log e ih => exact .log (ih h)
+  | erf e ih => exact .erf (ih h)
+  | sinh e ih => exact .sinh (ih h)
+  | cosh e ih => exact .cosh (ih h)
+  | tanh e ih => exact .tanh (ih h)
+  | sqrt e ih => exact .sqrt (ih h)
+  | namedConst c => exact .namedConst c
+  | inv _ _ | atan _ _ | arsinh _ _ | atanh _ _ | sinc _ _ =>
+      simp [Expr.checkSupportedCore] at h
+
+end LeanCert.Core
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Generic Taylor Series Abstractions
+
+This file provides generic Taylor series machinery for verified numerics.
+We wrap Mathlib's Taylor expansion theorems in forms convenient for our
+interval arithmetic framework.
+
+## Phase Status
+
+**This module is now Phase 3 complete.** The main Taylor remainder bound theorem
+`taylor_remainder_bound` is fully proved with no `sorry` markers.
+
+The theorem provides rigorous bounds on Taylor polynomial approximation errors
+using Lagrange's form of the remainder.
+
+## Main definitions
+
+* `TaylorApprox` - A Taylor approximation with explicit remainder bounds
+* Generic theorems for composing Taylor approximations
+
+## Design notes
+
+This file provides the abstract theory. Specific Taylor expansions for
+`exp`, `sin`, `cos`, etc. are built using this machinery in conjunction
+with Mathlib's calculus lemmas.
+-/
+
+/-! ### Taylor approximation structure -/
+
+@[expose] public section
+
+namespace LeanCert.Core
+
+/-- A Taylor approximation of a function on an interval.
+
+Given a function `f`, center `c`, and radius `r`, this represents:
+- A polynomial approximation `poly` of degree `n`
+- A remainder bound `R` such that for all `x` with `|x - c| ≤ r`:
+  `|f(x) - poly(x - c)| ≤ R`
+-/
+structure TaylorApprox where
+  /-- Degree of the polynomial -/
+  degree : ℕ
+  /-- Polynomial coefficients (Taylor coefficients at center) -/
+  coeffs : Fin (degree + 1) → ℚ
+  /-- Remainder bound -/
+  remainder : ℚ
+  /-- The remainder is non-negative -/
+  remainder_nonneg : 0 ≤ remainder
+
+namespace TaylorApprox
+
+/-- Evaluate the polynomial part at a point -/
+noncomputable def evalPoly (ta : TaylorApprox) (x : ℝ) : ℝ :=
+  ∑ i : Fin (ta.degree + 1), (ta.coeffs i : ℝ) * x ^ (i : ℕ)
+
+end TaylorApprox
+
+/-! ### Derivative bounds -/
+
+/-- A bound on the n-th derivative of a function on an interval -/
+structure DerivBound where
+  /-- The derivative order -/
+  order : ℕ
+  /-- Upper bound on |f^(n)(x)| for x in the interval -/
+  bound : ℚ
+  /-- The bound is non-negative -/
+  bound_nonneg : 0 ≤ bound
+
+/-! ### Taylor remainder bounds -/
+
+/-! ### Helper lemmas for iteratedDerivWithin conversion -/
+
+/-- Unique differentiability at the left endpoint of a closed interval. -/
+lemma uniqueDiffWithinAt_Icc_left {c x : ℝ} (hcx : c < x) :
+    UniqueDiffWithinAt ℝ (Set.Icc c x) c := by
+  apply uniqueDiffWithinAt_convex (convex_Icc c x)
+  · rw [interior_Icc]; exact Set.nonempty_Ioo.mpr hcx
+  · rw [closure_Icc]; exact Set.left_mem_Icc.mpr (le_of_lt hcx)
+
+/-- Unique differentiability at the right endpoint of a closed interval. -/
+lemma uniqueDiffWithinAt_Icc_right {c x : ℝ} (hcx : c < x) :
+    UniqueDiffWithinAt ℝ (Set.Icc c x) x := by
+  apply uniqueDiffWithinAt_convex (convex_Icc c x)
+  · rw [interior_Icc]; exact Set.nonempty_Ioo.mpr hcx
+  · rw [closure_Icc]; exact Set.right_mem_Icc.mpr (le_of_lt hcx)
+
+/-- At the left endpoint of [c, x], derivWithin equals deriv for differentiable functions. -/
+lemma derivWithin_Icc_left_eq_deriv {f : ℝ → ℝ} {c x : ℝ}
+    (hcx : c < x) (hf : DifferentiableAt ℝ f c) :
+    derivWithin f (Set.Icc c x) c = deriv f c := by
+  have hderiv := hf.hasDerivAt
+  have hderiv_within : HasDerivWithinAt f (deriv f c) (Set.Icc c x) c := hderiv.hasDerivWithinAt
+  exact hderiv_within.derivWithin (uniqueDiffWithinAt_Icc_left hcx)
+
+/-- At the right endpoint of [c, x], derivWithin equals deriv for differentiable functions. -/
+lemma derivWithin_Icc_right_eq_deriv {f : ℝ → ℝ} {c x : ℝ}
+    (hcx : c < x) (hf : DifferentiableAt ℝ f x) :
+    derivWithin f (Set.Icc c x) x = deriv f x := by
+  have hderiv := hf.hasDerivAt
+  have hderiv_within : HasDerivWithinAt f (deriv f x) (Set.Icc c x) x := hderiv.hasDerivWithinAt
+  exact hderiv_within.derivWithin (uniqueDiffWithinAt_Icc_right hcx)
+
+/-- Helper lemma: `iteratedDerivWithin` on `Icc a b` equals `iteratedDeriv` at interior points.
+
+This bridges Mathlib's `iteratedDerivWithin`-based Taylor theorems with global `iteratedDeriv`. -/
+lemma iteratedDerivWithin_Icc_interior_eq {f : ℝ → ℝ} {a b : ℝ} {x : ℝ} {n : ℕ}
+    (_hab : a < b) (hx : x ∈ Set.Ioo a b) :
+    iteratedDerivWithin n f (Set.Icc a b) x = iteratedDeriv n f x := by
+  open Set Filter Topology in
+  have h_nhds : Icc a b ∈ 𝓝 x := Icc_mem_nhds hx.1 hx.2
+  rw [iteratedDerivWithin_eq_iteratedFDerivWithin, iteratedDeriv_eq_iteratedFDeriv]
+  congr 1
+  have heq : (Set.Icc a b : Set ℝ) =ᶠ[𝓝 x] (Set.univ : Set ℝ) := by
+    filter_upwards [h_nhds] with y hy
+    exact propext ⟨fun _ => trivial, fun _ => hy⟩
+  rw [iteratedFDerivWithin_congr_set heq n, iteratedFDerivWithin_univ]
+
+/-- At the left endpoint of [c, x], iteratedDerivWithin equals iteratedDeriv for ContDiff functions.
+-/
+lemma iteratedDerivWithin_Icc_left_eq_iteratedDeriv {f : ℝ → ℝ} {c x : ℝ} {n : ℕ}
+    (hcx : c < x) (hf : ContDiff ℝ n f) :
+    iteratedDerivWithin n f (Set.Icc c x) c = iteratedDeriv n f c := by
+  open Set in
+  induction n generalizing f with
+  | zero => simp only [iteratedDerivWithin_zero, iteratedDeriv_zero]
+  | succ n ih =>
+    have hf_n : ContDiff ℝ n f := hf.of_le (by simp : (n : WithTop ℕ∞) ≤ (n + 1 : ℕ))
+    rw [iteratedDerivWithin_succ', iteratedDeriv_succ']
+    have h_ne_zero : ((n : ℕ) + 1 : WithTop ℕ∞) ≠ 0 := by simp
+    have h_eq_derivWithin : EqOn (derivWithin f (Icc c x)) (deriv f) (Icc c x) := by
+      intro y hy
+      rcases eq_or_ne y c with rfl | hne_c
+      · exact derivWithin_Icc_left_eq_deriv hcx
+            ((hf.differentiable h_ne_zero).differentiableAt)
+      · rcases eq_or_ne y x with rfl | hne_x
+        · exact derivWithin_Icc_right_eq_deriv hcx
+              ((hf.differentiable h_ne_zero).differentiableAt)
+        · apply derivWithin_of_mem_nhds
+          apply Icc_mem_nhds
+          · exact lt_of_le_of_ne hy.1 (Ne.symm hne_c)
+          · exact lt_of_le_of_ne hy.2 hne_x
+    have h_congr : EqOn (iteratedDerivWithin n (derivWithin f (Icc c x)) (Icc c x))
+                        (iteratedDerivWithin n (deriv f) (Icc c x)) (Icc c x) :=
+      iteratedDerivWithin_congr (n := n) h_eq_derivWithin
+    have hc_mem : c ∈ Icc c x := left_mem_Icc.mpr (le_of_lt hcx)
+    rw [h_congr hc_mem]
+    have hdf : ContDiff ℝ n (deriv f) := by
+      have h := hf.iterate_deriv' n 1
+      simp only [Function.iterate_one] at h
+      exact h
+    exact ih hdf
+
+/-- At the left endpoint of [c, x], iteratedDerivWithin equals iteratedDeriv for functions
+    that are ContDiffOn on an open set containing c.
+
+    This is useful for functions like log that are only smooth on (0, ∞). -/
+lemma iteratedDerivWithin_Icc_left_eq_iteratedDeriv_of_isOpen {f : ℝ → ℝ} {c x : ℝ} {n : ℕ} {U :
+  Set ℝ}
+    (hU_open : IsOpen U) (hcU : c ∈ U) (hcx : c < x) (hI_sub : Set.Icc c x ⊆ U)
+    (hf : ContDiffOn ℝ n f U) :
+    iteratedDerivWithin n f (Set.Icc c x) c = iteratedDeriv n f c := by
+  open Set in
+  induction n generalizing f with
+  | zero => simp only [iteratedDerivWithin_zero, iteratedDeriv_zero]
+  | succ n ih =>
+    have hf_n : ContDiffOn ℝ n f U := hf.of_le (by simp : (n : WithTop ℕ∞) ≤ (n + 1 : ℕ))
+    rw [iteratedDerivWithin_succ', iteratedDeriv_succ']
+    have h_ne_zero : ((n : ℕ) + 1 : WithTop ℕ∞) ≠ 0 := by simp
+    -- For each y in Icc c x, derivWithin f = deriv f
+    have h_eq_derivWithin : EqOn (derivWithin f (Icc c x)) (deriv f) (Icc c x) := by
+      intro y hy
+      rcases eq_or_ne y c with rfl | hne_c
+      · -- At c: use that f is differentiable at c (from ContDiffOn on open U)
+        exact derivWithin_Icc_left_eq_deriv hcx
+            ((hf.differentiableOn h_ne_zero).differentiableAt (hU_open.mem_nhds hcU))
+      · rcases eq_or_ne y x with rfl | hne_x
+        · -- At x: use right endpoint lemma
+          exact derivWithin_Icc_right_eq_deriv hcx
+              ((hf.differentiableOn h_ne_zero).differentiableAt (hU_open.mem_nhds (hI_sub hy)))
+        · -- Interior: Icc is a neighborhood
+          apply derivWithin_of_mem_nhds
+          apply Icc_mem_nhds
+          · exact lt_of_le_of_ne hy.1 (Ne.symm hne_c)
+          · exact lt_of_le_of_ne hy.2 hne_x
+    have h_congr : EqOn (iteratedDerivWithin n (derivWithin f (Icc c x)) (Icc c x))
+                        (iteratedDerivWithin n (deriv f) (Icc c x)) (Icc c x) :=
+      iteratedDerivWithin_congr (n := n) h_eq_derivWithin
+    have hc_mem : c ∈ Icc c x := left_mem_Icc.mpr (le_of_lt hcx)
+    rw [h_congr hc_mem]
+    have hdf : ContDiffOn ℝ n (deriv f) U := hf.deriv_of_isOpen hU_open le_rfl
+    exact ih hdf
+/-- Taylor remainder bound for c < x case with ContDiffOn hypothesis.
+
+    For functions that are ContDiffOn on an open set containing [a, b], this provides
+    the same Lagrange remainder bound as the global ContDiff version. -/
+theorem taylor_remainder_bound_on_c_lt_x {f : ℝ → ℝ} {a b c : ℝ} {m : ℕ} {M : ℝ} {U : Set ℝ}
+    (hU_open : IsOpen U) (hI_sub : Set.Icc a b ⊆ U)
+    (hca : a ≤ c)
+    (hf : ContDiffOn ℝ (m + 1) f U)
+    (hM : ∀ y ∈ Set.Icc a b, ‖iteratedDeriv (m + 1) f y‖ ≤ M)
+    (x : ℝ) (hx : x ∈ Set.Icc a b) (hcx : c < x) :
+    ‖f x - ∑ i ∈ Finset.range (m + 1), (iteratedDeriv i f c / i.factorial) * (x - c) ^ i‖
+        ≤ M * |x - c| ^ (m + 1) / (m + 1).factorial := by
+  open Set Finset in
+  have hcx_le : c ≤ x := le_of_lt hcx
+  have hcI : c ∈ Icc a b := ⟨hca, le_trans hcx_le hx.2⟩
+  have hI_sub' : Icc c x ⊆ Icc a b := Icc_subset_Icc hca hx.2
+  have hI_sub_U : Icc c x ⊆ U := fun y hy => hI_sub (hI_sub' hy)
+  have hcU : c ∈ U := hI_sub hcI
+  have hf_on : ContDiffOn ℝ (m + 1) f (Icc c x) := hf.mono hI_sub_U
+  have hf_on_m : ContDiffOn ℝ m f (Icc c x) := by
+    have hm_le : (m : WithTop ℕ∞) ≤ (m + 1 : ℕ) := by simp
+    exact hf_on.of_le hm_le
+  have hf_diff : DifferentiableOn ℝ (iteratedDerivWithin m f (Icc c x)) (Ioo c x) := by
+    have huniq := uniqueDiffOn_Icc hcx
+    have hm_lt : (m : WithTop ℕ∞) < (m + 1 : ℕ) := by
+      have : (m : ℕ) < m + 1 := Nat.lt_succ_self m
+      exact_mod_cast this
+    have hdiff := hf_on.differentiableOn_iteratedDerivWithin hm_lt huniq
+    exact hdiff.mono Ioo_subset_Icc_self
+  have hf_on_m' : ContDiffOn ℝ m f (uIcc c x) := by rwa [uIcc_of_le hcx_le]
+  have hf_diff' : DifferentiableOn ℝ (iteratedDerivWithin m f (uIcc c x)) (uIoo c x) := by
+    rwa [uIcc_of_le hcx_le, uIoo_of_lt hcx]
+  obtain ⟨ξ, hξ_mem, hLagrange⟩ :=
+    taylor_mean_remainder_lagrange (x₀ := c) (x := x) hcx.ne hf_on_m' hf_diff'
+  have hξ_mem_Ioo : ξ ∈ Ioo c x := by rwa [uIoo_of_lt hcx] at hξ_mem
+  rw [uIcc_of_le hcx_le] at hLagrange
+  have hξ_ab : ξ ∈ Icc a b := hI_sub' (Ioo_subset_Icc_self hξ_mem_Ioo)
+  have hderiv_ξ : iteratedDerivWithin (m + 1) f (Icc c x) ξ = iteratedDeriv (m + 1) f ξ :=
+    iteratedDerivWithin_Icc_interior_eq hcx hξ_mem_Ioo
+  have hsum_eq : ∑ i ∈ range (m + 1), (iteratedDeriv i f c / i.factorial) * (x - c) ^ i =
+                 taylorWithinEval f m (Icc c x) c x := by
+    rw [taylor_within_apply]
+    apply sum_congr rfl
+    intro i hi
+    have hderiv_c : iteratedDerivWithin i f (Icc c x) c = iteratedDeriv i f c := by
+      have hi_lt : i < m + 1 := mem_range.mp hi
+      have hi_le : (i : WithTop ℕ∞) ≤ (m + 1 : ℕ) := by
+        have : (i : ℕ) ≤ m := Nat.lt_succ_iff.mp hi_lt
+        exact le_of_lt (by exact_mod_cast Nat.lt_add_one_of_le this)
+      exact iteratedDerivWithin_Icc_left_eq_iteratedDeriv_of_isOpen hU_open hcU hcx hI_sub_U
+        (hf.of_le hi_le)
+    rw [hderiv_c]
+    simp only [smul_eq_mul]
+    ring
+  rw [hsum_eq, hLagrange, hderiv_ξ]
+  rw [norm_div, norm_mul, Real.norm_eq_abs, Real.norm_eq_abs, Real.norm_eq_abs]
+  have hfact_pos : (0 : ℝ) < ((m + 1).factorial : ℝ) := Nat.cast_pos.mpr (Nat.factorial_pos _)
+  rw [abs_of_pos hfact_pos]
+  have hxc_pos : 0 < x - c := sub_pos.mpr hcx
+  rw [abs_pow, abs_of_pos hxc_pos]
+  have hbound := hM ξ hξ_ab
+  have h_pow_fact_pos : 0 < (x - c) ^ (m + 1) / (m + 1).factorial := by
+    apply div_pos
+    · exact pow_pos hxc_pos _
+    · exact hfact_pos
+  calc |iteratedDeriv (m + 1) f ξ| * (x - c) ^ (m + 1) / ↑(m + 1).factorial
+      = |iteratedDeriv (m + 1) f ξ| * ((x - c) ^ (m + 1) / ↑(m + 1).factorial) := by ring
+    _ ≤ M * ((x - c) ^ (m + 1) / ↑(m + 1).factorial) := by
+        apply mul_le_mul_of_nonneg_right hbound (le_of_lt h_pow_fact_pos)
+    _ = M * (x - c) ^ (m + 1) / ↑(m + 1).factorial := by ring
+
+/-- Helper: translation invariance of iteratedDeriv for ContDiffOn on open sets. -/
+private lemma iteratedDeriv_translate_of_contDiffOn {f : ℝ → ℝ} {k : ℝ} {n : ℕ} {U : Set ℝ}
+    (hU_open : IsOpen U) (t : ℝ) (ht : t + k ∈ U)
+    (hf : ContDiffOn ℝ n f U) :
+    iteratedDeriv n (fun y => f (y + k)) t = iteratedDeriv n f (t + k) := by
+  induction n generalizing f with
+  | zero => simp only [iteratedDeriv_zero]
+  | succ n ih =>
+    rw [iteratedDeriv_succ', iteratedDeriv_succ']
+    -- Since t + k ∈ U and U is open, there's a neighborhood of t where y + k ∈ U for all y
+    have h_nhds_mem : ∀ᶠ y in nhds t, y + k ∈ U := by
+      have hcont : Continuous (fun y => y + k) := continuous_id.add continuous_const
+      have hU_nhds : U ∈ nhds (t + k) := hU_open.mem_nhds ht
+      exact hcont.continuousAt.preimage_mem_nhds hU_nhds
+    -- Derivative equality in a neighborhood of t
+    have h_deriv_eq_near : ∀ᶠ y in nhds t, deriv (fun z => f (z + k)) y = deriv f (y + k) := by
+      filter_upwards [h_nhds_mem] with y hy_mem
+      have hf' : ContDiffOn ℝ 1 f U := hf.of_le (by norm_cast; omega)
+      have hdiff : DifferentiableAt ℝ f (y + k) :=
+        (hf'.differentiableOn one_ne_zero).differentiableAt (hU_open.mem_nhds hy_mem)
+      have hlin : DifferentiableAt ℝ (fun z => z + k) y :=
+        differentiableAt_id.add (differentiableAt_const _)
+      have h := deriv_comp y hdiff hlin
+      simp only [deriv_add_const, deriv_id'', mul_one] at h
+      exact h
+    have h_iter_eq : iteratedDeriv n (deriv fun z => f (z + k)) t =
+                     iteratedDeriv n (fun y => deriv f (y + k)) t :=
+      Filter.EventuallyEq.iteratedDeriv_eq n h_deriv_eq_near
+    rw [h_iter_eq]
+    have hdf : ContDiffOn ℝ n (deriv f) U := hf.deriv_of_isOpen hU_open le_rfl
+    exact ih hdf
+
+/-- Key lemma for reflection: derivatives of g(t) = f(2c - t) when f is smooth on an open set.
+
+This is a local version of `iteratedDeriv_reflect` that works with ContDiffOn on an open set
+rather than global ContDiff. -/
+lemma iteratedDeriv_reflect_of_contDiffOn {f : ℝ → ℝ} {c : ℝ} {n : ℕ} {U : Set ℝ}
+    (hU_open : IsOpen U) (hf : ContDiffOn ℝ n f U) (t : ℝ) (ht : 2 * c - t ∈ U) :
+    iteratedDeriv n (fun s => f (2 * c - s)) t = (-1 : ℝ) ^ n * iteratedDeriv n f (2 * c - t) := by
+  -- Rewrite as f ∘ (· + 2c) ∘ (- ·)
+  have heq : (fun s => f (2 * c - s)) = (fun s => (fun y => f (y + 2 * c)) (-s)) := by
+    ext s; ring_nf
+  rw [heq]
+  rw [iteratedDeriv_comp_neg n (fun y => f (y + 2 * c)) t]
+  simp only [smul_eq_mul]
+  congr 1
+  -- Use the translation helper
+  have hmt : -t + 2 * c = 2 * c - t := by ring
+  rw [← hmt] at ht
+  convert iteratedDeriv_translate_of_contDiffOn hU_open (-t) ht hf using 1
+  rw [hmt]
+/-- Taylor remainder bound for x < c case with ContDiffOn hypothesis.
+
+    For functions that are ContDiffOn on an open set containing [a, b], this provides
+    the same Lagrange remainder bound as the global ContDiff version.
+
+    The proof uses reflection: define g(t) = f(2c - t), apply Lagrange to g on [c, 2c-x],
+    then convert back. -/
+theorem taylor_remainder_bound_on_x_lt_c {f : ℝ → ℝ} {a b c : ℝ} {m : ℕ} {M : ℝ} {U : Set ℝ}
+    (hU_open : IsOpen U) (hI_sub : Set.Icc a b ⊆ U)
+    (hcb : c ≤ b)
+    (hf : ContDiffOn ℝ (m + 1) f U)
+    (hM : ∀ y ∈ Set.Icc a b, ‖iteratedDeriv (m + 1) f y‖ ≤ M)
+    (x : ℝ) (hx : x ∈ Set.Icc a b) (hxc : x < c) :
+    ‖f x - ∑ i ∈ Finset.range (m + 1), (iteratedDeriv i f c / i.factorial) * (x - c) ^ i‖
+        ≤ M * |x - c| ^ (m + 1) / (m + 1).factorial := by
+  open Set Finset in
+  -- Define reflected function g(t) = f(2c - t)
+  set g : ℝ → ℝ := fun t => f (2 * c - t) with hg_def
+  -- Define V = preimage of U under reflection (the reflected open set)
+  set V : Set ℝ := (fun t => 2 * c - t) ⁻¹' U with hV_def
+  -- V is open (preimage of open set under continuous function)
+  have hV_open : IsOpen V := by
+    apply IsOpen.preimage _ hU_open
+    exact continuous_const.sub continuous_id
+  -- g is ContDiffOn V
+  have hg_ContDiffOn : ContDiffOn ℝ (m + 1) g V := by
+    have hlin : ContDiff ℝ ⊤ (fun t => 2 * c - t) := contDiff_const.sub contDiff_id
+    -- g = f ∘ (2c - ·) and (2c - ·) maps V into U
+    have hmaps : Set.MapsTo (fun t => 2 * c - t) V U := fun t ht => ht
+    have hlin_on : ContDiffOn ℝ (m + 1) (fun t => 2 * c - t) V :=
+      (hlin.contDiffOn.mono (Set.subset_univ _)).of_le le_top
+    exact ContDiffOn.comp hf hlin_on hmaps
+  -- The interval [c, 2c-x]
+  have h_interval_lt : c < 2 * c - x := by linarith
+  -- [c, 2c-x] maps under t ↦ 2c-t to [x, c] ⊆ [a,b] ⊆ U
+  -- So [c, 2c-x] ⊆ V
+  have hJ_sub_V : Icc c (2 * c - x) ⊆ V := by
+    intro t ht
+    -- 2c - t ∈ [x, c] ⊆ [a,b] ⊆ U
+    have ht_lower : x ≤ 2 * c - t := by linarith [ht.2]
+    have ht_upper : 2 * c - t ≤ c := by linarith [ht.1]
+    have h2ct_in_ab : 2 * c - t ∈ Icc a b := by
+      refine mem_Icc.mpr ⟨?_, ?_⟩
+      · exact le_trans hx.1 ht_lower
+      · exact le_trans ht_upper hcb
+    exact hI_sub h2ct_in_ab
+  -- c ∈ V since c maps to c ∈ [a,b] ⊆ U
+  have hcV : c ∈ V := by
+    have hc_ab : c ∈ Icc a b := ⟨le_trans hx.1 (le_of_lt hxc), hcb⟩
+    -- Need to show c ∈ V, i.e., (fun t => 2 * c - t) c ∈ U, i.e., 2c - c = c ∈ U
+    change c ∈ V
+    simp only [hV_def, Set.mem_preimage]
+    have h2cc : 2 * c - c = c := by ring
+    rw [h2cc]
+    exact hI_sub hc_ab
+  -- Helper: map [c, 2c-x] to [a,b]
+  have h_map_to_ab : ∀ t ∈ Icc c (2 * c - x), 2 * c - t ∈ Icc a b := by
+    intro t ht
+    have ht_lower : x ≤ 2 * c - t := by linarith [ht.2]
+    have ht_upper : 2 * c - t ≤ c := by linarith [ht.1]
+    exact ⟨le_trans hx.1 ht_lower, le_trans ht_upper hcb⟩
+  -- Derivative bound for g using reflection formula
+  have hg_deriv_bound : ∀ y ∈ Icc c (2 * c - x), ‖iteratedDeriv (m + 1) g y‖ ≤ M := by
+    intro y hy
+    have h2cy_ab : 2 * c - y ∈ Icc a b := h_map_to_ab y hy
+    have h2cy_U : 2 * c - y ∈ U := hI_sub h2cy_ab
+    -- Use the reflection formula
+    have hg_deriv : iteratedDeriv (m + 1) g y = (-1 : ℝ) ^ (m + 1) * iteratedDeriv (m + 1) f (2 *
+      c - y) :=
+      iteratedDeriv_reflect_of_contDiffOn hU_open hf y h2cy_U
+    rw [hg_deriv, norm_mul, Real.norm_eq_abs]
+    have h_neg_one : |(-1 : ℝ) ^ (m + 1)| = 1 := by
+      rw [abs_pow, abs_neg, abs_one, one_pow]
+    rw [h_neg_one, one_mul]
+    exact hM (2 * c - y) h2cy_ab
+  -- Now apply taylor_remainder_bound_on_c_lt_x to g
+  have hcV' : a ≤ c := le_trans hx.1 (le_of_lt hxc)
+  -- g is ContDiffOn V and Icc c (2c-x) ⊆ V
+  have h_taylor_g := taylor_remainder_bound_on_c_lt_x hV_open hJ_sub_V (le_refl c)
+      hg_ContDiffOn hg_deriv_bound (2 * c - x) ⟨le_of_lt h_interval_lt, le_refl _⟩ h_interval_lt
+  -- g(2c-x) = f(x)
+  have hg_at_2cmx : g (2 * c - x) = f x := by simp only [hg_def, sub_sub_cancel]
+  -- g^(i)(c) = (-1)^i * f^(i)(c)
+  have hg_deriv_c : ∀ i ≤ m + 1, iteratedDeriv i g c = (-1 : ℝ) ^ i * iteratedDeriv i f c := by
+    intro i hi
+    have hcU : c ∈ U := hI_sub ⟨le_trans hx.1 (le_of_lt hxc), hcb⟩
+    have h2cc : 2 * c - c = c := by ring
+    rw [← h2cc] at hcU
+    have hfi : ContDiffOn ℝ i f U := hf.of_le (by exact_mod_cast hi)
+    have hrefl := iteratedDeriv_reflect_of_contDiffOn hU_open hfi c hcU
+    simp only [h2cc] at hrefl
+    exact hrefl
+  -- Now convert the Taylor sum
+  -- ∑ (iteratedDeriv i g c / i!) * ((2c-x) - c)^i = ∑ (iteratedDeriv i f c / i!) * (x - c)^i
+  have hsum_eq : ∑ i ∈ range (m + 1), (iteratedDeriv i g c / i.factorial) * ((2 * c - x) - c) ^ i =
+                 ∑ i ∈ range (m + 1), (iteratedDeriv i f c / i.factorial) * (x - c) ^ i := by
+    apply sum_congr rfl
+    intro i hi
+    have hi_le : i ≤ m + 1 := le_of_lt (mem_range.mp hi)
+    rw [hg_deriv_c i hi_le]
+    have h_diff : (2 * c - x) - c = c - x := by ring
+    rw [h_diff]
+    have h_pow : (c - x : ℝ) ^ i = (-1 : ℝ) ^ i * (x - c) ^ i := by
+      have : c - x = -(x - c) := by ring
+      rw [this, neg_eq_neg_one_mul, mul_pow]
+    rw [h_pow]
+    -- Simplify: (-1)^i * f^(i)(c) / i! * ((-1)^i * (x-c)^i)
+    have h_neg_sq : (-1 : ℝ) ^ i * (-1 : ℝ) ^ i = 1 := by rw [← pow_add]; norm_num
+    -- Goal: (-1)^i * iteratedDeriv i f c / i! * ((-1)^i * (x-c)^i) = iteratedDeriv i f c / i! *
+    -- (x-c)^i
+    calc (-1 : ℝ) ^ i * iteratedDeriv i f c / ↑i.factorial * ((-1) ^ i * (x - c) ^ i)
+        = ((-1 : ℝ) ^ i * (-1) ^ i) * (iteratedDeriv i f c / ↑i.factorial * (x - c) ^ i) := by ring
+      _ = 1 * (iteratedDeriv i f c / ↑i.factorial * (x - c) ^ i) := by rw [h_neg_sq]
+      _ = iteratedDeriv i f c / ↑i.factorial * (x - c) ^ i := by ring
+  -- Use h_taylor_g with the conversions
+  rw [hg_at_2cmx, hsum_eq] at h_taylor_g
+  -- The RHS: M * |(2c-x) - c|^(m+1) / (m+1)! = M * |x - c|^(m+1) / (m+1)!
+  have h_abs_eq : |(2 * c - x) - c| = |x - c| := by
+    have h : (2 * c - x) - c = c - x := by ring
+    rw [h, abs_sub_comm]
+  rw [h_abs_eq] at h_taylor_g
+  exact h_taylor_g
+/-- Combined Taylor remainder bound with ContDiffOn hypothesis.
+
+    For functions that are ContDiffOn on an open set containing [a, b], this provides
+    the Lagrange remainder bound for any x ∈ [a, b] and center c ∈ [a, b]. -/
+theorem taylor_remainder_bound_on {f : ℝ → ℝ} {a b c : ℝ} {n : ℕ} {M : ℝ} {U : Set ℝ}
+    (hU_open : IsOpen U) (hI_sub : Set.Icc a b ⊆ U)
+    (hca : a ≤ c) (hcb : c ≤ b)
+    (hf : ContDiffOn ℝ n f U)
+    (hM : ∀ x ∈ Set.Icc a b, ‖iteratedDeriv n f x‖ ≤ M)
+    (_hMnonneg : 0 ≤ M) :
+    ∀ x ∈ Set.Icc a b,
+      ‖f x - ∑ i ∈ Finset.range n, (iteratedDeriv i f c / i.factorial) * (x - c) ^ i‖
+        ≤ M * |x - c| ^ n / n.factorial := by
+  open Set Finset in
+  intro x hx
+  cases n with
+  | zero =>
+    simp only [range_zero, sum_empty, sub_zero, Nat.factorial_zero, Nat.cast_one,
+      div_one, pow_zero, mul_one]
+    have h : f = iteratedDeriv 0 f := iteratedDeriv_zero.symm
+    rw [h]
+    exact hM x hx
+  | succ m =>
+    by_cases hxc : x = c
+    · -- Case x = c: both sides are 0
+      have hrhs : M * |x - c| ^ (m + 1) / ↑(m + 1).factorial = 0 := by
+        rw [hxc, sub_self, abs_zero, zero_pow (Nat.succ_ne_zero m), mul_zero, zero_div]
+      rw [hrhs]
+      have hsum : ∑ i ∈ range (m + 1), (iteratedDeriv i f c / i.factorial) * (x - c) ^ i = f c := by
+        rw [hxc]
+        simp only [sub_self]
+        rw [sum_eq_single 0]
+        · simp [iteratedDeriv_zero]
+        · intro i _ hi
+          have hpos : 0 < i := Nat.pos_of_ne_zero hi
+          have : (0 : ℝ) ^ i = 0 := zero_pow hpos.ne'
+          simp [this]
+        · simp
+      rw [hsum, hxc, sub_self, norm_zero]
+    · rcases lt_or_gt_of_ne hxc with hxc_lt | hcx_lt
+      · exact taylor_remainder_bound_on_x_lt_c hU_open hI_sub hcb hf hM x hx hxc_lt
+      · exact taylor_remainder_bound_on_c_lt_x hU_open hI_sub hca hf hM x hx hcx_lt
+
+/-- Key lemma: derivatives of reflected function g(t) = f(2c - t).
+
+For the `x < c` case, we use reflection: define g(t) = f(2c - t), apply Lagrange
+to g on [c, 2c-x], then convert back. This lemma shows how g's derivatives
+relate to f's derivatives. -/
+lemma iteratedDeriv_reflect {f : ℝ → ℝ} {c : ℝ} {n : ℕ} (hf : ContDiff ℝ n f) (t : ℝ) :
+    iteratedDeriv n (fun s => f (2 * c - s)) t = (-1 : ℝ) ^ n * iteratedDeriv n f (2 * c - t) := by
+  have heq : (fun s => f (2 * c - s)) = (fun s => (fun x => f (x + 2 * c)) (-s)) := by
+    ext s; ring_nf
+  rw [heq]
+  rw [iteratedDeriv_comp_neg n (fun x => f (x + 2 * c)) t]
+  simp only [smul_eq_mul]
+  congr 1
+  clear heq
+  induction n generalizing f with
+  | zero => simp only [iteratedDeriv_zero]; ring_nf
+  | succ n ih =>
+    rw [iteratedDeriv_succ', iteratedDeriv_succ']
+    have hf' : ContDiff ℝ (n + 1) f := hf
+    have hderiv_eq : deriv (fun x => f (x + 2 * c)) = fun x => deriv f (x + 2 * c) := by
+      ext x
+      have hdiff : DifferentiableAt ℝ f (x + 2 * c) :=
+        (hf'.differentiable (by simp : ((n : ℕ) + 1 : WithTop ℕ∞) ≠ 0)).differentiableAt
+      have hlin : DifferentiableAt ℝ (fun y => y + 2 * c) x :=
+        differentiableAt_id.add (differentiableAt_const _)
+      have h1 := deriv_comp x hdiff hlin
+      simp only [deriv_add_const, deriv_id'', mul_one] at h1
+      exact h1
+    rw [hderiv_eq]
+    have hdf : ContDiff ℝ n (deriv f) := by
+      have h := hf'.iterate_deriv' n 1
+      simp only [Function.iterate_one] at h
+      exact h
+    exact ih hdf
+
+/-- Auxiliary lemma: (-1)^n * (-1)^n = 1 for any natural n. -/
+lemma neg_one_pow_mul_self (n : ℕ) : (-1 : ℝ) ^ n * (-1) ^ n = 1 := by
+  rw [← pow_add]
+  norm_num
+/-- Taylor remainder bound for c < x case (Lagrange form).
+
+For `c < x`, we apply `taylor_mean_remainder_lagrange` on `[c, x]` and convert
+`iteratedDerivWithin` to `iteratedDeriv` using the helper lemmas above. -/
+theorem taylor_remainder_bound_c_lt_x {f : ℝ → ℝ} {a b c : ℝ} {m : ℕ} {M : ℝ}
+    (hca : a ≤ c)
+    (hf : ContDiff ℝ (m + 1) f)
+    (hM : ∀ y ∈ Set.Icc a b, ‖iteratedDeriv (m + 1) f y‖ ≤ M)
+    (x : ℝ) (hx : x ∈ Set.Icc a b) (hcx : c < x) :
+    ‖f x - ∑ i ∈ Finset.range (m + 1), (iteratedDeriv i f c / i.factorial) * (x - c) ^ i‖
+        ≤ M * |x - c| ^ (m + 1) / (m + 1).factorial := by
+  open Set Finset in
+  have hcx_le : c ≤ x := le_of_lt hcx
+  have hI_sub : Icc c x ⊆ Icc a b := Icc_subset_Icc hca hx.2
+  have hf_on : ContDiffOn ℝ (m + 1) f (Icc c x) := hf.contDiffOn.mono hI_sub
+  have hf_on_m : ContDiffOn ℝ m f (Icc c x) := by
+    have hm_le : (m : WithTop ℕ∞) ≤ (m + 1 : ℕ) := by simp
+    exact hf_on.of_le hm_le
+  have hf_diff : DifferentiableOn ℝ (iteratedDerivWithin m f (Icc c x)) (Ioo c x) := by
+    have huniq := uniqueDiffOn_Icc hcx
+    have hm_lt : (m : WithTop ℕ∞) < (m + 1 : ℕ) := by
+      have : (m : ℕ) < m + 1 := Nat.lt_succ_self m
+      exact_mod_cast this
+    have hdiff := hf_on.differentiableOn_iteratedDerivWithin hm_lt huniq
+    exact hdiff.mono Ioo_subset_Icc_self
+  have hf_on_m' : ContDiffOn ℝ m f (uIcc c x) := by rwa [uIcc_of_le hcx_le]
+  have hf_diff' : DifferentiableOn ℝ (iteratedDerivWithin m f (uIcc c x)) (uIoo c x) := by
+    rwa [uIcc_of_le hcx_le, uIoo_of_lt hcx]
+  obtain ⟨ξ, hξ_mem, hLagrange⟩ :=
+    taylor_mean_remainder_lagrange (x₀ := c) (x := x) hcx.ne hf_on_m' hf_diff'
+  have hξ_mem_Ioo : ξ ∈ Ioo c x := by rwa [uIoo_of_lt hcx] at hξ_mem
+  rw [uIcc_of_le hcx_le] at hLagrange
+  have hξ_ab : ξ ∈ Icc a b := hI_sub (Ioo_subset_Icc_self hξ_mem_Ioo)
+  have hderiv_ξ : iteratedDerivWithin (m + 1) f (Icc c x) ξ = iteratedDeriv (m + 1) f ξ :=
+    iteratedDerivWithin_Icc_interior_eq hcx hξ_mem_Ioo
+  have hsum_eq : ∑ i ∈ range (m + 1), (iteratedDeriv i f c / i.factorial) * (x - c) ^ i =
+                 taylorWithinEval f m (Icc c x) c x := by
+    rw [taylor_within_apply]
+    apply sum_congr rfl
+    intro i hi
+    have hderiv_c : iteratedDerivWithin i f (Icc c x) c = iteratedDeriv i f c := by
+      have hi_lt : i < m + 1 := mem_range.mp hi
+      have hi_le : (i : WithTop ℕ∞) ≤ (m + 1 : ℕ) := by
+        have : (i : ℕ) ≤ m := Nat.lt_succ_iff.mp hi_lt
+        exact le_of_lt (by exact_mod_cast Nat.lt_add_one_of_le this)
+      exact iteratedDerivWithin_Icc_left_eq_iteratedDeriv hcx (hf.of_le hi_le)
+    rw [hderiv_c]
+    simp only [smul_eq_mul]
+    ring
+  rw [hsum_eq, hLagrange, hderiv_ξ]
+  rw [norm_div, norm_mul, Real.norm_eq_abs, Real.norm_eq_abs, Real.norm_eq_abs]
+  have hfact_pos : (0 : ℝ) < ((m + 1).factorial : ℝ) := Nat.cast_pos.mpr (Nat.factorial_pos _)
+  rw [abs_of_pos hfact_pos]
+  have hxc_pos : 0 < x - c := sub_pos.mpr hcx
+  rw [abs_pow, abs_of_pos hxc_pos]
+  have hbound := hM ξ hξ_ab
+  have h_pow_fact_pos : 0 < (x - c) ^ (m + 1) / (m + 1).factorial := by
+    apply div_pos
+    · exact pow_pos hxc_pos _
+    · exact hfact_pos
+  calc |iteratedDeriv (m + 1) f ξ| * (x - c) ^ (m + 1) / ↑(m + 1).factorial
+      = |iteratedDeriv (m + 1) f ξ| * ((x - c) ^ (m + 1) / ↑(m + 1).factorial) := by ring
+    _ ≤ M * ((x - c) ^ (m + 1) / ↑(m + 1).factorial) := by
+        apply mul_le_mul_of_nonneg_right hbound (le_of_lt h_pow_fact_pos)
+    _ = M * (x - c) ^ (m + 1) / ↑(m + 1).factorial := by ring
+/-- Taylor remainder bound for x < c case (Lagrange form via reflection).
+
+For `x < c`, we define g(t) = f(2c - t), apply `taylor_mean_remainder_lagrange` on `[c, 2c-x]`,
+then use `iteratedDeriv_reflect` to convert back to f's derivatives. The key insight is that
+g's Taylor expansion at c, evaluated at 2c-x, equals f's Taylor expansion at c, evaluated at x,
+because the (-1)^i factors from the derivatives cancel with the (-1)^i factors from the powers. -/
+theorem taylor_remainder_bound_x_lt_c {f : ℝ → ℝ} {a b c : ℝ} {m : ℕ} {M : ℝ}
+    (hcb : c ≤ b)
+    (hf : ContDiff ℝ (m + 1) f)
+    (hM : ∀ y ∈ Set.Icc a b, ‖iteratedDeriv (m + 1) f y‖ ≤ M)
+    (x : ℝ) (hx : x ∈ Set.Icc a b) (hxc : x < c) :
+    ‖f x - ∑ i ∈ Finset.range (m + 1), (iteratedDeriv i f c / i.factorial) * (x - c) ^ i‖
+        ≤ M * |x - c| ^ (m + 1) / (m + 1).factorial := by
+  open Set Finset in
+  -- Define reflected function g(t) = f(2c - t)
+  set g : ℝ → ℝ := fun t => f (2 * c - t) with hg_def
+  -- g is smooth
+  have hg_smooth : ContDiff ℝ (m + 1) g := by
+    have hlin : ContDiff ℝ ⊤ (fun t => 2 * c - t) := contDiff_const.sub contDiff_id
+    exact hf.comp (hlin.of_le le_top)
+  -- The interval [c, 2c - x]
+  have h_interval_lt : c < 2 * c - x := by linarith
+  -- Apply taylor_mean_remainder_lagrange to g on [c, 2c - x]
+  have hg_on : ContDiffOn ℝ (m + 1) g (Icc c (2 * c - x)) := hg_smooth.contDiffOn
+  have h_interval_le : c ≤ 2 * c - x := le_of_lt h_interval_lt
+  have hg_on_m : ContDiffOn ℝ m g (Icc c (2 * c - x)) := by
+    have hm_le : (m : WithTop ℕ∞) ≤ (m + 1 : ℕ) := by norm_cast; omega
+    exact hg_on.of_le hm_le
+  have hg_diff : DifferentiableOn ℝ (iteratedDerivWithin m g (Icc c (2 * c - x))) (Ioo c (2 * c -
+    x)) := by
+    have huniq := uniqueDiffOn_Icc h_interval_lt
+    have hm_lt : (m : WithTop ℕ∞) < (m + 1 : ℕ) := by
+      have : (m : ℕ) < m + 1 := Nat.lt_succ_self m
+      exact_mod_cast this
+    have hdiff := hg_on.differentiableOn_iteratedDerivWithin hm_lt huniq
+    exact hdiff.mono Ioo_subset_Icc_self
+  have hg_on_m' : ContDiffOn ℝ m g (uIcc c (2 * c - x)) := by rwa [uIcc_of_le h_interval_le]
+  have hg_diff' : DifferentiableOn ℝ (iteratedDerivWithin m g (uIcc c (2 * c - x))) (uIoo c (2 * c
+    - x)) := by
+    rwa [uIcc_of_le h_interval_le, uIoo_of_lt h_interval_lt]
+  obtain ⟨ξ', hξ'_mem, hLagrange⟩ :=
+    taylor_mean_remainder_lagrange (x₀ := c) (x := 2 * c - x)
+      h_interval_lt.ne hg_on_m' hg_diff'
+  have hξ'_mem_Ioo : ξ' ∈ Ioo c (2 * c - x) := by
+    rwa [uIoo_of_lt h_interval_lt] at hξ'_mem
+  rw [uIcc_of_le h_interval_le] at hLagrange
+  -- ξ = 2c - ξ' ∈ (x, c)
+  set ξ := 2 * c - ξ' with hξ_def
+  have hξ'_lt_2cmx : ξ' < 2 * c - x := hξ'_mem_Ioo.2
+  have hξ'_gt_c : c < ξ' := hξ'_mem_Ioo.1
+  have hξ_gt_x : x < ξ := by simp only [hξ_def]; linarith
+  have hξ_lt_c : ξ < c := by simp only [hξ_def]; linarith
+  have hξ_ab : ξ ∈ Icc a b :=
+    ⟨le_trans hx.1 (le_of_lt hξ_gt_x), le_trans (le_of_lt hξ_lt_c) hcb⟩
+  have hg_at_2cmx : g (2 * c - x) = f x := by simp only [hg_def, sub_sub_cancel]
+  -- g^{(k)}(c) = (-1)^k * f^{(k)}(c)
+  have hg_deriv : ∀ k ≤ m + 1, iteratedDeriv k g c = (-1 : ℝ) ^ k * iteratedDeriv k f c := by
+    intro k hk
+    have hg_eq : g = fun s => f (2 * c - s) := hg_def
+    rw [hg_eq]
+    rw [iteratedDeriv_reflect (hf.of_le (by exact_mod_cast hk)) c]
+    congr 1
+    ring_nf
+  -- Key: taylorWithinEval g m (Icc c (2c-x)) c (2c-x) = ∑ ... of f's Taylor coeffs
+  have hTaylor_g_to_f : taylorWithinEval g m (Icc c (2 * c - x)) c (2 * c - x) =
+                        ∑ i ∈ range (m + 1), (iteratedDeriv i f c / i.factorial) * (x - c) ^ i := by
+    rw [taylor_within_apply]
+    apply sum_congr rfl
+    intro i hi
+    have hi_lt : i < m + 1 := mem_range.mp hi
+    have hi_le : i ≤ m + 1 := le_of_lt hi_lt
+    have hi_le' : (i : WithTop ℕ∞) ≤ (m + 1 : ℕ) := by
+      have : (i : ℕ) ≤ m := Nat.lt_succ_iff.mp hi_lt
+      exact le_of_lt (by exact_mod_cast Nat.lt_add_one_of_le this)
+    have hderiv_c : iteratedDerivWithin i g (Icc c (2 * c - x)) c = iteratedDeriv i g c :=
+      iteratedDerivWithin_Icc_left_eq_iteratedDeriv h_interval_lt (hg_smooth.of_le hi_le')
+    rw [hderiv_c, hg_deriv i hi_le]
+    simp only [smul_eq_mul]
+    have h_diff : (2 * c - x) - c = c - x := by ring
+    rw [h_diff]
+    have h_pow : (c - x : ℝ) ^ i = (-1 : ℝ) ^ i * (x - c) ^ i := by
+      have : c - x = -(x - c) := by ring
+      rw [this, neg_eq_neg_one_mul, mul_pow]
+    rw [h_pow]
+    have h_neg_sq : (-1 : ℝ) ^ i * (-1 : ℝ) ^ i = 1 := neg_one_pow_mul_self i
+    have hgoal : (↑i.factorial : ℝ)⁻¹ * ((-1 : ℝ) ^ i * (x - c) ^ i) * ((-1 : ℝ) ^ i *
+      iteratedDeriv i f c)
+               = iteratedDeriv i f c / ↑i.factorial * (x - c) ^ i := by
+      have h1 : (↑i.factorial : ℝ)⁻¹ * ((-1 : ℝ) ^ i * (x - c) ^ i) * ((-1 : ℝ) ^ i *
+        iteratedDeriv i f c)
+              = ((-1 : ℝ) ^ i * (-1 : ℝ) ^ i) * ((↑i.factorial : ℝ)⁻¹ * (x - c) ^ i *
+                iteratedDeriv i f c) := by ring
+      rw [h1, h_neg_sq, one_mul]
+      field_simp
+    exact hgoal
+  -- Now use Lagrange formula
+  rw [hg_at_2cmx] at hLagrange
+  rw [← hTaylor_g_to_f, hLagrange]
+  have h_pow_rhs : ((2 * c - x) - c : ℝ) ^ (m + 1) = (c - x) ^ (m + 1) := by ring
+  have hderiv_ξ' : iteratedDerivWithin (m + 1) g (Icc c (2 * c - x)) ξ' = iteratedDeriv (m + 1) g
+    ξ' :=
+    iteratedDerivWithin_Icc_interior_eq h_interval_lt hξ'_mem_Ioo
+  -- iteratedDeriv (m+1) g ξ' = (-1)^(m+1) * iteratedDeriv (m+1) f ξ
+  have hg_deriv_ξ' : iteratedDeriv (m + 1) g ξ' = (-1 : ℝ) ^ (m + 1) * iteratedDeriv (m + 1) f ξ
+    := by
+    have hg_eq : g = fun s => f (2 * c - s) := hg_def
+    have h := @iteratedDeriv_reflect f c (m + 1) hf ξ'
+    simp only [hg_eq, hξ_def, h]
+  rw [hderiv_ξ', hg_deriv_ξ', h_pow_rhs]
+  have h_pow_eq : (c - x : ℝ) ^ (m + 1) = (-1 : ℝ) ^ (m + 1) * (x - c) ^ (m + 1) := by
+    have : c - x = -(x - c) := by ring
+    rw [this, neg_eq_neg_one_mul, mul_pow]
+  rw [h_pow_eq]
+  -- Simplify
+  have h_simplify : (-1 : ℝ) ^ (m + 1) * iteratedDeriv (m + 1) f ξ * ((-1 : ℝ) ^ (m + 1) * (x - c)
+    ^ (m + 1)) / ↑(m + 1).factorial =
+                   iteratedDeriv (m + 1) f ξ * (x - c) ^ (m + 1) / ↑(m + 1).factorial := by
+    have h_neg_sq := neg_one_pow_mul_self (m + 1)
+    have h1 : (-1 : ℝ) ^ (m + 1) * iteratedDeriv (m + 1) f ξ * ((-1 : ℝ) ^ (m + 1) * (x - c) ^ (m
+      + 1)) / ↑(m + 1).factorial =
+              ((-1 : ℝ) ^ (m + 1) * (-1 : ℝ) ^ (m + 1)) * (iteratedDeriv (m + 1) f ξ * (x - c) ^
+                (m + 1)) / ↑(m + 1).factorial := by ring
+    rw [h1, h_neg_sq, one_mul]
+  rw [h_simplify]
+  -- Bound the norm
+  rw [norm_div, norm_mul, Real.norm_eq_abs, Real.norm_eq_abs, Real.norm_eq_abs]
+  have hfact_pos : (0 : ℝ) < ((m + 1).factorial : ℝ) := Nat.cast_pos.mpr (Nat.factorial_pos _)
+  rw [abs_of_pos hfact_pos]
+  rw [abs_pow]
+  have hbound := hM ξ hξ_ab
+  have hxc_ne : x - c ≠ 0 := by linarith
+  have h_pow_fact_pos : 0 < |x - c| ^ (m + 1) / (m + 1).factorial := by
+    apply div_pos
+    · apply pow_pos; exact abs_pos.mpr hxc_ne
+    · exact hfact_pos
+  calc |iteratedDeriv (m + 1) f ξ| * |x - c| ^ (m + 1) / ↑(m + 1).factorial
+      = |iteratedDeriv (m + 1) f ξ| * (|x - c| ^ (m + 1) / ↑(m + 1).factorial) := by ring
+    _ ≤ M * (|x - c| ^ (m + 1) / ↑(m + 1).factorial) := by
+        apply mul_le_mul_of_nonneg_right hbound (le_of_lt h_pow_fact_pos)
+    _ = M * |x - c| ^ (m + 1) / ↑(m + 1).factorial := by ring
+/-- Lagrange form remainder bound for Taylor approximation.
+
+If `|f^(n)(x)| ≤ M` for all x in [a, b], and `c ∈ [a, b]`, then the Taylor polynomial
+of degree `n-1` (sum over range n) satisfies: for any `x ∈ [a, b]`:
+  `|f(x) - T_{n-1}(x; c)| ≤ M * |x - c|^n / n!`
+
+Note: The sum `∑ i ∈ Finset.range n` gives terms 0..n-1 (degree n-1 polynomial).
+The remainder involves the n-th derivative, matching our bound on `iteratedDeriv n f`.
+
+All cases are fully proved:
+- **Case n = 0**: Direct bound. ✓
+- **Case n > 0, x = c**: Trivial (both sides zero). ✓
+- **Case n > 0, c < x**: Via `taylor_remainder_bound_c_lt_x`. ✓
+- **Case n > 0, x < c**: Via `taylor_remainder_bound_x_lt_c` (reflection). ✓
+-/
+theorem taylor_remainder_bound {f : ℝ → ℝ} {a b c : ℝ} {n : ℕ} {M : ℝ}
+    (_hab : a ≤ b) (hca : a ≤ c) (hcb : c ≤ b)
+    (hf : ContDiff ℝ n f)
+    (hM : ∀ x ∈ Set.Icc a b, ‖iteratedDeriv n f x‖ ≤ M)
+    (_hMnonneg : 0 ≤ M) :
+    ∀ x ∈ Set.Icc a b,
+      ‖f x - ∑ i ∈ Finset.range n, (iteratedDeriv i f c / i.factorial) * (x - c) ^ i‖
+        ≤ M * |x - c| ^ n / n.factorial := by
+  open Set Finset in
+  intro x hx
+  cases n with
+  | zero =>
+    -- Case n = 0: sum is empty, need ‖f x‖ ≤ M
+    simp only [range_zero, sum_empty, sub_zero, Nat.factorial_zero, Nat.cast_one,
+      div_one, pow_zero, mul_one]
+    have h : f = iteratedDeriv 0 f := iteratedDeriv_zero.symm
+    rw [h]
+    exact hM x hx
+  | succ m =>
+    by_cases hxc : x = c
+    · -- Case x = c: both sides are 0
+      have hrhs : M * |x - c| ^ (m + 1) / ↑(m + 1).factorial = 0 := by
+        rw [hxc, sub_self, abs_zero, zero_pow (Nat.succ_ne_zero m), mul_zero, zero_div]
+      rw [hrhs]
+      have hsum : ∑ i ∈ range (m + 1), (iteratedDeriv i f c / i.factorial) * (x - c) ^ i = f c := by
+        rw [hxc]
+        simp only [sub_self]
+        rw [sum_eq_single 0]
+        · simp [iteratedDeriv_zero]
+        · intro i _ hi
+          have hpos : 0 < i := Nat.pos_of_ne_zero hi
+          have : (0 : ℝ) ^ i = 0 := zero_pow hpos.ne'
+          simp [this]
+        · simp
+      rw [hsum, hxc, sub_self, norm_zero]
+    · -- Case x ≠ c
+      rcases lt_or_gt_of_ne hxc with hxc_lt | hcx_lt
+      · -- Case x < c: use reflection approach
+        exact taylor_remainder_bound_x_lt_c hcb hf hM x hx hxc_lt
+      · -- Case c < x: fully proved
+        exact taylor_remainder_bound_c_lt_x hca hf hM x hx hcx_lt
+
+/-! ### Common Taylor expansions -/
+
+/-- Taylor coefficients for exp at 0: 1/n! -/
+def expTaylorCoeff (n : ℕ) : ℚ := 1 / n.factorial
+
+/-- Taylor coefficients for sin at 0 -/
+def sinTaylorCoeff (n : ℕ) : ℚ :=
+  if n % 2 = 0 then 0
+  else if (n / 2) % 2 = 0 then 1 / n.factorial
+  else -1 / n.factorial
+
+/-- Taylor coefficients for cos at 0 -/
+def cosTaylorCoeff (n : ℕ) : ℚ :=
+  if n % 2 = 1 then 0
+  else if (n / 2) % 2 = 0 then 1 / n.factorial
+  else -1 / n.factorial
+
+/-! ### Derivative bounds for common functions -/
+
+/-- All derivatives of exp are bounded by exp on any bounded interval -/
+theorem exp_deriv_bound {a b : ℝ} (_hab : a ≤ b) (n : ℕ) :
+    ∀ x ∈ Set.Icc a b, ‖iteratedDeriv n Real.exp x‖ ≤ Real.exp b := by
+  intro x hx
+  rw [iteratedDeriv_eq_iterate]
+  rw [Real.iter_deriv_exp]
+  rw [Real.norm_eq_abs, abs_of_pos (Real.exp_pos x)]
+  exact Real.exp_le_exp.mpr hx.2
+
+/-- All derivatives of sin and cos are bounded by 1.
+    The derivatives cycle: sin → cos → -sin → -cos → sin → ... -/
+theorem sin_cos_deriv_bound (n : ℕ) :
+    ∀ x : ℝ, ‖iteratedDeriv n Real.sin x‖ ≤ 1 ∧ ‖iteratedDeriv n Real.cos x‖ ≤ 1 := by
+  intro x
+  -- Prove by induction that both bounds hold together
+  induction n with
+  | zero =>
+    simp only [iteratedDeriv_zero]
+    exact ⟨Real.abs_sin_le_one x, Real.abs_cos_le_one x⟩
+  | succ n ih =>
+    constructor
+    · -- sin case: d/dx sin = cos
+      rw [iteratedDeriv_succ']
+      have hderiv : deriv Real.sin = Real.cos := Real.deriv_sin
+      rw [hderiv]
+      exact ih.2
+    · -- cos case: d/dx cos = -sin
+      rw [iteratedDeriv_succ']
+      have hderiv : deriv Real.cos = fun y => -Real.sin y := Real.deriv_cos'
+      rw [hderiv]
+      rw [iteratedDeriv_fun_neg n Real.sin x, norm_neg]
+      exact ih.1
+
+/-- cosh x ≤ exp |x| for all x -/
+theorem cosh_le_exp_abs (x : ℝ) : Real.cosh x ≤ Real.exp |x| := by
+  rw [Real.cosh_eq]
+  have h1 : Real.exp x ≤ Real.exp |x| := Real.exp_le_exp.mpr (le_abs_self x)
+  have habs_neg : -x ≤ |x| := neg_le_abs x
+  have h2 : Real.exp (-x) ≤ Real.exp |x| := Real.exp_le_exp.mpr habs_neg
+  linarith
+
+/-- |sinh x| ≤ cosh x for all x -/
+theorem abs_sinh_le_cosh (x : ℝ) : |Real.sinh x| ≤ Real.cosh x := by
+  have hcosh_pos : 0 < Real.cosh x := Real.cosh_pos x
+  have hcosh_sq : Real.cosh x ^ 2 = Real.sinh x ^ 2 + 1 := Real.cosh_sq x
+  have hsinh_sq_le : Real.sinh x ^ 2 ≤ Real.cosh x ^ 2 := by linarith [sq_nonneg (Real.cosh x)]
+  rw [abs_le]
+  constructor
+  · -- -cosh x ≤ sinh x
+    have h : Real.cosh x ^ 2 - Real.sinh x ^ 2 = 1 := Real.cosh_sq_sub_sinh_sq x
+    nlinarith [sq_nonneg (Real.cosh x + Real.sinh x), sq_nonneg (Real.cosh x - Real.sinh x)]
+  · -- sinh x ≤ cosh x
+    have h : Real.cosh x ^ 2 - Real.sinh x ^ 2 = 1 := Real.cosh_sq_sub_sinh_sq x
+    nlinarith [sq_nonneg (Real.cosh x + Real.sinh x), sq_nonneg (Real.cosh x - Real.sinh x)]
+
+/-- All derivatives of sinh and cosh are bounded by exp(max(|a|, |b|)) on [a, b].
+    The derivatives cycle: sinh → cosh → sinh → cosh → ... -/
+theorem sinh_cosh_deriv_bound {a b : ℝ} (_hab : a ≤ b) (n : ℕ) :
+    ∀ x ∈ Set.Icc a b, ‖iteratedDeriv n Real.sinh x‖ ≤ Real.exp (max (|a|) (|b|)) ∧
+                       ‖iteratedDeriv n Real.cosh x‖ ≤ Real.exp (max (|a|) (|b|)) := by
+  intro x hx
+  set M := Real.exp (max (|a|) (|b|)) with hM_def
+  have habs_x_le : |x| ≤ max (|a|) (|b|) := by
+    rw [abs_le]
+    constructor
+    · calc -max (|a|) (|b|) ≤ -|a| := neg_le_neg (le_max_left _ _)
+        _ ≤ a := neg_abs_le a
+        _ ≤ x := hx.1
+    · calc x ≤ b := hx.2
+        _ ≤ |b| := le_abs_self b
+        _ ≤ max (|a|) (|b|) := le_max_right _ _
+  have hcosh_bound : Real.cosh x ≤ M := by
+    calc Real.cosh x ≤ Real.exp |x| := cosh_le_exp_abs x
+      _ ≤ Real.exp (max (|a|) (|b|)) := Real.exp_le_exp.mpr habs_x_le
+  have hsinh_bound : |Real.sinh x| ≤ M := by
+    calc |Real.sinh x| ≤ Real.cosh x := abs_sinh_le_cosh x
+      _ ≤ M := hcosh_bound
+  -- Prove by induction that both bounds hold
+  induction n with
+  | zero =>
+    simp only [iteratedDeriv_zero, Real.norm_eq_abs]
+    exact ⟨hsinh_bound, by rw [abs_of_pos (Real.cosh_pos x)]; exact hcosh_bound⟩
+  | succ n ih =>
+    constructor
+    · -- sinh case: d/dx sinh = cosh
+      rw [iteratedDeriv_succ']
+      have hderiv : deriv Real.sinh = Real.cosh := by
+        funext x
+        exact LeanCert.Core.DerivativeIntervals.sinh_deriv_eq x
+      rw [hderiv]
+      exact ih.2
+    · -- cosh case: d/dx cosh = sinh
+      rw [iteratedDeriv_succ']
+      have hderiv : deriv Real.cosh = Real.sinh := Real.deriv_cosh
+      rw [hderiv]
+      exact ih.1
+
+/- The n-th iterated derivative of log at x > 0 is (-1)^(n-1) * (n-1)! * x^(-n) for n ≥ 1. -/
+theorem iteratedDeriv_log {n : ℕ} (hn : n ≠ 0) {x : ℝ} (hx : 0 < x) :
+    iteratedDeriv n Real.log x = (-1)^(n-1) * (n-1).factorial * x^(-(n : ℤ)) := by
+    match n with
+    | 1 =>
+      -- n = 1: iteratedDeriv 1 log x = deriv log x = 1/x = x^(-1)
+      simp only [iteratedDeriv_one, Real.deriv_log', tsub_self, pow_zero, Nat.factorial_zero,
+        Nat.cast_one, mul_one, Int.reduceNeg, zpow_neg, zpow_one, one_mul]
+    | n + 2 =>
+      -- n + 2: use induction
+      -- iteratedDeriv (n+2) log = deriv (iteratedDeriv (n+1) log)
+      rw [iteratedDeriv_succ]
+      -- By IH: iteratedDeriv (n+1) log y = (-1)^n * n! * y^(-(n+1)) for y > 0
+      have h_eq : ∀ y : ℝ, 0 < y → iteratedDeriv (n + 1) Real.log y =
+          (-1 : ℝ)^n * n.factorial * y^(-(n + 1 : ℤ)) := fun y hy => iteratedDeriv_log
+            n.succ_ne_zero hy
+      have h_deriv_eq : deriv (iteratedDeriv (n + 1) Real.log) x =
+          deriv (fun y => (-1 : ℝ)^n * n.factorial * y^(-(n + 1 : ℤ))) x := by
+        apply Filter.EventuallyEq.deriv_eq
+        filter_upwards [eventually_gt_nhds hx] with y hy
+        exact h_eq y hy
+      rw [h_deriv_eq]
+      have hdiff : DifferentiableAt ℝ (fun y => y ^ (-(n + 1 : ℤ))) x := by
+        apply DifferentiableAt.zpow differentiableAt_id
+        left; exact hx.ne'
+      rw [deriv_const_mul _ hdiff, deriv_zpow (-(n + 1 : ℤ)) x]
+      simp only [neg_add_rev, Int.reduceNeg, Int.cast_add, Int.cast_neg, Int.cast_one,
+        Int.cast_natCast, Nat.add_one_sub_one, Nat.cast_add, Nat.cast_ofNat]
+      rw [pow_succ (-1 : ℝ) n, Nat.factorial_succ]
+      push_cast
+      calc
+        (-1) ^ n * n.factorial * ((-1 + -(n : ℤ)) * x ^ (-1 + -(n : ℤ) - 1))
+        _ = (-1) ^ n * n.factorial * ((-1 + -(n : ℤ)) * x ^ (-2 + -(n : ℤ))) := by congr 3; grind
+        _ = (-1) ^ n * -1 * ((↑n + 1) * n.factorial) * x ^ (-2 + -(n : ℤ)) := by simp; ring
+
+end LeanCert.Core
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Rational Endpoint Intervals - Computable Taylor Series
+
+This file provides computable interval enclosures for transcendental functions
+using Taylor series with rational coefficients and rigorous remainder bounds.
+
+## Main definitions
+
+* `ratFactorial` - Compute n! as a rational
+* `pow` - Compute interval power
+* `absInterval`, `maxAbs` - Absolute value helpers
+* `evalTaylorSeries` - Evaluate Taylor polynomial on an interval
+* `expComputable`, `sinComputable`, `cosComputable` - Computable transcendental functions
+* `sinhComputable`, `coshComputable` - Computable hyperbolic functions
+
+## Main theorems
+
+* `mem_pow` - FTIA for interval power
+* `mem_evalTaylorSeries` - General FTIA for Taylor series
+* `mem_expComputable`, `mem_sinComputable`, `mem_cosComputable` - FTIA for computable functions
+
+## Design notes
+
+All definitions in this file use only rational arithmetic and are fully computable.
+The proofs connect these to the real-valued functions via Taylor's theorem.
+-/
+
+/-! ### Computable Taylor series helpers -/
+
+@[expose] public section
+
+namespace LeanCert.Core
+
+namespace IntervalRat
+
+/-- Compute n! as a Rational -/
+def ratFactorial (n : ℕ) : ℚ := (Nat.factorial n : ℚ)
+
+/-- Compute the integer power of an interval using exponentiation by squaring.
+    O(log n) interval multiplications instead of O(n). -/
+def pow (I : IntervalRat) : ℕ → IntervalRat
+  | 0 => singleton 1
+  | 1 => I
+  | n + 2 =>
+    let half := pow I ((n + 2) / 2)
+    let sq := mul half half
+    if (n + 2) % 2 == 0 then sq else mul I sq
+
+/-- Compute the absolute value interval: |I| = [0, max(|lo|, |hi|)] if 0 ∈ I,
+    or [min(|lo|,|hi|), max(|lo|,|hi|)] otherwise -/
+def absInterval (I : IntervalRat) : IntervalRat :=
+  if h1 : I.lo ≥ 0 then
+    I
+  else if h2 : I.hi ≤ 0 then
+    neg I
+  else
+    ⟨0, max (-I.lo) I.hi, by
+      apply le_max_of_le_right
+      push Not at h1 h2
+      linarith⟩
+
+/-- Maximum absolute value of an interval -/
+def maxAbs (I : IntervalRat) : ℚ := max (|I.lo|) (|I.hi|)
+
+/-- Evaluate Taylor series ∑_{i=0}^{n} c_i * x^i at interval I using Horner's method.
+    Computes c₀ + I * (c₁ + I * (c₂ + ... + I * cₙ)), which is mathematically
+    equivalent to the direct sum but uses fewer operations and often gives tighter
+    bounds by reducing the dependency problem in interval arithmetic. -/
+def evalTaylorSeries (coeffs : List ℚ) (I : IntervalRat) : IntervalRat :=
+  coeffs.foldr (fun c acc => add (singleton c) (mul acc I)) (singleton 0)
+
+/-! ### Computable exp via Taylor series -/
+
+/-- Tail-recursive exp coefficient generator.
+    `expTaylorCoeffsAux n k c` produces `n+1` coefficients starting at index `k`,
+    assuming `c = 1 / k!`. -/
+def expTaylorCoeffsAux : ℕ → ℕ → ℚ → List ℚ
+  | 0, _, c => [c]
+  | (n + 1), k, c => c :: expTaylorCoeffsAux n (k + 1) (c / (k + 1))
+
+/-- Taylor coefficients for exp: 1/i! for i = 0, 1, ..., n.
+    Implemented iteratively to avoid repeated factorial recomputation. -/
+def expTaylorCoeffs (n : ℕ) : List ℚ :=
+  expTaylorCoeffsAux n 0 1
+
+/-- `ratFactorial` unfolds factorial as a product. -/
+private lemma ratFactorial_succ (k : ℕ) :
+    ratFactorial (k + 1) = ratFactorial k * (k + 1) := by
+  simp [ratFactorial, Nat.factorial_succ, Nat.cast_mul, mul_comm]
+
+private lemma map_range_succ {α : Type*} (f : ℕ → α) (n : ℕ) :
+    (List.range (n + 1)).map f = f 0 :: (List.range n).map (fun i => f (i + 1)) := by
+  induction n with
+  | zero =>
+      simp
+  | succ n ih =>
+      -- range (n+2) = range (n+1) ++ [n+1]
+      calc
+        (List.range (n + 2)).map f
+            = (List.range (n + 1) ++ [n + 1]).map f := by
+                simp [List.range_succ]
+        _ = (List.range (n + 1)).map f ++ [f (n + 1)] := by
+                simp [List.map_append]
+        _ = (f 0 :: (List.range n).map (fun i => f (i + 1))) ++ [f (n + 1)] := by
+                simp [ih]
+        _ = f 0 :: ((List.range n).map (fun i => f (i + 1)) ++ [f (n + 1)]) := by
+                rfl
+        _ = f 0 :: (List.range (n + 1)).map (fun i => f (i + 1)) := by
+                simp [List.range_succ, List.map_append]
+
+/-- Iterative exp coefficients match the standard `1 / i!` definition. -/
+private lemma expTaylorCoeffsAux_eq_range (n k : ℕ) :
+    expTaylorCoeffsAux n k (1 / ratFactorial k) =
+      (List.range (n + 1)).map (fun i => 1 / ratFactorial (i + k)) := by
+  induction n generalizing k with
+  | zero =>
+      simp [expTaylorCoeffsAux]
+  | succ n ih =>
+      have hstep : (ratFactorial k)⁻¹ / (k + 1) = (ratFactorial (k + 1))⁻¹ := by
+        -- (1/k!) / (k+1) = 1/(k+1)!
+        simp [ratFactorial_succ, div_eq_mul_inv, mul_comm]
+      calc
+        expTaylorCoeffsAux (n + 1) k (1 / ratFactorial k)
+            = (1 / ratFactorial k) ::
+                expTaylorCoeffsAux n (k + 1) ((1 / ratFactorial k) / (k + 1)) := by
+                  rfl
+        _ = (1 / ratFactorial k) ::
+                expTaylorCoeffsAux n (k + 1) ((ratFactorial k)⁻¹ / (k + 1)) := by
+                  simp [one_div]
+        _ = (1 / ratFactorial k) ::
+                expTaylorCoeffsAux n (k + 1) (ratFactorial (k + 1))⁻¹ := by
+                  simp [hstep]
+        _ = (1 / ratFactorial k) ::
+              (List.range (n + 1)).map (fun i => (ratFactorial (i + (k + 1)))⁻¹) := by
+                  simpa [one_div] using (ih (k + 1))
+        _ = (List.range (n + 2)).map (fun i => 1 / ratFactorial (i + k)) := by
+            -- unfold the range map at the front
+            have hmap := map_range_succ (fun i => 1 / ratFactorial (i + k)) (n + 1)
+            -- rewrite and simplify
+            simpa [one_div, Nat.add_assoc, Nat.add_left_comm, Nat.add_comm] using hmap.symm
+
+private lemma expTaylorCoeffs_eq_range_map (n : ℕ) :
+    expTaylorCoeffs n = (List.range (n + 1)).map (fun i => 1 / ratFactorial i) := by
+  have h0 : (1 / ratFactorial 0 : ℚ) = 1 := by simp [ratFactorial]
+  simpa [expTaylorCoeffs, h0] using (expTaylorCoeffsAux_eq_range n 0)
+
+/-- Computable exp remainder bound using rational arithmetic.
+    The Lagrange remainder is exp(ξ) * x^{n+1} / (n+1)! where ξ is between 0 and x.
+    We use e < 3, so e^r ≤ 3^(⌈r⌉+1) as a conservative bound.
+
+    Returns an interval [-R, R] where R bounds the remainder. -/
+def expRemainderBoundComputable (I : IntervalRat) (n : ℕ) : IntervalRat :=
+  let r := maxAbs I
+  -- Crude bound: e < 3, so e^r ≤ 3^(ceil(r)+1)
+  let expBound := (3 : ℚ) ^ (Nat.ceil r + 1)
+  let xBound := r ^ (n + 1)
+  let R := expBound * xBound / ratFactorial (n + 1)
+  ⟨-R, R, by
+    have hr : 0 ≤ r := le_max_of_le_left (abs_nonneg I.lo)
+    have hR : 0 ≤ R := by
+      apply div_nonneg
+      · apply mul_nonneg
+        · apply pow_nonneg; norm_num
+        · apply pow_nonneg hr
+      · exact Nat.cast_nonneg _
+    linarith⟩
+
+/-- The raw rational Taylor enclosure of the exponential at a point. -/
+def expPointComputableRaw (q : ℚ) (n : ℕ := 10) : IntervalRat :=
+  let I := singleton q
+  let coeffs := expTaylorCoeffs n
+  let polyVal := evalTaylorSeries coeffs I
+  let remainder := expRemainderBoundComputable I n
+  add polyVal remainder
+
+/-- Heuristic reduction factor for exp evaluation.
+    We choose `k = log2(ceil(|q|) + 1)`, so `2^k` grows roughly with |q|.
+    No correctness depends on the specific choice; it only affects performance. -/
+def expReduceK (q : ℚ) : ℕ :=
+  Nat.log2 (Nat.ceil |q| + 1)
+
+/-- Computable interval enclosure for exp at a single rational point.
+    Uses argument reduction: `exp(q) = exp(q/2^k)^(2^k)`. -/
+def expPointComputable (q : ℚ) (n : ℕ := 10) : IntervalRat :=
+  let k := expReduceK q
+  let m : ℕ := (2:ℕ)^k
+  let q' : ℚ := q / m
+  pow (expPointComputableRaw q' n) m
+
+/-- Point exponential using Taylor coefficients prepared for depth `n`. -/
+def expPointComputableWithCoeffs (q : ℚ) (n : ℕ) (coeffs : List ℚ) : IntervalRat :=
+  let k := expReduceK q
+  let m : ℕ := (2 : ℕ) ^ k
+  let q' : ℚ := q / m
+  let I := singleton q'
+  let polyVal := evalTaylorSeries coeffs I
+  let remainder := expRemainderBoundComputable I n
+  pow (add polyVal remainder) m
+
+theorem expPointComputableWithCoeffs_eq (q : ℚ) (n : ℕ) :
+    expPointComputableWithCoeffs q n (expTaylorCoeffs n) = expPointComputable q n := by
+  simp [expPointComputableWithCoeffs, expPointComputable, expPointComputableRaw]
+
+/-- Hull of two intervals: smallest interval containing both. -/
+def hull (I J : IntervalRat) : IntervalRat :=
+  ⟨min I.lo J.lo, max I.hi J.hi, le_trans (min_le_left _ _) (le_trans I.le (le_max_left _ _))⟩
+
+/-- Membership in hull -/
+theorem mem_hull_left {x : ℝ} {I J : IntervalRat} (hx : x ∈ I) : x ∈ hull I J := by
+  simp only [hull, mem_def, Rat.cast_min, Rat.cast_max]
+  constructor
+  · exact le_trans (min_le_left _ _) hx.1
+  · exact le_trans hx.2 (le_max_left _ _)
+
+theorem mem_hull_right {x : ℝ} {I J : IntervalRat} (hx : x ∈ J) : x ∈ hull I J := by
+  simp only [hull, mem_def, Rat.cast_min, Rat.cast_max]
+  constructor
+  · exact le_trans (min_le_right _ _) hx.1
+  · exact le_trans hx.2 (le_max_right _ _)
+
+/-- Computable interval enclosure for exp using Taylor series with monotonicity optimization.
+
+    exp(x) = ∑_{i=0}^{n} x^i/i! + R where |R| ≤ exp(|x|) * |x|^{n+1} / (n+1)!
+
+    For intervals not crossing 0, we use endpoint evaluation and take the hull,
+    which is tighter than direct Taylor evaluation due to interval widening.
+
+    This is fully computable using only rational arithmetic. -/
+def expComputable (I : IntervalRat) (n : ℕ := 10) : IntervalRat :=
+  if I.hi ≤ 0 ∨ 0 ≤ I.lo then
+    -- Interval doesn't cross 0: use endpoint evaluation for tighter bounds
+    let expLo := expPointComputable I.lo n
+    let expHi := expPointComputable I.hi n
+    hull expLo expHi
+  else
+    -- Interval crosses 0: use standard Taylor (can't avoid interval widening)
+    let coeffs := expTaylorCoeffs n
+    let polyVal := evalTaylorSeries coeffs I
+    let remainder := expRemainderBoundComputable I n
+    add polyVal remainder
+
+/-- Interval exponential using Taylor coefficients prepared for depth `n`. -/
+def expComputableWithCoeffs (I : IntervalRat) (n : ℕ) (coeffs : List ℚ) : IntervalRat :=
+  if I.hi ≤ 0 ∨ 0 ≤ I.lo then
+    let expLo := expPointComputableWithCoeffs I.lo n coeffs
+    let expHi := expPointComputableWithCoeffs I.hi n coeffs
+    hull expLo expHi
+  else
+    let polyVal := evalTaylorSeries coeffs I
+    let remainder := expRemainderBoundComputable I n
+    add polyVal remainder
+
+theorem expComputableWithCoeffs_eq (I : IntervalRat) (n : ℕ) :
+    expComputableWithCoeffs I n (expTaylorCoeffs n) = expComputable I n := by
+  simp [expComputableWithCoeffs, expComputable, expPointComputableWithCoeffs_eq]
+
+/-! ### Computable sin via Taylor series -/
+
+/-- Taylor coefficients for sin: 0, 1, 0, -1/6, 0, 1/120, ... -/
+def sinTaylorCoeffs (n : ℕ) : List ℚ :=
+  (List.range (n + 1)).map (fun i =>
+    if i % 2 = 1 then  -- odd terms only
+      ((-1 : ℚ) ^ ((i - 1) / 2)) / ratFactorial i
+    else 0)
+
+/-- Computable sin remainder bound.
+    Since |sin^{(k)}(x)| ≤ 1 for all k, x, the remainder is bounded by |x|^{n+1}/(n+1)! -/
+def sinRemainderBoundComputable (I : IntervalRat) (n : ℕ) : IntervalRat :=
+  let r := maxAbs I
+  let R := r ^ (n + 1) / ratFactorial (n + 1)
+  ⟨-R, R, by
+    have hr : 0 ≤ r := le_max_of_le_left (abs_nonneg I.lo)
+    have hR : 0 ≤ R := by
+      apply div_nonneg
+      · apply pow_nonneg hr
+      · exact Nat.cast_nonneg _
+    linarith⟩
+
+/-- Computable interval enclosure for sin using Taylor series.
+
+    sin(x) = ∑_{k=0}^{n/2} (-1)^k x^{2k+1}/(2k+1)! + R
+    where |R| ≤ |x|^{n+1}/(n+1)! since all derivatives of sin are bounded by 1.
+
+    We intersect with [-1, 1] for tighter bounds on small intervals. -/
+def sinComputable (I : IntervalRat) (n : ℕ := 10) : IntervalRat :=
+  let coeffs := sinTaylorCoeffs n
+  let polyVal := evalTaylorSeries coeffs I
+  let remainder := sinRemainderBoundComputable I n
+  let raw := add polyVal remainder
+  -- Intersect with global bound [-1, 1]
+  let globalBound : IntervalRat := ⟨-1, 1, by norm_num⟩
+  match intersect raw globalBound with
+  | some refined => refined
+  | none => raw  -- Should not happen for valid inputs
+
+/-- Interval sine using Taylor coefficients prepared for depth `n`. -/
+def sinComputableWithCoeffs (I : IntervalRat) (n : ℕ) (coeffs : List ℚ) : IntervalRat :=
+  let polyVal := evalTaylorSeries coeffs I
+  let remainder := sinRemainderBoundComputable I n
+  let raw := add polyVal remainder
+  let globalBound : IntervalRat := ⟨-1, 1, by norm_num⟩
+  match intersect raw globalBound with
+  | some refined => refined
+  | none => raw
+
+theorem sinComputableWithCoeffs_eq (I : IntervalRat) (n : ℕ) :
+    sinComputableWithCoeffs I n (sinTaylorCoeffs n) = sinComputable I n := by
+  simp [sinComputableWithCoeffs, sinComputable]
+
+/-! ### Computable cos via Taylor series -/
+
+/-- Taylor coefficients for cos: 1, 0, -1/2, 0, 1/24, 0, ... -/
+def cosTaylorCoeffs (n : ℕ) : List ℚ :=
+  (List.range (n + 1)).map (fun i =>
+    if i % 2 = 0 then  -- even terms only
+      ((-1 : ℚ) ^ (i / 2)) / ratFactorial i
+    else 0)
+
+/-- Computable cos remainder bound.
+    Since |cos^{(k)}(x)| ≤ 1 for all k, x, the remainder is bounded by |x|^{n+1}/(n+1)! -/
+def cosRemainderBoundComputable (I : IntervalRat) (n : ℕ) : IntervalRat :=
+  let r := maxAbs I
+  let R := r ^ (n + 1) / ratFactorial (n + 1)
+  ⟨-R, R, by
+    have hr : 0 ≤ r := le_max_of_le_left (abs_nonneg I.lo)
+    have hR : 0 ≤ R := by
+      apply div_nonneg
+      · apply pow_nonneg hr
+      · exact Nat.cast_nonneg _
+    linarith⟩
+
+/-- Computable interval enclosure for cos using Taylor series.
+
+    cos(x) = ∑_{k=0}^{n/2} (-1)^k x^{2k}/(2k)! + R
+    where |R| ≤ |x|^{n+1}/(n+1)! since all derivatives of cos are bounded by 1.
+
+    We intersect with [-1, 1] for tighter bounds on small intervals. -/
+def cosComputable (I : IntervalRat) (n : ℕ := 10) : IntervalRat :=
+  let coeffs := cosTaylorCoeffs n
+  let polyVal := evalTaylorSeries coeffs I
+  let remainder := cosRemainderBoundComputable I n
+  let raw := add polyVal remainder
+  -- Intersect with global bound [-1, 1]
+  let globalBound : IntervalRat := ⟨-1, 1, by norm_num⟩
+  match intersect raw globalBound with
+  | some refined => refined
+  | none => raw  -- Should not happen for valid inputs
+
+/-- Interval cosine using Taylor coefficients prepared for depth `n`. -/
+def cosComputableWithCoeffs (I : IntervalRat) (n : ℕ) (coeffs : List ℚ) : IntervalRat :=
+  let polyVal := evalTaylorSeries coeffs I
+  let remainder := cosRemainderBoundComputable I n
+  let raw := add polyVal remainder
+  let globalBound : IntervalRat := ⟨-1, 1, by norm_num⟩
+  match intersect raw globalBound with
+  | some refined => refined
+  | none => raw
+
+theorem cosComputableWithCoeffs_eq (I : IntervalRat) (n : ℕ) :
+    cosComputableWithCoeffs I n (cosTaylorCoeffs n) = cosComputable I n := by
+  simp [cosComputableWithCoeffs, cosComputable]
+
+/-! ### Computable sinh and cosh via exp -/
+
+/-- Computable interval enclosure for sinh at a single rational point.
+    Uses the definition sinh(q) = (exp(q) - exp(-q)) / 2. -/
+def sinhPointComputable (q : ℚ) (n : ℕ := 10) : IntervalRat :=
+  let expPos := expPointComputable q n
+  let expNeg := expPointComputable (-q) n
+  -- sinh(q) = (exp(q) - exp(-q)) / 2
+  -- Lower bound: (expPos.lo - expNeg.hi) / 2
+  -- Upper bound: (expPos.hi - expNeg.lo) / 2
+  let sinhLo := (expPos.lo - expNeg.hi) / 2
+  let sinhHi := (expPos.hi - expNeg.lo) / 2
+  if h : sinhLo ≤ sinhHi then
+    ⟨sinhLo, sinhHi, h⟩
+  else
+    ⟨min sinhLo sinhHi, max sinhLo sinhHi, @min_le_max _ _ sinhLo sinhHi⟩
+
+/-- Computable interval enclosure for cosh at a single rational point.
+    Uses the definition cosh(q) = (exp(q) + exp(-q)) / 2. -/
+def coshPointComputable (q : ℚ) (n : ℕ := 10) : IntervalRat :=
+  let expPos := expPointComputable q n
+  let expNeg := expPointComputable (-q) n
+  -- cosh(q) = (exp(q) + exp(-q)) / 2
+  -- Lower bound: (expPos.lo + expNeg.lo) / 2
+  -- Upper bound: (expPos.hi + expNeg.hi) / 2
+  let coshLo := (expPos.lo + expNeg.lo) / 2
+  let coshHi := (expPos.hi + expNeg.hi) / 2
+  -- cosh ≥ 1 always, so ensure lower bound is at least 1
+  let safeLo := max 1 coshLo
+  if h : safeLo ≤ coshHi then
+    ⟨safeLo, coshHi, h⟩
+  else
+    ⟨1, max 2 coshHi, by
+      have h1 : (1 : ℚ) ≤ 2 := by norm_num
+      exact le_trans h1 (le_max_left _ _)⟩
+
+/-- The lower bound of coshPointComputable is always at least 1. -/
+theorem coshPointComputable_lo_ge_one (q : ℚ) (n : ℕ) : 1 ≤ (coshPointComputable q n).lo := by
+  simp only [coshPointComputable]
+  split_ifs with h
+  · exact le_max_left 1 _
+  · exact le_refl 1
+
+/-- Computable interval enclosure for sinh using exp with endpoint evaluation.
+
+    sinh(x) = (exp(x) - exp(-x)) / 2
+    Since sinh is strictly monotone increasing, sinh([a,b]) = [sinh(a), sinh(b)].
+    We use endpoint evaluation for tight bounds. -/
+def sinhComputable (I : IntervalRat) (n : ℕ := 10) : IntervalRat :=
+  -- sinh is strictly monotone increasing, so evaluate at endpoints
+  let sinhLo := sinhPointComputable I.lo n
+  let sinhHi := sinhPointComputable I.hi n
+  hull sinhLo sinhHi
+
+/-- Computable interval enclosure for cosh using exp with endpoint evaluation.
+
+    cosh(x) = (exp(x) + exp(-x)) / 2
+    cosh has minimum 1 at x = 0, and is symmetric: cosh(-x) = cosh(x).
+    - cosh is decreasing on (-∞, 0]
+    - cosh is increasing on [0, ∞)
+
+    We use endpoint evaluation with monotonicity for tight bounds. -/
+def coshComputable (I : IntervalRat) (n : ℕ := 10) : IntervalRat :=
+  let coshLo := coshPointComputable I.lo n
+  let coshHi := coshPointComputable I.hi n
+  if 0 ≤ I.lo then
+    -- Interval is non-negative: cosh is increasing, so [cosh(lo), cosh(hi)]
+    hull coshLo coshHi
+  else if I.hi ≤ 0 then
+    -- Interval is non-positive: cosh is decreasing, so [cosh(hi), cosh(lo)]
+    hull coshHi coshLo
+  else
+    -- Interval contains 0: minimum is 1 at x=0, max is at whichever endpoint is farther
+    let maxEndpoint := hull coshLo coshHi
+    ⟨1, maxEndpoint.hi, by
+      -- coshPointComputable ensures lower bound ≥ 1 via max 1 _
+      have hlo_ge1 := coshPointComputable_lo_ge_one I.lo n
+      have hhi_ge1 := coshPointComputable_lo_ge_one I.hi n
+      calc (1 : ℚ) ≤ min (coshPointComputable I.lo n).lo (coshPointComputable I.hi n).lo :=
+          le_min hlo_ge1 hhi_ge1
+        _ = maxEndpoint.lo := rfl
+        _ ≤ maxEndpoint.hi := maxEndpoint.le⟩
+
+/-! ### FTIA for pow -/
+
+/-- FTIA for interval power (binary exponentiation version) -/
+theorem mem_pow {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (n : ℕ) :
+    x ^ n ∈ pow I n := by
+  match n with
+  | 0 =>
+    simp only [_root_.pow_zero, pow, mem_def, singleton]
+    norm_num
+  | 1 =>
+    simp only [_root_.pow_one, pow]
+    exact hx
+  | n + 2 =>
+    have ih_half := mem_pow hx ((n + 2) / 2)
+    have h_sq := mem_mul ih_half ih_half
+    change x ^ (n + 2) ∈ pow I (n + 2)
+    unfold pow
+    split
+    · -- even case: x^(n+2) = (x^half)^2
+      next heven =>
+      have := beq_iff_eq.mp heven
+      rw [show x ^ (n + 2) = x ^ ((n + 2) / 2) * x ^ ((n + 2) / 2) from by
+        rw [← _root_.pow_add]; congr 1; omega]
+      exact h_sq
+    · -- odd case: x^(n+2) = x * (x^half)^2
+      next heven =>
+      have hodd : (n + 2) % 2 ≠ 0 := fun h => heven (beq_iff_eq.mpr h)
+      rw [show x ^ (n + 2) = x * (x ^ ((n + 2) / 2) * x ^ ((n + 2) / 2)) from by
+        rw [← _root_.pow_add, ← mul_comm, ← _root_.pow_succ]; congr 1; omega]
+      exact mem_mul hx h_sq
+termination_by n
+
+/-! ### Helper lemmas for Taylor series membership -/
+
+/-- Any x in I has |x| ≤ maxAbs I -/
+theorem abs_le_maxAbs {x : ℝ} {I : IntervalRat} (hx : x ∈ I) : |x| ≤ maxAbs I := by
+  simp only [mem_def, maxAbs] at *
+  have hlo : -(max |I.lo| |I.hi|) ≤ I.lo := by
+    calc -(max |I.lo| |I.hi|) ≤ -|I.lo| := neg_le_neg (le_max_left _ _)
+      _ ≤ I.lo := neg_abs_le _
+  have hhi : I.hi ≤ max |I.lo| |I.hi| := le_trans (le_abs_self _) (le_max_right _ _)
+  rw [abs_le]
+  constructor
+  · calc (-(max |I.lo| |I.hi| : ℚ) : ℝ) ≤ I.lo := by exact_mod_cast hlo
+      _ ≤ x := hx.1
+  · calc x ≤ I.hi := hx.2
+      _ ≤ max |I.lo| |I.hi| := by exact_mod_cast hhi
+
+/-- If |x| ≤ R for nonnegative R, then x ∈ [-R, R].
+    This is the key micro-lemma for embedding Lagrange remainder bounds into intervals. -/
+theorem abs_le_mem_symmetric_interval {x : ℝ} {R : ℚ} (hR : 0 ≤ R) (h : |x| ≤ R) :
+    x ∈ (⟨-R, R, by linarith⟩ : IntervalRat) := by
+  simp only [mem_def, Rat.cast_neg]
+  constructor
+  · have := neg_abs_le x; linarith
+  · exact le_trans (le_abs_self x) h
+
+/-- Domain setup for Taylor theorem: if |x| ≤ r for nonnegative r,
+    then x ∈ [-r, r] as an Icc with the required inequalities. -/
+theorem domain_from_abs_bound {x : ℝ} {r : ℚ} (_hr : 0 ≤ r) (habs : |x| ≤ r) :
+    x ∈ Set.Icc ((-r : ℚ) : ℝ) (r : ℚ) := by
+  simp only [Set.mem_Icc, Rat.cast_neg]
+  exact abs_le.mp habs
+
+/-- Combined domain setup from interval membership. -/
+theorem domain_from_mem {x : ℝ} {I : IntervalRat} (hx : x ∈ I) :
+    let r := maxAbs I
+    (0 : ℝ) ≤ r ∧ |x| ≤ r ∧ x ∈ Set.Icc ((-r : ℚ) : ℝ) (r : ℚ) ∧
+    ((-r : ℚ) : ℝ) ≤ 0 ∧ (0 : ℝ) ≤ (r : ℚ) ∧ ((-r : ℚ) : ℝ) ≤ r := by
+  have hr_nonneg : 0 ≤ maxAbs I := le_max_of_le_left (abs_nonneg I.lo)
+  have habs_x := abs_le_maxAbs hx
+  have hr_nonneg_real : (0 : ℝ) ≤ maxAbs I := by exact_mod_cast hr_nonneg
+  have hdom := domain_from_abs_bound hr_nonneg habs_x
+  refine ⟨hr_nonneg_real, habs_x, hdom, ?_, hr_nonneg_real, ?_⟩
+  · simp only [Rat.cast_neg]; linarith
+  · simp only [Rat.cast_neg]; linarith
+
+/-- Convert an absolute value bound |v| ≤ R to interval membership v ∈ [-R, R].
+    This is the key micro-lemma for the final step of Taylor remainder bounds. -/
+theorem remainder_to_interval {v : ℝ} {R : ℚ} (hbound : |v| ≤ R) :
+    v ∈ (⟨-R, R, by
+      have h1 : 0 ≤ |v| := abs_nonneg v
+      have h2 : (0 : ℝ) ≤ (R : ℚ) := le_trans h1 hbound
+      linarith [Rat.cast_nonneg.mp h2]⟩ : IntervalRat) := by
+  simp only [mem_def, Rat.cast_neg]
+  exact abs_le.mp hbound
+
+/-- Key lemma: exp(ξ) ≤ 3^(⌈r⌉+1) for |ξ| ≤ r -/
+theorem exp_bound_by_pow3 {r : ℚ} (_hr : 0 ≤ r) {ξ : ℝ} (hξ : |ξ| ≤ r) :
+    Real.exp ξ ≤ (3 : ℝ) ^ (Nat.ceil r + 1) := by
+  -- e < 3, using Real.exp_one_lt_d9 which gives exp(1) < 2.7182818286
+  have h3 : Real.exp 1 < 3 := by
+    have h := Real.exp_one_lt_d9  -- exp(1) < 2.7182818286
+    have h2 : (2.7182818286 : ℝ) < 3 := by norm_num
+    exact lt_trans h h2
+  have hceil : (r : ℝ) ≤ Nat.ceil r := by
+    have h : r ≤ (Nat.ceil r : ℚ) := Nat.le_ceil r
+    exact_mod_cast h
+  calc Real.exp ξ ≤ Real.exp |ξ| := Real.exp_le_exp.mpr (le_abs_self ξ)
+    _ ≤ Real.exp r := Real.exp_le_exp.mpr hξ
+    _ ≤ Real.exp (Nat.ceil r) := Real.exp_le_exp.mpr hceil
+    _ = Real.exp 1 ^ (Nat.ceil r) := by rw [← Real.exp_nat_mul]; ring_nf
+    _ ≤ 3 ^ (Nat.ceil r) := by
+        rcases Nat.eq_zero_or_pos (Nat.ceil r) with hr0 | hrpos
+        · simp [hr0]
+        · exact le_of_lt (pow_lt_pow_left₀ h3 (Real.exp_pos 1).le (Nat.pos_iff_ne_zero.mp hrpos))
+    _ ≤ 3 ^ (Nat.ceil r + 1) := pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 3) (Nat.le_succ _)
+
+/-! ### Coefficient matching lemmas -/
+
+/-- For exp, all iterated derivatives at 0 equal 1. -/
+lemma iteratedDeriv_exp_zero (i : ℕ) : iteratedDeriv i Real.exp 0 = 1 := by
+  simp [iteratedDeriv_eq_iterate, Real.iter_deriv_exp]
+
+/-! ### Helper lemmas for Taylor series membership -/
+
+/-- Polynomial sum starting at exponent `n`. -/
+private def polySumFrom (coeffs : List ℚ) (x : ℝ) (n : ℕ) : ℝ :=
+  ((coeffs.zipIdx n).map (fun (c, i) => (c : ℝ) * x ^ i)).sum
+
+/-- Horner polynomial evaluation: the real value computed by Horner's method.
+    `hornerVal [c₀, c₁, c₂] x = c₀ + x * (c₁ + x * (c₂ + x * 0)) = c₀ + c₁*x + c₂*x²` -/
+private def hornerVal (coeffs : List ℚ) (x : ℝ) : ℝ :=
+  coeffs.foldr (fun c acc => (c : ℝ) + x * acc) 0
+
+/-- hornerVal equals polySumFrom at any starting index n, scaled by x^n.
+    Specifically: x^n * hornerVal coeffs x = polySumFrom coeffs x n -/
+private lemma hornerVal_mul_pow_eq_polySumFrom (coeffs : List ℚ) (x : ℝ) (n : ℕ) :
+    x ^ n * hornerVal coeffs x = polySumFrom coeffs x n := by
+  induction coeffs generalizing n with
+  | nil => simp [hornerVal, polySumFrom]
+  | cons c cs ih =>
+    have hstep : hornerVal (c :: cs) x = (c : ℝ) + x * hornerVal cs x := by
+      simp [hornerVal]
+    have hunfold : polySumFrom (c :: cs) x n =
+        (c : ℝ) * x ^ n + polySumFrom cs x (n + 1) := by
+      simp [polySumFrom, List.zipIdx_cons]
+    rw [hstep, mul_add, hunfold, ← ih (n + 1)]
+    ring
+
+/-- hornerVal equals polySumFrom at index 0. -/
+private lemma hornerVal_eq_polySumFrom (coeffs : List ℚ) (x : ℝ) :
+    hornerVal coeffs x = polySumFrom coeffs x 0 := by
+  rw [← hornerVal_mul_pow_eq_polySumFrom coeffs x 0]
+  simp
+
+/-- Auxiliary lemma for Horner-style evaluation: the foldr contains hornerVal. -/
+private lemma mem_horner_foldr {x : ℝ} {I : IntervalRat} (hx : x ∈ I)
+    (coeffs : List ℚ) :
+    hornerVal coeffs x ∈
+      coeffs.foldr (fun c acc => add (singleton c) (mul acc I)) (singleton 0) := by
+  induction coeffs with
+  | nil =>
+      simp [hornerVal, mem_def, singleton]
+  | cons c cs ih =>
+      -- hornerVal (c :: cs) x = ↑c + x * hornerVal cs x
+      -- foldr ... (c :: cs) = add (singleton c) (mul (foldr ... cs) I)
+      change (c : ℝ) + x * hornerVal cs x ∈
+        add (singleton c) (mul (cs.foldr (fun c acc => add (singleton c) (mul acc I)) (singleton
+          0)) I)
+      have hc : (c : ℝ) ∈ singleton c := by simp [mem_def, singleton]
+      have hmul : hornerVal cs x * x ∈
+        mul (cs.foldr (fun c acc => add (singleton c) (mul acc I)) (singleton 0)) I :=
+        mem_mul ih hx
+      have : (c : ℝ) + x * hornerVal cs x = (c : ℝ) + hornerVal cs x * x := by ring
+      rw [this]
+      exact mem_add hc hmul
+
+/-- General FTIA for evalTaylorSeries (Horner version): if coeffs has length n+1, then
+    ∑_{i=0}^{n} coeffs[i] * x^i ∈ evalTaylorSeries coeffs I for x ∈ I. -/
+theorem mem_evalTaylorSeries {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (coeffs : List ℚ) :
+    (coeffs.zipIdx.map (fun (c, i) => (c : ℝ) * x ^ i)).sum ∈ evalTaylorSeries coeffs I := by
+  have hmem := mem_horner_foldr hx coeffs
+  rw [hornerVal_eq_polySumFrom] at hmem
+  simp only [evalTaylorSeries, polySumFrom] at *
+  exact hmem
+
+/-- Helper: (List.range n).map f).sum = ∑ i ∈ Finset.range n, f i -/
+private lemma list_map_sum_eq_finset_sum {α : Type*} [AddCommMonoid α]
+    (f : ℕ → α) (n : ℕ) : ((List.range n).map f).sum = ∑ i ∈ Finset.range n, f i := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    simp only [List.range_succ, List.map_append, List.sum_append, List.map_singleton,
+      List.sum_singleton, Finset.sum_range_succ]
+    rw [ih, add_comm]
+
+/-- Helper: zipIdx of List.range just pairs each element with its index (which is itself) -/
+private lemma zipIdx_range_map {α : Type*} (f : ℕ → ℕ → α) (n : ℕ) :
+    (List.range n).zipIdx.map (fun p => f p.1 p.2) = (List.range n).map (fun i => f i i) := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    simp only [List.range_succ, List.zipIdx_append, List.map_append, List.length_range]
+    rw [ih]
+    simp only [List.zipIdx_singleton, List.map_singleton, zero_add]
+
+/-- The exp Taylor polynomial value matches our evalTaylorSeries.
+    The proof shows that our list-based polynomial evaluation produces the same
+    sum as the Finset.sum form used in Mathlib's Taylor theorem. -/
+theorem mem_evalTaylorSeries_exp {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (n : ℕ) :
+    ∑ i ∈ Finset.range (n + 1), (1 / i.factorial : ℝ) * x ^ i ∈
+      evalTaylorSeries (expTaylorCoeffs n) I := by
+  have hmem := mem_evalTaylorSeries hx (expTaylorCoeffs n)
+  have hmem' :
+      (((List.range (n + 1)).map (fun i => 1 / ratFactorial i)).zipIdx.map
+          (fun p => (p.1 : ℝ) * x ^ p.2)).sum ∈
+        evalTaylorSeries (expTaylorCoeffs n) I := by
+    simpa [expTaylorCoeffs_eq_range_map] using hmem
+  convert hmem' using 1
+  rw [List.zipIdx_map]
+  simp only [List.map_map]
+  rw [← list_map_sum_eq_finset_sum (fun i => (1 / i.factorial : ℝ) * x ^ i) (n + 1)]
+  -- The two list maps are equal: both compute [1/0! * x^0, 1/1! * x^1, ...]
+  -- LHS: (List.range (n+1)).map (fun i => 1/i! * x^i)
+  -- RHS: zipIdx.map with Prod.map composition
+  -- For List.range, zipIdx gives [(0,0), (1,1), ...], so they match
+  congr 1
+  symm
+  -- The RHS has type (ℚ × ℕ) from Prod.map applied to zipIdx pairs
+  -- Step 1: Simplify the composition
+  have h1 : (List.range (n + 1)).zipIdx.map
+        ((fun p : ℚ × ℕ => (p.1 : ℝ) * x ^ p.2) ∘ Prod.map (fun i => (1 : ℚ) / ratFactorial i) id) =
+      (List.range (n + 1)).zipIdx.map (fun p : ℕ × ℕ => ((1 : ℚ) / ratFactorial p.1 : ℝ) * x ^
+        p.2) := by
+    apply List.map_congr_left
+    intro ⟨a, b⟩ _
+    simp only [Function.comp_apply, Prod.map_apply, id_eq, Rat.cast_div, Rat.cast_one]
+  -- Step 2: Use zipIdx_range_map to eliminate zipIdx
+  have h2 : (List.range (n + 1)).zipIdx.map (fun p : ℕ × ℕ => ((1 : ℚ) / ratFactorial p.1 : ℝ) * x
+    ^ p.2) =
+      (List.range (n + 1)).map (fun i => ((1 : ℚ) / ratFactorial i : ℝ) * x ^ i) := by
+    convert! zipIdx_range_map (fun a b => ((1 : ℚ) / ratFactorial a : ℝ) * x ^ b) (n + 1) using 2
+  -- Step 3: Simplify the casts
+  have h3 : (List.range (n + 1)).map (fun i => ((1 : ℚ) / ratFactorial i : ℝ) * x ^ i) =
+      (List.range (n + 1)).map (fun i => (1 / i.factorial : ℝ) * x ^ i) := by
+    apply List.map_congr_left
+    intro i _
+    simp [ratFactorial]
+  rw [h1, h2, h3]
+
+/-- The iterated derivative of sin is sin, cos, -sin, -cos in a cycle of 4. -/
+private lemma iteratedDeriv_sin (n : ℕ) : iteratedDeriv n Real.sin =
+    if n % 4 = 0 then Real.sin
+    else if n % 4 = 1 then Real.cos
+    else if n % 4 = 2 then fun x => -Real.sin x
+    else fun x => -Real.cos x := by
+  induction n with
+  | zero =>
+    simp only [iteratedDeriv_zero, Nat.zero_mod, ↓reduceIte]
+  | succ n ih =>
+    rw [iteratedDeriv_succ, ih]
+    have h4 : n % 4 < 4 := Nat.mod_lt n (by norm_num)
+    rcases (by omega : n % 4 = 0 ∨ n % 4 = 1 ∨ n % 4 = 2 ∨ n % 4 = 3) with h0 | h1 | h2 | h3
+    · -- n % 4 = 0: deriv sin = cos
+      have hn1 : (n + 1) % 4 = 1 := by omega
+      simp only [h0, hn1, ↓reduceIte, Real.deriv_sin]; norm_num
+    · -- n % 4 = 1: deriv cos = -sin
+      have hn1 : (n + 1) % 4 = 2 := by omega
+      simp only [h1, hn1, ↓reduceIte]; norm_num
+    · -- n % 4 = 2: deriv (-sin) = -cos
+      have hn1 : (n + 1) % 4 = 3 := by omega
+      simp only [h2, hn1, ↓reduceIte]; norm_num
+    · -- n % 4 = 3: deriv (-cos) = sin
+      have hn1 : (n + 1) % 4 = 0 := by omega
+      simp only [h3, hn1, ↓reduceIte]; norm_num
+
+/-- The iterated derivative of sin at 0 follows the pattern 0, 1, 0, -1, 0, 1, 0, -1, ... -/
+private lemma iteratedDeriv_sin_zero (i : ℕ) : iteratedDeriv i Real.sin 0 =
+    if i % 4 = 0 then 0
+    else if i % 4 = 1 then 1
+    else if i % 4 = 2 then 0
+    else -1 := by
+  rw [iteratedDeriv_sin]
+  have h4 : i % 4 < 4 := Nat.mod_lt i (by norm_num)
+  rcases (by omega : i % 4 = 0 ∨ i % 4 = 1 ∨ i % 4 = 2 ∨ i % 4 = 3) with h0 | h1 | h2 | h3
+  · simp only [h0, ↓reduceIte, Real.sin_zero]
+  · simp only [h1, ↓reduceIte]; norm_num [Real.cos_zero]
+  · simp only [h2, ↓reduceIte]; norm_num [Real.sin_zero]
+  · simp only [h3]; norm_num [Real.cos_zero]
+
+/-- The sin Taylor polynomial value matches our evalTaylorSeries.
+    Key: iteratedDeriv i sin 0 = 0, 1, 0, -1, 0, 1, ... matches sinTaylorCoeffs. -/
+theorem mem_evalTaylorSeries_sin {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (n : ℕ) :
+    ∑ i ∈ Finset.range (n + 1), (iteratedDeriv i Real.sin 0 / i.factorial) * x ^ i ∈
+      evalTaylorSeries (sinTaylorCoeffs n) I := by
+  have hmem := mem_evalTaylorSeries hx (sinTaylorCoeffs n)
+  convert hmem using 1
+  simp only [sinTaylorCoeffs, ratFactorial]
+  rw [List.zipIdx_map]
+  simp only [List.map_map]
+  rw [← list_map_sum_eq_finset_sum (fun i => (iteratedDeriv i Real.sin 0 / i.factorial) * x ^ i)
+    (n + 1)]
+  congr 1
+  symm
+  -- Step 1: Simplify the RHS using zipIdx_range_map
+  have h1 : (List.range (n + 1)).zipIdx.map
+        ((fun p : ℚ × ℕ => (p.1 : ℝ) * x ^ p.2) ∘ Prod.map
+          (fun i => if i % 2 = 1 then (-1 : ℚ) ^ ((i - 1) / 2) / i.factorial else 0) id) =
+      (List.range (n + 1)).map (fun i =>
+        ((if i % 2 = 1 then (-1 : ℚ) ^ ((i - 1) / 2) / i.factorial else 0 : ℚ) : ℝ) * x ^ i) := by
+    convert! zipIdx_range_map
+      (fun a b => ((if a % 2 = 1 then (-1 : ℚ) ^ ((a - 1) / 2) / a.factorial else 0 : ℚ) : ℝ) * x
+        ^ b)
+      (n + 1) using 2
+  rw [h1]
+  -- Step 2: Show term-by-term equality
+  apply List.map_congr_left
+  intro i _
+  -- Need: (iteratedDeriv i sin 0 / i!) * x^i = ((sinCoeff i) : ℝ) * x^i
+  -- where sinCoeff i = if i % 2 = 1 then (-1)^((i-1)/2) / i! else 0
+  congr 1
+  -- Show iteratedDeriv i sin 0 / i! = sinCoeff i (as ℝ)
+  rw [iteratedDeriv_sin_zero]
+  have h4 : i % 4 < 4 := Nat.mod_lt i (by norm_num)
+  rcases (by omega : i % 4 = 0 ∨ i % 4 = 1 ∨ i % 4 = 2 ∨ i % 4 = 3) with h0 | h1 | h2 | h3
+  · -- i % 4 = 0: i is even, iteratedDeriv = 0
+    have hi_even : i % 2 = 0 := by omega
+    have hi_ne : i % 2 ≠ 1 := by omega
+    simp only [h0, ↓reduceIte, zero_div, ite_eq_right hi_ne, Rat.cast_zero]
+  · -- i % 4 = 1: i is odd, iteratedDeriv = 1, coefficient = (-1)^((i-1)/2) / i!
+    have hi_odd : i % 2 = 1 := by omega
+    simp only [h1, ↓reduceIte, ite_eq_left hi_odd]
+    simp only [Rat.cast_div, Rat.cast_pow, Rat.cast_neg, Rat.cast_one, Rat.cast_natCast]
+    congr 1
+    have heven : Even ((i - 1) / 2) := ⟨(i - 1) / 2 / 2, by omega⟩
+    exact heven.neg_one_pow
+  · -- i % 4 = 2: i is even, iteratedDeriv = 0
+    have hi_even : i % 2 = 0 := by omega
+    have hi_ne : i % 2 ≠ 1 := by omega
+    simp only [h2, ↓reduceIte, ite_eq_right hi_ne]
+    norm_num
+  · -- i % 4 = 3: i is odd, iteratedDeriv = -1, coefficient = (-1)^((i-1)/2) / i!
+    have hi_odd : i % 2 = 1 := by omega
+    simp only [h3, ite_eq_left hi_odd]
+    simp only [Rat.cast_div, Rat.cast_pow, Rat.cast_neg, Rat.cast_one, Rat.cast_natCast]
+    have hodd : Odd ((i - 1) / 2) := ⟨(i - 1) / 2 / 2, by omega⟩
+    rw [hodd.neg_one_pow]
+    norm_num
+
+/-- The iterated derivative of cos is cos, -sin, -cos, sin in a cycle of 4. -/
+private lemma iteratedDeriv_cos (n : ℕ) : iteratedDeriv n Real.cos =
+    if n % 4 = 0 then Real.cos
+    else if n % 4 = 1 then fun x => -Real.sin x
+    else if n % 4 = 2 then fun x => -Real.cos x
+    else Real.sin := by
+  induction n with
+  | zero =>
+    simp only [iteratedDeriv_zero, Nat.zero_mod, ↓reduceIte]
+  | succ n ih =>
+    rw [iteratedDeriv_succ, ih]
+    have h4 : n % 4 < 4 := Nat.mod_lt n (by norm_num)
+    rcases (by omega : n % 4 = 0 ∨ n % 4 = 1 ∨ n % 4 = 2 ∨ n % 4 = 3) with h0 | h1 | h2 | h3
+    · -- n % 4 = 0: deriv cos = -sin
+      have hn1 : (n + 1) % 4 = 1 := by omega
+      simp only [h0, hn1, ↓reduceIte]; norm_num
+    · -- n % 4 = 1: deriv (-sin) = -cos
+      have hn1 : (n + 1) % 4 = 2 := by omega
+      simp only [h1, hn1, ↓reduceIte]; norm_num
+    · -- n % 4 = 2: deriv (-cos) = sin
+      have hn1 : (n + 1) % 4 = 3 := by omega
+      simp only [h2, hn1, ↓reduceIte]; norm_num
+    · -- n % 4 = 3: deriv sin = cos
+      have hn1 : (n + 1) % 4 = 0 := by omega
+      simp only [h3, hn1, ↓reduceIte]; norm_num
+
+/-- The iterated derivative of cos at 0 follows the pattern 1, 0, -1, 0, 1, 0, ... -/
+private lemma iteratedDeriv_cos_zero (i : ℕ) : iteratedDeriv i Real.cos 0 =
+    if i % 4 = 0 then 1
+    else if i % 4 = 1 then 0
+    else if i % 4 = 2 then -1
+    else 0 := by
+  rw [iteratedDeriv_cos]
+  have h4 : i % 4 < 4 := Nat.mod_lt i (by norm_num)
+  rcases (by omega : i % 4 = 0 ∨ i % 4 = 1 ∨ i % 4 = 2 ∨ i % 4 = 3) with h0 | h1 | h2 | h3
+  · simp only [h0, ↓reduceIte, Real.cos_zero]
+  · simp only [h1, ↓reduceIte]; norm_num [Real.sin_zero]
+  · simp only [h2, ↓reduceIte]; norm_num [Real.cos_zero]
+  · simp only [h3]; norm_num [Real.sin_zero]
+
+/-- The cos Taylor polynomial value matches our evalTaylorSeries.
+    Key: iteratedDeriv i cos 0 = 1, 0, -1, 0, 1, 0, ... matches cosTaylorCoeffs. -/
+theorem mem_evalTaylorSeries_cos {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (n : ℕ) :
+    ∑ i ∈ Finset.range (n + 1), (iteratedDeriv i Real.cos 0 / i.factorial) * x ^ i ∈
+      evalTaylorSeries (cosTaylorCoeffs n) I := by
+  have hmem := mem_evalTaylorSeries hx (cosTaylorCoeffs n)
+  convert hmem using 1
+  simp only [cosTaylorCoeffs, ratFactorial]
+  rw [List.zipIdx_map]
+  simp only [List.map_map]
+  rw [← list_map_sum_eq_finset_sum (fun i => (iteratedDeriv i Real.cos 0 / i.factorial) * x ^ i)
+    (n + 1)]
+  congr 1
+  symm
+  -- Step 1: Simplify the RHS using zipIdx_range_map
+  have h1 : (List.range (n + 1)).zipIdx.map
+        ((fun p : ℚ × ℕ => (p.1 : ℝ) * x ^ p.2) ∘ Prod.map
+          (fun i => if i % 2 = 0 then (-1 : ℚ) ^ (i / 2) / i.factorial else 0) id) =
+      (List.range (n + 1)).map (fun i =>
+        ((if i % 2 = 0 then (-1 : ℚ) ^ (i / 2) / i.factorial else 0 : ℚ) : ℝ) * x ^ i) := by
+    convert! zipIdx_range_map
+      (fun a b => ((if a % 2 = 0 then (-1 : ℚ) ^ (a / 2) / a.factorial else 0 : ℚ) : ℝ) * x ^ b)
+      (n + 1) using 2
+  rw [h1]
+  -- Step 2: Show term-by-term equality
+  apply List.map_congr_left
+  intro i _
+  congr 1
+  -- Show iteratedDeriv i cos 0 / i! = cosCoeff i (as ℝ)
+  rw [iteratedDeriv_cos_zero]
+  have h4 : i % 4 < 4 := Nat.mod_lt i (by norm_num)
+  rcases (by omega : i % 4 = 0 ∨ i % 4 = 1 ∨ i % 4 = 2 ∨ i % 4 = 3) with h0 | h1 | h2 | h3
+  · -- i % 4 = 0: i is even, iteratedDeriv = 1, coefficient = (-1)^(i/2) / i!
+    have hi_even : i % 2 = 0 := by omega
+    simp only [h0, ↓reduceIte, one_div, ite_eq_left hi_even]
+    simp only [Rat.cast_div, Rat.cast_pow, Rat.cast_neg, Rat.cast_one, Rat.cast_natCast]
+    have heven : Even (i / 2) := ⟨i / 2 / 2, by omega⟩
+    rw [heven.neg_one_pow]
+    ring
+  · -- i % 4 = 1: i is odd, iteratedDeriv = 0
+    have hi_odd : i % 2 = 1 := by omega
+    have hi_ne : i % 2 ≠ 0 := by omega
+    simp only [h1, ↓reduceIte, ite_eq_right hi_ne]
+    norm_num
+  · -- i % 4 = 2: i is even, iteratedDeriv = -1, coefficient = (-1)^(i/2) / i!
+    have hi_even : i % 2 = 0 := by omega
+    simp only [h2, ite_eq_left hi_even]
+    simp only [Rat.cast_div, Rat.cast_pow, Rat.cast_neg, Rat.cast_one, Rat.cast_natCast]
+    have hodd : Odd (i / 2) := ⟨i / 2 / 2, by omega⟩
+    rw [hodd.neg_one_pow]
+    norm_num
+  · -- i % 4 = 3: i is odd, iteratedDeriv = 0
+    have hi_odd : i % 2 = 1 := by omega
+    have hi_ne : i % 2 ≠ 0 := by omega
+    simp only [h3, ite_eq_right hi_ne]
+    norm_num
+
+/-! ### Taylor remainder micro-lemmas -/
+
+/-- Unified Taylor remainder bound for exp: given x ∈ I with r = maxAbs I,
+    the Taylor remainder |exp x - poly(x)| ≤ 3^(⌈r⌉+1) * r^(n+1) / (n+1)!.
+    This encapsulates the domain setup and remainder calculation. -/
+theorem exp_taylor_remainder_in_interval {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (n : ℕ) :
+    Real.exp x - ∑ i ∈ Finset.range (n + 1), (iteratedDeriv i Real.exp 0 / i.factorial) * x ^ i
+    ∈ expRemainderBoundComputable I n := by
+  -- Extract domain info
+  have ⟨hr_nonneg, habs_x, hdom, h0a, h0b, hab⟩ := domain_from_mem hx
+  set r := maxAbs I
+  set R := ((3 : ℚ) ^ (Nat.ceil r + 1) * r ^ (n + 1) / ratFactorial (n + 1))
+  -- Apply Taylor theorem
+  have hexp_smooth : ContDiff ℝ (n + 1) Real.exp := Real.contDiff_exp.of_le le_top
+  have hderiv_bound : ∀ y ∈ Set.Icc ((-r : ℚ) : ℝ) (r : ℚ),
+      ‖iteratedDeriv (n + 1) Real.exp y‖ ≤ Real.exp r := by
+    intro y hy
+    rw [iteratedDeriv_eq_iterate, Real.iter_deriv_exp, Real.norm_eq_abs, abs_of_pos (Real.exp_pos
+      y)]
+    exact Real.exp_le_exp.mpr hy.2
+  have hTaylor := taylor_remainder_bound hab h0a h0b hexp_smooth hderiv_bound
+    (Real.exp_pos (r : ℚ)).le x hdom
+  -- Compute remainder bound
+  have hr_nonneg_rat : 0 ≤ r := le_max_of_le_left (abs_nonneg I.lo)
+  have hexp_r_bound : Real.exp (r : ℚ) ≤ (3 : ℝ) ^ (Nat.ceil r + 1) := by
+    apply exp_bound_by_pow3 hr_nonneg_rat
+    rw [abs_of_nonneg hr_nonneg]
+  have hx_pow_bound : |x| ^ (n + 1) ≤ (r : ℝ) ^ (n + 1) :=
+    pow_le_pow_left₀ (abs_nonneg x) habs_x _
+  have hfact_pos : (0 : ℝ) < (n + 1).factorial := Nat.cast_pos.mpr (Nat.factorial_pos _)
+  have hrem_bound : Real.exp (r : ℚ) * |x - 0| ^ (n + 1) / (n + 1).factorial ≤ (R : ℝ) := by
+    simp only [sub_zero]
+    calc Real.exp (r : ℚ) * |x| ^ (n + 1) / (n + 1).factorial
+        ≤ (3 : ℝ) ^ (Nat.ceil r + 1) * (r : ℝ) ^ (n + 1) / (n + 1).factorial := by
+          apply div_le_div_of_nonneg_right _ hfact_pos.le
+          apply mul_le_mul hexp_r_bound hx_pow_bound (pow_nonneg (abs_nonneg x) _)
+          apply pow_nonneg; norm_num
+      _ = (R : ℝ) := by
+          simp only [R, ratFactorial, Rat.cast_div, Rat.cast_mul, Rat.cast_pow,
+            Rat.cast_natCast, Rat.cast_ofNat]
+  -- Convert to interval membership
+  simp only [expRemainderBoundComputable, mem_def, ratFactorial]
+  have h := hTaylor
+  simp only [sub_zero] at h hrem_bound
+  rw [Real.norm_eq_abs] at h
+  have hbound : |Real.exp x - ∑ i ∈ Finset.range (n + 1),
+      (iteratedDeriv i Real.exp 0 / i.factorial) * x ^ i| ≤ (R : ℝ) :=
+    le_trans h hrem_bound
+  have habs := abs_le.mp hbound
+  simp only [R, Rat.cast_div, Rat.cast_mul, Rat.cast_pow, Rat.cast_natCast, Rat.cast_ofNat,
+    Rat.cast_neg] at habs ⊢
+  exact habs
+
+/-- Unified Taylor remainder bound for sin: given x ∈ I with r = maxAbs I,
+    the Taylor remainder |sin x - poly(x)| ≤ r^(n+1) / (n+1)!.
+    Uses the fact that |sin^(k)(x)| ≤ 1 for all k, x. -/
+theorem sin_taylor_remainder_in_interval {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (n : ℕ) :
+    Real.sin x - ∑ i ∈ Finset.range (n + 1), (iteratedDeriv i Real.sin 0 / i.factorial) * x ^ i
+    ∈ sinRemainderBoundComputable I n := by
+  -- Extract domain info
+  have ⟨hr_nonneg, habs_x, hdom, h0a, h0b, hab⟩ := domain_from_mem hx
+  set r := maxAbs I
+  set R := (r ^ (n + 1) / ratFactorial (n + 1))
+  -- Apply Taylor theorem with M = 1
+  have hsin_smooth : ContDiff ℝ (n + 1) Real.sin := Real.contDiff_sin.of_le le_top
+  have hderiv_bound : ∀ y ∈ Set.Icc ((-r : ℚ) : ℝ) (r : ℚ),
+      ‖iteratedDeriv (n + 1) Real.sin y‖ ≤ 1 := by
+    intro y _; exact (sin_cos_deriv_bound (n + 1) y).1
+  have hTaylor := taylor_remainder_bound hab h0a h0b hsin_smooth hderiv_bound
+    (by norm_num : (0 : ℝ) ≤ 1) x hdom
+  -- Compute remainder bound
+  have hx_pow_bound : |x| ^ (n + 1) ≤ (r : ℝ) ^ (n + 1) :=
+    pow_le_pow_left₀ (abs_nonneg x) habs_x _
+  have hfact_pos : (0 : ℝ) < (n + 1).factorial := Nat.cast_pos.mpr (Nat.factorial_pos _)
+  have hrem_bound : 1 * |x - 0| ^ (n + 1) / (n + 1).factorial ≤ (R : ℝ) := by
+    simp only [sub_zero, one_mul]
+    calc |x| ^ (n + 1) / (n + 1).factorial
+        ≤ (r : ℝ) ^ (n + 1) / (n + 1).factorial :=
+          div_le_div_of_nonneg_right hx_pow_bound hfact_pos.le
+      _ = (R : ℝ) := by simp only [R, ratFactorial, Rat.cast_div, Rat.cast_pow, Rat.cast_natCast]
+  -- Convert to interval membership
+  simp only [sinRemainderBoundComputable, mem_def, ratFactorial]
+  have h := hTaylor
+  simp only [sub_zero, one_mul] at h hrem_bound
+  rw [Real.norm_eq_abs] at h
+  have hbound : |Real.sin x - ∑ i ∈ Finset.range (n + 1),
+      (iteratedDeriv i Real.sin 0 / i.factorial) * x ^ i| ≤ (R : ℝ) :=
+    le_trans h hrem_bound
+  have habs := abs_le.mp hbound
+  simp only [R, Rat.cast_div, Rat.cast_pow, Rat.cast_natCast, Rat.cast_neg] at habs ⊢
+  exact habs
+
+/-- Unified Taylor remainder bound for cos: given x ∈ I with r = maxAbs I,
+    the Taylor remainder |cos x - poly(x)| ≤ r^(n+1) / (n+1)!.
+    Uses the fact that |cos^(k)(x)| ≤ 1 for all k, x. -/
+theorem cos_taylor_remainder_in_interval {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (n : ℕ) :
+    Real.cos x - ∑ i ∈ Finset.range (n + 1), (iteratedDeriv i Real.cos 0 / i.factorial) * x ^ i
+    ∈ cosRemainderBoundComputable I n := by
+  -- Extract domain info
+  have ⟨hr_nonneg, habs_x, hdom, h0a, h0b, hab⟩ := domain_from_mem hx
+  set r := maxAbs I
+  set R := (r ^ (n + 1) / ratFactorial (n + 1))
+  -- Apply Taylor theorem with M = 1
+  have hcos_smooth : ContDiff ℝ (n + 1) Real.cos := Real.contDiff_cos.of_le le_top
+  have hderiv_bound : ∀ y ∈ Set.Icc ((-r : ℚ) : ℝ) (r : ℚ),
+      ‖iteratedDeriv (n + 1) Real.cos y‖ ≤ 1 := by
+    intro y _; exact (sin_cos_deriv_bound (n + 1) y).2
+  have hTaylor := taylor_remainder_bound hab h0a h0b hcos_smooth hderiv_bound
+    (by norm_num : (0 : ℝ) ≤ 1) x hdom
+  -- Compute remainder bound
+  have hx_pow_bound : |x| ^ (n + 1) ≤ (r : ℝ) ^ (n + 1) :=
+    pow_le_pow_left₀ (abs_nonneg x) habs_x _
+  have hfact_pos : (0 : ℝ) < (n + 1).factorial := Nat.cast_pos.mpr (Nat.factorial_pos _)
+  have hrem_bound : 1 * |x - 0| ^ (n + 1) / (n + 1).factorial ≤ (R : ℝ) := by
+    simp only [sub_zero, one_mul]
+    calc |x| ^ (n + 1) / (n + 1).factorial
+        ≤ (r : ℝ) ^ (n + 1) / (n + 1).factorial :=
+          div_le_div_of_nonneg_right hx_pow_bound hfact_pos.le
+      _ = (R : ℝ) := by simp only [R, ratFactorial, Rat.cast_div, Rat.cast_pow, Rat.cast_natCast]
+  -- Convert to interval membership
+  simp only [cosRemainderBoundComputable, mem_def, ratFactorial]
+  have h := hTaylor
+  simp only [sub_zero, one_mul] at h hrem_bound
+  rw [Real.norm_eq_abs] at h
+  have hbound : |Real.cos x - ∑ i ∈ Finset.range (n + 1),
+      (iteratedDeriv i Real.cos 0 / i.factorial) * x ^ i| ≤ (R : ℝ) :=
+    le_trans h hrem_bound
+  have habs := abs_le.mp hbound
+  simp only [R, Rat.cast_div, Rat.cast_pow, Rat.cast_natCast, Rat.cast_neg] at habs ⊢
+  exact habs
+
+/-! ### FTIA for computable functions -/
+
+private theorem mem_expPointComputableRaw (q : ℚ) (n : ℕ) :
+    Real.exp q ∈ expPointComputableRaw q n := by
+  simp only [expPointComputableRaw]
+  have hq_mem : (q : ℝ) ∈ singleton q := mem_singleton q
+  -- Strategy: exp q = poly(q) + remainder, with both in their respective intervals
+  have hpoly_mem : ∑ i ∈ Finset.range (n + 1), (iteratedDeriv i Real.exp 0 / i.factorial) * (q :
+    ℝ) ^ i
+      ∈ evalTaylorSeries (expTaylorCoeffs n) (singleton q) := by
+    have hsum_eq : ∑ i ∈ Finset.range (n + 1), (iteratedDeriv i Real.exp 0 / i.factorial) * (q :
+      ℝ) ^ i =
+        ∑ i ∈ Finset.range (n + 1), (1 / i.factorial : ℝ) * (q : ℝ) ^ i := by
+      apply Finset.sum_congr rfl; intro i _; rw [iteratedDeriv_exp_zero, one_div]
+    rw [hsum_eq]; exact mem_evalTaylorSeries_exp hq_mem n
+  have hrem_mem := exp_taylor_remainder_in_interval hq_mem n
+  have heq : Real.exp q = (∑ i ∈ Finset.range (n + 1),
+      (iteratedDeriv i Real.exp 0 / i.factorial) * (q : ℝ) ^ i) +
+      (Real.exp q - ∑ i ∈ Finset.range (n + 1),
+        (iteratedDeriv i Real.exp 0 / i.factorial) * (q : ℝ) ^ i) := by ring
+  rw [heq]; exact mem_add hpoly_mem hrem_mem
+
+/-- FTIA for single-point exp: Real.exp q ∈ expPointComputable q n -/
+theorem mem_expPointComputable (q : ℚ) (n : ℕ) :
+    Real.exp q ∈ expPointComputable q n := by
+  simp only [expPointComputable]
+  set k := expReduceK q
+  set m : ℕ := (2:ℕ)^k
+  set q' : ℚ := q / m
+  have hbase : Real.exp q' ∈ expPointComputableRaw q' n :=
+    mem_expPointComputableRaw q' n
+  have hpow : (Real.exp q') ^ m ∈ pow (expPointComputableRaw q' n) m :=
+    mem_pow hbase m
+  have hm : (m:ℝ) ≠ 0 := by
+    -- m = 2^k > 0
+    exact_mod_cast (pow_ne_zero k (by decide : (2:ℕ) ≠ 0))
+  have hq : (q : ℝ) = (m:ℝ) * (q' : ℝ) := by
+    -- q' = q / m
+    have hm' : (m:ℝ) ≠ 0 := hm
+    calc
+      (q:ℝ) = (m:ℝ) * ((q:ℝ) / (m:ℝ)) := by
+        field_simp [hm']
+      _ = (m:ℝ) * (q' : ℝ) := by
+        simp [q', Rat.cast_div, Rat.cast_natCast]
+  have hexp : Real.exp q = (Real.exp q') ^ m := by
+    calc
+      Real.exp q = Real.exp ((m:ℝ) * (q' : ℝ)) := by simp [hq]
+      _ = (Real.exp (q' : ℝ)) ^ m := by
+            -- exp (m * q') = (exp q')^m
+            simpa [mul_comm] using (Real.exp_nat_mul (q' : ℝ) m)
+  -- rewrite and apply membership
+  simpa [hexp]
+
+theorem mem_expComputable {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (n : ℕ) :
+    Real.exp x ∈ expComputable I n := by
+  simp only [expComputable]
+  split_ifs with h
+  · -- Interval doesn't cross 0: use endpoint evaluation and monotonicity
+    -- exp is monotone increasing, so exp([lo, hi]) ⊆ hull(exp(lo), exp(hi))
+    have hlo_mem := mem_expPointComputable I.lo n
+    have hhi_mem := mem_expPointComputable I.hi n
+    -- Since x ∈ [lo, hi] and exp is monotone, exp(x) ∈ [exp(lo), exp(hi)]
+    -- The hull contains both exp(lo) and exp(hi), so it contains exp(x)
+    rcases h with ⟨hhi_neg⟩ | ⟨hlo_pos⟩
+    · -- Case: hi ≤ 0 (negative interval)
+      -- exp(lo) ≤ exp(x) ≤ exp(hi) by monotonicity
+      have hx_le_hi : x ≤ I.hi := hx.2
+      have hlo_le_x : (I.lo : ℝ) ≤ x := hx.1
+      have hexp_mono1 : Real.exp x ≤ Real.exp I.hi := Real.exp_le_exp.mpr hx_le_hi
+      have hexp_mono2 : Real.exp I.lo ≤ Real.exp x := Real.exp_le_exp.mpr hlo_le_x
+      -- exp(x) is between exp(lo) and exp(hi), both of which are in the hull
+      simp only [hull, mem_def, Rat.cast_min, Rat.cast_max]
+      constructor
+      · -- lower bound: min(expLo.lo, expHi.lo) ≤ exp(x)
+        calc (min (expPointComputable I.lo n).lo (expPointComputable I.hi n).lo : ℝ)
+            ≤ (expPointComputable I.lo n).lo := by exact_mod_cast min_le_left _ _
+          _ ≤ Real.exp I.lo := hlo_mem.1
+          _ ≤ Real.exp x := hexp_mono2
+      · -- upper bound: exp(x) ≤ max(expLo.hi, expHi.hi)
+        calc Real.exp x ≤ Real.exp I.hi := hexp_mono1
+          _ ≤ (expPointComputable I.hi n).hi := hhi_mem.2
+          _ ≤ max ((expPointComputable I.lo n).hi : ℝ) ((expPointComputable I.hi n).hi : ℝ) :=
+            le_max_right _ _
+    · -- Case: 0 ≤ lo (positive interval) - same argument
+      have hx_le_hi : x ≤ I.hi := hx.2
+      have hlo_le_x : (I.lo : ℝ) ≤ x := hx.1
+      have hexp_mono1 : Real.exp x ≤ Real.exp I.hi := Real.exp_le_exp.mpr hx_le_hi
+      have hexp_mono2 : Real.exp I.lo ≤ Real.exp x := Real.exp_le_exp.mpr hlo_le_x
+      simp only [hull, mem_def, Rat.cast_min, Rat.cast_max]
+      constructor
+      · calc (min (expPointComputable I.lo n).lo (expPointComputable I.hi n).lo : ℝ)
+            ≤ (expPointComputable I.lo n).lo := by exact_mod_cast min_le_left _ _
+          _ ≤ Real.exp I.lo := hlo_mem.1
+          _ ≤ Real.exp x := hexp_mono2
+      · calc Real.exp x ≤ Real.exp I.hi := hexp_mono1
+          _ ≤ (expPointComputable I.hi n).hi := hhi_mem.2
+          _ ≤ max ((expPointComputable I.lo n).hi : ℝ) ((expPointComputable I.hi n).hi : ℝ) :=
+            le_max_right _ _
+  · -- Interval crosses 0: use standard Taylor
+    -- Strategy: exp x = poly(x) + remainder, with both in their respective intervals
+    have hpoly_mem : ∑ i ∈ Finset.range (n + 1), (iteratedDeriv i Real.exp 0 / i.factorial) * x ^ i
+        ∈ evalTaylorSeries (expTaylorCoeffs n) I := by
+      have hsum_eq : ∑ i ∈ Finset.range (n + 1), (iteratedDeriv i Real.exp 0 / i.factorial) * x ^
+        i =
+          ∑ i ∈ Finset.range (n + 1), (1 / i.factorial : ℝ) * x ^ i := by
+        apply Finset.sum_congr rfl; intro i _; rw [iteratedDeriv_exp_zero, one_div]
+      rw [hsum_eq]; exact mem_evalTaylorSeries_exp hx n
+    have hrem_mem := exp_taylor_remainder_in_interval hx n
+    have heq : Real.exp x = (∑ i ∈ Finset.range (n + 1),
+        (iteratedDeriv i Real.exp 0 / i.factorial) * x ^ i) +
+        (Real.exp x - ∑ i ∈ Finset.range (n + 1),
+          (iteratedDeriv i Real.exp 0 / i.factorial) * x ^ i) := by ring
+    rw [heq]; exact mem_add hpoly_mem hrem_mem
+
+/-- FTIA for sinComputable: Real.sin x ∈ sinComputable I n for any x ∈ I.
+
+    The proof uses the Taylor remainder micro-lemma and the global bound sin ∈ [-1, 1]. -/
+theorem mem_sinComputable {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (n : ℕ) :
+    Real.sin x ∈ sinComputable I n := by
+  simp only [sinComputable]
+  -- Strategy: sin x = poly(x) + remainder, intersected with global bound [-1, 1]
+
+  -- Polynomial part ∈ evalTaylorSeries
+  have hpoly_mem : ∑ i ∈ Finset.range (n + 1), (iteratedDeriv i Real.sin 0 / i.factorial) * x ^ i
+      ∈ evalTaylorSeries (sinTaylorCoeffs n) I := mem_evalTaylorSeries_sin hx n
+  -- Remainder part ∈ sinRemainderBoundComputable (via micro-lemma)
+  have hrem_mem := sin_taylor_remainder_in_interval hx n
+  -- Raw interval membership: sin x ∈ poly + remainder
+  have hraw_mem : Real.sin x ∈ add (evalTaylorSeries (sinTaylorCoeffs n) I)
+      (sinRemainderBoundComputable I n) := by
+    have heq : Real.sin x = (∑ i ∈ Finset.range (n + 1),
+        (iteratedDeriv i Real.sin 0 / i.factorial) * x ^ i) +
+        (Real.sin x - ∑ i ∈ Finset.range (n + 1),
+          (iteratedDeriv i Real.sin 0 / i.factorial) * x ^ i) := by ring
+    rw [heq]; exact mem_add hpoly_mem hrem_mem
+  -- Global bound: sin x ∈ [-1, 1]
+  have hglobal_mem : Real.sin x ∈ (⟨-1, 1, by norm_num⟩ : IntervalRat) := by
+    simp only [mem_def]; constructor
+    · simp only [Rat.cast_neg, Rat.cast_one]; exact Real.neg_one_le_sin x
+    · simp only [Rat.cast_one]; exact Real.sin_le_one x
+  -- Intersect and conclude
+  have ⟨K, hK_eq, hK_mem⟩ := mem_intersect hraw_mem hglobal_mem
+  simp only [hK_eq]; exact hK_mem
+
+/-- FTIA for cosComputable: Real.cos x ∈ cosComputable I n for any x ∈ I.
+
+    The proof uses the Taylor remainder micro-lemma and the global bound cos ∈ [-1, 1]. -/
+theorem mem_cosComputable {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (n : ℕ) :
+    Real.cos x ∈ cosComputable I n := by
+  simp only [cosComputable]
+  -- Strategy: cos x = poly(x) + remainder, intersected with global bound [-1, 1]
+
+  -- Polynomial part ∈ evalTaylorSeries
+  have hpoly_mem : ∑ i ∈ Finset.range (n + 1), (iteratedDeriv i Real.cos 0 / i.factorial) * x ^ i
+      ∈ evalTaylorSeries (cosTaylorCoeffs n) I := mem_evalTaylorSeries_cos hx n
+  -- Remainder part ∈ cosRemainderBoundComputable (via micro-lemma)
+  have hrem_mem := cos_taylor_remainder_in_interval hx n
+  -- Raw interval membership: cos x ∈ poly + remainder
+  have hraw_mem : Real.cos x ∈ add (evalTaylorSeries (cosTaylorCoeffs n) I)
+      (cosRemainderBoundComputable I n) := by
+    have heq : Real.cos x = (∑ i ∈ Finset.range (n + 1),
+        (iteratedDeriv i Real.cos 0 / i.factorial) * x ^ i) +
+        (Real.cos x - ∑ i ∈ Finset.range (n + 1),
+          (iteratedDeriv i Real.cos 0 / i.factorial) * x ^ i) := by ring
+    rw [heq]; exact mem_add hpoly_mem hrem_mem
+  -- Global bound: cos x ∈ [-1, 1]
+  have hglobal_mem : Real.cos x ∈ (⟨-1, 1, by norm_num⟩ : IntervalRat) := by
+    simp only [mem_def]; constructor
+    · simp only [Rat.cast_neg, Rat.cast_one]; exact Real.neg_one_le_cos x
+    · simp only [Rat.cast_one]; exact Real.cos_le_one x
+  -- Intersect and conclude
+  have ⟨K, hK_eq, hK_mem⟩ := mem_intersect hraw_mem hglobal_mem
+  simp only [hK_eq]; exact hK_mem
+
+/-- FTIA for sinhPointComputable: Real.sinh q ∈ sinhPointComputable q n -/
+theorem mem_sinhPointComputable (q : ℚ) (n : ℕ) :
+    Real.sinh q ∈ sinhPointComputable q n := by
+  simp only [sinhPointComputable]
+  have hexp_pos := mem_expPointComputable q n
+  have hexp_neg := mem_expPointComputable (-q) n
+  rw [Real.sinh_eq]
+  simp only [Rat.cast_neg] at hexp_neg
+  simp only [mem_def] at hexp_pos hexp_neg ⊢
+  obtain ⟨hexp_pos_lo, hexp_pos_hi⟩ := hexp_pos
+  obtain ⟨hexp_neg_lo, hexp_neg_hi⟩ := hexp_neg
+  split_ifs with h
+  · constructor <;> { simp only [Rat.cast_div, Rat.cast_sub, Rat.cast_ofNat]; linarith }
+  · -- Fallback case: use min/max bounds
+    constructor
+    · simp only [Rat.cast_min, Rat.cast_div, Rat.cast_sub, Rat.cast_ofNat]
+      apply min_le_of_left_le; linarith
+    · simp only [Rat.cast_max, Rat.cast_div, Rat.cast_sub, Rat.cast_ofNat]
+      apply le_max_of_le_right; linarith
+
+/-- FTIA for coshPointComputable: Real.cosh q ∈ coshPointComputable q n -/
+theorem mem_coshPointComputable (q : ℚ) (n : ℕ) :
+    Real.cosh q ∈ coshPointComputable q n := by
+  simp only [coshPointComputable]
+  have hexp_pos := mem_expPointComputable q n
+  have hexp_neg := mem_expPointComputable (-q) n
+  rw [Real.cosh_eq]
+  simp only [Rat.cast_neg] at hexp_neg
+  simp only [mem_def] at hexp_pos hexp_neg ⊢
+  obtain ⟨hexp_pos_lo, hexp_pos_hi⟩ := hexp_pos
+  obtain ⟨hexp_neg_lo, hexp_neg_hi⟩ := hexp_neg
+  -- cosh q ≥ 1 always (AM-GM)
+  have hcosh_ge_one : 1 ≤ (Real.exp q + Real.exp (-(q : ℝ))) / 2 := by
+    have h1 : Real.exp q > 0 := Real.exp_pos q
+    have h2 : Real.exp (-(q : ℝ)) > 0 := Real.exp_pos (-(q : ℝ))
+    have hprod : Real.exp q * Real.exp (-(q : ℝ)) = 1 := by
+      rw [← Real.exp_add, add_neg_cancel, Real.exp_zero]
+    have ham : Real.exp q + Real.exp (-(q : ℝ)) ≥ 2 := by nlinarith [sq_nonneg (Real.exp q -
+      Real.exp (-(q : ℝ))), hprod]
+    linarith
+  split_ifs with h
+  · constructor
+    · -- Lower bound: max 1 coshLo ≤ cosh q
+      simp only [Rat.cast_max, Rat.cast_one, Rat.cast_div, Rat.cast_add, Rat.cast_ofNat]
+      apply max_le
+      · exact hcosh_ge_one
+      · linarith
+    · -- Upper bound
+      simp only [Rat.cast_div, Rat.cast_add, Rat.cast_ofNat]
+      linarith
+  · -- Fallback
+    constructor
+    · simp only [Rat.cast_one]
+      exact hcosh_ge_one
+    · simp only [Rat.cast_max, Rat.cast_ofNat, Rat.cast_div, Rat.cast_add]
+      apply le_max_of_le_right
+      linarith
+
+/-- FTIA for sinhComputable: Real.sinh x ∈ sinhComputable I n for any x ∈ I.
+
+    Uses endpoint evaluation and monotonicity of sinh. -/
+theorem mem_sinhComputable {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (n : ℕ) :
+    Real.sinh x ∈ sinhComputable I n := by
+  simp only [sinhComputable]
+  -- sinh is strictly monotone increasing
+  have hsinh_mono : StrictMono Real.sinh := Real.sinh_strictMono
+  have hlo_mem := mem_sinhPointComputable I.lo n
+  have hhi_mem := mem_sinhPointComputable I.hi n
+  -- x ∈ [lo, hi] implies sinh(lo) ≤ sinh(x) ≤ sinh(hi)
+  have hlo_le_x : (I.lo : ℝ) ≤ x := hx.1
+  have hx_le_hi : x ≤ (I.hi : ℝ) := hx.2
+  have hsinh_lo_le : Real.sinh I.lo ≤ Real.sinh x :=
+    hsinh_mono.monotone hlo_le_x
+  have hsinh_x_le_hi : Real.sinh x ≤ Real.sinh I.hi :=
+    hsinh_mono.monotone hx_le_hi
+  -- sinh x is between sinh(lo) and sinh(hi), which are in the hull
+  simp only [hull, mem_def, Rat.cast_min, Rat.cast_max]
+  constructor
+  · calc (min (sinhPointComputable I.lo n).lo (sinhPointComputable I.hi n).lo : ℝ)
+        ≤ (sinhPointComputable I.lo n).lo := by exact_mod_cast min_le_left _ _
+      _ ≤ Real.sinh I.lo := hlo_mem.1
+      _ ≤ Real.sinh x := hsinh_lo_le
+  · calc Real.sinh x ≤ Real.sinh I.hi := hsinh_x_le_hi
+      _ ≤ (sinhPointComputable I.hi n).hi := hhi_mem.2
+      _ ≤ max ((sinhPointComputable I.lo n).hi : ℝ) ((sinhPointComputable I.hi n).hi : ℝ) :=
+        le_max_right _ _
+
+/-- FTIA for coshComputable: Real.cosh x ∈ coshComputable I n for any x ∈ I.
+
+    Uses endpoint evaluation and monotonicity properties of cosh. -/
+theorem mem_coshComputable {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (n : ℕ) :
+    Real.cosh x ∈ coshComputable I n := by
+  simp only [coshComputable]
+  have hlo_mem := mem_coshPointComputable I.lo n
+  have hhi_mem := mem_coshPointComputable I.hi n
+  -- cosh x ≥ 1 always (AM-GM)
+  have hcosh_ge_one : 1 ≤ Real.cosh x := Real.one_le_cosh x
+  -- Key lemma: cosh a ≤ cosh b iff |a| ≤ |b|
+  split_ifs with h1 h2
+  · -- Case: 0 ≤ I.lo (non-negative interval, cosh is increasing)
+    have hlo_nonneg : (0 : ℝ) ≤ I.lo := by exact_mod_cast h1
+    have hx_nonneg : 0 ≤ x := le_trans hlo_nonneg hx.1
+    have hhi_nonneg : (0 : ℝ) ≤ I.hi := le_trans hlo_nonneg (by exact_mod_cast I.le)
+    -- For 0 ≤ a ≤ b: |a| = a ≤ b = |b|, so cosh(a) ≤ cosh(b)
+    have hcosh_lo_le : Real.cosh I.lo ≤ Real.cosh x := by
+      rw [Real.cosh_le_cosh]
+      rw [abs_of_nonneg hlo_nonneg, abs_of_nonneg hx_nonneg]
+      exact hx.1
+    have hcosh_x_le_hi : Real.cosh x ≤ Real.cosh I.hi := by
+      rw [Real.cosh_le_cosh]
+      rw [abs_of_nonneg hx_nonneg, abs_of_nonneg hhi_nonneg]
+      exact hx.2
+    simp only [hull, mem_def, Rat.cast_min, Rat.cast_max]
+    constructor
+    · calc (min (coshPointComputable I.lo n).lo (coshPointComputable I.hi n).lo : ℝ)
+          ≤ (coshPointComputable I.lo n).lo := by exact_mod_cast min_le_left _ _
+        _ ≤ Real.cosh I.lo := hlo_mem.1
+        _ ≤ Real.cosh x := hcosh_lo_le
+    · calc Real.cosh x ≤ Real.cosh I.hi := hcosh_x_le_hi
+        _ ≤ (coshPointComputable I.hi n).hi := hhi_mem.2
+        _ ≤ max ((coshPointComputable I.lo n).hi : ℝ) ((coshPointComputable I.hi n).hi : ℝ) :=
+          le_max_right _ _
+  · -- Case: I.hi ≤ 0 (non-positive interval, cosh is decreasing)
+    have hhi_nonpos : I.hi ≤ (0 : ℝ) := by exact_mod_cast h2
+    have hx_nonpos : x ≤ 0 := le_trans hx.2 hhi_nonpos
+    have hlo_nonpos : (I.lo : ℝ) ≤ 0 := le_trans (by exact_mod_cast I.le) hhi_nonpos
+    -- For a ≤ b ≤ 0: |a| = -a ≥ -b = |b|, so cosh(a) ≥ cosh(b)
+    have hcosh_hi_le : Real.cosh I.hi ≤ Real.cosh x := by
+      rw [Real.cosh_le_cosh]
+      rw [abs_of_nonpos hhi_nonpos, abs_of_nonpos hx_nonpos]
+      linarith [hx.2]
+    have hcosh_x_le_lo : Real.cosh x ≤ Real.cosh I.lo := by
+      rw [Real.cosh_le_cosh]
+      rw [abs_of_nonpos hx_nonpos, abs_of_nonpos hlo_nonpos]
+      linarith [hx.1]
+    simp only [hull, mem_def, Rat.cast_min, Rat.cast_max]
+    constructor
+    · calc (min (coshPointComputable I.hi n).lo (coshPointComputable I.lo n).lo : ℝ)
+          ≤ (coshPointComputable I.hi n).lo := by exact_mod_cast min_le_left _ _
+        _ ≤ Real.cosh I.hi := hhi_mem.1
+        _ ≤ Real.cosh x := hcosh_hi_le
+    · calc Real.cosh x ≤ Real.cosh I.lo := hcosh_x_le_lo
+        _ ≤ (coshPointComputable I.lo n).hi := hlo_mem.2
+        _ ≤ max ((coshPointComputable I.hi n).hi : ℝ) ((coshPointComputable I.lo n).hi : ℝ) :=
+          le_max_right _ _
+  · -- Case: interval contains 0, minimum is 1
+    simp only [mem_def, Rat.cast_one, hull, Rat.cast_max]
+    constructor
+    · exact hcosh_ge_one
+    · -- Upper bound is max of endpoint cosh values
+      push Not at h1 h2
+      have hhi_pos : (0 : ℝ) < I.hi := by exact_mod_cast h2
+      have hlo_neg : (I.lo : ℝ) < 0 := by exact_mod_cast h1
+      have hmax_bound : Real.cosh x ≤ max (Real.cosh I.lo) (Real.cosh I.hi) := by
+        -- x is between lo and hi, and interval contains 0
+        by_cases! hx_nonneg : 0 ≤ x
+        · -- x ≥ 0: cosh(x) ≤ cosh(hi) since 0 ≤ x ≤ hi means |x| ≤ |hi|
+          apply le_max_of_le_right
+          rw [Real.cosh_le_cosh]
+          rw [abs_of_nonneg hx_nonneg, abs_of_nonneg (le_of_lt hhi_pos)]
+          exact hx.2
+        · -- x < 0: cosh(x) ≤ cosh(lo) since lo ≤ x < 0 means |x| ≤ |lo|
+          apply le_max_of_le_left
+          rw [Real.cosh_le_cosh]
+          rw [abs_of_neg hx_nonneg, abs_of_neg hlo_neg]
+          linarith [hx.1]
+      calc Real.cosh x ≤ max (Real.cosh I.lo) (Real.cosh I.hi) := hmax_bound
+        _ ≤ max ((coshPointComputable I.lo n).hi : ℝ) ((coshPointComputable I.hi n).hi : ℝ) := by
+            apply max_le_max
+            · exact hlo_mem.2
+            · exact hhi_mem.2
+
+/-! ### Computable atanh via Taylor series
+
+For |y| < 1, atanh(y) = y + y³/3 + y⁵/5 + ...
+We compute this series for y ∈ [-1/3, 1/3] where it converges rapidly.
+-/
+
+/-- Taylor coefficients for atanh: 0, 1, 0, 1/3, 0, 1/5, ...
+    atanh(y) = Σ y^(2k+1)/(2k+1) = y + y³/3 + y⁵/5 + ... -/
+def atanhTaylorCoeffs (n : ℕ) : List ℚ :=
+  let f : ℕ → ℚ := fun i => if i % 2 = 1 then 1 / i else 0
+  (List.range (n + 1)).map f
+
+/-- Computable atanh remainder bound.
+    For |y| ≤ r < 1, the remainder after n terms is bounded by r^(n+1)/(1 - r²).
+    We use a conservative bound: r^(n+1) / ((n+1) * (1 - r)) for simplicity. -/
+def atanhRemainderBoundComputable (r : ℚ) (n : ℕ) : IntervalRat :=
+  let r' := |r|  -- Use absolute value to ensure non-negativity
+  if h : r' ≥ 1 then
+    ⟨-1000, 1000, by norm_num⟩  -- Fallback for bad input
+  else
+    let R := r' ^ (n + 1) / (1 - r')
+    ⟨-R, R, by
+      have hr : r' < 1 := not_le.mp h
+      have hr_nonneg : 0 ≤ r' := abs_nonneg r
+      have hdenom_pos : 0 < 1 - r' := by linarith
+      have hR_nonneg : 0 ≤ R := by
+        apply div_nonneg
+        · apply pow_nonneg hr_nonneg
+        · linarith
+      linarith⟩
+
+/-- Computable interval enclosure for atanh at a single rational point.
+    Requires |q| < 1 for convergence. For |q| ≤ 1/3, this is very accurate. -/
+def atanhPointComputable (q : ℚ) (n : ℕ := 15) : IntervalRat :=
+  let r := |q|
+  if r ≥ 1 then
+    ⟨-1000, 1000, by norm_num⟩  -- Fallback
+  else
+    let I := singleton q
+    let coeffs := atanhTaylorCoeffs n
+    let polyVal := evalTaylorSeries coeffs I
+    let remainder := atanhRemainderBoundComputable r n
+    add polyVal remainder
+
+/-- Point `atanh` using Taylor coefficients prepared for depth `n`. -/
+def atanhPointComputableWithCoeffs (q : ℚ) (n : ℕ)
+    (coeffs : List ℚ) : IntervalRat :=
+  let r := |q|
+  if r ≥ 1 then
+    ⟨-1000, 1000, by norm_num⟩
+  else
+    let I := singleton q
+    let polyVal := evalTaylorSeries coeffs I
+    let remainder := atanhRemainderBoundComputable r n
+    add polyVal remainder
+
+theorem atanhPointComputableWithCoeffs_eq (q : ℚ) (n : ℕ) :
+    atanhPointComputableWithCoeffs q n (atanhTaylorCoeffs n) =
+      atanhPointComputable q n := by
+  simp [atanhPointComputableWithCoeffs, atanhPointComputable]
+
+/-- The atanh series: atanh(x) = Σ_{k=0}^∞ x^(2k+1)/(2k+1) for |x| < 1.
+    Derived from Mathlib's hasSum_log_sub_log_of_abs_lt_one. -/
+theorem Real.atanh_hasSum' {x : ℝ} (hx : |x| < 1) :
+    HasSum (fun k : ℕ => x ^ (2 * k + 1) / (2 * k + 1)) (Real.atanh x) := by
+  have hlog := Real.hasSum_log_sub_log_of_abs_lt_one hx
+  have h1 : 0 < 1 + x := by linarith [(abs_lt.mp hx).1]
+  have h2 : 0 < 1 - x := by linarith [(abs_lt.mp hx).2]
+  have h_eq : Real.atanh x = (1 / 2) * (Real.log (1 + x) - Real.log (1 - x)) := by
+    rw [Real.atanh, Real.log_div (ne_of_gt h1) (ne_of_gt h2)]
+  rw [h_eq]
+  convert! hlog.mul_left (1 / 2) using 1
+  funext k
+  field_simp
+
+/-- The atanh Taylor polynomial membership: the partial sum of atanh coefficients at q
+    is in evalTaylorSeries (atanhTaylorCoeffs n) (singleton q). -/
+theorem mem_evalTaylorSeries_atanh {q : ℚ} (n : ℕ) :
+    ((atanhTaylorCoeffs n).zipIdx.map (fun (c, i) => (c : ℝ) * (q : ℝ) ^ i)).sum ∈
+      evalTaylorSeries (atanhTaylorCoeffs n) (singleton q) :=
+  mem_evalTaylorSeries (mem_singleton q) (atanhTaylorCoeffs n)
+
+/-- The atanh Taylor remainder bound: for |q| < 1, the remainder after n terms is bounded.
+    The remainder is |Σ_{k: 2k+1 > n} q^(2k+1)/(2k+1)| ≤ |q|^(n+1)/(1-|q|). -/
+private theorem atanhTaylorCoeffs_sum (q : ℚ) (n : ℕ) :
+    ((atanhTaylorCoeffs n).zipIdx.map (fun (c, i) ↦ (c : ℝ) * (q : ℝ) ^ i)).sum =
+      ∑ k ∈ Finset.range ((n + 1) / 2), (q : ℝ) ^ (2 * k + 1) / (2 * k + 1) := by
+  let term := fun k : ℕ ↦ (q : ℝ) ^ (2 * k + 1) / (2 * k + 1)
+  let m := (n + 1) / 2
+  -- Step 1: Convert the list sum to a Finset sum over indices.
+  have hlist :
+      ((atanhTaylorCoeffs n).zipIdx.map (fun (c, i) => (c : ℝ) * (q : ℝ) ^ i)).sum =
+      ∑ i ∈ Finset.range (n + 1),
+        ((if i % 2 = 1 then ((i : ℚ) : ℚ)⁻¹ else 0 : ℚ) : ℝ) * (q : ℝ) ^ i := by
+    simp only [atanhTaylorCoeffs, one_div]
+    rw [List.zipIdx_map]
+    simp only [List.map_map]
+    -- Simplify the composition after zipIdx_map.
+    have h1 :
+        (List.range (n + 1)).zipIdx.map
+            ((fun p : ℚ × ℕ => (p.1 : ℝ) * (q : ℝ) ^ p.2) ∘
+              Prod.map (fun i : ℕ => if i % 2 = 1 then ((i : ℚ) : ℚ)⁻¹ else 0) id) =
+          (List.range (n + 1)).zipIdx.map
+            (fun p : ℕ × ℕ =>
+              ((if p.1 % 2 = 1 then ((p.1 : ℚ) : ℚ)⁻¹ else 0 : ℚ) : ℝ) * (q : ℝ) ^ p.2) := by
+      apply List.map_congr_left
+      intro ⟨a, b⟩ _
+      simp [Function.comp, Prod.map_apply]
+    -- Replace zipIdx over range with a direct map on indices.
+    have h2 :
+        (List.range (n + 1)).zipIdx.map
+            (fun p : ℕ × ℕ =>
+              ((if p.1 % 2 = 1 then ((p.1 : ℚ) : ℚ)⁻¹ else 0 : ℚ) : ℝ) * (q : ℝ) ^ p.2) =
+          (List.range (n + 1)).map
+            (fun i : ℕ =>
+              ((if i % 2 = 1 then ((i : ℚ) : ℚ)⁻¹ else 0 : ℚ) : ℝ) * (q : ℝ) ^ i) := by
+      convert! zipIdx_range_map
+        (fun a b =>
+          ((if a % 2 = 1 then ((a : ℚ) : ℚ)⁻¹ else 0 : ℚ) : ℝ) * (q : ℝ) ^ b) (n + 1) using 2
+    rw [h1, h2]
+    exact list_map_sum_eq_finset_sum
+      (fun i : ℕ =>
+        ((if i % 2 = 1 then ((i : ℚ) : ℚ)⁻¹ else 0 : ℚ) : ℝ) * (q : ℝ) ^ i) (n + 1)
+  -- Step 2: Filter to odd indices (even indices contribute zero).
+  have hsum_filter :
+      ∑ i ∈ Finset.range (n + 1),
+          ((if i % 2 = 1 then ((i : ℚ) : ℚ)⁻¹ else 0 : ℚ) : ℝ) * (q : ℝ) ^ i =
+        ∑ i ∈ (Finset.range (n + 1)).filter (fun i => i % 2 = 1),
+          (q : ℝ) ^ i / i := by
+    have hterm :
+        ∀ i : ℕ,
+          ((if i % 2 = 1 then ((i : ℚ) : ℚ)⁻¹ else 0 : ℚ) : ℝ) * (q : ℝ) ^ i =
+            if i % 2 = 1 then (q : ℝ) ^ i / i else 0 := by
+      intro i
+      by_cases hodd : i % 2 = 1
+      · simp [hodd, div_eq_mul_inv, Rat.cast_inv, Rat.cast_natCast,
+          mul_comm]
+      · simp [hodd]
+    calc
+      ∑ i ∈ Finset.range (n + 1),
+          ((if i % 2 = 1 then ((i : ℚ) : ℚ)⁻¹ else 0 : ℚ) : ℝ) * (q : ℝ) ^ i
+          =
+          ∑ i ∈ Finset.range (n + 1),
+            (if i % 2 = 1 then (q : ℝ) ^ i / i else 0) := by
+              refine Finset.sum_congr rfl ?_
+              intro i hi
+              exact hterm i
+      _ =
+          ∑ i ∈ (Finset.range (n + 1)).filter (fun i => i % 2 = 1),
+            (q : ℝ) ^ i / i := by
+              symm
+              exact (Finset.sum_filter
+                (s := Finset.range (n + 1))
+                (f := fun i => (q : ℝ) ^ i / i)
+                (p := fun i => i % 2 = 1))
+  -- Step 3: Reindex odd indices by k with i = 2k+1.
+  have hsum_reindex :
+      ∑ i ∈ (Finset.range (n + 1)).filter (fun i => i % 2 = 1),
+          (q : ℝ) ^ i / i =
+        ∑ k ∈ Finset.range m, term k := by
+    refine (Finset.sum_nbij
+      (s := Finset.range m)
+      (t := (Finset.range (n + 1)).filter (fun i => i % 2 = 1))
+      (i := fun k => 2 * k + 1)
+      (f := term)
+      (g := fun i => (q : ℝ) ^ i / i)
+      ?_ ?_ ?_ ?_).symm
+    · intro k hk
+      apply Finset.mem_filter.2
+      constructor
+      · -- membership in range
+        have hk' : k < m := Finset.mem_range.1 hk
+        have hk_succ : Nat.succ k ≤ m := (Nat.succ_le_iff).2 hk'
+        have hk_succ' : k + 1 ≤ (n + 1) / 2 := by simpa [m] using hk_succ
+        have hk_mul : (k + 1) * 2 ≤ n + 1 :=
+          (Nat.le_div_iff_mul_le Nat.zero_lt_two).1 hk_succ'
+        have hk_mul' : 2 * k + 2 ≤ n + 1 := by
+          simpa [Nat.add_mul, Nat.mul_comm, Nat.mul_left_comm, Nat.mul_assoc] using hk_mul
+        have hk_lt : 2 * k + 1 < n + 1 := by
+          have hlt : 2 * k + 1 < 2 * k + 2 := Nat.lt_succ_self (2 * k + 1)
+          exact lt_of_lt_of_le hlt hk_mul'
+        exact Finset.mem_range.2 hk_lt
+      · -- oddness
+        have hodd : Odd (2 * k + 1) := ⟨k, rfl⟩
+        exact (Nat.odd_iff).1 hodd
+    · intro a ha b hb hEq
+      have hEq' : Nat.succ (2 * a) = Nat.succ (2 * b) := by
+        simpa [Nat.succ_eq_add_one] using hEq
+      have hEq'' : 2 * a = 2 * b := (Nat.succ_inj).1 hEq'
+      exact Nat.mul_left_cancel Nat.zero_lt_two hEq''
+    · intro i hi
+      simp only [Finset.coe_range, Set.mem_image, Set.mem_Iio, m]
+      use (i - 1) / 2
+      simp at hi
+      grind only
+    · intro k hk
+      simp [term]
+  -- Combine the steps.
+  calc
+    ((atanhTaylorCoeffs n).zipIdx.map (fun (c, i) => (c : ℝ) * (q : ℝ) ^ i)).sum
+        = ∑ i ∈ Finset.range (n + 1),
+            ((if i % 2 = 1 then ((i : ℚ) : ℚ)⁻¹ else 0 : ℚ) : ℝ) * (q : ℝ) ^ i := hlist
+    _ = ∑ i ∈ (Finset.range (n + 1)).filter (fun i => i % 2 = 1),
+          (q : ℝ) ^ i / i := hsum_filter
+    _ = ∑ k ∈ Finset.range m, term k := hsum_reindex
+
+
+theorem atanh_taylor_remainder_in_interval {q : ℚ} (hq : |(q : ℝ)| < 1) (n : ℕ) :
+    Real.atanh q - ((atanhTaylorCoeffs n).zipIdx.map (fun (c, i) => (c : ℝ) * (q : ℝ) ^ i)).sum
+    ∈ atanhRemainderBoundComputable |q| n := by
+  -- The remainder is the tail of the series
+  simp only [atanhRemainderBoundComputable, abs_abs]
+  have hq_abs_lt : |q| < 1 := by
+    have h := abs_lt.mp hq
+    rw [abs_lt]; constructor
+    · exact_mod_cast h.1
+    · exact_mod_cast h.2
+  have h_not_ge : ¬(|q| ≥ 1) := not_le.mpr hq_abs_lt
+  simp only [h_not_ge, mem_def]
+  -- The bound R = |q|^(n+1)/(1-|q|) in ℚ
+  set R_rat := |q| ^ (n + 1) / (1 - |q|) with hR_rat_def
+  -- Key bounds: |q| < 1 in ℚ
+  have hr_nonneg_rat : 0 ≤ |q| := abs_nonneg q
+  have hdenom_pos_rat : 0 < 1 - |q| := by linarith
+  have hR_nonneg_rat' : 0 ≤ R_rat := by
+    apply div_nonneg
+    · exact pow_nonneg hr_nonneg_rat (n + 1)
+    · linarith
+  have hR_nonneg : (0 : ℝ) ≤ R_rat := Rat.cast_nonneg.mpr hR_nonneg_rat'
+  -- The absolute value of the remainder is bounded
+  have habs : |Real.atanh q - ((atanhTaylorCoeffs n).zipIdx.map
+      (fun (c, i) => (c : ℝ) * (q : ℝ) ^ i)).sum| ≤ (R_rat : ℝ) := by
+    -- Strategy: The atanh series is Σ_k q^(2k+1)/(2k+1)
+    -- The polynomial computes Σ_{i≤n, i odd} q^i/i
+    -- The remainder is Σ_{k: 2k+1 > n} q^(2k+1)/(2k+1)
+    -- Bound: |remainder| ≤ Σ_{k: 2k+1 > n} |q|^(2k+1) ≤ |q|^(n+1)/(1-|q|)
+
+    -- Get the series representation
+    have hseries := Real.atanh_hasSum' hq
+    -- The polynomial sum equals the partial series sum for terms with index ≤ n
+    -- atanhTaylorCoeffs n = [0, 1, 0, 1/3, 0, 1/5, ...] for indices 0..n
+    -- So the sum is Σ_{i odd, i ≤ n} q^i/i = Σ_{k: 2k+1 ≤ n} q^(2k+1)/(2k+1)
+
+    -- For the bound, we use that each term |q^(2k+1)/(2k+1)| ≤ |q|^(2k+1)
+    -- and the geometric series Σ_{m≥n+1} |q|^m ≤ |q|^(n+1)/(1-|q|)
+
+    -- This is a non-trivial series argument - for now use the bound directly
+    -- The remainder terms satisfy |term_k| ≤ |q|^(2k+1) for 2k+1 > n
+    -- Summing: |remainder| ≤ |q|^(n+1) * (1 + |q| + |q|² + ...) = |q|^(n+1)/(1-|q|)
+    have hq_real_lt : |(q : ℝ)| < 1 := hq
+    have hq_abs_nonneg : 0 ≤ |(q : ℝ)| := abs_nonneg _
+    -- The remainder is bounded by a geometric series tail
+    -- |Σ_{k: 2k+1 > n} q^(2k+1)/(2k+1)| ≤ Σ_{k: 2k+1 > n} |q|^(2k+1)
+    --                                    ≤ |q|^(n+1) + |q|^(n+2) + ...
+    --                                    = |q|^(n+1) / (1 - |q|)
+    -- Define the series term and the split point m
+    let term := fun k : ℕ => (q : ℝ) ^ (2 * k + 1) / (2 * k + 1)
+    -- m = number of terms with 2k+1 ≤ n, i.e., k ≤ (n-1)/2
+    let m := (n + 1) / 2
+    -- Key: 2m ≥ n (so 2m+1 ≥ n+1)
+    have hm_bound : 2 * m ≥ n := by simp only [m]; omega
+    -- Step 1: Show polynomial sum equals partial series sum up to m terms
+    -- The polynomial atanhTaylorCoeffs n has coefficients:
+    -- index i: 1/i if i is odd and i ≤ n, else 0
+    -- So the sum gives Σ_{k: 2k+1 ≤ n} q^(2k+1)/(2k+1) = Σ_{k < m} term k
+    have h_poly_eq_partial := atanhTaylorCoeffs_sum q n
+    -- Step 2: The remainder is the tail of the series starting at m
+    have h_summable := hseries.summable
+    have h_tail_summable : Summable fun k => term (k + m) := (summable_nat_add_iff m).mpr h_summable
+    have h_tail_eq : Real.atanh q - ∑ k ∈ Finset.range m, term k = ∑' k, term (k + m) := by
+      have h_split := h_summable.sum_add_tsum_nat_add m
+      have h_tsum_eq : ∑' i, term i = Real.atanh q := hseries.tsum_eq
+      linarith [h_split, h_tsum_eq]
+    rw [h_poly_eq_partial, h_tail_eq]
+    -- Step 3: Bound the tail by geometric series
+    have hz_abs_sq : |(q : ℝ)| ^ 2 < 1 := by
+      have h1 : |(q : ℝ)| ^ 2 ≤ |(q : ℝ)| := by
+        have hn : 0 ≤ |(q : ℝ)| := abs_nonneg _
+        have hle : |(q : ℝ)| ≤ 1 := le_of_lt hq
+        nlinarith [sq_nonneg (|(q : ℝ)| - 1)]
+      linarith
+    have hz_abs_nonneg : 0 ≤ |(q : ℝ)| := abs_nonneg _
+    have hz_abs_le : |(q : ℝ)| ≤ 1 := le_of_lt hq
+    -- Define the dominating geometric term
+    let geo_term := fun k : ℕ => |(q : ℝ)| ^ (2 * m + 1) * (|(q : ℝ)| ^ 2) ^ k
+    have h_geo_summable : Summable geo_term := by
+      apply Summable.mul_left
+      exact summable_geometric_of_lt_one (sq_nonneg _) hz_abs_sq
+    have h_term_bound : ∀ k, |term (k + m)| ≤ geo_term k := by
+      intro k
+      simp only [term, geo_term]
+      rw [abs_div, abs_pow]
+      have h_pow_eq : |(q : ℝ)| ^ (2 * (k + m) + 1) = |(q : ℝ)| ^ (2 * m + 1) * (|(q : ℝ)| ^ 2) ^
+        k := by
+        have : 2 * (k + m) + 1 = 2 * m + 1 + 2 * k := by ring
+        rw [this, pow_add, pow_mul]
+      rw [h_pow_eq]
+      have h_denom_pos' : (0 : ℝ) < 2 * (k + m : ℕ) + 1 := by positivity
+      have h_denom_ge_one : (1 : ℝ) ≤ |(2 : ℝ) * (k + m : ℕ) + 1| := by
+        rw [abs_of_pos h_denom_pos']
+        have hk : (0 : ℝ) ≤ k := Nat.cast_nonneg k
+        have hm' : (0 : ℝ) ≤ m := Nat.cast_nonneg m
+        push_cast; linarith
+      calc |(q : ℝ)| ^ (2 * m + 1) * (|(q : ℝ)| ^ 2) ^ k / |(2 : ℝ) * (k + m : ℕ) + 1|
+          ≤ |(q : ℝ)| ^ (2 * m + 1) * (|(q : ℝ)| ^ 2) ^ k / 1 := by
+            apply div_le_div_of_nonneg_left _ (by positivity) h_denom_ge_one
+            positivity
+        _ = |(q : ℝ)| ^ (2 * m + 1) * (|(q : ℝ)| ^ 2) ^ k := by ring
+    have h_norm_sum : ‖∑' k, term (k + m)‖ ≤ ∑' k, ‖term (k + m)‖ :=
+      norm_tsum_le_tsum_norm h_tail_summable.norm
+    simp only [Real.norm_eq_abs] at h_norm_sum
+    calc |∑' k, term (k + m)|
+        ≤ ∑' k, |term (k + m)| := h_norm_sum
+      _ ≤ ∑' k, geo_term k := h_tail_summable.abs.tsum_le_tsum h_term_bound h_geo_summable
+      _ = |(q : ℝ)| ^ (2 * m + 1) * ∑' k, (|(q : ℝ)| ^ 2) ^ k := by
+          simp only [geo_term]; rw [tsum_mul_left]
+      _ = |(q : ℝ)| ^ (2 * m + 1) / (1 - |(q : ℝ)| ^ 2) := by
+          rw [tsum_geometric_of_lt_one (sq_nonneg _) hz_abs_sq]; ring
+      _ ≤ |(q : ℝ)| ^ (n + 1) / (1 - |(q : ℝ)| ^ 2) := by
+          -- 2m + 1 ≥ n + 1 since 2m ≥ n
+          have h_exp_le : n + 1 ≤ 2 * m + 1 := by omega
+          have h_pow_le : |(q : ℝ)| ^ (2 * m + 1) ≤ |(q : ℝ)| ^ (n + 1) :=
+            pow_le_pow_of_le_one hz_abs_nonneg hz_abs_le h_exp_le
+          have h_denom_pos : 0 < 1 - |(q : ℝ)| ^ 2 := by nlinarith [sq_nonneg (q : ℝ), sq_abs (q :
+            ℝ)]
+          exact div_le_div_of_nonneg_right h_pow_le h_denom_pos.le
+      _ ≤ |(q : ℝ)| ^ (n + 1) / (1 - |(q : ℝ)|) := by
+          -- 1 - |q|² ≥ 1 - |q| since |q|² ≤ |q| for |q| ≤ 1
+          have h1 : |(q : ℝ)| ^ 2 ≤ |(q : ℝ)| := by nlinarith [sq_nonneg (|(q : ℝ)| - 1)]
+          have h2 : 1 - |(q : ℝ)| ≤ 1 - |(q : ℝ)| ^ 2 := by linarith
+          have h3 : 0 < 1 - |(q : ℝ)| := by linarith
+          have h4 : 0 ≤ |(q : ℝ)| ^ (n + 1) := pow_nonneg hz_abs_nonneg _
+          exact div_le_div_of_nonneg_left h4 h3 h2
+      _ = (R_rat : ℝ) := by
+          -- R_rat = |q|^(n+1)/(1-|q|) in ℚ
+          simp only [hR_rat_def]
+          -- Show: |(q:ℝ)|^(n+1)/(1-|(q:ℝ)|) = (|q|^(n+1)/(1-|q|) : ℚ)
+          rw [← Rat.cast_abs q]
+          push_cast
+          ring
+  have h := abs_le.mp habs
+  constructor
+  · calc ((-R_rat : ℚ) : ℝ) = -((R_rat : ℚ) : ℝ) := by push_cast; ring
+      _ ≤ _ := h.1
+  · exact h.2
+
+/-- FTIA for atanhPointComputable: Real.atanh q ∈ atanhPointComputable q n for |q| < 1. -/
+theorem mem_atanhPointComputable (q : ℚ) (n : ℕ) (hq : |(q : ℝ)| < 1) :
+    Real.atanh q ∈ atanhPointComputable q n := by
+  simp only [atanhPointComputable]
+  -- Since |q| < 1, the condition |q| ≥ 1 is false
+  have hq_rat : |q| < 1 := by
+    have h := abs_lt.mp hq
+    rw [abs_lt]
+    constructor
+    · have : (-1 : ℝ) < q := h.1
+      exact_mod_cast this
+    · have : (q : ℝ) < 1 := h.2
+      exact_mod_cast this
+  have h_not_ge : ¬(|q| ≥ 1) := not_le.mpr hq_rat
+  simp only [h_not_ge, ↓reduceIte]
+  -- Polynomial part: the partial sum is in evalTaylorSeries
+  have hpoly := mem_evalTaylorSeries_atanh n (q := q)
+  -- Remainder part: the tail is in atanhRemainderBoundComputable
+  have hrem := atanh_taylor_remainder_in_interval hq n
+  -- Combine: atanh q = partial_sum + remainder
+  have heq : Real.atanh q = ((atanhTaylorCoeffs n).zipIdx.map
+      (fun (c, i) => (c : ℝ) * (q : ℝ) ^ i)).sum +
+      (Real.atanh q - ((atanhTaylorCoeffs n).zipIdx.map
+        (fun (c, i) => (c : ℝ) * (q : ℝ) ^ i)).sum) := by ring
+  rw [heq]
+  exact mem_add hpoly hrem
+
+/-! ### Computable ln(2) via atanh
+
+ln(2) = 2 * atanh(1/3), since:
+  2 = (1 + 1/3) / (1 - 1/3) = (4/3) / (2/3)
+  So atanh(1/3) = (1/2) * ln(2), giving ln(2) = 2 * atanh(1/3)
+-/
+
+/-- Compute ln(2) as an interval using 2 * atanh(1/3).
+    This converges rapidly since atanh series at 1/3 has |y| = 1/3. -/
+def ln2Computable (n : ℕ := 20) : IntervalRat :=
+  let atanh_third := atanhPointComputable (1/3) n
+  scale 2 atanh_third
+
+/-- FTIA for ln2Computable: Real.log 2 ∈ ln2Computable n.
+    Uses the identity log(2) = 2 * atanh(1/3) from log_via_atanh. -/
+theorem mem_ln2Computable (n : ℕ) : Real.log 2 ∈ ln2Computable n := by
+  simp only [ln2Computable]
+  -- log(2) = 2 * atanh(1/3) via log_via_atanh
+  have hlog_eq : Real.log 2 = 2 * Real.atanh (↑(1/3 : ℚ)) := by
+    have h := LogReduction.log_via_atanh (by norm_num : (0 : ℚ) < 2)
+    -- Convert: ((↑2 : ℝ) - 1) / ((↑2 : ℝ) + 1) = ↑(1/3 : ℚ)
+    have h_arg : ((↑(2 : ℚ) : ℝ) - 1) / (↑(2 : ℚ) + 1) = ↑(1/3 : ℚ) := by
+      simp only [Rat.cast_ofNat]
+      norm_num
+    rw [h_arg] at h
+    convert! h using 1
+  rw [hlog_eq]
+  -- atanh(1/3) ∈ atanhPointComputable (1/3) n by mem_atanhPointComputable
+  have h_third_lt : |(↑(1/3 : ℚ) : ℝ)| < 1 := by
+    rw [abs_lt]
+    constructor <;> norm_num
+  exact mem_scale 2 (mem_atanhPointComputable (1/3) n h_third_lt)
+
+/-! ### Computable log via argument reduction
+
+For q > 0, we compute:
+  1. Reduce q to m * 2^k where m ∈ [1/2, 2]
+  2. Compute log(m) = 2 * atanh((m-1)/(m+1)), which has |arg| ≤ 1/3
+  3. Result = log(m) + k * ln(2)
+-/
+
+/-- Reduction exponent k such that q * 2^(-k) ≈ 1 -/
+def logReductionExponent (q : ℚ) : ℤ :=
+  if q ≤ 0 then 0
+  else
+    let n_log := q.num.natAbs.log2
+    let d_log := q.den.log2
+    (n_log : ℤ) - (d_log : ℤ)
+
+/-- Reduced mantissa m = q * 2^(-k) -/
+def logReduceMantissa (q : ℚ) : ℚ :=
+  if q ≤ 0 then 1
+  else q * (2 : ℚ) ^ (-logReductionExponent q)
+
+/-- Computable log at a single rational point q > 0.
+    Returns log(q) = log(m) + k * ln(2) where m = q * 2^(-k). -/
+def logPointComputable (q : ℚ) (n : ℕ := 20) : IntervalRat :=
+  if q ≤ 0 then
+    ⟨-1000, 1000, by norm_num⟩  -- Fallback for non-positive
+  else
+    let k := logReductionExponent q
+    let m := logReduceMantissa q
+    -- log(m) = 2 * atanh((m-1)/(m+1))
+    let y := (m - 1) / (m + 1)
+    let atanh_y := atanhPointComputable y n
+    let log_m := scale 2 atanh_y
+    -- k * ln(2)
+    let k_ln2 := scale k (ln2Computable n)
+    add log_m k_ln2
+
+/-- Logarithm evaluation using all configuration-dependent data prepared for depth `n`. -/
+def logPointComputablePrepared (q : ℚ) (n : ℕ) (ln2 : IntervalRat)
+    (atanhCoeffs : List ℚ) : IntervalRat :=
+  if q ≤ 0 then
+    ⟨-1000, 1000, by norm_num⟩
+  else
+    let k := logReductionExponent q
+    let m := logReduceMantissa q
+    let y := (m - 1) / (m + 1)
+    let atanh_y := atanhPointComputableWithCoeffs y n atanhCoeffs
+    let log_m := scale 2 atanh_y
+    let k_ln2 := scale k ln2
+    add log_m k_ln2
+
+theorem logPointComputablePrepared_eq (q : ℚ) (n : ℕ) :
+    logPointComputablePrepared q n (ln2Computable n) (atanhTaylorCoeffs n) =
+      logPointComputable q n := by
+  simp [logPointComputablePrepared, logPointComputable,
+    atanhPointComputableWithCoeffs_eq]
+
+/-- Computable interval enclosure for log using endpoint evaluation.
+    Since log is strictly increasing on (0, ∞), we evaluate at endpoints. -/
+def logComputable (I : IntervalRat) (n : ℕ := 20) : IntervalRat :=
+  if I.lo ≤ 0 then
+    ⟨-1000, 1000, by norm_num⟩  -- Fallback for non-positive interval
+  else
+    -- log is monotone increasing, so log([lo, hi]) = [log(lo), log(hi)]
+    let logLo := logPointComputable I.lo n
+    let logHi := logPointComputable I.hi n
+    hull logLo logHi
+
+/-- Interval logarithm using all data prepared for depth `n`. -/
+def logComputablePrepared (I : IntervalRat) (n : ℕ) (ln2 : IntervalRat)
+    (atanhCoeffs : List ℚ) : IntervalRat :=
+  if I.lo ≤ 0 then
+    ⟨-1000, 1000, by norm_num⟩
+  else
+    let logLo := logPointComputablePrepared I.lo n ln2 atanhCoeffs
+    let logHi := logPointComputablePrepared I.hi n ln2 atanhCoeffs
+    hull logLo logHi
+
+theorem logComputablePrepared_eq (I : IntervalRat) (n : ℕ) :
+    logComputablePrepared I n (ln2Computable n) (atanhTaylorCoeffs n) =
+      logComputable I n := by
+  simp [logComputablePrepared, logComputable, logPointComputablePrepared_eq]
+
+/-- The local logReductionExponent equals LogReduction.reductionExponent for positive q -/
+private theorem logReductionExponent_eq {q : ℚ} (hq : 0 < q) :
+    logReductionExponent q = LogReduction.reductionExponent q := by
+  simp only [logReductionExponent, LogReduction.reductionExponent, not_le.mpr hq, ↓reduceIte]
+
+/-- The local logReduceMantissa equals LogReduction.reduceMantissa for positive q -/
+private theorem logReduceMantissa_eq {q : ℚ} (hq : 0 < q) :
+    logReduceMantissa q = LogReduction.reduceMantissa q := by
+  simp only [logReduceMantissa, LogReduction.reduceMantissa, not_le.mpr hq, ↓reduceIte,
+             logReductionExponent_eq hq]
+
+/-- FTIA for logPointComputable -/
+theorem mem_logPointComputable {q : ℚ} (hq : 0 < q) (n : ℕ) :
+    Real.log q ∈ logPointComputable q n := by
+  unfold logPointComputable
+  simp only [not_le.mpr hq, ↓reduceIte]
+  -- Get the reduced mantissa and exponent
+  set k := logReductionExponent q with hk_def
+  set m := logReduceMantissa q with hm_def
+  -- Show they equal the LogReduction definitions
+  have hk_eq : k = LogReduction.reductionExponent q := logReductionExponent_eq hq
+  have hm_eq : m = LogReduction.reduceMantissa q := logReduceMantissa_eq hq
+  -- Get bounds on m from LogReduction.reduced_bounds
+  have hm_bounds : 1/2 ≤ m ∧ m ≤ 2 := by rw [hm_eq]; exact LogReduction.reduced_bounds hq
+  have hm_pos : 0 < m := by rw [hm_eq]; exact LogReduction.reduced_pos hq
+  -- Compute y = (m-1)/(m+1) and get bounds from atanh_arg_bounds
+  set y := (m - 1) / (m + 1) with hy_def
+  have hy_bounds : (-1)/3 ≤ y ∧ y ≤ 1/3 := LogReduction.atanh_arg_bounds hm_bounds.1 hm_bounds.2
+  -- Key: |y| ≤ 1/3 < 1, so we can use mem_atanhPointComputable
+  have hy_abs_lt : |(y : ℝ)| < 1 := by
+    rw [abs_lt]
+    have hlo : ((-1 : ℚ) / 3 : ℝ) ≤ y := by exact_mod_cast hy_bounds.1
+    have hhi : (y : ℝ) ≤ (1 : ℚ) / 3 := by exact_mod_cast hy_bounds.2
+    constructor <;> linarith
+  -- Step 1: atanh(y) ∈ atanhPointComputable y n
+  have h_atanh := mem_atanhPointComputable y n hy_abs_lt
+  -- Step 2: log(m) = 2 * atanh(y), so log(m) ∈ scale 2 (atanhPointComputable y n)
+  have h_log_m_eq : Real.log m = 2 * Real.atanh y := by
+    have h := LogReduction.log_via_atanh hm_pos
+    -- h : Real.log m = 2 * Real.atanh ((m - 1) / (m + 1))
+    -- Need to show: ((m - 1) / (m + 1) : ℚ) : ℝ = ((m : ℝ) - 1) / ((m : ℝ) + 1)
+    have hcast : (y : ℝ) = ((m : ℝ) - 1) / ((m : ℝ) + 1) := by
+      simp only [hy_def]
+      push_cast
+      ring
+    rw [hcast]
+    exact h
+  have h_log_m := mem_scale 2 h_atanh
+  -- h_log_m : (2 : ℚ) * Real.atanh y ∈ scale 2 (atanhPointComputable y n)
+  -- h_log_m_eq : Real.log m = 2 * Real.atanh y
+  have h_log_m' : Real.log m ∈ scale 2 (atanhPointComputable y n) := by
+    rw [h_log_m_eq]
+    exact h_log_m
+  -- Step 3: log(2) ∈ ln2Computable n
+  have h_ln2 := mem_ln2Computable n
+  -- Step 4: k * log(2) ∈ scale k (ln2Computable n)
+  have h_k_ln2 := mem_scale k h_ln2
+  -- Step 5: log(q) = log(m) + k * log(2) by log_reduction
+  have h_log_eq : Real.log q = Real.log m + k * Real.log 2 := by
+    have h := LogReduction.log_reduction hq
+    exact h
+  -- Step 6: Combine using mem_add
+  rw [h_log_eq]
+  exact mem_add h_log_m' h_k_ln2
+
+/-- FTIA for logComputable: if x ∈ I and I.lo > 0, then log(x) ∈ logComputable I n -/
+theorem mem_logComputable {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (hpos : 0 < I.lo) (n : ℕ) :
+    Real.log x ∈ logComputable I n := by
+  simp only [logComputable, not_le.mpr hpos, ↓reduceIte]
+  -- Since log is strictly monotone and x ∈ [lo, hi] with lo > 0:
+  -- log(lo) ≤ log(x) ≤ log(hi)
+  have hlo_pos : (0 : ℝ) < I.lo := by exact_mod_cast hpos
+  have hx_pos : 0 < x := lt_of_lt_of_le hlo_pos hx.1
+  have hlo_mem := mem_logPointComputable hpos n
+  have hhi_pos : 0 < I.hi := lt_of_lt_of_le hpos I.le
+  have hhi_mem := mem_logPointComputable hhi_pos n
+  -- x ∈ [lo, hi] implies log(lo) ≤ log(x) ≤ log(hi) by monotonicity
+  have hlog_lo_le : Real.log I.lo ≤ Real.log x :=
+    Real.log_le_log hlo_pos hx.1
+  have hlog_x_le_hi : Real.log x ≤ Real.log I.hi :=
+    Real.log_le_log hx_pos hx.2
+  simp only [hull, mem_def, Rat.cast_min, Rat.cast_max]
+  constructor
+  · calc (min (logPointComputable I.lo n).lo (logPointComputable I.hi n).lo : ℝ)
+        ≤ (logPointComputable I.lo n).lo := by exact_mod_cast min_le_left _ _
+      _ ≤ Real.log I.lo := hlo_mem.1
+      _ ≤ Real.log x := hlog_lo_le
+  · calc Real.log x ≤ Real.log I.hi := hlog_x_le_hi
+      _ ≤ (logPointComputable I.hi n).hi := hhi_mem.2
+      _ ≤ max ((logPointComputable I.lo n).hi : ℝ) ((logPointComputable I.hi n).hi : ℝ) :=
+        le_max_right _ _
+
+/-- Conditional version of mem_logComputable for use in correctness proofs.
+    Requires I.lo > 0 so the log interval is well-defined and monotone. -/
+theorem mem_logComputable' {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (hpos : 0 < I.lo) (n : ℕ) :
+    Real.log x ∈ logComputable I n := by
+  exact mem_logComputable hx hpos n
+
+/-! ### Computable erf via Taylor series -/
+
+/-- Interval containing 2/√π ≈ 1.128379...
+    Used for erf calculations. 2/√π is in (1.128, 1.129). -/
+def twoDivSqrtPi : IntervalRat :=
+  ⟨1128/1000, 1129/1000, by norm_num⟩
+
+/-- Taylor coefficients for erf (without the 2/√π factor):
+    erf(x) = (2/√π) * Σ_{n=0}^∞ (-1)^n * x^(2n+1) / (n! * (2n+1))
+
+    So the coefficient of x^k is:
+    - 0 if k is even
+    - (-1)^((k-1)/2) / (((k-1)/2)! * k) if k is odd -/
+def erfTaylorCoeffs (n : ℕ) : List ℚ :=
+  (List.range (n + 1)).map (fun i =>
+    if i % 2 = 1 then  -- odd terms only: x, x³, x⁵, ...
+      let m : ℕ := (i - 1) / 2  -- for x^(2m+1), m = (i-1)/2
+      ((-1 : ℚ) ^ m) / (ratFactorial m * (i : ℚ))
+    else 0)
+
+/-- Computable erf remainder bound.
+    Since |erf^{(k)}(x)| ≤ (2/√π) * 2^k for all x (rough bound),
+    and erf is bounded by 1, we use a combination.
+
+    For the Taylor remainder centered at 0, we use:
+    |R_n(x)| ≤ sup|f^{(n+1)}(ξ)| * |x|^{n+1} / (n+1)!
+
+    A conservative bound: |erf^{(k)}(x)| ≤ (2/√π) * k! / (k/2)! ≤ 2 * k^{k/2}
+    But since |erf| ≤ 1, we can intersect with [-1, 1].
+
+    We use: remainder ≤ 2 * |x|^{n+1} / (n+1)! (very conservative). -/
+def erfRemainderBoundComputable (I : IntervalRat) (n : ℕ) : IntervalRat :=
+  let r := maxAbs I
+  -- Conservative bound: use 2 as the derivative bound (since 2/√π ≈ 1.13)
+  let R := 2 * r ^ (n + 1) / ratFactorial (n + 1)
+  ⟨-R, R, by
+    have hr : 0 ≤ r := le_max_of_le_left (abs_nonneg I.lo)
+    have hfact : (0 : ℚ) ≤ ratFactorial (n + 1) := by
+      simp only [ratFactorial]; exact Nat.cast_nonneg _
+    have hpow : (0 : ℚ) ≤ r ^ (n + 1) := pow_nonneg hr _
+    have hR : 0 ≤ R := div_nonneg (mul_nonneg (by norm_num : (0:ℚ) ≤ 2) hpow) hfact
+    linarith⟩
+
+/-- Computable interval enclosure for erf at a single rational point.
+    Uses sign-aware clipped linear bounds:
+    `|erf(q)| ≤ min(1, (2257/2000)*|q|)`.
+
+    Cases:
+    - `q < 0  => erf(q) ∈ [-1, 0]`
+    - `q = 0  => erf(q) = 0`
+    - `q > 0  => erf(q) ∈ [0, 1]`
+
+    with tighter near-zero magnitude via `(2257/2000)*|q|`. -/
+def erfPointComputable (q : ℚ) (n : ℕ := 15) : IntervalRat :=
+  let _ := n
+  let b : ℚ := min 1 ((2257 / 2000) * |q|)
+  if hq : q < 0 then
+    ⟨-b, 0, by
+      unfold b
+      have hb_nonneg : (0 : ℚ) ≤ min 1 ((2257 / 2000) * |q|) := by
+        apply le_min
+        · norm_num
+        · exact mul_nonneg (by norm_num) (abs_nonneg q)
+      linarith⟩
+  else if h0 : q = 0 then
+    ⟨0, 0, by norm_num⟩
+  else
+    ⟨0, b, by
+      unfold b
+      have hb_nonneg : (0 : ℚ) ≤ min 1 ((2257 / 2000) * |q|) := by
+        apply le_min
+        · norm_num
+        · exact mul_nonneg (by norm_num) (abs_nonneg q)
+      exact hb_nonneg⟩
+
+/-- Computable interval enclosure for erf using Taylor series with monotonicity.
+
+    erf(x) = (2/√π) * Σ_{n=0}^∞ (-1)^n * x^(2n+1) / (n! * (2n+1))
+
+    Since erf is strictly monotone increasing (erf'(x) = (2/√π)e^{-x²} > 0),
+    we use endpoint evaluation: erf([a,b]) ⊆ [erf(a), erf(b)].
+
+    We intersect with [-1, 1] for safety. -/
+def erfComputable (I : IntervalRat) (n : ℕ := 15) : IntervalRat :=
+  -- erf is strictly monotone increasing, so evaluate at endpoints
+  let erfLo := erfPointComputable I.lo n
+  let erfHi := erfPointComputable I.hi n
+  let raw := hull erfLo erfHi
+  -- Intersect with global bound [-1, 1]
+  let globalBound : IntervalRat := ⟨-1, 1, by norm_num⟩
+  match intersect raw globalBound with
+  | some refined => refined
+  | none => globalBound
+
+/-- 2/√π is in the interval twoDivSqrtPi.
+    2/√π ≈ 1.1283791670955126, which is in (1.128, 1.129).
+
+    This is a helper theorem for correctness proofs. The computation of
+    erfComputable is independent of this theorem. -/
+theorem two_div_sqrt_pi_mem : 2 / Real.sqrt Real.pi ∈ twoDivSqrtPi := by
+  simp only [twoDivSqrtPi, mem_def]
+  -- From π bounds: 3.1415 < π < 3.1416 (Real.pi_gt_d4, Real.pi_lt_d4)
+  -- So √π is between √3.1415 ≈ 1.7724 and √3.1416 ≈ 1.7725
+  -- Thus 2/√π is between 2/1.7725 ≈ 1.1283 and 2/1.7724 ≈ 1.1285
+  have hpi_lo : (3.1415 : ℝ) < Real.pi := Real.pi_gt_d4
+  have hpi_hi : Real.pi < (3.1416 : ℝ) := Real.pi_lt_d4
+  have hsqrt_lo : (1.7724 : ℝ) < Real.sqrt Real.pi := by
+    have h1 : (1.7724 : ℝ) ^ 2 < Real.pi := by
+      have : (1.7724 : ℝ) ^ 2 = 3.14140176 := by ring
+      linarith
+    have h2 : (0 : ℝ) ≤ 1.7724 := by norm_num
+    have h3 : (0 : ℝ) ≤ 1.7724 ^ 2 := by positivity
+    calc (1.7724 : ℝ) = Real.sqrt (1.7724 ^ 2) := (Real.sqrt_sq h2).symm
+      _ < Real.sqrt Real.pi := Real.sqrt_lt_sqrt h3 h1
+  have hsqrt_hi : Real.sqrt Real.pi < (1.7725 : ℝ) := by
+    have h1 : Real.pi < (1.7725 : ℝ) ^ 2 := by
+      have : (1.7725 : ℝ) ^ 2 = 3.14175625 := by ring
+      linarith
+    have hpi_pos : (0 : ℝ) < Real.pi := Real.pi_pos
+    rw [← Real.sqrt_sq (le_of_lt (by norm_num : (0 : ℝ) < 1.7725))]
+    exact Real.sqrt_lt_sqrt (le_of_lt hpi_pos) h1
+  constructor
+  · -- Goal: ↑(1128 / 1000) ≤ 2 / √π
+    have h1 : ((1128 / 1000 : ℚ) : ℝ) < 2 / 1.7725 := by norm_num
+    have h2 : (2 : ℝ) / 1.7725 < 2 / Real.sqrt Real.pi := by
+      apply div_lt_div_of_pos_left (by norm_num : (0 : ℝ) < 2)
+      · exact Real.sqrt_pos.mpr Real.pi_pos
+      · exact hsqrt_hi
+    exact le_of_lt (lt_trans h1 h2)
+  · -- Goal: 2 / √π ≤ ↑(1129 / 1000)
+    have h1 : 2 / Real.sqrt Real.pi < (2 : ℝ) / 1.7724 := by
+      apply div_lt_div_of_pos_left (by norm_num : (0 : ℝ) < 2)
+      · norm_num
+      · exact hsqrt_lo
+    have h2 : (2 : ℝ) / 1.7724 < ((1129 / 1000 : ℚ) : ℝ) := by norm_num
+    exact le_of_lt (lt_trans h1 h2)
+
+/-- The "inner" erf function without the 2/√π factor:
+    erfInner(x) = ∫₀ˣ exp(-t²) dt = (√π/2) * erf(x)
+
+    This is the function whose Taylor series coefficients are erfTaylorCoeffs. -/
+noncomputable def erfInner (x : ℝ) : ℝ := ∫ t in (0 : ℝ)..x, Real.exp (-(t^2))
+
+/-- erf(x) = (2/√π) * erfInner(x) -/
+theorem erf_eq_factor_mul_inner (x : ℝ) : Real.erf x = (2 / Real.sqrt Real.pi) * erfInner x := by
+  unfold Real.erf erfInner
+  ring
+
+/-- The derivatives of erfInner(x) = ∫₀ˣ exp(-t²) dt.
+    erfInner'(x) = exp(-x²)
+    erfInner''(x) = -2x * exp(-x²)
+    etc. -/
+theorem erfInner_deriv : deriv erfInner = fun x => Real.exp (-(x^2)) := by
+  ext x
+  unfold erfInner
+  have hcont : Continuous (fun t => Real.exp (-(t^2))) :=
+    Real.continuous_exp.comp (continuous_neg.comp (continuous_pow 2))
+  exact hcont.integral_hasStrictDerivAt 0 x |>.hasDerivAt.deriv
+
+/-- exp(-x²) is smooth. -/
+theorem contDiff_exp_neg_sq : ContDiff ℝ ⊤ (fun x => Real.exp (-(x^2))) :=
+  Real.contDiff_exp.comp (contDiff_neg.comp (contDiff_id.pow 2))
+
+/-- exp(-x²) is analytic everywhere. -/
+theorem analyticAt_exp_neg_sq (x : ℝ) : AnalyticAt ℝ (fun t => Real.exp (-(t^2))) x := by
+  have h1 : AnalyticAt ℝ (fun t : ℝ => t^2) x := analyticAt_id.pow 2
+  have h2 : AnalyticAt ℝ (fun t : ℝ => -(t^2)) x := h1.neg
+  exact h2.rexp
+
+/-- exp(-x²) is analytic on ℝ. -/
+theorem analyticOnNhd_exp_neg_sq : AnalyticOnNhd ℝ (fun t => Real.exp (-(t^2))) Set.univ :=
+  fun x _ => analyticAt_exp_neg_sq x
+
+/-- Helper: erfInner is C^n for all n. -/
+private theorem erfInner_contDiff_nat (n : ℕ) : ContDiff ℝ n erfInner := by
+  induction n with
+  | zero =>
+    -- C^0 = continuous
+    refine contDiff_zero.mpr ?_
+    have hcont : Continuous (fun t => Real.exp (-(t^2))) :=
+      Real.continuous_exp.comp (continuous_neg.comp (continuous_pow 2))
+    have h_int : ∀ a b : ℝ, IntervalIntegrable (fun t => Real.exp (-(t^2))) MeasureTheory.volume a
+      b :=
+      fun a b => hcont.intervalIntegrable a b
+    exact intervalIntegral.continuous_primitive h_int 0
+  | succ n _ih =>
+    -- C^(n+1) from C^n of derivative
+    have hdiff : Differentiable ℝ erfInner := fun x =>
+      (Real.continuous_exp.comp (continuous_neg.comp (continuous_pow
+        2))).integral_hasStrictDerivAt 0 x
+        |>.hasStrictFDerivAt.differentiableAt
+    have hderiv_smooth : ContDiff ℝ n (deriv erfInner) := by
+      simp only [erfInner_deriv]
+      exact contDiff_exp_neg_sq.of_le le_top
+    exact contDiff_succ_iff_deriv.mpr ⟨hdiff, by simp, hderiv_smooth⟩
+
+/-! ### Computable atanh interval via endpoint evaluation -/
+
+/-- Computable interval enclosure for atanh using endpoint evaluation.
+    Since atanh is strictly increasing on (-1, 1), we evaluate at endpoints.
+    Requires the interval to be strictly inside (-1, 1); returns a wide fallback otherwise. -/
+def atanhComputable (I : IntervalRat) (n : ℕ := 15) : IntervalRat :=
+  if I.lo ≤ -1 then
+    ⟨-1000, 1000, by norm_num⟩
+  else if 1 ≤ I.hi then
+    ⟨-1000, 1000, by norm_num⟩
+  else
+    let atanhLo := atanhPointComputable I.lo n
+    let atanhHi := atanhPointComputable I.hi n
+    hull atanhLo atanhHi
+
+/-- Interval `atanh` using Taylor coefficients prepared for depth `n`. -/
+def atanhComputableWithCoeffs (I : IntervalRat) (n : ℕ)
+    (coeffs : List ℚ) : IntervalRat :=
+  if I.lo ≤ -1 then
+    ⟨-1000, 1000, by norm_num⟩
+  else if 1 ≤ I.hi then
+    ⟨-1000, 1000, by norm_num⟩
+  else
+    let atanhLo := atanhPointComputableWithCoeffs I.lo n coeffs
+    let atanhHi := atanhPointComputableWithCoeffs I.hi n coeffs
+    hull atanhLo atanhHi
+
+theorem atanhComputableWithCoeffs_eq (I : IntervalRat) (n : ℕ) :
+    atanhComputableWithCoeffs I n (atanhTaylorCoeffs n) = atanhComputable I n := by
+  simp [atanhComputableWithCoeffs, atanhComputable, atanhPointComputableWithCoeffs_eq]
+
+/-- FTIA for atanhComputable: if x ∈ I and I ⊂ (-1, 1), then atanh(x) ∈ atanhComputable I n. -/
+theorem mem_atanhComputable {x : ℝ} {I : IntervalRat} (hx : x ∈ I)
+    (hlo : -1 < I.lo) (hhi : I.hi < 1) (n : ℕ) :
+    Real.atanh x ∈ atanhComputable I n := by
+  simp only [atanhComputable, not_le.mpr hlo, ↓reduceIte, not_le.mpr hhi, ↓reduceIte]
+  -- x ∈ I ⊂ (-1, 1), so |lo|, |hi|, |x| < 1
+  have hlo_real : (-1 : ℝ) < I.lo := by exact_mod_cast hlo
+  have hhi_real : (I.hi : ℝ) < 1 := by exact_mod_cast hhi
+  have hx_lo : -1 < x := by linarith [hx.1]
+  have hx_hi : x < 1 := by linarith [hx.2]
+  have hle : (I.lo : ℝ) ≤ I.hi := by exact_mod_cast I.le
+  have habs_lo : |(I.lo : ℝ)| < 1 := by rw [abs_lt]; constructor <;> linarith
+  have habs_hi : |(I.hi : ℝ)| < 1 := by rw [abs_lt]; constructor <;> linarith
+  have habs_x : |x| < 1 := by rw [abs_lt]; constructor <;> linarith
+  -- atanh at endpoints
+  have hlo_abs_rat : |(I.lo : ℝ)| < 1 := habs_lo
+  have hhi_abs_rat : |(I.hi : ℝ)| < 1 := habs_hi
+  have hlo_mem := mem_atanhPointComputable I.lo n hlo_abs_rat
+  have hhi_mem := mem_atanhPointComputable I.hi n hhi_abs_rat
+  -- Monotonicity: atanh(lo) ≤ atanh(x) ≤ atanh(hi)
+  have hatanh_lo_le : Real.atanh I.lo ≤ Real.atanh x :=
+    Real.atanh_mono habs_lo habs_x hx.1
+  have hatanh_x_le_hi : Real.atanh x ≤ Real.atanh I.hi :=
+    Real.atanh_mono habs_x habs_hi hx.2
+  simp only [hull, mem_def, Rat.cast_min, Rat.cast_max]
+  constructor
+  · calc (min (atanhPointComputable I.lo n).lo (atanhPointComputable I.hi n).lo : ℝ)
+        ≤ (atanhPointComputable I.lo n).lo := by exact_mod_cast min_le_left _ _
+      _ ≤ Real.atanh I.lo := hlo_mem.1
+      _ ≤ Real.atanh x := hatanh_lo_le
+  · calc Real.atanh x ≤ Real.atanh I.hi := hatanh_x_le_hi
+      _ ≤ (atanhPointComputable I.hi n).hi := hhi_mem.2
+      _ ≤ max ((atanhPointComputable I.lo n).hi : ℝ) ((atanhPointComputable I.hi n).hi : ℝ) :=
+        le_max_right _ _
+
+end IntervalRat
+
+end LeanCert.Core
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Rational Endpoint Intervals
+
+This file defines `IntervalRat`, a concrete interval type with rational endpoints
+suitable for computation. We prove the Fundamental Theorem of Interval Arithmetic
+(FTIA) for each operation.
+
+## Module Structure
+
+* `IntervalRat.Basic` - Core types: `IntervalRat`, membership, basic operations
+* `IntervalRat.Transcendental` - Noncomputable transcendental bounds (exp, log, sqrt, atanh)
+* `IntervalRat.Taylor` - Computable Taylor series evaluators
+
+## Main definitions
+
+* `LeanCert.Core.IntervalRat` - Intervals with rational endpoints
+* `LeanCert.Core.IntervalRat.toSet` - Semantic interpretation as a subset of ℝ
+* Operations: `add`, `neg`, `sub`, `mul`, `inv`, `div`
+* Computable: `expComputable`, `sinComputable`, `cosComputable`
+
+## Main theorems
+
+* `mem_add` - FTIA for addition
+* `mem_neg` - FTIA for negation
+* `mem_sub` - FTIA for subtraction
+* `mem_mul` - FTIA for multiplication
+* `mem_expComputable`, `mem_sinComputable`, `mem_cosComputable` - FTIA for computable functions
+
+## Design notes
+
+All operations maintain the invariant `lo ≤ hi`. Domain restrictions for partial
+operations (like `inv`) are encoded via separate types or explicit hypotheses.
+-/
+
+@[expose] public section
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2025 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Dyadic Intervals
+
+Intervals with Dyadic endpoints. These support "Outward Rounding", ensuring
+mathematical soundness even when we limit numerical precision.
+
+## Main definitions
+
+* `IntervalDyadic` - Interval with Dyadic endpoints
+* `IntervalDyadic.add`, `mul`, `neg` - Interval arithmetic operations
+* `IntervalDyadic.roundOut` - Outward rounding to control precision
+* `IntervalDyadic.toIntervalRat` - Convert to rational interval for verification
+
+## Design notes
+
+The key feature is `roundOut`: after each operation, we can enforce a minimum
+exponent to prevent precision explosion. When rounding:
+- Lower bounds are shifted down (toward -∞)
+- Upper bounds are shifted up (toward +∞)
+
+This maintains the containment invariant: the rounded interval always contains
+the original interval, which contains the true mathematical value.
+
+## Performance
+
+In v1.0, `Rat` multiplication of `1/3 * 1/3 * ... * 1/3` (10 times) creates
+a denominator of 3^10 = 59049. Each operation requires GCD computation.
+
+In v1.1, with `precision = -10`, the denominator stays fixed at 2^10 = 1024.
+The result is slightly less tight but computed significantly faster.
+-/
+
+@[expose] public section
+
+namespace LeanCert.Core
+
+/-- An interval with Dyadic endpoints.
+
+Maintains invariant: `lo.toRat ≤ hi.toRat` -/
+structure IntervalDyadic where
+  /-- Lower bound -/
+  lo : Dyadic
+  /-- Upper bound -/
+  hi : Dyadic
+  /-- Invariant: lower bound ≤ upper bound -/
+  le : lo.toRat ≤ hi.toRat
+  deriving Repr
+
+namespace IntervalDyadic
+
+/-! ### Membership and Sets -/
+
+/-- The set of reals contained in this interval -/
+def toSet (I : IntervalDyadic) : Set ℝ := Set.Icc (I.lo : ℝ) (I.hi : ℝ)
+
+/-- Membership in a Dyadic interval -/
+instance : Membership ℝ IntervalDyadic where
+  mem I x := (I.lo.toRat : ℝ) ≤ x ∧ x ≤ (I.hi.toRat : ℝ)
+
+@[simp]
+theorem mem_def (x : ℝ) (I : IntervalDyadic) :
+    x ∈ I ↔ (I.lo.toRat : ℝ) ≤ x ∧ x ≤ (I.hi.toRat : ℝ) := Iff.rfl
+
+/-! ### Conversion to IntervalRat -/
+
+/-- Convert to IntervalRat for verification with existing theorems -/
+def toIntervalRat (I : IntervalDyadic) : IntervalRat :=
+  ⟨I.lo.toRat, I.hi.toRat, I.le⟩
+
+/-- Membership is preserved by conversion to IntervalRat -/
+theorem mem_toIntervalRat {x : ℝ} {I : IntervalDyadic} :
+    x ∈ I ↔ x ∈ I.toIntervalRat := by
+  simp only [mem_def, toIntervalRat, IntervalRat.mem_def]
+
+/-! ### Construction -/
+
+/-- Create a singleton interval from a Dyadic -/
+def singleton (d : Dyadic) : IntervalDyadic := ⟨d, d, le_refl _⟩
+
+/-- A Dyadic value is in its singleton interval -/
+theorem mem_singleton (d : Dyadic) : (d.toRat : ℝ) ∈ singleton d := by
+  simp only [mem_def, singleton]
+  exact ⟨le_refl _, le_refl _⟩
+
+/-- Default interval [0, 0] -/
+instance : Inhabited IntervalDyadic where
+  default := singleton Dyadic.zero
+
+/-- Create an interval, checking the invariant -/
+def mk? (lo hi : Dyadic) : Option IntervalDyadic :=
+  if h : lo.toRat ≤ hi.toRat then some ⟨lo, hi, h⟩ else none
+
+/-- The width of an interval -/
+def width (I : IntervalDyadic) : Dyadic := I.hi.sub I.lo
+
+/-- The dyadic width denotes endpoint subtraction exactly. -/
+@[simp]
+theorem width_toRat (I : IntervalDyadic) :
+    I.width.toRat = I.hi.toRat - I.lo.toRat := by
+  simp only [width, Dyadic.sub, Dyadic.toRat_add, Dyadic.toRat_neg]
+  ring
+
+/-- Exact midpoint of a dyadic interval.
+
+Dyadics are closed under division by two, so midpoint construction must lower
+the exponent rather than shift (and potentially round) the mantissa. -/
+def midpoint (I : IntervalDyadic) : Dyadic :=
+  (I.lo.add I.hi).scale2 (-1)
+
+/-- The dyadic midpoint denotes the ordinary rational midpoint exactly. -/
+@[simp]
+theorem midpoint_toRat (I : IntervalDyadic) :
+    I.midpoint.toRat = (I.lo.toRat + I.hi.toRat) / 2 := by
+  simp only [midpoint, Dyadic.toRat_scale2, Dyadic.toRat_add]
+  norm_num [div_eq_mul_inv]
+
+/-- The lower endpoint is at most the exact midpoint. -/
+theorem lo_le_midpoint (I : IntervalDyadic) : I.lo.toRat ≤ I.midpoint.toRat := by
+  rw [midpoint_toRat]
+  linarith [I.le]
+
+/-- The exact midpoint is at most the upper endpoint. -/
+theorem midpoint_le_hi (I : IntervalDyadic) : I.midpoint.toRat ≤ I.hi.toRat := by
+  rw [midpoint_toRat]
+  linarith [I.le]
+
+/-- Split a dyadic interval into two exact closed halves. -/
+def bisect (I : IntervalDyadic) : IntervalDyadic × IntervalDyadic :=
+  (⟨I.lo, I.midpoint, I.lo_le_midpoint⟩,
+   ⟨I.midpoint, I.hi, I.midpoint_le_hi⟩)
+
+/-- A point in the original interval and left of the midpoint is in the left half. -/
+theorem mem_bisect_left {x : ℝ} {I : IntervalDyadic} (hx : x ∈ I)
+    (hm : x ≤ (I.midpoint.toRat : ℝ)) : x ∈ (I.bisect).1 := by
+  exact ⟨hx.1, hm⟩
+
+/-- A point in the original interval and right of the midpoint is in the right half. -/
+theorem mem_bisect_right {x : ℝ} {I : IntervalDyadic} (hx : x ∈ I)
+    (hm : (I.midpoint.toRat : ℝ) ≤ x) : x ∈ (I.bisect).2 := by
+  exact ⟨hm, hx.2⟩
+
+/-- Membership in the left half implies membership in the original interval. -/
+theorem mem_of_mem_bisect_left {x : ℝ} {I : IntervalDyadic}
+    (hx : x ∈ (I.bisect).1) : x ∈ I := by
+  refine ⟨hx.1, hx.2.trans ?_⟩
+  exact_mod_cast I.midpoint_le_hi
+
+/-- Membership in the right half implies membership in the original interval. -/
+theorem mem_of_mem_bisect_right {x : ℝ} {I : IntervalDyadic}
+    (hx : x ∈ (I.bisect).2) : x ∈ I := by
+  refine ⟨?_, hx.2⟩
+  exact (by exact_mod_cast I.lo_le_midpoint : (I.lo.toRat : ℝ) ≤ I.midpoint.toRat) |>.trans hx.1
+
+/-- The two closed halves cover the original interval, including their shared seam. -/
+theorem mem_bisect_or {x : ℝ} {I : IntervalDyadic} (hx : x ∈ I) :
+    x ∈ (I.bisect).1 ∨ x ∈ (I.bisect).2 := by
+  by_cases hm : x ≤ (I.midpoint.toRat : ℝ)
+  · exact Or.inl (mem_bisect_left hx hm)
+  · exact Or.inr (mem_bisect_right hx (le_of_lt (not_le.mp hm)))
+
+/-- Both closed halves contain the midpoint seam. -/
+theorem midpoint_mem_bisect_both (I : IntervalDyadic) :
+    (I.midpoint.toRat : ℝ) ∈ (I.bisect).1 ∧
+      (I.midpoint.toRat : ℝ) ∈ (I.bisect).2 := by
+  constructor
+  · exact ⟨by exact_mod_cast I.lo_le_midpoint, le_rfl⟩
+  · exact ⟨le_rfl, by exact_mod_cast I.midpoint_le_hi⟩
+
+/-- The overlap of the two closed children is exactly their midpoint seam. -/
+theorem bisect_intersection (I : IntervalDyadic) :
+    I.bisect.1.toSet ∩ I.bisect.2.toSet = {I.midpoint.toReal} := by
+  ext x
+  simp only [Set.mem_inter_iff, Set.mem_singleton_iff]
+  constructor
+  · intro hx
+    have hx' :
+        ((I.lo.toRat : ℝ) ≤ x ∧ x ≤ (I.midpoint.toRat : ℝ)) ∧
+          ((I.midpoint.toRat : ℝ) ≤ x ∧ x ≤ (I.hi.toRat : ℝ)) := by
+      simpa [toSet, bisect, Dyadic.toReal] using hx
+    exact le_antisymm hx'.1.2 hx'.2.1
+  · rintro rfl
+    simpa [toSet, bisect, Dyadic.toReal] using I.midpoint_mem_bisect_both
+
+/-- The left child has exactly half the semantic width of its parent. -/
+theorem bisect_left_width_toRat (I : IntervalDyadic) :
+    I.bisect.1.width.toRat = I.width.toRat / 2 := by
+  simp only [width_toRat, bisect, midpoint_toRat]
+  ring
+
+/-- The right child has exactly half the semantic width of its parent. -/
+theorem bisect_right_width_toRat (I : IntervalDyadic) :
+    I.bisect.2.width.toRat = I.width.toRat / 2 := by
+  simp only [width_toRat, bisect, midpoint_toRat]
+  ring
+
+/-! ### Outward Rounding -/
+
+/-- Outward rounding: enforces a maximum precision (minimum exponent).
+
+This is the key operation for preventing precision explosion.
+`minExp` is the minimum allowed exponent (higher = coarser precision).
+
+For example, `minExp = -10` ensures all values are multiples of 2^(-10) ≈ 0.001.
+-/
+def roundOut (I : IntervalDyadic) (minExp : Int) : IntervalDyadic :=
+  let newLo := I.lo.shiftDown minExp
+  let newHi := I.hi.shiftUp minExp
+  ⟨newLo, newHi, by
+    -- Proof: newLo ≤ I.lo ≤ I.hi ≤ newHi
+    have h1 : newLo.toRat ≤ I.lo.toRat := Dyadic.toRat_shiftDown_le I.lo minExp
+    have h2 : I.lo.toRat ≤ I.hi.toRat := I.le
+    have h3 : I.hi.toRat ≤ newHi.toRat := Dyadic.toRat_shiftUp_ge I.hi minExp
+    linarith⟩
+
+/-- roundOut produces an interval containing the original -/
+theorem roundOut_contains {x : ℝ} {I : IntervalDyadic} (hx : x ∈ I) (minExp : Int) :
+    x ∈ I.roundOut minExp := by
+  simp only [mem_def] at *
+  constructor
+  · have h := Dyadic.toRat_shiftDown_le I.lo minExp
+    have h' : ((I.lo.shiftDown minExp).toRat : ℝ) ≤ (I.lo.toRat : ℝ) := by exact_mod_cast h
+    calc ((I.roundOut minExp).lo.toRat : ℝ) = ((I.lo.shiftDown minExp).toRat : ℝ) := by rfl
+      _ ≤ (I.lo.toRat : ℝ) := h'
+      _ ≤ x := hx.1
+  · have h := Dyadic.toRat_shiftUp_ge I.hi minExp
+    have h' : (I.hi.toRat : ℝ) ≤ ((I.hi.shiftUp minExp).toRat : ℝ) := by exact_mod_cast h
+    calc x ≤ (I.hi.toRat : ℝ) := hx.2
+      _ ≤ ((I.hi.shiftUp minExp).toRat : ℝ) := h'
+      _ = ((I.roundOut minExp).hi.toRat : ℝ) := by rfl
+
+/-! ### Interval Negation -/
+
+/-- Negate an interval -/
+def neg (I : IntervalDyadic) : IntervalDyadic where
+  lo := I.hi.neg
+  hi := I.lo.neg
+  le := by
+    simp only [Dyadic.toRat_neg]
+    linarith [I.le]
+
+/-- FTIA for negation -/
+theorem mem_neg {x : ℝ} {I : IntervalDyadic} (hx : x ∈ I) : -x ∈ neg I := by
+  simp only [mem_def] at *
+  simp only [neg, Dyadic.toRat_neg, Rat.cast_neg]
+  constructor <;> linarith
+
+/-! ### Interval Addition -/
+
+/-- Add two intervals -/
+def add (I J : IntervalDyadic) : IntervalDyadic where
+  lo := I.lo.add J.lo
+  hi := I.hi.add J.hi
+  le := by
+    simp only [Dyadic.toRat_add]
+    linarith [I.le, J.le]
+
+/-- FTIA for addition -/
+theorem mem_add {x y : ℝ} {I J : IntervalDyadic} (hx : x ∈ I) (hy : y ∈ J) :
+    x + y ∈ add I J := by
+  simp only [mem_def] at *
+  simp only [add, Dyadic.toRat_add, Rat.cast_add]
+  constructor <;> linarith
+
+/-- Add with precision control -/
+def addRounded (I J : IntervalDyadic) (prec : Int := -53) : IntervalDyadic :=
+  (add I J).roundOut prec
+
+/-! ### Interval Subtraction -/
+
+/-- Subtract two intervals -/
+def sub (I J : IntervalDyadic) : IntervalDyadic := add I (neg J)
+
+/-- FTIA for subtraction -/
+theorem mem_sub {x y : ℝ} {I J : IntervalDyadic} (hx : x ∈ I) (hy : y ∈ J) :
+    x - y ∈ sub I J := by
+  simp only [sub_eq_add_neg]
+  exact mem_add hx (mem_neg hy)
+
+/-! ### Interval Multiplication -/
+
+/-- Multiply two intervals.
+
+Uses min/max of all four endpoint products to handle signs correctly.
+This is the exact multiplication - mantissas may grow. Use `mulRounded` or
+`mulNormalized` for controlled precision.
+
+Correctness: For x ∈ [a,b] and y ∈ [c,d], the product x*y lies in the interval
+[min(ac,ad,bc,bd), max(ac,ad,bc,bd)]. -/
+def mul (I J : IntervalDyadic) : IntervalDyadic :=
+  let v1 := I.lo.mul J.lo
+  let v2 := I.lo.mul J.hi
+  let v3 := I.hi.mul J.lo
+  let v4 := I.hi.mul J.hi
+  ⟨Dyadic.min4 v1 v2 v3 v4, Dyadic.max4 v1 v2 v3 v4, Dyadic.min4_le_max4 v1 v2 v3 v4⟩
+
+/-- Fast interval multiplication using sign-based case splitting.
+    Reduces from 4 multiplications + 12 comparisons to 2 multiplications
+    in the common case (both intervals positive or both negative).
+    Falls back to the full 4-way product for mixed-sign intervals. -/
+def mulFast (I J : IntervalDyadic) : IntervalDyadic :=
+  if hI : Dyadic.le 0 I.lo then
+    if hJ : Dyadic.le 0 J.lo then
+      ⟨I.lo.mul J.lo, I.hi.mul J.hi, by
+        simp only [Dyadic.toRat_mul]
+        have := (Dyadic.le_iff_toRat_le 0 I.lo).mp hI
+        have := (Dyadic.le_iff_toRat_le 0 J.lo).mp hJ
+        simp only [Dyadic.toRat_zero] at *
+        nlinarith [I.le, J.le]⟩
+    else if hJ2 : Dyadic.le J.hi 0 then
+      ⟨I.hi.mul J.lo, I.lo.mul J.hi, by
+        simp only [Dyadic.toRat_mul]
+        have := (Dyadic.le_iff_toRat_le 0 I.lo).mp hI
+        have := (Dyadic.le_iff_toRat_le J.hi 0).mp hJ2
+        simp only [Dyadic.toRat_zero] at *
+        nlinarith [I.le, J.le]⟩
+    else
+      ⟨I.hi.mul J.lo, I.hi.mul J.hi, by
+        simp only [Dyadic.toRat_mul]
+        have := (Dyadic.le_iff_toRat_le 0 I.lo).mp hI
+        simp only [Dyadic.toRat_zero] at *
+        nlinarith [I.le, J.le]⟩
+  else if hI2 : Dyadic.le I.hi 0 then
+    if hJ : Dyadic.le 0 J.lo then
+      ⟨I.lo.mul J.hi, I.hi.mul J.lo, by
+        simp only [Dyadic.toRat_mul]
+        have := (Dyadic.le_iff_toRat_le I.hi 0).mp hI2
+        have := (Dyadic.le_iff_toRat_le 0 J.lo).mp hJ
+        simp only [Dyadic.toRat_zero] at *
+        nlinarith [I.le, J.le]⟩
+    else if hJ2 : Dyadic.le J.hi 0 then
+      ⟨I.hi.mul J.hi, I.lo.mul J.lo, by
+        simp only [Dyadic.toRat_mul]
+        have := (Dyadic.le_iff_toRat_le I.hi 0).mp hI2
+        have := (Dyadic.le_iff_toRat_le J.hi 0).mp hJ2
+        simp only [Dyadic.toRat_zero] at *
+        nlinarith [I.le, J.le]⟩
+    else
+      ⟨I.lo.mul J.hi, I.lo.mul J.lo, by
+        simp only [Dyadic.toRat_mul]
+        have := (Dyadic.le_iff_toRat_le I.hi 0).mp hI2
+        simp only [Dyadic.toRat_zero] at *
+        nlinarith [I.le, J.le]⟩
+  else
+    if hJ : Dyadic.le 0 J.lo then
+      ⟨I.lo.mul J.hi, I.hi.mul J.hi, by
+        simp only [Dyadic.toRat_mul]
+        have := (Dyadic.le_iff_toRat_le 0 J.lo).mp hJ
+        simp only [Dyadic.toRat_zero] at *
+        nlinarith [I.le, J.le]⟩
+    else if hJ2 : Dyadic.le J.hi 0 then
+      ⟨I.hi.mul J.lo, I.lo.mul J.lo, by
+        simp only [Dyadic.toRat_mul]
+        have := (Dyadic.le_iff_toRat_le J.hi 0).mp hJ2
+        simp only [Dyadic.toRat_zero] at *
+        nlinarith [I.le, J.le]⟩
+    else
+      mul I J
+
+/- We intentionally do not attach `mulFast` as a native replacement here.
+   Native certificate checking should execute the same definition that the
+   kernel proofs reason about. Keep `mulFast` available only as an internal
+   candidate for future explicitly-audited runtime backends. -/
+
+/-- FTIA for multiplication -/
+theorem mem_mul {x y : ℝ} {I J : IntervalDyadic} (hx : x ∈ I) (hy : y ∈ J) :
+    x * y ∈ mul I J := by
+  -- Use IntervalRat.mem_mul and translate the bounds
+  have hxI := mem_toIntervalRat.mp hx
+  have hyJ := mem_toIntervalRat.mp hy
+  have hmul := IntervalRat.mem_mul hxI hyJ
+  simp only [mem_def, mul] at *
+  -- hmul has bounds from IntervalRat.mul which are min/max of 4 products
+  -- Our Dyadic.min4/max4 has the same structure with toRat preserving the products
+  constructor
+  · -- Lower bound: Dyadic.min4.toRat equals the rational min4 from hmul
+    simp only [IntervalRat.mem_def, IntervalRat.mul, toIntervalRat] at hmul
+    have heq : (Dyadic.min4 (I.lo.mul J.lo) (I.lo.mul J.hi) (I.hi.mul J.lo) (I.hi.mul J.hi)).toRat
+        = min (min (I.lo.toRat * J.lo.toRat) (I.lo.toRat * J.hi.toRat))
+              (min (I.hi.toRat * J.lo.toRat) (I.hi.toRat * J.hi.toRat)) := by
+      simp only [Dyadic.min4, Dyadic.toRat_mul, Dyadic.min_toRat]
+    rw [heq]
+    exact_mod_cast hmul.1
+  · -- Upper bound: Dyadic.max4.toRat equals the rational max4 from hmul
+    simp only [IntervalRat.mem_def, IntervalRat.mul, toIntervalRat] at hmul
+    have heq : (Dyadic.max4 (I.lo.mul J.lo) (I.lo.mul J.hi) (I.hi.mul J.lo) (I.hi.mul J.hi)).toRat
+        = max (max (I.lo.toRat * J.lo.toRat) (I.lo.toRat * J.hi.toRat))
+              (max (I.hi.toRat * J.lo.toRat) (I.hi.toRat * J.hi.toRat)) := by
+      simp only [Dyadic.max4, Dyadic.toRat_mul, Dyadic.max_toRat]
+    rw [heq]
+    exact_mod_cast hmul.2
+
+/-- Dyadic.min4 converts to rational min4 -/
+private theorem min4_toRat (a b c d : Dyadic) :
+    (Dyadic.min4 a b c d).toRat = IntervalRat.min4 a.toRat b.toRat c.toRat d.toRat := by
+  simp only [Dyadic.min4, IntervalRat.min4, Dyadic.min_toRat]
+
+/-- Dyadic.max4 converts to rational max4 -/
+private theorem max4_toRat (a b c d : Dyadic) :
+    (Dyadic.max4 a b c d).toRat = IntervalRat.max4 a.toRat b.toRat c.toRat d.toRat := by
+  simp only [Dyadic.max4, IntervalRat.max4, Dyadic.max_toRat]
+
+/-- Extract 0 ≤ toRat from Dyadic.le 0 d -/
+private theorem toRat_nonneg_of_le {d : Dyadic} (h : Dyadic.le 0 d) : 0 ≤ d.toRat := by
+  have := (Dyadic.le_iff_toRat_le 0 d).mp h; rwa [Dyadic.toRat_zero] at this
+
+/-- Extract toRat ≤ 0 from Dyadic.le d 0 -/
+private theorem toRat_nonpos_of_le {d : Dyadic} (h : Dyadic.le d 0) : d.toRat ≤ 0 := by
+  have := (Dyadic.le_iff_toRat_le d 0).mp h; rwa [Dyadic.toRat_zero] at this
+
+/-- Extract toRat < 0 from ¬ Dyadic.le 0 d -/
+private theorem toRat_neg_of_not_le {d : Dyadic} (h : ¬ Dyadic.le 0 d) : d.toRat < 0 := by
+  exact lt_of_not_ge fun h' => h ((Dyadic.le_iff_toRat_le 0 d).mpr (by rwa [Dyadic.toRat_zero]))
+
+/-- Extract 0 < toRat from ¬ Dyadic.le d 0 -/
+private theorem toRat_pos_of_not_le {d : Dyadic} (h : ¬ Dyadic.le d 0) : 0 < d.toRat := by
+  exact lt_of_not_ge fun h' => h ((Dyadic.le_iff_toRat_le d 0).mpr (by rwa [Dyadic.toRat_zero]))
+
+/-- mulFast endpoints match mul endpoints: lo -/
+private theorem mulFast_lo (I J : IntervalDyadic) : (mulFast I J).lo.toRat = (mul I J).lo.toRat :=
+  by
+  have hrhs : (mul I J).lo.toRat = IntervalRat.min4
+      (I.lo.toRat * J.lo.toRat) (I.lo.toRat * J.hi.toRat)
+      (I.hi.toRat * J.lo.toRat) (I.hi.toRat * J.hi.toRat) := by
+    simp only [mul, Dyadic.toRat_mul, min4_toRat]
+  rw [hrhs]; unfold mulFast
+  split
+  · rename_i hI
+    have hIlo := toRat_nonneg_of_le hI
+    split
+    · rename_i hJ
+      have hJlo := toRat_nonneg_of_le hJ
+      simp only [Dyadic.toRat_mul]
+      exact IntervalRat.eq_min4_of_le (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+        (by nlinarith [I.le, J.le])
+    · split
+      · rename_i _ hJ2
+        have hJhi := toRat_nonpos_of_le hJ2
+        simp only [Dyadic.toRat_mul]
+        exact IntervalRat.eq_min4_of_le3 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+      · rename_i hJ hJ2
+        have hJlo := le_of_lt (toRat_neg_of_not_le hJ)
+        have hJhi := le_of_lt (toRat_pos_of_not_le hJ2)
+        simp only [Dyadic.toRat_mul]
+        exact IntervalRat.eq_min4_of_le3 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+  · split
+    · rename_i _ hI2
+      have hIhi := toRat_nonpos_of_le hI2
+      split
+      · rename_i hJ
+        have hJlo := toRat_nonneg_of_le hJ
+        simp only [Dyadic.toRat_mul]
+        exact IntervalRat.eq_min4_of_le2 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+      · split
+        · rename_i _ hJ2
+          have hJhi := toRat_nonpos_of_le hJ2
+          simp only [Dyadic.toRat_mul]
+          exact IntervalRat.eq_min4_of_le4 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+            (by nlinarith [I.le, J.le])
+        · rename_i hJ hJ2
+          have hJlo := le_of_lt (toRat_neg_of_not_le hJ)
+          have hJhi := le_of_lt (toRat_pos_of_not_le hJ2)
+          simp only [Dyadic.toRat_mul]
+          exact IntervalRat.eq_min4_of_le2 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+            (by nlinarith [I.le, J.le])
+    · rename_i hI hI2
+      have hIlo := le_of_lt (toRat_neg_of_not_le hI)
+      have hIhi := le_of_lt (toRat_pos_of_not_le hI2)
+      split
+      · rename_i hJ
+        have hJlo := toRat_nonneg_of_le hJ
+        simp only [Dyadic.toRat_mul]
+        exact IntervalRat.eq_min4_of_le2 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+      · split
+        · rename_i _ hJ2
+          have hJhi := toRat_nonpos_of_le hJ2
+          simp only [Dyadic.toRat_mul]
+          exact IntervalRat.eq_min4_of_le3 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+            (by nlinarith [I.le, J.le])
+        · exact hrhs
+
+/-- mulFast endpoints match mul endpoints: hi -/
+private theorem mulFast_hi (I J : IntervalDyadic) : (mulFast I J).hi.toRat = (mul I J).hi.toRat :=
+  by
+  have hrhs : (mul I J).hi.toRat = IntervalRat.max4
+      (I.lo.toRat * J.lo.toRat) (I.lo.toRat * J.hi.toRat)
+      (I.hi.toRat * J.lo.toRat) (I.hi.toRat * J.hi.toRat) := by
+    simp only [mul, Dyadic.toRat_mul, max4_toRat]
+  rw [hrhs]; unfold mulFast
+  split
+  · rename_i hI
+    have hIlo := toRat_nonneg_of_le hI
+    split
+    · rename_i hJ
+      have hJlo := toRat_nonneg_of_le hJ
+      simp only [Dyadic.toRat_mul]
+      exact IntervalRat.eq_max4_of_ge4 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+        (by nlinarith [I.le, J.le])
+    · split
+      · rename_i _ hJ2
+        have hJhi := toRat_nonpos_of_le hJ2
+        simp only [Dyadic.toRat_mul]
+        exact IntervalRat.eq_max4_of_ge2 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+      · rename_i hJ hJ2
+        have hJlo := le_of_lt (toRat_neg_of_not_le hJ)
+        have hJhi := le_of_lt (toRat_pos_of_not_le hJ2)
+        simp only [Dyadic.toRat_mul]
+        exact IntervalRat.eq_max4_of_ge4 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+  · split
+    · rename_i _ hI2
+      have hIhi := toRat_nonpos_of_le hI2
+      split
+      · rename_i hJ
+        have hJlo := toRat_nonneg_of_le hJ
+        simp only [Dyadic.toRat_mul]
+        exact IntervalRat.eq_max4_of_ge3 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+      · split
+        · rename_i _ hJ2
+          have hJhi := toRat_nonpos_of_le hJ2
+          simp only [Dyadic.toRat_mul]
+          exact IntervalRat.eq_max4_of_ge (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+            (by nlinarith [I.le, J.le])
+        · rename_i hJ hJ2
+          have hJlo := le_of_lt (toRat_neg_of_not_le hJ)
+          have hJhi := le_of_lt (toRat_pos_of_not_le hJ2)
+          simp only [Dyadic.toRat_mul]
+          exact IntervalRat.eq_max4_of_ge (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+            (by nlinarith [I.le, J.le])
+    · rename_i hI hI2
+      have hIlo := le_of_lt (toRat_neg_of_not_le hI)
+      have hIhi := le_of_lt (toRat_pos_of_not_le hI2)
+      split
+      · rename_i hJ
+        have hJlo := toRat_nonneg_of_le hJ
+        simp only [Dyadic.toRat_mul]
+        exact IntervalRat.eq_max4_of_ge4 (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+          (by nlinarith [I.le, J.le])
+      · split
+        · rename_i _ hJ2
+          have hJhi := toRat_nonpos_of_le hJ2
+          simp only [Dyadic.toRat_mul]
+          exact IntervalRat.eq_max4_of_ge (by nlinarith [I.le, J.le]) (by nlinarith [I.le, J.le])
+            (by nlinarith [I.le, J.le])
+        · exact hrhs
+
+/-- `mulFast` preserves the containment property of `mul`.
+    This is retained as documentation and a future audited optimization hook;
+    production certificate checking currently uses `mul` directly. -/
+theorem mem_mulFast {x y : ℝ} {I J : IntervalDyadic} (hx : x ∈ I) (hy : y ∈ J) :
+    x * y ∈ mulFast I J := by
+  simp only [mem_def]
+  constructor
+  · rw [mulFast_lo]; exact (mem_mul hx hy).1
+  · rw [mulFast_hi]; exact (mem_mul hx hy).2
+
+/-- Multiply with precision control (outward rounding) -/
+def mulRounded (I J : IntervalDyadic) (prec : Int := -53) : IntervalDyadic :=
+  (mul I J).roundOut prec
+
+/-- Multiply with mantissa normalization (prevents bit explosion) -/
+def mulNormalized (I J : IntervalDyadic) (maxBits : Nat := 256) : IntervalDyadic :=
+  let result := mul I J
+  ⟨result.lo.normalizeDown maxBits, result.hi.normalizeUp maxBits, by
+    -- Normalization preserves ordering: lo.normalizeDown ≤ lo ≤ hi ≤ hi.normalizeUp
+    calc (result.lo.normalizeDown maxBits).toRat
+        ≤ result.lo.toRat := Dyadic.toRat_normalizeDown_le _ _
+      _ ≤ result.hi.toRat := result.le
+      _ ≤ (result.hi.normalizeUp maxBits).toRat := Dyadic.toRat_normalizeUp_ge _ _⟩
+
+/-! ### Interval Scaling -/
+
+/-- Scale an interval by a constant Dyadic -/
+def scale (I : IntervalDyadic) (c : Dyadic) : IntervalDyadic :=
+  if hc : Dyadic.le Dyadic.zero c then
+    ⟨I.lo.mul c, I.hi.mul c, by
+      rw [Dyadic.toRat_mul, Dyadic.toRat_mul]
+      have hcnn : 0 ≤ c.toRat := by
+        have := Dyadic.le_iff_toRat_le Dyadic.zero c |>.mp hc
+        have hz : Dyadic.zero.toRat = 0 := Dyadic.toRat_zero
+        rw [hz] at this; exact this
+      exact mul_le_mul_of_nonneg_right I.le hcnn⟩
+  else
+    ⟨I.hi.mul c, I.lo.mul c, by
+      rw [Dyadic.toRat_mul, Dyadic.toRat_mul]
+      have hcneg : c.toRat < 0 := by
+        have hf := Dyadic.le_iff_toRat_le Dyadic.zero c
+        have hz : Dyadic.zero.toRat = 0 := Dyadic.toRat_zero
+        rw [hz] at hf
+        by_contra! hge
+        exact hc (hf.mpr hge)
+      exact mul_le_mul_of_nonpos_right I.le (le_of_lt hcneg)⟩
+
+/-- Scale by a power of 2 (very efficient: just adjusts exponents) -/
+def scale2 (I : IntervalDyadic) (n : Int) : IntervalDyadic :=
+  ⟨I.lo.scale2 n, I.hi.scale2 n, Dyadic.toRat_scale2_le_scale2 I.lo I.hi n I.le⟩
+
+/-! ### Square Root -/
+
+/-- Square root of an interval.
+    Returns a conservative bound [0, max(hi, 1)].
+    This is sound because:
+    - sqrt(x) = 0 for x < 0 (by definition in Mathlib)
+    - sqrt(x) ≥ 0 for all x
+    - sqrt(x) ≤ max(x, 1) for x ≥ 0
+    Therefore for any x ∈ [lo, hi], sqrt(x) ∈ [0, max(hi, 1)]. -/
+def sqrt (I : IntervalDyadic) (_prec : Int := -53) : IntervalDyadic :=
+  -- Conservative bound: [0, max(hi, 1)]
+  -- sqrt(x) ≥ 0 for all x (including negative where sqrt returns 0)
+  -- sqrt(x) ≤ max(x, 1) for x ≥ 0
+  -- sqrt(x) = 0 ≤ max(hi, 1) for x < 0
+  let one := Dyadic.ofInt 1
+  let hi_bound := Dyadic.max I.hi one
+  ⟨Dyadic.zero, hi_bound, by
+    have hz : Dyadic.zero.toRat = 0 := Dyadic.toRat_zero
+    rw [Dyadic.max_toRat, Dyadic.toRat_ofInt, hz]
+    exact le_max_of_le_right (by norm_num : (0 : ℚ) ≤ 1)⟩
+
+/-- sqrt(x) ≤ max(x, 1) for all x ≥ 0 -/
+private theorem sqrt_le_max_one {x : ℝ} (hx : 0 ≤ x) : Real.sqrt x ≤ max x 1 := by
+  by_cases! h : x ≤ 1
+  · -- Case x ≤ 1: sqrt(x) ≤ 1 ≤ max(x, 1)
+    -- sqrt(x) ≤ 1 when x ≤ 1 because sqrt(x)^2 = x ≤ 1 and sqrt(x) ≥ 0
+    have hsqrt_le_one : Real.sqrt x ≤ 1 := by
+      rw [← Real.sqrt_one]
+      exact Real.sqrt_le_sqrt h
+    calc Real.sqrt x ≤ 1 := hsqrt_le_one
+      _ ≤ max x 1 := le_max_right x 1
+  · -- Case x > 1: sqrt(x) ≤ x = max(x, 1)
+    -- sqrt(x) ≤ x for x ≥ 1 follows from sqrt(x) * sqrt(x) = x and sqrt(x) ≥ 1
+    have h1 : 1 ≤ Real.sqrt x := by
+      rw [← Real.sqrt_one]
+      exact Real.sqrt_le_sqrt (le_of_lt h)
+    have hsqrt_le_x : Real.sqrt x ≤ x := by
+      calc Real.sqrt x = Real.sqrt x * 1 := by ring
+        _ ≤ Real.sqrt x * Real.sqrt x := mul_le_mul_of_nonneg_left h1 (Real.sqrt_nonneg x)
+        _ = x := by rw [← sq, Real.sq_sqrt hx]
+    calc Real.sqrt x ≤ x := hsqrt_le_x
+      _ ≤ max x 1 := le_max_left x 1
+
+/-- Soundness of interval sqrt: if x ∈ I, x ≥ 0, then Real.sqrt x ∈ sqrt I -/
+theorem mem_sqrt {x : ℝ} {I : IntervalDyadic} (hx : x ∈ I) (hx_nn : 0 ≤ x) (prec : Int) :
+    Real.sqrt x ∈ sqrt I prec := by
+  simp only [sqrt, mem_def, Dyadic.max_toRat, Dyadic.toRat_ofInt]
+  have hz : Dyadic.zero.toRat = 0 := Dyadic.toRat_zero
+  constructor
+  · -- Lower bound: 0 ≤ sqrt(x)
+    simp only [hz, Rat.cast_zero]
+    exact Real.sqrt_nonneg x
+  · -- Upper bound: sqrt(x) ≤ max(hi, 1)
+    have hhi_ge : x ≤ (I.hi.toRat : ℝ) := hx.2
+    calc Real.sqrt x
+        ≤ max x 1 := sqrt_le_max_one hx_nn
+      _ ≤ max (I.hi.toRat : ℝ) 1 := max_le_max_right 1 hhi_ge
+      _ = (↑(max I.hi.toRat 1) : ℝ) := by simp [Rat.cast_max]
+
+/-- General soundness of interval sqrt for any real input.
+    Handles both non-negative inputs and negative inputs (where Real.sqrt returns 0). -/
+theorem mem_sqrt' {x : ℝ} {I : IntervalDyadic} (hx : x ∈ I) (prec : Int) :
+    Real.sqrt x ∈ sqrt I prec := by
+  rcases le_or_gt 0 x with hx_nn | hx_neg
+  · -- Non-negative case
+    exact mem_sqrt hx hx_nn prec
+  · -- Negative case: sqrt(x) = 0 for x < 0
+    have hsqrt_zero : Real.sqrt x = 0 := Real.sqrt_eq_zero'.mpr (le_of_lt hx_neg)
+    simp only [sqrt, mem_def, Dyadic.max_toRat, Dyadic.toRat_ofInt, hsqrt_zero]
+    have hz : Dyadic.zero.toRat = 0 := Dyadic.toRat_zero
+    simp only [hz, Rat.cast_zero, Rat.cast_max]
+    constructor
+    · norm_num
+    · apply le_max_of_le_right
+      norm_num
+
+/-! ### Comparison and Containment -/
+
+/-- Check if interval contains zero -/
+def containsZero (I : IntervalDyadic) : Bool :=
+  Dyadic.le I.lo Dyadic.zero && Dyadic.le Dyadic.zero I.hi
+
+/-- Check if entire interval is positive -/
+def isPositive (I : IntervalDyadic) : Bool :=
+  Dyadic.lt Dyadic.zero I.lo
+
+/-- Check if entire interval is negative -/
+def isNegative (I : IntervalDyadic) : Bool :=
+  Dyadic.lt I.hi Dyadic.zero
+
+/-- Check if I ⊆ J -/
+def subset (I J : IntervalDyadic) : Bool :=
+  Dyadic.le J.lo I.lo && Dyadic.le I.hi J.hi
+
+/-- Check if the upper bound is ≤ a rational -/
+def upperBoundedBy (I : IntervalDyadic) (q : ℚ) : Bool :=
+  I.hi.toRat ≤ q
+
+/-- Check if a rational is ≤ the lower bound -/
+def lowerBoundedBy (I : IntervalDyadic) (q : ℚ) : Bool :=
+  q ≤ I.lo.toRat
+
+/-- Upper bound extraction from membership -/
+theorem le_hi_of_mem {x : ℝ} {I : IntervalDyadic} (hx : x ∈ I) :
+    x ≤ (I.hi.toRat : ℝ) := hx.2
+
+/-- Lower bound extraction from membership -/
+theorem lo_le_of_mem {x : ℝ} {I : IntervalDyadic} (hx : x ∈ I) :
+    (I.lo.toRat : ℝ) ≤ x := hx.1
+
+/-- What upperBoundedBy means: the interval's hi endpoint is ≤ q -/
+theorem upperBoundedBy_spec {I : IntervalDyadic} {q : ℚ}
+    (h : I.upperBoundedBy q = true) : (I.hi.toRat : ℝ) ≤ q := by
+  simp only [upperBoundedBy, decide_eq_true_eq] at h
+  exact_mod_cast h
+
+/-- What lowerBoundedBy means: q is ≤ the interval's lo endpoint -/
+theorem lowerBoundedBy_spec {I : IntervalDyadic} {q : ℚ}
+    (h : I.lowerBoundedBy q = true) : (q : ℝ) ≤ I.lo.toRat := by
+  simp only [lowerBoundedBy, decide_eq_true_eq] at h
+  exact_mod_cast h
+
+/-! ### Helper for Transcendentals -/
+
+/-- Convert from IntervalRat (for transcendental results) -/
+def ofIntervalRat (I : IntervalRat) (prec : Int := -53) : IntervalDyadic :=
+  -- Convert Rat endpoints to Dyadic with outward rounding
+  -- For simplicity, we use a conservative conversion
+  let lo := Dyadic.ofInt (Int.floor (I.lo * (2 ^ (-prec).toNat)))
+  let loD := lo.scale2 prec
+  let hi := Dyadic.ofInt (Int.ceil (I.hi * (2 ^ (-prec).toNat)))
+  let hiD := hi.scale2 prec
+  ⟨loD, hiD, by
+    -- Need: loD.toRat ≤ hiD.toRat
+    -- loD.toRat = floor(I.lo * 2^n) * 2^prec
+    -- hiD.toRat = ceil(I.hi * 2^n) * 2^prec
+    -- This follows from: floor(I.lo * 2^n) ≤ ceil(I.hi * 2^n)
+    -- Which is: floor(I.lo * 2^n) ≤ I.lo * 2^n ≤ I.hi * 2^n ≤ ceil(I.hi * 2^n)
+    apply Dyadic.toRat_scale2_le_scale2
+    rw [Dyadic.toRat_ofInt, Dyadic.toRat_ofInt]
+    have hle : I.lo ≤ I.hi := I.le
+    have hscale_pos : (0 : ℚ) < (2 : ℚ) ^ (-prec).toNat := pow_pos (by norm_num : (0 : ℚ) < 2) _
+    calc (⌊I.lo * (2 : ℚ) ^ (-prec).toNat⌋ : ℚ)
+        ≤ I.lo * (2 : ℚ) ^ (-prec).toNat := Int.floor_le _
+      _ ≤ I.hi * (2 : ℚ) ^ (-prec).toNat := mul_le_mul_of_nonneg_right hle (le_of_lt hscale_pos)
+      _ ≤ ⌈I.hi * (2 : ℚ) ^ (-prec).toNat⌉ := Int.le_ceil _⟩
+
+/-- If x ∈ IntervalRat I, then x ∈ ofIntervalRat I prec (outward rounding preserves membership).
+    Requires precision ≤ 0 (e.g. -53). -/
+theorem mem_ofIntervalRat {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (prec : Int)
+    (hprec : prec ≤ 0 := by norm_num) :
+    x ∈ ofIntervalRat I prec := by
+  -- The conversion uses floor/ceil which provides outward rounding:
+  -- floor(lo * 2^n) * 2^(-n) ≤ lo ≤ x ≤ hi ≤ ceil(hi * 2^n) * 2^(-n)
+  -- This follows from floor(a) ≤ a and a ≤ ceil(a), combined with
+  -- the fact that multiplying by 2^(-n) when prec = -n cancels the 2^n factor.
+  simp only [mem_def, ofIntervalRat, IntervalRat.mem_def] at *
+  have h2n_pos : (0 : ℚ) < (2 : ℚ) ^ (-prec).toNat := pow_pos (by norm_num) _
+  have hfloor : (⌊I.lo * (2 : ℚ) ^ (-prec).toNat⌋ : ℚ) ≤ I.lo * (2 : ℚ) ^ (-prec).toNat :=
+    Int.floor_le _
+  have hceil : I.hi * (2 : ℚ) ^ (-prec).toNat ≤ (⌈I.hi * (2 : ℚ) ^ (-prec).toNat⌉ : ℚ) :=
+    Int.le_ceil _
+  have hzpow_pos : (0 : ℚ) < (2 : ℚ) ^ prec := zpow_pos (by norm_num) _
+  -- Key: 2^(-prec).toNat * 2^prec = 1 when prec ≤ 0 (so (-prec).toNat = -prec)
+  have hcancel : (2 : ℚ) ^ (-prec).toNat * (2 : ℚ) ^ prec = 1 := by
+    set n := (-prec).toNat with hn_def
+    have hprec_eq : prec = -(n : ℤ) := by omega
+    rw [hprec_eq, zpow_neg, zpow_natCast]
+    exact mul_inv_cancel₀ (ne_of_gt h2n_pos)
+  constructor
+  · -- Lower bound
+    rw [Dyadic.toRat_scale2, Dyadic.toRat_ofInt]
+    have hle : (⌊I.lo * (2 : ℚ) ^ (-prec).toNat⌋ : ℚ) * (2 : ℚ) ^ prec ≤ I.lo := by
+      calc (⌊I.lo * (2 : ℚ) ^ (-prec).toNat⌋ : ℚ) * (2 : ℚ) ^ prec
+          ≤ (I.lo * (2 : ℚ) ^ (-prec).toNat) * (2 : ℚ) ^ prec := by
+            apply mul_le_mul_of_nonneg_right hfloor (le_of_lt hzpow_pos)
+        _ = I.lo * ((2 : ℚ) ^ (-prec).toNat * (2 : ℚ) ^ prec) := by ring
+        _ = I.lo * 1 := by rw [hcancel]
+        _ = I.lo := by ring
+    exact le_trans (by exact_mod_cast hle) hx.1
+  · -- Upper bound
+    rw [Dyadic.toRat_scale2, Dyadic.toRat_ofInt]
+    have hle : I.hi ≤ (⌈I.hi * (2 : ℚ) ^ (-prec).toNat⌉ : ℚ) * (2 : ℚ) ^ prec := by
+      calc I.hi = I.hi * 1 := by ring
+        _ = I.hi * ((2 : ℚ) ^ (-prec).toNat * (2 : ℚ) ^ prec) := by rw [hcancel]
+        _ = (I.hi * (2 : ℚ) ^ (-prec).toNat) * (2 : ℚ) ^ prec := by ring
+        _ ≤ (⌈I.hi * (2 : ℚ) ^ (-prec).toNat⌉ : ℚ) * (2 : ℚ) ^ prec := by
+            apply mul_le_mul_of_nonneg_right hceil (le_of_lt hzpow_pos)
+    exact le_trans hx.2 (by exact_mod_cast hle)
+
+end IntervalDyadic
+end LeanCert.Core
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Intervals with Real Endpoints
+
+This file defines `IntervalReal`, an interval type with real (ℝ) endpoints.
+Unlike `IntervalRat` (which has rational endpoints and is suitable for computation),
+`IntervalReal` allows us to represent intervals involving transcendental values
+like `[1, Real.exp 1]` or `[Real.log 2, π]`.
+
+## Main definitions
+
+* `IntervalReal` - Intervals with real endpoints
+* `expInterval` - Interval bound for exp
+* `logInterval` - Interval bound for log (on positive intervals)
+
+## Main theorems
+
+* `mem_expInterval` - FTIA for exp: if x ∈ I, then exp(x) ∈ expInterval(I)
+
+## Design notes
+
+This type complements `IntervalRat` for proving facts about transcendental functions.
+While `IntervalRat` is used for numerical computation (since rationals are computable),
+`IntervalReal` is used for correctness proofs involving exp, log, etc.
+
+Typical workflow:
+1. Use `IntervalRat` for actual interval arithmetic computations
+2. Convert to `IntervalReal` when proving facts about transcendental functions
+3. Use mathlib's analysis lemmas directly on real intervals
+-/
+
+@[expose] public section
+
+namespace LeanCert.Core
+
+/-- An interval with real endpoints -/
+structure IntervalReal where
+  /-- Lower endpoint of the interval. -/
+  lo : ℝ
+  /-- Upper endpoint of the interval. -/
+  hi : ℝ
+  le : lo ≤ hi
+
+namespace IntervalReal
+
+/-- The set of reals contained in this interval -/
+def toSet (I : IntervalReal) : Set ℝ := Set.Icc I.lo I.hi
+
+/-- Membership in an interval -/
+instance : Membership ℝ IntervalReal where
+  mem I x := I.lo ≤ x ∧ x ≤ I.hi
+
+@[simp]
+theorem mem_def (x : ℝ) (I : IntervalReal) : x ∈ I ↔ I.lo ≤ x ∧ x ≤ I.hi :=
+  Iff.rfl
+
+theorem mem_toSet_iff (x : ℝ) (I : IntervalReal) : x ∈ I.toSet ↔ x ∈ I := by
+  simp only [toSet, Set.mem_Icc, mem_def]
+
+/-- Create an interval from a single real -/
+def singleton (r : ℝ) : IntervalReal := ⟨r, r, le_refl r⟩
+
+theorem mem_singleton (r : ℝ) : r ∈ singleton r := ⟨le_refl r, le_refl r⟩
+
+/-- Convert from IntervalRat to IntervalReal -/
+def ofRat (I : IntervalRat) : IntervalReal where
+  lo := I.lo
+  hi := I.hi
+  le := by exact_mod_cast I.le
+
+theorem mem_ofRat {x : ℝ} {I : IntervalRat} (hx : x ∈ I) : x ∈ ofRat I := by
+  simp only [mem_def, ofRat] at *
+  exact hx
+
+/-! ### Interval addition -/
+
+/-- Add two intervals -/
+def add (I J : IntervalReal) : IntervalReal where
+  lo := I.lo + J.lo
+  hi := I.hi + J.hi
+  le := by linarith [I.le, J.le]
+
+/-- FTIA for addition -/
+theorem mem_add {x y : ℝ} {I J : IntervalReal} (hx : x ∈ I) (hy : y ∈ J) :
+    x + y ∈ add I J := by
+  simp only [mem_def] at *
+  simp only [add]
+  constructor <;> linarith
+
+/-! ### Interval negation -/
+
+/-- Negate an interval -/
+def neg (I : IntervalReal) : IntervalReal where
+  lo := -I.hi
+  hi := -I.lo
+  le := by linarith [I.le]
+
+/-- FTIA for negation -/
+theorem mem_neg {x : ℝ} {I : IntervalReal} (hx : x ∈ I) : -x ∈ neg I := by
+  simp only [mem_def] at *
+  simp only [neg]
+  constructor <;> linarith
+
+/-! ### Interval multiplication -/
+
+/-- The minimum of four real endpoints, used for interval multiplication. -/
+def min4 (a b c d : ℝ) : ℝ := min (min a b) (min c d)
+/-- The maximum of four real endpoints, used for interval multiplication. -/
+def max4 (a b c d : ℝ) : ℝ := max (max a b) (max c d)
+
+/-- Multiply two intervals -/
+def mul (I J : IntervalReal) : IntervalReal where
+  lo := min4 (I.lo * J.lo) (I.lo * J.hi) (I.hi * J.lo) (I.hi * J.hi)
+  hi := max4 (I.lo * J.lo) (I.lo * J.hi) (I.hi * J.lo) (I.hi * J.hi)
+  le := by
+    simp only [min4, max4]
+    exact le_trans (min_le_of_left_le (min_le_left _ _))
+                   (le_max_of_le_left (le_max_left _ _))
+
+/-! ### Exponential interval -/
+
+/-- Interval bound for exp.
+    Since exp is strictly increasing, exp([a,b]) = [exp(a), exp(b)]. -/
+noncomputable def expInterval (I : IntervalReal) : IntervalReal where
+  lo := Real.exp I.lo
+  hi := Real.exp I.hi
+  le := Real.exp_le_exp.mpr I.le
+
+/-- FTIA for exp: if x ∈ [a,b], then exp(x) ∈ [exp(a), exp(b)].
+    This is FULLY PROVED - no sorry, no axioms. -/
+theorem mem_expInterval {x : ℝ} {I : IntervalReal} (hx : x ∈ I) :
+    Real.exp x ∈ expInterval I := by
+  simp only [mem_def] at *
+  simp only [expInterval]
+  constructor
+  · exact Real.exp_le_exp.mpr hx.1
+  · exact Real.exp_le_exp.mpr hx.2
+
+/-! ### Logarithm interval (for positive intervals) -/
+
+/-- An interval that is strictly positive -/
+structure IntervalRealPos extends IntervalReal where
+  lo_pos : 0 < toIntervalReal.lo
+
+/-- Interval bound for log on positive intervals.
+    Since log is strictly increasing on (0, ∞), log([a,b]) = [log(a), log(b)] for a > 0. -/
+noncomputable def logInterval (I : IntervalRealPos) : IntervalReal where
+  lo := Real.log I.lo
+  hi := Real.log I.hi
+  le := Real.log_le_log I.lo_pos I.le
+
+/-- FTIA for log: if x ∈ [a,b] with a > 0, then log(x) ∈ [log(a), log(b)].
+    This is FULLY PROVED - no sorry, no axioms. -/
+theorem mem_logInterval {x : ℝ} {I : IntervalRealPos} (hx : x ∈ I.toIntervalReal) :
+    Real.log x ∈ logInterval I := by
+  simp only [mem_def] at *
+  simp only [logInterval]
+  have hx_pos : 0 < x := lt_of_lt_of_le I.lo_pos hx.1
+  constructor
+  · exact Real.log_le_log I.lo_pos hx.1
+  · exact Real.log_le_log hx_pos hx.2
+
+/-! ### Trigonometric intervals (global bounds) -/
+
+/-- Interval bound for sin. Since |sin x| ≤ 1 for all x, we use [-1, 1]. -/
+def sinInterval (_I : IntervalReal) : IntervalReal :=
+  ⟨-1, 1, by norm_num⟩
+
+/-- FTIA for sin -/
+theorem mem_sinInterval {x : ℝ} {I : IntervalReal} (_hx : x ∈ I) :
+    Real.sin x ∈ sinInterval I := by
+  simp only [mem_def, sinInterval]
+  have h := Real.sin_mem_Icc x
+  simp only [Set.mem_Icc] at h
+  exact h
+
+/-- Interval bound for cos. Since |cos x| ≤ 1 for all x, we use [-1, 1]. -/
+def cosInterval (_I : IntervalReal) : IntervalReal :=
+  ⟨-1, 1, by norm_num⟩
+
+/-- FTIA for cos -/
+theorem mem_cosInterval {x : ℝ} {I : IntervalReal} (_hx : x ∈ I) :
+    Real.cos x ∈ cosInterval I := by
+  simp only [mem_def, cosInterval]
+  have h := Real.cos_mem_Icc x
+  simp only [Set.mem_Icc] at h
+  exact h
+
+/-- Interval bound for atan. Since arctan x ∈ (-π/2, π/2) ⊂ [-2, 2] for all x. -/
+def atanInterval (_I : IntervalReal) : IntervalReal :=
+  ⟨-2, 2, by norm_num⟩
+
+/-- FTIA for atan -/
+theorem mem_atanInterval {x : ℝ} {I : IntervalReal} (_hx : x ∈ I) :
+    Real.arctan x ∈ atanInterval I := by
+  simp only [mem_def, atanInterval]
+  constructor
+  · have := Real.neg_pi_div_two_lt_arctan x
+    have hpi : Real.pi < 4 := Real.pi_lt_four
+    linarith
+  · have := Real.arctan_lt_pi_div_two x
+    have hpi : Real.pi < 4 := Real.pi_lt_four
+    linarith
+
+/-- The derivative of arsinh is bounded by 1: |arsinh'(x)| = 1/√(1+x²) ≤ 1 -/
+private theorem arsinh_deriv_abs_le_one (x : ℝ) : |deriv Real.arsinh x| ≤ 1 := by
+  have hderiv : deriv Real.arsinh x = (Real.sqrt (1 + x ^ 2))⁻¹ :=
+    (Real.hasDerivAt_arsinh x).deriv
+  rw [hderiv]
+  have hsqrt_ge_one : Real.sqrt (1 + x ^ 2) ≥ 1 := by
+    have h1 : 1 + x ^ 2 ≥ 1 := by nlinarith [sq_nonneg x]
+    calc Real.sqrt (1 + x ^ 2) ≥ Real.sqrt 1 := Real.sqrt_le_sqrt h1
+      _ = 1 := Real.sqrt_one
+  have hsqrt_pos : 0 < Real.sqrt (1 + x ^ 2) := by
+    apply Real.sqrt_pos.mpr
+    nlinarith [sq_nonneg x]
+  rw [abs_inv, abs_of_pos hsqrt_pos]
+  exact inv_le_one_of_one_le₀ hsqrt_ge_one
+
+/-- |arsinh x| ≤ |x| for all x. This follows from MVT and |arsinh'| ≤ 1. -/
+theorem abs_arsinh_le_abs (x : ℝ) : |Real.arsinh x| ≤ |x| := by
+  rcases le_or_gt 0 x with hx | hx
+  · -- Case x ≥ 0: arsinh x ≥ 0 by monotonicity of arsinh
+    rcases eq_or_lt_of_le hx with rfl | hx_pos
+    · -- x = 0
+      simp [Real.arsinh_zero]
+    · -- x > 0
+      have harsinh_nonneg : 0 ≤ Real.arsinh x := Real.arsinh_nonneg_iff.mpr (le_of_lt hx_pos)
+      have hcont : ContinuousOn Real.arsinh (Set.Icc 0 x) :=
+        Real.continuous_arsinh.continuousOn
+      have hdiff : DifferentiableOn ℝ Real.arsinh (Set.Ioo 0 x) :=
+        Real.differentiable_arsinh.differentiableOn
+      obtain ⟨c, ⟨_, _⟩, hc_eq⟩ := exists_deriv_eq_slope Real.arsinh hx_pos hcont hdiff
+      simp only [Real.arsinh_zero, sub_zero] at hc_eq
+      have hderiv_bound : |deriv Real.arsinh c| ≤ 1 := arsinh_deriv_abs_le_one c
+      rw [abs_of_nonneg harsinh_nonneg, abs_of_pos hx_pos]
+      have heq : Real.arsinh x = deriv Real.arsinh c * x := by rw [hc_eq]; field_simp
+      calc Real.arsinh x = deriv Real.arsinh c * x := heq
+        _ ≤ |deriv Real.arsinh c * x| := le_abs_self _
+        _ = |deriv Real.arsinh c| * |x| := abs_mul _ _
+        _ ≤ 1 * x := by rw [abs_of_pos hx_pos]; nlinarith [abs_nonneg x, hderiv_bound]
+        _ = x := one_mul _
+  · -- Case x < 0: use oddness arsinh(-x) = -arsinh(x)
+    have hx_neg : -x > 0 := by linarith
+    -- For x < 0, arsinh x < 0 (arsinh is strictly increasing, arsinh 0 = 0)
+    have harsinh_neg : Real.arsinh x < 0 := Real.arsinh_neg_iff.mpr hx
+    have harsinh_neg_nonneg : 0 ≤ Real.arsinh (-x) := Real.arsinh_nonneg_iff.mpr (by linarith)
+    -- Use the positive case on -x
+    have hcont : ContinuousOn Real.arsinh (Set.Icc 0 (-x)) :=
+      Real.continuous_arsinh.continuousOn
+    have hdiff : DifferentiableOn ℝ Real.arsinh (Set.Ioo 0 (-x)) :=
+      Real.differentiable_arsinh.differentiableOn
+    obtain ⟨c, ⟨_, _⟩, hc_eq⟩ := exists_deriv_eq_slope Real.arsinh hx_neg hcont hdiff
+    simp only [Real.arsinh_zero, sub_zero] at hc_eq
+    have hderiv_bound : |deriv Real.arsinh c| ≤ 1 := arsinh_deriv_abs_le_one c
+    rw [abs_of_neg harsinh_neg, abs_of_neg hx]
+    have heq : Real.arsinh (-x) = deriv Real.arsinh c * (-x) := by
+      have hne : (-x) ≠ 0 := ne_of_gt hx_neg
+      have : deriv Real.arsinh c * (-x) = (Real.arsinh (-x) / (-x)) * (-x) := by rw [hc_eq]
+      rw [this, div_mul_cancel₀ _ hne]
+    calc -Real.arsinh x = Real.arsinh (-x) := by rw [Real.arsinh_neg]
+      _ = deriv Real.arsinh c * (-x) := heq
+      _ ≤ |deriv Real.arsinh c * (-x)| := le_abs_self _
+      _ = |deriv Real.arsinh c| * |-x| := abs_mul _ _
+      _ ≤ 1 * (-x) := by rw [abs_of_pos hx_neg]; nlinarith [abs_nonneg (-x), hderiv_bound]
+      _ = -x := one_mul _
+
+/-- Interval bound for arsinh. Uses a conservative bound based on input interval. -/
+def arsinhInterval (I : IntervalReal) : IntervalReal :=
+  -- arsinh is monotonic, so we just apply it to endpoints
+  -- Using conservative bound: |arsinh x| ≤ |x|, plus margin of 1 for safety
+  let bound := max (|I.lo|) (|I.hi|) + 1
+  ⟨-bound, bound, by
+    have h3 : (0 : ℝ) ≤ max (|I.lo|) (|I.hi|) := le_max_of_le_left (abs_nonneg I.lo)
+    change -(max (|I.lo|) (|I.hi|) + 1) ≤ max (|I.lo|) (|I.hi|) + 1
+    linarith⟩
+
+/-- FTIA for arsinh -/
+theorem mem_arsinhInterval {x : ℝ} {I : IntervalReal} (hx : x ∈ I) :
+    Real.arsinh x ∈ arsinhInterval I := by
+  unfold arsinhInterval
+  simp only [mem_def]
+  have habs : |Real.arsinh x| ≤ max (|I.lo|) (|I.hi|) + 1 := by
+    have h1 : |Real.arsinh x| ≤ |x| := abs_arsinh_le_abs x
+    have h2 : |x| ≤ max (|I.lo|) (|I.hi|) := by
+      simp only [IntervalReal.mem_def] at hx
+      rcases le_or_gt x 0 with hxneg | hxpos
+      · rw [abs_of_nonpos hxneg]
+        calc -x ≤ -I.lo := by linarith [hx.1]
+          _ ≤ |I.lo| := neg_le_abs I.lo
+          _ ≤ max (|I.lo|) (|I.hi|) := le_max_left _ _
+      · rw [abs_of_pos hxpos]
+        calc x ≤ I.hi := hx.2
+          _ ≤ |I.hi| := le_abs_self I.hi
+          _ ≤ max (|I.lo|) (|I.hi|) := le_max_right _ _
+    linarith
+  constructor
+  · have := neg_abs_le (Real.arsinh x)
+    linarith
+  · have := le_abs_self (Real.arsinh x)
+    linarith
+
+/-! ### Hyperbolic function intervals -/
+
+/-- Interval bound for sinh.
+    Since sinh is strictly monotonic increasing, sinh([a,b]) = [sinh(a), sinh(b)]. -/
+noncomputable def sinhInterval (I : IntervalReal) : IntervalReal where
+  lo := Real.sinh I.lo
+  hi := Real.sinh I.hi
+  le := Real.sinh_le_sinh.mpr I.le
+
+/-- FTIA for sinh: if x ∈ [a,b], then sinh(x) ∈ [sinh(a), sinh(b)].
+    This is FULLY PROVED - no sorry, no axioms. -/
+theorem mem_sinhInterval {x : ℝ} {I : IntervalReal} (hx : x ∈ I) :
+    Real.sinh x ∈ sinhInterval I := by
+  simp only [mem_def] at *
+  simp only [sinhInterval]
+  exact ⟨Real.sinh_le_sinh.mpr hx.1, Real.sinh_le_sinh.mpr hx.2⟩
+
+/-- Interval bound for cosh.
+    cosh is convex with minimum at 0:
+    - If interval is all non-negative: cosh is increasing
+    - If interval is all non-positive: cosh is decreasing
+    - If interval contains 0: minimum is cosh(0) = 1, max is at endpoints -/
+noncomputable def coshInterval (I : IntervalReal) : IntervalReal :=
+  if h1 : 0 ≤ I.lo then
+    -- Interval is [a,b] with 0 ≤ a: cosh increasing, so [cosh(a), cosh(b)]
+    -- Since 0 ≤ a ≤ b, we have |a| ≤ |b|, so cosh(a) ≤ cosh(b)
+    ⟨Real.cosh I.lo, Real.cosh I.hi, by
+      rw [Real.cosh_le_cosh]
+      rw [abs_of_nonneg h1, abs_of_nonneg (le_trans h1 I.le)]
+      exact I.le⟩
+  else if h2 : I.hi ≤ 0 then
+    -- Interval is [a,b] with b ≤ 0: cosh decreasing, so [cosh(b), cosh(a)]
+    -- Since a ≤ b ≤ 0, we have |b| ≤ |a|, so cosh(b) ≤ cosh(a)
+    ⟨Real.cosh I.hi, Real.cosh I.lo, by
+      rw [Real.cosh_le_cosh]
+      have hlo_neg : I.lo < 0 := not_le.mp h1
+      rw [abs_of_nonpos h2, abs_of_nonpos (le_of_lt hlo_neg)]
+      linarith [I.le]⟩
+  else
+    -- Interval contains 0: min is 1, max is max(cosh(lo), cosh(hi))
+    ⟨1, max (Real.cosh I.lo) (Real.cosh I.hi), by
+      have h := Real.one_le_cosh I.lo
+      exact le_trans h (le_max_left _ _)⟩
+
+/-- FTIA for cosh: if x ∈ [a,b], then cosh(x) ∈ coshInterval([a,b]).
+    This is FULLY PROVED - no sorry, no axioms. -/
+theorem mem_coshInterval {x : ℝ} {I : IntervalReal} (hx : x ∈ I) :
+    Real.cosh x ∈ coshInterval I := by
+  simp only [mem_def] at hx
+  unfold coshInterval
+  split_ifs with h1 h2
+  · -- Case: 0 ≤ I.lo (cosh increasing on [0, ∞))
+    simp only [mem_def]
+    have hx_nonneg : 0 ≤ x := le_trans h1 hx.1
+    constructor
+    · rw [Real.cosh_le_cosh, abs_of_nonneg h1, abs_of_nonneg hx_nonneg]
+      exact hx.1
+    · rw [Real.cosh_le_cosh, abs_of_nonneg hx_nonneg, abs_of_nonneg (le_trans h1 I.le)]
+      exact hx.2
+  · -- Case: I.hi ≤ 0 (cosh decreasing on (-∞, 0])
+    simp only [mem_def]
+    have hx_nonpos : x ≤ 0 := le_trans hx.2 h2
+    have hlo_neg : I.lo < 0 := not_le.mp h1
+    constructor
+    · rw [Real.cosh_le_cosh, abs_of_nonpos h2, abs_of_nonpos hx_nonpos]
+      linarith
+    · rw [Real.cosh_le_cosh, abs_of_nonpos hx_nonpos, abs_of_nonpos (le_of_lt hlo_neg)]
+      linarith
+  · -- Case: interval contains 0
+    simp only [mem_def]
+    push Not at h1 h2
+    constructor
+    · exact Real.one_le_cosh x
+    · -- cosh(x) ≤ max(cosh(lo), cosh(hi))
+      by_cases! hx0 : 0 ≤ x
+      · -- x ≥ 0: cosh(x) ≤ cosh(hi) since |x| ≤ |hi|
+        have hle : Real.cosh x ≤ Real.cosh I.hi := by
+          rw [Real.cosh_le_cosh, abs_of_nonneg hx0, abs_of_pos h2]
+          exact hx.2
+        exact le_trans hle (le_max_right _ _)
+      · -- x < 0: cosh(x) ≤ cosh(lo) since |x| ≤ |lo|
+        have hx_nonpos : x ≤ 0 := le_of_lt hx0
+        have hle : Real.cosh x ≤ Real.cosh I.lo := by
+          rw [Real.cosh_le_cosh, abs_of_nonpos hx_nonpos, abs_of_neg h1]
+          linarith
+        exact le_trans hle (le_max_left _ _)
+
+/-! ### Square root interval -/
+
+/-- Interval bound for sqrt.
+    For any interval I, sqrt(x) ∈ [0, max(hi, 1)] for x ∈ I.
+    This is always sound because:
+    - sqrt(x) ≥ 0 for all x (Mathlib convention: sqrt(negative) = 0)
+    - sqrt(x) ≤ max(x, 1) for x ≥ 0, so sqrt(x) ≤ max(hi, 1) -/
+noncomputable def sqrtInterval (I : IntervalReal) : IntervalReal where
+  lo := 0
+  hi := max I.hi 1
+  le := le_max_of_le_right zero_le_one
+
+/-- Helper: sqrt(x) ≤ max(x, 1) for x ≥ 0 -/
+private theorem sqrt_le_max_one {x : ℝ} (hx : 0 ≤ x) : Real.sqrt x ≤ max x 1 := by
+  rcases le_or_gt x 1 with hle | hgt
+  · -- x ≤ 1: sqrt(x) ≤ 1 ≤ max(x, 1)
+    calc Real.sqrt x ≤ Real.sqrt 1 := Real.sqrt_le_sqrt hle
+      _ = 1 := Real.sqrt_one
+      _ ≤ max x 1 := le_max_right x 1
+  · -- x > 1: sqrt(x) < x ≤ max(x, 1)
+    have hx_pos : 0 < x := lt_trans zero_lt_one hgt
+    have hsqrt_pos : 0 < Real.sqrt x := Real.sqrt_pos.mpr hx_pos
+    have hsqrt_gt_one : 1 < Real.sqrt x := by
+      rw [← Real.sqrt_one]
+      exact Real.sqrt_lt_sqrt (by norm_num) hgt
+    have hsqrt_lt : Real.sqrt x < x := by
+      have h1 : Real.sqrt x * Real.sqrt x = x := Real.mul_self_sqrt hx
+      have h2 : Real.sqrt x * 1 < Real.sqrt x * Real.sqrt x :=
+        mul_lt_mul_of_pos_left hsqrt_gt_one hsqrt_pos
+      simp only [mul_one] at h2
+      linarith
+    calc Real.sqrt x ≤ x := le_of_lt hsqrt_lt
+      _ ≤ max x 1 := le_max_left x 1
+
+/-- FTIA for sqrt: if x ∈ I, then sqrt(x) ∈ sqrtInterval(I).
+    Works for all x including negative (where sqrt returns 0 by Mathlib convention). -/
+theorem mem_sqrtInterval {x : ℝ} {I : IntervalReal} (hx : x ∈ I) :
+    Real.sqrt x ∈ sqrtInterval I := by
+  simp only [mem_def, sqrtInterval]
+  constructor
+  · exact Real.sqrt_nonneg x
+  · rcases le_or_gt 0 x with hnn | hneg
+    · -- x ≥ 0: sqrt(x) ≤ max(x, 1) ≤ max(hi, 1)
+      calc Real.sqrt x ≤ max x 1 := sqrt_le_max_one hnn
+        _ ≤ max I.hi 1 := max_le_max_right 1 hx.2
+    · -- x < 0: sqrt(x) = 0 ≤ max(hi, 1)
+      have hsqrt_zero : Real.sqrt x = 0 := Real.sqrt_eq_zero'.mpr (le_of_lt hneg)
+      rw [hsqrt_zero]
+      exact le_max_of_le_right zero_le_one
+
+end IntervalReal
+
+end LeanCert.Core
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Shared Trigonometric Range Reduction
+
+Common rational `π` bounds and interval-shifting proofs used by IntervalRat and
+Taylor-model trigonometric evaluators.
+-/
+
+/-! ### Rational approximations of π and 2π -/
+
+@[expose] public section
+
+namespace LeanCert.Core.TrigReduction
+
+open LeanCert.Core
+
+/-- Lower bound for π: 314159265/100000000 < π -/
+def piRatLo : ℚ := 314159265/100000000
+
+/-- Upper bound for π: π < 355/113 -/
+def piRatHi : ℚ := 355/113
+
+/-- Lower bound for 2π -/
+def twoPiRatLo : ℚ := 2 * piRatLo
+
+/-- Upper bound for 2π -/
+def twoPiRatHi : ℚ := 2 * piRatHi
+
+/-- `piRatLo < π`, proved from Mathlib's decimal π bounds. -/
+theorem piRatLo_lt_pi : (piRatLo : ℝ) < Real.pi := by
+  have h₁ : (piRatLo : ℝ) < (3.14159265358979323846 : ℝ) := by
+    norm_num [piRatLo]
+  exact lt_trans h₁ Real.pi_gt_d20
+
+/-- `π < piRatHi`, proved from Mathlib's decimal π bounds. -/
+theorem pi_lt_piRatHi : Real.pi < (piRatHi : ℝ) := by
+  have h₁ : (3.14159265358979323847 : ℝ) < (piRatHi : ℝ) := by
+    norm_num [piRatHi]
+  exact lt_trans Real.pi_lt_d20 h₁
+
+/-- twoPiRatLo < 2π -/
+theorem twoPiRatLo_lt_two_pi : (twoPiRatLo : ℝ) < 2 * Real.pi := by
+  unfold twoPiRatLo
+  simp only [Rat.cast_mul, Rat.cast_ofNat]
+  have h := piRatLo_lt_pi
+  linarith
+
+/-- 2π < twoPiRatHi -/
+theorem two_pi_lt_twoPiRatHi : 2 * Real.pi < (twoPiRatHi : ℝ) := by
+  unfold twoPiRatHi
+  simp only [Rat.cast_mul, Rat.cast_ofNat]
+  have h := pi_lt_piRatHi
+  linarith
+
+/-- twoPiRatLo ≤ twoPiRatHi -/
+theorem twoPiRatLo_le_twoPiRatHi : twoPiRatLo ≤ twoPiRatHi := by
+  unfold twoPiRatLo twoPiRatHi piRatLo piRatHi
+  norm_num
+
+/-! ### Range reduction -/
+
+/-- Compute the shift amount k such that I.midpoint - k * 2π is approximately in [-π, π]. -/
+def computeShiftK (I : IntervalRat) : ℤ :=
+  let mid := I.midpoint
+  (mid / twoPiRatHi).floor
+
+/-- Shift an interval by subtracting k * 2π using rational bounds. -/
+def shiftInterval (I : IntervalRat) (k : ℤ) : IntervalRat :=
+  if hk : k ≥ 0 then
+    ⟨I.lo - k * twoPiRatHi, I.hi - k * twoPiRatLo,
+      by have h1 := twoPiRatLo_le_twoPiRatHi
+         have hk_cast : (0 : ℚ) ≤ k := Int.cast_nonneg hk
+         have h2 : (k : ℚ) * twoPiRatLo ≤ k * twoPiRatHi :=
+           mul_le_mul_of_nonneg_left h1 hk_cast
+         linarith [I.le]⟩
+  else
+    ⟨I.lo - k * twoPiRatLo, I.hi - k * twoPiRatHi,
+      by have h1 := twoPiRatLo_le_twoPiRatHi
+         have hkneg : k < 0 := Int.not_le.mp hk
+         have hk_cast : (k : ℚ) ≤ 0 := Int.cast_nonpos.mpr (le_of_lt hkneg)
+         have h2 : (k : ℚ) * twoPiRatHi ≤ k * twoPiRatLo :=
+           mul_le_mul_of_nonpos_left h1 hk_cast
+         linarith [I.le]⟩
+
+/-- Reduce interval to be approximately in [-π, π] by subtracting multiples of 2π. -/
+def reduceToMainPeriod (I : IntervalRat) : IntervalRat × ℤ :=
+  let k := computeShiftK I
+  (shiftInterval I k, k)
+
+/-! ### Correctness of range reduction -/
+
+/-- If x ∈ I, then x - 2πk ∈ shiftInterval I k. -/
+theorem mem_shiftInterval_of_mem {x : ℝ} {I : IntervalRat} {k : ℤ}
+    (hx : x ∈ I) : x - 2 * Real.pi * k ∈ shiftInterval I k := by
+  simp only [IntervalRat.mem_def] at hx ⊢
+  unfold shiftInterval
+  split_ifs with hk
+  · constructor
+    · have h1 : (I.lo : ℝ) ≤ x := hx.1
+      have h2 : 2 * Real.pi ≤ (twoPiRatHi : ℝ) := le_of_lt two_pi_lt_twoPiRatHi
+      have hk_cast : (0 : ℝ) ≤ k := Int.cast_nonneg hk
+      have h3 : 2 * Real.pi * k ≤ (twoPiRatHi : ℝ) * k :=
+        mul_le_mul_of_nonneg_right h2 hk_cast
+      simp only [Rat.cast_sub, Rat.cast_mul, Rat.cast_intCast]
+      linarith
+    · have h1 : x ≤ (I.hi : ℝ) := hx.2
+      have h2 : (twoPiRatLo : ℝ) ≤ 2 * Real.pi := le_of_lt twoPiRatLo_lt_two_pi
+      have hk_cast : (0 : ℝ) ≤ k := Int.cast_nonneg hk
+      have h3 : (twoPiRatLo : ℝ) * k ≤ 2 * Real.pi * k :=
+        mul_le_mul_of_nonneg_right h2 hk_cast
+      simp only [Rat.cast_sub, Rat.cast_mul, Rat.cast_intCast]
+      linarith
+  · have hkneg : k < 0 := Int.not_le.mp hk
+    have hk_cast : (k : ℝ) ≤ 0 := Int.cast_nonpos.mpr (le_of_lt hkneg)
+    constructor
+    · have h1 : (I.lo : ℝ) ≤ x := hx.1
+      have h2 : (twoPiRatLo : ℝ) ≤ 2 * Real.pi := le_of_lt twoPiRatLo_lt_two_pi
+      have h3 : (k : ℝ) * (2 * Real.pi) ≤ k * twoPiRatLo :=
+        mul_le_mul_of_nonpos_left h2 hk_cast
+      simp only [Rat.cast_sub, Rat.cast_mul, Rat.cast_intCast]
+      linarith
+    · have h1 : x ≤ (I.hi : ℝ) := hx.2
+      have h2 : 2 * Real.pi ≤ (twoPiRatHi : ℝ) := le_of_lt two_pi_lt_twoPiRatHi
+      have h3 : (k : ℝ) * twoPiRatHi ≤ k * (2 * Real.pi) :=
+        mul_le_mul_of_nonpos_left h2 hk_cast
+      simp only [Rat.cast_sub, Rat.cast_mul, Rat.cast_intCast]
+      linarith
+
+/-- Convenience form: if x ∈ I, then x - 2πk ∈ (reduceToMainPeriod I).1. -/
+theorem mem_reduceToMainPeriod_of_mem {x : ℝ} {I : IntervalRat} (hx : x ∈ I) :
+    x - 2 * Real.pi * (reduceToMainPeriod I).2 ∈ (reduceToMainPeriod I).1 := by
+  unfold reduceToMainPeriod
+  exact mem_shiftInterval_of_mem hx
+
+end LeanCert.Core.TrigReduction
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Computable Range Reduction for Trigonometric Functions
+
+This file implements range reduction for sin and cos to improve Taylor series
+convergence for intervals far from 0.
+
+## Problem
+
+Standard Taylor series for sin/cos centered at 0 have remainder bounds of
+`|x|^{n+1} / (n+1)!`. For intervals far from 0 (e.g., [10, 11]), this remainder
+is huge even for moderate n.
+
+## Solution: Range Reduction
+
+Use the periodicity of sin/cos:
+- `sin(x) = sin(x - 2πk)` for any integer k
+- `cos(x) = cos(x - 2πk)` for any integer k
+
+By choosing k to bring x - 2πk into [-π, π], the Taylor series converges much
+faster since |x - 2πk| ≤ π ≈ 3.14.
+
+## Main definitions
+
+* `twoPiRatLo`, `twoPiRatHi` - Rational bounds for 2π
+* `reduceToMainPeriod` - Shift interval to be near 0
+* `sinComputableReduced`, `cosComputableReduced` - Computable evaluation with range reduction
+-/
+
+/-! ### Shared range-reduction API -/
+
+@[expose] public section
+
+namespace LeanCert.Core.IntervalRat
+
+/-- The rational lower π bound from the trigonometric reduction implementation. -/
+def piRatLo : ℚ := LeanCert.Core.TrigReduction.piRatLo
+/-- The rational upper π bound from the trigonometric reduction implementation. -/
+def piRatHi : ℚ := LeanCert.Core.TrigReduction.piRatHi
+/-- The rational lower bound for twice π used in argument reduction. -/
+def twoPiRatLo : ℚ := LeanCert.Core.TrigReduction.twoPiRatLo
+/-- The rational upper bound for twice π used in argument reduction. -/
+def twoPiRatHi : ℚ := LeanCert.Core.TrigReduction.twoPiRatHi
+/-- Choose the integer number of periods used to reduce an input interval. -/
+def computeShiftK : IntervalRat → ℤ := LeanCert.Core.TrigReduction.computeShiftK
+/-- Shift an interval by the selected integer multiple of the period enclosure. -/
+def shiftInterval : IntervalRat → ℤ → IntervalRat := LeanCert.Core.TrigReduction.shiftInterval
+/-- Return the reduced interval and its integer period shift. -/
+def reduceToMainPeriod : IntervalRat → IntervalRat × ℤ :=
+  LeanCert.Core.TrigReduction.reduceToMainPeriod
+
+theorem piRatLo_lt_pi : (piRatLo : ℝ) < Real.pi :=
+  LeanCert.Core.TrigReduction.piRatLo_lt_pi
+
+theorem pi_lt_piRatHi : Real.pi < (piRatHi : ℝ) :=
+  LeanCert.Core.TrigReduction.pi_lt_piRatHi
+
+theorem twoPiRatLo_lt_two_pi : (twoPiRatLo : ℝ) < 2 * Real.pi :=
+  LeanCert.Core.TrigReduction.twoPiRatLo_lt_two_pi
+
+theorem two_pi_lt_twoPiRatHi : 2 * Real.pi < (twoPiRatHi : ℝ) :=
+  LeanCert.Core.TrigReduction.two_pi_lt_twoPiRatHi
+
+theorem twoPiRatLo_le_twoPiRatHi : twoPiRatLo ≤ twoPiRatHi :=
+  LeanCert.Core.TrigReduction.twoPiRatLo_le_twoPiRatHi
+
+theorem mem_shiftInterval_of_mem {x : ℝ} {I : IntervalRat} {k : ℤ}
+    (hx : x ∈ I) : x - 2 * Real.pi * k ∈ shiftInterval I k := by
+  change x - 2 * Real.pi * k ∈ LeanCert.Core.TrigReduction.shiftInterval I k
+  exact LeanCert.Core.TrigReduction.mem_shiftInterval_of_mem hx
+
+/-! ### Computable reduced evaluation -/
+
+/-- Computable sin evaluation with range reduction. -/
+def sinComputableReduced (I : IntervalRat) (n : ℕ := 10) : IntervalRat :=
+  let (Ired, _k) := reduceToMainPeriod I
+  if Ired.lo < -4 ∨ Ired.hi > 4 then
+    ⟨-1, 1, by norm_num⟩
+  else
+    sinComputable Ired n
+
+/-- Computable cos evaluation with range reduction. -/
+def cosComputableReduced (I : IntervalRat) (n : ℕ := 10) : IntervalRat :=
+  let (Ired, _k) := reduceToMainPeriod I
+  if Ired.lo < -4 ∨ Ired.hi > 4 then
+    ⟨-1, 1, by norm_num⟩
+  else
+    cosComputable Ired n
+
+/-- sin x ∈ sinComputableReduced I for all x ∈ I -/
+theorem mem_sinComputableReduced {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (n : ℕ := 10) :
+    Real.sin x ∈ sinComputableReduced I n := by
+  unfold sinComputableReduced reduceToMainPeriod
+  simp only []
+  set Ired := shiftInterval I (computeShiftK I) with hIred
+  set k := computeShiftK I with hk
+  have hxred : x - 2 * Real.pi * k ∈ Ired := by
+    rw [hIred, hk]
+    exact mem_shiftInterval_of_mem hx
+  have hsin_eq : Real.sin x = Real.sin (x - 2 * Real.pi * k) := by
+    have h := Real.sin_add_int_mul_two_pi x (-k)
+    simp only [Int.cast_neg, neg_mul] at h
+    rw [← h]
+    ring_nf
+  split_ifs with hwide
+  · simp only [IntervalRat.mem_def, Rat.cast_neg, Rat.cast_one]
+    exact ⟨Real.neg_one_le_sin x, Real.sin_le_one x⟩
+  · push Not at hwide
+    rw [hsin_eq]
+    exact mem_sinComputable hxred n
+
+/-- cos x ∈ cosComputableReduced I for all x ∈ I -/
+theorem mem_cosComputableReduced {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (n : ℕ := 10) :
+    Real.cos x ∈ cosComputableReduced I n := by
+  unfold cosComputableReduced reduceToMainPeriod
+  simp only []
+  set Ired := shiftInterval I (computeShiftK I) with hIred
+  set k := computeShiftK I with hk
+  have hxred : x - 2 * Real.pi * k ∈ Ired := by
+    rw [hIred, hk]
+    exact mem_shiftInterval_of_mem hx
+  have hcos_eq : Real.cos x = Real.cos (x - 2 * Real.pi * k) := by
+    have h := Real.cos_add_int_mul_two_pi x (-k)
+    simp only [Int.cast_neg, neg_mul] at h
+    rw [← h]
+    ring_nf
+  split_ifs with hwide
+  · simp only [IntervalRat.mem_def, Rat.cast_neg, Rat.cast_one]
+    exact ⟨Real.neg_one_le_cos x, Real.cos_le_one x⟩
+  · rw [hcos_eq]
+    exact mem_cosComputable hxred n
+
+end LeanCert.Core.IntervalRat
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Computable Interval Evaluation
+
+This file implements the computable interval evaluator for `LeanCert.Core.Expr`.
+Given an expression and intervals for its variables, we compute an interval
+guaranteed to contain all possible values.
+
+## Main definitions
+
+* `IntervalEnv` - Variable assignment as intervals
+* `EvalConfig` - Configuration for evaluation parameters (Taylor depth)
+* `LeanCert.Internal.Rational.evalTotalCore` - Computable interval evaluator for core expressions
+* `evalIntervalCore_correct` - Correctness theorem for core evaluation
+
+## Design notes
+
+This evaluator is COMPUTABLE, allowing use of `native_decide` for bound checking
+in tactics. The transcendental functions (exp, sin, cos) use Taylor series with
+configurable depth for precision control.
+
+For inv: computes bounds using `invInterval`, but correctness is not covered by
+`evalIntervalCore_correct`. Use `evalIntervalOption` for inv.
+-/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+open LeanCert.Core
+
+-- Re-export support predicates from Core for backward compatibility
+export LeanCert.Core (ExprSupportedCore ADSupported)
+
+/-! ### Interval bounds for transcendental functions -/
+
+/-- Simple interval bound for sin.
+    Since |sin x| ≤ 1 for all x, we use the global bound [-1, 1].
+    This is sound but not tight. -/
+def sinInterval (_I : IntervalRat) : IntervalRat :=
+  ⟨-1, 1, by norm_num⟩
+
+/-- Correctness of sin interval: sin x ∈ [-1, 1] for all x -/
+theorem mem_sinInterval {x : ℝ} {I : IntervalRat} (_hx : x ∈ I) :
+    Real.sin x ∈ sinInterval I := by
+  simp only [IntervalRat.mem_def, sinInterval]
+  have h := Real.sin_mem_Icc x
+  simp only [Set.mem_Icc] at h
+  simp only [Rat.cast_neg, Rat.cast_one]
+  exact h
+
+/-- Simple interval bound for cos.
+    Since |cos x| ≤ 1 for all x, we use the global bound [-1, 1].
+    This is sound but not tight. -/
+def cosInterval (_I : IntervalRat) : IntervalRat :=
+  ⟨-1, 1, by norm_num⟩
+
+/-- Correctness of cos interval: cos x ∈ [-1, 1] for all x -/
+theorem mem_cosInterval {x : ℝ} {I : IntervalRat} (_hx : x ∈ I) :
+    Real.cos x ∈ cosInterval I := by
+  simp only [IntervalRat.mem_def, cosInterval]
+  have h := Real.cos_mem_Icc x
+  simp only [Set.mem_Icc] at h
+  simp only [Rat.cast_neg, Rat.cast_one]
+  exact h
+
+/-- Simple interval bound for atan.
+    Since atan x ∈ (-π/2, π/2) for all x, we use the global bound [-2, 2].
+    This is sound but not tight (π/2 ≈ 1.57). -/
+def atanInterval (_I : IntervalRat) : IntervalRat :=
+  ⟨-2, 2, by norm_num⟩
+
+/-- Correctness of atan interval: arctan x ∈ [-2, 2] for all x -/
+theorem mem_atanInterval {x : ℝ} {I : IntervalRat} (_hx : x ∈ I) :
+    Real.arctan x ∈ atanInterval I := by
+  simp only [IntervalRat.mem_def, atanInterval, Rat.cast_neg, Rat.cast_ofNat]
+  -- arctan x ∈ (-π/2, π/2), and π/2 < 2 since π < 4
+  have h_lo : Real.arctan x > -Real.pi / 2 := by
+    have := Real.neg_pi_div_two_lt_arctan x; linarith
+  have h_hi : Real.arctan x < Real.pi / 2 := Real.arctan_lt_pi_div_two x
+  have hpi_lt_four : Real.pi < 4 := Real.pi_lt_four
+  have hpi_lo : -Real.pi / 2 > (-2 : ℝ) := by linarith
+  have hpi_hi : Real.pi / 2 < (2 : ℝ) := by linarith
+  constructor <;> linarith
+
+/-- Interval bound for erf using computable Taylor series.
+    erf(x) = (2/√π) * ∫₀ˣ exp(-t²) dt, strictly monotone increasing.
+    This computes tight bounds using verified Taylor series. -/
+def erfInterval (I : IntervalRat) (taylorDepth : ℕ := 15) : IntervalRat :=
+  IntervalRat.erfComputable I taylorDepth
+
+/-- Helper: The integral of exp(-t²) over (a, b] is positive when a < b.
+    This follows from exp(-t²) > 0 and the interval having positive measure. -/
+private theorem integral_exp_neg_sq_pos (a b : ℝ) (hab : a < b) :
+    0 < ∫ t in a..b, Real.exp (-(t^2)) := by
+  have hcont : Continuous (fun t => Real.exp (-(t^2))) :=
+    Real.continuous_exp.comp (continuous_neg.comp (continuous_pow 2))
+  apply intervalIntegral.integral_pos hab hcont.continuousOn
+  · intro x _; exact le_of_lt (Real.exp_pos _)
+  · exact ⟨a, Set.left_mem_Icc.mpr (le_of_lt hab), Real.exp_pos _⟩
+
+/-- erf is strictly monotone increasing.
+    Proof: erf'(x) = (2/√π) * exp(-x²) > 0 for all x.
+
+    We use that for a < b, the integral ∫_{a}^{b} exp(-t²) dt > 0 since the
+    integrand is strictly positive. -/
+theorem Real.strictMono_erf : StrictMono Real.erf := by
+  intro a b hab
+  unfold Real.erf
+  have hfactor : (0 : ℝ) < 2 / Real.sqrt Real.pi := by
+    apply div_pos (by norm_num : (0 : ℝ) < 2)
+    exact Real.sqrt_pos.mpr Real.pi_pos
+  have hcont : Continuous (fun t => Real.exp (-(t^2))) :=
+    Real.continuous_exp.comp (continuous_neg.comp (continuous_pow 2))
+  have hint : ∀ c d : ℝ, IntervalIntegrable (fun t => Real.exp (-(t^2))) MeasureTheory.volume c d :=
+    fun c d => hcont.intervalIntegrable c d
+  -- The integral ∫ t in a..b, exp(-t²) > 0 when a < b since exp(-t²) > 0
+  have hpos : 0 < ∫ x in a..b, Real.exp (-(x^2)) := integral_exp_neg_sq_pos a b hab
+  -- Split integral: ∫ 0..b = ∫ 0..a + ∫ a..b
+  have heq := (intervalIntegral.integral_add_adjacent_intervals (hint 0 a) (hint a b)).symm
+  -- Since ∫ a..b > 0, we have ∫ 0..a < ∫ 0..a + ∫ a..b = ∫ 0..b
+  have hineq : ∫ x in (0 : ℝ)..a, Real.exp (-(x^2)) < ∫ x in (0 : ℝ)..b, Real.exp (-(x^2)) := by
+    rw [heq]
+    linarith
+  -- Multiply both sides by positive factor
+  exact mul_lt_mul_of_pos_left hineq hfactor
+
+/-- Quantitative global bound: `|erf x| ≤ (2257/2000) * |x|`.
+    This is conservative but fully proved from the integral definition. -/
+private theorem Real.abs_erf_le_2257_div_2000_mul_abs (x : ℝ) :
+    |Real.erf x| ≤ ((2257 : ℝ) / 2000) * |x| := by
+  unfold Real.erf
+  have h_int :
+      ‖∫ t in (0 : ℝ)..x, Real.exp (-(t ^ 2))‖ ≤ |x| := by
+    have hnorm :
+        ∀ t ∈ Set.uIoc (0 : ℝ) x, ‖Real.exp (-(t ^ 2))‖ ≤ (1 : ℝ) := by
+      intro t _ht
+      have hsq : 0 ≤ t ^ 2 := sq_nonneg t
+      have hle : Real.exp (-(t ^ 2)) ≤ 1 := Real.exp_le_one_iff.mpr (by linarith)
+      have hnonneg : 0 ≤ Real.exp (-(t ^ 2)) := le_of_lt (Real.exp_pos _)
+      simpa [Real.norm_eq_abs, abs_of_nonneg hnonneg] using hle
+    have h := intervalIntegral.norm_integral_le_of_norm_le_const
+      (a := (0 : ℝ)) (b := x) (C := (1 : ℝ)) (f := fun t => Real.exp (-(t ^ 2))) hnorm
+    simpa [one_mul, sub_zero] using h
+  have hfactor_nonneg : 0 ≤ 2 / Real.sqrt Real.pi := by
+    exact div_nonneg (by norm_num) (le_of_lt (Real.sqrt_pos.mpr Real.pi_pos))
+  have hsqrt_ge_17724 : (17724 / 10000 : ℝ) ≤ Real.sqrt Real.pi := by
+    have hpi_ge : (3.1415 : ℝ) ≤ Real.pi := le_of_lt Real.pi_gt_d4
+    have hsq : (17724 / 10000 : ℝ) ^ 2 ≤ Real.pi := by
+      have hcalc : (17724 / 10000 : ℝ) ^ 2 = 3.14140176 := by norm_num
+      linarith
+    have hnonneg : (0 : ℝ) ≤ (17724 / 10000 : ℝ) := by norm_num
+    have hsqrt_sq : (17724 / 10000 : ℝ) = Real.sqrt ((17724 / 10000 : ℝ) ^ 2) := by
+      symm; exact Real.sqrt_sq hnonneg
+    calc
+      (17724 / 10000 : ℝ) = Real.sqrt ((17724 / 10000 : ℝ) ^ 2) := hsqrt_sq
+      _ ≤ Real.sqrt Real.pi := by exact Real.sqrt_le_sqrt hsq
+  have hfactor_le_2257_div_2000 : 2 / Real.sqrt Real.pi ≤ ((2257 : ℝ) / 2000) := by
+    have hmul : (2 : ℝ) ≤ ((2257 : ℝ) / 2000) * Real.sqrt Real.pi := by
+      calc
+        (2 : ℝ) ≤ ((2257 : ℝ) / 2000) * (17724 / 10000 : ℝ) := by norm_num
+        _ ≤ ((2257 : ℝ) / 2000) * Real.sqrt Real.pi := by
+              gcongr
+    exact (div_le_iff₀ (Real.sqrt_pos.mpr Real.pi_pos)).2 hmul
+  calc
+    |(2 / Real.sqrt Real.pi) * ∫ t in (0 : ℝ)..x, Real.exp (-(t ^ 2))|
+        = (2 / Real.sqrt Real.pi) * ‖∫ t in (0 : ℝ)..x, Real.exp (-(t ^ 2))‖ := by
+            rw [abs_mul, abs_of_nonneg hfactor_nonneg, Real.norm_eq_abs]
+    _ ≤ (2 / Real.sqrt Real.pi) * |x| := by
+          gcongr
+    _ ≤ ((2257 : ℝ) / 2000) * |x| := by
+          nlinarith [hfactor_nonneg, hfactor_le_2257_div_2000, abs_nonneg x]
+
+/-- Correctness of `erfPointComputable`.
+    The enclosure is sign-aware and uses monotonicity of erf with `erf(0)=0`. -/
+theorem mem_erfPointComputable (q : ℚ) (n : ℕ) :
+    Real.erf q ∈ IntervalRat.erfPointComputable q n := by
+  unfold IntervalRat.erfPointComputable
+  have h_abs1 : |Real.erf q| ≤ (1 : ℝ) := by
+    refine (abs_le.2 ?_)
+    exact_mod_cast (Real.erf_mem_Icc q)
+  have h_abs2 : |Real.erf q| ≤ ((2257 : ℝ) / 2000) * |(q : ℝ)| :=
+    Real.abs_erf_le_2257_div_2000_mul_abs q
+  by_cases hq : q < 0
+  · simp only [hq, ↓reduceDIte, IntervalRat.mem_def, Rat.cast_neg, Rat.cast_min, Rat.cast_one,
+    Rat.cast_mul, Rat.cast_div, Rat.cast_ofNat, Rat.cast_abs, Rat.cast_zero]
+    have hq_real : (q : ℝ) < 0 := by exact_mod_cast hq
+    have herf_lt_erf0 : Real.erf q < Real.erf 0 := Real.strictMono_erf hq_real
+    have herf_nonpos : Real.erf q ≤ 0 := by
+      simpa [Real.erf] using le_of_lt herf_lt_erf0
+    have hneg_min : -(min (1 : ℝ) (((2257 : ℝ) / 2000) * |(q : ℝ)|)) ≤ Real.erf q := by
+      exact (abs_le.mp (le_min h_abs1 h_abs2)).1
+    exact ⟨hneg_min, herf_nonpos⟩
+  · by_cases h0 : q = 0
+    · subst h0
+      simp [IntervalRat.mem_def, Real.erf]
+    · simp only [hq, ↓reduceDIte, h0, IntervalRat.mem_def, Rat.cast_zero,
+        Rat.cast_min, Rat.cast_one,
+      Rat.cast_mul, Rat.cast_div, Rat.cast_ofNat, Rat.cast_abs, le_inf_iff]
+      have hq_pos : 0 < q := lt_of_le_of_ne (le_of_not_gt hq) (by
+        intro hz
+        exact h0 hz.symm)
+      have hq_pos_real : (0 : ℝ) < q := by exact_mod_cast hq_pos
+      have herf0_lt : Real.erf 0 < Real.erf q := Real.strictMono_erf hq_pos_real
+      have herf_nonneg : 0 ≤ Real.erf q := by
+        simpa [Real.erf] using le_of_lt herf0_lt
+      exact ⟨herf_nonneg, (abs_le.mp h_abs1).2, (abs_le.mp h_abs2).2⟩
+
+/-- Correctness of erf interval using monotonicity and endpoint evaluation.
+
+    Since erf is strictly monotone increasing:
+    - For x ∈ [I.lo, I.hi], we have erf(I.lo) ≤ erf(x) ≤ erf(I.hi)
+    - erfPointComputable(I.lo) contains erf(I.lo)
+    - erfPointComputable(I.hi) contains erf(I.hi)
+    - hull of these intervals contains [erf(I.lo), erf(I.hi)]
+    - Therefore erf(x) ∈ hull(...) ∩ [-1, 1] -/
+theorem mem_erfInterval {x : ℝ} {I : IntervalRat} (hx : x ∈ I) (n : ℕ := 15) :
+    Real.erf x ∈ erfInterval I n := by
+  unfold erfInterval IntervalRat.erfComputable
+  simp only []
+  -- Get membership in endpoint intervals
+  have hlo_mem := mem_erfPointComputable I.lo n
+  have hhi_mem := mem_erfPointComputable I.hi n
+  -- Use monotonicity
+  have hx_bounds := hx
+  rw [IntervalRat.mem_def] at hx_bounds
+  have herf_lo : Real.erf I.lo ≤ Real.erf x :=
+    Real.strictMono_erf.monotone hx_bounds.1
+  have herf_hi : Real.erf x ≤ Real.erf I.hi :=
+    Real.strictMono_erf.monotone hx_bounds.2
+  -- erf x is in hull of endpoint intervals
+  have h_in_hull : Real.erf x ∈ IntervalRat.hull
+      (IntervalRat.erfPointComputable I.lo n) (IntervalRat.erfPointComputable I.hi n) := by
+    rw [IntervalRat.mem_def] at hlo_mem hhi_mem ⊢
+    unfold IntervalRat.hull
+    simp only []
+    constructor
+    · calc (↑(min (IntervalRat.erfPointComputable I.lo n).lo
+              (IntervalRat.erfPointComputable I.hi n).lo) : ℝ)
+          ≤ (IntervalRat.erfPointComputable I.lo n).lo := by
+            simp only [Rat.cast_min]; exact min_le_left _ _
+        _ ≤ Real.erf I.lo := hlo_mem.1
+        _ ≤ Real.erf x := herf_lo
+    · calc Real.erf x
+          ≤ Real.erf I.hi := herf_hi
+        _ ≤ (IntervalRat.erfPointComputable I.hi n).hi := hhi_mem.2
+        _ ≤ (↑(max (IntervalRat.erfPointComputable I.lo n).hi
+              (IntervalRat.erfPointComputable I.hi n).hi) : ℝ) := by
+            simp only [Rat.cast_max]; exact le_max_right _ _
+  -- erf x is in [-1, 1]
+  have h_global : Real.erf x ∈ ({lo := -1, hi := 1, le := by norm_num} : IntervalRat) := by
+    simp only [IntervalRat.mem_def, Rat.cast_neg, Rat.cast_one]
+    exact Real.erf_mem_Icc x
+  -- Combine via case analysis on intersection result
+  set erfLo := IntervalRat.erfPointComputable I.lo n
+  set erfHi := IntervalRat.erfPointComputable I.hi n
+  set raw := erfLo.hull erfHi
+  set globalBound : IntervalRat := ⟨-1, 1, by norm_num⟩
+  -- Use mem_intersect which says: if x ∈ I and x ∈ J, then ∃ K, intersect I J = some K ∧ x ∈ K
+  have ⟨K, hK_eq, hK_mem⟩ := IntervalRat.mem_intersect h_in_hull h_global
+  simp only [hK_eq]
+  exact hK_mem
+
+/-- Simple interval bound for arsinh.
+    arsinh is unbounded, so we use a very rough linear bound.
+    We use max(|lo|, |hi|) + 1 as a safe bound that always works. -/
+def arsinhInterval (I : IntervalRat) : IntervalRat :=
+  let bound := max (abs I.lo) (abs I.hi) + 1
+  ⟨-bound, bound, by
+    have h1 : (0 : ℚ) ≤ |I.lo| := abs_nonneg I.lo
+    have h2 : |I.lo| ≤ |I.lo| ⊔ |I.hi| := le_max_left _ _
+    have h3 : (0 : ℚ) ≤ |I.lo| ⊔ |I.hi| := le_trans h1 h2
+    have h4 : (0 : ℚ) ≤ |I.lo| ⊔ |I.hi| + 1 := by linarith
+    have h5 : -(|I.lo| ⊔ |I.hi| + 1) ≤ |I.lo| ⊔ |I.hi| + 1 := by linarith
+    exact h5⟩
+
+/-- Correctness of arsinh interval.
+    Uses the bound |arsinh x| ≤ |x| for all x (from IntervalRealEndpoints). -/
+theorem mem_arsinhInterval {x : ℝ} {I : IntervalRat} (hx : x ∈ I) :
+    Real.arsinh x ∈ arsinhInterval I := by
+  simp only [IntervalRat.mem_def, arsinhInterval]
+  -- Key bound: |arsinh x| ≤ |x| + 1
+  -- This is a well-known bound that follows from sinh y ≥ y for y ≥ 0
+  have hbound : |Real.arsinh x| ≤ |x| + 1 := by
+    -- Use the sharper bound |arsinh x| ≤ |x| from IntervalRealEndpoints
+    have h := IntervalReal.abs_arsinh_le_abs x
+    linarith
+  simp only [IntervalRat.mem_def] at hx
+  have hx_bound : |x| ≤ max (|I.lo|) (|I.hi|) := by
+    simp only [Rat.cast_abs, Rat.cast_max]
+    rcases le_or_gt x 0 with hxneg | hxpos
+    · simp only [abs_of_nonpos hxneg]
+      calc -x ≤ -(I.lo : ℝ) := by linarith [hx.1]
+        _ ≤ |↑I.lo| := neg_le_abs _
+        _ ≤ max |↑I.lo| |↑I.hi| := le_max_left _ _
+    · simp only [abs_of_pos hxpos]
+      calc x ≤ I.hi := hx.2
+        _ ≤ |↑I.hi| := le_abs_self _
+        _ ≤ max |↑I.lo| |↑I.hi| := le_max_right _ _
+  have habs_lo : -(|x| + 1) ≤ Real.arsinh x := (abs_le.mp hbound).1
+  have habs_hi : Real.arsinh x ≤ |x| + 1 := (abs_le.mp hbound).2
+  have hmax_cast : (max (|I.lo|) (|I.hi|) : ℝ) = max |↑I.lo| |↑I.hi| := by
+    rfl
+  constructor
+  · simp only [Rat.cast_neg, Rat.cast_add, Rat.cast_one, Rat.cast_max, Rat.cast_abs]
+    have h1 : -(|↑I.lo| ⊔ |↑I.hi| + 1) = -(max |↑I.lo| |↑I.hi|) - (1 : ℝ) := by ring
+    have h2 : -(max |↑I.lo| |↑I.hi|) - (1 : ℝ) ≤ -|x| - 1 := by
+      have hxb : |x| ≤ max |↑I.lo| |↑I.hi| := by
+        convert hx_bound using 1
+        simp [Rat.cast_max, Rat.cast_abs]
+      linarith
+    have h3 : -|x| - (1 : ℝ) ≤ Real.arsinh x := by
+      have : -(|x| + 1) = -|x| - 1 := by ring
+      rw [← this]; exact habs_lo
+    linarith
+  · simp only [Rat.cast_add, Rat.cast_one, Rat.cast_max, Rat.cast_abs]
+    have h1 : Real.arsinh x ≤ |x| + 1 := habs_hi
+    have h2 : |x| + (1 : ℝ) ≤ max |↑I.lo| |↑I.hi| + 1 := by
+      have hxb : |x| ≤ max |↑I.lo| |↑I.hi| := by
+        convert hx_bound using 1
+        simp [Rat.cast_max, Rat.cast_abs]
+      linarith
+    linarith
+
+/-- Interval bound for atanh.
+    atanh is defined for (-1, 1). If interval is within this range,
+    we compute tight bounds using monotonicity. Otherwise returns default. -/
+noncomputable def atanhInterval (I : IntervalRat) : IntervalRat :=
+  if h : -1 < I.lo ∧ I.hi < 1 then
+    let Iball : IntervalRat.IntervalRatInUnitBall := ⟨I.lo, I.hi, I.le, h.1, h.2⟩
+    IntervalRat.atanhIntervalComputed Iball
+  else
+    default
+
+/-- Correctness of atanh interval: when x ∈ I and I ⊂ (-1, 1), atanh(x) ∈ atanhInterval I. -/
+theorem mem_atanhInterval {x : ℝ} {I : IntervalRat} (hx : x ∈ I)
+    (hlo : -1 < I.lo) (hhi : I.hi < 1) :
+    Real.atanh x ∈ atanhInterval I := by
+  unfold atanhInterval
+  simp only [hlo, hhi, and_self, ↓reduceDIte]
+  let Iball : IntervalRat.IntervalRatInUnitBall := ⟨I.lo, I.hi, I.le, hlo, hhi⟩
+  have hx_ball : x ∈ Iball := by
+    simp only [Membership.mem]
+    exact hx
+  exact IntervalRat.mem_atanhIntervalComputed hx_ball
+
+/-- Tight interval bound for tanh.
+    Since tanh(x) ∈ (-1, 1) for all x ∈ ℝ, we use the global bound [-1, 1].
+    This avoids the interval explosion that occurs when desugaring to exp. -/
+def tanhInterval (_I : IntervalRat) : IntervalRat :=
+  ⟨-1, 1, by norm_num⟩
+
+/-- Correctness of tanh interval: tanh x ∈ [-1, 1] for all x -/
+theorem mem_tanhInterval {x : ℝ} {I : IntervalRat} (_hx : x ∈ I) :
+    Real.tanh x ∈ tanhInterval I := by
+  simp only [IntervalRat.mem_def, tanhInterval, Rat.cast_neg, Rat.cast_one]
+  constructor
+  -- tanh x ≥ -1 for all x: use that tanh = sinh/cosh and cosh > 0
+  · rw [Real.tanh_eq_sinh_div_cosh]
+    have hcosh : Real.cosh x > 0 := Real.cosh_pos x
+    -- -1 ≤ sinh/cosh iff -cosh ≤ sinh (since cosh > 0)
+    rw [le_div_iff₀ hcosh, neg_one_mul]
+    -- Need: -cosh x ≤ sinh x
+    -- sinh x = (exp x - exp(-x))/2, cosh x = (exp x + exp(-x))/2
+    -- sinh x + cosh x = exp x ≥ 0 ✓
+    rw [Real.sinh_eq, Real.cosh_eq]
+    have h1 : Real.exp x > 0 := Real.exp_pos x
+    linarith
+  -- tanh x ≤ 1 for all x: use that sinh ≤ cosh (since exp(-x) > 0)
+  · rw [Real.tanh_eq_sinh_div_cosh]
+    have hcosh : Real.cosh x > 0 := Real.cosh_pos x
+    rw [div_le_one₀ hcosh]
+    -- Need: sinh x ≤ cosh x
+    rw [Real.sinh_eq, Real.cosh_eq]
+    have h2 : Real.exp (-x) > 0 := Real.exp_pos (-x)
+    linarith
+
+/-- Interval enclosure for π.
+    Uses tight bounds from Mathlib's pi_gt_d20 and pi_lt_d20 which give 20 decimal digits.
+    3.14159265358979323846 < π < 3.14159265358979323847 -/
+def piInterval : IntervalRat :=
+  -- Use rational approximation: 31415926535897932/10000000000000000 ≤ pi ≤
+  -- 31415926535897933/10000000000000000
+  ⟨31415926535897932/10000000000000000, 31415926535897933/10000000000000000, by norm_num⟩
+
+/-- Correctness of pi interval: Real.pi ∈ piInterval -/
+theorem mem_piInterval : Real.pi ∈ piInterval := by
+  simp only [IntervalRat.mem_def, piInterval]
+  constructor
+  · -- Lower bound from pi_gt_d20
+    have h : (3.14159265358979323846 : ℝ) < Real.pi := Real.pi_gt_d20
+    have eq : (31415926535897932 : ℚ) / 10000000000000000 = (3.1415926535897932 : ℚ) := by norm_num
+    simp only [Rat.cast_div, Rat.cast_ofNat] at *
+    linarith
+  · -- Upper bound from pi_lt_d20
+    have h : Real.pi < (3.14159265358979323847 : ℝ) := Real.pi_lt_d20
+    have eq : (31415926535897933 : ℚ) / 10000000000000000 = (3.1415926535897933 : ℚ) := by norm_num
+    simp only [Rat.cast_div, Rat.cast_ofNat] at *
+    linarith
+
+/-- Interval enclosure for the Euler–Mascheroni constant γ.
+    Tight bounds derived from `eulerMascheroniSeq 100 < γ < eulerMascheroniSeq' 100`
+    combined with explicit Taylor partial sums for `log`. The proofs below are
+    axiom-free (no `native_decide`): all rational arithmetic is certified by
+    `norm_num`, and the only analytic inputs are Mathlib's
+    `abs_log_sub_add_sum_range_le` and the d9 bounds on `log 2`. -/
+def eulerMascheroniInterval : IntervalRat :=
+  ⟨5722/10000, 5823/10000, by norm_num⟩
+
+/-- `harmonic 100` as an explicit rational literal. -/
+private lemma harmonic_100_eq : harmonic 100 =
+    14466636279520351160221518043104131447711/2788815009188499086581352357412492142272 := by
+  norm_num [harmonic_succ, harmonic_zero]
+
+/-- Series bound for `log (5/4) = -log (1 - 1/5)`: nine Taylor terms with the
+    Mathlib tail estimate `|x|^(n+1)/(1-|x|)`. -/
+private lemma log_five_quarter_approx :
+    |Real.log (5/4) - 219656921/984375000| ≤ (1/7812500 : ℝ) := by
+  have h := Real.abs_log_sub_add_sum_range_le (x := (1/5 : ℝ))
+    (by rw [abs_of_nonneg] <;> norm_num) 9
+  have hsum : (∑ i ∈ Finset.range 9, (1/5 : ℝ) ^ (i + 1) / (i + 1)) = 219656921/984375000 := by
+    simp [Finset.sum_range_succ]
+    norm_num
+  have hlog : Real.log (1 - 1/5) = -Real.log (5/4) := by
+    rw [show (1 - 1/5 : ℝ) = (5/4)⁻¹ by norm_num, Real.log_inv]
+  rw [hsum, hlog, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 1/5)] at h
+  calc |Real.log (5/4) - 219656921/984375000|
+      = |(219656921/984375000 : ℝ) + -Real.log (5/4)| := by rw [← abs_neg]; ring_nf
+    _ ≤ (1/5 : ℝ) ^ 10 / (1 - 1/5) := h
+    _ ≤ (1/7812500 : ℝ) := by norm_num
+
+/-- Series bound for `log (101/100) = -log (1 - 1/101)`: three Taylor terms. -/
+private lemma log_ratio_101_approx :
+    |Real.log (101/100) - 61511/6181806| ≤ (1/103030100 : ℝ) := by
+  have h := Real.abs_log_sub_add_sum_range_le (x := (1/101 : ℝ))
+    (by rw [abs_of_nonneg] <;> norm_num) 3
+  have hsum : (∑ i ∈ Finset.range 3, (1/101 : ℝ) ^ (i + 1) / (i + 1)) = 61511/6181806 := by
+    simp [Finset.sum_range_succ]
+    norm_num
+  have hlog : Real.log (1 - 1/101) = -Real.log (101/100) := by
+    rw [show (1 - 1/101 : ℝ) = (101/100)⁻¹ by norm_num, Real.log_inv]
+  rw [hsum, hlog, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 1/101)] at h
+  calc |Real.log (101/100) - 61511/6181806|
+      = |(61511/6181806 : ℝ) + -Real.log (101/100)| := by rw [← abs_neg]; ring_nf
+    _ ≤ (1/101 : ℝ) ^ 4 / (1 - 1/101) := h
+    _ ≤ (1/103030100 : ℝ) := by norm_num
+
+/-- Decompose `log 100` over `log 2` and `log (5/4)`: `100 = 2^6 * (5/4)^2`. -/
+private lemma log_100_eq : Real.log 100 = 6 * Real.log 2 + 2 * Real.log (5/4) := by
+  rw [show (100 : ℝ) = 2^6 * (5/4)^2 by norm_num,
+      Real.log_mul (by norm_num) (by norm_num), Real.log_pow, Real.log_pow]
+  push_cast; ring
+
+/-- Decompose `log 101` as `log 100 + log (101/100)`. -/
+private lemma log_101_eq :
+    Real.log 101 = 6 * Real.log 2 + 2 * Real.log (5/4) + Real.log (101/100) := by
+  rw [← log_100_eq, show (101 : ℝ) = 100 * (101/100) by norm_num,
+      Real.log_mul (by norm_num) (by norm_num)]
+  norm_num
+
+/-- Helper: `5722/10000 ≤ harmonic 100 - log 101` via explicit series bounds. -/
+private theorem eulerMascheroniSeq_100_ge :
+    (5722/10000 : ℝ) ≤ Real.eulerMascheroniSeq 100 := by
+  have h54 := log_five_quarter_approx
+  have h101 := log_ratio_101_approx
+  have hlog2 := Real.log_two_lt_d9
+  have heq : Real.eulerMascheroniSeq 100 = ↑(harmonic 100 : ℚ) - Real.log 101 := by
+    simp [Real.eulerMascheroniSeq]; ring_nf
+  rw [heq, harmonic_100_eq, log_101_eq]
+  rw [abs_le] at h54 h101
+  push_cast
+  linarith [h54.1, h54.2, h101.1, h101.2]
+
+/-- Helper: `harmonic 100 - log 100 ≤ 5823/10000` via explicit series bounds. -/
+private theorem eulerMascheroniSeq'_100_le :
+    Real.eulerMascheroniSeq' 100 ≤ (5823/10000 : ℝ) := by
+  have h54 := log_five_quarter_approx
+  have hlog2 := Real.log_two_gt_d9
+  have heq : Real.eulerMascheroniSeq' 100 = ↑(harmonic 100 : ℚ) - Real.log 100 := by
+    simp [Real.eulerMascheroniSeq']
+  rw [heq, harmonic_100_eq, log_100_eq]
+  rw [abs_le] at h54
+  push_cast
+  linarith [h54.1, h54.2]
+
+/-- Correctness of Euler–Mascheroni interval: γ ∈ eulerMascheroniInterval -/
+theorem mem_eulerMascheroniInterval :
+    Real.eulerMascheroniConstant ∈ eulerMascheroniInterval := by
+  simp only [IntervalRat.mem_def, eulerMascheroniInterval]
+  constructor
+  · -- Lower: 5722/10000 ≤ eulerMascheroniSeq 100 < γ
+    have := Real.eulerMascheroniSeq_lt_eulerMascheroniConstant 100
+    linarith [eulerMascheroniSeq_100_ge]
+  · -- Upper: γ < eulerMascheroniSeq' 100 ≤ 5823/10000
+    have := Real.eulerMascheroniConstant_lt_eulerMascheroniSeq' 100
+    linarith [eulerMascheroniSeq'_100_le]
+
+end LeanCert.Engine
+
+namespace LeanCert.Core.MathConst
+open LeanCert.Engine
+
+/-- Centralized interval lookup for named mathematical constants.
+    Extending this table is the ONLY change needed to add a new constant. -/
+def interval : MathConst → IntervalRat
+  | .pi => piInterval
+  | .eulerMascheroni => eulerMascheroniInterval
+
+/-- Correctness: the real value of every named constant is in its interval. -/
+theorem mem_interval (c : MathConst) : c.toReal ∈ c.interval := by
+  cases c with
+  | pi => exact mem_piInterval
+  | eulerMascheroni => exact mem_eulerMascheroniInterval
+
+end LeanCert.Core.MathConst
+
+namespace LeanCert.Engine
+open LeanCert.Core
+
+/-- Interval bound for sinh using computable Taylor series for exp.
+    sinh(x) = (exp(x) - exp(-x)) / 2, and sinh is strictly monotonic.
+    This computes tight bounds using the verified exp implementation. -/
+def sinhInterval (I : IntervalRat) (taylorDepth : ℕ := 10) : IntervalRat :=
+  IntervalRat.sinhComputable I taylorDepth
+
+/-- Interval bound for cosh using computable Taylor series for exp.
+    cosh(x) = (exp(x) + exp(-x)) / 2, with minimum 1 at x = 0.
+    This computes tight bounds using the verified exp implementation. -/
+def coshInterval (I : IntervalRat) (taylorDepth : ℕ := 10) : IntervalRat :=
+  IntervalRat.coshComputable I taylorDepth
+
+/-! ### Interval inverse -/
+
+/-- Wide bound constant for when inverse is undefined (denominator contains 0) -/
+def invWideBound : ℚ := 10^30
+
+private theorem invWideBound_pos : (0 : ℝ) < invWideBound := by
+  simp only [invWideBound]
+  norm_num
+
+/-- Computable interval inverse.
+    For [a,b] with a > 0: returns [1/b, 1/a] (1/x is decreasing on positive reals)
+    For [a,b] with b < 0: returns [1/b, 1/a] (1/x is decreasing on negative reals)
+    For intervals containing 0: returns wide bounds [-M, M]. NOTE: this branch is
+    NOT a sound enclosure of x⁻¹ in general (1/x is unbounded near 0); the
+    correctness theorem `mem_invInterval` therefore requires the extra
+    hypothesis `|x⁻¹| ≤ invWideBound` in that case. -/
+def invInterval (I : IntervalRat) : IntervalRat :=
+  if h : I.lo > 0 then
+    -- Positive interval: 1/x is decreasing, so [1/b, 1/a]
+    let invHi := 1 / I.lo
+    let invLo := 1 / I.hi
+    if hle : invLo ≤ invHi then { lo := invLo, hi := invHi, le := hle }
+    else { lo := invHi, hi := invLo, le := by linarith }
+  else if h' : I.hi < 0 then
+    -- Negative interval: 1/x is decreasing, so [1/b, 1/a]
+    let invHi := 1 / I.lo
+    let invLo := 1 / I.hi
+    if hle : invLo ≤ invHi then { lo := invLo, hi := invHi, le := hle }
+    else { lo := invHi, hi := invLo, le := by linarith }
+  else
+    -- Interval contains zero: return wide bounds
+    ⟨-invWideBound, invWideBound, by simp only [invWideBound]; norm_num⟩
+
+/-- Correctness of invInterval when denominator interval is positive.
+    For x ∈ [a,b] with a > 0, we have 1/x ∈ [1/b, 1/a] -/
+theorem mem_invInterval_pos {x : ℝ} {I : IntervalRat}
+    (hx : x ∈ I) (hpos : I.lo > 0) :
+    x⁻¹ ∈ invInterval I := by
+  simp only [IntervalRat.mem_def] at hx ⊢
+  simp only [invInterval, hpos, ↓reduceDIte]
+  -- x > 0 since x ≥ I.lo > 0
+  have hx_pos : (x : ℝ) > 0 := by
+    have h1 : (I.lo : ℝ) ≤ x := hx.1
+    have h2 : (0 : ℝ) < I.lo := by exact_mod_cast hpos
+    linarith
+  have hx_ne : x ≠ 0 := ne_of_gt hx_pos
+  -- For positive x: 1/b ≤ 1/x ≤ 1/a when a ≤ x ≤ b
+  have hlo_pos : (0 : ℝ) < I.lo := by exact_mod_cast hpos
+  have hlo_ne : (I.lo : ℝ) ≠ 0 := ne_of_gt hlo_pos
+  have hhi_pos : (0 : ℝ) < I.hi := by
+    have := I.le
+    have h : (I.lo : ℝ) ≤ I.hi := by exact_mod_cast this
+    linarith
+  have hhi_ne : (I.hi : ℝ) ≠ 0 := ne_of_gt hhi_pos
+  split_ifs with hle
+  · -- Case: 1/hi ≤ 1/lo (normal ordering)
+    simp only [Rat.cast_inv, one_div]
+    constructor
+    · -- (I.hi)⁻¹ ≤ x⁻¹ follows from x ≤ I.hi
+      exact (inv_le_inv₀ hhi_pos hx_pos).mpr hx.2
+    · -- x⁻¹ ≤ (I.lo)⁻¹ follows from I.lo ≤ x
+      exact (inv_le_inv₀ hx_pos hlo_pos).mpr hx.1
+  · -- Case: 1/lo < 1/hi (impossible for positive lo ≤ hi, but we prove the goal anyway)
+    -- 1/hi ≤ 1/lo is always true when lo ≤ hi and both positive
+    have hhi_pos_q : (0 : ℚ) < I.hi := by
+      have := I.le
+      linarith
+    have hle' : (1 : ℚ) / I.hi ≤ 1 / I.lo := by
+      rw [div_le_div_iff₀ hhi_pos_q hpos]
+      simp only [one_mul]
+      exact I.le
+    exact absurd hle' hle
+
+/-- Correctness of invInterval when denominator interval is negative.
+    For x ∈ [a,b] with b < 0, we have 1/x ∈ [1/b, 1/a] -/
+theorem mem_invInterval_neg {x : ℝ} {I : IntervalRat}
+    (hx : x ∈ I) (hneg : I.hi < 0) :
+    x⁻¹ ∈ invInterval I := by
+  simp only [IntervalRat.mem_def] at hx ⊢
+  -- I.lo ≤ 0 since I.lo ≤ I.hi < 0
+  have hlo_le : ¬(I.lo > 0) := by
+    have := I.le
+    have hhi_neg : I.hi < 0 := hneg
+    intro h
+    have : (I.lo : ℚ) ≤ I.hi := this
+    linarith
+  simp only [invInterval, hlo_le, ↓reduceDIte, hneg]
+  -- x < 0 since x ≤ I.hi < 0
+  have hx_neg : (x : ℝ) < 0 := by
+    have h1 : x ≤ I.hi := hx.2
+    have h2 : (I.hi : ℝ) < 0 := by exact_mod_cast hneg
+    linarith
+  have hlo_neg : (I.lo : ℝ) < 0 := by
+    have h1 : I.lo ≤ I.hi := I.le
+    have h2 : (I.hi : ℝ) < 0 := by exact_mod_cast hneg
+    have h3 : (I.lo : ℝ) ≤ I.hi := by exact_mod_cast h1
+    linarith
+  have hhi_neg' : (I.hi : ℝ) < 0 := by exact_mod_cast hneg
+  -- For negative x: 1/x is still decreasing, so 1/b ≤ 1/x ≤ 1/a
+  split_ifs with hle
+  · simp only [Rat.cast_inv, one_div]
+    constructor
+    · -- (I.hi)⁻¹ ≤ x⁻¹ when x ≤ hi < 0 (since 1/x is decreasing for negatives)
+      exact (inv_le_inv_of_neg hhi_neg' hx_neg).mpr hx.2
+    · -- x⁻¹ ≤ (I.lo)⁻¹ when lo ≤ x < 0
+      exact (inv_le_inv_of_neg hx_neg hlo_neg).mpr hx.1
+  · -- Case: 1/lo < 1/hi (impossible for negative lo ≤ hi < 0)
+    -- For negative numbers: lo ≤ hi < 0 implies 1/hi ≤ 1/lo
+    -- Use: one_div_le_one_div_of_neg_of_le (hb : b < 0) (h : a ≤ b) : 1 / b ≤ 1 / a
+    have hle' : (1 : ℚ) / I.hi ≤ 1 / I.lo :=
+      one_div_le_one_div_of_neg_of_le hneg I.le
+    exact absurd hle' hle
+
+/-- Correctness of invInterval when denominator interval contains zero.
+    Requires a bound on |x⁻¹| to be provable, since x can be arbitrarily close to 0. -/
+theorem mem_invInterval_wide {x : ℝ} {I : IntervalRat}
+    (_hx : x ∈ I) (hlo : ¬(I.lo > 0)) (hhi : ¬(I.hi < 0))
+    (hbnd : |x⁻¹| ≤ invWideBound) :
+    x⁻¹ ∈ invInterval I := by
+  simp only [IntervalRat.mem_def, invInterval, hlo, hhi, ↓reduceDIte]
+  constructor
+  · -- -invWideBound ≤ x⁻¹
+    calc ((-invWideBound : ℚ) : ℝ) = -(invWideBound : ℝ) := by simp
+      _ ≤ -|x⁻¹| := neg_le_neg hbnd
+      _ ≤ x⁻¹ := neg_abs_le x⁻¹
+  · -- x⁻¹ ≤ invWideBound
+    calc x⁻¹ ≤ |x⁻¹| := le_abs_self x⁻¹
+      _ ≤ (invWideBound : ℝ) := hbnd
+
+/-- Main correctness theorem for invInterval.
+    Fully proved for intervals bounded away from zero.
+    For intervals containing zero, requires a bound on |x⁻¹|. -/
+theorem mem_invInterval {x : ℝ} {I : IntervalRat}
+    (hx : x ∈ I) (hbnd : |x⁻¹| ≤ invWideBound) :
+    x⁻¹ ∈ invInterval I := by
+  -- Case split based on the interval position
+  by_cases hpos : I.lo > 0
+  · exact mem_invInterval_pos hx hpos
+  · by_cases hneg : I.hi < 0
+    · exact mem_invInterval_neg hx hneg
+    · exact mem_invInterval_wide hx hpos hneg hbnd
+
+/-- Correctness of invInterval for intervals bounded away from zero (no extra hypothesis needed) -/
+theorem mem_invInterval_nonzero {x : ℝ} {I : IntervalRat}
+    (hx : x ∈ I) (hnonzero : I.lo > 0 ∨ I.hi < 0) :
+    x⁻¹ ∈ invInterval I := by
+  rcases hnonzero with hpos | hneg
+  · exact mem_invInterval_pos hx hpos
+  · exact mem_invInterval_neg hx hneg
+
+/-! ### Core interval evaluation (computable) -/
+
+/-- Variable assignment as intervals -/
+abbrev IntervalEnv := Nat → IntervalRat
+
+/-- Configuration for interval evaluation parameters.
+    This allows certificates to specify the required precision. -/
+structure EvalConfig where
+  /-- Number of Taylor series terms for transcendental functions -/
+  taylorDepth : ℕ := 10
+  deriving Repr, DecidableEq
+
+/-- Default evaluation configuration with 10 Taylor terms -/
+instance : Inhabited EvalConfig := ⟨{ taylorDepth := 10 }⟩
+
+end LeanCert.Engine
+
+namespace LeanCert.Internal.Rational
+
+open LeanCert.Core LeanCert.Engine
+
+/-- Internal total evaluator for the theorem-restricted core expression fragment.
+
+    For expressions in `ExprSupportedCore`, this computes correct interval
+    bounds with a fully-verified proof (given domain validity conditions).
+
+    For inv: computes bounds using `invInterval`, but correctness is not
+    covered by `evalIntervalCore_correct`. Use `evalIntervalOption` for inv.
+
+    For log: uses `logComputable` with Taylor series. Correctness requires
+    that the argument interval is positive (see `evalDomainValid`).
+
+    This evaluator is COMPUTABLE, allowing use of `native_decide` for
+    bound checking in tactics. The transcendental functions (exp, sin, cos, log)
+    use Taylor series with configurable depth for precision control. -/
+def evalTotalCore (e : Expr) (ρ : IntervalEnv) (cfg : EvalConfig := {}) : IntervalRat :=
+  match e with
+  | Expr.const q => IntervalRat.singleton q
+  | Expr.var idx => ρ idx
+  | Expr.add e₁ e₂ => IntervalRat.add (LeanCert.Internal.Rational.evalTotalCore e₁ ρ cfg)
+    (LeanCert.Internal.Rational.evalTotalCore e₂ ρ cfg)
+  | Expr.mul e₁ e₂ => IntervalRat.mul (LeanCert.Internal.Rational.evalTotalCore e₁ ρ cfg)
+    (LeanCert.Internal.Rational.evalTotalCore e₂ ρ cfg)
+  | Expr.neg e => IntervalRat.neg (LeanCert.Internal.Rational.evalTotalCore e ρ cfg)
+  | Expr.inv e => invInterval (LeanCert.Internal.Rational.evalTotalCore e ρ cfg)
+  | Expr.exp e => IntervalRat.expComputable (LeanCert.Internal.Rational.evalTotalCore e ρ cfg)
+    cfg.taylorDepth
+  | Expr.sin e => IntervalRat.sinComputableReduced (LeanCert.Internal.Rational.evalTotalCore e ρ
+    cfg) cfg.taylorDepth
+  | Expr.cos e => IntervalRat.cosComputableReduced (LeanCert.Internal.Rational.evalTotalCore e ρ
+    cfg) cfg.taylorDepth
+  | Expr.log e => IntervalRat.logComputable (LeanCert.Internal.Rational.evalTotalCore e ρ cfg)
+    cfg.taylorDepth
+  | Expr.atan e => atanInterval (LeanCert.Internal.Rational.evalTotalCore e ρ cfg)
+  | Expr.arsinh e => arsinhInterval (LeanCert.Internal.Rational.evalTotalCore e ρ cfg)
+  | Expr.atanh _ => default  -- Not in ExprSupportedCore; use evalIntervalOption for atanh
+  | Expr.sinc _ => ⟨-1, 1, by norm_num⟩  -- sinc is bounded by [-1, 1]
+  | Expr.erf e => erfInterval (LeanCert.Internal.Rational.evalTotalCore e ρ cfg) cfg.taylorDepth
+  | Expr.sinh e => sinhInterval (LeanCert.Internal.Rational.evalTotalCore e ρ cfg) cfg.taylorDepth
+  | Expr.cosh e => coshInterval (LeanCert.Internal.Rational.evalTotalCore e ρ cfg) cfg.taylorDepth
+  | Expr.tanh e => tanhInterval (LeanCert.Internal.Rational.evalTotalCore e ρ cfg)
+  | Expr.sqrt e => IntervalRat.sqrtIntervalTightPrec (LeanCert.Internal.Rational.evalTotalCore e ρ
+    cfg)
+  | Expr.namedConst c => c.interval
+
+end LeanCert.Internal.Rational
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+/-- A real environment is contained in an interval environment -/
+def envMem (ρ_real : Nat → ℝ) (ρ_int : IntervalEnv) : Prop :=
+  ∀ i, ρ_real i ∈ ρ_int i
+
+/-- Domain validity predicate for expressions with domain restrictions.
+
+    For log: requires the argument interval to be strictly positive.
+    This ensures that logComputable returns correct bounds.
+
+    For other expressions: always true (no domain restrictions). -/
+def evalDomainValid (e : Expr) (ρ : IntervalEnv) (cfg : EvalConfig := {}) : Prop :=
+  match e with
+  | Expr.const _ => True
+  | Expr.var _ => True
+  | Expr.add e₁ e₂ => evalDomainValid e₁ ρ cfg ∧ evalDomainValid e₂ ρ cfg
+  | Expr.mul e₁ e₂ => evalDomainValid e₁ ρ cfg ∧ evalDomainValid e₂ ρ cfg
+  | Expr.neg e => evalDomainValid e ρ cfg
+  | Expr.inv e => evalDomainValid e ρ cfg  -- Note: inv correctness not covered anyway
+  | Expr.exp e => evalDomainValid e ρ cfg
+  | Expr.sin e => evalDomainValid e ρ cfg
+  | Expr.cos e => evalDomainValid e ρ cfg
+  | Expr.log e => evalDomainValid e ρ cfg ∧ (LeanCert.Internal.Rational.evalTotalCore e ρ cfg).lo
+    > 0
+  | Expr.atan e => evalDomainValid e ρ cfg
+  | Expr.arsinh e => evalDomainValid e ρ cfg
+  | Expr.atanh e => evalDomainValid e ρ cfg
+  | Expr.sinc e => evalDomainValid e ρ cfg
+  | Expr.erf e => evalDomainValid e ρ cfg
+  | Expr.sinh e => evalDomainValid e ρ cfg
+  | Expr.cosh e => evalDomainValid e ρ cfg
+  | Expr.tanh e => evalDomainValid e ρ cfg
+  | Expr.sqrt e => evalDomainValid e ρ cfg
+  | Expr.namedConst _ => True
+
+/-- Single-variable domain validity -/
+def evalDomainValid1 (e : Expr) (I : IntervalRat) (cfg : EvalConfig := {}) : Prop :=
+  evalDomainValid e (fun _ => I) cfg
+
+/-- Computable (decidable) check for domain validity -/
+def checkDomainValid (e : Expr) (ρ : IntervalEnv) (cfg : EvalConfig := {}) : Bool :=
+  match e with
+  | Expr.const _ => true
+  | Expr.var _ => true
+  | Expr.add e₁ e₂ => checkDomainValid e₁ ρ cfg && checkDomainValid e₂ ρ cfg
+  | Expr.mul e₁ e₂ => checkDomainValid e₁ ρ cfg && checkDomainValid e₂ ρ cfg
+  | Expr.neg e => checkDomainValid e ρ cfg
+  | Expr.inv e => checkDomainValid e ρ cfg
+  | Expr.exp e => checkDomainValid e ρ cfg
+  | Expr.sin e => checkDomainValid e ρ cfg
+  | Expr.cos e => checkDomainValid e ρ cfg
+  | Expr.log e => checkDomainValid e ρ cfg && decide ((LeanCert.Internal.Rational.evalTotalCore e
+    ρ cfg).lo > 0)
+  | Expr.atan e => checkDomainValid e ρ cfg
+  | Expr.arsinh e => checkDomainValid e ρ cfg
+  | Expr.atanh e => checkDomainValid e ρ cfg
+  | Expr.sinc e => checkDomainValid e ρ cfg
+  | Expr.erf e => checkDomainValid e ρ cfg
+  | Expr.sinh e => checkDomainValid e ρ cfg
+  | Expr.cosh e => checkDomainValid e ρ cfg
+  | Expr.tanh e => checkDomainValid e ρ cfg
+  | Expr.sqrt e => checkDomainValid e ρ cfg
+  | Expr.namedConst _ => true
+
+/-- Single-variable domain check -/
+def checkDomainValid1 (e : Expr) (I : IntervalRat) (cfg : EvalConfig := {}) : Bool :=
+  checkDomainValid e (fun _ => I) cfg
+
+/-- checkDomainValid = true implies evalDomainValid -/
+theorem checkDomainValid_correct (e : Expr) (ρ : IntervalEnv) (cfg : EvalConfig)
+    (h : checkDomainValid e ρ cfg = true) : evalDomainValid e ρ cfg := by
+  induction e with
+  | const q => trivial
+  | var idx => trivial
+  | add e₁ e₂ ih₁ ih₂ =>
+    simp only [checkDomainValid, Bool.and_eq_true] at h
+    simp only [evalDomainValid]
+    exact ⟨ih₁ h.1, ih₂ h.2⟩
+  | mul e₁ e₂ ih₁ ih₂ =>
+    simp only [checkDomainValid, Bool.and_eq_true] at h
+    simp only [evalDomainValid]
+    exact ⟨ih₁ h.1, ih₂ h.2⟩
+  | neg e ih =>
+    simp only [checkDomainValid] at h
+    simp only [evalDomainValid]
+    exact ih h
+  | inv e ih =>
+    simp only [checkDomainValid] at h
+    simp only [evalDomainValid]
+    exact ih h
+  | exp e ih =>
+    simp only [checkDomainValid] at h
+    simp only [evalDomainValid]
+    exact ih h
+  | sin e ih =>
+    simp only [checkDomainValid] at h
+    simp only [evalDomainValid]
+    exact ih h
+  | cos e ih =>
+    simp only [checkDomainValid] at h
+    simp only [evalDomainValid]
+    exact ih h
+  | log e ih =>
+    simp only [checkDomainValid, Bool.and_eq_true, decide_eq_true_eq] at h
+    simp only [evalDomainValid]
+    exact ⟨ih h.1, h.2⟩
+  | atan e ih =>
+    simp only [checkDomainValid] at h
+    simp only [evalDomainValid]
+    exact ih h
+  | arsinh e ih =>
+    simp only [checkDomainValid] at h
+    simp only [evalDomainValid]
+    exact ih h
+  | atanh e ih =>
+    simp only [checkDomainValid] at h
+    simp only [evalDomainValid]
+    exact ih h
+  | sinc e ih =>
+    simp only [checkDomainValid] at h
+    simp only [evalDomainValid]
+    exact ih h
+  | erf e ih =>
+    simp only [checkDomainValid] at h
+    simp only [evalDomainValid]
+    exact ih h
+  | sinh e ih =>
+    simp only [checkDomainValid] at h
+    simp only [evalDomainValid]
+    exact ih h
+  | cosh e ih =>
+    simp only [checkDomainValid] at h
+    simp only [evalDomainValid]
+    exact ih h
+  | tanh e ih =>
+    simp only [checkDomainValid] at h
+    simp only [evalDomainValid]
+    exact ih h
+  | sqrt e ih =>
+    simp only [checkDomainValid] at h
+    simp only [evalDomainValid]
+    exact ih h
+  | namedConst _ => trivial
+
+/-- checkDomainValid1 = true implies evalDomainValid1 -/
+theorem checkDomainValid1_correct (e : Expr) (I : IntervalRat) (cfg : EvalConfig)
+    (h : checkDomainValid1 e I cfg = true) : evalDomainValid1 e I cfg :=
+  checkDomainValid_correct e (fun _ => I) cfg h
+
+/-- evalDomainValid is equivalent to checkDomainValid = true -/
+theorem evalDomainValid_iff_checkDomainValid (e : Expr) (ρ : IntervalEnv) (cfg : EvalConfig) :
+    evalDomainValid e ρ cfg ↔ checkDomainValid e ρ cfg = true := by
+  constructor
+  · -- evalDomainValid → checkDomainValid
+    intro h
+    induction e with
+    | const _ => rfl
+    | var _ => rfl
+    | add e₁ e₂ ih₁ ih₂ =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid, Bool.and_eq_true]
+      exact ⟨ih₁ h.1, ih₂ h.2⟩
+    | mul e₁ e₂ ih₁ ih₂ =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid, Bool.and_eq_true]
+      exact ⟨ih₁ h.1, ih₂ h.2⟩
+    | neg e ih =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid]
+      exact ih h
+    | inv e ih =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid]
+      exact ih h
+    | exp e ih =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid]
+      exact ih h
+    | sin e ih =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid]
+      exact ih h
+    | cos e ih =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid]
+      exact ih h
+    | log e ih =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid, Bool.and_eq_true, decide_eq_true_eq]
+      exact ⟨ih h.1, h.2⟩
+    | atan e ih =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid]
+      exact ih h
+    | arsinh e ih =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid]
+      exact ih h
+    | atanh e ih =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid]
+      exact ih h
+    | sinc e ih =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid]
+      exact ih h
+    | erf e ih =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid]
+      exact ih h
+    | sinh e ih =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid]
+      exact ih h
+    | cosh e ih =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid]
+      exact ih h
+    | tanh e ih =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid]
+      exact ih h
+    | sqrt e ih =>
+      simp only [evalDomainValid] at h
+      simp only [checkDomainValid]
+      exact ih h
+    | namedConst _ => rfl
+  · -- checkDomainValid → evalDomainValid
+    exact checkDomainValid_correct e ρ cfg
+
+/-- Decidability instance for domain validity -/
+instance decidableEvalDomainValid (e : Expr) (ρ : IntervalEnv) (cfg : EvalConfig) :
+    Decidable (evalDomainValid e ρ cfg) :=
+  decidable_of_iff' _ (evalDomainValid_iff_checkDomainValid e ρ cfg)
+
+/-- Decidability instance for single-variable domain validity -/
+instance decidableEvalDomainValid1 (e : Expr) (I : IntervalRat) (cfg : EvalConfig) :
+    Decidable (evalDomainValid1 e I cfg) :=
+  decidableEvalDomainValid e (fun _ => I) cfg
+
+/-- ADSupported expressions (which don't include log) always have valid domains.
+    This is because only log has domain restrictions (positive argument). -/
+theorem ADSupported.domainValid {e : Expr} (hsupp : ADSupported e)
+    (ρ : IntervalEnv) (cfg : EvalConfig) : evalDomainValid e ρ cfg := by
+  induction hsupp generalizing ρ cfg with
+  | const _ => trivial
+  | var _ => trivial
+  | add _ _ ih₁ ih₂ => exact ⟨ih₁ ρ cfg, ih₂ ρ cfg⟩
+  | mul _ _ ih₁ ih₂ => exact ⟨ih₁ ρ cfg, ih₂ ρ cfg⟩
+  | neg _ ih => exact ih ρ cfg
+  | sin _ ih => exact ih ρ cfg
+  | cos _ ih => exact ih ρ cfg
+  | exp _ ih => exact ih ρ cfg
+
+/-- Single-variable version of domainValid for ADSupported -/
+theorem exprSupported_domainValid1 {e : Expr} (hsupp : ADSupported e)
+    (I : IntervalRat) (cfg : EvalConfig) : evalDomainValid1 e I cfg :=
+  ADSupported.domainValid hsupp (fun _ => I) cfg
+
+/-- Fundamental correctness theorem for core evaluation.
+
+    This theorem is FULLY PROVED for core expressions (no sorry, no axioms).
+    The `hsupp` hypothesis ensures we only consider expressions in the
+    computable verified subset. The `hdom` hypothesis ensures domain validity
+    (e.g., log arguments are positive). Works for any Taylor depth. -/
+theorem evalIntervalCore_correct (e : Expr) (hsupp : ExprSupportedCore e)
+    (ρ_real : Nat → ℝ) (ρ_int : IntervalEnv) (hρ : envMem ρ_real ρ_int)
+    (cfg : EvalConfig := {})
+    (hdom : evalDomainValid e ρ_int cfg) :
+    Expr.eval ρ_real e ∈ LeanCert.Internal.Rational.evalTotalCore e ρ_int cfg := by
+  induction hsupp with
+  | const q =>
+    simp only [Expr.eval_const, LeanCert.Internal.Rational.evalTotalCore]
+    exact IntervalRat.mem_singleton q
+  | var idx =>
+    simp only [Expr.eval_var, LeanCert.Internal.Rational.evalTotalCore]
+    exact hρ idx
+  | add h₁ h₂ ih₁ ih₂ =>
+    simp only [evalDomainValid] at hdom
+    simp only [Expr.eval_add, LeanCert.Internal.Rational.evalTotalCore]
+    exact IntervalRat.mem_add (ih₁ hdom.1) (ih₂ hdom.2)
+  | mul h₁ h₂ ih₁ ih₂ =>
+    simp only [evalDomainValid] at hdom
+    simp only [Expr.eval_mul, LeanCert.Internal.Rational.evalTotalCore]
+    exact IntervalRat.mem_mul (ih₁ hdom.1) (ih₂ hdom.2)
+  | neg _ ih =>
+    simp only [evalDomainValid] at hdom
+    simp only [Expr.eval_neg, LeanCert.Internal.Rational.evalTotalCore]
+    exact IntervalRat.mem_neg (ih hdom)
+  | sin _ ih =>
+    simp only [evalDomainValid] at hdom
+    simp only [Expr.eval_sin, LeanCert.Internal.Rational.evalTotalCore]
+    exact IntervalRat.mem_sinComputableReduced (ih hdom) cfg.taylorDepth
+  | cos _ ih =>
+    simp only [evalDomainValid] at hdom
+    simp only [Expr.eval_cos, LeanCert.Internal.Rational.evalTotalCore]
+    exact IntervalRat.mem_cosComputableReduced (ih hdom) cfg.taylorDepth
+  | exp _ ih =>
+    simp only [evalDomainValid] at hdom
+    simp only [Expr.eval_exp, LeanCert.Internal.Rational.evalTotalCore]
+    exact IntervalRat.mem_expComputable (ih hdom) cfg.taylorDepth
+  | log _ ih =>
+    simp only [evalDomainValid] at hdom
+    simp only [Expr.eval_log, LeanCert.Internal.Rational.evalTotalCore]
+    exact IntervalRat.mem_logComputable (ih hdom.1) hdom.2 cfg.taylorDepth
+  | sqrt _ ih =>
+    simp only [evalDomainValid] at hdom
+    simp only [Expr.eval_sqrt, LeanCert.Internal.Rational.evalTotalCore]
+    exact IntervalRat.mem_sqrtIntervalTightPrec' (ih hdom)
+  | sinh _ ih =>
+    simp only [evalDomainValid] at hdom
+    simp only [Expr.eval_sinh, LeanCert.Internal.Rational.evalTotalCore, sinhInterval]
+    exact IntervalRat.mem_sinhComputable (ih hdom) cfg.taylorDepth
+  | cosh _ ih =>
+    simp only [evalDomainValid] at hdom
+    simp only [Expr.eval_cosh, LeanCert.Internal.Rational.evalTotalCore, coshInterval]
+    exact IntervalRat.mem_coshComputable (ih hdom) cfg.taylorDepth
+  | tanh _ ih =>
+    simp only [evalDomainValid] at hdom
+    simp only [Expr.eval_tanh, LeanCert.Internal.Rational.evalTotalCore, tanhInterval]
+    exact mem_tanhInterval (ih hdom)
+  | erf _ ih =>
+    simp only [evalDomainValid] at hdom
+    simp only [Expr.eval_erf, LeanCert.Internal.Rational.evalTotalCore]
+    exact mem_erfInterval (ih hdom) cfg.taylorDepth
+  | namedConst c =>
+    simp only [Expr.eval_namedConst, LeanCert.Internal.Rational.evalTotalCore]
+    exact c.mem_interval
+
+/-! ### Convenience functions -/
+
+end LeanCert.Engine
+
+namespace LeanCert.Internal.Rational
+
+open LeanCert.Core LeanCert.Engine
+
+/-- Computable single-variable evaluation for core expressions -/
+def evalTotalCore1 (e : Expr) (I : IntervalRat) (cfg : EvalConfig := {}) : IntervalRat :=
+  LeanCert.Internal.Rational.evalTotalCore e (fun _ => I) cfg
+
+end LeanCert.Internal.Rational
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+/-- Correctness for single-variable core evaluation -/
+theorem evalIntervalCore1_correct (e : Expr) (hsupp : ExprSupportedCore e)
+    (x : ℝ) (I : IntervalRat) (hx : x ∈ I)
+    (cfg : EvalConfig := {})
+    (hdom : evalDomainValid1 e I cfg) :
+    Expr.eval (fun _ => x) e ∈ LeanCert.Internal.Rational.evalTotalCore1 e I cfg :=
+  evalIntervalCore_correct e hsupp _ _ (fun _ => hx) cfg hdom
+
+/-! ### Smart constructors for supported expressions -/
+
+/-- Build a constant expression (always supported) -/
+def mkConst (q : ℚ) : { e : Expr // ADSupported e } :=
+  ⟨Expr.const q, ADSupported.const q⟩
+
+/-- Build a variable expression (always supported) -/
+def mkVar (idx : Nat) : { e : Expr // ADSupported e } :=
+  ⟨Expr.var idx, ADSupported.var idx⟩
+
+/-- Build an addition (supported if both operands are supported) -/
+def mkAdd (e₁ e₂ : { e : Expr // ADSupported e }) : { e : Expr // ADSupported e } :=
+  ⟨Expr.add e₁.val e₂.val, ADSupported.add e₁.property e₂.property⟩
+
+/-- Build a multiplication (supported if both operands are supported) -/
+def mkMul (e₁ e₂ : { e : Expr // ADSupported e }) : { e : Expr // ADSupported e } :=
+  ⟨Expr.mul e₁.val e₂.val, ADSupported.mul e₁.property e₂.property⟩
+
+/-- Build a negation (supported if operand is supported) -/
+def mkNeg (e : { e : Expr // ADSupported e }) : { e : Expr // ADSupported e } :=
+  ⟨Expr.neg e.val, ADSupported.neg e.property⟩
+
+/-- Build a sin (supported if operand is supported) -/
+def mkSin (e : { e : Expr // ADSupported e }) : { e : Expr // ADSupported e } :=
+  ⟨Expr.sin e.val, ADSupported.sin e.property⟩
+
+/-- Build a cos (supported if operand is supported) -/
+def mkCos (e : { e : Expr // ADSupported e }) : { e : Expr // ADSupported e } :=
+  ⟨Expr.cos e.val, ADSupported.cos e.property⟩
+
+/-- Build an exp (supported if operand is supported) -/
+def mkExp (e : { e : Expr // ADSupported e }) : { e : Expr // ADSupported e } :=
+  ⟨Expr.exp e.val, ADSupported.exp e.property⟩
+
+end LeanCert.Engine
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Checked evaluation results
+
+Certified evaluators return a finite enclosure only after all partial-domain
+conditions have been checked.  Failures are data: callers must propagate them
+instead of substituting a finite sentinel that could be mistaken for an
+enclosure.
+-/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+/-- Why a checked evaluator could not produce a finite certified enclosure. -/
+inductive EvalError where
+  | reciprocalContainsZero (interval : IntervalRat)
+  | logNonpositive (interval : IntervalRat)
+  | atanhOutsideUnitBall (interval : IntervalRat)
+  | unsupportedBackend (operation : String)
+  | unsupportedFeature (feature : String)
+  | invalidConfiguration (message : String)
+  | nestedFailure (operation : String) (cause : EvalError)
+  deriving Repr, DecidableEq
+
+/-- Result type used by checked evaluators and public computation APIs. -/
+abbrev EvalResult (α : Type) := Except EvalError α
+
+end LeanCert.Engine
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Extended (Noncomputable) Interval Evaluation
+
+This file implements the noncomputable interval evaluator for `LeanCert.Core.Expr`,
+supporting exp with floor/ceil bounds and partial evaluation for inv/log.
+
+## Main definitions
+
+* `evalInterval` - Noncomputable interval evaluator supporting exp
+* `evalInterval_correct` - Correctness theorem for extended evaluation
+* `evalIntervalOption` - Partial (Option-returning) evaluator with inv/log support
+* `evalIntervalOption_correct` - Correctness theorem for partial evaluation
+
+## Design notes
+
+The extended evaluator uses `Real.exp` with floor/ceil bounds, which requires
+noncomputability. For computability, use `LeanCert.Internal.Rational.evalTotalCore` instead.
+
+The partial evaluator `evalIntervalOption` returns `none` when:
+- The denominator interval for `inv` contains zero
+- The argument interval for `log` is not strictly positive
+- The argument for `atanh` is not in (-1, 1)
+
+When it returns `some I`, correctness is guaranteed.
+-/
+
+/-! ### Extended interval evaluation (noncomputable, supports exp) -/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+end LeanCert.Engine
+
+namespace LeanCert.Internal.Rational
+
+open LeanCert.Core LeanCert.Engine
+
+/-- Noncomputable interval evaluator supporting exp.
+
+    For supported expressions (const, var, add, mul, neg, sin, cos, exp), this
+    computes correct interval bounds with a fully-verified proof.
+
+    For unsupported expressions (inv, log), returns a default interval.
+    Do not rely on results for expressions containing inv or log.
+    Use evalIntervalOption for partial functions like inv and log.
+
+    This evaluator is NONCOMPUTABLE due to exp using Real.exp with floor/ceil. -/
+noncomputable def evalUnchecked (e : Expr) (ρ : IntervalEnv) : IntervalRat :=
+  match e with
+  | Expr.const q => IntervalRat.singleton q
+  | Expr.var idx => ρ idx
+  | Expr.add e₁ e₂ => IntervalRat.add (evalUnchecked e₁ ρ) (evalUnchecked e₂ ρ)
+  | Expr.mul e₁ e₂ => IntervalRat.mul (evalUnchecked e₁ ρ) (evalUnchecked e₂ ρ)
+  | Expr.neg e => IntervalRat.neg (evalUnchecked e ρ)
+  | Expr.inv _ => default  -- Not in ADSupported; safe default
+  | Expr.exp e => IntervalRat.expInterval (evalUnchecked e ρ)
+  | Expr.sin e => sinInterval (evalUnchecked e ρ)
+  | Expr.cos e => cosInterval (evalUnchecked e ρ)
+  | Expr.log _ => default  -- Not in ADSupported; use evalIntervalOption for log
+  | Expr.atan e => atanInterval (evalUnchecked e ρ)
+  | Expr.arsinh e => arsinhInterval (evalUnchecked e ρ)
+  | Expr.atanh _ => default  -- Not in ADSupported; use evalIntervalOption for atanh
+  | Expr.sinc _ => ⟨-1, 1, by norm_num⟩  -- sinc is bounded by [-1, 1]
+  | Expr.erf _ => ⟨-1, 1, by norm_num⟩  -- erf is bounded by [-1, 1]
+  | Expr.sinh _ => default  -- sinh unbounded; use LeanCert.Internal.Rational.evalTotalCore for
+    -- tight bounds
+  | Expr.cosh _ => default  -- cosh unbounded; use LeanCert.Internal.Rational.evalTotalCore for
+    -- tight bounds
+  | Expr.tanh _ => default  -- tanh bounded but not in ADSupported; use
+    -- LeanCert.Internal.Rational.evalTotalCore
+  | Expr.sqrt e => IntervalRat.sqrtInterval (evalUnchecked e ρ)
+  | Expr.namedConst c => c.interval
+
+end LeanCert.Internal.Rational
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+/-- Fundamental correctness theorem for extended evaluation.
+
+    This theorem is FULLY PROVED (no sorry, no axioms) for supported expressions.
+    The `hsupp` hypothesis ensures we only consider expressions in the verified subset. -/
+theorem evalInterval_correct (e : Expr) (hsupp : ADSupported e)
+    (ρ_real : Nat → ℝ) (ρ_int : IntervalEnv) (hρ : envMem ρ_real ρ_int) :
+    Expr.eval ρ_real e ∈ LeanCert.Internal.Rational.evalUnchecked e ρ_int := by
+  induction hsupp with
+  | const q =>
+    simp only [Expr.eval_const, LeanCert.Internal.Rational.evalUnchecked]
+    exact IntervalRat.mem_singleton q
+  | var idx =>
+    simp only [Expr.eval_var, LeanCert.Internal.Rational.evalUnchecked]
+    exact hρ idx
+  | add h₁ h₂ ih₁ ih₂ =>
+    simp only [Expr.eval_add, LeanCert.Internal.Rational.evalUnchecked]
+    exact IntervalRat.mem_add ih₁ ih₂
+  | mul h₁ h₂ ih₁ ih₂ =>
+    simp only [Expr.eval_mul, LeanCert.Internal.Rational.evalUnchecked]
+    exact IntervalRat.mem_mul ih₁ ih₂
+  | neg _ ih =>
+    simp only [Expr.eval_neg, LeanCert.Internal.Rational.evalUnchecked]
+    exact IntervalRat.mem_neg ih
+  | sin _ ih =>
+    simp only [Expr.eval_sin, LeanCert.Internal.Rational.evalUnchecked]
+    exact mem_sinInterval ih
+  | cos _ ih =>
+    simp only [Expr.eval_cos, LeanCert.Internal.Rational.evalUnchecked]
+    exact mem_cosInterval ih
+  | exp _ ih =>
+    simp only [Expr.eval_exp, LeanCert.Internal.Rational.evalUnchecked]
+    exact IntervalRat.mem_expInterval ih
+
+/-! ### Convenience functions
+
+Note: LeanCert.Internal.Rational.evalTotalCore now uses Taylor series for exp/sin/cos, which gives
+different (often tighter) intervals than evalInterval's floor/ceil bounds.
+Both are correct, but they are not necessarily equal.
+
+For purely algebraic expressions (const, var, add, mul, neg),
+both evaluators give identical results. -/
+
+end LeanCert.Engine
+
+namespace LeanCert.Internal.Rational
+
+open LeanCert.Core LeanCert.Engine
+
+/-- Internal single-variable wrapper around `evalUnchecked`. -/
+noncomputable def evalUnchecked1 (e : Expr) (I : IntervalRat) : IntervalRat :=
+  evalUnchecked e (fun _ => I)
+
+end LeanCert.Internal.Rational
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+/-- Correctness for single-variable extended evaluation -/
+theorem evalInterval1_correct (e : Expr) (hsupp : ADSupported e)
+    (x : ℝ) (I : IntervalRat) (hx : x ∈ I) :
+    Expr.eval (fun _ => x) e ∈ LeanCert.Internal.Rational.evalUnchecked1 e I :=
+  evalInterval_correct e hsupp _ _ (fun _ => hx)
+
+/-! ### Partial interval evaluation with inv support -/
+
+/-- Partial (Option-returning) interval evaluator supporting inv.
+
+    For expressions with inv, this evaluator returns `none` if the denominator
+    interval contains zero, and `some I` with a correct enclosure otherwise.
+
+    This allows safe interval evaluation of expressions like 1/x when we can
+    verify the denominator is bounded away from zero.
+
+    For expressions without inv, this always returns `some` with the same
+    result as `evalInterval`. -/
+def evalIntervalLogDepth : ℕ := 60
+
+/-- The Taylor truncation depth used by the checked exponential evaluator. -/
+def evalIntervalExpDepth : ℕ := 30
+
+/-- Taylor depth used by the strict `atanh` evaluator. -/
+def evalIntervalAtanhDepth : ℕ := 30
+
+/-- Evaluate an expression by rational intervals, failing when a domain check fails. -/
+def evalIntervalOption (e : Expr) (ρ : IntervalEnv) : Option IntervalRat :=
+  match e with
+  | Expr.const q => some (IntervalRat.singleton q)
+  | Expr.var idx => some (ρ idx)
+  | Expr.add e₁ e₂ =>
+      match evalIntervalOption e₁ ρ, evalIntervalOption e₂ ρ with
+      | some I₁, some I₂ => some (IntervalRat.add I₁ I₂)
+      | _, _ => none
+  | Expr.mul e₁ e₂ =>
+      match evalIntervalOption e₁ ρ, evalIntervalOption e₂ ρ with
+      | some I₁, some I₂ => some (IntervalRat.mul I₁ I₂)
+      | _, _ => none
+  | Expr.neg e =>
+      match evalIntervalOption e ρ with
+      | some I => some (IntervalRat.neg I)
+      | none => none
+  | Expr.inv e₁ =>
+      match evalIntervalOption e₁ ρ with
+      | none => none
+      | some J =>
+          if h : IntervalRat.containsZero J then
+            none
+          else
+            some (IntervalRat.invNonzero ⟨J, h⟩)
+  | Expr.exp e =>
+      match evalIntervalOption e ρ with
+      | some I => some (IntervalRat.expComputable I evalIntervalExpDepth)
+      | none => none
+  | Expr.sin e =>
+      match evalIntervalOption e ρ with
+      | some I => some (sinInterval I)
+      | none => none
+  | Expr.cos e =>
+      match evalIntervalOption e ρ with
+      | some I => some (cosInterval I)
+      | none => none
+  | Expr.log e =>
+      match evalIntervalOption e ρ with
+      | none => none
+      | some J =>
+          if h : IntervalRat.isPositive J then
+            some (IntervalRat.logComputable J evalIntervalLogDepth)
+          else
+            none
+  | Expr.atan e =>
+      match evalIntervalOption e ρ with
+      | some I => some (atanInterval I)
+      | none => none
+  | Expr.arsinh e =>
+      match evalIntervalOption e ρ with
+      | some I => some (arsinhInterval I)
+      | none => none
+  | Expr.atanh e =>
+      match evalIntervalOption e ρ with
+      | none => none
+      | some J =>
+          if -1 < J.lo ∧ J.hi < 1 then
+            some (IntervalRat.atanhComputable J evalIntervalAtanhDepth)
+          else
+            none
+  | Expr.sinc e =>
+      match evalIntervalOption e ρ with
+      | some _ => some ⟨-1, 1, by norm_num⟩  -- sinc is bounded by [-1, 1]
+      | none => none
+  | Expr.erf e =>
+      match evalIntervalOption e ρ with
+      | some _ => some ⟨-1, 1, by norm_num⟩  -- erf is bounded by [-1, 1]
+      | none => none
+  | Expr.sinh e =>
+      match evalIntervalOption e ρ with
+      | some I => some (sinhInterval I)
+      | none => none
+  | Expr.cosh e =>
+      match evalIntervalOption e ρ with
+      | some I => some (coshInterval I)
+      | none => none
+  | Expr.tanh e =>
+      match evalIntervalOption e ρ with
+      | some I => some (tanhInterval I)
+      | none => none
+  | Expr.sqrt e =>
+      match evalIntervalOption e ρ with
+      | some I => some (IntervalRat.sqrtInterval I)
+      | none => none
+  | Expr.namedConst c => some c.interval
+
+/-- Main correctness theorem for evalIntervalOption (approach 1 from plan).
+
+    When evalIntervalOption returns `some I`:
+    1. The expression evaluates to a value in I for all ρ_real ∈ ρ_int
+    2. All inv denominators along the evaluation are guaranteed nonzero
+       (because their intervals don't contain zero)
+
+    This follows your suggestion to keep ADSupported syntactic and add
+    separate semantic hypotheses. The key insight is that if evalIntervalOption
+    succeeds (returns Some), the interval arithmetic has already verified
+    that no denominator interval contains zero. -/
+theorem evalIntervalOption_correct (e : Expr)
+    (ρ_int : IntervalEnv) (I : IntervalRat)
+    (hsome : evalIntervalOption e ρ_int = some I)
+    (ρ_real : Nat → ℝ) (hρ : envMem ρ_real ρ_int) :
+    Expr.eval ρ_real e ∈ I := by
+  induction e generalizing I with
+  | const q =>
+    simp only [evalIntervalOption] at hsome
+    cases hsome
+    simp only [Expr.eval_const]
+    exact IntervalRat.mem_singleton q
+  | var idx =>
+    simp only [evalIntervalOption] at hsome
+    cases hsome
+    simp only [Expr.eval_var]
+    exact hρ idx
+  | add e₁ e₂ ih₁ ih₂ =>
+    simp only [evalIntervalOption] at hsome
+    cases heq₁ : evalIntervalOption e₁ ρ_int with
+    | none => simp only [heq₁] at hsome; contradiction
+    | some I₁ =>
+      cases heq₂ : evalIntervalOption e₂ ρ_int with
+      | none => simp only [heq₁, heq₂] at hsome; contradiction
+      | some I₂ =>
+        simp only [heq₁, heq₂] at hsome
+        cases hsome
+        simp only [Expr.eval_add]
+        exact IntervalRat.mem_add (ih₁ I₁ heq₁) (ih₂ I₂ heq₂)
+  | mul e₁ e₂ ih₁ ih₂ =>
+    simp only [evalIntervalOption] at hsome
+    cases heq₁ : evalIntervalOption e₁ ρ_int with
+    | none => simp only [heq₁] at hsome; contradiction
+    | some I₁ =>
+      cases heq₂ : evalIntervalOption e₂ ρ_int with
+      | none => simp only [heq₁, heq₂] at hsome; contradiction
+      | some I₂ =>
+        simp only [heq₁, heq₂] at hsome
+        cases hsome
+        simp only [Expr.eval_mul]
+        exact IntervalRat.mem_mul (ih₁ I₁ heq₁) (ih₂ I₂ heq₂)
+  | neg e ih =>
+    simp only [evalIntervalOption] at hsome
+    cases heq : evalIntervalOption e ρ_int with
+    | none => simp only [heq] at hsome; contradiction
+    | some I' =>
+      simp only [heq] at hsome
+      cases hsome
+      simp only [Expr.eval_neg]
+      exact IntervalRat.mem_neg (ih I' heq)
+  | inv e ih =>
+    simp only [evalIntervalOption] at hsome
+    cases heq : evalIntervalOption e ρ_int with
+    | none => simp only [heq] at hsome; contradiction
+    | some J =>
+      simp only [heq] at hsome
+      split at hsome
+      · contradiction
+      · rename_i hnonzero
+        cases hsome
+        simp only [Expr.eval_inv]
+        have hJ_mem := ih J heq
+        -- The denominator is nonzero because J doesn't contain zero and eval ∈ J
+        have heval_ne : Expr.eval ρ_real e ≠ 0 := by
+          intro heq_zero
+          rw [heq_zero] at hJ_mem
+          simp only [IntervalRat.mem_def] at hJ_mem
+          simp only [IntervalRat.containsZero, not_and, not_le] at hnonzero
+          rcases le_or_gt J.lo 0 with hlo | hlo
+          · have hhi_nonneg : (0 : ℚ) ≤ J.hi := by
+              have h : (0 : ℝ) ≤ J.hi := hJ_mem.2
+              exact_mod_cast h
+            exact absurd (hnonzero hlo) (not_lt.mpr hhi_nonneg)
+          · have hlo_pos : (0 : ℝ) < J.lo := by exact_mod_cast hlo
+            exact absurd hJ_mem.1 (not_le.mpr hlo_pos)
+        exact IntervalRat.mem_invNonzero hJ_mem heval_ne
+  | log e ih =>
+    simp only [evalIntervalOption] at hsome
+    cases heq : evalIntervalOption e ρ_int with
+    | none => simp only [heq] at hsome; contradiction
+    | some J =>
+      simp only [heq] at hsome
+      split at hsome
+      · rename_i hpos
+        cases hsome
+        simp only [Expr.eval_log]
+        have hJ_mem := ih J heq
+        -- The argument is positive because J.lo > 0 and eval ∈ J
+        exact IntervalRat.mem_logComputable hJ_mem hpos evalIntervalLogDepth
+      · contradiction
+  | atanh e ih =>
+    simp only [evalIntervalOption] at hsome
+    cases heq : evalIntervalOption e ρ_int with
+    | none => simp only [heq] at hsome; contradiction
+    | some J =>
+      simp only [heq] at hsome
+      split at hsome
+      · rename_i hunit
+        cases hsome
+        simp only [Expr.eval_atanh]
+        exact IntervalRat.mem_atanhComputable (ih J heq) hunit.1 hunit.2
+          evalIntervalAtanhDepth
+      · contradiction
+  | sinc e ih =>
+    simp only [evalIntervalOption] at hsome
+    cases heq : evalIntervalOption e ρ_int with
+    | none => simp only [heq] at hsome; contradiction
+    | some I' =>
+      simp only [heq] at hsome
+      cases hsome
+      simp only [Expr.eval_sinc]
+      -- sinc(x) ∈ [-1, 1] for all x, using Mathlib's Real.sinc lemmas
+      simp only [IntervalRat.mem_def]
+      simpa only [Set.mem_Icc, Rat.cast_neg, Rat.cast_one] using
+        (Real.sinc_mem_Icc (Expr.eval ρ_real e))
+  | erf e ih =>
+    simp only [evalIntervalOption] at hsome
+    cases heq : evalIntervalOption e ρ_int with
+    | none => simp only [heq] at hsome; contradiction
+    | some I' =>
+      simp only [heq] at hsome
+      cases hsome
+      simp only [Expr.eval_erf]
+      -- erf(x) ∈ [-1, 1] for all x
+      simp only [IntervalRat.mem_def]
+      simpa only [Set.mem_Icc, Rat.cast_neg, Rat.cast_one] using
+        (Real.erf_mem_Icc (Expr.eval ρ_real e))
+  | namedConst c =>
+    simp only [evalIntervalOption] at hsome
+    cases hsome
+    simp only [Expr.eval_namedConst]
+    exact c.mem_interval
+  | exp e ih | sin e ih | cos e ih | sinh e ih | cosh e ih | tanh e ih
+  | atan e ih | arsinh e ih | sqrt e ih =>
+    cases heq : evalIntervalOption e ρ_int with
+    | none => simp only [evalIntervalOption, heq] at hsome; contradiction
+    | some I' =>
+      simp only [evalIntervalOption, heq] at hsome
+      cases hsome
+      first
+      | exact IntervalRat.mem_expComputable (ih I' heq) evalIntervalExpDepth
+      | exact mem_sinInterval (ih I' heq)
+      | exact mem_cosInterval (ih I' heq)
+      | exact IntervalRat.mem_sinhComputable (ih I' heq) 10
+      | exact IntervalRat.mem_coshComputable (ih I' heq) 10
+      | exact mem_tanhInterval (ih I' heq)
+      | exact mem_atanInterval (ih I' heq)
+      | exact mem_arsinhInterval (ih I' heq)
+      | exact IntervalRat.mem_sqrtInterval' (ih I' heq)
+
+/-! ### Checked API with diagnostics -/
+
+/-- Diagnose the first partial-domain failure after `evalIntervalOption` returned
+`none`. This function is deliberately separate from the trusted computation:
+soundness depends only on successful `evalIntervalOption`, while diagnostics may be
+refined without changing the correctness theorem. -/
+def diagnoseEvalIntervalFailure (e : Expr) (ρ : IntervalEnv) : EvalError :=
+  match e with
+  | .add e₁ e₂ | .mul e₁ e₂ =>
+      if evalIntervalOption e₁ ρ = none then
+        .nestedFailure "left operand" (diagnoseEvalIntervalFailure e₁ ρ)
+      else
+        .nestedFailure "right operand" (diagnoseEvalIntervalFailure e₂ ρ)
+  | .neg e | .exp e | .sin e | .cos e | .atan e | .arsinh e | .sinc e |
+      .erf e | .sinh e | .cosh e | .tanh e | .sqrt e =>
+      .nestedFailure "unary operand" (diagnoseEvalIntervalFailure e ρ)
+  | .inv e =>
+      match evalIntervalOption e ρ with
+      | some I => .reciprocalContainsZero I
+      | none => .nestedFailure "reciprocal operand" (diagnoseEvalIntervalFailure e ρ)
+  | .log e =>
+      match evalIntervalOption e ρ with
+      | some I => .logNonpositive I
+      | none => .nestedFailure "logarithm operand" (diagnoseEvalIntervalFailure e ρ)
+  | .atanh e =>
+      match evalIntervalOption e ρ with
+      | some I => .atanhOutsideUnitBall I
+      | none => .nestedFailure "atanh operand" (diagnoseEvalIntervalFailure e ρ)
+  | .const _ | .var _ | .namedConst _ =>
+      .unsupportedBackend "internal: total expression unexpectedly failed"
+termination_by e
+
+/-- Checked rational evaluator. Every successful result is a certified finite
+enclosure; domain-invalid expressions return a structured error. -/
+def evalIntervalChecked (e : Expr) (ρ : IntervalEnv) : EvalResult IntervalRat :=
+  match evalIntervalOption e ρ with
+  | some I => .ok I
+  | none => .error (diagnoseEvalIntervalFailure e ρ)
+
+/-- Success of `evalIntervalChecked` is sufficient for enclosure correctness
+for every expression constructor. -/
+theorem evalIntervalChecked_correct (e : Expr) (ρ_int : IntervalEnv) (I : IntervalRat)
+    (hsuccess : evalIntervalChecked e ρ_int = .ok I)
+    (ρ_real : Nat → ℝ) (hρ : envMem ρ_real ρ_int) :
+    Expr.eval ρ_real e ∈ I := by
+  cases heval : evalIntervalOption e ρ_int with
+  | none =>
+    rw [evalIntervalChecked, heval] at hsuccess
+    contradiction
+  | some J =>
+    rw [evalIntervalChecked, heval] at hsuccess
+    injection hsuccess with hJI
+    subst I
+    exact evalIntervalOption_correct e ρ_int J heval ρ_real hρ
+
+/-- Checked Rational evaluation with a tight path for the verified computable
+core. Unsupported syntax or a failed core-domain check falls back to the
+general checked evaluator, preserving its structured domain errors. -/
+def evalIntervalTightChecked (e : Expr) (ρ : IntervalEnv)
+    (cfg : EvalConfig := {}) : EvalResult IntervalRat :=
+  if e.checkSupportedCore && checkDomainValid e ρ cfg then
+    .ok (LeanCert.Internal.Rational.evalTotalCore e ρ cfg)
+  else
+    evalIntervalChecked e ρ
+
+/-- Every successful tight checked Rational evaluation encloses the expression
+value. The core branch uses the core correctness theorem; the fallback uses
+the general checked evaluator theorem. -/
+theorem evalIntervalTightChecked_correct (e : Expr) (ρ_int : IntervalEnv)
+    (cfg : EvalConfig) (I : IntervalRat)
+    (hsuccess : evalIntervalTightChecked e ρ_int cfg = .ok I)
+    (ρ_real : Nat → ℝ) (hρ : envMem ρ_real ρ_int) :
+    Expr.eval ρ_real e ∈ I := by
+  by_cases hsupp : e.checkSupportedCore = true
+  · by_cases hdom : checkDomainValid e ρ_int cfg = true
+    · have hsupp' := Expr.checkSupportedCore_correct hsupp
+      have hdom' := checkDomainValid_correct e ρ_int cfg hdom
+      have hmem := evalIntervalCore_correct e hsupp' ρ_real ρ_int hρ cfg hdom'
+      simp only [ExceptT.stM_eq, evalIntervalTightChecked, hsupp, hdom, Bool.and_self, ↓reduceIte,
+        Except.ok.injEq] at hsuccess
+      subst I
+      exact hmem
+    · have hfallback :
+          evalIntervalTightChecked e ρ_int cfg = evalIntervalChecked e ρ_int := by
+        simp [evalIntervalTightChecked, hsupp, hdom]
+      rw [hfallback] at hsuccess
+      exact evalIntervalChecked_correct e ρ_int I hsuccess ρ_real hρ
+  · have hfallback :
+        evalIntervalTightChecked e ρ_int cfg = evalIntervalChecked e ρ_int := by
+      simp [evalIntervalTightChecked, hsupp]
+    rw [hfallback] at hsuccess
+    exact evalIntervalChecked_correct e ρ_int I hsuccess ρ_real hρ
+
+/-- Single-variable version of evalIntervalOption -/
+def evalIntervalOption1 (e : Expr) (I : IntervalRat) : Option IntervalRat :=
+  evalIntervalOption e (fun _ => I)
+
+/-- Correctness for single-variable partial evaluation -/
+theorem evalIntervalOption1_correct (e : Expr)
+    (I : IntervalRat) (J : IntervalRat)
+    (hsome : evalIntervalOption1 e I = some J)
+    (x : ℝ) (hx : x ∈ I) :
+    Expr.eval (fun _ => x) e ∈ J :=
+  evalIntervalOption_correct e _ J hsome _ (fun _ => hx)
+
+/-- When evalIntervalOption succeeds, we get bounds -/
+theorem evalIntervalOption_le_of_hi (e : Expr)
+    (I : IntervalRat) (J : IntervalRat) (c : ℚ)
+    (hsome : evalIntervalOption1 e I = some J)
+    (hhi : J.hi ≤ c) :
+    ∀ x ∈ I, Expr.eval (fun _ => x) e ≤ c := by
+  intro x hx
+  have hmem := evalIntervalOption1_correct e I J hsome x hx
+  simp only [IntervalRat.mem_def] at hmem
+  have heval_le_hi : Expr.eval (fun _ => x) e ≤ J.hi := hmem.2
+  have hhi_le_c : (J.hi : ℝ) ≤ c := by exact_mod_cast hhi
+  exact le_trans heval_le_hi hhi_le_c
+
+/-- When evalIntervalOption succeeds, we get lower bounds -/
+theorem evalIntervalOption_ge_of_lo (e : Expr)
+    (I : IntervalRat) (J : IntervalRat) (c : ℚ)
+    (hsome : evalIntervalOption1 e I = some J)
+    (hlo : c ≤ J.lo) :
+    ∀ x ∈ I, c ≤ Expr.eval (fun _ => x) e := by
+  intro x hx
+  have hmem := evalIntervalOption1_correct e I J hsome x hx
+  simp only [IntervalRat.mem_def] at hmem
+  have hlo_le_eval : J.lo ≤ Expr.eval (fun _ => x) e := hmem.1
+  have hc_le_lo : (c : ℝ) ≤ J.lo := by exact_mod_cast hlo
+  exact le_trans hc_le_lo hlo_le_eval
+
+end LeanCert.Engine
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Semantic Lemmas for Interval Bounds
+
+This file provides semantic lemmas for deriving real bounds from certified
+interval enclosures. Tactics consume these lemmas, but the definitions belong
+to the engine layer so checked programmatic APIs do not depend on tactic
+modules.
+
+## Main theorems
+
+### Core (computable) lemmas
+* `exprCore_le_of_interval_hi` - Upper bound from interval high endpoint
+* `exprCore_ge_of_interval_lo` - Lower bound from interval low endpoint
+* `exprCore_lt_of_interval_hi_lt` - Strict upper bound
+* `exprCore_gt_of_interval_lo_gt` - Strict lower bound
+
+### Extended (noncomputable) lemmas
+* `expr_le_of_interval_hi` - Upper bound from interval high endpoint
+* `expr_ge_of_interval_lo` - Lower bound from interval low endpoint
+* `expr_lt_of_interval_hi_lt` - Strict upper bound
+* `expr_gt_of_interval_lo_gt` - Strict lower bound
+* `expr_le_of_mem_interval` - Single-point variant
+* `expr_ge_of_mem_interval` - Single-point variant
+
+## Usage
+
+These lemmas are intended to be used by tactics like `certify_bound` and
+`interval_decide` to close goals of the form `∀ x ∈ I, f(x) ≤ c` or similar.
+
+The core lemmas use the computable evaluator and can work with `native_decide`.
+The extended lemmas use the noncomputable evaluator with floor/ceil bounds.
+-/
+
+/-! ### Tactic-facing lemmas for interval bounds (core, computable) -/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+/-- Upper bound lemma for core expressions (computable).
+    FULLY PROVED - no sorry, no axioms. Accepts configurable Taylor depth.
+    Requires domain validity (e.g., log arguments must be positive). -/
+theorem exprCore_le_of_interval_hi (e : Expr) (hsupp : ExprSupportedCore e)
+    (I : IntervalRat) (c : ℚ) (cfg : EvalConfig := {})
+    (hdom : evalDomainValid1 e I cfg)
+    (hhi : (LeanCert.Internal.Rational.evalTotalCore1 e I cfg).hi ≤ c) :
+    ∀ x ∈ I, Expr.eval (fun _ => x) e ≤ c := by
+  intro x hx
+  have hmem := evalIntervalCore1_correct e hsupp x I hx cfg hdom
+  simp only [IntervalRat.mem_def] at hmem
+  have heval_le_hi : Expr.eval (fun _ => x) e ≤ (LeanCert.Internal.Rational.evalTotalCore1 e I
+    cfg).hi := hmem.2
+  have hhi_le_c : ((LeanCert.Internal.Rational.evalTotalCore1 e I cfg).hi : ℝ) ≤ c := by
+    exact_mod_cast hhi
+  exact le_trans heval_le_hi hhi_le_c
+
+/-- Lower bound lemma for core expressions (computable).
+    FULLY PROVED - no sorry, no axioms. Accepts configurable Taylor depth.
+    Requires domain validity (e.g., log arguments must be positive). -/
+theorem exprCore_ge_of_interval_lo (e : Expr) (hsupp : ExprSupportedCore e)
+    (I : IntervalRat) (c : ℚ) (cfg : EvalConfig := {})
+    (hdom : evalDomainValid1 e I cfg)
+    (hlo : c ≤ (LeanCert.Internal.Rational.evalTotalCore1 e I cfg).lo) :
+    ∀ x ∈ I, c ≤ Expr.eval (fun _ => x) e := by
+  intro x hx
+  have hmem := evalIntervalCore1_correct e hsupp x I hx cfg hdom
+  simp only [IntervalRat.mem_def] at hmem
+  have hlo_le_eval : (LeanCert.Internal.Rational.evalTotalCore1 e I cfg).lo ≤ Expr.eval (fun _ =>
+    x) e := hmem.1
+  have hc_le_lo : (c : ℝ) ≤ (LeanCert.Internal.Rational.evalTotalCore1 e I cfg).lo := by
+    exact_mod_cast hlo
+  exact le_trans hc_le_lo hlo_le_eval
+
+/-- Strict upper bound for core expressions (computable).
+    FULLY PROVED - no sorry, no axioms. Accepts configurable Taylor depth.
+    Requires domain validity (e.g., log arguments must be positive). -/
+theorem exprCore_lt_of_interval_hi_lt (e : Expr) (hsupp : ExprSupportedCore e)
+    (I : IntervalRat) (c : ℚ) (cfg : EvalConfig := {})
+    (hdom : evalDomainValid1 e I cfg)
+    (hhi : (LeanCert.Internal.Rational.evalTotalCore1 e I cfg).hi < c) :
+    ∀ x ∈ I, Expr.eval (fun _ => x) e < c := by
+  intro x hx
+  have hmem := evalIntervalCore1_correct e hsupp x I hx cfg hdom
+  simp only [IntervalRat.mem_def] at hmem
+  have heval_le_hi : Expr.eval (fun _ => x) e ≤ (LeanCert.Internal.Rational.evalTotalCore1 e I
+    cfg).hi := hmem.2
+  have hhi_lt_c : ((LeanCert.Internal.Rational.evalTotalCore1 e I cfg).hi : ℝ) < c := by
+    exact_mod_cast hhi
+  exact lt_of_le_of_lt heval_le_hi hhi_lt_c
+
+/-- Strict lower bound for core expressions (computable).
+    FULLY PROVED - no sorry, no axioms. Accepts configurable Taylor depth.
+    Requires domain validity (e.g., log arguments must be positive). -/
+theorem exprCore_gt_of_interval_lo_gt (e : Expr) (hsupp : ExprSupportedCore e)
+    (I : IntervalRat) (c : ℚ) (cfg : EvalConfig := {})
+    (hdom : evalDomainValid1 e I cfg)
+    (hlo : c < (LeanCert.Internal.Rational.evalTotalCore1 e I cfg).lo) :
+    ∀ x ∈ I, c < Expr.eval (fun _ => x) e := by
+  intro x hx
+  have hmem := evalIntervalCore1_correct e hsupp x I hx cfg hdom
+  simp only [IntervalRat.mem_def] at hmem
+  have hlo_le_eval : (LeanCert.Internal.Rational.evalTotalCore1 e I cfg).lo ≤ Expr.eval (fun _ =>
+    x) e := hmem.1
+  have hc_lt_lo : (c : ℝ) < (LeanCert.Internal.Rational.evalTotalCore1 e I cfg).lo := by
+    exact_mod_cast hlo
+  exact lt_of_lt_of_le hc_lt_lo hlo_le_eval
+
+/-! ### Tactic-facing lemmas for interval bounds (extended, noncomputable) -/
+
+/-- Upper bound lemma for extended expressions.
+    FULLY PROVED - no sorry, no axioms. -/
+theorem expr_le_of_interval_hi (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (c : ℚ) (hhi : (LeanCert.Internal.Rational.evalUnchecked1 e I).hi ≤ c) :
+    ∀ x ∈ I, Expr.eval (fun _ => x) e ≤ c := by
+  intro x hx
+  have hmem := evalInterval1_correct e hsupp x I hx
+  simp only [IntervalRat.mem_def] at hmem
+  have heval_le_hi : Expr.eval (fun _ => x) e ≤ (LeanCert.Internal.Rational.evalUnchecked1 e I).hi
+    := hmem.2
+  have hhi_le_c : ((LeanCert.Internal.Rational.evalUnchecked1 e I).hi : ℝ) ≤ c := by
+    exact_mod_cast hhi
+  exact le_trans heval_le_hi hhi_le_c
+
+/-- Lower bound lemma for extended expressions.
+    FULLY PROVED - no sorry, no axioms. -/
+theorem expr_ge_of_interval_lo (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (c : ℚ) (hlo : c ≤ (LeanCert.Internal.Rational.evalUnchecked1 e I).lo) :
+    ∀ x ∈ I, c ≤ Expr.eval (fun _ => x) e := by
+  intro x hx
+  have hmem := evalInterval1_correct e hsupp x I hx
+  simp only [IntervalRat.mem_def] at hmem
+  have hlo_le_eval : (LeanCert.Internal.Rational.evalUnchecked1 e I).lo ≤ Expr.eval (fun _ => x) e
+    := hmem.1
+  have hc_le_lo : (c : ℝ) ≤ (LeanCert.Internal.Rational.evalUnchecked1 e I).lo := by
+    exact_mod_cast hlo
+  exact le_trans hc_le_lo hlo_le_eval
+
+/-- Strict upper bound for extended expressions.
+    FULLY PROVED - no sorry, no axioms. -/
+theorem expr_lt_of_interval_hi_lt (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (c : ℚ) (hhi : (LeanCert.Internal.Rational.evalUnchecked1 e I).hi < c) :
+    ∀ x ∈ I, Expr.eval (fun _ => x) e < c := by
+  intro x hx
+  have hmem := evalInterval1_correct e hsupp x I hx
+  simp only [IntervalRat.mem_def] at hmem
+  have heval_le_hi : Expr.eval (fun _ => x) e ≤ (LeanCert.Internal.Rational.evalUnchecked1 e I).hi
+    := hmem.2
+  have hhi_lt_c : ((LeanCert.Internal.Rational.evalUnchecked1 e I).hi : ℝ) < c := by
+    exact_mod_cast hhi
+  exact lt_of_le_of_lt heval_le_hi hhi_lt_c
+
+/-- Strict lower bound for extended expressions.
+    FULLY PROVED - no sorry, no axioms. -/
+theorem expr_gt_of_interval_lo_gt (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (c : ℚ) (hlo : c < (LeanCert.Internal.Rational.evalUnchecked1 e I).lo) :
+    ∀ x ∈ I, c < Expr.eval (fun _ => x) e := by
+  intro x hx
+  have hmem := evalInterval1_correct e hsupp x I hx
+  simp only [IntervalRat.mem_def] at hmem
+  have hlo_le_eval : (LeanCert.Internal.Rational.evalUnchecked1 e I).lo ≤ Expr.eval (fun _ => x) e
+    := hmem.1
+  have hc_lt_lo : (c : ℝ) < (LeanCert.Internal.Rational.evalUnchecked1 e I).lo := by
+    exact_mod_cast hlo
+  exact lt_of_lt_of_le hc_lt_lo hlo_le_eval
+
+/-- Variant for single point (extended). -/
+theorem expr_le_of_mem_interval (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (c : ℚ) (x : ℝ) (hx : x ∈ I)
+    (hhi : (LeanCert.Internal.Rational.evalUnchecked1 e I).hi ≤ c) :
+    Expr.eval (fun _ => x) e ≤ c :=
+  expr_le_of_interval_hi e hsupp I c hhi x hx
+
+/-- Variant for single point (extended). -/
+theorem expr_ge_of_mem_interval (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (c : ℚ) (x : ℝ) (hx : x ∈ I)
+    (hlo : c ≤ (LeanCert.Internal.Rational.evalUnchecked1 e I).lo) :
+    c ≤ Expr.eval (fun _ => x) e :=
+  expr_ge_of_interval_lo e hsupp I c hlo x hx
+
+end LeanCert.Engine
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Interval Evaluation of Expressions
+
+This file re-exports the interval evaluation infrastructure for `LeanCert.Core.Expr`.
+
+## Module structure
+
+The implementation is split across several files:
+
+* `LeanCert.Core.Support` - Expression support predicates for total core
+  evaluation and automatic differentiation
+
+* `LeanCert.Engine.Eval.Core` - Computable interval evaluator
+(`LeanCert.Internal.Rational.evalTotalCore`)
+  and transcendental interval bounds
+
+* `LeanCert.Engine.Eval.Extended` - Internal noncomputable evaluator and the
+  partial evaluator with inv/log support (`evalIntervalOption`)
+
+* `LeanCert.Engine.Bounds.Lemmas` - Semantic lemmas for deriving real bounds
+
+## Main definitions (re-exported)
+
+### Expression support predicates
+* `ExprSupportedCore` - Computable subset (const, var, add, mul, neg, sin, cos, exp, sqrt, sinh,
+cosh, tanh, pi)
+* `ADSupported` - Noncomputable AD subset (const, var, add, mul, neg, sin, cos, exp)
+
+### Evaluators
+* `LeanCert.Internal.Rational.evalTotalCore` - Computable interval evaluator (uses Taylor series)
+* `LeanCert.Internal.Rational.evalUnchecked` - Internal noncomputable evaluator
+* `evalIntervalOption` - Partial evaluator with inv/log support
+
+### Correctness theorems
+* `evalIntervalCore_correct` - Core evaluator correctness
+* `evalInterval_correct` - Extended evaluator correctness
+* `evalIntervalOption_correct` - Partial evaluator correctness
+
+### Tactic lemmas
+* `exprCore_le_of_interval_hi` / `exprCore_ge_of_interval_lo` - Core bounds
+* `expr_le_of_interval_hi` / `expr_ge_of_interval_lo` - Extended bounds
+
+## Design notes
+
+The evaluators are split by computability:
+- `LeanCert.Internal.Rational.evalTotalCore` is COMPUTABLE, enabling `native_decide` in tactics
+- `evalInterval` is NONCOMPUTABLE, using Real.exp with floor/ceil bounds
+- Both are fully verified (no sorry, no axioms)
+-/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+
+-- Re-export all definitions from component modules
+-- All imports are transitive, so users can continue to `import LeanCert.Engine.IntervalEval`
+-- and have access to all definitions as before.
+
+end LeanCert.Engine
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Automatic Differentiation - Basic Definitions
+
+This file provides the core types and algebraic operations for forward-mode
+automatic differentiation using interval arithmetic.
+
+## Main definitions
+
+* `DualInterval` - A pair of intervals representing (value, derivative)
+* `DualInterval.const` - Constant dual (derivative is zero)
+* `DualInterval.varActive` - Active variable (derivative is 1)
+* `DualInterval.varPassive` - Passive variable (derivative is 0)
+* `DualInterval.add` - Addition with sum rule
+* `DualInterval.mul` - Multiplication with product rule
+* `DualInterval.neg` - Negation
+-/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+/-- Dual number with interval components: represents (value, derivative) -/
+structure DualInterval where
+  /-- The interval enclosing the function value. -/
+  val : IntervalRat
+  /-- The interval enclosing the derivative. -/
+  der : IntervalRat
+  deriving Repr
+
+/-- Default DualInterval for unsupported expression branches -/
+instance : Inhabited DualInterval where
+  default := ⟨default, default⟩
+
+namespace DualInterval
+
+/-- Dual interval for a constant (derivative is zero) -/
+def const (q : ℚ) : DualInterval :=
+  { val := IntervalRat.singleton q
+    der := IntervalRat.singleton 0 }
+
+/-- Dual interval for the constant π (derivative is zero) -/
+def piConst : DualInterval :=
+  { val := piInterval
+    der := IntervalRat.singleton 0 }
+
+/-- Dual interval for the Euler–Mascheroni constant γ (derivative is zero) -/
+def eulerMascheroniConst : DualInterval :=
+  { val := eulerMascheroniInterval
+    der := IntervalRat.singleton 0 }
+
+/-- Dual interval for a named mathematical constant (derivative is zero) -/
+def ofMathConst (c : MathConst) : DualInterval :=
+  { val := c.interval
+    der := IntervalRat.singleton 0 }
+
+/-- Dual interval for the variable we're differentiating with respect to -/
+def varActive (I : IntervalRat) : DualInterval :=
+  { val := I
+    der := IntervalRat.singleton 1 }
+
+/-- Dual interval for a passive variable -/
+def varPassive (I : IntervalRat) : DualInterval :=
+  { val := I
+    der := IntervalRat.singleton 0 }
+
+/-- Add two dual intervals -/
+def add (d₁ d₂ : DualInterval) : DualInterval :=
+  { val := IntervalRat.add d₁.val d₂.val
+    der := IntervalRat.add d₁.der d₂.der }
+
+/-- Multiply two dual intervals (product rule) -/
+def mul (d₁ d₂ : DualInterval) : DualInterval :=
+  { val := IntervalRat.mul d₁.val d₂.val
+    -- d(f*g) = f'*g + f*g'
+    der := IntervalRat.add
+             (IntervalRat.mul d₁.der d₂.val)
+             (IntervalRat.mul d₁.val d₂.der) }
+
+/-- Negate a dual interval -/
+def neg (d : DualInterval) : DualInterval :=
+  { val := IntervalRat.neg d.val
+    der := IntervalRat.neg d.der }
+
+/-- Inverse of a dual interval (quotient rule: d(1/f) = -f'/f²)
+    Uses invInterval for the value component.
+    For intervals containing zero, returns wide bounds. -/
+def inv (d : DualInterval) : DualInterval :=
+  let inv_val := invInterval d.val
+  -- d(1/f) = -f'/f² = -f' * (1/f)²
+  let inv_sq := IntervalRat.mul inv_val inv_val
+  let der' := IntervalRat.neg (IntervalRat.mul d.der inv_sq)
+  { val := inv_val, der := der' }
+
+end DualInterval
+
+end LeanCert.Engine
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Automatic Differentiation - Transcendental Functions
+
+This file provides dual interval implementations for transcendental functions,
+implementing the chain rule for each.
+
+## Main definitions
+
+### Total functions (always succeed)
+* `DualInterval.sin` - Sine with chain rule
+* `DualInterval.cos` - Cosine with chain rule
+* `DualInterval.exp` - Exponential with chain rule
+* `DualInterval.atan` - Arctangent with chain rule
+* `DualInterval.arsinh` - Inverse hyperbolic sine
+* `DualInterval.sinh`, `cosh`, `tanh` - Hyperbolic functions
+* `DualInterval.erf` - Error function
+* `DualInterval.sqrt` - Square root (conservative derivative bound)
+* `DualInterval.sinc` - sinc function
+
+### Partial functions (return Option)
+* `DualInterval.atanhOption` - Inverse hyperbolic tangent (requires |x| < 1)
+* `DualInterval.inv?` - Inverse (requires nonzero)
+* `DualInterval.logOption` - Logarithm (requires positive)
+* `DualInterval.sqrt?` - Square root with tight bounds (requires positive)
+-/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+namespace DualInterval
+
+/-- Dual for sin (chain rule: d(sin f) = cos(f) * f') -/
+def sin (d : DualInterval) : DualInterval :=
+  { val := sinInterval d.val
+    der := IntervalRat.mul (cosInterval d.val) d.der }
+
+/-- Dual for cos (chain rule: d(cos f) = -sin(f) * f') -/
+def cos (d : DualInterval) : DualInterval :=
+  { val := cosInterval d.val
+    der := IntervalRat.mul (IntervalRat.neg (sinInterval d.val)) d.der }
+
+/-- Dual for exp (chain rule: d(exp f) = exp(f) * f') -/
+noncomputable def exp (d : DualInterval) : DualInterval :=
+  { val := IntervalRat.expInterval d.val
+    der := IntervalRat.mul (IntervalRat.expInterval d.val) d.der }
+
+/-- The interval [0, 1] used to bound derivative factors in (0, 1] -/
+def unitInterval : IntervalRat := ⟨0, 1, by norm_num⟩
+
+/-- The derivative factor for arctan: 1/(1+x²) is in [0, 1] -/
+theorem arctan_deriv_factor_mem_unitInterval (y : ℝ) :
+    1 / (1 + y ^ 2) ∈ unitInterval := by
+  simp only [unitInterval, IntervalRat.mem_def, Rat.cast_zero, Rat.cast_one]
+  have hpos : 0 < 1 + y ^ 2 := by nlinarith [sq_nonneg y]
+  constructor
+  · exact div_nonneg (by norm_num : (0 : ℝ) ≤ 1) (le_of_lt hpos)
+  · rw [div_le_one hpos]
+    nlinarith [sq_nonneg y]
+
+/-- The derivative factor for arsinh: 1/√(1+x²) is in [0, 1] -/
+theorem arsinh_deriv_factor_mem_unitInterval (y : ℝ) :
+    (Real.sqrt (1 + y ^ 2))⁻¹ ∈ unitInterval := by
+  simp only [unitInterval, IntervalRat.mem_def, Rat.cast_zero, Rat.cast_one]
+  have h1 : 1 + y ^ 2 ≥ 1 := by nlinarith [sq_nonneg y]
+  have hsqrt_ge_one : Real.sqrt (1 + y ^ 2) ≥ 1 := by
+    calc Real.sqrt (1 + y ^ 2) ≥ Real.sqrt 1 := Real.sqrt_le_sqrt h1
+      _ = 1 := Real.sqrt_one
+  have hsqrt_pos : 0 < Real.sqrt (1 + y ^ 2) := by
+    apply Real.sqrt_pos.mpr
+    nlinarith [sq_nonneg y]
+  constructor
+  · exact inv_nonneg.mpr (le_of_lt hsqrt_pos)
+  · exact inv_le_one_of_one_le₀ hsqrt_ge_one
+
+/-- Dual for atan (chain rule: d(atan f) = f' / (1 + f²)) -/
+def atan (d : DualInterval) : DualInterval :=
+  { val := atanInterval d.val
+    -- d(atan f) = f' / (1 + f²)
+    -- Since 0 < 1/(1+x²) ≤ 1 for all x, we multiply d.der by [0, 1]
+    der := IntervalRat.mul d.der unitInterval }
+
+/-- Dual for arsinh (chain rule: d(arsinh f) = f' / √(1 + f²)) -/
+def arsinh (d : DualInterval) : DualInterval :=
+  { val := arsinhInterval d.val
+    -- d(arsinh f) = f' / √(1 + f²)
+    -- Since 0 < 1/√(1+f²) ≤ 1, we multiply d.der by [0, 1]
+    der := IntervalRat.mul d.der unitInterval }
+
+/-- Dual for sinh (chain rule: d(sinh f) = cosh(f) * f') -/
+def sinh (d : DualInterval) : DualInterval :=
+  { val := sinhInterval d.val
+    -- sinh'(x) = cosh(x), so d(sinh f) = cosh(f) * f'
+    der := IntervalRat.mul (coshInterval d.val) d.der }
+
+/-- Dual for cosh (chain rule: d(cosh f) = sinh(f) * f') -/
+def cosh (d : DualInterval) : DualInterval :=
+  { val := coshInterval d.val
+    -- cosh'(x) = sinh(x), so d(cosh f) = sinh(f) * f'
+    der := IntervalRat.mul (sinhInterval d.val) d.der }
+
+/-- Dual for tanh (chain rule: d(tanh f) = sech²(f) * f' = (1 - tanh²(f)) * f')
+    Since sech²(x) = 1 - tanh²(x) ∈ (0, 1] for all x, we use [0, 1] as bound. -/
+def tanh (d : DualInterval) : DualInterval :=
+  { val := tanhInterval d.val
+    -- tanh'(x) = sech²(x) = 1 - tanh²(x), which is in (0, 1]
+    der := IntervalRat.mul d.der unitInterval }
+
+/-- Interval containing 2/√π ≈ 1.128379... -/
+def twoDivSqrtPi : IntervalRat :=
+  ⟨1128/1000, 1129/1000, by norm_num⟩
+
+/-- Dual for erf (chain rule: d(erf f) = (2/√π) * exp(-f²) * f')
+    erf'(x) = (2/√π) * exp(-x²), which is always positive and bounded by 2/√π ≈ 1.13 -/
+noncomputable def erf (d : DualInterval) : DualInterval :=
+  { val := ⟨-1, 1, by norm_num⟩  -- erf is bounded in [-1, 1]
+    der :=
+      let valSq := IntervalRat.mul d.val d.val
+      let negValSq := IntervalRat.neg valSq
+      let expPart := IntervalRat.expInterval negValSq
+      let factor := IntervalRat.mul twoDivSqrtPi expPart
+      IntervalRat.mul factor d.der }
+
+/-- Computable dual for erf using expComputable -/
+def erfCore (d : DualInterval) (n : ℕ := 10) : DualInterval :=
+  { val := ⟨-1, 1, by norm_num⟩  -- erf is bounded in [-1, 1]
+    der :=
+      let valSq := IntervalRat.mul d.val d.val
+      let negValSq := IntervalRat.neg valSq
+      let expPart := IntervalRat.expComputable negValSq n
+      let factor := IntervalRat.mul twoDivSqrtPi expPart
+      IntervalRat.mul factor d.der }
+
+/-- Dual for sqrt (chain rule: d(sqrt f) = f' / (2 * sqrt(f)))
+    sqrt'(x) = 1/(2*sqrt(x)) for x > 0, undefined at x = 0.
+    We use sqrtInterval for the value and a conservative bound for the derivative.
+    Note: The derivative blows up as x → 0+, so this uses a wide conservative bound. -/
+def sqrt (d : DualInterval) : DualInterval :=
+  { val := IntervalRat.sqrtInterval d.val
+    -- The derivative 1/(2*sqrt(x)) ∈ [0, ∞) for x > 0
+    -- We use a very conservative bound [-100, 100] * der
+    der := IntervalRat.mul d.der ⟨-100, 100, by norm_num⟩ }
+
+/-- Conservative derivative bound [-1, 1] for sinc derivative -/
+def sincDerivBound : IntervalRat := ⟨-1, 1, by norm_num⟩
+
+/-- Dual for sinc (chain rule: d(sinc f) = sinc'(f) * f')
+    sinc'(x) = (x cos x - sin x) / x² for x ≠ 0, limit 0 at x = 0.
+    We use conservative bound: |sinc'(x)| ≤ 1 for all x. -/
+def sinc (d : DualInterval) : DualInterval :=
+  { val := ⟨-1, 1, by norm_num⟩  -- sinc is bounded in [-1, 1]
+    -- sinc'(x) ∈ [-1, 1] (conservative bound), so d(sinc f) = sinc'(f) * f'
+    der := IntervalRat.mul sincDerivBound d.der }
+
+/-! ### Partial functions (domain-restricted) -/
+
+/-- Partial dual for atanh (chain rule: d(atanh f) = f' / (1 - f²))
+    Returns None if the value interval is not contained in (-1, 1). -/
+noncomputable def atanhOption (d : DualInterval) : Option DualInterval :=
+  -- For atanh to be defined, we need |val| < 1
+  -- We check if the interval is strictly inside (-1, 1)
+  if d.val.hi < 1 ∧ d.val.lo > -1 then
+    -- Very rough bound: atanh' = 1/(1-x²), which blows up as |x| → 1
+    -- For now we use [-100, 100] * der as a hugely conservative bound
+    let bound : ℚ := 100
+    some { val := atanhInterval d.val
+           der := ⟨-bound * (|d.der.lo| + |d.der.hi| + 1),
+                   bound * (|d.der.lo| + |d.der.hi| + 1),
+                   by
+                     have h1 : (0 : ℚ) ≤ |d.der.lo| := abs_nonneg _
+                     have h2 : (0 : ℚ) ≤ |d.der.hi| := abs_nonneg _
+                     have h : (0 : ℚ) ≤ bound * (|d.der.lo| + |d.der.hi| + 1) := by
+                       exact mul_nonneg (by norm_num) (by linarith)
+                     linarith⟩ }
+  else
+    none
+
+/-- Partial dual for inv (chain rule: d(1/f) = -f'/f²)
+    Returns None if the value interval contains zero. -/
+def inv? (d : DualInterval) : Option DualInterval :=
+  if h : IntervalRat.containsZero d.val then
+    none
+  else
+    let inv_val := IntervalRat.invNonzero ⟨d.val, h⟩
+    -- d(1/f) = -f'/f² = -f' * (1/f)²
+    let inv_sq := IntervalRat.mul inv_val inv_val
+    let der' := IntervalRat.neg (IntervalRat.mul d.der inv_sq)
+    some { val := inv_val, der := der' }
+
+/-- Partial dual for log (chain rule: d(log f) = f'/f)
+    Returns None if the value interval is not strictly positive. -/
+noncomputable def logOption (d : DualInterval) : Option DualInterval :=
+  if h : IntervalRat.isPositive d.val then
+    let log_val := IntervalRat.logInterval ⟨d.val, h⟩
+    -- d(log f) = f'/f
+    let inv_val := IntervalRat.invNonzero ⟨d.val, by
+      -- isPositive implies not containsZero
+      simp only [IntervalRat.isPositive, IntervalRat.containsZero] at h ⊢
+      intro ⟨hle, _⟩
+      exact absurd h (not_lt.mpr hle)⟩
+    let der' := IntervalRat.mul d.der inv_val
+    some { val := log_val, der := der' }
+  else
+    none
+
+/-- Compute a conservative upper bound on 1/(2*sqrt(lo)) for lo > 0.
+    Uses the fact that:
+    - For 0 < lo ≤ 1: sqrt(lo) ≥ lo, so 1/(2*sqrt(lo)) ≤ 1/(2*lo)
+    - For lo > 1: sqrt(lo) > 1, so 1/(2*sqrt(lo)) < 1/2 -/
+def sqrtDerivCoefBound (lo : ℚ) (hpos : 0 < lo) : IntervalRat :=
+  if lo ≤ 1 then
+    ⟨0, 1 / (2 * lo), by
+      exact div_nonneg (by norm_num) (mul_nonneg (by norm_num) hpos.le)⟩
+  else
+    ⟨0, 1 / 2, by norm_num⟩
+
+/-- For lo > 0, the derivative coefficient 1/(2*sqrt(x)) is bounded above for x ≥ lo. -/
+theorem sqrtDerivCoef_bound {x lo : ℝ} (hlo_pos : 0 < lo) (hx_ge : lo ≤ x) :
+    1 / (2 * Real.sqrt x) ≤ max (1 / (2 * lo)) (1 / 2) := by
+  have hx_pos : 0 < x := lt_of_lt_of_le hlo_pos hx_ge
+  have hsqrt_pos : 0 < Real.sqrt x := Real.sqrt_pos.mpr hx_pos
+  have hdenom_pos : 0 < 2 * Real.sqrt x := by positivity
+  -- sqrt(x) ≥ sqrt(lo) since sqrt is monotone
+  have hsqrt_mono : Real.sqrt lo ≤ Real.sqrt x := Real.sqrt_le_sqrt hx_ge
+  have hsqrt_lo_pos : 0 < Real.sqrt lo := Real.sqrt_pos.mpr hlo_pos
+  -- 1/(2*sqrt(x)) ≤ 1/(2*sqrt(lo))
+  have hcoef_le : 1 / (2 * Real.sqrt x) ≤ 1 / (2 * Real.sqrt lo) := by
+    apply div_le_div_of_nonneg_left (by norm_num : (0 : ℝ) ≤ 1)
+    · positivity
+    · linarith
+  -- Now bound 1/(2*sqrt(lo))
+  rcases le_or_gt lo 1 with hle | hgt
+  · -- lo ≤ 1: sqrt(lo) ≥ lo (since sqrt(x) ≥ x for 0 < x ≤ 1)
+    have hsqrt_ge_lo : lo ≤ Real.sqrt lo := by
+      have h1 : lo * lo ≤ lo := by nlinarith
+      have h2 : lo = Real.sqrt lo * Real.sqrt lo → lo ≤ Real.sqrt lo := by
+        intro heq
+        by_contra! hc
+        have : lo * lo > lo := by nlinarith
+        linarith
+      rw [← Real.sq_sqrt (le_of_lt hlo_pos)] at h1
+      rcases eq_or_lt_of_le (Real.sqrt_nonneg lo) with hsz | hsgt
+      · -- Case sqrt(lo) = 0 contradicts lo > 0
+        have hsqrt_zero : Real.sqrt lo = 0 := hsz.symm
+        have hlo_le_zero : lo ≤ 0 := Real.sqrt_eq_zero'.mp hsqrt_zero
+        linarith
+      · nlinarith [Real.sq_sqrt (le_of_lt hlo_pos)]
+    -- 1/(2*sqrt(lo)) ≤ 1/(2*lo)
+    have hbound : 1 / (2 * Real.sqrt lo) ≤ 1 / (2 * lo) := by
+      apply div_le_div_of_nonneg_left (by norm_num : (0 : ℝ) ≤ 1)
+      · positivity
+      · linarith
+    calc 1 / (2 * Real.sqrt x) ≤ 1 / (2 * Real.sqrt lo) := hcoef_le
+      _ ≤ 1 / (2 * lo) := hbound
+      _ ≤ max (1 / (2 * lo)) (1 / 2) := le_max_left _ _
+  · -- lo > 1: sqrt(lo) > 1
+    have hsqrt_gt_one : 1 < Real.sqrt lo := by
+      rw [← Real.sqrt_one]
+      exact Real.sqrt_lt_sqrt (by norm_num) hgt
+    -- 1/(2*sqrt(lo)) < 1/2
+    have hbound : 1 / (2 * Real.sqrt lo) < 1 / 2 := by
+      rw [div_lt_div_iff₀ (by positivity : (0 : ℝ) < 2 * Real.sqrt lo) (by norm_num : (0 : ℝ) < 2)]
+      ring_nf
+      linarith
+    calc 1 / (2 * Real.sqrt x) ≤ 1 / (2 * Real.sqrt lo) := hcoef_le
+      _ ≤ 1 / 2 := le_of_lt hbound
+      _ ≤ max (1 / (2 * lo)) (1 / 2) := le_max_right _ _
+
+/-- Partial dual for sqrt (chain rule: d(sqrt f) = f' / (2 * sqrt(f)))
+    Returns None if the value interval is not strictly positive. -/
+noncomputable def sqrt? (d : DualInterval) : Option DualInterval :=
+  if h : IntervalRat.isPositive d.val then
+    let sqrt_val := IntervalRat.sqrtInterval d.val
+    -- d(sqrt f) = f' / (2 * sqrt(f)) = f' * (1 / (2 * sqrt(f)))
+    let deriv_coef := sqrtDerivCoefBound d.val.lo h
+    let der' := IntervalRat.mul d.der deriv_coef
+    some { val := sqrt_val, der := der' }
+  else
+    none
+
+end DualInterval
+
+end LeanCert.Engine
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Automatic Differentiation - Evaluators
+
+This file provides the evaluation functions for automatic differentiation,
+mapping expressions to dual intervals.
+
+## Main definitions
+
+* `DualEnv` - Environment mapping variable indices to dual intervals
+* `LeanCert.Internal.AD.evalUnchecked` - Main evaluator for supported expressions (total)
+* `evalDualOption` - Partial evaluator supporting domain-checked functions (returns Option)
+* `evalDualOption1` - Single-variable version of evalDualOption
+* `mkDualEnv` - Create a dual environment for differentiation w.r.t. a variable
+* `evalWithDeriv` - Evaluate and differentiate w.r.t. a variable index
+* `derivInterval` - Get just the derivative interval
+* `evalWithDeriv1` - Single-variable evaluation and differentiation
+-/
+
+/-! ### Dual evaluation -/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+/-- Environment for dual evaluation -/
+abbrev DualEnv := Nat → DualInterval
+
+end LeanCert.Engine
+
+namespace LeanCert.Internal.AD
+
+open LeanCert.Core LeanCert.Engine
+
+/-- Evaluate expression in dual interval mode.
+
+    For supported expressions (const, var, add, mul, neg, sin, cos, exp), this
+    computes correct dual interval bounds with a fully-verified proof.
+
+    For unsupported expressions (inv, log), returns a default interval.
+    Do not rely on results for expressions containing inv or log.
+    Use evalDualOption for partial functions like inv and log. -/
+noncomputable def evalUnchecked (e : Expr) (ρ : DualEnv) : DualInterval :=
+  match e with
+  | Expr.const q => DualInterval.const q
+  | Expr.var idx => ρ idx
+  | Expr.add e₁ e₂ => DualInterval.add (LeanCert.Internal.AD.evalUnchecked e₁ ρ)
+    (LeanCert.Internal.AD.evalUnchecked e₂ ρ)
+  | Expr.mul e₁ e₂ => DualInterval.mul (LeanCert.Internal.AD.evalUnchecked e₁ ρ)
+    (LeanCert.Internal.AD.evalUnchecked e₂ ρ)
+  | Expr.neg e => DualInterval.neg (LeanCert.Internal.AD.evalUnchecked e ρ)
+  | Expr.inv _ => default  -- Not in ADSupported; safe default
+  | Expr.exp e => DualInterval.exp (LeanCert.Internal.AD.evalUnchecked e ρ)
+  | Expr.sin e => DualInterval.sin (LeanCert.Internal.AD.evalUnchecked e ρ)
+  | Expr.cos e => DualInterval.cos (LeanCert.Internal.AD.evalUnchecked e ρ)
+  | Expr.log _ => default  -- Not in ADSupported; use evalDualOption for log
+  | Expr.atan e => DualInterval.atan (LeanCert.Internal.AD.evalUnchecked e ρ)
+  | Expr.arsinh e => DualInterval.arsinh (LeanCert.Internal.AD.evalUnchecked e ρ)
+  | Expr.atanh _ => default  -- Partial function; use evalDualOption for atanh
+  | Expr.sinc e => DualInterval.sinc (LeanCert.Internal.AD.evalUnchecked e ρ)
+  | Expr.erf e => DualInterval.erf (LeanCert.Internal.AD.evalUnchecked e ρ)
+  | Expr.sinh e => DualInterval.sinh (LeanCert.Internal.AD.evalUnchecked e ρ)
+  | Expr.cosh e => DualInterval.cosh (LeanCert.Internal.AD.evalUnchecked e ρ)
+  | Expr.tanh e => DualInterval.tanh (LeanCert.Internal.AD.evalUnchecked e ρ)
+  | Expr.sqrt e => DualInterval.sqrt (LeanCert.Internal.AD.evalUnchecked e ρ)
+  | Expr.namedConst c => DualInterval.ofMathConst c
+
+end LeanCert.Internal.AD
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+/-! ### Partial dual evaluation -/
+
+/-- Partial dual evaluator that supports domain-checked functions.
+    Returns `none` if any domain error would occur:
+    - inv of an interval containing zero
+    - log of an interval not strictly positive
+    When it returns `some`, the result is guaranteed to be correct.
+
+    The total and computable dual evaluators support `tanh`, but this
+    Option-returning evaluator deliberately keeps `tanh` disabled until the
+    `evalDualOption`-specific value/differentiability/derivative correctness path
+    is wired for that constructor. -/
+noncomputable def evalDualOption (e : Expr) (ρ : DualEnv) : Option DualInterval :=
+  match e with
+  | Expr.const q => some (DualInterval.const q)
+  | Expr.var idx => some (ρ idx)
+  | Expr.add e₁ e₂ =>
+      match evalDualOption e₁ ρ, evalDualOption e₂ ρ with
+      | some d₁, some d₂ => some (DualInterval.add d₁ d₂)
+      | _, _ => none
+  | Expr.mul e₁ e₂ =>
+      match evalDualOption e₁ ρ, evalDualOption e₂ ρ with
+      | some d₁, some d₂ => some (DualInterval.mul d₁ d₂)
+      | _, _ => none
+  | Expr.neg e =>
+      match evalDualOption e ρ with
+      | some d => some (DualInterval.neg d)
+      | none => none
+  | Expr.inv e₁ =>
+      match evalDualOption e₁ ρ with
+      | none => none
+      | some d => DualInterval.inv? d
+  | Expr.exp e =>
+      match evalDualOption e ρ with
+      | some d => some (DualInterval.exp d)
+      | none => none
+  | Expr.sin e =>
+      match evalDualOption e ρ with
+      | some d => some (DualInterval.sin d)
+      | none => none
+  | Expr.cos e =>
+      match evalDualOption e ρ with
+      | some d => some (DualInterval.cos d)
+      | none => none
+  | Expr.log e =>
+      match evalDualOption e ρ with
+      | none => none
+      | some d => DualInterval.logOption d
+  | Expr.atan e =>
+      match evalDualOption e ρ with
+      | some d => some (DualInterval.atan d)
+      | none => none
+  | Expr.arsinh e =>
+      match evalDualOption e ρ with
+      | some d => some (DualInterval.arsinh d)
+      | none => none
+  | Expr.atanh _ =>
+      -- atanh is partial (defined only for |x| < 1) and requires complex bounds
+      -- We return none to avoid the complexity of proving atanh bounds
+      none
+  | Expr.sinc e =>
+      match evalDualOption e ρ with
+      | some d => some (DualInterval.sinc d)
+      | none => none
+  | Expr.erf e =>
+      match evalDualOption e ρ with
+      | some d => some (DualInterval.erf d)
+      | none => none
+  | Expr.sinh e =>
+      match evalDualOption e ρ with
+      | some d => some (DualInterval.sinh d)
+      | none => none
+  | Expr.cosh e =>
+      match evalDualOption e ρ with
+      | some d => some (DualInterval.cosh d)
+      | none => none
+  | Expr.tanh _ =>
+      -- See the docstring above:
+      -- `LeanCert.Internal.AD.evalUnchecked`/`LeanCert.Internal.AD.evalTotalCore` support tanh, but
+      -- the partial `evalDualOption` correctness theorem currently treats tanh as
+      -- outside this Option API.
+      none
+  | Expr.sqrt e =>
+      match evalDualOption e ρ with
+      | some d => DualInterval.sqrt? d
+      | none => none
+  | Expr.namedConst c => some (DualInterval.ofMathConst c)
+
+/-- Single-variable version of evalDualOption -/
+noncomputable def evalDualOption1 (e : Expr) (I : IntervalRat) : Option DualInterval :=
+  evalDualOption e (fun _ => DualInterval.varActive I)
+
+/-! ### Single variable differentiation -/
+
+/-- Create dual environment for differentiating with respect to variable `idx` -/
+def mkDualEnv (ρ : IntervalEnv) (idx : Nat) : DualEnv :=
+  fun i => if i = idx then DualInterval.varActive (ρ i) else DualInterval.varPassive (ρ i)
+
+/-- Evaluate and differentiate with respect to variable `idx` -/
+noncomputable def evalWithDeriv (e : Expr) (ρ : IntervalEnv) (idx : Nat) : DualInterval :=
+  LeanCert.Internal.AD.evalUnchecked e (mkDualEnv ρ idx)
+
+/-- Get just the derivative interval -/
+noncomputable def derivInterval (e : Expr) (ρ : IntervalEnv) (idx : Nat) : IntervalRat :=
+  (evalWithDeriv e ρ idx).der
+
+/-- Evaluate and differentiate a single-variable expression -/
+noncomputable def evalWithDeriv1 (e : Expr) (I : IntervalRat) : DualInterval :=
+  LeanCert.Internal.AD.evalUnchecked e (fun _ => DualInterval.varActive I)
+
+end LeanCert.Engine
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Automatic Differentiation - Correctness Theorems
+
+This file proves the correctness of forward-mode automatic differentiation
+for supported expressions (ADSupported).
+
+## Main theorems
+
+* `LeanCert.Engine.evalDualUnchecked_val_correct` - Value component is correct
+* `LeanCert.Engine.evalDualUnchecked_der_correct` - Derivative component is correct
+* `evalFunc1_differentiable` - Supported expressions are differentiable
+* `deriv_mem_dualDer` - Key theorem: computed derivative contains true derivative
+* `LeanCert.Engine.evalDualUnchecked_der_correct_idx` - n-variable derivative correctness
+* `evalWithDeriv1_correct` - Single-variable correctness
+
+## Design notes
+
+All theorems in this file are FULLY PROVED with no sorry or axioms.
+The correctness relies on the chain rule for each supported operation.
+-/
+
+/-! ### Correctness -/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+
+open LeanCert.Core Filter
+open scoped Topology
+
+/-- The value component is correct for supported expressions.
+
+    This theorem is FULLY PROVED (no sorry, no axioms) for supported expressions.
+    The `hsupp` hypothesis ensures we only consider expressions in the verified subset. -/
+theorem evalDualUnchecked_val_correct (e : Expr) (hsupp : ADSupported e)
+    (ρ_real : Nat → ℝ) (ρ_dual : DualEnv)
+    (hρ : ∀ i, ρ_real i ∈ (ρ_dual i).val) :
+    Expr.eval ρ_real e ∈ (LeanCert.Internal.AD.evalUnchecked e ρ_dual).val := by
+  induction hsupp with
+  | const q =>
+    simp only [Expr.eval_const, LeanCert.Internal.AD.evalUnchecked, DualInterval.const]
+    exact IntervalRat.mem_singleton q
+  | var idx =>
+    simp only [Expr.eval_var, LeanCert.Internal.AD.evalUnchecked]
+    exact hρ idx
+  | add _ _ ih₁ ih₂ =>
+    simp only [Expr.eval_add, LeanCert.Internal.AD.evalUnchecked, DualInterval.add]
+    exact IntervalRat.mem_add ih₁ ih₂
+  | mul _ _ ih₁ ih₂ =>
+    simp only [Expr.eval_mul, LeanCert.Internal.AD.evalUnchecked, DualInterval.mul]
+    exact IntervalRat.mem_mul ih₁ ih₂
+  | neg _ ih =>
+    simp only [Expr.eval_neg, LeanCert.Internal.AD.evalUnchecked, DualInterval.neg]
+    exact IntervalRat.mem_neg ih
+  | sin _ ih =>
+    simp only [Expr.eval_sin, LeanCert.Internal.AD.evalUnchecked, DualInterval.sin]
+    exact mem_sinInterval ih
+  | cos _ ih =>
+    simp only [Expr.eval_cos, LeanCert.Internal.AD.evalUnchecked, DualInterval.cos]
+    exact mem_cosInterval ih
+  | exp _ ih =>
+    simp only [Expr.eval_exp, LeanCert.Internal.AD.evalUnchecked, DualInterval.exp]
+    exact IntervalRat.mem_expInterval ih
+
+/-! ### Single-variable evaluation for derivative proofs -/
+
+/-- The function t ↦ Expr.eval (fun _ => t) e for single-variable expressions -/
+noncomputable abbrev evalFunc1 (e : Expr) : ℝ → ℝ :=
+  fun t => Expr.eval (fun _ => t) e
+
+/-! ### Differentiability of supported expressions -/
+
+/-- Supported expressions are differentiable as single-variable functions.
+    This theorem shows that for any supported expression,
+    the function `evalFunc1 e` is differentiable.
+
+    FULLY PROVED - no sorry, no axioms. -/
+theorem evalFunc1_differentiable (e : Expr) (hsupp : ADSupported e) :
+    Differentiable ℝ (evalFunc1 e) := by
+  induction hsupp with
+  | const q =>
+    exact differentiable_const _
+  | var _ =>
+    exact differentiable_id
+  | add _ _ ih₁ ih₂ =>
+    exact Differentiable.add ih₁ ih₂
+  | mul _ _ ih₁ ih₂ =>
+    exact Differentiable.mul ih₁ ih₂
+  | neg _ ih =>
+    exact Differentiable.neg ih
+  | sin _ ih =>
+    exact Real.differentiable_sin.comp ih
+  | cos _ ih =>
+    exact Real.differentiable_cos.comp ih
+  | exp _ ih =>
+    exact Real.differentiable_exp.comp ih
+
+/-! ### Core derivative correctness micro-lemmas -/
+
+/-- Real.cos y is always in cosInterval I (which is [-1, 1]).
+    This micro-lemma encapsulates the fact that cosInterval uses the global bound. -/
+theorem cos_mem_cosInterval_of_any (y : ℝ) (I : IntervalRat) :
+    Real.cos y ∈ cosInterval I := by
+  simp only [cosInterval, IntervalRat.mem_def, Rat.cast_neg, Rat.cast_one]
+  exact Real.cos_mem_Icc _
+
+/-- Real.sin y is always in sinInterval I (which is [-1, 1]).
+    This micro-lemma encapsulates the fact that sinInterval uses the global bound. -/
+theorem sin_mem_sinInterval_of_any (y : ℝ) (I : IntervalRat) :
+    Real.sin y ∈ sinInterval I := by
+  simp only [sinInterval, IntervalRat.mem_def, Rat.cast_neg, Rat.cast_one]
+  exact Real.sin_mem_Icc _
+
+/-- -Real.sin y is always in IntervalRat.neg (sinInterval I).
+    This combines the sin global bound with negation for the cos derivative rule. -/
+theorem neg_sin_mem_neg_sinInterval (y : ℝ) (I : IntervalRat) :
+    -Real.sin y ∈ IntervalRat.neg (sinInterval I) :=
+  IntervalRat.mem_neg (sin_mem_sinInterval_of_any y I)
+
+/-- The key lemma for n-variable AD: updateVar ρ_real idx x respects mkDualEnv ρ_int idx.
+    This encapsulates the repeated proof that appears in mul/exp cases. -/
+theorem updateVar_mem_mkDualEnv_val (ρ_real : Nat → ℝ) (ρ_int : IntervalEnv) (idx : Nat)
+    (x : ℝ) (hx : x ∈ ρ_int idx) (hρ : ∀ i, ρ_real i ∈ ρ_int i) :
+    ∀ i, Expr.updateVar ρ_real idx x i ∈ (mkDualEnv ρ_int idx i).val := by
+  intro i
+  by_cases hi : i = idx
+  · subst hi
+    simp only [Expr.updateVar_same, mkDualEnv, ↓reduceIte, DualInterval.varActive]
+    exact hx
+  · simp only [Expr.updateVar_other _ _ _ _ hi, mkDualEnv, ite_eq_right hi, DualInterval.varPassive]
+    exact hρ i
+
+/-! ### evalFunc1 unfolding lemmas -/
+
+/-- Helper lemma: evalFunc1 for addition unfolds correctly -/
+theorem evalFunc1_add (e₁ e₂ : Expr) :
+    evalFunc1 (Expr.add e₁ e₂) = fun t => evalFunc1 e₁ t + evalFunc1 e₂ t := rfl
+
+/-- Helper lemma: evalFunc1 for addition in Pi form -/
+theorem evalFunc1_add_pi (e₁ e₂ : Expr) :
+    evalFunc1 (Expr.add e₁ e₂) = evalFunc1 e₁ + evalFunc1 e₂ := rfl
+
+/-- Helper lemma: evalFunc1 for multiplication unfolds correctly -/
+theorem evalFunc1_mul (e₁ e₂ : Expr) :
+    evalFunc1 (Expr.mul e₁ e₂) = fun t => evalFunc1 e₁ t * evalFunc1 e₂ t := rfl
+
+/-- Helper lemma: evalFunc1 for multiplication in Pi form -/
+theorem evalFunc1_mul_pi (e₁ e₂ : Expr) :
+    evalFunc1 (Expr.mul e₁ e₂) = evalFunc1 e₁ * evalFunc1 e₂ := rfl
+
+/-- Helper lemma: evalFunc1 for negation unfolds correctly -/
+theorem evalFunc1_neg (e : Expr) :
+    evalFunc1 (Expr.neg e) = fun t => -(evalFunc1 e t) := rfl
+
+/-- Helper lemma: evalFunc1 for negation in Pi form -/
+theorem evalFunc1_neg_pi (e : Expr) :
+    evalFunc1 (Expr.neg e) = -evalFunc1 e := rfl
+
+/-- Helper lemma: evalFunc1 for sin unfolds correctly -/
+theorem evalFunc1_sin (e : Expr) :
+    evalFunc1 (Expr.sin e) = fun t => Real.sin (evalFunc1 e t) := rfl
+
+/-- Helper lemma: evalFunc1 for cos unfolds correctly -/
+theorem evalFunc1_cos (e : Expr) :
+    evalFunc1 (Expr.cos e) = fun t => Real.cos (evalFunc1 e t) := rfl
+
+/-- Helper lemma: evalFunc1 for exp unfolds correctly -/
+theorem evalFunc1_exp (e : Expr) :
+    evalFunc1 (Expr.exp e) = fun t => Real.exp (evalFunc1 e t) := rfl
+
+/-- Helper lemma: evalFunc1 for log unfolds correctly -/
+theorem evalFunc1_log (e : Expr) :
+    evalFunc1 (Expr.log e) = fun t => Real.log (evalFunc1 e t) := rfl
+
+/-- Helper lemma: evalFunc1 for atan unfolds correctly -/
+theorem evalFunc1_atan (e : Expr) :
+    evalFunc1 (Expr.atan e) = fun t => Real.arctan (evalFunc1 e t) := rfl
+
+/-- Helper lemma: evalFunc1 for arsinh unfolds correctly -/
+theorem evalFunc1_arsinh (e : Expr) :
+    evalFunc1 (Expr.arsinh e) = fun t => Real.arsinh (evalFunc1 e t) := rfl
+
+/-- Helper lemma: evalFunc1 for atanh unfolds correctly -/
+theorem evalFunc1_atanh (e : Expr) :
+    evalFunc1 (Expr.atanh e) = fun t => Real.atanh (evalFunc1 e t) := rfl
+
+/-- Helper lemma: evalFunc1 for sinc unfolds correctly -/
+theorem evalFunc1_sinc (e : Expr) :
+    evalFunc1 (Expr.sinc e) = fun t => Real.sinc (evalFunc1 e t) := rfl
+
+/-- Helper lemma: evalFunc1 for erf unfolds correctly -/
+theorem evalFunc1_erf (e : Expr) :
+    evalFunc1 (Expr.erf e) = fun t => Real.erf (evalFunc1 e t) := rfl
+
+/-- Helper lemma: evalFunc1 for sqrt unfolds correctly -/
+theorem evalFunc1_sqrt (e : Expr) :
+    evalFunc1 (Expr.sqrt e) = fun t => Real.sqrt (evalFunc1 e t) := rfl
+
+/-- Helper lemma: evalFunc1 for const -/
+@[simp]
+theorem evalFunc1_const (q : ℚ) : evalFunc1 (Expr.const q) = fun _ => (q : ℝ) := rfl
+
+/-- Helper lemma: evalFunc1 for var -/
+@[simp]
+theorem evalFunc1_var (i : ℕ) : evalFunc1 (Expr.var i) = id := rfl
+
+/-- Helper lemma: evalFunc1 for namedConst (constant function) -/
+@[simp]
+theorem evalFunc1_namedConst (c : MathConst) : evalFunc1 (Expr.namedConst c) = fun _ => c.toReal
+  := rfl
+
+/-- Helper lemma: evalFunc1 for pi (constant function) -/
+theorem evalFunc1_pi : evalFunc1 (Expr.namedConst .pi) = fun _ => Real.pi := rfl
+
+/-- Helper lemma: evalFunc1 for eulerMascheroni (constant function) -/
+theorem evalFunc1_eulerMascheroni : evalFunc1 (Expr.namedConst .eulerMascheroni) = fun _ =>
+  Real.eulerMascheroniConstant := rfl
+
+/-- Helper: the dual environment evaluation gives correct derivative.
+    This connects the interval-based AD to actual calculus derivatives.
+
+    FULLY PROVED - no sorry, no axioms. -/
+theorem deriv_mem_dualDer (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (x : ℝ) (hx : x ∈ I) :
+    deriv (evalFunc1 e) x ∈
+      (LeanCert.Internal.AD.evalUnchecked e (fun _ => DualInterval.varActive I)).der := by
+  induction hsupp generalizing x with
+  | const q =>
+    -- d/dx (const) = 0
+    simp only [LeanCert.Internal.AD.evalUnchecked, DualInterval.const, evalFunc1_const, deriv_const]
+    convert IntervalRat.mem_singleton 0 using 1
+    norm_cast
+  | var _ =>
+    -- d/dx (x) = 1
+    simp only [LeanCert.Internal.AD.evalUnchecked, DualInterval.varActive, evalFunc1_var]
+    rw [deriv_id]
+    convert IntervalRat.mem_singleton 1 using 1
+    norm_cast
+  | add h₁ h₂ ih₁ ih₂ =>
+    -- d/dx (f + g) = f' + g'
+    have hd₁ := evalFunc1_differentiable _ h₁
+    have hd₂ := evalFunc1_differentiable _ h₂
+    simp only [LeanCert.Internal.AD.evalUnchecked, DualInterval.add, evalFunc1_add_pi, deriv_add
+      (hd₁ x) (hd₂ x)]
+    exact IntervalRat.mem_add (ih₁ x hx) (ih₂ x hx)
+  | mul h₁ h₂ ih₁ ih₂ =>
+    -- d/dx (f * g) = f' * g + f * g'
+    have hd₁ := evalFunc1_differentiable _ h₁
+    have hd₂ := evalFunc1_differentiable _ h₂
+    simp only [LeanCert.Internal.AD.evalUnchecked, DualInterval.mul, evalFunc1_mul_pi, deriv_mul
+      (hd₁ x) (hd₂ x)]
+    -- Need: f'(x)*g(x) + f(x)*g'(x) ∈ der₁*val₂ + val₁*der₂
+    have hval₁ := LeanCert.Engine.evalDualUnchecked_val_correct _ h₁ (fun _ => x) (fun _ =>
+      DualInterval.varActive I)
+      (fun _ => hx)
+    have hval₂ := LeanCert.Engine.evalDualUnchecked_val_correct _ h₂ (fun _ => x) (fun _ =>
+      DualInterval.varActive I)
+      (fun _ => hx)
+    simp only [DualInterval.varActive] at hval₁ hval₂
+    exact IntervalRat.mem_add (IntervalRat.mem_mul (ih₁ x hx) hval₂) (IntervalRat.mem_mul hval₁
+      (ih₂ x hx))
+  | neg hs ih =>
+    -- d/dx (-f) = -f'
+    have hd := evalFunc1_differentiable _ hs
+    simp only [LeanCert.Internal.AD.evalUnchecked, DualInterval.neg, evalFunc1_neg_pi, deriv.neg]
+    exact IntervalRat.mem_neg (ih x hx)
+  | @sin e' hs ih =>
+    -- d/dx (sin f) = cos(f) * f'
+    have hd := evalFunc1_differentiable e' hs
+    simp only [LeanCert.Internal.AD.evalUnchecked, DualInterval.sin, evalFunc1_sin]
+    rw [deriv_sin (hd.differentiableAt)]
+    exact IntervalRat.mem_mul (cos_mem_cosInterval_of_any _ _) (ih x hx)
+  | @cos e' hs ih =>
+    -- d/dx (cos f) = -sin(f) * f'
+    have hd := evalFunc1_differentiable e' hs
+    simp only [LeanCert.Internal.AD.evalUnchecked, DualInterval.cos, evalFunc1_cos]
+    rw [deriv_cos (hd.differentiableAt)]
+    exact IntervalRat.mem_mul (neg_sin_mem_neg_sinInterval _ _) (ih x hx)
+  | @exp e' hs ih =>
+    -- d/dx (exp f) = exp(f) * f'
+    have hd := evalFunc1_differentiable e' hs
+    simp only [LeanCert.Internal.AD.evalUnchecked, DualInterval.exp, evalFunc1_exp]
+    rw [deriv_exp (hd.differentiableAt)]
+    -- exp(f(x)) ∈ expInterval and f'(x) ∈ der
+    have hval := LeanCert.Engine.evalDualUnchecked_val_correct e' hs (fun _ => x) (fun _ =>
+      DualInterval.varActive I)
+      (fun _ => hx)
+    simp only [DualInterval.varActive] at hval
+    have hexp := IntervalRat.mem_expInterval hval
+    exact IntervalRat.mem_mul hexp (ih x hx)
+
+/-- The derivative component of dual interval evaluation is correct.
+    For a supported expression evaluated at a point x in the interval I,
+    the derivative of the expression (as a function of x) lies in the
+    computed derivative interval.
+
+    This is the fundamental correctness theorem for forward-mode AD:
+    the derivative bounds computed by interval arithmetic contain the
+    true derivative at every point in the domain.
+
+    FULLY PROVED - no sorry, no axioms. -/
+theorem evalDualUnchecked_der_correct (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (x : ℝ) (hx : x ∈ I) :
+    deriv (fun t => Expr.eval (fun _ => t) e) x ∈
+      (LeanCert.Internal.AD.evalUnchecked e (fun _ => DualInterval.varActive I)).der :=
+  deriv_mem_dualDer e hsupp I x hx
+
+/-! ### Generalized n-variable derivative correctness -/
+
+/-- Helper: evalAlong is differentiable for supported expressions -/
+theorem evalAlong_differentiable (e : Expr) (hsupp : ADSupported e)
+    (ρ : Nat → ℝ) (idx : Nat) :
+    Differentiable ℝ (Expr.evalAlong e ρ idx) := by
+  induction hsupp with
+  | const q =>
+    simp only [Expr.evalAlong_const']
+    exact differentiable_const _
+  | var i =>
+    by_cases h : i = idx
+    · subst h
+      simp only [Expr.evalAlong_var_active]
+      exact differentiable_id
+    · simp only [Expr.evalAlong_var_passive _ _ _ h]
+      exact differentiable_const _
+  | add _ _ ih₁ ih₂ =>
+    simp only [Expr.evalAlong_add]
+    exact Differentiable.add ih₁ ih₂
+  | mul _ _ ih₁ ih₂ =>
+    simp only [Expr.evalAlong_mul]
+    exact Differentiable.mul ih₁ ih₂
+  | neg _ ih =>
+    simp only [Expr.evalAlong_neg]
+    exact Differentiable.neg ih
+  | sin _ ih =>
+    simp only [Expr.evalAlong_sin]
+    exact Real.differentiable_sin.comp ih
+  | cos _ ih =>
+    simp only [Expr.evalAlong_cos]
+    exact Real.differentiable_cos.comp ih
+  | exp _ ih =>
+    simp only [Expr.evalAlong_exp]
+    exact Real.differentiable_exp.comp ih
+
+/-- The derivative of evalAlong with respect to coordinate `idx` lies in the
+    computed derivative interval from mkDualEnv.
+
+    This is the fundamental n-variable derivative correctness theorem.
+    It shows that for any expression, if we differentiate along coordinate `idx`
+    while holding all other coordinates fixed according to `ρ`, the derivative
+    at any point `x` in the interval `ρ_int idx` lies in the computed interval.
+
+    FULLY PROVED - no sorry, no axioms. -/
+theorem evalDualUnchecked_der_correct_idx (e : Expr) (hsupp : ADSupported e)
+    (ρ_real : Nat → ℝ) (ρ_int : IntervalEnv) (idx : Nat)
+    (hρ : ∀ i, ρ_real i ∈ ρ_int i)
+    (x : ℝ) (hx : x ∈ ρ_int idx) :
+    deriv (Expr.evalAlong e ρ_real idx) x ∈
+      (LeanCert.Internal.AD.evalUnchecked e (mkDualEnv ρ_int idx)).der := by
+  induction hsupp generalizing x with
+  | const q =>
+    -- d/dt (const) = 0
+    simp only [Expr.evalAlong_const', deriv_const, LeanCert.Internal.AD.evalUnchecked,
+      DualInterval.const]
+    convert IntervalRat.mem_singleton 0 using 1
+    norm_cast
+  | var i =>
+    by_cases h : i = idx
+    · -- Active variable: d/dt (t) = 1
+      subst h
+      simp only [Expr.evalAlong_var_active, LeanCert.Internal.AD.evalUnchecked, mkDualEnv,
+        ↓reduceIte,
+        DualInterval.varActive, deriv_id]
+      convert IntervalRat.mem_singleton 1 using 1
+      norm_cast
+    · -- Passive variable: d/dt (const) = 0
+      simp only [Expr.evalAlong_var_passive _ _ _ h, deriv_const,
+        LeanCert.Internal.AD.evalUnchecked, mkDualEnv,
+        ite_eq_right h, DualInterval.varPassive]
+      convert IntervalRat.mem_singleton 0 using 1
+      norm_cast
+  | add h₁ h₂ ih₁ ih₂ =>
+    have hd₁ := evalAlong_differentiable _ h₁ ρ_real idx
+    have hd₂ := evalAlong_differentiable _ h₂ ρ_real idx
+    simp only [Expr.evalAlong_add_pi, deriv_add (hd₁ x) (hd₂ x),
+      LeanCert.Internal.AD.evalUnchecked, DualInterval.add]
+    exact IntervalRat.mem_add (ih₁ x hx) (ih₂ x hx)
+  | mul h₁ h₂ ih₁ ih₂ =>
+    have hd₁ := evalAlong_differentiable _ h₁ ρ_real idx
+    have hd₂ := evalAlong_differentiable _ h₂ ρ_real idx
+    simp only [Expr.evalAlong_mul_pi, deriv_mul (hd₁ x) (hd₂ x),
+      LeanCert.Internal.AD.evalUnchecked, DualInterval.mul]
+    have hmem := updateVar_mem_mkDualEnv_val ρ_real ρ_int idx x hx hρ
+    have hval₁ := LeanCert.Engine.evalDualUnchecked_val_correct _ h₁ _ _ hmem
+    have hval₂ := LeanCert.Engine.evalDualUnchecked_val_correct _ h₂ _ _ hmem
+    exact IntervalRat.mem_add (IntervalRat.mem_mul (ih₁ x hx) hval₂) (IntervalRat.mem_mul hval₁
+      (ih₂ x hx))
+  | neg hs ih =>
+    have hd := evalAlong_differentiable _ hs ρ_real idx
+    simp only [Expr.evalAlong_neg_pi, deriv.neg, LeanCert.Internal.AD.evalUnchecked,
+      DualInterval.neg]
+    exact IntervalRat.mem_neg (ih x hx)
+  | @sin e' hs ih =>
+    have hd := evalAlong_differentiable e' hs ρ_real idx
+    simp only [Expr.evalAlong_sin, deriv_sin (hd.differentiableAt),
+      LeanCert.Internal.AD.evalUnchecked, DualInterval.sin]
+    exact IntervalRat.mem_mul (cos_mem_cosInterval_of_any _ _) (ih x hx)
+  | @cos e' hs ih =>
+    have hd := evalAlong_differentiable e' hs ρ_real idx
+    simp only [Expr.evalAlong_cos, deriv_cos (hd.differentiableAt),
+      LeanCert.Internal.AD.evalUnchecked, DualInterval.cos]
+    exact IntervalRat.mem_mul (neg_sin_mem_neg_sinInterval _ _) (ih x hx)
+  | @exp e' hs ih =>
+    have hd := evalAlong_differentiable e' hs ρ_real idx
+    simp only [Expr.evalAlong_exp, deriv_exp (hd.differentiableAt),
+      LeanCert.Internal.AD.evalUnchecked, DualInterval.exp]
+    have hmem := updateVar_mem_mkDualEnv_val ρ_real ρ_int idx x hx hρ
+    have hval := LeanCert.Engine.evalDualUnchecked_val_correct e' hs _ _ hmem
+    exact IntervalRat.mem_mul (IntervalRat.mem_expInterval hval) (ih x hx)
+
+/-- Convenience theorem: derivInterval correctness for n-variable expressions.
+    The derivative of `evalAlong e ρ idx` at any point `x` in the interval
+    lies in `derivInterval e ρ_int idx`. -/
+theorem derivInterval_correct_idx (e : Expr) (hsupp : ADSupported e)
+    (ρ_real : Nat → ℝ) (ρ_int : IntervalEnv) (idx : Nat)
+    (hρ : ∀ i, ρ_real i ∈ ρ_int i)
+    (x : ℝ) (hx : x ∈ ρ_int idx) :
+    deriv (Expr.evalAlong e ρ_real idx) x ∈ derivInterval e ρ_int idx := by
+  simp only [derivInterval, evalWithDeriv]
+  exact LeanCert.Engine.evalDualUnchecked_der_correct_idx e hsupp ρ_real ρ_int idx hρ x hx
+
+/-! ### Single-variable expressions -/
+
+/-- Predicate: expression only uses variable index 0.
+    This is useful for proving that different environments give the same result
+    when they agree at index 0. -/
+inductive UsesOnlyVar0 : Expr → Prop where
+  | const (q : ℚ) : UsesOnlyVar0 (Expr.const q)
+  | var0 : UsesOnlyVar0 (Expr.var 0)
+  | add (e₁ e₂ : Expr) (h₁ : UsesOnlyVar0 e₁) (h₂ : UsesOnlyVar0 e₂) : UsesOnlyVar0 (Expr.add e₁ e₂)
+  | mul (e₁ e₂ : Expr) (h₁ : UsesOnlyVar0 e₁) (h₂ : UsesOnlyVar0 e₂) : UsesOnlyVar0 (Expr.mul e₁ e₂)
+  | neg (e : Expr) (h : UsesOnlyVar0 e) : UsesOnlyVar0 (Expr.neg e)
+  | inv (e : Expr) (h : UsesOnlyVar0 e) : UsesOnlyVar0 (Expr.inv e)
+  | sin (e : Expr) (h : UsesOnlyVar0 e) : UsesOnlyVar0 (Expr.sin e)
+  | cos (e : Expr) (h : UsesOnlyVar0 e) : UsesOnlyVar0 (Expr.cos e)
+  | exp (e : Expr) (h : UsesOnlyVar0 e) : UsesOnlyVar0 (Expr.exp e)
+  | log (e : Expr) (h : UsesOnlyVar0 e) : UsesOnlyVar0 (Expr.log e)
+  | atan (e : Expr) (h : UsesOnlyVar0 e) : UsesOnlyVar0 (Expr.atan e)
+  | arsinh (e : Expr) (h : UsesOnlyVar0 e) : UsesOnlyVar0 (Expr.arsinh e)
+  | atanh (e : Expr) (h : UsesOnlyVar0 e) : UsesOnlyVar0 (Expr.atanh e)
+  | sinc (e : Expr) (h : UsesOnlyVar0 e) : UsesOnlyVar0 (Expr.sinc e)
+  | erf (e : Expr) (h : UsesOnlyVar0 e) : UsesOnlyVar0 (Expr.erf e)
+  | sinh (e : Expr) (h : UsesOnlyVar0 e) : UsesOnlyVar0 (Expr.sinh e)
+  | cosh (e : Expr) (h : UsesOnlyVar0 e) : UsesOnlyVar0 (Expr.cosh e)
+  | tanh (e : Expr) (h : UsesOnlyVar0 e) : UsesOnlyVar0 (Expr.tanh e)
+  | sqrt (e : Expr) (h : UsesOnlyVar0 e) : UsesOnlyVar0 (Expr.sqrt e)
+  | namedConst (c : MathConst) : UsesOnlyVar0 (Expr.namedConst c)
+
+/-- Bridge between the canonical boolean predicate and the proof-carrying
+`UsesOnlyVar0` predicate used by AD correctness lemmas. -/
+theorem Expr.usesOnlyVar0_iff_UsesOnlyVar0 {e : Expr} :
+    e.usesOnlyVar0 = true ↔ UsesOnlyVar0 e := by
+  constructor
+  · intro he
+    induction e with
+    | const q => exact UsesOnlyVar0.const q
+    | var i =>
+      simp only [Expr.usesOnlyVar0] at he
+      have hi : i = 0 := beq_iff_eq.mp he
+      subst hi
+      exact UsesOnlyVar0.var0
+    | add e₁ e₂ ih₁ ih₂ =>
+      simp only [Expr.usesOnlyVar0, Bool.and_eq_true] at he
+      exact UsesOnlyVar0.add e₁ e₂ (ih₁ he.1) (ih₂ he.2)
+    | mul e₁ e₂ ih₁ ih₂ =>
+      simp only [Expr.usesOnlyVar0, Bool.and_eq_true] at he
+      exact UsesOnlyVar0.mul e₁ e₂ (ih₁ he.1) (ih₂ he.2)
+    | neg e ih =>
+      exact UsesOnlyVar0.neg e (ih he)
+    | inv e ih =>
+      exact UsesOnlyVar0.inv e (ih he)
+    | exp e ih =>
+      exact UsesOnlyVar0.exp e (ih he)
+    | sin e ih =>
+      exact UsesOnlyVar0.sin e (ih he)
+    | cos e ih =>
+      exact UsesOnlyVar0.cos e (ih he)
+    | log e ih =>
+      exact UsesOnlyVar0.log e (ih he)
+    | atan e ih =>
+      exact UsesOnlyVar0.atan e (ih he)
+    | arsinh e ih =>
+      exact UsesOnlyVar0.arsinh e (ih he)
+    | atanh e ih =>
+      exact UsesOnlyVar0.atanh e (ih he)
+    | sinc e ih =>
+      exact UsesOnlyVar0.sinc e (ih he)
+    | erf e ih =>
+      exact UsesOnlyVar0.erf e (ih he)
+    | sinh e ih =>
+      exact UsesOnlyVar0.sinh e (ih he)
+    | cosh e ih =>
+      exact UsesOnlyVar0.cosh e (ih he)
+    | tanh e ih =>
+      exact UsesOnlyVar0.tanh e (ih he)
+    | sqrt e ih =>
+      exact UsesOnlyVar0.sqrt e (ih he)
+    | namedConst c =>
+      exact UsesOnlyVar0.namedConst c
+  · intro h
+    induction h with
+    | const q => rfl
+    | var0 => rfl
+    | add _ _ _ _ ih₁ ih₂ =>
+      rw [Expr.usesOnlyVar0, ih₁, ih₂]
+      rfl
+    | mul _ _ _ _ ih₁ ih₂ =>
+      rw [Expr.usesOnlyVar0, ih₁, ih₂]
+      rfl
+    | neg _ _ ih | inv _ _ ih | sin _ _ ih | cos _ _ ih | exp _ _ ih | log _ _ ih
+    | atan _ _ ih | arsinh _ _ ih | atanh _ _ ih | sinc _ _ ih | erf _ _ ih
+    | sinh _ _ ih | cosh _ _ ih | tanh _ _ ih | sqrt _ _ ih =>
+      simpa only [Expr.usesOnlyVar0] using ih
+    | namedConst c => rfl
+
+/-- For expressions using only var 0, LeanCert.Internal.AD.evalUnchecked agrees for environments
+that match at 0 -/
+theorem evalDualUnchecked_congr_at_0 (e : Expr) (h : UsesOnlyVar0 e)
+    (ρ₁ ρ₂ : DualEnv) (heq : ρ₁ 0 = ρ₂ 0) :
+    LeanCert.Internal.AD.evalUnchecked e ρ₁ = LeanCert.Internal.AD.evalUnchecked e ρ₂ := by
+  induction h generalizing ρ₁ ρ₂ with
+  | const q => simp only [LeanCert.Internal.AD.evalUnchecked]
+  | var0 => simp only [LeanCert.Internal.AD.evalUnchecked, heq]
+  | add _ _ _ _ ih₁ ih₂ =>
+    simp only [LeanCert.Internal.AD.evalUnchecked, ih₁ ρ₁ ρ₂ heq, ih₂ ρ₁ ρ₂ heq]
+  | mul _ _ _ _ ih₁ ih₂ =>
+    simp only [LeanCert.Internal.AD.evalUnchecked, ih₁ ρ₁ ρ₂ heq, ih₂ ρ₁ ρ₂ heq]
+  | neg _ _ ih =>
+    simp only [LeanCert.Internal.AD.evalUnchecked, ih ρ₁ ρ₂ heq]
+  | inv _ _ ih =>
+    simp only [LeanCert.Internal.AD.evalUnchecked]
+  | sin _ _ ih =>
+    simp only [LeanCert.Internal.AD.evalUnchecked, ih ρ₁ ρ₂ heq]
+  | cos _ _ ih =>
+    simp only [LeanCert.Internal.AD.evalUnchecked, ih ρ₁ ρ₂ heq]
+  | exp _ _ ih =>
+    simp only [LeanCert.Internal.AD.evalUnchecked, ih ρ₁ ρ₂ heq]
+  | log _ _ ih =>
+    simp only [LeanCert.Internal.AD.evalUnchecked]
+  | atan _ _ ih =>
+    simp only [LeanCert.Internal.AD.evalUnchecked, ih ρ₁ ρ₂ heq]
+  | arsinh _ _ ih =>
+    simp only [LeanCert.Internal.AD.evalUnchecked, ih ρ₁ ρ₂ heq]
+  | atanh _ _ ih =>
+    simp only [LeanCert.Internal.AD.evalUnchecked]
+  | sinc _ _ ih =>
+    simp only [LeanCert.Internal.AD.evalUnchecked, ih ρ₁ ρ₂ heq]
+  | erf _ _ ih =>
+    simp only [LeanCert.Internal.AD.evalUnchecked, ih ρ₁ ρ₂ heq]
+  | sinh _ _ ih =>
+    simp only [LeanCert.Internal.AD.evalUnchecked, ih ρ₁ ρ₂ heq]
+  | cosh _ _ ih =>
+    simp only [LeanCert.Internal.AD.evalUnchecked, ih ρ₁ ρ₂ heq]
+  | tanh _ _ ih =>
+    simp only [LeanCert.Internal.AD.evalUnchecked, ih ρ₁ ρ₂ heq]
+  | sqrt _ _ ih =>
+    simp only [LeanCert.Internal.AD.evalUnchecked, ih ρ₁ ρ₂ heq]
+  | namedConst _ =>
+    simp only [LeanCert.Internal.AD.evalUnchecked]
+
+/-- mkDualEnv at index 0 equals varActive at 0 -/
+theorem mkDualEnv_at_0 (I : IntervalRat) :
+    mkDualEnv (fun _ => I) 0 0 = DualInterval.varActive I := by
+  simp only [mkDualEnv, ↓reduceIte]
+
+/-- For expressions using only var 0, derivInterval equals evalWithDeriv1 -/
+theorem derivInterval_eq_evalWithDeriv1_of_UsesOnlyVar0 (e : Expr) (h : UsesOnlyVar0 e)
+    (I : IntervalRat) :
+    derivInterval e (fun _ => I) 0 = (evalWithDeriv1 e I).der := by
+  simp only [derivInterval, evalWithDeriv, evalWithDeriv1]
+  -- Need to show: (LeanCert.Internal.AD.evalUnchecked e (mkDualEnv (fun _ => I) 0)).der =
+  -- (LeanCert.Internal.AD.evalUnchecked e (fun _ => varActive I)).der
+  have henv_eq : mkDualEnv (fun _ => I) 0 0 = (fun _ => DualInterval.varActive I) 0 := by
+    simp only [mkDualEnv_at_0]
+  have heq := LeanCert.Engine.evalDualUnchecked_congr_at_0 e h (mkDualEnv (fun _ => I) 0) (fun _
+    => DualInterval.varActive I) henv_eq
+  rw [heq]
+
+/-- Correctness of single-variable derivative bounds -/
+theorem evalWithDeriv1_correct (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (x : ℝ) (hx : x ∈ I) :
+    Expr.eval (fun _ => x) e ∈ (evalWithDeriv1 e I).val ∧
+    deriv (fun t => Expr.eval (fun _ => t) e) x ∈ (evalWithDeriv1 e I).der := by
+  constructor
+  · exact LeanCert.Engine.evalDualUnchecked_val_correct e hsupp (fun _ => x) _ (fun _ => hx)
+  · exact LeanCert.Engine.evalDualUnchecked_der_correct e hsupp I x hx
+
+end LeanCert.Engine
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Automatic Differentiation - Computable Evaluators
+
+This file provides computable dual evaluators using Taylor-based approximations
+for transcendental functions. This enables `native_decide` for derivative-based
+bound checking.
+
+## Main definitions
+
+* `DualInterval.expCore`, `sinCore`, `cosCore` - Taylor-based dual functions
+* `DualInterval.sinhCore`, `coshCore`, `tanhCore` - Hyperbolic Taylor-based duals
+* `LeanCert.Internal.AD.evalTotalCore` - Computable dual evaluator for ExprSupportedCore
+* `derivIntervalCore` - Computable single-variable derivative interval
+
+## Main theorems
+
+* `LeanCert.Engine.evalDualTotalCore_val_correct` - Value component is correct
+* `LeanCert.Engine.evalDualTotalCore_der_correct` - Derivative component is correct
+* `derivIntervalCore_correct` - Derivative interval correctness
+* `strictMonoOn_of_derivIntervalCore_pos` - Monotonicity from positive derivative
+* `strictAntiOn_of_derivIntervalCore_neg` - Antitonicity from negative derivative
+-/
+
+/-! ### Computable Dual Evaluation for ExprSupportedCore
+
+This section provides a fully computable dual evaluator that uses Taylor-based
+approximations for transcendental functions. This enables `native_decide` for
+derivative-based bound checking.
+-/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+
+open LeanCert.Core Filter
+open scoped Topology
+
+namespace DualInterval
+
+/-- Computable dual for exp using Taylor series (chain rule: d(exp f) = exp(f) * f') -/
+def expCore (d : DualInterval) (n : ℕ := 10) : DualInterval :=
+  let expVal := IntervalRat.expComputable d.val n
+  { val := expVal
+    der := IntervalRat.mul expVal d.der }
+
+/-- Computable dual for sin using Taylor series -/
+def sinCore (d : DualInterval) (n : ℕ := 10) : DualInterval :=
+  { val := IntervalRat.sinComputable d.val n
+    der := IntervalRat.mul (IntervalRat.cosComputable d.val n) d.der }
+
+/-- Computable dual for cos using Taylor series -/
+def cosCore (d : DualInterval) (n : ℕ := 10) : DualInterval :=
+  { val := IntervalRat.cosComputable d.val n
+    der := IntervalRat.mul (IntervalRat.neg (IntervalRat.sinComputable d.val n)) d.der }
+
+/-- Computable dual for log using Taylor series via atanh reduction.
+    Chain rule: d(log f) = f' / f -/
+def logCore (d : DualInterval) (n : ℕ := 20) : DualInterval :=
+  { val := IntervalRat.logComputable d.val n
+    -- der = d.der / d.val = d.der * (1/d.val)
+    der := IntervalRat.mul (invInterval d.val) d.der }
+
+/-- Computable dual for sinh using Taylor series (chain rule: d(sinh f) = cosh(f) * f') -/
+def sinhCore (d : DualInterval) (n : ℕ := 10) : DualInterval :=
+  { val := IntervalRat.sinhComputable d.val n
+    der := IntervalRat.mul (IntervalRat.coshComputable d.val n) d.der }
+
+/-- Computable dual for cosh using Taylor series (chain rule: d(cosh f) = sinh(f) * f') -/
+def coshCore (d : DualInterval) (n : ℕ := 10) : DualInterval :=
+  { val := IntervalRat.coshComputable d.val n
+    der := IntervalRat.mul (IntervalRat.sinhComputable d.val n) d.der }
+
+/-- Computable dual for tanh (chain rule: d(tanh f) = sech²(f) * f')
+    Since sech²(x) ∈ (0, 1], we use [0, 1] as a conservative bound. -/
+def tanhCore (d : DualInterval) (_n : ℕ := 10) : DualInterval :=
+  { val := tanhInterval d.val
+    -- tanh'(x) = sech²(x) ∈ (0, 1], use [0, 1] as bound
+    der := IntervalRat.mul d.der ⟨0, 1, by norm_num⟩ }
+
+end DualInterval
+
+end LeanCert.Engine
+
+namespace LeanCert.Internal.AD
+
+open LeanCert.Core LeanCert.Engine
+
+/-- Computable dual interval evaluator for ExprSupportedCore expressions.
+
+    This uses Taylor series approximations for transcendental functions,
+    making it fully computable and usable with `native_decide`.
+
+    For `inv` and `log`, use `evalDualChecked` or the bounded-denominator
+    `evalDualDyadicChecked`; they validate the actual input box before exposing
+    this total kernel's result. Correctness is not claimed for unchecked
+    partial-operation branches here. -/
+def evalTotalCore (e : Expr) (ρ : DualEnv) (cfg : EvalConfig := {}) : DualInterval :=
+  match e with
+  | Expr.const q => DualInterval.const q
+  | Expr.var idx => ρ idx
+  | Expr.add e₁ e₂ => DualInterval.add (LeanCert.Internal.AD.evalTotalCore e₁ ρ cfg)
+    (LeanCert.Internal.AD.evalTotalCore e₂ ρ cfg)
+  | Expr.mul e₁ e₂ => DualInterval.mul (LeanCert.Internal.AD.evalTotalCore e₁ ρ cfg)
+    (LeanCert.Internal.AD.evalTotalCore e₂ ρ cfg)
+  | Expr.neg e => DualInterval.neg (LeanCert.Internal.AD.evalTotalCore e ρ cfg)
+  | Expr.inv e => DualInterval.inv (LeanCert.Internal.AD.evalTotalCore e ρ cfg)
+  | Expr.exp e => DualInterval.expCore (LeanCert.Internal.AD.evalTotalCore e ρ cfg) cfg.taylorDepth
+  | Expr.sin e => DualInterval.sinCore (LeanCert.Internal.AD.evalTotalCore e ρ cfg) cfg.taylorDepth
+  | Expr.cos e => DualInterval.cosCore (LeanCert.Internal.AD.evalTotalCore e ρ cfg) cfg.taylorDepth
+  | Expr.log e => DualInterval.logCore (LeanCert.Internal.AD.evalTotalCore e ρ cfg) cfg.taylorDepth
+  | Expr.atan e => DualInterval.atan (LeanCert.Internal.AD.evalTotalCore e ρ cfg)
+  | Expr.arsinh e => DualInterval.arsinh (LeanCert.Internal.AD.evalTotalCore e ρ cfg)
+  | Expr.atanh _ => default
+  | Expr.sinc e => DualInterval.sinc (LeanCert.Internal.AD.evalTotalCore e ρ cfg)
+  | Expr.erf e => DualInterval.erfCore (LeanCert.Internal.AD.evalTotalCore e ρ cfg) cfg.taylorDepth
+  | Expr.sinh e => DualInterval.sinhCore (LeanCert.Internal.AD.evalTotalCore e ρ cfg)
+    cfg.taylorDepth
+  | Expr.cosh e => DualInterval.coshCore (LeanCert.Internal.AD.evalTotalCore e ρ cfg)
+    cfg.taylorDepth
+  | Expr.tanh e => DualInterval.tanhCore (LeanCert.Internal.AD.evalTotalCore e ρ cfg)
+    cfg.taylorDepth
+  | Expr.sqrt e => DualInterval.sqrt (LeanCert.Internal.AD.evalTotalCore e ρ cfg)
+  | Expr.namedConst c => DualInterval.ofMathConst c
+
+end LeanCert.Internal.AD
+
+namespace LeanCert.Engine
+
+open LeanCert.Core Filter
+open scoped Topology
+
+/-- Computable single-variable derivative interval -/
+def derivIntervalCore (e : Expr) (I : IntervalRat) (cfg : EvalConfig := {}) : IntervalRat :=
+  (LeanCert.Internal.AD.evalTotalCore e (fun _ => DualInterval.varActive I) cfg).der
+
+/-- Domain validity for dual evaluation.
+    This is defined directly in terms of LeanCert.Internal.AD.evalTotalCore to ensure compatibility.
+    For log, we require the argument interval to have positive lower bound. -/
+def evalDomainValidDual (e : Expr) (ρ : DualEnv) (cfg : EvalConfig := {}) : Prop :=
+  match e with
+  | Expr.const _ => True
+  | Expr.var _ => True
+  | Expr.add e₁ e₂ => evalDomainValidDual e₁ ρ cfg ∧ evalDomainValidDual e₂ ρ cfg
+  | Expr.mul e₁ e₂ => evalDomainValidDual e₁ ρ cfg ∧ evalDomainValidDual e₂ ρ cfg
+  | Expr.neg e => evalDomainValidDual e ρ cfg
+  | Expr.inv e => evalDomainValidDual e ρ cfg
+  | Expr.exp e => evalDomainValidDual e ρ cfg
+  | Expr.sin e => evalDomainValidDual e ρ cfg
+  | Expr.cos e => evalDomainValidDual e ρ cfg
+  | Expr.log e => evalDomainValidDual e ρ cfg ∧ (LeanCert.Internal.AD.evalTotalCore e ρ
+    cfg).val.lo > 0
+  | Expr.atan e => evalDomainValidDual e ρ cfg
+  | Expr.arsinh e => evalDomainValidDual e ρ cfg
+  | Expr.atanh e => evalDomainValidDual e ρ cfg
+  | Expr.sinc e => evalDomainValidDual e ρ cfg
+  | Expr.erf e => evalDomainValidDual e ρ cfg
+  | Expr.sinh e => evalDomainValidDual e ρ cfg
+  | Expr.cosh e => evalDomainValidDual e ρ cfg
+  | Expr.tanh e => evalDomainValidDual e ρ cfg
+  | Expr.sqrt e => evalDomainValidDual e ρ cfg
+  | Expr.namedConst _ => True
+
+/-- Correctness theorem for computable dual value component.
+
+    Note: Requires domain validity for log (positive argument interval). -/
+theorem evalDualTotalCore_val_correct (e : Expr) (hsupp : ExprSupportedCore e)
+    (ρ_real : Nat → ℝ) (ρ_dual : DualEnv) (cfg : EvalConfig)
+    (hρ : ∀ i, ρ_real i ∈ (ρ_dual i).val)
+    (hdom : evalDomainValidDual e ρ_dual cfg) :
+    Expr.eval ρ_real e ∈ (LeanCert.Internal.AD.evalTotalCore e ρ_dual cfg).val := by
+  induction hsupp with
+  | const q =>
+    simp only [Expr.eval_const, LeanCert.Internal.AD.evalTotalCore, DualInterval.const]
+    exact IntervalRat.mem_singleton q
+  | var idx =>
+    simp only [Expr.eval_var, LeanCert.Internal.AD.evalTotalCore]
+    exact hρ idx
+  | add _ _ ih₁ ih₂ =>
+    simp only [evalDomainValidDual] at hdom
+    simp only [Expr.eval_add, LeanCert.Internal.AD.evalTotalCore, DualInterval.add]
+    exact IntervalRat.mem_add (ih₁ hdom.1) (ih₂ hdom.2)
+  | mul _ _ ih₁ ih₂ =>
+    simp only [evalDomainValidDual] at hdom
+    simp only [Expr.eval_mul, LeanCert.Internal.AD.evalTotalCore, DualInterval.mul]
+    exact IntervalRat.mem_mul (ih₁ hdom.1) (ih₂ hdom.2)
+  | neg _ ih =>
+    simp only [evalDomainValidDual] at hdom
+    simp only [Expr.eval_neg, LeanCert.Internal.AD.evalTotalCore, DualInterval.neg]
+    exact IntervalRat.mem_neg (ih hdom)
+  | sin _ ih =>
+    simp only [evalDomainValidDual] at hdom
+    simp only [Expr.eval_sin, LeanCert.Internal.AD.evalTotalCore, DualInterval.sinCore]
+    exact IntervalRat.mem_sinComputable (ih hdom) cfg.taylorDepth
+  | cos _ ih =>
+    simp only [evalDomainValidDual] at hdom
+    simp only [Expr.eval_cos, LeanCert.Internal.AD.evalTotalCore, DualInterval.cosCore]
+    exact IntervalRat.mem_cosComputable (ih hdom) cfg.taylorDepth
+  | exp _ ih =>
+    simp only [evalDomainValidDual] at hdom
+    simp only [Expr.eval_exp, LeanCert.Internal.AD.evalTotalCore, DualInterval.expCore]
+    exact IntervalRat.mem_expComputable (ih hdom) cfg.taylorDepth
+  | sqrt _ ih =>
+    simp only [evalDomainValidDual] at hdom
+    simp only [Expr.eval_sqrt, LeanCert.Internal.AD.evalTotalCore, DualInterval.sqrt]
+    exact IntervalRat.mem_sqrtInterval' (ih hdom)
+  | sinh _ ih =>
+    simp only [evalDomainValidDual] at hdom
+    simp only [Expr.eval_sinh, LeanCert.Internal.AD.evalTotalCore, DualInterval.sinhCore]
+    exact IntervalRat.mem_sinhComputable (ih hdom) cfg.taylorDepth
+  | cosh _ ih =>
+    simp only [evalDomainValidDual] at hdom
+    simp only [Expr.eval_cosh, LeanCert.Internal.AD.evalTotalCore, DualInterval.coshCore]
+    exact IntervalRat.mem_coshComputable (ih hdom) cfg.taylorDepth
+  | tanh _ ih =>
+    simp only [evalDomainValidDual] at hdom
+    simp only [Expr.eval_tanh, LeanCert.Internal.AD.evalTotalCore, DualInterval.tanhCore]
+    exact mem_tanhInterval (ih hdom)
+  | erf _ ih =>
+    simp only [evalDomainValidDual] at hdom
+    simp only [Expr.eval_erf, LeanCert.Internal.AD.evalTotalCore, DualInterval.erfCore]
+    simp only [IntervalRat.mem_def, Rat.cast_neg, Rat.cast_one]
+    exact Real.erf_mem_Icc _
+  | log _ ih =>
+    simp only [evalDomainValidDual] at hdom
+    simp only [Expr.eval_log, LeanCert.Internal.AD.evalTotalCore, DualInterval.logCore]
+    exact IntervalRat.mem_logComputable (ih hdom.1) hdom.2 cfg.taylorDepth
+  | namedConst c =>
+    simp only [Expr.eval_namedConst, LeanCert.Internal.AD.evalTotalCore, DualInterval.ofMathConst]
+    exact c.mem_interval
+
+/-- For ADSupported expressions (which exclude log), domain validity is trivially true.
+    This is because ADSupported has no log constructor. -/
+theorem evalDomainValidDual_of_ExprSupported (e : Expr) (hsupp : ADSupported e)
+    (ρ : DualEnv) (cfg : EvalConfig) : evalDomainValidDual e ρ cfg := by
+  induction hsupp with
+  | const _ => trivial
+  | var _ => trivial
+  | add _ _ ih₁ ih₂ => exact ⟨ih₁, ih₂⟩
+  | mul _ _ ih₁ ih₂ => exact ⟨ih₁, ih₂⟩
+  | neg _ ih => exact ih
+  | sin _ ih => exact ih
+  | cos _ ih => exact ih
+  | exp _ ih => exact ih
+
+/-- Correctness theorem for computable dual derivative component.
+    Uses ADSupported since derivative correctness requires differentiability. -/
+theorem evalDualTotalCore_der_correct (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (x : ℝ) (hx : x ∈ I) (cfg : EvalConfig) :
+    deriv (evalFunc1 e) x ∈ (LeanCert.Internal.AD.evalTotalCore e (fun _ => DualInterval.varActive
+      I) cfg).der := by
+  induction hsupp generalizing x with
+  | const q =>
+    simp only [LeanCert.Internal.AD.evalTotalCore, DualInterval.const, evalFunc1_const, deriv_const]
+    convert IntervalRat.mem_singleton 0 using 1
+    norm_cast
+  | var _ =>
+    simp only [LeanCert.Internal.AD.evalTotalCore, DualInterval.varActive, evalFunc1_var]
+    rw [deriv_id]
+    convert IntervalRat.mem_singleton 1 using 1
+    norm_cast
+  | add h₁ h₂ ih₁ ih₂ =>
+    have hd₁ := evalFunc1_differentiable _ h₁
+    have hd₂ := evalFunc1_differentiable _ h₂
+    simp only [LeanCert.Internal.AD.evalTotalCore, DualInterval.add, evalFunc1_add_pi, deriv_add
+      (hd₁ x) (hd₂ x)]
+    exact IntervalRat.mem_add (ih₁ x hx) (ih₂ x hx)
+  | mul h₁ h₂ ih₁ ih₂ =>
+    have hd₁ := evalFunc1_differentiable _ h₁
+    have hd₂ := evalFunc1_differentiable _ h₂
+    simp only [LeanCert.Internal.AD.evalTotalCore, DualInterval.mul, evalFunc1_mul_pi, deriv_mul
+      (hd₁ x) (hd₂ x)]
+    have hdom₁ := evalDomainValidDual_of_ExprSupported _ h₁ (fun _ => DualInterval.varActive I) cfg
+    have hdom₂ := evalDomainValidDual_of_ExprSupported _ h₂ (fun _ => DualInterval.varActive I) cfg
+    have hval₁ := LeanCert.Engine.evalDualTotalCore_val_correct _ h₁.toCore (fun _ => x)
+      (fun _ => DualInterval.varActive I) cfg (fun _ => hx) hdom₁
+    have hval₂ := LeanCert.Engine.evalDualTotalCore_val_correct _ h₂.toCore (fun _ => x)
+      (fun _ => DualInterval.varActive I) cfg (fun _ => hx) hdom₂
+    exact IntervalRat.mem_add (IntervalRat.mem_mul (ih₁ x hx) hval₂) (IntervalRat.mem_mul hval₁
+      (ih₂ x hx))
+  | neg hs ih =>
+    have hd := evalFunc1_differentiable _ hs
+    simp only [LeanCert.Internal.AD.evalTotalCore, DualInterval.neg, evalFunc1_neg_pi, deriv.neg]
+    exact IntervalRat.mem_neg (ih x hx)
+  | @sin e' hs ih =>
+    have hd := evalFunc1_differentiable e' hs
+    simp only [LeanCert.Internal.AD.evalTotalCore, DualInterval.sinCore, evalFunc1_sin]
+    rw [deriv_sin (hd.differentiableAt)]
+    have hdom := evalDomainValidDual_of_ExprSupported e' hs (fun _ => DualInterval.varActive I) cfg
+    have hval := LeanCert.Engine.evalDualTotalCore_val_correct e' hs.toCore (fun _ => x)
+      (fun _ => DualInterval.varActive I) cfg (fun _ => hx) hdom
+    have hcos := IntervalRat.mem_cosComputable hval cfg.taylorDepth
+    exact IntervalRat.mem_mul hcos (ih x hx)
+  | @cos e' hs ih =>
+    have hd := evalFunc1_differentiable e' hs
+    simp only [LeanCert.Internal.AD.evalTotalCore, DualInterval.cosCore, evalFunc1_cos]
+    rw [deriv_cos (hd.differentiableAt)]
+    have hdom := evalDomainValidDual_of_ExprSupported e' hs (fun _ => DualInterval.varActive I) cfg
+    have hval := LeanCert.Engine.evalDualTotalCore_val_correct e' hs.toCore (fun _ => x)
+      (fun _ => DualInterval.varActive I) cfg (fun _ => hx) hdom
+    have hsin := IntervalRat.mem_sinComputable hval cfg.taylorDepth
+    have hnegsin := IntervalRat.mem_neg hsin
+    exact IntervalRat.mem_mul hnegsin (ih x hx)
+  | @exp e' hs ih =>
+    have hd := evalFunc1_differentiable e' hs
+    simp only [LeanCert.Internal.AD.evalTotalCore, DualInterval.expCore, evalFunc1_exp]
+    rw [deriv_exp (hd.differentiableAt)]
+    have hdom := evalDomainValidDual_of_ExprSupported e' hs (fun _ => DualInterval.varActive I) cfg
+    have hval := LeanCert.Engine.evalDualTotalCore_val_correct e' hs.toCore (fun _ => x)
+      (fun _ => DualInterval.varActive I) cfg (fun _ => hx) hdom
+    have hexp := IntervalRat.mem_expComputable hval cfg.taylorDepth
+    exact IntervalRat.mem_mul hexp (ih x hx)
+
+/-- Convenience theorem: derivIntervalCore correctness -/
+theorem derivIntervalCore_correct (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (x : ℝ) (hx : x ∈ I) (cfg : EvalConfig) :
+    deriv (evalFunc1 e) x ∈ derivIntervalCore e I cfg :=
+  LeanCert.Engine.evalDualTotalCore_der_correct e hsupp I x hx cfg
+
+/-- If derivIntervalCore doesn't contain zero, the derivative is nonzero everywhere on I.
+    This is a key theorem for Newton contraction analysis. -/
+theorem derivIntervalCore_nonzero_implies_deriv_nonzero (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (cfg : EvalConfig)
+    (h : ¬(derivIntervalCore e I cfg).containsZero) :
+    ∀ x ∈ I, deriv (evalFunc1 e) x ≠ 0 := by
+  intro x hx hcontra
+  have hmem := derivIntervalCore_correct e hsupp I x hx cfg
+  simp only [IntervalRat.mem_def] at hmem
+  simp only [IntervalRat.containsZero, not_and_or, not_le] at h
+  rw [hcontra] at hmem
+  rcases h with hlo | hhi
+  · exact absurd hmem.1 (not_le.mpr (by exact_mod_cast hlo))
+  · exact absurd hmem.2 (not_le.mpr (by exact_mod_cast hhi))
+
+/-- If derivIntervalCore.lo > 0, then the derivative is positive everywhere on I. -/
+theorem derivIntervalCore_pos_implies_deriv_pos (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (cfg : EvalConfig)
+    (h : 0 < (derivIntervalCore e I cfg).lo) :
+    ∀ x ∈ I, 0 < deriv (evalFunc1 e) x := by
+  intro x hx
+  have hmem := derivIntervalCore_correct e hsupp I x hx cfg
+  simp only [IntervalRat.mem_def] at hmem
+  calc (0 : ℝ) < (derivIntervalCore e I cfg).lo := by exact_mod_cast h
+    _ ≤ deriv (evalFunc1 e) x := hmem.1
+
+/-- If derivIntervalCore.hi < 0, then the derivative is negative everywhere on I. -/
+theorem derivIntervalCore_neg_implies_deriv_neg (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (cfg : EvalConfig)
+    (h : (derivIntervalCore e I cfg).hi < 0) :
+    ∀ x ∈ I, deriv (evalFunc1 e) x < 0 := by
+  intro x hx
+  have hmem := derivIntervalCore_correct e hsupp I x hx cfg
+  simp only [IntervalRat.mem_def] at hmem
+  calc deriv (evalFunc1 e) x ≤ (derivIntervalCore e I cfg).hi := hmem.2
+    _ < 0 := by exact_mod_cast h
+
+/-- Strictly positive derivative (via Core bounds) implies strict monotonicity -/
+theorem strictMonoOn_of_derivIntervalCore_pos (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (cfg : EvalConfig)
+    (hpos : 0 < (derivIntervalCore e I cfg).lo) :
+    StrictMonoOn (evalFunc1 e) (Set.Icc (I.lo : ℝ) (I.hi : ℝ)) := by
+  have hdiff := evalFunc1_differentiable e hsupp
+  have hderiv_pos := derivIntervalCore_pos_implies_deriv_pos e hsupp I cfg hpos
+  apply strictMonoOn_of_deriv_pos (convex_Icc _ _)
+  · exact hdiff.continuous.continuousOn
+  · intro x hx
+    rw [interior_Icc] at hx
+    have hx_mem : x ∈ I := by
+      simp only [IntervalRat.mem_def]
+      exact ⟨le_of_lt hx.1, le_of_lt hx.2⟩
+    exact hderiv_pos x hx_mem
+
+/-- Strictly negative derivative (via Core bounds) implies strict antitonicity -/
+theorem strictAntiOn_of_derivIntervalCore_neg (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (cfg : EvalConfig)
+    (hneg : (derivIntervalCore e I cfg).hi < 0) :
+    StrictAntiOn (evalFunc1 e) (Set.Icc (I.lo : ℝ) (I.hi : ℝ)) := by
+  have hdiff := evalFunc1_differentiable e hsupp
+  have hderiv_neg := derivIntervalCore_neg_implies_deriv_neg e hsupp I cfg hneg
+  apply strictAntiOn_of_deriv_neg (convex_Icc _ _)
+  · exact hdiff.continuous.continuousOn
+  · intro x hx
+    rw [interior_Icc] at hx
+    have hx_mem : x ∈ I := by
+      simp only [IntervalRat.mem_def]
+      exact ⟨le_of_lt hx.1, le_of_lt hx.2⟩
+    exact hderiv_neg x hx_mem
+
+end LeanCert.Engine
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Computable domain-aware automatic differentiation
+
+This module extends LeanCert's computable forward-mode AD path with checked
+reciprocal and logarithm nodes.  The existing `ADSupported` fragment remains
+the domain-free fast path.  Here, successful evaluation itself certifies both
+syntactic support and every partial-domain side condition.
+-/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+/-- The computable domain-aware AD fragment.  Unlike `ADSupported`, this is a
+box-dependent check because reciprocal and logarithm have partial domains. -/
+def checkADDomain (e : Expr) (ρ : DualEnv) (cfg : EvalConfig := {}) : Bool :=
+  match e with
+  | .const _ | .var _ => true
+  | .add a b | .mul a b => checkADDomain a ρ cfg && checkADDomain b ρ cfg
+  | .neg a | .exp a | .sin a | .cos a => checkADDomain a ρ cfg
+  | .inv a =>
+      checkADDomain a ρ cfg &&
+        decide (¬IntervalRat.containsZero
+          (LeanCert.Internal.AD.evalTotalCore a ρ cfg).val)
+  | .log a =>
+      checkADDomain a ρ cfg &&
+        decide (IntervalRat.isPositive
+          (LeanCert.Internal.AD.evalTotalCore a ρ cfg).val)
+  | _ => false
+
+/-- Best-effort structured explanation for a failed domain-aware AD check. -/
+def diagnoseADDomainFailure (e : Expr) (ρ : DualEnv) (cfg : EvalConfig := {}) : EvalError :=
+  match e with
+  | .add a b | .mul a b =>
+      if checkADDomain a ρ cfg then
+        .nestedFailure "right operand" (diagnoseADDomainFailure b ρ cfg)
+      else
+        .nestedFailure "left operand" (diagnoseADDomainFailure a ρ cfg)
+  | .neg a | .exp a | .sin a | .cos a =>
+      .nestedFailure "unary operand" (diagnoseADDomainFailure a ρ cfg)
+  | .inv a =>
+      if checkADDomain a ρ cfg then
+        .reciprocalContainsZero (LeanCert.Internal.AD.evalTotalCore a ρ cfg).val
+      else
+        .nestedFailure "reciprocal operand" (diagnoseADDomainFailure a ρ cfg)
+  | .log a =>
+      if checkADDomain a ρ cfg then
+        .logNonpositive (LeanCert.Internal.AD.evalTotalCore a ρ cfg).val
+      else
+        .nestedFailure "logarithm operand" (diagnoseADDomainFailure a ρ cfg)
+  | .const _ | .var _ => .invalidConfiguration "successful AD expression diagnosed as failure"
+  | _ => .unsupportedFeature "domain-aware automatic differentiation"
+
+/-- Computable checked dual evaluation.  No finite derivative enclosure is
+returned unless every reciprocal denominator excludes zero and every logarithm
+argument is strictly positive on the input box. -/
+def evalDualChecked (e : Expr) (ρ : DualEnv) (cfg : EvalConfig := {}) :
+    EvalResult DualInterval :=
+  if checkADDomain e ρ cfg then
+    .ok (LeanCert.Internal.AD.evalTotalCore e ρ cfg)
+  else
+    .error (diagnoseADDomainFailure e ρ cfg)
+
+/-- Checked value-and-derivative evaluation along coordinate `idx`. -/
+def evalWithDerivChecked (e : Expr) (ρ : IntervalEnv) (idx : Nat)
+    (cfg : EvalConfig := {}) : EvalResult DualInterval :=
+  evalDualChecked e (mkDualEnv ρ idx) cfg
+
+/-- Checked derivative enclosure along coordinate `idx`. -/
+def derivIntervalChecked (e : Expr) (ρ : IntervalEnv) (idx : Nat)
+    (cfg : EvalConfig := {}) : EvalResult IntervalRat :=
+  return (← evalWithDerivChecked e ρ idx cfg).der
+
+/-- Checked derivative enclosure for a single-variable expression. -/
+def derivIntervalChecked1 (e : Expr) (I : IntervalRat) (cfg : EvalConfig := {}) :
+    EvalResult IntervalRat :=
+  derivIntervalChecked e (fun _ => I) 0 cfg
+
+private theorem checkADDomain_of_evalDualChecked_ok {e : Expr} {ρ : DualEnv}
+    {cfg : EvalConfig} {D : DualInterval} (h : evalDualChecked e ρ cfg = .ok D) :
+    checkADDomain e ρ cfg = true := by
+  simp only [evalDualChecked] at h
+  split at h
+  · assumption
+  · contradiction
+
+private theorem evalDualChecked_ok_eq {e : Expr} {ρ : DualEnv}
+    {cfg : EvalConfig} {D : DualInterval} (h : evalDualChecked e ρ cfg = .ok D) :
+    D = LeanCert.Internal.AD.evalTotalCore e ρ cfg := by
+  simp only [evalDualChecked] at h
+  split at h
+  · exact (Except.ok.inj h).symm
+  · contradiction
+
+private theorem not_containsZero_cases {I : IntervalRat}
+    (h : ¬IntervalRat.containsZero I) : I.lo > 0 ∨ I.hi < 0 := by
+  simp only [IntervalRat.containsZero, not_and_or, not_le] at h
+  exact h
+
+private theorem evalTotalCore_val_correct_of_check (e : Expr)
+    (ρReal : Nat → ℝ) (ρDual : DualEnv) (cfg : EvalConfig)
+    (hρ : ∀ i, ρReal i ∈ (ρDual i).val)
+    (hcheck : checkADDomain e ρDual cfg = true) :
+    Expr.eval ρReal e ∈ (LeanCert.Internal.AD.evalTotalCore e ρDual cfg).val := by
+  induction e with
+  | const q =>
+      exact IntervalRat.mem_singleton q
+  | var i =>
+      exact hρ i
+  | add a b iha ihb =>
+      simp only [checkADDomain, Bool.and_eq_true] at hcheck
+      simp only [Expr.eval_add, LeanCert.Internal.AD.evalTotalCore, DualInterval.add]
+      exact IntervalRat.mem_add (iha hcheck.1) (ihb hcheck.2)
+  | mul a b iha ihb =>
+      simp only [checkADDomain, Bool.and_eq_true] at hcheck
+      simp only [Expr.eval_mul, LeanCert.Internal.AD.evalTotalCore, DualInterval.mul]
+      exact IntervalRat.mem_mul (iha hcheck.1) (ihb hcheck.2)
+  | neg a ih =>
+      simp only [checkADDomain] at hcheck
+      exact IntervalRat.mem_neg (ih hcheck)
+  | inv a ih =>
+      simp only [checkADDomain, Bool.and_eq_true, decide_eq_true_eq] at hcheck
+      simp only [Expr.eval_inv, LeanCert.Internal.AD.evalTotalCore, DualInterval.inv]
+      exact mem_invInterval_nonzero (ih hcheck.1) (not_containsZero_cases hcheck.2)
+  | exp a ih =>
+      simp only [checkADDomain] at hcheck
+      exact IntervalRat.mem_expComputable (ih hcheck) cfg.taylorDepth
+  | sin a ih =>
+      simp only [checkADDomain] at hcheck
+      exact IntervalRat.mem_sinComputable (ih hcheck) cfg.taylorDepth
+  | cos a ih =>
+      simp only [checkADDomain] at hcheck
+      exact IntervalRat.mem_cosComputable (ih hcheck) cfg.taylorDepth
+  | log a ih =>
+      simp only [checkADDomain, Bool.and_eq_true, decide_eq_true_eq] at hcheck
+      exact IntervalRat.mem_logComputable (ih hcheck.1) hcheck.2 cfg.taylorDepth
+  | atan | arsinh | atanh | sinc | erf | sinh | cosh | tanh | sqrt | namedConst =>
+      simp [checkADDomain] at hcheck
+
+/-- Successful checked dual evaluation encloses the expression value for every
+real environment represented by the dual environment. -/
+theorem evalDualChecked_val_correct (e : Expr)
+    (ρReal : Nat → ℝ) (ρDual : DualEnv) (cfg : EvalConfig) (D : DualInterval)
+    (hρ : ∀ i, ρReal i ∈ (ρDual i).val)
+    (hok : evalDualChecked e ρDual cfg = .ok D) :
+    Expr.eval ρReal e ∈ D.val := by
+  rw [evalDualChecked_ok_eq hok]
+  exact evalTotalCore_val_correct_of_check e ρReal ρDual cfg hρ
+    (checkADDomain_of_evalDualChecked_ok hok)
+
+private theorem evalAlong_differentiableAt_of_check (e : Expr)
+    (ρReal : Nat → ℝ) (ρInt : IntervalEnv) (idx : Nat) (cfg : EvalConfig)
+    (x : ℝ) (hx : x ∈ ρInt idx) (hρ : ∀ i, ρReal i ∈ ρInt i)
+    (hcheck : checkADDomain e (mkDualEnv ρInt idx) cfg = true) :
+    DifferentiableAt ℝ (Expr.evalAlong e ρReal idx) x := by
+  have hmem : ∀ i, Expr.updateVar ρReal idx x i ∈ (mkDualEnv ρInt idx i).val :=
+    updateVar_mem_mkDualEnv_val ρReal ρInt idx x hx hρ
+  induction e with
+  | const => exact differentiableAt_const _
+  | var i =>
+      by_cases hi : i = idx
+      · subst i
+        simpa only [Expr.evalAlong_var_active] using differentiableAt_id
+      · simpa only [Expr.evalAlong_var_passive _ _ _ hi] using differentiableAt_const (c := ρReal i)
+  | add a b iha ihb =>
+      simp only [checkADDomain, Bool.and_eq_true] at hcheck
+      simpa only [Expr.evalAlong_add_pi] using (iha hcheck.1).add (ihb hcheck.2)
+  | mul a b iha ihb =>
+      simp only [checkADDomain, Bool.and_eq_true] at hcheck
+      simpa only [Expr.evalAlong_mul_pi] using (iha hcheck.1).mul (ihb hcheck.2)
+  | neg a ih =>
+      simp only [checkADDomain] at hcheck
+      simpa only [Expr.evalAlong_neg_pi] using (ih hcheck).neg
+  | inv a ih =>
+      simp only [checkADDomain, Bool.and_eq_true, decide_eq_true_eq] at hcheck
+      have hval := evalTotalCore_val_correct_of_check a (Expr.updateVar ρReal idx x)
+        (mkDualEnv ρInt idx) cfg hmem hcheck.1
+      have hne : Expr.evalAlong a ρReal idx x ≠ 0 := by
+        rw [Expr.evalAlong_eq]
+        intro hz
+        rw [hz] at hval
+        exact hcheck.2 ⟨by exact_mod_cast hval.1, by exact_mod_cast hval.2⟩
+      exact DifferentiableAt.inv (ih hcheck.1) hne
+  | exp a ih =>
+      simp only [checkADDomain] at hcheck
+      exact Real.differentiableAt_exp.comp x (ih hcheck)
+  | sin a ih =>
+      simp only [checkADDomain] at hcheck
+      exact Real.differentiableAt_sin.comp x (ih hcheck)
+  | cos a ih =>
+      simp only [checkADDomain] at hcheck
+      exact Real.differentiableAt_cos.comp x (ih hcheck)
+  | log a ih =>
+      simp only [checkADDomain, Bool.and_eq_true, decide_eq_true_eq] at hcheck
+      have hval := evalTotalCore_val_correct_of_check a (Expr.updateVar ρReal idx x)
+        (mkDualEnv ρInt idx) cfg hmem hcheck.1
+      have hpos : 0 < Expr.evalAlong a ρReal idx x := by
+        rw [Expr.evalAlong_eq]
+        exact lt_of_lt_of_le (by exact_mod_cast hcheck.2) hval.1
+      exact (Real.differentiableAt_log (ne_of_gt hpos)).comp x (ih hcheck.1)
+  | atan | arsinh | atanh | sinc | erf | sinh | cosh | tanh | sqrt | namedConst =>
+      simp [checkADDomain] at hcheck
+
+/-- A successful checked evaluation proves differentiability at every point in
+the input box along the selected coordinate. -/
+theorem evalWithDerivChecked_differentiableAt (e : Expr)
+    (ρReal : Nat → ℝ) (ρInt : IntervalEnv) (idx : Nat) (cfg : EvalConfig)
+    (D : DualInterval) (x : ℝ) (hx : x ∈ ρInt idx)
+    (hρ : ∀ i, ρReal i ∈ ρInt i)
+    (hok : evalWithDerivChecked e ρInt idx cfg = .ok D) :
+    DifferentiableAt ℝ (Expr.evalAlong e ρReal idx) x := by
+  exact evalAlong_differentiableAt_of_check e ρReal ρInt idx cfg x hx hρ
+    (checkADDomain_of_evalDualChecked_ok hok)
+
+private theorem evalTotalCore_der_correct_idx_of_check (e : Expr)
+    (ρReal : Nat → ℝ) (ρInt : IntervalEnv) (idx : Nat) (cfg : EvalConfig)
+    (x : ℝ) (hx : x ∈ ρInt idx) (hρ : ∀ i, ρReal i ∈ ρInt i)
+    (hcheck : checkADDomain e (mkDualEnv ρInt idx) cfg = true) :
+    deriv (Expr.evalAlong e ρReal idx) x ∈
+      (LeanCert.Internal.AD.evalTotalCore e (mkDualEnv ρInt idx) cfg).der := by
+  have hmem : ∀ i, Expr.updateVar ρReal idx x i ∈ (mkDualEnv ρInt idx i).val :=
+    updateVar_mem_mkDualEnv_val ρReal ρInt idx x hx hρ
+  induction e with
+  | const q =>
+      simp only [Expr.evalAlong_const', deriv_const, LeanCert.Internal.AD.evalTotalCore,
+        DualInterval.const]
+      exact_mod_cast IntervalRat.mem_singleton 0
+  | var i =>
+      by_cases hi : i = idx
+      · subst i
+        simp only [Expr.evalAlong_var_active, deriv_id, LeanCert.Internal.AD.evalTotalCore,
+          mkDualEnv, ↓reduceIte, DualInterval.varActive]
+        exact_mod_cast IntervalRat.mem_singleton 1
+      · simp only [Expr.evalAlong_var_passive _ _ _ hi, deriv_const,
+          LeanCert.Internal.AD.evalTotalCore, mkDualEnv, ite_eq_right hi, DualInterval.varPassive]
+        exact_mod_cast IntervalRat.mem_singleton 0
+  | add a b iha ihb =>
+      simp only [checkADDomain, Bool.and_eq_true] at hcheck
+      have hda := evalAlong_differentiableAt_of_check a ρReal ρInt idx cfg x hx hρ hcheck.1
+      have hdb := evalAlong_differentiableAt_of_check b ρReal ρInt idx cfg x hx hρ hcheck.2
+      simp only [Expr.evalAlong_add_pi, deriv_add hda hdb, LeanCert.Internal.AD.evalTotalCore,
+        DualInterval.add]
+      exact IntervalRat.mem_add (iha hcheck.1) (ihb hcheck.2)
+  | mul a b iha ihb =>
+      simp only [checkADDomain, Bool.and_eq_true] at hcheck
+      have hda := evalAlong_differentiableAt_of_check a ρReal ρInt idx cfg x hx hρ hcheck.1
+      have hdb := evalAlong_differentiableAt_of_check b ρReal ρInt idx cfg x hx hρ hcheck.2
+      simp only [Expr.evalAlong_mul_pi, deriv_mul hda hdb, LeanCert.Internal.AD.evalTotalCore,
+        DualInterval.mul]
+      have hva := evalTotalCore_val_correct_of_check a (Expr.updateVar ρReal idx x)
+        (mkDualEnv ρInt idx) cfg hmem hcheck.1
+      have hvb := evalTotalCore_val_correct_of_check b (Expr.updateVar ρReal idx x)
+        (mkDualEnv ρInt idx) cfg hmem hcheck.2
+      exact IntervalRat.mem_add (IntervalRat.mem_mul (iha hcheck.1) hvb)
+        (IntervalRat.mem_mul hva (ihb hcheck.2))
+  | neg a ih =>
+      simp only [checkADDomain] at hcheck
+      have hd := evalAlong_differentiableAt_of_check a ρReal ρInt idx cfg x hx hρ hcheck
+      simp only [Expr.evalAlong_neg_pi, deriv.neg, LeanCert.Internal.AD.evalTotalCore,
+        DualInterval.neg]
+      exact IntervalRat.mem_neg (ih hcheck)
+  | inv a ih =>
+      simp only [checkADDomain, Bool.and_eq_true, decide_eq_true_eq] at hcheck
+      have hval := evalTotalCore_val_correct_of_check a (Expr.updateVar ρReal idx x)
+        (mkDualEnv ρInt idx) cfg hmem hcheck.1
+      have hval' : Expr.evalAlong a ρReal idx x ∈
+          (LeanCert.Internal.AD.evalTotalCore a (mkDualEnv ρInt idx) cfg).val := by
+        simpa only [Expr.evalAlong_eq] using hval
+      have hne : Expr.evalAlong a ρReal idx x ≠ 0 := by
+        intro hz
+        rw [hz] at hval'
+        exact hcheck.2 ⟨by exact_mod_cast hval'.1, by exact_mod_cast hval'.2⟩
+      have hd := evalAlong_differentiableAt_of_check a ρReal ρInt idx cfg x hx hρ hcheck.1
+      rw [Expr.evalAlong_inv]
+      have hcomp : (fun t => (Expr.evalAlong a ρReal idx t)⁻¹) =
+          (fun y : ℝ => y⁻¹) ∘ Expr.evalAlong a ρReal idx := rfl
+      rw [hcomp, deriv_comp x (hasDerivAt_inv hne).differentiableAt hd]
+      simp only [(hasDerivAt_inv hne).deriv, LeanCert.Internal.AD.evalTotalCore,
+        DualInterval.inv]
+      have hder := ih hcheck.1
+      have hinv := mem_invInterval_nonzero hval' (not_containsZero_cases hcheck.2)
+      have hneg := IntervalRat.mem_neg
+        (IntervalRat.mem_mul hder (IntervalRat.mem_mul hinv hinv))
+      convert hneg using 1
+      field_simp
+  | exp a ih =>
+      simp only [checkADDomain] at hcheck
+      have hd := evalAlong_differentiableAt_of_check a ρReal ρInt idx cfg x hx hρ hcheck
+      simp only [Expr.evalAlong_exp, deriv_exp hd, LeanCert.Internal.AD.evalTotalCore,
+        DualInterval.expCore]
+      have hval := evalTotalCore_val_correct_of_check a (Expr.updateVar ρReal idx x)
+        (mkDualEnv ρInt idx) cfg hmem hcheck
+      exact IntervalRat.mem_mul (IntervalRat.mem_expComputable hval cfg.taylorDepth) (ih hcheck)
+  | sin a ih =>
+      simp only [checkADDomain] at hcheck
+      have hd := evalAlong_differentiableAt_of_check a ρReal ρInt idx cfg x hx hρ hcheck
+      simp only [Expr.evalAlong_sin, deriv_sin hd, LeanCert.Internal.AD.evalTotalCore,
+        DualInterval.sinCore]
+      have hval := evalTotalCore_val_correct_of_check a (Expr.updateVar ρReal idx x)
+        (mkDualEnv ρInt idx) cfg hmem hcheck
+      exact IntervalRat.mem_mul (IntervalRat.mem_cosComputable hval cfg.taylorDepth) (ih hcheck)
+  | cos a ih =>
+      simp only [checkADDomain] at hcheck
+      have hd := evalAlong_differentiableAt_of_check a ρReal ρInt idx cfg x hx hρ hcheck
+      simp only [Expr.evalAlong_cos, deriv_cos hd, LeanCert.Internal.AD.evalTotalCore,
+        DualInterval.cosCore]
+      have hval := evalTotalCore_val_correct_of_check a (Expr.updateVar ρReal idx x)
+        (mkDualEnv ρInt idx) cfg hmem hcheck
+      exact IntervalRat.mem_mul
+        (IntervalRat.mem_neg (IntervalRat.mem_sinComputable hval cfg.taylorDepth)) (ih hcheck)
+  | log a ih =>
+      simp only [checkADDomain, Bool.and_eq_true, decide_eq_true_eq] at hcheck
+      have hval := evalTotalCore_val_correct_of_check a (Expr.updateVar ρReal idx x)
+        (mkDualEnv ρInt idx) cfg hmem hcheck.1
+      have hval' : Expr.evalAlong a ρReal idx x ∈
+          (LeanCert.Internal.AD.evalTotalCore a (mkDualEnv ρInt idx) cfg).val := by
+        simpa only [Expr.evalAlong_eq] using hval
+      have hpos : 0 < Expr.evalAlong a ρReal idx x :=
+        lt_of_lt_of_le (by exact_mod_cast hcheck.2) hval'.1
+      have hd := evalAlong_differentiableAt_of_check a ρReal ρInt idx cfg x hx hρ hcheck.1
+      rw [Expr.evalAlong_log]
+      have hcomp : (fun t => Real.log (Expr.evalAlong a ρReal idx t)) =
+          Real.log ∘ Expr.evalAlong a ρReal idx := rfl
+      rw [hcomp, deriv_comp x (Real.differentiableAt_log (ne_of_gt hpos)) hd]
+      simp only [Real.deriv_log, LeanCert.Internal.AD.evalTotalCore, DualInterval.logCore]
+      have hder := ih hcheck.1
+      have hnz : ¬IntervalRat.containsZero
+          (LeanCert.Internal.AD.evalTotalCore a (mkDualEnv ρInt idx) cfg).val := by
+        intro hz
+        exact (not_lt_of_ge hz.1) hcheck.2
+      have hinv := mem_invInterval_nonzero hval' (not_containsZero_cases hnz)
+      exact IntervalRat.mem_mul hinv hder
+  | atan | arsinh | atanh | sinc | erf | sinh | cosh | tanh | sqrt | namedConst =>
+      simp [checkADDomain] at hcheck
+
+/-- Successful checked indexed AD encloses the true partial derivative. -/
+theorem evalWithDerivChecked_der_correct (e : Expr)
+    (ρReal : Nat → ℝ) (ρInt : IntervalEnv) (idx : Nat) (cfg : EvalConfig)
+    (D : DualInterval) (x : ℝ) (hx : x ∈ ρInt idx)
+    (hρ : ∀ i, ρReal i ∈ ρInt i)
+    (hok : evalWithDerivChecked e ρInt idx cfg = .ok D) :
+    deriv (Expr.evalAlong e ρReal idx) x ∈ D.der := by
+  rw [evalDualChecked_ok_eq hok]
+  exact evalTotalCore_der_correct_idx_of_check e ρReal ρInt idx cfg x hx hρ
+    (checkADDomain_of_evalDualChecked_ok hok)
+
+/-- Golden soundness theorem for the checked derivative-only API. -/
+theorem derivIntervalChecked_correct (e : Expr)
+    (ρReal : Nat → ℝ) (ρInt : IntervalEnv) (idx : Nat) (cfg : EvalConfig)
+    (dI : IntervalRat) (x : ℝ) (hx : x ∈ ρInt idx)
+    (hρ : ∀ i, ρReal i ∈ ρInt i)
+    (hok : derivIntervalChecked e ρInt idx cfg = .ok dI) :
+    deriv (Expr.evalAlong e ρReal idx) x ∈ dI := by
+  simp only [derivIntervalChecked, bind, Except.bind] at hok
+  cases hdual : evalWithDerivChecked e ρInt idx cfg with
+  | error err => simp [hdual] at hok
+  | ok D =>
+      rw [hdual] at hok
+      simp only [pure, Except.pure, Except.ok.injEq] at hok
+      subst dI
+      exact evalWithDerivChecked_der_correct e ρReal ρInt idx cfg D x hx hρ hdual
+
+end LeanCert.Engine
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/Development/IntervalArithmetic/Foundations/Development002.lean b/LeanPool/MovingSofa/Development/IntervalArithmetic/Foundations/Development002.lean
new file mode 100644
index 0000000000..f1b0bd6235
--- /dev/null
+++ b/LeanPool/MovingSofa/Development/IntervalArithmetic/Foundations/Development002.lean
@@ -0,0 +1,5502 @@
+/-
+Copyright (c) 2026 Dean Cureton and contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dean Cureton, The Moving Sofa contributors
+-/
+module
+
+public import LeanPool.MovingSofa.Development.IntervalArithmetic.Foundations.Development001
+
+
+
+
+
+
+
+
+
+
+public import Mathlib.Analysis.Calculus.Deriv.Pi
+public import Mathlib.Analysis.Calculus.MeanValue
+public import Mathlib.Analysis.Matrix.Normed
+public import Mathlib.Analysis.SpecialFunctions.Arsinh
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
+public import Mathlib.Topology.MetricSpace.Contracting
+/-!
+# Moving sofa: related mathematical developments
+
+* `LeanCert.Engine.AD.PartialCorrectness`.
+* `LeanCert.Engine.IntervalEvalDyadic`.
+* `LeanCert.Engine.AD.Dyadic`.
+* `LeanCert.Engine.AD`.
+* `LeanCert.Engine.Optimization.Box`.
+* `LeanCert.Engine.Optimize`.
+* `LeanCert.Engine.Optimization.Gradient`.
+* `LeanCert.Engine.RootFinding.Krawczyk`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Automatic Differentiation - Partial Correctness Theorems
+
+This file proves the correctness of the partial dual evaluator `evalDualOption`
+which handles expressions with inv, log, and other domain-restricted functions.
+
+## Main theorems
+
+* `evalDualOption_val_correct` - Value component is correct when evalDualOption returns some
+* `evalDualOption1_val_correct` - Single-variable version
+* `evalFunc1_differentiableAt_of_evalDualOption` - Differentiability when evalDualOption succeeds
+* `evalDualOption_der_correct` - Derivative component is correct
+* `evalDualOption1_correct` - Combined correctness theorem
+
+## Design notes
+
+All theorems are FULLY PROVED with no sorry or axioms.
+The key insight is that when evalDualOption returns `some`, the domain
+constraints (nonzero for inv, positive for log) are satisfied.
+-/
+
+/-! ### Correctness of partial dual evaluator -/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+
+open LeanCert.Core Filter
+open scoped Topology
+
+/-- The value component of evalDualOption is correct when it returns some.
+    This theorem extends to expressions with inv. -/
+theorem evalDualOption_val_correct (e : Expr)
+    (ρ_real : Nat → ℝ) (ρ_dual : DualEnv) (D : DualInterval)
+    (hsome : evalDualOption e ρ_dual = some D)
+    (hρ : ∀ i, ρ_real i ∈ (ρ_dual i).val) :
+    Expr.eval ρ_real e ∈ D.val := by
+  induction e generalizing D with
+  | const q =>
+    simp only [evalDualOption, Option.some.injEq] at hsome
+    rw [← hsome]
+    simp only [Expr.eval_const, DualInterval.const]
+    exact IntervalRat.mem_singleton q
+  | var idx =>
+    simp only [evalDualOption, Option.some.injEq] at hsome
+    rw [← hsome]
+    exact hρ idx
+  | add e₁ e₂ ih₁ ih₂ =>
+    simp only [evalDualOption] at hsome
+    cases heq₁ : evalDualOption e₁ ρ_dual with
+    | none => simp only [heq₁, reduceCtorEq] at hsome
+    | some d₁ =>
+      cases heq₂ : evalDualOption e₂ ρ_dual with
+      | none => simp only [heq₁, heq₂, reduceCtorEq] at hsome
+      | some d₂ =>
+        simp only [heq₁, heq₂, Option.some.injEq] at hsome
+        rw [← hsome]
+        simp only [Expr.eval_add, DualInterval.add]
+        exact IntervalRat.mem_add (ih₁ d₁ heq₁) (ih₂ d₂ heq₂)
+  | mul e₁ e₂ ih₁ ih₂ =>
+    simp only [evalDualOption] at hsome
+    cases heq₁ : evalDualOption e₁ ρ_dual with
+    | none => simp only [heq₁, reduceCtorEq] at hsome
+    | some d₁ =>
+      cases heq₂ : evalDualOption e₂ ρ_dual with
+      | none => simp only [heq₁, heq₂, reduceCtorEq] at hsome
+      | some d₂ =>
+        simp only [heq₁, heq₂, Option.some.injEq] at hsome
+        rw [← hsome]
+        simp only [Expr.eval_mul, DualInterval.mul]
+        exact IntervalRat.mem_mul (ih₁ d₁ heq₁) (ih₂ d₂ heq₂)
+  | neg e' ih =>
+    simp only [evalDualOption] at hsome
+    cases heq : evalDualOption e' ρ_dual with
+    | none => simp only [heq, reduceCtorEq] at hsome
+    | some d =>
+      simp only [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [Expr.eval_neg, DualInterval.neg]
+      exact IntervalRat.mem_neg (ih d heq)
+  | inv e' ih =>
+    simp only [evalDualOption] at hsome
+    cases heq : evalDualOption e' ρ_dual with
+    | none => simp only [heq, reduceCtorEq] at hsome
+    | some d =>
+      simp only [heq, DualInterval.inv?] at hsome
+      split at hsome
+      · simp at hsome
+      · next hnotzero =>
+        simp only [Option.some.injEq] at hsome
+        rw [← hsome]
+        simp only [Expr.eval_inv]
+        have hmem := ih d heq
+        -- The denominator is nonzero because d.val doesn't contain zero and eval ∈ d.val
+        have heval_ne : Expr.eval ρ_real e' ≠ 0 := by
+          intro heq_zero
+          rw [heq_zero] at hmem
+          simp only [IntervalRat.mem_def] at hmem
+          simp only [IntervalRat.containsZero, not_and, not_le] at hnotzero
+          rcases le_or_gt d.val.lo 0 with hlo | hlo
+          · have hhi_nonneg : (0 : ℚ) ≤ d.val.hi := by
+              have h : (0 : ℝ) ≤ d.val.hi := hmem.2
+              exact_mod_cast h
+            exact absurd (hnotzero hlo) (not_lt.mpr hhi_nonneg)
+          · have hlo_pos : (0 : ℝ) < d.val.lo := by exact_mod_cast hlo
+            exact absurd hmem.1 (not_le.mpr hlo_pos)
+        exact IntervalRat.mem_invNonzero hmem heval_ne
+  | sin e' ih =>
+    simp only [evalDualOption] at hsome
+    cases heq : evalDualOption e' ρ_dual with
+    | none => simp only [heq, reduceCtorEq] at hsome
+    | some d =>
+      simp only [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [Expr.eval_sin, DualInterval.sin]
+      exact mem_sinInterval (ih d heq)
+  | cos e' ih =>
+    simp only [evalDualOption] at hsome
+    cases heq : evalDualOption e' ρ_dual with
+    | none => simp only [heq, reduceCtorEq] at hsome
+    | some d =>
+      simp only [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [Expr.eval_cos, DualInterval.cos]
+      exact mem_cosInterval (ih d heq)
+  | exp e' ih =>
+    simp only [evalDualOption] at hsome
+    cases heq : evalDualOption e' ρ_dual with
+    | none => simp only [heq, reduceCtorEq] at hsome
+    | some d =>
+      simp only [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [Expr.eval_exp, DualInterval.exp]
+      exact IntervalRat.mem_expInterval (ih d heq)
+  | sinh e' ih =>
+    simp only [evalDualOption] at hsome
+    cases heq : evalDualOption e' ρ_dual with
+    | none => simp only [heq, reduceCtorEq] at hsome
+    | some d =>
+      simp only [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [Expr.eval_sinh, DualInterval.sinh, sinhInterval]
+      exact IntervalRat.mem_sinhComputable (ih d heq) 10
+  | cosh e' ih =>
+    simp only [evalDualOption] at hsome
+    cases heq : evalDualOption e' ρ_dual with
+    | none => simp only [heq, reduceCtorEq] at hsome
+    | some d =>
+      simp only [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [Expr.eval_cosh, DualInterval.cosh, coshInterval]
+      exact IntervalRat.mem_coshComputable (ih d heq) 10
+  | tanh _ _ =>
+    simp only [evalDualOption] at hsome
+    cases hsome
+  | log e' ih =>
+    simp only [evalDualOption] at hsome
+    cases heq : evalDualOption e' ρ_dual with
+    | none => simp only [heq, reduceCtorEq] at hsome
+    | some d =>
+      simp only [heq, DualInterval.logOption] at hsome
+      split at hsome
+      · rename_i hpos
+        simp only [Option.some.injEq] at hsome
+        rw [← hsome]
+        simp only [Expr.eval_log]
+        have hJ_mem := ih d heq
+        exact IntervalRat.mem_logInterval hJ_mem
+      · contradiction
+  | atan e' ih =>
+    simp only [evalDualOption] at hsome
+    cases heq : evalDualOption e' ρ_dual with
+    | none => simp only [heq, reduceCtorEq] at hsome
+    | some d =>
+      simp only [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [Expr.eval_atan, DualInterval.atan]
+      exact mem_atanInterval (ih d heq)
+  | arsinh e' ih =>
+    simp only [evalDualOption] at hsome
+    cases heq : evalDualOption e' ρ_dual with
+    | none => simp only [heq, reduceCtorEq] at hsome
+    | some d =>
+      simp only [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [Expr.eval_arsinh, DualInterval.arsinh]
+      exact mem_arsinhInterval (ih d heq)
+  | atanh _ _ =>
+    -- evalDualOption returns none for atanh, so hsome : none = some D is a contradiction
+    simp only [evalDualOption] at hsome
+    contradiction
+  | sinc e' ih =>
+    simp only [evalDualOption] at hsome
+    cases heq : evalDualOption e' ρ_dual with
+    | none => simp only [heq, reduceCtorEq] at hsome
+    | some d =>
+      simp only [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [Expr.eval_sinc, DualInterval.sinc]
+      simp only [IntervalRat.mem_def, Rat.cast_neg, Rat.cast_one]
+      exact Real.sinc_mem_Icc _
+  | erf e' ih =>
+    simp only [evalDualOption] at hsome
+    cases heq : evalDualOption e' ρ_dual with
+    | none => simp only [heq, reduceCtorEq] at hsome
+    | some d =>
+      simp only [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [Expr.eval_erf, DualInterval.erf]
+      simp only [IntervalRat.mem_def, Rat.cast_neg, Rat.cast_one]
+      exact Real.erf_mem_Icc _
+  | sqrt e' ih =>
+    simp only [evalDualOption] at hsome
+    cases heq : evalDualOption e' ρ_dual with
+    | none => simp only [heq, reduceCtorEq] at hsome
+    | some d =>
+      simp only [heq] at hsome
+      -- hsome : DualInterval.sqrt? d = some D
+      simp only [DualInterval.sqrt?] at hsome
+      split at hsome
+      · next hpos =>
+        simp only [Option.some.injEq] at hsome
+        rw [← hsome]
+        simp only [Expr.eval_sqrt]
+        exact IntervalRat.mem_sqrtInterval' (ih d heq)
+      · exact absurd hsome (by simp)
+  | namedConst c =>
+    simp only [evalDualOption, Option.some.injEq] at hsome
+    rw [← hsome]
+    simp only [Expr.eval_namedConst, DualInterval.ofMathConst]
+    exact c.mem_interval
+
+/-- Single-variable version of evalDualOption_val_correct -/
+theorem evalDualOption1_val_correct (e : Expr)
+    (I : IntervalRat) (D : DualInterval) (x : ℝ) (hx : x ∈ I)
+    (hsome : evalDualOption1 e I = some D) :
+    Expr.eval (fun _ => x) e ∈ D.val := by
+  apply evalDualOption_val_correct e (fun _ => x) (fun _ => DualInterval.varActive I)
+  · exact hsome
+  · intro _; exact hx
+
+/-- Helper lemma: evalFunc1 for inv unfolds correctly -/
+theorem evalFunc1_inv (e : Expr) :
+    evalFunc1 (Expr.inv e) = fun t => (evalFunc1 e t)⁻¹ := rfl
+
+/-- Expressions with inv are differentiable when the denominator is nonzero. -/
+theorem evalFunc1_differentiableAt_of_evalDualOption (e : Expr)
+    (I : IntervalRat) (D : DualInterval) (x : ℝ) (hx : x ∈ I)
+    (hsome : evalDualOption1 e I = some D) :
+    DifferentiableAt ℝ (evalFunc1 e) x := by
+  induction e generalizing D with
+  | const q => exact differentiableAt_const _
+  | var _ => exact differentiableAt_id
+  | add e₁ e₂ ih₁ ih₂ =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq₁ : evalDualOption e₁ _ with
+    | none => rw [heq₁] at hsome; exact absurd hsome (by simp)
+    | some d₁ =>
+      cases heq₂ : evalDualOption e₂ _ with
+      | none => rw [heq₁, heq₂] at hsome; exact absurd hsome (by simp)
+      | some d₂ =>
+        exact DifferentiableAt.add (ih₁ d₁ heq₁) (ih₂ d₂ heq₂)
+  | mul e₁ e₂ ih₁ ih₂ =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq₁ : evalDualOption e₁ _ with
+    | none => rw [heq₁] at hsome; exact absurd hsome (by simp)
+    | some d₁ =>
+      cases heq₂ : evalDualOption e₂ _ with
+      | none => rw [heq₁, heq₂] at hsome; exact absurd hsome (by simp)
+      | some d₂ =>
+        exact DifferentiableAt.mul (ih₁ d₁ heq₁) (ih₂ d₂ heq₂)
+  | neg e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      exact DifferentiableAt.neg (ih d heq)
+  | inv e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      rw [heq] at hsome
+      simp only [DualInterval.inv?] at hsome
+      split at hsome
+      · exact absurd hsome (by simp)
+      · next hnotzero =>
+        simp only [Option.some.injEq] at hsome
+        have hval := evalDualOption1_val_correct e' I d x hx heq
+        -- Denominator is nonzero
+        have hne : evalFunc1 e' x ≠ 0 := by
+          intro heq_zero
+          -- hval has type: Expr.eval (fun x_1 => x) e' ∈ d.val
+          -- need to convert to: evalFunc1 e' x = 0
+          have hval' : evalFunc1 e' x ∈ d.val := hval
+          rw [heq_zero] at hval'
+          simp only [IntervalRat.mem_def] at hval'
+          simp only [IntervalRat.containsZero, not_and, not_le] at hnotzero
+          rcases le_or_gt d.val.lo 0 with hlo | hlo
+          · have hhi_nonneg : (0 : ℚ) ≤ d.val.hi := by
+              have h : (0 : ℝ) ≤ d.val.hi := hval'.2
+              exact_mod_cast h
+            exact absurd (hnotzero hlo) (not_lt.mpr hhi_nonneg)
+          · have hlo_pos : (0 : ℝ) < d.val.lo := by exact_mod_cast hlo
+            exact absurd hval'.1 (not_le.mpr hlo_pos)
+        exact DifferentiableAt.inv (ih d heq) hne
+  | sin e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      exact Real.differentiableAt_sin.comp x (ih d heq)
+  | cos e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      exact Real.differentiableAt_cos.comp x (ih d heq)
+  | exp e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      exact Real.differentiableAt_exp.comp x (ih d heq)
+  | sinh e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      exact (ih d heq).sinh
+  | cosh e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      exact (ih d heq).cosh
+  | tanh _ _ =>
+    unfold evalDualOption1 evalDualOption at hsome
+    contradiction
+  | log e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      rw [heq] at hsome
+      simp only [DualInterval.logOption] at hsome
+      split at hsome
+      · next hpos =>
+        simp only [Option.some.injEq] at hsome
+        have hval := evalDualOption1_val_correct e' I d x hx heq
+        -- The argument is positive, so log is differentiable
+        have hpos_x : 0 < evalFunc1 e' x := by
+          have hval' : evalFunc1 e' x ∈ d.val := hval
+          simp only [IntervalRat.mem_def] at hval'
+          simp only [IntervalRat.isPositive] at hpos
+          have hlo_pos : (0 : ℝ) < d.val.lo := by exact_mod_cast hpos
+          exact lt_of_lt_of_le hlo_pos hval'.1
+        exact Real.differentiableAt_log (ne_of_gt hpos_x) |>.comp x (ih d heq)
+      · exact absurd hsome (by simp)
+  | atan e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      exact (Real.differentiable_arctan _).comp x (ih d heq)
+  | arsinh e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      exact (Real.differentiable_arsinh _).comp x (ih d heq)
+  | atanh _ _ =>
+    -- evalDualOption returns none for atanh, so hsome is a contradiction
+    simp only [evalDualOption1, evalDualOption] at hsome
+    contradiction
+  | sinc e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      -- sinc is differentiable everywhere
+      -- sinc = dslope sin 0, which is differentiable:
+      -- - At x ≠ 0: sinc x = sin x / x is differentiable
+      -- - At x = 0: sinc is differentiable with derivative 0 (from sin's Taylor series)
+      have hsinc_diff : Differentiable ℝ Real.sinc := Real.differentiable_sinc
+      exact hsinc_diff.differentiableAt.comp x (ih d heq)
+  | erf e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      -- erf is differentiable everywhere using FTC
+      -- erf(x) = (2/√π) * ∫₀ˣ exp(-t²) dt
+      -- By FTC, the integral of a continuous function is differentiable
+      have herf_diff : Differentiable ℝ Real.erf := by
+        unfold Real.erf
+        apply Differentiable.const_mul
+        -- Use FTC: for continuous f, ∫ t in a..x, f t is differentiable in x
+        intro y
+        have hcont : Continuous (fun t => Real.exp (-(t^2))) :=
+          Real.continuous_exp.comp (continuous_neg.comp (continuous_pow 2))
+        exact (hcont.integral_hasStrictDerivAt 0 y).hasStrictFDerivAt.differentiableAt
+      exact herf_diff.differentiableAt.comp x (ih d heq)
+  | sqrt e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      rw [heq] at hsome
+      simp only [DualInterval.sqrt?] at hsome
+      split at hsome
+      · next hpos =>
+        simp only [Option.some.injEq] at hsome
+        have hval := evalDualOption1_val_correct e' I d x hx heq
+        -- The argument is positive, so sqrt is differentiable
+        have hpos_x : 0 < evalFunc1 e' x := by
+          have hval' : evalFunc1 e' x ∈ d.val := hval
+          simp only [IntervalRat.mem_def] at hval'
+          simp only [IntervalRat.isPositive] at hpos
+          have hlo_pos : (0 : ℝ) < d.val.lo := by exact_mod_cast hpos
+          exact lt_of_lt_of_le hlo_pos hval'.1
+        -- sqrt is differentiable at nonzero points
+        have hne : evalFunc1 e' x ≠ 0 := ne_of_gt hpos_x
+        exact (Real.hasDerivAt_sqrt hne).differentiableAt.comp x (ih d heq)
+      · exact absurd hsome (by simp)
+  | namedConst _ => exact differentiableAt_const _
+
+/-- The derivative component of evalDualOption is correct when it returns some.
+    For a supported expression (with inv) evaluated at a point x in the interval I,
+    the derivative of the expression lies in the computed derivative interval.
+
+    This extends the AD correctness theorem to expressions with inv. -/
+private theorem evalDual?_der_correct_inv (e' : Expr)
+    (I : IntervalRat) (D : DualInterval) (x : ℝ) (hx : x ∈ I)
+    (hsome : evalDualOption1 (Expr.inv e') I = some D)
+    (ih : ∀ D, evalDualOption1 e' I = some D → deriv (evalFunc1 e') x ∈ D.der) :
+    deriv (evalFunc1 (Expr.inv e')) x ∈ D.der := by
+  unfold evalDualOption1 evalDualOption at hsome
+  cases heq : evalDualOption e' _ with
+  | none => rw [heq] at hsome; exact absurd hsome (by simp)
+  | some d =>
+    rw [heq] at hsome
+    simp only [DualInterval.inv?] at hsome
+    split at hsome
+    · exact absurd hsome (by simp)
+    · next hnotzero =>
+      simp only [Option.some.injEq] at hsome
+      rw [← hsome]
+      -- d(1/f) = -f'/f² = -f' * (1/f)²
+      have hval := evalDualOption1_val_correct e' I d x hx heq
+      have hval' : evalFunc1 e' x ∈ d.val := hval
+      have hne : evalFunc1 e' x ≠ 0 := by
+        intro heq_zero
+        rw [heq_zero] at hval'
+        simp only [IntervalRat.mem_def] at hval'
+        simp only [IntervalRat.containsZero, not_and, not_le] at hnotzero
+        rcases le_or_gt d.val.lo 0 with hlo | hlo
+        · have hhi_nonneg : (0 : ℚ) ≤ d.val.hi := by
+            have h : (0 : ℝ) ≤ d.val.hi := hval'.2
+            exact_mod_cast h
+          exact absurd (hnotzero hlo) (not_lt.mpr hhi_nonneg)
+        · have hlo_pos : (0 : ℝ) < d.val.lo := by exact_mod_cast hlo
+          exact absurd hval'.1 (not_le.mpr hlo_pos)
+      have hd := evalFunc1_differentiableAt_of_evalDualOption e' I d x hx heq
+      -- deriv (1/f) = deriv(f⁻¹ ∘ f) = -(f')/(f²) using chain rule
+      -- We use: deriv (fun y => y⁻¹) (f x) * deriv f x = -(f x)⁻² * f'(x)
+      rw [evalFunc1_inv]
+      -- The goal is: deriv (fun t => (evalFunc1 e' t)⁻¹) x ∈ ...
+      -- First show this equals the composition derivative
+      have heq_fun : (fun t => (evalFunc1 e' t)⁻¹) = (fun y => y⁻¹) ∘ evalFunc1 e' := rfl
+      rw [heq_fun, deriv_comp x (hasDerivAt_inv hne).differentiableAt hd]
+      simp only [hasDerivAt_inv hne |>.deriv]
+      -- Now goal is: -(evalFunc1 e' x ^ 2)⁻¹ * deriv (evalFunc1 e') x ∈ ...
+      -- Which equals: -(deriv f * (1/f)²)
+      have hder := ih d heq
+      -- Construct the nonzero interval
+      let I_nz : IntervalRat.IntervalRatNonzero := ⟨d.val, hnotzero⟩
+      have hinv := IntervalRat.mem_invNonzero (I := I_nz) hval' hne
+      have hinv_sq := IntervalRat.mem_mul hinv hinv
+      have hprod := IntervalRat.mem_mul hder hinv_sq
+      have hneg := IntervalRat.mem_neg hprod
+      -- hneg : -(deriv (evalFunc1 e') x * ((evalFunc1 e' x)⁻¹ * (evalFunc1 e' x)⁻¹)) ∈ ...
+      -- goal : -(evalFunc1 e' x ^ 2)⁻¹ * deriv (evalFunc1 e') x ∈ ...
+      -- These are equal: -(a ^ 2)⁻¹ * b = -(b * (a⁻¹ * a⁻¹))
+      have heq_val : -(evalFunc1 e' x ^ 2)⁻¹ * deriv (evalFunc1 e') x =
+                     -(deriv (evalFunc1 e') x * ((evalFunc1 e' x)⁻¹ * (evalFunc1 e' x)⁻¹)) := by
+        have h1 : (evalFunc1 e' x ^ 2)⁻¹ = (evalFunc1 e' x)⁻¹ * (evalFunc1 e' x)⁻¹ := by
+          rw [sq, mul_inv]
+        rw [h1]
+        ring
+      rw [heq_val]
+      exact hneg
+
+private theorem evalDual?_der_correct_log (e' : Expr)
+    (I : IntervalRat) (D : DualInterval) (x : ℝ) (hx : x ∈ I)
+    (hsome : evalDualOption1 (Expr.log e') I = some D)
+    (ih : ∀ D, evalDualOption1 e' I = some D → deriv (evalFunc1 e') x ∈ D.der) :
+    deriv (evalFunc1 (Expr.log e')) x ∈ D.der := by
+  unfold evalDualOption1 evalDualOption at hsome
+  cases heq : evalDualOption e' _ with
+  | none => rw [heq] at hsome; exact absurd hsome (by simp)
+  | some d =>
+    rw [heq] at hsome
+    simp only [DualInterval.logOption] at hsome
+    split at hsome
+    · next hpos =>
+      simp only [Option.some.injEq] at hsome
+      rw [← hsome]
+      -- d(log f)/dx = f'/f
+      have hval := evalDualOption1_val_correct e' I d x hx heq
+      have hval' : evalFunc1 e' x ∈ d.val := hval
+      have hpos_x : 0 < evalFunc1 e' x := by
+        simp only [IntervalRat.mem_def] at hval'
+        simp only [IntervalRat.isPositive] at hpos
+        have hlo_pos : (0 : ℝ) < d.val.lo := by exact_mod_cast hpos
+        exact lt_of_lt_of_le hlo_pos hval'.1
+      have hne_x : evalFunc1 e' x ≠ 0 := ne_of_gt hpos_x
+      have hd := evalFunc1_differentiableAt_of_evalDualOption e' I d x hx heq
+      rw [evalFunc1_log]
+      have heq_fun : (fun t => Real.log (evalFunc1 e' t)) = Real.log ∘ evalFunc1 e' := rfl
+      rw [heq_fun, deriv_comp x (Real.differentiableAt_log hne_x) hd]
+      simp only [Real.deriv_log (evalFunc1 e' x)]
+      -- Goal: (evalFunc1 e' x)⁻¹ * deriv (evalFunc1 e') x ∈ ...
+      -- From DualInterval.logOption: der' := d.der * invNonzero d.val
+      have hder := ih d heq
+      -- Build the nonzero interval from the positive interval
+      have hnotzero : ¬IntervalRat.containsZero d.val := by
+        simp only [IntervalRat.containsZero, IntervalRat.isPositive] at hpos ⊢
+        intro ⟨hle, _⟩
+        exact absurd hpos (not_lt.mpr hle)
+      let I_nz : IntervalRat.IntervalRatNonzero := ⟨d.val, hnotzero⟩
+      have hinv := IntervalRat.mem_invNonzero (I := I_nz) hval' hne_x
+      -- Need to show: (evalFunc1 e' x)⁻¹ * deriv (evalFunc1 e') x ∈ d.der * invNonzero d.val
+      -- But mem_mul expects the arguments in different order, so use commutativity
+      have hmul := IntervalRat.mem_mul hder hinv
+      convert hmul using 1
+      ring
+    · exact absurd hsome (by simp)
+
+private theorem evalDual?_der_correct_erf (e' : Expr)
+    (I : IntervalRat) (D : DualInterval) (x : ℝ) (hx : x ∈ I)
+    (hsome : evalDualOption1 (Expr.erf e') I = some D)
+    (ih : ∀ D, evalDualOption1 e' I = some D → deriv (evalFunc1 e') x ∈ D.der) :
+    deriv (evalFunc1 (Expr.erf e')) x ∈ D.der := by
+  unfold evalDualOption1 evalDualOption at hsome
+  cases heq : evalDualOption e' _ with
+  | none => rw [heq] at hsome; exact absurd hsome (by simp)
+  | some d =>
+    rw [heq, Option.some.injEq] at hsome
+    rw [← hsome]
+    simp only [DualInterval.erf]
+    -- The derivative of erf ∘ f is (2/√π) * exp(-f(x)²) * f'(x)
+    have hd_inner := evalFunc1_differentiableAt_of_evalDualOption e' I d x hx heq
+    have herf_diff : Differentiable ℝ Real.erf := by
+      unfold Real.erf
+      apply Differentiable.const_mul
+      intro y
+      have hcont : Continuous (fun t => Real.exp (-(t^2))) :=
+        Real.continuous_exp.comp (continuous_neg.comp (continuous_pow 2))
+      exact (hcont.integral_hasStrictDerivAt 0 y).hasStrictFDerivAt.differentiableAt
+    have heq_comp : (fun t => Real.erf (evalFunc1 e' t)) = Real.erf ∘ evalFunc1 e' := rfl
+    rw [evalFunc1_erf, heq_comp, deriv_comp x herf_diff.differentiableAt hd_inner]
+    -- deriv erf y = (2/√π) * exp(-y²) by FTC
+    have hderiv_erf : ∀ y, deriv Real.erf y = (2 / Real.sqrt Real.pi) * Real.exp (-(y^2)) := by
+      intro y
+      unfold Real.erf
+      rw [deriv_const_mul]
+      · have hcont : Continuous (fun t => Real.exp (-(t^2))) :=
+          Real.continuous_exp.comp (continuous_neg.comp (continuous_pow 2))
+        rw [(hcont.integral_hasStrictDerivAt 0 y).hasDerivAt.deriv]
+      · have hcont : Continuous (fun t => Real.exp (-(t^2))) :=
+          Real.continuous_exp.comp (continuous_neg.comp (continuous_pow 2))
+        exact (hcont.integral_hasStrictDerivAt 0 y).hasStrictFDerivAt.differentiableAt
+    rw [hderiv_erf]
+    -- Now goal: (2/√π) * exp(-(f(x))²) * f'(x) ∈ twoDivSqrtPi * expInterval(-val²) * der
+    -- Factor: (2/√π) ∈ twoDivSqrtPi
+    -- 2/√π ≈ 1.1284, twoDivSqrtPi = [1.128, 1.129]
+    have hfactor : 2 / Real.sqrt Real.pi ∈ DualInterval.twoDivSqrtPi := by
+      simp only [DualInterval.twoDivSqrtPi, IntervalRat.mem_def]
+      -- From π bounds: 3.1415 < π < 3.1416 (Real.pi_gt_d4, Real.pi_lt_d4)
+      -- So √π is between √3.1415 ≈ 1.7724 and √3.1416 ≈ 1.7724
+      -- Thus 2/√π is between 2/1.7725 ≈ 1.1283 and 2/1.7723 ≈ 1.1285
+      have hpi_lo : (3.1415 : ℝ) < Real.pi := Real.pi_gt_d4
+      have hpi_hi : Real.pi < (3.1416 : ℝ) := Real.pi_lt_d4
+      have hsqrt_lo : (1.7724 : ℝ) < Real.sqrt Real.pi := by
+        have h1 : (1.7724 : ℝ) ^ 2 < Real.pi := by
+          have : (1.7724 : ℝ) ^ 2 = 3.14140176 := by ring
+          linarith
+        have h2 : (0 : ℝ) ≤ 1.7724 := by norm_num
+        have h3 : (0 : ℝ) ≤ 1.7724 ^ 2 := by positivity
+        calc (1.7724 : ℝ) = Real.sqrt (1.7724 ^ 2) := (Real.sqrt_sq h2).symm
+          _ < Real.sqrt Real.pi := Real.sqrt_lt_sqrt h3 h1
+      have hsqrt_hi : Real.sqrt Real.pi < (1.7725 : ℝ) := by
+        have h1 : Real.pi < (1.7725 : ℝ) ^ 2 := by
+          have : (1.7725 : ℝ) ^ 2 = 3.14175625 := by ring
+          linarith
+        have hpi_pos : (0 : ℝ) < Real.pi := Real.pi_pos
+        rw [← Real.sqrt_sq (le_of_lt (by norm_num : (0 : ℝ) < 1.7725))]
+        exact Real.sqrt_lt_sqrt (le_of_lt hpi_pos) h1
+      constructor
+      · -- Goal: ↑(1128 / 1000) ≤ 2 / √π
+        have h1 : ((1128 / 1000 : ℚ) : ℝ) < 2 / 1.7725 := by norm_num
+        have h2 : (2 : ℝ) / 1.7725 < 2 / Real.sqrt Real.pi := by
+          apply div_lt_div_of_pos_left (by norm_num : (0 : ℝ) < 2)
+          · exact Real.sqrt_pos.mpr Real.pi_pos
+          · exact hsqrt_hi
+        exact le_of_lt (lt_trans h1 h2)
+      · -- Goal: 2 / √π ≤ ↑(1129 / 1000)
+        have h1 : 2 / Real.sqrt Real.pi < (2 : ℝ) / 1.7724 := by
+          apply div_lt_div_of_pos_left (by norm_num : (0 : ℝ) < 2)
+          · norm_num
+          · exact hsqrt_lo
+        have h2 : (2 : ℝ) / 1.7724 < ((1129 / 1000 : ℚ) : ℝ) := by norm_num
+        exact le_of_lt (lt_trans h1 h2)
+    -- exp(-(f(x))²) ∈ expInterval(-val²)
+    have hval := evalDualOption1_val_correct e' I d x hx heq
+    have hval_sq := IntervalRat.mem_mul hval hval
+    have hneg_val_sq := IntervalRat.mem_neg hval_sq
+    have hexp := IntervalRat.mem_expInterval hneg_val_sq
+    -- f'(x) ∈ d.der
+    have hder := ih d heq
+    -- Combine: (2/√π) * exp(-f(x)²) ∈ twoDivSqrtPi * expInterval(-val²)
+    have hprod1 := IntervalRat.mem_mul hfactor hexp
+    -- Full product: ((2/√π) * exp(-f(x)²)) * f'(x) ∈ (twoDivSqrtPi * expInterval(-val²)) * der
+    have hprod2 := IntervalRat.mem_mul hprod1 hder
+    convert hprod2 using 1
+    ring_nf
+
+private theorem evalDual?_der_correct_sqrt (e' : Expr)
+    (I : IntervalRat) (D : DualInterval) (x : ℝ) (hx : x ∈ I)
+    (hsome : evalDualOption1 (Expr.sqrt e') I = some D)
+    (ih : ∀ D, evalDualOption1 e' I = some D → deriv (evalFunc1 e') x ∈ D.der) :
+    deriv (evalFunc1 (Expr.sqrt e')) x ∈ D.der := by
+  unfold evalDualOption1 evalDualOption at hsome
+  cases heq : evalDualOption e' _ with
+  | none => rw [heq] at hsome; exact absurd hsome (by simp)
+  | some d =>
+    rw [heq] at hsome
+    simp only [DualInterval.sqrt?] at hsome
+    split at hsome
+    · next hpos =>
+      simp only [Option.some.injEq] at hsome
+      rw [← hsome]
+      -- d(sqrt f)/dx = f'/(2*sqrt(f))
+      have hval := evalDualOption1_val_correct e' I d x hx heq
+      have hval' : evalFunc1 e' x ∈ d.val := hval
+      have hpos_x : 0 < evalFunc1 e' x := by
+        simp only [IntervalRat.mem_def] at hval'
+        simp only [IntervalRat.isPositive] at hpos
+        have hlo_pos : (0 : ℝ) < d.val.lo := by exact_mod_cast hpos
+        exact lt_of_lt_of_le hlo_pos hval'.1
+      have hne_x : evalFunc1 e' x ≠ 0 := ne_of_gt hpos_x
+      have hd := evalFunc1_differentiableAt_of_evalDualOption e' I d x hx heq
+      rw [evalFunc1_sqrt]
+      have heq_fun : (fun t => Real.sqrt (evalFunc1 e' t)) = Real.sqrt ∘ evalFunc1 e' := rfl
+      rw [heq_fun, deriv_comp x (Real.hasDerivAt_sqrt hne_x).differentiableAt hd]
+      rw [(Real.hasDerivAt_sqrt hne_x).deriv]
+      -- Goal: deriv (evalFunc1 e') x / (2 * √(evalFunc1 e' x)) ∈ d.der * sqrtDerivCoefBound
+      have hder := ih d heq
+      -- Show 1/(2*sqrt(f(x))) is in sqrtDerivCoefBound
+      have hlo_le_val : (d.val.lo : ℝ) ≤ evalFunc1 e' x := by
+        simp only [IntervalRat.mem_def] at hval'
+        exact hval'.1
+      have hlo_pos_real : (0 : ℝ) < (d.val.lo : ℚ) := by
+        simp only [IntervalRat.isPositive] at hpos
+        exact_mod_cast hpos
+      -- Use our sqrtDerivCoef_bound theorem
+      have hcoef_bound := DualInterval.sqrtDerivCoef_bound hlo_pos_real hlo_le_val
+      -- The coefficient 1/(2*sqrt(f(x))) is positive and bounded
+      have hcoef_pos : 0 ≤ 1 / (2 * Real.sqrt (evalFunc1 e' x)) := by
+        apply div_nonneg (by norm_num)
+        apply mul_nonneg (by norm_num)
+        exact le_of_lt (Real.sqrt_pos.mpr hpos_x)
+      have hcoef_mem : 1 / (2 * Real.sqrt (evalFunc1 e' x)) ∈ DualInterval.sqrtDerivCoefBound
+        d.val.lo hpos := by
+        simp only [DualInterval.sqrtDerivCoefBound, IntervalRat.mem_def]
+        split
+        · -- d.val.lo ≤ 1
+          next hle_one =>
+          constructor
+          · simp only [Rat.cast_zero]
+            exact hcoef_pos
+          · -- Show 1/(2*sqrt(f(x))) ≤ 1/(2*lo)
+            have hle_real : (d.val.lo : ℝ) ≤ 1 := by exact_mod_cast hle_one
+            have h1 : max (1 / (2 * (d.val.lo : ℝ))) (1 / 2) = 1 / (2 * (d.val.lo : ℝ)) := by
+              apply max_eq_left
+              -- 1/2 ≤ 1/(2*lo) when lo ≤ 1
+              apply div_le_div_of_nonneg_left (by norm_num) (by positivity)
+              calc 2 * (d.val.lo : ℝ) ≤ 2 * 1 := by nlinarith
+                _ = 2 := by ring
+            calc 1 / (2 * Real.sqrt (evalFunc1 e' x))
+                ≤ max (1 / (2 * (d.val.lo : ℝ))) (1 / 2) := hcoef_bound
+              _ = 1 / (2 * (d.val.lo : ℝ)) := h1
+              _ = ↑(1 / (2 * d.val.lo)) := by push_cast; ring
+        · -- d.val.lo > 1
+          next hgt_one =>
+          push Not at hgt_one
+          constructor
+          · simp only [Rat.cast_zero]
+            exact hcoef_pos
+          · -- Show 1/(2*sqrt(f(x))) ≤ 1/2
+            have hgt_real : 1 < (d.val.lo : ℝ) := by exact_mod_cast hgt_one
+            have h1 : max (1 / (2 * (d.val.lo : ℝ))) (1 / 2) = 1 / 2 := by
+              apply max_eq_right
+              -- 1/(2*lo) ≤ 1/2 when lo > 1
+              apply div_le_div_of_nonneg_left (by norm_num) (by norm_num : (0:ℝ) < 2)
+              calc (2 : ℝ) = 2 * 1 := by ring
+                _ ≤ 2 * d.val.lo := by nlinarith
+            calc 1 / (2 * Real.sqrt (evalFunc1 e' x))
+                ≤ max (1 / (2 * (d.val.lo : ℝ))) (1 / 2) := hcoef_bound
+              _ = 1 / 2 := h1
+              _ = ↑(1 / 2 : ℚ) := by norm_num
+      -- Combine: f'(x)/(2*sqrt(f(x))) = f'(x) * (1/(2*sqrt(f(x)))) ∈ d.der * sqrtDerivCoefBound
+      have hmul := IntervalRat.mem_mul hder hcoef_mem
+      convert hmul using 1
+      field_simp [ne_of_gt (Real.sqrt_pos.mpr hpos_x)]
+    · exact absurd hsome (by simp)
+
+theorem evalDualOption_der_correct (e : Expr)
+    (I : IntervalRat) (D : DualInterval) (x : ℝ) (hx : x ∈ I)
+    (hsome : evalDualOption1 e I = some D) :
+    deriv (evalFunc1 e) x ∈ D.der := by
+  induction e generalizing D with
+  | const q =>
+    simp only [evalDualOption1, evalDualOption, Option.some.injEq] at hsome
+    rw [← hsome]
+    simp only [evalFunc1_const, deriv_const, DualInterval.const]
+    convert IntervalRat.mem_singleton 0 using 1
+    norm_cast
+  | var _ =>
+    simp only [evalDualOption1, evalDualOption, Option.some.injEq] at hsome
+    rw [← hsome]
+    simp only [evalFunc1_var, deriv_id, DualInterval.varActive]
+    convert IntervalRat.mem_singleton 1 using 1
+    norm_cast
+  | add e₁ e₂ ih₁ ih₂ =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq₁ : evalDualOption e₁ _ with
+    | none => rw [heq₁] at hsome; exact absurd hsome (by simp)
+    | some d₁ =>
+      cases heq₂ : evalDualOption e₂ _ with
+      | none => rw [heq₁, heq₂] at hsome; exact absurd hsome (by simp)
+      | some d₂ =>
+        rw [heq₁, heq₂, Option.some.injEq] at hsome
+        rw [← hsome]
+        simp only [DualInterval.add]
+        have hd₁ := evalFunc1_differentiableAt_of_evalDualOption e₁ I d₁ x hx heq₁
+        have hd₂ := evalFunc1_differentiableAt_of_evalDualOption e₂ I d₂ x hx heq₂
+        simp only [evalFunc1_add_pi, deriv_add hd₁ hd₂]
+        exact IntervalRat.mem_add (ih₁ d₁ heq₁) (ih₂ d₂ heq₂)
+  | mul e₁ e₂ ih₁ ih₂ =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq₁ : evalDualOption e₁ _ with
+    | none => rw [heq₁] at hsome; exact absurd hsome (by simp)
+    | some d₁ =>
+      cases heq₂ : evalDualOption e₂ _ with
+      | none => rw [heq₁, heq₂] at hsome; exact absurd hsome (by simp)
+      | some d₂ =>
+        rw [heq₁, heq₂, Option.some.injEq] at hsome
+        rw [← hsome]
+        simp only [DualInterval.mul]
+        have hd₁ := evalFunc1_differentiableAt_of_evalDualOption e₁ I d₁ x hx heq₁
+        have hd₂ := evalFunc1_differentiableAt_of_evalDualOption e₂ I d₂ x hx heq₂
+        simp only [evalFunc1_mul_pi, deriv_mul hd₁ hd₂]
+        have hval₁ := evalDualOption1_val_correct e₁ I d₁ x hx heq₁
+        have hval₂ := evalDualOption1_val_correct e₂ I d₂ x hx heq₂
+        exact IntervalRat.mem_add (IntervalRat.mem_mul (ih₁ d₁ heq₁) hval₂)
+                                  (IntervalRat.mem_mul hval₁ (ih₂ d₂ heq₂))
+  | neg e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      rw [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [DualInterval.neg, evalFunc1_neg_pi, deriv.neg]
+      exact IntervalRat.mem_neg (ih d heq)
+  | inv e' ih => exact evalDual?_der_correct_inv e' I D x hx hsome ih
+  | sin e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      rw [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [DualInterval.sin]
+      have hd := evalFunc1_differentiableAt_of_evalDualOption e' I d x hx heq
+      rw [evalFunc1_sin, deriv_sin hd]
+      exact IntervalRat.mem_mul (cos_mem_cosInterval_of_any _ _) (ih d heq)
+  | cos e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      rw [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [DualInterval.cos]
+      have hd := evalFunc1_differentiableAt_of_evalDualOption e' I d x hx heq
+      rw [evalFunc1_cos, deriv_cos hd]
+      exact IntervalRat.mem_mul (neg_sin_mem_neg_sinInterval _ _) (ih d heq)
+  | exp e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      rw [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [DualInterval.exp]
+      have hd := evalFunc1_differentiableAt_of_evalDualOption e' I d x hx heq
+      rw [evalFunc1_exp, deriv_exp hd]
+      have hval := evalDualOption1_val_correct e' I d x hx heq
+      have hexp := IntervalRat.mem_expInterval hval
+      exact IntervalRat.mem_mul hexp (ih d heq)
+  | sinh e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      rw [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [DualInterval.sinh, coshInterval]
+      have hd := evalFunc1_differentiableAt_of_evalDualOption e' I d x hx heq
+      change deriv (fun t => Real.sinh (evalFunc1 e' t)) x ∈
+        IntervalRat.mul (IntervalRat.coshComputable d.val 10) d.der
+      have hder :
+          HasDerivAt (fun t => Real.sinh (evalFunc1 e' t))
+            (Real.cosh (evalFunc1 e' x) * deriv (evalFunc1 e') x) x :=
+        (Real.hasDerivAt_sinh (evalFunc1 e' x)).comp x hd.hasDerivAt
+      rw [hder.deriv]
+      have hval := evalDualOption1_val_correct e' I d x hx heq
+      have hcosh := IntervalRat.mem_coshComputable hval 10
+      exact IntervalRat.mem_mul hcosh (ih d heq)
+  | cosh e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      rw [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [DualInterval.cosh, sinhInterval]
+      have hd := evalFunc1_differentiableAt_of_evalDualOption e' I d x hx heq
+      change deriv (fun t => Real.cosh (evalFunc1 e' t)) x ∈
+        IntervalRat.mul (IntervalRat.sinhComputable d.val 10) d.der
+      have hder :
+          HasDerivAt (fun t => Real.cosh (evalFunc1 e' t))
+            (Real.sinh (evalFunc1 e' x) * deriv (evalFunc1 e') x) x :=
+        (Real.hasDerivAt_cosh (evalFunc1 e' x)).comp x hd.hasDerivAt
+      rw [hder.deriv]
+      have hval := evalDualOption1_val_correct e' I d x hx heq
+      have hsinh := IntervalRat.mem_sinhComputable hval 10
+      exact IntervalRat.mem_mul hsinh (ih d heq)
+  | tanh _ _ =>
+    unfold evalDualOption1 evalDualOption at hsome
+    contradiction
+  | log e' ih => exact evalDual?_der_correct_log e' I D x hx hsome ih
+  | atan e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      rw [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [DualInterval.atan]
+      have hd := evalFunc1_differentiableAt_of_evalDualOption e' I d x hx heq
+      rw [evalFunc1_atan]
+      -- deriv (arctan ∘ f) x = (1 / (1 + f(x)²)) * f'(x)
+      have heq_deriv := HasDerivAt.arctan (hd.hasDerivAt)
+      rw [heq_deriv.deriv, mul_comm]
+      -- The factor 1/(1+f(x)²) is in unitInterval
+      have hfactor := DualInterval.arctan_deriv_factor_mem_unitInterval (evalFunc1 e' x)
+      exact IntervalRat.mem_mul (ih d heq) hfactor
+  | arsinh e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      rw [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [DualInterval.arsinh]
+      have hd := evalFunc1_differentiableAt_of_evalDualOption e' I d x hx heq
+      rw [evalFunc1_arsinh]
+      -- deriv (arsinh ∘ f) x = (√(1 + f(x)²))⁻¹ • f'(x)
+      have heq_deriv := HasDerivAt.arsinh (hd.hasDerivAt)
+      rw [heq_deriv.deriv, smul_eq_mul, mul_comm]
+      -- The factor 1/√(1+f(x)²) is in unitInterval
+      have hfactor := DualInterval.arsinh_deriv_factor_mem_unitInterval (evalFunc1 e' x)
+      exact IntervalRat.mem_mul (ih d heq) hfactor
+  | atanh _ _ =>
+    -- evalDualOption returns none for atanh, so hsome is a contradiction
+    simp only [evalDualOption1, evalDualOption] at hsome
+    contradiction
+  | sinc e' ih =>
+    unfold evalDualOption1 evalDualOption at hsome
+    cases heq : evalDualOption e' _ with
+    | none => rw [heq] at hsome; exact absurd hsome (by simp)
+    | some d =>
+      rw [heq, Option.some.injEq] at hsome
+      rw [← hsome]
+      simp only [DualInterval.sinc]
+      -- The derivative of sinc ∘ f is sinc'(f(x)) * f'(x)
+      -- sinc'(y) ∈ [-1, 1] for all y, and f'(x) ∈ d.der
+      have hd_inner := evalFunc1_differentiableAt_of_evalDualOption e' I d x hx heq
+      have heq_comp : (fun t => Real.sinc (evalFunc1 e' t)) = Real.sinc ∘ evalFunc1 e' := rfl
+      rw [evalFunc1_sinc, heq_comp, deriv_comp x Real.differentiable_sinc.differentiableAt hd_inner]
+      -- deriv sinc (evalFunc1 e' x) * deriv (evalFunc1 e') x ∈ sincDerivBound * d.der
+      have hsinc_bound : deriv Real.sinc (evalFunc1 e' x) ∈ DualInterval.sincDerivBound := by
+        simp only [DualInterval.sincDerivBound, IntervalRat.mem_def, Rat.cast_neg, Rat.cast_one]
+        exact Real.deriv_sinc_mem_Icc (evalFunc1 e' x)
+      have hder := ih d heq
+      exact IntervalRat.mem_mul hsinc_bound hder
+  | erf e' ih => exact evalDual?_der_correct_erf e' I D x hx hsome ih
+  | sqrt e' ih => exact evalDual?_der_correct_sqrt e' I D x hx hsome ih
+  | namedConst c =>
+    simp only [evalDualOption1, evalDualOption, Option.some.injEq] at hsome
+    rw [← hsome]
+    simp only [evalFunc1_namedConst, deriv_const, DualInterval.ofMathConst]
+    convert IntervalRat.mem_singleton 0 using 1
+    norm_cast
+
+/-- Combined correctness theorem for evalDualOption1 -/
+theorem evalDualOption1_correct (e : Expr)
+    (I : IntervalRat) (D : DualInterval) (x : ℝ) (hx : x ∈ I)
+    (hsome : evalDualOption1 e I = some D) :
+    Expr.eval (fun _ => x) e ∈ D.val ∧ deriv (evalFunc1 e) x ∈ D.der :=
+  ⟨evalDualOption1_val_correct e I D x hx hsome, evalDualOption_der_correct e I D x hx hsome⟩
+
+end LeanCert.Engine
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2025 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# High-Performance Dyadic Interval Evaluator
+
+This evaluator replaces `Rat` with `Dyadic` to prevent denominator explosion.
+It is designed for complex expressions where the standard evaluator becomes slow.
+
+## Main definitions
+
+* `DyadicConfig` - Configuration for precision and Taylor depth
+* `LeanCert.Internal.Dyadic.evalUnchecked` - Dyadic interval evaluator for expressions
+* `evalIntervalDyadic_correct` - Correctness theorem
+
+## Performance
+
+In v1.0, every `Rat` multiplication required GCD normalization. For deep
+expressions (e.g., Taylor series with 20+ terms, or optimization with 100+
+iterations), denominators grow exponentially, causing timeouts.
+
+In v1.1, Dyadic arithmetic uses bit-shifts instead of GCD. With `roundOut`,
+we can enforce a maximum precision after each operation, keeping computation
+bounded regardless of expression depth.
+
+## Example
+
+Consider computing `sin(sin(sin(x)))` with 15-term Taylor series:
+- v1.0 (Rat): ~500ms per call, denominators grow to millions of digits
+- v1.1 (Dyadic): ~5ms per call, precision stays at 53 bits
+
+## Design notes
+
+For transcendental functions (sin, cos, exp), we delegate to the existing
+`IntervalRat` implementation with Taylor series, then convert the result
+to `IntervalDyadic` with outward rounding. This reuses verified code while
+gaining the performance benefits of Dyadic for polynomial operations.
+-/
+
+/-! ### Configuration -/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+/-- Configuration for Dyadic interval evaluation.
+
+* `precision` - Minimum exponent for outward rounding. A value of -53 gives
+  IEEE double-like precision (~15 decimal digits). Use -100 for higher precision.
+* `taylorDepth` - Number of Taylor terms for transcendental functions.
+-/
+structure DyadicConfig where
+  /-- Minimum exponent (higher = coarser). -53 ≈ IEEE double precision. -/
+  precision : Int := -53
+  /-- Number of Taylor series terms for transcendental functions -/
+  taylorDepth : Nat := 10
+  deriving Repr, DecidableEq
+
+/-- Default configuration with IEEE double-like precision -/
+instance : Inhabited DyadicConfig := ⟨{}⟩
+
+/-- High-precision configuration for critical calculations -/
+def DyadicConfig.highPrecision : DyadicConfig :=
+  { precision := -100, taylorDepth := 20 }
+
+/-- Fast configuration for rapid evaluation (lower precision) -/
+def DyadicConfig.fast : DyadicConfig :=
+  { precision := -30, taylorDepth := 8 }
+
+/-! ### Variable Environment -/
+
+/-- Variable assignment as Dyadic intervals -/
+abbrev IntervalDyadicEnv := Nat → IntervalDyadic
+
+/-- Convert a rational interval environment to Dyadic -/
+def toDyadicEnv (ρ : IntervalEnv) (prec : Int := -53) : IntervalDyadicEnv :=
+  fun i => IntervalDyadic.ofIntervalRat (ρ i) prec
+
+/-! ### Transcendental Function Wrappers -/
+
+/-- Compute sin interval using rational Taylor series, convert to Dyadic -/
+def sinIntervalDyadic (I : IntervalDyadic) (cfg : DyadicConfig) : IntervalDyadic :=
+  let Irat := I.toIntervalRat
+  let result := IntervalRat.sinComputable Irat cfg.taylorDepth
+  IntervalDyadic.ofIntervalRat result cfg.precision
+
+/-- Compute cos interval using rational Taylor series, convert to Dyadic -/
+def cosIntervalDyadic (I : IntervalDyadic) (cfg : DyadicConfig) : IntervalDyadic :=
+  let Irat := I.toIntervalRat
+  let result := IntervalRat.cosComputable Irat cfg.taylorDepth
+  IntervalDyadic.ofIntervalRat result cfg.precision
+
+/-- Compute exp interval using rational Taylor series, convert to Dyadic -/
+def expIntervalDyadic (I : IntervalDyadic) (cfg : DyadicConfig) : IntervalDyadic :=
+  let Irat := I.toIntervalRat
+  let result := IntervalRat.expComputable Irat cfg.taylorDepth
+  IntervalDyadic.ofIntervalRat result cfg.precision
+
+/-- Compute sinh interval using rational Taylor series, convert to Dyadic -/
+def sinhIntervalDyadic (I : IntervalDyadic) (cfg : DyadicConfig) : IntervalDyadic :=
+  let Irat := I.toIntervalRat
+  let result := IntervalRat.sinhComputable Irat cfg.taylorDepth
+  IntervalDyadic.ofIntervalRat result cfg.precision
+
+/-- Compute cosh interval using rational Taylor series, convert to Dyadic -/
+def coshIntervalDyadic (I : IntervalDyadic) (cfg : DyadicConfig) : IntervalDyadic :=
+  let Irat := I.toIntervalRat
+  let result := IntervalRat.coshComputable Irat cfg.taylorDepth
+  IntervalDyadic.ofIntervalRat result cfg.precision
+
+/-- atan interval: global bound [-2, 2] -/
+def atanIntervalDyadic (_I : IntervalDyadic) (_cfg : DyadicConfig) : IntervalDyadic :=
+  let neg2 := Core.Dyadic.ofInt (-2)
+  let pos2 := Core.Dyadic.ofInt 2
+  ⟨neg2, pos2, by rw [Dyadic.toRat_ofInt, Dyadic.toRat_ofInt]; norm_num⟩
+
+/-- tanh interval: global bound [-1, 1] -/
+def tanhIntervalDyadic (_I : IntervalDyadic) (_cfg : DyadicConfig) : IntervalDyadic :=
+  let neg1 := Core.Dyadic.ofInt (-1)
+  let pos1 := Core.Dyadic.ofInt 1
+  ⟨neg1, pos1, by rw [Dyadic.toRat_ofInt, Dyadic.toRat_ofInt]; norm_num⟩
+
+/-- arsinh interval: wide box bound via rational arsinhInterval -/
+def arsinhIntervalDyadic (I : IntervalDyadic) (cfg : DyadicConfig) : IntervalDyadic :=
+  let Irat := I.toIntervalRat
+  let result := arsinhInterval Irat
+  IntervalDyadic.ofIntervalRat result cfg.precision
+
+/-- atanh interval: computable Taylor series via rational atanhComputable -/
+def atanhIntervalDyadic (I : IntervalDyadic) (cfg : DyadicConfig) : IntervalDyadic :=
+  let Irat := I.toIntervalRat
+  let result := IntervalRat.atanhComputable Irat cfg.taylorDepth
+  IntervalDyadic.ofIntervalRat result cfg.precision
+
+/-- sinc interval: global bound [-1, 1] -/
+def sincIntervalDyadic (_I : IntervalDyadic) (_cfg : DyadicConfig) : IntervalDyadic :=
+  let neg1 := Core.Dyadic.ofInt (-1)
+  let pos1 := Core.Dyadic.ofInt 1
+  ⟨neg1, pos1, by rw [Dyadic.toRat_ofInt, Dyadic.toRat_ofInt]; norm_num⟩
+
+/-- erf interval: global bound [-1, 1] -/
+def erfIntervalDyadic (_I : IntervalDyadic) (_cfg : DyadicConfig) : IntervalDyadic :=
+  let neg1 := Core.Dyadic.ofInt (-1)
+  let pos1 := Core.Dyadic.ofInt 1
+  ⟨neg1, pos1, by rw [Dyadic.toRat_ofInt, Dyadic.toRat_ofInt]; norm_num⟩
+
+/-- Compute inv interval: convert to Rat, use invInterval, convert back to Dyadic -/
+def invIntervalDyadic (I : IntervalDyadic) (cfg : DyadicConfig) : IntervalDyadic :=
+  let Irat := I.toIntervalRat
+  let result := invInterval Irat
+  IntervalDyadic.ofIntervalRat result cfg.precision
+
+/-- sqrt interval: uses conservative bound [0, max(hi, 1)] -/
+def sqrtIntervalDyadic (I : IntervalDyadic) (cfg : DyadicConfig) : IntervalDyadic :=
+  IntervalDyadic.sqrt I cfg.precision
+
+/-- log interval: conservative global bound.
+    For any x > 0, log(x) ∈ (-∞, ∞), but we use a finite interval.
+    For x ∈ [lo, hi] with lo > 0:
+    - log is monotone, so log(x) ∈ [log(lo), log(hi)]
+    - Legacy total-evaluator fallback; strict callers reject this domain -/
+def logIntervalDyadic (I : IntervalDyadic) (cfg : DyadicConfig) : IntervalDyadic :=
+  -- Compute log using Taylor series via atanh reduction
+  -- Convert to rational, compute, convert back with outward rounding
+  let IRat := I.toIntervalRat
+  if IRat.lo > 0 then
+    let result := IntervalRat.logComputable IRat cfg.taylorDepth
+    IntervalDyadic.ofIntervalRat result cfg.precision
+  else
+    -- Legacy heuristic sentinel for invalid input. It is not a sound log
+    -- enclosure; `evalIntervalDyadicChecked` rejects this branch.
+    ⟨Core.Dyadic.ofInt (-1000), Core.Dyadic.ofInt 1000, by simp [Dyadic.toRat_ofInt]⟩
+
+/-- Real power from a cached logarithm interval.
+
+    If `Real.log x ∈ logBase`, this computes an interval for
+    `x ^ (p : ℝ)` as `exp(p * log x)` without recomputing `log x`.
+    This is the hot-path primitive for cached finite sums with many terms of
+    the form `x ^ q_k`. -/
+def rpowFromCachedLogDyadic (logBase : IntervalDyadic) (p : ℚ)
+    (cfg : DyadicConfig) : IntervalDyadic :=
+  let pInterval := IntervalDyadic.ofIntervalRat (IntervalRat.singleton p) cfg.precision
+  let pLogBase := (IntervalDyadic.mul pInterval logBase).roundOut cfg.precision
+  expIntervalDyadic pLogBase cfg
+
+/-- Real power x^p for x > 0 and rational p, computed via exp(p * log(x)).
+
+    For x ∈ [lo, hi] with lo > 0 and rational exponent p:
+    - x^p = exp(p * log(x))
+    - We compute log(x), multiply by p, then apply exp
+
+    This is the key operation for BKLNW-style sums where terms are x^(1/k - 1/3). -/
+def rpowIntervalDyadic (base : IntervalDyadic) (p : ℚ) (cfg : DyadicConfig) : IntervalDyadic :=
+  -- x^p = exp(p * log(x)) for x > 0
+  let logBase := logIntervalDyadic base cfg
+  rpowFromCachedLogDyadic logBase p cfg
+
+/-- Correctness of `rpowFromCachedLogDyadic`: a cached interval containing
+    `Real.log x` is enough to enclose `x ^ p`. -/
+theorem mem_rpowFromCachedLogDyadic {x : ℝ} {logBase : IntervalDyadic}
+    (hlog : Real.log x ∈ logBase) (hx_pos : 0 < x)
+    (p : ℚ) (cfg : DyadicConfig) (hprec : cfg.precision ≤ 0 := by norm_num) :
+    x ^ (p : ℝ) ∈ rpowFromCachedLogDyadic logBase p cfg := by
+  simp only [rpowFromCachedLogDyadic]
+  rw [Real.rpow_def_of_pos hx_pos, mul_comm]
+  have hp_mem : (p : ℝ) ∈
+      IntervalDyadic.ofIntervalRat (IntervalRat.singleton p) cfg.precision := by
+    apply IntervalDyadic.mem_ofIntervalRat
+    · exact IntervalRat.mem_singleton p
+    · exact hprec
+  have hmul := IntervalDyadic.mem_mul hp_mem hlog
+  have hmul_rounded := IntervalDyadic.roundOut_contains hmul cfg.precision
+  simp only [expIntervalDyadic]
+  have hrat := IntervalDyadic.mem_toIntervalRat.mp hmul_rounded
+  have hexp := IntervalRat.mem_expComputable hrat cfg.taylorDepth
+  exact IntervalDyadic.mem_ofIntervalRat hexp cfg.precision hprec
+
+/-- Correctness of rpowIntervalDyadic: if x ∈ base and base.lo > 0, then x^p ∈ result -/
+theorem mem_rpowIntervalDyadic {x : ℝ} {base : IntervalDyadic} (hx : x ∈ base)
+    (hpos : base.toIntervalRat.lo > 0) (p : ℚ) (cfg : DyadicConfig)
+    (hprec : cfg.precision ≤ 0 := by norm_num) :
+    x ^ (p : ℝ) ∈ rpowIntervalDyadic base p cfg := by
+  simp only [rpowIntervalDyadic]
+  have hx_pos : 0 < x := by
+    have hlo := hx.1
+    have hlo_pos : (0 : ℝ) < base.lo.toRat := by exact_mod_cast hpos
+    linarith
+  have hlog_mem : Real.log x ∈ logIntervalDyadic base cfg := by
+    simp only [logIntervalDyadic, hpos, ↓reduceIte]
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp hx
+    have hlog := IntervalRat.mem_logComputable hrat hpos cfg.taylorDepth
+    exact IntervalDyadic.mem_ofIntervalRat hlog cfg.precision hprec
+  exact mem_rpowFromCachedLogDyadic hlog_mem hx_pos p cfg hprec
+
+/-! ### Main Evaluator -/
+
+end LeanCert.Engine
+
+namespace LeanCert.Internal.Dyadic
+
+open LeanCert.Core LeanCert.Engine
+
+/-- High-performance Dyadic interval evaluator.
+
+This is the core function for v1.1. It evaluates expressions using Dyadic
+arithmetic for polynomial operations (add, mul, neg) and delegates to
+rational Taylor series for transcendentals.
+
+Returns an interval guaranteed to contain all possible values of the expression
+when `ExprSupportedCore` holds. For other expressions, it computes conservative
+fallbacks (e.g., inv/log), but the core correctness theorem does not apply. -/
+def evalUnchecked (e : Expr) (ρ : IntervalDyadicEnv) (cfg : DyadicConfig := {}) : IntervalDyadic :=
+  match e with
+  | Expr.const q =>
+      -- Convert rational constant to Dyadic interval with outward rounding
+      IntervalDyadic.ofIntervalRat (IntervalRat.singleton q) cfg.precision
+  | Expr.var idx => ρ idx
+  | Expr.add e₁ e₂ =>
+      let I₁ := LeanCert.Internal.Dyadic.evalUnchecked e₁ ρ cfg
+      let I₂ := LeanCert.Internal.Dyadic.evalUnchecked e₂ ρ cfg
+      (IntervalDyadic.add I₁ I₂).roundOut cfg.precision
+  | Expr.mul e₁ e₂ =>
+      let I₁ := LeanCert.Internal.Dyadic.evalUnchecked e₁ ρ cfg
+      let I₂ := LeanCert.Internal.Dyadic.evalUnchecked e₂ ρ cfg
+      (IntervalDyadic.mul I₁ I₂).roundOut cfg.precision
+  | Expr.neg e =>
+      let I := LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg
+      IntervalDyadic.neg I  -- Negation doesn't increase precision
+  | Expr.inv e => invIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg) cfg
+  | Expr.exp e => expIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg) cfg
+  | Expr.sin e => sinIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg) cfg
+  | Expr.cos e => cosIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg) cfg
+  | Expr.log e => logIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg) cfg
+  | Expr.atan e => atanIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg) cfg
+  | Expr.arsinh e => arsinhIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg) cfg
+  | Expr.atanh e => atanhIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg) cfg
+  | Expr.sinc e => sincIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg) cfg
+  | Expr.erf e => erfIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg) cfg
+  | Expr.sinh e => sinhIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg) cfg
+  | Expr.cosh e => coshIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg) cfg
+  | Expr.tanh e => tanhIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg) cfg
+  | Expr.sqrt e => sqrtIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg) cfg
+  | Expr.namedConst c => IntervalDyadic.ofIntervalRat c.interval cfg.precision
+
+end LeanCert.Internal.Dyadic
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+/-! ### Correctness -/
+
+/-- A real environment is contained in a Dyadic interval environment -/
+def envMemDyadic (ρ_real : Nat → ℝ) (ρ_dyad : IntervalDyadicEnv) : Prop :=
+  ∀ i, ρ_real i ∈ ρ_dyad i
+
+/-- Domain validity for Dyadic evaluation.
+    This is defined directly in terms of LeanCert.Internal.Dyadic.evalUnchecked to ensure
+    compatibility.
+    For log, we require the argument interval (converted to Rat) to have positive lower bound. -/
+def evalDomainValidDyadic (e : Expr) (ρ : IntervalDyadicEnv) (cfg : DyadicConfig := {}) : Prop :=
+  match e with
+  | Expr.const _ => True
+  | Expr.var _ => True
+  | Expr.add e₁ e₂ => evalDomainValidDyadic e₁ ρ cfg ∧ evalDomainValidDyadic e₂ ρ cfg
+  | Expr.mul e₁ e₂ => evalDomainValidDyadic e₁ ρ cfg ∧ evalDomainValidDyadic e₂ ρ cfg
+  | Expr.neg e => evalDomainValidDyadic e ρ cfg
+  | Expr.inv e => evalDomainValidDyadic e ρ cfg ∧
+      ((LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg).toIntervalRat.lo > 0 ∨
+       (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg).toIntervalRat.hi < 0)
+  | Expr.exp e => evalDomainValidDyadic e ρ cfg
+  | Expr.sin e => evalDomainValidDyadic e ρ cfg
+  | Expr.cos e => evalDomainValidDyadic e ρ cfg
+  | Expr.log e => evalDomainValidDyadic e ρ cfg ∧ (LeanCert.Internal.Dyadic.evalUnchecked e ρ
+    cfg).toIntervalRat.lo > 0
+  | Expr.atan e => evalDomainValidDyadic e ρ cfg
+  | Expr.arsinh e => evalDomainValidDyadic e ρ cfg
+  | Expr.atanh e => evalDomainValidDyadic e ρ cfg ∧
+      (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg).toIntervalRat.lo > -1 ∧
+      (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg).toIntervalRat.hi < 1
+  | Expr.sinc e => evalDomainValidDyadic e ρ cfg
+  | Expr.erf e => evalDomainValidDyadic e ρ cfg
+  | Expr.sinh e => evalDomainValidDyadic e ρ cfg
+  | Expr.cosh e => evalDomainValidDyadic e ρ cfg
+  | Expr.tanh e => evalDomainValidDyadic e ρ cfg
+  | Expr.sqrt e => evalDomainValidDyadic e ρ cfg
+  | Expr.namedConst _ => True
+
+/-- Computable (Bool) domain validity check for Dyadic evaluation. -/
+def checkDomainValidDyadic (e : Expr) (ρ : IntervalDyadicEnv) (cfg : DyadicConfig := {}) : Bool :=
+  match e with
+  | Expr.const _ => true
+  | Expr.var _ => true
+  | Expr.add e₁ e₂ => checkDomainValidDyadic e₁ ρ cfg && checkDomainValidDyadic e₂ ρ cfg
+  | Expr.mul e₁ e₂ => checkDomainValidDyadic e₁ ρ cfg && checkDomainValidDyadic e₂ ρ cfg
+  | Expr.neg e => checkDomainValidDyadic e ρ cfg
+  | Expr.inv e => checkDomainValidDyadic e ρ cfg &&
+      (decide ((LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg).toIntervalRat.lo > 0) ||
+       decide ((LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg).toIntervalRat.hi < 0))
+  | Expr.exp e => checkDomainValidDyadic e ρ cfg
+  | Expr.sin e => checkDomainValidDyadic e ρ cfg
+  | Expr.cos e => checkDomainValidDyadic e ρ cfg
+  | Expr.log e => checkDomainValidDyadic e ρ cfg &&
+      decide ((LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg).toIntervalRat.lo > 0)
+  | Expr.atan e => checkDomainValidDyadic e ρ cfg
+  | Expr.arsinh e => checkDomainValidDyadic e ρ cfg
+  | Expr.atanh e => checkDomainValidDyadic e ρ cfg &&
+      decide ((LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg).toIntervalRat.lo > -1) &&
+      decide ((LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg).toIntervalRat.hi < 1)
+  | Expr.sinc e => checkDomainValidDyadic e ρ cfg
+  | Expr.erf e => checkDomainValidDyadic e ρ cfg
+  | Expr.sinh e => checkDomainValidDyadic e ρ cfg
+  | Expr.cosh e => checkDomainValidDyadic e ρ cfg
+  | Expr.tanh e => checkDomainValidDyadic e ρ cfg
+  | Expr.sqrt e => checkDomainValidDyadic e ρ cfg
+  | Expr.namedConst _ => true
+
+theorem checkDomainValidDyadic_correct (e : Expr) (ρ : IntervalDyadicEnv) (cfg : DyadicConfig) :
+    checkDomainValidDyadic e ρ cfg = true → evalDomainValidDyadic e ρ cfg := by
+  induction e with
+  | const _ => intro; trivial
+  | var _ => intro; trivial
+  | add e₁ e₂ ih₁ ih₂ =>
+    simp only [checkDomainValidDyadic, Bool.and_eq_true, evalDomainValidDyadic]
+    intro ⟨h1, h2⟩; exact ⟨ih₁ h1, ih₂ h2⟩
+  | mul e₁ e₂ ih₁ ih₂ =>
+    simp only [checkDomainValidDyadic, Bool.and_eq_true, evalDomainValidDyadic]
+    intro ⟨h1, h2⟩; exact ⟨ih₁ h1, ih₂ h2⟩
+  | neg e ih =>
+    simp only [checkDomainValidDyadic, evalDomainValidDyadic]; exact ih
+  | inv e ih =>
+    simp only [checkDomainValidDyadic, Bool.and_eq_true, Bool.or_eq_true,
+      decide_eq_true_eq, evalDomainValidDyadic]
+    intro ⟨h1, h2⟩; exact ⟨ih h1, h2⟩
+  | exp e ih => simp only [checkDomainValidDyadic, evalDomainValidDyadic]; exact ih
+  | sin e ih => simp only [checkDomainValidDyadic, evalDomainValidDyadic]; exact ih
+  | cos e ih => simp only [checkDomainValidDyadic, evalDomainValidDyadic]; exact ih
+  | log e ih =>
+    simp only [checkDomainValidDyadic, Bool.and_eq_true, decide_eq_true_eq, evalDomainValidDyadic]
+    intro ⟨h1, h2⟩; exact ⟨ih h1, h2⟩
+  | atan e ih => simp only [checkDomainValidDyadic, evalDomainValidDyadic]; exact ih
+  | arsinh e ih => simp only [checkDomainValidDyadic, evalDomainValidDyadic]; exact ih
+  | atanh e ih =>
+    simp only [checkDomainValidDyadic, Bool.and_eq_true, decide_eq_true_eq, evalDomainValidDyadic]
+    intro ⟨⟨h1, h2⟩, h3⟩; exact ⟨ih h1, h2, h3⟩
+  | sinc e ih => simp only [checkDomainValidDyadic, evalDomainValidDyadic]; exact ih
+  | erf e ih => simp only [checkDomainValidDyadic, evalDomainValidDyadic]; exact ih
+  | sinh e ih => simp only [checkDomainValidDyadic, evalDomainValidDyadic]; exact ih
+  | cosh e ih => simp only [checkDomainValidDyadic, evalDomainValidDyadic]; exact ih
+  | tanh e ih => simp only [checkDomainValidDyadic, evalDomainValidDyadic]; exact ih
+  | sqrt e ih => simp only [checkDomainValidDyadic, evalDomainValidDyadic]; exact ih
+  | namedConst _ => intro; trivial
+
+end LeanCert.Engine
+
+namespace LeanCert.Internal.Dyadic
+
+open LeanCert.Core LeanCert.Engine
+
+/-- Static data prepared once for repeated Dyadic evaluation.
+
+The context is operation-independent: logarithm, exponential, trigonometric,
+and inverse-hyperbolic kernels share certified constants and coefficient
+families without changing the public evaluator configuration. -/
+structure PreparedContext where
+  /-- Precision and rounding settings shared by prepared interval computations. -/
+  cfg : DyadicConfig
+  /-- A precomputed rational enclosure of the natural logarithm of two. -/
+  ln2 : IntervalRat
+  /-- Precomputed rational coefficients for the exponential Taylor polynomial. -/
+  expCoeffs : List ℚ
+  /-- Precomputed rational coefficients for the sine Taylor polynomial. -/
+  sinCoeffs : List ℚ
+  /-- Precomputed rational coefficients for the cosine Taylor polynomial. -/
+  cosCoeffs : List ℚ
+  /-- Precomputed rational coefficients for the inverse hyperbolic tangent expansion. -/
+  atanhCoeffs : List ℚ
+  deriving Repr
+
+/-- Deterministically prepare all configuration-dependent numerical data. -/
+def prepareContext (cfg : DyadicConfig) : PreparedContext :=
+  { cfg
+    ln2 := IntervalRat.ln2Computable cfg.taylorDepth
+    expCoeffs := IntervalRat.expTaylorCoeffs cfg.taylorDepth
+    sinCoeffs := IntervalRat.sinTaylorCoeffs cfg.taylorDepth
+    cosCoeffs := IntervalRat.cosTaylorCoeffs cfg.taylorDepth
+    atanhCoeffs := IntervalRat.atanhTaylorCoeffs cfg.taylorDepth }
+
+@[simp] theorem prepareContext_cfg (cfg : DyadicConfig) :
+    (prepareContext cfg).cfg = cfg := rfl
+
+/-- Logarithm kernel using the context's prepared certified constants. -/
+def logIntervalPrepared (I : IntervalDyadic) (ctx : PreparedContext) : IntervalDyadic :=
+  let IRat := I.toIntervalRat
+  if IRat.lo > 0 then
+    let result := IntervalRat.logComputablePrepared IRat ctx.cfg.taylorDepth ctx.ln2
+      ctx.atanhCoeffs
+    IntervalDyadic.ofIntervalRat result ctx.cfg.precision
+  else
+    ⟨Core.Dyadic.ofInt (-1000), Core.Dyadic.ofInt 1000, by simp [Dyadic.toRat_ofInt]⟩
+
+theorem logIntervalPrepared_prepareContext (I : IntervalDyadic) (cfg : DyadicConfig) :
+    logIntervalPrepared I (prepareContext cfg) = logIntervalDyadic I cfg := by
+  simp [logIntervalPrepared, prepareContext, logIntervalDyadic,
+    IntervalRat.logComputablePrepared_eq]
+
+/-- Exponential kernel using the context's prepared Taylor coefficients. -/
+def expIntervalPrepared (I : IntervalDyadic) (ctx : PreparedContext) : IntervalDyadic :=
+  let result := IntervalRat.expComputableWithCoeffs I.toIntervalRat ctx.cfg.taylorDepth
+    ctx.expCoeffs
+  IntervalDyadic.ofIntervalRat result ctx.cfg.precision
+
+theorem expIntervalPrepared_prepareContext (I : IntervalDyadic) (cfg : DyadicConfig) :
+    expIntervalPrepared I (prepareContext cfg) = expIntervalDyadic I cfg := by
+  simp [expIntervalPrepared, prepareContext, expIntervalDyadic,
+    IntervalRat.expComputableWithCoeffs_eq]
+
+/-- Sine kernel using the context's prepared Taylor coefficients. -/
+def sinIntervalPrepared (I : IntervalDyadic) (ctx : PreparedContext) : IntervalDyadic :=
+  let result := IntervalRat.sinComputableWithCoeffs I.toIntervalRat ctx.cfg.taylorDepth
+    ctx.sinCoeffs
+  IntervalDyadic.ofIntervalRat result ctx.cfg.precision
+
+theorem sinIntervalPrepared_prepareContext (I : IntervalDyadic) (cfg : DyadicConfig) :
+    sinIntervalPrepared I (prepareContext cfg) = sinIntervalDyadic I cfg := by
+  simp [sinIntervalPrepared, prepareContext, sinIntervalDyadic,
+    IntervalRat.sinComputableWithCoeffs_eq]
+
+/-- Cosine kernel using the context's prepared Taylor coefficients. -/
+def cosIntervalPrepared (I : IntervalDyadic) (ctx : PreparedContext) : IntervalDyadic :=
+  let result := IntervalRat.cosComputableWithCoeffs I.toIntervalRat ctx.cfg.taylorDepth
+    ctx.cosCoeffs
+  IntervalDyadic.ofIntervalRat result ctx.cfg.precision
+
+theorem cosIntervalPrepared_prepareContext (I : IntervalDyadic) (cfg : DyadicConfig) :
+    cosIntervalPrepared I (prepareContext cfg) = cosIntervalDyadic I cfg := by
+  simp [cosIntervalPrepared, prepareContext, cosIntervalDyadic,
+    IntervalRat.cosComputableWithCoeffs_eq]
+
+/-- Inverse-hyperbolic-tangent kernel using prepared Taylor coefficients. -/
+def atanhIntervalPrepared (I : IntervalDyadic) (ctx : PreparedContext) : IntervalDyadic :=
+  let result := IntervalRat.atanhComputableWithCoeffs I.toIntervalRat ctx.cfg.taylorDepth
+    ctx.atanhCoeffs
+  IntervalDyadic.ofIntervalRat result ctx.cfg.precision
+
+theorem atanhIntervalPrepared_prepareContext (I : IntervalDyadic) (cfg : DyadicConfig) :
+    atanhIntervalPrepared I (prepareContext cfg) = atanhIntervalDyadic I cfg := by
+  simp [atanhIntervalPrepared, prepareContext, atanhIntervalDyadic,
+    IntervalRat.atanhComputableWithCoeffs_eq]
+
+/-- Evaluate an expression and its domain-validity bit in one traversal.
+The value component is exactly `LeanCert.Internal.Dyadic.evalUnchecked`; the validity component is
+exactly `checkDomainValidDyadic`. -/
+def evalCached (e : Expr) (ρ : IntervalDyadicEnv)
+    (cfg : DyadicConfig := {}) : IntervalDyadic × Bool :=
+  match e with
+  | .const q =>
+      (IntervalDyadic.ofIntervalRat (IntervalRat.singleton q) cfg.precision, true)
+  | .var idx => (ρ idx, true)
+  | .add e₁ e₂ =>
+      let r₁ := LeanCert.Internal.Dyadic.evalCached e₁ ρ cfg
+      let r₂ := LeanCert.Internal.Dyadic.evalCached e₂ ρ cfg
+      ((IntervalDyadic.add r₁.1 r₂.1).roundOut cfg.precision, r₁.2 && r₂.2)
+  | .mul e₁ e₂ =>
+      let r₁ := LeanCert.Internal.Dyadic.evalCached e₁ ρ cfg
+      let r₂ := LeanCert.Internal.Dyadic.evalCached e₂ ρ cfg
+      ((IntervalDyadic.mul r₁.1 r₂.1).roundOut cfg.precision, r₁.2 && r₂.2)
+  | .neg e =>
+      let r := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+      (IntervalDyadic.neg r.1, r.2)
+  | .inv e =>
+      let r := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+      let I := r.1.toIntervalRat
+      (invIntervalDyadic r.1 cfg, r.2 && (decide (I.lo > 0) || decide (I.hi < 0)))
+  | .exp e =>
+      let r := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+      (expIntervalDyadic r.1 cfg, r.2)
+  | .sin e =>
+      let r := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+      (sinIntervalDyadic r.1 cfg, r.2)
+  | .cos e =>
+      let r := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+      (cosIntervalDyadic r.1 cfg, r.2)
+  | .log e =>
+      let r := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+      (logIntervalDyadic r.1 cfg, r.2 && decide (r.1.toIntervalRat.lo > 0))
+  | .atan e =>
+      let r := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+      (atanIntervalDyadic r.1 cfg, r.2)
+  | .arsinh e =>
+      let r := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+      (arsinhIntervalDyadic r.1 cfg, r.2)
+  | .atanh e =>
+      let r := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+      let I := r.1.toIntervalRat
+      (atanhIntervalDyadic r.1 cfg,
+        r.2 && decide (I.lo > -1) && decide (I.hi < 1))
+  | .sinc e =>
+      let r := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+      (sincIntervalDyadic r.1 cfg, r.2)
+  | .erf e =>
+      let r := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+      (erfIntervalDyadic r.1 cfg, r.2)
+  | .sinh e =>
+      let r := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+      (sinhIntervalDyadic r.1 cfg, r.2)
+  | .cosh e =>
+      let r := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+      (coshIntervalDyadic r.1 cfg, r.2)
+  | .tanh e =>
+      let r := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+      (tanhIntervalDyadic r.1 cfg, r.2)
+  | .sqrt e =>
+      let r := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+      (sqrtIntervalDyadic r.1 cfg, r.2)
+  | .namedConst c => (IntervalDyadic.ofIntervalRat c.interval cfg.precision, true)
+
+/-- Prepared evaluator: identical certificate semantics to `evalCached`, but
+configuration-dependent data is shared across every evaluation. -/
+def evalPreparedCached (e : Expr) (ρ : IntervalDyadicEnv)
+    (ctx : PreparedContext) : IntervalDyadic × Bool :=
+  match e with
+  | .const q =>
+      (IntervalDyadic.ofIntervalRat (IntervalRat.singleton q) ctx.cfg.precision, true)
+  | .var idx => (ρ idx, true)
+  | .add e₁ e₂ =>
+      let r₁ := evalPreparedCached e₁ ρ ctx
+      let r₂ := evalPreparedCached e₂ ρ ctx
+      ((IntervalDyadic.add r₁.1 r₂.1).roundOut ctx.cfg.precision, r₁.2 && r₂.2)
+  | .mul e₁ e₂ =>
+      let r₁ := evalPreparedCached e₁ ρ ctx
+      let r₂ := evalPreparedCached e₂ ρ ctx
+      ((IntervalDyadic.mul r₁.1 r₂.1).roundOut ctx.cfg.precision, r₁.2 && r₂.2)
+  | .neg e =>
+      let r := evalPreparedCached e ρ ctx
+      (IntervalDyadic.neg r.1, r.2)
+  | .inv e =>
+      let r := evalPreparedCached e ρ ctx
+      let I := r.1.toIntervalRat
+      (invIntervalDyadic r.1 ctx.cfg, r.2 && (decide (I.lo > 0) || decide (I.hi < 0)))
+  | .exp e =>
+      let r := evalPreparedCached e ρ ctx
+      (expIntervalPrepared r.1 ctx, r.2)
+  | .sin e =>
+      let r := evalPreparedCached e ρ ctx
+      (sinIntervalPrepared r.1 ctx, r.2)
+  | .cos e =>
+      let r := evalPreparedCached e ρ ctx
+      (cosIntervalPrepared r.1 ctx, r.2)
+  | .log e =>
+      let r := evalPreparedCached e ρ ctx
+      (logIntervalPrepared r.1 ctx, r.2 && decide (r.1.toIntervalRat.lo > 0))
+  | .atan e =>
+      let r := evalPreparedCached e ρ ctx
+      (atanIntervalDyadic r.1 ctx.cfg, r.2)
+  | .arsinh e =>
+      let r := evalPreparedCached e ρ ctx
+      (arsinhIntervalDyadic r.1 ctx.cfg, r.2)
+  | .atanh e =>
+      let r := evalPreparedCached e ρ ctx
+      let I := r.1.toIntervalRat
+      (atanhIntervalPrepared r.1 ctx,
+        r.2 && decide (I.lo > -1) && decide (I.hi < 1))
+  | .sinc e =>
+      let r := evalPreparedCached e ρ ctx
+      (sincIntervalDyadic r.1 ctx.cfg, r.2)
+  | .erf e =>
+      let r := evalPreparedCached e ρ ctx
+      (erfIntervalDyadic r.1 ctx.cfg, r.2)
+  | .sinh e =>
+      let r := evalPreparedCached e ρ ctx
+      (sinhIntervalDyadic r.1 ctx.cfg, r.2)
+  | .cosh e =>
+      let r := evalPreparedCached e ρ ctx
+      (coshIntervalDyadic r.1 ctx.cfg, r.2)
+  | .tanh e =>
+      let r := evalPreparedCached e ρ ctx
+      (tanhIntervalDyadic r.1 ctx.cfg, r.2)
+  | .sqrt e =>
+      let r := evalPreparedCached e ρ ctx
+      (sqrtIntervalDyadic r.1 ctx.cfg, r.2)
+  | .namedConst c =>
+      (IntervalDyadic.ofIntervalRat c.interval ctx.cfg.precision, true)
+
+end LeanCert.Internal.Dyadic
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+theorem evalIntervalDyadicCached_fst (e : Expr) (ρ : IntervalDyadicEnv)
+    (cfg : DyadicConfig := {}) :
+    (LeanCert.Internal.Dyadic.evalCached e ρ cfg).1 = LeanCert.Internal.Dyadic.evalUnchecked e ρ
+      cfg := by
+  induction e <;> simp [LeanCert.Internal.Dyadic.evalCached,
+    LeanCert.Internal.Dyadic.evalUnchecked, *]
+
+theorem evalIntervalDyadicCached_snd (e : Expr) (ρ : IntervalDyadicEnv)
+    (cfg : DyadicConfig := {}) :
+    (LeanCert.Internal.Dyadic.evalCached e ρ cfg).2 = checkDomainValidDyadic e ρ cfg := by
+  induction e <;>
+    simp [LeanCert.Internal.Dyadic.evalCached, checkDomainValidDyadic,
+      evalIntervalDyadicCached_fst, *]
+
+theorem evalIntervalDyadicPreparedCached_fst (e : Expr) (ρ : IntervalDyadicEnv)
+    (cfg : DyadicConfig := {}) :
+    (LeanCert.Internal.Dyadic.evalPreparedCached e ρ
+      (LeanCert.Internal.Dyadic.prepareContext cfg)).1 =
+      LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg := by
+  induction e <;>
+    simp [LeanCert.Internal.Dyadic.evalPreparedCached,
+      LeanCert.Internal.Dyadic.evalUnchecked,
+      LeanCert.Internal.Dyadic.logIntervalPrepared_prepareContext,
+      LeanCert.Internal.Dyadic.expIntervalPrepared_prepareContext,
+      LeanCert.Internal.Dyadic.sinIntervalPrepared_prepareContext,
+      LeanCert.Internal.Dyadic.cosIntervalPrepared_prepareContext,
+      LeanCert.Internal.Dyadic.atanhIntervalPrepared_prepareContext, *]
+
+theorem evalIntervalDyadicPreparedCached_snd (e : Expr) (ρ : IntervalDyadicEnv)
+    (cfg : DyadicConfig := {}) :
+    (LeanCert.Internal.Dyadic.evalPreparedCached e ρ
+      (LeanCert.Internal.Dyadic.prepareContext cfg)).2 =
+      checkDomainValidDyadic e ρ cfg := by
+  induction e <;>
+    simp [LeanCert.Internal.Dyadic.evalPreparedCached,
+      checkDomainValidDyadic,
+      evalIntervalDyadicPreparedCached_fst, *]
+
+/-- Diagnose the first failed Dyadic domain check. -/
+def diagnoseEvalIntervalDyadicFailure (e : Expr) (ρ : IntervalDyadicEnv)
+    (cfg : DyadicConfig := {}) : EvalError :=
+  match e with
+  | .add e₁ e₂ | .mul e₁ e₂ =>
+      if checkDomainValidDyadic e₁ ρ cfg then
+        .nestedFailure "right operand" (diagnoseEvalIntervalDyadicFailure e₂ ρ cfg)
+      else
+        .nestedFailure "left operand" (diagnoseEvalIntervalDyadicFailure e₁ ρ cfg)
+  | .neg e | .exp e | .sin e | .cos e | .atan e | .arsinh e | .sinc e |
+      .erf e | .sinh e | .cosh e | .tanh e | .sqrt e =>
+      .nestedFailure "unary operand" (diagnoseEvalIntervalDyadicFailure e ρ cfg)
+  | .inv e =>
+      if checkDomainValidDyadic e ρ cfg then
+        .reciprocalContainsZero (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg).toIntervalRat
+      else
+        .nestedFailure "reciprocal operand" (diagnoseEvalIntervalDyadicFailure e ρ cfg)
+  | .log e =>
+      if checkDomainValidDyadic e ρ cfg then
+        .logNonpositive (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg).toIntervalRat
+      else
+        .nestedFailure "logarithm operand" (diagnoseEvalIntervalDyadicFailure e ρ cfg)
+  | .atanh e =>
+      if checkDomainValidDyadic e ρ cfg then
+        .atanhOutsideUnitBall (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg).toIntervalRat
+      else
+        .nestedFailure "atanh operand" (diagnoseEvalIntervalDyadicFailure e ρ cfg)
+  | .const _ | .var _ | .namedConst _ =>
+      .unsupportedBackend "internal: total Dyadic expression unexpectedly failed"
+termination_by e
+
+/-- Checked Dyadic evaluator. The finite fallback branches of the legacy total
+evaluator are never exposed after a failed domain check. -/
+def evalIntervalDyadicChecked (e : Expr) (ρ : IntervalDyadicEnv)
+    (cfg : DyadicConfig := {}) : EvalResult IntervalDyadic :=
+  let cached := LeanCert.Internal.Dyadic.evalCached e ρ cfg
+  if cached.2 then
+    .ok cached.1
+  else
+    .error (diagnoseEvalIntervalDyadicFailure e ρ cfg)
+
+/-- Domain validity is trivially true for ADSupported expressions (which exclude log). -/
+theorem evalDomainValidDyadic_of_ExprSupported {e : Expr} (hsupp : ADSupported e)
+    (ρ : IntervalDyadicEnv) (cfg : DyadicConfig := {}) : evalDomainValidDyadic e ρ cfg := by
+  induction hsupp with
+  | const _ => trivial
+  | var _ => trivial
+  | add _ _ ih1 ih2 => exact ⟨ih1, ih2⟩
+  | mul _ _ ih1 ih2 => exact ⟨ih1, ih2⟩
+  | neg _ ih => exact ih
+  | sin _ ih => exact ih
+  | cos _ ih => exact ih
+  | exp _ ih => exact ih
+
+/-- Fundamental correctness theorem for Dyadic evaluation.
+
+This theorem states that for any supported expression and any real values
+within the input intervals, the result of evaluating the expression is
+contained in the computed Dyadic interval.
+
+The proof follows the same structure as `evalIntervalCore_correct`, but
+with additional steps for handling Dyadic ↔ Rat conversions and rounding.
+Requires cfg.precision ≤ 0 (the default -53 satisfies this).
+
+Note: Requires domain validity for log (positive argument interval). -/
+theorem evalIntervalDyadic_correct (e : Expr) (hsupp : ExprSupportedCore e)
+    (ρ_real : Nat → ℝ) (ρ_dyad : IntervalDyadicEnv)
+    (hρ : envMemDyadic ρ_real ρ_dyad) (cfg : DyadicConfig := {})
+    (hprec : cfg.precision ≤ 0 := by norm_num)
+    (hdom : evalDomainValidDyadic e ρ_dyad cfg) :
+    Expr.eval ρ_real e ∈ LeanCert.Internal.Dyadic.evalUnchecked e ρ_dyad cfg := by
+  induction hsupp with
+  | const q =>
+    simp only [Expr.eval_const, LeanCert.Internal.Dyadic.evalUnchecked]
+    apply IntervalDyadic.mem_ofIntervalRat
+    · exact IntervalRat.mem_singleton q
+    · exact hprec
+  | var idx =>
+    simp only [Expr.eval_var, LeanCert.Internal.Dyadic.evalUnchecked]
+    exact hρ idx
+  | add _ _ ih₁ ih₂ =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_add, LeanCert.Internal.Dyadic.evalUnchecked]
+    have h := IntervalDyadic.mem_add (ih₁ hdom.1) (ih₂ hdom.2)
+    exact IntervalDyadic.roundOut_contains h cfg.precision
+  | mul _ _ ih₁ ih₂ =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_mul, LeanCert.Internal.Dyadic.evalUnchecked]
+    have h := IntervalDyadic.mem_mul (ih₁ hdom.1) (ih₂ hdom.2)
+    exact IntervalDyadic.roundOut_contains h cfg.precision
+  | neg _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_neg, LeanCert.Internal.Dyadic.evalUnchecked]
+    exact IntervalDyadic.mem_neg (ih hdom)
+  | sin _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_sin, LeanCert.Internal.Dyadic.evalUnchecked, sinIntervalDyadic]
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp (ih hdom)
+    have hsin := IntervalRat.mem_sinComputable hrat cfg.taylorDepth
+    exact IntervalDyadic.mem_ofIntervalRat hsin cfg.precision hprec
+  | cos _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_cos, LeanCert.Internal.Dyadic.evalUnchecked, cosIntervalDyadic]
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp (ih hdom)
+    have hcos := IntervalRat.mem_cosComputable hrat cfg.taylorDepth
+    exact IntervalDyadic.mem_ofIntervalRat hcos cfg.precision hprec
+  | exp _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_exp, LeanCert.Internal.Dyadic.evalUnchecked, expIntervalDyadic]
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp (ih hdom)
+    have hexp := IntervalRat.mem_expComputable hrat cfg.taylorDepth
+    exact IntervalDyadic.mem_ofIntervalRat hexp cfg.precision hprec
+  | sqrt _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_sqrt, LeanCert.Internal.Dyadic.evalUnchecked, sqrtIntervalDyadic]
+    exact IntervalDyadic.mem_sqrt' (ih hdom) cfg.precision
+  | sinh _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_sinh, LeanCert.Internal.Dyadic.evalUnchecked, sinhIntervalDyadic]
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp (ih hdom)
+    have hsinh := IntervalRat.mem_sinhComputable hrat cfg.taylorDepth
+    exact IntervalDyadic.mem_ofIntervalRat hsinh cfg.precision hprec
+  | cosh _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_cosh, LeanCert.Internal.Dyadic.evalUnchecked, coshIntervalDyadic]
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp (ih hdom)
+    have hcosh := IntervalRat.mem_coshComputable hrat cfg.taylorDepth
+    exact IntervalDyadic.mem_ofIntervalRat hcosh cfg.precision hprec
+  | @tanh e' _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_tanh, LeanCert.Internal.Dyadic.evalUnchecked, tanhIntervalDyadic]
+    -- tanh x ∈ [-1, 1] for all x
+    rw [IntervalDyadic.mem_def, Dyadic.toRat_ofInt, Dyadic.toRat_ofInt]
+    simp only [Int.cast_neg, Int.cast_one, Rat.cast_neg, Rat.cast_one]
+    set x := Expr.eval ρ_real e' with hx
+    constructor
+    -- tanh x ≥ -1: use tanh = sinh/cosh, cosh > 0, and -cosh ≤ sinh
+    · rw [Real.tanh_eq_sinh_div_cosh]
+      have hcosh : Real.cosh x > 0 := Real.cosh_pos x
+      rw [le_div_iff₀ hcosh, neg_one_mul]
+      rw [Real.sinh_eq, Real.cosh_eq]
+      have h1 : Real.exp x > 0 := Real.exp_pos x
+      linarith
+    -- tanh x ≤ 1: use sinh ≤ cosh (since exp(-x) > 0)
+    · rw [Real.tanh_eq_sinh_div_cosh]
+      have hcosh : Real.cosh x > 0 := Real.cosh_pos x
+      rw [div_le_one₀ hcosh]
+      rw [Real.sinh_eq, Real.cosh_eq]
+      have h2 : Real.exp (-x) > 0 := Real.exp_pos (-x)
+      linarith
+  | @erf e' _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_erf, LeanCert.Internal.Dyadic.evalUnchecked, erfIntervalDyadic]
+    -- erf x ∈ [-1, 1] for all x
+    rw [IntervalDyadic.mem_def, Dyadic.toRat_ofInt, Dyadic.toRat_ofInt]
+    simp only [Int.cast_neg, Int.cast_one, Rat.cast_neg, Rat.cast_one]
+    exact Real.erf_mem_Icc _
+  | log _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_log, LeanCert.Internal.Dyadic.evalUnchecked, logIntervalDyadic]
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp (ih hdom.1)
+    -- hdom.2 gives us the positivity condition: IRat.lo > 0
+    -- This makes the if-condition true, so we take the positive branch
+    have hpos : (LeanCert.Internal.Dyadic.evalUnchecked _ ρ_dyad cfg).toIntervalRat.lo > 0 := hdom.2
+    simp only [hpos, ↓reduceIte]
+    have hlog := IntervalRat.mem_logComputable hrat hpos cfg.taylorDepth
+    exact IntervalDyadic.mem_ofIntervalRat hlog cfg.precision hprec
+  | namedConst c =>
+    simp only [Expr.eval_namedConst, LeanCert.Internal.Dyadic.evalUnchecked]
+    exact IntervalDyadic.mem_ofIntervalRat c.mem_interval cfg.precision hprec
+
+/-- Correctness of Dyadic evaluation for every expression whose recursively
+checked domain conditions hold. -/
+theorem evalIntervalDyadic_correct_of_domain (e : Expr)
+    (ρ_real : Nat → ℝ) (ρ_dyad : IntervalDyadicEnv)
+    (hρ : envMemDyadic ρ_real ρ_dyad) (cfg : DyadicConfig := {})
+    (hprec : cfg.precision ≤ 0 := by norm_num)
+    (hdom : evalDomainValidDyadic e ρ_dyad cfg) :
+    Expr.eval ρ_real e ∈ LeanCert.Internal.Dyadic.evalUnchecked e ρ_dyad cfg := by
+  induction e with
+  | const q =>
+    simp only [Expr.eval_const, LeanCert.Internal.Dyadic.evalUnchecked]
+    apply IntervalDyadic.mem_ofIntervalRat
+    · exact IntervalRat.mem_singleton q
+    · exact hprec
+  | var idx =>
+    simp only [Expr.eval_var, LeanCert.Internal.Dyadic.evalUnchecked]
+    exact hρ idx
+  | add _ _ ih₁ ih₂ =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_add, LeanCert.Internal.Dyadic.evalUnchecked]
+    have h := IntervalDyadic.mem_add (ih₁ hdom.1) (ih₂ hdom.2)
+    exact IntervalDyadic.roundOut_contains h cfg.precision
+  | mul _ _ ih₁ ih₂ =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_mul, LeanCert.Internal.Dyadic.evalUnchecked]
+    have h := IntervalDyadic.mem_mul (ih₁ hdom.1) (ih₂ hdom.2)
+    exact IntervalDyadic.roundOut_contains h cfg.precision
+  | neg _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_neg, LeanCert.Internal.Dyadic.evalUnchecked]
+    exact IntervalDyadic.mem_neg (ih hdom)
+  | inv _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_inv, LeanCert.Internal.Dyadic.evalUnchecked, invIntervalDyadic]
+    have hinner := ih hdom.1
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp hinner
+    have hinv := mem_invInterval_nonzero hrat hdom.2
+    exact IntervalDyadic.mem_ofIntervalRat hinv cfg.precision hprec
+  | exp _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_exp, LeanCert.Internal.Dyadic.evalUnchecked, expIntervalDyadic]
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp (ih hdom)
+    have hexp := IntervalRat.mem_expComputable hrat cfg.taylorDepth
+    exact IntervalDyadic.mem_ofIntervalRat hexp cfg.precision hprec
+  | sin _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_sin, LeanCert.Internal.Dyadic.evalUnchecked, sinIntervalDyadic]
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp (ih hdom)
+    have hsin := IntervalRat.mem_sinComputable hrat cfg.taylorDepth
+    exact IntervalDyadic.mem_ofIntervalRat hsin cfg.precision hprec
+  | cos _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_cos, LeanCert.Internal.Dyadic.evalUnchecked, cosIntervalDyadic]
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp (ih hdom)
+    have hcos := IntervalRat.mem_cosComputable hrat cfg.taylorDepth
+    exact IntervalDyadic.mem_ofIntervalRat hcos cfg.precision hprec
+  | log _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_log, LeanCert.Internal.Dyadic.evalUnchecked, logIntervalDyadic]
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp (ih hdom.1)
+    have hpos : (LeanCert.Internal.Dyadic.evalUnchecked _ ρ_dyad cfg).toIntervalRat.lo > 0 := hdom.2
+    simp only [hpos, ↓reduceIte]
+    have hlog := IntervalRat.mem_logComputable hrat hpos cfg.taylorDepth
+    exact IntervalDyadic.mem_ofIntervalRat hlog cfg.precision hprec
+  | sinh _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_sinh, LeanCert.Internal.Dyadic.evalUnchecked, sinhIntervalDyadic]
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp (ih hdom)
+    have hsinh := IntervalRat.mem_sinhComputable hrat cfg.taylorDepth
+    exact IntervalDyadic.mem_ofIntervalRat hsinh cfg.precision hprec
+  | cosh _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_cosh, LeanCert.Internal.Dyadic.evalUnchecked, coshIntervalDyadic]
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp (ih hdom)
+    have hcosh := IntervalRat.mem_coshComputable hrat cfg.taylorDepth
+    exact IntervalDyadic.mem_ofIntervalRat hcosh cfg.precision hprec
+  | tanh e' ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_tanh, LeanCert.Internal.Dyadic.evalUnchecked, tanhIntervalDyadic]
+    rw [IntervalDyadic.mem_def, Dyadic.toRat_ofInt, Dyadic.toRat_ofInt]
+    simp only [Int.cast_neg, Int.cast_one, Rat.cast_neg, Rat.cast_one]
+    set x := Expr.eval ρ_real e' with hx
+    constructor
+    · rw [Real.tanh_eq_sinh_div_cosh]
+      have hcosh : Real.cosh x > 0 := Real.cosh_pos x
+      rw [le_div_iff₀ hcosh, neg_one_mul]
+      rw [Real.sinh_eq, Real.cosh_eq]
+      have h1 : Real.exp x > 0 := Real.exp_pos x
+      linarith
+    · rw [Real.tanh_eq_sinh_div_cosh]
+      have hcosh : Real.cosh x > 0 := Real.cosh_pos x
+      rw [div_le_one₀ hcosh]
+      rw [Real.sinh_eq, Real.cosh_eq]
+      have h2 : Real.exp (-x) > 0 := Real.exp_pos (-x)
+      linarith
+  | atan e' ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_atan, LeanCert.Internal.Dyadic.evalUnchecked, atanIntervalDyadic]
+    rw [IntervalDyadic.mem_def, Dyadic.toRat_ofInt, Dyadic.toRat_ofInt]
+    simp only [Int.cast_neg, Int.cast_ofNat, Rat.cast_neg, Rat.cast_ofNat]
+    set x := Expr.eval ρ_real e' with hx
+    constructor
+    · linarith [Real.neg_pi_div_two_lt_arctan x, Real.pi_lt_four]
+    · linarith [Real.arctan_lt_pi_div_two x, Real.pi_lt_four]
+  | sinc _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_sinc, LeanCert.Internal.Dyadic.evalUnchecked, sincIntervalDyadic]
+    rw [IntervalDyadic.mem_def, Dyadic.toRat_ofInt, Dyadic.toRat_ofInt]
+    simp only [Int.cast_neg, Int.cast_one, Rat.cast_neg, Rat.cast_one]
+    exact Real.sinc_mem_Icc _
+  | arsinh _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_arsinh, LeanCert.Internal.Dyadic.evalUnchecked, arsinhIntervalDyadic]
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp (ih hdom)
+    have harsinh := mem_arsinhInterval hrat
+    exact IntervalDyadic.mem_ofIntervalRat harsinh cfg.precision hprec
+  | atanh _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    obtain ⟨hdom_sub, hlo_gt, hhi_lt⟩ := hdom
+    simp only [Expr.eval_atanh, LeanCert.Internal.Dyadic.evalUnchecked, atanhIntervalDyadic]
+    have hrat := IntervalDyadic.mem_toIntervalRat.mp (ih hdom_sub)
+    have hatanh := IntervalRat.mem_atanhComputable hrat hlo_gt hhi_lt cfg.taylorDepth
+    exact IntervalDyadic.mem_ofIntervalRat hatanh cfg.precision hprec
+  | erf _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_erf, LeanCert.Internal.Dyadic.evalUnchecked, erfIntervalDyadic]
+    rw [IntervalDyadic.mem_def, Dyadic.toRat_ofInt, Dyadic.toRat_ofInt]
+    simp only [Int.cast_neg, Int.cast_one, Rat.cast_neg, Rat.cast_one]
+    exact Real.erf_mem_Icc _
+  | sqrt _ ih =>
+    simp only [evalDomainValidDyadic] at hdom
+    simp only [Expr.eval_sqrt, LeanCert.Internal.Dyadic.evalUnchecked, sqrtIntervalDyadic]
+    exact IntervalDyadic.mem_sqrt' (ih hdom) cfg.precision
+  | namedConst c =>
+    simp only [Expr.eval_namedConst, LeanCert.Internal.Dyadic.evalUnchecked]
+    exact IntervalDyadic.mem_ofIntervalRat c.mem_interval cfg.precision hprec
+
+/-- Successful checked Dyadic evaluation encloses the true value for every
+expression, provided outward-rounding precision is nonpositive. -/
+theorem evalIntervalDyadicChecked_correct (e : Expr)
+    (ρ_real : Nat → ℝ) (ρ_dyad : IntervalDyadicEnv)
+    (hρ : envMemDyadic ρ_real ρ_dyad) (cfg : DyadicConfig := {})
+    (hprec : cfg.precision ≤ 0 := by norm_num)
+    (result : IntervalDyadic)
+    (hsuccess : evalIntervalDyadicChecked e ρ_dyad cfg = .ok result) :
+    Expr.eval ρ_real e ∈ result := by
+  unfold evalIntervalDyadicChecked at hsuccess
+  let cached := LeanCert.Internal.Dyadic.evalCached e ρ_dyad cfg
+  cases hvalid : cached.2 with
+  | false =>
+    have : Except.error (diagnoseEvalIntervalDyadicFailure e ρ_dyad cfg) =
+        Except.ok result := by
+      simp [cached, hvalid] at hsuccess
+    contradiction
+  | true =>
+    have hsuccess' : (Except.ok cached.1 : EvalResult IntervalDyadic) = Except.ok result := by
+      simpa [cached, hvalid] using hsuccess
+    injection hsuccess' with hresult
+    subst result
+    have hcheck : checkDomainValidDyadic e ρ_dyad cfg = true := by
+      calc
+        checkDomainValidDyadic e ρ_dyad cfg =
+            (LeanCert.Internal.Dyadic.evalCached e ρ_dyad cfg).2 :=
+          (evalIntervalDyadicCached_snd e ρ_dyad cfg).symm
+        _ = cached.2 := rfl
+        _ = true := hvalid
+    have hsound := evalIntervalDyadic_correct_of_domain e
+      ρ_real ρ_dyad hρ cfg hprec (checkDomainValidDyadic_correct e ρ_dyad cfg hcheck)
+    change Expr.eval ρ_real e ∈ (LeanCert.Internal.Dyadic.evalCached e ρ_dyad cfg).1
+    rw [evalIntervalDyadicCached_fst e ρ_dyad cfg]
+    exact hsound
+
+/-! ### Verification Checkers -/
+
+/-- Check if expression is bounded above by q -/
+def checkUpperBoundDyadic (e : Expr) (ρ : IntervalDyadicEnv) (q : ℚ)
+    (cfg : DyadicConfig := {}) : Bool :=
+  (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg).upperBoundedBy q
+
+/-- Check if expression is bounded below by q -/
+def checkLowerBoundDyadic (e : Expr) (ρ : IntervalDyadicEnv) (q : ℚ)
+    (cfg : DyadicConfig := {}) : Bool :=
+  (LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg).lowerBoundedBy q
+
+/-- Check if expression is bounded in interval [lo, hi] -/
+def checkBoundsDyadic (e : Expr) (ρ : IntervalDyadicEnv) (lo hi : ℚ)
+    (cfg : DyadicConfig := {}) : Bool :=
+  let result := LeanCert.Internal.Dyadic.evalUnchecked e ρ cfg
+  result.lowerBoundedBy lo && result.upperBoundedBy hi
+
+end LeanCert.Engine
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Domain-aware automatic differentiation with Dyadic intervals
+
+This module is the bounded-denominator counterpart of `AD.DomainChecked`.
+Polynomial dual arithmetic is performed with Dyadic endpoints and outward
+rounding after every addition and multiplication.  Transcendental values use
+the same verified rational Taylor kernels as the Dyadic value evaluator and
+are rounded back to Dyadic endpoints.
+
+The public entry points are checked: unsupported syntax, invalid reciprocal or
+logarithm domains, and positive rounding precisions return `EvalError` rather
+than a finite interval.
+-/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+/-- A value interval and one directional derivative interval, both Dyadic. -/
+structure DualIntervalDyadic where
+  /-- The interval enclosing the function value. -/
+  val : IntervalDyadic
+  /-- The interval enclosing the derivative. -/
+  der : IntervalDyadic
+  deriving Repr
+
+instance : Inhabited DualIntervalDyadic where
+  default := ⟨default, default⟩
+
+/-- An environment of Dyadic dual intervals. -/
+abbrev DualDyadicEnv := Nat → DualIntervalDyadic
+
+namespace DualIntervalDyadic
+
+/-- The singleton dyadic interval at zero. -/
+def zero : IntervalDyadic := IntervalDyadic.singleton Core.Dyadic.zero
+/-- The singleton dyadic interval at one. -/
+def one : IntervalDyadic := IntervalDyadic.singleton (Core.Dyadic.ofInt 1)
+
+/-- Enclose a rational constant in a dyadic interval with zero derivative. -/
+def const (q : ℚ) (cfg : DyadicConfig := {}) : DualIntervalDyadic :=
+  ⟨IntervalDyadic.ofIntervalRat (IntervalRat.singleton q) cfg.precision, zero⟩
+
+/-- An input interval whose derivative is the singleton one. -/
+def varActive (I : IntervalDyadic) : DualIntervalDyadic := ⟨I, one⟩
+/-- An input interval held constant, with singleton zero derivative. -/
+def varPassive (I : IntervalDyadic) : DualIntervalDyadic := ⟨I, zero⟩
+
+/-- Addition with dyadic interval propagation of values and derivatives. -/
+def add (a b : DualIntervalDyadic) (cfg : DyadicConfig := {}) : DualIntervalDyadic :=
+  ⟨IntervalDyadic.addRounded a.val b.val cfg.precision,
+   IntervalDyadic.addRounded a.der b.der cfg.precision⟩
+
+/-- Multiplication with dyadic interval propagation of values and derivatives. -/
+def mul (a b : DualIntervalDyadic) (cfg : DyadicConfig := {}) : DualIntervalDyadic :=
+  ⟨IntervalDyadic.mulRounded a.val b.val cfg.precision,
+   IntervalDyadic.addRounded
+     (IntervalDyadic.mulRounded a.der b.val cfg.precision)
+     (IntervalDyadic.mulRounded a.val b.der cfg.precision)
+     cfg.precision⟩
+
+/-- Negation with dyadic interval propagation of values and derivatives. -/
+def neg (a : DualIntervalDyadic) : DualIntervalDyadic :=
+  ⟨IntervalDyadic.neg a.val, IntervalDyadic.neg a.der⟩
+
+/-- Reciprocal with dyadic interval propagation of values and derivatives. -/
+def inv (a : DualIntervalDyadic) (cfg : DyadicConfig := {}) : DualIntervalDyadic :=
+  let invVal := invIntervalDyadic a.val cfg
+  let invSq := IntervalDyadic.mulRounded invVal invVal cfg.precision
+  ⟨invVal, IntervalDyadic.neg (IntervalDyadic.mulRounded a.der invSq cfg.precision)⟩
+
+/-- Exponential with dyadic interval propagation of values and derivatives. -/
+def exp (a : DualIntervalDyadic) (cfg : DyadicConfig := {}) : DualIntervalDyadic :=
+  let value := expIntervalDyadic a.val cfg
+  ⟨value, IntervalDyadic.mulRounded value a.der cfg.precision⟩
+
+/-- Sine with dyadic interval propagation of values and derivatives. -/
+def sin (a : DualIntervalDyadic) (cfg : DyadicConfig := {}) : DualIntervalDyadic :=
+  ⟨sinIntervalDyadic a.val cfg,
+   IntervalDyadic.mulRounded (cosIntervalDyadic a.val cfg) a.der cfg.precision⟩
+
+/-- Cosine with dyadic interval propagation of values and derivatives. -/
+def cos (a : DualIntervalDyadic) (cfg : DyadicConfig := {}) : DualIntervalDyadic :=
+  ⟨cosIntervalDyadic a.val cfg,
+   IntervalDyadic.mulRounded (IntervalDyadic.neg (sinIntervalDyadic a.val cfg))
+     a.der cfg.precision⟩
+
+/-- Natural logarithm with dyadic interval propagation of values and derivatives. -/
+def log (a : DualIntervalDyadic) (cfg : DyadicConfig := {}) : DualIntervalDyadic :=
+  ⟨logIntervalDyadic a.val cfg,
+   IntervalDyadic.mulRounded (invIntervalDyadic a.val cfg) a.der cfg.precision⟩
+
+end DualIntervalDyadic
+end LeanCert.Engine
+
+namespace LeanCert.Internal.AD.Dyadic
+
+open LeanCert.Core LeanCert.Engine
+
+/-- Total computational kernel.  Sound public use goes through the checked
+entry points below. -/
+def evalTotal (e : Expr) (rho : DualDyadicEnv) (cfg : DyadicConfig := {}) :
+    DualIntervalDyadic :=
+  match e with
+  | .const q => DualIntervalDyadic.const q cfg
+  | .var i => rho i
+  | .add a b => DualIntervalDyadic.add (evalTotal a rho cfg) (evalTotal b rho cfg) cfg
+  | .mul a b => DualIntervalDyadic.mul (evalTotal a rho cfg) (evalTotal b rho cfg) cfg
+  | .neg a => DualIntervalDyadic.neg (evalTotal a rho cfg)
+  | .inv a => DualIntervalDyadic.inv (evalTotal a rho cfg) cfg
+  | .exp a => DualIntervalDyadic.exp (evalTotal a rho cfg) cfg
+  | .sin a => DualIntervalDyadic.sin (evalTotal a rho cfg) cfg
+  | .cos a => DualIntervalDyadic.cos (evalTotal a rho cfg) cfg
+  | .log a => DualIntervalDyadic.log (evalTotal a rho cfg) cfg
+  | _ => default
+
+end LeanCert.Internal.AD.Dyadic
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+/-- Value projection of a Dyadic dual environment. -/
+def DualDyadicEnv.values (rho : DualDyadicEnv) : IntervalDyadicEnv := fun i => (rho i).val
+
+/-- Dual environment selecting coordinate `idx`. -/
+def mkDualDyadicEnv (rho : IntervalDyadicEnv) (idx : Nat) : DualDyadicEnv :=
+  fun i => if i = idx then DualIntervalDyadic.varActive (rho i)
+    else DualIntervalDyadic.varPassive (rho i)
+
+@[simp] theorem mkDualDyadicEnv_values (rho : IntervalDyadicEnv) (idx : Nat) :
+    (mkDualDyadicEnv rho idx).values = rho := by
+  funext i
+  by_cases hi : i = idx <;>
+    simp [mkDualDyadicEnv, DualDyadicEnv.values, hi,
+      DualIntervalDyadic.varActive, DualIntervalDyadic.varPassive]
+
+/-- The deliberately small v1 Dyadic AD fragment. -/
+def checkDyadicADFragment : Expr → Bool
+  | .const _ | .var _ => true
+  | .add a b | .mul a b => checkDyadicADFragment a && checkDyadicADFragment b
+  | .neg a | .inv a | .exp a | .sin a | .cos a | .log a => checkDyadicADFragment a
+  | _ => false
+
+/-- Box-dependent domain check for the Dyadic AD kernel. -/
+def checkDyadicADDomain (e : Expr) (rho : DualDyadicEnv) (cfg : DyadicConfig := {}) : Bool :=
+  checkDyadicADFragment e && (LeanCert.Internal.Dyadic.evalCached e rho.values cfg).2
+
+/-- Explain the first failed support or domain check. -/
+def diagnoseDyadicADFailure (e : Expr) (rho : DualDyadicEnv)
+    (cfg : DyadicConfig := {}) : EvalError :=
+  if !checkDyadicADFragment e then
+    .unsupportedFeature "Dyadic automatic differentiation"
+  else
+    diagnoseEvalIntervalDyadicFailure e rho.values cfg
+
+/-- Checked Dyadic dual evaluation. -/
+def evalDualDyadicChecked (e : Expr) (rho : DualDyadicEnv) (cfg : DyadicConfig := {}) :
+    EvalResult DualIntervalDyadic :=
+  if cfg.precision ≤ 0 then
+    if checkDyadicADDomain e rho cfg then
+      .ok (LeanCert.Internal.AD.Dyadic.evalTotal e rho cfg)
+    else
+      .error (diagnoseDyadicADFailure e rho cfg)
+  else
+    .error (.invalidConfiguration "Dyadic AD precision must be nonpositive")
+
+/-- Checked value-and-derivative evaluation along coordinate `idx`. -/
+def evalWithDerivDyadicChecked (e : Expr) (rho : IntervalDyadicEnv) (idx : Nat)
+    (cfg : DyadicConfig := {}) : EvalResult DualIntervalDyadic :=
+  evalDualDyadicChecked e (mkDualDyadicEnv rho idx) cfg
+
+/-- Checked Dyadic derivative enclosure along coordinate `idx`. -/
+def derivIntervalDyadicChecked (e : Expr) (rho : IntervalDyadicEnv) (idx : Nat)
+    (cfg : DyadicConfig := {}) : EvalResult IntervalDyadic :=
+  return (← evalWithDerivDyadicChecked e rho idx cfg).der
+
+/-- Checked Dyadic derivative enclosure for a single-variable expression. -/
+def derivIntervalDyadicChecked1 (e : Expr) (I : IntervalDyadic)
+    (cfg : DyadicConfig := {}) : EvalResult IntervalDyadic :=
+  derivIntervalDyadicChecked e (fun _ => I) 0 cfg
+
+/-- Convenience boundary for callers whose boxes use rational endpoints. -/
+def derivIntervalDyadicCheckedOfRat (e : Expr) (rho : IntervalEnv) (idx : Nat)
+    (cfg : DyadicConfig := {}) : EvalResult IntervalDyadic :=
+  if cfg.precision ≤ 0 then
+    derivIntervalDyadicChecked e (toDyadicEnv rho cfg.precision) idx cfg
+  else
+    .error (.invalidConfiguration "Dyadic AD precision must be nonpositive")
+
+/-- Single-variable rational-input convenience boundary. -/
+def derivIntervalDyadicChecked1OfRat (e : Expr) (I : IntervalRat)
+    (cfg : DyadicConfig := {}) : EvalResult IntervalDyadic :=
+  derivIntervalDyadicCheckedOfRat e (fun _ => I) 0 cfg
+
+/-- The value projection of the dual kernel is exactly the ordinary Dyadic
+value kernel. -/
+theorem evalTotalDyadic_val_of_fragment (e : Expr) (rho : DualDyadicEnv)
+    (cfg : DyadicConfig) (hsupp : checkDyadicADFragment e = true) :
+    (LeanCert.Internal.AD.Dyadic.evalTotal e rho cfg).val =
+      LeanCert.Internal.Dyadic.evalUnchecked e rho.values cfg := by
+  induction e with
+  | const | var =>
+      simp [LeanCert.Internal.AD.Dyadic.evalTotal, LeanCert.Internal.Dyadic.evalUnchecked,
+        DualIntervalDyadic.const, DualDyadicEnv.values]
+  | add a b iha ihb | mul a b iha ihb =>
+      simp only [checkDyadicADFragment, Bool.and_eq_true] at hsupp
+      simp [LeanCert.Internal.AD.Dyadic.evalTotal, LeanCert.Internal.Dyadic.evalUnchecked,
+        DualIntervalDyadic.add, DualIntervalDyadic.mul, IntervalDyadic.addRounded,
+        IntervalDyadic.mulRounded, iha hsupp.1, ihb hsupp.2]
+  | neg a ih | inv a ih | exp a ih | sin a ih | cos a ih | log a ih =>
+      simp only [checkDyadicADFragment] at hsupp
+      simp [LeanCert.Internal.AD.Dyadic.evalTotal, LeanCert.Internal.Dyadic.evalUnchecked,
+        DualIntervalDyadic.neg, DualIntervalDyadic.inv, DualIntervalDyadic.exp,
+        DualIntervalDyadic.sin, DualIntervalDyadic.cos, DualIntervalDyadic.log, ih hsupp]
+  | atan | arsinh | atanh | sinc | erf | sinh | cosh | tanh | sqrt | namedConst =>
+      simp [checkDyadicADFragment] at hsupp
+
+private theorem evalDualDyadicChecked_facts {e : Expr} {rho : DualDyadicEnv}
+    {cfg : DyadicConfig} {D : DualIntervalDyadic}
+    (hok : evalDualDyadicChecked e rho cfg = .ok D) :
+    cfg.precision ≤ 0 ∧ checkDyadicADFragment e = true ∧
+      evalDomainValidDyadic e rho.values cfg ∧
+      D = LeanCert.Internal.AD.Dyadic.evalTotal e rho cfg := by
+  simp only [evalDualDyadicChecked] at hok
+  split at hok
+  · rename_i hprec
+    split at hok
+    · rename_i hcheck
+      have hparts : checkDyadicADFragment e = true ∧
+          (LeanCert.Internal.Dyadic.evalCached e rho.values cfg).2 = true := by
+        simpa only [checkDyadicADDomain, Bool.and_eq_true] using hcheck
+      have hdomainCheck : checkDomainValidDyadic e rho.values cfg = true := by
+        rw [← evalIntervalDyadicCached_snd e rho.values cfg]
+        exact hparts.2
+      exact ⟨hprec, hparts.1,
+        checkDomainValidDyadic_correct e rho.values cfg hdomainCheck,
+        (Except.ok.inj hok).symm⟩
+    · contradiction
+  · contradiction
+
+private theorem evalTotalDyadic_val_correct_of_facts (e : Expr)
+    (rhoReal : Nat → ℝ) (rho : DualDyadicEnv) (cfg : DyadicConfig)
+    (hprec : cfg.precision ≤ 0) (hsupp : checkDyadicADFragment e = true)
+    (hdom : evalDomainValidDyadic e rho.values cfg)
+    (hrho : ∀ i, rhoReal i ∈ (rho i).val) :
+    Expr.eval rhoReal e ∈ (LeanCert.Internal.AD.Dyadic.evalTotal e rho cfg).val := by
+  rw [evalTotalDyadic_val_of_fragment e rho cfg hsupp]
+  exact evalIntervalDyadic_correct_of_domain e rhoReal rho.values hrho cfg hprec hdom
+
+/-- Golden theorem for successful checked Dyadic dual evaluation. -/
+theorem evalDualDyadicChecked_val_correct (e : Expr)
+    (rhoReal : Nat → ℝ) (rho : DualDyadicEnv) (cfg : DyadicConfig)
+    (D : DualIntervalDyadic) (hrho : ∀ i, rhoReal i ∈ (rho i).val)
+    (hok : evalDualDyadicChecked e rho cfg = .ok D) :
+    Expr.eval rhoReal e ∈ D.val := by
+  obtain ⟨hprec, hsupp, hdom, rfl⟩ := evalDualDyadicChecked_facts hok
+  exact evalTotalDyadic_val_correct_of_facts e rhoReal rho cfg hprec hsupp hdom hrho
+
+private theorem updateVar_mem_mkDualDyadicEnv_val
+    (rhoReal : Nat → ℝ) (rho : IntervalDyadicEnv) (idx : Nat)
+    (x : ℝ) (hx : x ∈ rho idx) (hrho : ∀ i, rhoReal i ∈ rho i) :
+    ∀ i, Expr.updateVar rhoReal idx x i ∈ (mkDualDyadicEnv rho idx i).val := by
+  intro i
+  by_cases hi : i = idx
+  · subst i
+    simpa [Expr.updateVar, mkDualDyadicEnv, DualIntervalDyadic.varActive] using hx
+  · simpa [Expr.updateVar, mkDualDyadicEnv, hi, DualIntervalDyadic.varPassive] using hrho i
+
+private theorem evalAlong_differentiableAt_of_dyadic_facts (e : Expr)
+    (rhoReal : Nat → ℝ) (rho : IntervalDyadicEnv) (idx : Nat) (cfg : DyadicConfig)
+    (x : ℝ) (hx : x ∈ rho idx) (hrho : ∀ i, rhoReal i ∈ rho i)
+    (hprec : cfg.precision ≤ 0) (hsupp : checkDyadicADFragment e = true)
+    (hdom : evalDomainValidDyadic e rho cfg) :
+    DifferentiableAt ℝ (Expr.evalAlong e rhoReal idx) x := by
+  have hmem := updateVar_mem_mkDualDyadicEnv_val rhoReal rho idx x hx hrho
+  have hmemValue : envMemDyadic (Expr.updateVar rhoReal idx x) rho := by
+    intro i
+    by_cases hi : i = idx
+    · subst i
+      simpa [Expr.updateVar] using hx
+    · simpa [Expr.updateVar, hi] using hrho i
+  induction e with
+  | const => exact differentiableAt_const _
+  | var i =>
+      by_cases hi : i = idx
+      · subst i
+        simpa only [Expr.evalAlong_var_active] using differentiableAt_id
+      · simpa only [Expr.evalAlong_var_passive _ _ _ hi] using
+          differentiableAt_const (c := rhoReal i)
+  | add a b iha ihb =>
+      simp only [checkDyadicADFragment, Bool.and_eq_true] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      simpa only [Expr.evalAlong_add_pi] using
+        (iha hsupp.1 hdom.1).add (ihb hsupp.2 hdom.2)
+  | mul a b iha ihb =>
+      simp only [checkDyadicADFragment, Bool.and_eq_true] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      simpa only [Expr.evalAlong_mul_pi] using
+        (iha hsupp.1 hdom.1).mul (ihb hsupp.2 hdom.2)
+  | neg a ih =>
+      simp only [checkDyadicADFragment] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      simpa only [Expr.evalAlong_neg_pi] using (ih hsupp hdom).neg
+  | inv a ih =>
+      simp only [checkDyadicADFragment] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      have hval := evalIntervalDyadic_correct_of_domain a
+        (Expr.updateVar rhoReal idx x) rho hmemValue cfg hprec hdom.1
+      have hne : Expr.evalAlong a rhoReal idx x ≠ 0 := by
+        rw [Expr.evalAlong_eq]
+        intro hz
+        rw [hz] at hval
+        rcases hdom.2 with hpos | hneg
+        · exact (not_lt_of_ge (IntervalDyadic.mem_toIntervalRat.mp hval).1)
+            (by exact_mod_cast hpos)
+        · exact (not_lt_of_ge (IntervalDyadic.mem_toIntervalRat.mp hval).2)
+            (by exact_mod_cast hneg)
+      exact DifferentiableAt.inv (ih hsupp hdom.1) hne
+  | exp a ih =>
+      simp only [checkDyadicADFragment] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      exact Real.differentiableAt_exp.comp x (ih hsupp hdom)
+  | sin a ih =>
+      simp only [checkDyadicADFragment] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      exact Real.differentiableAt_sin.comp x (ih hsupp hdom)
+  | cos a ih =>
+      simp only [checkDyadicADFragment] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      exact Real.differentiableAt_cos.comp x (ih hsupp hdom)
+  | log a ih =>
+      simp only [checkDyadicADFragment] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      have hval := evalIntervalDyadic_correct_of_domain a
+        (Expr.updateVar rhoReal idx x) rho hmemValue cfg hprec hdom.1
+      have hpos : 0 < Expr.evalAlong a rhoReal idx x := by
+        rw [Expr.evalAlong_eq]
+        exact lt_of_lt_of_le (by exact_mod_cast hdom.2)
+          (IntervalDyadic.mem_toIntervalRat.mp hval).1
+      exact (Real.differentiableAt_log (ne_of_gt hpos)).comp x (ih hsupp hdom.1)
+  | atan | arsinh | atanh | sinc | erf | sinh | cosh | tanh | sqrt | namedConst =>
+      simp [checkDyadicADFragment] at hsupp
+
+/-- Successful checked Dyadic indexed AD proves differentiability throughout
+the selected input interval. -/
+theorem evalWithDerivDyadicChecked_differentiableAt (e : Expr)
+    (rhoReal : Nat → ℝ) (rho : IntervalDyadicEnv) (idx : Nat) (cfg : DyadicConfig)
+    (D : DualIntervalDyadic) (x : ℝ) (hx : x ∈ rho idx)
+    (hrho : ∀ i, rhoReal i ∈ rho i)
+    (hok : evalWithDerivDyadicChecked e rho idx cfg = .ok D) :
+    DifferentiableAt ℝ (Expr.evalAlong e rhoReal idx) x := by
+  obtain ⟨hprec, hsupp, hdom, _⟩ := evalDualDyadicChecked_facts hok
+  exact evalAlong_differentiableAt_of_dyadic_facts e rhoReal rho idx cfg x hx hrho
+    hprec hsupp (by simpa using hdom)
+private theorem evalTotalDyadic_der_correct_of_facts (e : Expr)
+    (rhoReal : Nat → ℝ) (rho : IntervalDyadicEnv) (idx : Nat) (cfg : DyadicConfig)
+    (x : ℝ) (hx : x ∈ rho idx) (hrho : ∀ i, rhoReal i ∈ rho i)
+    (hprec : cfg.precision ≤ 0) (hsupp : checkDyadicADFragment e = true)
+    (hdom : evalDomainValidDyadic e rho cfg) :
+    deriv (Expr.evalAlong e rhoReal idx) x ∈
+      (LeanCert.Internal.AD.Dyadic.evalTotal e (mkDualDyadicEnv rho idx) cfg).der := by
+  have hmemValue : envMemDyadic (Expr.updateVar rhoReal idx x) rho := by
+    intro i
+    by_cases hi : i = idx
+    · subst i
+      simpa [Expr.updateVar] using hx
+    · simpa [Expr.updateVar, hi] using hrho i
+  induction e with
+  | const q =>
+      simp only [Expr.evalAlong_const', deriv_const, LeanCert.Internal.AD.Dyadic.evalTotal,
+        DualIntervalDyadic.const]
+      norm_num [DualIntervalDyadic.zero, IntervalDyadic.mem_def, IntervalDyadic.singleton,
+        Core.Dyadic.zero, Core.Dyadic.toRat]
+  | var i =>
+      by_cases hi : i = idx
+      · subst i
+        simp only [Expr.evalAlong_var_active, deriv_id, LeanCert.Internal.AD.Dyadic.evalTotal,
+          mkDualDyadicEnv, ↓reduceIte, DualIntervalDyadic.varActive]
+        simpa [DualIntervalDyadic.one, Core.Dyadic.toRat_ofInt] using
+          IntervalDyadic.mem_singleton (Core.Dyadic.ofInt 1)
+      · simp only [Expr.evalAlong_var_passive _ _ _ hi, deriv_const,
+          LeanCert.Internal.AD.Dyadic.evalTotal, mkDualDyadicEnv, ite_eq_right hi,
+          DualIntervalDyadic.varPassive]
+        norm_num [DualIntervalDyadic.zero, IntervalDyadic.mem_def, IntervalDyadic.singleton,
+          Core.Dyadic.zero, Core.Dyadic.toRat]
+  | add a b iha ihb =>
+      simp only [checkDyadicADFragment, Bool.and_eq_true] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      have hda := evalAlong_differentiableAt_of_dyadic_facts a rhoReal rho idx cfg x hx hrho
+        hprec hsupp.1 hdom.1
+      have hdb := evalAlong_differentiableAt_of_dyadic_facts b rhoReal rho idx cfg x hx hrho
+        hprec hsupp.2 hdom.2
+      simp only [Expr.evalAlong_add_pi, deriv_add hda hdb,
+        LeanCert.Internal.AD.Dyadic.evalTotal, DualIntervalDyadic.add,
+        IntervalDyadic.addRounded]
+      exact IntervalDyadic.roundOut_contains
+        (IntervalDyadic.mem_add (iha hsupp.1 hdom.1) (ihb hsupp.2 hdom.2)) cfg.precision
+  | mul a b iha ihb =>
+      simp only [checkDyadicADFragment, Bool.and_eq_true] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      have hda := evalAlong_differentiableAt_of_dyadic_facts a rhoReal rho idx cfg x hx hrho
+        hprec hsupp.1 hdom.1
+      have hdb := evalAlong_differentiableAt_of_dyadic_facts b rhoReal rho idx cfg x hx hrho
+        hprec hsupp.2 hdom.2
+      have hmemDual := updateVar_mem_mkDualDyadicEnv_val rhoReal rho idx x hx hrho
+      have hva := evalTotalDyadic_val_correct_of_facts a
+        (Expr.updateVar rhoReal idx x) (mkDualDyadicEnv rho idx) cfg hprec hsupp.1
+        (by simpa using hdom.1) hmemDual
+      have hvb := evalTotalDyadic_val_correct_of_facts b
+        (Expr.updateVar rhoReal idx x) (mkDualDyadicEnv rho idx) cfg hprec hsupp.2
+        (by simpa using hdom.2) hmemDual
+      have hleft := IntervalDyadic.roundOut_contains
+        (IntervalDyadic.mem_mul (iha hsupp.1 hdom.1) hvb) cfg.precision
+      have hright := IntervalDyadic.roundOut_contains
+        (IntervalDyadic.mem_mul hva (ihb hsupp.2 hdom.2)) cfg.precision
+      simp only [Expr.evalAlong_mul_pi, deriv_mul hda hdb,
+        LeanCert.Internal.AD.Dyadic.evalTotal, DualIntervalDyadic.mul,
+        IntervalDyadic.mulRounded, IntervalDyadic.addRounded]
+      exact IntervalDyadic.roundOut_contains (IntervalDyadic.mem_add hleft hright) cfg.precision
+  | neg a ih =>
+      simp only [checkDyadicADFragment] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      have hd := evalAlong_differentiableAt_of_dyadic_facts a rhoReal rho idx cfg x hx hrho
+        hprec hsupp hdom
+      simp only [Expr.evalAlong_neg_pi, deriv.neg, LeanCert.Internal.AD.Dyadic.evalTotal,
+        DualIntervalDyadic.neg]
+      exact IntervalDyadic.mem_neg (ih hsupp hdom)
+  | inv a ih =>
+      simp only [checkDyadicADFragment] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      have hval := evalIntervalDyadic_correct_of_domain a
+        (Expr.updateVar rhoReal idx x) rho hmemValue cfg hprec hdom.1
+      have hval' : Expr.evalAlong a rhoReal idx x ∈
+          LeanCert.Internal.Dyadic.evalUnchecked a rho cfg := by
+        simpa only [Expr.evalAlong_eq] using hval
+      have hne : Expr.evalAlong a rhoReal idx x ≠ 0 := by
+        intro hz
+        rw [hz] at hval'
+        rcases hdom.2 with hpos | hneg
+        · exact (not_lt_of_ge (IntervalDyadic.mem_toIntervalRat.mp hval').1)
+            (by exact_mod_cast hpos)
+        · exact (not_lt_of_ge (IntervalDyadic.mem_toIntervalRat.mp hval').2)
+            (by exact_mod_cast hneg)
+      have hd := evalAlong_differentiableAt_of_dyadic_facts a rhoReal rho idx cfg x hx hrho
+        hprec hsupp hdom.1
+      rw [Expr.evalAlong_inv]
+      have hcomp : (fun t => (Expr.evalAlong a rhoReal idx t)⁻¹) =
+          (fun y : ℝ => y⁻¹) ∘ Expr.evalAlong a rhoReal idx := rfl
+      rw [hcomp, deriv_comp x (hasDerivAt_inv hne).differentiableAt hd]
+      simp only [(hasDerivAt_inv hne).deriv, LeanCert.Internal.AD.Dyadic.evalTotal,
+        DualIntervalDyadic.inv]
+      rw [evalTotalDyadic_val_of_fragment a (mkDualDyadicEnv rho idx) cfg hsupp]
+      simp only [mkDualDyadicEnv_values]
+      have hrat := IntervalDyadic.mem_toIntervalRat.mp hval'
+      have hinvRat := mem_invInterval_nonzero hrat hdom.2
+      have hinv : (Expr.evalAlong a rhoReal idx x)⁻¹ ∈
+          invIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked a rho cfg) cfg := by
+        exact IntervalDyadic.mem_ofIntervalRat hinvRat cfg.precision hprec
+      have hinvSq := IntervalDyadic.roundOut_contains
+        (IntervalDyadic.mem_mul hinv hinv) cfg.precision
+      have hprod := IntervalDyadic.roundOut_contains
+        (IntervalDyadic.mem_mul (ih hsupp hdom.1) hinvSq) cfg.precision
+      have hneg := IntervalDyadic.mem_neg hprod
+      simpa [IntervalDyadic.mulRounded, pow_two, mul_assoc, mul_comm, mul_left_comm] using hneg
+  | exp a ih =>
+      simp only [checkDyadicADFragment] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      have hd := evalAlong_differentiableAt_of_dyadic_facts a rhoReal rho idx cfg x hx hrho
+        hprec hsupp hdom
+      have hval := evalIntervalDyadic_correct_of_domain a
+        (Expr.updateVar rhoReal idx x) rho hmemValue cfg hprec hdom
+      have hexpRat := IntervalRat.mem_expComputable
+        (IntervalDyadic.mem_toIntervalRat.mp hval) cfg.taylorDepth
+      have hexp : Real.exp (Expr.evalAlong a rhoReal idx x) ∈
+          expIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked a rho cfg) cfg := by
+        simpa [Expr.evalAlong_eq, expIntervalDyadic] using
+          (IntervalDyadic.mem_ofIntervalRat hexpRat cfg.precision hprec)
+      simp only [Expr.evalAlong_exp, deriv_exp hd, LeanCert.Internal.AD.Dyadic.evalTotal,
+        DualIntervalDyadic.exp, IntervalDyadic.mulRounded]
+      rw [evalTotalDyadic_val_of_fragment a (mkDualDyadicEnv rho idx) cfg hsupp]
+      simp only [mkDualDyadicEnv_values]
+      exact IntervalDyadic.roundOut_contains
+        (IntervalDyadic.mem_mul hexp (ih hsupp hdom)) cfg.precision
+  | sin a ih =>
+      simp only [checkDyadicADFragment] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      have hd := evalAlong_differentiableAt_of_dyadic_facts a rhoReal rho idx cfg x hx hrho
+        hprec hsupp hdom
+      have hval := evalIntervalDyadic_correct_of_domain a
+        (Expr.updateVar rhoReal idx x) rho hmemValue cfg hprec hdom
+      have hcosRat := IntervalRat.mem_cosComputable
+        (IntervalDyadic.mem_toIntervalRat.mp hval) cfg.taylorDepth
+      have hcos : Real.cos (Expr.evalAlong a rhoReal idx x) ∈
+          cosIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked a rho cfg) cfg := by
+        simpa [Expr.evalAlong_eq, cosIntervalDyadic] using
+          (IntervalDyadic.mem_ofIntervalRat hcosRat cfg.precision hprec)
+      simp only [Expr.evalAlong_sin, deriv_sin hd, LeanCert.Internal.AD.Dyadic.evalTotal,
+        DualIntervalDyadic.sin, IntervalDyadic.mulRounded]
+      rw [evalTotalDyadic_val_of_fragment a (mkDualDyadicEnv rho idx) cfg hsupp]
+      simp only [mkDualDyadicEnv_values]
+      exact IntervalDyadic.roundOut_contains
+        (IntervalDyadic.mem_mul hcos (ih hsupp hdom)) cfg.precision
+  | cos a ih =>
+      simp only [checkDyadicADFragment] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      have hd := evalAlong_differentiableAt_of_dyadic_facts a rhoReal rho idx cfg x hx hrho
+        hprec hsupp hdom
+      have hval := evalIntervalDyadic_correct_of_domain a
+        (Expr.updateVar rhoReal idx x) rho hmemValue cfg hprec hdom
+      have hsinRat := IntervalRat.mem_sinComputable
+        (IntervalDyadic.mem_toIntervalRat.mp hval) cfg.taylorDepth
+      have hsin : Real.sin (Expr.evalAlong a rhoReal idx x) ∈
+          sinIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked a rho cfg) cfg := by
+        simpa [Expr.evalAlong_eq, sinIntervalDyadic] using
+          (IntervalDyadic.mem_ofIntervalRat hsinRat cfg.precision hprec)
+      simp only [Expr.evalAlong_cos, deriv_cos hd, LeanCert.Internal.AD.Dyadic.evalTotal,
+        DualIntervalDyadic.cos, IntervalDyadic.mulRounded]
+      rw [evalTotalDyadic_val_of_fragment a (mkDualDyadicEnv rho idx) cfg hsupp]
+      simp only [mkDualDyadicEnv_values]
+      exact IntervalDyadic.roundOut_contains
+        (IntervalDyadic.mem_mul (IntervalDyadic.mem_neg hsin) (ih hsupp hdom)) cfg.precision
+  | log a ih =>
+      simp only [checkDyadicADFragment] at hsupp
+      simp only [evalDomainValidDyadic] at hdom
+      have hval := evalIntervalDyadic_correct_of_domain a
+        (Expr.updateVar rhoReal idx x) rho hmemValue cfg hprec hdom.1
+      have hval' : Expr.evalAlong a rhoReal idx x ∈
+          LeanCert.Internal.Dyadic.evalUnchecked a rho cfg := by
+        simpa only [Expr.evalAlong_eq] using hval
+      have hpos : 0 < Expr.evalAlong a rhoReal idx x :=
+        lt_of_lt_of_le (by exact_mod_cast hdom.2)
+          (IntervalDyadic.mem_toIntervalRat.mp hval').1
+      have hd := evalAlong_differentiableAt_of_dyadic_facts a rhoReal rho idx cfg x hx hrho
+        hprec hsupp hdom.1
+      rw [Expr.evalAlong_log]
+      have hcomp : (fun t => Real.log (Expr.evalAlong a rhoReal idx t)) =
+          Real.log ∘ Expr.evalAlong a rhoReal idx := rfl
+      rw [hcomp, deriv_comp x (Real.differentiableAt_log (ne_of_gt hpos)) hd]
+      simp only [Real.deriv_log, LeanCert.Internal.AD.Dyadic.evalTotal,
+        DualIntervalDyadic.log, IntervalDyadic.mulRounded]
+      rw [evalTotalDyadic_val_of_fragment a (mkDualDyadicEnv rho idx) cfg hsupp]
+      simp only [mkDualDyadicEnv_values]
+      have hnz : (LeanCert.Internal.Dyadic.evalUnchecked a rho cfg).toIntervalRat.lo > 0 ∨
+          (LeanCert.Internal.Dyadic.evalUnchecked a rho cfg).toIntervalRat.hi < 0 := Or.inl hdom.2
+      have hinvRat := mem_invInterval_nonzero
+        (IntervalDyadic.mem_toIntervalRat.mp hval') hnz
+      have hinv : (Expr.evalAlong a rhoReal idx x)⁻¹ ∈
+          invIntervalDyadic (LeanCert.Internal.Dyadic.evalUnchecked a rho cfg) cfg :=
+        IntervalDyadic.mem_ofIntervalRat hinvRat cfg.precision hprec
+      exact IntervalDyadic.roundOut_contains
+        (IntervalDyadic.mem_mul hinv (ih hsupp hdom.1)) cfg.precision
+  | atan | arsinh | atanh | sinc | erf | sinh | cosh | tanh | sqrt | namedConst =>
+      simp [checkDyadicADFragment] at hsupp
+
+/-- Golden theorem: successful checked Dyadic indexed AD encloses the true
+partial derivative. -/
+theorem evalWithDerivDyadicChecked_der_correct (e : Expr)
+    (rhoReal : Nat → ℝ) (rho : IntervalDyadicEnv) (idx : Nat) (cfg : DyadicConfig)
+    (D : DualIntervalDyadic) (x : ℝ) (hx : x ∈ rho idx)
+    (hrho : ∀ i, rhoReal i ∈ rho i)
+    (hok : evalWithDerivDyadicChecked e rho idx cfg = .ok D) :
+    deriv (Expr.evalAlong e rhoReal idx) x ∈ D.der := by
+  obtain ⟨hprec, hsupp, hdom, rfl⟩ := evalDualDyadicChecked_facts hok
+  exact evalTotalDyadic_der_correct_of_facts e rhoReal rho idx cfg x hx hrho hprec hsupp
+    (by simpa using hdom)
+
+/-- Golden theorem for the derivative-only Dyadic API. -/
+theorem derivIntervalDyadicChecked_correct (e : Expr)
+    (rhoReal : Nat → ℝ) (rho : IntervalDyadicEnv) (idx : Nat) (cfg : DyadicConfig)
+    (dI : IntervalDyadic) (x : ℝ) (hx : x ∈ rho idx)
+    (hrho : ∀ i, rhoReal i ∈ rho i)
+    (hok : derivIntervalDyadicChecked e rho idx cfg = .ok dI) :
+    deriv (Expr.evalAlong e rhoReal idx) x ∈ dI := by
+  simp only [derivIntervalDyadicChecked, bind, Except.bind] at hok
+  cases hdual : evalWithDerivDyadicChecked e rho idx cfg with
+  | error err => simp [hdual] at hok
+  | ok D =>
+      rw [hdual] at hok
+      simp only [pure, Except.pure, Except.ok.injEq] at hok
+      subst dI
+      exact evalWithDerivDyadicChecked_der_correct e rhoReal rho idx cfg D x hx hrho hdual
+
+/-- Golden theorem for rational input boxes converted to the Dyadic backend. -/
+theorem derivIntervalDyadicCheckedOfRat_correct (e : Expr)
+    (rhoReal : Nat → ℝ) (rho : IntervalEnv) (idx : Nat) (cfg : DyadicConfig)
+    (dI : IntervalDyadic) (x : ℝ) (hx : x ∈ rho idx)
+    (hrho : ∀ i, rhoReal i ∈ rho i)
+    (hok : derivIntervalDyadicCheckedOfRat e rho idx cfg = .ok dI) :
+    deriv (Expr.evalAlong e rhoReal idx) x ∈ dI := by
+  simp only [derivIntervalDyadicCheckedOfRat] at hok
+  split at hok
+  · rename_i hprec
+    apply derivIntervalDyadicChecked_correct e rhoReal
+      (toDyadicEnv rho cfg.precision) idx cfg dI x
+    · exact IntervalDyadic.mem_ofIntervalRat hx cfg.precision hprec
+    · intro i
+      exact IntervalDyadic.mem_ofIntervalRat (hrho i) cfg.precision hprec
+    · exact hok
+  · contradiction
+
+/-- Compute the first `n` partial derivatives with the checked Dyadic backend. -/
+def gradientIntervalDyadicChecked (e : Expr) (rho : IntervalDyadicEnv) (n : Nat)
+    (cfg : DyadicConfig := {}) : EvalResult (List IntervalDyadic) :=
+  if cfg.precision ≤ 0 then
+    (List.range n).mapM fun i => derivIntervalDyadicChecked e rho i cfg
+  else
+    .error (.invalidConfiguration "Dyadic AD precision must be nonpositive")
+
+/-- Rational-input convenience boundary for the first `n` Dyadic partials. -/
+def gradientIntervalDyadicCheckedOfRat (e : Expr) (rho : IntervalEnv) (n : Nat)
+    (cfg : DyadicConfig := {}) : EvalResult (List IntervalDyadic) :=
+  if cfg.precision ≤ 0 then
+    gradientIntervalDyadicChecked e (toDyadicEnv rho cfg.precision) n cfg
+  else
+    .error (.invalidConfiguration "Dyadic AD precision must be nonpositive")
+
+private theorem derivIntervalsDyadicChecked_correct (e : Expr)
+    (rhoReal : Nat → ℝ) (rho : IntervalDyadicEnv) (cfg : DyadicConfig)
+    (hrho : ∀ i, rhoReal i ∈ rho i) (indices : List Nat)
+    (gradient : List IntervalDyadic)
+    (hok : indices.mapM (fun i => derivIntervalDyadicChecked e rho i cfg) = .ok gradient) :
+    List.Forall₂ (fun i dI => deriv (Expr.evalAlong e rhoReal i) (rhoReal i) ∈ dI)
+      indices gradient := by
+  induction indices generalizing gradient with
+  | nil =>
+      simp only [List.mapM_nil, pure, Except.pure, Except.ok.injEq] at hok
+      subst gradient
+      exact .nil
+  | cons i indices ih =>
+      simp only [List.mapM_cons] at hok
+      cases hdi : derivIntervalDyadicChecked e rho i cfg with
+      | error err =>
+          rw [hdi] at hok
+          simp only [bind, Except.bind] at hok
+          cases hok
+      | ok dI =>
+          rw [hdi] at hok
+          simp only [bind, Except.bind] at hok
+          cases htail : indices.mapM (fun j => derivIntervalDyadicChecked e rho j cfg) with
+          | error err =>
+              rw [htail] at hok
+              cases hok
+          | ok tail =>
+              rw [htail] at hok
+              simp only [pure, Except.pure, Except.ok.injEq] at hok
+              subst gradient
+              exact .cons
+                (derivIntervalDyadicChecked_correct e rhoReal rho i cfg dI
+                  (rhoReal i) (hrho i) hrho hdi)
+                (ih tail htail)
+
+/-- Golden theorem for the checked Dyadic gradient API. -/
+theorem gradientIntervalDyadicChecked_correct (e : Expr)
+    (rhoReal : Nat → ℝ) (rho : IntervalDyadicEnv) (n : Nat) (cfg : DyadicConfig)
+    (hrho : ∀ i, rhoReal i ∈ rho i) (gradient : List IntervalDyadic)
+    (hok : gradientIntervalDyadicChecked e rho n cfg = .ok gradient) :
+    List.Forall₂ (fun i dI => deriv (Expr.evalAlong e rhoReal i) (rhoReal i) ∈ dI)
+      (List.range n) gradient := by
+  simp only [gradientIntervalDyadicChecked] at hok
+  split at hok
+  · exact derivIntervalsDyadicChecked_correct e rhoReal rho cfg hrho
+      (List.range n) gradient hok
+  · contradiction
+
+/-- Golden theorem for checked Dyadic gradients over rational input boxes. -/
+theorem gradientIntervalDyadicCheckedOfRat_correct (e : Expr)
+    (rhoReal : Nat → ℝ) (rho : IntervalEnv) (n : Nat) (cfg : DyadicConfig)
+    (hrho : ∀ i, rhoReal i ∈ rho i) (gradient : List IntervalDyadic)
+    (hok : gradientIntervalDyadicCheckedOfRat e rho n cfg = .ok gradient) :
+    List.Forall₂ (fun i dI => deriv (Expr.evalAlong e rhoReal i) (rhoReal i) ∈ dI)
+      (List.range n) gradient := by
+  simp only [gradientIntervalDyadicCheckedOfRat] at hok
+  split at hok
+  · rename_i hprec
+    apply gradientIntervalDyadicChecked_correct e rhoReal
+      (toDyadicEnv rho cfg.precision) n cfg
+    · intro i
+      exact IntervalDyadic.mem_ofIntervalRat (hrho i) cfg.precision hprec
+    · exact hok
+  · contradiction
+
+end LeanCert.Engine
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Automatic Differentiation via Intervals
+
+This module provides forward-mode automatic differentiation using interval
+arithmetic. We compute both the value and derivative of an expression,
+with rigorous bounds on both.
+
+## Module Structure
+
+* `AD.Basic` - Core types: `DualInterval`, basic operations (`add`, `mul`, `neg`)
+* `AD.Transcendental` - Transcendental functions (`sin`, `cos`, `exp`, etc.)
+* `AD.Eval` - Evaluators: `LeanCert.Internal.AD.evalUnchecked`, `evalDualOption`, `derivInterval`
+* `AD.Correctness` - Correctness theorems for supported expressions
+* `AD.PartialCorrectness` - Correctness for partial functions (inv, log, sqrt)
+* `AD.Computable` - Taylor-based computable evaluators
+* `AD.DomainChecked` - Computable AD with box-dependent checks for inv and log
+* `AD.Dyadic` - Domain-aware AD with bounded-denominator Dyadic arithmetic
+
+## Main definitions
+
+* `DualInterval` - A pair of intervals representing (value, derivative)
+* `LeanCert.Internal.AD.evalUnchecked` - Evaluate expression to get value and derivative intervals
+* `evalDualOption` - Partial evaluator supporting inv, log, sqrt
+* `LeanCert.Internal.AD.evalTotalCore` - Computable evaluator for native_decide
+* `evalDualChecked`, `derivIntervalChecked` - Computable, structured-failure APIs for inv/log
+* `evalDualDyadicChecked`, `derivIntervalDyadicChecked` - Checked Dyadic counterparts
+
+## Main theorems
+
+* `LeanCert.Engine.evalDualUnchecked_val_correct` - Value component is correct for supported
+expressions
+* `LeanCert.Engine.evalDualUnchecked_der_correct` - Derivative component is correct for supported
+expressions
+* `evalDualOption_val_correct`, `evalDualOption_der_correct` - Correctness with domain checks
+* `LeanCert.Internal.AD.evalTotalCore_val_correct`, `LeanCert.Internal.AD.evalTotalCore_der_correct`
+- Computable correctness
+* `evalWithDerivChecked_der_correct`, `derivIntervalChecked_correct` - Golden theorems for
+  successful domain-aware computation
+
+All theorems are FULLY PROVED with no sorry or axioms.
+-/
+
+@[expose] public section
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# N-Dimensional Boxes for Global Optimization
+
+This file provides the `Box` type for representing n-dimensional domains
+as products of intervals, along with helper functions for branch-and-bound
+optimization.
+
+## Main definitions
+
+* `Box` - A list of intervals representing an n-dimensional box
+* `Box.toEnv` - Convert a box to an interval environment
+* `Box.mem` - Membership: a point (list of reals) is in a box
+* `Box.widestDim` - Find the dimension with the largest width (for splitting)
+* `Box.split` - Split a box along a given dimension
+* `Box.volume` - Product of interval widths (heuristic measure)
+
+## Design
+
+A `Box` is represented as `List IntervalRat`, where the i-th element is the
+interval for the i-th variable. This allows boxes of any dimension.
+
+For expressions with `n` variables (var 0 through var n-1), a box should
+have at least `n` intervals. The conversion `Box.toEnv` returns `default`
+for out-of-bounds indices.
+-/
+
+/-! ### Box type and basic operations -/
+
+@[expose] public section
+
+namespace LeanCert.Engine.Optimization
+
+open LeanCert.Core
+open LeanCert.Engine
+
+/-- An n-dimensional box as a list of intervals.
+    The i-th element is the interval for variable i. -/
+abbrev Box := List IntervalRat
+
+/-- A point in ℝⁿ represented as a list of reals -/
+abbrev Point := List ℝ
+
+/-- A rational point in ℚⁿ -/
+abbrev PointQ := List ℚ
+
+namespace Box
+
+/-- Create a one-dimensional box from a single interval. -/
+def ofInterval (I : IntervalRat) : Box := [I]
+
+/-- Convert a box to an interval environment.
+    Returns `default` for out-of-bounds indices (i.e., the whole real line approximation). -/
+def toEnv (B : Box) : IntervalEnv :=
+  fun i => B.getD i default
+
+/-- The dimension (number of intervals) of a box -/
+def dim (B : Box) : ℕ := B.length
+
+/-- Check if a point is in a box.
+    A point is in a box if each coordinate is in the corresponding interval. -/
+def mem (p : Point) (B : Box) : Prop :=
+  ∃ (h : p.length = B.length), ∀ i : Fin B.length, p[i.val]'(by rw [h]; exact i.isLt) ∈
+    B[i.val]'(by exact i.isLt)
+
+/-- Membership for a real function (environment) in a box.
+    This is the key notion for correctness: ρ ∈ B means ρ i ∈ B[i] for all i < B.length. -/
+def envMem (ρ : Nat → ℝ) (B : Box) : Prop :=
+  ∀ i : Fin B.length, ρ i.val ∈ B[i.val]'(by exact i.isLt)
+
+/-- envMem implies the general IntervalEnv membership for toEnv
+    NOTE: For indices beyond the box dimension, we use the default interval [0,0].
+    This means expressions should only use variables 0..B.length-1 for correct results. -/
+theorem envMem_toEnv (ρ : Nat → ℝ) (B : Box) (h : envMem ρ B)
+    (hzero : ∀ i, i ≥ B.length → ρ i = 0) :
+    LeanCert.Engine.envMem ρ (toEnv B) := by
+  intro i
+  unfold toEnv
+  by_cases hi : i < B.length
+  · simp only [List.getD, List.getElem?_eq_getElem hi, Option.getD]
+    exact h ⟨i, hi⟩
+  · have hge : B.length ≤ i := not_lt.mp hi
+    simp only [List.getD, List.getElem?_eq_none (not_lt.mp hi), Option.getD]
+    rw [IntervalRat.mem_default]
+    exact hzero i hge
+
+/-! ### Width and dimension selection -/
+
+/-- Width of an interval -/
+def intervalWidth (I : IntervalRat) : ℚ := I.hi - I.lo
+
+/-- Get the width of each dimension -/
+def widths (B : Box) : List ℚ := B.map intervalWidth
+
+/-- Find the index of the maximum element in a list (returns 0 for empty list) -/
+def maxIdx (xs : List ℚ) : Nat :=
+  match xs with
+  | [] => 0
+  | [_] => 0
+  | x :: y :: rest =>
+    let restMax := maxIdx (y :: rest)
+    if x ≥ (y :: rest).getD restMax 0 then 0 else restMax + 1
+
+/-- Find the dimension with the widest interval (heuristic for splitting).
+    Returns 0 if the box is empty. -/
+def widestDim (B : Box) : Nat :=
+  maxIdx (widths B)
+
+/-- Alternative: find widest dimension with explicit fold -/
+def widestDim' (B : Box) : Nat :=
+  let indexed := B.zipIdx.map fun (I, i) => (i, intervalWidth I)
+  match indexed.argmax (·.2) with
+  | some (i, _) => i
+  | none => 0
+
+/-! ### Box splitting -/
+
+/-- Split a box along a given dimension by bisecting that interval.
+    Returns the original box if the dimension is out of bounds. -/
+def split (B : Box) (dim : Nat) : Box × Box :=
+  if h : dim < B.length then
+    let I := B[dim]'h
+    let (I₁, I₂) := I.bisect
+    let B₁ := B.set dim I₁
+    let B₂ := B.set dim I₂
+    (B₁, B₂)
+  else
+    (B, B)
+
+/-- Split along the widest dimension -/
+def splitWidest (B : Box) : Box × Box :=
+  split B (widestDim B)
+
+/-- Splitting preserves the number of box coordinates. -/
+theorem split_length_eq (B : Box) (d : Nat) :
+    (split B d).1.length = B.length ∧ (split B d).2.length = B.length := by
+  simp only [split]
+  split_ifs with h
+  · constructor <;> simp only [List.length_set]
+  · exact ⟨rfl, rfl⟩
+
+/-- The lower child of a split is a semantic sub-box of its parent. -/
+theorem envMem_of_envMem_split (B : Box) (d : Nat) (ρ : Nat → ℝ) :
+    envMem ρ (split B d).1 → envMem ρ B := by
+  intro h
+  unfold split at h
+  split_ifs at h with hd
+  · intro ⟨i, hi⟩
+    have hi' : i < (B.set d (B[d].bisect.1)).length := by
+      simp only [List.length_set]
+      exact hi
+    have hmem := h ⟨i, hi'⟩
+    simp only [List.getElem_set] at hmem
+    split_ifs at hmem with heq
+    · subst heq
+      exact IntervalRat.mem_of_mem_bisect_left hmem
+    · exact hmem
+  · exact h
+
+/-- The upper child of a split is a semantic sub-box of its parent. -/
+theorem envMem_of_envMem_split_right (B : Box) (d : Nat) (ρ : Nat → ℝ) :
+    envMem ρ (split B d).2 → envMem ρ B := by
+  intro h
+  unfold split at h
+  split_ifs at h with hd
+  · intro ⟨i, hi⟩
+    have hi' : i < (B.set d (B[d].bisect.2)).length := by
+      simp only [List.length_set]
+      exact hi
+    have hmem := h ⟨i, hi'⟩
+    simp only [List.getElem_set] at hmem
+    split_ifs at hmem with heq
+    · subst heq
+      exact IntervalRat.mem_of_mem_bisect_right hmem
+    · exact hmem
+  · exact h
+
+/-! ### Volume and size heuristics -/
+
+/-- Volume of a box (product of widths).
+    Returns 0 for empty box. -/
+def volume (B : Box) : ℚ :=
+  (widths B).foldl (· * ·) 1
+
+/-- Maximum width across all dimensions -/
+def maxWidth (B : Box) : ℚ :=
+  (widths B).foldl max 0
+
+/-- Check if a box is "small" (max width below threshold) -/
+def isSmall (B : Box) (threshold : ℚ) : Bool :=
+  maxWidth B ≤ threshold
+
+/-! ### Box construction helpers -/
+
+/-- Create a box from a list of (lo, hi) pairs -/
+def ofPairs (pairs : List (ℚ × ℚ)) : Box :=
+  pairs.filterMap fun (lo, hi) =>
+    if h : lo ≤ hi then some ⟨lo, hi, h⟩ else none
+
+/-- Create a unit box [0,1]ⁿ -/
+def unit (n : Nat) : Box :=
+  List.replicate n ⟨0, 1, by norm_num⟩
+
+/-- Create a symmetric box [-1,1]ⁿ -/
+def symmetric (n : Nat) : Box :=
+  List.replicate n ⟨-1, 1, by norm_num⟩
+
+/-- Create a box from bounds: [lo₀, hi₀] × ... × [loₙ₋₁, hiₙ₋₁] -/
+def ofBounds (los his : List ℚ) (_h : los.length = his.length) : Box :=
+  (los.zip his).filterMap fun (lo, hi) =>
+    if hle : lo ≤ hi then some ⟨lo, hi, hle⟩ else none
+
+/-! ### Membership lemmas -/
+
+/-- If a point is in a box, its i-th coordinate is in the i-th interval -/
+theorem coord_mem_of_mem (p : Point) (B : Box) (h : mem p B) (i : Fin B.length) :
+    p[i.val]'(by obtain ⟨hlen, _⟩ := h; rw [hlen]; exact i.isLt) ∈ B[i.val]'(by exact i.isLt) := by
+  obtain ⟨hlen, hmem⟩ := h
+  exact hmem i
+
+/-- After splitting, any point in the original box is in one of the halves. -/
+theorem mem_split_cases (B : Box) (d : Nat) (hd : d < B.length)
+    (p : Point) (hp : mem p B) :
+    mem p (split B d).1 ∨ mem p (split B d).2 := by
+  simp only [split, hd, ↓reduceDIte]
+  obtain ⟨hp_len, hp_mem⟩ := hp
+  have hp_d : p[d]'(by rw [hp_len]; exact hd) ∈ B[d]'hd := hp_mem ⟨d, hd⟩
+  have hbisect := IntervalRat.mem_bisect_or hp_d
+  cases hbisect with
+  | inl hleft =>
+    left
+    have hlen1 : (B.set d (B[d].bisect.1)).length = B.length := List.length_set
+    use (by rw [hlen1]; exact hp_len)
+    intro ⟨i, hi⟩
+    have hi_orig : i < B.length := by rw [← hlen1]; exact hi
+    by_cases h_eq : i = d
+    · -- i = d: use the bisection result
+      simp only [h_eq, List.getElem_set_self]
+      convert hleft using 2
+    · -- i ≠ d: use original membership
+      have hne : d ≠ i := fun h => h_eq h.symm
+      simp only [List.getElem_set_ne hne hi]
+      have hp_i := hp_mem ⟨i, hi_orig⟩
+      convert hp_i using 2
+  | inr hright =>
+    right
+    have hlen2 : (B.set d (B[d].bisect.2)).length = B.length := List.length_set
+    use (by rw [hlen2]; exact hp_len)
+    intro ⟨i, hi⟩
+    have hi_orig : i < B.length := by rw [← hlen2]; exact hi
+    by_cases h_eq : i = d
+    · simp only [h_eq, List.getElem_set_self]
+      convert hright using 2
+    · have hne : d ≠ i := fun h => h_eq h.symm
+      simp only [List.getElem_set_ne hne hi]
+      have hp_i := hp_mem ⟨i, hi_orig⟩
+      convert hp_i using 2
+
+/-- Environment membership after splitting. -/
+theorem envMem_split_cases (B : Box) (d : Nat) (hd : d < B.length)
+    (ρ : Nat → ℝ) (hρ : envMem ρ B) :
+    envMem ρ (split B d).1 ∨ envMem ρ (split B d).2 := by
+  simp only [split, hd, ↓reduceDIte]
+  have hρ_d : ρ d ∈ B[d]'hd := hρ ⟨d, hd⟩
+  have hbisect := IntervalRat.mem_bisect_or hρ_d
+  -- After bisection, ρ d is in either the left or right half
+  cases hbisect with
+  | inl hleft =>
+    left
+    intro ⟨i, hi⟩
+    -- After B.set d I₁, length is preserved
+    have hlen : (B.set d (B[d].bisect.1)).length = B.length := List.length_set
+    have hi_orig : i < B.length := by rw [← hlen]; exact hi
+    by_cases h_eq : i = d
+    · -- i = d: use the bisection result
+      simp only [h_eq, List.getElem_set_self]
+      exact hleft
+    · -- i ≠ d: use original membership
+      have hne : d ≠ i := fun h => h_eq h.symm
+      simp only [List.getElem_set_ne hne hi]
+      exact hρ ⟨i, hi_orig⟩
+  | inr hright =>
+    right
+    intro ⟨i, hi⟩
+    have hlen : (B.set d (B[d].bisect.2)).length = B.length := List.length_set
+    have hi_orig : i < B.length := by rw [← hlen]; exact hi
+    by_cases h_eq : i = d
+    · simp only [h_eq, List.getElem_set_self]
+      exact hright
+    · have hne : d ≠ i := fun h => h_eq h.symm
+      simp only [List.getElem_set_ne hne hi]
+      exact hρ ⟨i, hi_orig⟩
+
+end Box
+
+end LeanCert.Engine.Optimization
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Global Optimization via Branch-and-Bound
+
+This file implements verified global optimization over intervals using
+branch-and-bound with interval arithmetic and derivative bounds.
+
+## Phase Status
+
+**This module is now largely Phase 1 (verified).** The core correctness theorems
+are fully proved for the ADSupported subset.
+
+## Main definitions
+
+* `minimizeInterval` - Find a lower bound on min f over an interval
+* `maximizeInterval` - Find an upper bound on max f over an interval
+* Correctness theorems (fully proved for ADSupported)
+
+## Algorithm
+
+Branch-and-bound with the following pruning rules:
+1. **Interval evaluation**: If f([a,b]) = [lo, hi], then min f ≥ lo
+2. **Monotonicity**: If f'([a,b]) > 0, f is increasing, so min is at a
+3. **Subdivision**: Split interval and recurse
+
+-/
+
+/-! ### Optimization result -/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+
+open LeanCert.Core
+
+/-- Result of an optimization computation -/
+structure OptResult where
+  /-- Interval containing the optimum value -/
+  valueBound : IntervalRat
+  /-- Interval containing an optimizing point (if found) -/
+  argBound : Option IntervalRat
+  /-- Depth of search performed -/
+  depth : ℕ
+
+/-- Combine two subinterval bounds, preserving their lower and upper bounds. -/
+noncomputable def OptResult.merge (r₁ r₂ : OptResult) : OptResult :=
+  if h : r₁.valueBound.lo ≤ r₂.valueBound.lo then
+    { valueBound :=
+        { lo := r₁.valueBound.lo
+          hi := min r₁.valueBound.hi r₂.valueBound.hi
+          le := by
+            apply le_min
+            · exact r₁.valueBound.le
+            · calc r₁.valueBound.lo ≤ r₂.valueBound.lo := h
+                _ ≤ r₂.valueBound.hi := r₂.valueBound.le }
+      argBound := r₁.argBound
+      depth := max r₁.depth r₂.depth }
+  else
+    { valueBound :=
+        { lo := r₂.valueBound.lo
+          hi := min r₁.valueBound.hi r₂.valueBound.hi
+          le := by
+            apply le_min
+            · calc r₂.valueBound.lo ≤ r₁.valueBound.lo := le_of_lt (not_le.mp h)
+                _ ≤ r₁.valueBound.hi := r₁.valueBound.le
+            · exact r₂.valueBound.le }
+      argBound := r₂.argBound
+      depth := max r₁.depth r₂.depth }
+
+/-! ### Minimization -/
+
+/-- Check if derivative interval is strictly positive -/
+noncomputable def derivStrictlyPositive (e : Expr) (I : IntervalRat) (varIdx : Nat) : Bool :=
+  let dI := derivInterval e (fun _ => I) varIdx
+  0 < dI.lo
+
+/-- Check if derivative interval is strictly negative -/
+noncomputable def derivStrictlyNegative (e : Expr) (I : IntervalRat) (varIdx : Nat) : Bool :=
+  let dI := derivInterval e (fun _ => I) varIdx
+  dI.hi < 0
+
+/-- Simple branch-and-bound minimization.
+
+Returns an interval [lo, hi] such that:
+- lo ≤ min_{x ∈ I} f(x)
+- hi ≥ min_{x ∈ I} f(x)
+-/
+noncomputable def minimizeInterval (e : Expr) (I : IntervalRat) (varIdx : Nat)
+    (maxDepth : ℕ) : OptResult :=
+  go I maxDepth
+where
+  /-- Recursive interval minimization with a decreasing subdivision budget. -/
+  go (J : IntervalRat) (depth : ℕ) : OptResult :=
+    if depth = 0 then
+      -- Base case: just use interval evaluation
+      { valueBound := LeanCert.Internal.Rational.evalUnchecked1 e J
+        argBound := some J
+        depth := maxDepth }
+    else
+      -- Check monotonicity
+      if derivStrictlyPositive e J varIdx then
+        -- f is increasing, minimum is at left endpoint
+        { valueBound := LeanCert.Internal.Rational.evalUnchecked1 e (IntervalRat.singleton J.lo)
+          argBound := some (IntervalRat.singleton J.lo)
+          depth := maxDepth - depth }
+      else if derivStrictlyNegative e J varIdx then
+        -- f is decreasing, minimum is at right endpoint
+        { valueBound := LeanCert.Internal.Rational.evalUnchecked1 e (IntervalRat.singleton J.hi)
+          argBound := some (IntervalRat.singleton J.hi)
+          depth := maxDepth - depth }
+      else
+        -- Bisect and take the better bound
+        let (J₁, J₂) := J.bisect
+        let r₁ := go J₁ (depth - 1)
+        let r₂ := go J₂ (depth - 1)
+        OptResult.merge r₁ r₂
+  termination_by depth
+  decreasing_by all_goals omega
+
+/-- Base case correctness: interval evaluation gives a valid lower bound.
+    This theorem is FULLY PROVED - no sorry, no axioms. -/
+theorem minimizeInterval_base_correct (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) :
+    ∀ x ∈ I, (LeanCert.Internal.Rational.evalUnchecked1 e I).lo ≤ Expr.eval (fun _ => x) e := by
+  intro x hx
+  have h := evalInterval1_correct e hsupp x I hx
+  simp only [IntervalRat.mem_def] at h
+  exact h.1
+
+/-! ### Derivative bounds and monotonicity -/
+
+/-- For single-variable expressions evaluated with (fun _ => I), derivative is correct.
+    We use evalWithDeriv1 which directly uses varActive for all variables. -/
+theorem derivInterval_correct_evalWithDeriv1 (e : Expr) (hsupp : ADSupported e)
+    (I : IntervalRat) (x : ℝ) (hx : x ∈ I) :
+    deriv (evalFunc1 e) x ∈ (evalWithDeriv1 e I).der := by
+  -- evalWithDeriv1 e I = LeanCert.Internal.AD.evalUnchecked e (fun _ => DualInterval.varActive I)
+  -- This matches exactly what LeanCert.Engine.evalDualUnchecked_der_correct expects
+  simp only [evalWithDeriv1]
+  exact LeanCert.Engine.evalDualUnchecked_der_correct e hsupp I x hx
+
+/-- Derivative is in derivInterval for expressions that only use var 0.
+    FULLY PROVED - no sorry. -/
+theorem derivInterval_correct_single (e : Expr) (hsupp : ADSupported e) (hvar0 : UsesOnlyVar0 e)
+    (I : IntervalRat) (x : ℝ) (hx : x ∈ I) :
+    deriv (evalFunc1 e) x ∈ derivInterval e (fun _ => I) 0 := by
+  rw [derivInterval_eq_evalWithDeriv1_of_UsesOnlyVar0 e hvar0]
+  exact derivInterval_correct_evalWithDeriv1 e hsupp I x hx
+
+/-- If the derivative interval is strictly positive, derivative is positive everywhere -/
+theorem deriv_pos_on_interval (e : Expr) (hsupp : ADSupported e) (hvar0 : UsesOnlyVar0 e)
+    (I : IntervalRat) (hpos : 0 < (derivInterval e (fun _ => I) 0).lo) :
+    ∀ x ∈ I, 0 < deriv (evalFunc1 e) x := by
+  intro x hx
+  have hmem := derivInterval_correct_single e hsupp hvar0 I x hx
+  simp only [IntervalRat.mem_def] at hmem
+  have hlo_cast : (0 : ℝ) < ((derivInterval e (fun _ => I) 0).lo : ℝ) := by exact_mod_cast hpos
+  calc (0 : ℝ) < ((derivInterval e (fun _ => I) 0).lo : ℝ) := hlo_cast
+    _ ≤ deriv (evalFunc1 e) x := hmem.1
+
+/-- If the derivative interval is strictly negative, derivative is negative everywhere -/
+theorem deriv_neg_on_interval (e : Expr) (hsupp : ADSupported e) (hvar0 : UsesOnlyVar0 e)
+    (I : IntervalRat) (hneg : (derivInterval e (fun _ => I) 0).hi < 0) :
+    ∀ x ∈ I, deriv (evalFunc1 e) x < 0 := by
+  intro x hx
+  have hmem := derivInterval_correct_single e hsupp hvar0 I x hx
+  simp only [IntervalRat.mem_def] at hmem
+  have hhi_cast : ((derivInterval e (fun _ => I) 0).hi : ℝ) < (0 : ℝ) := by exact_mod_cast hneg
+  calc deriv (evalFunc1 e) x ≤ ((derivInterval e (fun _ => I) 0).hi : ℝ) := hmem.2
+    _ < 0 := hhi_cast
+
+/-- Strictly positive derivative implies strict monotonicity -/
+theorem strictMonoOn_of_deriv_pos_interval (e : Expr) (hsupp : ADSupported e) (hvar0 :
+  UsesOnlyVar0 e)
+    (I : IntervalRat) (hpos : 0 < (derivInterval e (fun _ => I) 0).lo) :
+    StrictMonoOn (evalFunc1 e) (Set.Icc (I.lo : ℝ) (I.hi : ℝ)) := by
+  have hdiff := evalFunc1_differentiable e hsupp
+  have hderiv_pos := deriv_pos_on_interval e hsupp hvar0 I hpos
+  apply strictMonoOn_of_deriv_pos (convex_Icc _ _)
+  · exact hdiff.continuous.continuousOn
+  · intro x hx
+    -- Interior of Icc is Ioo, and Ioo ⊆ Icc
+    rw [interior_Icc] at hx
+    have hx_mem : x ∈ I := by
+      simp only [IntervalRat.mem_def]
+      exact ⟨le_of_lt hx.1, le_of_lt hx.2⟩
+    exact hderiv_pos x hx_mem
+
+/-- Strictly negative derivative implies strict antitonicity -/
+theorem strictAntiOn_of_deriv_neg_interval (e : Expr) (hsupp : ADSupported e) (hvar0 :
+  UsesOnlyVar0 e)
+    (I : IntervalRat) (hneg : (derivInterval e (fun _ => I) 0).hi < 0) :
+    StrictAntiOn (evalFunc1 e) (Set.Icc (I.lo : ℝ) (I.hi : ℝ)) := by
+  have hdiff := evalFunc1_differentiable e hsupp
+  have hderiv_neg := deriv_neg_on_interval e hsupp hvar0 I hneg
+  apply strictAntiOn_of_deriv_neg (convex_Icc _ _)
+  · exact hdiff.continuous.continuousOn
+  · intro x hx
+    -- Interior of Icc is Ioo
+    rw [interior_Icc] at hx
+    have hx_mem : x ∈ I := by
+      simp only [IntervalRat.mem_def]
+      exact ⟨le_of_lt hx.1, le_of_lt hx.2⟩
+    exact hderiv_neg x hx_mem
+
+/-! ### Monotonicity-based bounds -/
+
+/-- For increasing functions, minimum is at the left endpoint -/
+theorem increasing_min_at_left (e : Expr) (hsupp : ADSupported e) (hvar0 : UsesOnlyVar0 e)
+    (I : IntervalRat) (hpos : 0 < (derivInterval e (fun _ => I) 0).lo) :
+    ∀ x ∈ I, evalFunc1 e I.lo ≤ evalFunc1 e x := by
+  intro x hx
+  have hmono := strictMonoOn_of_deriv_pos_interval e hsupp hvar0 I hpos
+  simp only [IntervalRat.mem_def] at hx
+  by_cases heq : (I.lo : ℝ) = x
+  · rw [heq]
+  · apply le_of_lt
+    apply hmono
+    · simp only [Set.mem_Icc]
+      exact ⟨le_refl _, Rat.cast_le.mpr I.le⟩
+    · simp only [Set.mem_Icc]; exact hx
+    · exact lt_of_le_of_ne hx.1 heq
+
+/-- For decreasing functions, minimum is at the right endpoint -/
+theorem decreasing_min_at_right (e : Expr) (hsupp : ADSupported e) (hvar0 : UsesOnlyVar0 e)
+    (I : IntervalRat) (hneg : (derivInterval e (fun _ => I) 0).hi < 0) :
+    ∀ x ∈ I, evalFunc1 e I.hi ≤ evalFunc1 e x := by
+  intro x hx
+  have hmono := strictAntiOn_of_deriv_neg_interval e hsupp hvar0 I hneg
+  simp only [IntervalRat.mem_def] at hx
+  by_cases heq : x = (I.hi : ℝ)
+  · rw [heq]
+  · apply le_of_lt
+    apply hmono
+    · simp only [Set.mem_Icc]; exact hx
+    · simp only [Set.mem_Icc]; exact ⟨(Rat.cast_le.mpr I.le), le_refl _⟩
+    · exact lt_of_le_of_ne hx.2 heq
+
+/-- Interval at left endpoint gives valid lower bound for increasing functions -/
+theorem increasing_endpoint_bound (e : Expr) (hsupp : ADSupported e) (hvar0 : UsesOnlyVar0 e)
+    (I : IntervalRat) (hpos : 0 < (derivInterval e (fun _ => I) 0).lo) :
+    ∀ x ∈ I, (LeanCert.Internal.Rational.evalUnchecked1 e (IntervalRat.singleton I.lo)).lo ≤
+      Expr.eval (fun _ => x) e := by
+  intro x hx
+  have hlo_mem : (I.lo : ℝ) ∈ IntervalRat.singleton I.lo := IntervalRat.mem_singleton I.lo
+  have heval := evalInterval1_correct e hsupp I.lo (IntervalRat.singleton I.lo) hlo_mem
+  simp only [IntervalRat.mem_def] at heval
+  have hmin := increasing_min_at_left e hsupp hvar0 I hpos x hx
+  simp only [evalFunc1] at hmin
+  calc (LeanCert.Internal.Rational.evalUnchecked1 e (IntervalRat.singleton I.lo)).lo
+      ≤ Expr.eval (fun _ => I.lo) e := by exact_mod_cast heval.1
+    _ ≤ Expr.eval (fun _ => x) e := hmin
+
+/-- Interval at right endpoint gives valid lower bound for decreasing functions -/
+theorem decreasing_endpoint_bound (e : Expr) (hsupp : ADSupported e) (hvar0 : UsesOnlyVar0 e)
+    (I : IntervalRat) (hneg : (derivInterval e (fun _ => I) 0).hi < 0) :
+    ∀ x ∈ I, (LeanCert.Internal.Rational.evalUnchecked1 e (IntervalRat.singleton I.hi)).lo ≤
+      Expr.eval (fun _ => x) e := by
+  intro x hx
+  have hhi_mem : (I.hi : ℝ) ∈ IntervalRat.singleton I.hi := IntervalRat.mem_singleton I.hi
+  have heval := evalInterval1_correct e hsupp I.hi (IntervalRat.singleton I.hi) hhi_mem
+  simp only [IntervalRat.mem_def] at heval
+  have hmin := decreasing_min_at_right e hsupp hvar0 I hneg x hx
+  simp only [evalFunc1] at hmin
+  calc (LeanCert.Internal.Rational.evalUnchecked1 e (IntervalRat.singleton I.hi)).lo
+      ≤ Expr.eval (fun _ => I.hi) e := by exact_mod_cast heval.1
+    _ ≤ Expr.eval (fun _ => x) e := hmin
+
+/-! ### Full optimization correctness -/
+
+/-- Helper lemma for go correctness.
+    FULLY PROVED for varIdx = 0 and UsesOnlyVar0 expressions. -/
+theorem minimizeInterval_go_correct (e : Expr) (hsupp : ADSupported e) (hvar0 : UsesOnlyVar0 e)
+    (maxDepth : ℕ) (depth : ℕ) (J : IntervalRat) :
+    ∀ x ∈ J, (minimizeInterval.go e 0 maxDepth J depth).valueBound.lo
+             ≤ Expr.eval (fun _ => x) e := by
+  induction depth generalizing J with
+  | zero =>
+    -- Base case: depth = 0, just use interval evaluation
+    intro x hx
+    simp only [minimizeInterval.go, ↓reduceIte]
+    exact minimizeInterval_base_correct e hsupp J x hx
+  | succ n ih =>
+    intro x hx
+    -- Unfold the go definition for succ case
+    -- Note: n + 1 ≠ 0, so we skip the first branch
+    have hdepth : n + 1 ≠ 0 := Nat.succ_ne_zero n
+    rw [minimizeInterval.go]
+    simp only [hdepth, ↓reduceIte, OptResult.merge, Nat.add_sub_cancel]
+    -- Now we branch on the derivative conditions
+    split_ifs with hpos hneg hle
+    · -- Derivative strictly positive: increasing function
+      have hpos' : 0 < (derivInterval e (fun _ => J) 0).lo := by
+        simp only [derivStrictlyPositive, decide_eq_true_eq] at hpos
+        exact hpos
+      exact increasing_endpoint_bound e hsupp hvar0 J hpos' x hx
+    · -- Derivative strictly negative: decreasing function
+      have hneg' : (derivInterval e (fun _ => J) 0).hi < 0 := by
+        simp only [derivStrictlyNegative, decide_eq_true_eq] at hneg
+        exact hneg
+      exact decreasing_endpoint_bound e hsupp hvar0 J hneg' x hx
+    · -- Bisection case, hle (r₁.lo ≤ r₂.lo)
+      -- x is in one of the bisected intervals
+      have hcases := IntervalRat.mem_bisect_or hx
+      cases hcases with
+      | inl h1 => exact ih J.bisect.1 x h1
+      | inr h2 =>
+        have h := ih J.bisect.2 x h2
+        have hle_cast : ((minimizeInterval.go e 0 maxDepth (J.bisect.fst) n).valueBound.lo : ℝ)
+            ≤ (minimizeInterval.go e 0 maxDepth (J.bisect.snd) n).valueBound.lo := by
+              exact_mod_cast hle
+        calc ((minimizeInterval.go e 0 maxDepth (J.bisect.fst) n).valueBound.lo : ℝ)
+            ≤ (minimizeInterval.go e 0 maxDepth (J.bisect.snd) n).valueBound.lo := hle_cast
+          _ ≤ Expr.eval (fun _ => x) e := h
+    · -- Bisection case, ¬hle (r₁.lo > r₂.lo)
+      have hcases := IntervalRat.mem_bisect_or hx
+      cases hcases with
+      | inl h1 =>
+        have h := ih J.bisect.1 x h1
+        have hlt := not_le.mp hle
+        have hlt_cast : ((minimizeInterval.go e 0 maxDepth (J.bisect.snd) n).valueBound.lo : ℝ)
+            ≤ (minimizeInterval.go e 0 maxDepth (J.bisect.fst) n).valueBound.lo := by
+              exact_mod_cast (le_of_lt hlt)
+        calc ((minimizeInterval.go e 0 maxDepth (J.bisect.snd) n).valueBound.lo : ℝ)
+            ≤ (minimizeInterval.go e 0 maxDepth (J.bisect.fst) n).valueBound.lo := hlt_cast
+          _ ≤ Expr.eval (fun _ => x) e := h
+      | inr h2 => exact ih J.bisect.2 x h2
+
+/-- Correctness: the minimum is in the computed interval.
+    FULLY PROVED for varIdx = 0 and UsesOnlyVar0 expressions. -/
+theorem minimizeInterval_correct (e : Expr) (hsupp : ADSupported e) (hvar0 : UsesOnlyVar0 e)
+    (I : IntervalRat)
+    (maxDepth : ℕ) :
+    ∀ x ∈ I, (minimizeInterval e I 0 maxDepth).valueBound.lo ≤ Expr.eval (fun _ => x) e := by
+  simp only [minimizeInterval]
+  exact minimizeInterval_go_correct e hsupp hvar0 maxDepth maxDepth I
+
+/-! ### Maximization -/
+
+/-- Maximize by minimizing the negation -/
+noncomputable def maximizeInterval (e : Expr) (I : IntervalRat) (varIdx : Nat)
+    (maxDepth : ℕ) : OptResult :=
+  let negResult := minimizeInterval (Expr.neg e) I varIdx maxDepth
+  { valueBound := IntervalRat.neg negResult.valueBound
+    argBound := negResult.argBound
+    depth := negResult.depth }
+
+/-- Correctness: the maximum is in the computed interval.
+    FULLY PROVED for varIdx = 0 and UsesOnlyVar0 expressions. -/
+theorem maximizeInterval_correct (e : Expr) (hsupp : ADSupported e) (hvar0 : UsesOnlyVar0 e)
+    (I : IntervalRat)
+    (maxDepth : ℕ) :
+    ∀ x ∈ I, Expr.eval (fun _ => x) e ≤ (maximizeInterval e I 0 maxDepth).valueBound.hi := by
+  intro x hx
+  -- max f = -min(-f)
+  -- result.valueBound = neg (minimizeInterval (neg e) ...).valueBound
+  -- result.valueBound.hi = -(minimizeInterval (neg e) ...).valueBound.lo
+  -- We need: f(x) ≤ result.valueBound.hi
+  --        = -(minimizeInterval (neg e) ...).valueBound.lo
+  -- Equivalently: (minimizeInterval (neg e) ...).valueBound.lo ≤ -f(x)
+  have hneg_supp : ADSupported (Expr.neg e) := ADSupported.neg hsupp
+  have hneg_var0 : UsesOnlyVar0 (Expr.neg e) := UsesOnlyVar0.neg e hvar0
+  have hmin := minimizeInterval_correct (Expr.neg e) hneg_supp hneg_var0 I maxDepth x hx
+  simp only [Expr.eval_neg] at hmin
+  -- hmin: (minimizeInterval (neg e) ...).valueBound.lo ≤ -f(x)
+  -- Need: f(x) ≤ (maximizeInterval e ...).valueBound.hi
+  --     = -(minimizeInterval (neg e) ...).valueBound.lo
+  simp only [maximizeInterval, IntervalRat.neg]
+  -- Goal: f(x) ≤ ↑(-(minimizeInterval (neg e) ...).valueBound.lo)
+  -- From hmin: ↑(minimizeInterval (neg e) ...).valueBound.lo ≤ -f(x)
+  -- So: f(x) ≤ -(minimizeInterval (neg e) ...).valueBound.lo
+  have h : Expr.eval (fun _ => x) e ≤
+           -(((minimizeInterval (Expr.neg e) I 0 maxDepth).valueBound.lo) : ℝ) := by
+    have hcast : (((minimizeInterval (Expr.neg e) I 0 maxDepth).valueBound.lo) : ℝ)
+                 ≤ -Expr.eval (fun _ => x) e := hmin
+    linarith
+  convert h using 1
+  -- Need: ↑(-(minimizeInterval e.neg I 0 maxDepth).valueBound.lo) =
+  --       -↑(minimizeInterval e.neg I 0 maxDepth).valueBound.lo
+  simp only [Rat.cast_neg]
+
+/-! ### Bounds checking -/
+
+/-- Check if f(x) ≥ c for all x in I -/
+noncomputable def checkLowerBound (e : Expr) (I : IntervalRat) (c : ℚ) (varIdx : Nat)
+    (maxDepth : ℕ) : Bool :=
+  let result := minimizeInterval e I varIdx maxDepth
+  c ≤ result.valueBound.lo
+
+/-- Check if f(x) ≤ c for all x in I -/
+noncomputable def checkUpperBound (e : Expr) (I : IntervalRat) (c : ℚ) (varIdx : Nat)
+    (maxDepth : ℕ) : Bool :=
+  let result := maximizeInterval e I varIdx maxDepth
+  result.valueBound.hi ≤ c
+
+/-! ## N-Variable Optimization Infrastructure
+
+The following section provides generalized optimization infrastructure that works
+with arbitrary variable indices and multi-variable environments. This allows
+optimizing along any coordinate while holding other variables fixed.
+
+Key types:
+- `IntervalEnv := Nat → IntervalRat` - maps variable indices to intervals
+- `evalAlong e ρ idx` - evaluates `e` as a function of variable `idx`
+
+The main advantage is that we can now optimize expressions with multiple variables
+by fixing all but one variable and optimizing along that coordinate.
+-/
+
+/-! ### N-variable derivative bounds -/
+
+/-- If the derivative interval along `idx` is strictly positive, derivative is positive everywhere.
+    Generalized version that works with any variable index and environment. -/
+theorem deriv_pos_on_interval_idx (e : Expr) (hsupp : ADSupported e)
+    (ρ_real : Nat → ℝ) (ρ_int : IntervalEnv) (idx : Nat)
+    (hρ : ∀ i, ρ_real i ∈ ρ_int i)
+    (hpos : 0 < (derivInterval e ρ_int idx).lo) :
+    ∀ x ∈ ρ_int idx, 0 < deriv (Expr.evalAlong e ρ_real idx) x := by
+  intro x hx
+  have hmem := derivInterval_correct_idx e hsupp ρ_real ρ_int idx hρ x hx
+  simp only [IntervalRat.mem_def] at hmem
+  have hlo_cast : (0 : ℝ) < ((derivInterval e ρ_int idx).lo : ℝ) := by exact_mod_cast hpos
+  calc (0 : ℝ) < ((derivInterval e ρ_int idx).lo : ℝ) := hlo_cast
+    _ ≤ deriv (Expr.evalAlong e ρ_real idx) x := hmem.1
+
+/-- If the derivative interval along `idx` is strictly negative, derivative is negative everywhere.
+    Generalized version that works with any variable index and environment. -/
+theorem deriv_neg_on_interval_idx (e : Expr) (hsupp : ADSupported e)
+    (ρ_real : Nat → ℝ) (ρ_int : IntervalEnv) (idx : Nat)
+    (hρ : ∀ i, ρ_real i ∈ ρ_int i)
+    (hneg : (derivInterval e ρ_int idx).hi < 0) :
+    ∀ x ∈ ρ_int idx, deriv (Expr.evalAlong e ρ_real idx) x < 0 := by
+  intro x hx
+  have hmem := derivInterval_correct_idx e hsupp ρ_real ρ_int idx hρ x hx
+  simp only [IntervalRat.mem_def] at hmem
+  have hhi_cast : ((derivInterval e ρ_int idx).hi : ℝ) < (0 : ℝ) := by exact_mod_cast hneg
+  calc deriv (Expr.evalAlong e ρ_real idx) x ≤ ((derivInterval e ρ_int idx).hi : ℝ) := hmem.2
+    _ < 0 := hhi_cast
+
+/-- Strictly positive derivative along `idx` implies strict monotonicity.
+    Generalized n-variable version. -/
+theorem strictMonoOn_of_deriv_pos_interval_idx (e : Expr) (hsupp : ADSupported e)
+    (ρ_real : Nat → ℝ) (ρ_int : IntervalEnv) (idx : Nat)
+    (hρ : ∀ i, ρ_real i ∈ ρ_int i)
+    (hpos : 0 < (derivInterval e ρ_int idx).lo) :
+    StrictMonoOn (Expr.evalAlong e ρ_real idx)
+      (Set.Icc ((ρ_int idx).lo : ℝ) ((ρ_int idx).hi : ℝ)) := by
+  have hdiff := evalAlong_differentiable e hsupp ρ_real idx
+  have hderiv_pos := deriv_pos_on_interval_idx e hsupp ρ_real ρ_int idx hρ hpos
+  apply strictMonoOn_of_deriv_pos (convex_Icc _ _)
+  · exact hdiff.continuous.continuousOn
+  · intro x hx
+    rw [interior_Icc] at hx
+    have hx_mem : x ∈ ρ_int idx := by
+      simp only [IntervalRat.mem_def]
+      exact ⟨le_of_lt hx.1, le_of_lt hx.2⟩
+    exact hderiv_pos x hx_mem
+
+/-- Strictly negative derivative along `idx` implies strict antitonicity.
+    Generalized n-variable version. -/
+theorem strictAntiOn_of_deriv_neg_interval_idx (e : Expr) (hsupp : ADSupported e)
+    (ρ_real : Nat → ℝ) (ρ_int : IntervalEnv) (idx : Nat)
+    (hρ : ∀ i, ρ_real i ∈ ρ_int i)
+    (hneg : (derivInterval e ρ_int idx).hi < 0) :
+    StrictAntiOn (Expr.evalAlong e ρ_real idx)
+      (Set.Icc ((ρ_int idx).lo : ℝ) ((ρ_int idx).hi : ℝ)) := by
+  have hdiff := evalAlong_differentiable e hsupp ρ_real idx
+  have hderiv_neg := deriv_neg_on_interval_idx e hsupp ρ_real ρ_int idx hρ hneg
+  apply strictAntiOn_of_deriv_neg (convex_Icc _ _)
+  · exact hdiff.continuous.continuousOn
+  · intro x hx
+    rw [interior_Icc] at hx
+    have hx_mem : x ∈ ρ_int idx := by
+      simp only [IntervalRat.mem_def]
+      exact ⟨le_of_lt hx.1, le_of_lt hx.2⟩
+    exact hderiv_neg x hx_mem
+
+/-! ### N-variable monotonicity-based bounds -/
+
+/-- For increasing functions along `idx`, minimum is at the left endpoint.
+    Generalized n-variable version. -/
+theorem increasing_min_at_left_idx (e : Expr) (hsupp : ADSupported e)
+    (ρ_real : Nat → ℝ) (ρ_int : IntervalEnv) (idx : Nat)
+    (hρ : ∀ i, ρ_real i ∈ ρ_int i)
+    (hpos : 0 < (derivInterval e ρ_int idx).lo) :
+    ∀ x ∈ ρ_int idx,
+      Expr.evalAlong e ρ_real idx (ρ_int idx).lo ≤ Expr.evalAlong e ρ_real idx x := by
+  intro x hx
+  have hmono := strictMonoOn_of_deriv_pos_interval_idx e hsupp ρ_real ρ_int idx hρ hpos
+  simp only [IntervalRat.mem_def] at hx
+  by_cases heq : ((ρ_int idx).lo : ℝ) = x
+  · rw [heq]
+  · apply le_of_lt
+    apply hmono
+    · simp only [Set.mem_Icc]
+      exact ⟨le_refl _, Rat.cast_le.mpr (ρ_int idx).le⟩
+    · simp only [Set.mem_Icc]; exact hx
+    · exact lt_of_le_of_ne hx.1 heq
+
+/-- For decreasing functions along `idx`, minimum is at the right endpoint.
+    Generalized n-variable version. -/
+theorem decreasing_min_at_right_idx (e : Expr) (hsupp : ADSupported e)
+    (ρ_real : Nat → ℝ) (ρ_int : IntervalEnv) (idx : Nat)
+    (hρ : ∀ i, ρ_real i ∈ ρ_int i)
+    (hneg : (derivInterval e ρ_int idx).hi < 0) :
+    ∀ x ∈ ρ_int idx,
+      Expr.evalAlong e ρ_real idx (ρ_int idx).hi ≤ Expr.evalAlong e ρ_real idx x := by
+  intro x hx
+  have hmono := strictAntiOn_of_deriv_neg_interval_idx e hsupp ρ_real ρ_int idx hρ hneg
+  simp only [IntervalRat.mem_def] at hx
+  by_cases heq : x = ((ρ_int idx).hi : ℝ)
+  · rw [heq]
+  · apply le_of_lt
+    apply hmono
+    · simp only [Set.mem_Icc]; exact hx
+    · simp only [Set.mem_Icc]; exact ⟨(Rat.cast_le.mpr (ρ_int idx).le), le_refl _⟩
+    · exact lt_of_le_of_ne hx.2 heq
+
+/-! ### N-variable minimization algorithm -/
+
+/-- Check if derivative interval along `idx` is strictly positive -/
+noncomputable def derivStrictlyPositiveIdx (e : Expr) (ρ : IntervalEnv) (idx : Nat) : Bool :=
+  let dI := derivInterval e ρ idx
+  0 < dI.lo
+
+/-- Check if derivative interval along `idx` is strictly negative -/
+noncomputable def derivStrictlyNegativeIdx (e : Expr) (ρ : IntervalEnv) (idx : Nat) : Bool :=
+  let dI := derivInterval e ρ idx
+  dI.hi < 0
+
+/-- Update an interval environment at a specific index -/
+def updateIntervalEnv (ρ : IntervalEnv) (idx : Nat) (J : IntervalRat) : IntervalEnv :=
+  fun i => if i = idx then J else ρ i
+
+/-- N-variable branch-and-bound minimization along coordinate `idx`.
+
+Returns an interval [lo, hi] such that for any ρ_real with ρ_real i ∈ ρ i:
+- lo ≤ min_{t ∈ ρ idx} evalAlong e ρ_real idx t
+- hi ≥ min_{t ∈ ρ idx} evalAlong e ρ_real idx t
+-/
+noncomputable def minimizeIntervalIdx (e : Expr) (ρ : IntervalEnv) (idx : Nat)
+    (maxDepth : ℕ) : OptResult :=
+  go (ρ idx) maxDepth
+where
+  /-- Recursive coordinate minimization with a decreasing subdivision budget. -/
+  go (J : IntervalRat) (depth : ℕ) : OptResult :=
+    let ρ' := updateIntervalEnv ρ idx J
+    if depth = 0 then
+      -- Base case: just use interval evaluation
+      { valueBound := LeanCert.Internal.Rational.evalUnchecked e ρ'
+        argBound := some J
+        depth := maxDepth }
+    else
+      -- Check monotonicity along idx
+      if derivStrictlyPositiveIdx e ρ' idx then
+        -- f is increasing along idx, minimum is at left endpoint
+        let ρ_lo := updateIntervalEnv ρ idx (IntervalRat.singleton J.lo)
+        { valueBound := LeanCert.Internal.Rational.evalUnchecked e ρ_lo
+          argBound := some (IntervalRat.singleton J.lo)
+          depth := maxDepth - depth }
+      else if derivStrictlyNegativeIdx e ρ' idx then
+        -- f is decreasing along idx, minimum is at right endpoint
+        let ρ_hi := updateIntervalEnv ρ idx (IntervalRat.singleton J.hi)
+        { valueBound := LeanCert.Internal.Rational.evalUnchecked e ρ_hi
+          argBound := some (IntervalRat.singleton J.hi)
+          depth := maxDepth - depth }
+      else
+        -- Bisect and take the better bound
+        let (J₁, J₂) := J.bisect
+        let r₁ := go J₁ (depth - 1)
+        let r₂ := go J₂ (depth - 1)
+        OptResult.merge r₁ r₂
+  termination_by depth
+  decreasing_by all_goals omega
+
+/-! ### N-variable minimization correctness -/
+
+/-- Base case correctness for n-variable optimization.
+    Interval evaluation gives a valid lower bound. -/
+theorem minimizeIntervalIdx_base_correct (e : Expr) (hsupp : ADSupported e)
+    (ρ_int : IntervalEnv) :
+    ∀ ρ_real : Nat → ℝ, (∀ i, ρ_real i ∈ ρ_int i) →
+      (LeanCert.Internal.Rational.evalUnchecked e ρ_int).lo ≤ Expr.eval ρ_real e := by
+  intro ρ_real hρ
+  have h := evalInterval_correct e hsupp ρ_real ρ_int hρ
+  simp only [IntervalRat.mem_def] at h
+  exact h.1
+
+/-- Helper lemma for go correctness in n-variable setting -/
+theorem minimizeIntervalIdx_go_correct (e : Expr) (hsupp : ADSupported e)
+    (ρ_int : IntervalEnv) (idx : Nat) (maxDepth depth : ℕ) (J : IntervalRat)
+    (hJ_sub : ∀ t, t ∈ J → t ∈ ρ_int idx) :
+    ∀ ρ_real : Nat → ℝ, (∀ i, ρ_real i ∈ ρ_int i) →
+      ∀ t ∈ J,
+        (minimizeIntervalIdx.go e ρ_int idx maxDepth J depth).valueBound.lo
+          ≤ Expr.eval (Expr.updateVar ρ_real idx t) e := by
+  induction depth generalizing J with
+  | zero =>
+    intro ρ_real hρ t ht
+    rw [minimizeIntervalIdx.go]
+    simp only [↓reduceIte]
+    -- Base case: interval evaluation bounds all points
+    have hρ' : ∀ i, (Expr.updateVar ρ_real idx t) i ∈ (updateIntervalEnv ρ_int idx J) i := by
+      intro i
+      simp only [Expr.updateVar, updateIntervalEnv]
+      split_ifs with hi
+      · exact ht
+      · exact hρ i
+    have heval := evalInterval_correct e hsupp (Expr.updateVar ρ_real idx t) (updateIntervalEnv
+      ρ_int idx J) hρ'
+    simp only [IntervalRat.mem_def] at heval
+    exact heval.1
+  | succ n ih =>
+    intro ρ_real hρ t ht
+    have hdepth : n + 1 ≠ 0 := Nat.succ_ne_zero n
+    rw [minimizeIntervalIdx.go]
+    simp only [hdepth, ↓reduceIte, OptResult.merge, Nat.add_sub_cancel]
+    -- Note: ρ' in the go function is updateIntervalEnv ρ_int idx J
+    split_ifs with hpos hneg hle
+    · -- Derivative strictly positive: increasing function
+      -- hpos : derivStrictlyPositiveIdx e (updateIntervalEnv ρ_int idx J) idx = true
+      have hpos' : 0 < (derivInterval e (updateIntervalEnv ρ_int idx J) idx).lo := by
+        simp only [derivStrictlyPositiveIdx, decide_eq_true_eq] at hpos
+        exact hpos
+      have hρ_lo : ∀ i, (Expr.updateVar ρ_real idx J.lo) i ∈
+          (updateIntervalEnv ρ_int idx (IntervalRat.singleton J.lo)) i := by
+        intro i
+        simp only [Expr.updateVar, updateIntervalEnv]
+        split_ifs with hi
+        · exact IntervalRat.mem_singleton _
+        · exact hρ i
+      have heval := evalInterval_correct e hsupp (Expr.updateVar ρ_real idx J.lo)
+        (updateIntervalEnv ρ_int idx (IntervalRat.singleton J.lo)) hρ_lo
+      simp only [IntervalRat.mem_def] at heval
+      -- Build membership for the derivative bound
+      -- We use a modified ρ_real that has J.lo at index idx, since ρ_real idx is never used
+      -- in the derivative computation (it gets replaced by the variable of differentiation)
+      let ρ_real' := Expr.updateVar ρ_real idx (J.lo : ℝ)
+      have hρ'_deriv : ∀ i, ρ_real' i ∈ (updateIntervalEnv ρ_int idx J) i := by
+        intro i
+        simp only [ρ_real', Expr.updateVar, updateIntervalEnv]
+        split_ifs with hi
+        · -- At idx: J.lo ∈ J
+          simp only [IntervalRat.mem_def, Rat.cast_le]
+          exact ⟨le_refl _, J.le⟩
+        · exact hρ i
+      -- Since evalAlong replaces ρ_real' idx with the variable x, monotonicity for ρ_real'
+      -- is the same as for ρ_real
+      have hmono' := strictMonoOn_of_deriv_pos_interval_idx e hsupp ρ_real'
+        (updateIntervalEnv ρ_int idx J) idx hρ'_deriv hpos'
+      -- evalAlong e ρ_real idx = evalAlong e ρ_real' idx because only ρ(idx) differs
+      -- and evalAlong replaces it with the function argument
+      have heq_along : ∀ x, Expr.evalAlong e ρ_real idx x = Expr.evalAlong e ρ_real' idx x := by
+        intro x
+        simp only [Expr.evalAlong]
+        congr 1
+        funext i
+        simp only [Expr.updateVar, ρ_real']
+        split_ifs with h1
+        · rfl  -- i = idx: both give x
+        · rfl  -- i ≠ idx: both give ρ_real i
+      have hmono : StrictMonoOn (Expr.evalAlong e ρ_real idx)
+          (Set.Icc ((updateIntervalEnv ρ_int idx J idx).lo : ℝ)
+            ((updateIntervalEnv ρ_int idx J idx).hi : ℝ)) := by
+        intro x hx y hy hxy
+        rw [heq_along x, heq_along y]
+        exact hmono' hx hy hxy
+      simp only [IntervalRat.mem_def] at ht
+      by_cases heq : (J.lo : ℝ) = t
+      · calc (LeanCert.Internal.Rational.evalUnchecked e (updateIntervalEnv ρ_int idx
+        (IntervalRat.singleton J.lo))).lo
+            ≤ Expr.eval (Expr.updateVar ρ_real idx J.lo) e := heval.1
+          _ = Expr.eval (Expr.updateVar ρ_real idx t) e := by rw [← heq]
+      · have hlt : (J.lo : ℝ) < t := lt_of_le_of_ne ht.1 heq
+        have hJlo_mem : (J.lo : ℝ) ∈ Set.Icc ((updateIntervalEnv ρ_int idx J idx).lo : ℝ)
+            ((updateIntervalEnv ρ_int idx J idx).hi) := by
+          simp only [Set.mem_Icc, updateIntervalEnv, ↓reduceIte]
+          exact ⟨le_refl _, Rat.cast_le.mpr J.le⟩
+        have ht_mem : t ∈ Set.Icc ((updateIntervalEnv ρ_int idx J idx).lo : ℝ)
+            ((updateIntervalEnv ρ_int idx J idx).hi) := by
+          simp only [Set.mem_Icc, updateIntervalEnv, ↓reduceIte]
+          exact ht
+        have hmono_at := hmono hJlo_mem ht_mem hlt
+        simp only [Expr.evalAlong] at hmono_at
+        calc (LeanCert.Internal.Rational.evalUnchecked e (updateIntervalEnv ρ_int idx
+          (IntervalRat.singleton J.lo))).lo
+            ≤ Expr.eval (Expr.updateVar ρ_real idx J.lo) e := heval.1
+          _ ≤ Expr.eval (Expr.updateVar ρ_real idx t) e := le_of_lt hmono_at
+    · -- Derivative strictly negative: decreasing function
+      have hneg' : (derivInterval e (updateIntervalEnv ρ_int idx J) idx).hi < 0 := by
+        simp only [derivStrictlyNegativeIdx, decide_eq_true_eq] at hneg
+        exact hneg
+      have hρ_hi : ∀ i, (Expr.updateVar ρ_real idx J.hi) i ∈
+          (updateIntervalEnv ρ_int idx (IntervalRat.singleton J.hi)) i := by
+        intro i
+        simp only [Expr.updateVar, updateIntervalEnv]
+        split_ifs with hi
+        · exact IntervalRat.mem_singleton _
+        · exact hρ i
+      have heval := evalInterval_correct e hsupp (Expr.updateVar ρ_real idx J.hi)
+        (updateIntervalEnv ρ_int idx (IntervalRat.singleton J.hi)) hρ_hi
+      simp only [IntervalRat.mem_def] at heval
+      -- We use a modified ρ_real that has J.hi at index idx
+      let ρ_real' := Expr.updateVar ρ_real idx (J.hi : ℝ)
+      have hρ'_deriv : ∀ i, ρ_real' i ∈ (updateIntervalEnv ρ_int idx J) i := by
+        intro i
+        simp only [ρ_real', Expr.updateVar, updateIntervalEnv]
+        split_ifs with hi
+        · simp only [IntervalRat.mem_def, Rat.cast_le]
+          exact ⟨J.le, le_refl _⟩
+        · exact hρ i
+      have hmono' := strictAntiOn_of_deriv_neg_interval_idx e hsupp ρ_real'
+        (updateIntervalEnv ρ_int idx J) idx hρ'_deriv hneg'
+      have heq_along : ∀ x, Expr.evalAlong e ρ_real idx x = Expr.evalAlong e ρ_real' idx x := by
+        intro x
+        simp only [Expr.evalAlong]
+        congr 1
+        funext i
+        simp only [Expr.updateVar, ρ_real']
+        split_ifs with h1
+        · rfl
+        · rfl
+      have hmono : StrictAntiOn (Expr.evalAlong e ρ_real idx)
+          (Set.Icc ((updateIntervalEnv ρ_int idx J idx).lo : ℝ)
+            ((updateIntervalEnv ρ_int idx J idx).hi : ℝ)) := by
+        intro x hx y hy hxy
+        rw [heq_along x, heq_along y]
+        exact hmono' hx hy hxy
+      simp only [IntervalRat.mem_def] at ht
+      by_cases heq : t = (J.hi : ℝ)
+      · calc (LeanCert.Internal.Rational.evalUnchecked e (updateIntervalEnv ρ_int idx
+        (IntervalRat.singleton J.hi))).lo
+            ≤ Expr.eval (Expr.updateVar ρ_real idx J.hi) e := heval.1
+          _ = Expr.eval (Expr.updateVar ρ_real idx t) e := by rw [← heq]
+      · have hlt : t < (J.hi : ℝ) := lt_of_le_of_ne ht.2 heq
+        have ht_mem : t ∈ Set.Icc ((updateIntervalEnv ρ_int idx J idx).lo : ℝ)
+            ((updateIntervalEnv ρ_int idx J idx).hi) := by
+          simp only [Set.mem_Icc, updateIntervalEnv, ↓reduceIte]
+          exact ht
+        have hJhi_mem : (J.hi : ℝ) ∈ Set.Icc ((updateIntervalEnv ρ_int idx J idx).lo : ℝ)
+            ((updateIntervalEnv ρ_int idx J idx).hi) := by
+          simp only [Set.mem_Icc, updateIntervalEnv, ↓reduceIte]
+          exact ⟨Rat.cast_le.mpr J.le, le_refl _⟩
+        have hmono_at := hmono ht_mem hJhi_mem hlt
+        simp only [Expr.evalAlong] at hmono_at
+        calc (LeanCert.Internal.Rational.evalUnchecked e (updateIntervalEnv ρ_int idx
+          (IntervalRat.singleton J.hi))).lo
+            ≤ Expr.eval (Expr.updateVar ρ_real idx J.hi) e := heval.1
+          _ ≤ Expr.eval (Expr.updateVar ρ_real idx t) e := le_of_lt hmono_at
+    · -- Bisection case, hle (r₁.lo ≤ r₂.lo)
+      have hcases := IntervalRat.mem_bisect_or ht
+      cases hcases with
+      | inl h1 =>
+        have hsub1 : ∀ s, s ∈ J.bisect.1 → s ∈ ρ_int idx := fun s hs =>
+          hJ_sub s (IntervalRat.mem_of_mem_bisect_left hs)
+        exact ih J.bisect.1 hsub1 ρ_real hρ t h1
+      | inr h2 =>
+        have hsub2 : ∀ s, s ∈ J.bisect.2 → s ∈ ρ_int idx := fun s hs =>
+          hJ_sub s (IntervalRat.mem_of_mem_bisect_right hs)
+        have h := ih J.bisect.2 hsub2 ρ_real hρ t h2
+        have hle_cast : ((minimizeIntervalIdx.go e ρ_int idx maxDepth (J.bisect.fst)
+          n).valueBound.lo : ℝ)
+            ≤ (minimizeIntervalIdx.go e ρ_int idx maxDepth (J.bisect.snd) n).valueBound.lo := by
+              exact_mod_cast hle
+        calc ((minimizeIntervalIdx.go e ρ_int idx maxDepth (J.bisect.fst) n).valueBound.lo : ℝ)
+            ≤ (minimizeIntervalIdx.go e ρ_int idx maxDepth (J.bisect.snd) n).valueBound.lo :=
+              hle_cast
+          _ ≤ Expr.eval (Expr.updateVar ρ_real idx t) e := h
+    · -- Bisection case, ¬hle (r₁.lo > r₂.lo)
+      have hcases := IntervalRat.mem_bisect_or ht
+      cases hcases with
+      | inl h1 =>
+        have hsub1 : ∀ s, s ∈ J.bisect.1 → s ∈ ρ_int idx := fun s hs =>
+          hJ_sub s (IntervalRat.mem_of_mem_bisect_left hs)
+        have h := ih J.bisect.1 hsub1 ρ_real hρ t h1
+        have hlt := not_le.mp hle
+        have hlt_cast : ((minimizeIntervalIdx.go e ρ_int idx maxDepth (J.bisect.snd)
+          n).valueBound.lo : ℝ)
+            ≤ (minimizeIntervalIdx.go e ρ_int idx maxDepth (J.bisect.fst) n).valueBound.lo := by
+          exact_mod_cast (le_of_lt hlt)
+        calc ((minimizeIntervalIdx.go e ρ_int idx maxDepth (J.bisect.snd) n).valueBound.lo : ℝ)
+            ≤ (minimizeIntervalIdx.go e ρ_int idx maxDepth (J.bisect.fst) n).valueBound.lo :=
+              hlt_cast
+          _ ≤ Expr.eval (Expr.updateVar ρ_real idx t) e := h
+      | inr h2 =>
+        have hsub2 : ∀ s, s ∈ J.bisect.2 → s ∈ ρ_int idx := fun s hs =>
+          hJ_sub s (IntervalRat.mem_of_mem_bisect_right hs)
+        exact ih J.bisect.2 hsub2 ρ_real hρ t h2
+
+/-- Correctness theorem for n-variable minimization:
+    For any real environment ρ_real satisfying ρ_int, and any t in ρ_int idx,
+    the computed lower bound is valid.
+
+    FULLY PROVED - no sorry, no axioms. -/
+theorem minimizeIntervalIdx_correct (e : Expr) (hsupp : ADSupported e)
+    (ρ_int : IntervalEnv) (idx : Nat) (maxDepth : ℕ) :
+    ∀ ρ_real : Nat → ℝ, (∀ i, ρ_real i ∈ ρ_int i) →
+      ∀ t ∈ ρ_int idx,
+        (minimizeIntervalIdx e ρ_int idx maxDepth).valueBound.lo
+          ≤ Expr.eval (Expr.updateVar ρ_real idx t) e := by
+  intro ρ_real hρ t ht
+  simp only [minimizeIntervalIdx]
+  exact minimizeIntervalIdx_go_correct e hsupp ρ_int idx maxDepth maxDepth (ρ_int idx)
+    (fun s hs => hs) ρ_real hρ t ht
+
+/-- N-variable maximization via minimization of negation -/
+noncomputable def maximizeIntervalIdx (e : Expr) (ρ : IntervalEnv) (idx : Nat)
+    (maxDepth : ℕ) : OptResult :=
+  let negResult := minimizeIntervalIdx (Expr.neg e) ρ idx maxDepth
+  { valueBound := IntervalRat.neg negResult.valueBound
+    argBound := negResult.argBound
+    depth := negResult.depth }
+
+/-- Correctness theorem for n-variable maximization -/
+theorem maximizeIntervalIdx_correct (e : Expr) (hsupp : ADSupported e)
+    (ρ_int : IntervalEnv) (idx : Nat) (maxDepth : ℕ) :
+    ∀ ρ_real : Nat → ℝ, (∀ i, ρ_real i ∈ ρ_int i) →
+      ∀ t ∈ ρ_int idx,
+        Expr.eval (Expr.updateVar ρ_real idx t) e
+          ≤ (maximizeIntervalIdx e ρ_int idx maxDepth).valueBound.hi := by
+  intro ρ_real hρ t ht
+  have hneg_supp : ADSupported (Expr.neg e) := ADSupported.neg hsupp
+  have hmin := minimizeIntervalIdx_correct (Expr.neg e) hneg_supp ρ_int idx maxDepth ρ_real hρ t ht
+  simp only [Expr.eval_neg] at hmin
+  simp only [maximizeIntervalIdx, IntervalRat.neg]
+  have h : Expr.eval (Expr.updateVar ρ_real idx t) e ≤
+           -(((minimizeIntervalIdx (Expr.neg e) ρ_int idx maxDepth).valueBound.lo) : ℝ) := by
+    linarith
+  convert h using 1
+  simp only [Rat.cast_neg]
+
+end LeanCert.Engine
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2024 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Gradient Interval Computation for Optimization
+
+This file provides functions to compute interval bounds on the gradient ∇f(B)
+of an expression over a box B. This is used for monotonicity-based pruning
+in branch-and-bound global optimization.
+
+## Main definitions
+
+* `gradientInterval` - Compute interval bounds on all partial derivatives over a box
+* `gradientSignature` - Determine the sign of each partial derivative
+* `canPruneToLo` / `canPruneToHi` - Check if a coordinate can be pruned by monotonicity
+
+## Design
+
+The gradient is computed by running forward-mode AD (from AD.lean) for each
+coordinate direction. The result is a list of intervals, one per variable.
+
+Monotonicity pruning: If ∂f/∂xᵢ > 0 on the entire box B, then f is minimized
+when xᵢ = B[i].lo. We can shrink the box in that dimension to a point.
+-/
+
+/-! ### Gradient computation -/
+
+@[expose] public section
+
+namespace LeanCert.Engine.Optimization
+
+open LeanCert.Core
+open LeanCert.Engine
+
+/-- Compute the gradient interval: bounds on each partial derivative over a box.
+    Returns a list of intervals, where the i-th interval contains ∂f/∂xᵢ for all x ∈ B. -/
+noncomputable def gradientInterval (e : Expr) (B : Box) : List IntervalRat :=
+  List.ofFn fun (i : Fin B.length) => derivInterval e (Box.toEnv B) i.val
+
+/-- Compute gradient for n variables (explicit dimension) -/
+noncomputable def gradientIntervalN (e : Expr) (B : Box) (n : Nat) : List IntervalRat :=
+  (List.range n).map fun i => derivInterval e (Box.toEnv B) i
+
+/-! ### Computable versions -/
+
+/-- Create dual environment for differentiating with respect to variable `idx` (computable).
+    Active variable gets der = 1, passive variables get der = 0. -/
+def mkDualEnvCore (ρ : IntervalEnv) (idx : Nat) : DualEnv :=
+  fun i => if i = idx then DualInterval.varActive (ρ i) else DualInterval.varPassive (ρ i)
+
+/-- Evaluate with derivative with respect to variable `idx` (computable version) -/
+def evalWithDerivCore (e : Expr) (ρ : IntervalEnv) (idx : Nat) (cfg : EvalConfig := {}) :
+  DualInterval :=
+  LeanCert.Internal.AD.evalTotalCore e (mkDualEnvCore ρ idx) cfg
+
+/-- Computable derivative interval for multi-variable expressions.
+    Computes the interval containing ∂f/∂xᵢ over the box. -/
+def derivIntervalCoreN (e : Expr) (ρ : IntervalEnv) (idx : Nat) (cfg : EvalConfig := {}) :
+  IntervalRat :=
+  (evalWithDerivCore e ρ idx cfg).der
+
+/-- Correctness of the computable derivative evaluator for an arbitrary
+coordinate of a multivariate expression. -/
+theorem evalDualTotalCore_der_correct_idx (e : Expr) (hsupp : ADSupported e)
+    (ρ_real : Nat → ℝ) (ρ_int : IntervalEnv) (idx : Nat)
+    (hρ : ∀ i, ρ_real i ∈ ρ_int i) (x : ℝ) (hx : x ∈ ρ_int idx)
+    (cfg : EvalConfig) :
+    deriv (Expr.evalAlong e ρ_real idx) x ∈
+      (LeanCert.Internal.AD.evalTotalCore e (mkDualEnvCore ρ_int idx) cfg).der := by
+  have hmem : ∀ i, Expr.updateVar ρ_real idx x i ∈
+      (mkDualEnvCore ρ_int idx i).val := by
+    simpa only [mkDualEnvCore, mkDualEnv] using
+      (updateVar_mem_mkDualEnv_val ρ_real ρ_int idx x hx hρ)
+  induction hsupp generalizing x with
+  | const q =>
+      simp only [Expr.evalAlong_const', deriv_const, LeanCert.Internal.AD.evalTotalCore,
+        DualInterval.const]
+      exact_mod_cast IntervalRat.mem_singleton 0
+  | var i =>
+      by_cases hi : i = idx
+      · subst i
+        simp only [Expr.evalAlong_var_active, LeanCert.Internal.AD.evalTotalCore, mkDualEnvCore,
+          ↓reduceIte, DualInterval.varActive, deriv_id]
+        exact_mod_cast IntervalRat.mem_singleton 1
+      · simp only [Expr.evalAlong_var_passive _ _ _ hi, deriv_const,
+        LeanCert.Internal.AD.evalTotalCore,
+          mkDualEnvCore, ite_eq_right hi, DualInterval.varPassive]
+        exact_mod_cast IntervalRat.mem_singleton 0
+  | add h₁ h₂ ih₁ ih₂ =>
+      have hd₁ := evalAlong_differentiable _ h₁ ρ_real idx
+      have hd₂ := evalAlong_differentiable _ h₂ ρ_real idx
+      simp only [Expr.evalAlong_add_pi, deriv_add (hd₁ x) (hd₂ x),
+        LeanCert.Internal.AD.evalTotalCore,
+        DualInterval.add]
+      exact IntervalRat.mem_add (ih₁ x hx hmem) (ih₂ x hx hmem)
+  | mul h₁ h₂ ih₁ ih₂ =>
+      have hd₁ := evalAlong_differentiable _ h₁ ρ_real idx
+      have hd₂ := evalAlong_differentiable _ h₂ ρ_real idx
+      simp only [Expr.evalAlong_mul_pi, deriv_mul (hd₁ x) (hd₂ x),
+        LeanCert.Internal.AD.evalTotalCore,
+        DualInterval.mul]
+      have hdom₁ := evalDomainValidDual_of_ExprSupported _ h₁
+        (mkDualEnvCore ρ_int idx) cfg
+      have hdom₂ := evalDomainValidDual_of_ExprSupported _ h₂
+        (mkDualEnvCore ρ_int idx) cfg
+      have hval₁ := LeanCert.Engine.evalDualTotalCore_val_correct _ h₁.toCore
+        (Expr.updateVar ρ_real idx x) (mkDualEnvCore ρ_int idx) cfg hmem hdom₁
+      have hval₂ := LeanCert.Engine.evalDualTotalCore_val_correct _ h₂.toCore
+        (Expr.updateVar ρ_real idx x) (mkDualEnvCore ρ_int idx) cfg hmem hdom₂
+      exact IntervalRat.mem_add (IntervalRat.mem_mul (ih₁ x hx hmem) hval₂)
+        (IntervalRat.mem_mul hval₁ (ih₂ x hx hmem))
+  | neg hs ih =>
+      have hd := evalAlong_differentiable _ hs ρ_real idx
+      simp only [Expr.evalAlong_neg_pi, deriv.neg, LeanCert.Internal.AD.evalTotalCore,
+        DualInterval.neg]
+      exact IntervalRat.mem_neg (ih x hx hmem)
+  | @sin e' hs ih =>
+      have hd := evalAlong_differentiable e' hs ρ_real idx
+      simp only [Expr.evalAlong_sin, deriv_sin (hd.differentiableAt),
+        LeanCert.Internal.AD.evalTotalCore,
+        DualInterval.sinCore]
+      have hdom := evalDomainValidDual_of_ExprSupported e' hs
+        (mkDualEnvCore ρ_int idx) cfg
+      have hval := LeanCert.Engine.evalDualTotalCore_val_correct e' hs.toCore
+        (Expr.updateVar ρ_real idx x) (mkDualEnvCore ρ_int idx) cfg hmem hdom
+      exact IntervalRat.mem_mul
+        (IntervalRat.mem_cosComputable hval cfg.taylorDepth) (ih x hx hmem)
+  | @cos e' hs ih =>
+      have hd := evalAlong_differentiable e' hs ρ_real idx
+      simp only [Expr.evalAlong_cos, deriv_cos (hd.differentiableAt),
+        LeanCert.Internal.AD.evalTotalCore,
+        DualInterval.cosCore]
+      have hdom := evalDomainValidDual_of_ExprSupported e' hs
+        (mkDualEnvCore ρ_int idx) cfg
+      have hval := LeanCert.Engine.evalDualTotalCore_val_correct e' hs.toCore
+        (Expr.updateVar ρ_real idx x) (mkDualEnvCore ρ_int idx) cfg hmem hdom
+      exact IntervalRat.mem_mul
+        (IntervalRat.mem_neg (IntervalRat.mem_sinComputable hval cfg.taylorDepth)) (ih x hx hmem)
+  | @exp e' hs ih =>
+      have hd := evalAlong_differentiable e' hs ρ_real idx
+      simp only [Expr.evalAlong_exp, deriv_exp (hd.differentiableAt),
+        LeanCert.Internal.AD.evalTotalCore,
+        DualInterval.expCore]
+      have hdom := evalDomainValidDual_of_ExprSupported e' hs
+        (mkDualEnvCore ρ_int idx) cfg
+      have hval := LeanCert.Engine.evalDualTotalCore_val_correct e' hs.toCore
+        (Expr.updateVar ρ_real idx x) (mkDualEnvCore ρ_int idx) cfg hmem hdom
+      exact IntervalRat.mem_mul
+        (IntervalRat.mem_expComputable hval cfg.taylorDepth) (ih x hx hmem)
+
+/-- Computable version of gradient interval for Core expressions.
+    This can be used with `native_decide` for verified optimization. -/
+def gradientIntervalCore (e : Expr) (B : Box) (cfg : EvalConfig := {}) : List IntervalRat :=
+  (List.range B.length).map fun i => derivIntervalCoreN e (Box.toEnv B) i cfg
+
+/-- Compute every partial derivative using the domain-aware checked AD path.
+Unlike `gradientIntervalCore`, this rejects unsupported syntax, reciprocal
+arguments containing zero, and nonpositive logarithm arguments instead of
+returning a finite interval that could be mistaken for a certificate. -/
+def gradientIntervalChecked (e : Expr) (B : Box) (cfg : EvalConfig := {}) :
+    EvalResult (List IntervalRat) :=
+  (List.range B.length).mapM fun i =>
+    derivIntervalChecked e (Box.toEnv B) i cfg
+
+private theorem derivIntervalsChecked_correct (e : Expr) (B : Box)
+    (cfg : EvalConfig) (ρReal : Nat → ℝ)
+    (hρ : ∀ i, ρReal i ∈ Box.toEnv B i) (indices : List Nat)
+    (gradient : List IntervalRat)
+    (hok : indices.mapM (fun i => derivIntervalChecked e (Box.toEnv B) i cfg) =
+      .ok gradient) :
+    List.Forall₂ (fun i dI => deriv (Expr.evalAlong e ρReal i) (ρReal i) ∈ dI)
+      indices gradient := by
+  induction indices generalizing gradient with
+  | nil =>
+      simp only [List.mapM_nil, pure, Except.pure, Except.ok.injEq] at hok
+      subst gradient
+      exact .nil
+  | cons i indices ih =>
+      simp only [List.mapM_cons] at hok
+      cases hdi : derivIntervalChecked e (Box.toEnv B) i cfg with
+      | error err =>
+          rw [hdi] at hok
+          simp only [bind, Except.bind] at hok
+          cases hok
+      | ok dI =>
+          rw [hdi] at hok
+          simp only [bind, Except.bind] at hok
+          cases htail : List.mapM
+              (fun j => derivIntervalChecked e (Box.toEnv B) j cfg) indices with
+          | error err =>
+              rw [htail] at hok
+              cases hok
+          | ok tail =>
+              rw [htail] at hok
+              simp only [pure, Except.pure, Except.ok.injEq] at hok
+              subst gradient
+              exact .cons
+                (derivIntervalChecked_correct e ρReal (Box.toEnv B) i cfg dI
+                  (ρReal i) (hρ i) hρ hdi)
+                (ih tail htail)
+
+/-- Golden soundness theorem for a successfully computed checked gradient.
+The output list is aligned with coordinates `0, …, B.length - 1`. -/
+theorem gradientIntervalChecked_correct (e : Expr) (B : Box)
+    (cfg : EvalConfig) (ρReal : Nat → ℝ)
+    (hρ : Box.envMem ρReal B)
+    (hzero : ∀ i, i ≥ B.length → ρReal i = 0)
+    (gradient : List IntervalRat)
+    (hok : gradientIntervalChecked e B cfg = .ok gradient) :
+    List.Forall₂ (fun i dI => deriv (Expr.evalAlong e ρReal i) (ρReal i) ∈ dI)
+      (List.range B.length) gradient := by
+  exact derivIntervalsChecked_correct e B cfg ρReal
+    (Box.envMem_toEnv ρReal B hρ hzero) (List.range B.length) gradient hok
+
+/-! ### Sign classification -/
+
+/-- Classification of an interval's sign -/
+inductive IntervalSign where
+  | positive     -- lo > 0 (strictly positive)
+  | negative     -- hi < 0 (strictly negative)
+  | nonpositive  -- hi ≤ 0
+  | nonnegative  -- lo ≥ 0
+  | indefinite   -- contains zero in interior
+  deriving Repr, DecidableEq
+
+/-- Classify the sign of an interval -/
+def classifySign (I : IntervalRat) : IntervalSign :=
+  if I.lo > 0 then .positive
+  else if I.hi < 0 then .negative
+  else if I.hi ≤ 0 then .nonpositive
+  else if I.lo ≥ 0 then .nonnegative
+  else .indefinite
+
+/-- The gradient signature: sign of each partial derivative (noncomputable wrapper) -/
+noncomputable def gradientSignature (e : Expr) (B : Box) : List IntervalSign :=
+  (gradientIntervalN e B B.length).map classifySign
+
+/-! ### Monotonicity predicates -/
+
+/-- Check if interval is strictly positive -/
+def isStrictlyPositive (I : IntervalRat) : Bool := I.lo > 0
+
+/-- Check if interval is strictly negative -/
+def isStrictlyNegative (I : IntervalRat) : Bool := I.hi < 0
+
+/-- Check if interval is nonnegative -/
+def isNonnegative (I : IntervalRat) : Bool := I.lo ≥ 0
+
+/-- Check if interval is nonpositive -/
+def isNonpositive (I : IntervalRat) : Bool := I.hi ≤ 0
+
+/-! ### Pruning queries -/
+
+/-- Can we prune coordinate i to its low endpoint for minimization?
+    True if ∂f/∂xᵢ > 0 on B (f is increasing in xᵢ, so min is at lo). -/
+def canPruneToLo (deriv_i : IntervalRat) : Bool :=
+  isStrictlyPositive deriv_i
+
+/-- Can we prune coordinate i to its high endpoint for minimization?
+    True if ∂f/∂xᵢ < 0 on B (f is decreasing in xᵢ, so min is at hi). -/
+def canPruneToHi (deriv_i : IntervalRat) : Bool :=
+  isStrictlyNegative deriv_i
+
+/-- Prune a box for minimization by fixing monotonic coordinates.
+    Returns a (potentially smaller) box and a list of fixed coordinates. -/
+def pruneBoxForMin (B : Box) (grad : List IntervalRat) : Box × List Nat :=
+  let pruned := B.zipIdx.map fun (I, idx) =>
+    match grad[idx]? with
+    | some di =>
+      if canPruneToLo di then
+        -- ∂f/∂xᵢ > 0: fix xᵢ = I.lo
+        (IntervalRat.singleton I.lo, some idx)
+      else if canPruneToHi di then
+        -- ∂f/∂xᵢ < 0: fix xᵢ = I.hi
+        (IntervalRat.singleton I.hi, some idx)
+      else
+        (I, none)
+    | none => (I, none)
+  (pruned.map (·.1), pruned.filterMap (·.2))
+
+/-! ### Correctness theorems -/
+
+/-- The computed gradient interval contains the true partial derivatives.
+    This follows from evalDual_der_correct_idx in AD.lean. -/
+theorem gradientInterval_correct (e : Expr) (hsupp : ADSupported e)
+    (B : Box) (ρ : Nat → ℝ) (hρ : Box.envMem ρ B)
+    (hzero : ∀ i, i ≥ B.length → ρ i = 0)
+    (i : Fin B.length) :
+    deriv (Expr.evalAlong e ρ i.val) (ρ i.val) ∈
+      ((gradientIntervalN e B B.length)[i.val]?).getD default := by
+  -- The i-th element of gradientIntervalN is derivInterval e (toEnv B) i
+  have hget : (gradientIntervalN e B B.length)[i.val]? =
+      some (derivInterval e (Box.toEnv B) i.val) := by
+    simp only [gradientIntervalN]
+    rw [List.getElem?_map]
+    simp only [List.getElem?_range i.isLt, Option.map_some]
+  simp only [hget, Option.getD]
+  -- Apply the AD correctness theorem
+  have henv : ∀ j, ρ j ∈ Box.toEnv B j := Box.envMem_toEnv ρ B hρ hzero
+  exact derivInterval_correct_idx e hsupp ρ (Box.toEnv B) i.val henv (ρ i.val) (henv i.val)
+
+/-- If we prune a coordinate to lo because ∂f/∂xᵢ > 0, the minimum is preserved.
+    Informal: if f is increasing in xᵢ on B, then min{f(x) : x ∈ B} = min{f(x) : xᵢ = B[i].lo}.
+    NOTE: Requires ρ j = 0 for j ≥ B.length (standard assumption for box membership). -/
+theorem pruneToLo_preserves_min (e : Expr) (hsupp : ADSupported e)
+    (B : Box) (i : Fin B.length)
+    (hgrad : isStrictlyPositive (derivInterval e (Box.toEnv B) i.val) = true) :
+    ∀ (ρ : Nat → ℝ), Box.envMem ρ B → (∀ j, j ≥ B.length → ρ j = 0) →
+      ∃ (ρ' : Nat → ℝ), Box.envMem ρ' B ∧ (∀ j, j ≥ B.length → ρ' j = 0) ∧
+        ρ' i.val = B[i.val].lo ∧ Expr.eval ρ' e ≤ Expr.eval ρ e := by
+  intro ρ hρ hzero
+  -- The idea: if ∂f/∂xᵢ > 0 everywhere, then f(ρ) ≥ f(ρ[xᵢ := lo])
+  -- We construct ρ' by replacing coordinate i with the low endpoint
+  let ρ' : Nat → ℝ := fun j => if j = i.val then (B[i.val].lo : ℝ) else ρ j
+  use ρ'
+  constructor
+  · -- ρ' ∈ B
+    intro ⟨j, hj⟩
+    by_cases h : j = i.val
+    · subst h
+      simp only [↓reduceIte, ρ', IntervalRat.mem_def]
+      constructor
+      · exact le_refl _
+      · have := B[i.val].le
+        exact_mod_cast this
+    · simp only [ite_eq_right h, ρ']
+      exact hρ ⟨j, hj⟩
+  constructor
+  · -- ρ' j = 0 for j ≥ B.length
+    intro j hj
+    simp only [ρ']
+    have hne : j ≠ i.val := by
+      intro heq
+      rw [heq] at hj
+      exact absurd i.isLt (not_lt.mpr hj)
+    simp only [ite_eq_right hne]
+    exact hzero j hj
+  constructor
+  · -- ρ' i = B[i].lo
+    simp only [↓reduceIte, ρ']
+  · -- f(ρ') ≤ f(ρ)
+    -- Use the monotonicity theorem: if ∂f/∂xᵢ > 0, minimum is at left endpoint
+    -- Convert the boolean condition to a real inequality
+    simp only [isStrictlyPositive, decide_eq_true_eq] at hgrad
+    -- Build the interval environment from the box
+    let ρ_int : IntervalEnv := Box.toEnv B
+    -- Show ρ ∈ ρ_int (membership in the interval environment)
+    have hρ_int : ∀ j, ρ j ∈ ρ_int j := by
+      intro j
+      simp only [ρ_int, Box.toEnv, List.getD]
+      by_cases hj : j < B.length
+      · simp only [List.getElem?_eq_getElem hj, Option.getD]
+        exact hρ ⟨j, hj⟩
+      · simp only [List.getElem?_eq_none (not_lt.mp hj), Option.getD]
+        rw [IntervalRat.mem_default]
+        exact hzero j (not_lt.mp hj)
+    -- Apply the monotonicity theorem
+    have hmono := increasing_min_at_left_idx e hsupp ρ ρ_int i.val hρ_int hgrad
+    -- The key fact: ρ i.val ∈ ρ_int i.val = B[i.val]
+    have hρ_i_mem : ρ i.val ∈ ρ_int i.val := hρ_int i.val
+    have hmin := hmono (ρ i.val) hρ_i_mem
+    -- Now relate evalAlong to eval via ρ'
+    -- evalAlong e ρ i.val t = eval (updateVar ρ i.val t) e
+    simp only [Expr.evalAlong] at hmin
+    -- Need: (ρ_int i.val).lo = B[i.val].lo
+    have hlo_eq : (ρ_int i.val).lo = B[i.val].lo := by
+      simp only [ρ_int, Box.toEnv, List.getD, List.getElem?_eq_getElem i.isLt, Option.getD]
+    -- And: updateVar ρ i.val (ρ_int i.val).lo = ρ'
+    have hρ'_eq : Expr.updateVar ρ i.val ((ρ_int i.val).lo : ℝ) = ρ' := by
+      funext j
+      simp only [Expr.updateVar, ρ']
+      split_ifs with hj
+      · simp only [hlo_eq]
+      · rfl
+    -- And: updateVar ρ i.val (ρ i.val) = ρ
+    have hρ_eq : Expr.updateVar ρ i.val (ρ i.val) = ρ := Expr.updateVar_self ρ i.val
+    rw [hρ'_eq, hρ_eq] at hmin
+    exact hmin
+
+/-- If we prune a coordinate to hi because ∂f/∂xᵢ < 0, the minimum is preserved.
+    NOTE: Requires ρ j = 0 for j ≥ B.length (standard assumption for box membership). -/
+theorem pruneToHi_preserves_min (e : Expr) (hsupp : ADSupported e)
+    (B : Box) (i : Fin B.length)
+    (hgrad : isStrictlyNegative (derivInterval e (Box.toEnv B) i.val) = true) :
+    ∀ (ρ : Nat → ℝ), Box.envMem ρ B → (∀ j, j ≥ B.length → ρ j = 0) →
+      ∃ (ρ' : Nat → ℝ), Box.envMem ρ' B ∧ (∀ j, j ≥ B.length → ρ' j = 0) ∧
+        ρ' i.val = B[i.val].hi ∧ Expr.eval ρ' e ≤ Expr.eval ρ e := by
+  intro ρ hρ hzero
+  let ρ' : Nat → ℝ := fun j => if j = i.val then (B[i.val].hi : ℝ) else ρ j
+  use ρ'
+  constructor
+  · -- ρ' ∈ B
+    intro ⟨j, hj⟩
+    by_cases h : j = i.val
+    · subst h
+      simp only [↓reduceIte, ρ', IntervalRat.mem_def]
+      constructor
+      · have := B[i.val].le
+        exact_mod_cast this
+      · exact le_refl _
+    · simp only [ite_eq_right h, ρ']
+      exact hρ ⟨j, hj⟩
+  constructor
+  · -- ρ' j = 0 for j ≥ B.length
+    intro j hj
+    simp only [ρ']
+    have hne : j ≠ i.val := by
+      intro heq
+      rw [heq] at hj
+      exact absurd i.isLt (not_lt.mpr hj)
+    simp only [ite_eq_right hne]
+    exact hzero j hj
+  constructor
+  · simp only [↓reduceIte, ρ']
+  · -- f(ρ') ≤ f(ρ)
+    -- Use the monotonicity theorem: if ∂f/∂xᵢ < 0, minimum is at right endpoint
+    simp only [isStrictlyNegative, decide_eq_true_eq] at hgrad
+    -- Build the interval environment from the box
+    let ρ_int : IntervalEnv := Box.toEnv B
+    -- Show ρ ∈ ρ_int
+    have hρ_int : ∀ j, ρ j ∈ ρ_int j := by
+      intro j
+      simp only [ρ_int, Box.toEnv, List.getD]
+      by_cases hj : j < B.length
+      · simp only [List.getElem?_eq_getElem hj, Option.getD]
+        exact hρ ⟨j, hj⟩
+      · simp only [List.getElem?_eq_none (not_lt.mp hj), Option.getD]
+        rw [IntervalRat.mem_default]
+        exact hzero j (not_lt.mp hj)
+    -- Apply the monotonicity theorem for decreasing functions
+    have hmono := decreasing_min_at_right_idx e hsupp ρ ρ_int i.val hρ_int hgrad
+    -- The key fact: ρ i.val ∈ ρ_int i.val = B[i.val]
+    have hρ_i_mem : ρ i.val ∈ ρ_int i.val := hρ_int i.val
+    have hmin := hmono (ρ i.val) hρ_i_mem
+    -- Now relate evalAlong to eval via ρ'
+    simp only [Expr.evalAlong] at hmin
+    -- Need: (ρ_int i.val).hi = B[i.val].hi
+    have hhi_eq : (ρ_int i.val).hi = B[i.val].hi := by
+      simp only [ρ_int, Box.toEnv, List.getD, List.getElem?_eq_getElem i.isLt, Option.getD]
+    -- And: updateVar ρ i.val (ρ_int i.val).hi = ρ'
+    have hρ'_eq : Expr.updateVar ρ i.val ((ρ_int i.val).hi : ℝ) = ρ' := by
+      funext j
+      simp only [Expr.updateVar, ρ']
+      split_ifs with hj
+      · simp only [hhi_eq]
+      · rfl
+    -- And: updateVar ρ i.val (ρ i.val) = ρ
+    have hρ_eq : Expr.updateVar ρ i.val (ρ i.val) = ρ := Expr.updateVar_self ρ i.val
+    rw [hρ'_eq, hρ_eq] at hmin
+    exact hmin
+
+/-! ### Pruned box membership and correctness -/
+
+/-- Helper: membership in the pruned box implies membership in the original box.
+    The pruned box only shrinks coordinates, never expands them. -/
+theorem pruneBoxForMin_subset (B : Box) (grad : List IntervalRat) :
+    ∀ ρ, Box.envMem ρ (pruneBoxForMin B grad).1 → Box.envMem ρ B := by
+  intro ρ hρ'
+  let B' := (pruneBoxForMin B grad).1
+  intro ⟨j, hj⟩
+  have hj' : j < B'.length := by
+    simp only [B', pruneBoxForMin, List.length_map, List.length_zipIdx]
+    exact hj
+  have hρ'_mem : ρ j ∈ B'[j] := hρ' ⟨j, hj'⟩
+  -- Get the j-th element of B'
+  simp only [B', pruneBoxForMin] at hρ'_mem
+  rw [List.getElem_map, List.getElem_map] at hρ'_mem
+  have h_zipIdx_len : j < (B.zipIdx).length := by simp only [List.length_zipIdx]; exact hj
+  have h_zipIdx : (B.zipIdx)[j] = (B[j], j) := by
+    rw [List.getElem_zipIdx (h := h_zipIdx_len)]
+    simp only [Nat.zero_add]
+  simp only [h_zipIdx] at hρ'_mem
+  -- Now analyze the cases
+  cases hgrad_j : grad[j]? with
+  | none =>
+    simp only [hgrad_j] at hρ'_mem
+    exact hρ'_mem
+  | some di =>
+    simp only [hgrad_j] at hρ'_mem
+    by_cases hlo : canPruneToLo di
+    · simp only [hlo, ↓reduceIte] at hρ'_mem
+      simp only [IntervalRat.mem_def]
+      have hρ_eq : ρ j = B[j].lo := by
+        simp only [IntervalRat.singleton, IntervalRat.mem_def] at hρ'_mem
+        linarith [hρ'_mem.1, hρ'_mem.2]
+      rw [hρ_eq]
+      exact ⟨le_refl _, by exact_mod_cast B[j].le⟩
+    · by_cases hhi : canPruneToHi di
+      · simp only [hlo, Bool.false_eq_true, ↓reduceIte, hhi] at hρ'_mem
+        simp only [IntervalRat.mem_def]
+        have hρ_eq : ρ j = B[j].hi := by
+          simp only [IntervalRat.singleton, IntervalRat.mem_def] at hρ'_mem
+          linarith [hρ'_mem.1, hρ'_mem.2]
+        rw [hρ_eq]
+        exact ⟨by exact_mod_cast B[j].le, le_refl _⟩
+      · simp only [hlo, Bool.false_eq_true, ↓reduceIte, hhi] at hρ'_mem
+        exact hρ'_mem
+
+/-- The pruned box has the same length as the original box -/
+theorem pruneBoxForMin_length (B : Box) (grad : List IntervalRat) :
+    (pruneBoxForMin B grad).1.length = B.length := by
+  simp only [pruneBoxForMin, List.length_map, List.length_zipIdx]
+
+/-- A positive computable derivative enclosure makes the objective increasing
+along the selected coordinate. -/
+theorem increasing_min_at_left_idx_core (e : Expr) (hsupp : ADSupported e)
+    (ρ_real : Nat → ℝ) (ρ_int : IntervalEnv) (idx : Nat)
+    (hρ : ∀ i, ρ_real i ∈ ρ_int i) (cfg : EvalConfig)
+    (hpos : 0 < (derivIntervalCoreN e ρ_int idx cfg).lo) :
+    ∀ x ∈ ρ_int idx,
+      Expr.evalAlong e ρ_real idx (ρ_int idx).lo ≤ Expr.evalAlong e ρ_real idx x := by
+  have hdiff := evalAlong_differentiable e hsupp ρ_real idx
+  have hmono : StrictMonoOn (Expr.evalAlong e ρ_real idx)
+      (Set.Icc ((ρ_int idx).lo : ℝ) ((ρ_int idx).hi : ℝ)) := by
+    apply strictMonoOn_of_deriv_pos (convex_Icc _ _)
+    · exact hdiff.continuous.continuousOn
+    · intro x hx
+      rw [interior_Icc] at hx
+      have hx' : x ∈ ρ_int idx := ⟨le_of_lt hx.1, le_of_lt hx.2⟩
+      have hmem := evalDualTotalCore_der_correct_idx e hsupp ρ_real ρ_int idx hρ x hx' cfg
+      exact lt_of_lt_of_le (by exact_mod_cast hpos) ((IntervalRat.mem_def _ _).mp hmem).1
+  intro x hx
+  rcases hx with ⟨hlo, hhi⟩
+  by_cases heq : ((ρ_int idx).lo : ℝ) = x
+  · exact heq ▸ le_rfl
+  · exact le_of_lt (hmono
+      ⟨le_rfl, by exact_mod_cast (ρ_int idx).le⟩ ⟨hlo, hhi⟩
+      (lt_of_le_of_ne hlo heq))
+
+/-- A negative computable derivative enclosure makes the objective decreasing
+along the selected coordinate. -/
+theorem decreasing_min_at_right_idx_core (e : Expr) (hsupp : ADSupported e)
+    (ρ_real : Nat → ℝ) (ρ_int : IntervalEnv) (idx : Nat)
+    (hρ : ∀ i, ρ_real i ∈ ρ_int i) (cfg : EvalConfig)
+    (hneg : (derivIntervalCoreN e ρ_int idx cfg).hi < 0) :
+    ∀ x ∈ ρ_int idx,
+      Expr.evalAlong e ρ_real idx (ρ_int idx).hi ≤ Expr.evalAlong e ρ_real idx x := by
+  have hdiff := evalAlong_differentiable e hsupp ρ_real idx
+  have hmono : StrictAntiOn (Expr.evalAlong e ρ_real idx)
+      (Set.Icc ((ρ_int idx).lo : ℝ) ((ρ_int idx).hi : ℝ)) := by
+    apply strictAntiOn_of_deriv_neg (convex_Icc _ _)
+    · exact hdiff.continuous.continuousOn
+    · intro x hx
+      rw [interior_Icc] at hx
+      have hx' : x ∈ ρ_int idx := ⟨le_of_lt hx.1, le_of_lt hx.2⟩
+      have hmem := evalDualTotalCore_der_correct_idx e hsupp ρ_real ρ_int idx hρ x hx' cfg
+      exact lt_of_le_of_lt ((IntervalRat.mem_def _ _).mp hmem).2 (by exact_mod_cast hneg)
+  intro x hx
+  rcases hx with ⟨hlo, hhi⟩
+  by_cases heq : x = ((ρ_int idx).hi : ℝ)
+  · exact heq ▸ le_rfl
+  · exact le_of_lt (hmono ⟨hlo, hhi⟩
+      ⟨by exact_mod_cast (ρ_int idx).le, le_rfl⟩
+      (lt_of_le_of_ne hhi heq))
+
+/-- **Main correctness theorem for pruneBoxForMin:**
+
+    After pruning, for any point ρ in the original box B, there exists a point ρ'
+    in the pruned box B' such that f(ρ') ≤ f(ρ).
+
+    This means the minimum over B can be found by searching only in B'.
+
+    The proof constructs ρ' by moving each coordinate to its endpoint when the
+    gradient has a definite sign. For each coordinate:
+    - If ∂f/∂xᵢ > 0 on B, move xᵢ to B[i].lo (f is increasing, min at left)
+    - If ∂f/∂xᵢ < 0 on B, move xᵢ to B[i].hi (f is decreasing, min at right)
+    - Otherwise, keep xᵢ = ρ[i]
+
+    The proof then shows f(ρ') ≤ f(ρ) by induction on coordinates, using
+    the monotonicity lemmas `increasing_min_at_left_idx` and `decreasing_min_at_right_idx`.
+-/
+private def prunePrefixEnvironment (B : Box) (grad : List IntervalRat) (ρ : Nat → ℝ) :
+    Nat → (Nat → ℝ) := fun k j =>
+  if h : j < k ∧ j < B.length then
+    match grad[j]? with
+    | some di =>
+      if canPruneToLo di then (B[j].lo : ℝ)
+      else if canPruneToHi di then (B[j].hi : ℝ)
+      else ρ j
+    | none => ρ j
+  else if hj : j < B.length then ρ j
+  else 0
+
+private theorem prunePrefixEnvironment_mem (B : Box) (grad : List IntervalRat)
+    (ρ : Nat → ℝ) (m : ℕ)
+    (hρB : Box.envMem ρ B)
+    : Box.envMem (prunePrefixEnvironment B grad ρ m) B := by
+  let ρ_seq := prunePrefixEnvironment B grad ρ
+  intro ⟨j, hj⟩
+  simp only [prunePrefixEnvironment]
+  by_cases h1 : j < m ∧ j < B.length
+  · simp only [dite_eq_left h1]
+    cases hgrad_j : grad[j]? with
+    | none => exact hρB ⟨j, hj⟩
+    | some dj =>
+      simp only [IntervalRat.mem_def]
+      by_cases hlo_j : canPruneToLo dj
+      · simp only [hlo_j, ↓reduceIte]
+        exact ⟨le_refl _, by exact_mod_cast B[j].le⟩
+      · by_cases hhi_j : canPruneToHi dj
+        · simp only [hlo_j, Bool.false_eq_true, ↓reduceIte, hhi_j]
+          exact ⟨by exact_mod_cast B[j].le, le_refl _⟩
+        · simp only [hlo_j, Bool.false_eq_true, ↓reduceIte, hhi_j]
+          exact hρB ⟨j, hj⟩
+  · simp only [dite_eq_right h1, dite_eq_left hj]
+    exact hρB ⟨j, hj⟩
+
+private theorem prunePrefixEnvironment_zero (B : Box) (grad : List IntervalRat)
+    (ρ : Nat → ℝ) (m : ℕ)
+    : ∀ j, j ≥ B.length → prunePrefixEnvironment B grad ρ m j = 0 := by
+  let ρ_seq := prunePrefixEnvironment B grad ρ
+  intro j hjge
+  simp only [prunePrefixEnvironment]
+  have h1 : ¬(j < m ∧ j < B.length) := fun h => absurd h.2 (not_lt.mpr hjge)
+  simp only [dite_eq_right h1]
+  have h2 : ¬(j < B.length) := not_lt.mpr hjge
+  simp only [dite_eq_right h2]
+
+private theorem prunePrefixEnvironment_step (e : Expr) (hsupp : ADSupported e)
+    (B : Box) (cfg : EvalConfig) (ρ : Nat → ℝ) (hρB : Box.envMem ρ B) :
+    ∀ m < B.length,
+      Expr.eval (prunePrefixEnvironment B (gradientIntervalCore e B cfg) ρ (m + 1)) e ≤
+        Expr.eval (prunePrefixEnvironment B (gradientIntervalCore e B cfg) ρ m) e := by
+  let grad := gradientIntervalCore e B cfg
+  let ρ_seq := prunePrefixEnvironment B grad ρ
+  have hρ_seq_step : ∀ k j, k ≠ j → ρ_seq k j = ρ_seq (k + 1) j := by
+    intro k j hne
+    simp only [ρ_seq, prunePrefixEnvironment]
+    by_cases h1 : j < k ∧ j < B.length
+    · -- j < k and j < B.length: both difs are positive
+      have h2 : j < k + 1 ∧ j < B.length := ⟨Nat.lt_of_lt_of_le h1.1 (Nat.le_succ k), h1.2⟩
+      simp only [dite_eq_left h1, dite_eq_left h2]
+    · by_cases h2 : j < k + 1 ∧ j < B.length
+      · -- j < k + 1 but not (j < k ∧ j < B.length)
+        have hj_ge_k : k ≤ j := by
+          by_contra! hlt
+          exact h1 ⟨hlt, h2.2⟩
+        have hj_lt_k1 : j < k + 1 := h2.1
+        have hj_eq_k : j = k := Nat.eq_of_le_of_lt_succ hj_ge_k hj_lt_k1
+        exact absurd hj_eq_k.symm hne
+      · -- Neither condition holds
+        simp only [dite_eq_right h1, dite_eq_right h2]
+  intro m hm
+  change Expr.eval (ρ_seq (m + 1)) e ≤ Expr.eval (ρ_seq m) e
+  cases hgrad_m : grad[m]? with
+  | none =>
+    have heq : ∀ j, ρ_seq (m + 1) j = ρ_seq m j := by
+      intro j
+      by_cases hj_eq : j = m
+      · -- When j = m: both sides simplify to ρ m
+        simp only [ρ_seq, prunePrefixEnvironment, hgrad_m, hj_eq]
+        have h1 : ¬(m < m ∧ m < B.length) := fun h => Nat.lt_irrefl m h.1
+        have h2 : m < m + 1 ∧ m < B.length := ⟨Nat.lt_succ_self m, hm⟩
+        simp only [dite_eq_right h1, dite_eq_left h2, dite_eq_left hm]
+      · exact (hρ_seq_step m j (Ne.symm hj_eq)).symm
+    simp only [funext heq]
+    exact le_refl _
+  | some di =>
+    by_cases hlo : canPruneToLo di
+    · -- ∂f/∂x_m > 0, fixing x_m = lo decreases f
+      have hcoord_m_before : ρ_seq m m = ρ m := by
+        simp only [ρ_seq, prunePrefixEnvironment]
+        have h1 : ¬(m < m ∧ m < B.length) := fun h => Nat.lt_irrefl m h.1
+        simp only [dite_eq_right h1, dite_eq_left hm]
+      have hcoord_m_after : ρ_seq (m + 1) m = B[m].lo := by
+        simp only [ρ_seq, prunePrefixEnvironment, hgrad_m, hlo, ↓reduceIte]
+        have h2 : m < m + 1 ∧ m < B.length := ⟨Nat.lt_succ_self m, hm⟩
+        simp only [dite_eq_left h2]
+      have hρ_seq_m_mem := prunePrefixEnvironment_mem B grad ρ m hρB
+      have hρ_seq_m_zero := prunePrefixEnvironment_zero B grad ρ m
+      have hgrad_di : derivIntervalCoreN e (Box.toEnv B) m cfg = di := by
+        simp only [grad, gradientIntervalCore] at hgrad_m
+        rw [List.getElem?_map, List.getElem?_range hm] at hgrad_m
+        simp only [Option.map_some] at hgrad_m
+        exact Option.some.inj hgrad_m
+      let ρ_int : IntervalEnv := Box.toEnv B
+      have hρ_int : ∀ j, ρ_seq m j ∈ ρ_int j := by
+        intro j
+        simp only [ρ_int, Box.toEnv, List.getD]
+        by_cases hj : j < B.length
+        · simp only [List.getElem?_eq_getElem hj, Option.getD]
+          exact hρ_seq_m_mem ⟨j, hj⟩
+        · simp only [List.getElem?_eq_none (not_lt.mp hj), Option.getD]
+          rw [IntervalRat.mem_default]
+          exact hρ_seq_m_zero j (not_lt.mp hj)
+      have hpos : 0 < (derivIntervalCoreN e ρ_int m cfg).lo := by
+        simp only [ρ_int]
+        rw [hgrad_di]
+        simp only [isStrictlyPositive, canPruneToLo] at hlo
+        exact decide_eq_true_iff.mp hlo
+      have hmono := increasing_min_at_left_idx_core e hsupp (ρ_seq m) ρ_int m
+        hρ_int cfg hpos
+      have hρ_m_mem : ρ_seq m m ∈ ρ_int m := hρ_int m
+      have hmin := hmono (ρ_seq m m) hρ_m_mem
+      have hlo_eq : (ρ_int m).lo = B[m].lo := by
+        simp only [ρ_int, Box.toEnv, List.getD, List.getElem?_eq_getElem hm, Option.getD]
+      simp only [Expr.evalAlong] at hmin
+      have hupdate_self : Expr.updateVar (ρ_seq m) m (ρ_seq m m) = ρ_seq m :=
+        Expr.updateVar_self (ρ_seq m) m
+      have hρ_seq_update : ρ_seq (m + 1) = Expr.updateVar (ρ_seq m) m (B[m].lo : ℝ) := by
+        funext j
+        simp only [Expr.updateVar]
+        by_cases hj_eq : j = m
+        · subst hj_eq
+          simp only [↓reduceIte]
+          exact hcoord_m_after
+        · simp only [ite_eq_right hj_eq]
+          exact (hρ_seq_step m j (Ne.symm hj_eq)).symm
+      have hupdate_lo : Expr.updateVar (ρ_seq m) m ((ρ_int m).lo : ℝ) = ρ_seq (m + 1) := by
+        rw [hlo_eq, ← hρ_seq_update]
+      rw [hupdate_lo, hupdate_self] at hmin
+      exact hmin
+    · by_cases hhi : canPruneToHi di
+      · -- ∂f/∂x_m < 0, fixing x_m = hi decreases f
+        have hcoord_m_before : ρ_seq m m = ρ m := by
+          simp only [ρ_seq, prunePrefixEnvironment]
+          have h1 : ¬(m < m ∧ m < B.length) := fun h => Nat.lt_irrefl m h.1
+          simp only [dite_eq_right h1, dite_eq_left hm]
+        have hcoord_m_after : ρ_seq (m + 1) m = B[m].hi := by
+          simp only [ρ_seq, prunePrefixEnvironment, hgrad_m, hlo, Bool.false_eq_true,
+            ↓reduceIte, hhi]
+          have h2 : m < m + 1 ∧ m < B.length := ⟨Nat.lt_succ_self m, hm⟩
+          simp only [dite_eq_left h2]
+        have hρ_seq_m_mem := prunePrefixEnvironment_mem B grad ρ m hρB
+        have hρ_seq_m_zero := prunePrefixEnvironment_zero B grad ρ m
+        have hgrad_di : derivIntervalCoreN e (Box.toEnv B) m cfg = di := by
+          simp only [grad, gradientIntervalCore] at hgrad_m
+          rw [List.getElem?_map, List.getElem?_range hm] at hgrad_m
+          simp only [Option.map_some] at hgrad_m
+          exact Option.some.inj hgrad_m
+        let ρ_int : IntervalEnv := Box.toEnv B
+        have hρ_int : ∀ j, ρ_seq m j ∈ ρ_int j := by
+          intro j
+          simp only [ρ_int, Box.toEnv, List.getD]
+          by_cases hj : j < B.length
+          · simp only [List.getElem?_eq_getElem hj, Option.getD]
+            exact hρ_seq_m_mem ⟨j, hj⟩
+          · simp only [List.getElem?_eq_none (not_lt.mp hj), Option.getD]
+            rw [IntervalRat.mem_default]
+            exact hρ_seq_m_zero j (not_lt.mp hj)
+        have hneg : (derivIntervalCoreN e ρ_int m cfg).hi < 0 := by
+          simp only [ρ_int]
+          rw [hgrad_di]
+          simp only [isStrictlyNegative, canPruneToHi] at hhi
+          exact decide_eq_true_iff.mp hhi
+        have hmono := decreasing_min_at_right_idx_core e hsupp (ρ_seq m) ρ_int m
+          hρ_int cfg hneg
+        have hρ_m_mem : ρ_seq m m ∈ ρ_int m := hρ_int m
+        have hmin := hmono (ρ_seq m m) hρ_m_mem
+        have hhi_eq : (ρ_int m).hi = B[m].hi := by
+          simp only [ρ_int, Box.toEnv, List.getD, List.getElem?_eq_getElem hm, Option.getD]
+        simp only [Expr.evalAlong] at hmin
+        have hupdate_self : Expr.updateVar (ρ_seq m) m (ρ_seq m m) = ρ_seq m :=
+          Expr.updateVar_self (ρ_seq m) m
+        have hρ_seq_update : ρ_seq (m + 1) = Expr.updateVar (ρ_seq m) m (B[m].hi : ℝ) := by
+          funext j
+          simp only [Expr.updateVar]
+          by_cases hj_eq : j = m
+          · subst hj_eq
+            simp only [↓reduceIte]
+            exact hcoord_m_after
+          · simp only [ite_eq_right hj_eq]
+            exact (hρ_seq_step m j (Ne.symm hj_eq)).symm
+        have hupdate_hi : Expr.updateVar (ρ_seq m) m ((ρ_int m).hi : ℝ) = ρ_seq (m + 1) := by
+          rw [hhi_eq, ← hρ_seq_update]
+        rw [hupdate_hi, hupdate_self] at hmin
+        exact hmin
+      · -- No pruning at coord m: ρ_seq (m+1) = ρ_seq m at all coords
+        have heq : ∀ j, ρ_seq (m + 1) j = ρ_seq m j := by
+          intro j
+          by_cases hj_eq : j = m
+          · -- When j = m: need ρ_seq (m+1) m = ρ_seq m m
+            simp only [ρ_seq, prunePrefixEnvironment, hgrad_m, hlo, Bool.false_eq_true,
+              ↓reduceIte, hhi, hj_eq]
+            have h1 : ¬(m < m ∧ m < B.length) := fun h => Nat.lt_irrefl m h.1
+            have h2 : m < m + 1 ∧ m < B.length := ⟨Nat.lt_succ_self m, hm⟩
+            simp only [dite_eq_right h1, dite_eq_left h2, dite_eq_left hm]
+          · exact (hρ_seq_step m j (Ne.symm hj_eq)).symm
+        simp only [funext heq]
+        exact le_refl _
+
+theorem pruneBoxForMin_correct (e : Expr) (hsupp : ADSupported e) (B : Box)
+    (cfg : EvalConfig := {}) :
+    let grad := gradientIntervalCore e B cfg
+    let B' := (pruneBoxForMin B grad).1
+    ∀ (ρ : Nat → ℝ), Box.envMem ρ B → (∀ i, i ≥ B.length → ρ i = 0) →
+      ∃ (ρ' : Nat → ℝ), Box.envMem ρ' B' ∧ (∀ i, i ≥ B'.length → ρ' i = 0) ∧
+        Expr.eval ρ' e ≤ Expr.eval ρ e := by
+  intro grad B' ρ hρB hzero
+  let ρ' : Nat → ℝ := fun j =>
+    if h : j < B.length then
+      match grad[j]? with
+      | some di =>
+        if canPruneToLo di then (B[j].lo : ℝ)
+        else if canPruneToHi di then (B[j].hi : ℝ)
+        else ρ j
+      | none => ρ j
+    else 0
+  use ρ'
+  constructor
+  · -- ρ' ∈ B'
+    intro ⟨j, hj'⟩
+    have hB'_len : B'.length = B.length := pruneBoxForMin_length B grad
+    have hj : j < B.length := by rw [← hB'_len]; exact hj'
+    simp only [ρ', hj, ↓reduceDIte]
+    simp only [B', pruneBoxForMin]
+    rw [List.getElem_map, List.getElem_map]
+    have h_zipIdx_len : j < (B.zipIdx).length := by simp only [List.length_zipIdx]; exact hj
+    have h_zipIdx : (B.zipIdx)[j] = (B[j], j) := by
+      rw [List.getElem_zipIdx (h := h_zipIdx_len)]
+      simp only [Nat.zero_add]
+    simp only [h_zipIdx]
+    cases hgrad_j : grad[j]? with
+    | none =>
+      simp only [IntervalRat.mem_def]
+      exact hρB ⟨j, hj⟩
+    | some di =>
+      simp only [IntervalRat.mem_def]
+      by_cases hlo : canPruneToLo di
+      · simp only [hlo, ↓reduceIte]
+        exact IntervalRat.mem_singleton _
+      · by_cases hhi : canPruneToHi di
+        · simp only [hlo, Bool.false_eq_true, ↓reduceIte, hhi]
+          exact IntervalRat.mem_singleton _
+        · simp only [hlo, Bool.false_eq_true, ↓reduceIte, hhi]
+          exact hρB ⟨j, hj⟩
+  constructor
+  · -- ρ' i = 0 for i ≥ B'.length
+    intro i hi
+    have hB'_len : B'.length = B.length := pruneBoxForMin_length B grad
+    rw [hB'_len] at hi
+    simp only [ρ', not_lt.mpr hi, ↓reduceDIte]
+  · -- f(ρ') ≤ f(ρ)
+    let ρ_seq := prunePrefixEnvironment B grad ρ
+    have hρ_seq_zero : ∀ j, ρ_seq 0 j = if j < B.length then ρ j else 0 := by
+      intro j
+      simp only [ρ_seq, prunePrefixEnvironment]
+      have h_neg : ¬(j < 0 ∧ j < B.length) := fun h => Nat.not_lt_zero j h.1
+      simp only [dite_eq_right h_neg]
+      split_ifs <;> rfl
+    have hρ_seq_final : ∀ j, ρ_seq B.length j = ρ' j := by
+      intro j
+      simp only [ρ_seq, prunePrefixEnvironment, ρ']
+      by_cases hj : j < B.length
+      · -- j < B.length
+        have h_and : j < B.length ∧ j < B.length := ⟨hj, hj⟩
+        simp only [dite_eq_left h_and, dite_eq_left hj]
+      · -- j ≥ B.length
+        have h_nand : ¬(j < B.length ∧ j < B.length) := fun h => hj h.2
+        simp only [dite_eq_right h_nand, dite_eq_right hj]
+    have hstep := prunePrefixEnvironment_step e hsupp B cfg ρ hρB
+    have hchain : ∀ n ≤ B.length, Expr.eval (ρ_seq n) e ≤ Expr.eval (ρ_seq 0) e := by
+      intro n hn
+      induction n with
+      | zero => exact le_refl _
+      | succ m ih =>
+        have hm_lt : m < B.length := Nat.lt_of_succ_le hn
+        calc Expr.eval (ρ_seq (m + 1)) e ≤ Expr.eval (ρ_seq m) e := hstep m hm_lt
+          _ ≤ Expr.eval (ρ_seq 0) e := ih (Nat.le_of_lt hm_lt)
+    have hfinal_eq : Expr.eval (ρ_seq B.length) e = Expr.eval ρ' e := by
+      congr 1; funext j; exact hρ_seq_final j
+    have hρ_seq0_eq : Expr.eval (ρ_seq 0) e = Expr.eval ρ e := by
+      congr 1
+      funext j
+      rw [hρ_seq_zero]
+      split_ifs with hj
+      · rfl
+      · exact (hzero j (not_lt.mp hj)).symm
+    rw [← hfinal_eq, ← hρ_seq0_eq]
+    exact hchain B.length (le_refl _)
+
+end LeanCert.Engine.Optimization
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 LeanCert Contributors. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: LeanCert Contributors
+-/
+/-!
+# Certified roots of differentiable systems
+
+This module implements a strong, norm-form Krawczyk test for square systems in
+LeanCert's `ADSupported` expression fragment. The center and rational
+preconditioner are untrusted certificate data;
+`krawczykCheck` verifies AD support, center containment, invertibility,
+a strict infinity-norm contraction bound, and a strict self-map enclosure.
+
+The golden theorem `krawczykCheck_sound` turns a successful Boolean check into
+existence and uniqueness of a real root in the supplied box.
+-/
+
+@[expose] public section
+
+namespace LeanCert.Engine
+open LeanCert.Core
+
+/-- Use the singleton zero interval as the additive identity. -/
+local instance intervalRatZero : Zero IntervalRat := ⟨IntervalRat.singleton 0⟩
+/-- Use endpoint-wise interval addition for finite interval sums. -/
+local instance intervalRatAdd : Add IntervalRat := ⟨IntervalRat.add⟩
+
+private theorem intervalRat_ext {a b : IntervalRat} (hlo : a.lo = b.lo) (hhi : a.hi = b.hi) :
+    a = b := by
+  cases a
+  cases b
+  simp_all
+
+/-- Natural scalar multiplication by repeated interval addition. -/
+local instance : SMul Nat IntervalRat where
+  smul n a :=
+    { lo := (n : ℚ) * a.lo
+      hi := (n : ℚ) * a.hi
+      le := mul_le_mul_of_nonneg_left a.le (by positivity) }
+
+/-- The additive commutative monoid of rational intervals. -/
+local instance : AddCommMonoid IntervalRat :=
+  Function.Injective.addCommMonoid
+    (fun I : IntervalRat => (I.lo, I.hi))
+    (by intro a b h; exact intervalRat_ext (congrArg Prod.fst h) (congrArg Prod.snd h))
+    rfl (fun _ _ => rfl) (fun _ _ => rfl)
+
+/-- Extend a finite real coordinate vector to an expression environment with zero defaults. -/
+noncomputable def finEnv {n : Nat} (x : Fin n → ℝ) : Nat → ℝ :=
+  fun i => if h : i < n then x ⟨i, h⟩ else 0
+
+/-- Evaluate an expression in the environment of a finite real coordinate vector. -/
+noncomputable def evalFin {n : Nat} (e : Expr) (x : Fin n → ℝ) : ℝ :=
+  Expr.eval (finEnv x) e
+
+theorem evalFin_differentiable {n : Nat} (e : Expr) (h : ADSupported e) :
+    Differentiable ℝ (evalFin (n := n) e) := by
+  induction h with
+  | const q =>
+      rw [show evalFin (n := n) (.const q) = (fun _ : Fin n → ℝ => (q : ℝ)) from rfl]
+      exact differentiable_const _
+  | var i =>
+      by_cases hi : i < n
+      · rw [show evalFin (n := n) (.var i) = (fun x : Fin n → ℝ => x ⟨i, hi⟩) by
+          funext x; simp [evalFin, finEnv, hi]]
+        exact differentiable_apply _
+      · rw [show evalFin (n := n) (.var i) = (fun _ : Fin n → ℝ => 0) by
+          funext x; simp [evalFin, finEnv, hi]]
+        exact differentiable_const _
+  | add h₁ h₂ ih₁ ih₂ =>
+      rw [show evalFin (n := n) (.add _ _) = evalFin (n := n) _ + evalFin (n := n) _ from rfl]
+      exact ih₁.add ih₂
+  | mul h₁ h₂ ih₁ ih₂ =>
+      rw [show evalFin (n := n) (.mul _ _) = evalFin (n := n) _ * evalFin (n := n) _ from rfl]
+      exact ih₁.mul ih₂
+  | neg h ih =>
+      rw [show evalFin (n := n) (.neg _) = -evalFin (n := n) _ from rfl]
+      exact ih.neg
+  | sin h ih =>
+      rw [show evalFin (n := n) (.sin _) = Real.sin ∘ evalFin (n := n) _ from rfl]
+      exact Real.differentiable_sin.comp ih
+  | cos h ih =>
+      rw [show evalFin (n := n) (.cos _) = Real.cos ∘ evalFin (n := n) _ from rfl]
+      exact Real.differentiable_cos.comp ih
+  | exp h ih =>
+      rw [show evalFin (n := n) (.exp _) = Real.exp ∘ evalFin (n := n) _ from rfl]
+      exact Real.differentiable_exp.comp ih
+
+theorem finEnv_update {n : Nat} (x : Fin n → ℝ) (j : Fin n) (t : ℝ) :
+    finEnv (Function.update x j t) = Expr.updateVar (finEnv x) j.val t := by
+  funext i
+  by_cases hi : i < n
+  · by_cases hij : i = j.val
+    · subst i
+      simp [finEnv, Expr.updateVar]
+    · have hfin : (⟨i, hi⟩ : Fin n) ≠ j := by
+        intro h
+        exact hij (Fin.ext_iff.mp h)
+      simp [finEnv, Expr.updateVar, hi, hij, hfin]
+  · have hij : i ≠ j.val := by omega
+    simp [finEnv, Expr.updateVar, hi, hij]
+
+theorem fderiv_single_eq_deriv_evalAlong {n : Nat} (e : Expr) (h : ADSupported e)
+    (x : Fin n → ℝ) (j : Fin n) :
+    fderiv ℝ (evalFin (n := n) e) x (Pi.single j 1) =
+      deriv (Expr.evalAlong e (finEnv x) j.val) (x j) := by
+  have hout : HasFDerivAt (evalFin (n := n) e)
+      (fderiv ℝ (evalFin (n := n) e) (Function.update x j (x j)))
+      (Function.update x j (x j)) :=
+    (evalFin_differentiable (n := n) e h).differentiableAt.hasFDerivAt
+  have hcomp := hout.comp_hasDerivAt (x j) (hasDerivAt_update x j (x j))
+  rw [Function.update_eq_self] at hcomp
+  have heq : (evalFin (n := n) e ∘ Function.update x j) =
+      Expr.evalAlong e (finEnv x) j.val := by
+    funext t
+    simp only [Function.comp_apply, evalFin, Expr.evalAlong_eq]
+    rw [finEnv_update]
+  rw [heq] at hcomp
+  exact hcomp.deriv.symm
+
+/-- Evaluate all coordinate expressions of a finite system. -/
+noncomputable def systemEval {n : Nat} (F : Fin n → Expr) (x : Fin n → ℝ) : Fin n → ℝ :=
+  fun i => evalFin (F i) x
+
+theorem systemEval_differentiable {n : Nat} (F : Fin n → Expr)
+    (h : ∀ i, ADSupported (F i)) : Differentiable ℝ (systemEval F) := by
+  rw [differentiable_pi]
+  intro i
+  exact evalFin_differentiable (F i) (h i)
+
+/-- The matrix of coordinate partial derivatives of an expression system. -/
+noncomputable def jacobianAt {n : Nat} (F : Fin n → Expr) (x : Fin n → ℝ) :
+    Matrix (Fin n) (Fin n) ℝ :=
+  LinearMap.toMatrix' (fderiv ℝ (systemEval F) x).toLinearMap
+
+theorem jacobianAt_apply {n : Nat} (F : Fin n → Expr) (h : ∀ i, ADSupported (F i))
+    (x : Fin n → ℝ) (i j : Fin n) :
+    jacobianAt F x i j = deriv (Expr.evalAlong (F i) (finEnv x) j.val) (x j) := by
+  rw [jacobianAt, LinearMap.toMatrix'_apply]
+  change fderiv ℝ (systemEval F) x (Pi.single j 1) i = _
+  rw [show systemEval F = (fun x i => evalFin (F i) x) from rfl]
+  have hpi := fderiv_pi (𝕜 := ℝ) (x := x)
+    (φ := fun i => evalFin (F i)) (fun i => (evalFin_differentiable (F i) (h i)).differentiableAt)
+  rw [hpi]
+  simp only [ContinuousLinearMap.coe_pi']
+  exact fderiv_single_eq_deriv_evalAlong (F i) (h i) x j
+
+/-- Extend a finite interval box to the evaluator’s indexed environment. -/
+def finBoxEnv {n : Nat} (X : Fin n → IntervalRat) : IntervalEnv :=
+  fun i => if h : i < n then X ⟨i, h⟩ else IntervalRat.singleton 0
+
+/-- Coordinatewise membership of a real vector in a rational interval box. -/
+def FinBoxMem {n : Nat} (x : Fin n → ℝ) (X : Fin n → IntervalRat) : Prop :=
+  ∀ i, x i ∈ X i
+
+/-- Enclose the Jacobian entries by interval automatic differentiation. -/
+def intervalJacobian {n : Nat} (F : Fin n → Expr) (X : Fin n → IntervalRat)
+    (cfg : EvalConfig := {}) : Matrix (Fin n) (Fin n) IntervalRat :=
+  fun i j => Optimization.derivIntervalCoreN (F i) (finBoxEnv X) j.val cfg
+
+theorem finEnv_mem_finBoxEnv {n : Nat} {x : Fin n → ℝ} {X : Fin n → IntervalRat}
+    (hx : FinBoxMem x X) : ∀ i, finEnv x i ∈ finBoxEnv X i := by
+  intro i
+  by_cases hi : i < n
+  · simpa [finEnv, finBoxEnv, hi] using hx ⟨i, hi⟩
+  · simpa [finEnv, finBoxEnv, hi] using IntervalRat.mem_singleton 0
+
+theorem jacobianAt_mem_intervalJacobian {n : Nat} (F : Fin n → Expr)
+    (h : ∀ i, ADSupported (F i)) (X : Fin n → IntervalRat)
+    (x : Fin n → ℝ) (hx : FinBoxMem x X) (cfg : EvalConfig) (i j : Fin n) :
+    jacobianAt F x i j ∈ intervalJacobian F X cfg i j := by
+  rw [jacobianAt_apply F h x i j]
+  exact Optimization.evalDualTotalCore_der_correct_idx (F i) (h i) (finEnv x) (finBoxEnv X)
+    j.val (finEnv_mem_finBoxEnv hx) (x j) (by
+      simpa only [finBoxEnv, j.isLt, dite_true] using hx j) cfg
+
+/-- Interpret a real matrix as a continuous linear map on finite coordinate vectors. -/
+noncomputable def matrixCLM {n : Nat} (A : Matrix (Fin n) (Fin n) ℝ) :
+    (Fin n → ℝ) →L[ℝ] (Fin n → ℝ) :=
+  ContinuousLinearMap.mk (Matrix.mulVecLin A)
+
+/-- The preconditioned Newton map `x ↦ x - Y (F x)`. -/
+noncomputable def newtonMap {n : Nat} (Y : Matrix (Fin n) (Fin n) ℝ)
+    (F : Fin n → Expr) (x : Fin n → ℝ) : Fin n → ℝ :=
+  x - Matrix.mulVec Y (systemEval F x)
+
+theorem newtonMap_differentiable {n : Nat} (Y : Matrix (Fin n) (Fin n) ℝ)
+    (F : Fin n → Expr) (h : ∀ i, ADSupported (F i)) :
+    Differentiable ℝ (newtonMap Y F) := by
+  have hY : Differentiable ℝ (fun x => matrixCLM Y (systemEval F x)) :=
+    (matrixCLM Y).differentiable.comp (systemEval_differentiable F h)
+  rw [show newtonMap Y F = id - (fun x => matrixCLM Y (systemEval F x)) by
+    funext x
+    rfl]
+  exact differentiable_id.sub hY
+
+theorem newtonMap_fderiv_matrix {n : Nat} (Y : Matrix (Fin n) (Fin n) ℝ)
+    (F : Fin n → Expr) (h : ∀ i, ADSupported (F i)) (x : Fin n → ℝ) :
+    LinearMap.toMatrix' (fderiv ℝ (newtonMap Y F) x).toLinearMap =
+      1 - Y * jacobianAt F x := by
+  have hF : HasFDerivAt (systemEval F) (fderiv ℝ (systemEval F) x) x :=
+    (systemEval_differentiable F h).differentiableAt.hasFDerivAt
+  have hY := (matrixCLM Y).hasFDerivAt.comp x hF
+  have hg : HasFDerivAt (newtonMap Y F)
+      (ContinuousLinearMap.id ℝ _ - (matrixCLM Y).comp (fderiv ℝ (systemEval F) x)) x := by
+    change HasFDerivAt (fun z => z - matrixCLM Y (systemEval F z)) _ x
+    exact (hasFDerivAt_id x).sub hY
+  rw [hg.fderiv]
+  change LinearMap.toMatrix' ((ContinuousLinearMap.id ℝ _ -
+    (matrixCLM Y).comp (fderiv ℝ (systemEval F) x)).toLinearMap) = _
+  rw [ContinuousLinearMap.toLinearMap_sub]
+  change LinearMap.toMatrix' (1 -
+    (matrixCLM Y).toLinearMap.comp (fderiv ℝ (systemEval F) x).toLinearMap) = _
+  rw [map_sub, LinearMap.toMatrix'_one, LinearMap.toMatrix'_comp]
+  have hmatY : LinearMap.toMatrix' (matrixCLM Y).toLinearMap = Y := by
+    change LinearMap.toMatrix' (Matrix.mulVecLin Y) = Y
+    exact LinearMap.toMatrix'_toLin' Y
+  rw [hmatY]
+  rfl
+
+/-- The set of real vectors lying coordinatewise in the interval box. -/
+def finBoxSet {n : Nat} (X : Fin n → IntervalRat) : Set (Fin n → ℝ) :=
+  {x | FinBoxMem x X}
+
+theorem finBoxSet_closed {n : Nat} (X : Fin n → IntervalRat) : IsClosed (finBoxSet X) := by
+  have hi : ∀ i : Fin n, IsClosed {x : Fin n → ℝ | x i ∈ X i} := by
+    intro i
+    exact isClosed_Icc.preimage (continuous_apply i)
+  have hinter := isClosed_iInter hi
+  have heq : (⋂ i : Fin n, {x : Fin n → ℝ | x i ∈ X i}) = finBoxSet X := by
+    ext x
+    simp [finBoxSet, FinBoxMem]
+  rw [← heq]
+  exact hinter
+
+theorem finBoxSet_convex {n : Nat} (X : Fin n → IntervalRat) : Convex ℝ (finBoxSet X) := by
+  intro x hx y hy a b ha hb hab i
+  simp only [finBoxSet, Set.mem_ofPred_eq, FinBoxMem, IntervalRat.mem_def] at hx hy ⊢
+  specialize hx i
+  specialize hy i
+  simp only [Pi.add_apply, Pi.smul_apply, smul_eq_mul]
+  constructor
+  · calc
+      (X i).lo = a * (X i).lo + b * (X i).lo := by
+        calc
+          ((X i).lo : ℝ) = 1 * (X i).lo := by ring
+          _ = (a + b) * (X i).lo := by rw [hab]
+          _ = _ := by ring
+      _ ≤ a * x i + b * y i := add_le_add
+        (mul_le_mul_of_nonneg_left hx.1 ha) (mul_le_mul_of_nonneg_left hy.1 hb)
+  · calc
+      a * x i + b * y i ≤ a * (X i).hi + b * (X i).hi := add_le_add
+        (mul_le_mul_of_nonneg_left hx.2 ha) (mul_le_mul_of_nonneg_left hy.2 hb)
+      _ = (X i).hi := by
+        calc
+          a * (X i).hi + b * (X i).hi = (a + b) * (X i).hi := by ring
+          _ = 1 * (X i).hi := by rw [hab]
+          _ = _ := by ring
+
+theorem contraction_unique_fixedPoint_in_finBox {n : Nat}
+    (X : Fin n → IntervalRat) (m : Fin n → ℝ) (hm : FinBoxMem m X)
+    (g : (Fin n → ℝ) → (Fin n → ℝ)) (hmap : Set.MapsTo g (finBoxSet X) (finBoxSet X))
+    (q : ℝ) (hq0 : 0 ≤ q) (hq1 : q < 1)
+    (hdiff : ∀ x ∈ finBoxSet X, DifferentiableAt ℝ g x)
+    (hderiv : ∀ x ∈ finBoxSet X, ‖fderiv ℝ g x‖ ≤ q) :
+    ∃! x, FinBoxMem x X ∧ g x = x := by
+  let K : NNReal := ⟨q, hq0⟩
+  have hlipOn : LipschitzOnWith K g (finBoxSet X) := by
+    rw [lipschitzOnWith_iff_norm_sub_le]
+    intro x hx y hy
+    change ‖g x - g y‖ ≤ q * ‖x - y‖
+    simpa [norm_sub_rev] using
+      (finBoxSet_convex X).norm_image_sub_le_of_norm_fderiv_le hdiff hderiv hy hx
+  have hlipRestr : LipschitzWith K (hmap.restrict g (finBoxSet X) (finBoxSet X)) :=
+    lipschitzOnWith_iff_restrict.mp hlipOn
+  have hcontract : ContractingWith K (hmap.restrict g (finBoxSet X) (finBoxSet X)) := by
+    refine ⟨?_, hlipRestr⟩
+    exact hq1
+  obtain ⟨x, hx, hfix, _⟩ := hcontract.exists_fixedPoint'
+    (finBoxSet_closed X).isComplete hmap hm (edist_ne_top m (g m))
+  refine ⟨x, ⟨hx, hfix⟩, ?_⟩
+  intro y hy
+  have hl := (finBoxSet_convex X).norm_image_sub_le_of_norm_fderiv_le
+    hdiff hderiv hx hy.1
+  rw [hy.2, hfix] at hl
+  have hzero : ‖y - x‖ = 0 := by nlinarith [norm_nonneg (y - x)]
+  exact sub_eq_zero.mp (norm_eq_zero.mp hzero)
+
+open scoped Matrix.Norms.Operator
+
+/-- The identity matrix represented by singleton rational intervals. -/
+def intervalIdentity {n : Nat} : Matrix (Fin n) (Fin n) IntervalRat :=
+  fun i j => IntervalRat.singleton (if i = j then 1 else 0)
+
+/-- Multiply a rational matrix by an interval matrix using interval sums. -/
+def ratMulInterval {n : Nat} (Y : Matrix (Fin n) (Fin n) ℚ)
+    (J : Matrix (Fin n) (Fin n) IntervalRat) : Matrix (Fin n) (Fin n) IntervalRat :=
+  fun i j => ∑ k, IntervalRat.scale (Y i k) (J k j)
+
+/-- The interval matrix `I - Y J` enclosing the Newton-map derivative. -/
+def preconditionedJacobian {n : Nat} (Y : Matrix (Fin n) (Fin n) ℚ)
+    (J : Matrix (Fin n) (Fin n) IntervalRat) : Matrix (Fin n) (Fin n) IntervalRat :=
+  fun i j => IntervalRat.sub (intervalIdentity i j) (ratMulInterval Y J i j)
+
+theorem mem_interval_sum {ι : Type*} (s : Finset ι)
+    (x : ι → ℝ) (I : ι → IntervalRat) (h : ∀ i ∈ s, x i ∈ I i) :
+    (∑ i ∈ s, x i) ∈ ∑ i ∈ s, I i := by
+  classical
+  induction s using Finset.induction_on with
+  | empty =>
+      simp only [Finset.sum_empty]
+      have hz : (0 : IntervalRat) = IntervalRat.singleton 0 :=
+        intervalRat_ext rfl rfl
+      rw [hz]
+      norm_num [IntervalRat.mem_def, IntervalRat.singleton]
+  | @insert a s ha ih =>
+      simp only [Finset.mem_insert] at h
+      simp only [Finset.sum_insert ha]
+      exact IntervalRat.mem_add (h a (Or.inl rfl))
+        (ih fun i hi => h i (Or.inr hi))
+
+theorem mem_ratMulInterval {n : Nat} (Y : Matrix (Fin n) (Fin n) ℚ)
+    (Jreal : Matrix (Fin n) (Fin n) ℝ) (J : Matrix (Fin n) (Fin n) IntervalRat)
+    (hJ : ∀ i j, Jreal i j ∈ J i j) (i j : Fin n) :
+    (Y.map (fun q => (q : ℝ)) * Jreal) i j ∈ ratMulInterval Y J i j := by
+  simp only [Matrix.mul_apply, ratMulInterval]
+  exact mem_interval_sum Finset.univ (fun k => (Y i k : ℝ) * Jreal k j)
+    (fun k => IntervalRat.scale (Y i k) (J k j)) fun k _ =>
+      IntervalRat.mem_scale (Y i k) (hJ k j)
+
+theorem mem_preconditionedJacobian {n : Nat} (Y : Matrix (Fin n) (Fin n) ℚ)
+    (Jreal : Matrix (Fin n) (Fin n) ℝ) (J : Matrix (Fin n) (Fin n) IntervalRat)
+    (hJ : ∀ i j, Jreal i j ∈ J i j) (i j : Fin n) :
+    (1 - Y.map (fun q => (q : ℝ)) * Jreal) i j ∈ preconditionedJacobian Y J i j := by
+  apply IntervalRat.mem_sub
+  · rw [Matrix.one_apply]
+    unfold intervalIdentity
+    split
+    · norm_num [IntervalRat.mem_def, IntervalRat.singleton]
+    · norm_num [IntervalRat.mem_def, IntervalRat.singleton]
+  · exact mem_ratMulInterval Y Jreal J hJ i j
+
+/-- The larger absolute endpoint, bounding absolute values in the interval. -/
+def intervalAbsBound (I : IntervalRat) : ℚ := max |I.lo| |I.hi|
+
+/-- The sum of absolute interval bounds in a selected matrix row. -/
+def intervalMatrixRowBound {n : Nat} (A : Matrix (Fin n) (Fin n) IntervalRat)
+    (i : Fin n) : ℚ := ∑ j, intervalAbsBound (A i j)
+
+/-- The maximum interval row sum, bounding the matrix’s sup-norm operator norm. -/
+def intervalMatrixBound {n : Nat} (A : Matrix (Fin n) (Fin n) IntervalRat) : ℚ :=
+  (List.ofFn fun i => intervalMatrixRowBound A i).foldl max 0
+
+theorem abs_le_intervalAbsBound {x : ℝ} {I : IntervalRat} (hx : x ∈ I) :
+    |x| ≤ intervalAbsBound I := by
+  rw [abs_le]
+  constructor
+  · have hlo : ((I.lo : ℚ) : ℝ) ≤ x := hx.1
+    have habs : -|(I.lo : ℝ)| ≤ (I.lo : ℝ) := neg_abs_le _
+    have hmax : |(I.lo : ℝ)| ≤ (intervalAbsBound I : ℝ) := by
+      exact_mod_cast le_max_left |I.lo| |I.hi|
+    linarith [neg_le_neg hmax]
+  · have hhi : x ≤ ((I.hi : ℚ) : ℝ) := hx.2
+    have habs : (I.hi : ℝ) ≤ |(I.hi : ℝ)| := le_abs_self _
+    have hmax : |(I.hi : ℝ)| ≤ (intervalAbsBound I : ℝ) := by
+      exact_mod_cast le_max_right |I.lo| |I.hi|
+    linarith
+
+private theorem le_foldl_max (xs : List ℚ) (a : ℚ) : a ≤ xs.foldl max a := by
+  induction xs generalizing a with
+  | nil => simp
+  | cons x xs ih =>
+      simp only [List.foldl_cons]
+      exact le_trans (le_max_left _ _) (ih _)
+
+private theorem foldl_max_mono {xs : List ℚ} {a b : ℚ} (hab : a ≤ b) :
+    xs.foldl max a ≤ xs.foldl max b := by
+  induction xs generalizing a b with
+  | nil => simpa using hab
+  | cons x xs ih =>
+      simp only [List.foldl_cons]
+      exact ih (max_le_max_right x hab)
+
+private theorem le_foldl_max_of_mem {x : ℚ} {xs : List ℚ} (hx : x ∈ xs) :
+    x ≤ xs.foldl max 0 := by
+  induction xs with
+  | nil => simp at hx
+  | cons a as ih =>
+      simp only [List.mem_cons] at hx
+      simp only [List.foldl_cons]
+      rcases hx with rfl | hx
+      · exact le_trans (le_max_right _ _) (le_foldl_max as _)
+      · exact le_trans (ih hx) (foldl_max_mono (le_max_left 0 a))
+
+theorem intervalMatrixRowBound_le_bound {n : Nat} (A : Matrix (Fin n) (Fin n) IntervalRat)
+    (i : Fin n) : intervalMatrixRowBound A i ≤ intervalMatrixBound A := by
+  apply le_foldl_max_of_mem
+  exact List.mem_ofFn.mpr ⟨i, rfl⟩
+
+theorem matrix_norm_le_intervalMatrixBound {n : Nat}
+    (R : Matrix (Fin n) (Fin n) ℝ) (A : Matrix (Fin n) (Fin n) IntervalRat)
+    (hmem : ∀ i j, R i j ∈ A i j) :
+    ‖R‖ ≤ (intervalMatrixBound A : ℝ) := by
+  rw [Matrix.linfty_opNorm_def]
+  have hbound0 : 0 ≤ intervalMatrixBound A := le_foldl_max _ 0
+  let Q : NNReal := ⟨intervalMatrixBound A, by exact_mod_cast hbound0⟩
+  change (↑((Finset.univ : Finset (Fin n)).sup fun i => ∑ j, ‖R i j‖₊) : ℝ) ≤ ↑Q
+  apply NNReal.coe_le_coe.mpr
+  refine Finset.sup_le fun i _ => ?_
+  apply NNReal.coe_le_coe.mp
+  simp only [NNReal.coe_sum, coe_nnnorm]
+  change (∑ j, |R i j|) ≤ (intervalMatrixBound A : ℝ)
+  calc
+    (∑ j, |R i j|) ≤ ∑ j, (intervalAbsBound (A i j) : ℝ) :=
+      Finset.sum_le_sum fun j _ => abs_le_intervalAbsBound (hmem i j)
+    _ = (intervalMatrixRowBound A i : ℝ) := by
+      simp [intervalMatrixRowBound]
+    _ ≤ (intervalMatrixBound A : ℝ) := by
+      exact_mod_cast intervalMatrixRowBound_le_bound A i
+
+theorem newtonMap_fderiv_norm_le {n : Nat} (Y : Matrix (Fin n) (Fin n) ℚ)
+    (F : Fin n → Expr) (h : ∀ i, ADSupported (F i)) (X : Fin n → IntervalRat)
+    (x : Fin n → ℝ) (hx : FinBoxMem x X) (cfg : EvalConfig) :
+    ‖fderiv ℝ (newtonMap (Y.map fun q => (q : ℝ)) F) x‖ ≤
+      (intervalMatrixBound (preconditionedJacobian Y (intervalJacobian F X cfg)) : ℝ) := by
+  rw [← Matrix.linfty_opNorm_toMatrix]
+  rw [newtonMap_fderiv_matrix (Y.map fun q => (q : ℝ)) F h x]
+  apply matrix_norm_le_intervalMatrixBound
+  exact mem_preconditionedJacobian Y (jacobianAt F x) (intervalJacobian F X cfg)
+    (jacobianAt_mem_intervalJacobian F h X x hx cfg)
+
+/-- Represent a rational center by singleton intervals in the evaluation environment. -/
+def pointIntervalEnv {n : Nat} (m : Fin n → ℚ) : IntervalEnv :=
+  fun i => if h : i < n then IntervalRat.singleton (m ⟨i, h⟩) else IntervalRat.singleton 0
+
+/-- Enclose the expression system at a rational center by interval evaluation. -/
+def pointEvalIntervals {n : Nat} (F : Fin n → Expr) (m : Fin n → ℚ)
+    (cfg : EvalConfig := {}) : Fin n → IntervalRat :=
+  fun i => LeanCert.Internal.Rational.evalTotalCore (F i) (pointIntervalEnv m) cfg
+
+/-- Multiply a rational matrix by an interval vector. -/
+def intervalRatMatVec {n : Nat} (Y : Matrix (Fin n) (Fin n) ℚ)
+    (v : Fin n → IntervalRat) : Fin n → IntervalRat :=
+  fun i => ∑ j, IntervalRat.scale (Y i j) (v j)
+
+/-- An interval enclosure of the preconditioned Newton map at the chosen center. -/
+def newtonCenterInterval {n : Nat} (F : Fin n → Expr) (m : Fin n → ℚ)
+    (Y : Matrix (Fin n) (Fin n) ℚ) (cfg : EvalConfig := {}) : Fin n → IntervalRat :=
+  fun i => IntervalRat.sub (IntervalRat.singleton (m i))
+    (intervalRatMatVec Y (pointEvalIntervals F m cfg) i)
+
+theorem finEnv_ratCast_mem_pointIntervalEnv {n : Nat} (m : Fin n → ℚ) :
+    ∀ i, finEnv (fun j => (m j : ℝ)) i ∈ pointIntervalEnv m i := by
+  intro i
+  by_cases hi : i < n
+  · simpa [finEnv, pointIntervalEnv, hi] using IntervalRat.mem_singleton (m ⟨i, hi⟩)
+  · simpa [finEnv, pointIntervalEnv, hi] using IntervalRat.mem_singleton 0
+
+theorem systemEval_mem_pointEvalIntervals {n : Nat} (F : Fin n → Expr)
+    (h : ∀ i, ADSupported (F i)) (m : Fin n → ℚ) (cfg : EvalConfig) (i : Fin n) :
+    systemEval F (fun j => (m j : ℝ)) i ∈ pointEvalIntervals F m cfg i := by
+  exact evalIntervalCore_correct (F i) (h i).toCore _ _
+    (finEnv_ratCast_mem_pointIntervalEnv m) cfg
+    (LeanCert.Engine.ADSupported.domainValid (h i) _ cfg)
+
+theorem newtonMap_center_mem {n : Nat} (F : Fin n → Expr)
+    (h : ∀ i, ADSupported (F i)) (m : Fin n → ℚ)
+    (Y : Matrix (Fin n) (Fin n) ℚ) (cfg : EvalConfig) (i : Fin n) :
+    newtonMap (Y.map fun q => (q : ℝ)) F (fun j => (m j : ℝ)) i ∈
+      newtonCenterInterval F m Y cfg i := by
+  apply IntervalRat.mem_sub
+  · exact IntervalRat.mem_singleton (m i)
+  · simp only [Matrix.mulVec, dotProduct, intervalRatMatVec]
+    exact mem_interval_sum Finset.univ
+      (fun j => (Y i j : ℝ) * systemEval F (fun k => (m k : ℝ)) j)
+      (fun j => IntervalRat.scale (Y i j) (pointEvalIntervals F m cfg j))
+      (fun j _ => IntervalRat.mem_scale (Y i j)
+        (systemEval_mem_pointEvalIntervals F h m cfg j))
+
+/-- The greater endpoint distance from the center in a selected coordinate. -/
+def coordinateRadius {n : Nat} (X : Fin n → IntervalRat) (m : Fin n → ℚ)
+    (i : Fin n) : ℚ := max (m i - (X i).lo) ((X i).hi - m i)
+
+/-- The maximum coordinate radius of the box around the chosen center. -/
+def boxRadius {n : Nat} (X : Fin n → IntervalRat) (m : Fin n → ℚ) : ℚ :=
+  (List.ofFn fun i => coordinateRadius X m i).foldl max 0
+
+theorem coordinateRadius_le_boxRadius {n : Nat} (X : Fin n → IntervalRat)
+    (m : Fin n → ℚ) (i : Fin n) : coordinateRadius X m i ≤ boxRadius X m := by
+  apply le_foldl_max_of_mem
+  exact List.mem_ofFn.mpr ⟨i, rfl⟩
+
+theorem abs_sub_center_le_coordinateRadius {n : Nat} {X : Fin n → IntervalRat}
+    {m : Fin n → ℚ} {x : Fin n → ℝ} (hx : FinBoxMem x X) (i : Fin n) :
+    |x i - (m i : ℝ)| ≤ coordinateRadius X m i := by
+  rw [abs_le]
+  constructor
+  · have hlo := (hx i).1
+    have hmax : m i - (X i).lo ≤ coordinateRadius X m i := le_max_left _ _
+    have hmaxR : ((m i - (X i).lo : ℚ) : ℝ) ≤ (coordinateRadius X m i : ℝ) :=
+      by exact_mod_cast hmax
+    push_cast at hmaxR
+    linarith
+  · have hhi := (hx i).2
+    have hmax : (X i).hi - m i ≤ coordinateRadius X m i := le_max_right _ _
+    have hmaxR : (((X i).hi - m i : ℚ) : ℝ) ≤ (coordinateRadius X m i : ℝ) :=
+      by exact_mod_cast hmax
+    push_cast at hmaxR
+    linarith
+
+theorem boxRadius_nonneg {n : Nat} (X : Fin n → IntervalRat) (m : Fin n → ℚ) :
+    0 ≤ boxRadius X m := le_foldl_max _ 0
+
+theorem norm_sub_center_le_boxRadius {n : Nat} {X : Fin n → IntervalRat}
+    {m : Fin n → ℚ} {x : Fin n → ℝ} (hx : FinBoxMem x X) :
+    ‖x - (fun i => (m i : ℝ))‖ ≤ (boxRadius X m : ℝ) := by
+  rw [pi_norm_le_iff_of_nonneg]
+  · intro i
+    rw [Real.norm_eq_abs]
+    exact (abs_sub_center_le_coordinateRadius hx i).trans
+      (by exact_mod_cast coordinateRadius_le_boxRadius X m i)
+  · exact_mod_cast boxRadius_nonneg X m
+
+/-- The rational interval with endpoints `-d` and `d`. -/
+def symmetricInterval (d : ℚ) : IntervalRat :=
+  { lo := -|d|
+    hi := |d|
+    le := neg_nonpos.mpr (abs_nonneg d) |>.trans (abs_nonneg d) }
+
+/-- Enclose the Newton image using its center value and a uniform derivative bound. -/
+def newtonImageEnclosure {n : Nat} (F : Fin n → Expr) (X : Fin n → IntervalRat)
+    (m : Fin n → ℚ) (Y : Matrix (Fin n) (Fin n) ℚ) (cfg : EvalConfig := {}) :
+    Fin n → IntervalRat :=
+  let q := intervalMatrixBound (preconditionedJacobian Y (intervalJacobian F X cfg))
+  let r := boxRadius X m
+  fun i => IntervalRat.add (newtonCenterInterval F m Y cfg i) (symmetricInterval (q * r))
+
+/-- Test whether both endpoints lie strictly inside another interval. -/
+def intervalStrictInside (I X : IntervalRat) : Bool := X.lo < I.lo && I.hi < X.hi
+
+theorem intervalStrictInside_sound {I X : IntervalRat} (h : intervalStrictInside I X = true) :
+    ∀ x : ℝ, x ∈ I → x ∈ X := by
+  intro x hx
+  simp only [intervalStrictInside, Bool.and_eq_true, decide_eq_true_eq] at h
+  exact ⟨le_trans (by exact_mod_cast h.1.le) hx.1, le_trans hx.2 (by exact_mod_cast h.2.le)⟩
+
+theorem newtonMap_mapsTo_of_imageEnclosure {n : Nat} (F : Fin n → Expr)
+    (h : ∀ i, ADSupported (F i)) (X : Fin n → IntervalRat) (m : Fin n → ℚ)
+    (hm : FinBoxMem (fun i => (m i : ℝ)) X) (Y : Matrix (Fin n) (Fin n) ℚ)
+    (cfg : EvalConfig)
+    (hencl : ∀ i, intervalStrictInside (newtonImageEnclosure F X m Y cfg i) (X i) = true) :
+    Set.MapsTo (newtonMap (Y.map fun q => (q : ℝ)) F) (finBoxSet X) (finBoxSet X) := by
+  intro x hx i
+  let q : ℚ := intervalMatrixBound (preconditionedJacobian Y (intervalJacobian F X cfg))
+  let mr : Fin n → ℝ := fun i => (m i : ℝ)
+  have hdiff : ‖newtonMap (Y.map fun q => (q : ℝ)) F x -
+      newtonMap (Y.map fun q => (q : ℝ)) F mr‖ ≤ (q : ℝ) * ‖x - mr‖ := by
+    apply (finBoxSet_convex X).norm_image_sub_le_of_norm_fderiv_le (𝕜 := ℝ)
+    · intro z hz
+      exact (newtonMap_differentiable (Y.map fun q => (q : ℝ)) F h).differentiableAt
+    · intro z hz
+      exact newtonMap_fderiv_norm_le Y F h X z hz cfg
+    · exact hm
+    · exact hx
+  have hq0 : 0 ≤ q := le_foldl_max _ 0
+  have hr0 : 0 ≤ boxRadius X m := boxRadius_nonneg X m
+  have hnorm : ‖newtonMap (Y.map fun q => (q : ℝ)) F x -
+      newtonMap (Y.map fun q => (q : ℝ)) F mr‖ ≤
+      ((q * boxRadius X m : ℚ) : ℝ) := by
+    calc
+      _ ≤ (q : ℝ) * ‖x - mr‖ := hdiff
+      _ ≤ (q : ℝ) * (boxRadius X m : ℝ) :=
+        mul_le_mul_of_nonneg_left (norm_sub_center_le_boxRadius hx) (by exact_mod_cast hq0)
+      _ = _ := by push_cast; rfl
+  have hcoord : |(newtonMap (Y.map fun q => (q : ℝ)) F x -
+      newtonMap (Y.map fun q => (q : ℝ)) F mr) i| ≤
+      ((q * boxRadius X m : ℚ) : ℝ) :=
+    (norm_le_pi_norm _ i).trans hnorm
+  have herr : (newtonMap (Y.map fun q => (q : ℝ)) F x -
+      newtonMap (Y.map fun q => (q : ℝ)) F mr) i ∈
+      symmetricInterval (q * boxRadius X m) := by
+    rw [abs_le] at hcoord
+    simpa [symmetricInterval, IntervalRat.mem_def, abs_of_nonneg (mul_nonneg hq0 hr0)] using hcoord
+  have hcenter := newtonMap_center_mem F h m Y cfg i
+  have himage : newtonMap (Y.map fun q => (q : ℝ)) F x i ∈
+      newtonImageEnclosure F X m Y cfg i := by
+    have hadd := IntervalRat.mem_add hcenter herr
+    simpa [newtonImageEnclosure, q, mr, Pi.sub_apply] using hadd
+  exact intervalStrictInside_sound (hencl i) _ himage
+
+/-- Rational center and preconditioner data for a checkable Krawczyk certificate. -/
+structure KrawczykCert (n : Nat) where
+  /-- The rational center used by the Newton-image enclosure. -/
+  center : Fin n → ℚ
+  /-- The rational matrix used to precondition the equation system. -/
+  preconditioner : Matrix (Fin n) (Fin n) ℚ
+
+/-- Coordinatewise containment of the rational center in the interval box. -/
+def centerInside {n : Nat} (X : Fin n → IntervalRat) (m : Fin n → ℚ) : Prop :=
+  ∀ i, (X i).lo ≤ m i ∧ m i ≤ (X i).hi
+
+instance {n : Nat} (X : Fin n → IntervalRat) (m : Fin n → ℚ) :
+    Decidable (centerInside X m) := by
+  unfold centerInside
+  infer_instance
+
+/-- Check supported derivatives, center containment, invertibility, contraction and self-map
+bounds. -/
+def krawczykCheck {n : Nat} (F : Fin n → Expr) (X : Fin n → IntervalRat)
+    (cert : KrawczykCert n) (cfg : EvalConfig := {}) : Bool :=
+  decide (∀ i, (F i).checkADSupported = true) &&
+  decide (centerInside X cert.center) &&
+  decide (cert.preconditioner.det ≠ 0) &&
+  decide (intervalMatrixBound
+    (preconditionedJacobian cert.preconditioner (intervalJacobian F X cfg)) < 1) &&
+  decide (∀ i, intervalStrictInside
+    (newtonImageEnclosure F X cert.center cert.preconditioner cfg i) (X i) = true)
+
+/-- Every expression in the finite system evaluates to zero at the given point. -/
+def SystemZero {n : Nat} (F : Fin n → Expr) (x : Fin n → ℝ) : Prop :=
+  ∀ i, evalFin (F i) x = 0
+
+theorem centerInside_sound {n : Nat} {X : Fin n → IntervalRat} {m : Fin n → ℚ}
+    (h : centerInside X m) : FinBoxMem (fun i => (m i : ℝ)) X := by
+  intro i
+  constructor
+  · change ((X i).lo : ℝ) ≤ (m i : ℝ)
+    exact_mod_cast (h i).1
+  · change (m i : ℝ) ≤ ((X i).hi : ℝ)
+    exact_mod_cast (h i).2
+
+theorem fixedPoint_iff_systemZero {n : Nat} (F : Fin n → Expr)
+    (Y : Matrix (Fin n) (Fin n) ℚ) (hdet : Y.det ≠ 0) (x : Fin n → ℝ) :
+    newtonMap (Y.map fun q => (q : ℝ)) F x = x ↔ SystemZero F x := by
+  let Yr : Matrix (Fin n) (Fin n) ℝ := Y.map fun q => (q : ℝ)
+  have hdetR : Yr.det ≠ 0 := by
+    simp only [Yr]
+    exact_mod_cast hdet
+  have hinj : Function.Injective Yr.mulVec :=
+    Matrix.mulVec_injective_iff_isUnit.mpr
+      (Yr.isUnit_iff_isUnit_det.mpr (isUnit_iff_ne_zero.mpr hdetR))
+  constructor
+  · intro hfix
+    have hmul : Yr.mulVec (systemEval F x) = 0 := by
+      funext i
+      have hi := congrFun hfix i
+      change x i - Yr.mulVec (systemEval F x) i = x i at hi
+      exact sub_eq_self.mp hi
+    have hz : systemEval F x = 0 := hinj (by simpa using hmul)
+    intro i
+    exact congrFun hz i
+  · intro hz
+    have hs : systemEval F x = 0 := by
+      funext i
+      exact hz i
+    funext i
+    simp [newtonMap, hs]
+
+theorem krawczykCheck_sound {n : Nat} (F : Fin n → Expr) (X : Fin n → IntervalRat)
+    (cert : KrawczykCert n) (cfg : EvalConfig)
+    (hcheck : krawczykCheck F X cert cfg = true) :
+    ∃! x, FinBoxMem x X ∧ SystemZero F x := by
+  simp only [krawczykCheck, Bool.and_eq_true, decide_eq_true_eq] at hcheck
+  rcases hcheck with ⟨⟨⟨⟨hsupported, hcenter⟩, hdet⟩, hq⟩, hencl⟩
+  have hsupp : ∀ i, ADSupported (F i) := fun i =>
+    ((F i).checkADSupported_eq_true_iff).mp (hsupported i)
+  let q : ℚ := intervalMatrixBound
+    (preconditionedJacobian cert.preconditioner (intervalJacobian F X cfg))
+  have hq0 : 0 ≤ q := le_foldl_max _ 0
+  have hfixed := contraction_unique_fixedPoint_in_finBox X
+    (fun i => (cert.center i : ℝ)) (centerInside_sound hcenter)
+    (newtonMap (cert.preconditioner.map fun q => (q : ℝ)) F)
+    (newtonMap_mapsTo_of_imageEnclosure F hsupp X cert.center
+      (centerInside_sound hcenter) cert.preconditioner cfg hencl)
+    (q : ℝ) (by exact_mod_cast hq0) (by exact_mod_cast hq)
+    (fun x _ => (newtonMap_differentiable
+      (cert.preconditioner.map fun q => (q : ℝ)) F hsupp).differentiableAt)
+    (fun x hx => newtonMap_fderiv_norm_le cert.preconditioner F hsupp X x hx cfg)
+  refine ⟨hfixed.choose, ⟨hfixed.choose_spec.1.1,
+    (fixedPoint_iff_systemZero F cert.preconditioner hdet hfixed.choose).mp
+      hfixed.choose_spec.1.2⟩, ?_⟩
+  intro y hy
+  exact hfixed.unique ⟨hy.1,
+    (fixedPoint_iff_systemZero F cert.preconditioner hdet y).mpr hy.2⟩
+    hfixed.choose_spec.1
+
+end LeanCert.Engine
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle001.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle001.lean
new file mode 100644
index 0000000000..ca67023775
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle001.lean
@@ -0,0 +1,816 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import Mathlib.Algebra.BigOperators.Fin
+public import Mathlib.Basic.Real.Basic
+public import Mathlib.Data.Rat.Cast.Order
+public import Mathlib.Data.Rat.Floor
+public import Mathlib.Data.Set.Lattice.Bounded
+public import Mathlib.Data.Set.Lattice.Disjoint
+public import Mathlib.Data.Set.Lattice.Image
+public import Mathlib.Data.Set.Lattice.Indexed
+public import Mathlib.Data.Set.Lattice.Order
+public import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
+public import Mathlib.Tactic.NormNum
+public import Mathlib.Topology.Constructions
+public import Mathlib.Topology.Defs.Basic
+public import Mathlib.Topology.Instances.Real.Lemmas
+
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.Foundation.Batch001`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `Basic`.
+* `KrawczykSpec`.
+* `RationalInterval`.
+* `CertificateManifest`.
+* `ExactReplay`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Basic planar geometry for the moving-sofa problem
+
+This file fixes the exact hallway convention used by the manuscript.
+The inner quadrant is open, so contact with an inner wall is allowed.
+-/
+
+@[expose] public section
+
+namespace GerverSofa
+
+/-- Cartesian coordinates for the hallway and sofa geometry. -/
+abbrev Point := ℝ × ℝ
+
+/-- Horizontal unit-width arm `(-∞,1] × [0,1]`. -/
+def horizontalArm : Set Point :=
+  {p | p.1 ≤ 1 ∧ 0 ≤ p.2 ∧ p.2 ≤ 1}
+
+/-- Vertical unit-width arm `[0,1] × (-∞,1]`. -/
+def verticalArm : Set Point :=
+  {p | 0 ≤ p.1 ∧ p.1 ≤ 1 ∧ p.2 ≤ 1}
+
+/-- Closed outer quarter-plane `(-∞,1]²`. -/
+def outerQuarter : Set Point :=
+  {p | p.1 ≤ 1 ∧ p.2 ≤ 1}
+
+/-- Open inner quarter-plane `(-∞,0)²`. -/
+def innerQuarter : Set Point :=
+  {p | p.1 < 0 ∧ p.2 < 0}
+
+/-- The standard unit right-angled hallway. -/
+def hallway : Set Point := horizontalArm ∪ verticalArm
+
+/-- Quarter-plane presentation of the standard hallway. -/
+theorem hallway_eq_outer_diff_inner :
+    hallway = outerQuarter \ innerQuarter := by
+  ext p
+  constructor
+  · intro hp
+    rcases hp with hp | hp
+    · exact ⟨⟨hp.1, hp.2.2⟩, fun hn => (not_lt_of_ge hp.2.1) hn.2⟩
+    · exact ⟨⟨hp.2.1, hp.2.2⟩, fun hn => (not_lt_of_ge hp.1) hn.1⟩
+  · rintro ⟨hout, hnotin⟩
+    by_cases hx : 0 ≤ p.1
+    · exact Or.inr ⟨hx, hout.1, hout.2⟩
+    · have hxneg : p.1 < 0 := lt_of_not_ge hx
+      have hy : 0 ≤ p.2 := by
+        by_contra hy0
+        exact hnotin ⟨hxneg, lt_of_not_ge hy0⟩
+      exact Or.inl ⟨hout.1, hy, hout.2⟩
+
+@[simp] theorem mem_horizontalArm (p : Point) :
+    p ∈ horizontalArm ↔ p.1 ≤ 1 ∧ 0 ≤ p.2 ∧ p.2 ≤ 1 := Iff.rfl
+
+@[simp] theorem mem_verticalArm (p : Point) :
+    p ∈ verticalArm ↔ 0 ≤ p.1 ∧ p.1 ≤ 1 ∧ p.2 ≤ 1 := Iff.rfl
+
+@[simp] theorem mem_outerQuarter (p : Point) :
+    p ∈ outerQuarter ↔ p.1 ≤ 1 ∧ p.2 ≤ 1 := Iff.rfl
+
+@[simp] theorem mem_innerQuarter (p : Point) :
+    p ∈ innerQuarter ↔ p.1 < 0 ∧ p.2 < 0 := Iff.rfl
+
+end GerverSofa
+
+end
+
+end
+
+section
+
+/-!
+# Specification boundary for interval/Krawczyk certification
+
+The records below make the logical target explicit.  A completed numerical
+formalisation must construct these records from exact interval operations,
+Taylor bounds for `sin`/`cos`, a certified interval Jacobian and the general
+Krawczyk theorem.  No global axiom is introduced here.
+-/
+
+@[expose] public section
+
+namespace GerverSofa
+
+/-- A unique solution of a predicate inside an explicit set. -/
+structure CertifiedUniqueSolution {α : Type*} (P : α → Prop) (X : Set α) where
+  /-- The certified point satisfying the predicate in the specified domain. -/
+  solution : α
+  solution_mem : solution ∈ X
+  satisfies : P solution
+  unique_iff : ∀ y, y ∈ X → (P y ↔ y = solution)
+
+/-- Any other solution in the certified domain equals the recorded solution. -/
+theorem CertifiedUniqueSolution.unique {α : Type*} {P : α → Prop} {X : Set α}
+    (c : CertifiedUniqueSolution P X) (y : α) (hy : y ∈ X) (hP : P y) :
+    y = c.solution := (c.unique_iff y hy).mp hP
+
+/-- Real coordinate vectors indexed by a finite type. -/
+abbrev Vec (n : Nat) := Fin n → ℝ
+
+/-- A unique zero of a vector-valued function in a set. -/
+def CertifiedUniqueZero {n : Nat} (F : Vec n → Vec n) (X : Set (Vec n)) :=
+  CertifiedUniqueSolution (fun x => F x = 0) X
+
+end GerverSofa
+
+end
+
+end
+
+section
+
+/-!
+# Exact rational intervals
+
+This module is intentionally small.  It provides the decidable relations used
+to audit the published output manifest.  It does not assert that a particular
+transcendental expression is enclosed; that analytic soundness is a distinct
+proof obligation.
+-/
+
+@[expose] public section
+
+namespace GerverSofa
+
+/-- Rational endpoints used by exact interval computations; no ordering is assumed. -/
+structure RatInterval where
+  /-- Lower endpoint of the interval. -/
+  lo : ℚ
+  /-- Upper endpoint of the interval. -/
+  hi : ℚ
+  deriving Repr, DecidableEq
+
+namespace RatInterval
+
+/-- Strict inclusion in the interior of another interval. -/
+def strictInsideB (x outer : RatInterval) : Bool :=
+  decide (outer.lo < x.lo ∧ x.hi < outer.hi)
+
+/-- Point interval. -/
+def point (x : ℚ) : RatInterval := ⟨x, x⟩
+
+/-- Exact interval addition. -/
+def add (x y : RatInterval) : RatInterval :=
+  ⟨x.lo + y.lo, x.hi + y.hi⟩
+
+/-- Exact interval negation. -/
+def neg (x : RatInterval) : RatInterval :=
+  ⟨-x.hi, -x.lo⟩
+
+/-- Exact interval subtraction. -/
+def sub (x y : RatInterval) : RatInterval := add x (neg y)
+
+/-- Product hull of two rational intervals. -/
+def mul (x y : RatInterval) : RatInterval :=
+  let p₁ := x.lo * y.lo
+  let p₂ := x.lo * y.hi
+  let p₃ := x.hi * y.lo
+  let p₄ := x.hi * y.hi
+  ⟨min (min p₁ p₂) (min p₃ p₄), max (max p₁ p₂) (max p₃ p₄)⟩
+
+/-! ## Real semantics -/
+
+/-- Semantic membership of a real number in a rational interval. -/
+def Contains (z : RatInterval) (x : ℝ) : Prop :=
+  (z.lo : ℝ) ≤ x ∧ x ≤ (z.hi : ℝ)
+
+@[simp] theorem strictInsideB_eq_true_iff (x outer : RatInterval) :
+    strictInsideB x outer = true ↔ outer.lo < x.lo ∧ x.hi < outer.hi := by
+  simp [strictInsideB]
+
+@[simp] theorem contains_point_iff (q : ℚ) (x : ℝ) :
+    Contains (point q) x ↔ x = (q : ℝ) := by
+  constructor
+  · intro hx
+    exact le_antisymm hx.2 hx.1
+  · rintro rfl
+    exact ⟨le_rfl, le_rfl⟩
+
+/-- Exact interval addition is sound over the reals. -/
+theorem contains_add {a b : RatInterval} {x y : ℝ}
+    (hx : Contains a x) (hy : Contains b y) :
+    Contains (add a b) (x + y) := by
+  rcases hx with ⟨hxl, hxu⟩
+  rcases hy with ⟨hyl, hyu⟩
+  simpa [Contains, add] using
+    (And.intro (add_le_add hxl hyl) (add_le_add hxu hyu))
+
+/-- Exact interval negation is sound over the reals. -/
+theorem contains_neg {a : RatInterval} {x : ℝ}
+    (hx : Contains a x) : Contains (neg a) (-x) := by
+  rcases hx with ⟨hxl, hxu⟩
+  simpa [Contains, neg] using
+    (And.intro (neg_le_neg hxu) (neg_le_neg hxl))
+
+/-- Exact interval subtraction is sound over the reals. -/
+theorem contains_sub {a b : RatInterval} {x y : ℝ}
+    (hx : Contains a x) (hy : Contains b y) :
+    Contains (sub a b) (x - y) := by
+  simpa [sub, sub_eq_add_neg] using contains_add hx (contains_neg hy)
+
+/-- Closed interval inclusion transports semantic membership. -/
+theorem contains_of_subset
+    {x outer : RatInterval} {r : ℝ}
+    (hsub : outer.lo ≤ x.lo ∧ x.hi ≤ outer.hi)
+    (hr : Contains x r) : Contains outer r := by
+  constructor
+  · exact le_trans (by exact_mod_cast hsub.1) hr.1
+  · exact le_trans hr.2 (by exact_mod_cast hsub.2)
+
+/-! ## Multiplicative soundness -/
+
+private theorem min_mul_le_mul_right
+    {l u z c : ℝ} (hz : l ≤ z ∧ z ≤ u) :
+    min (l * c) (u * c) ≤ z * c := by
+  by_cases hc : 0 ≤ c
+  · exact le_trans (min_le_left _ _)
+      (mul_le_mul_of_nonneg_right hz.1 hc)
+  · have hc' : c ≤ 0 := le_of_lt (lt_of_not_ge hc)
+    exact le_trans (min_le_right _ _)
+      (mul_le_mul_of_nonpos_right hz.2 hc')
+
+private theorem mul_le_max_mul_right
+    {l u z c : ℝ} (hz : l ≤ z ∧ z ≤ u) :
+    z * c ≤ max (l * c) (u * c) := by
+  by_cases hc : 0 ≤ c
+  · exact le_trans (mul_le_mul_of_nonneg_right hz.2 hc)
+      (le_max_right _ _)
+  · have hc' : c ≤ 0 := le_of_lt (lt_of_not_ge hc)
+    exact le_trans (mul_le_mul_of_nonpos_right hz.1 hc')
+      (le_max_left _ _)
+
+private theorem min_mul_le_mul_left
+    {l u z c : ℝ} (hz : l ≤ z ∧ z ≤ u) :
+    min (c * l) (c * u) ≤ c * z := by
+  by_cases hc : 0 ≤ c
+  · exact le_trans (min_le_left _ _)
+      (mul_le_mul_of_nonneg_left hz.1 hc)
+  · have hc' : c ≤ 0 := le_of_lt (lt_of_not_ge hc)
+    exact le_trans (min_le_right _ _)
+      (mul_le_mul_of_nonpos_left hz.2 hc')
+
+private theorem mul_le_max_mul_left
+    {l u z c : ℝ} (hz : l ≤ z ∧ z ≤ u) :
+    c * z ≤ max (c * l) (c * u) := by
+  by_cases hc : 0 ≤ c
+  · exact le_trans (mul_le_mul_of_nonneg_left hz.2 hc)
+      (le_max_right _ _)
+  · have hc' : c ≤ 0 := le_of_lt (lt_of_not_ge hc)
+    exact le_trans (mul_le_mul_of_nonpos_left hz.1 hc')
+      (le_max_left _ _)
+
+/-- The four-corner product hull contains every real product of members. -/
+theorem contains_mul {a b : RatInterval} {x y : ℝ}
+    (hx : Contains a x) (hy : Contains b y) :
+    Contains (mul a b) (x * y) := by
+  let A : ℝ := a.lo
+  let B : ℝ := a.hi
+  let C : ℝ := b.lo
+  let D : ℝ := b.hi
+  have hx' : A ≤ x ∧ x ≤ B := by
+    simpa [A, B, Contains] using hx
+  have hy' : C ≤ y ∧ y ≤ D := by
+    simpa [C, D, Contains] using hy
+  have hLowC :
+      min (min (A * C) (A * D)) (min (B * C) (B * D)) ≤
+        min (A * C) (B * C) := by
+    apply le_min
+    · exact le_trans (min_le_left _ _) (min_le_left _ _)
+    · exact le_trans (min_le_right _ _) (min_le_left _ _)
+  have hLowD :
+      min (min (A * C) (A * D)) (min (B * C) (B * D)) ≤
+        min (A * D) (B * D) := by
+    apply le_min
+    · exact le_trans (min_le_left _ _) (min_le_right _ _)
+    · exact le_trans (min_le_right _ _) (min_le_right _ _)
+  have hLower :
+      min (min (A * C) (A * D)) (min (B * C) (B * D)) ≤ x * y := by
+    have hXC :
+        min (min (A * C) (A * D)) (min (B * C) (B * D)) ≤ x * C :=
+      le_trans hLowC (min_mul_le_mul_right hx')
+    have hXD :
+        min (min (A * C) (A * D)) (min (B * C) (B * D)) ≤ x * D :=
+      le_trans hLowD (min_mul_le_mul_right hx')
+    exact le_trans (le_min hXC hXD) (min_mul_le_mul_left hy')
+  have hUpC :
+      max (A * C) (B * C) ≤
+        max (max (A * C) (A * D)) (max (B * C) (B * D)) := by
+    apply max_le
+    · exact le_trans (le_max_left _ _) (le_max_left _ _)
+    · exact le_trans (le_max_left _ _) (le_max_right _ _)
+  have hUpD :
+      max (A * D) (B * D) ≤
+        max (max (A * C) (A * D)) (max (B * C) (B * D)) := by
+    apply max_le
+    · exact le_trans (le_max_right _ _) (le_max_left _ _)
+    · exact le_trans (le_max_right _ _) (le_max_right _ _)
+  have hUpper :
+      x * y ≤ max (max (A * C) (A * D)) (max (B * C) (B * D)) := by
+    have hXC : x * C ≤
+        max (max (A * C) (A * D)) (max (B * C) (B * D)) :=
+      le_trans (mul_le_max_mul_right hx') hUpC
+    have hXD : x * D ≤
+        max (max (A * C) (A * D)) (max (B * C) (B * D)) :=
+      le_trans (mul_le_max_mul_right hx') hUpD
+    exact le_trans (mul_le_max_mul_left hy') (max_le hXC hXD)
+  simpa [Contains, mul, A, B, C, D] using And.intro hLower hUpper
+
+/-- Rational scaling is a special case of sound interval multiplication. -/
+theorem contains_scale {a : ℚ} {z : RatInterval} {x : ℝ}
+    (hx : Contains z x) :
+    Contains (mul (point a) z) ((a : ℝ) * x) := by
+  exact contains_mul ((contains_point_iff a (a : ℝ)).2 rfl) hx
+
+/-- Semantic validity follows from the existence of a contained real point. -/
+theorem valid_of_contains {z : RatInterval} {x : ℝ} (hx : Contains z x) :
+    (z.lo : ℝ) ≤ (z.hi : ℝ) := le_trans hx.1 hx.2
+
+/-- A strict Boolean inclusion is, in particular, a closed semantic inclusion. -/
+theorem contains_of_strictInsideB
+    {x outer : RatInterval} {r : ℝ}
+    (hstrict : strictInsideB x outer = true)
+    (hr : Contains x r) : Contains outer r := by
+  have h := (strictInsideB_eq_true_iff x outer).1 hstrict
+  exact contains_of_subset
+    ⟨le_of_lt h.1, le_of_lt h.2⟩ hr
+
+end RatInterval
+end GerverSofa
+
+end
+
+end
+
+section
+
+/-!
+# Exact rational audit of the published certificate manifest
+
+This file contains no floating-point literals.  Every value is emitted as an
+integer numerator and a positive integer denominator from the deterministic
+Python `Fraction` replay.  kernel reduction therefore checks the strict box
+inclusions and every published rational margin in the Lean kernel/runtime.
+
+This manifest audit deliberately does **not** by itself prove the analytic
+soundness of the sine/cosine enclosures or the Krawczyk existence theorem;
+those proof obligations are represented separately in `KrawczykSpec.lean` and
+`GerverCertificate.lean`.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.CertificateManifest
+
+open GerverSofa
+
+/-- Reconstruct a natural number from base-10³⁵ chunks to share decimal elaboration. -/
+def naturalFromChunks (chunks : List ℕ) : ℕ :=
+  chunks.foldl (fun n part ↦ n * 10 ^ 35 + part) 0
+
+/-- Construct an exact rational number from its integer numerator and natural denominator. -/
+def q (n : Int) (d : Nat) : ℚ := (n : ℚ) / (d : ℚ)
+/-- The frozen rational enclosure of π used by the certificate manifest. -/
+def declaredPi : RatInterval := ⟨q 31415926535897932384626433832795028841971693993751
+  10000000000000000000000000000000000000000000000000, q
+  39269908169872415480783042290993786052464617492189
+  12500000000000000000000000000000000000000000000000⟩
+/-- The frozen interval obtained from the manifest’s Machin-formula computation. -/
+def machinPi : RatInterval := ⟨q (naturalFromChunks [295505634309898,
+  38894826843310944618692683285957161, 54840908697424127783146795390704820,
+  62285304765520132090285996624652812, 88395258878959894304877191976978068]) (naturalFromChunks
+  [94062364823852, 62383071716578914264051429490249517, 24107156289559560132134809009503737,
+  77692872755942719128485898305927492, 83511011526570655405521392822265625]), q (naturalFromChunks
+  [3375915467483141175, 2280823352893512868952395431804961, 33509141418627522640018221186493436,
+  5780061841563520292595907995044547, 84418309242836154390330033778013752]) (naturalFromChunks
+  [1074587268220657145, 16687904540832335376340782508534865, 44974883186326861534485086372601111,
+  18916938440811867648599426576863646, 86497881848481483757495880126953125])⟩
+/-- The four rational intervals specifying the reduced parameter box. -/
+def x4 : List RatInterval := [
+    ⟨q 1888531216873 20000000000000, q 4721328042183 50000000000000⟩,
+    ⟨q 69960186366677 50000000000000, q 34980093183339 25000000000000⟩,
+    ⟨q 122429264969 3125000000000, q 3917736479009 100000000000000⟩,
+    ⟨q 2129067216821 3125000000000, q 68130150938273 100000000000000⟩
+  ]
+/-- The twenty-two rational intervals specifying the full Romik parameter box. -/
+def z22 : List RatInterval := [
+    ⟨q (-21032242207268875141628571849) 100000000000000000000000000000, q
+      (-21032242207268875141608571849) 100000000000000000000000000000⟩,
+    ⟨q 2499999999999999999999 10000000000000000000000, q 2500000000000000000001
+      10000000000000000000000⟩,
+    ⟨q (-91917929277159332227479610289) 100000000000000000000000000000, q
+      (-91917929277159332227459610289) 100000000000000000000000000000⟩,
+    ⟨q 29525413734425341573853797657 62500000000000000000000000000, q
+      29525413734425341573866297657 62500000000000000000000000000⟩,
+    ⟨q (-15344080735756291713875357283) 25000000000000000000000000000, q
+      (-15344080735756291713870357283) 25000000000000000000000000000⟩,
+    ⟨q 17792529580064437214538861001 20000000000000000000000000000, q
+      17792529580064437214542861001 20000000000000000000000000000⟩,
+    ⟨q (-15417358304445500741761623987) 50000000000000000000000000000, q
+      (-15417358304445500741751623987) 50000000000000000000000000000⟩,
+    ⟨q 29525413734425341573853797657 62500000000000000000000000000, q
+      29525413734425341573866297657 62500000000000000000000000000⟩,
+    ⟨q (-20344080735756291713874857283) 20000000000000000000000000000, q
+      (-20344080735756291713870857283) 20000000000000000000000000000⟩,
+    ⟨q 2499999999999999999999 10000000000000000000000, q 2500000000000000000001
+      10000000000000000000000⟩,
+    ⟨q 2420644844145377502832171437 2000000000000000000000000000, q 2420644844145377502832571437
+      2000000000000000000000000000⟩,
+    ⟨q (-2500000000000000000001) 10000000000000000000000, q (-2499999999999999999999)
+      10000000000000000000000⟩,
+    ⟨q (-52762459802678462416060380937) 100000000000000000000000000000, q
+      (-52762459802678462416040380937) 100000000000000000000000000000⟩,
+    ⟨q 92025838516063762289360579501 100000000000000000000000000000, q
+      92025838516063762289380579501 100000000000000000000000000000⟩,
+    ⟨q 313022761424232933776114655193 500000000000000000000000000000, q
+      313022761424232933776214655193 500000000000000000000000000000⟩,
+    ⟨q (-151160128631428920268654781) 160000000000000000000000000, q
+      (-151160128631428920268622781) 160000000000000000000000000⟩,
+    ⟨q 1641278451780291167220080819 1250000000000000000000000000, q 1641278451780291167220330819
+      1250000000000000000000000000⟩,
+    ⟨q (-105076534082910887440587258861) 200000000000000000000000000000, q
+      (-105076534082910887440547258861) 200000000000000000000000000000⟩,
+    ⟨q 2420644844145377502832171437 2000000000000000000000000000, q 2420644844145377502832571437
+      2000000000000000000000000000⟩,
+    ⟨q 2499999999999999999999 10000000000000000000000, q 2500000000000000000001
+      10000000000000000000000⟩,
+    ⟨q 1958868239504182093160893749 50000000000000000000000000000, q 78354729580167283726435751
+      2000000000000000000000000000⟩,
+    ⟨q 34065075469136244723692787727 50000000000000000000000000000, q
+      34065075469136244723692787983 50000000000000000000000000000⟩
+  ]
+
+/-- Small kernel-checkable chunks of the frozen rational certificate.
+
+The previous one-shot package asked the kernel to normalize the whole executable
+replay in one enormous `decide +kernel`.  That is logically sound but can take
+hours.  Here the proof path uses the already frozen rational certificate and
+checks it in bounded, independent chunks.  The expensive executable replay is
+still retained in `ExactReplay.lean` as a diagnostic cross-check, but it is not
+recomputed while building the trusted certificate. -/
+def piCheck : Bool :=
+  RatInterval.strictInsideB machinPi declaredPi
+theorem piCheck_eq_true : piCheck = true := by
+  decide +kernel
+
+end GerverSofa.CertificateManifest
+
+end
+
+end
+
+section
+
+/-!
+# Executable exact-rational replay of the 4D and 22D Krawczyk inclusions
+
+This is a direct, floating-point-free transcription of the companion Python
+algorithm.  It computes with `ℚ`, interval automatic differentiation and the
+frozen rational preconditioners.  The analytic theorem saying that the Taylor
+intervals enclose the real `sin` and `cos`, and the abstract Krawczyk theorem,
+remain separate proof obligations; the arithmetic replay itself is decidable.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.ExactReplay
+
+open GerverSofa
+open RatInterval
+open scoped BigOperators
+
+/-- Construct an exact rational number from its integer numerator and natural denominator. -/
+def q (n : Int) (d : Nat := 1) : ℚ := (n : ℚ) / (d : ℚ)
+
+/-- The singleton rational interval at zero. -/
+def zeroI : RatInterval := point 0
+/-- The singleton rational interval at one. -/
+def oneI : RatInterval := point 1
+
+private def midpoint (x : RatInterval) : ℚ := (x.lo + x.hi) / 2
+
+/-- Multiply an interval by a rational singleton using exact interval arithmetic. -/
+def scale (a : ℚ) (x : RatInterval) : RatInterval :=
+  mul (point a) x
+
+/-- A factorial interpreted as an exact rational number. -/
+def factorialQ (n : Nat) : ℚ := (Nat.factorial n : ℚ)
+
+/-- An alternating Taylor term with the specified power and factorial denominator. -/
+def signedTerm (k : Nat) (x : ℚ) (power : Nat) : ℚ :=
+  let z := x ^ power / factorialQ power
+  if k % 2 = 0 then z else -z
+
+/-- The finite odd-power Taylor sum for sine at a rational argument. -/
+def sinPartial (x : ℚ) (terms : Nat) : ℚ :=
+  Finset.sum (Finset.range terms) (fun k => signedTerm k x (2 * k + 1))
+
+/-- The finite even-power Taylor sum for cosine at a rational argument. -/
+def cosPartial (x : ℚ) (terms : Nat) : ℚ :=
+  Finset.sum (Finset.range terms) (fun k => signedTerm k x (2 * k))
+
+/-- The interval between the nineteen- and twenty-term sine Taylor sums. -/
+def sinBound (x : ℚ) : RatInterval :=
+  let a := sinPartial x 19
+  let b := sinPartial x 20
+  ⟨min a b, max a b⟩
+
+/-- The interval between the nineteen- and twenty-term cosine Taylor sums. -/
+def cosBound (x : ℚ) : RatInterval :=
+  let a := cosPartial x 19
+  let b := cosPartial x 20
+  ⟨min a b, max a b⟩
+
+/-- The rational π enclosure used for trigonometric argument reduction. -/
+def piI : RatInterval :=
+  ⟨q 157079632679489661923132169163975144209858469968755
+    50000000000000000000000000000000000000000000000000,
+   q 78539816339744830961566084581987572104929234984378
+     25000000000000000000000000000000000000000000000000⟩
+
+/-- A finite alternating rational Taylor sum for arctangent. -/
+def atanPartial (x : ℚ) (terms : Nat) : ℚ :=
+  Finset.sum (Finset.range terms) (fun k =>
+    if k % 2 = 0 then
+      x ^ (2 * k + 1) / ((2 * k + 1 : Nat) : ℚ)
+    else
+      -(x ^ (2 * k + 1) / ((2 * k + 1 : Nat) : ℚ)))
+
+/-- The interval between two specified arctangent Taylor sums. -/
+def atanBound (x : ℚ) (lowTerms highTerms : Nat) : RatInterval :=
+  let a := atanPartial x lowTerms
+  let b := atanPartial x highTerms
+  ⟨min a b, max a b⟩
+
+/-- Evaluate the Machin expression `16 atan(1/5) - 4 atan(1/239)` by rational intervals. -/
+def machinPi : RatInterval :=
+  sub (scale 16 (atanBound (1 / 5) 43 44))
+      (scale 4 (atanBound (1 / 239) 13 14))
+
+/-- Exact downward rounding to a fixed number of decimal places. -/
+def floorDecimal (x : ℚ) (digits : Nat := 60) : ℚ :=
+  let scale : ℚ := (10 : ℚ) ^ digits
+  ((⌊x * scale⌋ : ℤ) : ℚ) / scale
+
+/-- Exact upward rounding to a fixed number of decimal places. -/
+def ceilDecimal (x : ℚ) (digits : Nat := 60) : ℚ :=
+  let scale : ℚ := (10 : ℚ) ^ digits
+  ((⌈x * scale⌉ : ℤ) : ℚ) / scale
+
+/-- The same 60-decimal outward rounding used by the submitted verifier. -/
+def outwardDecimal (z : RatInterval) : RatInterval :=
+  ⟨floorDecimal z.lo, ceilDecimal z.hi⟩
+
+/-- Evaluate the small-argument sine enclosure with outward decimal rounding. -/
+def sinSmall (x : RatInterval) : RatInterval :=
+  outwardDecimal ⟨(sinBound x.lo).lo, (sinBound x.hi).hi⟩
+
+/-- Evaluate the small-argument cosine enclosure with outward decimal rounding. -/
+def cosSmall (x : RatInterval) : RatInterval :=
+  outwardDecimal ⟨(cosBound x.hi).lo, (cosBound x.lo).hi⟩
+
+/-- Clamp an interval to the physical angular range used by the Gerver
+certificate.  If `x ∈ [0, π/2]` and `x` is enclosed by the input interval,
+then `x` is still enclosed after clamping once `piI` has been proved to
+contain `Real.pi`. -/
+def physicalClamp (x : RatInterval) : RatInterval :=
+  ⟨max 0 x.lo, min (piI.hi / 2) x.hi⟩
+
+/-- A fail-closed enclosure used only when an externally supplied interval is
+too wide for the small-argument Taylor/range-reduction evaluator.  Every
+certified Gerver call remains in one of the two sharp branches below, so this
+fallback does not alter the frozen replay. -/
+def universalTrigInterval : RatInterval := ⟨-1, 1⟩
+
+/-- Complementary interval for the identity `sin x = cos (π/2-x)` and
+`cos x = sin (π/2-x)`. -/
+def complementInterval (x : RatInterval) : RatInterval :=
+  ⟨max 0 (piI.lo / 2 - x.hi), piI.hi / 2 - x.lo⟩
+
+/-- Evaluate sine by small-argument bounds and complementary-angle reduction. -/
+def sinI (x : RatInterval) : RatInterval :=
+  let z := physicalClamp x
+  if z.hi ≤ 9 / 10 then sinSmall z
+  else
+    let y := complementInterval z
+    if y.hi ≤ 9 / 10 then cosSmall y else universalTrigInterval
+
+/-- Evaluate cosine by small-argument bounds and complementary-angle reduction. -/
+def cosI (x : RatInterval) : RatInterval :=
+  let z := physicalClamp x
+  if z.hi ≤ 9 / 10 then cosSmall z
+  else
+    let y := complementInterval z
+    if y.hi ≤ 9 / 10 then sinSmall y else universalTrigInterval
+
+/-- An interval value together with a list of interval partial derivatives. -/
+structure D where
+  /-- The interval enclosing the scalar value. -/
+  val : RatInterval
+  /-- Interval enclosures for the coordinate partial derivatives. -/
+  der : List RatInterval
+  deriving Repr
+
+namespace D
+
+/-- A constant interval with zero partial derivatives in every coordinate. -/
+def const (v : RatInterval) (n : Nat) : D :=
+  ⟨v, List.replicate n zeroI⟩
+
+/-- A rational singleton with zero partial derivatives. -/
+def pointConst (v : ℚ) (n : Nat) : D := const (point v) n
+
+/-- An interval variable with the selected coordinate derivative equal to one. -/
+def varD (v : RatInterval) (j n : Nat) : D :=
+  ⟨v, (List.range n).map (fun k => point (if k = j then 1 else 0))⟩
+
+/-- Addition with interval propagation of all coordinate derivatives. -/
+def addD (x y : D) : D :=
+  ⟨add x.val y.val, List.zipWith add x.der y.der⟩
+
+/-- Negation with interval propagation of all coordinate derivatives. -/
+def negD (x : D) : D :=
+  ⟨neg x.val, x.der.map neg⟩
+
+/-- Subtraction with interval propagation of all coordinate derivatives. -/
+def subD (x y : D) : D := addD x (negD y)
+
+/-- Multiplication with interval propagation of all coordinate derivatives. -/
+def mulD (x y : D) : D :=
+  ⟨mul x.val y.val,
+   List.zipWith (fun dx dy => add (mul dx y.val) (mul x.val dy)) x.der y.der⟩
+
+/-- Rational scaling with interval propagation of all coordinate derivatives. -/
+def scaleD (a : ℚ) (x : D) : D :=
+  ⟨scale a x.val, x.der.map (scale a)⟩
+
+/-- Sine with interval propagation of all coordinate derivatives. -/
+def sinD (x : D) : D :=
+  let sv := sinI x.val
+  let cv := cosI x.val
+  ⟨sv, x.der.map (mul cv)⟩
+
+/-- Cosine with interval propagation of all coordinate derivatives. -/
+def cosD (x : D) : D :=
+  let sv := sinI x.val
+  let cv := cosI x.val
+  ⟨cv, x.der.map (fun z => neg (mul sv z))⟩
+
+instance : Add D := ⟨addD⟩
+instance : Neg D := ⟨negD⟩
+instance : Sub D := ⟨subD⟩
+instance : Mul D := ⟨mulD⟩
+instance : HMul ℚ D D := ⟨scaleD⟩
+
+end D
+
+/-- Read an interval coordinate, returning the zero interval outside the list. -/
+def getI (xs : List RatInterval) (i : Nat) : RatInterval :=
+  xs.getD i zeroI
+
+/-- Read a rational coordinate, returning zero outside the list. -/
+def getQ (xs : List ℚ) (i : Nat) : ℚ := xs.getD i 0
+/-- Read a rational matrix row, returning the empty row outside the list. -/
+def getRow (m : List (List ℚ)) (i : Nat) : List ℚ := m.getD i []
+
+/-! ## Reduced 4D system -/
+
+private def x4 : List RatInterval := [
+  ⟨q 1888531216873 20000000000000, q 4721328042183 50000000000000⟩,
+  ⟨q 69960186366677 50000000000000, q 34980093183339 25000000000000⟩,
+  ⟨q 122429264969 3125000000000, q 3917736479009 100000000000000⟩,
+  ⟨q 2129067216821 3125000000000, q 68130150938273 100000000000000⟩
+]
+
+private def z22 : List RatInterval := [
+  ⟨q (-21032242207268875141628571849) 100000000000000000000000000000, q
+    (-21032242207268875141608571849) 100000000000000000000000000000⟩,
+  ⟨q 2499999999999999999999 10000000000000000000000, q 2500000000000000000001
+    10000000000000000000000⟩,
+  ⟨q (-91917929277159332227479610289) 100000000000000000000000000000, q
+    (-91917929277159332227459610289) 100000000000000000000000000000⟩,
+  ⟨q 29525413734425341573853797657 62500000000000000000000000000, q 29525413734425341573866297657
+    62500000000000000000000000000⟩,
+  ⟨q (-15344080735756291713875357283) 25000000000000000000000000000, q
+    (-15344080735756291713870357283) 25000000000000000000000000000⟩,
+  ⟨q 17792529580064437214538861001 20000000000000000000000000000, q 17792529580064437214542861001
+    20000000000000000000000000000⟩,
+  ⟨q (-15417358304445500741761623987) 50000000000000000000000000000, q
+    (-15417358304445500741751623987) 50000000000000000000000000000⟩,
+  ⟨q 29525413734425341573853797657 62500000000000000000000000000, q 29525413734425341573866297657
+    62500000000000000000000000000⟩,
+  ⟨q (-20344080735756291713874857283) 20000000000000000000000000000, q
+    (-20344080735756291713870857283) 20000000000000000000000000000⟩,
+  ⟨q 2499999999999999999999 10000000000000000000000, q 2500000000000000000001
+    10000000000000000000000⟩,
+  ⟨q 2420644844145377502832171437 2000000000000000000000000000, q 2420644844145377502832571437
+    2000000000000000000000000000⟩,
+  ⟨q (-2500000000000000000001) 10000000000000000000000, q (-2499999999999999999999)
+    10000000000000000000000⟩,
+  ⟨q (-52762459802678462416060380937) 100000000000000000000000000000, q
+    (-52762459802678462416040380937) 100000000000000000000000000000⟩,
+  ⟨q 92025838516063762289360579501 100000000000000000000000000000, q 92025838516063762289380579501
+    100000000000000000000000000000⟩,
+  ⟨q 313022761424232933776114655193 500000000000000000000000000000, q
+    313022761424232933776214655193 500000000000000000000000000000⟩,
+  ⟨q (-151160128631428920268654781) 160000000000000000000000000, q (-151160128631428920268622781)
+    160000000000000000000000000⟩,
+  ⟨q 1641278451780291167220080819 1250000000000000000000000000, q 1641278451780291167220330819
+    1250000000000000000000000000⟩,
+  ⟨q (-105076534082910887440587258861) 200000000000000000000000000000, q
+    (-105076534082910887440547258861) 200000000000000000000000000000⟩,
+  ⟨q 2420644844145377502832171437 2000000000000000000000000000, q 2420644844145377502832571437
+    2000000000000000000000000000⟩,
+  ⟨q 2499999999999999999999 10000000000000000000000, q 2500000000000000000001
+    10000000000000000000000⟩,
+  ⟨q 1958868239504182093160893749 50000000000000000000000000000, q 78354729580167283726435751
+    2000000000000000000000000000⟩,
+  ⟨q 34065075469136244723692787727 50000000000000000000000000000, q 34065075469136244723692787983
+    50000000000000000000000000000⟩
+]
+
+/-! ## Direct 22D Romik system -/
+
+/-! ## Public proof-carrying view and optional executable cross-check
+
+The full executable Krawczyk/grid replay above is intentionally retained, but
+normalizing it in one kernel reduction is prohibitively expensive.  The trusted
+proof path therefore consumes the frozen exact-rational certificate emitted by
+the independent replay and checks that certificate in `CertificateManifest`.
+This is the standard proof-carrying-data split: expensive certificate discovery
+is outside the kernel; small rational certificate verification is inside it.
+
+The executable wrappers prefixed by `executable` remain available for offline
+cross-checking and provenance. -/
+
+/-- The rational interval used internally by the executable replay for `Real.pi`. -/
+def declaredPiInterval : RatInterval := piI
+
+/-- Public wrapper around the executable sine enclosure. -/
+def sineInterval (x : RatInterval) : RatInterval := sinI x
+
+/-- Public wrapper around the executable cosine enclosure. -/
+def cosineInterval (x : RatInterval) : RatInterval := cosI x
+
+/-- Frozen reduced input box used by the trusted certificate. -/
+def reducedInputBox : List RatInterval := CertificateManifest.x4
+
+/-- Frozen direct-system input box used by the trusted certificate. -/
+def fullInputBox : List RatInterval := CertificateManifest.z22
+
+end GerverSofa.ExactReplay
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle002.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle002.lean
new file mode 100644
index 0000000000..e3f0fb9275
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle002.lean
@@ -0,0 +1,5428 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import Aesop
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle001
+public import Mathlib.Algebra.Order.Floor.Ring
+public import Mathlib.Analysis.Calculus.Deriv.Add
+public import Mathlib.Analysis.Calculus.Deriv.Basic
+public import Mathlib.Analysis.Calculus.Deriv.Mul
+public import Mathlib.Analysis.Calculus.Deriv.Slope
+public import Mathlib.Analysis.InnerProductSpace.PiL2
+public import Mathlib.Analysis.Normed.Affine.Isometry
+public import Mathlib.Analysis.Real.Pi.Bounds
+public import Mathlib.Analysis.SpecialFunctions.Complex.Arctan
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
+public import Mathlib.Analysis.SpecificLimits.Normed
+public import Mathlib.Data.List.GetD
+public import Mathlib.Data.List.Zip
+public import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
+public import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
+public import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
+public import Mathlib.Tactic
+public import Mathlib.Tactic.Linarith
+public import Mathlib.Tactic.LinearCombination
+public import Mathlib.Tactic.NormNum
+public import Mathlib.Topology.Algebra.ContinuousAffineMap.Topology
+public import Mathlib.Topology.Algebra.InfiniteSum.NatInt
+public import Mathlib.Topology.Instances.Real.Lemmas
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.Foundation.Batch002`.
+* `GerverSofa.KernelOnly.Foundation.Batch001`.
+* `GerverSofa.KernelOnly.PartF.Semantics.Batch001`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `SE2`.
+* `SupportingHallway`.
+* `Systems`.
+* `Boxes`.
+* `Geometry`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Orientation-preserving rigid motions of the plane
+
+An element is represented by `(c,s,tₓ,tᵧ)` with `c²+s²=1`.  Its linear
+part is the rotation matrix `[[c,-s],[s,c]]`, hence every value is an element
+of `SE(2)` and not an arbitrary affine equivalence.
+-/
+
+@[expose] public section
+
+namespace GerverSofa
+
+/-- An orientation-preserving planar rigid motion represented by rotation and translation. -/
+structure SE2 where
+  /-- The cosine coefficient of the rotation. -/
+  c : ℝ
+  /-- The sine coefficient of the rotation. -/
+  s : ℝ
+  /-- The horizontal translation component. -/
+  tx : ℝ
+  /-- The vertical translation component. -/
+  ty : ℝ
+  unit : c * c + s * s = 1
+
+namespace SE2
+
+/-- Action of an orientation-preserving rigid motion on the plane. -/
+def act (g : SE2) (p : Point) : Point :=
+  (g.c * p.1 - g.s * p.2 + g.tx,
+   g.s * p.1 + g.c * p.2 + g.ty)
+
+/-- Identity element. -/
+def one : SE2 where
+  c := 1
+  s := 0
+  tx := 0
+  ty := 0
+  unit := by norm_num
+
+/-- Inverse orientation-preserving rigid motion. -/
+def inv (g : SE2) : SE2 where
+  c := g.c
+  s := -g.s
+  tx := -(g.c * g.tx + g.s * g.ty)
+  ty := g.s * g.tx - g.c * g.ty
+  unit := by nlinarith [g.unit]
+
+@[simp] theorem one_act (p : Point) : one.act p = p := by
+  rcases p with ⟨x, y⟩
+  simp [one, act]
+
+@[simp] theorem inv_act_act (g : SE2) (p : Point) :
+    g.inv.act (g.act p) = p := by
+  rcases p with ⟨x, y⟩
+  apply Prod.ext
+  · dsimp [act, inv]
+    linear_combination x * g.unit
+  · dsimp [act, inv]
+    linear_combination y * g.unit
+
+@[simp] theorem act_inv_act (g : SE2) (p : Point) :
+    g.act (g.inv.act p) = p := by
+  rcases p with ⟨x, y⟩
+  apply Prod.ext
+  · dsimp [act, inv]
+    linear_combination (x - g.tx) * g.unit
+  · dsimp [act, inv]
+    linear_combination (y - g.ty) * g.unit
+
+/-- Componentwise continuity is the topology-free representation of a path
+in `SE(2)` used by the formal moving-sofa definition. -/
+def ContinuousPath (g : ℝ → SE2) : Prop :=
+  Continuous (fun t => (g t).c) ∧
+  Continuous (fun t => (g t).s) ∧
+  Continuous (fun t => (g t).tx) ∧
+  Continuous (fun t => (g t).ty)
+
+/-- Inversion preserves continuous `SE(2)` paths. -/
+theorem continuousPath_inv {g : ℝ → SE2} (hg : ContinuousPath g) :
+    ContinuousPath (fun t => (g t).inv) := by
+  rcases hg with ⟨hc, hs, htx, hty⟩
+  refine ⟨hc, hs.neg, ?_, ?_⟩
+  · exact ((hc.mul htx).add (hs.mul hty)).neg
+  · exact (hs.mul htx).sub (hc.mul hty)
+
+end SE2
+end GerverSofa
+
+end
+
+end
+
+section
+
+/-!
+# Supporting hallway and inverse motion
+-/
+
+@[expose] public section
+
+namespace GerverSofa
+
+/-- World-frame hallway obtained from the standard hallway by `frame`. -/
+def supportingHallway (frame : SE2) : Set Point :=
+  frame.act '' hallway
+
+/-- Membership in a supporting hallway is equivalent to standard-hallway
+membership after applying the inverse frame. -/
+theorem mem_supportingHallway_iff (frame : SE2) (q : Point) :
+    q ∈ supportingHallway frame ↔ frame.inv.act q ∈ hallway := by
+  constructor
+  · rintro ⟨p, hp, rfl⟩
+    simpa using hp
+  · intro hq
+    refine ⟨frame.inv.act q, hq, ?_⟩
+    exact frame.act_inv_act q
+
+/-- Set-level version of `mem_supportingHallway_iff`. -/
+theorem inv_image_subset_hallway_of_subset_supporting
+    (frame : SE2) (S : Set Point)
+    (hS : S ⊆ supportingHallway frame) :
+    frame.inv.act '' S ⊆ hallway := by
+  rintro p ⟨q, hqS, rfl⟩
+  exact (mem_supportingHallway_iff frame q).1 (hS hqS)
+
+end GerverSofa
+
+end
+
+end
+
+section
+
+/-!
+# The concrete reduced and 22-dimensional Romik systems
+
+These are direct Lean transcriptions of equations (F1)--(F4) and of the
+independent equations (27)--(39), (41), (43) used by the companion verifier.
+-/
+
+@[expose] public section
+
+namespace GerverSofa
+
+noncomputable section
+
+namespace Reduced
+
+/-- The four real parameters of the reduced Gerver equations. -/
+structure Params where
+  /-- The first scalar parameter in the reduced equations. -/
+  a : ℝ
+  /-- The second scalar parameter in the reduced equations. -/
+  b : ℝ
+  /-- The first switching angle. -/
+  phi : ℝ
+  /-- The second switching angle. -/
+  theta : ℝ
+
+/-- Four-dimensional reduced switching system. -/
+def system (p : Params) : Fin 4 → ℝ :=
+  let cp := Real.cos p.phi
+  let sp := Real.sin p.phi
+  let ct := Real.cos p.theta
+  let st := Real.sin p.theta
+  let delta := p.theta - p.phi
+  ![
+    p.a * (ct - cp) - 2 * p.b * sp + (delta - 1) * ct - st + cp + sp,
+    p.a * (3 * st + sp) - 2 * p.b * cp + 3 * (delta - 1) * st + 3 * ct - sp + cp,
+    p.a * cp - sp - (1 / 2 : ℝ) + (1 / 2 : ℝ) * cp - p.b * sp,
+    p.a + Real.pi / 2 - p.phi - p.theta - p.b
+      + (1 / 2 : ℝ) * delta * (1 + p.a) + (1 / 4 : ℝ) * delta * delta
+  ]
+
+/-- The proposition that all four reduced equations vanish. -/
+def Equations (p : Params) : Prop := system p = 0
+
+end Reduced
+
+namespace Romik
+
+/-- Translation, phase, and switching-angle parameters of the five Romik branches. -/
+structure Params where
+  /-- Horizontal translation coefficient for phase 1. -/
+  k11 : ℝ
+  /-- Vertical translation coefficient for phase 1. -/
+  k12 : ℝ
+  /-- Horizontal translation coefficient for phase 2. -/
+  k21 : ℝ
+  /-- Vertical translation coefficient for phase 2. -/
+  k22 : ℝ
+  /-- Horizontal translation coefficient for phase 3. -/
+  k31 : ℝ
+  /-- Vertical translation coefficient for phase 3. -/
+  k32 : ℝ
+  /-- Horizontal translation coefficient for phase 4. -/
+  k41 : ℝ
+  /-- Vertical translation coefficient for phase 4. -/
+  k42 : ℝ
+  /-- Horizontal translation coefficient for phase 5. -/
+  k51 : ℝ
+  /-- Vertical translation coefficient for phase 5. -/
+  k52 : ℝ
+  /-- Coefficient 1 in the phase 1 closed formula. -/
+  a1 : ℝ
+  /-- Coefficient 2 in the phase 1 closed formula. -/
+  a2 : ℝ
+  /-- Coefficient 1 in the phase 2 closed formula. -/
+  b1 : ℝ
+  /-- Coefficient 2 in the phase 2 closed formula. -/
+  b2 : ℝ
+  /-- Coefficient 1 in the phase 3 closed formula. -/
+  c1 : ℝ
+  /-- Coefficient 2 in the phase 3 closed formula. -/
+  c2 : ℝ
+  /-- Coefficient 1 in the phase 4 closed formula. -/
+  d1 : ℝ
+  /-- Coefficient 2 in the phase 4 closed formula. -/
+  d2 : ℝ
+  /-- Coefficient 1 in the phase 5 closed formula. -/
+  e1 : ℝ
+  /-- Coefficient 2 in the phase 5 closed formula. -/
+  e2 : ℝ
+  /-- First switching angle. -/
+  phi : ℝ
+  /-- Second switching angle. -/
+  theta : ℝ
+
+/-- Rotation of a body-frame vector into the world frame. -/
+def rot (t : ℝ) (z : Point) : Point :=
+  (Real.cos t * z.1 - Real.sin t * z.2,
+   Real.sin t * z.1 + Real.cos t * z.2)
+
+/-- Translate a point by the two specified coordinate offsets. -/
+def addK (r : Point) (kx ky : ℝ) : Point := (r.1 + kx, r.2 + ky)
+
+/-- Phase 1 of the five-phase Gerver path. -/
+def path1 (p : Params) (t : ℝ) : Point :=
+  let z : Point :=
+    (p.a1 * Real.cos t + p.a2 * Real.sin t - 1,
+     -p.a2 * Real.cos t + p.a1 * Real.sin t - 1 / 2)
+  addK (rot t z) p.k11 p.k12
+
+/-- Phase 2 of the five-phase Gerver path. -/
+def path2 (p : Params) (t : ℝ) : Point :=
+  let z : Point :=
+    (-(1 / 4 : ℝ) * t * t + p.b1 * t + p.b2,
+     (1 / 2 : ℝ) * t - p.b1 - 1)
+  addK (rot t z) p.k21 p.k22
+
+/-- Phase 3 of the five-phase Gerver path. -/
+def path3 (p : Params) (t : ℝ) : Point :=
+  addK (rot t (p.c1 - t, p.c2 + t)) p.k31 p.k32
+
+/-- Phase 4 of the five-phase Gerver path. -/
+def path4 (p : Params) (t : ℝ) : Point :=
+  let z : Point :=
+    (-(1 / 2 : ℝ) * t + p.d1 - 1,
+     -(1 / 4 : ℝ) * t * t + p.d1 * t + p.d2)
+  addK (rot t z) p.k41 p.k42
+
+/-- Phase 5 of the five-phase Gerver path. -/
+def path5 (p : Params) (t : ℝ) : Point :=
+  let z : Point :=
+    (p.e1 * Real.cos t + p.e2 * Real.sin t - 1 / 2,
+     -p.e2 * Real.cos t + p.e1 * Real.sin t - 1)
+  addK (rot t z) p.k51 p.k52
+
+/-- Body-frame derivative coefficients `(alpha,beta)` on phase 1. -/
+def alphaBeta1 (p : Params) (t : ℝ) : Point :=
+  (-2 * p.a1 * Real.sin t + 2 * p.a2 * Real.cos t + 1 / 2,
+   2 * p.a1 * Real.cos t + 2 * p.a2 * Real.sin t - 1)
+
+/-- Body-frame velocity coordinates for phase 2. -/
+def alphaBeta2 (p : Params) (t : ℝ) : Point :=
+  (1 + 2 * p.b1 - t,
+   -(1 / 4 : ℝ) * t * t + p.b1 * t + p.b2 + 1 / 2)
+
+/-- Body-frame velocity coordinates for phase 3. -/
+def alphaBeta3 (p : Params) (t : ℝ) : Point :=
+  (-1 - p.c2 - t, 1 + p.c1 - t)
+
+/-- Body-frame velocity coordinates for phase 4. -/
+def alphaBeta4 (p : Params) (t : ℝ) : Point :=
+  ((1 / 4 : ℝ) * t * t - p.d1 * t - p.d2 - 1 / 2,
+   2 * p.d1 - 1 - t)
+
+/-- Body-frame velocity coordinates for phase 5. -/
+def alphaBeta5 (p : Params) (t : ℝ) : Point :=
+  (1 - 2 * p.e1 * Real.sin t + 2 * p.e2 * Real.cos t,
+   2 * p.e1 * Real.cos t + 2 * p.e2 * Real.sin t - 1 / 2)
+
+/-- Rotate body-frame velocity coordinates into the world frame. -/
+def pathPrimeFromAB (t : ℝ) (ab : Point) : Point := rot t ab
+
+/-- World-frame velocity formula for phase 1. -/
+def pathPrime1 (p : Params) (t : ℝ) : Point := pathPrimeFromAB t (alphaBeta1 p t)
+/-- World-frame velocity formula for phase 2. -/
+def pathPrime2 (p : Params) (t : ℝ) : Point := pathPrimeFromAB t (alphaBeta2 p t)
+/-- World-frame velocity formula for phase 3. -/
+def pathPrime3 (p : Params) (t : ℝ) : Point := pathPrimeFromAB t (alphaBeta3 p t)
+
+/-- Romik's 22 independent scalar equations. -/
+def system (p : Params) : Fin 22 → ℝ :=
+  let halfPi := Real.pi / 2
+  let quarterPi := Real.pi / 4
+  let eta := halfPi - p.theta
+  let tau := halfPi - p.phi
+  let x1phi := path1 p p.phi
+  let x2phi := path2 p p.phi
+  let v1phi := pathPrime1 p p.phi
+  let v2phi := pathPrime2 p p.phi
+  let x2theta := path2 p p.theta
+  let x3theta := path3 p p.theta
+  let v2theta := pathPrime2 p p.theta
+  let v3theta := pathPrime3 p p.theta
+  let x3eta := path3 p eta
+  let x4eta := path4 p eta
+  let x4tau := path4 p tau
+  let x5tau := path5 p tau
+  let abEta := alphaBeta3 p eta
+  let bEta : Point :=
+    (x3eta.1 - abEta.1 * Real.sin eta,
+     x3eta.2 + abEta.1 * Real.cos eta)
+  ![
+    p.e1 - p.a1,
+    p.e2 + p.a2,
+    p.d1 + p.b1 - quarterPi,
+    p.d2 - p.b2 - quarterPi * (2 * p.b1 - quarterPi),
+    p.c2 - p.c1 + halfPi,
+    p.k11 - 1 + p.a1,
+    p.k12 - 1 / 4,
+    p.a2 + 1 / 4,
+    x1phi.1 - x2phi.1,
+    x1phi.2 - x2phi.2,
+    v1phi.1 - v2phi.1,
+    v1phi.2 - v2phi.2,
+    x2theta.1 - x3theta.1,
+    x2theta.2 - x3theta.2,
+    v2theta.1 - v3theta.1,
+    v2theta.2 - v3theta.2,
+    x3eta.1 - x4eta.1,
+    x3eta.2 - x4eta.2,
+    x4tau.1 - x5tau.1,
+    x4tau.2 - x5tau.2,
+    x1phi.1 - bEta.1,
+    x1phi.2 - bEta.2
+  ]
+
+/-- The proposition that all 22 independent equations vanish. -/
+def Equations (p : Params) : Prop := system p = 0
+
+/-- The physical five-phase path on `[0,π/2]`.  At a switching angle either
+adjacent formula may be chosen; the certified matching equations prove they
+coincide. -/
+def path (p : Params) (t : ℝ) : Point :=
+  let eta := Real.pi / 2 - p.theta
+  let tau := Real.pi / 2 - p.phi
+  if t ≤ p.phi then path1 p t
+  else if t ≤ p.theta then path2 p t
+  else if t ≤ eta then path3 p t
+  else if t ≤ tau then path4 p t
+  else path5 p t
+
+/-- Linear normalization from unit time to physical rotation angle. -/
+def angle (u : ℝ) : ℝ := u * (Real.pi / 2)
+
+/-- Standard-to-world frame `q ↦ x(t)+R_t q` for normalized time. -/
+def frame (p : Params) (u : ℝ) : SE2 :=
+  let t := angle u
+  let x := path p t
+  { c := Real.cos t
+    s := Real.sin t
+    tx := x.1
+    ty := x.2
+    unit := by nlinarith [Real.sin_sq_add_cos_sq t] }
+
+@[simp] theorem angle_zero : angle 0 = 0 := by
+  simp [angle]
+
+@[simp] theorem angle_one : angle 1 = Real.pi / 2 := by
+  simp [angle]
+
+/-- The normalized physical angle depends continuously on time. -/
+theorem continuous_angle : Continuous angle := by
+  unfold angle
+  fun_prop
+
+/-- Continuity of the five-phase path implies componentwise continuity of the
+supporting `SE(2)` frame. -/
+theorem continuousPath_frame_of_path
+    (p : Params) (hpath : Continuous (path p)) :
+    SE2.ContinuousPath (frame p) := by
+  refine ⟨?_, ?_, ?_, ?_⟩
+  · change Continuous (fun u : ℝ => Real.cos (angle u))
+    simpa [Function.comp_def] using (Real.continuous_cos.comp continuous_angle)
+  · change Continuous (fun u : ℝ => Real.sin (angle u))
+    simpa [Function.comp_def] using (Real.continuous_sin.comp continuous_angle)
+  · change Continuous (fun u : ℝ => (path p (angle u)).1)
+    have hp : Continuous (fun u : ℝ => path p (angle u)) := by
+      simpa [Function.comp_def] using (hpath.comp continuous_angle)
+    exact hp.fst
+  · change Continuous (fun u : ℝ => (path p (angle u)).2)
+    have hp : Continuous (fun u : ℝ => path p (angle u)) := by
+      simpa [Function.comp_def] using (hpath.comp continuous_angle)
+    exact hp.snd
+
+/-! ## Continuity from the independent matching equations -/
+
+/-- Ordering data needed to read the five branches in their intended order. -/
+structure SwitchOrder (p : Params) : Prop where
+  phi_le_theta : p.phi ≤ p.theta
+  theta_le_eta : p.theta ≤ Real.pi / 2 - p.theta
+  eta_le_tau : Real.pi / 2 - p.theta ≤ Real.pi / 2 - p.phi
+
+private theorem continuous_path1 (p : Params) : Continuous (path1 p) := by
+  unfold path1 addK rot
+  fun_prop
+
+private theorem continuous_path2 (p : Params) : Continuous (path2 p) := by
+  unfold path2 addK rot
+  fun_prop
+
+private theorem continuous_path3 (p : Params) : Continuous (path3 p) := by
+  unfold path3 addK rot
+  fun_prop
+
+private theorem continuous_path4 (p : Params) : Continuous (path4 p) := by
+  unfold path4 addK rot
+  fun_prop
+
+private theorem continuous_path5 (p : Params) : Continuous (path5 p) := by
+  unfold path5 addK rot
+  fun_prop
+
+/-- Equation (35), extracted from the direct 22D system. -/
+theorem match_path12_of_equations {p : Params} (heq : Equations p) :
+    path1 p p.phi = path2 p p.phi := by
+  have h8 := congrFun heq (8 : Fin 22)
+  have h9 := congrFun heq (9 : Fin 22)
+  simp [system] at h8 h9
+  apply Prod.ext <;> linarith
+
+/-- Equation (37), extracted from the direct 22D system. -/
+theorem match_path23_of_equations {p : Params} (heq : Equations p) :
+    path2 p p.theta = path3 p p.theta := by
+  have h12 := congrFun heq (12 : Fin 22)
+  have h13 := congrFun heq (13 : Fin 22)
+  simp [system] at h12 h13
+  apply Prod.ext <;> linarith
+
+/-- Equation (39), extracted from the direct 22D system. -/
+theorem match_path34_of_equations {p : Params} (heq : Equations p) :
+    path3 p (Real.pi / 2 - p.theta) =
+      path4 p (Real.pi / 2 - p.theta) := by
+  have h16 := congrFun heq (16 : Fin 22)
+  have h17 := congrFun heq (17 : Fin 22)
+  simp [system] at h16 h17
+  apply Prod.ext <;> linarith
+
+/-- Equation (41), extracted from the direct 22D system. -/
+theorem match_path45_of_equations {p : Params} (heq : Equations p) :
+    path4 p (Real.pi / 2 - p.phi) =
+      path5 p (Real.pi / 2 - p.phi) := by
+  have h18 := congrFun heq (18 : Fin 22)
+  have h19 := congrFun heq (19 : Fin 22)
+  simp [system] at h18 h19
+  apply Prod.ext <;> linarith
+
+/-- The four positional matching equations make the literal nested-`if`
+five-phase path continuous whenever its switches are ordered. -/
+theorem continuous_path_of_order_and_equations
+    {p : Params} (hord : SwitchOrder p) (heq : Equations p) :
+    Continuous (path p) := by
+  have h12 := match_path12_of_equations heq
+  have h23 := match_path23_of_equations heq
+  have h34 := match_path34_of_equations heq
+  have h45 := match_path45_of_equations heq
+  have hc45 : Continuous
+      (fun t : ℝ =>
+        if t ≤ Real.pi / 2 - p.phi then path4 p t else path5 p t) := by
+    exact (continuous_path4 p).if_le (continuous_path5 p)
+      continuous_id continuous_const (by
+        intro t ht
+        subst t
+        exact h45)
+  have hc345 : Continuous
+      (fun t : ℝ =>
+        if t ≤ Real.pi / 2 - p.theta then path3 p t
+        else if t ≤ Real.pi / 2 - p.phi then path4 p t else path5 p t) := by
+    exact (continuous_path3 p).if_le hc45
+      continuous_id continuous_const (by
+        intro t ht
+        subst t
+        rw [ite_eq_left hord.eta_le_tau]
+        exact h34)
+  have hc2345 : Continuous
+      (fun t : ℝ =>
+        if t ≤ p.theta then path2 p t
+        else if t ≤ Real.pi / 2 - p.theta then path3 p t
+        else if t ≤ Real.pi / 2 - p.phi then path4 p t else path5 p t) := by
+    exact (continuous_path2 p).if_le hc345
+      continuous_id continuous_const (by
+        intro t ht
+        subst t
+        rw [ite_eq_left hord.theta_le_eta]
+        exact h23)
+  have hc12345 : Continuous
+      (fun t : ℝ =>
+        if t ≤ p.phi then path1 p t
+        else if t ≤ p.theta then path2 p t
+        else if t ≤ Real.pi / 2 - p.theta then path3 p t
+        else if t ≤ Real.pi / 2 - p.phi then path4 p t else path5 p t) := by
+    exact (continuous_path1 p).if_le hc2345
+      continuous_id continuous_const (by
+        intro t ht
+        subst t
+        rw [ite_eq_left hord.phi_le_theta]
+        exact h12)
+  unfold path
+  exact hc12345
+
+end Romik
+
+end
+end GerverSofa
+
+end
+
+end
+
+section
+
+/-!
+# Exact real boxes used by the Krawczyk certificates
+
+Every endpoint is written as an exact integer quotient.  There are no binary
+floating-point constants in these definitions.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+
+/-- Interpret an integer numerator and natural denominator as a real quotient. -/
+def qR (n : Int) (d : Nat) : ℝ := (n : ℝ) / (d : ℝ)
+
+namespace Reduced
+
+/-- Input box `X_a × X_b × X_phi × X_theta`. -/
+def box : Set Params :=
+  {p |
+    qR 1888531216873 20000000000000 ≤ p.a ∧ p.a ≤ qR 4721328042183 50000000000000 ∧
+    qR 69960186366677 50000000000000 ≤ p.b ∧ p.b ≤ qR 34980093183339 25000000000000 ∧
+    qR 122429264969 3125000000000 ≤ p.phi ∧ p.phi ≤ qR 3917736479009 100000000000000 ∧
+    qR 2129067216821 3125000000000 ≤ p.theta ∧ p.theta ≤ qR 68130150938273 100000000000000}
+
+end Reduced
+
+namespace Romik
+
+/-- The direct 22-dimensional box `Y × Phi × Theta`. -/
+def box : Set Params :=
+  {p |
+    qR (-21032242207268875141628571849) 100000000000000000000000000000 ≤ p.k11 ∧ p.k11 ≤ qR
+      (-21032242207268875141608571849) 100000000000000000000000000000 ∧
+    qR 2499999999999999999999 10000000000000000000000 ≤ p.k12 ∧ p.k12 ≤ qR 2500000000000000000001
+      10000000000000000000000 ∧
+    qR (-91917929277159332227479610289) 100000000000000000000000000000 ≤ p.k21 ∧ p.k21 ≤ qR
+      (-91917929277159332227459610289) 100000000000000000000000000000 ∧
+    qR 29525413734425341573853797657 62500000000000000000000000000 ≤ p.k22 ∧ p.k22 ≤ qR
+      29525413734425341573866297657 62500000000000000000000000000 ∧
+    qR (-15344080735756291713875357283) 25000000000000000000000000000 ≤ p.k31 ∧ p.k31 ≤ qR
+      (-15344080735756291713870357283) 25000000000000000000000000000 ∧
+    qR 17792529580064437214538861001 20000000000000000000000000000 ≤ p.k32 ∧ p.k32 ≤ qR
+      17792529580064437214542861001 20000000000000000000000000000 ∧
+    qR (-15417358304445500741761623987) 50000000000000000000000000000 ≤ p.k41 ∧ p.k41 ≤ qR
+      (-15417358304445500741751623987) 50000000000000000000000000000 ∧
+    qR 29525413734425341573853797657 62500000000000000000000000000 ≤ p.k42 ∧ p.k42 ≤ qR
+      29525413734425341573866297657 62500000000000000000000000000 ∧
+    qR (-20344080735756291713874857283) 20000000000000000000000000000 ≤ p.k51 ∧ p.k51 ≤ qR
+      (-20344080735756291713870857283) 20000000000000000000000000000 ∧
+    qR 2499999999999999999999 10000000000000000000000 ≤ p.k52 ∧ p.k52 ≤ qR 2500000000000000000001
+      10000000000000000000000 ∧
+    qR 2420644844145377502832171437 2000000000000000000000000000 ≤ p.a1 ∧ p.a1 ≤ qR
+      2420644844145377502832571437 2000000000000000000000000000 ∧
+    qR (-2500000000000000000001) 10000000000000000000000 ≤ p.a2 ∧ p.a2 ≤ qR
+      (-2499999999999999999999) 10000000000000000000000 ∧
+    qR (-52762459802678462416060380937) 100000000000000000000000000000 ≤ p.b1 ∧ p.b1 ≤ qR
+      (-52762459802678462416040380937) 100000000000000000000000000000 ∧
+    qR 92025838516063762289360579501 100000000000000000000000000000 ≤ p.b2 ∧ p.b2 ≤ qR
+      92025838516063762289380579501 100000000000000000000000000000 ∧
+    qR 313022761424232933776114655193 500000000000000000000000000000 ≤ p.c1 ∧ p.c1 ≤ qR
+      313022761424232933776214655193 500000000000000000000000000000 ∧
+    qR (-151160128631428920268654781) 160000000000000000000000000 ≤ p.c2 ∧ p.c2 ≤ qR
+      (-151160128631428920268622781) 160000000000000000000000000 ∧
+    qR 1641278451780291167220080819 1250000000000000000000000000 ≤ p.d1 ∧ p.d1 ≤ qR
+      1641278451780291167220330819 1250000000000000000000000000 ∧
+    qR (-105076534082910887440587258861) 200000000000000000000000000000 ≤ p.d2 ∧ p.d2 ≤ qR
+      (-105076534082910887440547258861) 200000000000000000000000000000 ∧
+    qR 2420644844145377502832171437 2000000000000000000000000000 ≤ p.e1 ∧ p.e1 ≤ qR
+      2420644844145377502832571437 2000000000000000000000000000 ∧
+    qR 2499999999999999999999 10000000000000000000000 ≤ p.e2 ∧ p.e2 ≤ qR 2500000000000000000001
+      10000000000000000000000 ∧
+    qR 1958868239504182093160893749 50000000000000000000000000000 ≤ p.phi ∧ p.phi ≤ qR
+      78354729580167283726435751 2000000000000000000000000000 ∧
+    qR 34065075469136244723692787727 50000000000000000000000000000 ≤ p.theta ∧ p.theta ≤ qR
+      34065075469136244723692787983 50000000000000000000000000000}
+
+end Romik
+end GerverSofa
+
+namespace GerverSofa.Romik
+
+/-- Every point in the certified direct-system box has a positive first
+switching angle. -/
+theorem phi_pos_of_mem_box {p : Params} (hp : p ∈ box) : 0 < p.phi := by
+  dsimp [box, qR] at hp
+  have hlo :
+      ((1958868239504182093160893749 : ℝ) /
+        50000000000000000000000000000) ≤ p.phi := by
+    aesop
+  have hpositive :
+      (0 : ℝ) <
+        ((1958868239504182093160893749 : ℝ) /
+          50000000000000000000000000000) := by
+    norm_num
+  exact lt_of_lt_of_le hpositive hlo
+
+/-- Equations (32)--(34), together with the certified box, force the exact
+initial path normalisation `x(0)=(0,0)`. -/
+theorem path_zero_of_mem_box_and_equations
+    {p : Params} (hp : p ∈ box) (heq : Equations p) :
+    path p 0 = (0, 0) := by
+  have hphi : 0 < p.phi := phi_pos_of_mem_box hp
+  have h5 := congrFun heq (5 : Fin 22)
+  have h6 := congrFun heq (6 : Fin 22)
+  have h7 := congrFun heq (7 : Fin 22)
+  simp [system] at h5 h6 h7
+  have h0phi : (0 : ℝ) ≤ p.phi := le_of_lt hphi
+  apply Prod.ext <;>
+    simp [path, h0phi, path1, rot, addK] <;>
+    linarith
+
+/-- The exact direct-system box orders the four physical switching times. -/
+theorem switchOrder_of_mem_box {p : Params} (hp : p ∈ box) : SwitchOrder p := by
+  dsimp [box, qR] at hp
+  have hphiLo :
+      ((1958868239504182093160893749 : ℝ) /
+        50000000000000000000000000000) ≤ p.phi := by
+    aesop
+  have hphiHi :
+      p.phi ≤ ((78354729580167283726435751 : ℝ) /
+        2000000000000000000000000000) := by
+    aesop
+  have hthetaLo :
+      ((34065075469136244723692787727 : ℝ) /
+        50000000000000000000000000000) ≤ p.theta := by
+    aesop
+  have hthetaHi :
+      p.theta ≤ ((34065075469136244723692787983 : ℝ) /
+        50000000000000000000000000000) := by
+    aesop
+  have hphiTheta : p.phi ≤ p.theta := by
+    norm_num at hphiHi hthetaLo
+    linarith
+  have hthetaEta : p.theta ≤ Real.pi / 2 - p.theta := by
+    norm_num at hthetaHi
+    nlinarith [Real.pi_gt_three]
+  exact
+    { phi_le_theta := hphiTheta
+      theta_le_eta := hthetaEta
+      eta_le_tau := by linarith }
+
+/-- The certified box and the positional equations discharge the path
+continuity field needed by the moving-sofa assembly. -/
+theorem continuous_path_of_mem_box_and_equations
+    {p : Params} (hp : p ∈ box) (heq : Equations p) :
+    Continuous (path p) :=
+  continuous_path_of_order_and_equations (switchOrder_of_mem_box hp) heq
+
+/-- The direct-system box supplies the strict lower bound on `a₁` used in the
+area proposition. -/
+theorem a1_lower_bound_of_mem_box {p : Params} (hp : p ∈ box) :
+    ((2420644844145377502832171437 : ℝ) /
+      2000000000000000000000000000) ≤ p.a1 := by
+  dsimp [box, qR] at hp
+  aesop
+
+end GerverSofa.Romik
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Concrete Gerver cap, niche and fixed sofa
+
+The definitions follow the manuscript literally.  This file also proves the
+closedness part of the main topological certificate directly from the
+half-plane definitions; it does not use the numerical certificate or Baek's
+cap theory.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+
+/-- Euclidean scalar product in the fixed coordinate representation. -/
+def dot (p q : Point) : ℝ := p.1 * q.1 + p.2 * q.2
+
+/-- Rotating outer-wall normals. -/
+def u (t : ℝ) : Point := (Real.cos t, Real.sin t)
+/-- The vertical unit vector rotated counterclockwise through angle `t`. -/
+def v (t : ℝ) : Point := (-Real.sin t, Real.cos t)
+
+/-- The lower fan in the manuscript normalisation. -/
+def capFan : Set Point := {q | 0 ≤ q.2}
+
+/-- First rotating supporting half-plane. -/
+def supportHalfU (p : Romik.Params) (t : ℝ) : Set Point :=
+  {q | dot q (u t) ≤ dot (Romik.path p t) (u t) + 1}
+
+/-- Second rotating supporting half-plane. -/
+def supportHalfV (p : Romik.Params) (t : ℝ) : Set Point :=
+  {q | dot q (v t) ≤ dot (Romik.path p t) (v t) + 1}
+
+/-- The cap `K₀` reconstructed from the five-phase path. -/
+def Romik.K0 (p : Romik.Params) : Set Point :=
+  {q | 0 ≤ q.2 ∧
+    ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      q ∈ supportHalfU p t ∩ supportHalfV p t}
+
+/-- Open inner quadrant of the supporting hallway at physical angle `t`. -/
+def Romik.innerQuadrantAt (p : Romik.Params) (t : ℝ) : Set Point :=
+  let x := Romik.path p t
+  {q | dot (q.1 - x.1, q.2 - x.2) (u t) < 0 ∧
+       dot (q.1 - x.1, q.2 - x.2) (v t) < 0}
+
+/-- Union of all forbidden inner quadrants at interior rotation times. -/
+def Romik.innerUnion (p : Romik.Params) : Set Point :=
+  {q | ∃ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2), q ∈ Romik.innerQuadrantAt p t}
+
+/-- The niche removed from the cap. -/
+def Romik.niche (p : Romik.Params) : Set Point :=
+  capFan ∩ Romik.innerUnion p
+
+/-- The fixed Gerver candidate `G = K₀ \ N(K₀)`. -/
+def Romik.sofa (p : Romik.Params) : Set Point :=
+  Romik.K0 p \ Romik.niche p
+
+/-- Physical supporting hallway at normalized time `s`. -/
+def Romik.hallwayAt (p : Romik.Params) (s : ℝ) : Set Point :=
+  supportingHallway (Romik.frame p s)
+
+/-! ## Closedness facts independent of the numerical certificate -/
+
+private theorem continuous_dot_fixed (a : Point) :
+    Continuous (fun q : Point => dot q a) := by
+  unfold dot
+  fun_prop
+
+private theorem continuous_shifted_dot (x a : Point) :
+    Continuous (fun q : Point => dot (q.1 - x.1, q.2 - x.2) a) := by
+  unfold dot
+  fun_prop
+
+/-- A fixed supporting half-plane is closed. -/
+theorem isClosed_supportHalfU (p : Romik.Params) (t : ℝ) :
+    IsClosed (supportHalfU p t) := by
+  change IsClosed ((fun q : Point => dot q (u t)) ⁻¹'
+    Set.Iic (dot (Romik.path p t) (u t) + 1))
+  exact isClosed_Iic.preimage (continuous_dot_fixed (u t))
+
+/-- The other fixed supporting half-plane is closed. -/
+theorem isClosed_supportHalfV (p : Romik.Params) (t : ℝ) :
+    IsClosed (supportHalfV p t) := by
+  change IsClosed ((fun q : Point => dot q (v t)) ⁻¹'
+    Set.Iic (dot (Romik.path p t) (v t) + 1))
+  exact isClosed_Iic.preimage (continuous_dot_fixed (v t))
+
+/-- The fan constraint is closed. -/
+theorem isClosed_capFan : IsClosed capFan := by
+  change IsClosed ((fun q : Point => q.2) ⁻¹' Set.Ici 0)
+  exact isClosed_Ici.preimage continuous_snd
+
+/-- The bounded-quantifier definition of `K₀` as an explicit intersection. -/
+theorem Romik.K0_eq_inter_iInter (p : Romik.Params) :
+    Romik.K0 p = capFan ∩
+      ⋂ t : {t : ℝ // t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)},
+        (supportHalfU p t.1 ∩ supportHalfV p t.1) := by
+  ext q
+  constructor
+  · rintro ⟨hbase, hall⟩
+    refine ⟨hbase, Set.mem_iInter.mpr ?_⟩
+    intro t
+    exact hall t.1 t.2
+  · rintro ⟨hbase, hall⟩
+    refine ⟨hbase, ?_⟩
+    intro t ht
+    exact Set.mem_iInter.mp hall ⟨t, ht⟩
+
+/-- `K₀` is closed, before any support-maximisation or compactness argument. -/
+theorem Romik.isClosed_K0 (p : Romik.Params) : IsClosed (Romik.K0 p) := by
+  rw [Romik.K0_eq_inter_iInter]
+  exact isClosed_capFan.inter <|
+    isClosed_iInter fun t =>
+      (isClosed_supportHalfU p t.1).inter (isClosed_supportHalfV p t.1)
+
+/-- Every instantaneous inner quadrant is open. -/
+theorem Romik.isOpen_innerQuadrantAt (p : Romik.Params) (t : ℝ) :
+    IsOpen (Romik.innerQuadrantAt p t) := by
+  let x := Romik.path p t
+  have hu : IsOpen
+      ((fun q : Point => dot (q.1 - x.1, q.2 - x.2) (u t)) ⁻¹' Set.Iio 0) :=
+    isOpen_Iio.preimage (continuous_shifted_dot x (u t))
+  have hv : IsOpen
+      ((fun q : Point => dot (q.1 - x.1, q.2 - x.2) (v t)) ⁻¹' Set.Iio 0) :=
+    isOpen_Iio.preimage (continuous_shifted_dot x (v t))
+  change IsOpen
+    (((fun q : Point => dot (q.1 - x.1, q.2 - x.2) (u t)) ⁻¹' Set.Iio 0) ∩
+     ((fun q : Point => dot (q.1 - x.1, q.2 - x.2) (v t)) ⁻¹' Set.Iio 0))
+  exact hu.inter hv
+
+/-- The union of all interior-time inner quadrants is open. -/
+theorem Romik.isOpen_innerUnion (p : Romik.Params) :
+    IsOpen (Romik.innerUnion p) := by
+  have hrepr : Romik.innerUnion p =
+      ⋃ t : {t : ℝ // t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2)},
+        Romik.innerQuadrantAt p t.1 := by
+    ext q
+    constructor
+    · rintro ⟨t, ht, hq⟩
+      exact Set.mem_iUnion.mpr ⟨⟨t, ht⟩, hq⟩
+    · intro hq
+      rcases Set.mem_iUnion.mp hq with ⟨t, hq⟩
+      exact ⟨t.1, t.2, hq⟩
+  rw [hrepr]
+  exact isOpen_iUnion fun t => Romik.isOpen_innerQuadrantAt p t.1
+
+/-- The cap lies in its fan by definition. -/
+theorem Romik.K0_subset_capFan (p : Romik.Params) : Romik.K0 p ⊆ capFan := by
+  intro q hq
+  exact hq.1
+
+/-- Since `K₀ ⊆ capFan`, removing the niche is the same as removing the open
+union of forbidden quadrants. -/
+theorem Romik.sofa_eq_K0_diff_innerUnion (p : Romik.Params) :
+    Romik.sofa p = Romik.K0 p \ Romik.innerUnion p := by
+  ext q
+  constructor
+  · rintro ⟨hqK, hqN⟩
+    refine ⟨hqK, ?_⟩
+    intro hqU
+    exact hqN ⟨Romik.K0_subset_capFan p hqK, hqU⟩
+  · rintro ⟨hqK, hqU⟩
+    refine ⟨hqK, ?_⟩
+    rintro ⟨_hqFan, hqInner⟩
+    exact hqU hqInner
+
+/-- The concrete fixed sofa candidate is closed for every parameter vector. -/
+theorem Romik.isClosed_sofa (p : Romik.Params) : IsClosed (Romik.sofa p) := by
+  rw [Romik.sofa_eq_K0_diff_innerUnion]
+  simpa [Set.sdiff_eq] using
+    (Romik.isClosed_K0 p).inter (Romik.isOpen_innerUnion p).isClosed_compl
+
+/-! ## Algebraic hallway characterisation and endpoint assembly -/
+
+@[simp] theorem Romik.mem_supportHalfU (p : Romik.Params) (t : ℝ) (q : Point) :
+    q ∈ supportHalfU p t ↔
+      dot q (u t) ≤ dot (Romik.path p t) (u t) + 1 := Iff.rfl
+
+@[simp] theorem Romik.mem_supportHalfV (p : Romik.Params) (t : ℝ) (q : Point) :
+    q ∈ supportHalfV p t ↔
+      dot q (v t) ≤ dot (Romik.path p t) (v t) + 1 := Iff.rfl
+
+@[simp] theorem Romik.mem_K0 (p : Romik.Params) (q : Point) :
+    q ∈ Romik.K0 p ↔
+      0 ≤ q.2 ∧
+      ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+        q ∈ supportHalfU p t ∩ supportHalfV p t := Iff.rfl
+
+@[simp] theorem Romik.mem_innerUnion (p : Romik.Params) (q : Point) :
+    q ∈ Romik.innerUnion p ↔
+      ∃ t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2),
+        q ∈ Romik.innerQuadrantAt p t := Iff.rfl
+
+@[simp] theorem Romik.mem_niche (p : Romik.Params) (q : Point) :
+    q ∈ Romik.niche p ↔ q ∈ capFan ∧ q ∈ Romik.innerUnion p := Iff.rfl
+
+@[simp] theorem Romik.mem_sofa (p : Romik.Params) (q : Point) :
+    q ∈ Romik.sofa p ↔ q ∈ Romik.K0 p ∧ q ∉ Romik.niche p := Iff.rfl
+
+/-- The inverse supporting frame has exactly the two signed wall coordinates
+used in the manuscript. -/
+theorem Romik.frame_inv_act_formula (p : Romik.Params) (s : ℝ) (q : Point) :
+    (Romik.frame p s).inv.act q =
+      (dot (q.1 - (Romik.path p (Romik.angle s)).1,
+            q.2 - (Romik.path p (Romik.angle s)).2) (u (Romik.angle s)),
+       dot (q.1 - (Romik.path p (Romik.angle s)).1,
+            q.2 - (Romik.path p (Romik.angle s)).2) (v (Romik.angle s))) := by
+  apply Prod.ext <;>
+    simp [Romik.frame, SE2.inv, SE2.act, dot, u, v] <;> ring
+
+/-- A point is in the supporting hallway iff its two wall coordinates lie in
+`Q⁺` but not simultaneously in the open inner quadrant `Q⁻`. -/
+theorem Romik.mem_hallwayAt_iff_wall_coordinates
+    (p : Romik.Params) (s : ℝ) (q : Point) :
+    q ∈ Romik.hallwayAt p s ↔
+      let t := Romik.angle s
+      let x := Romik.path p t
+      (dot (q.1 - x.1, q.2 - x.2) (u t) ≤ 1 ∧
+       dot (q.1 - x.1, q.2 - x.2) (v t) ≤ 1) ∧
+      ¬ (dot (q.1 - x.1, q.2 - x.2) (u t) < 0 ∧
+         dot (q.1 - x.1, q.2 - x.2) (v t) < 0) := by
+  rw [Romik.hallwayAt, mem_supportingHallway_iff]
+  rw [Romik.frame_inv_act_formula]
+  rw [hallway_eq_outer_diff_inner]
+  simp [outerQuarter, innerQuarter]
+
+/-- A sofa point cannot lie in an instantaneous forbidden quadrant at an
+interior physical time. -/
+theorem Romik.not_mem_innerQuadrantAt_of_mem_sofa
+    (p : Romik.Params) {q : Point} (hq : q ∈ Romik.sofa p)
+    {t : ℝ} (ht : t ∈ Set.Ioo (0 : ℝ) (Real.pi / 2)) :
+    q ∉ Romik.innerQuadrantAt p t := by
+  intro hinner
+  exact hq.2 ⟨Romik.K0_subset_capFan p hq.1, ⟨t, ht, hinner⟩⟩
+
+/-- The lower-fan constraint excludes the initial inner quadrant once the
+physical path starts at the origin. -/
+theorem Romik.not_mem_initial_innerQuadrant
+    (p : Romik.Params) (hzero : Romik.path p 0 = (0, 0))
+    {q : Point} (hq : q ∈ Romik.sofa p) :
+    q ∉ Romik.innerQuadrantAt p 0 := by
+  intro hinner
+  have hbase : 0 ≤ q.2 := hq.1.1
+  have hneg : q.2 < 0 := by
+    simpa [Romik.innerQuadrantAt, dot, u, v, hzero] using hinner.2
+  exact (not_lt_of_ge hbase) hneg
+
+/-- The lower-fan constraint excludes the final inner quadrant once the final
+path point has second coordinate zero. -/
+theorem Romik.not_mem_final_innerQuadrant
+    (p : Romik.Params)
+    (hend : (Romik.path p (Real.pi / 2)).2 = 0)
+    {q : Point} (hq : q ∈ Romik.sofa p) :
+    q ∉ Romik.innerQuadrantAt p (Real.pi / 2) := by
+  intro hinner
+  have hbase : 0 ≤ q.2 := hq.1.1
+  have hneg : q.2 < 0 := by
+    simpa [Romik.innerQuadrantAt, dot, u, v, hend] using hinner.1
+  exact (not_lt_of_ge hbase) hneg
+
+/-- The concrete cap-minus-niche set lies in every supporting hallway.  The
+only endpoint input is the pair of path normalisations used in the paper. -/
+theorem Romik.sofa_subset_hallwayAt
+    (p : Romik.Params)
+    (hzero : Romik.path p 0 = (0, 0))
+    (hend : (Romik.path p (Real.pi / 2)).2 = 0) :
+    ∀ s ∈ Set.Icc (0 : ℝ) 1,
+      Romik.sofa p ⊆ Romik.hallwayAt p s := by
+  intro s hs q hq
+  rw [Romik.mem_hallwayAt_iff_wall_coordinates]
+  dsimp only
+  have hangle : Romik.angle s ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := by
+    constructor
+    · exact mul_nonneg hs.1 (by positivity)
+    · have hm := mul_le_mul_of_nonneg_right hs.2
+          (show 0 ≤ Real.pi / 2 by positivity)
+      simpa [Romik.angle] using hm
+  have hsupports := hq.1.2 (Romik.angle s) hangle
+  constructor
+  · constructor
+    · have hu := hsupports.1
+      change dot q (u (Romik.angle s)) ≤
+        dot (Romik.path p (Romik.angle s)) (u (Romik.angle s)) + 1 at hu
+      dsimp [dot] at hu ⊢
+      linarith
+    · have hv := hsupports.2
+      change dot q (v (Romik.angle s)) ≤
+        dot (Romik.path p (Romik.angle s)) (v (Romik.angle s)) + 1 at hv
+      dsimp [dot] at hv ⊢
+      linarith
+  · change q ∉ Romik.innerQuadrantAt p (Romik.angle s)
+    by_cases hs0 : s = 0
+    · subst s
+      simpa using Romik.not_mem_initial_innerQuadrant p hzero hq
+    by_cases hs1 : s = 1
+    · subst s
+      simpa using Romik.not_mem_final_innerQuadrant p hend hq
+    · have hsIoo : s ∈ Set.Ioo (0 : ℝ) 1 := by
+        exact ⟨lt_of_le_of_ne hs.1 (Ne.symm hs0),
+          lt_of_le_of_ne hs.2 hs1⟩
+      have hangleIoo : Romik.angle s ∈
+          Set.Ioo (0 : ℝ) (Real.pi / 2) := by
+        constructor
+        · exact mul_pos hsIoo.1 (by positivity)
+        · have hm := mul_lt_mul_of_pos_right hsIoo.2
+              (show 0 < Real.pi / 2 by positivity)
+          simpa [Romik.angle] using hm
+      exact Romik.not_mem_innerQuadrantAt_of_mem_sofa p hq hangleIoo
+
+/-- The inverse frame at time zero places the concrete sofa in the horizontal
+arm. -/
+theorem Romik.initial_arm_of_path_zero
+    (p : Romik.Params) (hzero : Romik.path p 0 = (0, 0)) :
+    ((Romik.frame p 0).inv.act '' Romik.sofa p) ⊆ horizontalArm := by
+  rintro y ⟨q, hq, rfl⟩
+  have htime : (0 : ℝ) ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := by
+    constructor <;> positivity
+  have hs := hq.1.2 0 htime
+  have hx : q.1 ≤ 1 := by
+    have hu := hs.1
+    simpa [supportHalfU, dot, u, hzero] using hu
+  have hy0 : 0 ≤ q.2 := hq.1.1
+  have hy1 : q.2 ≤ 1 := by
+    have hv := hs.2
+    simpa [supportHalfV, dot, v, hzero] using hv
+  rw [Romik.frame_inv_act_formula]
+  simpa [Romik.angle, hzero, dot, u, v, horizontalArm] using
+    (show q.1 ≤ 1 ∧ 0 ≤ q.2 ∧ q.2 ≤ 1 from ⟨hx, hy0, hy1⟩)
+
+/-- The inverse frame at time one places the concrete sofa in the vertical
+arm. -/
+theorem Romik.final_arm_of_path_end_y_zero
+    (p : Romik.Params)
+    (hend : (Romik.path p (Real.pi / 2)).2 = 0) :
+    ((Romik.frame p 1).inv.act '' Romik.sofa p) ⊆ verticalArm := by
+  rintro y ⟨q, hq, rfl⟩
+  have htime : Real.pi / 2 ∈ Set.Icc (0 : ℝ) (Real.pi / 2) := by
+    constructor
+    · positivity
+    · rfl
+  have hs := hq.1.2 (Real.pi / 2) htime
+  have hy0 : 0 ≤ q.2 := hq.1.1
+  have hy1 : q.2 ≤ 1 := by
+    have hu := hs.1
+    simpa [supportHalfU, dot, u, hend] using hu
+  have hv := hs.2
+  have hv1 :
+      dot (q.1 - (Romik.path p (Real.pi / 2)).1,
+           q.2 - (Romik.path p (Real.pi / 2)).2) (v (Real.pi / 2)) ≤ 1 := by
+    change dot q (v (Real.pi / 2)) ≤
+      dot (Romik.path p (Real.pi / 2)) (v (Real.pi / 2)) + 1 at hv
+    dsimp [dot] at hv ⊢
+    linarith
+  rw [Romik.frame_inv_act_formula]
+  simpa [Romik.angle, dot, u, v, hend, verticalArm] using
+    (show 0 ≤ q.2 ∧ q.2 ≤ 1 ∧
+      dot (q.1 - (Romik.path p (Real.pi / 2)).1,
+           q.2 - (Romik.path p (Real.pi / 2)).2) (v (Real.pi / 2)) ≤ 1
+      from ⟨hy0, hy1, hv1⟩)
+
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.AlternatingSeries`.
+* `KernelOnly.Coordinates`.
+* `KernelOnly.EndpointSymmetry`.
+* `KernelOnly.Identification`.
+* `KernelOnly.ReplayConsequences`.
+* `KernelOnly.SoundnessInterfaces`.
+* `KernelOnly.TranscendentalSoundness`.
+* `KernelOnly.ADCoreSoundness`.
+* `KernelOnly.ReducedADSoundness`.
+* `KernelOnly.FullADSoundness`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Alternating-series kernel for the executable transcendental layer
+
+This file connects the exact rational partial sums used by `ExactReplay` to
+Mathlib's real power-series theorems.  The Taylor evaluator is intentionally
+used only on arguments in `[0, 1]`; the range-reduction layer proves this
+precondition before calling these results.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.ExactReplay
+
+open scoped BigOperators
+open Filter Finset
+
+/-- Real magnitude of the `n`th sine-series term. -/
+def sinMagnitude (x : ℝ) (n : ℕ) : ℝ :=
+  x ^ (2 * n + 1) / (Nat.factorial (2 * n + 1) : ℝ)
+
+/-- Real magnitude of the `n`th cosine-series term. -/
+def cosMagnitude (x : ℝ) (n : ℕ) : ℝ :=
+  x ^ (2 * n) / (Nat.factorial (2 * n) : ℝ)
+
+/-- Real magnitude of the `n`th arctangent-series term. -/
+def atanMagnitude (x : ℝ) (n : ℕ) : ℝ :=
+  x ^ (2 * n + 1) / ((2 * n + 1 : ℕ) : ℝ)
+
+/-- On `[0,1]`, the unsigned sine Taylor terms decrease. -/
+theorem antitone_sinMagnitude {x : ℝ} (hx0 : 0 ≤ x) (hx1 : x ≤ 1) :
+    Antitone (sinMagnitude x) := by
+  apply antitone_nat_of_succ_le
+  intro n
+  simp only [sinMagnitude]
+  refine div_le_div₀ (pow_nonneg hx0 _) ?_ (by positivity) ?_
+  · exact pow_le_pow_of_le_one hx0 hx1 (by omega)
+  · exact_mod_cast Nat.factorial_le (by omega)
+
+/-- On `[0,1]`, the unsigned cosine Taylor terms decrease. -/
+theorem antitone_cosMagnitude {x : ℝ} (hx0 : 0 ≤ x) (hx1 : x ≤ 1) :
+    Antitone (cosMagnitude x) := by
+  apply antitone_nat_of_succ_le
+  intro n
+  simp only [cosMagnitude]
+  refine div_le_div₀ (pow_nonneg hx0 _) ?_ (by positivity) ?_
+  · exact pow_le_pow_of_le_one hx0 hx1 (by omega)
+  · exact_mod_cast Nat.factorial_le (by omega)
+
+/-- On `[0,1]`, the unsigned arctangent Taylor terms decrease. -/
+theorem antitone_atanMagnitude {x : ℝ} (hx0 : 0 ≤ x) (hx1 : x ≤ 1) :
+    Antitone (atanMagnitude x) := by
+  apply antitone_nat_of_succ_le
+  intro n
+  simp only [atanMagnitude]
+  refine div_le_div₀ (pow_nonneg hx0 _) ?_ (by positivity) ?_
+  · exact pow_le_pow_of_le_one hx0 hx1 (by omega)
+  · exact_mod_cast (show 2 * n + 1 ≤ 2 * (n + 1) + 1 by omega)
+
+/-- Casting the executable sine partial sum to `ℝ` gives the corresponding
+Mathlib finite Taylor sum. -/
+theorem coe_sinPartial (x : ℚ) (terms : ℕ) :
+    ((GerverSofa.ExactReplay.sinPartial x terms : ℚ) : ℝ) =
+      ∑ k ∈ Finset.range terms,
+        (-1 : ℝ) ^ k * (x : ℝ) ^ (2 * k + 1) /
+          (Nat.factorial (2 * k + 1) : ℝ) := by
+  simp only [sinPartial, signedTerm, factorialQ, Rat.cast_sum, neg_one_pow_eq_ite, Nat.even_iff,
+    ite_mul, one_mul, neg_mul]
+  refine Finset.sum_congr rfl ?_
+  intro k hk
+  by_cases h : k % 2 = 0 <;> simp [h, neg_div]
+
+/-- Casting the executable cosine partial sum to `ℝ` gives the corresponding
+Mathlib finite Taylor sum. -/
+theorem coe_cosPartial (x : ℚ) (terms : ℕ) :
+    ((GerverSofa.ExactReplay.cosPartial x terms : ℚ) : ℝ) =
+      ∑ k ∈ Finset.range terms,
+        (-1 : ℝ) ^ k * (x : ℝ) ^ (2 * k) /
+          (Nat.factorial (2 * k) : ℝ) := by
+  simp only [cosPartial, signedTerm, factorialQ, Rat.cast_sum, neg_one_pow_eq_ite, Nat.even_iff,
+    ite_mul, one_mul, neg_mul]
+  refine Finset.sum_congr rfl ?_
+  intro k hk
+  by_cases h : k % 2 = 0 <;> simp [h, neg_div]
+
+/-- Casting the executable arctangent partial sum to `ℝ` gives the
+corresponding Mathlib finite Taylor sum. -/
+theorem coe_atanPartial (x : ℚ) (terms : ℕ) :
+    ((GerverSofa.ExactReplay.atanPartial x terms : ℚ) : ℝ) =
+      ∑ k ∈ Finset.range terms,
+        (-1 : ℝ) ^ k * (x : ℝ) ^ (2 * k + 1) /
+          ((2 * k + 1 : ℕ) : ℝ) := by
+  simp only [atanPartial, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat, Nat.cast_one, Rat.cast_sum,
+    neg_one_pow_eq_ite, Nat.even_iff, ite_mul, one_mul, neg_mul]
+  refine Finset.sum_congr rfl ?_
+  intro k hk
+  by_cases h : k % 2 = 0 <;> simp [h, neg_div]
+
+/-- The 20-term sine partial sum is a lower bound and the 19-term partial sum
+is an upper bound for every rational argument in `[0,1]`. -/
+theorem sine_between_partials {x : ℚ}
+    (hx0 : (0 : ℚ) ≤ x) (hx1 : x ≤ 1) :
+    ((GerverSofa.ExactReplay.sinPartial x 20 : ℚ) : ℝ) ≤ Real.sin (x : ℝ) ∧
+      Real.sin (x : ℝ) ≤ ((GerverSofa.ExactReplay.sinPartial x 19 : ℚ) : ℝ) := by
+  have hx0r : (0 : ℝ) ≤ (x : ℝ) := by exact_mod_cast hx0
+  have hx1r : (x : ℝ) ≤ 1 := by exact_mod_cast hx1
+  have hanti : Antitone (sinMagnitude (x : ℝ)) :=
+    antitone_sinMagnitude hx0r hx1r
+  have htendRaw := (Real.hasSum_sin (x : ℝ)).tendsto_sum_nat
+  have htendMag :
+      Filter.Tendsto
+        (fun n : ℕ => ∑ i ∈ Finset.range n, (-1 : ℝ) ^ i * sinMagnitude (x : ℝ) i)
+        Filter.atTop (nhds (Real.sin (x : ℝ))) := by
+    simpa only [sinMagnitude, mul_div_assoc] using htendRaw
+  have hlower := Antitone.alternating_series_le_tendsto htendMag hanti 10
+  have hupper := Antitone.tendsto_le_alternating_series htendMag hanti 9
+  have h20 : (2 * 10 : ℕ) = 20 := by norm_num
+  have h19 : (2 * 9 + 1 : ℕ) = 19 := by norm_num
+  constructor
+  · rw [coe_sinPartial]
+    simpa only [sinMagnitude, mul_div_assoc, h20] using hlower
+  · rw [coe_sinPartial]
+    simpa only [sinMagnitude, mul_div_assoc, h19] using hupper
+
+/-- The 20-term cosine partial sum is a lower bound and the 19-term partial
+sum is an upper bound for every rational argument in `[0,1]`. -/
+theorem cosine_between_partials {x : ℚ}
+    (hx0 : (0 : ℚ) ≤ x) (hx1 : x ≤ 1) :
+    ((GerverSofa.ExactReplay.cosPartial x 20 : ℚ) : ℝ) ≤ Real.cos (x : ℝ) ∧
+      Real.cos (x : ℝ) ≤ ((GerverSofa.ExactReplay.cosPartial x 19 : ℚ) : ℝ) := by
+  have hx0r : (0 : ℝ) ≤ (x : ℝ) := by exact_mod_cast hx0
+  have hx1r : (x : ℝ) ≤ 1 := by exact_mod_cast hx1
+  have hanti : Antitone (cosMagnitude (x : ℝ)) :=
+    antitone_cosMagnitude hx0r hx1r
+  have htendRaw := (Real.hasSum_cos (x : ℝ)).tendsto_sum_nat
+  have htendMag :
+      Filter.Tendsto
+        (fun n : ℕ => ∑ i ∈ Finset.range n, (-1 : ℝ) ^ i * cosMagnitude (x : ℝ) i)
+        Filter.atTop (nhds (Real.cos (x : ℝ))) := by
+    simpa only [cosMagnitude, mul_div_assoc] using htendRaw
+  have hlower := Antitone.alternating_series_le_tendsto htendMag hanti 10
+  have hupper := Antitone.tendsto_le_alternating_series htendMag hanti 9
+  have h20 : (2 * 10 : ℕ) = 20 := by norm_num
+  have h19 : (2 * 9 + 1 : ℕ) = 19 := by norm_num
+  constructor
+  · rw [coe_cosPartial]
+    simpa only [cosMagnitude, mul_div_assoc, h20] using hlower
+  · rw [coe_cosPartial]
+    simpa only [cosMagnitude, mul_div_assoc, h19] using hupper
+
+/-- Even/odd arctangent partial sums provide certified lower/upper bounds. -/
+theorem arctan_between_partials {x : ℚ}
+    (hx0 : (0 : ℚ) ≤ x) (hx1 : x < 1) (k : ℕ) :
+    ((GerverSofa.ExactReplay.atanPartial x (2 * k + 2) : ℚ) : ℝ) ≤ Real.arctan (x : ℝ) ∧
+      Real.arctan (x : ℝ) ≤ ((GerverSofa.ExactReplay.atanPartial x (2 * k + 1) : ℚ) : ℝ) := by
+  have hx0r : (0 : ℝ) ≤ (x : ℝ) := by exact_mod_cast hx0
+  have hx1r : (x : ℝ) < 1 := by exact_mod_cast hx1
+  have hx1le : (x : ℝ) ≤ 1 := hx1r.le
+  have habs : ‖(x : ℝ)‖ < 1 := by
+    rw [Real.norm_eq_abs, abs_of_nonneg hx0r]
+    exact hx1r
+  have hanti : Antitone (atanMagnitude (x : ℝ)) :=
+    antitone_atanMagnitude hx0r hx1le
+  have htendRaw := (Real.hasSum_arctan habs).tendsto_sum_nat
+  have htendMag :
+      Filter.Tendsto
+        (fun n : ℕ => ∑ i ∈ Finset.range n, (-1 : ℝ) ^ i * atanMagnitude (x : ℝ) i)
+        Filter.atTop (nhds (Real.arctan (x : ℝ))) := by
+    simpa only [atanMagnitude, mul_div_assoc] using htendRaw
+  have hlower := Antitone.alternating_series_le_tendsto htendMag hanti (k + 1)
+  have hupper := Antitone.tendsto_le_alternating_series htendMag hanti k
+  constructor
+  · rw [coe_atanPartial]
+    have hidx : 2 * (k + 1) = 2 * k + 2 := by omega
+    simpa only [atanMagnitude, mul_div_assoc, hidx] using hlower
+  · rw [coe_atanPartial]
+    simpa only [atanMagnitude, mul_div_assoc] using hupper
+
+end GerverSofa.ExactReplay
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Coordinate equivalences for the certified systems
+
+The executable interval layer works with `Fin n → ℝ`, while the manuscript
+layer uses named parameter records.  These equivalences are the explicit,
+kernel-checked bridge between the two representations.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+
+namespace Reduced
+
+/-- Named reduced parameters as a four-vector in manuscript order. -/
+def coordEquiv : Params ≃ Vec 4 where
+  toFun p := ![p.a, p.b, p.phi, p.theta]
+  invFun x :=
+    { a := x 0
+      b := x 1
+      phi := x 2
+      theta := x 3 }
+  left_inv p := by
+    cases p
+    rfl
+  right_inv x := by
+    funext i
+    fin_cases i <;> rfl
+
+/-- Reduced system expressed in finite-vector coordinates. -/
+def vectorSystem (x : Vec 4) : Vec 4 :=
+  system (coordEquiv.symm x)
+
+/-- The reduced parameter box expressed in finite-vector coordinates. -/
+def vectorBox : Set (Vec 4) :=
+  {x | coordEquiv.symm x ∈ box}
+
+@[simp] theorem mem_vectorBox_iff (x : Vec 4) :
+    x ∈ vectorBox ↔ coordEquiv.symm x ∈ box := Iff.rfl
+
+/-- Transport a finite-vector uniqueness certificate back to named reduced
+parameters. -/
+def uniqueSolutionOfVector
+    (c : CertifiedUniqueZero vectorSystem vectorBox) :
+    CertifiedUniqueSolution Equations box where
+  solution := coordEquiv.symm c.solution
+  solution_mem := by
+    simpa [vectorBox] using c.solution_mem
+  satisfies := by
+    simpa [Equations, vectorSystem] using c.satisfies
+  unique_iff y hy := by
+    constructor
+    · intro hEq
+      have hyVec : coordEquiv y ∈ vectorBox := by
+        simpa [vectorBox] using hy
+      have hzero : vectorSystem (coordEquiv y) = 0 := by
+        simpa [Equations, vectorSystem] using hEq
+      have h := c.unique (coordEquiv y) hyVec hzero
+      apply coordEquiv.injective
+      simpa using h
+    · rintro rfl
+      simpa [Equations, vectorSystem] using c.satisfies
+
+end Reduced
+
+namespace Romik
+
+/-- Named Romik parameters as a 22-vector in verifier order. -/
+def coordEquiv : Params ≃ Vec 22 where
+  toFun p := ![
+    p.k11, p.k12, p.k21, p.k22, p.k31, p.k32, p.k41, p.k42, p.k51, p.k52,
+    p.a1, p.a2, p.b1, p.b2, p.c1, p.c2, p.d1, p.d2, p.e1, p.e2,
+    p.phi, p.theta
+  ]
+  invFun x :=
+    { k11 := x 0
+      k12 := x 1
+      k21 := x 2
+      k22 := x 3
+      k31 := x 4
+      k32 := x 5
+      k41 := x 6
+      k42 := x 7
+      k51 := x 8
+      k52 := x 9
+      a1 := x 10
+      a2 := x 11
+      b1 := x 12
+      b2 := x 13
+      c1 := x 14
+      c2 := x 15
+      d1 := x 16
+      d2 := x 17
+      e1 := x 18
+      e2 := x 19
+      phi := x 20
+      theta := x 21 }
+  left_inv p := by
+    cases p
+    rfl
+  right_inv x := by
+    funext i
+    fin_cases i <;> rfl
+
+/-- Direct Romik system expressed in finite-vector coordinates. -/
+def vectorSystem (x : Vec 22) : Vec 22 :=
+  system (coordEquiv.symm x)
+
+/-- The direct Romik box expressed in finite-vector coordinates. -/
+def vectorBox : Set (Vec 22) :=
+  {x | coordEquiv.symm x ∈ box}
+
+@[simp] theorem mem_vectorBox_iff (x : Vec 22) :
+    x ∈ vectorBox ↔ coordEquiv.symm x ∈ box := Iff.rfl
+
+/-- Transport a finite-vector uniqueness certificate back to named Romik
+parameters. -/
+def uniqueSolutionOfVector
+    (c : CertifiedUniqueZero vectorSystem vectorBox) :
+    CertifiedUniqueSolution Equations box where
+  solution := coordEquiv.symm c.solution
+  solution_mem := by
+    simpa [vectorBox] using c.solution_mem
+  satisfies := by
+    simpa [Equations, vectorSystem] using c.satisfies
+  unique_iff y hy := by
+    constructor
+    · intro hEq
+      have hyVec : coordEquiv y ∈ vectorBox := by
+        simpa [vectorBox] using hy
+      have hzero : vectorSystem (coordEquiv y) = 0 := by
+        simpa [Equations, vectorSystem] using hEq
+      have h := c.unique (coordEquiv y) hyVec hzero
+      apply coordEquiv.injective
+      simpa using h
+    · rintro rfl
+      simpa [Equations, vectorSystem] using c.satisfies
+
+end Romik
+
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Endpoint symmetry derived from the direct 22-dimensional system
+
+The manuscript obtains the terminal condition `x₂(π/2)=0` from the reflection
+symmetry of the five phases.  This file proves the required consequence
+without introducing a symmetry assumption and without using numerical
+approximations.
+
+FIX12 keeps the BATCH11 mathematics and public theorem statements unchanged,
+but factors the formula-level algebra into small coordinate identities.  This
+avoids asking `ring` to normalize the fully unfolded five-phase expressions in
+one large proof term.
+-/
+
+/-! ## Exact parameter consequences of equations 27--34 -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.Romik
+
+/-- Equation 27. -/
+theorem e1_eq_a1_of_equations {p : Params} (heq : Equations p) :
+    p.e1 = p.a1 := by
+  have h0 := congrFun heq (0 : Fin 22)
+  simp [system] at h0
+  linarith
+
+/-- Equation 28. -/
+theorem e2_eq_neg_a2_of_equations {p : Params} (heq : Equations p) :
+    p.e2 = -p.a2 := by
+  have h1 := congrFun heq (1 : Fin 22)
+  simp [system] at h1
+  linarith
+
+/-- Equation 29. -/
+theorem d1_eq_quarterPi_sub_b1_of_equations
+    {p : Params} (heq : Equations p) :
+    p.d1 = Real.pi / 4 - p.b1 := by
+  have h2 := congrFun heq (2 : Fin 22)
+  simp [system] at h2
+  linarith
+
+/-- Equation 30. -/
+theorem d2_eq_b2_add_quarterPi_correction_of_equations
+    {p : Params} (heq : Equations p) :
+    p.d2 = p.b2 + (Real.pi / 4) * (2 * p.b1 - Real.pi / 4) := by
+  have h3 := congrFun heq (3 : Fin 22)
+  simp [system] at h3
+  linarith
+
+/-- Equation 31. -/
+theorem c2_eq_c1_sub_halfPi_of_equations
+    {p : Params} (heq : Equations p) :
+    p.c2 = p.c1 - Real.pi / 2 := by
+  have h4 := congrFun heq (4 : Fin 22)
+  simp [system] at h4
+  linarith
+
+/-- Equation 33. -/
+theorem k12_eq_quarter_of_equations {p : Params} (heq : Equations p) :
+    p.k12 = (1 / 4 : ℝ) := by
+  have h6 := congrFun heq (6 : Fin 22)
+  simp [system] at h6
+  linarith
+
+/-- Equation 34. -/
+theorem a2_eq_neg_quarter_of_equations {p : Params} (heq : Equations p) :
+    p.a2 = -(1 / 4 : ℝ) := by
+  have h7 := congrFun heq (7 : Fin 22)
+  simp [system] at h7
+  linarith
+
+/-- Equations 28 and 34. -/
+theorem e2_eq_quarter_of_equations {p : Params} (heq : Equations p) :
+    p.e2 = (1 / 4 : ℝ) := by
+  rw [e2_eq_neg_a2_of_equations heq, a2_eq_neg_quarter_of_equations heq]
+  norm_num
+
+/-! ## Small definitional coordinate identities
+
+These are deliberately `rfl`: they expose only the second coordinate of one
+phase at a time.  Downstream algebra therefore operates on compact scalar
+expressions rather than on the fully unfolded `Point`/`rot`/`addK` terms. -/
+
+private theorem path1_snd_formula (p : Params) (t : ℝ) :
+    (path1 p t).2 =
+      Real.sin t *
+          (p.a1 * Real.cos t + p.a2 * Real.sin t - 1) +
+        Real.cos t *
+          (-p.a2 * Real.cos t + p.a1 * Real.sin t - 1 / 2) +
+        p.k12 := by
+  rfl
+
+private theorem path2_snd_formula (p : Params) (t : ℝ) :
+    (path2 p t).2 =
+      Real.sin t *
+          (-(1 / 4 : ℝ) * t * t + p.b1 * t + p.b2) +
+        Real.cos t *
+          ((1 / 2 : ℝ) * t - p.b1 - 1) +
+        p.k22 := by
+  rfl
+
+private theorem path3_snd_formula (p : Params) (t : ℝ) :
+    (path3 p t).2 =
+      Real.sin t * (p.c1 - t) +
+        Real.cos t * (p.c2 + t) +
+        p.k32 := by
+  rfl
+
+private theorem path4_snd_formula (p : Params) (t : ℝ) :
+    (path4 p t).2 =
+      Real.sin t *
+          (-(1 / 2 : ℝ) * t + p.d1 - 1) +
+        Real.cos t *
+          (-(1 / 4 : ℝ) * t * t + p.d1 * t + p.d2) +
+        p.k42 := by
+  rfl
+
+private theorem path5_snd_formula (p : Params) (t : ℝ) :
+    (path5 p t).2 =
+      Real.sin t *
+          (p.e1 * Real.cos t + p.e2 * Real.sin t - 1 / 2) +
+        Real.cos t *
+          (-p.e2 * Real.cos t + p.e1 * Real.sin t - 1) +
+        p.k52 := by
+  rfl
+
+/-! ## Formula-level reflection identities -/
+
+/-- The vertical phase-5 formula at reflected time differs from phase 1 only
+by its vertical translation constant. -/
+theorem phase15_vertical_reflection_of_equations
+    {p : Params} (heq : Equations p) (t : ℝ) :
+    (path5 p (Real.pi / 2 - t)).2 - p.k52 =
+      (path1 p t).2 - p.k12 := by
+  rw [path5_snd_formula, path1_snd_formula]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  rw [e1_eq_a1_of_equations heq, e2_eq_neg_a2_of_equations heq]
+  ring
+
+/-- The vertical phase-4 formula at reflected time differs from phase 2 only
+by its vertical translation constant. -/
+theorem phase24_vertical_reflection_of_equations
+    {p : Params} (heq : Equations p) (t : ℝ) :
+    (path4 p (Real.pi / 2 - t)).2 - p.k42 =
+      (path2 p t).2 - p.k22 := by
+  rw [path4_snd_formula, path2_snd_formula]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  rw [d1_eq_quarterPi_sub_b1_of_equations heq,
+    d2_eq_b2_add_quarterPi_correction_of_equations heq]
+  have hlinear :
+      -(1 / 2 : ℝ) * (Real.pi / 2 - t) +
+          (Real.pi / 4 - p.b1) - 1 =
+        (1 / 2 : ℝ) * t - p.b1 - 1 := by
+    ring
+  have hquadratic :
+      -(1 / 4 : ℝ) * (Real.pi / 2 - t) * (Real.pi / 2 - t) +
+          (Real.pi / 4 - p.b1) * (Real.pi / 2 - t) +
+          (p.b2 + (Real.pi / 4) * (2 * p.b1 - Real.pi / 4)) =
+        -(1 / 4 : ℝ) * t * t + p.b1 * t + p.b2 := by
+    ring
+  rw [hlinear, hquadratic]
+  ring
+
+/-- The middle phase has exact vertical reflection symmetry. -/
+theorem phase3_vertical_reflection_of_equations
+    {p : Params} (heq : Equations p) (t : ℝ) :
+    (path3 p (Real.pi / 2 - t)).2 = (path3 p t).2 := by
+  rw [path3_snd_formula, path3_snd_formula]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  have hc2 := c2_eq_c1_sub_halfPi_of_equations heq
+  have hleft :
+      p.c1 - (Real.pi / 2 - t) = p.c2 + t := by
+    rw [hc2]
+    ring
+  have hright :
+      p.c2 + (Real.pi / 2 - t) = p.c1 - t := by
+    rw [hc2]
+    ring
+  rw [hleft, hright]
+  ring
+
+/-! ## Translation constants forced by matching -/
+
+/-- Matching at `θ` and `π/2-θ`, together with middle-phase reflection,
+forces the phase-2 and phase-4 vertical translations to coincide. -/
+theorem k42_eq_k22_of_equations {p : Params} (heq : Equations p) :
+    p.k42 = p.k22 := by
+  have h23 : (path2 p p.theta).2 = (path3 p p.theta).2 :=
+    congrArg Prod.snd (match_path23_of_equations heq)
+  have h34 : (path3 p (Real.pi / 2 - p.theta)).2 =
+      (path4 p (Real.pi / 2 - p.theta)).2 :=
+    congrArg Prod.snd (match_path34_of_equations heq)
+  have h24 := phase24_vertical_reflection_of_equations heq p.theta
+  have h33 := phase3_vertical_reflection_of_equations heq p.theta
+  rw [← h34, h33, ← h23] at h24
+  linarith
+
+/-- Matching at `φ` and `π/2-φ` then forces the phase-1 and phase-5 vertical
+translations to coincide. -/
+theorem k52_eq_k12_of_equations {p : Params} (heq : Equations p) :
+    p.k52 = p.k12 := by
+  have h12 : (path1 p p.phi).2 = (path2 p p.phi).2 :=
+    congrArg Prod.snd (match_path12_of_equations heq)
+  have h45 : (path4 p (Real.pi / 2 - p.phi)).2 =
+      (path5 p (Real.pi / 2 - p.phi)).2 :=
+    congrArg Prod.snd (match_path45_of_equations heq)
+  have h24 := phase24_vertical_reflection_of_equations heq p.phi
+  have hk := k42_eq_k22_of_equations heq
+  rw [hk] at h24
+  have hy42 :
+      (path4 p (Real.pi / 2 - p.phi)).2 = (path2 p p.phi).2 := by
+    linarith
+  have hy51 :
+      (path5 p (Real.pi / 2 - p.phi)).2 = (path1 p p.phi).2 := by
+    calc
+      (path5 p (Real.pi / 2 - p.phi)).2 =
+          (path4 p (Real.pi / 2 - p.phi)).2 := by
+            exact h45.symm
+      _ = (path2 p p.phi).2 := hy42
+      _ = (path1 p p.phi).2 := h12.symm
+  have h15 := phase15_vertical_reflection_of_equations heq p.phi
+  rw [hy51] at h15
+  linarith
+
+/-- Equations 27--41 force the previously dependent coefficient `k₅₂=1/4`.
+It is not an independent hypothesis of the final certificate. -/
+theorem k52_eq_quarter_of_equations {p : Params} (heq : Equations p) :
+    p.k52 = (1 / 4 : ℝ) := by
+  rw [k52_eq_k12_of_equations heq, k12_eq_quarter_of_equations heq]
+
+/-! ## Terminal condition -/
+
+/-- The explicit fifth phase ends at vertical coordinate zero. -/
+theorem path5_end_y_zero_of_equations {p : Params} (heq : Equations p) :
+    (path5 p (Real.pi / 2)).2 = 0 := by
+  rw [path5_snd_formula]
+  simp [e2_eq_quarter_of_equations heq,
+    k52_eq_quarter_of_equations heq]; norm_num
+
+/-- The certified direct-system box places `π/2` strictly after the fourth
+switch, so the literal nested-`if` path uses phase 5 at the endpoint. -/
+theorem path_halfPi_eq_path5_of_mem_box {p : Params} (hp : p ∈ box) :
+    path p (Real.pi / 2) = path5 p (Real.pi / 2) := by
+  have hphi : 0 < p.phi := phi_pos_of_mem_box hp
+  have hord : SwitchOrder p := switchOrder_of_mem_box hp
+  have hthetaPos : 0 < p.theta := lt_of_lt_of_le hphi hord.phi_le_theta
+  have hthetaLt : p.theta < Real.pi / 2 := by
+    linarith [hord.theta_le_eta]
+  have hphiLt : p.phi < Real.pi / 2 :=
+    lt_of_le_of_lt hord.phi_le_theta hthetaLt
+  have hetaLt : Real.pi / 2 - p.theta < Real.pi / 2 := by
+    linarith
+  have htauLt : Real.pi / 2 - p.phi < Real.pi / 2 := by
+    linarith
+  simp [path, not_le.mpr hphiLt, not_le.mpr hthetaLt,
+    not_le.mpr hetaLt, not_le.mpr htauLt]
+
+/-- The terminal vertical normalisation is a theorem of the concrete box and
+22 equations.  No reflection hypothesis and no certificate field remain. -/
+theorem path_end_y_zero_of_mem_box_and_equations
+    {p : Params} (hp : p ∈ box) (heq : Equations p) :
+    (path p (Real.pi / 2)).2 = 0 := by
+  rw [path_halfPi_eq_path5_of_mem_box hp]
+  exact path5_end_y_zero_of_equations heq
+
+end GerverSofa.Romik
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Identification with Romik's hallway-intersection reconstruction
+
+This is the set-theoretic part of Proposition `prop:gerver`.  It uses only the
+literal definitions and the two endpoint normalisations; the eighteen-piece
+boundary statement remains a separate field of `FullArticleCertificate`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.Romik
+
+/-- Romik's fixed-frame reconstruction: initial arm, every supporting hallway,
+and the final transported vertical arm. -/
+def reconstructedSet (p : Params) : Set Point :=
+  {q | q ∈ horizontalArm ∧
+    (∀ s ∈ Set.Icc (0 : ℝ) 1, q ∈ hallwayAt p s) ∧
+    q ∈ (frame p 1).act '' verticalArm}
+
+/-- Convert a physical angle in `[0,π/2]` to normalized time. -/
+def normalizedTime (t : ℝ) : ℝ := t / (Real.pi / 2)
+
+@[simp] theorem angle_normalizedTime (t : ℝ) :
+    angle (normalizedTime t) = t := by
+  have hT : Real.pi / 2 ≠ 0 := by positivity
+  simp [angle, normalizedTime, hT]
+
+theorem normalizedTime_mem_unit {t : ℝ}
+    (ht : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2)) :
+    normalizedTime t ∈ Set.Icc (0 : ℝ) 1 := by
+  have hT : 0 < Real.pi / 2 := by positivity
+  constructor
+  · exact div_nonneg ht.1 (le_of_lt hT)
+  · exact (div_le_iff₀ hT).2 (by simpa using ht.2)
+
+/-- Every point of the cap-minus-niche set lies in Romik's hallway
+intersection reconstruction. -/
+theorem sofa_subset_reconstructedSet
+    (p : Params)
+    (hzero : path p 0 = (0, 0))
+    (hend : (path p (Real.pi / 2)).2 = 0) :
+    sofa p ⊆ reconstructedSet p := by
+  intro q hq
+  refine ⟨?_, ?_, ?_⟩
+  · have hi : (frame p 0).inv.act q ∈ horizontalArm :=
+      initial_arm_of_path_zero p hzero ⟨q, hq, rfl⟩
+    simpa [frame, angle, hzero, SE2.inv, SE2.act] using hi
+  · intro s hs
+    exact sofa_subset_hallwayAt p hzero hend s hs hq
+  · have hf : (frame p 1).inv.act q ∈ verticalArm :=
+      final_arm_of_path_end_y_zero p hend ⟨q, hq, rfl⟩
+    exact ⟨(frame p 1).inv.act q, hf, (frame p 1).act_inv_act q⟩
+
+/-- Membership in every physical supporting hallway gives all outer support
+inequalities in the cap definition. -/
+theorem mem_K0_of_mem_all_hallways
+    (p : Params) {q : Point}
+    (hbase : 0 ≤ q.2)
+    (hall : ∀ s ∈ Set.Icc (0 : ℝ) 1, q ∈ hallwayAt p s) :
+    q ∈ K0 p := by
+  refine ⟨hbase, ?_⟩
+  intro t ht
+  let s := normalizedTime t
+  have hs : s ∈ Set.Icc (0 : ℝ) 1 := normalizedTime_mem_unit ht
+  have hhall := hall s hs
+  have hwall :
+      (dot (q.1 - (path p t).1, q.2 - (path p t).2) (u t) ≤ 1 ∧
+       dot (q.1 - (path p t).1, q.2 - (path p t).2) (v t) ≤ 1) ∧
+      ¬ (dot (q.1 - (path p t).1, q.2 - (path p t).2) (u t) < 0 ∧
+         dot (q.1 - (path p t).1, q.2 - (path p t).2) (v t) < 0) := by
+    rw [mem_hallwayAt_iff_wall_coordinates] at hhall
+    simpa [s] using hhall
+  constructor
+  · change dot q (u t) ≤ dot (path p t) (u t) + 1
+    dsimp [dot] at hwall ⊢
+    linarith [hwall.1.1]
+  · change dot q (v t) ≤ dot (path p t) (v t) + 1
+    dsimp [dot] at hwall ⊢
+    linarith [hwall.1.2]
+
+/-- Membership in every physical hallway excludes every open interior-time
+inner quadrant. -/
+theorem not_mem_innerUnion_of_mem_all_hallways
+    (p : Params) {q : Point}
+    (hall : ∀ s ∈ Set.Icc (0 : ℝ) 1, q ∈ hallwayAt p s) :
+    q ∉ innerUnion p := by
+  rintro ⟨t, ht, hinner⟩
+  let s := normalizedTime t
+  have htIcc : t ∈ Set.Icc (0 : ℝ) (Real.pi / 2) :=
+    ⟨le_of_lt ht.1, le_of_lt ht.2⟩
+  have hs : s ∈ Set.Icc (0 : ℝ) 1 := normalizedTime_mem_unit htIcc
+  have hhall := hall s hs
+  rw [mem_hallwayAt_iff_wall_coordinates] at hhall
+  have hwall :
+      ¬ (dot (q.1 - (path p t).1, q.2 - (path p t).2) (u t) < 0 ∧
+         dot (q.1 - (path p t).1, q.2 - (path p t).2) (v t) < 0) := by
+    simpa [s] using hhall.2
+  exact hwall (by simpa [innerQuadrantAt] using hinner)
+
+/-- Conversely, Romik's hallway intersection lies in the concrete
+cap-minus-niche set. -/
+theorem reconstructedSet_subset_sofa (p : Params) :
+    reconstructedSet p ⊆ sofa p := by
+  intro q hq
+  have hbase : 0 ≤ q.2 := hq.1.2.1
+  have hK : q ∈ K0 p := mem_K0_of_mem_all_hallways p hbase hq.2.1
+  have hU : q ∉ innerUnion p :=
+    not_mem_innerUnion_of_mem_all_hallways p hq.2.1
+  refine ⟨hK, ?_⟩
+  rintro ⟨_hfan, hinner⟩
+  exact hU hinner
+
+/-- Set-theoretic identification `G = Sₓ`, conditional only on the endpoint
+normalisations already isolated by the main certificate. -/
+theorem sofa_eq_reconstructedSet
+    (p : Params)
+    (hzero : path p 0 = (0, 0))
+    (hend : (path p (Real.pi / 2)).2 = 0) :
+    sofa p = reconstructedSet p := by
+  apply Set.Subset.antisymm
+  · exact sofa_subset_reconstructedSet p hzero hend
+  · exact reconstructedSet_subset_sofa p
+
+end GerverSofa.Romik
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Named consequences of the frozen kernel-only certificate
+
+These handles intentionally reason about the proof-carrying rational data, not
+about re-running the expensive Krawczyk/grid search inside kernel reduction.
+The latter remains available under `ExactReplay.executable*` for independent
+diagnostic comparison.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.CertificateManifest
+theorem machin_inside_declared :
+    RatInterval.strictInsideB machinPi declaredPi = true := by
+  simpa [piCheck] using piCheck_eq_true
+
+end GerverSofa.CertificateManifest
+
+end
+
+end
+
+section
+
+/-!
+# Semantic interfaces for the executable interval certificate
+
+These definitions state, without hiding any mathematical assumption, the
+bridges that turn the frozen rational replay into facts about `Real.sin`,
+`Real.cos`, the two real systems, and their Jacobians.  Concrete proof terms
+for these interfaces are the remaining analytic part of the end-to-end
+certificate; no axiom is declared here.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+
+open RatInterval
+
+/-- A rational interval list encloses a finite real vector coordinatewise. -/
+def EnclosesVec {n : Nat} (box : List RatInterval) (x : Vec n) : Prop :=
+  box.length = n ∧
+    ∀ i : Fin n, Contains (box.getD i.1 (point 0)) (x i)
+
+/-- A one-dimensional real derivative certificate in the classical
+difference-quotient form.  This is the exact real specialization of the
+right-hand side of Mathlib's `hasDerivAt_iff_tendsto_slope_zero`: it states
+that `(f (x+t)-f x)/t` tends to `f'` as `t → 0`, `t ≠ 0`.
+
+Unlike storing raw `HasDerivAt`/`DifferentiableAt`, this proposition contains
+no hidden `AddCommGroup`/`Module` instance path for the codomain `ℝ`; this
+removes the instance diamond exposed by Lean 4.33 while retaining the full
+mathematical meaning of an actual derivative. -/
+def RealDerivativeAt (f : ℝ → ℝ) (f' x : ℝ) : Prop :=
+  Filter.Tendsto
+    (fun t : ℝ => t⁻¹ * (f (x + t) - f x))
+    (nhdsWithin 0 (({0} : Set ℝ)ᶜ))
+    (nhds f')
+
+/-- Exact analytic correctness required from the executable trigonometric
+layer.  The domain is the physical range used by the Gerver certificate. -/
+structure TranscendentalSoundness : Prop where
+  pi_mem : Contains ExactReplay.declaredPiInterval Real.pi
+  sine_mem : ∀ (z : RatInterval) (x : ℝ),
+    Contains z x → 0 ≤ x → x ≤ Real.pi / 2 →
+      Contains (ExactReplay.sineInterval z) (Real.sin x)
+  cosine_mem : ∀ (z : RatInterval) (x : ℝ),
+    Contains z x → 0 ≤ x → x ≤ Real.pi / 2 →
+      Contains (ExactReplay.cosineInterval z) (Real.cos x)
+
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Soundness of the exact transcendental interval evaluator
+
+This file proves the analytic trust bridge omitted by the executable replay:
+
+* the alternating rational arctangent sums enclose the two Machin terms;
+* the computed Machin interval encloses `Real.pi` and lies in the declared
+  interval;
+* fixed-decimal rounding is outward;
+* the small-argument Taylor intervals enclose `Real.sin` and `Real.cos`;
+* complementary-angle reduction is sound on `[0, π/2]`;
+* externally over-wide intervals fail closed to `[-1,1]` rather than silently
+  violating the Taylor precondition.
+
+No project axiom and no floating-point literal occurs in this file.
+-/
+
+/-! ## Partial-sum interval consequences -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.ExactReplay
+
+open RatInterval
+
+/-- The executable sine Taylor hull contains the exact real sine value on
+`[0,1]`. -/
+theorem sinBound_contains {x : ℚ} (hx0 : 0 ≤ x) (hx1 : x ≤ 1) :
+    Contains (sinBound x) (Real.sin (x : ℝ)) := by
+  have h := sine_between_partials hx0 hx1
+  constructor
+  · have hmin : min (sinPartial x 19) (sinPartial x 20) ≤ sinPartial x 20 :=
+      min_le_right _ _
+    exact le_trans (by exact_mod_cast hmin) h.1
+  · have hmax : sinPartial x 19 ≤ max (sinPartial x 19) (sinPartial x 20) :=
+      le_max_left _ _
+    exact le_trans h.2 (by exact_mod_cast hmax)
+
+/-- The executable cosine Taylor hull contains the exact real cosine value on
+`[0,1]`. -/
+theorem cosBound_contains {x : ℚ} (hx0 : 0 ≤ x) (hx1 : x ≤ 1) :
+    Contains (cosBound x) (Real.cos (x : ℝ)) := by
+  have h := cosine_between_partials hx0 hx1
+  constructor
+  · have hmin : min (cosPartial x 19) (cosPartial x 20) ≤ cosPartial x 20 :=
+      min_le_right _ _
+    exact le_trans (by exact_mod_cast hmin) h.1
+  · have hmax : cosPartial x 19 ≤ max (cosPartial x 19) (cosPartial x 20) :=
+      le_max_left _ _
+    exact le_trans h.2 (by exact_mod_cast hmax)
+
+/-- Consecutive odd/even arctangent sums form a semantic interval. -/
+theorem atanBound_contains {x : ℚ}
+    (hx0 : 0 ≤ x) (hx1 : x < 1) (k : ℕ) :
+    Contains (atanBound x (2 * k + 1) (2 * k + 2))
+      (Real.arctan (x : ℝ)) := by
+  have h := arctan_between_partials hx0 hx1 k
+  constructor
+  · have hmin :
+        min (atanPartial x (2 * k + 1)) (atanPartial x (2 * k + 2)) ≤
+          atanPartial x (2 * k + 2) := min_le_right _ _
+    exact le_trans (by exact_mod_cast hmin) h.1
+  · have hmax :
+        atanPartial x (2 * k + 1) ≤
+          max (atanPartial x (2 * k + 1)) (atanPartial x (2 * k + 2)) :=
+      le_max_left _ _
+    exact le_trans h.2 (by exact_mod_cast hmax)
+
+/-! ## Machin identity and the declared interval for π -/
+
+/-- The interval computed from the two exact arctangent Taylor certificates
+contains the true value of `π`. -/
+theorem machinPi_contains_pi : Contains machinPi Real.pi := by
+  have h5 : Contains (atanBound (1 / 5) 43 44)
+      (Real.arctan ((1 / 5 : ℚ) : ℝ)) := by
+    simpa using (atanBound_contains (x := (1 / 5 : ℚ)) (by norm_num) (by norm_num) 21)
+  have h239 : Contains (atanBound (1 / 239) 13 14)
+      (Real.arctan ((1 / 239 : ℚ) : ℝ)) := by
+    simpa using (atanBound_contains (x := (1 / 239 : ℚ)) (by norm_num) (by norm_num) 6)
+  have h16 := RatInterval.contains_scale (a := (16 : ℚ)) h5
+  have h4 := RatInterval.contains_scale (a := (4 : ℚ)) h239
+  have hsub := RatInterval.contains_sub h16 h4
+  have hMachin :
+      (16 : ℝ) * Real.arctan (1 / 5) -
+          4 * Real.arctan (1 / 239) = Real.pi := by
+    nlinarith [Real.four_mul_arctan_inv_5_sub_arctan_inv_239]
+  have hsub' : Contains machinPi
+      ((16 : ℝ) * Real.arctan (1 / 5) -
+        4 * Real.arctan (1 / 239)) := by
+    simpa [machinPi, scale] using hsub
+  exact hMachin ▸ hsub'
+
+/-- The executable and proof-carrying manifests contain byte-for-byte equal
+Machin intervals.  This is a finite rational normalization, not a numerical
+assumption. -/
+private theorem ratInterval_eq_of_endpoints {a b : RatInterval}
+    (hlo : a.lo = b.lo) (hhi : a.hi = b.hi) : a = b := by
+  cases a
+  cases b
+  simp_all
+
+/-- Exact lower endpoint agreement between the executable Machin evaluation
+and the frozen manifest.  This is a small bounded kernel computation (two
+arctangent sums), deliberately separated from the old monolithic replay. -/
+private theorem machinPi_lo_eq_manifest :
+    machinPi.lo = CertificateManifest.machinPi.lo := by
+    decide +kernel
+
+/-- Exact upper endpoint agreement between the executable Machin evaluation
+and the frozen manifest. -/
+private theorem machinPi_hi_eq_manifest :
+    machinPi.hi = CertificateManifest.machinPi.hi := by
+    decide +kernel
+
+theorem machinPi_eq_manifest :
+    machinPi = CertificateManifest.machinPi :=
+  ratInterval_eq_of_endpoints machinPi_lo_eq_manifest machinPi_hi_eq_manifest
+
+/-- The executable declared interval agrees with the frozen manifest. -/
+theorem piI_eq_manifest : piI = CertificateManifest.declaredPi := by
+  apply ratInterval_eq_of_endpoints
+  · decide +kernel
+  · decide +kernel
+
+/-- The exact interval used by every transcendental call encloses `Real.pi`. -/
+theorem piI_contains_pi : Contains piI Real.pi := by
+  have hm : Contains CertificateManifest.machinPi Real.pi := by
+    simpa [machinPi_eq_manifest] using machinPi_contains_pi
+  have hd : Contains CertificateManifest.declaredPi Real.pi :=
+    RatInterval.contains_of_strictInsideB
+      CertificateManifest.machin_inside_declared hm
+  simpa [piI_eq_manifest] using hd
+
+/-! ## Outward decimal rounding -/
+
+/-- Fixed-decimal floor rounding never exceeds the input rational. -/
+theorem floorDecimal_le (x : ℚ) (digits : ℕ := 60) :
+    floorDecimal x digits ≤ x := by
+  let s : ℚ := (10 : ℚ) ^ digits
+  have hs : 0 < s := by positivity
+  apply (div_le_iff₀ hs).2
+  have hf : (((⌊x * s⌋ : ℤ) : ℚ)) ≤ x * s := Int.floor_le _
+  simpa [floorDecimal, s] using hf
+
+/-- Fixed-decimal ceiling rounding never lies below the input rational. -/
+theorem le_ceilDecimal (x : ℚ) (digits : ℕ := 60) :
+    x ≤ ceilDecimal x digits := by
+  let s : ℚ := (10 : ℚ) ^ digits
+  have hs : 0 < s := by positivity
+  apply (le_div_iff₀ hs).2
+  have hc : x * s ≤ (((⌈x * s⌉ : ℤ) : ℚ)) := Int.le_ceil _
+  simpa [ceilDecimal, s] using hc
+
+/-- The exact 60-decimal conversion is outward in real semantics. -/
+theorem outwardDecimal_contains {z : RatInterval} {x : ℝ}
+    (hx : Contains z x) : Contains (outwardDecimal z) x := by
+  constructor
+  · exact le_trans (by exact_mod_cast floorDecimal_le z.lo (digits := 60)) hx.1
+  · exact le_trans hx.2 (by exact_mod_cast le_ceilDecimal z.hi (digits := 60))
+
+/-! ## Small-argument sine and cosine -/
+
+/-- The small sine evaluator is sound whenever its whole input lies in
+`[0, 9/10]`. -/
+theorem sinSmall_contains {z : RatInterval} {x : ℝ}
+    (hx : Contains z x) (hz0 : 0 ≤ z.lo) (hz9 : z.hi ≤ 9 / 10) :
+    Contains (sinSmall z) (Real.sin x) := by
+  have hzvalid : z.lo ≤ z.hi := by exact_mod_cast RatInterval.valid_of_contains hx
+  have hlo1 : z.lo ≤ 1 := le_trans hzvalid (le_trans hz9 (by norm_num))
+  have hhi0 : 0 ≤ z.hi := le_trans hz0 hzvalid
+  have hhi1 : z.hi ≤ 1 := le_trans hz9 (by norm_num)
+  have hloBound := sinBound_contains hz0 hlo1
+  have hhiBound := sinBound_contains hhi0 hhi1
+  have hpi : (9 / 10 : ℝ) ≤ Real.pi / 2 := by
+    nlinarith [Real.pi_gt_three]
+  have hloMem : -(Real.pi / 2) ≤ (z.lo : ℝ) := by
+    have : (0 : ℝ) ≤ (z.lo : ℝ) := by exact_mod_cast hz0
+    nlinarith [Real.pi_pos]
+  have hhiMem : (z.hi : ℝ) ≤ Real.pi / 2 := by
+    have hz9R0 : (z.hi : ℝ) ≤ (((9 / 10 : ℚ) : ℝ)) :=
+      (Rat.cast_le).2 hz9
+    have hz9R : (z.hi : ℝ) ≤ (9 / 10 : ℝ) := by
+      norm_num at hz9R0 ⊢
+      exact hz9R0
+    exact le_trans hz9R hpi
+  have hmonoLo : Real.sin (z.lo : ℝ) ≤ Real.sin x :=
+    Real.sin_le_sin_of_le_of_le_pi_div_two hloMem (le_trans hx.2 hhiMem) hx.1
+  have hxLower : -(Real.pi / 2) ≤ x := by
+    have hneg : -(Real.pi / 2) ≤ (0 : ℝ) := by
+      nlinarith [Real.pi_pos]
+    exact le_trans hneg (le_trans (by exact_mod_cast hz0) hx.1)
+  have hmonoHi : Real.sin x ≤ Real.sin (z.hi : ℝ) :=
+    Real.sin_le_sin_of_le_of_le_pi_div_two hxLower hhiMem hx.2
+  apply outwardDecimal_contains
+  constructor
+  · exact le_trans hloBound.1 hmonoLo
+  · exact le_trans hmonoHi hhiBound.2
+
+/-- The small cosine evaluator is sound whenever its whole input lies in
+`[0, 9/10]`. -/
+theorem cosSmall_contains {z : RatInterval} {x : ℝ}
+    (hx : Contains z x) (hz0 : 0 ≤ z.lo) (hz9 : z.hi ≤ 9 / 10) :
+    Contains (cosSmall z) (Real.cos x) := by
+  have hzvalid : z.lo ≤ z.hi := by exact_mod_cast RatInterval.valid_of_contains hx
+  have hlo1 : z.lo ≤ 1 := le_trans hzvalid (le_trans hz9 (by norm_num))
+  have hhi0 : 0 ≤ z.hi := le_trans hz0 hzvalid
+  have hhi1 : z.hi ≤ 1 := le_trans hz9 (by norm_num)
+  have hloBound := cosBound_contains hz0 hlo1
+  have hhiBound := cosBound_contains hhi0 hhi1
+  have hpi : (9 / 10 : ℝ) ≤ Real.pi := by
+    nlinarith [Real.pi_gt_three]
+  have hlo0 : (0 : ℝ) ≤ (z.lo : ℝ) := by exact_mod_cast hz0
+  have hhiPi : (z.hi : ℝ) ≤ Real.pi := by
+    have hz9R0 : (z.hi : ℝ) ≤ (((9 / 10 : ℚ) : ℝ)) :=
+      (Rat.cast_le).2 hz9
+    have hz9R : (z.hi : ℝ) ≤ (9 / 10 : ℝ) := by
+      norm_num at hz9R0 ⊢
+      exact hz9R0
+    exact le_trans hz9R hpi
+  have hx0 : (0 : ℝ) ≤ x := le_trans hlo0 hx.1
+  have hxPi : x ≤ Real.pi := le_trans hx.2 hhiPi
+  have hcosLower : Real.cos (z.hi : ℝ) ≤ Real.cos x :=
+    Real.cos_le_cos_of_nonneg_of_le_pi hx0 hhiPi hx.2
+  have hcosUpper : Real.cos x ≤ Real.cos (z.lo : ℝ) :=
+    Real.cos_le_cos_of_nonneg_of_le_pi hlo0 hxPi hx.1
+  apply outwardDecimal_contains
+  constructor
+  · exact le_trans hhiBound.1 hcosLower
+  · exact le_trans hcosUpper hloBound.2
+
+/-! ## Range reduction and fail-closed totality -/
+
+/-- Physical clamping preserves every enclosed angle in `[0,π/2]`. -/
+theorem physicalClamp_contains {z : RatInterval} {x : ℝ}
+    (hz : Contains z x) (hx0 : 0 ≤ x) (hxpi : x ≤ Real.pi / 2) :
+    Contains (physicalClamp z) x := by
+  have hpi := piI_contains_pi
+  constructor
+  · simpa [physicalClamp] using (max_le hx0 hz.1)
+  · have hupper : x ≤ ((piI.hi / 2 : ℚ) : ℝ) := by
+      have hhi : Real.pi ≤ (piI.hi : ℝ) := hpi.2
+      have hxhi : x ≤ (piI.hi : ℝ) / 2 := by
+        linarith [hxpi, hhi]
+      have hcast : (((piI.hi / 2 : ℚ) : ℝ)) = (piI.hi : ℝ) / 2 := by
+        norm_num
+      rw [hcast]
+      exact hxhi
+    simpa [physicalClamp] using (le_min hupper hz.2)
+
+/-- Complementary-angle range reduction encloses `π/2-x`. -/
+theorem complementInterval_contains {z : RatInterval} {x : ℝ}
+    (hz : Contains z x) (hxpi : x ≤ Real.pi / 2) :
+    Contains (complementInterval z) (Real.pi / 2 - x) := by
+  have hpi := piI_contains_pi
+  constructor
+  · have hzero : 0 ≤ Real.pi / 2 - x := by linarith
+    have hdiff : ((piI.lo / 2 - z.hi : ℚ) : ℝ) ≤ Real.pi / 2 - x := by
+      have hlo : (piI.lo : ℝ) ≤ Real.pi := hpi.1
+      have hdiff' : (piI.lo : ℝ) / 2 - (z.hi : ℝ) ≤ Real.pi / 2 - x := by
+        linarith [hlo, hz.2]
+      have hcast : (((piI.lo / 2 - z.hi : ℚ) : ℝ)) =
+          (piI.lo : ℝ) / 2 - (z.hi : ℝ) := by
+        norm_num
+      rw [hcast]
+      exact hdiff'
+    simpa [complementInterval] using (max_le hzero hdiff)
+  · have hdiff : Real.pi / 2 - x ≤ ((piI.hi / 2 - z.lo : ℚ) : ℝ) := by
+      have hhi : Real.pi ≤ (piI.hi : ℝ) := hpi.2
+      have hdiff' : Real.pi / 2 - x ≤ (piI.hi : ℝ) / 2 - (z.lo : ℝ) := by
+        linarith [hhi, hz.1]
+      have hcast : (((piI.hi / 2 - z.lo : ℚ) : ℝ)) =
+          (piI.hi : ℝ) / 2 - (z.lo : ℝ) := by
+        norm_num
+      rw [hcast]
+      exact hdiff'
+    simpa [complementInterval] using hdiff
+
+/-- Universal fallback for sine. -/
+theorem universal_contains_sin (x : ℝ) :
+    Contains universalTrigInterval (Real.sin x) := by
+  simpa [Contains, universalTrigInterval] using Real.sin_mem_Icc x
+
+/-- Universal fallback for cosine. -/
+theorem universal_contains_cos (x : ℝ) :
+    Contains universalTrigInterval (Real.cos x) := by
+  simpa [Contains, universalTrigInterval] using Real.cos_mem_Icc x
+
+/-- Full soundness of the executable sine interval on the physical angular
+range. -/
+theorem sinI_contains {z : RatInterval} {x : ℝ}
+    (hz : Contains z x) (hx0 : 0 ≤ x) (hxpi : x ≤ Real.pi / 2) :
+    Contains (sinI z) (Real.sin x) := by
+  let w := physicalClamp z
+  have hw : Contains w x := physicalClamp_contains hz hx0 hxpi
+  have hw0 : 0 ≤ w.lo := by simp [w, physicalClamp]
+  simp only [sinI]
+  split_ifs with hsmall hcomp
+  · exact sinSmall_contains hw hw0 hsmall
+  · let y := complementInterval w
+    have hy : Contains y (Real.pi / 2 - x) :=
+      complementInterval_contains hw hxpi
+    have hy0 : 0 ≤ y.lo := by simp [y, complementInterval]
+    have hcy := cosSmall_contains hy hy0 hcomp
+    simpa [Real.cos_pi_div_two_sub] using hcy
+  · exact universal_contains_sin x
+
+/-- Full soundness of the executable cosine interval on the physical angular
+range. -/
+theorem cosI_contains {z : RatInterval} {x : ℝ}
+    (hz : Contains z x) (hx0 : 0 ≤ x) (hxpi : x ≤ Real.pi / 2) :
+    Contains (cosI z) (Real.cos x) := by
+  let w := physicalClamp z
+  have hw : Contains w x := physicalClamp_contains hz hx0 hxpi
+  have hw0 : 0 ≤ w.lo := by simp [w, physicalClamp]
+  simp only [cosI]
+  split_ifs with hsmall hcomp
+  · exact cosSmall_contains hw hw0 hsmall
+  · let y := complementInterval w
+    have hy : Contains y (Real.pi / 2 - x) :=
+      complementInterval_contains hw hxpi
+    have hy0 : 0 ≤ y.lo := by simp [y, complementInterval]
+    have hsy := sinSmall_contains hy hy0 hcomp
+    simpa [Real.sin_pi_div_two_sub] using hsy
+  · exact universal_contains_cos x
+
+end GerverSofa.ExactReplay
+
+namespace GerverSofa
+
+open RatInterval
+
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Structural soundness of first-order interval automatic differentiation
+
+The executable `ExactReplay.D` object stores a value interval and one interval
+for every first partial derivative.  This file supplies the reusable semantic
+induction for all constructors actually used by the 4D and 22D certificates.
+The concrete systems are handled in a separate module by instantiating these
+constructor theorems.
+-/
+
+/-! ## Smooth scalar models -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+
+open RatInterval
+
+/-- A scalar function together with its coordinate gradient and a proof that
+those coordinates are the actual partial derivatives.  The derivative witness
+is stored as `RealDerivativeAt`, a first-principles real difference-quotient
+limit.  Constructor proofs temporarily move through Mathlib's `HasDerivAt`
+API and immediately return to this instance-stable semantic proposition. -/
+structure ScalarModel (n : Nat) where
+  /-- The real scalar function represented by the model. -/
+  value : Vec n → ℝ
+  /-- The coordinate gradient of the scalar function. -/
+  gradient : Vec n → Vec n
+  hasDeriv_update : ∀ x j,
+    RealDerivativeAt (fun t : ℝ => value (Function.update x j t))
+      (gradient x j) (x j)
+
+namespace ScalarModel
+
+/-- Constant scalar model. -/
+def const (n : Nat) (c : ℝ) : ScalarModel n where
+  value := fun _ => c
+  gradient := fun _ _ => 0
+  hasDeriv_update := by
+    intro x j
+    simpa only [RealDerivativeAt, smul_eq_mul] using
+      (hasDerivAt_const (x j) c).tendsto_slope_zero
+
+/-- Coordinate projection. -/
+def var (n : Nat) (k : Fin n) : ScalarModel n where
+  value := fun x => x k
+  gradient := fun _ j => if j = k then 1 else 0
+  hasDeriv_update := by
+    intro x j
+    by_cases h : j = k
+    · subst k
+      have hid : RealDerivativeAt (fun t : ℝ => t) 1 (x j) := by
+        simpa only [RealDerivativeAt, smul_eq_mul] using
+          (hasDerivAt_id' (x j)).tendsto_slope_zero
+      convert hid using 1 <;> simp
+    · have hkj : k ≠ j := Ne.symm h
+      simpa only [RealDerivativeAt, Function.update_of_ne hkj, ite_eq_right h,
+        smul_eq_mul] using (hasDerivAt_const (x j) (x k)).tendsto_slope_zero
+
+/-- Pointwise addition. -/
+def add {n : Nat} (f g : ScalarModel n) : ScalarModel n where
+  value := fun x => f.value x + g.value x
+  gradient := fun x j => f.gradient x j + g.gradient x j
+  hasDeriv_update := by
+    intro x j
+    have hfh :=
+      (hasDerivAt_iff_tendsto_slope_zero
+        (f := fun t : ℝ => f.value (Function.update x j t))
+        (f' := f.gradient x j) (x := x j)).2 (by
+          simpa only [RealDerivativeAt, smul_eq_mul] using
+            f.hasDeriv_update x j)
+    have hgh :=
+      (hasDerivAt_iff_tendsto_slope_zero
+        (f := fun t : ℝ => g.value (Function.update x j t))
+        (f' := g.gradient x j) (x := x j)).2 (by
+          simpa only [RealDerivativeAt, smul_eq_mul] using
+            g.hasDeriv_update x j)
+    simpa only [RealDerivativeAt, smul_eq_mul] using
+      (hfh.fun_add hgh).tendsto_slope_zero
+
+/-- Pointwise negation. -/
+def neg {n : Nat} (f : ScalarModel n) : ScalarModel n where
+  value := fun x => -f.value x
+  gradient := fun x j => -f.gradient x j
+  hasDeriv_update := by
+    intro x j
+    have hfh :=
+      (hasDerivAt_iff_tendsto_slope_zero
+        (f := fun t : ℝ => f.value (Function.update x j t))
+        (f' := f.gradient x j) (x := x j)).2 (by
+          simpa only [RealDerivativeAt, smul_eq_mul] using
+            f.hasDeriv_update x j)
+    simpa only [RealDerivativeAt, smul_eq_mul] using
+      hfh.fun_neg.tendsto_slope_zero
+
+/-- Pointwise subtraction. -/
+def sub {n : Nat} (f g : ScalarModel n) : ScalarModel n :=
+  add f (neg g)
+
+/-- Pointwise multiplication. -/
+def mul {n : Nat} (f g : ScalarModel n) : ScalarModel n where
+  value := fun x => f.value x * g.value x
+  gradient := fun x j =>
+    f.gradient x j * g.value x + f.value x * g.gradient x j
+  hasDeriv_update := by
+    intro x j
+    have hfh :=
+      (hasDerivAt_iff_tendsto_slope_zero
+        (f := fun t : ℝ => f.value (Function.update x j t))
+        (f' := f.gradient x j) (x := x j)).2 (by
+          simpa only [RealDerivativeAt, smul_eq_mul] using
+            f.hasDeriv_update x j)
+    have hgh :=
+      (hasDerivAt_iff_tendsto_slope_zero
+        (f := fun t : ℝ => g.value (Function.update x j t))
+        (f' := g.gradient x j) (x := x j)).2 (by
+          simpa only [RealDerivativeAt, smul_eq_mul] using
+            g.hasDeriv_update x j)
+    have hupd : Function.update x j (x j) = x :=
+      Function.update_eq_self j x
+    simpa only [RealDerivativeAt, smul_eq_mul, hupd] using
+      (hfh.fun_mul hgh).tendsto_slope_zero
+
+/-- Rational scaling. -/
+def scale {n : Nat} (a : ℚ) (f : ScalarModel n) : ScalarModel n where
+  value := fun x => (a : ℝ) * f.value x
+  gradient := fun x j => (a : ℝ) * f.gradient x j
+  hasDeriv_update := by
+    intro x j
+    have hfh :=
+      (hasDerivAt_iff_tendsto_slope_zero
+        (f := fun t : ℝ => f.value (Function.update x j t))
+        (f' := f.gradient x j) (x := x j)).2 (by
+          simpa only [RealDerivativeAt, smul_eq_mul] using
+            f.hasDeriv_update x j)
+    have hs := HasDerivAt.const_mul (a : ℝ) hfh
+    simpa only [RealDerivativeAt, smul_eq_mul] using hs.tendsto_slope_zero
+
+/-- Sine composition. -/
+def sin {n : Nat} (f : ScalarModel n) : ScalarModel n where
+  value := fun x => Real.sin (f.value x)
+  gradient := fun x j => Real.cos (f.value x) * f.gradient x j
+  hasDeriv_update := by
+    intro x j
+    have hfh :=
+      (hasDerivAt_iff_tendsto_slope_zero
+        (f := fun t : ℝ => f.value (Function.update x j t))
+        (f' := f.gradient x j) (x := x j)).2 (by
+          simpa only [RealDerivativeAt, smul_eq_mul] using
+            f.hasDeriv_update x j)
+    have hupd : Function.update x j (x j) = x :=
+      Function.update_eq_self j x
+    simpa only [RealDerivativeAt, smul_eq_mul, hupd] using
+      hfh.sin.tendsto_slope_zero
+
+/-- Cosine composition. -/
+def cos {n : Nat} (f : ScalarModel n) : ScalarModel n where
+  value := fun x => Real.cos (f.value x)
+  gradient := fun x j => -Real.sin (f.value x) * f.gradient x j
+  hasDeriv_update := by
+    intro x j
+    have hfh :=
+      (hasDerivAt_iff_tendsto_slope_zero
+        (f := fun t : ℝ => f.value (Function.update x j t))
+        (f' := f.gradient x j) (x := x j)).2 (by
+          simpa only [RealDerivativeAt, smul_eq_mul] using
+            f.hasDeriv_update x j)
+    have hupd : Function.update x j (x j) = x :=
+      Function.update_eq_self j x
+    simpa only [RealDerivativeAt, smul_eq_mul, hupd] using
+      hfh.cos.tendsto_slope_zero
+
+instance {n : Nat} : Add (ScalarModel n) := ⟨add⟩
+instance {n : Nat} : Neg (ScalarModel n) := ⟨neg⟩
+instance {n : Nat} : Sub (ScalarModel n) := ⟨sub⟩
+instance {n : Nat} : Mul (ScalarModel n) := ⟨mul⟩
+instance {n : Nat} : HMul ℚ (ScalarModel n) (ScalarModel n) := ⟨scale⟩
+
+/-! The executable systems are written with notation, while constructor-level
+    soundness lemmas produce the named operations above.  These tiny simp
+    bridges make that definitional equality explicit without unfolding the
+    proof-carrying structures themselves. -/
+@[simp] theorem add_notation {n : Nat} (f g : ScalarModel n) :
+    f + g = add f g := rfl
+
+@[simp] theorem neg_notation {n : Nat} (f : ScalarModel n) :
+    -f = neg f := rfl
+
+@[simp] theorem sub_notation {n : Nat} (f g : ScalarModel n) :
+    f - g = sub f g := rfl
+
+@[simp] theorem mul_notation {n : Nat} (f g : ScalarModel n) :
+    f * g = mul f g := rfl
+
+@[simp] theorem scale_notation {n : Nat} (a : ℚ) (f : ScalarModel n) :
+    a * f = scale a f := rfl
+
+@[simp] theorem add_value {n : Nat} (f g : ScalarModel n) (x : Vec n) :
+    (add f g).value x = f.value x + g.value x := rfl
+
+@[simp] theorem neg_value {n : Nat} (f : ScalarModel n) (x : Vec n) :
+    (neg f).value x = -f.value x := rfl
+
+@[simp] theorem sub_value {n : Nat} (f g : ScalarModel n) (x : Vec n) :
+    (sub f g).value x = f.value x - g.value x := rfl
+
+@[simp] theorem mul_value {n : Nat} (f g : ScalarModel n) (x : Vec n) :
+    (mul f g).value x = f.value x * g.value x := rfl
+
+@[simp] theorem scale_value {n : Nat} (a : ℚ) (f : ScalarModel n) (x : Vec n) :
+    (scale a f).value x = (a : ℝ) * f.value x := rfl
+
+@[simp] theorem sin_value {n : Nat} (f : ScalarModel n) (x : Vec n) :
+    (sin f).value x = Real.sin (f.value x) := rfl
+
+@[simp] theorem cos_value {n : Nat} (f : ScalarModel n) (x : Vec n) :
+    (cos f).value x = Real.cos (f.value x) := rfl
+
+end ScalarModel
+
+namespace ExactReplay.D
+
+/-! Matching notation bridges for the executable dual intervals. -/
+@[simp] theorem add_notation (x y : ExactReplay.D) :
+    x + y = addD x y := rfl
+
+@[simp] theorem neg_notation (x : ExactReplay.D) :
+    -x = negD x := rfl
+
+@[simp] theorem sub_notation (x y : ExactReplay.D) :
+    x - y = subD x y := rfl
+
+@[simp] theorem mul_notation (x y : ExactReplay.D) :
+    x * y = mulD x y := rfl
+
+@[simp] theorem scale_notation (a : ℚ) (x : ExactReplay.D) :
+    a * x = scaleD a x := rfl
+
+end ExactReplay.D
+
+/-! ## Semantic relation for the executable dual interval -/
+
+/-- An executable dual interval encloses a smooth scalar model on a set. -/
+structure DSoundOn {n : Nat} (X : Set (Vec n))
+    (d : ExactReplay.D) (f : ScalarModel n) : Prop where
+  value_sound : ∀ x ∈ X, Contains d.val (f.value x)
+  derivative_length : d.der.length = n
+  derivative_sound : ∀ x ∈ X, ∀ j : Fin n,
+    Contains (d.der.getD j.1 ExactReplay.zeroI) (f.gradient x j)
+
+/-! ## List access lemmas used by the AD constructors -/
+
+private theorem getD_map_of_lt
+    {α β : Type*} (f : α → β) (xs : List α)
+    (i : Nat) (h : i < xs.length) (da : α) (db : β) :
+    (xs.map f).getD i db = f (xs.getD i da) := by
+  have hmap : i < (xs.map f).length := by simpa using h
+  rw [List.getD_eq_getElem (xs.map f) db hmap]
+  rw [List.getD_eq_getElem xs da h]
+  simp
+
+private theorem getD_zipWith_of_lt
+    {α β γ : Type*} (f : α → β → γ)
+    (xs : List α) (ys : List β) (i : Nat)
+    (hx : i < xs.length) (hy : i < ys.length)
+    (da : α) (db : β) (dc : γ) :
+    (List.zipWith f xs ys).getD i dc =
+      f (xs.getD i da) (ys.getD i db) := by
+  have hz : i < (List.zipWith f xs ys).length := by
+    simpa only [List.length_zipWith] using (lt_min hx hy)
+  rw [List.getD_eq_getElem (List.zipWith f xs ys) dc hz]
+  rw [List.getD_eq_getElem xs da hx]
+  rw [List.getD_eq_getElem ys db hy]
+  simp
+
+/-! ## Constructor soundness -/
+
+/-- Exact interval constant. -/
+theorem DSoundOn.const {n : Nat} {X : Set (Vec n)}
+    {z : RatInterval} {c : ℝ} (hc : Contains z c) :
+    DSoundOn X (ExactReplay.D.const z n) (ScalarModel.const n c) := by
+  refine ⟨?_, ?_, ?_⟩
+  · intro x hx
+    exact hc
+  · simp [ExactReplay.D.const]
+  · intro x hx j
+    simp [ExactReplay.D.const, ExactReplay.zeroI, ScalarModel.const,
+      RatInterval.Contains, RatInterval.point]
+
+/-- Rational point constant. -/
+theorem DSoundOn.pointConst {n : Nat} {X : Set (Vec n)} (q : ℚ) :
+    DSoundOn X (ExactReplay.D.pointConst q n)
+      (ScalarModel.const n (q : ℝ)) := by
+  apply DSoundOn.const
+  exact (RatInterval.contains_point_iff q (q : ℝ)).2 rfl
+
+/-- Coordinate variable read from an enclosing input interval. -/
+theorem DSoundOn.varD {n : Nat} {X : Set (Vec n)}
+    (input : RatInterval) (k : Fin n)
+    (hinput : ∀ x ∈ X, Contains input (x k)) :
+    DSoundOn X (ExactReplay.D.varD input k.1 n) (ScalarModel.var n k) := by
+  refine ⟨hinput, ?_, ?_⟩
+  · simp [ExactReplay.D.varD]
+  · intro x hx j
+    have hj : j.1 < (List.range n).length := by simp
+    change Contains
+      (((List.range n).map
+        (fun m => RatInterval.point (if m = k.1 then 1 else 0))).getD
+          j.1 ExactReplay.zeroI)
+      (if j = k then 1 else 0)
+    rw [getD_map_of_lt (fun m => RatInterval.point (if m = k.1 then 1 else 0))
+      (List.range n) j.1 hj 0 ExactReplay.zeroI]
+    have hjrange : (List.range n).getD j.1 0 = j.1 := by
+      rw [List.getD_eq_getElem (List.range n) 0 hj]
+      simp
+    rw [hjrange]
+    by_cases hjk : j = k
+    · subst k
+      simp [RatInterval.Contains, RatInterval.point]
+    · have hval : j.1 ≠ k.1 := by
+        intro hval
+        exact hjk (Fin.ext hval)
+      simp [hjk, hval, RatInterval.Contains, RatInterval.point]
+
+/-- Addition constructor. -/
+theorem DSoundOn.add {n : Nat} {X : Set (Vec n)}
+    {dx dy : ExactReplay.D} {f g : ScalarModel n}
+    (hx : DSoundOn X dx f) (hy : DSoundOn X dy g) :
+    DSoundOn X (ExactReplay.D.addD dx dy) (ScalarModel.add f g) := by
+  refine ⟨?_, ?_, ?_⟩
+  · intro x hX
+    exact RatInterval.contains_add (hx.value_sound x hX) (hy.value_sound x hX)
+  · simp [ExactReplay.D.addD, hx.derivative_length, hy.derivative_length]
+  · intro x hX j
+    have hjx : j.1 < dx.der.length := by simp [hx.derivative_length]
+    have hjy : j.1 < dy.der.length := by simp [hy.derivative_length]
+    change Contains
+      ((List.zipWith RatInterval.add dx.der dy.der).getD j.1 ExactReplay.zeroI)
+      (f.gradient x j + g.gradient x j)
+    rw [getD_zipWith_of_lt RatInterval.add dx.der dy.der j.1 hjx hjy
+      ExactReplay.zeroI ExactReplay.zeroI ExactReplay.zeroI]
+    exact RatInterval.contains_add
+      (hx.derivative_sound x hX j) (hy.derivative_sound x hX j)
+
+/-- Negation constructor. -/
+theorem DSoundOn.neg {n : Nat} {X : Set (Vec n)}
+    {d : ExactReplay.D} {f : ScalarModel n}
+    (h : DSoundOn X d f) :
+    DSoundOn X (ExactReplay.D.negD d) (ScalarModel.neg f) := by
+  refine ⟨?_, ?_, ?_⟩
+  · intro x hX
+    exact RatInterval.contains_neg (h.value_sound x hX)
+  · simp [ExactReplay.D.negD, h.derivative_length]
+  · intro x hX j
+    have hj : j.1 < d.der.length := by simp [h.derivative_length]
+    change Contains
+      ((d.der.map RatInterval.neg).getD j.1 ExactReplay.zeroI)
+      (-f.gradient x j)
+    rw [getD_map_of_lt RatInterval.neg d.der j.1 hj
+      ExactReplay.zeroI ExactReplay.zeroI]
+    exact RatInterval.contains_neg (h.derivative_sound x hX j)
+
+/-- Subtraction constructor. -/
+theorem DSoundOn.sub {n : Nat} {X : Set (Vec n)}
+    {dx dy : ExactReplay.D} {f g : ScalarModel n}
+    (hx : DSoundOn X dx f) (hy : DSoundOn X dy g) :
+    DSoundOn X (ExactReplay.D.subD dx dy) (ScalarModel.sub f g) := by
+  simpa [ExactReplay.D.subD, ScalarModel.sub] using hx.add hy.neg
+
+/-- Multiplication constructor and product rule. -/
+theorem DSoundOn.mul {n : Nat} {X : Set (Vec n)}
+    {dx dy : ExactReplay.D} {f g : ScalarModel n}
+    (hx : DSoundOn X dx f) (hy : DSoundOn X dy g) :
+    DSoundOn X (ExactReplay.D.mulD dx dy) (ScalarModel.mul f g) := by
+  refine ⟨?_, ?_, ?_⟩
+  · intro x hX
+    exact RatInterval.contains_mul (hx.value_sound x hX) (hy.value_sound x hX)
+  · simp [ExactReplay.D.mulD, hx.derivative_length, hy.derivative_length]
+  · intro x hX j
+    have hjx : j.1 < dx.der.length := by simp [hx.derivative_length]
+    have hjy : j.1 < dy.der.length := by simp [hy.derivative_length]
+    change Contains
+      ((List.zipWith
+        (fun ddx ddy => RatInterval.add (RatInterval.mul ddx dy.val)
+          (RatInterval.mul dx.val ddy))
+        dx.der dy.der).getD j.1 ExactReplay.zeroI)
+      (f.gradient x j * g.value x + f.value x * g.gradient x j)
+    rw [getD_zipWith_of_lt
+      (fun ddx ddy => RatInterval.add (RatInterval.mul ddx dy.val)
+        (RatInterval.mul dx.val ddy))
+      dx.der dy.der j.1 hjx hjy ExactReplay.zeroI ExactReplay.zeroI ExactReplay.zeroI]
+    exact RatInterval.contains_add
+      (RatInterval.contains_mul (hx.derivative_sound x hX j)
+        (hy.value_sound x hX))
+      (RatInterval.contains_mul (hx.value_sound x hX)
+        (hy.derivative_sound x hX j))
+
+/-- Rational scaling constructor. -/
+theorem DSoundOn.scale {n : Nat} {X : Set (Vec n)}
+    (a : ℚ) {d : ExactReplay.D} {f : ScalarModel n}
+    (h : DSoundOn X d f) :
+    DSoundOn X (ExactReplay.D.scaleD a d) (ScalarModel.scale a f) := by
+  refine ⟨?_, ?_, ?_⟩
+  · intro x hX
+    exact RatInterval.contains_scale (h.value_sound x hX)
+  · simp [ExactReplay.D.scaleD, h.derivative_length]
+  · intro x hX j
+    have hj : j.1 < d.der.length := by simp [h.derivative_length]
+    change Contains
+      ((d.der.map (ExactReplay.scale a)).getD j.1 ExactReplay.zeroI)
+      ((a : ℝ) * f.gradient x j)
+    rw [getD_map_of_lt (ExactReplay.scale a) d.der j.1 hj
+      ExactReplay.zeroI ExactReplay.zeroI]
+    exact RatInterval.contains_scale (h.derivative_sound x hX j)
+
+/-- Sine constructor and chain rule. -/
+theorem DSoundOn.sin {n : Nat} {X : Set (Vec n)}
+    {d : ExactReplay.D} {f : ScalarModel n}
+    (h : DSoundOn X d f)
+    (hphysical : ∀ x ∈ X, 0 ≤ f.value x ∧ f.value x ≤ Real.pi / 2) :
+    DSoundOn X (ExactReplay.D.sinD d) (ScalarModel.sin f) := by
+  refine ⟨?_, ?_, ?_⟩
+  · intro x hX
+    exact ExactReplay.sinI_contains (h.value_sound x hX)
+      (hphysical x hX).1 (hphysical x hX).2
+  · simp [ExactReplay.D.sinD, h.derivative_length]
+  · intro x hX j
+    have hj : j.1 < d.der.length := by simp [h.derivative_length]
+    change Contains
+      ((d.der.map (RatInterval.mul (ExactReplay.cosI d.val))).getD
+        j.1 ExactReplay.zeroI)
+      (Real.cos (f.value x) * f.gradient x j)
+    rw [getD_map_of_lt (RatInterval.mul (ExactReplay.cosI d.val)) d.der
+      j.1 hj ExactReplay.zeroI ExactReplay.zeroI]
+    exact RatInterval.contains_mul
+      (ExactReplay.cosI_contains (h.value_sound x hX)
+        (hphysical x hX).1 (hphysical x hX).2)
+      (h.derivative_sound x hX j)
+
+/-- Cosine constructor and chain rule. -/
+theorem DSoundOn.cos {n : Nat} {X : Set (Vec n)}
+    {d : ExactReplay.D} {f : ScalarModel n}
+    (h : DSoundOn X d f)
+    (hphysical : ∀ x ∈ X, 0 ≤ f.value x ∧ f.value x ≤ Real.pi / 2) :
+    DSoundOn X (ExactReplay.D.cosD d) (ScalarModel.cos f) := by
+  refine ⟨?_, ?_, ?_⟩
+  · intro x hX
+    exact ExactReplay.cosI_contains (h.value_sound x hX)
+      (hphysical x hX).1 (hphysical x hX).2
+  · simp [ExactReplay.D.cosD, h.derivative_length]
+  · intro x hX j
+    have hj : j.1 < d.der.length := by simp [h.derivative_length]
+    change Contains
+      ((d.der.map (fun z => RatInterval.neg
+        (RatInterval.mul (ExactReplay.sinI d.val) z))).getD
+        j.1 ExactReplay.zeroI)
+      (-Real.sin (f.value x) * f.gradient x j)
+    rw [getD_map_of_lt (fun z => RatInterval.neg
+      (RatInterval.mul (ExactReplay.sinI d.val) z)) d.der
+      j.1 hj ExactReplay.zeroI ExactReplay.zeroI]
+    simpa only [neg_mul] using
+      RatInterval.contains_neg (RatInterval.contains_mul
+        (ExactReplay.sinI_contains (h.value_sound x hX)
+          (hphysical x hX).1 (hphysical x hX).2)
+        (h.derivative_sound x hX j))
+
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Concrete interval-AD soundness for the reduced 4D system
+
+This module instantiates the constructor-level AD theorem with equations
+(F1)--(F4), proves that the frozen rational list is exactly the manuscript box,
+and identifies the four smooth scalar models with `Reduced.vectorSystem`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+
+open RatInterval
+
+namespace Reduced
+
+/-- The four frozen rational intervals are exactly the named reduced box. -/
+theorem inputBox_exact (x : Vec 4) :
+    x ∈ vectorBox ↔ EnclosesVec ExactReplay.reducedInputBox x := by
+  constructor
+  · intro hx
+    change coordEquiv.symm x ∈ box at hx
+    dsimp [box, qR, coordEquiv] at hx
+    refine ⟨?_, ?_⟩
+    · norm_num [ExactReplay.reducedInputBox, CertificateManifest.x4]
+    · intro i
+      fin_cases i <;>
+        simp [ExactReplay.reducedInputBox, CertificateManifest.x4,
+          CertificateManifest.q, RatInterval.Contains] <;>
+        aesop
+  · rintro ⟨hlen, hx⟩
+    change coordEquiv.symm x ∈ box
+    dsimp [box, qR, coordEquiv]
+    have h0 := hx (0 : Fin 4)
+    have h1 := hx (1 : Fin 4)
+    have h2 := hx (2 : Fin 4)
+    have h3 := hx (3 : Fin 4)
+    simp [ExactReplay.reducedInputBox, CertificateManifest.x4,
+      CertificateManifest.q, RatInterval.Contains] at h0 h1 h2 h3
+    aesop
+
+/-- Scalar models matching the four reduced equations. -/
+def models : Fin 4 → ScalarModel 4 :=
+  let a := ScalarModel.var 4 (0 : Fin 4)
+  let b := ScalarModel.var 4 (1 : Fin 4)
+  let phi := ScalarModel.var 4 (2 : Fin 4)
+  let theta := ScalarModel.var 4 (3 : Fin 4)
+  let one := ScalarModel.const 4 1
+  let half := ScalarModel.const 4 (1 / 2 : ℝ)
+  let quarter := ScalarModel.const 4 (1 / 4 : ℝ)
+  let piM := ScalarModel.const 4 Real.pi
+  let cp := ScalarModel.cos phi
+  let sp := ScalarModel.sin phi
+  let ct := ScalarModel.cos theta
+  let st := ScalarModel.sin theta
+  let delta := theta - phi
+  let f1 := a * (ct - cp) - (2 : ℚ) * b * sp + (delta - one) * ct - st + cp + sp
+  let f2 := a * ((3 : ℚ) * st + sp) - (2 : ℚ) * b * cp
+    + (3 : ℚ) * (delta - one) * st + (3 : ℚ) * ct - sp + cp
+  let f3 := a * cp - sp - half + half * cp - b * sp
+  let f4 := a + (1 / 2 : ℚ) * piM - phi - theta - b
+    + half * delta * (one + a) + quarter * delta * delta
+  ![f1, f2, f3, f4]
+
+/-- The model values are definitionally the manuscript reduced system after
+coordinate conversion. -/
+theorem vectorSystem_eq_models (x : Vec 4) :
+    vectorSystem x = fun i => (models i).value x := by
+  funext i
+  fin_cases i <;>
+    simp [vectorSystem, coordEquiv, system, models, ScalarModel.var,
+      ScalarModel.const, ScalarModel.add, ScalarModel.neg, ScalarModel.sub,
+      ScalarModel.mul, ScalarModel.scale, ScalarModel.sin, ScalarModel.cos] <;>
+    ring
+
+end Reduced
+
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Concrete interval-AD soundness for the direct 22D Romik system
+
+The direct system contains five trigonometric path pieces, three derivative
+pieces and six matching pairs.  To avoid 22 unrelated derivative proofs, this
+file evaluates every expression in a proof-carrying dual object.  Its first
+projection is the exact executable `ExactReplay.D`; its second projection is a
+smooth real scalar model; and its third field is the constructor-level
+soundness theorem from `ADCoreSoundness`.
+-/
+
+/-! ## Exact identification of the 22D input box
+
+The 22 coordinates are kept as separate tiny lemmas.  This is deliberately
+chunked: expanding all 44 rational endpoint inequalities in one `simp` call
+exhausts the default heartbeat budget even though every coordinate identity is
+individually trivial. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+
+open RatInterval
+
+namespace Romik
+
+private theorem fullInputBox_coord_0_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 0) (x 0) ↔
+      qR (-21032242207268875141628571849) 100000000000000000000000000000 ≤ x 0 ∧
+      x 0 ≤ qR (-21032242207268875141608571849) 100000000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_1_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 1) (x 1) ↔
+      qR 2499999999999999999999 10000000000000000000000 ≤ x 1 ∧
+      x 1 ≤ qR 2500000000000000000001 10000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_2_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 2) (x 2) ↔
+      qR (-91917929277159332227479610289) 100000000000000000000000000000 ≤ x 2 ∧
+      x 2 ≤ qR (-91917929277159332227459610289) 100000000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_3_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 3) (x 3) ↔
+      qR 29525413734425341573853797657 62500000000000000000000000000 ≤ x 3 ∧
+      x 3 ≤ qR 29525413734425341573866297657 62500000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_4_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 4) (x 4) ↔
+      qR (-15344080735756291713875357283) 25000000000000000000000000000 ≤ x 4 ∧
+      x 4 ≤ qR (-15344080735756291713870357283) 25000000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_5_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 5) (x 5) ↔
+      qR 17792529580064437214538861001 20000000000000000000000000000 ≤ x 5 ∧
+      x 5 ≤ qR 17792529580064437214542861001 20000000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_6_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 6) (x 6) ↔
+      qR (-15417358304445500741761623987) 50000000000000000000000000000 ≤ x 6 ∧
+      x 6 ≤ qR (-15417358304445500741751623987) 50000000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_7_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 7) (x 7) ↔
+      qR 29525413734425341573853797657 62500000000000000000000000000 ≤ x 7 ∧
+      x 7 ≤ qR 29525413734425341573866297657 62500000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_8_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 8) (x 8) ↔
+      qR (-20344080735756291713874857283) 20000000000000000000000000000 ≤ x 8 ∧
+      x 8 ≤ qR (-20344080735756291713870857283) 20000000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_9_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 9) (x 9) ↔
+      qR 2499999999999999999999 10000000000000000000000 ≤ x 9 ∧
+      x 9 ≤ qR 2500000000000000000001 10000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_10_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 10) (x 10) ↔
+      qR 2420644844145377502832171437 2000000000000000000000000000 ≤ x 10 ∧
+      x 10 ≤ qR 2420644844145377502832571437 2000000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_11_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 11) (x 11) ↔
+      qR (-2500000000000000000001) 10000000000000000000000 ≤ x 11 ∧
+      x 11 ≤ qR (-2499999999999999999999) 10000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_12_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 12) (x 12) ↔
+      qR (-52762459802678462416060380937) 100000000000000000000000000000 ≤ x 12 ∧
+      x 12 ≤ qR (-52762459802678462416040380937) 100000000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_13_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 13) (x 13) ↔
+      qR 92025838516063762289360579501 100000000000000000000000000000 ≤ x 13 ∧
+      x 13 ≤ qR 92025838516063762289380579501 100000000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_14_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 14) (x 14) ↔
+      qR 313022761424232933776114655193 500000000000000000000000000000 ≤ x 14 ∧
+      x 14 ≤ qR 313022761424232933776214655193 500000000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_15_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 15) (x 15) ↔
+      qR (-151160128631428920268654781) 160000000000000000000000000 ≤ x 15 ∧
+      x 15 ≤ qR (-151160128631428920268622781) 160000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_16_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 16) (x 16) ↔
+      qR 1641278451780291167220080819 1250000000000000000000000000 ≤ x 16 ∧
+      x 16 ≤ qR 1641278451780291167220330819 1250000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_17_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 17) (x 17) ↔
+      qR (-105076534082910887440587258861) 200000000000000000000000000000 ≤ x 17 ∧
+      x 17 ≤ qR (-105076534082910887440547258861) 200000000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_18_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 18) (x 18) ↔
+      qR 2420644844145377502832171437 2000000000000000000000000000 ≤ x 18 ∧
+      x 18 ≤ qR 2420644844145377502832571437 2000000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_19_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 19) (x 19) ↔
+      qR 2499999999999999999999 10000000000000000000000 ≤ x 19 ∧
+      x 19 ≤ qR 2500000000000000000001 10000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_20_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 20) (x 20) ↔
+      qR 1958868239504182093160893749 50000000000000000000000000000 ≤ x 20 ∧
+      x 20 ≤ qR 78354729580167283726435751 2000000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+private theorem fullInputBox_coord_21_iff (x : Vec 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 21) (x 21) ↔
+      qR 34065075469136244723692787727 50000000000000000000000000000 ≤ x 21 ∧
+      x 21 ≤ qR 34065075469136244723692787983 50000000000000000000000000000 := by
+  simp [ExactReplay.fullInputBox, CertificateManifest.z22,
+    CertificateManifest.q, ExactReplay.getI, RatInterval.Contains, qR]
+
+/-- The 22 frozen rational intervals are exactly the named direct-system box. -/
+theorem inputBox_exact (x : Vec 22) :
+    x ∈ vectorBox ↔ EnclosesVec ExactReplay.fullInputBox x := by
+  constructor
+  · intro hx
+    change coordEquiv.symm x ∈ box at hx
+    dsimp [box, coordEquiv] at hx
+    rcases hx with ⟨h0lo, h0hi, h1lo, h1hi, h2lo, h2hi, h3lo, h3hi, h4lo, h4hi, h5lo, h5hi, h6lo,
+      h6hi, h7lo, h7hi, h8lo, h8hi, h9lo, h9hi, h10lo, h10hi, h11lo, h11hi, h12lo, h12hi, h13lo,
+      h13hi, h14lo, h14hi, h15lo, h15hi, h16lo, h16hi, h17lo, h17hi, h18lo, h18hi, h19lo, h19hi,
+      h20lo, h20hi, h21lo, h21hi⟩
+    refine ⟨?_, ?_⟩
+    · norm_num [ExactReplay.fullInputBox, CertificateManifest.z22]
+    · intro i
+      fin_cases i
+      · exact (fullInputBox_coord_0_iff x).2 ⟨h0lo, h0hi⟩
+      · exact (fullInputBox_coord_1_iff x).2 ⟨h1lo, h1hi⟩
+      · exact (fullInputBox_coord_2_iff x).2 ⟨h2lo, h2hi⟩
+      · exact (fullInputBox_coord_3_iff x).2 ⟨h3lo, h3hi⟩
+      · exact (fullInputBox_coord_4_iff x).2 ⟨h4lo, h4hi⟩
+      · exact (fullInputBox_coord_5_iff x).2 ⟨h5lo, h5hi⟩
+      · exact (fullInputBox_coord_6_iff x).2 ⟨h6lo, h6hi⟩
+      · exact (fullInputBox_coord_7_iff x).2 ⟨h7lo, h7hi⟩
+      · exact (fullInputBox_coord_8_iff x).2 ⟨h8lo, h8hi⟩
+      · exact (fullInputBox_coord_9_iff x).2 ⟨h9lo, h9hi⟩
+      · exact (fullInputBox_coord_10_iff x).2 ⟨h10lo, h10hi⟩
+      · exact (fullInputBox_coord_11_iff x).2 ⟨h11lo, h11hi⟩
+      · exact (fullInputBox_coord_12_iff x).2 ⟨h12lo, h12hi⟩
+      · exact (fullInputBox_coord_13_iff x).2 ⟨h13lo, h13hi⟩
+      · exact (fullInputBox_coord_14_iff x).2 ⟨h14lo, h14hi⟩
+      · exact (fullInputBox_coord_15_iff x).2 ⟨h15lo, h15hi⟩
+      · exact (fullInputBox_coord_16_iff x).2 ⟨h16lo, h16hi⟩
+      · exact (fullInputBox_coord_17_iff x).2 ⟨h17lo, h17hi⟩
+      · exact (fullInputBox_coord_18_iff x).2 ⟨h18lo, h18hi⟩
+      · exact (fullInputBox_coord_19_iff x).2 ⟨h19lo, h19hi⟩
+      · exact (fullInputBox_coord_20_iff x).2 ⟨h20lo, h20hi⟩
+      · exact (fullInputBox_coord_21_iff x).2 ⟨h21lo, h21hi⟩
+  · rintro ⟨_hlen, hx⟩
+    change coordEquiv.symm x ∈ box
+    dsimp [box, coordEquiv]
+    have h0 := (fullInputBox_coord_0_iff x).1 (hx (0 : Fin 22))
+    have h1 := (fullInputBox_coord_1_iff x).1 (hx (1 : Fin 22))
+    have h2 := (fullInputBox_coord_2_iff x).1 (hx (2 : Fin 22))
+    have h3 := (fullInputBox_coord_3_iff x).1 (hx (3 : Fin 22))
+    have h4 := (fullInputBox_coord_4_iff x).1 (hx (4 : Fin 22))
+    have h5 := (fullInputBox_coord_5_iff x).1 (hx (5 : Fin 22))
+    have h6 := (fullInputBox_coord_6_iff x).1 (hx (6 : Fin 22))
+    have h7 := (fullInputBox_coord_7_iff x).1 (hx (7 : Fin 22))
+    have h8 := (fullInputBox_coord_8_iff x).1 (hx (8 : Fin 22))
+    have h9 := (fullInputBox_coord_9_iff x).1 (hx (9 : Fin 22))
+    have h10 := (fullInputBox_coord_10_iff x).1 (hx (10 : Fin 22))
+    have h11 := (fullInputBox_coord_11_iff x).1 (hx (11 : Fin 22))
+    have h12 := (fullInputBox_coord_12_iff x).1 (hx (12 : Fin 22))
+    have h13 := (fullInputBox_coord_13_iff x).1 (hx (13 : Fin 22))
+    have h14 := (fullInputBox_coord_14_iff x).1 (hx (14 : Fin 22))
+    have h15 := (fullInputBox_coord_15_iff x).1 (hx (15 : Fin 22))
+    have h16 := (fullInputBox_coord_16_iff x).1 (hx (16 : Fin 22))
+    have h17 := (fullInputBox_coord_17_iff x).1 (hx (17 : Fin 22))
+    have h18 := (fullInputBox_coord_18_iff x).1 (hx (18 : Fin 22))
+    have h19 := (fullInputBox_coord_19_iff x).1 (hx (19 : Fin 22))
+    have h20 := (fullInputBox_coord_20_iff x).1 (hx (20 : Fin 22))
+    have h21 := (fullInputBox_coord_21_iff x).1 (hx (21 : Fin 22))
+    exact ⟨h0.1, h0.2, h1.1, h1.2, h2.1, h2.2, h3.1, h3.2, h4.1, h4.2, h5.1, h5.2, h6.1, h6.2,
+      h7.1, h7.2, h8.1, h8.2, h9.1, h9.2, h10.1, h10.2, h11.1, h11.2, h12.1, h12.2, h13.1, h13.2,
+      h14.1, h14.2, h15.1, h15.2, h16.1, h16.2, h17.1, h17.2, h18.1, h18.2, h19.1, h19.2, h20.1,
+      h20.2, h21.1, h21.2⟩
+
+/-- All four switching times used by the direct evaluator lie in `[0,π/2]`. -/
+theorem full_switches_physical {x : Vec 22} (hx : x ∈ vectorBox) :
+    (0 ≤ x 20 ∧ x 20 ≤ Real.pi / 2) ∧
+    (0 ≤ x 21 ∧ x 21 ≤ Real.pi / 2) ∧
+    (0 ≤ Real.pi / 2 - x 21 ∧ Real.pi / 2 - x 21 ≤ Real.pi / 2) ∧
+    (0 ≤ Real.pi / 2 - x 20 ∧ Real.pi / 2 - x 20 ≤ Real.pi / 2) := by
+  let p := coordEquiv.symm x
+  have hp : p ∈ box := hx
+  have hphiPos : 0 < p.phi := phi_pos_of_mem_box hp
+  have hord : SwitchOrder p := switchOrder_of_mem_box hp
+  have htheta0 : 0 ≤ p.theta :=
+    le_trans hphiPos.le hord.phi_le_theta
+  have hthetaHalf : p.theta ≤ Real.pi / 2 := by
+    linarith [hord.theta_le_eta]
+  have hphiHalf : p.phi ≤ Real.pi / 2 :=
+    le_trans hord.phi_le_theta hthetaHalf
+  change
+    (0 ≤ p.phi ∧ p.phi ≤ Real.pi / 2) ∧
+    (0 ≤ p.theta ∧ p.theta ≤ Real.pi / 2) ∧
+    (0 ≤ Real.pi / 2 - p.theta ∧
+      Real.pi / 2 - p.theta ≤ Real.pi / 2) ∧
+    (0 ≤ Real.pi / 2 - p.phi ∧
+      Real.pi / 2 - p.phi ≤ Real.pi / 2)
+  constructor
+  · exact ⟨hphiPos.le, hphiHalf⟩
+  constructor
+  · exact ⟨htheta0, hthetaHalf⟩
+  constructor
+  · constructor <;> linarith
+  · constructor <;> linarith
+
+/-! ## Proof-carrying dual expressions -/
+
+/-- One executable interval dual paired with its real semantic model. -/
+structure SoundDual (n : Nat) (X : Set (Vec n)) where
+  /-- The interval value and derivative data being certified. -/
+  d : ExactReplay.D
+  /-- The real scalar function and gradient represented by the interval data. -/
+  model : ScalarModel n
+  sound : DSoundOn X d model
+
+namespace SoundDual
+
+variable {n : Nat} {X : Set (Vec n)}
+
+/-- A constant scalar model with a certified interval enclosure. -/
+def const (z : RatInterval) (c : ℝ) (h : Contains z c) : SoundDual n X :=
+  ⟨ExactReplay.D.const z n, ScalarModel.const n c, DSoundOn.const h⟩
+
+/-- A rational constant represented by a singleton interval and zero gradient. -/
+def pointConst (q : ℚ) : SoundDual n X :=
+  ⟨ExactReplay.D.pointConst q n, ScalarModel.const n (q : ℝ),
+    DSoundOn.pointConst q⟩
+
+/-- A coordinate projection with its certified input interval. -/
+def var (input : RatInterval) (k : Fin n)
+    (h : ∀ x ∈ X, Contains input (x k)) : SoundDual n X :=
+  ⟨ExactReplay.D.varD input k.1 n, ScalarModel.var n k,
+    DSoundOn.varD input k h⟩
+
+/-- Addition with certified interval value and gradient enclosures. -/
+def add (a b : SoundDual n X) : SoundDual n X :=
+  ⟨ExactReplay.D.addD a.d b.d, ScalarModel.add a.model b.model,
+    a.sound.add b.sound⟩
+
+/-- Negation with certified interval value and gradient enclosures. -/
+def neg (a : SoundDual n X) : SoundDual n X :=
+  ⟨ExactReplay.D.negD a.d, ScalarModel.neg a.model, a.sound.neg⟩
+
+/-- Subtraction with certified interval value and gradient enclosures. -/
+def sub (a b : SoundDual n X) : SoundDual n X :=
+  ⟨ExactReplay.D.subD a.d b.d, ScalarModel.sub a.model b.model,
+    a.sound.sub b.sound⟩
+
+/-- Multiplication with certified interval value and gradient enclosures. -/
+def mul (a b : SoundDual n X) : SoundDual n X :=
+  ⟨ExactReplay.D.mulD a.d b.d, ScalarModel.mul a.model b.model,
+    a.sound.mul b.sound⟩
+
+/-- Rational scaling with certified interval value and gradient enclosures. -/
+def scale (q : ℚ) (a : SoundDual n X) : SoundDual n X :=
+  ⟨ExactReplay.D.scaleD q a.d, ScalarModel.scale q a.model,
+    DSoundOn.scale q a.sound⟩
+
+/-- Sine with certified interval value and gradient enclosures. -/
+def sin (a : SoundDual n X)
+    (h : ∀ x ∈ X, 0 ≤ a.model.value x ∧
+      a.model.value x ≤ Real.pi / 2) : SoundDual n X :=
+  ⟨ExactReplay.D.sinD a.d, ScalarModel.sin a.model, a.sound.sin h⟩
+
+/-- Cosine with certified interval value and gradient enclosures. -/
+def cos (a : SoundDual n X)
+    (h : ∀ x ∈ X, 0 ≤ a.model.value x ∧
+      a.model.value x ≤ Real.pi / 2) : SoundDual n X :=
+  ⟨ExactReplay.D.cosD a.d, ScalarModel.cos a.model, a.sound.cos h⟩
+
+instance : Add (SoundDual n X) := ⟨add⟩
+instance : Neg (SoundDual n X) := ⟨neg⟩
+instance : Sub (SoundDual n X) := ⟨sub⟩
+instance : Mul (SoundDual n X) := ⟨mul⟩
+instance : HMul ℚ (SoundDual n X) (SoundDual n X) := ⟨scale⟩
+
+/-! Small projection lemmas keep the simplifier away from the proof fields of
+`SoundDual`.  All are definitional equalities. -/
+@[simp] theorem const_model_value (z : RatInterval) (c : ℝ)
+    (h : Contains z c) (x : Vec n) :
+    (const z c h : SoundDual n X).model.value x = c := rfl
+
+@[simp] theorem pointConst_model_value (q : ℚ) (x : Vec n) :
+    (pointConst q : SoundDual n X).model.value x = (q : ℝ) := rfl
+
+@[simp] theorem var_model_value (input : RatInterval) (k : Fin n)
+    (h : ∀ x ∈ X, Contains input (x k)) (x : Vec n) :
+    (var input k h : SoundDual n X).model.value x = x k := rfl
+
+@[simp] theorem add_model_value (a b : SoundDual n X) (x : Vec n) :
+    (a + b).model.value x = a.model.value x + b.model.value x := rfl
+
+@[simp] theorem neg_model_value (a : SoundDual n X) (x : Vec n) :
+    (-a).model.value x = -a.model.value x := rfl
+
+@[simp] theorem sub_model_value (a b : SoundDual n X) (x : Vec n) :
+    (a - b).model.value x = a.model.value x - b.model.value x := rfl
+
+@[simp] theorem mul_model_value (a b : SoundDual n X) (x : Vec n) :
+    (a * b).model.value x = a.model.value x * b.model.value x := rfl
+
+@[simp] theorem scale_model_value (q : ℚ) (a : SoundDual n X) (x : Vec n) :
+    (q * a).model.value x = (q : ℝ) * a.model.value x := rfl
+
+@[simp] theorem sin_model_value (a : SoundDual n X)
+    (h : ∀ x ∈ X, 0 ≤ a.model.value x ∧ a.model.value x ≤ Real.pi / 2)
+    (x : Vec n) :
+    (sin a h).model.value x = Real.sin (a.model.value x) := rfl
+
+@[simp] theorem cos_model_value (a : SoundDual n X)
+    (h : ∀ x ∈ X, 0 ≤ a.model.value x ∧ a.model.value x ≤ Real.pi / 2)
+    (x : Vec n) :
+    (cos a h).model.value x = Real.cos (a.model.value x) := rfl
+
+end SoundDual
+
+/-- A certified physical angle, used to justify every sine/cosine constructor. -/
+structure AngleDual (n : Nat) (X : Set (Vec n)) where
+  /-- The certified scalar model for the angle variable. -/
+  dual : SoundDual n X
+  physical : ∀ x ∈ X, 0 ≤ dual.model.value x ∧
+    dual.model.value x ≤ Real.pi / 2
+
+namespace AngleDual
+
+variable {n : Nat} {X : Set (Vec n)}
+
+/-- Sine with certified interval value and gradient enclosures. -/
+def sin (t : AngleDual n X) : SoundDual n X :=
+  SoundDual.sin t.dual t.physical
+
+/-- Cosine with certified interval value and gradient enclosures. -/
+def cos (t : AngleDual n X) : SoundDual n X :=
+  SoundDual.cos t.dual t.physical
+
+@[simp] theorem sin_model_value (t : AngleDual n X) (x : Vec n) :
+    t.sin.model.value x = Real.sin (t.dual.model.value x) := rfl
+
+@[simp] theorem cos_model_value (t : AngleDual n X) (x : Vec n) :
+    t.cos.model.value x = Real.cos (t.dual.model.value x) := rfl
+
+end AngleDual
+
+/-- Named proof-carrying versions of all direct-system variables. -/
+structure FullVars (X : Set (Vec 22)) where
+  /-- Horizontal translation coefficient for phase 1 as a certified dual interval. -/
+  k11 : SoundDual 22 X
+  /-- Vertical translation coefficient for phase 1 as a certified dual interval. -/
+  k12 : SoundDual 22 X
+  /-- Horizontal translation coefficient for phase 2 as a certified dual interval. -/
+  k21 : SoundDual 22 X
+  /-- Vertical translation coefficient for phase 2 as a certified dual interval. -/
+  k22 : SoundDual 22 X
+  /-- Horizontal translation coefficient for phase 3 as a certified dual interval. -/
+  k31 : SoundDual 22 X
+  /-- Vertical translation coefficient for phase 3 as a certified dual interval. -/
+  k32 : SoundDual 22 X
+  /-- Horizontal translation coefficient for phase 4 as a certified dual interval. -/
+  k41 : SoundDual 22 X
+  /-- Vertical translation coefficient for phase 4 as a certified dual interval. -/
+  k42 : SoundDual 22 X
+  /-- Horizontal translation coefficient for phase 5 as a certified dual interval. -/
+  k51 : SoundDual 22 X
+  /-- Vertical translation coefficient for phase 5 as a certified dual interval. -/
+  k52 : SoundDual 22 X
+  /-- Coefficient 1 in the phase 1 closed formula as a certified dual interval. -/
+  a1 : SoundDual 22 X
+  /-- Coefficient 2 in the phase 1 closed formula as a certified dual interval. -/
+  a2 : SoundDual 22 X
+  /-- Coefficient 1 in the phase 2 closed formula as a certified dual interval. -/
+  b1 : SoundDual 22 X
+  /-- Coefficient 2 in the phase 2 closed formula as a certified dual interval. -/
+  b2 : SoundDual 22 X
+  /-- Coefficient 1 in the phase 3 closed formula as a certified dual interval. -/
+  c1 : SoundDual 22 X
+  /-- Coefficient 2 in the phase 3 closed formula as a certified dual interval. -/
+  c2 : SoundDual 22 X
+  /-- Coefficient 1 in the phase 4 closed formula as a certified dual interval. -/
+  d1 : SoundDual 22 X
+  /-- Coefficient 2 in the phase 4 closed formula as a certified dual interval. -/
+  d2 : SoundDual 22 X
+  /-- Coefficient 1 in the phase 5 closed formula as a certified dual interval. -/
+  e1 : SoundDual 22 X
+  /-- Coefficient 2 in the phase 5 closed formula as a certified dual interval. -/
+  e2 : SoundDual 22 X
+
+/-- The 22 coordinate variables equipped with their input-enclosure proofs. -/
+def inputDual (i : Fin 22) : SoundDual 22 vectorBox :=
+  SoundDual.var (ExactReplay.getI ExactReplay.fullInputBox i.1) i (by
+    intro x hx
+    exact ((inputBox_exact x).1 hx).2 i)
+
+@[simp] theorem inputDual_model_value (i : Fin 22) (x : Vec 22) :
+    (inputDual i).model.value x = x i := rfl
+
+/-- Named first twenty variables in verifier order. -/
+def fullVars : FullVars vectorBox where
+  k11 := inputDual 0
+  k12 := inputDual 1
+  k21 := inputDual 2
+  k22 := inputDual 3
+  k31 := inputDual 4
+  k32 := inputDual 5
+  k41 := inputDual 6
+  k42 := inputDual 7
+  k51 := inputDual 8
+  k52 := inputDual 9
+  a1 := inputDual 10
+  a2 := inputDual 11
+  b1 := inputDual 12
+  b2 := inputDual 13
+  c1 := inputDual 14
+  c2 := inputDual 15
+  d1 := inputDual 16
+  d2 := inputDual 17
+  e1 := inputDual 18
+  e2 := inputDual 19
+
+/-- Exact proof-carrying π constant. -/
+def piDual : SoundDual 22 vectorBox :=
+  SoundDual.const ExactReplay.piI Real.pi ExactReplay.piI_contains_pi
+
+@[simp] theorem piDual_model_value (x : Vec 22) :
+    piDual.model.value x = Real.pi := rfl
+
+/-- First switching angle. -/
+def phiDual : AngleDual 22 vectorBox where
+  dual := inputDual 20
+  physical := by
+    intro x hx
+    simpa only [inputDual_model_value] using
+      (full_switches_physical hx).1
+
+@[simp] theorem phiDual_model_value (x : Vec 22) :
+    phiDual.dual.model.value x = x 20 := rfl
+
+/-- Second switching angle. -/
+def thetaDual : AngleDual 22 vectorBox where
+  dual := inputDual 21
+  physical := by
+    intro x hx
+    simpa only [inputDual_model_value] using
+      (full_switches_physical hx).2.1
+
+@[simp] theorem thetaDual_model_value (x : Vec 22) :
+    thetaDual.dual.model.value x = x 21 := rfl
+
+private theorem eta_raw_model_value (x : Vec 22) :
+    (((1 / 2 : ℚ) * piDual - thetaDual.dual).model.value x) =
+      Real.pi / 2 - x 21 := by
+  simp; ring
+
+/-- Reflected third switching angle `π/2-θ`. -/
+def etaDual : AngleDual 22 vectorBox where
+  dual := (1 / 2 : ℚ) * piDual - thetaDual.dual
+  physical := by
+    intro x hx
+    rw [eta_raw_model_value]
+    exact (full_switches_physical hx).2.2.1
+
+@[simp] theorem etaDual_model_value (x : Vec 22) :
+    etaDual.dual.model.value x = Real.pi / 2 - x 21 := by
+  exact eta_raw_model_value x
+
+private theorem tau_raw_model_value (x : Vec 22) :
+    (((1 / 2 : ℚ) * piDual - phiDual.dual).model.value x) =
+      Real.pi / 2 - x 20 := by
+  simp; ring
+
+/-- Reflected fourth switching angle `π/2-φ`. -/
+def tauDual : AngleDual 22 vectorBox where
+  dual := (1 / 2 : ℚ) * piDual - phiDual.dual
+  physical := by
+    intro x hx
+    rw [tau_raw_model_value]
+    exact (full_switches_physical hx).2.2.2
+
+@[simp] theorem tauDual_model_value (x : Vec 22) :
+    tauDual.dual.model.value x = Real.pi / 2 - x 20 := by
+  exact tau_raw_model_value x
+
+/-- Rotation of a proof-carrying body-frame vector. -/
+def rotDual (t : AngleDual 22 vectorBox)
+    (z1 z2 : SoundDual 22 vectorBox) :
+    SoundDual 22 vectorBox × SoundDual 22 vectorBox :=
+  let ct := t.cos
+  let st := t.sin
+  (ct * z1 - st * z2, st * z1 + ct * z2)
+
+/-- One proof-carrying path piece. -/
+def pathPieceDual (j : Nat) (t : AngleDual 22 vectorBox)
+    (p : FullVars vectorBox := fullVars) :
+    SoundDual 22 vectorBox × SoundDual 22 vectorBox :=
+  let one : SoundDual 22 vectorBox := SoundDual.pointConst 1
+  let half : SoundDual 22 vectorBox := SoundDual.pointConst (1 / 2)
+  let quarter : SoundDual 22 vectorBox := SoundDual.pointConst (1 / 4)
+  let ct := t.cos
+  let st := t.sin
+  let data : SoundDual 22 vectorBox × SoundDual 22 vectorBox ×
+      SoundDual 22 vectorBox × SoundDual 22 vectorBox :=
+    if j = 1 then
+      (p.a1 * ct + p.a2 * st - one,
+       -p.a2 * ct + p.a1 * st - half,
+       p.k11, p.k12)
+    else if j = 2 then
+      (-quarter * t.dual * t.dual + p.b1 * t.dual + p.b2,
+       half * t.dual - p.b1 - one,
+       p.k21, p.k22)
+    else if j = 3 then
+      (p.c1 - t.dual, p.c2 + t.dual, p.k31, p.k32)
+    else if j = 4 then
+      (-half * t.dual + p.d1 - one,
+       -quarter * t.dual * t.dual + p.d1 * t.dual + p.d2,
+       p.k41, p.k42)
+    else
+      (p.e1 * ct + p.e2 * st - half,
+       -p.e2 * ct + p.e1 * st - one,
+       p.k51, p.k52)
+  let rr := rotDual t data.1 data.2.1
+  (rr.1 + data.2.2.1, rr.2 + data.2.2.2)
+
+/-- Body-frame derivative coefficients for one phase. -/
+def alphaBetaDual (j : Nat) (t : AngleDual 22 vectorBox)
+    (p : FullVars vectorBox := fullVars) :
+    SoundDual 22 vectorBox × SoundDual 22 vectorBox :=
+  let one : SoundDual 22 vectorBox := SoundDual.pointConst 1
+  let half : SoundDual 22 vectorBox := SoundDual.pointConst (1 / 2)
+  let quarter : SoundDual 22 vectorBox := SoundDual.pointConst (1 / 4)
+  let ct := t.cos
+  let st := t.sin
+  if j = 1 then
+    (-(2 : ℚ) * p.a1 * st + (2 : ℚ) * p.a2 * ct + half,
+     (2 : ℚ) * p.a1 * ct + (2 : ℚ) * p.a2 * st - one)
+  else if j = 2 then
+    (one + (2 : ℚ) * p.b1 - t.dual,
+     -quarter * t.dual * t.dual + p.b1 * t.dual + p.b2 + half)
+  else if j = 3 then
+    (-one - p.c2 - t.dual, one + p.c1 - t.dual)
+  else if j = 4 then
+    (quarter * t.dual * t.dual - p.d1 * t.dual - p.d2 - half,
+     (2 : ℚ) * p.d1 - one - t.dual)
+  else
+    (one - (2 : ℚ) * p.e1 * st + (2 : ℚ) * p.e2 * ct,
+     (2 : ℚ) * p.e1 * ct + (2 : ℚ) * p.e2 * st - half)
+
+/-- World-frame derivative piece. -/
+def pathPrimeDual (j : Nat) (t : AngleDual 22 vectorBox)
+    (p : FullVars vectorBox := fullVars) :
+    SoundDual 22 vectorBox × SoundDual 22 vectorBox :=
+  let ab := alphaBetaDual j t p
+  rotDual t ab.1 ab.2
+
+/-- The complete proof-carrying direct system in manuscript order. -/
+def fullDualOutput : List (SoundDual 22 vectorBox) :=
+  let p := fullVars
+  let halfPi := (1 / 2 : ℚ) * piDual
+  let quarterPi := (1 / 4 : ℚ) * piDual
+  let one : SoundDual 22 vectorBox := SoundDual.pointConst 1
+  let quarter : SoundDual 22 vectorBox := SoundDual.pointConst (1 / 4)
+  let first : List (SoundDual 22 vectorBox) := [
+    p.e1 - p.a1,
+    p.e2 + p.a2,
+    p.d1 + p.b1 - quarterPi,
+    p.d2 - p.b2 - quarterPi * ((2 : ℚ) * p.b1 - quarterPi),
+    p.c2 - p.c1 + halfPi,
+    p.k11 - one + p.a1,
+    p.k12 - quarter,
+    p.a2 + quarter
+  ]
+  let pairs := [
+    (pathPieceDual 1 phiDual p, pathPieceDual 2 phiDual p),
+    (pathPrimeDual 1 phiDual p, pathPrimeDual 2 phiDual p),
+    (pathPieceDual 2 thetaDual p, pathPieceDual 3 thetaDual p),
+    (pathPrimeDual 2 thetaDual p, pathPrimeDual 3 thetaDual p),
+    (pathPieceDual 3 etaDual p, pathPieceDual 4 etaDual p),
+    (pathPieceDual 4 tauDual p, pathPieceDual 5 tauDual p)
+  ]
+  let matchEqs := pairs.flatMap (fun lr =>
+    [lr.1.1 - lr.2.1, lr.1.2 - lr.2.2])
+  let lhs := pathPieceDual 1 phiDual p
+  let xe := pathPieceDual 3 etaDual p
+  let ae := (alphaBetaDual 3 etaDual p).1
+  let be :=
+    (xe.1 - ae * etaDual.sin, xe.2 + ae * etaDual.cos)
+  first ++ matchEqs ++ [lhs.1 - be.1, lhs.2 - be.2]
+
+/-! Public-shape unfold lemmas for the three derivative path pieces.
+
+`Systems.lean` deliberately hides the helper `pathPrimeFromAB`.  When `system`
+is unfolded outside that file, the private helper survives as an inaccessible
+constant, so `ring` cannot see that the right-hand side is just a rotation.
+These `rfl` lemmas expose exactly the public normal form needed by the four
+derivative-matching equations. -/
+private theorem pathPrime1_eq_rot (p : Params) (t : ℝ) :
+    pathPrime1 p t = rot t (alphaBeta1 p t) := rfl
+
+private theorem pathPrime2_eq_rot (p : Params) (t : ℝ) :
+    pathPrime2 p t = rot t (alphaBeta2 p t) := rfl
+
+private theorem pathPrime3_eq_rot (p : Params) (t : ℝ) :
+    pathPrime3 p t = rot t (alphaBeta3 p t) := rfl
+
+/-! The semantic identification is split coordinatewise so each normalization
+gets its own heartbeat budget.  A single 22-way `fin_cases <;> simp <;> ring`
+command is mathematically fine but deterministically exhausts 200000 heartbeats. -/
+
+private theorem fullDualOutput_model_eq_0 (x : Vec 22) :
+    ((fullDualOutput.getD 0 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (0 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_1 (x : Vec 22) :
+    ((fullDualOutput.getD 1 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (1 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_2 (x : Vec 22) :
+    ((fullDualOutput.getD 2 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (2 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]; ring
+
+private theorem fullDualOutput_model_eq_3 (x : Vec 22) :
+    ((fullDualOutput.getD 3 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (3 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]; ring
+
+private theorem fullDualOutput_model_eq_4 (x : Vec 22) :
+    ((fullDualOutput.getD 4 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (4 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]; ring
+
+private theorem fullDualOutput_model_eq_5 (x : Vec 22) :
+    ((fullDualOutput.getD 5 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (5 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_6 (x : Vec 22) :
+    ((fullDualOutput.getD 6 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (6 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_7 (x : Vec 22) :
+    ((fullDualOutput.getD 7 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (7 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_8 (x : Vec 22) :
+    ((fullDualOutput.getD 8 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (8 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_9 (x : Vec 22) :
+    ((fullDualOutput.getD 9 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (9 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_10 (x : Vec 22) :
+    ((fullDualOutput.getD 10 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (10 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1_eq_rot,
+    pathPrime2_eq_rot, pathPrime3_eq_rot, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_11 (x : Vec 22) :
+    ((fullDualOutput.getD 11 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (11 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1_eq_rot,
+    pathPrime2_eq_rot, pathPrime3_eq_rot, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_12 (x : Vec 22) :
+    ((fullDualOutput.getD 12 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (12 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_13 (x : Vec 22) :
+    ((fullDualOutput.getD 13 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (13 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_14 (x : Vec 22) :
+    ((fullDualOutput.getD 14 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (14 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1_eq_rot,
+    pathPrime2_eq_rot, pathPrime3_eq_rot, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_15 (x : Vec 22) :
+    ((fullDualOutput.getD 15 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (15 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1_eq_rot,
+    pathPrime2_eq_rot, pathPrime3_eq_rot, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_16 (x : Vec 22) :
+    ((fullDualOutput.getD 16 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (16 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_17 (x : Vec 22) :
+    ((fullDualOutput.getD 17 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (17 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_18 (x : Vec 22) :
+    ((fullDualOutput.getD 18 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (18 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_19 (x : Vec 22) :
+    ((fullDualOutput.getD 19 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (19 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_20 (x : Vec 22) :
+    ((fullDualOutput.getD 20 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (20 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]
+
+private theorem fullDualOutput_model_eq_21 (x : Vec 22) :
+    ((fullDualOutput.getD 21 (SoundDual.pointConst 0)).model.value x) =
+      vectorSystem x (21 : Fin 22) := by
+  simp [fullDualOutput, fullVars, rotDual, pathPieceDual,
+    alphaBetaDual, pathPrimeDual, vectorSystem, coordEquiv, system, rot,
+    addK, path1, path2, path3, path4, path5, pathPrime1, pathPrime2,
+    pathPrime3, alphaBeta1, alphaBeta2, alphaBeta3]
+
+/-- The real models carried by `fullDualOutput` are exactly the 22 manuscript
+functions in finite-vector coordinates. -/
+theorem fullDualOutput_model_eq (x : Vec 22) :
+    (fun i : Fin 22 =>
+      ((fullDualOutput.getD i.1
+        (SoundDual.pointConst 0)).model.value x)) = vectorSystem x := by
+  funext i
+  fin_cases i
+  · exact fullDualOutput_model_eq_0 x
+  · exact fullDualOutput_model_eq_1 x
+  · exact fullDualOutput_model_eq_2 x
+  · exact fullDualOutput_model_eq_3 x
+  · exact fullDualOutput_model_eq_4 x
+  · exact fullDualOutput_model_eq_5 x
+  · exact fullDualOutput_model_eq_6 x
+  · exact fullDualOutput_model_eq_7 x
+  · exact fullDualOutput_model_eq_8 x
+  · exact fullDualOutput_model_eq_9 x
+  · exact fullDualOutput_model_eq_10 x
+  · exact fullDualOutput_model_eq_11 x
+  · exact fullDualOutput_model_eq_12 x
+  · exact fullDualOutput_model_eq_13 x
+  · exact fullDualOutput_model_eq_14 x
+  · exact fullDualOutput_model_eq_15 x
+  · exact fullDualOutput_model_eq_16 x
+  · exact fullDualOutput_model_eq_17 x
+  · exact fullDualOutput_model_eq_18 x
+  · exact fullDualOutput_model_eq_19 x
+  · exact fullDualOutput_model_eq_20 x
+  · exact fullDualOutput_model_eq_21 x
+
+end Romik
+
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartF.F01AffineOrder`.
+* `KernelOnly.PartF.F01PhaseAlgebra`.
+* `KernelOnly.PartF.F01SetMotion`.
+* `KernelOnly.PartF.F02Coordinates`.
+* `KernelOnly.PartF.F04IntegralPrimitives`.
+* `KernelOnly.PartF.F04IntegralEvaluation`.
+* `KernelOnly.PartF.F05DictionaryEquations`.
+* `KernelOnly.PartF.F06ReverseSystem`.
+* `KernelOnly.PartF.F06FullReconstruction`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# F01: the affine-order bridge for issue #5270
+
+The two constructors are deliberately given different names. No upstream
+definition is changed or imported, and no theorem from the conjecture file
+is used. Both application conventions and the conversion between them are
+proved for an arbitrary linear isometry equivalence.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartF
+
+/-- The Euclidean plane used for the continuous rigid-motion formulation. -/
+abbrev Plane := EuclideanSpace ℝ (Fin 2)
+/-- Affine isometries of the Euclidean plane. -/
+abbrev Rigid := Plane ≃ᵃⁱ[ℝ] Plane
+
+/-- The ordered coordinate-basis orientation used by the pinned upstream
+`FormalConjecturesForMathlib/Geometry/2d.lean`. Importing `Mathlib` alone
+does not install that project's plane-orientation instance. -/
+instance planeOriented : Module.Oriented ℝ Plane (Fin 2) :=
+  ⟨Module.Basis.orientation <| PiLp.basisFun 2 _ _⟩
+
+/-- The second instance supplied by the same upstream plane helper. -/
+instance planeFactFinrank : Fact (Module.finrank ℝ Plane = 2) :=
+  ⟨finrank_euclideanSpace_fin⟩
+
+/-- The topology used by the pinned `MovingSofa.IsMovingSofa` definition. -/
+instance rigidTopology : TopologicalSpace Rigid :=
+  .induced (fun e => e.toAffineIsometry.toContinuousAffineMap) inferInstance
+
+/-- The composition currently implemented upstream: `q ↦ R q + p`. -/
+def rotateThenTranslate (R : Plane ≃ₗᵢ[ℝ] Plane) (p : Plane) : Rigid :=
+  R.toAffineIsometryEquiv.trans (AffineIsometryEquiv.vaddConst ℝ p)
+
+/-- The documented composition: `q ↦ R (q + p)`. -/
+def translateThenRotate (R : Plane ≃ₗᵢ[ℝ] Plane) (p : Plane) : Rigid :=
+  (AffineIsometryEquiv.vaddConst ℝ p).trans R.toAffineIsometryEquiv
+
+@[simp] theorem rotateThenTranslate_apply
+    (R : Plane ≃ₗᵢ[ℝ] Plane) (p q : Plane) :
+    rotateThenTranslate R p q = R q + p := rfl
+
+@[simp] theorem translateThenRotate_apply
+    (R : Plane ≃ₗᵢ[ℝ] Plane) (p q : Plane) :
+    translateThenRotate R p q = R (q + p) := rfl
+
+/-- Keeping the old composition is equivalent to rotating the translation. -/
+theorem translateThenRotate_eq_rotateThenTranslate
+    (R : Plane ≃ₗᵢ[ℝ] Plane) (p : Plane) :
+    translateThenRotate R p = rotateThenTranslate R (R p) := by
+  ext q
+  simp only [translateThenRotate_apply, rotateThenTranslate_apply, map_add]
+
+/-- Counterclockwise rotation of the oriented plane through angle `t`. -/
+def rotation (t : ℝ) : Plane ≃ₗᵢ[ℝ] Plane :=
+  EuclideanGeometry.o.rotation (t : Real.Angle)
+
+/-- Translate by the body-frame offset and then rotate through `t`. -/
+def bodyFrame (t : ℝ) (p : Plane) : Rigid :=
+  translateThenRotate (rotation t) p
+
+/-- Rotate through `t` and then translate by the world position. -/
+def worldFrame (t : ℝ) (x : Plane) : Rigid :=
+  rotateThenTranslate (rotation t) x
+
+theorem bodyFrame_eq_worldFrame (t : ℝ) (p : Plane) :
+    bodyFrame t p = worldFrame t (rotation t p) :=
+  translateThenRotate_eq_rotateThenTranslate (rotation t) p
+
+end GerverSofa.PartF
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# F01: algebraic half of the integral-to-five-phase representation
+
+The coefficient dictionary and the five explicit body-frame branches come
+from the 4 September integral-motion excerpt. They are compared to the
+actual `Romik.path1` ... `Romik.path5` definitions, then assembled using the
+same `if` boundaries as `Romik.path`.
+
+This module does NOT evaluate the integrals: `closedPath` is explicitly
+named as a closed-form candidate. Proving that the pinned integral path
+equals it, and identifying `dictionary d` with the certified 22D tuple,
+remain separate obligations. No such equality is assumed here.
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace GerverSofa.PartF.Phases
+
+/-- The terminal rotation angle π/2. -/
+def T : ℝ := Real.pi / 2
+/-- The auxiliary height parameter `(a + θ - φ - 1) / 2`. -/
+def hStar (d : Reduced.Params) : ℝ := (d.a + d.theta - d.phi - 1) / 2
+/-- Phase 4 integration constant for the vertical primitive. -/
+def U4 (d : Reduced.Params) : ℝ :=
+  (1 + d.a) / 2 * Real.cos d.phi - (d.b + 1 / 2) * Real.sin d.phi
+/-- Phase 4 integration constant for the horizontal primitive. -/
+def V4 (d : Reduced.Params) : ℝ :=
+  1 - (d.b + 1 / 2) * Real.cos d.phi - (1 + d.a) / 2 * Real.sin d.phi
+/-- Phase 3 integration constant for the vertical primitive. -/
+def U3 (d : Reduced.Params) : ℝ :=
+  U4 d + 1 / 2 * Real.sin d.theta - hStar d * Real.cos d.theta
+/-- Phase 3 integration constant for the horizontal primitive. -/
+def V3 (d : Reduced.Params) : ℝ :=
+  V4 d + 1 / 2 * Real.cos d.theta + hStar d * Real.sin d.theta
+/-- Phase 2 integration constant for the vertical primitive. -/
+def U2 (d : Reduced.Params) : ℝ := U4 d
+/-- Phase 2 integration constant for the horizontal primitive. -/
+def V2 (d : Reduced.Params) : ℝ :=
+  V4 d + Real.cos d.theta + 2 * hStar d * Real.sin d.theta
+/-- Phase 1 integration constant for the vertical primitive. -/
+def U1 (d : Reduced.Params) : ℝ :=
+  U2 d + d.a / 2 * Real.cos d.phi - 1 / 2 * Real.sin d.phi
+/-- Phase 1 integration constant for the horizontal primitive. -/
+def V1 (d : Reduced.Params) : ℝ :=
+  V2 d + d.a / 2 * Real.sin d.phi + 1 / 2 * Real.cos d.phi
+/-- The first-phase offset determined by `4 V1 - 2`. -/
+def ell (d : Reduced.Params) : ℝ := 4 * V1 d - 2
+
+theorem U1_eq_half (d : Reduced.Params) (hd : Reduced.Equations d) :
+    U1 d = 1 / 2 := by
+  change Reduced.system d = 0 at hd
+  have h := congrFun hd (2 : Fin 4)
+  change d.a * Real.cos d.phi - Real.sin d.phi - 1 / 2 +
+    1 / 2 * Real.cos d.phi - d.b * Real.sin d.phi = 0 at h
+  dsimp [U1, U2, U4]
+  linear_combination h
+
+theorem ell_eq (d : Reduced.Params) (hd : Reduced.Equations d) :
+    ell d = V2 d + V4 d := by
+  change Reduced.system d = 0 at hd
+  have h := congrFun hd (1 : Fin 4)
+  change d.a * (3 * Real.sin d.theta + Real.sin d.phi) -
+    2 * d.b * Real.cos d.phi +
+    3 * (d.theta - d.phi - 1) * Real.sin d.theta +
+    3 * Real.cos d.theta - Real.sin d.phi + Real.cos d.phi = 0 at h
+  dsimp [ell, V1, V2, V4, hStar]
+  linear_combination h
+
+theorem ell_eq_two_V3 (d : Reduced.Params) (hd : Reduced.Equations d) :
+    ell d = 2 * V3 d := by
+  rw [ell_eq d hd]
+  dsimp [V2, V3]
+  ring
+
+/-- Express all twenty-two Romik parameters in terms of the four reduced parameters. -/
+def dictionary (d : Reduced.Params) : Romik.Params where
+  k11 := 1 - 3 / 2 * (1 - V1 d)
+  k12 := 1 / 4
+  k21 := V4 d
+  k22 := U4 d
+  k31 := V3 d
+  k32 := U3 d
+  k41 := V2 d
+  k42 := U4 d
+  k51 := V1 d - 3 / 2 * (1 - V1 d)
+  k52 := 1 / 4
+  a1 := 3 / 2 * (1 - V1 d)
+  a2 := -1 / 4
+  b1 := (d.phi - 1 - d.a) / 2
+  b2 := d.b - 1 / 2 + (1 + d.a) / 2 * d.phi - d.phi ^ 2 / 4
+  c1 := d.a + T - d.phi - 1
+  c2 := d.a - d.phi - 1
+  d1 := (T - d.phi + 1 + d.a) / 2
+  d2 := d.b - 1 / 2 - (1 + d.a) / 2 * (T - d.phi) - (T - d.phi)^2 / 4
+  e1 := 3 / 2 * (1 - V1 d)
+  e2 := 1 / 4
+  phi := d.phi
+  theta := d.theta
+
+@[simp] theorem dictionary_phi (d : Reduced.Params) : (dictionary d).phi = d.phi := rfl
+@[simp] theorem dictionary_theta (d : Reduced.Params) : (dictionary d).theta = d.theta := rfl
+
+/-- The polynomial profile for phase 2 of the path. -/
+def g2 (d : Reduced.Params) (t : ℝ) : ℝ := (1 + d.a + t - d.phi) / 2
+/-- The polynomial profile for phase 3 of the path. -/
+def g3 (d : Reduced.Params) (t : ℝ) : ℝ := d.a + t - d.phi
+/-- The polynomial profile for phase 4 of the path. -/
+def g4 (d : Reduced.Params) (t : ℝ) : ℝ :=
+  d.b + 1 / 2 - (T - d.phi - t) * (1 + d.a) / 2 - (T - d.phi - t)^2 / 4
+
+/-- The closed rotation-path formula for phase 1. -/
+def closed1 (d : Reduced.Params) (t : ℝ) : Point :=
+  (Real.cos t - 1,
+    -1 / 2 + 1 / 2 * Real.cos t + (V1 d - ell d) * Real.sin t)
+
+/-- The closed rotation-path formula for phase 2. -/
+def closed2 (d : Reduced.Params) (t : ℝ) : Point :=
+  (g4 d (T - t) + U4 d * Real.sin t + V4 d * Real.cos t - 1,
+    g2 d t + U4 d * Real.cos t + (V2 d - ell d) * Real.sin t - 1)
+
+/-- The closed rotation-path formula for phase 3. -/
+def closed3 (d : Reduced.Params) (t : ℝ) : Point :=
+  (g3 d (T - t) + U3 d * Real.sin t + V3 d * Real.cos t - 1,
+    g3 d t + U3 d * Real.cos t + (V3 d - ell d) * Real.sin t - 1)
+
+/-- The closed rotation-path formula for phase 4. -/
+def closed4 (d : Reduced.Params) (t : ℝ) : Point :=
+  (g2 d (T - t) + U4 d * Real.sin t + V2 d * Real.cos t - 1,
+    g4 d t + U4 d * Real.cos t + (V4 d - ell d) * Real.sin t - 1)
+
+/-- The closed rotation-path formula for phase 5. -/
+def closed5 (d : Reduced.Params) (t : ℝ) : Point :=
+  (-1 / 2 + V1 d * Real.cos t + 1 / 2 * Real.sin t,
+    (1 - ell d) * Real.sin t - 1)
+
+theorem phase1 (d : Reduced.Params) (t : ℝ) :
+    Romik.rot t (closed1 d t) = Romik.path1 (dictionary d) t := by
+  apply Prod.ext
+  · dsimp [Romik.rot, closed1, Romik.path1, Romik.addK, dictionary, ell]
+    linear_combination (1 - 3 / 2 * (1 - V1 d)) * Real.sin_sq_add_cos_sq t
+  · dsimp [Romik.rot, closed1, Romik.path1, Romik.addK, dictionary, ell]
+    linear_combination (1 / 4 : ℝ) * Real.sin_sq_add_cos_sq t
+
+theorem phase2 (d : Reduced.Params) (hd : Reduced.Equations d) (t : ℝ) :
+    Romik.rot t (closed2 d t) = Romik.path2 (dictionary d) t := by
+  unfold closed2
+  rw [ell_eq d hd]
+  apply Prod.ext
+  · dsimp [Romik.rot, Romik.path2, Romik.addK, dictionary, g2, g4, T]
+    linear_combination V4 d * Real.sin_sq_add_cos_sq t
+  · dsimp [Romik.rot, Romik.path2, Romik.addK, dictionary, g2, g4, T]
+    linear_combination U4 d * Real.sin_sq_add_cos_sq t
+
+theorem phase3 (d : Reduced.Params) (hd : Reduced.Equations d) (t : ℝ) :
+    Romik.rot t (closed3 d t) = Romik.path3 (dictionary d) t := by
+  unfold closed3
+  rw [ell_eq_two_V3 d hd]
+  apply Prod.ext
+  · dsimp [Romik.rot, Romik.path3, Romik.addK, dictionary, g3, T]
+    linear_combination V3 d * Real.sin_sq_add_cos_sq t
+  · dsimp [Romik.rot, Romik.path3, Romik.addK, dictionary, g3, T]
+    linear_combination U3 d * Real.sin_sq_add_cos_sq t
+
+theorem phase4 (d : Reduced.Params) (hd : Reduced.Equations d) (t : ℝ) :
+    Romik.rot t (closed4 d t) = Romik.path4 (dictionary d) t := by
+  unfold closed4
+  rw [ell_eq d hd]
+  apply Prod.ext
+  · dsimp [Romik.rot, Romik.path4, Romik.addK, dictionary, g2, g4, T]
+    linear_combination V2 d * Real.sin_sq_add_cos_sq t
+  · dsimp [Romik.rot, Romik.path4, Romik.addK, dictionary, g2, g4, T]
+    linear_combination U4 d * Real.sin_sq_add_cos_sq t
+
+theorem phase5 (d : Reduced.Params) (t : ℝ) :
+    Romik.rot t (closed5 d t) = Romik.path5 (dictionary d) t := by
+  apply Prod.ext
+  · dsimp [Romik.rot, closed5, Romik.path5, Romik.addK, dictionary, ell]
+    linear_combination (V1 d - 3 / 2 * (1 - V1 d)) * Real.sin_sq_add_cos_sq t
+  · dsimp [Romik.rot, closed5, Romik.path5, Romik.addK, dictionary, ell]
+    linear_combination (1 / 4 : ℝ) * Real.sin_sq_add_cos_sq t
+
+/-- The five-branch closed formula for the rotation path. -/
+def closedPath (d : Reduced.Params) (t : ℝ) : Point :=
+  if t ≤ d.phi then closed1 d t
+  else if t ≤ d.theta then closed2 d t
+  else if t ≤ Real.pi / 2 - d.theta then closed3 d t
+  else if t ≤ Real.pi / 2 - d.phi then closed4 d t
+  else closed5 d t
+
+/-- Exact assembly for the explicit closed path, including every switching
+endpoint. This is not yet the corresponding theorem for the integral path. -/
+theorem fivePhaseRepresentation (d : Reduced.Params)
+    (hd : Reduced.Equations d) (t : ℝ) :
+    Romik.rot t (closedPath d t) = Romik.path (dictionary d) t := by
+  -- Expose both sets of conditions, including the local `eta` and `tau`
+  -- definitions inside `Romik.path`, before choosing a branch.
+  change Romik.rot t
+      (if t ≤ d.phi then closed1 d t
+       else if t ≤ d.theta then closed2 d t
+       else if t ≤ Real.pi / 2 - d.theta then closed3 d t
+       else if t ≤ Real.pi / 2 - d.phi then closed4 d t
+       else closed5 d t) =
+    (if t ≤ d.phi then Romik.path1 (dictionary d) t
+     else if t ≤ d.theta then Romik.path2 (dictionary d) t
+     else if t ≤ Real.pi / 2 - d.theta then Romik.path3 (dictionary d) t
+     else if t ≤ Real.pi / 2 - d.phi then Romik.path4 (dictionary d) t
+     else Romik.path5 (dictionary d) t)
+  by_cases h1 : t ≤ d.phi
+  · simpa only [ite_eq_left h1] using phase1 d t
+  by_cases h2 : t ≤ d.theta
+  · simpa only [ite_eq_right h1, ite_eq_left h2] using phase2 d hd t
+  by_cases h3 : t ≤ Real.pi / 2 - d.theta
+  · simpa only [ite_eq_right h1, ite_eq_right h2, ite_eq_left h3] using phase3 d hd t
+  by_cases h4 : t ≤ Real.pi / 2 - d.phi
+  · simpa only [ite_eq_right h1, ite_eq_right h2, ite_eq_right h3, ite_eq_left h4] using phase4 d
+      hd t
+  · simpa only [ite_eq_right h1, ite_eq_right h2, ite_eq_right h3, ite_eq_right h4] using phase5 d t
+
+end GerverSofa.PartF.Phases
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# F01: supporting intersections and their continuous inverse motion
+
+`Model.IsMovingSofa` has the seven fields and the rigid-motion topology of
+the upstream definition. This local, independently named model avoids
+importing upstream conjectures with unfinished proof terms. Its concrete
+instantiation for the integral Gerver path still needs the analytic bridge.
+-/
+
+@[expose] public section
+
+noncomputable section
+open scoped unitInterval
+
+namespace GerverSofa.PartF
+
+/-- Intersect the endpoint hallway arms and all intermediate moving hallways. -/
+def frameIntersection (F : I → Rigid) (H V L : Set Plane) : Set Plane :=
+  F 0 '' H ∩ F 1 '' V ∩ ⋂ s, F s '' L
+
+/-- The hallway intersection parametrized by angles from zero to π/2. -/
+def angleIntersection (F : ℝ → Rigid) (H V L : Set Plane) : Set Plane :=
+  F 0 '' H ∩ F (Real.pi / 2) '' V ∩
+    ⋂ t ∈ Set.Icc 0 (Real.pi / 2), F t '' L
+
+/-- The hallway intersection generated by a path in body-frame coordinates. -/
+def bodySofa (p : ℝ → Plane) (H V L : Set Plane) : Set Plane :=
+  angleIntersection (fun t => bodyFrame t (p t)) H V L
+
+/-- The hallway intersection generated by a path in world-frame coordinates. -/
+def worldSofa (x : ℝ → Plane) (H V L : Set Plane) : Set Plane :=
+  angleIntersection (fun t => worldFrame t (x t)) H V L
+
+theorem bodySofa_eq_worldSofa_of_path_identity
+    (p x : ℝ → Plane) (H V L : Set Plane)
+    (hpath : ∀ t ∈ Set.Icc 0 (Real.pi / 2), x t = rotation t (p t)) :
+    bodySofa p H V L = worldSofa x H V L := by
+  have hT : (0 : ℝ) ≤ Real.pi / 2 := by positivity
+  have h0 := hpath 0 ⟨le_rfl, hT⟩
+  have h1 := hpath (Real.pi / 2) ⟨hT, le_rfl⟩
+  -- Expose the applications before rewriting under the image coercions.
+  change
+    bodyFrame 0 (p 0) '' H ∩
+      bodyFrame (Real.pi / 2) (p (Real.pi / 2)) '' V ∩
+        (⋂ t ∈ Set.Icc 0 (Real.pi / 2), bodyFrame t (p t) '' L) =
+    worldFrame 0 (x 0) '' H ∩
+      worldFrame (Real.pi / 2) (x (Real.pi / 2)) '' V ∩
+        (⋂ t ∈ Set.Icc 0 (Real.pi / 2), worldFrame t (x t) '' L)
+  rw [bodyFrame_eq_worldFrame, ← h0, bodyFrame_eq_worldFrame, ← h1]
+  congr 1
+  apply Set.iInter_congr
+  intro t
+  apply Set.iInter_congr
+  intro ht
+  rw [bodyFrame_eq_worldFrame, ← hpath t ht]
+
+/-- Normalizing the physical angle does not change the intersection. -/
+theorem angleIntersection_eq_frameIntersection
+    (F : ℝ → Rigid) (H V L : Set Plane) :
+    angleIntersection F H V L =
+      frameIntersection (fun s : I => F ((s : ℝ) * (Real.pi / 2))) H V L := by
+  have hT : (0 : ℝ) < Real.pi / 2 := by positivity
+  have hz : ((0 : I) : ℝ) = 0 := rfl
+  have ho : ((1 : I) : ℝ) = 1 := rfl
+  ext q
+  simp only [angleIntersection, frameIntersection, Set.mem_inter_iff,
+    Set.mem_iInter, hz, ho, zero_mul, one_mul]
+  constructor
+  · rintro ⟨hends, hall⟩
+    refine ⟨hends, ?_⟩
+    intro s
+    apply hall
+    exact ⟨mul_nonneg s.property.1 hT.le,
+      by nlinarith [s.property.2]⟩
+  · rintro ⟨hends, hall⟩
+    refine ⟨hends, ?_⟩
+    intro t ht
+    let s : I := ⟨t / (Real.pi / 2), div_nonneg ht.1 hT.le,
+      (div_le_iff₀ hT).2 (by simpa using ht.2)⟩
+    have h := hall s
+    simpa only [s, div_mul_cancel₀ t (ne_of_gt hT)] using h
+
+namespace Model
+
+/-- The horizontal unit-width hallway arm extending to the left. -/
+def horizontalHallway : Set Plane := {q | q 0 ≤ 1 ∧ 0 ≤ q 1 ∧ q 1 ≤ 1}
+/-- The vertical unit-width hallway arm extending downwards. -/
+def verticalHallway : Set Plane := {q | 0 ≤ q 0 ∧ q 0 ≤ 1 ∧ q 1 ≤ 1}
+/-- The union of the two perpendicular unit-width hallway arms. -/
+def hallway : Set Plane := horizontalHallway ∪ verticalHallway
+
+/-- A nonempty closed connected set moving continuously between the hallway arms. -/
+structure IsMovingSofa (S : Set Plane) (m : I → Rigid) : Prop where
+  isConnected : IsConnected S
+  isClosed : IsClosed S
+  continuous : Continuous m
+  zero : m 0 = AffineIsometryEquiv.refl ℝ Plane
+  initial : S ⊆ horizontalHallway
+  subset_hallway : ∀ s, m s '' S ⊆ hallway
+  final : m 1 '' S ⊆ verticalHallway
+
+end Model
+
+end GerverSofa.PartF
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# F02: the coordinate homeomorphism
+
+The product `Point = ℝ × ℝ` and `Plane = EuclideanSpace ℝ (Fin 2)`
+have the same coordinates and topology. Their norms differ. Accordingly,
+the coordinate identification below is a linear equivalence and a
+homeomorphism. Euclidean isometries are constructed separately.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartF.Coordinates
+
+/-- Convert a pair of real coordinates to a Euclidean vector. -/
+def toPlane (q : Point) : Plane := WithLp.toLp 2 ![q.1, q.2]
+
+/-- Extract the two real coordinates of a Euclidean vector. -/
+def fromPlane (q : Plane) : Point := (q 0, q 1)
+
+@[simp] theorem toPlane_zero_coord (q : Point) : toPlane q 0 = q.1 := rfl
+@[simp] theorem toPlane_one_coord (q : Point) : toPlane q 1 = q.2 := rfl
+
+theorem plane_ext {q r : Plane} (h0 : q 0 = r 0) (h1 : q 1 = r 1) : q = r := by
+  ext i
+  fin_cases i
+  · exact h0
+  · exact h1
+
+@[simp] theorem fromPlane_toPlane (q : Point) : fromPlane (toPlane q) = q := rfl
+
+@[simp] theorem toPlane_fromPlane (q : Plane) : toPlane (fromPlane q) = q := by
+  exact plane_ext rfl rfl
+
+theorem toPlane_add (q r : Point) : toPlane (q + r) = toPlane q + toPlane r := by
+  exact plane_ext rfl rfl
+
+theorem toPlane_smul (c : ℝ) (q : Point) : toPlane (c • q) = c • toPlane q := by
+  exact plane_ext rfl rfl
+
+/-- The coordinate equivalence, without any assertion that the two norms agree. -/
+def linearEquiv : Point ≃ₗ[ℝ] Plane where
+  toFun := toPlane
+  invFun := fromPlane
+  left_inv := fromPlane_toPlane
+  right_inv := toPlane_fromPlane
+  map_add' := toPlane_add
+  map_smul' := toPlane_smul
+
+/-- The coordinate homeomorphism between real pairs and the Euclidean plane. -/
+def homeomorph : Point ≃ₜ Plane := linearEquiv.toContinuousLinearEquiv.toHomeomorph
+
+theorem continuous_toPlane : Continuous toPlane :=
+  linearEquiv.toContinuousLinearEquiv.continuous
+
+theorem continuous_fromPlane : Continuous fromPlane :=
+  linearEquiv.toContinuousLinearEquiv.symm.continuous
+
+theorem image_eq_preimage (S : Set Point) :
+    toPlane '' S = fromPlane ⁻¹' S := by
+  ext q
+  constructor
+  · rintro ⟨p, hp, rfl⟩
+    exact hp
+  · intro hq
+    exact ⟨fromPlane q, hq, toPlane_fromPlane q⟩
+
+theorem isClosed_image {S : Set Point} (hS : IsClosed S) : IsClosed (toPlane '' S) := by
+  rw [image_eq_preimage]
+  exact hS.preimage continuous_fromPlane
+
+theorem isConnected_image {S : Set Point} (hS : IsConnected S) :
+    IsConnected (toPlane '' S) :=
+  hS.image toPlane continuous_toPlane.continuousOn
+
+theorem horizontal_image : toPlane '' horizontalArm = Model.horizontalHallway := by
+  ext q
+  constructor
+  · rintro ⟨p, hp, rfl⟩
+    exact hp
+  · intro hq
+    exact ⟨fromPlane q, hq, toPlane_fromPlane q⟩
+
+theorem vertical_image : toPlane '' verticalArm = Model.verticalHallway := by
+  ext q
+  constructor
+  · rintro ⟨p, hp, rfl⟩
+    exact hp
+  · intro hq
+    exact ⟨fromPlane q, hq, toPlane_fromPlane q⟩
+
+theorem hallway_image : toPlane '' GerverSofa.hallway = Model.hallway := by
+  simp only [GerverSofa.hallway, Model.hallway, Set.image_union,
+    horizontal_image, vertical_image]
+
+end GerverSofa.PartF.Coordinates
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# F04: literal integral data and its branch primitives
+
+The parameterized definitions below are the pinned upstream scalar integrals.
+The phase boundaries are preserved. Integrating across the jumps will use
+equality on open intervals, not an incorrect global continuity assertion for r.
+-/
+
+@[expose] public section
+
+noncomputable section
+open MeasureTheory
+namespace GerverSofa.PartF.Integrals
+
+open Phases
+
+/-- The reflected second switching angle `π/2 - θ`. -/
+def eta (d : Reduced.Params) : ℝ := T - d.theta
+/-- The reflected first switching angle `π/2 - φ`. -/
+def tau (d : Reduced.Params) : ℝ := T - d.phi
+/-- The switching-angle condition `0 ≤ φ ≤ θ ≤ π/4`. -/
+def Ordered (d : Reduced.Params) : Prop :=
+  0 ≤ d.phi ∧ d.phi ≤ d.theta ∧ d.theta ≤ Real.pi / 4
+
+theorem ordered_knots (d : Reduced.Params) (ho : Ordered d) :
+    0 ≤ d.phi ∧ d.phi ≤ d.theta ∧ d.theta ≤ eta d ∧
+      eta d ≤ tau d ∧ tau d ≤ T := by
+  rcases ho with ⟨h0, h1, h2⟩
+  dsimp [eta, tau, T]
+  exact ⟨h0, h1, by linarith, by linarith, by linarith⟩
+
+/-- The integrand profile on phase 1. -/
+def r1 (_d : Reduced.Params) (_t : ℝ) : ℝ := 1 / 2
+/-- The integrand profile on phase 2. -/
+def r2 (d : Reduced.Params) (t : ℝ) : ℝ := g2 d t
+/-- The integrand profile on phase 3. -/
+def r3 (d : Reduced.Params) (t : ℝ) : ℝ := g3 d t
+/-- The integrand profile on phase 4. -/
+def r4 (d : Reduced.Params) (t : ℝ) : ℝ :=
+  d.b - (T - t - d.phi) * (1 + d.a) / 2 - (T - t - d.phi) ^ 2 / 4
+
+/-- The piecewise phase profile used in the integral representation. -/
+def r (d : Reduced.Params) (t : ℝ) : ℝ :=
+  if t ≤ d.phi then r1 d t
+  else if t ≤ d.theta then r2 d t
+  else if t ≤ eta d then r3 d t
+  else if t ≤ tau d then r4 d t
+  else 0
+
+/-- One minus the cosine-weighted profile integral from `t` to the last switching angle. -/
+def xi (d : Reduced.Params) (t : ℝ) : ℝ :=
+  1 - ∫ s in t..tau d, r d s * Real.cos s
+/-- The sine-weighted profile integral from `t` to the last switching angle. -/
+def zeta (d : Reduced.Params) (t : ℝ) : ℝ :=
+  ∫ s in t..tau d, r d s * Real.sin s
+
+/-- The rotation path expressed using the two profile integrals. -/
+def path (d : Reduced.Params) (t : ℝ) : Point :=
+  (if t ≤ d.phi then Real.cos t - 1
+   else xi d (T - t) * Real.cos t + zeta d (T - t) * Real.sin t - 1,
+   if t ≤ tau d then
+     zeta d t * Real.cos t - (4 * xi d 0 - 2 - xi d t) * Real.sin t - 1
+   else -(4 * xi d 0 - 3) * Real.sin t - 1)
+
+/-- The derivative formula for the fourth phase profile. -/
+def dg4 (d : Reduced.Params) (t : ℝ) : ℝ :=
+  (1 + d.a) / 2 + (T - d.phi - t) / 2
+
+/-- The horizontal primitive assembled from a profile, its derivative, and an offset. -/
+def primitiveX (V : ℝ) (g gp : ℝ → ℝ) (t : ℝ) : ℝ :=
+  V + g t * Real.sin t + gp t * Real.cos t
+/-- The vertical primitive assembled from a profile, its derivative, and an offset. -/
+def primitiveY (U : ℝ) (g gp : ℝ → ℝ) (t : ℝ) : ℝ :=
+  U + g t * Real.cos t - gp t * Real.sin t
+
+/-- Horizontal primitive specialized to phase 1. -/
+def X1 (d : Reduced.Params) : ℝ → ℝ := primitiveX (V1 d) (r1 d) (fun _ => 0)
+/-- Vertical primitive specialized to phase 1. -/
+def Y1 (d : Reduced.Params) : ℝ → ℝ := primitiveY (U1 d) (r1 d) (fun _ => 0)
+/-- Horizontal primitive specialized to phase 2. -/
+def X2 (d : Reduced.Params) : ℝ → ℝ := primitiveX (V2 d) (g2 d) (fun _ => 1 / 2)
+/-- Vertical primitive specialized to phase 2. -/
+def Y2 (d : Reduced.Params) : ℝ → ℝ := primitiveY (U2 d) (g2 d) (fun _ => 1 / 2)
+/-- Horizontal primitive specialized to phase 3. -/
+def X3 (d : Reduced.Params) : ℝ → ℝ := primitiveX (V3 d) (g3 d) (fun _ => 1)
+/-- Vertical primitive specialized to phase 3. -/
+def Y3 (d : Reduced.Params) : ℝ → ℝ := primitiveY (U3 d) (g3 d) (fun _ => 1)
+/-- Horizontal primitive specialized to phase 4. -/
+def X4 (d : Reduced.Params) : ℝ → ℝ := primitiveX (V4 d) (g4 d) (dg4 d)
+/-- Vertical primitive specialized to phase 4. -/
+def Y4 (d : Reduced.Params) : ℝ → ℝ := primitiveY (U4 d) (g4 d) (dg4 d)
+
+/-- F05 repair: normalize only the scalar derivative, never typeclass arguments. -/
+theorem primitiveX_hasDerivAt (V : ℝ) {g gp : ℝ → ℝ} {t gpp : ℝ}
+    (hg : HasDerivAt g (gp t) t) (hgp : HasDerivAt gp gpp t) :
+    HasDerivAt (primitiveX V g gp) ((g t + gpp) * Real.cos t) t := by
+  change HasDerivAt (fun s => V + g s * Real.sin s + gp s * Real.cos s) _ t
+  have h := ((hasDerivAt_const t V).fun_add
+    (hg.fun_mul (Real.hasDerivAt_sin t))).fun_add
+    (hgp.fun_mul (Real.hasDerivAt_cos t))
+  exact h.congr_deriv (by ring)
+
+theorem primitiveY_hasDerivAt (U : ℝ) {g gp : ℝ → ℝ} {t gpp : ℝ}
+    (hg : HasDerivAt g (gp t) t) (hgp : HasDerivAt gp gpp t) :
+    HasDerivAt (primitiveY U g gp) (-(g t + gpp) * Real.sin t) t := by
+  change HasDerivAt (fun s => U + g s * Real.cos s - gp s * Real.sin s) _ t
+  have h := ((hasDerivAt_const t U).fun_add
+    (hg.fun_mul (Real.hasDerivAt_cos t))).fun_sub
+    (hgp.fun_mul (Real.hasDerivAt_sin t))
+  exact h.congr_deriv (by ring)
+
+theorem g2_hasDerivAt (d : Reduced.Params) (t : ℝ) :
+    HasDerivAt (g2 d) (1 / 2) t := by
+  change HasDerivAt (fun s => (1 + d.a + s - d.phi) / 2) _ t
+  have h := (((hasDerivAt_const t (1 + d.a)).fun_add
+    (hasDerivAt_id t)).fun_sub (hasDerivAt_const t d.phi)).div_const 2
+  exact h.congr_deriv (by ring)
+
+theorem g3_hasDerivAt (d : Reduced.Params) (t : ℝ) :
+    HasDerivAt (g3 d) 1 t := by
+  change HasDerivAt (fun s => d.a + s - d.phi) _ t
+  have h := ((hasDerivAt_const t d.a).fun_add
+    (hasDerivAt_id t)).fun_sub (hasDerivAt_const t d.phi)
+  exact h.congr_deriv (by ring)
+
+theorem g4_hasDerivAt (d : Reduced.Params) (t : ℝ) :
+    HasDerivAt (g4 d) (dg4 d t) t := by
+  change HasDerivAt (fun s => d.b + 1 / 2 -
+    (T - d.phi - s) * (1 + d.a) / 2 - (T - d.phi - s)^2 / 4) _ t
+  have h := (hasDerivAt_const t (T - d.phi)).fun_sub (hasDerivAt_id t)
+  have hfull := ((hasDerivAt_const t (d.b + 1 / 2)).fun_sub
+    ((h.mul_const (1 + d.a)).div_const 2)).fun_sub ((h.pow 2).div_const 4)
+  exact hfull.congr_deriv (by dsimp [dg4]; ring)
+
+theorem dg4_hasDerivAt (d : Reduced.Params) (t : ℝ) :
+    HasDerivAt (dg4 d) (-1 / 2) t := by
+  change HasDerivAt (fun s => (1 + d.a) / 2 + (T - d.phi - s) / 2) _ t
+  have h := (hasDerivAt_const t ((1 + d.a) / 2)).fun_add
+    (((hasDerivAt_const t (T - d.phi)).fun_sub (hasDerivAt_id t)).div_const 2)
+  exact h.congr_deriv (by ring)
+
+theorem X1_hasDerivAt (d : Reduced.Params) (t : ℝ) :
+    HasDerivAt (X1 d) (r1 d t * Real.cos t) t := by
+  have h := primitiveX_hasDerivAt (V1 d) (g := r1 d) (gp := fun _ => 0)
+    (hasDerivAt_const t (1 / 2 : ℝ)) (hasDerivAt_const t (0 : ℝ))
+  exact h.congr_deriv (by dsimp [r1, r2, r3, r4, g2, g3, g4]; ring)
+
+theorem Y1_hasDerivAt (d : Reduced.Params) (t : ℝ) :
+    HasDerivAt (Y1 d) (-r1 d t * Real.sin t) t := by
+  have h := primitiveY_hasDerivAt (U1 d) (g := r1 d) (gp := fun _ => 0)
+    (hasDerivAt_const t (1 / 2 : ℝ)) (hasDerivAt_const t (0 : ℝ))
+  exact h.congr_deriv (by dsimp [r1, r2, r3, r4, g2, g3, g4]; ring)
+
+theorem X2_hasDerivAt (d : Reduced.Params) (t : ℝ) :
+    HasDerivAt (X2 d) (r2 d t * Real.cos t) t := by
+  have h := primitiveX_hasDerivAt (V2 d) (g := g2 d) (gp := fun _ => 1 / 2)
+    (g2_hasDerivAt d t) (hasDerivAt_const t (1 / 2 : ℝ))
+  exact h.congr_deriv (by dsimp [r1, r2, r3, r4, g2, g3, g4]; ring)
+
+theorem Y2_hasDerivAt (d : Reduced.Params) (t : ℝ) :
+    HasDerivAt (Y2 d) (-r2 d t * Real.sin t) t := by
+  have h := primitiveY_hasDerivAt (U2 d) (g := g2 d) (gp := fun _ => 1 / 2)
+    (g2_hasDerivAt d t) (hasDerivAt_const t (1 / 2 : ℝ))
+  exact h.congr_deriv (by dsimp [r1, r2, r3, r4, g2, g3, g4]; ring)
+
+theorem X3_hasDerivAt (d : Reduced.Params) (t : ℝ) :
+    HasDerivAt (X3 d) (r3 d t * Real.cos t) t := by
+  have h := primitiveX_hasDerivAt (V3 d) (g := g3 d) (gp := fun _ => 1)
+    (g3_hasDerivAt d t) (hasDerivAt_const t (1 : ℝ))
+  exact h.congr_deriv (by dsimp [r1, r2, r3, r4, g2, g3, g4]; ring)
+
+theorem Y3_hasDerivAt (d : Reduced.Params) (t : ℝ) :
+    HasDerivAt (Y3 d) (-r3 d t * Real.sin t) t := by
+  have h := primitiveY_hasDerivAt (U3 d) (g := g3 d) (gp := fun _ => 1)
+    (g3_hasDerivAt d t) (hasDerivAt_const t (1 : ℝ))
+  exact h.congr_deriv (by dsimp [r1, r2, r3, r4, g2, g3, g4]; ring)
+
+theorem X4_hasDerivAt (d : Reduced.Params) (t : ℝ) :
+    HasDerivAt (X4 d) (r4 d t * Real.cos t) t := by
+  have h := primitiveX_hasDerivAt (V4 d) (g := g4 d) (gp := dg4 d)
+    (g4_hasDerivAt d t) (dg4_hasDerivAt d t)
+  exact h.congr_deriv (by dsimp [r1, r2, r3, r4, g2, g3, g4]; ring)
+
+theorem Y4_hasDerivAt (d : Reduced.Params) (t : ℝ) :
+    HasDerivAt (Y4 d) (-r4 d t * Real.sin t) t := by
+  have h := primitiveY_hasDerivAt (U4 d) (g := g4 d) (gp := dg4 d)
+    (g4_hasDerivAt d t) (dg4_hasDerivAt d t)
+  exact h.congr_deriv (by dsimp [r1, r2, r3, r4, g2, g3, g4]; ring)
+
+theorem X1_join (d : Reduced.Params) :
+    X1 d d.phi = X2 d d.phi := by
+  dsimp [X1, X2, primitiveX, r1, g2, g3,
+    U1, U2, U3, V1, V2, V3, hStar]
+  ring
+
+theorem Y1_join (d : Reduced.Params) :
+    Y1 d d.phi = Y2 d d.phi := by
+  dsimp [Y1, Y2, primitiveY, r1, g2, g3,
+    U1, U2, U3, V1, V2, V3, hStar]
+  ring
+
+theorem X2_join (d : Reduced.Params) :
+    X2 d d.theta = X3 d d.theta := by
+  dsimp [X2, X3, primitiveX, r1, g2, g3,
+    U1, U2, U3, V1, V2, V3, hStar]
+  ring
+
+theorem Y2_join (d : Reduced.Params) :
+    Y2 d d.theta = Y3 d d.theta := by
+  dsimp [Y2, Y3, primitiveY, r1, g2, g3,
+    U1, U2, U3, V1, V2, V3, hStar]
+  ring
+
+theorem fourth_equation (d : Reduced.Params) (hd : Reduced.Equations d) :
+    d.a + T - d.phi - d.theta - d.b +
+      (1 / 2 : ℝ) * (d.theta - d.phi) * (1 + d.a) +
+      (1 / 4 : ℝ) * (d.theta - d.phi) * (d.theta - d.phi) = 0 := by
+  exact congrFun hd (3 : Fin 4)
+
+theorem X3_join (d : Reduced.Params) (hd : Reduced.Equations d) :
+    X3 d (eta d) = X4 d (eta d) := by
+  have he := fourth_equation d hd
+  dsimp [X3, X4, primitiveX, eta, g3, g4, dg4, U3, V3, hStar, T] at *
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  linear_combination Real.cos d.theta * he
+
+theorem Y3_join (d : Reduced.Params) (hd : Reduced.Equations d) :
+    Y3 d (eta d) = Y4 d (eta d) := by
+  have he := fourth_equation d hd
+  dsimp [Y3, Y4, primitiveY, eta, g3, g4, dg4, U3, V3, hStar, T] at *
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  linear_combination Real.sin d.theta * he
+
+theorem X4_terminal (d : Reduced.Params) :
+    X4 d (tau d) = 1 := by
+  dsimp [X4, primitiveX, tau, g4, dg4, U4, V4, T]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  ring
+
+theorem Y4_terminal (d : Reduced.Params) :
+    Y4 d (tau d) = 0 := by
+  dsimp [Y4, primitiveY, tau, g4, dg4, U4, V4, T]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  ring
+
+theorem r_phase1 (d : Reduced.Params) {t : ℝ} (ht : t ≤ d.phi) :
+    r d t = r1 d t := by simp only [r, ite_eq_left ht]
+
+theorem r_phase2 (d : Reduced.Params) {t : ℝ}
+    (hlo : d.phi < t) (hhi : t ≤ d.theta) : r d t = r2 d t := by
+  simp only [r, ite_eq_right (not_le.mpr hlo), ite_eq_left hhi]
+
+theorem r_phase3 (d : Reduced.Params) (ho : Ordered d) {t : ℝ}
+    (hlo : d.theta < t) (hhi : t ≤ eta d) : r d t = r3 d t := by
+  have hphi : d.phi < t := lt_of_le_of_lt ho.2.1 hlo
+  simp only [r, ite_eq_right (not_le.mpr hphi), ite_eq_right (not_le.mpr hlo), ite_eq_left hhi]
+
+theorem r_phase4 (d : Reduced.Params) (ho : Ordered d) {t : ℝ}
+    (hlo : eta d < t) (hhi : t ≤ tau d) : r d t = r4 d t := by
+  have htheta : d.theta < t := lt_of_le_of_lt (ordered_knots d ho).2.2.1 hlo
+  have hphi : d.phi < t := lt_of_le_of_lt ho.2.1 htheta
+  simp only [r, ite_eq_right (not_le.mpr hphi), ite_eq_right (not_le.mpr htheta),
+    ite_eq_right (not_le.mpr hlo), ite_eq_left hhi]
+
+end GerverSofa.PartF.Integrals
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# F04: evaluation of the literal integrals on all four closed intervals
+
+The fundamental theorem is applied to smooth branch primitives. Equality with
+the discontinuous integrand is required only on the open interval. Adjacent
+integrals are then added, using the proved matching values at every switch.
+-/
+
+@[expose] public section
+
+noncomputable section
+open MeasureTheory
+namespace GerverSofa.PartF.Integrals
+
+open Phases
+
+theorem integrate_branch (f f' P : ℝ → ℝ) (a b : ℝ)
+    (hab : a ≤ b) (hf' : Continuous f')
+    (hderiv : ∀ t, HasDerivAt P (f' t) t)
+    (heq : ∀ t ∈ Set.Ioo a b, f t = f' t) :
+    IntervalIntegrable f volume a b ∧
+      (∫ t in a..b, f t) = P b - P a := by
+  have hi : IntervalIntegrable f' volume a b := hf'.intervalIntegrable a b
+  have hf : IntervalIntegrable f volume a b := hi.congr_uIoo (by
+    rw [Set.uIoo_of_le hab]
+    intro t ht
+    exact (heq t ht).symm)
+  refine ⟨hf, ?_⟩
+  calc
+    (∫ t in a..b, f t) = ∫ t in a..b, f' t :=
+      intervalIntegral.integral_congr_Ioo_of_le hab heq
+    _ = P b - P a :=
+      intervalIntegral.integral_eq_sub_of_hasDerivAt (fun t _ => hderiv t) hi
+
+theorem cos_tail4 (d : Reduced.Params) (ho : Ordered d)
+    (t : ℝ) (ht : t ∈ Set.Icc (eta d) (tau d)) :
+    IntervalIntegrable (fun u => r d u * Real.cos u) volume t (tau d) ∧
+      (∫ u in t..tau d, r d u * Real.cos u) = 1 - X4 d t := by
+  have hpiece := integrate_branch
+    (fun u => r d u * Real.cos u) (fun u => r4 d u * Real.cos u)
+    (X4 d) t (tau d) ht.2
+    (by dsimp [r1, r2, r3, r4, g2, g3]; fun_prop)
+    (by
+      intro u
+      exact X4_hasDerivAt d u)
+    (by
+      intro u hu
+      rw [r_phase4 d ho (lt_of_le_of_lt ht.1 hu.1) (le_of_lt hu.2)])
+  refine ⟨hpiece.1, ?_⟩
+  rw [hpiece.2]
+  rw [X4_terminal]
+
+theorem sin_tail4 (d : Reduced.Params) (ho : Ordered d)
+    (t : ℝ) (ht : t ∈ Set.Icc (eta d) (tau d)) :
+    IntervalIntegrable (fun u => r d u * Real.sin u) volume t (tau d) ∧
+      (∫ u in t..tau d, r d u * Real.sin u) = Y4 d t := by
+  have hpiece := integrate_branch
+    (fun u => r d u * Real.sin u) (fun u => r4 d u * Real.sin u)
+    (fun u => -Y4 d u) t (tau d) ht.2
+    (by dsimp [r1, r2, r3, r4, g2, g3]; fun_prop)
+    (by
+      intro u
+      exact (Y4_hasDerivAt d u).fun_neg.congr_deriv (by ring))
+    (by
+      intro u hu
+      rw [r_phase4 d ho (lt_of_le_of_lt ht.1 hu.1) (le_of_lt hu.2)])
+  refine ⟨hpiece.1, ?_⟩
+  rw [hpiece.2]
+  rw [Y4_terminal]; ring
+
+theorem cos_tail3 (d : Reduced.Params) (ho : Ordered d) (hd : Reduced.Equations d)
+    (t : ℝ) (ht : t ∈ Set.Icc (d.theta) (eta d)) :
+    IntervalIntegrable (fun u => r d u * Real.cos u) volume t (tau d) ∧
+      (∫ u in t..tau d, r d u * Real.cos u) = 1 - X3 d t := by
+  have hpiece := integrate_branch
+    (fun u => r d u * Real.cos u) (fun u => r3 d u * Real.cos u)
+    (X3 d) t (eta d) ht.2
+    (by dsimp [r1, r2, r3, r4, g2, g3]; fun_prop)
+    (by
+      intro u
+      exact X3_hasDerivAt d u)
+    (by
+      intro u hu
+      rw [r_phase3 d ho (lt_of_le_of_lt ht.1 hu.1) (le_of_lt hu.2)])
+  have hrest := cos_tail4 d ho (eta d) ⟨le_rfl, (ordered_knots d ho).2.2.2.1⟩
+  refine ⟨hpiece.1.trans hrest.1, ?_⟩
+  rw [← intervalIntegral.integral_add_adjacent_intervals hpiece.1 hrest.1,
+    hpiece.2, hrest.2]
+  rw [X3_join d hd]; ring
+
+theorem sin_tail3 (d : Reduced.Params) (ho : Ordered d) (hd : Reduced.Equations d)
+    (t : ℝ) (ht : t ∈ Set.Icc (d.theta) (eta d)) :
+    IntervalIntegrable (fun u => r d u * Real.sin u) volume t (tau d) ∧
+      (∫ u in t..tau d, r d u * Real.sin u) = Y3 d t := by
+  have hpiece := integrate_branch
+    (fun u => r d u * Real.sin u) (fun u => r3 d u * Real.sin u)
+    (fun u => -Y3 d u) t (eta d) ht.2
+    (by dsimp [r1, r2, r3, r4, g2, g3]; fun_prop)
+    (by
+      intro u
+      exact (Y3_hasDerivAt d u).fun_neg.congr_deriv (by ring))
+    (by
+      intro u hu
+      rw [r_phase3 d ho (lt_of_le_of_lt ht.1 hu.1) (le_of_lt hu.2)])
+  have hrest := sin_tail4 d ho (eta d) ⟨le_rfl, (ordered_knots d ho).2.2.2.1⟩
+  refine ⟨hpiece.1.trans hrest.1, ?_⟩
+  rw [← intervalIntegral.integral_add_adjacent_intervals hpiece.1 hrest.1,
+    hpiece.2, hrest.2]
+  rw [Y3_join d hd]; ring
+
+theorem cos_tail2 (d : Reduced.Params) (ho : Ordered d) (hd : Reduced.Equations d)
+    (t : ℝ) (ht : t ∈ Set.Icc (d.phi) (d.theta)) :
+    IntervalIntegrable (fun u => r d u * Real.cos u) volume t (tau d) ∧
+      (∫ u in t..tau d, r d u * Real.cos u) = 1 - X2 d t := by
+  have hpiece := integrate_branch
+    (fun u => r d u * Real.cos u) (fun u => r2 d u * Real.cos u)
+    (X2 d) t (d.theta) ht.2
+    (by dsimp [r1, r2, r3, r4, g2, g3]; fun_prop)
+    (by
+      intro u
+      exact X2_hasDerivAt d u)
+    (by
+      intro u hu
+      rw [r_phase2 d (lt_of_le_of_lt ht.1 hu.1) (le_of_lt hu.2)])
+  have hrest := cos_tail3 d ho hd (d.theta) ⟨le_rfl, (ordered_knots d ho).2.2.1⟩
+  refine ⟨hpiece.1.trans hrest.1, ?_⟩
+  rw [← intervalIntegral.integral_add_adjacent_intervals hpiece.1 hrest.1,
+    hpiece.2, hrest.2]
+  rw [X2_join d]; ring
+
+theorem sin_tail2 (d : Reduced.Params) (ho : Ordered d) (hd : Reduced.Equations d)
+    (t : ℝ) (ht : t ∈ Set.Icc (d.phi) (d.theta)) :
+    IntervalIntegrable (fun u => r d u * Real.sin u) volume t (tau d) ∧
+      (∫ u in t..tau d, r d u * Real.sin u) = Y2 d t := by
+  have hpiece := integrate_branch
+    (fun u => r d u * Real.sin u) (fun u => r2 d u * Real.sin u)
+    (fun u => -Y2 d u) t (d.theta) ht.2
+    (by dsimp [r1, r2, r3, r4, g2, g3]; fun_prop)
+    (by
+      intro u
+      exact (Y2_hasDerivAt d u).fun_neg.congr_deriv (by ring))
+    (by
+      intro u hu
+      rw [r_phase2 d (lt_of_le_of_lt ht.1 hu.1) (le_of_lt hu.2)])
+  have hrest := sin_tail3 d ho hd (d.theta) ⟨le_rfl, (ordered_knots d ho).2.2.1⟩
+  refine ⟨hpiece.1.trans hrest.1, ?_⟩
+  rw [← intervalIntegral.integral_add_adjacent_intervals hpiece.1 hrest.1,
+    hpiece.2, hrest.2]
+  rw [Y2_join d]; ring
+
+theorem cos_tail1 (d : Reduced.Params) (ho : Ordered d) (hd : Reduced.Equations d)
+    (t : ℝ) (ht : t ∈ Set.Icc (0) (d.phi)) :
+    IntervalIntegrable (fun u => r d u * Real.cos u) volume t (tau d) ∧
+      (∫ u in t..tau d, r d u * Real.cos u) = 1 - X1 d t := by
+  have hpiece := integrate_branch
+    (fun u => r d u * Real.cos u) (fun u => r1 d u * Real.cos u)
+    (X1 d) t (d.phi) ht.2
+    (by dsimp [r1, r2, r3, r4, g2, g3]; fun_prop)
+    (by
+      intro u
+      exact X1_hasDerivAt d u)
+    (by
+      intro u hu
+      rw [r_phase1 d (le_of_lt hu.2)])
+  have hrest := cos_tail2 d ho hd (d.phi) ⟨le_rfl, ho.2.1⟩
+  refine ⟨hpiece.1.trans hrest.1, ?_⟩
+  rw [← intervalIntegral.integral_add_adjacent_intervals hpiece.1 hrest.1,
+    hpiece.2, hrest.2]
+  rw [X1_join d]; ring
+
+theorem sin_tail1 (d : Reduced.Params) (ho : Ordered d) (hd : Reduced.Equations d)
+    (t : ℝ) (ht : t ∈ Set.Icc (0) (d.phi)) :
+    IntervalIntegrable (fun u => r d u * Real.sin u) volume t (tau d) ∧
+      (∫ u in t..tau d, r d u * Real.sin u) = Y1 d t := by
+  have hpiece := integrate_branch
+    (fun u => r d u * Real.sin u) (fun u => r1 d u * Real.sin u)
+    (fun u => -Y1 d u) t (d.phi) ht.2
+    (by dsimp [r1, r2, r3, r4, g2, g3]; fun_prop)
+    (by
+      intro u
+      exact (Y1_hasDerivAt d u).fun_neg.congr_deriv (by ring))
+    (by
+      intro u hu
+      rw [r_phase1 d (le_of_lt hu.2)])
+  have hrest := sin_tail2 d ho hd (d.phi) ⟨le_rfl, ho.2.1⟩
+  refine ⟨hpiece.1.trans hrest.1, ?_⟩
+  rw [← intervalIntegral.integral_add_adjacent_intervals hpiece.1 hrest.1,
+    hpiece.2, hrest.2]
+  rw [Y1_join d]; ring
+
+theorem xi_zeta_phase1 (d : Reduced.Params) (ho : Ordered d) (hd : Reduced.Equations d)
+    (t : ℝ) (ht : t ∈ Set.Icc (0) (d.phi)) :
+    xi d t = X1 d t ∧ zeta d t = Y1 d t := by
+  constructor
+  · unfold xi
+    rw [(cos_tail1 d ho hd t ht).2]
+    ring
+  · exact (sin_tail1 d ho hd t ht).2
+
+theorem xi_zeta_phase2 (d : Reduced.Params) (ho : Ordered d) (hd : Reduced.Equations d)
+    (t : ℝ) (ht : t ∈ Set.Icc (d.phi) (d.theta)) :
+    xi d t = X2 d t ∧ zeta d t = Y2 d t := by
+  constructor
+  · unfold xi
+    rw [(cos_tail2 d ho hd t ht).2]
+    ring
+  · exact (sin_tail2 d ho hd t ht).2
+
+theorem xi_zeta_phase3 (d : Reduced.Params) (ho : Ordered d) (hd : Reduced.Equations d)
+    (t : ℝ) (ht : t ∈ Set.Icc (d.theta) (eta d)) :
+    xi d t = X3 d t ∧ zeta d t = Y3 d t := by
+  constructor
+  · unfold xi
+    rw [(cos_tail3 d ho hd t ht).2]
+    ring
+  · exact (sin_tail3 d ho hd t ht).2
+
+theorem xi_zeta_phase4 (d : Reduced.Params) (ho : Ordered d)
+    (t : ℝ) (ht : t ∈ Set.Icc (eta d) (tau d)) :
+    xi d t = X4 d t ∧ zeta d t = Y4 d t := by
+  constructor
+  · unfold xi
+    rw [(cos_tail4 d ho t ht).2]
+    ring
+  · exact (sin_tail4 d ho t ht).2
+
+theorem xi_zero (d : Reduced.Params) (ho : Ordered d)
+    (hd : Reduced.Equations d) : xi d 0 = V1 d := by
+  have h := (xi_zeta_phase1 d ho hd 0 ⟨le_rfl, ho.1⟩).1
+  simpa [X1, primitiveX] using h
+
+/-- The projection `ξ sin t + ζ cos t` used in the integral identities. -/
+def W (d : Reduced.Params) (t : ℝ) : ℝ :=
+  xi d t * Real.sin t + zeta d t * Real.cos t
+
+theorem primitive_W (g gp U V t : ℝ) :
+    (V + g * Real.sin t + gp * Real.cos t) * Real.sin t +
+      (U + g * Real.cos t - gp * Real.sin t) * Real.cos t =
+      g + U * Real.cos t + V * Real.sin t := by
+  linear_combination g * Real.sin_sq_add_cos_sq t
+
+theorem W_phase1 (d : Reduced.Params) (ho : Ordered d) (hd : Reduced.Equations d)
+    (t : ℝ) (ht : t ∈ Set.Icc (0) (d.phi)) :
+    W d t = r1 d t + U1 d * Real.cos t + V1 d * Real.sin t := by
+  rcases xi_zeta_phase1 d ho hd t ht with ⟨hx, hy⟩
+  unfold W
+  rw [hx, hy]
+  exact primitive_W (r1 d t) (0) (U1 d) (V1 d) t
+
+theorem W_phase2 (d : Reduced.Params) (ho : Ordered d) (hd : Reduced.Equations d)
+    (t : ℝ) (ht : t ∈ Set.Icc (d.phi) (d.theta)) :
+    W d t = g2 d t + U2 d * Real.cos t + V2 d * Real.sin t := by
+  rcases xi_zeta_phase2 d ho hd t ht with ⟨hx, hy⟩
+  unfold W
+  rw [hx, hy]
+  exact primitive_W (g2 d t) (1 / 2) (U2 d) (V2 d) t
+
+theorem W_phase3 (d : Reduced.Params) (ho : Ordered d) (hd : Reduced.Equations d)
+    (t : ℝ) (ht : t ∈ Set.Icc (d.theta) (eta d)) :
+    W d t = g3 d t + U3 d * Real.cos t + V3 d * Real.sin t := by
+  rcases xi_zeta_phase3 d ho hd t ht with ⟨hx, hy⟩
+  unfold W
+  rw [hx, hy]
+  exact primitive_W (g3 d t) (1) (U3 d) (V3 d) t
+
+theorem W_phase4 (d : Reduced.Params) (ho : Ordered d)
+    (t : ℝ) (ht : t ∈ Set.Icc (eta d) (tau d)) :
+    W d t = g4 d t + U4 d * Real.cos t + V4 d * Real.sin t := by
+  rcases xi_zeta_phase4 d ho t ht with ⟨hx, hy⟩
+  unfold W
+  rw [hx, hy]
+  exact primitive_W (g4 d t) (dg4 d t) (U4 d) (V4 d) t
+
+end GerverSofa.PartF.Integrals
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# F05: the four-parameter dictionary satisfies the full 22 equations
+
+These polynomial certificates use the four reduced equations and the two
+trigonometric circle identities. No box membership or uniqueness is assumed.
+They do not identify the dictionary with the independently certified 22D root.
+The module is independent of the F04 integral evaluation.
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace GerverSofa.PartF.Phases
+
+theorem dictionary_equations (d : Reduced.Params) (hd : Reduced.Equations d) :
+    Romik.Equations (dictionary d) := by
+  have h1 := congrFun hd (0 : Fin 4)
+  have h2 := congrFun hd (1 : Fin 4)
+  have h3 := congrFun hd (2 : Fin 4)
+  have h4 := congrFun hd (3 : Fin 4)
+  change d.a * (Real.cos d.theta - Real.cos d.phi) -
+    2 * d.b * Real.sin d.phi + (d.theta - d.phi - 1) * Real.cos d.theta -
+    Real.sin d.theta + Real.cos d.phi + Real.sin d.phi = 0 at h1
+  change d.a * (3 * Real.sin d.theta + Real.sin d.phi) -
+    2 * d.b * Real.cos d.phi + 3 * (d.theta - d.phi - 1) * Real.sin d.theta +
+    3 * Real.cos d.theta - Real.sin d.phi + Real.cos d.phi = 0 at h2
+  change d.a * Real.cos d.phi - Real.sin d.phi - 1 / 2 +
+    1 / 2 * Real.cos d.phi - d.b * Real.sin d.phi = 0 at h3
+  change d.a + Real.pi / 2 - d.phi - d.theta - d.b +
+    1 / 2 * (d.theta - d.phi) * (1 + d.a) +
+    1 / 4 * (d.theta - d.phi) * (d.theta - d.phi) = 0 at h4
+  have hcphi := Real.sin_sq_add_cos_sq d.phi
+  have hctheta := Real.sin_sq_add_cos_sq d.theta
+  change Romik.system (dictionary d) = 0
+  funext i
+  fin_cases i
+  -- Row 0: exact scalar certificate.
+  · change (dictionary d).e1 - (dictionary d).a1 = 0
+    dsimp [dictionary, T]; ring
+  -- Row 1: exact scalar certificate.
+  · change (dictionary d).e2 + (dictionary d).a2 = 0
+    dsimp [dictionary, T]; ring
+  -- Row 2: exact scalar certificate.
+  · change (dictionary d).d1 + (dictionary d).b1 - Real.pi / 4 = 0
+    dsimp [dictionary, T]; ring
+  -- Row 3: exact scalar certificate.
+  · change (dictionary d).d2 - (dictionary d).b2 - Real.pi / 4 * (2 * (dictionary d).b1 - Real.pi
+    / 4) = 0
+    dsimp [dictionary, T]; ring
+  -- Row 4: exact scalar certificate.
+  · change (dictionary d).c2 - (dictionary d).c1 + Real.pi / 2 = 0
+    dsimp [dictionary, T]; ring
+  -- Row 5: exact scalar certificate.
+  · change (dictionary d).k11 - 1 + (dictionary d).a1 = 0
+    dsimp [dictionary, T]; ring
+  -- Row 6: exact scalar certificate.
+  · change (dictionary d).k12 - 1 / 4 = 0
+    dsimp [dictionary, T]; ring
+  -- Row 7: exact scalar certificate.
+  · change (dictionary d).a2 + 1 / 4 = 0
+    dsimp [dictionary, T]; ring
+  -- Row 8: exact scalar certificate.
+  · change (Romik.path1 (dictionary d) (d.phi)).1 - (Romik.path2 (dictionary d) (d.phi)).1 = 0
+    dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.path4, Romik.path5,
+      Romik.rot, Romik.addK, Romik.alphaBeta1, Romik.alphaBeta2, Romik.alphaBeta3,
+      dictionary, U1, U2, U3, U4, V1, V2, V3, V4, hStar, T]
+    linear_combination ((Real.sin d.phi) ^ 2) * h2 +
+      ((Real.sin d.phi) * (Real.cos d.phi)) * h3 +
+      (-1 * (d.a) * (Real.sin d.phi) + (-3 / 2 : ℝ) * (d.a) * (Real.sin d.theta) + (3 / 2 : ℝ) *
+        (d.b) * (Real.cos d.phi) + (3 / 2 : ℝ) * (d.phi) * (Real.sin d.theta) + (-3 / 2 : ℝ) *
+        (d.theta) * (Real.sin d.theta) + (1 / 4 : ℝ) * (Real.sin d.phi) + (3 / 2 : ℝ) * (Real.sin
+        d.theta) + (-3 / 2 : ℝ) * (Real.cos d.theta)) * hcphi
+  -- Row 9: exact scalar certificate.
+  · change (Romik.path1 (dictionary d) (d.phi)).2 - (Romik.path2 (dictionary d) (d.phi)).2 = 0
+    dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.path4, Romik.path5,
+      Romik.rot, Romik.addK, Romik.alphaBeta1, Romik.alphaBeta2, Romik.alphaBeta3,
+      dictionary, U1, U2, U3, U4, V1, V2, V3, V4, hStar, T]
+    linear_combination (-1 * (Real.sin d.phi) * (Real.cos d.phi)) * h2 +
+      ((Real.sin d.phi) ^ 2 + -1) * h3 +
+      ((d.b) * (Real.sin d.phi) + (Real.sin d.phi) + (1 / 4 : ℝ)) * hcphi
+  -- Row 10: exact scalar certificate.
+  · change (Romik.rot (d.phi) (Romik.alphaBeta1 (dictionary d) (d.phi))).1 - (Romik.rot (d.phi)
+    (Romik.alphaBeta2 (dictionary d) (d.phi))).1 = 0
+    dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.path4, Romik.path5,
+      Romik.rot, Romik.addK, Romik.alphaBeta1, Romik.alphaBeta2, Romik.alphaBeta3,
+      dictionary, U1, U2, U3, U4, V1, V2, V3, V4, hStar, T]
+    linear_combination (2 * (Real.sin d.phi) * (Real.cos d.phi)) * h2 +
+      (-2 * (Real.sin d.phi) ^ 2 + 1) * h3 +
+      (-2 * (d.b) * (Real.sin d.phi) + -2 * (Real.sin d.phi) + (-1 / 2 : ℝ)) * hcphi
+  -- Row 11: exact scalar certificate.
+  · change (Romik.rot (d.phi) (Romik.alphaBeta1 (dictionary d) (d.phi))).2 - (Romik.rot (d.phi)
+    (Romik.alphaBeta2 (dictionary d) (d.phi))).2 = 0
+    dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.path4, Romik.path5,
+      Romik.rot, Romik.addK, Romik.alphaBeta1, Romik.alphaBeta2, Romik.alphaBeta3,
+      dictionary, U1, U2, U3, U4, V1, V2, V3, V4, hStar, T]
+    linear_combination (2 * (Real.sin d.phi) ^ 2 + -1) * h2 +
+      (2 * (Real.sin d.phi) * (Real.cos d.phi)) * h3 +
+      (-2 * (d.a) * (Real.sin d.phi) + -3 * (d.a) * (Real.sin d.theta) + 3 * (d.b) * (Real.cos
+        d.phi) + 3 * (d.phi) * (Real.sin d.theta) + -3 * (d.theta) * (Real.sin d.theta) + (1 / 2 :
+        ℝ) * (Real.sin d.phi) + 3 * (Real.sin d.theta) + -3 * (Real.cos d.theta)) * hcphi
+  -- Row 12: exact scalar certificate.
+  · change (Romik.path2 (dictionary d) (d.theta)).1 - (Romik.path3 (dictionary d) (d.theta)).1 = 0
+    dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.path4, Romik.path5,
+      Romik.rot, Romik.addK, Romik.alphaBeta1, Romik.alphaBeta2, Romik.alphaBeta3,
+      dictionary, U1, U2, U3, U4, V1, V2, V3, V4, hStar, T]
+    linear_combination (-1 * (Real.cos d.theta)) * h4
+  -- Row 13: exact scalar certificate.
+  · change (Romik.path2 (dictionary d) (d.theta)).2 - (Romik.path3 (dictionary d) (d.theta)).2 = 0
+    dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.path4, Romik.path5,
+      Romik.rot, Romik.addK, Romik.alphaBeta1, Romik.alphaBeta2, Romik.alphaBeta3,
+      dictionary, U1, U2, U3, U4, V1, V2, V3, V4, hStar, T]
+    linear_combination (-1 * (Real.sin d.theta)) * h4
+  -- Row 14: exact scalar certificate.
+  · change (Romik.rot (d.theta) (Romik.alphaBeta2 (dictionary d) (d.theta))).1 - (Romik.rot
+    (d.theta) (Romik.alphaBeta3 (dictionary d) (d.theta))).1 = 0
+    dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.path4, Romik.path5,
+      Romik.rot, Romik.addK, Romik.alphaBeta1, Romik.alphaBeta2, Romik.alphaBeta3,
+      dictionary, U1, U2, U3, U4, V1, V2, V3, V4, hStar, T]
+    linear_combination ((Real.sin d.theta)) * h4
+  -- Row 15: exact scalar certificate.
+  · change (Romik.rot (d.theta) (Romik.alphaBeta2 (dictionary d) (d.theta))).2 - (Romik.rot
+    (d.theta) (Romik.alphaBeta3 (dictionary d) (d.theta))).2 = 0
+    dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.path4, Romik.path5,
+      Romik.rot, Romik.addK, Romik.alphaBeta1, Romik.alphaBeta2, Romik.alphaBeta3,
+      dictionary, U1, U2, U3, U4, V1, V2, V3, V4, hStar, T]
+    linear_combination (-1 * (Real.cos d.theta)) * h4
+  -- Row 16: exact scalar certificate.
+  · change (Romik.path3 (dictionary d) (Real.pi / 2 - d.theta)).1 - (Romik.path4 (dictionary d)
+    (Real.pi / 2 - d.theta)).1 = 0
+    dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.path4, Romik.path5,
+      Romik.rot, Romik.addK, Romik.alphaBeta1, Romik.alphaBeta2, Romik.alphaBeta3,
+      dictionary, U1, U2, U3, U4, V1, V2, V3, V4, hStar, T]
+    simp only [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+    linear_combination (-1 * (Real.cos d.theta)) * h4
+  -- Row 17: exact scalar certificate.
+  · change (Romik.path3 (dictionary d) (Real.pi / 2 - d.theta)).2 - (Romik.path4 (dictionary d)
+    (Real.pi / 2 - d.theta)).2 = 0
+    dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.path4, Romik.path5,
+      Romik.rot, Romik.addK, Romik.alphaBeta1, Romik.alphaBeta2, Romik.alphaBeta3,
+      dictionary, U1, U2, U3, U4, V1, V2, V3, V4, hStar, T]
+    simp only [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+    linear_combination ((Real.sin d.theta)) * h4
+  -- Row 18: exact scalar certificate.
+  · change (Romik.path4 (dictionary d) (Real.pi / 2 - d.phi)).1 - (Romik.path5 (dictionary d)
+    (Real.pi / 2 - d.phi)).1 = 0
+    dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.path4, Romik.path5,
+      Romik.rot, Romik.addK, Romik.alphaBeta1, Romik.alphaBeta2, Romik.alphaBeta3,
+      dictionary, U1, U2, U3, U4, V1, V2, V3, V4, hStar, T]
+    simp only [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+    linear_combination ((Real.sin d.phi) ^ 2 + -1) * h2 +
+      ((Real.sin d.phi) * (Real.cos d.phi)) * h3 +
+      (-1 * (d.a) * (Real.sin d.phi) + (-3 / 2 : ℝ) * (d.a) * (Real.sin d.theta) + (3 / 2 : ℝ) *
+        (d.b) * (Real.cos d.phi) + (3 / 2 : ℝ) * (d.phi) * (Real.sin d.theta) + (-3 / 2 : ℝ) *
+        (d.theta) * (Real.sin d.theta) + (1 / 4 : ℝ) * (Real.sin d.phi) + (3 / 2 : ℝ) * (Real.sin
+        d.theta) + (-3 / 2 : ℝ) * (Real.cos d.theta)) * hcphi
+  -- Row 19: exact scalar certificate.
+  · change (Romik.path4 (dictionary d) (Real.pi / 2 - d.phi)).2 - (Romik.path5 (dictionary d)
+    (Real.pi / 2 - d.phi)).2 = 0
+    dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.path4, Romik.path5,
+      Romik.rot, Romik.addK, Romik.alphaBeta1, Romik.alphaBeta2, Romik.alphaBeta3,
+      dictionary, U1, U2, U3, U4, V1, V2, V3, V4, hStar, T]
+    simp only [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+    linear_combination ((Real.sin d.phi) * (Real.cos d.phi)) * h2 +
+      (-1 * (Real.sin d.phi) ^ 2 + 1) * h3 +
+      (-1 * (d.b) * (Real.sin d.phi) + -1 * (Real.sin d.phi) + (-1 / 4 : ℝ)) * hcphi
+  -- Row 20: exact scalar certificate.
+  · change (Romik.path1 (dictionary d) d.phi).1 - ((Romik.path3 (dictionary d) (Real.pi / 2 -
+    d.theta)).1 - (Romik.alphaBeta3 (dictionary d) (Real.pi / 2 - d.theta)).1 * Real.sin (Real.pi
+    / 2 - d.theta)) = 0
+    dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.path4, Romik.path5,
+      Romik.rot, Romik.addK, Romik.alphaBeta1, Romik.alphaBeta2, Romik.alphaBeta3,
+      dictionary, U1, U2, U3, U4, V1, V2, V3, V4, hStar, T]
+    simp only [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+    linear_combination ((Real.sin d.phi) ^ 2 + (-1 / 2 : ℝ)) * h2 +
+      ((Real.sin d.phi) * (Real.cos d.phi)) * h3 +
+      (-1 * (d.a) * (Real.sin d.phi) + (-3 / 2 : ℝ) * (d.a) * (Real.sin d.theta) + (3 / 2 : ℝ) *
+        (d.b) * (Real.cos d.phi) + (3 / 2 : ℝ) * (d.phi) * (Real.sin d.theta) + (-3 / 2 : ℝ) *
+        (d.theta) * (Real.sin d.theta) + (1 / 4 : ℝ) * (Real.sin d.phi) + (3 / 2 : ℝ) * (Real.sin
+        d.theta) + (-3 / 2 : ℝ) * (Real.cos d.theta)) * hcphi
+  -- Row 21: exact scalar certificate.
+  · change (Romik.path1 (dictionary d) d.phi).2 - ((Romik.path3 (dictionary d) (Real.pi / 2 -
+    d.theta)).2 + (Romik.alphaBeta3 (dictionary d) (Real.pi / 2 - d.theta)).1 * Real.cos (Real.pi
+    / 2 - d.theta)) = 0
+    dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.path4, Romik.path5,
+      Romik.rot, Romik.addK, Romik.alphaBeta1, Romik.alphaBeta2, Romik.alphaBeta3,
+      dictionary, U1, U2, U3, U4, V1, V2, V3, V4, hStar, T]
+    simp only [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+    linear_combination ((-1 / 2 : ℝ)) * h1 +
+      (-1 * (Real.sin d.phi) * (Real.cos d.phi)) * h2 +
+      ((Real.sin d.phi) ^ 2 + -1) * h3 +
+      ((d.b) * (Real.sin d.phi) + (Real.sin d.phi) + (1 / 4 : ℝ)) * hcphi
+
+/-- Read back the four free parameters from the 22D representation. -/
+def undictionary (p : Romik.Params) : Reduced.Params where
+  a := p.phi - 1 - 2 * p.b1
+  b := p.b2 + 1 / 2 - (1 + (p.phi - 1 - 2 * p.b1)) * p.phi / 2 + p.phi ^ 2 / 4
+  phi := p.phi
+  theta := p.theta
+
+theorem undictionary_dictionary (d : Reduced.Params) :
+    undictionary (dictionary d) = d := by
+  rcases d with ⟨a, b, phi, theta⟩
+  dsimp [undictionary, dictionary]
+  congr 1 <;> ring
+
+end GerverSofa.PartF.Phases
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# F06: reverse reduction from the full Romik equations
+
+This direction uses velocity matching and the two contact equations. It does
+not assume membership in either numerical box or any strict angle inequality.
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace GerverSofa.PartF.Phases
+
+theorem rot_injective (t : ℝ) : Function.Injective (Romik.rot t) := by
+  intro z w h
+  have hx := congrArg Prod.fst h
+  have hy := congrArg Prod.snd h
+  dsimp [Romik.rot] at hx hy
+  apply Prod.ext
+  · linear_combination Real.cos t * hx + Real.sin t * hy -
+      (z.1 - w.1) * Real.sin_sq_add_cos_sq t
+  · linear_combination -Real.sin t * hx + Real.cos t * hy -
+      (z.2 - w.2) * Real.sin_sq_add_cos_sq t
+
+theorem alphaBeta12_eq {p : Romik.Params} (hp : Romik.Equations p) :
+    Romik.alphaBeta1 p p.phi = Romik.alphaBeta2 p p.phi := by
+  apply rot_injective p.phi
+  apply Prod.ext
+  · have h := congrFun hp (10 : Fin 22)
+    change (Romik.rot p.phi (Romik.alphaBeta1 p p.phi)).1 -
+      (Romik.rot p.phi (Romik.alphaBeta2 p p.phi)).1 = 0 at h
+    exact sub_eq_zero.mp h
+  · have h := congrFun hp (11 : Fin 22)
+    change (Romik.rot p.phi (Romik.alphaBeta1 p p.phi)).2 -
+      (Romik.rot p.phi (Romik.alphaBeta2 p p.phi)).2 = 0 at h
+    exact sub_eq_zero.mp h
+
+theorem alphaBeta23_eq {p : Romik.Params} (hp : Romik.Equations p) :
+    Romik.alphaBeta2 p p.theta = Romik.alphaBeta3 p p.theta := by
+  apply rot_injective p.theta
+  apply Prod.ext
+  · have h := congrFun hp (14 : Fin 22)
+    change (Romik.rot p.theta (Romik.alphaBeta2 p p.theta)).1 -
+      (Romik.rot p.theta (Romik.alphaBeta3 p p.theta)).1 = 0 at h
+    exact sub_eq_zero.mp h
+  · have h := congrFun hp (15 : Fin 22)
+    change (Romik.rot p.theta (Romik.alphaBeta2 p p.theta)).2 -
+      (Romik.rot p.theta (Romik.alphaBeta3 p p.theta)).2 = 0 at h
+    exact sub_eq_zero.mp h
+
+theorem c2_eq_of_full {p : Romik.Params} (hp : Romik.Equations p) :
+    p.c2 = -2 - 2 * p.b1 := by
+  have h := congrArg Prod.fst (alphaBeta23_eq hp)
+  dsimp [Romik.alphaBeta2, Romik.alphaBeta3] at h
+  linarith only [h]
+
+theorem c1_eq_of_full {p : Romik.Params} (hp : Romik.Equations p) :
+    p.c1 = Real.pi / 2 - 2 - 2 * p.b1 := by
+  have hs := Romik.c2_eq_c1_sub_halfPi_of_equations hp
+  have hc := c2_eq_of_full hp
+  linarith only [hs, hc]
+
+theorem reverse_equation4 {p : Romik.Params} (hp : Romik.Equations p) :
+    Reduced.system (undictionary p) 3 = 0 := by
+  have h := congrArg Prod.snd (alphaBeta23_eq hp)
+  dsimp [Romik.alphaBeta2, Romik.alphaBeta3] at h
+  rw [c1_eq_of_full hp] at h
+  change (undictionary p).a + Real.pi / 2 - (undictionary p).phi -
+    (undictionary p).theta - (undictionary p).b +
+    1 / 2 * ((undictionary p).theta - (undictionary p).phi) *
+      (1 + (undictionary p).a) +
+    1 / 4 * ((undictionary p).theta - (undictionary p).phi) *
+      ((undictionary p).theta - (undictionary p).phi) = 0
+  dsimp [undictionary]
+  linear_combination -h
+
+theorem reverse_equation3 {p : Romik.Params} (hp : Romik.Equations p) :
+    Reduced.system (undictionary p) 2 = 0 := by
+  have ha := congrArg Prod.fst (alphaBeta12_eq hp)
+  have hb := congrArg Prod.snd (alphaBeta12_eq hp)
+  dsimp [Romik.alphaBeta1, Romik.alphaBeta2] at ha hb
+  rw [Romik.a2_eq_neg_quarter_of_equations hp] at ha hb
+  change (undictionary p).a * Real.cos (undictionary p).phi -
+    Real.sin (undictionary p).phi - 1 / 2 +
+    1 / 2 * Real.cos (undictionary p).phi -
+    (undictionary p).b * Real.sin (undictionary p).phi = 0
+  dsimp [undictionary]
+  linear_combination Real.cos p.phi * ha + Real.sin p.phi * hb +
+    (1 / 2 : ℝ) * Real.sin_sq_add_cos_sq p.phi
+
+theorem reverse_equation1 {p : Romik.Params} (hp : Romik.Equations p) :
+    Reduced.system (undictionary p) 0 = 0 := by
+  have h9 := congrFun hp (9 : Fin 22)
+  have h13 := congrFun hp (13 : Fin 22)
+  have h21 := congrFun hp (21 : Fin 22)
+  change (Romik.path1 p p.phi).2 - (Romik.path2 p p.phi).2 = 0 at h9
+  change (Romik.path2 p p.theta).2 - (Romik.path3 p p.theta).2 = 0 at h13
+  change (Romik.path1 p p.phi).2 -
+    ((Romik.path3 p (Real.pi / 2 - p.theta)).2 +
+      (Romik.alphaBeta3 p (Real.pi / 2 - p.theta)).1 *
+        Real.cos (Real.pi / 2 - p.theta)) = 0 at h21
+  dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.rot, Romik.addK,
+    Romik.alphaBeta3] at h9 h13 h21
+  simp only [c1_eq_of_full hp, c2_eq_of_full hp] at h13 h21
+  simp only [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub] at h21
+  have h4 := reverse_equation4 hp
+  change (undictionary p).a + Real.pi / 2 - (undictionary p).phi -
+    (undictionary p).theta - (undictionary p).b +
+    1 / 2 * ((undictionary p).theta - (undictionary p).phi) *
+      (1 + (undictionary p).a) +
+    1 / 4 * ((undictionary p).theta - (undictionary p).phi) *
+      ((undictionary p).theta - (undictionary p).phi) = 0 at h4
+  change (undictionary p).a * (Real.cos (undictionary p).theta -
+    Real.cos (undictionary p).phi) - 2 * (undictionary p).b *
+    Real.sin (undictionary p).phi +
+    ((undictionary p).theta - (undictionary p).phi - 1) *
+    Real.cos (undictionary p).theta - Real.sin (undictionary p).theta +
+    Real.cos (undictionary p).phi + Real.sin (undictionary p).phi = 0
+  dsimp [undictionary] at h4 ⊢
+  linear_combination -2 * h21 + 2 * h9 + 2 * h13 + 2 * Real.sin p.theta * h4
+
+theorem reverse_equation2 {p : Romik.Params} (hp : Romik.Equations p) :
+    Reduced.system (undictionary p) 1 = 0 := by
+  have h8 := congrFun hp (8 : Fin 22)
+  have h12 := congrFun hp (12 : Fin 22)
+  have h20 := congrFun hp (20 : Fin 22)
+  change (Romik.path1 p p.phi).1 - (Romik.path2 p p.phi).1 = 0 at h8
+  change (Romik.path2 p p.theta).1 - (Romik.path3 p p.theta).1 = 0 at h12
+  change (Romik.path1 p p.phi).1 -
+    ((Romik.path3 p (Real.pi / 2 - p.theta)).1 -
+      (Romik.alphaBeta3 p (Real.pi / 2 - p.theta)).1 *
+        Real.sin (Real.pi / 2 - p.theta)) = 0 at h20
+  dsimp [Romik.path1, Romik.path2, Romik.path3, Romik.rot, Romik.addK,
+    Romik.alphaBeta3] at h8 h12 h20
+  simp only [c1_eq_of_full hp, c2_eq_of_full hp] at h12 h20
+  simp only [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub] at h20
+  have h4 := reverse_equation4 hp
+  change (undictionary p).a + Real.pi / 2 - (undictionary p).phi -
+    (undictionary p).theta - (undictionary p).b +
+    1 / 2 * ((undictionary p).theta - (undictionary p).phi) *
+      (1 + (undictionary p).a) +
+    1 / 4 * ((undictionary p).theta - (undictionary p).phi) *
+      ((undictionary p).theta - (undictionary p).phi) = 0 at h4
+  change (undictionary p).a * (3 * Real.sin (undictionary p).theta +
+    Real.sin (undictionary p).phi) - 2 * (undictionary p).b *
+    Real.cos (undictionary p).phi +
+    3 * ((undictionary p).theta - (undictionary p).phi - 1) *
+    Real.sin (undictionary p).theta + 3 * Real.cos (undictionary p).theta -
+    Real.sin (undictionary p).phi + Real.cos (undictionary p).phi = 0
+  dsimp [undictionary] at h4 ⊢
+  linear_combination -2 * h20 + 2 * h8 + 2 * h12 + 2 * Real.cos p.theta * h4
+
+theorem undictionary_equations {p : Romik.Params} (hp : Romik.Equations p) :
+    Reduced.Equations (undictionary p) := by
+  change Reduced.system (undictionary p) = 0
+  funext i
+  fin_cases i
+  · exact reverse_equation1 hp
+  · exact reverse_equation2 hp
+  · exact reverse_equation3 hp
+  · exact reverse_equation4 hp
+
+end GerverSofa.PartF.Phases
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# F06: the reverse parameters determine the full solution
+
+Velocity matching determines the shape coefficients. Positional matching
+then determines each successive translation. No numerical enclosure is used.
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace GerverSofa.PartF.Phases
+
+theorem full_equations_injective {p q : Romik.Params}
+    (hp : Romik.Equations p) (hq : Romik.Equations q)
+    (h : undictionary p = undictionary q) : p = q := by
+  have hphi : p.phi = q.phi := congrArg Reduced.Params.phi h
+  have htheta : p.theta = q.theta := congrArg Reduced.Params.theta h
+  have ha := congrArg Reduced.Params.a h
+  have hb := congrArg Reduced.Params.b h
+  dsimp [undictionary] at ha hb
+  have hb1 : p.b1 = q.b1 := by
+    rw [hphi] at ha
+    linarith only [ha]
+  have hb2 : p.b2 = q.b2 := by
+    simp only [hphi, hb1] at hb
+    linarith only [hb]
+  have ha2 : p.a2 = q.a2 := by
+    rw [Romik.a2_eq_neg_quarter_of_equations hp,
+      Romik.a2_eq_neg_quarter_of_equations hq]
+  have ha1 : p.a1 = q.a1 := by
+    have hpa := congrArg Prod.fst (alphaBeta12_eq hp)
+    have hpb := congrArg Prod.snd (alphaBeta12_eq hp)
+    have hqa := congrArg Prod.fst (alphaBeta12_eq hq)
+    have hqb := congrArg Prod.snd (alphaBeta12_eq hq)
+    dsimp [Romik.alphaBeta1, Romik.alphaBeta2] at hpa hpb hqa hqb
+    simp only [hphi, hb1, hb2, ha2] at hpa hpb
+    linear_combination (-Real.sin q.phi / 2) * hpa +
+      (Real.sin q.phi / 2) * hqa + (Real.cos q.phi / 2) * hpb -
+      (Real.cos q.phi / 2) * hqb -
+      (p.a1 - q.a1) * Real.sin_sq_add_cos_sq q.phi
+  have hc1 : p.c1 = q.c1 := by
+    rw [c1_eq_of_full hp, c1_eq_of_full hq, hb1]
+  have hc2 : p.c2 = q.c2 := by
+    rw [c2_eq_of_full hp, c2_eq_of_full hq, hb1]
+  have hd1 : p.d1 = q.d1 := by
+    rw [Romik.d1_eq_quarterPi_sub_b1_of_equations hp,
+      Romik.d1_eq_quarterPi_sub_b1_of_equations hq, hb1]
+  have hd2 : p.d2 = q.d2 := by
+    rw [Romik.d2_eq_b2_add_quarterPi_correction_of_equations hp,
+      Romik.d2_eq_b2_add_quarterPi_correction_of_equations hq, hb1, hb2]
+  have he1 : p.e1 = q.e1 := by
+    rw [Romik.e1_eq_a1_of_equations hp, Romik.e1_eq_a1_of_equations hq, ha1]
+  have he2 : p.e2 = q.e2 := by
+    rw [Romik.e2_eq_neg_a2_of_equations hp, Romik.e2_eq_neg_a2_of_equations hq, ha2]
+  have hk11 : p.k11 = q.k11 := by
+    have h5p := congrFun hp (5 : Fin 22)
+    have h5q := congrFun hq (5 : Fin 22)
+    change p.k11 - 1 + p.a1 = 0 at h5p
+    change q.k11 - 1 + q.a1 = 0 at h5q
+    linarith only [h5p, h5q, ha1]
+  have hk12 : p.k12 = q.k12 := by
+    rw [Romik.k12_eq_quarter_of_equations hp, Romik.k12_eq_quarter_of_equations hq]
+  have hpath1 (t : ℝ) : Romik.path1 p t = Romik.path1 q t := by
+    simp only [Romik.path1, ha1, ha2, hk11, hk12]
+  have hj2 : Romik.path1 p (q.phi) = Romik.path2 p (q.phi) := by
+    simpa only [hphi] using Romik.match_path12_of_equations hp
+  have hx2 : Romik.path2 p (q.phi) = Romik.path2 q (q.phi) :=
+    hj2.symm.trans ((hpath1 (q.phi)).trans (Romik.match_path12_of_equations hq))
+  dsimp [Romik.path2, Romik.addK] at hx2
+  simp only [hb1, hb2] at hx2
+  have hk21 : p.k21 = q.k21 := add_left_cancel (congrArg Prod.fst hx2)
+  have hk22 : p.k22 = q.k22 := add_left_cancel (congrArg Prod.snd hx2)
+  have hpath2 (t : ℝ) : Romik.path2 p t = Romik.path2 q t := by
+    simp only [Romik.path2, hb1, hb2, hk21, hk22]
+  have hj3 : Romik.path2 p (q.theta) = Romik.path3 p (q.theta) := by
+    simpa only [htheta] using Romik.match_path23_of_equations hp
+  have hx3 : Romik.path3 p (q.theta) = Romik.path3 q (q.theta) :=
+    hj3.symm.trans ((hpath2 (q.theta)).trans (Romik.match_path23_of_equations hq))
+  dsimp [Romik.path3, Romik.addK] at hx3
+  simp only [hc1, hc2] at hx3
+  have hk31 : p.k31 = q.k31 := add_left_cancel (congrArg Prod.fst hx3)
+  have hk32 : p.k32 = q.k32 := add_left_cancel (congrArg Prod.snd hx3)
+  have hpath3 (t : ℝ) : Romik.path3 p t = Romik.path3 q t := by
+    simp only [Romik.path3, hc1, hc2, hk31, hk32]
+  have hj4 : Romik.path3 p (Real.pi / 2 - q.theta) = Romik.path4 p (Real.pi / 2 - q.theta) := by
+    simpa only [htheta] using Romik.match_path34_of_equations hp
+  have hx4 : Romik.path4 p (Real.pi / 2 - q.theta) = Romik.path4 q (Real.pi / 2 - q.theta) :=
+    hj4.symm.trans ((hpath3 (Real.pi / 2 - q.theta)).trans (Romik.match_path34_of_equations hq))
+  dsimp [Romik.path4, Romik.addK] at hx4
+  simp only [hd1, hd2] at hx4
+  have hk41 : p.k41 = q.k41 := add_left_cancel (congrArg Prod.fst hx4)
+  have hk42 : p.k42 = q.k42 := add_left_cancel (congrArg Prod.snd hx4)
+  have hpath4 (t : ℝ) : Romik.path4 p t = Romik.path4 q t := by
+    simp only [Romik.path4, hd1, hd2, hk41, hk42]
+  have hj5 : Romik.path4 p (Real.pi / 2 - q.phi) = Romik.path5 p (Real.pi / 2 - q.phi) := by
+    simpa only [hphi] using Romik.match_path45_of_equations hp
+  have hx5 : Romik.path5 p (Real.pi / 2 - q.phi) = Romik.path5 q (Real.pi / 2 - q.phi) :=
+    hj5.symm.trans ((hpath4 (Real.pi / 2 - q.phi)).trans (Romik.match_path45_of_equations hq))
+  dsimp [Romik.path5, Romik.addK] at hx5
+  simp only [he1, he2] at hx5
+  have hk51 : p.k51 = q.k51 := add_left_cancel (congrArg Prod.fst hx5)
+  have hk52 : p.k52 = q.k52 := add_left_cancel (congrArg Prod.snd hx5)
+  cases p
+  cases q
+  congr 1
+
+theorem dictionary_undictionary_of_equations {p : Romik.Params}
+    (hp : Romik.Equations p) : dictionary (undictionary p) = p := by
+  apply full_equations_injective (dictionary_equations _ (undictionary_equations hp)) hp
+  exact undictionary_dictionary (undictionary p)
+
+end GerverSofa.PartF.Phases
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle003.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle003.lean
new file mode 100644
index 0000000000..7954bf2c24
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle003.lean
@@ -0,0 +1,7766 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle001
+public import Mathlib.Tactic.FinCases
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartB.Semantics.Batch001`.
+* `GerverSofa.KernelOnly.PartB.Certificates.Batch004`.
+* `GerverSofa.KernelOnly.PartB.Certificates.Batch003`.
+* `GerverSofa.KernelOnly.PartB.Certificates.Batch002`.
+* `GerverSofa.KernelOnly.PartB.Certificates.Batch001`.
+* `GerverSofa.KernelOnly.PartB.Semantics.Batch002`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartB.Arithmetic`.
+* `KernelOnly.PartB.CellCache`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Part B exact product-cell arithmetic
+
+This module is deliberately independent of the noncomputable Part A root.
+It contains only exact rational interval data, so all 64 diagnostic and
+production rows can be kernel-reduced without rebuilding semantic layers.
+-/
+
+@[expose] public section
+
+namespace GerverSofa
+namespace PartB
+
+open RatInterval
+
+/-- Exact target `0.171` used by the continuum theorem. -/
+def targetQ : ℚ := 171 / 1000
+
+/-- The 64 closed mesh cells determined by the 65 nodes `iπ/128`. -/
+abbrev Cell := Fin 64
+
+/-- Exact rational mesh coefficient. -/
+def nodeCoeff (i : Nat) : ℚ := (i : ℚ) / 128
+
+/-- Hull of two rational intervals. -/
+def intervalHull (a b : RatInterval) : RatInterval :=
+  ⟨min a.lo b.lo, max a.hi b.hi⟩
+
+/-- Componentwise hull of two planar interval boxes. -/
+def pointHull (a b : RatInterval × RatInterval) : RatInterval × RatInterval :=
+  (intervalHull a.1 b.1, intervalHull a.2 b.2)
+
+/-- Exact interval enclosure of one physical time cell. -/
+def cellTimeInterval (i : Cell) : RatInterval :=
+  intervalHull
+    (ExactReplay.scale (nodeCoeff i.1) ExactReplay.piI)
+    (ExactReplay.scale (nodeCoeff (i.1 + 1)) ExactReplay.piI)
+
+/-- Exact interval evaluation of one path phase on one complete mesh cell. -/
+def cellPieceInterval (j : Nat) (i : Cell) : RatInterval × RatInterval :=
+  let t := cellTimeInterval i
+  let k11 := ExactReplay.getI ExactReplay.fullInputBox 0
+  let k12 := ExactReplay.getI ExactReplay.fullInputBox 1
+  let k21 := ExactReplay.getI ExactReplay.fullInputBox 2
+  let k22 := ExactReplay.getI ExactReplay.fullInputBox 3
+  let k31 := ExactReplay.getI ExactReplay.fullInputBox 4
+  let k32 := ExactReplay.getI ExactReplay.fullInputBox 5
+  let k41 := ExactReplay.getI ExactReplay.fullInputBox 6
+  let k42 := ExactReplay.getI ExactReplay.fullInputBox 7
+  let k51 := ExactReplay.getI ExactReplay.fullInputBox 8
+  let k52 := ExactReplay.getI ExactReplay.fullInputBox 9
+  let a1 := ExactReplay.getI ExactReplay.fullInputBox 10
+  let a2 := ExactReplay.getI ExactReplay.fullInputBox 11
+  let b1 := ExactReplay.getI ExactReplay.fullInputBox 12
+  let b2 := ExactReplay.getI ExactReplay.fullInputBox 13
+  let c1 := ExactReplay.getI ExactReplay.fullInputBox 14
+  let c2 := ExactReplay.getI ExactReplay.fullInputBox 15
+  let d1 := ExactReplay.getI ExactReplay.fullInputBox 16
+  let d2 := ExactReplay.getI ExactReplay.fullInputBox 17
+  let e1 := ExactReplay.getI ExactReplay.fullInputBox 18
+  let e2 := ExactReplay.getI ExactReplay.fullInputBox 19
+  let one := ExactReplay.oneI
+  let half := RatInterval.point (1 / 2)
+  let quarter := RatInterval.point (1 / 4)
+  let ct := ExactReplay.cosineInterval t
+  let st := ExactReplay.sineInterval t
+  let data : RatInterval × RatInterval × RatInterval × RatInterval :=
+    if j = 1 then
+      (RatInterval.sub
+          (RatInterval.add (RatInterval.mul a1 ct) (RatInterval.mul a2 st)) one,
+       RatInterval.sub
+          (RatInterval.add (RatInterval.mul (RatInterval.neg a2) ct)
+            (RatInterval.mul a1 st)) half,
+       k11, k12)
+    else if j = 2 then
+      (RatInterval.add
+          (RatInterval.add
+            (RatInterval.mul (RatInterval.neg quarter) (RatInterval.mul t t))
+            (RatInterval.mul b1 t)) b2,
+       RatInterval.sub
+          (RatInterval.sub (RatInterval.mul half t) b1) one,
+       k21, k22)
+    else if j = 3 then
+      (RatInterval.sub c1 t, RatInterval.add c2 t, k31, k32)
+    else if j = 4 then
+      (RatInterval.sub
+          (RatInterval.add (RatInterval.mul (RatInterval.neg half) t) d1) one,
+       RatInterval.add
+          (RatInterval.add
+            (RatInterval.mul (RatInterval.neg quarter) (RatInterval.mul t t))
+            (RatInterval.mul d1 t)) d2,
+       k41, k42)
+    else
+      (RatInterval.sub
+          (RatInterval.add (RatInterval.mul e1 ct) (RatInterval.mul e2 st)) half,
+       RatInterval.sub
+          (RatInterval.add (RatInterval.mul (RatInterval.neg e2) ct)
+            (RatInterval.mul e1 st)) one,
+       k51, k52)
+  let r1 := RatInterval.sub (RatInterval.mul ct data.1)
+      (RatInterval.mul st data.2.1)
+  let r2 := RatInterval.add (RatInterval.mul st data.1)
+      (RatInterval.mul ct data.2.1)
+  (RatInterval.add r1 data.2.2.1, RatInterval.add r2 data.2.2.2)
+
+/-- The path enclosure for a cell.  Four cells contain a switching angle and
+therefore take the hull of the two adjacent analytic phases. -/
+def cellPathInterval (i : Cell) : RatInterval × RatInterval :=
+  if i.1 = 0 then cellPieceInterval 1 i
+  else if i.1 = 1 then pointHull (cellPieceInterval 1 i) (cellPieceInterval 2 i)
+  else if i.1 ≤ 26 then cellPieceInterval 2 i
+  else if i.1 = 27 then pointHull (cellPieceInterval 2 i) (cellPieceInterval 3 i)
+  else if i.1 ≤ 35 then cellPieceInterval 3 i
+  else if i.1 = 36 then pointHull (cellPieceInterval 3 i) (cellPieceInterval 4 i)
+  else if i.1 ≤ 61 then cellPieceInterval 4 i
+  else if i.1 = 62 then pointHull (cellPieceInterval 4 i) (cellPieceInterval 5 i)
+  else cellPieceInterval 5 i
+
+/-- Exact interval image of `Gu` on one product cell. -/
+def guCellInterval (i j : Cell) : RatInterval :=
+  let xi := cellPathInterval i
+  let xj := cellPathInterval j
+  let c := ExactReplay.cosineInterval (cellTimeInterval i)
+  let s := ExactReplay.sineInterval (cellTimeInterval i)
+  let dx := RatInterval.sub xi.1 xj.1
+  let dy := RatInterval.sub xi.2 xj.2
+  RatInterval.add
+    (RatInterval.add ExactReplay.oneI (RatInterval.mul dx c))
+    (RatInterval.mul dy s)
+
+/-- Exact interval image of `Gv` on one product cell. -/
+def gvCellInterval (i j : Cell) : RatInterval :=
+  let xi := cellPathInterval i
+  let xj := cellPathInterval j
+  let c := ExactReplay.cosineInterval (cellTimeInterval i)
+  let s := ExactReplay.sineInterval (cellTimeInterval i)
+  let dx := RatInterval.sub xi.1 xj.1
+  let dy := RatInterval.sub xi.2 xj.2
+  RatInterval.add
+    (RatInterval.add ExactReplay.oneI
+      (RatInterval.mul dx (RatInterval.neg s)))
+    (RatInterval.mul dy c)
+
+end PartB
+end GerverSofa
+
+end
+
+end
+
+section
+
+/-!
+# Shared exact interval evaluations for the product-cell certificates
+
+Each table is checked against the original evaluator by the Lean kernel.
+Sharing these evaluations avoids repeating the same Taylor computations in every row.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+
+/-- Reconstruct a natural number from base-10³⁵ chunks to share decimal elaboration. -/
+def naturalFromChunks (chunks : List ℕ) : ℕ :=
+  chunks.foldl (fun value digit => value * 10 ^ 35 + digit) 0
+
+/-- The exact cached path-coordinate pair for cell 0, sharing kernel reduction. -/
+def cachedPath00 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      32806,
+      10584377830325517359742700908850139,
+      81607231342652381947323845793090454,
+      96144429388381831087652413032396797,
+      36868186593732552082470650873461363]) : ℤ))
+      (naturalFromChunks [
+      5000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      7671443713161115292750432506903126,
+      43012829623794223200639225310598163,
+      33075928858326908145305686159237480,
+      91804944677611174042315416975718857])
+      (naturalFromChunks [
+      12,
+      50000000000000000000000000000000000,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      37647662,
+      97447248562927873266328560474815312,
+      52671058691422530540628864810962801]) : ℤ))
+      (naturalFromChunks [
+      500000000000,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      87326,
+      79792061110285957775373564882726820,
+      13026632299000909904799014917975,
+      23616446106846274741656489466558196,
+      56629420125081855603683656880487509])
+      (naturalFromChunks [
+      2500000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 1, sharing kernel reduction. -/
+def cachedPath01 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      397036559637444138821,
+      23429525818379333822931949222691947,
+      56024914729271489239510386186014461,
+      63013117025166082450127814194603137,
+      19506445506199262500255354407394047]) : ℤ))
+      (naturalFromChunks [
+      25600000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      67381694628068574712,
+      83394745669405697834748999450389856,
+      81605402638451307158023210957921285,
+      62191351641107813192015354106152141,
+      69296959622689866676384082977573983])
+      (naturalFromChunks [
+      6553600000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      342928,
+      50341465024201468247210548101732037,
+      79271707896851698787958705410829626,
+      75662558918120062032679167808769808,
+      7365225904837141520082684788612007])
+      (naturalFromChunks [
+      10000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2281520274224607845462,
+      45701440614950960933322900164023200,
+      89547874173976888161305214283996027,
+      10849698557208910987992676093182092,
+      50000036718501707078589374865104737])
+      (naturalFromChunks [
+      32768000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 2, sharing kernel reduction. -/
+def cachedPath02 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      257772707128381117139,
+      53978082056034263614025552654969234,
+      5677716471911388018466159748398086,
+      55737085031605040111489958464383129,
+      43581452883468530887685734453022763]) : ℤ))
+      (naturalFromChunks [
+      12800000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      105594279288160461622,
+      23118192145018609615577389422239556,
+      10464295176902166486759014940652048,
+      74082432524857500494799764093759241,
+      94830813003930941592709856379344343])
+      (naturalFromChunks [
+      16384000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1398712699947913978358,
+      25805078325706303648518730929878394,
+      20081081063356190170462867234790298,
+      57907457058855645071830435777098150,
+      86042787547446458393668863106067779])
+      (naturalFromChunks [
+      20480000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      26566583229087814189,
+      33178357444382229822963276002699166,
+      73942154990466666177652686009829343,
+      30487308499161687512543927329346219,
+      54195585922384154538050876594980787])
+      (naturalFromChunks [
+      256000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 3, sharing kernel reduction. -/
+def cachedPath03 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      167431482098179646296,
+      27653758568720291948081335555214563,
+      13535285732814069244132527171883621,
+      20204239016170134246262736347286126,
+      34014896056281309646512058126667913]) : ℤ))
+      (naturalFromChunks [
+      6400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      38729481078125875109,
+      76637089920329856098896184724691039,
+      95799959139430198336179084055884012,
+      55820193666305563558980105845381763,
+      76477890508403675181138314083052333])
+      (naturalFromChunks [
+      32768000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      651119810786289015763,
+      4114142687924407599428325716088078,
+      61023984528169290192365705819202379,
+      79260090486694955629447642207579405,
+      43485043701296945996538579373026391])
+      (naturalFromChunks [
+      6400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9005771742771812930505,
+      58946786492959638175245597174788749,
+      13038617100112156254240943432178320,
+      21142311146930295085709411441844239,
+      67644010607726716023327914535386407])
+      (naturalFromChunks [
+      65536000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 4, sharing kernel reduction. -/
+def cachedPath04 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      137437287975512062338,
+      70919967300170333811485565966232643,
+      10638242570981596591505205507888429,
+      52601132651448521777311208605664925,
+      56192875748372588280815659249114093]) : ℤ))
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      22467360245297988108,
+      56112342793820765380264380655133334,
+      29531591438965372171000682985755594,
+      45429999022724786122338458184914204,
+      45847365690044714883433796662013467]) : ℤ))
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      275747586818673826219,
+      50625546074936342703565113798977182,
+      1146546306727773560019166956028854,
+      3932337771064777894467575156849641,
+      36243655997364990168367093676435219])
+      (naturalFromChunks [
+      2048000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      698348227654038481169,
+      70667851612610394756475484262548393,
+      90457228968935196947491747574871989,
+      62939162869168395373032340806845914,
+      87252404329967669574965606841769861])
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 5, sharing kernel reduction. -/
+def cachedPath05 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      1082669277612280165857,
+      22213585892920254341856704042520998,
+      56859783556261704480209291502237127,
+      44933671459361595388973986850463799,
+      46427798661081607985721723295531127]) : ℤ))
+      (naturalFromChunks [
+      25600000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      17736738199344183985,
+      44229400551172330353320805277303073,
+      90697109383486804175230843143445183,
+      11805592692696122400824278116777688,
+      52807939621165628837900571803012337]) : ℤ))
+      (naturalFromChunks [
+      1310720000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2137037759635821945951,
+      15731002209590382227717230893097918,
+      41621198581896007480883971564401711,
+      45389410335857388480574935897589974,
+      99696444501811673621546840076749761])
+      (naturalFromChunks [
+      12800000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      10640683123606004173,
+      82310372426003510408455238278908779,
+      81388355897557796731545048927062756,
+      16302258963323942165116560400676029,
+      18203798106858477556707665657063091])
+      (naturalFromChunks [
+      52428800000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 6, sharing kernel reduction. -/
+def cachedPath06 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      2680170847027654534774,
+      61684040855844559981322023906641975,
+      56610652358746122042253325013736687,
+      6277299733388226031141299438197790,
+      40974691037601367059759116049734507]) : ℤ))
+      (naturalFromChunks [
+      51200000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      93930487064265178796,
+      29716699436248786974627997210335582,
+      63237469195814457886919109558296907,
+      84856482043037459793656525431470394,
+      93931613502494613889086677778825469]) : ℤ))
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      20339129419442060876594,
+      1391626348646451668662347055303391,
+      15465887621692242482729366937872609,
+      66857945034095006437916010249540579,
+      30871376576739564690425626507123041])
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      3846022310136996615290,
+      27002240885343835803255878584829161,
+      14458217213200498088882103192466379,
+      33396394683931457590206731343397247,
+      73912213202174365346865034585117389])
+      (naturalFromChunks [
+      16384000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 7, sharing kernel reduction. -/
+def cachedPath07 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      20381017934146979245,
+      61693320989470503371401442326964127,
+      91673893309935437692126346083375172,
+      67450320294182700510700928248389603,
+      58833253908103808060331507300338921]) : ℤ))
+      (naturalFromChunks [
+      320000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      2205767220636735668142,
+      35940810189556752352642392711344277,
+      26762691955107816441077933703341457,
+      20749704589015203587956213954524896,
+      54736979393155779517911320839349317]) : ℤ))
+      (naturalFromChunks [
+      65536000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      367350088712004395952,
+      39766286536720862570011349737953538,
+      31538324792015689931670428716135632,
+      68644132135603915097918340468769638,
+      55268171291805020647080139170596203])
+      (naturalFromChunks [
+      1600000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      8709907405773836844111,
+      50502012136984439007331574618703806,
+      49469165334826194083182637786951896,
+      72907450548893762289861020992783930,
+      87470163690321514353102027409415497])
+      (naturalFromChunks [
+      32768000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 8, sharing kernel reduction. -/
+def cachedPath08 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      7812229105725993755149,
+      97641031873286603147948844413615301,
+      75796343540237770318921365820474347,
+      19977088575061253290482682834293765,
+      45995749618422983633275842386474821]) : ℤ))
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      46773151077659169260,
+      97287040064071377465512343272931470,
+      21573777092767783318958310731734855,
+      93471337492423523767974706447344462,
+      75704022560099374876133741254428909]) : ℤ))
+      (naturalFromChunks [
+      1024000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      26604782435194208738668,
+      80872704614358068781074789513746518,
+      94493941070370043337611991993687952,
+      40842667694072401880457855491441986,
+      85621910315436639878553104013171707])
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      151598740593416817709,
+      9226378357809752850127442420102858,
+      55734093541596926198555674243503923,
+      73808143245022699455357008191806776,
+      98485060194256457186362628699029373])
+      (naturalFromChunks [
+      512000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 9, sharing kernel reduction. -/
+def cachedPath09 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      23069638736666002967,
+      12877861193895208738983408374202765,
+      64588438122218512545763457152858773,
+      22073903643433764383209779706672121,
+      94801059121046692794847678062741701]) : ℤ))
+      (naturalFromChunks [
+      256000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      3863909614874605471117,
+      40648359854104395342155670334436315,
+      67926626152545763385582714167881168,
+      97456953543625620753899127559258247,
+      82607008735167379153577507820221821]) : ℤ))
+      (naturalFromChunks [
+      65536000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      296170256433636930537,
+      66630312236551833493561182688790199,
+      6592420113866089103834981083121999,
+      69467143296029617178237254333184173,
+      33713656595485136675368018626081107])
+      (naturalFromChunks [
+      1024000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      333363201323075817867,
+      31687272918839090125789612300645001,
+      1166954169132719044158493534968013,
+      62890785944223099129582144205997474,
+      90230433777095280334927491780464439])
+      (naturalFromChunks [
+      1024000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 10, sharing kernel reduction. -/
+def cachedPath10 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      10765216446638226850857,
+      59299744066641916576448686407520110,
+      84702330161375927295680856336342208,
+      4516745507330992498489976621197065,
+      81511408194208121910810941333273883]) : ℤ))
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      6018505024411921763,
+      17030749444711649401090166160001354,
+      83388115444377136493465757028048465,
+      72372814606718744507039061226771621,
+      85261206054916356321021061778285257]) : ℤ))
+      (naturalFromChunks [
+      81920000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      650840451921177077031,
+      93038030793606040741970645022740278,
+      42989991280547562838022103397091788,
+      61237015064879275543251033445908778,
+      10927714242048705550957807694300573])
+      (naturalFromChunks [
+      2048000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      58020991744810199843,
+      68507999215275411867922016848468589,
+      36207505591585407971144829603117203,
+      25831506287041077221720129610627196,
+      15847080038091018522379721036807589])
+      (naturalFromChunks [
+      163840000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 11, sharing kernel reduction. -/
+def cachedPath11 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      388141736688336868895,
+      66984257106958467182590409196082226,
+      95481003145049932367296596182226025,
+      25677986066847416526982958324964912,
+      41008523163620647626192920678433497]) : ℤ))
+      (naturalFromChunks [
+      3200000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      1168758992281440224547,
+      16287102208851664666399704791712721,
+      47676942061864743051552440361261081,
+      90874044232132797985332200650670144,
+      580541931498526622366134765199423]) : ℤ))
+      (naturalFromChunks [
+      13107200000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      221092429879427027330,
+      88608008438198821562909998043439665,
+      13047336253194869013295956424692359,
+      53363047135120315900082011173485363,
+      47675313688007582100846275507645983])
+      (naturalFromChunks [
+      640000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      25020942119207618609007,
+      65662086365274136324062902241613617,
+      20230407829829238053233158295393339,
+      19732129575996048464524852170204763,
+      56414247462160268260580912159592919])
+      (naturalFromChunks [
+      65536000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 12, sharing kernel reduction. -/
+def cachedPath12 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      1773731627147483191230,
+      36175209239618255945472513069837148,
+      86686865313617304064171377235314437,
+      89468174345549440081679584819172702,
+      69303600230459694542080597360087611]) : ℤ))
+      (naturalFromChunks [
+      12800000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      434274526001465855850,
+      40140375888482141794938444560415681,
+      81298170686723382039329372006731002,
+      36272032346813362557734848497153891,
+      23649817013654428840899432451078853]) : ℤ))
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      38110474596107811052997,
+      22567448013227543795816574190045428,
+      6148621739332637984916400807260212,
+      3105445367900174250407320778460984,
+      67721687845924257486180275393586281])
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1673113596354495819892,
+      99663859850762056945110856967538721,
+      15315187268085034040526245941768117,
+      88868877078658470159320662446010791,
+      40266793041155812902529175745836119])
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 13, sharing kernel reduction. -/
+def cachedPath13 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      2008628765280496405999,
+      88041818710889025150991536031988718,
+      67802598540852776870906963033442949,
+      32376771739553337244108350073120327,
+      8101166361721646485967471924171109]) : ℤ))
+      (naturalFromChunks [
+      12800000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      2031497294957303917014,
+      84974101791982474935481011792204520,
+      98426372616984371284109496076401980,
+      65348067744465036717049064033141828,
+      55012639527231088679188109392017203]) : ℤ))
+      (naturalFromChunks [
+      16384000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      10186092982599889011385,
+      75445809284625668857234827446401256,
+      9572922591564378539877611180619063,
+      59863876662292155909697277164309157,
+      49029987172795481332882145234705061])
+      (naturalFromChunks [
+      25600000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      28452052022163414126520,
+      3188857407044636447299600183732073,
+      64570549278856232420072930604538885,
+      83164904112139175558683749277586688,
+      98727427902645257279661989958565199])
+      (naturalFromChunks [
+      65536000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 14, sharing kernel reduction. -/
+def cachedPath14 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      722150173434592909301,
+      58851046991353602091763548730853537,
+      98680631631649365122309591599360436,
+      72445330859530973155614348395162504,
+      30169567161732563058636612092586367]) : ℤ))
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      1171732991930521869823,
+      37343757013793546654818827806379467,
+      58313561677554029029079042684963043,
+      1394828477275913674129295740057125,
+      21868219111420349550885807779387851]) : ℤ))
+      (naturalFromChunks [
+      8192000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      346175364735067842297,
+      48277590864265851194901038690959438,
+      40969866803779673682446409520025577,
+      41104078294823259265316314539343406,
+      37139012554037842050137668908985367])
+      (naturalFromChunks [
+      819200000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      3758096417306143734854,
+      49157986947996196645973836218009980,
+      81801442549508901343781373774936220,
+      62988791699570044998146115095030841,
+      28693784905060628582225160964862971])
+      (naturalFromChunks [
+      8192000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 15, sharing kernel reduction. -/
+def cachedPath15 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      1573401899376989857,
+      61390673445606358559943249262846074,
+      40485303902441957060259762088802928,
+      29980996385309511029707993723591671,
+      20430162592414247590508160529451147]) : ℤ))
+      (naturalFromChunks [
+      8000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      85513475642321595311,
+      1611388138227603773581264109809037,
+      29910490424832874624738082855515607,
+      97815095887290868761373883257814986,
+      83728519998198042073066687725457571]) : ℤ))
+      (naturalFromChunks [
+      524288000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      178471541349866931552,
+      82080798306619057628694172953525409,
+      51461411668440376719160952009274494,
+      64011786424290545832853943329622757,
+      49291117201164234567807063159973151])
+      (naturalFromChunks [
+      400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1264208077804334994646,
+      45984056187109252142191193421006168,
+      21654621402168520232087262712011627,
+      31900409802051276593209813112534137,
+      86912007640780707299131977850470871])
+      (naturalFromChunks [
+      2621440000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 16, sharing kernel reduction. -/
+def cachedPath16 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      2232175522607699188225,
+      91190032953322708632477580711179517,
+      18198603757775366862712918851175636,
+      52554716962569962340204476268256643,
+      48709358495140720560234482012956223]) : ℤ))
+      (naturalFromChunks [
+      10240000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      47144573986484561700,
+      20393882355647221312919724009336554,
+      18935012311525313090963907184081000,
+      8639326835990205916958039686816462,
+      78321641706534454926480214352213933]) : ℤ))
+      (naturalFromChunks [
+      256000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      47990508228491983736890,
+      82777188360891801519454862611393458,
+      97118251705832496709509992986480042,
+      96040507250019094751055079215951112,
+      85280886769706192295013734428061719])
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      25836464706022605256,
+      22509887965423031993705386895346402,
+      37478877332775466019282535233622298,
+      15550952670072662534870817639365081,
+      8902297904708943185834023894469691])
+      (naturalFromChunks [
+      51200000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 17, sharing kernel reduction. -/
+def cachedPath17 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      6148896072700788812949,
+      53043421946390568404884368544093008,
+      78958785753223498876364649105232574,
+      71290632492409024233963674357743240,
+      45501385942277855477573264461473913]) : ℤ))
+      (naturalFromChunks [
+      25600000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      6755149746078905205046,
+      28811037024431407510219365078312336,
+      76658383922927573992298460543320860,
+      53378925909911150600872596743289554,
+      15594343672769671009892433651174151]) : ℤ))
+      (naturalFromChunks [
+      32768000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      313582638106586757705,
+      87372461386231909729904051768169255,
+      93832339306618535561602351534722481,
+      17396715823787651057097928500086174,
+      80862350746552740550298161258652403])
+      (naturalFromChunks [
+      640000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1722931461148887366510,
+      46409158181583098086856385010171943,
+      65411799477445493270226484209591504,
+      52504702741971018918801514136857410,
+      10006424262474222625926664492887733])
+      (naturalFromChunks [
+      3276800000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 18, sharing kernel reduction. -/
+def cachedPath18 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      26956076469151142512381,
+      49866755369027077396387222904166349,
+      64686367075487352613288857348431751,
+      22591177282038582248518256987764101,
+      77830896854799858276659869424597243]) : ℤ))
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      3752476427667790480364,
+      70645451745216788284098881576421063,
+      58888219615823033835494844744798409,
+      84305029134523970870708510923637953,
+      18092139045718317015403128802538637]) : ℤ))
+      (naturalFromChunks [
+      16384000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      52232948799621723373687,
+      97740482995519195309788825469980642,
+      4716579278934641305838724815833601,
+      15259250362978229305147952254142814,
+      23930779926923549256356963352108949])
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4470827035596881257958,
+      86961292595799761390584647742694265,
+      26232252001435539485088086711004455,
+      89975776386155454364118256180248451,
+      89268163566843973969280851164826807])
+      (naturalFromChunks [
+      8192000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 19, sharing kernel reduction. -/
+def cachedPath19 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      73495325945139818809,
+      8306371050718563749998178032974320,
+      33501377488505376525542559208090777,
+      49474525788223448235415596534377789,
+      96013070524635323176719248601235113]) : ℤ))
+      (naturalFromChunks [
+      256000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      16564588421182096012176,
+      29273864383361591221571777624871588,
+      20641942462953251634759374321949101,
+      23911874168868967341875742603229148,
+      16128655578108919788171346684036691]) : ℤ))
+      (naturalFromChunks [
+      65536000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      27082979284001047506,
+      79559520492811177479762935518293604,
+      63949262813513831368097858266552329,
+      65635176247659198607447324192713447,
+      26455732627678590531196400719831429])
+      (naturalFromChunks [
+      51200000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      289002363685836856899,
+      17527026093291510038295223648736736,
+      85271027410740226881270884576939535,
+      83954866806232751564042949776919242,
+      97837979159043726499299779485472231])
+      (naturalFromChunks [
+      512000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 20, sharing kernel reduction. -/
+def cachedPath20 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      6383300875517254031192,
+      29131494420264809031195032822421161,
+      82700393794748046111940805113088212,
+      41845293725036682366573040277997876,
+      85112712985090380754514866912717999]) : ℤ))
+      (naturalFromChunks [
+      20480000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      45427533799981145858,
+      17953586887851602939012078609143271,
+      12949960118367039099434387728043027,
+      41958000804686948437166022867217953,
+      29216245456445894150979303716934211]) : ℤ))
+      (naturalFromChunks [
+      163840000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      13992176165349168127972,
+      83856438909825257690323117211977158,
+      71687832787594844182089566373415837,
+      5686253870847922247895428866751799,
+      47367000195086438325264253866482631])
+      (naturalFromChunks [
+      25600000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      47666841691201280669,
+      55695454142107537182683935505704275,
+      18006844203933817590497298518268947,
+      68498735306032288499334897180644104,
+      72115191142903246332754968347275817])
+      (naturalFromChunks [
+      81920000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 21, sharing kernel reduction. -/
+def cachedPath21 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      431322785493072023787,
+      56599427004958405722851738981481554,
+      3384495675567098461980779425235168,
+      5188460070626790897620710851323643,
+      18763100099683038942106162392653023]) : ℤ))
+      (naturalFromChunks [
+      1280000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      3965151587818684362262,
+      94751912373538557491256839435816527,
+      71181151810458439208200293840752877,
+      36536264412921590864906946192054462,
+      53257345090778891246698910753912169]) : ℤ))
+      (naturalFromChunks [
+      13107200000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      7204728572088139345589,
+      79024545077048721515620566775608790,
+      74705170721863916571618142077828695,
+      48213159886109202965179325337537991,
+      97490990118691632549838940501890353])
+      (naturalFromChunks [
+      12800000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9797009015704454333044,
+      79531328010772785133822223168203504,
+      55555474198942598713408721457461837,
+      12673798127526764858314469522941493,
+      61383687717026893520715691959243299])
+      (naturalFromChunks [
+      16384000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 22, sharing kernel reduction. -/
+def cachedPath22 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      7432116517734618500798,
+      93762291373445073759496636164725261,
+      48065959727405143181300388384062330,
+      90025766802837099493707177047597270,
+      41875785710555023943980535691796973]) : ℤ))
+      (naturalFromChunks [
+      20480000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      2690664215580521882457,
+      8621210847716537766938969675171070,
+      42829090155427542736073778285756218,
+      32035517854597147101577588497096984,
+      98899547702258424270356703570598241]) : ℤ))
+      (naturalFromChunks [
+      8192000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      59170165059194991306530,
+      62513249769526416741865203352728492,
+      98237134454343858181410934094382458,
+      19014127533375934479135993784516078,
+      11676782152962256292537976344249187])
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      401540232308970637289,
+      71957584818052195563077915556536227,
+      50857577687287708582644416149596137,
+      98501216961056532077278540375472866,
+      62043058645309070670626166162915149])
+      (naturalFromChunks [
+      655360000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 23, sharing kernel reduction. -/
+def cachedPath23 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      311524683689077264076,
+      81865832439147061117839984178997685,
+      50586424143718922702154526372735043,
+      65002669484641632410342495227830606,
+      68183280954699394374852966183255519]) : ℤ))
+      (naturalFromChunks [
+      800000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      1454130764090289837339,
+      80087964315861009545481612666624919,
+      79757576261319729592487442773874219,
+      88594085886580568798190957097675234,
+      48573394307380205278064762344761647]) : ℤ))
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      29571653763547763974,
+      37519747249989126270084668660719189,
+      47928550782524185138035854584515490,
+      24783589875666482940303231198041709,
+      36720058144278890294853221355302129])
+      (naturalFromChunks [
+      50000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      41029595577706048530349,
+      68340607272369226115598963398150844,
+      41502464298175621172978543202541514,
+      9716158378904356395067473110130294,
+      46472184805117273614742091299973859])
+      (naturalFromChunks [
+      65536000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 24, sharing kernel reduction. -/
+def cachedPath24 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      68230104524419658859,
+      3710018708085601961211588192614519,
+      84719400461408904752799797939644498,
+      73402860115551976871940859828566932,
+      65156912367605957567924221548286637]) : ℤ))
+      (naturalFromChunks [
+      163840000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      97829798387851889725,
+      59693060005088178349348158273886876,
+      5983367253201647876129139448652466,
+      81855465756118828867565142653836040,
+      73895300861281584358526124751061481]) : ℤ))
+      (naturalFromChunks [
+      256000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      98900495310502982360,
+      29947821290084573791962464607669732,
+      87953102054601526846580069211511667,
+      1365590475649153559773357988948831,
+      81931142070375210687381955343034307])
+      (naturalFromChunks [
+      163840000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      81666103229825782382,
+      44915206810025307319170317505802327,
+      32432750801747532264219951780762419,
+      34121066514689279195213992138260298,
+      16053739472581819338905830015984959])
+      (naturalFromChunks [
+      128000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 25, sharing kernel reduction. -/
+def cachedPath25 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      2273035225321697952023,
+      80779446263128223074618891179067779,
+      75772731684221422477462013180320621,
+      28702581762506875003691180955447801,
+      67549939675737803575799401258684339]) : ℤ))
+      (naturalFromChunks [
+      5120000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      21485268119185817861,
+      7629542580321023468912116859403396,
+      29257650911934346579814917546276952,
+      40836213232775692138446652035476381,
+      30958237619418358021179764206263731]) : ℤ))
+      (naturalFromChunks [
+      52428800000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      7864724439432997294651,
+      65660774943433334157277551864757897,
+      26549610611022141140843763350181339,
+      93105276316741746818811720060394398,
+      30984780313361542265556932244536979])
+      (naturalFromChunks [
+      12800000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      68004474492341669463,
+      26122830130631861841938880657614502,
+      70424812979247918609585316059100120,
+      22590215519708530283359232084542963,
+      49173423578136672004622928618214973])
+      (naturalFromChunks [
+      104857600000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 26, sharing kernel reduction. -/
+def cachedPath26 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      386559050735951328428,
+      93860253269995947760667068660819041,
+      55832417161228116891869017173574073,
+      50866365672816566287113469711448389,
+      54678889780552157407435810450991959]) : ℤ))
+      (naturalFromChunks [
+      819200000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      1434937896783106078200,
+      59837873105880821029753500423892350,
+      46883270872549022778880941131262379,
+      32073849385708394435928305921846717,
+      87395432974093925440676251287975507]) : ℤ))
+      (naturalFromChunks [
+      3276800000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      63875358401001909129984,
+      53931875469103674470904999909821618,
+      20280316461203537408057328231977250,
+      46775415376695493454704764317128611,
+      25074892422465531369740415912627057])
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      10774353128052990596664,
+      86416535155033626126722231447149571,
+      60117012165770299347635001926095565,
+      48528175529105605724925051284711865,
+      58625931480763357258191677872631807])
+      (naturalFromChunks [
+      16384000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 27, sharing kernel reduction. -/
+def cachedPath27 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      100710,
+      71722422228052535036734160717951499,
+      42069078062123627069531182627031188,
+      60431569588115081772421647447206221]) : ℤ))
+      (naturalFromChunks [
+      200000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      18525,
+      51271337754643255148382898432441303,
+      34014299681814911264804038180437217,
+      55598380712549504288585707822915881]) : ℤ))
+      (naturalFromChunks [
+      40000,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      4022053,
+      77416366696858139556873172408424940,
+      79924851481875114027458417550002170,
+      34057053097440188372375809026152099])
+      (naturalFromChunks [
+      6400000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4275306,
+      20792989124941816891545250537159438,
+      71482972875293744656267212392707172,
+      74890077329955155725363858706138837])
+      (naturalFromChunks [
+      6400000,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 28, sharing kernel reduction. -/
+def cachedPath28 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      106444,
+      71744953679696011782194578058509761,
+      28035999862186441429009740137354524,
+      55943708561620240215135912314082437]) : ℤ))
+      (naturalFromChunks [
+      200000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      15740,
+      8776750732905718839524000408044492,
+      99632166219777093229462388845919239,
+      70796128024550735631231214963854567]) : ℤ))
+      (naturalFromChunks [
+      32000,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      253825,
+      64058481427245968592862495904428802,
+      74663502867928339362798909724833356,
+      11912099252696050870015849165168127])
+      (naturalFromChunks [
+      400000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      539518,
+      65313367049853022234902972594033713,
+      8806996846566129895374134291782278,
+      17196073080542759291747212267703029])
+      (naturalFromChunks [
+      800000,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 29, sharing kernel reduction. -/
+def cachedPath29 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      89777,
+      55072345144042668244602106782333374,
+      94731070144562590808464065036350402,
+      29965644846821098710715917405854081]) : ℤ))
+      (naturalFromChunks [
+      160000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      666645,
+      1171114820531568089470279569241904,
+      54693932395102532299204730188616454,
+      52407016948570446094514425857577827]) : ℤ))
+      (naturalFromChunks [
+      1280000,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      818124,
+      19421896421628635293000871486288819,
+      76388621254444552878242434497475770,
+      38060824003142928695543674529583673])
+      (naturalFromChunks [
+      1280000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      217341,
+      31536488008256338931217996649598166,
+      30981014634211552498131485054543784,
+      62955665460692692088623299896984261])
+      (naturalFromChunks [
+      320000,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 30, sharing kernel reduction. -/
+def cachedPath30 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      472122,
+      27345801840505864683752565458625043,
+      340765445130149541613146265145124,
+      25488894246904112308449638971592661]) : ℤ))
+      (naturalFromChunks [
+      800000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      70386,
+      51776643102131349176530360857896298,
+      79131142730744996451010861490057546,
+      22223031733375280889558616807561627]) : ℤ))
+      (naturalFromChunks [
+      128000,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      256890,
+      54877857413562604335920363182883048,
+      56113696613416722526159793535269617,
+      80194738117956331104947460921687699])
+      (naturalFromChunks [
+      400000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      545912,
+      38121625321411199307989203679242465,
+      8310504212414468291325709056123391,
+      336853041636457322733126202333951])
+      (naturalFromChunks [
+      800000,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 31, sharing kernel reduction. -/
+def cachedPath31 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      1548,
+      23451984757618522483743068889538775,
+      2467044525233695220466329701764808,
+      13030059661837812627229581950797903]) : ℤ))
+      (naturalFromChunks [
+      2500,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      741188,
+      10428913847327982246365640472463594,
+      37223266166343662856610535550156216,
+      38829505429891035371828895235774909]) : ℤ))
+      (naturalFromChunks [
+      1280000,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      4120068,
+      88587496638281270192440342247918330,
+      852068397896095040199119774396422,
+      79684037861133912448924813205371683])
+      (naturalFromChunks [
+      6400000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4377541,
+      82957527272248733233379179755066150,
+      71402981636095666726216332865018091,
+      20707916364849404817370397513298871])
+      (naturalFromChunks [
+      6400000,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 32, sharing kernel reduction. -/
+def cachedPath32 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      1037557,
+      20381538224777594765266198263867591,
+      17355344825998510312127185975386422,
+      19010784226342049402859228835237643]) : ℤ))
+      (naturalFromChunks [
+      1600000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      6082,
+      32650921472863019893610369745587324,
+      87027570329937475358145268319359999,
+      67679932552720075109023094295453559]) : ℤ))
+      (naturalFromChunks [
+      10000,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      206003,
+      44429374831914063509622015054768728,
+      55276994246074772276377295948882947,
+      38179827455483014586016497717007069])
+      (naturalFromChunks [
+      320000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1094385,
+      45739381818062183308344784394971508,
+      83458021763788356837982250938260234,
+      49082253023200486240710766338626717])
+      (naturalFromChunks [
+      1600000,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 33, sharing kernel reduction. -/
+def cachedPath33 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      27105,
+      27155241043932392696664843446877106,
+      50354511670250788480004983656385382,
+      56990910931227646574861427579691257]) : ℤ))
+      (naturalFromChunks [
+      40000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      815838,
+      22980861482340674564574493531771712,
+      94800799591790395897536460124431280,
+      15716935301933724021892220250268897]) : ℤ))
+      (naturalFromChunks [
+      1280000,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      822049,
+      75609143723400333874945137498657250,
+      73729894201134539091820954320725837,
+      38537948222578738727711283411367437])
+      (naturalFromChunks [
+      1280000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4367299,
+      4973002571289594463913502933804126,
+      765491218844505831266316844074914,
+      50292332714913081064053457711124621])
+      (naturalFromChunks [
+      6400000,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 34, sharing kernel reduction. -/
+def cachedPath34 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      226147,
+      21390757401654622492277079674082469,
+      30163024462876945293415825142433941,
+      64363150663990668544077633154881581]) : ℤ))
+      (naturalFromChunks [
+      320000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      426506,
+      73077691637404356050880871492442174,
+      72191980451911471149162489950729374,
+      79179700669322613733030819794165769]) : ℤ))
+      (naturalFromChunks [
+      640000,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      409062,
+      9710948210814317646500415179265414,
+      739276934016533677155708834300459,
+      38717100103909717517868281743284367])
+      (naturalFromChunks [
+      640000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      10867,
+      6576824400412816946560899305607678,
+      948756152874816408148517313783160,
+      64178612778337557259232429562576461])
+      (naturalFromChunks [
+      16000,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 35, sharing kernel reduction. -/
+def cachedPath35 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      18391,
+      21790314748260181728078487908628352,
+      13221599473533026755052808740851754,
+      42819733087419604316361358437656289]) : ℤ))
+      (naturalFromChunks [
+      25000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      177997,
+      53513288175419116530506659247824450,
+      50634414250675474978678305401995017,
+      62828540394903469587928314344297931]) : ℤ))
+      (naturalFromChunks [
+      256000,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2030605,
+      12467851417967748742899823374996283,
+      74418666746470078202535116915796006,
+      79808973208004414171127008899367409])
+      (naturalFromChunks [
+      3200000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2158074,
+      61253468199412088939611742940784153,
+      38263522231899227350465390975518044,
+      66441686576767014574220099412652673])
+      (naturalFromChunks [
+      3200000,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 36, sharing kernel reduction. -/
+def cachedPath36 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      157650260878099320243,
+      77877183835544861117453291204457878,
+      85408826503185095076290249776113065,
+      56338317715966295383843926844190705,
+      54063379152704294116189504951559771]) : ℤ))
+      (naturalFromChunks [
+      204800000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      2944527374203468901935,
+      73418658657466729713988915032841429,
+      27452335493026361108700336829728293,
+      2352474002776145182751284289312870,
+      74349856918070432957517775957957047]) : ℤ))
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      31995384474996649030920,
+      30982370731303358225852993965188042,
+      57786581883320588009296983500155874,
+      78279139001959409928903376240709002,
+      71736359927529675093500875116198311])
+      (naturalFromChunks [
+      51200000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2753291354746011280217,
+      92137331557403889246327384849983287,
+      87643888223871324018867139235900343,
+      39952227844095517663775256005116018,
+      55803892166959366738336404579319193])
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 37, sharing kernel reduction. -/
+def cachedPath37 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      13085741654074134269776,
+      1707740814580816308858624218147761,
+      10586530483574567377783177782449970,
+      85464263175550897348181033552398662,
+      55101101011774827779589093218772079]) : ℤ))
+      (naturalFromChunks [
+      16384000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      9558378885962762387558,
+      92529579844214364227344025520042798,
+      53025129524144499922517865350529380,
+      70255581650820681934789214371770310,
+      25610589354628624289781311565119499]) : ℤ))
+      (naturalFromChunks [
+      12800000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      15796594000328278354811,
+      4806842944676444889692661186813701,
+      17188049668106790986154225185769722,
+      51290900371701866475946425238545154,
+      19537735839780775892090405221462671])
+      (naturalFromChunks [
+      25600000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      21776409721634122234757,
+      85765475947541595980081949119086403,
+      36410766429321241702119140339133286,
+      18158245202580975878649676448499776,
+      69968545950001021423433392167593871])
+      (naturalFromChunks [
+      32768000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 38, sharing kernel reduction. -/
+def cachedPath38 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      3388245014781300975675,
+      62257780396155028107572725815834243,
+      79605526765556607257159114066075160,
+      53458880396600943367472701738966522,
+      99811072926611660865191915110308823]) : ℤ))
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      15856849343285038121097,
+      21335660177827266289372394922475410,
+      32353301978731273911042184450596208,
+      52565989780522737971663069774584053,
+      55958672018940696425292920070853233]) : ℤ))
+      (naturalFromChunks [
+      20480000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      62234059691033561205202,
+      72041491975324071202667201957781561,
+      40488269595395087680097267140261562,
+      41944073624348685550025871544939687,
+      51518162853490671913188410094879277])
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2147775196323453072477,
+      12080543567950122671501329245521788,
+      70551249522013375517987908675331295,
+      84559345963399853657792798252786655,
+      52654409187657027218125608568912411])
+      (naturalFromChunks [
+      3276800000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 39, sharing kernel reduction. -/
+def cachedPath39 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      7006380179505396340401,
+      26591034675492979700690500474508639,
+      72925296017885390679453357256060938,
+      20774394996248112906989115061316339,
+      71566095562074281428499310135888971]) : ℤ))
+      (naturalFromChunks [
+      8192000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      5128677963200454349,
+      42652589724043692268324409396339717,
+      67644942894425804576402185090826651,
+      37901527079252080862568026861272062,
+      42301357834304252210273270595382177]) : ℤ))
+      (naturalFromChunks [
+      6400000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      597030955646730910,
+      4518838282687858670956353430855641,
+      2837082848479040788814812047680624,
+      80215994492834686912530087027551208,
+      24034863560506258813513243395950341])
+      (naturalFromChunks [
+      1000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      42262001779426669296770,
+      83370691679274566788010351722341093,
+      25529337445357597648696356226010463,
+      32948024434572673961575707866688863,
+      56067163497323040407316462812425513])
+      (naturalFromChunks [
+      65536000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 40, sharing kernel reduction. -/
+def cachedPath40 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      36160306118857710882,
+      5637364715452318846181583026935750,
+      65994035601950159937052876687844090,
+      66620789810865566670384141816717892,
+      42889012909875150037302349889175209]) : ℤ))
+      (naturalFromChunks [
+      40960000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      42392471525259002272765,
+      98497625433815176813013519398764759,
+      79997446803030558650552850096755195,
+      40429479525172061992291781353866063,
+      92414253188762505333920506346965707]) : ℤ))
+      (naturalFromChunks [
+      51200000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      11978893942808972830606,
+      36739675436558411121279413558673518,
+      92338319843915562704176182731675689,
+      69886701125460575500497427411269169,
+      8797979497755839010486854871716501])
+      (naturalFromChunks [
+      20480000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      12960543596711541739,
+      68716410404648112988552785205398062,
+      92144207255726863406181658992288210,
+      25803750077843375985321827072164013,
+      62254430007761006508875765434759847])
+      (naturalFromChunks [
+      20480000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 41, sharing kernel reduction. -/
+def cachedPath41 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      59624491974566283933102,
+      82547220559936315181955057616233508,
+      86429527491057069340402326136849956,
+      92268404683545009733566424318178944,
+      72678650790114615252734176824129619]) : ℤ))
+      (naturalFromChunks [
+      65536000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      10932093047356330136971,
+      9696759512862818178266377510251090,
+      32134671497289089611679498261973339,
+      30178901499109653244739872845361679,
+      88390514781426934848327838990545399]) : ℤ))
+      (naturalFromChunks [
+      12800000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      7314016486177454451940,
+      95937799647973067723765081269277736,
+      26983114173056975045466954039518194,
+      13748801857053647035388932557238911,
+      86147570199010292928060413920106303])
+      (naturalFromChunks [
+      12800000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      10148070266148323980082,
+      18023060885212154755876893073643680,
+      45450828997018946703879364114695361,
+      74817629414783409651610861273106753,
+      51717065852373876182065250581819883])
+      (naturalFromChunks [
+      16384000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 42, sharing kernel reduction. -/
+def cachedPath42 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      7668918286564322679689,
+      9352599576170392709293563228818052,
+      30179754261523247850192644312643956,
+      52362071461116693780421485311512892,
+      89279188531735054053690373980524333]) : ℤ))
+      (naturalFromChunks [
+      8192000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      90068570023718357663162,
+      65524347277540766694871527812740147,
+      99333196312274416093445656564252617,
+      72690538562106212467477997881839128,
+      51086064377181974217262512486170671]) : ℤ))
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      28495867128803959300327,
+      26163041370325280163063847273374332,
+      3209929424656591702633152350833764,
+      68656394173078496353596419806842287,
+      45868761308054199345187540466626551])
+      (naturalFromChunks [
+      51200000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9904831318221862012534,
+      50905794324561024451615961636933478,
+      79207383237093218573890891722618781,
+      38125139494542634661757462080484819,
+      55303660877500939781949538329617873])
+      (naturalFromChunks [
+      16384000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 43, sharing kernel reduction. -/
+def cachedPath43 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      7879178388149477673746,
+      23714673916456777669685937995870527,
+      10545171267936721742260093570952531,
+      83095828373908876865982289055582915,
+      65730978145220111651927232365375147]) : ℤ))
+      (naturalFromChunks [
+      8192000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      1157685667064592245728,
+      49828259997210687750039367447408573,
+      88604682248001454789638479975678175,
+      86060921589098163162873101250266023,
+      56250931209511387448426599233937467]) : ℤ))
+      (naturalFromChunks [
+      1280000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      216157248190062554479,
+      71785053818265652223979546824576430,
+      76538378090938234336923342683059768,
+      14250195768537990973036583127359109,
+      18479952534475768060907467061622801])
+      (naturalFromChunks [
+      400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      38556703137517870211117,
+      28241724684927075716250640065144502,
+      57720915238635422114559816095869955,
+      96042046875150346981580176867951899,
+      35505717856204925869704657360521887])
+      (naturalFromChunks [
+      65536000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 44, sharing kernel reduction. -/
+def cachedPath44 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      505212415317868294538,
+      50211467016962255326024614023257326,
+      66981110742508224011555354486321043,
+      56476770122811380947738989290375097,
+      77582920687818381268977858681743971]) : ℤ))
+      (naturalFromChunks [
+      512000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      3803605970039735401272,
+      72145347326947618237545474684673909,
+      56928945374755982658264895716101313,
+      31803122109492817865433432527235195,
+      77794107227009722635424036662968813]) : ℤ))
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2141954827079049538394,
+      27639236917832832044482705046802756,
+      77253700858583690067036036199108468,
+      88790303745620361460471883435053948,
+      52663687011200192143889685329791313])
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      584474609222468956590,
+      58905073657357212545258259266027752,
+      26942537469885083847676325235661881,
+      78856820038651330485557574638873692,
+      46292422660751355963068226877845179])
+      (naturalFromChunks [
+      1024000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 45, sharing kernel reduction. -/
+def cachedPath45 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      2649966872698085003432,
+      99273626753215562186566571228357382,
+      44211785454541797674607812071472944,
+      64593253396544484079453272548864777,
+      6138136071829232871078471374530409]) : ℤ))
+      (naturalFromChunks [
+      2621440000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      24372285137753689855826,
+      86409123090265534706672834197570492,
+      80495670025600713919364203079150146,
+      46909900536807392155913195672728113,
+      23013681544175995100353056511267931]) : ℤ))
+      (naturalFromChunks [
+      25600000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      12908236509867806135982,
+      24405113202977992091996386183512872,
+      7347215579436704568971982509727907,
+      51917697256365602987645567880147497,
+      20723926776211771775388129903317057])
+      (naturalFromChunks [
+      25600000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      723408550138949514911,
+      51054859033676080415227491800102356,
+      1853793193088089386983222768366205,
+      67933245721963854464881836676503061,
+      78864150468167902422453221477050403])
+      (naturalFromChunks [
+      1310720000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 46, sharing kernel reduction. -/
+def cachedPath46 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      8472000394770787861028,
+      2665536894607534478167324040939957,
+      90085702472351983988977353805875365,
+      40372119564336809430230185470490439,
+      80620696047447523449649303610982639]) : ℤ))
+      (naturalFromChunks [
+      8192000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      99806644543929202256362,
+      91556188147150478662833102530468523,
+      9753752197400458595687344685903683,
+      37579992291173423103390901427802553,
+      69404480549133793329741677048669969]) : ℤ))
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      49592034369823818580422,
+      34667425687438318739634370691310607,
+      68547011140961530710105652862218337,
+      44359651950619047364576791908808164,
+      79006900321058864968721496978135957])
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      272273988974513452729,
+      76664278446186546275125135494686871,
+      85014858795928474065043429435580073,
+      72931283928705849307596020749146835,
+      32772759217420412222349767430958437])
+      (naturalFromChunks [
+      512000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 47, sharing kernel reduction. -/
+def cachedPath47 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      17311100997464031881445,
+      1512344220772055950560395149915447,
+      16116332505401731781570101989325900,
+      20897977391904366721180889633953509,
+      70906311868029514423129786747076429]) : ℤ))
+      (naturalFromChunks [
+      16384000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      39858444972835500800,
+      57018352083868226106314877565159042,
+      16832487704410403187344799405406145,
+      37596833154004124356891301865804278,
+      39076000321679708747565563543634747]) : ℤ))
+      (naturalFromChunks [
+      40000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      37054585605281853954,
+      94915339884404304339119805910159131,
+      36396236373999104738310437920257421,
+      12437246976726412883316301516375859,
+      88718647538119505660283673844534681])
+      (naturalFromChunks [
+      80000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      16725317056024294295559,
+      77034318343739309400978069538019864,
+      95615820602912051382268769443385630,
+      90296848652607106485154043933012904,
+      45892241938224594282749322505985729])
+      (naturalFromChunks [
+      32768000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 48, sharing kernel reduction. -/
+def cachedPath48 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      275981217681435378862,
+      86619130979711065865052112704629290,
+      84634547757417395005440610964719260,
+      82934756496722231909479808215220524,
+      12122757196464645819591082423394779]) : ℤ))
+      (naturalFromChunks [
+      256000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      13022146108334202659600,
+      61687366393398644740428741388576365,
+      87356394020849788304998433341014883,
+      73243083608818824479047713621000212,
+      36880174613750086299745333662297679]) : ℤ))
+      (naturalFromChunks [
+      12800000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      45150361306281488147735,
+      94224690673439145410284198439667332,
+      67694304580166197059409903448335114,
+      52859201600001340710079618836587823,
+      48280367670439693521216150398367587])
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4995556961198896662,
+      76610657888273821370146691095609852,
+      41621384411657579053042638689649666,
+      37077681604945017607115854269328256,
+      77774981353087221497454858704972097])
+      (naturalFromChunks [
+      10240000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 49, sharing kernel reduction. -/
+def cachedPath49 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      71993284149943212616658,
+      97099490456612209913860422673534614,
+      97964111359791215592675078898571154,
+      28204840341701991238585237533028204,
+      4665316128572012761507436991339023]) : ℤ))
+      (naturalFromChunks [
+      65536000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      42488220324628514009,
+      35738382166792831036049248306933591,
+      95681369709997401681406271575757441,
+      95481092383957032676419720956763770,
+      58074866801741818163395943711773391]) : ℤ))
+      (naturalFromChunks [
+      40960000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      4275758936002020188,
+      63375988869887781459431726228500339,
+      44628594077556808294419735106435159,
+      81216750618248984389030957476883755,
+      5627553881252710983505717429428701])
+      (naturalFromChunks [
+      10240000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      15208210740232113699496,
+      68493925350317657180141097639558021,
+      99944416878430663157971220944245843,
+      76917173107151541378507664010532663,
+      76154619888679762733143406056433321])
+      (naturalFromChunks [
+      32768000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 50, sharing kernel reduction. -/
+def cachedPath50 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      29307081218630724415,
+      80430592264357168198119289665091910,
+      12034387977291587567826251842157099,
+      93110922645113395920887621734524031,
+      38930225913411758723380740217608283]) : ℤ))
+      (naturalFromChunks [
+      26214400000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      108163180183189475643827,
+      25549384594879428175351783936800175,
+      56466592596719639860898979988625645,
+      24417204935529484086378145612724161,
+      47289449639005726591578868873234177]) : ℤ))
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      20127898947040552111019,
+      16213136189845430568926329312694187,
+      71304509393861978260060093665311672,
+      26501755357712753992712139300445255,
+      15315969047664909556137812603846479])
+      (naturalFromChunks [
+      51200000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      5757574789943071342,
+      13393811773274506708685529580597747,
+      76739018115844315809213323172749689,
+      19701315615203688644934297900707542,
+      17454811849992515369678321983529931])
+      (naturalFromChunks [
+      13107200000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 51, sharing kernel reduction. -/
+def cachedPath51 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      1861789442860452858052,
+      37525670196665161410601596030472595,
+      74195387808184084220935328394729863,
+      58360803492354401315992514247453685,
+      44044378386369864244309468289867267]) : ℤ))
+      (naturalFromChunks [
+      1638400000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      6875039738607663470993,
+      68055020551644687732198428785777321,
+      99128782006745908568032578798351692,
+      82902667640438661627154410285780134,
+      78933966096920581771998575869931393]) : ℤ))
+      (naturalFromChunks [
+      6400000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2353086842178169546350,
+      34182377096921935559552742894014476,
+      39201710421975555257206922422686458,
+      97562116398243036094073361828868317,
+      6534469126379371925997495371079853])
+      (naturalFromChunks [
+      6400000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      13544348377143453411646,
+      18177972451653669979422730866805892,
+      72597721025586188040230125583174036,
+      52766344123903491694231185064347674,
+      53216195051668262858705627367822151])
+      (naturalFromChunks [
+      32768000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 52, sharing kernel reduction. -/
+def cachedPath52 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      2362567135595058373067,
+      79146255403379597026328502579953079,
+      68421356139909629911851834511996783,
+      45719972557172012366800070712023143,
+      58820123729708530443484881700403439]) : ℤ))
+      (naturalFromChunks [
+      2048000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      13966083178290418210331,
+      96126169696089460106517877766753560,
+      85248533127098065360131991933703069,
+      854203439712620730209737427584982,
+      59179652572347895401779369488827973]) : ℤ))
+      (naturalFromChunks [
+      12800000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      6988583791290071667195,
+      49928742147790508645321146377384150,
+      75426375740615352401675989428196315,
+      42601120959721244617955537273518334,
+      72740970265481089186556127197612667])
+      (naturalFromChunks [
+      20480000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1582610411666785026245,
+      95288658565463839607726946171383040,
+      22896685612447778725819104221928348,
+      67913818017450811474166838137633790,
+      12850677809462307682975440381537859])
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 53, sharing kernel reduction. -/
+def cachedPath53 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      19164192308501195176730,
+      49730122551557770670473019875089675,
+      30691583688146243136579503846280768,
+      64728109512060421590601469029854805,
+      15797315631581253513621188725261721]) : ℤ))
+      (naturalFromChunks [
+      16384000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      1416789883055958606153,
+      54065828925476856714357397251893026,
+      78368486010663540401489265647580244,
+      17966558365555546532205866471159716,
+      1317816718938404836221035893849017]) : ℤ))
+      (naturalFromChunks [
+      1280000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      8035271800160913109973,
+      38500510846430161546365406016310841,
+      44742895495566306637806878690220600,
+      6640410991488991474351745313570233,
+      54767044531080527081559116378247541])
+      (naturalFromChunks [
+      25600000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      23490058084259180072820,
+      1516488527900369035709955826877710,
+      23991563544490884038658282362514016,
+      89288770638214613208139798889965964,
+      68691802325192982237427355700859099])
+      (naturalFromChunks [
+      65536000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 54, sharing kernel reduction. -/
+def cachedPath54 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      19408227985754104559635,
+      8396324328521525928134306385747216,
+      87139152456209524296278577264112978,
+      37474120520085124558449065157410788,
+      60355706230284863020459636448069033]) : ℤ))
+      (naturalFromChunks [
+      16384000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      35887597459077945473,
+      59035212923305063461215666296455650,
+      92005749006915165436833750201274608,
+      57460797406582414793243334993166877,
+      89172572889006383593751743229969143]) : ℤ))
+      (naturalFromChunks [
+      32000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1169949385766368642493,
+      15619656802541891056022752247159386,
+      36991108023943947733087845255336015,
+      84726614262493445545688641342819049,
+      80209377524264534094747826981637327])
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      5399159943562120680457,
+      47477558985996245565212371678229073,
+      98596008763272863910814866964505341,
+      61043410630469080719053047038055160,
+      29276213552181227154041549550355293])
+      (naturalFromChunks [
+      16384000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 55, sharing kernel reduction. -/
+def cachedPath55 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      3141126728033232139300,
+      64433709273365123641076652943456198,
+      38237652440255962047204367230007711,
+      10962127756509147014114748288147563,
+      6638630291687101485472293632323533]) : ℤ))
+      (naturalFromChunks [
+      2621440000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      907939944361995515657,
+      62104033544065297279192978383106796,
+      26824475805742319319759258436661448,
+      84800465760849910768572062524018572,
+      99150079727612104064270838751869299]) : ℤ))
+      (naturalFromChunks [
+      800000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      410481792508244094258,
+      37437527439361368769059261307906714,
+      28558060276409424485467883250747904,
+      77054668705147774032497475256810071,
+      65436848316672757445121948944023519])
+      (naturalFromChunks [
+      1600000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      392893398411529813972,
+      29422772376653529920942085353092201,
+      59006719482609845780199983468598448,
+      36267306121028127479704529278197436,
+      56827690490780014032667461219173243])
+      (naturalFromChunks [
+      1310720000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 56, sharing kernel reduction. -/
+def cachedPath56 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      247938289843843481706,
+      87972967728677912801209050476587008,
+      73996445291204324188419887357768491,
+      10675039180128071640268490241818327,
+      50236178533167617809229070901382781]) : ℤ))
+      (naturalFromChunks [
+      204800000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      14683458153830003704904,
+      69646805199971480692106625580355812,
+      6523328941128642934926690503153143,
+      9272242886390845438931150544463141,
+      54684817919758799937296519304779131]) : ℤ))
+      (naturalFromChunks [
+      12800000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      11606243571831659676745,
+      63916517908471891481559286566749837,
+      87629564580066460276723997717623838,
+      67642068872609652266539598575592867,
+      54111965568185015171691497722774567])
+      (naturalFromChunks [
+      51200000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      137792127168248826715,
+      95529132471145098400729089727287115,
+      41567730914177240086907325208714506,
+      33917053708245402416328235106112739,
+      17773058008440909251614960282670999])
+      (naturalFromChunks [
+      512000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 57, sharing kernel reduction. -/
+def cachedPath57 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      80067006011017004461286,
+      1796391837515470644568159494326366,
+      30955463005646480206256370460884923,
+      61330182251842024652621971430645627,
+      66963203290414428546674718193511439]) : ℤ))
+      (naturalFromChunks [
+      65536000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      7411939527323301962829,
+      99810774970572544608625822283237291,
+      93343113147993040686831867199618724,
+      57921030378128294595617217251247964,
+      74808189927179389982575801908700929]) : ℤ))
+      (naturalFromChunks [
+      6400000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      5019727765136790482313,
+      23217892944513224113929013455933640,
+      72507099784439513525018974913485235,
+      85194440258981594059741435461348076,
+      83936762323367696968108607654401011])
+      (naturalFromChunks [
+      25600000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      7789065877333308284904,
+      54737189541097368458695265351802718,
+      41135777746979912138288052060510952,
+      27066361141533621168199516304976736,
+      2847598125027115477645371663448341])
+      (naturalFromChunks [
+      32768000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 58, sharing kernel reduction. -/
+def cachedPath58 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      10088299575335840431240,
+      55205804665072096782588160606672331,
+      11638969101672594796819427669537107,
+      98050308332084005555475167798696904,
+      96865471851587102571918776114254837]) : ℤ))
+      (naturalFromChunks [
+      8192000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      119583273905248130666943,
+      13019329116091235392628306054606371,
+      59964469300072616270621376592449125,
+      14311517803419467558922559890197347,
+      32313454234762690790445530286233781]) : ℤ))
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2109429760060368627470,
+      80006764042218937252484565613892903,
+      93412665360575265295705165935041889,
+      3651628705288727530696183298924961,
+      70209506322442573399541021334135201])
+      (naturalFromChunks [
+      12800000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      841893124854572354980,
+      42397658068461024358164265043485336,
+      45538646147866940676041643609537972,
+      53401481845321185738987124597585745,
+      81786377887782996222358894912070487])
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 59, sharing kernel reduction. -/
+def cachedPath59 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      40628261855695108521476,
+      98436164360080019039896744190127201,
+      47601975957783876214227478621905782,
+      25576246974829156663735797899602996,
+      45420038345741960605410511808728001]) : ℤ))
+      (naturalFromChunks [
+      32768000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      301103621620064197898,
+      8299094629563441193986484979507147,
+      89737269524841028484629942536317194,
+      74517547697049172540089371671692020,
+      88609728189593713036286665297731631]) : ℤ))
+      (naturalFromChunks [
+      256000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      34018762971280009037,
+      16038881456561924907038935837437297,
+      99960461060024166826417223242565572,
+      16254863570643640444066946522242964,
+      81343029253984610677309885333180499])
+      (naturalFromChunks [
+      256000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      11317340890168309697721,
+      45410299028241762657703443089265939,
+      79771779002626572528856603660265125,
+      54970491515034613284867040238729879,
+      5521750114139824355078337721894631])
+      (naturalFromChunks [
+      65536000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 60, sharing kernel reduction. -/
+def cachedPath60 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      204289048264916622360,
+      25303690206038581779430164991556215,
+      2192542270261213781441669906981730,
+      66383526667398392301139367816128553,
+      15726067403295889577864319623494443]) : ℤ))
+      (naturalFromChunks [
+      163840000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      121162818757260980189505,
+      73632300228112868338850596005476251,
+      35119747346774774138799209858922082,
+      30292570526512098617456899222946956,
+      95047460575534287969219970635556017]) : ℤ))
+      (naturalFromChunks [
+      102400000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1285080943834146982381,
+      94076033656129489289430988656901334,
+      22865595832567104132331138343971778,
+      2985720029198170062456256798785702,
+      33723476089459091675404318918498647])
+      (naturalFromChunks [
+      12800000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2280706495280212676,
+      29625833186939974161218473680947791,
+      60244908883737083223098208681189621,
+      77514689582140903829783894757156308,
+      75484925373905506392385720664182509])
+      (naturalFromChunks [
+      16384000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 61, sharing kernel reduction. -/
+def cachedPath61 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      5130128137297982199970,
+      11979015270500852222911160591749029,
+      25387714147347173771511135197567510,
+      99264800091974613863810433868205371,
+      70937418602639320365290984537430863]) : ℤ))
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      30436213158943509395870,
+      14842680304673036196000346367541529,
+      1194030108900438413113319737277935,
+      18710598814112320770476596937507304,
+      54248570701731552372197427932900391]) : ℤ))
+      (naturalFromChunks [
+      25600000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1725122804722919108622,
+      65996905569999348341980100555845890,
+      48926926223974220807794570992152633,
+      6524284593208549896091570936524320,
+      74595623604981686149735450315547597])
+      (naturalFromChunks [
+      25600000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      6890349842948582395353,
+      39086950416118534041198169122654079,
+      31423384727865696078286774376217932,
+      59251993932395900187461249530151928,
+      16286338519958681309858881941110863])
+      (naturalFromChunks [
+      65536000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 62, sharing kernel reduction. -/
+def cachedPath62 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      643393124505926708537,
+      38552685670926548460089453816778488,
+      38544509865162488014208150769602241,
+      85686048365271887104508360608569377,
+      12303532657538062832233408151059051]) : ℤ))
+      (naturalFromChunks [
+      512000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      7636576749161744317076,
+      85235683889594245840246913456978817,
+      46167670878428799642138256405623192,
+      46371516784914811403710767791822485,
+      31039193548587898073356345094381007]) : ℤ))
+      (naturalFromChunks [
+      6400000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (naturalFromChunks [
+      86818643584218515155,
+      3965904627296979473505833857292664,
+      31216863896850199025975956944204168,
+      64713996069642754926766697377185979,
+      84282986450372156536682514815769571])
+      (naturalFromChunks [
+      2560000000000000000000,
+      0,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      288973460328803356995,
+      84320300751959621006429197034719980,
+      50460615055939948234575415871664412,
+      61757624002882346781705050908845772,
+      32582892504375119601395101704657703])
+      (naturalFromChunks [
+      4096000000000000000000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for cell 63, sharing kernel reduction. -/
+def cachedPath63 : RatInterval × RatInterval :=
+  (⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      15,
+      42079517288790286680036160806903126,
+      43012829623794531003067991380792405,
+      28809926010512734426230880545237976,
+      61731630191804635017229301491936027]) : ℤ))
+      (naturalFromChunks [
+      12,
+      50000000000000000000000000000000000,
+      0,
+      0,
+      0])),
+    (ExactReplay.q
+      (-((naturalFromChunks [
+      1526206,
+      54711468459557349232894324772787465,
+      4598192129324489394972717423811205,
+      95446486344966956824293103245753667,
+      87711918370437577208529341465376517]) : ℤ))
+      (naturalFromChunks [
+      1250000,
+      0,
+      0,
+      0,
+      0]))⟩,
+    ⟨(ExactReplay.q
+      (-((naturalFromChunks [
+      18823831,
+      48723624281463936633164280237407656,
+      26335680331785015270314432345086971]) : ℤ))
+      (naturalFromChunks [
+      250000000000,
+      0,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      87326,
+      79792061110285957775373564882726820,
+      13026983046313166892255834128622,
+      16157419760387029253635163962055685,
+      70086123399049940050711975911460799])
+      (naturalFromChunks [
+      2500000,
+      0,
+      0,
+      0,
+      0]))⟩)
+
+/-- The exact cached path-coordinate pair for the selected angle cell. -/
+def cachedPath (i : Cell) : RatInterval × RatInterval :=
+  match i.val with
+  | 0 => cachedPath00
+  | 1 => cachedPath01
+  | 2 => cachedPath02
+  | 3 => cachedPath03
+  | 4 => cachedPath04
+  | 5 => cachedPath05
+  | 6 => cachedPath06
+  | 7 => cachedPath07
+  | 8 => cachedPath08
+  | 9 => cachedPath09
+  | 10 => cachedPath10
+  | 11 => cachedPath11
+  | 12 => cachedPath12
+  | 13 => cachedPath13
+  | 14 => cachedPath14
+  | 15 => cachedPath15
+  | 16 => cachedPath16
+  | 17 => cachedPath17
+  | 18 => cachedPath18
+  | 19 => cachedPath19
+  | 20 => cachedPath20
+  | 21 => cachedPath21
+  | 22 => cachedPath22
+  | 23 => cachedPath23
+  | 24 => cachedPath24
+  | 25 => cachedPath25
+  | 26 => cachedPath26
+  | 27 => cachedPath27
+  | 28 => cachedPath28
+  | 29 => cachedPath29
+  | 30 => cachedPath30
+  | 31 => cachedPath31
+  | 32 => cachedPath32
+  | 33 => cachedPath33
+  | 34 => cachedPath34
+  | 35 => cachedPath35
+  | 36 => cachedPath36
+  | 37 => cachedPath37
+  | 38 => cachedPath38
+  | 39 => cachedPath39
+  | 40 => cachedPath40
+  | 41 => cachedPath41
+  | 42 => cachedPath42
+  | 43 => cachedPath43
+  | 44 => cachedPath44
+  | 45 => cachedPath45
+  | 46 => cachedPath46
+  | 47 => cachedPath47
+  | 48 => cachedPath48
+  | 49 => cachedPath49
+  | 50 => cachedPath50
+  | 51 => cachedPath51
+  | 52 => cachedPath52
+  | 53 => cachedPath53
+  | 54 => cachedPath54
+  | 55 => cachedPath55
+  | 56 => cachedPath56
+  | 57 => cachedPath57
+  | 58 => cachedPath58
+  | 59 => cachedPath59
+  | 60 => cachedPath60
+  | 61 => cachedPath61
+  | 62 => cachedPath62
+  | _ => cachedPath63
+
+/-- The exact cached cosine interval for cell 0, sharing kernel reduction. -/
+def cachedCosine00 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      4998494093481021100578828,
+      24833086098425030540628864810962801])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      1
+      1)⟩
+
+/-- The exact cached cosine interval for cell 1, sharing kernel reduction. -/
+def cachedCosine01 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9987954562051723927147716,
+      4759100694443203614704611783472007])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      999698818696204220115765,
+      64966617219685006108125772962576017])
+      (naturalFromChunks [
+      1000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 2, sharing kernel reduction. -/
+def cachedCosine02 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1246613070848362770169496,
+      42522820977646461084895777668595067])
+      (naturalFromChunks [
+      1250000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9987954562051723927147716,
+      4759100694443203614704611798805657])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 3, sharing kernel reduction. -/
+def cachedCosine03 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9951847266721968862448369,
+      53109479921575474868729857018402753])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4986452283393451080677985,
+      70091283910585844339583110691621963])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 4, sharing kernel reduction. -/
+def cachedCosine04 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9924795345987099981567672,
+      51661117820010820654634159478691333])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9951847266721968862448369,
+      53109479921575474868729857079663467])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 5, sharing kernel reduction. -/
+def cachedCosine05 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9891765099647809734516737,
+      38016243063983689533336907303495943])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4962397672993549990783836,
+      25830558910005410327317079787162337])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 6, sharing kernel reduction. -/
+def cachedCosine06 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      4926388211944706223870092,
+      16589273893580064577906407370938883])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2472941274911952433629184,
+      34504060765995922383334226860263941])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 7, sharing kernel reduction. -/
+def cachedCosine07 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1961570560806460898252364,
+      47226847807394786746178667184423201])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9852776423889412447740184,
+      33178547787160129155812814928867333])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 8, sharing kernel reduction. -/
+def cachedCosine08 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9757021300385285444603957,
+      66419527971644012265792042946977421])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9807852804032304491261822,
+      36134239036973933730893336165978909])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 9, sharing kernel reduction. -/
+def cachedCosine09 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2425078132986359981509960,
+      51821525062864216490562018457956381])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9757021300385285444603957,
+      66419527971644012265792043255088541])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 10, sharing kernel reduction. -/
+def cachedCosine10 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9637760657954398666864643,
+      55507835153663083848826632379316763])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1212539066493179990754980,
+      25910762531432108245281009276435257])
+      (naturalFromChunks [
+      1250000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 11, sharing kernel reduction. -/
+def cachedCosine11 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      4784701678661044324678989,
+      43490134984741424602815018437715273])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1927552131590879733372928,
+      71101567030732616769765326567545863])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 12, sharing kernel reduction. -/
+def cachedCosine12 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1186910225741295833994920,
+      9273668128531528019229051300090539])
+      (naturalFromChunks [
+      1250000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9569403357322088649357978,
+      86980269969482849205630037419714317])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 13, sharing kernel reduction. -/
+def cachedCosine13 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      4707720325915103892062547,
+      1299751178592794897912591172543821])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2373820451482591667989840,
+      18547336257063056038458102759472587])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 14, sharing kernel reduction. -/
+def cachedCosine14 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9329927988347388877116602,
+      55543302498295015520512294450889103])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4707720325915103892062547,
+      1299751178592794897912591541017099])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 15, sharing kernel reduction. -/
+def cachedCosine15 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      184775906502257351225636,
+      63787935765736448332517272836157107])
+      (naturalFromChunks [
+      200000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1865985597669477775423320,
+      51108660499659003104102459058878619])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 16, sharing kernel reduction. -/
+def cachedCosine16 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      914209755703530654635014,
+      82939357740104469111568217623822567])
+      (naturalFromChunks [
+      1000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9238795325112867561281831,
+      89396788286822416625863642764563933])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 17, sharing kernel reduction. -/
+def cachedCosine17 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9039892931234433315862002,
+      97230537048710132025050607197151713])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4571048778517653273175074,
+      14696788700522345557841088657323959])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 18, sharing kernel reduction. -/
+def cachedCosine18 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      8932243011955153203424164,
+      47493397978000625588998871843311403])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9039892931234433315862002,
+      97230537048710132025050608399650477])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 19, sharing kernel reduction. -/
+def cachedCosine19 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      8819212643483550297127568,
+      63660388349508442620674726936265953])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      8932243011955153203424164,
+      47493397978000625588998873178095531])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 20, sharing kernel reduction. -/
+def cachedCosine20 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1740173982217422837304584,
+      80896769768782165557905964868101359])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      8819212643483550297127568,
+      63660388349508442620674728409384211])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 21, sharing kernel reduction. -/
+def cachedCosine21 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      428864305000136034951134,
+      99214238506852124539971686638286303])
+      (naturalFromChunks [
+      500000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1740173982217422837304584,
+      80896769768782165557905965191570209])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 22, sharing kernel reduction. -/
+def cachedCosine22 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1689707130499414146519142,
+      41020991419543957196277821452719797])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4288643050001360349511349,
+      92142385068521245399716867266541721])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 23, sharing kernel reduction. -/
+def cachedCosine23 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      4157348061512726185393941,
+      88808952878369280405993624243212271])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      528033478281066920787232,
+      319059818607486623836819324184143])
+      (naturalFromChunks [
+      625000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 24, sharing kernel reduction. -/
+def cachedCosine24 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      8175848131515836965049208,
+      84130633809471042517566912499545797])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2078674030756363092696970,
+      94404476439184640202996812643334609])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 25, sharing kernel reduction. -/
+def cachedCosine25 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1606415062961289819613353,
+      2592628384775913885434091823855011])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4087924065757918482524604,
+      42065316904735521258783457382713731])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 26, sharing kernel reduction. -/
+def cachedCosine26 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      63067714210128500960733,
+      17642877514261251945097197174484151])
+      (naturalFromChunks [
+      80000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1606415062961289819613353,
+      2592628384775913885434092322625003])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 27, sharing kernel reduction. -/
+def cachedCosine27 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      7730104533627369608109066,
+      9758469800971041292900807641309649])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      7883464276266062620091647,
+      5359689282656493137149649715782547])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 28, sharing kernel reduction. -/
+def cachedCosine28 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      7572088465064845475754640,
+      53605784473040433715731614750260323])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      7730104533627369608109066,
+      9758469800971041292900811650456043])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 29, sharing kernel reduction. -/
+def cachedCosine29 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      7409511253549590911756168,
+      97495162729728955309309087808108197])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      7572088465064845475754640,
+      53605784473040433715731622390746331])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 30, sharing kernel reduction. -/
+def cachedCosine30 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      7242470829514669209410692,
+      43290553167483093004800434490403453])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1852377813387397727939042,
+      24373790682432238827327276982699053])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 31, sharing kernel reduction. -/
+def cachedCosine31 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      7071067811865475244008443,
+      62104849039284835937688471452121003])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      7242470829514669209410692,
+      43290553167483093004800496842348879])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 32, sharing kernel reduction. -/
+def cachedCosine32 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      6895405447370669246167306,
+      29957484702845536844279120311458241])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      7071067811865475244008443,
+      62104849039284835937688672182256779])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 33, sharing kernel reduction. -/
+def cachedCosine33 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      6715589548470184006253768,
+      50427421803228750632199790827325159])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      6895405447370669246167306,
+      29957484702845536844279758975438619])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 34, sharing kernel reduction. -/
+def cachedCosine34 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      6531728429537767640842030,
+      13656305415076860023714154803176483])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      6715589548470184006253768,
+      50427421803228750632201768979747313])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 35, sharing kernel reduction. -/
+def cachedCosine35 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      396495802602278436384482,
+      25826593335667230443427606866067141])
+      (naturalFromChunks [
+      625000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      653172842953776764084203,
+      1365630541507686002372009887695613])
+      (naturalFromChunks [
+      1000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 36, sharing kernel reduction. -/
+def cachedCosine36 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      3076157952903134227424567,
+      81706992138829715003882209350922361])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      6343932841636454982151716,
+      13225493370675687094841728373967047])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 37, sharing kernel reduction. -/
+def cachedCosine37 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      5956993044924333434670365,
+      28829969889511926338437499068151809])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      3076157952903134227424567,
+      81706992138829715003882215574100041])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 38, sharing kernel reduction. -/
+def cachedCosine38 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      5758081914178453007459724,
+      53815730841776008455314090690525803])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1191398608984866686934073,
+      5765993977902385267687502374095949])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 39, sharing kernel reduction. -/
+def cachedCosine39 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      86807848909312847616067,
+      31467945826162108393605543717839779])
+      (naturalFromChunks [
+      156250000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      5758081914178453007459724,
+      53815730841776008455314103848793047])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 40, sharing kernel reduction. -/
+def cachedCosine40 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1069995239774194421326153,
+      80927403583112053138434378794486499])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2777851165098011123714154,
+      6974266437187468595377405726590153])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 41, sharing kernel reduction. -/
+def cachedCosine41 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2570513720966108632968469,
+      19484407886304024560208105658651777])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1337494049717743026657692,
+      26159254478890066423042976958330357])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 42, sharing kernel reduction. -/
+def cachedCosine42 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2464490961148920184365133,
+      44379404634119843653274178452139081])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      5141027441932217265936938,
+      38968815772608049120416225523435447])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 43, sharing kernel reduction. -/
+def cachedCosine43 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      294622960516248530347742,
+      26619078398603591269933279609753839])
+      (naturalFromChunks [
+      625000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4928981922297840368730266,
+      88758809268239687306548371451045337])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 44, sharing kernel reduction. -/
+def cachedCosine44 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      4496113296546066000462945,
+      79424227075883187048377850210090887])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1178491842064994121390969,
+      6476313594414365079733122159620701])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 45, sharing kernel reduction. -/
+def cachedCosine45 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      4275550934302820943209668,
+      56888798534304578629342451401959127])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2248056648273033000231472,
+      89712113537941593524188932711408637])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 46, sharing kernel reduction. -/
+def cachedCosine46 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      4052413140049898709084813,
+      5505052466511947754105081593303987])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      133610966696963154475302,
+      14277774954197018082166952091852347])
+      (naturalFromChunks [
+      312500000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 47, sharing kernel reduction. -/
+def cachedCosine47 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      765366864730179543456919,
+      96806079773352268912497123888020471])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2026206570024949354542406,
+      52752526233255973877052548724564719])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 48, sharing kernel reduction. -/
+def cachedCosine48 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      3598950365349881487751045,
+      72326756420202317421129018070100053])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      153073372946035908691383,
+      99361215954670453782499425424319767])
+      (naturalFromChunks [
+      400000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 49, sharing kernel reduction. -/
+def cachedCosine49 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      842224633480550126723133,
+      3154786892619441694917803582196619])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1799475182674940743875522,
+      86163378210101158710564517271627079])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 50, sharing kernel reduction. -/
+def cachedCosine50 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1568408701994457383282394,
+      22997050154996688754728269328643201])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1684449266961100253446266,
+      6309573785238883389835615550020069])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 51, sharing kernel reduction. -/
+def cachedCosine51 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2902846772544623676361923,
+      75817395274691476278324142814062403])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1568408701994457383282394,
+      22997050154996688754728277859560449])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 52, sharing kernel reduction. -/
+def cachedCosine52 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      533425514949796772650573,
+      3023287278808423397671229548961797])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2902846772544623676361923,
+      75817395274691476278324160158605989])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 53, sharing kernel reduction. -/
+def cachedCosine53 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2429801799032638899482741,
+      62077471118320990783283823500268651])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2667127574748983863252865,
+      15116436394042116988356165363840189])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 54, sharing kernel reduction. -/
+def cachedCosine54 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2191012401568697972277375,
+      47497357798848360796705583324219953])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2429801799032638899482741,
+      62077471118320990783283841385219883])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 55, sharing kernel reduction. -/
+def cachedCosine55 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1950903220161282678482848,
+      68477022240927691617751945869062889])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1095506200784348986138687,
+      73748678899424180398352800733090717])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 56, sharing kernel reduction. -/
+def cachedCosine56 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      854809443801506131818211,
+      78604131765983164529572290956241377])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      975451610080641339241424,
+      34238511120463845808875982129393449])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 57, sharing kernel reduction. -/
+def cachedCosine57 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1467304744553617516588501,
+      29646717819706215316529382832580349])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      854809443801506131818211,
+      78604131765983164529572300270194091])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 58, sharing kernel reduction. -/
+def cachedCosine58 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      153013343999020248123380,
+      59268868223446902951135637234185599])
+      (naturalFromChunks [
+      1250000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      366826186138404379147125,
+      32411679454926553829132350422189393])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 59, sharing kernel reduction. -/
+def cachedCosine59 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      980171403295606019941955,
+      63888641845861136673167491056370909])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1224106751992161984987044,
+      74150945787575223609085116947700849])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 60, sharing kernel reduction. -/
+def cachedCosine60 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      91955704499584279411832,
+      2696904290226662408209734700340633])
+      (naturalFromChunks [
+      1250000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      98017140329560601994195,
+      56388864184586113667316751033807499])
+      (naturalFromChunks [
+      1000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 61, sharing kernel reduction. -/
+def cachedCosine61 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      490676743274180142549549,
+      76942682658314745363025743153535963])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      735645635996674235294656,
+      21575234321813299265677897081054297])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 62, sharing kernel reduction. -/
+def cachedCosine62 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      6135307130728072007933,
+      63236482073126636652980986039783221])
+      (naturalFromChunks [
+      250000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      122669185818545035637387,
+      44235670664578686340756440704330377])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for cell 63, sharing kernel reduction. -/
+def cachedCosine63 : RatInterval :=
+  ⟨(ExactReplay.q
+      0
+      1),
+    (ExactReplay.q
+      (naturalFromChunks [
+      122706142614561440158672,
+      64729641462532733059619730714551137])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached cosine interval for the selected angle cell. -/
+def cachedCosine (i : Cell) : RatInterval :=
+  match i.val with
+  | 0 => cachedCosine00
+  | 1 => cachedCosine01
+  | 2 => cachedCosine02
+  | 3 => cachedCosine03
+  | 4 => cachedCosine04
+  | 5 => cachedCosine05
+  | 6 => cachedCosine06
+  | 7 => cachedCosine07
+  | 8 => cachedCosine08
+  | 9 => cachedCosine09
+  | 10 => cachedCosine10
+  | 11 => cachedCosine11
+  | 12 => cachedCosine12
+  | 13 => cachedCosine13
+  | 14 => cachedCosine14
+  | 15 => cachedCosine15
+  | 16 => cachedCosine16
+  | 17 => cachedCosine17
+  | 18 => cachedCosine18
+  | 19 => cachedCosine19
+  | 20 => cachedCosine20
+  | 21 => cachedCosine21
+  | 22 => cachedCosine22
+  | 23 => cachedCosine23
+  | 24 => cachedCosine24
+  | 25 => cachedCosine25
+  | 26 => cachedCosine26
+  | 27 => cachedCosine27
+  | 28 => cachedCosine28
+  | 29 => cachedCosine29
+  | 30 => cachedCosine30
+  | 31 => cachedCosine31
+  | 32 => cachedCosine32
+  | 33 => cachedCosine33
+  | 34 => cachedCosine34
+  | 35 => cachedCosine35
+  | 36 => cachedCosine36
+  | 37 => cachedCosine37
+  | 38 => cachedCosine38
+  | 39 => cachedCosine39
+  | 40 => cachedCosine40
+  | 41 => cachedCosine41
+  | 42 => cachedCosine42
+  | 43 => cachedCosine43
+  | 44 => cachedCosine44
+  | 45 => cachedCosine45
+  | 46 => cachedCosine46
+  | 47 => cachedCosine47
+  | 48 => cachedCosine48
+  | 49 => cachedCosine49
+  | 50 => cachedCosine50
+  | 51 => cachedCosine51
+  | 52 => cachedCosine52
+  | 53 => cachedCosine53
+  | 54 => cachedCosine54
+  | 55 => cachedCosine55
+  | 56 => cachedCosine56
+  | 57 => cachedCosine57
+  | 58 => cachedCosine58
+  | 59 => cachedCosine59
+  | 60 => cachedCosine60
+  | 61 => cachedCosine61
+  | 62 => cachedCosine62
+  | _ => cachedCosine63
+
+/-- The exact cached sine interval for cell 0, sharing kernel reduction. -/
+def cachedSine00 : RatInterval :=
+  ⟨(ExactReplay.q
+      0
+      1),
+    (ExactReplay.q
+      (naturalFromChunks [
+      61353071307280720079336,
+      32364820731266366529809862897079257])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 1, sharing kernel reduction. -/
+def cachedSine01 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      122706142614561440158672,
+      64729641462532733059619725716057043])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      245338371637090071274774,
+      88471341329157372681512876570745263])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 2, sharing kernel reduction. -/
+def cachedSine02 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      98135348654836028509909,
+      95388536531662949072605150565873389])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      11494463062448034926479,
+      337113036278332801026216993369213])
+      (naturalFromChunks [
+      156250000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 3, sharing kernel reduction. -/
+def cachedSine03 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      735645635996674235294656,
+      21575234321813299265677887108149729])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      980171403295606019941955,
+      63888641845861136673167501008218177])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 4, sharing kernel reduction. -/
+def cachedSine04 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      490085701647803009970977,
+      81944320922930568336583750193113861])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1224106751992161984987044,
+      74150945787575223609085107798280139])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 5, sharing kernel reduction. -/
+def cachedSine05 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      612053375996080992493522,
+      37075472893787611804542553511452751])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      29346094891072350331770,
+      2592934356394124306330587854486909])
+      (naturalFromChunks [
+      200000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 6, sharing kernel reduction. -/
+def cachedSine06 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1467304744553617516588501,
+      29646717819706215316529391796992471])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1709618887603012263636423,
+      57208263531966329059144591765259179])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 7, sharing kernel reduction. -/
+def cachedSine07 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1709618887603012263636423,
+      57208263531966329059144590687611757])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      975451610080641339241424,
+      34238511120463845808875977838457847])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 8, sharing kernel reduction. -/
+def cachedSine08 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1950903220161282678482848,
+      68477022240927691617751954450934093])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1095506200784348986138687,
+      73748678899424180398352796540620627])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 9, sharing kernel reduction. -/
+def cachedSine09 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2191012401568697972277375,
+      47497357798848360796705591709160133])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      37965653109884982804417,
+      83782460486223765480988809893759081])
+      (naturalFromChunks [
+      156250000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 10, sharing kernel reduction. -/
+def cachedSine10 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      48596035980652777989654,
+      83241549422366419815665676633698147])
+      (naturalFromChunks [
+      200000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      666781893687245965813216,
+      28779109098510529247089039345642411])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 11, sharing kernel reduction. -/
+def cachedSine11 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      266712757474898386325286,
+      51511643639404211698835615572607953])
+      (naturalFromChunks [
+      1000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2902846772544623676361923,
+      75817395274691476278324152383465761])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 12, sharing kernel reduction. -/
+def cachedSine12 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2902846772544623676361923,
+      75817395274691476278324150589202631])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      3136817403988914766564788,
+      45994100309993377509456548152568209])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 13, sharing kernel reduction. -/
+def cachedSine13 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      3136817403988914766564788,
+      45994100309993377509456546223839091])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      3368898533922200506892532,
+      12619147570477766779671223744227129])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 14, sharing kernel reduction. -/
+def cachedSine14 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      673779706784440101378506,
+      42523829514095553355934244336919897])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1799475182674940743875522,
+      86163378210101158710564513700014021])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 15, sharing kernel reduction. -/
+def cachedSine15 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      3598950365349881487751045,
+      72326756420202317421129025213326169])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      3826834323650897717284599,
+      84030398866761344562485628678897681])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 16, sharing kernel reduction. -/
+def cachedSine16 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      3826834323650897717284599,
+      84030398866761344562485626369198849])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      810482628009979741816962,
+      61101010493302389550821018147080309])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 17, sharing kernel reduction. -/
+def cachedSine17 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      101310328501247467727120,
+      32637626311662798693852627207675797])
+      (naturalFromChunks [
+      250000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      213777546715141047160483,
+      42844439926715228931467123022092603])
+      (naturalFromChunks [
+      500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 18, sharing kernel reduction. -/
+def cachedSine18 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      4275550934302820943209668,
+      56888798534304578629342457899382171])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2248056648273033000231472,
+      89712113537941593524188929571166953])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 19, sharing kernel reduction. -/
+def cachedSine19 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      899222659309213200092589,
+      15884845415176637409675571298114851])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      36827870064531066293467,
+      78327384799825448908741660020119329])
+      (naturalFromChunks [
+      78125000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 20, sharing kernel reduction. -/
+def cachedSine20 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1178491842064994121390969,
+      6476313594414365079733119954817529])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2464490961148920184365133,
+      44379404634119843653274182802574183])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 21, sharing kernel reduction. -/
+def cachedSine21 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2464490961148920184365133,
+      44379404634119843653274181375087567])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1285256860483054316484234,
+      59742203943152012280104054973647861])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 22, sharing kernel reduction. -/
+def cachedSine22 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      5141027441932217265936938,
+      38968815772608049120416216946147557])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      213999047954838884265230,
+      76185480716622410627686876096839131])
+      (naturalFromChunks [
+      400000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 23, sharing kernel reduction. -/
+def cachedSine23 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      167186756214717878332211,
+      53269906809861258302880371855774239])
+      (naturalFromChunks [
+      312500000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      5555702330196022247428308,
+      13948532874374937190754806256495229])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 24, sharing kernel reduction. -/
+def cachedSine24 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      5555702330196022247428308,
+      13948532874374937190754803138430933])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      719760239272306625932465,
+      56726966355222001056914262358329449])
+      (naturalFromChunks [
+      1250000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 25, sharing kernel reduction. -/
+def cachedSine25 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2879040957089226503729862,
+      26907865420888004227657047836341629])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      5956993044924333434670365,
+      28829969889511926338437507101435027])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 26, sharing kernel reduction. -/
+def cachedSine26 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      5956993044924333434670365,
+      28829969889511926338437503837196527])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      6152315905806268454849135,
+      63413984277659430007764426590572297])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 27, sharing kernel reduction. -/
+def cachedSine27 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      6152315905806268454849135,
+      63413984277659430007764423259472507])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      6343932841636454982151716,
+      13225493370675687094841724025783247])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 28, sharing kernel reduction. -/
+def cachedSine28 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      6343932841636454982151716,
+      13225493370675687094841720622123773])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      6531728429537767640842030,
+      13656305415076860023714163065397987])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 29, sharing kernel reduction. -/
+def cachedSine29 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1632932107384441910210507,
+      53414076353769215005928539887217203])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1678897387117546001563442,
+      12606855450807187658049949320398029])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 30, sharing kernel reduction. -/
+def cachedSine30 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      3357794774235092003126884,
+      25213710901614375316099896743952173])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      3447702723685334623083653,
+      14978742351422768422139563431888753])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 31, sharing kernel reduction. -/
+def cachedSine31 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      6895405447370669246167306,
+      29957484702845536844279122204435661])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      3535533905932737622004221,
+      81052424519642417968844240256401493])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 32, sharing kernel reduction. -/
+def cachedSine32 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      353553390593273762200422,
+      18105242451964241796884423650304123])
+      (naturalFromChunks [
+      500000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1448494165902933841882138,
+      48658110633496618600960090516316199])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 33, sharing kernel reduction. -/
+def cachedSine33 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      7242470829514669209410692,
+      43290553167483093004800435840085161])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1157736133367123579961,
+      90140233619176520149267079552312427])
+      (naturalFromChunks [
+      1562500000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 34, sharing kernel reduction. -/
+def cachedSine34 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      3704755626774795455878084,
+      48747581364864477654654544494722267])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1893022116266211368938660,
+      13401446118260108428932937539722931])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 35, sharing kernel reduction. -/
+def cachedSine35 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      3786044232532422737877320,
+      26802892236520216857865807875785847])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      7730104533627369608109066,
+      9758469800971041292901204478665627])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 36, sharing kernel reduction. -/
+def cachedSine36 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      309204181345094784324362,
+      64390338792038841651716032162913897])
+      (naturalFromChunks [
+      400000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      394173213813303131004582,
+      35267984464132824656857482663629509])
+      (naturalFromChunks [
+      500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 37, sharing kernel reduction. -/
+def cachedSine37 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      3941732138133031310045823,
+      52679844641328246568574821626855621])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2008018828701612274516691,
+      28240785480969892356792616287522409])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 38, sharing kernel reduction. -/
+def cachedSine38 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1606415062961289819613353,
+      2592628384775913885434091116462087])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2043962032878959241262302,
+      21032658452367760629391729568564657])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 39, sharing kernel reduction. -/
+def cachedSine39 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      817584813151583696504920,
+      88413063380947104251756690899071463])
+      (naturalFromChunks [
+      1000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1039337015378181546348485,
+      47202238219592320101498406755706549])
+      (naturalFromChunks [
+      1250000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 40, sharing kernel reduction. -/
+def cachedSine40 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      4157348061512726185393941,
+      88808952878369280405993622507055293])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      8448535652497070732595712,
+      5104957097719785981389112614274791])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 41, sharing kernel reduction. -/
+def cachedSine41 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      4224267826248535366297856,
+      2552478548859892990694551918135241])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      8577286100002720699022699,
+      84284770137042490799433737906882701])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 42, sharing kernel reduction. -/
+def cachedSine42 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      8577286100002720699022699,
+      84284770137042490799433729391926801])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4350434955543557093261462,
+      2241924421955413894764914634755387])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 43, sharing kernel reduction. -/
+def cachedSine43 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1740173982217422837304584,
+      80896769768782165557905964205769413])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1102401580435443787140946,
+      7957548543688555327584341456279597])
+      (naturalFromChunks [
+      1250000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 44, sharing kernel reduction. -/
+def cachedSine44 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      8819212643483550297127568,
+      63660388349508442620674723695413387])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1116530376494394150428020,
+      55936674747250078198624859542428149])
+      (naturalFromChunks [
+      1250000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 45, sharing kernel reduction. -/
+def cachedSine45 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      8932243011955153203424164,
+      47493397978000625588998868681981741])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9039892931234433315862002,
+      97230537048710132025050611472702711])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 46, sharing kernel reduction. -/
+def cachedSine46 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9039892931234433315862002,
+      97230537048710132025050604124099479])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4571048778517653273175074,
+      14696788700522345557841090145319409])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 47, sharing kernel reduction. -/
+def cachedSine47 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      914209755703530654635014,
+      82939357740104469111568217326223477])
+      (naturalFromChunks [
+      1000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2309698831278216890320457,
+      97349197071705604156465911408672419])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 48, sharing kernel reduction. -/
+def cachedSine48 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1154849415639108445160228,
+      98674598535852802078232954867216201])
+      (naturalFromChunks [
+      1250000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9329927988347388877116602,
+      55543302498295015520512298049839469])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 49, sharing kernel reduction. -/
+def cachedSine49 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9329927988347388877116602,
+      55543302498295015520512291695442729])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9415440651830207784125094,
+      2599502357185589795825185713986177])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 50, sharing kernel reduction. -/
+def cachedSine50 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9415440651830207784125094,
+      2599502357185589795825179713135663])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9495281805930366671959360,
+      74189345028252224153832413537541717])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 51, sharing kernel reduction. -/
+def cachedSine51 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9495281805930366671959360,
+      74189345028252224153832407901072943])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      239235083933052216233949,
+      47174506749237071230140750994456933])
+      (naturalFromChunks [
+      250000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 52, sharing kernel reduction. -/
+def cachedSine52 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1196175419665261081169747,
+      35872533746185356150703754314608443])
+      (naturalFromChunks [
+      1250000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4818880328977199333432321,
+      77753917576831541924413317523222169])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 53, sharing kernel reduction. -/
+def cachedSine53 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      481888032897719933343232,
+      17775391757683154192441331508530087])
+      (naturalFromChunks [
+      500000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      2425078132986359981509960,
+      51821525062864216490562019065406831])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 54, sharing kernel reduction. -/
+def cachedSine54 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      75783691655823749422186,
+      26619422658214506765330063060794377])
+      (naturalFromChunks [
+      78125000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9757021300385285444603957,
+      66419527971644012265792045137989823])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 55, sharing kernel reduction. -/
+def cachedSine55 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      4878510650192642722301978,
+      83209763985822006132896020532038069])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9807852804032304491261822,
+      36134239036973933730893337873019227])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 56, sharing kernel reduction. -/
+def cachedSine56 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      1225981600504038061407727,
+      79516779879621741716361666776884461])
+      (naturalFromChunks [
+      1250000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      1970555284777882489548036,
+      86635709557432025831162563290299331])
+      (naturalFromChunks [
+      2000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 57, sharing kernel reduction. -/
+def cachedSine57 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2463194105972353111935046,
+      8294636946790032288953203304812111])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9891765099647809734516737,
+      38016243063983689533336908770800689])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 58, sharing kernel reduction. -/
+def cachedSine58 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9891765099647809734516737,
+      38016243063983689533336905973751019])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4962397672993549990783836,
+      25830558910005410327317080351399043])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 59, sharing kernel reduction. -/
+def cachedSine59 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9924795345987099981567672,
+      51661117820010820654634158350217921])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      4975923633360984431224184,
+      76554739960787737434364928999287079])
+      (naturalFromChunks [
+      5000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 60, sharing kernel reduction. -/
+def cachedSine60 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9951847266721968862448369,
+      53109479921575474868729856099492063])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      9972904566786902161355971,
+      40182567821171688679166222084406173])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 61, sharing kernel reduction. -/
+def cachedSine61 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      9972904566786902161355971,
+      40182567821171688679166220647598289])
+      (naturalFromChunks [
+      10000000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      624247160128232745446732,
+      25297443793402700225919038267134297])
+      (naturalFromChunks [
+      625000000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 62, sharing kernel reduction. -/
+def cachedSine62 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      624247160128232745446732,
+      25297443793402700225919038206758057])
+      (naturalFromChunks [
+      625000000000000000000000,
+      0])),
+    (ExactReplay.q
+      (naturalFromChunks [
+      312405880842563818786176,
+      76552067881151564408789304058354309])
+      (naturalFromChunks [
+      312500000000000000000000,
+      0]))⟩
+
+/-- The exact cached sine interval for cell 63, sharing kernel reduction. -/
+def cachedSine63 : RatInterval :=
+  ⟨(ExactReplay.q
+      (naturalFromChunks [
+      2499247046740510550289414,
+      12416543049212515270314432345086971])
+      (naturalFromChunks [
+      2500000000000000000000000,
+      0])),
+    (ExactReplay.q
+      1
+      1)⟩
+
+/-- The exact cached sine interval for the selected angle cell. -/
+def cachedSine (i : Cell) : RatInterval :=
+  match i.val with
+  | 0 => cachedSine00
+  | 1 => cachedSine01
+  | 2 => cachedSine02
+  | 3 => cachedSine03
+  | 4 => cachedSine04
+  | 5 => cachedSine05
+  | 6 => cachedSine06
+  | 7 => cachedSine07
+  | 8 => cachedSine08
+  | 9 => cachedSine09
+  | 10 => cachedSine10
+  | 11 => cachedSine11
+  | 12 => cachedSine12
+  | 13 => cachedSine13
+  | 14 => cachedSine14
+  | 15 => cachedSine15
+  | 16 => cachedSine16
+  | 17 => cachedSine17
+  | 18 => cachedSine18
+  | 19 => cachedSine19
+  | 20 => cachedSine20
+  | 21 => cachedSine21
+  | 22 => cachedSine22
+  | 23 => cachedSine23
+  | 24 => cachedSine24
+  | 25 => cachedSine25
+  | 26 => cachedSine26
+  | 27 => cachedSine27
+  | 28 => cachedSine28
+  | 29 => cachedSine29
+  | 30 => cachedSine30
+  | 31 => cachedSine31
+  | 32 => cachedSine32
+  | 33 => cachedSine33
+  | 34 => cachedSine34
+  | 35 => cachedSine35
+  | 36 => cachedSine36
+  | 37 => cachedSine37
+  | 38 => cachedSine38
+  | 39 => cachedSine39
+  | 40 => cachedSine40
+  | 41 => cachedSine41
+  | 42 => cachedSine42
+  | 43 => cachedSine43
+  | 44 => cachedSine44
+  | 45 => cachedSine45
+  | 46 => cachedSine46
+  | 47 => cachedSine47
+  | 48 => cachedSine48
+  | 49 => cachedSine49
+  | 50 => cachedSine50
+  | 51 => cachedSine51
+  | 52 => cachedSine52
+  | 53 => cachedSine53
+  | 54 => cachedSine54
+  | 55 => cachedSine55
+  | 56 => cachedSine56
+  | 57 => cachedSine57
+  | 58 => cachedSine58
+  | 59 => cachedSine59
+  | 60 => cachedSine60
+  | 61 => cachedSine61
+  | 62 => cachedSine62
+  | _ => cachedSine63
+
+theorem cachedPath_eq : ∀ i : Cell, cellPathInterval i = cachedPath i := by
+  decide +kernel
+
+theorem cachedCosine_eq : ∀ i : Cell,
+    ExactReplay.cosineInterval (cellTimeInterval i) = cachedCosine i := by
+  decide +kernel
+
+theorem cachedSine_eq : ∀ i : Cell,
+    ExactReplay.sineInterval (cellTimeInterval i) = cachedSine i := by
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartB.CellRows.Row48`.
+* `KernelOnly.PartB.CellRows.Row49`.
+* `KernelOnly.PartB.CellRows.Row50`.
+* `KernelOnly.PartB.CellRows.Row51`.
+* `KernelOnly.PartB.CellRows.Row52`.
+* `KernelOnly.PartB.CellRows.Row53`.
+* `KernelOnly.PartB.CellRows.Row54`.
+* `KernelOnly.PartB.CellRows.Row55`.
+* `KernelOnly.PartB.CellRows.Row56`.
+* `KernelOnly.PartB.CellRows.Row57`.
+* `KernelOnly.PartB.CellRows.Row58`.
+* `KernelOnly.PartB.CellRows.Row59`.
+* `KernelOnly.PartB.CellRows.Row60`.
+* `KernelOnly.PartB.CellRows.Row61`.
+* `KernelOnly.PartB.CellRows.Row62`.
+* `KernelOnly.PartB.CellRows.Row63`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 48.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_48 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (48 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (48 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 49.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_49 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (49 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (49 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 50.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_50 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (50 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (50 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 51.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_51 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (51 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (51 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 52.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_52 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (52 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (52 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 53.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_53 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (53 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (53 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 54.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_54 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (54 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (54 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 55.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_55 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (55 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (55 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 56.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_56 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (56 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (56 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 57.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_57 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (57 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (57 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 58.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_58 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (58 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (58 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 59.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_59 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (59 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (59 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 60.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_60 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (60 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (60 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 61.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_61 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (61 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (61 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 62.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_62 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (62 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (62 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 63.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_63 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (63 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (63 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartB.CellRows.Row32`.
+* `KernelOnly.PartB.CellRows.Row33`.
+* `KernelOnly.PartB.CellRows.Row34`.
+* `KernelOnly.PartB.CellRows.Row35`.
+* `KernelOnly.PartB.CellRows.Row36`.
+* `KernelOnly.PartB.CellRows.Row37`.
+* `KernelOnly.PartB.CellRows.Row38`.
+* `KernelOnly.PartB.CellRows.Row39`.
+* `KernelOnly.PartB.CellRows.Row40`.
+* `KernelOnly.PartB.CellRows.Row41`.
+* `KernelOnly.PartB.CellRows.Row42`.
+* `KernelOnly.PartB.CellRows.Row43`.
+* `KernelOnly.PartB.CellRows.Row44`.
+* `KernelOnly.PartB.CellRows.Row45`.
+* `KernelOnly.PartB.CellRows.Row46`.
+* `KernelOnly.PartB.CellRows.Row47`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 32.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_32 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (32 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (32 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 33.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_33 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (33 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (33 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 34.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_34 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (34 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (34 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 35.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_35 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (35 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (35 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 36.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_36 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (36 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (36 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 37.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_37 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (37 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (37 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 38.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_38 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (38 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (38 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 39.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_39 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (39 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (39 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 40.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_40 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (40 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (40 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 41.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_41 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (41 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (41 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 42.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_42 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (42 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (42 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 43.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_43 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (43 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (43 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 44.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_44 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (44 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (44 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 45.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_45 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (45 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (45 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 46.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_46 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (46 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (46 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 47.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_47 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (47 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (47 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartB.CellRows.Row16`.
+* `KernelOnly.PartB.CellRows.Row17`.
+* `KernelOnly.PartB.CellRows.Row18`.
+* `KernelOnly.PartB.CellRows.Row19`.
+* `KernelOnly.PartB.CellRows.Row20`.
+* `KernelOnly.PartB.CellRows.Row21`.
+* `KernelOnly.PartB.CellRows.Row22`.
+* `KernelOnly.PartB.CellRows.Row23`.
+* `KernelOnly.PartB.CellRows.Row24`.
+* `KernelOnly.PartB.CellRows.Row25`.
+* `KernelOnly.PartB.CellRows.Row26`.
+* `KernelOnly.PartB.CellRows.Row27`.
+* `KernelOnly.PartB.CellRows.Row28`.
+* `KernelOnly.PartB.CellRows.Row29`.
+* `KernelOnly.PartB.CellRows.Row30`.
+* `KernelOnly.PartB.CellRows.Row31`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 16.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_16 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (16 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (16 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 17.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_17 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (17 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (17 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 18.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_18 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (18 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (18 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 19.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_19 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (19 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (19 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 20.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_20 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (20 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (20 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 21.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_21 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (21 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (21 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 22.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_22 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (22 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (22 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 23.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_23 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (23 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (23 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 24.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_24 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (24 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (24 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 25.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_25 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (25 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (25 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 26.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_26 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (26 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (26 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 27.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_27 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (27 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (27 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 28.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_28 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (28 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (28 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 29.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_29 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (29 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (29 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 30.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_30 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (30 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (30 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 31.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_31 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (31 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (31 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartB.CellRows.Row00`.
+* `KernelOnly.PartB.CellRows.Row01`.
+* `KernelOnly.PartB.CellRows.Row02`.
+* `KernelOnly.PartB.CellRows.Row03`.
+* `KernelOnly.PartB.CellRows.Row04`.
+* `KernelOnly.PartB.CellRows.Row05`.
+* `KernelOnly.PartB.CellRows.Row06`.
+* `KernelOnly.PartB.CellRows.Row07`.
+* `KernelOnly.PartB.CellRows.Row08`.
+* `KernelOnly.PartB.CellRows.Row09`.
+* `KernelOnly.PartB.CellRows.Row10`.
+* `KernelOnly.PartB.CellRows.Row11`.
+* `KernelOnly.PartB.CellRows.Row12`.
+* `KernelOnly.PartB.CellRows.Row13`.
+* `KernelOnly.PartB.CellRows.Row14`.
+* `KernelOnly.PartB.CellRows.Row15`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 00.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_00 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (0 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (0 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 01.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_01 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (1 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (1 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 02.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_02 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (2 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (2 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 03.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_03 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (3 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (3 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 04.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_04 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (4 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (4 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 05.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_05 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (5 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (5 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 06.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_06 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (6 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (6 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 07.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_07 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (7 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (7 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 08.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_08 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (8 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (8 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 09.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_09 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (9 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (9 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 10.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_10 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (10 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (10 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 11.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_11 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (11 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (11 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 12.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_12 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (12 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (12 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 13.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_13 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (13 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (13 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 14.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_14 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (14 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (14 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+section
+
+/-!
+Independent exact product-cell lower bounds for row 15.
+Build all 64 row modules sequentially before the final Part B assembly.
+-/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+theorem cell_row_15 :
+    ∀ j : Cell,
+      targetQ < (guCellInterval (15 : Cell) j).lo ∧
+      targetQ < (gvCellInterval (15 : Cell) j).lo := by
+  simp only [guCellInterval, gvCellInterval, cachedPath_eq, cachedCosine_eq, cachedSine_eq]
+  decide +kernel
+
+end GerverSofa.PartB
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartB.AllRows`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! Assembly of the 64 independently kernel-checked product-cell rows. -/
+
+@[expose] public section
+
+namespace GerverSofa.PartB
+
+theorem all_cell_lower (i j : Cell) :
+    targetQ < (guCellInterval i j).lo ∧
+    targetQ < (gvCellInterval i j).lo := by
+  fin_cases i
+  · exact cell_row_00 j
+  · exact cell_row_01 j
+  · exact cell_row_02 j
+  · exact cell_row_03 j
+  · exact cell_row_04 j
+  · exact cell_row_05 j
+  · exact cell_row_06 j
+  · exact cell_row_07 j
+  · exact cell_row_08 j
+  · exact cell_row_09 j
+  · exact cell_row_10 j
+  · exact cell_row_11 j
+  · exact cell_row_12 j
+  · exact cell_row_13 j
+  · exact cell_row_14 j
+  · exact cell_row_15 j
+  · exact cell_row_16 j
+  · exact cell_row_17 j
+  · exact cell_row_18 j
+  · exact cell_row_19 j
+  · exact cell_row_20 j
+  · exact cell_row_21 j
+  · exact cell_row_22 j
+  · exact cell_row_23 j
+  · exact cell_row_24 j
+  · exact cell_row_25 j
+  · exact cell_row_26 j
+  · exact cell_row_27 j
+  · exact cell_row_28 j
+  · exact cell_row_29 j
+  · exact cell_row_30 j
+  · exact cell_row_31 j
+  · exact cell_row_32 j
+  · exact cell_row_33 j
+  · exact cell_row_34 j
+  · exact cell_row_35 j
+  · exact cell_row_36 j
+  · exact cell_row_37 j
+  · exact cell_row_38 j
+  · exact cell_row_39 j
+  · exact cell_row_40 j
+  · exact cell_row_41 j
+  · exact cell_row_42 j
+  · exact cell_row_43 j
+  · exact cell_row_44 j
+  · exact cell_row_45 j
+  · exact cell_row_46 j
+  · exact cell_row_47 j
+  · exact cell_row_48 j
+  · exact cell_row_49 j
+  · exact cell_row_50 j
+  · exact cell_row_51 j
+  · exact cell_row_52 j
+  · exact cell_row_53 j
+  · exact cell_row_54 j
+  · exact cell_row_55 j
+  · exact cell_row_56 j
+  · exact cell_row_57 j
+  · exact cell_row_58 j
+  · exact cell_row_59 j
+  · exact cell_row_60 j
+  · exact cell_row_61 j
+  · exact cell_row_62 j
+  · exact cell_row_63 j
+
+end GerverSofa.PartB
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle004.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle004.lean
new file mode 100644
index 0000000000..71ed7fbab2
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle004.lean
@@ -0,0 +1,2382 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.Development.IntervalArithmetic.Foundations.Development002
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle002
+public import Mathlib.Analysis.Normed.Operator.Banach
+public import Mathlib.LinearAlgebra.FiniteDimensional.Basic
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.Foundation.Batch002`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.LeanCertGerverData`.
+* `KernelOnly.LeanCertNamedConstAD`.
+* `KernelOnly.LeanCertGerverExpr`.
+* `KernelOnly.LeanCertGerverCorrespondence`.
+* `KernelOnly.LeanCertNoDetKrawczyk`.
+* `KernelOnly.LeanCertGerverNumericsCore`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Part A final adapter data
+
+The original Gerver boxes are affinely normalized to `[-1,1]^n`.
+If `x = m + D u`, then the preconditioner is transformed from `C` to
+`D⁻¹ C`.  Consequently
+
+`I - (D⁻¹ C) (J_F(x) D) = D⁻¹ (I - C J_F(x)) D`,
+
+so the old weighted sup-norm contraction becomes the ordinary infinity norm
+used by LeanCert.
+
+The preconditioner literals below are exact public copies of the frozen
+rational data already used by `ExactReplay`.  This avoids exposing private
+helpers during simplification.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+
+/-- Construct an exact rational from an integer numerator and natural denominator. -/
+def q (n : Int) (d : Nat) : ℚ := (n : ℚ) / (d : ℚ)
+
+/-- The rational preconditioner table for the four reduced equations. -/
+def reducedCData : List (List ℚ) := [
+  [q (-6807128249614081174405751646233895883477) 10000000000000000000000000000000000000000, q
+    (-1668645298254898823756872685901589912999) 5000000000000000000000000000000000000000, q
+    142655469136580991746408771201083165281 250000000000000000000000000000000000000, q
+    6979196385375907434754729126542237618887 10000000000000000000000000000000000000000],
+  [q (-1166509731502349263126377264338437200033) 2500000000000000000000000000000000000000, q
+    12885386748281427252440540606364884831 80000000000000000000000000000000000000, q
+    33867580317340694704135157308931896467 25000000000000000000000000000000000000, q
+    (-669198169890925059603288141447454387733) 500000000000000000000000000000000000000],
+  [q (-2734456716994644402025329684152137828373) 10000000000000000000000000000000000000000, q
+    (-8773012441591382751224116906117915601) 62500000000000000000000000000000000000, q
+    (-31169129168086947221083235454859786381) 156250000000000000000000000000000000000, q
+    3097544571982855798592484554650267326733 10000000000000000000000000000000000000000],
+  [q 5937522534788503855395693041282403727707 10000000000000000000000000000000000000000, q
+    (-12894234217809317884645394772127946998229) 5000000000000000000000000000000000000000, q
+    (-8661560217304957931983485334759974222533) 5000000000000000000000000000000000000000, q
+    51750746732148285574274925461942534764817 10000000000000000000000000000000000000000]
+]
+
+/-- The rational preconditioner table for the twenty-two full equations. -/
+def fullCData : List (List ℚ) := [
+  [q 97524664435861749 500000000000000000000000000000000000, q 0 1, q 0 1, q 0 1, q
+    101771747759019027 100000000000000000, q 1 1, q 0 1, q 63813611575753437 50000000000000000, q
+    (-388441807573624079) 5000000000000000000, q 19675950003915553 31250000000000000, q
+    (-298520460968330237) 500000000000000000, q (-548415754484802953) 1000000000000000000, q
+    (-388441807573624079) 5000000000000000000, q 19675950003915553 31250000000000000, q
+    20287597594763393 100000000000000000, q (-10119107089800079) 10000000000000000, q 0 1, q 0 1,
+    q 0 1, q 0 1, q 388441807573624079 5000000000000000000, q (-19675950003915553)
+    31250000000000000],
+  [q 0 1, q 0 1, q 0 1, q 0 1, q 0 1, q 0 1, q 1 1, q 0 1, q 0 1, q 0 1, q 0 1, q 0 1, q 0 1, q 0
+    1, q 0 1, q 0 1, q 0 1, q 0 1, q 0 1, q 0 1, q 0 1, q 0 1],
+  [q (-803432388936903217) 10000000000000000000000000000000000000, q 0 1, q 0 1, q 0 1, q
+    100605379900828451 50000000000000000, q 1 1, q 0 1, q 31765043860672787 12500000000000000, q
+    (-4704777879791199) 4000000000000000, q 121750850637699193 100000000000000000, q
+    (-122756605381830597) 100000000000000000, q (-30600542365273823) 312500000000000000, q
+    (-88097234973899999) 500000000000000000, q 121750850637699193 100000000000000000, q
+    396110376767452643 1000000000000000000, q (-40111351612012509) 20000000000000000, q 0 1, q 0
+    1, q 0 1, q 0 1, q 88097234973899999 500000000000000000, q (-121750850637699193)
+    100000000000000000],
+  [q 348543935143050227 50000000000000000000000000000000000, q 0 1, q 0 1, q 0 1, q
+    48663249173571721 200000000000000000, q 0 1, q 1 1, q (-229748851313700347)
+    1000000000000000000, q 38538669009926041 200000000000000000, q (-148514169684624997)
+    250000000000000000, q (-77095605529021527) 200000000000000000, q 899174721266506223
+    100000000000000000000, q 38538669009926041 200000000000000000, q 101485830315375003
+    250000000000000000, q 475834949279834579 5000000000000000000, q (-48935313448682273)
+    250000000000000000, q 0 1, q 0 1, q 0 1, q 0 1, q (-38538669009926041) 200000000000000000, q
+    (-101485830315375003) 250000000000000000],
+  [q (-27668786810187381) 62500000000000000000000000000000000, q 0 1, q 0 1, q 0 1, q
+    156689147323938593 100000000000000000, q 1 1, q 0 1, q 6806785234747033 4000000000000000, q
+    (-770251148686299847) 1000000000000000000, q 839507200167063927 1000000000000000000, q
+    (-32364013024529501) 40000000000000000, q (-325328497901119923) 5000000000000000000, q
+    (-770251148686299847) 1000000000000000000, q 839507200167063927 1000000000000000000, q
+    107433644407840839 1000000000000000000, q (-203691379660578653) 250000000000000000, q 0 1, q 0
+    1, q 0 1, q 0 1, q (-229748851313700181) 1000000000000000000, q (-839507200167063927)
+    1000000000000000000],
+  [q 667707347369355603 100000000000000000000000000000000000, q 0 1, q 0 1, q 0 1, q
+    39302260617100937 200000000000000000, q 0 1, q 1 1, q 810045947451984161 10000000000000000000,
+    q 770773380198521041 1000000000000000000, q (-376226714953999619) 1000000000000000000, q
+    (-67547253722371331) 125000000000000000, q (-317030024240986261) 100000000000000000000, q
+    770773380198521041 1000000000000000000, q (-376226714953999619) 1000000000000000000, q
+    (-50799288029152201) 50000000000000000, q (-293762044611903417) 1000000000000000000, q 0 1, q
+    0 1, q 0 1, q 0 1, q (-770773380198521041) 1000000000000000000, q (-623773285046000381)
+    1000000000000000000],
+  [q (-27668786810187381) 62500000000000000000000000000000000, q 0 1, q 611138637517184627
+    10000000000000000000, q 48547105024120657 62500000000000000, q (-142441420640562477)
+    500000000000000000, q 1 1, q 0 1, q 107773638564961749 125000000000000000, q
+    (-91076956856199917) 250000000000000000, q 461505893957135871 1000000000000000000, q
+    (-195317298704084513) 500000000000000000, q (-32209663591571687) 1000000000000000000, q
+    (-91076956856199917) 250000000000000000, q 461505893957135871 1000000000000000000, q
+    (-45310771987942769) 250000000000000000, q 376036543315996397 1000000000000000000, q (-1) 1, q
+    0 1, q 0 1, q 0 1, q (-635692172575200387) 1000000000000000000, q (-461505893957135871)
+    1000000000000000000],
+  [q 667707347369355603 100000000000000000000000000000000000, q 0 1, q (-133696152744215091)
+    100000000000000000, q (-31490225467869859) 50000000000000000, q 12045884354915673
+    125000000000000000, q 0 1, q 1 1, q (-229748851313700347) 1000000000000000000, q
+    12043334065601893 62500000000000000, q (-594056678738499877) 1000000000000000000, q
+    (-385478027645107413) 1000000000000000000, q 449587360633251377 50000000000000000000, q
+    12043334065601893 62500000000000000, q (-594056678738499877) 1000000000000000000, q
+    951669898559669991 10000000000000000000, q (-195741253794729203) 1000000000000000000, q 0 1, q
+    (-1) 1, q 0 1, q 0 1, q (-12043334065601893) 62500000000000000, q (-405943321261500123)
+    1000000000000000000],
+  [q 498465919198540053 500000000000000000, q (-391372890930701059) 5000000000000000000, q
+    (-8938515563507099) 6250000000000000, q (-55619746202997561) 250000000000000000, q
+    354753639572627011 500000000000000000, q 1 1, q 0 1, q 106356019292922399 50000000000000000, q
+    (-18512557434314989) 40000000000000000, q 52469200010441497 50000000000000000, q
+    (-102115972928981447) 100000000000000000, q 26142772207772183 62500000000000000, q
+    (-18512557434314989) 40000000000000000, q 52469200010441497 50000000000000000, q
+    29978282170118999 2500000000000000000, q (-308810164152310551) 500000000000000000, q (-1) 1, q
+    0 1, q (-1) 1, q 0 1, q (-53718606414212533) 100000000000000000, q (-52469200010441497)
+    50000000000000000],
+  [q (-782745781861401979) 10000000000000000000, q (-498465919198540053) 500000000000000000, q
+    (-277739416158080399) 1000000000000000000, q (-295318582899254667) 500000000000000000, q
+    (-18368646378566663) 125000000000000000, q 0 1, q 1 1, q (-133419044772954399)
+    250000000000000000000000000000000, q 14163773178071801 250000000000000000000000000000000, q
+    (-1) 1, q 79939871278032317 250000000000000000000000000000000, q (-2418968094552299)
+    125000000000000000000000000000000, q 14163773178071801 250000000000000000000000000000000, q
+    (-1) 1, q 470629071136429359 5000000000000000000000000000000000, q (-223711639524848453)
+    5000000000000000000000000000000000, q 0 1, q (-1) 1, q 0 1, q (-1) 1, q (-14163773178071801)
+    250000000000000000000000000000000, q (-21329108080229053) 625000000000000000000000000000000],
+  [q (-97524664435861749) 500000000000000000000000000000000000, q 0 1, q 0 1, q 0 1, q
+    (-101771747759019027) 100000000000000000, q 0 1, q 0 1, q (-63813611575753437)
+    50000000000000000, q 388441807573624079 5000000000000000000, q (-19675950003915553)
+    31250000000000000, q 298520460968330237 500000000000000000, q 548415754484802953
+    1000000000000000000, q 388441807573624079 5000000000000000000, q (-19675950003915553)
+    31250000000000000, q (-20287597594763393) 100000000000000000, q 10119107089800079
+    10000000000000000, q 0 1, q 0 1, q 0 1, q 0 1, q (-388441807573624079) 5000000000000000000, q
+    19675950003915553 31250000000000000],
+  [q (-349465940599428639) 50000000000000000000000000000000000, q 0 1, q 0 1, q 0 1, q
+    (-25253489731359111) 2500000000000000000000000000000000, q 0 1, q 0 1, q 50000000000000011
+    50000000000000000, q (-353674953581839119) 5000000000000000000000000000000000, q
+    94007553259298153 2000000000000000000000000000000000, q (-5928343202681263)
+    25000000000000000000000000000000, q (-173331002701677877) 10000000000000000000000000000000000,
+    q (-353674953581839119) 5000000000000000000000000000000000, q 94007553259298153
+    2000000000000000000000000000000000, q (-288351357559389053)
+    50000000000000000000000000000000000, q (-899167441101047079)
+    10000000000000000000000000000000000, q 0 1, q 0 1, q 0 1, q 0 1, q 353674953581839119
+    5000000000000000000000000000000000, q (-94007553259298153) 2000000000000000000000000000000000],
+  [q 199958107491582403 10000000000000000000000000000000000000000000000000000, q 0 1, q 0 1, q 0
+    1, q 122021605728424423 500000000000000000, q 0 1, q 0 1, q 770104303222080677
+    1000000000000000000, q 193360860585517491 1000000000000000000, q 407267153261943571
+    1000000000000000000, q (-383870749384345733) 1000000000000000000, q (-150698973735099509)
+    5000000000000000000, q 193360860585517491 1000000000000000000, q 407267153261943571
+    1000000000000000000, q 59669749873825801 625000000000000000, q (-98153004518806597)
+    500000000000000000, q 0 1, q 0 1, q 0 1, q 0 1, q (-193360860585517491) 1000000000000000000, q
+    (-407267153261943571) 1000000000000000000],
+  [q (-466502263483670063) 1000000000000000000000000000000000000, q 0 1, q 0 1, q 0 1, q
+    (-50741366468655913) 25000000000000000, q 0 1, q 0 1, q (-50425164328510137)
+    20000000000000000, q 80468307190360827 500000000000000000, q (-124842964508786847)
+    100000000000000000, q 6283806471937059 5000000000000000, q 493376058472616841
+    5000000000000000000, q 80468307190360827 500000000000000000, q (-124842964508786847)
+    100000000000000000, q (-403274191369431989) 1000000000000000000, q 10096930279303431
+    5000000000000000, q 0 1, q 0 1, q 0 1, q 0 1, q (-80468307190360827) 500000000000000000, q
+    124842964508786847 100000000000000000],
+  [q 0 1, q 0 1, q 0 1, q 0 1, q (-74404321145684893) 50000000000000000, q 0 1, q 0 1, q
+    (-30804172128883227) 20000000000000000, q (-386721721171035093) 1000000000000000000, q
+    (-814534306523887031) 1000000000000000000, q 153548299753738271 200000000000000000, q
+    150698973735099509 2500000000000000000, q (-386721721171035093) 1000000000000000000, q
+    (-814534306523887031) 1000000000000000000, q 585810480789687893 1000000000000000000, q
+    20448330548652467 20000000000000000, q 0 1, q 0 1, q 0 1, q 0 1, q 386721721171035093
+    1000000000000000000, q 814534306523887031 1000000000000000000],
+  [q 0 1, q 0 1, q 0 1, q 0 1, q (-488086422913697859) 1000000000000000000, q 0 1, q 0 1, q
+    (-30804172128883227) 20000000000000000, q (-386721721171035093) 1000000000000000000, q
+    (-814534306523887031) 1000000000000000000, q 153548299753738271 200000000000000000, q
+    150698973735099509 2500000000000000000, q (-386721721171035093) 1000000000000000000, q
+    (-814534306523887031) 1000000000000000000, q 585810480789687893 1000000000000000000, q
+    20448330548652467 20000000000000000, q 0 1, q 0 1, q 0 1, q 0 1, q 386721721171035093
+    1000000000000000000, q 814534306523887031 1000000000000000000],
+  [q 0 1, q 0 1, q 1 1, q 0 1, q (-244043211456848957) 1000000000000000000, q 0 1, q 0 1, q
+    (-770104303222080677) 1000000000000000000, q (-193360860585517547) 1000000000000000000, q
+    (-407267153261943571) 1000000000000000000, q 191935374692172811 500000000000000000, q
+    301397947470199087 10000000000000000000, q (-193360860585517547) 1000000000000000000, q
+    (-407267153261943571) 1000000000000000000, q (-477357998990606547) 5000000000000000000, q
+    98153004518806597 500000000000000000, q 0 1, q 0 1, q 0 1, q 0 1, q 193360860585517547
+    1000000000000000000, q 407267153261943571 1000000000000000000],
+  [q 0 1, q 0 1, q 0 1, q 1 1, q (-32926249572211761) 20000000000000000, q 0 1, q 0 1, q
+    (-8197382535470743) 6250000000000000, q 29041696495897041 62500000000000000, q
+    (-304347948359896703) 500000000000000000, q 326889265645238569 500000000000000000, q
+    513317328155524291 10000000000000000000, q 29041696495897041 62500000000000000, q
+    (-304347948359896703) 500000000000000000, q (-63326938273327693) 250000000000000000, q
+    42775732448415943 25000000000000000, q 0 1, q 0 1, q 0 1, q 0 1, q (-29041696495897041)
+    62500000000000000, q 304347948359896703 500000000000000000],
+  [q 1 1, q 0 1, q 0 1, q 0 1, q (-101771747759019027) 100000000000000000, q 0 1, q 0 1, q
+    (-31906805787876713) 25000000000000000, q 388441807573624287 5000000000000000000, q
+    (-314815200062648959) 500000000000000000, q 597040921936660363 1000000000000000000, q
+    548415754484802953 1000000000000000000, q 388441807573624287 5000000000000000000, q
+    (-314815200062648959) 500000000000000000, q (-12679748496727131) 62500000000000000, q
+    10119107089800079 10000000000000000, q 0 1, q 0 1, q 0 1, q 0 1, q (-388441807573624287)
+    5000000000000000000, q 314815200062648959 500000000000000000],
+  [q 0 1, q 1 1, q 0 1, q 0 1, q 0 1, q 0 1, q 0 1, q (-1) 1, q 0 1, q 0 1, q 0 1, q 0 1, q 0 1, q
+    0 1, q 0 1, q 0 1, q 0 1, q 0 1, q 0 1, q 0 1, q 0 1, q 0 1],
+  [q (-498591875083207211) 50000000000000000000000000000000000, q 0 1, q 0 1, q 0 1, q
+    (-29185445604164767) 100000000000000000, q 0 1, q 0 1, q 79792970670302621 200000000000000000,
+    q (-280736398130924159) 1000000000000000000, q (-546891343398928753) 1000000000000000000, q
+    (-19887038235377269) 100000000000000000, q (-31228805607184943) 2000000000000000000, q
+    (-280736398130924159) 1000000000000000000, q (-546891343398928753) 1000000000000000000, q
+    (-62553783297256041) 500000000000000000, q 55986183994322411 250000000000000000, q 0 1, q 0 1,
+    q 0 1, q 0 1, q 280736398130924159 1000000000000000000, q 546891343398928753
+    1000000000000000000],
+  [q 354149563585840613 100000000000000000000000000000000000, q 0 1, q 0 1, q 0 1, q
+    142456721371719919 50000000000000000, q 0 1, q 0 1, q 173231204346099177 50000000000000000, q
+    (-515769368712372689) 100000000000000000, q 59375225347885019 50000000000000000, q
+    (-5396865719265527) 3125000000000000, q (-135595994488679511) 1000000000000000000, q
+    (-515769368712372689) 100000000000000000, q 59375225347885019 50000000000000000, q
+    55842855499643207 25000000000000000, q (-656462321731497211) 1000000000000000000, q 0 1, q 0
+    1, q 0 1, q 0 1, q 515769368712372689 100000000000000000, q (-59375225347885019)
+    50000000000000000]
+]
+
+/-- Interpret rational row lists as a square matrix, using zero for missing entries. -/
+def listMatrix {n : Nat} (rows : List (List ℚ)) :
+    Matrix (Fin n) (Fin n) ℚ :=
+  fun i j => ExactReplay.getQ (ExactReplay.getRow rows i.1) j.1
+
+/-- The rational midpoint of a selected interval coordinate. -/
+def boxMid (box : List RatInterval) (i : Nat) : ℚ :=
+  let z := ExactReplay.getI box i
+  (z.lo + z.hi) / 2
+
+/-- Half the width of a selected rational interval coordinate. -/
+def boxRad (box : List RatInterval) (i : Nat) : ℚ :=
+  let z := ExactReplay.getI box i
+  (z.hi - z.lo) / 2
+
+/-- Map normalized coordinates to the given box using its midpoints and radii. -/
+def affineFromBox {n : Nat} (box : List RatInterval)
+    (u : Fin n → ℝ) : Fin n → ℝ :=
+  fun i => (boxMid box i.1 : ℝ) + (boxRad box i.1 : ℝ) * u i
+
+/-- Subtract each box midpoint and divide by its coordinate radius. -/
+def normalizeToBox {n : Nat} (box : List RatInterval)
+    (x : Fin n → ℝ) : Fin n → ℝ :=
+  fun i => (x i - (boxMid box i.1 : ℝ)) / (boxRad box i.1 : ℝ)
+
+/-- The coordinate box with interval `[-1, 1]` in every dimension. -/
+def unitBox {n : Nat} : Fin n → IntervalRat :=
+  fun _ =>
+    { lo := -1
+      hi := 1
+      le := by norm_num }
+
+/-- The rational origin used as the normalized Newton center. -/
+def zeroCenter {n : Nat} : Fin n → ℚ := fun _ => 0
+
+/-- Rescale each preconditioner row by the inverse box radius. -/
+def scaledPreconditioner {n : Nat}
+    (box : List RatInterval) (rows : List (List ℚ)) :
+    Matrix (Fin n) (Fin n) ℚ :=
+  fun i j => (boxRad box i.1)⁻¹ * listMatrix rows i j
+
+/-- The preconditioner for the normalized four-dimensional system. -/
+def reducedY : Matrix (Fin 4) (Fin 4) ℚ :=
+  scaledPreconditioner ExactReplay.reducedInputBox reducedCData
+
+/-- The preconditioner for the normalized twenty-two-dimensional system. -/
+def fullY : Matrix (Fin 22) (Fin 22) ℚ :=
+  scaledPreconditioner ExactReplay.fullInputBox fullCData
+
+/-- Map the normalized unit box into the reduced parameter box. -/
+def reducedAffine : (Fin 4 → ℝ) → Fin 4 → ℝ :=
+  affineFromBox ExactReplay.reducedInputBox
+
+/-- Map the normalized unit box into the full Romik parameter box. -/
+def fullAffine : (Fin 22 → ℝ) → Fin 22 → ℝ :=
+  affineFromBox ExactReplay.fullInputBox
+
+/-- Convert reduced parameter coordinates to normalized box coordinates. -/
+def reducedNormalize : (Fin 4 → ℝ) → Fin 4 → ℝ :=
+  normalizeToBox ExactReplay.reducedInputBox
+
+/-- A deliberately generous verified contraction target.
+The legacy exact bounds are ~5.2e-13 (4D) and ~8.6e-11 (22D), so `1/100`
+leaves many orders of magnitude of slack while keeping the normalized self-map
+well inside the unit box. -/
+def qTarget : ℚ := 1 / 100
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Minimal named-constant extension of LeanCert's AD soundness layer
+
+LeanCert's computable total dual evaluator already evaluates
+`Expr.namedConst c` as `DualInterval.ofMathConst c`, whose derivative component
+is the singleton interval `{0}`.  Its public `ADSupported` predicate, however,
+does not currently include `namedConst`.
+
+The Gerver systems necessarily contain the exact constant `Real.pi`.  This file
+adds exactly one missing syntactic case -- differentiable named mathematical
+constants -- while reusing LeanCert's existing evaluator, interval Jacobian,
+matrix norm machinery, Newton map, and contraction theorem unchanged.
+
+No interval arithmetic is reimplemented here.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+open LeanCert.Engine.Optimization
+
+/-- The LeanCert AD fragment plus named mathematical constants.
+Every constructor here is everywhere differentiable. -/
+inductive ADConstSupported : Expr → Prop where
+  | const (q : ℚ) : ADConstSupported (.const q)
+  | var (idx : Nat) : ADConstSupported (.var idx)
+  | add {a b : Expr} :
+      ADConstSupported a → ADConstSupported b → ADConstSupported (.add a b)
+  | mul {a b : Expr} :
+      ADConstSupported a → ADConstSupported b → ADConstSupported (.mul a b)
+  | neg {a : Expr} : ADConstSupported a → ADConstSupported (.neg a)
+  | exp {a : Expr} : ADConstSupported a → ADConstSupported (.exp a)
+  | sin {a : Expr} : ADConstSupported a → ADConstSupported (.sin a)
+  | cos {a : Expr} : ADConstSupported a → ADConstSupported (.cos a)
+  | namedConst (c : MathConst) : ADConstSupported (.namedConst c)
+
+/-- Computable recognition of the exact fragment used by the Gerver models. -/
+def checkADConstSupported : Expr → Bool
+  | .const _ | .var _ | .namedConst _ => true
+  | .add a b | .mul a b => checkADConstSupported a && checkADConstSupported b
+  | .neg a | .exp a | .sin a | .cos a => checkADConstSupported a
+  | _ => false
+
+theorem checkADConstSupported_correct {e : Expr}
+    (h : checkADConstSupported e = true) : ADConstSupported e := by
+  induction e with
+  | const q => exact .const q
+  | var i => exact .var i
+  | add a b iha ihb =>
+      simp only [checkADConstSupported, Bool.and_eq_true] at h
+      exact .add (iha h.1) (ihb h.2)
+  | mul a b iha ihb =>
+      simp only [checkADConstSupported, Bool.and_eq_true] at h
+      exact .mul (iha h.1) (ihb h.2)
+  | neg a ih =>
+      exact .neg (ih h)
+  | exp a ih =>
+      exact .exp (ih h)
+  | sin a ih =>
+      exact .sin (ih h)
+  | cos a ih =>
+      exact .cos (ih h)
+  | namedConst c =>
+      exact .namedConst c
+  | inv _ _ | log _ _ | atan _ _ | arsinh _ _ | atanh _ _ | sinc _ _
+    | erf _ _ | sinh _ _ | cosh _ _ | tanh _ _ | sqrt _ _ =>
+      simp [checkADConstSupported] at h
+
+theorem ADConstSupported.toCore {e : Expr} (h : ADConstSupported e) :
+    ExprSupportedCore e := by
+  induction h with
+  | const q => exact .const q
+  | var i => exact .var i
+  | add _ _ iha ihb => exact .add iha ihb
+  | mul _ _ iha ihb => exact .mul iha ihb
+  | neg _ ih => exact .neg ih
+  | exp _ ih => exact .exp ih
+  | sin _ ih => exact .sin ih
+  | cos _ ih => exact .cos ih
+  | namedConst c => exact .namedConst c
+
+/-- Dual-evaluator domain validity is likewise automatic for this fragment. -/
+theorem ADConstSupported.domainValidDual {e : Expr} (h : ADConstSupported e)
+    (ρ : DualEnv) (cfg : EvalConfig) : evalDomainValidDual e ρ cfg := by
+  induction h with
+  | const _ => trivial
+  | var _ => trivial
+  | add _ _ iha ihb => exact ⟨iha, ihb⟩
+  | mul _ _ iha ihb => exact ⟨iha, ihb⟩
+  | neg _ ih => exact ih
+  | exp _ ih => exact ih
+  | sin _ ih => exact ih
+  | cos _ ih => exact ih
+  | namedConst _ => trivial
+
+/-! ## Calculus layer -/
+
+theorem evalFin_differentiable_const {n : Nat} (e : Expr)
+    (h : ADConstSupported e) :
+    Differentiable ℝ (evalFin (n := n) e) := by
+  induction h with
+  | const q =>
+      rw [show evalFin (n := n) (.const q) =
+        (fun _ : Fin n → ℝ => (q : ℝ)) from rfl]
+      exact differentiable_const _
+  | var i =>
+      by_cases hi : i < n
+      · rw [show evalFin (n := n) (.var i) =
+          (fun x : Fin n → ℝ => x ⟨i, hi⟩) by
+            funext x
+            simp [evalFin, finEnv, hi]]
+        exact differentiable_apply _
+      · rw [show evalFin (n := n) (.var i) =
+          (fun _ : Fin n → ℝ => 0) by
+            funext x
+            simp [evalFin, finEnv, hi]]
+        exact differentiable_const _
+  | add _ _ iha ihb =>
+      rw [show evalFin (n := n) (.add _ _) =
+        evalFin (n := n) _ + evalFin (n := n) _ from rfl]
+      exact iha.add ihb
+  | mul _ _ iha ihb =>
+      rw [show evalFin (n := n) (.mul _ _) =
+        evalFin (n := n) _ * evalFin (n := n) _ from rfl]
+      exact iha.mul ihb
+  | neg _ ih =>
+      rw [show evalFin (n := n) (.neg _) = -evalFin (n := n) _ from rfl]
+      exact ih.neg
+  | exp _ ih =>
+      rw [show evalFin (n := n) (.exp _) =
+        Real.exp ∘ evalFin (n := n) _ from rfl]
+      exact Real.differentiable_exp.comp ih
+  | sin _ ih =>
+      rw [show evalFin (n := n) (.sin _) =
+        Real.sin ∘ evalFin (n := n) _ from rfl]
+      exact Real.differentiable_sin.comp ih
+  | cos _ ih =>
+      rw [show evalFin (n := n) (.cos _) =
+        Real.cos ∘ evalFin (n := n) _ from rfl]
+      exact Real.differentiable_cos.comp ih
+  | namedConst c =>
+      rw [show evalFin (n := n) (.namedConst c) =
+        (fun _ : Fin n → ℝ => c.toReal) from rfl]
+      exact differentiable_const _
+
+theorem evalAlong_differentiable_const (e : Expr) (h : ADConstSupported e)
+    (ρ : Nat → ℝ) (idx : Nat) :
+    Differentiable ℝ (Expr.evalAlong e ρ idx) := by
+  induction h with
+  | const q =>
+      simp only [Expr.evalAlong_const']
+      exact differentiable_const _
+  | var i =>
+      by_cases hi : i = idx
+      · subst i
+        simp only [Expr.evalAlong_var_active]
+        exact differentiable_id
+      · simp only [Expr.evalAlong_var_passive _ _ _ hi]
+        exact differentiable_const _
+  | add _ _ iha ihb =>
+      simp only [Expr.evalAlong_add]
+      exact iha.add ihb
+  | mul _ _ iha ihb =>
+      simp only [Expr.evalAlong_mul]
+      exact iha.mul ihb
+  | neg _ ih =>
+      simp only [Expr.evalAlong_neg]
+      exact ih.neg
+  | exp _ ih =>
+      simp only [Expr.evalAlong_exp]
+      exact Real.differentiable_exp.comp ih
+  | sin _ ih =>
+      simp only [Expr.evalAlong_sin]
+      exact Real.differentiable_sin.comp ih
+  | cos _ ih =>
+      simp only [Expr.evalAlong_cos]
+      exact Real.differentiable_cos.comp ih
+  | namedConst c =>
+      rw [show Expr.evalAlong (.namedConst c) ρ idx =
+        (fun _ : ℝ => c.toReal) from rfl]
+      exact differentiable_const _
+
+/-- LeanCert's computable total AD derivative theorem, extended by the single
+missing `namedConst` case. The computed interval is unchanged. -/
+theorem evalDualTotalCore_der_correct_idx_const
+    (e : Expr) (h : ADConstSupported e)
+    (ρReal : Nat → ℝ) (ρInt : IntervalEnv) (idx : Nat)
+    (hρ : ∀ i, ρReal i ∈ ρInt i)
+    (x : ℝ) (hx : x ∈ ρInt idx) (cfg : EvalConfig) :
+    deriv (Expr.evalAlong e ρReal idx) x ∈
+      (LeanCert.Internal.AD.evalTotalCore e
+        (mkDualEnvCore ρInt idx) cfg).der := by
+  have hmem : ∀ i, Expr.updateVar ρReal idx x i ∈
+      (mkDualEnvCore ρInt idx i).val := by
+    simpa only [mkDualEnvCore, mkDualEnv] using
+      (updateVar_mem_mkDualEnv_val ρReal ρInt idx x hx hρ)
+  induction h generalizing x with
+  | const q =>
+      simp only [Expr.evalAlong_const', deriv_const,
+        LeanCert.Internal.AD.evalTotalCore, DualInterval.const]
+      exact_mod_cast IntervalRat.mem_singleton 0
+  | var i =>
+      by_cases hi : i = idx
+      · subst i
+        simp only [Expr.evalAlong_var_active, LeanCert.Internal.AD.evalTotalCore,
+          mkDualEnvCore, ↓reduceIte, DualInterval.varActive, deriv_id]
+        exact_mod_cast IntervalRat.mem_singleton 1
+      · simp only [Expr.evalAlong_var_passive _ _ _ hi, deriv_const,
+          LeanCert.Internal.AD.evalTotalCore, mkDualEnvCore, ite_eq_right hi,
+          DualInterval.varPassive]
+        exact_mod_cast IntervalRat.mem_singleton 0
+  | add ha hb iha ihb =>
+      have hda := evalAlong_differentiable_const _ ha ρReal idx
+      have hdb := evalAlong_differentiable_const _ hb ρReal idx
+      simp only [Expr.evalAlong_add_pi, deriv_add (hda x) (hdb x),
+        LeanCert.Internal.AD.evalTotalCore, DualInterval.add]
+      exact IntervalRat.mem_add (iha x hx hmem) (ihb x hx hmem)
+  | mul ha hb iha ihb =>
+      have hda := evalAlong_differentiable_const _ ha ρReal idx
+      have hdb := evalAlong_differentiable_const _ hb ρReal idx
+      simp only [Expr.evalAlong_mul_pi, deriv_mul (hda x) (hdb x),
+        LeanCert.Internal.AD.evalTotalCore, DualInterval.mul]
+      have hvalA := LeanCert.Engine.evalDualTotalCore_val_correct _ ha.toCore
+        (Expr.updateVar ρReal idx x) (mkDualEnvCore ρInt idx) cfg hmem
+        (ha.domainValidDual (mkDualEnvCore ρInt idx) cfg)
+      have hvalB := LeanCert.Engine.evalDualTotalCore_val_correct _ hb.toCore
+        (Expr.updateVar ρReal idx x) (mkDualEnvCore ρInt idx) cfg hmem
+        (hb.domainValidDual (mkDualEnvCore ρInt idx) cfg)
+      exact IntervalRat.mem_add
+        (IntervalRat.mem_mul (iha x hx hmem) hvalB)
+        (IntervalRat.mem_mul hvalA (ihb x hx hmem))
+  | neg ha ih =>
+      have hd := evalAlong_differentiable_const _ ha ρReal idx
+      simp only [Expr.evalAlong_neg_pi, deriv.neg,
+        LeanCert.Internal.AD.evalTotalCore, DualInterval.neg]
+      exact IntervalRat.mem_neg (ih x hx hmem)
+  | @exp a ha ih =>
+      have hd := evalAlong_differentiable_const a ha ρReal idx
+      simp only [Expr.evalAlong_exp, deriv_exp (hd.differentiableAt),
+        LeanCert.Internal.AD.evalTotalCore, DualInterval.expCore]
+      have hval := LeanCert.Engine.evalDualTotalCore_val_correct a ha.toCore
+        (Expr.updateVar ρReal idx x) (mkDualEnvCore ρInt idx) cfg hmem
+        (ha.domainValidDual (mkDualEnvCore ρInt idx) cfg)
+      exact IntervalRat.mem_mul
+        (IntervalRat.mem_expComputable hval cfg.taylorDepth) (ih x hx hmem)
+  | @sin a ha ih =>
+      have hd := evalAlong_differentiable_const a ha ρReal idx
+      simp only [Expr.evalAlong_sin, deriv_sin (hd.differentiableAt),
+        LeanCert.Internal.AD.evalTotalCore, DualInterval.sinCore]
+      have hval := LeanCert.Engine.evalDualTotalCore_val_correct a ha.toCore
+        (Expr.updateVar ρReal idx x) (mkDualEnvCore ρInt idx) cfg hmem
+        (ha.domainValidDual (mkDualEnvCore ρInt idx) cfg)
+      exact IntervalRat.mem_mul
+        (IntervalRat.mem_cosComputable hval cfg.taylorDepth) (ih x hx hmem)
+  | @cos a ha ih =>
+      have hd := evalAlong_differentiable_const a ha ρReal idx
+      simp only [Expr.evalAlong_cos, deriv_cos (hd.differentiableAt),
+        LeanCert.Internal.AD.evalTotalCore, DualInterval.cosCore]
+      have hval := LeanCert.Engine.evalDualTotalCore_val_correct a ha.toCore
+        (Expr.updateVar ρReal idx x) (mkDualEnvCore ρInt idx) cfg hmem
+        (ha.domainValidDual (mkDualEnvCore ρInt idx) cfg)
+      exact IntervalRat.mem_mul
+        (IntervalRat.mem_neg
+          (IntervalRat.mem_sinComputable hval cfg.taylorDepth))
+        (ih x hx hmem)
+  | namedConst c =>
+      rw [show Expr.evalAlong (.namedConst c) ρReal idx =
+        (fun _ : ℝ => c.toReal) from rfl]
+      simpa [LeanCert.Internal.AD.evalTotalCore, DualInterval.ofMathConst] using
+        (IntervalRat.mem_singleton (0 : ℚ))
+
+/-! ## Krawczyk calculus adapters retaining LeanCert's data path -/
+
+theorem fderiv_single_eq_deriv_evalAlong_const {n : Nat}
+    (e : Expr) (h : ADConstSupported e)
+    (x : Fin n → ℝ) (j : Fin n) :
+    fderiv ℝ (evalFin (n := n) e) x (Pi.single j 1) =
+      deriv (Expr.evalAlong e (finEnv x) j.val) (x j) := by
+  have hout : HasFDerivAt (evalFin (n := n) e)
+      (fderiv ℝ (evalFin (n := n) e) (Function.update x j (x j)))
+      (Function.update x j (x j)) :=
+    (evalFin_differentiable_const (n := n) e h).differentiableAt.hasFDerivAt
+  have hcomp := hout.comp_hasDerivAt (x j) (hasDerivAt_update x j (x j))
+  rw [Function.update_eq_self] at hcomp
+  have heq : (evalFin (n := n) e ∘ Function.update x j) =
+      Expr.evalAlong e (finEnv x) j.val := by
+    funext t
+    simp only [Function.comp_apply, evalFin, Expr.evalAlong_eq]
+    rw [finEnv_update]
+  rw [heq] at hcomp
+  exact hcomp.deriv.symm
+
+theorem systemEval_differentiable_const {n : Nat}
+    (F : Fin n → Expr) (h : ∀ i, ADConstSupported (F i)) :
+    Differentiable ℝ (systemEval F) := by
+  rw [differentiable_pi]
+  intro i
+  exact evalFin_differentiable_const (F i) (h i)
+
+theorem jacobianAt_apply_const {n : Nat}
+    (F : Fin n → Expr) (h : ∀ i, ADConstSupported (F i))
+    (x : Fin n → ℝ) (i j : Fin n) :
+    jacobianAt F x i j =
+      deriv (Expr.evalAlong (F i) (finEnv x) j.val) (x j) := by
+  rw [jacobianAt, LinearMap.toMatrix'_apply]
+  change fderiv ℝ (systemEval F) x (Pi.single j 1) i = _
+  rw [show systemEval F = (fun x i => evalFin (F i) x) from rfl]
+  have hpi := fderiv_pi (𝕜 := ℝ) (x := x)
+    (φ := fun i => evalFin (F i))
+    (fun i => (evalFin_differentiable_const (F i) (h i)).differentiableAt)
+  rw [hpi]
+  simp only [ContinuousLinearMap.coe_pi']
+  exact fderiv_single_eq_deriv_evalAlong_const (F i) (h i) x j
+
+theorem jacobianAt_mem_intervalJacobian_const {n : Nat}
+    (F : Fin n → Expr) (h : ∀ i, ADConstSupported (F i))
+    (X : Fin n → IntervalRat) (x : Fin n → ℝ)
+    (hx : FinBoxMem x X) (cfg : EvalConfig) (i j : Fin n) :
+    jacobianAt F x i j ∈ intervalJacobian F X cfg i j := by
+  rw [jacobianAt_apply_const F h x i j]
+  exact evalDualTotalCore_der_correct_idx_const (F i) (h i)
+    (finEnv x) (finBoxEnv X) j.val (finEnv_mem_finBoxEnv hx)
+    (x j) (by
+      simpa only [finBoxEnv, j.isLt, dite_true] using hx j) cfg
+
+theorem newtonMap_differentiable_const {n : Nat}
+    (Y : Matrix (Fin n) (Fin n) ℝ)
+    (F : Fin n → Expr) (h : ∀ i, ADConstSupported (F i)) :
+    Differentiable ℝ (newtonMap Y F) := by
+  have hY : Differentiable ℝ (fun x => matrixCLM Y (systemEval F x)) :=
+    (matrixCLM Y).differentiable.comp (systemEval_differentiable_const F h)
+  rw [show newtonMap Y F =
+      id - (fun x => matrixCLM Y (systemEval F x)) by
+        funext x
+        rfl]
+  exact differentiable_id.sub hY
+
+theorem newtonMap_fderiv_matrix_const {n : Nat}
+    (Y : Matrix (Fin n) (Fin n) ℝ)
+    (F : Fin n → Expr) (h : ∀ i, ADConstSupported (F i))
+    (x : Fin n → ℝ) :
+    LinearMap.toMatrix' (fderiv ℝ (newtonMap Y F) x).toLinearMap =
+      1 - Y * jacobianAt F x := by
+  have hF : HasFDerivAt (systemEval F) (fderiv ℝ (systemEval F) x) x :=
+    (systemEval_differentiable_const F h).differentiableAt.hasFDerivAt
+  have hY := (matrixCLM Y).hasFDerivAt.comp x hF
+  have hg : HasFDerivAt (newtonMap Y F)
+      (ContinuousLinearMap.id ℝ _ -
+        (matrixCLM Y).comp (fderiv ℝ (systemEval F) x)) x := by
+    change HasFDerivAt (fun z => z - matrixCLM Y (systemEval F z)) _ x
+    exact (hasFDerivAt_id x).sub hY
+  rw [hg.fderiv]
+  change LinearMap.toMatrix' ((ContinuousLinearMap.id ℝ _ -
+    (matrixCLM Y).comp (fderiv ℝ (systemEval F) x)).toLinearMap) = _
+  rw [ContinuousLinearMap.toLinearMap_sub]
+  change LinearMap.toMatrix' (1 -
+    (matrixCLM Y).toLinearMap.comp
+      (fderiv ℝ (systemEval F) x).toLinearMap) = _
+  rw [map_sub, LinearMap.toMatrix'_one, LinearMap.toMatrix'_comp]
+  have hmatY : LinearMap.toMatrix' (matrixCLM Y).toLinearMap = Y := by
+    change LinearMap.toMatrix' (Matrix.mulVecLin Y) = Y
+    exact LinearMap.toMatrix'_toLin' Y
+  rw [hmatY]
+  rfl
+
+theorem newtonMap_fderiv_norm_le_const {n : Nat}
+    (Y : Matrix (Fin n) (Fin n) ℚ)
+    (F : Fin n → Expr) (h : ∀ i, ADConstSupported (F i))
+    (X : Fin n → IntervalRat) (x : Fin n → ℝ)
+    (hx : FinBoxMem x X) (cfg : EvalConfig) :
+    ‖fderiv ℝ (newtonMap (Y.map fun q => (q : ℝ)) F) x‖ ≤
+      (intervalMatrixBound
+        (preconditionedJacobian Y (intervalJacobian F X cfg)) : ℝ) := by
+  rw [← Matrix.linfty_opNorm_toMatrix]
+  rw [newtonMap_fderiv_matrix_const
+    (Y.map fun q => (q : ℝ)) F h x]
+  apply matrix_norm_le_intervalMatrixBound
+  exact mem_preconditionedJacobian Y (jacobianAt F x)
+    (intervalJacobian F X cfg)
+    (jacobianAt_mem_intervalJacobian_const F h X x hx cfg)
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# LeanCert expression models for the normalized Gerver systems
+
+These expressions use LeanCert's differentiable AD fragment plus exact
+named mathematical constants.  `LeanCertNamedConstAD` supplies the one
+missing soundness case for `namedConst` (derivative zero).
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+
+/-- The expression syntax interpreted by the certified interval evaluator. -/
+abbrev E := Expr
+
+/-- A rational constant expression. -/
+def ec (r : ℚ) : E := .const r
+/-- An indexed variable expression. -/
+def ev (i : Nat) : E := .var i
+/-- Construct the sum of two expressions. -/
+def eadd (a b : E) : E := .add a b
+/-- Construct the negation of an expression. -/
+def eneg (a : E) : E := .neg a
+/-- Construct a difference using addition and negation. -/
+def esub (a b : E) : E := .add a (.neg b)
+/-- Construct the product of two expressions. -/
+def emul (a b : E) : E := .mul a b
+/-- Multiply an expression by a rational constant. -/
+def escale (r : ℚ) (a : E) : E := .mul (.const r) a
+/-- Apply sine in the expression syntax. -/
+def esin (a : E) : E := .sin a
+/-- Apply cosine in the expression syntax. -/
+def ecos (a : E) : E := .cos a
+/-- The named π constant in the expression syntax. -/
+def epi : E := .namedConst .pi
+
+/-- An expression for a box coordinate in terms of its normalized variable. -/
+def scaledVar (box : List RatInterval) (i : Nat) : E :=
+  eadd (ec (boxMid box i)) (emul (ec (boxRad box i)) (ev i))
+
+/-! ## Reduced 4D -/
+
+/-- The four reduced equations encoded as evaluator expressions. -/
+def reducedExprList : List E :=
+  let a := scaledVar ExactReplay.reducedInputBox 0
+  let b := scaledVar ExactReplay.reducedInputBox 1
+  let phi := scaledVar ExactReplay.reducedInputBox 2
+  let theta := scaledVar ExactReplay.reducedInputBox 3
+  let one := ec 1
+  let half := ec (1 / 2)
+  let quarter := ec (1 / 4)
+  let cp := ecos phi
+  let sp := esin phi
+  let ct := ecos theta
+  let st := esin theta
+  let delta := esub theta phi
+  -- Keep the mathematical value exactly; AST need not match the old D object.
+  let f1 :=
+    eadd
+      (esub
+        (eadd
+          (esub (emul a (esub ct cp)) (emul (escale 2 b) sp))
+          (emul (esub delta one) ct))
+        st)
+      (eadd cp sp)
+  let f2 :=
+    eadd
+      (esub
+        (eadd
+          (eadd
+            (esub (emul a (eadd (escale 3 st) sp))
+              (emul (escale 2 b) cp))
+            (emul (escale 3 (esub delta one)) st))
+          (escale 3 ct))
+        sp)
+      cp
+  let f3 :=
+    esub
+      (eadd
+        (esub (esub (emul a cp) sp) half)
+        (emul half cp))
+      (emul b sp)
+  let f4 :=
+    eadd
+      (eadd
+        (esub
+          (esub
+            (esub
+              (esub (eadd a (escale (1 / 2) epi)) phi)
+              theta)
+            b)
+          (ec 0))
+        (emul (emul half delta) (eadd one a)))
+      (emul (emul quarter delta) delta)
+  List.cons f1 (List.cons f2 (List.cons f3 (List.cons f4 List.nil)))
+
+/-- Select an equation from the reduced four-dimensional expression system. -/
+def reducedExpr (i : Fin 4) : E :=
+  reducedExprList.getD i.1 (ec 0)
+
+/-! ## Direct 22D -/
+
+/-- A full Romik parameter encoded as a normalized variable expression. -/
+def fv (i : Nat) : E := scaledVar ExactReplay.fullInputBox i
+
+/-- The expression for the first switching angle φ. -/
+def phiE : E := fv 20
+/-- The expression for the second switching angle θ. -/
+def thetaE : E := fv 21
+/-- The expression for the reflected angle `π/2 - θ`. -/
+def etaE : E := esub (escale (1 / 2) epi) thetaE
+/-- The expression for the reflected angle `π/2 - φ`. -/
+def tauE : E := esub (escale (1 / 2) epi) phiE
+
+/-- Rotate a pair of coordinate expressions by an angle expression. -/
+def rotE (t z1 z2 : E) : E × E :=
+  let ct := ecos t
+  let st := esin t
+  (esub (emul ct z1) (emul st z2),
+   eadd (emul st z1) (emul ct z2))
+
+/-- Encode a selected closed path branch as two coordinate expressions. -/
+def pathPieceE (j : Nat) (t : E) : E × E :=
+  let one := ec 1
+  let half := ec (1 / 2)
+  let quarter := ec (1 / 4)
+  let ct := ecos t
+  let st := esin t
+  let data : E × E × E × E :=
+    if j = 1 then
+      (esub (eadd (emul (fv 10) ct) (emul (fv 11) st)) one,
+       esub (eadd (emul (eneg (fv 11)) ct) (emul (fv 10) st)) half,
+       fv 0, fv 1)
+    else if j = 2 then
+      (eadd (eadd (emul (emul (eneg quarter) t) t) (emul (fv 12) t)) (fv 13),
+       esub (esub (emul half t) (fv 12)) one,
+       fv 2, fv 3)
+    else if j = 3 then
+      (esub (fv 14) t, eadd (fv 15) t, fv 4, fv 5)
+    else if j = 4 then
+      (esub (eadd (emul (eneg half) t) (fv 16)) one,
+       eadd (eadd (emul (emul (eneg quarter) t) t) (emul (fv 16) t)) (fv 17),
+       fv 6, fv 7)
+    else
+      (esub (eadd (emul (fv 18) ct) (emul (fv 19) st)) half,
+       esub (eadd (emul (eneg (fv 19)) ct) (emul (fv 18) st)) one,
+       fv 8, fv 9)
+  let rr := rotE t data.1 data.2.1
+  (eadd rr.1 data.2.2.1, eadd rr.2 data.2.2.2)
+
+/-- Encode the body-frame velocity of a selected path branch. -/
+def alphaBetaE (j : Nat) (t : E) : E × E :=
+  let one := ec 1
+  let half := ec (1 / 2)
+  let quarter := ec (1 / 4)
+  let ct := ecos t
+  let st := esin t
+  if j = 1 then
+    (eadd (eadd (emul (escale (-2) (fv 10)) st)
+                 (emul (escale 2 (fv 11)) ct)) half,
+     esub (eadd (emul (escale 2 (fv 10)) ct)
+                 (emul (escale 2 (fv 11)) st)) one)
+  else if j = 2 then
+    (esub (eadd one (escale 2 (fv 12))) t,
+     eadd (eadd (eadd (emul (emul (eneg quarter) t) t)
+                       (emul (fv 12) t)) (fv 13)) half)
+  else if j = 3 then
+    (esub (esub (eneg one) (fv 15)) t,
+     esub (eadd one (fv 14)) t)
+  else if j = 4 then
+    (esub (esub (esub (emul (emul quarter t) t)
+                       (emul (fv 16) t)) (fv 17)) half,
+     esub (esub (escale 2 (fv 16)) one) t)
+  else
+    (eadd (esub one (emul (escale 2 (fv 18)) st))
+          (emul (escale 2 (fv 19)) ct),
+     esub (eadd (emul (escale 2 (fv 18)) ct)
+                 (emul (escale 2 (fv 19)) st)) half)
+
+/-- Encode the world-frame derivative of a selected path branch. -/
+def pathPrimeE (j : Nat) (t : E) : E × E :=
+  let ab := alphaBetaE j t
+  rotE t ab.1 ab.2
+
+/-- The twenty-two full equations encoded as evaluator expressions. -/
+def fullExprList : List E :=
+  let halfPi := escale (1 / 2) epi
+  let quarterPi := escale (1 / 4) epi
+  let one := ec 1
+  let quarter := ec (1 / 4)
+  let first : List E := [
+    esub (fv 18) (fv 10),
+    eadd (fv 19) (fv 11),
+    esub (eadd (fv 16) (fv 12)) quarterPi,
+    esub (esub (fv 17) (fv 13))
+      (emul quarterPi (esub (escale 2 (fv 12)) quarterPi)),
+    eadd (esub (fv 15) (fv 14)) halfPi,
+    eadd (esub (fv 0) one) (fv 10),
+    esub (fv 1) quarter,
+    eadd (fv 11) quarter
+  ]
+  let pairs : List ((E × E) × (E × E)) := [
+    (pathPieceE 1 phiE, pathPieceE 2 phiE),
+    (pathPrimeE 1 phiE, pathPrimeE 2 phiE),
+    (pathPieceE 2 thetaE, pathPieceE 3 thetaE),
+    (pathPrimeE 2 thetaE, pathPrimeE 3 thetaE),
+    (pathPieceE 3 etaE, pathPieceE 4 etaE),
+    (pathPieceE 4 tauE, pathPieceE 5 tauE)
+  ]
+  let matchEqs : List E := pairs.flatMap (fun lr =>
+    List.cons (esub lr.1.1 lr.2.1)
+      (List.cons (esub lr.1.2 lr.2.2) List.nil))
+  let lhs := pathPieceE 1 phiE
+  let xe := pathPieceE 3 etaE
+  let ae := (alphaBetaE 3 etaE).1
+  let be :=
+    (esub xe.1 (emul ae (esin etaE)),
+     eadd xe.2 (emul ae (ecos etaE)))
+  first ++ matchEqs ++
+    List.cons (esub lhs.1 be.1) (List.cons (esub lhs.2 be.2) List.nil)
+
+/-- Select one equation from the full Romik expression system. -/
+def fullExpr (i : Fin 22) : E :=
+  fullExprList.getD i.1 (ec 0)
+
+/-- Purely syntactic support checks for the Gerver expression fragment. -/
+theorem reducedExpr_supported :
+    ∀ i : Fin 4, ADConstSupported (reducedExpr i) := by
+  intro i
+  apply checkADConstSupported_correct
+  fin_cases i <;> decide
+
+theorem fullExpr_supported :
+    ∀ i : Fin 22, ADConstSupported (fullExpr i) := by
+  intro i
+  apply checkADConstSupported_correct
+  fin_cases i <;> decide
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Lean Cert Gerver Correspondence
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+
+/-- LeanCert's named π has exactly Mathlib's real value. -/
+@[simp] theorem mathConst_pi_toReal :
+    MathConst.pi.toReal = Real.pi := rfl
+
+/-! Reduced semantic correspondence. -/
+theorem reducedExpr_eval_0 (u : Fin 4 → ℝ) :
+    evalFin (reducedExpr (0 : Fin 4)) u =
+      Reduced.vectorSystem (reducedAffine u) (0 : Fin 4) := by
+  have hm := congrFun (Reduced.vectorSystem_eq_models (reducedAffine u)) (0 : Fin 4)
+  rw [hm]
+  simp [reducedExpr, reducedExprList, reducedAffine, affineFromBox,
+    scaledVar, boxMid, boxRad, eadd, esub, emul, escale, esin, ecos,
+    ec, ev, epi, evalFin, finEnv, Reduced.models, ScalarModel.var,
+    ScalarModel.const, ScalarModel.add, ScalarModel.neg, ScalarModel.sub,
+    ScalarModel.mul, ScalarModel.scale, ScalarModel.sin, ScalarModel.cos]; ring
+theorem reducedExpr_eval_1 (u : Fin 4 → ℝ) :
+    evalFin (reducedExpr (1 : Fin 4)) u =
+      Reduced.vectorSystem (reducedAffine u) (1 : Fin 4) := by
+  have hm := congrFun (Reduced.vectorSystem_eq_models (reducedAffine u)) (1 : Fin 4)
+  rw [hm]
+  simp [reducedExpr, reducedExprList, reducedAffine, affineFromBox,
+    scaledVar, boxMid, boxRad, eadd, esub, emul, escale, esin, ecos,
+    ec, ev, epi, evalFin, finEnv, Reduced.models, ScalarModel.var,
+    ScalarModel.const, ScalarModel.add, ScalarModel.neg, ScalarModel.sub,
+    ScalarModel.mul, ScalarModel.scale, ScalarModel.sin, ScalarModel.cos]
+
+theorem reducedExpr_eval_2 (u : Fin 4 → ℝ) :
+    evalFin (reducedExpr (2 : Fin 4)) u =
+      Reduced.vectorSystem (reducedAffine u) (2 : Fin 4) := by
+  have hm := congrFun (Reduced.vectorSystem_eq_models (reducedAffine u)) (2 : Fin 4)
+  rw [hm]
+  simp [reducedExpr, reducedExprList, reducedAffine, affineFromBox,
+    scaledVar, boxMid, boxRad, eadd, esub, emul, escale, esin, ecos,
+    ec, ev, epi, evalFin, finEnv, Reduced.models, ScalarModel.var,
+    ScalarModel.const, ScalarModel.add, ScalarModel.neg, ScalarModel.sub,
+    ScalarModel.mul, ScalarModel.scale, ScalarModel.sin, ScalarModel.cos]
+
+theorem reducedExpr_eval_3 (u : Fin 4 → ℝ) :
+    evalFin (reducedExpr (3 : Fin 4)) u =
+      Reduced.vectorSystem (reducedAffine u) (3 : Fin 4) := by
+  have hm := congrFun (Reduced.vectorSystem_eq_models (reducedAffine u)) (3 : Fin 4)
+  rw [hm]
+  simp [reducedExpr, reducedExprList, reducedAffine, affineFromBox,
+    scaledVar, boxMid, boxRad, eadd, esub, emul, escale, esin, ecos,
+    ec, ev, epi, evalFin, finEnv, Reduced.models, ScalarModel.var,
+    ScalarModel.const, ScalarModel.add, ScalarModel.neg, ScalarModel.sub,
+    ScalarModel.mul, ScalarModel.scale, ScalarModel.sin, ScalarModel.cos]
+
+theorem reduced_systemEval_eq (u : Fin 4 → ℝ) :
+    systemEval reducedExpr u = Reduced.vectorSystem (reducedAffine u) := by
+  funext i
+  fin_cases i
+  · exact reducedExpr_eval_0 u
+  · exact reducedExpr_eval_1 u
+  · exact reducedExpr_eval_2 u
+  · exact reducedExpr_eval_3 u
+
+/-! Full 22D semantic correspondence through the already proved public
+`fullDualOutput_model_eq`.  This avoids unfolding the private pathPrime helper. -/
+theorem fullExpr_eval_0 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (0 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (0 : Fin 22) := by
+  calc
+    evalFin (fullExpr (0 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 0 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring
+    _ = Romik.vectorSystem (fullAffine u) (0 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (0 : Fin 22)
+theorem fullExpr_eval_1 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (1 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (1 : Fin 22) := by
+  calc
+    evalFin (fullExpr (1 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 1 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]
+    _ = Romik.vectorSystem (fullAffine u) (1 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (1 : Fin 22)
+theorem fullExpr_eval_2 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (2 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (2 : Fin 22) := by
+  calc
+    evalFin (fullExpr (2 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 2 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring
+    _ = Romik.vectorSystem (fullAffine u) (2 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (2 : Fin 22)
+theorem fullExpr_eval_3 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (3 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (3 : Fin 22) := by
+  calc
+    evalFin (fullExpr (3 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 3 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring
+    _ = Romik.vectorSystem (fullAffine u) (3 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (3 : Fin 22)
+theorem fullExpr_eval_4 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (4 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (4 : Fin 22) := by
+  calc
+    evalFin (fullExpr (4 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 4 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring
+    _ = Romik.vectorSystem (fullAffine u) (4 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (4 : Fin 22)
+theorem fullExpr_eval_5 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (5 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (5 : Fin 22) := by
+  calc
+    evalFin (fullExpr (5 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 5 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring
+    _ = Romik.vectorSystem (fullAffine u) (5 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (5 : Fin 22)
+theorem fullExpr_eval_6 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (6 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (6 : Fin 22) := by
+  calc
+    evalFin (fullExpr (6 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 6 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring
+    _ = Romik.vectorSystem (fullAffine u) (6 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (6 : Fin 22)
+theorem fullExpr_eval_7 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (7 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (7 : Fin 22) := by
+  calc
+    evalFin (fullExpr (7 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 7 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]
+    _ = Romik.vectorSystem (fullAffine u) (7 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (7 : Fin 22)
+theorem fullExpr_eval_8 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (8 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (8 : Fin 22) := by
+  calc
+    evalFin (fullExpr (8 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 8 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring
+    _ = Romik.vectorSystem (fullAffine u) (8 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (8 : Fin 22)
+theorem fullExpr_eval_9 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (9 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (9 : Fin 22) := by
+  calc
+    evalFin (fullExpr (9 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 9 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring
+    _ = Romik.vectorSystem (fullAffine u) (9 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (9 : Fin 22)
+theorem fullExpr_eval_10 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (10 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (10 : Fin 22) := by
+  calc
+    evalFin (fullExpr (10 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 10 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring
+    _ = Romik.vectorSystem (fullAffine u) (10 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (10 : Fin 22)
+theorem fullExpr_eval_11 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (11 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (11 : Fin 22) := by
+  calc
+    evalFin (fullExpr (11 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 11 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring
+    _ = Romik.vectorSystem (fullAffine u) (11 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (11 : Fin 22)
+theorem fullExpr_eval_12 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (12 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (12 : Fin 22) := by
+  calc
+    evalFin (fullExpr (12 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 12 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring
+    _ = Romik.vectorSystem (fullAffine u) (12 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (12 : Fin 22)
+theorem fullExpr_eval_13 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (13 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (13 : Fin 22) := by
+  calc
+    evalFin (fullExpr (13 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 13 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring
+    _ = Romik.vectorSystem (fullAffine u) (13 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (13 : Fin 22)
+theorem fullExpr_eval_14 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (14 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (14 : Fin 22) := by
+  calc
+    evalFin (fullExpr (14 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 14 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring
+    _ = Romik.vectorSystem (fullAffine u) (14 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (14 : Fin 22)
+theorem fullExpr_eval_15 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (15 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (15 : Fin 22) := by
+  calc
+    evalFin (fullExpr (15 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 15 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring
+    _ = Romik.vectorSystem (fullAffine u) (15 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (15 : Fin 22)
+theorem fullExpr_eval_16 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (16 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (16 : Fin 22) := by
+  calc
+    evalFin (fullExpr (16 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 16 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring_nf
+    _ = Romik.vectorSystem (fullAffine u) (16 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (16 : Fin 22)
+theorem fullExpr_eval_17 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (17 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (17 : Fin 22) := by
+  calc
+    evalFin (fullExpr (17 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 17 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring_nf
+    _ = Romik.vectorSystem (fullAffine u) (17 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (17 : Fin 22)
+theorem fullExpr_eval_18 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (18 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (18 : Fin 22) := by
+  calc
+    evalFin (fullExpr (18 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 18 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring_nf
+    _ = Romik.vectorSystem (fullAffine u) (18 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (18 : Fin 22)
+theorem fullExpr_eval_19 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (19 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (19 : Fin 22) := by
+  calc
+    evalFin (fullExpr (19 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 19 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring_nf
+    _ = Romik.vectorSystem (fullAffine u) (19 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (19 : Fin 22)
+theorem fullExpr_eval_20 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (20 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (20 : Fin 22) := by
+  calc
+    evalFin (fullExpr (20 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 20 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring_nf
+    _ = Romik.vectorSystem (fullAffine u) (20 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (20 : Fin 22)
+theorem fullExpr_eval_21 (u : Fin 22 → ℝ) :
+    evalFin (fullExpr (21 : Fin 22)) u =
+      Romik.vectorSystem (fullAffine u) (21 : Fin 22) := by
+  calc
+    evalFin (fullExpr (21 : Fin 22)) u =
+        ((Romik.fullDualOutput.getD 21 (Romik.SoundDual.pointConst 0)).model.value
+          (fullAffine u)) := by
+      simp [fullExpr, fullExprList, fv, phiE, thetaE, etaE, tauE,
+        rotE, pathPieceE, alphaBetaE, pathPrimeE,
+        fullAffine, affineFromBox, scaledVar, boxMid, boxRad,
+        eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+        evalFin, finEnv,
+        Romik.fullDualOutput, Romik.fullVars, Romik.inputDual, Romik.piDual,
+        Romik.phiDual, Romik.thetaDual, Romik.etaDual, Romik.tauDual,
+        Romik.rotDual, Romik.pathPieceDual, Romik.alphaBetaDual,
+        Romik.pathPrimeDual]; ring_nf
+    _ = Romik.vectorSystem (fullAffine u) (21 : Fin 22) := by
+      exact congrFun (Romik.fullDualOutput_model_eq (fullAffine u)) (21 : Fin 22)
+
+theorem full_systemEval_eq (u : Fin 22 → ℝ) :
+    systemEval fullExpr u = Romik.vectorSystem (fullAffine u) := by
+  funext i
+  fin_cases i
+  · exact fullExpr_eval_0 u
+  · exact fullExpr_eval_1 u
+  · exact fullExpr_eval_2 u
+  · exact fullExpr_eval_3 u
+  · exact fullExpr_eval_4 u
+  · exact fullExpr_eval_5 u
+  · exact fullExpr_eval_6 u
+  · exact fullExpr_eval_7 u
+  · exact fullExpr_eval_8 u
+  · exact fullExpr_eval_9 u
+  · exact fullExpr_eval_10 u
+  · exact fullExpr_eval_11 u
+  · exact fullExpr_eval_12 u
+  · exact fullExpr_eval_13 u
+  · exact fullExpr_eval_14 u
+  · exact fullExpr_eval_15 u
+  · exact fullExpr_eval_16 u
+  · exact fullExpr_eval_17 u
+  · exact fullExpr_eval_18 u
+  · exact fullExpr_eval_19 u
+  · exact fullExpr_eval_20 u
+  · exact fullExpr_eval_21 u
+
+/-! ## Box transport -/
+
+theorem reduced_box_strict :
+    ∀ i : Fin 4,
+      (ExactReplay.getI ExactReplay.reducedInputBox i.1).lo <
+      (ExactReplay.getI ExactReplay.reducedInputBox i.1).hi := by
+  intro i
+  fin_cases i <;>
+    norm_num [ExactReplay.reducedInputBox, CertificateManifest.x4,
+      CertificateManifest.q, ExactReplay.getI]
+
+theorem full_box_strict :
+    ∀ i : Fin 22,
+      (ExactReplay.getI ExactReplay.fullInputBox i.1).lo <
+      (ExactReplay.getI ExactReplay.fullInputBox i.1).hi := by
+  intro i
+  fin_cases i <;>
+    norm_num [ExactReplay.fullInputBox, CertificateManifest.z22,
+      CertificateManifest.q, ExactReplay.getI]
+
+theorem affine_mem_interval
+    (z : RatInterval) (hz : z.lo < z.hi) (u : ℝ)
+    (hu : (-1 : ℝ) ≤ u ∧ u ≤ 1) :
+    RatInterval.Contains z
+      ((((z.lo + z.hi) / 2 : ℚ) : ℝ) +
+       ((((z.hi - z.lo) / 2 : ℚ) : ℝ) * u)) := by
+  have hzR : (z.lo : ℝ) < (z.hi : ℝ) := by
+    exact_mod_cast hz
+  have hrad : 0 ≤ ((z.hi : ℝ) - (z.lo : ℝ)) / 2 := by
+    linarith
+  constructor
+  · push_cast
+    calc
+      (z.lo : ℝ) =
+          ((z.lo : ℝ) + (z.hi : ℝ)) / 2 +
+            (((z.hi : ℝ) - (z.lo : ℝ)) / 2) * (-1) := by ring
+      _ ≤ ((z.lo : ℝ) + (z.hi : ℝ)) / 2 +
+            (((z.hi : ℝ) - (z.lo : ℝ)) / 2) * u := by
+        exact add_le_add_right
+          (mul_le_mul_of_nonneg_left hu.1 hrad)
+          (((z.lo : ℝ) + (z.hi : ℝ)) / 2)
+  · push_cast
+    calc
+      ((z.lo : ℝ) + (z.hi : ℝ)) / 2 +
+            (((z.hi : ℝ) - (z.lo : ℝ)) / 2) * u
+          ≤ ((z.lo : ℝ) + (z.hi : ℝ)) / 2 +
+            (((z.hi : ℝ) - (z.lo : ℝ)) / 2) * 1 := by
+        exact add_le_add_right
+          (mul_le_mul_of_nonneg_left hu.2 hrad)
+          (((z.lo : ℝ) + (z.hi : ℝ)) / 2)
+      _ = (z.hi : ℝ) := by ring
+
+theorem normalize_mem_unit
+    (z : RatInterval) (hz : z.lo < z.hi) (x : ℝ)
+    (hx : RatInterval.Contains z x) :
+    ((x - ((((z.lo + z.hi) / 2 : ℚ) : ℝ))) /
+      ((((z.hi - z.lo) / 2 : ℚ) : ℝ))) ∈
+      (unitBox (n := 1) (0 : Fin 1)) := by
+  have hrQ : (0 : ℚ) < (z.hi - z.lo) / 2 := by
+    linarith
+  have hr : (0 : ℝ) < ((((z.hi - z.lo) / 2 : ℚ) : ℝ)) := by
+    exact_mod_cast hrQ
+  have hbounds :
+      (-1 : ℝ) ≤
+          (x - ((((z.lo + z.hi) / 2 : ℚ) : ℝ))) /
+            ((((z.hi - z.lo) / 2 : ℚ) : ℝ)) ∧
+        (x - ((((z.lo + z.hi) / 2 : ℚ) : ℝ))) /
+            ((((z.hi - z.lo) / 2 : ℚ) : ℝ)) ≤ 1 := by
+    constructor
+    · rw [le_div_iff₀ hr]
+      push_cast
+      have hxlo := hx.1
+      linarith
+    · rw [div_le_iff₀ hr]
+      push_cast
+      have hxhi := hx.2
+      linarith
+  simpa [unitBox, IntervalRat.mem_def] using hbounds
+
+theorem affine_normalize_coord
+    (z : RatInterval) (hz : z.lo < z.hi) (x : ℝ) :
+    ((((z.lo + z.hi) / 2 : ℚ) : ℝ) +
+      ((((z.hi - z.lo) / 2 : ℚ) : ℝ) *
+        ((x - ((((z.lo + z.hi) / 2 : ℚ) : ℝ))) /
+          ((((z.hi - z.lo) / 2 : ℚ) : ℝ))))) = x := by
+  have hrQ : ((z.hi - z.lo) / 2 : ℚ) ≠ 0 := by
+    have : (0 : ℚ) < (z.hi - z.lo) / 2 := by linarith
+    exact ne_of_gt this
+  have hr : ((((z.hi - z.lo) / 2 : ℚ) : ℝ)) ≠ 0 := by
+    exact_mod_cast hrQ
+  field_simp
+  ring
+
+theorem reducedAffine_mem {u : Fin 4 → ℝ}
+    (hu : FinBoxMem u (unitBox (n := 4))) :
+    reducedAffine u ∈ Reduced.vectorBox := by
+  apply (Reduced.inputBox_exact _).2
+  refine ⟨by norm_num [ExactReplay.reducedInputBox, CertificateManifest.x4], ?_⟩
+  intro i
+  simpa [reducedAffine, affineFromBox, boxMid, boxRad,
+    ExactReplay.getI, ExactReplay.zeroI, RatInterval.point] using
+    affine_mem_interval
+      (ExactReplay.getI ExactReplay.reducedInputBox i.1)
+      (reduced_box_strict i) (u i) (by
+        simpa [unitBox, IntervalRat.mem_def] using hu i)
+
+theorem fullAffine_mem {u : Fin 22 → ℝ}
+    (hu : FinBoxMem u (unitBox (n := 22))) :
+    fullAffine u ∈ Romik.vectorBox := by
+  apply (Romik.inputBox_exact _).2
+  refine ⟨by norm_num [ExactReplay.fullInputBox, CertificateManifest.z22], ?_⟩
+  intro i
+  simpa [fullAffine, affineFromBox, boxMid, boxRad,
+    ExactReplay.getI, ExactReplay.zeroI, RatInterval.point] using
+    affine_mem_interval
+      (ExactReplay.getI ExactReplay.fullInputBox i.1)
+      (full_box_strict i) (u i) (by
+        simpa [unitBox, IntervalRat.mem_def] using hu i)
+
+theorem reducedNormalize_mem {x : Fin 4 → ℝ}
+    (hx : x ∈ Reduced.vectorBox) :
+    FinBoxMem (reducedNormalize x) (unitBox (n := 4)) := by
+  have henc := (Reduced.inputBox_exact x).1 hx
+  intro i
+  have hi := henc.2 i
+  have h := normalize_mem_unit
+    (ExactReplay.getI ExactReplay.reducedInputBox i.1)
+    (reduced_box_strict i) (x i) hi
+  simpa [reducedNormalize, normalizeToBox, boxMid, boxRad, unitBox,
+    IntervalRat.mem_def] using h
+
+theorem reducedAffine_normalize (x : Fin 4 → ℝ) :
+    reducedAffine (reducedNormalize x) = x := by
+  funext i
+  simpa [reducedAffine, reducedNormalize, affineFromBox, normalizeToBox,
+    boxMid, boxRad] using
+      affine_normalize_coord
+        (ExactReplay.getI ExactReplay.reducedInputBox i.1)
+        (reduced_box_strict i) (x i)
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Determinant-free checked Krawczyk endgame
+
+For a square system, the usual explicit determinant test on the preconditioner
+is redundant once `‖I - YJ‖ < 1` is known: at any point in the box,
+`YJ = 1 - T` with `‖T‖ < 1`, hence `YJ` is a unit by the Neumann-series
+theorem.  Therefore `Y` is surjective, hence injective in finite dimension.
+
+This matters computationally in dimension 22 because Mathlib's generic
+Leibniz determinant is the wrong algorithm for a huge exact rational matrix.
+-/
+
+/-!
+## Kernel-reducible finite interval sums
+
+LeanCert's public `intervalRatMatVec` uses `Finset.sum` with a local
+proof-transported `AddCommMonoid IntervalRat`.  That is semantically sound, but
+closed `decide +kernel` computations can get stuck reducing the transported
+typeclass instance.
+
+We keep LeanCert's interval operations and mathematical semantics, but package
+this one finite sum by structural recursion on `Fin n`.  Mathlib's existing
+`Fin.sum_univ_succ` is the semantic bridge to the ordinary real finite sum.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+
+/-- A finite interval sum that reduces by structural recursion, with no
+`AddCommMonoid IntervalRat` instance involved in computation. -/
+def kernelIntervalFinSum : {n : Nat} → (Fin n → IntervalRat) → IntervalRat
+  | 0, _ => IntervalRat.singleton 0
+  | n + 1, f =>
+      IntervalRat.add (f 0)
+        (kernelIntervalFinSum (fun j : Fin n => f j.succ))
+
+/-- Semantic soundness of the kernel-reducible finite interval sum. -/
+theorem mem_kernelIntervalFinSum :
+    ∀ (n : Nat) (x : Fin n → ℝ) (I : Fin n → IntervalRat),
+      (∀ i, x i ∈ I i) →
+      (∑ i, x i) ∈ kernelIntervalFinSum I := by
+  intro n
+  induction n with
+  | zero =>
+      intro x I h
+      simp [kernelIntervalFinSum, IntervalRat.mem_def, IntervalRat.singleton]
+  | succ n ih =>
+      intro x I h
+      rw [Fin.sum_univ_succ]
+      exact IntervalRat.mem_add (h 0)
+        (ih (fun j : Fin n => x j.succ)
+          (fun j : Fin n => I j.succ)
+          (fun j => h j.succ))
+
+/-- Kernel-reducible exact interval matrix-vector product. -/
+def kernelIntervalRatMatVec {n : Nat}
+    (Y : Matrix (Fin n) (Fin n) ℚ)
+    (v : Fin n → IntervalRat) : Fin n → IntervalRat :=
+  fun i =>
+    kernelIntervalFinSum
+      (fun j => IntervalRat.scale (Y i j) (v j))
+
+/-!
+## Kernel-reducible point values at the Newton center
+
+LeanCert's ordinary rational evaluator uses `sinComputableReduced` and
+`cosComputableReduced`.  Those are mathematically excellent general-purpose
+routines, but their period-reduction step computes a rational floor.  On the
+closed Gerver certificates that floor can prevent `decide +kernel` from
+normalizing an otherwise entirely rational proposition.
+
+For the exact Gerver expression fragment (`ADConstSupported`) we therefore use
+a tiny point evaluator that is identical on algebraic operations and named
+constants, but calls the already-proved *unreduced* Taylor enclosures
+`sinComputable` and `cosComputable`.  These enclosures are globally sound for
+arbitrary real arguments; argument reduction is an optimization, not a
+soundness requirement.  This keeps the Newton-center calculation purely in
+kernel-reducible rational arithmetic without changing the mathematical map.
+-/
+
+/-- A kernel-reducible interval for π, using the exact project interval
+whose semantic soundness is already proved in `TranscendentalSoundness`.
+
+The generic LeanCert `MathConst.interval` route is mathematically sound, but
+its named-constant machinery does not always normalize far enough for closed
+`decide +kernel` comparisons.  Reusing the project's already-certified exact
+rational π endpoints removes only that reduction bottleneck; it does not
+change the represented real constant or weaken any enclosure. -/
+def kernelPiInterval : IntervalRat :=
+  { lo := ExactReplay.piI.lo
+    hi := ExactReplay.piI.hi
+    le := by
+      norm_num [ExactReplay.piI, ExactReplay.q] }
+
+/-- The true real π lies in the kernel-reducible π interval. -/
+theorem real_pi_mem_kernelPiInterval : Real.pi ∈ kernelPiInterval := by
+  simpa [kernelPiInterval, IntervalRat.mem_def, RatInterval.Contains] using
+    ExactReplay.piI_contains_pi
+
+/-- Named constants for the point evaluator.  π takes the specialized
+kernel-reducible path; every other LeanCert named constant keeps LeanCert's
+public certified interval unchanged. -/
+def kernelNamedConstInterval (c : MathConst) : IntervalRat :=
+  if c = .pi then kernelPiInterval else c.interval
+
+/-- Point evaluator for the everywhere-defined Gerver expression fragment.
+Unsupported constructors are deliberately mapped to `{0}`; the soundness
+lemma below is only stated for `ADConstSupported`, whose constructors are all
+handled explicitly. -/
+def kernelPointEvalCore (e : Expr) (ρ : IntervalEnv)
+    (depth : Nat) : IntervalRat :=
+  match e with
+  | .const q => IntervalRat.singleton q
+  | .var idx => ρ idx
+  | .add a b => IntervalRat.add
+      (kernelPointEvalCore a ρ depth) (kernelPointEvalCore b ρ depth)
+  | .mul a b => IntervalRat.mul
+      (kernelPointEvalCore a ρ depth) (kernelPointEvalCore b ρ depth)
+  | .neg a => IntervalRat.neg (kernelPointEvalCore a ρ depth)
+  | .exp a => IntervalRat.expComputable
+      (kernelPointEvalCore a ρ depth) depth
+  | .sin a => IntervalRat.sinComputable
+      (kernelPointEvalCore a ρ depth) depth
+  | .cos a => IntervalRat.cosComputable
+      (kernelPointEvalCore a ρ depth) depth
+  | .namedConst c => kernelNamedConstInterval c
+  | _ => IntervalRat.singleton 0
+
+/-- Soundness of the kernel-reducible point evaluator on exactly the Gerver
+fragment.  In the sine/cosine cases this uses LeanCert's global Taylor
+correctness theorems directly, so no period-reduction hypothesis is needed. -/
+theorem eval_mem_kernelPointEvalCore {e : Expr}
+    (h : ADConstSupported e)
+    (ρReal : Nat → ℝ) (ρInt : IntervalEnv)
+    (hmem : envMem ρReal ρInt) (depth : Nat) :
+    Expr.eval ρReal e ∈ kernelPointEvalCore e ρInt depth := by
+  induction h with
+  | const q =>
+      simp only [Expr.eval_const, kernelPointEvalCore]
+      exact IntervalRat.mem_singleton q
+  | var idx =>
+      simp only [Expr.eval_var, kernelPointEvalCore]
+      exact hmem idx
+  | add _ _ iha ihb =>
+      simp only [Expr.eval_add, kernelPointEvalCore]
+      exact IntervalRat.mem_add iha ihb
+  | mul _ _ iha ihb =>
+      simp only [Expr.eval_mul, kernelPointEvalCore]
+      exact IntervalRat.mem_mul iha ihb
+  | neg _ ih =>
+      simp only [Expr.eval_neg, kernelPointEvalCore]
+      exact IntervalRat.mem_neg ih
+  | exp _ ih =>
+      simp only [Expr.eval_exp, kernelPointEvalCore]
+      exact IntervalRat.mem_expComputable ih depth
+  | sin _ ih =>
+      simp only [Expr.eval_sin, kernelPointEvalCore]
+      exact IntervalRat.mem_sinComputable ih depth
+  | cos _ ih =>
+      simp only [Expr.eval_cos, kernelPointEvalCore]
+      exact IntervalRat.mem_cosComputable ih depth
+  | namedConst c =>
+      simp only [Expr.eval_namedConst, kernelPointEvalCore]
+      by_cases hc : c = .pi
+      · subst c
+        simp only [kernelNamedConstInterval]
+        simpa using real_pi_mem_kernelPiInterval
+      · simp only [kernelNamedConstInterval, ite_eq_right hc]
+        exact c.mem_interval
+
+/-- Taylor depth used only for the Newton-center point values.
+The reduced system has coordinates at scale about `10^-15`; the 22D system
+contains substantially thinner coordinates.  Depth 26 is the smallest retained value after an
+exact-rational replay of all
+22 normalized Newton-center images that still leaves the direct self-map
+strictly inside the unit box.  It materially reduces kernel rational size
+compared with depth 27/34 while leaving the Jacobian evaluator and its
+already-passing certificates untouched. -/
+def kernelCenterTaylorDepth (n : Nat) : Nat :=
+  if n ≤ 4 then 20 else 26
+
+/-- Point-value enclosures for a square system at a rational center. -/
+def pointEvalIntervalsKernel {n : Nat}
+    (F : Fin n → Expr) (m : Fin n → ℚ)
+    (_cfg : EvalConfig := {}) : Fin n → IntervalRat :=
+  fun i => kernelPointEvalCore (F i) (pointIntervalEnv m)
+    (kernelCenterTaylorDepth n)
+
+/-- Every real system coordinate at the rational center lies in the
+kernel-reducible point enclosure. -/
+theorem systemEval_mem_pointEvalIntervalsKernel_const {n : Nat}
+    (F : Fin n → Expr) (h : ∀ i, ADConstSupported (F i))
+    (m : Fin n → ℚ) (cfg : EvalConfig) (i : Fin n) :
+    systemEval F (fun j => (m j : ℝ)) i ∈
+      pointEvalIntervalsKernel F m cfg i := by
+  change Expr.eval (finEnv (fun j => (m j : ℝ))) (F i) ∈
+    kernelPointEvalCore (F i) (pointIntervalEnv m)
+      (kernelCenterTaylorDepth n)
+  exact eval_mem_kernelPointEvalCore (h i)
+    (finEnv (fun j => (m j : ℝ))) (pointIntervalEnv m)
+    (finEnv_ratCast_mem_pointIntervalEnv m) (kernelCenterTaylorDepth n)
+
+/-- Kernel-reducible version of LeanCert's Newton-center enclosure. -/
+def kernelNewtonCenterInterval {n : Nat}
+    (F : Fin n → Expr) (m : Fin n → ℚ)
+    (Y : Matrix (Fin n) (Fin n) ℚ)
+    (cfg : EvalConfig := {}) : Fin n → IntervalRat :=
+  fun i =>
+    IntervalRat.sub (IntervalRat.singleton (m i))
+      (kernelIntervalRatMatVec Y (pointEvalIntervalsKernel F m cfg) i)
+
+/-- The real Newton center lies in the kernel-reducible enclosure. -/
+theorem newtonMap_center_mem_kernel_const {n : Nat}
+    (F : Fin n → Expr) (h : ∀ i, ADConstSupported (F i))
+    (m : Fin n → ℚ) (Y : Matrix (Fin n) (Fin n) ℚ)
+    (cfg : EvalConfig) (i : Fin n) :
+    newtonMap (Y.map fun q => (q : ℝ)) F
+        (fun j => (m j : ℝ)) i ∈
+      kernelNewtonCenterInterval F m Y cfg i := by
+  apply IntervalRat.mem_sub
+  · exact IntervalRat.mem_singleton (m i)
+  · simp only [Matrix.mulVec, dotProduct, kernelIntervalRatMatVec]
+    exact mem_kernelIntervalFinSum n
+      (fun j => (Y i j : ℝ) * systemEval F (fun k => (m k : ℝ)) j)
+      (fun j => IntervalRat.scale (Y i j) (pointEvalIntervalsKernel F m cfg j))
+      (fun j =>
+        IntervalRat.mem_scale (Y i j)
+          (systemEval_mem_pointEvalIntervalsKernel_const F h m cfg j))
+
+/-- Same checked self-map enclosure as before, now with a kernel-reducible
+finite interval sum at the Newton center. -/
+def imageEnclosureWithQ {n : Nat}
+    (F : Fin n → Expr) (X : Fin n → IntervalRat)
+    (m : Fin n → ℚ) (Y : Matrix (Fin n) (Fin n) ℚ)
+    (cfg : EvalConfig) (q : ℚ) : Fin n → IntervalRat :=
+  fun i => IntervalRat.add (kernelNewtonCenterInterval F m Y cfg i)
+    (symmetricInterval (q * boxRadius X m))
+
+theorem mapsTo_of_bound_and_enclosure {n : Nat}
+    (F : Fin n → Expr) (hsupp : ∀ i, ADConstSupported (F i))
+    (X : Fin n → IntervalRat) (m : Fin n → ℚ)
+    (hm : FinBoxMem (fun i => (m i : ℝ)) X)
+    (Y : Matrix (Fin n) (Fin n) ℚ) (cfg : EvalConfig)
+    (q : ℚ) (hq0 : 0 ≤ q)
+    (hbound :
+      intervalMatrixBound
+        (preconditionedJacobian Y (intervalJacobian F X cfg)) ≤ q)
+    (hencl : ∀ i,
+      intervalStrictInside (imageEnclosureWithQ F X m Y cfg q i) (X i) = true) :
+    Set.MapsTo (newtonMap (Y.map fun r => (r : ℝ)) F)
+      (finBoxSet X) (finBoxSet X) := by
+  intro x hx i
+  let mr : Fin n → ℝ := fun i => (m i : ℝ)
+  have hdiff :
+      ‖newtonMap (Y.map fun r => (r : ℝ)) F x -
+          newtonMap (Y.map fun r => (r : ℝ)) F mr‖
+        ≤ (q : ℝ) * ‖x - mr‖ := by
+    apply (finBoxSet_convex X).norm_image_sub_le_of_norm_fderiv_le (𝕜 := ℝ)
+    · intro z hz
+      exact (newtonMap_differentiable_const (Y.map fun r => (r : ℝ)) F hsupp).differentiableAt
+    · intro z hz
+      calc
+        ‖fderiv ℝ (newtonMap (Y.map fun r => (r : ℝ)) F) z‖
+            ≤ (intervalMatrixBound
+                (preconditionedJacobian Y (intervalJacobian F X cfg)) : ℝ) :=
+          newtonMap_fderiv_norm_le_const Y F hsupp X z hz cfg
+        _ ≤ (q : ℝ) := by exact_mod_cast hbound
+    · exact hm
+    · exact hx
+  have hr0 : 0 ≤ boxRadius X m := boxRadius_nonneg X m
+  have hnorm :
+      ‖newtonMap (Y.map fun r => (r : ℝ)) F x -
+          newtonMap (Y.map fun r => (r : ℝ)) F mr‖
+        ≤ ((q * boxRadius X m : ℚ) : ℝ) := by
+    calc
+      _ ≤ (q : ℝ) * ‖x - mr‖ := hdiff
+      _ ≤ (q : ℝ) * (boxRadius X m : ℝ) :=
+        mul_le_mul_of_nonneg_left (norm_sub_center_le_boxRadius hx)
+          (by exact_mod_cast hq0)
+      _ = _ := by push_cast; rfl
+  have hcoord :
+      |(newtonMap (Y.map fun r => (r : ℝ)) F x -
+          newtonMap (Y.map fun r => (r : ℝ)) F mr) i|
+        ≤ ((q * boxRadius X m : ℚ) : ℝ) :=
+    (norm_le_pi_norm _ i).trans hnorm
+  have herr :
+      (newtonMap (Y.map fun r => (r : ℝ)) F x -
+          newtonMap (Y.map fun r => (r : ℝ)) F mr) i ∈
+        symmetricInterval (q * boxRadius X m) := by
+    rw [abs_le] at hcoord
+    simpa [symmetricInterval, IntervalRat.mem_def,
+      abs_of_nonneg (mul_nonneg hq0 hr0)] using hcoord
+  have hcenter := newtonMap_center_mem_kernel_const F hsupp m Y cfg i
+  have himage :
+      newtonMap (Y.map fun r => (r : ℝ)) F x i ∈
+        imageEnclosureWithQ F X m Y cfg q i := by
+    have hadd := IntervalRat.mem_add hcenter herr
+    simpa [imageEnclosureWithQ, mr, Pi.sub_apply] using hadd
+  exact intervalStrictInside_sound (hencl i) _ himage
+
+theorem preconditioner_injective_of_bound_lt_one {n : Nat}
+    (F : Fin n → Expr) (hsupp : ∀ i, ADConstSupported (F i))
+    (X : Fin n → IntervalRat) (m : Fin n → ℚ)
+    (hm : FinBoxMem (fun i => (m i : ℝ)) X)
+    (Y : Matrix (Fin n) (Fin n) ℚ) (cfg : EvalConfig)
+    (hq :
+      intervalMatrixBound
+        (preconditionedJacobian Y (intervalJacobian F X cfg)) < 1) :
+    Function.Injective
+      (fun v : Fin n → ℝ => (Y.map fun r => (r : ℝ)).mulVec v) := by
+  let Yr : Matrix (Fin n) (Fin n) ℝ := Y.map fun r => (r : ℝ)
+  let mr : Fin n → ℝ := fun i => (m i : ℝ)
+  let J : (Fin n → ℝ) →L[ℝ] (Fin n → ℝ) :=
+    fderiv ℝ (systemEval F) mr
+  let A : (Fin n → ℝ) →L[ℝ] (Fin n → ℝ) := matrixCLM Yr
+  let T : (Fin n → ℝ) →L[ℝ] (Fin n → ℝ) :=
+    fderiv ℝ (newtonMap Yr F) mr
+  have hTle :
+      ‖T‖ ≤
+        (intervalMatrixBound
+          (preconditionedJacobian Y (intervalJacobian F X cfg)) : ℝ) := by
+    exact newtonMap_fderiv_norm_le_const Y F hsupp X mr hm cfg
+  have hTlt : ‖T‖ < 1 := lt_of_le_of_lt hTle (by exact_mod_cast hq)
+  let S : (Fin n → ℝ) →L[ℝ] (Fin n → ℝ) := 1 - T
+  have hunitS : IsUnit S := by
+    dsimp [S]
+    exact isUnit_one_sub_of_norm_lt_one hTlt
+  have hbijS : Function.Bijective ⇑S :=
+    (ContinuousLinearMap.isUnit_iff_bijective).mp hunitS
+  have hF : HasFDerivAt (systemEval F) J mr :=
+    (systemEval_differentiable_const F hsupp).differentiableAt.hasFDerivAt
+  have hAJ : HasFDerivAt (fun x => A (systemEval F x)) (A.comp J) mr :=
+    A.hasFDerivAt.comp mr hF
+  have hg : HasFDerivAt (newtonMap Yr F)
+      (ContinuousLinearMap.id ℝ _ - A.comp J) mr := by
+    change HasFDerivAt (fun z => z - A (systemEval F z)) _ mr
+    exact (hasFDerivAt_id mr).sub hAJ
+  have hT :
+      T = ContinuousLinearMap.id ℝ _ - A.comp J := hg.fderiv
+  have hS : S = A.comp J := by
+    dsimp [S]
+    rw [hT, ContinuousLinearMap.one_def]
+    abel
+  have hsurjComp : Function.Surjective ⇑(A.comp J) := by
+    rw [← hS]
+    exact hbijS.2
+  have hsurjA : Function.Surjective ⇑A := by
+    intro y
+    obtain ⟨x, hx⟩ := hsurjComp y
+    refine ⟨J x, ?_⟩
+    simpa only [ContinuousLinearMap.comp_apply] using hx
+  have hsurjLin : Function.Surjective ⇑A.toLinearMap := hsurjA
+  have hinjLin : Function.Injective ⇑A.toLinearMap :=
+    (LinearMap.injective_iff_surjective (f := A.toLinearMap)).2 hsurjLin
+  intro x y hxy
+  apply hinjLin
+  change (Y.map fun r => (r : ℝ)).mulVec x =
+    (Y.map fun r => (r : ℝ)).mulVec y
+  exact hxy
+
+theorem fixedPoint_iff_systemZero_of_injective {n : Nat}
+    (F : Fin n → Expr) (Y : Matrix (Fin n) (Fin n) ℚ)
+    (hinj : Function.Injective
+      (fun v : Fin n → ℝ => (Y.map fun r => (r : ℝ)).mulVec v))
+    (x : Fin n → ℝ) :
+    newtonMap (Y.map fun r => (r : ℝ)) F x = x ↔ SystemZero F x := by
+  let Yr : Matrix (Fin n) (Fin n) ℝ := Y.map fun r => (r : ℝ)
+  constructor
+  · intro hfix
+    have hmul : Yr.mulVec (systemEval F x) = 0 := by
+      funext i
+      have hi := congrFun hfix i
+      change x i - Yr.mulVec (systemEval F x) i = x i at hi
+      exact sub_eq_self.mp hi
+    have hz : systemEval F x = 0 := hinj (by simpa [Yr] using hmul)
+    intro i
+    exact congrFun hz i
+  · intro hz
+    have hs : systemEval F x = 0 := by
+      funext i
+      exact hz i
+    funext i
+    simp [newtonMap, hs]
+
+theorem uniqueSystemZero_of_certified_contraction {n : Nat}
+    (F : Fin n → Expr) (hsupp : ∀ i, ADConstSupported (F i))
+    (X : Fin n → IntervalRat) (m : Fin n → ℚ)
+    (hm : FinBoxMem (fun i => (m i : ℝ)) X)
+    (Y : Matrix (Fin n) (Fin n) ℚ) (cfg : EvalConfig)
+    (q : ℚ) (hq0 : 0 ≤ q) (hq1 : q < 1)
+    (hbound :
+      intervalMatrixBound
+        (preconditionedJacobian Y (intervalJacobian F X cfg)) ≤ q)
+    (hencl : ∀ i,
+      intervalStrictInside (imageEnclosureWithQ F X m Y cfg q i) (X i) = true) :
+    ∃! x, FinBoxMem x X ∧ SystemZero F x := by
+  have hmap := mapsTo_of_bound_and_enclosure
+    F hsupp X m hm Y cfg q hq0 hbound hencl
+  have hactualLt :
+      intervalMatrixBound
+        (preconditionedJacobian Y (intervalJacobian F X cfg)) < 1 :=
+    lt_of_le_of_lt hbound hq1
+  have hinj := preconditioner_injective_of_bound_lt_one
+    F hsupp X m hm Y cfg hactualLt
+  have hfixed := contraction_unique_fixedPoint_in_finBox X
+    (fun i => (m i : ℝ)) hm
+    (newtonMap (Y.map fun r => (r : ℝ)) F)
+    hmap (q : ℝ) (by exact_mod_cast hq0) (by exact_mod_cast hq1)
+    (fun x _ =>
+      (newtonMap_differentiable_const (Y.map fun r => (r : ℝ)) F hsupp).differentiableAt)
+    (fun x hx => by
+      calc
+        ‖fderiv ℝ (newtonMap (Y.map fun r => (r : ℝ)) F) x‖
+            ≤ (intervalMatrixBound
+                (preconditionedJacobian Y (intervalJacobian F X cfg)) : ℝ) :=
+          newtonMap_fderiv_norm_le_const Y F hsupp X x hx cfg
+        _ ≤ (q : ℝ) := by exact_mod_cast hbound)
+  refine ⟨hfixed.choose, ⟨hfixed.choose_spec.1.1,
+    (fixedPoint_iff_systemZero_of_injective F Y hinj hfixed.choose).mp
+      hfixed.choose_spec.1.2⟩, ?_⟩
+  intro y hy
+  exact hfixed.unique ⟨hy.1,
+    (fixedPoint_iff_systemZero_of_injective F Y hinj y).mpr hy.2⟩
+    hfixed.choose_spec.1
+
+/-!
+## Cached Newton-center values for large closed systems
+
+For the direct 22D Gerver certificate, recomputing all 22 transcendental
+point values inside every image-row proposition creates a very large kernel
+reduction.  The following variant accepts independently checked point-value
+enclosures.  It changes no mathematics: each cached interval is accompanied
+by a kernel proof that the corresponding real system value lies in it.
+-/
+
+/-- Endpoint inclusion between two rational intervals. -/
+def intervalContained (I J : IntervalRat) : Prop :=
+  J.lo ≤ I.lo ∧ I.hi ≤ J.hi
+
+theorem mem_of_intervalContained {x : ℝ} {I J : IntervalRat}
+    (hx : x ∈ I) (hIJ : intervalContained I J) : x ∈ J := by
+  change J.lo ≤ I.lo ∧ I.hi ≤ J.hi at hIJ
+  rw [IntervalRat.mem_def] at hx ⊢
+  have hlo : (J.lo : ℝ) ≤ (I.lo : ℝ) := by
+    exact_mod_cast hIJ.1
+  have hhi : (I.hi : ℝ) ≤ (J.hi : ℝ) := by
+    exact_mod_cast hIJ.2
+  exact ⟨hlo.trans hx.1, hx.2.trans hhi⟩
+
+/-- Newton-center enclosure from externally supplied, proof-carrying point
+value intervals. -/
+def kernelNewtonCenterIntervalWithValues {n : Nat}
+    (m : Fin n → ℚ) (Y : Matrix (Fin n) (Fin n) ℚ)
+    (V : Fin n → IntervalRat) : Fin n → IntervalRat :=
+  fun i =>
+    IntervalRat.sub (IntervalRat.singleton (m i))
+      (kernelIntervalRatMatVec Y V i)
+
+theorem newtonMap_center_mem_kernel_values {n : Nat}
+    (F : Fin n → Expr) (m : Fin n → ℚ)
+    (Y : Matrix (Fin n) (Fin n) ℚ)
+    (V : Fin n → IntervalRat)
+    (hV : ∀ j, systemEval F (fun k => (m k : ℝ)) j ∈ V j)
+    (i : Fin n) :
+    newtonMap (Y.map fun q => (q : ℝ)) F
+        (fun j => (m j : ℝ)) i ∈
+      kernelNewtonCenterIntervalWithValues m Y V i := by
+  apply IntervalRat.mem_sub
+  · exact IntervalRat.mem_singleton (m i)
+  · simp only [Matrix.mulVec, dotProduct, kernelIntervalRatMatVec]
+    exact mem_kernelIntervalFinSum n
+      (fun j => (Y i j : ℝ) * systemEval F (fun k => (m k : ℝ)) j)
+      (fun j => IntervalRat.scale (Y i j) (V j))
+      (fun j => IntervalRat.mem_scale (Y i j) (hV j))
+
+/-- Self-map enclosure built from independently certified center values. -/
+def imageEnclosureWithValuesQ {n : Nat}
+    (X : Fin n → IntervalRat) (m : Fin n → ℚ)
+    (Y : Matrix (Fin n) (Fin n) ℚ) (V : Fin n → IntervalRat)
+    (q : ℚ) : Fin n → IntervalRat :=
+  fun i => IntervalRat.add (kernelNewtonCenterIntervalWithValues m Y V i)
+    (symmetricInterval (q * boxRadius X m))
+
+theorem mapsTo_of_bound_and_values_enclosure {n : Nat}
+    (F : Fin n → Expr) (hsupp : ∀ i, ADConstSupported (F i))
+    (X : Fin n → IntervalRat) (m : Fin n → ℚ)
+    (hm : FinBoxMem (fun i => (m i : ℝ)) X)
+    (Y : Matrix (Fin n) (Fin n) ℚ) (cfg : EvalConfig)
+    (q : ℚ) (hq0 : 0 ≤ q)
+    (hbound :
+      intervalMatrixBound
+        (preconditionedJacobian Y (intervalJacobian F X cfg)) ≤ q)
+    (V : Fin n → IntervalRat)
+    (hV : ∀ j, systemEval F (fun k => (m k : ℝ)) j ∈ V j)
+    (hencl : ∀ i,
+      intervalStrictInside (imageEnclosureWithValuesQ X m Y V q i) (X i) = true) :
+    Set.MapsTo (newtonMap (Y.map fun r => (r : ℝ)) F)
+      (finBoxSet X) (finBoxSet X) := by
+  intro x hx i
+  let mr : Fin n → ℝ := fun i => (m i : ℝ)
+  have hdiff :
+      ‖newtonMap (Y.map fun r => (r : ℝ)) F x -
+          newtonMap (Y.map fun r => (r : ℝ)) F mr‖
+        ≤ (q : ℝ) * ‖x - mr‖ := by
+    apply (finBoxSet_convex X).norm_image_sub_le_of_norm_fderiv_le (𝕜 := ℝ)
+    · intro z hz
+      exact (newtonMap_differentiable_const (Y.map fun r => (r : ℝ)) F hsupp).differentiableAt
+    · intro z hz
+      calc
+        ‖fderiv ℝ (newtonMap (Y.map fun r => (r : ℝ)) F) z‖
+            ≤ (intervalMatrixBound
+                (preconditionedJacobian Y (intervalJacobian F X cfg)) : ℝ) :=
+          newtonMap_fderiv_norm_le_const Y F hsupp X z hz cfg
+        _ ≤ (q : ℝ) := by exact_mod_cast hbound
+    · exact hm
+    · exact hx
+  have hr0 : 0 ≤ boxRadius X m := boxRadius_nonneg X m
+  have hnorm :
+      ‖newtonMap (Y.map fun r => (r : ℝ)) F x -
+          newtonMap (Y.map fun r => (r : ℝ)) F mr‖
+        ≤ ((q * boxRadius X m : ℚ) : ℝ) := by
+    calc
+      _ ≤ (q : ℝ) * ‖x - mr‖ := hdiff
+      _ ≤ (q : ℝ) * (boxRadius X m : ℝ) :=
+        mul_le_mul_of_nonneg_left (norm_sub_center_le_boxRadius hx)
+          (by exact_mod_cast hq0)
+      _ = _ := by push_cast; rfl
+  have hcoord :
+      |(newtonMap (Y.map fun r => (r : ℝ)) F x -
+          newtonMap (Y.map fun r => (r : ℝ)) F mr) i|
+        ≤ ((q * boxRadius X m : ℚ) : ℝ) :=
+    (norm_le_pi_norm _ i).trans hnorm
+  have herr :
+      (newtonMap (Y.map fun r => (r : ℝ)) F x -
+          newtonMap (Y.map fun r => (r : ℝ)) F mr) i ∈
+        symmetricInterval (q * boxRadius X m) := by
+    rw [abs_le] at hcoord
+    simpa [symmetricInterval, IntervalRat.mem_def,
+      abs_of_nonneg (mul_nonneg hq0 hr0)] using hcoord
+  have hcenter := newtonMap_center_mem_kernel_values F m Y V hV i
+  have himage :
+      newtonMap (Y.map fun r => (r : ℝ)) F x i ∈
+        imageEnclosureWithValuesQ X m Y V q i := by
+    have hadd := IntervalRat.mem_add hcenter herr
+    simpa [imageEnclosureWithValuesQ, mr, Pi.sub_apply] using hadd
+  exact intervalStrictInside_sound (hencl i) _ himage
+
+theorem uniqueSystemZero_of_certified_contraction_values {n : Nat}
+    (F : Fin n → Expr) (hsupp : ∀ i, ADConstSupported (F i))
+    (X : Fin n → IntervalRat) (m : Fin n → ℚ)
+    (hm : FinBoxMem (fun i => (m i : ℝ)) X)
+    (Y : Matrix (Fin n) (Fin n) ℚ) (cfg : EvalConfig)
+    (q : ℚ) (hq0 : 0 ≤ q) (hq1 : q < 1)
+    (hbound :
+      intervalMatrixBound
+        (preconditionedJacobian Y (intervalJacobian F X cfg)) ≤ q)
+    (V : Fin n → IntervalRat)
+    (hV : ∀ j, systemEval F (fun k => (m k : ℝ)) j ∈ V j)
+    (hencl : ∀ i,
+      intervalStrictInside (imageEnclosureWithValuesQ X m Y V q i) (X i) = true) :
+    ∃! x, FinBoxMem x X ∧ SystemZero F x := by
+  have hmap := mapsTo_of_bound_and_values_enclosure
+    F hsupp X m hm Y cfg q hq0 hbound V hV hencl
+  have hactualLt :
+      intervalMatrixBound
+        (preconditionedJacobian Y (intervalJacobian F X cfg)) < 1 :=
+    lt_of_le_of_lt hbound hq1
+  have hinj := preconditioner_injective_of_bound_lt_one
+    F hsupp X m hm Y cfg hactualLt
+  have hfixed := contraction_unique_fixedPoint_in_finBox X
+    (fun i => (m i : ℝ)) hm
+    (newtonMap (Y.map fun r => (r : ℝ)) F)
+    hmap (q : ℝ) (by exact_mod_cast hq0) (by exact_mod_cast hq1)
+    (fun x _ =>
+      (newtonMap_differentiable_const (Y.map fun r => (r : ℝ)) F hsupp).differentiableAt)
+    (fun x hx => by
+      calc
+        ‖fderiv ℝ (newtonMap (Y.map fun r => (r : ℝ)) F) x‖
+            ≤ (intervalMatrixBound
+                (preconditionedJacobian Y (intervalJacobian F X cfg)) : ℝ) :=
+          newtonMap_fderiv_norm_le_const Y F hsupp X x hx cfg
+        _ ≤ (q : ℝ) := by exact_mod_cast hbound)
+  refine ⟨hfixed.choose, ⟨hfixed.choose_spec.1.1,
+    (fixedPoint_iff_systemZero_of_injective F Y hinj hfixed.choose).mp
+      hfixed.choose_spec.1.2⟩, ?_⟩
+  intro y hy
+  exact hfixed.unique ⟨hy.1,
+    (fixedPoint_iff_systemZero_of_injective F Y hinj y).mpr hy.2⟩
+    hfixed.choose_spec.1
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# LeanCert Gerver numerical core
+
+Only shared definitions and lightweight list lemmas live here.
+The expensive 22D kernel checks are split into one Lake module per row.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+
+/-- The default evaluation configuration used for the numerical certificates. -/
+def cfg : EvalConfig := {}
+
+/-- The interval enclosure of the normalized reduced Newton-map derivative. -/
+def reducedPJ : Matrix (Fin 4) (Fin 4) IntervalRat :=
+  preconditionedJacobian reducedY
+    (intervalJacobian reducedExpr (unitBox (n := 4)) cfg)
+
+/-- The interval enclosure of the normalized full Newton-map derivative. -/
+def fullPJ : Matrix (Fin 22) (Fin 22) IntervalRat :=
+  preconditionedJacobian fullY
+    (intervalJacobian fullExpr (unitBox (n := 22)) cfg)
+
+theorem qTarget_pos : (0 : ℚ) < qTarget := by norm_num [qTarget]
+theorem qTarget_nonneg : (0 : ℚ) ≤ qTarget := qTarget_pos.le
+theorem qTarget_lt_one : qTarget < (1 : ℚ) := by norm_num [qTarget]
+
+/-- Assemble a bound from independently kernel-checked rows without re-running AD. -/
+theorem foldl_max_lt {xs : List ℚ} {a q : ℚ}
+    (ha : a < q) (hxs : ∀ x ∈ xs, x < q) :
+    xs.foldl max a < q := by
+  induction xs generalizing a with
+  | nil => simpa using ha
+  | cons x xs ih =>
+      simp only [List.foldl_cons]
+      apply ih
+      · exact max_lt ha (hxs x (by simp))
+      · intro y hy
+        exact hxs y (by simp [hy])
+
+theorem matrixBound_lt_of_rows {n : Nat}
+    (A : Matrix (Fin n) (Fin n) IntervalRat) (q : ℚ)
+    (hq0 : 0 < q)
+    (hrows : ∀ i : Fin n, intervalMatrixRowBound A i < q) :
+    intervalMatrixBound A < q := by
+  unfold intervalMatrixBound
+  apply foldl_max_lt hq0
+  intro x hx
+  rcases List.mem_ofFn.mp hx with ⟨i, rfl⟩
+  exact hrows i
+
+/-! ## Proof-carrying cache for the 22 Newton-center residuals
+
+These deliberately simple rational intervals are much wider than the actual
+point residuals, but still narrow enough after the scaled preconditioner to
+leave a large self-map margin.  Each coordinate is checked independently in a
+separate module before it is used by the final contraction theorem.
+-/
+/-- Cached interval evaluations of the full system at the normalized center. -/
+def fullPointCacheList : List IntervalRat := [
+  ⟨q (-1) 100000000000000000000000000000, q 1 100000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 100000000000000000000000000000, q 1 100000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q 0 1, q 3 100000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 50000000000000000000000000000, q 1 100000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 100000000000000000000000000000, q 1 50000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q 0 1, q 1 50000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 100000000000000000000000000000, q 1 100000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 100000000000000000000000000000, q 1 100000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 100000000000000000000000000000, q 1 50000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 100000000000000000000000000000, q 1 50000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 50000000000000000000000000000, q 1 100000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q 1 100000000000000000000000000000, q 1 25000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q 0 1, q 3 100000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 50000000000000000000000000000, q 1 100000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 100000000000000000000000000000, q 1 50000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 100000000000000000000000000000, q 1 50000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 50000000000000000000000000000, q 1 100000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 50000000000000000000000000000, q 1 100000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 25000000000000000000000, q 1 20000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 20000000000000000000000, q 1 25000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-1) 100000000000000000000000000000, q 3 100000000000000000000000000000, by norm_num [q]⟩,
+  ⟨q (-3) 100000000000000000000000000000, q 1 100000000000000000000000000000, by norm_num [q]⟩
+]
+
+/-- Select a cached center evaluation of a full-system equation. -/
+def fullPointCache (i : Fin 22) : IntervalRat :=
+  fullPointCacheList.getD i.1 (IntervalRat.singleton 0)
+
+/-- The full Newton image enclosure computed with cached center evaluations. -/
+def fullImageCachedQ (i : Fin 22) : IntervalRat :=
+  imageEnclosureWithValuesQ (unitBox (n := 22)) (zeroCenter (n := 22))
+    fullY fullPointCache qTarget i
+
+/-- The reduced Newton image enclosure on the normalized unit box. -/
+def reducedImageQ (i : Fin 4) : IntervalRat :=
+  imageEnclosureWithQ reducedExpr (unitBox (n := 4))
+    (zeroCenter (n := 4)) reducedY cfg qTarget i
+
+theorem reduced_center_mem :
+    FinBoxMem (fun i : Fin 4 => ((zeroCenter (n := 4) i : ℚ) : ℝ))
+      (unitBox (n := 4)) := by
+  intro i
+  simp [zeroCenter, unitBox, IntervalRat.mem_def]
+
+theorem full_center_mem :
+    FinBoxMem (fun i : Fin 22 => ((zeroCenter (n := 22) i : ℚ) : ℝ))
+      (unitBox (n := 22)) := by
+  intro i
+  simp [zeroCenter, unitBox, IntervalRat.mem_def]
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle005.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle005.lean
new file mode 100644
index 0000000000..cde177c757
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle005.lean
@@ -0,0 +1,10349 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle002
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle004
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle003
+
+public import Mathlib.Data.Nat.Find
+public import Mathlib.Tactic
+public import Mathlib.Tactic.FieldSimp
+public import Mathlib.Tactic.FinCases
+public import Mathlib.Tactic.Ring
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.Foundation.Batch003`.
+* `GerverSofa.KernelOnly.Numerics.Batch001`.
+* `GerverSofa.KernelOnly.Numerics.Batch002`.
+* `GerverSofa.KernelOnly.Numerics.Batch003`.
+* `GerverSofa.KernelOnly.Numerics.Batch004`.
+* `GerverSofa.KernelOnly.Numerics.Batch005`.
+* `GerverSofa.KernelOnly.Foundation.Batch004`.
+* `GerverSofa.KernelOnly.PartE.Semantics.Batch001`.
+* `GerverSofa.KernelOnly.PartB.Semantics.Batch003`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.JacobianCache`.
+* `KernelOnly.LeanCertGerverNumericsReduced`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Shared exact interval Jacobian for the 22-dimensional certificate
+
+The kernel verifies every matrix entry against the original automatic differentiation
+evaluator. Matrix row bounds can then reuse the certified entries.
+-/
+
+@[expose] public section
+namespace GerverSofa.PartALeanCert
+open LeanCert.Core LeanCert.Engine
+
+/-- Reconstruct a natural number from base-10³⁵ chunks for decimal elaboration. -/
+def naturalFromChunks (chunks : List ℕ) : ℕ :=
+  chunks.foldl (fun value digit => value * 10 ^ 35 + digit) 0
+
+/-- Exact cached Jacobian interval number 0, shared by equal entries. -/
+def jacobianEntry000 : IntervalRat :=
+  ⟨(q
+      0
+      1),
+    (q
+      0
+      1), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 1, shared by equal entries. -/
+def jacobianEntry001 : IntervalRat :=
+  ⟨(q
+      (-(1 : ℤ))
+      10000000000000000000000),
+    (q
+      (-(1 : ℤ))
+      10000000000000000000000), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 2, shared by equal entries. -/
+def jacobianEntry002 : IntervalRat :=
+  ⟨(q
+      1
+      10000000000000000000000),
+    (q
+      1
+      10000000000000000000000), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 3, shared by equal entries. -/
+def jacobianEntry003 : IntervalRat :=
+  ⟨(q
+      (-(31415926535897933 : ℤ))
+      (naturalFromChunks [
+      2000,
+      0])),
+    (q
+      (-(7853981633974483 : ℤ))
+      (naturalFromChunks [
+      500,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 4, shared by equal entries. -/
+def jacobianEntry004 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      7302291171534437540739059211270,
+      62603078071522934985237770352870131,
+      65836645916281358406641371202899231,
+      53726555807649881687955070859646221,
+      28324204846900711954852999095288560,
+      74723762903619705246321908744465090,
+      64826457649890704414497430491211281,
+      20574610373540063477933619663935914,
+      80242971244164644799995665107793557,
+      6086704947498027804967631838853576,
+      62481503746522252138415644597421765,
+      32033537775236312858538976911582807,
+      40311379023805556405510702050965361,
+      77540705318547318829968145740509773,
+      17367742985344688946603311510163578,
+      74909356280402076830989205440282523,
+      6995813836602201925464533090592071])
+      (naturalFromChunks [
+      732476473344000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      507103553578780384773563442541417,
+      94698791213826119093186876572859864,
+      15548305859588655267023070389345830,
+      33159838526908442543211837120728759,
+      3865445100998990097784146416535036,
+      64642752897093102844923553384615171,
+      68763366624136086130719360401948367,
+      619291325796097507385113031956620,
+      19067116153975375531560230982057442,
+      92910533440132892667508087960607232,
+      10839074800776283949308197382802296,
+      66457159709219595865536845126743026,
+      34469108831014037616961089980351793,
+      26133430441883788061713970279416234,
+      21778463327947902555078192692702636,
+      91002976914649816729672745776196632,
+      38062025338495397123749885159021881])
+      (naturalFromChunks [
+      50866421760000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 5, shared by equal entries. -/
+def jacobianEntry005 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      26155443047017164924,
+      51948774198841577898573116771393516,
+      40492442572889628469879882033256716,
+      74655549198190939008143206760585064,
+      81437387381446629703018736147636028,
+      89476350149712987515166932908391254,
+      86210131010838527615254636680772364,
+      14418167525844757550395013766238555,
+      58785462379224720453071008471370354,
+      20265150396760286139513154614339561,
+      67644720387271822398769553018517054,
+      16756002041840783466098940078636727,
+      15776017768574755885160693218947784,
+      49406689895231013099621174747218230,
+      18705223576009121622005064353559890,
+      38638014191096243625012399666628208,
+      73505208703653573295616036495840233,
+      89953867437106799895512969220274001])
+      (naturalFromChunks [
+      33414990,
+      73277460480000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      26155443047017164924,
+      53109182283427083433072861240124059,
+      23126507692524839291624930472864879,
+      72341749128136688648611150016306326,
+      44139489352178350868323375137797958,
+      65590161521538195435503240261540309,
+      35051120974370767103797318127249504,
+      7323238045333273918434772577727849,
+      8062969635695436059523728331659876,
+      9411507113629570449186098067951242,
+      82358573374363972295006778826115466,
+      92254742191300297818748796835597741,
+      83167490748934545082649491651932510,
+      38857585646488763229576985079482408,
+      4442337792553033022243475041749642,
+      61368240185514979758553782063506707,
+      60389259661333090014056246622771847,
+      38781140546001429783512969220274001])
+      (naturalFromChunks [
+      33414990,
+      73277460480000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 6, shared by equal entries. -/
+def jacobianEntry006 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      12804383003350103328,
+      63307894022272997926538740311484949,
+      25196721501196286822405177768269239,
+      82390931751685070431278135581129012,
+      57276281642207501766192210160813629,
+      43155082000316152942557215709730655,
+      60260303521391232922052999203904374,
+      76502717447660000145505841759109266,
+      4612015658335133293806642573132171,
+      12428298655865671194597563325604001]) : ℤ))
+      (naturalFromChunks [
+      16349921,
+      28000000000000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      29100870462159325746,
+      88690408519371071042552548825676558,
+      7151971153710758093004047524014875,
+      57778246443332979957500192691404948,
+      13830430762981650893880915068665480,
+      44366481210286091359786109795397712,
+      35368953204907053050532703318844226,
+      30845169156305955909356081423954341,
+      37163402169222139733626194195214059,
+      59269530634977972716063063008513591]) : ℤ))
+      (naturalFromChunks [
+      37158912,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 7, shared by equal entries. -/
+def jacobianEntry007 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      8168687708195300209523685201,
+      51466733258999157794562262879739361,
+      83044118018001634139480470724425976,
+      3124995890782157382952812227479795,
+      79691021612686864121793062218595028,
+      61848622605893783127624967686605586,
+      37563139673860619382511127033019145,
+      95717533889869613908008887710180867,
+      61214949413130194370370491393605751]) : ℤ))
+      (naturalFromChunks [
+      817496064000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      8168687708195300209523548508,
+      262084693744252290618237845077936,
+      59630898042252737007119539862573770,
+      85670741929566438838418210096502502,
+      68746015112854582979695067224582614,
+      601569332966945250398739051546917,
+      36111245854814052753836553329544049,
+      75810951854582975130593161890447049,
+      87296345979597779517629508606394249]) : ℤ))
+      (naturalFromChunks [
+      817496064000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 8, shared by equal entries. -/
+def jacobianEntry008 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      4351836038171981263012981260,
+      76017600016021480743208388315632944,
+      11059251508958509065408380879317143,
+      17207561234128667472718121598587959,
+      77890635486980056142496917042239998,
+      68914016068855974869102476150899260,
+      61291516587791258017470197705861923,
+      75450006095166098923841042297078292,
+      98737604596196464211229335895301237,
+      22932866990655735600974232255961840,
+      16153008731703206338486341650497171,
+      98729543000784849015390340426058108,
+      68171485621218593874174140251422992,
+      4958239564296464343197683438164893,
+      52788849768402637056113713231700226,
+      45073579579069554108872126778739708,
+      99045958204655518403989095572552812,
+      45924517960050625410242642699422681]) : ℤ))
+      (naturalFromChunks [
+      3341499,
+      7327746048000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      4003588609101864096620301357,
+      54033188319852210103648514686085193,
+      24581453046159760205867563659589489,
+      96451152674805880241114754614814231,
+      74693101752113895664848438931085207,
+      26771314933477010177981076459428154,
+      21316221310561555250942325250311776,
+      33148371225775893519263090604991158,
+      26172684688928662912950639001936885,
+      35478157868322140978363434983717300,
+      16559283188850556724662197227110021,
+      45126752726317960859820799533549302,
+      52417900864638374263706646541159920,
+      72699857435174057099230395775600834,
+      65327674957856151672412107355643413,
+      2342514632318889702282044195493814,
+      39788224412818939694489250424650159,
+      7942965715558610045757357300577319])
+      (naturalFromChunks [
+      3341499,
+      7327746048000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 9, shared by equal entries. -/
+def jacobianEntry009 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      507103553578780384773563442541417,
+      94698791213826119093186876572859864,
+      15548305859588655267023070389345830,
+      33159838526908442543211837120728759,
+      3865445100998990097784146416535036,
+      64642752897093102844923553384615171,
+      68763366624136086130719360401948367,
+      619291325796097507385113031956620,
+      19067116153975375531560230982057442,
+      92910533440132892667508087960607232,
+      10839074800776283949308197382802296,
+      66457159709219595865536845126743026,
+      34469108831014037616961089980351793,
+      26133430441883788061713970279416234,
+      21778463327947902555078192692702636,
+      91002976914649816729672745776196632,
+      38062025338495397123749885159021881]) : ℤ))
+      (naturalFromChunks [
+      50866421760000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      7302291171534437540739059211270,
+      62603078071522934985237770352870131,
+      65836645916281358406641371202899231,
+      53726555807649881687955070859646221,
+      28324204846900711954852999095288560,
+      74723762903619705246321908744465090,
+      64826457649890704414497430491211281,
+      20574610373540063477933619663935914,
+      80242971244164644799995665107793557,
+      6086704947498027804967631838853576,
+      62481503746522252138415644597421765,
+      32033537775236312858538976911582807,
+      40311379023805556405510702050965361,
+      77540705318547318829968145740509773,
+      17367742985344688946603311510163578,
+      74909356280402076830989205440282523,
+      6995813836602201925464533090592071]) : ℤ))
+      (naturalFromChunks [
+      732476473344000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 10, shared by equal entries. -/
+def jacobianEntry010 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      163122868986236350200,
+      13212104989835737702967996320253347,
+      13343446698691791948896941422365475,
+      8096241524554184648945198975238842,
+      55534909982015434708979369629464764,
+      76573441012851358138234190261693558,
+      79427907664974237935799314423674117,
+      4953829431951288217980965287083443,
+      95618746533545362476016957876485554,
+      44154543264308318693402436674395999])
+      (naturalFromChunks [
+      16349921,
+      28000000000000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      370733793150537159545,
+      76127734659590509217418760566762798,
+      83500695503852348111812708082675358,
+      8167915944020144671694939034466768,
+      70108246297247634239713857794350318,
+      48346428376057315228605372168926753,
+      77052008275409499730746022341858967,
+      30494332824458242394752971556556394,
+      26480558256939393278929373508455480,
+      5642901791267147283936936991486409])
+      (naturalFromChunks [
+      37158912,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 11, shared by equal entries. -/
+def jacobianEntry011 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      320191491967265573299622687,
+      53013595983521471973089342170075396,
+      64210432087316798696539851495592012,
+      18605199481029466524389064552432920,
+      91322680546566024998863058319218327,
+      25520274592961360039367904724384559,
+      37639720591938304020705346465799237,
+      90794994241578641575695128019032725,
+      32079403795705900482370491393605751]) : ℤ))
+      (naturalFromChunks [
+      817496064000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      320191491967265573299485989,
+      93256458762401047215486481839485294,
+      66387420606013932458366418942019752,
+      75158597834282975620530000652601944,
+      81258519302743687820429638412903277,
+      26525331629470118664578696517478146,
+      37125905076490006801321755179098797,
+      2827449942997481909038311980967274,
+      67920596204294099517629508606394249]) : ℤ))
+      (naturalFromChunks [
+      817496064000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 12, shared by equal entries. -/
+def jacobianEntry012 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      3420031750228409906252618489,
+      90931929615262590050819024036832147,
+      21506571044326610764292459863905479,
+      74538770850235934189940647549294463,
+      29621768507857295177364391873262079,
+      10933450678245914120167591207427285,
+      49181137297721373459800467154721233,
+      89787178661856429371317394337935251,
+      65843424654096570705147958715768220,
+      63615469750503758464529825528864113,
+      66409373661200684292384252948301388,
+      32939891107539306601569814693951796,
+      74322536731541153535575038767603862,
+      44440620881220284070119322114887784,
+      91668814424924291631841697402988320,
+      32219074607272143865331607160174037,
+      50460727063802276122996533532669802,
+      13969508541598381181757357300577319]) : ℤ))
+      (naturalFromChunks [
+      3341499,
+      7327746048000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      3332542732427061890511872752,
+      47000681977616353407805735599142210,
+      94344814107689418558967999766690207,
+      42694687435724596552986916073706462,
+      91145540018505849404450452414800312,
+      62191047578086169648923996108922988,
+      90916961106701487614400715766740886,
+      99743074794088195900952245813912149,
+      572460563594575054478438406055569,
+      63828687932987171491220304569694473,
+      9634735810764848166091054799398721,
+      66738387738006625412848379545388946,
+      876281196732513526586105195911404,
+      43530939696670331083413526607388632,
+      52575071804007568732157139880458185,
+      57448568521393973246195227052776783,
+      27582531650396302574779440591735431,
+      78361975134010854274242642699422681])
+      (naturalFromChunks [
+      3341499,
+      7327746048000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 13, shared by equal entries. -/
+def jacobianEntry013 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      26155443047017164924,
+      53109182283427083433072861240124059,
+      23126507692524839291624930472864879,
+      72341749128136688648611150016306326,
+      44139489352178350868323375137797958,
+      65590161521538195435503240261540309,
+      35051120974370767103797318127249504,
+      7323238045333273918434772577727849,
+      8062969635695436059523728331659876,
+      9411507113629570449186098067951242,
+      82358573374363972295006778826115466,
+      92254742191300297818748796835597741,
+      83167490748934545082649491651932510,
+      38857585646488763229576985079482408,
+      4442337792553033022243475041749642,
+      61368240185514979758553782063506707,
+      60389259661333090014056246622771847,
+      38781140546001429783512969220274001]) : ℤ))
+      (naturalFromChunks [
+      16707495,
+      36638730240000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      26155443047017164924,
+      51948774198841577898573116771393516,
+      40492442572889628469879882033256716,
+      74655549198190939008143206760585064,
+      81437387381446629703018736147636028,
+      89476350149712987515166932908391254,
+      86210131010838527615254636680772364,
+      14418167525844757550395013766238555,
+      58785462379224720453071008471370354,
+      20265150396760286139513154614339561,
+      67644720387271822398769553018517054,
+      16756002041840783466098940078636727,
+      15776017768574755885160693218947784,
+      49406689895231013099621174747218230,
+      18705223576009121622005064353559890,
+      38638014191096243625012399666628208,
+      73505208703653573295616036495840233,
+      89953867437106799895512969220274001]) : ℤ))
+      (naturalFromChunks [
+      16707495,
+      36638730240000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 14, shared by equal entries. -/
+def jacobianEntry014 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      7302291171534437540739059211270,
+      62603078071522934985237770352870131,
+      65836645916281358406641371202899231,
+      53726555807649881687955070859646221,
+      28324204846900711954852999095288560,
+      74723762903619705246321908744465090,
+      64826457649890704414497430491211281,
+      20574610373540063477933619663935914,
+      80242971244164644799995665107793557,
+      6086704947498027804967631838853576,
+      62481503746522252138415644597421765,
+      32033537775236312858538976911582807,
+      40311379023805556405510702050965361,
+      77540705318547318829968145740509773,
+      17367742985344688946603311510163578,
+      74909356280402076830989205440282523,
+      6995813836602201925464533090592071])
+      (naturalFromChunks [
+      366238236672000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      507103553578780384773563442541417,
+      94698791213826119093186876572859864,
+      15548305859588655267023070389345830,
+      33159838526908442543211837120728759,
+      3865445100998990097784146416535036,
+      64642752897093102844923553384615171,
+      68763366624136086130719360401948367,
+      619291325796097507385113031956620,
+      19067116153975375531560230982057442,
+      92910533440132892667508087960607232,
+      10839074800776283949308197382802296,
+      66457159709219595865536845126743026,
+      34469108831014037616961089980351793,
+      26133430441883788061713970279416234,
+      21778463327947902555078192692702636,
+      91002976914649816729672745776196632,
+      38062025338495397123749885159021881])
+      (naturalFromChunks [
+      25433210880000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 15, shared by equal entries. -/
+def jacobianEntry015 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      8162415578753558859765,
+      21665156988862230891185263515522227,
+      5475742064434838230756858649664301,
+      11145876946719536779723020855540569,
+      45180499115234782980284496634384612,
+      61562143060941770461550657670678134,
+      77851409811074333968515937016012156,
+      86394958927807611277089641126564192,
+      329521433808032633320293778255977,
+      42359469703938620123306306906350499]) : ℤ))
+      (naturalFromChunks [
+      408748032,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      108832207716714118130,
+      20103040331692477192617680710872701,
+      34499773297103250931317227188252090,
+      77747927745989674413424658581201014,
+      82846638452311912322023676407788703,
+      8579556341627581599191031756556113,
+      27507247462399718344026789833468394,
+      54680642982723441769880155966315560,
+      58837615426490760689324182743025242,
+      18981013754769439564467478891465333]) : ℤ))
+      (naturalFromChunks [
+      5449973,
+      76000000000000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 16, shared by equal entries. -/
+def jacobianEntry016 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      320191491967265573299485989,
+      93256458762401047215486481839485294,
+      66387420606013932458366418942019752,
+      75158597834282975620530000652601944,
+      81258519302743687820429638412903277,
+      26525331629470118664578696517478146,
+      37125905076490006801321755179098797,
+      2827449942997481909038311980967274,
+      67920596204294099517629508606394249])
+      (naturalFromChunks [
+      817496064000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      320191491967265573299622687,
+      53013595983521471973089342170075396,
+      64210432087316798696539851495592012,
+      18605199481029466524389064552432920,
+      91322680546566024998863058319218327,
+      25520274592961360039367904724384559,
+      37639720591938304020705346465799237,
+      90794994241578641575695128019032725,
+      32079403795705900482370491393605751])
+      (naturalFromChunks [
+      817496064000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 17, shared by equal entries. -/
+def jacobianEntry017 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      151634312,
+      99668462617431325434924986957764715,
+      38315082266817589430283894717677095,
+      91549475369686381133287033133555911,
+      75876109123654603111324496463071341,
+      61064827155573427181097189016633488,
+      85681524817888904508842338212217589,
+      72675186999137526039138098013588344,
+      5255150243958148600212313159803052,
+      34131427677854616396547911665043624,
+      47176259434611004442151332683134006,
+      4578465050974687057903727292714271,
+      34203900707994397580821218261750773,
+      79564513611835155373207464454168863,
+      72538969294172836248080372278010559,
+      15171635065118638502549526772181162,
+      80370610047525248386619024623513609,
+      82193934464217261030954945210498359,
+      59164853490080518634357239863562013]) : ℤ))
+      (naturalFromChunks [
+      4,
+      17687384159682560000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      151634312,
+      99668462617416978481107742412336288,
+      27061273687144305340763248099942418,
+      66472128684716957417589220526464924,
+      11716352479633381087922121005932875,
+      1659185385004219149455490139931359,
+      26671873583124798776596643903348241,
+      51898790226105682616793173713373460,
+      33007661002417084715966542914381025,
+      13652365332265279873898886620915624,
+      15150983906130322213181695343945582,
+      55791040746026127822866989331461882,
+      62778000991423073680992127673149718,
+      67423346041116510446078407330606718,
+      52870955179725239729519256588904336,
+      67033883199464819170875651586192127,
+      83155782680642200937115120313076282,
+      59476081743970139085199978805464650,
+      68034783235751598055931400136437987]) : ℤ))
+      (naturalFromChunks [
+      4,
+      17687384159682560000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 18, shared by equal entries. -/
+def jacobianEntry018 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      26155443047017164924,
+      51948774198841577898573116771393516,
+      40492442572889628469879882033256716,
+      74655549198190939008143206760585064,
+      81437387381446629703018736147636028,
+      89476350149712987515166932908391254,
+      86210131010838527615254636680772364,
+      14418167525844757550395013766238555,
+      58785462379224720453071008471370354,
+      20265150396760286139513154614339561,
+      67644720387271822398769553018517054,
+      16756002041840783466098940078636727,
+      15776017768574755885160693218947784,
+      49406689895231013099621174747218230,
+      18705223576009121622005064353559890,
+      38638014191096243625012399666628208,
+      73505208703653573295616036495840233,
+      89953867437106799895512969220274001])
+      (naturalFromChunks [
+      16707495,
+      36638730240000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      26155443047017164924,
+      53109182283427083433072861240124059,
+      23126507692524839291624930472864879,
+      72341749128136688648611150016306326,
+      44139489352178350868323375137797958,
+      65590161521538195435503240261540309,
+      35051120974370767103797318127249504,
+      7323238045333273918434772577727849,
+      8062969635695436059523728331659876,
+      9411507113629570449186098067951242,
+      82358573374363972295006778826115466,
+      92254742191300297818748796835597741,
+      83167490748934545082649491651932510,
+      38857585646488763229576985079482408,
+      4442337792553033022243475041749642,
+      61368240185514979758553782063506707,
+      60389259661333090014056246622771847,
+      38781140546001429783512969220274001])
+      (naturalFromChunks [
+      16707495,
+      36638730240000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 19, shared by equal entries. -/
+def jacobianEntry019 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      6402737614231804931,
+      54184423027514972532322726591090597,
+      55115838261801642856247050566355423,
+      24524985374596353350622488689636767,
+      87380707822887037565837570046024931,
+      93863912170240548267261472165696250,
+      3009807424728548253606359872670434,
+      69367064421186628325692891091003942,
+      14225822099974403069308217406847332,
+      99879933551955223731532521108534667]) : ℤ))
+      (naturalFromChunks [
+      5449973,
+      76000000000000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      480205321067385369865,
+      49893990341311162668601644825683058,
+      52936415884528642030010751947373102,
+      36548348455961696673989929870454755,
+      82231979022057811204533975970139491,
+      33195156575812819692706540038723094,
+      62965803816930121800863137168164078,
+      94236712133090486501657850182653745,
+      57161157821741639216937894962108958,
+      71677286984757699876693693093649501]) : ℤ))
+      (naturalFromChunks [
+      408748032,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 20, shared by equal entries. -/
+def jacobianEntry020 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      246109182533,
+      63689055212495349844348455603740526,
+      2459665425556941806640787160098397,
+      11270491871680120158217013599123496,
+      66445409117884011348990144753541055,
+      82506355901505301651215645152779516,
+      47134236083428643784023976624455135,
+      84574353764108143116719381874126648,
+      44893312590349192655134993659785375,
+      95003490014442919984607271514730456,
+      98104254324749327817987197614854755,
+      90973124714442734213821944183162994,
+      74410925451472791561088694529710817,
+      48831698317442489424799148900490262,
+      31813301634820340377618334876356607,
+      37586912956214654189485401270870441,
+      82561104235738048120696057098145611,
+      37593003366534348715354835966369263,
+      51581067049589316696246150566891409]) : ℤ))
+      (naturalFromChunks [
+      92819,
+      41870215168000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      2214982642802,
+      73201493225300966841097223910816439,
+      16463987998273083717738236625958625,
+      90442503703288354254191954688229478,
+      76037541601454949005935848779731582,
+      78709483599527540687207532308343764,
+      60923823608200002470591613505767552,
+      77822847198303311411748421153740031,
+      25427395199376513930956996404577040,
+      86315368278883883977338450914883035,
+      78411783013296250792276932159953039,
+      36088998744013857335500585055966756,
+      19639790244902352874810113946088499,
+      35948411286810057474015964561512623,
+      13189450925765156926809532464558295,
+      21149086210374540235685034274315927,
+      57812206231928416870078433837854017,
+      41763164161878274216848841975509772,
+      63668822760479365098269930296822681]) : ℤ))
+      (naturalFromChunks [
+      835374,
+      76831936512000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 21, shared by equal entries. -/
+def jacobianEntry021 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      37,
+      64964932916807993933992739614852565,
+      73425747625329018471695404071919474,
+      32428598921433773523276536986136716,
+      95804936725122105032338314478045544,
+      37839005332906669519148382444918613,
+      93910509049666464822504678557122067,
+      47563294676419806542477618269298479,
+      20057194249093442371341925670977569,
+      83735790938670159845954730089687134,
+      97739845371556534324558878348085397])
+      (naturalFromChunks [
+      324843750000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      112,
+      94894810802129538499495336548811395,
+      30627346704914725170313587692932357,
+      71856848395017812308204077540266185,
+      46110065952096461758306574901708328,
+      54964034329753530228877898686780806,
+      93612167582613954419042710857760045,
+      27573090730890605253666184909411466,
+      41466664660821619052535510154677044,
+      12164592087685865488315347374867414,
+      85111645614031567298658147323581761])
+      (naturalFromChunks [
+      974531250000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 22, shared by equal entries. -/
+def jacobianEntry022 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      50464715,
+      64732670758294711200924782925221777,
+      98385684290077839487301182380997546,
+      53610107900326691496911028225237884,
+      82456569428218072679478834100193025,
+      10063572333607529684946179543893525,
+      13470787246553804150266810892380951,
+      21872306604849333638776947219394501,
+      54380505433679613411774213979274707,
+      18467897289415753684541230752125211])
+      (naturalFromChunks [
+      649687500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      151394147,
+      8534174455720092134237275166603186,
+      16840640244550300415844583544502225,
+      12894405273196777971379918291978794,
+      4506348409732893778736073442501460,
+      54756683995003484397907923306619480,
+      27034357940592422468567779837099716,
+      93382309438500237515411019035358590,
+      70067448925761052854406226725970958,
+      16662408131752738946376307743624367])
+      (naturalFromChunks [
+      1949062500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 23, shared by equal entries. -/
+def jacobianEntry023 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      151394147,
+      8534174455720092134237275166603186,
+      16840640244550300415844583544502225,
+      12894405273196777971379918291978794,
+      4506348409732893778736073442501460,
+      54756683995003484397907923306619480,
+      27034357940592422468567779837099716,
+      93382309438500237515411019035358590,
+      70067448925761052854406226725970958,
+      16662408131752738946376307743624367]) : ℤ))
+      (naturalFromChunks [
+      1949062500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      50464715,
+      64732670758294711200924782925221777,
+      98385684290077839487301182380997546,
+      53610107900326691496911028225237884,
+      82456569428218072679478834100193025,
+      10063572333607529684946179543893525,
+      13470787246553804150266810892380951,
+      21872306604849333638776947219394501,
+      54380505433679613411774213979274707,
+      18467897289415753684541230752125211]) : ℤ))
+      (naturalFromChunks [
+      649687500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 24, shared by equal entries. -/
+def jacobianEntry024 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      1515467,
+      10063860960057925900960894489197103,
+      66168917845173991972096672824527974,
+      53183908241597557284573812277137549,
+      69640192145790299690329009864755874,
+      48003263610387117627231916521224306,
+      49002445145814704036244459302136418,
+      52127853237031002786897475766547098,
+      81067701652781485466240570252843736,
+      13145612862570953840168193731560193])
+      (naturalFromChunks [
+      24062500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      5845373,
+      10928995235501331264236167522763103,
+      2827494537651899868515205360977465,
+      59035800219581348449874311656188945,
+      85789624445699996804775633486439122,
+      59044586474521916916533288165489581,
+      82103179922346046789035128073019526,
+      33880376764493741963357186245379313,
+      20795421034628556058786371881888446,
+      35009778958654892330779824178267827])
+      (naturalFromChunks [
+      92812500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 25, shared by equal entries. -/
+def jacobianEntry025 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      485091274100409627536,
+      13901544392852629819418868825090398,
+      35450433787827185612016598182037754,
+      5557708859197826275518814695136589,
+      98534250869383383955967122947444557,
+      83051016282114999111800152645156444,
+      94051377464808728780631414730120654,
+      20449081665850328703595286977030564,
+      80319078339700008695123242082940053,
+      11376622677708865854065652102189644,
+      79582174573488767294650349188727457]) : ℤ))
+      (naturalFromChunks [
+      1208496093750000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      4365821466903686647860,
+      88877680303907831714098684624218164,
+      73233505317830833297096253249348548,
+      91698411880449546229923050680040289,
+      49365738783344007659430652207853027,
+      99599067937200731781200114966439453,
+      48169942696227672029010618751339327,
+      19182588669537476646476988824059052,
+      94524946691928014519702481161264369,
+      61261519466972078491402374373043157,
+      53840789798821459410646967230091689])
+      (naturalFromChunks [
+      10876464843750000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 26, shared by equal entries. -/
+def jacobianEntry026 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      33,
+      88122760773053616496499903379466451,
+      5096077917318477688114054076305538,
+      66937289792503507838052086551881692,
+      94567719696477485778768802524360075,
+      30155624493727131320408201099332102,
+      10729506593912846967972547381167997,
+      53258972054123314253208391418671093,
+      31444869785293900831980717333511578,
+      38657798557831769524160962916922611,
+      1909813885330397026323364955743809]) : ℤ))
+      (naturalFromChunks [
+      974531250000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      11,
+      29374249573782686599660933109279673,
+      77799022607584671204943003755369587,
+      39911281305734336515928231949106631,
+      53334218904798949134269278401553318,
+      4415711043391639552362372265268,
+      48459612534507903238809762376445765,
+      24175194930226652146003865268126180,
+      66486347960967454364101507365219009,
+      70781124931533619480101266683596603,
+      61839501461989477567113950892139413]) : ℤ))
+      (naturalFromChunks [
+      324843750000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 27, shared by equal entries. -/
+def jacobianEntry027 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      5845373,
+      10928995235501331264236167522763103,
+      2827494537651899868515205360977465,
+      59035800219581348449874311656188945,
+      85789624445699996804775633486439122,
+      59044586474521916916533288165489581,
+      82103179922346046789035128073019526,
+      33880376764493741963357186245379313,
+      20795421034628556058786371881888446,
+      35009778958654892330779824178267827]) : ℤ))
+      (naturalFromChunks [
+      92812500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      1515467,
+      10063860960057925900960894489197103,
+      66168917845173991972096672824527974,
+      53183908241597557284573812277137549,
+      69640192145790299690329009864755874,
+      48003263610387117627231916521224306,
+      49002445145814704036244459302136418,
+      52127853237031002786897475766547098,
+      81067701652781485466240570252843736,
+      13145612862570953840168193731560193]) : ℤ))
+      (naturalFromChunks [
+      24062500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 28, shared by equal entries. -/
+def jacobianEntry028 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      30560750268325715259578,
+      45059249755923892730152515121737000,
+      58021311014309865864504834518098863,
+      4409058016588930549734277954327549,
+      54632230606525788025255003949808604,
+      58201999332763125181834068907814752,
+      1845299535612137125670146205250468,
+      39494269092926646227582315363058128,
+      66208368007310371308944498944717222,
+      28663989853683689734397219349462828,
+      57376478591750215874528770610641823]) : ℤ))
+      (naturalFromChunks [
+      76135253906250000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      10186916756108571753295,
+      20813014449914099973949523518285637,
+      25879790067798983771189226304697433,
+      13287263925990852675053183577173353,
+      24175154909157233857386045489373007,
+      10054497976011609597655857661546764,
+      80552341052976620573582613155341240,
+      5533115486344029076499006209232462,
+      32191053785896760019321063619776548,
+      23396935955731872188083995486212636,
+      28700416043264113187657332963276597])
+      (naturalFromChunks [
+      25378417968750000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 29, shared by equal entries. -/
+def jacobianEntry029 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      12,
+      17536677313372688582338844523890378,
+      79629307266808938747954249135320259,
+      29929536036929428893091326801540248,
+      5076153377694554414435206046990122,
+      69885223580943186147772401615118346,
+      81045289756633842171903945836617070,
+      47740918622126651523556894885533263,
+      45403808383740057067969404417441999,
+      78172161388494088062659369651403993,
+      22700017153996492522371316964046471])
+      (naturalFromChunks [
+      108281250000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      109,
+      57830115040140844356545970498104034,
+      35255386337638599963264261998597310,
+      92048546239706144436441072241840838,
+      93507421949651690645215691892396796,
+      55228651872069128822150400053293755,
+      41703520111091817264183781665057916,
+      8244818745278033503327149124180610,
+      99374404819018363712507144536624941,
+      37205706889035835400530436105076482,
+      83128163885330397026323364955743809])
+      (naturalFromChunks [
+      974531250000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 30, shared by equal entries. -/
+def jacobianEntry030 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      1616957444096510759899354277419,
+      74887375716016938866746843920307243,
+      51157643195695921477281203192066356,
+      65616641824260524729119407276033009,
+      19453254735736653206166309707726957,
+      25101192785985169104390988325685347,
+      38655013556252709961311577586809900,
+      18240984014065070799873049505704412,
+      76422211541039454430245645876015181,
+      27318904695997226642375849205843618,
+      93945888129792339562971998889829791]) : ℤ))
+      (naturalFromChunks [
+      76135253906250000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      230993905776237202179782526637,
+      46380081887227801251185026370458557,
+      36299359151163811902965995716516440,
+      34142872270834534835273241258812516,
+      99228338057946677951716964654906042,
+      15793633184532460378322558399258845,
+      25333391287807285014026792617205859,
+      40553149297417714112305457641343041,
+      93069716940229523227333746374619791,
+      85092810503043596757341376611646048,
+      26199040201178540589353032769908311]) : ℤ))
+      (naturalFromChunks [
+      10876464843750000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 31, shared by equal entries. -/
+def jacobianEntry031 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      8,
+      30120788396990136387998914655275452,
+      84216527272245704045798146768143229,
+      35027745272027862155757125533228922,
+      61642432409673724743054876274239816,
+      94038507625280772820748050010827378,
+      63817752940382276188715595408188949,
+      47389756914634072351573605912174475,
+      36948655523295107845913674274096569,
+      3254224057777686364833081765757501,
+      57714342195936647760651268335440771])
+      (naturalFromChunks [
+      46406250000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      174,
+      32536575556579511263473611028570384,
+      31885526393607370515262207102588647,
+      98195718270920117912362801220538575,
+      44503216743153141608273025045859936,
+      16042754297911512709005522286306544,
+      57673079666003139052533995726604811,
+      98075746474846632437956800159867042,
+      89745533012742482652374127411581803,
+      95033259690364931364684820563021286,
+      66329995614031567298658147323581761])
+      (naturalFromChunks [
+      974531250000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 32, shared by equal entries. -/
+def jacobianEntry032 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      1994234009759722264901457439969,
+      49360039213559102366964122375663351,
+      26276931489110799309059606173493625,
+      17307770798047303551297700165907731,
+      74465689568175033235651048932898327,
+      60190081655045440563869917472971320,
+      76463266348926871010277778288675026,
+      70886987244004625827398580947934584,
+      46339734348312336507039774433488912,
+      62898915938966790889384172277141695,
+      62273561870207660437028001110170209])
+      (naturalFromChunks [
+      76135253906250000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      31654509736866231897509356301,
+      96398575588059848678301250606225753,
+      4628880409120348398733370847936584,
+      47437997395792112352641432414198774,
+      34711976823280843741667151149709780,
+      34210209603989770096373385264530149,
+      27939894135903299690825094113084142,
+      37773425439440608770757404602060859,
+      47282318700833837675881509717267934,
+      57091083563816549717105934641946221,
+      65395412199869051045627440803343521])
+      (naturalFromChunks [
+      1208496093750000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 33, shared by equal entries. -/
+def jacobianEntry033 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      17536119,
+      10084661563291164598531904096705187,
+      46246089116201486859339585521089185,
+      47186683144289407157503065803807234,
+      2736998198785441838380689680013748,
+      14445096024031011027047707460337333,
+      8143972684978169436762250440322623,
+      64373987713694619207980776799913826,
+      80013137739028021233752253233721137,
+      37578161140524796131380045368913689])
+      (naturalFromChunks [
+      278437500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      40917612,
+      13306087550417405412114869598177685,
+      61426539310448489769736824177739239,
+      65005504077220973363480503874133275,
+      7758904769637556336167078526384687,
+      13681138997322924765112313260520347,
+      95053470579008545798388028033874490,
+      59531717903887874867560469954283919,
+      81699651147333165609137637996749039,
+      36339657338775475693446560805868059])
+      (naturalFromChunks [
+      649687500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 34, shared by equal entries. -/
+def jacobianEntry034 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      16821572,
+      18935248922107409779011936861546892,
+      71376476958348548561460683376481257,
+      85061682616827543129062393687586261,
+      19405532908626417149486715056543736,
+      30904983944941244570869968909097209,
+      84428116516805737364712773753692006,
+      7330986321138140492228688272409182,
+      66949715050890374381608090773085069,
+      24267769112925158564482186935289353]) : ℤ))
+      (naturalFromChunks [
+      216562500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      50464715,
+      67030536116960140322895629400942777,
+      62809145236700557323210904616674083,
+      3998159701527047864350336713655997,
+      48772477919508938158243888925739049,
+      85589032778862507796065820479243490,
+      95333264316105385148079429867168206,
+      70254532318534439970587147833422747,
+      89754962979828876855175727680744792,
+      27196692661224524306553439194131941]) : ℤ))
+      (naturalFromChunks [
+      649687500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 35, shared by equal entries. -/
+def jacobianEntry035 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      7352760958989911093002129779957971,
+      38299626901833229408593441362376647,
+      92593236747330642672779646852860252,
+      73430341751574594566568257632400528,
+      53456098225869269864160115756019539,
+      71054459678265101317888910028801934,
+      92929446877184370292402003967286189,
+      91542092966319163073734525547102562,
+      83104794391129222903067344677697336,
+      85339746954367425306185809042916111])
+      (naturalFromChunks [
+      12031250000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      5,
+      95573892122875938165955167205300950,
+      45474966287466331105510345585053465,
+      20454107543288661900942349570220890,
+      16795478239478380724974877073695132,
+      34387002344909475926872312131100829,
+      94733980912638861602057403153906644,
+      20190737994589786780104393410056966,
+      26725777363866819408909926349358299,
+      72939016476447914490644182542565658,
+      16488233725009926776350666430464321])
+      (naturalFromChunks [
+      974531250000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 36, shared by equal entries. -/
+def jacobianEntry036 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      50464715,
+      67030536116960140322895629400942777,
+      62809145236700557323210904616674083,
+      3998159701527047864350336713655997,
+      48772477919508938158243888925739049,
+      85589032778862507796065820479243490,
+      95333264316105385148079429867168206,
+      70254532318534439970587147833422747,
+      89754962979828876855175727680744792,
+      27196692661224524306553439194131941])
+      (naturalFromChunks [
+      649687500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      16821572,
+      18935248922107409779011936861546892,
+      71376476958348548561460683376481257,
+      85061682616827543129062393687586261,
+      19405532908626417149486715056543736,
+      30904983944941244570869968909097209,
+      84428116516805737364712773753692006,
+      7330986321138140492228688272409182,
+      66949715050890374381608090773085069,
+      24267769112925158564482186935289353])
+      (naturalFromChunks [
+      216562500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 37, shared by equal entries. -/
+def jacobianEntry037 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      82018298446083648441095,
+      10965710589111349810828174202906852,
+      54057918251696895929389124203075439,
+      51503111143266982888961939226985620,
+      56532227134084956283945894303401769,
+      27155206676930610540553187776425173,
+      44182607927942638851725840040513037,
+      77100986836809465552248365631933610,
+      26396926506923914412488108739234273,
+      89724086546600335428749301776520308,
+      9073434067037848400695748454857241]) : ℤ))
+      (naturalFromChunks [
+      10876464843750000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      191376029713746975942689,
+      71427720680276115950984118452646937,
+      76990984290896808332157074694965771,
+      23166720757868465448572539517937869,
+      66650687737106865486375290154560759,
+      46736331001143431974909521514872549,
+      7241292628428576370337512262443261,
+      83520945849443779692405052115115290,
+      11381496595280236344831737232564080,
+      30961872224334936764543012880487464,
+      2018216484530971445961951504455909])
+      (naturalFromChunks [
+      25378417968750000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 38, shared by equal entries. -/
+def jacobianEntry038 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      43,
+      43036025064825739724446855858177309,
+      17377584136198853420641235028599370,
+      89501802582762793756629937648601420,
+      41957346250886858179423278261662359,
+      74434390456173978902661864456680827,
+      17700294853974925183359977921745825,
+      96215418133771769643550042298776660,
+      25438591223545544179560895655700338,
+      53423854591549230289207107890943880,
+      73786294575003308925450222143488107]) : ℤ))
+      (naturalFromChunks [
+      324843750000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      130,
+      29107820749784137890280387454738648,
+      8259215646414634522137312737256813,
+      60191491061502203103569543518883330,
+      14451579793330521252460742504994390,
+      36489859432554132088150258773391840,
+      15297323233866240025550377276113097,
+      24117980008544563686397198039037574,
+      63567756076579199446657768682298758,
+      89366910210193362190565397570327059,
+      41889353303761449801050532476204991]) : ℤ))
+      (naturalFromChunks [
+      974531250000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 39, shared by equal entries. -/
+def jacobianEntry039 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      40917612,
+      13306087550417405412114869598177685,
+      61426539310448489769736824177739239,
+      65005504077220973363480503874133275,
+      7758904769637556336167078526384687,
+      13681138997322924765112313260520347,
+      95053470579008545798388028033874490,
+      59531717903887874867560469954283919,
+      81699651147333165609137637996749039,
+      36339657338775475693446560805868059]) : ℤ))
+      (naturalFromChunks [
+      649687500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      17536119,
+      10084661563291164598531904096705187,
+      46246089116201486859339585521089185,
+      47186683144289407157503065803807234,
+      2736998198785441838380689680013748,
+      14445096024031011027047707460337333,
+      8143972684978169436762250440322623,
+      64373987713694619207980776799913826,
+      80013137739028021233752253233721137,
+      37578161140524796131380045368913689]) : ℤ))
+      (naturalFromChunks [
+      278437500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 40, shared by equal entries. -/
+def jacobianEntry040 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      191376029714356954554402,
+      23695053132912262710199406214725767,
+      66427307139850131414137123653076340,
+      33959991347605774688110272085705446,
+      51884051757771194724224599960175901,
+      71880779059571397762173050272077633,
+      10880729126112322936891309361298248,
+      72928571032892669105059550213046544,
+      25480559037944012052870474799985087,
+      22845714475780468121837713853728585,
+      59363696156421646268290079728000229]) : ℤ))
+      (naturalFromChunks [
+      25378417968750000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      82018298452297005939724,
+      74867343575117313914636059574760046,
+      72573132086062988228579157302169401,
+      60312928033574625036509098431768199,
+      86446711137163797818119119390626448,
+      79949613416832221972615865545749037,
+      44605357245011970259260573291765301,
+      34130509668188161235121698086735352,
+      73286042215300953320329745822260505,
+      53652023047698947080917291385304288,
+      54425942779084702048269407787623961])
+      (naturalFromChunks [
+      10876464843750000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 41, shared by equal entries. -/
+def jacobianEntry041 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      62913700311502602125493655458169539,
+      29500598271144577883408071150512484,
+      38588748274724302762788183944538169,
+      90960694599198179364454091211434112,
+      23236725889133969923537385602541213,
+      42390933092632254551785633794958687,
+      97930992394419849553536416060336039,
+      82550767915670116956169079835065789,
+      65813602491809827545933067864284466,
+      37344540435161992730930961295145671]) : ℤ))
+      (naturalFromChunks [
+      4218750000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      83045315832353848792064018417284445,
+      43007317033146046945486727064539853,
+      59853819844081438900829397436916557,
+      31455881781688608387356961202702700,
+      36843301032989766817980058155882325,
+      4892890807983502209632446042416553,
+      25852552828266117983633830033688742,
+      50544872829900710946324458660501892,
+      37345816084159602973065242351186303,
+      64891668628515642011874473787550643]) : ℤ))
+      (naturalFromChunks [
+      5568750000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 42, shared by equal entries. -/
+def jacobianEntry042 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      9274178,
+      39628007903092105676910274471642625,
+      8460682123929382678966274409151638,
+      2525267524709556731345482139467976,
+      38852543140445848589701338604826101,
+      95812158128401783022006918391741233,
+      18238086367602360448440440499288897,
+      27346283579310163722933960955767787,
+      3936496585231591577070586698710788,
+      87144021379327657050998090608058631]) : ℤ))
+      (naturalFromChunks [
+      92812500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      27822383,
+      39292601818031452402622978687728903,
+      38970945000507503648122650927477903,
+      69476662625930791291279624486197360,
+      56124412275420953602991918245391930,
+      52899540562556606508030739850561998,
+      78047214983164602784752394541057797,
+      62909855508282933504913871496950963,
+      23810432119305225268788239903867633,
+      38567935862017028847005728175824107]) : ℤ))
+      (naturalFromChunks [
+      278437500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 43, shared by equal entries. -/
+def jacobianEntry043 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      31977577063,
+      64493014863610428794579711755834698,
+      6126787268670042666449841663631995,
+      24384138840484890806488912784787953,
+      50992658956985079916312647524515686,
+      97355377530910807256457070938504953,
+      34303413771707733090562630520701533,
+      13471419750799962662137281315302664,
+      99034325567934596434472930778715937,
+      47151189077650338686546784172871320,
+      73141847581964488288853012252696877,
+      76007467476980728364856719735165672,
+      84522116324710919696566340292662435,
+      72495628465203378061100147351737729,
+      25467852101977704648798957730703065,
+      54630333100353909405568400695090052,
+      56748302483955548620912385059624897,
+      34266013882014103432069821931722411])
+      (naturalFromChunks [
+      320760000000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      19986212645115,
+      23778882643476778456146946607351543,
+      43173188821542934522492500154177809,
+      87754129015722280107768014539338804,
+      95084408166129275918896274614798198,
+      27389034044922172893118366449494479,
+      47774882350879642754431995975862456,
+      43428392856024354125631190738456078,
+      70592814072682791156886664203453679,
+      50457412439856964435370485944710880,
+      13929713045539262589176056857131808,
+      80476371182240578165560053023642244,
+      17617173661931480200830790142960965,
+      636599137259771434818751181840303,
+      24285444011045926403160849987993491,
+      31512432755204999220816364868239660,
+      12943061642567295603821880849534881,
+      29322569212096131691590313864126589])
+      (naturalFromChunks [
+      2,
+      475000000000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 44, shared by equal entries. -/
+def jacobianEntry044 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      10114716257576,
+      56413508374272350231687714888574862,
+      97655693948583087969196018310566635,
+      85834636257667451953823857500193972,
+      78415440053177866992849542488690255,
+      78738038619906540843919983247546758,
+      88507065683221705207687683392888001,
+      58738931364878689422778014064700625,
+      61631021283312825624980061862150459,
+      12507820869076756066914229059071991,
+      99169257319113724846023470986627803,
+      82826954504778091142371581316279233,
+      35842638838713248752786409939193615,
+      92192463206003419680446242201086476,
+      43415318165703827946900819015603592,
+      8151015042163267002949180193293454,
+      60482210303372048065448857443359327,
+      46769533217684967682513944330393515,
+      16001686632148675843294636200782483]) : ℤ))
+      (naturalFromChunks [
+      12,
+      92124023437500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      1123694810930,
+      24822732348723027650303556253854653,
+      2041673063789141106471056791484407,
+      9330580287717921553199706197573873,
+      16481515400947420206215274511965484,
+      8959970028143335483504893249895287,
+      3488161900912211925053005592308207,
+      23781757072930561565286313603438609,
+      16336491973918019441434630812682870,
+      29836718896886107276218454909750608,
+      79067054064968790090703023469462762,
+      96579683912943848998341156456796582,
+      44049463152100658027684157834369236,
+      38069367260558529429092675523967210,
+      11729947761038490364995760596262582,
+      28344150671241673754956441284643967,
+      37949192468712990483505118533892011,
+      90345085560248653974654261702043494,
+      69659909625794297315921626244531387]) : ℤ))
+      (naturalFromChunks [
+      1,
+      43569335937500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 45, shared by equal entries. -/
+def jacobianEntry045 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      2091,
+      90777255471720937398394745777629880,
+      78166682342265933289604573880938523,
+      11480811393385331129320572957045471,
+      89007220641067008157843714773645641,
+      88521651750216504600638858288802763,
+      90911516946512321335555612773529250,
+      93374111032720125785090276745277869,
+      60895366385043969566639293279164567,
+      10518458321538694046569451794934025,
+      80847834589216329111283875389905451,
+      45373689546482777382117193736636776,
+      76655216957945450951939163707856090,
+      71255742777756629091942518363673072,
+      94613679419212902916905662817994740,
+      12440668941133310476132819508103563,
+      98290451238975048191476378822482342,
+      69546300316038622663375184535759494,
+      16855922675677089846079277814551568,
+      6290055276994395936665074066384769]) : ℤ))
+      (naturalFromChunks [
+      3,
+      87637207031250000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      645,
+      41931621879594859728836761549152771,
+      57857630478148993208399521258353240,
+      24974507765034502042152131794922811,
+      14683800671394121539682019830780558,
+      62685994636562229261564017585937314,
+      87582118311124833698222674173018182,
+      23995305753644246962344688274871229,
+      89945999903416874656232019509858811,
+      5054282779120229240307810526011454,
+      93173234925913135070337406167366833,
+      13780593649663500176963028506087291,
+      80857419284754149643901258806550864,
+      63976541179667701706938880489585792,
+      91025690555698814145647202980509945,
+      30479890887379797298058278519655533,
+      70934496011914864976070883880054546,
+      95722991192613003741194627416450948,
+      32366924374968737121919280776211076,
+      66132481574335201354444975311205077])
+      (naturalFromChunks [
+      1,
+      29212402343750000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 46, shared by equal entries. -/
+def jacobianEntry046 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      58985167275757503076276755659046857,
+      71123181475386544333799129533751529,
+      243715573358998910061773181794391,
+      73420184357698372063295604316131492,
+      68462567024524818571792791737689042,
+      6477477324196413846582613631659908,
+      78751454466354401074771661512307366,
+      35198904767889893862668328862756068,
+      96552807071098214494083183438091054,
+      23485418628515642011874473787550643])
+      (naturalFromChunks [
+      5568750000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      1,
+      34058945124061560426716003005897462,
+      84952279676144560450474964148804356,
+      28503085313873976697890923503425077,
+      69329361670215117402207170533692067,
+      60145926986326804712405449452110559,
+      32833207753458591867212007934301405,
+      27365008025147038887392469398543564,
+      19350123197441227202753315280369313,
+      72223349477062861908523743920381109,
+      8127371305485978192792883885437013])
+      (naturalFromChunks [
+      12656250000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 47, shared by equal entries. -/
+def jacobianEntry047 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      40388,
+      15780304261501021507681257122839594,
+      58410858435645254869378236303040131,
+      61978945724661011746312555278703146,
+      28899189191944818645637495334759181,
+      27962745849574308028529813434567295,
+      22571121116919363213350801237568330,
+      60283174409494608053608437502590864,
+      86600895734459452651972972822434061,
+      84848580957852482549889101043549041])
+      (naturalFromChunks [
+      10312500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      1090632,
+      5659636967863010604568293350439534,
+      58428238055044706927599558776990465,
+      23781803431995170938158942718107700,
+      84080332136460093039771740728331505,
+      34598883984711842144516491544529454,
+      68808892097483108729150825208058316,
+      38997552012719541411474119385775894,
+      31372171644725255166487518259636639,
+      7525814137982971152994271824175893])
+      (naturalFromChunks [
+      278437500000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 48, shared by equal entries. -/
+def jacobianEntry048 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      19986212645115,
+      23778882643476778456146946607351543,
+      43173188821542934522492500154177809,
+      87754129015722280107768014539338804,
+      95084408166129275918896274614798198,
+      27389034044922172893118366449494479,
+      47774882350879642754431995975862456,
+      43428392856024354125631190738456078,
+      70592814072682791156886664203453679,
+      50457412439856964435370485944710880,
+      13929713045539262589176056857131808,
+      80476371182240578165560053023642244,
+      17617173661931480200830790142960965,
+      636599137259771434818751181840303,
+      24285444011045926403160849987993491,
+      31512432755204999220816364868239660,
+      12943061642567295603821880849534881,
+      29322569212096131691590313864126589]) : ℤ))
+      (naturalFromChunks [
+      2,
+      475000000000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      31977577063,
+      64493014863610428794579711755834698,
+      6126787268670042666449841663631995,
+      24384138840484890806488912784787953,
+      50992658956985079916312647524515686,
+      97355377530910807256457070938504953,
+      34303413771707733090562630520701533,
+      13471419750799962662137281315302664,
+      99034325567934596434472930778715937,
+      47151189077650338686546784172871320,
+      73141847581964488288853012252696877,
+      76007467476980728364856719735165672,
+      84522116324710919696566340292662435,
+      72495628465203378061100147351737729,
+      25467852101977704648798957730703065,
+      54630333100353909405568400695090052,
+      56748302483955548620912385059624897,
+      34266013882014103432069821931722411]) : ℤ))
+      (naturalFromChunks [
+      320760000000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 49, shared by equal entries. -/
+def jacobianEntry049 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      894,
+      66271478398461557537071194378393090,
+      55183902304062782329489987965879027,
+      40943337127025536872573596477075292,
+      45090309561234043998649294396692798,
+      60870883304058602541285360539104900,
+      79457415417858045939557208987736973,
+      12578139961036170601399857722161467,
+      90030554509993207855720137299089487,
+      83536433708476808617532043034810799,
+      48972077614064918920958208338324105,
+      40784569972972586593916739341191923,
+      6346393655368575067585084952842386,
+      30109659646748424951147051310721408,
+      4052876343767096799421109351046534,
+      82933220023334525129751523855071861,
+      60625709555571525179656963963752562,
+      12374512377157072857240839248821428,
+      86736708593857040933444457591221532,
+      32305018425664798645555024688794923]) : ℤ))
+      (naturalFromChunks [
+      1,
+      29212402343750000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      1344,
+      17757686254046876488035583261556641,
+      78650055084389974652045253544446324,
+      87444266832178408986033878898091884,
+      57840378716251994984550282875733531,
+      90171622725133182081026108811497017,
+      22601008722988299108099406265272679,
+      4894553128059246184381855578277596,
+      78304617687127006541222489906684996,
+      88830936420610317676278814283869126,
+      49801109652730054647084748232974216,
+      19381577242482056974018785915512477,
+      9525604935436321019038552769825964,
+      16071076048406921003320368502013828,
+      33458946401814437708844894158624628,
+      39254790958666032858391137273938453,
+      54432553271902024830401459040558056,
+      80626609933560802713034251587777435,
+      16686023144012178411503747369520201,
+      7772444723005604063334925933615231])
+      (naturalFromChunks [
+      3,
+      87637207031250000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 50, shared by equal entries. -/
+def jacobianEntry050 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      44887605324681044507070090591848950,
+      25660142819172544180585572756384845,
+      25593444074477790761418422174551393,
+      4722060403185156645108128121946079,
+      53562959527980612958272043124024069,
+      28975542617155913473029445696953905,
+      75869213322439990753049458491902400,
+      5547091086421123144824272319255207,
+      72803307338775475693446560805868059]) : ℤ))
+      (naturalFromChunks [
+      324843750000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      44887605324681044507070090591848950,
+      25660142819172544180585572756384845,
+      25593444074477790761418422174551393,
+      4722060403185156645108128121946079,
+      53562959527980612958272043124024069,
+      28975542617155913473029445696953905,
+      75869213322439990753049458491902400,
+      5547091086421123144824272319255207,
+      72803307338775475693446560805868059])
+      (naturalFromChunks [
+      324843750000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 51, shared by equal entries. -/
+def jacobianEntry051 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      12958624537686,
+      3411955604200248628537498878931409,
+      49964257633083246338658489719080626,
+      1059736373559872021203039742230195,
+      23450331873295340539312674961708827,
+      6166062580590247393436451534768442,
+      17841265684237176674649739904330005,
+      45374322973612922328625375437434767,
+      36674052302390468485597935658095276,
+      10463225556386694007300163247335867,
+      14646687719797409692324352968355344,
+      85602818651703206338486341650497171,
+      98729543000784849015390340426058108,
+      68171485621218593874174140251422992,
+      4958239564296464343197683438164893,
+      52788849768402637056113713231700226,
+      45073579579069554108872126778739708,
+      99045958204655518403989095572552812,
+      45924517960050625410242642699422681]) : ℤ))
+      (naturalFromChunks [
+      3341499,
+      7327746048000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      12958624537686,
+      3411951140534475048444696848148017,
+      70323934752057596763868527237977638,
+      12772172004940854896605310490281237,
+      12236995020814456370184196890045501,
+      54114802261359873677414058685386936,
+      61283306753299767786210496864503533,
+      76173485512064918020142368216058323,
+      58668505991971025064779049450337336,
+      91641774237735595741183649664361991,
+      43250608161528561287679690742416900,
+      43777998244749443275337802772889978,
+      54873247273682039140179200466450697,
+      47582099135361625736293353458840079,
+      27300142564825942900769604224399165,
+      34672325042143848327587892644356586,
+      97657485367681110297717955804506185,
+      60211775587181060305510749575349840,
+      92057034284441389954242642699422681]) : ℤ))
+      (naturalFromChunks [
+      3341499,
+      7327746048000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 52, shared by equal entries. -/
+def jacobianEntry052 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      199,
+      42337922293587149037102343768136308,
+      25921537238296178274377734707614062,
+      30854886327910292822772796631537074,
+      96015254260274561579626703197638098,
+      72577453252666732232931549106824202,
+      10760903073611646435874334103299175,
+      20879823725305028196056192376499755,
+      67097728956153505088339683389872668,
+      38093102944346985303629873571427245,
+      38173056714334370883169614740284191])
+      (naturalFromChunks [
+      380676269531250000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      199,
+      42343663574472357813613206851607195,
+      74466926423157249498029889935253632,
+      6226363067310336734062800117422978,
+      50653761683863254200122848695935101,
+      9529087591502531252943714659385561,
+      7957790597042837565500288598614318,
+      5467066308168839135364868911101570,
+      74421796319432774584780565647697841,
+      14675386325469931234150978464302653,
+      66076982272908831164147605559795169])
+      (naturalFromChunks [
+      380676269531250000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 53, shared by equal entries. -/
+def jacobianEntry053 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      1,
+      34662815854107032023310640058798372,
+      30278497058967530646916686942796710,
+      32354865110818534993960025497874593,
+      52058863458707288069918203909528912,
+      19963872411875848553001493779599856,
+      14076301471089225668914165509682553,
+      13988619857900645073407986131727485,
+      92503494634401674095573570677099578,
+      22985922016326427080339682417604177]) : ℤ))
+      (naturalFromChunks [
+      974531250000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      1,
+      34662815854107032023310640058798372,
+      30278497058967530646916686942796710,
+      32354865110818534993960025497874593,
+      52058863458707288069918203909528912,
+      19963872411875848553001493779599856,
+      14076301471089225668914165509682553,
+      13988619857900645073407986131727485,
+      92503494634401674095573570677099578,
+      22985922016326427080339682417604177])
+      (naturalFromChunks [
+      974531250000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 54, shared by equal entries. -/
+def jacobianEntry054 : IntervalRat :=
+  ⟨(q
+      (naturalFromChunks [
+      121146792163966,
+      31566016439952163144557534488385169,
+      5217032132931414440907470672743303,
+      74020617385281094664118685498184083,
+      74229180346417659277865752141558727,
+      3093295849137281312998248418142639,
+      98452353292999508126048579050066448,
+      17390001012980894263766730296203347,
+      95634577277379975862219676731727365,
+      25069763558549783859622607506205925,
+      460822967832849253390195359638086,
+      93927907772399315707615747051698611,
+      67060108892460693398430185306048203,
+      25677463268458846464424961232396137,
+      55559379118779715929880677885112215,
+      8331185575075708368158302597011679,
+      67780925392727856134668392839825962,
+      49539272936197723877003466467330197,
+      86030491458401618818242642699422681])
+      (naturalFromChunks [
+      3341499,
+      7327746048000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (naturalFromChunks [
+      121146792163966,
+      31566020903606026329779437565677673,
+      76852309176823775608249235125454854,
+      91533516022034347759759452382249680,
+      6302649633124527938423627484572530,
+      14575051052128363230557765073290711,
+      28961297958582541548365605332025690,
+      84373133080906120133344034671380725,
+      44073991579243734065714874538488525,
+      82055748030000834243540185726150196,
+      16719844692655035596477369282087977,
+      79084545730764848166091054799398721,
+      66738387738006625412848379545388946,
+      876281196732513526586105195911404,
+      43530939696670331083413526607388632,
+      52575071804007568732157139880458185,
+      57448568521393973246195227052776783,
+      27582531650396302574779440591735431,
+      78361975134010854274242642699422681])
+      (naturalFromChunks [
+      3341499,
+      7327746048000000000000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Exact cached Jacobian interval number 55, shared by equal entries. -/
+def jacobianEntry055 : IntervalRat :=
+  ⟨(q
+      (-((naturalFromChunks [
+      161,
+      69576770785652954124268059928092552,
+      41125461964680753035908064823083298,
+      3398530467978541744421246575512057,
+      5887481091326438268740209211073076,
+      51569255847826610076224083379157727,
+      6008219627226671767979426206742030,
+      28007405389022218092447449490845533,
+      72256749128638655842408278496656361,
+      60078660246637482188947712873293451,
+      56898806281413363132602792026408929]) : ℤ))
+      (naturalFromChunks [
+      380676269531250000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])),
+    (q
+      (-((naturalFromChunks [
+      161,
+      69571029504767345282873690874819234,
+      29127143311790006180748587793323213,
+      1104883701748548388871833193850885,
+      18596991957433403390259605204944387,
+      50512313948589461443873177352519423,
+      69153055437698746771232688006830315,
+      40954776535535326676927122783315447,
+      27950902226399756515328192552327982,
+      41289491566442480259836576851168953,
+      89538880722838902851624801206897951]) : ℤ))
+      (naturalFromChunks [
+      380676269531250000000000000000,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0,
+      0])), by decide +kernel⟩
+
+/-- Cached interval row 0 of the full Jacobian. -/
+def jacobianRow00 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry001
+  | 11 => jacobianEntry000
+  | 12 => jacobianEntry000
+  | 13 => jacobianEntry000
+  | 14 => jacobianEntry000
+  | 15 => jacobianEntry000
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry002
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry000
+  | _ => jacobianEntry000
+
+/-- Cached interval row 1 of the full Jacobian. -/
+def jacobianRow01 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry000
+  | 11 => jacobianEntry002
+  | 12 => jacobianEntry000
+  | 13 => jacobianEntry000
+  | 14 => jacobianEntry000
+  | 15 => jacobianEntry000
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry002
+  | 20 => jacobianEntry000
+  | _ => jacobianEntry000
+
+/-- Cached interval row 2 of the full Jacobian. -/
+def jacobianRow02 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry000
+  | 11 => jacobianEntry000
+  | 12 => jacobianEntry002
+  | 13 => jacobianEntry000
+  | 14 => jacobianEntry000
+  | 15 => jacobianEntry000
+  | 16 => jacobianEntry002
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry000
+  | _ => jacobianEntry000
+
+/-- Cached interval row 3 of the full Jacobian. -/
+def jacobianRow03 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry000
+  | 11 => jacobianEntry000
+  | 12 => jacobianEntry003
+  | 13 => jacobianEntry001
+  | 14 => jacobianEntry000
+  | 15 => jacobianEntry000
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry002
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry000
+  | _ => jacobianEntry000
+
+/-- Cached interval row 4 of the full Jacobian. -/
+def jacobianRow04 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry000
+  | 11 => jacobianEntry000
+  | 12 => jacobianEntry000
+  | 13 => jacobianEntry000
+  | 14 => jacobianEntry001
+  | 15 => jacobianEntry002
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry000
+  | _ => jacobianEntry000
+
+/-- Cached interval row 5 of the full Jacobian. -/
+def jacobianRow05 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry002
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry002
+  | 11 => jacobianEntry000
+  | 12 => jacobianEntry000
+  | 13 => jacobianEntry000
+  | 14 => jacobianEntry000
+  | 15 => jacobianEntry000
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry000
+  | _ => jacobianEntry000
+
+/-- Cached interval row 6 of the full Jacobian. -/
+def jacobianRow06 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry002
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry000
+  | 11 => jacobianEntry000
+  | 12 => jacobianEntry000
+  | 13 => jacobianEntry000
+  | 14 => jacobianEntry000
+  | 15 => jacobianEntry000
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry000
+  | _ => jacobianEntry000
+
+/-- Cached interval row 7 of the full Jacobian. -/
+def jacobianRow07 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry000
+  | 11 => jacobianEntry002
+  | 12 => jacobianEntry000
+  | 13 => jacobianEntry000
+  | 14 => jacobianEntry000
+  | 15 => jacobianEntry000
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry000
+  | _ => jacobianEntry000
+
+/-- Cached interval row 8 of the full Jacobian. -/
+def jacobianRow08 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry002
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry001
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry004
+  | 11 => jacobianEntry005
+  | 12 => jacobianEntry006
+  | 13 => jacobianEntry007
+  | 14 => jacobianEntry000
+  | 15 => jacobianEntry000
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry008
+  | _ => jacobianEntry000
+
+/-- Cached interval row 9 of the full Jacobian. -/
+def jacobianRow09 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry002
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry001
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry005
+  | 11 => jacobianEntry009
+  | 12 => jacobianEntry010
+  | 13 => jacobianEntry011
+  | 14 => jacobianEntry000
+  | 15 => jacobianEntry000
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry012
+  | _ => jacobianEntry000
+
+/-- Cached interval row 10 of the full Jacobian. -/
+def jacobianRow10 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry013
+  | 11 => jacobianEntry014
+  | 12 => jacobianEntry015
+  | 13 => jacobianEntry016
+  | 14 => jacobianEntry000
+  | 15 => jacobianEntry000
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry017
+  | _ => jacobianEntry000
+
+/-- Cached interval row 11 of the full Jacobian. -/
+def jacobianRow11 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry014
+  | 11 => jacobianEntry018
+  | 12 => jacobianEntry019
+  | 13 => jacobianEntry007
+  | 14 => jacobianEntry000
+  | 15 => jacobianEntry000
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry020
+  | _ => jacobianEntry000
+
+/-- Cached interval row 12 of the full Jacobian. -/
+def jacobianRow12 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry002
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry001
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry000
+  | 11 => jacobianEntry000
+  | 12 => jacobianEntry021
+  | 13 => jacobianEntry022
+  | 14 => jacobianEntry023
+  | 15 => jacobianEntry024
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry000
+  | _ => jacobianEntry025
+
+/-- Cached interval row 13 of the full Jacobian. -/
+def jacobianRow13 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry002
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry001
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry000
+  | 11 => jacobianEntry000
+  | 12 => jacobianEntry026
+  | 13 => jacobianEntry024
+  | 14 => jacobianEntry027
+  | 15 => jacobianEntry023
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry000
+  | _ => jacobianEntry028
+
+/-- Cached interval row 14 of the full Jacobian. -/
+def jacobianRow14 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry000
+  | 11 => jacobianEntry000
+  | 12 => jacobianEntry029
+  | 13 => jacobianEntry027
+  | 14 => jacobianEntry024
+  | 15 => jacobianEntry022
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry000
+  | _ => jacobianEntry030
+
+/-- Cached interval row 15 of the full Jacobian. -/
+def jacobianRow15 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry000
+  | 11 => jacobianEntry000
+  | 12 => jacobianEntry031
+  | 13 => jacobianEntry022
+  | 14 => jacobianEntry023
+  | 15 => jacobianEntry024
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry000
+  | _ => jacobianEntry032
+
+/-- Cached interval row 16 of the full Jacobian. -/
+def jacobianRow16 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry002
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry001
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry000
+  | 11 => jacobianEntry000
+  | 12 => jacobianEntry000
+  | 13 => jacobianEntry000
+  | 14 => jacobianEntry033
+  | 15 => jacobianEntry034
+  | 16 => jacobianEntry035
+  | 17 => jacobianEntry036
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry000
+  | _ => jacobianEntry037
+
+/-- Cached interval row 17 of the full Jacobian. -/
+def jacobianRow17 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry002
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry001
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry000
+  | 11 => jacobianEntry000
+  | 12 => jacobianEntry000
+  | 13 => jacobianEntry000
+  | 14 => jacobianEntry036
+  | 15 => jacobianEntry033
+  | 16 => jacobianEntry038
+  | 17 => jacobianEntry039
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry000
+  | _ => jacobianEntry040
+
+/-- Cached interval row 18 of the full Jacobian. -/
+def jacobianRow18 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry002
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry001
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry000
+  | 11 => jacobianEntry000
+  | 12 => jacobianEntry000
+  | 13 => jacobianEntry000
+  | 14 => jacobianEntry000
+  | 15 => jacobianEntry000
+  | 16 => jacobianEntry041
+  | 17 => jacobianEntry042
+  | 18 => jacobianEntry043
+  | 19 => jacobianEntry044
+  | 20 => jacobianEntry045
+  | _ => jacobianEntry000
+
+/-- Cached interval row 19 of the full Jacobian. -/
+def jacobianRow19 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry002
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry001
+  | 10 => jacobianEntry000
+  | 11 => jacobianEntry000
+  | 12 => jacobianEntry000
+  | 13 => jacobianEntry000
+  | 14 => jacobianEntry000
+  | 15 => jacobianEntry000
+  | 16 => jacobianEntry046
+  | 17 => jacobianEntry047
+  | 18 => jacobianEntry044
+  | 19 => jacobianEntry048
+  | 20 => jacobianEntry049
+  | _ => jacobianEntry000
+
+/-- Cached interval row 20 of the full Jacobian. -/
+def jacobianRow20 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry002
+  | 1 => jacobianEntry000
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry001
+  | 5 => jacobianEntry000
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry004
+  | 11 => jacobianEntry005
+  | 12 => jacobianEntry000
+  | 13 => jacobianEntry000
+  | 14 => jacobianEntry039
+  | 15 => jacobianEntry050
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry051
+  | _ => jacobianEntry052
+
+/-- Cached interval row 21 of the full Jacobian. -/
+def jacobianRow21 (j : Fin 22) : IntervalRat :=
+  match j.val with
+  | 0 => jacobianEntry000
+  | 1 => jacobianEntry002
+  | 2 => jacobianEntry000
+  | 3 => jacobianEntry000
+  | 4 => jacobianEntry000
+  | 5 => jacobianEntry001
+  | 6 => jacobianEntry000
+  | 7 => jacobianEntry000
+  | 8 => jacobianEntry000
+  | 9 => jacobianEntry000
+  | 10 => jacobianEntry005
+  | 11 => jacobianEntry009
+  | 12 => jacobianEntry000
+  | 13 => jacobianEntry000
+  | 14 => jacobianEntry034
+  | 15 => jacobianEntry053
+  | 16 => jacobianEntry000
+  | 17 => jacobianEntry000
+  | 18 => jacobianEntry000
+  | 19 => jacobianEntry000
+  | 20 => jacobianEntry054
+  | _ => jacobianEntry055
+
+/-- Select an exact cached interval from the full interval Jacobian. -/
+def jacobianCache (i j : Fin 22) : IntervalRat :=
+  match i.val with
+  | 0 => jacobianRow00 j
+  | 1 => jacobianRow01 j
+  | 2 => jacobianRow02 j
+  | 3 => jacobianRow03 j
+  | 4 => jacobianRow04 j
+  | 5 => jacobianRow05 j
+  | 6 => jacobianRow06 j
+  | 7 => jacobianRow07 j
+  | 8 => jacobianRow08 j
+  | 9 => jacobianRow09 j
+  | 10 => jacobianRow10 j
+  | 11 => jacobianRow11 j
+  | 12 => jacobianRow12 j
+  | 13 => jacobianRow13 j
+  | 14 => jacobianRow14 j
+  | 15 => jacobianRow15 j
+  | 16 => jacobianRow16 j
+  | 17 => jacobianRow17 j
+  | 18 => jacobianRow18 j
+  | 19 => jacobianRow19 j
+  | 20 => jacobianRow20 j
+  | _ => jacobianRow21 j
+
+theorem jacobianRow00_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (0 : Fin 22) j =
+      jacobianCache (0 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow01_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (1 : Fin 22) j =
+      jacobianCache (1 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow02_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (2 : Fin 22) j =
+      jacobianCache (2 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow03_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (3 : Fin 22) j =
+      jacobianCache (3 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow04_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (4 : Fin 22) j =
+      jacobianCache (4 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow05_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (5 : Fin 22) j =
+      jacobianCache (5 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow06_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (6 : Fin 22) j =
+      jacobianCache (6 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow07_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (7 : Fin 22) j =
+      jacobianCache (7 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow08_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (8 : Fin 22) j =
+      jacobianCache (8 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow09_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (9 : Fin 22) j =
+      jacobianCache (9 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow10_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (10 : Fin 22) j =
+      jacobianCache (10 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow11_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (11 : Fin 22) j =
+      jacobianCache (11 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow12_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (12 : Fin 22) j =
+      jacobianCache (12 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow13_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (13 : Fin 22) j =
+      jacobianCache (13 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow14_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (14 : Fin 22) j =
+      jacobianCache (14 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow15_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (15 : Fin 22) j =
+      jacobianCache (15 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow16_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (16 : Fin 22) j =
+      jacobianCache (16 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow17_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (17 : Fin 22) j =
+      jacobianCache (17 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow18_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (18 : Fin 22) j =
+      jacobianCache (18 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow19_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (19 : Fin 22) j =
+      jacobianCache (19 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow20_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (20 : Fin 22) j =
+      jacobianCache (20 : Fin 22) j := by decide +kernel
+
+theorem jacobianRow21_eq : ∀ j : Fin 22,
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg (21 : Fin 22) j =
+      jacobianCache (21 : Fin 22) j := by decide +kernel
+
+theorem jacobianCache_eq :
+    intervalJacobian fullExpr (unitBox (n := 22)) cfg = jacobianCache := by
+  funext i j
+  fin_cases i
+  · exact jacobianRow00_eq j
+  · exact jacobianRow01_eq j
+  · exact jacobianRow02_eq j
+  · exact jacobianRow03_eq j
+  · exact jacobianRow04_eq j
+  · exact jacobianRow05_eq j
+  · exact jacobianRow06_eq j
+  · exact jacobianRow07_eq j
+  · exact jacobianRow08_eq j
+  · exact jacobianRow09_eq j
+  · exact jacobianRow10_eq j
+  · exact jacobianRow11_eq j
+  · exact jacobianRow12_eq j
+  · exact jacobianRow13_eq j
+  · exact jacobianRow14_eq j
+  · exact jacobianRow15_eq j
+  · exact jacobianRow16_eq j
+  · exact jacobianRow17_eq j
+  · exact jacobianRow18_eq j
+  · exact jacobianRow19_eq j
+  · exact jacobianRow20_eq j
+  · exact jacobianRow21_eq j
+
+end GerverSofa.PartALeanCert
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Lean Cert Gerver Numerics Reduced
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem reduced_row_0_lt :
+    intervalMatrixRowBound reducedPJ (0 : Fin 4) < qTarget := by
+  decide +kernel
+theorem reduced_row_1_lt :
+    intervalMatrixRowBound reducedPJ (1 : Fin 4) < qTarget := by
+  decide +kernel
+theorem reduced_row_2_lt :
+    intervalMatrixRowBound reducedPJ (2 : Fin 4) < qTarget := by
+  decide +kernel
+theorem reduced_row_3_lt :
+    intervalMatrixRowBound reducedPJ (3 : Fin 4) < qTarget := by
+  decide +kernel
+
+theorem reduced_bound_lt : intervalMatrixBound reducedPJ < qTarget := by
+  apply matrixBound_lt_of_rows reducedPJ qTarget qTarget_pos
+  intro i
+  fin_cases i
+  · exact reduced_row_0_lt
+  · exact reduced_row_1_lt
+  · exact reduced_row_2_lt
+  · exact reduced_row_3_lt
+
+theorem reduced_bound_le : intervalMatrixBound reducedPJ ≤ qTarget :=
+  reduced_bound_lt.le
+theorem reduced_image_0_inside :
+    intervalStrictInside (reducedImageQ (0 : Fin 4))
+      (unitBox (n := 4) (0 : Fin 4)) = true := by
+  decide +kernel
+theorem reduced_image_1_inside :
+    intervalStrictInside (reducedImageQ (1 : Fin 4))
+      (unitBox (n := 4) (1 : Fin 4)) = true := by
+  decide +kernel
+theorem reduced_image_2_inside :
+    intervalStrictInside (reducedImageQ (2 : Fin 4))
+      (unitBox (n := 4) (2 : Fin 4)) = true := by
+  decide +kernel
+theorem reduced_image_3_inside :
+    intervalStrictInside (reducedImageQ (3 : Fin 4))
+      (unitBox (n := 4) (3 : Fin 4)) = true := by
+  decide +kernel
+
+theorem reduced_images_inside :
+    ∀ i : Fin 4,
+      intervalStrictInside (reducedImageQ i) (unitBox (n := 4) i) = true := by
+  intro i
+  fin_cases i
+  · exact reduced_image_0_inside
+  · exact reduced_image_1_inside
+  · exact reduced_image_2_inside
+  · exact reduced_image_3_inside
+
+theorem reduced_unique_scaled :
+    ∃! u, FinBoxMem u (unitBox (n := 4)) ∧ SystemZero reducedExpr u := by
+  exact uniqueSystemZero_of_certified_contraction
+    reducedExpr reducedExpr_supported (unitBox (n := 4)) (zeroCenter (n := 4))
+    reduced_center_mem reducedY cfg qTarget qTarget_nonneg qTarget_lt_one
+    (by simpa [reducedPJ] using reduced_bound_le)
+    reduced_images_inside
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.LeanCertNumericsRows.Full00`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint00`.
+* `KernelOnly.LeanCertNumericsRows.FullImage00`.
+* `KernelOnly.LeanCertNumericsRows.Full01`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint01`.
+* `KernelOnly.LeanCertNumericsRows.FullImage01`.
+* `KernelOnly.LeanCertNumericsRows.Full02`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint02`.
+* `KernelOnly.LeanCertNumericsRows.FullImage02`.
+* `KernelOnly.LeanCertNumericsRows.Full03`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint03`.
+* `KernelOnly.LeanCertNumericsRows.FullImage03`.
+* `KernelOnly.LeanCertNumericsRows.Full04`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint04`.
+* `KernelOnly.LeanCertNumericsRows.FullImage04`.
+* `KernelOnly.LeanCertNumericsRows.Full05`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Direct 22D certificate, row 0: Jacobian bound
+
+The row modules form a deliberate dependency chain beginning after the
+already-certified reduced 4D module.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_0_lt :
+    intervalMatrixRowBound fullPJ (0 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 0
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_0_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (0 : Fin 22))
+      (fullPointCache (0 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_0_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (0 : Fin 22) ∈
+      fullPointCache (0 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (0 : Fin 22))
+    full_point_0_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 0: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_0_inside :
+    intervalStrictInside (fullImageCachedQ (0 : Fin 22))
+      (unitBox (n := 22) (0 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 1: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_1_lt :
+    intervalMatrixRowBound fullPJ (1 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 1
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_1_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (1 : Fin 22))
+      (fullPointCache (1 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_1_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (1 : Fin 22) ∈
+      fullPointCache (1 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (1 : Fin 22))
+    full_point_1_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 1: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_1_inside :
+    intervalStrictInside (fullImageCachedQ (1 : Fin 22))
+      (unitBox (n := 22) (1 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 2: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_2_lt :
+    intervalMatrixRowBound fullPJ (2 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 2
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_2_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (2 : Fin 22))
+      (fullPointCache (2 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_2_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (2 : Fin 22) ∈
+      fullPointCache (2 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (2 : Fin 22))
+    full_point_2_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 2: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_2_inside :
+    intervalStrictInside (fullImageCachedQ (2 : Fin 22))
+      (unitBox (n := 22) (2 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 3: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_3_lt :
+    intervalMatrixRowBound fullPJ (3 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 3
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_3_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (3 : Fin 22))
+      (fullPointCache (3 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_3_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (3 : Fin 22) ∈
+      fullPointCache (3 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (3 : Fin 22))
+    full_point_3_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 3: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_3_inside :
+    intervalStrictInside (fullImageCachedQ (3 : Fin 22))
+      (unitBox (n := 22) (3 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 4: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_4_lt :
+    intervalMatrixRowBound fullPJ (4 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 4
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_4_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (4 : Fin 22))
+      (fullPointCache (4 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_4_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (4 : Fin 22) ∈
+      fullPointCache (4 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (4 : Fin 22))
+    full_point_4_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 4: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_4_inside :
+    intervalStrictInside (fullImageCachedQ (4 : Fin 22))
+      (unitBox (n := 22) (4 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 5: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_5_lt :
+    intervalMatrixRowBound fullPJ (5 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.LeanCertNumericsRows.FullPoint05`.
+* `KernelOnly.LeanCertNumericsRows.FullImage05`.
+* `KernelOnly.LeanCertNumericsRows.Full06`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint06`.
+* `KernelOnly.LeanCertNumericsRows.FullImage06`.
+* `KernelOnly.LeanCertNumericsRows.Full07`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint07`.
+* `KernelOnly.LeanCertNumericsRows.FullImage07`.
+* `KernelOnly.LeanCertNumericsRows.Full08`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint08`.
+* `KernelOnly.LeanCertNumericsRows.FullImage08`.
+* `KernelOnly.LeanCertNumericsRows.Full09`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint09`.
+* `KernelOnly.LeanCertNumericsRows.FullImage09`.
+* `KernelOnly.LeanCertNumericsRows.Full10`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint10`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Direct 22D certificate, point residual 5
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_5_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (5 : Fin 22))
+      (fullPointCache (5 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_5_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (5 : Fin 22) ∈
+      fullPointCache (5 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (5 : Fin 22))
+    full_point_5_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 5: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_5_inside :
+    intervalStrictInside (fullImageCachedQ (5 : Fin 22))
+      (unitBox (n := 22) (5 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 6: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_6_lt :
+    intervalMatrixRowBound fullPJ (6 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 6
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_6_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (6 : Fin 22))
+      (fullPointCache (6 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_6_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (6 : Fin 22) ∈
+      fullPointCache (6 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (6 : Fin 22))
+    full_point_6_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 6: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_6_inside :
+    intervalStrictInside (fullImageCachedQ (6 : Fin 22))
+      (unitBox (n := 22) (6 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 7: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_7_lt :
+    intervalMatrixRowBound fullPJ (7 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 7
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_7_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (7 : Fin 22))
+      (fullPointCache (7 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_7_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (7 : Fin 22) ∈
+      fullPointCache (7 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (7 : Fin 22))
+    full_point_7_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 7: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_7_inside :
+    intervalStrictInside (fullImageCachedQ (7 : Fin 22))
+      (unitBox (n := 22) (7 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 8: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_8_lt :
+    intervalMatrixRowBound fullPJ (8 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 8
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_8_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (8 : Fin 22))
+      (fullPointCache (8 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_8_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (8 : Fin 22) ∈
+      fullPointCache (8 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (8 : Fin 22))
+    full_point_8_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 8: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_8_inside :
+    intervalStrictInside (fullImageCachedQ (8 : Fin 22))
+      (unitBox (n := 22) (8 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 9: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_9_lt :
+    intervalMatrixRowBound fullPJ (9 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 9
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_9_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (9 : Fin 22))
+      (fullPointCache (9 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_9_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (9 : Fin 22) ∈
+      fullPointCache (9 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (9 : Fin 22))
+    full_point_9_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 9: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_9_inside :
+    intervalStrictInside (fullImageCachedQ (9 : Fin 22))
+      (unitBox (n := 22) (9 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 10: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_10_lt :
+    intervalMatrixRowBound fullPJ (10 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 10
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_10_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (10 : Fin 22))
+      (fullPointCache (10 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_10_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (10 : Fin 22) ∈
+      fullPointCache (10 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (10 : Fin 22))
+    full_point_10_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.LeanCertNumericsRows.FullImage10`.
+* `KernelOnly.LeanCertNumericsRows.Full11`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint11`.
+* `KernelOnly.LeanCertNumericsRows.FullImage11`.
+* `KernelOnly.LeanCertNumericsRows.Full12`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint12`.
+* `KernelOnly.LeanCertNumericsRows.FullImage12`.
+* `KernelOnly.LeanCertNumericsRows.Full13`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint13`.
+* `KernelOnly.LeanCertNumericsRows.FullImage13`.
+* `KernelOnly.LeanCertNumericsRows.Full14`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint14`.
+* `KernelOnly.LeanCertNumericsRows.FullImage14`.
+* `KernelOnly.LeanCertNumericsRows.Full15`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint15`.
+* `KernelOnly.LeanCertNumericsRows.FullImage15`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Direct 22D certificate, row 10: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_10_inside :
+    intervalStrictInside (fullImageCachedQ (10 : Fin 22))
+      (unitBox (n := 22) (10 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 11: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_11_lt :
+    intervalMatrixRowBound fullPJ (11 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 11
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_11_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (11 : Fin 22))
+      (fullPointCache (11 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_11_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (11 : Fin 22) ∈
+      fullPointCache (11 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (11 : Fin 22))
+    full_point_11_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 11: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_11_inside :
+    intervalStrictInside (fullImageCachedQ (11 : Fin 22))
+      (unitBox (n := 22) (11 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 12: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_12_lt :
+    intervalMatrixRowBound fullPJ (12 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 12
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_12_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (12 : Fin 22))
+      (fullPointCache (12 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_12_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (12 : Fin 22) ∈
+      fullPointCache (12 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (12 : Fin 22))
+    full_point_12_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 12: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_12_inside :
+    intervalStrictInside (fullImageCachedQ (12 : Fin 22))
+      (unitBox (n := 22) (12 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 13: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_13_lt :
+    intervalMatrixRowBound fullPJ (13 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 13
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_13_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (13 : Fin 22))
+      (fullPointCache (13 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_13_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (13 : Fin 22) ∈
+      fullPointCache (13 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (13 : Fin 22))
+    full_point_13_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 13: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_13_inside :
+    intervalStrictInside (fullImageCachedQ (13 : Fin 22))
+      (unitBox (n := 22) (13 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 14: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_14_lt :
+    intervalMatrixRowBound fullPJ (14 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 14
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_14_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (14 : Fin 22))
+      (fullPointCache (14 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_14_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (14 : Fin 22) ∈
+      fullPointCache (14 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (14 : Fin 22))
+    full_point_14_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 14: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_14_inside :
+    intervalStrictInside (fullImageCachedQ (14 : Fin 22))
+      (unitBox (n := 22) (14 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 15: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_15_lt :
+    intervalMatrixRowBound fullPJ (15 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 15
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_15_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (15 : Fin 22))
+      (fullPointCache (15 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_15_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (15 : Fin 22) ∈
+      fullPointCache (15 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (15 : Fin 22))
+    full_point_15_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 15: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_15_inside :
+    intervalStrictInside (fullImageCachedQ (15 : Fin 22))
+      (unitBox (n := 22) (15 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.LeanCertNumericsRows.Full16`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint16`.
+* `KernelOnly.LeanCertNumericsRows.FullImage16`.
+* `KernelOnly.LeanCertNumericsRows.Full17`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint17`.
+* `KernelOnly.LeanCertNumericsRows.FullImage17`.
+* `KernelOnly.LeanCertNumericsRows.Full18`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint18`.
+* `KernelOnly.LeanCertNumericsRows.FullImage18`.
+* `KernelOnly.LeanCertNumericsRows.Full19`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint19`.
+* `KernelOnly.LeanCertNumericsRows.FullImage19`.
+* `KernelOnly.LeanCertNumericsRows.Full20`.
+* `KernelOnly.LeanCertNumericsRows.FullPoint20`.
+* `KernelOnly.LeanCertNumericsRows.FullImage20`.
+* `KernelOnly.LeanCertNumericsRows.Full21`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Direct 22D certificate, row 16: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_16_lt :
+    intervalMatrixRowBound fullPJ (16 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 16
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_16_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (16 : Fin 22))
+      (fullPointCache (16 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_16_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (16 : Fin 22) ∈
+      fullPointCache (16 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (16 : Fin 22))
+    full_point_16_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 16: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_16_inside :
+    intervalStrictInside (fullImageCachedQ (16 : Fin 22))
+      (unitBox (n := 22) (16 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 17: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_17_lt :
+    intervalMatrixRowBound fullPJ (17 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 17
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_17_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (17 : Fin 22))
+      (fullPointCache (17 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_17_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (17 : Fin 22) ∈
+      fullPointCache (17 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (17 : Fin 22))
+    full_point_17_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 17: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_17_inside :
+    intervalStrictInside (fullImageCachedQ (17 : Fin 22))
+      (unitBox (n := 22) (17 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 18: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_18_lt :
+    intervalMatrixRowBound fullPJ (18 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 18
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_18_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (18 : Fin 22))
+      (fullPointCache (18 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_18_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (18 : Fin 22) ∈
+      fullPointCache (18 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (18 : Fin 22))
+    full_point_18_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 18: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_18_inside :
+    intervalStrictInside (fullImageCachedQ (18 : Fin 22))
+      (unitBox (n := 22) (18 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 19: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_19_lt :
+    intervalMatrixRowBound fullPJ (19 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 19
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_19_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (19 : Fin 22))
+      (fullPointCache (19 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_19_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (19 : Fin 22) ∈
+      fullPointCache (19 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (19 : Fin 22))
+    full_point_19_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 19: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_19_inside :
+    intervalStrictInside (fullImageCachedQ (19 : Fin 22))
+      (unitBox (n := 22) (19 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 20: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_20_lt :
+    intervalMatrixRowBound fullPJ (20 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, point residual 20
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_20_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (20 : Fin 22))
+      (fullPointCache (20 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_20_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (20 : Fin 22) ∈
+      fullPointCache (20 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (20 : Fin 22))
+    full_point_20_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 20: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_20_inside :
+    intervalStrictInside (fullImageCachedQ (20 : Fin 22))
+      (unitBox (n := 22) (20 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 21: Jacobian bound
+
+The row modules form a deliberate dependency chain.  Lake therefore checks
+only one expensive closed kernel proposition at a time instead of launching
+all 22 rows concurrently and exhausting RAM.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_row_21_lt :
+    intervalMatrixRowBound fullPJ (21 : Fin 22) < qTarget := by
+  unfold fullPJ
+  rw [jacobianCache_eq]
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.LeanCertNumericsRows.FullPoint21`.
+* `KernelOnly.LeanCertNumericsRows.FullImage21`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Direct 22D certificate, point residual 21
+
+The expensive transcendental evaluation is checked once for this single
+coordinate and then replaced by a small rational cache interval downstream.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_point_21_contained :
+    intervalContained
+      (pointEvalIntervalsKernel fullExpr (zeroCenter (n := 22)) cfg (21 : Fin 22))
+      (fullPointCache (21 : Fin 22)) := by
+  constructor <;> decide +kernel
+
+theorem full_point_21_mem :
+    systemEval fullExpr
+        (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) (21 : Fin 22) ∈
+      fullPointCache (21 : Fin 22) :=
+  mem_of_intervalContained
+    (systemEval_mem_pointEvalIntervalsKernel_const
+      fullExpr fullExpr_supported (zeroCenter (n := 22)) cfg (21 : Fin 22))
+    full_point_21_contained
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct 22D certificate, row 21: strict self-map image
+
+This theorem is isolated from the Jacobian-row theorem so each Lean process
+checks one heavy proposition and then releases its memory before the next
+module in the serial chain begins.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+theorem full_image_21_inside :
+    intervalStrictInside (fullImageCachedQ (21 : Fin 22))
+      (unitBox (n := 22) (21 : Fin 22)) = true := by
+  decide +kernel
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.LeanCertGerverNumerics`.
+* `KernelOnly.ConcreteUniqueZeros`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Final LeanCert numerical certificate for Gerver
+
+All expensive 22D checks are already cached in a serial dependency chain
+ending at `FullImage21`.  This module only assembles them into the global
+norm/self-map facts and the unique-root theorem.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartALeanCert
+
+open LeanCert.Core
+open LeanCert.Engine
+
+theorem full_bound_lt : intervalMatrixBound fullPJ < qTarget := by
+  apply matrixBound_lt_of_rows fullPJ qTarget qTarget_pos
+  intro i
+  fin_cases i
+  · exact full_row_0_lt
+  · exact full_row_1_lt
+  · exact full_row_2_lt
+  · exact full_row_3_lt
+  · exact full_row_4_lt
+  · exact full_row_5_lt
+  · exact full_row_6_lt
+  · exact full_row_7_lt
+  · exact full_row_8_lt
+  · exact full_row_9_lt
+  · exact full_row_10_lt
+  · exact full_row_11_lt
+  · exact full_row_12_lt
+  · exact full_row_13_lt
+  · exact full_row_14_lt
+  · exact full_row_15_lt
+  · exact full_row_16_lt
+  · exact full_row_17_lt
+  · exact full_row_18_lt
+  · exact full_row_19_lt
+  · exact full_row_20_lt
+  · exact full_row_21_lt
+
+theorem full_bound_le : intervalMatrixBound fullPJ ≤ qTarget :=
+  full_bound_lt.le
+
+theorem full_point_values_mem :
+    ∀ j : Fin 22,
+      systemEval fullExpr
+          (fun k => ((zeroCenter (n := 22) k : ℚ) : ℝ)) j ∈
+        fullPointCache j := by
+  intro j
+  fin_cases j
+  · exact full_point_0_mem
+  · exact full_point_1_mem
+  · exact full_point_2_mem
+  · exact full_point_3_mem
+  · exact full_point_4_mem
+  · exact full_point_5_mem
+  · exact full_point_6_mem
+  · exact full_point_7_mem
+  · exact full_point_8_mem
+  · exact full_point_9_mem
+  · exact full_point_10_mem
+  · exact full_point_11_mem
+  · exact full_point_12_mem
+  · exact full_point_13_mem
+  · exact full_point_14_mem
+  · exact full_point_15_mem
+  · exact full_point_16_mem
+  · exact full_point_17_mem
+  · exact full_point_18_mem
+  · exact full_point_19_mem
+  · exact full_point_20_mem
+  · exact full_point_21_mem
+
+theorem full_images_inside :
+    ∀ i : Fin 22,
+      intervalStrictInside (fullImageCachedQ i) (unitBox (n := 22) i) = true := by
+  intro i
+  fin_cases i
+  · exact full_image_0_inside
+  · exact full_image_1_inside
+  · exact full_image_2_inside
+  · exact full_image_3_inside
+  · exact full_image_4_inside
+  · exact full_image_5_inside
+  · exact full_image_6_inside
+  · exact full_image_7_inside
+  · exact full_image_8_inside
+  · exact full_image_9_inside
+  · exact full_image_10_inside
+  · exact full_image_11_inside
+  · exact full_image_12_inside
+  · exact full_image_13_inside
+  · exact full_image_14_inside
+  · exact full_image_15_inside
+  · exact full_image_16_inside
+  · exact full_image_17_inside
+  · exact full_image_18_inside
+  · exact full_image_19_inside
+  · exact full_image_20_inside
+  · exact full_image_21_inside
+
+theorem full_unique_scaled :
+    ∃! u, FinBoxMem u (unitBox (n := 22)) ∧ SystemZero fullExpr u := by
+  exact uniqueSystemZero_of_certified_contraction_values
+    fullExpr fullExpr_supported (unitBox (n := 22)) (zeroCenter (n := 22))
+    full_center_mem fullY cfg qTarget qTarget_nonneg qTarget_lt_one
+    (by simpa [fullPJ] using full_bound_le)
+    fullPointCache full_point_values_mem full_images_inside
+
+end PartALeanCert
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Concrete Unique Zeros
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+
+open LeanCert.Engine
+
+namespace PartALeanCert
+
+/-- The unique normalized reduced-system zero selected from its existence certificate. -/
+def reducedRootU : Fin 4 → ℝ := Classical.choose reduced_unique_scaled
+/-- The unique normalized full-system zero selected from its existence certificate. -/
+def fullRootU : Fin 22 → ℝ := Classical.choose full_unique_scaled
+
+theorem reducedRootU_spec :
+    FinBoxMem reducedRootU (unitBox (n := 4)) ∧
+      SystemZero reducedExpr reducedRootU :=
+  (Classical.choose_spec reduced_unique_scaled).1
+
+theorem fullRootU_spec :
+    FinBoxMem fullRootU (unitBox (n := 22)) ∧
+      SystemZero fullExpr fullRootU :=
+  (Classical.choose_spec full_unique_scaled).1
+
+/-- The certified reduced-system root in the original box coordinates. -/
+def reducedRoot : Vec 4 := reducedAffine reducedRootU
+/-- The certified full-system root in the original Romik parameter coordinates. -/
+def fullRoot : Vec 22 := fullAffine fullRootU
+
+theorem reducedRoot_mem : reducedRoot ∈ Reduced.vectorBox :=
+  reducedAffine_mem reducedRootU_spec.1
+
+theorem fullRoot_mem : fullRoot ∈ Romik.vectorBox :=
+  fullAffine_mem fullRootU_spec.1
+
+theorem reducedRoot_zero : Reduced.vectorSystem reducedRoot = 0 := by
+  have hz : systemEval reducedExpr reducedRootU = 0 := by
+    funext i
+    exact reducedRootU_spec.2 i
+  rw [reduced_systemEval_eq] at hz
+  exact hz
+
+theorem fullRoot_zero : Romik.vectorSystem fullRoot = 0 := by
+  have hz : systemEval fullExpr fullRootU = 0 := by
+    funext i
+    exact fullRootU_spec.2 i
+  rw [full_systemEval_eq] at hz
+  exact hz
+
+/-- Concrete reduced 4D unique zero — no assumptions. -/
+def reducedCertifiedUniqueZero :
+    CertifiedUniqueZero Reduced.vectorSystem Reduced.vectorBox where
+  solution := reducedRoot
+  solution_mem := reducedRoot_mem
+  satisfies := reducedRoot_zero
+  unique_iff := by
+    intro y hy
+    constructor
+    · intro hzero
+      let uy := reducedNormalize y
+      have huy : FinBoxMem uy (unitBox (n := 4)) := reducedNormalize_mem hy
+      have hsys : SystemZero reducedExpr uy := by
+        intro i
+        change systemEval reducedExpr uy i = 0
+        calc
+          systemEval reducedExpr uy i =
+              Reduced.vectorSystem (reducedAffine uy) i :=
+            congrFun (reduced_systemEval_eq uy) i
+          _ = Reduced.vectorSystem y i := by
+            rw [show reducedAffine uy = y by
+              simpa [uy] using reducedAffine_normalize y]
+          _ = 0 := congrFun hzero i
+      have huEq : uy = reducedRootU := by
+        exact reduced_unique_scaled.unique ⟨huy, hsys⟩ reducedRootU_spec
+      calc
+        y = reducedAffine uy := by symm; exact reducedAffine_normalize y
+        _ = reducedAffine reducedRootU := by rw [huEq]
+        _ = reducedRoot := rfl
+    · rintro rfl
+      exact reducedRoot_zero
+
+/-- Existing manuscript-level unique-solution interfaces are now discharged
+by concrete numerical certificates rather than assumptions. -/
+def reducedCertifiedUniqueSolution :
+    CertifiedUniqueSolution Reduced.Equations Reduced.box :=
+  Reduced.uniqueSolutionOfVector reducedCertifiedUniqueZero
+
+end PartALeanCert
+
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.DeepMindABPhiThetaBridge`.
+* `KernelOnly.PartE.TwoAngleReduction`.
+* `KernelOnly.PartE.ScaledResidualInterval`.
+* `KernelOnly.PartE.FiniteCoverReplay`.
+* `KernelOnly.PartE.LocalTwoAngleKrawczykClosure`.
+* `KernelOnly.PartE.AdaptiveGlobalCoverClosure`.
+* `KernelOnly.PartE.E24AlignedRegionFoundation`.
+* `KernelOnly.PartE.E24AlignedLogic`.
+* `KernelOnly.PartE.E24KC2ProofHelpers`.
+* `KernelOnly.PartE.E24PhiBelowKernelHL`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Part E01: bridge to DeepMind's Gerver-constant specification
+
+This module mirrors the four equations and the physical domain used by
+`GerversSofa.ABφθSpec` in `google-deepmind/formal-conjectures`.  It proves that
+the four displayed equations are exactly the already certified reduced Gerver
+system, proves that the certified rational box lies in the physical domain,
+and reduces the tuple-shaped global uniqueness statement to one explicit
+global enclosure target.
+
+The enclosure target is a proposition passed as an ordinary theorem argument.
+E01 does not claim that the global exclusion step has already been proved.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+/-- Reduced parameters assembled in the order `(A, B, phi, theta)`. -/
+def reducedParams (A B phi theta : ℝ) : Reduced.Params :=
+  { a := A, b := B, phi := phi, theta := theta }
+
+/-- The physical domain appearing in DeepMind's `ABφθSpec`. -/
+def PhysicalDomain (p : Reduced.Params) : Prop :=
+  0 ≤ p.phi ∧ p.phi ≤ p.theta ∧ p.theta ≤ Real.pi / 4 ∧
+    0 ≤ p.a ∧ 0 ≤ p.b
+
+/-- The four displayed equations in DeepMind's `ABφθSpec`. -/
+def DeepMindEquations (A B phi theta : ℝ) : Prop :=
+  A * (Real.cos theta - Real.cos phi) - 2 * B * Real.sin phi
+      + (theta - phi - 1) * Real.cos theta - Real.sin theta
+      + Real.cos phi + Real.sin phi = 0 ∧
+  A * (3 * Real.sin theta + Real.sin phi) - 2 * B * Real.cos phi
+      + 3 * (theta - phi - 1) * Real.sin theta + 3 * Real.cos theta
+      - Real.sin phi + Real.cos phi = 0 ∧
+  A * Real.cos phi
+      - (Real.sin phi + 1 / 2 - Real.cos phi / 2 + B * Real.sin phi) = 0 ∧
+  (A + Real.pi / 2 - phi - theta)
+      - (B - (theta - phi) * (1 + A) / 2 - (theta - phi) ^ 2 / 4) = 0
+
+/-- A local mirror of DeepMind's complete four-constant specification. -/
+def DeepMindABPhiThetaSpec (A B phi theta : ℝ) : Prop :=
+  PhysicalDomain (reducedParams A B phi theta) ∧
+    DeepMindEquations A B phi theta
+
+/-- Explicit equivalence between DeepMind's four equations and the certified
+reduced Gerver system. -/
+theorem deepMindEquations_iff_reducedEquations (A B phi theta : ℝ) :
+    DeepMindEquations A B phi theta ↔
+      Reduced.Equations (reducedParams A B phi theta) := by
+  constructor
+  · rintro ⟨h0, h1, h2, h3⟩
+    change Reduced.system (reducedParams A B phi theta) = 0
+    funext i
+    fin_cases i
+    · change
+        A * (Real.cos theta - Real.cos phi) - 2 * B * Real.sin phi
+            + (theta - phi - 1) * Real.cos theta - Real.sin theta
+            + Real.cos phi + Real.sin phi = 0
+      exact h0
+    · change
+        A * (3 * Real.sin theta + Real.sin phi) - 2 * B * Real.cos phi
+            + 3 * (theta - phi - 1) * Real.sin theta + 3 * Real.cos theta
+            - Real.sin phi + Real.cos phi = 0
+      exact h1
+    · change
+        A * Real.cos phi - Real.sin phi - 1 / 2
+            + 1 / 2 * Real.cos phi - B * Real.sin phi = 0
+      ring_nf at h2 ⊢
+      exact h2
+    · change
+        A + Real.pi / 2 - phi - theta - B
+            + 1 / 2 * (theta - phi) * (1 + A)
+            + 1 / 4 * (theta - phi) * (theta - phi) = 0
+      ring_nf at h3 ⊢
+      exact h3
+  · intro h
+    change Reduced.system (reducedParams A B phi theta) = 0 at h
+    have h0 := congrFun h (0 : Fin 4)
+    have h1 := congrFun h (1 : Fin 4)
+    have h2 := congrFun h (2 : Fin 4)
+    have h3 := congrFun h (3 : Fin 4)
+    change
+      A * (Real.cos theta - Real.cos phi) - 2 * B * Real.sin phi
+          + (theta - phi - 1) * Real.cos theta - Real.sin theta
+          + Real.cos phi + Real.sin phi = 0 at h0
+    change
+      A * (3 * Real.sin theta + Real.sin phi) - 2 * B * Real.cos phi
+          + 3 * (theta - phi - 1) * Real.sin theta + 3 * Real.cos theta
+          - Real.sin phi + Real.cos phi = 0 at h1
+    change
+      A * Real.cos phi - Real.sin phi - 1 / 2
+          + 1 / 2 * Real.cos phi - B * Real.sin phi = 0 at h2
+    change
+      A + Real.pi / 2 - phi - theta - B
+          + 1 / 2 * (theta - phi) * (1 + A)
+          + 1 / 4 * (theta - phi) * (theta - phi) = 0 at h3
+    refine ⟨h0, h1, ?_, ?_⟩
+    · ring_nf at h2 ⊢
+      exact h2
+    · ring_nf at h3 ⊢
+      exact h3
+
+/-- Every point of the certified reduced box satisfies DeepMind's broad
+physical-domain inequalities. -/
+theorem physicalDomain_of_mem_reducedBox {p : Reduced.Params}
+    (hp : p ∈ Reduced.box) : PhysicalDomain p := by
+  dsimp [Reduced.box, qR] at hp
+  rcases hp with
+    ⟨haLo, _haHi, hbLo, _hbHi, hphiLo, hphiHi, hthetaLo, hthetaHi⟩
+  have hphiNonneg : 0 ≤ p.phi := by
+    have hconst :
+        (0 : ℝ) ≤ (122429264969 : ℝ) / 3125000000000 := by
+      norm_num
+    exact hconst.trans hphiLo
+  have hphiTheta : p.phi ≤ p.theta := by
+    have hgap :
+        (3917736479009 : ℝ) / 100000000000000 ≤
+          (2129067216821 : ℝ) / 3125000000000 := by
+      norm_num
+    exact hphiHi.trans (hgap.trans hthetaLo)
+  have hthetaPi : p.theta ≤ Real.pi / 4 := by
+    have hrat :
+        (68130150938273 : ℝ) / 100000000000000 < (3 : ℝ) / 4 := by
+      norm_num
+    have hthree : (3 : ℝ) / 4 < Real.pi / 4 := by
+      nlinarith [Real.pi_gt_three]
+    exact hthetaHi.trans (le_of_lt (hrat.trans hthree))
+  have haNonneg : 0 ≤ p.a := by
+    have hconst :
+        (0 : ℝ) ≤ (1888531216873 : ℝ) / 20000000000000 := by
+      norm_num
+    exact hconst.trans haLo
+  have hbNonneg : 0 ≤ p.b := by
+    have hconst :
+        (0 : ℝ) ≤ (69960186366677 : ℝ) / 50000000000000 := by
+      norm_num
+    exact hconst.trans hbLo
+  exact ⟨hphiNonneg, hphiTheta, hthetaPi, haNonneg, hbNonneg⟩
+
+/-- The mirrored DeepMind specification is precisely physical-domain
+membership plus the already named reduced equations. -/
+theorem deepMindSpec_iff_physicalDomain_and_reducedEquations
+    (A B phi theta : ℝ) :
+    DeepMindABPhiThetaSpec A B phi theta ↔
+      PhysicalDomain (reducedParams A B phi theta) ∧
+        Reduced.Equations (reducedParams A B phi theta) := by
+  unfold DeepMindABPhiThetaSpec
+  rw [deepMindEquations_iff_reducedEquations]
+
+/-- A certified-box solution of the reduced system is automatically a
+solution of the mirrored DeepMind specification. -/
+theorem deepMindSpec_of_mem_reducedBox_and_equations {p : Reduced.Params}
+    (hp : p ∈ Reduced.box) (heq : Reduced.Equations p) :
+    DeepMindABPhiThetaSpec p.a p.b p.phi p.theta := by
+  apply
+    (deepMindSpec_iff_physicalDomain_and_reducedEquations
+      p.a p.b p.phi p.theta).2
+  simpa [reducedParams] using
+    And.intro (physicalDomain_of_mem_reducedBox hp) heq
+
+/-- The sole new mathematical target left after E01: every physical solution
+of the reduced equations lies in the already certified rational box. -/
+def GlobalEnclosureTarget : Prop :=
+  ∀ p : Reduced.Params,
+    PhysicalDomain p → Reduced.Equations p → p ∈ Reduced.box
+
+/-- Coordinate equivalence between DeepMind's nested tuple and the named
+reduced-parameter record. -/
+def tupleEquiv : (ℝ × ℝ × ℝ × ℝ) ≃ Reduced.Params where
+  toFun x := reducedParams x.1 x.2.1 x.2.2.1 x.2.2.2
+  invFun p := (p.a, p.b, p.phi, p.theta)
+  left_inv x := by
+    rcases x with ⟨A, B, phi, theta⟩
+    rfl
+  right_inv p := by
+    cases p
+    rfl
+
+/-- Once the explicit global enclosure theorem is supplied, the existing
+kernel-checked local certificate yields the exact tuple-shaped uniqueness
+statement required by DeepMind. -/
+theorem deepMindABPhiTheta_existsUnique_of_globalEnclosure
+    (hglobal : GlobalEnclosureTarget) :
+    ∃! ABphiTheta : ℝ × ℝ × ℝ × ℝ,
+      DeepMindABPhiThetaSpec
+        ABphiTheta.1 ABphiTheta.2.1 ABphiTheta.2.2.1 ABphiTheta.2.2.2 := by
+  let cert := PartALeanCert.reducedCertifiedUniqueSolution
+  refine ⟨tupleEquiv.symm cert.solution, ?_, ?_⟩
+  · change
+      DeepMindABPhiThetaSpec
+        cert.solution.a cert.solution.b cert.solution.phi cert.solution.theta
+    exact
+      deepMindSpec_of_mem_reducedBox_and_equations
+        cert.solution_mem cert.satisfies
+  · intro y hy
+    apply tupleEquiv.injective
+    rw [tupleEquiv.apply_symm_apply]
+    apply cert.unique (tupleEquiv y)
+    · apply hglobal (tupleEquiv y)
+      · exact
+          ((deepMindSpec_iff_physicalDomain_and_reducedEquations
+            y.1 y.2.1 y.2.2.1 y.2.2.2).1 hy).1
+      · exact
+          ((deepMindSpec_iff_physicalDomain_and_reducedEquations
+            y.1 y.2.1 y.2.2.1 y.2.2.2).1 hy).2
+    · exact
+        ((deepMindSpec_iff_physicalDomain_and_reducedEquations
+          y.1 y.2.1 y.2.2.1 y.2.2.2).1 hy).2
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part E02: exact reduction of the DeepMind system to two angles
+
+E01 identified the sole missing mathematical input as a global enclosure of
+all physical solutions of the four-variable reduced system.  E02 eliminates
+the two linear variables `A` and `B` exactly.
+
+The third and fourth Gerver equations first give `B` as an affine expression
+in `A`, and then give `A` as a quotient depending only on `(phi, theta)`.  The
+denominator is proved nonzero for every physical solution; this is a theorem,
+not an additional assumption.  Consequently existence of a physical
+four-variable solution is equivalent to a two-angle specification.
+
+The remaining enclosure target quantifies only over
+`0 <= phi <= theta <= pi/4`.  It is still an ordinary theorem argument: E02
+does not declare the interval branch-and-bound conclusion as an axiom.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+/-- Difference of the two switching angles. -/
+def angleDelta (phi theta : ℝ) : ℝ := theta - phi
+
+/-- Constant part of the fourth equation after solving it for `B`. -/
+def bBase (phi theta : ℝ) : ℝ :=
+  Real.pi / 2 - phi - theta + angleDelta phi theta / 2 +
+    angleDelta phi theta ^ 2 / 4
+
+/-- The value of `B` forced by the fourth equation once `A` is fixed. -/
+def bFromA (A phi theta : ℝ) : ℝ :=
+  A * (1 + angleDelta phi theta / 2) + bBase phi theta
+
+/-- Denominator obtained from the third equation after eliminating `B`. -/
+def angleDenominator (phi theta : ℝ) : ℝ :=
+  Real.cos phi - (1 + angleDelta phi theta / 2) * Real.sin phi
+
+/-- Numerator obtained from the third equation after eliminating `B`. -/
+def angleNumerator (phi theta : ℝ) : ℝ :=
+  Real.sin phi + 1 / 2 - Real.cos phi / 2 +
+    bBase phi theta * Real.sin phi
+
+/-- Reconstructed value of `A`, depending only on the two angles. -/
+def reconstructedA (phi theta : ℝ) : ℝ :=
+  angleNumerator phi theta / angleDenominator phi theta
+
+/-- Reconstructed value of `B`, depending only on the two angles. -/
+def reconstructedB (phi theta : ℝ) : ℝ :=
+  bFromA (reconstructedA phi theta) phi theta
+
+/-- The triangular physical domain for the two switching angles. -/
+def PhysicalAngleDomain (phi theta : ℝ) : Prop :=
+  0 ≤ phi ∧ phi ≤ theta ∧ theta ≤ Real.pi / 4
+
+/-- The last two displayed equations of the DeepMind system. -/
+def ThirdFourthEquations (A B phi theta : ℝ) : Prop :=
+  A * Real.cos phi
+      - (Real.sin phi + 1 / 2 - Real.cos phi / 2 + B * Real.sin phi) = 0 ∧
+  (A + Real.pi / 2 - phi - theta)
+      - (B - (theta - phi) * (1 + A) / 2 - (theta - phi) ^ 2 / 4) = 0
+
+/-- The fourth equation is exactly the affine reconstruction formula for `B`. -/
+theorem fourthEquation_iff_b_eq_bFromA (A B phi theta : ℝ) :
+    (A + Real.pi / 2 - phi - theta)
+          - (B - (theta - phi) * (1 + A) / 2 -
+            (theta - phi) ^ 2 / 4) = 0 ↔
+      B = bFromA A phi theta := by
+  constructor <;> intro h <;>
+    dsimp [bFromA, bBase, angleDelta] at h ⊢ <;>
+    ring_nf at h ⊢ <;>
+    linarith
+
+/-- After the fourth equation has reconstructed `B`, the third equation is
+exactly `A * denominator = numerator`. -/
+theorem thirdEquation_iff_mul_denominator_eq_numerator
+    (A B phi theta : ℝ) (hB : B = bFromA A phi theta) :
+    A * Real.cos phi
+          - (Real.sin phi + 1 / 2 - Real.cos phi / 2 +
+            B * Real.sin phi) = 0 ↔
+      A * angleDenominator phi theta = angleNumerator phi theta := by
+  subst B
+  constructor <;> intro h <;>
+    dsimp [bFromA, bBase, angleDelta, angleDenominator,
+      angleNumerator] at h ⊢ <;>
+    ring_nf at h ⊢ <;>
+    linarith
+
+/-- Exact simultaneous elimination statement for equations three and four. -/
+theorem thirdFourthEquations_iff_eliminated (A B phi theta : ℝ) :
+    ThirdFourthEquations A B phi theta ↔
+      B = bFromA A phi theta ∧
+        A * angleDenominator phi theta = angleNumerator phi theta := by
+  constructor
+  · rintro ⟨h3, h4⟩
+    have hB := (fourthEquation_iff_b_eq_bFromA A B phi theta).1 h4
+    exact ⟨hB,
+      (thirdEquation_iff_mul_denominator_eq_numerator
+        A B phi theta hB).1 h3⟩
+  · rintro ⟨hB, hA⟩
+    exact ⟨
+      (thirdEquation_iff_mul_denominator_eq_numerator
+        A B phi theta hB).2 hA,
+      (fourthEquation_iff_b_eq_bFromA A B phi theta).2 hB⟩
+
+/-- The physical four-variable domain projects to the triangular angle domain. -/
+theorem physicalAngleDomain_of_physicalDomain {A B phi theta : ℝ}
+    (h : PhysicalDomain (reducedParams A B phi theta)) :
+    PhysicalAngleDomain phi theta := by
+  exact ⟨h.1, h.2.1, h.2.2.1⟩
+
+/-- The constant part of the reconstructed `B` is nonnegative throughout the
+physical angle triangle. -/
+theorem bBase_nonneg_of_physicalAngleDomain {phi theta : ℝ}
+    (h : PhysicalAngleDomain phi theta) : 0 ≤ bBase phi theta := by
+  rcases h with ⟨hphi0, hphiTheta, htheta⟩
+  have hdelta : 0 ≤ theta - phi := sub_nonneg.mpr hphiTheta
+  have hsum : phi + theta ≤ Real.pi / 2 := by
+    nlinarith
+  have hsquare : 0 ≤ (theta - phi) ^ 2 := sq_nonneg (theta - phi)
+  dsimp [bBase, angleDelta]
+  nlinarith
+
+/-- The sine of the first switching angle is nonnegative in the physical
+triangle. -/
+theorem sin_phi_nonneg_of_physicalAngleDomain {phi theta : ℝ}
+    (h : PhysicalAngleDomain phi theta) : 0 ≤ Real.sin phi := by
+  rcases h with ⟨hphi0, hphiTheta, htheta⟩
+  apply Real.sin_nonneg_of_nonneg_of_le_pi hphi0
+  nlinarith [Real.pi_gt_three]
+
+/-- The cosine of the first switching angle is strictly positive in the
+physical triangle. -/
+theorem cos_phi_pos_of_physicalAngleDomain {phi theta : ℝ}
+    (h : PhysicalAngleDomain phi theta) : 0 < Real.cos phi := by
+  rcases h with ⟨hphi0, hphiTheta, htheta⟩
+  apply Real.cos_pos_of_mem_Ioo
+  constructor <;> nlinarith [Real.pi_pos]
+
+/-- No physical solution of all four equations can hit the apparent zero
+denominator of the two-angle reconstruction. -/
+theorem angleDenominator_ne_zero_of_physical_and_equations
+    {A B phi theta : ℝ}
+    (hdom : PhysicalDomain (reducedParams A B phi theta))
+    (heq : DeepMindEquations A B phi theta) :
+    angleDenominator phi theta ≠ 0 := by
+  have hang : PhysicalAngleDomain phi theta :=
+    physicalAngleDomain_of_physicalDomain hdom
+  have helim :
+      B = bFromA A phi theta ∧
+        A * angleDenominator phi theta = angleNumerator phi theta :=
+    (thirdFourthEquations_iff_eliminated A B phi theta).1
+      ⟨heq.2.2.1, heq.2.2.2⟩
+  intro hden
+  have hnumZero : angleNumerator phi theta = 0 := by
+    rw [← helim.2, hden, mul_zero]
+  have hsinNonneg : 0 ≤ Real.sin phi :=
+    sin_phi_nonneg_of_physicalAngleDomain hang
+  have hcosPos : 0 < Real.cos phi :=
+    cos_phi_pos_of_physicalAngleDomain hang
+  have hsinNe : Real.sin phi ≠ 0 := by
+    intro hsin
+    have hcosZero : Real.cos phi = 0 := by
+      dsimp [angleDenominator, angleDelta] at hden
+      rw [hsin, mul_zero, sub_zero] at hden
+      exact hden
+    exact (ne_of_gt hcosPos) hcosZero
+  have hsinPos : 0 < Real.sin phi :=
+    lt_of_le_of_ne hsinNonneg hsinNe.symm
+  have hbaseNonneg : 0 ≤ bBase phi theta :=
+    bBase_nonneg_of_physicalAngleDomain hang
+  have hcosGap : 0 ≤ 1 / 2 - Real.cos phi / 2 := by
+    nlinarith [Real.cos_le_one phi]
+  have hbaseSin : 0 ≤ bBase phi theta * Real.sin phi :=
+    mul_nonneg hbaseNonneg hsinNonneg
+  have hnumPos : 0 < angleNumerator phi theta := by
+    dsimp [angleNumerator]
+    nlinarith
+  exact (ne_of_gt hnumPos) hnumZero
+
+/-- Every physical four-variable solution has the reconstructed value of `A`. -/
+theorem a_eq_reconstructed_of_physical_and_equations
+    {A B phi theta : ℝ}
+    (hdom : PhysicalDomain (reducedParams A B phi theta))
+    (heq : DeepMindEquations A B phi theta) :
+    A = reconstructedA phi theta := by
+  have hden :=
+    angleDenominator_ne_zero_of_physical_and_equations hdom heq
+  have helim :=
+    (thirdFourthEquations_iff_eliminated A B phi theta).1
+      ⟨heq.2.2.1, heq.2.2.2⟩
+  exact (eq_div_iff hden).2 helim.2
+
+/-- Every physical four-variable solution has the reconstructed value of `B`. -/
+theorem b_eq_reconstructed_of_physical_and_equations
+    {A B phi theta : ℝ}
+    (hdom : PhysicalDomain (reducedParams A B phi theta))
+    (heq : DeepMindEquations A B phi theta) :
+    B = reconstructedB phi theta := by
+  have hA := a_eq_reconstructed_of_physical_and_equations hdom heq
+  have hB :=
+    ((thirdFourthEquations_iff_eliminated A B phi theta).1
+      ⟨heq.2.2.1, heq.2.2.2⟩).1
+  calc
+    B = bFromA A phi theta := hB
+    _ = bFromA (reconstructedA phi theta) phi theta := by rw [hA]
+    _ = reconstructedB phi theta := rfl
+
+/-- The first two equations after exact reconstruction of `A` and `B`. -/
+def TwoAngleEquations (phi theta : ℝ) : Prop :=
+  let A := reconstructedA phi theta
+  let B := reconstructedB phi theta
+  A * (Real.cos theta - Real.cos phi) - 2 * B * Real.sin phi
+      + (theta - phi - 1) * Real.cos theta - Real.sin theta
+      + Real.cos phi + Real.sin phi = 0 ∧
+  A * (3 * Real.sin theta + Real.sin phi) - 2 * B * Real.cos phi
+      + 3 * (theta - phi - 1) * Real.sin theta + 3 * Real.cos theta
+      - Real.sin phi + Real.cos phi = 0
+
+/-- Complete two-dimensional specification equivalent to existence of a
+physical solution with the given two angles. -/
+def TwoAngleSpec (phi theta : ℝ) : Prop :=
+  PhysicalAngleDomain phi theta ∧
+    angleDenominator phi theta ≠ 0 ∧
+    0 ≤ reconstructedA phi theta ∧
+    0 ≤ reconstructedB phi theta ∧
+    TwoAngleEquations phi theta
+
+/-- The reconstructed parameters assembled as a reduced-system record. -/
+def reconstructedParams (phi theta : ℝ) : Reduced.Params :=
+  reducedParams (reconstructedA phi theta) (reconstructedB phi theta) phi theta
+
+/-- Two-angle data reconstruct a physical solution of all four displayed
+DeepMind equations. -/
+theorem deepMindSpec_of_twoAngleSpec {phi theta : ℝ}
+    (h : TwoAngleSpec phi theta) :
+    DeepMindABPhiThetaSpec
+      (reconstructedA phi theta) (reconstructedB phi theta) phi theta := by
+  rcases h with ⟨hang, hden, hA0, hB0, h12⟩
+  have h34 : ThirdFourthEquations
+      (reconstructedA phi theta) (reconstructedB phi theta) phi theta := by
+    apply (thirdFourthEquations_iff_eliminated
+      (reconstructedA phi theta) (reconstructedB phi theta) phi theta).2
+    constructor
+    · rfl
+    · simp [reconstructedA, hden]
+  refine ⟨?_, ?_⟩
+  · exact ⟨hang.1, hang.2.1, hang.2.2, hA0, hB0⟩
+  · rcases h12 with ⟨h1, h2⟩
+    exact ⟨h1, h2, h34.1, h34.2⟩
+
+/-- Exact dimension reduction: for fixed angles, a physical four-variable
+solution exists if and only if the reconstructed two-angle specification
+holds. -/
+theorem exists_deepMindSpec_iff_twoAngleSpec (phi theta : ℝ) :
+    (∃ A B : ℝ, DeepMindABPhiThetaSpec A B phi theta) ↔
+      TwoAngleSpec phi theta := by
+  constructor
+  · rintro ⟨A, B, hspec⟩
+    have hden := angleDenominator_ne_zero_of_physical_and_equations
+      hspec.1 hspec.2
+    have hA := a_eq_reconstructed_of_physical_and_equations hspec.1 hspec.2
+    have hB := b_eq_reconstructed_of_physical_and_equations hspec.1 hspec.2
+    refine ⟨physicalAngleDomain_of_physicalDomain hspec.1, hden, ?_, ?_, ?_⟩
+    · have hA0 : 0 ≤ A := by
+        simpa only [reducedParams] using hspec.1.2.2.2.1
+      rw [hA] at hA0
+      exact hA0
+    · have hB0 : 0 ≤ B := by
+        simpa only [reducedParams] using hspec.1.2.2.2.2
+      rw [hB] at hB0
+      exact hB0
+    · rcases hspec.2 with ⟨h1, h2, _h3, _h4⟩
+      simpa [TwoAngleEquations, hA, hB] using And.intro h1 h2
+  · intro h
+    exact ⟨reconstructedA phi theta, reconstructedB phi theta,
+      deepMindSpec_of_twoAngleSpec h⟩
+
+/-- The sole mathematical target left after E02.  Unlike E01's four-variable
+target, this quantifies only over the compact triangle of the two angles. -/
+def TwoAngleEnclosureTarget : Prop :=
+  ∀ phi theta : ℝ,
+    TwoAngleSpec phi theta → reconstructedParams phi theta ∈ Reduced.box
+
+/-- A proof of the two-angle enclosure target yields E01's full global
+enclosure target. -/
+theorem globalEnclosureTarget_of_twoAngleEnclosure
+    (hangle : TwoAngleEnclosureTarget) : GlobalEnclosureTarget := by
+  intro p hdom heq
+  have hdeep : DeepMindEquations p.a p.b p.phi p.theta :=
+    (deepMindEquations_iff_reducedEquations
+      p.a p.b p.phi p.theta).2 (by simpa [reducedParams] using heq)
+  have hspec : DeepMindABPhiThetaSpec p.a p.b p.phi p.theta :=
+    ⟨by simpa [reducedParams] using hdom, hdeep⟩
+  have htwo : TwoAngleSpec p.phi p.theta :=
+    (exists_deepMindSpec_iff_twoAngleSpec p.phi p.theta).1
+      ⟨p.a, p.b, hspec⟩
+  have hmem := hangle p.phi p.theta htwo
+  have hA := a_eq_reconstructed_of_physical_and_equations hspec.1 hspec.2
+  have hB := b_eq_reconstructed_of_physical_and_equations hspec.1 hspec.2
+  change
+    reducedParams
+      (reconstructedA p.phi p.theta) (reconstructedB p.phi p.theta)
+      p.phi p.theta ∈ Reduced.box at hmem
+  rw [← hA, ← hB] at hmem
+  simpa [reconstructedParams, reducedParams] using hmem
+
+/-- Terminal E02 bridge: the exact DeepMind-shaped uniqueness theorem now
+requires only the compact two-angle enclosure theorem. -/
+theorem deepMindABPhiTheta_existsUnique_of_twoAngleEnclosure
+    (hangle : TwoAngleEnclosureTarget) :
+    ∃! ABphiTheta : ℝ × ℝ × ℝ × ℝ,
+      DeepMindABPhiThetaSpec
+        ABphiTheta.1 ABphiTheta.2.1 ABphiTheta.2.2.1 ABphiTheta.2.2.2 :=
+  deepMindABPhiTheta_existsUnique_of_globalEnclosure
+    (globalEnclosureTarget_of_twoAngleEnclosure hangle)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part E03: division-free two-angle residuals and interval rejection kernel
+
+E02 reduced the four-variable DeepMind system to two angles, but its
+reconstructed values contain a quotient.  Direct interval evaluation of that
+quotient is unnecessarily singular near the corner where its denominator can
+vanish.
+
+This module clears the denominator exactly.  It proves that, whenever the E02
+denominator is nonzero, the two original reconstructed equations are
+equivalent to two smooth residuals containing only addition, multiplication,
+`sin`, `cos`, and the named constant `pi`.
+
+The same residuals are encoded as LeanCert expressions.  The final theorem is
+a reusable, executable cell-rejection kernel: if certified interval evaluation
+of either residual excludes zero on a rational rectangle, no common zero can
+lie in that rectangle.  E03 intentionally does not postulate a global cover;
+the finite branch-and-bound cover is the next data layer.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+open LeanCert.Core
+open LeanCert.Engine
+open PartALeanCert
+
+/-- First reconstructed E02 equation, written as a residual. -/
+def firstReconstructedResidual (phi theta : ℝ) : ℝ :=
+  reconstructedA phi theta * (Real.cos theta - Real.cos phi)
+    - 2 * reconstructedB phi theta * Real.sin phi
+    + (theta - phi - 1) * Real.cos theta - Real.sin theta
+    + Real.cos phi + Real.sin phi
+
+/-- Second reconstructed E02 equation, written as a residual. -/
+def secondReconstructedResidual (phi theta : ℝ) : ℝ :=
+  reconstructedA phi theta * (3 * Real.sin theta + Real.sin phi)
+    - 2 * reconstructedB phi theta * Real.cos phi
+    + 3 * (theta - phi - 1) * Real.sin theta + 3 * Real.cos theta
+    - Real.sin phi + Real.cos phi
+
+/-- Numerator of `denominator * reconstructedB`, with no division. -/
+def scaledB (phi theta : ℝ) : ℝ :=
+  angleNumerator phi theta * (1 + angleDelta phi theta / 2)
+    + bBase phi theta * angleDenominator phi theta
+
+/-- Division-free first residual. -/
+def scaledResidualOne (phi theta : ℝ) : ℝ :=
+  angleNumerator phi theta * (Real.cos theta - Real.cos phi)
+    - 2 * scaledB phi theta * Real.sin phi
+    + (angleDelta phi theta - 1) * Real.cos theta
+        * angleDenominator phi theta
+    - Real.sin theta * angleDenominator phi theta
+    + Real.cos phi * angleDenominator phi theta
+    + Real.sin phi * angleDenominator phi theta
+
+/-- Division-free second residual. -/
+def scaledResidualTwo (phi theta : ℝ) : ℝ :=
+  angleNumerator phi theta * (3 * Real.sin theta + Real.sin phi)
+    - 2 * scaledB phi theta * Real.cos phi
+    + 3 * (angleDelta phi theta - 1) * Real.sin theta
+        * angleDenominator phi theta
+    + 3 * Real.cos theta * angleDenominator phi theta
+    - Real.sin phi * angleDenominator phi theta
+    + Real.cos phi * angleDenominator phi theta
+
+/-- The cleared numerator is exactly `denominator * reconstructedB`. -/
+theorem denominator_mul_reconstructedB_eq_scaledB
+    (phi theta : ℝ) (hden : angleDenominator phi theta ≠ 0) :
+    angleDenominator phi theta * reconstructedB phi theta =
+      scaledB phi theta := by
+  dsimp [reconstructedB, bFromA, reconstructedA, scaledB]
+  field_simp [hden]
+
+/-- The first smooth residual is the original residual multiplied by the
+E02 denominator. -/
+theorem scaledResidualOne_eq_denominator_mul
+    (phi theta : ℝ) (hden : angleDenominator phi theta ≠ 0) :
+    scaledResidualOne phi theta =
+      angleDenominator phi theta * firstReconstructedResidual phi theta := by
+  have hA : angleDenominator phi theta * reconstructedA phi theta =
+      angleNumerator phi theta := by
+    dsimp [reconstructedA]
+    field_simp [hden]
+  have hB := denominator_mul_reconstructedB_eq_scaledB phi theta hden
+  unfold scaledResidualOne firstReconstructedResidual
+  rw [← hA, ← hB]
+  dsimp [angleDelta]
+  ring
+
+/-- The second smooth residual is the original residual multiplied by the
+E02 denominator. -/
+theorem scaledResidualTwo_eq_denominator_mul
+    (phi theta : ℝ) (hden : angleDenominator phi theta ≠ 0) :
+    scaledResidualTwo phi theta =
+      angleDenominator phi theta * secondReconstructedResidual phi theta := by
+  have hA : angleDenominator phi theta * reconstructedA phi theta =
+      angleNumerator phi theta := by
+    dsimp [reconstructedA]
+    field_simp [hden]
+  have hB := denominator_mul_reconstructedB_eq_scaledB phi theta hden
+  unfold scaledResidualTwo secondReconstructedResidual
+  rw [← hA, ← hB]
+  dsimp [angleDelta]
+  ring
+
+/-- E02's two equations are exactly the vanishing of the two ordinary
+reconstructed residuals. -/
+theorem twoAngleEquations_iff_reconstructedResiduals_zero
+    (phi theta : ℝ) :
+    TwoAngleEquations phi theta ↔
+      firstReconstructedResidual phi theta = 0 ∧
+      secondReconstructedResidual phi theta = 0 := by
+  rfl
+
+/-- Exact denominator-clearing equivalence used by the interval layer. -/
+theorem twoAngleEquations_iff_scaledResiduals_zero
+    (phi theta : ℝ) (hden : angleDenominator phi theta ≠ 0) :
+    TwoAngleEquations phi theta ↔
+      scaledResidualOne phi theta = 0 ∧
+      scaledResidualTwo phi theta = 0 := by
+  rw [twoAngleEquations_iff_reconstructedResiduals_zero]
+  rw [scaledResidualOne_eq_denominator_mul phi theta hden]
+  rw [scaledResidualTwo_eq_denominator_mul phi theta hden]
+  constructor
+  · rintro ⟨h1, h2⟩
+    simp [h1, h2]
+  · rintro ⟨h1, h2⟩
+    exact ⟨(mul_eq_zero.mp h1).resolve_left hden,
+      (mul_eq_zero.mp h2).resolve_left hden⟩
+
+/-- The complete E02 specification with only smooth equations in its final
+conjunct. -/
+def ScaledTwoAngleSpec (phi theta : ℝ) : Prop :=
+  PhysicalAngleDomain phi theta ∧
+    angleDenominator phi theta ≠ 0 ∧
+    0 ≤ reconstructedA phi theta ∧
+    0 ≤ reconstructedB phi theta ∧
+    scaledResidualOne phi theta = 0 ∧
+    scaledResidualTwo phi theta = 0
+
+/-- No mathematical information is lost by clearing the denominator. -/
+theorem twoAngleSpec_iff_scaledTwoAngleSpec (phi theta : ℝ) :
+    TwoAngleSpec phi theta ↔ ScaledTwoAngleSpec phi theta := by
+  constructor
+  · rintro ⟨hdom, hden, hA, hB, heq⟩
+    exact ⟨hdom, hden, hA, hB,
+      (twoAngleEquations_iff_scaledResiduals_zero phi theta hden).1 heq⟩
+  · rintro ⟨hdom, hden, hA, hB, heq⟩
+    exact ⟨hdom, hden, hA, hB,
+      (twoAngleEquations_iff_scaledResiduals_zero phi theta hden).2 heq⟩
+
+/-- E02's remaining enclosure target, now stated over division-free
+equations. -/
+def ScaledResidualEnclosureTarget : Prop :=
+  ∀ phi theta : ℝ,
+    ScaledTwoAngleSpec phi theta → reconstructedParams phi theta ∈ Reduced.box
+
+/-- The smooth-residual enclosure target is exactly the E02 target. -/
+theorem scaledResidualEnclosureTarget_iff_twoAngleEnclosureTarget :
+    ScaledResidualEnclosureTarget ↔ TwoAngleEnclosureTarget := by
+  constructor
+  · intro h phi theta hspec
+    exact h phi theta ((twoAngleSpec_iff_scaledTwoAngleSpec phi theta).1 hspec)
+  · intro h phi theta hspec
+    exact h phi theta ((twoAngleSpec_iff_scaledTwoAngleSpec phi theta).2 hspec)
+
+/-! ## LeanCert expression model -/
+
+/-- LeanCert AST for the two division-free residuals.  Variable 0 is `phi`
+and variable 1 is `theta`. -/
+def scaledResidualExprList : List Expr :=
+  let phi := ev 0
+  let theta := ev 1
+  let one := ec 1
+  let half := ec (1 / 2)
+  let delta := esub theta phi
+  let cp := ecos phi
+  let sp := esin phi
+  let ct := ecos theta
+  let st := esin theta
+  let base :=
+    eadd
+      (eadd (esub (esub (escale (1 / 2) epi) phi) theta)
+        (escale (1 / 2) delta))
+      (escale (1 / 4) (emul delta delta))
+  let den := esub cp (emul (eadd one (escale (1 / 2) delta)) sp)
+  let num :=
+    eadd
+      (eadd (eadd sp half) (eneg (escale (1 / 2) cp)))
+      (emul base sp)
+  let numB :=
+    eadd (emul num (eadd one (escale (1 / 2) delta))) (emul base den)
+  let r1 :=
+    eadd
+      (eadd
+        (eadd
+          (eadd
+            (esub (emul num (esub ct cp)) (emul (escale 2 numB) sp))
+            (emul (emul (esub delta one) ct) den))
+          (eneg (emul st den)))
+        (emul cp den))
+      (emul sp den)
+  let r2 :=
+    eadd
+      (eadd
+        (eadd
+          (eadd
+            (esub (emul num (eadd (escale 3 st) sp))
+              (emul (escale 2 numB) cp))
+            (emul (emul (escale 3 (esub delta one)) st) den))
+          (emul (escale 3 ct) den))
+        (eneg (emul sp den)))
+      (emul cp den)
+  r1 :: r2 :: []
+
+/-- Select one of the two scaled residual expressions for the angle system. -/
+def scaledResidualExpr (i : Fin 2) : Expr :=
+  scaledResidualExprList.getD i.1 (ec 0)
+
+/-- Both ASTs belong to LeanCert's fully proved core and AD fragment. -/
+theorem scaledResidualExpr_supported :
+    ∀ i : Fin 2, ADConstSupported (scaledResidualExpr i) := by
+  intro i
+  apply checkADConstSupported_correct
+  fin_cases i <;> decide
+theorem scaledResidualExpr_eval_zero (phi theta : ℝ) :
+    evalFin (scaledResidualExpr (0 : Fin 2)) ![phi, theta] =
+      scaledResidualOne phi theta := by
+  simp [scaledResidualExpr, scaledResidualExprList, scaledResidualOne, scaledB,
+    angleNumerator, angleDenominator, bBase, angleDelta,
+    eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+    evalFin, finEnv]; ring
+theorem scaledResidualExpr_eval_one (phi theta : ℝ) :
+    evalFin (scaledResidualExpr (1 : Fin 2)) ![phi, theta] =
+      scaledResidualTwo phi theta := by
+  simp [scaledResidualExpr, scaledResidualExprList, scaledResidualTwo, scaledB,
+    angleNumerator, angleDenominator, bBase, angleDelta,
+    eadd, eneg, esub, emul, escale, esin, ecos, ec, ev, epi,
+    evalFin, finEnv]; ring
+
+/-- Rational interval environment for the two angle variables. -/
+def angleIntervalEnv (phiI thetaI : IntervalRat) : IntervalEnv
+  | 0 => phiI
+  | 1 => thetaI
+  | _ => default
+
+/-- Executable test that an interval lies strictly on one side of zero. -/
+def intervalExcludesZero (I : IntervalRat) : Bool :=
+  decide (I.hi < 0 ∨ 0 < I.lo)
+
+/-- The executable zero-exclusion test is sound over real interval
+membership. -/
+theorem intervalExcludesZero_sound (I : IntervalRat)
+    (h : intervalExcludesZero I = true) : (0 : ℝ) ∉ I := by
+  have hrat : I.hi < 0 ∨ 0 < I.lo := by
+    exact of_decide_eq_true h
+  intro hmem
+  rw [IntervalRat.mem_def] at hmem
+  rcases hrat with hhi | hlo
+  · have hhi' : (I.hi : ℝ) < 0 := by exact_mod_cast hhi
+    linarith
+  · have hlo' : 0 < (I.lo : ℝ) := by exact_mod_cast hlo
+    linarith
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part E04: finite-cover replay foundation
+
+E03 supplied a sound, executable rejection theorem for one rational rectangle
+in the `(phi, theta)` plane.  This module lifts that kernel to finite lists of
+rectangles and states the exact two remaining data obligations:
+
+1. a finite rejected cover of the physical angle triangle outside the angle
+   projection of `Reduced.box`;
+2. local reconstruction into `Reduced.box` inside that angle projection.
+
+Their conjunction yields `TwoAngleEnclosureTarget` and therefore the exact
+DeepMind-shaped uniqueness theorem.  No cover or local enclosure is assumed as
+an axiom: both remain ordinary theorem arguments.
+
+The module also replays one nontrivial pilot rectangle near the origin.  This
+checks the complete path from rational cell data through LeanCert interval
+evaluation to the no-common-zero theorem before the large cover is generated.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+open LeanCert.Core
+open LeanCert.Engine
+open PartALeanCert
+
+/-- Kernel-reducible interval evaluation for one smooth residual.  This uses
+the already certified project-specialized interval for `pi`, avoiding the
+generic named-constant normalization bottleneck in closed `decide` replays. -/
+def scaledResidualKernelInterval (i : Fin 2)
+    (phiI thetaI : IntervalRat) (cfg : EvalConfig := {}) : IntervalRat :=
+  kernelPointEvalCore (scaledResidualExpr i)
+    (angleIntervalEnv phiI thetaI) cfg.taylorDepth
+
+/-- Soundness of the kernel-reducible residual evaluator. -/
+theorem scaledResidual_mem_kernelInterval (i : Fin 2)
+    (phi theta : ℝ) (phiI thetaI : IntervalRat)
+    (hphi : phi ∈ phiI) (htheta : theta ∈ thetaI)
+    (cfg : EvalConfig := {}) :
+    (if i = 0 then scaledResidualOne phi theta
+      else scaledResidualTwo phi theta) ∈
+      scaledResidualKernelInterval i phiI thetaI cfg := by
+  let u : Fin 2 → ℝ := ![phi, theta]
+  have henv : envMem (finEnv u) (angleIntervalEnv phiI thetaI) := by
+    intro n
+    rcases n with (_ | _ | n)
+    · simpa [u, finEnv, angleIntervalEnv] using hphi
+    · simpa [u, finEnv, angleIntervalEnv] using htheta
+    · change finEnv u (Nat.succ (Nat.succ n)) ∈ (default : IntervalRat)
+      rw [IntervalRat.mem_default]
+      simp [finEnv]
+  have hcore := eval_mem_kernelPointEvalCore
+    (scaledResidualExpr_supported i) (finEnv u)
+    (angleIntervalEnv phiI thetaI) henv cfg.taylorDepth
+  change evalFin (scaledResidualExpr i) u ∈
+    scaledResidualKernelInterval i phiI thetaI cfg at hcore
+  fin_cases i
+  · simpa [u, scaledResidualExpr_eval_zero] using hcore
+  · simpa [u, scaledResidualExpr_eval_one] using hcore
+
+/-- A rational rectangle in the two-angle plane. -/
+structure AngleCell where
+  /-- The rational interval for the first switching angle φ. -/
+  phiI : IntervalRat
+  /-- The rational interval for the second switching angle θ. -/
+  thetaI : IntervalRat
+
+namespace AngleCell
+
+/-- Real point membership in a rational angle cell. -/
+def Contains (cell : AngleCell) (phi theta : ℝ) : Prop :=
+  phi ∈ cell.phiI ∧ theta ∈ cell.thetaI
+
+/-- Executable E03 rejection test for a complete angle cell. -/
+def rejected (cell : AngleCell) (cfg : EvalConfig := {}) : Bool :=
+  intervalExcludesZero
+      (scaledResidualKernelInterval (0 : Fin 2) cell.phiI cell.thetaI cfg) ||
+    intervalExcludesZero
+      (scaledResidualKernelInterval (1 : Fin 2) cell.phiI cell.thetaI cfg)
+
+/-- A cell accepted by the executable checker contains no common zero of the
+two smooth residuals. -/
+theorem no_common_zero_of_rejected
+    (cell : AngleCell) (phi theta : ℝ)
+    (hmem : cell.Contains phi theta)
+    (cfg : EvalConfig := {})
+    (hreject : cell.rejected cfg = true) :
+    ¬ (scaledResidualOne phi theta = 0 ∧
+      scaledResidualTwo phi theta = 0) := by
+  have hreject' :
+      intervalExcludesZero
+          (scaledResidualKernelInterval
+            (0 : Fin 2) cell.phiI cell.thetaI cfg) = true ∨
+        intervalExcludesZero
+          (scaledResidualKernelInterval
+            (1 : Fin 2) cell.phiI cell.thetaI cfg) = true := by
+    simpa only [rejected, Bool.or_eq_true] using hreject
+  rintro ⟨hzero1, hzero2⟩
+  rcases hreject' with hreject' | hreject'
+  · have hmem' := scaledResidual_mem_kernelInterval (0 : Fin 2)
+      phi theta cell.phiI cell.thetaI hmem.1 hmem.2 cfg
+    have hnot := intervalExcludesZero_sound _ hreject'
+    apply hnot
+    simpa [hzero1] using hmem'
+  · have hmem' := scaledResidual_mem_kernelInterval (1 : Fin 2)
+      phi theta cell.phiI cell.thetaI hmem.1 hmem.2 cfg
+    have hnot := intervalExcludesZero_sound _ hreject'
+    apply hnot
+    simpa [hzero2] using hmem'
+
+end AngleCell
+
+/-! ## Pilot replay -/
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part E21F: local two-angle Krawczyk closure repair
+
+The residual-only cover becomes inefficient close to the certified Gerver
+zero.  This module replaces arbitrarily deep subdivision there by a single
+two-dimensional contraction certificate on a deliberately wider rational
+box.  The wide box contains both the exact projection of `Reduced.box` and
+the complete unresolved E19 tail.
+
+The resulting theorem identifies every smooth residual zero in the wide box
+with the already certified Part A reduced solution.  A terminal composition
+theorem therefore needs interval rejection only outside this wide local box.
+-/
+
+/-! ## Concrete two-dimensional contraction data -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+open LeanCert.Core
+open LeanCert.Engine
+open PartALeanCert
+
+/-- Local angle box containing the complete unresolved E20 tail and the exact
+angle projection of `Reduced.box`.  E21F narrows the exploratory E21 box to
+the region actually required by the recorded E20 extrema. -/
+def localAngleX : Fin 2 → IntervalRat := ![
+  ⟨391 / 10000, 157 / 4000, by norm_num⟩,
+  ⟨68113 / 100000, 34069 / 50000, by norm_num⟩
+]
+
+/-- Rational center near the already certified Gerver zero. -/
+def localAngleCenter : Fin 2 → ℚ := ![
+  3917736479 / 100000000000,
+  681301509383 / 1000000000000
+]
+
+/-- Rational approximation to the inverse Jacobian of the two scaled
+residuals at `localAngleCenter`. -/
+def localAngleY : Matrix (Fin 2) (Fin 2) ℚ := ![
+  ![-2886 / 10000, -1481 / 10000],
+  ![6267 / 10000, -27218 / 10000]
+]
+
+/-- The default interval-evaluation configuration for local angle certification. -/
+def localAngleCfg : EvalConfig := {}
+
+/-- Contraction constant used by the checked local uniqueness theorem. -/
+def localAngleQ : ℚ := 3 / 10
+
+/-- Enclose the preconditioned Newton derivative on the local two-angle box. -/
+def localAnglePJ : Matrix (Fin 2) (Fin 2) IntervalRat :=
+  preconditionedJacobian localAngleY
+    (intervalJacobian scaledResidualExpr localAngleX localAngleCfg)
+
+/-- Enclose the local Newton image using the certified contraction bound. -/
+def localAngleImage (i : Fin 2) : IntervalRat :=
+  imageEnclosureWithQ scaledResidualExpr localAngleX localAngleCenter
+    localAngleY localAngleCfg localAngleQ i
+
+theorem localAngleQ_nonneg : (0 : ℚ) ≤ localAngleQ := by
+  norm_num [localAngleQ]
+
+theorem localAngleQ_lt_one : localAngleQ < (1 : ℚ) := by
+  norm_num [localAngleQ]
+
+theorem localAngleCenter_mem :
+    FinBoxMem (fun i : Fin 2 => (localAngleCenter i : ℝ)) localAngleX := by
+  intro i
+  fin_cases i <;>
+    simp [localAngleCenter, localAngleX, IntervalRat.mem_def] <;>
+    norm_num
+theorem localAngle_bound_lt :
+    intervalMatrixBound localAnglePJ < localAngleQ := by
+  decide +kernel
+
+theorem localAngle_bound_le :
+    intervalMatrixBound localAnglePJ ≤ localAngleQ :=
+  localAngle_bound_lt.le
+theorem localAngle_image_zero_inside :
+    intervalStrictInside (localAngleImage (0 : Fin 2))
+      (localAngleX (0 : Fin 2)) = true := by
+  decide +kernel
+theorem localAngle_image_one_inside :
+    intervalStrictInside (localAngleImage (1 : Fin 2))
+      (localAngleX (1 : Fin 2)) = true := by
+  decide +kernel
+
+theorem localAngle_images_inside :
+    ∀ i : Fin 2,
+      intervalStrictInside (localAngleImage i) (localAngleX i) = true := by
+  intro i
+  fin_cases i
+  · exact localAngle_image_zero_inside
+  · exact localAngle_image_one_inside
+
+/-- Kernel-checked existence and uniqueness of a common scaled-residual zero
+throughout the complete wide local box. -/
+theorem localAngle_unique_scaled :
+    ∃! u, FinBoxMem u localAngleX ∧ SystemZero scaledResidualExpr u := by
+  exact uniqueSystemZero_of_certified_contraction
+    scaledResidualExpr scaledResidualExpr_supported
+    localAngleX localAngleCenter localAngleCenter_mem
+    localAngleY localAngleCfg localAngleQ
+    localAngleQ_nonneg localAngleQ_lt_one
+    (by simpa [localAnglePJ] using localAngle_bound_le)
+    (by simpa [localAngleImage] using localAngle_images_inside)
+
+/-! ## Semantic bridge to named angles and the Part A solution -/
+
+/-- The two-angle cell corresponding to the certified local interval box. -/
+def localAngleCell : AngleCell :=
+  ⟨localAngleX (0 : Fin 2), localAngleX (1 : Fin 2)⟩
+
+/-- Package the two switching angles as a two-coordinate real vector. -/
+def localAngleVector (phi theta : ℝ) : Fin 2 → ℝ := ![phi, theta]
+
+theorem localAngleVector_mem_iff (phi theta : ℝ) :
+    FinBoxMem (localAngleVector phi theta) localAngleX ↔
+      localAngleCell.Contains phi theta := by
+  constructor
+  · intro h
+    exact ⟨by simpa [localAngleVector, localAngleCell] using h (0 : Fin 2),
+      by simpa [localAngleVector, localAngleCell] using h (1 : Fin 2)⟩
+  · rintro ⟨hphi, htheta⟩ i
+    fin_cases i
+    · simpa [localAngleVector, localAngleCell] using hphi
+    · simpa [localAngleVector, localAngleCell] using htheta
+
+theorem localAngleVector_systemZero_iff (phi theta : ℝ) :
+    SystemZero scaledResidualExpr (localAngleVector phi theta) ↔
+      scaledResidualOne phi theta = 0 ∧ scaledResidualTwo phi theta = 0 := by
+  constructor
+  · intro h
+    constructor
+    · have hzero := h (0 : Fin 2)
+      change evalFin (scaledResidualExpr (0 : Fin 2))
+        (localAngleVector phi theta) = 0 at hzero
+      simpa [localAngleVector, scaledResidualExpr_eval_zero] using hzero
+    · have hzero := h (1 : Fin 2)
+      change evalFin (scaledResidualExpr (1 : Fin 2))
+        (localAngleVector phi theta) = 0 at hzero
+      simpa [localAngleVector, scaledResidualExpr_eval_one] using hzero
+  · rintro ⟨hzeroOne, hzeroTwo⟩ i
+    fin_cases i
+    · change evalFin (scaledResidualExpr (0 : Fin 2))
+        (localAngleVector phi theta) = 0
+      simpa [localAngleVector, scaledResidualExpr_eval_zero] using hzeroOne
+    · change evalFin (scaledResidualExpr (1 : Fin 2))
+        (localAngleVector phi theta) = 0
+      simpa [localAngleVector, scaledResidualExpr_eval_one] using hzeroTwo
+
+/-- The reduced parameter tuple selected by the concrete uniqueness certificate. -/
+def certifiedReducedSolution : Reduced.Params :=
+  PartALeanCert.reducedCertifiedUniqueSolution.solution
+
+theorem certifiedReducedSolution_mem :
+    certifiedReducedSolution ∈ Reduced.box := by
+  exact PartALeanCert.reducedCertifiedUniqueSolution.solution_mem
+
+theorem certifiedReducedSolution_satisfies :
+    Reduced.Equations certifiedReducedSolution := by
+  exact PartALeanCert.reducedCertifiedUniqueSolution.satisfies
+
+/-- The Part A solution's angle pair lies strictly inside the wide local
+box, by exact rational endpoint comparison. -/
+theorem certifiedAngles_mem_localAngleCell :
+    localAngleCell.Contains
+      certifiedReducedSolution.phi certifiedReducedSolution.theta := by
+  have hp := certifiedReducedSolution_mem
+  dsimp [Reduced.box, qR] at hp
+  rcases hp with
+    ⟨_haLo, _haHi, _hbLo, _hbHi,
+      hphiLo, hphiHi, hthetaLo, hthetaHi⟩
+  change
+    certifiedReducedSolution.phi ∈ localAngleX (0 : Fin 2) ∧
+      certifiedReducedSolution.theta ∈ localAngleX (1 : Fin 2)
+  constructor
+  · change
+      (((391 / 10000 : ℚ) : ℝ) ≤ certifiedReducedSolution.phi ∧
+        certifiedReducedSolution.phi ≤ ((157 / 4000 : ℚ) : ℝ))
+    constructor
+    · norm_num at hphiLo ⊢
+      linarith
+    · norm_num at hphiHi ⊢
+      linarith
+  · change
+      (((68113 / 100000 : ℚ) : ℝ) ≤ certifiedReducedSolution.theta ∧
+        certifiedReducedSolution.theta ≤ ((34069 / 50000 : ℚ) : ℝ))
+    constructor
+    · norm_num at hthetaLo ⊢
+      linarith
+    · norm_num at hthetaHi ⊢
+      linarith
+
+theorem certifiedAngles_mem_localAngleX :
+    FinBoxMem
+      (localAngleVector certifiedReducedSolution.phi
+        certifiedReducedSolution.theta) localAngleX :=
+  (localAngleVector_mem_iff _ _).2 certifiedAngles_mem_localAngleCell
+
+theorem certifiedAngles_systemZero :
+    SystemZero scaledResidualExpr
+      (localAngleVector certifiedReducedSolution.phi
+        certifiedReducedSolution.theta) := by
+  have hdeep : DeepMindABPhiThetaSpec
+      certifiedReducedSolution.a certifiedReducedSolution.b
+      certifiedReducedSolution.phi certifiedReducedSolution.theta :=
+    deepMindSpec_of_mem_reducedBox_and_equations
+      certifiedReducedSolution_mem certifiedReducedSolution_satisfies
+  have htwo : TwoAngleSpec
+      certifiedReducedSolution.phi certifiedReducedSolution.theta :=
+    (exists_deepMindSpec_iff_twoAngleSpec
+      certifiedReducedSolution.phi certifiedReducedSolution.theta).1
+      ⟨certifiedReducedSolution.a, certifiedReducedSolution.b, hdeep⟩
+  have hscaled :=
+    (twoAngleSpec_iff_scaledTwoAngleSpec
+      certifiedReducedSolution.phi certifiedReducedSolution.theta).1 htwo
+  rcases hscaled with ⟨_hdom, _hden, _hA, _hB, hzeroOne, hzeroTwo⟩
+  exact (localAngleVector_systemZero_iff _ _).2 ⟨hzeroOne, hzeroTwo⟩
+
+/-- Every smooth physical solution in the wide local cell reconstructs to
+the already certified Part A solution and hence belongs to `Reduced.box`. -/
+theorem reconstructedParams_mem_reducedBox_of_mem_localAngleCell
+    {phi theta : ℝ} (hspec : ScaledTwoAngleSpec phi theta)
+    (hlocal : localAngleCell.Contains phi theta) :
+    reconstructedParams phi theta ∈ Reduced.box := by
+  have hmem : FinBoxMem (localAngleVector phi theta) localAngleX :=
+    (localAngleVector_mem_iff phi theta).2 hlocal
+  have hzero : SystemZero scaledResidualExpr (localAngleVector phi theta) :=
+    (localAngleVector_systemZero_iff phi theta).2
+      ⟨hspec.2.2.2.2.1, hspec.2.2.2.2.2⟩
+  have heq :
+      localAngleVector phi theta =
+        localAngleVector certifiedReducedSolution.phi
+          certifiedReducedSolution.theta :=
+    localAngle_unique_scaled.unique ⟨hmem, hzero⟩
+      ⟨certifiedAngles_mem_localAngleX, certifiedAngles_systemZero⟩
+  have hphi : phi = certifiedReducedSolution.phi := by
+    have h := congrFun heq (0 : Fin 2)
+    simpa [localAngleVector] using h
+  have htheta : theta = certifiedReducedSolution.theta := by
+    have h := congrFun heq (1 : Fin 2)
+    simpa [localAngleVector] using h
+  have hdeep : DeepMindABPhiThetaSpec
+      certifiedReducedSolution.a certifiedReducedSolution.b
+      certifiedReducedSolution.phi certifiedReducedSolution.theta :=
+    deepMindSpec_of_mem_reducedBox_and_equations
+      certifiedReducedSolution_mem certifiedReducedSolution_satisfies
+  have hA := a_eq_reconstructed_of_physical_and_equations hdeep.1 hdeep.2
+  have hB := b_eq_reconstructed_of_physical_and_equations hdeep.1 hdeep.2
+  rw [hphi, htheta]
+  change reducedParams
+      (reconstructedA certifiedReducedSolution.phi
+        certifiedReducedSolution.theta)
+      (reconstructedB certifiedReducedSolution.phi
+        certifiedReducedSolution.theta)
+      certifiedReducedSolution.phi certifiedReducedSolution.theta ∈ Reduced.box
+  rw [← hA, ← hB]
+  simpa [reducedParams] using certifiedReducedSolution_mem
+
+/-! ## Terminal composition with rejection only outside the wide box -/
+
+/-- Solutions in the local angle cell reconstruct into the reduced parameter box. -/
+def WideLocalReconstructionEnclosureTarget : Prop :=
+  ∀ phi theta : ℝ,
+    ScaledTwoAngleSpec phi theta →
+    localAngleCell.Contains phi theta →
+    reconstructedParams phi theta ∈ Reduced.box
+
+theorem wideLocalReconstructionEnclosureTarget :
+    WideLocalReconstructionEnclosureTarget := by
+  intro phi theta hspec hlocal
+  exact reconstructedParams_mem_reducedBox_of_mem_localAngleCell hspec hlocal
+
+/-! Machine-readable replay markers consumed by the E21 runner. -/
+
+/- Exact diagnostic values are printed even if a later repair run stops at a
+closed Boolean theorem. -/
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part E22F foundation: adaptive global cover
+
+This module replaces the probe-only upper-wedge files by one kernel-reducible
+adaptive checker.  Starting from the rational square `[0, 4/5]^2`, it prunes
+cells which are outside the physical triangle, cells wholly contained in the
+wide E21 local box, and cells rejected by the certified E03 interval kernel.
+Every remaining cell is split into four exact rational children.  Depth 18 is
+the depth reached by the final E20 refinement around the Gerver root.
+
+The soundness theorem is independent of the closed computation.  E22F compiles
+the depth-18 Boolean certificate in sixty-four independent depth-15 modules;
+`ParallelAdaptiveGlobalCoverClosure` recombines them without recomputation.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+open LeanCert.Core
+
+/-- The rational midpoint of an angle interval. -/
+def angleMid (i : IntervalRat) : ℚ := (i.lo + i.hi) / 2
+
+/-- The closed lower half of a rational interval. -/
+def intervalLow (i : IntervalRat) : IntervalRat :=
+  ⟨i.lo, angleMid i, by
+    dsimp [angleMid]
+    linarith [i.le]⟩
+
+/-- The closed upper half of a rational interval. -/
+def intervalHigh (i : IntervalRat) : IntervalRat :=
+  ⟨angleMid i, i.hi, by
+    dsimp [angleMid]
+    linarith [i.le]⟩
+
+/-- Bisect both angle intervals and select the lower φ half and lower θ half. -/
+def childLL (cell : AngleCell) : AngleCell :=
+  ⟨intervalLow cell.phiI, intervalLow cell.thetaI⟩
+
+/-- Bisect both angle intervals and select the lower φ half and upper θ half. -/
+def childLH (cell : AngleCell) : AngleCell :=
+  ⟨intervalLow cell.phiI, intervalHigh cell.thetaI⟩
+
+/-- Bisect both angle intervals and select the upper φ half and lower θ half. -/
+def childHL (cell : AngleCell) : AngleCell :=
+  ⟨intervalHigh cell.phiI, intervalLow cell.thetaI⟩
+
+/-- Bisect both angle intervals and select the upper φ half and upper θ half. -/
+def childHH (cell : AngleCell) : AngleCell :=
+  ⟨intervalHigh cell.phiI, intervalHigh cell.thetaI⟩
+
+/-- A rational cell lies strictly above the physical half-plane `phi ≤ theta`.
+The strict comparison deliberately keeps all cells touching the diagonal. -/
+def physicallyIrrelevant (cell : AngleCell) : Bool :=
+  decide (cell.thetaI.hi < cell.phiI.lo)
+
+/-- Every point of the rational cell lies in the wide E21 local box. -/
+def cellInsideLocal (cell : AngleCell) : Bool :=
+  decide (
+    localAngleCell.phiI.lo ≤ cell.phiI.lo ∧
+    cell.phiI.hi ≤ localAngleCell.phiI.hi ∧
+    localAngleCell.thetaI.lo ≤ cell.thetaI.lo ∧
+    cell.thetaI.hi ≤ localAngleCell.thetaI.hi)
+
+/-- Adaptive four-way replay.  A node closes when it is irrelevant, local, or
+rejected.  Otherwise all four rational midpoint children must close. -/
+def adaptiveCoverCheck : Nat → AngleCell → Bool
+  | 0, cell =>
+      if physicallyIrrelevant cell = true then true
+      else if cellInsideLocal cell = true then true
+      else cell.rejected
+  | depth + 1, cell =>
+      if physicallyIrrelevant cell = true then true
+      else if cellInsideLocal cell = true then true
+      else if cell.rejected = true then true
+      else
+        adaptiveCoverCheck depth (childLL cell) &&
+          (adaptiveCoverCheck depth (childLH cell) &&
+            (adaptiveCoverCheck depth (childHL cell) &&
+              adaptiveCoverCheck depth (childHH cell)))
+
+theorem physicallyIrrelevant_no_physical_point
+    (cell : AngleCell) (phi theta : ℝ)
+    (hirr : physicallyIrrelevant cell = true)
+    (hdom : PhysicalAngleDomain phi theta)
+    (hmem : cell.Contains phi theta) : False := by
+  have hrat : cell.thetaI.hi < cell.phiI.lo := by
+    exact of_decide_eq_true hirr
+  have hreal : (cell.thetaI.hi : ℝ) < (cell.phiI.lo : ℝ) := by
+    exact_mod_cast hrat
+  change phi ∈ cell.phiI ∧ theta ∈ cell.thetaI at hmem
+  simp only [IntervalRat.mem_def] at hmem
+  linarith [hdom.2.1, hmem.1.1, hmem.2.2]
+
+theorem cellInsideLocal_sound
+    (cell : AngleCell) (phi theta : ℝ)
+    (hlocal : cellInsideLocal cell = true)
+    (hmem : cell.Contains phi theta) :
+    localAngleCell.Contains phi theta := by
+  have hb :
+      localAngleCell.phiI.lo ≤ cell.phiI.lo ∧
+      cell.phiI.hi ≤ localAngleCell.phiI.hi ∧
+      localAngleCell.thetaI.lo ≤ cell.thetaI.lo ∧
+      cell.thetaI.hi ≤ localAngleCell.thetaI.hi := by
+    exact of_decide_eq_true hlocal
+  change phi ∈ cell.phiI ∧ theta ∈ cell.thetaI at hmem
+  change phi ∈ localAngleCell.phiI ∧ theta ∈ localAngleCell.thetaI
+  simp only [IntervalRat.mem_def] at hmem ⊢
+  have hpLo : (localAngleCell.phiI.lo : ℝ) ≤ (cell.phiI.lo : ℝ) := by
+    exact_mod_cast hb.1
+  have hpHi : (cell.phiI.hi : ℝ) ≤ (localAngleCell.phiI.hi : ℝ) := by
+    exact_mod_cast hb.2.1
+  have htLo : (localAngleCell.thetaI.lo : ℝ) ≤ (cell.thetaI.lo : ℝ) := by
+    exact_mod_cast hb.2.2.1
+  have htHi : (cell.thetaI.hi : ℝ) ≤ (localAngleCell.thetaI.hi : ℝ) := by
+    exact_mod_cast hb.2.2.2
+  exact ⟨⟨hpLo.trans hmem.1.1, hmem.1.2.trans hpHi⟩,
+    ⟨htLo.trans hmem.2.1, hmem.2.2.trans htHi⟩⟩
+
+/-- Every real point in a parent cell belongs to at least one of its four
+closed midpoint children. -/
+theorem contains_some_midpoint_child
+    (cell : AngleCell) (phi theta : ℝ)
+    (hmem : cell.Contains phi theta) :
+    (childLL cell).Contains phi theta ∨
+      (childLH cell).Contains phi theta ∨
+      (childHL cell).Contains phi theta ∨
+      (childHH cell).Contains phi theta := by
+  change phi ∈ cell.phiI ∧ theta ∈ cell.thetaI at hmem
+  simp only [IntervalRat.mem_def] at hmem
+  by_cases hp : phi ≤ (angleMid cell.phiI : ℝ)
+  · by_cases ht : theta ≤ (angleMid cell.thetaI : ℝ)
+    · left
+      change phi ∈ intervalLow cell.phiI ∧ theta ∈ intervalLow cell.thetaI
+      simp only [IntervalRat.mem_def, intervalLow]
+      exact ⟨⟨hmem.1.1, hp⟩, ⟨hmem.2.1, ht⟩⟩
+    · right; left
+      have ht' : (angleMid cell.thetaI : ℝ) ≤ theta :=
+        le_of_lt (lt_of_not_ge ht)
+      change phi ∈ intervalLow cell.phiI ∧ theta ∈ intervalHigh cell.thetaI
+      simp only [IntervalRat.mem_def, intervalLow, intervalHigh]
+      exact ⟨⟨hmem.1.1, hp⟩, ⟨ht', hmem.2.2⟩⟩
+  · have hp' : (angleMid cell.phiI : ℝ) ≤ phi :=
+      le_of_lt (lt_of_not_ge hp)
+    by_cases ht : theta ≤ (angleMid cell.thetaI : ℝ)
+    · right; right; left
+      change phi ∈ intervalHigh cell.phiI ∧ theta ∈ intervalLow cell.thetaI
+      simp only [IntervalRat.mem_def, intervalLow, intervalHigh]
+      exact ⟨⟨hp', hmem.1.2⟩, ⟨hmem.2.1, ht⟩⟩
+    · right; right; right
+      have ht' : (angleMid cell.thetaI : ℝ) ≤ theta :=
+        le_of_lt (lt_of_not_ge ht)
+      change phi ∈ intervalHigh cell.phiI ∧ theta ∈ intervalHigh cell.thetaI
+      simp only [IntervalRat.mem_def, intervalHigh]
+      exact ⟨⟨hp', hmem.1.2⟩, ⟨ht', hmem.2.2⟩⟩
+
+/-- Semantic soundness of the adaptive Boolean replay. -/
+theorem adaptiveCoverCheck_no_common_zero
+    (depth : Nat) (cell : AngleCell)
+    (hcheck : adaptiveCoverCheck depth cell = true)
+    (phi theta : ℝ)
+    (hdom : PhysicalAngleDomain phi theta)
+    (hmem : cell.Contains phi theta)
+    (houtside : ¬ localAngleCell.Contains phi theta) :
+    ¬ (scaledResidualOne phi theta = 0 ∧
+      scaledResidualTwo phi theta = 0) := by
+  induction depth generalizing cell with
+  | zero =>
+      by_cases hirr : physicallyIrrelevant cell = true
+      · exact (physicallyIrrelevant_no_physical_point
+          cell phi theta hirr hdom hmem).elim
+      by_cases hlocal : cellInsideLocal cell = true
+      · exact (houtside (cellInsideLocal_sound
+          cell phi theta hlocal hmem)).elim
+      have hrejected : cell.rejected = true := by
+        simpa [adaptiveCoverCheck, hirr, hlocal] using hcheck
+      exact AngleCell.no_common_zero_of_rejected
+        cell phi theta hmem (cfg := {}) hrejected
+  | succ depth ih =>
+      by_cases hirr : physicallyIrrelevant cell = true
+      · exact (physicallyIrrelevant_no_physical_point
+          cell phi theta hirr hdom hmem).elim
+      by_cases hlocal : cellInsideLocal cell = true
+      · exact (houtside (cellInsideLocal_sound
+          cell phi theta hlocal hmem)).elim
+      by_cases hrejected : cell.rejected = true
+      · exact AngleCell.no_common_zero_of_rejected
+          cell phi theta hmem (cfg := {}) hrejected
+      have hchildren :
+          adaptiveCoverCheck depth (childLL cell) = true ∧
+          (adaptiveCoverCheck depth (childLH cell) = true ∧
+          (adaptiveCoverCheck depth (childHL cell) = true ∧
+            adaptiveCoverCheck depth (childHH cell) = true)) := by
+        simpa [adaptiveCoverCheck, hirr, hlocal, hrejected,
+          Bool.and_eq_true] using hcheck
+      rcases contains_some_midpoint_child cell phi theta hmem with
+        hll | hlh | hhl | hhh
+      · exact ih (cell := childLL cell) hchildren.1 hll
+      · exact ih (cell := childLH cell) hchildren.2.1 hlh
+      · exact ih (cell := childHL cell) hchildren.2.2.1 hhl
+      · exact ih (cell := childHH cell) hchildren.2.2.2 hhh
+
+/-- Rational root containing the complete physical angle triangle. -/
+def globalAngleRoot : AngleCell :=
+  ⟨⟨0, 4 / 5, by norm_num⟩, ⟨0, 4 / 5, by norm_num⟩⟩
+
+theorem physicalAngleDomain_mem_globalAngleRoot
+    (phi theta : ℝ) (hdom : PhysicalAngleDomain phi theta) :
+    globalAngleRoot.Contains phi theta := by
+  have hpi := ExactReplay.piI_contains_pi
+  have hhi : Real.pi ≤ (ExactReplay.piI.hi : ℝ) := hpi.2
+  have h32 : (ExactReplay.piI.hi : ℝ) < (16 / 5 : ℝ) := by
+    norm_num [ExactReplay.piI, ExactReplay.q]
+  have htheta : theta < (4 / 5 : ℝ) := by
+    linarith [hdom.2.2]
+  change phi ∈ (⟨0, 4 / 5, by norm_num⟩ : IntervalRat) ∧
+    theta ∈ (⟨0, 4 / 5, by norm_num⟩ : IntervalRat)
+  simp only [IntervalRat.mem_def]
+  norm_num at ⊢
+  exact ⟨⟨hdom.1, (hdom.2.1.trans htheta.le)⟩,
+    ⟨hdom.1.trans hdom.2.1, htheta.le⟩⟩
+
+/-- A parent closes whenever its four children close.  This lemma permits the
+depth-18 computation to be compiled in independent shards without changing
+the kernel proposition that is certified. -/
+theorem adaptiveCoverCheck_succ_of_children
+    (depth : Nat) (cell : AngleCell)
+    (hLL : adaptiveCoverCheck depth (childLL cell) = true)
+    (hLH : adaptiveCoverCheck depth (childLH cell) = true)
+    (hHL : adaptiveCoverCheck depth (childHL cell) = true)
+    (hHH : adaptiveCoverCheck depth (childHH cell) = true) :
+    adaptiveCoverCheck (depth + 1) cell = true := by
+  by_cases hirr : physicallyIrrelevant cell = true
+  · simp [adaptiveCoverCheck, hirr]
+  by_cases hlocal : cellInsideLocal cell = true
+  · simp [adaptiveCoverCheck, hirr, hlocal]
+  by_cases hrejected : cell.rejected = true
+  · simp [adaptiveCoverCheck, hirr, hlocal, hrejected]
+  simp [adaptiveCoverCheck, hirr, hlocal, hrejected, hLL, hLH, hHL, hHH]
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# E24 aligned region foundation
+
+Four rational rectangles are aligned exactly with the four sides of the
+wide E21 local angle cell.  Their union covers every point of the global
+physical angle root that is not in the local cell.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+/-- The exclusion root cell below the local φ interval, with `φ ≤ 391/10000`. -/
+def e24PhiBelowRoot : AngleCell :=
+  ⟨⟨0, 391 / 10000, by norm_num⟩,
+    ⟨0, 4 / 5, by norm_num⟩⟩
+
+/-- The exclusion root cell above the local φ interval, with `157/4000 ≤ φ`. -/
+def e24PhiAboveRoot : AngleCell :=
+  ⟨⟨157 / 4000, 4 / 5, by norm_num⟩,
+    ⟨0, 4 / 5, by norm_num⟩⟩
+
+/-- The exclusion root cell below the local θ interval, with `θ ≤ 68113/100000`. -/
+def e24ThetaBelowRoot : AngleCell :=
+  ⟨⟨0, 4 / 5, by norm_num⟩,
+    ⟨0, 68113 / 100000, by norm_num⟩⟩
+
+/-- The exclusion root cell above the local θ interval, with `34069/50000 ≤ θ`. -/
+def e24ThetaAboveRoot : AngleCell :=
+  ⟨⟨0, 4 / 5, by norm_num⟩,
+    ⟨34069 / 50000, 4 / 5, by norm_num⟩⟩
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# E24 aligned-region semantic closure
+
+This file contains no closed heavy computation.  It proves that the four
+aligned rectangles cover the complement of the E21 local cell inside the
+physical triangle and turns four Boolean adaptive-cover certificates into the
+terminal DeepMind-shaped uniqueness theorem.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+open LeanCert.Core
+
+theorem e24AlignedRoots_cover_outside_local
+    (phi theta : ℝ)
+    (hdom : PhysicalAngleDomain phi theta)
+    (houtside : ¬ localAngleCell.Contains phi theta) :
+    e24PhiBelowRoot.Contains phi theta ∨
+      e24PhiAboveRoot.Contains phi theta ∨
+      e24ThetaBelowRoot.Contains phi theta ∨
+      e24ThetaAboveRoot.Contains phi theta := by
+  have hglobal :=
+    physicalAngleDomain_mem_globalAngleRoot phi theta hdom
+  change
+    phi ∈ globalAngleRoot.phiI ∧ theta ∈ globalAngleRoot.thetaI at hglobal
+  simp only [globalAngleRoot, IntervalRat.mem_def] at hglobal
+  by_cases hpLo :
+      phi ≤ (((391 / 10000 : ℚ) : ℝ))
+  · left
+    change
+      phi ∈ e24PhiBelowRoot.phiI ∧ theta ∈ e24PhiBelowRoot.thetaI
+    simp only [e24PhiBelowRoot, IntervalRat.mem_def]
+    exact ⟨⟨hglobal.1.1, hpLo⟩, hglobal.2⟩
+  by_cases hpHi :
+      (((157 / 4000 : ℚ) : ℝ)) ≤ phi
+  · right; left
+    change
+      phi ∈ e24PhiAboveRoot.phiI ∧ theta ∈ e24PhiAboveRoot.thetaI
+    simp only [e24PhiAboveRoot, IntervalRat.mem_def]
+    exact ⟨⟨hpHi, hglobal.1.2⟩, hglobal.2⟩
+  by_cases htLo :
+      theta ≤ (((68113 / 100000 : ℚ) : ℝ))
+  · right; right; left
+    change
+      phi ∈ e24ThetaBelowRoot.phiI ∧ theta ∈ e24ThetaBelowRoot.thetaI
+    simp only [e24ThetaBelowRoot, IntervalRat.mem_def]
+    exact ⟨hglobal.1, ⟨hglobal.2.1, htLo⟩⟩
+  by_cases htHi :
+      (((34069 / 50000 : ℚ) : ℝ)) ≤ theta
+  · right; right; right
+    change
+      phi ∈ e24ThetaAboveRoot.phiI ∧ theta ∈ e24ThetaAboveRoot.thetaI
+    simp only [e24ThetaAboveRoot, IntervalRat.mem_def]
+    exact ⟨hglobal.1, ⟨htHi, hglobal.2.2⟩⟩
+  have hphiLo :
+      (((391 / 10000 : ℚ) : ℝ)) < phi :=
+    lt_of_not_ge hpLo
+  have hphiHi :
+      phi < (((157 / 4000 : ℚ) : ℝ)) :=
+    lt_of_not_ge hpHi
+  have hthetaLo :
+      (((68113 / 100000 : ℚ) : ℝ)) < theta :=
+    lt_of_not_ge htLo
+  have hthetaHi :
+      theta < (((34069 / 50000 : ℚ) : ℝ)) :=
+    lt_of_not_ge htHi
+  exfalso
+  apply houtside
+  change
+    phi ∈ localAngleCell.phiI ∧ theta ∈ localAngleCell.thetaI
+  constructor
+  · change phi ∈ localAngleX (0 : Fin 2)
+    simp only [localAngleX, IntervalRat.mem_def]
+    exact ⟨hphiLo.le, hphiHi.le⟩
+  · change theta ∈ localAngleX (1 : Fin 2)
+    simp only [localAngleX, IntervalRat.mem_def]
+    exact ⟨hthetaLo.le, hthetaHi.le⟩
+
+theorem e24NoCommonZeroOutsideLocal_of_alignedChecks
+    (hPhiBelow : adaptiveCoverCheck 14 e24PhiBelowRoot = true)
+    (hPhiAbove : adaptiveCoverCheck 16 e24PhiAboveRoot = true)
+    (hThetaBelow : adaptiveCoverCheck 18 e24ThetaBelowRoot = true)
+    (hThetaAbove : adaptiveCoverCheck 19 e24ThetaAboveRoot = true)
+    (phi theta : ℝ)
+    (hdom : PhysicalAngleDomain phi theta)
+    (houtside : ¬ localAngleCell.Contains phi theta) :
+    ¬ (scaledResidualOne phi theta = 0 ∧
+      scaledResidualTwo phi theta = 0) := by
+  rcases e24AlignedRoots_cover_outside_local phi theta hdom houtside with
+    hmem | hmem | hmem | hmem
+  · exact adaptiveCoverCheck_no_common_zero
+      14 e24PhiBelowRoot hPhiBelow phi theta hdom hmem houtside
+  · exact adaptiveCoverCheck_no_common_zero
+      16 e24PhiAboveRoot hPhiAbove phi theta hdom hmem houtside
+  · exact adaptiveCoverCheck_no_common_zero
+      18 e24ThetaBelowRoot hThetaBelow phi theta hdom hmem houtside
+  · exact adaptiveCoverCheck_no_common_zero
+      19 e24ThetaAboveRoot hThetaAbove phi theta hdom hmem houtside
+
+theorem e24ScaledResidualEnclosureTarget_of_alignedChecks
+    (hPhiBelow : adaptiveCoverCheck 14 e24PhiBelowRoot = true)
+    (hPhiAbove : adaptiveCoverCheck 16 e24PhiAboveRoot = true)
+    (hThetaBelow : adaptiveCoverCheck 18 e24ThetaBelowRoot = true)
+    (hThetaAbove : adaptiveCoverCheck 19 e24ThetaAboveRoot = true) :
+    ScaledResidualEnclosureTarget := by
+  intro phi theta hspec
+  by_cases hin : localAngleCell.Contains phi theta
+  · exact wideLocalReconstructionEnclosureTarget phi theta hspec hin
+  · have hnozero :=
+      e24NoCommonZeroOutsideLocal_of_alignedChecks
+        hPhiBelow hPhiAbove hThetaBelow hThetaAbove
+        phi theta hspec.1 hin
+    exact (hnozero ⟨hspec.2.2.2.2.1, hspec.2.2.2.2.2⟩).elim
+
+theorem deepMindABPhiTheta_existsUnique_of_alignedChecks
+    (hPhiBelow : adaptiveCoverCheck 14 e24PhiBelowRoot = true)
+    (hPhiAbove : adaptiveCoverCheck 16 e24PhiAboveRoot = true)
+    (hThetaBelow : adaptiveCoverCheck 18 e24ThetaBelowRoot = true)
+    (hThetaAbove : adaptiveCoverCheck 19 e24ThetaAboveRoot = true) :
+    ∃! ABphiTheta : ℝ × ℝ × ℝ × ℝ,
+      DeepMindABPhiThetaSpec
+        ABphiTheta.1 ABphiTheta.2.1 ABphiTheta.2.2.1 ABphiTheta.2.2.2 :=
+  deepMindABPhiTheta_existsUnique_of_twoAngleEnclosure
+    ((scaledResidualEnclosureTarget_iff_twoAngleEnclosureTarget).1
+      (e24ScaledResidualEnclosureTarget_of_alignedChecks
+        hPhiBelow hPhiAbove hThetaBelow hThetaAbove))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# E24KC2 proof helpers
+
+The discovery phase is intentionally outside the trusted chain.  It only chooses
+where to stop splitting and which terminal reason to claim.
+
+Every claimed leaf is then checked by the Lean kernel:
+* physically irrelevant leaf -> `physicallyIrrelevant cell = true`;
+* local leaf -> `cellInsideLocal cell = true`;
+* interval-rejected leaf -> `cell.rejected = true`;
+* unresolved frontier -> the old adaptive checker, but with remaining depth <= 9.
+
+These lemmas lift a primitive terminal fact to an `adaptiveCoverCheck` fact at
+arbitrary remaining depth without asking the kernel to explore the subtree.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+theorem adaptiveCoverCheck_true_of_physicallyIrrelevant
+    (depth : Nat) (cell : AngleCell)
+    (h : physicallyIrrelevant cell = true) :
+    adaptiveCoverCheck depth cell = true := by
+  cases depth <;> simp [adaptiveCoverCheck, h]
+
+theorem adaptiveCoverCheck_true_of_rejected
+    (depth : Nat) (cell : AngleCell)
+    (h : cell.rejected = true) :
+    adaptiveCoverCheck depth cell = true := by
+  by_cases hirr : physicallyIrrelevant cell = true
+  · cases depth <;> simp [adaptiveCoverCheck, hirr]
+  · by_cases hlocal : cellInsideLocal cell = true
+    · cases depth <;> simp [adaptiveCoverCheck, hirr, hlocal]
+    · cases depth <;> simp [adaptiveCoverCheck, hirr, hlocal, h]
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24 kernel child certificate: PhiBelow/HL, remaining depth 13. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+namespace CoverCertificateeccce3bad5
+
+private abbrev cellRoot : AngleCell :=
+  (childHL e24PhiBelowRoot)
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+private abbrev cell1130 : AngleCell :=
+  childLL cell113
+
+private abbrev cell1131 : AngleCell :=
+  childLH cell113
+
+private abbrev cell1132 : AngleCell :=
+  childHL cell113
+
+private abbrev cell1133 : AngleCell :=
+  childHH cell113
+
+private abbrev cell1310 : AngleCell :=
+  childLL cell131
+
+private abbrev cell1311 : AngleCell :=
+  childLH cell131
+
+private abbrev cell1312 : AngleCell :=
+  childHL cell131
+
+private abbrev cell1313 : AngleCell :=
+  childHH cell131
+
+private abbrev cell1330 : AngleCell :=
+  childLL cell133
+
+private abbrev cell1331 : AngleCell :=
+  childLH cell133
+
+private abbrev cell1332 : AngleCell :=
+  childHL cell133
+
+private abbrev cell1333 : AngleCell :=
+  childHH cell133
+
+private abbrev cell3110 : AngleCell :=
+  childLL cell311
+
+private abbrev cell3111 : AngleCell :=
+  childLH cell311
+
+private abbrev cell3112 : AngleCell :=
+  childHL cell311
+
+private abbrev cell3113 : AngleCell :=
+  childHH cell311
+
+private abbrev cell3130 : AngleCell :=
+  childLL cell313
+
+private abbrev cell3131 : AngleCell :=
+  childLH cell313
+
+private abbrev cell3132 : AngleCell :=
+  childHL cell313
+
+private abbrev cell3133 : AngleCell :=
+  childHH cell313
+
+private abbrev cell3310 : AngleCell :=
+  childLL cell331
+
+private abbrev cell3311 : AngleCell :=
+  childLH cell331
+
+private abbrev cell3312 : AngleCell :=
+  childHL cell331
+
+private abbrev cell3313 : AngleCell :=
+  childHH cell331
+
+private abbrev cell3330 : AngleCell :=
+  childLL cell333
+
+private abbrev cell3331 : AngleCell :=
+  childLH cell333
+
+private abbrev cell3332 : AngleCell :=
+  childHL cell333
+
+private abbrev cell3333 : AngleCell :=
+  childHH cell333
+
+private theorem checked1130 : adaptiveCoverCheck 9 cell1130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell1130 (by decide +kernel)
+
+private theorem checked1131 : adaptiveCoverCheck 9 cell1131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell1131 (by decide +kernel)
+
+private theorem checked1132 : adaptiveCoverCheck 9 cell1132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell1132 (by decide +kernel)
+
+private theorem checked1133 : adaptiveCoverCheck 9 cell1133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell1133 (by decide +kernel)
+
+private theorem checked1310 : adaptiveCoverCheck 9 cell1310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell1310 (by decide +kernel)
+
+private theorem checked1311 : adaptiveCoverCheck 9 cell1311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell1311 (by decide +kernel)
+
+private theorem checked1312 : adaptiveCoverCheck 9 cell1312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell1312 (by decide +kernel)
+
+private theorem checked1313 : adaptiveCoverCheck 9 cell1313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell1313 (by decide +kernel)
+
+private theorem checked1330 : adaptiveCoverCheck 9 cell1330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell1330 (by decide +kernel)
+
+private theorem checked1331 : adaptiveCoverCheck 9 cell1331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell1331 (by decide +kernel)
+
+private theorem checked1332 : adaptiveCoverCheck 9 cell1332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell1332 (by decide +kernel)
+
+private theorem checked1333 : adaptiveCoverCheck 9 cell1333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell1333 (by decide +kernel)
+
+private theorem checked3110 : adaptiveCoverCheck 9 cell3110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3110 (by decide +kernel)
+
+private theorem checked3111 : adaptiveCoverCheck 9 cell3111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3111 (by decide +kernel)
+
+private theorem checked3112 : adaptiveCoverCheck 9 cell3112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3112 (by decide +kernel)
+
+private theorem checked3113 : adaptiveCoverCheck 9 cell3113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3113 (by decide +kernel)
+
+private theorem checked3130 : adaptiveCoverCheck 9 cell3130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3130 (by decide +kernel)
+
+private theorem checked3131 : adaptiveCoverCheck 9 cell3131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3131 (by decide +kernel)
+
+private theorem checked3132 : adaptiveCoverCheck 9 cell3132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3132 (by decide +kernel)
+
+private theorem checked3133 : adaptiveCoverCheck 9 cell3133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3133 (by decide +kernel)
+
+private theorem checked3310 : adaptiveCoverCheck 9 cell3310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3310 (by decide +kernel)
+
+private theorem checked3311 : adaptiveCoverCheck 9 cell3311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3311 (by decide +kernel)
+
+private theorem checked3312 : adaptiveCoverCheck 9 cell3312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3312 (by decide +kernel)
+
+private theorem checked3313 : adaptiveCoverCheck 9 cell3313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3313 (by decide +kernel)
+
+private theorem checked3330 : adaptiveCoverCheck 9 cell3330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3330 (by decide +kernel)
+
+private theorem checked3331 : adaptiveCoverCheck 9 cell3331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3331 (by decide +kernel)
+
+private theorem checked3332 : adaptiveCoverCheck 9 cell3332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3332 (by decide +kernel)
+
+private theorem checked3333 : adaptiveCoverCheck 9 cell3333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3333 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 10 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 10 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 10 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 10 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 10 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 10 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 10 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 10 cell113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell113
+    checked1130 checked1131 checked1132 checked1133
+
+private theorem checked120 : adaptiveCoverCheck 10 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 10 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 10 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 10 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 10 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 10 cell131 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell131
+    checked1310 checked1311 checked1312 checked1313
+
+private theorem checked132 : adaptiveCoverCheck 10 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 10 cell133 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell133
+    checked1330 checked1331 checked1332 checked1333
+
+private theorem checked300 : adaptiveCoverCheck 10 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 10 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 10 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 10 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 10 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 10 cell311 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell311
+    checked3110 checked3111 checked3112 checked3113
+
+private theorem checked312 : adaptiveCoverCheck 10 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 10 cell313 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell313
+    checked3130 checked3131 checked3132 checked3133
+
+private theorem checked320 : adaptiveCoverCheck 10 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 10 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 10 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 10 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 10 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 10 cell331 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell331
+    checked3310 checked3311 checked3312 checked3313
+
+private theorem checked332 : adaptiveCoverCheck 10 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 10 cell333 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell333
+    checked3330 checked3331 checked3332 checked3333
+
+private theorem checked00 : adaptiveCoverCheck 11 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 11 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 11 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 11 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 11 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 11 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 11 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 11 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 11 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 11 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 11 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 11 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 11 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 11 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 11 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 11 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 12 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 11 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 12 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 11 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 12 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 11 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 12 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 11 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 13 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 12 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateeccce3bad5
+
+theorem e24PhiBelowKernelHL :
+    adaptiveCoverCheck 13 (childHL e24PhiBelowRoot) = true := by
+  exact CoverCertificateeccce3bad5.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartB.Definitions`.
+* `KernelOnly.PartB.Parameters`.
+* `KernelOnly.PartB.CellSemanticsCore`.
+* `KernelOnly.PartB.Phase1Semantics`.
+* `KernelOnly.PartB.Phase2Semantics`.
+* `KernelOnly.PartB.Phase3Semantics`.
+* `KernelOnly.PartB.Phase4Semantics`.
+* `KernelOnly.PartB.Phase5Semantics`.
+* `KernelOnly.PartB.CellSemantics`.
+* `KernelOnly.PartB.CellCover`.
+* `KernelOnly.PartB.Continuum`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Part B semantic definitions
+
+The numerical layer from Part A is frozen.  This file names the certified
+parameter vector and the two real support functions whose continuum lower
+bounds are established by the exact cell certificate.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartB
+
+open RatInterval
+
+/-- The unique direct-system parameter vector certified in Part A. -/
+def params : Romik.Params :=
+  Romik.coordEquiv.symm PartALeanCert.fullRoot
+
+/-- The physical parameter interval. -/
+def physicalInterval : Set ℝ := Set.Icc (0 : ℝ) (Real.pi / 2)
+
+/-- The two global support functions used in the manuscript's grid lemma. -/
+def Gu (s t : ℝ) : ℝ :=
+  1 + dot (Romik.path params s - Romik.path params t) (u s)
+
+/-- The vertical support slack between two path positions in the frame at `s`. -/
+def Gv (s t : ℝ) : ℝ :=
+  1 + dot (Romik.path params s - Romik.path params t) (v s)
+
+/-- The real version of the exact rational target. -/
+def target : ℝ := (targetQ : ℝ)
+
+/-- Physical mesh node `iπ/128`. -/
+def nodeTime (i : Nat) : ℝ := (nodeCoeff i : ℝ) * Real.pi
+
+/-- Physical closed cell. -/
+def cellSet (i : Cell) : Set ℝ :=
+  Set.Icc (nodeTime i.1) (nodeTime (i.1 + 1))
+
+/-- Semantic containment in a planar interval box. -/
+def PointContains (z : RatInterval × RatInterval) (p : Point) : Prop :=
+  RatInterval.Contains z.1 p.1 ∧ RatInterval.Contains z.2 p.2
+
+end PartB
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part B parameter, matching and regularity certificate
+
+Every theorem in this file is a direct consequence of the concrete Part A
+unique zero.  No numerical computation is repeated.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartB
+
+/-- The certified parameter vector lies in the direct Romik box. -/
+theorem params_mem : params ∈ Romik.box := by
+  simpa [params, Romik.vectorBox] using PartALeanCert.fullRoot_mem
+
+/-- The certified parameter vector satisfies all 22 direct equations. -/
+theorem params_equations : Romik.Equations params := by
+  simpa [params, Romik.Equations, Romik.vectorSystem] using
+    PartALeanCert.fullRoot_zero
+
+/-- The four physical switches are correctly ordered. -/
+theorem params_switchOrder : Romik.SwitchOrder params :=
+  Romik.switchOrder_of_mem_box params_mem
+
+/-- Positional matching at `φ`. -/
+theorem match12 : Romik.path1 params params.phi = Romik.path2 params params.phi :=
+  Romik.match_path12_of_equations params_equations
+
+/-- Positional matching at `θ`. -/
+theorem match23 : Romik.path2 params params.theta = Romik.path3 params params.theta :=
+  Romik.match_path23_of_equations params_equations
+
+/-- Positional matching at `π/2-θ`. -/
+theorem match34 :
+    Romik.path3 params (Real.pi / 2 - params.theta) =
+      Romik.path4 params (Real.pi / 2 - params.theta) :=
+  Romik.match_path34_of_equations params_equations
+
+/-- Positional matching at `π/2-φ`. -/
+theorem match45 :
+    Romik.path4 params (Real.pi / 2 - params.phi) =
+      Romik.path5 params (Real.pi / 2 - params.phi) :=
+  Romik.match_path45_of_equations params_equations
+
+/-- Global continuity of the literal five-phase path. -/
+theorem continuous_path : Continuous (Romik.path params) :=
+  Romik.continuous_path_of_mem_box_and_equations params_mem params_equations
+
+/-- Global continuity of the associated rigid frame. -/
+theorem continuous_frame : SE2.ContinuousPath (Romik.frame params) :=
+  Romik.continuousPath_frame_of_path params continuous_path
+
+/-- Exact initial normalization. -/
+theorem path_zero : Romik.path params 0 = (0, 0) :=
+  Romik.path_zero_of_mem_box_and_equations params_mem params_equations
+
+/-- Clean exact bounds for the two independent switching angles. -/
+theorem phi_bounds :
+    ((1958868239504182093160893749 : ℝ) /
+      50000000000000000000000000000) ≤ params.phi ∧
+    params.phi ≤
+      ((78354729580167283726435751 : ℝ) /
+        2000000000000000000000000000) := by
+  have hp := params_mem
+  dsimp [params, Romik.vectorBox, Romik.box, qR] at hp ⊢
+  aesop
+
+theorem theta_bounds :
+    ((34065075469136244723692787727 : ℝ) /
+      50000000000000000000000000000) ≤ params.theta ∧
+    params.theta ≤
+      ((34065075469136244723692787983 : ℝ) /
+        50000000000000000000000000000) := by
+  have hp := params_mem
+  dsimp [params, Romik.vectorBox, Romik.box, qR] at hp ⊢
+  aesop
+
+end PartB
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Core semantic infrastructure for the Part B cell certificate
+
+Hull, time-cell, certified-root-coordinate and trigonometric containment lemmas
+are isolated here so the five analytic phases can be compiled and diagnosed
+independently.
+-/
+
+/-! ## Hull and time-cell semantics -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartB
+
+open RatInterval
+
+theorem contains_intervalHull_left {a b : RatInterval} {x : ℝ}
+    (hx : Contains a x) : Contains (intervalHull a b) x := by
+  constructor
+  · have hmin : min a.lo b.lo ≤ a.lo := min_le_left _ _
+    exact le_trans (by exact_mod_cast hmin) hx.1
+  · have hmax : a.hi ≤ max a.hi b.hi := le_max_left _ _
+    exact le_trans hx.2 (by exact_mod_cast hmax)
+
+theorem contains_intervalHull_right {a b : RatInterval} {x : ℝ}
+    (hx : Contains b x) : Contains (intervalHull a b) x := by
+  constructor
+  · have hmin : min a.lo b.lo ≤ b.lo := min_le_right _ _
+    exact le_trans (by exact_mod_cast hmin) hx.1
+  · have hmax : b.hi ≤ max a.hi b.hi := le_max_right _ _
+    exact le_trans hx.2 (by exact_mod_cast hmax)
+
+theorem pointContains_hull_left {a b : RatInterval × RatInterval} {x : Point}
+    (hx : PointContains a x) : PointContains (pointHull a b) x := by
+  exact ⟨contains_intervalHull_left hx.1,
+    contains_intervalHull_left hx.2⟩
+
+theorem pointContains_hull_right {a b : RatInterval × RatInterval} {x : Point}
+    (hx : PointContains b x) : PointContains (pointHull a b) x := by
+  exact ⟨contains_intervalHull_right hx.1,
+    contains_intervalHull_right hx.2⟩
+
+theorem nodeTime_mono {i j : Nat} (hij : i ≤ j) :
+    nodeTime i ≤ nodeTime j := by
+  unfold nodeTime nodeCoeff
+  apply mul_le_mul_of_nonneg_right _ Real.pi_pos.le
+  norm_num
+  have hijR : (i : ℝ) ≤ (j : ℝ) := by
+    exact_mod_cast hij
+  exact div_le_div_of_nonneg_right hijR (by norm_num)
+
+theorem cellTimeInterval_contains_start (i : Cell) :
+    Contains (cellTimeInterval i) (nodeTime i.1) := by
+  apply contains_intervalHull_left
+  simpa [nodeTime, nodeCoeff, ExactReplay.scale] using
+    (RatInterval.contains_scale (a := nodeCoeff i.1)
+      ExactReplay.piI_contains_pi)
+
+theorem cellTimeInterval_contains_end (i : Cell) :
+    Contains (cellTimeInterval i) (nodeTime (i.1 + 1)) := by
+  apply contains_intervalHull_right
+  simpa [nodeTime, nodeCoeff, ExactReplay.scale] using
+    (RatInterval.contains_scale (a := nodeCoeff (i.1 + 1))
+      ExactReplay.piI_contains_pi)
+
+theorem cellTimeInterval_contains {i : Cell} {t : ℝ}
+    (ht : t ∈ cellSet i) : Contains (cellTimeInterval i) t := by
+  exact ⟨le_trans (cellTimeInterval_contains_start i).1 ht.1,
+    le_trans ht.2 (cellTimeInterval_contains_end i).2⟩
+
+/-! ## The Part A root is enclosed coordinatewise -/
+
+theorem root_coord_contains (k : Fin 22) :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox k.1)
+      (PartALeanCert.fullRoot k) := by
+  exact ((Romik.inputBox_exact PartALeanCert.fullRoot).1
+    PartALeanCert.fullRoot_mem).2 k
+
+theorem k11_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 0) params.k11 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 0)
+    (PartALeanCert.fullRoot (0 : Fin 22))
+  exact root_coord_contains (0 : Fin 22)
+theorem k12_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 1) params.k12 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 1)
+    (PartALeanCert.fullRoot (1 : Fin 22))
+  exact root_coord_contains (1 : Fin 22)
+theorem k21_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 2) params.k21 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 2)
+    (PartALeanCert.fullRoot (2 : Fin 22))
+  exact root_coord_contains (2 : Fin 22)
+theorem k22_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 3) params.k22 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 3)
+    (PartALeanCert.fullRoot (3 : Fin 22))
+  exact root_coord_contains (3 : Fin 22)
+theorem k31_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 4) params.k31 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 4)
+    (PartALeanCert.fullRoot (4 : Fin 22))
+  exact root_coord_contains (4 : Fin 22)
+theorem k32_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 5) params.k32 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 5)
+    (PartALeanCert.fullRoot (5 : Fin 22))
+  exact root_coord_contains (5 : Fin 22)
+theorem k41_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 6) params.k41 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 6)
+    (PartALeanCert.fullRoot (6 : Fin 22))
+  exact root_coord_contains (6 : Fin 22)
+theorem k42_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 7) params.k42 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 7)
+    (PartALeanCert.fullRoot (7 : Fin 22))
+  exact root_coord_contains (7 : Fin 22)
+theorem k51_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 8) params.k51 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 8)
+    (PartALeanCert.fullRoot (8 : Fin 22))
+  exact root_coord_contains (8 : Fin 22)
+theorem k52_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 9) params.k52 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 9)
+    (PartALeanCert.fullRoot (9 : Fin 22))
+  exact root_coord_contains (9 : Fin 22)
+theorem a1_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 10) params.a1 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 10)
+    (PartALeanCert.fullRoot (10 : Fin 22))
+  exact root_coord_contains (10 : Fin 22)
+theorem a2_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 11) params.a2 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 11)
+    (PartALeanCert.fullRoot (11 : Fin 22))
+  exact root_coord_contains (11 : Fin 22)
+theorem b1_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 12) params.b1 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 12)
+    (PartALeanCert.fullRoot (12 : Fin 22))
+  exact root_coord_contains (12 : Fin 22)
+theorem b2_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 13) params.b2 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 13)
+    (PartALeanCert.fullRoot (13 : Fin 22))
+  exact root_coord_contains (13 : Fin 22)
+theorem c1_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 14) params.c1 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 14)
+    (PartALeanCert.fullRoot (14 : Fin 22))
+  exact root_coord_contains (14 : Fin 22)
+theorem c2_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 15) params.c2 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 15)
+    (PartALeanCert.fullRoot (15 : Fin 22))
+  exact root_coord_contains (15 : Fin 22)
+theorem d1_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 16) params.d1 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 16)
+    (PartALeanCert.fullRoot (16 : Fin 22))
+  exact root_coord_contains (16 : Fin 22)
+theorem d2_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 17) params.d2 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 17)
+    (PartALeanCert.fullRoot (17 : Fin 22))
+  exact root_coord_contains (17 : Fin 22)
+theorem e1_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 18) params.e1 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 18)
+    (PartALeanCert.fullRoot (18 : Fin 22))
+  exact root_coord_contains (18 : Fin 22)
+theorem e2_contains :
+    Contains (ExactReplay.getI ExactReplay.fullInputBox 19) params.e2 := by
+  change Contains (ExactReplay.getI ExactReplay.fullInputBox 19)
+    (PartALeanCert.fullRoot (19 : Fin 22))
+  exact root_coord_contains (19 : Fin 22)
+
+theorem point_contains (q : ℚ) : Contains (RatInterval.point q) (q : ℝ) :=
+  (RatInterval.contains_point_iff q (q : ℝ)).2 rfl
+
+theorem trig_contains {i : Cell} {t : ℝ}
+    (htCell : t ∈ cellSet i) (htPhys : t ∈ physicalInterval) :
+    Contains (ExactReplay.cosineInterval (cellTimeInterval i)) (Real.cos t) ∧
+    Contains (ExactReplay.sineInterval (cellTimeInterval i)) (Real.sin t) := by
+  have htI := cellTimeInterval_contains htCell
+  exact ⟨ExactReplay.cosI_contains htI htPhys.1 htPhys.2,
+    ExactReplay.sinI_contains htI htPhys.1 htPhys.2⟩
+
+end PartB
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Semantic enclosure for analytic path phase 1. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartB
+
+open RatInterval
+
+theorem cellPiece1_contains {i : Cell} {t : ℝ}
+    (htCell : t ∈ cellSet i) (htPhys : t ∈ physicalInterval) :
+    PointContains (cellPieceInterval 1 i) (Romik.path1 params t) := by
+  let ti := cellTimeInterval i
+  have htI : Contains ti t := cellTimeInterval_contains htCell
+  have htr := trig_contains htCell htPhys
+  have hc := htr.1
+  have hs := htr.2
+  have hone := point_contains 1
+  have hhalf := point_contains (1 / 2)
+  have hz1 := RatInterval.contains_sub
+    (RatInterval.contains_add (RatInterval.contains_mul a1_contains hc)
+      (RatInterval.contains_mul a2_contains hs)) hone
+  have hz2 := RatInterval.contains_sub
+    (RatInterval.contains_add
+      (RatInterval.contains_mul (RatInterval.contains_neg a2_contains) hc)
+      (RatInterval.contains_mul a1_contains hs)) hhalf
+  have hr1 := RatInterval.contains_sub
+    (RatInterval.contains_mul hc hz1) (RatInterval.contains_mul hs hz2)
+  have hr2 := RatInterval.contains_add
+    (RatInterval.contains_mul hs hz1) (RatInterval.contains_mul hc hz2)
+  constructor
+  · simpa [cellPieceInterval, ti, Romik.path1, Romik.rot, Romik.addK, ExactReplay.oneI] using
+      RatInterval.contains_add hr1 k11_contains
+  · simpa [cellPieceInterval, ti, Romik.path1, Romik.rot, Romik.addK, ExactReplay.oneI] using
+      RatInterval.contains_add hr2 k12_contains
+
+end PartB
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Semantic enclosure for analytic path phase 2. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartB
+
+open RatInterval
+
+theorem cellPiece2_contains {i : Cell} {t : ℝ}
+    (htCell : t ∈ cellSet i) (htPhys : t ∈ physicalInterval) :
+    PointContains (cellPieceInterval 2 i) (Romik.path2 params t) := by
+  let ti := cellTimeInterval i
+  have htI : Contains ti t := cellTimeInterval_contains htCell
+  have htr := trig_contains htCell htPhys
+  have hc := htr.1
+  have hs := htr.2
+  have hone := point_contains 1
+  have hhalf := point_contains (1 / 2)
+  have hquarter := point_contains (1 / 4)
+  have htt := RatInterval.contains_mul htI htI
+  have hz1 := RatInterval.contains_add
+    (RatInterval.contains_add
+      (RatInterval.contains_mul (RatInterval.contains_neg hquarter) htt)
+      (RatInterval.contains_mul b1_contains htI)) b2_contains
+  have hz2 := RatInterval.contains_sub
+    (RatInterval.contains_sub (RatInterval.contains_mul hhalf htI) b1_contains)
+    hone
+  have hr1 := RatInterval.contains_sub
+    (RatInterval.contains_mul hc hz1) (RatInterval.contains_mul hs hz2)
+  have hr2 := RatInterval.contains_add
+    (RatInterval.contains_mul hs hz1) (RatInterval.contains_mul hc hz2)
+  constructor
+  · simpa [cellPieceInterval, ti, Romik.path2, Romik.rot, Romik.addK,
+      ExactReplay.oneI, mul_assoc] using RatInterval.contains_add hr1 k21_contains
+  · simpa [cellPieceInterval, ti, Romik.path2, Romik.rot, Romik.addK,
+      ExactReplay.oneI, mul_assoc] using RatInterval.contains_add hr2 k22_contains
+
+end PartB
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Semantic enclosure for analytic path phase 3. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartB
+
+open RatInterval
+
+theorem cellPiece3_contains {i : Cell} {t : ℝ}
+    (htCell : t ∈ cellSet i) (htPhys : t ∈ physicalInterval) :
+    PointContains (cellPieceInterval 3 i) (Romik.path3 params t) := by
+  let ti := cellTimeInterval i
+  have htI : Contains ti t := cellTimeInterval_contains htCell
+  have htr := trig_contains htCell htPhys
+  have hc := htr.1
+  have hs := htr.2
+  have hz1 := RatInterval.contains_sub c1_contains htI
+  have hz2 := RatInterval.contains_add c2_contains htI
+  have hr1 := RatInterval.contains_sub
+    (RatInterval.contains_mul hc hz1) (RatInterval.contains_mul hs hz2)
+  have hr2 := RatInterval.contains_add
+    (RatInterval.contains_mul hs hz1) (RatInterval.contains_mul hc hz2)
+  constructor
+  · simpa [cellPieceInterval, ti, Romik.path3, Romik.rot, Romik.addK] using
+      RatInterval.contains_add hr1 k31_contains
+  · simpa [cellPieceInterval, ti, Romik.path3, Romik.rot, Romik.addK] using
+      RatInterval.contains_add hr2 k32_contains
+
+end PartB
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Semantic enclosure for analytic path phase 4. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartB
+
+open RatInterval
+
+theorem cellPiece4_contains {i : Cell} {t : ℝ}
+    (htCell : t ∈ cellSet i) (htPhys : t ∈ physicalInterval) :
+    PointContains (cellPieceInterval 4 i) (Romik.path4 params t) := by
+  let ti := cellTimeInterval i
+  have htI : Contains ti t := cellTimeInterval_contains htCell
+  have htr := trig_contains htCell htPhys
+  have hc := htr.1
+  have hs := htr.2
+  have hone := point_contains 1
+  have hhalf := point_contains (1 / 2)
+  have hquarter := point_contains (1 / 4)
+  have htt := RatInterval.contains_mul htI htI
+  have hz1 := RatInterval.contains_sub
+    (RatInterval.contains_add
+      (RatInterval.contains_mul (RatInterval.contains_neg hhalf) htI)
+      d1_contains) hone
+  have hz2 := RatInterval.contains_add
+    (RatInterval.contains_add
+      (RatInterval.contains_mul (RatInterval.contains_neg hquarter) htt)
+      (RatInterval.contains_mul d1_contains htI)) d2_contains
+  have hr1 := RatInterval.contains_sub
+    (RatInterval.contains_mul hc hz1) (RatInterval.contains_mul hs hz2)
+  have hr2 := RatInterval.contains_add
+    (RatInterval.contains_mul hs hz1) (RatInterval.contains_mul hc hz2)
+  constructor
+  · simpa [cellPieceInterval, ti, Romik.path4, Romik.rot, Romik.addK,
+      ExactReplay.oneI, mul_assoc] using RatInterval.contains_add hr1 k41_contains
+  · simpa [cellPieceInterval, ti, Romik.path4, Romik.rot, Romik.addK,
+      ExactReplay.oneI, mul_assoc] using RatInterval.contains_add hr2 k42_contains
+
+end PartB
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Semantic enclosure for analytic path phase 5. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartB
+
+open RatInterval
+
+theorem cellPiece5_contains {i : Cell} {t : ℝ}
+    (htCell : t ∈ cellSet i) (htPhys : t ∈ physicalInterval) :
+    PointContains (cellPieceInterval 5 i) (Romik.path5 params t) := by
+  let ti := cellTimeInterval i
+  have htr := trig_contains htCell htPhys
+  have hc := htr.1
+  have hs := htr.2
+  have hone := point_contains 1
+  have hhalf := point_contains (1 / 2)
+  have hz1 := RatInterval.contains_sub
+    (RatInterval.contains_add (RatInterval.contains_mul e1_contains hc)
+      (RatInterval.contains_mul e2_contains hs)) hhalf
+  have hz2 := RatInterval.contains_sub
+    (RatInterval.contains_add
+      (RatInterval.contains_mul (RatInterval.contains_neg e2_contains) hc)
+      (RatInterval.contains_mul e1_contains hs)) hone
+  have hr1 := RatInterval.contains_sub
+    (RatInterval.contains_mul hc hz1) (RatInterval.contains_mul hs hz2)
+  have hr2 := RatInterval.contains_add
+    (RatInterval.contains_mul hs hz1) (RatInterval.contains_mul hc hz2)
+  constructor
+  · simpa [cellPieceInterval, ti, Romik.path5, Romik.rot, Romik.addK, ExactReplay.oneI] using
+      RatInterval.contains_add hr1 k51_contains
+  · simpa [cellPieceInterval, ti, Romik.path5, Romik.rot, Romik.addK, ExactReplay.oneI] using
+      RatInterval.contains_add hr2 k52_contains
+
+end PartB
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Assembly of semantic soundness for the Part B cell certificate
+
+The five phase enclosures are independent modules.  This file classifies the
+four switching cells, selects the appropriate phase/hull, and proves semantic
+soundness of the two product-cell support expressions.
+-/
+
+/-! ## Switches lie in exactly four mesh cells -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartB
+
+open RatInterval
+
+private theorem node1_lt_phi : nodeTime 1 < params.phi := by
+  have hpi : Real.pi ≤ (ExactReplay.piI.hi : ℝ) :=
+    ExactReplay.piI_contains_pi.2
+  have hp := phi_bounds.1
+  unfold nodeTime nodeCoeff
+  norm_num [ExactReplay.piI, ExactReplay.q] at hpi hp ⊢
+  nlinarith
+
+private theorem phi_lt_node2 : params.phi < nodeTime 2 := by
+  have hpi : (ExactReplay.piI.lo : ℝ) ≤ Real.pi :=
+    ExactReplay.piI_contains_pi.1
+  have hp := phi_bounds.2
+  unfold nodeTime nodeCoeff
+  norm_num [ExactReplay.piI, ExactReplay.q] at hpi hp ⊢
+  nlinarith
+
+private theorem node27_lt_theta : nodeTime 27 < params.theta := by
+  have hpi : Real.pi ≤ (ExactReplay.piI.hi : ℝ) :=
+    ExactReplay.piI_contains_pi.2
+  have hp := theta_bounds.1
+  unfold nodeTime nodeCoeff
+  norm_num [ExactReplay.piI, ExactReplay.q] at hpi hp ⊢
+  nlinarith
+
+private theorem theta_lt_node28 : params.theta < nodeTime 28 := by
+  have hpi : (ExactReplay.piI.lo : ℝ) ≤ Real.pi :=
+    ExactReplay.piI_contains_pi.1
+  have hp := theta_bounds.2
+  unfold nodeTime nodeCoeff
+  norm_num [ExactReplay.piI, ExactReplay.q] at hpi hp ⊢
+  nlinarith
+
+private theorem node36_lt_eta :
+    nodeTime 36 < Real.pi / 2 - params.theta := by
+  have h := theta_lt_node28
+  unfold nodeTime nodeCoeff at h ⊢
+  norm_num at h ⊢
+  linarith
+
+private theorem eta_lt_node37 :
+    Real.pi / 2 - params.theta < nodeTime 37 := by
+  have h := node27_lt_theta
+  unfold nodeTime nodeCoeff at h ⊢
+  norm_num at h ⊢
+  linarith
+
+private theorem node62_lt_tau :
+    nodeTime 62 < Real.pi / 2 - params.phi := by
+  have h := phi_lt_node2
+  unfold nodeTime nodeCoeff at h ⊢
+  norm_num at h ⊢
+  linarith
+
+private theorem tau_lt_node63 :
+    Real.pi / 2 - params.phi < nodeTime 63 := by
+  have h := node1_lt_phi
+  unfold nodeTime nodeCoeff at h ⊢
+  norm_num at h ⊢
+  linarith
+
+private theorem phase1_cell {i : Cell} {t : ℝ}
+    (ht : t ∈ cellSet i) (hphase : t ≤ params.phi) : i.1 ≤ 1 := by
+  by_contra hnot
+  have hi : 2 ≤ i.1 := by omega
+  have h2i : nodeTime 2 ≤ nodeTime i.1 := nodeTime_mono hi
+  linarith [phi_lt_node2, ht.1]
+
+private theorem phase2_cell {i : Cell} {t : ℝ}
+    (ht : t ∈ cellSet i) (hlo : params.phi < t)
+    (hhi : t ≤ params.theta) : 1 ≤ i.1 ∧ i.1 ≤ 27 := by
+  constructor
+  · by_contra hnot
+    have hi0 : i.1 = 0 := by omega
+    have hend : t ≤ nodeTime 1 := by simpa [cellSet, hi0] using ht.2
+    linarith [node1_lt_phi]
+  · by_contra hnot
+    have hi : 28 ≤ i.1 := by omega
+    have h28i : nodeTime 28 ≤ nodeTime i.1 := nodeTime_mono hi
+    linarith [theta_lt_node28, ht.1]
+
+private theorem phase3_cell {i : Cell} {t : ℝ}
+    (ht : t ∈ cellSet i) (hlo : params.theta < t)
+    (hhi : t ≤ Real.pi / 2 - params.theta) :
+    27 ≤ i.1 ∧ i.1 ≤ 36 := by
+  constructor
+  · by_contra hnot
+    have hi : i.1 + 1 ≤ 27 := by omega
+    have hend : nodeTime (i.1 + 1) ≤ nodeTime 27 := nodeTime_mono hi
+    linarith [node27_lt_theta, ht.2]
+  · by_contra hnot
+    have hi : 37 ≤ i.1 := by omega
+    have h37i : nodeTime 37 ≤ nodeTime i.1 := nodeTime_mono hi
+    linarith [eta_lt_node37, ht.1]
+
+private theorem phase4_cell {i : Cell} {t : ℝ}
+    (ht : t ∈ cellSet i)
+    (hlo : Real.pi / 2 - params.theta < t)
+    (hhi : t ≤ Real.pi / 2 - params.phi) :
+    36 ≤ i.1 ∧ i.1 ≤ 62 := by
+  constructor
+  · by_contra hnot
+    have hi : i.1 + 1 ≤ 36 := by omega
+    have hend : nodeTime (i.1 + 1) ≤ nodeTime 36 := nodeTime_mono hi
+    linarith [node36_lt_eta, ht.2]
+  · by_contra hnot
+    have hi : 63 ≤ i.1 := by omega
+    have h63i : nodeTime 63 ≤ nodeTime i.1 := nodeTime_mono hi
+    linarith [tau_lt_node63, ht.1]
+
+private theorem phase5_cell {i : Cell} {t : ℝ}
+    (ht : t ∈ cellSet i)
+    (hlo : Real.pi / 2 - params.phi < t) : 62 ≤ i.1 := by
+  by_contra hnot
+  have hi : i.1 + 1 ≤ 62 := by omega
+  have hend : nodeTime (i.1 + 1) ≤ nodeTime 62 := nodeTime_mono hi
+  linarith [node62_lt_tau, ht.2]
+
+/-! ## Every literal path branch is contained in the selected cell hull -/
+
+private theorem cellPath_contains_piece1 {i : Cell} {t : ℝ}
+    (htCell : t ∈ cellSet i) (htPhys : t ∈ physicalInterval)
+    (hi : i.1 ≤ 1) :
+    PointContains (cellPathInterval i) (Romik.path1 params t) := by
+  have h := cellPiece1_contains htCell htPhys
+  by_cases h0 : i.1 = 0
+  · simpa [cellPathInterval, h0] using h
+  · have h1 : i.1 = 1 := by omega
+    simpa [cellPathInterval, h0, h1] using pointContains_hull_left h
+
+private theorem cellPath_contains_piece2 {i : Cell} {t : ℝ}
+    (htCell : t ∈ cellSet i) (htPhys : t ∈ physicalInterval)
+    (hi : 1 ≤ i.1 ∧ i.1 ≤ 27) :
+    PointContains (cellPathInterval i) (Romik.path2 params t) := by
+  have h := cellPiece2_contains htCell htPhys
+  by_cases h1 : i.1 = 1
+  · have h0 : i.1 ≠ 0 := by omega
+    simpa [cellPathInterval, h0, h1] using pointContains_hull_right h
+  by_cases h27 : i.1 = 27
+  · have h0 : i.1 ≠ 0 := by omega
+    have h26 : ¬ i.1 ≤ 26 := by omega
+    simpa [cellPathInterval, h0, h1, h26, h27] using pointContains_hull_left h
+  · have h0 : i.1 ≠ 0 := by omega
+    have h26 : i.1 ≤ 26 := by omega
+    simpa [cellPathInterval, h0, h1, h26] using h
+
+private theorem cellPath_contains_piece3 {i : Cell} {t : ℝ}
+    (htCell : t ∈ cellSet i) (htPhys : t ∈ physicalInterval)
+    (hi : 27 ≤ i.1 ∧ i.1 ≤ 36) :
+    PointContains (cellPathInterval i) (Romik.path3 params t) := by
+  have h := cellPiece3_contains htCell htPhys
+  by_cases h27 : i.1 = 27
+  · have h0 : i.1 ≠ 0 := by omega
+    have h1 : i.1 ≠ 1 := by omega
+    have h26 : ¬ i.1 ≤ 26 := by omega
+    simpa [cellPathInterval, h0, h1, h26, h27] using pointContains_hull_right h
+  by_cases h36 : i.1 = 36
+  · have h0 : i.1 ≠ 0 := by omega
+    have h1 : i.1 ≠ 1 := by omega
+    have h26 : ¬ i.1 ≤ 26 := by omega
+    have h27n : i.1 ≠ 27 := by omega
+    have h35 : ¬ i.1 ≤ 35 := by omega
+    simpa [cellPathInterval, h0, h1, h26, h27n, h35, h36] using
+      pointContains_hull_left h
+  · have h0 : i.1 ≠ 0 := by omega
+    have h1 : i.1 ≠ 1 := by omega
+    have h26 : ¬ i.1 ≤ 26 := by omega
+    have h27n : i.1 ≠ 27 := by omega
+    have h35 : i.1 ≤ 35 := by omega
+    simpa [cellPathInterval, h0, h1, h26, h27n, h35] using h
+
+private theorem cellPath_contains_piece4 {i : Cell} {t : ℝ}
+    (htCell : t ∈ cellSet i) (htPhys : t ∈ physicalInterval)
+    (hi : 36 ≤ i.1 ∧ i.1 ≤ 62) :
+    PointContains (cellPathInterval i) (Romik.path4 params t) := by
+  have h := cellPiece4_contains htCell htPhys
+  by_cases h36 : i.1 = 36
+  · have h0 : i.1 ≠ 0 := by omega
+    have h1 : i.1 ≠ 1 := by omega
+    have h26 : ¬ i.1 ≤ 26 := by omega
+    have h27 : i.1 ≠ 27 := by omega
+    have h35 : ¬ i.1 ≤ 35 := by omega
+    simpa [cellPathInterval, h0, h1, h26, h27, h35, h36] using
+      pointContains_hull_right h
+  by_cases h62 : i.1 = 62
+  · have h0 : i.1 ≠ 0 := by omega
+    have h1 : i.1 ≠ 1 := by omega
+    have h26 : ¬ i.1 ≤ 26 := by omega
+    have h27 : i.1 ≠ 27 := by omega
+    have h35 : ¬ i.1 ≤ 35 := by omega
+    have h36n : i.1 ≠ 36 := by omega
+    have h61 : ¬ i.1 ≤ 61 := by omega
+    simpa [cellPathInterval, h0, h1, h26, h27, h35, h36n, h61, h62] using
+      pointContains_hull_left h
+  · have h0 : i.1 ≠ 0 := by omega
+    have h1 : i.1 ≠ 1 := by omega
+    have h26 : ¬ i.1 ≤ 26 := by omega
+    have h27 : i.1 ≠ 27 := by omega
+    have h35 : ¬ i.1 ≤ 35 := by omega
+    have h36n : i.1 ≠ 36 := by omega
+    have h61 : i.1 ≤ 61 := by omega
+    simpa [cellPathInterval, h0, h1, h26, h27, h35, h36n, h61] using h
+
+private theorem cellPath_contains_piece5 {i : Cell} {t : ℝ}
+    (htCell : t ∈ cellSet i) (htPhys : t ∈ physicalInterval)
+    (hi : 62 ≤ i.1) :
+    PointContains (cellPathInterval i) (Romik.path5 params t) := by
+  have h := cellPiece5_contains htCell htPhys
+  by_cases h62 : i.1 = 62
+  · have h0 : i.1 ≠ 0 := by omega
+    have h1 : i.1 ≠ 1 := by omega
+    have h26 : ¬ i.1 ≤ 26 := by omega
+    have h27 : i.1 ≠ 27 := by omega
+    have h35 : ¬ i.1 ≤ 35 := by omega
+    have h36 : i.1 ≠ 36 := by omega
+    have h61 : ¬ i.1 ≤ 61 := by omega
+    simpa [cellPathInterval, h0, h1, h26, h27, h35, h36, h61, h62] using
+      pointContains_hull_right h
+  · have h63 : i.1 = 63 := by omega
+    have h0 : i.1 ≠ 0 := by omega
+    have h1 : i.1 ≠ 1 := by omega
+    have h26 : ¬ i.1 ≤ 26 := by omega
+    have h27 : i.1 ≠ 27 := by omega
+    have h35 : ¬ i.1 ≤ 35 := by omega
+    have h36 : i.1 ≠ 36 := by omega
+    have h61 : ¬ i.1 ≤ 61 := by omega
+    have h62n : i.1 ≠ 62 := by omega
+    simpa [cellPathInterval, h0, h1, h26, h27, h35, h36, h61, h62n,
+      h63] using h
+
+/-- The interval selected for a cell contains the literal five-phase path at
+every physical time in that cell. -/
+theorem cellPath_contains {i : Cell} {t : ℝ}
+    (htCell : t ∈ cellSet i) (htPhys : t ∈ physicalInterval) :
+    PointContains (cellPathInterval i) (Romik.path params t) := by
+  unfold Romik.path
+  by_cases h1 : t ≤ params.phi
+  · rw [ite_eq_left h1]
+    exact cellPath_contains_piece1 htCell htPhys (phase1_cell htCell h1)
+  · rw [ite_eq_right h1]
+    have h1' : params.phi < t := lt_of_not_ge h1
+    by_cases h2 : t ≤ params.theta
+    · rw [ite_eq_left h2]
+      exact cellPath_contains_piece2 htCell htPhys
+        (phase2_cell htCell h1' h2)
+    · rw [ite_eq_right h2]
+      have h2' : params.theta < t := lt_of_not_ge h2
+      by_cases h3 : t ≤ Real.pi / 2 - params.theta
+      · rw [ite_eq_left h3]
+        exact cellPath_contains_piece3 htCell htPhys
+          (phase3_cell htCell h2' h3)
+      · rw [ite_eq_right h3]
+        have h3' : Real.pi / 2 - params.theta < t := lt_of_not_ge h3
+        by_cases h4 : t ≤ Real.pi / 2 - params.phi
+        · rw [ite_eq_left h4]
+          exact cellPath_contains_piece4 htCell htPhys
+            (phase4_cell htCell h3' h4)
+        · rw [ite_eq_right h4]
+          have h4' : Real.pi / 2 - params.phi < t := lt_of_not_ge h4
+          exact cellPath_contains_piece5 htCell htPhys
+            (phase5_cell htCell h4')
+
+/-! ## Semantic soundness of the two product-cell expressions -/
+
+theorem guCell_contains {i j : Cell} {s t : ℝ}
+    (hsCell : s ∈ cellSet i) (htCell : t ∈ cellSet j)
+    (hsPhys : s ∈ physicalInterval) (htPhys : t ∈ physicalInterval) :
+    Contains (guCellInterval i j) (Gu s t) := by
+  have hxs := cellPath_contains hsCell hsPhys
+  have hxt := cellPath_contains htCell htPhys
+  have htr := trig_contains hsCell hsPhys
+  have hdx := RatInterval.contains_sub hxs.1 hxt.1
+  have hdy := RatInterval.contains_sub hxs.2 hxt.2
+  have hone := point_contains 1
+  have h := RatInterval.contains_add
+    (RatInterval.contains_add hone (RatInterval.contains_mul hdx htr.1))
+    (RatInterval.contains_mul hdy htr.2)
+  simpa [guCellInterval, Gu, dot, u, sub_eq_add_neg, ExactReplay.oneI, add_assoc] using h
+
+theorem gvCell_contains {i j : Cell} {s t : ℝ}
+    (hsCell : s ∈ cellSet i) (htCell : t ∈ cellSet j)
+    (hsPhys : s ∈ physicalInterval) (htPhys : t ∈ physicalInterval) :
+    Contains (gvCellInterval i j) (Gv s t) := by
+  have hxs := cellPath_contains hsCell hsPhys
+  have hxt := cellPath_contains htCell htPhys
+  have htr := trig_contains hsCell hsPhys
+  have hdx := RatInterval.contains_sub hxs.1 hxt.1
+  have hdy := RatInterval.contains_sub hxs.2 hxt.2
+  have hone := point_contains 1
+  have h := RatInterval.contains_add
+    (RatInterval.contains_add hone
+      (RatInterval.contains_mul hdx (RatInterval.contains_neg htr.2)))
+    (RatInterval.contains_mul hdy htr.1)
+  simpa [gvCellInterval, Gv, dot, v, sub_eq_add_neg, ExactReplay.oneI, add_assoc] using h
+
+end PartB
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Coverage by the 64 exact mesh cells
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartB
+
+/-- Every physical angle belongs to one of the 64 closed cells between the
+65 nodes `iπ/128`. -/
+theorem exists_cell_cover {t : ℝ} (ht : t ∈ physicalInterval) :
+    ∃ i : Cell, t ∈ cellSet i := by
+  let P : Nat → Prop := fun n => t ≤ nodeTime (n + 1)
+  have hnode64 : nodeTime 64 = Real.pi / 2 := by
+    norm_num [nodeTime, nodeCoeff]; ring
+  have h63 : P 63 := by
+    change t ≤ nodeTime 64
+    simpa [hnode64] using ht.2
+  have hex : ∃ n : Nat, P n := ⟨63, h63⟩
+  let n : Nat := Nat.find hex
+  have hn63 : n ≤ 63 := Nat.find_min' hex h63
+  have hn64 : n < 64 := by omega
+  refine ⟨⟨n, hn64⟩, ?_⟩
+  constructor
+  · by_cases hn0 : n = 0
+    · have hnode0 : nodeTime n = 0 := by
+        rw [hn0]
+        norm_num [nodeTime, nodeCoeff]
+      simpa [cellSet, hnode0] using ht.1
+    · have hpred : n - 1 < n := Nat.sub_one_lt hn0
+      have hnot : ¬ P (n - 1) := Nat.find_min hex hpred
+      have hlt : nodeTime ((n - 1) + 1) < t := lt_of_not_ge hnot
+      have hsucc : (n - 1) + 1 = n := Nat.sub_add_cancel (Nat.one_le_iff_ne_zero.2 hn0)
+      simpa [cellSet, hsucc] using hlt.le
+  · exact Nat.find_spec hex
+
+end PartB
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Exact continuum support margins
+
+The 64×64 cell certificate is now transported to every pair of physical
+angles.  This is the semantic conclusion needed from Part B.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartB
+
+/-- First global support inequality on the complete square. -/
+theorem Gu_gt_target :
+    ∀ s ∈ physicalInterval, ∀ t ∈ physicalInterval, target < Gu s t := by
+  intro s hs t ht
+  rcases exists_cell_cover hs with ⟨i, hsi⟩
+  rcases exists_cell_cover ht with ⟨j, htj⟩
+  have hsem := guCell_contains hsi htj hs ht
+  have hrat := (all_cell_lower i j).1
+  have hcast : (targetQ : ℝ) < ((guCellInterval i j).lo : ℝ) := by
+    exact_mod_cast hrat
+  simpa [target] using lt_of_lt_of_le hcast hsem.1
+
+/-- Second global support inequality on the complete square. -/
+theorem Gv_gt_target :
+    ∀ s ∈ physicalInterval, ∀ t ∈ physicalInterval, target < Gv s t := by
+  intro s hs t ht
+  rcases exists_cell_cover hs with ⟨i, hsi⟩
+  rcases exists_cell_cover ht with ⟨j, htj⟩
+  have hsem := gvCell_contains hsi htj hs ht
+  have hrat := (all_cell_lower i j).2
+  have hcast : (targetQ : ℝ) < ((gvCellInterval i j).lo : ℝ) := by
+    exact_mod_cast hrat
+  simpa [target] using lt_of_lt_of_le hcast hsem.1
+
+/-- Manuscript form of the two `0.171` inequalities. -/
+theorem global_half_plane_margins :
+    (∀ s ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+        (171 : ℝ) / 1000 < Gu s t) ∧
+    (∀ s ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+      ∀ t ∈ Set.Icc (0 : ℝ) (Real.pi / 2),
+        (171 : ℝ) / 1000 < Gv s t) := by
+  constructor
+  · simpa [physicalInterval, target, targetQ] using Gu_gt_target
+  · simpa [physicalInterval, target, targetQ] using Gv_gt_target
+
+end PartB
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle006.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle006.lean
new file mode 100644
index 0000000000..41591796f3
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle006.lean
@@ -0,0 +1,9406 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle002
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+public import Mathlib.Analysis.Calculus.Deriv.MeanValue
+public import Mathlib.Analysis.Calculus.Deriv.Pow
+public import Mathlib.Analysis.Calculus.Deriv.Prod
+public import Mathlib.Analysis.Convex.Deriv
+public import Mathlib.Analysis.Convex.PathConnected
+public import Mathlib.Topology.Connected.Basic
+public import Mathlib.Topology.Connected.Clopen
+public import Mathlib.Topology.Constructions.SumProd
+public import Mathlib.Topology.Order.Compact
+public import Mathlib.Topology.Order.DenselyOrdered
+public import Mathlib.Topology.Order.IntermediateValue
+public import Mathlib.Topology.Order.MonotoneContinuity
+public import Mathlib.Topology.Order.ProjIcc
+public import Mathlib.Topology.Separation.Hausdorff
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartC.Semantics.Batch001`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartC.Definitions`.
+* `KernelOnly.PartC.Parameters`.
+* `KernelOnly.PartC.ContactAlgebra`.
+* `KernelOnly.PartC.GlobalSupport`.
+* `KernelOnly.PartC.BaseGeometry`.
+* `KernelOnly.PartC.TopologySpec`.
+* `KernelOnly.PartC.GeometrySpec`.
+* `KernelOnly.PartC.Stage2.MeshFacts`.
+* `KernelOnly.PartC.Stage2.EnvelopeAlgebra`.
+* `KernelOnly.PartC.Stage2.NoHiddenCore`.
+* `KernelOnly.PartC.Stage2.SupportPhaseDerivatives`.
+* `KernelOnly.PartC.Stage3.SupportDirect`.
+* `KernelOnly.PartC.Stage3.FinalClosureDirect`.
+* `KernelOnly.PartC.Stage4.PathDifferential`.
+* `KernelOnly.PartC.Stage4.ExactGeometryFacts`.
+* `KernelOnly.PartC.Stage4.NoHiddenMatchFacts`.
+* `KernelOnly.PartC.Stage4.NoHiddenContinuityFacts`.
+* `KernelOnly.PartC.Stage4.NoHiddenSignFacts`.
+* `KernelOnly.PartC.Stage4.NoHiddenTurningFacts`.
+* `KernelOnly.PartC.Stage4.NoHiddenDifferentialFacts`.
+* `KernelOnly.PartC.Stage4.NoHiddenReflectionFacts`.
+* `KernelOnly.PartC.Stage4.NoHiddenDirect`.
+* `KernelOnly.PartC.Stage4.NicheEnvelopeSupport`.
+* `KernelOnly.PartC.Stage4.SofaConvexityFacts`.
+* `KernelOnly.PartC.Stage4.VerticalFillTopology`.
+* `KernelOnly.PartC.Stage4.WallGraphs`.
+* `KernelOnly.PartC.Stage4.NicheMembershipFacts`.
+* `KernelOnly.PartC.Stage4.NicheFrontierTopology`.
+* `KernelOnly.PartC.Stage4.NicheConnectedDirect`.
+* `KernelOnly.PartC.Stage4.SofaFiberTopology`.
+* `KernelOnly.PartC.Stage4.SofaTopologyDirect`.
+* `KernelOnly.PartC.Stage4.FinalTopologyClosure`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Part C concrete objects
+
+This file fixes the exact objects used throughout Part C.  The numerical root,
+regularity and global two-variable support margins are inherited from the
+closed Parts A and B.  No alternative parameter vector or geometric set is
+introduced here.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+
+/-- The certified twenty-two dimensional Gerver parameter vector. -/
+abbrev params : Romik.Params := PartB.params
+
+/-- Physical terminal angle. -/
+def T : ℝ := Real.pi / 2
+
+/-- Reflected switching angles. -/
+def eta : ℝ := T - params.theta
+/-- The reflected switching angle `π/2 - φ` for the certified parameters. -/
+def tau : ℝ := T - params.phi
+
+/-- The concrete cap, fixed sofa, reconstructed Romik set and frame. -/
+abbrev K : Set Point := Romik.K0 params
+/-- The fixed Gerver sofa at the certified Romik parameters. -/
+abbrev G : Set Point := Romik.sofa params
+/-- The reconstructed contact-curve set at the certified parameters. -/
+abbrev Sx : Set Point := Romik.reconstructedSet params
+/-- The rigid frame path determined by the certified Romik parameters. -/
+abbrev frame : ℝ → SE2 := Romik.frame params
+
+/-- Endpoint support point used for the direct nonemptiness proof. -/
+def anchor : Point := (1, 0)
+
+/-- Piecewise body-frame derivative coefficients.  This definition is local to
+Part C so that the geometric layer does not import the later Part D article
+claims. -/
+def alphaBetaAt (t : ℝ) : Point :=
+  if t ≤ params.phi then Romik.alphaBeta1 params t
+  else if t ≤ params.theta then Romik.alphaBeta2 params t
+  else if t ≤ eta then Romik.alphaBeta3 params t
+  else if t ≤ tau then Romik.alphaBeta4 params t
+  else Romik.alphaBeta5 params t
+
+/-- The horizontal body-frame velocity coefficient. -/
+def alpha (t : ℝ) : ℝ := (alphaBetaAt t).1
+/-- The vertical body-frame velocity coefficient. -/
+def beta (t : ℝ) : ℝ := (alphaBetaAt t).2
+
+/-- The four standard contact curves. -/
+def A (t : ℝ) : Point :=
+  let x := Romik.path params t
+  let a := alpha t
+  (x.1 + a * (v t).1 + (u t).1,
+   x.2 + a * (v t).2 + (u t).2)
+
+/-- The inner contact curve `x + α v`. -/
+def B (t : ℝ) : Point :=
+  let x := Romik.path params t
+  let a := alpha t
+  (x.1 + a * (v t).1,
+   x.2 + a * (v t).2)
+
+/-- The outer contact curve `x - β u + v`. -/
+def C (t : ℝ) : Point :=
+  let x := Romik.path params t
+  let b := beta t
+  (x.1 - b * (u t).1 + (v t).1,
+   x.2 - b * (u t).2 + (v t).2)
+
+/-- The inner contact curve `x - β u`. -/
+def D (t : ℝ) : Point :=
+  let x := Romik.path params t
+  let b := beta t
+  (x.1 - b * (u t).1,
+   x.2 - b * (u t).2)
+
+/-- Image of a parametrized curve over a time set. -/
+def curveImage (f : ℝ → Point) (I : Set ℝ) : Set Point := f '' I
+
+/-- A closed line segment, written without depending on a specialized convex
+geometry API. -/
+def lineSegment (a b : Point) : Set Point :=
+  {q | ∃ r ∈ Set.Icc (0 : ℝ) 1,
+    q = ((1 - r) * a.1 + r * b.1,
+         (1 - r) * a.2 + r * b.2)}
+
+/-- Set-valued form of the niche boundary described in the manuscript. -/
+def claimedNicheBoundary : Set Point :=
+  lineSegment (D 0) (B T) ∪
+  curveImage B (Set.Icc eta T) ∪
+  curveImage (Romik.path params) (Set.Icc params.phi tau) ∪
+  curveImage D (Set.Icc 0 params.theta)
+
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Frozen parameter and endpoint consequences for Part C
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+
+theorem params_mem : params ∈ Romik.box := PartB.params_mem
+
+theorem params_equations : Romik.Equations params := PartB.params_equations
+
+theorem switchOrder : Romik.SwitchOrder params := PartB.params_switchOrder
+
+theorem pathContinuous : Continuous (Romik.path params) := PartB.continuous_path
+
+theorem frameContinuous : SE2.ContinuousPath frame := PartB.continuous_frame
+
+theorem pathZero : Romik.path params 0 = (0, 0) := PartB.path_zero
+
+theorem pathEndYZero : (Romik.path params T).2 = 0 := by
+  simpa [T] using
+    (Romik.path_end_y_zero_of_mem_box_and_equations params_mem params_equations)
+
+theorem phi_bounds :
+    ((1958868239504182093160893749 : ℝ) /
+      50000000000000000000000000000) ≤ params.phi ∧
+    params.phi ≤
+      ((78354729580167283726435751 : ℝ) /
+        2000000000000000000000000000) :=
+  PartB.phi_bounds
+
+theorem theta_bounds :
+    ((34065075469136244723692787727 : ℝ) /
+      50000000000000000000000000000) ≤ params.theta ∧
+    params.theta ≤
+      ((34065075469136244723692787983 : ℝ) /
+        50000000000000000000000000000) :=
+  PartB.theta_bounds
+
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Exact contact-curve algebra
+
+These identities are independent of every interval estimate.  They record
+that `A` and `C` lie on the two outer supporting lines, while `B` and `D` lie
+on the corresponding inner-wall lines.  Later geometric work only has to prove
+that the relevant contact points belong to the cap and that no hidden crossing
+changes the boundary envelope.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+
+/-- `A(t)` lies on the first outer support line. -/
+theorem A_support_identity (t : ℝ) :
+    dot (A t) (u t) = dot (Romik.path params t) (u t) + 1 := by
+  simp only [A, dot, u, v]
+  nlinarith [Real.sin_sq_add_cos_sq t]
+
+/-- `C(t)` lies on the second outer support line. -/
+theorem C_support_identity (t : ℝ) :
+    dot (C t) (v t) = dot (Romik.path params t) (v t) + 1 := by
+  simp only [C, dot, u, v]
+  nlinarith [Real.sin_sq_add_cos_sq t]
+
+/-- `B(t)` lies on the first inner wall through the corner path. -/
+theorem B_inner_u_identity (t : ℝ) :
+    dot (B t - Romik.path params t) (u t) = 0 := by
+  change
+    ((B t).1 - (Romik.path params t).1) * Real.cos t +
+      ((B t).2 - (Romik.path params t).2) * Real.sin t = 0
+  simp [B, v]; ring
+
+/-- `D(t)` lies on the second inner wall through the corner path. -/
+theorem D_inner_v_identity (t : ℝ) :
+    dot (D t - Romik.path params t) (v t) = 0 := by
+  change
+    ((D t).1 - (Romik.path params t).1) * (-Real.sin t) +
+      ((D t).2 - (Romik.path params t).2) * Real.cos t = 0
+  simp [D, u]; ring
+
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Consequences of the certified Part B continuum margins
+
+The two strict inequalities already certified on the complete square imply
+that every point of the Gerver rotation path satisfies every outer supporting
+half-plane inequality.  This is one of the main bridges from Part B into the
+global geometry of Part C.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+
+theorem Gu_positive {s t : ℝ}
+    (hs : s ∈ Set.Icc (0 : ℝ) T)
+    (ht : t ∈ Set.Icc (0 : ℝ) T) :
+    0 < PartB.Gu s t := by
+  have h := PartB.global_half_plane_margins.1 s (by simpa [T] using hs)
+    t (by simpa [T] using ht)
+  linarith
+
+theorem Gv_positive {s t : ℝ}
+    (hs : s ∈ Set.Icc (0 : ℝ) T)
+    (ht : t ∈ Set.Icc (0 : ℝ) T) :
+    0 < PartB.Gv s t := by
+  have h := PartB.global_half_plane_margins.2 s (by simpa [T] using hs)
+    t (by simpa [T] using ht)
+  linarith
+
+/-- Every path point lies in every first outer supporting half-plane. -/
+theorem path_mem_supportHalfU {s t : ℝ}
+    (hs : s ∈ Set.Icc (0 : ℝ) T)
+    (ht : t ∈ Set.Icc (0 : ℝ) T) :
+    Romik.path params t ∈ supportHalfU params s := by
+  have h := Gu_positive hs ht
+  change dot (Romik.path params t) (u s) ≤
+    dot (Romik.path params s) (u s) + 1
+  dsimp [PartB.Gu, dot] at h ⊢
+  linarith
+
+/-- Every path point lies in every second outer supporting half-plane. -/
+theorem path_mem_supportHalfV {s t : ℝ}
+    (hs : s ∈ Set.Icc (0 : ℝ) T)
+    (ht : t ∈ Set.Icc (0 : ℝ) T) :
+    Romik.path params t ∈ supportHalfV params s := by
+  have h := Gv_positive hs ht
+  change dot (Romik.path params t) (v s) ≤
+    dot (Romik.path params s) (v s) + 1
+  dsimp [PartB.Gv, dot] at h ⊢
+  linarith
+
+theorem path_mem_outer_supports {s t : ℝ}
+    (hs : s ∈ Set.Icc (0 : ℝ) T)
+    (ht : t ∈ Set.Icc (0 : ℝ) T) :
+    Romik.path params t ∈
+      supportHalfU params s ∩ supportHalfV params s :=
+  ⟨path_mem_supportHalfU hs ht, path_mem_supportHalfV hs ht⟩
+
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C geometric consequences already closed by the existing source
+
+Closedness, supporting-hallway containment, endpoint arms, motion continuity
+and the set-theoretic identification with Romik's reconstruction require no new
+numerical replay.  They are collected here for the concrete certified
+parameter vector.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+
+theorem closed_G : IsClosed G := Romik.isClosed_sofa params
+
+theorem G_subset_hallway :
+    ∀ s ∈ Set.Icc (0 : ℝ) 1, G ⊆ Romik.hallwayAt params s :=
+  Romik.sofa_subset_hallwayAt params pathZero pathEndYZero
+
+theorem initialArm : ((frame 0).inv.act '' G) ⊆ horizontalArm :=
+  Romik.initial_arm_of_path_zero params pathZero
+
+theorem finalArm : ((frame 1).inv.act '' G) ⊆ verticalArm :=
+  Romik.final_arm_of_path_end_y_zero params pathEndYZero
+
+theorem G_eq_Sx : G = Sx :=
+  Romik.sofa_eq_reconstructedSet params pathZero pathEndYZero
+
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Exact topology target for the concrete Gerver set
+
+This file states, but does not postulate, the two genuinely remaining
+set-theoretic obligations.  A later concrete proof must construct a value of
+`TopologyCertificate`; until then Part C cannot close.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+
+/-- Direct topology certificate with the manuscript's explicit endpoint
+support witness, rather than an anonymous existential. -/
+structure TopologyCertificate : Prop where
+  anchor_mem : anchor ∈ G
+  connected : IsConnected G
+
+namespace TopologyCertificate
+
+theorem nonempty (cert : TopologyCertificate) : G.Nonempty :=
+  ⟨anchor, cert.anchor_mem⟩
+
+end TopologyCertificate
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Global support and niche-topology certificate interfaces
+
+The statements here encode the exact Part C geometry that is not allowed to be
+hidden inside an arbitrary `connected` field: contact support, the two
+no-hidden-crossing inequalities, and the claimed boundary of the niche.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+
+/-- First no-hidden-crossing inequality from the manuscript. -/
+def NoHiddenCrossingU : Prop :=
+  ∀ r ∈ Set.Icc params.phi T,
+    ∀ t ∈ Set.Icc r T,
+      0 ≤ dot (Romik.path params r - Romik.path params t) (u t)
+
+/-- Reflected no-hidden-crossing inequality. -/
+def NoHiddenCrossingV : Prop :=
+  ∀ t ∈ Set.Icc 0 tau,
+    ∀ r ∈ Set.Icc t tau,
+      0 ≤ dot (Romik.path params r - Romik.path params t) (v t)
+
+/-- Proof-carrying niche topology.  The boundary equality is set-valued; the
+orientation and eighteen-piece count belong to the later Part D boundary
+certificate. -/
+structure NicheTopologyCertificate : Prop where
+  noHiddenU : NoHiddenCrossingU
+  noHiddenV : NoHiddenCrossingV
+  boundary_eq : frontier (Romik.niche params) = claimedNicheBoundary
+  niche_connected : IsConnected (Romik.niche params)
+
+/-- Complete genuinely new geometry required in Part C. -/
+structure GlobalGeometryCertificate : Prop where
+  supportA : ∀ t ∈ Set.Icc (0 : ℝ) T, A t ∈ K
+  supportC : ∀ t ∈ Set.Icc (0 : ℝ) T, C t ∈ K
+  nicheTopology : NicheTopologyCertificate
+  topology : TopologyCertificate
+
+namespace GlobalGeometryCertificate
+
+theorem nonempty (cert : GlobalGeometryCertificate) : G.Nonempty :=
+  cert.topology.nonempty
+
+theorem connected (cert : GlobalGeometryCertificate) : IsConnected G :=
+  cert.topology.connected
+
+end GlobalGeometryCertificate
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 2 public mesh and branch facts
+
+This module exposes the switch-cell classification already used internally in
+Part B.  It is deliberately proved again from the frozen angle boxes so that
+Part C contact and no-hidden-crossing certificates can reuse the exact 64-cell
+mesh without depending on private declarations.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage2
+
+open RatInterval
+
+theorem node1_lt_phi : PartB.nodeTime 1 < params.phi := by
+  have hpi : Real.pi ≤ (ExactReplay.piI.hi : ℝ) :=
+    ExactReplay.piI_contains_pi.2
+  have hp := phi_bounds.1
+  unfold PartB.nodeTime PartB.nodeCoeff
+  norm_num [ExactReplay.piI, ExactReplay.q] at hpi hp ⊢
+  nlinarith
+
+theorem phi_lt_node2 : params.phi < PartB.nodeTime 2 := by
+  have hpi : (ExactReplay.piI.lo : ℝ) ≤ Real.pi :=
+    ExactReplay.piI_contains_pi.1
+  have hp := phi_bounds.2
+  unfold PartB.nodeTime PartB.nodeCoeff
+  norm_num [ExactReplay.piI, ExactReplay.q] at hpi hp ⊢
+  nlinarith
+
+theorem node27_lt_theta : PartB.nodeTime 27 < params.theta := by
+  have hpi : Real.pi ≤ (ExactReplay.piI.hi : ℝ) :=
+    ExactReplay.piI_contains_pi.2
+  have hp := theta_bounds.1
+  unfold PartB.nodeTime PartB.nodeCoeff
+  norm_num [ExactReplay.piI, ExactReplay.q] at hpi hp ⊢
+  nlinarith
+
+theorem theta_lt_node28 : params.theta < PartB.nodeTime 28 := by
+  have hpi : (ExactReplay.piI.lo : ℝ) ≤ Real.pi :=
+    ExactReplay.piI_contains_pi.1
+  have hp := theta_bounds.2
+  unfold PartB.nodeTime PartB.nodeCoeff
+  norm_num [ExactReplay.piI, ExactReplay.q] at hpi hp ⊢
+  nlinarith
+
+theorem node36_lt_eta : PartB.nodeTime 36 < eta := by
+  change PartB.nodeTime 36 < Real.pi / 2 - params.theta
+  have h := theta_lt_node28
+  unfold PartB.nodeTime PartB.nodeCoeff at h ⊢
+  norm_num at h ⊢
+  linarith
+
+theorem eta_lt_node37 : eta < PartB.nodeTime 37 := by
+  change Real.pi / 2 - params.theta < PartB.nodeTime 37
+  have h := node27_lt_theta
+  unfold PartB.nodeTime PartB.nodeCoeff at h ⊢
+  norm_num at h ⊢
+  linarith
+
+theorem node62_lt_tau : PartB.nodeTime 62 < tau := by
+  change PartB.nodeTime 62 < Real.pi / 2 - params.phi
+  have h := phi_lt_node2
+  unfold PartB.nodeTime PartB.nodeCoeff at h ⊢
+  norm_num at h ⊢
+  linarith
+
+theorem tau_lt_node63 : tau < PartB.nodeTime 63 := by
+  change Real.pi / 2 - params.phi < PartB.nodeTime 63
+  have h := node1_lt_phi
+  unfold PartB.nodeTime PartB.nodeCoeff at h ⊢
+  norm_num at h ⊢
+  linarith
+
+theorem phase1_cell {i : PartB.Cell} {t : ℝ}
+    (ht : t ∈ PartB.cellSet i) (hphase : t ≤ params.phi) : i.1 ≤ 1 := by
+  by_contra hnot
+  have hi : 2 ≤ i.1 := by omega
+  have h2i : PartB.nodeTime 2 ≤ PartB.nodeTime i.1 :=
+    PartB.nodeTime_mono hi
+  linarith [phi_lt_node2, ht.1]
+
+theorem phase2_cell {i : PartB.Cell} {t : ℝ}
+    (ht : t ∈ PartB.cellSet i)
+    (hlo : params.phi < t) (hhi : t ≤ params.theta) :
+    1 ≤ i.1 ∧ i.1 ≤ 27 := by
+  constructor
+  · by_contra hnot
+    have hi : i.1 + 1 ≤ 1 := by omega
+    have hend : PartB.nodeTime (i.1 + 1) ≤ PartB.nodeTime 1 :=
+      PartB.nodeTime_mono hi
+    linarith [node1_lt_phi, ht.2]
+  · by_contra hnot
+    have hi : 28 ≤ i.1 := by omega
+    have h28i : PartB.nodeTime 28 ≤ PartB.nodeTime i.1 :=
+      PartB.nodeTime_mono hi
+    linarith [theta_lt_node28, ht.1]
+
+theorem phase3_cell {i : PartB.Cell} {t : ℝ}
+    (ht : t ∈ PartB.cellSet i)
+    (hlo : params.theta < t) (hhi : t ≤ eta) :
+    27 ≤ i.1 ∧ i.1 ≤ 36 := by
+  constructor
+  · by_contra hnot
+    have hi : i.1 + 1 ≤ 27 := by omega
+    have hend : PartB.nodeTime (i.1 + 1) ≤ PartB.nodeTime 27 :=
+      PartB.nodeTime_mono hi
+    linarith [node27_lt_theta, ht.2]
+  · by_contra hnot
+    have hi : 37 ≤ i.1 := by omega
+    have h37i : PartB.nodeTime 37 ≤ PartB.nodeTime i.1 :=
+      PartB.nodeTime_mono hi
+    linarith [eta_lt_node37, ht.1]
+
+theorem phase4_cell {i : PartB.Cell} {t : ℝ}
+    (ht : t ∈ PartB.cellSet i)
+    (hlo : eta < t) (hhi : t ≤ tau) :
+    36 ≤ i.1 ∧ i.1 ≤ 62 := by
+  constructor
+  · by_contra hnot
+    have hi : i.1 + 1 ≤ 36 := by omega
+    have hend : PartB.nodeTime (i.1 + 1) ≤ PartB.nodeTime 36 :=
+      PartB.nodeTime_mono hi
+    linarith [node36_lt_eta, ht.2]
+  · by_contra hnot
+    have hi : 63 ≤ i.1 := by omega
+    have h63i : PartB.nodeTime 63 ≤ PartB.nodeTime i.1 :=
+      PartB.nodeTime_mono hi
+    linarith [tau_lt_node63, ht.1]
+
+theorem phase5_cell {i : PartB.Cell} {t : ℝ}
+    (ht : t ∈ PartB.cellSet i) (hlo : tau < t) : 62 ≤ i.1 := by
+  by_contra hnot
+  have hi : i.1 + 1 ≤ 62 := by omega
+  have hend : PartB.nodeTime (i.1 + 1) ≤ PartB.nodeTime 62 :=
+    PartB.nodeTime_mono hi
+  linarith [node62_lt_tau, ht.2]
+
+theorem phi_nonneg : 0 ≤ params.phi :=
+  (Romik.phi_pos_of_mem_box params_mem).le
+
+theorem tau_le_T : tau ≤ T := by
+  unfold tau
+  linarith [phi_nonneg]
+
+theorem zero_le_theta : 0 ≤ params.theta := by
+  linarith [phi_nonneg, switchOrder.phi_le_theta]
+
+theorem eta_le_tau : eta ≤ tau := by
+  simpa [eta, tau, T] using switchOrder.eta_le_tau
+
+theorem theta_le_eta : params.theta ≤ eta := by
+  simpa [eta, T] using switchOrder.theta_le_eta
+
+end Stage2
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 2 redesign: exact envelope algebra
+
+The first 64x64 contact-box attempt was intentionally fail-closed but too
+coarse at contact/equality cells: interval dependency destroys exact
+cancellation.  This module switches to the analytic envelope coefficients.
+
+For the outer `u`-contact curve `A`, the phasewise velocity is a nonnegative
+scalar multiple of `v`; for the outer `v`-contact curve `C`, the velocity is a
+nonpositive scalar multiple of `u`.  The scalar coefficients below are the
+five exact algebraic pieces.  No numerical root is recomputed here.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage2
+
+/-- Phasewise scalar multiplying `v(t)` in the derivative of `A`. -/
+def rhoA (t : ℝ) : ℝ :=
+  if t ≤ params.phi then 0
+  else if t ≤ params.theta then
+    -(1 / 4 : ℝ) * t * t + params.b1 * t + params.b2 + 1 / 2
+  else if t ≤ eta then
+    1 + params.c1 - t
+  else if t ≤ tau then
+    params.d1 - t / 2
+  else
+    1 / 2
+
+/-- Phasewise nonnegative scalar for `C'(t) = -rhoC(t) * u(t)`. -/
+def rhoC (t : ℝ) : ℝ :=
+  if t ≤ params.phi then 1 / 2
+  else if t ≤ params.theta then
+    t / 2 - params.b1
+  else if t ≤ eta then
+    1 + params.c2 + t
+  else if t ≤ tau then
+    -(1 / 4 : ℝ) * t * t + params.d1 * t + params.d2 + 1 / 2
+  else
+    0
+
+private theorem b1_lower : (-53 / 100 : ℝ) ≤ params.b1 := by
+  have hp := params_mem
+  dsimp [Romik.box, qR] at hp
+  have h :
+      ((-52762459802678462416060380937 : ℝ) /
+        100000000000000000000000000000) ≤ params.b1 := by
+    aesop
+  norm_num at h ⊢
+  linarith
+
+private theorem b1_upper : params.b1 ≤ (-1 / 2 : ℝ) := by
+  have hp := params_mem
+  dsimp [Romik.box, qR] at hp
+  have h : params.b1 ≤
+      ((-52762459802678462416040380937 : ℝ) /
+        100000000000000000000000000000) := by
+    aesop
+  norm_num at h ⊢
+  linarith
+
+private theorem b2_lower : (9 / 10 : ℝ) ≤ params.b2 := by
+  have hp := params_mem
+  dsimp [Romik.box, qR] at hp
+  have h :
+      ((92025838516063762289360579501 : ℝ) /
+        100000000000000000000000000000) ≤ params.b2 := by
+    aesop
+  norm_num at h ⊢
+  linarith
+
+private theorem c1_lower : (3 / 5 : ℝ) ≤ params.c1 := by
+  have hp := params_mem
+  dsimp [Romik.box, qR] at hp
+  have h :
+      ((313022761424232933776114655193 : ℝ) /
+        500000000000000000000000000000) ≤ params.c1 := by
+    aesop
+  norm_num at h ⊢
+  linarith
+
+private theorem c2_lower : (-1 : ℝ) ≤ params.c2 := by
+  have hp := params_mem
+  dsimp [Romik.box, qR] at hp
+  have h :
+      ((-151160128631428920268654781 : ℝ) /
+        160000000000000000000000000) ≤ params.c2 := by
+    aesop
+  norm_num at h ⊢
+  linarith
+
+private theorem d1_lower : (13 / 10 : ℝ) ≤ params.d1 := by
+  have hp := params_mem
+  dsimp [Romik.box, qR] at hp
+  have h :
+      ((1641278451780291167220080819 : ℝ) /
+        1250000000000000000000000000) ≤ params.d1 := by
+    aesop
+  norm_num at h ⊢
+  linarith
+
+private theorem d2_lower : (-53 / 100 : ℝ) ≤ params.d2 := by
+  have hp := params_mem
+  dsimp [Romik.box, qR] at hp
+  have h :
+      ((-105076534082910887440587258861 : ℝ) /
+        200000000000000000000000000000) ≤ params.d2 := by
+    aesop
+  norm_num at h ⊢
+  linarith
+
+private theorem theta_lower_crude : (2 / 3 : ℝ) ≤ params.theta := by
+  have h := theta_bounds.1
+  norm_num at h ⊢
+  linarith
+
+private theorem theta_upper_crude : params.theta ≤ (7 / 10 : ℝ) := by
+  have h := theta_bounds.2
+  norm_num at h ⊢
+  linarith
+
+private theorem T_lt_two : T < 2 := by
+  have hpi := ExactReplay.piI_contains_pi
+  have hhi : Real.pi ≤ (ExactReplay.piI.hi : ℝ) := hpi.2
+  have hfour : (ExactReplay.piI.hi : ℝ) < 4 := by
+    norm_num [ExactReplay.piI, ExactReplay.q]
+  dsimp [T]
+  linarith
+
+private theorem three_halves_lt_T : (3 / 2 : ℝ) < T := by
+  have hpi := ExactReplay.piI_contains_pi
+  have hlo : (ExactReplay.piI.lo : ℝ) ≤ Real.pi := hpi.1
+  have hthree : (3 : ℝ) < (ExactReplay.piI.lo : ℝ) := by
+    norm_num [ExactReplay.piI, ExactReplay.q]
+  dsimp [T]
+  linarith
+
+private theorem eta_lt_four_thirds : eta < (4 / 3 : ℝ) := by
+  dsimp [eta]
+  nlinarith [T_lt_two, theta_lower_crude]
+
+private theorem four_fifths_lt_eta : (4 / 5 : ℝ) < eta := by
+  dsimp [eta]
+  nlinarith [three_halves_lt_T, theta_upper_crude]
+
+private theorem rhoA_phase2_nonneg
+    {t : ℝ} (ht0 : 0 ≤ t) (htθ : t ≤ params.theta) :
+    0 ≤ -(1 / 4 : ℝ) * t * t + params.b1 * t + params.b2 + 1 / 2 := by
+  have ht07 : t ≤ (7 / 10 : ℝ) := le_trans htθ theta_upper_crude
+  have ht1 : t ≤ 1 := by linarith
+  have hbt : (-53 / 100 : ℝ) * t ≤ params.b1 * t :=
+    mul_le_mul_of_nonneg_right b1_lower ht0
+  have hquad : 0 ≤ t * (1 - t) := mul_nonneg ht0 (by linarith)
+  nlinarith [b2_lower]
+
+private theorem rhoA_phase3_nonneg
+    {t : ℝ} (_ht0 : 0 ≤ t) (htη : t ≤ eta) :
+    0 ≤ 1 + params.c1 - t := by
+  have ht : t < (4 / 3 : ℝ) := lt_of_le_of_lt htη eta_lt_four_thirds
+  nlinarith [c1_lower]
+
+private theorem rhoA_phase4_nonneg
+    {t : ℝ} (_ht0 : 0 ≤ t) (htτ : t ≤ tau) :
+    0 ≤ params.d1 - t / 2 := by
+  have htT : t ≤ T := le_trans htτ tau_le_T
+  have ht2 : t < 2 := lt_of_le_of_lt htT T_lt_two
+  nlinarith [d1_lower]
+
+private theorem rhoC_phase2_nonneg
+    {t : ℝ} (ht0 : 0 ≤ t) :
+    0 ≤ t / 2 - params.b1 := by
+  nlinarith [b1_upper]
+
+private theorem rhoC_phase3_nonneg
+    {t : ℝ} (ht0 : 0 ≤ t) :
+    0 ≤ 1 + params.c2 + t := by
+  nlinarith [c2_lower]
+
+private theorem rhoC_phase4_nonneg
+    {t : ℝ} (hηt : eta ≤ t) (htτ : t ≤ tau) :
+    0 ≤ -(1 / 4 : ℝ) * t * t + params.d1 * t + params.d2 + 1 / 2 := by
+  have ht0 : 0 ≤ t := by
+    have : (0 : ℝ) < eta := lt_trans (by norm_num) four_fifths_lt_eta
+    linarith
+  have htT : t ≤ T := le_trans htτ tau_le_T
+  have ht2 : t < 2 := lt_of_le_of_lt htT T_lt_two
+  have hmul : (13 / 10 : ℝ) * t ≤ params.d1 * t :=
+    mul_le_mul_of_nonneg_right d1_lower ht0
+  have hquad : 0 ≤ t * (2 - t) := mul_nonneg ht0 (by linarith)
+  nlinarith [d2_lower, four_fifths_lt_eta]
+
+/-- The `A` envelope coefficient is nonnegative on the full physical range. -/
+theorem rhoA_nonneg {t : ℝ} (ht : t ∈ Set.Icc (0 : ℝ) T) :
+    0 ≤ rhoA t := by
+  unfold rhoA
+  split_ifs with hφ hθ hη hτ
+  · norm_num
+  · exact rhoA_phase2_nonneg ht.1 hθ
+  · exact rhoA_phase3_nonneg ht.1 hη
+  · exact rhoA_phase4_nonneg ht.1 hτ
+  · norm_num
+
+/-- The reflected `C` envelope coefficient is nonnegative on the full physical range. -/
+theorem rhoC_nonneg {t : ℝ} (ht : t ∈ Set.Icc (0 : ℝ) T) :
+    0 ≤ rhoC t := by
+  unfold rhoC
+  split_ifs with hφ hθ hη hτ
+  · norm_num
+  · exact rhoC_phase2_nonneg ht.1
+  · exact rhoC_phase3_nonneg ht.1
+  · have hηt : eta ≤ t := le_of_lt (lt_of_not_ge hη)
+    exact rhoC_phase4_nonneg hηt hτ
+  · norm_num
+
+/-- The first outer contact starts at the exact fan endpoint `(1,0)`. -/
+theorem A_zero_eq_anchor : A 0 = anchor := by
+  have hφ : (0 : ℝ) ≤ params.phi := le_of_lt (Romik.phi_pos_of_mem_box params_mem)
+  have ha2 := Romik.a2_eq_neg_quarter_of_equations params_equations
+  have halpha0 : alpha 0 = 0 := by
+    unfold alpha alphaBetaAt
+    simp [hφ, Romik.alphaBeta1, ha2]; norm_num
+  simp [A, anchor, pathZero, halpha0, u, v]
+
+/-- The second outer contact ends exactly on the fan boundary. -/
+theorem C_T_snd_zero : (C T).2 = 0 := by
+  have hφpos : 0 < params.phi := Romik.phi_pos_of_mem_box params_mem
+  have hτT : tau < T := by
+    dsimp [tau]
+    linarith
+  have hητ : eta ≤ tau := switchOrder.eta_le_tau
+  have hθη : params.theta ≤ eta := switchOrder.theta_le_eta
+  have hφθ : params.phi ≤ params.theta := switchOrder.phi_le_theta
+  have hφT : params.phi < T := lt_of_le_of_lt (le_trans hφθ (le_trans hθη hητ)) hτT
+  have hθT : params.theta < T := lt_of_le_of_lt (le_trans hθη hητ) hτT
+  have hηT : eta < T := lt_of_le_of_lt hητ hτT
+  have he2 := Romik.e2_eq_quarter_of_equations params_equations
+  have hABT : alphaBetaAt T = Romik.alphaBeta5 params T := by
+    unfold alphaBetaAt
+    rw [ite_eq_right (not_le.mpr hφT)]
+    rw [ite_eq_right (not_le.mpr hθT)]
+    rw [ite_eq_right (not_le.mpr hηT)]
+    rw [ite_eq_right (not_le.mpr hτT)]
+  have hbetaT : beta T = 0 := by
+    rw [beta, hABT]
+    simp [Romik.alphaBeta5, T, he2]; norm_num
+  change (Romik.path params T).2 - beta T * (u T).2 + (v T).2 = 0
+  rw [pathEndYZero, hbetaT]
+  simp [u, v, T]
+
+end Stage2
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Exact cell reduction of the two no-hidden-crossing inequalities
+
+Off-diagonal mesh cells are reduced to executable upper bounds for the already
+sound Part B intervals `Gu` and `Gv`.  The only analytic remainder after all
+128 row certificates pass is the ordered triangle inside each single mesh
+cell, named `SameCellU` and `SameCellV` below.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage2
+
+/-- Manuscript `U(r,t)` in the orientation used by `NoHiddenCrossingU`. -/
+def UValue (r t : ℝ) : ℝ :=
+  dot (Romik.path params r - Romik.path params t) (u t)
+
+/-- Reflected manuscript `V(t,r)`. -/
+def VValue (t r : ℝ) : ℝ :=
+  dot (Romik.path params r - Romik.path params t) (v t)
+
+end Stage2
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# C10: source-clean phase derivative identities — residual batch fix
+
+This revision keeps the C09 no-`convert` architecture and fixes the complete residual class from the
+C09 clean build.
+It uses eta-expanded derivative combinators (`fun_add`, `fun_sub`, `fun_mul`,
+`fun_neg`, `fun_smul`) so the function carried by `HasDerivAt` is in the desired
+shape from the start.  The proof remains organized in body coordinates and the
+accompanying runner forces a fresh build of this module.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage2
+
+/-- Phase-local outer `A` contact, written in vector form. -/
+def phaseA1 (t : ℝ) : Point :=
+  Romik.path1 params t + (Romik.alphaBeta1 params t).1 • v t + u t
+
+/-- The phase 2 formula for the outer A contact curve. -/
+def phaseA2 (t : ℝ) : Point :=
+  Romik.path2 params t + (Romik.alphaBeta2 params t).1 • v t + u t
+
+/-- The phase 3 formula for the outer A contact curve. -/
+def phaseA3 (t : ℝ) : Point :=
+  Romik.path3 params t + (Romik.alphaBeta3 params t).1 • v t + u t
+
+/-- The phase 4 formula for the outer A contact curve. -/
+def phaseA4 (t : ℝ) : Point :=
+  Romik.path4 params t + (Romik.alphaBeta4 params t).1 • v t + u t
+
+/-- The phase 5 formula for the outer A contact curve. -/
+def phaseA5 (t : ℝ) : Point :=
+  Romik.path5 params t + (Romik.alphaBeta5 params t).1 • v t + u t
+
+/-- Phase-local outer `C` contact, written in vector form. -/
+def phaseC1 (t : ℝ) : Point :=
+  Romik.path1 params t - (Romik.alphaBeta1 params t).2 • u t + v t
+
+/-- The phase 2 formula for the outer C contact curve. -/
+def phaseC2 (t : ℝ) : Point :=
+  Romik.path2 params t - (Romik.alphaBeta2 params t).2 • u t + v t
+
+/-- The phase 3 formula for the outer C contact curve. -/
+def phaseC3 (t : ℝ) : Point :=
+  Romik.path3 params t - (Romik.alphaBeta3 params t).2 • u t + v t
+
+/-- The phase 4 formula for the outer C contact curve. -/
+def phaseC4 (t : ℝ) : Point :=
+  Romik.path4 params t - (Romik.alphaBeta4 params t).2 • u t + v t
+
+/-- The phase 5 formula for the outer C contact curve. -/
+def phaseC5 (t : ℝ) : Point :=
+  Romik.path5 params t - (Romik.alphaBeta5 params t).2 • u t + v t
+
+/-- The vertical frame vector scaled by the contact coefficient `r`. -/
+def vecA (r t : ℝ) : Point :=
+  (-r * Real.sin t, r * Real.cos t)
+
+/-- The negative horizontal frame vector scaled by the contact coefficient `r`. -/
+def vecC (r t : ℝ) : Point :=
+  (-r * Real.cos t, -r * Real.sin t)
+
+/-- Derivative of the scalar square in the additive form used by the phase formulas. -/
+theorem square_hasDerivAt (t : ℝ) :
+    HasDerivAt (fun s : ℝ => s * s) (t + t) t := by
+  have h := (hasDerivAt_id t).fun_mul (hasDerivAt_id t)
+  simpa only [id_eq, one_mul, mul_one] using h
+
+/-- The horizontal frame vector has derivative equal to the vertical frame vector. -/
+theorem u_hasDerivAt (t : ℝ) : HasDerivAt u (v t) t := by
+  change HasDerivAt (fun s : ℝ => (Real.cos s, Real.sin s))
+    (-Real.sin t, Real.cos t) t
+  exact (Real.hasDerivAt_cos t).prodMk (Real.hasDerivAt_sin t)
+
+/-- The vertical frame vector has derivative equal to the negative horizontal vector. -/
+theorem v_hasDerivAt (t : ℝ) : HasDerivAt v (-u t) t := by
+  change HasDerivAt (fun s : ℝ => (-Real.sin s, Real.cos s))
+    (-Real.cos t, -Real.sin t) t
+  exact (Real.hasDerivAt_sin t).fun_neg.prodMk (Real.hasDerivAt_cos t)
+
+/-- Derivative of `addK (rot t (z1 t,z2 t))` in body coordinates. -/
+theorem rotAddK_hasDerivAt
+    {z1 z2 : ℝ → ℝ} {z1' z2' k1 k2 : ℝ} (t : ℝ)
+    (hz1 : HasDerivAt z1 z1' t) (hz2 : HasDerivAt z2 z2' t) :
+    HasDerivAt
+      (fun s => Romik.addK (Romik.rot s (z1 s, z2 s)) k1 k2)
+      (Romik.rot t (z1' - z2 t, z2' + z1 t)) t := by
+  apply HasDerivAt.prodMk
+  · change HasDerivAt
+      (fun s => Real.cos s * z1 s - Real.sin s * z2 s + k1)
+      (Real.cos t * (z1' - z2 t) - Real.sin t * (z2' + z1 t)) t
+    have h1 := (Real.hasDerivAt_cos t).fun_mul hz1
+    have h2 := (Real.hasDerivAt_sin t).fun_mul hz2
+    have h := (h1.fun_sub h2).fun_add (hasDerivAt_const t k1)
+    exact h.congr_deriv (by ring)
+  · change HasDerivAt
+      (fun s => Real.sin s * z1 s + Real.cos s * z2 s + k2)
+      (Real.sin t * (z1' - z2 t) + Real.cos t * (z2' + z1 t)) t
+    have h1 := (Real.hasDerivAt_sin t).fun_mul hz1
+    have h2 := (Real.hasDerivAt_cos t).fun_mul hz2
+    have h := (h1.fun_add h2).fun_add (hasDerivAt_const t k2)
+    exact h.congr_deriv (by ring)
+
+/-- The phase 1 path has the specified velocity in the rotating frame. -/
+theorem path1_hasDerivAt (t : ℝ) :
+    HasDerivAt (Romik.path1 params)
+      (Romik.rot t (Romik.alphaBeta1 params t)) t := by
+  let z1 : ℝ → ℝ := fun s =>
+    params.a1 * Real.cos s + params.a2 * Real.sin s - 1
+  let z2 : ℝ → ℝ := fun s =>
+    -params.a2 * Real.cos s + params.a1 * Real.sin s - 1 / 2
+  have hz1 : HasDerivAt z1
+      (-params.a1 * Real.sin t + params.a2 * Real.cos t) t := by
+    have h1 := HasDerivAt.const_mul params.a1 (Real.hasDerivAt_cos t)
+    have h2 := HasDerivAt.const_mul params.a2 (Real.hasDerivAt_sin t)
+    have h := (h1.fun_add h2).fun_sub (hasDerivAt_const t (1 : ℝ))
+    dsimp [z1]
+    exact h.congr_deriv (by ring)
+  have hz2 : HasDerivAt z2
+      (params.a2 * Real.sin t + params.a1 * Real.cos t) t := by
+    have h1 := HasDerivAt.const_mul (-params.a2) (Real.hasDerivAt_cos t)
+    have h2 := HasDerivAt.const_mul params.a1 (Real.hasDerivAt_sin t)
+    have h := (h1.fun_add h2).fun_sub (hasDerivAt_const t (1 / 2 : ℝ))
+    dsimp [z2]
+    exact h.congr_deriv (by ring)
+  change HasDerivAt
+    (fun s => Romik.addK (Romik.rot s (z1 s, z2 s)) params.k11 params.k12)
+    (Romik.rot t (Romik.alphaBeta1 params t)) t
+  have h := rotAddK_hasDerivAt (k1 := params.k11) (k2 := params.k12) t hz1 hz2
+  refine h.congr_deriv ?_
+  apply congrArg (Romik.rot t)
+  ext <;> dsimp [z1, z2, Romik.alphaBeta1] <;> ring
+
+/-- The phase 2 path has the specified velocity in the rotating frame. -/
+theorem path2_hasDerivAt (t : ℝ) :
+    HasDerivAt (Romik.path2 params)
+      (Romik.rot t (Romik.alphaBeta2 params t)) t := by
+  let z1 : ℝ → ℝ := fun s =>
+    -(1 / 4 : ℝ) * s * s + params.b1 * s + params.b2
+  let z2 : ℝ → ℝ := fun s =>
+    (1 / 2 : ℝ) * s - params.b1 - 1
+  have hz1 : HasDerivAt z1 (-(1 / 2 : ℝ) * t + params.b1) t := by
+    have h1 := HasDerivAt.const_mul (-(1 / 4 : ℝ)) (square_hasDerivAt t)
+    have h2 := HasDerivAt.const_mul params.b1 (hasDerivAt_id t)
+    have hraw := (h1.fun_add h2).fun_add (hasDerivAt_const t params.b2)
+    have heq : (-(1 / 4 : ℝ)) * (t + t) + params.b1 * 1 + 0 =
+        -(1 / 2 : ℝ) * t + params.b1 := by ring
+    rw [← heq]
+    simpa only [z1, id_eq, mul_assoc] using hraw
+  have hz2 : HasDerivAt z2 (1 / 2 : ℝ) t := by
+    have h1 := HasDerivAt.const_mul (1 / 2 : ℝ) (hasDerivAt_id t)
+    have h := (h1.fun_sub (hasDerivAt_const t params.b1)).fun_sub
+      (hasDerivAt_const t (1 : ℝ))
+    dsimp [z2]
+    exact h.congr_deriv (by ring)
+  change HasDerivAt
+    (fun s => Romik.addK (Romik.rot s (z1 s, z2 s)) params.k21 params.k22)
+    (Romik.rot t (Romik.alphaBeta2 params t)) t
+  have h := rotAddK_hasDerivAt (k1 := params.k21) (k2 := params.k22) t hz1 hz2
+  refine h.congr_deriv ?_
+  apply congrArg (Romik.rot t)
+  ext <;> dsimp [z1, z2, Romik.alphaBeta2] <;> ring
+
+/-- The phase 3 path has the specified velocity in the rotating frame. -/
+theorem path3_hasDerivAt (t : ℝ) :
+    HasDerivAt (Romik.path3 params)
+      (Romik.rot t (Romik.alphaBeta3 params t)) t := by
+  let z1 : ℝ → ℝ := fun s => params.c1 - s
+  let z2 : ℝ → ℝ := fun s => params.c2 + s
+  have hz1 : HasDerivAt z1 (-1) t := by
+    have h := (hasDerivAt_const t params.c1).fun_sub (hasDerivAt_id t)
+    dsimp [z1]
+    exact h.congr_deriv (by ring)
+  have hz2 : HasDerivAt z2 1 t := by
+    have h := (hasDerivAt_const t params.c2).fun_add (hasDerivAt_id t)
+    dsimp [z2]
+    exact h.congr_deriv (by ring)
+  change HasDerivAt
+    (fun s => Romik.addK (Romik.rot s (z1 s, z2 s)) params.k31 params.k32)
+    (Romik.rot t (Romik.alphaBeta3 params t)) t
+  have h := rotAddK_hasDerivAt (k1 := params.k31) (k2 := params.k32) t hz1 hz2
+  refine h.congr_deriv ?_
+  apply congrArg (Romik.rot t)
+  ext <;> dsimp [z1, z2, Romik.alphaBeta3] <;> ring
+
+/-- The phase 4 path has the specified velocity in the rotating frame. -/
+theorem path4_hasDerivAt (t : ℝ) :
+    HasDerivAt (Romik.path4 params)
+      (Romik.rot t (Romik.alphaBeta4 params t)) t := by
+  let z1 : ℝ → ℝ := fun s => -(1 / 2 : ℝ) * s + params.d1 - 1
+  let z2 : ℝ → ℝ := fun s =>
+    -(1 / 4 : ℝ) * s * s + params.d1 * s + params.d2
+  have hz1 : HasDerivAt z1 (-(1 / 2 : ℝ)) t := by
+    have h1 := HasDerivAt.const_mul (-(1 / 2 : ℝ)) (hasDerivAt_id t)
+    have h := (h1.fun_add (hasDerivAt_const t params.d1)).fun_sub
+      (hasDerivAt_const t (1 : ℝ))
+    dsimp [z1]
+    exact h.congr_deriv (by ring)
+  have hz2 : HasDerivAt z2 (-(1 / 2 : ℝ) * t + params.d1) t := by
+    have h1 := HasDerivAt.const_mul (-(1 / 4 : ℝ)) (square_hasDerivAt t)
+    have h2 := HasDerivAt.const_mul params.d1 (hasDerivAt_id t)
+    have hraw := (h1.fun_add h2).fun_add (hasDerivAt_const t params.d2)
+    have heq : (-(1 / 4 : ℝ)) * (t + t) + params.d1 * 1 + 0 =
+        -(1 / 2 : ℝ) * t + params.d1 := by ring
+    rw [← heq]
+    simpa only [z2, id_eq, mul_assoc] using hraw
+  change HasDerivAt
+    (fun s => Romik.addK (Romik.rot s (z1 s, z2 s)) params.k41 params.k42)
+    (Romik.rot t (Romik.alphaBeta4 params t)) t
+  have h := rotAddK_hasDerivAt (k1 := params.k41) (k2 := params.k42) t hz1 hz2
+  refine h.congr_deriv ?_
+  apply congrArg (Romik.rot t)
+  ext <;> dsimp [z1, z2, Romik.alphaBeta4] <;> ring
+
+/-- The phase 5 path has the specified velocity in the rotating frame. -/
+theorem path5_hasDerivAt (t : ℝ) :
+    HasDerivAt (Romik.path5 params)
+      (Romik.rot t (Romik.alphaBeta5 params t)) t := by
+  let z1 : ℝ → ℝ := fun s =>
+    params.e1 * Real.cos s + params.e2 * Real.sin s - 1 / 2
+  let z2 : ℝ → ℝ := fun s =>
+    -params.e2 * Real.cos s + params.e1 * Real.sin s - 1
+  have hz1 : HasDerivAt z1
+      (-params.e1 * Real.sin t + params.e2 * Real.cos t) t := by
+    have h1 := HasDerivAt.const_mul params.e1 (Real.hasDerivAt_cos t)
+    have h2 := HasDerivAt.const_mul params.e2 (Real.hasDerivAt_sin t)
+    have h := (h1.fun_add h2).fun_sub (hasDerivAt_const t (1 / 2 : ℝ))
+    dsimp [z1]
+    exact h.congr_deriv (by ring)
+  have hz2 : HasDerivAt z2
+      (params.e2 * Real.sin t + params.e1 * Real.cos t) t := by
+    have h1 := HasDerivAt.const_mul (-params.e2) (Real.hasDerivAt_cos t)
+    have h2 := HasDerivAt.const_mul params.e1 (Real.hasDerivAt_sin t)
+    have h := (h1.fun_add h2).fun_sub (hasDerivAt_const t (1 : ℝ))
+    dsimp [z2]
+    exact h.congr_deriv (by ring)
+  change HasDerivAt
+    (fun s => Romik.addK (Romik.rot s (z1 s, z2 s)) params.k51 params.k52)
+    (Romik.rot t (Romik.alphaBeta5 params t)) t
+  have h := rotAddK_hasDerivAt (k1 := params.k51) (k2 := params.k52) t hz1 hz2
+  refine h.congr_deriv ?_
+  apply congrArg (Romik.rot t)
+  ext <;> dsimp [z1, z2, Romik.alphaBeta5] <;> ring
+
+/-- Generic outer-A envelope identity from `x'=R_t(alpha,beta)`. -/
+private theorem contactA_hasDerivAt
+    {x : ℝ → Point} {ab : ℝ → Point} {a' : ℝ} (t : ℝ)
+    (hx : HasDerivAt x (Romik.rot t (ab t)) t)
+    (ha : HasDerivAt (fun s => (ab s).1) a' t) :
+    HasDerivAt
+      (fun s => x s + (ab s).1 • v s + u s)
+      (vecA ((ab t).2 + a' + 1) t) t := by
+  have hscaled := ha.fun_smul (v_hasDerivAt t)
+  have h := (hx.fun_add hscaled).fun_add (u_hasDerivAt t)
+  refine h.congr_deriv ?_
+  ext <;> simp [Romik.rot, u, v, vecA] <;> ring
+
+/-- Generic outer-C envelope identity from `x'=R_t(alpha,beta)`. -/
+private theorem contactC_hasDerivAt
+    {x : ℝ → Point} {ab : ℝ → Point} {b' : ℝ} (t : ℝ)
+    (hx : HasDerivAt x (Romik.rot t (ab t)) t)
+    (hb : HasDerivAt (fun s => (ab s).2) b' t) :
+    HasDerivAt
+      (fun s => x s - (ab s).2 • u s + v s)
+      (vecC (b' + 1 - (ab t).1) t) t := by
+  have hscaled := hb.fun_smul (u_hasDerivAt t)
+  have h := (hx.fun_sub hscaled).fun_add (v_hasDerivAt t)
+  refine h.congr_deriv ?_
+  ext <;> simp [Romik.rot, u, v, vecC] <;> ring
+
+private theorem alpha1_fst_hasDerivAt (t : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta1 params s).1)
+      (-2 * params.a1 * Real.cos t - 2 * params.a2 * Real.sin t) t := by
+  have h1 := HasDerivAt.const_mul (-2 * params.a1) (Real.hasDerivAt_sin t)
+  have h2 := HasDerivAt.const_mul (2 * params.a2) (Real.hasDerivAt_cos t)
+  have h := (h1.fun_add h2).fun_add (hasDerivAt_const t (1 / 2 : ℝ))
+  dsimp [Romik.alphaBeta1]
+  exact h.congr_deriv (by ring)
+
+private theorem alpha2_fst_hasDerivAt (t : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta2 params s).1) (-1) t := by
+  have h := (hasDerivAt_const t (1 + 2 * params.b1)).fun_sub (hasDerivAt_id t)
+  dsimp [Romik.alphaBeta2]
+  exact h.congr_deriv (by ring)
+
+private theorem alpha3_fst_hasDerivAt (t : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta3 params s).1) (-1) t := by
+  have h := (hasDerivAt_const t (-1 - params.c2)).fun_sub (hasDerivAt_id t)
+  dsimp [Romik.alphaBeta3]
+  exact h.congr_deriv (by ring)
+
+private theorem alpha4_fst_hasDerivAt (t : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta4 params s).1)
+      ((1 / 2 : ℝ) * t - params.d1) t := by
+  have h1 := HasDerivAt.const_mul (1 / 4 : ℝ) (square_hasDerivAt t)
+  have h2 := HasDerivAt.const_mul params.d1 (hasDerivAt_id t)
+  have hraw := (((h1.fun_sub h2).fun_sub (hasDerivAt_const t params.d2)).fun_sub
+    (hasDerivAt_const t (1 / 2 : ℝ)))
+  have heq : (1 / 4 : ℝ) * (t + t) - params.d1 * 1 - 0 - 0 =
+      (1 / 2 : ℝ) * t - params.d1 := by ring
+  rw [← heq]
+  simpa only [Romik.alphaBeta4, id_eq, mul_assoc] using hraw
+
+private theorem alpha5_fst_hasDerivAt (t : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta5 params s).1)
+      (-2 * params.e1 * Real.cos t - 2 * params.e2 * Real.sin t) t := by
+  have h1 := HasDerivAt.const_mul (2 * params.e1) (Real.hasDerivAt_sin t)
+  have h2 := HasDerivAt.const_mul (2 * params.e2) (Real.hasDerivAt_cos t)
+  have h := ((hasDerivAt_const t (1 : ℝ)).fun_sub h1).fun_add h2
+  dsimp [Romik.alphaBeta5]
+  exact h.congr_deriv (by ring)
+
+private theorem beta1_hasDerivAt (t : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta1 params s).2)
+      (-2 * params.a1 * Real.sin t + 2 * params.a2 * Real.cos t) t := by
+  have h1 := HasDerivAt.const_mul (2 * params.a1) (Real.hasDerivAt_cos t)
+  have h2 := HasDerivAt.const_mul (2 * params.a2) (Real.hasDerivAt_sin t)
+  have h := (h1.fun_add h2).fun_sub (hasDerivAt_const t (1 : ℝ))
+  dsimp [Romik.alphaBeta1]
+  exact h.congr_deriv (by ring)
+
+private theorem beta2_hasDerivAt (t : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta2 params s).2)
+      (-(1 / 2 : ℝ) * t + params.b1) t := by
+  have h1 := HasDerivAt.const_mul (-(1 / 4 : ℝ)) (square_hasDerivAt t)
+  have h2 := HasDerivAt.const_mul params.b1 (hasDerivAt_id t)
+  have hraw := ((h1.fun_add h2).fun_add (hasDerivAt_const t params.b2)).fun_add
+    (hasDerivAt_const t (1 / 2 : ℝ))
+  have heq : (-(1 / 4 : ℝ)) * (t + t) + params.b1 * 1 + 0 + 0 =
+      -(1 / 2 : ℝ) * t + params.b1 := by ring
+  rw [← heq]
+  simpa only [Romik.alphaBeta2, id_eq, mul_assoc] using hraw
+
+private theorem beta3_hasDerivAt (t : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta3 params s).2) (-1) t := by
+  have h := (hasDerivAt_const t (1 + params.c1)).fun_sub (hasDerivAt_id t)
+  dsimp [Romik.alphaBeta3]
+  exact h.congr_deriv (by ring)
+
+private theorem beta4_hasDerivAt (t : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta4 params s).2) (-1) t := by
+  have h := (hasDerivAt_const t (2 * params.d1 - 1)).fun_sub (hasDerivAt_id t)
+  dsimp [Romik.alphaBeta4]
+  exact h.congr_deriv (by ring)
+
+private theorem beta5_hasDerivAt (t : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta5 params s).2)
+      (-2 * params.e1 * Real.sin t + 2 * params.e2 * Real.cos t) t := by
+  have h1 := HasDerivAt.const_mul (2 * params.e1) (Real.hasDerivAt_cos t)
+  have h2 := HasDerivAt.const_mul (2 * params.e2) (Real.hasDerivAt_sin t)
+  have h := (h1.fun_add h2).fun_sub (hasDerivAt_const t (1 / 2 : ℝ))
+  dsimp [Romik.alphaBeta5]
+  exact h.congr_deriv (by ring)
+
+/-- Phase 1: `A' = 0 * v`. -/
+theorem A1_hasDerivAt (t : ℝ) :
+    HasDerivAt phaseA1 (vecA 0 t) t := by
+  change HasDerivAt
+    (fun s => Romik.path1 params s + (Romik.alphaBeta1 params s).1 • v s + u s)
+    (vecA 0 t) t
+  have h := contactA_hasDerivAt t (path1_hasDerivAt t) (alpha1_fst_hasDerivAt t)
+  refine h.congr_deriv ?_
+  apply congrArg (fun r => vecA r t)
+  dsimp [Romik.alphaBeta1]
+  ring
+
+/-- Phase 2: `A' = beta_2 * v`. -/
+theorem A2_hasDerivAt (t : ℝ) :
+    HasDerivAt phaseA2
+      (vecA (-(1 / 4 : ℝ) * t * t + params.b1 * t + params.b2 + 1 / 2) t) t := by
+  change HasDerivAt
+    (fun s => Romik.path2 params s + (Romik.alphaBeta2 params s).1 • v s + u s)
+    (vecA (-(1 / 4 : ℝ) * t * t + params.b1 * t + params.b2 + 1 / 2) t) t
+  have h := contactA_hasDerivAt t (path2_hasDerivAt t) (alpha2_fst_hasDerivAt t)
+  refine h.congr_deriv ?_
+  apply congrArg (fun r => vecA r t)
+  dsimp [Romik.alphaBeta2]
+  ring
+
+/-- Phase 3: `A' = beta_3 * v`. -/
+theorem A3_hasDerivAt (t : ℝ) :
+    HasDerivAt phaseA3 (vecA (1 + params.c1 - t) t) t := by
+  change HasDerivAt
+    (fun s => Romik.path3 params s + (Romik.alphaBeta3 params s).1 • v s + u s)
+    (vecA (1 + params.c1 - t) t) t
+  have h := contactA_hasDerivAt t (path3_hasDerivAt t) (alpha3_fst_hasDerivAt t)
+  refine h.congr_deriv ?_
+  apply congrArg (fun r => vecA r t)
+  dsimp [Romik.alphaBeta3]
+  ring
+
+/-- Phase 4: `A' = (d1-t/2) * v`. -/
+theorem A4_hasDerivAt (t : ℝ) :
+    HasDerivAt phaseA4 (vecA (params.d1 - t / 2) t) t := by
+  change HasDerivAt
+    (fun s => Romik.path4 params s + (Romik.alphaBeta4 params s).1 • v s + u s)
+    (vecA (params.d1 - t / 2) t) t
+  have h := contactA_hasDerivAt t (path4_hasDerivAt t) (alpha4_fst_hasDerivAt t)
+  refine h.congr_deriv ?_
+  apply congrArg (fun r => vecA r t)
+  dsimp [Romik.alphaBeta4]
+  ring
+
+/-- Phase 5: `A' = (1/2) * v`. -/
+theorem A5_hasDerivAt (t : ℝ) :
+    HasDerivAt phaseA5 (vecA (1 / 2 : ℝ) t) t := by
+  change HasDerivAt
+    (fun s => Romik.path5 params s + (Romik.alphaBeta5 params s).1 • v s + u s)
+    (vecA (1 / 2 : ℝ) t) t
+  have h := contactA_hasDerivAt t (path5_hasDerivAt t) (alpha5_fst_hasDerivAt t)
+  refine h.congr_deriv ?_
+  apply congrArg (fun r => vecA r t)
+  dsimp [Romik.alphaBeta5]
+  ring
+
+/-- Phase 1: `C' = -(1/2) * u`. -/
+theorem C1_hasDerivAt (t : ℝ) :
+    HasDerivAt phaseC1 (vecC (1 / 2 : ℝ) t) t := by
+  change HasDerivAt
+    (fun s => Romik.path1 params s - (Romik.alphaBeta1 params s).2 • u s + v s)
+    (vecC (1 / 2 : ℝ) t) t
+  have h := contactC_hasDerivAt t (path1_hasDerivAt t) (beta1_hasDerivAt t)
+  refine h.congr_deriv ?_
+  apply congrArg (fun r => vecC r t)
+  dsimp [Romik.alphaBeta1]
+  ring
+
+/-- Phase 2: `C' = -(t/2-b1) * u`. -/
+theorem C2_hasDerivAt (t : ℝ) :
+    HasDerivAt phaseC2 (vecC (t / 2 - params.b1) t) t := by
+  change HasDerivAt
+    (fun s => Romik.path2 params s - (Romik.alphaBeta2 params s).2 • u s + v s)
+    (vecC (t / 2 - params.b1) t) t
+  have h := contactC_hasDerivAt t (path2_hasDerivAt t) (beta2_hasDerivAt t)
+  refine h.congr_deriv ?_
+  apply congrArg (fun r => vecC r t)
+  dsimp [Romik.alphaBeta2]
+  ring
+
+/-- Phase 3: `C' = -(1+c2+t) * u`. -/
+theorem C3_hasDerivAt (t : ℝ) :
+    HasDerivAt phaseC3 (vecC (1 + params.c2 + t) t) t := by
+  change HasDerivAt
+    (fun s => Romik.path3 params s - (Romik.alphaBeta3 params s).2 • u s + v s)
+    (vecC (1 + params.c2 + t) t) t
+  have h := contactC_hasDerivAt t (path3_hasDerivAt t) (beta3_hasDerivAt t)
+  refine h.congr_deriv ?_
+  apply congrArg (fun r => vecC r t)
+  dsimp [Romik.alphaBeta3]
+  ring
+
+/-- Phase 4: `C' = -rhoC_4 * u`. -/
+theorem C4_hasDerivAt (t : ℝ) :
+    HasDerivAt phaseC4
+      (vecC (-(1 / 4 : ℝ) * t * t + params.d1 * t + params.d2 + 1 / 2) t) t := by
+  change HasDerivAt
+    (fun s => Romik.path4 params s - (Romik.alphaBeta4 params s).2 • u s + v s)
+    (vecC (-(1 / 4 : ℝ) * t * t + params.d1 * t + params.d2 + 1 / 2) t) t
+  have h := contactC_hasDerivAt t (path4_hasDerivAt t) (beta4_hasDerivAt t)
+  refine h.congr_deriv ?_
+  apply congrArg (fun r => vecC r t)
+  dsimp [Romik.alphaBeta4]
+  ring
+
+/-- Phase 5: `C' = 0 * u`. -/
+theorem C5_hasDerivAt (t : ℝ) :
+    HasDerivAt phaseC5 (vecC 0 t) t := by
+  change HasDerivAt
+    (fun s => Romik.path5 params s - (Romik.alphaBeta5 params s).2 • u s + v s)
+    (vecC 0 t) t
+  have h := contactC_hasDerivAt t (path5_hasDerivAt t) (beta5_hasDerivAt t)
+  refine h.congr_deriv ?_
+  apply congrArg (fun r => vecC r t)
+  dsimp [Romik.alphaBeta5]
+  ring
+
+/-! ## Public derivative shapes for downstream Part C closure
+
+The source-clean phase proofs above use private `vecA`/`vecC` helpers.
+These public lemmas expose exactly the stable geometric derivative shapes used
+by the direct support proof.  The transport is derivative-only: the function
+itself is unchanged, and the remaining pair equality is elementary algebra.
+-/
+
+theorem A1_hasDerivAt_public (t : ℝ) :
+    HasDerivAt phaseA1 ((0 : ℝ) • v t) t := by
+  refine (A1_hasDerivAt t).congr_deriv ?_
+  apply Prod.ext <;> simp [vecA, v]
+
+theorem A2_hasDerivAt_public (t : ℝ) :
+    HasDerivAt phaseA2
+      ((-(1 / 4 : ℝ) * t * t + params.b1 * t + params.b2 + 1 / 2) • v t) t := by
+  refine (A2_hasDerivAt t).congr_deriv ?_
+  apply Prod.ext <;> simp [vecA, v]; ring
+
+theorem A3_hasDerivAt_public (t : ℝ) :
+    HasDerivAt phaseA3 ((1 + params.c1 - t) • v t) t := by
+  refine (A3_hasDerivAt t).congr_deriv ?_
+  apply Prod.ext <;> simp [vecA, v]; ring
+
+theorem A4_hasDerivAt_public (t : ℝ) :
+    HasDerivAt phaseA4 ((params.d1 - t / 2) • v t) t := by
+  refine (A4_hasDerivAt t).congr_deriv ?_
+  apply Prod.ext <;> simp [vecA, v]; ring
+
+theorem A5_hasDerivAt_public (t : ℝ) :
+    HasDerivAt phaseA5 ((1 / 2 : ℝ) • v t) t := by
+  refine (A5_hasDerivAt t).congr_deriv ?_
+  apply Prod.ext <;> simp [vecA, v]
+
+theorem C1_hasDerivAt_public (t : ℝ) :
+    HasDerivAt phaseC1 ((-(1 / 2 : ℝ)) • u t) t := by
+  refine (C1_hasDerivAt t).congr_deriv ?_
+  apply Prod.ext <;> simp [vecC, u]
+
+theorem C2_hasDerivAt_public (t : ℝ) :
+    HasDerivAt phaseC2 ((-(t / 2 - params.b1)) • u t) t := by
+  refine (C2_hasDerivAt t).congr_deriv ?_
+  apply Prod.ext <;> simp [vecC, u]
+
+theorem C3_hasDerivAt_public (t : ℝ) :
+    HasDerivAt phaseC3 ((-(1 + params.c2 + t)) • u t) t := by
+  refine (C3_hasDerivAt t).congr_deriv ?_
+  apply Prod.ext <;> simp [vecC, u]
+
+theorem C4_hasDerivAt_public (t : ℝ) :
+    HasDerivAt phaseC4
+      ((-(-(1 / 4 : ℝ) * t * t + params.d1 * t + params.d2 + 1 / 2)) • u t) t := by
+  refine (C4_hasDerivAt t).congr_deriv ?_
+  apply Prod.ext <;> simp [vecC, u]
+
+theorem C5_hasDerivAt_public (t : ℝ) :
+    HasDerivAt phaseC5 ((0 : ℝ) • u t) t := by
+  refine (C5_hasDerivAt t).congr_deriv ?_
+  apply Prod.ext <;> simp [vecC, u]
+
+end Stage2
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Direct outer-support closure for Part C
+
+This proof uses the five source-clean phase derivative identities and glues
+monotonicity across the four certified switching times.  It does not require a
+globally differentiable contact parametrisation at the speed-change junctions.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Set
+
+namespace GerverSofa
+namespace PartC
+namespace Stage3
+
+open Stage2
+
+private theorem T_nonneg : (0 : ℝ) ≤ T := by
+  dsimp [T]
+  positivity
+
+private theorem T_le_pi : T ≤ Real.pi := by
+  dsimp [T]
+  nlinarith [Real.pi_pos]
+
+private theorem cos_nonneg_physical {x y : ℝ}
+    (hx : x ∈ Icc (0 : ℝ) T) (hy : y ∈ Icc (0 : ℝ) T) :
+    0 ≤ Real.cos (x - y) := by
+  rcases hx with ⟨hx0, hxT⟩
+  rcases hy with ⟨hy0, hyT⟩
+  apply Real.cos_nonneg_of_neg_pi_div_two_le_of_le
+  · dsimp [T] at hxT hyT ⊢
+    linarith
+  · dsimp [T] at hxT hyT ⊢
+    linarith
+
+private theorem sin_nonneg_diff {x y : ℝ}
+    (hx : x ∈ Icc (0 : ℝ) T) (hy : y ∈ Icc (0 : ℝ) T) (hxy : x ≤ y) :
+    0 ≤ Real.sin (y - x) := by
+  rcases hx with ⟨hx0, hxT⟩
+  rcases hy with ⟨hy0, hyT⟩
+  apply Real.sin_nonneg_of_nonneg_of_le_pi
+  · linarith
+  · have hT : y - x ≤ T := by linarith
+    exact le_trans hT T_le_pi
+
+private theorem dot_hasDerivAt_fixed {f : ℝ → Point} {df : Point} {t : ℝ}
+    (h : HasDerivAt f df t) (w : Point) :
+    HasDerivAt (fun s => dot (f s) w) (dot df w) t := by
+  have h1 := HasDerivAt.const_mul w.1 h.fst
+  have h2 := HasDerivAt.const_mul w.2 h.snd
+  have hs := h1.fun_add h2
+  simpa [dot, mul_comm] using hs
+
+private theorem glue_mono {f : ℝ → ℝ} {a b c : ℝ}
+    (hab : a ≤ b) (hbc : b ≤ c)
+    (h1 : MonotoneOn f (Icc a b)) (h2 : MonotoneOn f (Icc b c)) :
+    MonotoneOn f (Icc a c) := by
+  intro x hx y hy hxy
+  by_cases hyb : y ≤ b
+  · exact h1 ⟨hx.1, le_trans hxy hyb⟩ ⟨hy.1, hyb⟩ hxy
+  by_cases hbx : b ≤ x
+  · exact h2 ⟨hbx, le_trans hxy hy.2⟩ ⟨le_trans hbx hxy, hy.2⟩ hxy
+  · have hxb : x ≤ b := le_of_not_ge hbx
+    have hby : b ≤ y := le_of_lt (lt_of_not_ge hyb)
+    exact le_trans
+      (h1 ⟨hx.1, hxb⟩ ⟨hab, le_rfl⟩ hxb)
+      (h2 ⟨le_rfl, hbc⟩ ⟨hby, hy.2⟩ hby)
+
+private theorem glue_anti {f : ℝ → ℝ} {a b c : ℝ}
+    (hab : a ≤ b) (hbc : b ≤ c)
+    (h1 : AntitoneOn f (Icc a b)) (h2 : AntitoneOn f (Icc b c)) :
+    AntitoneOn f (Icc a c) := by
+  intro x hx y hy hxy
+  by_cases hyb : y ≤ b
+  · exact h1 ⟨hx.1, le_trans hxy hyb⟩ ⟨hy.1, hyb⟩ hxy
+  by_cases hbx : b ≤ x
+  · exact h2 ⟨hbx, le_trans hxy hy.2⟩ ⟨le_trans hbx hxy, hy.2⟩ hxy
+  · have hxb : x ≤ b := le_of_not_ge hbx
+    have hby : b ≤ y := le_of_lt (lt_of_not_ge hyb)
+    exact le_trans
+      (h2 ⟨le_rfl, hbc⟩ ⟨hby, hy.2⟩ hby)
+      (h1 ⟨hx.1, hxb⟩ ⟨hab, le_rfl⟩ hxb)
+
+private theorem restrict_mono {f : ℝ → ℝ} {a b c d : ℝ}
+    (h : MonotoneOn f (Icc a b)) (hac : a ≤ c) (hdb : d ≤ b) :
+    MonotoneOn f (Icc c d) := by
+  intro x hx y hy hxy
+  exact h ⟨le_trans hac hx.1, le_trans hx.2 hdb⟩
+    ⟨le_trans hac hy.1, le_trans hy.2 hdb⟩ hxy
+
+private def a1c (_t : ℝ) : ℝ := 0
+private def a2c (t : ℝ) : ℝ := -(1 / 4 : ℝ) * t * t + params.b1 * t + params.b2 + 1 / 2
+private def a3c (t : ℝ) : ℝ := 1 + params.c1 - t
+private def a4c (t : ℝ) : ℝ := params.d1 - t / 2
+private def a5c (_t : ℝ) : ℝ := 1 / 2
+
+private def c1c (_t : ℝ) : ℝ := 1 / 2
+private def c2c (t : ℝ) : ℝ := t / 2 - params.b1
+private def c3c (t : ℝ) : ℝ := 1 + params.c2 + t
+private def c4c (t : ℝ) : ℝ := -(1 / 4 : ℝ) * t * t + params.d1 * t + params.d2 + 1 / 2
+private def c5c (_t : ℝ) : ℝ := 0
+
+private theorem a1_nonneg (t : ℝ) : 0 ≤ a1c t := by simp [a1c]
+private theorem a2_nonneg {t : ℝ} (ht : t ∈ Ioo params.phi params.theta) : 0 ≤ a2c t := by
+  have hp : t ∈ Icc (0 : ℝ) T := by
+    exact ⟨le_trans phi_nonneg (le_of_lt ht.1),
+      le_trans (le_of_lt ht.2) (le_trans theta_le_eta (le_trans eta_le_tau tau_le_T))⟩
+  have h := rhoA_nonneg hp
+  simpa [rhoA, a2c, not_le.mpr ht.1, le_of_lt ht.2] using h
+private theorem a3_nonneg {t : ℝ} (ht : t ∈ Ioo params.theta eta) : 0 ≤ a3c t := by
+  have hp : t ∈ Icc (0 : ℝ) T := ⟨le_trans zero_le_theta (le_of_lt ht.1),
+    le_trans (le_of_lt ht.2) (le_trans eta_le_tau tau_le_T)⟩
+  have h := rhoA_nonneg hp
+  have hphi : params.phi < t := lt_of_le_of_lt switchOrder.phi_le_theta ht.1
+  simpa [rhoA, a3c, not_le.mpr hphi, not_le.mpr ht.1, le_of_lt ht.2] using h
+private theorem a4_nonneg {t : ℝ} (ht : t ∈ Ioo eta tau) : 0 ≤ a4c t := by
+  have hp : t ∈ Icc (0 : ℝ) T := ⟨le_trans (le_trans zero_le_theta theta_le_eta) (le_of_lt ht.1),
+    le_trans (le_of_lt ht.2) tau_le_T⟩
+  have h := rhoA_nonneg hp
+  have htheta : params.theta < t := lt_of_le_of_lt theta_le_eta ht.1
+  have hphi : params.phi < t := lt_of_le_of_lt switchOrder.phi_le_theta htheta
+  simpa [rhoA, a4c, not_le.mpr hphi, not_le.mpr htheta, not_le.mpr ht.1,
+    le_of_lt ht.2] using h
+private theorem a5_nonneg (t : ℝ) : 0 ≤ a5c t := by norm_num [a5c]
+
+private theorem c1_nonneg (t : ℝ) : 0 ≤ c1c t := by norm_num [c1c]
+private theorem c2_nonneg {t : ℝ} (ht : t ∈ Ioo params.phi params.theta) : 0 ≤ c2c t := by
+  have hp : t ∈ Icc (0 : ℝ) T := ⟨le_trans phi_nonneg (le_of_lt ht.1),
+    le_trans (le_of_lt ht.2) (le_trans theta_le_eta (le_trans eta_le_tau tau_le_T))⟩
+  have h := rhoC_nonneg hp
+  simpa [rhoC, c2c, not_le.mpr ht.1, le_of_lt ht.2] using h
+private theorem c3_nonneg {t : ℝ} (ht : t ∈ Ioo params.theta eta) : 0 ≤ c3c t := by
+  have hp : t ∈ Icc (0 : ℝ) T := ⟨le_trans zero_le_theta (le_of_lt ht.1),
+    le_trans (le_of_lt ht.2) (le_trans eta_le_tau tau_le_T)⟩
+  have h := rhoC_nonneg hp
+  have hphi : params.phi < t := lt_of_le_of_lt switchOrder.phi_le_theta ht.1
+  simpa [rhoC, c3c, not_le.mpr hphi, not_le.mpr ht.1, le_of_lt ht.2] using h
+private theorem c4_nonneg {t : ℝ} (ht : t ∈ Ioo eta tau) : 0 ≤ c4c t := by
+  have hp : t ∈ Icc (0 : ℝ) T := ⟨le_trans (le_trans zero_le_theta theta_le_eta) (le_of_lt ht.1),
+    le_trans (le_of_lt ht.2) tau_le_T⟩
+  have h := rhoC_nonneg hp
+  have htheta : params.theta < t := lt_of_le_of_lt theta_le_eta ht.1
+  have hphi : params.phi < t := lt_of_le_of_lt switchOrder.phi_le_theta htheta
+  simpa [rhoC, c4c, not_le.mpr hphi, not_le.mpr htheta, not_le.mpr ht.1,
+    le_of_lt ht.2] using h
+private theorem c5_nonneg (t : ℝ) : 0 ≤ c5c t := by simp [c5c]
+
+/-- The horizontal rotating-frame projection recovers the first body coordinate. -/
+theorem rot_dot_u (t : ℝ) (z : Point) : dot (Romik.rot t z) (u t) = z.1 := by
+  dsimp [dot, Romik.rot, u]
+  calc
+    (Real.cos t * z.1 - Real.sin t * z.2) * Real.cos t +
+        (Real.sin t * z.1 + Real.cos t * z.2) * Real.sin t
+        = z.1 * (Real.sin t ^ 2 + Real.cos t ^ 2) := by ring
+    _ = z.1 := by rw [Real.sin_sq_add_cos_sq]; ring
+
+/-- The vertical rotating-frame projection recovers the second body coordinate. -/
+theorem rot_dot_v (t : ℝ) (z : Point) : dot (Romik.rot t z) (v t) = z.2 := by
+  dsimp [dot, Romik.rot, v]
+  calc
+    (Real.cos t * z.1 - Real.sin t * z.2) * (-Real.sin t) +
+        (Real.sin t * z.1 + Real.cos t * z.2) * Real.cos t
+        = z.2 * (Real.sin t ^ 2 + Real.cos t ^ 2) := by ring
+    _ = z.2 := by rw [Real.sin_sq_add_cos_sq]; ring
+
+/-- Rotation in body coordinates is injective. -/
+theorem rot_injective (t : ℝ) : Function.Injective (Romik.rot t) := by
+  intro x y h
+  have hu := congrArg (fun z => dot z (u t)) h
+  have hv := congrArg (fun z => dot z (v t)) h
+  rw [rot_dot_u, rot_dot_u] at hu
+  rw [rot_dot_v, rot_dot_v] at hv
+  exact Prod.ext hu hv
+
+private theorem pathPrime1_eq_rot (p : Romik.Params) (t : ℝ) :
+    Romik.pathPrime1 p t = Romik.rot t (Romik.alphaBeta1 p t) := rfl
+private theorem pathPrime2_eq_rot (p : Romik.Params) (t : ℝ) :
+    Romik.pathPrime2 p t = Romik.rot t (Romik.alphaBeta2 p t) := rfl
+private theorem pathPrime3_eq_rot (p : Romik.Params) (t : ℝ) :
+    Romik.pathPrime3 p t = Romik.rot t (Romik.alphaBeta3 p t) := rfl
+
+private theorem matchPrime12 : Romik.pathPrime1 params params.phi = Romik.pathPrime2 params
+  params.phi := by
+  have h10 := congrFun params_equations (10 : Fin 22)
+  have h11 := congrFun params_equations (11 : Fin 22)
+  simp [Romik.system] at h10 h11
+  apply Prod.ext <;> linarith
+private theorem matchPrime23 : Romik.pathPrime2 params params.theta = Romik.pathPrime3 params
+  params.theta := by
+  have h14 := congrFun params_equations (14 : Fin 22)
+  have h15 := congrFun params_equations (15 : Fin 22)
+  simp [Romik.system] at h14 h15
+  apply Prod.ext <;> linarith
+private theorem matchAB12 : Romik.alphaBeta1 params params.phi = Romik.alphaBeta2 params
+  params.phi := by
+  apply rot_injective params.phi
+  simpa [pathPrime1_eq_rot, pathPrime2_eq_rot] using matchPrime12
+private theorem matchAB23 : Romik.alphaBeta2 params params.theta = Romik.alphaBeta3 params
+  params.theta := by
+  apply rot_injective params.theta
+  simpa [pathPrime2_eq_rot, pathPrime3_eq_rot] using matchPrime23
+
+private def reflAB (z : Point) : Point := (-z.2, -z.1)
+private theorem ab4_reflect_ab2 (t : ℝ) : Romik.alphaBeta4 params (T - t) = reflAB
+  (Romik.alphaBeta2 params t) := by
+  dsimp [T, reflAB, Romik.alphaBeta4, Romik.alphaBeta2]
+  rw [Romik.d1_eq_quarterPi_sub_b1_of_equations params_equations,
+      Romik.d2_eq_b2_add_quarterPi_correction_of_equations params_equations]
+  ring_nf
+private theorem ab3_reflect (t : ℝ) : Romik.alphaBeta3 params (T - t) = reflAB (Romik.alphaBeta3
+  params t) := by
+  dsimp [T, reflAB, Romik.alphaBeta3]
+  rw [Romik.c2_eq_c1_sub_halfPi_of_equations params_equations]
+  ring_nf
+private theorem ab5_reflect_ab1 (t : ℝ) : Romik.alphaBeta5 params (T - t) = reflAB
+  (Romik.alphaBeta1 params t) := by
+  dsimp [T, reflAB, Romik.alphaBeta5, Romik.alphaBeta1]
+  rw [Romik.e1_eq_a1_of_equations params_equations,
+      Romik.e2_eq_neg_a2_of_equations params_equations]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  ring_nf
+private theorem matchAB34 : Romik.alphaBeta3 params eta = Romik.alphaBeta4 params eta := by
+  have h3 := ab3_reflect params.theta
+  have h4 := ab4_reflect_ab2 params.theta
+  have hm := congrArg reflAB matchAB23
+  simpa [eta] using h3.trans (hm.symm.trans h4.symm)
+private theorem matchAB45 : Romik.alphaBeta4 params tau = Romik.alphaBeta5 params tau := by
+  have h4 := ab4_reflect_ab2 params.phi
+  have h5 := ab5_reflect_ab1 params.phi
+  have hm := congrArg reflAB matchAB12
+  simpa [tau] using h4.trans (hm.symm.trans h5.symm)
+
+private theorem A_eq_phase1 {t : ℝ} (ht : t ∈ Icc (0 : ℝ) params.phi) : A t = phaseA1 t := by
+  apply Prod.ext <;> simp [A, alpha, alphaBetaAt, Romik.path, phaseA1, ht.2]
+
+private theorem A_eq_phase2 {t : ℝ} (ht : t ∈ Icc params.phi params.theta) : A t = phaseA2 t := by
+  by_cases h : t = params.phi
+  · subst t
+    calc
+      A params.phi = phaseA1 params.phi := A_eq_phase1 ⟨phi_nonneg, le_rfl⟩
+      _ = phaseA2 params.phi := by
+        unfold phaseA1 phaseA2
+        rw [PartB.match12, matchAB12]
+  · have hp : params.phi < t := lt_of_le_of_ne ht.1 (Ne.symm h)
+    apply Prod.ext <;> simp [A, alpha, alphaBetaAt, Romik.path, phaseA2, not_le.mpr hp, ht.2]
+
+private theorem A_eq_phase3 {t : ℝ} (ht : t ∈ Icc params.theta eta) : A t = phaseA3 t := by
+  by_cases h : t = params.theta
+  · subst t
+    calc
+      A params.theta = phaseA2 params.theta := A_eq_phase2 ⟨switchOrder.phi_le_theta, le_rfl⟩
+      _ = phaseA3 params.theta := by
+        unfold phaseA2 phaseA3
+        rw [PartB.match23, matchAB23]
+  · have htheta : params.theta < t := lt_of_le_of_ne ht.1 (Ne.symm h)
+    have hphi : params.phi < t := lt_of_le_of_lt switchOrder.phi_le_theta htheta
+    have hηraw : t ≤ Real.pi / 2 - params.theta := by
+      simpa [eta, T] using ht.2
+    apply Prod.ext <;> simp [A, alpha, alphaBetaAt, Romik.path, phaseA3, not_le.mpr hphi,
+      not_le.mpr htheta, hηraw, eta, T]
+
+private theorem A_eq_phase4 {t : ℝ} (ht : t ∈ Icc eta tau) : A t = phaseA4 t := by
+  by_cases h : t = eta
+  · subst t
+    have hm : Romik.path3 params eta = Romik.path4 params eta := by
+      simpa [eta, T] using PartB.match34
+    calc
+      A eta = phaseA3 eta := A_eq_phase3 ⟨theta_le_eta, le_rfl⟩
+      _ = phaseA4 eta := by
+        unfold phaseA3 phaseA4
+        rw [hm, matchAB34]
+  · have he : eta < t := lt_of_le_of_ne ht.1 (Ne.symm h)
+    have htheta : params.theta < t := lt_of_le_of_lt theta_le_eta he
+    have hphi : params.phi < t := lt_of_le_of_lt switchOrder.phi_le_theta htheta
+    have hηraw : Real.pi / 2 - params.theta < t := by
+      simpa [eta, T] using he
+    have hτraw : t ≤ Real.pi / 2 - params.phi := by
+      simpa [tau, T] using ht.2
+    apply Prod.ext <;> simp [A, alpha, alphaBetaAt, Romik.path, phaseA4, not_le.mpr hphi,
+      not_le.mpr htheta, not_le.mpr hηraw, hτraw, eta, tau, T]
+
+private theorem A_eq_phase5 {t : ℝ} (ht : t ∈ Icc tau T) : A t = phaseA5 t := by
+  by_cases h : t = tau
+  · subst t
+    have hm : Romik.path4 params tau = Romik.path5 params tau := by
+      simpa [tau, T] using PartB.match45
+    calc
+      A tau = phaseA4 tau := A_eq_phase4 ⟨eta_le_tau, le_rfl⟩
+      _ = phaseA5 tau := by
+        unfold phaseA4 phaseA5
+        rw [hm, matchAB45]
+  · have htau : tau < t := lt_of_le_of_ne ht.1 (Ne.symm h)
+    have heta : eta < t := lt_of_le_of_lt eta_le_tau htau
+    have htheta : params.theta < t := lt_of_le_of_lt theta_le_eta heta
+    have hphi : params.phi < t := lt_of_le_of_lt switchOrder.phi_le_theta htheta
+    have hηraw : Real.pi / 2 - params.theta < t := by
+      simpa [eta, T] using heta
+    have hτraw : Real.pi / 2 - params.phi < t := by
+      simpa [tau, T] using htau
+    apply Prod.ext <;> simp [A, alpha, alphaBetaAt, Romik.path, phaseA5, not_le.mpr hphi,
+      not_le.mpr htheta, not_le.mpr hηraw, not_le.mpr hτraw, eta, tau, T]
+
+private theorem C_eq_phase1 {t : ℝ} (ht : t ∈ Icc (0 : ℝ) params.phi) : C t = phaseC1 t := by
+  apply Prod.ext <;> simp [C, beta, alphaBetaAt, Romik.path, phaseC1, ht.2]
+
+private theorem C_eq_phase2 {t : ℝ} (ht : t ∈ Icc params.phi params.theta) : C t = phaseC2 t := by
+  by_cases h : t = params.phi
+  · subst t
+    calc
+      C params.phi = phaseC1 params.phi := C_eq_phase1 ⟨phi_nonneg, le_rfl⟩
+      _ = phaseC2 params.phi := by
+        unfold phaseC1 phaseC2
+        rw [PartB.match12, matchAB12]
+  · have hp : params.phi < t := lt_of_le_of_ne ht.1 (Ne.symm h)
+    apply Prod.ext <;> simp [C, beta, alphaBetaAt, Romik.path, phaseC2, not_le.mpr hp, ht.2]
+
+private theorem C_eq_phase3 {t : ℝ} (ht : t ∈ Icc params.theta eta) : C t = phaseC3 t := by
+  by_cases h : t = params.theta
+  · subst t
+    calc
+      C params.theta = phaseC2 params.theta := C_eq_phase2 ⟨switchOrder.phi_le_theta, le_rfl⟩
+      _ = phaseC3 params.theta := by
+        unfold phaseC2 phaseC3
+        rw [PartB.match23, matchAB23]
+  · have htheta : params.theta < t := lt_of_le_of_ne ht.1 (Ne.symm h)
+    have hphi : params.phi < t := lt_of_le_of_lt switchOrder.phi_le_theta htheta
+    have hηraw : t ≤ Real.pi / 2 - params.theta := by
+      simpa [eta, T] using ht.2
+    apply Prod.ext <;> simp [C, beta, alphaBetaAt, Romik.path, phaseC3, not_le.mpr hphi,
+      not_le.mpr htheta, hηraw, eta, T]
+
+private theorem C_eq_phase4 {t : ℝ} (ht : t ∈ Icc eta tau) : C t = phaseC4 t := by
+  by_cases h : t = eta
+  · subst t
+    have hm : Romik.path3 params eta = Romik.path4 params eta := by
+      simpa [eta, T] using PartB.match34
+    calc
+      C eta = phaseC3 eta := C_eq_phase3 ⟨theta_le_eta, le_rfl⟩
+      _ = phaseC4 eta := by
+        unfold phaseC3 phaseC4
+        rw [hm, matchAB34]
+  · have he : eta < t := lt_of_le_of_ne ht.1 (Ne.symm h)
+    have htheta : params.theta < t := lt_of_le_of_lt theta_le_eta he
+    have hphi : params.phi < t := lt_of_le_of_lt switchOrder.phi_le_theta htheta
+    have hηraw : Real.pi / 2 - params.theta < t := by
+      simpa [eta, T] using he
+    have hτraw : t ≤ Real.pi / 2 - params.phi := by
+      simpa [tau, T] using ht.2
+    apply Prod.ext <;> simp [C, beta, alphaBetaAt, Romik.path, phaseC4, not_le.mpr hphi,
+      not_le.mpr htheta, not_le.mpr hηraw, hτraw, eta, tau, T]
+
+private theorem C_eq_phase5 {t : ℝ} (ht : t ∈ Icc tau T) : C t = phaseC5 t := by
+  by_cases h : t = tau
+  · subst t
+    have hm : Romik.path4 params tau = Romik.path5 params tau := by
+      simpa [tau, T] using PartB.match45
+    calc
+      C tau = phaseC4 tau := C_eq_phase4 ⟨eta_le_tau, le_rfl⟩
+      _ = phaseC5 tau := by
+        unfold phaseC4 phaseC5
+        rw [hm, matchAB45]
+  · have htau : tau < t := lt_of_le_of_ne ht.1 (Ne.symm h)
+    have heta : eta < t := lt_of_le_of_lt eta_le_tau htau
+    have htheta : params.theta < t := lt_of_le_of_lt theta_le_eta heta
+    have hphi : params.phi < t := lt_of_le_of_lt switchOrder.phi_le_theta htheta
+    have hηraw : Real.pi / 2 - params.theta < t := by
+      simpa [eta, T] using heta
+    have hτraw : Real.pi / 2 - params.phi < t := by
+      simpa [tau, T] using htau
+    apply Prod.ext <;> simp [C, beta, alphaBetaAt, Romik.path, phaseC5, not_le.mpr hphi,
+      not_le.mpr htheta, not_le.mpr hηraw, not_le.mpr hτraw, eta, tau, T]
+
+private theorem C5_hasDerivAt_coeff (t : ℝ) :
+    HasDerivAt phaseC5 ((-(c5c t)) • u t) t := by
+  simpa [c5c] using C5_hasDerivAt_public t
+
+private theorem transfer_mono_A {F : ℝ → Point} {a b : ℝ} {w : Point}
+    (heq : ∀ t ∈ Icc a b, A t = F t)
+    (hm : MonotoneOn (fun t => dot (F t) w) (Icc a b)) :
+    MonotoneOn (fun t => dot (A t) w) (Icc a b) := by
+  intro x hx y hy hxy
+  simpa [heq x hx, heq y hy] using hm hx hy hxy
+private theorem transfer_anti_A {F : ℝ → Point} {a b : ℝ} {w : Point}
+    (heq : ∀ t ∈ Icc a b, A t = F t)
+    (hm : AntitoneOn (fun t => dot (F t) w) (Icc a b)) :
+    AntitoneOn (fun t => dot (A t) w) (Icc a b) := by
+  intro x hx y hy hxy
+  simpa [heq x hx, heq y hy] using hm hx hy hxy
+private theorem transfer_mono_C {F : ℝ → Point} {a b : ℝ} {w : Point}
+    (heq : ∀ t ∈ Icc a b, C t = F t)
+    (hm : MonotoneOn (fun t => dot (F t) w) (Icc a b)) :
+    MonotoneOn (fun t => dot (C t) w) (Icc a b) := by
+  intro x hx y hy hxy
+  simpa [heq x hx, heq y hy] using hm hx hy hxy
+private theorem transfer_anti_C {F : ℝ → Point} {a b : ℝ} {w : Point}
+    (heq : ∀ t ∈ Icc a b, C t = F t)
+    (hm : AntitoneOn (fun t => dot (F t) w) (Icc a b)) :
+    AntitoneOn (fun t => dot (C t) w) (Icc a b) := by
+  intro x hx y hy hxy
+  simpa [heq x hx, heq y hy] using hm hx hy hxy
+
+private theorem phase_A_own_mono {F : ℝ → Point} {r : ℝ → ℝ} {a b s : ℝ}
+    (hder : ∀ t, HasDerivAt F ((r t) • v t) t)
+    (hr : ∀ t ∈ Ioo a b, 0 ≤ r t)
+    (ha0 : 0 ≤ a) (hbs : b ≤ s) (hsT : s ≤ T) :
+    MonotoneOn (fun t => dot (F t) (u s)) (Icc a b) := by
+  refine monotoneOn_of_hasDerivWithinAt_nonneg (f' := fun t => dot ((r t) • v t) (u s))
+    (convex_Icc a b) ?_ ?_ ?_
+  · intro t ht; exact (dot_hasDerivAt_fixed (hder t) (u s)).continuousAt.continuousWithinAt
+  · intro t ht; exact (dot_hasDerivAt_fixed (hder t) (u s)).hasDerivWithinAt
+  · intro t ht
+    have hi : t ∈ Ioo a b := by simpa only [interior_Icc] using ht
+    have hrt := hr t hi
+    have ht0 : 0 ≤ t := le_trans ha0 (le_of_lt hi.1)
+    have hsin := sin_nonneg_diff ⟨ht0, le_trans (le_of_lt hi.2) (le_trans hbs hsT)⟩
+      ⟨le_trans ht0 (le_trans (le_of_lt hi.2) hbs), hsT⟩ (le_trans (le_of_lt hi.2) hbs)
+    have heq : dot ((r t) • v t) (u s) = r t * Real.sin (s - t) := by
+      simp [dot, u, v, Real.sin_sub]
+      ring
+    rw [heq]
+    exact mul_nonneg hrt hsin
+
+private theorem phase_A_own_anti {F : ℝ → Point} {r : ℝ → ℝ} {a b s : ℝ}
+    (hder : ∀ t, HasDerivAt F ((r t) • v t) t)
+    (hr : ∀ t ∈ Ioo a b, 0 ≤ r t)
+    (hs0 : 0 ≤ s) (hsa : s ≤ a) (hbT : b ≤ T) :
+    AntitoneOn (fun t => dot (F t) (u s)) (Icc a b) := by
+  refine antitoneOn_of_hasDerivWithinAt_nonpos (f' := fun t => dot ((r t) • v t) (u s))
+    (convex_Icc a b) ?_ ?_ ?_
+  · intro t ht; exact (dot_hasDerivAt_fixed (hder t) (u s)).continuousAt.continuousWithinAt
+  · intro t ht; exact (dot_hasDerivAt_fixed (hder t) (u s)).hasDerivWithinAt
+  · intro t ht
+    have hi : t ∈ Ioo a b := by simpa only [interior_Icc] using ht
+    have hrt := hr t hi
+    have htT : t ≤ T := le_trans (le_of_lt hi.2) hbT
+    have hsin0 : Real.sin (s - t) ≤ 0 := by
+      have hnon : s - t ≤ 0 := by linarith [hsa, hi.1]
+      have hnegpi : -Real.pi ≤ s - t := by
+        have : t - s ≤ T := by linarith
+        nlinarith [T_le_pi]
+      exact Real.sin_nonpos_of_nonpos_of_neg_pi_le hnon hnegpi
+    have heq : dot ((r t) • v t) (u s) = r t * Real.sin (s - t) := by
+      simp [dot, u, v, Real.sin_sub]
+      ring
+    rw [heq]
+    exact mul_nonpos_of_nonneg_of_nonpos hrt hsin0
+
+private theorem phase_C_own_mono {F : ℝ → Point} {r : ℝ → ℝ} {a b s : ℝ}
+    (hder : ∀ t, HasDerivAt F ((-(r t)) • u t) t)
+    (hr : ∀ t ∈ Ioo a b, 0 ≤ r t)
+    (ha0 : 0 ≤ a) (hbs : b ≤ s) (hsT : s ≤ T) :
+    MonotoneOn (fun t => dot (F t) (v s)) (Icc a b) := by
+  refine monotoneOn_of_hasDerivWithinAt_nonneg (f' := fun t => dot ((-(r t)) • u t) (v s))
+    (convex_Icc a b) ?_ ?_ ?_
+  · intro t ht; exact (dot_hasDerivAt_fixed (hder t) (v s)).continuousAt.continuousWithinAt
+  · intro t ht; exact (dot_hasDerivAt_fixed (hder t) (v s)).hasDerivWithinAt
+  · intro t ht
+    have hi : t ∈ Ioo a b := by simpa only [interior_Icc] using ht
+    have hrt := hr t hi
+    have ht0 : 0 ≤ t := le_trans ha0 (le_of_lt hi.1)
+    have hsin := sin_nonneg_diff ⟨ht0, le_trans (le_of_lt hi.2) (le_trans hbs hsT)⟩
+      ⟨le_trans ht0 (le_trans (le_of_lt hi.2) hbs), hsT⟩ (le_trans (le_of_lt hi.2) hbs)
+    have heq : dot ((-(r t)) • u t) (v s) = r t * Real.sin (s - t) := by
+      simp [dot, u, v, Real.sin_sub]
+      ring
+    rw [heq]
+    exact mul_nonneg hrt hsin
+
+private theorem phase_C_own_anti {F : ℝ → Point} {r : ℝ → ℝ} {a b s : ℝ}
+    (hder : ∀ t, HasDerivAt F ((-(r t)) • u t) t)
+    (hr : ∀ t ∈ Ioo a b, 0 ≤ r t)
+    (hs0 : 0 ≤ s) (hsa : s ≤ a) (hbT : b ≤ T) :
+    AntitoneOn (fun t => dot (F t) (v s)) (Icc a b) := by
+  refine antitoneOn_of_hasDerivWithinAt_nonpos (f' := fun t => dot ((-(r t)) • u t) (v s))
+    (convex_Icc a b) ?_ ?_ ?_
+  · intro t ht; exact (dot_hasDerivAt_fixed (hder t) (v s)).continuousAt.continuousWithinAt
+  · intro t ht; exact (dot_hasDerivAt_fixed (hder t) (v s)).hasDerivWithinAt
+  · intro t ht
+    have hi : t ∈ Ioo a b := by simpa only [interior_Icc] using ht
+    have hrt := hr t hi
+    have hsin0 : Real.sin (s - t) ≤ 0 := by
+      have hnon : s - t ≤ 0 := by linarith [hsa, hi.1]
+      have hnegpi : -Real.pi ≤ s - t := by
+        have : t - s ≤ T := by linarith [hbT, hi.2, hs0]
+        nlinarith [T_le_pi]
+      exact Real.sin_nonpos_of_nonpos_of_neg_pi_le hnon hnegpi
+    have heq : dot ((-(r t)) • u t) (v s) = r t * Real.sin (s - t) := by
+      simp [dot, u, v, Real.sin_sub]
+      ring
+    rw [heq]
+    exact mul_nonpos_of_nonneg_of_nonpos hrt hsin0
+
+private theorem phase_A_cross_mono {F : ℝ → Point} {r : ℝ → ℝ} {a b s : ℝ}
+    (hder : ∀ t, HasDerivAt F ((r t) • v t) t)
+    (hr : ∀ t ∈ Ioo a b, 0 ≤ r t)
+    (ha0 : 0 ≤ a) (hbT : b ≤ T) (hs : s ∈ Icc (0 : ℝ) T) :
+    MonotoneOn (fun t => dot (F t) (v s)) (Icc a b) := by
+  refine monotoneOn_of_hasDerivWithinAt_nonneg (f' := fun t => dot ((r t) • v t) (v s))
+    (convex_Icc a b) ?_ ?_ ?_
+  · intro t ht; exact (dot_hasDerivAt_fixed (hder t) (v s)).continuousAt.continuousWithinAt
+  · intro t ht; exact (dot_hasDerivAt_fixed (hder t) (v s)).hasDerivWithinAt
+  · intro t ht
+    have hi : t ∈ Ioo a b := by simpa only [interior_Icc] using ht
+    have hp : t ∈ Icc (0 : ℝ) T := ⟨le_trans ha0 (le_of_lt hi.1), le_trans (le_of_lt hi.2) hbT⟩
+    have hc := cos_nonneg_physical hp hs
+    have heq : dot ((r t) • v t) (v s) = r t * Real.cos (t - s) := by
+      simp [dot, v, Real.cos_sub]
+      ring
+    rw [heq]
+    exact mul_nonneg (hr t hi) hc
+
+private theorem phase_C_cross_anti {F : ℝ → Point} {r : ℝ → ℝ} {a b s : ℝ}
+    (hder : ∀ t, HasDerivAt F ((-(r t)) • u t) t)
+    (hr : ∀ t ∈ Ioo a b, 0 ≤ r t)
+    (ha0 : 0 ≤ a) (hbT : b ≤ T) (hs : s ∈ Icc (0 : ℝ) T) :
+    AntitoneOn (fun t => dot (F t) (u s)) (Icc a b) := by
+  refine antitoneOn_of_hasDerivWithinAt_nonpos (f' := fun t => dot ((-(r t)) • u t) (u s))
+    (convex_Icc a b) ?_ ?_ ?_
+  · intro t ht; exact (dot_hasDerivAt_fixed (hder t) (u s)).continuousAt.continuousWithinAt
+  · intro t ht; exact (dot_hasDerivAt_fixed (hder t) (u s)).hasDerivWithinAt
+  · intro t ht
+    have hi : t ∈ Ioo a b := by simpa only [interior_Icc] using ht
+    have hp : t ∈ Icc (0 : ℝ) T := ⟨le_trans ha0 (le_of_lt hi.1), le_trans (le_of_lt hi.2) hbT⟩
+    have hc := cos_nonneg_physical hp hs
+    have heq : dot ((-(r t)) • u t) (u s) = -(r t * Real.cos (t - s)) := by
+      simp [dot, u, Real.cos_sub]
+      ring
+    rw [heq]
+    exact neg_nonpos.mpr (mul_nonneg (hr t hi) hc)
+
+private theorem phase_A_y_mono {F : ℝ → Point} {r : ℝ → ℝ} {a b : ℝ}
+    (hder : ∀ t, HasDerivAt F ((r t) • v t) t)
+    (hr : ∀ t ∈ Ioo a b, 0 ≤ r t)
+    (ha0 : 0 ≤ a) (hbT : b ≤ T) :
+    MonotoneOn (fun t => dot (F t) (0, 1)) (Icc a b) := by
+  refine monotoneOn_of_hasDerivWithinAt_nonneg
+    (f' := fun t => dot ((r t) • v t) (0, 1))
+    (convex_Icc a b) ?_ ?_ ?_
+  · intro t ht
+    exact (dot_hasDerivAt_fixed (hder t) (0, 1)).continuousAt.continuousWithinAt
+  · intro t ht
+    exact (dot_hasDerivAt_fixed (hder t) (0, 1)).hasDerivWithinAt
+  · intro t ht
+    have hi : t ∈ Ioo a b := by simpa only [interior_Icc] using ht
+    have ht0 : 0 ≤ t := le_trans ha0 (le_of_lt hi.1)
+    have htT : t ≤ T := le_trans (le_of_lt hi.2) hbT
+    have hc : 0 ≤ Real.cos t := by
+      apply Real.cos_nonneg_of_neg_pi_div_two_le_of_le
+      · nlinarith [Real.pi_pos, ht0]
+      · simpa [T] using htT
+    have heq : dot ((r t) • v t) (0, 1) = r t * Real.cos t := by
+      simp [dot, v]
+    rw [heq]
+    exact mul_nonneg (hr t hi) hc
+
+private theorem phase_C_y_anti {F : ℝ → Point} {r : ℝ → ℝ} {a b : ℝ}
+    (hder : ∀ t, HasDerivAt F ((-(r t)) • u t) t)
+    (hr : ∀ t ∈ Ioo a b, 0 ≤ r t)
+    (ha0 : 0 ≤ a) (hbT : b ≤ T) :
+    AntitoneOn (fun t => dot (F t) (0, 1)) (Icc a b) := by
+  refine antitoneOn_of_hasDerivWithinAt_nonpos
+    (f' := fun t => dot ((-(r t)) • u t) (0, 1))
+    (convex_Icc a b) ?_ ?_ ?_
+  · intro t ht
+    exact (dot_hasDerivAt_fixed (hder t) (0, 1)).continuousAt.continuousWithinAt
+  · intro t ht
+    exact (dot_hasDerivAt_fixed (hder t) (0, 1)).hasDerivWithinAt
+  · intro t ht
+    have hi : t ∈ Ioo a b := by simpa only [interior_Icc] using ht
+    have ht0 : 0 ≤ t := le_trans ha0 (le_of_lt hi.1)
+    have hsin : 0 ≤ Real.sin t :=
+      Real.sin_nonneg_of_nonneg_of_le_pi ht0
+        (le_trans (le_trans (le_of_lt hi.2) hbT) T_le_pi)
+    have heq : dot ((-(r t)) • u t) (0, 1) = -(r t * Real.sin t) := by
+      simp [dot, u]
+    rw [heq]
+    exact neg_nonpos.mpr (mul_nonneg (hr t hi) hsin)
+
+-- Full physical-phase monotonicities for cross supports and vertical coordinates.
+private theorem A_v_global_mono (s : ℝ) (hs : s ∈ Icc (0 : ℝ) T) :
+    MonotoneOn (fun t => dot (A t) (v s)) (Icc (0 : ℝ) T) := by
+  have h1 := transfer_mono_A (w:=v s) (fun t ht => A_eq_phase1 ht)
+    (phase_A_cross_mono (r:=a1c) A1_hasDerivAt_public (fun t ht => a1_nonneg t)
+      (by rfl) (le_trans switchOrder.phi_le_theta (le_trans theta_le_eta (le_trans eta_le_tau
+        tau_le_T))) hs)
+  have h2 := transfer_mono_A (w:=v s) (fun t ht => A_eq_phase2 ht)
+    (phase_A_cross_mono (r:=a2c) A2_hasDerivAt_public (fun t ht => a2_nonneg ht)
+      phi_nonneg (le_trans theta_le_eta (le_trans eta_le_tau tau_le_T)) hs)
+  have h3 := transfer_mono_A (w:=v s) (fun t ht => A_eq_phase3 ht)
+    (phase_A_cross_mono (r:=a3c) A3_hasDerivAt_public (fun t ht => a3_nonneg ht)
+      zero_le_theta (le_trans eta_le_tau tau_le_T) hs)
+  have h4 := transfer_mono_A (w:=v s) (fun t ht => A_eq_phase4 ht)
+    (phase_A_cross_mono (r:=a4c) A4_hasDerivAt_public (fun t ht => a4_nonneg ht)
+      (le_trans zero_le_theta theta_le_eta) tau_le_T hs)
+  have h5 := transfer_mono_A (w:=v s) (fun t ht => A_eq_phase5 ht)
+    (phase_A_cross_mono (r:=a5c) A5_hasDerivAt_public (fun t ht => a5_nonneg t)
+      (le_trans (le_trans zero_le_theta theta_le_eta) eta_le_tau) (by rfl) hs)
+  have h12 := glue_mono phi_nonneg switchOrder.phi_le_theta h1 h2
+  have h123 := glue_mono zero_le_theta theta_le_eta h12 h3
+  have h1234 := glue_mono (le_trans zero_le_theta theta_le_eta) eta_le_tau h123 h4
+  exact glue_mono
+    (le_trans (le_trans zero_le_theta theta_le_eta) eta_le_tau) tau_le_T h1234 h5
+
+private theorem C_u_global_anti (s : ℝ) (hs : s ∈ Icc (0 : ℝ) T) :
+    AntitoneOn (fun t => dot (C t) (u s)) (Icc (0 : ℝ) T) := by
+  have h1 := transfer_anti_C (w:=u s) (fun t ht => C_eq_phase1 ht)
+    (phase_C_cross_anti (r:=c1c) C1_hasDerivAt_public (fun t ht => c1_nonneg t)
+      (by rfl) (le_trans switchOrder.phi_le_theta (le_trans theta_le_eta (le_trans eta_le_tau
+        tau_le_T))) hs)
+  have h2 := transfer_anti_C (w:=u s) (fun t ht => C_eq_phase2 ht)
+    (phase_C_cross_anti (r:=c2c) C2_hasDerivAt_public (fun t ht => c2_nonneg ht)
+      phi_nonneg (le_trans theta_le_eta (le_trans eta_le_tau tau_le_T)) hs)
+  have h3 := transfer_anti_C (w:=u s) (fun t ht => C_eq_phase3 ht)
+    (phase_C_cross_anti (r:=c3c) C3_hasDerivAt_public (fun t ht => c3_nonneg ht)
+      zero_le_theta (le_trans eta_le_tau tau_le_T) hs)
+  have h4 := transfer_anti_C (w:=u s) (fun t ht => C_eq_phase4 ht)
+    (phase_C_cross_anti (r:=c4c) C4_hasDerivAt_public (fun t ht => c4_nonneg ht)
+      (le_trans zero_le_theta theta_le_eta) tau_le_T hs)
+  have h5 := transfer_anti_C (w:=u s) (fun t ht => C_eq_phase5 ht)
+    (phase_C_cross_anti (r:=c5c) C5_hasDerivAt_coeff (fun t ht => c5_nonneg t)
+      (le_trans (le_trans zero_le_theta theta_le_eta) eta_le_tau) (by rfl) hs)
+  have h12 := glue_anti phi_nonneg switchOrder.phi_le_theta h1 h2
+  have h123 := glue_anti zero_le_theta theta_le_eta h12 h3
+  have h1234 := glue_anti (le_trans zero_le_theta theta_le_eta) eta_le_tau h123 h4
+  exact glue_anti
+    (le_trans (le_trans zero_le_theta theta_le_eta) eta_le_tau) tau_le_T h1234 h5
+
+private theorem A_y_global_mono :
+    MonotoneOn (fun t => dot (A t) (0, 1)) (Icc (0 : ℝ) T) := by
+  have h1 := transfer_mono_A (w:=(0,1)) (fun t ht => A_eq_phase1 ht)
+    (phase_A_y_mono (r:=a1c) A1_hasDerivAt_public (fun t ht => a1_nonneg t)
+      (by rfl) (le_trans switchOrder.phi_le_theta (le_trans theta_le_eta (le_trans eta_le_tau
+        tau_le_T))))
+  have h2 := transfer_mono_A (w:=(0,1)) (fun t ht => A_eq_phase2 ht)
+    (phase_A_y_mono (r:=a2c) A2_hasDerivAt_public (fun t ht => a2_nonneg ht)
+      phi_nonneg (le_trans theta_le_eta (le_trans eta_le_tau tau_le_T)))
+  have h3 := transfer_mono_A (w:=(0,1)) (fun t ht => A_eq_phase3 ht)
+    (phase_A_y_mono (r:=a3c) A3_hasDerivAt_public (fun t ht => a3_nonneg ht)
+      zero_le_theta (le_trans eta_le_tau tau_le_T))
+  have h4 := transfer_mono_A (w:=(0,1)) (fun t ht => A_eq_phase4 ht)
+    (phase_A_y_mono (r:=a4c) A4_hasDerivAt_public (fun t ht => a4_nonneg ht)
+      (le_trans zero_le_theta theta_le_eta) tau_le_T)
+  have h5 := transfer_mono_A (w:=(0,1)) (fun t ht => A_eq_phase5 ht)
+    (phase_A_y_mono (r:=a5c) A5_hasDerivAt_public (fun t ht => a5_nonneg t)
+      (le_trans (le_trans zero_le_theta theta_le_eta) eta_le_tau) (by rfl))
+  have h12 := glue_mono phi_nonneg switchOrder.phi_le_theta h1 h2
+  have h123 := glue_mono zero_le_theta theta_le_eta h12 h3
+  have h1234 := glue_mono (le_trans zero_le_theta theta_le_eta) eta_le_tau h123 h4
+  exact glue_mono
+    (le_trans (le_trans zero_le_theta theta_le_eta) eta_le_tau) tau_le_T h1234 h5
+
+private theorem C_y_global_anti :
+    AntitoneOn (fun t => dot (C t) (0, 1)) (Icc (0 : ℝ) T) := by
+  have h1 := transfer_anti_C (w:=(0,1)) (fun t ht => C_eq_phase1 ht)
+    (phase_C_y_anti (r:=c1c) C1_hasDerivAt_public (fun t ht => c1_nonneg t)
+      (by rfl) (le_trans switchOrder.phi_le_theta (le_trans theta_le_eta (le_trans eta_le_tau
+        tau_le_T))))
+  have h2 := transfer_anti_C (w:=(0,1)) (fun t ht => C_eq_phase2 ht)
+    (phase_C_y_anti (r:=c2c) C2_hasDerivAt_public (fun t ht => c2_nonneg ht)
+      phi_nonneg (le_trans theta_le_eta (le_trans eta_le_tau tau_le_T)))
+  have h3 := transfer_anti_C (w:=(0,1)) (fun t ht => C_eq_phase3 ht)
+    (phase_C_y_anti (r:=c3c) C3_hasDerivAt_public (fun t ht => c3_nonneg ht)
+      zero_le_theta (le_trans eta_le_tau tau_le_T))
+  have h4 := transfer_anti_C (w:=(0,1)) (fun t ht => C_eq_phase4 ht)
+    (phase_C_y_anti (r:=c4c) C4_hasDerivAt_public (fun t ht => c4_nonneg ht)
+      (le_trans zero_le_theta theta_le_eta) tau_le_T)
+  have h5 := transfer_anti_C (w:=(0,1)) (fun t ht => C_eq_phase5 ht)
+    (phase_C_y_anti (r:=c5c) C5_hasDerivAt_coeff (fun t ht => c5_nonneg t)
+      (le_trans (le_trans zero_le_theta theta_le_eta) eta_le_tau) (by rfl))
+  have h12 := glue_anti phi_nonneg switchOrder.phi_le_theta h1 h2
+  have h123 := glue_anti zero_le_theta theta_le_eta h12 h3
+  have h1234 := glue_anti (le_trans zero_le_theta theta_le_eta) eta_le_tau h123 h4
+  exact glue_anti
+    (le_trans (le_trans zero_le_theta theta_le_eta) eta_le_tau) tau_le_T h1234 h5
+
+private theorem A_u_left_mono (s : ℝ) (hs : s ∈ Icc (0 : ℝ) T) :
+    MonotoneOn (fun t => dot (A t) (u s)) (Icc (0 : ℝ) s) := by
+  by_cases h1 : s ≤ params.phi
+  · apply transfer_mono_A (w := u s)
+    · intro t ht
+      exact A_eq_phase1 ⟨ht.1, le_trans ht.2 h1⟩
+    · exact phase_A_own_mono (r := a1c) A1_hasDerivAt_public
+        (fun t _ => a1_nonneg t) (by rfl) (by rfl) hs.2
+  have hphi : params.phi ≤ s := le_of_lt (lt_of_not_ge h1)
+  by_cases h2 : s ≤ params.theta
+  · have m1 : MonotoneOn (fun t => dot (A t) (u s)) (Icc (0 : ℝ) params.phi) :=
+      transfer_mono_A (w := u s) (fun t ht => A_eq_phase1 ht)
+        (phase_A_own_mono (r := a1c) A1_hasDerivAt_public
+          (fun t _ => a1_nonneg t) (by rfl) hphi hs.2)
+    have m2 : MonotoneOn (fun t => dot (A t) (u s)) (Icc params.phi s) := by
+      apply transfer_mono_A (w := u s)
+      · intro t ht
+        exact A_eq_phase2 ⟨ht.1, le_trans ht.2 h2⟩
+      · exact phase_A_own_mono (r := a2c) A2_hasDerivAt_public
+          (fun t ht => a2_nonneg ⟨ht.1, lt_of_lt_of_le ht.2 h2⟩)
+          phi_nonneg (by rfl) hs.2
+    exact glue_mono phi_nonneg hphi m1 m2
+  have htheta : params.theta ≤ s := le_of_lt (lt_of_not_ge h2)
+  by_cases h3 : s ≤ eta
+  · have m12 : MonotoneOn (fun t => dot (A t) (u s)) (Icc (0 : ℝ) params.theta) := by
+      have m1 := transfer_mono_A (w := u s) (fun t ht => A_eq_phase1 ht)
+        (phase_A_own_mono (r := a1c) A1_hasDerivAt_public
+          (fun t _ => a1_nonneg t) (by rfl) (le_trans switchOrder.phi_le_theta htheta) hs.2)
+      have m2 := transfer_mono_A (w := u s) (fun t ht => A_eq_phase2 ht)
+        (phase_A_own_mono (r := a2c) A2_hasDerivAt_public
+          (fun t ht => a2_nonneg ht) phi_nonneg htheta hs.2)
+      exact glue_mono phi_nonneg switchOrder.phi_le_theta m1 m2
+    have m3 : MonotoneOn (fun t => dot (A t) (u s)) (Icc params.theta s) := by
+      apply transfer_mono_A (w := u s)
+      · intro t ht
+        exact A_eq_phase3 ⟨ht.1, le_trans ht.2 h3⟩
+      · exact phase_A_own_mono (r := a3c) A3_hasDerivAt_public
+          (fun t ht => a3_nonneg ⟨ht.1, lt_of_lt_of_le ht.2 h3⟩)
+          zero_le_theta (by rfl) hs.2
+    exact glue_mono zero_le_theta htheta m12 m3
+  have heta : eta ≤ s := le_of_lt (lt_of_not_ge h3)
+  by_cases h4 : s ≤ tau
+  · have m123 : MonotoneOn (fun t => dot (A t) (u s)) (Icc (0 : ℝ) eta) := by
+      have m1 := transfer_mono_A (w := u s) (fun t ht => A_eq_phase1 ht)
+        (phase_A_own_mono (r := a1c) A1_hasDerivAt_public
+          (fun t _ => a1_nonneg t) (by rfl)
+          (le_trans switchOrder.phi_le_theta (le_trans theta_le_eta heta)) hs.2)
+      have m2 := transfer_mono_A (w := u s) (fun t ht => A_eq_phase2 ht)
+        (phase_A_own_mono (r := a2c) A2_hasDerivAt_public
+          (fun t ht => a2_nonneg ht) phi_nonneg (le_trans theta_le_eta heta) hs.2)
+      have m3 := transfer_mono_A (w := u s) (fun t ht => A_eq_phase3 ht)
+        (phase_A_own_mono (r := a3c) A3_hasDerivAt_public
+          (fun t ht => a3_nonneg ht) zero_le_theta heta hs.2)
+      have m12 := glue_mono phi_nonneg switchOrder.phi_le_theta m1 m2
+      exact glue_mono zero_le_theta theta_le_eta m12 m3
+    have m4 : MonotoneOn (fun t => dot (A t) (u s)) (Icc eta s) := by
+      apply transfer_mono_A (w := u s)
+      · intro t ht
+        exact A_eq_phase4 ⟨ht.1, le_trans ht.2 h4⟩
+      · exact phase_A_own_mono (r := a4c) A4_hasDerivAt_public
+          (fun t ht => a4_nonneg ⟨ht.1, lt_of_lt_of_le ht.2 h4⟩)
+          (le_trans zero_le_theta theta_le_eta) (by rfl) hs.2
+    exact glue_mono (le_trans zero_le_theta theta_le_eta) heta m123 m4
+  have htau : tau ≤ s := le_of_lt (lt_of_not_ge h4)
+  have mpre : MonotoneOn (fun t => dot (A t) (u s)) (Icc (0 : ℝ) tau) := by
+    have m1 := transfer_mono_A (w := u s) (fun t ht => A_eq_phase1 ht)
+      (phase_A_own_mono (r := a1c) A1_hasDerivAt_public
+        (fun t _ => a1_nonneg t) (by rfl)
+        (le_trans switchOrder.phi_le_theta (le_trans theta_le_eta (le_trans eta_le_tau htau))) hs.2)
+    have m2 := transfer_mono_A (w := u s) (fun t ht => A_eq_phase2 ht)
+      (phase_A_own_mono (r := a2c) A2_hasDerivAt_public
+        (fun t ht => a2_nonneg ht) phi_nonneg
+        (le_trans theta_le_eta (le_trans eta_le_tau htau)) hs.2)
+    have m3 := transfer_mono_A (w := u s) (fun t ht => A_eq_phase3 ht)
+      (phase_A_own_mono (r := a3c) A3_hasDerivAt_public
+        (fun t ht => a3_nonneg ht) zero_le_theta (le_trans eta_le_tau htau) hs.2)
+    have m4 := transfer_mono_A (w := u s) (fun t ht => A_eq_phase4 ht)
+      (phase_A_own_mono (r := a4c) A4_hasDerivAt_public
+        (fun t ht => a4_nonneg ht) (le_trans zero_le_theta theta_le_eta) htau hs.2)
+    have m12 := glue_mono phi_nonneg switchOrder.phi_le_theta m1 m2
+    have m123 := glue_mono zero_le_theta theta_le_eta m12 m3
+    exact glue_mono (le_trans zero_le_theta theta_le_eta) eta_le_tau m123 m4
+  have m5 : MonotoneOn (fun t => dot (A t) (u s)) (Icc tau s) := by
+    apply transfer_mono_A (w := u s)
+    · intro t ht
+      exact A_eq_phase5 ⟨ht.1, le_trans ht.2 hs.2⟩
+    · exact phase_A_own_mono (r := a5c) A5_hasDerivAt_public
+        (fun t _ => a5_nonneg t) (le_trans (le_trans zero_le_theta theta_le_eta) eta_le_tau)
+        (by rfl) hs.2
+  exact glue_mono (le_trans (le_trans zero_le_theta theta_le_eta) eta_le_tau) htau mpre m5
+
+private theorem A_u_right_anti (s : ℝ) (hs : s ∈ Icc (0 : ℝ) T) :
+    AntitoneOn (fun t => dot (A t) (u s)) (Icc s T) := by
+  by_cases h1 : s ≤ params.phi
+  · have m1 : AntitoneOn (fun t => dot (A t) (u s)) (Icc s params.phi) := by
+      apply transfer_anti_A (w := u s)
+      · intro t ht
+        exact A_eq_phase1 ⟨le_trans hs.1 ht.1, ht.2⟩
+      · exact phase_A_own_anti (r := a1c) A1_hasDerivAt_public
+          (fun t _ => a1_nonneg t) hs.1 (by rfl)
+          (le_trans switchOrder.phi_le_theta (le_trans theta_le_eta (le_trans eta_le_tau tau_le_T)))
+    have m2 := transfer_anti_A (w := u s) (fun t ht => A_eq_phase2 ht)
+      (phase_A_own_anti (r := a2c) A2_hasDerivAt_public
+        (fun t ht => a2_nonneg ht) hs.1 h1
+        (le_trans theta_le_eta (le_trans eta_le_tau tau_le_T)))
+    have m3 := transfer_anti_A (w := u s) (fun t ht => A_eq_phase3 ht)
+      (phase_A_own_anti (r := a3c) A3_hasDerivAt_public
+        (fun t ht => a3_nonneg ht) hs.1 (le_trans h1 switchOrder.phi_le_theta)
+        (le_trans eta_le_tau tau_le_T))
+    have m4 := transfer_anti_A (w := u s) (fun t ht => A_eq_phase4 ht)
+      (phase_A_own_anti (r := a4c) A4_hasDerivAt_public
+        (fun t ht => a4_nonneg ht) hs.1
+        (le_trans (le_trans h1 switchOrder.phi_le_theta) theta_le_eta) tau_le_T)
+    have m5 := transfer_anti_A (w := u s) (fun t ht => A_eq_phase5 ht)
+      (phase_A_own_anti (r := a5c) A5_hasDerivAt_public
+        (fun t _ => a5_nonneg t) hs.1
+        (le_trans (le_trans (le_trans h1 switchOrder.phi_le_theta) theta_le_eta) eta_le_tau)
+        (by rfl))
+    have m23 := glue_anti switchOrder.phi_le_theta theta_le_eta m2 m3
+    have m234 := glue_anti (le_trans switchOrder.phi_le_theta theta_le_eta) eta_le_tau m23 m4
+    have m2345 := glue_anti
+      (le_trans (le_trans switchOrder.phi_le_theta theta_le_eta) eta_le_tau) tau_le_T m234 m5
+    exact glue_anti h1
+      (le_trans (le_trans (le_trans switchOrder.phi_le_theta theta_le_eta) eta_le_tau) tau_le_T)
+      m1 m2345
+  have hphi : params.phi ≤ s := le_of_lt (lt_of_not_ge h1)
+  by_cases h2 : s ≤ params.theta
+  · have m2 : AntitoneOn (fun t => dot (A t) (u s)) (Icc s params.theta) := by
+      apply transfer_anti_A (w := u s)
+      · intro t ht
+        exact A_eq_phase2 ⟨le_trans hphi ht.1, ht.2⟩
+      · exact phase_A_own_anti (r := a2c) A2_hasDerivAt_public
+          (fun t ht => a2_nonneg ⟨lt_of_le_of_lt hphi ht.1, ht.2⟩)
+          hs.1 (by rfl) (le_trans theta_le_eta (le_trans eta_le_tau tau_le_T))
+    have m3 := transfer_anti_A (w := u s) (fun t ht => A_eq_phase3 ht)
+      (phase_A_own_anti (r := a3c) A3_hasDerivAt_public
+        (fun t ht => a3_nonneg ht) hs.1 h2 (le_trans eta_le_tau tau_le_T))
+    have m4 := transfer_anti_A (w := u s) (fun t ht => A_eq_phase4 ht)
+      (phase_A_own_anti (r := a4c) A4_hasDerivAt_public
+        (fun t ht => a4_nonneg ht) hs.1 (le_trans h2 theta_le_eta) tau_le_T)
+    have m5 := transfer_anti_A (w := u s) (fun t ht => A_eq_phase5 ht)
+      (phase_A_own_anti (r := a5c) A5_hasDerivAt_public
+        (fun t _ => a5_nonneg t) hs.1 (le_trans (le_trans h2 theta_le_eta) eta_le_tau) (by rfl))
+    have m34 := glue_anti theta_le_eta eta_le_tau m3 m4
+    have m345 := glue_anti (le_trans theta_le_eta eta_le_tau) tau_le_T m34 m5
+    exact glue_anti h2
+      (le_trans (le_trans theta_le_eta eta_le_tau) tau_le_T) m2 m345
+  have htheta : params.theta ≤ s := le_of_lt (lt_of_not_ge h2)
+  by_cases h3 : s ≤ eta
+  · have m3 : AntitoneOn (fun t => dot (A t) (u s)) (Icc s eta) := by
+      apply transfer_anti_A (w := u s)
+      · intro t ht
+        exact A_eq_phase3 ⟨le_trans htheta ht.1, ht.2⟩
+      · exact phase_A_own_anti (r := a3c) A3_hasDerivAt_public
+          (fun t ht => a3_nonneg ⟨lt_of_le_of_lt htheta ht.1, ht.2⟩)
+          hs.1 (by rfl) (le_trans eta_le_tau tau_le_T)
+    have m4 := transfer_anti_A (w := u s) (fun t ht => A_eq_phase4 ht)
+      (phase_A_own_anti (r := a4c) A4_hasDerivAt_public
+        (fun t ht => a4_nonneg ht) hs.1 h3 tau_le_T)
+    have m5 := transfer_anti_A (w := u s) (fun t ht => A_eq_phase5 ht)
+      (phase_A_own_anti (r := a5c) A5_hasDerivAt_public
+        (fun t _ => a5_nonneg t) hs.1 (le_trans h3 eta_le_tau) (by rfl))
+    have m45 := glue_anti eta_le_tau tau_le_T m4 m5
+    exact glue_anti h3 (le_trans eta_le_tau tau_le_T) m3 m45
+  have heta : eta ≤ s := le_of_lt (lt_of_not_ge h3)
+  by_cases h4 : s ≤ tau
+  · have m4 : AntitoneOn (fun t => dot (A t) (u s)) (Icc s tau) := by
+      apply transfer_anti_A (w := u s)
+      · intro t ht
+        exact A_eq_phase4 ⟨le_trans heta ht.1, ht.2⟩
+      · exact phase_A_own_anti (r := a4c) A4_hasDerivAt_public
+          (fun t ht => a4_nonneg ⟨lt_of_le_of_lt heta ht.1, ht.2⟩)
+          hs.1 (by rfl) tau_le_T
+    have m5 := transfer_anti_A (w := u s) (fun t ht => A_eq_phase5 ht)
+      (phase_A_own_anti (r := a5c) A5_hasDerivAt_public
+        (fun t _ => a5_nonneg t) hs.1 h4 (by rfl))
+    exact glue_anti h4 tau_le_T m4 m5
+  have htau : tau ≤ s := le_of_lt (lt_of_not_ge h4)
+  apply transfer_anti_A (w := u s)
+  · intro t ht
+    exact A_eq_phase5 ⟨le_trans htau ht.1, ht.2⟩
+  · exact phase_A_own_anti (r := a5c) A5_hasDerivAt_public
+      (fun t _ => a5_nonneg t) hs.1 (by rfl) (by rfl)
+
+private theorem C_v_left_mono (s : ℝ) (hs : s ∈ Icc (0 : ℝ) T) :
+    MonotoneOn (fun t => dot (C t) (v s)) (Icc (0 : ℝ) s) := by
+  by_cases h1 : s ≤ params.phi
+  · apply transfer_mono_C (w := v s)
+    · intro t ht
+      exact C_eq_phase1 ⟨ht.1, le_trans ht.2 h1⟩
+    · exact phase_C_own_mono (r := c1c) C1_hasDerivAt_public
+        (fun t _ => c1_nonneg t) (by rfl) (by rfl) hs.2
+  have hphi : params.phi ≤ s := le_of_lt (lt_of_not_ge h1)
+  by_cases h2 : s ≤ params.theta
+  · have m1 := transfer_mono_C (w := v s) (fun t ht => C_eq_phase1 ht)
+      (phase_C_own_mono (r := c1c) C1_hasDerivAt_public
+        (fun t _ => c1_nonneg t) (by rfl) hphi hs.2)
+    have m2 : MonotoneOn (fun t => dot (C t) (v s)) (Icc params.phi s) := by
+      apply transfer_mono_C (w := v s)
+      · intro t ht
+        exact C_eq_phase2 ⟨ht.1, le_trans ht.2 h2⟩
+      · exact phase_C_own_mono (r := c2c) C2_hasDerivAt_public
+          (fun t ht => c2_nonneg ⟨ht.1, lt_of_lt_of_le ht.2 h2⟩)
+          phi_nonneg (by rfl) hs.2
+    exact glue_mono phi_nonneg hphi m1 m2
+  have htheta : params.theta ≤ s := le_of_lt (lt_of_not_ge h2)
+  by_cases h3 : s ≤ eta
+  · have m12 : MonotoneOn (fun t => dot (C t) (v s)) (Icc (0 : ℝ) params.theta) := by
+      have m1 := transfer_mono_C (w := v s) (fun t ht => C_eq_phase1 ht)
+        (phase_C_own_mono (r := c1c) C1_hasDerivAt_public
+          (fun t _ => c1_nonneg t) (by rfl) (le_trans switchOrder.phi_le_theta htheta) hs.2)
+      have m2 := transfer_mono_C (w := v s) (fun t ht => C_eq_phase2 ht)
+        (phase_C_own_mono (r := c2c) C2_hasDerivAt_public
+          (fun t ht => c2_nonneg ht) phi_nonneg htheta hs.2)
+      exact glue_mono phi_nonneg switchOrder.phi_le_theta m1 m2
+    have m3 : MonotoneOn (fun t => dot (C t) (v s)) (Icc params.theta s) := by
+      apply transfer_mono_C (w := v s)
+      · intro t ht
+        exact C_eq_phase3 ⟨ht.1, le_trans ht.2 h3⟩
+      · exact phase_C_own_mono (r := c3c) C3_hasDerivAt_public
+          (fun t ht => c3_nonneg ⟨ht.1, lt_of_lt_of_le ht.2 h3⟩)
+          zero_le_theta (by rfl) hs.2
+    exact glue_mono zero_le_theta htheta m12 m3
+  have heta : eta ≤ s := le_of_lt (lt_of_not_ge h3)
+  by_cases h4 : s ≤ tau
+  · have m123 : MonotoneOn (fun t => dot (C t) (v s)) (Icc (0 : ℝ) eta) := by
+      have m1 := transfer_mono_C (w := v s) (fun t ht => C_eq_phase1 ht)
+        (phase_C_own_mono (r := c1c) C1_hasDerivAt_public
+          (fun t _ => c1_nonneg t) (by rfl)
+          (le_trans switchOrder.phi_le_theta (le_trans theta_le_eta heta)) hs.2)
+      have m2 := transfer_mono_C (w := v s) (fun t ht => C_eq_phase2 ht)
+        (phase_C_own_mono (r := c2c) C2_hasDerivAt_public
+          (fun t ht => c2_nonneg ht) phi_nonneg (le_trans theta_le_eta heta) hs.2)
+      have m3 := transfer_mono_C (w := v s) (fun t ht => C_eq_phase3 ht)
+        (phase_C_own_mono (r := c3c) C3_hasDerivAt_public
+          (fun t ht => c3_nonneg ht) zero_le_theta heta hs.2)
+      have m12 := glue_mono phi_nonneg switchOrder.phi_le_theta m1 m2
+      exact glue_mono zero_le_theta theta_le_eta m12 m3
+    have m4 : MonotoneOn (fun t => dot (C t) (v s)) (Icc eta s) := by
+      apply transfer_mono_C (w := v s)
+      · intro t ht
+        exact C_eq_phase4 ⟨ht.1, le_trans ht.2 h4⟩
+      · exact phase_C_own_mono (r := c4c) C4_hasDerivAt_public
+          (fun t ht => c4_nonneg ⟨ht.1, lt_of_lt_of_le ht.2 h4⟩)
+          (le_trans zero_le_theta theta_le_eta) (by rfl) hs.2
+    exact glue_mono (le_trans zero_le_theta theta_le_eta) heta m123 m4
+  have htau : tau ≤ s := le_of_lt (lt_of_not_ge h4)
+  have mpre : MonotoneOn (fun t => dot (C t) (v s)) (Icc (0 : ℝ) tau) := by
+    have m1 := transfer_mono_C (w := v s) (fun t ht => C_eq_phase1 ht)
+      (phase_C_own_mono (r := c1c) C1_hasDerivAt_public
+        (fun t _ => c1_nonneg t) (by rfl)
+        (le_trans switchOrder.phi_le_theta (le_trans theta_le_eta (le_trans eta_le_tau htau))) hs.2)
+    have m2 := transfer_mono_C (w := v s) (fun t ht => C_eq_phase2 ht)
+      (phase_C_own_mono (r := c2c) C2_hasDerivAt_public
+        (fun t ht => c2_nonneg ht) phi_nonneg
+        (le_trans theta_le_eta (le_trans eta_le_tau htau)) hs.2)
+    have m3 := transfer_mono_C (w := v s) (fun t ht => C_eq_phase3 ht)
+      (phase_C_own_mono (r := c3c) C3_hasDerivAt_public
+        (fun t ht => c3_nonneg ht) zero_le_theta (le_trans eta_le_tau htau) hs.2)
+    have m4 := transfer_mono_C (w := v s) (fun t ht => C_eq_phase4 ht)
+      (phase_C_own_mono (r := c4c) C4_hasDerivAt_public
+        (fun t ht => c4_nonneg ht) (le_trans zero_le_theta theta_le_eta) htau hs.2)
+    have m12 := glue_mono phi_nonneg switchOrder.phi_le_theta m1 m2
+    have m123 := glue_mono zero_le_theta theta_le_eta m12 m3
+    exact glue_mono (le_trans zero_le_theta theta_le_eta) eta_le_tau m123 m4
+  have m5 : MonotoneOn (fun t => dot (C t) (v s)) (Icc tau s) := by
+    apply transfer_mono_C (w := v s)
+    · intro t ht
+      exact C_eq_phase5 ⟨ht.1, le_trans ht.2 hs.2⟩
+    · exact phase_C_own_mono (r := c5c) C5_hasDerivAt_coeff
+        (fun t _ => c5_nonneg t) (le_trans (le_trans zero_le_theta theta_le_eta) eta_le_tau)
+        (by rfl) hs.2
+  exact glue_mono (le_trans (le_trans zero_le_theta theta_le_eta) eta_le_tau) htau mpre m5
+
+private theorem C_v_right_anti (s : ℝ) (hs : s ∈ Icc (0 : ℝ) T) :
+    AntitoneOn (fun t => dot (C t) (v s)) (Icc s T) := by
+  by_cases h1 : s ≤ params.phi
+  · have m1 : AntitoneOn (fun t => dot (C t) (v s)) (Icc s params.phi) := by
+      apply transfer_anti_C (w := v s)
+      · intro t ht
+        exact C_eq_phase1 ⟨le_trans hs.1 ht.1, ht.2⟩
+      · exact phase_C_own_anti (r := c1c) C1_hasDerivAt_public
+          (fun t _ => c1_nonneg t) hs.1 (by rfl)
+          (le_trans switchOrder.phi_le_theta (le_trans theta_le_eta (le_trans eta_le_tau tau_le_T)))
+    have m2 := transfer_anti_C (w := v s) (fun t ht => C_eq_phase2 ht)
+      (phase_C_own_anti (r := c2c) C2_hasDerivAt_public
+        (fun t ht => c2_nonneg ht) hs.1 h1
+        (le_trans theta_le_eta (le_trans eta_le_tau tau_le_T)))
+    have m3 := transfer_anti_C (w := v s) (fun t ht => C_eq_phase3 ht)
+      (phase_C_own_anti (r := c3c) C3_hasDerivAt_public
+        (fun t ht => c3_nonneg ht) hs.1 (le_trans h1 switchOrder.phi_le_theta)
+        (le_trans eta_le_tau tau_le_T))
+    have m4 := transfer_anti_C (w := v s) (fun t ht => C_eq_phase4 ht)
+      (phase_C_own_anti (r := c4c) C4_hasDerivAt_public
+        (fun t ht => c4_nonneg ht) hs.1
+        (le_trans (le_trans h1 switchOrder.phi_le_theta) theta_le_eta) tau_le_T)
+    have m5 := transfer_anti_C (w := v s) (fun t ht => C_eq_phase5 ht)
+      (phase_C_own_anti (r := c5c) C5_hasDerivAt_coeff
+        (fun t _ => c5_nonneg t) hs.1
+        (le_trans (le_trans (le_trans h1 switchOrder.phi_le_theta) theta_le_eta) eta_le_tau)
+        (by rfl))
+    have m23 := glue_anti switchOrder.phi_le_theta theta_le_eta m2 m3
+    have m234 := glue_anti (le_trans switchOrder.phi_le_theta theta_le_eta) eta_le_tau m23 m4
+    have m2345 := glue_anti
+      (le_trans (le_trans switchOrder.phi_le_theta theta_le_eta) eta_le_tau) tau_le_T m234 m5
+    exact glue_anti h1
+      (le_trans (le_trans (le_trans switchOrder.phi_le_theta theta_le_eta) eta_le_tau) tau_le_T)
+      m1 m2345
+  have hphi : params.phi ≤ s := le_of_lt (lt_of_not_ge h1)
+  by_cases h2 : s ≤ params.theta
+  · have m2 : AntitoneOn (fun t => dot (C t) (v s)) (Icc s params.theta) := by
+      apply transfer_anti_C (w := v s)
+      · intro t ht
+        exact C_eq_phase2 ⟨le_trans hphi ht.1, ht.2⟩
+      · exact phase_C_own_anti (r := c2c) C2_hasDerivAt_public
+          (fun t ht => c2_nonneg ⟨lt_of_le_of_lt hphi ht.1, ht.2⟩)
+          hs.1 (by rfl) (le_trans theta_le_eta (le_trans eta_le_tau tau_le_T))
+    have m3 := transfer_anti_C (w := v s) (fun t ht => C_eq_phase3 ht)
+      (phase_C_own_anti (r := c3c) C3_hasDerivAt_public
+        (fun t ht => c3_nonneg ht) hs.1 h2 (le_trans eta_le_tau tau_le_T))
+    have m4 := transfer_anti_C (w := v s) (fun t ht => C_eq_phase4 ht)
+      (phase_C_own_anti (r := c4c) C4_hasDerivAt_public
+        (fun t ht => c4_nonneg ht) hs.1 (le_trans h2 theta_le_eta) tau_le_T)
+    have m5 := transfer_anti_C (w := v s) (fun t ht => C_eq_phase5 ht)
+      (phase_C_own_anti (r := c5c) C5_hasDerivAt_coeff
+        (fun t _ => c5_nonneg t) hs.1 (le_trans (le_trans h2 theta_le_eta) eta_le_tau) (by rfl))
+    have m34 := glue_anti theta_le_eta eta_le_tau m3 m4
+    have m345 := glue_anti (le_trans theta_le_eta eta_le_tau) tau_le_T m34 m5
+    exact glue_anti h2
+      (le_trans (le_trans theta_le_eta eta_le_tau) tau_le_T) m2 m345
+  have htheta : params.theta ≤ s := le_of_lt (lt_of_not_ge h2)
+  by_cases h3 : s ≤ eta
+  · have m3 : AntitoneOn (fun t => dot (C t) (v s)) (Icc s eta) := by
+      apply transfer_anti_C (w := v s)
+      · intro t ht
+        exact C_eq_phase3 ⟨le_trans htheta ht.1, ht.2⟩
+      · exact phase_C_own_anti (r := c3c) C3_hasDerivAt_public
+          (fun t ht => c3_nonneg ⟨lt_of_le_of_lt htheta ht.1, ht.2⟩)
+          hs.1 (by rfl) (le_trans eta_le_tau tau_le_T)
+    have m4 := transfer_anti_C (w := v s) (fun t ht => C_eq_phase4 ht)
+      (phase_C_own_anti (r := c4c) C4_hasDerivAt_public
+        (fun t ht => c4_nonneg ht) hs.1 h3 tau_le_T)
+    have m5 := transfer_anti_C (w := v s) (fun t ht => C_eq_phase5 ht)
+      (phase_C_own_anti (r := c5c) C5_hasDerivAt_coeff
+        (fun t _ => c5_nonneg t) hs.1 (le_trans h3 eta_le_tau) (by rfl))
+    have m45 := glue_anti eta_le_tau tau_le_T m4 m5
+    exact glue_anti h3 (le_trans eta_le_tau tau_le_T) m3 m45
+  have heta : eta ≤ s := le_of_lt (lt_of_not_ge h3)
+  by_cases h4 : s ≤ tau
+  · have m4 : AntitoneOn (fun t => dot (C t) (v s)) (Icc s tau) := by
+      apply transfer_anti_C (w := v s)
+      · intro t ht
+        exact C_eq_phase4 ⟨le_trans heta ht.1, ht.2⟩
+      · exact phase_C_own_anti (r := c4c) C4_hasDerivAt_public
+          (fun t ht => c4_nonneg ⟨lt_of_le_of_lt heta ht.1, ht.2⟩)
+          hs.1 (by rfl) tau_le_T
+    have m5 := transfer_anti_C (w := v s) (fun t ht => C_eq_phase5 ht)
+      (phase_C_own_anti (r := c5c) C5_hasDerivAt_coeff
+        (fun t _ => c5_nonneg t) hs.1 h4 (by rfl))
+    exact glue_anti h4 tau_le_T m4 m5
+  have htau : tau ≤ s := le_of_lt (lt_of_not_ge h4)
+  apply transfer_anti_C (w := v s)
+  · intro t ht
+    exact C_eq_phase5 ⟨le_trans htau ht.1, ht.2⟩
+  · exact phase_C_own_anti (r := c5c) C5_hasDerivAt_coeff
+      (fun t _ => c5_nonneg t) hs.1 (by rfl) (by rfl)
+
+theorem A_own_max (s : ℝ) (hs : s ∈ Icc (0 : ℝ) T) :
+    ∀ t ∈ Icc (0 : ℝ) T, dot (A t) (u s) ≤ dot (A s) (u s) := by
+  intro t ht
+  by_cases hts : t ≤ s
+  · exact A_u_left_mono s hs ⟨ht.1, hts⟩ ⟨hs.1, le_rfl⟩ hts
+  · have hst : s ≤ t := le_of_lt (lt_of_not_ge hts)
+    exact A_u_right_anti s hs ⟨le_rfl, hs.2⟩ ⟨hst, ht.2⟩ hst
+
+theorem C_own_max (s : ℝ) (hs : s ∈ Icc (0 : ℝ) T) :
+    ∀ t ∈ Icc (0 : ℝ) T, dot (C t) (v s) ≤ dot (C s) (v s) := by
+  intro t ht
+  by_cases hts : t ≤ s
+  · exact C_v_left_mono s hs ⟨ht.1, hts⟩ ⟨hs.1, le_rfl⟩ hts
+  · have hst : s ≤ t := le_of_lt (lt_of_not_ge hts)
+    exact C_v_right_anti s hs ⟨le_rfl, hs.2⟩ ⟨hst, ht.2⟩ hst
+
+private theorem a1_lower : (6 / 5 : ℝ) ≤ params.a1 := by
+  have h := Romik.a1_lower_bound_of_mem_box params_mem
+  norm_num at h ⊢
+  linarith
+
+private theorem k51_lower : (-11 / 10 : ℝ) ≤ params.k51 := by
+  have hp := params_mem
+  dsimp [Romik.box] at hp
+  have hlo :
+      qR (-20344080735756291713874857283) 20000000000000000000000000000 ≤ params.k51 := by
+    aesop
+  norm_num [qR] at hlo ⊢
+  linarith
+
+private theorem endpointGap_nonneg : 0 ≤ 3 * params.a1 + params.k51 - 1 := by
+  nlinarith [a1_lower, k51_lower]
+
+private theorem path5_fst_formula (p : Romik.Params) (t : ℝ) :
+    (Romik.path5 p t).1 =
+      Real.cos t * (p.e1 * Real.cos t + p.e2 * Real.sin t - 1 / 2) -
+        Real.sin t * (-p.e2 * Real.cos t + p.e1 * Real.sin t - 1) + p.k51 := by
+  rfl
+
+private theorem C_zero_formula : C 0 = (1 - 2 * params.a1, 1) := by
+  have hphi : (0 : ℝ) ≤ params.phi := phi_nonneg
+  have ha2 := Romik.a2_eq_neg_quarter_of_equations params_equations
+  change
+    ((Romik.path params 0).1 - beta 0 * (u 0).1 + (v 0).1,
+     (Romik.path params 0).2 - beta 0 * (u 0).2 + (v 0).2) =
+      (1 - 2 * params.a1, 1)
+  rw [pathZero]
+  apply Prod.ext <;> simp [beta, alphaBetaAt, hphi, Romik.alphaBeta1, u, v, ha2]
+
+private theorem A_T_formula : A T = (params.a1 + params.k51, 1) := by
+  have he1 := Romik.e1_eq_a1_of_equations params_equations
+  have he2 := Romik.e2_eq_quarter_of_equations params_equations
+  have hphase : A T = phaseA5 T := A_eq_phase5 ⟨tau_le_T, le_rfl⟩
+  rw [hphase]
+  apply Prod.ext
+  · simp [phaseA5, path5_fst_formula, Romik.alphaBeta5, u, v, T, he1, he2]; ring
+  · have hpath : Romik.path params T = Romik.path5 params T := by
+      simpa [T] using Romik.path_halfPi_eq_path5_of_mem_box params_mem
+    have hy : (Romik.path5 params T).2 = 0 := by
+      rw [← hpath]
+      exact pathEndYZero
+    have hyHalfPi : (Romik.path5 params (Real.pi / 2)).2 = 0 := by
+      simpa [T] using hy
+    simp [phaseA5, hyHalfPi, Romik.alphaBeta5, u, v, T, he1, he2]
+
+private theorem endpoint_relation : A T = C 0 + (3 * params.a1 + params.k51 - 1) • (1, 0) := by
+  rw [A_T_formula, C_zero_formula]
+  apply Prod.ext <;> simp; ring
+
+private theorem A_T_v_le_C_zero_v (s : ℝ) (hs : s ∈ Icc (0 : ℝ) T) :
+    dot (A T) (v s) ≤ dot (C 0) (v s) := by
+  rw [endpoint_relation]
+  have hsin : 0 ≤ Real.sin s := Real.sin_nonneg_of_nonneg_of_le_pi hs.1 (le_trans hs.2 T_le_pi)
+  simp [dot, v]
+  nlinarith [endpointGap_nonneg]
+
+private theorem C_zero_u_le_A_T_u (s : ℝ) (hs : s ∈ Icc (0 : ℝ) T) :
+    dot (C 0) (u s) ≤ dot (A T) (u s) := by
+  rw [endpoint_relation]
+  rcases hs with ⟨hs0, hsT⟩
+  have hc : 0 ≤ Real.cos s := by
+    apply Real.cos_nonneg_of_neg_pi_div_two_le_of_le
+    · dsimp [T] at hsT ⊢
+      nlinarith [Real.pi_pos]
+    · simpa [T] using hsT
+  simp [dot, u]
+  nlinarith [endpointGap_nonneg]
+
+theorem A_base_nonneg : ∀ t ∈ Icc (0 : ℝ) T, 0 ≤ (A t).2 := by
+  intro t ht
+  have hm := A_y_global_mono ⟨le_rfl, T_nonneg⟩ ht ht.1
+  have h0 : dot (A 0) (0, 1) = 0 := by
+    rw [A_zero_eq_anchor]
+    simp [dot, anchor]
+  have ht' : dot (A t) (0, 1) = (A t).2 := by simp [dot]
+  change dot (A 0) (0, 1) ≤ dot (A t) (0, 1) at hm
+  rw [h0, ht'] at hm
+  exact hm
+
+theorem C_base_nonneg : ∀ t ∈ Icc (0 : ℝ) T, 0 ≤ (C t).2 := by
+  intro t ht
+  have hm := C_y_global_anti ht ⟨T_nonneg, le_rfl⟩ ht.2
+  have hT : dot (C T) (0, 1) = 0 := by
+    simp [dot, C_T_snd_zero]
+  have ht' : dot (C t) (0, 1) = (C t).2 := by simp [dot]
+  change dot (C T) (0, 1) ≤ dot (C t) (0, 1) at hm
+  rw [hT, ht'] at hm
+  exact hm
+
+theorem supportA_direct : ∀ t ∈ Icc (0 : ℝ) T, A t ∈ K := by
+  intro t ht
+  rw [Romik.mem_K0]
+  refine ⟨A_base_nonneg t ht, ?_⟩
+  intro s hs
+  have hsT : s ∈ Icc (0 : ℝ) T := by simpa [T] using hs
+  constructor
+  · change dot (A t) (u s) ≤ dot (Romik.path params s) (u s) + 1
+    have hmax := A_own_max s hsT t ht
+    rw [A_support_identity] at hmax
+    exact hmax
+  · change dot (A t) (v s) ≤ dot (Romik.path params s) (v s) + 1
+    have hmono := A_v_global_mono s hsT ht ⟨T_nonneg, le_rfl⟩ ht.2
+    have hsep := A_T_v_le_C_zero_v s hsT
+    have hmax := C_own_max s hsT 0 ⟨le_rfl, T_nonneg⟩
+    have hid := C_support_identity s
+    linarith
+
+theorem supportC_direct : ∀ t ∈ Icc (0 : ℝ) T, C t ∈ K := by
+  intro t ht
+  rw [Romik.mem_K0]
+  refine ⟨C_base_nonneg t ht, ?_⟩
+  intro s hs
+  have hsT : s ∈ Icc (0 : ℝ) T := by simpa [T] using hs
+  constructor
+  · change dot (C t) (u s) ≤ dot (Romik.path params s) (u s) + 1
+    have hanti := C_u_global_anti s hsT ⟨le_rfl, T_nonneg⟩ ht ht.1
+    have hsep := C_zero_u_le_A_T_u s hsT
+    have hmax := A_own_max s hsT T ⟨T_nonneg, le_rfl⟩
+    have hid := A_support_identity s
+    linarith
+  · change dot (C t) (v s) ≤ dot (Romik.path params s) (v s) + 1
+    have hmax := C_own_max s hsT t ht
+    rw [C_support_identity] at hmax
+    exact hmax
+
+end Stage3
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C direct final assembly
+
+This module bypasses the earlier conditional calculus/branch certificate layers.
+The two outer-support fields are now concrete theorems.  What remains here is
+exactly the six substantive topology/no-hidden facts; no additional certificate
+wrapper is introduced.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage3
+
+/-- Direct global geometry from the concrete support proof and the six exact
+remaining topology/no-hidden statements. -/
+theorem globalGeometryDirect
+    (hU : NoHiddenCrossingU)
+    (hV : NoHiddenCrossingV)
+    (hboundary : frontier (Romik.niche params) = claimedNicheBoundary)
+    (hniche : IsConnected (Romik.niche params))
+    (hanchor : anchor ∈ G)
+    (hG : IsConnected G) : GlobalGeometryCertificate :=
+  { supportA := supportA_direct
+    supportC := supportC_direct
+    nicheTopology :=
+      { noHiddenU := hU
+        noHiddenV := hV
+        boundary_eq := hboundary
+        niche_connected := hniche }
+    topology :=
+      { anchor_mem := hanchor
+        connected := hG } }
+
+end Stage3
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 4: public path differential layer
+
+Source-clean phase derivatives of the five Gerver path pieces.  These are
+provided here as compatibility wrappers around the shared Stage 2 proofs.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+theorem path1_hasDerivAt_public (t : ℝ) :
+    HasDerivAt (Romik.path1 params)
+      (Romik.rot t (Romik.alphaBeta1 params t)) t :=
+  Stage2.path1_hasDerivAt t
+
+theorem path2_hasDerivAt_public (t : ℝ) :
+    HasDerivAt (Romik.path2 params)
+      (Romik.rot t (Romik.alphaBeta2 params t)) t :=
+  Stage2.path2_hasDerivAt t
+
+theorem path3_hasDerivAt_public (t : ℝ) :
+    HasDerivAt (Romik.path3 params)
+      (Romik.rot t (Romik.alphaBeta3 params t)) t :=
+  Stage2.path3_hasDerivAt t
+
+theorem path4_hasDerivAt_public (t : ℝ) :
+    HasDerivAt (Romik.path4 params)
+      (Romik.rot t (Romik.alphaBeta4 params t)) t :=
+  Stage2.path4_hasDerivAt t
+
+theorem path5_hasDerivAt_public (t : ℝ) :
+    HasDerivAt (Romik.path5 params)
+      (Romik.rot t (Romik.alphaBeta5 params t)) t :=
+  Stage2.path5_hasDerivAt t
+
+/-! The five public phase derivative theorems above are the complete differential
+interface used by Stage 4.  We intentionally do not assert a global `HasDerivAt`
+for the nested-if path at switching times: such a theorem requires a separate
+matching-of-derivatives argument and is neither needed nor used by the Part C
+closure. -/
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 4: concrete geometry facts
+
+C21: source-clean root closure.  In particular, branch decisions for the
+literal nested-if path are made before endpoint abbreviations are unfolded.
+The core-path nonnegativity statement is transported from the already sound
+Part B cell enclosure instead of being reproved by a large transcendental
+`nlinarith` call.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Set
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Stage2
+
+private theorem T_nonneg : (0 : ℝ) ≤ T := by
+  dsimp [T]
+  positivity
+
+private theorem T_le_pi : T ≤ Real.pi := by
+  dsimp [T]
+  nlinarith [Real.pi_pos]
+
+private theorem rat_nonneg_cast {q : ℚ} (h : 0 ≤ q) : (0 : ℝ) ≤ (q : ℝ) := by
+  exact_mod_cast h
+
+private theorem rat_pos_cast {q : ℚ} (h : 0 < q) : (0 : ℝ) < (q : ℝ) := by
+  exact_mod_cast h
+
+theorem phi_pos : 0 < params.phi := Romik.phi_pos_of_mem_box params_mem
+
+theorem phi_lt_theta : params.phi < params.theta := by
+  have hphi := phi_bounds.2
+  have htheta := theta_bounds.1
+  norm_num at hphi htheta ⊢
+  linarith
+
+theorem theta_pos : 0 < params.theta := lt_trans phi_pos phi_lt_theta
+
+theorem theta_lt_eta : params.theta < eta := by
+  have htheta := theta_bounds.2
+  norm_num at htheta
+  dsimp [eta, T]
+  nlinarith [Real.pi_gt_three]
+
+theorem eta_lt_tau : eta < tau := by
+  dsimp [eta, tau]
+  linarith [phi_lt_theta]
+
+theorem eta_lt_T : eta < T := by
+  dsimp [eta]
+  linarith [theta_pos]
+
+theorem tau_lt_T : tau < T := by
+  dsimp [tau]
+  linarith [phi_pos]
+
+theorem theta_lt_T : params.theta < T := lt_trans theta_lt_eta eta_lt_T
+
+theorem phi_lt_T : params.phi < T := lt_trans phi_lt_theta theta_lt_T
+
+theorem phi_lt_eta : params.phi < eta := lt_trans phi_lt_theta theta_lt_eta
+
+theorem theta_lt_tau : params.theta < tau :=
+  lt_trans theta_lt_eta eta_lt_tau
+
+private theorem path_at_phi :
+    Romik.path params params.phi = Romik.path1 params params.phi := by
+  simp only [Romik.path, ite_eq_left (le_refl params.phi)]
+
+private theorem path_at_theta :
+    Romik.path params params.theta = Romik.path2 params params.theta := by
+  simp only [Romik.path, ite_eq_right (not_le.mpr phi_lt_theta),
+    ite_eq_left (le_refl params.theta)]
+
+private theorem path_at_eta :
+    Romik.path params eta = Romik.path3 params eta := by
+  have hraw : eta ≤ Real.pi / 2 - params.theta := by
+    rfl
+  simp only [Romik.path, ite_eq_right (not_le.mpr phi_lt_eta),
+    ite_eq_right (not_le.mpr theta_lt_eta), ite_eq_left hraw]
+
+private theorem path_at_tau :
+    Romik.path params tau = Romik.path4 params tau := by
+  have hphi : ¬ tau ≤ params.phi := not_le.mpr (lt_trans phi_lt_theta theta_lt_tau)
+  have htheta : ¬ tau ≤ params.theta := not_le.mpr theta_lt_tau
+  have heta : ¬ tau ≤ Real.pi / 2 - params.theta := by
+    simpa [eta, T] using (not_le.mpr eta_lt_tau)
+  have htau : tau ≤ Real.pi / 2 - params.phi := by
+    rfl
+  simp only [Romik.path, ite_eq_right hphi, ite_eq_right htheta, ite_eq_right heta, ite_eq_left
+    htau]
+
+/-! ## C23: minimal exact reflection bridge
+
+C22 tried to replace the whole scalar layer at once and regressed badly.  C23
+rolls back to the stable C21 file and cherry-picks only the reflection bridge
+needed for `D_theta_eq_path_tau`. -/
+
+private def hReflect (z : Point) : Point :=
+  (2 * params.k31 - z.1, z.2)
+
+private theorem hReflect_involutive (z : Point) : hReflect (hReflect z) = z := by
+  ext <;> simp [hReflect]
+
+private theorem phase24_horizontal_reflection (t : ℝ) :
+    (Romik.path4 params (T - t)).1 + (Romik.path2 params t).1 =
+      params.k41 + params.k21 := by
+  have hd1 := Romik.d1_eq_quarterPi_sub_b1_of_equations params_equations
+  have hd2 := Romik.d2_eq_b2_add_quarterPi_correction_of_equations params_equations
+  dsimp [T, Romik.path4, Romik.path2, Romik.rot, Romik.addK]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub, hd1, hd2]
+  ring
+
+private theorem phase3_horizontal_reflection (t : ℝ) :
+    (Romik.path3 params (T - t)).1 + (Romik.path3 params t).1 =
+      2 * params.k31 := by
+  have hc2 := Romik.c2_eq_c1_sub_halfPi_of_equations params_equations
+  dsimp [T, Romik.path3, Romik.rot, Romik.addK]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub, hc2]
+  ring
+
+private theorem k41_add_k21_eq_two_k31 :
+    params.k41 + params.k21 = 2 * params.k31 := by
+  have h24 := phase24_horizontal_reflection params.theta
+  have h33 := phase3_horizontal_reflection params.theta
+  have h23 := congrArg Prod.fst PartB.match23
+  have h34 := congrArg Prod.fst PartB.match34
+  simp only [T] at h24 h33 h34
+  linarith
+
+private theorem phase24_reflect (t : ℝ) :
+    Romik.path4 params (T - t) = hReflect (Romik.path2 params t) := by
+  apply Prod.ext
+  · have h := phase24_horizontal_reflection t
+    rw [k41_add_k21_eq_two_k31] at h
+    dsimp [hReflect]
+    linarith
+  · have h := Romik.phase24_vertical_reflection_of_equations params_equations t
+    have hk := Romik.k42_eq_k22_of_equations params_equations
+    rw [hk] at h
+    have h' : (Romik.path4 params (T - t)).2 = (Romik.path2 params t).2 := by
+      dsimp [T] at ⊢
+      linarith [h]
+    simpa [hReflect] using h'
+
+private theorem phase3_reflect (t : ℝ) :
+    Romik.path3 params (T - t) = hReflect (Romik.path3 params t) := by
+  apply Prod.ext
+  · have h := phase3_horizontal_reflection t
+    dsimp [hReflect]
+    linarith
+  · have h := Romik.phase3_vertical_reflection_of_equations params_equations t
+    simpa [T, hReflect] using h
+
+private theorem rot_dot_u (t : ℝ) (z : Point) :
+    dot (Romik.rot t z) (u t) = z.1 :=
+  Stage3.rot_dot_u t z
+
+private theorem rot_dot_v (t : ℝ) (z : Point) :
+    dot (Romik.rot t z) (v t) = z.2 :=
+  Stage3.rot_dot_v t z
+
+private theorem rot_injective (t : ℝ) : Function.Injective (Romik.rot t) :=
+  Stage3.rot_injective t
+
+private theorem matchPrime23 :
+    Romik.pathPrime2 params params.theta =
+      Romik.pathPrime3 params params.theta := by
+  have h14 := congrFun params_equations (14 : Fin 22)
+  have h15 := congrFun params_equations (15 : Fin 22)
+  simp [Romik.system] at h14 h15
+  apply Prod.ext <;> linarith
+
+private theorem pathPrime2_eq_rot (p : Romik.Params) (t : ℝ) :
+    Romik.pathPrime2 p t = Romik.rot t (Romik.alphaBeta2 p t) := rfl
+
+private theorem pathPrime3_eq_rot (p : Romik.Params) (t : ℝ) :
+    Romik.pathPrime3 p t = Romik.rot t (Romik.alphaBeta3 p t) := rfl
+
+private theorem matchAB23 :
+    Romik.alphaBeta2 params params.theta =
+      Romik.alphaBeta3 params params.theta := by
+  apply rot_injective params.theta
+  simpa [pathPrime2_eq_rot, pathPrime3_eq_rot] using matchPrime23
+
+private def reflAB (z : Point) : Point := (-z.2, -z.1)
+
+private theorem ab3_reflect (t : ℝ) :
+    Romik.alphaBeta3 params (T - t) = reflAB (Romik.alphaBeta3 params t) := by
+  dsimp [T, reflAB, Romik.alphaBeta3]
+  rw [Romik.c2_eq_c1_sub_halfPi_of_equations params_equations]
+  ring_nf
+
+/-- Equation pair 20--21 is exactly the first niche contact. -/
+theorem B_eta_eq_path_phi : B eta = Romik.path params params.phi := by
+  have h20 := congrFun params_equations (20 : Fin 22)
+  have h21 := congrFun params_equations (21 : Fin 22)
+  simp only [Romik.system, one_div, Fin.isValue, Matrix.cons_val, Pi.zero_apply] at h20 h21
+  have h20' :
+      (Romik.path1 params params.phi).1 -
+          ((Romik.path3 params eta).1 -
+            (Romik.alphaBeta3 params eta).1 * Real.sin eta) = 0 := by
+    simpa [eta, T] using h20
+  have h21' :
+      (Romik.path1 params params.phi).2 -
+          ((Romik.path3 params eta).2 +
+            (Romik.alphaBeta3 params eta).1 * Real.cos eta) = 0 := by
+    simpa [eta, T] using h21
+  rw [path_at_phi]
+  apply Prod.ext
+  · change (Romik.path params eta).1 + alpha eta * (v eta).1 =
+      (Romik.path1 params params.phi).1
+    rw [path_at_eta]
+    have ha : alpha eta = (Romik.alphaBeta3 params eta).1 := by
+      simp only [alpha, alphaBetaAt, ite_eq_right (not_le.mpr phi_lt_eta),
+        ite_eq_right (not_le.mpr theta_lt_eta), ite_eq_left (le_refl eta)]
+    rw [ha]
+    simp only [v]
+    linarith
+  · change (Romik.path params eta).2 + alpha eta * (v eta).2 =
+      (Romik.path1 params params.phi).2
+    rw [path_at_eta]
+    have ha : alpha eta = (Romik.alphaBeta3 params eta).1 := by
+      simp only [alpha, alphaBetaAt, ite_eq_right (not_le.mpr phi_lt_eta),
+        ite_eq_right (not_le.mpr theta_lt_eta), ite_eq_left (le_refl eta)]
+    rw [ha]
+    simp only [v]
+    linarith
+
+/-- Reflected contact at the other end of the core. -/
+theorem D_theta_eq_path_tau : D params.theta = Romik.path params tau := by
+  have hb : beta params.theta = (Romik.alphaBeta2 params params.theta).2 := by
+    simp only [beta, alphaBetaAt, ite_eq_right (not_le.mpr phi_lt_theta),
+      ite_eq_left (le_refl params.theta)]
+  have ha : alpha eta = (Romik.alphaBeta3 params eta).1 := by
+    simp only [alpha, alphaBetaAt, ite_eq_right (not_le.mpr phi_lt_eta),
+      ite_eq_right (not_le.mpr theta_lt_eta), ite_eq_left (le_refl eta)]
+  have h3 : Romik.path3 params eta = hReflect (Romik.path2 params params.theta) := by
+    calc
+      Romik.path3 params eta = hReflect (Romik.path3 params params.theta) := by
+        simpa [eta] using phase3_reflect params.theta
+      _ = hReflect (Romik.path2 params params.theta) := by
+        exact congrArg hReflect PartB.match23.symm
+  have hab0 : (Romik.alphaBeta3 params eta).1 =
+      -(Romik.alphaBeta3 params params.theta).2 := by
+    have h := congrArg Prod.fst (ab3_reflect params.theta)
+    simpa [eta, reflAB] using h
+  have hab : (Romik.alphaBeta3 params eta).1 =
+      -(Romik.alphaBeta2 params params.theta).2 := by
+    calc
+      (Romik.alphaBeta3 params eta).1 =
+          -(Romik.alphaBeta3 params params.theta).2 := hab0
+      _ = -(Romik.alphaBeta2 params params.theta).2 := by
+        rw [← congrArg Prod.snd matchAB23]
+  have hDB : hReflect (D params.theta) = B eta := by
+    rw [D, B, path_at_theta, path_at_eta, hb, ha, h3, hab]
+    apply Prod.ext
+    · simp [hReflect, u, v, eta, T, Real.sin_pi_div_two_sub,
+        Real.cos_pi_div_two_sub]
+      ring
+    · simp [hReflect, u, v, eta, T, Real.sin_pi_div_two_sub,
+        Real.cos_pi_div_two_sub]
+      ring
+  have hTau : Romik.path params tau = hReflect (Romik.path params params.phi) := by
+    rw [path_at_tau, path_at_phi]
+    calc
+      Romik.path4 params tau = hReflect (Romik.path2 params params.phi) := by
+        simpa [tau] using phase24_reflect params.phi
+      _ = hReflect (Romik.path1 params params.phi) := by
+        exact congrArg hReflect PartB.match12.symm
+  have hDB' := congrArg hReflect hDB
+  rw [hReflect_involutive] at hDB'
+  calc
+    D params.theta = hReflect (B eta) := hDB'
+    _ = hReflect (Romik.path params params.phi) := congrArg hReflect B_eta_eq_path_phi
+    _ = Romik.path params tau := hTau.symm
+
+/-- The two base endpoints of the claimed niche boundary lie on `y=0`. -/
+theorem B_T_y_zero : (B T).2 = 0 := by
+  change (Romik.path params T).2 + alpha T * Real.cos T = 0
+  have hEnd : (Romik.path params T).2 = 0 := by
+    simpa [T] using pathEndYZero
+  rw [hEnd]
+  simp [T]
+
+theorem D_zero_y_zero : (D 0).2 = 0 := by
+  change (Romik.path params 0).2 - beta 0 * Real.sin 0 = 0
+  rw [pathZero]
+  norm_num
+
+/-! Exact mesh transport for the literal core path.  Cells 1--62 are uniformly
+above the base.  Cells 0 and 63 are intentionally excluded: their interval
+hulls contain the terminal base contacts and have a tiny negative lower hull. -/
+private theorem core_path_cell_lo_nonneg :
+    ∀ i : PartB.Cell, 1 ≤ i.1 → i.1 ≤ 62 →
+      (0 : ℚ) ≤ (PartB.cellPathInterval i).2.lo := by
+  decide +kernel
+
+/-! C30 strengthens the already kernel-checked cell statement from nonnegativity
+to strict positivity on every core cell.  The computation is the same finite
+exact-rational decision problem; only the target relation is stronger. -/
+private theorem core_path_cell_lo_pos :
+    ∀ i : PartB.Cell, 1 ≤ i.1 → i.1 ≤ 62 →
+      (0 : ℚ) < (PartB.cellPathInterval i).2.lo := by
+  decide +kernel
+private theorem cell1_path_y_lo_pos :
+    (0 : ℚ) < (PartB.cellPathInterval (1 : PartB.Cell)).2.lo := by
+  decide +kernel
+
+/-- The first core contact is strictly above the base. -/
+theorem B_eta_y_pos : 0 < (B eta).2 := by
+  rw [B_eta_eq_path_phi]
+  have hcell : params.phi ∈ PartB.cellSet (1 : PartB.Cell) :=
+    ⟨Stage2.node1_lt_phi.le, Stage2.phi_lt_node2.le⟩
+  have hphys : params.phi ∈ PartB.physicalInterval := by
+    constructor
+    · exact phi_pos.le
+    · simpa [PartB.physicalInterval, T] using phi_lt_T.le
+  have hcontains := PartB.cellPath_contains hcell hphys
+  exact lt_of_lt_of_le (rat_pos_cast cell1_path_y_lo_pos) hcontains.2.1
+
+/-- The reflected core contact has the same strictly positive height. -/
+theorem D_theta_y_pos : 0 < (D params.theta).2 := by
+  rw [D_theta_eq_path_tau, path_at_tau]
+  have h24 := Romik.phase24_vertical_reflection_of_equations params_equations params.phi
+  have h24' :
+      (Romik.path4 params tau).2 - params.k42 =
+        (Romik.path2 params params.phi).2 - params.k22 := by
+    simpa [tau, T] using h24
+  have hk := Romik.k42_eq_k22_of_equations params_equations
+  have hm12y := congrArg Prod.snd PartB.match12
+  have h := B_eta_y_pos
+  rw [B_eta_eq_path_phi, path_at_phi] at h
+  nlinarith
+
+/-- Nonnegativity of the literal Gerver path on the core interval, transported
+from the already proved Part B interval semantics. -/
+theorem path_core_y_nonneg {t : ℝ} (ht : t ∈ Icc params.phi tau) :
+    0 ≤ (Romik.path params t).2 := by
+  have htPhys : t ∈ PartB.physicalInterval := by
+    constructor
+    · exact le_trans phi_pos.le ht.1
+    · have htT : t ≤ T := le_trans ht.2 tau_lt_T.le
+      simpa [PartB.physicalInterval, T] using htT
+  rcases PartB.exists_cell_cover htPhys with ⟨i, hi⟩
+  have hi1 : 1 ≤ i.1 := by
+    by_contra hnot
+    have hend : PartB.nodeTime (i.1 + 1) ≤ PartB.nodeTime 1 :=
+      PartB.nodeTime_mono (by omega)
+    linarith [Stage2.node1_lt_phi, hi.2, ht.1]
+  have hi62 : i.1 ≤ 62 := by
+    by_contra hnot
+    have h63 : 63 ≤ i.1 := by omega
+    have hstart : PartB.nodeTime 63 ≤ PartB.nodeTime i.1 :=
+      PartB.nodeTime_mono h63
+    linarith [Stage2.tau_lt_node63, hstart, hi.1, ht.2]
+  have hcontains := PartB.cellPath_contains hi htPhys
+  exact le_trans (rat_nonneg_cast (core_path_cell_lo_nonneg i hi1 hi62)) hcontains.2.1
+
+/-- Strict positivity of the literal Gerver path height on the complete core
+interval `[phi,tau]`.  C30 transports the exact strict lower hull proved above
+through the existing Part B interval-containment theorem. -/
+theorem path_core_y_pos {t : ℝ} (ht : t ∈ Icc params.phi tau) :
+    0 < (Romik.path params t).2 := by
+  have htPhys : t ∈ PartB.physicalInterval := by
+    constructor
+    · exact le_trans phi_pos.le ht.1
+    · have htT : t ≤ T := le_trans ht.2 tau_lt_T.le
+      simpa [PartB.physicalInterval, T] using htT
+  rcases PartB.exists_cell_cover htPhys with ⟨i, hi⟩
+  have hi1 : 1 ≤ i.1 := by
+    by_contra hnot
+    have hend : PartB.nodeTime (i.1 + 1) ≤ PartB.nodeTime 1 :=
+      PartB.nodeTime_mono (by omega)
+    linarith [Stage2.node1_lt_phi, hi.2, ht.1]
+  have hi62 : i.1 ≤ 62 := by
+    by_contra hnot
+    have h63 : 63 ≤ i.1 := by omega
+    have hstart : PartB.nodeTime 63 ≤ PartB.nodeTime i.1 :=
+      PartB.nodeTime_mono h63
+    linarith [Stage2.tau_lt_node63, hstart, hi.1, ht.2]
+  have hcontains := PartB.cellPath_contains hi htPhys
+  exact lt_of_lt_of_le (rat_pos_cast (core_path_cell_lo_pos i hi1 hi62)) hcontains.2.1
+
+/-! A compact phase-2 lower bound for the reflected contact `D`.  It is also
+used, by the exact phase-2/phase-4 reflection, to control the late `B` arc. -/
+private theorem b1_lower_crude : (-53 / 100 : ℝ) ≤ params.b1 := by
+  have hb := PartB.b1_contains.1
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at hb ⊢
+  linarith
+
+private theorem theta_upper_crude : params.theta ≤ (7 / 10 : ℝ) := by
+  have h := theta_bounds.2
+  norm_num at h ⊢
+  linarith
+
+private theorem k22_sub_b1_sub_one_nonneg :
+    0 ≤ params.k22 - params.b1 - 1 := by
+  have hk := PartB.k22_contains.1
+  have hb := PartB.b1_contains.2
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at hk hb ⊢
+  linarith
+
+private theorem phase2_D_y_nonneg {t : ℝ}
+    (ht0 : 0 ≤ t) (htθ : t ≤ params.theta) :
+    0 ≤ (Romik.path2 params t).2 -
+      (Romik.alphaBeta2 params t).2 * Real.sin t := by
+  have hz : t / 2 - params.b1 - 1 ≤ 0 := by
+    nlinarith [b1_lower_crude, theta_upper_crude]
+  have hzc : t / 2 - params.b1 - 1 ≤
+      (t / 2 - params.b1 - 1) * Real.cos t := by
+    have hprod : 0 ≤
+        (t / 2 - params.b1 - 1) * (Real.cos t - 1) :=
+      mul_nonneg_of_nonpos_of_nonpos hz (sub_nonpos.mpr (Real.cos_le_one t))
+    nlinarith
+  have hs : Real.sin t ≤ t := Real.sin_le ht0
+  simp [Romik.path2, Romik.rot, Romik.addK, Romik.alphaBeta2]
+  nlinarith [hzc, hs, k22_sub_b1_sub_one_nonneg]
+
+/-- Nonnegative height of the reflected early contact arc. -/
+theorem D_y_nonneg {t : ℝ} (ht : t ∈ Icc (0 : ℝ) params.theta) : 0 ≤ (D t).2 := by
+  by_cases hphi : t ≤ params.phi
+  · have hpath : Romik.path params t = Romik.path1 params t := by
+      simp only [Romik.path, ite_eq_left hphi]
+    have hb : beta t = (Romik.alphaBeta1 params t).2 := by
+      simp only [beta, alphaBetaAt, ite_eq_left hphi]
+    change 0 ≤ (Romik.path params t).2 - beta t * Real.sin t
+    rw [hpath, hb]
+    have ha2 := Romik.a2_eq_neg_quarter_of_equations params_equations
+    have hk12 := Romik.k12_eq_quarter_of_equations params_equations
+    simp [Romik.path1, Romik.rot, Romik.addK, Romik.alphaBeta1, ha2, hk12]
+    nlinarith [Real.sin_sq_add_cos_sq t, Real.cos_le_one t]
+  · have hphi' : params.phi < t := lt_of_not_ge hphi
+    have hpath : Romik.path params t = Romik.path2 params t := by
+      simp only [Romik.path, ite_eq_right hphi, ite_eq_left ht.2]
+    have hb : beta t = (Romik.alphaBeta2 params t).2 := by
+      simp only [beta, alphaBetaAt, ite_eq_right hphi, ite_eq_left ht.2]
+    change 0 ≤ (Romik.path params t).2 - beta t * Real.sin t
+    rw [hpath, hb]
+    exact phase2_D_y_nonneg ht.1 ht.2
+
+/-- Nonnegative height of the reflected late contact arc. -/
+theorem B_y_nonneg {t : ℝ} (ht : t ∈ Icc eta T) : 0 ≤ (B t).2 := by
+  by_cases hetaEq : t = eta
+  · subst t
+    exact B_eta_y_pos.le
+  have hetaT : eta < t := lt_of_le_of_ne ht.1 (Ne.symm hetaEq)
+  by_cases htau : t ≤ tau
+  · have hphiT : params.phi < t := lt_trans phi_lt_eta hetaT
+    have hthetaT : params.theta < t := lt_trans theta_lt_eta hetaT
+    have hpath : Romik.path params t = Romik.path4 params t := by
+      have hetaRaw : ¬ t ≤ Real.pi / 2 - params.theta := by
+        simpa [eta, T] using (not_le.mpr hetaT)
+      have htauRaw : t ≤ Real.pi / 2 - params.phi := by
+        simpa [tau, T] using htau
+      simp only [Romik.path, ite_eq_right (not_le.mpr hphiT),
+        ite_eq_right (not_le.mpr hthetaT), ite_eq_right hetaRaw, ite_eq_left htauRaw]
+    have ha : alpha t = (Romik.alphaBeta4 params t).1 := by
+      simp only [alpha, alphaBetaAt, ite_eq_right (not_le.mpr hphiT),
+        ite_eq_right (not_le.mpr hthetaT), ite_eq_right (not_le.mpr hetaT), ite_eq_left htau]
+    change 0 ≤ (Romik.path params t).2 + alpha t * Real.cos t
+    rw [hpath, ha]
+    let s : ℝ := T - t
+    have hs0 : 0 ≤ s := by dsimp [s]; linarith [ht.2]
+    have hsθ : s ≤ params.theta := by
+      dsimp [s, eta] at *
+      linarith
+    have hD2 := phase2_D_y_nonneg hs0 hsθ
+    have hd1 := Romik.d1_eq_quarterPi_sub_b1_of_equations params_equations
+    have hk42 := Romik.k42_eq_k22_of_equations params_equations
+    have hreflect :
+        (Romik.path4 params t).2 +
+            (Romik.alphaBeta4 params t).1 * Real.cos t =
+          (Romik.path2 params s).2 -
+            (Romik.alphaBeta2 params s).2 * Real.sin s := by
+      dsimp [s, T]
+      simp [Romik.path4, Romik.path2, Romik.rot, Romik.addK,
+        Romik.alphaBeta4, Romik.alphaBeta2, hd1, hk42,
+        Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+      ring
+    rw [hreflect]
+    exact hD2
+  · have htauT : tau < t := lt_of_not_ge htau
+    have hphiT : params.phi < t := lt_trans (lt_trans phi_lt_theta theta_lt_tau) htauT
+    have hthetaT : params.theta < t := lt_trans theta_lt_tau htauT
+    have hetaT' : eta < t := lt_trans eta_lt_tau htauT
+    have hpath : Romik.path params t = Romik.path5 params t := by
+      have hetaRaw : ¬ t ≤ Real.pi / 2 - params.theta := by
+        simpa [eta, T] using (not_le.mpr hetaT')
+      have htauRaw : ¬ t ≤ Real.pi / 2 - params.phi := by
+        simpa [tau, T] using (not_le.mpr htauT)
+      simp only [Romik.path, ite_eq_right (not_le.mpr hphiT),
+        ite_eq_right (not_le.mpr hthetaT), ite_eq_right hetaRaw, ite_eq_right htauRaw]
+    have ha : alpha t = (Romik.alphaBeta5 params t).1 := by
+      simp only [alpha, alphaBetaAt, ite_eq_right (not_le.mpr hphiT),
+        ite_eq_right (not_le.mpr hthetaT), ite_eq_right (not_le.mpr hetaT'), ite_eq_right htau]
+    change 0 ≤ (Romik.path params t).2 + alpha t * Real.cos t
+    rw [hpath, ha]
+    have he2 := Romik.e2_eq_quarter_of_equations params_equations
+    have hk52 := Romik.k52_eq_quarter_of_equations params_equations
+    simp [Romik.path5, Romik.rot, Romik.addK, Romik.alphaBeta5, he2, hk52]
+    nlinarith [Real.sin_sq_add_cos_sq t, Real.sin_le_one t]
+
+/-! C31: strict positivity on the two open contact tails.  The base endpoints
+`D 0` and `B T` have height exactly zero, so the natural domains are `Ioc`
+and `Ico`.  These statements are used only for topology of the strict vertical
+fills; the existing closed-interval nonnegativity theorems remain unchanged. -/
+private theorem k22_sub_b1_sub_one_pos :
+    0 < params.k22 - params.b1 - 1 := by
+  have hk := PartB.k22_contains.1
+  have hb := PartB.b1_contains.2
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at hk hb ⊢
+  linarith
+
+private theorem phase2_D_y_pos {t : ℝ}
+    (ht0 : 0 ≤ t) (htθ : t ≤ params.theta) :
+    0 < (Romik.path2 params t).2 -
+      (Romik.alphaBeta2 params t).2 * Real.sin t := by
+  have hz : t / 2 - params.b1 - 1 ≤ 0 := by
+    nlinarith [b1_lower_crude, theta_upper_crude]
+  have hzc : t / 2 - params.b1 - 1 ≤
+      (t / 2 - params.b1 - 1) * Real.cos t := by
+    have hprod : 0 ≤
+        (t / 2 - params.b1 - 1) * (Real.cos t - 1) :=
+      mul_nonneg_of_nonpos_of_nonpos hz (sub_nonpos.mpr (Real.cos_le_one t))
+    nlinarith
+  have hs : Real.sin t ≤ t := Real.sin_le ht0
+  simp [Romik.path2, Romik.rot, Romik.addK, Romik.alphaBeta2]
+  nlinarith [hzc, hs, k22_sub_b1_sub_one_pos]
+
+/-- Strict height of the early reflected contact arc away from its base endpoint. -/
+theorem D_y_pos {t : ℝ} (ht : t ∈ Ioc (0 : ℝ) params.theta) :
+    0 < (D t).2 := by
+  by_cases hphi : t ≤ params.phi
+  · have hpath : Romik.path params t = Romik.path1 params t := by
+      simp only [Romik.path, ite_eq_left hphi]
+    have hb : beta t = (Romik.alphaBeta1 params t).2 := by
+      simp only [beta, alphaBetaAt, ite_eq_left hphi]
+    change 0 < (Romik.path params t).2 - beta t * Real.sin t
+    rw [hpath, hb]
+    have ha2 := Romik.a2_eq_neg_quarter_of_equations params_equations
+    have hk12 := Romik.k12_eq_quarter_of_equations params_equations
+    have hcos : Real.cos t < 1 := by
+      have hc := Real.cos_lt_cos_of_nonneg_of_le_pi
+        (x := (0 : ℝ)) (y := t) (by norm_num)
+        (le_trans ht.2 (le_trans theta_lt_T.le (by
+          dsimp [T]
+          linarith [Real.pi_pos]))) ht.1
+      simpa using hc
+    simp [Romik.path1, Romik.rot, Romik.addK, Romik.alphaBeta1, ha2, hk12]
+    nlinarith [Real.sin_sq_add_cos_sq t, hcos]
+  · have hpath : Romik.path params t = Romik.path2 params t := by
+      simp only [Romik.path, ite_eq_right hphi, ite_eq_left ht.2]
+    have hb : beta t = (Romik.alphaBeta2 params t).2 := by
+      simp only [beta, alphaBetaAt, ite_eq_right hphi, ite_eq_left ht.2]
+    change 0 < (Romik.path params t).2 - beta t * Real.sin t
+    rw [hpath, hb]
+    exact phase2_D_y_pos ht.1.le ht.2
+
+/-- Strict height of the late contact arc away from its base endpoint. -/
+theorem B_y_pos {t : ℝ} (ht : t ∈ Ico eta T) :
+    0 < (B t).2 := by
+  by_cases hetaEq : t = eta
+  · subst t
+    exact B_eta_y_pos
+  have hetaT : eta < t := lt_of_le_of_ne ht.1 (Ne.symm hetaEq)
+  by_cases htau : t ≤ tau
+  · have hphiT : params.phi < t := lt_trans phi_lt_eta hetaT
+    have hthetaT : params.theta < t := lt_trans theta_lt_eta hetaT
+    have hpath : Romik.path params t = Romik.path4 params t := by
+      have hetaRaw : ¬ t ≤ Real.pi / 2 - params.theta := by
+        simpa [eta, T] using (not_le.mpr hetaT)
+      have htauRaw : t ≤ Real.pi / 2 - params.phi := by
+        simpa [tau, T] using htau
+      simp only [Romik.path, ite_eq_right (not_le.mpr hphiT),
+        ite_eq_right (not_le.mpr hthetaT), ite_eq_right hetaRaw, ite_eq_left htauRaw]
+    have ha : alpha t = (Romik.alphaBeta4 params t).1 := by
+      simp only [alpha, alphaBetaAt, ite_eq_right (not_le.mpr hphiT),
+        ite_eq_right (not_le.mpr hthetaT), ite_eq_right (not_le.mpr hetaT), ite_eq_left htau]
+    change 0 < (Romik.path params t).2 + alpha t * Real.cos t
+    rw [hpath, ha]
+    let s : ℝ := T - t
+    have hs0 : 0 ≤ s := by
+      dsimp [s]
+      exact sub_nonneg.mpr ht.2.le
+    have hsθ : s ≤ params.theta := by
+      dsimp [s, eta] at *
+      linarith
+    have hD2 := phase2_D_y_pos hs0 hsθ
+    have hd1 := Romik.d1_eq_quarterPi_sub_b1_of_equations params_equations
+    have hk42 := Romik.k42_eq_k22_of_equations params_equations
+    have hreflect :
+        (Romik.path4 params t).2 +
+            (Romik.alphaBeta4 params t).1 * Real.cos t =
+          (Romik.path2 params s).2 -
+            (Romik.alphaBeta2 params s).2 * Real.sin s := by
+      dsimp [s, T]
+      simp [Romik.path4, Romik.path2, Romik.rot, Romik.addK,
+        Romik.alphaBeta4, Romik.alphaBeta2, hd1, hk42,
+        Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+      ring
+    rw [hreflect]
+    exact hD2
+  · have htauT : tau < t := lt_of_not_ge htau
+    have hphiT : params.phi < t := lt_trans (lt_trans phi_lt_theta theta_lt_tau) htauT
+    have hthetaT : params.theta < t := lt_trans theta_lt_tau htauT
+    have hetaT' : eta < t := lt_trans eta_lt_tau htauT
+    have hpath : Romik.path params t = Romik.path5 params t := by
+      have hetaRaw : ¬ t ≤ Real.pi / 2 - params.theta := by
+        simpa [eta, T] using (not_le.mpr hetaT')
+      have htauRaw : ¬ t ≤ Real.pi / 2 - params.phi := by
+        simpa [tau, T] using (not_le.mpr htauT)
+      simp only [Romik.path, ite_eq_right (not_le.mpr hphiT),
+        ite_eq_right (not_le.mpr hthetaT), ite_eq_right hetaRaw, ite_eq_right htauRaw]
+    have ha : alpha t = (Romik.alphaBeta5 params t).1 := by
+      simp only [alpha, alphaBetaAt, ite_eq_right (not_le.mpr hphiT),
+        ite_eq_right (not_le.mpr hthetaT), ite_eq_right (not_le.mpr hetaT'), ite_eq_right htau]
+    change 0 < (Romik.path params t).2 + alpha t * Real.cos t
+    rw [hpath, ha]
+    have he2 := Romik.e2_eq_quarter_of_equations params_equations
+    have hk52 := Romik.k52_eq_quarter_of_equations params_equations
+    have hsin : Real.sin t < 1 := by
+      have hs := Real.sin_lt_sin_of_lt_of_le_pi_div_two
+        (x := t) (y := T)
+        (by
+          have ht0 : 0 ≤ t :=
+            le_trans theta_pos.le (le_trans theta_lt_eta.le ht.1)
+          nlinarith [Real.pi_pos])
+        (by simp [T]) ht.2
+      simpa [T] using hs
+    simp [Romik.path5, Romik.rot, Romik.addK, Romik.alphaBeta5, he2, hk52]
+    nlinarith [Real.sin_sq_add_cos_sq t, hsin]
+
+/-! ## C24: fixed-endpoint separation by a phase-2 Taylor bound and contact monotonicity
+
+C23 reduced the root gate to four scalar `U(phi,t)` residuals.  This replacement
+removes the large raw `nlinarith` calls.  Phase 2 is reduced to one variable
+`d=t-phi` and certified by low-order alternating Taylor bounds.  Phases 3--5
+use the exact contact identity `B(eta)=path(phi)` and the sign of the derivative
+of the phase-local `B` contact projected on the fixed normal `u(t)`.
+-/
+
+private theorem phi_lower_crude : (39 / 1000 : ℝ) ≤ params.phi := by
+  have h := phi_bounds.1
+  norm_num at h ⊢
+  linarith
+
+private theorem phi_upper_crude : params.phi ≤ (1 / 20 : ℝ) := by
+  have h := phi_bounds.2
+  norm_num at h ⊢
+  linarith
+
+private theorem theta_upper_689 : params.theta ≤ (689 / 1000 : ℝ) := by
+  have h := theta_bounds.2
+  norm_num at h ⊢
+  linarith
+
+private theorem b1_upper_crude : params.b1 ≤ (-527 / 1000 : ℝ) := by
+  have h := PartB.b1_contains.2
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at h ⊢
+  linarith
+
+private theorem b2_lower_crude : (9 / 10 : ℝ) ≤ params.b2 := by
+  have h := PartB.b2_contains.1
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at h ⊢
+  linarith
+
+private theorem b2_upper_921 : params.b2 ≤ (921 / 1000 : ℝ) := by
+  have h := PartB.b2_contains.2
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at h ⊢
+  linarith
+
+open scoped BigOperators
+open Filter Finset
+
+private theorem sin_taylor5_upper {x : ℝ} (hx0 : 0 ≤ x) (hx1 : x ≤ 1) :
+    Real.sin x ≤ x - x ^ 3 / 6 + x ^ 5 / 120 := by
+  have hanti : Antitone (ExactReplay.sinMagnitude x) :=
+    ExactReplay.antitone_sinMagnitude hx0 hx1
+  have htendRaw := (Real.hasSum_sin x).tendsto_sum_nat
+  have htendMag :
+      Filter.Tendsto
+        (fun n : ℕ => ∑ i ∈ Finset.range n,
+          (-1 : ℝ) ^ i * ExactReplay.sinMagnitude x i)
+        Filter.atTop (nhds (Real.sin x)) := by
+    simpa only [ExactReplay.sinMagnitude, mul_div_assoc] using htendRaw
+  have hupper := Antitone.tendsto_le_alternating_series htendMag hanti 1
+  have h3 : (2 * 1 + 1 : ℕ) = 3 := by norm_num
+  rw [h3] at hupper
+  have hsum :
+      (∑ i ∈ Finset.range 3,
+          (-1 : ℝ) ^ i * ExactReplay.sinMagnitude x i) =
+        x - x ^ 3 / 6 + x ^ 5 / 120 := by
+    norm_num [ExactReplay.sinMagnitude, Finset.sum_range_succ]; ring
+  rw [hsum] at hupper
+  exact hupper
+
+private theorem cos_taylor6_lower {x : ℝ} (hx0 : 0 ≤ x) (hx1 : x ≤ 1) :
+    1 - x ^ 2 / 2 + x ^ 4 / 24 - x ^ 6 / 720 ≤ Real.cos x := by
+  have hanti : Antitone (ExactReplay.cosMagnitude x) :=
+    ExactReplay.antitone_cosMagnitude hx0 hx1
+  have htendRaw := (Real.hasSum_cos x).tendsto_sum_nat
+  have htendMag :
+      Filter.Tendsto
+        (fun n : ℕ => ∑ i ∈ Finset.range n,
+          (-1 : ℝ) ^ i * ExactReplay.cosMagnitude x i)
+        Filter.atTop (nhds (Real.cos x)) := by
+    simpa only [ExactReplay.cosMagnitude, mul_div_assoc] using htendRaw
+  have hlower := Antitone.alternating_series_le_tendsto htendMag hanti 2
+  have h4 : (2 * 2 : ℕ) = 4 := by norm_num
+  rw [h4] at hlower
+  have hsum :
+      (∑ i ∈ Finset.range 4,
+          (-1 : ℝ) ^ i * ExactReplay.cosMagnitude x i) =
+        1 - x ^ 2 / 2 + x ^ 4 / 24 - x ^ 6 / 720 := by
+    norm_num [ExactReplay.cosMagnitude, Finset.sum_range_succ]; ring
+  rw [hsum] at hlower
+  exact hlower
+
+private theorem phase2_U_phi_formula {t : ℝ}
+    (hphiT : params.phi < t) (htheta : t ≤ params.theta) :
+    Stage2.UValue params.phi t =
+      (-(1 / 4 : ℝ) * params.phi * params.phi +
+          params.b1 * params.phi + params.b2) *
+        (Real.cos (t - params.phi) - 1) +
+      (params.phi / 2 - params.b1 - 1) * Real.sin (t - params.phi) +
+      (params.phi / 2 - params.b1) * (t - params.phi) +
+      (t - params.phi) * (t - params.phi) / 4 := by
+  have hPhi2 : Romik.path params params.phi =
+      Romik.path2 params params.phi := path_at_phi.trans PartB.match12
+  have hPathT : Romik.path params t = Romik.path2 params t := by
+    simp only [Romik.path, ite_eq_right (not_le.mpr hphiT), ite_eq_left htheta]
+  unfold Stage2.UValue
+  rw [hPhi2, hPathT]
+  simp [dot, u, Romik.path2, Romik.rot, Romik.addK,
+    Real.cos_sub, Real.sin_sub]
+  ring_nf
+  have hunit : Real.cos t ^ 2 + Real.sin t ^ 2 = 1 := by
+    nlinarith [Real.sin_sq_add_cos_sq t]
+  have hb1collapse :
+      params.b1 * Real.cos t ^ 2 * t + params.b1 * t * Real.sin t ^ 2 =
+        params.b1 * t := by
+    calc
+      params.b1 * Real.cos t ^ 2 * t + params.b1 * t * Real.sin t ^ 2 =
+          params.b1 * t * (Real.cos t ^ 2 + Real.sin t ^ 2) := by ring
+      _ = params.b1 * t := by rw [hunit]; ring
+  have hb2collapse :
+      params.b2 * Real.cos t ^ 2 + params.b2 * Real.sin t ^ 2 =
+        params.b2 := by
+    calc
+      params.b2 * Real.cos t ^ 2 + params.b2 * Real.sin t ^ 2 =
+          params.b2 * (Real.cos t ^ 2 + Real.sin t ^ 2) := by ring
+      _ = params.b2 := by rw [hunit]; ring
+  have ht2collapse :
+      Real.cos t ^ 2 * t ^ 2 * (1 / 4 : ℝ) +
+          t ^ 2 * Real.sin t ^ 2 * (1 / 4 : ℝ) =
+        t ^ 2 * (1 / 4 : ℝ) := by
+    calc
+      Real.cos t ^ 2 * t ^ 2 * (1 / 4 : ℝ) +
+          t ^ 2 * Real.sin t ^ 2 * (1 / 4 : ℝ) =
+          t ^ 2 * (1 / 4 : ℝ) * (Real.cos t ^ 2 + Real.sin t ^ 2) := by ring
+      _ = t ^ 2 * (1 / 4 : ℝ) := by rw [hunit]; ring
+  linarith [hb1collapse, hb2collapse, ht2collapse]
+private theorem phase2_U_phi_nonneg {t : ℝ}
+    (ht : t ∈ Icc params.phi params.theta) :
+    0 ≤ Stage2.UValue params.phi t := by
+  by_cases heq : t = params.phi
+  · subst t
+    simp [Stage2.UValue, dot]
+  have hphiT : params.phi < t := lt_of_le_of_ne ht.1 (Ne.symm heq)
+  let d : ℝ := t - params.phi
+  let z1 : ℝ :=
+    -(1 / 4 : ℝ) * params.phi * params.phi +
+      params.b1 * params.phi + params.b2
+  let z2 : ℝ := params.phi / 2 - params.b1 - 1
+  let c : ℝ := params.phi / 2 - params.b1
+  have hd0 : 0 ≤ d := by dsimp [d]; linarith
+  have hdA : d ≤ (13 / 20 : ℝ) := by
+    dsimp [d]
+    linarith [theta_upper_689, phi_lower_crude, ht.2]
+  have hd1 : d ≤ 1 := by linarith
+  have hp0 : 0 ≤ params.phi := phi_pos.le
+  have hb1lo := b1_lower_crude
+  have hb1hi := b1_upper_crude
+  have hz1nonneg : 0 ≤ z1 := by
+    have hbp : 0 ≤ (params.b1 + 53 / 100) * params.phi :=
+      mul_nonneg (by linarith) hp0
+    have hpp : 0 ≤ params.phi * (1 / 20 - params.phi) :=
+      mul_nonneg hp0 (by linarith [phi_upper_crude])
+    dsimp [z1]
+    nlinarith [b2_lower_crude, hbp, hpp]
+  have hz1le : z1 ≤ (91 / 100 : ℝ) := by
+    have hbp : params.b1 * params.phi ≤ (-527 / 1000 : ℝ) * (39 / 1000 : ℝ) := by
+      have h1 := mul_le_mul_of_nonneg_right b1_upper_crude hp0
+      have h2 := mul_le_mul_of_nonpos_left phi_lower_crude (by norm_num : (-527 / 1000 : ℝ) ≤ 0)
+      exact le_trans h1 h2
+    have hsquare : 0 ≤ params.phi * params.phi := mul_self_nonneg _
+    dsimp [z1]
+    nlinarith only [b2_upper_921, hbp, hsquare]
+  have hz2lo : (-227 / 500 : ℝ) ≤ z2 := by
+    dsimp [z2]
+    linarith [phi_lower_crude, b1_upper_crude]
+  have hz2hi : z2 ≤ 0 := by
+    dsimp [z2]
+    linarith [phi_upper_crude, hb1lo]
+  have hclo : (273 / 500 : ℝ) ≤ c := by
+    dsimp [c]
+    linarith [phi_lower_crude, b1_upper_crude]
+  have hsinU := sin_taylor5_upper hd0 hd1
+  have hcosL := cos_taylor6_lower hd0 hd1
+  let SU : ℝ := d - d ^ 3 / 6 + d ^ 5 / 120
+  let CL : ℝ := -(d ^ 2) / 2 + d ^ 4 / 24 - d ^ 6 / 720
+  have hd2le1 : d ^ 2 ≤ 1 := by
+    have hprod : 0 ≤ d * (1 - d) := mul_nonneg hd0 (sub_nonneg.mpr hd1)
+    nlinarith only [hprod, hd1]
+  have hSU0 : 0 ≤ SU := by
+    have hcube : d ^ 3 ≤ d := by
+      have hprod : 0 ≤ d * (1 - d ^ 2) :=
+        mul_nonneg hd0 (sub_nonneg.mpr hd2le1)
+      nlinarith only [hprod]
+    have hd5 : 0 ≤ d ^ 5 := by positivity
+    dsimp [SU]
+    nlinarith only [hcube, hd5, hd0]
+  have hCL0 : CL ≤ 0 := by
+    have hd2 : 0 ≤ d ^ 2 := sq_nonneg d
+    have hd4le : d ^ 4 ≤ d ^ 2 := by
+      have h := mul_nonneg hd2 (sub_nonneg.mpr hd2le1)
+      nlinarith
+    dsimp [CL]
+    nlinarith only [sq_nonneg (d ^ 3), hd4le, hd2]
+  have hcosm1 : CL ≤ Real.cos d - 1 := by
+    dsimp [CL]
+    linarith only [hcosL]
+  have hsinSU : Real.sin d ≤ SU := by
+    dsimp [SU]
+    exact hsinU
+  have hz1CL : (91 / 100 : ℝ) * CL ≤ z1 * CL := by
+    exact mul_le_mul_of_nonpos_right hz1le hCL0
+  have hz1cos : z1 * CL ≤ z1 * (Real.cos d - 1) := by
+    exact mul_le_mul_of_nonneg_left hcosm1 hz1nonneg
+  have hz2sin : z2 * SU ≤ z2 * Real.sin d := by
+    exact mul_le_mul_of_nonpos_left hsinSU hz2hi
+  have hz2SU : (-227 / 500 : ℝ) * SU ≤ z2 * SU := by
+    exact mul_le_mul_of_nonneg_right hz2lo hSU0
+  have hcD : (273 / 500 : ℝ) * d ≤ c * d := by
+    exact mul_le_mul_of_nonneg_right hclo hd0
+  let Q : ℝ :=
+    (23 / 250 : ℝ) - (41 / 200) * d + (227 / 3000) * d ^ 2 +
+      (91 / 2400) * d ^ 3 - (227 / 60000) * d ^ 4 -
+      (91 / 72000) * d ^ 5
+  let R : ℝ :=
+    (14560000 * d ^ 4 + 53048000 * d ^ 3 - 402318800 * d ^ 2 -
+      1133187220 * d + 1625028307 : ℝ) / 11520000000
+  have hd2A : d ^ 2 ≤ (13 / 20 : ℝ) ^ 2 := by nlinarith
+  have hR : 0 ≤ R := by
+    dsimp [R]
+    have h3 : 0 ≤ d ^ 3 := by positivity
+    have h4 : 0 ≤ d ^ 4 := by positivity
+    nlinarith only [hdA, hd2A, h3, h4]
+  have hQend : 0 < (71432009 / 230400000000 : ℝ) := by norm_num
+  have hQfactor :
+      Q - (71432009 / 230400000000 : ℝ) = ((13 / 20 : ℝ) - d) * R := by
+    dsimp [Q, R]
+    ring
+  have hQ : 0 ≤ Q := by
+    have hprod : 0 ≤ ((13 / 20 : ℝ) - d) * R :=
+      mul_nonneg (sub_nonneg.mpr hdA) hR
+    linarith only [hprod, hQfactor, hQend]
+  have hL :
+      0 ≤ (91 / 100 : ℝ) * CL + (-227 / 500 : ℝ) * SU +
+        (273 / 500 : ℝ) * d + d * d / 4 := by
+    have hid :
+        (91 / 100 : ℝ) * CL + (-227 / 500 : ℝ) * SU +
+            (273 / 500 : ℝ) * d + d * d / 4 = d * Q := by
+      dsimp [CL, SU, Q]
+      ring
+    rw [hid]
+    exact mul_nonneg hd0 hQ
+  have hformula' :
+      Stage2.UValue params.phi t =
+        z1 * (Real.cos d - 1) + z2 * Real.sin d + c * d + d * d / 4 := by
+    simpa [d, z1, z2, c] using phase2_U_phi_formula hphiT ht.2
+  rw [hformula']
+  linarith only [hL, hz1CL, hz1cos, hz2sin, hz2SU, hcD]
+
+private theorem pathPrime1_eq_rot (p : Romik.Params) (t : ℝ) :
+    Romik.pathPrime1 p t = Romik.rot t (Romik.alphaBeta1 p t) := rfl
+
+private theorem matchPrime12 :
+    Romik.pathPrime1 params params.phi = Romik.pathPrime2 params params.phi := by
+  have h10 := congrFun params_equations (10 : Fin 22)
+  have h11 := congrFun params_equations (11 : Fin 22)
+  simp [Romik.system] at h10 h11
+  apply Prod.ext <;> linarith
+
+private theorem matchAB12 :
+    Romik.alphaBeta1 params params.phi = Romik.alphaBeta2 params params.phi := by
+  apply rot_injective params.phi
+  simpa [pathPrime1_eq_rot, pathPrime2_eq_rot] using matchPrime12
+
+private theorem ab4_reflect_ab2 (t : ℝ) :
+    Romik.alphaBeta4 params (T - t) = reflAB (Romik.alphaBeta2 params t) := by
+  dsimp [T, reflAB, Romik.alphaBeta4, Romik.alphaBeta2]
+  rw [Romik.d1_eq_quarterPi_sub_b1_of_equations params_equations,
+      Romik.d2_eq_b2_add_quarterPi_correction_of_equations params_equations]
+  ring_nf
+
+private theorem ab5_reflect_ab1 (t : ℝ) :
+    Romik.alphaBeta5 params (T - t) = reflAB (Romik.alphaBeta1 params t) := by
+  dsimp [T, reflAB, Romik.alphaBeta5, Romik.alphaBeta1]
+  rw [Romik.e1_eq_a1_of_equations params_equations,
+      Romik.e2_eq_neg_a2_of_equations params_equations]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  ring_nf
+
+private theorem matchAB34 :
+    Romik.alphaBeta3 params eta = Romik.alphaBeta4 params eta := by
+  have h3 := ab3_reflect params.theta
+  have h4 := ab4_reflect_ab2 params.theta
+  have hm := congrArg reflAB matchAB23
+  simpa [eta] using h3.trans (hm.symm.trans h4.symm)
+
+private theorem matchAB45 :
+    Romik.alphaBeta4 params tau = Romik.alphaBeta5 params tau := by
+  have h4 := ab4_reflect_ab2 params.phi
+  have h5 := ab5_reflect_ab1 params.phi
+  have hm := congrArg reflAB matchAB12
+  simpa [tau] using h4.trans (hm.symm.trans h5.symm)
+
+private def phaseB3 (t : ℝ) : Point :=
+  Romik.path3 params t + (Romik.alphaBeta3 params t).1 • v t
+private def phaseB4_m71dda23 (t : ℝ) : Point :=
+  Romik.path4 params t + (Romik.alphaBeta4 params t).1 • v t
+private def phaseB5_m71dda23 (t : ℝ) : Point :=
+  Romik.path5 params t + (Romik.alphaBeta5 params t).1 • v t
+
+private theorem u_hasDerivAt_local (t : ℝ) : HasDerivAt u (v t) t :=
+  Stage2.u_hasDerivAt t
+
+private theorem phaseB3_eq_A3_sub_u (t : ℝ) :
+    phaseB3 t = Stage2.phaseA3 t - u t := by
+  apply Prod.ext <;> simp [phaseB3, Stage2.phaseA3]
+
+private theorem phaseB4_eq_A4_sub_u (t : ℝ) :
+    phaseB4_m71dda23 t = Stage2.phaseA4 t - u t := by
+  apply Prod.ext <;> simp [phaseB4_m71dda23, Stage2.phaseA4]
+
+private theorem phaseB5_eq_A5_sub_u (t : ℝ) :
+    phaseB5_m71dda23 t = Stage2.phaseA5 t - u t := by
+  apply Prod.ext <;> simp [phaseB5_m71dda23, Stage2.phaseA5]
+
+private theorem B_eq_phaseB3 {t : ℝ} (ht : t ∈ Icc params.theta eta) :
+    B t = phaseB3 t := by
+  by_cases h : t = params.theta
+  · subst t
+    rw [B, path_at_theta]
+    have ha : alpha params.theta = (Romik.alphaBeta2 params params.theta).1 := by
+      simp only [alpha, alphaBetaAt, ite_eq_right (not_le.mpr phi_lt_theta),
+        ite_eq_left (le_refl params.theta)]
+    rw [ha]
+    have hm := PartB.match23
+    rw [hm, matchAB23]
+    apply Prod.ext <;> simp [phaseB3, v]
+  · have htheta : params.theta < t := lt_of_le_of_ne ht.1 (Ne.symm h)
+    have hphi : params.phi < t := lt_trans phi_lt_theta htheta
+    have hetaRaw : t ≤ Real.pi / 2 - params.theta := by
+      simpa [eta, T] using ht.2
+    apply Prod.ext <;> simp [B, alpha, alphaBetaAt, Romik.path, phaseB3,
+      not_le.mpr hphi, not_le.mpr htheta, hetaRaw, eta, T]
+
+private theorem B_eq_phaseB4 {t : ℝ} (ht : t ∈ Icc eta tau) :
+    B t = phaseB4_m71dda23 t := by
+  by_cases h : t = eta
+  · subst t
+    have hB3 := B_eq_phaseB3 (t := eta) ⟨theta_lt_eta.le, le_rfl⟩
+    have hm : Romik.path3 params eta = Romik.path4 params eta := by
+      simpa [eta, T] using PartB.match34
+    calc
+      B eta = phaseB3 eta := hB3
+      _ = phaseB4_m71dda23 eta := by
+        unfold phaseB3 phaseB4_m71dda23
+        rw [hm, matchAB34]
+  · have hetaT : eta < t := lt_of_le_of_ne ht.1 (Ne.symm h)
+    have htheta : params.theta < t := lt_trans theta_lt_eta hetaT
+    have hphi : params.phi < t := lt_trans phi_lt_theta htheta
+    have hetaRaw : Real.pi / 2 - params.theta < t := by
+      simpa [eta, T] using hetaT
+    have htauRaw : t ≤ Real.pi / 2 - params.phi := by
+      simpa [tau, T] using ht.2
+    apply Prod.ext <;> simp [B, alpha, alphaBetaAt, Romik.path, phaseB4_m71dda23,
+      not_le.mpr hphi, not_le.mpr htheta, not_le.mpr hetaRaw, htauRaw,
+      eta, tau, T]
+
+private theorem B_eq_phaseB5 {t : ℝ} (ht : t ∈ Icc tau T) :
+    B t = phaseB5_m71dda23 t := by
+  by_cases h : t = tau
+  · subst t
+    have hB4 := B_eq_phaseB4 (t := tau) ⟨eta_lt_tau.le, le_rfl⟩
+    have hm : Romik.path4 params tau = Romik.path5 params tau := by
+      simpa [tau, T] using PartB.match45
+    calc
+      B tau = phaseB4_m71dda23 tau := hB4
+      _ = phaseB5_m71dda23 tau := by
+        unfold phaseB4_m71dda23 phaseB5_m71dda23
+        rw [hm, matchAB45]
+  · have htauT : tau < t := lt_of_le_of_ne ht.1 (Ne.symm h)
+    have hetaT : eta < t := lt_trans eta_lt_tau htauT
+    have htheta : params.theta < t := lt_trans theta_lt_eta hetaT
+    have hphi : params.phi < t := lt_trans phi_lt_theta htheta
+    have hetaRaw : Real.pi / 2 - params.theta < t := by
+      simpa [eta, T] using hetaT
+    have htauRaw : Real.pi / 2 - params.phi < t := by
+      simpa [tau, T] using htauT
+    apply Prod.ext <;> simp [B, alpha, alphaBetaAt, Romik.path, phaseB5_m71dda23,
+      not_le.mpr hphi, not_le.mpr htheta, not_le.mpr hetaRaw,
+      not_le.mpr htauRaw, eta, tau, T]
+
+private theorem c1_le_theta : params.c1 ≤ params.theta := by
+  have hc := PartB.c1_contains.2
+  have ht := theta_bounds.1
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at hc ht ⊢
+  linarith
+
+private theorem eta_lower_four_fifths : (4 / 5 : ℝ) ≤ eta := by
+  dsimp [eta, T]
+  nlinarith [Real.pi_gt_three, theta_upper_crude]
+
+private theorem d1_upper_33_25 : params.d1 ≤ (33 / 25 : ℝ) := by
+  have hd := PartB.d1_contains.2
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at hd ⊢
+  linarith
+
+private theorem phaseB3_coeff_nonpos {q : ℝ} (hq : params.theta ≤ q) :
+    params.c1 - q ≤ 0 := by linarith [c1_le_theta]
+
+private theorem phaseB4_coeff_nonpos {q : ℝ} (hq : eta ≤ q) :
+    params.d1 - q / 2 - 1 ≤ 0 := by
+  nlinarith [d1_upper_33_25, eta_lower_four_fifths]
+
+private theorem dot_hasDerivAt_fixed_local {f : ℝ → Point} {df : Point} {t : ℝ}
+    (h : HasDerivAt f df t) (w : Point) :
+    HasDerivAt (fun s => dot (f s) w) (dot df w) t := by
+  have h1 := HasDerivAt.const_mul w.1 h.fst
+  have h2 := HasDerivAt.const_mul w.2 h.snd
+  have hs := h1.fun_add h2
+  simpa [dot, mul_comm] using hs
+
+private theorem phaseB3_dot_hasDerivAt (s t : ℝ) :
+    HasDerivAt (fun q => dot (phaseB3 q) (u s))
+      (dot ((params.c1 - t) • v t) (u s)) t := by
+  have h := (Stage2.A3_hasDerivAt_public t).fun_sub (u_hasDerivAt_local t)
+  have hd : ((1 + params.c1 - t) • v t) - v t =
+      (params.c1 - t) • v t := by
+    apply Prod.ext <;> simp [v] <;> ring
+  rw [hd] at h
+  have hdot := dot_hasDerivAt_fixed_local h (u s)
+  simpa only [phaseB3_eq_A3_sub_u] using hdot
+
+private theorem phaseB4_dot_hasDerivAt (s t : ℝ) :
+    HasDerivAt (fun q => dot (phaseB4_m71dda23 q) (u s))
+      (dot ((params.d1 - t / 2 - 1) • v t) (u s)) t := by
+  have h := (Stage2.A4_hasDerivAt_public t).fun_sub (u_hasDerivAt_local t)
+  have hd : ((params.d1 - t / 2) • v t) - v t =
+      (params.d1 - t / 2 - 1) • v t := by
+    apply Prod.ext <;> simp [v] <;> ring
+  rw [hd] at h
+  have hdot := dot_hasDerivAt_fixed_local h (u s)
+  simpa only [phaseB4_eq_A4_sub_u] using hdot
+
+private theorem phaseB5_dot_hasDerivAt (s t : ℝ) :
+    HasDerivAt (fun q => dot (phaseB5_m71dda23 q) (u s))
+      (dot ((-(1 / 2 : ℝ)) • v t) (u s)) t := by
+  have h := (Stage2.A5_hasDerivAt_public t).fun_sub (u_hasDerivAt_local t)
+  have hd : ((1 / 2 : ℝ) • v t) - v t = (-(1 / 2 : ℝ)) • v t := by
+    apply Prod.ext <;> simp [v] <;> ring
+  rw [hd] at h
+  have hdot := dot_hasDerivAt_fixed_local h (u s)
+  simpa only [phaseB5_eq_A5_sub_u] using hdot
+
+private theorem phaseB_mono_left
+    {F : ℝ → Point} {r : ℝ → ℝ} {a b s : ℝ}
+    (hder : ∀ q,
+      HasDerivAt (fun x => dot (F x) (u s))
+        (dot ((r q) • v q) (u s)) q)
+    (hr : ∀ q ∈ Ioo a b, r q ≤ 0)
+    (hs0 : 0 ≤ s) (hsa : s ≤ a) (hbT : b ≤ T) :
+    MonotoneOn (fun q => dot (F q) (u s)) (Icc a b) := by
+  refine monotoneOn_of_hasDerivWithinAt_nonneg
+    (f' := fun q => dot ((r q) • v q) (u s)) (convex_Icc a b) ?_ ?_ ?_
+  · intro q hq
+    exact (hder q).continuousAt.continuousWithinAt
+  · intro q hq
+    exact (hder q).hasDerivWithinAt
+  · intro q hq
+    have hi : q ∈ Ioo a b := by simpa only [interior_Icc] using hq
+    have hsin : Real.sin (s - q) ≤ 0 := by
+      have hnon : s - q ≤ 0 := by linarith [hsa, hi.1]
+      have hnegpi : -Real.pi ≤ s - q := by
+        have hqT : q ≤ T := le_trans (le_of_lt hi.2) hbT
+        have hqs : q - s ≤ T := by linarith
+        nlinarith [T_le_pi]
+      exact Real.sin_nonpos_of_nonpos_of_neg_pi_le hnon hnegpi
+    have heq : dot ((r q) • v q) (u s) = r q * Real.sin (s - q) := by
+      simp [dot, u, v, Real.sin_sub]
+      ring
+    rw [heq]
+    exact mul_nonneg_of_nonpos_of_nonpos (hr q hi) hsin
+
+private theorem phaseB_anti_right
+    {F : ℝ → Point} {r : ℝ → ℝ} {a b s : ℝ}
+    (hder : ∀ q,
+      HasDerivAt (fun x => dot (F x) (u s))
+        (dot ((r q) • v q) (u s)) q)
+    (hr : ∀ q ∈ Ioo a b, r q ≤ 0)
+    (ha0 : 0 ≤ a) (hbs : b ≤ s) (hsT : s ≤ T) :
+    AntitoneOn (fun q => dot (F q) (u s)) (Icc a b) := by
+  refine antitoneOn_of_hasDerivWithinAt_nonpos
+    (f' := fun q => dot ((r q) • v q) (u s)) (convex_Icc a b) ?_ ?_ ?_
+  · intro q hq
+    exact (hder q).continuousAt.continuousWithinAt
+  · intro q hq
+    exact (hder q).hasDerivWithinAt
+  · intro q hq
+    have hi : q ∈ Ioo a b := by simpa only [interior_Icc] using hq
+    have hq0 : 0 ≤ q := le_trans ha0 (le_of_lt hi.1)
+    have hdelta0 : 0 ≤ s - q := by linarith [hi.2, hbs]
+    have hdeltaPi : s - q ≤ Real.pi := by
+      have : s - q ≤ T := by linarith [hsT, hq0]
+      exact le_trans this T_le_pi
+    have hsin : 0 ≤ Real.sin (s - q) :=
+      Real.sin_nonneg_of_nonneg_of_le_pi hdelta0 hdeltaPi
+    have heq : dot ((r q) • v q) (u s) = r q * Real.sin (s - q) := by
+      simp [dot, u, v, Real.sin_sub]
+      ring
+    rw [heq]
+    exact mul_nonpos_of_nonpos_of_nonneg (hr q hi) hsin
+
+private theorem criticalU_eq_Bdiff (t : ℝ) :
+    Stage2.UValue params.phi t = dot (B eta - B t) (u t) := by
+  unfold Stage2.UValue
+  rw [← B_eta_eq_path_phi]
+  have hinner := B_inner_u_identity t
+  unfold dot at hinner ⊢
+  dsimp at hinner ⊢
+  linarith
+
+private theorem phase3_U_phi_nonneg {t : ℝ}
+    (ht : t ∈ Icc params.theta eta) :
+    0 ≤ Stage2.UValue params.phi t := by
+  have hm := phaseB_mono_left (F := phaseB3)
+    (r := fun q => params.c1 - q) (a := t) (b := eta) (s := t)
+    (phaseB3_dot_hasDerivAt t)
+    (by intro q hq; exact phaseB3_coeff_nonpos (le_trans ht.1 (le_of_lt hq.1)))
+    (le_trans theta_pos.le ht.1) (le_rfl) eta_lt_T.le
+  have hcomp := hm ⟨le_rfl, ht.2⟩ ⟨ht.2, le_rfl⟩ ht.2
+  have hBt := B_eq_phaseB3 ht
+  have hBe := B_eq_phaseB3 (t := eta) ⟨theta_lt_eta.le, le_rfl⟩
+  rw [criticalU_eq_Bdiff, hBt, hBe]
+  unfold dot at hcomp ⊢
+  dsimp at hcomp ⊢
+  linarith
+
+private theorem phase4_U_phi_nonneg {t : ℝ}
+    (ht : t ∈ Icc eta tau) :
+    0 ≤ Stage2.UValue params.phi t := by
+  have hm := phaseB_anti_right (F := phaseB4_m71dda23)
+    (r := fun q => params.d1 - q / 2 - 1) (a := eta) (b := t) (s := t)
+    (phaseB4_dot_hasDerivAt t)
+    (by intro q hq; exact phaseB4_coeff_nonpos (le_of_lt hq.1))
+    (le_trans theta_pos.le theta_lt_eta.le) (le_rfl) (le_trans ht.2 tau_lt_T.le)
+  have hcomp := hm ⟨le_rfl, ht.1⟩ ⟨ht.1, le_rfl⟩ ht.1
+  have hBt := B_eq_phaseB4 ht
+  have hBe := B_eq_phaseB4 (t := eta) ⟨le_rfl, eta_lt_tau.le⟩
+  rw [criticalU_eq_Bdiff, hBt, hBe]
+  unfold dot at hcomp ⊢
+  dsimp at hcomp ⊢
+  linarith
+
+private theorem phase5_U_phi_nonneg {t : ℝ}
+    (ht : t ∈ Icc tau T) :
+    0 ≤ Stage2.UValue params.phi t := by
+  have hm4 := phaseB_anti_right (F := phaseB4_m71dda23)
+    (r := fun q => params.d1 - q / 2 - 1) (a := eta) (b := tau) (s := t)
+    (phaseB4_dot_hasDerivAt t)
+    (by intro q hq; exact phaseB4_coeff_nonpos (le_of_lt hq.1))
+    (le_trans theta_pos.le theta_lt_eta.le) ht.1 ht.2
+  have h4 := hm4 ⟨le_rfl, eta_lt_tau.le⟩ ⟨eta_lt_tau.le, le_rfl⟩ eta_lt_tau.le
+  have hm5 := phaseB_anti_right (F := phaseB5_m71dda23)
+    (r := fun _ => (-(1 / 2 : ℝ))) (a := tau) (b := t) (s := t)
+    (phaseB5_dot_hasDerivAt t)
+    (by intro q hq; norm_num)
+    (le_trans theta_pos.le (lt_trans theta_lt_eta eta_lt_tau).le) (le_rfl) ht.2
+  have h5 := hm5 ⟨le_rfl, ht.1⟩ ⟨ht.1, le_rfl⟩ ht.1
+  have hBetaEta := B_eq_phaseB4 (t := eta) ⟨le_rfl, eta_lt_tau.le⟩
+  have hBetaTau4 := B_eq_phaseB4 (t := tau) ⟨eta_lt_tau.le, le_rfl⟩
+  have hBetaTau5 := B_eq_phaseB5 (t := tau) ⟨le_rfl, tau_lt_T.le⟩
+  have hBetaT := B_eq_phaseB5 ht
+  rw [criticalU_eq_Bdiff, hBetaEta, hBetaT]
+  have hmatch : phaseB4_m71dda23 tau = phaseB5_m71dda23 tau := by
+    rw [← hBetaTau4, ← hBetaTau5]
+  change dot (phaseB4_m71dda23 tau) (u t) ≤ dot (phaseB4_m71dda23 eta) (u t) at h4
+  rw [hmatch] at h4
+  unfold dot at h4 h5 ⊢
+  dsimp at h4 h5 ⊢
+  linarith
+
+/-- Fixed-endpoint separation required by the direct no-hidden-crossing proof. -/
+theorem U_phi_nonneg {t : ℝ} (ht : t ∈ Icc params.phi T) :
+    0 ≤ Stage2.UValue params.phi t := by
+  by_cases h2 : t ≤ params.theta
+  · exact phase2_U_phi_nonneg ⟨ht.1, h2⟩
+  have htheta : params.theta < t := lt_of_not_ge h2
+  by_cases h3 : t ≤ eta
+  · exact phase3_U_phi_nonneg ⟨htheta.le, h3⟩
+  have hetaT : eta < t := lt_of_not_ge h3
+  by_cases h4 : t ≤ tau
+  · exact phase4_U_phi_nonneg ⟨hetaT.le, h4⟩
+  · exact phase5_U_phi_nonneg ⟨(lt_of_not_ge h4).le, ht.2⟩
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part C / Stage4 / No Hidden Match Facts
+-/
+
+@[expose] public section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Set
+open Stage2
+
+noncomputable section
+
+/-- Reflection on the `(alpha,beta)` coefficient plane induced by `t ↦ T-t`. -/
+def noHiddenReflAB (z : Point) : Point := (-z.2, -z.1)
+
+private theorem rot_dot_u_noHidden (t : ℝ) (z : Point) :
+    dot (Romik.rot t z) (u t) = z.1 :=
+  Stage3.rot_dot_u t z
+
+private theorem rot_dot_v_noHidden (t : ℝ) (z : Point) :
+    dot (Romik.rot t z) (v t) = z.2 :=
+  Stage3.rot_dot_v t z
+
+private theorem rot_injective_noHidden (t : ℝ) {z w : Point}
+    (h : Romik.rot t z = Romik.rot t w) : z = w :=
+  Stage3.rot_injective t h
+
+private theorem matchPrime12_noHidden :
+    Romik.pathPrime1 params params.phi = Romik.pathPrime2 params params.phi := by
+  have h10 := congrFun params_equations (10 : Fin 22)
+  have h11 := congrFun params_equations (11 : Fin 22)
+  simp [Romik.system] at h10 h11
+  exact Prod.ext (by linarith) (by linarith)
+
+private theorem matchPrime23_noHidden :
+    Romik.pathPrime2 params params.theta = Romik.pathPrime3 params params.theta := by
+  have h14 := congrFun params_equations (14 : Fin 22)
+  have h15 := congrFun params_equations (15 : Fin 22)
+  simp [Romik.system] at h14 h15
+  exact Prod.ext (by linarith) (by linarith)
+
+private theorem pathPrime1_eq_rot_noHidden (p : Romik.Params) (t : ℝ) :
+    Romik.pathPrime1 p t = Romik.rot t (Romik.alphaBeta1 p t) := rfl
+
+private theorem pathPrime2_eq_rot_noHidden (p : Romik.Params) (t : ℝ) :
+    Romik.pathPrime2 p t = Romik.rot t (Romik.alphaBeta2 p t) := rfl
+
+private theorem pathPrime3_eq_rot_noHidden (p : Romik.Params) (t : ℝ) :
+    Romik.pathPrime3 p t = Romik.rot t (Romik.alphaBeta3 p t) := rfl
+
+/-- Exact matching of the coefficient pair at the first switch. -/
+theorem alphaBeta_match12_direct :
+    Romik.alphaBeta1 params params.phi = Romik.alphaBeta2 params params.phi := by
+  apply rot_injective_noHidden params.phi
+  simpa [pathPrime1_eq_rot_noHidden, pathPrime2_eq_rot_noHidden] using matchPrime12_noHidden
+
+/-- Exact matching of the coefficient pair at the second switch. -/
+theorem alphaBeta_match23_direct :
+    Romik.alphaBeta2 params params.theta = Romik.alphaBeta3 params params.theta := by
+  apply rot_injective_noHidden params.theta
+  simpa [pathPrime2_eq_rot_noHidden, pathPrime3_eq_rot_noHidden] using matchPrime23_noHidden
+
+/-- Phase four is the reflected phase two coefficient pair. -/
+theorem alphaBeta4_reflect2_direct (s : ℝ) :
+    Romik.alphaBeta4 params (T - s) = noHiddenReflAB (Romik.alphaBeta2 params s) := by
+  dsimp [T, noHiddenReflAB, Romik.alphaBeta4, Romik.alphaBeta2]
+  rw [Romik.d1_eq_quarterPi_sub_b1_of_equations params_equations,
+      Romik.d2_eq_b2_add_quarterPi_correction_of_equations params_equations]
+  ring_nf
+
+/-- Phase three has the same coefficient reflection symmetry. -/
+theorem alphaBeta3_reflect_direct (s : ℝ) :
+    Romik.alphaBeta3 params (T - s) = noHiddenReflAB (Romik.alphaBeta3 params s) := by
+  dsimp [T, noHiddenReflAB, Romik.alphaBeta3]
+  rw [Romik.c2_eq_c1_sub_halfPi_of_equations params_equations]
+  ring_nf
+
+/-- Phase five is the reflected phase one coefficient pair. -/
+theorem alphaBeta5_reflect1_direct (s : ℝ) :
+    Romik.alphaBeta5 params (T - s) = noHiddenReflAB (Romik.alphaBeta1 params s) := by
+  dsimp [T, noHiddenReflAB, Romik.alphaBeta5, Romik.alphaBeta1]
+  rw [Romik.e1_eq_a1_of_equations params_equations,
+      Romik.e2_eq_neg_a2_of_equations params_equations]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  ring_nf
+
+/-- Exact matching of the coefficient pair at the third switch. -/
+theorem alphaBeta_match34_direct :
+    Romik.alphaBeta3 params eta = Romik.alphaBeta4 params eta := by
+  have h3 := alphaBeta3_reflect_direct params.theta
+  have h4 := alphaBeta4_reflect2_direct params.theta
+  have hm := congrArg noHiddenReflAB alphaBeta_match23_direct
+  simpa [eta] using h3.trans (hm.symm.trans h4.symm)
+
+/-- Exact matching of the coefficient pair at the fourth switch. -/
+theorem alphaBeta_match45_direct :
+    Romik.alphaBeta4 params tau = Romik.alphaBeta5 params tau := by
+  have h4 := alphaBeta4_reflect2_direct params.phi
+  have h5 := alphaBeta5_reflect1_direct params.phi
+  have hm := congrArg noHiddenReflAB alphaBeta_match12_direct
+  simpa [tau] using h4.trans (hm.symm.trans h5.symm)
+
+end
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 4: continuity of the matched velocity coefficients
+
+This file turns the five exact coefficient matching statements into a public
+continuity interface for `alphaBetaAt`, `alpha`, `beta`, `B`, and `D`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+private theorem alphaBeta1_continuous :
+    Continuous (Romik.alphaBeta1 params) := by
+  unfold Romik.alphaBeta1
+  fun_prop
+
+private theorem alphaBeta2_continuous :
+    Continuous (Romik.alphaBeta2 params) := by
+  unfold Romik.alphaBeta2
+  fun_prop
+
+private theorem alphaBeta3_continuous :
+    Continuous (Romik.alphaBeta3 params) := by
+  unfold Romik.alphaBeta3
+  fun_prop
+
+private theorem alphaBeta4_continuous :
+    Continuous (Romik.alphaBeta4 params) := by
+  unfold Romik.alphaBeta4
+  fun_prop
+
+private theorem alphaBeta5_continuous :
+    Continuous (Romik.alphaBeta5 params) := by
+  unfold Romik.alphaBeta5
+  fun_prop
+
+/-- The literal nested-if velocity coefficient is continuous across all four
+switches. -/
+theorem alphaBetaAt_continuous : Continuous alphaBetaAt := by
+  have h45 : Continuous
+      (fun t : ℝ =>
+        if t ≤ tau then Romik.alphaBeta4 params t
+        else Romik.alphaBeta5 params t) := by
+    exact alphaBeta4_continuous.if_le alphaBeta5_continuous
+      continuous_id continuous_const (by
+        intro t ht
+        subst t
+        exact alphaBeta_match45_direct)
+  have h345 : Continuous
+      (fun t : ℝ =>
+        if t ≤ eta then Romik.alphaBeta3 params t
+        else if t ≤ tau then Romik.alphaBeta4 params t
+        else Romik.alphaBeta5 params t) := by
+    exact alphaBeta3_continuous.if_le h45
+      continuous_id continuous_const (by
+        intro t ht
+        subst t
+        rw [ite_eq_left eta_lt_tau.le]
+        exact alphaBeta_match34_direct)
+  have h2345 : Continuous
+      (fun t : ℝ =>
+        if t ≤ params.theta then Romik.alphaBeta2 params t
+        else if t ≤ eta then Romik.alphaBeta3 params t
+        else if t ≤ tau then Romik.alphaBeta4 params t
+        else Romik.alphaBeta5 params t) := by
+    exact alphaBeta2_continuous.if_le h345
+      continuous_id continuous_const (by
+        intro t ht
+        subst t
+        rw [ite_eq_left theta_lt_eta.le]
+        exact alphaBeta_match23_direct)
+  have h12345 : Continuous
+      (fun t : ℝ =>
+        if t ≤ params.phi then Romik.alphaBeta1 params t
+        else if t ≤ params.theta then Romik.alphaBeta2 params t
+        else if t ≤ eta then Romik.alphaBeta3 params t
+        else if t ≤ tau then Romik.alphaBeta4 params t
+        else Romik.alphaBeta5 params t) := by
+    exact alphaBeta1_continuous.if_le h2345
+      continuous_id continuous_const (by
+        intro t ht
+        subst t
+        rw [ite_eq_left phi_lt_theta.le]
+        exact alphaBeta_match12_direct)
+  unfold alphaBetaAt
+  exact h12345
+
+theorem alpha_continuous : Continuous alpha := by
+  unfold alpha
+  exact continuous_fst.comp alphaBetaAt_continuous
+
+theorem beta_continuous : Continuous beta := by
+  unfold beta
+  exact continuous_snd.comp alphaBetaAt_continuous
+
+private theorem u_continuous : Continuous u := by
+  unfold u
+  fun_prop
+
+private theorem v_continuous : Continuous v := by
+  unfold v
+  fun_prop
+
+/-- Public continuity of the early reflected contact curve. -/
+theorem D_continuous : Continuous D := by
+  have hx := pathContinuous
+  have hfirst := (continuous_fst.comp hx).sub
+    (beta_continuous.mul (continuous_fst.comp u_continuous))
+  have hsecond := (continuous_snd.comp hx).sub
+    (beta_continuous.mul (continuous_snd.comp u_continuous))
+  change Continuous (fun t : ℝ =>
+    ((Romik.path params t).1 - beta t * (u t).1,
+     (Romik.path params t).2 - beta t * (u t).2))
+  exact hfirst.prodMk hsecond
+
+/-- Public continuity of the late reflected contact curve. -/
+theorem B_continuous : Continuous B := by
+  have hx := pathContinuous
+  have hfirst := (continuous_fst.comp hx).add
+    (alpha_continuous.mul (continuous_fst.comp v_continuous))
+  have hsecond := (continuous_snd.comp hx).add
+    (alpha_continuous.mul (continuous_snd.comp v_continuous))
+  change Continuous (fun t : ℝ =>
+    ((Romik.path params t).1 + alpha t * (v t).1,
+     (Romik.path params t).2 + alpha t * (v t).2))
+  exact hfirst.prodMk hsecond
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part C / Stage4 / No Hidden Sign Facts
+-/
+
+@[expose] public section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Set
+open Stage2
+
+noncomputable section
+
+private theorem phi_upper_twentieth : params.phi ≤ (1 / 20 : ℝ) := by
+  have h := phi_bounds.2
+  norm_num at h ⊢
+  linarith
+
+private theorem a1_lower_six_fifths : (6 / 5 : ℝ) ≤ params.a1 := by
+  have h := Romik.a1_lower_bound_of_mem_box params_mem
+  norm_num at h ⊢
+  linarith
+
+private theorem b1_upper_neg_half : params.b1 ≤ (-1 / 2 : ℝ) := by
+  have h := PartB.b1_contains.2
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at h ⊢
+  linarith
+
+private theorem d1_lower_thirteen_tenths : (13 / 10 : ℝ) ≤ params.d1 := by
+  have h := PartB.d1_contains.1
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at h ⊢
+  linarith
+
+private theorem T_lt_eight_fifths : T < (8 / 5 : ℝ) := by
+  have hp := ExactReplay.piI_contains_pi
+  have hhi : Real.pi ≤ (ExactReplay.piI.hi : ℝ) := hp.2
+  have h32 : (ExactReplay.piI.hi : ℝ) < (16 / 5 : ℝ) := by
+    norm_num [ExactReplay.piI, ExactReplay.q]
+  dsimp [T]
+  linarith
+
+private theorem alphaBeta1_fst_nonpos_small {s : ℝ}
+    (hs0 : 0 ≤ s) (hs20 : s ≤ (1 / 20 : ℝ)) :
+    (Romik.alphaBeta1 params s).1 ≤ 0 := by
+  have hpi : s ≤ Real.pi := by
+    nlinarith [Real.pi_gt_three]
+  have hsin0 : 0 ≤ Real.sin s :=
+    Real.sin_nonneg_of_nonneg_of_le_pi hs0 hpi
+  have hsinLower := Real.sin_ge_sub_cube hs0
+  have hcosLower : 1 - s ^ 2 / 2 ≤ Real.cos s :=
+    Real.one_sub_sq_div_two_le_cos
+  have hquad : 0 ≤ s * ((1 / 20 : ℝ) - s) :=
+    mul_nonneg hs0 (sub_nonneg.mpr hs20)
+  have hcube : 0 ≤ s ^ 2 * ((1 / 20 : ℝ) - s) :=
+    mul_nonneg (sq_nonneg s) (sub_nonneg.mpr hs20)
+  have hcoef : (12 / 5 : ℝ) ≤ 2 * params.a1 := by
+    nlinarith [a1_lower_six_fifths]
+  have hmul : (12 / 5 : ℝ) * Real.sin s ≤ 2 * params.a1 * Real.sin s :=
+    mul_le_mul_of_nonneg_right hcoef hsin0
+  have ha2 := Romik.a2_eq_neg_quarter_of_equations params_equations
+  dsimp [Romik.alphaBeta1]
+  rw [ha2]
+  nlinarith [hsinLower, hcosLower, hquad, hcube, hmul]
+
+private theorem alphaBeta1_snd_nonneg_small {s : ℝ}
+    (hs0 : 0 ≤ s) (hs20 : s ≤ (1 / 20 : ℝ)) :
+    0 ≤ (Romik.alphaBeta1 params s).2 := by
+  have hpi : s ≤ Real.pi := by
+    nlinarith [Real.pi_gt_three]
+  have hsin0 : 0 ≤ Real.sin s :=
+    Real.sin_nonneg_of_nonneg_of_le_pi hs0 hpi
+  have hsinUpper := Real.sin_le hs0
+  have hcosLower : 1 - s ^ 2 / 2 ≤ Real.cos s :=
+    Real.one_sub_sq_div_two_le_cos
+  have hcos0 : 0 ≤ Real.cos s := by
+    apply Real.cos_nonneg_of_mem_Icc
+    constructor <;> nlinarith [Real.pi_gt_three]
+  have hquad : 0 ≤ s * ((1 / 20 : ℝ) - s) :=
+    mul_nonneg hs0 (sub_nonneg.mpr hs20)
+  have hcoef : (12 / 5 : ℝ) ≤ 2 * params.a1 := by
+    nlinarith [a1_lower_six_fifths]
+  have hmul : (12 / 5 : ℝ) * Real.cos s ≤ 2 * params.a1 * Real.cos s :=
+    mul_le_mul_of_nonneg_right hcoef hcos0
+  have ha2 := Romik.a2_eq_neg_quarter_of_equations params_equations
+  dsimp [Romik.alphaBeta1]
+  rw [ha2]
+  nlinarith [hsinUpper, hcosLower, hquad, hmul]
+
+private theorem alphaBeta5_fst_nonpos_tail {t : ℝ}
+    (ht : t ∈ Icc tau T) :
+    (Romik.alphaBeta5 params t).1 ≤ 0 := by
+  let s : ℝ := T - t
+  have hs0 : 0 ≤ s := by
+    dsimp [s]
+    linarith [ht.2]
+  have hsphi : s ≤ params.phi := by
+    have hTtau : T - tau = params.phi := by
+      dsimp [tau]
+      ring
+    dsimp [s]
+    rw [← hTtau]
+    linarith [ht.1]
+  have hs20 : s ≤ (1 / 20 : ℝ) := le_trans hsphi phi_upper_twentieth
+  have hb := alphaBeta1_snd_nonneg_small hs0 hs20
+  have href := congrArg Prod.fst (alphaBeta5_reflect1_direct s)
+  have hTs : T - s = t := by
+    dsimp [s]
+    ring
+  rw [hTs] at href
+  have heq : (Romik.alphaBeta5 params t).1 = -(Romik.alphaBeta1 params s).2 := by
+    simpa [noHiddenReflAB] using href
+  rw [heq]
+  exact neg_nonpos.mpr hb
+
+private theorem alphaBeta5_snd_nonneg_tail {t : ℝ}
+    (ht : t ∈ Icc tau T) :
+    0 ≤ (Romik.alphaBeta5 params t).2 := by
+  let s : ℝ := T - t
+  have hs0 : 0 ≤ s := by
+    dsimp [s]
+    linarith [ht.2]
+  have hsphi : s ≤ params.phi := by
+    have hTtau : T - tau = params.phi := by
+      dsimp [tau]
+      ring
+    dsimp [s]
+    rw [← hTtau]
+    linarith [ht.1]
+  have hs20 : s ≤ (1 / 20 : ℝ) := le_trans hsphi phi_upper_twentieth
+  have ha := alphaBeta1_fst_nonpos_small hs0 hs20
+  have href := congrArg Prod.snd (alphaBeta5_reflect1_direct s)
+  have hTs : T - s = t := by
+    dsimp [s]
+    ring
+  rw [hTs] at href
+  have heq : (Romik.alphaBeta5 params t).2 = -(Romik.alphaBeta1 params s).1 := by
+    simpa [noHiddenReflAB] using href
+  rw [heq]
+  exact neg_nonneg.mpr ha
+
+/-- The `u_t` coefficient of the Gerver velocity is nonpositive on the
+entire no-hidden `U` domain. -/
+theorem alpha_nonpos {t : ℝ} (ht : t ∈ Icc params.phi T) :
+    alpha t ≤ 0 := by
+  have htPhysical : t ∈ Icc (0 : ℝ) T :=
+    ⟨le_trans phi_nonneg ht.1, ht.2⟩
+  by_cases hphi : t ≤ params.phi
+  · have heq : t = params.phi := le_antisymm hphi ht.1
+    subst t
+    have h := alphaBeta1_fst_nonpos_small phi_nonneg phi_upper_twentieth
+    simpa [alpha, alphaBetaAt] using h
+  · by_cases htheta : t ≤ params.theta
+    · simp only [alpha, alphaBetaAt, ite_eq_right hphi, ite_eq_left htheta]
+      dsimp [Romik.alphaBeta2]
+      have ht0 : 0 ≤ t := le_trans phi_nonneg ht.1
+      have hb : 1 + 2 * params.b1 ≤ 0 := by
+        nlinarith [b1_upper_neg_half]
+      nlinarith
+    · by_cases heta : t ≤ eta
+      · have hrho := Stage2.rhoC_nonneg htPhysical
+        unfold Stage2.rhoC at hrho
+        rw [ite_eq_right hphi, ite_eq_right htheta, ite_eq_left heta] at hrho
+        simp only [alpha, alphaBetaAt, ite_eq_right hphi, ite_eq_right htheta, ite_eq_left heta]
+        dsimp [Romik.alphaBeta3]
+        nlinarith
+      · by_cases htau : t ≤ tau
+        · have hrho := Stage2.rhoC_nonneg htPhysical
+          unfold Stage2.rhoC at hrho
+          rw [ite_eq_right hphi, ite_eq_right htheta, ite_eq_right heta, ite_eq_left htau] at hrho
+          simp only [alpha, alphaBetaAt, ite_eq_right hphi, ite_eq_right htheta, ite_eq_right
+            heta, ite_eq_left htau]
+          dsimp [Romik.alphaBeta4]
+          nlinarith
+        · have ht5 : t ∈ Icc tau T := ⟨le_of_lt (lt_of_not_ge htau), ht.2⟩
+          have h5 := alphaBeta5_fst_nonpos_tail ht5
+          simpa [alpha, alphaBetaAt, hphi, htheta, heta, htau] using h5
+
+/-- The `v_t` coefficient of the Gerver velocity is nonnegative on the
+entire no-hidden `U` domain. -/
+theorem beta_nonneg {t : ℝ} (ht : t ∈ Icc params.phi T) :
+    0 ≤ beta t := by
+  have htPhysical : t ∈ Icc (0 : ℝ) T :=
+    ⟨le_trans phi_nonneg ht.1, ht.2⟩
+  by_cases hphi : t ≤ params.phi
+  · have heq : t = params.phi := le_antisymm hphi ht.1
+    subst t
+    have h := alphaBeta1_snd_nonneg_small phi_nonneg phi_upper_twentieth
+    simpa [beta, alphaBetaAt] using h
+  · by_cases htheta : t ≤ params.theta
+    · have hrho := Stage2.rhoA_nonneg htPhysical
+      unfold Stage2.rhoA at hrho
+      rw [ite_eq_right hphi, ite_eq_left htheta] at hrho
+      simp only [beta, alphaBetaAt, ite_eq_right hphi, ite_eq_left htheta]
+      dsimp [Romik.alphaBeta2]
+      exact hrho
+    · by_cases heta : t ≤ eta
+      · have hrho := Stage2.rhoA_nonneg htPhysical
+        unfold Stage2.rhoA at hrho
+        rw [ite_eq_right hphi, ite_eq_right htheta, ite_eq_left heta] at hrho
+        simp only [beta, alphaBetaAt, ite_eq_right hphi, ite_eq_right htheta, ite_eq_left heta]
+        dsimp [Romik.alphaBeta3]
+        exact hrho
+      · by_cases htau : t ≤ tau
+        · simp only [beta, alphaBetaAt, ite_eq_right hphi, ite_eq_right htheta, ite_eq_right heta,
+          ite_eq_left htau]
+          dsimp [Romik.alphaBeta4]
+          nlinarith [d1_lower_thirteen_tenths, T_lt_eight_fifths, ht.2]
+        · have ht5 : t ∈ Icc tau T := ⟨le_of_lt (lt_of_not_ge htau), ht.2⟩
+          have h5 := alphaBeta5_snd_nonneg_tail ht5
+          simpa [beta, alphaBetaAt, hphi, htheta, heta, htau] using h5
+
+end
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 4: positive turning determinant on phases 2--5
+
+The determinant is the signed turning numerator of the nonzero path velocity.
+These phasewise facts are the analytic input for the remaining one-turn chord
+argument; they do not assert the no-hidden conclusion by themselves.
+-/
+
+@[expose] public section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Set
+open Stage2
+
+noncomputable section
+
+private theorem b1_upper_neg_half_turn : params.b1 ≤ (-1 / 2 : ℝ) := by
+  have h := PartB.b1_contains.2
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at h ⊢
+  linarith
+
+private theorem d1_lower_thirteen_tenths_turn : (13 / 10 : ℝ) ≤ params.d1 := by
+  have h := PartB.d1_contains.1
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at h ⊢
+  linarith
+
+private theorem T_lt_eight_fifths_turn : T < (8 / 5 : ℝ) := by
+  have hp := ExactReplay.piI_contains_pi
+  have hhi : Real.pi ≤ (ExactReplay.piI.hi : ℝ) := hp.2
+  have h32 : (ExactReplay.piI.hi : ℝ) < (16 / 5 : ℝ) := by
+    norm_num [ExactReplay.piI, ExactReplay.q]
+  dsimp [T]
+  linarith
+
+/-! The next six statements expose the coefficient-derivative signs used in
+the manuscript's tangent-angle argument.  They are deliberately stated for
+the explicit smooth-phase formulae; no derivative is assigned at a switch. -/
+
+theorem phase2_betaPrime_neg {r : ℝ}
+    (hr : r ∈ Ioo params.phi params.theta) :
+    -(1 / 2 : ℝ) * r + params.b1 < 0 := by
+  have hr0 : 0 ≤ r := le_trans phi_nonneg hr.1.le
+  nlinarith [b1_upper_neg_half_turn]
+
+theorem phase4_alphaPrime_neg {r : ℝ} (hr : r ∈ Ioo eta tau) :
+    (1 / 2 : ℝ) * r - params.d1 < 0 := by
+  have hrT : r < T := lt_trans hr.2 tau_lt_T
+  nlinarith [d1_lower_thirteen_tenths_turn, T_lt_eight_fifths_turn]
+
+private theorem a1_lower_six_fifths_turn : (6 / 5 : ℝ) ≤ params.a1 := by
+  have h := Romik.a1_lower_bound_of_mem_box params_mem
+  norm_num at h ⊢
+  linarith
+
+private theorem phi_upper_twentieth_turn : params.phi ≤ (1 / 20 : ℝ) := by
+  have h := phi_bounds.2
+  norm_num at h ⊢
+  linarith
+
+private theorem alpha1_prime_nonpos_small_turn {s : ℝ}
+    (hs0 : 0 ≤ s) (hs20 : s ≤ (1 / 20 : ℝ)) :
+    -2 * params.a1 * Real.cos s - 2 * params.a2 * Real.sin s ≤ 0 := by
+  have hsinUpper := Real.sin_le hs0
+  have hcosLower : 1 - s ^ 2 / 2 ≤ Real.cos s :=
+    Real.one_sub_sq_div_two_le_cos
+  have hcos0 : 0 ≤ Real.cos s := by
+    apply Real.cos_nonneg_of_mem_Icc
+    constructor <;> nlinarith [Real.pi_gt_three]
+  have hcoef : (12 / 5 : ℝ) ≤ 2 * params.a1 := by
+    nlinarith [a1_lower_six_fifths_turn]
+  have hmul : (12 / 5 : ℝ) * Real.cos s ≤ 2 * params.a1 * Real.cos s :=
+    mul_le_mul_of_nonneg_right hcoef hcos0
+  have ha2 := Romik.a2_eq_neg_quarter_of_equations params_equations
+  rw [ha2]
+  have hquad : 0 ≤ s * ((1 / 20 : ℝ) - s) :=
+    mul_nonneg hs0 (sub_nonneg.mpr hs20)
+  nlinarith [hsinUpper, hcosLower, hmul, hquad]
+
+private theorem beta1_prime_nonpos_small_turn {s : ℝ}
+    (hs0 : 0 ≤ s) (hs20 : s ≤ (1 / 20 : ℝ)) :
+    -2 * params.a1 * Real.sin s + 2 * params.a2 * Real.cos s ≤ 0 := by
+  have hpi : s ≤ Real.pi := by nlinarith [Real.pi_gt_three]
+  have hsin0 : 0 ≤ Real.sin s :=
+    Real.sin_nonneg_of_nonneg_of_le_pi hs0 hpi
+  have hcos0 : 0 ≤ Real.cos s := by
+    apply Real.cos_nonneg_of_mem_Icc
+    constructor <;> nlinarith [Real.pi_gt_three]
+  have ha1 : 0 ≤ params.a1 := le_trans (by norm_num : (0 : ℝ) ≤ 6 / 5)
+    a1_lower_six_fifths_turn
+  have ha2 := Romik.a2_eq_neg_quarter_of_equations params_equations
+  rw [ha2]
+  nlinarith [mul_nonneg ha1 hsin0, hcos0]
+
+private theorem phase5_derivative_signs_turn {t : ℝ}
+    (ht : t ∈ Ioo tau T) :
+    (-2 * params.e1 * Real.cos t - 2 * params.e2 * Real.sin t ≤ 0) ∧
+    (-2 * params.e1 * Real.sin t + 2 * params.e2 * Real.cos t ≤ 0) := by
+  let s : ℝ := T - t
+  have hs0 : 0 ≤ s := by dsimp [s]; linarith [ht.2]
+  have hsphi : s ≤ params.phi := by
+    have hTtau : T - tau = params.phi := by
+      dsimp [tau]
+      ring
+    dsimp [s]
+    rw [← hTtau]
+    linarith [ht.1]
+  have hs20 : s ≤ (1 / 20 : ℝ) := le_trans hsphi phi_upper_twentieth_turn
+  have hap1 := alpha1_prime_nonpos_small_turn hs0 hs20
+  have hbp1 := beta1_prime_nonpos_small_turn hs0 hs20
+  have he1 := Romik.e1_eq_a1_of_equations params_equations
+  have he2 := Romik.e2_eq_neg_a2_of_equations params_equations
+  have hsin := Real.sin_pi_div_two_sub s
+  have hcos := Real.cos_pi_div_two_sub s
+  have hts : t = T - s := by dsimp [s]; ring
+  constructor
+  · rw [hts]
+    dsimp [T]
+    rw [he1, he2, hcos, hsin]
+    simpa [mul_add, add_mul] using hbp1
+  · rw [hts]
+    dsimp [T]
+    rw [he1, he2, hsin, hcos]
+    simpa [mul_add, add_mul] using hap1
+
+/-- Phase-five coefficient derivatives are nonpositive.  This is the exact
+formula-level input used below; strict turning follows from the strict
+nonvanishing of the first coefficient. -/
+theorem phase5_coefficientPrime_nonpos {t : ℝ}
+    (ht : t ∈ Ioo tau T) :
+    (-2 * params.e1 * Real.cos t - 2 * params.e2 * Real.sin t ≤ 0) ∧
+    (-2 * params.e1 * Real.sin t + 2 * params.e2 * Real.cos t ≤ 0) :=
+  phase5_derivative_signs_turn ht
+
+end
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 4: global path derivative and one-turn velocity monotonicity
+
+The path derivative is glued across the four switches using equality of both
+the path values and the body-frame velocity coefficients.  The coefficient
+derivatives themselves need not match at a switch.  Consequently the fixed
+projection is proved antitone phase by phase and then glued order-theoretically.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Set
+open Stage2
+
+private theorem hasDerivAt_if_le_point
+    {f g f' g' : ℝ → Point} {c x : ℝ}
+    (hf : HasDerivAt f (f' x) x)
+    (hg : HasDerivAt g (g' x) x)
+    (hvalue : f c = g c)
+    (hderiv : f' c = g' c) :
+    HasDerivAt (fun y => if y ≤ c then f y else g y)
+      (if x ≤ c then f' x else g' x) x := by
+  rcases lt_trichotomy x c with hxc | hxc | hxc
+  · rw [ite_eq_left hxc.le]
+    apply hf.congr_of_eventuallyEq
+    filter_upwards [Iio_mem_nhds hxc] with y hy
+    have hyc : y < c := show y < c from hy
+    simp [hyc.le]
+  · subst x
+    rw [ite_eq_left le_rfl]
+    have hleft : HasDerivWithinAt
+        (fun y => if y ≤ c then f y else g y) (f' c) (Iic c) c := by
+      exact hf.hasDerivWithinAt.congr
+        (by
+          intro y hy
+          have hyc : y ≤ c := show y ≤ c from hy
+          simp [hyc])
+        (by simp)
+    have hright : HasDerivWithinAt
+        (fun y => if y ≤ c then f y else g y) (f' c) (Ici c) c := by
+      have hg' : HasDerivWithinAt g (f' c) (Ici c) c :=
+        hg.hasDerivWithinAt.congr_deriv hderiv.symm
+      exact hg'.congr
+        (by
+          intro y hy
+          have hcy : c ≤ y := show c ≤ y from hy
+          rcases hcy.eq_or_lt with h | h
+          · subst y
+            simpa using hvalue
+          · simp [not_le.mpr h])
+        (by simpa using hvalue)
+    simpa only [Iic_union_Ici, hasDerivWithinAt_univ] using
+      hleft.union hright
+  · rw [ite_eq_right (not_le.mpr hxc)]
+    apply hg.congr_of_eventuallyEq
+    filter_upwards [Ioi_mem_nhds hxc] with y hy
+    have hcy : c < y := show c < y from hy
+    simp [not_le.mpr hcy]
+
+/-- The literal five-piece Gerver path is differentiable at the switches as
+well as in the phase interiors. -/
+theorem path_hasDerivAt_noHidden (x : ℝ) :
+    HasDerivAt (Romik.path params)
+      (Romik.rot x (alphaBetaAt x)) x := by
+  let p45 : ℝ → Point := fun y =>
+    if y ≤ tau then Romik.path4 params y else Romik.path5 params y
+  let d45 : ℝ → Point := fun y =>
+    if y ≤ tau then Romik.rot y (Romik.alphaBeta4 params y)
+    else Romik.rot y (Romik.alphaBeta5 params y)
+  have hp45 (y : ℝ) : HasDerivAt p45 (d45 y) y := by
+    apply hasDerivAt_if_le_point
+      (path4_hasDerivAt_public y) (path5_hasDerivAt_public y)
+    · simpa [tau, T] using PartB.match45
+    · exact congrArg (Romik.rot tau) alphaBeta_match45_direct
+  let p345 : ℝ → Point := fun y =>
+    if y ≤ eta then Romik.path3 params y else p45 y
+  let d345 : ℝ → Point := fun y =>
+    if y ≤ eta then Romik.rot y (Romik.alphaBeta3 params y) else d45 y
+  have hp345 (y : ℝ) : HasDerivAt p345 (d345 y) y := by
+    apply hasDerivAt_if_le_point (path3_hasDerivAt_public y) (hp45 y)
+    · simp only [p45, ite_eq_left eta_lt_tau.le]
+      simpa [eta, T] using PartB.match34
+    · simp only [d45, ite_eq_left eta_lt_tau.le]
+      exact congrArg (Romik.rot eta) alphaBeta_match34_direct
+  let p2345 : ℝ → Point := fun y =>
+    if y ≤ params.theta then Romik.path2 params y else p345 y
+  let d2345 : ℝ → Point := fun y =>
+    if y ≤ params.theta then Romik.rot y (Romik.alphaBeta2 params y)
+    else d345 y
+  have hp2345 (y : ℝ) : HasDerivAt p2345 (d2345 y) y := by
+    apply hasDerivAt_if_le_point (path2_hasDerivAt_public y) (hp345 y)
+    · simp only [p345, ite_eq_left theta_lt_eta.le]
+      exact PartB.match23
+    · simp only [d345, ite_eq_left theta_lt_eta.le]
+      exact congrArg (Romik.rot params.theta) alphaBeta_match23_direct
+  let p12345 : ℝ → Point := fun y =>
+    if y ≤ params.phi then Romik.path1 params y else p2345 y
+  let d12345 : ℝ → Point := fun y =>
+    if y ≤ params.phi then Romik.rot y (Romik.alphaBeta1 params y)
+    else d2345 y
+  have hp12345 : HasDerivAt p12345 (d12345 x) x := by
+    apply hasDerivAt_if_le_point (path1_hasDerivAt_public x) (hp2345 x)
+    · simp only [p2345, ite_eq_left phi_lt_theta.le]
+      exact PartB.match12
+    · simp only [d2345, ite_eq_left phi_lt_theta.le]
+      exact congrArg (Romik.rot params.phi) alphaBeta_match12_direct
+  have hpEq : p12345 = Romik.path params := by
+    funext y
+    simp only [p45, p345, p2345, p12345, Romik.path, eta, tau, T]
+    rfl
+  have hdEq : d12345 x = Romik.rot x (alphaBetaAt x) := by
+    simp only [d45, d345, d2345, d12345, alphaBetaAt]
+    by_cases h1 : x ≤ params.phi
+    · simp only [ite_eq_left h1]
+    · by_cases h2 : x ≤ params.theta
+      · simp only [ite_eq_right h1, ite_eq_left h2]
+      · by_cases h3 : x ≤ eta
+        · simp only [ite_eq_right h1, ite_eq_right h2, ite_eq_left h3]
+        · by_cases h4 : x ≤ tau
+          · simp only [ite_eq_right h1, ite_eq_right h2, ite_eq_right h3, ite_eq_left h4]
+          · simp only [ite_eq_right h1, ite_eq_right h2, ite_eq_right h3, ite_eq_right h4]
+  rw [hpEq] at hp12345
+  exact hp12345.congr_deriv hdEq
+
+/-- Scalar projection of the corner velocity on the fixed vector `u(t)`. -/
+def uVelocity (t r : ℝ) : ℝ :=
+  alpha r * Real.cos (t - r) + beta r * Real.sin (t - r)
+
+private def uVelocityPiece (ab : ℝ → Point) (t r : ℝ) : ℝ :=
+  (ab r).1 * Real.cos (t - r) + (ab r).2 * Real.sin (t - r)
+
+private theorem square_hasDerivAt_noHidden (r : ℝ) :
+    HasDerivAt (fun s : ℝ => s * s) (r + r) r :=
+  Stage2.square_hasDerivAt r
+
+private theorem alpha2_hasDerivAt (r : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta2 params s).1) (-1) r := by
+  dsimp [Romik.alphaBeta2]
+  have h := (hasDerivAt_const r (1 + 2 * params.b1)).fun_sub
+    (hasDerivAt_id r)
+  exact h.congr_deriv (by ring)
+
+private theorem beta2_hasDerivAt (r : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta2 params s).2)
+      (-(1 / 2 : ℝ) * r + params.b1) r := by
+  dsimp [Romik.alphaBeta2]
+  have hsq := HasDerivAt.const_mul (-(1 / 4 : ℝ))
+    (square_hasDerivAt_noHidden r)
+  have hlin := HasDerivAt.const_mul params.b1 (hasDerivAt_id r)
+  have hraw := (((hsq.fun_add hlin).fun_add (hasDerivAt_const r params.b2)).fun_add
+    (hasDerivAt_const r (1 / 2 : ℝ)))
+  simpa only [id_eq, mul_assoc] using hraw.congr_deriv (by ring)
+
+private theorem alpha3_hasDerivAt (r : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta3 params s).1) (-1) r := by
+  dsimp [Romik.alphaBeta3]
+  have h := (hasDerivAt_const r (-1 - params.c2)).fun_sub
+    (hasDerivAt_id r)
+  exact h.congr_deriv (by ring)
+
+private theorem beta3_hasDerivAt (r : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta3 params s).2) (-1) r := by
+  dsimp [Romik.alphaBeta3]
+  have h := (hasDerivAt_const r (1 + params.c1)).fun_sub
+    (hasDerivAt_id r)
+  exact h.congr_deriv (by ring)
+
+private theorem alpha4_hasDerivAt (r : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta4 params s).1)
+      ((1 / 2 : ℝ) * r - params.d1) r := by
+  dsimp [Romik.alphaBeta4]
+  have hsq := HasDerivAt.const_mul (1 / 4 : ℝ)
+    (square_hasDerivAt_noHidden r)
+  have hlin := HasDerivAt.const_mul params.d1 (hasDerivAt_id r)
+  have hraw := (((hsq.fun_sub hlin).fun_sub (hasDerivAt_const r params.d2)).fun_sub
+    (hasDerivAt_const r (1 / 2 : ℝ)))
+  simpa only [id_eq, mul_assoc] using hraw.congr_deriv (by ring)
+
+private theorem beta4_hasDerivAt (r : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta4 params s).2) (-1) r := by
+  dsimp [Romik.alphaBeta4]
+  have h := (hasDerivAt_const r (2 * params.d1 - 1)).fun_sub
+    (hasDerivAt_id r)
+  exact h.congr_deriv (by ring)
+
+private theorem alpha5_hasDerivAt (r : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta5 params s).1)
+      (-2 * params.e1 * Real.cos r - 2 * params.e2 * Real.sin r) r := by
+  dsimp [Romik.alphaBeta5]
+  have hsin := HasDerivAt.const_mul (2 * params.e1)
+    (Real.hasDerivAt_sin r)
+  have hcos := HasDerivAt.const_mul (2 * params.e2)
+    (Real.hasDerivAt_cos r)
+  have hraw := ((hasDerivAt_const r (1 : ℝ)).fun_sub hsin).fun_add hcos
+  exact hraw.congr_deriv (by ring)
+
+private theorem beta5_hasDerivAt (r : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta5 params s).2)
+      (-2 * params.e1 * Real.sin r + 2 * params.e2 * Real.cos r) r := by
+  dsimp [Romik.alphaBeta5]
+  have hcos := HasDerivAt.const_mul (2 * params.e1)
+    (Real.hasDerivAt_cos r)
+  have hsin := HasDerivAt.const_mul (2 * params.e2)
+    (Real.hasDerivAt_sin r)
+  have hraw := (hcos.fun_add hsin).fun_sub
+    (hasDerivAt_const r (1 / 2 : ℝ))
+  exact hraw.congr_deriv (by ring)
+
+private theorem alphaBetaAt_eq_piece2 {r : ℝ}
+    (hr : r ∈ Icc params.phi params.theta) :
+    alphaBetaAt r = Romik.alphaBeta2 params r := by
+  by_cases hphi : r ≤ params.phi
+  · have hrphi : r = params.phi := le_antisymm hphi hr.1
+    subst r
+    simpa [alphaBetaAt] using alphaBeta_match12_direct
+  · simp [alphaBetaAt, hphi, hr.2]
+
+private theorem alphaBetaAt_eq_piece3 {r : ℝ}
+    (hr : r ∈ Icc params.theta eta) :
+    alphaBetaAt r = Romik.alphaBeta3 params r := by
+  by_cases htheta : r ≤ params.theta
+  · have hrtheta : r = params.theta := le_antisymm htheta hr.1
+    subst r
+    simp only [alphaBetaAt, ite_eq_right (not_le.mpr phi_lt_theta), ite_eq_left le_rfl]
+    exact alphaBeta_match23_direct
+  · have hphi : ¬ r ≤ params.phi :=
+      not_le.mpr (lt_trans phi_lt_theta (lt_of_not_ge htheta))
+    simp [alphaBetaAt, hphi, htheta, hr.2]
+
+private theorem alphaBetaAt_eq_piece4 {r : ℝ}
+    (hr : r ∈ Icc eta tau) :
+    alphaBetaAt r = Romik.alphaBeta4 params r := by
+  by_cases heta : r ≤ eta
+  · have hreta : r = eta := le_antisymm heta hr.1
+    subst r
+    simp only [alphaBetaAt, ite_eq_right (not_le.mpr phi_lt_eta),
+      ite_eq_right (not_le.mpr theta_lt_eta), ite_eq_left le_rfl]
+    exact alphaBeta_match34_direct
+  · have htheta : ¬ r ≤ params.theta :=
+      not_le.mpr (lt_trans theta_lt_eta (lt_of_not_ge heta))
+    have hphi : ¬ r ≤ params.phi :=
+      not_le.mpr (lt_trans phi_lt_theta (lt_of_not_ge htheta))
+    simp [alphaBetaAt, hphi, htheta, heta, hr.2]
+
+private theorem alphaBetaAt_eq_piece5 {r : ℝ}
+    (hr : r ∈ Icc tau T) :
+    alphaBetaAt r = Romik.alphaBeta5 params r := by
+  by_cases htau : r ≤ tau
+  · have hrtau : r = tau := le_antisymm htau hr.1
+    subst r
+    simp only [alphaBetaAt, ite_eq_right (not_le.mpr
+      (lt_trans phi_lt_theta (lt_trans theta_lt_eta eta_lt_tau))),
+      ite_eq_right (not_le.mpr (lt_trans theta_lt_eta eta_lt_tau)),
+      ite_eq_right (not_le.mpr eta_lt_tau), ite_eq_left le_rfl]
+    exact alphaBeta_match45_direct
+  · have heta : ¬ r ≤ eta :=
+      not_le.mpr (lt_trans eta_lt_tau (lt_of_not_ge htau))
+    have htheta : ¬ r ≤ params.theta :=
+      not_le.mpr (lt_trans theta_lt_eta (lt_of_not_ge heta))
+    have hphi : ¬ r ≤ params.phi :=
+      not_le.mpr (lt_trans phi_lt_theta (lt_of_not_ge htheta))
+    simp [alphaBetaAt, hphi, htheta, heta, htau]
+
+private theorem uVelocityPiece_hasDerivAt
+    {a b : ℝ → ℝ} {ap bp t r : ℝ}
+    (ha : HasDerivAt a ap r) (hb : HasDerivAt b bp r) :
+    HasDerivAt
+      (fun s => a s * Real.cos (t - s) + b s * Real.sin (t - s))
+      ((ap - b r) * Real.cos (t - r) +
+        (a r + bp) * Real.sin (t - r)) r := by
+  have hsub : HasDerivAt (fun s : ℝ => t - s) (-1) r :=
+    by simpa only [id_eq, zero_sub] using
+      ((hasDerivAt_const r t).fun_sub (hasDerivAt_id r))
+  have hcos : HasDerivAt (fun s : ℝ => Real.cos (t - s))
+      (Real.sin (t - r)) r := by
+    have h := (Real.hasDerivAt_cos (t - r)).comp r hsub
+    exact h.congr_deriv (by ring)
+  have hsin : HasDerivAt (fun s : ℝ => Real.sin (t - s))
+      (-Real.cos (t - r)) r := by
+    have h := (Real.hasDerivAt_sin (t - r)).comp r hsub
+    exact h.congr_deriv (by ring)
+  have h := (ha.fun_mul hcos).fun_add (hb.fun_mul hsin)
+  exact h.congr_deriv (by ring)
+
+private theorem uVelocity_eq_piece2 {t r : ℝ}
+    (hr : r ∈ Icc params.phi params.theta) :
+    uVelocity t r = uVelocityPiece (Romik.alphaBeta2 params) t r := by
+  unfold uVelocity uVelocityPiece
+  have h := alphaBetaAt_eq_piece2 hr
+  simp only [alpha, beta, h]
+
+private theorem uVelocity_eq_piece3 {t r : ℝ}
+    (hr : r ∈ Icc params.theta eta) :
+    uVelocity t r = uVelocityPiece (Romik.alphaBeta3 params) t r := by
+  unfold uVelocity uVelocityPiece
+  have h := alphaBetaAt_eq_piece3 hr
+  simp only [alpha, beta, h]
+
+private theorem uVelocity_eq_piece4 {t r : ℝ}
+    (hr : r ∈ Icc eta tau) :
+    uVelocity t r = uVelocityPiece (Romik.alphaBeta4 params) t r := by
+  unfold uVelocity uVelocityPiece
+  have h := alphaBetaAt_eq_piece4 hr
+  simp only [alpha, beta, h]
+
+private theorem uVelocity_eq_piece5 {t r : ℝ}
+    (hr : r ∈ Icc tau T) :
+    uVelocity t r = uVelocityPiece (Romik.alphaBeta5 params) t r := by
+  unfold uVelocity uVelocityPiece
+  have h := alphaBetaAt_eq_piece5 hr
+  simp only [alpha, beta, h]
+
+private theorem piece_signs
+    {ab : ℝ → Point} {r : ℝ} (hrT : r ∈ Icc params.phi T)
+    (heq : alphaBetaAt r = ab r) :
+    (ab r).1 ≤ 0 ∧ 0 ≤ (ab r).2 := by
+  have ha := alpha_nonpos hrT
+  have hb := beta_nonneg hrT
+  change (alphaBetaAt r).1 ≤ 0 at ha
+  change 0 ≤ (alphaBetaAt r).2 at hb
+  rw [heq] at ha hb
+  exact ⟨ha, hb⟩
+
+private theorem uVelocity_continuous (t : ℝ) : Continuous (uVelocity t) := by
+  unfold uVelocity
+  have hsub : Continuous (fun r : ℝ => t - r) :=
+    continuous_const.sub continuous_id
+  have ha : Continuous alpha := alpha_continuous
+  have hb : Continuous beta := beta_continuous
+  exact (ha.mul (Real.continuous_cos.comp hsub)).add
+    (hb.mul (Real.continuous_sin.comp hsub))
+
+private theorem uVelocity_hasDerivAt_phase2 {t r : ℝ}
+    (hr : r ∈ Ioo params.phi params.theta) :
+    HasDerivAt (uVelocity t)
+      (((-1 : ℝ) - (Romik.alphaBeta2 params r).2) * Real.cos (t-r) +
+      ((Romik.alphaBeta2 params r).1 +
+        (-(1/2 : ℝ) * r + params.b1)) * Real.sin (t-r)) r := by
+  have hd := uVelocityPiece_hasDerivAt (t := t)
+    (alpha2_hasDerivAt r) (beta2_hasDerivAt r)
+  apply hd.congr_of_eventuallyEq
+  filter_upwards [Ioo_mem_nhds hr.1 hr.2] with s hs
+  exact uVelocity_eq_piece2 (t := t) ⟨hs.1.le, hs.2.le⟩
+
+private theorem uVelocity_hasDerivAt_phase3 {t r : ℝ}
+    (hr : r ∈ Ioo params.theta eta) :
+    HasDerivAt (uVelocity t)
+      (((-1 : ℝ) - (Romik.alphaBeta3 params r).2) * Real.cos (t-r) +
+      ((Romik.alphaBeta3 params r).1 - 1) * Real.sin (t-r)) r := by
+  have hd := uVelocityPiece_hasDerivAt (t := t)
+    (alpha3_hasDerivAt r) (beta3_hasDerivAt r)
+  apply hd.congr_of_eventuallyEq
+  filter_upwards [Ioo_mem_nhds hr.1 hr.2] with s hs
+  exact uVelocity_eq_piece3 (t := t) ⟨hs.1.le, hs.2.le⟩
+
+private theorem uVelocity_hasDerivAt_phase4 {t r : ℝ}
+    (hr : r ∈ Ioo eta tau) :
+    HasDerivAt (uVelocity t)
+      ((((1/2 : ℝ) * r - params.d1) -
+        (Romik.alphaBeta4 params r).2) * Real.cos (t-r) +
+      ((Romik.alphaBeta4 params r).1 - 1) * Real.sin (t-r)) r := by
+  have hd := uVelocityPiece_hasDerivAt (t := t)
+    (alpha4_hasDerivAt r) (beta4_hasDerivAt r)
+  apply hd.congr_of_eventuallyEq
+  filter_upwards [Ioo_mem_nhds hr.1 hr.2] with s hs
+  exact uVelocity_eq_piece4 (t := t) ⟨hs.1.le, hs.2.le⟩
+
+private theorem uVelocity_hasDerivAt_phase5 {t r : ℝ}
+    (hr : r ∈ Ioo tau T) :
+    HasDerivAt (uVelocity t)
+      (((-2 * params.e1 * Real.cos r - 2 * params.e2 * Real.sin r) -
+          (Romik.alphaBeta5 params r).2) * Real.cos (t-r) +
+      ((Romik.alphaBeta5 params r).1 +
+          (-2 * params.e1 * Real.sin r + 2 * params.e2 * Real.cos r)) *
+        Real.sin (t-r)) r := by
+  have hd := uVelocityPiece_hasDerivAt (t := t)
+    (alpha5_hasDerivAt r) (beta5_hasDerivAt r)
+  apply hd.congr_of_eventuallyEq
+  filter_upwards [Ioo_mem_nhds hr.1 hr.2] with s hs
+  exact uVelocity_eq_piece5 (t := t) ⟨hs.1.le, hs.2.le⟩
+
+private theorem uVelocity_phase2_anti {t hi : ℝ}
+    (hhi : hi ∈ Icc params.phi params.theta) (hit : hi ≤ t) (htT : t ≤ T) :
+    AntitoneOn (uVelocity t) (Icc params.phi hi) := by
+  let f' : ℝ → ℝ := fun r =>
+    (((-1 : ℝ) - (Romik.alphaBeta2 params r).2) * Real.cos (t-r) +
+      ((Romik.alphaBeta2 params r).1 +
+        (-(1/2 : ℝ) * r + params.b1)) * Real.sin (t-r))
+  apply antitoneOn_of_hasDerivWithinAt_nonpos
+    (f' := f') (convex_Icc params.phi hi)
+  · exact (uVelocity_continuous t).continuousOn
+  · intro r hr
+    have hir : r ∈ Ioo params.phi hi := by simpa only [interior_Icc] using hr
+    have hrphase : r ∈ Ioo params.phi params.theta :=
+      ⟨hir.1, lt_of_lt_of_le hir.2 hhi.2⟩
+    exact (uVelocity_hasDerivAt_phase2 (t := t) hrphase).hasDerivWithinAt
+  · intro r hr
+    have hir : r ∈ Ioo params.phi hi := by simpa only [interior_Icc] using hr
+    have hrphase : r ∈ Ioo params.phi params.theta :=
+      ⟨hir.1, lt_of_lt_of_le hir.2 hhi.2⟩
+    have hs := piece_signs
+      (show r ∈ Icc params.phi T from
+        ⟨hir.1.le, le_trans (le_trans hir.2.le hhi.2) theta_lt_T.le⟩)
+      (alphaBetaAt_eq_piece2 ⟨hir.1.le,
+        le_trans hir.2.le hhi.2⟩)
+    have hbp := (phase2_betaPrime_neg hrphase).le
+    have hr0 : 0 ≤ r := le_trans phi_pos.le hir.1.le
+    have hd0 : 0 ≤ t-r := by linarith [hir.2, hit]
+    have hdT : t-r ≤ T := by linarith [htT, hr0]
+    have hsin : 0 ≤ Real.sin (t-r) := Real.sin_nonneg_of_nonneg_of_le_pi hd0
+      (le_trans hdT (by dsimp [T]; linarith [Real.pi_pos]))
+    have hcos : 0 ≤ Real.cos (t-r) := Real.cos_nonneg_of_mem_Icc
+      ⟨by linarith [Real.pi_pos, hd0], by simpa only [T] using hdT⟩
+    dsimp only [f']
+    exact add_nonpos
+      (mul_nonpos_of_nonpos_of_nonneg (by linarith [hs.2]) hcos)
+      (mul_nonpos_of_nonpos_of_nonneg (by linarith [hs.1, hbp]) hsin)
+
+private theorem uVelocity_phase3_anti {t hi : ℝ}
+    (hhi : hi ∈ Icc params.theta eta) (hit : hi ≤ t) (htT : t ≤ T) :
+    AntitoneOn (uVelocity t) (Icc params.theta hi) := by
+  let f' : ℝ → ℝ := fun r =>
+    (((-1 : ℝ) - (Romik.alphaBeta3 params r).2) * Real.cos (t-r) +
+      ((Romik.alphaBeta3 params r).1 - 1) * Real.sin (t-r))
+  apply antitoneOn_of_hasDerivWithinAt_nonpos
+    (f' := f') (convex_Icc params.theta hi)
+  · exact (uVelocity_continuous t).continuousOn
+  · intro r hr
+    have hir : r ∈ Ioo params.theta hi := by simpa only [interior_Icc] using hr
+    have hrphase : r ∈ Ioo params.theta eta :=
+      ⟨hir.1, lt_of_lt_of_le hir.2 hhi.2⟩
+    exact (uVelocity_hasDerivAt_phase3 (t := t) hrphase).hasDerivWithinAt
+  · intro r hr
+    have hir : r ∈ Ioo params.theta hi := by simpa only [interior_Icc] using hr
+    have hrphase : r ∈ Ioo params.theta eta :=
+      ⟨hir.1, lt_of_lt_of_le hir.2 hhi.2⟩
+    have hs := piece_signs
+      (show r ∈ Icc params.phi T from
+        ⟨(lt_trans phi_lt_theta hir.1).le,
+          le_trans (le_trans hir.2.le hhi.2) eta_lt_T.le⟩)
+      (alphaBetaAt_eq_piece3 ⟨hir.1.le, le_trans hir.2.le hhi.2⟩)
+    have hr0 : 0 ≤ r := le_trans theta_pos.le hir.1.le
+    have hd0 : 0 ≤ t-r := by linarith [hir.2, hit]
+    have hdT : t-r ≤ T := by linarith [htT, hr0]
+    have hsin : 0 ≤ Real.sin (t-r) := Real.sin_nonneg_of_nonneg_of_le_pi hd0
+      (le_trans hdT (by dsimp [T]; linarith [Real.pi_pos]))
+    have hcos : 0 ≤ Real.cos (t-r) := Real.cos_nonneg_of_mem_Icc
+      ⟨by linarith [Real.pi_pos, hd0], by simpa only [T] using hdT⟩
+    dsimp only [f']
+    exact add_nonpos
+      (mul_nonpos_of_nonpos_of_nonneg (by linarith [hs.2]) hcos)
+      (mul_nonpos_of_nonpos_of_nonneg (by linarith [hs.1]) hsin)
+
+private theorem uVelocity_phase4_anti {t hi : ℝ}
+    (hhi : hi ∈ Icc eta tau) (hit : hi ≤ t) (htT : t ≤ T) :
+    AntitoneOn (uVelocity t) (Icc eta hi) := by
+  let f' : ℝ → ℝ := fun r =>
+    ((((1/2 : ℝ) * r - params.d1) -
+        (Romik.alphaBeta4 params r).2) * Real.cos (t-r) +
+      ((Romik.alphaBeta4 params r).1 - 1) * Real.sin (t-r))
+  apply antitoneOn_of_hasDerivWithinAt_nonpos
+    (f' := f') (convex_Icc eta hi)
+  · exact (uVelocity_continuous t).continuousOn
+  · intro r hr
+    have hir : r ∈ Ioo eta hi := by simpa only [interior_Icc] using hr
+    have hrphase : r ∈ Ioo eta tau :=
+      ⟨hir.1, lt_of_lt_of_le hir.2 hhi.2⟩
+    exact (uVelocity_hasDerivAt_phase4 (t := t) hrphase).hasDerivWithinAt
+  · intro r hr
+    have hir : r ∈ Ioo eta hi := by simpa only [interior_Icc] using hr
+    have hrphase : r ∈ Ioo eta tau :=
+      ⟨hir.1, lt_of_lt_of_le hir.2 hhi.2⟩
+    have hs := piece_signs
+      (show r ∈ Icc params.phi T from
+        ⟨(lt_trans phi_lt_eta hir.1).le,
+          le_trans (le_trans hir.2.le hhi.2) tau_lt_T.le⟩)
+      (alphaBetaAt_eq_piece4 ⟨hir.1.le, le_trans hir.2.le hhi.2⟩)
+    have hap := (phase4_alphaPrime_neg hrphase).le
+    have hr0 : 0 ≤ r := le_trans theta_pos.le (lt_trans theta_lt_eta hir.1).le
+    have hd0 : 0 ≤ t-r := by linarith [hir.2, hit]
+    have hdT : t-r ≤ T := by linarith [htT, hr0]
+    have hsin : 0 ≤ Real.sin (t-r) := Real.sin_nonneg_of_nonneg_of_le_pi hd0
+      (le_trans hdT (by dsimp [T]; linarith [Real.pi_pos]))
+    have hcos : 0 ≤ Real.cos (t-r) := Real.cos_nonneg_of_mem_Icc
+      ⟨by linarith [Real.pi_pos, hd0], by simpa only [T] using hdT⟩
+    dsimp only [f']
+    exact add_nonpos
+      (mul_nonpos_of_nonpos_of_nonneg (by linarith [hs.2, hap]) hcos)
+      (mul_nonpos_of_nonpos_of_nonneg (by linarith [hs.1]) hsin)
+
+private theorem uVelocity_phase5_anti {t hi : ℝ}
+    (hhi : hi ∈ Icc tau T) (hit : hi ≤ t) (htT : t ≤ T) :
+    AntitoneOn (uVelocity t) (Icc tau hi) := by
+  let f' : ℝ → ℝ := fun r =>
+    (((-2 * params.e1 * Real.cos r - 2 * params.e2 * Real.sin r) -
+        (Romik.alphaBeta5 params r).2) * Real.cos (t-r) +
+      ((Romik.alphaBeta5 params r).1 +
+        (-2 * params.e1 * Real.sin r + 2 * params.e2 * Real.cos r)) *
+        Real.sin (t-r))
+  apply antitoneOn_of_hasDerivWithinAt_nonpos
+    (f' := f') (convex_Icc tau hi)
+  · exact (uVelocity_continuous t).continuousOn
+  · intro r hr
+    have hir : r ∈ Ioo tau hi := by simpa only [interior_Icc] using hr
+    have hrphase : r ∈ Ioo tau T :=
+      ⟨hir.1, lt_of_lt_of_le hir.2 hhi.2⟩
+    exact (uVelocity_hasDerivAt_phase5 (t := t) hrphase).hasDerivWithinAt
+  · intro r hr
+    have hir : r ∈ Ioo tau hi := by simpa only [interior_Icc] using hr
+    have hrphase : r ∈ Ioo tau T :=
+      ⟨hir.1, lt_of_lt_of_le hir.2 hhi.2⟩
+    have hs := piece_signs
+      (show r ∈ Icc params.phi T from
+        ⟨(lt_trans (lt_trans phi_lt_eta eta_lt_tau) hir.1).le,
+          le_trans hir.2.le hhi.2⟩)
+      (alphaBetaAt_eq_piece5 ⟨hir.1.le, le_trans hir.2.le hhi.2⟩)
+    obtain ⟨hap, hbp⟩ := phase5_coefficientPrime_nonpos hrphase
+    have hr0 : 0 ≤ r := le_trans phi_pos.le
+      (lt_trans (lt_trans phi_lt_eta eta_lt_tau) hir.1).le
+    have hd0 : 0 ≤ t-r := by linarith [hir.2, hit]
+    have hdT : t-r ≤ T := by linarith [htT, hr0]
+    have hsin : 0 ≤ Real.sin (t-r) := Real.sin_nonneg_of_nonneg_of_le_pi hd0
+      (le_trans hdT (by dsimp [T]; linarith [Real.pi_pos]))
+    have hcos : 0 ≤ Real.cos (t-r) := Real.cos_nonneg_of_mem_Icc
+      ⟨by linarith [Real.pi_pos, hd0], by simpa only [T] using hdT⟩
+    dsimp only [f']
+    exact add_nonpos
+      (mul_nonpos_of_nonpos_of_nonneg (by linarith [hs.2, hap]) hcos)
+      (mul_nonpos_of_nonpos_of_nonneg (by linarith [hs.1, hbp]) hsin)
+
+private theorem antitoneOn_Icc_glue {f : ℝ → ℝ} {a b c : ℝ}
+    (hab : a ≤ b)
+    (hleft : AntitoneOn f (Icc a b))
+    (hright : AntitoneOn f (Icc b c)) :
+    AntitoneOn f (Icc a c) := by
+  intro x hx y hy hxy
+  by_cases hyb : y ≤ b
+  · exact hleft ⟨hx.1, le_trans hxy hyb⟩ ⟨hy.1, hyb⟩ hxy
+  by_cases hbx : b ≤ x
+  · exact hright ⟨hbx, le_trans hxy hy.2⟩ ⟨le_trans hbx hxy, hy.2⟩ hxy
+  have hxb : x ≤ b := le_of_not_ge hbx
+  have hby : b ≤ y := le_of_not_ge hyb
+  exact le_trans
+    (hright ⟨le_rfl, le_trans hby hy.2⟩ ⟨hby, hy.2⟩ hby)
+    (hleft ⟨hx.1, hxb⟩ ⟨hab, le_rfl⟩ hxb)
+
+/-- For fixed terminal time, the projected path velocity is antitone on the
+complete ordered no-hidden interval. -/
+theorem uVelocity_antitone {t : ℝ} (ht : t ∈ Icc params.phi T) :
+    AntitoneOn (uVelocity t) (Icc params.phi t) := by
+  by_cases h2 : t ≤ params.theta
+  · exact uVelocity_phase2_anti ⟨ht.1, h2⟩ le_rfl ht.2
+  have ht2 : params.theta ≤ t := (lt_of_not_ge h2).le
+  have hphase2 := uVelocity_phase2_anti
+    (t := t) (hi := params.theta) ⟨phi_lt_theta.le, le_rfl⟩ ht2 ht.2
+  by_cases h3 : t ≤ eta
+  · exact antitoneOn_Icc_glue phi_lt_theta.le hphase2
+      (uVelocity_phase3_anti ⟨ht2, h3⟩ le_rfl ht.2)
+  have ht3 : eta ≤ t := (lt_of_not_ge h3).le
+  have hphase3 := uVelocity_phase3_anti
+    (t := t) (hi := eta) ⟨theta_lt_eta.le, le_rfl⟩ ht3 ht.2
+  have h23 := antitoneOn_Icc_glue phi_lt_theta.le
+    hphase2 hphase3
+  by_cases h4 : t ≤ tau
+  · exact antitoneOn_Icc_glue phi_lt_eta.le h23
+      (uVelocity_phase4_anti ⟨ht3, h4⟩ le_rfl ht.2)
+  have ht4 : tau ≤ t := (lt_of_not_ge h4).le
+  have hphase4 := uVelocity_phase4_anti
+    (t := t) (hi := tau) ⟨eta_lt_tau.le, le_rfl⟩ ht4 ht.2
+  have h234 := antitoneOn_Icc_glue phi_lt_eta.le h23 hphase4
+  exact antitoneOn_Icc_glue
+    (lt_trans phi_lt_eta eta_lt_tau).le h234
+    (uVelocity_phase5_anti ⟨ht4, ht.2⟩ le_rfl ht.2)
+
+private theorem dot_fixed_hasDerivAt {f : ℝ → Point} {f' : Point}
+    {r : ℝ} (hf : HasDerivAt f f' r) (w : Point) :
+    HasDerivAt (fun s => dot (f s) w) (dot f' w) r := by
+  have h1 := HasDerivAt.const_mul w.1 hf.fst
+  have h2 := HasDerivAt.const_mul w.2 hf.snd
+  simpa [dot, mul_comm] using h1.fun_add h2
+
+/-- Derivative of `U(r,t)` in its first variable. -/
+theorem UValue_hasDerivAt_noHidden (r t : ℝ) :
+    HasDerivAt (fun s => Stage2.UValue s t) (uVelocity t r) r := by
+  have hpRaw := (path_hasDerivAt_noHidden r).fun_sub
+    (hasDerivAt_const r (Romik.path params t))
+  have hp : HasDerivAt
+      (fun s => Romik.path params s - Romik.path params t)
+      (Romik.rot r (alphaBetaAt r)) r := by
+    exact hpRaw.congr_deriv (by simp)
+  have hd := dot_fixed_hasDerivAt hp (u t)
+  have hrot : dot (Romik.rot r (alphaBetaAt r)) (u t) = uVelocity t r := by
+    dsimp [dot, Romik.rot, u, uVelocity, alpha, beta]
+    rw [Real.cos_sub, Real.sin_sub]
+    ring
+  exact hd.congr_deriv hrot
+
+/-- Concavity form of the manuscript's one-turn tangent argument. -/
+theorem U_concave_on (t : ℝ) (ht : t ∈ Icc params.phi T) :
+    ConcaveOn ℝ (Icc params.phi t) (fun r => Stage2.UValue r t) := by
+  have hanti := uVelocity_antitone ht
+  have hantiDeriv : AntitoneOn
+      (deriv (fun r => Stage2.UValue r t)) (interior (Icc params.phi t)) := by
+    intro x hx y hy hxy
+    rw [(UValue_hasDerivAt_noHidden x t).deriv,
+      (UValue_hasDerivAt_noHidden y t).deriv]
+    exact hanti (interior_subset hx) (interior_subset hy) hxy
+  exact hantiDeriv.concaveOn_of_deriv (convex_Icc params.phi t)
+    (by
+      intro r hr
+      exact (UValue_hasDerivAt_noHidden r t).continuousAt.continuousWithinAt)
+    (by
+      intro r hr
+      exact (UValue_hasDerivAt_noHidden r t).differentiableAt.differentiableWithinAt)
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 4: exact reflection bridge for the second no-hidden inequality
+
+The affine reflection is derived phase by phase from the certified matching
+equations.  No global symmetry hypothesis is introduced.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Set
+open Stage2
+
+/-- Reflect a point horizontally about the axis `x = k31`. -/
+def noHiddenHReflect (z : Point) : Point :=
+  (2 * params.k31 - z.1, z.2)
+
+@[simp] theorem noHiddenHReflect_involutive (z : Point) :
+    noHiddenHReflect (noHiddenHReflect z) = z := by
+  ext <;> simp [noHiddenHReflect]
+
+private theorem phase24_horizontal_reflection_noHidden (s : ℝ) :
+    (Romik.path4 params (T - s)).1 + (Romik.path2 params s).1 =
+      params.k41 + params.k21 := by
+  have hd1 := Romik.d1_eq_quarterPi_sub_b1_of_equations params_equations
+  have hd2 :=
+    Romik.d2_eq_b2_add_quarterPi_correction_of_equations params_equations
+  dsimp [T, Romik.path4, Romik.path2, Romik.rot, Romik.addK]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub, hd1, hd2]
+  ring
+
+private theorem phase3_horizontal_reflection_noHidden (s : ℝ) :
+    (Romik.path3 params (T - s)).1 + (Romik.path3 params s).1 =
+      2 * params.k31 := by
+  have hc2 := Romik.c2_eq_c1_sub_halfPi_of_equations params_equations
+  dsimp [T, Romik.path3, Romik.rot, Romik.addK]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub, hc2]
+  ring
+
+private theorem k41_add_k21_eq_two_k31_noHidden :
+    params.k41 + params.k21 = 2 * params.k31 := by
+  have h24 := phase24_horizontal_reflection_noHidden params.theta
+  have h33 := phase3_horizontal_reflection_noHidden params.theta
+  have h23 := congrArg Prod.fst PartB.match23
+  have h34 := congrArg Prod.fst PartB.match34
+  simp only [T] at h24 h33 h34
+  linarith
+
+private theorem phase24_reflect_noHidden (s : ℝ) :
+    Romik.path4 params (T - s) =
+      noHiddenHReflect (Romik.path2 params s) := by
+  apply Prod.ext
+  · have h := phase24_horizontal_reflection_noHidden s
+    rw [k41_add_k21_eq_two_k31_noHidden] at h
+    dsimp [noHiddenHReflect]
+    linarith
+  · have h :=
+      Romik.phase24_vertical_reflection_of_equations params_equations s
+    have hk := Romik.k42_eq_k22_of_equations params_equations
+    rw [hk] at h
+    dsimp [T] at h ⊢
+    dsimp [noHiddenHReflect]
+    linarith
+
+private theorem phase42_reflect_noHidden (s : ℝ) :
+    Romik.path2 params (T - s) =
+      noHiddenHReflect (Romik.path4 params s) := by
+  have h := congrArg noHiddenHReflect
+    (phase24_reflect_noHidden (T - s))
+  have hT : T - (T - s) = s := by ring
+  rw [hT, noHiddenHReflect_involutive] at h
+  exact h.symm
+
+private theorem phase3_reflect_noHidden (s : ℝ) :
+    Romik.path3 params (T - s) =
+      noHiddenHReflect (Romik.path3 params s) := by
+  apply Prod.ext
+  · have h := phase3_horizontal_reflection_noHidden s
+    dsimp [noHiddenHReflect]
+    linarith
+  · have h :=
+      Romik.phase3_vertical_reflection_of_equations params_equations s
+    simpa [T, noHiddenHReflect] using h
+
+private theorem phase15_horizontal_reflection_noHidden (s : ℝ) :
+    (Romik.path5 params (T - s)).1 + (Romik.path1 params s).1 =
+      params.k51 + params.k11 := by
+  have he1 := Romik.e1_eq_a1_of_equations params_equations
+  have he2 := Romik.e2_eq_neg_a2_of_equations params_equations
+  dsimp [T, Romik.path5, Romik.path1, Romik.rot, Romik.addK]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub, he1, he2]
+  ring
+
+private theorem k51_add_k11_eq_two_k31_noHidden :
+    params.k51 + params.k11 = 2 * params.k31 := by
+  have h15 := phase15_horizontal_reflection_noHidden params.phi
+  have h24 := phase24_horizontal_reflection_noHidden params.phi
+  have h12 := congrArg Prod.fst PartB.match12
+  have h45 := congrArg Prod.fst PartB.match45
+  rw [k41_add_k21_eq_two_k31_noHidden] at h24
+  simp only [T] at h15 h24 h45
+  linarith
+
+private theorem phase15_reflect_noHidden (s : ℝ) :
+    Romik.path5 params (T - s) =
+      noHiddenHReflect (Romik.path1 params s) := by
+  apply Prod.ext
+  · have h := phase15_horizontal_reflection_noHidden s
+    rw [k51_add_k11_eq_two_k31_noHidden] at h
+    dsimp [noHiddenHReflect]
+    linarith
+  · have h :=
+      Romik.phase15_vertical_reflection_of_equations params_equations s
+    have hk := Romik.k52_eq_k12_of_equations params_equations
+    rw [hk] at h
+    dsimp [T] at h ⊢
+    dsimp [noHiddenHReflect]
+    linarith
+
+private theorem phase51_reflect_noHidden (s : ℝ) :
+    Romik.path1 params (T - s) =
+      noHiddenHReflect (Romik.path5 params s) := by
+  have h := congrArg noHiddenHReflect
+    (phase15_reflect_noHidden (T - s))
+  have hT : T - (T - s) = s := by ring
+  rw [hT, noHiddenHReflect_involutive] at h
+  exact h.symm
+
+private theorem path_eq_phase1_noHidden {s : ℝ} (hs : s ≤ params.phi) :
+    Romik.path params s = Romik.path1 params s := by
+  simp only [Romik.path, ite_eq_left hs]
+
+private theorem path_eq_phase2_noHidden {s : ℝ}
+    (hphi : params.phi < s) (hs : s ≤ params.theta) :
+    Romik.path params s = Romik.path2 params s := by
+  simp only [Romik.path, ite_eq_right (not_le.mpr hphi), ite_eq_left hs]
+
+private theorem path_eq_phase3_noHidden {s : ℝ}
+    (htheta : params.theta < s) (hs : s ≤ eta) :
+    Romik.path params s = Romik.path3 params s := by
+  have hphi : params.phi < s := lt_trans phi_lt_theta htheta
+  simpa only [eta, T] using
+    (show Romik.path params s = Romik.path3 params s by
+      simp only [Romik.path, ite_eq_right (not_le.mpr hphi),
+        ite_eq_right (not_le.mpr htheta), ite_eq_left (by simpa [eta, T] using hs)])
+
+private theorem path_eq_phase4_noHidden {s : ℝ}
+    (heta : eta < s) (hs : s ≤ tau) :
+    Romik.path params s = Romik.path4 params s := by
+  have htheta : params.theta < s := lt_trans theta_lt_eta heta
+  have hphi : params.phi < s := lt_trans phi_lt_theta htheta
+  have hnotEta : ¬ s ≤ Real.pi / 2 - params.theta := by
+    apply not_le.mpr
+    simpa only [eta, T] using heta
+  have hleTau : s ≤ Real.pi / 2 - params.phi := by
+    simpa only [tau, T] using hs
+  simp only [Romik.path, ite_eq_right (not_le.mpr hphi),
+    ite_eq_right (not_le.mpr htheta),
+    ite_eq_right hnotEta, ite_eq_left hleTau]
+
+private theorem path_eq_phase5_noHidden {s : ℝ} (hs : tau < s) :
+    Romik.path params s = Romik.path5 params s := by
+  have heta : eta < s := lt_trans eta_lt_tau hs
+  have htheta : params.theta < s := lt_trans theta_lt_eta heta
+  have hphi : params.phi < s := lt_trans phi_lt_theta htheta
+  have hnotEta : ¬ s ≤ Real.pi / 2 - params.theta := by
+    apply not_le.mpr
+    simpa only [eta, T] using heta
+  have hnotTau : ¬ s ≤ Real.pi / 2 - params.phi := by
+    apply not_le.mpr
+    simpa only [tau, T] using hs
+  simp only [Romik.path, ite_eq_right (not_le.mpr hphi),
+    ite_eq_right (not_le.mpr htheta),
+    ite_eq_right hnotEta, ite_eq_right hnotTau]
+
+/-- Exact global horizontal reflection of the literal five-piece path. -/
+theorem path_reflect_noHidden {s : ℝ} (hs : s ∈ Icc (0 : ℝ) T) :
+    Romik.path params (T - s) =
+      noHiddenHReflect (Romik.path params s) := by
+  by_cases h1 : s ≤ params.phi
+  · by_cases he : s = params.phi
+    · subst s
+      rw [show T - params.phi = tau by rfl,
+        path_eq_phase4_noHidden eta_lt_tau le_rfl,
+        path_eq_phase1_noHidden le_rfl]
+      exact (phase24_reflect_noHidden params.phi).trans
+        (congrArg noHiddenHReflect PartB.match12.symm)
+    · have hsphi : s < params.phi := lt_of_le_of_ne h1 he
+      have href : tau < T - s := by dsimp [tau]; linarith
+      rw [path_eq_phase5_noHidden href, path_eq_phase1_noHidden h1]
+      exact phase15_reflect_noHidden s
+  · have hphi : params.phi < s := lt_of_not_ge h1
+    by_cases h2 : s ≤ params.theta
+    · by_cases he : s = params.theta
+      · subst s
+        rw [show T - params.theta = eta by rfl,
+          path_eq_phase3_noHidden theta_lt_eta le_rfl,
+          path_eq_phase2_noHidden phi_lt_theta le_rfl]
+        exact (phase3_reflect_noHidden params.theta).trans
+          (congrArg noHiddenHReflect PartB.match23.symm)
+      · have hstheta : s < params.theta := lt_of_le_of_ne h2 he
+        have hetaRef : eta < T - s := by dsimp [eta]; linarith
+        have htauRef : T - s ≤ tau := by dsimp [tau]; linarith
+        rw [path_eq_phase4_noHidden hetaRef htauRef,
+          path_eq_phase2_noHidden hphi h2]
+        exact phase24_reflect_noHidden s
+    · have htheta : params.theta < s := lt_of_not_ge h2
+      by_cases h3 : s ≤ eta
+      · by_cases he : s = eta
+        · subst s
+          rw [show T - eta = params.theta by
+                dsimp [eta]; ring,
+            path_eq_phase2_noHidden phi_lt_theta le_rfl,
+            path_eq_phase3_noHidden theta_lt_eta le_rfl]
+          have h := congrArg noHiddenHReflect
+            (phase3_reflect_noHidden params.theta)
+          rw [noHiddenHReflect_involutive] at h
+          simpa [eta] using (PartB.match23.trans h.symm)
+        · have hseta : s < eta := lt_of_le_of_ne h3 he
+          have hthetaRef : params.theta < T - s := by
+            dsimp [eta] at hseta
+            linarith
+          have hetaRef : T - s ≤ eta := by dsimp [eta]; linarith
+          rw [path_eq_phase3_noHidden hthetaRef hetaRef,
+            path_eq_phase3_noHidden htheta h3]
+          exact phase3_reflect_noHidden s
+      · have hetaS : eta < s := lt_of_not_ge h3
+        by_cases h4 : s ≤ tau
+        · by_cases he : s = tau
+          · subst s
+            rw [show T - tau = params.phi by
+                  dsimp [tau]; ring,
+              path_eq_phase1_noHidden le_rfl,
+              path_eq_phase4_noHidden eta_lt_tau le_rfl]
+            have h := congrArg noHiddenHReflect
+              (phase24_reflect_noHidden params.phi)
+            rw [noHiddenHReflect_involutive] at h
+            simpa [tau] using (PartB.match12.trans h.symm)
+          · have hstau : s < tau := lt_of_le_of_ne h4 he
+            have hphiRef : params.phi < T - s := by
+              dsimp [tau] at hstau
+              linarith
+            have hthetaRef : T - s ≤ params.theta := by
+              dsimp [eta] at hetaS
+              linarith
+            rw [path_eq_phase2_noHidden hphiRef hthetaRef,
+              path_eq_phase4_noHidden hetaS h4]
+            exact phase42_reflect_noHidden s
+        · have htauS : tau < s := lt_of_not_ge h4
+          have href0 : 0 ≤ T - s := by linarith [hs.2]
+          have hrefphi : T - s ≤ params.phi := by
+            dsimp [tau] at htauS
+            linarith
+          rw [path_eq_phase1_noHidden hrefphi,
+            path_eq_phase5_noHidden htauS]
+          exact phase51_reflect_noHidden s
+
+/-- Reflection transports the second no-hidden scalar to the first one. -/
+theorem VValue_eq_reflected_UValue {t r : ℝ}
+    (ht : t ∈ Icc (0 : ℝ) T) (hr : r ∈ Icc (0 : ℝ) T) :
+    Stage2.VValue t r = Stage2.UValue (T-r) (T-t) := by
+  rw [Stage2.VValue, Stage2.UValue,
+    path_reflect_noHidden hr, path_reflect_noHidden ht]
+  dsimp [dot, noHiddenHReflect, u, v, T]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  ring
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 4: direct no-hidden-crossing closure
+
+The first inequality follows from concavity and the two endpoint values.  The
+second is its exact phase-derived horizontal reflection.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Set
+open Stage2
+
+/-- Concrete first no-hidden-crossing theorem. -/
+theorem noHiddenU_direct : NoHiddenCrossingU := by
+  intro r hr t ht
+  change 0 ≤ Stage2.UValue r t
+  have htPhys : t ∈ Icc params.phi T :=
+    ⟨le_trans hr.1 ht.1, ht.2⟩
+  have hconc := U_concave_on t htPhys
+  have hphi := U_phi_nonneg htPhys
+  have htt : Stage2.UValue t t = 0 := by
+    simp [Stage2.UValue, dot]
+  by_cases hrt : r = t
+  · subst r
+    exact le_of_eq htt.symm
+  · have hrtlt : r < t := lt_of_le_of_ne ht.1 hrt
+    have hphit : params.phi < t := lt_of_le_of_lt hr.1 hrtlt
+    let a : ℝ := (t-r) / (t-params.phi)
+    let b : ℝ := (r-params.phi) / (t-params.phi)
+    have ha : 0 ≤ a := by
+      dsimp [a]
+      exact div_nonneg (sub_nonneg.mpr ht.1) (sub_nonneg.mpr hphit.le)
+    have hb : 0 ≤ b := by
+      dsimp [b]
+      exact div_nonneg (sub_nonneg.mpr hr.1) (sub_nonneg.mpr hphit.le)
+    have hab : a + b = 1 := by
+      dsimp [a, b]
+      field_simp [sub_ne_zero.mpr (ne_of_gt hphit)]
+      ring
+    have hrcomb : a • params.phi + b • t = r := by
+      dsimp [a, b]
+      field_simp [sub_ne_zero.mpr (ne_of_gt hphit)]
+      ring
+    have hJ := hconc.2
+      (show params.phi ∈ Icc params.phi t from ⟨le_rfl, hphit.le⟩)
+      (show t ∈ Icc params.phi t from ⟨hphit.le, le_rfl⟩)
+      ha hb hab
+    rw [hrcomb] at hJ
+    change a • Stage2.UValue params.phi t + b • Stage2.UValue t t ≤
+      Stage2.UValue r t at hJ
+    rw [htt] at hJ
+    simp only [smul_eq_mul, mul_zero, add_zero] at hJ
+    nlinarith
+
+/-- Concrete reflected no-hidden-crossing theorem. -/
+theorem noHiddenV_direct : NoHiddenCrossingV := by
+  intro t ht r hr
+  change 0 ≤ Stage2.VValue t r
+  have htT : t ∈ Icc (0 : ℝ) T :=
+    ⟨ht.1, le_trans ht.2 Stage2.tau_le_T⟩
+  have hrT : r ∈ Icc (0 : ℝ) T :=
+    ⟨le_trans ht.1 hr.1, le_trans hr.2 Stage2.tau_le_T⟩
+  have hleft : T-r ∈ Icc params.phi T := by
+    constructor
+    · dsimp [tau] at hr
+      linarith [hr.2]
+    · linarith [hrT.1]
+  have hright : T-t ∈ Icc (T-r) T := by
+    constructor
+    · linarith [hr.1]
+    · linarith [ht.1]
+  have hU := noHiddenU_direct (T-r) hleft (T-t) hright
+  change 0 ≤ Stage2.UValue (T-r) (T-t) at hU
+  rw [← VValue_eq_reflected_UValue htT hrT] at hU
+  exact hU
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part C / Stage4 / Niche Envelope Support
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Set
+
+private theorem phi_le_one_twentieth_diag :
+    params.phi ≤ (1 / 20 : ℝ) := by
+  have h := phi_bounds.2
+  norm_num at h ⊢
+  linarith
+
+private theorem a1_lower_diag : (6 / 5 : ℝ) ≤ params.a1 := by
+  have h := Romik.a1_lower_bound_of_mem_box params_mem
+  norm_num at h ⊢
+  linarith
+
+private theorem a1_upper_diag : params.a1 ≤ (5 / 4 : ℝ) := by
+  have h := PartB.a1_contains.2
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at h ⊢
+  linarith
+
+private theorem phase1_alpha_lower {s : ℝ}
+    (hs0 : 0 ≤ s) (hs20 : s ≤ (1 / 20 : ℝ)) :
+    (-1 / 8 : ℝ) ≤ (Romik.alphaBeta1 params s).1 := by
+  have hpi : s ≤ Real.pi := by
+    nlinarith [Real.pi_gt_three]
+  have hsin0 : 0 ≤ Real.sin s :=
+    Real.sin_nonneg_of_nonneg_of_le_pi hs0 hpi
+  have hsinUpper := Real.sin_le hs0
+  have hcosUpper := Real.cos_le_one s
+  have ha2 := Romik.a2_eq_neg_quarter_of_equations params_equations
+  have hmul : 2 * params.a1 * Real.sin s ≤ (1 / 8 : ℝ) := by
+    have h1 : 2 * params.a1 ≤ (5 / 2 : ℝ) := by
+      nlinarith [a1_upper_diag]
+    have h2 : Real.sin s ≤ (1 / 20 : ℝ) :=
+      le_trans hsinUpper hs20
+    nlinarith [mul_nonneg hsin0 (sub_nonneg.mpr h1),
+      mul_nonneg (sub_nonneg.mpr h2) (show 0 ≤ (5 / 2 : ℝ) by norm_num)]
+  dsimp [Romik.alphaBeta1]
+  rw [ha2]
+  nlinarith
+
+private theorem phase1_alpha_nonpos_diag {s : ℝ}
+    (hs0 : 0 ≤ s) (hs20 : s ≤ (1 / 20 : ℝ)) :
+    (Romik.alphaBeta1 params s).1 ≤ 0 := by
+  have hpi : s ≤ Real.pi := by
+    nlinarith [Real.pi_gt_three]
+  have hsin0 : 0 ≤ Real.sin s :=
+    Real.sin_nonneg_of_nonneg_of_le_pi hs0 hpi
+  have hsinLower := Real.sin_ge_sub_cube hs0
+  have hcosLower : 1 - s ^ 2 / 2 ≤ Real.cos s :=
+    Real.one_sub_sq_div_two_le_cos
+  have hquad : 0 ≤ s * ((1 / 20 : ℝ) - s) :=
+    mul_nonneg hs0 (sub_nonneg.mpr hs20)
+  have hcube : 0 ≤ s ^ 2 * ((1 / 20 : ℝ) - s) :=
+    mul_nonneg (sq_nonneg s) (sub_nonneg.mpr hs20)
+  have hcoef : (12 / 5 : ℝ) ≤ 2 * params.a1 := by
+    nlinarith [a1_lower_diag]
+  have hmul : (12 / 5 : ℝ) * Real.sin s ≤
+      2 * params.a1 * Real.sin s :=
+    mul_le_mul_of_nonneg_right hcoef hsin0
+  have ha2 := Romik.a2_eq_neg_quarter_of_equations params_equations
+  dsimp [Romik.alphaBeta1]
+  rw [ha2]
+  nlinarith [hsinLower, hcosLower, hquad, hcube, hmul]
+
+private theorem phase1_beta_lower {s : ℝ}
+    (hs0 : 0 ≤ s) (hs20 : s ≤ (1 / 20 : ℝ)) :
+    (13 / 10 : ℝ) ≤ (Romik.alphaBeta1 params s).2 := by
+  have hpi : s ≤ Real.pi := by
+    nlinarith [Real.pi_gt_three]
+  have hsin0 : 0 ≤ Real.sin s :=
+    Real.sin_nonneg_of_nonneg_of_le_pi hs0 hpi
+  have hsinUpper := Real.sin_le hs0
+  have hcosLower : 1 - s ^ 2 / 2 ≤ Real.cos s :=
+    Real.one_sub_sq_div_two_le_cos
+  have hcosCrude : (799 / 800 : ℝ) ≤ Real.cos s := by
+    nlinarith [hcosLower, sq_nonneg s,
+      mul_nonneg hs0 (sub_nonneg.mpr hs20)]
+  have hcos0 : 0 ≤ Real.cos s := by linarith
+  have hcoef : (12 / 5 : ℝ) ≤ 2 * params.a1 := by
+    nlinarith [a1_lower_diag]
+  have hmul : (12 / 5 : ℝ) * Real.cos s ≤
+      2 * params.a1 * Real.cos s :=
+    mul_le_mul_of_nonneg_right hcoef hcos0
+  have ha2 := Romik.a2_eq_neg_quarter_of_equations params_equations
+  dsimp [Romik.alphaBeta1]
+  rw [ha2]
+  nlinarith [hsinUpper, hmul]
+
+private theorem dot_rot_phase1_diag (s t : ℝ) :
+    dot (Romik.rot s (Romik.alphaBeta1 params s)) (u t + v t) =
+      (Romik.alphaBeta1 params s).1 *
+          (Real.cos (s - t) + Real.sin (s - t)) +
+      (Romik.alphaBeta1 params s).2 *
+          (Real.cos (s - t) - Real.sin (s - t)) := by
+  simp [dot, Romik.rot, u, v, Real.cos_sub, Real.sin_sub]
+  ring
+
+private theorem phase1_diag_deriv_nonneg {t s : ℝ}
+    (ht0 : 0 ≤ t) (hts : t ≤ s) (hsphi : s ≤ params.phi) :
+    0 ≤ dot (Romik.rot s (Romik.alphaBeta1 params s)) (u t + v t) := by
+  have hs0 : 0 ≤ s := le_trans ht0 hts
+  have hs20 : s ≤ (1 / 20 : ℝ) :=
+    le_trans hsphi phi_le_one_twentieth_diag
+  have hd0 : 0 ≤ s - t := sub_nonneg.mpr hts
+  have hd20 : s - t ≤ (1 / 20 : ℝ) := by linarith
+  have hdpi : s - t ≤ Real.pi := by nlinarith [Real.pi_gt_three]
+  have hsin0 : 0 ≤ Real.sin (s - t) :=
+    Real.sin_nonneg_of_nonneg_of_le_pi hd0 hdpi
+  have hsinUpper := Real.sin_le hd0
+  have hcosLower : 1 - (s - t) ^ 2 / 2 ≤ Real.cos (s - t) :=
+    Real.one_sub_sq_div_two_le_cos
+  have hcosUpper := Real.cos_le_one (s - t)
+  have hdiff : (9 / 10 : ℝ) ≤
+      Real.cos (s - t) - Real.sin (s - t) := by
+    nlinarith [hcosLower, hsinUpper, sq_nonneg (s - t),
+      mul_nonneg hd0 (sub_nonneg.mpr hd20)]
+  have hsum0 : 0 ≤ Real.cos (s - t) + Real.sin (s - t) := by
+    nlinarith
+  have hsumUpper : Real.cos (s - t) + Real.sin (s - t) ≤ (21 / 20 : ℝ) := by
+    nlinarith
+  have haLo := phase1_alpha_lower hs0 hs20
+  have haHi := phase1_alpha_nonpos_diag hs0 hs20
+  have hbLo := phase1_beta_lower hs0 hs20
+  have haProd : (-21 / 160 : ℝ) ≤
+      (Romik.alphaBeta1 params s).1 *
+        (Real.cos (s - t) + Real.sin (s - t)) := by
+    have hmul1 := mul_le_mul_of_nonpos_left hsumUpper haHi
+    have hmul2 := mul_le_mul_of_nonneg_right haLo
+      (show 0 ≤ (21 / 20 : ℝ) by norm_num)
+    nlinarith
+  have hbProd : (117 / 100 : ℝ) ≤
+      (Romik.alphaBeta1 params s).2 *
+        (Real.cos (s - t) - Real.sin (s - t)) := by
+    have hdiff0 : 0 ≤ Real.cos (s - t) - Real.sin (s - t) := by linarith
+    have hb0 : 0 ≤ (Romik.alphaBeta1 params s).2 := by linarith
+    nlinarith [mul_nonneg
+      ((show 0 ≤ (Romik.alphaBeta1 params s).2 - 13 / 10 by linarith)) hdiff0,
+      mul_nonneg (show 0 ≤ (13 / 10 : ℝ) by norm_num)
+        (show 0 ≤ Real.cos (s - t) - Real.sin (s - t) - 9 / 10 by linarith)]
+  rw [dot_rot_phase1_diag]
+  linarith
+
+private theorem dot_path1_diag_hasDerivAt (t s : ℝ) :
+    HasDerivAt (fun r => dot (Romik.path1 params r) (u t + v t))
+      (dot (Romik.rot s (Romik.alphaBeta1 params s)) (u t + v t)) s := by
+  have h := path1_hasDerivAt_public s
+  have h1 := HasDerivAt.const_mul (u t + v t).1 h.fst
+  have h2 := HasDerivAt.const_mul (u t + v t).2 h.snd
+  simpa [dot, mul_comm] using h1.fun_add h2
+
+theorem early_endpoint_diagonal_nonneg {t : ℝ}
+    (ht : t ∈ Icc (0 : ℝ) params.phi) :
+    0 ≤ dot (Romik.path params params.phi - Romik.path params t) (u t + v t) := by
+  have hmono : MonotoneOn
+      (fun s => dot (Romik.path1 params s) (u t + v t))
+      (Icc t params.phi) := by
+    refine monotoneOn_of_hasDerivWithinAt_nonneg
+      (f' := fun s => dot (Romik.rot s (Romik.alphaBeta1 params s)) (u t + v t))
+      (convex_Icc t params.phi) ?_ ?_ ?_
+    · intro s hs
+      exact (dot_path1_diag_hasDerivAt t s).continuousAt.continuousWithinAt
+    · intro s hs
+      exact (dot_path1_diag_hasDerivAt t s).hasDerivWithinAt
+    · intro s hs
+      have hi : s ∈ Ioo t params.phi := by
+        simpa only [interior_Icc] using hs
+      exact phase1_diag_deriv_nonneg ht.1 hi.1.le hi.2.le
+  have hle := hmono
+    (show t ∈ Icc t params.phi from ⟨le_rfl, ht.2⟩)
+    (show params.phi ∈ Icc t params.phi from ⟨ht.2, le_rfl⟩) ht.2
+  have hpathT : Romik.path params t = Romik.path1 params t := by
+    simp only [Romik.path, ite_eq_left ht.2]
+  have hpathPhi : Romik.path params params.phi =
+      Romik.path1 params params.phi := by
+    simp only [Romik.path, ite_eq_left le_rfl]
+  rw [hpathT, hpathPhi]
+  unfold dot at hle ⊢
+  dsimp at hle ⊢
+  linarith
+
+private def diagPhaseB4 (r : ℝ) : Point :=
+  Stage2.phaseA4 r - u r
+
+private def diagPhaseB5 (r : ℝ) : Point :=
+  Stage2.phaseA5 r - u r
+
+private theorem u_hasDerivAt_diag (r : ℝ) : HasDerivAt u (v r) r :=
+  Stage2.u_hasDerivAt r
+
+private theorem diagPhaseB4_hasDerivAt (r : ℝ) :
+    HasDerivAt diagPhaseB4 ((params.d1 - r / 2 - 1) • v r) r := by
+  have h := (Stage2.A4_hasDerivAt_public r).fun_sub (u_hasDerivAt_diag r)
+  refine h.congr_deriv ?_
+  apply Prod.ext <;> simp [v] <;> ring
+
+private theorem diagPhaseB5_hasDerivAt (r : ℝ) :
+    HasDerivAt diagPhaseB5 ((-(1 / 2 : ℝ)) • v r) r := by
+  have h := (Stage2.A5_hasDerivAt_public r).fun_sub (u_hasDerivAt_diag r)
+  refine h.congr_deriv ?_
+  apply Prod.ext <;> simp [v] <;> ring
+
+private theorem diagPhaseB4_eq_B {r : ℝ} (hr : r ∈ Icc eta tau) :
+    diagPhaseB4 r = B r := by
+  rcases hr.1.eq_or_lt with h | h
+  · subst r
+    have hm : Romik.path3 params eta = Romik.path4 params eta := by
+      simpa [eta, T] using PartB.match34
+    have hdiag : diagPhaseB4 eta =
+        Romik.path4 params eta +
+          (Romik.alphaBeta4 params eta).1 • v eta := by
+      apply Prod.ext <;> simp [diagPhaseB4, Stage2.phaseA4]
+    have hetaRaw : eta ≤ Real.pi / 2 - params.theta := by rfl
+    have hB : B eta = Romik.path3 params eta +
+        (Romik.alphaBeta3 params eta).1 • v eta := by
+      apply Prod.ext <;>
+        simp [B, alpha, alphaBetaAt, Romik.path,
+          not_le.mpr phi_lt_eta, not_le.mpr theta_lt_eta, hetaRaw]
+    calc
+      diagPhaseB4 eta = Romik.path4 params eta +
+          (Romik.alphaBeta4 params eta).1 • v eta := hdiag
+      _ = Romik.path3 params eta +
+          (Romik.alphaBeta3 params eta).1 • v eta := by
+            rw [hm, alphaBeta_match34_direct]
+      _ = B eta := hB.symm
+  · have hetaRaw : Real.pi / 2 - params.theta < r := by
+      simpa [eta, T] using h
+    have htauRaw : r ≤ Real.pi / 2 - params.phi := by
+      simpa [tau, T] using hr.2
+    apply Prod.ext <;>
+      simp [diagPhaseB4, Stage2.phaseA4, B, alpha, alphaBetaAt,
+        Romik.path, not_le.mpr (lt_trans phi_lt_eta h),
+        not_le.mpr (lt_trans theta_lt_eta h), not_le.mpr hetaRaw, htauRaw, eta, tau, T]
+
+private theorem diagPhaseB5_eq_B {r : ℝ} (hr : r ∈ Icc tau T) :
+    diagPhaseB5 r = B r := by
+  rcases hr.1.eq_or_lt with h | h
+  · subst r
+    have hm : Romik.path4 params tau = Romik.path5 params tau := by
+      simpa [tau, T] using PartB.match45
+    have hdiag : diagPhaseB5 tau =
+        Romik.path5 params tau +
+          (Romik.alphaBeta5 params tau).1 • v tau := by
+      apply Prod.ext <;> simp [diagPhaseB5, Stage2.phaseA5]
+    have htauRaw : tau ≤ Real.pi / 2 - params.phi := by rfl
+    have hetaRaw : ¬ tau ≤ Real.pi / 2 - params.theta := by
+      exact not_le.mpr (by simpa [eta, T] using eta_lt_tau)
+    have hB : B tau = Romik.path4 params tau +
+        (Romik.alphaBeta4 params tau).1 • v tau := by
+      apply Prod.ext <;>
+        simp [B, alpha, alphaBetaAt, Romik.path,
+          not_le.mpr (lt_trans phi_lt_eta eta_lt_tau),
+          not_le.mpr (lt_trans theta_lt_eta eta_lt_tau),
+          not_le.mpr eta_lt_tau, hetaRaw, htauRaw]
+    calc
+      diagPhaseB5 tau = Romik.path5 params tau +
+          (Romik.alphaBeta5 params tau).1 • v tau := hdiag
+      _ = Romik.path4 params tau +
+          (Romik.alphaBeta4 params tau).1 • v tau := by
+            rw [hm, alphaBeta_match45_direct]
+      _ = B tau := hB.symm
+  · have hetaRaw : Real.pi / 2 - params.theta < r := by
+      simpa [eta, T] using lt_trans eta_lt_tau h
+    have htauRaw : Real.pi / 2 - params.phi < r := by
+      simpa [tau, T] using h
+    apply Prod.ext <;>
+      simp [diagPhaseB5, Stage2.phaseA5, B, alpha, alphaBetaAt,
+        Romik.path,
+        not_le.mpr (lt_trans (lt_trans phi_lt_eta eta_lt_tau) h),
+        not_le.mpr (lt_trans (lt_trans theta_lt_eta eta_lt_tau) h),
+        not_le.mpr hetaRaw, not_le.mpr htauRaw, eta, tau, T]
+
+private theorem theta_upper_diag : params.theta ≤ (689 / 1000 : ℝ) := by
+  have h := theta_bounds.2
+  norm_num at h ⊢
+  linarith
+
+private theorem halfT_le_eta_sub_phi : T / 2 ≤ eta - params.phi := by
+  have hp : (157 / 50 : ℝ) < Real.pi := by
+    have h := Real.pi_gt_d2
+    norm_num at h ⊢
+    exact h
+  dsimp [eta, T]
+  nlinarith [theta_upper_diag, phi_le_one_twentieth_diag]
+
+private theorem d1_upper_diag : params.d1 ≤ (33 / 25 : ℝ) := by
+  have h := PartB.d1_contains.2
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at h ⊢
+  linarith
+
+private theorem eta_lower_diag : (4 / 5 : ℝ) ≤ eta := by
+  dsimp [eta, T]
+  nlinarith [Real.pi_gt_three, theta_upper_diag]
+
+private theorem phaseB4_coeff_nonpos_diag {r : ℝ} (hr : eta ≤ r) :
+    params.d1 - r / 2 - 1 ≤ 0 := by
+  nlinarith [d1_upper_diag, eta_lower_diag]
+
+private theorem late_cos_sub_sin_nonpos {t r : ℝ}
+    (ht : t ∈ Icc (0 : ℝ) params.phi) (hr : r ∈ Icc eta T) :
+    Real.cos (r - t) - Real.sin (r - t) ≤ 0 := by
+  have hhalf : T / 2 ≤ r - t := by
+    nlinarith [halfT_le_eta_sub_phi, ht.2, hr.1]
+  have hdeltaT : r - t ≤ T := by linarith [hr.2, ht.1]
+  have hlo : -(Real.pi / 2) ≤ T - (r - t) := by
+    dsimp [T] at hdeltaT ⊢
+    nlinarith [Real.pi_pos]
+  have hhi : r - t ≤ Real.pi / 2 := by simpa [T] using hdeltaT
+  have horder : T - (r - t) ≤ r - t := by linarith
+  have hs := Real.sin_le_sin_of_le_of_le_pi_div_two hlo hhi horder
+  have htrig : Real.sin (T - (r - t)) = Real.cos (r - t) := by
+    change Real.sin (Real.pi / 2 - (r - t)) = Real.cos (r - t)
+    exact Real.sin_pi_div_two_sub (r - t)
+  rw [htrig] at hs
+  linarith
+
+private theorem dot_fixed_hasDerivAt_diag
+    {f : ℝ → Point} {df w : Point} {r : ℝ}
+    (h : HasDerivAt f df r) :
+    HasDerivAt (fun s => dot (f s) w) (dot df w) r := by
+  have h1 := HasDerivAt.const_mul w.1 h.fst
+  have h2 := HasDerivAt.const_mul w.2 h.snd
+  simpa [dot, mul_comm] using h1.fun_add h2
+
+private theorem phaseB4_diag_monotone (t : ℝ)
+    (ht : t ∈ Icc (0 : ℝ) params.phi) :
+    MonotoneOn (fun r => dot (diagPhaseB4 r) (u t + v t)) (Icc eta tau) := by
+  refine monotoneOn_of_hasDerivWithinAt_nonneg
+    (f' := fun r => dot ((params.d1 - r / 2 - 1) • v r) (u t + v t))
+    (convex_Icc eta tau) ?_ ?_ ?_
+  · intro r hr
+    exact (dot_fixed_hasDerivAt_diag (w := u t + v t)
+      (diagPhaseB4_hasDerivAt r)).continuousAt.continuousWithinAt
+  · intro r hr
+    exact (dot_fixed_hasDerivAt_diag (w := u t + v t)
+      (diagPhaseB4_hasDerivAt r)).hasDerivWithinAt
+  · intro r hr
+    have hi : r ∈ Ioo eta tau := by simpa only [interior_Icc] using hr
+    have hcoef := phaseB4_coeff_nonpos_diag hi.1.le
+    have htrig := late_cos_sub_sin_nonpos ht
+      ⟨hi.1.le, le_trans hi.2.le tau_lt_T.le⟩
+    have heq :
+        dot ((params.d1 - r / 2 - 1) • v r) (u t + v t) =
+          (params.d1 - r / 2 - 1) *
+            (Real.cos (r - t) - Real.sin (r - t)) := by
+      simp [dot, u, v, Real.cos_sub, Real.sin_sub]
+      ring
+    rw [heq]
+    exact mul_nonneg_of_nonpos_of_nonpos hcoef htrig
+
+private theorem phaseB5_diag_monotone (t : ℝ)
+    (ht : t ∈ Icc (0 : ℝ) params.phi) :
+    MonotoneOn (fun r => dot (diagPhaseB5 r) (u t + v t)) (Icc tau T) := by
+  refine monotoneOn_of_hasDerivWithinAt_nonneg
+    (f' := fun r => dot ((-(1 / 2 : ℝ)) • v r) (u t + v t))
+    (convex_Icc tau T) ?_ ?_ ?_
+  · intro r hr
+    exact (dot_fixed_hasDerivAt_diag (w := u t + v t)
+      (diagPhaseB5_hasDerivAt r)).continuousAt.continuousWithinAt
+  · intro r hr
+    exact (dot_fixed_hasDerivAt_diag (w := u t + v t)
+      (diagPhaseB5_hasDerivAt r)).hasDerivWithinAt
+  · intro r hr
+    have hi : r ∈ Ioo tau T := by simpa only [interior_Icc] using hr
+    have htrig := late_cos_sub_sin_nonpos ht
+      ⟨le_trans eta_lt_tau.le hi.1.le, hi.2.le⟩
+    have heq : dot ((-(1 / 2 : ℝ)) • v r) (u t + v t) =
+        (-(1 / 2 : ℝ)) *
+          (Real.cos (r - t) - Real.sin (r - t)) := by
+      simp [dot, u, v, Real.cos_sub, Real.sin_sub]
+      ring
+    rw [heq]
+    exact mul_nonneg_of_nonpos_of_nonpos (by norm_num) htrig
+
+private theorem B_diag_mono_from_eta {t r : ℝ}
+    (ht : t ∈ Icc (0 : ℝ) params.phi) (hr : r ∈ Icc eta T) :
+    dot (B eta) (u t + v t) ≤ dot (B r) (u t + v t) := by
+  by_cases hrtau : r ≤ tau
+  · have hm := phaseB4_diag_monotone t ht
+    have heta : eta ∈ Icc eta tau := ⟨le_rfl, eta_lt_tau.le⟩
+    have hrr : r ∈ Icc eta tau := ⟨hr.1, hrtau⟩
+    have h := hm heta hrr hr.1
+    change dot (diagPhaseB4 eta) (u t + v t) ≤
+      dot (diagPhaseB4 r) (u t + v t) at h
+    rw [diagPhaseB4_eq_B heta, diagPhaseB4_eq_B hrr] at h
+    exact h
+  · have htaur : tau ≤ r := le_of_lt (lt_of_not_ge hrtau)
+    have heta : eta ∈ Icc eta tau := ⟨le_rfl, eta_lt_tau.le⟩
+    have htau4 : tau ∈ Icc eta tau := ⟨eta_lt_tau.le, le_rfl⟩
+    have htau5 : tau ∈ Icc tau T := ⟨le_rfl, tau_lt_T.le⟩
+    have hrr : r ∈ Icc tau T := ⟨htaur, hr.2⟩
+    have h4 := phaseB4_diag_monotone t ht heta htau4 eta_lt_tau.le
+    have h5 := phaseB5_diag_monotone t ht htau5 hrr htaur
+    change dot (diagPhaseB4 eta) (u t + v t) ≤
+      dot (diagPhaseB4 tau) (u t + v t) at h4
+    change dot (diagPhaseB5 tau) (u t + v t) ≤
+      dot (diagPhaseB5 r) (u t + v t) at h5
+    rw [diagPhaseB4_eq_B heta, diagPhaseB4_eq_B htau4] at h4
+    rw [diagPhaseB5_eq_B htau5, diagPhaseB5_eq_B hrr] at h5
+    exact le_trans h4 h5
+
+theorem B_early_diagonal_nonneg {t r : ℝ}
+    (ht : t ∈ Icc (0 : ℝ) params.phi) (hr : r ∈ Icc eta T) :
+    0 ≤ dot (B r - Romik.path params t) (u t + v t) := by
+  have hstart := early_endpoint_diagonal_nonneg ht
+  have hmono := B_diag_mono_from_eta ht hr
+  rw [B_eta_eq_path_phi] at hmono
+  unfold dot at hstart hmono ⊢
+  dsimp at hstart hmono ⊢
+  linarith
+
+theorem B_early_boundary_outside {t r : ℝ}
+    (ht : t ∈ Icc (0 : ℝ) params.phi) (hr : r ∈ Icc eta T) :
+    0 ≤ dot (B r - Romik.path params t) (u t) ∨
+      0 ≤ dot (B r - Romik.path params t) (v t) := by
+  have hdiag := B_early_diagonal_nonneg ht hr
+  by_contra h
+  push Not at h
+  have hadd :
+      dot (B r - Romik.path params t) (u t + v t) =
+        dot (B r - Romik.path params t) (u t) +
+          dot (B r - Romik.path params t) (v t) := by
+    unfold dot
+    dsimp [u, v]
+    ring
+  rw [hadd] at hdiag
+  linarith
+
+private theorem T_le_pi_diag : T ≤ Real.pi := by
+  dsimp [T]
+  nlinarith [Real.pi_pos]
+
+private theorem contact_u_mono_after
+    {F : ℝ → Point} {c : ℝ → ℝ} {a b t : ℝ}
+    (hder : ∀ q, HasDerivAt F ((c q) • v q) q)
+    (hc : ∀ q ∈ Ioo a b, c q ≤ 0)
+    (ht0 : 0 ≤ t) (hta : t ≤ a) (hbT : b ≤ T) :
+    MonotoneOn (fun q => dot (F q) (u t)) (Icc a b) := by
+  refine monotoneOn_of_hasDerivWithinAt_nonneg
+    (f' := fun q => dot ((c q) • v q) (u t))
+    (convex_Icc a b) ?_ ?_ ?_
+  · intro q hq
+    exact (dot_fixed_hasDerivAt_diag (w := u t) (hder q)).continuousAt.continuousWithinAt
+  · intro q hq
+    exact (dot_fixed_hasDerivAt_diag (w := u t) (hder q)).hasDerivWithinAt
+  · intro q hq
+    have hi : q ∈ Ioo a b := by simpa only [interior_Icc] using hq
+    have hsin : Real.sin (t - q) ≤ 0 := by
+      apply Real.sin_nonpos_of_nonpos_of_neg_pi_le
+      · linarith [hta, hi.1]
+      · have hqT : q ≤ T := le_trans hi.2.le hbT
+        have : q - t ≤ T := by linarith
+        nlinarith [T_le_pi_diag]
+    have heq : dot ((c q) • v q) (u t) = c q * Real.sin (t - q) := by
+      simp [dot, u, v, Real.sin_sub]
+      ring
+    rw [heq]
+    exact mul_nonneg_of_nonpos_of_nonpos (hc q hi) hsin
+
+private theorem contact_u_anti_before
+    {F : ℝ → Point} {c : ℝ → ℝ} {a b t : ℝ}
+    (hder : ∀ q, HasDerivAt F ((c q) • v q) q)
+    (hc : ∀ q ∈ Ioo a b, c q ≤ 0)
+    (ha0 : 0 ≤ a) (hbt : b ≤ t) (htT : t ≤ T) :
+    AntitoneOn (fun q => dot (F q) (u t)) (Icc a b) := by
+  refine antitoneOn_of_hasDerivWithinAt_nonpos
+    (f' := fun q => dot ((c q) • v q) (u t))
+    (convex_Icc a b) ?_ ?_ ?_
+  · intro q hq
+    exact (dot_fixed_hasDerivAt_diag (w := u t) (hder q)).continuousAt.continuousWithinAt
+  · intro q hq
+    exact (dot_fixed_hasDerivAt_diag (w := u t) (hder q)).hasDerivWithinAt
+  · intro q hq
+    have hi : q ∈ Ioo a b := by simpa only [interior_Icc] using hq
+    have hsin : 0 ≤ Real.sin (t - q) := by
+      apply Real.sin_nonneg_of_nonneg_of_le_pi
+      · linarith [hi.2, hbt]
+      · have hq0 : 0 ≤ q := le_trans ha0 hi.1.le
+        have : t - q ≤ T := by linarith
+        exact le_trans this T_le_pi_diag
+    have heq : dot ((c q) • v q) (u t) = c q * Real.sin (t - q) := by
+      simp [dot, u, v, Real.sin_sub]
+      ring
+    rw [heq]
+    exact mul_nonpos_of_nonpos_of_nonneg (hc q hi) hsin
+
+private theorem phase4_u_mono_after {a b t : ℝ}
+    (ha : eta ≤ a) (hb : b ≤ tau) (ht0 : 0 ≤ t) (hta : t ≤ a) :
+    MonotoneOn (fun q => dot (diagPhaseB4 q) (u t)) (Icc a b) := by
+  exact contact_u_mono_after
+    (F := diagPhaseB4) (c := fun q => params.d1 - q / 2 - 1)
+    diagPhaseB4_hasDerivAt
+    (by intro q hq; exact phaseB4_coeff_nonpos_diag (le_trans ha hq.1.le))
+    ht0 hta (le_trans hb tau_lt_T.le)
+
+private theorem phase4_u_anti_before {a b t : ℝ}
+    (ha : eta ≤ a) (hbt : b ≤ t) (htT : t ≤ T) :
+    AntitoneOn (fun q => dot (diagPhaseB4 q) (u t)) (Icc a b) := by
+  exact contact_u_anti_before
+    (F := diagPhaseB4) (c := fun q => params.d1 - q / 2 - 1)
+    diagPhaseB4_hasDerivAt
+    (by intro q hq; exact phaseB4_coeff_nonpos_diag (le_trans ha hq.1.le))
+    (le_trans (by norm_num : (0 : ℝ) ≤ 4 / 5)
+      (le_trans eta_lower_diag ha)) hbt htT
+
+private theorem phase5_u_mono_after {a b t : ℝ}
+    (hb : b ≤ T) (ht0 : 0 ≤ t) (hta : t ≤ a) :
+    MonotoneOn (fun q => dot (diagPhaseB5 q) (u t)) (Icc a b) := by
+  exact contact_u_mono_after
+    (F := diagPhaseB5) (c := fun _ => (-(1 / 2 : ℝ)))
+    diagPhaseB5_hasDerivAt (by intro q hq; norm_num)
+    ht0 hta hb
+
+private theorem phase5_u_anti_before {a b t : ℝ}
+    (ha : tau ≤ a) (hbt : b ≤ t) (htT : t ≤ T) :
+    AntitoneOn (fun q => dot (diagPhaseB5 q) (u t)) (Icc a b) := by
+  exact contact_u_anti_before
+    (F := diagPhaseB5) (c := fun _ => (-(1 / 2 : ℝ)))
+    diagPhaseB5_hasDerivAt (by intro q hq; norm_num)
+    (le_trans (by norm_num : (0 : ℝ) ≤ 4 / 5)
+      (le_trans eta_lower_diag (le_trans eta_lt_tau.le ha))) hbt htT
+
+private theorem B_u_ge_eta {t r : ℝ}
+    (ht : t ∈ Icc (0 : ℝ) eta) (hr : r ∈ Icc eta T) :
+    dot (B eta) (u t) ≤ dot (B r) (u t) := by
+  by_cases hrtau : r ≤ tau
+  · have heta : eta ∈ Icc eta r := ⟨le_rfl, hr.1⟩
+    have hrr : r ∈ Icc eta r := ⟨hr.1, le_rfl⟩
+    have hm := phase4_u_mono_after (a := eta) (b := r) (t := t)
+      le_rfl hrtau ht.1 ht.2
+    have h := hm heta hrr hr.1
+    change dot (diagPhaseB4 eta) (u t) ≤ dot (diagPhaseB4 r) (u t) at h
+    rw [diagPhaseB4_eq_B ⟨le_rfl, eta_lt_tau.le⟩,
+      diagPhaseB4_eq_B ⟨hr.1, hrtau⟩] at h
+    exact h
+  · have htaur : tau ≤ r := le_of_lt (lt_of_not_ge hrtau)
+    have heta4 : eta ∈ Icc eta tau := ⟨le_rfl, eta_lt_tau.le⟩
+    have htau4 : tau ∈ Icc eta tau := ⟨eta_lt_tau.le, le_rfl⟩
+    have htau5 : tau ∈ Icc tau r := ⟨le_rfl, htaur⟩
+    have hrr5 : r ∈ Icc tau r := ⟨htaur, le_rfl⟩
+    have h4 := (phase4_u_mono_after (a := eta) (b := tau) (t := t)
+      le_rfl le_rfl ht.1 ht.2)
+      heta4 htau4 eta_lt_tau.le
+    have h5 := (phase5_u_mono_after (a := tau) (b := r) (t := t)
+      hr.2 ht.1 (le_trans ht.2 eta_lt_tau.le))
+      htau5 hrr5 htaur
+    change dot (diagPhaseB4 eta) (u t) ≤ dot (diagPhaseB4 tau) (u t) at h4
+    change dot (diagPhaseB5 tau) (u t) ≤ dot (diagPhaseB5 r) (u t) at h5
+    rw [diagPhaseB4_eq_B heta4, diagPhaseB4_eq_B htau4] at h4
+    rw [diagPhaseB5_eq_B ⟨le_rfl, tau_lt_T.le⟩,
+      diagPhaseB5_eq_B ⟨htaur, hr.2⟩] at h5
+    exact le_trans h4 h5
+
+private theorem B_u_global_min {t r : ℝ}
+    (ht : t ∈ Icc eta T) (hr : r ∈ Icc eta T) :
+    dot (B t) (u t) ≤ dot (B r) (u t) := by
+  have ht0 : 0 ≤ t := by nlinarith [eta_lower_diag, ht.1]
+  by_cases httau : t ≤ tau
+  · have ht4 : t ∈ Icc eta tau := ⟨ht.1, httau⟩
+    by_cases hrtau : r ≤ tau
+    · have hr4 : r ∈ Icc eta tau := ⟨hr.1, hrtau⟩
+      by_cases hrt : r ≤ t
+      · have hrr : r ∈ Icc r t := ⟨le_rfl, hrt⟩
+        have htt : t ∈ Icc r t := ⟨hrt, le_rfl⟩
+        have ha := phase4_u_anti_before (a := r) (b := t) (t := t)
+          hr.1 le_rfl ht.2
+        have h := ha hrr htt hrt
+        change dot (diagPhaseB4 t) (u t) ≤ dot (diagPhaseB4 r) (u t) at h
+        rw [diagPhaseB4_eq_B hr4, diagPhaseB4_eq_B ht4] at h
+        exact h
+      · have htr : t ≤ r := le_of_lt (lt_of_not_ge hrt)
+        have htt : t ∈ Icc t r := ⟨le_rfl, htr⟩
+        have hrr : r ∈ Icc t r := ⟨htr, le_rfl⟩
+        have hm := phase4_u_mono_after (a := t) (b := r) (t := t)
+          ht.1 hrtau ht0 le_rfl
+        have h := hm htt hrr htr
+        change dot (diagPhaseB4 t) (u t) ≤ dot (diagPhaseB4 r) (u t) at h
+        rw [diagPhaseB4_eq_B ht4, diagPhaseB4_eq_B hr4] at h
+        exact h
+    · have htaur : tau ≤ r := le_of_lt (lt_of_not_ge hrtau)
+      have htt4 : t ∈ Icc t tau := ⟨le_rfl, httau⟩
+      have htau4' : tau ∈ Icc t tau := ⟨httau, le_rfl⟩
+      have h4 := (phase4_u_mono_after (a := t) (b := tau) (t := t)
+        ht.1 le_rfl ht0 le_rfl)
+        htt4 htau4' httau
+      have htau5' : tau ∈ Icc tau r := ⟨le_rfl, htaur⟩
+      have hrr5 : r ∈ Icc tau r := ⟨htaur, le_rfl⟩
+      have h5 := (phase5_u_mono_after (a := tau) (b := r) (t := t)
+        hr.2 ht0 httau)
+        htau5' hrr5 htaur
+      change dot (diagPhaseB4 t) (u t) ≤ dot (diagPhaseB4 tau) (u t) at h4
+      change dot (diagPhaseB5 tau) (u t) ≤ dot (diagPhaseB5 r) (u t) at h5
+      rw [diagPhaseB4_eq_B ht4,
+        diagPhaseB4_eq_B ⟨eta_lt_tau.le, le_rfl⟩] at h4
+      rw [diagPhaseB5_eq_B ⟨le_rfl, tau_lt_T.le⟩,
+        diagPhaseB5_eq_B ⟨htaur, hr.2⟩] at h5
+      exact le_trans h4 h5
+  · have htaut : tau ≤ t := le_of_lt (lt_of_not_ge httau)
+    have ht5 : t ∈ Icc tau T := ⟨htaut, ht.2⟩
+    by_cases hrtau : r ≤ tau
+    · have hr4 : r ∈ Icc eta tau := ⟨hr.1, hrtau⟩
+      have hrr4 : r ∈ Icc r tau := ⟨le_rfl, hrtau⟩
+      have htau4' : tau ∈ Icc r tau := ⟨hrtau, le_rfl⟩
+      have h4 := (phase4_u_anti_before (a := r) (b := tau) (t := t)
+        hr.1 htaut ht.2) hrr4 htau4' hrtau
+      have htau5' : tau ∈ Icc tau t := ⟨le_rfl, htaut⟩
+      have htt5 : t ∈ Icc tau t := ⟨htaut, le_rfl⟩
+      have h5 := (phase5_u_anti_before (a := tau) (b := t) (t := t)
+        le_rfl le_rfl ht.2) htau5' htt5 htaut
+      change dot (diagPhaseB4 tau) (u t) ≤ dot (diagPhaseB4 r) (u t) at h4
+      change dot (diagPhaseB5 t) (u t) ≤ dot (diagPhaseB5 tau) (u t) at h5
+      rw [diagPhaseB4_eq_B hr4,
+        diagPhaseB4_eq_B ⟨eta_lt_tau.le, le_rfl⟩] at h4
+      rw [diagPhaseB5_eq_B ⟨le_rfl, tau_lt_T.le⟩,
+        diagPhaseB5_eq_B ht5] at h5
+      exact le_trans h5 h4
+    · have htaur : tau ≤ r := le_of_lt (lt_of_not_ge hrtau)
+      have hr5 : r ∈ Icc tau T := ⟨htaur, hr.2⟩
+      by_cases hrt : r ≤ t
+      · have hrr : r ∈ Icc r t := ⟨le_rfl, hrt⟩
+        have htt : t ∈ Icc r t := ⟨hrt, le_rfl⟩
+        have ha := phase5_u_anti_before (a := r) (b := t) (t := t)
+          htaur le_rfl ht.2
+        have h := ha hrr htt hrt
+        change dot (diagPhaseB5 t) (u t) ≤ dot (diagPhaseB5 r) (u t) at h
+        rw [diagPhaseB5_eq_B hr5, diagPhaseB5_eq_B ht5] at h
+        exact h
+      · have htr : t ≤ r := le_of_lt (lt_of_not_ge hrt)
+        have htt : t ∈ Icc t r := ⟨le_rfl, htr⟩
+        have hrr : r ∈ Icc t r := ⟨htr, le_rfl⟩
+        have hm := phase5_u_mono_after (a := t) (b := r) (t := t)
+          hr.2 ht0 le_rfl
+        have h := hm htt hrr htr
+        change dot (diagPhaseB5 t) (u t) ≤ dot (diagPhaseB5 r) (u t) at h
+        rw [diagPhaseB5_eq_B ht5, diagPhaseB5_eq_B hr5] at h
+        exact h
+
+theorem B_late_u_nonneg {t r : ℝ}
+    (ht : t ∈ Icc params.phi T) (hr : r ∈ Icc eta T) :
+    0 ≤ dot (B r - Romik.path params t) (u t) := by
+  by_cases hteta : t ≤ eta
+  · have hstart := U_phi_nonneg ht
+    have hmono := B_u_ge_eta
+      (show t ∈ Icc (0 : ℝ) eta from ⟨le_trans phi_pos.le ht.1, hteta⟩) hr
+    rw [B_eta_eq_path_phi] at hmono
+    unfold Stage2.UValue at hstart
+    unfold dot at hstart hmono ⊢
+    dsimp at hstart hmono ⊢
+    linarith
+  · have heta : eta ≤ t := le_of_lt (lt_of_not_ge hteta)
+    have hmin := B_u_global_min (show t ∈ Icc eta T from ⟨heta, ht.2⟩) hr
+    have hcontact := B_inner_u_identity t
+    unfold dot at hmin hcontact ⊢
+    dsimp at hmin hcontact ⊢
+    linarith
+
+theorem B_boundary_outside {t r : ℝ}
+    (ht : t ∈ Ioo (0 : ℝ) T) (hr : r ∈ Icc eta T) :
+    0 ≤ dot (B r - Romik.path params t) (u t) ∨
+      0 ≤ dot (B r - Romik.path params t) (v t) := by
+  by_cases htphi : t ≤ params.phi
+  · exact B_early_boundary_outside ⟨ht.1.le, htphi⟩ hr
+  · exact Or.inl (B_late_u_nonneg
+      ⟨le_of_lt (lt_of_not_ge htphi), ht.2.le⟩ hr)
+
+private theorem phase1_beta_upper {s : ℝ}
+    (hs0 : 0 ≤ s) (hs20 : s ≤ (1 / 20 : ℝ)) :
+    (Romik.alphaBeta1 params s).2 ≤ (3 / 2 : ℝ) := by
+  have hpi : s ≤ Real.pi := by nlinarith [Real.pi_gt_three]
+  have hsin0 : 0 ≤ Real.sin s :=
+    Real.sin_nonneg_of_nonneg_of_le_pi hs0 hpi
+  have hcosUpper := Real.cos_le_one s
+  have hcos0 : 0 ≤ Real.cos s := by
+    have hlow := Real.one_sub_sq_div_two_le_cos (x := s)
+    nlinarith [hlow, sq_nonneg s, mul_nonneg hs0 (sub_nonneg.mpr hs20)]
+  have ha2 := Romik.a2_eq_neg_quarter_of_equations params_equations
+  have hmul : 2 * params.a1 * Real.cos s ≤ (5 / 2 : ℝ) := by
+    have hc := mul_le_mul_of_nonneg_left hcosUpper
+      (show 0 ≤ 2 * params.a1 by nlinarith [a1_lower_diag])
+    have ha : 2 * params.a1 ≤ (5 / 2 : ℝ) := by nlinarith [a1_upper_diag]
+    nlinarith
+  dsimp [Romik.alphaBeta1]
+  rw [ha2]
+  nlinarith
+
+private theorem path1_fst_antitone :
+    AntitoneOn (fun s => (Romik.path1 params s).1)
+      (Icc (0 : ℝ) params.phi) := by
+  refine antitoneOn_of_hasDerivWithinAt_nonpos
+    (f' := fun s => (Romik.rot s (Romik.alphaBeta1 params s)).1)
+    (convex_Icc (0 : ℝ) params.phi) ?_ ?_ ?_
+  · intro s hs
+    have h := dot_fixed_hasDerivAt_diag (w := ((1 : ℝ), 0))
+      (path1_hasDerivAt_public s)
+    simpa [dot] using h.continuousAt.continuousWithinAt
+  · intro s hs
+    have h := dot_fixed_hasDerivAt_diag (w := ((1 : ℝ), 0))
+      (path1_hasDerivAt_public s)
+    simpa [dot] using h.hasDerivWithinAt
+  · intro s hs
+    have hi : s ∈ Ioo (0 : ℝ) params.phi := by
+      simpa only [interior_Icc] using hs
+    have hs20 := le_trans hi.2.le phi_le_one_twentieth_diag
+    have ha := phase1_alpha_nonpos_diag hi.1.le hs20
+    have hb := phase1_beta_lower hi.1.le hs20
+    have hsin : 0 ≤ Real.sin s := by
+      exact Real.sin_nonneg_of_nonneg_of_le_pi hi.1.le
+        (by nlinarith [Real.pi_gt_three, hs20])
+    have hcos : 0 ≤ Real.cos s := by
+      have hlow := Real.one_sub_sq_div_two_le_cos (x := s)
+      nlinarith [hlow, sq_nonneg s,
+        mul_nonneg hi.1.le (sub_nonneg.mpr hs20)]
+    dsimp [Romik.rot]
+    nlinarith [mul_nonpos_of_nonpos_of_nonneg ha hcos,
+      mul_nonneg (show 0 ≤ (Romik.alphaBeta1 params s).2 by linarith) hsin]
+
+private theorem path1_snd_slope_upper :
+    AntitoneOn (fun s => (Romik.path1 params s).2 - (3 / 2 : ℝ) * s)
+      (Icc (0 : ℝ) params.phi) := by
+  refine antitoneOn_of_hasDerivWithinAt_nonpos
+    (f' := fun s => (Romik.rot s (Romik.alphaBeta1 params s)).2 - 3 / 2)
+    (convex_Icc (0 : ℝ) params.phi) ?_ ?_ ?_
+  · intro s hs
+    have hp := dot_fixed_hasDerivAt_diag (w := ((0 : ℝ), 1))
+      (path1_hasDerivAt_public s)
+    have h := hp.fun_sub
+      (HasDerivAt.const_mul (3 / 2 : ℝ) (hasDerivAt_id s))
+    simpa [dot] using h.continuousAt.continuousWithinAt
+  · intro s hs
+    have hp := dot_fixed_hasDerivAt_diag (w := ((0 : ℝ), 1))
+      (path1_hasDerivAt_public s)
+    have h := hp.fun_sub
+      (HasDerivAt.const_mul (3 / 2 : ℝ) (hasDerivAt_id s))
+    simpa [dot] using h.hasDerivWithinAt
+  · intro s hs
+    have hi : s ∈ Ioo (0 : ℝ) params.phi := by
+      simpa only [interior_Icc] using hs
+    have hs20 := le_trans hi.2.le phi_le_one_twentieth_diag
+    have ha := phase1_alpha_nonpos_diag hi.1.le hs20
+    have hbLo := phase1_beta_lower hi.1.le hs20
+    have hbHi := phase1_beta_upper hi.1.le hs20
+    have hsin : 0 ≤ Real.sin s := by
+      exact Real.sin_nonneg_of_nonneg_of_le_pi hi.1.le
+        (by nlinarith [Real.pi_gt_three, hs20])
+    have hcos0 : 0 ≤ Real.cos s := by
+      have hlow := Real.one_sub_sq_div_two_le_cos (x := s)
+      nlinarith [hlow, sq_nonneg s,
+        mul_nonneg hi.1.le (sub_nonneg.mpr hs20)]
+    have hcos1 := Real.cos_le_one s
+    dsimp [Romik.rot]
+    have haSin := mul_nonpos_of_nonpos_of_nonneg ha hsin
+    have hbCos : (Romik.alphaBeta1 params s).2 * Real.cos s ≤ 3 / 2 := by
+      have h1 := mul_le_mul_of_nonneg_right hbHi hcos0
+      have h2 := mul_le_mul_of_nonneg_left hcos1
+        (show 0 ≤ (3 / 2 : ℝ) by norm_num)
+      nlinarith
+    nlinarith
+
+private theorem phase1_path_x_nonpos {t : ℝ}
+    (ht : t ∈ Icc (0 : ℝ) params.phi) :
+    (Romik.path params t).1 ≤ 0 := by
+  have h0 : (0 : ℝ) ∈ Icc (0 : ℝ) params.phi := ⟨le_rfl, phi_pos.le⟩
+  have h := path1_fst_antitone h0 ht ht.1
+  have hp0 := congrArg Prod.fst pathZero
+  have hpt : Romik.path params t = Romik.path1 params t := by
+    simp [Romik.path, ht.2]
+  have hzero : Romik.path params 0 = Romik.path1 params 0 := by
+    simp [Romik.path, phi_pos.le]
+  change (Romik.path1 params t).1 ≤ (Romik.path1 params 0).1 at h
+  rw [← congrArg Prod.fst hzero, ← congrArg Prod.fst hpt] at h
+  simpa [hp0] using h
+
+private theorem phase1_path_y_upper {t : ℝ}
+    (ht : t ∈ Icc (0 : ℝ) params.phi) :
+    (Romik.path params t).2 ≤ (3 / 40 : ℝ) := by
+  have h0 : (0 : ℝ) ∈ Icc (0 : ℝ) params.phi := ⟨le_rfl, phi_pos.le⟩
+  have h := path1_snd_slope_upper h0 ht ht.1
+  have hp0 := congrArg Prod.snd pathZero
+  have hpt : Romik.path params t = Romik.path1 params t := by
+    simp [Romik.path, ht.2]
+  have hzero : Romik.path params 0 = Romik.path1 params 0 := by
+    simp [Romik.path, phi_pos.le]
+  change (Romik.path1 params t).2 - 3 / 2 * t ≤
+    (Romik.path1 params 0).2 - 3 / 2 * 0 at h
+  rw [← congrArg Prod.snd hzero, ← congrArg Prod.snd hpt] at h
+  nlinarith [phi_le_one_twentieth_diag, ht.2, hp0]
+
+private theorem B_T_x_formula : (B T).1 = params.k51 + params.e1 := by
+  have hetaRaw : ¬ T ≤ Real.pi / 2 - params.theta := by
+    exact not_le.mpr (by simpa [eta, T] using eta_lt_T)
+  have htauRaw : ¬ T ≤ Real.pi / 2 - params.phi := by
+    exact not_le.mpr (by simpa [tau, T] using tau_lt_T)
+  have hpath : Romik.path params T = Romik.path5 params T := by
+    simp only [Romik.path, ite_eq_right (not_le.mpr phi_lt_T),
+      ite_eq_right (not_le.mpr theta_lt_T), ite_eq_right hetaRaw, ite_eq_right htauRaw]
+  have halpha : alpha T = (Romik.alphaBeta5 params T).1 := by
+    simp only [alpha, alphaBetaAt, ite_eq_right (not_le.mpr phi_lt_T),
+      ite_eq_right (not_le.mpr theta_lt_T), ite_eq_right (not_le.mpr eta_lt_T),
+      ite_eq_right (not_le.mpr tau_lt_T)]
+  rw [B, hpath, halpha]
+  simp [Romik.path5, Romik.alphaBeta5, Romik.rot, Romik.addK,
+    v, T]
+  ring
+
+private theorem B_T_x_lower : (19 / 100 : ℝ) ≤ (B T).1 := by
+  have hk := PartB.k51_contains.1
+  have he := PartB.e1_contains.1
+  rw [B_T_x_formula]
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at hk he ⊢
+  linarith
+
+theorem B_T_u_nonneg_early {t : ℝ}
+    (ht : t ∈ Icc (0 : ℝ) params.phi) :
+    0 ≤ dot (B T - Romik.path params t) (u t) := by
+  have hx := phase1_path_x_nonpos ht
+  have hy := phase1_path_y_upper ht
+  have hBTx := B_T_x_lower
+  have hBTy := B_T_y_zero
+  have ht20 := le_trans ht.2 phi_le_one_twentieth_diag
+  have hsin0 : 0 ≤ Real.sin t := Real.sin_nonneg_of_nonneg_of_le_pi
+    ht.1 (by nlinarith [Real.pi_gt_three, ht20])
+  have hsin1 := Real.sin_le ht.1
+  have hcosLower : (799 / 800 : ℝ) ≤ Real.cos t := by
+    have h := Real.one_sub_sq_div_two_le_cos (x := t)
+    nlinarith [h, sq_nonneg t, mul_nonneg ht.1 (sub_nonneg.mpr ht20)]
+  have hcos0 : 0 ≤ Real.cos t := by linarith
+  have hdx : (19 / 100 : ℝ) ≤ (B T).1 - (Romik.path params t).1 := by
+    linarith
+  have hdxCos : (19 / 100 : ℝ) * (799 / 800) ≤
+      ((B T).1 - (Romik.path params t).1) * Real.cos t := by
+    have h1 := mul_le_mul_of_nonneg_right hdx hcos0
+    have h2 := mul_le_mul_of_nonneg_left hcosLower
+      (show 0 ≤ (19 / 100 : ℝ) by norm_num)
+    exact le_trans h2 h1
+  have hySin : (Romik.path params t).2 * Real.sin t ≤
+      (3 / 40 : ℝ) * (1 / 20) := by
+    have h1 := mul_le_mul_of_nonneg_right hy hsin0
+    have h2 : (3 / 40 : ℝ) * Real.sin t ≤ (3 / 40) * (1 / 20) := by
+      apply mul_le_mul_of_nonneg_left
+      · exact le_trans hsin1 ht20
+      · norm_num
+    exact le_trans h1 h2
+  unfold dot
+  dsimp [u]
+  rw [hBTy]
+  nlinarith
+
+theorem B_T_u_nonneg {t : ℝ} (ht : t ∈ Ioo (0 : ℝ) T) :
+    0 ≤ dot (B T - Romik.path params t) (u t) := by
+  by_cases htphi : t ≤ params.phi
+  · exact B_T_u_nonneg_early ⟨ht.1.le, htphi⟩
+  · exact B_late_u_nonneg
+      ⟨le_of_lt (lt_of_not_ge htphi), ht.2.le⟩
+      ⟨eta_lt_T.le, le_rfl⟩
+
+/-! ## Exact reflected forms
+
+Only the early half of the coefficient reflection is needed here.  Its image
+under `r ↦ T-r` is exactly the late `B` interval.  Keeping the endpoint cases
+explicit avoids relying on definitional reduction across the four switching
+equalities of `alphaBetaAt`.
+-/
+
+private theorem alpha_reflect_beta_early {r : ℝ}
+    (hr : r ∈ Icc (0 : ℝ) params.theta) :
+    alpha (T - r) = -beta r := by
+  by_cases hrphi : r < params.phi
+  · have hsphi : ¬ T - r ≤ params.phi := by
+      have hphiTau := lt_trans phi_lt_eta eta_lt_tau
+      exact not_le.mpr (by
+        dsimp [T, tau] at hphiTau ⊢
+        nlinarith [hrphi])
+    have hstheta : ¬ T - r ≤ params.theta := by
+      exact not_le.mpr (lt_trans theta_lt_tau (by dsimp [tau]; linarith))
+    have hseta : ¬ T - r ≤ eta := by
+      exact not_le.mpr (by dsimp [eta]; linarith [hrphi, phi_lt_theta])
+    have hstau : ¬ T - r ≤ tau := by
+      exact not_le.mpr (by dsimp [tau]; linarith)
+    have href := congrArg Prod.fst (alphaBeta5_reflect1_direct r)
+    simpa [alpha, beta, alphaBetaAt, noHiddenReflAB, hrphi.le,
+      hsphi, hstheta, hseta, hstau] using href
+  · have hphir : params.phi ≤ r := le_of_not_gt hrphi
+    by_cases hre : r = params.phi
+    · subst r
+      have href := congrArg Prod.fst
+        (alphaBeta4_reflect2_direct params.phi)
+      have hmatch := congrArg Prod.snd alphaBeta_match12_direct
+      have hTphi : T - params.phi = tau := by
+        rfl
+      rw [hTphi] at href
+      rw [hTphi]
+      simp only [alpha, beta, alphaBetaAt,
+        ite_eq_right (not_le.mpr (lt_trans phi_lt_eta eta_lt_tau)),
+        ite_eq_right (not_le.mpr theta_lt_tau),
+        ite_eq_right (not_le.mpr eta_lt_tau),
+        ite_eq_left le_rfl]
+      dsimp [noHiddenReflAB] at href
+      rw [hmatch]
+      exact href
+    · have hphir' : params.phi < r := lt_of_le_of_ne hphir (Ne.symm hre)
+      by_cases hretheta : r = params.theta
+      · subst r
+        have href := congrArg Prod.fst
+          (alphaBeta3_reflect_direct params.theta)
+        have hmatch := congrArg Prod.snd alphaBeta_match23_direct
+        simp only [alpha, beta, alphaBetaAt,
+          ite_eq_right (not_le.mpr phi_lt_theta), show T - params.theta = eta by rfl,
+          ite_eq_right (not_le.mpr phi_lt_eta),
+          ite_eq_right (not_le.mpr theta_lt_eta), ite_eq_left le_rfl]
+        dsimp [eta] at href ⊢
+        dsimp [noHiddenReflAB] at href
+        rw [hmatch]
+        exact href
+      · have hrtheta : r < params.theta :=
+          lt_of_le_of_ne hr.2 hretheta
+        have hsphi : ¬ T - r ≤ params.phi := by
+          have hthetaTau := theta_lt_tau
+          exact not_le.mpr (by
+            dsimp [T, tau] at hthetaTau ⊢
+            linarith [hrtheta])
+        have hstheta : ¬ T - r ≤ params.theta := by
+          have hthetaEta := theta_lt_eta
+          exact not_le.mpr (by
+            dsimp [T, eta] at hthetaEta ⊢
+            linarith [hrtheta])
+        have hseta : ¬ T - r ≤ eta := by
+          exact not_le.mpr (by dsimp [eta]; linarith)
+        have hstau : T - r ≤ tau := by
+          dsimp [tau]
+          linarith
+        have href := congrArg Prod.fst (alphaBeta4_reflect2_direct r)
+        simpa [alpha, beta, alphaBetaAt, noHiddenReflAB,
+          not_le.mpr hphir', hrtheta.le,
+          hsphi, hstheta, hseta, hstau] using href
+
+/-- The late `B` contact arc is the exact horizontal reflection of the early
+`D` contact arc. -/
+theorem B_reflect_D {r : ℝ} (hr : r ∈ Icc (0 : ℝ) params.theta) :
+    B (T - r) = noHiddenHReflect (D r) := by
+  have hp := path_reflect_noHidden
+    (show r ∈ Icc (0 : ℝ) T from
+      ⟨hr.1, le_trans hr.2 theta_lt_T.le⟩)
+  have ha := alpha_reflect_beta_early hr
+  rw [B, D, hp, ha]
+  apply Prod.ext
+  · dsimp [noHiddenHReflect, u, v, T]
+    rw [Real.sin_pi_div_two_sub]
+    ring
+  · dsimp [noHiddenHReflect, u, v, T]
+    rw [Real.cos_pi_div_two_sub]
+    ring
+
+private theorem B_reflect_u_dot_eq_D_v {r t : ℝ}
+    (hr : r ∈ Icc (0 : ℝ) params.theta)
+    (ht : t ∈ Icc (0 : ℝ) T) :
+    dot (B (T - r) - Romik.path params (T - t)) (u (T - t)) =
+      dot (D r - Romik.path params t) (v t) := by
+  rw [B_reflect_D hr, path_reflect_noHidden ht]
+  dsimp [dot, noHiddenHReflect, u, v, T]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  ring
+
+private theorem B_reflect_v_dot_eq_D_u {r t : ℝ}
+    (hr : r ∈ Icc (0 : ℝ) params.theta)
+    (ht : t ∈ Icc (0 : ℝ) T) :
+    dot (B (T - r) - Romik.path params (T - t)) (v (T - t)) =
+      dot (D r - Romik.path params t) (u t) := by
+  rw [B_reflect_D hr, path_reflect_noHidden ht]
+  dsimp [dot, noHiddenHReflect, u, v, T]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  ring
+
+/-- Every point of the early `D` contact arc lies outside at least one of the
+two moving inner walls. -/
+theorem D_boundary_outside {t r : ℝ}
+    (ht : t ∈ Ioo (0 : ℝ) T)
+    (hr : r ∈ Icc (0 : ℝ) params.theta) :
+    0 ≤ dot (D r - Romik.path params t) (u t) ∨
+      0 ≤ dot (D r - Romik.path params t) (v t) := by
+  have htRef : T - t ∈ Ioo (0 : ℝ) T := by
+    have ht0 := ht.1
+    have htT := ht.2
+    constructor <;> dsimp [T] at ht0 htT ⊢ <;> linarith
+  have hrRef : T - r ∈ Icc eta T := by
+    constructor
+    · dsimp [eta]
+      linarith [hr.2]
+    · linarith [hr.1]
+  have htClosed : t ∈ Icc (0 : ℝ) T := ⟨ht.1.le, ht.2.le⟩
+  rcases B_boundary_outside htRef hrRef with hu | hv
+  · right
+    rw [B_reflect_u_dot_eq_D_v hr htClosed] at hu
+    exact hu
+  · left
+    rw [B_reflect_v_dot_eq_D_u hr htClosed] at hv
+    exact hv
+
+/-- Reflected endpoint separator used for the strict lower horizontal range of
+the niche. -/
+theorem D_zero_v_nonneg {t : ℝ} (ht : t ∈ Ioo (0 : ℝ) T) :
+    0 ≤ dot (D 0 - Romik.path params t) (v t) := by
+  have htRef : T - t ∈ Ioo (0 : ℝ) T := by
+    have ht0 := ht.1
+    have htT := ht.2
+    constructor <;> dsimp [T] at ht0 htT ⊢ <;> linarith
+  have h := B_T_u_nonneg htRef
+  have htransport := B_reflect_u_dot_eq_D_v
+    (t := t) (r := 0)
+    (show (0 : ℝ) ∈ Icc (0 : ℝ) params.theta from
+      ⟨le_rfl, theta_pos.le⟩)
+    (show t ∈ Icc (0 : ℝ) T from ⟨ht.1.le, ht.2.le⟩)
+  have htransport' :
+      dot (B T - Romik.path params (T - t)) (u (T - t)) =
+        dot (D 0 - Romik.path params t) (v t) := by
+    simpa using htransport
+  rw [htransport'] at h
+  exact h
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 4: independent cap convexity and anchor foundation
+
+This module separates the already direct cap geometry from the later theorem
+that removing the downward niche preserves connectedness.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Set
+
+/-- The concrete endpoint anchor belongs to the reconstructed cap. -/
+theorem anchor_mem_K_direct : anchor ∈ K := by
+  have hA0 : A 0 = anchor := Stage2.A_zero_eq_anchor
+  rw [← hA0]
+  exact Stage3.supportA_direct 0 (by
+    constructor
+    · norm_num
+    · dsimp [T]
+      positivity)
+
+/-- Convexity of the literal cap follows directly from its half-plane
+definition. -/
+theorem K_convex_direct : Convex ℝ K := by
+  intro x hx y hy a b ha hb hab
+  refine ⟨?_, ?_⟩
+  · change 0 ≤ a * x.2 + b * y.2
+    exact add_nonneg (mul_nonneg ha hx.1) (mul_nonneg hb hy.1)
+  · intro t ht
+    constructor
+    · change dot (a • x + b • y) (u t) ≤
+        dot (Romik.path params t) (u t) + 1
+      have hxU := (hx.2 t ht).1
+      have hyU := (hy.2 t ht).1
+      dsimp [supportHalfU, dot] at hxU hyU ⊢
+      have hax := mul_le_mul_of_nonneg_left hxU ha
+      have hby := mul_le_mul_of_nonneg_left hyU hb
+      calc
+        (a * x.1 + b * y.1) * (u t).1 +
+            (a * x.2 + b * y.2) * (u t).2 =
+            a * (x.1 * (u t).1 + x.2 * (u t).2) +
+              b * (y.1 * (u t).1 + y.2 * (u t).2) := by ring
+        _ ≤ a * ((Romik.path params t).1 * (u t).1 +
+              (Romik.path params t).2 * (u t).2 + 1) +
+            b * ((Romik.path params t).1 * (u t).1 +
+              (Romik.path params t).2 * (u t).2 + 1) := add_le_add hax hby
+        _ = (Romik.path params t).1 * (u t).1 +
+              (Romik.path params t).2 * (u t).2 + 1 := by
+            rw [← add_mul, hab]
+            ring
+    · change dot (a • x + b • y) (v t) ≤
+        dot (Romik.path params t) (v t) + 1
+      have hxV := (hx.2 t ht).2
+      have hyV := (hy.2 t ht).2
+      dsimp [supportHalfV, dot] at hxV hyV ⊢
+      have hax := mul_le_mul_of_nonneg_left hxV ha
+      have hby := mul_le_mul_of_nonneg_left hyV hb
+      calc
+        (a * x.1 + b * y.1) * (v t).1 +
+            (a * x.2 + b * y.2) * (v t).2 =
+            a * (x.1 * (v t).1 + x.2 * (v t).2) +
+              b * (y.1 * (v t).1 + y.2 * (v t).2) := by ring
+        _ ≤ a * ((Romik.path params t).1 * (v t).1 +
+              (Romik.path params t).2 * (v t).2 + 1) +
+            b * ((Romik.path params t).1 * (v t).1 +
+              (Romik.path params t).2 * (v t).2 + 1) := add_le_add hax hby
+        _ = (Romik.path params t).1 * (v t).1 +
+              (Romik.path params t).2 * (v t).2 + 1 := by
+            rw [← add_mul, hab]
+            ring
+
+/-- The nonempty convex cap is connected. -/
+theorem K_connected_direct : IsConnected K :=
+  K_convex_direct.isConnected ⟨anchor, anchor_mem_K_direct⟩
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 4: independent vertical-fill topology
+
+This module isolates the connectedness half of the niche argument from the
+upper-envelope/frontier equality.  It can therefore be kernel-built even while
+the boundary module is still under repair.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Set
+
+/-- Vertical fill below a parametrized graph. -/
+def verticalFill (f : ℝ → Point) (I : Set ℝ) : Set Point :=
+  {q | ∃ t ∈ I, q.1 = (f t).1 ∧ 0 ≤ q.2 ∧ q.2 < (f t).2}
+
+/-- The three certified upper-envelope pieces as strict vertical fills. -/
+def certifiedNicheRegion : Set Point :=
+  verticalFill D (Icc (0 : ℝ) params.theta) ∪
+  verticalFill (Romik.path params) (Icc params.phi tau) ∪
+  verticalFill B (Icc eta T)
+
+/-- Consecutive endpoint incidences of the three graph pieces. -/
+theorem niche_piece_endpoints :
+    B eta = Romik.path params params.phi ∧
+    D params.theta = Romik.path params tau ∧
+    (B T).2 = 0 ∧ (D 0).2 = 0 :=
+  ⟨B_eta_eq_path_phi, D_theta_eq_path_tau,
+    B_T_y_zero, D_zero_y_zero⟩
+
+/-- A strict positive vertical fill is a continuous image of a connected
+product. -/
+theorem verticalFill_strict_isConnected
+    {f : ℝ → Point} {I : Set ℝ}
+    (hI : IsConnected I)
+    (hf : ContinuousOn f I)
+    (hy : ∀ t ∈ I, 0 < (f t).2) :
+    IsConnected (verticalFill f I) := by
+  let P : Set (ℝ × ℝ) := I ×ˢ Ico (0 : ℝ) 1
+  have hP : IsConnected P := by
+    dsimp [P]
+    exact hI.prod (isConnected_Ico (by norm_num))
+  let F : (ℝ × ℝ) → Point := fun z =>
+    ((f z.1).1, z.2 * (f z.1).2)
+  have hfcomp : ContinuousOn (fun z : ℝ × ℝ => f z.1) P := by
+    exact hf.comp continuous_fst.continuousOn (by
+      intro z hz
+      exact hz.1)
+  have hF : ContinuousOn F P := by
+    dsimp [F]
+    exact hfcomp.fst.prodMk (continuous_snd.continuousOn.mul hfcomp.snd)
+  have himage : F '' P = verticalFill f I := by
+    ext q
+    constructor
+    · rintro ⟨z, hz, rfl⟩
+      have hlt := mul_lt_mul_of_pos_right hz.2.2 (hy z.1 hz.1)
+      exact ⟨z.1, hz.1, rfl,
+        mul_nonneg hz.2.1 (hy z.1 hz.1).le, by simpa [F] using hlt⟩
+    · rintro ⟨t, ht, hx, hy0, hylt⟩
+      have htop := hy t ht
+      let s : ℝ := q.2 / (f t).2
+      have hs : s ∈ Ico (0 : ℝ) 1 := by
+        constructor
+        · exact div_nonneg hy0 htop.le
+        · exact (div_lt_one htop).2 hylt
+      refine ⟨(t, s), ⟨ht, hs⟩, ?_⟩
+      apply Prod.ext
+      · simpa [F] using hx.symm
+      · dsimp [F, s]
+        field_simp [ne_of_gt htop]
+  rw [← himage]
+  exact hP.image F hF
+
+theorem D_fill_Icc_eq_Ioc :
+    verticalFill D (Icc (0 : ℝ) params.theta) =
+      verticalFill D (Ioc (0 : ℝ) params.theta) := by
+  ext q
+  constructor
+  · rintro ⟨t, ht, hx, hy0, hylt⟩
+    have htpos : 0 < t := by
+      by_contra hnot
+      have ht0 : t = 0 := le_antisymm (le_of_not_gt hnot) ht.1
+      subst t
+      rw [D_zero_y_zero] at hylt
+      linarith
+    exact ⟨t, ⟨htpos, ht.2⟩, hx, hy0, hylt⟩
+  · rintro ⟨t, ht, hx, hy0, hylt⟩
+    exact ⟨t, ⟨ht.1.le, ht.2⟩, hx, hy0, hylt⟩
+
+theorem B_fill_Icc_eq_Ico :
+    verticalFill B (Icc eta T) = verticalFill B (Ico eta T) := by
+  ext q
+  constructor
+  · rintro ⟨t, ht, hx, hy0, hylt⟩
+    have htlt : t < T := by
+      by_contra hnot
+      have htT : t = T := le_antisymm ht.2 (le_of_not_gt hnot)
+      subst t
+      rw [B_T_y_zero] at hylt
+      linarith
+    exact ⟨t, ⟨ht.1, htlt⟩, hx, hy0, hylt⟩
+  · rintro ⟨t, ht, hx, hy0, hylt⟩
+    exact ⟨t, ⟨ht.1, ht.2.le⟩, hx, hy0, hylt⟩
+
+private theorem D_fill_connected :
+    IsConnected (verticalFill D (Ioc (0 : ℝ) params.theta)) := by
+  exact verticalFill_strict_isConnected
+    (isConnected_Ioc theta_pos) D_continuous.continuousOn
+    (by intro t ht; exact D_y_pos ht)
+
+private theorem path_fill_connected :
+    IsConnected
+      (verticalFill (Romik.path params) (Icc params.phi tau)) := by
+  exact verticalFill_strict_isConnected
+    (isConnected_Icc (le_trans phi_lt_theta.le
+      (le_trans theta_lt_eta.le eta_lt_tau.le)))
+    pathContinuous.continuousOn (by
+      intro t ht
+      exact path_core_y_pos ht)
+
+private theorem B_fill_connected :
+    IsConnected (verticalFill B (Ico eta T)) := by
+  exact verticalFill_strict_isConnected
+    (isConnected_Ico eta_lt_T) B_continuous.continuousOn
+    (by intro t ht; exact B_y_pos ht)
+
+theorem niche_fills_overlap :
+    (verticalFill D (Ioc (0 : ℝ) params.theta) ∩
+      verticalFill (Romik.path params) (Icc params.phi tau)).Nonempty ∧
+    (verticalFill (Romik.path params) (Icc params.phi tau) ∩
+      verticalFill B (Ico eta T)).Nonempty := by
+  refine ⟨?_, ?_⟩
+  · let q : Point := ((D params.theta).1, (D params.theta).2 / 2)
+    have hy0 : 0 ≤ q.2 := by
+      dsimp [q]
+      nlinarith [D_theta_y_pos]
+    have hylt : q.2 < (D params.theta).2 := by
+      dsimp [q]
+      nlinarith [D_theta_y_pos]
+    refine ⟨q, ?_, ?_⟩
+    · exact ⟨params.theta, ⟨theta_pos, le_rfl⟩, rfl, hy0, hylt⟩
+    · refine ⟨tau, ⟨?_, le_rfl⟩, ?_, ?_, ?_⟩
+      · exact le_trans phi_lt_theta.le (le_trans theta_lt_eta.le eta_lt_tau.le)
+      · dsimp [q]
+        rw [← D_theta_eq_path_tau]
+      · exact hy0
+      · simpa [D_theta_eq_path_tau] using hylt
+  · let q : Point := ((B eta).1, (B eta).2 / 2)
+    have hy0 : 0 ≤ q.2 := by
+      dsimp [q]
+      nlinarith [B_eta_y_pos]
+    have hylt : q.2 < (B eta).2 := by
+      dsimp [q]
+      nlinarith [B_eta_y_pos]
+    refine ⟨q, ?_, ?_⟩
+    · refine ⟨params.phi, ⟨le_rfl, ?_⟩, ?_, ?_, ?_⟩
+      · exact le_trans phi_lt_theta.le (le_trans theta_lt_eta.le eta_lt_tau.le)
+      · dsimp [q]
+        rw [← B_eta_eq_path_phi]
+      · exact hy0
+      · simpa [B_eta_eq_path_phi] using hylt
+    · exact ⟨eta, ⟨le_rfl, eta_lt_T⟩, rfl, hy0, hylt⟩
+
+/-- Connectedness of the certified three-piece vertical-fill region. -/
+theorem certifiedNicheRegion_connected : IsConnected certifiedNicheRegion := by
+  unfold certifiedNicheRegion
+  rw [D_fill_Icc_eq_Ioc, B_fill_Icc_eq_Ico]
+  rcases niche_fills_overlap with ⟨hDX, hXB⟩
+  have hDXc := D_fill_connected.union hDX path_fill_connected
+  exact hDXc.union (by
+    rcases hXB with ⟨q, hqX, hqB⟩
+    exact ⟨q, Or.inr hqX, hqB⟩) B_fill_connected
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 4: inner-wall graph identities
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Set
+
+/-- First inner wall as a graph over horizontal coordinate. -/
+def bRoof (t X : ℝ) : ℝ :=
+  (dot (Romik.path params t) (u t) - X * Real.cos t) / Real.sin t
+
+/-- Second inner wall as a graph over horizontal coordinate. -/
+def dRoof (t X : ℝ) : ℝ :=
+  (dot (Romik.path params t) (v t) + X * Real.sin t) / Real.cos t
+
+/-- Vertical roof of one instantaneous open inner quadrant. -/
+def instantRoof (t X : ℝ) : ℝ := min (bRoof t X) (dRoof t X)
+
+/-- A point of nonnegative height is in the instantaneous inner quadrant iff it
+lies strictly below both graph roofs. -/
+theorem mem_innerQuadrant_iff_roofs {t X Y : ℝ}
+    (ht : t ∈ Set.Ioo (0 : ℝ) T) :
+    (X,Y) ∈ Romik.innerQuadrantAt params t ↔
+      Y < bRoof t X ∧ Y < dRoof t X := by
+  have hs : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi ht.1 (lt_trans ht.2 (by
+    dsimp [T]; linarith [Real.pi_pos]))
+  have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo ⟨by
+    linarith [Real.pi_pos, ht.1], by simpa [T] using ht.2⟩
+  simp only [Romik.innerQuadrantAt, dot, u, v, mul_neg, neg_add_lt_iff_lt_add, add_zero,
+    mem_ofPred_eq, bRoof, dRoof]
+  constructor
+  · rintro ⟨hu,hv⟩
+    constructor
+    · apply (lt_div_iff₀ hs).2
+      linarith
+    · apply (lt_div_iff₀ hc).2
+      linarith
+  · rintro ⟨hb,hd⟩
+    constructor
+    · have := (lt_div_iff₀ hs).1 hb
+      linarith
+    · have := (lt_div_iff₀ hc).1 hd
+      linarith
+
+/-- The two walls agree at their apex. -/
+theorem roofs_at_path (t : ℝ) (ht : t ∈ Set.Ioo (0 : ℝ) T) :
+    bRoof t (Romik.path params t).1 = (Romik.path params t).2 ∧
+    dRoof t (Romik.path params t).1 = (Romik.path params t).2 := by
+  have hs : Real.sin t ≠ 0 := ne_of_gt (Real.sin_pos_of_pos_of_lt_pi ht.1 (lt_trans ht.2 (by
+    dsimp [T]; linarith [Real.pi_pos])))
+  have hc : Real.cos t ≠ 0 := ne_of_gt (Real.cos_pos_of_mem_Ioo ⟨by
+    linarith [Real.pi_pos, ht.1], by simpa [T] using ht.2⟩)
+  constructor <;> simp [bRoof, dRoof, dot, u, v, hs, hc]
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 4: direct membership of the certified vertical fills
+
+These are the three concrete reverse-inclusion facts needed to identify the
+literal niche with its certified vertical-fill region.  They are proved from
+the literal open-quadrant definition, not assumed through a Stage 2 API.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Set
+open Stage2
+
+private theorem sin_pos_physical {t : ℝ} (ht : t ∈ Ioo (0 : ℝ) T) :
+    0 < Real.sin t :=
+  Real.sin_pos_of_pos_of_lt_pi ht.1
+    (lt_trans ht.2 (by dsimp [T]; linarith [Real.pi_pos]))
+
+private theorem cos_pos_physical {t : ℝ} (ht : t ∈ Ioo (0 : ℝ) T) :
+    0 < Real.cos t :=
+  Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, ht.1], by simpa [T] using ht.2⟩
+
+private theorem phi_upper_twentieth_membership :
+    params.phi ≤ (1 / 20 : ℝ) := by
+  have h := phi_bounds.2
+  norm_num at h ⊢
+  linarith
+
+private theorem a1_lower_six_fifths_membership :
+    (6 / 5 : ℝ) ≤ params.a1 := by
+  have h := Romik.a1_lower_bound_of_mem_box params_mem
+  norm_num at h ⊢
+  linarith
+
+private theorem alphaBeta1_snd_nonneg_early {s : ℝ}
+    (hs0 : 0 ≤ s) (hs20 : s ≤ (1 / 20 : ℝ)) :
+    0 ≤ (Romik.alphaBeta1 params s).2 := by
+  have hpi : s ≤ Real.pi := by
+    nlinarith [Real.pi_gt_three]
+  have hsin0 : 0 ≤ Real.sin s :=
+    Real.sin_nonneg_of_nonneg_of_le_pi hs0 hpi
+  have hsinUpper := Real.sin_le hs0
+  have hcosLower : 1 - s ^ 2 / 2 ≤ Real.cos s :=
+    Real.one_sub_sq_div_two_le_cos
+  have hcos0 : 0 ≤ Real.cos s := by
+    apply Real.cos_nonneg_of_mem_Icc
+    constructor <;> nlinarith [Real.pi_gt_three]
+  have hquad : 0 ≤ s * ((1 / 20 : ℝ) - s) :=
+    mul_nonneg hs0 (sub_nonneg.mpr hs20)
+  have hcoef : (12 / 5 : ℝ) ≤ 2 * params.a1 := by
+    nlinarith [a1_lower_six_fifths_membership]
+  have hmul : (12 / 5 : ℝ) * Real.cos s ≤
+      2 * params.a1 * Real.cos s :=
+    mul_le_mul_of_nonneg_right hcoef hcos0
+  have ha2 := Romik.a2_eq_neg_quarter_of_equations params_equations
+  dsimp [Romik.alphaBeta1]
+  rw [ha2]
+  nlinarith [hsinUpper, hcosLower, hquad, hmul]
+
+private theorem beta_nonneg_early {t : ℝ}
+    (ht : t ∈ Icc (0 : ℝ) params.theta) : 0 ≤ beta t := by
+  by_cases hphi : t ≤ params.phi
+  · have hs20 : t ≤ (1 / 20 : ℝ) :=
+      le_trans hphi phi_upper_twentieth_membership
+    have h := alphaBeta1_snd_nonneg_early ht.1 hs20
+    simpa [beta, alphaBetaAt, hphi] using h
+  · have htPhysical : t ∈ Icc (0 : ℝ) T :=
+      ⟨ht.1, le_trans ht.2 (le_trans theta_lt_eta.le
+        (le_trans eta_lt_tau.le tau_le_T))⟩
+    have hrho := Stage2.rhoA_nonneg htPhysical
+    unfold Stage2.rhoA at hrho
+    rw [ite_eq_right hphi, ite_eq_left ht.2] at hrho
+    simp only [beta, alphaBetaAt, ite_eq_right hphi, ite_eq_left ht.2]
+    dsimp [Romik.alphaBeta2]
+    exact hrho
+
+private theorem B_bRoof_eq {t : ℝ} (ht : t ∈ Ioo (0 : ℝ) T) :
+    bRoof t (B t).1 = (B t).2 := by
+  have hs0 : Real.sin t ≠ 0 := ne_of_gt (sin_pos_physical ht)
+  have hcontact := B_inner_u_identity t
+  unfold bRoof
+  apply (div_eq_iff hs0).2
+  dsimp [dot, u] at hcontact ⊢
+  nlinarith
+
+private theorem B_le_dRoof {t : ℝ}
+    (ht : t ∈ Ioo (0 : ℝ) T) (htPhi : t ∈ Icc params.phi T) :
+    (B t).2 ≤ dRoof t (B t).1 := by
+  have hc := cos_pos_physical ht
+  have ha := alpha_nonpos htPhi
+  have hv : dot (B t - Romik.path params t) (v t) = alpha t := by
+    simp only [B, dot, v]
+    simp
+    nlinarith [Real.sin_sq_add_cos_sq t]
+  unfold dRoof
+  apply (le_div_iff₀ hc).2
+  dsimp [dot, v] at hv ⊢
+  nlinarith
+
+private theorem D_dRoof_eq {t : ℝ} (ht : t ∈ Ioo (0 : ℝ) T) :
+    dRoof t (D t).1 = (D t).2 := by
+  have hc0 : Real.cos t ≠ 0 := ne_of_gt (cos_pos_physical ht)
+  have hcontact := D_inner_v_identity t
+  unfold dRoof
+  apply (div_eq_iff hc0).2
+  dsimp [dot, v] at hcontact ⊢
+  nlinarith
+
+private theorem D_le_bRoof {t : ℝ}
+    (ht : t ∈ Ioo (0 : ℝ) T)
+    (htTheta : t ∈ Icc (0 : ℝ) params.theta) :
+    (D t).2 ≤ bRoof t (D t).1 := by
+  have hs := sin_pos_physical ht
+  have hb := beta_nonneg_early htTheta
+  have hu : dot (D t - Romik.path params t) (u t) = -beta t := by
+    simp only [D, dot, u]
+    simp
+    nlinarith [Real.sin_sq_add_cos_sq t]
+  unfold bRoof
+  apply (le_div_iff₀ hs).2
+  dsimp [dot, u] at hu ⊢
+  nlinarith
+
+/-- Every strict vertical point below a core-path point belongs to the literal
+niche. -/
+theorem vertical_below_path_mem_niche {r Y : ℝ}
+    (hr : r ∈ Icc params.phi tau) (hY0 : 0 ≤ Y)
+    (hY : Y < (Romik.path params r).2) :
+    ((Romik.path params r).1, Y) ∈ Romik.niche params := by
+  have hrIoo : r ∈ Ioo (0 : ℝ) T :=
+    ⟨lt_of_lt_of_le phi_pos hr.1,
+      lt_of_le_of_lt hr.2 tau_lt_T⟩
+  refine ⟨by simpa [capFan] using hY0, ⟨r, hrIoo, ?_⟩⟩
+  apply (mem_innerQuadrant_iff_roofs hrIoo).2
+  have hroof := roofs_at_path r hrIoo
+  rw [hroof.1, hroof.2]
+  exact ⟨hY, hY⟩
+
+/-- Every strict vertical point below the late `B` contact belongs to the
+literal niche. -/
+theorem vertical_below_B_mem_niche {r Y : ℝ}
+    (hr : r ∈ Icc eta T) (hY0 : 0 ≤ Y) (hY : Y < (B r).2) :
+    ((B r).1, Y) ∈ Romik.niche params := by
+  have hrlt : r < T := by
+    by_contra hnot
+    have hrT : r = T := le_antisymm hr.2 (le_of_not_gt hnot)
+    subst r
+    rw [B_T_y_zero] at hY
+    linarith
+  have hrIoo : r ∈ Ioo (0 : ℝ) T :=
+    ⟨lt_of_lt_of_le (lt_trans theta_pos theta_lt_eta) hr.1, hrlt⟩
+  have hrPhi : r ∈ Icc params.phi T :=
+    ⟨le_trans (le_trans phi_lt_theta.le theta_lt_eta.le) hr.1, hr.2⟩
+  refine ⟨by simpa [capFan] using hY0, ⟨r, hrIoo, ?_⟩⟩
+  apply (mem_innerQuadrant_iff_roofs hrIoo).2
+  refine ⟨?_, ?_⟩
+  · rw [B_bRoof_eq hrIoo]
+    exact hY
+  · exact lt_of_lt_of_le hY (B_le_dRoof hrIoo hrPhi)
+
+/-- Every strict vertical point below the early `D` contact belongs to the
+literal niche. -/
+theorem vertical_below_D_mem_niche {r Y : ℝ}
+    (hr : r ∈ Icc (0 : ℝ) params.theta) (hY0 : 0 ≤ Y)
+    (hY : Y < (D r).2) :
+    ((D r).1, Y) ∈ Romik.niche params := by
+  have hrpos : 0 < r := by
+    by_contra hnot
+    have hr0 : r = 0 := le_antisymm (le_of_not_gt hnot) hr.1
+    subst r
+    rw [D_zero_y_zero] at hY
+    linarith
+  have hrIoo : r ∈ Ioo (0 : ℝ) T :=
+    ⟨hrpos, lt_of_le_of_lt hr.2 theta_lt_T⟩
+  refine ⟨by simpa [capFan] using hY0, ⟨r, hrIoo, ?_⟩⟩
+  apply (mem_innerQuadrant_iff_roofs hrIoo).2
+  refine ⟨?_, ?_⟩
+  · exact lt_of_lt_of_le hY (D_le_bRoof hrIoo hr)
+  · rw [D_dRoof_eq hrIoo]
+    exact hY
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 4: frontier of a strict vertical subgraph
+
+This file contains the topological calculation used by the concrete Gerver
+niche.  The generic lemma is intentionally independent of the old Stage 2
+topology sketches: it computes the closure and the interior of a strict
+vertical subgraph directly.
+-/
+
+/-! ## The three Gerver roof arcs are graphs over horizontal position -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Set
+
+private theorem u_hasDerivAt_frontier (t : ℝ) : HasDerivAt u (v t) t :=
+  Stage2.u_hasDerivAt t
+
+private theorem v_hasDerivAt_frontier (t : ℝ) : HasDerivAt v (-u t) t :=
+  Stage2.v_hasDerivAt t
+
+private theorem square_hasDerivAt_frontier (t : ℝ) :
+    HasDerivAt (fun s : ℝ => s * s) (t + t) t :=
+  Stage2.square_hasDerivAt t
+
+private theorem beta1_hasDerivAt_frontier (t : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta1 params s).2)
+      (-2 * params.a1 * Real.sin t + 2 * params.a2 * Real.cos t) t := by
+  have h1 := HasDerivAt.const_mul (2 * params.a1) (Real.hasDerivAt_cos t)
+  have h2 := HasDerivAt.const_mul (2 * params.a2) (Real.hasDerivAt_sin t)
+  have h := (h1.fun_add h2).fun_sub (hasDerivAt_const t (1 : ℝ))
+  refine h.congr_deriv ?_
+  ring
+
+private theorem beta2_hasDerivAt_frontier (t : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta2 params s).2)
+      (-(1 / 2 : ℝ) * t + params.b1) t :=
+  beta2_hasDerivAt t
+
+private theorem alpha4_hasDerivAt_frontier (t : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta4 params s).1)
+      ((1 / 2 : ℝ) * t - params.d1) t :=
+  alpha4_hasDerivAt t
+
+private theorem alpha5_hasDerivAt_frontier (t : ℝ) :
+    HasDerivAt (fun s => (Romik.alphaBeta5 params s).1)
+      (-2 * params.e1 * Real.cos t - 2 * params.e2 * Real.sin t) t :=
+  alpha5_hasDerivAt t
+
+private theorem contactD_hasDerivAt
+    {x : ℝ → Point} {ab : ℝ → Point} {bp : ℝ} (t : ℝ)
+    (hx : HasDerivAt x (Romik.rot t (ab t)) t)
+    (hb : HasDerivAt (fun s => (ab s).2) bp t) :
+    HasDerivAt (fun s => x s - (ab s).2 • u s)
+      (((ab t).1 - bp) • u t) t := by
+  have h := hx.fun_sub (hb.fun_smul (u_hasDerivAt_frontier t))
+  refine h.congr_deriv ?_
+  ext <;> simp [Romik.rot, u, v] <;> ring
+
+private theorem contactB_hasDerivAt
+    {x : ℝ → Point} {ab : ℝ → Point} {ap : ℝ} (t : ℝ)
+    (hx : HasDerivAt x (Romik.rot t (ab t)) t)
+    (ha : HasDerivAt (fun s => (ab s).1) ap t) :
+    HasDerivAt (fun s => x s + (ab s).1 • v s)
+      (((ab t).2 + ap) • v t) t := by
+  have h := hx.fun_add (ha.fun_smul (v_hasDerivAt_frontier t))
+  refine h.congr_deriv ?_
+  ext <;> simp [Romik.rot, u, v] <;> ring
+
+private def phaseD1 (t : ℝ) : Point :=
+  Romik.path1 params t - (Romik.alphaBeta1 params t).2 • u t
+
+private def phaseD2 (t : ℝ) : Point :=
+  Romik.path2 params t - (Romik.alphaBeta2 params t).2 • u t
+
+private def phaseB4_m421765a (t : ℝ) : Point :=
+  Romik.path4 params t + (Romik.alphaBeta4 params t).1 • v t
+
+private def phaseB5_m421765a (t : ℝ) : Point :=
+  Romik.path5 params t + (Romik.alphaBeta5 params t).1 • v t
+
+private theorem phaseD1_hasDerivAt (t : ℝ) :
+    HasDerivAt phaseD1 ((1 / 2 : ℝ) • u t) t := by
+  have h := contactD_hasDerivAt t (path1_hasDerivAt_public t)
+    (beta1_hasDerivAt_frontier t)
+  exact h.congr_deriv (by
+    apply congrArg (fun c : ℝ => c • u t)
+    dsimp [Romik.alphaBeta1]
+    ring)
+
+private theorem phaseD2_hasDerivAt (t : ℝ) :
+    HasDerivAt phaseD2 ((1 + params.b1 - t / 2) • u t) t := by
+  have h := contactD_hasDerivAt t (path2_hasDerivAt_public t)
+    (beta2_hasDerivAt_frontier t)
+  exact h.congr_deriv (by
+    apply congrArg (fun c : ℝ => c • u t)
+    dsimp [Romik.alphaBeta2]
+    ring)
+
+private theorem phaseB4_hasDerivAt (t : ℝ) :
+    HasDerivAt phaseB4_m421765a ((params.d1 - 1 - t / 2) • v t) t := by
+  have h := contactB_hasDerivAt t (path4_hasDerivAt_public t)
+    (alpha4_hasDerivAt_frontier t)
+  exact h.congr_deriv (by
+    apply congrArg (fun c : ℝ => c • v t)
+    dsimp [Romik.alphaBeta4]
+    ring)
+
+private theorem phaseB5_hasDerivAt (t : ℝ) :
+    HasDerivAt phaseB5_m421765a ((-(1 / 2 : ℝ)) • v t) t := by
+  have h := contactB_hasDerivAt t (path5_hasDerivAt_public t)
+    (alpha5_hasDerivAt_frontier t)
+  exact h.congr_deriv (by
+    apply congrArg (fun c : ℝ => c • v t)
+    dsimp [Romik.alphaBeta5]
+    ring)
+
+private theorem phaseD1_continuous : Continuous phaseD1 := by
+  exact continuous_iff_continuousAt.2 fun t => (phaseD1_hasDerivAt t).continuousAt
+
+private theorem phaseD2_continuous : Continuous phaseD2 := by
+  exact continuous_iff_continuousAt.2 fun t => (phaseD2_hasDerivAt t).continuousAt
+
+private theorem phaseB4_continuous : Continuous phaseB4_m421765a := by
+  exact continuous_iff_continuousAt.2 fun t => (phaseB4_hasDerivAt t).continuousAt
+
+private theorem phaseB5_continuous : Continuous phaseB5_m421765a := by
+  exact continuous_iff_continuousAt.2 fun t => (phaseB5_hasDerivAt t).continuousAt
+
+private theorem dot_fixed_hasDerivAt_frontier
+    {f : ℝ → Point} {df w : Point} {t : ℝ}
+    (h : HasDerivAt f df t) :
+    HasDerivAt (fun s => dot (f s) w) (dot df w) t :=
+  dot_fixed_hasDerivAt h w
+
+private theorem b1_lower_frontier : (-53 / 100 : ℝ) ≤ params.b1 := by
+  have h := PartB.b1_contains.1
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at h ⊢
+  linarith
+
+private theorem theta_upper_frontier : params.theta ≤ (689 / 1000 : ℝ) := by
+  have h := theta_bounds.2
+  norm_num at h ⊢
+  linarith
+
+private theorem d1_lower_frontier : (13 / 10 : ℝ) ≤ params.d1 := by
+  have h := PartB.d1_contains.1
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at h ⊢
+  linarith
+
+private theorem d1_upper_frontier : params.d1 ≤ (33 / 25 : ℝ) := by
+  have h := PartB.d1_contains.2
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at h ⊢
+  linarith
+
+private theorem eta_lower_frontier : (4 / 5 : ℝ) ≤ eta := by
+  dsimp [eta, T]
+  nlinarith [Real.pi_gt_three, theta_upper_frontier]
+
+private theorem T_upper_frontier : T < (8 / 5 : ℝ) := by
+  have hp := ExactReplay.piI_contains_pi
+  have hhi : Real.pi ≤ (ExactReplay.piI.hi : ℝ) := hp.2
+  have h32 : (ExactReplay.piI.hi : ℝ) < (16 / 5 : ℝ) := by
+    norm_num [ExactReplay.piI, ExactReplay.q]
+  dsimp [T]
+  linarith
+
+private theorem phaseD1_fst_strictMono :
+    StrictMonoOn (fun t => (phaseD1 t).1) (Icc (0 : ℝ) params.phi) := by
+  apply strictMonoOn_of_deriv_pos (convex_Icc _ _) phaseD1_continuous.fst.continuousOn
+  intro t ht
+  rw [interior_Icc] at ht
+  have hder : HasDerivAt (fun s => (phaseD1 s).1)
+      (((1 / 2 : ℝ) • u t).1) t := by
+    have h := dot_fixed_hasDerivAt_frontier (w := ((1 : ℝ), 0))
+      (phaseD1_hasDerivAt t)
+    simpa [dot] using h
+  rw [hder.deriv]
+  have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, ht.1], by
+      have htT : t < T := lt_trans ht.2 phi_lt_T
+      simpa [T] using htT⟩
+  simpa [u] using mul_pos (by norm_num : (0 : ℝ) < 1 / 2) hc
+
+private theorem phaseD2_fst_strictMono :
+    StrictMonoOn (fun t => (phaseD2 t).1) (Icc params.phi params.theta) := by
+  apply strictMonoOn_of_deriv_pos (convex_Icc _ _) phaseD2_continuous.fst.continuousOn
+  intro t ht
+  rw [interior_Icc] at ht
+  have hder : HasDerivAt (fun s => (phaseD2 s).1)
+      (((1 + params.b1 - t / 2) • u t).1) t := by
+    have h := dot_fixed_hasDerivAt_frontier (w := ((1 : ℝ), 0))
+      (phaseD2_hasDerivAt t)
+    simpa [dot] using h
+  rw [hder.deriv]
+  have hcoef : 0 < 1 + params.b1 - t / 2 := by
+    nlinarith [b1_lower_frontier, theta_upper_frontier, ht.2]
+  have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, phi_pos, ht.1], by
+      have htT : t < T := lt_trans ht.2 theta_lt_T
+      simpa [T] using htT⟩
+  simpa [u] using mul_pos hcoef hc
+
+private theorem phaseB4_fst_strictMono :
+    StrictMonoOn (fun t => (phaseB4_m421765a t).1) (Icc eta tau) := by
+  apply strictMonoOn_of_deriv_pos (convex_Icc _ _) phaseB4_continuous.fst.continuousOn
+  intro t ht
+  rw [interior_Icc] at ht
+  have hder : HasDerivAt (fun s => (phaseB4_m421765a s).1)
+      (((params.d1 - 1 - t / 2) • v t).1) t := by
+    have h := dot_fixed_hasDerivAt_frontier (w := ((1 : ℝ), 0))
+      (phaseB4_hasDerivAt t)
+    simpa [dot] using h
+  rw [hder.deriv]
+  have hcoef : params.d1 - 1 - t / 2 < 0 := by
+    nlinarith [d1_upper_frontier, eta_lower_frontier, ht.1]
+  have hs : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi
+    (lt_trans (lt_trans theta_pos theta_lt_eta) ht.1)
+    (lt_trans (lt_trans ht.2 tau_lt_T) (by
+      dsimp [T]; nlinarith [Real.pi_pos]))
+  have hm : 0 < (params.d1 - 1 - t / 2) * (-Real.sin t) :=
+    mul_pos_of_neg_of_neg hcoef (neg_neg_of_pos hs)
+  simpa [v] using hm
+
+private theorem phaseB5_fst_strictMono :
+    StrictMonoOn (fun t => (phaseB5_m421765a t).1) (Icc tau T) := by
+  apply strictMonoOn_of_deriv_pos (convex_Icc _ _) phaseB5_continuous.fst.continuousOn
+  intro t ht
+  rw [interior_Icc] at ht
+  have hder : HasDerivAt (fun s => (phaseB5_m421765a s).1)
+      (((-(1 / 2 : ℝ)) • v t).1) t := by
+    have h := dot_fixed_hasDerivAt_frontier (w := ((1 : ℝ), 0))
+      (phaseB5_hasDerivAt t)
+    simpa [dot] using h
+  rw [hder.deriv]
+  have hs : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi
+    (lt_trans (lt_trans (lt_trans theta_pos theta_lt_eta) eta_lt_tau) ht.1)
+    (lt_trans ht.2 (by dsimp [T]; nlinarith [Real.pi_pos]))
+  simpa [v] using mul_pos_of_neg_of_neg (by norm_num : (-(1 / 2 : ℝ)) < 0)
+    (neg_neg_of_pos hs)
+
+private theorem phaseD1_eq_D {t : ℝ} (ht : t ∈ Icc (0 : ℝ) params.phi) :
+    phaseD1 t = D t := by
+  apply Prod.ext <;>
+    simp [phaseD1, D, beta, alphaBetaAt, Romik.path, ht.2]
+
+private theorem phaseD2_eq_D {t : ℝ} (ht : t ∈ Icc params.phi params.theta) :
+    phaseD2 t = D t := by
+  rcases ht.1.eq_or_lt with h | h
+  · subst t
+    apply Prod.ext <;>
+      simp [phaseD2, D, beta, alphaBetaAt, Romik.path, PartB.match12, alphaBeta_match12_direct]
+  · apply Prod.ext <;>
+      simp [phaseD2, D, beta, alphaBetaAt, Romik.path, not_le.mpr h, ht.2]
+
+private theorem phaseB4_eq_B {t : ℝ} (ht : t ∈ Icc eta tau) :
+    phaseB4_m421765a t = B t := by
+  rcases ht.1.eq_or_lt with h | h
+  · subst t
+    have hm : Romik.path3 params eta = Romik.path4 params eta := by
+      simpa [eta, T] using PartB.match34
+    have hphase : phaseB4_m421765a eta = Romik.path4 params eta +
+        (Romik.alphaBeta4 params eta).1 • v eta := by
+      rfl
+    have hetaRaw : eta ≤ Real.pi / 2 - params.theta := by rfl
+    have hB : B eta = Romik.path3 params eta +
+        (Romik.alphaBeta3 params eta).1 • v eta := by
+      apply Prod.ext <;>
+        simp [B, alpha, alphaBetaAt, Romik.path,
+          not_le.mpr phi_lt_eta, not_le.mpr theta_lt_eta, hetaRaw]
+    calc
+      phaseB4_m421765a eta = Romik.path4 params eta +
+          (Romik.alphaBeta4 params eta).1 • v eta := hphase
+      _ = Romik.path3 params eta +
+          (Romik.alphaBeta3 params eta).1 • v eta := by
+            rw [hm, alphaBeta_match34_direct]
+      _ = B eta := hB.symm
+  · have hetaRaw : Real.pi / 2 - params.theta < t := by
+      simpa [eta, T] using h
+    have htauRaw : t ≤ Real.pi / 2 - params.phi := by
+      simpa [tau, T] using ht.2
+    apply Prod.ext <;>
+      simp [phaseB4_m421765a, B, alpha, alphaBetaAt, Romik.path,
+        not_le.mpr (lt_trans phi_lt_eta h),
+        not_le.mpr (lt_trans theta_lt_eta h), not_le.mpr hetaRaw, htauRaw, eta, tau, T]
+
+private theorem phaseB5_eq_B {t : ℝ} (ht : t ∈ Icc tau T) :
+    phaseB5_m421765a t = B t := by
+  rcases ht.1.eq_or_lt with h | h
+  · subst t
+    have hm : Romik.path4 params tau = Romik.path5 params tau := by
+      simpa [tau, T] using PartB.match45
+    have hphase : phaseB5_m421765a tau = Romik.path5 params tau +
+        (Romik.alphaBeta5 params tau).1 • v tau := by
+      rfl
+    have htauRaw : tau ≤ Real.pi / 2 - params.phi := by rfl
+    have hetaRaw : ¬ tau ≤ Real.pi / 2 - params.theta := by
+      exact not_le.mpr (by simpa [eta, T] using eta_lt_tau)
+    have hB : B tau = Romik.path4 params tau +
+        (Romik.alphaBeta4 params tau).1 • v tau := by
+      apply Prod.ext <;>
+        simp [B, alpha, alphaBetaAt, Romik.path,
+          not_le.mpr (lt_trans phi_lt_eta eta_lt_tau),
+          not_le.mpr (lt_trans theta_lt_eta eta_lt_tau),
+          not_le.mpr eta_lt_tau, hetaRaw, htauRaw]
+    calc
+      phaseB5_m421765a tau = Romik.path5 params tau +
+          (Romik.alphaBeta5 params tau).1 • v tau := hphase
+      _ = Romik.path4 params tau +
+          (Romik.alphaBeta4 params tau).1 • v tau := by
+            rw [hm, alphaBeta_match45_direct]
+      _ = B tau := hB.symm
+  · have hetaRaw : Real.pi / 2 - params.theta < t := by
+      simpa [eta, T] using lt_trans eta_lt_tau h
+    have htauRaw : Real.pi / 2 - params.phi < t := by
+      simpa [tau, T] using h
+    apply Prod.ext <;>
+      simp [phaseB5_m421765a, B, alpha, alphaBetaAt, Romik.path,
+        not_le.mpr (lt_trans (lt_trans phi_lt_eta eta_lt_tau) h),
+        not_le.mpr (lt_trans (lt_trans theta_lt_eta eta_lt_tau) h),
+        not_le.mpr hetaRaw, not_le.mpr htauRaw, eta, tau, T]
+
+/-- The early contact arc has strictly increasing horizontal projection. -/
+theorem D_fst_strictMono :
+    StrictMonoOn (fun t => (D t).1) (Icc (0 : ℝ) params.theta) := by
+  intro x hx y hy hxy
+  by_cases hyphi : y ≤ params.phi
+  · have hxphi : x ∈ Icc (0 : ℝ) params.phi :=
+      ⟨hx.1, le_trans hxy.le hyphi⟩
+    have hyphi' : y ∈ Icc (0 : ℝ) params.phi := ⟨hy.1, hyphi⟩
+    simpa [phaseD1_eq_D hxphi, phaseD1_eq_D hyphi'] using
+      phaseD1_fst_strictMono hxphi hyphi' hxy
+  · have hphiy : params.phi < y := lt_of_not_ge hyphi
+    by_cases hxphi : x ≤ params.phi
+    · have hx1 : x ∈ Icc (0 : ℝ) params.phi := ⟨hx.1, hxphi⟩
+      have hphi1 : params.phi ∈ Icc (0 : ℝ) params.phi :=
+        ⟨phi_pos.le, le_rfl⟩
+      have hphi2 : params.phi ∈ Icc params.phi params.theta :=
+        ⟨le_rfl, phi_lt_theta.le⟩
+      have hy2 : y ∈ Icc params.phi params.theta :=
+        ⟨hphiy.le, hy.2⟩
+      have hleft := phaseD1_fst_strictMono.monotoneOn hx1 hphi1 hxphi
+      have hright := phaseD2_fst_strictMono hphi2 hy2 hphiy
+      calc
+        (D x).1 = (phaseD1 x).1 := congrArg Prod.fst (phaseD1_eq_D hx1).symm
+        _ ≤ (phaseD1 params.phi).1 := hleft
+        _ = (D params.phi).1 := congrArg Prod.fst (phaseD1_eq_D hphi1)
+        _ = (phaseD2 params.phi).1 := congrArg Prod.fst (phaseD2_eq_D hphi2).symm
+        _ < (phaseD2 y).1 := hright
+        _ = (D y).1 := congrArg Prod.fst (phaseD2_eq_D hy2)
+    · have hphix : params.phi < x := lt_of_not_ge hxphi
+      have hx2 : x ∈ Icc params.phi params.theta := ⟨hphix.le, hx.2⟩
+      have hy2 : y ∈ Icc params.phi params.theta := ⟨hphiy.le, hy.2⟩
+      simpa [phaseD2_eq_D hx2, phaseD2_eq_D hy2] using
+        phaseD2_fst_strictMono hx2 hy2 hxy
+
+/-- The late contact arc has strictly increasing horizontal projection. -/
+theorem B_fst_strictMono :
+    StrictMonoOn (fun t => (B t).1) (Icc eta T) := by
+  intro x hx y hy hxy
+  by_cases hytau : y ≤ tau
+  · have hx4 : x ∈ Icc eta tau := ⟨hx.1, le_trans hxy.le hytau⟩
+    have hy4 : y ∈ Icc eta tau := ⟨hy.1, hytau⟩
+    simpa [phaseB4_eq_B hx4, phaseB4_eq_B hy4] using
+      phaseB4_fst_strictMono hx4 hy4 hxy
+  · have htauy : tau < y := lt_of_not_ge hytau
+    by_cases hxtau : x ≤ tau
+    · have hx4 : x ∈ Icc eta tau := ⟨hx.1, hxtau⟩
+      have htau4 : tau ∈ Icc eta tau := ⟨eta_lt_tau.le, le_rfl⟩
+      have htau5 : tau ∈ Icc tau T := ⟨le_rfl, tau_lt_T.le⟩
+      have hy5 : y ∈ Icc tau T := ⟨htauy.le, hy.2⟩
+      have hleft := phaseB4_fst_strictMono.monotoneOn hx4 htau4 hxtau
+      have hright := phaseB5_fst_strictMono htau5 hy5 htauy
+      calc
+        (B x).1 = (phaseB4_m421765a x).1 := congrArg Prod.fst (phaseB4_eq_B hx4).symm
+        _ ≤ (phaseB4_m421765a tau).1 := hleft
+        _ = (B tau).1 := congrArg Prod.fst (phaseB4_eq_B htau4)
+        _ = (phaseB5_m421765a tau).1 := congrArg Prod.fst (phaseB5_eq_B htau5).symm
+        _ < (phaseB5_m421765a y).1 := hright
+        _ = (B y).1 := congrArg Prod.fst (phaseB5_eq_B hy5)
+    · have htaux : tau < x := lt_of_not_ge hxtau
+      have hx5 : x ∈ Icc tau T := ⟨htaux.le, hx.2⟩
+      have hy5 : y ∈ Icc tau T := ⟨htauy.le, hy.2⟩
+      simpa [phaseB5_eq_B hx5, phaseB5_eq_B hy5] using
+        phaseB5_fst_strictMono hx5 hy5 hxy
+
+private theorem b1_upper_frontier : params.b1 ≤ (-527 / 1000 : ℝ) := by
+  have h := PartB.b1_contains.2
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at h ⊢
+  linarith
+
+private theorem c2_lower_frontier : (-1 : ℝ) ≤ params.c2 := by
+  have h := PartB.c2_contains.1
+  norm_num [RatInterval.Contains, ExactReplay.fullInputBox, ExactReplay.getI,
+    CertificateManifest.z22, CertificateManifest.q] at h ⊢
+  linarith
+
+private theorem path_fst_deriv_neg {t : ℝ} (ht : t ∈ Ioo params.phi tau) :
+    deriv (fun s => (Romik.path params s).1) t < 0 := by
+  have hder : HasDerivAt (fun s => (Romik.path params s).1)
+      ((Romik.rot t (alphaBetaAt t)).1) t := by
+    have h := dot_fixed_hasDerivAt_frontier (w := ((1 : ℝ), 0))
+      (path_hasDerivAt_noHidden t)
+    simpa [dot] using h
+  rw [hder.deriv]
+  have htPhys : t ∈ Icc params.phi T := ⟨ht.1.le, le_trans ht.2.le tau_lt_T.le⟩
+  have ha := alpha_nonpos htPhys
+  have hb := beta_nonneg htPhys
+  have hs : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi
+    (lt_trans phi_pos ht.1) (lt_trans (lt_trans ht.2 tau_lt_T)
+      (by dsimp [T]; nlinarith [Real.pi_pos]))
+  have hc : 0 < Real.cos t := Real.cos_pos_of_mem_Ioo
+    ⟨by linarith [Real.pi_pos, phi_pos, ht.1], by
+      have htT := lt_trans ht.2 tau_lt_T
+      simpa [T] using htT⟩
+  by_cases htheta : t ≤ params.theta
+  · have hphi : ¬ t ≤ params.phi := not_le.mpr ht.1
+    have has : alpha t < 0 := by
+      simp only [alpha, alphaBetaAt, ite_eq_right hphi, ite_eq_left htheta]
+      dsimp [Romik.alphaBeta2]
+      nlinarith [b1_upper_frontier, ht.1, phi_pos]
+    dsimp [Romik.rot]
+    change Real.cos t * alpha t - Real.sin t * beta t < 0
+    nlinarith [mul_neg_of_neg_of_pos has hc, mul_nonneg hb hs.le]
+  · by_cases heta : t ≤ eta
+    · have hphi : ¬ t ≤ params.phi := not_le.mpr ht.1
+      have has : alpha t < 0 := by
+        simp only [alpha, alphaBetaAt, ite_eq_right hphi, ite_eq_right htheta, ite_eq_left heta]
+        dsimp [Romik.alphaBeta3]
+        nlinarith [c2_lower_frontier, theta_pos, lt_of_not_ge htheta]
+      dsimp [Romik.rot]
+      change Real.cos t * alpha t - Real.sin t * beta t < 0
+      nlinarith [mul_neg_of_neg_of_pos has hc, mul_nonneg hb hs.le]
+    · have hphi : ¬ t ≤ params.phi := not_le.mpr ht.1
+      have hbp : 0 < beta t := by
+        simp only [beta, alphaBetaAt, ite_eq_right hphi, ite_eq_right htheta, ite_eq_right heta,
+          ite_eq_left ht.2.le]
+        dsimp [Romik.alphaBeta4]
+        nlinarith [d1_lower_frontier, T_upper_frontier,
+          lt_trans ht.2 tau_lt_T]
+      dsimp [Romik.rot]
+      change Real.cos t * alpha t - Real.sin t * beta t < 0
+      nlinarith [mul_nonpos_of_nonpos_of_nonneg ha hc.le,
+        mul_pos hbp hs]
+
+/-- The core path runs strictly from right to left in horizontal projection. -/
+theorem core_path_fst_strictAnti :
+    StrictAntiOn (fun t => (Romik.path params t).1) (Icc params.phi tau) := by
+  apply strictAntiOn_of_deriv_neg (convex_Icc _ _) pathContinuous.fst.continuousOn
+  intro t ht
+  rw [interior_Icc] at ht
+  exact path_fst_deriv_neg ht
+
+/-! ## A single left-to-right parametrization of the three roof arcs -/
+
+/-- The affine reversal which runs through the core path from `tau` to `phi`
+while the auxiliary parameter runs from `theta` to `eta`. -/
+def coreReverseTime (s : ℝ) : ℝ :=
+  tau - ((tau - params.phi) / (eta - params.theta)) *
+    (s - params.theta)
+
+private theorem coreReverseTime_theta :
+    coreReverseTime params.theta = tau := by
+  simp [coreReverseTime]
+
+private theorem coreReverseTime_eta : coreReverseTime eta = params.phi := by
+  have hne : eta - params.theta ≠ 0 := sub_ne_zero.mpr theta_lt_eta.ne'
+  dsimp [coreReverseTime]
+  field_simp [hne]
+  ring
+
+private theorem coreReverseTime_strictAnti : StrictAnti coreReverseTime := by
+  intro x y hxy
+  have hc : 0 < (tau - params.phi) / (eta - params.theta) :=
+    div_pos (sub_pos.mpr (lt_trans phi_lt_theta
+      (lt_trans theta_lt_eta eta_lt_tau))) (sub_pos.mpr theta_lt_eta)
+  have hprod := mul_pos hc (sub_pos.mpr hxy)
+  dsimp [coreReverseTime]
+  nlinarith [hprod]
+
+private theorem coreReverseTime_mem {s : ℝ}
+    (hs : s ∈ Icc params.theta eta) :
+    coreReverseTime s ∈ Icc params.phi tau := by
+  constructor
+  · rw [← coreReverseTime_eta]
+    exact coreReverseTime_strictAnti.antitone hs.2
+  · rw [← coreReverseTime_theta]
+    exact coreReverseTime_strictAnti.antitone hs.1
+
+private theorem coreReverseTime_continuous : Continuous coreReverseTime := by
+  unfold coreReverseTime
+  fun_prop
+
+/-- A single continuous parametrization of the complete upper niche arc. -/
+def nicheTopArc (s : ℝ) : Point :=
+  if s ≤ params.theta then D s
+  else if s ≤ eta then Romik.path params (coreReverseTime s)
+  else B s
+
+private theorem nicheTopArc_of_le_theta {s : ℝ} (hs : s ≤ params.theta) :
+    nicheTopArc s = D s := by
+  simp [nicheTopArc, hs]
+
+private theorem nicheTopArc_of_middle {s : ℝ}
+    (hθ : params.theta < s) (hη : s ≤ eta) :
+    nicheTopArc s = Romik.path params (coreReverseTime s) := by
+  simp [nicheTopArc, not_le.mpr hθ, hη]
+
+private theorem nicheTopArc_of_late {s : ℝ} (hη : eta < s) :
+    nicheTopArc s = B s := by
+  simp [nicheTopArc, not_le.mpr (lt_trans theta_lt_eta hη),
+    not_le.mpr hη]
+
+theorem nicheTopArc_continuous : Continuous nicheTopArc := by
+  have hcore : Continuous
+      (fun s : ℝ => Romik.path params (coreReverseTime s)) :=
+    pathContinuous.comp coreReverseTime_continuous
+  have hcoreB : Continuous (fun s : ℝ =>
+      if s ≤ eta then Romik.path params (coreReverseTime s) else B s) := by
+    exact hcore.if_le B_continuous continuous_id continuous_const (by
+      intro s hs
+      subst s
+      rw [coreReverseTime_eta]
+      exact B_eta_eq_path_phi.symm)
+  exact D_continuous.if_le hcoreB continuous_id continuous_const (by
+    intro s hs
+    subst s
+    rw [ite_eq_left theta_lt_eta.le, coreReverseTime_theta]
+    exact D_theta_eq_path_tau)
+
+/-- Horizontal projection of the complete upper arc is strictly increasing. -/
+theorem nicheTopArc_fst_strictMono :
+    StrictMonoOn (fun s => (nicheTopArc s).1) (Icc (0 : ℝ) T) := by
+  intro x hx y hy hxy
+  change (nicheTopArc x).1 < (nicheTopArc y).1
+  by_cases hyθ : y ≤ params.theta
+  · have hxD : x ∈ Icc (0 : ℝ) params.theta :=
+      ⟨hx.1, le_trans hxy.le hyθ⟩
+    have hyD : y ∈ Icc (0 : ℝ) params.theta := ⟨hy.1, hyθ⟩
+    rw [nicheTopArc_of_le_theta hxD.2, nicheTopArc_of_le_theta hyθ]
+    exact D_fst_strictMono hxD hyD hxy
+  · have hθy : params.theta < y := lt_of_not_ge hyθ
+    by_cases hxθ : x ≤ params.theta
+    · have hxD : x ∈ Icc (0 : ℝ) params.theta := ⟨hx.1, hxθ⟩
+      have hθD : params.theta ∈ Icc (0 : ℝ) params.theta :=
+        ⟨theta_pos.le, le_rfl⟩
+      have hleft := D_fst_strictMono.monotoneOn hxD hθD hxθ
+      rw [nicheTopArc_of_le_theta hxθ]
+      by_cases hyη : y ≤ eta
+      · have hyM : y ∈ Icc params.theta eta := ⟨hθy.le, hyη⟩
+        have hcy := coreReverseTime_mem hyM
+        have hcytau : coreReverseTime y < tau := by
+          rw [← coreReverseTime_theta]
+          exact coreReverseTime_strictAnti hθy
+        have hright := core_path_fst_strictAnti hcy
+          (right_mem_Icc.2 (lt_trans phi_lt_theta
+            (lt_trans theta_lt_eta eta_lt_tau)).le) hcytau
+        rw [nicheTopArc_of_middle hθy hyη]
+        calc
+          (D x).1 ≤ (D params.theta).1 := hleft
+          _ = (Romik.path params tau).1 := congrArg Prod.fst D_theta_eq_path_tau
+          _ < (Romik.path params (coreReverseTime y)).1 := hright
+      · have hηy : eta < y := lt_of_not_ge hyη
+        have hyB : y ∈ Icc eta T := ⟨hηy.le, hy.2⟩
+        have hηB : eta ∈ Icc eta T := ⟨le_rfl, eta_lt_T.le⟩
+        have hB := B_fst_strictMono hηB hyB hηy
+        have hcore := core_path_fst_strictAnti
+          (left_mem_Icc.2 (lt_trans phi_lt_theta
+            (lt_trans theta_lt_eta eta_lt_tau)).le)
+          (right_mem_Icc.2 (lt_trans phi_lt_theta
+            (lt_trans theta_lt_eta eta_lt_tau)).le)
+          (lt_trans phi_lt_theta (lt_trans theta_lt_eta eta_lt_tau))
+        rw [nicheTopArc_of_late hηy]
+        calc
+          (D x).1 ≤ (D params.theta).1 := hleft
+          _ = (Romik.path params tau).1 := congrArg Prod.fst D_theta_eq_path_tau
+          _ < (Romik.path params params.phi).1 := hcore
+          _ = (B eta).1 := congrArg Prod.fst B_eta_eq_path_phi.symm
+          _ < (B y).1 := hB
+    · have hθx : params.theta < x := lt_of_not_ge hxθ
+      by_cases hyη : y ≤ eta
+      · have hxM : x ∈ Icc params.theta eta :=
+          ⟨hθx.le, le_trans hxy.le hyη⟩
+        have hyM : y ∈ Icc params.theta eta := ⟨hθy.le, hyη⟩
+        have hcyx : coreReverseTime y < coreReverseTime x :=
+          coreReverseTime_strictAnti hxy
+        rw [nicheTopArc_of_middle hθx hxM.2,
+          nicheTopArc_of_middle hθy hyη]
+        exact core_path_fst_strictAnti (coreReverseTime_mem hyM)
+          (coreReverseTime_mem hxM) hcyx
+      · have hηy : eta < y := lt_of_not_ge hyη
+        rw [nicheTopArc_of_late hηy]
+        by_cases hxη : x ≤ eta
+        · have hxM : x ∈ Icc params.theta eta := ⟨hθx.le, hxη⟩
+          have hcx := coreReverseTime_mem hxM
+          have hpath : (Romik.path params (coreReverseTime x)).1 ≤
+              (Romik.path params params.phi).1 :=
+            core_path_fst_strictAnti.antitoneOn
+              (left_mem_Icc.2 (lt_trans phi_lt_theta
+                (lt_trans theta_lt_eta eta_lt_tau)).le)
+              hcx hcx.1
+          have hyB : y ∈ Icc eta T := ⟨hηy.le, hy.2⟩
+          have hB := B_fst_strictMono
+            (left_mem_Icc.2 eta_lt_T.le) hyB hηy
+          rw [nicheTopArc_of_middle hθx hxη]
+          calc
+            (Romik.path params (coreReverseTime x)).1 ≤
+                (Romik.path params params.phi).1 := hpath
+            _ = (B eta).1 := congrArg Prod.fst B_eta_eq_path_phi.symm
+            _ < (B y).1 := hB
+        · have hηx : eta < x := lt_of_not_ge hxη
+          have hxB : x ∈ Icc eta T := ⟨hηx.le, hx.2⟩
+          have hyB : y ∈ Icc eta T := ⟨hηy.le, hy.2⟩
+          rw [nicheTopArc_of_late hηx]
+          exact B_fst_strictMono hxB hyB hxy
+
+theorem nicheTopArc_y_nonneg {s : ℝ} (hs : s ∈ Icc (0 : ℝ) T) :
+    0 ≤ (nicheTopArc s).2 := by
+  by_cases hθ : s ≤ params.theta
+  · rw [nicheTopArc_of_le_theta hθ]
+    exact D_y_nonneg ⟨hs.1, hθ⟩
+  · have hθs : params.theta < s := lt_of_not_ge hθ
+    by_cases hη : s ≤ eta
+    · rw [nicheTopArc_of_middle hθs hη]
+      exact path_core_y_nonneg (coreReverseTime_mem ⟨hθs.le, hη⟩)
+    · have hηs : eta < s := lt_of_not_ge hη
+      rw [nicheTopArc_of_late hηs]
+      exact B_y_nonneg ⟨hηs.le, hs.2⟩
+
+theorem nicheTopArc_y_pos {s : ℝ} (hs : s ∈ Ioo (0 : ℝ) T) :
+    0 < (nicheTopArc s).2 := by
+  by_cases hθ : s ≤ params.theta
+  · rw [nicheTopArc_of_le_theta hθ]
+    exact D_y_pos ⟨hs.1, hθ⟩
+  · have hθs : params.theta < s := lt_of_not_ge hθ
+    by_cases hη : s ≤ eta
+    · rw [nicheTopArc_of_middle hθs hη]
+      exact path_core_y_pos (coreReverseTime_mem ⟨hθs.le, hη⟩)
+    · have hηs : eta < s := lt_of_not_ge hη
+      rw [nicheTopArc_of_late hηs]
+      exact B_y_pos ⟨hηs.le, hs.2⟩
+
+theorem nicheTopArc_zero : nicheTopArc 0 = D 0 := by
+  exact nicheTopArc_of_le_theta theta_pos.le
+
+theorem nicheTopArc_T : nicheTopArc T = B T := by
+  exact nicheTopArc_of_late eta_lt_T
+
+theorem coreReverseTime_image :
+    coreReverseTime '' Icc params.theta eta = Icc params.phi tau := by
+  simpa [coreReverseTime_theta, coreReverseTime_eta] using
+    coreReverseTime_continuous.continuousOn.image_Icc_of_antitoneOn
+      theta_lt_eta.le
+      (coreReverseTime_strictAnti.antitone.antitoneOn (Icc params.theta eta))
+
+/-- The glued arc has exactly the three curve images, with no additional
+points introduced by the reparametrization. -/
+theorem nicheTopArc_image :
+    curveImage nicheTopArc (Icc (0 : ℝ) T) =
+      curveImage D (Icc (0 : ℝ) params.theta) ∪
+      curveImage (Romik.path params) (Icc params.phi tau) ∪
+      curveImage B (Icc eta T) := by
+  classical
+  ext p
+  constructor
+  · rintro ⟨s, hs, rfl⟩
+    by_cases hθ : s ≤ params.theta
+    · left; left
+      exact ⟨s, ⟨hs.1, hθ⟩, (nicheTopArc_of_le_theta hθ).symm⟩
+    · have hθs : params.theta < s := lt_of_not_ge hθ
+      by_cases hη : s ≤ eta
+      · left; right
+        exact ⟨coreReverseTime s, coreReverseTime_mem ⟨hθs.le, hη⟩,
+          (nicheTopArc_of_middle hθs hη).symm⟩
+      · right
+        have hηs : eta < s := lt_of_not_ge hη
+        exact ⟨s, ⟨hηs.le, hs.2⟩, (nicheTopArc_of_late hηs).symm⟩
+  · rintro ((⟨s, hs, rfl⟩ | ⟨t, ht, rfl⟩) | ⟨s, hs, rfl⟩)
+    · exact ⟨s, ⟨hs.1, le_trans hs.2 theta_lt_T.le⟩,
+        nicheTopArc_of_le_theta hs.2⟩
+    · rw [← coreReverseTime_image] at ht
+      rcases ht with ⟨s, hs, rfl⟩
+      by_cases hst : s = params.theta
+      · subst s
+        refine ⟨params.theta, ⟨theta_pos.le, theta_lt_T.le⟩, ?_⟩
+        rw [nicheTopArc_of_le_theta le_rfl, coreReverseTime_theta]
+        exact D_theta_eq_path_tau
+      · have hθs : params.theta < s := lt_of_le_of_ne hs.1 (Ne.symm hst)
+        exact ⟨s, ⟨le_trans theta_pos.le hs.1,
+          le_trans hs.2 eta_lt_T.le⟩,
+          nicheTopArc_of_middle hθs hs.2⟩
+    · by_cases hse : s = eta
+      · subst s
+        refine ⟨eta, ⟨le_trans theta_pos.le theta_lt_eta.le, eta_lt_T.le⟩, ?_⟩
+        rw [nicheTopArc_of_middle theta_lt_eta le_rfl, coreReverseTime_eta]
+        exact B_eta_eq_path_phi.symm
+      · have hηs : eta < s := lt_of_le_of_ne hs.1 (Ne.symm hse)
+        exact ⟨s, ⟨le_trans (le_trans theta_pos.le theta_lt_eta.le) hs.1, hs.2⟩,
+          nicheTopArc_of_late hηs⟩
+
+/-- A vertically filled strict subgraph over a compact interval. -/
+def strictSubgraphRegion (a b : ℝ) (H : ℝ → ℝ) : Set Point :=
+  {p | p.1 ∈ Icc a b ∧ 0 ≤ p.2 ∧ p.2 < H p.1}
+
+/-- The closed vertical fill associated with `strictSubgraphRegion`. -/
+def closedSubgraphRegion (a b : ℝ) (H : ℝ → ℝ) : Set Point :=
+  {p | p.1 ∈ Icc a b ∧ 0 ≤ p.2 ∧ p.2 ≤ H p.1}
+
+/-- The base together with the graph of the roof. -/
+def strictSubgraphBoundary (a b : ℝ) (H : ℝ → ℝ) : Set Point :=
+  {p | p.1 ∈ Icc a b ∧ (p.2 = 0 ∨ p.2 = H p.1)}
+
+private theorem strictSubgraphRegion_as_image
+    {a b : ℝ} {H : ℝ → ℝ}
+    (hHpos : ∀ x ∈ Ioo a b, 0 < H x)
+    (ha : H a = 0) (hb : H b = 0) :
+    (fun z : ℝ × ℝ => (z.1, z.2 * H z.1)) ''
+        (Ioo a b ×ˢ Ico (0 : ℝ) 1) =
+      strictSubgraphRegion a b H := by
+  classical
+  ext p
+  constructor
+  · rintro ⟨z, ⟨hzx, hzs⟩, rfl⟩
+    have htop := hHpos z.1 hzx
+    refine ⟨⟨hzx.1.le, hzx.2.le⟩, ?_, ?_⟩
+    · exact mul_nonneg hzs.1 htop.le
+    · exact mul_lt_of_lt_one_left htop hzs.2
+  · rintro ⟨hx, hy0, hy⟩
+    have hxa : a < p.1 := by
+      rcases hx.1.eq_or_lt with h | h
+      · rw [← h, ha] at hy
+        linarith
+      · exact h
+    have hxb : p.1 < b := by
+      rcases hx.2.eq_or_lt with h | h
+      · rw [h, hb] at hy
+        linarith
+      · exact h
+    have htop := hHpos p.1 ⟨hxa, hxb⟩
+    let s : ℝ := p.2 / H p.1
+    have hs : s ∈ Ico (0 : ℝ) 1 := by
+      constructor
+      · exact div_nonneg hy0 htop.le
+      · exact (div_lt_one htop).2 hy
+    refine ⟨(p.1, s), ⟨⟨hxa, hxb⟩, hs⟩, ?_⟩
+    apply Prod.ext
+    · rfl
+    · dsimp [s]
+      field_simp [ne_of_gt htop]
+
+private theorem closedSubgraphRegion_as_image
+    {a b : ℝ} {H : ℝ → ℝ}
+    (hHnonneg : ∀ x ∈ Icc a b, 0 ≤ H x) :
+    (fun z : ℝ × ℝ => (z.1, z.2 * H z.1)) ''
+        (Icc a b ×ˢ Icc (0 : ℝ) 1) =
+      closedSubgraphRegion a b H := by
+  classical
+  ext p
+  constructor
+  · rintro ⟨z, ⟨hzx, hzs⟩, rfl⟩
+    have htop := hHnonneg z.1 hzx
+    refine ⟨hzx, mul_nonneg hzs.1 htop, ?_⟩
+    exact mul_le_of_le_one_left htop hzs.2
+  · rintro ⟨hx, hy0, hy⟩
+    have htop := hHnonneg p.1 hx
+    by_cases hzero : H p.1 = 0
+    · have hpy : p.2 = 0 := by linarith
+      refine ⟨(p.1, 0), ⟨hx, by norm_num⟩, ?_⟩
+      apply Prod.ext
+      · rfl
+      · simpa only [zero_mul] using hpy.symm
+    · have htop' : 0 < H p.1 := lt_of_le_of_ne htop (Ne.symm hzero)
+      let s : ℝ := p.2 / H p.1
+      have hs : s ∈ Icc (0 : ℝ) 1 := by
+        constructor
+        · exact div_nonneg hy0 htop'.le
+        · exact (div_le_one htop').2 hy
+      refine ⟨(p.1, s), ⟨hx, hs⟩, ?_⟩
+      apply Prod.ext
+      · rfl
+      · dsimp [s]
+        field_simp [hzero]
+
+theorem closure_strictSubgraphRegion
+    {a b : ℝ} {H : ℝ → ℝ} (hab : a < b)
+    (hH : Continuous H)
+    (hHnonneg : ∀ x ∈ Icc a b, 0 ≤ H x)
+    (hHpos : ∀ x ∈ Ioo a b, 0 < H x)
+    (ha : H a = 0) (hb : H b = 0) :
+    closure (strictSubgraphRegion a b H) = closedSubgraphRegion a b H := by
+  let P : Set (ℝ × ℝ) := Ioo a b ×ˢ Ico (0 : ℝ) 1
+  let Q : Set (ℝ × ℝ) := Icc a b ×ˢ Icc (0 : ℝ) 1
+  let F : ℝ × ℝ → Point := fun z => (z.1, z.2 * H z.1)
+  have hclP : closure P = Q := by
+    simp only [P, Q, closure_prod_eq, closure_Ioo hab.ne,
+      closure_Ico (by norm_num : (0 : ℝ) ≠ 1)]
+  have hQcompact : IsCompact Q := by
+    exact isCompact_Icc.prod isCompact_Icc
+  have hF : Continuous F := by
+    dsimp [F]
+    exact continuous_fst.prodMk (continuous_snd.mul (hH.comp continuous_fst))
+  have hPcompact : IsCompact (closure P) := by
+    rw [hclP]
+    exact hQcompact
+  calc
+    closure (strictSubgraphRegion a b H) = closure (F '' P) := by
+      rw [strictSubgraphRegion_as_image hHpos ha hb]
+    _ = F '' closure P :=
+      (image_closure_of_isCompact hPcompact hF.continuousOn).symm
+    _ = F '' Q := by rw [hclP]
+    _ = closedSubgraphRegion a b H :=
+      closedSubgraphRegion_as_image hHnonneg
+
+theorem interior_strictSubgraphRegion
+    {a b : ℝ} {H : ℝ → ℝ} (hH : Continuous H) :
+    interior (strictSubgraphRegion a b H) =
+      {p | p.1 ∈ Ioo a b ∧ 0 < p.2 ∧ p.2 < H p.1} := by
+  let O : Set Point := {p | p.1 ∈ Ioo a b ∧ 0 < p.2 ∧ p.2 < H p.1}
+  have hOopen : IsOpen O := by
+    dsimp [O]
+    exact (isOpen_Ioo.preimage continuous_fst).inter
+      ((isOpen_Ioi.preimage continuous_snd).inter
+        (isOpen_lt continuous_snd (hH.comp continuous_fst)))
+  apply Set.Subset.antisymm
+  · intro p hp
+    have hpS := interior_subset hp
+    have hxmono : strictSubgraphRegion a b H ⊆ Icc a b ×ˢ (Set.univ : Set ℝ) := by
+      intro q hq
+      exact ⟨hq.1, Set.mem_univ _⟩
+    have hymono : strictSubgraphRegion a b H ⊆ (Set.univ : Set ℝ) ×ˢ Ici 0 := by
+      intro q hq
+      exact ⟨Set.mem_univ _, hq.2.1⟩
+    have hx := interior_mono hxmono hp
+    have hy := interior_mono hymono hp
+    rw [interior_prod_eq, interior_Icc, interior_univ] at hx
+    rw [interior_prod_eq, interior_univ, interior_Ici] at hy
+    exact ⟨hx.1, hy.2, hpS.2.2⟩
+  · apply interior_maximal
+    · intro p hp
+      exact ⟨⟨hp.1.1.le, hp.1.2.le⟩, hp.2.1.le, hp.2.2⟩
+    · exact hOopen
+
+/-- The frontier of a positive strict vertical subgraph consists exactly of
+its roof and its base. -/
+theorem frontier_strictSubgraphRegion
+    {a b : ℝ} {H : ℝ → ℝ} (hab : a < b)
+    (hH : Continuous H)
+    (hHnonneg : ∀ x ∈ Icc a b, 0 ≤ H x)
+    (hHpos : ∀ x ∈ Ioo a b, 0 < H x)
+    (ha : H a = 0) (hb : H b = 0) :
+    frontier (strictSubgraphRegion a b H) = strictSubgraphBoundary a b H := by
+  rw [frontier, closure_strictSubgraphRegion hab hH hHnonneg hHpos ha hb,
+    interior_strictSubgraphRegion hH]
+  ext p
+  simp only [Set.mem_sdiff, closedSubgraphRegion, strictSubgraphBoundary,
+    Set.mem_ofPred_eq]
+  constructor
+  · rintro ⟨⟨hx, hy0, hyH⟩, hnot⟩
+    refine ⟨hx, ?_⟩
+    by_cases hya : p.2 = 0
+    · exact Or.inl hya
+    right
+    by_contra hyne
+    apply hnot
+    refine ⟨?_, lt_of_le_of_ne hy0 (Ne.symm hya), lt_of_le_of_ne hyH hyne⟩
+    constructor
+    · apply lt_of_le_of_ne hx.1
+      intro heq
+      have hp0 : p.2 = 0 := by
+        rw [← heq, ha] at hyH
+        linarith
+      exact hya hp0
+    · apply lt_of_le_of_ne hx.2
+      intro heq
+      have hp0 : p.2 = 0 := by
+        rw [heq, hb] at hyH
+        linarith
+      exact hya hp0
+  · rintro ⟨hx, hbase | hroof⟩
+    · refine ⟨⟨hx, ?_, ?_⟩, ?_⟩
+      · simp [hbase]
+      · simpa [hbase] using hHnonneg p.1 hx
+      · intro hstrict
+        rw [hbase] at hstrict
+        exact (lt_irrefl (0 : ℝ)) hstrict.2.1
+    · refine ⟨⟨hx, ?_, hroof.le⟩, ?_⟩
+      · rw [hroof]
+        exact hHnonneg p.1 hx
+      · intro hstrict
+        exact (lt_irrefl (H p.1)) (hroof ▸ hstrict.2.2)
+
+/-! ## Turning a monotone roof arc into a global continuous graph -/
+
+/-- Identify a strictly horizontally increasing arc with its horizontal coordinate interval. -/
+noncomputable def horizontalOrderIso
+    (γ : ℝ → Point) {a b : ℝ} (hab : a < b)
+    (hγ : ContinuousOn γ (Icc a b))
+    (hx : StrictMonoOn (fun t ↦ (γ t).1) (Icc a b)) :
+    Icc a b ≃o Icc (γ a).1 (γ b).1 := by
+  have himage :
+      (fun t ↦ (γ t).1) '' Icc a b = Icc (γ a).1 (γ b).1 :=
+    hγ.fst.image_Icc_of_monotoneOn hab.le hx.monotoneOn
+  exact
+    (StrictMonoOn.orderIso (fun t ↦ (γ t).1) (Icc a b) hx).trans
+      (Set.orderIsoOfEq _ _ himage)
+
+@[simp]
+theorem horizontalOrderIso_apply_val
+    (γ : ℝ → Point) {a b : ℝ} (hab : a < b)
+    (hγ : ContinuousOn γ (Icc a b))
+    (hx : StrictMonoOn (fun t ↦ (γ t).1) (Icc a b))
+    (t : Icc a b) :
+    ((horizontalOrderIso γ hab hγ hx t : Icc (γ a).1 (γ b).1) : ℝ) =
+      (γ t.1).1 := by
+  rfl
+
+/-- Express the arc’s vertical coordinate as a function of its horizontal coordinate. -/
+noncomputable def roofOnHorizontalRange
+    (γ : ℝ → Point) {a b : ℝ} (hab : a < b)
+    (hγ : ContinuousOn γ (Icc a b))
+    (hx : StrictMonoOn (fun t ↦ (γ t).1) (Icc a b)) :
+    Icc (γ a).1 (γ b).1 → ℝ :=
+  fun X ↦ (γ ((horizontalOrderIso γ hab hγ hx).symm X).1).2
+
+theorem continuous_roofOnHorizontalRange
+    (γ : ℝ → Point) {a b : ℝ} (hab : a < b)
+    (hγ : ContinuousOn γ (Icc a b))
+    (hx : StrictMonoOn (fun t ↦ (γ t).1) (Icc a b)) :
+    Continuous (roofOnHorizontalRange γ hab hγ hx) := by
+  exact continuous_snd.comp
+    (hγ.domRestrict.comp (horizontalOrderIso γ hab hγ hx).symm.continuous)
+
+theorem horizontal_endpoints_le
+    (γ : ℝ → Point) {a b : ℝ} (hab : a < b)
+    (hx : StrictMonoOn (fun t ↦ (γ t).1) (Icc a b)) :
+    (γ a).1 ≤ (γ b).1 :=
+  (hx (left_mem_Icc.2 hab.le) (right_mem_Icc.2 hab.le) hab).le
+
+/-- Extend the roof graph to the real line by clamping to its endpoint interval. -/
+noncomputable def monotoneCurveRoof
+    (γ : ℝ → Point) {a b : ℝ} (hab : a < b)
+    (hγ : ContinuousOn γ (Icc a b))
+    (hx : StrictMonoOn (fun t ↦ (γ t).1) (Icc a b)) : ℝ → ℝ :=
+  Set.IccExtend (horizontal_endpoints_le γ hab hx)
+    (roofOnHorizontalRange γ hab hγ hx)
+
+theorem continuous_monotoneCurveRoof
+    (γ : ℝ → Point) {a b : ℝ} (hab : a < b)
+    (hγ : ContinuousOn γ (Icc a b))
+    (hx : StrictMonoOn (fun t ↦ (γ t).1) (Icc a b)) :
+    Continuous (monotoneCurveRoof γ hab hγ hx) := by
+  exact (continuous_roofOnHorizontalRange γ hab hγ hx).Icc_extend'
+
+private theorem horizontal_mem_range
+    (γ : ℝ → Point) {a b t : ℝ} (hab : a < b)
+    (hx : StrictMonoOn (fun t ↦ (γ t).1) (Icc a b))
+    (ht : t ∈ Icc a b) :
+    (γ t).1 ∈ Icc (γ a).1 (γ b).1 := by
+  exact ⟨hx.monotoneOn (left_mem_Icc.2 hab.le) ht ht.1,
+    hx.monotoneOn ht (right_mem_Icc.2 hab.le) ht.2⟩
+
+theorem monotoneCurveRoof_at
+    (γ : ℝ → Point) {a b t : ℝ} (hab : a < b)
+    (hγ : ContinuousOn γ (Icc a b))
+    (hx : StrictMonoOn (fun t ↦ (γ t).1) (Icc a b))
+    (ht : t ∈ Icc a b) :
+    monotoneCurveRoof γ hab hγ hx (γ t).1 = (γ t).2 := by
+  let e := horizontalOrderIso γ hab hγ hx
+  have htr := horizontal_mem_range γ hab hx ht
+  have heq : e ⟨t, ht⟩ = ⟨(γ t).1, htr⟩ := by
+    apply Subtype.ext
+    rfl
+  have hinv : e.symm ⟨(γ t).1, htr⟩ = ⟨t, ht⟩ := by
+    rw [← heq, e.symm_apply_apply]
+  rw [monotoneCurveRoof, Set.IccExtend_of_mem _ _ htr]
+  change (γ (e.symm ⟨(γ t).1, htr⟩).1).2 = (γ t).2
+  rw [hinv]
+
+theorem verticalFill_eq_strictSubgraphRegion
+    (γ : ℝ → Point) {a b : ℝ} (hab : a < b)
+    (hγ : ContinuousOn γ (Icc a b))
+    (hx : StrictMonoOn (fun t ↦ (γ t).1) (Icc a b)) :
+    verticalFill γ (Icc a b) =
+      strictSubgraphRegion (γ a).1 (γ b).1
+        (monotoneCurveRoof γ hab hγ hx) := by
+  ext p
+  constructor
+  · rintro ⟨t, ht, hxt, hy0, hy⟩
+    have htr := horizontal_mem_range γ hab hx ht
+    refine ⟨?_, hy0, ?_⟩
+    · simpa [hxt] using htr
+    · rw [hxt, monotoneCurveRoof_at γ hab hγ hx ht]
+      exact hy
+  · rintro ⟨hpx, hy0, hy⟩
+    let e := horizontalOrderIso γ hab hγ hx
+    let ts : Icc a b := e.symm ⟨p.1, hpx⟩
+    have heval := congrArg Subtype.val
+      ((horizontalOrderIso γ hab hγ hx).apply_symm_apply ⟨p.1, hpx⟩)
+    have hxt : (γ ts.1).1 = p.1 := by
+      change
+        (γ ((horizontalOrderIso γ hab hγ hx).symm ⟨p.1, hpx⟩).1).1 = p.1
+      simpa only [horizontalOrderIso_apply_val] using heval
+    refine ⟨ts.1, ts.2, hxt.symm, hy0, ?_⟩
+    rw [← hxt, monotoneCurveRoof_at γ hab hγ hx ts.2] at hy
+    exact hy
+
+theorem curveImage_eq_roofGraph
+    (γ : ℝ → Point) {a b : ℝ} (hab : a < b)
+    (hγ : ContinuousOn γ (Icc a b))
+    (hx : StrictMonoOn (fun t ↦ (γ t).1) (Icc a b)) :
+    curveImage γ (Icc a b) =
+      {p | p.1 ∈ Icc (γ a).1 (γ b).1 ∧
+        p.2 = monotoneCurveRoof γ hab hγ hx p.1} := by
+  ext p
+  constructor
+  · rintro ⟨t, ht, rfl⟩
+    exact ⟨horizontal_mem_range γ hab hx ht,
+      (monotoneCurveRoof_at γ hab hγ hx ht).symm⟩
+  · rintro ⟨hpx, hpy⟩
+    let e := horizontalOrderIso γ hab hγ hx
+    let ts : Icc a b := e.symm ⟨p.1, hpx⟩
+    have heval := congrArg Subtype.val
+      ((horizontalOrderIso γ hab hγ hx).apply_symm_apply ⟨p.1, hpx⟩)
+    have hxt : (γ ts.1).1 = p.1 := by
+      change
+        (γ ((horizontalOrderIso γ hab hγ hx).symm ⟨p.1, hpx⟩).1).1 = p.1
+      simpa only [horizontalOrderIso_apply_val] using heval
+    refine ⟨ts.1, ts.2, ?_⟩
+    apply Prod.ext
+    · exact hxt
+    · calc
+        (γ ts.1).2 = monotoneCurveRoof γ hab hγ hx (γ ts.1).1 :=
+          (monotoneCurveRoof_at γ hab hγ hx ts.2).symm
+        _ = monotoneCurveRoof γ hab hγ hx p.1 := by rw [hxt]
+        _ = p.2 := hpy.symm
+
+theorem T_pos_frontier : 0 < T := by
+  dsimp [T]
+  nlinarith [Real.pi_pos]
+
+/-- The continuous roof function determined by the complete niche top arc. -/
+noncomputable def nicheRoof : ℝ → ℝ :=
+  monotoneCurveRoof nicheTopArc T_pos_frontier
+    nicheTopArc_continuous.continuousOn nicheTopArc_fst_strictMono
+
+theorem nicheRoof_continuous : Continuous nicheRoof :=
+  continuous_monotoneCurveRoof nicheTopArc T_pos_frontier
+    nicheTopArc_continuous.continuousOn nicheTopArc_fst_strictMono
+
+theorem nicheRoof_at_topArc {s : ℝ} (hs : s ∈ Icc (0 : ℝ) T) :
+    nicheRoof (nicheTopArc s).1 = (nicheTopArc s).2 :=
+  monotoneCurveRoof_at nicheTopArc T_pos_frontier
+    nicheTopArc_continuous.continuousOn nicheTopArc_fst_strictMono hs
+
+theorem nicheRoof_nonneg {X : ℝ}
+    (hX : X ∈ Icc (D 0).1 (B T).1) : 0 ≤ nicheRoof X := by
+  have hends : Icc (D 0).1 (B T).1 =
+      Icc (nicheTopArc 0).1 (nicheTopArc T).1 := by
+    rw [nicheTopArc_zero, nicheTopArc_T]
+  rw [hends] at hX
+  let e := horizontalOrderIso nicheTopArc T_pos_frontier
+    nicheTopArc_continuous.continuousOn nicheTopArc_fst_strictMono
+  let s : Icc (0 : ℝ) T := e.symm ⟨X, hX⟩
+  have heval := congrArg Subtype.val
+    ((horizontalOrderIso nicheTopArc T_pos_frontier
+      nicheTopArc_continuous.continuousOn
+      nicheTopArc_fst_strictMono).apply_symm_apply ⟨X, hX⟩)
+  have hxarc : (nicheTopArc s.1).1 = X := by
+    change
+      (nicheTopArc ((horizontalOrderIso nicheTopArc T_pos_frontier
+        nicheTopArc_continuous.continuousOn
+        nicheTopArc_fst_strictMono).symm ⟨X, hX⟩).1).1 = X
+    simpa only [horizontalOrderIso_apply_val] using heval
+  rw [← hxarc, nicheRoof_at_topArc s.2]
+  exact nicheTopArc_y_nonneg s.2
+
+theorem nicheRoof_pos {X : ℝ}
+    (hX : X ∈ Ioo (D 0).1 (B T).1) : 0 < nicheRoof X := by
+  have hends : Ioo (D 0).1 (B T).1 =
+      Ioo (nicheTopArc 0).1 (nicheTopArc T).1 := by
+    rw [nicheTopArc_zero, nicheTopArc_T]
+  rw [hends] at hX
+  let e := horizontalOrderIso nicheTopArc T_pos_frontier
+    nicheTopArc_continuous.continuousOn nicheTopArc_fst_strictMono
+  have hXcc : X ∈ Icc (nicheTopArc 0).1 (nicheTopArc T).1 :=
+    ⟨hX.1.le, hX.2.le⟩
+  let s : Icc (0 : ℝ) T := e.symm ⟨X, hXcc⟩
+  have heval := congrArg Subtype.val
+    ((horizontalOrderIso nicheTopArc T_pos_frontier
+      nicheTopArc_continuous.continuousOn
+      nicheTopArc_fst_strictMono).apply_symm_apply ⟨X, hXcc⟩)
+  have hxarc : (nicheTopArc s.1).1 = X := by
+    change
+      (nicheTopArc ((horizontalOrderIso nicheTopArc T_pos_frontier
+        nicheTopArc_continuous.continuousOn
+        nicheTopArc_fst_strictMono).symm ⟨X, hXcc⟩).1).1 = X
+    simpa only [horizontalOrderIso_apply_val] using heval
+  have hs0 : 0 < s.1 := by
+    by_contra hn
+    have hszero : s.1 = 0 := le_antisymm (le_of_not_gt hn) s.2.1
+    have hxzero : (nicheTopArc 0).1 = X := by
+      simpa [hszero] using hxarc
+    exact (ne_of_lt hX.1) hxzero
+  have hsT : s.1 < T := by
+    by_contra hn
+    have hsT' : s.1 = T := le_antisymm s.2.2 (le_of_not_gt hn)
+    have hxT : (nicheTopArc T).1 = X := by
+      simpa [hsT'] using hxarc
+    exact (ne_of_gt hX.2) hxT
+  rw [← hxarc, nicheRoof_at_topArc s.2]
+  exact nicheTopArc_y_pos ⟨hs0, hsT⟩
+
+theorem nicheRoof_left : nicheRoof (D 0).1 = 0 := by
+  rw [← nicheTopArc_zero, nicheRoof_at_topArc (left_mem_Icc.2 T_pos_frontier.le),
+    nicheTopArc_zero]
+  exact D_zero_y_zero
+
+theorem nicheRoof_right : nicheRoof (B T).1 = 0 := by
+  rw [← nicheTopArc_T, nicheRoof_at_topArc (right_mem_Icc.2 T_pos_frontier.le),
+    nicheTopArc_T]
+  exact B_T_y_zero
+
+private theorem nicheTopArc_verticalFill :
+    verticalFill nicheTopArc (Icc (0 : ℝ) T) = certifiedNicheRegion := by
+  classical
+  ext q
+  constructor
+  · rintro ⟨s, hs, hx, hy0, hy⟩
+    have htop : nicheTopArc s ∈
+        curveImage D (Icc (0 : ℝ) params.theta) ∪
+        curveImage (Romik.path params) (Icc params.phi tau) ∪
+        curveImage B (Icc eta T) := by
+      rw [← nicheTopArc_image]
+      exact ⟨s, hs, rfl⟩
+    rcases htop with (⟨r, hr, her⟩ | ⟨r, hr, her⟩) | ⟨r, hr, her⟩
+    · exact Or.inl (Or.inl ⟨r, hr, by simpa [her] using hx,
+        hy0, by simpa [her] using hy⟩)
+    · exact Or.inl (Or.inr ⟨r, hr, by simpa [her] using hx,
+        hy0, by simpa [her] using hy⟩)
+    · exact Or.inr ⟨r, hr, by simpa [her] using hx,
+        hy0, by simpa [her] using hy⟩
+  · intro hq
+    rcases hq with (⟨r, hr, hx, hy0, hy⟩ | ⟨r, hr, hx, hy0, hy⟩) |
+      ⟨r, hr, hx, hy0, hy⟩
+    · have htop : D r ∈ curveImage nicheTopArc (Icc (0 : ℝ) T) := by
+        rw [nicheTopArc_image]
+        exact Or.inl (Or.inl ⟨r, hr, rfl⟩)
+      rcases htop with ⟨s, hs, her⟩
+      exact ⟨s, hs, by simpa [her] using hx,
+        hy0, by simpa [her] using hy⟩
+    · have htop : Romik.path params r ∈
+          curveImage nicheTopArc (Icc (0 : ℝ) T) := by
+        rw [nicheTopArc_image]
+        exact Or.inl (Or.inr ⟨r, hr, rfl⟩)
+      rcases htop with ⟨s, hs, her⟩
+      exact ⟨s, hs, by simpa [her] using hx,
+        hy0, by simpa [her] using hy⟩
+    · have htop : B r ∈ curveImage nicheTopArc (Icc (0 : ℝ) T) := by
+        rw [nicheTopArc_image]
+        exact Or.inr ⟨r, hr, rfl⟩
+      rcases htop with ⟨s, hs, her⟩
+      exact ⟨s, hs, by simpa [her] using hx,
+        hy0, by simpa [her] using hy⟩
+
+/-- The three certified fills form one strict subgraph. -/
+theorem certifiedNicheRegion_eq_strictSubgraph :
+    certifiedNicheRegion =
+      strictSubgraphRegion (D 0).1 (B T).1 nicheRoof := by
+  rw [← nicheTopArc_verticalFill,
+    verticalFill_eq_strictSubgraphRegion nicheTopArc T_pos_frontier
+      nicheTopArc_continuous.continuousOn nicheTopArc_fst_strictMono,
+    nicheTopArc_zero, nicheTopArc_T]
+  rfl
+
+private theorem lineSegment_horizontal
+    {a b : Point} (hab : a.1 < b.1) (ha : a.2 = 0) (hb : b.2 = 0) :
+    lineSegment a b = {p | p.1 ∈ Icc a.1 b.1 ∧ p.2 = 0} := by
+  classical
+  ext p
+  constructor
+  · rintro ⟨r, hr, rfl⟩
+    constructor
+    · have hba : 0 ≤ b.1 - a.1 := sub_nonneg.mpr hab.le
+      constructor
+      · have hmul : 0 ≤ r * (b.1 - a.1) := mul_nonneg hr.1 hba
+        nlinarith [hmul]
+      · have hmul : 0 ≤ (1 - r) * (b.1 - a.1) :=
+          mul_nonneg (sub_nonneg.mpr hr.2) hba
+        nlinarith [hmul]
+    · simp [ha, hb]
+  · rintro ⟨hx, hy⟩
+    let r : ℝ := (p.1 - a.1) / (b.1 - a.1)
+    have hden : 0 < b.1 - a.1 := sub_pos.mpr hab
+    have hr : r ∈ Icc (0 : ℝ) 1 := by
+      constructor
+      · exact div_nonneg (sub_nonneg.mpr hx.1) hden.le
+      · exact (div_le_one hden).2 (by linarith [hx.2])
+    refine ⟨r, hr, ?_⟩
+    apply Prod.ext
+    · dsimp [r]
+      field_simp [ne_of_gt hden]
+      ring
+    · simp [ha, hb, hy]
+
+private theorem niche_horizontal_endpoints : (D 0).1 < (B T).1 := by
+  have h := nicheTopArc_fst_strictMono
+    (left_mem_Icc.2 T_pos_frontier.le)
+    (right_mem_Icc.2 T_pos_frontier.le) T_pos_frontier
+  simpa [nicheTopArc_zero, nicheTopArc_T] using h
+
+theorem claimedNicheBoundary_eq_subgraphBoundary :
+    claimedNicheBoundary =
+      strictSubgraphBoundary (D 0).1 (B T).1 nicheRoof := by
+  have hbase := lineSegment_horizontal niche_horizontal_endpoints
+    D_zero_y_zero B_T_y_zero
+  have hroof := curveImage_eq_roofGraph nicheTopArc T_pos_frontier
+    nicheTopArc_continuous.continuousOn nicheTopArc_fst_strictMono
+  rw [nicheTopArc_zero, nicheTopArc_T] at hroof
+  ext p
+  constructor
+  · intro hp
+    change p ∈ lineSegment (D 0) (B T) ∪
+      curveImage B (Icc eta T) ∪
+      curveImage (Romik.path params) (Icc params.phi tau) ∪
+      curveImage D (Icc 0 params.theta) at hp
+    have hp' : p ∈ lineSegment (D 0) (B T) ∨
+        p ∈ curveImage nicheTopArc (Icc (0 : ℝ) T) := by
+      rw [nicheTopArc_image]
+      rcases hp with ((hb | hB) | hpath) | hD
+      · exact Or.inl hb
+      · exact Or.inr (Or.inr hB)
+      · exact Or.inr (Or.inl (Or.inr hpath))
+      · exact Or.inr (Or.inl (Or.inl hD))
+    rcases hp' with hb | hr
+    · rw [hbase] at hb
+      exact ⟨hb.1, Or.inl hb.2⟩
+    · rw [hroof] at hr
+      exact ⟨hr.1, Or.inr hr.2⟩
+  · rintro ⟨hx, hbase' | hroof'⟩
+    · have hb : p ∈ lineSegment (D 0) (B T) := by
+        rw [hbase]
+        exact ⟨hx, hbase'⟩
+      change p ∈ lineSegment (D 0) (B T) ∪
+        curveImage B (Icc eta T) ∪
+        curveImage (Romik.path params) (Icc params.phi tau) ∪
+        curveImage D (Icc 0 params.theta)
+      exact Or.inl (Or.inl (Or.inl hb))
+    · have hr : p ∈ curveImage nicheTopArc (Icc (0 : ℝ) T) := by
+        rw [hroof]
+        exact ⟨hx, hroof'⟩
+      rw [nicheTopArc_image] at hr
+      change p ∈ lineSegment (D 0) (B T) ∪
+        curveImage B (Icc eta T) ∪
+        curveImage (Romik.path params) (Icc params.phi tau) ∪
+        curveImage D (Icc 0 params.theta)
+      rcases hr with (hD | hpath) | hB
+      · exact Or.inr hD
+      · exact Or.inl (Or.inr hpath)
+      · exact Or.inl (Or.inl (Or.inr hB))
+
+/-- Direct frontier calculation for the certified three-fill region. -/
+theorem frontier_certifiedNicheRegion :
+    frontier certifiedNicheRegion = claimedNicheBoundary := by
+  rw [certifiedNicheRegion_eq_strictSubgraph,
+    frontier_strictSubgraphRegion niche_horizontal_endpoints nicheRoof_continuous
+      (by intro X hX; exact nicheRoof_nonneg hX)
+      (by intro X hX; exact nicheRoof_pos hX)
+      nicheRoof_left nicheRoof_right,
+    ← claimedNicheBoundary_eq_subgraphBoundary]
+
+/-! ## Literal instantaneous wedges lie below the certified graph -/
+
+private theorem sin_pos_frontier {t : ℝ} (ht : t ∈ Ioo (0 : ℝ) T) :
+    0 < Real.sin t :=
+  Real.sin_pos_of_pos_of_lt_pi ht.1
+    (lt_trans ht.2 (by dsimp [T]; nlinarith [Real.pi_pos]))
+
+private theorem cos_pos_frontier {t : ℝ} (ht : t ∈ Ioo (0 : ℝ) T) :
+    0 < Real.cos t :=
+  Real.cos_pos_of_mem_Ioo
+    ⟨by nlinarith [Real.pi_pos, ht.1], by simpa [T] using ht.2⟩
+
+private theorem instantRoof_le_of_u_outside {t X Y : ℝ}
+    (ht : t ∈ Ioo (0 : ℝ) T)
+    (hu : 0 ≤ dot ((X, Y) - Romik.path params t) (u t)) :
+    instantRoof t X ≤ Y := by
+  apply le_trans (min_le_left _ _)
+  unfold bRoof
+  apply (div_le_iff₀ (sin_pos_frontier ht)).2
+  dsimp [dot, u] at hu ⊢
+  nlinarith
+
+private theorem instantRoof_le_of_v_outside {t X Y : ℝ}
+    (ht : t ∈ Ioo (0 : ℝ) T)
+    (hv : 0 ≤ dot ((X, Y) - Romik.path params t) (v t)) :
+    instantRoof t X ≤ Y := by
+  apply le_trans (min_le_right _ _)
+  unfold dRoof
+  apply (div_le_iff₀ (cos_pos_frontier ht)).2
+  dsimp [dot, v] at hv ⊢
+  nlinarith
+
+/-- A core path point dominates every instantaneous roof at its horizontal
+coordinate.  This is the exact geometric content of the two no-hidden
+inequalities. -/
+theorem core_is_upper_envelope {r : ℝ} (hr : r ∈ Icc params.phi tau) :
+    ∀ t ∈ Ioo (0 : ℝ) T,
+      instantRoof t (Romik.path params r).1 ≤ (Romik.path params r).2 := by
+  intro t ht
+  by_cases htr : r ≤ t
+  · have hU : 0 ≤ dot (Romik.path params r - Romik.path params t) (u t) :=
+      noHiddenU_direct r ⟨hr.1, le_trans hr.2 tau_lt_T.le⟩ t
+        ⟨htr, ht.2.le⟩
+    exact instantRoof_le_of_u_outside ht hU
+  · have htr' : t ≤ r := le_of_not_ge htr
+    have hV : 0 ≤ dot (Romik.path params r - Romik.path params t) (v t) :=
+      noHiddenV_direct t ⟨ht.1.le, le_trans htr' hr.2⟩ r ⟨htr', hr.2⟩
+    exact instantRoof_le_of_v_outside ht hV
+
+/-- Every point of the glued certified roof dominates every instantaneous
+wall roof at its horizontal coordinate. -/
+theorem nicheTopArc_is_upper_envelope {s t : ℝ}
+    (hs : s ∈ Icc (0 : ℝ) T) (ht : t ∈ Ioo (0 : ℝ) T) :
+    instantRoof t (nicheTopArc s).1 ≤ (nicheTopArc s).2 := by
+  by_cases hθ : s ≤ params.theta
+  · rw [nicheTopArc_of_le_theta hθ]
+    rcases D_boundary_outside ht ⟨hs.1, hθ⟩ with hu | hv
+    · exact instantRoof_le_of_u_outside ht hu
+    · exact instantRoof_le_of_v_outside ht hv
+  · have hθs : params.theta < s := lt_of_not_ge hθ
+    by_cases hη : s ≤ eta
+    · rw [nicheTopArc_of_middle hθs hη]
+      exact core_is_upper_envelope (coreReverseTime_mem ⟨hθs.le, hη⟩) t ht
+    · have hηs : eta < s := lt_of_not_ge hη
+      rw [nicheTopArc_of_late hηs]
+      rcases B_boundary_outside ht ⟨hηs.le, hs.2⟩ with hu | hv
+      · exact instantRoof_le_of_u_outside ht hu
+      · exact instantRoof_le_of_v_outside ht hv
+
+private theorem niche_point_horizontal_range {q : Point}
+    (hq : q ∈ Romik.niche params) : q.1 ∈ Ioo (D 0).1 (B T).1 := by
+  rcases hq with ⟨hqFan, t, ht, hquad⟩
+  have hq0 : 0 ≤ q.2 := by simpa [capFan] using hqFan
+  have hwalls := hquad
+  change
+    dot (q - Romik.path params t) (u t) < 0 ∧
+      dot (q - Romik.path params t) (v t) < 0 at hwalls
+  have hBT := B_T_u_nonneg ht
+  have hqBT : dot (q - B T) (u t) < 0 := by
+    unfold dot at hwalls hBT ⊢
+    dsimp [u] at hwalls hBT ⊢
+    linarith
+  have hright : q.1 < (B T).1 := by
+    by_contra hn
+    have hdx : 0 ≤ q.1 - (B T).1 := sub_nonneg.mpr (le_of_not_gt hn)
+    have hxterm : 0 ≤ (q.1 - (B T).1) * Real.cos t :=
+      mul_nonneg hdx (cos_pos_frontier ht).le
+    have hyterm : 0 ≤ (q.2 - (B T).2) * Real.sin t := by
+      rw [B_T_y_zero]
+      simpa using mul_nonneg hq0 (sin_pos_frontier ht).le
+    dsimp [dot, u] at hqBT
+    linarith
+  have hD := D_zero_v_nonneg ht
+  have hqD : dot (q - D 0) (v t) < 0 := by
+    unfold dot at hwalls hD ⊢
+    dsimp [v] at hwalls hD ⊢
+    linarith
+  have hleft : (D 0).1 < q.1 := by
+    by_contra hn
+    have hdx : q.1 - (D 0).1 ≤ 0 := sub_nonpos.mpr (le_of_not_gt hn)
+    have hxterm : 0 ≤ (q.1 - (D 0).1) * (-Real.sin t) :=
+      mul_nonneg_of_nonpos_of_nonpos hdx (neg_nonpos.mpr (sin_pos_frontier ht).le)
+    have hyterm : 0 ≤ (q.2 - (D 0).2) * Real.cos t := by
+      rw [D_zero_y_zero]
+      simpa using mul_nonneg hq0 (cos_pos_frontier ht).le
+    dsimp [dot, v] at hqD
+    linarith
+  exact ⟨hleft, hright⟩
+
+private theorem niche_subset_certifiedNicheRegion :
+    Romik.niche params ⊆ certifiedNicheRegion := by
+  intro q hq
+  rcases hq with ⟨hqFan, t, ht, hquad⟩
+  have hqAll : q ∈ Romik.niche params := ⟨hqFan, t, ht, hquad⟩
+  have hXopen := niche_point_horizontal_range hqAll
+  have hX : q.1 ∈ Icc (nicheTopArc 0).1 (nicheTopArc T).1 := by
+    simpa [nicheTopArc_zero, nicheTopArc_T] using
+      (show q.1 ∈ Icc (D 0).1 (B T).1 from ⟨hXopen.1.le, hXopen.2.le⟩)
+  let e := horizontalOrderIso nicheTopArc T_pos_frontier
+    nicheTopArc_continuous.continuousOn nicheTopArc_fst_strictMono
+  let s : Icc (0 : ℝ) T := e.symm ⟨q.1, hX⟩
+  have heval := congrArg Subtype.val
+    ((horizontalOrderIso nicheTopArc T_pos_frontier
+      nicheTopArc_continuous.continuousOn
+      nicheTopArc_fst_strictMono).apply_symm_apply ⟨q.1, hX⟩)
+  have hxarc : (nicheTopArc s.1).1 = q.1 := by
+    change
+      (nicheTopArc ((horizontalOrderIso nicheTopArc T_pos_frontier
+        nicheTopArc_continuous.continuousOn
+        nicheTopArc_fst_strictMono).symm ⟨q.1, hX⟩).1).1 = q.1
+    simpa only [horizontalOrderIso_apply_val] using heval
+  have hinstant : q.2 < instantRoof t q.1 := by
+    have hw := (mem_innerQuadrant_iff_roofs ht).1 hquad
+    simpa [instantRoof, lt_min_iff] using hw
+  have hupper := nicheTopArc_is_upper_envelope s.2 ht
+  have hroofAt := nicheRoof_at_topArc s.2
+  have hq0 : 0 ≤ q.2 := by simpa [capFan] using hqFan
+  rw [certifiedNicheRegion_eq_strictSubgraph]
+  refine ⟨by simpa [nicheTopArc_zero, nicheTopArc_T] using hX, hq0, ?_⟩
+  calc
+    q.2 < instantRoof t q.1 := hinstant
+    _ ≤ (nicheTopArc s.1).2 := by simpa [hxarc] using hupper
+    _ = nicheRoof q.1 := by rw [← hroofAt, hxarc]
+
+private theorem certifiedNicheRegion_subset_niche :
+    certifiedNicheRegion ⊆ Romik.niche params := by
+  intro q hq
+  rcases hq with (⟨r, hr, hx, hy0, hy⟩ | ⟨r, hr, hx, hy0, hy⟩) |
+    ⟨r, hr, hx, hy0, hy⟩
+  · have hmem := vertical_below_D_mem_niche hr hy0 hy
+    have heq : q = ((D r).1, q.2) := by
+      apply Prod.ext
+      · exact hx
+      · rfl
+    rw [heq]
+    exact hmem
+  · have hmem := vertical_below_path_mem_niche hr hy0 hy
+    have heq : q = ((Romik.path params r).1, q.2) := by
+      apply Prod.ext
+      · exact hx
+      · rfl
+    rw [heq]
+    exact hmem
+  · have hmem := vertical_below_B_mem_niche hr hy0 hy
+    have heq : q = ((B r).1, q.2) := by
+      apply Prod.ext
+      · exact hx
+      · rfl
+    rw [heq]
+    exact hmem
+
+/-- Literal niche equals the three concrete strict vertical fills. -/
+theorem niche_eq_certifiedNicheRegion :
+    Romik.niche params = certifiedNicheRegion :=
+  Set.Subset.antisymm niche_subset_certifiedNicheRegion
+    certifiedNicheRegion_subset_niche
+
+/-- Final exact set-valued boundary equality required by Part C. -/
+theorem niche_frontier_direct :
+    frontier (Romik.niche params) = claimedNicheBoundary := by
+  rw [niche_eq_certifiedNicheRegion]
+  exact frontier_certifiedNicheRegion
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 4: connectedness of the literal niche
+
+The independent vertical-fill module proves connectedness of the certified
+three-piece region.  Once the boundary layer identifies that region with the
+literal niche, connectedness is immediate.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Set
+
+/-- Concrete connectedness of the literal niche. -/
+theorem niche_connected_direct : IsConnected (Romik.niche params) := by
+  rw [niche_eq_certifiedNicheRegion]
+  exact certifiedNicheRegion_connected
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 4: vertical-fibre topology of the fixed sofa
+
+The argument in this file is independent of any Jordan-curve theorem.  The
+cap is compact and convex.  Its removed niche is a strict vertical subgraph.
+Consequently every nonempty vertical section of the complement is an interval.
+Moreover, a highest point of each cap section survives: if it belonged to the
+strict subgraph, one of the certified graph points would be a still higher cap
+point.  Thus the horizontal projections of the cap and sofa agree.  A compact
+map with connected fibres over that connected projection closes connectedness.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Set
+open Stage2
+
+private theorem physical_cos_diff_nonneg {s t : ℝ}
+    (hs : s ∈ Icc (0 : ℝ) T) (ht : t ∈ Icc (0 : ℝ) T) :
+    0 ≤ Real.cos (t - s) := by
+  apply Real.cos_nonneg_of_neg_pi_div_two_le_of_le
+  · have hsT := hs.2
+    have ht0 := ht.1
+    dsimp [T] at hsT
+    linarith
+  · have hs0 := hs.1
+    have htT := ht.2
+    dsimp [T] at htT
+    linarith
+
+private theorem phi_upper_twentieth_fiber :
+    params.phi ≤ (1 / 20 : ℝ) := by
+  have h := phi_bounds.2
+  norm_num at h ⊢
+  linarith
+
+private theorem a1_lower_six_fifths_fiber :
+    (6 / 5 : ℝ) ≤ params.a1 := by
+  have h := Romik.a1_lower_bound_of_mem_box params_mem
+  norm_num at h ⊢
+  linarith
+
+private theorem alphaBeta1_snd_nonneg_fiber {s : ℝ}
+    (hs0 : 0 ≤ s) (hs20 : s ≤ (1 / 20 : ℝ)) :
+    0 ≤ (Romik.alphaBeta1 params s).2 := by
+  have hpi : s ≤ Real.pi := by
+    nlinarith [Real.pi_gt_three]
+  have hsin0 : 0 ≤ Real.sin s :=
+    Real.sin_nonneg_of_nonneg_of_le_pi hs0 hpi
+  have hsinUpper := Real.sin_le hs0
+  have hcosLower : 1 - s ^ 2 / 2 ≤ Real.cos s :=
+    Real.one_sub_sq_div_two_le_cos
+  have hcos0 : 0 ≤ Real.cos s := by
+    apply Real.cos_nonneg_of_mem_Icc
+    constructor <;> nlinarith [Real.pi_gt_three]
+  have hquad : 0 ≤ s * ((1 / 20 : ℝ) - s) :=
+    mul_nonneg hs0 (sub_nonneg.mpr hs20)
+  have hcoef : (12 / 5 : ℝ) ≤ 2 * params.a1 := by
+    nlinarith [a1_lower_six_fifths_fiber]
+  have hmul : (12 / 5 : ℝ) * Real.cos s ≤
+      2 * params.a1 * Real.cos s :=
+    mul_le_mul_of_nonneg_right hcoef hcos0
+  have ha2 := Romik.a2_eq_neg_quarter_of_equations params_equations
+  dsimp [Romik.alphaBeta1]
+  rw [ha2]
+  nlinarith [hsinUpper, hcosLower, hquad, hmul]
+
+/-- The early inner-contact coefficient is nonnegative on the whole `D`
+parameter interval. -/
+theorem beta_nonneg_on_D {t : ℝ}
+    (ht : t ∈ Icc (0 : ℝ) params.theta) : 0 ≤ beta t := by
+  by_cases hphi : t ≤ params.phi
+  · have hs20 : t ≤ (1 / 20 : ℝ) :=
+      le_trans hphi phi_upper_twentieth_fiber
+    have h := alphaBeta1_snd_nonneg_fiber ht.1 hs20
+    simpa [beta, alphaBetaAt, hphi] using h
+  · have htPhysical : t ∈ Icc (0 : ℝ) T :=
+      ⟨ht.1, le_trans ht.2 (le_trans theta_lt_eta.le
+        (le_trans eta_lt_tau.le tau_le_T))⟩
+    have hrho := Stage2.rhoA_nonneg htPhysical
+    unfold Stage2.rhoA at hrho
+    rw [ite_eq_right hphi, ite_eq_left ht.2] at hrho
+    simp only [beta, alphaBetaAt, ite_eq_right hphi, ite_eq_left ht.2]
+    dsimp [Romik.alphaBeta2]
+    exact hrho
+
+/-- Core path points are cap points. -/
+theorem core_path_mem_K {t : ℝ} (ht : t ∈ Icc params.phi tau) :
+    Romik.path params t ∈ K := by
+  rw [Romik.mem_K0]
+  refine ⟨path_core_y_nonneg ht, ?_⟩
+  intro s hs
+  have hs' : s ∈ Icc (0 : ℝ) T := by simpa [T] using hs
+  have ht' : t ∈ Icc (0 : ℝ) T :=
+    ⟨le_trans phi_pos.le ht.1, le_trans ht.2 tau_le_T⟩
+  exact path_mem_outer_supports hs' ht'
+
+/-- Every certified early `D` roof point lies in the cap. -/
+theorem D_mem_K {t : ℝ} (ht : t ∈ Icc (0 : ℝ) params.theta) :
+    D t ∈ K := by
+  rw [Romik.mem_K0]
+  refine ⟨D_y_nonneg ht, ?_⟩
+  intro s hs
+  have hs' : s ∈ Icc (0 : ℝ) T := by simpa [T] using hs
+  have ht' : t ∈ Icc (0 : ℝ) T :=
+    ⟨ht.1, le_trans ht.2 (le_trans theta_lt_eta.le
+      (le_trans eta_lt_tau.le tau_le_T))⟩
+  have hcos := physical_cos_diff_nonneg hs' ht'
+  have hbeta := beta_nonneg_on_D ht
+  have hp := path_mem_outer_supports hs' ht'
+  have hC := Stage3.supportC_direct t ht'
+  constructor
+  · have hpU := hp.1
+    change dot (D t) (u s) ≤ dot (Romik.path params s) (u s) + 1
+    change dot (Romik.path params t) (u s) ≤
+      dot (Romik.path params s) (u s) + 1 at hpU
+    have hrel : dot (D t) (u s) =
+        dot (Romik.path params t) (u s) - beta t * Real.cos (t - s) := by
+      simp [D, dot, u, Real.cos_sub]
+      ring
+    rw [hrel]
+    nlinarith [mul_nonneg hbeta hcos]
+  · have hCV := (hC.2 s hs).2
+    change dot (D t) (v s) ≤ dot (Romik.path params s) (v s) + 1
+    change dot (C t) (v s) ≤ dot (Romik.path params s) (v s) + 1 at hCV
+    have hrel : dot (C t) (v s) =
+        dot (D t) (v s) + Real.cos (t - s) := by
+      simp [C, D, dot, u, v, Real.cos_sub]
+      ring
+    rw [hrel] at hCV
+    linarith
+
+/-- Every certified late `B` roof point lies in the cap. -/
+theorem B_mem_K {t : ℝ} (ht : t ∈ Icc eta T) : B t ∈ K := by
+  rw [Romik.mem_K0]
+  refine ⟨B_y_nonneg ht, ?_⟩
+  intro s hs
+  have hs' : s ∈ Icc (0 : ℝ) T := by simpa [T] using hs
+  have ht' : t ∈ Icc (0 : ℝ) T :=
+    ⟨le_trans phi_pos.le (le_trans phi_lt_eta.le ht.1), ht.2⟩
+  have hcos := physical_cos_diff_nonneg hs' ht'
+  have halpha := alpha_nonpos ⟨le_trans phi_lt_eta.le ht.1, ht.2⟩
+  have hp := path_mem_outer_supports hs' ht'
+  have hA := Stage3.supportA_direct t ht'
+  constructor
+  · have hAU := (hA.2 s hs).1
+    change dot (B t) (u s) ≤ dot (Romik.path params s) (u s) + 1
+    change dot (A t) (u s) ≤ dot (Romik.path params s) (u s) + 1 at hAU
+    have hrel : dot (A t) (u s) =
+        dot (B t) (u s) + Real.cos (t - s) := by
+      simp [A, B, dot, u, v, Real.cos_sub]
+      ring
+    rw [hrel] at hAU
+    linarith
+  · have hpV := hp.2
+    change dot (B t) (v s) ≤ dot (Romik.path params s) (v s) + 1
+    change dot (Romik.path params t) (v s) ≤
+      dot (Romik.path params s) (v s) + 1 at hpV
+    have hrel : dot (B t) (v s) =
+        dot (Romik.path params t) (v s) + alpha t * Real.cos (t - s) := by
+      simp [B, dot, v, Real.cos_sub]
+      ring
+    rw [hrel]
+    nlinarith [mul_nonpos_of_nonpos_of_nonneg halpha hcos]
+
+/-- Every graph which forms the certified strict niche roof consists of cap
+points. -/
+theorem certified_roof_mem_K :
+    (∀ t ∈ Icc (0 : ℝ) params.theta, D t ∈ K) ∧
+    (∀ t ∈ Icc params.phi tau, Romik.path params t ∈ K) ∧
+    (∀ t ∈ Icc eta T, B t ∈ K) :=
+  ⟨fun _ ht => D_mem_K ht,
+    fun _ ht => core_path_mem_K ht,
+    fun _ ht => B_mem_K ht⟩
+
+/-- Strict vertical fills are downward closed along every nonnegative fibre. -/
+theorem verticalFill_downward
+    {f : ℝ → Point} {I : Set ℝ} {p q : Point}
+    (hx : p.1 = q.1) (hp0 : 0 ≤ p.2) (hy : p.2 ≤ q.2)
+    (hq : q ∈ verticalFill f I) : p ∈ verticalFill f I := by
+  rcases hq with ⟨t, ht, hqx, hq0, hqtop⟩
+  exact ⟨t, ht, hx.trans hqx, hp0, lt_of_le_of_lt hy hqtop⟩
+
+/-- The union of the three strict graph fills is downward closed in each
+nonnegative vertical fibre. -/
+theorem certifiedNicheRegion_downward {p q : Point}
+    (hx : p.1 = q.1) (hp0 : 0 ≤ p.2) (hy : p.2 ≤ q.2)
+    (hq : q ∈ certifiedNicheRegion) : p ∈ certifiedNicheRegion := by
+  rcases hq with (hD | hP) | hB
+  · exact Or.inl (Or.inl (verticalFill_downward hx hp0 hy hD))
+  · exact Or.inl (Or.inr (verticalFill_downward hx hp0 hy hP))
+  · exact Or.inr (verticalFill_downward hx hp0 hy hB)
+
+/-- The Gerver cap is compact. -/
+theorem K_compact_direct : IsCompact K := by
+  let box : Set Point :=
+    Icc ((Romik.path params T).1 - 1) 1 ×ˢ Icc (0 : ℝ) 1
+  have hbox : IsCompact box := isCompact_Icc.prod isCompact_Icc
+  apply hbox.of_isClosed_subset (Romik.isClosed_K0 params)
+  intro q hq
+  have h0 : (0 : ℝ) ∈ Icc (0 : ℝ) (Real.pi / 2) := by
+    constructor
+    · exact le_rfl
+    · positivity
+  have hT : T ∈ Icc (0 : ℝ) (Real.pi / 2) := by
+    exact ⟨by dsimp [T]; positivity, by dsimp [T]; exact le_rfl⟩
+  have hs0 := hq.2 0 h0
+  have hsT := hq.2 T hT
+  have hxhi : q.1 ≤ 1 := by
+    have h := hs0.1
+    simpa [supportHalfU, dot, u, pathZero] using h
+  have hyhi : q.2 ≤ 1 := by
+    have h := hs0.2
+    simpa [supportHalfV, dot, v, pathZero] using h
+  have hxlo : (Romik.path params T).1 - 1 ≤ q.1 := by
+    have h := hsT.2
+    simp [supportHalfV, dot, v, T] at h
+    dsimp [T] at ⊢
+    linarith
+  exact ⟨⟨hxlo, hxhi⟩, ⟨hq.1, hyhi⟩⟩
+
+/-- The fixed sofa is compact, as the difference of the compact cap and the
+open union of forbidden quadrants. -/
+theorem G_compact_direct : IsCompact G := by
+  change IsCompact (Romik.sofa params)
+  rw [Romik.sofa_eq_K0_diff_innerUnion]
+  exact K_compact_direct.diff (Romik.isOpen_innerUnion params)
+
+private def KVerticalSection (x : ℝ) : Set Point :=
+  K ∩ {q | q.1 = x}
+
+private theorem KVerticalSection_compact (x : ℝ) :
+    IsCompact (KVerticalSection x) := by
+  apply K_compact_direct.inter_right
+  exact isClosed_singleton.preimage continuous_fst
+
+/-- Every cap fibre has a highest point, and that point survives removal of
+the strict niche. -/
+theorem exists_G_point_over_K {q : Point} (hq : q ∈ K) :
+    ∃ p ∈ G, p.1 = q.1 := by
+  let F := KVerticalSection q.1
+  have hFcompact : IsCompact F := KVerticalSection_compact q.1
+  have hFne : F.Nonempty := ⟨q, hq, rfl⟩
+  rcases hFcompact.exists_isMaxOn hFne continuous_snd.continuousOn with
+    ⟨p, hpF, hpmax⟩
+  have hpK : p ∈ K := hpF.1
+  have hpx : p.1 = q.1 := hpF.2
+  refine ⟨p, ⟨hpK, ?_⟩, hpx⟩
+  intro hpN
+  rw [niche_eq_certifiedNicheRegion] at hpN
+  rcases hpN with (hD | hP) | hB
+  · rcases hD with ⟨t, ht, hx, _hy0, hylt⟩
+    have hwK := D_mem_K ht
+    have hwF : D t ∈ F := ⟨hwK, hx.symm.trans hpx⟩
+    exact (not_lt_of_ge (hpmax hwF)) hylt
+  · rcases hP with ⟨t, ht, hx, _hy0, hylt⟩
+    have hwK := core_path_mem_K ht
+    have hwF : Romik.path params t ∈ F := ⟨hwK, hx.symm.trans hpx⟩
+    exact (not_lt_of_ge (hpmax hwF)) hylt
+  · rcases hB with ⟨t, ht, hx, _hy0, hylt⟩
+    have hwK := B_mem_K ht
+    have hwF : B t ∈ F := ⟨hwK, hx.symm.trans hpx⟩
+    exact (not_lt_of_ge (hpmax hwF)) hylt
+
+/-- Removing the strict niche does not change the horizontal projection. -/
+theorem fst_image_G_eq_fst_image_K :
+    Prod.fst '' G = Prod.fst '' K := by
+  apply Set.Subset.antisymm
+  · rintro x ⟨q, hq, rfl⟩
+    exact ⟨q, hq.1, rfl⟩
+  · rintro x ⟨q, hq, rfl⟩
+    rcases exists_G_point_over_K hq with ⟨p, hp, hpx⟩
+    exact ⟨p, hp, hpx⟩
+
+/-- A vertical section of the fixed sofa. -/
+def GVerticalSection (x : ℝ) : Set Point :=
+  {q | q ∈ G ∧ q.1 = x}
+
+/-- Every vertical section of `G` is convex. -/
+theorem GVerticalSection_convex (x : ℝ) :
+    Convex ℝ (GVerticalSection x) := by
+  intro p hp q hq a b ha hb hab
+  have hpK : p ∈ K := hp.1.1
+  have hqK : q ∈ K := hq.1.1
+  have hzK := K_convex_direct hpK hqK ha hb hab
+  have hzx : (a • p + b • q).1 = x := by
+    change a * p.1 + b * q.1 = x
+    rw [hp.2, hq.2]
+    calc
+      a * x + b * x = (a + b) * x := by ring
+      _ = x := by rw [hab, one_mul]
+  refine ⟨⟨hzK, ?_⟩, hzx⟩
+  intro hzN
+  have hzR : a • p + b • q ∈ certifiedNicheRegion := by
+    rw [← niche_eq_certifiedNicheRegion]
+    exact hzN
+  rcases le_total p.2 q.2 with hpq | hqp
+  · have hpz : p.2 ≤ (a • p + b • q).2 := by
+      change p.2 ≤ a * p.2 + b * q.2
+      calc
+        p.2 = a * p.2 + b * p.2 := by
+          calc
+            p.2 = (a + b) * p.2 := by rw [hab, one_mul]
+            _ = a * p.2 + b * p.2 := by ring
+        _ ≤ a * p.2 + b * q.2 :=
+          add_le_add le_rfl (mul_le_mul_of_nonneg_left hpq hb)
+    have hpR := certifiedNicheRegion_downward
+      (hp.2.trans hzx.symm) hpK.1 hpz hzR
+    exact hp.1.2 (by
+      rw [niche_eq_certifiedNicheRegion]
+      exact hpR)
+  · have hqz : q.2 ≤ (a • p + b • q).2 := by
+      change q.2 ≤ a * p.2 + b * q.2
+      calc
+        q.2 = a * q.2 + b * q.2 := by
+          calc
+            q.2 = (a + b) * q.2 := by rw [hab, one_mul]
+            _ = a * q.2 + b * q.2 := by ring
+        _ ≤ a * p.2 + b * q.2 :=
+          add_le_add (mul_le_mul_of_nonneg_left hqp ha) le_rfl
+    have hqR := certifiedNicheRegion_downward
+      (hq.2.trans hzx.symm) hqK.1 hqz hzR
+    exact hq.1.2 (by
+      rw [niche_eq_certifiedNicheRegion]
+      exact hqR)
+
+/-- Each nonempty vertical section is connected. -/
+theorem GVerticalSection_connected {x : ℝ}
+    (hne : (GVerticalSection x).Nonempty) :
+    IsConnected (GVerticalSection x) :=
+  (GVerticalSection_convex x).isConnected hne
+
+private def GProjection : Set ℝ := Prod.fst '' G
+
+private def projectG : G → GProjection := fun q =>
+  ⟨q.1.1, ⟨q.1, q.2, rfl⟩⟩
+
+private theorem projectG_continuous : Continuous projectG := by
+  exact (continuous_fst.comp continuous_subtype_val).subtype_mk _
+
+private theorem projectG_surjective : Function.Surjective projectG := by
+  rintro ⟨x, q, hq, hqx⟩
+  refine ⟨⟨q, hq⟩, ?_⟩
+  apply Subtype.ext
+  exact hqx
+
+private theorem projectG_fiber_connected (x : GProjection) :
+    IsConnected (projectG ⁻¹' {x}) := by
+  have hsectionNe : (GVerticalSection x.1).Nonempty := by
+    rcases x.2 with ⟨q, hq, hqx⟩
+    exact ⟨q, hq, hqx⟩
+  have hsection := GVerticalSection_connected hsectionNe
+  have : ConnectedSpace (GVerticalSection x.1) :=
+    isConnected_iff_connectedSpace.mp hsection
+  let e : GVerticalSection x.1 → G := fun q => ⟨q.1, q.2.1⟩
+  have he : Continuous e := by
+    exact continuous_subtype_val.subtype_mk _
+  have hrange : Set.range e = projectG ⁻¹' {x} := by
+    ext q
+    constructor
+    · rintro ⟨z, rfl⟩
+      apply Set.mem_preimage.mpr
+      apply Set.mem_singleton_iff.mpr
+      apply Subtype.ext
+      exact z.2.2
+    · intro hq
+      have hx : q.1.1 = x.1 := by
+        have := Set.mem_singleton_iff.mp hq
+        exact congrArg Subtype.val this
+      exact ⟨⟨q.1, q.2, hx⟩, rfl⟩
+  rw [← hrange]
+  exact isConnected_range he
+
+/-- Connectedness of the fixed Gerver sofa by compact connected fibres. -/
+theorem G_connected_by_vertical_fibers : IsConnected G := by
+  have hprojK : IsConnected (Prod.fst '' K) :=
+    K_connected_direct.image Prod.fst continuous_fst.continuousOn
+  have hproj : IsConnected GProjection := by
+    simpa [GProjection, fst_image_G_eq_fst_image_K] using hprojK
+  have : CompactSpace G := isCompact_iff_compactSpace.mp G_compact_direct
+  have : ConnectedSpace GProjection := isConnected_iff_connectedSpace.mp hproj
+  have hquot : Topology.IsQuotientMap projectG :=
+    Topology.IsQuotientMap.of_surjective_continuous
+      projectG_surjective projectG_continuous
+  have hpre : IsConnected (projectG ⁻¹' (Set.univ : Set GProjection)) :=
+    Topology.IsCoinducing.isConnected_preimage_of_isClosed
+      projectG_fiber_connected hquot.isCoinducing isClosed_univ isConnected_univ
+  have huniv : IsConnected (Set.univ : Set G) := by simpa using hpre
+  have hspace : ConnectedSpace G := connectedSpace_iff_univ.mpr huniv
+  exact isConnected_iff_connectedSpace.mpr hspace
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C Stage 4: topology of the concrete fixed set
+
+No new certificate structure is introduced here.  This file is intended to
+close the two literal topology fields left by `RemainingTopologyTarget`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+open Set
+
+/-- The right endpoint of the niche roof still satisfies the zero-angle
+horizontal support inequality of the cap. -/
+private theorem B_T_fst_le_one : (B T).1 ≤ 1 := by
+  have hBTK : B T ∈ K := B_mem_K ⟨eta_lt_T.le, le_rfl⟩
+  rw [Romik.mem_K0] at hBTK
+  have h0 : (0 : ℝ) ∈ Icc (0 : ℝ) (Real.pi / 2) := by
+    constructor
+    · exact le_rfl
+    · positivity
+  have h := (hBTK.2 0 h0).1
+  simpa [supportHalfU, dot, u, pathZero] using h
+
+/-- The left endpoint of the zero-angle support face is the concrete anchor
+and it is not removed by the niche. -/
+theorem anchor_mem_direct : anchor ∈ G := by
+  have hA0 : A 0 = anchor := Stage2.A_zero_eq_anchor
+  have hK : anchor ∈ K := by
+    rw [← hA0]
+    exact Stage3.supportA_direct 0 (by
+      constructor
+      · norm_num
+      · dsimp [T]; positivity)
+  refine ⟨hK, ?_⟩
+  intro hn
+  rw [niche_eq_certifiedNicheRegion] at hn
+  rcases hn with (hD | hx) | hB
+  · rcases hD with ⟨t, ht, hx1, _hy0, _hylt⟩
+    have htT : t < T := lt_of_le_of_lt ht.2 (lt_trans theta_lt_eta eta_lt_T)
+    have htPhysical : t ∈ Icc (0 : ℝ) T := ⟨ht.1, htT.le⟩
+    have hroof := nicheTopArc_fst_strictMono htPhysical
+      (right_mem_Icc.2 (show (0 : ℝ) ≤ T by dsimp [T]; positivity)) htT
+    have hleft : nicheTopArc t = D t := by
+      simp [nicheTopArc, ht.2]
+    change (nicheTopArc t).1 < (nicheTopArc T).1 at hroof
+    rw [hleft, nicheTopArc_T] at hroof
+    change (1 : ℝ) = (D t).1 at hx1
+    linarith [B_T_fst_le_one]
+  · rcases hx with ⟨t, ht, hx1, _hy0, _hylt⟩
+    have hpath : (Romik.path params t).1 ≤
+        (Romik.path params params.phi).1 :=
+      core_path_fst_strictAnti.antitoneOn
+        (left_mem_Icc.2 (lt_trans phi_lt_theta
+          (lt_trans theta_lt_eta eta_lt_tau)).le)
+        ht ht.1
+    have hlate : (B eta).1 < (B T).1 :=
+      B_fst_strictMono (left_mem_Icc.2 eta_lt_T.le)
+        (right_mem_Icc.2 eta_lt_T.le) eta_lt_T
+    have hroof : (Romik.path params t).1 < (B T).1 := by
+      calc
+        (Romik.path params t).1 ≤
+            (Romik.path params params.phi).1 := hpath
+        _ = (B eta).1 := congrArg Prod.fst B_eta_eq_path_phi.symm
+        _ < (B T).1 := hlate
+    change (1 : ℝ) = (Romik.path params t).1 at hx1
+    linarith [B_T_fst_le_one]
+  · rcases hB with ⟨t, ht, hx1, _hy0, _hylt⟩
+    by_cases hT : t = T
+    · subst t
+      change (0 : ℝ) < (B T).2 at _hylt
+      rw [B_T_y_zero] at _hylt
+      exact (lt_irrefl 0 _hylt)
+    · have htT : t < T := lt_of_le_of_ne ht.2 hT
+      have hroof : (B t).1 < (B T).1 :=
+        B_fst_strictMono ht (right_mem_Icc.2 eta_lt_T.le) htT
+      change (1 : ℝ) = (B t).1 at hx1
+      linarith [B_T_fst_le_one]
+
+/-- Direct connectedness of the cap-minus-niche set, obtained from the
+compact connected horizontal projection and the connected vertical fibres. -/
+theorem G_connected_direct : IsConnected G :=
+  G_connected_by_vertical_fibers
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Part C final unconditional closure
+
+This is the only terminal assembly for Part C.  It contains no payload and no
+new certificate interface.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartC
+namespace Stage4
+
+/-- Concrete global geometry certificate. -/
+theorem globalGeometryCertificate : GlobalGeometryCertificate :=
+  Stage3.globalGeometryDirect
+    noHiddenU_direct noHiddenV_direct
+    niche_frontier_direct niche_connected_direct
+    anchor_mem_direct G_connected_direct
+
+end Stage4
+end PartC
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle007.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle007.lean
new file mode 100644
index 0000000000..145884a738
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle007.lean
@@ -0,0 +1,1582 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Semantics.Batch002`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24PhiBelowKernelLH`.
+* `KernelOnly.PartE.E24PhiBelowKernelLL`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24 kernel child certificate: PhiBelow/LH, remaining depth 13. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+namespace CoverCertificatee18e89e753
+
+private abbrev cellRoot : AngleCell :=
+  (childLH e24PhiBelowRoot)
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+private abbrev cell0000 : AngleCell :=
+  childLL cell000
+
+private abbrev cell0001 : AngleCell :=
+  childLH cell000
+
+private abbrev cell0002 : AngleCell :=
+  childHL cell000
+
+private abbrev cell0003 : AngleCell :=
+  childHH cell000
+
+private abbrev cell0020 : AngleCell :=
+  childLL cell002
+
+private abbrev cell0021 : AngleCell :=
+  childLH cell002
+
+private abbrev cell0022 : AngleCell :=
+  childHL cell002
+
+private abbrev cell0023 : AngleCell :=
+  childHH cell002
+
+private abbrev cell0200 : AngleCell :=
+  childLL cell020
+
+private abbrev cell0201 : AngleCell :=
+  childLH cell020
+
+private abbrev cell0202 : AngleCell :=
+  childHL cell020
+
+private abbrev cell0203 : AngleCell :=
+  childHH cell020
+
+private abbrev cell0210 : AngleCell :=
+  childLL cell021
+
+private abbrev cell0211 : AngleCell :=
+  childLH cell021
+
+private abbrev cell0212 : AngleCell :=
+  childHL cell021
+
+private abbrev cell0213 : AngleCell :=
+  childHH cell021
+
+private abbrev cell0220 : AngleCell :=
+  childLL cell022
+
+private abbrev cell0221 : AngleCell :=
+  childLH cell022
+
+private abbrev cell0222 : AngleCell :=
+  childHL cell022
+
+private abbrev cell0223 : AngleCell :=
+  childHH cell022
+
+private abbrev cell0230 : AngleCell :=
+  childLL cell023
+
+private abbrev cell0231 : AngleCell :=
+  childLH cell023
+
+private abbrev cell0232 : AngleCell :=
+  childHL cell023
+
+private abbrev cell0233 : AngleCell :=
+  childHH cell023
+
+private abbrev cell0320 : AngleCell :=
+  childLL cell032
+
+private abbrev cell0321 : AngleCell :=
+  childLH cell032
+
+private abbrev cell0322 : AngleCell :=
+  childHL cell032
+
+private abbrev cell0323 : AngleCell :=
+  childHH cell032
+
+private abbrev cell2000 : AngleCell :=
+  childLL cell200
+
+private abbrev cell2001 : AngleCell :=
+  childLH cell200
+
+private abbrev cell2002 : AngleCell :=
+  childHL cell200
+
+private abbrev cell2003 : AngleCell :=
+  childHH cell200
+
+private abbrev cell2010 : AngleCell :=
+  childLL cell201
+
+private abbrev cell2011 : AngleCell :=
+  childLH cell201
+
+private abbrev cell2012 : AngleCell :=
+  childHL cell201
+
+private abbrev cell2013 : AngleCell :=
+  childHH cell201
+
+private abbrev cell2020 : AngleCell :=
+  childLL cell202
+
+private abbrev cell2021 : AngleCell :=
+  childLH cell202
+
+private abbrev cell2022 : AngleCell :=
+  childHL cell202
+
+private abbrev cell2023 : AngleCell :=
+  childHH cell202
+
+private abbrev cell2030 : AngleCell :=
+  childLL cell203
+
+private abbrev cell2031 : AngleCell :=
+  childLH cell203
+
+private abbrev cell2032 : AngleCell :=
+  childHL cell203
+
+private abbrev cell2033 : AngleCell :=
+  childHH cell203
+
+private abbrev cell2100 : AngleCell :=
+  childLL cell210
+
+private abbrev cell2101 : AngleCell :=
+  childLH cell210
+
+private abbrev cell2102 : AngleCell :=
+  childHL cell210
+
+private abbrev cell2103 : AngleCell :=
+  childHH cell210
+
+private abbrev cell2120 : AngleCell :=
+  childLL cell212
+
+private abbrev cell2121 : AngleCell :=
+  childLH cell212
+
+private abbrev cell2122 : AngleCell :=
+  childHL cell212
+
+private abbrev cell2123 : AngleCell :=
+  childHH cell212
+
+private abbrev cell2130 : AngleCell :=
+  childLL cell213
+
+private abbrev cell2131 : AngleCell :=
+  childLH cell213
+
+private abbrev cell2132 : AngleCell :=
+  childHL cell213
+
+private abbrev cell2133 : AngleCell :=
+  childHH cell213
+
+private abbrev cell2200 : AngleCell :=
+  childLL cell220
+
+private abbrev cell2201 : AngleCell :=
+  childLH cell220
+
+private abbrev cell2202 : AngleCell :=
+  childHL cell220
+
+private abbrev cell2203 : AngleCell :=
+  childHH cell220
+
+private abbrev cell2210 : AngleCell :=
+  childLL cell221
+
+private abbrev cell2211 : AngleCell :=
+  childLH cell221
+
+private abbrev cell2212 : AngleCell :=
+  childHL cell221
+
+private abbrev cell2213 : AngleCell :=
+  childHH cell221
+
+private abbrev cell2220 : AngleCell :=
+  childLL cell222
+
+private abbrev cell2221 : AngleCell :=
+  childLH cell222
+
+private abbrev cell2222 : AngleCell :=
+  childHL cell222
+
+private abbrev cell2223 : AngleCell :=
+  childHH cell222
+
+private abbrev cell2230 : AngleCell :=
+  childLL cell223
+
+private abbrev cell2231 : AngleCell :=
+  childLH cell223
+
+private abbrev cell2232 : AngleCell :=
+  childHL cell223
+
+private abbrev cell2233 : AngleCell :=
+  childHH cell223
+
+private abbrev cell2300 : AngleCell :=
+  childLL cell230
+
+private abbrev cell2301 : AngleCell :=
+  childLH cell230
+
+private abbrev cell2302 : AngleCell :=
+  childHL cell230
+
+private abbrev cell2303 : AngleCell :=
+  childHH cell230
+
+private abbrev cell2310 : AngleCell :=
+  childLL cell231
+
+private abbrev cell2311 : AngleCell :=
+  childLH cell231
+
+private abbrev cell2312 : AngleCell :=
+  childHL cell231
+
+private abbrev cell2313 : AngleCell :=
+  childHH cell231
+
+private abbrev cell2320 : AngleCell :=
+  childLL cell232
+
+private abbrev cell2321 : AngleCell :=
+  childLH cell232
+
+private abbrev cell2322 : AngleCell :=
+  childHL cell232
+
+private abbrev cell2323 : AngleCell :=
+  childHH cell232
+
+private abbrev cell2330 : AngleCell :=
+  childLL cell233
+
+private abbrev cell2331 : AngleCell :=
+  childLH cell233
+
+private abbrev cell2332 : AngleCell :=
+  childHL cell233
+
+private abbrev cell2333 : AngleCell :=
+  childHH cell233
+
+private abbrev cell3200 : AngleCell :=
+  childLL cell320
+
+private abbrev cell3201 : AngleCell :=
+  childLH cell320
+
+private abbrev cell3202 : AngleCell :=
+  childHL cell320
+
+private abbrev cell3203 : AngleCell :=
+  childHH cell320
+
+private abbrev cell3220 : AngleCell :=
+  childLL cell322
+
+private abbrev cell3221 : AngleCell :=
+  childLH cell322
+
+private abbrev cell3222 : AngleCell :=
+  childHL cell322
+
+private abbrev cell3223 : AngleCell :=
+  childHH cell322
+
+private abbrev cell3230 : AngleCell :=
+  childLL cell323
+
+private abbrev cell3231 : AngleCell :=
+  childLH cell323
+
+private abbrev cell3232 : AngleCell :=
+  childHL cell323
+
+private abbrev cell3233 : AngleCell :=
+  childHH cell323
+
+private abbrev cell23200 : AngleCell :=
+  childLL cell2320
+
+private abbrev cell23201 : AngleCell :=
+  childLH cell2320
+
+private abbrev cell23202 : AngleCell :=
+  childHL cell2320
+
+private abbrev cell23203 : AngleCell :=
+  childHH cell2320
+
+private abbrev cell23220 : AngleCell :=
+  childLL cell2322
+
+private abbrev cell23221 : AngleCell :=
+  childLH cell2322
+
+private abbrev cell23222 : AngleCell :=
+  childHL cell2322
+
+private abbrev cell23223 : AngleCell :=
+  childHH cell2322
+
+private theorem checked23200 : adaptiveCoverCheck 8 cell23200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell23200 (by decide +kernel)
+
+private theorem checked23201 : adaptiveCoverCheck 8 cell23201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell23201 (by decide +kernel)
+
+private theorem checked23202 : adaptiveCoverCheck 8 cell23202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell23202 (by decide +kernel)
+
+private theorem checked23203 : adaptiveCoverCheck 8 cell23203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell23203 (by decide +kernel)
+
+private theorem checked23220 : adaptiveCoverCheck 8 cell23220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell23220 (by decide +kernel)
+
+private theorem checked23221 : adaptiveCoverCheck 8 cell23221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell23221 (by decide +kernel)
+
+private theorem checked23222 : adaptiveCoverCheck 8 cell23222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell23222 (by decide +kernel)
+
+private theorem checked23223 : adaptiveCoverCheck 8 cell23223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell23223 (by decide +kernel)
+
+private theorem checked0000 : adaptiveCoverCheck 9 cell0000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0000 (by decide +kernel)
+
+private theorem checked0001 : adaptiveCoverCheck 9 cell0001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0001 (by decide +kernel)
+
+private theorem checked0002 : adaptiveCoverCheck 9 cell0002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0002 (by decide +kernel)
+
+private theorem checked0003 : adaptiveCoverCheck 9 cell0003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0003 (by decide +kernel)
+
+private theorem checked0020 : adaptiveCoverCheck 9 cell0020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0020 (by decide +kernel)
+
+private theorem checked0021 : adaptiveCoverCheck 9 cell0021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0021 (by decide +kernel)
+
+private theorem checked0022 : adaptiveCoverCheck 9 cell0022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0022 (by decide +kernel)
+
+private theorem checked0023 : adaptiveCoverCheck 9 cell0023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0023 (by decide +kernel)
+
+private theorem checked0200 : adaptiveCoverCheck 9 cell0200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0200 (by decide +kernel)
+
+private theorem checked0201 : adaptiveCoverCheck 9 cell0201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0201 (by decide +kernel)
+
+private theorem checked0202 : adaptiveCoverCheck 9 cell0202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0202 (by decide +kernel)
+
+private theorem checked0203 : adaptiveCoverCheck 9 cell0203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0203 (by decide +kernel)
+
+private theorem checked0210 : adaptiveCoverCheck 9 cell0210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0210 (by decide +kernel)
+
+private theorem checked0211 : adaptiveCoverCheck 9 cell0211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0211 (by decide +kernel)
+
+private theorem checked0212 : adaptiveCoverCheck 9 cell0212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0212 (by decide +kernel)
+
+private theorem checked0213 : adaptiveCoverCheck 9 cell0213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0213 (by decide +kernel)
+
+private theorem checked0220 : adaptiveCoverCheck 9 cell0220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0220 (by decide +kernel)
+
+private theorem checked0221 : adaptiveCoverCheck 9 cell0221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0221 (by decide +kernel)
+
+private theorem checked0222 : adaptiveCoverCheck 9 cell0222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0222 (by decide +kernel)
+
+private theorem checked0223 : adaptiveCoverCheck 9 cell0223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0223 (by decide +kernel)
+
+private theorem checked0230 : adaptiveCoverCheck 9 cell0230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0230 (by decide +kernel)
+
+private theorem checked0231 : adaptiveCoverCheck 9 cell0231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0231 (by decide +kernel)
+
+private theorem checked0232 : adaptiveCoverCheck 9 cell0232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0232 (by decide +kernel)
+
+private theorem checked0233 : adaptiveCoverCheck 9 cell0233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0233 (by decide +kernel)
+
+private theorem checked0320 : adaptiveCoverCheck 9 cell0320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0320 (by decide +kernel)
+
+private theorem checked0321 : adaptiveCoverCheck 9 cell0321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0321 (by decide +kernel)
+
+private theorem checked0322 : adaptiveCoverCheck 9 cell0322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0322 (by decide +kernel)
+
+private theorem checked0323 : adaptiveCoverCheck 9 cell0323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell0323 (by decide +kernel)
+
+private theorem checked2000 : adaptiveCoverCheck 9 cell2000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2000 (by decide +kernel)
+
+private theorem checked2001 : adaptiveCoverCheck 9 cell2001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2001 (by decide +kernel)
+
+private theorem checked2002 : adaptiveCoverCheck 9 cell2002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2002 (by decide +kernel)
+
+private theorem checked2003 : adaptiveCoverCheck 9 cell2003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2003 (by decide +kernel)
+
+private theorem checked2010 : adaptiveCoverCheck 9 cell2010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2010 (by decide +kernel)
+
+private theorem checked2011 : adaptiveCoverCheck 9 cell2011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2011 (by decide +kernel)
+
+private theorem checked2012 : adaptiveCoverCheck 9 cell2012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2012 (by decide +kernel)
+
+private theorem checked2013 : adaptiveCoverCheck 9 cell2013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2013 (by decide +kernel)
+
+private theorem checked2020 : adaptiveCoverCheck 9 cell2020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2020 (by decide +kernel)
+
+private theorem checked2021 : adaptiveCoverCheck 9 cell2021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2021 (by decide +kernel)
+
+private theorem checked2022 : adaptiveCoverCheck 9 cell2022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2022 (by decide +kernel)
+
+private theorem checked2023 : adaptiveCoverCheck 9 cell2023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2023 (by decide +kernel)
+
+private theorem checked2030 : adaptiveCoverCheck 9 cell2030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2030 (by decide +kernel)
+
+private theorem checked2031 : adaptiveCoverCheck 9 cell2031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2031 (by decide +kernel)
+
+private theorem checked2032 : adaptiveCoverCheck 9 cell2032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2032 (by decide +kernel)
+
+private theorem checked2033 : adaptiveCoverCheck 9 cell2033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2033 (by decide +kernel)
+
+private theorem checked2100 : adaptiveCoverCheck 9 cell2100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2100 (by decide +kernel)
+
+private theorem checked2101 : adaptiveCoverCheck 9 cell2101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2101 (by decide +kernel)
+
+private theorem checked2102 : adaptiveCoverCheck 9 cell2102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2102 (by decide +kernel)
+
+private theorem checked2103 : adaptiveCoverCheck 9 cell2103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2103 (by decide +kernel)
+
+private theorem checked2120 : adaptiveCoverCheck 9 cell2120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2120 (by decide +kernel)
+
+private theorem checked2121 : adaptiveCoverCheck 9 cell2121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2121 (by decide +kernel)
+
+private theorem checked2122 : adaptiveCoverCheck 9 cell2122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2122 (by decide +kernel)
+
+private theorem checked2123 : adaptiveCoverCheck 9 cell2123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2123 (by decide +kernel)
+
+private theorem checked2130 : adaptiveCoverCheck 9 cell2130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2130 (by decide +kernel)
+
+private theorem checked2131 : adaptiveCoverCheck 9 cell2131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2131 (by decide +kernel)
+
+private theorem checked2132 : adaptiveCoverCheck 9 cell2132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2132 (by decide +kernel)
+
+private theorem checked2133 : adaptiveCoverCheck 9 cell2133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2133 (by decide +kernel)
+
+private theorem checked2200 : adaptiveCoverCheck 9 cell2200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2200 (by decide +kernel)
+
+private theorem checked2201 : adaptiveCoverCheck 9 cell2201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2201 (by decide +kernel)
+
+private theorem checked2202 : adaptiveCoverCheck 9 cell2202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2202 (by decide +kernel)
+
+private theorem checked2203 : adaptiveCoverCheck 9 cell2203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2203 (by decide +kernel)
+
+private theorem checked2210 : adaptiveCoverCheck 9 cell2210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2210 (by decide +kernel)
+
+private theorem checked2211 : adaptiveCoverCheck 9 cell2211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2211 (by decide +kernel)
+
+private theorem checked2212 : adaptiveCoverCheck 9 cell2212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2212 (by decide +kernel)
+
+private theorem checked2213 : adaptiveCoverCheck 9 cell2213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2213 (by decide +kernel)
+
+private theorem checked2220 : adaptiveCoverCheck 9 cell2220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2220 (by decide +kernel)
+
+private theorem checked2221 : adaptiveCoverCheck 9 cell2221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2221 (by decide +kernel)
+
+private theorem checked2222 : adaptiveCoverCheck 9 cell2222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2222 (by decide +kernel)
+
+private theorem checked2223 : adaptiveCoverCheck 9 cell2223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2223 (by decide +kernel)
+
+private theorem checked2230 : adaptiveCoverCheck 9 cell2230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2230 (by decide +kernel)
+
+private theorem checked2231 : adaptiveCoverCheck 9 cell2231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2231 (by decide +kernel)
+
+private theorem checked2232 : adaptiveCoverCheck 9 cell2232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2232 (by decide +kernel)
+
+private theorem checked2233 : adaptiveCoverCheck 9 cell2233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2233 (by decide +kernel)
+
+private theorem checked2300 : adaptiveCoverCheck 9 cell2300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2300 (by decide +kernel)
+
+private theorem checked2301 : adaptiveCoverCheck 9 cell2301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2301 (by decide +kernel)
+
+private theorem checked2302 : adaptiveCoverCheck 9 cell2302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2302 (by decide +kernel)
+
+private theorem checked2303 : adaptiveCoverCheck 9 cell2303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2303 (by decide +kernel)
+
+private theorem checked2310 : adaptiveCoverCheck 9 cell2310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2310 (by decide +kernel)
+
+private theorem checked2311 : adaptiveCoverCheck 9 cell2311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2311 (by decide +kernel)
+
+private theorem checked2312 : adaptiveCoverCheck 9 cell2312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2312 (by decide +kernel)
+
+private theorem checked2313 : adaptiveCoverCheck 9 cell2313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2313 (by decide +kernel)
+
+private theorem checked2320 : adaptiveCoverCheck 9 cell2320 = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cell2320
+    checked23200 checked23201 checked23202 checked23203
+
+private theorem checked2321 : adaptiveCoverCheck 9 cell2321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2321 (by decide +kernel)
+
+private theorem checked2322 : adaptiveCoverCheck 9 cell2322 = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cell2322
+    checked23220 checked23221 checked23222 checked23223
+
+private theorem checked2323 : adaptiveCoverCheck 9 cell2323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2323 (by decide +kernel)
+
+private theorem checked2330 : adaptiveCoverCheck 9 cell2330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2330 (by decide +kernel)
+
+private theorem checked2331 : adaptiveCoverCheck 9 cell2331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2331 (by decide +kernel)
+
+private theorem checked2332 : adaptiveCoverCheck 9 cell2332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2332 (by decide +kernel)
+
+private theorem checked2333 : adaptiveCoverCheck 9 cell2333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell2333 (by decide +kernel)
+
+private theorem checked3200 : adaptiveCoverCheck 9 cell3200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3200 (by decide +kernel)
+
+private theorem checked3201 : adaptiveCoverCheck 9 cell3201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3201 (by decide +kernel)
+
+private theorem checked3202 : adaptiveCoverCheck 9 cell3202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3202 (by decide +kernel)
+
+private theorem checked3203 : adaptiveCoverCheck 9 cell3203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3203 (by decide +kernel)
+
+private theorem checked3220 : adaptiveCoverCheck 9 cell3220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3220 (by decide +kernel)
+
+private theorem checked3221 : adaptiveCoverCheck 9 cell3221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3221 (by decide +kernel)
+
+private theorem checked3222 : adaptiveCoverCheck 9 cell3222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3222 (by decide +kernel)
+
+private theorem checked3223 : adaptiveCoverCheck 9 cell3223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3223 (by decide +kernel)
+
+private theorem checked3230 : adaptiveCoverCheck 9 cell3230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3230 (by decide +kernel)
+
+private theorem checked3231 : adaptiveCoverCheck 9 cell3231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3231 (by decide +kernel)
+
+private theorem checked3232 : adaptiveCoverCheck 9 cell3232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3232 (by decide +kernel)
+
+private theorem checked3233 : adaptiveCoverCheck 9 cell3233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 9 cell3233 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 10 cell000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell000
+    checked0000 checked0001 checked0002 checked0003
+
+private theorem checked001 : adaptiveCoverCheck 10 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 10 cell002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell002
+    checked0020 checked0021 checked0022 checked0023
+
+private theorem checked003 : adaptiveCoverCheck 10 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 10 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 10 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 10 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 10 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 10 cell020 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell020
+    checked0200 checked0201 checked0202 checked0203
+
+private theorem checked021 : adaptiveCoverCheck 10 cell021 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell021
+    checked0210 checked0211 checked0212 checked0213
+
+private theorem checked022 : adaptiveCoverCheck 10 cell022 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell022
+    checked0220 checked0221 checked0222 checked0223
+
+private theorem checked023 : adaptiveCoverCheck 10 cell023 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell023
+    checked0230 checked0231 checked0232 checked0233
+
+private theorem checked030 : adaptiveCoverCheck 10 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 10 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 10 cell032 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell032
+    checked0320 checked0321 checked0322 checked0323
+
+private theorem checked033 : adaptiveCoverCheck 10 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 10 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 10 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 10 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 10 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell103 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 10 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 10 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 10 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 10 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 10 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 10 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 10 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 10 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 10 cell200 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell200
+    checked2000 checked2001 checked2002 checked2003
+
+private theorem checked201 : adaptiveCoverCheck 10 cell201 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell201
+    checked2010 checked2011 checked2012 checked2013
+
+private theorem checked202 : adaptiveCoverCheck 10 cell202 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell202
+    checked2020 checked2021 checked2022 checked2023
+
+private theorem checked203 : adaptiveCoverCheck 10 cell203 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell203
+    checked2030 checked2031 checked2032 checked2033
+
+private theorem checked210 : adaptiveCoverCheck 10 cell210 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell210
+    checked2100 checked2101 checked2102 checked2103
+
+private theorem checked211 : adaptiveCoverCheck 10 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 10 cell212 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell212
+    checked2120 checked2121 checked2122 checked2123
+
+private theorem checked213 : adaptiveCoverCheck 10 cell213 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell213
+    checked2130 checked2131 checked2132 checked2133
+
+private theorem checked220 : adaptiveCoverCheck 10 cell220 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell220
+    checked2200 checked2201 checked2202 checked2203
+
+private theorem checked221 : adaptiveCoverCheck 10 cell221 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell221
+    checked2210 checked2211 checked2212 checked2213
+
+private theorem checked222 : adaptiveCoverCheck 10 cell222 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell222
+    checked2220 checked2221 checked2222 checked2223
+
+private theorem checked223 : adaptiveCoverCheck 10 cell223 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell223
+    checked2230 checked2231 checked2232 checked2233
+
+private theorem checked230 : adaptiveCoverCheck 10 cell230 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell230
+    checked2300 checked2301 checked2302 checked2303
+
+private theorem checked231 : adaptiveCoverCheck 10 cell231 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell231
+    checked2310 checked2311 checked2312 checked2313
+
+private theorem checked232 : adaptiveCoverCheck 10 cell232 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell232
+    checked2320 checked2321 checked2322 checked2323
+
+private theorem checked233 : adaptiveCoverCheck 10 cell233 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell233
+    checked2330 checked2331 checked2332 checked2333
+
+private theorem checked300 : adaptiveCoverCheck 10 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 10 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 10 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 10 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 10 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 10 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 10 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 10 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 10 cell320 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell320
+    checked3200 checked3201 checked3202 checked3203
+
+private theorem checked321 : adaptiveCoverCheck 10 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 10 cell322 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell322
+    checked3220 checked3221 checked3222 checked3223
+
+private theorem checked323 : adaptiveCoverCheck 10 cell323 = true := by
+  exact adaptiveCoverCheck_succ_of_children 9 cell323
+    checked3230 checked3231 checked3232 checked3233
+
+private theorem checked330 : adaptiveCoverCheck 10 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 10 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 10 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 10 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 11 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 11 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 11 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 11 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 11 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 11 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 11 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 11 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 11 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 11 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 11 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 11 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 11 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 11 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 11 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 11 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 12 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 11 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 12 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 11 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 12 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 11 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 12 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 11 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 13 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 12 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee18e89e753
+
+theorem e24PhiBelowKernelLH :
+    adaptiveCoverCheck 13 (childLH e24PhiBelowRoot) = true := by
+  exact CoverCertificatee18e89e753.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24 kernel child certificate: PhiBelow/LL, remaining depth 13. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+namespace CoverCertificate4615b9c889
+
+private abbrev cellRoot : AngleCell :=
+  (childLL e24PhiBelowRoot)
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+private theorem checked110 : adaptiveCoverCheck 10 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 10 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 10 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 10 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 10 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 10 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 10 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 10 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 10 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 10 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 10 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 10 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell133 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 10 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 10 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 10 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 10 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 10 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 10 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 10 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 10 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 10 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 10 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 10 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 10 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 10 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 10 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 10 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 10 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 10 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 11 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 11 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 11 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 11 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 11 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 11 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 11 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 11 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 11 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 11 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 11 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 11 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 11 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 11 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 11 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 11 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 11 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 10 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 12 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 11 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 12 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 11 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 12 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 11 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 12 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 11 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 13 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 12 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4615b9c889
+
+theorem e24PhiBelowKernelLL :
+    adaptiveCoverCheck 13 (childLL e24PhiBelowRoot) = true := by
+  exact CoverCertificate4615b9c889.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle008.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle008.lean
new file mode 100644
index 0000000000..21f4ca8b07
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/Core/Bundle008.lean
@@ -0,0 +1,868 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle006
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle032
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle002
+public import Mathlib.Analysis.Normed.Affine.ContinuousAffineMap
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartF.Semantics.Batch002`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartF.F01ExistingMotion`.
+* `KernelOnly.PartF.F01ParameterChoice`.
+* `KernelOnly.PartF.F02EuclideanMotion`.
+* `KernelOnly.PartF.F03CanonicalRotation`.
+* `KernelOnly.PartF.F03SetIdentification`.
+* `KernelOnly.PartF.F04IntegralRepresentation`.
+* `KernelOnly.PartF.F06ParameterIdentification`.
+* `KernelOnly.PartF.F06IntegralMotion`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# F01: exact identity normalization of the already certified motion
+
+The original project asks for an initial translation. For its concrete
+Gerver path the inverse frame is exactly the identity, which is the
+stronger normalization required by DeepMind. These statements refer to
+the existing `ℝ × ℝ` coordinate model, not to an unproved isometry between
+the product norm and the Euclidean norm.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartF.Existing
+
+/-- Invert the hallway frame to obtain the sofa’s rigid motion. -/
+def motion (s : ℝ) : SE2 := (PartC.frame s).inv
+
+theorem motion_zero : motion 0 = SE2.one := by
+  change (Romik.frame PartC.params 0).inv = SE2.one
+  simp [Romik.frame, Romik.angle, PartC.pathZero, SE2.inv, SE2.one]
+
+theorem motion_continuous : SE2.ContinuousPath motion :=
+  SE2.continuousPath_inv PartC.frameContinuous
+
+theorem initial : PartC.G ⊆ horizontalArm := by
+  intro q hq
+  have h := PartC.initialArm ⟨q, hq, rfl⟩
+  change (motion 0).act q ∈ horizontalArm at h
+  rw [motion_zero, SE2.one_act] at h
+  exact h
+
+theorem all_times :
+    ∀ s ∈ Set.Icc (0 : ℝ) 1, (motion s).act '' PartC.G ⊆ hallway := by
+  intro s hs
+  exact inv_image_subset_hallway_of_subset_supporting
+    (PartC.frame s) PartC.G (PartC.G_subset_hallway s hs)
+
+theorem final : (motion 1).act '' PartC.G ⊆ verticalArm := PartC.finalArm
+
+end GerverSofa.PartF.Existing
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# F01: parameter selection is independent of the existence proof
+
+This module reuses the frozen E24KC6 theorem. It does not repeat numerical
+exclusion. The literal upstream order of conjunctions is checked against
+the existing specification, and its selected tuple is identified with the
+certified reduced solution. No identification with the full 22D tuple is
+claimed by this module.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartF.Parameters
+
+/-- The nonnegative parameter and ordered-angle equations used by the upstream canonical
+definition. -/
+def upstreamSpec (A B φ θ : ℝ) : Prop :=
+  0 ≤ φ ∧ φ ≤ θ ∧ θ ≤ Real.pi / 4 ∧ 0 ≤ A ∧ 0 ≤ B ∧
+  A * (θ.cos - φ.cos) - 2 * B * φ.sin
+    + (θ - φ - 1) * θ.cos - θ.sin + φ.cos + φ.sin = 0 ∧
+  A * (3 * θ.sin + φ.sin) - 2 * B * φ.cos
+    + 3 * (θ - φ - 1) * θ.sin + 3 * θ.cos - φ.sin + φ.cos = 0 ∧
+  A * φ.cos - (φ.sin + 1 / 2 - φ.cos / 2 + B * φ.sin) = 0 ∧
+  (A + Real.pi / 2 - φ - θ) -
+    (B - (θ - φ) * (1 + A) / 2 - (θ - φ)^2 / 4) = 0
+
+theorem upstreamSpec_iff (A B φ θ : ℝ) :
+    upstreamSpec A B φ θ ↔ PartE.DeepMindABPhiThetaSpec A B φ θ := by
+  unfold upstreamSpec PartE.DeepMindABPhiThetaSpec PartE.PhysicalDomain
+    PartE.reducedParams PartE.DeepMindEquations
+  dsimp only
+  tauto
+
+/-- Package the upstream four-parameter specification as a predicate on a nested tuple. -/
+def TupleSpec (q : ℝ × ℝ × ℝ × ℝ) : Prop :=
+  upstreamSpec q.1 q.2.1 q.2.2.1 q.2.2.2
+
+theorem existsUnique : ∃! q, TupleSpec q := by
+  simpa only [TupleSpec, upstreamSpec_iff] using PartE.deepMindABPhiTheta_existsUnique
+
+/-- The certified reduced solution converted to the upstream four-parameter tuple. -/
+def certified : ℝ × ℝ × ℝ × ℝ :=
+  PartE.tupleEquiv.symm PartALeanCert.reducedCertifiedUniqueSolution.solution
+
+theorem certified_spec : TupleSpec certified := by
+  apply (upstreamSpec_iff _ _ _ _).2
+  exact PartE.deepMindSpec_of_mem_reducedBox_and_equations
+    PartALeanCert.reducedCertifiedUniqueSolution.solution_mem
+    PartALeanCert.reducedCertifiedUniqueSolution.satisfies
+
+theorem any_solution_eq_certified (q : ℝ × ℝ × ℝ × ℝ) (hq : TupleSpec q) :
+    q = certified := existsUnique.unique hq certified_spec
+
+/-- This applies to any eventual upstream proof of the *same* specification. -/
+theorem choice_independent (h : ∃! q, TupleSpec q) : h.choose = certified :=
+  any_solution_eq_certified h.choose h.choose_spec.1
+
+end GerverSofa.PartF.Parameters
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# F02: the certified sofa as a set with Euclidean rigid motion
+
+Every existing `SE2` action is realized by an affine isometry of `Plane`.
+Norm preservation is proved in `Plane` from the equation `c*c+s*s=1`.
+Continuity is proved in the induced continuous-affine-map topology of F01.
+The final theorem transfers all seven motion fields for the already
+certified set, without adding geometric hypotheses.
+
+This module does not identify the set with the literal upstream integral
+definition. Its rotation matrices are explicit; their comparison with
+`F01.rotation` and the integral representation are separate bridge steps.
+-/
+
+@[expose] public section
+
+noncomputable section
+open scoped unitInterval
+namespace GerverSofa.PartF.EuclideanMotion
+
+open Coordinates
+
+/-- The standard rotation matrix associated with an existing `SE2` value. -/
+def linearPart (g : SE2) : Plane ≃ₗ[ℝ] Plane where
+  toFun := fun q => toPlane (g.c * q 0 - g.s * q 1, g.s * q 0 + g.c * q 1)
+  invFun := fun q => toPlane (g.c * q 0 + g.s * q 1, -g.s * q 0 + g.c * q 1)
+  left_inv := by
+    intro q
+    apply plane_ext
+    · change g.c * (g.c * q 0 - g.s * q 1) +
+        g.s * (g.s * q 0 + g.c * q 1) = q 0
+      linear_combination (q 0) * g.unit
+    · change -g.s * (g.c * q 0 - g.s * q 1) +
+        g.c * (g.s * q 0 + g.c * q 1) = q 1
+      linear_combination (q 1) * g.unit
+  right_inv := by
+    intro q
+    apply plane_ext
+    · change g.c * (g.c * q 0 + g.s * q 1) -
+        g.s * (-g.s * q 0 + g.c * q 1) = q 0
+      linear_combination (q 0) * g.unit
+    · change g.s * (g.c * q 0 + g.s * q 1) +
+        g.c * (-g.s * q 0 + g.c * q 1) = q 1
+      linear_combination (q 1) * g.unit
+  map_add' := by
+    intro q r
+    apply plane_ext
+    · change g.c * (q 0 + r 0) - g.s * (q 1 + r 1) =
+        (g.c * q 0 - g.s * q 1) + (g.c * r 0 - g.s * r 1)
+      ring
+    · change g.s * (q 0 + r 0) + g.c * (q 1 + r 1) =
+        (g.s * q 0 + g.c * q 1) + (g.s * r 0 + g.c * r 1)
+      ring
+  map_smul' := by
+    intro c q
+    apply plane_ext
+    · change g.c * (c * q 0) - g.s * (c * q 1) = c * (g.c * q 0 - g.s * q 1)
+      ring
+    · change g.s * (c * q 0) + g.c * (c * q 1) = c * (g.s * q 0 + g.c * q 1)
+      ring
+
+@[simp] theorem linearPart_zero_coord (g : SE2) (q : Plane) :
+    linearPart g q 0 = g.c * q 0 - g.s * q 1 := rfl
+
+@[simp] theorem linearPart_one_coord (g : SE2) (q : Plane) :
+    linearPart g q 1 = g.s * q 0 + g.c * q 1 := rfl
+
+theorem linearPart_norm_sq (g : SE2) (q : Plane) :
+    ‖linearPart g q‖ ^ 2 = ‖q‖ ^ 2 := by
+  rw [EuclideanSpace.real_norm_sq_eq, EuclideanSpace.real_norm_sq_eq]
+  simp only [Fin.sum_univ_two]
+  change (g.c * q 0 - g.s * q 1) ^ 2 + (g.s * q 0 + g.c * q 1) ^ 2 =
+    (q 0) ^ 2 + (q 1) ^ 2
+  linear_combination ((q 0) ^ 2 + (q 1) ^ 2) * g.unit
+
+theorem linearPart_norm (g : SE2) (q : Plane) : ‖linearPart g q‖ = ‖q‖ :=
+  (sq_eq_sq₀ (norm_nonneg _) (norm_nonneg _)).mp (linearPart_norm_sq g q)
+
+/-- The linear isometry determined by the rotation coefficients of an SE2 motion. -/
+def linearIsometry (g : SE2) : Plane ≃ₗᵢ[ℝ] Plane where
+  toLinearEquiv := linearPart g
+  norm_map' := linearPart_norm g
+
+/-- A genuine Euclidean affine isometry with the original coordinate action. -/
+def ofSE2 (g : SE2) : Rigid :=
+  rotateThenTranslate (linearIsometry g) (toPlane (g.tx, g.ty))
+
+theorem ofSE2_apply (g : SE2) (q : Plane) :
+    ofSE2 g q = toPlane (g.act (fromPlane q)) := by
+  exact plane_ext rfl rfl
+
+theorem ofSE2_apply_toPlane (g : SE2) (q : Point) :
+    ofSE2 g (toPlane q) = toPlane (g.act q) := by
+  rw [ofSE2_apply, fromPlane_toPlane]
+
+theorem ofSE2_one : ofSE2 SE2.one = AffineIsometryEquiv.refl ℝ Plane := by
+  apply AffineIsometryEquiv.ext
+  intro q
+  change ofSE2 SE2.one q = q
+  rw [ofSE2_apply, SE2.one_act, toPlane_fromPlane]
+
+theorem image_action (g : SE2) (S : Set Point) :
+    ofSE2 g '' (toPlane '' S) = toPlane '' (g.act '' S) := by
+  simp only [Set.image_image, ofSE2_apply_toPlane]
+
+/-- The coordinate quarter-turn, used to express continuous matrix coefficients. -/
+def quarterTurnLinear : Plane →ₗ[ℝ] Plane where
+  toFun := fun q => toPlane (-q 1, q 0)
+  map_add' := by
+    intro q r
+    apply plane_ext
+    · change -(q 1 + r 1) = -q 1 + -r 1
+      ring
+    · rfl
+  map_smul' := by
+    intro c q
+    apply plane_ext
+    · change -(c * q 1) = c * (-q 1)
+      ring
+    · rfl
+
+/-- The quarter-turn rotation bundled as a continuous linear map. -/
+def quarterTurn : Plane →L[ℝ] Plane := quarterTurnLinear.toContinuousLinearMap
+
+/-- Componentwise continuous `SE2` coefficients give continuity in the
+actual topology on affine isometry equivalences. -/
+theorem continuous_ofSE2 {X : Type*} [TopologicalSpace X]
+    (g : X → SE2)
+    (hc : Continuous (fun t => (g t).c))
+    (hs : Continuous (fun t => (g t).s))
+    (htx : Continuous (fun t => (g t).tx))
+    (hty : Continuous (fun t => (g t).ty)) :
+    Continuous (fun t => ofSE2 (g t)) := by
+  let lin : X → (Plane →L[ℝ] Plane) := fun t =>
+    (g t).c • ContinuousLinearMap.id ℝ Plane + (g t).s • quarterTurn
+  let offset : X → Plane := fun t => toPlane ((g t).tx, (g t).ty)
+  let D := ContinuousAffineMap.decompLinearIsometryEquiv ℝ ℝ Plane Plane
+  have hlin : Continuous lin :=
+    (hc.smul continuous_const).add (hs.smul continuous_const)
+  have hoffset : Continuous offset := continuous_toPlane.comp (htx.prodMk hty)
+  have hD : Continuous (fun t => D.symm (offset t, lin t)) :=
+    D.symm.continuous.comp (hoffset.prodMk hlin)
+  apply continuous_induced_rng.mpr
+  change Continuous (fun t => (ofSE2 (g t)).toAffineIsometry.toContinuousAffineMap)
+  have heq :
+      (fun t => (ofSE2 (g t)).toAffineIsometry.toContinuousAffineMap) =
+      (fun t => D.symm (offset t, lin t)) := by
+    funext t
+    apply ContinuousAffineMap.ext
+    intro q
+    change ofSE2 (g t) q = D.symm (offset t, lin t) q
+    dsimp only [D]
+    rw [ContinuousAffineMap.decompLinearIsometryEquiv_symm_apply, ofSE2_apply]
+    apply plane_ext
+    · change (g t).c * q 0 - (g t).s * q 1 + (g t).tx =
+        (g t).c * q 0 + (g t).s * (-q 1) + (g t).tx
+      ring
+    · change (g t).s * q 0 + (g t).c * q 1 + (g t).ty =
+        (g t).c * q 1 + (g t).s * q 0 + (g t).ty
+      ring
+  rw [heq]
+  exact hD
+
+theorem continuous_ofSE2_path (g : ℝ → SE2) (hg : SE2.ContinuousPath g) :
+    Continuous (fun t => ofSE2 (g t)) := by
+  rcases hg with ⟨hc, hs, htx, hty⟩
+  exact continuous_ofSE2 g hc hs htx hty
+
+/-- The coordinate image of the certified Gerver sofa in the Euclidean plane. -/
+def sofa : Set Plane := toPlane '' PartC.G
+
+/-- The certified sofa motion as affine isometries indexed by the unit interval. -/
+def motion (s : I) : Rigid := ofSE2 (Existing.motion (s : ℝ))
+
+theorem motion_continuous : Continuous motion :=
+  (continuous_ofSE2_path Existing.motion Existing.motion_continuous).comp continuous_subtype_val
+
+/-- All seven motion requirements hold for the coordinate image of the
+already certified Gerver set. No additional geometric assumptions occur. -/
+theorem concrete_movingSofa : Model.IsMovingSofa sofa motion := by
+  refine ⟨?_, ?_, motion_continuous, ?_, ?_, ?_, ?_⟩
+  · exact isConnected_image PartC.Stage4.globalGeometryCertificate.connected
+  · exact isClosed_image PartC.closed_G
+  · change ofSE2 (Existing.motion 0) = AffineIsometryEquiv.refl ℝ Plane
+    rw [Existing.motion_zero, ofSE2_one]
+  · change toPlane '' PartC.G ⊆ Model.horizontalHallway
+    rw [← horizontal_image]
+    exact Set.image_mono Existing.initial
+  · intro s
+    change ofSE2 (Existing.motion (s : ℝ)) '' (toPlane '' PartC.G) ⊆ Model.hallway
+    rw [image_action, ← hallway_image]
+    exact Set.image_mono (Existing.all_times (s : ℝ) s.property)
+  · change ofSE2 (Existing.motion 1) '' (toPlane '' PartC.G) ⊆ Model.verticalHallway
+    rw [image_action, ← vertical_image]
+    exact Set.image_mono Existing.final
+
+end GerverSofa.PartF.EuclideanMotion
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# F03: the upstream-oriented rotation in the certified coordinates
+
+The orientation is the ordered coordinate-basis orientation installed in F01,
+with the same definition as the pinned upstream plane helper. We compute its
+area form, then its right-angle rotation, before comparing rotation matrices.
+No replacement orientation or additional orientation hypothesis is introduced.
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace GerverSofa.PartF.CanonicalRotation
+
+open Coordinates EuclideanMotion
+
+/-- The sign is fixed by the ordered orthonormal coordinate basis. -/
+theorem areaForm_coordinates (q r : Plane) :
+    EuclideanGeometry.o.areaForm q r = q 0 * r 1 - q 1 * r 0 := by
+  rw [Orientation.areaForm_to_volumeForm,
+    EuclideanGeometry.o.volumeForm_robust (EuclideanSpace.basisFun (Fin 2) ℝ) rfl,
+    Module.Basis.det_apply]
+  simp [Module.Basis.toMatrix, Matrix.det_fin_two, mul_comm]
+
+/-- The positive quarter-turn is exactly `(x,y) ↦ (-y,x)`. -/
+theorem rightAngleRotation_apply (q : Plane) :
+    EuclideanGeometry.o.rightAngleRotation q = toPlane (-q 1, q 0) := by
+  apply plane_ext
+  · have h := EuclideanGeometry.o.inner_rightAngleRotation_left q
+      (EuclideanSpace.basisFun (Fin 2) ℝ 0)
+    rw [EuclideanSpace.inner_basisFun_real, areaForm_coordinates] at h
+    simpa [EuclideanSpace.basisFun_apply, PiLp.single_apply] using h
+  · have h := EuclideanGeometry.o.inner_rightAngleRotation_left q
+      (EuclideanSpace.basisFun (Fin 2) ℝ 1)
+    rw [EuclideanSpace.inner_basisFun_real, areaForm_coordinates] at h
+    simpa [EuclideanSpace.basisFun_apply, PiLp.single_apply] using h
+
+theorem rotation_apply (t : ℝ) (q : Plane) :
+    rotation t q = toPlane (Romik.rot t (fromPlane q)) := by
+  change EuclideanGeometry.o.rotation (t : Real.Angle) q = _
+  rw [Orientation.rotation_apply, rightAngleRotation_apply]
+  simp only [Real.Angle.cos_coe, Real.Angle.sin_coe]
+  apply plane_ext
+  · change Real.cos t * q 0 + Real.sin t * (-q 1) =
+      Real.cos t * q 0 - Real.sin t * q 1
+    ring
+  · change Real.cos t * q 1 + Real.sin t * q 0 =
+      Real.sin t * q 0 + Real.cos t * q 1
+    ring
+
+theorem rotation_toPlane (t : ℝ) (q : Point) :
+    rotation t (toPlane q) = toPlane (Romik.rot t q) := by
+  rw [rotation_apply, fromPlane_toPlane]
+
+theorem linearIsometry_eq_rotation (g : SE2) (t : ℝ)
+    (hc : g.c = Real.cos t) (hs : g.s = Real.sin t) :
+    linearIsometry g = rotation t := by
+  apply LinearIsometryEquiv.ext
+  intro q
+  rw [rotation_apply]
+  change toPlane (g.c * q 0 - g.s * q 1, g.s * q 0 + g.c * q 1) =
+    toPlane (Real.cos t * q 0 - Real.sin t * q 1,
+      Real.sin t * q 0 + Real.cos t * q 1)
+  rw [hc, hs]
+
+theorem ofSE2_eq_worldFrame (g : SE2) (t : ℝ)
+    (hc : g.c = Real.cos t) (hs : g.s = Real.sin t) :
+    ofSE2 g = worldFrame t (toPlane (g.tx, g.ty)) := by
+  unfold ofSE2 worldFrame
+  rw [linearIsometry_eq_rotation g t hc hs]
+
+/-- The physical angle is `s * (π/2)`, including both endpoints. -/
+theorem romikFrame_eq_worldFrame (p : Romik.Params) (s : ℝ) :
+    ofSE2 (Romik.frame p s) =
+      worldFrame (Romik.angle s) (toPlane (Romik.path p (Romik.angle s))) := by
+  exact ofSE2_eq_worldFrame (Romik.frame p s) (Romik.angle s) rfl rfl
+
+theorem romikFrame_zero_act (p : Romik.Params)
+    (hzero : Romik.path p 0 = (0, 0)) (q : Point) :
+    (Romik.frame p 0).act q = q := by
+  simp [Romik.frame, Romik.angle, hzero, SE2.act]
+
+/-- The already proved five-phase algebra now uses the canonical rotation. -/
+theorem closedPath_rotation (d : Reduced.Params)
+    (hd : Reduced.Equations d) (t : ℝ) :
+    rotation t (toPlane (Phases.closedPath d t)) =
+      toPlane (Romik.path (Phases.dictionary d) t) := by
+  rw [rotation_toPlane, Phases.fivePhaseRepresentation d hd t]
+
+end GerverSofa.PartF.CanonicalRotation
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# F03: the certified set as a canonical hallway intersection
+
+We transport the existing reconstruction theorem, including the endpoint arms,
+to the Euclidean world-frame convention. The final conditional interfaces state
+the outstanding path/dictionary equalities explicitly. They do not prove the
+literal integral representation or identify the certified 22D tuple.
+-/
+
+@[expose] public section
+
+noncomputable section
+open scoped unitInterval
+namespace GerverSofa.PartF.SetIdentification
+
+open Coordinates EuclideanMotion CanonicalRotation
+
+theorem mem_coordinate_image (S : Set Point) (q : Plane) :
+    q ∈ toPlane '' S ↔ fromPlane q ∈ S := by
+  rw [image_eq_preimage]
+  rfl
+
+theorem mem_ofSE2_image (g : SE2) (S : Set Point) (q : Plane) :
+    q ∈ ofSE2 g '' (toPlane '' S) ↔ fromPlane q ∈ g.act '' S := by
+  rw [image_action, mem_coordinate_image]
+
+/-- The change from real unit time to subtype unit time loses no constraint. -/
+theorem reconstructedSet_image (p : Romik.Params)
+    (hzero : Romik.path p 0 = (0, 0)) :
+    toPlane '' Romik.reconstructedSet p =
+      frameIntersection (fun s : I => ofSE2 (Romik.frame p (s : ℝ)))
+        Model.horizontalHallway Model.verticalHallway Model.hallway := by
+  have hstart : (Romik.frame p 0).act '' horizontalArm = horizontalArm := by
+    have hfun : (Romik.frame p 0).act = id :=
+      funext (romikFrame_zero_act p hzero)
+    rw [hfun, Set.image_id]
+  ext q
+  simp only [frameIntersection, ← horizontal_image, ← vertical_image,
+    ← hallway_image, Set.mem_inter_iff, Set.mem_iInter,
+    mem_ofSE2_image, mem_coordinate_image]
+  change (fromPlane q ∈ horizontalArm ∧
+      (∀ s ∈ Set.Icc (0 : ℝ) 1,
+        fromPlane q ∈ (Romik.frame p s).act '' hallway) ∧
+      fromPlane q ∈ (Romik.frame p 1).act '' verticalArm) ↔
+    (fromPlane q ∈ (Romik.frame p 0).act '' horizontalArm ∧
+      fromPlane q ∈ (Romik.frame p 1).act '' verticalArm) ∧
+      ∀ s : I, fromPlane q ∈ (Romik.frame p (s : ℝ)).act '' hallway
+  rw [hstart]
+  constructor
+  · rintro ⟨hi, hall, hf⟩
+    exact ⟨⟨hi, hf⟩, fun s => hall (s : ℝ) s.property⟩
+  · rintro ⟨⟨hi, hf⟩, hall⟩
+    exact ⟨hi, fun s hs => hall ⟨s, hs⟩, hf⟩
+
+theorem reconstructedSet_image_eq_worldSofa (p : Romik.Params)
+    (hzero : Romik.path p 0 = (0, 0)) :
+    toPlane '' Romik.reconstructedSet p =
+      worldSofa (fun t => toPlane (Romik.path p t))
+        Model.horizontalHallway Model.verticalHallway Model.hallway := by
+  rw [reconstructedSet_image p hzero, worldSofa, angleIntersection_eq_frameIntersection]
+  have hframes :
+      (fun s : I => ofSE2 (Romik.frame p (s : ℝ))) =
+      (fun s : I => worldFrame ((s : ℝ) * (Real.pi / 2))
+        (toPlane (Romik.path p ((s : ℝ) * (Real.pi / 2))))) := by
+    funext s
+    exact romikFrame_eq_worldFrame p (s : ℝ)
+  exact congrArg (fun F => frameIntersection F
+    Model.horizontalHallway Model.verticalHallway Model.hallway) hframes
+
+/-- An unconditional identification of the concrete, already certified set. -/
+theorem sofa_eq_worldSofa :
+    EuclideanMotion.sofa =
+      worldSofa (fun t => toPlane (Romik.path PartC.params t))
+        Model.horizontalHallway Model.verticalHallway Model.hallway := by
+  change toPlane '' PartC.G = _
+  rw [PartC.G_eq_Sx]
+  exact reconstructedSet_image_eq_worldSofa PartC.params PartC.pathZero
+
+end GerverSofa.PartF.SetIdentification
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# F04: the literal integral path equals the closed five-phase path
+
+All integral identities are proved in F04IntegralEvaluation. Reflection of the
+closed intervals then gives the same five branches and the same endpoint choices
+as F01. The final identification with PartC.params still requires the 22D
+dictionary theorem; it remains an explicit hypothesis in the two last results.
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace GerverSofa.PartF.Integrals
+
+open Phases Coordinates
+
+theorem path_first_of_le (d : Reduced.Params) (t : ℝ) (ht : t ≤ d.phi) :
+    (path d t).1 = Real.cos t - 1 := by
+  simp only [path, ite_eq_left ht]
+
+theorem path_first_of_gt (d : Reduced.Params) (t : ℝ) (ht : d.phi < t) :
+    (path d t).1 = W d (T - t) - 1 := by
+  simp only [path, ite_eq_right (not_le.mpr ht)]
+  unfold W
+  dsimp only [T]
+  rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+
+theorem path_second_of_le (d : Reduced.Params) (ho : Ordered d)
+    (hd : Reduced.Equations d) (t : ℝ) (ht : t ≤ tau d) :
+    (path d t).2 = W d t - ell d * Real.sin t - 1 := by
+  simp only [path, ite_eq_left ht]
+  rw [xi_zero d ho hd]
+  unfold W ell
+  ring
+
+theorem path_second_of_gt (d : Reduced.Params) (ho : Ordered d)
+    (hd : Reduced.Equations d) (t : ℝ) (ht : tau d < t) :
+    (path d t).2 = (1 - ell d) * Real.sin t - 1 := by
+  simp only [path, ite_eq_right (not_le.mpr ht)]
+  rw [xi_zero d ho hd]
+  unfold ell
+  ring
+
+theorem path_phase1 (d : Reduced.Params) (ho : Ordered d)
+    (hd : Reduced.Equations d) (t : ℝ) (ht : t ∈ Set.Icc 0 d.phi) :
+    path d t = closed1 d t := by
+  have hk := ordered_knots d ho
+  have htau : t ≤ tau d := ht.2.trans (hk.2.1.trans (hk.2.2.1.trans hk.2.2.2.1))
+  apply Prod.ext
+  · exact path_first_of_le d t ht.2
+  · rw [path_second_of_le d ho hd t htau, W_phase1 d ho hd t ht, U1_eq_half d hd]
+    dsimp [closed1, r1]
+    ring
+
+theorem path_phase2 (d : Reduced.Params) (ho : Ordered d)
+    (hd : Reduced.Equations d) (t : ℝ)
+    (hlo : d.phi < t) (hhi : t ≤ d.theta) : path d t = closed2 d t := by
+  have hk := ordered_knots d ho
+  have hr : T - t ∈ Set.Icc (eta d) (tau d) := by
+    dsimp [eta, tau]
+    constructor <;> linarith
+  have htau : t ≤ tau d := hhi.trans (hk.2.2.1.trans hk.2.2.2.1)
+  apply Prod.ext
+  · rw [path_first_of_gt d t hlo, W_phase4 d ho (T - t) hr]
+    simp only [closed2, T, Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  · rw [path_second_of_le d ho hd t htau, W_phase2 d ho hd t ⟨hlo.le, hhi⟩]
+    dsimp [closed2, U2]
+    ring
+
+theorem path_phase3 (d : Reduced.Params) (ho : Ordered d)
+    (hd : Reduced.Equations d) (t : ℝ)
+    (hlo : d.theta < t) (hhi : t ≤ eta d) : path d t = closed3 d t := by
+  have hk := ordered_knots d ho
+  have hphi : d.phi < t := lt_of_le_of_lt ho.2.1 hlo
+  have hr : T - t ∈ Set.Icc d.theta (eta d) := by
+    dsimp [eta] at *
+    constructor <;> linarith
+  have htau : t ≤ tau d := hhi.trans hk.2.2.2.1
+  apply Prod.ext
+  · rw [path_first_of_gt d t hphi, W_phase3 d ho hd (T - t) hr]
+    simp only [closed3, T, Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  · rw [path_second_of_le d ho hd t htau, W_phase3 d ho hd t ⟨hlo.le, hhi⟩]
+    dsimp [closed3]
+    ring
+
+theorem path_phase4 (d : Reduced.Params) (ho : Ordered d)
+    (hd : Reduced.Equations d) (t : ℝ)
+    (hlo : eta d < t) (hhi : t ≤ tau d) : path d t = closed4 d t := by
+  have hk := ordered_knots d ho
+  have hphi : d.phi < t := lt_of_le_of_lt (ho.2.1.trans hk.2.2.1) hlo
+  have hr : T - t ∈ Set.Icc d.phi d.theta := by
+    dsimp [eta, tau] at *
+    constructor <;> linarith
+  apply Prod.ext
+  · rw [path_first_of_gt d t hphi, W_phase2 d ho hd (T - t) hr]
+    simp only [closed4, U2, T, Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+  · rw [path_second_of_le d ho hd t hhi, W_phase4 d ho t ⟨hlo.le, hhi⟩]
+    dsimp [closed4]
+    ring
+
+theorem path_phase5 (d : Reduced.Params) (ho : Ordered d)
+    (hd : Reduced.Equations d) (t : ℝ)
+    (hlo : tau d < t) (hhi : t ≤ T) : path d t = closed5 d t := by
+  have hk := ordered_knots d ho
+  have hphi : d.phi < t :=
+    lt_of_le_of_lt (ho.2.1.trans (hk.2.2.1.trans hk.2.2.2.1)) hlo
+  have hr : T - t ∈ Set.Icc 0 d.phi := by
+    dsimp [tau] at *
+    constructor <;> linarith
+  apply Prod.ext
+  · rw [path_first_of_gt d t hphi, W_phase1 d ho hd (T - t) hr, U1_eq_half d hd]
+    dsimp only [closed5, r1, T]
+    rw [Real.sin_pi_div_two_sub, Real.cos_pi_div_two_sub]
+    ring
+  · exact path_second_of_gt d ho hd t hlo
+
+/-- The literal integral path, not just its formal candidate, has these phases. -/
+theorem path_eq_closedPath (d : Reduced.Params) (ho : Ordered d)
+    (hd : Reduced.Equations d) (t : ℝ) (ht : t ∈ Set.Icc 0 T) :
+    path d t = closedPath d t := by
+  change path d t =
+    (if t ≤ d.phi then closed1 d t
+     else if t ≤ d.theta then closed2 d t
+     else if t ≤ eta d then closed3 d t
+     else if t ≤ tau d then closed4 d t
+     else closed5 d t)
+  by_cases h1 : t ≤ d.phi
+  · simpa only [ite_eq_left h1] using path_phase1 d ho hd t ⟨ht.1, h1⟩
+  by_cases h2 : t ≤ d.theta
+  · simpa only [ite_eq_right h1, ite_eq_left h2] using path_phase2 d ho hd t (lt_of_not_ge h1) h2
+  by_cases h3 : t ≤ eta d
+  · simpa only [ite_eq_right h1, ite_eq_right h2, ite_eq_left h3] using
+      path_phase3 d ho hd t (lt_of_not_ge h2) h3
+  by_cases h4 : t ≤ tau d
+  · simpa only [ite_eq_right h1, ite_eq_right h2, ite_eq_right h3, ite_eq_left h4] using
+      path_phase4 d ho hd t (lt_of_not_ge h3) h4
+  · simpa only [ite_eq_right h1, ite_eq_right h2, ite_eq_right h3, ite_eq_right h4] using
+      path_phase5 d ho hd t (lt_of_not_ge h4) ht.2
+
+theorem integral_rotation (d : Reduced.Params) (ho : Ordered d)
+    (hd : Reduced.Equations d) (t : ℝ) (ht : t ∈ Set.Icc 0 T) :
+    rotation t (toPlane (path d t)) = toPlane (Romik.path (dictionary d) t) := by
+  rw [path_eq_closedPath d ho hd t ht, CanonicalRotation.closedPath_rotation d hd t]
+
+theorem integral_sofa_eq_worldSofa (d : Reduced.Params) (ho : Ordered d)
+    (hd : Reduced.Equations d) :
+    bodySofa (fun t => toPlane (path d t))
+        Model.horizontalHallway Model.verticalHallway Model.hallway =
+      worldSofa (fun t => toPlane (Romik.path (dictionary d) t))
+        Model.horizontalHallway Model.verticalHallway Model.hallway := by
+  apply bodySofa_eq_worldSofa_of_path_identity
+  intro t ht
+  exact (integral_rotation d ho hd t ht).symm
+
+/-- The reduced parameters selected by the concrete uniqueness certificate. -/
+def certified : Reduced.Params := PartALeanCert.reducedCertifiedUniqueSolution.solution
+
+theorem certified_ordered : Ordered certified := by
+  have h := PartE.physicalDomain_of_mem_reducedBox
+    PartALeanCert.reducedCertifiedUniqueSolution.solution_mem
+  exact ⟨h.1, h.2.1, h.2.2.1⟩
+
+theorem certified_equations : Reduced.Equations certified :=
+  PartALeanCert.reducedCertifiedUniqueSolution.satisfies
+
+/-- An unconditional representation theorem for the certified reduced tuple. -/
+theorem certified_integral_rotation (t : ℝ) (ht : t ∈ Set.Icc 0 T) :
+    rotation t (toPlane (path certified t)) =
+      toPlane (Romik.path (dictionary certified) t) :=
+  integral_rotation certified certified_ordered certified_equations t ht
+
+/-- Remaining obligation: identify the full dictionary with the independent 22D root. -/
+theorem integral_sofa_eq_certified_of_dictionary
+    (hdictionary : dictionary certified = PartC.params) :
+    bodySofa (fun t => toPlane (path certified t))
+        Model.horizontalHallway Model.verticalHallway Model.hallway =
+      EuclideanMotion.sofa := by
+  rw [integral_sofa_eq_worldSofa certified certified_ordered certified_equations,
+    hdictionary, ← SetIdentification.sofa_eq_worldSofa]
+
+end GerverSofa.PartF.Integrals
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# F06: identify the two independently certified parameter choices
+
+The 22D box supplies only coarse physical inequalities for its reverse
+parameters. Global 4D uniqueness from Part E then identifies the reduced root.
+The full algebraic reconstruction closes the 22D identity without a new
+Krawczyk run or a forward enclosure into the narrow full box.
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace GerverSofa.PartF.Parameters
+
+open Phases
+
+theorem undictionary_physical_of_mem_full_box {p : Romik.Params}
+    (hp : p ∈ Romik.box) : PartE.PhysicalDomain (undictionary p) := by
+  dsimp [Romik.box, qR] at hp
+  rcases hp with ⟨_, _, _, _, _, _, _, _,
+    _, _, _, _, _, _, _, _,
+    _, _, _, _, _, _, _, _,
+    hb1lo, hb1hi, hb2lo, _, _, _, _, _,
+    _, _, _, _, _, _, _, _,
+    hphilo, hphihi, hthetalo, hthetahi⟩
+  norm_num at hb1lo hb1hi hb2lo hphilo hphihi hthetalo hthetahi
+  have hphi0 : 0 ≤ p.phi := by linarith only [hphilo]
+  have hphi1 : p.phi ≤ 1 / 20 := by linarith only [hphihi]
+  have htheta0 : (3 / 5 : ℝ) ≤ p.theta := by linarith only [hthetalo]
+  have htheta1 : p.theta ≤ 7 / 10 := by linarith only [hthetahi]
+  change 0 ≤ p.phi ∧ p.phi ≤ p.theta ∧ p.theta ≤ Real.pi / 4 ∧
+    0 ≤ p.phi - 1 - 2 * p.b1 ∧
+    0 ≤ p.b2 + 1 / 2 - (1 + (p.phi - 1 - 2 * p.b1)) * p.phi / 2 + p.phi ^ 2 / 4
+  refine ⟨hphi0, ?_, ?_, ?_, ?_⟩
+  · linarith only [hphi1, htheta0]
+  · nlinarith only [htheta1, Real.pi_gt_three]
+  · linarith only [hphi0, hb1hi]
+  · have hprod : 0 ≤ (p.b1 + 3 / 5) * p.phi :=
+      mul_nonneg (by linarith only [hb1lo]) hphi0
+    have hsq : 0 ≤ p.phi * (1 / 20 - p.phi) :=
+      mul_nonneg hphi0 (by linarith only [hphi1])
+    nlinarith only [hb2lo, hprod, hsq, hphi0, hphi1]
+
+theorem undictionary_eq_reduced_certified {p : Romik.Params}
+    (hp : p ∈ Romik.box) (heq : Romik.Equations p) :
+    undictionary p = PartALeanCert.reducedCertifiedUniqueSolution.solution := by
+  let d := undictionary p
+  have hdom : PartE.PhysicalDomain d := undictionary_physical_of_mem_full_box hp
+  have he : Reduced.Equations d := undictionary_equations heq
+  have hspec : PartE.DeepMindABPhiThetaSpec d.a d.b d.phi d.theta :=
+    (PartE.deepMindSpec_iff_physicalDomain_and_reducedEquations _ _ _ _).2 ⟨hdom, he⟩
+  let q : ℝ × ℝ × ℝ × ℝ := (d.a, d.b, d.phi, d.theta)
+  have hq : TupleSpec q := (upstreamSpec_iff _ _ _ _).2 hspec
+  have htuple := any_solution_eq_certified q hq
+  have hparams := congrArg PartE.tupleEquiv htuple
+  rw [certified, PartE.tupleEquiv.apply_symm_apply] at hparams
+  exact hparams
+
+theorem dictionary_eq_full_certified :
+    dictionary PartALeanCert.reducedCertifiedUniqueSolution.solution = PartB.params := by
+  have hu := undictionary_eq_reduced_certified PartB.params_mem PartB.params_equations
+  calc
+    dictionary PartALeanCert.reducedCertifiedUniqueSolution.solution =
+        dictionary (undictionary PartB.params) := congrArg dictionary hu.symm
+    _ = PartB.params := dictionary_undictionary_of_equations PartB.params_equations
+
+end GerverSofa.PartF.Parameters
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# F06: unconditional motion bridge for the literal integral construction
+
+This is the translate-then-rotate body model defined in Part F, with the
+canonical Euclidean orientation. It does not silently replace the current
+upstream rotateTranslate definition discussed in issue #5270.
+-/
+
+@[expose] public section
+
+noncomputable section
+namespace GerverSofa.PartF.Integrals
+
+open Phases Coordinates
+
+theorem certified_dictionary_eq_params : dictionary certified = PartC.params :=
+  Parameters.dictionary_eq_full_certified
+
+theorem certified_integral_rotation_to_full (t : ℝ) (ht : t ∈ Set.Icc 0 T) :
+    rotation t (toPlane (path certified t)) = toPlane (Romik.path PartC.params t) := by
+  rw [certified_integral_rotation t ht, certified_dictionary_eq_params]
+
+theorem integral_sofa_eq_certified :
+    bodySofa (fun t => toPlane (path certified t))
+        Model.horizontalHallway Model.verticalHallway Model.hallway =
+      EuclideanMotion.sofa :=
+  integral_sofa_eq_certified_of_dictionary certified_dictionary_eq_params
+
+end GerverSofa.PartF.Integrals
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle001.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle001.lean
new file mode 100644
index 0000000000..41448d2064
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle001.lean
@@ -0,0 +1,15130 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch001`.
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch016`.
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch004`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CoverCertificate7eaf46f70e
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate7eaf46f70e
+
+namespace GerverSofa.PartE.CoverCertificate3307b34685
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate3307b34685
+
+namespace GerverSofa.PartE.CoverCertificatea0fa380a93
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatea0fa380a93
+
+namespace GerverSofa.PartE.CoverCertificate5f1fd442d3
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate5f1fd442d3
+
+namespace GerverSofa.PartE.CoverCertificated6f9cbc5d6
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificated6f9cbc5d6
+
+namespace GerverSofa.PartE.CoverCertificatecaa270a0ec
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childLH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatecaa270a0ec
+
+namespace GerverSofa.PartE.CoverCertificate0254ae442a
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childLH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate0254ae442a
+
+namespace GerverSofa.PartE.CoverCertificate44197a14bc
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childLH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate44197a14bc
+
+namespace GerverSofa.PartE.CoverCertificateaa4818b079
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childLH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificateaa4818b079
+
+namespace GerverSofa.PartE.CoverCertificateade660b261
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childLH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificateade660b261
+
+namespace GerverSofa.PartE.CoverCertificateead4291aaf
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childLH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificateead4291aaf
+
+namespace GerverSofa.PartE.CoverCertificate76deff8421
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childLH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate76deff8421
+
+namespace GerverSofa.PartE.CoverCertificatef9c3dd592d
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHL (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatef9c3dd592d
+
+namespace GerverSofa.PartE.CoverCertificate55e26a37bf
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHL (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate55e26a37bf
+
+namespace GerverSofa.PartE.CoverCertificated546542719
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childHL (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificated546542719
+
+namespace GerverSofa.PartE.CoverCertificate09704881ea
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childHL (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate09704881ea
+
+namespace GerverSofa.PartE.CoverCertificate89f8bf996f
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate89f8bf996f
+
+namespace GerverSofa.PartE.CoverCertificate072d32638e
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate072d32638e
+
+namespace GerverSofa.PartE.CoverCertificate5c3ff9507c
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate5c3ff9507c
+
+namespace GerverSofa.PartE.CoverCertificate0e6fd3d794
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate0e6fd3d794
+
+namespace GerverSofa.PartE.CoverCertificate1d8d5ed6e8
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate1d8d5ed6e8
+
+namespace GerverSofa.PartE.CoverCertificatead0c92c0d4
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatead0c92c0d4
+
+namespace GerverSofa.PartE.CoverCertificate4e6964ff18
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate4e6964ff18
+
+namespace GerverSofa.PartE.CoverCertificate14906b8c69
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate14906b8c69
+
+namespace GerverSofa.PartE.CoverCertificate4614f242b3
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate4614f242b3
+
+namespace GerverSofa.PartE.CoverCertificatef3c458cae3
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatef3c458cae3
+
+namespace GerverSofa.PartE.CoverCertificateeb6553a302
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificateeb6553a302
+
+namespace GerverSofa.PartE.CoverCertificatefa7152b230
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatefa7152b230
+
+namespace GerverSofa.PartE.CoverCertificate8afa5ed490
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate8afa5ed490
+
+namespace GerverSofa.PartE.CoverCertificated9debec79d
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+end GerverSofa.PartE.CoverCertificated9debec79d
+
+namespace GerverSofa.PartE.CoverCertificate67d4072331
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate67d4072331
+
+namespace GerverSofa.PartE.CoverCertificatebed93df298
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childHH (childLL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatebed93df298
+
+namespace GerverSofa.PartE.CoverCertificatefd2c3f3274
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHL (childLH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatefd2c3f3274
+
+namespace GerverSofa.PartE.CoverCertificate1ffa851843
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHL (childLH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate1ffa851843
+
+namespace GerverSofa.PartE.CoverCertificated8ac44ac8f
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHL (childLH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificated8ac44ac8f
+
+namespace GerverSofa.PartE.CoverCertificate3caeb3dd1d
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childHL (childLH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate3caeb3dd1d
+
+namespace GerverSofa.PartE.CoverCertificate3195681328
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childHL (childLH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate3195681328
+
+namespace GerverSofa.PartE.CoverCertificate24d91bb18c
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childHL (childLH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate24d91bb18c
+
+namespace GerverSofa.PartE.CoverCertificatea5eff6d169
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childHL (childLH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatea5eff6d169
+
+namespace GerverSofa.PartE.CoverCertificateabfc719760
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childHL (childLH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificateabfc719760
+
+namespace GerverSofa.PartE.CoverCertificate0a514de912
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLL (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate0a514de912
+
+namespace GerverSofa.PartE.CoverCertificate7ed638f543
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate7ed638f543
+
+namespace GerverSofa.PartE.CoverCertificatebdb1950cb6
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childLL (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatebdb1950cb6
+
+namespace GerverSofa.PartE.CoverCertificate5fb7a54ee1
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childLL (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate5fb7a54ee1
+
+namespace GerverSofa.PartE.CoverCertificated58a87e9b8
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificated58a87e9b8
+
+namespace GerverSofa.PartE.CoverCertificate3ddf17b582
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate3ddf17b582
+
+namespace GerverSofa.PartE.CoverCertificated26b7e0a55
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificated26b7e0a55
+
+namespace GerverSofa.PartE.CoverCertificatea04fbf8169
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatea04fbf8169
+
+namespace GerverSofa.PartE.CoverCertificate3ba43c403b
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate3ba43c403b
+
+namespace GerverSofa.PartE.CoverCertificate1430173dea
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate1430173dea
+
+namespace GerverSofa.PartE.CoverCertificate2756e7c1ab
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate2756e7c1ab
+
+namespace GerverSofa.PartE.CoverCertificate8e43edbf8f
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate8e43edbf8f
+
+namespace GerverSofa.PartE.CoverCertificated1acdb7cc1
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificated1acdb7cc1
+
+namespace GerverSofa.PartE.CoverCertificated9d4fcb914
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificated9d4fcb914
+
+namespace GerverSofa.PartE.CoverCertificate18d654774a
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate18d654774a
+
+namespace GerverSofa.PartE.CoverCertificate3c1ddcbe4a
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate3c1ddcbe4a
+
+namespace GerverSofa.PartE.CoverCertificatecaa9842c01
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatecaa9842c01
+
+namespace GerverSofa.PartE.CoverCertificate8a64f55483
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate8a64f55483
+
+namespace GerverSofa.PartE.CoverCertificatea45749320c
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatea45749320c
+
+namespace GerverSofa.PartE.CoverCertificate96200cdbc3
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childLH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate96200cdbc3
+
+namespace GerverSofa.PartE.CoverCertificate1e0715601d
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLH (childLL (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate1e0715601d
+
+namespace GerverSofa.PartE.CoverCertificate645c2a2fa6
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLL (childLH (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificate645c2a2fa6
+
+namespace GerverSofa.PartE.CoverCertificate22a5b25f41
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLL (childLH (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificate22a5b25f41
+
+namespace GerverSofa.PartE.CoverCertificate052935fbc6
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLL (childLH (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell1010 : AngleCell :=
+  childLL cell101
+
+private abbrev cell1011 : AngleCell :=
+  childLH cell101
+
+private abbrev cell1012 : AngleCell :=
+  childHL cell101
+
+private abbrev cell1013 : AngleCell :=
+  childHH cell101
+
+private abbrev cell1100 : AngleCell :=
+  childLL cell110
+
+private abbrev cell1101 : AngleCell :=
+  childLH cell110
+
+private abbrev cell1102 : AngleCell :=
+  childHL cell110
+
+private abbrev cell1103 : AngleCell :=
+  childHH cell110
+
+private abbrev cell1110 : AngleCell :=
+  childLL cell111
+
+private abbrev cell1111 : AngleCell :=
+  childLH cell111
+
+private abbrev cell1112 : AngleCell :=
+  childHL cell111
+
+private abbrev cell1113 : AngleCell :=
+  childHH cell111
+
+end GerverSofa.PartE.CoverCertificate052935fbc6
+
+namespace GerverSofa.PartE.CoverCertificateda1b0b5031
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLL (childLH (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell0000 : AngleCell :=
+  childLL cell000
+
+private abbrev cell0001 : AngleCell :=
+  childLH cell000
+
+private abbrev cell0002 : AngleCell :=
+  childHL cell000
+
+private abbrev cell0003 : AngleCell :=
+  childHH cell000
+
+private abbrev cell0010 : AngleCell :=
+  childLL cell001
+
+private abbrev cell0011 : AngleCell :=
+  childLH cell001
+
+private abbrev cell0012 : AngleCell :=
+  childHL cell001
+
+private abbrev cell0013 : AngleCell :=
+  childHH cell001
+
+private abbrev cell0100 : AngleCell :=
+  childLL cell010
+
+private abbrev cell0101 : AngleCell :=
+  childLH cell010
+
+private abbrev cell0102 : AngleCell :=
+  childHL cell010
+
+private abbrev cell0103 : AngleCell :=
+  childHH cell010
+
+private abbrev cell0110 : AngleCell :=
+  childLL cell011
+
+private abbrev cell0111 : AngleCell :=
+  childLH cell011
+
+private abbrev cell0112 : AngleCell :=
+  childHL cell011
+
+private abbrev cell0113 : AngleCell :=
+  childHH cell011
+
+private abbrev cell1000 : AngleCell :=
+  childLL cell100
+
+private abbrev cell1001 : AngleCell :=
+  childLH cell100
+
+private abbrev cell1002 : AngleCell :=
+  childHL cell100
+
+private abbrev cell1003 : AngleCell :=
+  childHH cell100
+
+private abbrev cell1010 : AngleCell :=
+  childLL cell101
+
+private abbrev cell1011 : AngleCell :=
+  childLH cell101
+
+private abbrev cell1012 : AngleCell :=
+  childHL cell101
+
+private abbrev cell1013 : AngleCell :=
+  childHH cell101
+
+private abbrev cell1100 : AngleCell :=
+  childLL cell110
+
+private abbrev cell1101 : AngleCell :=
+  childLH cell110
+
+private abbrev cell1102 : AngleCell :=
+  childHL cell110
+
+private abbrev cell1103 : AngleCell :=
+  childHH cell110
+
+private abbrev cell1110 : AngleCell :=
+  childLL cell111
+
+private abbrev cell1111 : AngleCell :=
+  childLH cell111
+
+private abbrev cell1112 : AngleCell :=
+  childHL cell111
+
+private abbrev cell1113 : AngleCell :=
+  childHH cell111
+
+private abbrev cell11110 : AngleCell :=
+  childLL cell1111
+
+private abbrev cell11111 : AngleCell :=
+  childLH cell1111
+
+private abbrev cell11112 : AngleCell :=
+  childHL cell1111
+
+private abbrev cell11113 : AngleCell :=
+  childHH cell1111
+
+end GerverSofa.PartE.CoverCertificateda1b0b5031
+
+namespace GerverSofa.PartE.CoverCertificate87253a3a53
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childLH (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate87253a3a53
+
+namespace GerverSofa.PartE.CoverCertificate3f791a2e61
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childLH (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate3f791a2e61
+
+namespace GerverSofa.PartE.CertificateCells68c1109b36
+
+/-- Subcell `3000` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3000 : AngleCell :=
+  childLL (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells68c1109b36
+
+namespace GerverSofa.PartE.CertificateCells3eb225f04c
+
+/-- Subcell `3000` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3000 : AngleCell :=
+  childLL (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells3eb225f04c
+
+namespace GerverSofa.PartE.CertificateCells50ca4e3af4
+
+/-- Subcell `3000` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3000 : AngleCell :=
+  childLL (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells50ca4e3af4
+
+namespace GerverSofa.PartE.CertificateCellsfa60677f23
+
+/-- Subcell `3000` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3000 : AngleCell :=
+  childLL (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsfa60677f23
+
+namespace GerverSofa.PartE.CertificateCellsadacfb8de1
+
+/-- Subcell `3001` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3001 : AngleCell :=
+  childLH (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsadacfb8de1
+
+namespace GerverSofa.PartE.CertificateCells2e1d52b748
+
+/-- Subcell `3001` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3001 : AngleCell :=
+  childLH (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells2e1d52b748
+
+namespace GerverSofa.PartE.CertificateCells7e7c37fb1b
+
+/-- Subcell `3001` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3001 : AngleCell :=
+  childLH (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells7e7c37fb1b
+
+namespace GerverSofa.PartE.CertificateCellsa8cebe99f5
+
+/-- Subcell `3001` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3001 : AngleCell :=
+  childLH (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa8cebe99f5
+
+namespace GerverSofa.PartE.CertificateCells7730b0e091
+
+/-- Subcell `3002` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3002 : AngleCell :=
+  childHL (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells7730b0e091
+
+namespace GerverSofa.PartE.CertificateCellse8a24af0c4
+
+/-- Subcell `3002` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3002 : AngleCell :=
+  childHL (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellse8a24af0c4
+
+namespace GerverSofa.PartE.CertificateCellse2e04a6320
+
+/-- Subcell `3002` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3002 : AngleCell :=
+  childHL (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellse2e04a6320
+
+namespace GerverSofa.PartE.CertificateCells1fb6bd85e4
+
+/-- Subcell `3002` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3002 : AngleCell :=
+  childHL (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells1fb6bd85e4
+
+namespace GerverSofa.PartE.CertificateCellsf0432d10ef
+
+/-- Subcell `3003` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3003 : AngleCell :=
+  childHH (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsf0432d10ef
+
+namespace GerverSofa.PartE.CertificateCells283f971137
+
+/-- Subcell `3003` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3003 : AngleCell :=
+  childHH (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells283f971137
+
+namespace GerverSofa.PartE.CertificateCellsad7da039d5
+
+/-- Subcell `3003` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3003 : AngleCell :=
+  childHH (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsad7da039d5
+
+namespace GerverSofa.PartE.CertificateCells1e800a73bb
+
+/-- Subcell `3003` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3003 : AngleCell :=
+  childHH (childLL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells1e800a73bb
+
+namespace GerverSofa.PartE.CertificateCellsb9d63d60b8
+
+/-- Subcell `3010` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3010 : AngleCell :=
+  childLL (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsb9d63d60b8
+
+namespace GerverSofa.PartE.CertificateCells73ece5e183
+
+/-- Subcell `3010` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3010 : AngleCell :=
+  childLL (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells73ece5e183
+
+namespace GerverSofa.PartE.CertificateCellsb58c03cf78
+
+/-- Subcell `3010` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3010 : AngleCell :=
+  childLL (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsb58c03cf78
+
+namespace GerverSofa.PartE.CertificateCells3c73252ed6
+
+/-- Subcell `3010` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3010 : AngleCell :=
+  childLL (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells3c73252ed6
+
+namespace GerverSofa.PartE.CertificateCells75abf59285
+
+/-- Subcell `3011` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3011 : AngleCell :=
+  childLH (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells75abf59285
+
+namespace GerverSofa.PartE.CertificateCellsc77278a9b0
+
+/-- Subcell `3011` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3011 : AngleCell :=
+  childLH (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsc77278a9b0
+
+namespace GerverSofa.PartE.CertificateCells80b449154c
+
+/-- Subcell `3011` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3011 : AngleCell :=
+  childLH (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells80b449154c
+
+namespace GerverSofa.PartE.CertificateCells19cd4580af
+
+/-- Subcell `3011` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3011 : AngleCell :=
+  childLH (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells19cd4580af
+
+namespace GerverSofa.PartE.CertificateCellsad60f51a47
+
+/-- Subcell `3012` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3012 : AngleCell :=
+  childHL (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsad60f51a47
+
+namespace GerverSofa.PartE.CertificateCells59722ddf21
+
+/-- Subcell `3012` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3012 : AngleCell :=
+  childHL (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells59722ddf21
+
+namespace GerverSofa.PartE.CertificateCells38add90453
+
+/-- Subcell `3012` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3012 : AngleCell :=
+  childHL (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells38add90453
+
+namespace GerverSofa.PartE.CertificateCellsa2fd436678
+
+/-- Subcell `3012` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3012 : AngleCell :=
+  childHL (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa2fd436678
+
+namespace GerverSofa.PartE.CertificateCells4036e8788a
+
+/-- Subcell `3013` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3013 : AngleCell :=
+  childHH (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells4036e8788a
+
+namespace GerverSofa.PartE.CertificateCellse8e082f5ed
+
+/-- Subcell `3013` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3013 : AngleCell :=
+  childHH (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellse8e082f5ed
+
+namespace GerverSofa.PartE.CertificateCells33c2a4ae09
+
+/-- Subcell `3013` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3013 : AngleCell :=
+  childHH (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells33c2a4ae09
+
+namespace GerverSofa.PartE.CertificateCells8acf2cf43b
+
+/-- Subcell `3013` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3013 : AngleCell :=
+  childHH (childLH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells8acf2cf43b
+
+namespace GerverSofa.PartE.CertificateCells4c5b439e46
+
+/-- Subcell `3020` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3020 : AngleCell :=
+  childLL (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells4c5b439e46
+
+namespace GerverSofa.PartE.CertificateCells1d890d537b
+
+/-- Subcell `3020` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3020 : AngleCell :=
+  childLL (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells1d890d537b
+
+namespace GerverSofa.PartE.CertificateCells3dfd612f88
+
+/-- Subcell `3020` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3020 : AngleCell :=
+  childLL (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells3dfd612f88
+
+namespace GerverSofa.PartE.CertificateCellsb73c54b84b
+
+/-- Subcell `3020` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3020 : AngleCell :=
+  childLL (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsb73c54b84b
+
+namespace GerverSofa.PartE.CertificateCellsa2d944a3cb
+
+/-- Subcell `3021` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3021 : AngleCell :=
+  childLH (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa2d944a3cb
+
+namespace GerverSofa.PartE.CertificateCellsa175b3d804
+
+/-- Subcell `3021` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3021 : AngleCell :=
+  childLH (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa175b3d804
+
+namespace GerverSofa.PartE.CertificateCells2377a41132
+
+/-- Subcell `3021` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3021 : AngleCell :=
+  childLH (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells2377a41132
+
+namespace GerverSofa.PartE.CertificateCells141b81b7b9
+
+/-- Subcell `3021` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3021 : AngleCell :=
+  childLH (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells141b81b7b9
+
+namespace GerverSofa.PartE.CertificateCells53ed2717a4
+
+/-- Subcell `3022` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3022 : AngleCell :=
+  childHL (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells53ed2717a4
+
+namespace GerverSofa.PartE.CertificateCells6cc17280cd
+
+/-- Subcell `3022` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3022 : AngleCell :=
+  childHL (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells6cc17280cd
+
+namespace GerverSofa.PartE.CertificateCellscd48c30df5
+
+/-- Subcell `3022` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3022 : AngleCell :=
+  childHL (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellscd48c30df5
+
+namespace GerverSofa.PartE.CertificateCells6a90ac3f5f
+
+/-- Subcell `3022` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3022 : AngleCell :=
+  childHL (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells6a90ac3f5f
+
+namespace GerverSofa.PartE.CertificateCellscb8a79a1e7
+
+/-- Subcell `3023` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3023 : AngleCell :=
+  childHH (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellscb8a79a1e7
+
+namespace GerverSofa.PartE.CertificateCellsa17655a9cd
+
+/-- Subcell `3023` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3023 : AngleCell :=
+  childHH (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa17655a9cd
+
+namespace GerverSofa.PartE.CertificateCellsea417091fc
+
+/-- Subcell `3023` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3023 : AngleCell :=
+  childHH (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsea417091fc
+
+namespace GerverSofa.PartE.CertificateCells58f12e8647
+
+/-- Subcell `3023` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3023 : AngleCell :=
+  childHH (childHL (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells58f12e8647
+
+namespace GerverSofa.PartE.CertificateCells5256e12156
+
+/-- Subcell `3030` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3030 : AngleCell :=
+  childLL (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells5256e12156
+
+namespace GerverSofa.PartE.CertificateCellsa7b064ca3f
+
+/-- Subcell `3030` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3030 : AngleCell :=
+  childLL (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa7b064ca3f
+
+namespace GerverSofa.PartE.CertificateCellsa6d5038417
+
+/-- Subcell `3030` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3030 : AngleCell :=
+  childLL (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa6d5038417
+
+namespace GerverSofa.PartE.CertificateCellsa40437554b
+
+/-- Subcell `3030` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3030 : AngleCell :=
+  childLL (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa40437554b
+
+namespace GerverSofa.PartE.CertificateCells99d0a4be54
+
+/-- Subcell `3031` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3031 : AngleCell :=
+  childLH (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells99d0a4be54
+
+namespace GerverSofa.PartE.CertificateCells473a273fc9
+
+/-- Subcell `3031` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3031 : AngleCell :=
+  childLH (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells473a273fc9
+
+namespace GerverSofa.PartE.CertificateCells3c757ee420
+
+/-- Subcell `3031` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3031 : AngleCell :=
+  childLH (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells3c757ee420
+
+namespace GerverSofa.PartE.CertificateCellsd8b0ff734e
+
+/-- Subcell `3031` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3031 : AngleCell :=
+  childLH (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsd8b0ff734e
+
+namespace GerverSofa.PartE.CertificateCells03aa6fa886
+
+/-- Subcell `3032` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3032 : AngleCell :=
+  childHL (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells03aa6fa886
+
+namespace GerverSofa.PartE.CertificateCellsf82ea84544
+
+/-- Subcell `3032` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3032 : AngleCell :=
+  childHL (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsf82ea84544
+
+namespace GerverSofa.PartE.CertificateCellsa1f10633df
+
+/-- Subcell `3032` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3032 : AngleCell :=
+  childHL (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa1f10633df
+
+namespace GerverSofa.PartE.CertificateCells92aeaddf18
+
+/-- Subcell `3032` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3032 : AngleCell :=
+  childHL (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells92aeaddf18
+
+namespace GerverSofa.PartE.CertificateCells4d2736f1ee
+
+/-- Subcell `3033` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3033 : AngleCell :=
+  childHH (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells4d2736f1ee
+
+namespace GerverSofa.PartE.CertificateCellsf1c569afeb
+
+/-- Subcell `3033` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3033 : AngleCell :=
+  childHH (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsf1c569afeb
+
+namespace GerverSofa.PartE.CertificateCells899e3e87cc
+
+/-- Subcell `3033` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3033 : AngleCell :=
+  childHH (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells899e3e87cc
+
+namespace GerverSofa.PartE.CertificateCells8cea727797
+
+/-- Subcell `3033` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3033 : AngleCell :=
+  childHH (childHH (childLL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells8cea727797
+
+namespace GerverSofa.PartE.CertificateCellsdf788d056e
+
+/-- Subcell `3100` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3100 : AngleCell :=
+  childLL (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsdf788d056e
+
+namespace GerverSofa.PartE.CertificateCells83252066d4
+
+/-- Subcell `3100` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3100 : AngleCell :=
+  childLL (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells83252066d4
+
+namespace GerverSofa.PartE.CertificateCellsd15521ecc4
+
+/-- Subcell `3100` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3100 : AngleCell :=
+  childLL (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsd15521ecc4
+
+namespace GerverSofa.PartE.CertificateCells9ebe05244a
+
+/-- Subcell `3100` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3100 : AngleCell :=
+  childLL (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells9ebe05244a
+
+namespace GerverSofa.PartE.CertificateCells21fc209558
+
+/-- Subcell `3101` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3101 : AngleCell :=
+  childLH (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells21fc209558
+
+namespace GerverSofa.PartE.CertificateCellsc545b9b00b
+
+/-- Subcell `3101` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3101 : AngleCell :=
+  childLH (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsc545b9b00b
+
+namespace GerverSofa.PartE.CertificateCellsa64e976caf
+
+/-- Subcell `3101` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3101 : AngleCell :=
+  childLH (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa64e976caf
+
+namespace GerverSofa.PartE.CertificateCellsf9eb6beb81
+
+/-- Subcell `3101` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3101 : AngleCell :=
+  childLH (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsf9eb6beb81
+
+namespace GerverSofa.PartE.CertificateCells1d67e12ba0
+
+/-- Subcell `3102` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3102 : AngleCell :=
+  childHL (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells1d67e12ba0
+
+namespace GerverSofa.PartE.CertificateCellsa2d5114e5e
+
+/-- Subcell `3102` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3102 : AngleCell :=
+  childHL (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa2d5114e5e
+
+namespace GerverSofa.PartE.CertificateCells414314cff6
+
+/-- Subcell `3102` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3102 : AngleCell :=
+  childHL (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells414314cff6
+
+namespace GerverSofa.PartE.CertificateCellsf9010539ff
+
+/-- Subcell `3102` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3102 : AngleCell :=
+  childHL (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsf9010539ff
+
+namespace GerverSofa.PartE.CertificateCellse34346aa4a
+
+/-- Subcell `3103` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3103 : AngleCell :=
+  childHH (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellse34346aa4a
+
+namespace GerverSofa.PartE.CertificateCellse191f15fec
+
+/-- Subcell `3103` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3103 : AngleCell :=
+  childHH (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellse191f15fec
+
+namespace GerverSofa.PartE.CertificateCellsf92808a3cc
+
+/-- Subcell `3103` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3103 : AngleCell :=
+  childHH (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsf92808a3cc
+
+namespace GerverSofa.PartE.CertificateCells897e4339a9
+
+/-- Subcell `3103` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3103 : AngleCell :=
+  childHH (childLL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells897e4339a9
+
+namespace GerverSofa.PartE.CertificateCellsf4227a46c1
+
+/-- Subcell `3110` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3110 : AngleCell :=
+  childLL (childLH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsf4227a46c1
+
+namespace GerverSofa.PartE.CertificateCellsa2ba14667f
+
+/-- Subcell `3111` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3111 : AngleCell :=
+  childLH (childLH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa2ba14667f
+
+namespace GerverSofa.PartE.CertificateCells172a47199c
+
+/-- Subcell `3112` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3112 : AngleCell :=
+  childHL (childLH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells172a47199c
+
+namespace GerverSofa.PartE.CertificateCells503a496e12
+
+/-- Subcell `3112` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3112 : AngleCell :=
+  childHL (childLH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells503a496e12
+
+namespace GerverSofa.PartE.CertificateCells4f35c82776
+
+/-- Subcell `3112` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3112 : AngleCell :=
+  childHL (childLH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells4f35c82776
+
+namespace GerverSofa.PartE.CertificateCellsddf26a1af9
+
+/-- Subcell `3112` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3112 : AngleCell :=
+  childHL (childLH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsddf26a1af9
+
+namespace GerverSofa.PartE.CertificateCells725d74e81a
+
+/-- Subcell `3113` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3113 : AngleCell :=
+  childHH (childLH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells725d74e81a
+
+namespace GerverSofa.PartE.CertificateCells3b33327e8a
+
+/-- Subcell `3120` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3120 : AngleCell :=
+  childLL (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells3b33327e8a
+
+namespace GerverSofa.PartE.CertificateCells7bfd1f464a
+
+/-- Subcell `3120` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3120 : AngleCell :=
+  childLL (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells7bfd1f464a
+
+namespace GerverSofa.PartE.CertificateCellsdd644ededa
+
+/-- Subcell `3120` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3120 : AngleCell :=
+  childLL (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsdd644ededa
+
+namespace GerverSofa.PartE.CertificateCellsdd54c72360
+
+/-- Subcell `3120` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3120 : AngleCell :=
+  childLL (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsdd54c72360
+
+namespace GerverSofa.PartE.CertificateCellsf0f5b1fc54
+
+/-- Subcell `3121` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3121 : AngleCell :=
+  childLH (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsf0f5b1fc54
+
+namespace GerverSofa.PartE.CertificateCells2b0822f9b0
+
+/-- Subcell `3121` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3121 : AngleCell :=
+  childLH (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells2b0822f9b0
+
+namespace GerverSofa.PartE.CertificateCells05e3ee5af2
+
+/-- Subcell `3121` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3121 : AngleCell :=
+  childLH (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells05e3ee5af2
+
+namespace GerverSofa.PartE.CertificateCells8f81fc559c
+
+/-- Subcell `3121` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3121 : AngleCell :=
+  childLH (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells8f81fc559c
+
+namespace GerverSofa.PartE.CertificateCells509a6074b0
+
+/-- Subcell `3122` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3122 : AngleCell :=
+  childHL (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells509a6074b0
+
+namespace GerverSofa.PartE.CertificateCells6ac6e4ee1a
+
+/-- Subcell `3122` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3122 : AngleCell :=
+  childHL (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells6ac6e4ee1a
+
+namespace GerverSofa.PartE.CertificateCells2e92118453
+
+/-- Subcell `3122` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3122 : AngleCell :=
+  childHL (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells2e92118453
+
+namespace GerverSofa.PartE.CertificateCells5761c4811e
+
+/-- Subcell `3122` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3122 : AngleCell :=
+  childHL (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells5761c4811e
+
+namespace GerverSofa.PartE.CertificateCellsf228466209
+
+/-- Subcell `3123` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3123 : AngleCell :=
+  childHH (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsf228466209
+
+namespace GerverSofa.PartE.CertificateCells4924d89385
+
+/-- Subcell `3123` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3123 : AngleCell :=
+  childHH (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells4924d89385
+
+namespace GerverSofa.PartE.CertificateCells75bbd1e667
+
+/-- Subcell `3123` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3123 : AngleCell :=
+  childHH (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells75bbd1e667
+
+namespace GerverSofa.PartE.CertificateCells131513700f
+
+/-- Subcell `3123` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3123 : AngleCell :=
+  childHH (childHL (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells131513700f
+
+namespace GerverSofa.PartE.CertificateCells0436aa31cb
+
+/-- Subcell `3130` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3130 : AngleCell :=
+  childLL (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells0436aa31cb
+
+namespace GerverSofa.PartE.CertificateCells4cca5ef5ea
+
+/-- Subcell `3130` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3130 : AngleCell :=
+  childLL (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells4cca5ef5ea
+
+namespace GerverSofa.PartE.CertificateCells2133748f57
+
+/-- Subcell `3130` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3130 : AngleCell :=
+  childLL (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells2133748f57
+
+namespace GerverSofa.PartE.CertificateCells2d9432dbe0
+
+/-- Subcell `3130` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3130 : AngleCell :=
+  childLL (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells2d9432dbe0
+
+namespace GerverSofa.PartE.CertificateCells28839ca767
+
+/-- Subcell `3131` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3131 : AngleCell :=
+  childLH (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells28839ca767
+
+namespace GerverSofa.PartE.CertificateCells1b0bf20c51
+
+/-- Subcell `3131` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3131 : AngleCell :=
+  childLH (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells1b0bf20c51
+
+namespace GerverSofa.PartE.CertificateCells9fb8ae7256
+
+/-- Subcell `3131` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3131 : AngleCell :=
+  childLH (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells9fb8ae7256
+
+namespace GerverSofa.PartE.CertificateCells10b8b658d1
+
+/-- Subcell `3131` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3131 : AngleCell :=
+  childLH (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells10b8b658d1
+
+namespace GerverSofa.PartE.CertificateCells372671dc1b
+
+/-- Subcell `3132` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3132 : AngleCell :=
+  childHL (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells372671dc1b
+
+namespace GerverSofa.PartE.CertificateCells91a123c697
+
+/-- Subcell `3132` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3132 : AngleCell :=
+  childHL (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells91a123c697
+
+namespace GerverSofa.PartE.CertificateCellsce14c27bce
+
+/-- Subcell `3132` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3132 : AngleCell :=
+  childHL (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsce14c27bce
+
+namespace GerverSofa.PartE.CertificateCellsabc89a203a
+
+/-- Subcell `3132` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3132 : AngleCell :=
+  childHL (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsabc89a203a
+
+namespace GerverSofa.PartE.CertificateCells6f99a95071
+
+/-- Subcell `3133` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3133 : AngleCell :=
+  childHH (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells6f99a95071
+
+namespace GerverSofa.PartE.CertificateCells4acdbba755
+
+/-- Subcell `3133` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3133 : AngleCell :=
+  childHH (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells4acdbba755
+
+namespace GerverSofa.PartE.CertificateCells45607104df
+
+/-- Subcell `3133` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3133 : AngleCell :=
+  childHH (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells45607104df
+
+namespace GerverSofa.PartE.CertificateCellsf2e66397f6
+
+/-- Subcell `3133` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3133 : AngleCell :=
+  childHH (childHH (childLH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsf2e66397f6
+
+namespace GerverSofa.PartE.CertificateCellsf60a68912f
+
+/-- Subcell `3200` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3200 : AngleCell :=
+  childLL (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsf60a68912f
+
+namespace GerverSofa.PartE.CertificateCellsdb667fab49
+
+/-- Subcell `3200` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3200 : AngleCell :=
+  childLL (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsdb667fab49
+
+namespace GerverSofa.PartE.CertificateCells378433963d
+
+/-- Subcell `3200` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3200 : AngleCell :=
+  childLL (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells378433963d
+
+namespace GerverSofa.PartE.CertificateCells3c30c58615
+
+/-- Subcell `3200` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3200 : AngleCell :=
+  childLL (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells3c30c58615
+
+namespace GerverSofa.PartE.CertificateCellscbc2cb8c4d
+
+/-- Subcell `3201` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3201 : AngleCell :=
+  childLH (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellscbc2cb8c4d
+
+namespace GerverSofa.PartE.CertificateCells0ae5e3afe8
+
+/-- Subcell `3201` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3201 : AngleCell :=
+  childLH (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells0ae5e3afe8
+
+namespace GerverSofa.PartE.CertificateCells72905f39fb
+
+/-- Subcell `3201` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3201 : AngleCell :=
+  childLH (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells72905f39fb
+
+namespace GerverSofa.PartE.CertificateCells36ded03275
+
+/-- Subcell `3201` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3201 : AngleCell :=
+  childLH (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells36ded03275
+
+namespace GerverSofa.PartE.CertificateCells055eee17e7
+
+/-- Subcell `3202` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3202 : AngleCell :=
+  childHL (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells055eee17e7
+
+namespace GerverSofa.PartE.CertificateCells86a7b10d51
+
+/-- Subcell `3202` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3202 : AngleCell :=
+  childHL (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells86a7b10d51
+
+namespace GerverSofa.PartE.CertificateCells62973479ba
+
+/-- Subcell `3202` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3202 : AngleCell :=
+  childHL (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells62973479ba
+
+namespace GerverSofa.PartE.CertificateCellse26adcc478
+
+/-- Subcell `3202` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3202 : AngleCell :=
+  childHL (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellse26adcc478
+
+namespace GerverSofa.PartE.CertificateCellsd471241912
+
+/-- Subcell `3203` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3203 : AngleCell :=
+  childHH (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsd471241912
+
+namespace GerverSofa.PartE.CertificateCells89c6c55f33
+
+/-- Subcell `3203` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3203 : AngleCell :=
+  childHH (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells89c6c55f33
+
+namespace GerverSofa.PartE.CertificateCells371a4c38c5
+
+/-- Subcell `3203` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3203 : AngleCell :=
+  childHH (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells371a4c38c5
+
+namespace GerverSofa.PartE.CertificateCells96fe0c4e84
+
+/-- Subcell `3203` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3203 : AngleCell :=
+  childHH (childLL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells96fe0c4e84
+
+namespace GerverSofa.PartE.CertificateCells9ae3b0f784
+
+/-- Subcell `3210` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3210 : AngleCell :=
+  childLL (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells9ae3b0f784
+
+namespace GerverSofa.PartE.CertificateCells9614d1d005
+
+/-- Subcell `3210` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3210 : AngleCell :=
+  childLL (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells9614d1d005
+
+namespace GerverSofa.PartE.CertificateCells27f26686ef
+
+/-- Subcell `3210` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3210 : AngleCell :=
+  childLL (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells27f26686ef
+
+namespace GerverSofa.PartE.CertificateCells9dc2121909
+
+/-- Subcell `3210` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3210 : AngleCell :=
+  childLL (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells9dc2121909
+
+namespace GerverSofa.PartE.CertificateCellsc35310d566
+
+/-- Subcell `3211` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3211 : AngleCell :=
+  childLH (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsc35310d566
+
+namespace GerverSofa.PartE.CertificateCellsa20e4da390
+
+/-- Subcell `3211` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3211 : AngleCell :=
+  childLH (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa20e4da390
+
+namespace GerverSofa.PartE.CertificateCells3a90cbf635
+
+/-- Subcell `3211` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3211 : AngleCell :=
+  childLH (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells3a90cbf635
+
+namespace GerverSofa.PartE.CertificateCells214a11de68
+
+/-- Subcell `3211` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3211 : AngleCell :=
+  childLH (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells214a11de68
+
+namespace GerverSofa.PartE.CertificateCells032b91f5bc
+
+/-- Subcell `3212` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3212 : AngleCell :=
+  childHL (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells032b91f5bc
+
+namespace GerverSofa.PartE.CertificateCellsaa8e493e23
+
+/-- Subcell `3212` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3212 : AngleCell :=
+  childHL (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsaa8e493e23
+
+namespace GerverSofa.PartE.CertificateCells6bd715f340
+
+/-- Subcell `3212` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3212 : AngleCell :=
+  childHL (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells6bd715f340
+
+namespace GerverSofa.PartE.CertificateCellsa228f232aa
+
+/-- Subcell `3212` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3212 : AngleCell :=
+  childHL (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa228f232aa
+
+namespace GerverSofa.PartE.CertificateCells4dea68b576
+
+/-- Subcell `3213` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3213 : AngleCell :=
+  childHH (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells4dea68b576
+
+namespace GerverSofa.PartE.CertificateCells3e707fd6af
+
+/-- Subcell `3213` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3213 : AngleCell :=
+  childHH (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells3e707fd6af
+
+namespace GerverSofa.PartE.CertificateCells46e40ef0a9
+
+/-- Subcell `3213` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3213 : AngleCell :=
+  childHH (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells46e40ef0a9
+
+namespace GerverSofa.PartE.CertificateCells9651c4a476
+
+/-- Subcell `3213` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3213 : AngleCell :=
+  childHH (childLH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells9651c4a476
+
+namespace GerverSofa.PartE.CertificateCells2fa7cd8552
+
+/-- Subcell `3220` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3220 : AngleCell :=
+  childLL (childHL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells2fa7cd8552
+
+namespace GerverSofa.PartE.CertificateCellsec4b7eff4d
+
+/-- Subcell `3220` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3220 : AngleCell :=
+  childLL (childHL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsec4b7eff4d
+
+namespace GerverSofa.PartE.CertificateCells434ca1e74d
+
+/-- Subcell `3220` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3220 : AngleCell :=
+  childLL (childHL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells434ca1e74d
+
+namespace GerverSofa.PartE.CertificateCells886952c236
+
+/-- Subcell `3220` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3220 : AngleCell :=
+  childLL (childHL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells886952c236
+
+namespace GerverSofa.PartE.CertificateCells969a7c12d6
+
+/-- Subcell `3221` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3221 : AngleCell :=
+  childLH (childHL (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells969a7c12d6
+
+namespace GerverSofa.PartE.CertificateCells7aa82bee5f
+
+/-- Subcell `1100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1100 : AngleCell :=
+  childLL (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells7aa82bee5f
+
+namespace GerverSofa.PartE.CertificateCellsf8d478d36a
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCellsf8d478d36a
+
+namespace GerverSofa.PartE.CertificateCellsce54db6860
+
+/-- Subcell `1023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1023 : AngleCell :=
+  childHH (childHL (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1030 : AngleCell :=
+  childLL (childHH (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1031 : AngleCell :=
+  childLH (childHH (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1032 : AngleCell :=
+  childHL (childHH (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1033 : AngleCell :=
+  childHH (childHH (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1100 : AngleCell :=
+  childLL (childLL (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1102 : AngleCell :=
+  childHL (childLL (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1103 : AngleCell :=
+  childHH (childLL (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11000320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11000320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaBelowCell1100)))
+
+/-- Subcell `11000321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11000321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaBelowCell1100)))
+
+/-- Subcell `11000322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11000322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaBelowCell1100)))
+
+/-- Subcell `11000323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11000323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaBelowCell1100)))
+
+/-- Subcell `11000330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11000330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaBelowCell1100)))
+
+/-- Subcell `11000331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11000331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaBelowCell1100)))
+
+/-- Subcell `11000332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11000332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaBelowCell1100)))
+
+/-- Subcell `11000333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11000333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaBelowCell1100)))
+
+/-- Subcell `11001220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaBelowCell1100)))
+
+/-- Subcell `11001221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaBelowCell1100)))
+
+/-- Subcell `11001222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaBelowCell1100)))
+
+/-- Subcell `11001223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaBelowCell1100)))
+
+/-- Subcell `11001230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaBelowCell1100)))
+
+/-- Subcell `11001231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaBelowCell1100)))
+
+/-- Subcell `11001232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaBelowCell1100)))
+
+/-- Subcell `11001233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaBelowCell1100)))
+
+/-- Subcell `11001320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaBelowCell1100)))
+
+/-- Subcell `11001321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaBelowCell1100)))
+
+/-- Subcell `11001322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaBelowCell1100)))
+
+/-- Subcell `11001323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaBelowCell1100)))
+
+/-- Subcell `11001330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaBelowCell1100)))
+
+/-- Subcell `11001331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaBelowCell1100)))
+
+/-- Subcell `11001332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaBelowCell1100)))
+
+/-- Subcell `11001333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11001333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaBelowCell1100)))
+
+/-- Subcell `11002000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11002133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaBelowCell1100)))
+
+/-- Subcell `11003000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11003313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaBelowCell1100)))
+
+/-- Subcell `11010220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11010220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaBelowCell1101)))
+
+/-- Subcell `11010221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11010221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaBelowCell1101)))
+
+/-- Subcell `11010222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11010222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaBelowCell1101)))
+
+/-- Subcell `11010223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11010223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaBelowCell1101)))
+
+/-- Subcell `11010230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11010230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaBelowCell1101)))
+
+/-- Subcell `11010231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11010231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaBelowCell1101)))
+
+/-- Subcell `11010232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11010232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaBelowCell1101)))
+
+/-- Subcell `11010233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11010233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaBelowCell1101)))
+
+/-- Subcell `11012000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11013000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11102000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11103000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaBelowCell1110)))
+
+end GerverSofa.PartE.CertificateCellsce54db6860
+
+namespace GerverSofa.PartE.CoverCertificate446b5129d2
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate446b5129d2
+
+namespace GerverSofa.PartE.CoverCertificate3d29d6c185
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate3d29d6c185
+
+namespace GerverSofa.PartE.CoverCertificate5443a01fd5
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate5443a01fd5
+
+namespace GerverSofa.PartE.CoverCertificateef75f67166
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificateef75f67166
+
+namespace GerverSofa.PartE.CoverCertificate77769ce963
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate77769ce963
+
+namespace GerverSofa.PartE.CoverCertificateaaf4f15552
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificateaaf4f15552
+
+namespace GerverSofa.PartE.CertificateCells5c0c39e876
+
+/-- Subcell `3330` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3330 : AngleCell :=
+  childLL (childHH (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells5c0c39e876
+
+namespace GerverSofa.PartE.CertificateCellsa1414ae513
+
+/-- Subcell `3330` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3330 : AngleCell :=
+  childLL (childHH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3331` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3331 : AngleCell :=
+  childLH (childHH (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa1414ae513
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30000`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30001`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30002`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30003`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30010`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30011`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30012`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30013`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30020`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30021`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30022`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30023`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30030`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30031`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30032`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30033`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30100`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30101`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30102`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30103`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30110`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30111`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30112`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30113`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30120`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30121`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30122`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30123`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30130`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30131`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30132`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30133`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30200`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30201`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30202`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30203`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30210`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30211`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30212`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30213`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30220`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30221`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30222`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30223`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30230`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30231`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30232`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30233`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30300`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30301`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30302`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30303`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30310`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30311`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30312`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30313`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30320`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30321`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30322`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30323`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30330`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30331`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30332`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf30333`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31000`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31001`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31002`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31003`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31010`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31011`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31012`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31013`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31020`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31021`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31022`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31023`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31030`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31031`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31032`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31033`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf3110`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf3111`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31120`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31121`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31122`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31123`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf3113`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31200`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31201`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31202`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31203`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31210`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31211`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31212`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31213`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31220`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31221`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31222`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31223`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31230`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31231`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31232`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31233`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31300`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31301`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31302`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31303`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31310`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31311`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31312`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31313`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31320`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31321`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31322`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31323`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31330`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31331`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31332`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf31333`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32000`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32001`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32002`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32003`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32010`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32011`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32012`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32013`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32020`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32021`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32022`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32023`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32030`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32031`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32032`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32033`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32100`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32101`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32102`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32103`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32110`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32111`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32112`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32113`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32120`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32121`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32122`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32123`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32130`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32131`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32132`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32133`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32200`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32201`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32202`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32203`.
+* `KernelOnly.PartE.E24KC2PhiBelowLeaf32210`.
+* `KernelOnly.PartE.E24KC4BatchPhiAboveF9C1LLL0006`.
+* `KernelOnly.PartE.E24KC4BatchPhiAboveF9C1LLRLL0009`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30000 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells68c1109b36
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells68c1109b36
+
+open CertificateCells68c1109b36
+theorem e24KC2PhiBelowLeaf30000 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3000) = true := by
+  have h : ((childLL phiBelowCell3000)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3000) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30001 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3eb225f04c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3eb225f04c
+
+open CertificateCells3eb225f04c
+theorem e24KC2PhiBelowLeaf30001 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3000) = true := by
+  have h : ((childLH phiBelowCell3000)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3000) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30002 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells50ca4e3af4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells50ca4e3af4
+
+open CertificateCells50ca4e3af4
+theorem e24KC2PhiBelowLeaf30002 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3000) = true := by
+  have h : ((childHL phiBelowCell3000)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3000) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30003 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsfa60677f23
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsfa60677f23
+
+open CertificateCellsfa60677f23
+theorem e24KC2PhiBelowLeaf30003 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3000) = true := by
+  have h : ((childHH phiBelowCell3000)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3000) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30010 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsadacfb8de1
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsadacfb8de1
+
+open CertificateCellsadacfb8de1
+theorem e24KC2PhiBelowLeaf30010 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3001) = true := by
+  have h : ((childLL phiBelowCell3001)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3001) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30011 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2e1d52b748
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2e1d52b748
+
+open CertificateCells2e1d52b748
+theorem e24KC2PhiBelowLeaf30011 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3001) = true := by
+  have h : ((childLH phiBelowCell3001)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3001) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30012 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells7e7c37fb1b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells7e7c37fb1b
+
+open CertificateCells7e7c37fb1b
+theorem e24KC2PhiBelowLeaf30012 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3001) = true := by
+  have h : ((childHL phiBelowCell3001)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3001) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30013 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa8cebe99f5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa8cebe99f5
+
+open CertificateCellsa8cebe99f5
+theorem e24KC2PhiBelowLeaf30013 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3001) = true := by
+  have h : ((childHH phiBelowCell3001)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3001) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30020 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells7730b0e091
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells7730b0e091
+
+open CertificateCells7730b0e091
+theorem e24KC2PhiBelowLeaf30020 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3002) = true := by
+  have h : ((childLL phiBelowCell3002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3002) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30021 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse8a24af0c4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse8a24af0c4
+
+open CertificateCellse8a24af0c4
+theorem e24KC2PhiBelowLeaf30021 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3002) = true := by
+  have h : ((childLH phiBelowCell3002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3002) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30022 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse2e04a6320
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse2e04a6320
+
+open CertificateCellse2e04a6320
+theorem e24KC2PhiBelowLeaf30022 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3002) = true := by
+  have h : ((childHL phiBelowCell3002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3002) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30023 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1fb6bd85e4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1fb6bd85e4
+
+open CertificateCells1fb6bd85e4
+theorem e24KC2PhiBelowLeaf30023 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3002) = true := by
+  have h : ((childHH phiBelowCell3002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3002) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30030 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf0432d10ef
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf0432d10ef
+
+open CertificateCellsf0432d10ef
+theorem e24KC2PhiBelowLeaf30030 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3003) = true := by
+  have h : ((childLL phiBelowCell3003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3003) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30031 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells283f971137
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells283f971137
+
+open CertificateCells283f971137
+theorem e24KC2PhiBelowLeaf30031 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3003) = true := by
+  have h : ((childLH phiBelowCell3003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3003) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30032 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsad7da039d5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsad7da039d5
+
+open CertificateCellsad7da039d5
+theorem e24KC2PhiBelowLeaf30032 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3003) = true := by
+  have h : ((childHL phiBelowCell3003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3003) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30033 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1e800a73bb
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1e800a73bb
+
+open CertificateCells1e800a73bb
+theorem e24KC2PhiBelowLeaf30033 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3003) = true := by
+  have h : ((childHH phiBelowCell3003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3003) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30100 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb9d63d60b8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb9d63d60b8
+
+open CertificateCellsb9d63d60b8
+namespace CoverCertificate7eaf46f70e
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate7eaf46f70e
+
+theorem e24KC2PhiBelowLeaf30100 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3010) = true := by
+  exact CoverCertificate7eaf46f70e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30101 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells73ece5e183
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells73ece5e183
+
+open CertificateCells73ece5e183
+namespace CoverCertificate3307b34685
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3307b34685
+
+theorem e24KC2PhiBelowLeaf30101 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3010) = true := by
+  exact CoverCertificate3307b34685.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30102 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb58c03cf78
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb58c03cf78
+
+open CertificateCellsb58c03cf78
+namespace CoverCertificatea0fa380a93
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea0fa380a93
+
+theorem e24KC2PhiBelowLeaf30102 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3010) = true := by
+  exact CoverCertificatea0fa380a93.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30103 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3c73252ed6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3c73252ed6
+
+open CertificateCells3c73252ed6
+namespace CoverCertificate5f1fd442d3
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5f1fd442d3
+
+theorem e24KC2PhiBelowLeaf30103 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3010) = true := by
+  exact CoverCertificate5f1fd442d3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30110 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells75abf59285
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells75abf59285
+
+open CertificateCells75abf59285
+theorem e24KC2PhiBelowLeaf30110 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3011) = true := by
+  have h : ((childLL phiBelowCell3011)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3011) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30111 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc77278a9b0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc77278a9b0
+
+open CertificateCellsc77278a9b0
+theorem e24KC2PhiBelowLeaf30111 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3011) = true := by
+  have h : ((childLH phiBelowCell3011)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3011) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30112 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells80b449154c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells80b449154c
+
+open CertificateCells80b449154c
+namespace CoverCertificated6f9cbc5d6
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated6f9cbc5d6
+
+theorem e24KC2PhiBelowLeaf30112 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3011) = true := by
+  exact CoverCertificated6f9cbc5d6.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30113 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells19cd4580af
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells19cd4580af
+
+open CertificateCells19cd4580af
+theorem e24KC2PhiBelowLeaf30113 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3011) = true := by
+  have h : ((childHH phiBelowCell3011)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3011) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30120 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsad60f51a47
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsad60f51a47
+
+open CertificateCellsad60f51a47
+namespace CoverCertificatecaa270a0ec
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatecaa270a0ec
+
+theorem e24KC2PhiBelowLeaf30120 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3012) = true := by
+  exact CoverCertificatecaa270a0ec.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30121 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells59722ddf21
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells59722ddf21
+
+open CertificateCells59722ddf21
+namespace CoverCertificate0254ae442a
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0254ae442a
+
+theorem e24KC2PhiBelowLeaf30121 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3012) = true := by
+  exact CoverCertificate0254ae442a.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30122 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells38add90453
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells38add90453
+
+open CertificateCells38add90453
+namespace CoverCertificate44197a14bc
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate44197a14bc
+
+theorem e24KC2PhiBelowLeaf30122 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3012) = true := by
+  exact CoverCertificate44197a14bc.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30123 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa2fd436678
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa2fd436678
+
+open CertificateCellsa2fd436678
+namespace CoverCertificateaa4818b079
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateaa4818b079
+
+theorem e24KC2PhiBelowLeaf30123 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3012) = true := by
+  exact CoverCertificateaa4818b079.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30130 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4036e8788a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4036e8788a
+
+open CertificateCells4036e8788a
+namespace CoverCertificateade660b261
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateade660b261
+
+theorem e24KC2PhiBelowLeaf30130 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3013) = true := by
+  exact CoverCertificateade660b261.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30131 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse8e082f5ed
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse8e082f5ed
+
+open CertificateCellse8e082f5ed
+theorem e24KC2PhiBelowLeaf30131 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3013) = true := by
+  have h : ((childLH phiBelowCell3013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3013) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30132 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells33c2a4ae09
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells33c2a4ae09
+
+open CertificateCells33c2a4ae09
+namespace CoverCertificateead4291aaf
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateead4291aaf
+
+theorem e24KC2PhiBelowLeaf30132 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3013) = true := by
+  exact CoverCertificateead4291aaf.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30133 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8acf2cf43b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8acf2cf43b
+
+open CertificateCells8acf2cf43b
+namespace CoverCertificate76deff8421
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate76deff8421
+
+theorem e24KC2PhiBelowLeaf30133 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3013) = true := by
+  exact CoverCertificate76deff8421.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30200 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4c5b439e46
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4c5b439e46
+
+open CertificateCells4c5b439e46
+theorem e24KC2PhiBelowLeaf30200 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3020) = true := by
+  have h : ((childLL phiBelowCell3020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3020) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30201 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1d890d537b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1d890d537b
+
+open CertificateCells1d890d537b
+theorem e24KC2PhiBelowLeaf30201 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3020) = true := by
+  have h : ((childLH phiBelowCell3020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3020) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30202 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3dfd612f88
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3dfd612f88
+
+open CertificateCells3dfd612f88
+theorem e24KC2PhiBelowLeaf30202 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3020) = true := by
+  have h : ((childHL phiBelowCell3020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3020) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30203 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb73c54b84b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb73c54b84b
+
+open CertificateCellsb73c54b84b
+theorem e24KC2PhiBelowLeaf30203 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3020) = true := by
+  have h : ((childHH phiBelowCell3020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3020) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30210 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa2d944a3cb
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa2d944a3cb
+
+open CertificateCellsa2d944a3cb
+theorem e24KC2PhiBelowLeaf30210 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3021) = true := by
+  have h : ((childLL phiBelowCell3021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3021) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30211 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa175b3d804
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa175b3d804
+
+open CertificateCellsa175b3d804
+namespace CoverCertificatef9c3dd592d
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef9c3dd592d
+
+theorem e24KC2PhiBelowLeaf30211 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3021) = true := by
+  exact CoverCertificatef9c3dd592d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30212 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2377a41132
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2377a41132
+
+open CertificateCells2377a41132
+theorem e24KC2PhiBelowLeaf30212 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3021) = true := by
+  have h : ((childHL phiBelowCell3021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3021) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30213 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells141b81b7b9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells141b81b7b9
+
+open CertificateCells141b81b7b9
+namespace CoverCertificate55e26a37bf
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate55e26a37bf
+
+theorem e24KC2PhiBelowLeaf30213 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3021) = true := by
+  exact CoverCertificate55e26a37bf.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30220 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells53ed2717a4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells53ed2717a4
+
+open CertificateCells53ed2717a4
+theorem e24KC2PhiBelowLeaf30220 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3022) = true := by
+  have h : ((childLL phiBelowCell3022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3022) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30221 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6cc17280cd
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6cc17280cd
+
+open CertificateCells6cc17280cd
+theorem e24KC2PhiBelowLeaf30221 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3022) = true := by
+  have h : ((childLH phiBelowCell3022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3022) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30222 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellscd48c30df5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellscd48c30df5
+
+open CertificateCellscd48c30df5
+theorem e24KC2PhiBelowLeaf30222 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3022) = true := by
+  have h : ((childHL phiBelowCell3022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3022) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30223 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6a90ac3f5f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6a90ac3f5f
+
+open CertificateCells6a90ac3f5f
+theorem e24KC2PhiBelowLeaf30223 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3022) = true := by
+  have h : ((childHH phiBelowCell3022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3022) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30230 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellscb8a79a1e7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellscb8a79a1e7
+
+open CertificateCellscb8a79a1e7
+theorem e24KC2PhiBelowLeaf30230 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3023) = true := by
+  have h : ((childLL phiBelowCell3023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3023) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30231 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa17655a9cd
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa17655a9cd
+
+open CertificateCellsa17655a9cd
+namespace CoverCertificated546542719
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated546542719
+
+theorem e24KC2PhiBelowLeaf30231 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3023) = true := by
+  exact CoverCertificated546542719.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30232 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsea417091fc
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsea417091fc
+
+open CertificateCellsea417091fc
+theorem e24KC2PhiBelowLeaf30232 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3023) = true := by
+  have h : ((childHL phiBelowCell3023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3023) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30233 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells58f12e8647
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells58f12e8647
+
+open CertificateCells58f12e8647
+namespace CoverCertificate09704881ea
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate09704881ea
+
+theorem e24KC2PhiBelowLeaf30233 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3023) = true := by
+  exact CoverCertificate09704881ea.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30300 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5256e12156
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5256e12156
+
+open CertificateCells5256e12156
+namespace CoverCertificate89f8bf996f
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate89f8bf996f
+
+theorem e24KC2PhiBelowLeaf30300 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3030) = true := by
+  exact CoverCertificate89f8bf996f.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30301 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa7b064ca3f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa7b064ca3f
+
+open CertificateCellsa7b064ca3f
+namespace CoverCertificate072d32638e
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate072d32638e
+
+theorem e24KC2PhiBelowLeaf30301 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3030) = true := by
+  exact CoverCertificate072d32638e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30302 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa6d5038417
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa6d5038417
+
+open CertificateCellsa6d5038417
+namespace CoverCertificate5c3ff9507c
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5c3ff9507c
+
+theorem e24KC2PhiBelowLeaf30302 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3030) = true := by
+  exact CoverCertificate5c3ff9507c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30303 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa40437554b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa40437554b
+
+open CertificateCellsa40437554b
+namespace CoverCertificate0e6fd3d794
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0e6fd3d794
+
+theorem e24KC2PhiBelowLeaf30303 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3030) = true := by
+  exact CoverCertificate0e6fd3d794.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30310 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells99d0a4be54
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells99d0a4be54
+
+open CertificateCells99d0a4be54
+namespace CoverCertificate1d8d5ed6e8
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1d8d5ed6e8
+
+theorem e24KC2PhiBelowLeaf30310 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3031) = true := by
+  exact CoverCertificate1d8d5ed6e8.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30311 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells473a273fc9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells473a273fc9
+
+open CertificateCells473a273fc9
+namespace CoverCertificatead0c92c0d4
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatead0c92c0d4
+
+theorem e24KC2PhiBelowLeaf30311 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3031) = true := by
+  exact CoverCertificatead0c92c0d4.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30312 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3c757ee420
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3c757ee420
+
+open CertificateCells3c757ee420
+namespace CoverCertificate4e6964ff18
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4e6964ff18
+
+theorem e24KC2PhiBelowLeaf30312 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3031) = true := by
+  exact CoverCertificate4e6964ff18.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30313 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd8b0ff734e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd8b0ff734e
+
+open CertificateCellsd8b0ff734e
+namespace CoverCertificate14906b8c69
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate14906b8c69
+
+theorem e24KC2PhiBelowLeaf30313 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3031) = true := by
+  exact CoverCertificate14906b8c69.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30320 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells03aa6fa886
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells03aa6fa886
+
+open CertificateCells03aa6fa886
+namespace CoverCertificate4614f242b3
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4614f242b3
+
+theorem e24KC2PhiBelowLeaf30320 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3032) = true := by
+  exact CoverCertificate4614f242b3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30321 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf82ea84544
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf82ea84544
+
+open CertificateCellsf82ea84544
+namespace CoverCertificatef3c458cae3
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef3c458cae3
+
+theorem e24KC2PhiBelowLeaf30321 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3032) = true := by
+  exact CoverCertificatef3c458cae3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30322 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa1f10633df
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa1f10633df
+
+open CertificateCellsa1f10633df
+namespace CoverCertificateeb6553a302
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateeb6553a302
+
+theorem e24KC2PhiBelowLeaf30322 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3032) = true := by
+  exact CoverCertificateeb6553a302.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30323 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells92aeaddf18
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells92aeaddf18
+
+open CertificateCells92aeaddf18
+namespace CoverCertificatefa7152b230
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefa7152b230
+
+theorem e24KC2PhiBelowLeaf30323 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3032) = true := by
+  exact CoverCertificatefa7152b230.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30330 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4d2736f1ee
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4d2736f1ee
+
+open CertificateCells4d2736f1ee
+namespace CoverCertificate8afa5ed490
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate8afa5ed490
+
+theorem e24KC2PhiBelowLeaf30330 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3033) = true := by
+  exact CoverCertificate8afa5ed490.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30331 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf1c569afeb
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf1c569afeb
+
+open CertificateCellsf1c569afeb
+namespace CoverCertificated9debec79d
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated9debec79d
+
+theorem e24KC2PhiBelowLeaf30331 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3033) = true := by
+  exact CoverCertificated9debec79d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30332 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells899e3e87cc
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells899e3e87cc
+
+open CertificateCells899e3e87cc
+namespace CoverCertificate67d4072331
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate67d4072331
+
+theorem e24KC2PhiBelowLeaf30332 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3033) = true := by
+  exact CoverCertificate67d4072331.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=30333 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8cea727797
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8cea727797
+
+open CertificateCells8cea727797
+namespace CoverCertificatebed93df298
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatebed93df298
+
+theorem e24KC2PhiBelowLeaf30333 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3033) = true := by
+  exact CoverCertificatebed93df298.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31000 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdf788d056e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdf788d056e
+
+open CertificateCellsdf788d056e
+theorem e24KC2PhiBelowLeaf31000 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3100) = true := by
+  have h : ((childLL phiBelowCell3100)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3100) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31001 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells83252066d4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells83252066d4
+
+open CertificateCells83252066d4
+theorem e24KC2PhiBelowLeaf31001 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3100) = true := by
+  have h : ((childLH phiBelowCell3100)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3100) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31002 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd15521ecc4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd15521ecc4
+
+open CertificateCellsd15521ecc4
+theorem e24KC2PhiBelowLeaf31002 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3100) = true := by
+  have h : ((childHL phiBelowCell3100)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3100) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31003 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9ebe05244a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9ebe05244a
+
+open CertificateCells9ebe05244a
+theorem e24KC2PhiBelowLeaf31003 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3100) = true := by
+  have h : ((childHH phiBelowCell3100)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3100) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31010 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells21fc209558
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells21fc209558
+
+open CertificateCells21fc209558
+theorem e24KC2PhiBelowLeaf31010 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3101) = true := by
+  have h : ((childLL phiBelowCell3101)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3101) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31011 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc545b9b00b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc545b9b00b
+
+open CertificateCellsc545b9b00b
+theorem e24KC2PhiBelowLeaf31011 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3101) = true := by
+  have h : ((childLH phiBelowCell3101)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3101) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31012 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa64e976caf
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa64e976caf
+
+open CertificateCellsa64e976caf
+theorem e24KC2PhiBelowLeaf31012 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3101) = true := by
+  have h : ((childHL phiBelowCell3101)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3101) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31013 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf9eb6beb81
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf9eb6beb81
+
+open CertificateCellsf9eb6beb81
+theorem e24KC2PhiBelowLeaf31013 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3101) = true := by
+  have h : ((childHH phiBelowCell3101)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3101) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31020 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1d67e12ba0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1d67e12ba0
+
+open CertificateCells1d67e12ba0
+theorem e24KC2PhiBelowLeaf31020 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3102) = true := by
+  have h : ((childLL phiBelowCell3102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3102) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31021 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa2d5114e5e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa2d5114e5e
+
+open CertificateCellsa2d5114e5e
+theorem e24KC2PhiBelowLeaf31021 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3102) = true := by
+  have h : ((childLH phiBelowCell3102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3102) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31022 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells414314cff6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells414314cff6
+
+open CertificateCells414314cff6
+theorem e24KC2PhiBelowLeaf31022 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3102) = true := by
+  have h : ((childHL phiBelowCell3102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3102) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31023 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf9010539ff
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf9010539ff
+
+open CertificateCellsf9010539ff
+theorem e24KC2PhiBelowLeaf31023 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3102) = true := by
+  have h : ((childHH phiBelowCell3102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3102) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31030 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse34346aa4a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse34346aa4a
+
+open CertificateCellse34346aa4a
+theorem e24KC2PhiBelowLeaf31030 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3103) = true := by
+  have h : ((childLL phiBelowCell3103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3103) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31031 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse191f15fec
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse191f15fec
+
+open CertificateCellse191f15fec
+theorem e24KC2PhiBelowLeaf31031 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3103) = true := by
+  have h : ((childLH phiBelowCell3103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3103) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31032 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf92808a3cc
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf92808a3cc
+
+open CertificateCellsf92808a3cc
+theorem e24KC2PhiBelowLeaf31032 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3103) = true := by
+  have h : ((childHL phiBelowCell3103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3103) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31033 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells897e4339a9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells897e4339a9
+
+open CertificateCells897e4339a9
+theorem e24KC2PhiBelowLeaf31033 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3103) = true := by
+  have h : ((childHH phiBelowCell3103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3103) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=3110 remaining=10 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf4227a46c1
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf4227a46c1
+
+open CertificateCellsf4227a46c1
+theorem e24KC2PhiBelowLeaf3110 :
+    adaptiveCoverCheck 10 phiBelowCell3110 = true := by
+  have h : (phiBelowCell3110).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 phiBelowCell3110 h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=3111 remaining=10 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa2ba14667f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa2ba14667f
+
+open CertificateCellsa2ba14667f
+theorem e24KC2PhiBelowLeaf3111 :
+    adaptiveCoverCheck 10 phiBelowCell3111 = true := by
+  have h : (phiBelowCell3111).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 phiBelowCell3111 h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31120 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells172a47199c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells172a47199c
+
+open CertificateCells172a47199c
+theorem e24KC2PhiBelowLeaf31120 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3112) = true := by
+  have h : ((childLL phiBelowCell3112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3112) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31121 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells503a496e12
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells503a496e12
+
+open CertificateCells503a496e12
+theorem e24KC2PhiBelowLeaf31121 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3112) = true := by
+  have h : ((childLH phiBelowCell3112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3112) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31122 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4f35c82776
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4f35c82776
+
+open CertificateCells4f35c82776
+theorem e24KC2PhiBelowLeaf31122 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3112) = true := by
+  have h : ((childHL phiBelowCell3112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3112) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31123 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsddf26a1af9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsddf26a1af9
+
+open CertificateCellsddf26a1af9
+theorem e24KC2PhiBelowLeaf31123 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3112) = true := by
+  have h : ((childHH phiBelowCell3112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3112) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=3113 remaining=10 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells725d74e81a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells725d74e81a
+
+open CertificateCells725d74e81a
+theorem e24KC2PhiBelowLeaf3113 :
+    adaptiveCoverCheck 10 phiBelowCell3113 = true := by
+  have h : (phiBelowCell3113).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 phiBelowCell3113 h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31200 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3b33327e8a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3b33327e8a
+
+open CertificateCells3b33327e8a
+namespace CoverCertificatefd2c3f3274
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefd2c3f3274
+
+theorem e24KC2PhiBelowLeaf31200 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3120) = true := by
+  exact CoverCertificatefd2c3f3274.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31201 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells7bfd1f464a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells7bfd1f464a
+
+open CertificateCells7bfd1f464a
+theorem e24KC2PhiBelowLeaf31201 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3120) = true := by
+  have h : ((childLH phiBelowCell3120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3120) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31202 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdd644ededa
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdd644ededa
+
+open CertificateCellsdd644ededa
+namespace CoverCertificate1ffa851843
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1ffa851843
+
+theorem e24KC2PhiBelowLeaf31202 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3120) = true := by
+  exact CoverCertificate1ffa851843.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31203 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdd54c72360
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdd54c72360
+
+open CertificateCellsdd54c72360
+namespace CoverCertificated8ac44ac8f
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated8ac44ac8f
+
+theorem e24KC2PhiBelowLeaf31203 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3120) = true := by
+  exact CoverCertificated8ac44ac8f.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31210 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf0f5b1fc54
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf0f5b1fc54
+
+open CertificateCellsf0f5b1fc54
+theorem e24KC2PhiBelowLeaf31210 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3121) = true := by
+  have h : ((childLL phiBelowCell3121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3121) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31211 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2b0822f9b0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2b0822f9b0
+
+open CertificateCells2b0822f9b0
+theorem e24KC2PhiBelowLeaf31211 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3121) = true := by
+  have h : ((childLH phiBelowCell3121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3121) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31212 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells05e3ee5af2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells05e3ee5af2
+
+open CertificateCells05e3ee5af2
+theorem e24KC2PhiBelowLeaf31212 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3121) = true := by
+  have h : ((childHL phiBelowCell3121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3121) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31213 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8f81fc559c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8f81fc559c
+
+open CertificateCells8f81fc559c
+theorem e24KC2PhiBelowLeaf31213 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3121) = true := by
+  have h : ((childHH phiBelowCell3121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3121) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31220 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells509a6074b0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells509a6074b0
+
+open CertificateCells509a6074b0
+namespace CoverCertificate3caeb3dd1d
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3caeb3dd1d
+
+theorem e24KC2PhiBelowLeaf31220 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3122) = true := by
+  exact CoverCertificate3caeb3dd1d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31221 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6ac6e4ee1a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6ac6e4ee1a
+
+open CertificateCells6ac6e4ee1a
+namespace CoverCertificate3195681328
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3195681328
+
+theorem e24KC2PhiBelowLeaf31221 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3122) = true := by
+  exact CoverCertificate3195681328.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31222 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2e92118453
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2e92118453
+
+open CertificateCells2e92118453
+namespace CoverCertificate24d91bb18c
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate24d91bb18c
+
+theorem e24KC2PhiBelowLeaf31222 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3122) = true := by
+  exact CoverCertificate24d91bb18c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31223 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5761c4811e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5761c4811e
+
+open CertificateCells5761c4811e
+namespace CoverCertificatea5eff6d169
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea5eff6d169
+
+theorem e24KC2PhiBelowLeaf31223 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3122) = true := by
+  exact CoverCertificatea5eff6d169.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31230 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf228466209
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf228466209
+
+open CertificateCellsf228466209
+theorem e24KC2PhiBelowLeaf31230 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3123) = true := by
+  have h : ((childLL phiBelowCell3123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3123) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31231 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4924d89385
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4924d89385
+
+open CertificateCells4924d89385
+theorem e24KC2PhiBelowLeaf31231 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3123) = true := by
+  have h : ((childLH phiBelowCell3123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3123) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31232 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells75bbd1e667
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells75bbd1e667
+
+open CertificateCells75bbd1e667
+namespace CoverCertificateabfc719760
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateabfc719760
+
+theorem e24KC2PhiBelowLeaf31232 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3123) = true := by
+  exact CoverCertificateabfc719760.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31233 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells131513700f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells131513700f
+
+open CertificateCells131513700f
+theorem e24KC2PhiBelowLeaf31233 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3123) = true := by
+  have h : ((childHH phiBelowCell3123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3123) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31300 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0436aa31cb
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0436aa31cb
+
+open CertificateCells0436aa31cb
+theorem e24KC2PhiBelowLeaf31300 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3130) = true := by
+  have h : ((childLL phiBelowCell3130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3130) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31301 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4cca5ef5ea
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4cca5ef5ea
+
+open CertificateCells4cca5ef5ea
+theorem e24KC2PhiBelowLeaf31301 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3130) = true := by
+  have h : ((childLH phiBelowCell3130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3130) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31302 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2133748f57
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2133748f57
+
+open CertificateCells2133748f57
+theorem e24KC2PhiBelowLeaf31302 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3130) = true := by
+  have h : ((childHL phiBelowCell3130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3130) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31303 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2d9432dbe0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2d9432dbe0
+
+open CertificateCells2d9432dbe0
+theorem e24KC2PhiBelowLeaf31303 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3130) = true := by
+  have h : ((childHH phiBelowCell3130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3130) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31310 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells28839ca767
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells28839ca767
+
+open CertificateCells28839ca767
+theorem e24KC2PhiBelowLeaf31310 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3131) = true := by
+  have h : ((childLL phiBelowCell3131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3131) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31311 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1b0bf20c51
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1b0bf20c51
+
+open CertificateCells1b0bf20c51
+theorem e24KC2PhiBelowLeaf31311 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3131) = true := by
+  have h : ((childLH phiBelowCell3131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3131) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31312 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9fb8ae7256
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9fb8ae7256
+
+open CertificateCells9fb8ae7256
+theorem e24KC2PhiBelowLeaf31312 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3131) = true := by
+  have h : ((childHL phiBelowCell3131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3131) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31313 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells10b8b658d1
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells10b8b658d1
+
+open CertificateCells10b8b658d1
+theorem e24KC2PhiBelowLeaf31313 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3131) = true := by
+  have h : ((childHH phiBelowCell3131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3131) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31320 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells372671dc1b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells372671dc1b
+
+open CertificateCells372671dc1b
+theorem e24KC2PhiBelowLeaf31320 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3132) = true := by
+  have h : ((childLL phiBelowCell3132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3132) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31321 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells91a123c697
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells91a123c697
+
+open CertificateCells91a123c697
+theorem e24KC2PhiBelowLeaf31321 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3132) = true := by
+  have h : ((childLH phiBelowCell3132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3132) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31322 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsce14c27bce
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsce14c27bce
+
+open CertificateCellsce14c27bce
+theorem e24KC2PhiBelowLeaf31322 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3132) = true := by
+  have h : ((childHL phiBelowCell3132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3132) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31323 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsabc89a203a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsabc89a203a
+
+open CertificateCellsabc89a203a
+theorem e24KC2PhiBelowLeaf31323 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3132) = true := by
+  have h : ((childHH phiBelowCell3132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3132) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31330 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6f99a95071
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6f99a95071
+
+open CertificateCells6f99a95071
+theorem e24KC2PhiBelowLeaf31330 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3133) = true := by
+  have h : ((childLL phiBelowCell3133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3133) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31331 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4acdbba755
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4acdbba755
+
+open CertificateCells4acdbba755
+theorem e24KC2PhiBelowLeaf31331 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3133) = true := by
+  have h : ((childLH phiBelowCell3133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3133) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31332 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells45607104df
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells45607104df
+
+open CertificateCells45607104df
+theorem e24KC2PhiBelowLeaf31332 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3133) = true := by
+  have h : ((childHL phiBelowCell3133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3133) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=31333 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf2e66397f6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf2e66397f6
+
+open CertificateCellsf2e66397f6
+theorem e24KC2PhiBelowLeaf31333 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3133) = true := by
+  have h : ((childHH phiBelowCell3133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3133) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32000 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf60a68912f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf60a68912f
+
+open CertificateCellsf60a68912f
+theorem e24KC2PhiBelowLeaf32000 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3200) = true := by
+  have h : ((childLL phiBelowCell3200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3200) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32001 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdb667fab49
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdb667fab49
+
+open CertificateCellsdb667fab49
+theorem e24KC2PhiBelowLeaf32001 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3200) = true := by
+  have h : ((childLH phiBelowCell3200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3200) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32002 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells378433963d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells378433963d
+
+open CertificateCells378433963d
+theorem e24KC2PhiBelowLeaf32002 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3200) = true := by
+  have h : ((childHL phiBelowCell3200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3200) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32003 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3c30c58615
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3c30c58615
+
+open CertificateCells3c30c58615
+theorem e24KC2PhiBelowLeaf32003 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3200) = true := by
+  have h : ((childHH phiBelowCell3200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3200) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32010 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellscbc2cb8c4d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellscbc2cb8c4d
+
+open CertificateCellscbc2cb8c4d
+theorem e24KC2PhiBelowLeaf32010 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3201) = true := by
+  have h : ((childLL phiBelowCell3201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3201) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32011 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0ae5e3afe8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0ae5e3afe8
+
+open CertificateCells0ae5e3afe8
+namespace CoverCertificate0a514de912
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0a514de912
+
+theorem e24KC2PhiBelowLeaf32011 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3201) = true := by
+  exact CoverCertificate0a514de912.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32012 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells72905f39fb
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells72905f39fb
+
+open CertificateCells72905f39fb
+theorem e24KC2PhiBelowLeaf32012 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3201) = true := by
+  have h : ((childHL phiBelowCell3201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3201) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32013 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells36ded03275
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells36ded03275
+
+open CertificateCells36ded03275
+namespace CoverCertificate7ed638f543
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate7ed638f543
+
+theorem e24KC2PhiBelowLeaf32013 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3201) = true := by
+  exact CoverCertificate7ed638f543.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32020 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells055eee17e7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells055eee17e7
+
+open CertificateCells055eee17e7
+theorem e24KC2PhiBelowLeaf32020 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3202) = true := by
+  have h : ((childLL phiBelowCell3202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3202) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32021 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells86a7b10d51
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells86a7b10d51
+
+open CertificateCells86a7b10d51
+theorem e24KC2PhiBelowLeaf32021 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3202) = true := by
+  have h : ((childLH phiBelowCell3202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3202) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32022 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells62973479ba
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells62973479ba
+
+open CertificateCells62973479ba
+theorem e24KC2PhiBelowLeaf32022 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3202) = true := by
+  have h : ((childHL phiBelowCell3202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3202) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32023 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse26adcc478
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse26adcc478
+
+open CertificateCellse26adcc478
+theorem e24KC2PhiBelowLeaf32023 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3202) = true := by
+  have h : ((childHH phiBelowCell3202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3202) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32030 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd471241912
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd471241912
+
+open CertificateCellsd471241912
+theorem e24KC2PhiBelowLeaf32030 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3203) = true := by
+  have h : ((childLL phiBelowCell3203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3203) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32031 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells89c6c55f33
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells89c6c55f33
+
+open CertificateCells89c6c55f33
+namespace CoverCertificatebdb1950cb6
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatebdb1950cb6
+
+theorem e24KC2PhiBelowLeaf32031 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3203) = true := by
+  exact CoverCertificatebdb1950cb6.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32032 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells371a4c38c5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells371a4c38c5
+
+open CertificateCells371a4c38c5
+theorem e24KC2PhiBelowLeaf32032 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3203) = true := by
+  have h : ((childHL phiBelowCell3203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3203) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32033 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells96fe0c4e84
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells96fe0c4e84
+
+open CertificateCells96fe0c4e84
+namespace CoverCertificate5fb7a54ee1
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5fb7a54ee1
+
+theorem e24KC2PhiBelowLeaf32033 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3203) = true := by
+  exact CoverCertificate5fb7a54ee1.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32100 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9ae3b0f784
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9ae3b0f784
+
+open CertificateCells9ae3b0f784
+namespace CoverCertificated58a87e9b8
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated58a87e9b8
+
+theorem e24KC2PhiBelowLeaf32100 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3210) = true := by
+  exact CoverCertificated58a87e9b8.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32101 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9614d1d005
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9614d1d005
+
+open CertificateCells9614d1d005
+namespace CoverCertificate3ddf17b582
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3ddf17b582
+
+theorem e24KC2PhiBelowLeaf32101 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3210) = true := by
+  exact CoverCertificate3ddf17b582.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32102 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells27f26686ef
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells27f26686ef
+
+open CertificateCells27f26686ef
+namespace CoverCertificated26b7e0a55
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated26b7e0a55
+
+theorem e24KC2PhiBelowLeaf32102 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3210) = true := by
+  exact CoverCertificated26b7e0a55.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32103 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9dc2121909
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9dc2121909
+
+open CertificateCells9dc2121909
+namespace CoverCertificatea04fbf8169
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea04fbf8169
+
+theorem e24KC2PhiBelowLeaf32103 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3210) = true := by
+  exact CoverCertificatea04fbf8169.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32110 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc35310d566
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc35310d566
+
+open CertificateCellsc35310d566
+namespace CoverCertificate3ba43c403b
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3ba43c403b
+
+theorem e24KC2PhiBelowLeaf32110 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3211) = true := by
+  exact CoverCertificate3ba43c403b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32111 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa20e4da390
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa20e4da390
+
+open CertificateCellsa20e4da390
+namespace CoverCertificate1430173dea
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1430173dea
+
+theorem e24KC2PhiBelowLeaf32111 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3211) = true := by
+  exact CoverCertificate1430173dea.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32112 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3a90cbf635
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3a90cbf635
+
+open CertificateCells3a90cbf635
+namespace CoverCertificate2756e7c1ab
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate2756e7c1ab
+
+theorem e24KC2PhiBelowLeaf32112 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3211) = true := by
+  exact CoverCertificate2756e7c1ab.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32113 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells214a11de68
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells214a11de68
+
+open CertificateCells214a11de68
+namespace CoverCertificate8e43edbf8f
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate8e43edbf8f
+
+theorem e24KC2PhiBelowLeaf32113 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3211) = true := by
+  exact CoverCertificate8e43edbf8f.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32120 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells032b91f5bc
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells032b91f5bc
+
+open CertificateCells032b91f5bc
+namespace CoverCertificated1acdb7cc1
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated1acdb7cc1
+
+theorem e24KC2PhiBelowLeaf32120 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3212) = true := by
+  exact CoverCertificated1acdb7cc1.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32121 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsaa8e493e23
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsaa8e493e23
+
+open CertificateCellsaa8e493e23
+namespace CoverCertificated9d4fcb914
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated9d4fcb914
+
+theorem e24KC2PhiBelowLeaf32121 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3212) = true := by
+  exact CoverCertificated9d4fcb914.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32122 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6bd715f340
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6bd715f340
+
+open CertificateCells6bd715f340
+namespace CoverCertificate18d654774a
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate18d654774a
+
+theorem e24KC2PhiBelowLeaf32122 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3212) = true := by
+  exact CoverCertificate18d654774a.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32123 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa228f232aa
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa228f232aa
+
+open CertificateCellsa228f232aa
+namespace CoverCertificate3c1ddcbe4a
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3c1ddcbe4a
+
+theorem e24KC2PhiBelowLeaf32123 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3212) = true := by
+  exact CoverCertificate3c1ddcbe4a.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32130 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4dea68b576
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4dea68b576
+
+open CertificateCells4dea68b576
+namespace CoverCertificatecaa9842c01
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatecaa9842c01
+
+theorem e24KC2PhiBelowLeaf32130 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3213) = true := by
+  exact CoverCertificatecaa9842c01.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32131 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3e707fd6af
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3e707fd6af
+
+open CertificateCells3e707fd6af
+namespace CoverCertificate8a64f55483
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate8a64f55483
+
+theorem e24KC2PhiBelowLeaf32131 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3213) = true := by
+  exact CoverCertificate8a64f55483.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32132 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells46e40ef0a9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells46e40ef0a9
+
+open CertificateCells46e40ef0a9
+namespace CoverCertificatea45749320c
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea45749320c
+
+theorem e24KC2PhiBelowLeaf32132 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3213) = true := by
+  exact CoverCertificatea45749320c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32133 remaining=9 kind=F reason=ADAPTIVE.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9651c4a476
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9651c4a476
+
+open CertificateCells9651c4a476
+namespace CoverCertificate96200cdbc3
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate96200cdbc3
+
+theorem e24KC2PhiBelowLeaf32133 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3213) = true := by
+  exact CoverCertificate96200cdbc3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32200 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2fa7cd8552
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2fa7cd8552
+
+open CertificateCells2fa7cd8552
+theorem e24KC2PhiBelowLeaf32200 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3220) = true := by
+  have h : ((childLL phiBelowCell3220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3220) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32201 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsec4b7eff4d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsec4b7eff4d
+
+open CertificateCellsec4b7eff4d
+theorem e24KC2PhiBelowLeaf32201 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3220) = true := by
+  have h : ((childLH phiBelowCell3220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3220) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32202 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells434ca1e74d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells434ca1e74d
+
+open CertificateCells434ca1e74d
+theorem e24KC2PhiBelowLeaf32202 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3220) = true := by
+  have h : ((childHL phiBelowCell3220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3220) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32203 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells886952c236
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells886952c236
+
+open CertificateCells886952c236
+theorem e24KC2PhiBelowLeaf32203 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3220) = true := by
+  have h : ((childHH phiBelowCell3220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3220) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! Generated E24KC2 kernel leaf. Discovery claim: PB path=32210 remaining=9 kind=T reason=R. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells969a7c12d6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells969a7c12d6
+
+open CertificateCells969a7c12d6
+theorem e24KC2PhiBelowLeaf32210 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3221) = true := by
+  have h : ((childLL phiBelowCell3221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3221) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC4 auto-tuned batched kernel certificates. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells7aa82bee5f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells7aa82bee5f
+
+open CertificateCells7aa82bee5f
+namespace CoverCertificate1e0715601d
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1e0715601d
+
+theorem e24KC2PhiAboveLeaf1100111 :
+    adaptiveCoverCheck 9 (childLH (childLH (childLH phiAboveCell1100))) = true := by
+  exact CoverCertificate1e0715601d.checkedRoot
+namespace CoverCertificate645c2a2fa6
+
+private theorem checked010 : adaptiveCoverCheck 6 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 6 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 6 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 6 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 6 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 6 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 6 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 6 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 6 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 6 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 6 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 6 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate645c2a2fa6
+
+theorem e24KC2PhiAboveLeaf1101000 :
+    adaptiveCoverCheck 9 (childLL (childLL (childLL phiAboveCell1101))) = true := by
+  exact CoverCertificate645c2a2fa6.checkedRoot
+namespace CoverCertificate22a5b25f41
+
+private theorem checked000 : adaptiveCoverCheck 6 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 6 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 6 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 6 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 6 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 6 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 6 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 6 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 6 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 6 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 6 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 6 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 6 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 6 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 6 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 6 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate22a5b25f41
+
+theorem e24KC2PhiAboveLeaf1101001 :
+    adaptiveCoverCheck 9 (childLH (childLL (childLL phiAboveCell1101))) = true := by
+  exact CoverCertificate22a5b25f41.checkedRoot
+namespace CoverCertificate052935fbc6
+
+private theorem checked1010 : adaptiveCoverCheck 5 cell1010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1010 (by decide +kernel)
+
+private theorem checked1011 : adaptiveCoverCheck 5 cell1011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1011 (by decide +kernel)
+
+private theorem checked1012 : adaptiveCoverCheck 5 cell1012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1012 (by decide +kernel)
+
+private theorem checked1013 : adaptiveCoverCheck 5 cell1013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1013 (by decide +kernel)
+
+private theorem checked1100 : adaptiveCoverCheck 5 cell1100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1100 (by decide +kernel)
+
+private theorem checked1101 : adaptiveCoverCheck 5 cell1101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1101 (by decide +kernel)
+
+private theorem checked1102 : adaptiveCoverCheck 5 cell1102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1102 (by decide +kernel)
+
+private theorem checked1103 : adaptiveCoverCheck 5 cell1103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1103 (by decide +kernel)
+
+private theorem checked1110 : adaptiveCoverCheck 5 cell1110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1110 (by decide +kernel)
+
+private theorem checked1111 : adaptiveCoverCheck 5 cell1111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1111 (by decide +kernel)
+
+private theorem checked1112 : adaptiveCoverCheck 5 cell1112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1112 (by decide +kernel)
+
+private theorem checked1113 : adaptiveCoverCheck 5 cell1113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1113 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 6 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 6 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 6 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 6 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 6 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 6 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 6 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 6 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 6 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 6 cell101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell101
+    checked1010 checked1011 checked1012 checked1013
+
+private theorem checked102 : adaptiveCoverCheck 6 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 6 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 6 cell110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell110
+    checked1100 checked1101 checked1102 checked1103
+
+private theorem checked111 : adaptiveCoverCheck 6 cell111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell111
+    checked1110 checked1111 checked1112 checked1113
+
+private theorem checked112 : adaptiveCoverCheck 6 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 6 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate052935fbc6
+
+theorem e24KC2PhiAboveLeaf1101010 :
+    adaptiveCoverCheck 9 (childLL (childLH (childLL phiAboveCell1101))) = true := by
+  exact CoverCertificate052935fbc6.checkedRoot
+namespace CoverCertificateda1b0b5031
+
+private theorem checked11110 : adaptiveCoverCheck 4 cell11110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11110 (by decide +kernel)
+
+private theorem checked11111 : adaptiveCoverCheck 4 cell11111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11111 (by decide +kernel)
+
+private theorem checked11112 : adaptiveCoverCheck 4 cell11112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11112 (by decide +kernel)
+
+private theorem checked11113 : adaptiveCoverCheck 4 cell11113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11113 (by decide +kernel)
+
+private theorem checked0000 : adaptiveCoverCheck 5 cell0000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0000 (by decide +kernel)
+
+private theorem checked0001 : adaptiveCoverCheck 5 cell0001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0001 (by decide +kernel)
+
+private theorem checked0002 : adaptiveCoverCheck 5 cell0002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0002 (by decide +kernel)
+
+private theorem checked0003 : adaptiveCoverCheck 5 cell0003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0003 (by decide +kernel)
+
+private theorem checked0010 : adaptiveCoverCheck 5 cell0010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0010 (by decide +kernel)
+
+private theorem checked0011 : adaptiveCoverCheck 5 cell0011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0011 (by decide +kernel)
+
+private theorem checked0012 : adaptiveCoverCheck 5 cell0012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0012 (by decide +kernel)
+
+private theorem checked0013 : adaptiveCoverCheck 5 cell0013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0013 (by decide +kernel)
+
+private theorem checked0100 : adaptiveCoverCheck 5 cell0100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0100 (by decide +kernel)
+
+private theorem checked0101 : adaptiveCoverCheck 5 cell0101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0101 (by decide +kernel)
+
+private theorem checked0102 : adaptiveCoverCheck 5 cell0102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0102 (by decide +kernel)
+
+private theorem checked0103 : adaptiveCoverCheck 5 cell0103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0103 (by decide +kernel)
+
+private theorem checked0110 : adaptiveCoverCheck 5 cell0110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0110 (by decide +kernel)
+
+private theorem checked0111 : adaptiveCoverCheck 5 cell0111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0111 (by decide +kernel)
+
+private theorem checked0112 : adaptiveCoverCheck 5 cell0112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0112 (by decide +kernel)
+
+private theorem checked0113 : adaptiveCoverCheck 5 cell0113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0113 (by decide +kernel)
+
+private theorem checked1000 : adaptiveCoverCheck 5 cell1000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1000 (by decide +kernel)
+
+private theorem checked1001 : adaptiveCoverCheck 5 cell1001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1001 (by decide +kernel)
+
+private theorem checked1002 : adaptiveCoverCheck 5 cell1002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1002 (by decide +kernel)
+
+private theorem checked1003 : adaptiveCoverCheck 5 cell1003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1003 (by decide +kernel)
+
+private theorem checked1010 : adaptiveCoverCheck 5 cell1010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1010 (by decide +kernel)
+
+private theorem checked1011 : adaptiveCoverCheck 5 cell1011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1011 (by decide +kernel)
+
+private theorem checked1012 : adaptiveCoverCheck 5 cell1012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1012 (by decide +kernel)
+
+private theorem checked1013 : adaptiveCoverCheck 5 cell1013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1013 (by decide +kernel)
+
+private theorem checked1100 : adaptiveCoverCheck 5 cell1100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1100 (by decide +kernel)
+
+private theorem checked1101 : adaptiveCoverCheck 5 cell1101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1101 (by decide +kernel)
+
+private theorem checked1102 : adaptiveCoverCheck 5 cell1102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1102 (by decide +kernel)
+
+private theorem checked1103 : adaptiveCoverCheck 5 cell1103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1103 (by decide +kernel)
+
+private theorem checked1110 : adaptiveCoverCheck 5 cell1110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1110 (by decide +kernel)
+
+private theorem checked1111 : adaptiveCoverCheck 5 cell1111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1111
+    checked11110 checked11111 checked11112 checked11113
+
+private theorem checked1112 : adaptiveCoverCheck 5 cell1112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1112 (by decide +kernel)
+
+private theorem checked1113 : adaptiveCoverCheck 5 cell1113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1113 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 6 cell000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell000
+    checked0000 checked0001 checked0002 checked0003
+
+private theorem checked001 : adaptiveCoverCheck 6 cell001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell001
+    checked0010 checked0011 checked0012 checked0013
+
+private theorem checked002 : adaptiveCoverCheck 6 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 6 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 6 cell010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell010
+    checked0100 checked0101 checked0102 checked0103
+
+private theorem checked011 : adaptiveCoverCheck 6 cell011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell011
+    checked0110 checked0111 checked0112 checked0113
+
+private theorem checked012 : adaptiveCoverCheck 6 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 6 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 6 cell100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell100
+    checked1000 checked1001 checked1002 checked1003
+
+private theorem checked101 : adaptiveCoverCheck 6 cell101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell101
+    checked1010 checked1011 checked1012 checked1013
+
+private theorem checked102 : adaptiveCoverCheck 6 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 6 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 6 cell110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell110
+    checked1100 checked1101 checked1102 checked1103
+
+private theorem checked111 : adaptiveCoverCheck 6 cell111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell111
+    checked1110 checked1111 checked1112 checked1113
+
+private theorem checked112 : adaptiveCoverCheck 6 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 6 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateda1b0b5031
+
+theorem e24KC2PhiAboveLeaf1101011 :
+    adaptiveCoverCheck 9 (childLH (childLH (childLL phiAboveCell1101))) = true := by
+  exact CoverCertificateda1b0b5031.checkedRoot
+namespace CoverCertificate87253a3a53
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate87253a3a53
+
+theorem e24KC2PhiAboveLeaf1101012 :
+    adaptiveCoverCheck 9 (childHL (childLH (childLL phiAboveCell1101))) = true := by
+  exact CoverCertificate87253a3a53.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC4 auto-tuned batched kernel certificates. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf8d478d36a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf8d478d36a
+
+open CertificateCellsf8d478d36a
+namespace CoverCertificate3f791a2e61
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3f791a2e61
+
+theorem e24KC2PhiAboveLeaf1101013 :
+    adaptiveCoverCheck 9 (childHH (childLH (childLL phiAboveCell1101))) = true := by
+  exact CoverCertificate3f791a2e61.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC5TerminalBatchT665600016`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsce54db6860
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsce54db6860
+
+open CertificateCellsce54db6860
+theorem e24KC2ThetaBelowLeaf1023 :
+    adaptiveCoverCheck 14 thetaBelowCell1023 = true := by
+  have h : (thetaBelowCell1023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1023 h
+theorem e24KC2ThetaBelowLeaf1030 :
+    adaptiveCoverCheck 14 thetaBelowCell1030 = true := by
+  have h : (thetaBelowCell1030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1030 h
+theorem e24KC2ThetaBelowLeaf1031 :
+    adaptiveCoverCheck 14 thetaBelowCell1031 = true := by
+  have h : (thetaBelowCell1031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1031 h
+theorem e24KC2ThetaBelowLeaf1032 :
+    adaptiveCoverCheck 14 thetaBelowCell1032 = true := by
+  have h : (thetaBelowCell1032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1032 h
+theorem e24KC2ThetaBelowLeaf1033 :
+    adaptiveCoverCheck 14 thetaBelowCell1033 = true := by
+  have h : (thetaBelowCell1033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1033 h
+theorem e24KC2ThetaBelowLeaf110000 :
+    adaptiveCoverCheck 12 (childLL (childLL thetaBelowCell1100)) = true := by
+  have h : ((childLL (childLL thetaBelowCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL thetaBelowCell1100)) h
+theorem e24KC2ThetaBelowLeaf110001 :
+    adaptiveCoverCheck 12 (childLH (childLL thetaBelowCell1100)) = true := by
+  have h : ((childLH (childLL thetaBelowCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL thetaBelowCell1100)) h
+theorem e24KC2ThetaBelowLeaf1100020 :
+    adaptiveCoverCheck 11 (childLL (childHL (childLL thetaBelowCell1100))) = true := by
+  have h : ((childLL (childHL (childLL thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childLL thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100021 :
+    adaptiveCoverCheck 11 (childLH (childHL (childLL thetaBelowCell1100))) = true := by
+  have h : ((childLH (childHL (childLL thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childLL thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100022 :
+    adaptiveCoverCheck 11 (childHL (childHL (childLL thetaBelowCell1100))) = true := by
+  have h : ((childHL (childHL (childLL thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childLL thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100023 :
+    adaptiveCoverCheck 11 (childHH (childHL (childLL thetaBelowCell1100))) = true := by
+  have h : ((childHH (childHL (childLL thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childLL thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100030 :
+    adaptiveCoverCheck 11 (childLL (childHH (childLL thetaBelowCell1100))) = true := by
+  have h : ((childLL (childHH (childLL thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childLL thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100031 :
+    adaptiveCoverCheck 11 (childLH (childHH (childLL thetaBelowCell1100))) = true := by
+  have h : ((childLH (childHH (childLL thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childLL thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf11000320 :
+    adaptiveCoverCheck 10 thetaBelowCell11000320 = true := by
+  have h : (thetaBelowCell11000320).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11000320 h
+theorem e24KC2ThetaBelowLeaf11000321 :
+    adaptiveCoverCheck 10 thetaBelowCell11000321 = true := by
+  have h : (thetaBelowCell11000321).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11000321 h
+theorem e24KC2ThetaBelowLeaf11000322 :
+    adaptiveCoverCheck 10 thetaBelowCell11000322 = true := by
+  have h : (thetaBelowCell11000322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11000322 h
+theorem e24KC2ThetaBelowLeaf11000323 :
+    adaptiveCoverCheck 10 thetaBelowCell11000323 = true := by
+  have h : (thetaBelowCell11000323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11000323 h
+theorem e24KC2ThetaBelowLeaf11000330 :
+    adaptiveCoverCheck 10 thetaBelowCell11000330 = true := by
+  have h : (thetaBelowCell11000330).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11000330 h
+theorem e24KC2ThetaBelowLeaf11000331 :
+    adaptiveCoverCheck 10 thetaBelowCell11000331 = true := by
+  have h : (thetaBelowCell11000331).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11000331 h
+theorem e24KC2ThetaBelowLeaf11000332 :
+    adaptiveCoverCheck 10 thetaBelowCell11000332 = true := by
+  have h : (thetaBelowCell11000332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11000332 h
+theorem e24KC2ThetaBelowLeaf11000333 :
+    adaptiveCoverCheck 10 thetaBelowCell11000333 = true := by
+  have h : (thetaBelowCell11000333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11000333 h
+theorem e24KC2ThetaBelowLeaf110010 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell1100)) = true := by
+  have h : ((childLL (childLH thetaBelowCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH thetaBelowCell1100)) h
+theorem e24KC2ThetaBelowLeaf110011 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell1100)) = true := by
+  have h : ((childLH (childLH thetaBelowCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH thetaBelowCell1100)) h
+theorem e24KC2ThetaBelowLeaf1100120 :
+    adaptiveCoverCheck 11 (childLL (childHL (childLH thetaBelowCell1100))) = true := by
+  have h : ((childLL (childHL (childLH thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childLH thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100121 :
+    adaptiveCoverCheck 11 (childLH (childHL (childLH thetaBelowCell1100))) = true := by
+  have h : ((childLH (childHL (childLH thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childLH thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf11001220 :
+    adaptiveCoverCheck 10 thetaBelowCell11001220 = true := by
+  have h : (thetaBelowCell11001220).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001220 h
+theorem e24KC2ThetaBelowLeaf11001221 :
+    adaptiveCoverCheck 10 thetaBelowCell11001221 = true := by
+  have h : (thetaBelowCell11001221).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001221 h
+theorem e24KC2ThetaBelowLeaf11001222 :
+    adaptiveCoverCheck 10 thetaBelowCell11001222 = true := by
+  have h : (thetaBelowCell11001222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001222 h
+theorem e24KC2ThetaBelowLeaf11001223 :
+    adaptiveCoverCheck 10 thetaBelowCell11001223 = true := by
+  have h : (thetaBelowCell11001223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001223 h
+theorem e24KC2ThetaBelowLeaf11001230 :
+    adaptiveCoverCheck 10 thetaBelowCell11001230 = true := by
+  have h : (thetaBelowCell11001230).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001230 h
+theorem e24KC2ThetaBelowLeaf11001231 :
+    adaptiveCoverCheck 10 thetaBelowCell11001231 = true := by
+  have h : (thetaBelowCell11001231).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001231 h
+theorem e24KC2ThetaBelowLeaf11001232 :
+    adaptiveCoverCheck 10 thetaBelowCell11001232 = true := by
+  have h : (thetaBelowCell11001232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001232 h
+theorem e24KC2ThetaBelowLeaf11001233 :
+    adaptiveCoverCheck 10 thetaBelowCell11001233 = true := by
+  have h : (thetaBelowCell11001233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001233 h
+theorem e24KC2ThetaBelowLeaf1100130 :
+    adaptiveCoverCheck 11 (childLL (childHH (childLH thetaBelowCell1100))) = true := by
+  have h : ((childLL (childHH (childLH thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childLH thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100131 :
+    adaptiveCoverCheck 11 (childLH (childHH (childLH thetaBelowCell1100))) = true := by
+  have h : ((childLH (childHH (childLH thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childLH thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf11001320 :
+    adaptiveCoverCheck 10 thetaBelowCell11001320 = true := by
+  have h : (thetaBelowCell11001320).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001320 h
+theorem e24KC2ThetaBelowLeaf11001321 :
+    adaptiveCoverCheck 10 thetaBelowCell11001321 = true := by
+  have h : (thetaBelowCell11001321).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001321 h
+theorem e24KC2ThetaBelowLeaf11001322 :
+    adaptiveCoverCheck 10 thetaBelowCell11001322 = true := by
+  have h : (thetaBelowCell11001322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001322 h
+theorem e24KC2ThetaBelowLeaf11001323 :
+    adaptiveCoverCheck 10 thetaBelowCell11001323 = true := by
+  have h : (thetaBelowCell11001323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001323 h
+theorem e24KC2ThetaBelowLeaf11001330 :
+    adaptiveCoverCheck 10 thetaBelowCell11001330 = true := by
+  have h : (thetaBelowCell11001330).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001330 h
+theorem e24KC2ThetaBelowLeaf11001331 :
+    adaptiveCoverCheck 10 thetaBelowCell11001331 = true := by
+  have h : (thetaBelowCell11001331).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001331 h
+theorem e24KC2ThetaBelowLeaf11001332 :
+    adaptiveCoverCheck 10 thetaBelowCell11001332 = true := by
+  have h : (thetaBelowCell11001332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001332 h
+theorem e24KC2ThetaBelowLeaf11001333 :
+    adaptiveCoverCheck 10 thetaBelowCell11001333 = true := by
+  have h : (thetaBelowCell11001333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11001333 h
+theorem e24KC2ThetaBelowLeaf11002000 :
+    adaptiveCoverCheck 10 thetaBelowCell11002000 = true := by
+  have h : (thetaBelowCell11002000).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002000 h
+theorem e24KC2ThetaBelowLeaf11002001 :
+    adaptiveCoverCheck 10 thetaBelowCell11002001 = true := by
+  have h : (thetaBelowCell11002001).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002001 h
+theorem e24KC2ThetaBelowLeaf11002002 :
+    adaptiveCoverCheck 10 thetaBelowCell11002002 = true := by
+  have h : (thetaBelowCell11002002).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002002 h
+theorem e24KC2ThetaBelowLeaf11002003 :
+    adaptiveCoverCheck 10 thetaBelowCell11002003 = true := by
+  have h : (thetaBelowCell11002003).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002003 h
+theorem e24KC2ThetaBelowLeaf11002010 :
+    adaptiveCoverCheck 10 thetaBelowCell11002010 = true := by
+  have h : (thetaBelowCell11002010).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002010 h
+theorem e24KC2ThetaBelowLeaf11002011 :
+    adaptiveCoverCheck 10 thetaBelowCell11002011 = true := by
+  have h : (thetaBelowCell11002011).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002011 h
+theorem e24KC2ThetaBelowLeaf11002012 :
+    adaptiveCoverCheck 10 thetaBelowCell11002012 = true := by
+  have h : (thetaBelowCell11002012).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002012 h
+theorem e24KC2ThetaBelowLeaf11002013 :
+    adaptiveCoverCheck 10 thetaBelowCell11002013 = true := by
+  have h : (thetaBelowCell11002013).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002013 h
+theorem e24KC2ThetaBelowLeaf11002020 :
+    adaptiveCoverCheck 10 thetaBelowCell11002020 = true := by
+  have h : (thetaBelowCell11002020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002020 h
+theorem e24KC2ThetaBelowLeaf11002021 :
+    adaptiveCoverCheck 10 thetaBelowCell11002021 = true := by
+  have h : (thetaBelowCell11002021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002021 h
+theorem e24KC2ThetaBelowLeaf11002022 :
+    adaptiveCoverCheck 10 thetaBelowCell11002022 = true := by
+  have h : (thetaBelowCell11002022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002022 h
+theorem e24KC2ThetaBelowLeaf11002023 :
+    adaptiveCoverCheck 10 thetaBelowCell11002023 = true := by
+  have h : (thetaBelowCell11002023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002023 h
+theorem e24KC2ThetaBelowLeaf11002030 :
+    adaptiveCoverCheck 10 thetaBelowCell11002030 = true := by
+  have h : (thetaBelowCell11002030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002030 h
+theorem e24KC2ThetaBelowLeaf11002031 :
+    adaptiveCoverCheck 10 thetaBelowCell11002031 = true := by
+  have h : (thetaBelowCell11002031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002031 h
+theorem e24KC2ThetaBelowLeaf11002032 :
+    adaptiveCoverCheck 10 thetaBelowCell11002032 = true := by
+  have h : (thetaBelowCell11002032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002032 h
+theorem e24KC2ThetaBelowLeaf11002033 :
+    adaptiveCoverCheck 10 thetaBelowCell11002033 = true := by
+  have h : (thetaBelowCell11002033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002033 h
+theorem e24KC2ThetaBelowLeaf11002100 :
+    adaptiveCoverCheck 10 thetaBelowCell11002100 = true := by
+  have h : (thetaBelowCell11002100).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002100 h
+theorem e24KC2ThetaBelowLeaf11002101 :
+    adaptiveCoverCheck 10 thetaBelowCell11002101 = true := by
+  have h : (thetaBelowCell11002101).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002101 h
+theorem e24KC2ThetaBelowLeaf11002102 :
+    adaptiveCoverCheck 10 thetaBelowCell11002102 = true := by
+  have h : (thetaBelowCell11002102).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002102 h
+theorem e24KC2ThetaBelowLeaf11002103 :
+    adaptiveCoverCheck 10 thetaBelowCell11002103 = true := by
+  have h : (thetaBelowCell11002103).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002103 h
+theorem e24KC2ThetaBelowLeaf11002110 :
+    adaptiveCoverCheck 10 thetaBelowCell11002110 = true := by
+  have h : (thetaBelowCell11002110).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002110 h
+theorem e24KC2ThetaBelowLeaf11002111 :
+    adaptiveCoverCheck 10 thetaBelowCell11002111 = true := by
+  have h : (thetaBelowCell11002111).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002111 h
+theorem e24KC2ThetaBelowLeaf11002112 :
+    adaptiveCoverCheck 10 thetaBelowCell11002112 = true := by
+  have h : (thetaBelowCell11002112).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002112 h
+theorem e24KC2ThetaBelowLeaf11002113 :
+    adaptiveCoverCheck 10 thetaBelowCell11002113 = true := by
+  have h : (thetaBelowCell11002113).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002113 h
+theorem e24KC2ThetaBelowLeaf11002120 :
+    adaptiveCoverCheck 10 thetaBelowCell11002120 = true := by
+  have h : (thetaBelowCell11002120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002120 h
+theorem e24KC2ThetaBelowLeaf11002121 :
+    adaptiveCoverCheck 10 thetaBelowCell11002121 = true := by
+  have h : (thetaBelowCell11002121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002121 h
+theorem e24KC2ThetaBelowLeaf11002122 :
+    adaptiveCoverCheck 10 thetaBelowCell11002122 = true := by
+  have h : (thetaBelowCell11002122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002122 h
+theorem e24KC2ThetaBelowLeaf11002123 :
+    adaptiveCoverCheck 10 thetaBelowCell11002123 = true := by
+  have h : (thetaBelowCell11002123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002123 h
+theorem e24KC2ThetaBelowLeaf11002130 :
+    adaptiveCoverCheck 10 thetaBelowCell11002130 = true := by
+  have h : (thetaBelowCell11002130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002130 h
+theorem e24KC2ThetaBelowLeaf11002131 :
+    adaptiveCoverCheck 10 thetaBelowCell11002131 = true := by
+  have h : (thetaBelowCell11002131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002131 h
+theorem e24KC2ThetaBelowLeaf11002132 :
+    adaptiveCoverCheck 10 thetaBelowCell11002132 = true := by
+  have h : (thetaBelowCell11002132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002132 h
+theorem e24KC2ThetaBelowLeaf11002133 :
+    adaptiveCoverCheck 10 thetaBelowCell11002133 = true := by
+  have h : (thetaBelowCell11002133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11002133 h
+theorem e24KC2ThetaBelowLeaf1100220 :
+    adaptiveCoverCheck 11 (childLL (childHL (childHL thetaBelowCell1100))) = true := by
+  have h : ((childLL (childHL (childHL thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childHL thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100221 :
+    adaptiveCoverCheck 11 (childLH (childHL (childHL thetaBelowCell1100))) = true := by
+  have h : ((childLH (childHL (childHL thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childHL thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100222 :
+    adaptiveCoverCheck 11 (childHL (childHL (childHL thetaBelowCell1100))) = true := by
+  have h : ((childHL (childHL (childHL thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childHL thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100223 :
+    adaptiveCoverCheck 11 (childHH (childHL (childHL thetaBelowCell1100))) = true := by
+  have h : ((childHH (childHL (childHL thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childHL thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100230 :
+    adaptiveCoverCheck 11 (childLL (childHH (childHL thetaBelowCell1100))) = true := by
+  have h : ((childLL (childHH (childHL thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childHL thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100231 :
+    adaptiveCoverCheck 11 (childLH (childHH (childHL thetaBelowCell1100))) = true := by
+  have h : ((childLH (childHH (childHL thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childHL thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100232 :
+    adaptiveCoverCheck 11 (childHL (childHH (childHL thetaBelowCell1100))) = true := by
+  have h : ((childHL (childHH (childHL thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childHL thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100233 :
+    adaptiveCoverCheck 11 (childHH (childHH (childHL thetaBelowCell1100))) = true := by
+  have h : ((childHH (childHH (childHL thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childHL thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf11003000 :
+    adaptiveCoverCheck 10 thetaBelowCell11003000 = true := by
+  have h : (thetaBelowCell11003000).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003000 h
+theorem e24KC2ThetaBelowLeaf11003001 :
+    adaptiveCoverCheck 10 thetaBelowCell11003001 = true := by
+  have h : (thetaBelowCell11003001).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003001 h
+theorem e24KC2ThetaBelowLeaf11003002 :
+    adaptiveCoverCheck 10 thetaBelowCell11003002 = true := by
+  have h : (thetaBelowCell11003002).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003002 h
+theorem e24KC2ThetaBelowLeaf11003003 :
+    adaptiveCoverCheck 10 thetaBelowCell11003003 = true := by
+  have h : (thetaBelowCell11003003).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003003 h
+theorem e24KC2ThetaBelowLeaf11003010 :
+    adaptiveCoverCheck 10 thetaBelowCell11003010 = true := by
+  have h : (thetaBelowCell11003010).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003010 h
+theorem e24KC2ThetaBelowLeaf11003011 :
+    adaptiveCoverCheck 10 thetaBelowCell11003011 = true := by
+  have h : (thetaBelowCell11003011).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003011 h
+theorem e24KC2ThetaBelowLeaf11003012 :
+    adaptiveCoverCheck 10 thetaBelowCell11003012 = true := by
+  have h : (thetaBelowCell11003012).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003012 h
+theorem e24KC2ThetaBelowLeaf11003013 :
+    adaptiveCoverCheck 10 thetaBelowCell11003013 = true := by
+  have h : (thetaBelowCell11003013).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003013 h
+theorem e24KC2ThetaBelowLeaf11003020 :
+    adaptiveCoverCheck 10 thetaBelowCell11003020 = true := by
+  have h : (thetaBelowCell11003020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003020 h
+theorem e24KC2ThetaBelowLeaf11003021 :
+    adaptiveCoverCheck 10 thetaBelowCell11003021 = true := by
+  have h : (thetaBelowCell11003021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003021 h
+theorem e24KC2ThetaBelowLeaf11003022 :
+    adaptiveCoverCheck 10 thetaBelowCell11003022 = true := by
+  have h : (thetaBelowCell11003022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003022 h
+theorem e24KC2ThetaBelowLeaf11003023 :
+    adaptiveCoverCheck 10 thetaBelowCell11003023 = true := by
+  have h : (thetaBelowCell11003023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003023 h
+theorem e24KC2ThetaBelowLeaf11003030 :
+    adaptiveCoverCheck 10 thetaBelowCell11003030 = true := by
+  have h : (thetaBelowCell11003030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003030 h
+theorem e24KC2ThetaBelowLeaf11003031 :
+    adaptiveCoverCheck 10 thetaBelowCell11003031 = true := by
+  have h : (thetaBelowCell11003031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003031 h
+theorem e24KC2ThetaBelowLeaf11003032 :
+    adaptiveCoverCheck 10 thetaBelowCell11003032 = true := by
+  have h : (thetaBelowCell11003032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003032 h
+theorem e24KC2ThetaBelowLeaf11003033 :
+    adaptiveCoverCheck 10 thetaBelowCell11003033 = true := by
+  have h : (thetaBelowCell11003033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003033 h
+theorem e24KC2ThetaBelowLeaf11003100 :
+    adaptiveCoverCheck 10 thetaBelowCell11003100 = true := by
+  have h : (thetaBelowCell11003100).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003100 h
+theorem e24KC2ThetaBelowLeaf11003101 :
+    adaptiveCoverCheck 10 thetaBelowCell11003101 = true := by
+  have h : (thetaBelowCell11003101).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003101 h
+theorem e24KC2ThetaBelowLeaf11003102 :
+    adaptiveCoverCheck 10 thetaBelowCell11003102 = true := by
+  have h : (thetaBelowCell11003102).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003102 h
+theorem e24KC2ThetaBelowLeaf11003103 :
+    adaptiveCoverCheck 10 thetaBelowCell11003103 = true := by
+  have h : (thetaBelowCell11003103).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003103 h
+theorem e24KC2ThetaBelowLeaf11003110 :
+    adaptiveCoverCheck 10 thetaBelowCell11003110 = true := by
+  have h : (thetaBelowCell11003110).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003110 h
+theorem e24KC2ThetaBelowLeaf11003111 :
+    adaptiveCoverCheck 10 thetaBelowCell11003111 = true := by
+  have h : (thetaBelowCell11003111).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003111 h
+theorem e24KC2ThetaBelowLeaf11003112 :
+    adaptiveCoverCheck 10 thetaBelowCell11003112 = true := by
+  have h : (thetaBelowCell11003112).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003112 h
+theorem e24KC2ThetaBelowLeaf11003113 :
+    adaptiveCoverCheck 10 thetaBelowCell11003113 = true := by
+  have h : (thetaBelowCell11003113).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003113 h
+theorem e24KC2ThetaBelowLeaf11003120 :
+    adaptiveCoverCheck 10 thetaBelowCell11003120 = true := by
+  have h : (thetaBelowCell11003120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003120 h
+theorem e24KC2ThetaBelowLeaf11003121 :
+    adaptiveCoverCheck 10 thetaBelowCell11003121 = true := by
+  have h : (thetaBelowCell11003121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003121 h
+theorem e24KC2ThetaBelowLeaf11003122 :
+    adaptiveCoverCheck 10 thetaBelowCell11003122 = true := by
+  have h : (thetaBelowCell11003122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003122 h
+theorem e24KC2ThetaBelowLeaf11003123 :
+    adaptiveCoverCheck 10 thetaBelowCell11003123 = true := by
+  have h : (thetaBelowCell11003123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003123 h
+theorem e24KC2ThetaBelowLeaf11003130 :
+    adaptiveCoverCheck 10 thetaBelowCell11003130 = true := by
+  have h : (thetaBelowCell11003130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003130 h
+theorem e24KC2ThetaBelowLeaf11003131 :
+    adaptiveCoverCheck 10 thetaBelowCell11003131 = true := by
+  have h : (thetaBelowCell11003131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003131 h
+theorem e24KC2ThetaBelowLeaf11003132 :
+    adaptiveCoverCheck 10 thetaBelowCell11003132 = true := by
+  have h : (thetaBelowCell11003132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003132 h
+theorem e24KC2ThetaBelowLeaf11003133 :
+    adaptiveCoverCheck 10 thetaBelowCell11003133 = true := by
+  have h : (thetaBelowCell11003133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003133 h
+theorem e24KC2ThetaBelowLeaf1100320 :
+    adaptiveCoverCheck 11 (childLL (childHL (childHH thetaBelowCell1100))) = true := by
+  have h : ((childLL (childHL (childHH thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childHH thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100321 :
+    adaptiveCoverCheck 11 (childLH (childHL (childHH thetaBelowCell1100))) = true := by
+  have h : ((childLH (childHL (childHH thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childHH thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100322 :
+    adaptiveCoverCheck 11 (childHL (childHL (childHH thetaBelowCell1100))) = true := by
+  have h : ((childHL (childHL (childHH thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childHH thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100323 :
+    adaptiveCoverCheck 11 (childHH (childHL (childHH thetaBelowCell1100))) = true := by
+  have h : ((childHH (childHL (childHH thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childHH thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf11003300 :
+    adaptiveCoverCheck 10 thetaBelowCell11003300 = true := by
+  have h : (thetaBelowCell11003300).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003300 h
+theorem e24KC2ThetaBelowLeaf11003301 :
+    adaptiveCoverCheck 10 thetaBelowCell11003301 = true := by
+  have h : (thetaBelowCell11003301).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003301 h
+theorem e24KC2ThetaBelowLeaf11003302 :
+    adaptiveCoverCheck 10 thetaBelowCell11003302 = true := by
+  have h : (thetaBelowCell11003302).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003302 h
+theorem e24KC2ThetaBelowLeaf11003303 :
+    adaptiveCoverCheck 10 thetaBelowCell11003303 = true := by
+  have h : (thetaBelowCell11003303).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003303 h
+theorem e24KC2ThetaBelowLeaf11003310 :
+    adaptiveCoverCheck 10 thetaBelowCell11003310 = true := by
+  have h : (thetaBelowCell11003310).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003310 h
+theorem e24KC2ThetaBelowLeaf11003311 :
+    adaptiveCoverCheck 10 thetaBelowCell11003311 = true := by
+  have h : (thetaBelowCell11003311).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003311 h
+theorem e24KC2ThetaBelowLeaf11003312 :
+    adaptiveCoverCheck 10 thetaBelowCell11003312 = true := by
+  have h : (thetaBelowCell11003312).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003312 h
+theorem e24KC2ThetaBelowLeaf11003313 :
+    adaptiveCoverCheck 10 thetaBelowCell11003313 = true := by
+  have h : (thetaBelowCell11003313).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11003313 h
+theorem e24KC2ThetaBelowLeaf1100332 :
+    adaptiveCoverCheck 11 (childHL (childHH (childHH thetaBelowCell1100))) = true := by
+  have h : ((childHL (childHH (childHH thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childHH thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf1100333 :
+    adaptiveCoverCheck 11 (childHH (childHH (childHH thetaBelowCell1100))) = true := by
+  have h : ((childHH (childHH (childHH thetaBelowCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childHH thetaBelowCell1100))) h
+theorem e24KC2ThetaBelowLeaf110100 :
+    adaptiveCoverCheck 12 (childLL (childLL thetaBelowCell1101)) = true := by
+  have h : ((childLL (childLL thetaBelowCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL thetaBelowCell1101)) h
+theorem e24KC2ThetaBelowLeaf110101 :
+    adaptiveCoverCheck 12 (childLH (childLL thetaBelowCell1101)) = true := by
+  have h : ((childLH (childLL thetaBelowCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL thetaBelowCell1101)) h
+theorem e24KC2ThetaBelowLeaf1101020 :
+    adaptiveCoverCheck 11 (childLL (childHL (childLL thetaBelowCell1101))) = true := by
+  have h : ((childLL (childHL (childLL thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childLL thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf1101021 :
+    adaptiveCoverCheck 11 (childLH (childHL (childLL thetaBelowCell1101))) = true := by
+  have h : ((childLH (childHL (childLL thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childLL thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf11010220 :
+    adaptiveCoverCheck 10 thetaBelowCell11010220 = true := by
+  have h : (thetaBelowCell11010220).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11010220 h
+theorem e24KC2ThetaBelowLeaf11010221 :
+    adaptiveCoverCheck 10 thetaBelowCell11010221 = true := by
+  have h : (thetaBelowCell11010221).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11010221 h
+theorem e24KC2ThetaBelowLeaf11010222 :
+    adaptiveCoverCheck 10 thetaBelowCell11010222 = true := by
+  have h : (thetaBelowCell11010222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11010222 h
+theorem e24KC2ThetaBelowLeaf11010223 :
+    adaptiveCoverCheck 10 thetaBelowCell11010223 = true := by
+  have h : (thetaBelowCell11010223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11010223 h
+theorem e24KC2ThetaBelowLeaf11010230 :
+    adaptiveCoverCheck 10 thetaBelowCell11010230 = true := by
+  have h : (thetaBelowCell11010230).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11010230 h
+theorem e24KC2ThetaBelowLeaf11010231 :
+    adaptiveCoverCheck 10 thetaBelowCell11010231 = true := by
+  have h : (thetaBelowCell11010231).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11010231 h
+theorem e24KC2ThetaBelowLeaf11010232 :
+    adaptiveCoverCheck 10 thetaBelowCell11010232 = true := by
+  have h : (thetaBelowCell11010232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11010232 h
+theorem e24KC2ThetaBelowLeaf11010233 :
+    adaptiveCoverCheck 10 thetaBelowCell11010233 = true := by
+  have h : (thetaBelowCell11010233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11010233 h
+theorem e24KC2ThetaBelowLeaf1101030 :
+    adaptiveCoverCheck 11 (childLL (childHH (childLL thetaBelowCell1101))) = true := by
+  have h : ((childLL (childHH (childLL thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childLL thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf1101031 :
+    adaptiveCoverCheck 11 (childLH (childHH (childLL thetaBelowCell1101))) = true := by
+  have h : ((childLH (childHH (childLL thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childLL thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf1101032 :
+    adaptiveCoverCheck 11 (childHL (childHH (childLL thetaBelowCell1101))) = true := by
+  have h : ((childHL (childHH (childLL thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childLL thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf1101033 :
+    adaptiveCoverCheck 11 (childHH (childHH (childLL thetaBelowCell1101))) = true := by
+  have h : ((childHH (childHH (childLL thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childLL thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf110110 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell1101)) = true := by
+  have h : ((childLL (childLH thetaBelowCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH thetaBelowCell1101)) h
+theorem e24KC2ThetaBelowLeaf110111 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell1101)) = true := by
+  have h : ((childLH (childLH thetaBelowCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH thetaBelowCell1101)) h
+theorem e24KC2ThetaBelowLeaf1101120 :
+    adaptiveCoverCheck 11 (childLL (childHL (childLH thetaBelowCell1101))) = true := by
+  have h : ((childLL (childHL (childLH thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childLH thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf1101121 :
+    adaptiveCoverCheck 11 (childLH (childHL (childLH thetaBelowCell1101))) = true := by
+  have h : ((childLH (childHL (childLH thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childLH thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf1101122 :
+    adaptiveCoverCheck 11 (childHL (childHL (childLH thetaBelowCell1101))) = true := by
+  have h : ((childHL (childHL (childLH thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childLH thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf1101123 :
+    adaptiveCoverCheck 11 (childHH (childHL (childLH thetaBelowCell1101))) = true := by
+  have h : ((childHH (childHL (childLH thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childLH thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf1101130 :
+    adaptiveCoverCheck 11 (childLL (childHH (childLH thetaBelowCell1101))) = true := by
+  have h : ((childLL (childHH (childLH thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childLH thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf1101131 :
+    adaptiveCoverCheck 11 (childLH (childHH (childLH thetaBelowCell1101))) = true := by
+  have h : ((childLH (childHH (childLH thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childLH thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf1101132 :
+    adaptiveCoverCheck 11 (childHL (childHH (childLH thetaBelowCell1101))) = true := by
+  have h : ((childHL (childHH (childLH thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childLH thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf1101133 :
+    adaptiveCoverCheck 11 (childHH (childHH (childLH thetaBelowCell1101))) = true := by
+  have h : ((childHH (childHH (childLH thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childLH thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf11012000 :
+    adaptiveCoverCheck 10 thetaBelowCell11012000 = true := by
+  have h : (thetaBelowCell11012000).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012000 h
+theorem e24KC2ThetaBelowLeaf11012001 :
+    adaptiveCoverCheck 10 thetaBelowCell11012001 = true := by
+  have h : (thetaBelowCell11012001).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012001 h
+theorem e24KC2ThetaBelowLeaf11012002 :
+    adaptiveCoverCheck 10 thetaBelowCell11012002 = true := by
+  have h : (thetaBelowCell11012002).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012002 h
+theorem e24KC2ThetaBelowLeaf11012003 :
+    adaptiveCoverCheck 10 thetaBelowCell11012003 = true := by
+  have h : (thetaBelowCell11012003).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012003 h
+theorem e24KC2ThetaBelowLeaf11012010 :
+    adaptiveCoverCheck 10 thetaBelowCell11012010 = true := by
+  have h : (thetaBelowCell11012010).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012010 h
+theorem e24KC2ThetaBelowLeaf11012011 :
+    adaptiveCoverCheck 10 thetaBelowCell11012011 = true := by
+  have h : (thetaBelowCell11012011).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012011 h
+theorem e24KC2ThetaBelowLeaf11012012 :
+    adaptiveCoverCheck 10 thetaBelowCell11012012 = true := by
+  have h : (thetaBelowCell11012012).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012012 h
+theorem e24KC2ThetaBelowLeaf11012013 :
+    adaptiveCoverCheck 10 thetaBelowCell11012013 = true := by
+  have h : (thetaBelowCell11012013).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012013 h
+theorem e24KC2ThetaBelowLeaf11012020 :
+    adaptiveCoverCheck 10 thetaBelowCell11012020 = true := by
+  have h : (thetaBelowCell11012020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012020 h
+theorem e24KC2ThetaBelowLeaf11012021 :
+    adaptiveCoverCheck 10 thetaBelowCell11012021 = true := by
+  have h : (thetaBelowCell11012021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012021 h
+theorem e24KC2ThetaBelowLeaf11012022 :
+    adaptiveCoverCheck 10 thetaBelowCell11012022 = true := by
+  have h : (thetaBelowCell11012022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012022 h
+theorem e24KC2ThetaBelowLeaf11012023 :
+    adaptiveCoverCheck 10 thetaBelowCell11012023 = true := by
+  have h : (thetaBelowCell11012023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012023 h
+theorem e24KC2ThetaBelowLeaf11012030 :
+    adaptiveCoverCheck 10 thetaBelowCell11012030 = true := by
+  have h : (thetaBelowCell11012030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012030 h
+theorem e24KC2ThetaBelowLeaf11012031 :
+    adaptiveCoverCheck 10 thetaBelowCell11012031 = true := by
+  have h : (thetaBelowCell11012031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012031 h
+theorem e24KC2ThetaBelowLeaf11012032 :
+    adaptiveCoverCheck 10 thetaBelowCell11012032 = true := by
+  have h : (thetaBelowCell11012032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012032 h
+theorem e24KC2ThetaBelowLeaf11012033 :
+    adaptiveCoverCheck 10 thetaBelowCell11012033 = true := by
+  have h : (thetaBelowCell11012033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012033 h
+theorem e24KC2ThetaBelowLeaf11012100 :
+    adaptiveCoverCheck 10 thetaBelowCell11012100 = true := by
+  have h : (thetaBelowCell11012100).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012100 h
+theorem e24KC2ThetaBelowLeaf11012101 :
+    adaptiveCoverCheck 10 thetaBelowCell11012101 = true := by
+  have h : (thetaBelowCell11012101).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012101 h
+theorem e24KC2ThetaBelowLeaf11012102 :
+    adaptiveCoverCheck 10 thetaBelowCell11012102 = true := by
+  have h : (thetaBelowCell11012102).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012102 h
+theorem e24KC2ThetaBelowLeaf11012103 :
+    adaptiveCoverCheck 10 thetaBelowCell11012103 = true := by
+  have h : (thetaBelowCell11012103).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012103 h
+theorem e24KC2ThetaBelowLeaf11012110 :
+    adaptiveCoverCheck 10 thetaBelowCell11012110 = true := by
+  have h : (thetaBelowCell11012110).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012110 h
+theorem e24KC2ThetaBelowLeaf11012111 :
+    adaptiveCoverCheck 10 thetaBelowCell11012111 = true := by
+  have h : (thetaBelowCell11012111).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012111 h
+theorem e24KC2ThetaBelowLeaf11012112 :
+    adaptiveCoverCheck 10 thetaBelowCell11012112 = true := by
+  have h : (thetaBelowCell11012112).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012112 h
+theorem e24KC2ThetaBelowLeaf11012113 :
+    adaptiveCoverCheck 10 thetaBelowCell11012113 = true := by
+  have h : (thetaBelowCell11012113).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012113 h
+theorem e24KC2ThetaBelowLeaf11012120 :
+    adaptiveCoverCheck 10 thetaBelowCell11012120 = true := by
+  have h : (thetaBelowCell11012120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012120 h
+theorem e24KC2ThetaBelowLeaf11012121 :
+    adaptiveCoverCheck 10 thetaBelowCell11012121 = true := by
+  have h : (thetaBelowCell11012121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012121 h
+theorem e24KC2ThetaBelowLeaf11012122 :
+    adaptiveCoverCheck 10 thetaBelowCell11012122 = true := by
+  have h : (thetaBelowCell11012122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012122 h
+theorem e24KC2ThetaBelowLeaf11012123 :
+    adaptiveCoverCheck 10 thetaBelowCell11012123 = true := by
+  have h : (thetaBelowCell11012123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012123 h
+theorem e24KC2ThetaBelowLeaf11012130 :
+    adaptiveCoverCheck 10 thetaBelowCell11012130 = true := by
+  have h : (thetaBelowCell11012130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012130 h
+theorem e24KC2ThetaBelowLeaf11012131 :
+    adaptiveCoverCheck 10 thetaBelowCell11012131 = true := by
+  have h : (thetaBelowCell11012131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012131 h
+theorem e24KC2ThetaBelowLeaf110121320 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11012132) = true := by
+  have h : ((childLL thetaBelowCell11012132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11012132) h
+theorem e24KC2ThetaBelowLeaf110121321 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11012132) = true := by
+  have h : ((childLH thetaBelowCell11012132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11012132) h
+theorem e24KC2ThetaBelowLeaf110121322 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11012132) = true := by
+  have h : ((childHL thetaBelowCell11012132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11012132) h
+theorem e24KC2ThetaBelowLeaf110121323 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11012132) = true := by
+  have h : ((childHH thetaBelowCell11012132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11012132) h
+theorem e24KC2ThetaBelowLeaf110121330 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11012133) = true := by
+  have h : ((childLL thetaBelowCell11012133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11012133) h
+theorem e24KC2ThetaBelowLeaf110121331 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11012133) = true := by
+  have h : ((childLH thetaBelowCell11012133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11012133) h
+theorem e24KC2ThetaBelowLeaf110121332 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11012133) = true := by
+  have h : ((childHL thetaBelowCell11012133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11012133) h
+theorem e24KC2ThetaBelowLeaf110121333 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11012133) = true := by
+  have h : ((childHH thetaBelowCell11012133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11012133) h
+theorem e24KC2ThetaBelowLeaf11012200 :
+    adaptiveCoverCheck 10 thetaBelowCell11012200 = true := by
+  have h : (thetaBelowCell11012200).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012200 h
+theorem e24KC2ThetaBelowLeaf11012201 :
+    adaptiveCoverCheck 10 thetaBelowCell11012201 = true := by
+  have h : (thetaBelowCell11012201).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012201 h
+theorem e24KC2ThetaBelowLeaf11012202 :
+    adaptiveCoverCheck 10 thetaBelowCell11012202 = true := by
+  have h : (thetaBelowCell11012202).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012202 h
+theorem e24KC2ThetaBelowLeaf11012203 :
+    adaptiveCoverCheck 10 thetaBelowCell11012203 = true := by
+  have h : (thetaBelowCell11012203).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012203 h
+theorem e24KC2ThetaBelowLeaf11012210 :
+    adaptiveCoverCheck 10 thetaBelowCell11012210 = true := by
+  have h : (thetaBelowCell11012210).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012210 h
+theorem e24KC2ThetaBelowLeaf11012211 :
+    adaptiveCoverCheck 10 thetaBelowCell11012211 = true := by
+  have h : (thetaBelowCell11012211).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012211 h
+theorem e24KC2ThetaBelowLeaf11012212 :
+    adaptiveCoverCheck 10 thetaBelowCell11012212 = true := by
+  have h : (thetaBelowCell11012212).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012212 h
+theorem e24KC2ThetaBelowLeaf11012213 :
+    adaptiveCoverCheck 10 thetaBelowCell11012213 = true := by
+  have h : (thetaBelowCell11012213).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012213 h
+theorem e24KC2ThetaBelowLeaf1101222 :
+    adaptiveCoverCheck 11 (childHL (childHL (childHL thetaBelowCell1101))) = true := by
+  have h : ((childHL (childHL (childHL thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childHL thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf1101223 :
+    adaptiveCoverCheck 11 (childHH (childHL (childHL thetaBelowCell1101))) = true := by
+  have h : ((childHH (childHL (childHL thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childHL thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf11012300 :
+    adaptiveCoverCheck 10 thetaBelowCell11012300 = true := by
+  have h : (thetaBelowCell11012300).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012300 h
+theorem e24KC2ThetaBelowLeaf11012301 :
+    adaptiveCoverCheck 10 thetaBelowCell11012301 = true := by
+  have h : (thetaBelowCell11012301).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012301 h
+theorem e24KC2ThetaBelowLeaf11012302 :
+    adaptiveCoverCheck 10 thetaBelowCell11012302 = true := by
+  have h : (thetaBelowCell11012302).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012302 h
+theorem e24KC2ThetaBelowLeaf11012303 :
+    adaptiveCoverCheck 10 thetaBelowCell11012303 = true := by
+  have h : (thetaBelowCell11012303).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012303 h
+theorem e24KC2ThetaBelowLeaf11012310 :
+    adaptiveCoverCheck 10 thetaBelowCell11012310 = true := by
+  have h : (thetaBelowCell11012310).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012310 h
+theorem e24KC2ThetaBelowLeaf11012311 :
+    adaptiveCoverCheck 10 thetaBelowCell11012311 = true := by
+  have h : (thetaBelowCell11012311).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012311 h
+theorem e24KC2ThetaBelowLeaf11012312 :
+    adaptiveCoverCheck 10 thetaBelowCell11012312 = true := by
+  have h : (thetaBelowCell11012312).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012312 h
+theorem e24KC2ThetaBelowLeaf11012313 :
+    adaptiveCoverCheck 10 thetaBelowCell11012313 = true := by
+  have h : (thetaBelowCell11012313).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11012313 h
+theorem e24KC2ThetaBelowLeaf1101232 :
+    adaptiveCoverCheck 11 (childHL (childHH (childHL thetaBelowCell1101))) = true := by
+  have h : ((childHL (childHH (childHL thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childHL thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf1101233 :
+    adaptiveCoverCheck 11 (childHH (childHH (childHL thetaBelowCell1101))) = true := by
+  have h : ((childHH (childHH (childHL thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childHL thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf11013000 :
+    adaptiveCoverCheck 10 thetaBelowCell11013000 = true := by
+  have h : (thetaBelowCell11013000).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013000 h
+theorem e24KC2ThetaBelowLeaf11013001 :
+    adaptiveCoverCheck 10 thetaBelowCell11013001 = true := by
+  have h : (thetaBelowCell11013001).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013001 h
+theorem e24KC2ThetaBelowLeaf11013002 :
+    adaptiveCoverCheck 10 thetaBelowCell11013002 = true := by
+  have h : (thetaBelowCell11013002).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013002 h
+theorem e24KC2ThetaBelowLeaf11013003 :
+    adaptiveCoverCheck 10 thetaBelowCell11013003 = true := by
+  have h : (thetaBelowCell11013003).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013003 h
+theorem e24KC2ThetaBelowLeaf11013010 :
+    adaptiveCoverCheck 10 thetaBelowCell11013010 = true := by
+  have h : (thetaBelowCell11013010).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013010 h
+theorem e24KC2ThetaBelowLeaf11013011 :
+    adaptiveCoverCheck 10 thetaBelowCell11013011 = true := by
+  have h : (thetaBelowCell11013011).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013011 h
+theorem e24KC2ThetaBelowLeaf11013012 :
+    adaptiveCoverCheck 10 thetaBelowCell11013012 = true := by
+  have h : (thetaBelowCell11013012).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013012 h
+theorem e24KC2ThetaBelowLeaf110130130 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013013) = true := by
+  have h : ((childLL thetaBelowCell11013013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013013) h
+theorem e24KC2ThetaBelowLeaf110130131 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013013) = true := by
+  have h : ((childLH thetaBelowCell11013013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013013) h
+theorem e24KC2ThetaBelowLeaf110130132 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013013) = true := by
+  have h : ((childHL thetaBelowCell11013013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013013) h
+theorem e24KC2ThetaBelowLeaf110130133 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013013) = true := by
+  have h : ((childHH thetaBelowCell11013013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013013) h
+theorem e24KC2ThetaBelowLeaf11013020 :
+    adaptiveCoverCheck 10 thetaBelowCell11013020 = true := by
+  have h : (thetaBelowCell11013020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013020 h
+theorem e24KC2ThetaBelowLeaf110130210 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013021) = true := by
+  have h : ((childLL thetaBelowCell11013021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013021) h
+theorem e24KC2ThetaBelowLeaf110130211 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013021) = true := by
+  have h : ((childLH thetaBelowCell11013021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013021) h
+theorem e24KC2ThetaBelowLeaf110130212 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013021) = true := by
+  have h : ((childHL thetaBelowCell11013021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013021) h
+theorem e24KC2ThetaBelowLeaf110130213 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013021) = true := by
+  have h : ((childHH thetaBelowCell11013021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013021) h
+theorem e24KC2ThetaBelowLeaf110130220 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013022) = true := by
+  have h : ((childLL thetaBelowCell11013022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013022) h
+theorem e24KC2ThetaBelowLeaf110130221 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013022) = true := by
+  have h : ((childLH thetaBelowCell11013022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013022) h
+theorem e24KC2ThetaBelowLeaf110130222 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013022) = true := by
+  have h : ((childHL thetaBelowCell11013022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013022) h
+theorem e24KC2ThetaBelowLeaf110130223 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013022) = true := by
+  have h : ((childHH thetaBelowCell11013022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013022) h
+theorem e24KC2ThetaBelowLeaf110130230 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013023) = true := by
+  have h : ((childLL thetaBelowCell11013023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013023) h
+theorem e24KC2ThetaBelowLeaf110130231 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013023) = true := by
+  have h : ((childLH thetaBelowCell11013023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013023) h
+theorem e24KC2ThetaBelowLeaf110130232 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013023) = true := by
+  have h : ((childHL thetaBelowCell11013023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013023) h
+theorem e24KC2ThetaBelowLeaf110130233 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013023) = true := by
+  have h : ((childHH thetaBelowCell11013023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013023) h
+theorem e24KC2ThetaBelowLeaf110130300 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013030) = true := by
+  have h : ((childLL thetaBelowCell11013030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013030) h
+theorem e24KC2ThetaBelowLeaf110130301 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013030) = true := by
+  have h : ((childLH thetaBelowCell11013030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013030) h
+theorem e24KC2ThetaBelowLeaf110130302 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013030) = true := by
+  have h : ((childHL thetaBelowCell11013030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013030) h
+theorem e24KC2ThetaBelowLeaf110130303 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013030) = true := by
+  have h : ((childHH thetaBelowCell11013030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013030) h
+theorem e24KC2ThetaBelowLeaf110130310 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013031) = true := by
+  have h : ((childLL thetaBelowCell11013031)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013031) h
+theorem e24KC2ThetaBelowLeaf110130311 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013031) = true := by
+  have h : ((childLH thetaBelowCell11013031)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013031) h
+theorem e24KC2ThetaBelowLeaf110130312 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013031) = true := by
+  have h : ((childHL thetaBelowCell11013031)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013031) h
+theorem e24KC2ThetaBelowLeaf110130313 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013031) = true := by
+  have h : ((childHH thetaBelowCell11013031)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013031) h
+theorem e24KC2ThetaBelowLeaf110130320 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013032) = true := by
+  have h : ((childLL thetaBelowCell11013032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013032) h
+theorem e24KC2ThetaBelowLeaf110130321 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013032) = true := by
+  have h : ((childLH thetaBelowCell11013032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013032) h
+theorem e24KC2ThetaBelowLeaf110130322 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013032) = true := by
+  have h : ((childHL thetaBelowCell11013032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013032) h
+theorem e24KC2ThetaBelowLeaf110130323 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013032) = true := by
+  have h : ((childHH thetaBelowCell11013032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013032) h
+theorem e24KC2ThetaBelowLeaf110130330 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013033) = true := by
+  have h : ((childLL thetaBelowCell11013033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013033) h
+theorem e24KC2ThetaBelowLeaf110130331 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013033) = true := by
+  have h : ((childLH thetaBelowCell11013033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013033) h
+theorem e24KC2ThetaBelowLeaf110130332 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013033) = true := by
+  have h : ((childHL thetaBelowCell11013033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013033) h
+theorem e24KC2ThetaBelowLeaf110130333 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013033) = true := by
+  have h : ((childHH thetaBelowCell11013033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013033) h
+theorem e24KC2ThetaBelowLeaf11013100 :
+    adaptiveCoverCheck 10 thetaBelowCell11013100 = true := by
+  have h : (thetaBelowCell11013100).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013100 h
+theorem e24KC2ThetaBelowLeaf11013101 :
+    adaptiveCoverCheck 10 thetaBelowCell11013101 = true := by
+  have h : (thetaBelowCell11013101).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013101 h
+theorem e24KC2ThetaBelowLeaf110131020 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013102) = true := by
+  have h : ((childLL thetaBelowCell11013102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013102) h
+theorem e24KC2ThetaBelowLeaf110131021 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013102) = true := by
+  have h : ((childLH thetaBelowCell11013102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013102) h
+theorem e24KC2ThetaBelowLeaf110131022 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013102) = true := by
+  have h : ((childHL thetaBelowCell11013102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013102) h
+theorem e24KC2ThetaBelowLeaf110131023 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013102) = true := by
+  have h : ((childHH thetaBelowCell11013102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013102) h
+theorem e24KC2ThetaBelowLeaf110131030 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013103) = true := by
+  have h : ((childLL thetaBelowCell11013103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013103) h
+theorem e24KC2ThetaBelowLeaf110131031 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013103) = true := by
+  have h : ((childLH thetaBelowCell11013103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013103) h
+theorem e24KC2ThetaBelowLeaf110131032 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013103) = true := by
+  have h : ((childHL thetaBelowCell11013103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013103) h
+theorem e24KC2ThetaBelowLeaf110131033 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013103) = true := by
+  have h : ((childHH thetaBelowCell11013103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013103) h
+theorem e24KC2ThetaBelowLeaf11013110 :
+    adaptiveCoverCheck 10 thetaBelowCell11013110 = true := by
+  have h : (thetaBelowCell11013110).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013110 h
+theorem e24KC2ThetaBelowLeaf11013111 :
+    adaptiveCoverCheck 10 thetaBelowCell11013111 = true := by
+  have h : (thetaBelowCell11013111).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013111 h
+theorem e24KC2ThetaBelowLeaf110131120 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013112) = true := by
+  have h : ((childLL thetaBelowCell11013112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013112) h
+theorem e24KC2ThetaBelowLeaf110131121 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013112) = true := by
+  have h : ((childLH thetaBelowCell11013112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013112) h
+theorem e24KC2ThetaBelowLeaf110131122 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013112) = true := by
+  have h : ((childHL thetaBelowCell11013112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013112) h
+theorem e24KC2ThetaBelowLeaf110131123 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013112) = true := by
+  have h : ((childHH thetaBelowCell11013112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013112) h
+theorem e24KC2ThetaBelowLeaf110131130 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013113) = true := by
+  have h : ((childLL thetaBelowCell11013113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013113) h
+theorem e24KC2ThetaBelowLeaf110131131 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013113) = true := by
+  have h : ((childLH thetaBelowCell11013113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013113) h
+theorem e24KC2ThetaBelowLeaf110131132 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013113) = true := by
+  have h : ((childHL thetaBelowCell11013113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013113) h
+theorem e24KC2ThetaBelowLeaf110131133 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013113) = true := by
+  have h : ((childHH thetaBelowCell11013113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013113) h
+theorem e24KC2ThetaBelowLeaf110131200 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013120) = true := by
+  have h : ((childLL thetaBelowCell11013120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013120) h
+theorem e24KC2ThetaBelowLeaf110131201 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013120) = true := by
+  have h : ((childLH thetaBelowCell11013120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013120) h
+theorem e24KC2ThetaBelowLeaf110131202 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013120) = true := by
+  have h : ((childHL thetaBelowCell11013120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013120) h
+theorem e24KC2ThetaBelowLeaf110131203 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013120) = true := by
+  have h : ((childHH thetaBelowCell11013120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013120) h
+theorem e24KC2ThetaBelowLeaf110131210 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013121) = true := by
+  have h : ((childLL thetaBelowCell11013121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013121) h
+theorem e24KC2ThetaBelowLeaf110131211 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013121) = true := by
+  have h : ((childLH thetaBelowCell11013121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013121) h
+theorem e24KC2ThetaBelowLeaf110131212 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013121) = true := by
+  have h : ((childHL thetaBelowCell11013121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013121) h
+theorem e24KC2ThetaBelowLeaf110131213 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013121) = true := by
+  have h : ((childHH thetaBelowCell11013121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013121) h
+theorem e24KC2ThetaBelowLeaf110131220 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013122) = true := by
+  have h : ((childLL thetaBelowCell11013122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013122) h
+theorem e24KC2ThetaBelowLeaf110131221 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013122) = true := by
+  have h : ((childLH thetaBelowCell11013122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013122) h
+theorem e24KC2ThetaBelowLeaf110131222 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013122) = true := by
+  have h : ((childHL thetaBelowCell11013122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013122) h
+theorem e24KC2ThetaBelowLeaf110131223 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013122) = true := by
+  have h : ((childHH thetaBelowCell11013122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013122) h
+theorem e24KC2ThetaBelowLeaf110131230 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013123) = true := by
+  have h : ((childLL thetaBelowCell11013123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013123) h
+theorem e24KC2ThetaBelowLeaf110131231 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013123) = true := by
+  have h : ((childLH thetaBelowCell11013123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013123) h
+theorem e24KC2ThetaBelowLeaf110131232 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013123) = true := by
+  have h : ((childHL thetaBelowCell11013123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013123) h
+theorem e24KC2ThetaBelowLeaf110131233 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013123) = true := by
+  have h : ((childHH thetaBelowCell11013123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013123) h
+theorem e24KC2ThetaBelowLeaf110131300 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013130) = true := by
+  have h : ((childLL thetaBelowCell11013130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013130) h
+theorem e24KC2ThetaBelowLeaf110131301 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013130) = true := by
+  have h : ((childLH thetaBelowCell11013130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013130) h
+theorem e24KC2ThetaBelowLeaf110131302 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013130) = true := by
+  have h : ((childHL thetaBelowCell11013130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013130) h
+theorem e24KC2ThetaBelowLeaf110131303 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013130) = true := by
+  have h : ((childHH thetaBelowCell11013130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013130) h
+theorem e24KC2ThetaBelowLeaf110131310 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013131) = true := by
+  have h : ((childLL thetaBelowCell11013131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013131) h
+theorem e24KC2ThetaBelowLeaf110131311 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013131) = true := by
+  have h : ((childLH thetaBelowCell11013131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013131) h
+theorem e24KC2ThetaBelowLeaf110131312 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013131) = true := by
+  have h : ((childHL thetaBelowCell11013131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013131) h
+theorem e24KC2ThetaBelowLeaf110131313 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013131) = true := by
+  have h : ((childHH thetaBelowCell11013131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013131) h
+theorem e24KC2ThetaBelowLeaf110131320 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013132) = true := by
+  have h : ((childLL thetaBelowCell11013132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013132) h
+theorem e24KC2ThetaBelowLeaf110131321 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013132) = true := by
+  have h : ((childLH thetaBelowCell11013132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013132) h
+theorem e24KC2ThetaBelowLeaf110131322 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013132) = true := by
+  have h : ((childHL thetaBelowCell11013132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013132) h
+theorem e24KC2ThetaBelowLeaf110131323 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013132) = true := by
+  have h : ((childHH thetaBelowCell11013132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013132) h
+theorem e24KC2ThetaBelowLeaf110131330 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11013133) = true := by
+  have h : ((childLL thetaBelowCell11013133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11013133) h
+theorem e24KC2ThetaBelowLeaf110131331 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11013133) = true := by
+  have h : ((childLH thetaBelowCell11013133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11013133) h
+theorem e24KC2ThetaBelowLeaf110131332 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11013133) = true := by
+  have h : ((childHL thetaBelowCell11013133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11013133) h
+theorem e24KC2ThetaBelowLeaf110131333 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11013133) = true := by
+  have h : ((childHH thetaBelowCell11013133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11013133) h
+theorem e24KC2ThetaBelowLeaf11013200 :
+    adaptiveCoverCheck 10 thetaBelowCell11013200 = true := by
+  have h : (thetaBelowCell11013200).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013200 h
+theorem e24KC2ThetaBelowLeaf11013201 :
+    adaptiveCoverCheck 10 thetaBelowCell11013201 = true := by
+  have h : (thetaBelowCell11013201).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013201 h
+theorem e24KC2ThetaBelowLeaf11013202 :
+    adaptiveCoverCheck 10 thetaBelowCell11013202 = true := by
+  have h : (thetaBelowCell11013202).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013202 h
+theorem e24KC2ThetaBelowLeaf11013203 :
+    adaptiveCoverCheck 10 thetaBelowCell11013203 = true := by
+  have h : (thetaBelowCell11013203).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013203 h
+theorem e24KC2ThetaBelowLeaf11013210 :
+    adaptiveCoverCheck 10 thetaBelowCell11013210 = true := by
+  have h : (thetaBelowCell11013210).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013210 h
+theorem e24KC2ThetaBelowLeaf11013211 :
+    adaptiveCoverCheck 10 thetaBelowCell11013211 = true := by
+  have h : (thetaBelowCell11013211).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013211 h
+theorem e24KC2ThetaBelowLeaf11013212 :
+    adaptiveCoverCheck 10 thetaBelowCell11013212 = true := by
+  have h : (thetaBelowCell11013212).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013212 h
+theorem e24KC2ThetaBelowLeaf11013213 :
+    adaptiveCoverCheck 10 thetaBelowCell11013213 = true := by
+  have h : (thetaBelowCell11013213).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013213 h
+theorem e24KC2ThetaBelowLeaf1101322 :
+    adaptiveCoverCheck 11 (childHL (childHL (childHH thetaBelowCell1101))) = true := by
+  have h : ((childHL (childHL (childHH thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childHH thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf1101323 :
+    adaptiveCoverCheck 11 (childHH (childHL (childHH thetaBelowCell1101))) = true := by
+  have h : ((childHH (childHL (childHH thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childHH thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf11013300 :
+    adaptiveCoverCheck 10 thetaBelowCell11013300 = true := by
+  have h : (thetaBelowCell11013300).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013300 h
+theorem e24KC2ThetaBelowLeaf11013301 :
+    adaptiveCoverCheck 10 thetaBelowCell11013301 = true := by
+  have h : (thetaBelowCell11013301).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013301 h
+theorem e24KC2ThetaBelowLeaf11013302 :
+    adaptiveCoverCheck 10 thetaBelowCell11013302 = true := by
+  have h : (thetaBelowCell11013302).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013302 h
+theorem e24KC2ThetaBelowLeaf11013303 :
+    adaptiveCoverCheck 10 thetaBelowCell11013303 = true := by
+  have h : (thetaBelowCell11013303).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013303 h
+theorem e24KC2ThetaBelowLeaf11013310 :
+    adaptiveCoverCheck 10 thetaBelowCell11013310 = true := by
+  have h : (thetaBelowCell11013310).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013310 h
+theorem e24KC2ThetaBelowLeaf11013311 :
+    adaptiveCoverCheck 10 thetaBelowCell11013311 = true := by
+  have h : (thetaBelowCell11013311).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013311 h
+theorem e24KC2ThetaBelowLeaf11013312 :
+    adaptiveCoverCheck 10 thetaBelowCell11013312 = true := by
+  have h : (thetaBelowCell11013312).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013312 h
+theorem e24KC2ThetaBelowLeaf11013313 :
+    adaptiveCoverCheck 10 thetaBelowCell11013313 = true := by
+  have h : (thetaBelowCell11013313).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11013313 h
+theorem e24KC2ThetaBelowLeaf1101332 :
+    adaptiveCoverCheck 11 (childHL (childHH (childHH thetaBelowCell1101))) = true := by
+  have h : ((childHL (childHH (childHH thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childHH thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf1101333 :
+    adaptiveCoverCheck 11 (childHH (childHH (childHH thetaBelowCell1101))) = true := by
+  have h : ((childHH (childHH (childHH thetaBelowCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childHH thetaBelowCell1101))) h
+theorem e24KC2ThetaBelowLeaf110200 :
+    adaptiveCoverCheck 12 (childLL (childLL thetaBelowCell1102)) = true := by
+  have h : ((childLL (childLL thetaBelowCell1102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL thetaBelowCell1102)) h
+theorem e24KC2ThetaBelowLeaf110201 :
+    adaptiveCoverCheck 12 (childLH (childLL thetaBelowCell1102)) = true := by
+  have h : ((childLH (childLL thetaBelowCell1102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL thetaBelowCell1102)) h
+theorem e24KC2ThetaBelowLeaf110202 :
+    adaptiveCoverCheck 12 (childHL (childLL thetaBelowCell1102)) = true := by
+  have h : ((childHL (childLL thetaBelowCell1102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL thetaBelowCell1102)) h
+theorem e24KC2ThetaBelowLeaf110203 :
+    adaptiveCoverCheck 12 (childHH (childLL thetaBelowCell1102)) = true := by
+  have h : ((childHH (childLL thetaBelowCell1102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL thetaBelowCell1102)) h
+theorem e24KC2ThetaBelowLeaf110210 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell1102)) = true := by
+  have h : ((childLL (childLH thetaBelowCell1102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH thetaBelowCell1102)) h
+theorem e24KC2ThetaBelowLeaf110211 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell1102)) = true := by
+  have h : ((childLH (childLH thetaBelowCell1102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH thetaBelowCell1102)) h
+theorem e24KC2ThetaBelowLeaf110212 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell1102)) = true := by
+  have h : ((childHL (childLH thetaBelowCell1102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH thetaBelowCell1102)) h
+theorem e24KC2ThetaBelowLeaf110213 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell1102)) = true := by
+  have h : ((childHH (childLH thetaBelowCell1102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH thetaBelowCell1102)) h
+theorem e24KC2ThetaBelowLeaf11022 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1102) = true := by
+  have h : ((childHL thetaBelowCell1102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell1102) h
+theorem e24KC2ThetaBelowLeaf11023 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1102) = true := by
+  have h : ((childHH thetaBelowCell1102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell1102) h
+theorem e24KC2ThetaBelowLeaf110300 :
+    adaptiveCoverCheck 12 (childLL (childLL thetaBelowCell1103)) = true := by
+  have h : ((childLL (childLL thetaBelowCell1103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL thetaBelowCell1103)) h
+theorem e24KC2ThetaBelowLeaf110301 :
+    adaptiveCoverCheck 12 (childLH (childLL thetaBelowCell1103)) = true := by
+  have h : ((childLH (childLL thetaBelowCell1103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL thetaBelowCell1103)) h
+theorem e24KC2ThetaBelowLeaf110302 :
+    adaptiveCoverCheck 12 (childHL (childLL thetaBelowCell1103)) = true := by
+  have h : ((childHL (childLL thetaBelowCell1103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL thetaBelowCell1103)) h
+theorem e24KC2ThetaBelowLeaf110303 :
+    adaptiveCoverCheck 12 (childHH (childLL thetaBelowCell1103)) = true := by
+  have h : ((childHH (childLL thetaBelowCell1103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL thetaBelowCell1103)) h
+theorem e24KC2ThetaBelowLeaf110310 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell1103)) = true := by
+  have h : ((childLL (childLH thetaBelowCell1103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH thetaBelowCell1103)) h
+theorem e24KC2ThetaBelowLeaf110311 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell1103)) = true := by
+  have h : ((childLH (childLH thetaBelowCell1103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH thetaBelowCell1103)) h
+theorem e24KC2ThetaBelowLeaf110312 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell1103)) = true := by
+  have h : ((childHL (childLH thetaBelowCell1103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH thetaBelowCell1103)) h
+theorem e24KC2ThetaBelowLeaf110313 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell1103)) = true := by
+  have h : ((childHH (childLH thetaBelowCell1103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH thetaBelowCell1103)) h
+theorem e24KC2ThetaBelowLeaf11032 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1103) = true := by
+  have h : ((childHL thetaBelowCell1103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell1103) h
+theorem e24KC2ThetaBelowLeaf11033 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1103) = true := by
+  have h : ((childHH thetaBelowCell1103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell1103) h
+theorem e24KC2ThetaBelowLeaf111000 :
+    adaptiveCoverCheck 12 (childLL (childLL thetaBelowCell1110)) = true := by
+  have h : ((childLL (childLL thetaBelowCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL thetaBelowCell1110)) h
+theorem e24KC2ThetaBelowLeaf111001 :
+    adaptiveCoverCheck 12 (childLH (childLL thetaBelowCell1110)) = true := by
+  have h : ((childLH (childLL thetaBelowCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL thetaBelowCell1110)) h
+theorem e24KC2ThetaBelowLeaf1110020 :
+    adaptiveCoverCheck 11 (childLL (childHL (childLL thetaBelowCell1110))) = true := by
+  have h : ((childLL (childHL (childLL thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childLL thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110021 :
+    adaptiveCoverCheck 11 (childLH (childHL (childLL thetaBelowCell1110))) = true := by
+  have h : ((childLH (childHL (childLL thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childLL thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110022 :
+    adaptiveCoverCheck 11 (childHL (childHL (childLL thetaBelowCell1110))) = true := by
+  have h : ((childHL (childHL (childLL thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childLL thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110023 :
+    adaptiveCoverCheck 11 (childHH (childHL (childLL thetaBelowCell1110))) = true := by
+  have h : ((childHH (childHL (childLL thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childLL thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110030 :
+    adaptiveCoverCheck 11 (childLL (childHH (childLL thetaBelowCell1110))) = true := by
+  have h : ((childLL (childHH (childLL thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childLL thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110031 :
+    adaptiveCoverCheck 11 (childLH (childHH (childLL thetaBelowCell1110))) = true := by
+  have h : ((childLH (childHH (childLL thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childLL thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110032 :
+    adaptiveCoverCheck 11 (childHL (childHH (childLL thetaBelowCell1110))) = true := by
+  have h : ((childHL (childHH (childLL thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childLL thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110033 :
+    adaptiveCoverCheck 11 (childHH (childHH (childLL thetaBelowCell1110))) = true := by
+  have h : ((childHH (childHH (childLL thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childLL thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf111010 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell1110)) = true := by
+  have h : ((childLL (childLH thetaBelowCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH thetaBelowCell1110)) h
+theorem e24KC2ThetaBelowLeaf111011 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell1110)) = true := by
+  have h : ((childLH (childLH thetaBelowCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH thetaBelowCell1110)) h
+theorem e24KC2ThetaBelowLeaf1110120 :
+    adaptiveCoverCheck 11 (childLL (childHL (childLH thetaBelowCell1110))) = true := by
+  have h : ((childLL (childHL (childLH thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childLH thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110121 :
+    adaptiveCoverCheck 11 (childLH (childHL (childLH thetaBelowCell1110))) = true := by
+  have h : ((childLH (childHL (childLH thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childLH thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110122 :
+    adaptiveCoverCheck 11 (childHL (childHL (childLH thetaBelowCell1110))) = true := by
+  have h : ((childHL (childHL (childLH thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childLH thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110123 :
+    adaptiveCoverCheck 11 (childHH (childHL (childLH thetaBelowCell1110))) = true := by
+  have h : ((childHH (childHL (childLH thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childLH thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110130 :
+    adaptiveCoverCheck 11 (childLL (childHH (childLH thetaBelowCell1110))) = true := by
+  have h : ((childLL (childHH (childLH thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childLH thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110131 :
+    adaptiveCoverCheck 11 (childLH (childHH (childLH thetaBelowCell1110))) = true := by
+  have h : ((childLH (childHH (childLH thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childLH thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110132 :
+    adaptiveCoverCheck 11 (childHL (childHH (childLH thetaBelowCell1110))) = true := by
+  have h : ((childHL (childHH (childLH thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childLH thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110133 :
+    adaptiveCoverCheck 11 (childHH (childHH (childLH thetaBelowCell1110))) = true := by
+  have h : ((childHH (childHH (childLH thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childLH thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf11102000 :
+    adaptiveCoverCheck 10 thetaBelowCell11102000 = true := by
+  have h : (thetaBelowCell11102000).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102000 h
+theorem e24KC2ThetaBelowLeaf11102001 :
+    adaptiveCoverCheck 10 thetaBelowCell11102001 = true := by
+  have h : (thetaBelowCell11102001).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102001 h
+theorem e24KC2ThetaBelowLeaf111020020 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102002) = true := by
+  have h : ((childLL thetaBelowCell11102002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102002) h
+theorem e24KC2ThetaBelowLeaf111020021 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102002) = true := by
+  have h : ((childLH thetaBelowCell11102002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102002) h
+theorem e24KC2ThetaBelowLeaf111020022 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102002) = true := by
+  have h : ((childHL thetaBelowCell11102002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102002) h
+theorem e24KC2ThetaBelowLeaf111020023 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102002) = true := by
+  have h : ((childHH thetaBelowCell11102002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102002) h
+theorem e24KC2ThetaBelowLeaf111020030 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102003) = true := by
+  have h : ((childLL thetaBelowCell11102003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102003) h
+theorem e24KC2ThetaBelowLeaf111020031 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102003) = true := by
+  have h : ((childLH thetaBelowCell11102003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102003) h
+theorem e24KC2ThetaBelowLeaf111020032 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102003) = true := by
+  have h : ((childHL thetaBelowCell11102003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102003) h
+theorem e24KC2ThetaBelowLeaf111020033 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102003) = true := by
+  have h : ((childHH thetaBelowCell11102003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102003) h
+theorem e24KC2ThetaBelowLeaf11102010 :
+    adaptiveCoverCheck 10 thetaBelowCell11102010 = true := by
+  have h : (thetaBelowCell11102010).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102010 h
+theorem e24KC2ThetaBelowLeaf11102011 :
+    adaptiveCoverCheck 10 thetaBelowCell11102011 = true := by
+  have h : (thetaBelowCell11102011).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102011 h
+theorem e24KC2ThetaBelowLeaf111020120 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102012) = true := by
+  have h : ((childLL thetaBelowCell11102012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102012) h
+theorem e24KC2ThetaBelowLeaf111020121 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102012) = true := by
+  have h : ((childLH thetaBelowCell11102012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102012) h
+theorem e24KC2ThetaBelowLeaf111020122 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102012) = true := by
+  have h : ((childHL thetaBelowCell11102012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102012) h
+theorem e24KC2ThetaBelowLeaf111020123 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102012) = true := by
+  have h : ((childHH thetaBelowCell11102012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102012) h
+theorem e24KC2ThetaBelowLeaf111020130 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102013) = true := by
+  have h : ((childLL thetaBelowCell11102013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102013) h
+theorem e24KC2ThetaBelowLeaf111020131 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102013) = true := by
+  have h : ((childLH thetaBelowCell11102013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102013) h
+theorem e24KC2ThetaBelowLeaf111020132 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102013) = true := by
+  have h : ((childHL thetaBelowCell11102013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102013) h
+theorem e24KC2ThetaBelowLeaf111020133 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102013) = true := by
+  have h : ((childHH thetaBelowCell11102013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102013) h
+theorem e24KC2ThetaBelowLeaf111020200 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102020) = true := by
+  have h : ((childLL thetaBelowCell11102020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102020) h
+theorem e24KC2ThetaBelowLeaf111020201 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102020) = true := by
+  have h : ((childLH thetaBelowCell11102020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102020) h
+theorem e24KC2ThetaBelowLeaf111020202 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102020) = true := by
+  have h : ((childHL thetaBelowCell11102020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102020) h
+theorem e24KC2ThetaBelowLeaf111020203 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102020) = true := by
+  have h : ((childHH thetaBelowCell11102020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102020) h
+theorem e24KC2ThetaBelowLeaf111020210 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102021) = true := by
+  have h : ((childLL thetaBelowCell11102021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102021) h
+theorem e24KC2ThetaBelowLeaf111020211 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102021) = true := by
+  have h : ((childLH thetaBelowCell11102021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102021) h
+theorem e24KC2ThetaBelowLeaf111020212 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102021) = true := by
+  have h : ((childHL thetaBelowCell11102021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102021) h
+theorem e24KC2ThetaBelowLeaf111020213 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102021) = true := by
+  have h : ((childHH thetaBelowCell11102021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102021) h
+theorem e24KC2ThetaBelowLeaf111020220 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102022) = true := by
+  have h : ((childLL thetaBelowCell11102022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102022) h
+theorem e24KC2ThetaBelowLeaf111020221 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102022) = true := by
+  have h : ((childLH thetaBelowCell11102022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102022) h
+theorem e24KC2ThetaBelowLeaf111020222 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102022) = true := by
+  have h : ((childHL thetaBelowCell11102022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102022) h
+theorem e24KC2ThetaBelowLeaf111020223 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102022) = true := by
+  have h : ((childHH thetaBelowCell11102022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102022) h
+theorem e24KC2ThetaBelowLeaf111020230 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102023) = true := by
+  have h : ((childLL thetaBelowCell11102023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102023) h
+theorem e24KC2ThetaBelowLeaf111020231 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102023) = true := by
+  have h : ((childLH thetaBelowCell11102023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102023) h
+theorem e24KC2ThetaBelowLeaf111020232 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102023) = true := by
+  have h : ((childHL thetaBelowCell11102023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102023) h
+theorem e24KC2ThetaBelowLeaf111020233 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102023) = true := by
+  have h : ((childHH thetaBelowCell11102023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102023) h
+theorem e24KC2ThetaBelowLeaf111020300 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102030) = true := by
+  have h : ((childLL thetaBelowCell11102030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102030) h
+theorem e24KC2ThetaBelowLeaf111020301 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102030) = true := by
+  have h : ((childLH thetaBelowCell11102030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102030) h
+theorem e24KC2ThetaBelowLeaf111020302 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102030) = true := by
+  have h : ((childHL thetaBelowCell11102030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102030) h
+theorem e24KC2ThetaBelowLeaf111020303 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102030) = true := by
+  have h : ((childHH thetaBelowCell11102030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102030) h
+theorem e24KC2ThetaBelowLeaf111020310 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102031) = true := by
+  have h : ((childLL thetaBelowCell11102031)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102031) h
+theorem e24KC2ThetaBelowLeaf111020311 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102031) = true := by
+  have h : ((childLH thetaBelowCell11102031)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102031) h
+theorem e24KC2ThetaBelowLeaf111020312 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102031) = true := by
+  have h : ((childHL thetaBelowCell11102031)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102031) h
+theorem e24KC2ThetaBelowLeaf111020313 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102031) = true := by
+  have h : ((childHH thetaBelowCell11102031)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102031) h
+theorem e24KC2ThetaBelowLeaf111020320 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102032) = true := by
+  have h : ((childLL thetaBelowCell11102032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102032) h
+theorem e24KC2ThetaBelowLeaf111020321 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102032) = true := by
+  have h : ((childLH thetaBelowCell11102032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102032) h
+theorem e24KC2ThetaBelowLeaf111020322 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102032) = true := by
+  have h : ((childHL thetaBelowCell11102032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102032) h
+theorem e24KC2ThetaBelowLeaf111020323 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102032) = true := by
+  have h : ((childHH thetaBelowCell11102032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102032) h
+theorem e24KC2ThetaBelowLeaf111020330 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102033) = true := by
+  have h : ((childLL thetaBelowCell11102033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102033) h
+theorem e24KC2ThetaBelowLeaf111020331 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102033) = true := by
+  have h : ((childLH thetaBelowCell11102033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102033) h
+theorem e24KC2ThetaBelowLeaf111020332 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102033) = true := by
+  have h : ((childHL thetaBelowCell11102033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102033) h
+theorem e24KC2ThetaBelowLeaf111020333 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102033) = true := by
+  have h : ((childHH thetaBelowCell11102033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102033) h
+theorem e24KC2ThetaBelowLeaf11102100 :
+    adaptiveCoverCheck 10 thetaBelowCell11102100 = true := by
+  have h : (thetaBelowCell11102100).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102100 h
+theorem e24KC2ThetaBelowLeaf11102101 :
+    adaptiveCoverCheck 10 thetaBelowCell11102101 = true := by
+  have h : (thetaBelowCell11102101).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102101 h
+theorem e24KC2ThetaBelowLeaf11102102 :
+    adaptiveCoverCheck 10 thetaBelowCell11102102 = true := by
+  have h : (thetaBelowCell11102102).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102102 h
+theorem e24KC2ThetaBelowLeaf11102103 :
+    adaptiveCoverCheck 10 thetaBelowCell11102103 = true := by
+  have h : (thetaBelowCell11102103).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102103 h
+theorem e24KC2ThetaBelowLeaf11102110 :
+    adaptiveCoverCheck 10 thetaBelowCell11102110 = true := by
+  have h : (thetaBelowCell11102110).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102110 h
+theorem e24KC2ThetaBelowLeaf11102111 :
+    adaptiveCoverCheck 10 thetaBelowCell11102111 = true := by
+  have h : (thetaBelowCell11102111).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102111 h
+theorem e24KC2ThetaBelowLeaf11102112 :
+    adaptiveCoverCheck 10 thetaBelowCell11102112 = true := by
+  have h : (thetaBelowCell11102112).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102112 h
+theorem e24KC2ThetaBelowLeaf11102113 :
+    adaptiveCoverCheck 10 thetaBelowCell11102113 = true := by
+  have h : (thetaBelowCell11102113).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102113 h
+theorem e24KC2ThetaBelowLeaf111021200 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102120) = true := by
+  have h : ((childLL thetaBelowCell11102120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102120) h
+theorem e24KC2ThetaBelowLeaf111021201 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102120) = true := by
+  have h : ((childLH thetaBelowCell11102120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102120) h
+theorem e24KC2ThetaBelowLeaf111021202 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102120) = true := by
+  have h : ((childHL thetaBelowCell11102120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102120) h
+theorem e24KC2ThetaBelowLeaf111021203 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102120) = true := by
+  have h : ((childHH thetaBelowCell11102120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102120) h
+theorem e24KC2ThetaBelowLeaf111021210 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102121) = true := by
+  have h : ((childLL thetaBelowCell11102121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102121) h
+theorem e24KC2ThetaBelowLeaf111021211 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102121) = true := by
+  have h : ((childLH thetaBelowCell11102121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102121) h
+theorem e24KC2ThetaBelowLeaf111021212 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102121) = true := by
+  have h : ((childHL thetaBelowCell11102121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102121) h
+theorem e24KC2ThetaBelowLeaf111021213 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102121) = true := by
+  have h : ((childHH thetaBelowCell11102121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102121) h
+theorem e24KC2ThetaBelowLeaf111021220 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102122) = true := by
+  have h : ((childLL thetaBelowCell11102122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102122) h
+theorem e24KC2ThetaBelowLeaf111021221 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102122) = true := by
+  have h : ((childLH thetaBelowCell11102122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102122) h
+theorem e24KC2ThetaBelowLeaf111021222 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102122) = true := by
+  have h : ((childHL thetaBelowCell11102122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102122) h
+theorem e24KC2ThetaBelowLeaf111021223 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102122) = true := by
+  have h : ((childHH thetaBelowCell11102122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102122) h
+theorem e24KC2ThetaBelowLeaf111021230 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102123) = true := by
+  have h : ((childLL thetaBelowCell11102123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102123) h
+theorem e24KC2ThetaBelowLeaf111021231 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102123) = true := by
+  have h : ((childLH thetaBelowCell11102123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102123) h
+theorem e24KC2ThetaBelowLeaf111021232 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102123) = true := by
+  have h : ((childHL thetaBelowCell11102123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102123) h
+theorem e24KC2ThetaBelowLeaf111021233 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102123) = true := by
+  have h : ((childHH thetaBelowCell11102123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102123) h
+theorem e24KC2ThetaBelowLeaf111021300 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102130) = true := by
+  have h : ((childLL thetaBelowCell11102130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102130) h
+theorem e24KC2ThetaBelowLeaf111021301 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102130) = true := by
+  have h : ((childLH thetaBelowCell11102130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102130) h
+theorem e24KC2ThetaBelowLeaf111021302 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102130) = true := by
+  have h : ((childHL thetaBelowCell11102130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102130) h
+theorem e24KC2ThetaBelowLeaf111021303 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102130) = true := by
+  have h : ((childHH thetaBelowCell11102130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102130) h
+theorem e24KC2ThetaBelowLeaf111021310 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102131) = true := by
+  have h : ((childLL thetaBelowCell11102131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102131) h
+theorem e24KC2ThetaBelowLeaf111021311 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102131) = true := by
+  have h : ((childLH thetaBelowCell11102131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102131) h
+theorem e24KC2ThetaBelowLeaf111021312 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102131) = true := by
+  have h : ((childHL thetaBelowCell11102131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102131) h
+theorem e24KC2ThetaBelowLeaf111021313 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102131) = true := by
+  have h : ((childHH thetaBelowCell11102131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102131) h
+theorem e24KC2ThetaBelowLeaf111021320 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102132) = true := by
+  have h : ((childLL thetaBelowCell11102132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102132) h
+theorem e24KC2ThetaBelowLeaf111021321 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102132) = true := by
+  have h : ((childLH thetaBelowCell11102132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102132) h
+theorem e24KC2ThetaBelowLeaf111021322 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102132) = true := by
+  have h : ((childHL thetaBelowCell11102132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102132) h
+theorem e24KC2ThetaBelowLeaf111021323 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102132) = true := by
+  have h : ((childHH thetaBelowCell11102132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102132) h
+theorem e24KC2ThetaBelowLeaf111021330 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102133) = true := by
+  have h : ((childLL thetaBelowCell11102133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102133) h
+theorem e24KC2ThetaBelowLeaf111021331 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102133) = true := by
+  have h : ((childLH thetaBelowCell11102133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102133) h
+theorem e24KC2ThetaBelowLeaf111021332 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102133) = true := by
+  have h : ((childHL thetaBelowCell11102133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102133) h
+theorem e24KC2ThetaBelowLeaf111021333 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102133) = true := by
+  have h : ((childHH thetaBelowCell11102133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102133) h
+theorem e24KC2ThetaBelowLeaf111022000 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102200) = true := by
+  have h : ((childLL thetaBelowCell11102200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102200) h
+theorem e24KC2ThetaBelowLeaf111022001 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102200) = true := by
+  have h : ((childLH thetaBelowCell11102200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102200) h
+theorem e24KC2ThetaBelowLeaf111022002 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102200) = true := by
+  have h : ((childHL thetaBelowCell11102200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102200) h
+theorem e24KC2ThetaBelowLeaf111022003 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102200) = true := by
+  have h : ((childHH thetaBelowCell11102200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102200) h
+theorem e24KC2ThetaBelowLeaf111022010 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102201) = true := by
+  have h : ((childLL thetaBelowCell11102201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102201) h
+theorem e24KC2ThetaBelowLeaf111022011 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102201) = true := by
+  have h : ((childLH thetaBelowCell11102201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102201) h
+theorem e24KC2ThetaBelowLeaf111022012 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102201) = true := by
+  have h : ((childHL thetaBelowCell11102201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102201) h
+theorem e24KC2ThetaBelowLeaf111022013 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102201) = true := by
+  have h : ((childHH thetaBelowCell11102201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102201) h
+theorem e24KC2ThetaBelowLeaf11102202 :
+    adaptiveCoverCheck 10 thetaBelowCell11102202 = true := by
+  have h : (thetaBelowCell11102202).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102202 h
+theorem e24KC2ThetaBelowLeaf11102203 :
+    adaptiveCoverCheck 10 thetaBelowCell11102203 = true := by
+  have h : (thetaBelowCell11102203).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102203 h
+theorem e24KC2ThetaBelowLeaf111022100 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102210) = true := by
+  have h : ((childLL thetaBelowCell11102210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102210) h
+theorem e24KC2ThetaBelowLeaf111022101 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102210) = true := by
+  have h : ((childLH thetaBelowCell11102210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102210) h
+theorem e24KC2ThetaBelowLeaf111022102 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102210) = true := by
+  have h : ((childHL thetaBelowCell11102210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102210) h
+theorem e24KC2ThetaBelowLeaf111022103 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102210) = true := by
+  have h : ((childHH thetaBelowCell11102210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102210) h
+theorem e24KC2ThetaBelowLeaf111022110 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102211) = true := by
+  have h : ((childLL thetaBelowCell11102211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102211) h
+theorem e24KC2ThetaBelowLeaf111022111 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102211) = true := by
+  have h : ((childLH thetaBelowCell11102211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102211) h
+theorem e24KC2ThetaBelowLeaf111022112 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102211) = true := by
+  have h : ((childHL thetaBelowCell11102211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102211) h
+theorem e24KC2ThetaBelowLeaf111022113 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102211) = true := by
+  have h : ((childHH thetaBelowCell11102211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102211) h
+theorem e24KC2ThetaBelowLeaf11102212 :
+    adaptiveCoverCheck 10 thetaBelowCell11102212 = true := by
+  have h : (thetaBelowCell11102212).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102212 h
+theorem e24KC2ThetaBelowLeaf11102213 :
+    adaptiveCoverCheck 10 thetaBelowCell11102213 = true := by
+  have h : (thetaBelowCell11102213).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102213 h
+theorem e24KC2ThetaBelowLeaf1110222 :
+    adaptiveCoverCheck 11 (childHL (childHL (childHL thetaBelowCell1110))) = true := by
+  have h : ((childHL (childHL (childHL thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childHL thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110223 :
+    adaptiveCoverCheck 11 (childHH (childHL (childHL thetaBelowCell1110))) = true := by
+  have h : ((childHH (childHL (childHL thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childHL thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf111023000 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102300) = true := by
+  have h : ((childLL thetaBelowCell11102300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102300) h
+theorem e24KC2ThetaBelowLeaf111023001 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102300) = true := by
+  have h : ((childLH thetaBelowCell11102300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102300) h
+theorem e24KC2ThetaBelowLeaf111023002 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102300) = true := by
+  have h : ((childHL thetaBelowCell11102300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102300) h
+theorem e24KC2ThetaBelowLeaf111023003 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102300) = true := by
+  have h : ((childHH thetaBelowCell11102300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102300) h
+theorem e24KC2ThetaBelowLeaf111023010 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102301) = true := by
+  have h : ((childLL thetaBelowCell11102301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102301) h
+theorem e24KC2ThetaBelowLeaf111023011 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102301) = true := by
+  have h : ((childLH thetaBelowCell11102301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102301) h
+theorem e24KC2ThetaBelowLeaf111023012 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102301) = true := by
+  have h : ((childHL thetaBelowCell11102301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102301) h
+theorem e24KC2ThetaBelowLeaf111023013 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102301) = true := by
+  have h : ((childHH thetaBelowCell11102301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102301) h
+theorem e24KC2ThetaBelowLeaf11102302 :
+    adaptiveCoverCheck 10 thetaBelowCell11102302 = true := by
+  have h : (thetaBelowCell11102302).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102302 h
+theorem e24KC2ThetaBelowLeaf11102303 :
+    adaptiveCoverCheck 10 thetaBelowCell11102303 = true := by
+  have h : (thetaBelowCell11102303).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102303 h
+theorem e24KC2ThetaBelowLeaf111023100 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102310) = true := by
+  have h : ((childLL thetaBelowCell11102310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102310) h
+theorem e24KC2ThetaBelowLeaf111023101 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102310) = true := by
+  have h : ((childLH thetaBelowCell11102310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102310) h
+theorem e24KC2ThetaBelowLeaf111023102 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102310) = true := by
+  have h : ((childHL thetaBelowCell11102310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102310) h
+theorem e24KC2ThetaBelowLeaf111023103 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102310) = true := by
+  have h : ((childHH thetaBelowCell11102310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102310) h
+theorem e24KC2ThetaBelowLeaf111023110 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11102311) = true := by
+  have h : ((childLL thetaBelowCell11102311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11102311) h
+theorem e24KC2ThetaBelowLeaf111023111 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11102311) = true := by
+  have h : ((childLH thetaBelowCell11102311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11102311) h
+theorem e24KC2ThetaBelowLeaf111023112 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11102311) = true := by
+  have h : ((childHL thetaBelowCell11102311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11102311) h
+theorem e24KC2ThetaBelowLeaf111023113 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11102311) = true := by
+  have h : ((childHH thetaBelowCell11102311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11102311) h
+theorem e24KC2ThetaBelowLeaf11102312 :
+    adaptiveCoverCheck 10 thetaBelowCell11102312 = true := by
+  have h : (thetaBelowCell11102312).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102312 h
+theorem e24KC2ThetaBelowLeaf11102313 :
+    adaptiveCoverCheck 10 thetaBelowCell11102313 = true := by
+  have h : (thetaBelowCell11102313).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11102313 h
+theorem e24KC2ThetaBelowLeaf1110232 :
+    adaptiveCoverCheck 11 (childHL (childHH (childHL thetaBelowCell1110))) = true := by
+  have h : ((childHL (childHH (childHL thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childHL thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110233 :
+    adaptiveCoverCheck 11 (childHH (childHH (childHL thetaBelowCell1110))) = true := by
+  have h : ((childHH (childHH (childHL thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childHL thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf11103000 :
+    adaptiveCoverCheck 10 thetaBelowCell11103000 = true := by
+  have h : (thetaBelowCell11103000).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103000 h
+theorem e24KC2ThetaBelowLeaf11103001 :
+    adaptiveCoverCheck 10 thetaBelowCell11103001 = true := by
+  have h : (thetaBelowCell11103001).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103001 h
+theorem e24KC2ThetaBelowLeaf11103002 :
+    adaptiveCoverCheck 10 thetaBelowCell11103002 = true := by
+  have h : (thetaBelowCell11103002).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103002 h
+theorem e24KC2ThetaBelowLeaf11103003 :
+    adaptiveCoverCheck 10 thetaBelowCell11103003 = true := by
+  have h : (thetaBelowCell11103003).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103003 h
+theorem e24KC2ThetaBelowLeaf11103010 :
+    adaptiveCoverCheck 10 thetaBelowCell11103010 = true := by
+  have h : (thetaBelowCell11103010).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103010 h
+theorem e24KC2ThetaBelowLeaf11103011 :
+    adaptiveCoverCheck 10 thetaBelowCell11103011 = true := by
+  have h : (thetaBelowCell11103011).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103011 h
+theorem e24KC2ThetaBelowLeaf11103012 :
+    adaptiveCoverCheck 10 thetaBelowCell11103012 = true := by
+  have h : (thetaBelowCell11103012).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103012 h
+theorem e24KC2ThetaBelowLeaf11103013 :
+    adaptiveCoverCheck 10 thetaBelowCell11103013 = true := by
+  have h : (thetaBelowCell11103013).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103013 h
+theorem e24KC2ThetaBelowLeaf111030200 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103020) = true := by
+  have h : ((childLL thetaBelowCell11103020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11103020) h
+theorem e24KC2ThetaBelowLeaf111030201 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103020) = true := by
+  have h : ((childLH thetaBelowCell11103020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11103020) h
+theorem e24KC2ThetaBelowLeaf111030202 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103020) = true := by
+  have h : ((childHL thetaBelowCell11103020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11103020) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC5FrontierBatchF96LR00464`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF96R00465`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5c0c39e876
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5c0c39e876
+
+open CertificateCells5c0c39e876
+namespace CoverCertificate446b5129d2
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate446b5129d2
+
+theorem e24KC2PhiBelowLeaf33301 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3330) = true := by
+  exact CoverCertificate446b5129d2.checkedRoot
+namespace CoverCertificate3d29d6c185
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3d29d6c185
+
+theorem e24KC2PhiBelowLeaf33302 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3330) = true := by
+  exact CoverCertificate3d29d6c185.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa1414ae513
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa1414ae513
+
+open CertificateCellsa1414ae513
+namespace CoverCertificate5443a01fd5
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5443a01fd5
+
+theorem e24KC2PhiBelowLeaf33303 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3330) = true := by
+  exact CoverCertificate5443a01fd5.checkedRoot
+namespace CoverCertificateef75f67166
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateef75f67166
+
+theorem e24KC2PhiBelowLeaf33310 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3331) = true := by
+  exact CoverCertificateef75f67166.checkedRoot
+namespace CoverCertificate77769ce963
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate77769ce963
+
+theorem e24KC2PhiBelowLeaf33311 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3331) = true := by
+  exact CoverCertificate77769ce963.checkedRoot
+namespace CoverCertificateaaf4f15552
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateaaf4f15552
+
+theorem e24KC2PhiBelowLeaf33312 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3331) = true := by
+  exact CoverCertificateaaf4f15552.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle002.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle002.lean
new file mode 100644
index 0000000000..c7fcc8a3ed
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle002.lean
@@ -0,0 +1,12859 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch002`.
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch014`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CoverCertificate6cd78cf99c
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLL (childLH (childLH (childLL (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate6cd78cf99c
+
+namespace GerverSofa.PartE.CoverCertificate3b2bb200ff
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLH (childLH (childLH (childLL (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate3b2bb200ff
+
+namespace GerverSofa.PartE.CoverCertificate416e1e96e0
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLH (childLH (childLH (childLL (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate416e1e96e0
+
+namespace GerverSofa.PartE.CoverCertificate2a65321173
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLH (childLH (childLH (childLL (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate2a65321173
+
+namespace GerverSofa.PartE.CoverCertificatecc929b68fe
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLH (childLH (childLH (childLL (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatecc929b68fe
+
+namespace GerverSofa.PartE.CoverCertificate82c27f8e4d
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLL (childLL (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate82c27f8e4d
+
+namespace GerverSofa.PartE.CoverCertificatecbf8beb65d
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLL (childLL (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatecbf8beb65d
+
+namespace GerverSofa.PartE.CoverCertificate319b5d79a7
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLL (childLL (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate319b5d79a7
+
+namespace GerverSofa.PartE.CoverCertificate52a2adcf18
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLL (childLL (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate52a2adcf18
+
+namespace GerverSofa.PartE.CoverCertificatef66846141d
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLH (childLL (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificatef66846141d
+
+namespace GerverSofa.PartE.CoverCertificate5cd1561dfb
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLH (childLL (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate5cd1561dfb
+
+namespace GerverSofa.PartE.CoverCertificate43ce0d7d08
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLH (childLL (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate43ce0d7d08
+
+namespace GerverSofa.PartE.CoverCertificate3c4ee9d54f
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childLH (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate3c4ee9d54f
+
+namespace GerverSofa.PartE.CoverCertificate5ea8c0bac2
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childLH (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate5ea8c0bac2
+
+namespace GerverSofa.PartE.CoverCertificatec619bc60a1
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLH (childLH (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatec619bc60a1
+
+namespace GerverSofa.PartE.CoverCertificate5746fe2caf
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLH (childLH (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate5746fe2caf
+
+namespace GerverSofa.PartE.CoverCertificate98b9778203
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate98b9778203
+
+namespace GerverSofa.PartE.CoverCertificate7f98aa56d1
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+end GerverSofa.PartE.CoverCertificate7f98aa56d1
+
+namespace GerverSofa.PartE.CoverCertificateb650783453
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificateb650783453
+
+namespace GerverSofa.PartE.CoverCertificate05dde891b1
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+private abbrev cell2220 : AngleCell :=
+  childLL cell222
+
+private abbrev cell2221 : AngleCell :=
+  childLH cell222
+
+private abbrev cell2222 : AngleCell :=
+  childHL cell222
+
+private abbrev cell2223 : AngleCell :=
+  childHH cell222
+
+end GerverSofa.PartE.CoverCertificate05dde891b1
+
+namespace GerverSofa.PartE.CoverCertificatedd1cfe6f59
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+end GerverSofa.PartE.CoverCertificatedd1cfe6f59
+
+namespace GerverSofa.PartE.CoverCertificatefe5dc96009
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatefe5dc96009
+
+namespace GerverSofa.PartE.CoverCertificate7ba2433252
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate7ba2433252
+
+namespace GerverSofa.PartE.CoverCertificate7633ba69a5
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate7633ba69a5
+
+namespace GerverSofa.PartE.CoverCertificate8e17b594b9
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+end GerverSofa.PartE.CoverCertificate8e17b594b9
+
+namespace GerverSofa.PartE.CoverCertificate2ad8ea30c5
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childLL (childHL (childHL (childLL (childLL (childLL (childLL
+    (e24ThetaAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate2ad8ea30c5
+
+namespace GerverSofa.PartE.CoverCertificate5fed50edc5
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childLL (childHL (childHL (childLL (childLL (childLL (childLL
+    (e24ThetaAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate5fed50edc5
+
+namespace GerverSofa.PartE.CoverCertificatebeb942e0ae
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childLL (childHL (childHL (childLL (childLL (childLL (childLL
+    (e24ThetaAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatebeb942e0ae
+
+namespace GerverSofa.PartE.CoverCertificate4aac8bf510
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childLL (childHL (childHL (childLL (childLL (childLL (childLL
+    (e24ThetaAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate4aac8bf510
+
+namespace GerverSofa.PartE.CoverCertificate018ca4eb00
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLH (childLL (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificate018ca4eb00
+
+namespace GerverSofa.PartE.CoverCertificatefbeddf9072
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childLL (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatefbeddf9072
+
+namespace GerverSofa.PartE.CoverCertificatee6bd46d219
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childLL (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificatee6bd46d219
+
+namespace GerverSofa.PartE.CoverCertificate2a0a30dc9d
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLH (childLL (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate2a0a30dc9d
+
+namespace GerverSofa.PartE.CoverCertificateff98d0df99
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLH (childLL (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificateff98d0df99
+
+namespace GerverSofa.PartE.CoverCertificate63e0e0ed96
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLH (childLL (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate63e0e0ed96
+
+namespace GerverSofa.PartE.CoverCertificate1d20eb40b2
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLH (childLL (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate1d20eb40b2
+
+namespace GerverSofa.PartE.CoverCertificate49ab43c4dd
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLL (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate49ab43c4dd
+
+namespace GerverSofa.PartE.CoverCertificatede0603de49
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLL (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatede0603de49
+
+namespace GerverSofa.PartE.CoverCertificate90d2e1946c
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate90d2e1946c
+
+namespace GerverSofa.PartE.CoverCertificatebf3761efb7
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificatebf3761efb7
+
+namespace GerverSofa.PartE.CoverCertificate48ab78c281
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLL (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate48ab78c281
+
+namespace GerverSofa.PartE.CoverCertificate5e935aba69
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLL (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate5e935aba69
+
+namespace GerverSofa.PartE.CoverCertificateea859f6e20
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificateea859f6e20
+
+namespace GerverSofa.PartE.CoverCertificatedc87a79519
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificatedc87a79519
+
+namespace GerverSofa.PartE.CoverCertificatefd3c81c65d
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLH (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatefd3c81c65d
+
+namespace GerverSofa.PartE.CoverPhiAbove1101100
+
+/-- The root angle cell for the expanded adaptive-cover certificate below `phiAboveCell1101`. -/
+abbrev coverCell : AngleCell :=
+  (childLL (childLL (childLH (childLH (childLL (childLH (childLH (e24PhiAboveRoot))))))))
+
+/-- Cached adaptive-cover cell 0; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0 : AngleCell := childLL coverCell
+
+/-- Cached adaptive-cover cell 1; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1 : AngleCell := childLH coverCell
+
+/-- Cached adaptive-cover cell 2; its parent subdivisions specify the exact interval. -/
+abbrev coverCell2 : AngleCell := childHL coverCell
+
+/-- Cached adaptive-cover cell 3; its parent subdivisions specify the exact interval. -/
+abbrev coverCell3 : AngleCell := childHH coverCell
+
+/-- Cached adaptive-cover cell 00; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00 : AngleCell := childLL coverCell0
+
+/-- Cached adaptive-cover cell 01; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01 : AngleCell := childLH coverCell0
+
+/-- Cached adaptive-cover cell 02; its parent subdivisions specify the exact interval. -/
+abbrev coverCell02 : AngleCell := childHL coverCell0
+
+/-- Cached adaptive-cover cell 03; its parent subdivisions specify the exact interval. -/
+abbrev coverCell03 : AngleCell := childHH coverCell0
+
+/-- Cached adaptive-cover cell 10; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10 : AngleCell := childLL coverCell1
+
+/-- Cached adaptive-cover cell 11; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11 : AngleCell := childLH coverCell1
+
+/-- Cached adaptive-cover cell 12; its parent subdivisions specify the exact interval. -/
+abbrev coverCell12 : AngleCell := childHL coverCell1
+
+/-- Cached adaptive-cover cell 13; its parent subdivisions specify the exact interval. -/
+abbrev coverCell13 : AngleCell := childHH coverCell1
+
+/-- Cached adaptive-cover cell 20; its parent subdivisions specify the exact interval. -/
+abbrev coverCell20 : AngleCell := childLL coverCell2
+
+/-- Cached adaptive-cover cell 21; its parent subdivisions specify the exact interval. -/
+abbrev coverCell21 : AngleCell := childLH coverCell2
+
+/-- Cached adaptive-cover cell 22; its parent subdivisions specify the exact interval. -/
+abbrev coverCell22 : AngleCell := childHL coverCell2
+
+/-- Cached adaptive-cover cell 23; its parent subdivisions specify the exact interval. -/
+abbrev coverCell23 : AngleCell := childHH coverCell2
+
+/-- Cached adaptive-cover cell 30; its parent subdivisions specify the exact interval. -/
+abbrev coverCell30 : AngleCell := childLL coverCell3
+
+/-- Cached adaptive-cover cell 31; its parent subdivisions specify the exact interval. -/
+abbrev coverCell31 : AngleCell := childLH coverCell3
+
+/-- Cached adaptive-cover cell 32; its parent subdivisions specify the exact interval. -/
+abbrev coverCell32 : AngleCell := childHL coverCell3
+
+/-- Cached adaptive-cover cell 33; its parent subdivisions specify the exact interval. -/
+abbrev coverCell33 : AngleCell := childHH coverCell3
+
+/-- Cached adaptive-cover cell 000; its parent subdivisions specify the exact interval. -/
+abbrev coverCell000 : AngleCell := childLL coverCell00
+
+/-- Cached adaptive-cover cell 001; its parent subdivisions specify the exact interval. -/
+abbrev coverCell001 : AngleCell := childLH coverCell00
+
+/-- Cached adaptive-cover cell 002; its parent subdivisions specify the exact interval. -/
+abbrev coverCell002 : AngleCell := childHL coverCell00
+
+/-- Cached adaptive-cover cell 003; its parent subdivisions specify the exact interval. -/
+abbrev coverCell003 : AngleCell := childHH coverCell00
+
+/-- Cached adaptive-cover cell 010; its parent subdivisions specify the exact interval. -/
+abbrev coverCell010 : AngleCell := childLL coverCell01
+
+/-- Cached adaptive-cover cell 011; its parent subdivisions specify the exact interval. -/
+abbrev coverCell011 : AngleCell := childLH coverCell01
+
+/-- Cached adaptive-cover cell 012; its parent subdivisions specify the exact interval. -/
+abbrev coverCell012 : AngleCell := childHL coverCell01
+
+/-- Cached adaptive-cover cell 013; its parent subdivisions specify the exact interval. -/
+abbrev coverCell013 : AngleCell := childHH coverCell01
+
+/-- Cached adaptive-cover cell 030; its parent subdivisions specify the exact interval. -/
+abbrev coverCell030 : AngleCell := childLL coverCell03
+
+/-- Cached adaptive-cover cell 031; its parent subdivisions specify the exact interval. -/
+abbrev coverCell031 : AngleCell := childLH coverCell03
+
+/-- Cached adaptive-cover cell 032; its parent subdivisions specify the exact interval. -/
+abbrev coverCell032 : AngleCell := childHL coverCell03
+
+/-- Cached adaptive-cover cell 033; its parent subdivisions specify the exact interval. -/
+abbrev coverCell033 : AngleCell := childHH coverCell03
+
+/-- Cached adaptive-cover cell 100; its parent subdivisions specify the exact interval. -/
+abbrev coverCell100 : AngleCell := childLL coverCell10
+
+/-- Cached adaptive-cover cell 101; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101 : AngleCell := childLH coverCell10
+
+/-- Cached adaptive-cover cell 102; its parent subdivisions specify the exact interval. -/
+abbrev coverCell102 : AngleCell := childHL coverCell10
+
+/-- Cached adaptive-cover cell 103; its parent subdivisions specify the exact interval. -/
+abbrev coverCell103 : AngleCell := childHH coverCell10
+
+/-- Cached adaptive-cover cell 110; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110 : AngleCell := childLL coverCell11
+
+/-- Cached adaptive-cover cell 111; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111 : AngleCell := childLH coverCell11
+
+/-- Cached adaptive-cover cell 112; its parent subdivisions specify the exact interval. -/
+abbrev coverCell112 : AngleCell := childHL coverCell11
+
+/-- Cached adaptive-cover cell 113; its parent subdivisions specify the exact interval. -/
+abbrev coverCell113 : AngleCell := childHH coverCell11
+
+/-- Cached adaptive-cover cell 120; its parent subdivisions specify the exact interval. -/
+abbrev coverCell120 : AngleCell := childLL coverCell12
+
+/-- Cached adaptive-cover cell 121; its parent subdivisions specify the exact interval. -/
+abbrev coverCell121 : AngleCell := childLH coverCell12
+
+/-- Cached adaptive-cover cell 122; its parent subdivisions specify the exact interval. -/
+abbrev coverCell122 : AngleCell := childHL coverCell12
+
+/-- Cached adaptive-cover cell 123; its parent subdivisions specify the exact interval. -/
+abbrev coverCell123 : AngleCell := childHH coverCell12
+
+/-- Cached adaptive-cover cell 130; its parent subdivisions specify the exact interval. -/
+abbrev coverCell130 : AngleCell := childLL coverCell13
+
+/-- Cached adaptive-cover cell 131; its parent subdivisions specify the exact interval. -/
+abbrev coverCell131 : AngleCell := childLH coverCell13
+
+/-- Cached adaptive-cover cell 132; its parent subdivisions specify the exact interval. -/
+abbrev coverCell132 : AngleCell := childHL coverCell13
+
+/-- Cached adaptive-cover cell 133; its parent subdivisions specify the exact interval. -/
+abbrev coverCell133 : AngleCell := childHH coverCell13
+
+/-- Cached adaptive-cover cell 0000; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0000 : AngleCell := childLL coverCell000
+
+/-- Cached adaptive-cover cell 0001; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0001 : AngleCell := childLH coverCell000
+
+/-- Cached adaptive-cover cell 0002; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0002 : AngleCell := childHL coverCell000
+
+/-- Cached adaptive-cover cell 0003; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0003 : AngleCell := childHH coverCell000
+
+/-- Cached adaptive-cover cell 0010; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0010 : AngleCell := childLL coverCell001
+
+/-- Cached adaptive-cover cell 0011; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0011 : AngleCell := childLH coverCell001
+
+/-- Cached adaptive-cover cell 0012; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0012 : AngleCell := childHL coverCell001
+
+/-- Cached adaptive-cover cell 0013; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0013 : AngleCell := childHH coverCell001
+
+/-- Cached adaptive-cover cell 0100; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0100 : AngleCell := childLL coverCell010
+
+/-- Cached adaptive-cover cell 0101; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0101 : AngleCell := childLH coverCell010
+
+/-- Cached adaptive-cover cell 0102; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0102 : AngleCell := childHL coverCell010
+
+/-- Cached adaptive-cover cell 0103; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0103 : AngleCell := childHH coverCell010
+
+/-- Cached adaptive-cover cell 0110; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0110 : AngleCell := childLL coverCell011
+
+/-- Cached adaptive-cover cell 0111; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0111 : AngleCell := childLH coverCell011
+
+/-- Cached adaptive-cover cell 0112; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0112 : AngleCell := childHL coverCell011
+
+/-- Cached adaptive-cover cell 0113; its parent subdivisions specify the exact interval. -/
+abbrev coverCell0113 : AngleCell := childHH coverCell011
+
+/-- Cached adaptive-cover cell 1000; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1000 : AngleCell := childLL coverCell100
+
+/-- Cached adaptive-cover cell 1001; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1001 : AngleCell := childLH coverCell100
+
+/-- Cached adaptive-cover cell 1002; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1002 : AngleCell := childHL coverCell100
+
+/-- Cached adaptive-cover cell 1003; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1003 : AngleCell := childHH coverCell100
+
+/-- Cached adaptive-cover cell 1010; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1010 : AngleCell := childLL coverCell101
+
+/-- Cached adaptive-cover cell 1011; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1011 : AngleCell := childLH coverCell101
+
+/-- Cached adaptive-cover cell 1012; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1012 : AngleCell := childHL coverCell101
+
+/-- Cached adaptive-cover cell 1013; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1013 : AngleCell := childHH coverCell101
+
+/-- Cached adaptive-cover cell 1100; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1100 : AngleCell := childLL coverCell110
+
+/-- Cached adaptive-cover cell 1101; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1101 : AngleCell := childLH coverCell110
+
+/-- Cached adaptive-cover cell 1102; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1102 : AngleCell := childHL coverCell110
+
+/-- Cached adaptive-cover cell 1103; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1103 : AngleCell := childHH coverCell110
+
+/-- Cached adaptive-cover cell 1110; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1110 : AngleCell := childLL coverCell111
+
+/-- Cached adaptive-cover cell 1111; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111 : AngleCell := childLH coverCell111
+
+/-- Cached adaptive-cover cell 1112; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1112 : AngleCell := childHL coverCell111
+
+/-- Cached adaptive-cover cell 1113; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1113 : AngleCell := childHH coverCell111
+
+/-- Cached adaptive-cover cell 1120; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1120 : AngleCell := childLL coverCell112
+
+/-- Cached adaptive-cover cell 1121; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1121 : AngleCell := childLH coverCell112
+
+/-- Cached adaptive-cover cell 1122; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1122 : AngleCell := childHL coverCell112
+
+/-- Cached adaptive-cover cell 1123; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1123 : AngleCell := childHH coverCell112
+
+/-- Cached adaptive-cover cell 1130; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1130 : AngleCell := childLL coverCell113
+
+/-- Cached adaptive-cover cell 1131; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1131 : AngleCell := childLH coverCell113
+
+/-- Cached adaptive-cover cell 1132; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1132 : AngleCell := childHL coverCell113
+
+/-- Cached adaptive-cover cell 1133; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1133 : AngleCell := childHH coverCell113
+
+/-- Cached adaptive-cover cell 00000; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00000 : AngleCell := childLL coverCell0000
+
+/-- Cached adaptive-cover cell 00001; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00001 : AngleCell := childLH coverCell0000
+
+/-- Cached adaptive-cover cell 00002; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00002 : AngleCell := childHL coverCell0000
+
+/-- Cached adaptive-cover cell 00003; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00003 : AngleCell := childHH coverCell0000
+
+/-- Cached adaptive-cover cell 00010; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00010 : AngleCell := childLL coverCell0001
+
+/-- Cached adaptive-cover cell 00011; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00011 : AngleCell := childLH coverCell0001
+
+/-- Cached adaptive-cover cell 00012; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00012 : AngleCell := childHL coverCell0001
+
+/-- Cached adaptive-cover cell 00013; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00013 : AngleCell := childHH coverCell0001
+
+/-- Cached adaptive-cover cell 00100; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00100 : AngleCell := childLL coverCell0010
+
+/-- Cached adaptive-cover cell 00101; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00101 : AngleCell := childLH coverCell0010
+
+/-- Cached adaptive-cover cell 00102; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00102 : AngleCell := childHL coverCell0010
+
+/-- Cached adaptive-cover cell 00103; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00103 : AngleCell := childHH coverCell0010
+
+/-- Cached adaptive-cover cell 00110; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00110 : AngleCell := childLL coverCell0011
+
+/-- Cached adaptive-cover cell 00111; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00111 : AngleCell := childLH coverCell0011
+
+/-- Cached adaptive-cover cell 00112; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00112 : AngleCell := childHL coverCell0011
+
+/-- Cached adaptive-cover cell 00113; its parent subdivisions specify the exact interval. -/
+abbrev coverCell00113 : AngleCell := childHH coverCell0011
+
+/-- Cached adaptive-cover cell 01000; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01000 : AngleCell := childLL coverCell0100
+
+/-- Cached adaptive-cover cell 01001; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01001 : AngleCell := childLH coverCell0100
+
+/-- Cached adaptive-cover cell 01002; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01002 : AngleCell := childHL coverCell0100
+
+/-- Cached adaptive-cover cell 01003; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01003 : AngleCell := childHH coverCell0100
+
+/-- Cached adaptive-cover cell 01010; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01010 : AngleCell := childLL coverCell0101
+
+/-- Cached adaptive-cover cell 01011; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01011 : AngleCell := childLH coverCell0101
+
+/-- Cached adaptive-cover cell 01012; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01012 : AngleCell := childHL coverCell0101
+
+/-- Cached adaptive-cover cell 01013; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01013 : AngleCell := childHH coverCell0101
+
+/-- Cached adaptive-cover cell 01100; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01100 : AngleCell := childLL coverCell0110
+
+/-- Cached adaptive-cover cell 01101; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01101 : AngleCell := childLH coverCell0110
+
+/-- Cached adaptive-cover cell 01102; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01102 : AngleCell := childHL coverCell0110
+
+/-- Cached adaptive-cover cell 01103; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01103 : AngleCell := childHH coverCell0110
+
+/-- Cached adaptive-cover cell 01110; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01110 : AngleCell := childLL coverCell0111
+
+/-- Cached adaptive-cover cell 01111; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01111 : AngleCell := childLH coverCell0111
+
+/-- Cached adaptive-cover cell 01112; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01112 : AngleCell := childHL coverCell0111
+
+/-- Cached adaptive-cover cell 01113; its parent subdivisions specify the exact interval. -/
+abbrev coverCell01113 : AngleCell := childHH coverCell0111
+
+/-- Cached adaptive-cover cell 10000; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10000 : AngleCell := childLL coverCell1000
+
+/-- Cached adaptive-cover cell 10001; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10001 : AngleCell := childLH coverCell1000
+
+/-- Cached adaptive-cover cell 10002; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10002 : AngleCell := childHL coverCell1000
+
+/-- Cached adaptive-cover cell 10003; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10003 : AngleCell := childHH coverCell1000
+
+/-- Cached adaptive-cover cell 10010; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10010 : AngleCell := childLL coverCell1001
+
+/-- Cached adaptive-cover cell 10011; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10011 : AngleCell := childLH coverCell1001
+
+/-- Cached adaptive-cover cell 10012; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10012 : AngleCell := childHL coverCell1001
+
+/-- Cached adaptive-cover cell 10013; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10013 : AngleCell := childHH coverCell1001
+
+/-- Cached adaptive-cover cell 10100; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10100 : AngleCell := childLL coverCell1010
+
+/-- Cached adaptive-cover cell 10101; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10101 : AngleCell := childLH coverCell1010
+
+/-- Cached adaptive-cover cell 10102; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10102 : AngleCell := childHL coverCell1010
+
+/-- Cached adaptive-cover cell 10103; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10103 : AngleCell := childHH coverCell1010
+
+/-- Cached adaptive-cover cell 10110; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10110 : AngleCell := childLL coverCell1011
+
+/-- Cached adaptive-cover cell 10111; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10111 : AngleCell := childLH coverCell1011
+
+/-- Cached adaptive-cover cell 10112; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10112 : AngleCell := childHL coverCell1011
+
+/-- Cached adaptive-cover cell 10113; its parent subdivisions specify the exact interval. -/
+abbrev coverCell10113 : AngleCell := childHH coverCell1011
+
+/-- Cached adaptive-cover cell 11000; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11000 : AngleCell := childLL coverCell1100
+
+/-- Cached adaptive-cover cell 11001; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11001 : AngleCell := childLH coverCell1100
+
+/-- Cached adaptive-cover cell 11002; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11002 : AngleCell := childHL coverCell1100
+
+/-- Cached adaptive-cover cell 11003; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11003 : AngleCell := childHH coverCell1100
+
+/-- Cached adaptive-cover cell 11010; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11010 : AngleCell := childLL coverCell1101
+
+/-- Cached adaptive-cover cell 11011; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11011 : AngleCell := childLH coverCell1101
+
+/-- Cached adaptive-cover cell 11012; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11012 : AngleCell := childHL coverCell1101
+
+/-- Cached adaptive-cover cell 11013; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11013 : AngleCell := childHH coverCell1101
+
+/-- Cached adaptive-cover cell 11100; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11100 : AngleCell := childLL coverCell1110
+
+/-- Cached adaptive-cover cell 11101; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11101 : AngleCell := childLH coverCell1110
+
+/-- Cached adaptive-cover cell 11102; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11102 : AngleCell := childHL coverCell1110
+
+/-- Cached adaptive-cover cell 11103; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11103 : AngleCell := childHH coverCell1110
+
+/-- Cached adaptive-cover cell 11110; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11110 : AngleCell := childLL coverCell1111
+
+/-- Cached adaptive-cover cell 11111; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11111 : AngleCell := childLH coverCell1111
+
+/-- Cached adaptive-cover cell 11112; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11112 : AngleCell := childHL coverCell1111
+
+/-- Cached adaptive-cover cell 11113; its parent subdivisions specify the exact interval. -/
+abbrev coverCell11113 : AngleCell := childHH coverCell1111
+
+/-- Cached adaptive-cover cell 100100; its parent subdivisions specify the exact interval. -/
+abbrev coverCell100100 : AngleCell := childLL coverCell10010
+
+/-- Cached adaptive-cover cell 100101; its parent subdivisions specify the exact interval. -/
+abbrev coverCell100101 : AngleCell := childLH coverCell10010
+
+/-- Cached adaptive-cover cell 100102; its parent subdivisions specify the exact interval. -/
+abbrev coverCell100102 : AngleCell := childHL coverCell10010
+
+/-- Cached adaptive-cover cell 100103; its parent subdivisions specify the exact interval. -/
+abbrev coverCell100103 : AngleCell := childHH coverCell10010
+
+/-- Cached adaptive-cover cell 100110; its parent subdivisions specify the exact interval. -/
+abbrev coverCell100110 : AngleCell := childLL coverCell10011
+
+/-- Cached adaptive-cover cell 100111; its parent subdivisions specify the exact interval. -/
+abbrev coverCell100111 : AngleCell := childLH coverCell10011
+
+/-- Cached adaptive-cover cell 100112; its parent subdivisions specify the exact interval. -/
+abbrev coverCell100112 : AngleCell := childHL coverCell10011
+
+/-- Cached adaptive-cover cell 100113; its parent subdivisions specify the exact interval. -/
+abbrev coverCell100113 : AngleCell := childHH coverCell10011
+
+/-- Cached adaptive-cover cell 101000; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101000 : AngleCell := childLL coverCell10100
+
+/-- Cached adaptive-cover cell 101001; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101001 : AngleCell := childLH coverCell10100
+
+/-- Cached adaptive-cover cell 101002; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101002 : AngleCell := childHL coverCell10100
+
+/-- Cached adaptive-cover cell 101003; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101003 : AngleCell := childHH coverCell10100
+
+/-- Cached adaptive-cover cell 101010; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101010 : AngleCell := childLL coverCell10101
+
+/-- Cached adaptive-cover cell 101011; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101011 : AngleCell := childLH coverCell10101
+
+/-- Cached adaptive-cover cell 101012; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101012 : AngleCell := childHL coverCell10101
+
+/-- Cached adaptive-cover cell 101013; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101013 : AngleCell := childHH coverCell10101
+
+/-- Cached adaptive-cover cell 101100; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101100 : AngleCell := childLL coverCell10110
+
+/-- Cached adaptive-cover cell 101101; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101101 : AngleCell := childLH coverCell10110
+
+/-- Cached adaptive-cover cell 101102; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101102 : AngleCell := childHL coverCell10110
+
+/-- Cached adaptive-cover cell 101103; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101103 : AngleCell := childHH coverCell10110
+
+/-- Cached adaptive-cover cell 101110; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101110 : AngleCell := childLL coverCell10111
+
+/-- Cached adaptive-cover cell 101111; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101111 : AngleCell := childLH coverCell10111
+
+/-- Cached adaptive-cover cell 101112; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101112 : AngleCell := childHL coverCell10111
+
+/-- Cached adaptive-cover cell 101113; its parent subdivisions specify the exact interval. -/
+abbrev coverCell101113 : AngleCell := childHH coverCell10111
+
+/-- Cached adaptive-cover cell 110000; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110000 : AngleCell := childLL coverCell11000
+
+/-- Cached adaptive-cover cell 110001; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110001 : AngleCell := childLH coverCell11000
+
+/-- Cached adaptive-cover cell 110002; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110002 : AngleCell := childHL coverCell11000
+
+/-- Cached adaptive-cover cell 110003; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110003 : AngleCell := childHH coverCell11000
+
+/-- Cached adaptive-cover cell 110010; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110010 : AngleCell := childLL coverCell11001
+
+/-- Cached adaptive-cover cell 110011; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110011 : AngleCell := childLH coverCell11001
+
+/-- Cached adaptive-cover cell 110012; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110012 : AngleCell := childHL coverCell11001
+
+/-- Cached adaptive-cover cell 110013; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110013 : AngleCell := childHH coverCell11001
+
+/-- Cached adaptive-cover cell 110100; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110100 : AngleCell := childLL coverCell11010
+
+/-- Cached adaptive-cover cell 110101; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110101 : AngleCell := childLH coverCell11010
+
+/-- Cached adaptive-cover cell 110102; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110102 : AngleCell := childHL coverCell11010
+
+/-- Cached adaptive-cover cell 110103; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110103 : AngleCell := childHH coverCell11010
+
+/-- Cached adaptive-cover cell 110110; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110110 : AngleCell := childLL coverCell11011
+
+/-- Cached adaptive-cover cell 110111; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110111 : AngleCell := childLH coverCell11011
+
+/-- Cached adaptive-cover cell 110112; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110112 : AngleCell := childHL coverCell11011
+
+/-- Cached adaptive-cover cell 110113; its parent subdivisions specify the exact interval. -/
+abbrev coverCell110113 : AngleCell := childHH coverCell11011
+
+/-- Cached adaptive-cover cell 111000; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111000 : AngleCell := childLL coverCell11100
+
+/-- Cached adaptive-cover cell 111001; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111001 : AngleCell := childLH coverCell11100
+
+/-- Cached adaptive-cover cell 111002; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111002 : AngleCell := childHL coverCell11100
+
+/-- Cached adaptive-cover cell 111003; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111003 : AngleCell := childHH coverCell11100
+
+/-- Cached adaptive-cover cell 111010; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111010 : AngleCell := childLL coverCell11101
+
+/-- Cached adaptive-cover cell 111011; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111011 : AngleCell := childLH coverCell11101
+
+/-- Cached adaptive-cover cell 111012; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111012 : AngleCell := childHL coverCell11101
+
+/-- Cached adaptive-cover cell 111013; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111013 : AngleCell := childHH coverCell11101
+
+/-- Cached adaptive-cover cell 111100; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111100 : AngleCell := childLL coverCell11110
+
+/-- Cached adaptive-cover cell 111101; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111101 : AngleCell := childLH coverCell11110
+
+/-- Cached adaptive-cover cell 111102; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111102 : AngleCell := childHL coverCell11110
+
+/-- Cached adaptive-cover cell 111103; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111103 : AngleCell := childHH coverCell11110
+
+/-- Cached adaptive-cover cell 111110; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111110 : AngleCell := childLL coverCell11111
+
+/-- Cached adaptive-cover cell 111111; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111111 : AngleCell := childLH coverCell11111
+
+/-- Cached adaptive-cover cell 111112; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111112 : AngleCell := childHL coverCell11111
+
+/-- Cached adaptive-cover cell 111113; its parent subdivisions specify the exact interval. -/
+abbrev coverCell111113 : AngleCell := childHH coverCell11111
+
+/-- Cached adaptive-cover cell 1110010; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1110010 : AngleCell := childLL coverCell111001
+
+/-- Cached adaptive-cover cell 1110011; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1110011 : AngleCell := childLH coverCell111001
+
+/-- Cached adaptive-cover cell 1110012; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1110012 : AngleCell := childHL coverCell111001
+
+/-- Cached adaptive-cover cell 1110013; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1110013 : AngleCell := childHH coverCell111001
+
+/-- Cached adaptive-cover cell 1110100; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1110100 : AngleCell := childLL coverCell111010
+
+/-- Cached adaptive-cover cell 1110101; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1110101 : AngleCell := childLH coverCell111010
+
+/-- Cached adaptive-cover cell 1110102; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1110102 : AngleCell := childHL coverCell111010
+
+/-- Cached adaptive-cover cell 1110103; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1110103 : AngleCell := childHH coverCell111010
+
+/-- Cached adaptive-cover cell 1110110; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1110110 : AngleCell := childLL coverCell111011
+
+/-- Cached adaptive-cover cell 1110111; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1110111 : AngleCell := childLH coverCell111011
+
+/-- Cached adaptive-cover cell 1110112; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1110112 : AngleCell := childHL coverCell111011
+
+/-- Cached adaptive-cover cell 1110113; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1110113 : AngleCell := childHH coverCell111011
+
+/-- Cached adaptive-cover cell 1111000; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111000 : AngleCell := childLL coverCell111100
+
+/-- Cached adaptive-cover cell 1111001; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111001 : AngleCell := childLH coverCell111100
+
+/-- Cached adaptive-cover cell 1111002; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111002 : AngleCell := childHL coverCell111100
+
+/-- Cached adaptive-cover cell 1111003; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111003 : AngleCell := childHH coverCell111100
+
+/-- Cached adaptive-cover cell 1111010; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111010 : AngleCell := childLL coverCell111101
+
+/-- Cached adaptive-cover cell 1111011; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111011 : AngleCell := childLH coverCell111101
+
+/-- Cached adaptive-cover cell 1111012; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111012 : AngleCell := childHL coverCell111101
+
+/-- Cached adaptive-cover cell 1111013; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111013 : AngleCell := childHH coverCell111101
+
+/-- Cached adaptive-cover cell 1111100; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111100 : AngleCell := childLL coverCell111110
+
+/-- Cached adaptive-cover cell 1111101; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111101 : AngleCell := childLH coverCell111110
+
+/-- Cached adaptive-cover cell 1111102; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111102 : AngleCell := childHL coverCell111110
+
+/-- Cached adaptive-cover cell 1111103; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111103 : AngleCell := childHH coverCell111110
+
+/-- Cached adaptive-cover cell 1111110; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111110 : AngleCell := childLL coverCell111111
+
+/-- Cached adaptive-cover cell 1111111; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111111 : AngleCell := childLH coverCell111111
+
+/-- Cached adaptive-cover cell 1111112; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111112 : AngleCell := childHL coverCell111111
+
+/-- Cached adaptive-cover cell 1111113; its parent subdivisions specify the exact interval. -/
+abbrev coverCell1111113 : AngleCell := childHH coverCell111111
+
+end GerverSofa.PartE.CoverPhiAbove1101100
+
+namespace GerverSofa.PartE.CoverPhiAbove1101100.CertificateCellsdb86b7498a
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CoverPhiAbove1101100.CertificateCellsdb86b7498a
+
+namespace GerverSofa.PartE.CertificateCellsf80409361d
+
+/-- Subcell `1011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1011 : AngleCell :=
+  childLH (childLH (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1100 : AngleCell :=
+  childLL (childLL (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCellsf80409361d
+
+namespace GerverSofa.PartE.CertificateCells93b6719794
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells93b6719794
+
+namespace GerverSofa.PartE.CertificateCellsdf243d4011
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCellsdf243d4011
+
+namespace GerverSofa.PartE.CertificateCells88c20ab319
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells88c20ab319
+
+namespace GerverSofa.PartE.CertificateCells2e53b5dd0c
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells2e53b5dd0c
+
+namespace GerverSofa.PartE.CertificateCellsce9330d4cd
+
+/-- Subcell `3331` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3331 : AngleCell :=
+  childLH (childHH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3332` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3332 : AngleCell :=
+  childHL (childHH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3333` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3333 : AngleCell :=
+  childHH (childHH (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsce9330d4cd
+
+namespace GerverSofa.PartE.CertificateCells9612bd9bbf
+
+/-- Subcell `3333` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3333 : AngleCell :=
+  childHH (childHH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+end GerverSofa.PartE.CertificateCells9612bd9bbf
+
+namespace GerverSofa.PartE.CertificateCellsd19e562ded
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+end GerverSofa.PartE.CertificateCellsd19e562ded
+
+namespace GerverSofa.PartE.CertificateCellse3e967073d
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCellse3e967073d
+
+namespace GerverSofa.PartE.CertificateCells23b3ec373b
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells23b3ec373b
+
+namespace GerverSofa.PartE.CertificateCells9b25d0fd5e
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells9b25d0fd5e
+
+namespace GerverSofa.PartE.CertificateCells486c3dc7d8
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells486c3dc7d8
+
+namespace GerverSofa.PartE.CertificateCellsdd72fc553d
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCellsdd72fc553d
+
+namespace GerverSofa.PartE.CertificateCells308b6c24e8
+
+/-- Subcell `1111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells308b6c24e8
+
+namespace GerverSofa.PartE.CertificateCells48ddfaabb3
+
+/-- Subcell `1111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells48ddfaabb3
+
+namespace GerverSofa.PartE.CertificateCells4368513ed3
+
+/-- Subcell `1111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells4368513ed3
+
+namespace GerverSofa.PartE.CertificateCellsd48a6bc390
+
+/-- Subcell `1111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCellsd48a6bc390
+
+namespace GerverSofa.PartE.CertificateCells3620ccdfd7
+
+/-- Subcell `1100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1100 : AngleCell :=
+  childLL (childLL (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1102 : AngleCell :=
+  childHL (childLL (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1103 : AngleCell :=
+  childHH (childLL (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `11003330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1111)))
+
+end GerverSofa.PartE.CertificateCells3620ccdfd7
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC4BatchPhiAboveF9C1LLRLRL0011`.
+* `KernelOnly.PartE.E24KC4PilotPhiAboveF90002`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF0LLR00112`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF0LRL00114`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF0RLR00169`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF0RRL00171`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF10400466`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF112L00468`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF112RLL00471`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF16LLL00214`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF16LLR00215`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF16LR00216`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF16RL00218`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF16RR00219`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF24LL00222`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF24LR00223`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF24RL00225`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF24RR00226`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC4 auto-tuned batched kernel certificates. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CoverPhiAbove1101100
+
+namespace CertificateCellsdb86b7498a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdb86b7498a
+
+open CertificateCellsdb86b7498a
+
+theorem checked1110010 : adaptiveCoverCheck 2 coverCell1110010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1110010 (by decide +kernel)
+
+theorem checked1110011 : adaptiveCoverCheck 2 coverCell1110011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1110011 (by decide +kernel)
+
+theorem checked1110012 : adaptiveCoverCheck 2 coverCell1110012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1110012 (by decide +kernel)
+
+theorem checked1110013 : adaptiveCoverCheck 2 coverCell1110013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1110013 (by decide +kernel)
+
+theorem checked1110100 : adaptiveCoverCheck 2 coverCell1110100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1110100 (by decide +kernel)
+
+theorem checked1110101 : adaptiveCoverCheck 2 coverCell1110101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1110101 (by decide +kernel)
+
+theorem checked1110102 : adaptiveCoverCheck 2 coverCell1110102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1110102 (by decide +kernel)
+
+theorem checked1110103 : adaptiveCoverCheck 2 coverCell1110103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1110103 (by decide +kernel)
+
+theorem checked1110110 : adaptiveCoverCheck 2 coverCell1110110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1110110 (by decide +kernel)
+
+theorem checked1110111 : adaptiveCoverCheck 2 coverCell1110111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1110111 (by decide +kernel)
+
+theorem checked1110112 : adaptiveCoverCheck 2 coverCell1110112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1110112 (by decide +kernel)
+
+theorem checked1110113 : adaptiveCoverCheck 2 coverCell1110113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1110113 (by decide +kernel)
+
+theorem checked1111000 : adaptiveCoverCheck 2 coverCell1111000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111000 (by decide +kernel)
+
+theorem checked1111001 : adaptiveCoverCheck 2 coverCell1111001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111001 (by decide +kernel)
+
+theorem checked1111002 : adaptiveCoverCheck 2 coverCell1111002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111002 (by decide +kernel)
+
+theorem checked1111003 : adaptiveCoverCheck 2 coverCell1111003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111003 (by decide +kernel)
+
+theorem checked1111010 : adaptiveCoverCheck 2 coverCell1111010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111010 (by decide +kernel)
+
+theorem checked1111011 : adaptiveCoverCheck 2 coverCell1111011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111011 (by decide +kernel)
+
+theorem checked1111012 : adaptiveCoverCheck 2 coverCell1111012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111012 (by decide +kernel)
+
+theorem checked1111013 : adaptiveCoverCheck 2 coverCell1111013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111013 (by decide +kernel)
+
+theorem checked1111100 : adaptiveCoverCheck 2 coverCell1111100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111100 (by decide +kernel)
+
+theorem checked1111101 : adaptiveCoverCheck 2 coverCell1111101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111101 (by decide +kernel)
+
+theorem checked1111102 : adaptiveCoverCheck 2 coverCell1111102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111102 (by decide +kernel)
+
+theorem checked1111103 : adaptiveCoverCheck 2 coverCell1111103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111103 (by decide +kernel)
+
+theorem checked1111110 : adaptiveCoverCheck 2 coverCell1111110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111110 (by decide +kernel)
+
+theorem checked1111111 : adaptiveCoverCheck 2 coverCell1111111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111111 (by decide +kernel)
+
+theorem checked1111112 : adaptiveCoverCheck 2 coverCell1111112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111112 (by decide +kernel)
+
+theorem checked1111113 : adaptiveCoverCheck 2 coverCell1111113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 coverCell1111113 (by decide +kernel)
+
+theorem checked100100 : adaptiveCoverCheck 3 coverCell100100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell100100 (by decide +kernel)
+
+theorem checked100101 : adaptiveCoverCheck 3 coverCell100101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell100101 (by decide +kernel)
+
+theorem checked100102 : adaptiveCoverCheck 3 coverCell100102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell100102 (by decide +kernel)
+
+theorem checked100103 : adaptiveCoverCheck 3 coverCell100103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell100103 (by decide +kernel)
+
+theorem checked100110 : adaptiveCoverCheck 3 coverCell100110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell100110 (by decide +kernel)
+
+theorem checked100111 : adaptiveCoverCheck 3 coverCell100111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell100111 (by decide +kernel)
+
+theorem checked100112 : adaptiveCoverCheck 3 coverCell100112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell100112 (by decide +kernel)
+
+theorem checked100113 : adaptiveCoverCheck 3 coverCell100113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell100113 (by decide +kernel)
+
+theorem checked101000 : adaptiveCoverCheck 3 coverCell101000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101000 (by decide +kernel)
+
+theorem checked101001 : adaptiveCoverCheck 3 coverCell101001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101001 (by decide +kernel)
+
+theorem checked101002 : adaptiveCoverCheck 3 coverCell101002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101002 (by decide +kernel)
+
+theorem checked101003 : adaptiveCoverCheck 3 coverCell101003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101003 (by decide +kernel)
+
+theorem checked101010 : adaptiveCoverCheck 3 coverCell101010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101010 (by decide +kernel)
+
+theorem checked101011 : adaptiveCoverCheck 3 coverCell101011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101011 (by decide +kernel)
+
+theorem checked101012 : adaptiveCoverCheck 3 coverCell101012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101012 (by decide +kernel)
+
+theorem checked101013 : adaptiveCoverCheck 3 coverCell101013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101013 (by decide +kernel)
+
+theorem checked101100 : adaptiveCoverCheck 3 coverCell101100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101100 (by decide +kernel)
+
+theorem checked101101 : adaptiveCoverCheck 3 coverCell101101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101101 (by decide +kernel)
+
+theorem checked101102 : adaptiveCoverCheck 3 coverCell101102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101102 (by decide +kernel)
+
+theorem checked101103 : adaptiveCoverCheck 3 coverCell101103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101103 (by decide +kernel)
+
+theorem checked101110 : adaptiveCoverCheck 3 coverCell101110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101110 (by decide +kernel)
+
+theorem checked101111 : adaptiveCoverCheck 3 coverCell101111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101111 (by decide +kernel)
+
+theorem checked101112 : adaptiveCoverCheck 3 coverCell101112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101112 (by decide +kernel)
+
+theorem checked101113 : adaptiveCoverCheck 3 coverCell101113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell101113 (by decide +kernel)
+
+theorem checked110000 : adaptiveCoverCheck 3 coverCell110000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110000 (by decide +kernel)
+
+theorem checked110001 : adaptiveCoverCheck 3 coverCell110001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110001 (by decide +kernel)
+
+theorem checked110002 : adaptiveCoverCheck 3 coverCell110002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110002 (by decide +kernel)
+
+theorem checked110003 : adaptiveCoverCheck 3 coverCell110003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110003 (by decide +kernel)
+
+theorem checked110010 : adaptiveCoverCheck 3 coverCell110010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110010 (by decide +kernel)
+
+theorem checked110011 : adaptiveCoverCheck 3 coverCell110011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110011 (by decide +kernel)
+
+theorem checked110012 : adaptiveCoverCheck 3 coverCell110012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110012 (by decide +kernel)
+
+theorem checked110013 : adaptiveCoverCheck 3 coverCell110013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110013 (by decide +kernel)
+
+theorem checked110100 : adaptiveCoverCheck 3 coverCell110100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110100 (by decide +kernel)
+
+theorem checked110101 : adaptiveCoverCheck 3 coverCell110101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110101 (by decide +kernel)
+
+theorem checked110102 : adaptiveCoverCheck 3 coverCell110102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110102 (by decide +kernel)
+
+theorem checked110103 : adaptiveCoverCheck 3 coverCell110103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110103 (by decide +kernel)
+
+theorem checked110110 : adaptiveCoverCheck 3 coverCell110110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110110 (by decide +kernel)
+
+theorem checked110111 : adaptiveCoverCheck 3 coverCell110111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110111 (by decide +kernel)
+
+theorem checked110112 : adaptiveCoverCheck 3 coverCell110112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110112 (by decide +kernel)
+
+theorem checked110113 : adaptiveCoverCheck 3 coverCell110113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell110113 (by decide +kernel)
+
+theorem checked111000 : adaptiveCoverCheck 3 coverCell111000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell111000 (by decide +kernel)
+
+theorem checked111001 : adaptiveCoverCheck 3 coverCell111001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 coverCell111001
+    checked1110010 checked1110011 checked1110012 checked1110013
+
+theorem checked111002 : adaptiveCoverCheck 3 coverCell111002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell111002 (by decide +kernel)
+
+theorem checked111003 : adaptiveCoverCheck 3 coverCell111003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell111003 (by decide +kernel)
+
+theorem checked111010 : adaptiveCoverCheck 3 coverCell111010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 coverCell111010
+    checked1110100 checked1110101 checked1110102 checked1110103
+
+theorem checked111011 : adaptiveCoverCheck 3 coverCell111011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 coverCell111011
+    checked1110110 checked1110111 checked1110112 checked1110113
+
+theorem checked111012 : adaptiveCoverCheck 3 coverCell111012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell111012 (by decide +kernel)
+
+theorem checked111013 : adaptiveCoverCheck 3 coverCell111013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell111013 (by decide +kernel)
+
+theorem checked111100 : adaptiveCoverCheck 3 coverCell111100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 coverCell111100
+    checked1111000 checked1111001 checked1111002 checked1111003
+
+theorem checked111101 : adaptiveCoverCheck 3 coverCell111101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 coverCell111101
+    checked1111010 checked1111011 checked1111012 checked1111013
+
+theorem checked111102 : adaptiveCoverCheck 3 coverCell111102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell111102 (by decide +kernel)
+
+theorem checked111103 : adaptiveCoverCheck 3 coverCell111103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell111103 (by decide +kernel)
+
+theorem checked111110 : adaptiveCoverCheck 3 coverCell111110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 coverCell111110
+    checked1111100 checked1111101 checked1111102 checked1111103
+
+theorem checked111111 : adaptiveCoverCheck 3 coverCell111111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 coverCell111111
+    checked1111110 checked1111111 checked1111112 checked1111113
+
+theorem checked111112 : adaptiveCoverCheck 3 coverCell111112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell111112 (by decide +kernel)
+
+theorem checked111113 : adaptiveCoverCheck 3 coverCell111113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 coverCell111113 (by decide +kernel)
+
+theorem checked00000 : adaptiveCoverCheck 4 coverCell00000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00000 (by decide +kernel)
+
+theorem checked00001 : adaptiveCoverCheck 4 coverCell00001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00001 (by decide +kernel)
+
+theorem checked00002 : adaptiveCoverCheck 4 coverCell00002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00002 (by decide +kernel)
+
+theorem checked00003 : adaptiveCoverCheck 4 coverCell00003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00003 (by decide +kernel)
+
+theorem checked00010 : adaptiveCoverCheck 4 coverCell00010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00010 (by decide +kernel)
+
+theorem checked00011 : adaptiveCoverCheck 4 coverCell00011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00011 (by decide +kernel)
+
+theorem checked00012 : adaptiveCoverCheck 4 coverCell00012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00012 (by decide +kernel)
+
+theorem checked00013 : adaptiveCoverCheck 4 coverCell00013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00013 (by decide +kernel)
+
+theorem checked00100 : adaptiveCoverCheck 4 coverCell00100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00100 (by decide +kernel)
+
+theorem checked00101 : adaptiveCoverCheck 4 coverCell00101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00101 (by decide +kernel)
+
+theorem checked00102 : adaptiveCoverCheck 4 coverCell00102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00102 (by decide +kernel)
+
+theorem checked00103 : adaptiveCoverCheck 4 coverCell00103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00103 (by decide +kernel)
+
+theorem checked00110 : adaptiveCoverCheck 4 coverCell00110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00110 (by decide +kernel)
+
+theorem checked00111 : adaptiveCoverCheck 4 coverCell00111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00111 (by decide +kernel)
+
+theorem checked00112 : adaptiveCoverCheck 4 coverCell00112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00112 (by decide +kernel)
+
+theorem checked00113 : adaptiveCoverCheck 4 coverCell00113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell00113 (by decide +kernel)
+
+theorem checked01000 : adaptiveCoverCheck 4 coverCell01000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01000 (by decide +kernel)
+
+theorem checked01001 : adaptiveCoverCheck 4 coverCell01001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01001 (by decide +kernel)
+
+theorem checked01002 : adaptiveCoverCheck 4 coverCell01002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01002 (by decide +kernel)
+
+theorem checked01003 : adaptiveCoverCheck 4 coverCell01003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01003 (by decide +kernel)
+
+theorem checked01010 : adaptiveCoverCheck 4 coverCell01010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01010 (by decide +kernel)
+
+theorem checked01011 : adaptiveCoverCheck 4 coverCell01011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01011 (by decide +kernel)
+
+theorem checked01012 : adaptiveCoverCheck 4 coverCell01012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01012 (by decide +kernel)
+
+theorem checked01013 : adaptiveCoverCheck 4 coverCell01013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01013 (by decide +kernel)
+
+theorem checked01100 : adaptiveCoverCheck 4 coverCell01100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01100 (by decide +kernel)
+
+theorem checked01101 : adaptiveCoverCheck 4 coverCell01101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01101 (by decide +kernel)
+
+theorem checked01102 : adaptiveCoverCheck 4 coverCell01102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01102 (by decide +kernel)
+
+theorem checked01103 : adaptiveCoverCheck 4 coverCell01103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01103 (by decide +kernel)
+
+theorem checked01110 : adaptiveCoverCheck 4 coverCell01110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01110 (by decide +kernel)
+
+theorem checked01111 : adaptiveCoverCheck 4 coverCell01111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01111 (by decide +kernel)
+
+theorem checked01112 : adaptiveCoverCheck 4 coverCell01112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01112 (by decide +kernel)
+
+theorem checked01113 : adaptiveCoverCheck 4 coverCell01113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell01113 (by decide +kernel)
+
+theorem checked10000 : adaptiveCoverCheck 4 coverCell10000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell10000 (by decide +kernel)
+
+theorem checked10001 : adaptiveCoverCheck 4 coverCell10001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell10001 (by decide +kernel)
+
+theorem checked10002 : adaptiveCoverCheck 4 coverCell10002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell10002 (by decide +kernel)
+
+theorem checked10003 : adaptiveCoverCheck 4 coverCell10003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell10003 (by decide +kernel)
+
+theorem checked10010 : adaptiveCoverCheck 4 coverCell10010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 coverCell10010
+    checked100100 checked100101 checked100102 checked100103
+
+theorem checked10011 : adaptiveCoverCheck 4 coverCell10011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 coverCell10011
+    checked100110 checked100111 checked100112 checked100113
+
+theorem checked10012 : adaptiveCoverCheck 4 coverCell10012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell10012 (by decide +kernel)
+
+theorem checked10013 : adaptiveCoverCheck 4 coverCell10013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell10013 (by decide +kernel)
+
+theorem checked10100 : adaptiveCoverCheck 4 coverCell10100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 coverCell10100
+    checked101000 checked101001 checked101002 checked101003
+
+theorem checked10101 : adaptiveCoverCheck 4 coverCell10101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 coverCell10101
+    checked101010 checked101011 checked101012 checked101013
+
+theorem checked10102 : adaptiveCoverCheck 4 coverCell10102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell10102 (by decide +kernel)
+
+theorem checked10103 : adaptiveCoverCheck 4 coverCell10103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell10103 (by decide +kernel)
+
+theorem checked10110 : adaptiveCoverCheck 4 coverCell10110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 coverCell10110
+    checked101100 checked101101 checked101102 checked101103
+
+theorem checked10111 : adaptiveCoverCheck 4 coverCell10111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 coverCell10111
+    checked101110 checked101111 checked101112 checked101113
+
+theorem checked10112 : adaptiveCoverCheck 4 coverCell10112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell10112 (by decide +kernel)
+
+theorem checked10113 : adaptiveCoverCheck 4 coverCell10113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell10113 (by decide +kernel)
+
+theorem checked11000 : adaptiveCoverCheck 4 coverCell11000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 coverCell11000
+    checked110000 checked110001 checked110002 checked110003
+
+theorem checked11001 : adaptiveCoverCheck 4 coverCell11001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 coverCell11001
+    checked110010 checked110011 checked110012 checked110013
+
+theorem checked11002 : adaptiveCoverCheck 4 coverCell11002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell11002 (by decide +kernel)
+
+theorem checked11003 : adaptiveCoverCheck 4 coverCell11003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell11003 (by decide +kernel)
+
+theorem checked11010 : adaptiveCoverCheck 4 coverCell11010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 coverCell11010
+    checked110100 checked110101 checked110102 checked110103
+
+theorem checked11011 : adaptiveCoverCheck 4 coverCell11011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 coverCell11011
+    checked110110 checked110111 checked110112 checked110113
+
+theorem checked11012 : adaptiveCoverCheck 4 coverCell11012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell11012 (by decide +kernel)
+
+theorem checked11013 : adaptiveCoverCheck 4 coverCell11013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell11013 (by decide +kernel)
+
+theorem checked11100 : adaptiveCoverCheck 4 coverCell11100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 coverCell11100
+    checked111000 checked111001 checked111002 checked111003
+
+theorem checked11101 : adaptiveCoverCheck 4 coverCell11101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 coverCell11101
+    checked111010 checked111011 checked111012 checked111013
+
+theorem checked11102 : adaptiveCoverCheck 4 coverCell11102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell11102 (by decide +kernel)
+
+theorem checked11103 : adaptiveCoverCheck 4 coverCell11103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell11103 (by decide +kernel)
+
+theorem checked11110 : adaptiveCoverCheck 4 coverCell11110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 coverCell11110
+    checked111100 checked111101 checked111102 checked111103
+
+theorem checked11111 : adaptiveCoverCheck 4 coverCell11111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 coverCell11111
+    checked111110 checked111111 checked111112 checked111113
+
+theorem checked11112 : adaptiveCoverCheck 4 coverCell11112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell11112 (by decide +kernel)
+
+theorem checked11113 : adaptiveCoverCheck 4 coverCell11113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 coverCell11113 (by decide +kernel)
+
+theorem checked0000 : adaptiveCoverCheck 5 coverCell0000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell0000
+    checked00000 checked00001 checked00002 checked00003
+
+theorem checked0001 : adaptiveCoverCheck 5 coverCell0001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell0001
+    checked00010 checked00011 checked00012 checked00013
+
+theorem checked0002 : adaptiveCoverCheck 5 coverCell0002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell0002 (by decide +kernel)
+
+theorem checked0003 : adaptiveCoverCheck 5 coverCell0003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell0003 (by decide +kernel)
+
+theorem checked0010 : adaptiveCoverCheck 5 coverCell0010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell0010
+    checked00100 checked00101 checked00102 checked00103
+
+theorem checked0011 : adaptiveCoverCheck 5 coverCell0011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell0011
+    checked00110 checked00111 checked00112 checked00113
+
+theorem checked0012 : adaptiveCoverCheck 5 coverCell0012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell0012 (by decide +kernel)
+
+theorem checked0013 : adaptiveCoverCheck 5 coverCell0013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell0013 (by decide +kernel)
+
+theorem checked0100 : adaptiveCoverCheck 5 coverCell0100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell0100
+    checked01000 checked01001 checked01002 checked01003
+
+theorem checked0101 : adaptiveCoverCheck 5 coverCell0101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell0101
+    checked01010 checked01011 checked01012 checked01013
+
+theorem checked0102 : adaptiveCoverCheck 5 coverCell0102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell0102 (by decide +kernel)
+
+theorem checked0103 : adaptiveCoverCheck 5 coverCell0103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell0103 (by decide +kernel)
+
+theorem checked0110 : adaptiveCoverCheck 5 coverCell0110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell0110
+    checked01100 checked01101 checked01102 checked01103
+
+theorem checked0111 : adaptiveCoverCheck 5 coverCell0111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell0111
+    checked01110 checked01111 checked01112 checked01113
+
+theorem checked0112 : adaptiveCoverCheck 5 coverCell0112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell0112 (by decide +kernel)
+
+theorem checked0113 : adaptiveCoverCheck 5 coverCell0113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell0113 (by decide +kernel)
+
+theorem checked1000 : adaptiveCoverCheck 5 coverCell1000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell1000
+    checked10000 checked10001 checked10002 checked10003
+
+theorem checked1001 : adaptiveCoverCheck 5 coverCell1001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell1001
+    checked10010 checked10011 checked10012 checked10013
+
+theorem checked1002 : adaptiveCoverCheck 5 coverCell1002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1002 (by decide +kernel)
+
+theorem checked1003 : adaptiveCoverCheck 5 coverCell1003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1003 (by decide +kernel)
+
+theorem checked1010 : adaptiveCoverCheck 5 coverCell1010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell1010
+    checked10100 checked10101 checked10102 checked10103
+
+theorem checked1011 : adaptiveCoverCheck 5 coverCell1011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell1011
+    checked10110 checked10111 checked10112 checked10113
+
+theorem checked1012 : adaptiveCoverCheck 5 coverCell1012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1012 (by decide +kernel)
+
+theorem checked1013 : adaptiveCoverCheck 5 coverCell1013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1013 (by decide +kernel)
+
+theorem checked1100 : adaptiveCoverCheck 5 coverCell1100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell1100
+    checked11000 checked11001 checked11002 checked11003
+
+theorem checked1101 : adaptiveCoverCheck 5 coverCell1101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell1101
+    checked11010 checked11011 checked11012 checked11013
+
+theorem checked1102 : adaptiveCoverCheck 5 coverCell1102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1102 (by decide +kernel)
+
+theorem checked1103 : adaptiveCoverCheck 5 coverCell1103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1103 (by decide +kernel)
+
+theorem checked1110 : adaptiveCoverCheck 5 coverCell1110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell1110
+    checked11100 checked11101 checked11102 checked11103
+
+theorem checked1111 : adaptiveCoverCheck 5 coverCell1111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 coverCell1111
+    checked11110 checked11111 checked11112 checked11113
+
+theorem checked1112 : adaptiveCoverCheck 5 coverCell1112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1112 (by decide +kernel)
+
+theorem checked1113 : adaptiveCoverCheck 5 coverCell1113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1113 (by decide +kernel)
+
+theorem checked1120 : adaptiveCoverCheck 5 coverCell1120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1120 (by decide +kernel)
+
+theorem checked1121 : adaptiveCoverCheck 5 coverCell1121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1121 (by decide +kernel)
+
+theorem checked1122 : adaptiveCoverCheck 5 coverCell1122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1122 (by decide +kernel)
+
+theorem checked1123 : adaptiveCoverCheck 5 coverCell1123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1123 (by decide +kernel)
+
+theorem checked1130 : adaptiveCoverCheck 5 coverCell1130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1130 (by decide +kernel)
+
+theorem checked1131 : adaptiveCoverCheck 5 coverCell1131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1131 (by decide +kernel)
+
+theorem checked1132 : adaptiveCoverCheck 5 coverCell1132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1132 (by decide +kernel)
+
+theorem checked1133 : adaptiveCoverCheck 5 coverCell1133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 coverCell1133 (by decide +kernel)
+
+theorem checked000 : adaptiveCoverCheck 6 coverCell000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 coverCell000
+    checked0000 checked0001 checked0002 checked0003
+
+theorem checked001 : adaptiveCoverCheck 6 coverCell001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 coverCell001
+    checked0010 checked0011 checked0012 checked0013
+
+theorem checked002 : adaptiveCoverCheck 6 coverCell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell002 (by decide +kernel)
+
+theorem checked003 : adaptiveCoverCheck 6 coverCell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell003 (by decide +kernel)
+
+theorem checked010 : adaptiveCoverCheck 6 coverCell010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 coverCell010
+    checked0100 checked0101 checked0102 checked0103
+
+theorem checked011 : adaptiveCoverCheck 6 coverCell011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 coverCell011
+    checked0110 checked0111 checked0112 checked0113
+
+theorem checked012 : adaptiveCoverCheck 6 coverCell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell012 (by decide +kernel)
+
+theorem checked013 : adaptiveCoverCheck 6 coverCell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell013 (by decide +kernel)
+
+theorem checked030 : adaptiveCoverCheck 6 coverCell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell030 (by decide +kernel)
+
+theorem checked031 : adaptiveCoverCheck 6 coverCell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell031 (by decide +kernel)
+
+theorem checked032 : adaptiveCoverCheck 6 coverCell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell032 (by decide +kernel)
+
+theorem checked033 : adaptiveCoverCheck 6 coverCell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell033 (by decide +kernel)
+
+theorem checked100 : adaptiveCoverCheck 6 coverCell100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 coverCell100
+    checked1000 checked1001 checked1002 checked1003
+
+theorem checked101 : adaptiveCoverCheck 6 coverCell101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 coverCell101
+    checked1010 checked1011 checked1012 checked1013
+
+theorem checked102 : adaptiveCoverCheck 6 coverCell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell102 (by decide +kernel)
+
+theorem checked103 : adaptiveCoverCheck 6 coverCell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell103 (by decide +kernel)
+
+theorem checked110 : adaptiveCoverCheck 6 coverCell110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 coverCell110
+    checked1100 checked1101 checked1102 checked1103
+
+theorem checked111 : adaptiveCoverCheck 6 coverCell111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 coverCell111
+    checked1110 checked1111 checked1112 checked1113
+
+theorem checked112 : adaptiveCoverCheck 6 coverCell112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 coverCell112
+    checked1120 checked1121 checked1122 checked1123
+
+theorem checked113 : adaptiveCoverCheck 6 coverCell113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 coverCell113
+    checked1130 checked1131 checked1132 checked1133
+
+theorem checked120 : adaptiveCoverCheck 6 coverCell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell120 (by decide +kernel)
+
+theorem checked121 : adaptiveCoverCheck 6 coverCell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell121 (by decide +kernel)
+
+theorem checked122 : adaptiveCoverCheck 6 coverCell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell122 (by decide +kernel)
+
+theorem checked123 : adaptiveCoverCheck 6 coverCell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell123 (by decide +kernel)
+
+theorem checked130 : adaptiveCoverCheck 6 coverCell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell130 (by decide +kernel)
+
+theorem checked131 : adaptiveCoverCheck 6 coverCell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell131 (by decide +kernel)
+
+theorem checked132 : adaptiveCoverCheck 6 coverCell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell132 (by decide +kernel)
+
+theorem checked133 : adaptiveCoverCheck 6 coverCell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 coverCell133 (by decide +kernel)
+
+theorem checked00 : adaptiveCoverCheck 7 coverCell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 coverCell00
+    checked000 checked001 checked002 checked003
+
+theorem checked01 : adaptiveCoverCheck 7 coverCell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 coverCell01
+    checked010 checked011 checked012 checked013
+
+theorem checked02 : adaptiveCoverCheck 7 coverCell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 coverCell02 (by decide +kernel)
+
+theorem checked03 : adaptiveCoverCheck 7 coverCell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 coverCell03
+    checked030 checked031 checked032 checked033
+
+theorem checked10 : adaptiveCoverCheck 7 coverCell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 coverCell10
+    checked100 checked101 checked102 checked103
+
+theorem checked11 : adaptiveCoverCheck 7 coverCell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 coverCell11
+    checked110 checked111 checked112 checked113
+
+theorem checked12 : adaptiveCoverCheck 7 coverCell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 coverCell12
+    checked120 checked121 checked122 checked123
+
+theorem checked13 : adaptiveCoverCheck 7 coverCell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 coverCell13
+    checked130 checked131 checked132 checked133
+
+theorem checked20 : adaptiveCoverCheck 7 coverCell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 coverCell20 (by decide +kernel)
+
+theorem checked21 : adaptiveCoverCheck 7 coverCell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 coverCell21 (by decide +kernel)
+
+theorem checked22 : adaptiveCoverCheck 7 coverCell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 coverCell22 (by decide +kernel)
+
+theorem checked23 : adaptiveCoverCheck 7 coverCell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 coverCell23 (by decide +kernel)
+
+theorem checked30 : adaptiveCoverCheck 7 coverCell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 coverCell30 (by decide +kernel)
+
+theorem checked31 : adaptiveCoverCheck 7 coverCell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 coverCell31 (by decide +kernel)
+
+theorem checked32 : adaptiveCoverCheck 7 coverCell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 coverCell32 (by decide +kernel)
+
+theorem checked33 : adaptiveCoverCheck 7 coverCell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 coverCell33 (by decide +kernel)
+
+theorem checked0 : adaptiveCoverCheck 8 coverCell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 coverCell0
+    checked00 checked01 checked02 checked03
+
+theorem checked1 : adaptiveCoverCheck 8 coverCell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 coverCell1
+    checked10 checked11 checked12 checked13
+
+theorem checked2 : adaptiveCoverCheck 8 coverCell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 coverCell2
+    checked20 checked21 checked22 checked23
+
+theorem checked3 : adaptiveCoverCheck 8 coverCell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 coverCell3
+    checked30 checked31 checked32 checked33
+
+theorem checkedRoot : adaptiveCoverCheck 9 coverCell = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 coverCell
+    checked0 checked1 checked2 checked3
+
+end CoverPhiAbove1101100
+
+theorem e24KC2PhiAboveLeaf1101100 :
+    adaptiveCoverCheck 9 CoverPhiAbove1101100.coverCell = true :=
+      CoverPhiAbove1101100.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC4 auto-tuned batched kernel certificates. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf80409361d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf80409361d
+
+open CertificateCellsf80409361d
+namespace CoverCertificate6cd78cf99c
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate6cd78cf99c
+
+theorem e24KC2PhiAboveLeaf1011011 :
+    adaptiveCoverCheck 9 (childLH (childLH (childLL phiAboveCell1011))) = true := by
+  exact CoverCertificate6cd78cf99c.checkedRoot
+namespace CoverCertificate3b2bb200ff
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3b2bb200ff
+
+theorem e24KC2PhiAboveLeaf1011100 :
+    adaptiveCoverCheck 9 (childLL (childLL (childLH phiAboveCell1011))) = true := by
+  exact CoverCertificate3b2bb200ff.checkedRoot
+namespace CoverCertificate416e1e96e0
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate416e1e96e0
+
+theorem e24KC2PhiAboveLeaf1011101 :
+    adaptiveCoverCheck 9 (childLH (childLL (childLH phiAboveCell1011))) = true := by
+  exact CoverCertificate416e1e96e0.checkedRoot
+namespace CoverCertificate2a65321173
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate2a65321173
+
+theorem e24KC2PhiAboveLeaf1011110 :
+    adaptiveCoverCheck 9 (childLL (childLH (childLH phiAboveCell1011))) = true := by
+  exact CoverCertificate2a65321173.checkedRoot
+namespace CoverCertificatecc929b68fe
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatecc929b68fe
+
+theorem e24KC2PhiAboveLeaf1011111 :
+    adaptiveCoverCheck 9 (childLH (childLH (childLH phiAboveCell1011))) = true := by
+  exact CoverCertificatecc929b68fe.checkedRoot
+namespace CoverCertificate82c27f8e4d
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate82c27f8e4d
+
+theorem e24KC2PhiAboveLeaf1100000 :
+    adaptiveCoverCheck 9 (childLL (childLL (childLL phiAboveCell1100))) = true := by
+  exact CoverCertificate82c27f8e4d.checkedRoot
+namespace CoverCertificatecbf8beb65d
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatecbf8beb65d
+
+theorem e24KC2PhiAboveLeaf1100001 :
+    adaptiveCoverCheck 9 (childLH (childLL (childLL phiAboveCell1100))) = true := by
+  exact CoverCertificatecbf8beb65d.checkedRoot
+namespace CoverCertificate319b5d79a7
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate319b5d79a7
+
+theorem e24KC2PhiAboveLeaf1100010 :
+    adaptiveCoverCheck 9 (childLL (childLH (childLL phiAboveCell1100))) = true := by
+  exact CoverCertificate319b5d79a7.checkedRoot
+namespace CoverCertificate52a2adcf18
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate52a2adcf18
+
+theorem e24KC2PhiAboveLeaf1100011 :
+    adaptiveCoverCheck 9 (childLH (childLH (childLL phiAboveCell1100))) = true := by
+  exact CoverCertificate52a2adcf18.checkedRoot
+namespace CoverCertificatef66846141d
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef66846141d
+
+theorem e24KC2PhiAboveLeaf1100100 :
+    adaptiveCoverCheck 9 (childLL (childLL (childLH phiAboveCell1100))) = true := by
+  exact CoverCertificatef66846141d.checkedRoot
+namespace CoverCertificate5cd1561dfb
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5cd1561dfb
+
+theorem e24KC2PhiAboveLeaf1100101 :
+    adaptiveCoverCheck 9 (childLH (childLL (childLH phiAboveCell1100))) = true := by
+  exact CoverCertificate5cd1561dfb.checkedRoot
+namespace CoverCertificate43ce0d7d08
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate43ce0d7d08
+
+theorem e24KC2PhiAboveLeaf1100110 :
+    adaptiveCoverCheck 9 (childLL (childLH (childLH phiAboveCell1100))) = true := by
+  exact CoverCertificate43ce0d7d08.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells93b6719794
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells93b6719794
+
+open CertificateCells93b6719794
+namespace CoverCertificate3c4ee9d54f
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3c4ee9d54f
+
+theorem e24KC2PhiAboveLeaf1101102 :
+    adaptiveCoverCheck 9 (childHL (childLL (childLH phiAboveCell1101))) = true := by
+  exact CoverCertificate3c4ee9d54f.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdf243d4011
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdf243d4011
+
+open CertificateCellsdf243d4011
+namespace CoverCertificate5ea8c0bac2
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5ea8c0bac2
+
+theorem e24KC2PhiAboveLeaf1101103 :
+    adaptiveCoverCheck 9 (childHH (childLL (childLH phiAboveCell1101))) = true := by
+  exact CoverCertificate5ea8c0bac2.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells88c20ab319
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells88c20ab319
+
+open CertificateCells88c20ab319
+namespace CoverCertificatec619bc60a1
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec619bc60a1
+
+theorem e24KC2PhiAboveLeaf1101112 :
+    adaptiveCoverCheck 9 (childHL (childLH (childLH phiAboveCell1101))) = true := by
+  exact CoverCertificatec619bc60a1.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2e53b5dd0c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2e53b5dd0c
+
+open CertificateCells2e53b5dd0c
+namespace CoverCertificate5746fe2caf
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5746fe2caf
+
+theorem e24KC2PhiAboveLeaf1101113 :
+    adaptiveCoverCheck 9 (childHH (childLH (childLH phiAboveCell1101))) = true := by
+  exact CoverCertificate5746fe2caf.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsce9330d4cd
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsce9330d4cd
+
+open CertificateCellsce9330d4cd
+namespace CoverCertificate98b9778203
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate98b9778203
+
+theorem e24KC2PhiBelowLeaf33313 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3331) = true := by
+  exact CoverCertificate98b9778203.checkedRoot
+namespace CoverCertificate7f98aa56d1
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate7f98aa56d1
+
+theorem e24KC2PhiBelowLeaf33320 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3332) = true := by
+  exact CoverCertificate7f98aa56d1.checkedRoot
+namespace CoverCertificateb650783453
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateb650783453
+
+theorem e24KC2PhiBelowLeaf33321 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3332) = true := by
+  exact CoverCertificateb650783453.checkedRoot
+namespace CoverCertificate05dde891b1
+
+private theorem checked2220 : adaptiveCoverCheck 5 cell2220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2220 (by decide +kernel)
+
+private theorem checked2221 : adaptiveCoverCheck 5 cell2221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2221 (by decide +kernel)
+
+private theorem checked2222 : adaptiveCoverCheck 5 cell2222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2222 (by decide +kernel)
+
+private theorem checked2223 : adaptiveCoverCheck 5 cell2223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2223 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 6 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 6 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 6 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 6 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 6 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 6 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 6 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 6 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell222
+    checked2220 checked2221 checked2222 checked2223
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate05dde891b1
+
+theorem e24KC2PhiBelowLeaf33322 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3332) = true := by
+  exact CoverCertificate05dde891b1.checkedRoot
+namespace CoverCertificatedd1cfe6f59
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatedd1cfe6f59
+
+theorem e24KC2PhiBelowLeaf33323 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3332) = true := by
+  exact CoverCertificatedd1cfe6f59.checkedRoot
+namespace CoverCertificatefe5dc96009
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefe5dc96009
+
+theorem e24KC2PhiBelowLeaf33330 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3333) = true := by
+  exact CoverCertificatefe5dc96009.checkedRoot
+namespace CoverCertificate7ba2433252
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate7ba2433252
+
+theorem e24KC2PhiBelowLeaf33331 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3333) = true := by
+  exact CoverCertificate7ba2433252.checkedRoot
+namespace CoverCertificate7633ba69a5
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate7633ba69a5
+
+theorem e24KC2PhiBelowLeaf33332 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3333) = true := by
+  exact CoverCertificate7633ba69a5.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9612bd9bbf
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9612bd9bbf
+
+open CertificateCells9612bd9bbf
+namespace CoverCertificate8e17b594b9
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate8e17b594b9
+
+theorem e24KC2PhiBelowLeaf33333 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3333) = true := by
+  exact CoverCertificate8e17b594b9.checkedRoot
+namespace CoverCertificate2ad8ea30c5
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate2ad8ea30c5
+
+theorem e24KC2ThetaAboveLeaf0000220002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00002200)) = true := by
+  exact CoverCertificate2ad8ea30c5.checkedRoot
+namespace CoverCertificate5fed50edc5
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5fed50edc5
+
+theorem e24KC2ThetaAboveLeaf0000220003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00002200)) = true := by
+  exact CoverCertificate5fed50edc5.checkedRoot
+namespace CoverCertificatebeb942e0ae
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatebeb942e0ae
+
+theorem e24KC2ThetaAboveLeaf0000220012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00002200)) = true := by
+  exact CoverCertificatebeb942e0ae.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd19e562ded
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd19e562ded
+
+open CertificateCellsd19e562ded
+namespace CoverCertificate4aac8bf510
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4aac8bf510
+
+theorem e24KC2ThetaAboveLeaf0000220013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00002200)) = true := by
+  exact CoverCertificate4aac8bf510.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse3e967073d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse3e967073d
+
+open CertificateCellse3e967073d
+namespace CoverCertificate018ca4eb00
+
+private theorem checked000 : adaptiveCoverCheck 6 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 6 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 6 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 6 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 6 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 6 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 6 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 6 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 6 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 6 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 6 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 6 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 6 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 6 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 6 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 6 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate018ca4eb00
+
+theorem e24KC2PhiAboveLeaf1110101 :
+    adaptiveCoverCheck 9 (childLH (childLL (childLH phiAboveCell1110))) = true := by
+  exact CoverCertificate018ca4eb00.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells23b3ec373b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells23b3ec373b
+
+open CertificateCells23b3ec373b
+namespace CoverCertificatefbeddf9072
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefbeddf9072
+
+theorem e24KC2PhiAboveLeaf1110102 :
+    adaptiveCoverCheck 9 (childHL (childLL (childLH phiAboveCell1110))) = true := by
+  exact CoverCertificatefbeddf9072.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9b25d0fd5e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9b25d0fd5e
+
+open CertificateCells9b25d0fd5e
+namespace CoverCertificatee6bd46d219
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee6bd46d219
+
+theorem e24KC2PhiAboveLeaf1110103 :
+    adaptiveCoverCheck 9 (childHH (childLL (childLH phiAboveCell1110))) = true := by
+  exact CoverCertificatee6bd46d219.checkedRoot
+namespace CoverCertificate2a0a30dc9d
+
+private theorem checked000 : adaptiveCoverCheck 6 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 6 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 6 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 6 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 6 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 6 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 6 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 6 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 6 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 6 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 6 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 6 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 6 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 6 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 6 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 6 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate2a0a30dc9d
+
+theorem e24KC2PhiAboveLeaf1110110 :
+    adaptiveCoverCheck 9 (childLL (childLH (childLH phiAboveCell1110))) = true := by
+  exact CoverCertificate2a0a30dc9d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells486c3dc7d8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells486c3dc7d8
+
+open CertificateCells486c3dc7d8
+namespace CoverCertificateff98d0df99
+
+private theorem checked000 : adaptiveCoverCheck 6 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 6 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 6 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 6 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 6 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 6 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 6 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 6 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 6 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 6 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 6 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 6 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 6 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 6 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 6 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 6 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateff98d0df99
+
+theorem e24KC2PhiAboveLeaf1110111 :
+    adaptiveCoverCheck 9 (childLH (childLH (childLH phiAboveCell1110))) = true := by
+  exact CoverCertificateff98d0df99.checkedRoot
+namespace CoverCertificate63e0e0ed96
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate63e0e0ed96
+
+theorem e24KC2PhiAboveLeaf1110112 :
+    adaptiveCoverCheck 9 (childHL (childLH (childLH phiAboveCell1110))) = true := by
+  exact CoverCertificate63e0e0ed96.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdd72fc553d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdd72fc553d
+
+open CertificateCellsdd72fc553d
+namespace CoverCertificate1d20eb40b2
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1d20eb40b2
+
+theorem e24KC2PhiAboveLeaf1110113 :
+    adaptiveCoverCheck 9 (childHH (childLH (childLH phiAboveCell1110))) = true := by
+  exact CoverCertificate1d20eb40b2.checkedRoot
+namespace CoverCertificate49ab43c4dd
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate49ab43c4dd
+
+theorem e24KC2PhiAboveLeaf1111000 :
+    adaptiveCoverCheck 9 (childLL (childLL (childLL phiAboveCell1111))) = true := by
+  exact CoverCertificate49ab43c4dd.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells308b6c24e8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells308b6c24e8
+
+open CertificateCells308b6c24e8
+namespace CoverCertificatede0603de49
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatede0603de49
+
+theorem e24KC2PhiAboveLeaf1111001 :
+    adaptiveCoverCheck 9 (childLH (childLL (childLL phiAboveCell1111))) = true := by
+  exact CoverCertificatede0603de49.checkedRoot
+namespace CoverCertificate90d2e1946c
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate90d2e1946c
+
+theorem e24KC2PhiAboveLeaf1111002 :
+    adaptiveCoverCheck 9 (childHL (childLL (childLL phiAboveCell1111))) = true := by
+  exact CoverCertificate90d2e1946c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells48ddfaabb3
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells48ddfaabb3
+
+open CertificateCells48ddfaabb3
+namespace CoverCertificatebf3761efb7
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatebf3761efb7
+
+theorem e24KC2PhiAboveLeaf1111003 :
+    adaptiveCoverCheck 9 (childHH (childLL (childLL phiAboveCell1111))) = true := by
+  exact CoverCertificatebf3761efb7.checkedRoot
+namespace CoverCertificate48ab78c281
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate48ab78c281
+
+theorem e24KC2PhiAboveLeaf1111010 :
+    adaptiveCoverCheck 9 (childLL (childLH (childLL phiAboveCell1111))) = true := by
+  exact CoverCertificate48ab78c281.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4368513ed3
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4368513ed3
+
+open CertificateCells4368513ed3
+namespace CoverCertificate5e935aba69
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5e935aba69
+
+theorem e24KC2PhiAboveLeaf1111011 :
+    adaptiveCoverCheck 9 (childLH (childLH (childLL phiAboveCell1111))) = true := by
+  exact CoverCertificate5e935aba69.checkedRoot
+namespace CoverCertificateea859f6e20
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateea859f6e20
+
+theorem e24KC2PhiAboveLeaf1111012 :
+    adaptiveCoverCheck 9 (childHL (childLH (childLL phiAboveCell1111))) = true := by
+  exact CoverCertificateea859f6e20.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd48a6bc390
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd48a6bc390
+
+open CertificateCellsd48a6bc390
+namespace CoverCertificatedc87a79519
+
+private theorem checked110 : adaptiveCoverCheck 6 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 6 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 6 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 6 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatedc87a79519
+
+theorem e24KC2PhiAboveLeaf1111013 :
+    adaptiveCoverCheck 9 (childHH (childLH (childLL phiAboveCell1111))) = true := by
+  exact CoverCertificatedc87a79519.checkedRoot
+namespace CoverCertificatefd3c81c65d
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefd3c81c65d
+
+theorem e24KC2PhiAboveLeaf1111100 :
+    adaptiveCoverCheck 9 (childLL (childLL (childLH phiAboveCell1111))) = true := by
+  exact CoverCertificatefd3c81c65d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC5TerminalBatchT563200014`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3620ccdfd7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3620ccdfd7
+
+open CertificateCells3620ccdfd7
+theorem e24KC2ThetaAboveLeaf110033301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003330) = true := by
+  have h : ((childLH thetaAboveCell11003330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003330) h
+theorem e24KC2ThetaAboveLeaf110033302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003330) = true := by
+  have h : ((childHL thetaAboveCell11003330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003330) h
+theorem e24KC2ThetaAboveLeaf110033303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003330) = true := by
+  have h : ((childHH thetaAboveCell11003330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003330) h
+theorem e24KC2ThetaAboveLeaf110033310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003331) = true := by
+  have h : ((childLL thetaAboveCell11003331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003331) h
+theorem e24KC2ThetaAboveLeaf110033311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003331) = true := by
+  have h : ((childLH thetaAboveCell11003331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003331) h
+theorem e24KC2ThetaAboveLeaf110033312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003331) = true := by
+  have h : ((childHL thetaAboveCell11003331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003331) h
+theorem e24KC2ThetaAboveLeaf110033313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003331) = true := by
+  have h : ((childHH thetaAboveCell11003331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003331) h
+theorem e24KC2ThetaAboveLeaf11003332 :
+    adaptiveCoverCheck 11 thetaAboveCell11003332 = true := by
+  have h : (thetaAboveCell11003332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003332 h
+theorem e24KC2ThetaAboveLeaf11003333 :
+    adaptiveCoverCheck 11 thetaAboveCell11003333 = true := by
+  have h : (thetaAboveCell11003333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003333 h
+theorem e24KC2ThetaAboveLeaf110100 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1101)) = true := by
+  have h : ((childLL (childLL thetaAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell1101)) h
+theorem e24KC2ThetaAboveLeaf110101 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1101)) = true := by
+  have h : ((childLH (childLL thetaAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell1101)) h
+theorem e24KC2ThetaAboveLeaf110102 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1101)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1101)) h
+theorem e24KC2ThetaAboveLeaf110103 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1101)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1101)) h
+theorem e24KC2ThetaAboveLeaf110110 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1101)) = true := by
+  have h : ((childLL (childLH thetaAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell1101)) h
+theorem e24KC2ThetaAboveLeaf110111 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1101)) = true := by
+  have h : ((childLH (childLH thetaAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell1101)) h
+theorem e24KC2ThetaAboveLeaf110112 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1101)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1101)) h
+theorem e24KC2ThetaAboveLeaf110113 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1101)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1101)) h
+theorem e24KC2ThetaAboveLeaf1101200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell1101))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf1101201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell1101))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf1101202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell1101))) = true := by
+  have h : ((childHL (childLL (childHL thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childHL thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf1101203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell1101))) = true := by
+  have h : ((childHH (childLL (childHL thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childHL thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf1101210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell1101))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf1101211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell1101))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf1101212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell1101))) = true := by
+  have h : ((childHL (childLH (childHL thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childHL thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf1101213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell1101))) = true := by
+  have h : ((childHH (childLH (childHL thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childHL thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf11012200 :
+    adaptiveCoverCheck 11 thetaAboveCell11012200 = true := by
+  have h : (thetaAboveCell11012200).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012200 h
+theorem e24KC2ThetaAboveLeaf11012201 :
+    adaptiveCoverCheck 11 thetaAboveCell11012201 = true := by
+  have h : (thetaAboveCell11012201).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012201 h
+theorem e24KC2ThetaAboveLeaf110122020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012202) = true := by
+  have h : ((childLL thetaAboveCell11012202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012202) h
+theorem e24KC2ThetaAboveLeaf110122021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012202) = true := by
+  have h : ((childLH thetaAboveCell11012202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012202) h
+theorem e24KC2ThetaAboveLeaf110122022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012202) = true := by
+  have h : ((childHL thetaAboveCell11012202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012202) h
+theorem e24KC2ThetaAboveLeaf110122023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012202) = true := by
+  have h : ((childHH thetaAboveCell11012202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012202) h
+theorem e24KC2ThetaAboveLeaf110122030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012203) = true := by
+  have h : ((childLL thetaAboveCell11012203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012203) h
+theorem e24KC2ThetaAboveLeaf110122031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012203) = true := by
+  have h : ((childLH thetaAboveCell11012203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012203) h
+theorem e24KC2ThetaAboveLeaf110122032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012203) = true := by
+  have h : ((childHL thetaAboveCell11012203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012203) h
+theorem e24KC2ThetaAboveLeaf110122033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012203) = true := by
+  have h : ((childHH thetaAboveCell11012203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012203) h
+theorem e24KC2ThetaAboveLeaf11012210 :
+    adaptiveCoverCheck 11 thetaAboveCell11012210 = true := by
+  have h : (thetaAboveCell11012210).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012210 h
+theorem e24KC2ThetaAboveLeaf11012211 :
+    adaptiveCoverCheck 11 thetaAboveCell11012211 = true := by
+  have h : (thetaAboveCell11012211).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012211 h
+theorem e24KC2ThetaAboveLeaf110122120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012212) = true := by
+  have h : ((childLL thetaAboveCell11012212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012212) h
+theorem e24KC2ThetaAboveLeaf110122121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012212) = true := by
+  have h : ((childLH thetaAboveCell11012212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012212) h
+theorem e24KC2ThetaAboveLeaf110122122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012212) = true := by
+  have h : ((childHL thetaAboveCell11012212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012212) h
+theorem e24KC2ThetaAboveLeaf110122123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012212) = true := by
+  have h : ((childHH thetaAboveCell11012212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012212) h
+theorem e24KC2ThetaAboveLeaf110122130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012213) = true := by
+  have h : ((childLL thetaAboveCell11012213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012213) h
+theorem e24KC2ThetaAboveLeaf110122131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012213) = true := by
+  have h : ((childLH thetaAboveCell11012213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012213) h
+theorem e24KC2ThetaAboveLeaf110122132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012213) = true := by
+  have h : ((childHL thetaAboveCell11012213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012213) h
+theorem e24KC2ThetaAboveLeaf110122133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012213) = true := by
+  have h : ((childHH thetaAboveCell11012213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012213) h
+theorem e24KC2ThetaAboveLeaf110122200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012220) = true := by
+  have h : ((childLL thetaAboveCell11012220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012220) h
+theorem e24KC2ThetaAboveLeaf110122201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012220) = true := by
+  have h : ((childLH thetaAboveCell11012220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012220) h
+theorem e24KC2ThetaAboveLeaf110122202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012220) = true := by
+  have h : ((childHL thetaAboveCell11012220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012220) h
+theorem e24KC2ThetaAboveLeaf110122203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012220) = true := by
+  have h : ((childHH thetaAboveCell11012220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012220) h
+theorem e24KC2ThetaAboveLeaf110122210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012221) = true := by
+  have h : ((childLL thetaAboveCell11012221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012221) h
+theorem e24KC2ThetaAboveLeaf110122211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012221) = true := by
+  have h : ((childLH thetaAboveCell11012221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012221) h
+theorem e24KC2ThetaAboveLeaf110122212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012221) = true := by
+  have h : ((childHL thetaAboveCell11012221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012221) h
+theorem e24KC2ThetaAboveLeaf110122213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012221) = true := by
+  have h : ((childHH thetaAboveCell11012221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012221) h
+theorem e24KC2ThetaAboveLeaf11012222 :
+    adaptiveCoverCheck 11 thetaAboveCell11012222 = true := by
+  have h : (thetaAboveCell11012222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012222 h
+theorem e24KC2ThetaAboveLeaf11012223 :
+    adaptiveCoverCheck 11 thetaAboveCell11012223 = true := by
+  have h : (thetaAboveCell11012223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012223 h
+theorem e24KC2ThetaAboveLeaf110122300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012230) = true := by
+  have h : ((childLL thetaAboveCell11012230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012230) h
+theorem e24KC2ThetaAboveLeaf110122301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012230) = true := by
+  have h : ((childLH thetaAboveCell11012230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012230) h
+theorem e24KC2ThetaAboveLeaf110122302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012230) = true := by
+  have h : ((childHL thetaAboveCell11012230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012230) h
+theorem e24KC2ThetaAboveLeaf110122303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012230) = true := by
+  have h : ((childHH thetaAboveCell11012230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012230) h
+theorem e24KC2ThetaAboveLeaf110122310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012231) = true := by
+  have h : ((childLL thetaAboveCell11012231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012231) h
+theorem e24KC2ThetaAboveLeaf110122311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012231) = true := by
+  have h : ((childLH thetaAboveCell11012231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012231) h
+theorem e24KC2ThetaAboveLeaf110122312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012231) = true := by
+  have h : ((childHL thetaAboveCell11012231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012231) h
+theorem e24KC2ThetaAboveLeaf110122313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012231) = true := by
+  have h : ((childHH thetaAboveCell11012231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012231) h
+theorem e24KC2ThetaAboveLeaf11012232 :
+    adaptiveCoverCheck 11 thetaAboveCell11012232 = true := by
+  have h : (thetaAboveCell11012232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012232 h
+theorem e24KC2ThetaAboveLeaf11012233 :
+    adaptiveCoverCheck 11 thetaAboveCell11012233 = true := by
+  have h : (thetaAboveCell11012233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012233 h
+theorem e24KC2ThetaAboveLeaf11012300 :
+    adaptiveCoverCheck 11 thetaAboveCell11012300 = true := by
+  have h : (thetaAboveCell11012300).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012300 h
+theorem e24KC2ThetaAboveLeaf11012301 :
+    adaptiveCoverCheck 11 thetaAboveCell11012301 = true := by
+  have h : (thetaAboveCell11012301).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012301 h
+theorem e24KC2ThetaAboveLeaf110123020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012302) = true := by
+  have h : ((childLL thetaAboveCell11012302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012302) h
+theorem e24KC2ThetaAboveLeaf110123021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012302) = true := by
+  have h : ((childLH thetaAboveCell11012302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012302) h
+theorem e24KC2ThetaAboveLeaf110123022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012302) = true := by
+  have h : ((childHL thetaAboveCell11012302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012302) h
+theorem e24KC2ThetaAboveLeaf110123023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012302) = true := by
+  have h : ((childHH thetaAboveCell11012302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012302) h
+theorem e24KC2ThetaAboveLeaf110123030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012303) = true := by
+  have h : ((childLL thetaAboveCell11012303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012303) h
+theorem e24KC2ThetaAboveLeaf110123031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012303) = true := by
+  have h : ((childLH thetaAboveCell11012303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012303) h
+theorem e24KC2ThetaAboveLeaf110123032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012303) = true := by
+  have h : ((childHL thetaAboveCell11012303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012303) h
+theorem e24KC2ThetaAboveLeaf110123033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012303) = true := by
+  have h : ((childHH thetaAboveCell11012303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012303) h
+theorem e24KC2ThetaAboveLeaf11012310 :
+    adaptiveCoverCheck 11 thetaAboveCell11012310 = true := by
+  have h : (thetaAboveCell11012310).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012310 h
+theorem e24KC2ThetaAboveLeaf11012311 :
+    adaptiveCoverCheck 11 thetaAboveCell11012311 = true := by
+  have h : (thetaAboveCell11012311).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012311 h
+theorem e24KC2ThetaAboveLeaf110123120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012312) = true := by
+  have h : ((childLL thetaAboveCell11012312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012312) h
+theorem e24KC2ThetaAboveLeaf110123121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012312) = true := by
+  have h : ((childLH thetaAboveCell11012312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012312) h
+theorem e24KC2ThetaAboveLeaf110123122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012312) = true := by
+  have h : ((childHL thetaAboveCell11012312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012312) h
+theorem e24KC2ThetaAboveLeaf110123123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012312) = true := by
+  have h : ((childHH thetaAboveCell11012312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012312) h
+theorem e24KC2ThetaAboveLeaf110123130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012313) = true := by
+  have h : ((childLL thetaAboveCell11012313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012313) h
+theorem e24KC2ThetaAboveLeaf110123131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012313) = true := by
+  have h : ((childLH thetaAboveCell11012313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012313) h
+theorem e24KC2ThetaAboveLeaf110123132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012313) = true := by
+  have h : ((childHL thetaAboveCell11012313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012313) h
+theorem e24KC2ThetaAboveLeaf110123133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012313) = true := by
+  have h : ((childHH thetaAboveCell11012313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012313) h
+theorem e24KC2ThetaAboveLeaf110123200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012320) = true := by
+  have h : ((childLL thetaAboveCell11012320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012320) h
+theorem e24KC2ThetaAboveLeaf110123201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012320) = true := by
+  have h : ((childLH thetaAboveCell11012320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012320) h
+theorem e24KC2ThetaAboveLeaf110123202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012320) = true := by
+  have h : ((childHL thetaAboveCell11012320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012320) h
+theorem e24KC2ThetaAboveLeaf110123203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012320) = true := by
+  have h : ((childHH thetaAboveCell11012320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012320) h
+theorem e24KC2ThetaAboveLeaf110123210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012321) = true := by
+  have h : ((childLL thetaAboveCell11012321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012321) h
+theorem e24KC2ThetaAboveLeaf110123211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012321) = true := by
+  have h : ((childLH thetaAboveCell11012321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012321) h
+theorem e24KC2ThetaAboveLeaf110123212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012321) = true := by
+  have h : ((childHL thetaAboveCell11012321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012321) h
+theorem e24KC2ThetaAboveLeaf110123213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012321) = true := by
+  have h : ((childHH thetaAboveCell11012321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012321) h
+theorem e24KC2ThetaAboveLeaf11012322 :
+    adaptiveCoverCheck 11 thetaAboveCell11012322 = true := by
+  have h : (thetaAboveCell11012322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012322 h
+theorem e24KC2ThetaAboveLeaf11012323 :
+    adaptiveCoverCheck 11 thetaAboveCell11012323 = true := by
+  have h : (thetaAboveCell11012323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012323 h
+theorem e24KC2ThetaAboveLeaf110123300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012330) = true := by
+  have h : ((childLL thetaAboveCell11012330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012330) h
+theorem e24KC2ThetaAboveLeaf110123301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012330) = true := by
+  have h : ((childLH thetaAboveCell11012330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012330) h
+theorem e24KC2ThetaAboveLeaf110123302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012330) = true := by
+  have h : ((childHL thetaAboveCell11012330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012330) h
+theorem e24KC2ThetaAboveLeaf110123303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012330) = true := by
+  have h : ((childHH thetaAboveCell11012330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012330) h
+theorem e24KC2ThetaAboveLeaf110123310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11012331) = true := by
+  have h : ((childLL thetaAboveCell11012331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11012331) h
+theorem e24KC2ThetaAboveLeaf110123311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11012331) = true := by
+  have h : ((childLH thetaAboveCell11012331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11012331) h
+theorem e24KC2ThetaAboveLeaf110123312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11012331) = true := by
+  have h : ((childHL thetaAboveCell11012331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11012331) h
+theorem e24KC2ThetaAboveLeaf110123313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11012331) = true := by
+  have h : ((childHH thetaAboveCell11012331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11012331) h
+theorem e24KC2ThetaAboveLeaf11012332 :
+    adaptiveCoverCheck 11 thetaAboveCell11012332 = true := by
+  have h : (thetaAboveCell11012332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012332 h
+theorem e24KC2ThetaAboveLeaf11012333 :
+    adaptiveCoverCheck 11 thetaAboveCell11012333 = true := by
+  have h : (thetaAboveCell11012333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11012333 h
+theorem e24KC2ThetaAboveLeaf1101300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell1101))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf1101301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell1101))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf1101302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell1101))) = true := by
+  have h : ((childHL (childLL (childHH thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childHH thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf1101303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell1101))) = true := by
+  have h : ((childHH (childLL (childHH thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childHH thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf1101310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell1101))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf1101311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell1101))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf1101312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell1101))) = true := by
+  have h : ((childHL (childLH (childHH thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childHH thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf1101313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell1101))) = true := by
+  have h : ((childHH (childLH (childHH thetaAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childHH thetaAboveCell1101))) h
+theorem e24KC2ThetaAboveLeaf11013200 :
+    adaptiveCoverCheck 11 thetaAboveCell11013200 = true := by
+  have h : (thetaAboveCell11013200).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013200 h
+theorem e24KC2ThetaAboveLeaf11013201 :
+    adaptiveCoverCheck 11 thetaAboveCell11013201 = true := by
+  have h : (thetaAboveCell11013201).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013201 h
+theorem e24KC2ThetaAboveLeaf110132020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013202) = true := by
+  have h : ((childLL thetaAboveCell11013202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013202) h
+theorem e24KC2ThetaAboveLeaf110132021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013202) = true := by
+  have h : ((childLH thetaAboveCell11013202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013202) h
+theorem e24KC2ThetaAboveLeaf110132022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013202) = true := by
+  have h : ((childHL thetaAboveCell11013202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013202) h
+theorem e24KC2ThetaAboveLeaf110132023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013202) = true := by
+  have h : ((childHH thetaAboveCell11013202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013202) h
+theorem e24KC2ThetaAboveLeaf110132030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013203) = true := by
+  have h : ((childLL thetaAboveCell11013203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013203) h
+theorem e24KC2ThetaAboveLeaf110132031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013203) = true := by
+  have h : ((childLH thetaAboveCell11013203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013203) h
+theorem e24KC2ThetaAboveLeaf110132032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013203) = true := by
+  have h : ((childHL thetaAboveCell11013203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013203) h
+theorem e24KC2ThetaAboveLeaf110132033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013203) = true := by
+  have h : ((childHH thetaAboveCell11013203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013203) h
+theorem e24KC2ThetaAboveLeaf11013210 :
+    adaptiveCoverCheck 11 thetaAboveCell11013210 = true := by
+  have h : (thetaAboveCell11013210).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013210 h
+theorem e24KC2ThetaAboveLeaf11013211 :
+    adaptiveCoverCheck 11 thetaAboveCell11013211 = true := by
+  have h : (thetaAboveCell11013211).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013211 h
+theorem e24KC2ThetaAboveLeaf110132120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013212) = true := by
+  have h : ((childLL thetaAboveCell11013212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013212) h
+theorem e24KC2ThetaAboveLeaf110132121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013212) = true := by
+  have h : ((childLH thetaAboveCell11013212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013212) h
+theorem e24KC2ThetaAboveLeaf110132122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013212) = true := by
+  have h : ((childHL thetaAboveCell11013212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013212) h
+theorem e24KC2ThetaAboveLeaf110132123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013212) = true := by
+  have h : ((childHH thetaAboveCell11013212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013212) h
+theorem e24KC2ThetaAboveLeaf110132130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013213) = true := by
+  have h : ((childLL thetaAboveCell11013213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013213) h
+theorem e24KC2ThetaAboveLeaf110132131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013213) = true := by
+  have h : ((childLH thetaAboveCell11013213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013213) h
+theorem e24KC2ThetaAboveLeaf110132132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013213) = true := by
+  have h : ((childHL thetaAboveCell11013213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013213) h
+theorem e24KC2ThetaAboveLeaf110132133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013213) = true := by
+  have h : ((childHH thetaAboveCell11013213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013213) h
+theorem e24KC2ThetaAboveLeaf110132200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013220) = true := by
+  have h : ((childLL thetaAboveCell11013220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013220) h
+theorem e24KC2ThetaAboveLeaf110132201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013220) = true := by
+  have h : ((childLH thetaAboveCell11013220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013220) h
+theorem e24KC2ThetaAboveLeaf110132202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013220) = true := by
+  have h : ((childHL thetaAboveCell11013220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013220) h
+theorem e24KC2ThetaAboveLeaf110132203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013220) = true := by
+  have h : ((childHH thetaAboveCell11013220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013220) h
+theorem e24KC2ThetaAboveLeaf110132210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013221) = true := by
+  have h : ((childLL thetaAboveCell11013221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013221) h
+theorem e24KC2ThetaAboveLeaf110132211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013221) = true := by
+  have h : ((childLH thetaAboveCell11013221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013221) h
+theorem e24KC2ThetaAboveLeaf110132212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013221) = true := by
+  have h : ((childHL thetaAboveCell11013221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013221) h
+theorem e24KC2ThetaAboveLeaf110132213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013221) = true := by
+  have h : ((childHH thetaAboveCell11013221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013221) h
+theorem e24KC2ThetaAboveLeaf11013222 :
+    adaptiveCoverCheck 11 thetaAboveCell11013222 = true := by
+  have h : (thetaAboveCell11013222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013222 h
+theorem e24KC2ThetaAboveLeaf11013223 :
+    adaptiveCoverCheck 11 thetaAboveCell11013223 = true := by
+  have h : (thetaAboveCell11013223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013223 h
+theorem e24KC2ThetaAboveLeaf110132300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013230) = true := by
+  have h : ((childLL thetaAboveCell11013230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013230) h
+theorem e24KC2ThetaAboveLeaf110132301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013230) = true := by
+  have h : ((childLH thetaAboveCell11013230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013230) h
+theorem e24KC2ThetaAboveLeaf110132302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013230) = true := by
+  have h : ((childHL thetaAboveCell11013230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013230) h
+theorem e24KC2ThetaAboveLeaf110132303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013230) = true := by
+  have h : ((childHH thetaAboveCell11013230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013230) h
+theorem e24KC2ThetaAboveLeaf110132310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013231) = true := by
+  have h : ((childLL thetaAboveCell11013231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013231) h
+theorem e24KC2ThetaAboveLeaf110132311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013231) = true := by
+  have h : ((childLH thetaAboveCell11013231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013231) h
+theorem e24KC2ThetaAboveLeaf110132312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013231) = true := by
+  have h : ((childHL thetaAboveCell11013231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013231) h
+theorem e24KC2ThetaAboveLeaf110132313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013231) = true := by
+  have h : ((childHH thetaAboveCell11013231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013231) h
+theorem e24KC2ThetaAboveLeaf11013232 :
+    adaptiveCoverCheck 11 thetaAboveCell11013232 = true := by
+  have h : (thetaAboveCell11013232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013232 h
+theorem e24KC2ThetaAboveLeaf11013233 :
+    adaptiveCoverCheck 11 thetaAboveCell11013233 = true := by
+  have h : (thetaAboveCell11013233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013233 h
+theorem e24KC2ThetaAboveLeaf11013300 :
+    adaptiveCoverCheck 11 thetaAboveCell11013300 = true := by
+  have h : (thetaAboveCell11013300).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013300 h
+theorem e24KC2ThetaAboveLeaf11013301 :
+    adaptiveCoverCheck 11 thetaAboveCell11013301 = true := by
+  have h : (thetaAboveCell11013301).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013301 h
+theorem e24KC2ThetaAboveLeaf110133020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013302) = true := by
+  have h : ((childLL thetaAboveCell11013302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013302) h
+theorem e24KC2ThetaAboveLeaf110133021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013302) = true := by
+  have h : ((childLH thetaAboveCell11013302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013302) h
+theorem e24KC2ThetaAboveLeaf110133022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013302) = true := by
+  have h : ((childHL thetaAboveCell11013302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013302) h
+theorem e24KC2ThetaAboveLeaf110133023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013302) = true := by
+  have h : ((childHH thetaAboveCell11013302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013302) h
+theorem e24KC2ThetaAboveLeaf110133030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013303) = true := by
+  have h : ((childLL thetaAboveCell11013303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013303) h
+theorem e24KC2ThetaAboveLeaf110133031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013303) = true := by
+  have h : ((childLH thetaAboveCell11013303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013303) h
+theorem e24KC2ThetaAboveLeaf110133032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013303) = true := by
+  have h : ((childHL thetaAboveCell11013303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013303) h
+theorem e24KC2ThetaAboveLeaf110133033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013303) = true := by
+  have h : ((childHH thetaAboveCell11013303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013303) h
+theorem e24KC2ThetaAboveLeaf11013310 :
+    adaptiveCoverCheck 11 thetaAboveCell11013310 = true := by
+  have h : (thetaAboveCell11013310).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013310 h
+theorem e24KC2ThetaAboveLeaf11013311 :
+    adaptiveCoverCheck 11 thetaAboveCell11013311 = true := by
+  have h : (thetaAboveCell11013311).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013311 h
+theorem e24KC2ThetaAboveLeaf110133120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013312) = true := by
+  have h : ((childLL thetaAboveCell11013312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013312) h
+theorem e24KC2ThetaAboveLeaf110133121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013312) = true := by
+  have h : ((childLH thetaAboveCell11013312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013312) h
+theorem e24KC2ThetaAboveLeaf110133122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013312) = true := by
+  have h : ((childHL thetaAboveCell11013312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013312) h
+theorem e24KC2ThetaAboveLeaf110133123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013312) = true := by
+  have h : ((childHH thetaAboveCell11013312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013312) h
+theorem e24KC2ThetaAboveLeaf110133130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013313) = true := by
+  have h : ((childLL thetaAboveCell11013313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013313) h
+theorem e24KC2ThetaAboveLeaf110133131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013313) = true := by
+  have h : ((childLH thetaAboveCell11013313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013313) h
+theorem e24KC2ThetaAboveLeaf110133132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013313) = true := by
+  have h : ((childHL thetaAboveCell11013313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013313) h
+theorem e24KC2ThetaAboveLeaf110133133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013313) = true := by
+  have h : ((childHH thetaAboveCell11013313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013313) h
+theorem e24KC2ThetaAboveLeaf110133200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013320) = true := by
+  have h : ((childLL thetaAboveCell11013320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013320) h
+theorem e24KC2ThetaAboveLeaf110133201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013320) = true := by
+  have h : ((childLH thetaAboveCell11013320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013320) h
+theorem e24KC2ThetaAboveLeaf110133202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013320) = true := by
+  have h : ((childHL thetaAboveCell11013320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013320) h
+theorem e24KC2ThetaAboveLeaf110133203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013320) = true := by
+  have h : ((childHH thetaAboveCell11013320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013320) h
+theorem e24KC2ThetaAboveLeaf110133210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013321) = true := by
+  have h : ((childLL thetaAboveCell11013321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013321) h
+theorem e24KC2ThetaAboveLeaf110133211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013321) = true := by
+  have h : ((childLH thetaAboveCell11013321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013321) h
+theorem e24KC2ThetaAboveLeaf110133212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013321) = true := by
+  have h : ((childHL thetaAboveCell11013321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013321) h
+theorem e24KC2ThetaAboveLeaf110133213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013321) = true := by
+  have h : ((childHH thetaAboveCell11013321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013321) h
+theorem e24KC2ThetaAboveLeaf11013322 :
+    adaptiveCoverCheck 11 thetaAboveCell11013322 = true := by
+  have h : (thetaAboveCell11013322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013322 h
+theorem e24KC2ThetaAboveLeaf11013323 :
+    adaptiveCoverCheck 11 thetaAboveCell11013323 = true := by
+  have h : (thetaAboveCell11013323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013323 h
+theorem e24KC2ThetaAboveLeaf110133300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013330) = true := by
+  have h : ((childLL thetaAboveCell11013330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013330) h
+theorem e24KC2ThetaAboveLeaf110133301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013330) = true := by
+  have h : ((childLH thetaAboveCell11013330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013330) h
+theorem e24KC2ThetaAboveLeaf110133302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013330) = true := by
+  have h : ((childHL thetaAboveCell11013330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013330) h
+theorem e24KC2ThetaAboveLeaf110133303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013330) = true := by
+  have h : ((childHH thetaAboveCell11013330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013330) h
+theorem e24KC2ThetaAboveLeaf110133310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11013331) = true := by
+  have h : ((childLL thetaAboveCell11013331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11013331) h
+theorem e24KC2ThetaAboveLeaf110133311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11013331) = true := by
+  have h : ((childLH thetaAboveCell11013331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11013331) h
+theorem e24KC2ThetaAboveLeaf110133312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11013331) = true := by
+  have h : ((childHL thetaAboveCell11013331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11013331) h
+theorem e24KC2ThetaAboveLeaf110133313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11013331) = true := by
+  have h : ((childHH thetaAboveCell11013331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11013331) h
+theorem e24KC2ThetaAboveLeaf11013332 :
+    adaptiveCoverCheck 11 thetaAboveCell11013332 = true := by
+  have h : (thetaAboveCell11013332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013332 h
+theorem e24KC2ThetaAboveLeaf11013333 :
+    adaptiveCoverCheck 11 thetaAboveCell11013333 = true := by
+  have h : (thetaAboveCell11013333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11013333 h
+theorem e24KC2ThetaAboveLeaf1102000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell1102))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf1102001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell1102))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf1102002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell1102))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf1102003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell1102))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf1102010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell1102))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf1102011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell1102))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf1102012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell1102))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf1102013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell1102))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf110202 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1102)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1102)) h
+theorem e24KC2ThetaAboveLeaf110203 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1102)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1102)) h
+theorem e24KC2ThetaAboveLeaf1102100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell1102))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf1102101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell1102))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf1102102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell1102))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf1102103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell1102))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf1102110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell1102))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf1102111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell1102))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf1102112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell1102))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf1102113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell1102))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell1102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell1102))) h
+theorem e24KC2ThetaAboveLeaf110212 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1102)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1102)) h
+theorem e24KC2ThetaAboveLeaf110213 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1102)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1102)) h
+theorem e24KC2ThetaAboveLeaf11022 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1102) = true := by
+  have h : ((childHL thetaAboveCell1102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell1102) h
+theorem e24KC2ThetaAboveLeaf11023 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1102) = true := by
+  have h : ((childHH thetaAboveCell1102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell1102) h
+theorem e24KC2ThetaAboveLeaf1103000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell1103))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf1103001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell1103))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf1103002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell1103))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf1103003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell1103))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf1103010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell1103))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf1103011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell1103))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf1103012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell1103))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf1103013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell1103))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf110302 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1103)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1103)) h
+theorem e24KC2ThetaAboveLeaf110303 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1103)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1103)) h
+theorem e24KC2ThetaAboveLeaf1103100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell1103))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf1103101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell1103))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf1103102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell1103))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf1103103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell1103))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf1103110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell1103))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf1103111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell1103))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf1103112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell1103))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf1103113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell1103))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell1103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell1103))) h
+theorem e24KC2ThetaAboveLeaf110312 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1103)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1103)) h
+theorem e24KC2ThetaAboveLeaf110313 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1103)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1103)) h
+theorem e24KC2ThetaAboveLeaf11032 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1103) = true := by
+  have h : ((childHL thetaAboveCell1103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell1103) h
+theorem e24KC2ThetaAboveLeaf11033 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1103) = true := by
+  have h : ((childHH thetaAboveCell1103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell1103) h
+theorem e24KC2ThetaAboveLeaf111000 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1110)) = true := by
+  have h : ((childLL (childLL thetaAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell1110)) h
+theorem e24KC2ThetaAboveLeaf111001 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1110)) = true := by
+  have h : ((childLH (childLL thetaAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell1110)) h
+theorem e24KC2ThetaAboveLeaf111002 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1110)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1110)) h
+theorem e24KC2ThetaAboveLeaf111003 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1110)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1110)) h
+theorem e24KC2ThetaAboveLeaf111010 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1110)) = true := by
+  have h : ((childLL (childLH thetaAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell1110)) h
+theorem e24KC2ThetaAboveLeaf111011 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1110)) = true := by
+  have h : ((childLH (childLH thetaAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell1110)) h
+theorem e24KC2ThetaAboveLeaf111012 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1110)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1110)) h
+theorem e24KC2ThetaAboveLeaf111013 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1110)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1110)) h
+theorem e24KC2ThetaAboveLeaf1110200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell1110))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf1110201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell1110))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf1110202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell1110))) = true := by
+  have h : ((childHL (childLL (childHL thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childHL thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf1110203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell1110))) = true := by
+  have h : ((childHH (childLL (childHL thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childHL thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf1110210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell1110))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf1110211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell1110))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf1110212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell1110))) = true := by
+  have h : ((childHL (childLH (childHL thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childHL thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf1110213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell1110))) = true := by
+  have h : ((childHH (childLH (childHL thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childHL thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf11102200 :
+    adaptiveCoverCheck 11 thetaAboveCell11102200 = true := by
+  have h : (thetaAboveCell11102200).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102200 h
+theorem e24KC2ThetaAboveLeaf11102201 :
+    adaptiveCoverCheck 11 thetaAboveCell11102201 = true := by
+  have h : (thetaAboveCell11102201).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102201 h
+theorem e24KC2ThetaAboveLeaf111022020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102202) = true := by
+  have h : ((childLL thetaAboveCell11102202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102202) h
+theorem e24KC2ThetaAboveLeaf111022021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102202) = true := by
+  have h : ((childLH thetaAboveCell11102202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102202) h
+theorem e24KC2ThetaAboveLeaf111022022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102202) = true := by
+  have h : ((childHL thetaAboveCell11102202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102202) h
+theorem e24KC2ThetaAboveLeaf111022023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102202) = true := by
+  have h : ((childHH thetaAboveCell11102202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102202) h
+theorem e24KC2ThetaAboveLeaf111022030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102203) = true := by
+  have h : ((childLL thetaAboveCell11102203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102203) h
+theorem e24KC2ThetaAboveLeaf111022031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102203) = true := by
+  have h : ((childLH thetaAboveCell11102203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102203) h
+theorem e24KC2ThetaAboveLeaf111022032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102203) = true := by
+  have h : ((childHL thetaAboveCell11102203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102203) h
+theorem e24KC2ThetaAboveLeaf111022033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102203) = true := by
+  have h : ((childHH thetaAboveCell11102203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102203) h
+theorem e24KC2ThetaAboveLeaf11102210 :
+    adaptiveCoverCheck 11 thetaAboveCell11102210 = true := by
+  have h : (thetaAboveCell11102210).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102210 h
+theorem e24KC2ThetaAboveLeaf11102211 :
+    adaptiveCoverCheck 11 thetaAboveCell11102211 = true := by
+  have h : (thetaAboveCell11102211).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102211 h
+theorem e24KC2ThetaAboveLeaf111022120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102212) = true := by
+  have h : ((childLL thetaAboveCell11102212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102212) h
+theorem e24KC2ThetaAboveLeaf111022121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102212) = true := by
+  have h : ((childLH thetaAboveCell11102212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102212) h
+theorem e24KC2ThetaAboveLeaf111022122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102212) = true := by
+  have h : ((childHL thetaAboveCell11102212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102212) h
+theorem e24KC2ThetaAboveLeaf111022123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102212) = true := by
+  have h : ((childHH thetaAboveCell11102212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102212) h
+theorem e24KC2ThetaAboveLeaf111022130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102213) = true := by
+  have h : ((childLL thetaAboveCell11102213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102213) h
+theorem e24KC2ThetaAboveLeaf111022131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102213) = true := by
+  have h : ((childLH thetaAboveCell11102213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102213) h
+theorem e24KC2ThetaAboveLeaf111022132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102213) = true := by
+  have h : ((childHL thetaAboveCell11102213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102213) h
+theorem e24KC2ThetaAboveLeaf111022133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102213) = true := by
+  have h : ((childHH thetaAboveCell11102213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102213) h
+theorem e24KC2ThetaAboveLeaf111022200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102220) = true := by
+  have h : ((childLL thetaAboveCell11102220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102220) h
+theorem e24KC2ThetaAboveLeaf111022201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102220) = true := by
+  have h : ((childLH thetaAboveCell11102220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102220) h
+theorem e24KC2ThetaAboveLeaf111022202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102220) = true := by
+  have h : ((childHL thetaAboveCell11102220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102220) h
+theorem e24KC2ThetaAboveLeaf111022203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102220) = true := by
+  have h : ((childHH thetaAboveCell11102220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102220) h
+theorem e24KC2ThetaAboveLeaf111022210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102221) = true := by
+  have h : ((childLL thetaAboveCell11102221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102221) h
+theorem e24KC2ThetaAboveLeaf111022211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102221) = true := by
+  have h : ((childLH thetaAboveCell11102221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102221) h
+theorem e24KC2ThetaAboveLeaf111022212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102221) = true := by
+  have h : ((childHL thetaAboveCell11102221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102221) h
+theorem e24KC2ThetaAboveLeaf111022213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102221) = true := by
+  have h : ((childHH thetaAboveCell11102221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102221) h
+theorem e24KC2ThetaAboveLeaf11102222 :
+    adaptiveCoverCheck 11 thetaAboveCell11102222 = true := by
+  have h : (thetaAboveCell11102222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102222 h
+theorem e24KC2ThetaAboveLeaf11102223 :
+    adaptiveCoverCheck 11 thetaAboveCell11102223 = true := by
+  have h : (thetaAboveCell11102223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102223 h
+theorem e24KC2ThetaAboveLeaf111022300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102230) = true := by
+  have h : ((childLL thetaAboveCell11102230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102230) h
+theorem e24KC2ThetaAboveLeaf111022301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102230) = true := by
+  have h : ((childLH thetaAboveCell11102230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102230) h
+theorem e24KC2ThetaAboveLeaf111022302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102230) = true := by
+  have h : ((childHL thetaAboveCell11102230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102230) h
+theorem e24KC2ThetaAboveLeaf111022303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102230) = true := by
+  have h : ((childHH thetaAboveCell11102230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102230) h
+theorem e24KC2ThetaAboveLeaf111022310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102231) = true := by
+  have h : ((childLL thetaAboveCell11102231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102231) h
+theorem e24KC2ThetaAboveLeaf111022311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102231) = true := by
+  have h : ((childLH thetaAboveCell11102231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102231) h
+theorem e24KC2ThetaAboveLeaf111022312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102231) = true := by
+  have h : ((childHL thetaAboveCell11102231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102231) h
+theorem e24KC2ThetaAboveLeaf111022313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102231) = true := by
+  have h : ((childHH thetaAboveCell11102231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102231) h
+theorem e24KC2ThetaAboveLeaf11102232 :
+    adaptiveCoverCheck 11 thetaAboveCell11102232 = true := by
+  have h : (thetaAboveCell11102232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102232 h
+theorem e24KC2ThetaAboveLeaf11102233 :
+    adaptiveCoverCheck 11 thetaAboveCell11102233 = true := by
+  have h : (thetaAboveCell11102233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102233 h
+theorem e24KC2ThetaAboveLeaf11102300 :
+    adaptiveCoverCheck 11 thetaAboveCell11102300 = true := by
+  have h : (thetaAboveCell11102300).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102300 h
+theorem e24KC2ThetaAboveLeaf11102301 :
+    adaptiveCoverCheck 11 thetaAboveCell11102301 = true := by
+  have h : (thetaAboveCell11102301).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102301 h
+theorem e24KC2ThetaAboveLeaf111023020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102302) = true := by
+  have h : ((childLL thetaAboveCell11102302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102302) h
+theorem e24KC2ThetaAboveLeaf111023021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102302) = true := by
+  have h : ((childLH thetaAboveCell11102302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102302) h
+theorem e24KC2ThetaAboveLeaf111023022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102302) = true := by
+  have h : ((childHL thetaAboveCell11102302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102302) h
+theorem e24KC2ThetaAboveLeaf111023023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102302) = true := by
+  have h : ((childHH thetaAboveCell11102302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102302) h
+theorem e24KC2ThetaAboveLeaf111023030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102303) = true := by
+  have h : ((childLL thetaAboveCell11102303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102303) h
+theorem e24KC2ThetaAboveLeaf111023031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102303) = true := by
+  have h : ((childLH thetaAboveCell11102303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102303) h
+theorem e24KC2ThetaAboveLeaf111023032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102303) = true := by
+  have h : ((childHL thetaAboveCell11102303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102303) h
+theorem e24KC2ThetaAboveLeaf111023033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102303) = true := by
+  have h : ((childHH thetaAboveCell11102303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102303) h
+theorem e24KC2ThetaAboveLeaf11102310 :
+    adaptiveCoverCheck 11 thetaAboveCell11102310 = true := by
+  have h : (thetaAboveCell11102310).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102310 h
+theorem e24KC2ThetaAboveLeaf11102311 :
+    adaptiveCoverCheck 11 thetaAboveCell11102311 = true := by
+  have h : (thetaAboveCell11102311).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102311 h
+theorem e24KC2ThetaAboveLeaf111023120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102312) = true := by
+  have h : ((childLL thetaAboveCell11102312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102312) h
+theorem e24KC2ThetaAboveLeaf111023121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102312) = true := by
+  have h : ((childLH thetaAboveCell11102312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102312) h
+theorem e24KC2ThetaAboveLeaf111023122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102312) = true := by
+  have h : ((childHL thetaAboveCell11102312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102312) h
+theorem e24KC2ThetaAboveLeaf111023123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102312) = true := by
+  have h : ((childHH thetaAboveCell11102312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102312) h
+theorem e24KC2ThetaAboveLeaf111023130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102313) = true := by
+  have h : ((childLL thetaAboveCell11102313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102313) h
+theorem e24KC2ThetaAboveLeaf111023131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102313) = true := by
+  have h : ((childLH thetaAboveCell11102313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102313) h
+theorem e24KC2ThetaAboveLeaf111023132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102313) = true := by
+  have h : ((childHL thetaAboveCell11102313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102313) h
+theorem e24KC2ThetaAboveLeaf111023133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102313) = true := by
+  have h : ((childHH thetaAboveCell11102313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102313) h
+theorem e24KC2ThetaAboveLeaf111023200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102320) = true := by
+  have h : ((childLL thetaAboveCell11102320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102320) h
+theorem e24KC2ThetaAboveLeaf111023201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102320) = true := by
+  have h : ((childLH thetaAboveCell11102320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102320) h
+theorem e24KC2ThetaAboveLeaf111023202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102320) = true := by
+  have h : ((childHL thetaAboveCell11102320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102320) h
+theorem e24KC2ThetaAboveLeaf111023203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102320) = true := by
+  have h : ((childHH thetaAboveCell11102320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102320) h
+theorem e24KC2ThetaAboveLeaf111023210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102321) = true := by
+  have h : ((childLL thetaAboveCell11102321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102321) h
+theorem e24KC2ThetaAboveLeaf111023211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102321) = true := by
+  have h : ((childLH thetaAboveCell11102321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102321) h
+theorem e24KC2ThetaAboveLeaf111023212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102321) = true := by
+  have h : ((childHL thetaAboveCell11102321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102321) h
+theorem e24KC2ThetaAboveLeaf111023213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102321) = true := by
+  have h : ((childHH thetaAboveCell11102321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102321) h
+theorem e24KC2ThetaAboveLeaf11102322 :
+    adaptiveCoverCheck 11 thetaAboveCell11102322 = true := by
+  have h : (thetaAboveCell11102322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102322 h
+theorem e24KC2ThetaAboveLeaf11102323 :
+    adaptiveCoverCheck 11 thetaAboveCell11102323 = true := by
+  have h : (thetaAboveCell11102323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102323 h
+theorem e24KC2ThetaAboveLeaf111023300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102330) = true := by
+  have h : ((childLL thetaAboveCell11102330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102330) h
+theorem e24KC2ThetaAboveLeaf111023301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102330) = true := by
+  have h : ((childLH thetaAboveCell11102330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102330) h
+theorem e24KC2ThetaAboveLeaf111023302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102330) = true := by
+  have h : ((childHL thetaAboveCell11102330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102330) h
+theorem e24KC2ThetaAboveLeaf111023303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102330) = true := by
+  have h : ((childHH thetaAboveCell11102330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102330) h
+theorem e24KC2ThetaAboveLeaf111023310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11102331) = true := by
+  have h : ((childLL thetaAboveCell11102331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11102331) h
+theorem e24KC2ThetaAboveLeaf111023311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11102331) = true := by
+  have h : ((childLH thetaAboveCell11102331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11102331) h
+theorem e24KC2ThetaAboveLeaf111023312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11102331) = true := by
+  have h : ((childHL thetaAboveCell11102331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11102331) h
+theorem e24KC2ThetaAboveLeaf111023313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11102331) = true := by
+  have h : ((childHH thetaAboveCell11102331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11102331) h
+theorem e24KC2ThetaAboveLeaf11102332 :
+    adaptiveCoverCheck 11 thetaAboveCell11102332 = true := by
+  have h : (thetaAboveCell11102332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102332 h
+theorem e24KC2ThetaAboveLeaf11102333 :
+    adaptiveCoverCheck 11 thetaAboveCell11102333 = true := by
+  have h : (thetaAboveCell11102333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11102333 h
+theorem e24KC2ThetaAboveLeaf1110300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell1110))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf1110301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell1110))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf1110302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell1110))) = true := by
+  have h : ((childHL (childLL (childHH thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childHH thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf1110303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell1110))) = true := by
+  have h : ((childHH (childLL (childHH thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childHH thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf1110310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell1110))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf1110311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell1110))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf1110312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell1110))) = true := by
+  have h : ((childHL (childLH (childHH thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childHH thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf1110313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell1110))) = true := by
+  have h : ((childHH (childLH (childHH thetaAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childHH thetaAboveCell1110))) h
+theorem e24KC2ThetaAboveLeaf11103200 :
+    adaptiveCoverCheck 11 thetaAboveCell11103200 = true := by
+  have h : (thetaAboveCell11103200).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103200 h
+theorem e24KC2ThetaAboveLeaf11103201 :
+    adaptiveCoverCheck 11 thetaAboveCell11103201 = true := by
+  have h : (thetaAboveCell11103201).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103201 h
+theorem e24KC2ThetaAboveLeaf111032020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103202) = true := by
+  have h : ((childLL thetaAboveCell11103202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103202) h
+theorem e24KC2ThetaAboveLeaf111032021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103202) = true := by
+  have h : ((childLH thetaAboveCell11103202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103202) h
+theorem e24KC2ThetaAboveLeaf111032022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103202) = true := by
+  have h : ((childHL thetaAboveCell11103202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103202) h
+theorem e24KC2ThetaAboveLeaf111032023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103202) = true := by
+  have h : ((childHH thetaAboveCell11103202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103202) h
+theorem e24KC2ThetaAboveLeaf111032030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103203) = true := by
+  have h : ((childLL thetaAboveCell11103203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103203) h
+theorem e24KC2ThetaAboveLeaf111032031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103203) = true := by
+  have h : ((childLH thetaAboveCell11103203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103203) h
+theorem e24KC2ThetaAboveLeaf111032032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103203) = true := by
+  have h : ((childHL thetaAboveCell11103203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103203) h
+theorem e24KC2ThetaAboveLeaf111032033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103203) = true := by
+  have h : ((childHH thetaAboveCell11103203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103203) h
+theorem e24KC2ThetaAboveLeaf11103210 :
+    adaptiveCoverCheck 11 thetaAboveCell11103210 = true := by
+  have h : (thetaAboveCell11103210).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103210 h
+theorem e24KC2ThetaAboveLeaf11103211 :
+    adaptiveCoverCheck 11 thetaAboveCell11103211 = true := by
+  have h : (thetaAboveCell11103211).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103211 h
+theorem e24KC2ThetaAboveLeaf111032120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103212) = true := by
+  have h : ((childLL thetaAboveCell11103212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103212) h
+theorem e24KC2ThetaAboveLeaf111032121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103212) = true := by
+  have h : ((childLH thetaAboveCell11103212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103212) h
+theorem e24KC2ThetaAboveLeaf111032122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103212) = true := by
+  have h : ((childHL thetaAboveCell11103212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103212) h
+theorem e24KC2ThetaAboveLeaf111032123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103212) = true := by
+  have h : ((childHH thetaAboveCell11103212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103212) h
+theorem e24KC2ThetaAboveLeaf111032130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103213) = true := by
+  have h : ((childLL thetaAboveCell11103213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103213) h
+theorem e24KC2ThetaAboveLeaf111032131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103213) = true := by
+  have h : ((childLH thetaAboveCell11103213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103213) h
+theorem e24KC2ThetaAboveLeaf111032132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103213) = true := by
+  have h : ((childHL thetaAboveCell11103213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103213) h
+theorem e24KC2ThetaAboveLeaf111032133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103213) = true := by
+  have h : ((childHH thetaAboveCell11103213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103213) h
+theorem e24KC2ThetaAboveLeaf111032200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103220) = true := by
+  have h : ((childLL thetaAboveCell11103220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103220) h
+theorem e24KC2ThetaAboveLeaf111032201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103220) = true := by
+  have h : ((childLH thetaAboveCell11103220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103220) h
+theorem e24KC2ThetaAboveLeaf111032202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103220) = true := by
+  have h : ((childHL thetaAboveCell11103220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103220) h
+theorem e24KC2ThetaAboveLeaf111032203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103220) = true := by
+  have h : ((childHH thetaAboveCell11103220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103220) h
+theorem e24KC2ThetaAboveLeaf111032210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103221) = true := by
+  have h : ((childLL thetaAboveCell11103221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103221) h
+theorem e24KC2ThetaAboveLeaf111032211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103221) = true := by
+  have h : ((childLH thetaAboveCell11103221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103221) h
+theorem e24KC2ThetaAboveLeaf111032212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103221) = true := by
+  have h : ((childHL thetaAboveCell11103221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103221) h
+theorem e24KC2ThetaAboveLeaf111032213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103221) = true := by
+  have h : ((childHH thetaAboveCell11103221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103221) h
+theorem e24KC2ThetaAboveLeaf11103222 :
+    adaptiveCoverCheck 11 thetaAboveCell11103222 = true := by
+  have h : (thetaAboveCell11103222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103222 h
+theorem e24KC2ThetaAboveLeaf11103223 :
+    adaptiveCoverCheck 11 thetaAboveCell11103223 = true := by
+  have h : (thetaAboveCell11103223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103223 h
+theorem e24KC2ThetaAboveLeaf111032300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103230) = true := by
+  have h : ((childLL thetaAboveCell11103230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103230) h
+theorem e24KC2ThetaAboveLeaf111032301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103230) = true := by
+  have h : ((childLH thetaAboveCell11103230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103230) h
+theorem e24KC2ThetaAboveLeaf111032302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103230) = true := by
+  have h : ((childHL thetaAboveCell11103230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103230) h
+theorem e24KC2ThetaAboveLeaf111032303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103230) = true := by
+  have h : ((childHH thetaAboveCell11103230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103230) h
+theorem e24KC2ThetaAboveLeaf111032310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103231) = true := by
+  have h : ((childLL thetaAboveCell11103231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103231) h
+theorem e24KC2ThetaAboveLeaf111032311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103231) = true := by
+  have h : ((childLH thetaAboveCell11103231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103231) h
+theorem e24KC2ThetaAboveLeaf111032312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103231) = true := by
+  have h : ((childHL thetaAboveCell11103231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103231) h
+theorem e24KC2ThetaAboveLeaf111032313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103231) = true := by
+  have h : ((childHH thetaAboveCell11103231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103231) h
+theorem e24KC2ThetaAboveLeaf11103232 :
+    adaptiveCoverCheck 11 thetaAboveCell11103232 = true := by
+  have h : (thetaAboveCell11103232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103232 h
+theorem e24KC2ThetaAboveLeaf11103233 :
+    adaptiveCoverCheck 11 thetaAboveCell11103233 = true := by
+  have h : (thetaAboveCell11103233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103233 h
+theorem e24KC2ThetaAboveLeaf11103300 :
+    adaptiveCoverCheck 11 thetaAboveCell11103300 = true := by
+  have h : (thetaAboveCell11103300).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103300 h
+theorem e24KC2ThetaAboveLeaf11103301 :
+    adaptiveCoverCheck 11 thetaAboveCell11103301 = true := by
+  have h : (thetaAboveCell11103301).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103301 h
+theorem e24KC2ThetaAboveLeaf111033020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103302) = true := by
+  have h : ((childLL thetaAboveCell11103302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103302) h
+theorem e24KC2ThetaAboveLeaf111033021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103302) = true := by
+  have h : ((childLH thetaAboveCell11103302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103302) h
+theorem e24KC2ThetaAboveLeaf111033022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103302) = true := by
+  have h : ((childHL thetaAboveCell11103302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103302) h
+theorem e24KC2ThetaAboveLeaf111033023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103302) = true := by
+  have h : ((childHH thetaAboveCell11103302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103302) h
+theorem e24KC2ThetaAboveLeaf111033030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103303) = true := by
+  have h : ((childLL thetaAboveCell11103303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103303) h
+theorem e24KC2ThetaAboveLeaf111033031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103303) = true := by
+  have h : ((childLH thetaAboveCell11103303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103303) h
+theorem e24KC2ThetaAboveLeaf111033032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103303) = true := by
+  have h : ((childHL thetaAboveCell11103303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103303) h
+theorem e24KC2ThetaAboveLeaf111033033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103303) = true := by
+  have h : ((childHH thetaAboveCell11103303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103303) h
+theorem e24KC2ThetaAboveLeaf11103310 :
+    adaptiveCoverCheck 11 thetaAboveCell11103310 = true := by
+  have h : (thetaAboveCell11103310).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103310 h
+theorem e24KC2ThetaAboveLeaf11103311 :
+    adaptiveCoverCheck 11 thetaAboveCell11103311 = true := by
+  have h : (thetaAboveCell11103311).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103311 h
+theorem e24KC2ThetaAboveLeaf111033120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103312) = true := by
+  have h : ((childLL thetaAboveCell11103312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103312) h
+theorem e24KC2ThetaAboveLeaf111033121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103312) = true := by
+  have h : ((childLH thetaAboveCell11103312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103312) h
+theorem e24KC2ThetaAboveLeaf111033122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103312) = true := by
+  have h : ((childHL thetaAboveCell11103312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103312) h
+theorem e24KC2ThetaAboveLeaf111033123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103312) = true := by
+  have h : ((childHH thetaAboveCell11103312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103312) h
+theorem e24KC2ThetaAboveLeaf111033130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103313) = true := by
+  have h : ((childLL thetaAboveCell11103313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103313) h
+theorem e24KC2ThetaAboveLeaf111033131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103313) = true := by
+  have h : ((childLH thetaAboveCell11103313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103313) h
+theorem e24KC2ThetaAboveLeaf111033132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103313) = true := by
+  have h : ((childHL thetaAboveCell11103313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103313) h
+theorem e24KC2ThetaAboveLeaf111033133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103313) = true := by
+  have h : ((childHH thetaAboveCell11103313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103313) h
+theorem e24KC2ThetaAboveLeaf111033200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103320) = true := by
+  have h : ((childLL thetaAboveCell11103320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103320) h
+theorem e24KC2ThetaAboveLeaf111033201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103320) = true := by
+  have h : ((childLH thetaAboveCell11103320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103320) h
+theorem e24KC2ThetaAboveLeaf111033202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103320) = true := by
+  have h : ((childHL thetaAboveCell11103320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103320) h
+theorem e24KC2ThetaAboveLeaf111033203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103320) = true := by
+  have h : ((childHH thetaAboveCell11103320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103320) h
+theorem e24KC2ThetaAboveLeaf111033210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103321) = true := by
+  have h : ((childLL thetaAboveCell11103321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103321) h
+theorem e24KC2ThetaAboveLeaf111033211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103321) = true := by
+  have h : ((childLH thetaAboveCell11103321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103321) h
+theorem e24KC2ThetaAboveLeaf111033212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103321) = true := by
+  have h : ((childHL thetaAboveCell11103321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103321) h
+theorem e24KC2ThetaAboveLeaf111033213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103321) = true := by
+  have h : ((childHH thetaAboveCell11103321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103321) h
+theorem e24KC2ThetaAboveLeaf11103322 :
+    adaptiveCoverCheck 11 thetaAboveCell11103322 = true := by
+  have h : (thetaAboveCell11103322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103322 h
+theorem e24KC2ThetaAboveLeaf11103323 :
+    adaptiveCoverCheck 11 thetaAboveCell11103323 = true := by
+  have h : (thetaAboveCell11103323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103323 h
+theorem e24KC2ThetaAboveLeaf111033300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103330) = true := by
+  have h : ((childLL thetaAboveCell11103330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103330) h
+theorem e24KC2ThetaAboveLeaf111033301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103330) = true := by
+  have h : ((childLH thetaAboveCell11103330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103330) h
+theorem e24KC2ThetaAboveLeaf111033302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103330) = true := by
+  have h : ((childHL thetaAboveCell11103330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103330) h
+theorem e24KC2ThetaAboveLeaf111033303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103330) = true := by
+  have h : ((childHH thetaAboveCell11103330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103330) h
+theorem e24KC2ThetaAboveLeaf111033310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11103331) = true := by
+  have h : ((childLL thetaAboveCell11103331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11103331) h
+theorem e24KC2ThetaAboveLeaf111033311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11103331) = true := by
+  have h : ((childLH thetaAboveCell11103331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11103331) h
+theorem e24KC2ThetaAboveLeaf111033312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11103331) = true := by
+  have h : ((childHL thetaAboveCell11103331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11103331) h
+theorem e24KC2ThetaAboveLeaf111033313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11103331) = true := by
+  have h : ((childHH thetaAboveCell11103331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11103331) h
+theorem e24KC2ThetaAboveLeaf11103332 :
+    adaptiveCoverCheck 11 thetaAboveCell11103332 = true := by
+  have h : (thetaAboveCell11103332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103332 h
+theorem e24KC2ThetaAboveLeaf11103333 :
+    adaptiveCoverCheck 11 thetaAboveCell11103333 = true := by
+  have h : (thetaAboveCell11103333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11103333 h
+theorem e24KC2ThetaAboveLeaf111100 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1111)) = true := by
+  have h : ((childLL (childLL thetaAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell1111)) h
+theorem e24KC2ThetaAboveLeaf111101 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1111)) = true := by
+  have h : ((childLH (childLL thetaAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell1111)) h
+theorem e24KC2ThetaAboveLeaf111102 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1111)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1111)) h
+theorem e24KC2ThetaAboveLeaf111103 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1111)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1111)) h
+theorem e24KC2ThetaAboveLeaf111110 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1111)) = true := by
+  have h : ((childLL (childLH thetaAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell1111)) h
+theorem e24KC2ThetaAboveLeaf111111 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1111)) = true := by
+  have h : ((childLH (childLH thetaAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell1111)) h
+theorem e24KC2ThetaAboveLeaf111112 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1111)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1111)) h
+theorem e24KC2ThetaAboveLeaf111113 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1111)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1111)) h
+theorem e24KC2ThetaAboveLeaf1111200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell1111))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf1111201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell1111))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf1111202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell1111))) = true := by
+  have h : ((childHL (childLL (childHL thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childHL thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf1111203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell1111))) = true := by
+  have h : ((childHH (childLL (childHL thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childHL thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf1111210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell1111))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf1111211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell1111))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf1111212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell1111))) = true := by
+  have h : ((childHL (childLH (childHL thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childHL thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf1111213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell1111))) = true := by
+  have h : ((childHH (childLH (childHL thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childHL thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf11112200 :
+    adaptiveCoverCheck 11 thetaAboveCell11112200 = true := by
+  have h : (thetaAboveCell11112200).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11112200 h
+theorem e24KC2ThetaAboveLeaf11112201 :
+    adaptiveCoverCheck 11 thetaAboveCell11112201 = true := by
+  have h : (thetaAboveCell11112201).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11112201 h
+theorem e24KC2ThetaAboveLeaf111122020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112202) = true := by
+  have h : ((childLL thetaAboveCell11112202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112202) h
+theorem e24KC2ThetaAboveLeaf111122021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112202) = true := by
+  have h : ((childLH thetaAboveCell11112202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112202) h
+theorem e24KC2ThetaAboveLeaf111122022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112202) = true := by
+  have h : ((childHL thetaAboveCell11112202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112202) h
+theorem e24KC2ThetaAboveLeaf111122023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112202) = true := by
+  have h : ((childHH thetaAboveCell11112202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112202) h
+theorem e24KC2ThetaAboveLeaf111122030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112203) = true := by
+  have h : ((childLL thetaAboveCell11112203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112203) h
+theorem e24KC2ThetaAboveLeaf111122031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112203) = true := by
+  have h : ((childLH thetaAboveCell11112203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112203) h
+theorem e24KC2ThetaAboveLeaf111122032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112203) = true := by
+  have h : ((childHL thetaAboveCell11112203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112203) h
+theorem e24KC2ThetaAboveLeaf111122033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112203) = true := by
+  have h : ((childHH thetaAboveCell11112203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112203) h
+theorem e24KC2ThetaAboveLeaf11112210 :
+    adaptiveCoverCheck 11 thetaAboveCell11112210 = true := by
+  have h : (thetaAboveCell11112210).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11112210 h
+theorem e24KC2ThetaAboveLeaf11112211 :
+    adaptiveCoverCheck 11 thetaAboveCell11112211 = true := by
+  have h : (thetaAboveCell11112211).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11112211 h
+theorem e24KC2ThetaAboveLeaf111122120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112212) = true := by
+  have h : ((childLL thetaAboveCell11112212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112212) h
+theorem e24KC2ThetaAboveLeaf111122121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112212) = true := by
+  have h : ((childLH thetaAboveCell11112212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112212) h
+theorem e24KC2ThetaAboveLeaf111122122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112212) = true := by
+  have h : ((childHL thetaAboveCell11112212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112212) h
+theorem e24KC2ThetaAboveLeaf111122123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112212) = true := by
+  have h : ((childHH thetaAboveCell11112212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112212) h
+theorem e24KC2ThetaAboveLeaf111122130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112213) = true := by
+  have h : ((childLL thetaAboveCell11112213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112213) h
+theorem e24KC2ThetaAboveLeaf111122131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112213) = true := by
+  have h : ((childLH thetaAboveCell11112213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112213) h
+theorem e24KC2ThetaAboveLeaf111122132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112213) = true := by
+  have h : ((childHL thetaAboveCell11112213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112213) h
+theorem e24KC2ThetaAboveLeaf111122133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112213) = true := by
+  have h : ((childHH thetaAboveCell11112213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112213) h
+theorem e24KC2ThetaAboveLeaf111122200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112220) = true := by
+  have h : ((childLL thetaAboveCell11112220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112220) h
+theorem e24KC2ThetaAboveLeaf111122201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112220) = true := by
+  have h : ((childLH thetaAboveCell11112220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112220) h
+theorem e24KC2ThetaAboveLeaf111122202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112220) = true := by
+  have h : ((childHL thetaAboveCell11112220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112220) h
+theorem e24KC2ThetaAboveLeaf111122203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112220) = true := by
+  have h : ((childHH thetaAboveCell11112220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112220) h
+theorem e24KC2ThetaAboveLeaf111122210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112221) = true := by
+  have h : ((childLL thetaAboveCell11112221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112221) h
+theorem e24KC2ThetaAboveLeaf111122211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112221) = true := by
+  have h : ((childLH thetaAboveCell11112221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112221) h
+theorem e24KC2ThetaAboveLeaf111122212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112221) = true := by
+  have h : ((childHL thetaAboveCell11112221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112221) h
+theorem e24KC2ThetaAboveLeaf111122213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112221) = true := by
+  have h : ((childHH thetaAboveCell11112221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112221) h
+theorem e24KC2ThetaAboveLeaf11112222 :
+    adaptiveCoverCheck 11 thetaAboveCell11112222 = true := by
+  have h : (thetaAboveCell11112222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11112222 h
+theorem e24KC2ThetaAboveLeaf11112223 :
+    adaptiveCoverCheck 11 thetaAboveCell11112223 = true := by
+  have h : (thetaAboveCell11112223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11112223 h
+theorem e24KC2ThetaAboveLeaf111122300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112230) = true := by
+  have h : ((childLL thetaAboveCell11112230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112230) h
+theorem e24KC2ThetaAboveLeaf111122301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112230) = true := by
+  have h : ((childLH thetaAboveCell11112230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112230) h
+theorem e24KC2ThetaAboveLeaf111122302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112230) = true := by
+  have h : ((childHL thetaAboveCell11112230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112230) h
+theorem e24KC2ThetaAboveLeaf111122303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112230) = true := by
+  have h : ((childHH thetaAboveCell11112230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112230) h
+theorem e24KC2ThetaAboveLeaf111122310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112231) = true := by
+  have h : ((childLL thetaAboveCell11112231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112231) h
+theorem e24KC2ThetaAboveLeaf111122311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112231) = true := by
+  have h : ((childLH thetaAboveCell11112231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112231) h
+theorem e24KC2ThetaAboveLeaf111122312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112231) = true := by
+  have h : ((childHL thetaAboveCell11112231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112231) h
+theorem e24KC2ThetaAboveLeaf111122313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112231) = true := by
+  have h : ((childHH thetaAboveCell11112231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112231) h
+theorem e24KC2ThetaAboveLeaf11112232 :
+    adaptiveCoverCheck 11 thetaAboveCell11112232 = true := by
+  have h : (thetaAboveCell11112232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11112232 h
+theorem e24KC2ThetaAboveLeaf11112233 :
+    adaptiveCoverCheck 11 thetaAboveCell11112233 = true := by
+  have h : (thetaAboveCell11112233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11112233 h
+theorem e24KC2ThetaAboveLeaf11112300 :
+    adaptiveCoverCheck 11 thetaAboveCell11112300 = true := by
+  have h : (thetaAboveCell11112300).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11112300 h
+theorem e24KC2ThetaAboveLeaf11112301 :
+    adaptiveCoverCheck 11 thetaAboveCell11112301 = true := by
+  have h : (thetaAboveCell11112301).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11112301 h
+theorem e24KC2ThetaAboveLeaf111123020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112302) = true := by
+  have h : ((childLL thetaAboveCell11112302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112302) h
+theorem e24KC2ThetaAboveLeaf111123021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112302) = true := by
+  have h : ((childLH thetaAboveCell11112302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112302) h
+theorem e24KC2ThetaAboveLeaf111123022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112302) = true := by
+  have h : ((childHL thetaAboveCell11112302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112302) h
+theorem e24KC2ThetaAboveLeaf111123023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112302) = true := by
+  have h : ((childHH thetaAboveCell11112302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112302) h
+theorem e24KC2ThetaAboveLeaf111123030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112303) = true := by
+  have h : ((childLL thetaAboveCell11112303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112303) h
+theorem e24KC2ThetaAboveLeaf111123031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112303) = true := by
+  have h : ((childLH thetaAboveCell11112303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112303) h
+theorem e24KC2ThetaAboveLeaf111123032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112303) = true := by
+  have h : ((childHL thetaAboveCell11112303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112303) h
+theorem e24KC2ThetaAboveLeaf111123033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112303) = true := by
+  have h : ((childHH thetaAboveCell11112303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112303) h
+theorem e24KC2ThetaAboveLeaf11112310 :
+    adaptiveCoverCheck 11 thetaAboveCell11112310 = true := by
+  have h : (thetaAboveCell11112310).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11112310 h
+theorem e24KC2ThetaAboveLeaf11112311 :
+    adaptiveCoverCheck 11 thetaAboveCell11112311 = true := by
+  have h : (thetaAboveCell11112311).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11112311 h
+theorem e24KC2ThetaAboveLeaf111123120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112312) = true := by
+  have h : ((childLL thetaAboveCell11112312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112312) h
+theorem e24KC2ThetaAboveLeaf111123121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112312) = true := by
+  have h : ((childLH thetaAboveCell11112312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112312) h
+theorem e24KC2ThetaAboveLeaf111123122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112312) = true := by
+  have h : ((childHL thetaAboveCell11112312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112312) h
+theorem e24KC2ThetaAboveLeaf111123123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112312) = true := by
+  have h : ((childHH thetaAboveCell11112312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112312) h
+theorem e24KC2ThetaAboveLeaf111123130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112313) = true := by
+  have h : ((childLL thetaAboveCell11112313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112313) h
+theorem e24KC2ThetaAboveLeaf111123131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112313) = true := by
+  have h : ((childLH thetaAboveCell11112313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112313) h
+theorem e24KC2ThetaAboveLeaf111123132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112313) = true := by
+  have h : ((childHL thetaAboveCell11112313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112313) h
+theorem e24KC2ThetaAboveLeaf111123133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112313) = true := by
+  have h : ((childHH thetaAboveCell11112313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112313) h
+theorem e24KC2ThetaAboveLeaf111123200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112320) = true := by
+  have h : ((childLL thetaAboveCell11112320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112320) h
+theorem e24KC2ThetaAboveLeaf111123201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112320) = true := by
+  have h : ((childLH thetaAboveCell11112320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112320) h
+theorem e24KC2ThetaAboveLeaf111123202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112320) = true := by
+  have h : ((childHL thetaAboveCell11112320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112320) h
+theorem e24KC2ThetaAboveLeaf111123203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112320) = true := by
+  have h : ((childHH thetaAboveCell11112320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112320) h
+theorem e24KC2ThetaAboveLeaf111123210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112321) = true := by
+  have h : ((childLL thetaAboveCell11112321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112321) h
+theorem e24KC2ThetaAboveLeaf111123211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112321) = true := by
+  have h : ((childLH thetaAboveCell11112321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112321) h
+theorem e24KC2ThetaAboveLeaf111123212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112321) = true := by
+  have h : ((childHL thetaAboveCell11112321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112321) h
+theorem e24KC2ThetaAboveLeaf111123213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112321) = true := by
+  have h : ((childHH thetaAboveCell11112321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112321) h
+theorem e24KC2ThetaAboveLeaf11112322 :
+    adaptiveCoverCheck 11 thetaAboveCell11112322 = true := by
+  have h : (thetaAboveCell11112322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11112322 h
+theorem e24KC2ThetaAboveLeaf11112323 :
+    adaptiveCoverCheck 11 thetaAboveCell11112323 = true := by
+  have h : (thetaAboveCell11112323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11112323 h
+theorem e24KC2ThetaAboveLeaf111123300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112330) = true := by
+  have h : ((childLL thetaAboveCell11112330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112330) h
+theorem e24KC2ThetaAboveLeaf111123301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112330) = true := by
+  have h : ((childLH thetaAboveCell11112330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112330) h
+theorem e24KC2ThetaAboveLeaf111123302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112330) = true := by
+  have h : ((childHL thetaAboveCell11112330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112330) h
+theorem e24KC2ThetaAboveLeaf111123303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112330) = true := by
+  have h : ((childHH thetaAboveCell11112330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112330) h
+theorem e24KC2ThetaAboveLeaf111123310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112331) = true := by
+  have h : ((childLL thetaAboveCell11112331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112331) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle003.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle003.lean
new file mode 100644
index 0000000000..0079b266b1
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle003.lean
@@ -0,0 +1,12868 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch003`.
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch012`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CoverCertificate12be39bbad
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLH (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificate12be39bbad
+
+namespace GerverSofa.PartE.CoverCertificatec4afef5d19
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+end GerverSofa.PartE.CoverCertificatec4afef5d19
+
+namespace GerverSofa.PartE.CoverCertificate62151508e5
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate62151508e5
+
+namespace GerverSofa.PartE.CoverCertificate2dddad520f
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLH (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+end GerverSofa.PartE.CoverCertificate2dddad520f
+
+namespace GerverSofa.PartE.CoverCertificate4f8bf27ed0
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLH (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate4f8bf27ed0
+
+namespace GerverSofa.PartE.CoverCertificate60aadf3941
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLH (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate60aadf3941
+
+namespace GerverSofa.PartE.CoverCertificate19be7723d5
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLH (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate19be7723d5
+
+namespace GerverSofa.PartE.CoverCertificatea0740027fc
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childLH (childLH (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatea0740027fc
+
+namespace GerverSofa.PartE.CoverCertificate069fa78a71
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHL (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate069fa78a71
+
+namespace GerverSofa.PartE.CoverCertificate1d00d9f69c
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHL (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate1d00d9f69c
+
+namespace GerverSofa.PartE.CoverCertificate01063797cc
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childHL (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate01063797cc
+
+namespace GerverSofa.PartE.CoverCertificatef842354b5b
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatef842354b5b
+
+namespace GerverSofa.PartE.CoverCertificate0e9db7bf30
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate0e9db7bf30
+
+namespace GerverSofa.PartE.CoverCertificate1dc6ae7126
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate1dc6ae7126
+
+namespace GerverSofa.PartE.CoverCertificate3c9114c82d
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate3c9114c82d
+
+namespace GerverSofa.PartE.CoverCertificateec279abdfe
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificateec279abdfe
+
+namespace GerverSofa.PartE.CoverCertificate4e21575659
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate4e21575659
+
+namespace GerverSofa.PartE.CoverCertificated2df3f1b8b
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificated2df3f1b8b
+
+namespace GerverSofa.PartE.CoverCertificatefd72248a13
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatefd72248a13
+
+namespace GerverSofa.PartE.CoverCertificate182948158b
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate182948158b
+
+namespace GerverSofa.PartE.CoverCertificate2c60bab4c6
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate2c60bab4c6
+
+namespace GerverSofa.PartE.CoverCertificatefc5d468444
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatefc5d468444
+
+namespace GerverSofa.PartE.CoverCertificatee3c94e2667
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatee3c94e2667
+
+namespace GerverSofa.PartE.CoverCertificatee175b9aa59
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatee175b9aa59
+
+namespace GerverSofa.PartE.CoverCertificate1efa344902
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate1efa344902
+
+namespace GerverSofa.PartE.CoverCertificatea932254ffe
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatea932254ffe
+
+namespace GerverSofa.PartE.CoverCertificate4b644f2af3
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childHH (childHL (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate4b644f2af3
+
+namespace GerverSofa.PartE.CoverCertificate27ea8c6704
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate27ea8c6704
+
+namespace GerverSofa.PartE.CoverCertificate7519fa3e77
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate7519fa3e77
+
+namespace GerverSofa.PartE.CoverCertificate79beaf8121
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate79beaf8121
+
+namespace GerverSofa.PartE.CoverCertificatef1c8d80a5e
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+end GerverSofa.PartE.CoverCertificatef1c8d80a5e
+
+namespace GerverSofa.PartE.CoverCertificateffa95a2bce
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificateffa95a2bce
+
+namespace GerverSofa.PartE.CoverCertificatea622d1d6d8
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatea622d1d6d8
+
+namespace GerverSofa.PartE.CoverCertificate6808a28f96
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate6808a28f96
+
+namespace GerverSofa.PartE.CoverCertificate64b5caf113
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate64b5caf113
+
+namespace GerverSofa.PartE.CoverCertificate8d399165ed
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+end GerverSofa.PartE.CoverCertificate8d399165ed
+
+namespace GerverSofa.PartE.CoverCertificateb129667c78
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificateb129667c78
+
+namespace GerverSofa.PartE.CoverCertificatee5db5e1b6d
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatee5db5e1b6d
+
+namespace GerverSofa.PartE.CoverCertificatee9fa971b95
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+end GerverSofa.PartE.CoverCertificatee9fa971b95
+
+namespace GerverSofa.PartE.CoverCertificated4cb65cefa
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+end GerverSofa.PartE.CoverCertificated4cb65cefa
+
+namespace GerverSofa.PartE.CoverCertificate75e4a1640f
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate75e4a1640f
+
+namespace GerverSofa.PartE.CoverCertificateb422ba7d7a
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificateb422ba7d7a
+
+namespace GerverSofa.PartE.CoverCertificate348be683b7
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childLL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+end GerverSofa.PartE.CoverCertificate348be683b7
+
+namespace GerverSofa.PartE.CoverCertificatee69a3e135e
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatee69a3e135e
+
+namespace GerverSofa.PartE.CoverCertificate7cfce934a3
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childLH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate7cfce934a3
+
+namespace GerverSofa.PartE.CoverCertificatee6e5d7b950
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childLH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatee6e5d7b950
+
+namespace GerverSofa.PartE.CoverCertificatedbde18308c
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childLH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatedbde18308c
+
+namespace GerverSofa.PartE.CoverCertificate78624e9f2e
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childLH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate78624e9f2e
+
+namespace GerverSofa.PartE.CoverCertificate86b7e57f5c
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childLH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate86b7e57f5c
+
+namespace GerverSofa.PartE.CoverCertificate81d94c7da5
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate81d94c7da5
+
+namespace GerverSofa.PartE.CoverCertificatecaeab30c1b
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childHL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatecaeab30c1b
+
+namespace GerverSofa.PartE.CoverCertificatecee0fdb8c5
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatecee0fdb8c5
+
+namespace GerverSofa.PartE.CoverCertificate92301c5291
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childHL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+end GerverSofa.PartE.CoverCertificate92301c5291
+
+namespace GerverSofa.PartE.CoverCertificate16ba550865
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate16ba550865
+
+namespace GerverSofa.PartE.CoverCertificatebb39ac0110
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+private abbrev cell2020 : AngleCell :=
+  childLL cell202
+
+private abbrev cell2021 : AngleCell :=
+  childLH cell202
+
+private abbrev cell2022 : AngleCell :=
+  childHL cell202
+
+private abbrev cell2023 : AngleCell :=
+  childHH cell202
+
+private abbrev cell2200 : AngleCell :=
+  childLL cell220
+
+private abbrev cell2201 : AngleCell :=
+  childLH cell220
+
+private abbrev cell2202 : AngleCell :=
+  childHL cell220
+
+private abbrev cell2203 : AngleCell :=
+  childHH cell220
+
+private abbrev cell2210 : AngleCell :=
+  childLL cell221
+
+private abbrev cell2211 : AngleCell :=
+  childLH cell221
+
+private abbrev cell2212 : AngleCell :=
+  childHL cell221
+
+private abbrev cell2213 : AngleCell :=
+  childHH cell221
+
+private abbrev cell2220 : AngleCell :=
+  childLL cell222
+
+private abbrev cell2221 : AngleCell :=
+  childLH cell222
+
+private abbrev cell2222 : AngleCell :=
+  childHL cell222
+
+private abbrev cell2223 : AngleCell :=
+  childHH cell222
+
+private abbrev cell2230 : AngleCell :=
+  childLL cell223
+
+private abbrev cell2231 : AngleCell :=
+  childLH cell223
+
+private abbrev cell2232 : AngleCell :=
+  childHL cell223
+
+private abbrev cell2233 : AngleCell :=
+  childHH cell223
+
+private abbrev cell2320 : AngleCell :=
+  childLL cell232
+
+private abbrev cell2321 : AngleCell :=
+  childLH cell232
+
+private abbrev cell2322 : AngleCell :=
+  childHL cell232
+
+private abbrev cell2323 : AngleCell :=
+  childHH cell232
+
+end GerverSofa.PartE.CoverCertificatebb39ac0110
+
+namespace GerverSofa.PartE.CoverCertificatef2ad9b0e27
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+end GerverSofa.PartE.CoverCertificatef2ad9b0e27
+
+namespace GerverSofa.PartE.CoverCertificate73093b0a81
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childHL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate73093b0a81
+
+namespace GerverSofa.PartE.CoverCertificate74c1ca1637
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childHL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificate74c1ca1637
+
+namespace GerverSofa.PartE.CoverCertificateec9c3e1faf
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+private abbrev cell2200 : AngleCell :=
+  childLL cell220
+
+private abbrev cell2201 : AngleCell :=
+  childLH cell220
+
+private abbrev cell2202 : AngleCell :=
+  childHL cell220
+
+private abbrev cell2203 : AngleCell :=
+  childHH cell220
+
+private abbrev cell2220 : AngleCell :=
+  childLL cell222
+
+private abbrev cell2221 : AngleCell :=
+  childLH cell222
+
+private abbrev cell2222 : AngleCell :=
+  childHL cell222
+
+private abbrev cell2223 : AngleCell :=
+  childHH cell222
+
+private abbrev cell2230 : AngleCell :=
+  childLL cell223
+
+private abbrev cell2231 : AngleCell :=
+  childLH cell223
+
+private abbrev cell2232 : AngleCell :=
+  childHL cell223
+
+private abbrev cell2233 : AngleCell :=
+  childHH cell223
+
+end GerverSofa.PartE.CoverCertificateec9c3e1faf
+
+namespace GerverSofa.PartE.CoverCertificate144616495b
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate144616495b
+
+namespace GerverSofa.PartE.CoverCertificatef96351225f
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatef96351225f
+
+namespace GerverSofa.PartE.CoverCertificateed7fe84102
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificateed7fe84102
+
+namespace GerverSofa.PartE.CoverCertificateb1ce232ebf
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificateb1ce232ebf
+
+namespace GerverSofa.PartE.CoverCertificate3e6f5d1050
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLH (childLL (childLH (childLH (childLH (e24PhiAboveRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificate3e6f5d1050
+
+namespace GerverSofa.PartE.CoverCertificate3c635b07e8
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHH (childHH (childHH (e24PhiBelowRoot))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate3c635b07e8
+
+namespace GerverSofa.PartE.CertificateCells976ad3f816
+
+/-- Subcell `1111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells976ad3f816
+
+namespace GerverSofa.PartE.CertificateCells83c065d08f
+
+/-- Subcell `3221` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3221 : AngleCell :=
+  childLH (childHL (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3223 : AngleCell :=
+  childHH (childHL (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3230` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3230 : AngleCell :=
+  childLL (childHH (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3231` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3231 : AngleCell :=
+  childLH (childHH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells83c065d08f
+
+namespace GerverSofa.PartE.CertificateCells9044f599c8
+
+/-- Subcell `3231` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3231 : AngleCell :=
+  childLH (childHH (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3232` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3232 : AngleCell :=
+  childHL (childHH (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3233` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3233 : AngleCell :=
+  childHH (childHH (childHL (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells9044f599c8
+
+namespace GerverSofa.PartE.CertificateCellse87459d2a7
+
+/-- Subcell `3233` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3233 : AngleCell :=
+  childHH (childHH (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3300` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3300 : AngleCell :=
+  childLL (childLL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3301` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3301 : AngleCell :=
+  childLH (childLL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellse87459d2a7
+
+namespace GerverSofa.PartE.CertificateCells5c2e184d25
+
+/-- Subcell `3301` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3301 : AngleCell :=
+  childLH (childLL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3302` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3302 : AngleCell :=
+  childHL (childLL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3303` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3303 : AngleCell :=
+  childHH (childLL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells5c2e184d25
+
+namespace GerverSofa.PartE.CertificateCells32f77991dd
+
+/-- Subcell `3303` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3303 : AngleCell :=
+  childHH (childLL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3310` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3310 : AngleCell :=
+  childLL (childLH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3312` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3312 : AngleCell :=
+  childHL (childLH (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells32f77991dd
+
+namespace GerverSofa.PartE.CertificateCells6c9e156424
+
+/-- Subcell `3313` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3313 : AngleCell :=
+  childHH (childLH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3320` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3320 : AngleCell :=
+  childLL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells6c9e156424
+
+namespace GerverSofa.PartE.CertificateCells613b88d08f
+
+/-- Subcell `3320` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3320 : AngleCell :=
+  childLL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells613b88d08f
+
+namespace GerverSofa.PartE.CertificateCells14a7ef0a1a
+
+/-- Subcell `3320` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3320 : AngleCell :=
+  childLL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells14a7ef0a1a
+
+namespace GerverSofa.PartE.CertificateCells756d9e6827
+
+/-- Subcell `3321` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3321 : AngleCell :=
+  childLH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells756d9e6827
+
+namespace GerverSofa.PartE.CertificateCells9052ca8de4
+
+/-- Subcell `3321` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3321 : AngleCell :=
+  childLH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells9052ca8de4
+
+namespace GerverSofa.PartE.CertificateCells8466e30288
+
+/-- Subcell `3321` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3321 : AngleCell :=
+  childLH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells8466e30288
+
+namespace GerverSofa.PartE.CertificateCells960370a0be
+
+/-- Subcell `3321` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3321 : AngleCell :=
+  childLH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3322` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3322 : AngleCell :=
+  childHL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells960370a0be
+
+namespace GerverSofa.PartE.CertificateCellsd434aeb685
+
+/-- Subcell `3322` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3322 : AngleCell :=
+  childHL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsd434aeb685
+
+namespace GerverSofa.PartE.CertificateCells720ec0d149
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells720ec0d149
+
+namespace GerverSofa.PartE.CertificateCells9005545168
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells9005545168
+
+namespace GerverSofa.PartE.CertificateCellse586487728
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCellse586487728
+
+namespace GerverSofa.PartE.CertificateCellsce87a9d03d
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCellsce87a9d03d
+
+namespace GerverSofa.PartE.CertificateCellsce3c5151af
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCellsce3c5151af
+
+namespace GerverSofa.PartE.CertificateCells7c64a780f8
+
+/-- Subcell `3330` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3330 : AngleCell :=
+  childLL (childHH (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells7c64a780f8
+
+namespace GerverSofa.PartE.CertificateCells168aeb56e4
+
+/-- Subcell `1010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1010 : AngleCell :=
+  childLL (childLH (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1011 : AngleCell :=
+  childLH (childLH (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1012 : AngleCell :=
+  childHL (childLH (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1013 : AngleCell :=
+  childHH (childLH (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1020 : AngleCell :=
+  childLL (childHL (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1021 : AngleCell :=
+  childLH (childHL (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1022 : AngleCell :=
+  childHL (childHL (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1023 : AngleCell :=
+  childHH (childHL (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1030 : AngleCell :=
+  childLL (childHH (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1031 : AngleCell :=
+  childLH (childHH (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1032 : AngleCell :=
+  childHL (childHH (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1033 : AngleCell :=
+  childHH (childHH (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1100 : AngleCell :=
+  childLL (childLL (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `10103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `11002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1100)))
+
+end GerverSofa.PartE.CertificateCells168aeb56e4
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC5FrontierBatchF3200227`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF4000228`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF4800229`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF5600230`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF6400231`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF7200232`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF80LL00235`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF80LRL00237`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF80LRR00238`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF80RLR00247`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF80RRL00249`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF80RRR00250`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF88LL00253`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF88LRR00261`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF88RRL00287`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF8LLR00192`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF8LRL00194`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF8RLR00209`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF8RR00210`.
+* `KernelOnly.PartE.E24KC5FrontierBatchF96LLR00463`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells976ad3f816
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells976ad3f816
+
+open CertificateCells976ad3f816
+namespace CoverCertificate12be39bbad
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate12be39bbad
+
+theorem e24KC2PhiAboveLeaf1111101 :
+    adaptiveCoverCheck 9 (childLH (childLL (childLH phiAboveCell1111))) = true := by
+  exact CoverCertificate12be39bbad.checkedRoot
+namespace CoverCertificatec4afef5d19
+
+private theorem checked000 : adaptiveCoverCheck 6 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 6 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 6 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 6 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 6 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 6 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 6 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 6 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 6 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 6 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 6 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 6 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell103 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec4afef5d19
+
+theorem e24KC2PhiAboveLeaf1111102 :
+    adaptiveCoverCheck 9 (childHL (childLL (childLH phiAboveCell1111))) = true := by
+  exact CoverCertificatec4afef5d19.checkedRoot
+namespace CoverCertificate62151508e5
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate62151508e5
+
+theorem e24KC2PhiAboveLeaf1111103 :
+    adaptiveCoverCheck 9 (childHH (childLL (childLH phiAboveCell1111))) = true := by
+  exact CoverCertificate62151508e5.checkedRoot
+namespace CoverCertificate2dddad520f
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate2dddad520f
+
+theorem e24KC2PhiAboveLeaf1111110 :
+    adaptiveCoverCheck 9 (childLL (childLH (childLH phiAboveCell1111))) = true := by
+  exact CoverCertificate2dddad520f.checkedRoot
+namespace CoverCertificate4f8bf27ed0
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4f8bf27ed0
+
+theorem e24KC2PhiAboveLeaf1111111 :
+    adaptiveCoverCheck 9 (childLH (childLH (childLH phiAboveCell1111))) = true := by
+  exact CoverCertificate4f8bf27ed0.checkedRoot
+namespace CoverCertificate60aadf3941
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate60aadf3941
+
+theorem e24KC2PhiAboveLeaf1111112 :
+    adaptiveCoverCheck 9 (childHL (childLH (childLH phiAboveCell1111))) = true := by
+  exact CoverCertificate60aadf3941.checkedRoot
+namespace CoverCertificate19be7723d5
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate19be7723d5
+
+theorem e24KC2PhiAboveLeaf1111113 :
+    adaptiveCoverCheck 9 (childHH (childLH (childLH phiAboveCell1111))) = true := by
+  exact CoverCertificate19be7723d5.checkedRoot
+namespace CoverCertificatea0740027fc
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea0740027fc
+
+theorem e24KC2PhiAboveLeaf1111131 :
+    adaptiveCoverCheck 9 (childLH (childHH (childLH phiAboveCell1111))) = true := by
+  exact CoverCertificatea0740027fc.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells83c065d08f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells83c065d08f
+
+open CertificateCells83c065d08f
+namespace CoverCertificate069fa78a71
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate069fa78a71
+
+theorem e24KC2PhiBelowLeaf32211 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3221) = true := by
+  exact CoverCertificate069fa78a71.checkedRoot
+namespace CoverCertificate1d00d9f69c
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1d00d9f69c
+
+theorem e24KC2PhiBelowLeaf32213 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3221) = true := by
+  exact CoverCertificate1d00d9f69c.checkedRoot
+namespace CoverCertificate01063797cc
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate01063797cc
+
+theorem e24KC2PhiBelowLeaf32231 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3223) = true := by
+  exact CoverCertificate01063797cc.checkedRoot
+namespace CoverCertificatef842354b5b
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef842354b5b
+
+theorem e24KC2PhiBelowLeaf32300 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3230) = true := by
+  exact CoverCertificatef842354b5b.checkedRoot
+namespace CoverCertificate0e9db7bf30
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0e9db7bf30
+
+theorem e24KC2PhiBelowLeaf32301 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3230) = true := by
+  exact CoverCertificate0e9db7bf30.checkedRoot
+namespace CoverCertificate1dc6ae7126
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1dc6ae7126
+
+theorem e24KC2PhiBelowLeaf32302 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3230) = true := by
+  exact CoverCertificate1dc6ae7126.checkedRoot
+namespace CoverCertificate3c9114c82d
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3c9114c82d
+
+theorem e24KC2PhiBelowLeaf32303 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3230) = true := by
+  exact CoverCertificate3c9114c82d.checkedRoot
+namespace CoverCertificateec279abdfe
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateec279abdfe
+
+theorem e24KC2PhiBelowLeaf32310 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3231) = true := by
+  exact CoverCertificateec279abdfe.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9044f599c8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9044f599c8
+
+open CertificateCells9044f599c8
+namespace CoverCertificate4e21575659
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4e21575659
+
+theorem e24KC2PhiBelowLeaf32311 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3231) = true := by
+  exact CoverCertificate4e21575659.checkedRoot
+namespace CoverCertificated2df3f1b8b
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated2df3f1b8b
+
+theorem e24KC2PhiBelowLeaf32312 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3231) = true := by
+  exact CoverCertificated2df3f1b8b.checkedRoot
+namespace CoverCertificatefd72248a13
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefd72248a13
+
+theorem e24KC2PhiBelowLeaf32313 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3231) = true := by
+  exact CoverCertificatefd72248a13.checkedRoot
+namespace CoverCertificate182948158b
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate182948158b
+
+theorem e24KC2PhiBelowLeaf32320 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3232) = true := by
+  exact CoverCertificate182948158b.checkedRoot
+namespace CoverCertificate2c60bab4c6
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate2c60bab4c6
+
+theorem e24KC2PhiBelowLeaf32321 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3232) = true := by
+  exact CoverCertificate2c60bab4c6.checkedRoot
+namespace CoverCertificatefc5d468444
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefc5d468444
+
+theorem e24KC2PhiBelowLeaf32322 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3232) = true := by
+  exact CoverCertificatefc5d468444.checkedRoot
+namespace CoverCertificatee3c94e2667
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee3c94e2667
+
+theorem e24KC2PhiBelowLeaf32323 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3232) = true := by
+  exact CoverCertificatee3c94e2667.checkedRoot
+namespace CoverCertificatee175b9aa59
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee175b9aa59
+
+theorem e24KC2PhiBelowLeaf32330 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3233) = true := by
+  exact CoverCertificatee175b9aa59.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse87459d2a7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse87459d2a7
+
+open CertificateCellse87459d2a7
+namespace CoverCertificate1efa344902
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1efa344902
+
+theorem e24KC2PhiBelowLeaf32331 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3233) = true := by
+  exact CoverCertificate1efa344902.checkedRoot
+namespace CoverCertificatea932254ffe
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea932254ffe
+
+theorem e24KC2PhiBelowLeaf32332 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3233) = true := by
+  exact CoverCertificatea932254ffe.checkedRoot
+namespace CoverCertificate4b644f2af3
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4b644f2af3
+
+theorem e24KC2PhiBelowLeaf32333 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3233) = true := by
+  exact CoverCertificate4b644f2af3.checkedRoot
+namespace CoverCertificate27ea8c6704
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate27ea8c6704
+
+theorem e24KC2PhiBelowLeaf33000 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3300) = true := by
+  exact CoverCertificate27ea8c6704.checkedRoot
+namespace CoverCertificate7519fa3e77
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate7519fa3e77
+
+theorem e24KC2PhiBelowLeaf33001 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3300) = true := by
+  exact CoverCertificate7519fa3e77.checkedRoot
+namespace CoverCertificate79beaf8121
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate79beaf8121
+
+theorem e24KC2PhiBelowLeaf33002 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3300) = true := by
+  exact CoverCertificate79beaf8121.checkedRoot
+namespace CoverCertificatef1c8d80a5e
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef1c8d80a5e
+
+theorem e24KC2PhiBelowLeaf33003 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3300) = true := by
+  exact CoverCertificatef1c8d80a5e.checkedRoot
+namespace CoverCertificateffa95a2bce
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateffa95a2bce
+
+theorem e24KC2PhiBelowLeaf33010 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3301) = true := by
+  exact CoverCertificateffa95a2bce.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5c2e184d25
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5c2e184d25
+
+open CertificateCells5c2e184d25
+namespace CoverCertificatea622d1d6d8
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea622d1d6d8
+
+theorem e24KC2PhiBelowLeaf33011 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3301) = true := by
+  exact CoverCertificatea622d1d6d8.checkedRoot
+namespace CoverCertificate6808a28f96
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate6808a28f96
+
+theorem e24KC2PhiBelowLeaf33012 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3301) = true := by
+  exact CoverCertificate6808a28f96.checkedRoot
+namespace CoverCertificate64b5caf113
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate64b5caf113
+
+theorem e24KC2PhiBelowLeaf33013 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3301) = true := by
+  exact CoverCertificate64b5caf113.checkedRoot
+namespace CoverCertificate8d399165ed
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate8d399165ed
+
+theorem e24KC2PhiBelowLeaf33020 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3302) = true := by
+  exact CoverCertificate8d399165ed.checkedRoot
+namespace CoverCertificateb129667c78
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateb129667c78
+
+theorem e24KC2PhiBelowLeaf33021 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3302) = true := by
+  exact CoverCertificateb129667c78.checkedRoot
+namespace CoverCertificatee5db5e1b6d
+
+private theorem checked100 : adaptiveCoverCheck 6 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 6 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 6 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 6 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 6 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 6 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 6 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 6 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee5db5e1b6d
+
+theorem e24KC2PhiBelowLeaf33022 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3302) = true := by
+  exact CoverCertificatee5db5e1b6d.checkedRoot
+namespace CoverCertificatee9fa971b95
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee9fa971b95
+
+theorem e24KC2PhiBelowLeaf33023 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3302) = true := by
+  exact CoverCertificatee9fa971b95.checkedRoot
+namespace CoverCertificated4cb65cefa
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated4cb65cefa
+
+theorem e24KC2PhiBelowLeaf33030 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3303) = true := by
+  exact CoverCertificated4cb65cefa.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells32f77991dd
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells32f77991dd
+
+open CertificateCells32f77991dd
+namespace CoverCertificate75e4a1640f
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate75e4a1640f
+
+theorem e24KC2PhiBelowLeaf33031 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3303) = true := by
+  exact CoverCertificate75e4a1640f.checkedRoot
+namespace CoverCertificateb422ba7d7a
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateb422ba7d7a
+
+theorem e24KC2PhiBelowLeaf33032 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3303) = true := by
+  exact CoverCertificateb422ba7d7a.checkedRoot
+namespace CoverCertificate348be683b7
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate348be683b7
+
+theorem e24KC2PhiBelowLeaf33033 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3303) = true := by
+  exact CoverCertificate348be683b7.checkedRoot
+namespace CoverCertificatee69a3e135e
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee69a3e135e
+
+theorem e24KC2PhiBelowLeaf33102 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3310) = true := by
+  exact CoverCertificatee69a3e135e.checkedRoot
+namespace CoverCertificate7cfce934a3
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate7cfce934a3
+
+theorem e24KC2PhiBelowLeaf33120 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3312) = true := by
+  exact CoverCertificate7cfce934a3.checkedRoot
+namespace CoverCertificatee6e5d7b950
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee6e5d7b950
+
+theorem e24KC2PhiBelowLeaf33121 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3312) = true := by
+  exact CoverCertificatee6e5d7b950.checkedRoot
+namespace CoverCertificatedbde18308c
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatedbde18308c
+
+theorem e24KC2PhiBelowLeaf33122 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3312) = true := by
+  exact CoverCertificatedbde18308c.checkedRoot
+namespace CoverCertificate78624e9f2e
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate78624e9f2e
+
+theorem e24KC2PhiBelowLeaf33123 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3312) = true := by
+  exact CoverCertificate78624e9f2e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6c9e156424
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6c9e156424
+
+open CertificateCells6c9e156424
+namespace CoverCertificate86b7e57f5c
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate86b7e57f5c
+
+theorem e24KC2PhiBelowLeaf33132 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3313) = true := by
+  exact CoverCertificate86b7e57f5c.checkedRoot
+namespace CoverCertificate81d94c7da5
+
+private theorem checked100 : adaptiveCoverCheck 6 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 6 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 6 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 6 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 6 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 6 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 6 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 6 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate81d94c7da5
+
+theorem e24KC2PhiBelowLeaf33200 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3320) = true := by
+  exact CoverCertificate81d94c7da5.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells613b88d08f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells613b88d08f
+
+open CertificateCells613b88d08f
+namespace CoverCertificatecaeab30c1b
+
+private theorem checked000 : adaptiveCoverCheck 6 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 6 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 6 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 6 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 6 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 6 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 6 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 6 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 6 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 6 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 6 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 6 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell103 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatecaeab30c1b
+
+theorem e24KC2PhiBelowLeaf33201 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3320) = true := by
+  exact CoverCertificatecaeab30c1b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells14a7ef0a1a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells14a7ef0a1a
+
+open CertificateCells14a7ef0a1a
+namespace CoverCertificatecee0fdb8c5
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 6 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 6 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 6 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 6 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 6 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 6 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 6 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 6 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatecee0fdb8c5
+
+theorem e24KC2PhiBelowLeaf33202 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3320) = true := by
+  exact CoverCertificatecee0fdb8c5.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells756d9e6827
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells756d9e6827
+
+open CertificateCells756d9e6827
+namespace CoverCertificate92301c5291
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate92301c5291
+
+theorem e24KC2PhiBelowLeaf33210 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3321) = true := by
+  exact CoverCertificate92301c5291.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9052ca8de4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9052ca8de4
+
+open CertificateCells9052ca8de4
+namespace CoverCertificate16ba550865
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate16ba550865
+
+theorem e24KC2PhiBelowLeaf33211 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3321) = true := by
+  exact CoverCertificate16ba550865.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8466e30288
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8466e30288
+
+open CertificateCells8466e30288
+namespace CoverCertificatebb39ac0110
+
+private theorem checked2020 : adaptiveCoverCheck 5 cell2020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2020 (by decide +kernel)
+
+private theorem checked2021 : adaptiveCoverCheck 5 cell2021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2021 (by decide +kernel)
+
+private theorem checked2022 : adaptiveCoverCheck 5 cell2022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2022 (by decide +kernel)
+
+private theorem checked2023 : adaptiveCoverCheck 5 cell2023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2023 (by decide +kernel)
+
+private theorem checked2200 : adaptiveCoverCheck 5 cell2200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2200 (by decide +kernel)
+
+private theorem checked2201 : adaptiveCoverCheck 5 cell2201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2201 (by decide +kernel)
+
+private theorem checked2202 : adaptiveCoverCheck 5 cell2202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2202 (by decide +kernel)
+
+private theorem checked2203 : adaptiveCoverCheck 5 cell2203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2203 (by decide +kernel)
+
+private theorem checked2210 : adaptiveCoverCheck 5 cell2210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2210 (by decide +kernel)
+
+private theorem checked2211 : adaptiveCoverCheck 5 cell2211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2211 (by decide +kernel)
+
+private theorem checked2212 : adaptiveCoverCheck 5 cell2212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2212 (by decide +kernel)
+
+private theorem checked2213 : adaptiveCoverCheck 5 cell2213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2213 (by decide +kernel)
+
+private theorem checked2220 : adaptiveCoverCheck 5 cell2220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2220 (by decide +kernel)
+
+private theorem checked2221 : adaptiveCoverCheck 5 cell2221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2221 (by decide +kernel)
+
+private theorem checked2222 : adaptiveCoverCheck 5 cell2222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2222 (by decide +kernel)
+
+private theorem checked2223 : adaptiveCoverCheck 5 cell2223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2223 (by decide +kernel)
+
+private theorem checked2230 : adaptiveCoverCheck 5 cell2230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2230 (by decide +kernel)
+
+private theorem checked2231 : adaptiveCoverCheck 5 cell2231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2231 (by decide +kernel)
+
+private theorem checked2232 : adaptiveCoverCheck 5 cell2232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2232 (by decide +kernel)
+
+private theorem checked2233 : adaptiveCoverCheck 5 cell2233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2233 (by decide +kernel)
+
+private theorem checked2320 : adaptiveCoverCheck 5 cell2320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2320 (by decide +kernel)
+
+private theorem checked2321 : adaptiveCoverCheck 5 cell2321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2321 (by decide +kernel)
+
+private theorem checked2322 : adaptiveCoverCheck 5 cell2322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2322 (by decide +kernel)
+
+private theorem checked2323 : adaptiveCoverCheck 5 cell2323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2323 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 6 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 6 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 6 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 6 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 6 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 6 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 6 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 6 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell202
+    checked2020 checked2021 checked2022 checked2023
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell220
+    checked2200 checked2201 checked2202 checked2203
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell221
+    checked2210 checked2211 checked2212 checked2213
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell222
+    checked2220 checked2221 checked2222 checked2223
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell223
+    checked2230 checked2231 checked2232 checked2233
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell232
+    checked2320 checked2321 checked2322 checked2323
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatebb39ac0110
+
+theorem e24KC2PhiBelowLeaf33212 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3321) = true := by
+  exact CoverCertificatebb39ac0110.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells960370a0be
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells960370a0be
+
+open CertificateCells960370a0be
+namespace CoverCertificatef2ad9b0e27
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell223 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef2ad9b0e27
+
+theorem e24KC2PhiBelowLeaf33213 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3321) = true := by
+  exact CoverCertificatef2ad9b0e27.checkedRoot
+namespace CoverCertificate73093b0a81
+
+private theorem checked010 : adaptiveCoverCheck 6 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 6 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 6 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 6 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell013 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 6 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 6 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 6 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 6 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 6 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 6 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 6 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 6 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate73093b0a81
+
+theorem e24KC2PhiBelowLeaf33220 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3322) = true := by
+  exact CoverCertificate73093b0a81.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd434aeb685
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd434aeb685
+
+open CertificateCellsd434aeb685
+namespace CoverCertificate74c1ca1637
+
+private theorem checked110 : adaptiveCoverCheck 6 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 6 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 6 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 6 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell113 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate74c1ca1637
+
+theorem e24KC2PhiBelowLeaf33222 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3322) = true := by
+  exact CoverCertificate74c1ca1637.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells720ec0d149
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells720ec0d149
+
+open CertificateCells720ec0d149
+namespace CoverCertificateec9c3e1faf
+
+private theorem checked2200 : adaptiveCoverCheck 5 cell2200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2200 (by decide +kernel)
+
+private theorem checked2201 : adaptiveCoverCheck 5 cell2201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2201 (by decide +kernel)
+
+private theorem checked2202 : adaptiveCoverCheck 5 cell2202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2202 (by decide +kernel)
+
+private theorem checked2203 : adaptiveCoverCheck 5 cell2203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2203 (by decide +kernel)
+
+private theorem checked2220 : adaptiveCoverCheck 5 cell2220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2220 (by decide +kernel)
+
+private theorem checked2221 : adaptiveCoverCheck 5 cell2221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2221 (by decide +kernel)
+
+private theorem checked2222 : adaptiveCoverCheck 5 cell2222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2222 (by decide +kernel)
+
+private theorem checked2223 : adaptiveCoverCheck 5 cell2223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2223 (by decide +kernel)
+
+private theorem checked2230 : adaptiveCoverCheck 5 cell2230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2230 (by decide +kernel)
+
+private theorem checked2231 : adaptiveCoverCheck 5 cell2231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2231 (by decide +kernel)
+
+private theorem checked2232 : adaptiveCoverCheck 5 cell2232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2232 (by decide +kernel)
+
+private theorem checked2233 : adaptiveCoverCheck 5 cell2233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2233 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 6 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 6 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 6 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 6 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 6 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 6 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 6 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 6 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 6 cell220 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell220
+    checked2200 checked2201 checked2202 checked2203
+
+private theorem checked221 : adaptiveCoverCheck 6 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 6 cell222 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell222
+    checked2220 checked2221 checked2222 checked2223
+
+private theorem checked223 : adaptiveCoverCheck 6 cell223 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell223
+    checked2230 checked2231 checked2232 checked2233
+
+private theorem checked230 : adaptiveCoverCheck 6 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 6 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 6 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 6 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 6 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 6 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 6 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 6 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 6 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 6 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 6 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 6 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateec9c3e1faf
+
+theorem e24KC2PhiBelowLeaf33231 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3323) = true := by
+  exact CoverCertificateec9c3e1faf.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9005545168
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9005545168
+
+open CertificateCells9005545168
+namespace CoverCertificate144616495b
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate144616495b
+
+theorem e24KC2PhiAboveLeaf1110002 :
+    adaptiveCoverCheck 9 (childHL (childLL (childLL phiAboveCell1110))) = true := by
+  exact CoverCertificate144616495b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse586487728
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse586487728
+
+open CertificateCellse586487728
+namespace CoverCertificatef96351225f
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef96351225f
+
+theorem e24KC2PhiAboveLeaf1110003 :
+    adaptiveCoverCheck 9 (childHH (childLL (childLL phiAboveCell1110))) = true := by
+  exact CoverCertificatef96351225f.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsce87a9d03d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsce87a9d03d
+
+open CertificateCellsce87a9d03d
+namespace CoverCertificateed7fe84102
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateed7fe84102
+
+theorem e24KC2PhiAboveLeaf1110012 :
+    adaptiveCoverCheck 9 (childHL (childLH (childLL phiAboveCell1110))) = true := by
+  exact CoverCertificateed7fe84102.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsce3c5151af
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsce3c5151af
+
+open CertificateCellsce3c5151af
+namespace CoverCertificateb1ce232ebf
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateb1ce232ebf
+
+theorem e24KC2PhiAboveLeaf1110013 :
+    adaptiveCoverCheck 9 (childHH (childLH (childLL phiAboveCell1110))) = true := by
+  exact CoverCertificateb1ce232ebf.checkedRoot
+namespace CoverCertificate3e6f5d1050
+
+private theorem checked000 : adaptiveCoverCheck 6 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 6 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 6 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 6 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 6 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 6 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 6 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 6 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 6 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 6 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 6 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 6 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 6 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 6 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 6 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 6 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 6 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 6 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 6 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 6 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 6 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 6 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 6 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 6 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 6 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 6 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 6 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 6 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 6 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 6 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 6 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 6 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 6 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 6 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 6 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 6 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 6 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 6 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 6 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 6 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 6 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 6 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 6 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 6 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 6 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 6 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 6 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 6 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 7 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 7 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 7 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 7 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 7 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 7 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 7 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 7 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 7 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 7 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 7 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 7 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 7 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 7 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 7 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 7 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3e6f5d1050
+
+theorem e24KC2PhiAboveLeaf1110100 :
+    adaptiveCoverCheck 9 (childLL (childLL (childLH phiAboveCell1110))) = true := by
+  exact CoverCertificate3e6f5d1050.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells7c64a780f8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells7c64a780f8
+
+open CertificateCells7c64a780f8
+namespace CoverCertificate3c635b07e8
+
+private theorem checked0 : adaptiveCoverCheck 8 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 8 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 8 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 8 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 8 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 9 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3c635b07e8
+
+theorem e24KC2PhiBelowLeaf33300 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3330) = true := by
+  exact CoverCertificate3c635b07e8.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC5TerminalBatchT512000013`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells168aeb56e4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells168aeb56e4
+
+open CertificateCells168aeb56e4
+theorem e24KC2ThetaAboveLeaf101033002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103300) = true := by
+  have h : ((childHL thetaAboveCell10103300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103300) h
+theorem e24KC2ThetaAboveLeaf101033003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103300) = true := by
+  have h : ((childHH thetaAboveCell10103300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103300) h
+theorem e24KC2ThetaAboveLeaf101033010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103301) = true := by
+  have h : ((childLL thetaAboveCell10103301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103301) h
+theorem e24KC2ThetaAboveLeaf101033011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103301) = true := by
+  have h : ((childLH thetaAboveCell10103301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103301) h
+theorem e24KC2ThetaAboveLeaf101033012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103301) = true := by
+  have h : ((childHL thetaAboveCell10103301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103301) h
+theorem e24KC2ThetaAboveLeaf101033013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103301) = true := by
+  have h : ((childHH thetaAboveCell10103301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103301) h
+theorem e24KC2ThetaAboveLeaf1010330200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10103302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10103302)) h
+theorem e24KC2ThetaAboveLeaf1010330201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10103302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10103302)) h
+theorem e24KC2ThetaAboveLeaf1010330202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10103302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10103302)) h
+theorem e24KC2ThetaAboveLeaf1010330203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10103302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10103302)) h
+theorem e24KC2ThetaAboveLeaf101033021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103302) = true := by
+  have h : ((childLH thetaAboveCell10103302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103302) h
+theorem e24KC2ThetaAboveLeaf101033022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103302) = true := by
+  have h : ((childHL thetaAboveCell10103302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103302) h
+theorem e24KC2ThetaAboveLeaf101033023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103302) = true := by
+  have h : ((childHH thetaAboveCell10103302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103302) h
+theorem e24KC2ThetaAboveLeaf101033030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103303) = true := by
+  have h : ((childLL thetaAboveCell10103303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103303) h
+theorem e24KC2ThetaAboveLeaf101033031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103303) = true := by
+  have h : ((childLH thetaAboveCell10103303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103303) h
+theorem e24KC2ThetaAboveLeaf101033032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103303) = true := by
+  have h : ((childHL thetaAboveCell10103303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103303) h
+theorem e24KC2ThetaAboveLeaf101033033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103303) = true := by
+  have h : ((childHH thetaAboveCell10103303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103303) h
+theorem e24KC2ThetaAboveLeaf101033100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103310) = true := by
+  have h : ((childLL thetaAboveCell10103310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103310) h
+theorem e24KC2ThetaAboveLeaf101033101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103310) = true := by
+  have h : ((childLH thetaAboveCell10103310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103310) h
+theorem e24KC2ThetaAboveLeaf101033102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103310) = true := by
+  have h : ((childHL thetaAboveCell10103310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103310) h
+theorem e24KC2ThetaAboveLeaf101033103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103310) = true := by
+  have h : ((childHH thetaAboveCell10103310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103310) h
+theorem e24KC2ThetaAboveLeaf101033110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103311) = true := by
+  have h : ((childLL thetaAboveCell10103311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103311) h
+theorem e24KC2ThetaAboveLeaf101033111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103311) = true := by
+  have h : ((childLH thetaAboveCell10103311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103311) h
+theorem e24KC2ThetaAboveLeaf101033112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103311) = true := by
+  have h : ((childHL thetaAboveCell10103311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103311) h
+theorem e24KC2ThetaAboveLeaf101033113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103311) = true := by
+  have h : ((childHH thetaAboveCell10103311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103311) h
+theorem e24KC2ThetaAboveLeaf101033120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103312) = true := by
+  have h : ((childLL thetaAboveCell10103312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103312) h
+theorem e24KC2ThetaAboveLeaf101033121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103312) = true := by
+  have h : ((childLH thetaAboveCell10103312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103312) h
+theorem e24KC2ThetaAboveLeaf101033122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103312) = true := by
+  have h : ((childHL thetaAboveCell10103312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103312) h
+theorem e24KC2ThetaAboveLeaf101033123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103312) = true := by
+  have h : ((childHH thetaAboveCell10103312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103312) h
+theorem e24KC2ThetaAboveLeaf101033130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103313) = true := by
+  have h : ((childLL thetaAboveCell10103313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103313) h
+theorem e24KC2ThetaAboveLeaf101033131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103313) = true := by
+  have h : ((childLH thetaAboveCell10103313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103313) h
+theorem e24KC2ThetaAboveLeaf101033132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103313) = true := by
+  have h : ((childHL thetaAboveCell10103313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103313) h
+theorem e24KC2ThetaAboveLeaf101033133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103313) = true := by
+  have h : ((childHH thetaAboveCell10103313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103313) h
+theorem e24KC2ThetaAboveLeaf101033200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103320) = true := by
+  have h : ((childLL thetaAboveCell10103320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103320) h
+theorem e24KC2ThetaAboveLeaf101033201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103320) = true := by
+  have h : ((childLH thetaAboveCell10103320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103320) h
+theorem e24KC2ThetaAboveLeaf101033202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103320) = true := by
+  have h : ((childHL thetaAboveCell10103320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103320) h
+theorem e24KC2ThetaAboveLeaf101033203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103320) = true := by
+  have h : ((childHH thetaAboveCell10103320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103320) h
+theorem e24KC2ThetaAboveLeaf101033210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103321) = true := by
+  have h : ((childLL thetaAboveCell10103321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103321) h
+theorem e24KC2ThetaAboveLeaf101033211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103321) = true := by
+  have h : ((childLH thetaAboveCell10103321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103321) h
+theorem e24KC2ThetaAboveLeaf101033212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103321) = true := by
+  have h : ((childHL thetaAboveCell10103321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103321) h
+theorem e24KC2ThetaAboveLeaf101033213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103321) = true := by
+  have h : ((childHH thetaAboveCell10103321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103321) h
+theorem e24KC2ThetaAboveLeaf10103322 :
+    adaptiveCoverCheck 11 thetaAboveCell10103322 = true := by
+  have h : (thetaAboveCell10103322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10103322 h
+theorem e24KC2ThetaAboveLeaf10103323 :
+    adaptiveCoverCheck 11 thetaAboveCell10103323 = true := by
+  have h : (thetaAboveCell10103323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10103323 h
+theorem e24KC2ThetaAboveLeaf101033300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103330) = true := by
+  have h : ((childLL thetaAboveCell10103330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103330) h
+theorem e24KC2ThetaAboveLeaf101033301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103330) = true := by
+  have h : ((childLH thetaAboveCell10103330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103330) h
+theorem e24KC2ThetaAboveLeaf101033302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103330) = true := by
+  have h : ((childHL thetaAboveCell10103330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103330) h
+theorem e24KC2ThetaAboveLeaf101033303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103330) = true := by
+  have h : ((childHH thetaAboveCell10103330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103330) h
+theorem e24KC2ThetaAboveLeaf101033310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103331) = true := by
+  have h : ((childLL thetaAboveCell10103331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103331) h
+theorem e24KC2ThetaAboveLeaf101033311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103331) = true := by
+  have h : ((childLH thetaAboveCell10103331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103331) h
+theorem e24KC2ThetaAboveLeaf101033312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103331) = true := by
+  have h : ((childHL thetaAboveCell10103331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103331) h
+theorem e24KC2ThetaAboveLeaf101033313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103331) = true := by
+  have h : ((childHH thetaAboveCell10103331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103331) h
+theorem e24KC2ThetaAboveLeaf10103332 :
+    adaptiveCoverCheck 11 thetaAboveCell10103332 = true := by
+  have h : (thetaAboveCell10103332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10103332 h
+theorem e24KC2ThetaAboveLeaf10103333 :
+    adaptiveCoverCheck 11 thetaAboveCell10103333 = true := by
+  have h : (thetaAboveCell10103333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10103333 h
+theorem e24KC2ThetaAboveLeaf101100 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1011)) = true := by
+  have h : ((childLL (childLL thetaAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell1011)) h
+theorem e24KC2ThetaAboveLeaf101101 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1011)) = true := by
+  have h : ((childLH (childLL thetaAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell1011)) h
+theorem e24KC2ThetaAboveLeaf101102 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1011)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1011)) h
+theorem e24KC2ThetaAboveLeaf101103 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1011)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1011)) h
+theorem e24KC2ThetaAboveLeaf101110 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1011)) = true := by
+  have h : ((childLL (childLH thetaAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell1011)) h
+theorem e24KC2ThetaAboveLeaf101111 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1011)) = true := by
+  have h : ((childLH (childLH thetaAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell1011)) h
+theorem e24KC2ThetaAboveLeaf101112 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1011)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1011)) h
+theorem e24KC2ThetaAboveLeaf101113 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1011)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1011)) h
+theorem e24KC2ThetaAboveLeaf1011200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell1011))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf1011201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell1011))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf1011202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell1011))) = true := by
+  have h : ((childHL (childLL (childHL thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childHL thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf1011203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell1011))) = true := by
+  have h : ((childHH (childLL (childHL thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childHL thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf1011210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell1011))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf1011211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell1011))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf1011212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell1011))) = true := by
+  have h : ((childHL (childLH (childHL thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childHL thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf1011213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell1011))) = true := by
+  have h : ((childHH (childLH (childHL thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childHL thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf101122000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112200) = true := by
+  have h : ((childLL thetaAboveCell10112200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112200) h
+theorem e24KC2ThetaAboveLeaf101122001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112200) = true := by
+  have h : ((childLH thetaAboveCell10112200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112200) h
+theorem e24KC2ThetaAboveLeaf101122002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112200) = true := by
+  have h : ((childHL thetaAboveCell10112200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112200) h
+theorem e24KC2ThetaAboveLeaf101122003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112200) = true := by
+  have h : ((childHH thetaAboveCell10112200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112200) h
+theorem e24KC2ThetaAboveLeaf101122010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112201) = true := by
+  have h : ((childLL thetaAboveCell10112201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112201) h
+theorem e24KC2ThetaAboveLeaf101122011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112201) = true := by
+  have h : ((childLH thetaAboveCell10112201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112201) h
+theorem e24KC2ThetaAboveLeaf101122012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112201) = true := by
+  have h : ((childHL thetaAboveCell10112201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112201) h
+theorem e24KC2ThetaAboveLeaf101122013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112201) = true := by
+  have h : ((childHH thetaAboveCell10112201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112201) h
+theorem e24KC2ThetaAboveLeaf101122020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112202) = true := by
+  have h : ((childLL thetaAboveCell10112202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112202) h
+theorem e24KC2ThetaAboveLeaf101122021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112202) = true := by
+  have h : ((childLH thetaAboveCell10112202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112202) h
+theorem e24KC2ThetaAboveLeaf101122022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112202) = true := by
+  have h : ((childHL thetaAboveCell10112202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112202) h
+theorem e24KC2ThetaAboveLeaf101122023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112202) = true := by
+  have h : ((childHH thetaAboveCell10112202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112202) h
+theorem e24KC2ThetaAboveLeaf101122030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112203) = true := by
+  have h : ((childLL thetaAboveCell10112203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112203) h
+theorem e24KC2ThetaAboveLeaf101122031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112203) = true := by
+  have h : ((childLH thetaAboveCell10112203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112203) h
+theorem e24KC2ThetaAboveLeaf101122032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112203) = true := by
+  have h : ((childHL thetaAboveCell10112203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112203) h
+theorem e24KC2ThetaAboveLeaf101122033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112203) = true := by
+  have h : ((childHH thetaAboveCell10112203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112203) h
+theorem e24KC2ThetaAboveLeaf101122100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112210) = true := by
+  have h : ((childLL thetaAboveCell10112210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112210) h
+theorem e24KC2ThetaAboveLeaf101122101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112210) = true := by
+  have h : ((childLH thetaAboveCell10112210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112210) h
+theorem e24KC2ThetaAboveLeaf101122102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112210) = true := by
+  have h : ((childHL thetaAboveCell10112210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112210) h
+theorem e24KC2ThetaAboveLeaf101122103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112210) = true := by
+  have h : ((childHH thetaAboveCell10112210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112210) h
+theorem e24KC2ThetaAboveLeaf101122110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112211) = true := by
+  have h : ((childLL thetaAboveCell10112211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112211) h
+theorem e24KC2ThetaAboveLeaf101122111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112211) = true := by
+  have h : ((childLH thetaAboveCell10112211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112211) h
+theorem e24KC2ThetaAboveLeaf101122112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112211) = true := by
+  have h : ((childHL thetaAboveCell10112211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112211) h
+theorem e24KC2ThetaAboveLeaf101122113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112211) = true := by
+  have h : ((childHH thetaAboveCell10112211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112211) h
+theorem e24KC2ThetaAboveLeaf101122120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112212) = true := by
+  have h : ((childLL thetaAboveCell10112212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112212) h
+theorem e24KC2ThetaAboveLeaf101122121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112212) = true := by
+  have h : ((childLH thetaAboveCell10112212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112212) h
+theorem e24KC2ThetaAboveLeaf101122122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112212) = true := by
+  have h : ((childHL thetaAboveCell10112212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112212) h
+theorem e24KC2ThetaAboveLeaf101122123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112212) = true := by
+  have h : ((childHH thetaAboveCell10112212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112212) h
+theorem e24KC2ThetaAboveLeaf101122130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112213) = true := by
+  have h : ((childLL thetaAboveCell10112213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112213) h
+theorem e24KC2ThetaAboveLeaf101122131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112213) = true := by
+  have h : ((childLH thetaAboveCell10112213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112213) h
+theorem e24KC2ThetaAboveLeaf101122132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112213) = true := by
+  have h : ((childHL thetaAboveCell10112213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112213) h
+theorem e24KC2ThetaAboveLeaf101122133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112213) = true := by
+  have h : ((childHH thetaAboveCell10112213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112213) h
+theorem e24KC2ThetaAboveLeaf101122200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112220) = true := by
+  have h : ((childLL thetaAboveCell10112220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112220) h
+theorem e24KC2ThetaAboveLeaf101122201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112220) = true := by
+  have h : ((childLH thetaAboveCell10112220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112220) h
+theorem e24KC2ThetaAboveLeaf101122202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112220) = true := by
+  have h : ((childHL thetaAboveCell10112220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112220) h
+theorem e24KC2ThetaAboveLeaf101122203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112220) = true := by
+  have h : ((childHH thetaAboveCell10112220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112220) h
+theorem e24KC2ThetaAboveLeaf101122210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112221) = true := by
+  have h : ((childLL thetaAboveCell10112221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112221) h
+theorem e24KC2ThetaAboveLeaf101122211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112221) = true := by
+  have h : ((childLH thetaAboveCell10112221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112221) h
+theorem e24KC2ThetaAboveLeaf101122212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112221) = true := by
+  have h : ((childHL thetaAboveCell10112221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112221) h
+theorem e24KC2ThetaAboveLeaf101122213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112221) = true := by
+  have h : ((childHH thetaAboveCell10112221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112221) h
+theorem e24KC2ThetaAboveLeaf10112222 :
+    adaptiveCoverCheck 11 thetaAboveCell10112222 = true := by
+  have h : (thetaAboveCell10112222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10112222 h
+theorem e24KC2ThetaAboveLeaf10112223 :
+    adaptiveCoverCheck 11 thetaAboveCell10112223 = true := by
+  have h : (thetaAboveCell10112223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10112223 h
+theorem e24KC2ThetaAboveLeaf101122300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112230) = true := by
+  have h : ((childLL thetaAboveCell10112230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112230) h
+theorem e24KC2ThetaAboveLeaf101122301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112230) = true := by
+  have h : ((childLH thetaAboveCell10112230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112230) h
+theorem e24KC2ThetaAboveLeaf101122302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112230) = true := by
+  have h : ((childHL thetaAboveCell10112230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112230) h
+theorem e24KC2ThetaAboveLeaf101122303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112230) = true := by
+  have h : ((childHH thetaAboveCell10112230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112230) h
+theorem e24KC2ThetaAboveLeaf101122310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112231) = true := by
+  have h : ((childLL thetaAboveCell10112231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112231) h
+theorem e24KC2ThetaAboveLeaf101122311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112231) = true := by
+  have h : ((childLH thetaAboveCell10112231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112231) h
+theorem e24KC2ThetaAboveLeaf101122312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112231) = true := by
+  have h : ((childHL thetaAboveCell10112231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112231) h
+theorem e24KC2ThetaAboveLeaf101122313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112231) = true := by
+  have h : ((childHH thetaAboveCell10112231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112231) h
+theorem e24KC2ThetaAboveLeaf10112232 :
+    adaptiveCoverCheck 11 thetaAboveCell10112232 = true := by
+  have h : (thetaAboveCell10112232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10112232 h
+theorem e24KC2ThetaAboveLeaf10112233 :
+    adaptiveCoverCheck 11 thetaAboveCell10112233 = true := by
+  have h : (thetaAboveCell10112233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10112233 h
+theorem e24KC2ThetaAboveLeaf101123000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112300) = true := by
+  have h : ((childLL thetaAboveCell10112300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112300) h
+theorem e24KC2ThetaAboveLeaf101123001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112300) = true := by
+  have h : ((childLH thetaAboveCell10112300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112300) h
+theorem e24KC2ThetaAboveLeaf101123002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112300) = true := by
+  have h : ((childHL thetaAboveCell10112300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112300) h
+theorem e24KC2ThetaAboveLeaf101123003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112300) = true := by
+  have h : ((childHH thetaAboveCell10112300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112300) h
+theorem e24KC2ThetaAboveLeaf101123010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112301) = true := by
+  have h : ((childLL thetaAboveCell10112301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112301) h
+theorem e24KC2ThetaAboveLeaf101123011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112301) = true := by
+  have h : ((childLH thetaAboveCell10112301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112301) h
+theorem e24KC2ThetaAboveLeaf101123012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112301) = true := by
+  have h : ((childHL thetaAboveCell10112301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112301) h
+theorem e24KC2ThetaAboveLeaf101123013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112301) = true := by
+  have h : ((childHH thetaAboveCell10112301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112301) h
+theorem e24KC2ThetaAboveLeaf101123020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112302) = true := by
+  have h : ((childLL thetaAboveCell10112302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112302) h
+theorem e24KC2ThetaAboveLeaf101123021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112302) = true := by
+  have h : ((childLH thetaAboveCell10112302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112302) h
+theorem e24KC2ThetaAboveLeaf101123022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112302) = true := by
+  have h : ((childHL thetaAboveCell10112302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112302) h
+theorem e24KC2ThetaAboveLeaf101123023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112302) = true := by
+  have h : ((childHH thetaAboveCell10112302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112302) h
+theorem e24KC2ThetaAboveLeaf101123030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112303) = true := by
+  have h : ((childLL thetaAboveCell10112303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112303) h
+theorem e24KC2ThetaAboveLeaf101123031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112303) = true := by
+  have h : ((childLH thetaAboveCell10112303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112303) h
+theorem e24KC2ThetaAboveLeaf101123032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112303) = true := by
+  have h : ((childHL thetaAboveCell10112303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112303) h
+theorem e24KC2ThetaAboveLeaf101123033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112303) = true := by
+  have h : ((childHH thetaAboveCell10112303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112303) h
+theorem e24KC2ThetaAboveLeaf101123100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112310) = true := by
+  have h : ((childLL thetaAboveCell10112310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112310) h
+theorem e24KC2ThetaAboveLeaf101123101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112310) = true := by
+  have h : ((childLH thetaAboveCell10112310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112310) h
+theorem e24KC2ThetaAboveLeaf101123102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112310) = true := by
+  have h : ((childHL thetaAboveCell10112310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112310) h
+theorem e24KC2ThetaAboveLeaf101123103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112310) = true := by
+  have h : ((childHH thetaAboveCell10112310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112310) h
+theorem e24KC2ThetaAboveLeaf101123110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112311) = true := by
+  have h : ((childLL thetaAboveCell10112311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112311) h
+theorem e24KC2ThetaAboveLeaf101123111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112311) = true := by
+  have h : ((childLH thetaAboveCell10112311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112311) h
+theorem e24KC2ThetaAboveLeaf101123112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112311) = true := by
+  have h : ((childHL thetaAboveCell10112311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112311) h
+theorem e24KC2ThetaAboveLeaf101123113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112311) = true := by
+  have h : ((childHH thetaAboveCell10112311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112311) h
+theorem e24KC2ThetaAboveLeaf101123120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112312) = true := by
+  have h : ((childLL thetaAboveCell10112312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112312) h
+theorem e24KC2ThetaAboveLeaf101123121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112312) = true := by
+  have h : ((childLH thetaAboveCell10112312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112312) h
+theorem e24KC2ThetaAboveLeaf101123122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112312) = true := by
+  have h : ((childHL thetaAboveCell10112312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112312) h
+theorem e24KC2ThetaAboveLeaf101123123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112312) = true := by
+  have h : ((childHH thetaAboveCell10112312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112312) h
+theorem e24KC2ThetaAboveLeaf101123130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112313) = true := by
+  have h : ((childLL thetaAboveCell10112313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112313) h
+theorem e24KC2ThetaAboveLeaf101123131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112313) = true := by
+  have h : ((childLH thetaAboveCell10112313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112313) h
+theorem e24KC2ThetaAboveLeaf101123132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112313) = true := by
+  have h : ((childHL thetaAboveCell10112313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112313) h
+theorem e24KC2ThetaAboveLeaf101123133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112313) = true := by
+  have h : ((childHH thetaAboveCell10112313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112313) h
+theorem e24KC2ThetaAboveLeaf101123200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112320) = true := by
+  have h : ((childLL thetaAboveCell10112320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112320) h
+theorem e24KC2ThetaAboveLeaf101123201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112320) = true := by
+  have h : ((childLH thetaAboveCell10112320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112320) h
+theorem e24KC2ThetaAboveLeaf101123202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112320) = true := by
+  have h : ((childHL thetaAboveCell10112320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112320) h
+theorem e24KC2ThetaAboveLeaf101123203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112320) = true := by
+  have h : ((childHH thetaAboveCell10112320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112320) h
+theorem e24KC2ThetaAboveLeaf101123210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112321) = true := by
+  have h : ((childLL thetaAboveCell10112321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112321) h
+theorem e24KC2ThetaAboveLeaf101123211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112321) = true := by
+  have h : ((childLH thetaAboveCell10112321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112321) h
+theorem e24KC2ThetaAboveLeaf101123212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112321) = true := by
+  have h : ((childHL thetaAboveCell10112321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112321) h
+theorem e24KC2ThetaAboveLeaf101123213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112321) = true := by
+  have h : ((childHH thetaAboveCell10112321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112321) h
+theorem e24KC2ThetaAboveLeaf10112322 :
+    adaptiveCoverCheck 11 thetaAboveCell10112322 = true := by
+  have h : (thetaAboveCell10112322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10112322 h
+theorem e24KC2ThetaAboveLeaf10112323 :
+    adaptiveCoverCheck 11 thetaAboveCell10112323 = true := by
+  have h : (thetaAboveCell10112323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10112323 h
+theorem e24KC2ThetaAboveLeaf101123300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112330) = true := by
+  have h : ((childLL thetaAboveCell10112330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112330) h
+theorem e24KC2ThetaAboveLeaf101123301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112330) = true := by
+  have h : ((childLH thetaAboveCell10112330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112330) h
+theorem e24KC2ThetaAboveLeaf101123302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112330) = true := by
+  have h : ((childHL thetaAboveCell10112330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112330) h
+theorem e24KC2ThetaAboveLeaf101123303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112330) = true := by
+  have h : ((childHH thetaAboveCell10112330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112330) h
+theorem e24KC2ThetaAboveLeaf101123310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10112331) = true := by
+  have h : ((childLL thetaAboveCell10112331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10112331) h
+theorem e24KC2ThetaAboveLeaf101123311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10112331) = true := by
+  have h : ((childLH thetaAboveCell10112331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10112331) h
+theorem e24KC2ThetaAboveLeaf101123312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10112331) = true := by
+  have h : ((childHL thetaAboveCell10112331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10112331) h
+theorem e24KC2ThetaAboveLeaf101123313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10112331) = true := by
+  have h : ((childHH thetaAboveCell10112331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10112331) h
+theorem e24KC2ThetaAboveLeaf10112332 :
+    adaptiveCoverCheck 11 thetaAboveCell10112332 = true := by
+  have h : (thetaAboveCell10112332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10112332 h
+theorem e24KC2ThetaAboveLeaf10112333 :
+    adaptiveCoverCheck 11 thetaAboveCell10112333 = true := by
+  have h : (thetaAboveCell10112333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10112333 h
+theorem e24KC2ThetaAboveLeaf1011300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell1011))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf1011301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell1011))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf1011302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell1011))) = true := by
+  have h : ((childHL (childLL (childHH thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childHH thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf1011303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell1011))) = true := by
+  have h : ((childHH (childLL (childHH thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childHH thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf1011310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell1011))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf1011311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell1011))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf1011312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell1011))) = true := by
+  have h : ((childHL (childLH (childHH thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childHH thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf1011313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell1011))) = true := by
+  have h : ((childHH (childLH (childHH thetaAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childHH thetaAboveCell1011))) h
+theorem e24KC2ThetaAboveLeaf101132000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113200) = true := by
+  have h : ((childLL thetaAboveCell10113200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113200) h
+theorem e24KC2ThetaAboveLeaf101132001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113200) = true := by
+  have h : ((childLH thetaAboveCell10113200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113200) h
+theorem e24KC2ThetaAboveLeaf101132002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113200) = true := by
+  have h : ((childHL thetaAboveCell10113200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113200) h
+theorem e24KC2ThetaAboveLeaf101132003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113200) = true := by
+  have h : ((childHH thetaAboveCell10113200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113200) h
+theorem e24KC2ThetaAboveLeaf101132010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113201) = true := by
+  have h : ((childLL thetaAboveCell10113201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113201) h
+theorem e24KC2ThetaAboveLeaf101132011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113201) = true := by
+  have h : ((childLH thetaAboveCell10113201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113201) h
+theorem e24KC2ThetaAboveLeaf101132012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113201) = true := by
+  have h : ((childHL thetaAboveCell10113201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113201) h
+theorem e24KC2ThetaAboveLeaf101132013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113201) = true := by
+  have h : ((childHH thetaAboveCell10113201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113201) h
+theorem e24KC2ThetaAboveLeaf101132020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113202) = true := by
+  have h : ((childLL thetaAboveCell10113202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113202) h
+theorem e24KC2ThetaAboveLeaf101132021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113202) = true := by
+  have h : ((childLH thetaAboveCell10113202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113202) h
+theorem e24KC2ThetaAboveLeaf101132022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113202) = true := by
+  have h : ((childHL thetaAboveCell10113202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113202) h
+theorem e24KC2ThetaAboveLeaf101132023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113202) = true := by
+  have h : ((childHH thetaAboveCell10113202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113202) h
+theorem e24KC2ThetaAboveLeaf101132030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113203) = true := by
+  have h : ((childLL thetaAboveCell10113203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113203) h
+theorem e24KC2ThetaAboveLeaf101132031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113203) = true := by
+  have h : ((childLH thetaAboveCell10113203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113203) h
+theorem e24KC2ThetaAboveLeaf101132032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113203) = true := by
+  have h : ((childHL thetaAboveCell10113203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113203) h
+theorem e24KC2ThetaAboveLeaf101132033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113203) = true := by
+  have h : ((childHH thetaAboveCell10113203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113203) h
+theorem e24KC2ThetaAboveLeaf101132100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113210) = true := by
+  have h : ((childLL thetaAboveCell10113210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113210) h
+theorem e24KC2ThetaAboveLeaf101132101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113210) = true := by
+  have h : ((childLH thetaAboveCell10113210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113210) h
+theorem e24KC2ThetaAboveLeaf101132102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113210) = true := by
+  have h : ((childHL thetaAboveCell10113210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113210) h
+theorem e24KC2ThetaAboveLeaf101132103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113210) = true := by
+  have h : ((childHH thetaAboveCell10113210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113210) h
+theorem e24KC2ThetaAboveLeaf101132110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113211) = true := by
+  have h : ((childLL thetaAboveCell10113211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113211) h
+theorem e24KC2ThetaAboveLeaf101132111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113211) = true := by
+  have h : ((childLH thetaAboveCell10113211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113211) h
+theorem e24KC2ThetaAboveLeaf101132112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113211) = true := by
+  have h : ((childHL thetaAboveCell10113211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113211) h
+theorem e24KC2ThetaAboveLeaf101132113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113211) = true := by
+  have h : ((childHH thetaAboveCell10113211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113211) h
+theorem e24KC2ThetaAboveLeaf101132120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113212) = true := by
+  have h : ((childLL thetaAboveCell10113212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113212) h
+theorem e24KC2ThetaAboveLeaf101132121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113212) = true := by
+  have h : ((childLH thetaAboveCell10113212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113212) h
+theorem e24KC2ThetaAboveLeaf101132122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113212) = true := by
+  have h : ((childHL thetaAboveCell10113212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113212) h
+theorem e24KC2ThetaAboveLeaf101132123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113212) = true := by
+  have h : ((childHH thetaAboveCell10113212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113212) h
+theorem e24KC2ThetaAboveLeaf101132130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113213) = true := by
+  have h : ((childLL thetaAboveCell10113213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113213) h
+theorem e24KC2ThetaAboveLeaf101132131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113213) = true := by
+  have h : ((childLH thetaAboveCell10113213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113213) h
+theorem e24KC2ThetaAboveLeaf101132132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113213) = true := by
+  have h : ((childHL thetaAboveCell10113213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113213) h
+theorem e24KC2ThetaAboveLeaf101132133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113213) = true := by
+  have h : ((childHH thetaAboveCell10113213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113213) h
+theorem e24KC2ThetaAboveLeaf101132200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113220) = true := by
+  have h : ((childLL thetaAboveCell10113220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113220) h
+theorem e24KC2ThetaAboveLeaf101132201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113220) = true := by
+  have h : ((childLH thetaAboveCell10113220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113220) h
+theorem e24KC2ThetaAboveLeaf101132202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113220) = true := by
+  have h : ((childHL thetaAboveCell10113220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113220) h
+theorem e24KC2ThetaAboveLeaf101132203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113220) = true := by
+  have h : ((childHH thetaAboveCell10113220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113220) h
+theorem e24KC2ThetaAboveLeaf101132210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113221) = true := by
+  have h : ((childLL thetaAboveCell10113221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113221) h
+theorem e24KC2ThetaAboveLeaf101132211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113221) = true := by
+  have h : ((childLH thetaAboveCell10113221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113221) h
+theorem e24KC2ThetaAboveLeaf101132212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113221) = true := by
+  have h : ((childHL thetaAboveCell10113221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113221) h
+theorem e24KC2ThetaAboveLeaf101132213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113221) = true := by
+  have h : ((childHH thetaAboveCell10113221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113221) h
+theorem e24KC2ThetaAboveLeaf10113222 :
+    adaptiveCoverCheck 11 thetaAboveCell10113222 = true := by
+  have h : (thetaAboveCell10113222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10113222 h
+theorem e24KC2ThetaAboveLeaf10113223 :
+    adaptiveCoverCheck 11 thetaAboveCell10113223 = true := by
+  have h : (thetaAboveCell10113223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10113223 h
+theorem e24KC2ThetaAboveLeaf101132300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113230) = true := by
+  have h : ((childLL thetaAboveCell10113230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113230) h
+theorem e24KC2ThetaAboveLeaf101132301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113230) = true := by
+  have h : ((childLH thetaAboveCell10113230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113230) h
+theorem e24KC2ThetaAboveLeaf101132302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113230) = true := by
+  have h : ((childHL thetaAboveCell10113230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113230) h
+theorem e24KC2ThetaAboveLeaf101132303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113230) = true := by
+  have h : ((childHH thetaAboveCell10113230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113230) h
+theorem e24KC2ThetaAboveLeaf101132310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113231) = true := by
+  have h : ((childLL thetaAboveCell10113231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113231) h
+theorem e24KC2ThetaAboveLeaf101132311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113231) = true := by
+  have h : ((childLH thetaAboveCell10113231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113231) h
+theorem e24KC2ThetaAboveLeaf101132312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113231) = true := by
+  have h : ((childHL thetaAboveCell10113231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113231) h
+theorem e24KC2ThetaAboveLeaf101132313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113231) = true := by
+  have h : ((childHH thetaAboveCell10113231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113231) h
+theorem e24KC2ThetaAboveLeaf10113232 :
+    adaptiveCoverCheck 11 thetaAboveCell10113232 = true := by
+  have h : (thetaAboveCell10113232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10113232 h
+theorem e24KC2ThetaAboveLeaf10113233 :
+    adaptiveCoverCheck 11 thetaAboveCell10113233 = true := by
+  have h : (thetaAboveCell10113233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10113233 h
+theorem e24KC2ThetaAboveLeaf101133000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113300) = true := by
+  have h : ((childLL thetaAboveCell10113300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113300) h
+theorem e24KC2ThetaAboveLeaf101133001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113300) = true := by
+  have h : ((childLH thetaAboveCell10113300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113300) h
+theorem e24KC2ThetaAboveLeaf101133002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113300) = true := by
+  have h : ((childHL thetaAboveCell10113300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113300) h
+theorem e24KC2ThetaAboveLeaf101133003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113300) = true := by
+  have h : ((childHH thetaAboveCell10113300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113300) h
+theorem e24KC2ThetaAboveLeaf101133010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113301) = true := by
+  have h : ((childLL thetaAboveCell10113301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113301) h
+theorem e24KC2ThetaAboveLeaf101133011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113301) = true := by
+  have h : ((childLH thetaAboveCell10113301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113301) h
+theorem e24KC2ThetaAboveLeaf101133012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113301) = true := by
+  have h : ((childHL thetaAboveCell10113301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113301) h
+theorem e24KC2ThetaAboveLeaf101133013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113301) = true := by
+  have h : ((childHH thetaAboveCell10113301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113301) h
+theorem e24KC2ThetaAboveLeaf101133020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113302) = true := by
+  have h : ((childLL thetaAboveCell10113302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113302) h
+theorem e24KC2ThetaAboveLeaf101133021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113302) = true := by
+  have h : ((childLH thetaAboveCell10113302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113302) h
+theorem e24KC2ThetaAboveLeaf101133022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113302) = true := by
+  have h : ((childHL thetaAboveCell10113302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113302) h
+theorem e24KC2ThetaAboveLeaf101133023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113302) = true := by
+  have h : ((childHH thetaAboveCell10113302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113302) h
+theorem e24KC2ThetaAboveLeaf101133030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113303) = true := by
+  have h : ((childLL thetaAboveCell10113303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113303) h
+theorem e24KC2ThetaAboveLeaf101133031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113303) = true := by
+  have h : ((childLH thetaAboveCell10113303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113303) h
+theorem e24KC2ThetaAboveLeaf101133032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113303) = true := by
+  have h : ((childHL thetaAboveCell10113303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113303) h
+theorem e24KC2ThetaAboveLeaf101133033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113303) = true := by
+  have h : ((childHH thetaAboveCell10113303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113303) h
+theorem e24KC2ThetaAboveLeaf101133100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113310) = true := by
+  have h : ((childLL thetaAboveCell10113310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113310) h
+theorem e24KC2ThetaAboveLeaf101133101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113310) = true := by
+  have h : ((childLH thetaAboveCell10113310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113310) h
+theorem e24KC2ThetaAboveLeaf101133102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113310) = true := by
+  have h : ((childHL thetaAboveCell10113310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113310) h
+theorem e24KC2ThetaAboveLeaf101133103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113310) = true := by
+  have h : ((childHH thetaAboveCell10113310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113310) h
+theorem e24KC2ThetaAboveLeaf101133110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113311) = true := by
+  have h : ((childLL thetaAboveCell10113311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113311) h
+theorem e24KC2ThetaAboveLeaf101133111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113311) = true := by
+  have h : ((childLH thetaAboveCell10113311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113311) h
+theorem e24KC2ThetaAboveLeaf101133112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113311) = true := by
+  have h : ((childHL thetaAboveCell10113311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113311) h
+theorem e24KC2ThetaAboveLeaf101133113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113311) = true := by
+  have h : ((childHH thetaAboveCell10113311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113311) h
+theorem e24KC2ThetaAboveLeaf101133120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113312) = true := by
+  have h : ((childLL thetaAboveCell10113312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113312) h
+theorem e24KC2ThetaAboveLeaf101133121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113312) = true := by
+  have h : ((childLH thetaAboveCell10113312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113312) h
+theorem e24KC2ThetaAboveLeaf101133122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113312) = true := by
+  have h : ((childHL thetaAboveCell10113312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113312) h
+theorem e24KC2ThetaAboveLeaf101133123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113312) = true := by
+  have h : ((childHH thetaAboveCell10113312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113312) h
+theorem e24KC2ThetaAboveLeaf101133130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113313) = true := by
+  have h : ((childLL thetaAboveCell10113313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113313) h
+theorem e24KC2ThetaAboveLeaf101133131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113313) = true := by
+  have h : ((childLH thetaAboveCell10113313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113313) h
+theorem e24KC2ThetaAboveLeaf101133132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113313) = true := by
+  have h : ((childHL thetaAboveCell10113313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113313) h
+theorem e24KC2ThetaAboveLeaf101133133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113313) = true := by
+  have h : ((childHH thetaAboveCell10113313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113313) h
+theorem e24KC2ThetaAboveLeaf101133200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113320) = true := by
+  have h : ((childLL thetaAboveCell10113320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113320) h
+theorem e24KC2ThetaAboveLeaf101133201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113320) = true := by
+  have h : ((childLH thetaAboveCell10113320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113320) h
+theorem e24KC2ThetaAboveLeaf101133202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113320) = true := by
+  have h : ((childHL thetaAboveCell10113320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113320) h
+theorem e24KC2ThetaAboveLeaf101133203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113320) = true := by
+  have h : ((childHH thetaAboveCell10113320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113320) h
+theorem e24KC2ThetaAboveLeaf101133210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113321) = true := by
+  have h : ((childLL thetaAboveCell10113321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113321) h
+theorem e24KC2ThetaAboveLeaf101133211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113321) = true := by
+  have h : ((childLH thetaAboveCell10113321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113321) h
+theorem e24KC2ThetaAboveLeaf101133212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113321) = true := by
+  have h : ((childHL thetaAboveCell10113321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113321) h
+theorem e24KC2ThetaAboveLeaf101133213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113321) = true := by
+  have h : ((childHH thetaAboveCell10113321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113321) h
+theorem e24KC2ThetaAboveLeaf10113322 :
+    adaptiveCoverCheck 11 thetaAboveCell10113322 = true := by
+  have h : (thetaAboveCell10113322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10113322 h
+theorem e24KC2ThetaAboveLeaf10113323 :
+    adaptiveCoverCheck 11 thetaAboveCell10113323 = true := by
+  have h : (thetaAboveCell10113323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10113323 h
+theorem e24KC2ThetaAboveLeaf101133300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113330) = true := by
+  have h : ((childLL thetaAboveCell10113330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113330) h
+theorem e24KC2ThetaAboveLeaf101133301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113330) = true := by
+  have h : ((childLH thetaAboveCell10113330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113330) h
+theorem e24KC2ThetaAboveLeaf101133302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113330) = true := by
+  have h : ((childHL thetaAboveCell10113330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113330) h
+theorem e24KC2ThetaAboveLeaf101133303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113330) = true := by
+  have h : ((childHH thetaAboveCell10113330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113330) h
+theorem e24KC2ThetaAboveLeaf101133310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10113331) = true := by
+  have h : ((childLL thetaAboveCell10113331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10113331) h
+theorem e24KC2ThetaAboveLeaf101133311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10113331) = true := by
+  have h : ((childLH thetaAboveCell10113331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10113331) h
+theorem e24KC2ThetaAboveLeaf101133312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10113331) = true := by
+  have h : ((childHL thetaAboveCell10113331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10113331) h
+theorem e24KC2ThetaAboveLeaf101133313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10113331) = true := by
+  have h : ((childHH thetaAboveCell10113331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10113331) h
+theorem e24KC2ThetaAboveLeaf10113332 :
+    adaptiveCoverCheck 11 thetaAboveCell10113332 = true := by
+  have h : (thetaAboveCell10113332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10113332 h
+theorem e24KC2ThetaAboveLeaf10113333 :
+    adaptiveCoverCheck 11 thetaAboveCell10113333 = true := by
+  have h : (thetaAboveCell10113333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10113333 h
+theorem e24KC2ThetaAboveLeaf1012000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell1012))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf1012001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell1012))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf1012002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell1012))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf1012003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell1012))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf1012010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell1012))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf1012011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell1012))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf1012012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell1012))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf1012013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell1012))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf101202 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1012)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1012)) h
+theorem e24KC2ThetaAboveLeaf101203 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1012)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1012)) h
+theorem e24KC2ThetaAboveLeaf1012100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell1012))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf1012101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell1012))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf1012102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell1012))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf1012103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell1012))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf1012110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell1012))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf1012111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell1012))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf1012112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell1012))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf1012113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell1012))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell1012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell1012))) h
+theorem e24KC2ThetaAboveLeaf101212 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1012)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1012)) h
+theorem e24KC2ThetaAboveLeaf101213 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1012)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1012)) h
+theorem e24KC2ThetaAboveLeaf10122 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1012) = true := by
+  have h : ((childHL thetaAboveCell1012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell1012) h
+theorem e24KC2ThetaAboveLeaf10123 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1012) = true := by
+  have h : ((childHH thetaAboveCell1012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell1012) h
+theorem e24KC2ThetaAboveLeaf1013000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell1013))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf1013001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell1013))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf1013002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell1013))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf1013003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell1013))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf1013010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell1013))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf1013011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell1013))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf1013012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell1013))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf1013013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell1013))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf101302 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1013)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1013)) h
+theorem e24KC2ThetaAboveLeaf101303 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1013)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1013)) h
+theorem e24KC2ThetaAboveLeaf1013100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell1013))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf1013101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell1013))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf1013102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell1013))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf1013103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell1013))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf1013110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell1013))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf1013111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell1013))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf1013112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell1013))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf1013113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell1013))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell1013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell1013))) h
+theorem e24KC2ThetaAboveLeaf101312 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1013)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1013)) h
+theorem e24KC2ThetaAboveLeaf101313 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1013)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1013)) h
+theorem e24KC2ThetaAboveLeaf10132 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1013) = true := by
+  have h : ((childHL thetaAboveCell1013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell1013) h
+theorem e24KC2ThetaAboveLeaf10133 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1013) = true := by
+  have h : ((childHH thetaAboveCell1013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell1013) h
+theorem e24KC2ThetaAboveLeaf1020 :
+    adaptiveCoverCheck 15 thetaAboveCell1020 = true := by
+  have h : (thetaAboveCell1020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1020 h
+theorem e24KC2ThetaAboveLeaf1021 :
+    adaptiveCoverCheck 15 thetaAboveCell1021 = true := by
+  have h : (thetaAboveCell1021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1021 h
+theorem e24KC2ThetaAboveLeaf1022 :
+    adaptiveCoverCheck 15 thetaAboveCell1022 = true := by
+  have h : (thetaAboveCell1022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1022 h
+theorem e24KC2ThetaAboveLeaf1023 :
+    adaptiveCoverCheck 15 thetaAboveCell1023 = true := by
+  have h : (thetaAboveCell1023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1023 h
+theorem e24KC2ThetaAboveLeaf1030 :
+    adaptiveCoverCheck 15 thetaAboveCell1030 = true := by
+  have h : (thetaAboveCell1030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1030 h
+theorem e24KC2ThetaAboveLeaf1031 :
+    adaptiveCoverCheck 15 thetaAboveCell1031 = true := by
+  have h : (thetaAboveCell1031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1031 h
+theorem e24KC2ThetaAboveLeaf1032 :
+    adaptiveCoverCheck 15 thetaAboveCell1032 = true := by
+  have h : (thetaAboveCell1032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1032 h
+theorem e24KC2ThetaAboveLeaf1033 :
+    adaptiveCoverCheck 15 thetaAboveCell1033 = true := by
+  have h : (thetaAboveCell1033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1033 h
+theorem e24KC2ThetaAboveLeaf110000 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1100)) = true := by
+  have h : ((childLL (childLL thetaAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell1100)) h
+theorem e24KC2ThetaAboveLeaf110001 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1100)) = true := by
+  have h : ((childLH (childLL thetaAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell1100)) h
+theorem e24KC2ThetaAboveLeaf110002 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1100)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1100)) h
+theorem e24KC2ThetaAboveLeaf110003 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1100)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1100)) h
+theorem e24KC2ThetaAboveLeaf110010 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1100)) = true := by
+  have h : ((childLL (childLH thetaAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell1100)) h
+theorem e24KC2ThetaAboveLeaf110011 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1100)) = true := by
+  have h : ((childLH (childLH thetaAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell1100)) h
+theorem e24KC2ThetaAboveLeaf110012 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1100)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1100)) h
+theorem e24KC2ThetaAboveLeaf110013 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1100)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1100)) h
+theorem e24KC2ThetaAboveLeaf1100200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell1100))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf1100201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell1100))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf1100202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell1100))) = true := by
+  have h : ((childHL (childLL (childHL thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childHL thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf1100203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell1100))) = true := by
+  have h : ((childHH (childLL (childHL thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childHL thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf1100210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell1100))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf1100211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell1100))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf1100212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell1100))) = true := by
+  have h : ((childHL (childLH (childHL thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childHL thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf1100213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell1100))) = true := by
+  have h : ((childHH (childLH (childHL thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childHL thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf11002200 :
+    adaptiveCoverCheck 11 thetaAboveCell11002200 = true := by
+  have h : (thetaAboveCell11002200).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002200 h
+theorem e24KC2ThetaAboveLeaf11002201 :
+    adaptiveCoverCheck 11 thetaAboveCell11002201 = true := by
+  have h : (thetaAboveCell11002201).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002201 h
+theorem e24KC2ThetaAboveLeaf110022020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002202) = true := by
+  have h : ((childLL thetaAboveCell11002202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002202) h
+theorem e24KC2ThetaAboveLeaf110022021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002202) = true := by
+  have h : ((childLH thetaAboveCell11002202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002202) h
+theorem e24KC2ThetaAboveLeaf110022022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002202) = true := by
+  have h : ((childHL thetaAboveCell11002202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002202) h
+theorem e24KC2ThetaAboveLeaf110022023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002202) = true := by
+  have h : ((childHH thetaAboveCell11002202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002202) h
+theorem e24KC2ThetaAboveLeaf110022030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002203) = true := by
+  have h : ((childLL thetaAboveCell11002203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002203) h
+theorem e24KC2ThetaAboveLeaf110022031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002203) = true := by
+  have h : ((childLH thetaAboveCell11002203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002203) h
+theorem e24KC2ThetaAboveLeaf110022032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002203) = true := by
+  have h : ((childHL thetaAboveCell11002203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002203) h
+theorem e24KC2ThetaAboveLeaf110022033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002203) = true := by
+  have h : ((childHH thetaAboveCell11002203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002203) h
+theorem e24KC2ThetaAboveLeaf11002210 :
+    adaptiveCoverCheck 11 thetaAboveCell11002210 = true := by
+  have h : (thetaAboveCell11002210).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002210 h
+theorem e24KC2ThetaAboveLeaf11002211 :
+    adaptiveCoverCheck 11 thetaAboveCell11002211 = true := by
+  have h : (thetaAboveCell11002211).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002211 h
+theorem e24KC2ThetaAboveLeaf110022120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002212) = true := by
+  have h : ((childLL thetaAboveCell11002212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002212) h
+theorem e24KC2ThetaAboveLeaf110022121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002212) = true := by
+  have h : ((childLH thetaAboveCell11002212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002212) h
+theorem e24KC2ThetaAboveLeaf110022122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002212) = true := by
+  have h : ((childHL thetaAboveCell11002212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002212) h
+theorem e24KC2ThetaAboveLeaf110022123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002212) = true := by
+  have h : ((childHH thetaAboveCell11002212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002212) h
+theorem e24KC2ThetaAboveLeaf110022130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002213) = true := by
+  have h : ((childLL thetaAboveCell11002213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002213) h
+theorem e24KC2ThetaAboveLeaf110022131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002213) = true := by
+  have h : ((childLH thetaAboveCell11002213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002213) h
+theorem e24KC2ThetaAboveLeaf110022132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002213) = true := by
+  have h : ((childHL thetaAboveCell11002213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002213) h
+theorem e24KC2ThetaAboveLeaf110022133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002213) = true := by
+  have h : ((childHH thetaAboveCell11002213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002213) h
+theorem e24KC2ThetaAboveLeaf110022200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002220) = true := by
+  have h : ((childLL thetaAboveCell11002220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002220) h
+theorem e24KC2ThetaAboveLeaf110022201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002220) = true := by
+  have h : ((childLH thetaAboveCell11002220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002220) h
+theorem e24KC2ThetaAboveLeaf110022202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002220) = true := by
+  have h : ((childHL thetaAboveCell11002220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002220) h
+theorem e24KC2ThetaAboveLeaf110022203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002220) = true := by
+  have h : ((childHH thetaAboveCell11002220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002220) h
+theorem e24KC2ThetaAboveLeaf110022210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002221) = true := by
+  have h : ((childLL thetaAboveCell11002221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002221) h
+theorem e24KC2ThetaAboveLeaf110022211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002221) = true := by
+  have h : ((childLH thetaAboveCell11002221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002221) h
+theorem e24KC2ThetaAboveLeaf110022212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002221) = true := by
+  have h : ((childHL thetaAboveCell11002221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002221) h
+theorem e24KC2ThetaAboveLeaf110022213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002221) = true := by
+  have h : ((childHH thetaAboveCell11002221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002221) h
+theorem e24KC2ThetaAboveLeaf11002222 :
+    adaptiveCoverCheck 11 thetaAboveCell11002222 = true := by
+  have h : (thetaAboveCell11002222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002222 h
+theorem e24KC2ThetaAboveLeaf11002223 :
+    adaptiveCoverCheck 11 thetaAboveCell11002223 = true := by
+  have h : (thetaAboveCell11002223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002223 h
+theorem e24KC2ThetaAboveLeaf110022300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002230) = true := by
+  have h : ((childLL thetaAboveCell11002230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002230) h
+theorem e24KC2ThetaAboveLeaf110022301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002230) = true := by
+  have h : ((childLH thetaAboveCell11002230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002230) h
+theorem e24KC2ThetaAboveLeaf110022302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002230) = true := by
+  have h : ((childHL thetaAboveCell11002230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002230) h
+theorem e24KC2ThetaAboveLeaf110022303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002230) = true := by
+  have h : ((childHH thetaAboveCell11002230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002230) h
+theorem e24KC2ThetaAboveLeaf110022310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002231) = true := by
+  have h : ((childLL thetaAboveCell11002231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002231) h
+theorem e24KC2ThetaAboveLeaf110022311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002231) = true := by
+  have h : ((childLH thetaAboveCell11002231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002231) h
+theorem e24KC2ThetaAboveLeaf110022312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002231) = true := by
+  have h : ((childHL thetaAboveCell11002231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002231) h
+theorem e24KC2ThetaAboveLeaf110022313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002231) = true := by
+  have h : ((childHH thetaAboveCell11002231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002231) h
+theorem e24KC2ThetaAboveLeaf11002232 :
+    adaptiveCoverCheck 11 thetaAboveCell11002232 = true := by
+  have h : (thetaAboveCell11002232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002232 h
+theorem e24KC2ThetaAboveLeaf11002233 :
+    adaptiveCoverCheck 11 thetaAboveCell11002233 = true := by
+  have h : (thetaAboveCell11002233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002233 h
+theorem e24KC2ThetaAboveLeaf11002300 :
+    adaptiveCoverCheck 11 thetaAboveCell11002300 = true := by
+  have h : (thetaAboveCell11002300).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002300 h
+theorem e24KC2ThetaAboveLeaf11002301 :
+    adaptiveCoverCheck 11 thetaAboveCell11002301 = true := by
+  have h : (thetaAboveCell11002301).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002301 h
+theorem e24KC2ThetaAboveLeaf110023020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002302) = true := by
+  have h : ((childLL thetaAboveCell11002302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002302) h
+theorem e24KC2ThetaAboveLeaf110023021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002302) = true := by
+  have h : ((childLH thetaAboveCell11002302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002302) h
+theorem e24KC2ThetaAboveLeaf110023022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002302) = true := by
+  have h : ((childHL thetaAboveCell11002302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002302) h
+theorem e24KC2ThetaAboveLeaf110023023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002302) = true := by
+  have h : ((childHH thetaAboveCell11002302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002302) h
+theorem e24KC2ThetaAboveLeaf110023030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002303) = true := by
+  have h : ((childLL thetaAboveCell11002303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002303) h
+theorem e24KC2ThetaAboveLeaf110023031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002303) = true := by
+  have h : ((childLH thetaAboveCell11002303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002303) h
+theorem e24KC2ThetaAboveLeaf110023032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002303) = true := by
+  have h : ((childHL thetaAboveCell11002303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002303) h
+theorem e24KC2ThetaAboveLeaf110023033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002303) = true := by
+  have h : ((childHH thetaAboveCell11002303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002303) h
+theorem e24KC2ThetaAboveLeaf11002310 :
+    adaptiveCoverCheck 11 thetaAboveCell11002310 = true := by
+  have h : (thetaAboveCell11002310).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002310 h
+theorem e24KC2ThetaAboveLeaf11002311 :
+    adaptiveCoverCheck 11 thetaAboveCell11002311 = true := by
+  have h : (thetaAboveCell11002311).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002311 h
+theorem e24KC2ThetaAboveLeaf110023120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002312) = true := by
+  have h : ((childLL thetaAboveCell11002312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002312) h
+theorem e24KC2ThetaAboveLeaf110023121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002312) = true := by
+  have h : ((childLH thetaAboveCell11002312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002312) h
+theorem e24KC2ThetaAboveLeaf110023122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002312) = true := by
+  have h : ((childHL thetaAboveCell11002312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002312) h
+theorem e24KC2ThetaAboveLeaf110023123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002312) = true := by
+  have h : ((childHH thetaAboveCell11002312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002312) h
+theorem e24KC2ThetaAboveLeaf110023130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002313) = true := by
+  have h : ((childLL thetaAboveCell11002313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002313) h
+theorem e24KC2ThetaAboveLeaf110023131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002313) = true := by
+  have h : ((childLH thetaAboveCell11002313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002313) h
+theorem e24KC2ThetaAboveLeaf110023132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002313) = true := by
+  have h : ((childHL thetaAboveCell11002313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002313) h
+theorem e24KC2ThetaAboveLeaf110023133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002313) = true := by
+  have h : ((childHH thetaAboveCell11002313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002313) h
+theorem e24KC2ThetaAboveLeaf110023200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002320) = true := by
+  have h : ((childLL thetaAboveCell11002320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002320) h
+theorem e24KC2ThetaAboveLeaf110023201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002320) = true := by
+  have h : ((childLH thetaAboveCell11002320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002320) h
+theorem e24KC2ThetaAboveLeaf110023202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002320) = true := by
+  have h : ((childHL thetaAboveCell11002320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002320) h
+theorem e24KC2ThetaAboveLeaf110023203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002320) = true := by
+  have h : ((childHH thetaAboveCell11002320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002320) h
+theorem e24KC2ThetaAboveLeaf110023210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002321) = true := by
+  have h : ((childLL thetaAboveCell11002321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002321) h
+theorem e24KC2ThetaAboveLeaf110023211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002321) = true := by
+  have h : ((childLH thetaAboveCell11002321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002321) h
+theorem e24KC2ThetaAboveLeaf110023212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002321) = true := by
+  have h : ((childHL thetaAboveCell11002321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002321) h
+theorem e24KC2ThetaAboveLeaf110023213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002321) = true := by
+  have h : ((childHH thetaAboveCell11002321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002321) h
+theorem e24KC2ThetaAboveLeaf11002322 :
+    adaptiveCoverCheck 11 thetaAboveCell11002322 = true := by
+  have h : (thetaAboveCell11002322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002322 h
+theorem e24KC2ThetaAboveLeaf11002323 :
+    adaptiveCoverCheck 11 thetaAboveCell11002323 = true := by
+  have h : (thetaAboveCell11002323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002323 h
+theorem e24KC2ThetaAboveLeaf110023300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002330) = true := by
+  have h : ((childLL thetaAboveCell11002330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002330) h
+theorem e24KC2ThetaAboveLeaf110023301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002330) = true := by
+  have h : ((childLH thetaAboveCell11002330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002330) h
+theorem e24KC2ThetaAboveLeaf110023302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002330) = true := by
+  have h : ((childHL thetaAboveCell11002330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002330) h
+theorem e24KC2ThetaAboveLeaf110023303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002330) = true := by
+  have h : ((childHH thetaAboveCell11002330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002330) h
+theorem e24KC2ThetaAboveLeaf110023310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11002331) = true := by
+  have h : ((childLL thetaAboveCell11002331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11002331) h
+theorem e24KC2ThetaAboveLeaf110023311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11002331) = true := by
+  have h : ((childLH thetaAboveCell11002331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11002331) h
+theorem e24KC2ThetaAboveLeaf110023312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11002331) = true := by
+  have h : ((childHL thetaAboveCell11002331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11002331) h
+theorem e24KC2ThetaAboveLeaf110023313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11002331) = true := by
+  have h : ((childHH thetaAboveCell11002331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11002331) h
+theorem e24KC2ThetaAboveLeaf11002332 :
+    adaptiveCoverCheck 11 thetaAboveCell11002332 = true := by
+  have h : (thetaAboveCell11002332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002332 h
+theorem e24KC2ThetaAboveLeaf11002333 :
+    adaptiveCoverCheck 11 thetaAboveCell11002333 = true := by
+  have h : (thetaAboveCell11002333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11002333 h
+theorem e24KC2ThetaAboveLeaf1100300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell1100))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf1100301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell1100))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf1100302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell1100))) = true := by
+  have h : ((childHL (childLL (childHH thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childHH thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf1100303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell1100))) = true := by
+  have h : ((childHH (childLL (childHH thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childHH thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf1100310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell1100))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf1100311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell1100))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf1100312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell1100))) = true := by
+  have h : ((childHL (childLH (childHH thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childHH thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf1100313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell1100))) = true := by
+  have h : ((childHH (childLH (childHH thetaAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childHH thetaAboveCell1100))) h
+theorem e24KC2ThetaAboveLeaf11003200 :
+    adaptiveCoverCheck 11 thetaAboveCell11003200 = true := by
+  have h : (thetaAboveCell11003200).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003200 h
+theorem e24KC2ThetaAboveLeaf11003201 :
+    adaptiveCoverCheck 11 thetaAboveCell11003201 = true := by
+  have h : (thetaAboveCell11003201).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003201 h
+theorem e24KC2ThetaAboveLeaf110032020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003202) = true := by
+  have h : ((childLL thetaAboveCell11003202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003202) h
+theorem e24KC2ThetaAboveLeaf110032021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003202) = true := by
+  have h : ((childLH thetaAboveCell11003202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003202) h
+theorem e24KC2ThetaAboveLeaf110032022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003202) = true := by
+  have h : ((childHL thetaAboveCell11003202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003202) h
+theorem e24KC2ThetaAboveLeaf110032023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003202) = true := by
+  have h : ((childHH thetaAboveCell11003202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003202) h
+theorem e24KC2ThetaAboveLeaf110032030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003203) = true := by
+  have h : ((childLL thetaAboveCell11003203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003203) h
+theorem e24KC2ThetaAboveLeaf110032031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003203) = true := by
+  have h : ((childLH thetaAboveCell11003203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003203) h
+theorem e24KC2ThetaAboveLeaf110032032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003203) = true := by
+  have h : ((childHL thetaAboveCell11003203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003203) h
+theorem e24KC2ThetaAboveLeaf110032033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003203) = true := by
+  have h : ((childHH thetaAboveCell11003203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003203) h
+theorem e24KC2ThetaAboveLeaf11003210 :
+    adaptiveCoverCheck 11 thetaAboveCell11003210 = true := by
+  have h : (thetaAboveCell11003210).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003210 h
+theorem e24KC2ThetaAboveLeaf11003211 :
+    adaptiveCoverCheck 11 thetaAboveCell11003211 = true := by
+  have h : (thetaAboveCell11003211).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003211 h
+theorem e24KC2ThetaAboveLeaf110032120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003212) = true := by
+  have h : ((childLL thetaAboveCell11003212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003212) h
+theorem e24KC2ThetaAboveLeaf110032121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003212) = true := by
+  have h : ((childLH thetaAboveCell11003212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003212) h
+theorem e24KC2ThetaAboveLeaf110032122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003212) = true := by
+  have h : ((childHL thetaAboveCell11003212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003212) h
+theorem e24KC2ThetaAboveLeaf110032123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003212) = true := by
+  have h : ((childHH thetaAboveCell11003212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003212) h
+theorem e24KC2ThetaAboveLeaf110032130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003213) = true := by
+  have h : ((childLL thetaAboveCell11003213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003213) h
+theorem e24KC2ThetaAboveLeaf110032131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003213) = true := by
+  have h : ((childLH thetaAboveCell11003213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003213) h
+theorem e24KC2ThetaAboveLeaf110032132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003213) = true := by
+  have h : ((childHL thetaAboveCell11003213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003213) h
+theorem e24KC2ThetaAboveLeaf110032133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003213) = true := by
+  have h : ((childHH thetaAboveCell11003213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003213) h
+theorem e24KC2ThetaAboveLeaf110032200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003220) = true := by
+  have h : ((childLL thetaAboveCell11003220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003220) h
+theorem e24KC2ThetaAboveLeaf110032201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003220) = true := by
+  have h : ((childLH thetaAboveCell11003220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003220) h
+theorem e24KC2ThetaAboveLeaf110032202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003220) = true := by
+  have h : ((childHL thetaAboveCell11003220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003220) h
+theorem e24KC2ThetaAboveLeaf110032203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003220) = true := by
+  have h : ((childHH thetaAboveCell11003220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003220) h
+theorem e24KC2ThetaAboveLeaf110032210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003221) = true := by
+  have h : ((childLL thetaAboveCell11003221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003221) h
+theorem e24KC2ThetaAboveLeaf110032211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003221) = true := by
+  have h : ((childLH thetaAboveCell11003221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003221) h
+theorem e24KC2ThetaAboveLeaf110032212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003221) = true := by
+  have h : ((childHL thetaAboveCell11003221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003221) h
+theorem e24KC2ThetaAboveLeaf110032213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003221) = true := by
+  have h : ((childHH thetaAboveCell11003221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003221) h
+theorem e24KC2ThetaAboveLeaf11003222 :
+    adaptiveCoverCheck 11 thetaAboveCell11003222 = true := by
+  have h : (thetaAboveCell11003222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003222 h
+theorem e24KC2ThetaAboveLeaf11003223 :
+    adaptiveCoverCheck 11 thetaAboveCell11003223 = true := by
+  have h : (thetaAboveCell11003223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003223 h
+theorem e24KC2ThetaAboveLeaf110032300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003230) = true := by
+  have h : ((childLL thetaAboveCell11003230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003230) h
+theorem e24KC2ThetaAboveLeaf110032301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003230) = true := by
+  have h : ((childLH thetaAboveCell11003230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003230) h
+theorem e24KC2ThetaAboveLeaf110032302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003230) = true := by
+  have h : ((childHL thetaAboveCell11003230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003230) h
+theorem e24KC2ThetaAboveLeaf110032303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003230) = true := by
+  have h : ((childHH thetaAboveCell11003230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003230) h
+theorem e24KC2ThetaAboveLeaf110032310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003231) = true := by
+  have h : ((childLL thetaAboveCell11003231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003231) h
+theorem e24KC2ThetaAboveLeaf110032311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003231) = true := by
+  have h : ((childLH thetaAboveCell11003231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003231) h
+theorem e24KC2ThetaAboveLeaf110032312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003231) = true := by
+  have h : ((childHL thetaAboveCell11003231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003231) h
+theorem e24KC2ThetaAboveLeaf110032313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003231) = true := by
+  have h : ((childHH thetaAboveCell11003231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003231) h
+theorem e24KC2ThetaAboveLeaf11003232 :
+    adaptiveCoverCheck 11 thetaAboveCell11003232 = true := by
+  have h : (thetaAboveCell11003232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003232 h
+theorem e24KC2ThetaAboveLeaf11003233 :
+    adaptiveCoverCheck 11 thetaAboveCell11003233 = true := by
+  have h : (thetaAboveCell11003233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003233 h
+theorem e24KC2ThetaAboveLeaf11003300 :
+    adaptiveCoverCheck 11 thetaAboveCell11003300 = true := by
+  have h : (thetaAboveCell11003300).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003300 h
+theorem e24KC2ThetaAboveLeaf11003301 :
+    adaptiveCoverCheck 11 thetaAboveCell11003301 = true := by
+  have h : (thetaAboveCell11003301).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003301 h
+theorem e24KC2ThetaAboveLeaf110033020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003302) = true := by
+  have h : ((childLL thetaAboveCell11003302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003302) h
+theorem e24KC2ThetaAboveLeaf110033021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003302) = true := by
+  have h : ((childLH thetaAboveCell11003302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003302) h
+theorem e24KC2ThetaAboveLeaf110033022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003302) = true := by
+  have h : ((childHL thetaAboveCell11003302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003302) h
+theorem e24KC2ThetaAboveLeaf110033023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003302) = true := by
+  have h : ((childHH thetaAboveCell11003302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003302) h
+theorem e24KC2ThetaAboveLeaf110033030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003303) = true := by
+  have h : ((childLL thetaAboveCell11003303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003303) h
+theorem e24KC2ThetaAboveLeaf110033031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003303) = true := by
+  have h : ((childLH thetaAboveCell11003303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003303) h
+theorem e24KC2ThetaAboveLeaf110033032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003303) = true := by
+  have h : ((childHL thetaAboveCell11003303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003303) h
+theorem e24KC2ThetaAboveLeaf110033033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003303) = true := by
+  have h : ((childHH thetaAboveCell11003303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003303) h
+theorem e24KC2ThetaAboveLeaf11003310 :
+    adaptiveCoverCheck 11 thetaAboveCell11003310 = true := by
+  have h : (thetaAboveCell11003310).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003310 h
+theorem e24KC2ThetaAboveLeaf11003311 :
+    adaptiveCoverCheck 11 thetaAboveCell11003311 = true := by
+  have h : (thetaAboveCell11003311).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003311 h
+theorem e24KC2ThetaAboveLeaf110033120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003312) = true := by
+  have h : ((childLL thetaAboveCell11003312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003312) h
+theorem e24KC2ThetaAboveLeaf110033121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003312) = true := by
+  have h : ((childLH thetaAboveCell11003312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003312) h
+theorem e24KC2ThetaAboveLeaf110033122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003312) = true := by
+  have h : ((childHL thetaAboveCell11003312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003312) h
+theorem e24KC2ThetaAboveLeaf110033123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003312) = true := by
+  have h : ((childHH thetaAboveCell11003312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003312) h
+theorem e24KC2ThetaAboveLeaf110033130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003313) = true := by
+  have h : ((childLL thetaAboveCell11003313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003313) h
+theorem e24KC2ThetaAboveLeaf110033131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003313) = true := by
+  have h : ((childLH thetaAboveCell11003313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003313) h
+theorem e24KC2ThetaAboveLeaf110033132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003313) = true := by
+  have h : ((childHL thetaAboveCell11003313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003313) h
+theorem e24KC2ThetaAboveLeaf110033133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003313) = true := by
+  have h : ((childHH thetaAboveCell11003313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003313) h
+theorem e24KC2ThetaAboveLeaf110033200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003320) = true := by
+  have h : ((childLL thetaAboveCell11003320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003320) h
+theorem e24KC2ThetaAboveLeaf110033201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003320) = true := by
+  have h : ((childLH thetaAboveCell11003320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003320) h
+theorem e24KC2ThetaAboveLeaf110033202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003320) = true := by
+  have h : ((childHL thetaAboveCell11003320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003320) h
+theorem e24KC2ThetaAboveLeaf110033203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003320) = true := by
+  have h : ((childHH thetaAboveCell11003320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003320) h
+theorem e24KC2ThetaAboveLeaf110033210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003321) = true := by
+  have h : ((childLL thetaAboveCell11003321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003321) h
+theorem e24KC2ThetaAboveLeaf110033211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11003321) = true := by
+  have h : ((childLH thetaAboveCell11003321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11003321) h
+theorem e24KC2ThetaAboveLeaf110033212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11003321) = true := by
+  have h : ((childHL thetaAboveCell11003321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11003321) h
+theorem e24KC2ThetaAboveLeaf110033213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11003321) = true := by
+  have h : ((childHH thetaAboveCell11003321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11003321) h
+theorem e24KC2ThetaAboveLeaf11003322 :
+    adaptiveCoverCheck 11 thetaAboveCell11003322 = true := by
+  have h : (thetaAboveCell11003322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003322 h
+theorem e24KC2ThetaAboveLeaf11003323 :
+    adaptiveCoverCheck 11 thetaAboveCell11003323 = true := by
+  have h : (thetaAboveCell11003323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11003323 h
+theorem e24KC2ThetaAboveLeaf110033300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11003330) = true := by
+  have h : ((childLL thetaAboveCell11003330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11003330) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle004.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle004.lean
new file mode 100644
index 0000000000..9ccd225f1d
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle004.lean
@@ -0,0 +1,10654 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch005`.
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch027`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCells0d991a929b
+
+/-- Subcell `0000` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0001` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0001 : AngleCell :=
+  childLH (childLL (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0002` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0002 : AngleCell :=
+  childHL (childLL (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0003` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0003 : AngleCell :=
+  childHH (childLL (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0010 : AngleCell :=
+  childLL (childLH (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0011 : AngleCell :=
+  childLH (childLH (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0012` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0012 : AngleCell :=
+  childHL (childLH (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0013` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0013 : AngleCell :=
+  childHH (childLH (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0030` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0030 : AngleCell :=
+  childLL (childHH (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0031` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0031 : AngleCell :=
+  childLH (childHH (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0032` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0032 : AngleCell :=
+  childHL (childHH (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0033` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0033 : AngleCell :=
+  childHH (childHH (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0100 : AngleCell :=
+  childLL (childLL (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0101 : AngleCell :=
+  childLH (childLL (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0102` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0102 : AngleCell :=
+  childHL (childLL (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0103` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0103 : AngleCell :=
+  childHH (childLL (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0110 : AngleCell :=
+  childLL (childLH (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0111 : AngleCell :=
+  childLH (childLH (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0112` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0112 : AngleCell :=
+  childHL (childLH (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0113` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0113 : AngleCell :=
+  childHH (childLH (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0120` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0120 : AngleCell :=
+  childLL (childHL (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0121` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0121 : AngleCell :=
+  childLH (childHL (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0122` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0122 : AngleCell :=
+  childHL (childHL (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0123` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0123 : AngleCell :=
+  childHH (childHL (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0130` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0130 : AngleCell :=
+  childLL (childHH (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0131` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0131 : AngleCell :=
+  childLH (childHH (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0132` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0132 : AngleCell :=
+  childHL (childHH (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0133` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0133 : AngleCell :=
+  childHH (childHH (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `1000` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1000 : AngleCell :=
+  childLL (childLL (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1001` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1001 : AngleCell :=
+  childLH (childLL (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1002` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1002 : AngleCell :=
+  childHL (childLL (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1003` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1003 : AngleCell :=
+  childHH (childLL (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1010 : AngleCell :=
+  childLL (childLH (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1011 : AngleCell :=
+  childLH (childLH (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1012` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1012 : AngleCell :=
+  childHL (childLH (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1013` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1013 : AngleCell :=
+  childHH (childLH (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1020` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1020 : AngleCell :=
+  childLL (childHL (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1021` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1021 : AngleCell :=
+  childLH (childHL (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1022` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1022 : AngleCell :=
+  childHL (childHL (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1023` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1023 : AngleCell :=
+  childHH (childHL (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1030` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1030 : AngleCell :=
+  childLL (childHH (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1031` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1031 : AngleCell :=
+  childLH (childHH (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1032` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1032 : AngleCell :=
+  childHL (childHH (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1033` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1033 : AngleCell :=
+  childHH (childHH (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1100 : AngleCell :=
+  childLL (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1102` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1102 : AngleCell :=
+  childHL (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1103` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1103 : AngleCell :=
+  childHH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1112` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1112 : AngleCell :=
+  childHL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1113` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1113 : AngleCell :=
+  childHH (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1120` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1120 : AngleCell :=
+  childLL (childHL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1121` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1121 : AngleCell :=
+  childLH (childHL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1122` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1122 : AngleCell :=
+  childHL (childHL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1123` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1123 : AngleCell :=
+  childHH (childHL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1130` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1130 : AngleCell :=
+  childLL (childHH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1131` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1131 : AngleCell :=
+  childLH (childHH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1132` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1132 : AngleCell :=
+  childHL (childHH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1133` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1133 : AngleCell :=
+  childHH (childHH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `3221` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3221 : AngleCell :=
+  childLH (childHL (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3222` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3222 : AngleCell :=
+  childHL (childHL (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3223 : AngleCell :=
+  childHH (childHL (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3310` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3310 : AngleCell :=
+  childLL (childLH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3311` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3311 : AngleCell :=
+  childLH (childLH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3313` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3313 : AngleCell :=
+  childHH (childLH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell0000)))
+
+end GerverSofa.PartE.CertificateCells0d991a929b
+
+namespace GerverSofa.PartE.CertificateCells802ef10e9d
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `000023102000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00002310)))
+
+/-- Subcell `000023103002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00002310)))
+
+/-- Subcell `000023110220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023110221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023110222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023110223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023110230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023110231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023110232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023110233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023110320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023110321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023110322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023110323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023110330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023110331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023110332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023110333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023110333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaAboveCell00002311)))
+
+/-- Subcell `000023111220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023111221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023111222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023111223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023111230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023111231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023111232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023111233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023111320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023111321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023111322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023111323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023111330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023111331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023111332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023111333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023111333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaAboveCell00002311)))
+
+/-- Subcell `000023112002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00002311)))
+
+/-- Subcell `000023113002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000023113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00002311)))
+
+/-- Subcell `000032000220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032000221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032000222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032000223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032000230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032000231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032000232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032000233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032000320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032000321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032000322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032000323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032000330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032000331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032000332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032000333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032000333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaAboveCell00003200)))
+
+/-- Subcell `000032001220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032001220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaAboveCell00003200)))
+
+/-- Subcell `000032001221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032001221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaAboveCell00003200)))
+
+/-- Subcell `000032001222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032001222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaAboveCell00003200)))
+
+/-- Subcell `000032001223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032001223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaAboveCell00003200)))
+
+/-- Subcell `000032002002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00003200)))
+
+/-- Subcell `000032003000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00003200)))
+
+/-- Subcell `000032003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00003200)))
+
+end GerverSofa.PartE.CertificateCells802ef10e9d
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC5TerminalBatchT000001`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0d991a929b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0d991a929b
+
+open CertificateCells0d991a929b
+
+theorem e24KC2PhiAboveLeaf00000 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell0000) = true := by
+  have h : physicallyIrrelevant (childLL phiAboveCell0000) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 11 (childLL phiAboveCell0000) h
+theorem e24KC2PhiAboveLeaf00001 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell0000) = true := by
+  have h : ((childLH phiAboveCell0000)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH phiAboveCell0000) h
+
+theorem e24KC2PhiAboveLeaf00002 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell0000) = true := by
+  have h : physicallyIrrelevant (childHL phiAboveCell0000) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 11 (childHL phiAboveCell0000) h
+
+theorem e24KC2PhiAboveLeaf00003 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell0000) = true := by
+  have h : physicallyIrrelevant (childHH phiAboveCell0000) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 11 (childHH phiAboveCell0000) h
+theorem e24KC2PhiAboveLeaf00010 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell0001) = true := by
+  have h : ((childLL phiAboveCell0001)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL phiAboveCell0001) h
+theorem e24KC2PhiAboveLeaf00011 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell0001) = true := by
+  have h : ((childLH phiAboveCell0001)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH phiAboveCell0001) h
+theorem e24KC2PhiAboveLeaf00012 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell0001) = true := by
+  have h : ((childHL phiAboveCell0001)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell0001) h
+theorem e24KC2PhiAboveLeaf00013 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell0001) = true := by
+  have h : ((childHH phiAboveCell0001)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell0001) h
+
+theorem e24KC2PhiAboveLeaf0002 :
+    adaptiveCoverCheck 12 phiAboveCell0002 = true := by
+  have h : physicallyIrrelevant phiAboveCell0002 = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 12 phiAboveCell0002 h
+theorem e24KC2PhiAboveLeaf0003 :
+    adaptiveCoverCheck 12 phiAboveCell0003 = true := by
+  have h : (phiAboveCell0003).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0003 h
+theorem e24KC2PhiAboveLeaf00100 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell0010) = true := by
+  have h : ((childLL phiAboveCell0010)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL phiAboveCell0010) h
+theorem e24KC2PhiAboveLeaf00101 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell0010) = true := by
+  have h : ((childLH phiAboveCell0010)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH phiAboveCell0010) h
+theorem e24KC2PhiAboveLeaf00102 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell0010) = true := by
+  have h : ((childHL phiAboveCell0010)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell0010) h
+theorem e24KC2PhiAboveLeaf00103 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell0010) = true := by
+  have h : ((childHH phiAboveCell0010)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell0010) h
+theorem e24KC2PhiAboveLeaf00110 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell0011) = true := by
+  have h : ((childLL phiAboveCell0011)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL phiAboveCell0011) h
+theorem e24KC2PhiAboveLeaf00111 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell0011) = true := by
+  have h : ((childLH phiAboveCell0011)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH phiAboveCell0011) h
+theorem e24KC2PhiAboveLeaf00112 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell0011) = true := by
+  have h : ((childHL phiAboveCell0011)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell0011) h
+theorem e24KC2PhiAboveLeaf00113 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell0011) = true := by
+  have h : ((childHH phiAboveCell0011)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell0011) h
+theorem e24KC2PhiAboveLeaf0012 :
+    adaptiveCoverCheck 12 phiAboveCell0012 = true := by
+  have h : (phiAboveCell0012).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0012 h
+theorem e24KC2PhiAboveLeaf0013 :
+    adaptiveCoverCheck 12 phiAboveCell0013 = true := by
+  have h : (phiAboveCell0013).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0013 h
+
+theorem e24KC2PhiAboveLeaf002 :
+    adaptiveCoverCheck 13 (childHL (childLL (childLL e24PhiAboveRoot))) = true := by
+  have h : physicallyIrrelevant (childHL (childLL (childLL e24PhiAboveRoot))) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 13 (childHL (childLL (childLL
+    e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf0030 :
+    adaptiveCoverCheck 12 phiAboveCell0030 = true := by
+  have h : (phiAboveCell0030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0030 h
+theorem e24KC2PhiAboveLeaf0031 :
+    adaptiveCoverCheck 12 phiAboveCell0031 = true := by
+  have h : (phiAboveCell0031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0031 h
+
+theorem e24KC2PhiAboveLeaf0032 :
+    adaptiveCoverCheck 12 phiAboveCell0032 = true := by
+  have h : physicallyIrrelevant phiAboveCell0032 = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 12 phiAboveCell0032 h
+theorem e24KC2PhiAboveLeaf0033 :
+    adaptiveCoverCheck 12 phiAboveCell0033 = true := by
+  have h : (phiAboveCell0033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0033 h
+theorem e24KC2PhiAboveLeaf01000 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell0100) = true := by
+  have h : ((childLL phiAboveCell0100)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL phiAboveCell0100) h
+theorem e24KC2PhiAboveLeaf01001 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell0100) = true := by
+  have h : ((childLH phiAboveCell0100)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH phiAboveCell0100) h
+theorem e24KC2PhiAboveLeaf01002 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell0100) = true := by
+  have h : ((childHL phiAboveCell0100)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell0100) h
+theorem e24KC2PhiAboveLeaf01003 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell0100) = true := by
+  have h : ((childHH phiAboveCell0100)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell0100) h
+theorem e24KC2PhiAboveLeaf01010 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell0101) = true := by
+  have h : ((childLL phiAboveCell0101)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL phiAboveCell0101) h
+theorem e24KC2PhiAboveLeaf010110 :
+    adaptiveCoverCheck 10 (childLL (childLH phiAboveCell0101)) = true := by
+  have h : ((childLL (childLH phiAboveCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childLH phiAboveCell0101)) h
+theorem e24KC2PhiAboveLeaf010111 :
+    adaptiveCoverCheck 10 (childLH (childLH phiAboveCell0101)) = true := by
+  have h : ((childLH (childLH phiAboveCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childLH phiAboveCell0101)) h
+theorem e24KC2PhiAboveLeaf010112 :
+    adaptiveCoverCheck 10 (childHL (childLH phiAboveCell0101)) = true := by
+  have h : ((childHL (childLH phiAboveCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childLH phiAboveCell0101)) h
+theorem e24KC2PhiAboveLeaf010113 :
+    adaptiveCoverCheck 10 (childHH (childLH phiAboveCell0101)) = true := by
+  have h : ((childHH (childLH phiAboveCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childLH phiAboveCell0101)) h
+theorem e24KC2PhiAboveLeaf01012 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell0101) = true := by
+  have h : ((childHL phiAboveCell0101)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell0101) h
+theorem e24KC2PhiAboveLeaf01013 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell0101) = true := by
+  have h : ((childHH phiAboveCell0101)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell0101) h
+theorem e24KC2PhiAboveLeaf0102 :
+    adaptiveCoverCheck 12 phiAboveCell0102 = true := by
+  have h : (phiAboveCell0102).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0102 h
+theorem e24KC2PhiAboveLeaf0103 :
+    adaptiveCoverCheck 12 phiAboveCell0103 = true := by
+  have h : (phiAboveCell0103).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0103 h
+theorem e24KC2PhiAboveLeaf011000 :
+    adaptiveCoverCheck 10 (childLL (childLL phiAboveCell0110)) = true := by
+  have h : ((childLL (childLL phiAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childLL phiAboveCell0110)) h
+theorem e24KC2PhiAboveLeaf011001 :
+    adaptiveCoverCheck 10 (childLH (childLL phiAboveCell0110)) = true := by
+  have h : ((childLH (childLL phiAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childLL phiAboveCell0110)) h
+theorem e24KC2PhiAboveLeaf011002 :
+    adaptiveCoverCheck 10 (childHL (childLL phiAboveCell0110)) = true := by
+  have h : ((childHL (childLL phiAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childLL phiAboveCell0110)) h
+theorem e24KC2PhiAboveLeaf011003 :
+    adaptiveCoverCheck 10 (childHH (childLL phiAboveCell0110)) = true := by
+  have h : ((childHH (childLL phiAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childLL phiAboveCell0110)) h
+theorem e24KC2PhiAboveLeaf011010 :
+    adaptiveCoverCheck 10 (childLL (childLH phiAboveCell0110)) = true := by
+  have h : ((childLL (childLH phiAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childLH phiAboveCell0110)) h
+theorem e24KC2PhiAboveLeaf011011 :
+    adaptiveCoverCheck 10 (childLH (childLH phiAboveCell0110)) = true := by
+  have h : ((childLH (childLH phiAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childLH phiAboveCell0110)) h
+theorem e24KC2PhiAboveLeaf011012 :
+    adaptiveCoverCheck 10 (childHL (childLH phiAboveCell0110)) = true := by
+  have h : ((childHL (childLH phiAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childLH phiAboveCell0110)) h
+theorem e24KC2PhiAboveLeaf011013 :
+    adaptiveCoverCheck 10 (childHH (childLH phiAboveCell0110)) = true := by
+  have h : ((childHH (childLH phiAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childLH phiAboveCell0110)) h
+theorem e24KC2PhiAboveLeaf01102 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell0110) = true := by
+  have h : ((childHL phiAboveCell0110)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell0110) h
+theorem e24KC2PhiAboveLeaf01103 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell0110) = true := by
+  have h : ((childHH phiAboveCell0110)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell0110) h
+theorem e24KC2PhiAboveLeaf011100 :
+    adaptiveCoverCheck 10 (childLL (childLL phiAboveCell0111)) = true := by
+  have h : ((childLL (childLL phiAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childLL phiAboveCell0111)) h
+theorem e24KC2PhiAboveLeaf011101 :
+    adaptiveCoverCheck 10 (childLH (childLL phiAboveCell0111)) = true := by
+  have h : ((childLH (childLL phiAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childLL phiAboveCell0111)) h
+theorem e24KC2PhiAboveLeaf011102 :
+    adaptiveCoverCheck 10 (childHL (childLL phiAboveCell0111)) = true := by
+  have h : ((childHL (childLL phiAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childLL phiAboveCell0111)) h
+theorem e24KC2PhiAboveLeaf011103 :
+    adaptiveCoverCheck 10 (childHH (childLL phiAboveCell0111)) = true := by
+  have h : ((childHH (childLL phiAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childLL phiAboveCell0111)) h
+theorem e24KC2PhiAboveLeaf011110 :
+    adaptiveCoverCheck 10 (childLL (childLH phiAboveCell0111)) = true := by
+  have h : ((childLL (childLH phiAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childLH phiAboveCell0111)) h
+theorem e24KC2PhiAboveLeaf011111 :
+    adaptiveCoverCheck 10 (childLH (childLH phiAboveCell0111)) = true := by
+  have h : ((childLH (childLH phiAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childLH phiAboveCell0111)) h
+theorem e24KC2PhiAboveLeaf011112 :
+    adaptiveCoverCheck 10 (childHL (childLH phiAboveCell0111)) = true := by
+  have h : ((childHL (childLH phiAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childLH phiAboveCell0111)) h
+theorem e24KC2PhiAboveLeaf011113 :
+    adaptiveCoverCheck 10 (childHH (childLH phiAboveCell0111)) = true := by
+  have h : ((childHH (childLH phiAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childLH phiAboveCell0111)) h
+theorem e24KC2PhiAboveLeaf01112 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell0111) = true := by
+  have h : ((childHL phiAboveCell0111)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell0111) h
+theorem e24KC2PhiAboveLeaf01113 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell0111) = true := by
+  have h : ((childHH phiAboveCell0111)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell0111) h
+theorem e24KC2PhiAboveLeaf0112 :
+    adaptiveCoverCheck 12 phiAboveCell0112 = true := by
+  have h : (phiAboveCell0112).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0112 h
+theorem e24KC2PhiAboveLeaf0113 :
+    adaptiveCoverCheck 12 phiAboveCell0113 = true := by
+  have h : (phiAboveCell0113).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0113 h
+theorem e24KC2PhiAboveLeaf0120 :
+    adaptiveCoverCheck 12 phiAboveCell0120 = true := by
+  have h : (phiAboveCell0120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0120 h
+theorem e24KC2PhiAboveLeaf0121 :
+    adaptiveCoverCheck 12 phiAboveCell0121 = true := by
+  have h : (phiAboveCell0121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0121 h
+theorem e24KC2PhiAboveLeaf0122 :
+    adaptiveCoverCheck 12 phiAboveCell0122 = true := by
+  have h : (phiAboveCell0122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0122 h
+theorem e24KC2PhiAboveLeaf0123 :
+    adaptiveCoverCheck 12 phiAboveCell0123 = true := by
+  have h : (phiAboveCell0123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0123 h
+theorem e24KC2PhiAboveLeaf0130 :
+    adaptiveCoverCheck 12 phiAboveCell0130 = true := by
+  have h : (phiAboveCell0130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0130 h
+theorem e24KC2PhiAboveLeaf0131 :
+    adaptiveCoverCheck 12 phiAboveCell0131 = true := by
+  have h : (phiAboveCell0131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0131 h
+theorem e24KC2PhiAboveLeaf0132 :
+    adaptiveCoverCheck 12 phiAboveCell0132 = true := by
+  have h : (phiAboveCell0132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0132 h
+theorem e24KC2PhiAboveLeaf0133 :
+    adaptiveCoverCheck 12 phiAboveCell0133 = true := by
+  have h : (phiAboveCell0133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell0133 h
+
+theorem e24KC2PhiAboveLeaf02 :
+    adaptiveCoverCheck 14 (childHL (childLL e24PhiAboveRoot)) = true := by
+  have h : physicallyIrrelevant (childHL (childLL e24PhiAboveRoot)) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 14 (childHL (childLL e24PhiAboveRoot)) h
+theorem e24KC2PhiAboveLeaf030 :
+    adaptiveCoverCheck 13 (childLL (childHH (childLL e24PhiAboveRoot))) = true := by
+  have h : ((childLL (childHH (childLL e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childHH (childLL e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf031 :
+    adaptiveCoverCheck 13 (childLH (childHH (childLL e24PhiAboveRoot))) = true := by
+  have h : ((childLH (childHH (childLL e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childHH (childLL e24PhiAboveRoot))) h
+
+theorem e24KC2PhiAboveLeaf032 :
+    adaptiveCoverCheck 13 (childHL (childHH (childLL e24PhiAboveRoot))) = true := by
+  have h : physicallyIrrelevant (childHL (childHH (childLL e24PhiAboveRoot))) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 13 (childHL (childHH (childLL
+    e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf033 :
+    adaptiveCoverCheck 13 (childHH (childHH (childLL e24PhiAboveRoot))) = true := by
+  have h : ((childHH (childHH (childLL e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childHH (childLL e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf100000 :
+    adaptiveCoverCheck 10 (childLL (childLL phiAboveCell1000)) = true := by
+  have h : ((childLL (childLL phiAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childLL phiAboveCell1000)) h
+theorem e24KC2PhiAboveLeaf100001 :
+    adaptiveCoverCheck 10 (childLH (childLL phiAboveCell1000)) = true := by
+  have h : ((childLH (childLL phiAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childLL phiAboveCell1000)) h
+theorem e24KC2PhiAboveLeaf100002 :
+    adaptiveCoverCheck 10 (childHL (childLL phiAboveCell1000)) = true := by
+  have h : ((childHL (childLL phiAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childLL phiAboveCell1000)) h
+theorem e24KC2PhiAboveLeaf100003 :
+    adaptiveCoverCheck 10 (childHH (childLL phiAboveCell1000)) = true := by
+  have h : ((childHH (childLL phiAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childLL phiAboveCell1000)) h
+theorem e24KC2PhiAboveLeaf100010 :
+    adaptiveCoverCheck 10 (childLL (childLH phiAboveCell1000)) = true := by
+  have h : ((childLL (childLH phiAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childLH phiAboveCell1000)) h
+theorem e24KC2PhiAboveLeaf100011 :
+    adaptiveCoverCheck 10 (childLH (childLH phiAboveCell1000)) = true := by
+  have h : ((childLH (childLH phiAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childLH phiAboveCell1000)) h
+theorem e24KC2PhiAboveLeaf100012 :
+    adaptiveCoverCheck 10 (childHL (childLH phiAboveCell1000)) = true := by
+  have h : ((childHL (childLH phiAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childLH phiAboveCell1000)) h
+theorem e24KC2PhiAboveLeaf100013 :
+    adaptiveCoverCheck 10 (childHH (childLH phiAboveCell1000)) = true := by
+  have h : ((childHH (childLH phiAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childLH phiAboveCell1000)) h
+theorem e24KC2PhiAboveLeaf10002 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1000) = true := by
+  have h : ((childHL phiAboveCell1000)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell1000) h
+theorem e24KC2PhiAboveLeaf10003 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1000) = true := by
+  have h : ((childHH phiAboveCell1000)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell1000) h
+theorem e24KC2PhiAboveLeaf100100 :
+    adaptiveCoverCheck 10 (childLL (childLL phiAboveCell1001)) = true := by
+  have h : ((childLL (childLL phiAboveCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childLL phiAboveCell1001)) h
+theorem e24KC2PhiAboveLeaf1001010 :
+    adaptiveCoverCheck 9 (childLL (childLH (childLL phiAboveCell1001))) = true := by
+  have h : ((childLL (childLH (childLL phiAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH (childLL phiAboveCell1001))) h
+theorem e24KC2PhiAboveLeaf1001011 :
+    adaptiveCoverCheck 9 (childLH (childLH (childLL phiAboveCell1001))) = true := by
+  have h : ((childLH (childLH (childLL phiAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH (childLL phiAboveCell1001))) h
+theorem e24KC2PhiAboveLeaf1001012 :
+    adaptiveCoverCheck 9 (childHL (childLH (childLL phiAboveCell1001))) = true := by
+  have h : ((childHL (childLH (childLL phiAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH (childLL phiAboveCell1001))) h
+theorem e24KC2PhiAboveLeaf1001013 :
+    adaptiveCoverCheck 9 (childHH (childLH (childLL phiAboveCell1001))) = true := by
+  have h : ((childHH (childLH (childLL phiAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH (childLL phiAboveCell1001))) h
+theorem e24KC2PhiAboveLeaf100102 :
+    adaptiveCoverCheck 10 (childHL (childLL phiAboveCell1001)) = true := by
+  have h : ((childHL (childLL phiAboveCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childLL phiAboveCell1001)) h
+theorem e24KC2PhiAboveLeaf100103 :
+    adaptiveCoverCheck 10 (childHH (childLL phiAboveCell1001)) = true := by
+  have h : ((childHH (childLL phiAboveCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childLL phiAboveCell1001)) h
+theorem e24KC2PhiAboveLeaf1001100 :
+    adaptiveCoverCheck 9 (childLL (childLL (childLH phiAboveCell1001))) = true := by
+  have h : ((childLL (childLL (childLH phiAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL (childLH phiAboveCell1001))) h
+theorem e24KC2PhiAboveLeaf1001101 :
+    adaptiveCoverCheck 9 (childLH (childLL (childLH phiAboveCell1001))) = true := by
+  have h : ((childLH (childLL (childLH phiAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL (childLH phiAboveCell1001))) h
+theorem e24KC2PhiAboveLeaf1001102 :
+    adaptiveCoverCheck 9 (childHL (childLL (childLH phiAboveCell1001))) = true := by
+  have h : ((childHL (childLL (childLH phiAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL (childLH phiAboveCell1001))) h
+theorem e24KC2PhiAboveLeaf1001103 :
+    adaptiveCoverCheck 9 (childHH (childLL (childLH phiAboveCell1001))) = true := by
+  have h : ((childHH (childLL (childLH phiAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL (childLH phiAboveCell1001))) h
+theorem e24KC2PhiAboveLeaf1001110 :
+    adaptiveCoverCheck 9 (childLL (childLH (childLH phiAboveCell1001))) = true := by
+  have h : ((childLL (childLH (childLH phiAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH (childLH phiAboveCell1001))) h
+theorem e24KC2PhiAboveLeaf1001111 :
+    adaptiveCoverCheck 9 (childLH (childLH (childLH phiAboveCell1001))) = true := by
+  have h : ((childLH (childLH (childLH phiAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH (childLH phiAboveCell1001))) h
+theorem e24KC2PhiAboveLeaf1001112 :
+    adaptiveCoverCheck 9 (childHL (childLH (childLH phiAboveCell1001))) = true := by
+  have h : ((childHL (childLH (childLH phiAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH (childLH phiAboveCell1001))) h
+theorem e24KC2PhiAboveLeaf1001113 :
+    adaptiveCoverCheck 9 (childHH (childLH (childLH phiAboveCell1001))) = true := by
+  have h : ((childHH (childLH (childLH phiAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH (childLH phiAboveCell1001))) h
+theorem e24KC2PhiAboveLeaf100112 :
+    adaptiveCoverCheck 10 (childHL (childLH phiAboveCell1001)) = true := by
+  have h : ((childHL (childLH phiAboveCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childLH phiAboveCell1001)) h
+theorem e24KC2PhiAboveLeaf100113 :
+    adaptiveCoverCheck 10 (childHH (childLH phiAboveCell1001)) = true := by
+  have h : ((childHH (childLH phiAboveCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childLH phiAboveCell1001)) h
+theorem e24KC2PhiAboveLeaf10012 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1001) = true := by
+  have h : ((childHL phiAboveCell1001)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell1001) h
+theorem e24KC2PhiAboveLeaf10013 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1001) = true := by
+  have h : ((childHH phiAboveCell1001)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell1001) h
+theorem e24KC2PhiAboveLeaf10020 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1002) = true := by
+  have h : ((childLL phiAboveCell1002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL phiAboveCell1002) h
+theorem e24KC2PhiAboveLeaf10021 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1002) = true := by
+  have h : ((childLH phiAboveCell1002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH phiAboveCell1002) h
+theorem e24KC2PhiAboveLeaf10022 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1002) = true := by
+  have h : ((childHL phiAboveCell1002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell1002) h
+theorem e24KC2PhiAboveLeaf10023 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1002) = true := by
+  have h : ((childHH phiAboveCell1002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell1002) h
+theorem e24KC2PhiAboveLeaf10030 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1003) = true := by
+  have h : ((childLL phiAboveCell1003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL phiAboveCell1003) h
+theorem e24KC2PhiAboveLeaf10031 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1003) = true := by
+  have h : ((childLH phiAboveCell1003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH phiAboveCell1003) h
+theorem e24KC2PhiAboveLeaf10032 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1003) = true := by
+  have h : ((childHL phiAboveCell1003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell1003) h
+theorem e24KC2PhiAboveLeaf10033 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1003) = true := by
+  have h : ((childHH phiAboveCell1003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell1003) h
+theorem e24KC2PhiAboveLeaf1010000 :
+    adaptiveCoverCheck 9 (childLL (childLL (childLL phiAboveCell1010))) = true := by
+  have h : ((childLL (childLL (childLL phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL (childLL phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf1010001 :
+    adaptiveCoverCheck 9 (childLH (childLL (childLL phiAboveCell1010))) = true := by
+  have h : ((childLH (childLL (childLL phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL (childLL phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf1010002 :
+    adaptiveCoverCheck 9 (childHL (childLL (childLL phiAboveCell1010))) = true := by
+  have h : ((childHL (childLL (childLL phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL (childLL phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf1010003 :
+    adaptiveCoverCheck 9 (childHH (childLL (childLL phiAboveCell1010))) = true := by
+  have h : ((childHH (childLL (childLL phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL (childLL phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf1010010 :
+    adaptiveCoverCheck 9 (childLL (childLH (childLL phiAboveCell1010))) = true := by
+  have h : ((childLL (childLH (childLL phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH (childLL phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf1010011 :
+    adaptiveCoverCheck 9 (childLH (childLH (childLL phiAboveCell1010))) = true := by
+  have h : ((childLH (childLH (childLL phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH (childLL phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf1010012 :
+    adaptiveCoverCheck 9 (childHL (childLH (childLL phiAboveCell1010))) = true := by
+  have h : ((childHL (childLH (childLL phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH (childLL phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf1010013 :
+    adaptiveCoverCheck 9 (childHH (childLH (childLL phiAboveCell1010))) = true := by
+  have h : ((childHH (childLH (childLL phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH (childLL phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf101002 :
+    adaptiveCoverCheck 10 (childHL (childLL phiAboveCell1010)) = true := by
+  have h : ((childHL (childLL phiAboveCell1010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childLL phiAboveCell1010)) h
+theorem e24KC2PhiAboveLeaf101003 :
+    adaptiveCoverCheck 10 (childHH (childLL phiAboveCell1010)) = true := by
+  have h : ((childHH (childLL phiAboveCell1010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childLL phiAboveCell1010)) h
+theorem e24KC2PhiAboveLeaf1010100 :
+    adaptiveCoverCheck 9 (childLL (childLL (childLH phiAboveCell1010))) = true := by
+  have h : ((childLL (childLL (childLH phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL (childLH phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf1010101 :
+    adaptiveCoverCheck 9 (childLH (childLL (childLH phiAboveCell1010))) = true := by
+  have h : ((childLH (childLL (childLH phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL (childLH phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf1010102 :
+    adaptiveCoverCheck 9 (childHL (childLL (childLH phiAboveCell1010))) = true := by
+  have h : ((childHL (childLL (childLH phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL (childLH phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf1010103 :
+    adaptiveCoverCheck 9 (childHH (childLL (childLH phiAboveCell1010))) = true := by
+  have h : ((childHH (childLL (childLH phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL (childLH phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf1010110 :
+    adaptiveCoverCheck 9 (childLL (childLH (childLH phiAboveCell1010))) = true := by
+  have h : ((childLL (childLH (childLH phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH (childLH phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf1010111 :
+    adaptiveCoverCheck 9 (childLH (childLH (childLH phiAboveCell1010))) = true := by
+  have h : ((childLH (childLH (childLH phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH (childLH phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf1010112 :
+    adaptiveCoverCheck 9 (childHL (childLH (childLH phiAboveCell1010))) = true := by
+  have h : ((childHL (childLH (childLH phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH (childLH phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf1010113 :
+    adaptiveCoverCheck 9 (childHH (childLH (childLH phiAboveCell1010))) = true := by
+  have h : ((childHH (childLH (childLH phiAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH (childLH phiAboveCell1010))) h
+theorem e24KC2PhiAboveLeaf101012 :
+    adaptiveCoverCheck 10 (childHL (childLH phiAboveCell1010)) = true := by
+  have h : ((childHL (childLH phiAboveCell1010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childLH phiAboveCell1010)) h
+theorem e24KC2PhiAboveLeaf101013 :
+    adaptiveCoverCheck 10 (childHH (childLH phiAboveCell1010)) = true := by
+  have h : ((childHH (childLH phiAboveCell1010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childLH phiAboveCell1010)) h
+theorem e24KC2PhiAboveLeaf10102 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1010) = true := by
+  have h : ((childHL phiAboveCell1010)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell1010) h
+theorem e24KC2PhiAboveLeaf10103 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1010) = true := by
+  have h : ((childHH phiAboveCell1010)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell1010) h
+theorem e24KC2PhiAboveLeaf1011000 :
+    adaptiveCoverCheck 9 (childLL (childLL (childLL phiAboveCell1011))) = true := by
+  have h : ((childLL (childLL (childLL phiAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL (childLL phiAboveCell1011))) h
+theorem e24KC2PhiAboveLeaf1011001 :
+    adaptiveCoverCheck 9 (childLH (childLL (childLL phiAboveCell1011))) = true := by
+  have h : ((childLH (childLL (childLL phiAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL (childLL phiAboveCell1011))) h
+theorem e24KC2PhiAboveLeaf1011002 :
+    adaptiveCoverCheck 9 (childHL (childLL (childLL phiAboveCell1011))) = true := by
+  have h : ((childHL (childLL (childLL phiAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL (childLL phiAboveCell1011))) h
+theorem e24KC2PhiAboveLeaf1011003 :
+    adaptiveCoverCheck 9 (childHH (childLL (childLL phiAboveCell1011))) = true := by
+  have h : ((childHH (childLL (childLL phiAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL (childLL phiAboveCell1011))) h
+theorem e24KC2PhiAboveLeaf1011010 :
+    adaptiveCoverCheck 9 (childLL (childLH (childLL phiAboveCell1011))) = true := by
+  have h : ((childLL (childLH (childLL phiAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH (childLL phiAboveCell1011))) h
+theorem e24KC2PhiAboveLeaf1011012 :
+    adaptiveCoverCheck 9 (childHL (childLH (childLL phiAboveCell1011))) = true := by
+  have h : ((childHL (childLH (childLL phiAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH (childLL phiAboveCell1011))) h
+theorem e24KC2PhiAboveLeaf1011013 :
+    adaptiveCoverCheck 9 (childHH (childLH (childLL phiAboveCell1011))) = true := by
+  have h : ((childHH (childLH (childLL phiAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH (childLL phiAboveCell1011))) h
+theorem e24KC2PhiAboveLeaf101102 :
+    adaptiveCoverCheck 10 (childHL (childLL phiAboveCell1011)) = true := by
+  have h : ((childHL (childLL phiAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childLL phiAboveCell1011)) h
+theorem e24KC2PhiAboveLeaf101103 :
+    adaptiveCoverCheck 10 (childHH (childLL phiAboveCell1011)) = true := by
+  have h : ((childHH (childLL phiAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childLL phiAboveCell1011)) h
+theorem e24KC2PhiAboveLeaf1011102 :
+    adaptiveCoverCheck 9 (childHL (childLL (childLH phiAboveCell1011))) = true := by
+  have h : ((childHL (childLL (childLH phiAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL (childLH phiAboveCell1011))) h
+theorem e24KC2PhiAboveLeaf1011103 :
+    adaptiveCoverCheck 9 (childHH (childLL (childLH phiAboveCell1011))) = true := by
+  have h : ((childHH (childLL (childLH phiAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL (childLH phiAboveCell1011))) h
+theorem e24KC2PhiAboveLeaf1011112 :
+    adaptiveCoverCheck 9 (childHL (childLH (childLH phiAboveCell1011))) = true := by
+  have h : ((childHL (childLH (childLH phiAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH (childLH phiAboveCell1011))) h
+theorem e24KC2PhiAboveLeaf1011113 :
+    adaptiveCoverCheck 9 (childHH (childLH (childLH phiAboveCell1011))) = true := by
+  have h : ((childHH (childLH (childLH phiAboveCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH (childLH phiAboveCell1011))) h
+theorem e24KC2PhiAboveLeaf101112 :
+    adaptiveCoverCheck 10 (childHL (childLH phiAboveCell1011)) = true := by
+  have h : ((childHL (childLH phiAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childLH phiAboveCell1011)) h
+theorem e24KC2PhiAboveLeaf101113 :
+    adaptiveCoverCheck 10 (childHH (childLH phiAboveCell1011)) = true := by
+  have h : ((childHH (childLH phiAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childLH phiAboveCell1011)) h
+theorem e24KC2PhiAboveLeaf10112 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1011) = true := by
+  have h : ((childHL phiAboveCell1011)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell1011) h
+theorem e24KC2PhiAboveLeaf101130 :
+    adaptiveCoverCheck 10 (childLL (childHH phiAboveCell1011)) = true := by
+  have h : ((childLL (childHH phiAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childHH phiAboveCell1011)) h
+theorem e24KC2PhiAboveLeaf101131 :
+    adaptiveCoverCheck 10 (childLH (childHH phiAboveCell1011)) = true := by
+  have h : ((childLH (childHH phiAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childHH phiAboveCell1011)) h
+theorem e24KC2PhiAboveLeaf101132 :
+    adaptiveCoverCheck 10 (childHL (childHH phiAboveCell1011)) = true := by
+  have h : ((childHL (childHH phiAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childHH phiAboveCell1011)) h
+theorem e24KC2PhiAboveLeaf101133 :
+    adaptiveCoverCheck 10 (childHH (childHH phiAboveCell1011)) = true := by
+  have h : ((childHH (childHH phiAboveCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childHH phiAboveCell1011)) h
+theorem e24KC2PhiAboveLeaf10120 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1012) = true := by
+  have h : ((childLL phiAboveCell1012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL phiAboveCell1012) h
+theorem e24KC2PhiAboveLeaf10121 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1012) = true := by
+  have h : ((childLH phiAboveCell1012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH phiAboveCell1012) h
+theorem e24KC2PhiAboveLeaf10122 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1012) = true := by
+  have h : ((childHL phiAboveCell1012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell1012) h
+theorem e24KC2PhiAboveLeaf10123 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1012) = true := by
+  have h : ((childHH phiAboveCell1012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell1012) h
+theorem e24KC2PhiAboveLeaf10130 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1013) = true := by
+  have h : ((childLL phiAboveCell1013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL phiAboveCell1013) h
+theorem e24KC2PhiAboveLeaf10131 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1013) = true := by
+  have h : ((childLH phiAboveCell1013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH phiAboveCell1013) h
+theorem e24KC2PhiAboveLeaf10132 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1013) = true := by
+  have h : ((childHL phiAboveCell1013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell1013) h
+theorem e24KC2PhiAboveLeaf10133 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1013) = true := by
+  have h : ((childHH phiAboveCell1013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell1013) h
+theorem e24KC2PhiAboveLeaf1020 :
+    adaptiveCoverCheck 12 phiAboveCell1020 = true := by
+  have h : (phiAboveCell1020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1020 h
+theorem e24KC2PhiAboveLeaf1021 :
+    adaptiveCoverCheck 12 phiAboveCell1021 = true := by
+  have h : (phiAboveCell1021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1021 h
+theorem e24KC2PhiAboveLeaf1022 :
+    adaptiveCoverCheck 12 phiAboveCell1022 = true := by
+  have h : (phiAboveCell1022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1022 h
+theorem e24KC2PhiAboveLeaf1023 :
+    adaptiveCoverCheck 12 phiAboveCell1023 = true := by
+  have h : (phiAboveCell1023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1023 h
+theorem e24KC2PhiAboveLeaf1030 :
+    adaptiveCoverCheck 12 phiAboveCell1030 = true := by
+  have h : (phiAboveCell1030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1030 h
+theorem e24KC2PhiAboveLeaf1031 :
+    adaptiveCoverCheck 12 phiAboveCell1031 = true := by
+  have h : (phiAboveCell1031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1031 h
+theorem e24KC2PhiAboveLeaf1032 :
+    adaptiveCoverCheck 12 phiAboveCell1032 = true := by
+  have h : (phiAboveCell1032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1032 h
+theorem e24KC2PhiAboveLeaf1033 :
+    adaptiveCoverCheck 12 phiAboveCell1033 = true := by
+  have h : (phiAboveCell1033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1033 h
+theorem e24KC2PhiAboveLeaf1100002 :
+    adaptiveCoverCheck 9 (childHL (childLL (childLL phiAboveCell1100))) = true := by
+  have h : ((childHL (childLL (childLL phiAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL (childLL phiAboveCell1100))) h
+theorem e24KC2PhiAboveLeaf1100003 :
+    adaptiveCoverCheck 9 (childHH (childLL (childLL phiAboveCell1100))) = true := by
+  have h : ((childHH (childLL (childLL phiAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL (childLL phiAboveCell1100))) h
+theorem e24KC2PhiAboveLeaf1100012 :
+    adaptiveCoverCheck 9 (childHL (childLH (childLL phiAboveCell1100))) = true := by
+  have h : ((childHL (childLH (childLL phiAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH (childLL phiAboveCell1100))) h
+theorem e24KC2PhiAboveLeaf1100013 :
+    adaptiveCoverCheck 9 (childHH (childLH (childLL phiAboveCell1100))) = true := by
+  have h : ((childHH (childLH (childLL phiAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH (childLL phiAboveCell1100))) h
+theorem e24KC2PhiAboveLeaf110002 :
+    adaptiveCoverCheck 10 (childHL (childLL phiAboveCell1100)) = true := by
+  have h : ((childHL (childLL phiAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childLL phiAboveCell1100)) h
+theorem e24KC2PhiAboveLeaf110003 :
+    adaptiveCoverCheck 10 (childHH (childLL phiAboveCell1100)) = true := by
+  have h : ((childHH (childLL phiAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childLL phiAboveCell1100)) h
+theorem e24KC2PhiAboveLeaf1100102 :
+    adaptiveCoverCheck 9 (childHL (childLL (childLH phiAboveCell1100))) = true := by
+  have h : ((childHL (childLL (childLH phiAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL (childLH phiAboveCell1100))) h
+theorem e24KC2PhiAboveLeaf1100103 :
+    adaptiveCoverCheck 9 (childHH (childLL (childLH phiAboveCell1100))) = true := by
+  have h : ((childHH (childLL (childLH phiAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL (childLH phiAboveCell1100))) h
+theorem e24KC2PhiAboveLeaf1100112 :
+    adaptiveCoverCheck 9 (childHL (childLH (childLH phiAboveCell1100))) = true := by
+  have h : ((childHL (childLH (childLH phiAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH (childLH phiAboveCell1100))) h
+theorem e24KC2PhiAboveLeaf1100113 :
+    adaptiveCoverCheck 9 (childHH (childLH (childLH phiAboveCell1100))) = true := by
+  have h : ((childHH (childLH (childLH phiAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH (childLH phiAboveCell1100))) h
+theorem e24KC2PhiAboveLeaf110012 :
+    adaptiveCoverCheck 10 (childHL (childLH phiAboveCell1100)) = true := by
+  have h : ((childHL (childLH phiAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childLH phiAboveCell1100)) h
+theorem e24KC2PhiAboveLeaf1100130 :
+    adaptiveCoverCheck 9 (childLL (childHH (childLH phiAboveCell1100))) = true := by
+  have h : ((childLL (childHH (childLH phiAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH (childLH phiAboveCell1100))) h
+theorem e24KC2PhiAboveLeaf1100131 :
+    adaptiveCoverCheck 9 (childLH (childHH (childLH phiAboveCell1100))) = true := by
+  have h : ((childLH (childHH (childLH phiAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH (childLH phiAboveCell1100))) h
+theorem e24KC2PhiAboveLeaf1100132 :
+    adaptiveCoverCheck 9 (childHL (childHH (childLH phiAboveCell1100))) = true := by
+  have h : ((childHL (childHH (childLH phiAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH (childLH phiAboveCell1100))) h
+theorem e24KC2PhiAboveLeaf1100133 :
+    adaptiveCoverCheck 9 (childHH (childHH (childLH phiAboveCell1100))) = true := by
+  have h : ((childHH (childHH (childLH phiAboveCell1100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH (childLH phiAboveCell1100))) h
+theorem e24KC2PhiAboveLeaf110020 :
+    adaptiveCoverCheck 10 (childLL (childHL phiAboveCell1100)) = true := by
+  have h : ((childLL (childHL phiAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childHL phiAboveCell1100)) h
+theorem e24KC2PhiAboveLeaf110021 :
+    adaptiveCoverCheck 10 (childLH (childHL phiAboveCell1100)) = true := by
+  have h : ((childLH (childHL phiAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childHL phiAboveCell1100)) h
+theorem e24KC2PhiAboveLeaf110022 :
+    adaptiveCoverCheck 10 (childHL (childHL phiAboveCell1100)) = true := by
+  have h : ((childHL (childHL phiAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childHL phiAboveCell1100)) h
+theorem e24KC2PhiAboveLeaf110023 :
+    adaptiveCoverCheck 10 (childHH (childHL phiAboveCell1100)) = true := by
+  have h : ((childHH (childHL phiAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childHL phiAboveCell1100)) h
+theorem e24KC2PhiAboveLeaf110030 :
+    adaptiveCoverCheck 10 (childLL (childHH phiAboveCell1100)) = true := by
+  have h : ((childLL (childHH phiAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childHH phiAboveCell1100)) h
+theorem e24KC2PhiAboveLeaf110031 :
+    adaptiveCoverCheck 10 (childLH (childHH phiAboveCell1100)) = true := by
+  have h : ((childLH (childHH phiAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childHH phiAboveCell1100)) h
+theorem e24KC2PhiAboveLeaf110032 :
+    adaptiveCoverCheck 10 (childHL (childHH phiAboveCell1100)) = true := by
+  have h : ((childHL (childHH phiAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childHH phiAboveCell1100)) h
+theorem e24KC2PhiAboveLeaf110033 :
+    adaptiveCoverCheck 10 (childHH (childHH phiAboveCell1100)) = true := by
+  have h : ((childHH (childHH phiAboveCell1100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childHH phiAboveCell1100)) h
+theorem e24KC2PhiAboveLeaf1101002 :
+    adaptiveCoverCheck 9 (childHL (childLL (childLL phiAboveCell1101))) = true := by
+  have h : ((childHL (childLL (childLL phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL (childLL phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101003 :
+    adaptiveCoverCheck 9 (childHH (childLL (childLL phiAboveCell1101))) = true := by
+  have h : ((childHH (childLL (childLL phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL (childLL phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101020 :
+    adaptiveCoverCheck 9 (childLL (childHL (childLL phiAboveCell1101))) = true := by
+  have h : ((childLL (childHL (childLL phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL (childLL phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101021 :
+    adaptiveCoverCheck 9 (childLH (childHL (childLL phiAboveCell1101))) = true := by
+  have h : ((childLH (childHL (childLL phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL (childLL phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101022 :
+    adaptiveCoverCheck 9 (childHL (childHL (childLL phiAboveCell1101))) = true := by
+  have h : ((childHL (childHL (childLL phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL (childLL phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101023 :
+    adaptiveCoverCheck 9 (childHH (childHL (childLL phiAboveCell1101))) = true := by
+  have h : ((childHH (childHL (childLL phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL (childLL phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101030 :
+    adaptiveCoverCheck 9 (childLL (childHH (childLL phiAboveCell1101))) = true := by
+  have h : ((childLL (childHH (childLL phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH (childLL phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101031 :
+    adaptiveCoverCheck 9 (childLH (childHH (childLL phiAboveCell1101))) = true := by
+  have h : ((childLH (childHH (childLL phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH (childLL phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101032 :
+    adaptiveCoverCheck 9 (childHL (childHH (childLL phiAboveCell1101))) = true := by
+  have h : ((childHL (childHH (childLL phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH (childLL phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101033 :
+    adaptiveCoverCheck 9 (childHH (childHH (childLL phiAboveCell1101))) = true := by
+  have h : ((childHH (childHH (childLL phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH (childLL phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101120 :
+    adaptiveCoverCheck 9 (childLL (childHL (childLH phiAboveCell1101))) = true := by
+  have h : ((childLL (childHL (childLH phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL (childLH phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101121 :
+    adaptiveCoverCheck 9 (childLH (childHL (childLH phiAboveCell1101))) = true := by
+  have h : ((childLH (childHL (childLH phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL (childLH phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101122 :
+    adaptiveCoverCheck 9 (childHL (childHL (childLH phiAboveCell1101))) = true := by
+  have h : ((childHL (childHL (childLH phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL (childLH phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101123 :
+    adaptiveCoverCheck 9 (childHH (childHL (childLH phiAboveCell1101))) = true := by
+  have h : ((childHH (childHL (childLH phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL (childLH phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101130 :
+    adaptiveCoverCheck 9 (childLL (childHH (childLH phiAboveCell1101))) = true := by
+  have h : ((childLL (childHH (childLH phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH (childLH phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101131 :
+    adaptiveCoverCheck 9 (childLH (childHH (childLH phiAboveCell1101))) = true := by
+  have h : ((childLH (childHH (childLH phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH (childLH phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101132 :
+    adaptiveCoverCheck 9 (childHL (childHH (childLH phiAboveCell1101))) = true := by
+  have h : ((childHL (childHH (childLH phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH (childLH phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf1101133 :
+    adaptiveCoverCheck 9 (childHH (childHH (childLH phiAboveCell1101))) = true := by
+  have h : ((childHH (childHH (childLH phiAboveCell1101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH (childLH phiAboveCell1101))) h
+theorem e24KC2PhiAboveLeaf110120 :
+    adaptiveCoverCheck 10 (childLL (childHL phiAboveCell1101)) = true := by
+  have h : ((childLL (childHL phiAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childHL phiAboveCell1101)) h
+theorem e24KC2PhiAboveLeaf110121 :
+    adaptiveCoverCheck 10 (childLH (childHL phiAboveCell1101)) = true := by
+  have h : ((childLH (childHL phiAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childHL phiAboveCell1101)) h
+theorem e24KC2PhiAboveLeaf110122 :
+    adaptiveCoverCheck 10 (childHL (childHL phiAboveCell1101)) = true := by
+  have h : ((childHL (childHL phiAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childHL phiAboveCell1101)) h
+theorem e24KC2PhiAboveLeaf110123 :
+    adaptiveCoverCheck 10 (childHH (childHL phiAboveCell1101)) = true := by
+  have h : ((childHH (childHL phiAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childHL phiAboveCell1101)) h
+theorem e24KC2PhiAboveLeaf110130 :
+    adaptiveCoverCheck 10 (childLL (childHH phiAboveCell1101)) = true := by
+  have h : ((childLL (childHH phiAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childHH phiAboveCell1101)) h
+theorem e24KC2PhiAboveLeaf110131 :
+    adaptiveCoverCheck 10 (childLH (childHH phiAboveCell1101)) = true := by
+  have h : ((childLH (childHH phiAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childHH phiAboveCell1101)) h
+theorem e24KC2PhiAboveLeaf110132 :
+    adaptiveCoverCheck 10 (childHL (childHH phiAboveCell1101)) = true := by
+  have h : ((childHL (childHH phiAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childHH phiAboveCell1101)) h
+theorem e24KC2PhiAboveLeaf110133 :
+    adaptiveCoverCheck 10 (childHH (childHH phiAboveCell1101)) = true := by
+  have h : ((childHH (childHH phiAboveCell1101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childHH phiAboveCell1101)) h
+theorem e24KC2PhiAboveLeaf11020 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1102) = true := by
+  have h : ((childLL phiAboveCell1102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL phiAboveCell1102) h
+theorem e24KC2PhiAboveLeaf11021 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1102) = true := by
+  have h : ((childLH phiAboveCell1102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH phiAboveCell1102) h
+theorem e24KC2PhiAboveLeaf11022 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1102) = true := by
+  have h : ((childHL phiAboveCell1102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell1102) h
+theorem e24KC2PhiAboveLeaf11023 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1102) = true := by
+  have h : ((childHH phiAboveCell1102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell1102) h
+theorem e24KC2PhiAboveLeaf11030 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1103) = true := by
+  have h : ((childLL phiAboveCell1103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL phiAboveCell1103) h
+theorem e24KC2PhiAboveLeaf11031 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1103) = true := by
+  have h : ((childLH phiAboveCell1103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH phiAboveCell1103) h
+theorem e24KC2PhiAboveLeaf11032 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1103) = true := by
+  have h : ((childHL phiAboveCell1103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell1103) h
+theorem e24KC2PhiAboveLeaf11033 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1103) = true := by
+  have h : ((childHH phiAboveCell1103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell1103) h
+theorem e24KC2PhiAboveLeaf1110020 :
+    adaptiveCoverCheck 9 (childLL (childHL (childLL phiAboveCell1110))) = true := by
+  have h : ((childLL (childHL (childLL phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL (childLL phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf1110021 :
+    adaptiveCoverCheck 9 (childLH (childHL (childLL phiAboveCell1110))) = true := by
+  have h : ((childLH (childHL (childLL phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL (childLL phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf1110022 :
+    adaptiveCoverCheck 9 (childHL (childHL (childLL phiAboveCell1110))) = true := by
+  have h : ((childHL (childHL (childLL phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL (childLL phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf1110023 :
+    adaptiveCoverCheck 9 (childHH (childHL (childLL phiAboveCell1110))) = true := by
+  have h : ((childHH (childHL (childLL phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL (childLL phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf1110030 :
+    adaptiveCoverCheck 9 (childLL (childHH (childLL phiAboveCell1110))) = true := by
+  have h : ((childLL (childHH (childLL phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH (childLL phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf1110031 :
+    adaptiveCoverCheck 9 (childLH (childHH (childLL phiAboveCell1110))) = true := by
+  have h : ((childLH (childHH (childLL phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH (childLL phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf1110032 :
+    adaptiveCoverCheck 9 (childHL (childHH (childLL phiAboveCell1110))) = true := by
+  have h : ((childHL (childHH (childLL phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH (childLL phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf1110033 :
+    adaptiveCoverCheck 9 (childHH (childHH (childLL phiAboveCell1110))) = true := by
+  have h : ((childHH (childHH (childLL phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH (childLL phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf1110120 :
+    adaptiveCoverCheck 9 (childLL (childHL (childLH phiAboveCell1110))) = true := by
+  have h : ((childLL (childHL (childLH phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL (childLH phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf1110121 :
+    adaptiveCoverCheck 9 (childLH (childHL (childLH phiAboveCell1110))) = true := by
+  have h : ((childLH (childHL (childLH phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL (childLH phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf1110122 :
+    adaptiveCoverCheck 9 (childHL (childHL (childLH phiAboveCell1110))) = true := by
+  have h : ((childHL (childHL (childLH phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL (childLH phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf1110123 :
+    adaptiveCoverCheck 9 (childHH (childHL (childLH phiAboveCell1110))) = true := by
+  have h : ((childHH (childHL (childLH phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL (childLH phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf1110130 :
+    adaptiveCoverCheck 9 (childLL (childHH (childLH phiAboveCell1110))) = true := by
+  have h : ((childLL (childHH (childLH phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH (childLH phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf1110131 :
+    adaptiveCoverCheck 9 (childLH (childHH (childLH phiAboveCell1110))) = true := by
+  have h : ((childLH (childHH (childLH phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH (childLH phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf1110132 :
+    adaptiveCoverCheck 9 (childHL (childHH (childLH phiAboveCell1110))) = true := by
+  have h : ((childHL (childHH (childLH phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH (childLH phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf1110133 :
+    adaptiveCoverCheck 9 (childHH (childHH (childLH phiAboveCell1110))) = true := by
+  have h : ((childHH (childHH (childLH phiAboveCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH (childLH phiAboveCell1110))) h
+theorem e24KC2PhiAboveLeaf111020 :
+    adaptiveCoverCheck 10 (childLL (childHL phiAboveCell1110)) = true := by
+  have h : ((childLL (childHL phiAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childHL phiAboveCell1110)) h
+theorem e24KC2PhiAboveLeaf111021 :
+    adaptiveCoverCheck 10 (childLH (childHL phiAboveCell1110)) = true := by
+  have h : ((childLH (childHL phiAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childHL phiAboveCell1110)) h
+theorem e24KC2PhiAboveLeaf111022 :
+    adaptiveCoverCheck 10 (childHL (childHL phiAboveCell1110)) = true := by
+  have h : ((childHL (childHL phiAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childHL phiAboveCell1110)) h
+theorem e24KC2PhiAboveLeaf111023 :
+    adaptiveCoverCheck 10 (childHH (childHL phiAboveCell1110)) = true := by
+  have h : ((childHH (childHL phiAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childHL phiAboveCell1110)) h
+theorem e24KC2PhiAboveLeaf111030 :
+    adaptiveCoverCheck 10 (childLL (childHH phiAboveCell1110)) = true := by
+  have h : ((childLL (childHH phiAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childHH phiAboveCell1110)) h
+theorem e24KC2PhiAboveLeaf111031 :
+    adaptiveCoverCheck 10 (childLH (childHH phiAboveCell1110)) = true := by
+  have h : ((childLH (childHH phiAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childHH phiAboveCell1110)) h
+theorem e24KC2PhiAboveLeaf111032 :
+    adaptiveCoverCheck 10 (childHL (childHH phiAboveCell1110)) = true := by
+  have h : ((childHL (childHH phiAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childHH phiAboveCell1110)) h
+theorem e24KC2PhiAboveLeaf111033 :
+    adaptiveCoverCheck 10 (childHH (childHH phiAboveCell1110)) = true := by
+  have h : ((childHH (childHH phiAboveCell1110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childHH phiAboveCell1110)) h
+theorem e24KC2PhiAboveLeaf1111020 :
+    adaptiveCoverCheck 9 (childLL (childHL (childLL phiAboveCell1111))) = true := by
+  have h : ((childLL (childHL (childLL phiAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL (childLL phiAboveCell1111))) h
+theorem e24KC2PhiAboveLeaf1111021 :
+    adaptiveCoverCheck 9 (childLH (childHL (childLL phiAboveCell1111))) = true := by
+  have h : ((childLH (childHL (childLL phiAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL (childLL phiAboveCell1111))) h
+theorem e24KC2PhiAboveLeaf1111022 :
+    adaptiveCoverCheck 9 (childHL (childHL (childLL phiAboveCell1111))) = true := by
+  have h : ((childHL (childHL (childLL phiAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL (childLL phiAboveCell1111))) h
+theorem e24KC2PhiAboveLeaf1111023 :
+    adaptiveCoverCheck 9 (childHH (childHL (childLL phiAboveCell1111))) = true := by
+  have h : ((childHH (childHL (childLL phiAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL (childLL phiAboveCell1111))) h
+theorem e24KC2PhiAboveLeaf1111030 :
+    adaptiveCoverCheck 9 (childLL (childHH (childLL phiAboveCell1111))) = true := by
+  have h : ((childLL (childHH (childLL phiAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH (childLL phiAboveCell1111))) h
+theorem e24KC2PhiAboveLeaf1111031 :
+    adaptiveCoverCheck 9 (childLH (childHH (childLL phiAboveCell1111))) = true := by
+  have h : ((childLH (childHH (childLL phiAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH (childLL phiAboveCell1111))) h
+theorem e24KC2PhiAboveLeaf1111032 :
+    adaptiveCoverCheck 9 (childHL (childHH (childLL phiAboveCell1111))) = true := by
+  have h : ((childHL (childHH (childLL phiAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH (childLL phiAboveCell1111))) h
+theorem e24KC2PhiAboveLeaf1111033 :
+    adaptiveCoverCheck 9 (childHH (childHH (childLL phiAboveCell1111))) = true := by
+  have h : ((childHH (childHH (childLL phiAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH (childLL phiAboveCell1111))) h
+theorem e24KC2PhiAboveLeaf1111120 :
+    adaptiveCoverCheck 9 (childLL (childHL (childLH phiAboveCell1111))) = true := by
+  have h : ((childLL (childHL (childLH phiAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL (childLH phiAboveCell1111))) h
+theorem e24KC2PhiAboveLeaf1111121 :
+    adaptiveCoverCheck 9 (childLH (childHL (childLH phiAboveCell1111))) = true := by
+  have h : ((childLH (childHL (childLH phiAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL (childLH phiAboveCell1111))) h
+theorem e24KC2PhiAboveLeaf1111122 :
+    adaptiveCoverCheck 9 (childHL (childHL (childLH phiAboveCell1111))) = true := by
+  have h : ((childHL (childHL (childLH phiAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL (childLH phiAboveCell1111))) h
+theorem e24KC2PhiAboveLeaf1111123 :
+    adaptiveCoverCheck 9 (childHH (childHL (childLH phiAboveCell1111))) = true := by
+  have h : ((childHH (childHL (childLH phiAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL (childLH phiAboveCell1111))) h
+theorem e24KC2PhiAboveLeaf1111130 :
+    adaptiveCoverCheck 9 (childLL (childHH (childLH phiAboveCell1111))) = true := by
+  have h : ((childLL (childHH (childLH phiAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH (childLH phiAboveCell1111))) h
+theorem e24KC2PhiAboveLeaf1111132 :
+    adaptiveCoverCheck 9 (childHL (childHH (childLH phiAboveCell1111))) = true := by
+  have h : ((childHL (childHH (childLH phiAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH (childLH phiAboveCell1111))) h
+theorem e24KC2PhiAboveLeaf1111133 :
+    adaptiveCoverCheck 9 (childHH (childHH (childLH phiAboveCell1111))) = true := by
+  have h : ((childHH (childHH (childLH phiAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH (childLH phiAboveCell1111))) h
+theorem e24KC2PhiAboveLeaf111120 :
+    adaptiveCoverCheck 10 (childLL (childHL phiAboveCell1111)) = true := by
+  have h : ((childLL (childHL phiAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childHL phiAboveCell1111)) h
+theorem e24KC2PhiAboveLeaf111121 :
+    adaptiveCoverCheck 10 (childLH (childHL phiAboveCell1111)) = true := by
+  have h : ((childLH (childHL phiAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childHL phiAboveCell1111)) h
+theorem e24KC2PhiAboveLeaf111122 :
+    adaptiveCoverCheck 10 (childHL (childHL phiAboveCell1111)) = true := by
+  have h : ((childHL (childHL phiAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childHL phiAboveCell1111)) h
+theorem e24KC2PhiAboveLeaf111123 :
+    adaptiveCoverCheck 10 (childHH (childHL phiAboveCell1111)) = true := by
+  have h : ((childHH (childHL phiAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childHL phiAboveCell1111)) h
+theorem e24KC2PhiAboveLeaf111130 :
+    adaptiveCoverCheck 10 (childLL (childHH phiAboveCell1111)) = true := by
+  have h : ((childLL (childHH phiAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL (childHH phiAboveCell1111)) h
+theorem e24KC2PhiAboveLeaf111131 :
+    adaptiveCoverCheck 10 (childLH (childHH phiAboveCell1111)) = true := by
+  have h : ((childLH (childHH phiAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH (childHH phiAboveCell1111)) h
+theorem e24KC2PhiAboveLeaf111132 :
+    adaptiveCoverCheck 10 (childHL (childHH phiAboveCell1111)) = true := by
+  have h : ((childHL (childHH phiAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL (childHH phiAboveCell1111)) h
+theorem e24KC2PhiAboveLeaf111133 :
+    adaptiveCoverCheck 10 (childHH (childHH phiAboveCell1111)) = true := by
+  have h : ((childHH (childHH phiAboveCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH (childHH phiAboveCell1111)) h
+theorem e24KC2PhiAboveLeaf11120 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1112) = true := by
+  have h : ((childLL phiAboveCell1112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL phiAboveCell1112) h
+theorem e24KC2PhiAboveLeaf11121 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1112) = true := by
+  have h : ((childLH phiAboveCell1112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH phiAboveCell1112) h
+theorem e24KC2PhiAboveLeaf11122 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1112) = true := by
+  have h : ((childHL phiAboveCell1112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell1112) h
+theorem e24KC2PhiAboveLeaf11123 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1112) = true := by
+  have h : ((childHH phiAboveCell1112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell1112) h
+theorem e24KC2PhiAboveLeaf11130 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1113) = true := by
+  have h : ((childLL phiAboveCell1113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL phiAboveCell1113) h
+theorem e24KC2PhiAboveLeaf11131 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1113) = true := by
+  have h : ((childLH phiAboveCell1113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH phiAboveCell1113) h
+theorem e24KC2PhiAboveLeaf11132 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1113) = true := by
+  have h : ((childHL phiAboveCell1113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL phiAboveCell1113) h
+theorem e24KC2PhiAboveLeaf11133 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1113) = true := by
+  have h : ((childHH phiAboveCell1113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH phiAboveCell1113) h
+theorem e24KC2PhiAboveLeaf1120 :
+    adaptiveCoverCheck 12 phiAboveCell1120 = true := by
+  have h : (phiAboveCell1120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1120 h
+theorem e24KC2PhiAboveLeaf1121 :
+    adaptiveCoverCheck 12 phiAboveCell1121 = true := by
+  have h : (phiAboveCell1121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1121 h
+theorem e24KC2PhiAboveLeaf1122 :
+    adaptiveCoverCheck 12 phiAboveCell1122 = true := by
+  have h : (phiAboveCell1122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1122 h
+theorem e24KC2PhiAboveLeaf1123 :
+    adaptiveCoverCheck 12 phiAboveCell1123 = true := by
+  have h : (phiAboveCell1123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1123 h
+theorem e24KC2PhiAboveLeaf1130 :
+    adaptiveCoverCheck 12 phiAboveCell1130 = true := by
+  have h : (phiAboveCell1130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1130 h
+theorem e24KC2PhiAboveLeaf1131 :
+    adaptiveCoverCheck 12 phiAboveCell1131 = true := by
+  have h : (phiAboveCell1131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1131 h
+theorem e24KC2PhiAboveLeaf1132 :
+    adaptiveCoverCheck 12 phiAboveCell1132 = true := by
+  have h : (phiAboveCell1132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1132 h
+theorem e24KC2PhiAboveLeaf1133 :
+    adaptiveCoverCheck 12 phiAboveCell1133 = true := by
+  have h : (phiAboveCell1133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 phiAboveCell1133 h
+theorem e24KC2PhiAboveLeaf120 :
+    adaptiveCoverCheck 13 (childLL (childHL (childLH e24PhiAboveRoot))) = true := by
+  have h : ((childLL (childHL (childLH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childHL (childLH e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf121 :
+    adaptiveCoverCheck 13 (childLH (childHL (childLH e24PhiAboveRoot))) = true := by
+  have h : ((childLH (childHL (childLH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childHL (childLH e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf122 :
+    adaptiveCoverCheck 13 (childHL (childHL (childLH e24PhiAboveRoot))) = true := by
+  have h : ((childHL (childHL (childLH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childHL (childLH e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf123 :
+    adaptiveCoverCheck 13 (childHH (childHL (childLH e24PhiAboveRoot))) = true := by
+  have h : ((childHH (childHL (childLH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childHL (childLH e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf130 :
+    adaptiveCoverCheck 13 (childLL (childHH (childLH e24PhiAboveRoot))) = true := by
+  have h : ((childLL (childHH (childLH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childHH (childLH e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf131 :
+    adaptiveCoverCheck 13 (childLH (childHH (childLH e24PhiAboveRoot))) = true := by
+  have h : ((childLH (childHH (childLH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childHH (childLH e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf132 :
+    adaptiveCoverCheck 13 (childHL (childHH (childLH e24PhiAboveRoot))) = true := by
+  have h : ((childHL (childHH (childLH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childHH (childLH e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf133 :
+    adaptiveCoverCheck 13 (childHH (childHH (childLH e24PhiAboveRoot))) = true := by
+  have h : ((childHH (childHH (childLH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childHH (childLH e24PhiAboveRoot))) h
+
+theorem e24KC2PhiAboveLeaf2 :
+    adaptiveCoverCheck 15 (childHL e24PhiAboveRoot) = true := by
+  have h : physicallyIrrelevant (childHL e24PhiAboveRoot) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 15 (childHL e24PhiAboveRoot) h
+theorem e24KC2PhiAboveLeaf300 :
+    adaptiveCoverCheck 13 (childLL (childLL (childHH e24PhiAboveRoot))) = true := by
+  have h : ((childLL (childLL (childHH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL (childHH e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf301 :
+    adaptiveCoverCheck 13 (childLH (childLL (childHH e24PhiAboveRoot))) = true := by
+  have h : ((childLH (childLL (childHH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL (childHH e24PhiAboveRoot))) h
+
+theorem e24KC2PhiAboveLeaf302 :
+    adaptiveCoverCheck 13 (childHL (childLL (childHH e24PhiAboveRoot))) = true := by
+  have h : physicallyIrrelevant (childHL (childLL (childHH e24PhiAboveRoot))) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 13 (childHL (childLL (childHH
+    e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf303 :
+    adaptiveCoverCheck 13 (childHH (childLL (childHH e24PhiAboveRoot))) = true := by
+  have h : ((childHH (childLL (childHH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL (childHH e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf310 :
+    adaptiveCoverCheck 13 (childLL (childLH (childHH e24PhiAboveRoot))) = true := by
+  have h : ((childLL (childLH (childHH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH (childHH e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf311 :
+    adaptiveCoverCheck 13 (childLH (childLH (childHH e24PhiAboveRoot))) = true := by
+  have h : ((childLH (childLH (childHH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH (childHH e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf312 :
+    adaptiveCoverCheck 13 (childHL (childLH (childHH e24PhiAboveRoot))) = true := by
+  have h : ((childHL (childLH (childHH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH (childHH e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf313 :
+    adaptiveCoverCheck 13 (childHH (childLH (childHH e24PhiAboveRoot))) = true := by
+  have h : ((childHH (childLH (childHH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH (childHH e24PhiAboveRoot))) h
+
+theorem e24KC2PhiAboveLeaf32 :
+    adaptiveCoverCheck 14 (childHL (childHH e24PhiAboveRoot)) = true := by
+  have h : physicallyIrrelevant (childHL (childHH e24PhiAboveRoot)) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 14 (childHL (childHH e24PhiAboveRoot)) h
+theorem e24KC2PhiAboveLeaf330 :
+    adaptiveCoverCheck 13 (childLL (childHH (childHH e24PhiAboveRoot))) = true := by
+  have h : ((childLL (childHH (childHH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childHH (childHH e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf331 :
+    adaptiveCoverCheck 13 (childLH (childHH (childHH e24PhiAboveRoot))) = true := by
+  have h : ((childLH (childHH (childHH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childHH (childHH e24PhiAboveRoot))) h
+
+theorem e24KC2PhiAboveLeaf332 :
+    adaptiveCoverCheck 13 (childHL (childHH (childHH e24PhiAboveRoot))) = true := by
+  have h : physicallyIrrelevant (childHL (childHH (childHH e24PhiAboveRoot))) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 13 (childHL (childHH (childHH
+    e24PhiAboveRoot))) h
+theorem e24KC2PhiAboveLeaf333 :
+    adaptiveCoverCheck 13 (childHH (childHH (childHH e24PhiAboveRoot))) = true := by
+  have h : ((childHH (childHH (childHH e24PhiAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childHH (childHH e24PhiAboveRoot))) h
+theorem e24KC2PhiBelowLeaf32212 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3221) = true := by
+  have h : ((childHL phiBelowCell3221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3221) h
+theorem e24KC2PhiBelowLeaf32220 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3222) = true := by
+  have h : ((childLL phiBelowCell3222)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3222) h
+theorem e24KC2PhiBelowLeaf32221 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3222) = true := by
+  have h : ((childLH phiBelowCell3222)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3222) h
+theorem e24KC2PhiBelowLeaf32222 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3222) = true := by
+  have h : ((childHL phiBelowCell3222)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3222) h
+theorem e24KC2PhiBelowLeaf32223 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3222) = true := by
+  have h : ((childHH phiBelowCell3222)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3222) h
+theorem e24KC2PhiBelowLeaf32230 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3223) = true := by
+  have h : ((childLL phiBelowCell3223)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3223) h
+theorem e24KC2PhiBelowLeaf32232 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3223) = true := by
+  have h : ((childHL phiBelowCell3223)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3223) h
+theorem e24KC2PhiBelowLeaf32233 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3223) = true := by
+  have h : ((childHH phiBelowCell3223)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3223) h
+theorem e24KC2PhiBelowLeaf33100 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3310) = true := by
+  have h : ((childLL phiBelowCell3310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3310) h
+theorem e24KC2PhiBelowLeaf33101 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3310) = true := by
+  have h : ((childLH phiBelowCell3310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3310) h
+theorem e24KC2PhiBelowLeaf33103 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3310) = true := by
+  have h : ((childHH phiBelowCell3310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3310) h
+theorem e24KC2PhiBelowLeaf33110 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3311) = true := by
+  have h : ((childLL phiBelowCell3311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3311) h
+theorem e24KC2PhiBelowLeaf33111 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3311) = true := by
+  have h : ((childLH phiBelowCell3311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3311) h
+theorem e24KC2PhiBelowLeaf33112 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3311) = true := by
+  have h : ((childHL phiBelowCell3311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL phiBelowCell3311) h
+theorem e24KC2PhiBelowLeaf33113 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3311) = true := by
+  have h : ((childHH phiBelowCell3311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3311) h
+theorem e24KC2PhiBelowLeaf33130 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3313) = true := by
+  have h : ((childLL phiBelowCell3313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL phiBelowCell3313) h
+theorem e24KC2PhiBelowLeaf33131 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3313) = true := by
+  have h : ((childLH phiBelowCell3313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH phiBelowCell3313) h
+theorem e24KC2PhiBelowLeaf33133 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3313) = true := by
+  have h : ((childHH phiBelowCell3313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH phiBelowCell3313) h
+theorem e24KC2ThetaAboveLeaf000000 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0000)) = true := by
+  have h : ((childLL (childLL thetaAboveCell0000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell0000)) h
+theorem e24KC2ThetaAboveLeaf000001 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0000)) = true := by
+  have h : ((childLH (childLL thetaAboveCell0000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell0000)) h
+theorem e24KC2ThetaAboveLeaf000002 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0000)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0000)) h
+theorem e24KC2ThetaAboveLeaf000003 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0000)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0000)) h
+theorem e24KC2ThetaAboveLeaf000010 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0000)) = true := by
+  have h : ((childLL (childLH thetaAboveCell0000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell0000)) h
+theorem e24KC2ThetaAboveLeaf000011 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0000)) = true := by
+  have h : ((childLH (childLH thetaAboveCell0000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell0000)) h
+theorem e24KC2ThetaAboveLeaf000012 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0000)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0000)) h
+theorem e24KC2ThetaAboveLeaf000013 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0000)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0000)) h
+theorem e24KC2ThetaAboveLeaf0000200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell0000))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell0000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell0000))) h
+theorem e24KC2ThetaAboveLeaf0000201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell0000))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell0000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell0000))) h
+theorem e24KC2ThetaAboveLeaf00002020 :
+    adaptiveCoverCheck 11 thetaAboveCell00002020 = true := by
+  have h : (thetaAboveCell00002020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002020 h
+theorem e24KC2ThetaAboveLeaf00002021 :
+    adaptiveCoverCheck 11 thetaAboveCell00002021 = true := by
+  have h : (thetaAboveCell00002021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002021 h
+theorem e24KC2ThetaAboveLeaf000020220 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002022) = true := by
+  have h : ((childLL thetaAboveCell00002022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00002022) h
+theorem e24KC2ThetaAboveLeaf000020221 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002022) = true := by
+  have h : ((childLH thetaAboveCell00002022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00002022) h
+theorem e24KC2ThetaAboveLeaf000020222 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002022) = true := by
+  have h : ((childHL thetaAboveCell00002022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002022) h
+theorem e24KC2ThetaAboveLeaf000020223 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002022) = true := by
+  have h : ((childHH thetaAboveCell00002022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002022) h
+theorem e24KC2ThetaAboveLeaf000020230 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002023) = true := by
+  have h : ((childLL thetaAboveCell00002023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00002023) h
+theorem e24KC2ThetaAboveLeaf000020231 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002023) = true := by
+  have h : ((childLH thetaAboveCell00002023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00002023) h
+theorem e24KC2ThetaAboveLeaf000020232 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002023) = true := by
+  have h : ((childHL thetaAboveCell00002023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002023) h
+theorem e24KC2ThetaAboveLeaf000020233 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002023) = true := by
+  have h : ((childHH thetaAboveCell00002023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002023) h
+theorem e24KC2ThetaAboveLeaf00002030 :
+    adaptiveCoverCheck 11 thetaAboveCell00002030 = true := by
+  have h : (thetaAboveCell00002030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002030 h
+theorem e24KC2ThetaAboveLeaf00002031 :
+    adaptiveCoverCheck 11 thetaAboveCell00002031 = true := by
+  have h : (thetaAboveCell00002031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002031 h
+theorem e24KC2ThetaAboveLeaf000020320 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002032) = true := by
+  have h : ((childLL thetaAboveCell00002032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00002032) h
+theorem e24KC2ThetaAboveLeaf000020321 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002032) = true := by
+  have h : ((childLH thetaAboveCell00002032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00002032) h
+theorem e24KC2ThetaAboveLeaf000020322 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002032) = true := by
+  have h : ((childHL thetaAboveCell00002032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002032) h
+theorem e24KC2ThetaAboveLeaf000020323 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002032) = true := by
+  have h : ((childHH thetaAboveCell00002032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002032) h
+theorem e24KC2ThetaAboveLeaf000020330 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002033) = true := by
+  have h : ((childLL thetaAboveCell00002033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00002033) h
+theorem e24KC2ThetaAboveLeaf000020331 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002033) = true := by
+  have h : ((childLH thetaAboveCell00002033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00002033) h
+theorem e24KC2ThetaAboveLeaf000020332 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002033) = true := by
+  have h : ((childHL thetaAboveCell00002033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002033) h
+theorem e24KC2ThetaAboveLeaf000020333 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002033) = true := by
+  have h : ((childHH thetaAboveCell00002033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002033) h
+theorem e24KC2ThetaAboveLeaf0000210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell0000))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell0000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell0000))) h
+theorem e24KC2ThetaAboveLeaf0000211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell0000))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell0000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell0000))) h
+theorem e24KC2ThetaAboveLeaf00002120 :
+    adaptiveCoverCheck 11 thetaAboveCell00002120 = true := by
+  have h : (thetaAboveCell00002120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002120 h
+theorem e24KC2ThetaAboveLeaf00002121 :
+    adaptiveCoverCheck 11 thetaAboveCell00002121 = true := by
+  have h : (thetaAboveCell00002121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002121 h
+theorem e24KC2ThetaAboveLeaf000021220 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002122) = true := by
+  have h : ((childLL thetaAboveCell00002122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00002122) h
+theorem e24KC2ThetaAboveLeaf000021221 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002122) = true := by
+  have h : ((childLH thetaAboveCell00002122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00002122) h
+theorem e24KC2ThetaAboveLeaf000021222 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002122) = true := by
+  have h : ((childHL thetaAboveCell00002122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002122) h
+theorem e24KC2ThetaAboveLeaf000021223 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002122) = true := by
+  have h : ((childHH thetaAboveCell00002122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002122) h
+theorem e24KC2ThetaAboveLeaf000021230 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002123) = true := by
+  have h : ((childLL thetaAboveCell00002123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00002123) h
+theorem e24KC2ThetaAboveLeaf000021231 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002123) = true := by
+  have h : ((childLH thetaAboveCell00002123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00002123) h
+theorem e24KC2ThetaAboveLeaf000021232 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002123) = true := by
+  have h : ((childHL thetaAboveCell00002123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002123) h
+theorem e24KC2ThetaAboveLeaf000021233 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002123) = true := by
+  have h : ((childHH thetaAboveCell00002123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002123) h
+theorem e24KC2ThetaAboveLeaf00002130 :
+    adaptiveCoverCheck 11 thetaAboveCell00002130 = true := by
+  have h : (thetaAboveCell00002130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002130 h
+theorem e24KC2ThetaAboveLeaf00002131 :
+    adaptiveCoverCheck 11 thetaAboveCell00002131 = true := by
+  have h : (thetaAboveCell00002131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002131 h
+theorem e24KC2ThetaAboveLeaf000021320 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002132) = true := by
+  have h : ((childLL thetaAboveCell00002132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00002132) h
+theorem e24KC2ThetaAboveLeaf000021321 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002132) = true := by
+  have h : ((childLH thetaAboveCell00002132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00002132) h
+theorem e24KC2ThetaAboveLeaf000021322 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002132) = true := by
+  have h : ((childHL thetaAboveCell00002132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002132) h
+theorem e24KC2ThetaAboveLeaf000021323 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002132) = true := by
+  have h : ((childHH thetaAboveCell00002132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002132) h
+theorem e24KC2ThetaAboveLeaf000021330 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002133) = true := by
+  have h : ((childLL thetaAboveCell00002133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00002133) h
+theorem e24KC2ThetaAboveLeaf000021331 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002133) = true := by
+  have h : ((childLH thetaAboveCell00002133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00002133) h
+theorem e24KC2ThetaAboveLeaf000021332 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002133) = true := by
+  have h : ((childHL thetaAboveCell00002133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002133) h
+theorem e24KC2ThetaAboveLeaf000021333 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002133) = true := by
+  have h : ((childHH thetaAboveCell00002133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002133) h
+theorem e24KC2ThetaAboveLeaf0000220000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00002200)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00002200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00002200)) h
+theorem e24KC2ThetaAboveLeaf0000220001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00002200)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00002200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00002200)) h
+theorem e24KC2ThetaAboveLeaf0000220010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00002200)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00002200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00002200)) h
+theorem e24KC2ThetaAboveLeaf0000220011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00002200)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00002200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00002200)) h
+theorem e24KC2ThetaAboveLeaf0000220100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00002201)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00002201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00002201)) h
+theorem e24KC2ThetaAboveLeaf0000220101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00002201)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00002201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00002201)) h
+theorem e24KC2ThetaAboveLeaf0000220110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00002201)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00002201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00002201)) h
+theorem e24KC2ThetaAboveLeaf0000220111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00002201)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00002201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00002201)) h
+theorem e24KC2ThetaAboveLeaf000022020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002202) = true := by
+  have h : ((childLL thetaAboveCell00002202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00002202) h
+theorem e24KC2ThetaAboveLeaf000022021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002202) = true := by
+  have h : ((childLH thetaAboveCell00002202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00002202) h
+theorem e24KC2ThetaAboveLeaf000022022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002202) = true := by
+  have h : ((childHL thetaAboveCell00002202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002202) h
+theorem e24KC2ThetaAboveLeaf000022023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002202) = true := by
+  have h : ((childHH thetaAboveCell00002202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002202) h
+theorem e24KC2ThetaAboveLeaf000022030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002203) = true := by
+  have h : ((childLL thetaAboveCell00002203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00002203) h
+theorem e24KC2ThetaAboveLeaf000022031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002203) = true := by
+  have h : ((childLH thetaAboveCell00002203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00002203) h
+theorem e24KC2ThetaAboveLeaf000022032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002203) = true := by
+  have h : ((childHL thetaAboveCell00002203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002203) h
+theorem e24KC2ThetaAboveLeaf000022033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002203) = true := by
+  have h : ((childHH thetaAboveCell00002203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002203) h
+theorem e24KC2ThetaAboveLeaf0000221000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00002210)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00002210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00002210)) h
+theorem e24KC2ThetaAboveLeaf0000221001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00002210)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00002210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00002210)) h
+theorem e24KC2ThetaAboveLeaf0000221010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00002210)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00002210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00002210)) h
+theorem e24KC2ThetaAboveLeaf0000221011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00002210)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00002210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00002210)) h
+theorem e24KC2ThetaAboveLeaf0000221100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00002211)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00002211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00002211)) h
+theorem e24KC2ThetaAboveLeaf0000221101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00002211)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00002211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00002211)) h
+theorem e24KC2ThetaAboveLeaf0000221110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00002211)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00002211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00002211)) h
+theorem e24KC2ThetaAboveLeaf0000221111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00002211)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00002211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00002211)) h
+theorem e24KC2ThetaAboveLeaf000022120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002212) = true := by
+  have h : ((childLL thetaAboveCell00002212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00002212) h
+theorem e24KC2ThetaAboveLeaf000022121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002212) = true := by
+  have h : ((childLH thetaAboveCell00002212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00002212) h
+theorem e24KC2ThetaAboveLeaf000022122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002212) = true := by
+  have h : ((childHL thetaAboveCell00002212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002212) h
+theorem e24KC2ThetaAboveLeaf000022123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002212) = true := by
+  have h : ((childHH thetaAboveCell00002212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002212) h
+theorem e24KC2ThetaAboveLeaf000022130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002213) = true := by
+  have h : ((childLL thetaAboveCell00002213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00002213) h
+theorem e24KC2ThetaAboveLeaf0000221310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00002213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00002213)) h
+theorem e24KC2ThetaAboveLeaf0000221311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00002213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00002213)) h
+theorem e24KC2ThetaAboveLeaf0000221312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00002213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00002213)) h
+theorem e24KC2ThetaAboveLeaf0000221313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00002213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00002213)) h
+theorem e24KC2ThetaAboveLeaf000022132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002213) = true := by
+  have h : ((childHL thetaAboveCell00002213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002213) h
+theorem e24KC2ThetaAboveLeaf000022133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002213) = true := by
+  have h : ((childHH thetaAboveCell00002213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002213) h
+theorem e24KC2ThetaAboveLeaf00002220 :
+    adaptiveCoverCheck 11 thetaAboveCell00002220 = true := by
+  have h : (thetaAboveCell00002220).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002220 h
+theorem e24KC2ThetaAboveLeaf00002221 :
+    adaptiveCoverCheck 11 thetaAboveCell00002221 = true := by
+  have h : (thetaAboveCell00002221).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002221 h
+theorem e24KC2ThetaAboveLeaf00002222 :
+    adaptiveCoverCheck 11 thetaAboveCell00002222 = true := by
+  have h : (thetaAboveCell00002222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002222 h
+theorem e24KC2ThetaAboveLeaf00002223 :
+    adaptiveCoverCheck 11 thetaAboveCell00002223 = true := by
+  have h : (thetaAboveCell00002223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002223 h
+theorem e24KC2ThetaAboveLeaf00002230 :
+    adaptiveCoverCheck 11 thetaAboveCell00002230 = true := by
+  have h : (thetaAboveCell00002230).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002230 h
+theorem e24KC2ThetaAboveLeaf00002231 :
+    adaptiveCoverCheck 11 thetaAboveCell00002231 = true := by
+  have h : (thetaAboveCell00002231).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002231 h
+theorem e24KC2ThetaAboveLeaf00002232 :
+    adaptiveCoverCheck 11 thetaAboveCell00002232 = true := by
+  have h : (thetaAboveCell00002232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002232 h
+theorem e24KC2ThetaAboveLeaf00002233 :
+    adaptiveCoverCheck 11 thetaAboveCell00002233 = true := by
+  have h : (thetaAboveCell00002233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002233 h
+theorem e24KC2ThetaAboveLeaf0000230000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00002300)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00002300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00002300)) h
+theorem e24KC2ThetaAboveLeaf0000230001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00002300)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00002300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00002300)) h
+theorem e24KC2ThetaAboveLeaf0000230010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00002300)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00002300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00002300)) h
+theorem e24KC2ThetaAboveLeaf0000230011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00002300)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00002300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00002300)) h
+theorem e24KC2ThetaAboveLeaf0000230100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00002301)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00002301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00002301)) h
+theorem e24KC2ThetaAboveLeaf0000230101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00002301)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00002301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00002301)) h
+theorem e24KC2ThetaAboveLeaf0000230110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00002301)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00002301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00002301)) h
+theorem e24KC2ThetaAboveLeaf0000230111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00002301)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00002301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00002301)) h
+theorem e24KC2ThetaAboveLeaf0000230200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00002302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00002302)) h
+theorem e24KC2ThetaAboveLeaf0000230201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00002302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00002302)) h
+theorem e24KC2ThetaAboveLeaf0000230202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00002302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00002302)) h
+theorem e24KC2ThetaAboveLeaf0000230203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00002302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00002302)) h
+theorem e24KC2ThetaAboveLeaf0000230210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00002302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00002302)) h
+theorem e24KC2ThetaAboveLeaf0000230211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00002302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00002302)) h
+theorem e24KC2ThetaAboveLeaf0000230212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00002302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00002302)) h
+theorem e24KC2ThetaAboveLeaf0000230213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00002302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00002302)) h
+theorem e24KC2ThetaAboveLeaf000023022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002302) = true := by
+  have h : ((childHL thetaAboveCell00002302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002302) h
+theorem e24KC2ThetaAboveLeaf000023023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002302) = true := by
+  have h : ((childHH thetaAboveCell00002302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002302) h
+theorem e24KC2ThetaAboveLeaf0000230300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00002303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00002303)) h
+theorem e24KC2ThetaAboveLeaf0000230301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00002303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00002303)) h
+theorem e24KC2ThetaAboveLeaf0000230302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00002303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00002303)) h
+theorem e24KC2ThetaAboveLeaf0000230303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00002303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00002303)) h
+theorem e24KC2ThetaAboveLeaf0000230310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00002303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00002303)) h
+theorem e24KC2ThetaAboveLeaf0000230311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00002303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00002303)) h
+theorem e24KC2ThetaAboveLeaf0000230312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00002303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00002303)) h
+theorem e24KC2ThetaAboveLeaf0000230313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00002303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00002303)) h
+theorem e24KC2ThetaAboveLeaf000023032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002303) = true := by
+  have h : ((childHL thetaAboveCell00002303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002303) h
+theorem e24KC2ThetaAboveLeaf000023033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002303) = true := by
+  have h : ((childHH thetaAboveCell00002303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002303) h
+theorem e24KC2ThetaAboveLeaf0000231000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00002310)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00002310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00002310)) h
+theorem e24KC2ThetaAboveLeaf0000231001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00002310)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00002310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00002310)) h
+theorem e24KC2ThetaAboveLeaf0000231010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00002310)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00002310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00002310)) h
+theorem e24KC2ThetaAboveLeaf0000231011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00002310)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00002310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00002310)) h
+theorem e24KC2ThetaAboveLeaf0000231100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00002311)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00002311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00002311)) h
+theorem e24KC2ThetaAboveLeaf0000231101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00002311)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00002311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00002311)) h
+theorem e24KC2ThetaAboveLeaf0000231110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00002311)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00002311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00002311)) h
+theorem e24KC2ThetaAboveLeaf0000231111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00002311)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00002311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00002311)) h
+theorem e24KC2ThetaAboveLeaf0000231200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00002312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00002312)) h
+theorem e24KC2ThetaAboveLeaf0000231201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00002312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00002312)) h
+theorem e24KC2ThetaAboveLeaf0000231202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00002312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00002312)) h
+theorem e24KC2ThetaAboveLeaf0000231203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00002312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00002312)) h
+theorem e24KC2ThetaAboveLeaf0000231210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00002312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00002312)) h
+theorem e24KC2ThetaAboveLeaf0000231211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00002312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00002312)) h
+theorem e24KC2ThetaAboveLeaf0000231212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00002312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00002312)) h
+theorem e24KC2ThetaAboveLeaf0000231213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00002312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00002312)) h
+theorem e24KC2ThetaAboveLeaf000023122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002312) = true := by
+  have h : ((childHL thetaAboveCell00002312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002312) h
+theorem e24KC2ThetaAboveLeaf000023123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002312) = true := by
+  have h : ((childHH thetaAboveCell00002312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002312) h
+theorem e24KC2ThetaAboveLeaf0000231300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00002313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00002313)) h
+theorem e24KC2ThetaAboveLeaf0000231301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00002313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00002313)) h
+theorem e24KC2ThetaAboveLeaf0000231302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00002313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00002313)) h
+theorem e24KC2ThetaAboveLeaf0000231303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00002313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00002313)) h
+theorem e24KC2ThetaAboveLeaf0000231310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00002313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00002313)) h
+theorem e24KC2ThetaAboveLeaf0000231311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00002313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00002313)) h
+theorem e24KC2ThetaAboveLeaf0000231312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00002313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00002313)) h
+theorem e24KC2ThetaAboveLeaf0000231313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00002313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00002313)) h
+theorem e24KC2ThetaAboveLeaf000023132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002313) = true := by
+  have h : ((childHL thetaAboveCell00002313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00002313) h
+theorem e24KC2ThetaAboveLeaf000023133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002313) = true := by
+  have h : ((childHH thetaAboveCell00002313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00002313) h
+theorem e24KC2ThetaAboveLeaf00002320 :
+    adaptiveCoverCheck 11 thetaAboveCell00002320 = true := by
+  have h : (thetaAboveCell00002320).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002320 h
+theorem e24KC2ThetaAboveLeaf00002321 :
+    adaptiveCoverCheck 11 thetaAboveCell00002321 = true := by
+  have h : (thetaAboveCell00002321).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002321 h
+theorem e24KC2ThetaAboveLeaf00002322 :
+    adaptiveCoverCheck 11 thetaAboveCell00002322 = true := by
+  have h : (thetaAboveCell00002322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002322 h
+theorem e24KC2ThetaAboveLeaf00002323 :
+    adaptiveCoverCheck 11 thetaAboveCell00002323 = true := by
+  have h : (thetaAboveCell00002323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002323 h
+theorem e24KC2ThetaAboveLeaf00002330 :
+    adaptiveCoverCheck 11 thetaAboveCell00002330 = true := by
+  have h : (thetaAboveCell00002330).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002330 h
+theorem e24KC2ThetaAboveLeaf00002331 :
+    adaptiveCoverCheck 11 thetaAboveCell00002331 = true := by
+  have h : (thetaAboveCell00002331).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002331 h
+theorem e24KC2ThetaAboveLeaf00002332 :
+    adaptiveCoverCheck 11 thetaAboveCell00002332 = true := by
+  have h : (thetaAboveCell00002332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002332 h
+theorem e24KC2ThetaAboveLeaf00002333 :
+    adaptiveCoverCheck 11 thetaAboveCell00002333 = true := by
+  have h : (thetaAboveCell00002333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00002333 h
+theorem e24KC2ThetaAboveLeaf0000300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell0000))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell0000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell0000))) h
+theorem e24KC2ThetaAboveLeaf0000301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell0000))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell0000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell0000))) h
+theorem e24KC2ThetaAboveLeaf00003020 :
+    adaptiveCoverCheck 11 thetaAboveCell00003020 = true := by
+  have h : (thetaAboveCell00003020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003020 h
+theorem e24KC2ThetaAboveLeaf00003021 :
+    adaptiveCoverCheck 11 thetaAboveCell00003021 = true := by
+  have h : (thetaAboveCell00003021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003021 h
+theorem e24KC2ThetaAboveLeaf000030220 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003022) = true := by
+  have h : ((childLL thetaAboveCell00003022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00003022) h
+theorem e24KC2ThetaAboveLeaf000030221 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003022) = true := by
+  have h : ((childLH thetaAboveCell00003022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00003022) h
+theorem e24KC2ThetaAboveLeaf000030222 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003022) = true := by
+  have h : ((childHL thetaAboveCell00003022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003022) h
+theorem e24KC2ThetaAboveLeaf000030223 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003022) = true := by
+  have h : ((childHH thetaAboveCell00003022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003022) h
+theorem e24KC2ThetaAboveLeaf000030230 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003023) = true := by
+  have h : ((childLL thetaAboveCell00003023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00003023) h
+theorem e24KC2ThetaAboveLeaf000030231 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003023) = true := by
+  have h : ((childLH thetaAboveCell00003023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00003023) h
+theorem e24KC2ThetaAboveLeaf000030232 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003023) = true := by
+  have h : ((childHL thetaAboveCell00003023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003023) h
+theorem e24KC2ThetaAboveLeaf000030233 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003023) = true := by
+  have h : ((childHH thetaAboveCell00003023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003023) h
+theorem e24KC2ThetaAboveLeaf00003030 :
+    adaptiveCoverCheck 11 thetaAboveCell00003030 = true := by
+  have h : (thetaAboveCell00003030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003030 h
+theorem e24KC2ThetaAboveLeaf00003031 :
+    adaptiveCoverCheck 11 thetaAboveCell00003031 = true := by
+  have h : (thetaAboveCell00003031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003031 h
+theorem e24KC2ThetaAboveLeaf000030320 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003032) = true := by
+  have h : ((childLL thetaAboveCell00003032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00003032) h
+theorem e24KC2ThetaAboveLeaf000030321 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003032) = true := by
+  have h : ((childLH thetaAboveCell00003032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00003032) h
+theorem e24KC2ThetaAboveLeaf000030322 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003032) = true := by
+  have h : ((childHL thetaAboveCell00003032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003032) h
+theorem e24KC2ThetaAboveLeaf000030323 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003032) = true := by
+  have h : ((childHH thetaAboveCell00003032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003032) h
+theorem e24KC2ThetaAboveLeaf000030330 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003033) = true := by
+  have h : ((childLL thetaAboveCell00003033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00003033) h
+theorem e24KC2ThetaAboveLeaf000030331 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003033) = true := by
+  have h : ((childLH thetaAboveCell00003033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00003033) h
+theorem e24KC2ThetaAboveLeaf000030332 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003033) = true := by
+  have h : ((childHL thetaAboveCell00003033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003033) h
+theorem e24KC2ThetaAboveLeaf000030333 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003033) = true := by
+  have h : ((childHH thetaAboveCell00003033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003033) h
+theorem e24KC2ThetaAboveLeaf0000310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell0000))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell0000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell0000))) h
+theorem e24KC2ThetaAboveLeaf0000311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell0000))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell0000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell0000))) h
+theorem e24KC2ThetaAboveLeaf00003120 :
+    adaptiveCoverCheck 11 thetaAboveCell00003120 = true := by
+  have h : (thetaAboveCell00003120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003120 h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6ProofBatchB39d17cf54d27511`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells802ef10e9d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells802ef10e9d
+
+open CertificateCells802ef10e9d
+theorem cover_subtree_f27aed8d7652 :
+    adaptiveCoverCheck 7 thetaAboveCell000023102000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102000
+    (by
+      have h : ((childLL thetaAboveCell000023102000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102000) h)
+    (by
+      have h : ((childLH thetaAboveCell000023102000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102000) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023102000)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023102000)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023102000)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023102000)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023102000)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023102000)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023102000)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023102000)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023102000)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023102000)) h))
+
+theorem cover_subtree_005f989c43cb :
+    adaptiveCoverCheck 7 thetaAboveCell000023102001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102001
+    (by
+      have h : ((childLL thetaAboveCell000023102001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102001) h)
+    (by
+      have h : ((childLH thetaAboveCell000023102001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102001) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023102001)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023102001)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023102001)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023102001)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023102001)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023102001)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023102001)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023102001)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023102001)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023102001)) h))
+
+theorem cover_subtree_9ea21eaf3506 :
+    adaptiveCoverCheck 7 thetaAboveCell000023102002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102002
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023102002)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023102002)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023102002)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023102002)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023102002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023102002)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023102002)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023102002)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023102002)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023102002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023102002)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023102002)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023102002)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023102002)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023102002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023102002)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023102002)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023102002)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023102002)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023102002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023102002)) h))
+
+theorem cover_subtree_aecd625dd7fb :
+    adaptiveCoverCheck 7 thetaAboveCell000023102003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102003
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023102003)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023102003)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023102003)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023102003)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023102003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023102003)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023102003)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023102003)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023102003)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023102003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023102003)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023102003)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023102003)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023102003)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023102003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023102003)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023102003)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023102003)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023102003)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023102003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023102003)) h))
+
+theorem cover_subtree_77a434675cac :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00002310)))
+    cover_subtree_f27aed8d7652
+    cover_subtree_005f989c43cb
+    cover_subtree_9ea21eaf3506
+    cover_subtree_aecd625dd7fb
+
+theorem cover_subtree_422f11b7507e :
+    adaptiveCoverCheck 7 thetaAboveCell000023102010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102010
+    (by
+      have h : ((childLL thetaAboveCell000023102010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102010) h)
+    (by
+      have h : ((childLH thetaAboveCell000023102010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102010) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023102010)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023102010)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023102010)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023102010)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023102010)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023102010)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023102010)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023102010)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023102010)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023102010)) h))
+
+theorem cover_subtree_9333e7948d29 :
+    adaptiveCoverCheck 7 thetaAboveCell000023102011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102011
+    (by
+      have h : ((childLL thetaAboveCell000023102011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102011) h)
+    (by
+      have h : ((childLH thetaAboveCell000023102011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102011) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023102011)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023102011)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023102011)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023102011)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023102011)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023102011)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023102011)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023102011)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023102011)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023102011)) h))
+
+theorem cover_subtree_0e2f446f3080 :
+    adaptiveCoverCheck 7 thetaAboveCell000023102012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102012
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023102012)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023102012)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023102012)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023102012)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023102012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023102012)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023102012)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023102012)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023102012)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023102012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023102012)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023102012)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023102012)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023102012)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023102012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023102012)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023102012)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023102012)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023102012)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023102012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023102012)) h))
+
+theorem cover_subtree_07cb153dceed :
+    adaptiveCoverCheck 7 thetaAboveCell000023102013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102013
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023102013)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023102013)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023102013)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023102013)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023102013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023102013)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023102013)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023102013)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023102013)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023102013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023102013)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023102013)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023102013)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023102013)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023102013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023102013)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023102013)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023102013)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023102013)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023102013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023102013)) h))
+
+theorem cover_subtree_334813029f45 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00002310)))
+    cover_subtree_422f11b7507e
+    cover_subtree_9333e7948d29
+    cover_subtree_0e2f446f3080
+    cover_subtree_07cb153dceed
+
+theorem cover_subtree_c82ade123eba :
+    adaptiveCoverCheck 8 (childHL (childLL (childHL thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL thetaAboveCell00002310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102020
+        (by
+          have h : ((childLL thetaAboveCell000023102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102020) h)
+        (by
+          have h : ((childLH thetaAboveCell000023102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102020) h)
+        (by
+          have h : ((childHL thetaAboveCell000023102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023102020) h)
+        (by
+          have h : ((childHH thetaAboveCell000023102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023102020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102021
+        (by
+          have h : ((childLL thetaAboveCell000023102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102021) h)
+        (by
+          have h : ((childLH thetaAboveCell000023102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102021) h)
+        (by
+          have h : ((childHL thetaAboveCell000023102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023102021) h)
+        (by
+          have h : ((childHH thetaAboveCell000023102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023102021) h))
+    (by
+      have h : (thetaAboveCell000023102022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023102022 h)
+    (by
+      have h : (thetaAboveCell000023102023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023102023 h)
+
+theorem cover_subtree_b2c2dad45076 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHL thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL thetaAboveCell00002310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102030
+        (by
+          have h : ((childLL thetaAboveCell000023102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102030) h)
+        (by
+          have h : ((childLH thetaAboveCell000023102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102030) h)
+        (by
+          have h : ((childHL thetaAboveCell000023102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023102030) h)
+        (by
+          have h : ((childHH thetaAboveCell000023102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023102030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102031
+        (by
+          have h : ((childLL thetaAboveCell000023102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102031) h)
+        (by
+          have h : ((childLH thetaAboveCell000023102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102031) h)
+        (by
+          have h : ((childHL thetaAboveCell000023102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023102031) h)
+        (by
+          have h : ((childHH thetaAboveCell000023102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023102031) h))
+    (by
+      have h : (thetaAboveCell000023102032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023102032 h)
+    (by
+      have h : (thetaAboveCell000023102033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023102033 h)
+
+theorem e24KC2ThetaAboveLeaf0000231020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00002310))
+    cover_subtree_77a434675cac
+    cover_subtree_334813029f45
+    cover_subtree_c82ade123eba
+    cover_subtree_b2c2dad45076
+theorem cover_subtree_d0ee7bfc8527 :
+    adaptiveCoverCheck 7 thetaAboveCell000023102100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102100
+    (by
+      have h : ((childLL thetaAboveCell000023102100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102100) h)
+    (by
+      have h : ((childLH thetaAboveCell000023102100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102100) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023102100)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023102100)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023102100)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023102100)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023102100)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023102100)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023102100)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023102100)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023102100)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023102100)) h))
+
+theorem cover_subtree_58a66e346033 :
+    adaptiveCoverCheck 7 thetaAboveCell000023102101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102101
+    (by
+      have h : ((childLL thetaAboveCell000023102101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102101) h)
+    (by
+      have h : ((childLH thetaAboveCell000023102101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102101) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023102101)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023102101)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023102101)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023102101)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023102101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023102101)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023102101)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023102101)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023102101)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023102101)) h))
+
+theorem cover_subtree_c17867007896 :
+    adaptiveCoverCheck 7 thetaAboveCell000023102102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102102
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023102102)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023102102)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023102102)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023102102)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023102102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023102102)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023102102)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023102102)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023102102)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023102102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023102102)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023102102)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023102102)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023102102)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023102102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023102102)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023102102)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023102102)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023102102)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023102102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023102102)) h))
+
+theorem cover_subtree_70f834eb3976 :
+    adaptiveCoverCheck 7 thetaAboveCell000023102103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102103
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023102103)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023102103)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023102103)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023102103)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023102103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023102103)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023102103)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023102103)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023102103)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023102103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023102103)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023102103)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023102103)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023102103)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023102103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023102103)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023102103)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023102103)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023102103)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023102103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023102103)) h))
+
+theorem cover_subtree_a226aa051b7b :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00002310)))
+    cover_subtree_d0ee7bfc8527
+    cover_subtree_58a66e346033
+    cover_subtree_c17867007896
+    cover_subtree_70f834eb3976
+
+theorem cover_subtree_71a49837eadf :
+    adaptiveCoverCheck 7 thetaAboveCell000023102112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102112
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023102112)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023102112)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023102112)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023102112)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023102112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023102112)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023102112)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023102112)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023102112)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023102112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023102112)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023102112)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023102112)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023102112)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023102112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023102112)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023102112)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023102112)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023102112)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023102112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023102112)) h))
+
+theorem cover_subtree_5a86501aa891 :
+    adaptiveCoverCheck 7 thetaAboveCell000023102113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102113
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023102113)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023102113)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023102113)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023102113)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023102113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023102113)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023102113)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023102113)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023102113)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023102113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023102113)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023102113)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023102113)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023102113)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023102113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023102113)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023102113)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023102113)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023102113)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023102113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023102113)) h))
+
+theorem cover_subtree_1d4344731b0b :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00002310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102110
+        (by
+          have h : ((childLL thetaAboveCell000023102110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102110) h)
+        (by
+          have h : ((childLH thetaAboveCell000023102110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102110) h)
+        (by
+          have h : ((childHL thetaAboveCell000023102110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023102110) h)
+        (by
+          have h : ((childHH thetaAboveCell000023102110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023102110) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102111
+        (by
+          have h : ((childLL thetaAboveCell000023102111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102111) h)
+        (by
+          have h : ((childLH thetaAboveCell000023102111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102111) h)
+        (by
+          have h : ((childHL thetaAboveCell000023102111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023102111) h)
+        (by
+          have h : ((childHH thetaAboveCell000023102111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023102111) h))
+    cover_subtree_71a49837eadf
+    cover_subtree_5a86501aa891
+
+theorem cover_subtree_3bdba90bbb0c :
+    adaptiveCoverCheck 8 (childHL (childLH (childHL thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL thetaAboveCell00002310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102120
+        (by
+          have h : ((childLL thetaAboveCell000023102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102120) h)
+        (by
+          have h : ((childLH thetaAboveCell000023102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102120) h)
+        (by
+          have h : ((childHL thetaAboveCell000023102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023102120) h)
+        (by
+          have h : ((childHH thetaAboveCell000023102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023102120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102121
+        (by
+          have h : ((childLL thetaAboveCell000023102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102121) h)
+        (by
+          have h : ((childLH thetaAboveCell000023102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102121) h)
+        (by
+          have h : ((childHL thetaAboveCell000023102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023102121) h)
+        (by
+          have h : ((childHH thetaAboveCell000023102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023102121) h))
+    (by
+      have h : (thetaAboveCell000023102122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023102122 h)
+    (by
+      have h : (thetaAboveCell000023102123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023102123 h)
+
+theorem cover_subtree_f8da85ca8892 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHL thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL thetaAboveCell00002310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102130
+        (by
+          have h : ((childLL thetaAboveCell000023102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102130) h)
+        (by
+          have h : ((childLH thetaAboveCell000023102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102130) h)
+        (by
+          have h : ((childHL thetaAboveCell000023102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023102130) h)
+        (by
+          have h : ((childHH thetaAboveCell000023102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023102130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023102131
+        (by
+          have h : ((childLL thetaAboveCell000023102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023102131) h)
+        (by
+          have h : ((childLH thetaAboveCell000023102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023102131) h)
+        (by
+          have h : ((childHL thetaAboveCell000023102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023102131) h)
+        (by
+          have h : ((childHH thetaAboveCell000023102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023102131) h))
+    (by
+      have h : (thetaAboveCell000023102132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023102132 h)
+    (by
+      have h : (thetaAboveCell000023102133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023102133 h)
+
+theorem e24KC2ThetaAboveLeaf0000231021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00002310))
+    cover_subtree_a226aa051b7b
+    cover_subtree_1d4344731b0b
+    cover_subtree_3bdba90bbb0c
+    cover_subtree_f8da85ca8892
+theorem e24KC2ThetaAboveLeaf0000231022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00002310))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00002310))) h)
+theorem e24KC2ThetaAboveLeaf0000231023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00002310))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00002310))) h)
+theorem cover_subtree_786f313d71e4 :
+    adaptiveCoverCheck 7 thetaAboveCell000023103002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103002
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023103002)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023103002)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023103002)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023103002)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023103002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023103002)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023103002)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023103002)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023103002)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023103002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023103002)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023103002)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023103002)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023103002)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023103002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023103002)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023103002)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023103002)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023103002)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023103002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023103002)) h))
+
+theorem cover_subtree_7c7cbd1b1de9 :
+    adaptiveCoverCheck 7 thetaAboveCell000023103003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103003
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023103003)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023103003)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023103003)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023103003)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023103003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023103003)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023103003)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023103003)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023103003)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023103003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023103003)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023103003)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023103003)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023103003)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023103003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023103003)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023103003)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023103003)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023103003)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023103003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023103003)) h))
+
+theorem cover_subtree_e95dc4fe2400 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00002310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103000
+        (by
+          have h : ((childLL thetaAboveCell000023103000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103000) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103000) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103000) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103000) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103001
+        (by
+          have h : ((childLL thetaAboveCell000023103001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103001) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103001) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103001) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103001) h))
+    cover_subtree_786f313d71e4
+    cover_subtree_7c7cbd1b1de9
+
+theorem cover_subtree_d1ef0fda9310 :
+    adaptiveCoverCheck 7 thetaAboveCell000023103012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103012
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023103012)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023103012)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023103012)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023103012)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023103012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023103012)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023103012)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023103012)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023103012)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023103012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023103012)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023103012)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023103012)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023103012)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023103012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023103012)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023103012)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023103012)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023103012)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023103012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023103012)) h))
+
+theorem cover_subtree_334bbbdc1a85 :
+    adaptiveCoverCheck 7 thetaAboveCell000023103013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103013
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023103013)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023103013)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023103013)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023103013)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023103013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023103013)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023103013)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023103013)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023103013)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023103013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023103013)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023103013)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023103013)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023103013)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023103013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023103013)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023103013)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023103013)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023103013)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023103013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023103013)) h))
+
+theorem cover_subtree_1b4b2c805488 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00002310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103010
+        (by
+          have h : ((childLL thetaAboveCell000023103010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103010) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103010) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103010) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103010) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103011
+        (by
+          have h : ((childLL thetaAboveCell000023103011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103011) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103011) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103011) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103011) h))
+    cover_subtree_d1ef0fda9310
+    cover_subtree_334bbbdc1a85
+
+theorem cover_subtree_d023ef13133b :
+    adaptiveCoverCheck 8 (childHL (childLL (childHH thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH thetaAboveCell00002310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103020
+        (by
+          have h : ((childLL thetaAboveCell000023103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103020) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103020) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103020) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103021
+        (by
+          have h : ((childLL thetaAboveCell000023103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103021) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103021) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103021) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103021) h))
+    (by
+      have h : (thetaAboveCell000023103022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023103022 h)
+    (by
+      have h : (thetaAboveCell000023103023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023103023 h)
+
+theorem cover_subtree_8107f33bf217 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHH thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH thetaAboveCell00002310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103030
+        (by
+          have h : ((childLL thetaAboveCell000023103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103030) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103030) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103030) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103031
+        (by
+          have h : ((childLL thetaAboveCell000023103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103031) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103031) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103031) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103031) h))
+    (by
+      have h : (thetaAboveCell000023103032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023103032 h)
+    (by
+      have h : (thetaAboveCell000023103033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023103033 h)
+
+theorem e24KC2ThetaAboveLeaf0000231030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00002310))
+    cover_subtree_e95dc4fe2400
+    cover_subtree_1b4b2c805488
+    cover_subtree_d023ef13133b
+    cover_subtree_8107f33bf217
+theorem cover_subtree_e7898c3dadb7 :
+    adaptiveCoverCheck 7 thetaAboveCell000023103102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103102
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023103102)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023103102)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023103102)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023103102)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023103102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023103102)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023103102)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023103102)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023103102)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023103102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023103102)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023103102)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023103102)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023103102)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023103102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023103102)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023103102)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023103102)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023103102)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023103102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023103102)) h))
+
+theorem cover_subtree_76677b2fe65f :
+    adaptiveCoverCheck 7 thetaAboveCell000023103103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103103
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023103103)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023103103)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023103103)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023103103)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023103103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023103103)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023103103)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023103103)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023103103)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023103103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023103103)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023103103)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023103103)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023103103)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023103103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023103103)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023103103)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023103103)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023103103)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023103103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023103103)) h))
+
+theorem cover_subtree_b293a3c8540f :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00002310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103100
+        (by
+          have h : ((childLL thetaAboveCell000023103100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103100) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103100) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103100) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103100) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103101
+        (by
+          have h : ((childLL thetaAboveCell000023103101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103101) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103101) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103101) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103101) h))
+    cover_subtree_e7898c3dadb7
+    cover_subtree_76677b2fe65f
+
+theorem cover_subtree_a17e19b0274e :
+    adaptiveCoverCheck 7 thetaAboveCell000023103112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103112
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023103112)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023103112)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023103112)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023103112)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023103112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023103112)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023103112)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023103112)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023103112)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023103112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023103112)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023103112)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023103112)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023103112)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023103112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023103112)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023103112)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023103112)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023103112)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023103112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023103112)) h))
+
+theorem cover_subtree_e650c47410b7 :
+    adaptiveCoverCheck 7 thetaAboveCell000023103113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103113
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023103113)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023103113)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023103113)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023103113)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023103113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023103113)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023103113)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023103113)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023103113)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023103113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023103113)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023103113)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023103113)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023103113)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023103113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023103113)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023103113)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023103113)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023103113)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023103113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023103113)) h))
+
+theorem cover_subtree_d40c83c06c06 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00002310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103110
+        (by
+          have h : ((childLL thetaAboveCell000023103110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103110) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103110) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103110) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103110) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103111
+        (by
+          have h : ((childLL thetaAboveCell000023103111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103111) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103111) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103111) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103111) h))
+    cover_subtree_a17e19b0274e
+    cover_subtree_e650c47410b7
+
+theorem cover_subtree_759d6e2f1f21 :
+    adaptiveCoverCheck 8 (childHL (childLH (childHH thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH thetaAboveCell00002310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103120
+        (by
+          have h : ((childLL thetaAboveCell000023103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103120) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103120) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103120) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103121
+        (by
+          have h : ((childLL thetaAboveCell000023103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103121) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103121) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103121) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103121) h))
+    (by
+      have h : (thetaAboveCell000023103122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023103122 h)
+    (by
+      have h : (thetaAboveCell000023103123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023103123 h)
+
+theorem cover_subtree_695ff4e0f9a2 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHH thetaAboveCell00002310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH thetaAboveCell00002310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103130
+        (by
+          have h : ((childLL thetaAboveCell000023103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103130) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103130) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103130) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023103131
+        (by
+          have h : ((childLL thetaAboveCell000023103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023103131) h)
+        (by
+          have h : ((childLH thetaAboveCell000023103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023103131) h)
+        (by
+          have h : ((childHL thetaAboveCell000023103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023103131) h)
+        (by
+          have h : ((childHH thetaAboveCell000023103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023103131) h))
+    (by
+      have h : (thetaAboveCell000023103132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023103132 h)
+    (by
+      have h : (thetaAboveCell000023103133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023103133 h)
+
+theorem e24KC2ThetaAboveLeaf0000231031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00002310))
+    cover_subtree_b293a3c8540f
+    cover_subtree_d40c83c06c06
+    cover_subtree_759d6e2f1f21
+    cover_subtree_695ff4e0f9a2
+theorem e24KC2ThetaAboveLeaf0000231032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00002310))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00002310))) h)
+theorem e24KC2ThetaAboveLeaf0000231033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00002310))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00002310))) h)
+theorem e24KC2ThetaAboveLeaf0000231102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00002311))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00002311))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLL
+        thetaAboveCell00002311)))
+        (by
+          have h : (thetaAboveCell000023110220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110220 h)
+        (by
+          have h : (thetaAboveCell000023110221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110221 h)
+        (by
+          have h : (thetaAboveCell000023110222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110222 h)
+        (by
+          have h : (thetaAboveCell000023110223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childLL
+        thetaAboveCell00002311)))
+        (by
+          have h : (thetaAboveCell000023110230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110230 h)
+        (by
+          have h : (thetaAboveCell000023110231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110231 h)
+        (by
+          have h : (thetaAboveCell000023110232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110232 h)
+        (by
+          have h : (thetaAboveCell000023110233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110233 h))
+theorem e24KC2ThetaAboveLeaf0000231103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00002311))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00002311))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childLL
+        thetaAboveCell00002311)))
+        (by
+          have h : (thetaAboveCell000023110320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110320 h)
+        (by
+          have h : (thetaAboveCell000023110321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110321 h)
+        (by
+          have h : (thetaAboveCell000023110322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110322 h)
+        (by
+          have h : (thetaAboveCell000023110323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childLL
+        thetaAboveCell00002311)))
+        (by
+          have h : (thetaAboveCell000023110330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110330 h)
+        (by
+          have h : (thetaAboveCell000023110331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110331 h)
+        (by
+          have h : (thetaAboveCell000023110332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110332 h)
+        (by
+          have h : (thetaAboveCell000023110333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023110333 h))
+theorem e24KC2ThetaAboveLeaf0000231112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00002311))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00002311))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLH
+        thetaAboveCell00002311)))
+        (by
+          have h : (thetaAboveCell000023111220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111220 h)
+        (by
+          have h : (thetaAboveCell000023111221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111221 h)
+        (by
+          have h : (thetaAboveCell000023111222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111222 h)
+        (by
+          have h : (thetaAboveCell000023111223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childLH
+        thetaAboveCell00002311)))
+        (by
+          have h : (thetaAboveCell000023111230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111230 h)
+        (by
+          have h : (thetaAboveCell000023111231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111231 h)
+        (by
+          have h : (thetaAboveCell000023111232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111232 h)
+        (by
+          have h : (thetaAboveCell000023111233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111233 h))
+theorem e24KC2ThetaAboveLeaf0000231113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00002311))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00002311))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childLH
+        thetaAboveCell00002311)))
+        (by
+          have h : (thetaAboveCell000023111320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111320 h)
+        (by
+          have h : (thetaAboveCell000023111321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111321 h)
+        (by
+          have h : (thetaAboveCell000023111322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111322 h)
+        (by
+          have h : (thetaAboveCell000023111323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childLH
+        thetaAboveCell00002311)))
+        (by
+          have h : (thetaAboveCell000023111330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111330 h)
+        (by
+          have h : (thetaAboveCell000023111331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111331 h)
+        (by
+          have h : (thetaAboveCell000023111332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111332 h)
+        (by
+          have h : (thetaAboveCell000023111333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023111333 h))
+theorem cover_subtree_b3ebd2cd2770 :
+    adaptiveCoverCheck 7 thetaAboveCell000023112002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112002
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023112002)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023112002)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023112002)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023112002)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023112002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023112002)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023112002)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023112002)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023112002)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023112002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023112002)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023112002)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023112002)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023112002)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023112002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023112002)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023112002)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023112002)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023112002)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023112002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023112002)) h))
+
+theorem cover_subtree_f788d0e2a5ae :
+    adaptiveCoverCheck 7 thetaAboveCell000023112003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112003
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023112003)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023112003)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023112003)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023112003)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023112003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023112003)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023112003)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023112003)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023112003)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023112003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023112003)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023112003)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023112003)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023112003)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023112003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023112003)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023112003)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023112003)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023112003)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023112003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023112003)) h))
+
+theorem cover_subtree_7f7afb8d0a4d :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112000
+        (by
+          have h : ((childLL thetaAboveCell000023112000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112000) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112000) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112000) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112000) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112001
+        (by
+          have h : ((childLL thetaAboveCell000023112001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112001) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112001) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112001) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112001) h))
+    cover_subtree_b3ebd2cd2770
+    cover_subtree_f788d0e2a5ae
+
+theorem cover_subtree_074cd1f5f52c :
+    adaptiveCoverCheck 7 thetaAboveCell000023112012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112012
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023112012)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023112012)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023112012)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023112012)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023112012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023112012)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023112012)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023112012)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023112012)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023112012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023112012)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023112012)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023112012)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023112012)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023112012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023112012)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023112012)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023112012)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023112012)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023112012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023112012)) h))
+
+theorem cover_subtree_dfbe90b65883 :
+    adaptiveCoverCheck 7 thetaAboveCell000023112013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112013
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023112013)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023112013)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023112013)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023112013)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023112013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023112013)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023112013)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023112013)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023112013)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023112013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023112013)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023112013)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023112013)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023112013)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023112013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023112013)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023112013)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023112013)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023112013)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023112013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023112013)) h))
+
+theorem cover_subtree_20dd499bd364 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112010
+        (by
+          have h : ((childLL thetaAboveCell000023112010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112010) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112010) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112010) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112010) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112011
+        (by
+          have h : ((childLL thetaAboveCell000023112011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112011) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112011) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112011) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112011) h))
+    cover_subtree_074cd1f5f52c
+    cover_subtree_dfbe90b65883
+
+theorem cover_subtree_97430f27dc25 :
+    adaptiveCoverCheck 8 (childHL (childLL (childHL thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112020
+        (by
+          have h : ((childLL thetaAboveCell000023112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112020) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112020) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112020) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112021
+        (by
+          have h : ((childLL thetaAboveCell000023112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112021) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112021) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112021) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112021) h))
+    (by
+      have h : (thetaAboveCell000023112022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023112022 h)
+    (by
+      have h : (thetaAboveCell000023112023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023112023 h)
+
+theorem cover_subtree_8ea35ca64562 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHL thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112030
+        (by
+          have h : ((childLL thetaAboveCell000023112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112030) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112030) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112030) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112031
+        (by
+          have h : ((childLL thetaAboveCell000023112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112031) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112031) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112031) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112031) h))
+    (by
+      have h : (thetaAboveCell000023112032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023112032 h)
+    (by
+      have h : (thetaAboveCell000023112033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023112033 h)
+
+theorem e24KC2ThetaAboveLeaf0000231120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00002311))
+    cover_subtree_7f7afb8d0a4d
+    cover_subtree_20dd499bd364
+    cover_subtree_97430f27dc25
+    cover_subtree_8ea35ca64562
+theorem cover_subtree_bb7934ab6a6d :
+    adaptiveCoverCheck 7 thetaAboveCell000023112102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112102
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023112102)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023112102)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023112102)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023112102)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023112102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023112102)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023112102)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023112102)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023112102)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023112102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023112102)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023112102)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023112102)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023112102)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023112102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023112102)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023112102)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023112102)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023112102)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023112102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023112102)) h))
+
+theorem cover_subtree_f960ce5080b6 :
+    adaptiveCoverCheck 7 thetaAboveCell000023112103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112103
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023112103)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023112103)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023112103)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023112103)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023112103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023112103)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023112103)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023112103)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023112103)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023112103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023112103)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023112103)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023112103)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023112103)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023112103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023112103)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023112103)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023112103)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023112103)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023112103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023112103)) h))
+
+theorem cover_subtree_be8b495598ef :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112100
+        (by
+          have h : ((childLL thetaAboveCell000023112100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112100) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112100) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112100) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112100) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112101
+        (by
+          have h : ((childLL thetaAboveCell000023112101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112101) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112101) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112101) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112101) h))
+    cover_subtree_bb7934ab6a6d
+    cover_subtree_f960ce5080b6
+
+theorem cover_subtree_c24cbad7c860 :
+    adaptiveCoverCheck 7 thetaAboveCell000023112112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112112
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023112112)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023112112)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023112112)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023112112)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023112112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023112112)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023112112)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023112112)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023112112)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023112112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023112112)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023112112)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023112112)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023112112)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023112112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023112112)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023112112)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023112112)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023112112)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023112112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023112112)) h))
+
+theorem cover_subtree_5030364bf316 :
+    adaptiveCoverCheck 7 thetaAboveCell000023112113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112113
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023112113)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023112113)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023112113)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023112113)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023112113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023112113)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023112113)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023112113)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023112113)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023112113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023112113)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023112113)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023112113)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023112113)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023112113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023112113)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023112113)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023112113)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023112113)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023112113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023112113)) h))
+
+theorem cover_subtree_b9cce4a71b20 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112110
+        (by
+          have h : ((childLL thetaAboveCell000023112110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112110) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112110) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112110) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112110) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112111
+        (by
+          have h : ((childLL thetaAboveCell000023112111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112111) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112111) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112111) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112111) h))
+    cover_subtree_c24cbad7c860
+    cover_subtree_5030364bf316
+
+theorem cover_subtree_f376dd8d0235 :
+    adaptiveCoverCheck 8 (childHL (childLH (childHL thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112120
+        (by
+          have h : ((childLL thetaAboveCell000023112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112120) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112120) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112120) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112121
+        (by
+          have h : ((childLL thetaAboveCell000023112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112121) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112121) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112121) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112121) h))
+    (by
+      have h : (thetaAboveCell000023112122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023112122 h)
+    (by
+      have h : (thetaAboveCell000023112123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023112123 h)
+
+theorem cover_subtree_a9dfb54a5534 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHL thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112130
+        (by
+          have h : ((childLL thetaAboveCell000023112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112130) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112130) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112130) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023112131
+        (by
+          have h : ((childLL thetaAboveCell000023112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023112131) h)
+        (by
+          have h : ((childLH thetaAboveCell000023112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023112131) h)
+        (by
+          have h : ((childHL thetaAboveCell000023112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023112131) h)
+        (by
+          have h : ((childHH thetaAboveCell000023112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023112131) h))
+    (by
+      have h : (thetaAboveCell000023112132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023112132 h)
+    (by
+      have h : (thetaAboveCell000023112133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023112133 h)
+
+theorem e24KC2ThetaAboveLeaf0000231121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00002311))
+    cover_subtree_be8b495598ef
+    cover_subtree_b9cce4a71b20
+    cover_subtree_f376dd8d0235
+    cover_subtree_a9dfb54a5534
+theorem e24KC2ThetaAboveLeaf0000231122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00002311))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00002311))) h)
+theorem e24KC2ThetaAboveLeaf0000231123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00002311))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00002311))) h)
+theorem cover_subtree_d1cb21971feb :
+    adaptiveCoverCheck 7 thetaAboveCell000023113002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113002
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023113002)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023113002)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023113002)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023113002)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023113002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023113002)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023113002)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023113002)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023113002)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023113002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023113002)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023113002)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023113002)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023113002)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023113002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023113002)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023113002)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023113002)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023113002)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023113002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023113002)) h))
+
+theorem cover_subtree_3efb79a6f9e0 :
+    adaptiveCoverCheck 7 thetaAboveCell000023113003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113003
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023113003)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023113003)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023113003)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023113003)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023113003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023113003)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023113003)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023113003)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023113003)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023113003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023113003)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023113003)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023113003)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023113003)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023113003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023113003)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023113003)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023113003)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023113003)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023113003)) h))
+
+theorem cover_subtree_f6e1fff83561 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113000
+        (by
+          have h : ((childLL thetaAboveCell000023113000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113000) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113000) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113000) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113000) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113001
+        (by
+          have h : ((childLL thetaAboveCell000023113001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113001) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113001) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113001) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113001) h))
+    cover_subtree_d1cb21971feb
+    cover_subtree_3efb79a6f9e0
+
+theorem cover_subtree_4cb56c792c1b :
+    adaptiveCoverCheck 7 thetaAboveCell000023113012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113012
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023113012)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023113012)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023113012)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023113012)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023113012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023113012)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023113012)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023113012)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023113012)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023113012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023113012)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023113012)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023113012)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023113012)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023113012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023113012)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023113012)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023113012)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023113012)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023113012)) h))
+
+theorem cover_subtree_d84633da1a8a :
+    adaptiveCoverCheck 7 thetaAboveCell000023113013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113013
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023113013)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023113013)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023113013)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023113013)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023113013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023113013)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023113013)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023113013)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023113013)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023113013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023113013)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023113013)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023113013)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023113013)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023113013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023113013)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023113013)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023113013)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023113013)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023113013)) h))
+
+theorem cover_subtree_5c601c4361a5 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113010
+        (by
+          have h : ((childLL thetaAboveCell000023113010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113010) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113010) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113010) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113010) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113011
+        (by
+          have h : ((childLL thetaAboveCell000023113011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113011) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113011) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113011) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113011) h))
+    cover_subtree_4cb56c792c1b
+    cover_subtree_d84633da1a8a
+
+theorem cover_subtree_7c8020e781ad :
+    adaptiveCoverCheck 8 (childHL (childLL (childHH thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113020
+        (by
+          have h : ((childLL thetaAboveCell000023113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113020) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113020) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113020) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113021
+        (by
+          have h : ((childLL thetaAboveCell000023113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113021) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113021) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113021) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113021) h))
+    (by
+      have h : (thetaAboveCell000023113022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023113022 h)
+    (by
+      have h : (thetaAboveCell000023113023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023113023 h)
+
+theorem cover_subtree_01c951b510fe :
+    adaptiveCoverCheck 8 (childHH (childLL (childHH thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113030
+        (by
+          have h : ((childLL thetaAboveCell000023113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113030) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113030) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113030) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113031
+        (by
+          have h : ((childLL thetaAboveCell000023113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113031) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113031) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113031) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113031) h))
+    (by
+      have h : (thetaAboveCell000023113032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023113032 h)
+    (by
+      have h : (thetaAboveCell000023113033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023113033 h)
+
+theorem e24KC2ThetaAboveLeaf0000231130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00002311))
+    cover_subtree_f6e1fff83561
+    cover_subtree_5c601c4361a5
+    cover_subtree_7c8020e781ad
+    cover_subtree_01c951b510fe
+theorem cover_subtree_195c372ae99b :
+    adaptiveCoverCheck 7 thetaAboveCell000023113102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113102
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023113102)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023113102)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023113102)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023113102)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023113102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023113102)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023113102)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023113102)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023113102)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023113102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023113102)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023113102)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023113102)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023113102)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023113102)) h))
+    (by
+      have h : ((childHH thetaAboveCell000023113102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113102) h)
+
+theorem cover_subtree_d1cce1dbdb65 :
+    adaptiveCoverCheck 7 thetaAboveCell000023113103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113103
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023113103)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023113103)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023113103)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023113103)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023113103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023113103)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023113103)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023113103)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023113103)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023113103)) h))
+    (by
+      have h : ((childHL thetaAboveCell000023113103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113103) h)
+    (by
+      have h : ((childHH thetaAboveCell000023113103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113103) h)
+
+theorem cover_subtree_f77ed6807865 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113100
+        (by
+          have h : ((childLL thetaAboveCell000023113100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113100) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113100) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113100) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113100) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113101
+        (by
+          have h : ((childLL thetaAboveCell000023113101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113101) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113101) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113101) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113101) h))
+    cover_subtree_195c372ae99b
+    cover_subtree_d1cce1dbdb65
+
+theorem cover_subtree_93cb889e1841 :
+    adaptiveCoverCheck 7 thetaAboveCell000023113112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113112
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023113112)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023113112)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023113112)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023113112)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023113112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023113112)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023113112)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023113112)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023113112)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023113112)) h))
+    (by
+      have h : ((childHL thetaAboveCell000023113112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113112) h)
+    (by
+      have h : ((childHH thetaAboveCell000023113112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113112) h)
+
+theorem cover_subtree_dd2e8b0050d0 :
+    adaptiveCoverCheck 7 thetaAboveCell000023113113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113113
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023113113)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023113113)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023113113)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023113113)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023113113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023113113)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023113113)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023113113)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023113113)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023113113)) h))
+    (by
+      have h : ((childHL thetaAboveCell000023113113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113113) h)
+    (by
+      have h : ((childHH thetaAboveCell000023113113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113113) h)
+
+theorem cover_subtree_9bd401d941dd :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113110
+        (by
+          have h : ((childLL thetaAboveCell000023113110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113110) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113110) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113110) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113110) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113111
+        (by
+          have h : ((childLL thetaAboveCell000023113111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113111) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113111) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113111) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113111) h))
+    cover_subtree_93cb889e1841
+    cover_subtree_dd2e8b0050d0
+
+theorem cover_subtree_17ce5f0eb4e5 :
+    adaptiveCoverCheck 8 (childHL (childLH (childHH thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113120
+        (by
+          have h : ((childLL thetaAboveCell000023113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113120) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113120) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113120) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113121
+        (by
+          have h : ((childLL thetaAboveCell000023113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113121) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113121) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113121) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113121) h))
+    (by
+      have h : (thetaAboveCell000023113122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023113122 h)
+    (by
+      have h : (thetaAboveCell000023113123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023113123 h)
+
+theorem cover_subtree_167fbbc5bd30 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHH thetaAboveCell00002311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH thetaAboveCell00002311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113130
+        (by
+          have h : ((childLL thetaAboveCell000023113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113130) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113130) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113130) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023113131
+        (by
+          have h : ((childLL thetaAboveCell000023113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023113131) h)
+        (by
+          have h : ((childLH thetaAboveCell000023113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023113131) h)
+        (by
+          have h : ((childHL thetaAboveCell000023113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023113131) h)
+        (by
+          have h : ((childHH thetaAboveCell000023113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023113131) h))
+    (by
+      have h : (thetaAboveCell000023113132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023113132 h)
+    (by
+      have h : (thetaAboveCell000023113133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023113133 h)
+
+theorem e24KC2ThetaAboveLeaf0000231131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00002311))
+    cover_subtree_f77ed6807865
+    cover_subtree_9bd401d941dd
+    cover_subtree_17ce5f0eb4e5
+    cover_subtree_167fbbc5bd30
+theorem e24KC2ThetaAboveLeaf0000231132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00002311))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00002311))) h)
+theorem e24KC2ThetaAboveLeaf0000231133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00002311))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00002311))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00002311))) h)
+theorem e24KC2ThetaAboveLeaf0000320002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00003200))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00003200))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLL
+        thetaAboveCell00003200)))
+        (by
+          have h : (thetaAboveCell000032000220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000220 h)
+        (by
+          have h : (thetaAboveCell000032000221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000221 h)
+        (by
+          have h : (thetaAboveCell000032000222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000222 h)
+        (by
+          have h : (thetaAboveCell000032000223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childLL
+        thetaAboveCell00003200)))
+        (by
+          have h : (thetaAboveCell000032000230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000230 h)
+        (by
+          have h : (thetaAboveCell000032000231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000231 h)
+        (by
+          have h : (thetaAboveCell000032000232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000232 h)
+        (by
+          have h : (thetaAboveCell000032000233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000233 h))
+theorem e24KC2ThetaAboveLeaf0000320003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00003200))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00003200))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childLL
+        thetaAboveCell00003200)))
+        (by
+          have h : (thetaAboveCell000032000320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000320 h)
+        (by
+          have h : (thetaAboveCell000032000321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000321 h)
+        (by
+          have h : (thetaAboveCell000032000322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000322 h)
+        (by
+          have h : (thetaAboveCell000032000323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childLL
+        thetaAboveCell00003200)))
+        (by
+          have h : (thetaAboveCell000032000330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000330 h)
+        (by
+          have h : (thetaAboveCell000032000331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000331 h)
+        (by
+          have h : (thetaAboveCell000032000332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000332 h)
+        (by
+          have h : (thetaAboveCell000032000333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032000333 h))
+theorem e24KC2ThetaAboveLeaf0000320012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00003200))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00003200))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLH
+        thetaAboveCell00003200)))
+        (by
+          have h : (thetaAboveCell000032001220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032001220 h)
+        (by
+          have h : (thetaAboveCell000032001221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032001221 h)
+        (by
+          have h : (thetaAboveCell000032001222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032001222 h)
+        (by
+          have h : (thetaAboveCell000032001223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032001223 h))
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell00003200))) h)
+theorem e24KC2ThetaAboveLeaf0000320013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00003200))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell00003200))) h)
+theorem cover_subtree_dc923db2d444 :
+    adaptiveCoverCheck 7 thetaAboveCell000032002002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002002
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000032002002)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000032002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000032002002)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000032002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000032002002)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000032002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000032002002)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000032002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000032002002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000032002002)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000032002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000032002002)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000032002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000032002002)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000032002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000032002002)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000032002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000032002002)) h))
+    (by
+      have h : ((childHL thetaAboveCell000032002002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002002) h)
+    (by
+      have h : ((childHH thetaAboveCell000032002002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002002) h)
+
+theorem cover_subtree_3d16a69d20d7 :
+    adaptiveCoverCheck 7 thetaAboveCell000032002003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002003
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000032002003)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000032002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000032002003)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000032002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000032002003)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000032002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000032002003)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000032002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000032002003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000032002003)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000032002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000032002003)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000032002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000032002003)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000032002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000032002003)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000032002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000032002003)) h))
+    (by
+      have h : ((childHL thetaAboveCell000032002003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002003) h)
+    (by
+      have h : ((childHH thetaAboveCell000032002003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002003) h)
+
+theorem cover_subtree_80a354cc93c8 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002000
+        (by
+          have h : ((childLL thetaAboveCell000032002000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002000) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002000) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002000) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002000) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002001
+        (by
+          have h : ((childLL thetaAboveCell000032002001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002001) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002001) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002001) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002001) h))
+    cover_subtree_dc923db2d444
+    cover_subtree_3d16a69d20d7
+
+theorem cover_subtree_0a679df94e2d :
+    adaptiveCoverCheck 7 thetaAboveCell000032002012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002012
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000032002012)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000032002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000032002012)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000032002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000032002012)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000032002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000032002012)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000032002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000032002012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000032002012)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000032002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000032002012)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000032002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000032002012)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000032002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000032002012)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000032002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000032002012)) h))
+    (by
+      have h : ((childHL thetaAboveCell000032002012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002012) h)
+    (by
+      have h : ((childHH thetaAboveCell000032002012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002012) h)
+
+theorem cover_subtree_a9518d4839ee :
+    adaptiveCoverCheck 7 thetaAboveCell000032002013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002013
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000032002013)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000032002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000032002013)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000032002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000032002013)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000032002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000032002013)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000032002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000032002013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000032002013)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000032002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000032002013)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000032002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000032002013)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000032002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000032002013)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000032002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000032002013)) h))
+    (by
+      have h : ((childHL thetaAboveCell000032002013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002013) h)
+    (by
+      have h : ((childHH thetaAboveCell000032002013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002013) h)
+
+theorem cover_subtree_69eb32740a9f :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002010
+        (by
+          have h : ((childLL thetaAboveCell000032002010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002010) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002010) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002010) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002010) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002011
+        (by
+          have h : ((childLL thetaAboveCell000032002011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002011) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002011) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002011) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002011) h))
+    cover_subtree_0a679df94e2d
+    cover_subtree_a9518d4839ee
+
+theorem cover_subtree_0afe6e641e2b :
+    adaptiveCoverCheck 8 (childHL (childLL (childHL thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002020
+        (by
+          have h : ((childLL thetaAboveCell000032002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002020) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002020) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002020) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002021
+        (by
+          have h : ((childLL thetaAboveCell000032002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002021) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002021) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002021) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002021) h))
+    (by
+      have h : (thetaAboveCell000032002022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032002022 h)
+    (by
+      have h : (thetaAboveCell000032002023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032002023 h)
+
+theorem cover_subtree_13f8c37ba985 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHL thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002030
+        (by
+          have h : ((childLL thetaAboveCell000032002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002030) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002030) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002030) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002031
+        (by
+          have h : ((childLL thetaAboveCell000032002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002031) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002031) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002031) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002031) h))
+    (by
+      have h : (thetaAboveCell000032002032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032002032 h)
+    (by
+      have h : (thetaAboveCell000032002033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032002033 h)
+
+theorem e24KC2ThetaAboveLeaf0000320020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00003200))
+    cover_subtree_80a354cc93c8
+    cover_subtree_69eb32740a9f
+    cover_subtree_0afe6e641e2b
+    cover_subtree_13f8c37ba985
+theorem cover_subtree_6185ee76a9ce :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002100
+        (by
+          have h : ((childLL thetaAboveCell000032002100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002100) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002100) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002100) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002100) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002101
+        (by
+          have h : ((childLL thetaAboveCell000032002101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002101) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002101) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002101) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002101) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002102
+        (by
+          have h : ((childLL thetaAboveCell000032002102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002102) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002102) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002102) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002103
+        (by
+          have h : ((childLL thetaAboveCell000032002103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002103) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002103) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002103) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002103) h))
+
+theorem cover_subtree_d6e21884e5c8 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002110
+        (by
+          have h : ((childLL thetaAboveCell000032002110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002110) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002110) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002110) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002110) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002111
+        (by
+          have h : ((childLL thetaAboveCell000032002111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002111) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002111) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002111) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002111) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002112
+        (by
+          have h : ((childLL thetaAboveCell000032002112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002112) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002112) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002112) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002113
+        (by
+          have h : ((childLL thetaAboveCell000032002113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002113) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002113) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002113) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002113) h))
+
+theorem cover_subtree_d4ae0081215d :
+    adaptiveCoverCheck 8 (childHL (childLH (childHL thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002120
+        (by
+          have h : ((childLL thetaAboveCell000032002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002120) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002120) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002120) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002121
+        (by
+          have h : ((childLL thetaAboveCell000032002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002121) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002121) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002121) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002121) h))
+    (by
+      have h : (thetaAboveCell000032002122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032002122 h)
+    (by
+      have h : (thetaAboveCell000032002123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032002123 h)
+
+theorem cover_subtree_14b21c11f4bb :
+    adaptiveCoverCheck 8 (childHH (childLH (childHL thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002130
+        (by
+          have h : ((childLL thetaAboveCell000032002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002130) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002130) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002130) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032002131
+        (by
+          have h : ((childLL thetaAboveCell000032002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032002131) h)
+        (by
+          have h : ((childLH thetaAboveCell000032002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032002131) h)
+        (by
+          have h : ((childHL thetaAboveCell000032002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032002131) h)
+        (by
+          have h : ((childHH thetaAboveCell000032002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032002131) h))
+    (by
+      have h : (thetaAboveCell000032002132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032002132 h)
+    (by
+      have h : (thetaAboveCell000032002133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032002133 h)
+
+theorem e24KC2ThetaAboveLeaf0000320021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00003200))
+    cover_subtree_6185ee76a9ce
+    cover_subtree_d6e21884e5c8
+    cover_subtree_d4ae0081215d
+    cover_subtree_14b21c11f4bb
+theorem e24KC2ThetaAboveLeaf0000320022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00003200))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00003200))) h)
+theorem e24KC2ThetaAboveLeaf0000320023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00003200))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00003200))) h)
+theorem cover_subtree_0c3f9f5d2a22 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003000
+        (by
+          have h : ((childLL thetaAboveCell000032003000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003000) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003000) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003000) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003000) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003001
+        (by
+          have h : ((childLL thetaAboveCell000032003001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003001) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003001) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003001) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003001) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003002
+        (by
+          have h : ((childLL thetaAboveCell000032003002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003002) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003002) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003002) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003003
+        (by
+          have h : ((childLL thetaAboveCell000032003003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003003) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003003) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003003) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003003) h))
+
+theorem cover_subtree_e68bbaf0142c :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003010
+        (by
+          have h : ((childLL thetaAboveCell000032003010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003010) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003010) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003010) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003010) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003011
+        (by
+          have h : ((childLL thetaAboveCell000032003011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003011) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003011) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003011) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003011) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003012
+        (by
+          have h : ((childLL thetaAboveCell000032003012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003012) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003012) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003012) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003013
+        (by
+          have h : ((childLL thetaAboveCell000032003013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003013) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003013) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003013) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003013) h))
+
+theorem cover_subtree_80b59cc08427 :
+    adaptiveCoverCheck 8 (childHL (childLL (childHH thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003020
+        (by
+          have h : ((childLL thetaAboveCell000032003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003020) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003020) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003020) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003021
+        (by
+          have h : ((childLL thetaAboveCell000032003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003021) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003021) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003021) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003021) h))
+    (by
+      have h : (thetaAboveCell000032003022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032003022 h)
+    (by
+      have h : (thetaAboveCell000032003023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032003023 h)
+
+theorem cover_subtree_45a4c3e7665d :
+    adaptiveCoverCheck 8 (childHH (childLL (childHH thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003030
+        (by
+          have h : ((childLL thetaAboveCell000032003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003030) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003030) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003030) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003031
+        (by
+          have h : ((childLL thetaAboveCell000032003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003031) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003031) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003031) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003031) h))
+    (by
+      have h : (thetaAboveCell000032003032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032003032 h)
+    (by
+      have h : (thetaAboveCell000032003033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032003033 h)
+
+theorem e24KC2ThetaAboveLeaf0000320030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00003200))
+    cover_subtree_0c3f9f5d2a22
+    cover_subtree_e68bbaf0142c
+    cover_subtree_80b59cc08427
+    cover_subtree_45a4c3e7665d
+theorem cover_subtree_cabec6ebeaec :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003100
+        (by
+          have h : ((childLL thetaAboveCell000032003100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003100) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003100) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003100) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003100) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003101
+        (by
+          have h : ((childLL thetaAboveCell000032003101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003101) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003101) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003101) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003101) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003102
+        (by
+          have h : ((childLL thetaAboveCell000032003102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003102) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003102) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003102) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003103
+        (by
+          have h : ((childLL thetaAboveCell000032003103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003103) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003103) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003103) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003103) h))
+
+theorem cover_subtree_f94fc5a27c16 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003110
+        (by
+          have h : ((childLL thetaAboveCell000032003110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003110) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003110) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003110) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003110) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003111
+        (by
+          have h : ((childLL thetaAboveCell000032003111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003111) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003111) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003111) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003111) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003112
+        (by
+          have h : ((childLL thetaAboveCell000032003112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003112) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003112) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003112) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003113
+        (by
+          have h : ((childLL thetaAboveCell000032003113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003113) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003113) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003113) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003113) h))
+
+theorem cover_subtree_5a58662c5bc0 :
+    adaptiveCoverCheck 8 (childHL (childLH (childHH thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003120
+        (by
+          have h : ((childLL thetaAboveCell000032003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003120) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003120) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003120) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003121
+        (by
+          have h : ((childLL thetaAboveCell000032003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003121) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003121) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003121) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003121) h))
+    (by
+      have h : (thetaAboveCell000032003122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032003122 h)
+    (by
+      have h : (thetaAboveCell000032003123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032003123 h)
+
+theorem cover_subtree_38aaf5d7c82e :
+    adaptiveCoverCheck 8 (childHH (childLH (childHH thetaAboveCell00003200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH thetaAboveCell00003200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003130
+        (by
+          have h : ((childLL thetaAboveCell000032003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003130) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003130) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003130) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032003131
+        (by
+          have h : ((childLL thetaAboveCell000032003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032003131) h)
+        (by
+          have h : ((childLH thetaAboveCell000032003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032003131) h)
+        (by
+          have h : ((childHL thetaAboveCell000032003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032003131) h)
+        (by
+          have h : ((childHH thetaAboveCell000032003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032003131) h))
+    (by
+      have h : (thetaAboveCell000032003132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032003132 h)
+    (by
+      have h : (thetaAboveCell000032003133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032003133 h)
+
+theorem e24KC2ThetaAboveLeaf0000320031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00003200))
+    cover_subtree_cabec6ebeaec
+    cover_subtree_f94fc5a27c16
+    cover_subtree_5a58662c5bc0
+    cover_subtree_38aaf5d7c82e
+theorem e24KC2ThetaAboveLeaf0000320032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00003200))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00003200))) h)
+theorem e24KC2ThetaAboveLeaf0000320033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00003200))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00003200))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00003200))) h)
+theorem e24KC2ThetaAboveLeaf0000320102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00003201))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell00003201))) h)
+theorem e24KC2ThetaAboveLeaf0000320103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00003201))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHL (childHH (childLL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLL
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHH (childHH (childLL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLL
+        thetaAboveCell00003201))) h)
+theorem e24KC2ThetaAboveLeaf0000320112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00003201))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHL (childHL (childLH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLH
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell00003201))) h)
+theorem e24KC2ThetaAboveLeaf0000320113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00003201))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell00003201))) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle005.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle005.lean
new file mode 100644
index 0000000000..0108fe293b
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle005.lean
@@ -0,0 +1,11109 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch006`.
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch028`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCells3b1d8a0c5a
+
+/-- Subcell `0001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0001 : AngleCell :=
+  childLH (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0002 : AngleCell :=
+  childHL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0003 : AngleCell :=
+  childHH (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0010 : AngleCell :=
+  childLL (childLH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0011 : AngleCell :=
+  childLH (childLH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0011)))
+
+end GerverSofa.PartE.CertificateCells3b1d8a0c5a
+
+namespace GerverSofa.PartE.CertificateCells3658ca4972
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111132101020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101020 : AngleCell :=
+  childLL (childHL (childLL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101021 : AngleCell :=
+  childLH (childHL (childLL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101022 : AngleCell :=
+  childHL (childHL (childLL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101023 : AngleCell :=
+  childHH (childHL (childLL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101030 : AngleCell :=
+  childLL (childHH (childLL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101031 : AngleCell :=
+  childLH (childHH (childLL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101032 : AngleCell :=
+  childHL (childHH (childLL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101033 : AngleCell :=
+  childHH (childHH (childLL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101120 : AngleCell :=
+  childLL (childHL (childLH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101121 : AngleCell :=
+  childLH (childHL (childLH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101122 : AngleCell :=
+  childHL (childHL (childLH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101123 : AngleCell :=
+  childHH (childHL (childLH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101130 : AngleCell :=
+  childLL (childHH (childLH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101131 : AngleCell :=
+  childLH (childHH (childLH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101132 : AngleCell :=
+  childHL (childHH (childLH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101133 : AngleCell :=
+  childHH (childHH (childLH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101200 : AngleCell :=
+  childLL (childLL (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101201 : AngleCell :=
+  childLH (childLL (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101202 : AngleCell :=
+  childHL (childLL (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101203 : AngleCell :=
+  childHH (childLL (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101210 : AngleCell :=
+  childLL (childLH (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101211 : AngleCell :=
+  childLH (childLH (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101212 : AngleCell :=
+  childHL (childLH (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101213 : AngleCell :=
+  childHH (childLH (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101300 : AngleCell :=
+  childLL (childLL (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101301 : AngleCell :=
+  childLH (childLL (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101302 : AngleCell :=
+  childHL (childLL (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101303 : AngleCell :=
+  childHH (childLL (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101310 : AngleCell :=
+  childLL (childLH (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101311 : AngleCell :=
+  childLH (childLH (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101312 : AngleCell :=
+  childHL (childLH (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101313 : AngleCell :=
+  childHH (childLH (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132101333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132101333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaBelowCell11113210)))
+
+/-- Subcell `111132110020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110020 : AngleCell :=
+  childLL (childHL (childLL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110021 : AngleCell :=
+  childLH (childHL (childLL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110022 : AngleCell :=
+  childHL (childHL (childLL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110023 : AngleCell :=
+  childHH (childHL (childLL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110030 : AngleCell :=
+  childLL (childHH (childLL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110031 : AngleCell :=
+  childLH (childHH (childLL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110032 : AngleCell :=
+  childHL (childHH (childLL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110033 : AngleCell :=
+  childHH (childHH (childLL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110120 : AngleCell :=
+  childLL (childHL (childLH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110121 : AngleCell :=
+  childLH (childHL (childLH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110122 : AngleCell :=
+  childHL (childHL (childLH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110123 : AngleCell :=
+  childHH (childHL (childLH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110130 : AngleCell :=
+  childLL (childHH (childLH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110131 : AngleCell :=
+  childLH (childHH (childLH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110132 : AngleCell :=
+  childHL (childHH (childLH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110133 : AngleCell :=
+  childHH (childHH (childLH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110200 : AngleCell :=
+  childLL (childLL (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110201 : AngleCell :=
+  childLH (childLL (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110202 : AngleCell :=
+  childHL (childLL (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110203 : AngleCell :=
+  childHH (childLL (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110210 : AngleCell :=
+  childLL (childLH (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110211 : AngleCell :=
+  childLH (childLH (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110212 : AngleCell :=
+  childHL (childLH (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110213 : AngleCell :=
+  childHH (childLH (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110300 : AngleCell :=
+  childLL (childLL (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110301 : AngleCell :=
+  childLH (childLL (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110302 : AngleCell :=
+  childHL (childLL (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110303 : AngleCell :=
+  childHH (childLL (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110310 : AngleCell :=
+  childLL (childLH (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110311 : AngleCell :=
+  childLH (childLH (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110312 : AngleCell :=
+  childHL (childLH (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110313 : AngleCell :=
+  childHH (childLH (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132110333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132110333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaBelowCell11113211)))
+
+/-- Subcell `111132111020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111020 : AngleCell :=
+  childLL (childHL (childLL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111021 : AngleCell :=
+  childLH (childHL (childLL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111022 : AngleCell :=
+  childHL (childHL (childLL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111023 : AngleCell :=
+  childHH (childHL (childLL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111030 : AngleCell :=
+  childLL (childHH (childLL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111031 : AngleCell :=
+  childLH (childHH (childLL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111032 : AngleCell :=
+  childHL (childHH (childLL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111033 : AngleCell :=
+  childHH (childHH (childLL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111120 : AngleCell :=
+  childLL (childHL (childLH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111121 : AngleCell :=
+  childLH (childHL (childLH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111122 : AngleCell :=
+  childHL (childHL (childLH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111123 : AngleCell :=
+  childHH (childHL (childLH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111130 : AngleCell :=
+  childLL (childHH (childLH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111131 : AngleCell :=
+  childLH (childHH (childLH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111132 : AngleCell :=
+  childHL (childHH (childLH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111133 : AngleCell :=
+  childHH (childHH (childLH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111200 : AngleCell :=
+  childLL (childLL (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111201 : AngleCell :=
+  childLH (childLL (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111202 : AngleCell :=
+  childHL (childLL (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111203 : AngleCell :=
+  childHH (childLL (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111210 : AngleCell :=
+  childLL (childLH (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111211 : AngleCell :=
+  childLH (childLH (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111212 : AngleCell :=
+  childHL (childLH (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111213 : AngleCell :=
+  childHH (childLH (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111300 : AngleCell :=
+  childLL (childLL (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111301 : AngleCell :=
+  childLH (childLL (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111302 : AngleCell :=
+  childHL (childLL (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111303 : AngleCell :=
+  childHH (childLL (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111310 : AngleCell :=
+  childLL (childLH (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111311 : AngleCell :=
+  childLH (childLH (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111312 : AngleCell :=
+  childHL (childLH (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111313 : AngleCell :=
+  childHH (childLH (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111132111333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132111333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaBelowCell11113211)))
+
+/-- Subcell `111133000020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000020 : AngleCell :=
+  childLL (childHL (childLL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000021 : AngleCell :=
+  childLH (childHL (childLL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000022 : AngleCell :=
+  childHL (childHL (childLL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000023 : AngleCell :=
+  childHH (childHL (childLL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000030 : AngleCell :=
+  childLL (childHH (childLL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000031 : AngleCell :=
+  childLH (childHH (childLL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000032 : AngleCell :=
+  childHL (childHH (childLL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000033 : AngleCell :=
+  childHH (childHH (childLL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000120 : AngleCell :=
+  childLL (childHL (childLH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000121 : AngleCell :=
+  childLH (childHL (childLH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000122 : AngleCell :=
+  childHL (childHL (childLH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000123 : AngleCell :=
+  childHH (childHL (childLH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000130 : AngleCell :=
+  childLL (childHH (childLH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000131 : AngleCell :=
+  childLH (childHH (childLH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000132 : AngleCell :=
+  childHL (childHH (childLH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000133 : AngleCell :=
+  childHH (childHH (childLH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000200 : AngleCell :=
+  childLL (childLL (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000201 : AngleCell :=
+  childLH (childLL (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000202 : AngleCell :=
+  childHL (childLL (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000203 : AngleCell :=
+  childHH (childLL (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000210 : AngleCell :=
+  childLL (childLH (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000211 : AngleCell :=
+  childLH (childLH (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000212 : AngleCell :=
+  childHL (childLH (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000213 : AngleCell :=
+  childHH (childLH (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000300 : AngleCell :=
+  childLL (childLL (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000301 : AngleCell :=
+  childLH (childLL (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000302 : AngleCell :=
+  childHL (childLL (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000303 : AngleCell :=
+  childHH (childLL (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000310 : AngleCell :=
+  childLL (childLH (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000311 : AngleCell :=
+  childLH (childLH (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000312 : AngleCell :=
+  childHL (childLH (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000313 : AngleCell :=
+  childHH (childLH (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133000333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133000333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaBelowCell11113300)))
+
+/-- Subcell `111133001020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001020 : AngleCell :=
+  childLL (childHL (childLL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001021 : AngleCell :=
+  childLH (childHL (childLL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001022 : AngleCell :=
+  childHL (childHL (childLL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001023 : AngleCell :=
+  childHH (childHL (childLL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001030 : AngleCell :=
+  childLL (childHH (childLL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001031 : AngleCell :=
+  childLH (childHH (childLL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001032 : AngleCell :=
+  childHL (childHH (childLL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001033 : AngleCell :=
+  childHH (childHH (childLL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001200 : AngleCell :=
+  childLL (childLL (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001201 : AngleCell :=
+  childLH (childLL (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001202 : AngleCell :=
+  childHL (childLL (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001203 : AngleCell :=
+  childHH (childLL (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001210 : AngleCell :=
+  childLL (childLH (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001211 : AngleCell :=
+  childLH (childLH (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001212 : AngleCell :=
+  childHL (childLH (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001213 : AngleCell :=
+  childHH (childLH (childHL (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001310 : AngleCell :=
+  childLL (childLH (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001311 : AngleCell :=
+  childLH (childLH (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001312 : AngleCell :=
+  childHL (childLH (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001313 : AngleCell :=
+  childHH (childLH (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001300 : AngleCell :=
+  childLL (childLL (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001301 : AngleCell :=
+  childLH (childLL (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001302 : AngleCell :=
+  childHL (childLL (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001303 : AngleCell :=
+  childHH (childLL (childHH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001120 : AngleCell :=
+  childLL (childHL (childLH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001121 : AngleCell :=
+  childLH (childHL (childLH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001122 : AngleCell :=
+  childHL (childHL (childLH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133001123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133001123 : AngleCell :=
+  childHH (childHL (childLH (childLH thetaBelowCell11113300)))
+
+/-- Subcell `111133002100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaBelowCell11113300)))
+
+/-- Subcell `111133002101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaBelowCell11113300)))
+
+/-- Subcell `111133002102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133002102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaBelowCell11113300)))
+
+/-- Subcell `111133002103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133002103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaBelowCell11113300)))
+
+/-- Subcell `111133002110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaBelowCell11113300)))
+
+/-- Subcell `111133002111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaBelowCell11113300)))
+
+/-- Subcell `111133002112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133002112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaBelowCell11113300)))
+
+/-- Subcell `111133002113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133002113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaBelowCell11113300)))
+
+/-- Subcell `111133003000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133003001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133003002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133003003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133003010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133003011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133003012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133003013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133003100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133003101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133003102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133003103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133003110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133003111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133003112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133003113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133003113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaBelowCell11113300)))
+
+/-- Subcell `111133010200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010200 : AngleCell :=
+  childLL (childLL (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010201 : AngleCell :=
+  childLH (childLL (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010202 : AngleCell :=
+  childHL (childLL (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010203 : AngleCell :=
+  childHH (childLL (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010210 : AngleCell :=
+  childLL (childLH (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010211 : AngleCell :=
+  childLH (childLH (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010212 : AngleCell :=
+  childHL (childLH (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010213 : AngleCell :=
+  childHH (childLH (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010300 : AngleCell :=
+  childLL (childLL (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010301 : AngleCell :=
+  childLH (childLL (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010302 : AngleCell :=
+  childHL (childLL (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010303 : AngleCell :=
+  childHH (childLL (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010310 : AngleCell :=
+  childLL (childLH (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010311 : AngleCell :=
+  childLH (childLH (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010312 : AngleCell :=
+  childHL (childLH (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010313 : AngleCell :=
+  childHH (childLH (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133010333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133010333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaBelowCell11113301)))
+
+/-- Subcell `111133011200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011200 : AngleCell :=
+  childLL (childLL (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011201 : AngleCell :=
+  childLH (childLL (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011202 : AngleCell :=
+  childHL (childLL (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011203 : AngleCell :=
+  childHH (childLL (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011210 : AngleCell :=
+  childLL (childLH (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011211 : AngleCell :=
+  childLH (childLH (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011212 : AngleCell :=
+  childHL (childLH (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011213 : AngleCell :=
+  childHH (childLH (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011300 : AngleCell :=
+  childLL (childLL (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011301 : AngleCell :=
+  childLH (childLL (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011302 : AngleCell :=
+  childHL (childLL (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011303 : AngleCell :=
+  childHH (childLL (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011310 : AngleCell :=
+  childLL (childLH (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011311 : AngleCell :=
+  childLH (childLH (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011312 : AngleCell :=
+  childHL (childLH (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011313 : AngleCell :=
+  childHH (childLH (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133011333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133011333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaBelowCell11113301)))
+
+/-- Subcell `111133012000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133012001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133012002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133012003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133012010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133012011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133012012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133012013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133012100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133012101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133012102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133012103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133012110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133012111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133012112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133012113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133012113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaBelowCell11113301)))
+
+/-- Subcell `111133013000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133013001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133013002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133013003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133013010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133013011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133013012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133013013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133013100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133013101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133013102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133013103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133013110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133013111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133013112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133013113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133013113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaBelowCell11113301)))
+
+/-- Subcell `111133100200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100200 : AngleCell :=
+  childLL (childLL (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100201 : AngleCell :=
+  childLH (childLL (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100202 : AngleCell :=
+  childHL (childLL (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100203 : AngleCell :=
+  childHH (childLL (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100210 : AngleCell :=
+  childLL (childLH (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100211 : AngleCell :=
+  childLH (childLH (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100212 : AngleCell :=
+  childHL (childLH (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100213 : AngleCell :=
+  childHH (childLH (childHL (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100300 : AngleCell :=
+  childLL (childLL (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100301 : AngleCell :=
+  childLH (childLL (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100302 : AngleCell :=
+  childHL (childLL (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100303 : AngleCell :=
+  childHH (childLL (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100310 : AngleCell :=
+  childLL (childLH (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100311 : AngleCell :=
+  childLH (childLH (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100312 : AngleCell :=
+  childHL (childLH (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133100313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133100313 : AngleCell :=
+  childHH (childLH (childHH (childLL thetaBelowCell11113310)))
+
+/-- Subcell `111133101220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101200 : AngleCell :=
+  childLL (childLL (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101201 : AngleCell :=
+  childLH (childLL (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101202 : AngleCell :=
+  childHL (childLL (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101203 : AngleCell :=
+  childHH (childLL (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101210 : AngleCell :=
+  childLL (childLH (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101211 : AngleCell :=
+  childLH (childLH (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101212 : AngleCell :=
+  childHL (childLH (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101213 : AngleCell :=
+  childHH (childLH (childHL (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101300 : AngleCell :=
+  childLL (childLL (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101301 : AngleCell :=
+  childLH (childLL (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101302 : AngleCell :=
+  childHL (childLL (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101303 : AngleCell :=
+  childHH (childLL (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101310 : AngleCell :=
+  childLL (childLH (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101311 : AngleCell :=
+  childLH (childLH (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101312 : AngleCell :=
+  childHL (childLH (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133101313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133101313 : AngleCell :=
+  childHH (childLH (childHH (childLH thetaBelowCell11113310)))
+
+/-- Subcell `111133102000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133102001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133102002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133102003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133102010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133102011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133102012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133102013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133102100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133102101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133102102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133102103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133102110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133102111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133102112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133102113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133102113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaBelowCell11113310)))
+
+/-- Subcell `111133103001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaBelowCell11113310)))
+
+/-- Subcell `111133103000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaBelowCell11113310)))
+
+/-- Subcell `111133103002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaBelowCell11113310)))
+
+/-- Subcell `111133103003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaBelowCell11113310)))
+
+/-- Subcell `111133103010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaBelowCell11113310)))
+
+/-- Subcell `111133103011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaBelowCell11113310)))
+
+/-- Subcell `111133103012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaBelowCell11113310)))
+
+/-- Subcell `111133103013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaBelowCell11113310)))
+
+/-- Subcell `111133103100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaBelowCell11113310)))
+
+/-- Subcell `111133103101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaBelowCell11113310)))
+
+/-- Subcell `111133103102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaBelowCell11113310)))
+
+/-- Subcell `111133103103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaBelowCell11113310)))
+
+/-- Subcell `111133103110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell11113310)))
+
+/-- Subcell `111133103111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113310)))
+
+/-- Subcell `111133103112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaBelowCell11113310)))
+
+/-- Subcell `111133103113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133103113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaBelowCell11113310)))
+
+end GerverSofa.PartE.CertificateCells3658ca4972
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC5TerminalBatchT102400005`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3b1d8a0c5a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3b1d8a0c5a
+
+open CertificateCells3b1d8a0c5a
+theorem e24KC2ThetaAboveLeaf0001331202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013312)) h
+theorem e24KC2ThetaAboveLeaf0001331203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013312)) h
+theorem e24KC2ThetaAboveLeaf0001331210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00013312)) h
+theorem e24KC2ThetaAboveLeaf0001331211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00013312)) h
+theorem e24KC2ThetaAboveLeaf0001331212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013312)) h
+theorem e24KC2ThetaAboveLeaf0001331213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013312)) h
+theorem e24KC2ThetaAboveLeaf000133122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013312) = true := by
+  have h : ((childHL thetaAboveCell00013312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013312) h
+theorem e24KC2ThetaAboveLeaf000133123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013312) = true := by
+  have h : ((childHH thetaAboveCell00013312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013312) h
+theorem e24KC2ThetaAboveLeaf0001331300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013313)) h
+theorem e24KC2ThetaAboveLeaf0001331302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013313)) h
+theorem e24KC2ThetaAboveLeaf0001331303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013313)) h
+theorem e24KC2ThetaAboveLeaf0001331312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013313)) h
+theorem e24KC2ThetaAboveLeaf0001331313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013313)) h
+theorem e24KC2ThetaAboveLeaf000133132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013313) = true := by
+  have h : ((childHL thetaAboveCell00013313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013313) h
+theorem e24KC2ThetaAboveLeaf000133133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013313) = true := by
+  have h : ((childHH thetaAboveCell00013313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013313) h
+theorem e24KC2ThetaAboveLeaf00013320 :
+    adaptiveCoverCheck 11 thetaAboveCell00013320 = true := by
+  have h : (thetaAboveCell00013320).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013320 h
+theorem e24KC2ThetaAboveLeaf00013321 :
+    adaptiveCoverCheck 11 thetaAboveCell00013321 = true := by
+  have h : (thetaAboveCell00013321).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013321 h
+theorem e24KC2ThetaAboveLeaf00013322 :
+    adaptiveCoverCheck 11 thetaAboveCell00013322 = true := by
+  have h : (thetaAboveCell00013322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013322 h
+theorem e24KC2ThetaAboveLeaf00013323 :
+    adaptiveCoverCheck 11 thetaAboveCell00013323 = true := by
+  have h : (thetaAboveCell00013323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013323 h
+theorem e24KC2ThetaAboveLeaf00013330 :
+    adaptiveCoverCheck 11 thetaAboveCell00013330 = true := by
+  have h : (thetaAboveCell00013330).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013330 h
+theorem e24KC2ThetaAboveLeaf00013331 :
+    adaptiveCoverCheck 11 thetaAboveCell00013331 = true := by
+  have h : (thetaAboveCell00013331).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013331 h
+theorem e24KC2ThetaAboveLeaf00013332 :
+    adaptiveCoverCheck 11 thetaAboveCell00013332 = true := by
+  have h : (thetaAboveCell00013332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013332 h
+theorem e24KC2ThetaAboveLeaf00013333 :
+    adaptiveCoverCheck 11 thetaAboveCell00013333 = true := by
+  have h : (thetaAboveCell00013333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013333 h
+theorem e24KC2ThetaAboveLeaf0002000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell0002))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf0002001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell0002))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf0002002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell0002))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf0002003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell0002))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf0002010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell0002))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf0002011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell0002))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf0002012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell0002))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf0002013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell0002))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf000202 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0002)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0002))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0002)) h
+theorem e24KC2ThetaAboveLeaf000203 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0002)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0002))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0002)) h
+theorem e24KC2ThetaAboveLeaf0002100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell0002))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf0002101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell0002))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf0002102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell0002))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf0002103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell0002))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf0002110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell0002))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf0002111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell0002))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf0002112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell0002))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf0002113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell0002))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell0002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell0002))) h
+theorem e24KC2ThetaAboveLeaf000212 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0002)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0002))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0002)) h
+theorem e24KC2ThetaAboveLeaf000213 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0002)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0002))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0002)) h
+theorem e24KC2ThetaAboveLeaf00022 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0002) = true := by
+  have h : ((childHL thetaAboveCell0002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell0002) h
+theorem e24KC2ThetaAboveLeaf00023 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0002) = true := by
+  have h : ((childHH thetaAboveCell0002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell0002) h
+theorem e24KC2ThetaAboveLeaf0003000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell0003))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf0003001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell0003))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf0003002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell0003))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf0003003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell0003))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf0003010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell0003))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf0003011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell0003))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf0003012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell0003))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf0003013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell0003))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf000302 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0003)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0003))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0003)) h
+theorem e24KC2ThetaAboveLeaf000303 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0003)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0003))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0003)) h
+theorem e24KC2ThetaAboveLeaf0003100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell0003))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf0003101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell0003))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf0003102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell0003))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf0003103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell0003))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf0003110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell0003))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf0003111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell0003))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf0003112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell0003))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf0003113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell0003))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell0003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell0003))) h
+theorem e24KC2ThetaAboveLeaf000312 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0003)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0003))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0003)) h
+theorem e24KC2ThetaAboveLeaf000313 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0003)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0003))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0003)) h
+theorem e24KC2ThetaAboveLeaf00032 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0003) = true := by
+  have h : ((childHL thetaAboveCell0003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell0003) h
+theorem e24KC2ThetaAboveLeaf00033 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0003) = true := by
+  have h : ((childHH thetaAboveCell0003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell0003) h
+theorem e24KC2ThetaAboveLeaf001000 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0010)) = true := by
+  have h : ((childLL (childLL thetaAboveCell0010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell0010)) h
+theorem e24KC2ThetaAboveLeaf001001 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0010)) = true := by
+  have h : ((childLH (childLL thetaAboveCell0010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell0010)) h
+theorem e24KC2ThetaAboveLeaf001002 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0010)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0010)) h
+theorem e24KC2ThetaAboveLeaf001003 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0010)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0010)) h
+theorem e24KC2ThetaAboveLeaf001010 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0010)) = true := by
+  have h : ((childLL (childLH thetaAboveCell0010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell0010)) h
+theorem e24KC2ThetaAboveLeaf001011 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0010)) = true := by
+  have h : ((childLH (childLH thetaAboveCell0010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell0010)) h
+theorem e24KC2ThetaAboveLeaf001012 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0010)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0010)) h
+theorem e24KC2ThetaAboveLeaf001013 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0010)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0010)) h
+theorem e24KC2ThetaAboveLeaf0010200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell0010))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell0010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell0010))) h
+theorem e24KC2ThetaAboveLeaf0010201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell0010))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell0010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell0010))) h
+theorem e24KC2ThetaAboveLeaf00102020 :
+    adaptiveCoverCheck 11 thetaAboveCell00102020 = true := by
+  have h : (thetaAboveCell00102020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102020 h
+theorem e24KC2ThetaAboveLeaf00102021 :
+    adaptiveCoverCheck 11 thetaAboveCell00102021 = true := by
+  have h : (thetaAboveCell00102021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102021 h
+theorem e24KC2ThetaAboveLeaf001020220 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102022) = true := by
+  have h : ((childLL thetaAboveCell00102022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00102022) h
+theorem e24KC2ThetaAboveLeaf001020221 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102022) = true := by
+  have h : ((childLH thetaAboveCell00102022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00102022) h
+theorem e24KC2ThetaAboveLeaf001020222 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102022) = true := by
+  have h : ((childHL thetaAboveCell00102022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102022) h
+theorem e24KC2ThetaAboveLeaf001020223 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102022) = true := by
+  have h : ((childHH thetaAboveCell00102022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102022) h
+theorem e24KC2ThetaAboveLeaf001020230 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102023) = true := by
+  have h : ((childLL thetaAboveCell00102023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00102023) h
+theorem e24KC2ThetaAboveLeaf001020231 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102023) = true := by
+  have h : ((childLH thetaAboveCell00102023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00102023) h
+theorem e24KC2ThetaAboveLeaf001020232 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102023) = true := by
+  have h : ((childHL thetaAboveCell00102023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102023) h
+theorem e24KC2ThetaAboveLeaf001020233 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102023) = true := by
+  have h : ((childHH thetaAboveCell00102023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102023) h
+theorem e24KC2ThetaAboveLeaf00102030 :
+    adaptiveCoverCheck 11 thetaAboveCell00102030 = true := by
+  have h : (thetaAboveCell00102030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102030 h
+theorem e24KC2ThetaAboveLeaf00102031 :
+    adaptiveCoverCheck 11 thetaAboveCell00102031 = true := by
+  have h : (thetaAboveCell00102031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102031 h
+theorem e24KC2ThetaAboveLeaf001020320 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102032) = true := by
+  have h : ((childLL thetaAboveCell00102032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00102032) h
+theorem e24KC2ThetaAboveLeaf001020321 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102032) = true := by
+  have h : ((childLH thetaAboveCell00102032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00102032) h
+theorem e24KC2ThetaAboveLeaf001020322 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102032) = true := by
+  have h : ((childHL thetaAboveCell00102032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102032) h
+theorem e24KC2ThetaAboveLeaf001020323 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102032) = true := by
+  have h : ((childHH thetaAboveCell00102032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102032) h
+theorem e24KC2ThetaAboveLeaf001020330 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102033) = true := by
+  have h : ((childLL thetaAboveCell00102033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00102033) h
+theorem e24KC2ThetaAboveLeaf001020331 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102033) = true := by
+  have h : ((childLH thetaAboveCell00102033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00102033) h
+theorem e24KC2ThetaAboveLeaf001020332 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102033) = true := by
+  have h : ((childHL thetaAboveCell00102033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102033) h
+theorem e24KC2ThetaAboveLeaf001020333 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102033) = true := by
+  have h : ((childHH thetaAboveCell00102033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102033) h
+theorem e24KC2ThetaAboveLeaf0010210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell0010))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell0010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell0010))) h
+theorem e24KC2ThetaAboveLeaf0010211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell0010))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell0010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell0010))) h
+theorem e24KC2ThetaAboveLeaf00102120 :
+    adaptiveCoverCheck 11 thetaAboveCell00102120 = true := by
+  have h : (thetaAboveCell00102120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102120 h
+theorem e24KC2ThetaAboveLeaf00102121 :
+    adaptiveCoverCheck 11 thetaAboveCell00102121 = true := by
+  have h : (thetaAboveCell00102121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102121 h
+theorem e24KC2ThetaAboveLeaf001021220 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102122) = true := by
+  have h : ((childLL thetaAboveCell00102122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00102122) h
+theorem e24KC2ThetaAboveLeaf001021221 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102122) = true := by
+  have h : ((childLH thetaAboveCell00102122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00102122) h
+theorem e24KC2ThetaAboveLeaf001021222 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102122) = true := by
+  have h : ((childHL thetaAboveCell00102122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102122) h
+theorem e24KC2ThetaAboveLeaf001021223 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102122) = true := by
+  have h : ((childHH thetaAboveCell00102122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102122) h
+theorem e24KC2ThetaAboveLeaf001021230 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102123) = true := by
+  have h : ((childLL thetaAboveCell00102123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00102123) h
+theorem e24KC2ThetaAboveLeaf001021231 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102123) = true := by
+  have h : ((childLH thetaAboveCell00102123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00102123) h
+theorem e24KC2ThetaAboveLeaf001021232 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102123) = true := by
+  have h : ((childHL thetaAboveCell00102123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102123) h
+theorem e24KC2ThetaAboveLeaf001021233 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102123) = true := by
+  have h : ((childHH thetaAboveCell00102123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102123) h
+theorem e24KC2ThetaAboveLeaf00102130 :
+    adaptiveCoverCheck 11 thetaAboveCell00102130 = true := by
+  have h : (thetaAboveCell00102130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102130 h
+theorem e24KC2ThetaAboveLeaf00102131 :
+    adaptiveCoverCheck 11 thetaAboveCell00102131 = true := by
+  have h : (thetaAboveCell00102131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102131 h
+theorem e24KC2ThetaAboveLeaf001021320 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102132) = true := by
+  have h : ((childLL thetaAboveCell00102132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00102132) h
+theorem e24KC2ThetaAboveLeaf001021321 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102132) = true := by
+  have h : ((childLH thetaAboveCell00102132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00102132) h
+theorem e24KC2ThetaAboveLeaf001021322 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102132) = true := by
+  have h : ((childHL thetaAboveCell00102132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102132) h
+theorem e24KC2ThetaAboveLeaf001021323 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102132) = true := by
+  have h : ((childHH thetaAboveCell00102132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102132) h
+theorem e24KC2ThetaAboveLeaf001021330 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102133) = true := by
+  have h : ((childLL thetaAboveCell00102133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00102133) h
+theorem e24KC2ThetaAboveLeaf001021331 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102133) = true := by
+  have h : ((childLH thetaAboveCell00102133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00102133) h
+theorem e24KC2ThetaAboveLeaf001021332 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102133) = true := by
+  have h : ((childHL thetaAboveCell00102133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102133) h
+theorem e24KC2ThetaAboveLeaf001021333 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102133) = true := by
+  have h : ((childHH thetaAboveCell00102133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102133) h
+theorem e24KC2ThetaAboveLeaf0010220000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102200)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00102200)) h
+theorem e24KC2ThetaAboveLeaf0010220001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102200)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00102200)) h
+theorem e24KC2ThetaAboveLeaf0010220002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102200)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102200)) h
+theorem e24KC2ThetaAboveLeaf0010220003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102200)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102200)) h
+theorem e24KC2ThetaAboveLeaf0010220010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102200)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00102200)) h
+theorem e24KC2ThetaAboveLeaf0010220011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102200)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00102200)) h
+theorem e24KC2ThetaAboveLeaf0010220012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102200)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102200)) h
+theorem e24KC2ThetaAboveLeaf0010220013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102200)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102200)) h
+theorem e24KC2ThetaAboveLeaf0010220100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102201)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00102201)) h
+theorem e24KC2ThetaAboveLeaf0010220101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102201)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00102201)) h
+theorem e24KC2ThetaAboveLeaf0010220102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102201)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102201)) h
+theorem e24KC2ThetaAboveLeaf0010220103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102201)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102201)) h
+theorem e24KC2ThetaAboveLeaf0010220110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102201)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00102201)) h
+theorem e24KC2ThetaAboveLeaf0010220111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102201)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00102201)) h
+theorem e24KC2ThetaAboveLeaf0010220112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102201)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102201)) h
+theorem e24KC2ThetaAboveLeaf0010220113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102201)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102201)) h
+theorem e24KC2ThetaAboveLeaf0010220202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102202)) h
+theorem e24KC2ThetaAboveLeaf0010220203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102202)) h
+theorem e24KC2ThetaAboveLeaf0010220212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102202)) h
+theorem e24KC2ThetaAboveLeaf0010220213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102202)) h
+theorem e24KC2ThetaAboveLeaf001022022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102202) = true := by
+  have h : ((childHL thetaAboveCell00102202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102202) h
+theorem e24KC2ThetaAboveLeaf001022023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102202) = true := by
+  have h : ((childHH thetaAboveCell00102202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102202) h
+theorem e24KC2ThetaAboveLeaf0010220302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102203)) h
+theorem e24KC2ThetaAboveLeaf0010220303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102203)) h
+theorem e24KC2ThetaAboveLeaf0010220312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102203)) h
+theorem e24KC2ThetaAboveLeaf0010220313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102203)) h
+theorem e24KC2ThetaAboveLeaf001022032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102203) = true := by
+  have h : ((childHL thetaAboveCell00102203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102203) h
+theorem e24KC2ThetaAboveLeaf001022033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102203) = true := by
+  have h : ((childHH thetaAboveCell00102203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102203) h
+theorem e24KC2ThetaAboveLeaf0010221000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102210)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00102210)) h
+theorem e24KC2ThetaAboveLeaf0010221001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102210)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00102210)) h
+theorem e24KC2ThetaAboveLeaf0010221002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102210)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102210)) h
+theorem e24KC2ThetaAboveLeaf0010221003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102210)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102210)) h
+theorem e24KC2ThetaAboveLeaf0010221010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102210)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00102210)) h
+theorem e24KC2ThetaAboveLeaf0010221011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102210)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00102210)) h
+theorem e24KC2ThetaAboveLeaf0010221012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102210)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102210)) h
+theorem e24KC2ThetaAboveLeaf0010221013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102210)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102210)) h
+theorem e24KC2ThetaAboveLeaf0010221100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102211)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00102211)) h
+theorem e24KC2ThetaAboveLeaf0010221101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102211)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00102211)) h
+theorem e24KC2ThetaAboveLeaf0010221102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102211)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102211)) h
+theorem e24KC2ThetaAboveLeaf0010221103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102211)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102211)) h
+theorem e24KC2ThetaAboveLeaf0010221110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102211)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00102211)) h
+theorem e24KC2ThetaAboveLeaf0010221111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102211)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00102211)) h
+theorem e24KC2ThetaAboveLeaf0010221112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102211)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102211)) h
+theorem e24KC2ThetaAboveLeaf0010221113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102211)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102211)) h
+theorem e24KC2ThetaAboveLeaf0010221202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102212)) h
+theorem e24KC2ThetaAboveLeaf0010221203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102212)) h
+theorem e24KC2ThetaAboveLeaf0010221212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102212)) h
+theorem e24KC2ThetaAboveLeaf0010221213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102212)) h
+theorem e24KC2ThetaAboveLeaf001022122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102212) = true := by
+  have h : ((childHL thetaAboveCell00102212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102212) h
+theorem e24KC2ThetaAboveLeaf001022123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102212) = true := by
+  have h : ((childHH thetaAboveCell00102212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102212) h
+theorem e24KC2ThetaAboveLeaf0010221302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102213)) h
+theorem e24KC2ThetaAboveLeaf0010221303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102213)) h
+theorem e24KC2ThetaAboveLeaf0010221312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102213)) h
+theorem e24KC2ThetaAboveLeaf0010221313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102213)) h
+theorem e24KC2ThetaAboveLeaf001022132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102213) = true := by
+  have h : ((childHL thetaAboveCell00102213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102213) h
+theorem e24KC2ThetaAboveLeaf001022133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102213) = true := by
+  have h : ((childHH thetaAboveCell00102213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102213) h
+theorem e24KC2ThetaAboveLeaf00102220 :
+    adaptiveCoverCheck 11 thetaAboveCell00102220 = true := by
+  have h : (thetaAboveCell00102220).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102220 h
+theorem e24KC2ThetaAboveLeaf00102221 :
+    adaptiveCoverCheck 11 thetaAboveCell00102221 = true := by
+  have h : (thetaAboveCell00102221).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102221 h
+theorem e24KC2ThetaAboveLeaf00102222 :
+    adaptiveCoverCheck 11 thetaAboveCell00102222 = true := by
+  have h : (thetaAboveCell00102222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102222 h
+theorem e24KC2ThetaAboveLeaf00102223 :
+    adaptiveCoverCheck 11 thetaAboveCell00102223 = true := by
+  have h : (thetaAboveCell00102223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102223 h
+theorem e24KC2ThetaAboveLeaf00102230 :
+    adaptiveCoverCheck 11 thetaAboveCell00102230 = true := by
+  have h : (thetaAboveCell00102230).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102230 h
+theorem e24KC2ThetaAboveLeaf00102231 :
+    adaptiveCoverCheck 11 thetaAboveCell00102231 = true := by
+  have h : (thetaAboveCell00102231).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102231 h
+theorem e24KC2ThetaAboveLeaf00102232 :
+    adaptiveCoverCheck 11 thetaAboveCell00102232 = true := by
+  have h : (thetaAboveCell00102232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102232 h
+theorem e24KC2ThetaAboveLeaf00102233 :
+    adaptiveCoverCheck 11 thetaAboveCell00102233 = true := by
+  have h : (thetaAboveCell00102233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102233 h
+theorem e24KC2ThetaAboveLeaf0010230000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102300)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00102300)) h
+theorem e24KC2ThetaAboveLeaf0010230001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102300)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00102300)) h
+theorem e24KC2ThetaAboveLeaf0010230002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102300)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102300)) h
+theorem e24KC2ThetaAboveLeaf0010230003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102300)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102300)) h
+theorem e24KC2ThetaAboveLeaf0010230010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102300)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00102300)) h
+theorem e24KC2ThetaAboveLeaf0010230011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102300)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00102300)) h
+theorem e24KC2ThetaAboveLeaf0010230012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102300)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102300)) h
+theorem e24KC2ThetaAboveLeaf0010230013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102300)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102300)) h
+theorem e24KC2ThetaAboveLeaf0010230100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102301)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00102301)) h
+theorem e24KC2ThetaAboveLeaf0010230101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102301)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00102301)) h
+theorem e24KC2ThetaAboveLeaf0010230102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102301)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102301)) h
+theorem e24KC2ThetaAboveLeaf0010230103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102301)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102301)) h
+theorem e24KC2ThetaAboveLeaf0010230110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102301)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00102301)) h
+theorem e24KC2ThetaAboveLeaf0010230111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102301)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00102301)) h
+theorem e24KC2ThetaAboveLeaf0010230112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102301)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102301)) h
+theorem e24KC2ThetaAboveLeaf0010230113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102301)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102301)) h
+theorem e24KC2ThetaAboveLeaf0010230202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102302)) h
+theorem e24KC2ThetaAboveLeaf0010230203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102302)) h
+theorem e24KC2ThetaAboveLeaf0010230212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102302)) h
+theorem e24KC2ThetaAboveLeaf0010230213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102302)) h
+theorem e24KC2ThetaAboveLeaf001023022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102302) = true := by
+  have h : ((childHL thetaAboveCell00102302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102302) h
+theorem e24KC2ThetaAboveLeaf001023023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102302) = true := by
+  have h : ((childHH thetaAboveCell00102302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102302) h
+theorem e24KC2ThetaAboveLeaf0010230302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102303)) h
+theorem e24KC2ThetaAboveLeaf0010230303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102303)) h
+theorem e24KC2ThetaAboveLeaf0010230312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102303)) h
+theorem e24KC2ThetaAboveLeaf0010230313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102303)) h
+theorem e24KC2ThetaAboveLeaf001023032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102303) = true := by
+  have h : ((childHL thetaAboveCell00102303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102303) h
+theorem e24KC2ThetaAboveLeaf001023033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102303) = true := by
+  have h : ((childHH thetaAboveCell00102303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102303) h
+theorem e24KC2ThetaAboveLeaf0010231000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102310)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00102310)) h
+theorem e24KC2ThetaAboveLeaf0010231001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102310)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00102310)) h
+theorem e24KC2ThetaAboveLeaf0010231002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102310)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102310)) h
+theorem e24KC2ThetaAboveLeaf0010231003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102310)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102310)) h
+theorem e24KC2ThetaAboveLeaf0010231010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102310)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00102310)) h
+theorem e24KC2ThetaAboveLeaf0010231011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102310)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00102310)) h
+theorem e24KC2ThetaAboveLeaf0010231012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102310)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102310)) h
+theorem e24KC2ThetaAboveLeaf0010231013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102310)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102310)) h
+theorem e24KC2ThetaAboveLeaf0010231100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102311)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00102311)) h
+theorem e24KC2ThetaAboveLeaf0010231101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102311)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00102311)) h
+theorem e24KC2ThetaAboveLeaf0010231102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102311)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102311)) h
+theorem e24KC2ThetaAboveLeaf0010231103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102311)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102311)) h
+theorem e24KC2ThetaAboveLeaf0010231110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102311)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00102311)) h
+theorem e24KC2ThetaAboveLeaf0010231111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102311)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00102311)) h
+theorem e24KC2ThetaAboveLeaf0010231112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102311)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102311)) h
+theorem e24KC2ThetaAboveLeaf0010231113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102311)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102311)) h
+theorem e24KC2ThetaAboveLeaf0010231202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102312)) h
+theorem e24KC2ThetaAboveLeaf0010231203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102312)) h
+theorem e24KC2ThetaAboveLeaf0010231212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102312)) h
+theorem e24KC2ThetaAboveLeaf0010231213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102312)) h
+theorem e24KC2ThetaAboveLeaf001023122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102312) = true := by
+  have h : ((childHL thetaAboveCell00102312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102312) h
+theorem e24KC2ThetaAboveLeaf001023123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102312) = true := by
+  have h : ((childHH thetaAboveCell00102312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102312) h
+theorem e24KC2ThetaAboveLeaf0010231302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00102313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00102313)) h
+theorem e24KC2ThetaAboveLeaf0010231303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00102313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00102313)) h
+theorem e24KC2ThetaAboveLeaf0010231312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00102313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00102313)) h
+theorem e24KC2ThetaAboveLeaf0010231313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00102313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00102313)) h
+theorem e24KC2ThetaAboveLeaf001023132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102313) = true := by
+  have h : ((childHL thetaAboveCell00102313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00102313) h
+theorem e24KC2ThetaAboveLeaf001023133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102313) = true := by
+  have h : ((childHH thetaAboveCell00102313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00102313) h
+theorem e24KC2ThetaAboveLeaf00102320 :
+    adaptiveCoverCheck 11 thetaAboveCell00102320 = true := by
+  have h : (thetaAboveCell00102320).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102320 h
+theorem e24KC2ThetaAboveLeaf00102321 :
+    adaptiveCoverCheck 11 thetaAboveCell00102321 = true := by
+  have h : (thetaAboveCell00102321).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102321 h
+theorem e24KC2ThetaAboveLeaf00102322 :
+    adaptiveCoverCheck 11 thetaAboveCell00102322 = true := by
+  have h : (thetaAboveCell00102322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102322 h
+theorem e24KC2ThetaAboveLeaf00102323 :
+    adaptiveCoverCheck 11 thetaAboveCell00102323 = true := by
+  have h : (thetaAboveCell00102323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102323 h
+theorem e24KC2ThetaAboveLeaf00102330 :
+    adaptiveCoverCheck 11 thetaAboveCell00102330 = true := by
+  have h : (thetaAboveCell00102330).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102330 h
+theorem e24KC2ThetaAboveLeaf00102331 :
+    adaptiveCoverCheck 11 thetaAboveCell00102331 = true := by
+  have h : (thetaAboveCell00102331).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102331 h
+theorem e24KC2ThetaAboveLeaf00102332 :
+    adaptiveCoverCheck 11 thetaAboveCell00102332 = true := by
+  have h : (thetaAboveCell00102332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102332 h
+theorem e24KC2ThetaAboveLeaf00102333 :
+    adaptiveCoverCheck 11 thetaAboveCell00102333 = true := by
+  have h : (thetaAboveCell00102333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00102333 h
+theorem e24KC2ThetaAboveLeaf0010300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell0010))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell0010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell0010))) h
+theorem e24KC2ThetaAboveLeaf0010301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell0010))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell0010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell0010))) h
+theorem e24KC2ThetaAboveLeaf00103020 :
+    adaptiveCoverCheck 11 thetaAboveCell00103020 = true := by
+  have h : (thetaAboveCell00103020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103020 h
+theorem e24KC2ThetaAboveLeaf00103021 :
+    adaptiveCoverCheck 11 thetaAboveCell00103021 = true := by
+  have h : (thetaAboveCell00103021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103021 h
+theorem e24KC2ThetaAboveLeaf00103022 :
+    adaptiveCoverCheck 11 thetaAboveCell00103022 = true := by
+  have h : (thetaAboveCell00103022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103022 h
+theorem e24KC2ThetaAboveLeaf00103023 :
+    adaptiveCoverCheck 11 thetaAboveCell00103023 = true := by
+  have h : (thetaAboveCell00103023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103023 h
+theorem e24KC2ThetaAboveLeaf00103030 :
+    adaptiveCoverCheck 11 thetaAboveCell00103030 = true := by
+  have h : (thetaAboveCell00103030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103030 h
+theorem e24KC2ThetaAboveLeaf00103031 :
+    adaptiveCoverCheck 11 thetaAboveCell00103031 = true := by
+  have h : (thetaAboveCell00103031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103031 h
+theorem e24KC2ThetaAboveLeaf00103032 :
+    adaptiveCoverCheck 11 thetaAboveCell00103032 = true := by
+  have h : (thetaAboveCell00103032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103032 h
+theorem e24KC2ThetaAboveLeaf00103033 :
+    adaptiveCoverCheck 11 thetaAboveCell00103033 = true := by
+  have h : (thetaAboveCell00103033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103033 h
+theorem e24KC2ThetaAboveLeaf0010310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell0010))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell0010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell0010))) h
+theorem e24KC2ThetaAboveLeaf0010311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell0010))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell0010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell0010))) h
+theorem e24KC2ThetaAboveLeaf00103120 :
+    adaptiveCoverCheck 11 thetaAboveCell00103120 = true := by
+  have h : (thetaAboveCell00103120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103120 h
+theorem e24KC2ThetaAboveLeaf00103121 :
+    adaptiveCoverCheck 11 thetaAboveCell00103121 = true := by
+  have h : (thetaAboveCell00103121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103121 h
+theorem e24KC2ThetaAboveLeaf00103122 :
+    adaptiveCoverCheck 11 thetaAboveCell00103122 = true := by
+  have h : (thetaAboveCell00103122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103122 h
+theorem e24KC2ThetaAboveLeaf00103123 :
+    adaptiveCoverCheck 11 thetaAboveCell00103123 = true := by
+  have h : (thetaAboveCell00103123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103123 h
+theorem e24KC2ThetaAboveLeaf00103130 :
+    adaptiveCoverCheck 11 thetaAboveCell00103130 = true := by
+  have h : (thetaAboveCell00103130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103130 h
+theorem e24KC2ThetaAboveLeaf00103131 :
+    adaptiveCoverCheck 11 thetaAboveCell00103131 = true := by
+  have h : (thetaAboveCell00103131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103131 h
+theorem e24KC2ThetaAboveLeaf00103132 :
+    adaptiveCoverCheck 11 thetaAboveCell00103132 = true := by
+  have h : (thetaAboveCell00103132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103132 h
+theorem e24KC2ThetaAboveLeaf00103133 :
+    adaptiveCoverCheck 11 thetaAboveCell00103133 = true := by
+  have h : (thetaAboveCell00103133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103133 h
+theorem e24KC2ThetaAboveLeaf0010320000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103200)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00103200)) h
+theorem e24KC2ThetaAboveLeaf0010320001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103200)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00103200)) h
+theorem e24KC2ThetaAboveLeaf0010320002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103200)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103200)) h
+theorem e24KC2ThetaAboveLeaf0010320003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103200)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103200)) h
+theorem e24KC2ThetaAboveLeaf0010320010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103200)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00103200)) h
+theorem e24KC2ThetaAboveLeaf0010320011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103200)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00103200)) h
+theorem e24KC2ThetaAboveLeaf0010320012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103200)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103200)) h
+theorem e24KC2ThetaAboveLeaf0010320013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103200)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103200)) h
+theorem e24KC2ThetaAboveLeaf0010320100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103201)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00103201)) h
+theorem e24KC2ThetaAboveLeaf0010320101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103201)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00103201)) h
+theorem e24KC2ThetaAboveLeaf0010320102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103201)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103201)) h
+theorem e24KC2ThetaAboveLeaf0010320103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103201)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103201)) h
+theorem e24KC2ThetaAboveLeaf0010320110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103201)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00103201)) h
+theorem e24KC2ThetaAboveLeaf0010320111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103201)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00103201)) h
+theorem e24KC2ThetaAboveLeaf0010320112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103201)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103201)) h
+theorem e24KC2ThetaAboveLeaf0010320113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103201)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103201)) h
+theorem e24KC2ThetaAboveLeaf0010320202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103202)) h
+theorem e24KC2ThetaAboveLeaf0010320203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103202)) h
+theorem e24KC2ThetaAboveLeaf0010320212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103202)) h
+theorem e24KC2ThetaAboveLeaf0010320213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103202)) h
+theorem e24KC2ThetaAboveLeaf001032022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103202) = true := by
+  have h : ((childHL thetaAboveCell00103202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00103202) h
+theorem e24KC2ThetaAboveLeaf001032023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103202) = true := by
+  have h : ((childHH thetaAboveCell00103202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00103202) h
+theorem e24KC2ThetaAboveLeaf0010320302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103203)) h
+theorem e24KC2ThetaAboveLeaf0010320303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103203)) h
+theorem e24KC2ThetaAboveLeaf0010320312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103203)) h
+theorem e24KC2ThetaAboveLeaf0010320313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103203)) h
+theorem e24KC2ThetaAboveLeaf001032032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103203) = true := by
+  have h : ((childHL thetaAboveCell00103203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00103203) h
+theorem e24KC2ThetaAboveLeaf001032033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103203) = true := by
+  have h : ((childHH thetaAboveCell00103203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00103203) h
+theorem e24KC2ThetaAboveLeaf0010321000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103210)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00103210)) h
+theorem e24KC2ThetaAboveLeaf0010321001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103210)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00103210)) h
+theorem e24KC2ThetaAboveLeaf0010321002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103210)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103210)) h
+theorem e24KC2ThetaAboveLeaf0010321003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103210)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103210)) h
+theorem e24KC2ThetaAboveLeaf0010321010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103210)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00103210)) h
+theorem e24KC2ThetaAboveLeaf0010321011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103210)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00103210)) h
+theorem e24KC2ThetaAboveLeaf0010321012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103210)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103210)) h
+theorem e24KC2ThetaAboveLeaf0010321013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103210)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103210)) h
+theorem e24KC2ThetaAboveLeaf0010321100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103211)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00103211)) h
+theorem e24KC2ThetaAboveLeaf0010321101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103211)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00103211)) h
+theorem e24KC2ThetaAboveLeaf0010321102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103211)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103211)) h
+theorem e24KC2ThetaAboveLeaf0010321103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103211)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103211)) h
+theorem e24KC2ThetaAboveLeaf0010321110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103211)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00103211)) h
+theorem e24KC2ThetaAboveLeaf0010321111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103211)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00103211)) h
+theorem e24KC2ThetaAboveLeaf0010321112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103211)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103211)) h
+theorem e24KC2ThetaAboveLeaf0010321113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103211)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103211)) h
+theorem e24KC2ThetaAboveLeaf0010321202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103212)) h
+theorem e24KC2ThetaAboveLeaf0010321203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103212)) h
+theorem e24KC2ThetaAboveLeaf0010321212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103212)) h
+theorem e24KC2ThetaAboveLeaf0010321213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103212)) h
+theorem e24KC2ThetaAboveLeaf001032122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103212) = true := by
+  have h : ((childHL thetaAboveCell00103212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00103212) h
+theorem e24KC2ThetaAboveLeaf001032123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103212) = true := by
+  have h : ((childHH thetaAboveCell00103212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00103212) h
+theorem e24KC2ThetaAboveLeaf0010321302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103213)) h
+theorem e24KC2ThetaAboveLeaf0010321303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103213)) h
+theorem e24KC2ThetaAboveLeaf0010321312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103213)) h
+theorem e24KC2ThetaAboveLeaf0010321313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103213)) h
+theorem e24KC2ThetaAboveLeaf001032132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103213) = true := by
+  have h : ((childHL thetaAboveCell00103213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00103213) h
+theorem e24KC2ThetaAboveLeaf001032133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103213) = true := by
+  have h : ((childHH thetaAboveCell00103213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00103213) h
+theorem e24KC2ThetaAboveLeaf00103220 :
+    adaptiveCoverCheck 11 thetaAboveCell00103220 = true := by
+  have h : (thetaAboveCell00103220).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103220 h
+theorem e24KC2ThetaAboveLeaf00103221 :
+    adaptiveCoverCheck 11 thetaAboveCell00103221 = true := by
+  have h : (thetaAboveCell00103221).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103221 h
+theorem e24KC2ThetaAboveLeaf00103222 :
+    adaptiveCoverCheck 11 thetaAboveCell00103222 = true := by
+  have h : (thetaAboveCell00103222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103222 h
+theorem e24KC2ThetaAboveLeaf00103223 :
+    adaptiveCoverCheck 11 thetaAboveCell00103223 = true := by
+  have h : (thetaAboveCell00103223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103223 h
+theorem e24KC2ThetaAboveLeaf00103230 :
+    adaptiveCoverCheck 11 thetaAboveCell00103230 = true := by
+  have h : (thetaAboveCell00103230).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103230 h
+theorem e24KC2ThetaAboveLeaf00103231 :
+    adaptiveCoverCheck 11 thetaAboveCell00103231 = true := by
+  have h : (thetaAboveCell00103231).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103231 h
+theorem e24KC2ThetaAboveLeaf00103232 :
+    adaptiveCoverCheck 11 thetaAboveCell00103232 = true := by
+  have h : (thetaAboveCell00103232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103232 h
+theorem e24KC2ThetaAboveLeaf00103233 :
+    adaptiveCoverCheck 11 thetaAboveCell00103233 = true := by
+  have h : (thetaAboveCell00103233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103233 h
+theorem e24KC2ThetaAboveLeaf0010330000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103300)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00103300)) h
+theorem e24KC2ThetaAboveLeaf0010330001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103300)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00103300)) h
+theorem e24KC2ThetaAboveLeaf0010330002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103300)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103300)) h
+theorem e24KC2ThetaAboveLeaf0010330003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103300)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103300)) h
+theorem e24KC2ThetaAboveLeaf0010330010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103300)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00103300)) h
+theorem e24KC2ThetaAboveLeaf0010330011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103300)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00103300)) h
+theorem e24KC2ThetaAboveLeaf0010330012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103300)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103300)) h
+theorem e24KC2ThetaAboveLeaf0010330013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103300)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103300)) h
+theorem e24KC2ThetaAboveLeaf0010330100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103301)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00103301)) h
+theorem e24KC2ThetaAboveLeaf0010330101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103301)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00103301)) h
+theorem e24KC2ThetaAboveLeaf0010330102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103301)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103301)) h
+theorem e24KC2ThetaAboveLeaf0010330103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103301)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103301)) h
+theorem e24KC2ThetaAboveLeaf0010330110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103301)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00103301)) h
+theorem e24KC2ThetaAboveLeaf0010330111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103301)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00103301)) h
+theorem e24KC2ThetaAboveLeaf0010330112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103301)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103301)) h
+theorem e24KC2ThetaAboveLeaf0010330113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103301)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103301)) h
+theorem e24KC2ThetaAboveLeaf0010330202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103302)) h
+theorem e24KC2ThetaAboveLeaf0010330203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103302)) h
+theorem e24KC2ThetaAboveLeaf0010330212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103302)) h
+theorem e24KC2ThetaAboveLeaf0010330213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103302)) h
+theorem e24KC2ThetaAboveLeaf001033022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103302) = true := by
+  have h : ((childHL thetaAboveCell00103302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00103302) h
+theorem e24KC2ThetaAboveLeaf001033023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103302) = true := by
+  have h : ((childHH thetaAboveCell00103302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00103302) h
+theorem e24KC2ThetaAboveLeaf0010330302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103303)) h
+theorem e24KC2ThetaAboveLeaf0010330303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103303)) h
+theorem e24KC2ThetaAboveLeaf0010330312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103303)) h
+theorem e24KC2ThetaAboveLeaf0010330313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103303)) h
+theorem e24KC2ThetaAboveLeaf001033032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103303) = true := by
+  have h : ((childHL thetaAboveCell00103303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00103303) h
+theorem e24KC2ThetaAboveLeaf001033033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103303) = true := by
+  have h : ((childHH thetaAboveCell00103303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00103303) h
+theorem e24KC2ThetaAboveLeaf0010331000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103310)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00103310)) h
+theorem e24KC2ThetaAboveLeaf0010331001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103310)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00103310)) h
+theorem e24KC2ThetaAboveLeaf0010331002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103310)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103310)) h
+theorem e24KC2ThetaAboveLeaf0010331003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103310)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103310)) h
+theorem e24KC2ThetaAboveLeaf0010331010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103310)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00103310)) h
+theorem e24KC2ThetaAboveLeaf0010331011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103310)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00103310)) h
+theorem e24KC2ThetaAboveLeaf0010331012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103310)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103310)) h
+theorem e24KC2ThetaAboveLeaf0010331013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103310)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103310)) h
+theorem e24KC2ThetaAboveLeaf0010331100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103311)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00103311)) h
+theorem e24KC2ThetaAboveLeaf0010331101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103311)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00103311)) h
+theorem e24KC2ThetaAboveLeaf0010331102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103311)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103311)) h
+theorem e24KC2ThetaAboveLeaf0010331103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103311)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103311)) h
+theorem e24KC2ThetaAboveLeaf0010331110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103311)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00103311)) h
+theorem e24KC2ThetaAboveLeaf0010331111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103311)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00103311)) h
+theorem e24KC2ThetaAboveLeaf0010331112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103311)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103311)) h
+theorem e24KC2ThetaAboveLeaf0010331113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103311)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103311)) h
+theorem e24KC2ThetaAboveLeaf0010331202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103312)) h
+theorem e24KC2ThetaAboveLeaf0010331203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103312)) h
+theorem e24KC2ThetaAboveLeaf0010331212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103312)) h
+theorem e24KC2ThetaAboveLeaf0010331213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103312)) h
+theorem e24KC2ThetaAboveLeaf001033122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103312) = true := by
+  have h : ((childHL thetaAboveCell00103312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00103312) h
+theorem e24KC2ThetaAboveLeaf001033123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103312) = true := by
+  have h : ((childHH thetaAboveCell00103312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00103312) h
+theorem e24KC2ThetaAboveLeaf0010331302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00103313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00103313)) h
+theorem e24KC2ThetaAboveLeaf0010331303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00103313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00103313)) h
+theorem e24KC2ThetaAboveLeaf0010331312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00103313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00103313)) h
+theorem e24KC2ThetaAboveLeaf0010331313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00103313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00103313)) h
+theorem e24KC2ThetaAboveLeaf001033132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103313) = true := by
+  have h : ((childHL thetaAboveCell00103313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00103313) h
+theorem e24KC2ThetaAboveLeaf001033133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103313) = true := by
+  have h : ((childHH thetaAboveCell00103313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00103313) h
+theorem e24KC2ThetaAboveLeaf00103320 :
+    adaptiveCoverCheck 11 thetaAboveCell00103320 = true := by
+  have h : (thetaAboveCell00103320).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103320 h
+theorem e24KC2ThetaAboveLeaf00103321 :
+    adaptiveCoverCheck 11 thetaAboveCell00103321 = true := by
+  have h : (thetaAboveCell00103321).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103321 h
+theorem e24KC2ThetaAboveLeaf00103322 :
+    adaptiveCoverCheck 11 thetaAboveCell00103322 = true := by
+  have h : (thetaAboveCell00103322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103322 h
+theorem e24KC2ThetaAboveLeaf00103323 :
+    adaptiveCoverCheck 11 thetaAboveCell00103323 = true := by
+  have h : (thetaAboveCell00103323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103323 h
+theorem e24KC2ThetaAboveLeaf00103330 :
+    adaptiveCoverCheck 11 thetaAboveCell00103330 = true := by
+  have h : (thetaAboveCell00103330).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103330 h
+theorem e24KC2ThetaAboveLeaf00103331 :
+    adaptiveCoverCheck 11 thetaAboveCell00103331 = true := by
+  have h : (thetaAboveCell00103331).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103331 h
+theorem e24KC2ThetaAboveLeaf00103332 :
+    adaptiveCoverCheck 11 thetaAboveCell00103332 = true := by
+  have h : (thetaAboveCell00103332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103332 h
+theorem e24KC2ThetaAboveLeaf00103333 :
+    adaptiveCoverCheck 11 thetaAboveCell00103333 = true := by
+  have h : (thetaAboveCell00103333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00103333 h
+theorem e24KC2ThetaAboveLeaf001100 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0011)) = true := by
+  have h : ((childLL (childLL thetaAboveCell0011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell0011)) h
+theorem e24KC2ThetaAboveLeaf001101 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0011)) = true := by
+  have h : ((childLH (childLL thetaAboveCell0011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell0011)) h
+theorem e24KC2ThetaAboveLeaf001102 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0011)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0011)) h
+theorem e24KC2ThetaAboveLeaf001103 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0011)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0011)) h
+theorem e24KC2ThetaAboveLeaf001110 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0011)) = true := by
+  have h : ((childLL (childLH thetaAboveCell0011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell0011)) h
+theorem e24KC2ThetaAboveLeaf001111 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0011)) = true := by
+  have h : ((childLH (childLH thetaAboveCell0011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell0011)) h
+theorem e24KC2ThetaAboveLeaf001112 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0011)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0011)) h
+theorem e24KC2ThetaAboveLeaf001113 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0011)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0011)) h
+theorem e24KC2ThetaAboveLeaf0011200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell0011))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell0011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell0011))) h
+theorem e24KC2ThetaAboveLeaf0011201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell0011))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell0011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell0011))) h
+theorem e24KC2ThetaAboveLeaf00112020 :
+    adaptiveCoverCheck 11 thetaAboveCell00112020 = true := by
+  have h : (thetaAboveCell00112020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112020 h
+theorem e24KC2ThetaAboveLeaf00112021 :
+    adaptiveCoverCheck 11 thetaAboveCell00112021 = true := by
+  have h : (thetaAboveCell00112021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112021 h
+theorem e24KC2ThetaAboveLeaf00112022 :
+    adaptiveCoverCheck 11 thetaAboveCell00112022 = true := by
+  have h : (thetaAboveCell00112022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112022 h
+theorem e24KC2ThetaAboveLeaf00112023 :
+    adaptiveCoverCheck 11 thetaAboveCell00112023 = true := by
+  have h : (thetaAboveCell00112023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112023 h
+theorem e24KC2ThetaAboveLeaf00112030 :
+    adaptiveCoverCheck 11 thetaAboveCell00112030 = true := by
+  have h : (thetaAboveCell00112030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112030 h
+theorem e24KC2ThetaAboveLeaf00112031 :
+    adaptiveCoverCheck 11 thetaAboveCell00112031 = true := by
+  have h : (thetaAboveCell00112031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112031 h
+theorem e24KC2ThetaAboveLeaf00112032 :
+    adaptiveCoverCheck 11 thetaAboveCell00112032 = true := by
+  have h : (thetaAboveCell00112032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112032 h
+theorem e24KC2ThetaAboveLeaf00112033 :
+    adaptiveCoverCheck 11 thetaAboveCell00112033 = true := by
+  have h : (thetaAboveCell00112033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112033 h
+theorem e24KC2ThetaAboveLeaf0011210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell0011))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell0011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell0011))) h
+theorem e24KC2ThetaAboveLeaf0011211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell0011))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell0011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell0011))) h
+theorem e24KC2ThetaAboveLeaf00112120 :
+    adaptiveCoverCheck 11 thetaAboveCell00112120 = true := by
+  have h : (thetaAboveCell00112120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112120 h
+theorem e24KC2ThetaAboveLeaf00112121 :
+    adaptiveCoverCheck 11 thetaAboveCell00112121 = true := by
+  have h : (thetaAboveCell00112121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112121 h
+theorem e24KC2ThetaAboveLeaf00112122 :
+    adaptiveCoverCheck 11 thetaAboveCell00112122 = true := by
+  have h : (thetaAboveCell00112122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112122 h
+theorem e24KC2ThetaAboveLeaf00112123 :
+    adaptiveCoverCheck 11 thetaAboveCell00112123 = true := by
+  have h : (thetaAboveCell00112123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112123 h
+theorem e24KC2ThetaAboveLeaf00112130 :
+    adaptiveCoverCheck 11 thetaAboveCell00112130 = true := by
+  have h : (thetaAboveCell00112130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112130 h
+theorem e24KC2ThetaAboveLeaf00112131 :
+    adaptiveCoverCheck 11 thetaAboveCell00112131 = true := by
+  have h : (thetaAboveCell00112131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112131 h
+theorem e24KC2ThetaAboveLeaf00112132 :
+    adaptiveCoverCheck 11 thetaAboveCell00112132 = true := by
+  have h : (thetaAboveCell00112132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112132 h
+theorem e24KC2ThetaAboveLeaf00112133 :
+    adaptiveCoverCheck 11 thetaAboveCell00112133 = true := by
+  have h : (thetaAboveCell00112133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112133 h
+theorem e24KC2ThetaAboveLeaf0011220000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00112200)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00112200)) h
+theorem e24KC2ThetaAboveLeaf0011220001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00112200)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00112200)) h
+theorem e24KC2ThetaAboveLeaf0011220002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00112200)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00112200)) h
+theorem e24KC2ThetaAboveLeaf0011220003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00112200)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00112200)) h
+theorem e24KC2ThetaAboveLeaf0011220010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00112200)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00112200)) h
+theorem e24KC2ThetaAboveLeaf0011220011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00112200)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00112200)) h
+theorem e24KC2ThetaAboveLeaf0011220012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00112200)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00112200)) h
+theorem e24KC2ThetaAboveLeaf0011220013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00112200)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00112200)) h
+theorem e24KC2ThetaAboveLeaf0011220100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00112201)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00112201)) h
+theorem e24KC2ThetaAboveLeaf0011220101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00112201)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00112201)) h
+theorem e24KC2ThetaAboveLeaf0011220102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00112201)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00112201)) h
+theorem e24KC2ThetaAboveLeaf0011220103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00112201)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00112201)) h
+theorem e24KC2ThetaAboveLeaf0011220110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00112201)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00112201)) h
+theorem e24KC2ThetaAboveLeaf0011220111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00112201)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00112201)) h
+theorem e24KC2ThetaAboveLeaf0011220112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00112201)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00112201)) h
+theorem e24KC2ThetaAboveLeaf0011220113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00112201)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00112201)) h
+theorem e24KC2ThetaAboveLeaf0011220202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00112202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00112202)) h
+theorem e24KC2ThetaAboveLeaf0011220203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00112202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00112202)) h
+theorem e24KC2ThetaAboveLeaf0011220212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00112202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00112202)) h
+theorem e24KC2ThetaAboveLeaf0011220213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00112202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00112202)) h
+theorem e24KC2ThetaAboveLeaf001122022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112202) = true := by
+  have h : ((childHL thetaAboveCell00112202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00112202) h
+theorem e24KC2ThetaAboveLeaf001122023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112202) = true := by
+  have h : ((childHH thetaAboveCell00112202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00112202) h
+theorem e24KC2ThetaAboveLeaf0011220302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00112203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00112203)) h
+theorem e24KC2ThetaAboveLeaf0011220303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00112203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00112203)) h
+theorem e24KC2ThetaAboveLeaf0011220312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00112203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00112203)) h
+theorem e24KC2ThetaAboveLeaf0011220313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00112203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00112203)) h
+theorem e24KC2ThetaAboveLeaf001122032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112203) = true := by
+  have h : ((childHL thetaAboveCell00112203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00112203) h
+theorem e24KC2ThetaAboveLeaf001122033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112203) = true := by
+  have h : ((childHH thetaAboveCell00112203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00112203) h
+theorem e24KC2ThetaAboveLeaf0011221000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00112210)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00112210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00112210)) h
+theorem e24KC2ThetaAboveLeaf0011221001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00112210)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00112210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00112210)) h
+theorem e24KC2ThetaAboveLeaf0011221002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00112210)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00112210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00112210)) h
+theorem e24KC2ThetaAboveLeaf0011221003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00112210)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00112210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00112210)) h
+theorem e24KC2ThetaAboveLeaf001122101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112210) = true := by
+  have h : ((childLH thetaAboveCell00112210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00112210) h
+theorem e24KC2ThetaAboveLeaf001122110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112211) = true := by
+  have h : ((childLL thetaAboveCell00112211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00112211) h
+theorem e24KC2ThetaAboveLeaf001122111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112211) = true := by
+  have h : ((childLH thetaAboveCell00112211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00112211) h
+theorem e24KC2ThetaAboveLeaf0011221202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00112212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00112212)) h
+theorem e24KC2ThetaAboveLeaf0011221203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00112212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00112212)) h
+theorem e24KC2ThetaAboveLeaf0011221212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00112212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00112212)) h
+theorem e24KC2ThetaAboveLeaf0011221213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00112212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00112212)) h
+theorem e24KC2ThetaAboveLeaf001122122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112212) = true := by
+  have h : ((childHL thetaAboveCell00112212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00112212) h
+theorem e24KC2ThetaAboveLeaf001122123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112212) = true := by
+  have h : ((childHH thetaAboveCell00112212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00112212) h
+theorem e24KC2ThetaAboveLeaf0011221302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00112213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00112213)) h
+theorem e24KC2ThetaAboveLeaf0011221303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00112213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00112213)) h
+theorem e24KC2ThetaAboveLeaf0011221312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00112213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00112213)) h
+theorem e24KC2ThetaAboveLeaf0011221313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00112213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00112213)) h
+theorem e24KC2ThetaAboveLeaf001122132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112213) = true := by
+  have h : ((childHL thetaAboveCell00112213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00112213) h
+theorem e24KC2ThetaAboveLeaf001122133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112213) = true := by
+  have h : ((childHH thetaAboveCell00112213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00112213) h
+theorem e24KC2ThetaAboveLeaf00112220 :
+    adaptiveCoverCheck 11 thetaAboveCell00112220 = true := by
+  have h : (thetaAboveCell00112220).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112220 h
+theorem e24KC2ThetaAboveLeaf00112221 :
+    adaptiveCoverCheck 11 thetaAboveCell00112221 = true := by
+  have h : (thetaAboveCell00112221).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112221 h
+theorem e24KC2ThetaAboveLeaf00112222 :
+    adaptiveCoverCheck 11 thetaAboveCell00112222 = true := by
+  have h : (thetaAboveCell00112222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112222 h
+theorem e24KC2ThetaAboveLeaf00112223 :
+    adaptiveCoverCheck 11 thetaAboveCell00112223 = true := by
+  have h : (thetaAboveCell00112223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112223 h
+theorem e24KC2ThetaAboveLeaf00112230 :
+    adaptiveCoverCheck 11 thetaAboveCell00112230 = true := by
+  have h : (thetaAboveCell00112230).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112230 h
+theorem e24KC2ThetaAboveLeaf00112231 :
+    adaptiveCoverCheck 11 thetaAboveCell00112231 = true := by
+  have h : (thetaAboveCell00112231).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112231 h
+theorem e24KC2ThetaAboveLeaf00112232 :
+    adaptiveCoverCheck 11 thetaAboveCell00112232 = true := by
+  have h : (thetaAboveCell00112232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112232 h
+theorem e24KC2ThetaAboveLeaf00112233 :
+    adaptiveCoverCheck 11 thetaAboveCell00112233 = true := by
+  have h : (thetaAboveCell00112233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112233 h
+theorem e24KC2ThetaAboveLeaf001123000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112300) = true := by
+  have h : ((childLL thetaAboveCell00112300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00112300) h
+theorem e24KC2ThetaAboveLeaf001123001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112300) = true := by
+  have h : ((childLH thetaAboveCell00112300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00112300) h
+theorem e24KC2ThetaAboveLeaf001123010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112301) = true := by
+  have h : ((childLL thetaAboveCell00112301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00112301) h
+theorem e24KC2ThetaAboveLeaf001123011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112301) = true := by
+  have h : ((childLH thetaAboveCell00112301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00112301) h
+theorem e24KC2ThetaAboveLeaf0011230202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00112302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00112302)) h
+theorem e24KC2ThetaAboveLeaf0011230203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00112302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00112302)) h
+theorem e24KC2ThetaAboveLeaf0011230212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00112302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00112302)) h
+theorem e24KC2ThetaAboveLeaf0011230213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00112302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00112302)) h
+theorem e24KC2ThetaAboveLeaf001123022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112302) = true := by
+  have h : ((childHL thetaAboveCell00112302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00112302) h
+theorem e24KC2ThetaAboveLeaf001123023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112302) = true := by
+  have h : ((childHH thetaAboveCell00112302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00112302) h
+theorem e24KC2ThetaAboveLeaf0011230302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00112303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00112303)) h
+theorem e24KC2ThetaAboveLeaf0011230303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00112303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00112303)) h
+theorem e24KC2ThetaAboveLeaf0011230312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00112303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00112303)) h
+theorem e24KC2ThetaAboveLeaf0011230313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00112303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00112303)) h
+theorem e24KC2ThetaAboveLeaf001123032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112303) = true := by
+  have h : ((childHL thetaAboveCell00112303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00112303) h
+theorem e24KC2ThetaAboveLeaf001123033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112303) = true := by
+  have h : ((childHH thetaAboveCell00112303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00112303) h
+theorem e24KC2ThetaAboveLeaf001123100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112310) = true := by
+  have h : ((childLL thetaAboveCell00112310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00112310) h
+theorem e24KC2ThetaAboveLeaf001123101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112310) = true := by
+  have h : ((childLH thetaAboveCell00112310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00112310) h
+theorem e24KC2ThetaAboveLeaf001123110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112311) = true := by
+  have h : ((childLL thetaAboveCell00112311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00112311) h
+theorem e24KC2ThetaAboveLeaf001123111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112311) = true := by
+  have h : ((childLH thetaAboveCell00112311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00112311) h
+theorem e24KC2ThetaAboveLeaf0011231121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00112311)) = true := by
+  have h : ((childLH (childHL thetaAboveCell00112311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell00112311)) h
+theorem e24KC2ThetaAboveLeaf0011231130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00112311)) = true := by
+  have h : ((childLL (childHH thetaAboveCell00112311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell00112311)) h
+theorem e24KC2ThetaAboveLeaf0011231131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00112311)) = true := by
+  have h : ((childLH (childHH thetaAboveCell00112311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell00112311)) h
+theorem e24KC2ThetaAboveLeaf0011231202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00112312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00112312)) h
+theorem e24KC2ThetaAboveLeaf0011231203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00112312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00112312)) h
+theorem e24KC2ThetaAboveLeaf0011231212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00112312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00112312)) h
+theorem e24KC2ThetaAboveLeaf0011231213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00112312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00112312)) h
+theorem e24KC2ThetaAboveLeaf001123122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112312) = true := by
+  have h : ((childHL thetaAboveCell00112312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00112312) h
+theorem e24KC2ThetaAboveLeaf001123123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112312) = true := by
+  have h : ((childHH thetaAboveCell00112312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00112312) h
+theorem e24KC2ThetaAboveLeaf0011231302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00112313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00112313)) h
+theorem e24KC2ThetaAboveLeaf0011231303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00112313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00112313)) h
+theorem e24KC2ThetaAboveLeaf0011231312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00112313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00112313)) h
+theorem e24KC2ThetaAboveLeaf0011231313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00112313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00112313)) h
+theorem e24KC2ThetaAboveLeaf001123132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112313) = true := by
+  have h : ((childHL thetaAboveCell00112313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00112313) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6ProofBatchC0ea1b4a6af3d1dc`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3658ca4972
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3658ca4972
+
+open CertificateCells3658ca4972
+theorem cover_subtree_c250c0508e67 :
+    adaptiveCoverCheck 8 (childLL (childLH thetaBelowCell11113210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH thetaBelowCell11113210))
+    (by
+      have h : ((childLL (childLL (childLH thetaBelowCell11113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLH
+        thetaBelowCell11113210))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaBelowCell11113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLH
+        thetaBelowCell11113210))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLL (childLH
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132101020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101020 h)
+        (by
+          have h : (thetaBelowCell111132101021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101021 h)
+        (by
+          have h : (thetaBelowCell111132101022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101022 h)
+        (by
+          have h : (thetaBelowCell111132101023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLL (childLH
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132101030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101030 h)
+        (by
+          have h : (thetaBelowCell111132101031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101031 h)
+        (by
+          have h : (thetaBelowCell111132101032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101032 h)
+        (by
+          have h : (thetaBelowCell111132101033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101033 h))
+
+theorem cover_subtree_37f607119182 :
+    adaptiveCoverCheck 8 (childLH (childLH thetaBelowCell11113210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH thetaBelowCell11113210))
+    (by
+      have h : ((childLL (childLH (childLH thetaBelowCell11113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLH
+        thetaBelowCell11113210))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaBelowCell11113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLH
+        thetaBelowCell11113210))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLH (childLH
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132101120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101120 h)
+        (by
+          have h : (thetaBelowCell111132101121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101121 h)
+        (by
+          have h : (thetaBelowCell111132101122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101122 h)
+        (by
+          have h : (thetaBelowCell111132101123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLH (childLH
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132101130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101130 h)
+        (by
+          have h : (thetaBelowCell111132101131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101131 h)
+        (by
+          have h : (thetaBelowCell111132101132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101132 h)
+        (by
+          have h : (thetaBelowCell111132101133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101133 h))
+
+theorem cover_subtree_9b56cbca582c :
+    adaptiveCoverCheck 8 (childHL (childLH thetaBelowCell11113210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH thetaBelowCell11113210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHL (childLH
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132101200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101200 h)
+        (by
+          have h : (thetaBelowCell111132101201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101201 h)
+        (by
+          have h : (thetaBelowCell111132101202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101202 h)
+        (by
+          have h : (thetaBelowCell111132101203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHL (childLH
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132101210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101210 h)
+        (by
+          have h : (thetaBelowCell111132101211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101211 h)
+        (by
+          have h : (thetaBelowCell111132101212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101212 h)
+        (by
+          have h : (thetaBelowCell111132101213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHL (childLH
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132101220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101220 h)
+        (by
+          have h : (thetaBelowCell111132101221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101221 h)
+        (by
+          have h : (thetaBelowCell111132101222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101222 h)
+        (by
+          have h : (thetaBelowCell111132101223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHL (childLH
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132101230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101230 h)
+        (by
+          have h : (thetaBelowCell111132101231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101231 h)
+        (by
+          have h : (thetaBelowCell111132101232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101232 h)
+        (by
+          have h : (thetaBelowCell111132101233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101233 h))
+
+theorem cover_subtree_321b06e53dc8 :
+    adaptiveCoverCheck 8 (childHH (childLH thetaBelowCell11113210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH thetaBelowCell11113210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHH (childLH
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132101300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101300 h)
+        (by
+          have h : (thetaBelowCell111132101301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101301 h)
+        (by
+          have h : (thetaBelowCell111132101302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101302 h)
+        (by
+          have h : (thetaBelowCell111132101303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHH (childLH
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132101310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101310 h)
+        (by
+          have h : (thetaBelowCell111132101311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101311 h)
+        (by
+          have h : (thetaBelowCell111132101312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101312 h)
+        (by
+          have h : (thetaBelowCell111132101313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHH (childLH
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132101320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101320 h)
+        (by
+          have h : (thetaBelowCell111132101321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101321 h)
+        (by
+          have h : (thetaBelowCell111132101322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101322 h)
+        (by
+          have h : (thetaBelowCell111132101323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHH (childLH
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132101330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101330 h)
+        (by
+          have h : (thetaBelowCell111132101331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101331 h)
+        (by
+          have h : (thetaBelowCell111132101332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101332 h)
+        (by
+          have h : (thetaBelowCell111132101333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132101333 h))
+
+theorem e24KC2ThetaBelowLeaf111132101 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113210) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11113210)
+    cover_subtree_c250c0508e67
+    cover_subtree_37f607119182
+    cover_subtree_9b56cbca582c
+    cover_subtree_321b06e53dc8
+theorem e24KC2ThetaBelowLeaf111132102 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113210) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113210)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL thetaBelowCell11113210))
+        (by
+          have h : ((childLL (childLL (childHL thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childHL
+            thetaBelowCell11113210))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childHL
+            thetaBelowCell11113210))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childHL
+            thetaBelowCell11113210))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childHL
+            thetaBelowCell11113210))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL thetaBelowCell11113210))
+        (by
+          have h : ((childLL (childLH (childHL thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childHL
+            thetaBelowCell11113210))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childHL
+            thetaBelowCell11113210))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childHL
+            thetaBelowCell11113210))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childHL
+            thetaBelowCell11113210))) h))
+    (by
+      have h : ((childHL (childHL thetaBelowCell11113210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11113210)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11113210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11113210)) h)
+theorem e24KC2ThetaBelowLeaf111132103 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113210) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113210)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH thetaBelowCell11113210))
+        (by
+          have h : ((childLL (childLL (childHH thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childHH
+            thetaBelowCell11113210))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childHH
+            thetaBelowCell11113210))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childHH
+            thetaBelowCell11113210))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childHH
+            thetaBelowCell11113210))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH thetaBelowCell11113210))
+        (by
+          have h : ((childLL (childLH (childHH thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childHH
+            thetaBelowCell11113210))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childHH
+            thetaBelowCell11113210))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childHH
+            thetaBelowCell11113210))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaBelowCell11113210)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childHH
+            thetaBelowCell11113210))) h))
+    (by
+      have h : ((childHL (childHH thetaBelowCell11113210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11113210)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11113210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11113210)) h)
+theorem cover_subtree_c437be4b8029 :
+    adaptiveCoverCheck 8 (childLL (childLL thetaBelowCell11113211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL thetaBelowCell11113211))
+    (by
+      have h : ((childLL (childLL (childLL thetaBelowCell11113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLL
+        thetaBelowCell11113211))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaBelowCell11113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLL
+        thetaBelowCell11113211))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLL (childLL
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132110020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110020 h)
+        (by
+          have h : (thetaBelowCell111132110021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110021 h)
+        (by
+          have h : (thetaBelowCell111132110022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110022 h)
+        (by
+          have h : (thetaBelowCell111132110023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLL (childLL
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132110030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110030 h)
+        (by
+          have h : (thetaBelowCell111132110031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110031 h)
+        (by
+          have h : (thetaBelowCell111132110032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110032 h)
+        (by
+          have h : (thetaBelowCell111132110033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110033 h))
+
+theorem cover_subtree_e1c2184f45cd :
+    adaptiveCoverCheck 8 (childLH (childLL thetaBelowCell11113211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL thetaBelowCell11113211))
+    (by
+      have h : ((childLL (childLH (childLL thetaBelowCell11113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLL
+        thetaBelowCell11113211))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaBelowCell11113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLL
+        thetaBelowCell11113211))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLH (childLL
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132110120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110120 h)
+        (by
+          have h : (thetaBelowCell111132110121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110121 h)
+        (by
+          have h : (thetaBelowCell111132110122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110122 h)
+        (by
+          have h : (thetaBelowCell111132110123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLH (childLL
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132110130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110130 h)
+        (by
+          have h : (thetaBelowCell111132110131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110131 h)
+        (by
+          have h : (thetaBelowCell111132110132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110132 h)
+        (by
+          have h : (thetaBelowCell111132110133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110133 h))
+
+theorem cover_subtree_baaa428341e9 :
+    adaptiveCoverCheck 8 (childHL (childLL thetaBelowCell11113211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL thetaBelowCell11113211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHL (childLL
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132110200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110200 h)
+        (by
+          have h : (thetaBelowCell111132110201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110201 h)
+        (by
+          have h : (thetaBelowCell111132110202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110202 h)
+        (by
+          have h : (thetaBelowCell111132110203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHL (childLL
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132110210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110210 h)
+        (by
+          have h : (thetaBelowCell111132110211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110211 h)
+        (by
+          have h : (thetaBelowCell111132110212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110212 h)
+        (by
+          have h : (thetaBelowCell111132110213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHL (childLL
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132110220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110220 h)
+        (by
+          have h : (thetaBelowCell111132110221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110221 h)
+        (by
+          have h : (thetaBelowCell111132110222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110222 h)
+        (by
+          have h : (thetaBelowCell111132110223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHL (childLL
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132110230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110230 h)
+        (by
+          have h : (thetaBelowCell111132110231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110231 h)
+        (by
+          have h : (thetaBelowCell111132110232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110232 h)
+        (by
+          have h : (thetaBelowCell111132110233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110233 h))
+
+theorem cover_subtree_fa56c52c3951 :
+    adaptiveCoverCheck 8 (childHH (childLL thetaBelowCell11113211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL thetaBelowCell11113211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHH (childLL
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132110300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110300 h)
+        (by
+          have h : (thetaBelowCell111132110301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110301 h)
+        (by
+          have h : (thetaBelowCell111132110302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110302 h)
+        (by
+          have h : (thetaBelowCell111132110303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHH (childLL
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132110310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110310 h)
+        (by
+          have h : (thetaBelowCell111132110311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110311 h)
+        (by
+          have h : (thetaBelowCell111132110312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110312 h)
+        (by
+          have h : (thetaBelowCell111132110313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHH (childLL
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132110320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110320 h)
+        (by
+          have h : (thetaBelowCell111132110321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110321 h)
+        (by
+          have h : (thetaBelowCell111132110322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110322 h)
+        (by
+          have h : (thetaBelowCell111132110323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHH (childLL
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132110330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110330 h)
+        (by
+          have h : (thetaBelowCell111132110331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110331 h)
+        (by
+          have h : (thetaBelowCell111132110332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110332 h)
+        (by
+          have h : (thetaBelowCell111132110333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132110333 h))
+
+theorem e24KC2ThetaBelowLeaf111132110 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113211) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11113211)
+    cover_subtree_c437be4b8029
+    cover_subtree_e1c2184f45cd
+    cover_subtree_baaa428341e9
+    cover_subtree_fa56c52c3951
+theorem cover_subtree_3dbc5ee0205c :
+    adaptiveCoverCheck 8 (childLL (childLH thetaBelowCell11113211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH thetaBelowCell11113211))
+    (by
+      have h : ((childLL (childLL (childLH thetaBelowCell11113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLH
+        thetaBelowCell11113211))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaBelowCell11113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLH
+        thetaBelowCell11113211))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLL (childLH
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132111020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111020 h)
+        (by
+          have h : (thetaBelowCell111132111021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111021 h)
+        (by
+          have h : (thetaBelowCell111132111022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111022 h)
+        (by
+          have h : (thetaBelowCell111132111023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLL (childLH
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132111030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111030 h)
+        (by
+          have h : (thetaBelowCell111132111031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111031 h)
+        (by
+          have h : (thetaBelowCell111132111032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111032 h)
+        (by
+          have h : (thetaBelowCell111132111033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111033 h))
+
+theorem cover_subtree_63676dfa48d8 :
+    adaptiveCoverCheck 8 (childLH (childLH thetaBelowCell11113211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH thetaBelowCell11113211))
+    (by
+      have h : ((childLL (childLH (childLH thetaBelowCell11113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLH
+        thetaBelowCell11113211))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaBelowCell11113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLH
+        thetaBelowCell11113211))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLH (childLH
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132111120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111120 h)
+        (by
+          have h : (thetaBelowCell111132111121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111121 h)
+        (by
+          have h : (thetaBelowCell111132111122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111122 h)
+        (by
+          have h : (thetaBelowCell111132111123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLH (childLH
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132111130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111130 h)
+        (by
+          have h : (thetaBelowCell111132111131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111131 h)
+        (by
+          have h : (thetaBelowCell111132111132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111132 h)
+        (by
+          have h : (thetaBelowCell111132111133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111133 h))
+
+theorem cover_subtree_27023d7e3233 :
+    adaptiveCoverCheck 8 (childHL (childLH thetaBelowCell11113211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH thetaBelowCell11113211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHL (childLH
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132111200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111200 h)
+        (by
+          have h : (thetaBelowCell111132111201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111201 h)
+        (by
+          have h : (thetaBelowCell111132111202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111202 h)
+        (by
+          have h : (thetaBelowCell111132111203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHL (childLH
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132111210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111210 h)
+        (by
+          have h : (thetaBelowCell111132111211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111211 h)
+        (by
+          have h : (thetaBelowCell111132111212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111212 h)
+        (by
+          have h : (thetaBelowCell111132111213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHL (childLH
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132111220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111220 h)
+        (by
+          have h : (thetaBelowCell111132111221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111221 h)
+        (by
+          have h : (thetaBelowCell111132111222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111222 h)
+        (by
+          have h : (thetaBelowCell111132111223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHL (childLH
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132111230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111230 h)
+        (by
+          have h : (thetaBelowCell111132111231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111231 h)
+        (by
+          have h : (thetaBelowCell111132111232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111232 h)
+        (by
+          have h : (thetaBelowCell111132111233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111233 h))
+
+theorem cover_subtree_f383ba4cac40 :
+    adaptiveCoverCheck 8 (childHH (childLH thetaBelowCell11113211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH thetaBelowCell11113211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHH (childLH
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132111300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111300 h)
+        (by
+          have h : (thetaBelowCell111132111301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111301 h)
+        (by
+          have h : (thetaBelowCell111132111302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111302 h)
+        (by
+          have h : (thetaBelowCell111132111303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHH (childLH
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132111310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111310 h)
+        (by
+          have h : (thetaBelowCell111132111311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111311 h)
+        (by
+          have h : (thetaBelowCell111132111312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111312 h)
+        (by
+          have h : (thetaBelowCell111132111313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHH (childLH
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132111320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111320 h)
+        (by
+          have h : (thetaBelowCell111132111321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111321 h)
+        (by
+          have h : (thetaBelowCell111132111322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111322 h)
+        (by
+          have h : (thetaBelowCell111132111323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHH (childLH
+        thetaBelowCell11113211)))
+        (by
+          have h : (thetaBelowCell111132111330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111330 h)
+        (by
+          have h : (thetaBelowCell111132111331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111331 h)
+        (by
+          have h : (thetaBelowCell111132111332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111332 h)
+        (by
+          have h : (thetaBelowCell111132111333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132111333 h))
+
+theorem e24KC2ThetaBelowLeaf111132111 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113211) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11113211)
+    cover_subtree_3dbc5ee0205c
+    cover_subtree_63676dfa48d8
+    cover_subtree_27023d7e3233
+    cover_subtree_f383ba4cac40
+theorem e24KC2ThetaBelowLeaf111132112 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113211) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113211)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL thetaBelowCell11113211))
+        (by
+          have h : ((childLL (childLL (childHL thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childHL
+            thetaBelowCell11113211))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childHL
+            thetaBelowCell11113211))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childHL
+            thetaBelowCell11113211))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childHL
+            thetaBelowCell11113211))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL thetaBelowCell11113211))
+        (by
+          have h : ((childLL (childLH (childHL thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childHL
+            thetaBelowCell11113211))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childHL
+            thetaBelowCell11113211))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childHL
+            thetaBelowCell11113211))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childHL
+            thetaBelowCell11113211))) h))
+    (by
+      have h : ((childHL (childHL thetaBelowCell11113211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11113211)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11113211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11113211)) h)
+theorem e24KC2ThetaBelowLeaf111132113 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113211) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113211)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH thetaBelowCell11113211))
+        (by
+          have h : ((childLL (childLL (childHH thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childHH
+            thetaBelowCell11113211))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childHH
+            thetaBelowCell11113211))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childHH
+            thetaBelowCell11113211))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childHH
+            thetaBelowCell11113211))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH thetaBelowCell11113211))
+        (by
+          have h : ((childLL (childLH (childHH thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childHH
+            thetaBelowCell11113211))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childHH
+            thetaBelowCell11113211))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childHH
+            thetaBelowCell11113211))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaBelowCell11113211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childHH
+            thetaBelowCell11113211))) h))
+    (by
+      have h : ((childHL (childHH thetaBelowCell11113211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11113211)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11113211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11113211)) h)
+theorem cover_subtree_2d104ae8510a :
+    adaptiveCoverCheck 8 (childLL (childLL thetaBelowCell11113300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL thetaBelowCell11113300))
+    (by
+      have h : ((childLL (childLL (childLL thetaBelowCell11113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLL
+        thetaBelowCell11113300))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaBelowCell11113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLL
+        thetaBelowCell11113300))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLL (childLL
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133000020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000020 h)
+        (by
+          have h : (thetaBelowCell111133000021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000021 h)
+        (by
+          have h : (thetaBelowCell111133000022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000022 h)
+        (by
+          have h : (thetaBelowCell111133000023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLL (childLL
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133000030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000030 h)
+        (by
+          have h : (thetaBelowCell111133000031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000031 h)
+        (by
+          have h : (thetaBelowCell111133000032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000032 h)
+        (by
+          have h : (thetaBelowCell111133000033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000033 h))
+
+theorem cover_subtree_73e80951f42f :
+    adaptiveCoverCheck 8 (childLH (childLL thetaBelowCell11113300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL thetaBelowCell11113300))
+    (by
+      have h : ((childLL (childLH (childLL thetaBelowCell11113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLL
+        thetaBelowCell11113300))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaBelowCell11113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLL
+        thetaBelowCell11113300))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLH (childLL
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133000120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000120 h)
+        (by
+          have h : (thetaBelowCell111133000121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000121 h)
+        (by
+          have h : (thetaBelowCell111133000122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000122 h)
+        (by
+          have h : (thetaBelowCell111133000123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLH (childLL
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133000130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000130 h)
+        (by
+          have h : (thetaBelowCell111133000131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000131 h)
+        (by
+          have h : (thetaBelowCell111133000132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000132 h)
+        (by
+          have h : (thetaBelowCell111133000133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000133 h))
+
+theorem cover_subtree_788fa0ec2ac5 :
+    adaptiveCoverCheck 8 (childHL (childLL thetaBelowCell11113300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL thetaBelowCell11113300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHL (childLL
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133000200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000200 h)
+        (by
+          have h : (thetaBelowCell111133000201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000201 h)
+        (by
+          have h : (thetaBelowCell111133000202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000202 h)
+        (by
+          have h : (thetaBelowCell111133000203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHL (childLL
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133000210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000210 h)
+        (by
+          have h : (thetaBelowCell111133000211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000211 h)
+        (by
+          have h : (thetaBelowCell111133000212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000212 h)
+        (by
+          have h : (thetaBelowCell111133000213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHL (childLL
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133000220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000220 h)
+        (by
+          have h : (thetaBelowCell111133000221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000221 h)
+        (by
+          have h : (thetaBelowCell111133000222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000222 h)
+        (by
+          have h : (thetaBelowCell111133000223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHL (childLL
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133000230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000230 h)
+        (by
+          have h : (thetaBelowCell111133000231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000231 h)
+        (by
+          have h : (thetaBelowCell111133000232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000232 h)
+        (by
+          have h : (thetaBelowCell111133000233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000233 h))
+
+theorem cover_subtree_a895dbe70eef :
+    adaptiveCoverCheck 8 (childHH (childLL thetaBelowCell11113300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL thetaBelowCell11113300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHH (childLL
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133000300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000300 h)
+        (by
+          have h : (thetaBelowCell111133000301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000301 h)
+        (by
+          have h : (thetaBelowCell111133000302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000302 h)
+        (by
+          have h : (thetaBelowCell111133000303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHH (childLL
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133000310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000310 h)
+        (by
+          have h : (thetaBelowCell111133000311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000311 h)
+        (by
+          have h : (thetaBelowCell111133000312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000312 h)
+        (by
+          have h : (thetaBelowCell111133000313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHH (childLL
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133000320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000320 h)
+        (by
+          have h : (thetaBelowCell111133000321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000321 h)
+        (by
+          have h : (thetaBelowCell111133000322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000322 h)
+        (by
+          have h : (thetaBelowCell111133000323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHH (childLL
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133000330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000330 h)
+        (by
+          have h : (thetaBelowCell111133000331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000331 h)
+        (by
+          have h : (thetaBelowCell111133000332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000332 h)
+        (by
+          have h : (thetaBelowCell111133000333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133000333 h))
+
+theorem e24KC2ThetaBelowLeaf111133000 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113300) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11113300)
+    cover_subtree_2d104ae8510a
+    cover_subtree_73e80951f42f
+    cover_subtree_788fa0ec2ac5
+    cover_subtree_a895dbe70eef
+theorem cover_subtree_f0725b4a5e12 :
+    adaptiveCoverCheck 8 (childLL (childLH thetaBelowCell11113300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH thetaBelowCell11113300))
+    (by
+      have h : ((childLL (childLL (childLH thetaBelowCell11113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLH
+        thetaBelowCell11113300))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaBelowCell11113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLH
+        thetaBelowCell11113300))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLL (childLH
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133001020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001020 h)
+        (by
+          have h : (thetaBelowCell111133001021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001021 h)
+        (by
+          have h : (thetaBelowCell111133001022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001022 h)
+        (by
+          have h : (thetaBelowCell111133001023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLL (childLH
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133001030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001030 h)
+        (by
+          have h : (thetaBelowCell111133001031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001031 h)
+        (by
+          have h : (thetaBelowCell111133001032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001032 h)
+        (by
+          have h : (thetaBelowCell111133001033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001033 h))
+
+theorem cover_subtree_cedab414430b :
+    adaptiveCoverCheck 7 (childHL (childHL (childLH thetaBelowCell11113300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHL (childLH thetaBelowCell11113300)))
+    (by
+      have h : (thetaBelowCell111133001220).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001220 h)
+    (by
+      have h : (thetaBelowCell111133001221).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001221 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001222
+        (by
+          have h : ((childLL thetaBelowCell111133001222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001222) h)
+        (by
+          have h : ((childLH thetaBelowCell111133001222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001222) h)
+        (by
+          have h : ((childHL thetaBelowCell111133001222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001222) h)
+        (by
+          have h : ((childHH thetaBelowCell111133001222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001222) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001223
+        (by
+          have h : ((childLL thetaBelowCell111133001223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001223) h)
+        (by
+          have h : ((childLH thetaBelowCell111133001223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001223) h)
+        (by
+          have h : ((childHL thetaBelowCell111133001223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001223) h)
+        (by
+          have h : ((childHH thetaBelowCell111133001223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001223) h))
+
+theorem cover_subtree_15d931d0b831 :
+    adaptiveCoverCheck 7 (childHH (childHL (childLH thetaBelowCell11113300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHL (childLH thetaBelowCell11113300)))
+    (by
+      have h : (thetaBelowCell111133001230).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001230 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001231
+        (by
+          have h : ((childLL thetaBelowCell111133001231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001231) h)
+        (by
+          have h : ((childLH thetaBelowCell111133001231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001231) h)
+        (by
+          have h : ((childHL thetaBelowCell111133001231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001231) h)
+        (by
+          have h : ((childHH thetaBelowCell111133001231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001231) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001232
+        (by
+          have h : ((childLL thetaBelowCell111133001232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001232) h)
+        (by
+          have h : ((childLH thetaBelowCell111133001232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001232) h)
+        (by
+          have h : ((childHL thetaBelowCell111133001232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001232) h)
+        (by
+          have h : ((childHH thetaBelowCell111133001232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001232) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001233
+        (by
+          have h : ((childLL thetaBelowCell111133001233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001233) h)
+        (by
+          have h : ((childLH thetaBelowCell111133001233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001233) h)
+        (by
+          have h : ((childHL thetaBelowCell111133001233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001233) h)
+        (by
+          have h : ((childHH thetaBelowCell111133001233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001233) h))
+
+theorem cover_subtree_4398f73b5a5d :
+    adaptiveCoverCheck 8 (childHL (childLH thetaBelowCell11113300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH thetaBelowCell11113300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHL (childLH
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133001200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001200 h)
+        (by
+          have h : (thetaBelowCell111133001201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001201 h)
+        (by
+          have h : (thetaBelowCell111133001202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001202 h)
+        (by
+          have h : (thetaBelowCell111133001203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHL (childLH
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133001210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001210 h)
+        (by
+          have h : (thetaBelowCell111133001211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001211 h)
+        (by
+          have h : (thetaBelowCell111133001212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001212 h)
+        (by
+          have h : (thetaBelowCell111133001213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001213 h))
+    cover_subtree_cedab414430b
+    cover_subtree_15d931d0b831
+
+theorem cover_subtree_947341870459 :
+    adaptiveCoverCheck 7 (childLH (childHH (childLH thetaBelowCell11113300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHH (childLH thetaBelowCell11113300)))
+    (by
+      have h : (thetaBelowCell111133001310).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001310 h)
+    (by
+      have h : (thetaBelowCell111133001311).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001311 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001312
+        (by
+          have h : ((childLL thetaBelowCell111133001312)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001312) h)
+        (by
+          have h : ((childLH thetaBelowCell111133001312)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001312) h)
+        (by
+          have h : ((childHL thetaBelowCell111133001312)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001312) h)
+        (by
+          have h : ((childHH thetaBelowCell111133001312)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001312) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001313
+        (by
+          have h : ((childLL thetaBelowCell111133001313)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001313) h)
+        (by
+          have h : ((childLH thetaBelowCell111133001313)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001313) h)
+        (by
+          have h : ((childHL thetaBelowCell111133001313)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001313) h)
+        (by
+          have h : ((childHH thetaBelowCell111133001313)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001313) h))
+
+theorem cover_subtree_28ced15c6953 :
+    adaptiveCoverCheck 7 (childHL (childHH (childLH thetaBelowCell11113300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHH (childLH thetaBelowCell11113300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001320
+        (by
+          have h : ((childLL thetaBelowCell111133001320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001320) h)
+        (by
+          have h : ((childLH thetaBelowCell111133001320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001320) h)
+        (by
+          have h : ((childHL thetaBelowCell111133001320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001320) h)
+        (by
+          have h : ((childHH thetaBelowCell111133001320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001320) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001321
+        (by
+          have h : ((childLL thetaBelowCell111133001321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001321) h)
+        (by
+          have h : ((childLH thetaBelowCell111133001321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001321) h)
+        (by
+          have h : ((childHL thetaBelowCell111133001321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001321) h)
+        (by
+          have h : ((childHH thetaBelowCell111133001321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001321) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001322
+        (by
+          have h : ((childLL thetaBelowCell111133001322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001322) h)
+        (by
+          have h : ((childLH thetaBelowCell111133001322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001322) h)
+        (by
+          have h : ((childHL thetaBelowCell111133001322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001322) h)
+        (by
+          have h : ((childHH thetaBelowCell111133001322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001322) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001323
+        (by
+          have h : ((childLL thetaBelowCell111133001323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001323) h)
+        (by
+          have h : ((childLH thetaBelowCell111133001323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001323) h)
+        (by
+          have h : ((childHL thetaBelowCell111133001323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001323) h)
+        (by
+          have h : ((childHH thetaBelowCell111133001323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001323) h))
+
+theorem cover_subtree_440561389a75 :
+    adaptiveCoverCheck 7 (childHH (childHH (childLH thetaBelowCell11113300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHH (childLH thetaBelowCell11113300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001330
+        (by
+          have h : ((childLL thetaBelowCell111133001330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001330) h)
+        (by
+          have h : ((childLH thetaBelowCell111133001330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001330) h)
+        (by
+          have h : ((childHL thetaBelowCell111133001330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001330) h)
+        (by
+          have h : ((childHH thetaBelowCell111133001330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001330) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001331
+        (by
+          have h : ((childLL thetaBelowCell111133001331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001331) h)
+        (by
+          have h : ((childLH thetaBelowCell111133001331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001331) h)
+        (by
+          have h : ((childHL thetaBelowCell111133001331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001331) h)
+        (by
+          have h : ((childHH thetaBelowCell111133001331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001331) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001332
+        (by
+          have h : ((childLL thetaBelowCell111133001332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001332) h)
+        (by
+          have h : ((childLH thetaBelowCell111133001332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001332) h)
+        (by
+          have h : ((childHL thetaBelowCell111133001332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001332) h)
+        (by
+          have h : ((childHH thetaBelowCell111133001332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001332) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001333
+        (by
+          have h : ((childLL thetaBelowCell111133001333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001333) h)
+        (by
+          have h : ((childLH thetaBelowCell111133001333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001333) h)
+        (by
+          have h : ((childHL thetaBelowCell111133001333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001333) h)
+        (by
+          have h : ((childHH thetaBelowCell111133001333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001333) h))
+
+theorem cover_subtree_42d3bf5965e9 :
+    adaptiveCoverCheck 8 (childHH (childLH thetaBelowCell11113300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH thetaBelowCell11113300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHH (childLH
+        thetaBelowCell11113300)))
+        (by
+          have h : (thetaBelowCell111133001300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001300 h)
+        (by
+          have h : (thetaBelowCell111133001301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001301 h)
+        (by
+          have h : (thetaBelowCell111133001302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001302 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133001303
+            (by
+              have h : ((childLL thetaBelowCell111133001303)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133001303) h)
+            (by
+              have h : ((childLH thetaBelowCell111133001303)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133001303) h)
+            (by
+              have h : ((childHL thetaBelowCell111133001303)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133001303) h)
+            (by
+              have h : ((childHH thetaBelowCell111133001303)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133001303) h)))
+    cover_subtree_947341870459
+    cover_subtree_28ced15c6953
+    cover_subtree_440561389a75
+
+theorem e24KC2ThetaBelowLeaf111133001 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113300) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11113300)
+    cover_subtree_f0725b4a5e12
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH thetaBelowCell11113300))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell11113300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLH
+            thetaBelowCell11113300))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell11113300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLH
+            thetaBelowCell11113300))) h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLH (childLH
+            thetaBelowCell11113300)))
+            (by
+              have h : (thetaBelowCell111133001120).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001120 h)
+            (by
+              have h : (thetaBelowCell111133001121).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001121 h)
+            (by
+              have h : (thetaBelowCell111133001122).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001122 h)
+            (by
+              have h : (thetaBelowCell111133001123).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133001123 h))
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell11113300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLH
+            thetaBelowCell11113300))) h))
+    cover_subtree_4398f73b5a5d
+    cover_subtree_42d3bf5965e9
+theorem e24KC2ThetaBelowLeaf111133002 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113300) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113300)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL thetaBelowCell11113300))
+        (by
+          have h : ((childLL (childLL (childHL thetaBelowCell11113300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childHL
+            thetaBelowCell11113300))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaBelowCell11113300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childHL
+            thetaBelowCell11113300))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaBelowCell11113300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childHL
+            thetaBelowCell11113300))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaBelowCell11113300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childHL
+            thetaBelowCell11113300))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL thetaBelowCell11113300))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLH (childHL
+            thetaBelowCell11113300)))
+            (by
+              have h : (thetaBelowCell111133002100).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133002100 h)
+            (by
+              have h : (thetaBelowCell111133002101).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133002101 h)
+            (by
+              have h : (thetaBelowCell111133002102).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133002102 h)
+            (by
+              have h : (thetaBelowCell111133002103).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133002103 h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childLH (childLH (childHL
+            thetaBelowCell11113300)))
+            (by
+              have h : (thetaBelowCell111133002110).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133002110 h)
+            (by
+              have h : (thetaBelowCell111133002111).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133002111 h)
+            (by
+              have h : (thetaBelowCell111133002112).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133002112 h)
+            (by
+              have h : (thetaBelowCell111133002113).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133002113 h))
+        (by
+          have h : ((childHL (childLH (childHL thetaBelowCell11113300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childHL
+            thetaBelowCell11113300))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaBelowCell11113300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childHL
+            thetaBelowCell11113300))) h))
+    (by
+      have h : ((childHL (childHL thetaBelowCell11113300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11113300)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11113300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11113300)) h)
+theorem e24KC2ThetaBelowLeaf111133003 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113300) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113300)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH thetaBelowCell11113300))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLL (childHH
+            thetaBelowCell11113300)))
+            (by
+              have h : (thetaBelowCell111133003000).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003000 h)
+            (by
+              have h : (thetaBelowCell111133003001).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003001 h)
+            (by
+              have h : (thetaBelowCell111133003002).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003002 h)
+            (by
+              have h : (thetaBelowCell111133003003).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003003 h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childLH (childLL (childHH
+            thetaBelowCell11113300)))
+            (by
+              have h : (thetaBelowCell111133003010).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003010 h)
+            (by
+              have h : (thetaBelowCell111133003011).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003011 h)
+            (by
+              have h : (thetaBelowCell111133003012).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003012 h)
+            (by
+              have h : (thetaBelowCell111133003013).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003013 h))
+        (by
+          have h : ((childHL (childLL (childHH thetaBelowCell11113300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childHH
+            thetaBelowCell11113300))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaBelowCell11113300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childHH
+            thetaBelowCell11113300))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH thetaBelowCell11113300))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLH (childHH
+            thetaBelowCell11113300)))
+            (by
+              have h : (thetaBelowCell111133003100).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003100 h)
+            (by
+              have h : (thetaBelowCell111133003101).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003101 h)
+            (by
+              have h : (thetaBelowCell111133003102).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003102 h)
+            (by
+              have h : (thetaBelowCell111133003103).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003103 h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childLH (childLH (childHH
+            thetaBelowCell11113300)))
+            (by
+              have h : (thetaBelowCell111133003110).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003110 h)
+            (by
+              have h : (thetaBelowCell111133003111).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003111 h)
+            (by
+              have h : (thetaBelowCell111133003112).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003112 h)
+            (by
+              have h : (thetaBelowCell111133003113).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133003113 h))
+        (by
+          have h : ((childHL (childLH (childHH thetaBelowCell11113300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childHH
+            thetaBelowCell11113300))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaBelowCell11113300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childHH
+            thetaBelowCell11113300))) h))
+    (by
+      have h : ((childHL (childHH thetaBelowCell11113300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11113300)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11113300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11113300)) h)
+theorem cover_subtree_37babc6363e9 :
+    adaptiveCoverCheck 7 (childLL (childHL (childLL thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHL (childLL thetaBelowCell11113301)))
+    (by
+      have h : (thetaBelowCell111133010200).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133010200 h)
+    (by
+      have h : (thetaBelowCell111133010201).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133010201 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010202
+        (by
+          have h : ((childLL thetaBelowCell111133010202)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010202) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010202)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010202) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010202)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010202) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010202)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010202) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010203
+        (by
+          have h : ((childLL thetaBelowCell111133010203)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010203) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010203)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010203) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010203)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010203) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010203)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010203) h))
+
+theorem cover_subtree_507281a99f13 :
+    adaptiveCoverCheck 7 (childLH (childHL (childLL thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHL (childLL thetaBelowCell11113301)))
+    (by
+      have h : (thetaBelowCell111133010210).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133010210 h)
+    (by
+      have h : (thetaBelowCell111133010211).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133010211 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010212
+        (by
+          have h : ((childLL thetaBelowCell111133010212)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010212) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010212)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010212) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010212)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010212) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010212)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010212) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010213
+        (by
+          have h : ((childLL thetaBelowCell111133010213)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010213) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010213)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010213) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010213)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010213) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010213)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010213) h))
+
+theorem cover_subtree_a5c13fd746d7 :
+    adaptiveCoverCheck 7 (childHL (childHL (childLL thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHL (childLL thetaBelowCell11113301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010220
+        (by
+          have h : ((childLL thetaBelowCell111133010220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010220) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010220) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010220) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010220) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010221
+        (by
+          have h : ((childLL thetaBelowCell111133010221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010221) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010221) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010221) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010221) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010222
+        (by
+          have h : ((childLL thetaBelowCell111133010222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010222) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010222) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010222) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010222) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010223
+        (by
+          have h : ((childLL thetaBelowCell111133010223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010223) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010223) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010223) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010223) h))
+
+theorem cover_subtree_6d4cbcd8b0bf :
+    adaptiveCoverCheck 7 (childHH (childHL (childLL thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHL (childLL thetaBelowCell11113301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010230
+        (by
+          have h : ((childLL thetaBelowCell111133010230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010230) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010230) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010230) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010230) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010231
+        (by
+          have h : ((childLL thetaBelowCell111133010231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010231) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010231) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010231) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010231) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010232
+        (by
+          have h : ((childLL thetaBelowCell111133010232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010232) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010232) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010232) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010232) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010233
+        (by
+          have h : ((childLL thetaBelowCell111133010233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010233) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010233) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010233) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010233) h))
+
+theorem cover_subtree_3b5ec2cd76c2 :
+    adaptiveCoverCheck 8 (childHL (childLL thetaBelowCell11113301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL thetaBelowCell11113301))
+    cover_subtree_37babc6363e9
+    cover_subtree_507281a99f13
+    cover_subtree_a5c13fd746d7
+    cover_subtree_6d4cbcd8b0bf
+
+theorem cover_subtree_a6f74a199082 :
+    adaptiveCoverCheck 7 (childLL (childHH (childLL thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHH (childLL thetaBelowCell11113301)))
+    (by
+      have h : (thetaBelowCell111133010300).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133010300 h)
+    (by
+      have h : (thetaBelowCell111133010301).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133010301 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010302
+        (by
+          have h : ((childLL thetaBelowCell111133010302)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010302) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010302)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010302) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010302)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010302) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010302)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010302) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010303
+        (by
+          have h : ((childLL thetaBelowCell111133010303)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010303) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010303)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010303) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010303)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010303) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010303)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010303) h))
+
+theorem cover_subtree_9f6cabcef004 :
+    adaptiveCoverCheck 7 (childLH (childHH (childLL thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHH (childLL thetaBelowCell11113301)))
+    (by
+      have h : (thetaBelowCell111133010310).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133010310 h)
+    (by
+      have h : (thetaBelowCell111133010311).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133010311 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010312
+        (by
+          have h : ((childLL thetaBelowCell111133010312)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010312) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010312)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010312) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010312)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010312) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010312)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010312) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010313
+        (by
+          have h : ((childLL thetaBelowCell111133010313)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010313) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010313)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010313) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010313)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010313) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010313)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010313) h))
+
+theorem cover_subtree_66d25ab1916a :
+    adaptiveCoverCheck 7 (childHL (childHH (childLL thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHH (childLL thetaBelowCell11113301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010320
+        (by
+          have h : ((childLL thetaBelowCell111133010320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010320) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010320) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010320) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010320) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010321
+        (by
+          have h : ((childLL thetaBelowCell111133010321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010321) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010321) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010321) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010321) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010322
+        (by
+          have h : ((childLL thetaBelowCell111133010322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010322) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010322) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010322) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010322) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010323
+        (by
+          have h : ((childLL thetaBelowCell111133010323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010323) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010323) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010323) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010323) h))
+
+theorem cover_subtree_7a16ff7db42a :
+    adaptiveCoverCheck 7 (childHH (childHH (childLL thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHH (childLL thetaBelowCell11113301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010330
+        (by
+          have h : ((childLL thetaBelowCell111133010330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010330) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010330) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010330) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010330) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010331
+        (by
+          have h : ((childLL thetaBelowCell111133010331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010331) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010331) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010331) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010331) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010332
+        (by
+          have h : ((childLL thetaBelowCell111133010332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010332) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010332) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010332) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010332) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133010333
+        (by
+          have h : ((childLL thetaBelowCell111133010333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133010333) h)
+        (by
+          have h : ((childLH thetaBelowCell111133010333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133010333) h)
+        (by
+          have h : ((childHL thetaBelowCell111133010333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133010333) h)
+        (by
+          have h : ((childHH thetaBelowCell111133010333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133010333) h))
+
+theorem cover_subtree_c3eeff28e7cb :
+    adaptiveCoverCheck 8 (childHH (childLL thetaBelowCell11113301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL thetaBelowCell11113301))
+    cover_subtree_a6f74a199082
+    cover_subtree_9f6cabcef004
+    cover_subtree_66d25ab1916a
+    cover_subtree_7a16ff7db42a
+
+theorem e24KC2ThetaBelowLeaf111133010 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113301) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11113301)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL thetaBelowCell11113301))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLL
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLL
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLL
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLL
+            thetaBelowCell11113301))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL thetaBelowCell11113301))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLL
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLL
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLL
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLL
+            thetaBelowCell11113301))) h))
+    cover_subtree_3b5ec2cd76c2
+    cover_subtree_c3eeff28e7cb
+theorem cover_subtree_1ae28d021e20 :
+    adaptiveCoverCheck 7 (childLL (childHL (childLH thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHL (childLH thetaBelowCell11113301)))
+    (by
+      have h : (thetaBelowCell111133011200).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133011200 h)
+    (by
+      have h : (thetaBelowCell111133011201).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133011201 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011202
+        (by
+          have h : ((childLL thetaBelowCell111133011202)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011202) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011202)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011202) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011202)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011202) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011202)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011202) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011203
+        (by
+          have h : ((childLL thetaBelowCell111133011203)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011203) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011203)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011203) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011203)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011203) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011203)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011203) h))
+
+theorem cover_subtree_40cb4ec2b408 :
+    adaptiveCoverCheck 7 (childLH (childHL (childLH thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHL (childLH thetaBelowCell11113301)))
+    (by
+      have h : (thetaBelowCell111133011210).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133011210 h)
+    (by
+      have h : (thetaBelowCell111133011211).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133011211 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011212
+        (by
+          have h : ((childLL thetaBelowCell111133011212)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011212) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011212)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011212) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011212)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011212) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011212)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011212) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011213
+        (by
+          have h : ((childLL thetaBelowCell111133011213)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011213) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011213)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011213) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011213)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011213) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011213)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011213) h))
+
+theorem cover_subtree_b87befd4bfbe :
+    adaptiveCoverCheck 7 (childHL (childHL (childLH thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHL (childLH thetaBelowCell11113301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011220
+        (by
+          have h : ((childLL thetaBelowCell111133011220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011220) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011220) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011220) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011220) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011221
+        (by
+          have h : ((childLL thetaBelowCell111133011221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011221) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011221) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011221) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011221) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011222
+        (by
+          have h : ((childLL thetaBelowCell111133011222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011222) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011222) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011222) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011222) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011223
+        (by
+          have h : ((childLL thetaBelowCell111133011223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011223) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011223) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011223) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011223) h))
+
+theorem cover_subtree_58527e6cb5e3 :
+    adaptiveCoverCheck 7 (childHH (childHL (childLH thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHL (childLH thetaBelowCell11113301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011230
+        (by
+          have h : ((childLL thetaBelowCell111133011230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011230) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011230) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011230) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011230) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011231
+        (by
+          have h : ((childLL thetaBelowCell111133011231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011231) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011231) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011231) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011231) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011232
+        (by
+          have h : ((childLL thetaBelowCell111133011232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011232) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011232) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011232) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011232) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011233
+        (by
+          have h : ((childLL thetaBelowCell111133011233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011233) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011233) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011233) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011233) h))
+
+theorem cover_subtree_c56ac23b21bd :
+    adaptiveCoverCheck 8 (childHL (childLH thetaBelowCell11113301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH thetaBelowCell11113301))
+    cover_subtree_1ae28d021e20
+    cover_subtree_40cb4ec2b408
+    cover_subtree_b87befd4bfbe
+    cover_subtree_58527e6cb5e3
+
+theorem cover_subtree_644ffb1207bc :
+    adaptiveCoverCheck 7 (childLL (childHH (childLH thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHH (childLH thetaBelowCell11113301)))
+    (by
+      have h : (thetaBelowCell111133011300).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133011300 h)
+    (by
+      have h : (thetaBelowCell111133011301).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133011301 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011302
+        (by
+          have h : ((childLL thetaBelowCell111133011302)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011302) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011302)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011302) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011302)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011302) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011302)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011302) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011303
+        (by
+          have h : ((childLL thetaBelowCell111133011303)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011303) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011303)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011303) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011303)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011303) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011303)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011303) h))
+
+theorem cover_subtree_15868ac019bc :
+    adaptiveCoverCheck 7 (childLH (childHH (childLH thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHH (childLH thetaBelowCell11113301)))
+    (by
+      have h : (thetaBelowCell111133011310).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133011310 h)
+    (by
+      have h : (thetaBelowCell111133011311).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133011311 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011312
+        (by
+          have h : ((childLL thetaBelowCell111133011312)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011312) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011312)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011312) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011312)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011312) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011312)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011312) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011313
+        (by
+          have h : ((childLL thetaBelowCell111133011313)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011313) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011313)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011313) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011313)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011313) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011313)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011313) h))
+
+theorem cover_subtree_c4a4ddefa402 :
+    adaptiveCoverCheck 7 (childHL (childHH (childLH thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHH (childLH thetaBelowCell11113301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011320
+        (by
+          have h : ((childLL thetaBelowCell111133011320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011320) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011320) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011320) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011320) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011321
+        (by
+          have h : ((childLL thetaBelowCell111133011321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011321) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011321) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011321) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011321) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011322
+        (by
+          have h : ((childLL thetaBelowCell111133011322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011322) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011322) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011322) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011322) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011323
+        (by
+          have h : ((childLL thetaBelowCell111133011323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011323) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011323) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011323) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011323) h))
+
+theorem cover_subtree_2e2978a517d6 :
+    adaptiveCoverCheck 7 (childHH (childHH (childLH thetaBelowCell11113301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHH (childLH thetaBelowCell11113301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011330
+        (by
+          have h : ((childLL thetaBelowCell111133011330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011330) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011330) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011330) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011330) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011331
+        (by
+          have h : ((childLL thetaBelowCell111133011331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011331) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011331) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011331) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011331) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011332
+        (by
+          have h : ((childLL thetaBelowCell111133011332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011332) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011332) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011332) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011332) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133011333
+        (by
+          have h : ((childLL thetaBelowCell111133011333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133011333) h)
+        (by
+          have h : ((childLH thetaBelowCell111133011333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133011333) h)
+        (by
+          have h : ((childHL thetaBelowCell111133011333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133011333) h)
+        (by
+          have h : ((childHH thetaBelowCell111133011333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133011333) h))
+
+theorem cover_subtree_f27cd76bf084 :
+    adaptiveCoverCheck 8 (childHH (childLH thetaBelowCell11113301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH thetaBelowCell11113301))
+    cover_subtree_644ffb1207bc
+    cover_subtree_15868ac019bc
+    cover_subtree_c4a4ddefa402
+    cover_subtree_2e2978a517d6
+
+theorem e24KC2ThetaBelowLeaf111133011 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113301) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11113301)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH thetaBelowCell11113301))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLH
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLH
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLH
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLH
+            thetaBelowCell11113301))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH thetaBelowCell11113301))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLH
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLH
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLH
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLH
+            thetaBelowCell11113301))) h))
+    cover_subtree_c56ac23b21bd
+    cover_subtree_f27cd76bf084
+theorem e24KC2ThetaBelowLeaf111133012 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113301) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113301)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL thetaBelowCell11113301))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLL (childHL
+            thetaBelowCell11113301)))
+            (by
+              have h : (thetaBelowCell111133012000).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012000 h)
+            (by
+              have h : (thetaBelowCell111133012001).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012001 h)
+            (by
+              have h : (thetaBelowCell111133012002).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012002 h)
+            (by
+              have h : (thetaBelowCell111133012003).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012003 h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childLH (childLL (childHL
+            thetaBelowCell11113301)))
+            (by
+              have h : (thetaBelowCell111133012010).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012010 h)
+            (by
+              have h : (thetaBelowCell111133012011).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012011 h)
+            (by
+              have h : (thetaBelowCell111133012012).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012012 h)
+            (by
+              have h : (thetaBelowCell111133012013).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012013 h))
+        (by
+          have h : ((childHL (childLL (childHL thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childHL
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childHL
+            thetaBelowCell11113301))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL thetaBelowCell11113301))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLH (childHL
+            thetaBelowCell11113301)))
+            (by
+              have h : (thetaBelowCell111133012100).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012100 h)
+            (by
+              have h : (thetaBelowCell111133012101).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012101 h)
+            (by
+              have h : (thetaBelowCell111133012102).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012102 h)
+            (by
+              have h : (thetaBelowCell111133012103).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012103 h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childLH (childLH (childHL
+            thetaBelowCell11113301)))
+            (by
+              have h : (thetaBelowCell111133012110).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012110 h)
+            (by
+              have h : (thetaBelowCell111133012111).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012111 h)
+            (by
+              have h : (thetaBelowCell111133012112).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012112 h)
+            (by
+              have h : (thetaBelowCell111133012113).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133012113 h))
+        (by
+          have h : ((childHL (childLH (childHL thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childHL
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childHL
+            thetaBelowCell11113301))) h))
+    (by
+      have h : ((childHL (childHL thetaBelowCell11113301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11113301)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11113301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11113301)) h)
+theorem e24KC2ThetaBelowLeaf111133013 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113301) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113301)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH thetaBelowCell11113301))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLL (childHH
+            thetaBelowCell11113301)))
+            (by
+              have h : (thetaBelowCell111133013000).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133013000 h)
+            (by
+              have h : (thetaBelowCell111133013001).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133013001 h)
+            (by
+              have h : (thetaBelowCell111133013002).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133013002 h)
+            (by
+              have h : (thetaBelowCell111133013003).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133013003 h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childLH (childLL (childHH
+            thetaBelowCell11113301)))
+            (by
+              have h : (thetaBelowCell111133013010).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133013010 h)
+            (by
+              have h : (thetaBelowCell111133013011).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133013011 h)
+            (by
+              have h : (thetaBelowCell111133013012).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133013012 h)
+            (by
+              have h : (thetaBelowCell111133013013).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133013013 h))
+        (by
+          have h : ((childHL (childLL (childHH thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childHH
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childHH
+            thetaBelowCell11113301))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH thetaBelowCell11113301))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLH (childHH
+            thetaBelowCell11113301)))
+            (by
+              have h : (thetaBelowCell111133013100).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133013100 h)
+            (by
+              exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133013101
+                (by
+                  have h : ((childLL thetaBelowCell111133013101)).rejected = true := by
+                    decide +kernel
+                  exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133013101)
+                    h)
+                (by
+                  have h : ((childLH thetaBelowCell111133013101)).rejected = true := by
+                    decide +kernel
+                  exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133013101)
+                    h)
+                (by
+                  have h : ((childHL thetaBelowCell111133013101)).rejected = true := by
+                    decide +kernel
+                  exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133013101)
+                    h)
+                (by
+                  have h : ((childHH thetaBelowCell111133013101)).rejected = true := by
+                    decide +kernel
+                  exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133013101)
+                    h))
+            (by
+              have h : (thetaBelowCell111133013102).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133013102 h)
+            (by
+              have h : (thetaBelowCell111133013103).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133013103 h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childLH (childLH (childHH
+            thetaBelowCell11113301)))
+            (by
+              exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133013110
+                (by
+                  have h : ((childLL thetaBelowCell111133013110)).rejected = true := by
+                    decide +kernel
+                  exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133013110)
+                    h)
+                (by
+                  have h : ((childLH thetaBelowCell111133013110)).rejected = true := by
+                    decide +kernel
+                  exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133013110)
+                    h)
+                (by
+                  have h : ((childHL thetaBelowCell111133013110)).rejected = true := by
+                    decide +kernel
+                  exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133013110)
+                    h)
+                (by
+                  have h : ((childHH thetaBelowCell111133013110)).rejected = true := by
+                    decide +kernel
+                  exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133013110)
+                    h))
+            (by
+              exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133013111
+                (by
+                  have h : ((childLL thetaBelowCell111133013111)).rejected = true := by
+                    decide +kernel
+                  exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133013111)
+                    h)
+                (by
+                  have h : ((childLH thetaBelowCell111133013111)).rejected = true := by
+                    decide +kernel
+                  exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133013111)
+                    h)
+                (by
+                  have h : ((childHL thetaBelowCell111133013111)).rejected = true := by
+                    decide +kernel
+                  exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133013111)
+                    h)
+                (by
+                  have h : ((childHH thetaBelowCell111133013111)).rejected = true := by
+                    decide +kernel
+                  exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133013111)
+                    h))
+            (by
+              have h : (thetaBelowCell111133013112).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133013112 h)
+            (by
+              have h : (thetaBelowCell111133013113).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133013113 h))
+        (by
+          have h : ((childHL (childLH (childHH thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childHH
+            thetaBelowCell11113301))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaBelowCell11113301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childHH
+            thetaBelowCell11113301))) h))
+    (by
+      have h : ((childHL (childHH thetaBelowCell11113301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11113301)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11113301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11113301)) h)
+theorem cover_subtree_e85bd45b4e1d :
+    adaptiveCoverCheck 7 (childLL (childHL (childLL thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHL (childLL thetaBelowCell11113310)))
+    (by
+      have h : (thetaBelowCell111133100200).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133100200 h)
+    (by
+      have h : (thetaBelowCell111133100201).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133100201 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100202
+        (by
+          have h : ((childLL thetaBelowCell111133100202)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100202) h)
+        (by
+          have h : ((childLH thetaBelowCell111133100202)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100202) h)
+        (by
+          have h : ((childHL thetaBelowCell111133100202)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133100202) h)
+        (by
+          have h : ((childHH thetaBelowCell111133100202)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133100202) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100203
+        (by
+          have h : ((childLL thetaBelowCell111133100203)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100203) h)
+        (by
+          have h : ((childLH thetaBelowCell111133100203)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100203) h)
+        (by
+          have h : ((childHL thetaBelowCell111133100203)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133100203) h)
+        (by
+          have h : ((childHH thetaBelowCell111133100203)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133100203) h))
+
+theorem cover_subtree_0f8d2b9c96c3 :
+    adaptiveCoverCheck 7 (childHL (childHL (childLL thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHL (childLL thetaBelowCell11113310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100220
+        (by
+          have h : ((childLL thetaBelowCell111133100220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100220) h)
+        (by
+          have h : ((childLH thetaBelowCell111133100220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100220) h)
+        (by
+          have h : ((childHL thetaBelowCell111133100220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133100220) h)
+        (by
+          have h : ((childHH thetaBelowCell111133100220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133100220) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100221
+        (by
+          have h : ((childLL thetaBelowCell111133100221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100221) h)
+        (by
+          have h : ((childLH thetaBelowCell111133100221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100221) h)
+        (by
+          have h : ((childHL thetaBelowCell111133100221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133100221) h)
+        (by
+          have h : ((childHH thetaBelowCell111133100221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133100221) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100222
+        (by
+          have h : ((childLL thetaBelowCell111133100222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100222) h)
+        (by
+          have h : ((childLH thetaBelowCell111133100222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100222) h)
+        (by
+          have h : ((childHL thetaBelowCell111133100222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133100222) h)
+        (by
+          have h : ((childHH thetaBelowCell111133100222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133100222) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100223
+        (by
+          have h : ((childLL thetaBelowCell111133100223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100223) h)
+        (by
+          have h : ((childLH thetaBelowCell111133100223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100223) h)
+        (by
+          have h : ((childHL thetaBelowCell111133100223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133100223) h)
+        (by
+          have h : ((childHH thetaBelowCell111133100223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133100223) h))
+
+theorem cover_subtree_8e5ba23f7a1f :
+    adaptiveCoverCheck 7 (childHH (childHL (childLL thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHL (childLL thetaBelowCell11113310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100230
+        (by
+          have h : ((childLL thetaBelowCell111133100230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100230) h)
+        (by
+          have h : ((childLH thetaBelowCell111133100230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100230) h)
+        (by
+          have h : ((childHL thetaBelowCell111133100230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133100230) h)
+        (by
+          have h : ((childHH thetaBelowCell111133100230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133100230) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100231
+        (by
+          have h : ((childLL thetaBelowCell111133100231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100231) h)
+        (by
+          have h : ((childLH thetaBelowCell111133100231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100231) h)
+        (by
+          have h : ((childHL thetaBelowCell111133100231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133100231) h)
+        (by
+          have h : ((childHH thetaBelowCell111133100231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133100231) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100232
+        (by
+          have h : ((childLL thetaBelowCell111133100232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100232) h)
+        (by
+          have h : ((childLH thetaBelowCell111133100232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100232) h)
+        (by
+          have h : ((childHL thetaBelowCell111133100232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133100232) h)
+        (by
+          have h : ((childHH thetaBelowCell111133100232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133100232) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100233
+        (by
+          have h : ((childLL thetaBelowCell111133100233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100233) h)
+        (by
+          have h : ((childLH thetaBelowCell111133100233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100233) h)
+        (by
+          have h : ((childHL thetaBelowCell111133100233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133100233) h)
+        (by
+          have h : ((childHH thetaBelowCell111133100233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133100233) h))
+
+theorem cover_subtree_dfa4585c7b57 :
+    adaptiveCoverCheck 8 (childHL (childLL thetaBelowCell11113310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL thetaBelowCell11113310))
+    cover_subtree_e85bd45b4e1d
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHL (childLL
+        thetaBelowCell11113310)))
+        (by
+          have h : (thetaBelowCell111133100210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133100210 h)
+        (by
+          have h : (thetaBelowCell111133100211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133100211 h)
+        (by
+          have h : (thetaBelowCell111133100212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133100212 h)
+        (by
+          have h : (thetaBelowCell111133100213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133100213 h))
+    cover_subtree_0f8d2b9c96c3
+    cover_subtree_8e5ba23f7a1f
+
+theorem cover_subtree_c078044ffcf3 :
+    adaptiveCoverCheck 6 thetaBelowCell111133100323 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100323
+    (by
+      have h : ((childLL thetaBelowCell111133100323)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100323) h)
+    (by
+      have h : ((childLH thetaBelowCell111133100323)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100323) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133100323)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133100323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133100323)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133100323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133100323)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133100323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133100323)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133100323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133100323)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133100323)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133100323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133100323)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133100323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133100323)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133100323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133100323)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133100323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133100323)) h))
+
+theorem cover_subtree_4936ea81b96a :
+    adaptiveCoverCheck 7 (childHL (childHH (childLL thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHH (childLL thetaBelowCell11113310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100320
+        (by
+          have h : ((childLL thetaBelowCell111133100320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100320) h)
+        (by
+          have h : ((childLH thetaBelowCell111133100320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100320) h)
+        (by
+          have h : ((childHL thetaBelowCell111133100320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133100320) h)
+        (by
+          have h : ((childHH thetaBelowCell111133100320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133100320) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100321
+        (by
+          have h : ((childLL thetaBelowCell111133100321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100321) h)
+        (by
+          have h : ((childLH thetaBelowCell111133100321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100321) h)
+        (by
+          have h : ((childHL thetaBelowCell111133100321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133100321) h)
+        (by
+          have h : ((childHH thetaBelowCell111133100321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133100321) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100322
+        (by
+          have h : ((childLL thetaBelowCell111133100322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100322) h)
+        (by
+          have h : ((childLH thetaBelowCell111133100322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100322) h)
+        (by
+          have h : ((childHL thetaBelowCell111133100322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133100322) h)
+        (by
+          have h : ((childHH thetaBelowCell111133100322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133100322) h))
+    cover_subtree_c078044ffcf3
+
+theorem cover_subtree_4aef0c949cec :
+    adaptiveCoverCheck 6 thetaBelowCell111133100332 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100332
+    (by
+      have h : ((childLL thetaBelowCell111133100332)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100332) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133100332)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133100332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133100332)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133100332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133100332)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133100332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133100332)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133100332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133100332)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133100332)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133100332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133100332)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133100332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133100332)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133100332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133100332)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133100332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133100332)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133100332)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133100332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133100332)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133100332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133100332)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133100332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133100332)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133100332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133100332)) h))
+
+theorem cover_subtree_4d117a1fae58 :
+    adaptiveCoverCheck 6 thetaBelowCell111133100333 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100333
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133100333)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133100333)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133100333)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133100333)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133100333)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133100333)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133100333)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133100333)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133100333)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133100333)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133100333)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133100333)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133100333)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133100333)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133100333)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133100333)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133100333)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133100333)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133100333)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133100333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133100333)) h))
+
+theorem cover_subtree_d8cdaaae2631 :
+    adaptiveCoverCheck 7 (childHH (childHH (childLL thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHH (childLL thetaBelowCell11113310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100330
+        (by
+          have h : ((childLL thetaBelowCell111133100330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100330) h)
+        (by
+          have h : ((childLH thetaBelowCell111133100330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100330) h)
+        (by
+          have h : ((childHL thetaBelowCell111133100330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133100330) h)
+        (by
+          have h : ((childHH thetaBelowCell111133100330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133100330) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133100331
+        (by
+          have h : ((childLL thetaBelowCell111133100331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133100331) h)
+        (by
+          have h : ((childLH thetaBelowCell111133100331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133100331) h)
+        (by
+          have h : ((childHL thetaBelowCell111133100331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133100331) h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133100331)
+            (by
+              have h : ((childLL (childHH thetaBelowCell111133100331))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+                thetaBelowCell111133100331)) h)
+            (by
+              have h : ((childLH (childHH thetaBelowCell111133100331))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+                thetaBelowCell111133100331)) h)
+            (by
+              have h : ((childHL (childHH thetaBelowCell111133100331))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+                thetaBelowCell111133100331)) h)
+            (by
+              have h : ((childHH (childHH thetaBelowCell111133100331))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+                thetaBelowCell111133100331)) h)))
+    cover_subtree_4aef0c949cec
+    cover_subtree_4d117a1fae58
+
+theorem cover_subtree_04c9e6a73353 :
+    adaptiveCoverCheck 8 (childHH (childLL thetaBelowCell11113310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL thetaBelowCell11113310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHH (childLL
+        thetaBelowCell11113310)))
+        (by
+          have h : (thetaBelowCell111133100300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133100300 h)
+        (by
+          have h : (thetaBelowCell111133100301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133100301 h)
+        (by
+          have h : (thetaBelowCell111133100302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133100302 h)
+        (by
+          have h : (thetaBelowCell111133100303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133100303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHH (childLL
+        thetaBelowCell11113310)))
+        (by
+          have h : (thetaBelowCell111133100310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133100310 h)
+        (by
+          have h : (thetaBelowCell111133100311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133100311 h)
+        (by
+          have h : (thetaBelowCell111133100312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133100312 h)
+        (by
+          have h : (thetaBelowCell111133100313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133100313 h))
+    cover_subtree_4936ea81b96a
+    cover_subtree_d8cdaaae2631
+
+theorem e24KC2ThetaBelowLeaf111133100 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113310) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11113310)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL thetaBelowCell11113310))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLL
+            thetaBelowCell11113310))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLL
+            thetaBelowCell11113310))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLL
+            thetaBelowCell11113310))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLL
+            thetaBelowCell11113310))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL thetaBelowCell11113310))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLL
+            thetaBelowCell11113310))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLL
+            thetaBelowCell11113310))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLL
+            thetaBelowCell11113310))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLL
+            thetaBelowCell11113310))) h))
+    cover_subtree_dfa4585c7b57
+    cover_subtree_04c9e6a73353
+theorem cover_subtree_ac282f2600dc :
+    adaptiveCoverCheck 6 thetaBelowCell111133101220 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101220
+    (by
+      have h : ((childLL thetaBelowCell111133101220)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133101220) h)
+    (by
+      have h : ((childLH thetaBelowCell111133101220)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133101220) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133101220)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133101220))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133101220)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133101220))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133101220)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133101220))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133101220)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133101220))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133101220)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133101220)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133101220))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133101220)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133101220))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133101220)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133101220))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133101220)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133101220))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133101220)) h))
+
+theorem cover_subtree_073de9109fcf :
+    adaptiveCoverCheck 6 thetaBelowCell111133101221 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101221
+    (by
+      have h : ((childLL thetaBelowCell111133101221)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133101221) h)
+    (by
+      have h : ((childLH thetaBelowCell111133101221)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133101221) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133101221)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133101221))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133101221)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133101221))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133101221)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133101221))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133101221)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133101221))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133101221)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133101221)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133101221))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133101221)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133101221))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133101221)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133101221))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133101221)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133101221))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133101221)) h))
+
+theorem cover_subtree_09bb9f7fb348 :
+    adaptiveCoverCheck 6 thetaBelowCell111133101222 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101222
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133101222)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133101222)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133101222)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133101222)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133101222)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133101222)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133101222)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133101222)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133101222)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133101222)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133101222)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133101222)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133101222)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133101222)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133101222)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133101222)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133101222)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133101222)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133101222)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133101222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133101222)) h))
+
+theorem cover_subtree_616efb7da0d5 :
+    adaptiveCoverCheck 6 thetaBelowCell111133101223 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101223
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133101223)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133101223)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133101223)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133101223)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133101223)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133101223)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133101223)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133101223)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133101223)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133101223)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133101223)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133101223)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133101223)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133101223)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133101223)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133101223)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133101223)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133101223)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133101223)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133101223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133101223)) h))
+
+theorem cover_subtree_76fb46ac376b :
+    adaptiveCoverCheck 7 (childHL (childHL (childLH thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHL (childLH thetaBelowCell11113310)))
+    cover_subtree_ac282f2600dc
+    cover_subtree_073de9109fcf
+    cover_subtree_09bb9f7fb348
+    cover_subtree_616efb7da0d5
+
+theorem cover_subtree_2ee17850d55a :
+    adaptiveCoverCheck 6 thetaBelowCell111133101230 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101230
+    (by
+      have h : ((childLL thetaBelowCell111133101230)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133101230) h)
+    (by
+      have h : ((childLH thetaBelowCell111133101230)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133101230) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133101230)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133101230))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133101230)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133101230))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133101230)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133101230))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133101230)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133101230))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133101230)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133101230)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133101230))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133101230)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133101230))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133101230)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133101230))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133101230)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133101230))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133101230)) h))
+
+theorem cover_subtree_7a3325b5d632 :
+    adaptiveCoverCheck 6 thetaBelowCell111133101232 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101232
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133101232)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133101232)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133101232)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133101232)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133101232)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133101232)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133101232)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133101232)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133101232)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133101232)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133101232)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133101232)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133101232)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133101232)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133101232)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133101232)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133101232)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133101232)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133101232)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133101232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133101232)) h))
+
+theorem cover_subtree_03b1ead1eef9 :
+    adaptiveCoverCheck 6 thetaBelowCell111133101233 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101233
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133101233)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133101233)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133101233)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133101233)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133101233)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133101233)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133101233)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133101233)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133101233)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133101233)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133101233)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133101233)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133101233)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133101233)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133101233)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133101233)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133101233)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133101233)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133101233)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133101233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133101233)) h))
+
+theorem cover_subtree_3534513f2dea :
+    adaptiveCoverCheck 7 (childHH (childHL (childLH thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHL (childLH thetaBelowCell11113310)))
+    cover_subtree_2ee17850d55a
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101231
+        (by
+          have h : ((childLL thetaBelowCell111133101231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133101231) h)
+        (by
+          have h : ((childLH thetaBelowCell111133101231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133101231) h)
+        (by
+          have h : ((childHL thetaBelowCell111133101231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133101231) h)
+        (by
+          have h : ((childHH thetaBelowCell111133101231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133101231) h))
+    cover_subtree_7a3325b5d632
+    cover_subtree_03b1ead1eef9
+
+theorem cover_subtree_fee8f6a2356e :
+    adaptiveCoverCheck 8 (childHL (childLH thetaBelowCell11113310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH thetaBelowCell11113310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHL (childLH
+        thetaBelowCell11113310)))
+        (by
+          have h : (thetaBelowCell111133101200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101200 h)
+        (by
+          have h : (thetaBelowCell111133101201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101201 h)
+        (by
+          have h : (thetaBelowCell111133101202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101202 h)
+        (by
+          have h : (thetaBelowCell111133101203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHL (childLH
+        thetaBelowCell11113310)))
+        (by
+          have h : (thetaBelowCell111133101210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101210 h)
+        (by
+          have h : (thetaBelowCell111133101211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101211 h)
+        (by
+          have h : (thetaBelowCell111133101212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101212 h)
+        (by
+          have h : (thetaBelowCell111133101213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101213 h))
+    cover_subtree_76fb46ac376b
+    cover_subtree_3534513f2dea
+
+theorem cover_subtree_5721355f2386 :
+    adaptiveCoverCheck 6 thetaBelowCell111133101322 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101322
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133101322)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133101322)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133101322)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133101322)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133101322)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133101322)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133101322)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133101322)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133101322)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133101322)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133101322)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133101322)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133101322)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133101322)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133101322)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133101322)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133101322)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133101322)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133101322)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133101322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133101322)) h))
+
+theorem cover_subtree_c66d1685818d :
+    adaptiveCoverCheck 6 thetaBelowCell111133101323 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101323
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133101323)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133101323)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133101323)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133101323)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133101323)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133101323)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133101323)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133101323)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133101323)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133101323)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133101323)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133101323)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133101323)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133101323)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133101323)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133101323)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133101323)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133101323)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133101323)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133101323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133101323)) h))
+
+theorem cover_subtree_8703d282848e :
+    adaptiveCoverCheck 7 (childHL (childHH (childLH thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHH (childLH thetaBelowCell11113310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101320
+        (by
+          have h : ((childLL thetaBelowCell111133101320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133101320) h)
+        (by
+          have h : ((childLH thetaBelowCell111133101320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133101320) h)
+        (by
+          have h : ((childHL thetaBelowCell111133101320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133101320) h)
+        (by
+          have h : ((childHH thetaBelowCell111133101320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133101320) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101321
+        (by
+          have h : ((childLL thetaBelowCell111133101321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133101321) h)
+        (by
+          have h : ((childLH thetaBelowCell111133101321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133101321) h)
+        (by
+          have h : ((childHL thetaBelowCell111133101321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133101321) h)
+        (by
+          have h : ((childHH thetaBelowCell111133101321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133101321) h))
+    cover_subtree_5721355f2386
+    cover_subtree_c66d1685818d
+
+theorem cover_subtree_e6b72831ee5a :
+    adaptiveCoverCheck 6 thetaBelowCell111133101332 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101332
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133101332)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133101332)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133101332)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133101332)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133101332)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133101332)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133101332)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133101332)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133101332)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133101332)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133101332)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133101332)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133101332)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133101332)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133101332)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133101332)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133101332)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133101332)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133101332)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133101332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133101332)) h))
+
+theorem cover_subtree_58f9acea628e :
+    adaptiveCoverCheck 6 thetaBelowCell111133101333 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101333
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133101333)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133101333)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133101333)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133101333)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133101333)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133101333)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133101333)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133101333)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133101333)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133101333)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133101333)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133101333)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133101333)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133101333)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133101333)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133101333)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133101333)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133101333)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133101333)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133101333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133101333)) h))
+
+theorem cover_subtree_72f2137bba25 :
+    adaptiveCoverCheck 7 (childHH (childHH (childLH thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHH (childLH thetaBelowCell11113310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101330
+        (by
+          have h : ((childLL thetaBelowCell111133101330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133101330) h)
+        (by
+          have h : ((childLH thetaBelowCell111133101330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133101330) h)
+        (by
+          have h : ((childHL thetaBelowCell111133101330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133101330) h)
+        (by
+          have h : ((childHH thetaBelowCell111133101330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133101330) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133101331
+        (by
+          have h : ((childLL thetaBelowCell111133101331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133101331) h)
+        (by
+          have h : ((childLH thetaBelowCell111133101331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133101331) h)
+        (by
+          have h : ((childHL thetaBelowCell111133101331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133101331) h)
+        (by
+          have h : ((childHH thetaBelowCell111133101331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133101331) h))
+    cover_subtree_e6b72831ee5a
+    cover_subtree_58f9acea628e
+
+theorem cover_subtree_d0cab3230f27 :
+    adaptiveCoverCheck 8 (childHH (childLH thetaBelowCell11113310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH thetaBelowCell11113310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHH (childLH
+        thetaBelowCell11113310)))
+        (by
+          have h : (thetaBelowCell111133101300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101300 h)
+        (by
+          have h : (thetaBelowCell111133101301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101301 h)
+        (by
+          have h : (thetaBelowCell111133101302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101302 h)
+        (by
+          have h : (thetaBelowCell111133101303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHH (childLH
+        thetaBelowCell11113310)))
+        (by
+          have h : (thetaBelowCell111133101310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101310 h)
+        (by
+          have h : (thetaBelowCell111133101311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101311 h)
+        (by
+          have h : (thetaBelowCell111133101312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101312 h)
+        (by
+          have h : (thetaBelowCell111133101313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133101313 h))
+    cover_subtree_8703d282848e
+    cover_subtree_72f2137bba25
+
+theorem e24KC2ThetaBelowLeaf111133101 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113310) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11113310)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH thetaBelowCell11113310))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLH
+            thetaBelowCell11113310))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLH
+            thetaBelowCell11113310))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLH
+            thetaBelowCell11113310))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLH
+            thetaBelowCell11113310))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH thetaBelowCell11113310))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLH
+            thetaBelowCell11113310))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLH
+            thetaBelowCell11113310))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLH
+            thetaBelowCell11113310))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLH
+            thetaBelowCell11113310))) h))
+    cover_subtree_fee8f6a2356e
+    cover_subtree_d0cab3230f27
+theorem cover_subtree_b25dcd0a7904 :
+    adaptiveCoverCheck 7 (childLL (childLL (childHL thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLL (childHL thetaBelowCell11113310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133102000
+        (by
+          have h : ((childLL thetaBelowCell111133102000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133102000) h)
+        (by
+          have h : ((childLH thetaBelowCell111133102000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133102000) h)
+        (by
+          have h : ((childHL thetaBelowCell111133102000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133102000) h)
+        (by
+          have h : ((childHH thetaBelowCell111133102000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133102000) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133102001
+        (by
+          have h : ((childLL thetaBelowCell111133102001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133102001) h)
+        (by
+          have h : ((childLH thetaBelowCell111133102001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133102001) h)
+        (by
+          have h : ((childHL thetaBelowCell111133102001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133102001) h)
+        (by
+          have h : ((childHH thetaBelowCell111133102001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133102001) h))
+    (by
+      have h : (thetaBelowCell111133102002).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133102002 h)
+    (by
+      have h : (thetaBelowCell111133102003).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133102003 h)
+
+theorem cover_subtree_b136e2f967cb :
+    adaptiveCoverCheck 7 (childLH (childLL (childHL thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLH (childLL (childHL thetaBelowCell11113310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133102010
+        (by
+          have h : ((childLL thetaBelowCell111133102010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133102010) h)
+        (by
+          have h : ((childLH thetaBelowCell111133102010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133102010) h)
+        (by
+          have h : ((childHL thetaBelowCell111133102010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133102010) h)
+        (by
+          have h : ((childHH thetaBelowCell111133102010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133102010) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133102011
+        (by
+          have h : ((childLL thetaBelowCell111133102011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133102011) h)
+        (by
+          have h : ((childLH thetaBelowCell111133102011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133102011) h)
+        (by
+          have h : ((childHL thetaBelowCell111133102011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133102011) h)
+        (by
+          have h : ((childHH thetaBelowCell111133102011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133102011) h))
+    (by
+      have h : (thetaBelowCell111133102012).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133102012 h)
+    (by
+      have h : (thetaBelowCell111133102013).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133102013 h)
+
+theorem cover_subtree_2f872975a74f :
+    adaptiveCoverCheck 8 (childLL (childHL thetaBelowCell11113310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL thetaBelowCell11113310))
+    cover_subtree_b25dcd0a7904
+    cover_subtree_b136e2f967cb
+    (by
+      have h : ((childHL (childLL (childHL thetaBelowCell11113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childHL
+        thetaBelowCell11113310))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaBelowCell11113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childHL
+        thetaBelowCell11113310))) h)
+
+theorem cover_subtree_d85bc7893c75 :
+    adaptiveCoverCheck 7 (childLL (childLH (childHL thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLH (childHL thetaBelowCell11113310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133102100
+        (by
+          have h : ((childLL thetaBelowCell111133102100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133102100) h)
+        (by
+          have h : ((childLH thetaBelowCell111133102100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133102100) h)
+        (by
+          have h : ((childHL thetaBelowCell111133102100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133102100) h)
+        (by
+          have h : ((childHH thetaBelowCell111133102100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133102100) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133102101
+        (by
+          have h : ((childLL thetaBelowCell111133102101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133102101) h)
+        (by
+          have h : ((childLH thetaBelowCell111133102101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133102101) h)
+        (by
+          have h : ((childHL thetaBelowCell111133102101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133102101) h)
+        (by
+          have h : ((childHH thetaBelowCell111133102101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133102101) h))
+    (by
+      have h : (thetaBelowCell111133102102).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133102102 h)
+    (by
+      have h : (thetaBelowCell111133102103).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133102103 h)
+
+theorem cover_subtree_16d826fbb67f :
+    adaptiveCoverCheck 7 (childLH (childLH (childHL thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLH (childLH (childHL thetaBelowCell11113310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133102110
+        (by
+          have h : ((childLL thetaBelowCell111133102110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133102110) h)
+        (by
+          have h : ((childLH thetaBelowCell111133102110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133102110) h)
+        (by
+          have h : ((childHL thetaBelowCell111133102110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133102110) h)
+        (by
+          have h : ((childHH thetaBelowCell111133102110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133102110) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133102111
+        (by
+          have h : ((childLL thetaBelowCell111133102111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133102111) h)
+        (by
+          have h : ((childLH thetaBelowCell111133102111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133102111) h)
+        (by
+          have h : ((childHL thetaBelowCell111133102111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133102111) h)
+        (by
+          have h : ((childHH thetaBelowCell111133102111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133102111) h))
+    (by
+      have h : (thetaBelowCell111133102112).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133102112 h)
+    (by
+      have h : (thetaBelowCell111133102113).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133102113 h)
+
+theorem cover_subtree_94cd8267c854 :
+    adaptiveCoverCheck 8 (childLH (childHL thetaBelowCell11113310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL thetaBelowCell11113310))
+    cover_subtree_d85bc7893c75
+    cover_subtree_16d826fbb67f
+    (by
+      have h : ((childHL (childLH (childHL thetaBelowCell11113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childHL
+        thetaBelowCell11113310))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaBelowCell11113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childHL
+        thetaBelowCell11113310))) h)
+
+theorem e24KC2ThetaBelowLeaf111133102 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113310) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113310)
+    cover_subtree_2f872975a74f
+    cover_subtree_94cd8267c854
+    (by
+      have h : ((childHL (childHL thetaBelowCell11113310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11113310)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11113310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11113310)) h)
+theorem cover_subtree_8bd8067ea850 :
+    adaptiveCoverCheck 6 thetaBelowCell111133103001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133103001
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133103001)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133103001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133103001)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133103001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133103001)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133103001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133103001)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133103001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133103001)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133103001)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133103001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133103001)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133103001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133103001)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133103001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133103001)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133103001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133103001)) h))
+    (by
+      have h : ((childHL thetaBelowCell111133103001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133103001) h)
+    (by
+      have h : ((childHH thetaBelowCell111133103001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133103001) h)
+
+theorem cover_subtree_e3f60d1c040d :
+    adaptiveCoverCheck 7 (childLL (childLL (childHH thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLL (childHH thetaBelowCell11113310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133103000
+        (by
+          have h : ((childLL thetaBelowCell111133103000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133103000) h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133103000)
+            (by
+              have h : ((childLL (childLH thetaBelowCell111133103000))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+                thetaBelowCell111133103000)) h)
+            (by
+              have h : ((childLH (childLH thetaBelowCell111133103000))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+                thetaBelowCell111133103000)) h)
+            (by
+              have h : ((childHL (childLH thetaBelowCell111133103000))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+                thetaBelowCell111133103000)) h)
+            (by
+              have h : ((childHH (childLH thetaBelowCell111133103000))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+                thetaBelowCell111133103000)) h))
+        (by
+          have h : ((childHL thetaBelowCell111133103000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133103000) h)
+        (by
+          have h : ((childHH thetaBelowCell111133103000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133103000) h))
+    cover_subtree_8bd8067ea850
+    (by
+      have h : (thetaBelowCell111133103002).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133103002 h)
+    (by
+      have h : (thetaBelowCell111133103003).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133103003 h)
+
+theorem cover_subtree_542375e93ae0 :
+    adaptiveCoverCheck 6 thetaBelowCell111133103010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133103010
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133103010)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133103010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133103010)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133103010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133103010)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133103010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133103010)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133103010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133103010)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133103010)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133103010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133103010)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133103010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133103010)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133103010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133103010)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133103010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133103010)) h))
+    (by
+      have h : ((childHL thetaBelowCell111133103010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133103010) h)
+    (by
+      have h : ((childHH thetaBelowCell111133103010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133103010) h)
+
+theorem cover_subtree_31a24cecd31f :
+    adaptiveCoverCheck 6 thetaBelowCell111133103011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133103011
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133103011)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133103011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133103011)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133103011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133103011)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133103011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133103011)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133103011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133103011)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133103011)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133103011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133103011)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133103011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133103011)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133103011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133103011)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133103011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133103011)) h))
+    (by
+      have h : ((childHL thetaBelowCell111133103011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133103011) h)
+    (by
+      have h : ((childHH thetaBelowCell111133103011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133103011) h)
+
+theorem cover_subtree_785b55e573a2 :
+    adaptiveCoverCheck 7 (childLH (childLL (childHH thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLH (childLL (childHH thetaBelowCell11113310)))
+    cover_subtree_542375e93ae0
+    cover_subtree_31a24cecd31f
+    (by
+      have h : (thetaBelowCell111133103012).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133103012 h)
+    (by
+      have h : (thetaBelowCell111133103013).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133103013 h)
+
+theorem cover_subtree_5ea908f6482f :
+    adaptiveCoverCheck 8 (childLL (childHH thetaBelowCell11113310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH thetaBelowCell11113310))
+    cover_subtree_e3f60d1c040d
+    cover_subtree_785b55e573a2
+    (by
+      have h : ((childHL (childLL (childHH thetaBelowCell11113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childHH
+        thetaBelowCell11113310))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaBelowCell11113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childHH
+        thetaBelowCell11113310))) h)
+
+theorem cover_subtree_90ee2a55599b :
+    adaptiveCoverCheck 6 thetaBelowCell111133103100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133103100
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133103100)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133103100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133103100)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133103100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133103100)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133103100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133103100)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133103100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133103100)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133103100)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133103100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133103100)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133103100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133103100)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133103100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133103100)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133103100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133103100)) h))
+    (by
+      have h : ((childHL thetaBelowCell111133103100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133103100) h)
+    (by
+      have h : ((childHH thetaBelowCell111133103100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133103100) h)
+
+theorem cover_subtree_846f12d54937 :
+    adaptiveCoverCheck 6 thetaBelowCell111133103101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133103101
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133103101)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133103101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133103101)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133103101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133103101)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133103101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133103101)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133103101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133103101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133103101)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133103101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133103101)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133103101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133103101)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133103101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133103101)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133103101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133103101)) h))
+    (by
+      have h : ((childHL thetaBelowCell111133103101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133103101) h)
+    (by
+      have h : ((childHH thetaBelowCell111133103101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133103101) h)
+
+theorem cover_subtree_c6f98b640f2b :
+    adaptiveCoverCheck 7 (childLL (childLH (childHH thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLH (childHH thetaBelowCell11113310)))
+    cover_subtree_90ee2a55599b
+    cover_subtree_846f12d54937
+    (by
+      have h : (thetaBelowCell111133103102).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133103102 h)
+    (by
+      have h : (thetaBelowCell111133103103).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133103103 h)
+
+theorem cover_subtree_83af3b43e370 :
+    adaptiveCoverCheck 6 thetaBelowCell111133103110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133103110
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133103110)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133103110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133103110)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133103110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133103110)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133103110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133103110)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133103110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133103110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133103110)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133103110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133103110)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133103110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133103110)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133103110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133103110)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133103110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133103110)) h))
+    (by
+      have h : ((childHL thetaBelowCell111133103110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133103110) h)
+    (by
+      have h : ((childHH thetaBelowCell111133103110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133103110) h)
+
+theorem cover_subtree_6c43321af990 :
+    adaptiveCoverCheck 6 thetaBelowCell111133103111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133103111
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133103111)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133103111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133103111)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133103111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133103111)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133103111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133103111)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133103111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133103111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133103111)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133103111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133103111)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133103111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133103111)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133103111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133103111)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133103111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133103111)) h))
+    (by
+      have h : ((childHL thetaBelowCell111133103111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133103111) h)
+    (by
+      have h : ((childHH thetaBelowCell111133103111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133103111) h)
+
+theorem cover_subtree_ad23d1ccf385 :
+    adaptiveCoverCheck 7 (childLH (childLH (childHH thetaBelowCell11113310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLH (childLH (childHH thetaBelowCell11113310)))
+    cover_subtree_83af3b43e370
+    cover_subtree_6c43321af990
+    (by
+      have h : (thetaBelowCell111133103112).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133103112 h)
+    (by
+      have h : (thetaBelowCell111133103113).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133103113 h)
+
+theorem cover_subtree_5edd590faea9 :
+    adaptiveCoverCheck 8 (childLH (childHH thetaBelowCell11113310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH thetaBelowCell11113310))
+    cover_subtree_c6f98b640f2b
+    cover_subtree_ad23d1ccf385
+    (by
+      have h : ((childHL (childLH (childHH thetaBelowCell11113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childHH
+        thetaBelowCell11113310))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaBelowCell11113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childHH
+        thetaBelowCell11113310))) h)
+
+theorem e24KC2ThetaBelowLeaf111133103 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113310) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113310)
+    cover_subtree_5ea908f6482f
+    cover_subtree_5edd590faea9
+    (by
+      have h : ((childHL (childHH thetaBelowCell11113310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11113310)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH thetaBelowCell11113310))
+        (by
+          have h : ((childLL (childHH (childHH thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHH
+            thetaBelowCell11113310))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHH
+            thetaBelowCell11113310))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHH
+            thetaBelowCell11113310))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaBelowCell11113310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHH
+            thetaBelowCell11113310))) h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle006.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle006.lean
new file mode 100644
index 0000000000..18d5f1e746
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle006.lean
@@ -0,0 +1,10386 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch007`.
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch023`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCells502d119e99
+
+/-- Subcell `0011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0011 : AngleCell :=
+  childLH (childLH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0012 : AngleCell :=
+  childHL (childLH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0013 : AngleCell :=
+  childHH (childLH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0020 : AngleCell :=
+  childLL (childHL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0021 : AngleCell :=
+  childLH (childHL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0022 : AngleCell :=
+  childHL (childHL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0023 : AngleCell :=
+  childHH (childHL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0030 : AngleCell :=
+  childLL (childHH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0031 : AngleCell :=
+  childLH (childHH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0032 : AngleCell :=
+  childHL (childHH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0033 : AngleCell :=
+  childHH (childHH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0100 : AngleCell :=
+  childLL (childLL (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `01002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0100)))
+
+end GerverSofa.PartE.CertificateCells502d119e99
+
+namespace GerverSofa.PartE.CertificateCells66bed457b5
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `000023002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00002300)))
+
+/-- Subcell `000023003000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00002300)))
+
+/-- Subcell `000023010220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023010221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023010222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023010223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023010230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023010231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023010232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023010233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023010320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023010321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023010322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023010323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023010330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023010331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023010332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023010333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023010333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaAboveCell00002301)))
+
+/-- Subcell `000023011220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023011221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023011222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023011223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023011230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023011231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023011232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023011233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023011320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023011321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023011322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023011323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023011330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023011331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023011332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023011333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023011333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaAboveCell00002301)))
+
+/-- Subcell `000023012000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00002301)))
+
+/-- Subcell `000023013000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00002301)))
+
+/-- Subcell `000023100220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023100221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023100222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023100223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023100230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023100231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023100232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023100233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023100320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023100321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023100322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023100323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023100330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023100331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023100332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023100333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023100333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaAboveCell00002310)))
+
+/-- Subcell `000023101220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaAboveCell00002310)))
+
+/-- Subcell `000023101221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaAboveCell00002310)))
+
+/-- Subcell `000023101222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaAboveCell00002310)))
+
+/-- Subcell `000023101223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaAboveCell00002310)))
+
+/-- Subcell `000023101230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaAboveCell00002310)))
+
+/-- Subcell `000023101231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaAboveCell00002310)))
+
+/-- Subcell `000023101232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaAboveCell00002310)))
+
+/-- Subcell `000023101233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaAboveCell00002310)))
+
+/-- Subcell `000023101320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaAboveCell00002310)))
+
+/-- Subcell `000023101321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaAboveCell00002310)))
+
+/-- Subcell `000023101322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaAboveCell00002310)))
+
+/-- Subcell `000023101323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaAboveCell00002310)))
+
+/-- Subcell `000023101330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaAboveCell00002310)))
+
+/-- Subcell `000023101331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaAboveCell00002310)))
+
+/-- Subcell `000023101332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaAboveCell00002310)))
+
+/-- Subcell `000023101333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023101333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaAboveCell00002310)))
+
+end GerverSofa.PartE.CertificateCells66bed457b5
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC5TerminalBatchT153600006`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells502d119e99
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells502d119e99
+
+open CertificateCells502d119e99
+theorem e24KC2ThetaAboveLeaf001123133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112313) = true := by
+  have h : ((childHH thetaAboveCell00112313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00112313) h
+theorem e24KC2ThetaAboveLeaf00112320 :
+    adaptiveCoverCheck 11 thetaAboveCell00112320 = true := by
+  have h : (thetaAboveCell00112320).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112320 h
+theorem e24KC2ThetaAboveLeaf00112321 :
+    adaptiveCoverCheck 11 thetaAboveCell00112321 = true := by
+  have h : (thetaAboveCell00112321).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112321 h
+theorem e24KC2ThetaAboveLeaf00112322 :
+    adaptiveCoverCheck 11 thetaAboveCell00112322 = true := by
+  have h : (thetaAboveCell00112322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112322 h
+theorem e24KC2ThetaAboveLeaf00112323 :
+    adaptiveCoverCheck 11 thetaAboveCell00112323 = true := by
+  have h : (thetaAboveCell00112323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112323 h
+theorem e24KC2ThetaAboveLeaf00112330 :
+    adaptiveCoverCheck 11 thetaAboveCell00112330 = true := by
+  have h : (thetaAboveCell00112330).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112330 h
+theorem e24KC2ThetaAboveLeaf00112331 :
+    adaptiveCoverCheck 11 thetaAboveCell00112331 = true := by
+  have h : (thetaAboveCell00112331).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112331 h
+theorem e24KC2ThetaAboveLeaf00112332 :
+    adaptiveCoverCheck 11 thetaAboveCell00112332 = true := by
+  have h : (thetaAboveCell00112332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112332 h
+theorem e24KC2ThetaAboveLeaf00112333 :
+    adaptiveCoverCheck 11 thetaAboveCell00112333 = true := by
+  have h : (thetaAboveCell00112333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00112333 h
+theorem e24KC2ThetaAboveLeaf0011300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell0011))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell0011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell0011))) h
+theorem e24KC2ThetaAboveLeaf0011301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell0011))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell0011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell0011))) h
+theorem e24KC2ThetaAboveLeaf00113020 :
+    adaptiveCoverCheck 11 thetaAboveCell00113020 = true := by
+  have h : (thetaAboveCell00113020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113020 h
+theorem e24KC2ThetaAboveLeaf00113021 :
+    adaptiveCoverCheck 11 thetaAboveCell00113021 = true := by
+  have h : (thetaAboveCell00113021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113021 h
+theorem e24KC2ThetaAboveLeaf00113022 :
+    adaptiveCoverCheck 11 thetaAboveCell00113022 = true := by
+  have h : (thetaAboveCell00113022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113022 h
+theorem e24KC2ThetaAboveLeaf00113023 :
+    adaptiveCoverCheck 11 thetaAboveCell00113023 = true := by
+  have h : (thetaAboveCell00113023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113023 h
+theorem e24KC2ThetaAboveLeaf00113030 :
+    adaptiveCoverCheck 11 thetaAboveCell00113030 = true := by
+  have h : (thetaAboveCell00113030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113030 h
+theorem e24KC2ThetaAboveLeaf00113031 :
+    adaptiveCoverCheck 11 thetaAboveCell00113031 = true := by
+  have h : (thetaAboveCell00113031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113031 h
+theorem e24KC2ThetaAboveLeaf00113032 :
+    adaptiveCoverCheck 11 thetaAboveCell00113032 = true := by
+  have h : (thetaAboveCell00113032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113032 h
+theorem e24KC2ThetaAboveLeaf00113033 :
+    adaptiveCoverCheck 11 thetaAboveCell00113033 = true := by
+  have h : (thetaAboveCell00113033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113033 h
+theorem e24KC2ThetaAboveLeaf0011310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell0011))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell0011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell0011))) h
+theorem e24KC2ThetaAboveLeaf0011311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell0011))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell0011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell0011))) h
+theorem e24KC2ThetaAboveLeaf00113120 :
+    adaptiveCoverCheck 11 thetaAboveCell00113120 = true := by
+  have h : (thetaAboveCell00113120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113120 h
+theorem e24KC2ThetaAboveLeaf00113121 :
+    adaptiveCoverCheck 11 thetaAboveCell00113121 = true := by
+  have h : (thetaAboveCell00113121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113121 h
+theorem e24KC2ThetaAboveLeaf00113122 :
+    adaptiveCoverCheck 11 thetaAboveCell00113122 = true := by
+  have h : (thetaAboveCell00113122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113122 h
+theorem e24KC2ThetaAboveLeaf00113123 :
+    adaptiveCoverCheck 11 thetaAboveCell00113123 = true := by
+  have h : (thetaAboveCell00113123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113123 h
+theorem e24KC2ThetaAboveLeaf00113130 :
+    adaptiveCoverCheck 11 thetaAboveCell00113130 = true := by
+  have h : (thetaAboveCell00113130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113130 h
+theorem e24KC2ThetaAboveLeaf00113131 :
+    adaptiveCoverCheck 11 thetaAboveCell00113131 = true := by
+  have h : (thetaAboveCell00113131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113131 h
+theorem e24KC2ThetaAboveLeaf00113132 :
+    adaptiveCoverCheck 11 thetaAboveCell00113132 = true := by
+  have h : (thetaAboveCell00113132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113132 h
+theorem e24KC2ThetaAboveLeaf00113133 :
+    adaptiveCoverCheck 11 thetaAboveCell00113133 = true := by
+  have h : (thetaAboveCell00113133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113133 h
+theorem e24KC2ThetaAboveLeaf001132000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113200) = true := by
+  have h : ((childLL thetaAboveCell00113200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00113200) h
+theorem e24KC2ThetaAboveLeaf001132001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113200) = true := by
+  have h : ((childLH thetaAboveCell00113200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00113200) h
+theorem e24KC2ThetaAboveLeaf0011320020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00113200)) = true := by
+  have h : ((childLL (childHL thetaAboveCell00113200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell00113200)) h
+theorem e24KC2ThetaAboveLeaf0011320021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00113200)) = true := by
+  have h : ((childLH (childHL thetaAboveCell00113200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell00113200)) h
+theorem e24KC2ThetaAboveLeaf0011320030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00113200)) = true := by
+  have h : ((childLL (childHH thetaAboveCell00113200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell00113200)) h
+theorem e24KC2ThetaAboveLeaf0011320031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00113200)) = true := by
+  have h : ((childLH (childHH thetaAboveCell00113200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell00113200)) h
+theorem e24KC2ThetaAboveLeaf001132010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113201) = true := by
+  have h : ((childLL thetaAboveCell00113201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00113201) h
+theorem e24KC2ThetaAboveLeaf001132011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113201) = true := by
+  have h : ((childLH thetaAboveCell00113201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00113201) h
+theorem e24KC2ThetaAboveLeaf0011320120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00113201)) = true := by
+  have h : ((childLL (childHL thetaAboveCell00113201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell00113201)) h
+theorem e24KC2ThetaAboveLeaf0011320121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00113201)) = true := by
+  have h : ((childLH (childHL thetaAboveCell00113201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell00113201)) h
+theorem e24KC2ThetaAboveLeaf0011320130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00113201)) = true := by
+  have h : ((childLL (childHH thetaAboveCell00113201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell00113201)) h
+theorem e24KC2ThetaAboveLeaf0011320131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00113201)) = true := by
+  have h : ((childLH (childHH thetaAboveCell00113201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell00113201)) h
+theorem e24KC2ThetaAboveLeaf0011320202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00113202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00113202)) h
+theorem e24KC2ThetaAboveLeaf0011320203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00113202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00113202)) h
+theorem e24KC2ThetaAboveLeaf0011320212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00113202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00113202)) h
+theorem e24KC2ThetaAboveLeaf0011320213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00113202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00113202)) h
+theorem e24KC2ThetaAboveLeaf001132022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113202) = true := by
+  have h : ((childHL thetaAboveCell00113202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00113202) h
+theorem e24KC2ThetaAboveLeaf001132023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113202) = true := by
+  have h : ((childHH thetaAboveCell00113202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00113202) h
+theorem e24KC2ThetaAboveLeaf0011320302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00113203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00113203)) h
+theorem e24KC2ThetaAboveLeaf0011320303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00113203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00113203)) h
+theorem e24KC2ThetaAboveLeaf0011320312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00113203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00113203)) h
+theorem e24KC2ThetaAboveLeaf0011320313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00113203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00113203)) h
+theorem e24KC2ThetaAboveLeaf001132032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113203) = true := by
+  have h : ((childHL thetaAboveCell00113203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00113203) h
+theorem e24KC2ThetaAboveLeaf001132033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113203) = true := by
+  have h : ((childHH thetaAboveCell00113203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00113203) h
+theorem e24KC2ThetaAboveLeaf001132100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113210) = true := by
+  have h : ((childLL thetaAboveCell00113210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00113210) h
+theorem e24KC2ThetaAboveLeaf001132101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113210) = true := by
+  have h : ((childLH thetaAboveCell00113210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00113210) h
+theorem e24KC2ThetaAboveLeaf0011321020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00113210)) = true := by
+  have h : ((childLL (childHL thetaAboveCell00113210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell00113210)) h
+theorem e24KC2ThetaAboveLeaf0011321021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00113210)) = true := by
+  have h : ((childLH (childHL thetaAboveCell00113210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell00113210)) h
+theorem e24KC2ThetaAboveLeaf0011321030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00113210)) = true := by
+  have h : ((childLL (childHH thetaAboveCell00113210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell00113210)) h
+theorem e24KC2ThetaAboveLeaf0011321031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00113210)) = true := by
+  have h : ((childLH (childHH thetaAboveCell00113210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell00113210)) h
+theorem e24KC2ThetaAboveLeaf001132110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113211) = true := by
+  have h : ((childLL thetaAboveCell00113211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00113211) h
+theorem e24KC2ThetaAboveLeaf001132111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113211) = true := by
+  have h : ((childLH thetaAboveCell00113211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00113211) h
+theorem e24KC2ThetaAboveLeaf0011321120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00113211)) = true := by
+  have h : ((childLL (childHL thetaAboveCell00113211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell00113211)) h
+theorem e24KC2ThetaAboveLeaf0011321121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00113211)) = true := by
+  have h : ((childLH (childHL thetaAboveCell00113211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell00113211)) h
+theorem e24KC2ThetaAboveLeaf0011321130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00113211)) = true := by
+  have h : ((childLL (childHH thetaAboveCell00113211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell00113211)) h
+theorem e24KC2ThetaAboveLeaf0011321131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00113211)) = true := by
+  have h : ((childLH (childHH thetaAboveCell00113211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell00113211)) h
+theorem e24KC2ThetaAboveLeaf0011321202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00113212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00113212)) h
+theorem e24KC2ThetaAboveLeaf0011321203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00113212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00113212)) h
+theorem e24KC2ThetaAboveLeaf0011321212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00113212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00113212)) h
+theorem e24KC2ThetaAboveLeaf0011321213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00113212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00113212)) h
+theorem e24KC2ThetaAboveLeaf001132122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113212) = true := by
+  have h : ((childHL thetaAboveCell00113212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00113212) h
+theorem e24KC2ThetaAboveLeaf001132123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113212) = true := by
+  have h : ((childHH thetaAboveCell00113212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00113212) h
+theorem e24KC2ThetaAboveLeaf0011321302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00113213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00113213)) h
+theorem e24KC2ThetaAboveLeaf0011321303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00113213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00113213)) h
+theorem e24KC2ThetaAboveLeaf0011321312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00113213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00113213)) h
+theorem e24KC2ThetaAboveLeaf0011321313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00113213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00113213)) h
+theorem e24KC2ThetaAboveLeaf001132132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113213) = true := by
+  have h : ((childHL thetaAboveCell00113213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00113213) h
+theorem e24KC2ThetaAboveLeaf001132133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113213) = true := by
+  have h : ((childHH thetaAboveCell00113213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00113213) h
+theorem e24KC2ThetaAboveLeaf00113220 :
+    adaptiveCoverCheck 11 thetaAboveCell00113220 = true := by
+  have h : (thetaAboveCell00113220).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113220 h
+theorem e24KC2ThetaAboveLeaf00113221 :
+    adaptiveCoverCheck 11 thetaAboveCell00113221 = true := by
+  have h : (thetaAboveCell00113221).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113221 h
+theorem e24KC2ThetaAboveLeaf00113222 :
+    adaptiveCoverCheck 11 thetaAboveCell00113222 = true := by
+  have h : (thetaAboveCell00113222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113222 h
+theorem e24KC2ThetaAboveLeaf00113223 :
+    adaptiveCoverCheck 11 thetaAboveCell00113223 = true := by
+  have h : (thetaAboveCell00113223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113223 h
+theorem e24KC2ThetaAboveLeaf00113230 :
+    adaptiveCoverCheck 11 thetaAboveCell00113230 = true := by
+  have h : (thetaAboveCell00113230).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113230 h
+theorem e24KC2ThetaAboveLeaf00113231 :
+    adaptiveCoverCheck 11 thetaAboveCell00113231 = true := by
+  have h : (thetaAboveCell00113231).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113231 h
+theorem e24KC2ThetaAboveLeaf00113232 :
+    adaptiveCoverCheck 11 thetaAboveCell00113232 = true := by
+  have h : (thetaAboveCell00113232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113232 h
+theorem e24KC2ThetaAboveLeaf00113233 :
+    adaptiveCoverCheck 11 thetaAboveCell00113233 = true := by
+  have h : (thetaAboveCell00113233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113233 h
+theorem e24KC2ThetaAboveLeaf001133000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113300) = true := by
+  have h : ((childLL thetaAboveCell00113300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00113300) h
+theorem e24KC2ThetaAboveLeaf001133001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113300) = true := by
+  have h : ((childLH thetaAboveCell00113300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00113300) h
+theorem e24KC2ThetaAboveLeaf0011330020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00113300)) = true := by
+  have h : ((childLL (childHL thetaAboveCell00113300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell00113300)) h
+theorem e24KC2ThetaAboveLeaf0011330021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00113300)) = true := by
+  have h : ((childLH (childHL thetaAboveCell00113300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell00113300)) h
+theorem e24KC2ThetaAboveLeaf0011330030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00113300)) = true := by
+  have h : ((childLL (childHH thetaAboveCell00113300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell00113300)) h
+theorem e24KC2ThetaAboveLeaf0011330031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00113300)) = true := by
+  have h : ((childLH (childHH thetaAboveCell00113300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell00113300)) h
+theorem e24KC2ThetaAboveLeaf001133010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113301) = true := by
+  have h : ((childLL thetaAboveCell00113301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00113301) h
+theorem e24KC2ThetaAboveLeaf001133011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113301) = true := by
+  have h : ((childLH thetaAboveCell00113301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00113301) h
+theorem e24KC2ThetaAboveLeaf0011330120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00113301)) = true := by
+  have h : ((childLL (childHL thetaAboveCell00113301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell00113301)) h
+theorem e24KC2ThetaAboveLeaf0011330121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00113301)) = true := by
+  have h : ((childLH (childHL thetaAboveCell00113301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell00113301)) h
+theorem e24KC2ThetaAboveLeaf0011330130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00113301)) = true := by
+  have h : ((childLL (childHH thetaAboveCell00113301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell00113301)) h
+theorem e24KC2ThetaAboveLeaf0011330131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00113301)) = true := by
+  have h : ((childLH (childHH thetaAboveCell00113301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell00113301)) h
+theorem e24KC2ThetaAboveLeaf0011330202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00113302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00113302)) h
+theorem e24KC2ThetaAboveLeaf0011330203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00113302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00113302)) h
+theorem e24KC2ThetaAboveLeaf0011330212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00113302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00113302)) h
+theorem e24KC2ThetaAboveLeaf0011330213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00113302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00113302)) h
+theorem e24KC2ThetaAboveLeaf001133022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113302) = true := by
+  have h : ((childHL thetaAboveCell00113302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00113302) h
+theorem e24KC2ThetaAboveLeaf001133023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113302) = true := by
+  have h : ((childHH thetaAboveCell00113302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00113302) h
+theorem e24KC2ThetaAboveLeaf0011330302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00113303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00113303)) h
+theorem e24KC2ThetaAboveLeaf0011330303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00113303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00113303)) h
+theorem e24KC2ThetaAboveLeaf0011330312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00113303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00113303)) h
+theorem e24KC2ThetaAboveLeaf0011330313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00113303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00113303)) h
+theorem e24KC2ThetaAboveLeaf001133032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113303) = true := by
+  have h : ((childHL thetaAboveCell00113303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00113303) h
+theorem e24KC2ThetaAboveLeaf001133033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113303) = true := by
+  have h : ((childHH thetaAboveCell00113303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00113303) h
+theorem e24KC2ThetaAboveLeaf001133100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113310) = true := by
+  have h : ((childLL thetaAboveCell00113310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00113310) h
+theorem e24KC2ThetaAboveLeaf001133101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113310) = true := by
+  have h : ((childLH thetaAboveCell00113310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00113310) h
+theorem e24KC2ThetaAboveLeaf0011331020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00113310)) = true := by
+  have h : ((childLL (childHL thetaAboveCell00113310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell00113310)) h
+theorem e24KC2ThetaAboveLeaf0011331021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00113310)) = true := by
+  have h : ((childLH (childHL thetaAboveCell00113310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell00113310)) h
+theorem e24KC2ThetaAboveLeaf0011331030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00113310)) = true := by
+  have h : ((childLL (childHH thetaAboveCell00113310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell00113310)) h
+theorem e24KC2ThetaAboveLeaf0011331031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00113310)) = true := by
+  have h : ((childLH (childHH thetaAboveCell00113310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell00113310)) h
+theorem e24KC2ThetaAboveLeaf001133110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113311) = true := by
+  have h : ((childLL thetaAboveCell00113311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00113311) h
+theorem e24KC2ThetaAboveLeaf001133111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113311) = true := by
+  have h : ((childLH thetaAboveCell00113311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00113311) h
+theorem e24KC2ThetaAboveLeaf0011331120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00113311)) = true := by
+  have h : ((childLL (childHL thetaAboveCell00113311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell00113311)) h
+theorem e24KC2ThetaAboveLeaf0011331121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00113311)) = true := by
+  have h : ((childLH (childHL thetaAboveCell00113311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell00113311)) h
+theorem e24KC2ThetaAboveLeaf0011331130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00113311)) = true := by
+  have h : ((childLL (childHH thetaAboveCell00113311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell00113311)) h
+theorem e24KC2ThetaAboveLeaf0011331131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00113311)) = true := by
+  have h : ((childLH (childHH thetaAboveCell00113311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell00113311)) h
+theorem e24KC2ThetaAboveLeaf0011331202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00113312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00113312)) h
+theorem e24KC2ThetaAboveLeaf0011331203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00113312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00113312)) h
+theorem e24KC2ThetaAboveLeaf0011331212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00113312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00113312)) h
+theorem e24KC2ThetaAboveLeaf0011331213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00113312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00113312)) h
+theorem e24KC2ThetaAboveLeaf001133122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113312) = true := by
+  have h : ((childHL thetaAboveCell00113312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00113312) h
+theorem e24KC2ThetaAboveLeaf001133123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113312) = true := by
+  have h : ((childHH thetaAboveCell00113312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00113312) h
+theorem e24KC2ThetaAboveLeaf0011331302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00113313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00113313)) h
+theorem e24KC2ThetaAboveLeaf0011331303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00113313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00113313)) h
+theorem e24KC2ThetaAboveLeaf0011331312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00113313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00113313)) h
+theorem e24KC2ThetaAboveLeaf0011331313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00113313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00113313)) h
+theorem e24KC2ThetaAboveLeaf001133132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113313) = true := by
+  have h : ((childHL thetaAboveCell00113313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00113313) h
+theorem e24KC2ThetaAboveLeaf001133133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113313) = true := by
+  have h : ((childHH thetaAboveCell00113313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00113313) h
+theorem e24KC2ThetaAboveLeaf00113320 :
+    adaptiveCoverCheck 11 thetaAboveCell00113320 = true := by
+  have h : (thetaAboveCell00113320).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113320 h
+theorem e24KC2ThetaAboveLeaf00113321 :
+    adaptiveCoverCheck 11 thetaAboveCell00113321 = true := by
+  have h : (thetaAboveCell00113321).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113321 h
+theorem e24KC2ThetaAboveLeaf00113322 :
+    adaptiveCoverCheck 11 thetaAboveCell00113322 = true := by
+  have h : (thetaAboveCell00113322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113322 h
+theorem e24KC2ThetaAboveLeaf00113323 :
+    adaptiveCoverCheck 11 thetaAboveCell00113323 = true := by
+  have h : (thetaAboveCell00113323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113323 h
+theorem e24KC2ThetaAboveLeaf00113330 :
+    adaptiveCoverCheck 11 thetaAboveCell00113330 = true := by
+  have h : (thetaAboveCell00113330).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113330 h
+theorem e24KC2ThetaAboveLeaf00113331 :
+    adaptiveCoverCheck 11 thetaAboveCell00113331 = true := by
+  have h : (thetaAboveCell00113331).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113331 h
+theorem e24KC2ThetaAboveLeaf00113332 :
+    adaptiveCoverCheck 11 thetaAboveCell00113332 = true := by
+  have h : (thetaAboveCell00113332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113332 h
+theorem e24KC2ThetaAboveLeaf00113333 :
+    adaptiveCoverCheck 11 thetaAboveCell00113333 = true := by
+  have h : (thetaAboveCell00113333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00113333 h
+theorem e24KC2ThetaAboveLeaf0012000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell0012))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf0012001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell0012))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf0012002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell0012))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf0012003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell0012))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf0012010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell0012))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf0012011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell0012))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf0012012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell0012))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf0012013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell0012))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf001202 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0012)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0012)) h
+theorem e24KC2ThetaAboveLeaf001203 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0012)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0012)) h
+theorem e24KC2ThetaAboveLeaf0012100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell0012))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf0012101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell0012))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf0012102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell0012))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf0012103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell0012))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf0012110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell0012))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf0012111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell0012))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf0012112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell0012))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf0012113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell0012))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell0012)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell0012))) h
+theorem e24KC2ThetaAboveLeaf001212 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0012)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0012)) h
+theorem e24KC2ThetaAboveLeaf001213 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0012)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0012)) h
+theorem e24KC2ThetaAboveLeaf00122 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0012) = true := by
+  have h : ((childHL thetaAboveCell0012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell0012) h
+theorem e24KC2ThetaAboveLeaf00123 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0012) = true := by
+  have h : ((childHH thetaAboveCell0012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell0012) h
+theorem e24KC2ThetaAboveLeaf0013000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell0013))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf0013001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell0013))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf0013002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell0013))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf0013003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell0013))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf0013010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell0013))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf0013011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell0013))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf0013012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell0013))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf0013013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell0013))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf001302 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0013)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0013)) h
+theorem e24KC2ThetaAboveLeaf001303 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0013)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0013)) h
+theorem e24KC2ThetaAboveLeaf0013100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell0013))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf0013101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell0013))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf0013102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell0013))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf0013103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell0013))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf0013110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell0013))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf0013111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell0013))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf0013112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell0013))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf0013113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell0013))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell0013)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell0013))) h
+theorem e24KC2ThetaAboveLeaf001312 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0013)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0013)) h
+theorem e24KC2ThetaAboveLeaf001313 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0013)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0013)) h
+theorem e24KC2ThetaAboveLeaf00132 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0013) = true := by
+  have h : ((childHL thetaAboveCell0013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell0013) h
+theorem e24KC2ThetaAboveLeaf00133 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0013) = true := by
+  have h : ((childHH thetaAboveCell0013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell0013) h
+theorem e24KC2ThetaAboveLeaf0020 :
+    adaptiveCoverCheck 15 thetaAboveCell0020 = true := by
+  have h : (thetaAboveCell0020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0020 h
+theorem e24KC2ThetaAboveLeaf0021 :
+    adaptiveCoverCheck 15 thetaAboveCell0021 = true := by
+  have h : (thetaAboveCell0021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0021 h
+theorem e24KC2ThetaAboveLeaf0022 :
+    adaptiveCoverCheck 15 thetaAboveCell0022 = true := by
+  have h : (thetaAboveCell0022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0022 h
+theorem e24KC2ThetaAboveLeaf0023 :
+    adaptiveCoverCheck 15 thetaAboveCell0023 = true := by
+  have h : (thetaAboveCell0023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0023 h
+theorem e24KC2ThetaAboveLeaf0030 :
+    adaptiveCoverCheck 15 thetaAboveCell0030 = true := by
+  have h : (thetaAboveCell0030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0030 h
+theorem e24KC2ThetaAboveLeaf0031 :
+    adaptiveCoverCheck 15 thetaAboveCell0031 = true := by
+  have h : (thetaAboveCell0031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0031 h
+theorem e24KC2ThetaAboveLeaf0032 :
+    adaptiveCoverCheck 15 thetaAboveCell0032 = true := by
+  have h : (thetaAboveCell0032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0032 h
+theorem e24KC2ThetaAboveLeaf0033 :
+    adaptiveCoverCheck 15 thetaAboveCell0033 = true := by
+  have h : (thetaAboveCell0033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0033 h
+theorem e24KC2ThetaAboveLeaf010000 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0100)) = true := by
+  have h : ((childLL (childLL thetaAboveCell0100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell0100)) h
+theorem e24KC2ThetaAboveLeaf010001 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0100)) = true := by
+  have h : ((childLH (childLL thetaAboveCell0100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell0100)) h
+theorem e24KC2ThetaAboveLeaf010002 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0100)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0100)) h
+theorem e24KC2ThetaAboveLeaf010003 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0100)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0100)) h
+theorem e24KC2ThetaAboveLeaf010010 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0100)) = true := by
+  have h : ((childLL (childLH thetaAboveCell0100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell0100)) h
+theorem e24KC2ThetaAboveLeaf010011 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0100)) = true := by
+  have h : ((childLH (childLH thetaAboveCell0100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell0100)) h
+theorem e24KC2ThetaAboveLeaf010012 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0100)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0100)) h
+theorem e24KC2ThetaAboveLeaf010013 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0100)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0100))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0100)) h
+theorem e24KC2ThetaAboveLeaf0100200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell0100))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell0100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell0100))) h
+theorem e24KC2ThetaAboveLeaf0100201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell0100))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell0100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell0100))) h
+theorem e24KC2ThetaAboveLeaf01002020 :
+    adaptiveCoverCheck 11 thetaAboveCell01002020 = true := by
+  have h : (thetaAboveCell01002020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002020 h
+theorem e24KC2ThetaAboveLeaf01002021 :
+    adaptiveCoverCheck 11 thetaAboveCell01002021 = true := by
+  have h : (thetaAboveCell01002021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002021 h
+theorem e24KC2ThetaAboveLeaf01002022 :
+    adaptiveCoverCheck 11 thetaAboveCell01002022 = true := by
+  have h : (thetaAboveCell01002022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002022 h
+theorem e24KC2ThetaAboveLeaf01002023 :
+    adaptiveCoverCheck 11 thetaAboveCell01002023 = true := by
+  have h : (thetaAboveCell01002023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002023 h
+theorem e24KC2ThetaAboveLeaf01002030 :
+    adaptiveCoverCheck 11 thetaAboveCell01002030 = true := by
+  have h : (thetaAboveCell01002030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002030 h
+theorem e24KC2ThetaAboveLeaf01002031 :
+    adaptiveCoverCheck 11 thetaAboveCell01002031 = true := by
+  have h : (thetaAboveCell01002031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002031 h
+theorem e24KC2ThetaAboveLeaf01002032 :
+    adaptiveCoverCheck 11 thetaAboveCell01002032 = true := by
+  have h : (thetaAboveCell01002032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002032 h
+theorem e24KC2ThetaAboveLeaf01002033 :
+    adaptiveCoverCheck 11 thetaAboveCell01002033 = true := by
+  have h : (thetaAboveCell01002033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002033 h
+theorem e24KC2ThetaAboveLeaf0100210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell0100))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell0100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell0100))) h
+theorem e24KC2ThetaAboveLeaf0100211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell0100))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell0100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell0100))) h
+theorem e24KC2ThetaAboveLeaf01002120 :
+    adaptiveCoverCheck 11 thetaAboveCell01002120 = true := by
+  have h : (thetaAboveCell01002120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002120 h
+theorem e24KC2ThetaAboveLeaf01002121 :
+    adaptiveCoverCheck 11 thetaAboveCell01002121 = true := by
+  have h : (thetaAboveCell01002121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002121 h
+theorem e24KC2ThetaAboveLeaf01002122 :
+    adaptiveCoverCheck 11 thetaAboveCell01002122 = true := by
+  have h : (thetaAboveCell01002122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002122 h
+theorem e24KC2ThetaAboveLeaf01002123 :
+    adaptiveCoverCheck 11 thetaAboveCell01002123 = true := by
+  have h : (thetaAboveCell01002123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002123 h
+theorem e24KC2ThetaAboveLeaf01002130 :
+    adaptiveCoverCheck 11 thetaAboveCell01002130 = true := by
+  have h : (thetaAboveCell01002130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002130 h
+theorem e24KC2ThetaAboveLeaf01002131 :
+    adaptiveCoverCheck 11 thetaAboveCell01002131 = true := by
+  have h : (thetaAboveCell01002131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002131 h
+theorem e24KC2ThetaAboveLeaf01002132 :
+    adaptiveCoverCheck 11 thetaAboveCell01002132 = true := by
+  have h : (thetaAboveCell01002132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002132 h
+theorem e24KC2ThetaAboveLeaf01002133 :
+    adaptiveCoverCheck 11 thetaAboveCell01002133 = true := by
+  have h : (thetaAboveCell01002133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002133 h
+theorem e24KC2ThetaAboveLeaf010022000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002200) = true := by
+  have h : ((childLL thetaAboveCell01002200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01002200) h
+theorem e24KC2ThetaAboveLeaf010022001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002200) = true := by
+  have h : ((childLH thetaAboveCell01002200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01002200) h
+theorem e24KC2ThetaAboveLeaf0100220020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01002200)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01002200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01002200)) h
+theorem e24KC2ThetaAboveLeaf0100220021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01002200)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01002200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01002200)) h
+theorem e24KC2ThetaAboveLeaf0100220030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01002200)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01002200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01002200)) h
+theorem e24KC2ThetaAboveLeaf0100220031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01002200)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01002200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01002200)) h
+theorem e24KC2ThetaAboveLeaf010022010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002201) = true := by
+  have h : ((childLL thetaAboveCell01002201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01002201) h
+theorem e24KC2ThetaAboveLeaf010022011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002201) = true := by
+  have h : ((childLH thetaAboveCell01002201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01002201) h
+theorem e24KC2ThetaAboveLeaf0100220120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01002201)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01002201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01002201)) h
+theorem e24KC2ThetaAboveLeaf0100220121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01002201)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01002201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01002201)) h
+theorem e24KC2ThetaAboveLeaf0100220130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01002201)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01002201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01002201)) h
+theorem e24KC2ThetaAboveLeaf0100220131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01002201)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01002201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01002201)) h
+theorem e24KC2ThetaAboveLeaf0100220202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01002202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01002202)) h
+theorem e24KC2ThetaAboveLeaf0100220203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01002202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01002202)) h
+theorem e24KC2ThetaAboveLeaf010022022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002202) = true := by
+  have h : ((childHL thetaAboveCell01002202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01002202) h
+theorem e24KC2ThetaAboveLeaf010022023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002202) = true := by
+  have h : ((childHH thetaAboveCell01002202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01002202) h
+theorem e24KC2ThetaAboveLeaf010022032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002203) = true := by
+  have h : ((childHL thetaAboveCell01002203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01002203) h
+theorem e24KC2ThetaAboveLeaf010022033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002203) = true := by
+  have h : ((childHH thetaAboveCell01002203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01002203) h
+theorem e24KC2ThetaAboveLeaf010022100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002210) = true := by
+  have h : ((childLL thetaAboveCell01002210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01002210) h
+theorem e24KC2ThetaAboveLeaf010022101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002210) = true := by
+  have h : ((childLH thetaAboveCell01002210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01002210) h
+theorem e24KC2ThetaAboveLeaf0100221020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01002210)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01002210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01002210)) h
+theorem e24KC2ThetaAboveLeaf0100221021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01002210)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01002210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01002210)) h
+theorem e24KC2ThetaAboveLeaf0100221030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01002210)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01002210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01002210)) h
+theorem e24KC2ThetaAboveLeaf0100221031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01002210)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01002210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01002210)) h
+theorem e24KC2ThetaAboveLeaf010022110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002211) = true := by
+  have h : ((childLL thetaAboveCell01002211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01002211) h
+theorem e24KC2ThetaAboveLeaf010022111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002211) = true := by
+  have h : ((childLH thetaAboveCell01002211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01002211) h
+theorem e24KC2ThetaAboveLeaf0100221120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01002211)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01002211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01002211)) h
+theorem e24KC2ThetaAboveLeaf0100221121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01002211)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01002211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01002211)) h
+theorem e24KC2ThetaAboveLeaf0100221130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01002211)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01002211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01002211)) h
+theorem e24KC2ThetaAboveLeaf0100221131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01002211)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01002211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01002211)) h
+theorem e24KC2ThetaAboveLeaf0100221203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01002212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01002212)) h
+theorem e24KC2ThetaAboveLeaf0100221212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01002212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01002212)) h
+theorem e24KC2ThetaAboveLeaf0100221213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01002212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01002212)) h
+theorem e24KC2ThetaAboveLeaf010022122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002212) = true := by
+  have h : ((childHL thetaAboveCell01002212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01002212) h
+theorem e24KC2ThetaAboveLeaf010022123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002212) = true := by
+  have h : ((childHH thetaAboveCell01002212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01002212) h
+theorem e24KC2ThetaAboveLeaf0100221302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01002213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01002213)) h
+theorem e24KC2ThetaAboveLeaf0100221303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01002213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01002213)) h
+theorem e24KC2ThetaAboveLeaf0100221312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01002213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01002213)) h
+theorem e24KC2ThetaAboveLeaf0100221313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01002213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01002213)) h
+theorem e24KC2ThetaAboveLeaf010022132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002213) = true := by
+  have h : ((childHL thetaAboveCell01002213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01002213) h
+theorem e24KC2ThetaAboveLeaf010022133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002213) = true := by
+  have h : ((childHH thetaAboveCell01002213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01002213) h
+theorem e24KC2ThetaAboveLeaf01002220 :
+    adaptiveCoverCheck 11 thetaAboveCell01002220 = true := by
+  have h : (thetaAboveCell01002220).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002220 h
+theorem e24KC2ThetaAboveLeaf01002221 :
+    adaptiveCoverCheck 11 thetaAboveCell01002221 = true := by
+  have h : (thetaAboveCell01002221).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002221 h
+theorem e24KC2ThetaAboveLeaf01002222 :
+    adaptiveCoverCheck 11 thetaAboveCell01002222 = true := by
+  have h : (thetaAboveCell01002222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002222 h
+theorem e24KC2ThetaAboveLeaf01002223 :
+    adaptiveCoverCheck 11 thetaAboveCell01002223 = true := by
+  have h : (thetaAboveCell01002223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002223 h
+theorem e24KC2ThetaAboveLeaf01002230 :
+    adaptiveCoverCheck 11 thetaAboveCell01002230 = true := by
+  have h : (thetaAboveCell01002230).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002230 h
+theorem e24KC2ThetaAboveLeaf01002231 :
+    adaptiveCoverCheck 11 thetaAboveCell01002231 = true := by
+  have h : (thetaAboveCell01002231).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002231 h
+theorem e24KC2ThetaAboveLeaf01002232 :
+    adaptiveCoverCheck 11 thetaAboveCell01002232 = true := by
+  have h : (thetaAboveCell01002232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002232 h
+theorem e24KC2ThetaAboveLeaf01002233 :
+    adaptiveCoverCheck 11 thetaAboveCell01002233 = true := by
+  have h : (thetaAboveCell01002233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002233 h
+theorem e24KC2ThetaAboveLeaf010023000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002300) = true := by
+  have h : ((childLL thetaAboveCell01002300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01002300) h
+theorem e24KC2ThetaAboveLeaf010023001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002300) = true := by
+  have h : ((childLH thetaAboveCell01002300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01002300) h
+theorem e24KC2ThetaAboveLeaf0100230020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01002300)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01002300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01002300)) h
+theorem e24KC2ThetaAboveLeaf0100230021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01002300)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01002300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01002300)) h
+theorem e24KC2ThetaAboveLeaf0100230030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01002300)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01002300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01002300)) h
+theorem e24KC2ThetaAboveLeaf0100230031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01002300)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01002300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01002300)) h
+theorem e24KC2ThetaAboveLeaf010023010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002301) = true := by
+  have h : ((childLL thetaAboveCell01002301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01002301) h
+theorem e24KC2ThetaAboveLeaf010023011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002301) = true := by
+  have h : ((childLH thetaAboveCell01002301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01002301) h
+theorem e24KC2ThetaAboveLeaf0100230120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01002301)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01002301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01002301)) h
+theorem e24KC2ThetaAboveLeaf0100230121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01002301)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01002301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01002301)) h
+theorem e24KC2ThetaAboveLeaf0100230130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01002301)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01002301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01002301)) h
+theorem e24KC2ThetaAboveLeaf0100230131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01002301)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01002301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01002301)) h
+theorem e24KC2ThetaAboveLeaf0100230202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01002302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01002302)) h
+theorem e24KC2ThetaAboveLeaf0100230203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01002302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01002302)) h
+theorem e24KC2ThetaAboveLeaf0100230212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01002302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01002302)) h
+theorem e24KC2ThetaAboveLeaf0100230213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01002302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01002302)) h
+theorem e24KC2ThetaAboveLeaf010023022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002302) = true := by
+  have h : ((childHL thetaAboveCell01002302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01002302) h
+theorem e24KC2ThetaAboveLeaf010023023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002302) = true := by
+  have h : ((childHH thetaAboveCell01002302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01002302) h
+theorem e24KC2ThetaAboveLeaf0100230302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01002303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01002303)) h
+theorem e24KC2ThetaAboveLeaf0100230303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01002303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01002303)) h
+theorem e24KC2ThetaAboveLeaf0100230312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01002303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01002303)) h
+theorem e24KC2ThetaAboveLeaf0100230313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01002303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01002303)) h
+theorem e24KC2ThetaAboveLeaf0100230320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01002303)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01002303)) h
+theorem e24KC2ThetaAboveLeaf0100230321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01002303)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01002303)) h
+theorem e24KC2ThetaAboveLeaf0100230322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01002303)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01002303)) h
+theorem e24KC2ThetaAboveLeaf0100230323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01002303)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01002303)) h
+theorem e24KC2ThetaAboveLeaf0100230330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01002303)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01002303)) h
+theorem e24KC2ThetaAboveLeaf0100230331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01002303)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01002303)) h
+theorem e24KC2ThetaAboveLeaf0100230332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01002303)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01002303)) h
+theorem e24KC2ThetaAboveLeaf0100230333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01002303)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01002303)) h
+theorem e24KC2ThetaAboveLeaf010023100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002310) = true := by
+  have h : ((childLL thetaAboveCell01002310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01002310) h
+theorem e24KC2ThetaAboveLeaf010023101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002310) = true := by
+  have h : ((childLH thetaAboveCell01002310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01002310) h
+theorem e24KC2ThetaAboveLeaf0100231020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01002310)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01002310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01002310)) h
+theorem e24KC2ThetaAboveLeaf0100231021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01002310)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01002310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01002310)) h
+theorem e24KC2ThetaAboveLeaf0100231030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01002310)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01002310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01002310)) h
+theorem e24KC2ThetaAboveLeaf0100231031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01002310)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01002310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01002310)) h
+theorem e24KC2ThetaAboveLeaf010023110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002311) = true := by
+  have h : ((childLL thetaAboveCell01002311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01002311) h
+theorem e24KC2ThetaAboveLeaf010023111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002311) = true := by
+  have h : ((childLH thetaAboveCell01002311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01002311) h
+theorem e24KC2ThetaAboveLeaf0100231120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01002311)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01002311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01002311)) h
+theorem e24KC2ThetaAboveLeaf0100231121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01002311)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01002311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01002311)) h
+theorem e24KC2ThetaAboveLeaf0100231130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01002311)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01002311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01002311)) h
+theorem e24KC2ThetaAboveLeaf0100231131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01002311)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01002311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01002311)) h
+theorem e24KC2ThetaAboveLeaf0100231200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01002312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01002312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01002312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01002312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01002312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01002312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01002312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01002312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01002312)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01002312)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01002312)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01002312)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01002312)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01002312)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01002312)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01002312)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01002312)) h
+theorem e24KC2ThetaAboveLeaf0100231300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01002313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf0100231301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01002313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf0100231302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01002313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf0100231303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01002313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf0100231310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01002313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf0100231311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01002313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf0100231312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01002313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf0100231313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01002313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf0100231320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01002313)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf0100231321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01002313)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf0100231322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01002313)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf0100231323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01002313)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf0100231330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01002313)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf0100231331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01002313)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf0100231332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01002313)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf0100231333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01002313)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01002313)) h
+theorem e24KC2ThetaAboveLeaf01002320 :
+    adaptiveCoverCheck 11 thetaAboveCell01002320 = true := by
+  have h : (thetaAboveCell01002320).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002320 h
+theorem e24KC2ThetaAboveLeaf01002321 :
+    adaptiveCoverCheck 11 thetaAboveCell01002321 = true := by
+  have h : (thetaAboveCell01002321).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002321 h
+theorem e24KC2ThetaAboveLeaf01002322 :
+    adaptiveCoverCheck 11 thetaAboveCell01002322 = true := by
+  have h : (thetaAboveCell01002322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002322 h
+theorem e24KC2ThetaAboveLeaf01002323 :
+    adaptiveCoverCheck 11 thetaAboveCell01002323 = true := by
+  have h : (thetaAboveCell01002323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002323 h
+theorem e24KC2ThetaAboveLeaf01002330 :
+    adaptiveCoverCheck 11 thetaAboveCell01002330 = true := by
+  have h : (thetaAboveCell01002330).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002330 h
+theorem e24KC2ThetaAboveLeaf01002331 :
+    adaptiveCoverCheck 11 thetaAboveCell01002331 = true := by
+  have h : (thetaAboveCell01002331).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002331 h
+theorem e24KC2ThetaAboveLeaf01002332 :
+    adaptiveCoverCheck 11 thetaAboveCell01002332 = true := by
+  have h : (thetaAboveCell01002332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002332 h
+theorem e24KC2ThetaAboveLeaf01002333 :
+    adaptiveCoverCheck 11 thetaAboveCell01002333 = true := by
+  have h : (thetaAboveCell01002333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01002333 h
+theorem e24KC2ThetaAboveLeaf0100300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell0100))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell0100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell0100))) h
+theorem e24KC2ThetaAboveLeaf0100301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell0100))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell0100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell0100))) h
+theorem e24KC2ThetaAboveLeaf01003020 :
+    adaptiveCoverCheck 11 thetaAboveCell01003020 = true := by
+  have h : (thetaAboveCell01003020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003020 h
+theorem e24KC2ThetaAboveLeaf01003021 :
+    adaptiveCoverCheck 11 thetaAboveCell01003021 = true := by
+  have h : (thetaAboveCell01003021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003021 h
+theorem e24KC2ThetaAboveLeaf01003022 :
+    adaptiveCoverCheck 11 thetaAboveCell01003022 = true := by
+  have h : (thetaAboveCell01003022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003022 h
+theorem e24KC2ThetaAboveLeaf01003023 :
+    adaptiveCoverCheck 11 thetaAboveCell01003023 = true := by
+  have h : (thetaAboveCell01003023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003023 h
+theorem e24KC2ThetaAboveLeaf01003030 :
+    adaptiveCoverCheck 11 thetaAboveCell01003030 = true := by
+  have h : (thetaAboveCell01003030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003030 h
+theorem e24KC2ThetaAboveLeaf01003031 :
+    adaptiveCoverCheck 11 thetaAboveCell01003031 = true := by
+  have h : (thetaAboveCell01003031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003031 h
+theorem e24KC2ThetaAboveLeaf01003032 :
+    adaptiveCoverCheck 11 thetaAboveCell01003032 = true := by
+  have h : (thetaAboveCell01003032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003032 h
+theorem e24KC2ThetaAboveLeaf01003033 :
+    adaptiveCoverCheck 11 thetaAboveCell01003033 = true := by
+  have h : (thetaAboveCell01003033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003033 h
+theorem e24KC2ThetaAboveLeaf0100310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell0100))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell0100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell0100))) h
+theorem e24KC2ThetaAboveLeaf0100311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell0100))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell0100)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell0100))) h
+theorem e24KC2ThetaAboveLeaf01003120 :
+    adaptiveCoverCheck 11 thetaAboveCell01003120 = true := by
+  have h : (thetaAboveCell01003120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003120 h
+theorem e24KC2ThetaAboveLeaf01003121 :
+    adaptiveCoverCheck 11 thetaAboveCell01003121 = true := by
+  have h : (thetaAboveCell01003121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003121 h
+theorem e24KC2ThetaAboveLeaf01003122 :
+    adaptiveCoverCheck 11 thetaAboveCell01003122 = true := by
+  have h : (thetaAboveCell01003122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003122 h
+theorem e24KC2ThetaAboveLeaf01003123 :
+    adaptiveCoverCheck 11 thetaAboveCell01003123 = true := by
+  have h : (thetaAboveCell01003123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003123 h
+theorem e24KC2ThetaAboveLeaf01003130 :
+    adaptiveCoverCheck 11 thetaAboveCell01003130 = true := by
+  have h : (thetaAboveCell01003130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003130 h
+theorem e24KC2ThetaAboveLeaf01003131 :
+    adaptiveCoverCheck 11 thetaAboveCell01003131 = true := by
+  have h : (thetaAboveCell01003131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003131 h
+theorem e24KC2ThetaAboveLeaf01003132 :
+    adaptiveCoverCheck 11 thetaAboveCell01003132 = true := by
+  have h : (thetaAboveCell01003132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003132 h
+theorem e24KC2ThetaAboveLeaf01003133 :
+    adaptiveCoverCheck 11 thetaAboveCell01003133 = true := by
+  have h : (thetaAboveCell01003133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003133 h
+theorem e24KC2ThetaAboveLeaf010032000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003200) = true := by
+  have h : ((childLL thetaAboveCell01003200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01003200) h
+theorem e24KC2ThetaAboveLeaf010032001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003200) = true := by
+  have h : ((childLH thetaAboveCell01003200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01003200) h
+theorem e24KC2ThetaAboveLeaf0100320020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003200)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003200)) h
+theorem e24KC2ThetaAboveLeaf0100320021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003200)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003200)) h
+theorem e24KC2ThetaAboveLeaf0100320030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003200)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003200)) h
+theorem e24KC2ThetaAboveLeaf0100320031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003200)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003200)) h
+theorem e24KC2ThetaAboveLeaf010032010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003201) = true := by
+  have h : ((childLL thetaAboveCell01003201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01003201) h
+theorem e24KC2ThetaAboveLeaf010032011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003201) = true := by
+  have h : ((childLH thetaAboveCell01003201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01003201) h
+theorem e24KC2ThetaAboveLeaf0100320120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003201)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003201)) h
+theorem e24KC2ThetaAboveLeaf0100320121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003201)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003201)) h
+theorem e24KC2ThetaAboveLeaf0100320130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003201)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003201)) h
+theorem e24KC2ThetaAboveLeaf0100320131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003201)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003201)) h
+theorem e24KC2ThetaAboveLeaf0100320200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01003202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01003202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01003202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01003202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01003202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01003202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01003202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01003202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003202)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003202)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003202)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003202)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003202)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003202)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003202)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003202)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01003202)) h
+theorem e24KC2ThetaAboveLeaf0100320300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01003203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf0100320301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01003203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf0100320302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01003203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf0100320303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01003203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf0100320310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01003203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf0100320311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01003203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf0100320312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01003203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf0100320313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01003203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf0100320320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003203)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf0100320321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003203)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf0100320322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003203)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf0100320323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003203)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf0100320330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003203)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf0100320331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003203)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf0100320332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003203)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf0100320333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003203)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01003203)) h
+theorem e24KC2ThetaAboveLeaf010032100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003210) = true := by
+  have h : ((childLL thetaAboveCell01003210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01003210) h
+theorem e24KC2ThetaAboveLeaf010032101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003210) = true := by
+  have h : ((childLH thetaAboveCell01003210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01003210) h
+theorem e24KC2ThetaAboveLeaf0100321020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003210)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003210)) h
+theorem e24KC2ThetaAboveLeaf0100321021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003210)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003210)) h
+theorem e24KC2ThetaAboveLeaf0100321022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003210)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01003210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01003210)) h
+theorem e24KC2ThetaAboveLeaf0100321023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003210)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01003210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01003210)) h
+theorem e24KC2ThetaAboveLeaf0100321030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003210)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003210)) h
+theorem e24KC2ThetaAboveLeaf0100321031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003210)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003210)) h
+theorem e24KC2ThetaAboveLeaf0100321032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003210)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01003210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01003210)) h
+theorem e24KC2ThetaAboveLeaf0100321033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003210)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01003210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01003210)) h
+theorem e24KC2ThetaAboveLeaf010032110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003211) = true := by
+  have h : ((childLL thetaAboveCell01003211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01003211) h
+theorem e24KC2ThetaAboveLeaf010032111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003211) = true := by
+  have h : ((childLH thetaAboveCell01003211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01003211) h
+theorem e24KC2ThetaAboveLeaf0100321120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003211)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003211)) h
+theorem e24KC2ThetaAboveLeaf0100321121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003211)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003211)) h
+theorem e24KC2ThetaAboveLeaf0100321122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003211)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01003211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01003211)) h
+theorem e24KC2ThetaAboveLeaf0100321123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003211)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01003211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01003211)) h
+theorem e24KC2ThetaAboveLeaf0100321130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003211)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003211)) h
+theorem e24KC2ThetaAboveLeaf0100321131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003211)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003211)) h
+theorem e24KC2ThetaAboveLeaf0100321132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003211)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01003211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01003211)) h
+theorem e24KC2ThetaAboveLeaf0100321133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003211)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01003211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01003211)) h
+theorem e24KC2ThetaAboveLeaf0100321200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01003212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01003212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01003212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01003212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01003212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01003212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01003212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01003212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003212)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003212)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003212)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003212)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003212)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003212)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003212)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003212)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01003212)) h
+theorem e24KC2ThetaAboveLeaf0100321300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01003213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf0100321301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01003213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf0100321302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01003213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf0100321303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01003213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf0100321310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01003213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf0100321311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01003213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf0100321312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01003213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf0100321313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01003213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf0100321320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003213)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf0100321321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003213)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf0100321322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003213)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf0100321323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003213)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf0100321330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003213)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf0100321331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003213)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf0100321332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003213)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf0100321333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003213)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01003213)) h
+theorem e24KC2ThetaAboveLeaf01003220 :
+    adaptiveCoverCheck 11 thetaAboveCell01003220 = true := by
+  have h : (thetaAboveCell01003220).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003220 h
+theorem e24KC2ThetaAboveLeaf01003221 :
+    adaptiveCoverCheck 11 thetaAboveCell01003221 = true := by
+  have h : (thetaAboveCell01003221).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003221 h
+theorem e24KC2ThetaAboveLeaf01003222 :
+    adaptiveCoverCheck 11 thetaAboveCell01003222 = true := by
+  have h : (thetaAboveCell01003222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003222 h
+theorem e24KC2ThetaAboveLeaf01003223 :
+    adaptiveCoverCheck 11 thetaAboveCell01003223 = true := by
+  have h : (thetaAboveCell01003223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003223 h
+theorem e24KC2ThetaAboveLeaf01003230 :
+    adaptiveCoverCheck 11 thetaAboveCell01003230 = true := by
+  have h : (thetaAboveCell01003230).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003230 h
+theorem e24KC2ThetaAboveLeaf01003231 :
+    adaptiveCoverCheck 11 thetaAboveCell01003231 = true := by
+  have h : (thetaAboveCell01003231).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003231 h
+theorem e24KC2ThetaAboveLeaf01003232 :
+    adaptiveCoverCheck 11 thetaAboveCell01003232 = true := by
+  have h : (thetaAboveCell01003232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003232 h
+theorem e24KC2ThetaAboveLeaf01003233 :
+    adaptiveCoverCheck 11 thetaAboveCell01003233 = true := by
+  have h : (thetaAboveCell01003233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003233 h
+theorem e24KC2ThetaAboveLeaf010033000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003300) = true := by
+  have h : ((childLL thetaAboveCell01003300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01003300) h
+theorem e24KC2ThetaAboveLeaf010033001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003300) = true := by
+  have h : ((childLH thetaAboveCell01003300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01003300) h
+theorem e24KC2ThetaAboveLeaf0100330020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003300)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003300)) h
+theorem e24KC2ThetaAboveLeaf0100330021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003300)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003300)) h
+theorem e24KC2ThetaAboveLeaf0100330022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003300)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01003300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01003300)) h
+theorem e24KC2ThetaAboveLeaf0100330023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003300)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01003300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01003300)) h
+theorem e24KC2ThetaAboveLeaf0100330030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003300)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003300)) h
+theorem e24KC2ThetaAboveLeaf0100330031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003300)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003300)) h
+theorem e24KC2ThetaAboveLeaf0100330032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003300)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01003300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01003300)) h
+theorem e24KC2ThetaAboveLeaf0100330033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003300)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01003300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01003300)) h
+theorem e24KC2ThetaAboveLeaf010033010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003301) = true := by
+  have h : ((childLL thetaAboveCell01003301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01003301) h
+theorem e24KC2ThetaAboveLeaf010033011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003301) = true := by
+  have h : ((childLH thetaAboveCell01003301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01003301) h
+theorem e24KC2ThetaAboveLeaf0100330120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003301)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003301)) h
+theorem e24KC2ThetaAboveLeaf0100330121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003301)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003301)) h
+theorem e24KC2ThetaAboveLeaf0100330122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003301)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01003301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01003301)) h
+theorem e24KC2ThetaAboveLeaf0100330123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003301)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01003301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01003301)) h
+theorem e24KC2ThetaAboveLeaf0100330130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003301)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003301)) h
+theorem e24KC2ThetaAboveLeaf0100330131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003301)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003301)) h
+theorem e24KC2ThetaAboveLeaf0100330132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003301)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01003301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01003301)) h
+theorem e24KC2ThetaAboveLeaf0100330133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003301)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01003301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01003301)) h
+theorem e24KC2ThetaAboveLeaf0100330200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01003302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01003302)) h
+theorem e24KC2ThetaAboveLeaf0100330201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01003302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01003302)) h
+theorem e24KC2ThetaAboveLeaf0100330202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01003302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01003302)) h
+theorem e24KC2ThetaAboveLeaf0100330203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01003302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01003302)) h
+theorem e24KC2ThetaAboveLeaf0100330210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01003302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01003302)) h
+theorem e24KC2ThetaAboveLeaf0100330211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01003302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01003302)) h
+theorem e24KC2ThetaAboveLeaf0100330212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01003302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01003302)) h
+theorem e24KC2ThetaAboveLeaf0100330213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01003302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01003302)) h
+theorem e24KC2ThetaAboveLeaf0100330220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003302)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003302)) h
+theorem e24KC2ThetaAboveLeaf0100330221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003302)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003302)) h
+theorem e24KC2ThetaAboveLeaf0100330222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003302)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01003302)) h
+theorem e24KC2ThetaAboveLeaf0100330223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003302)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01003302)) h
+theorem e24KC2ThetaAboveLeaf0100330230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003302)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003302)) h
+theorem e24KC2ThetaAboveLeaf0100330231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003302)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003302)) h
+theorem e24KC2ThetaAboveLeaf0100330232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003302)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01003302)) h
+theorem e24KC2ThetaAboveLeaf0100330233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003302)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01003302)) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6ProofBatch6018780a0c691fa2`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells66bed457b5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells66bed457b5
+
+open CertificateCells66bed457b5
+theorem cover_subtree_6415e703c807 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000023002000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002000)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000023002000))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000023002000)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000023002000))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000023002000)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000023002000))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000023002000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000023002000))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000023002000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000023002000))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000023002000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000023002000))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000023002000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000023002000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000023002000))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000023002000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000023002000))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000023002000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000023002000))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000023002000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000023002000))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000023002000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000023002000))) h))
+
+theorem cover_subtree_22d363d7acf8 :
+    adaptiveCoverCheck 7 thetaAboveCell000023002000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002000
+    (by
+      have h : ((childLL thetaAboveCell000023002000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023002000) h)
+    (by
+      have h : ((childLH thetaAboveCell000023002000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023002000) h)
+    cover_subtree_6415e703c807
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002000)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002000)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002000)) h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000023002000))
+            (by
+              have h : ((childLL (childHL (childHH thetaAboveCell000023002000)))).rejected = true
+                := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+                thetaAboveCell000023002000))) h)
+            (by
+              have h : ((childLH (childHL (childHH thetaAboveCell000023002000)))).rejected = true
+                := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+                thetaAboveCell000023002000))) h)
+            (by
+              have h : ((childHL (childHL (childHH thetaAboveCell000023002000)))).rejected = true
+                := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+                thetaAboveCell000023002000))) h)
+            (by
+              have h : ((childHH (childHL (childHH thetaAboveCell000023002000)))).rejected = true
+                := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+                thetaAboveCell000023002000))) h))
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002000)) h))
+
+theorem cover_subtree_549010c79ddd :
+    adaptiveCoverCheck 7 thetaAboveCell000023002001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002001
+    (by
+      have h : ((childLL thetaAboveCell000023002001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023002001) h)
+    (by
+      have h : ((childLH thetaAboveCell000023002001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023002001) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002001)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023002001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023002001)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023002001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023002001)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023002001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023002001)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023002001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023002001)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002001)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002001)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002001)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023002001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023002001)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002001)) h))
+
+theorem cover_subtree_cee31392fc88 :
+    adaptiveCoverCheck 7 thetaAboveCell000023002002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002002
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023002002)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023002002)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023002002)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023002002)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023002002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023002002)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023002002)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023002002)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023002002)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023002002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002002)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023002002)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023002002)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023002002)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023002002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002002)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002002)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002002)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023002002)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002002)) h))
+
+theorem cover_subtree_6e289b6e590a :
+    adaptiveCoverCheck 7 thetaAboveCell000023002003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002003
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023002003)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023002003)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023002003)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023002003)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023002003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023002003)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023002003)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023002003)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023002003)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023002003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002003)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023002003)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023002003)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023002003)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023002003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002003)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002003)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002003)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023002003)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002003)) h))
+
+theorem cover_subtree_682a52ae1e8e :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00002300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00002300)))
+    cover_subtree_22d363d7acf8
+    cover_subtree_549010c79ddd
+    cover_subtree_cee31392fc88
+    cover_subtree_6e289b6e590a
+
+theorem cover_subtree_f7dd1bf145ad :
+    adaptiveCoverCheck 7 thetaAboveCell000023002010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002010
+    (by
+      have h : ((childLL thetaAboveCell000023002010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023002010) h)
+    (by
+      have h : ((childLH thetaAboveCell000023002010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023002010) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002010)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023002010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023002010)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023002010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023002010)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023002010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023002010)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023002010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023002010)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002010)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002010)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002010)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023002010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023002010)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002010)) h))
+
+theorem cover_subtree_ef225adab279 :
+    adaptiveCoverCheck 7 thetaAboveCell000023002011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002011
+    (by
+      have h : ((childLL thetaAboveCell000023002011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023002011) h)
+    (by
+      have h : ((childLH thetaAboveCell000023002011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023002011) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002011)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023002011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023002011)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023002011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023002011)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023002011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023002011)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023002011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023002011)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002011)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002011)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002011)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023002011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023002011)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002011)) h))
+
+theorem cover_subtree_8107e804592b :
+    adaptiveCoverCheck 7 thetaAboveCell000023002012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002012
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023002012)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023002012)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023002012)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023002012)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023002012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023002012)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023002012)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023002012)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023002012)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023002012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002012)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023002012)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023002012)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023002012)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023002012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002012)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002012)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002012)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023002012)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002012)) h))
+
+theorem cover_subtree_ce4b7ef4c4ef :
+    adaptiveCoverCheck 7 thetaAboveCell000023002013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002013
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023002013)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023002013)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023002013)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023002013)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023002013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023002013)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023002013)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023002013)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023002013)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023002013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002013)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023002013)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023002013)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023002013)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023002013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002013)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002013)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002013)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023002013)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002013)) h))
+
+theorem cover_subtree_1c0e76b133f5 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00002300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00002300)))
+    cover_subtree_f7dd1bf145ad
+    cover_subtree_ef225adab279
+    cover_subtree_8107e804592b
+    cover_subtree_ce4b7ef4c4ef
+
+theorem cover_subtree_eef2c1bd32f7 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHL thetaAboveCell00002300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL thetaAboveCell00002300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002030
+        (by
+          have h : ((childLL thetaAboveCell000023002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023002030) h)
+        (by
+          have h : ((childLH thetaAboveCell000023002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023002030) h)
+        (by
+          have h : ((childHL thetaAboveCell000023002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023002030) h)
+        (by
+          have h : ((childHH thetaAboveCell000023002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023002030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002031
+        (by
+          have h : ((childLL thetaAboveCell000023002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023002031) h)
+        (by
+          have h : ((childLH thetaAboveCell000023002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023002031) h)
+        (by
+          have h : ((childHL thetaAboveCell000023002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023002031) h)
+        (by
+          have h : ((childHH thetaAboveCell000023002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023002031) h))
+    (by
+      have h : (thetaAboveCell000023002032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023002032 h)
+    (by
+      have h : (thetaAboveCell000023002033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023002033 h)
+
+theorem e24KC2ThetaAboveLeaf0000230020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00002300))
+    cover_subtree_682a52ae1e8e
+    cover_subtree_1c0e76b133f5
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00002300)))
+        (by
+          have h : (thetaAboveCell000023002020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023002020 h)
+        (by
+          have h : (thetaAboveCell000023002021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023002021 h)
+        (by
+          have h : (thetaAboveCell000023002022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023002022 h)
+        (by
+          have h : (thetaAboveCell000023002023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023002023 h))
+    cover_subtree_eef2c1bd32f7
+theorem cover_subtree_681c28bf7d84 :
+    adaptiveCoverCheck 7 thetaAboveCell000023002100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002100
+    (by
+      have h : ((childLL thetaAboveCell000023002100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023002100) h)
+    (by
+      have h : ((childLH thetaAboveCell000023002100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023002100) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002100)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023002100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023002100)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023002100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023002100)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023002100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023002100)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023002100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023002100)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002100)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002100)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002100)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023002100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023002100)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002100)) h))
+
+theorem cover_subtree_779aeebb5c73 :
+    adaptiveCoverCheck 7 thetaAboveCell000023002101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002101
+    (by
+      have h : ((childLL thetaAboveCell000023002101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023002101) h)
+    (by
+      have h : ((childLH thetaAboveCell000023002101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023002101) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002101)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023002101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023002101)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023002101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023002101)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023002101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023002101)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023002101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023002101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002101)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002101)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002101)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023002101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023002101)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002101)) h))
+
+theorem cover_subtree_ef8a630852fd :
+    adaptiveCoverCheck 7 thetaAboveCell000023002102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002102
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023002102)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023002102)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023002102)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023002102)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023002102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023002102)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023002102)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023002102)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023002102)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023002102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002102)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023002102)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023002102)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023002102)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023002102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002102)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002102)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002102)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023002102)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002102)) h))
+
+theorem cover_subtree_315ebd923bb4 :
+    adaptiveCoverCheck 7 thetaAboveCell000023002103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002103
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023002103)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023002103)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023002103)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023002103)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023002103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023002103)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023002103)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023002103)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023002103)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023002103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002103)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023002103)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023002103)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023002103)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023002103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002103)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002103)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002103)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023002103)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002103)) h))
+
+theorem cover_subtree_27e55c8a3d8e :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00002300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00002300)))
+    cover_subtree_681c28bf7d84
+    cover_subtree_779aeebb5c73
+    cover_subtree_ef8a630852fd
+    cover_subtree_315ebd923bb4
+
+theorem cover_subtree_6d8f95e98e1d :
+    adaptiveCoverCheck 7 thetaAboveCell000023002110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002110
+    (by
+      have h : ((childLL thetaAboveCell000023002110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023002110) h)
+    (by
+      have h : ((childLH thetaAboveCell000023002110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023002110) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002110)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023002110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023002110)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023002110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023002110)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023002110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023002110)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023002110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023002110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002110)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002110)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002110)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023002110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023002110)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002110)) h))
+
+theorem cover_subtree_d2993040295e :
+    adaptiveCoverCheck 7 thetaAboveCell000023002111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002111
+    (by
+      have h : ((childLL thetaAboveCell000023002111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023002111) h)
+    (by
+      have h : ((childLH thetaAboveCell000023002111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023002111) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002111)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023002111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023002111)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023002111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023002111)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023002111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023002111)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023002111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023002111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002111)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002111)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002111)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023002111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023002111)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002111)) h))
+
+theorem cover_subtree_eb2399542077 :
+    adaptiveCoverCheck 7 thetaAboveCell000023002112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002112
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023002112)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023002112)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023002112)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023002112)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023002112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023002112)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023002112)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023002112)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023002112)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023002112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002112)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023002112)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023002112)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023002112)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023002112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002112)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002112)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002112)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023002112)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002112)) h))
+
+theorem cover_subtree_da276d5ec562 :
+    adaptiveCoverCheck 7 thetaAboveCell000023002113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002113
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023002113)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023002113)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023002113)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023002113)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023002113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023002113)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023002113)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023002113)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023002113)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023002113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023002113)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023002113)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023002113)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023002113)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023002113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023002113)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023002113)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023002113)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023002113)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023002113)) h))
+
+theorem cover_subtree_0fc1de09da1d :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00002300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00002300)))
+    cover_subtree_6d8f95e98e1d
+    cover_subtree_d2993040295e
+    cover_subtree_eb2399542077
+    cover_subtree_da276d5ec562
+
+theorem cover_subtree_b179888799ba :
+    adaptiveCoverCheck 8 (childHL (childLH (childHL thetaAboveCell00002300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL thetaAboveCell00002300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002120
+        (by
+          have h : ((childLL thetaAboveCell000023002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023002120) h)
+        (by
+          have h : ((childLH thetaAboveCell000023002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023002120) h)
+        (by
+          have h : ((childHL thetaAboveCell000023002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023002120) h)
+        (by
+          have h : ((childHH thetaAboveCell000023002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023002120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002121
+        (by
+          have h : ((childLL thetaAboveCell000023002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023002121) h)
+        (by
+          have h : ((childLH thetaAboveCell000023002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023002121) h)
+        (by
+          have h : ((childHL thetaAboveCell000023002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023002121) h)
+        (by
+          have h : ((childHH thetaAboveCell000023002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023002121) h))
+    (by
+      have h : (thetaAboveCell000023002122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023002122 h)
+    (by
+      have h : (thetaAboveCell000023002123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023002123 h)
+
+theorem cover_subtree_dfa114599465 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHL thetaAboveCell00002300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL thetaAboveCell00002300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002130
+        (by
+          have h : ((childLL thetaAboveCell000023002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023002130) h)
+        (by
+          have h : ((childLH thetaAboveCell000023002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023002130) h)
+        (by
+          have h : ((childHL thetaAboveCell000023002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023002130) h)
+        (by
+          have h : ((childHH thetaAboveCell000023002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023002130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023002131
+        (by
+          have h : ((childLL thetaAboveCell000023002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023002131) h)
+        (by
+          have h : ((childLH thetaAboveCell000023002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023002131) h)
+        (by
+          have h : ((childHL thetaAboveCell000023002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023002131) h)
+        (by
+          have h : ((childHH thetaAboveCell000023002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023002131) h))
+    (by
+      have h : (thetaAboveCell000023002132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023002132 h)
+    (by
+      have h : (thetaAboveCell000023002133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023002133 h)
+
+theorem e24KC2ThetaAboveLeaf0000230021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00002300))
+    cover_subtree_27e55c8a3d8e
+    cover_subtree_0fc1de09da1d
+    cover_subtree_b179888799ba
+    cover_subtree_dfa114599465
+theorem e24KC2ThetaAboveLeaf0000230022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00002300))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00002300))) h)
+theorem e24KC2ThetaAboveLeaf0000230023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00002300))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00002300))) h)
+theorem cover_subtree_889f5da0dfde :
+    adaptiveCoverCheck 7 thetaAboveCell000023003000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003000
+    (by
+      have h : ((childLL thetaAboveCell000023003000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003000) h)
+    (by
+      have h : ((childLH thetaAboveCell000023003000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003000) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003000)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003000)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003000)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003000)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003000)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003000)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003000)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003000)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003000)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003000)) h))
+
+theorem cover_subtree_a942590dff68 :
+    adaptiveCoverCheck 7 thetaAboveCell000023003001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003001
+    (by
+      have h : ((childLL thetaAboveCell000023003001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003001) h)
+    (by
+      have h : ((childLH thetaAboveCell000023003001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003001) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003001)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003001)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003001)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003001)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003001)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003001)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003001)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003001)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003001)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003001)) h))
+
+theorem cover_subtree_34efc5e5d537 :
+    adaptiveCoverCheck 7 thetaAboveCell000023003002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003002
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023003002)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023003002)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023003002)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023003002)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023003002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023003002)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023003002)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023003002)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023003002)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023003002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003002)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003002)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003002)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003002)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003002)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003002)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003002)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003002)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003002)) h))
+
+theorem cover_subtree_808bd571537c :
+    adaptiveCoverCheck 7 thetaAboveCell000023003003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003003
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023003003)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023003003)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023003003)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023003003)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023003003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023003003)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023003003)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023003003)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023003003)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023003003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003003)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003003)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003003)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003003)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003003)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003003)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003003)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003003)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003003)) h))
+
+theorem cover_subtree_d6772732d30e :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00002300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00002300)))
+    cover_subtree_889f5da0dfde
+    cover_subtree_a942590dff68
+    cover_subtree_34efc5e5d537
+    cover_subtree_808bd571537c
+
+theorem cover_subtree_32b068109428 :
+    adaptiveCoverCheck 7 thetaAboveCell000023003010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003010
+    (by
+      have h : ((childLL thetaAboveCell000023003010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003010) h)
+    (by
+      have h : ((childLH thetaAboveCell000023003010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003010) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003010)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003010)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003010)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003010)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003010)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003010)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003010)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003010)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003010)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003010)) h))
+
+theorem cover_subtree_a0bb05b8d428 :
+    adaptiveCoverCheck 7 thetaAboveCell000023003011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003011
+    (by
+      have h : ((childLL thetaAboveCell000023003011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003011) h)
+    (by
+      have h : ((childLH thetaAboveCell000023003011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003011) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003011)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003011)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003011)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003011)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003011)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003011)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003011)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003011)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003011)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003011)) h))
+
+theorem cover_subtree_70a4687b5e1a :
+    adaptiveCoverCheck 7 thetaAboveCell000023003012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003012
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023003012)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023003012)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023003012)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023003012)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023003012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023003012)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023003012)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023003012)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023003012)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023003012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003012)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003012)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003012)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003012)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003012)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003012)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003012)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003012)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003012)) h))
+
+theorem cover_subtree_4ff3df89e6e6 :
+    adaptiveCoverCheck 7 thetaAboveCell000023003013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003013
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023003013)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023003013)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023003013)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023003013)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023003013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023003013)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023003013)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023003013)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023003013)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023003013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003013)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003013)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003013)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003013)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003013)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003013)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003013)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003013)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003013)) h))
+
+theorem cover_subtree_a0bd78863117 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00002300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00002300)))
+    cover_subtree_32b068109428
+    cover_subtree_a0bb05b8d428
+    cover_subtree_70a4687b5e1a
+    cover_subtree_4ff3df89e6e6
+
+theorem cover_subtree_f223ac82735f :
+    adaptiveCoverCheck 8 (childHL (childLL (childHH thetaAboveCell00002300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH thetaAboveCell00002300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003020
+        (by
+          have h : ((childLL thetaAboveCell000023003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003020) h)
+        (by
+          have h : ((childLH thetaAboveCell000023003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003020) h)
+        (by
+          have h : ((childHL thetaAboveCell000023003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023003020) h)
+        (by
+          have h : ((childHH thetaAboveCell000023003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023003020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003021
+        (by
+          have h : ((childLL thetaAboveCell000023003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003021) h)
+        (by
+          have h : ((childLH thetaAboveCell000023003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003021) h)
+        (by
+          have h : ((childHL thetaAboveCell000023003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023003021) h)
+        (by
+          have h : ((childHH thetaAboveCell000023003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023003021) h))
+    (by
+      have h : (thetaAboveCell000023003022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023003022 h)
+    (by
+      have h : (thetaAboveCell000023003023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023003023 h)
+
+theorem cover_subtree_e57b220fb2b2 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHH thetaAboveCell00002300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH thetaAboveCell00002300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003030
+        (by
+          have h : ((childLL thetaAboveCell000023003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003030) h)
+        (by
+          have h : ((childLH thetaAboveCell000023003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003030) h)
+        (by
+          have h : ((childHL thetaAboveCell000023003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023003030) h)
+        (by
+          have h : ((childHH thetaAboveCell000023003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023003030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003031
+        (by
+          have h : ((childLL thetaAboveCell000023003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003031) h)
+        (by
+          have h : ((childLH thetaAboveCell000023003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003031) h)
+        (by
+          have h : ((childHL thetaAboveCell000023003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023003031) h)
+        (by
+          have h : ((childHH thetaAboveCell000023003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023003031) h))
+    (by
+      have h : (thetaAboveCell000023003032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023003032 h)
+    (by
+      have h : (thetaAboveCell000023003033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023003033 h)
+
+theorem e24KC2ThetaAboveLeaf0000230030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00002300))
+    cover_subtree_d6772732d30e
+    cover_subtree_a0bd78863117
+    cover_subtree_f223ac82735f
+    cover_subtree_e57b220fb2b2
+theorem cover_subtree_6632af055f6f :
+    adaptiveCoverCheck 7 thetaAboveCell000023003100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003100
+    (by
+      have h : ((childLL thetaAboveCell000023003100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003100) h)
+    (by
+      have h : ((childLH thetaAboveCell000023003100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003100) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003100)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003100)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003100)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003100)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003100)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003100)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003100)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003100)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003100)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003100)) h))
+
+theorem cover_subtree_0ed63a4c90ba :
+    adaptiveCoverCheck 7 thetaAboveCell000023003101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003101
+    (by
+      have h : ((childLL thetaAboveCell000023003101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003101) h)
+    (by
+      have h : ((childLH thetaAboveCell000023003101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003101) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003101)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003101)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003101)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003101)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003101)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003101)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003101)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003101)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003101)) h))
+
+theorem cover_subtree_540084271e05 :
+    adaptiveCoverCheck 7 thetaAboveCell000023003102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003102
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023003102)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023003102)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023003102)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023003102)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023003102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023003102)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023003102)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023003102)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023003102)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023003102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003102)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003102)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003102)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003102)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003102)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003102)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003102)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003102)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003102)) h))
+
+theorem cover_subtree_17438f0398bf :
+    adaptiveCoverCheck 7 thetaAboveCell000023003103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003103
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023003103)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023003103)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023003103)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023003103)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023003103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023003103)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023003103)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023003103)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023003103)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023003103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003103)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003103)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003103)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003103)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003103)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003103)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003103)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003103)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003103)) h))
+
+theorem cover_subtree_694b687a682b :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00002300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00002300)))
+    cover_subtree_6632af055f6f
+    cover_subtree_0ed63a4c90ba
+    cover_subtree_540084271e05
+    cover_subtree_17438f0398bf
+
+theorem cover_subtree_58d260377aaf :
+    adaptiveCoverCheck 7 thetaAboveCell000023003110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003110
+    (by
+      have h : ((childLL thetaAboveCell000023003110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003110) h)
+    (by
+      have h : ((childLH thetaAboveCell000023003110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003110) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003110)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003110)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003110)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003110)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003110)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003110)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003110)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003110)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003110)) h))
+
+theorem cover_subtree_914c54a898af :
+    adaptiveCoverCheck 7 thetaAboveCell000023003111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003111
+    (by
+      have h : ((childLL thetaAboveCell000023003111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003111) h)
+    (by
+      have h : ((childLH thetaAboveCell000023003111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003111) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003111)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003111)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003111)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003111)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003111)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003111)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003111)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003111)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003111)) h))
+
+theorem cover_subtree_338e908ef3ba :
+    adaptiveCoverCheck 7 thetaAboveCell000023003112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003112
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023003112)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023003112)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023003112)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023003112)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023003112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023003112)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023003112)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023003112)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023003112)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023003112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003112)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003112)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003112)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003112)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003112)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003112)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003112)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003112)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003112)) h))
+
+theorem cover_subtree_81566ac30c71 :
+    adaptiveCoverCheck 7 thetaAboveCell000023003113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003113
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023003113)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023003113)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023003113)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023003113)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023003113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023003113)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023003113)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023003113)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023003113)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023003113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023003113)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023003113)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023003113)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023003113)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023003113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023003113)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023003113)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023003113)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023003113)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023003113)) h))
+
+theorem cover_subtree_d79efead209d :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00002300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00002300)))
+    cover_subtree_58d260377aaf
+    cover_subtree_914c54a898af
+    cover_subtree_338e908ef3ba
+    cover_subtree_81566ac30c71
+
+theorem cover_subtree_9f7611e92013 :
+    adaptiveCoverCheck 8 (childHL (childLH (childHH thetaAboveCell00002300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH thetaAboveCell00002300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003120
+        (by
+          have h : ((childLL thetaAboveCell000023003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003120) h)
+        (by
+          have h : ((childLH thetaAboveCell000023003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003120) h)
+        (by
+          have h : ((childHL thetaAboveCell000023003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023003120) h)
+        (by
+          have h : ((childHH thetaAboveCell000023003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023003120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003121
+        (by
+          have h : ((childLL thetaAboveCell000023003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003121) h)
+        (by
+          have h : ((childLH thetaAboveCell000023003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003121) h)
+        (by
+          have h : ((childHL thetaAboveCell000023003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023003121) h)
+        (by
+          have h : ((childHH thetaAboveCell000023003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023003121) h))
+    (by
+      have h : (thetaAboveCell000023003122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023003122 h)
+    (by
+      have h : (thetaAboveCell000023003123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023003123 h)
+
+theorem cover_subtree_5a6e4c62f563 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHH thetaAboveCell00002300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH thetaAboveCell00002300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003130
+        (by
+          have h : ((childLL thetaAboveCell000023003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003130) h)
+        (by
+          have h : ((childLH thetaAboveCell000023003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003130) h)
+        (by
+          have h : ((childHL thetaAboveCell000023003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023003130) h)
+        (by
+          have h : ((childHH thetaAboveCell000023003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023003130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023003131
+        (by
+          have h : ((childLL thetaAboveCell000023003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023003131) h)
+        (by
+          have h : ((childLH thetaAboveCell000023003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023003131) h)
+        (by
+          have h : ((childHL thetaAboveCell000023003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023003131) h)
+        (by
+          have h : ((childHH thetaAboveCell000023003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023003131) h))
+    (by
+      have h : (thetaAboveCell000023003132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023003132 h)
+    (by
+      have h : (thetaAboveCell000023003133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023003133 h)
+
+theorem e24KC2ThetaAboveLeaf0000230031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00002300))
+    cover_subtree_694b687a682b
+    cover_subtree_d79efead209d
+    cover_subtree_9f7611e92013
+    cover_subtree_5a6e4c62f563
+theorem e24KC2ThetaAboveLeaf0000230032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00002300))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00002300))) h)
+theorem e24KC2ThetaAboveLeaf0000230033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00002300))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00002300))) h)
+theorem e24KC2ThetaAboveLeaf0000230102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00002301))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00002301))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLL
+        thetaAboveCell00002301)))
+        (by
+          have h : (thetaAboveCell000023010220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010220 h)
+        (by
+          have h : (thetaAboveCell000023010221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010221 h)
+        (by
+          have h : (thetaAboveCell000023010222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010222 h)
+        (by
+          have h : (thetaAboveCell000023010223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childLL
+        thetaAboveCell00002301)))
+        (by
+          have h : (thetaAboveCell000023010230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010230 h)
+        (by
+          have h : (thetaAboveCell000023010231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010231 h)
+        (by
+          have h : (thetaAboveCell000023010232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010232 h)
+        (by
+          have h : (thetaAboveCell000023010233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010233 h))
+theorem e24KC2ThetaAboveLeaf0000230103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00002301))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00002301))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childLL
+        thetaAboveCell00002301)))
+        (by
+          have h : (thetaAboveCell000023010320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010320 h)
+        (by
+          have h : (thetaAboveCell000023010321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010321 h)
+        (by
+          have h : (thetaAboveCell000023010322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010322 h)
+        (by
+          have h : (thetaAboveCell000023010323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childLL
+        thetaAboveCell00002301)))
+        (by
+          have h : (thetaAboveCell000023010330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010330 h)
+        (by
+          have h : (thetaAboveCell000023010331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010331 h)
+        (by
+          have h : (thetaAboveCell000023010332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010332 h)
+        (by
+          have h : (thetaAboveCell000023010333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023010333 h))
+theorem e24KC2ThetaAboveLeaf0000230112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00002301))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00002301))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLH
+        thetaAboveCell00002301)))
+        (by
+          have h : (thetaAboveCell000023011220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011220 h)
+        (by
+          have h : (thetaAboveCell000023011221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011221 h)
+        (by
+          have h : (thetaAboveCell000023011222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011222 h)
+        (by
+          have h : (thetaAboveCell000023011223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childLH
+        thetaAboveCell00002301)))
+        (by
+          have h : (thetaAboveCell000023011230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011230 h)
+        (by
+          have h : (thetaAboveCell000023011231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011231 h)
+        (by
+          have h : (thetaAboveCell000023011232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011232 h)
+        (by
+          have h : (thetaAboveCell000023011233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011233 h))
+theorem e24KC2ThetaAboveLeaf0000230113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00002301))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00002301))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childLH
+        thetaAboveCell00002301)))
+        (by
+          have h : (thetaAboveCell000023011320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011320 h)
+        (by
+          have h : (thetaAboveCell000023011321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011321 h)
+        (by
+          have h : (thetaAboveCell000023011322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011322 h)
+        (by
+          have h : (thetaAboveCell000023011323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childLH
+        thetaAboveCell00002301)))
+        (by
+          have h : (thetaAboveCell000023011330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011330 h)
+        (by
+          have h : (thetaAboveCell000023011331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011331 h)
+        (by
+          have h : (thetaAboveCell000023011332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011332 h)
+        (by
+          have h : (thetaAboveCell000023011333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023011333 h))
+theorem cover_subtree_97a6e5a0f9c0 :
+    adaptiveCoverCheck 7 thetaAboveCell000023012000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012000
+    (by
+      have h : ((childLL thetaAboveCell000023012000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012000) h)
+    (by
+      have h : ((childLH thetaAboveCell000023012000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012000) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012000)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012000)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012000)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012000)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012000)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012000)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012000)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012000)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012000)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012000)) h))
+
+theorem cover_subtree_b0cb05399587 :
+    adaptiveCoverCheck 7 thetaAboveCell000023012001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012001
+    (by
+      have h : ((childLL thetaAboveCell000023012001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012001) h)
+    (by
+      have h : ((childLH thetaAboveCell000023012001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012001) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012001)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012001)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012001)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012001)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012001)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012001)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012001)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012001)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012001)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012001)) h))
+
+theorem cover_subtree_ec160a01c291 :
+    adaptiveCoverCheck 7 thetaAboveCell000023012002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012002
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023012002)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023012002)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023012002)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023012002)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023012002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023012002)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023012002)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023012002)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023012002)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023012002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012002)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012002)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012002)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012002)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012002)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012002)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012002)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012002)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012002)) h))
+
+theorem cover_subtree_01b50c333c69 :
+    adaptiveCoverCheck 7 thetaAboveCell000023012003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012003
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023012003)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023012003)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023012003)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023012003)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023012003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023012003)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023012003)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023012003)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023012003)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023012003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012003)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012003)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012003)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012003)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012003)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012003)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012003)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012003)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012003)) h))
+
+theorem cover_subtree_672879a40a99 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00002301)))
+    cover_subtree_97a6e5a0f9c0
+    cover_subtree_b0cb05399587
+    cover_subtree_ec160a01c291
+    cover_subtree_01b50c333c69
+
+theorem cover_subtree_d4f66ebe8c34 :
+    adaptiveCoverCheck 7 thetaAboveCell000023012010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012010
+    (by
+      have h : ((childLL thetaAboveCell000023012010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012010) h)
+    (by
+      have h : ((childLH thetaAboveCell000023012010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012010) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012010)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012010)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012010)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012010)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012010)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012010)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012010)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012010)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012010)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012010)) h))
+
+theorem cover_subtree_2daeb5f0aede :
+    adaptiveCoverCheck 7 thetaAboveCell000023012011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012011
+    (by
+      have h : ((childLL thetaAboveCell000023012011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012011) h)
+    (by
+      have h : ((childLH thetaAboveCell000023012011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012011) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012011)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012011)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012011)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012011)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012011)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012011)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012011)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012011)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012011)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012011)) h))
+
+theorem cover_subtree_0a19cf9c168e :
+    adaptiveCoverCheck 7 thetaAboveCell000023012012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012012
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023012012)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023012012)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023012012)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023012012)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023012012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023012012)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023012012)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023012012)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023012012)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023012012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012012)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012012)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012012)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012012)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012012)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012012)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012012)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012012)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012012)) h))
+
+theorem cover_subtree_5397c7c8c770 :
+    adaptiveCoverCheck 7 thetaAboveCell000023012013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012013
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023012013)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023012013)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023012013)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023012013)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023012013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023012013)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023012013)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023012013)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023012013)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023012013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012013)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012013)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012013)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012013)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012013)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012013)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012013)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012013)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012013)) h))
+
+theorem cover_subtree_857a1aa148c2 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00002301)))
+    cover_subtree_d4f66ebe8c34
+    cover_subtree_2daeb5f0aede
+    cover_subtree_0a19cf9c168e
+    cover_subtree_5397c7c8c770
+
+theorem cover_subtree_334f666b102f :
+    adaptiveCoverCheck 8 (childHL (childLL (childHL thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL thetaAboveCell00002301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012020
+        (by
+          have h : ((childLL thetaAboveCell000023012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012020) h)
+        (by
+          have h : ((childLH thetaAboveCell000023012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012020) h)
+        (by
+          have h : ((childHL thetaAboveCell000023012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023012020) h)
+        (by
+          have h : ((childHH thetaAboveCell000023012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023012020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012021
+        (by
+          have h : ((childLL thetaAboveCell000023012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012021) h)
+        (by
+          have h : ((childLH thetaAboveCell000023012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012021) h)
+        (by
+          have h : ((childHL thetaAboveCell000023012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023012021) h)
+        (by
+          have h : ((childHH thetaAboveCell000023012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023012021) h))
+    (by
+      have h : (thetaAboveCell000023012022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023012022 h)
+    (by
+      have h : (thetaAboveCell000023012023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023012023 h)
+
+theorem cover_subtree_f871df99dda8 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHL thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL thetaAboveCell00002301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012030
+        (by
+          have h : ((childLL thetaAboveCell000023012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012030) h)
+        (by
+          have h : ((childLH thetaAboveCell000023012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012030) h)
+        (by
+          have h : ((childHL thetaAboveCell000023012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023012030) h)
+        (by
+          have h : ((childHH thetaAboveCell000023012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023012030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012031
+        (by
+          have h : ((childLL thetaAboveCell000023012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012031) h)
+        (by
+          have h : ((childLH thetaAboveCell000023012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012031) h)
+        (by
+          have h : ((childHL thetaAboveCell000023012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023012031) h)
+        (by
+          have h : ((childHH thetaAboveCell000023012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023012031) h))
+    (by
+      have h : (thetaAboveCell000023012032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023012032 h)
+    (by
+      have h : (thetaAboveCell000023012033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023012033 h)
+
+theorem e24KC2ThetaAboveLeaf0000230120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00002301))
+    cover_subtree_672879a40a99
+    cover_subtree_857a1aa148c2
+    cover_subtree_334f666b102f
+    cover_subtree_f871df99dda8
+theorem cover_subtree_82ae04dee4d4 :
+    adaptiveCoverCheck 7 thetaAboveCell000023012100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012100
+    (by
+      have h : ((childLL thetaAboveCell000023012100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012100) h)
+    (by
+      have h : ((childLH thetaAboveCell000023012100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012100) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012100)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012100)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012100)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012100)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012100)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012100)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012100)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012100)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012100)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012100)) h))
+
+theorem cover_subtree_e5b18c272c0b :
+    adaptiveCoverCheck 7 thetaAboveCell000023012101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012101
+    (by
+      have h : ((childLL thetaAboveCell000023012101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012101) h)
+    (by
+      have h : ((childLH thetaAboveCell000023012101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012101) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012101)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012101)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012101)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012101)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012101)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012101)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012101)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012101)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012101)) h))
+
+theorem cover_subtree_9c247030adc1 :
+    adaptiveCoverCheck 7 thetaAboveCell000023012102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012102
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023012102)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023012102)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023012102)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023012102)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023012102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023012102)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023012102)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023012102)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023012102)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023012102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012102)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012102)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012102)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012102)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012102)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012102)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012102)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012102)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012102)) h))
+
+theorem cover_subtree_058bb465c064 :
+    adaptiveCoverCheck 7 thetaAboveCell000023012103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012103
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023012103)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023012103)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023012103)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023012103)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023012103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023012103)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023012103)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023012103)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023012103)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023012103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012103)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012103)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012103)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012103)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012103)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012103)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012103)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012103)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012103)) h))
+
+theorem cover_subtree_9278d082fba6 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00002301)))
+    cover_subtree_82ae04dee4d4
+    cover_subtree_e5b18c272c0b
+    cover_subtree_9c247030adc1
+    cover_subtree_058bb465c064
+
+theorem cover_subtree_8e71329f2820 :
+    adaptiveCoverCheck 7 thetaAboveCell000023012110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012110
+    (by
+      have h : ((childLL thetaAboveCell000023012110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012110) h)
+    (by
+      have h : ((childLH thetaAboveCell000023012110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012110) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012110)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012110)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012110)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012110)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012110)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012110)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012110)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012110)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012110)) h))
+
+theorem cover_subtree_31c6f84c8564 :
+    adaptiveCoverCheck 7 thetaAboveCell000023012111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012111
+    (by
+      have h : ((childLL thetaAboveCell000023012111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012111) h)
+    (by
+      have h : ((childLH thetaAboveCell000023012111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012111) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012111)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012111)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012111)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012111)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012111)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012111)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012111)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012111)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012111)) h))
+
+theorem cover_subtree_c08a134eb452 :
+    adaptiveCoverCheck 7 thetaAboveCell000023012112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012112
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023012112)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023012112)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023012112)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023012112)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023012112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023012112)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023012112)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023012112)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023012112)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023012112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012112)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012112)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012112)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012112)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012112)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012112)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012112)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012112)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012112)) h))
+
+theorem cover_subtree_90cbd7801a03 :
+    adaptiveCoverCheck 7 thetaAboveCell000023012113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012113
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023012113)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023012113)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023012113)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023012113)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023012113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023012113)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023012113)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023012113)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023012113)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023012113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023012113)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023012113)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023012113)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023012113)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023012113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023012113)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023012113)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023012113)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023012113)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023012113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023012113)) h))
+
+theorem cover_subtree_f310f207f337 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00002301)))
+    cover_subtree_8e71329f2820
+    cover_subtree_31c6f84c8564
+    cover_subtree_c08a134eb452
+    cover_subtree_90cbd7801a03
+
+theorem cover_subtree_81898220a27c :
+    adaptiveCoverCheck 8 (childHL (childLH (childHL thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL thetaAboveCell00002301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012120
+        (by
+          have h : ((childLL thetaAboveCell000023012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012120) h)
+        (by
+          have h : ((childLH thetaAboveCell000023012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012120) h)
+        (by
+          have h : ((childHL thetaAboveCell000023012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023012120) h)
+        (by
+          have h : ((childHH thetaAboveCell000023012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023012120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012121
+        (by
+          have h : ((childLL thetaAboveCell000023012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012121) h)
+        (by
+          have h : ((childLH thetaAboveCell000023012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012121) h)
+        (by
+          have h : ((childHL thetaAboveCell000023012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023012121) h)
+        (by
+          have h : ((childHH thetaAboveCell000023012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023012121) h))
+    (by
+      have h : (thetaAboveCell000023012122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023012122 h)
+    (by
+      have h : (thetaAboveCell000023012123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023012123 h)
+
+theorem cover_subtree_d001d1c55e17 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHL thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL thetaAboveCell00002301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012130
+        (by
+          have h : ((childLL thetaAboveCell000023012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012130) h)
+        (by
+          have h : ((childLH thetaAboveCell000023012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012130) h)
+        (by
+          have h : ((childHL thetaAboveCell000023012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023012130) h)
+        (by
+          have h : ((childHH thetaAboveCell000023012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023012130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023012131
+        (by
+          have h : ((childLL thetaAboveCell000023012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023012131) h)
+        (by
+          have h : ((childLH thetaAboveCell000023012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023012131) h)
+        (by
+          have h : ((childHL thetaAboveCell000023012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023012131) h)
+        (by
+          have h : ((childHH thetaAboveCell000023012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023012131) h))
+    (by
+      have h : (thetaAboveCell000023012132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023012132 h)
+    (by
+      have h : (thetaAboveCell000023012133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023012133 h)
+
+theorem e24KC2ThetaAboveLeaf0000230121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00002301))
+    cover_subtree_9278d082fba6
+    cover_subtree_f310f207f337
+    cover_subtree_81898220a27c
+    cover_subtree_d001d1c55e17
+theorem e24KC2ThetaAboveLeaf0000230122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00002301))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00002301))) h)
+theorem e24KC2ThetaAboveLeaf0000230123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00002301))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00002301))) h)
+theorem cover_subtree_1956171ef0f4 :
+    adaptiveCoverCheck 7 thetaAboveCell000023013000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013000
+    (by
+      have h : ((childLL thetaAboveCell000023013000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013000) h)
+    (by
+      have h : ((childLH thetaAboveCell000023013000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013000) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013000)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013000)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013000)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013000)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013000)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013000)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013000)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013000)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013000)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013000)) h))
+
+theorem cover_subtree_d3d741223709 :
+    adaptiveCoverCheck 7 thetaAboveCell000023013001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013001
+    (by
+      have h : ((childLL thetaAboveCell000023013001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013001) h)
+    (by
+      have h : ((childLH thetaAboveCell000023013001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013001) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013001)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013001)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013001)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013001)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013001)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013001)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013001)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013001)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013001)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013001)) h))
+
+theorem cover_subtree_c4ce6eecc651 :
+    adaptiveCoverCheck 7 thetaAboveCell000023013002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013002
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023013002)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023013002)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023013002)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023013002)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023013002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023013002)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023013002)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023013002)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023013002)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023013002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013002)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013002)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013002)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013002)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013002)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013002)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013002)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013002)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013002)) h))
+
+theorem cover_subtree_764c189c1428 :
+    adaptiveCoverCheck 7 thetaAboveCell000023013003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013003
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023013003)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023013003)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023013003)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023013003)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023013003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023013003)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023013003)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023013003)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023013003)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023013003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013003)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013003)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013003)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013003)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013003)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013003)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013003)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013003)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013003)) h))
+
+theorem cover_subtree_d1b1d3400177 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00002301)))
+    cover_subtree_1956171ef0f4
+    cover_subtree_d3d741223709
+    cover_subtree_c4ce6eecc651
+    cover_subtree_764c189c1428
+
+theorem cover_subtree_7d1a2c16b724 :
+    adaptiveCoverCheck 7 thetaAboveCell000023013010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013010
+    (by
+      have h : ((childLL thetaAboveCell000023013010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013010) h)
+    (by
+      have h : ((childLH thetaAboveCell000023013010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013010) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013010)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013010)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013010)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013010)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013010)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013010)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013010)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013010)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013010)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013010)) h))
+
+theorem cover_subtree_7a802d18a6f2 :
+    adaptiveCoverCheck 7 thetaAboveCell000023013011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013011
+    (by
+      have h : ((childLL thetaAboveCell000023013011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013011) h)
+    (by
+      have h : ((childLH thetaAboveCell000023013011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013011) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013011)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013011)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013011)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013011)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013011)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013011)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013011)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013011)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013011)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013011)) h))
+
+theorem cover_subtree_7e8d1889716b :
+    adaptiveCoverCheck 7 thetaAboveCell000023013012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013012
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023013012)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023013012)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023013012)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023013012)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023013012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023013012)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023013012)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023013012)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023013012)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023013012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013012)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013012)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013012)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013012)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013012)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013012)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013012)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013012)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013012)) h))
+
+theorem cover_subtree_6291f06ec5a0 :
+    adaptiveCoverCheck 7 thetaAboveCell000023013013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013013
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023013013)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023013013)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023013013)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023013013)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023013013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023013013)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023013013)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023013013)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023013013)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023013013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013013)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013013)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013013)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013013)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013013)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013013)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013013)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013013)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013013)) h))
+
+theorem cover_subtree_60b54bf255b1 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00002301)))
+    cover_subtree_7d1a2c16b724
+    cover_subtree_7a802d18a6f2
+    cover_subtree_7e8d1889716b
+    cover_subtree_6291f06ec5a0
+
+theorem cover_subtree_1e3ac8710eca :
+    adaptiveCoverCheck 8 (childHL (childLL (childHH thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH thetaAboveCell00002301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013020
+        (by
+          have h : ((childLL thetaAboveCell000023013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013020) h)
+        (by
+          have h : ((childLH thetaAboveCell000023013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013020) h)
+        (by
+          have h : ((childHL thetaAboveCell000023013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023013020) h)
+        (by
+          have h : ((childHH thetaAboveCell000023013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023013020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013021
+        (by
+          have h : ((childLL thetaAboveCell000023013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013021) h)
+        (by
+          have h : ((childLH thetaAboveCell000023013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013021) h)
+        (by
+          have h : ((childHL thetaAboveCell000023013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023013021) h)
+        (by
+          have h : ((childHH thetaAboveCell000023013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023013021) h))
+    (by
+      have h : (thetaAboveCell000023013022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023013022 h)
+    (by
+      have h : (thetaAboveCell000023013023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023013023 h)
+
+theorem cover_subtree_7c913110e3c1 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHH thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH thetaAboveCell00002301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013030
+        (by
+          have h : ((childLL thetaAboveCell000023013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013030) h)
+        (by
+          have h : ((childLH thetaAboveCell000023013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013030) h)
+        (by
+          have h : ((childHL thetaAboveCell000023013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023013030) h)
+        (by
+          have h : ((childHH thetaAboveCell000023013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023013030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013031
+        (by
+          have h : ((childLL thetaAboveCell000023013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013031) h)
+        (by
+          have h : ((childLH thetaAboveCell000023013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013031) h)
+        (by
+          have h : ((childHL thetaAboveCell000023013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023013031) h)
+        (by
+          have h : ((childHH thetaAboveCell000023013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023013031) h))
+    (by
+      have h : (thetaAboveCell000023013032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023013032 h)
+    (by
+      have h : (thetaAboveCell000023013033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023013033 h)
+
+theorem e24KC2ThetaAboveLeaf0000230130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00002301))
+    cover_subtree_d1b1d3400177
+    cover_subtree_60b54bf255b1
+    cover_subtree_1e3ac8710eca
+    cover_subtree_7c913110e3c1
+theorem cover_subtree_45e055a15e46 :
+    adaptiveCoverCheck 7 thetaAboveCell000023013100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013100
+    (by
+      have h : ((childLL thetaAboveCell000023013100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013100) h)
+    (by
+      have h : ((childLH thetaAboveCell000023013100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013100) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013100)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013100)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013100)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013100)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013100)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013100)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013100)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013100)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013100)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013100)) h))
+
+theorem cover_subtree_027a6b743c2f :
+    adaptiveCoverCheck 7 thetaAboveCell000023013101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013101
+    (by
+      have h : ((childLL thetaAboveCell000023013101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013101) h)
+    (by
+      have h : ((childLH thetaAboveCell000023013101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013101) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013101)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013101)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013101)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013101)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013101)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013101)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013101)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013101)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013101)) h))
+
+theorem cover_subtree_2814d469dde6 :
+    adaptiveCoverCheck 7 thetaAboveCell000023013102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013102
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023013102)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023013102)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023013102)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023013102)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023013102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023013102)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023013102)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023013102)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023013102)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023013102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013102)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013102)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013102)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013102)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013102)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013102)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013102)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013102)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013102)) h))
+
+theorem cover_subtree_f25d3ae916f4 :
+    adaptiveCoverCheck 7 thetaAboveCell000023013103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013103
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023013103)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023013103)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023013103)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023013103)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023013103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023013103)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023013103)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023013103)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023013103)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023013103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013103)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013103)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013103)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013103)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013103)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013103)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013103)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013103)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013103)) h))
+
+theorem cover_subtree_8f3c6cde6196 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00002301)))
+    cover_subtree_45e055a15e46
+    cover_subtree_027a6b743c2f
+    cover_subtree_2814d469dde6
+    cover_subtree_f25d3ae916f4
+
+theorem cover_subtree_6d98447fe707 :
+    adaptiveCoverCheck 7 thetaAboveCell000023013110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013110
+    (by
+      have h : ((childLL thetaAboveCell000023013110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013110) h)
+    (by
+      have h : ((childLH thetaAboveCell000023013110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013110) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013110)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013110)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013110)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013110)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013110)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013110)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013110)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013110)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013110)) h))
+
+theorem cover_subtree_d1177bcdcb0d :
+    adaptiveCoverCheck 7 thetaAboveCell000023013111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013111
+    (by
+      have h : ((childLL thetaAboveCell000023013111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013111) h)
+    (by
+      have h : ((childLH thetaAboveCell000023013111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013111) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013111)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013111)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013111)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013111)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013111)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013111)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013111)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013111)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013111)) h))
+
+theorem cover_subtree_6f6899221514 :
+    adaptiveCoverCheck 7 thetaAboveCell000023013112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013112
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023013112)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023013112)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023013112)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023013112)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023013112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023013112)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023013112)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023013112)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023013112)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023013112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013112)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013112)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013112)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013112)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013112)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013112)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013112)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013112)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013112)) h))
+
+theorem cover_subtree_b4799a6c1d22 :
+    adaptiveCoverCheck 7 thetaAboveCell000023013113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013113
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000023013113)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000023013113)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000023013113)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000023013113)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000023013113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000023013113)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000023013113)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000023013113)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000023013113)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000023013113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000023013113)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000023013113)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000023013113)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000023013113)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000023013113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000023013113)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000023013113)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000023013113)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000023013113)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000023013113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000023013113)) h))
+
+theorem cover_subtree_adb76ba4b661 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00002301)))
+    cover_subtree_6d98447fe707
+    cover_subtree_d1177bcdcb0d
+    cover_subtree_6f6899221514
+    cover_subtree_b4799a6c1d22
+
+theorem cover_subtree_8b60b15919f9 :
+    adaptiveCoverCheck 8 (childHL (childLH (childHH thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH thetaAboveCell00002301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013120
+        (by
+          have h : ((childLL thetaAboveCell000023013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013120) h)
+        (by
+          have h : ((childLH thetaAboveCell000023013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013120) h)
+        (by
+          have h : ((childHL thetaAboveCell000023013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023013120) h)
+        (by
+          have h : ((childHH thetaAboveCell000023013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023013120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013121
+        (by
+          have h : ((childLL thetaAboveCell000023013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013121) h)
+        (by
+          have h : ((childLH thetaAboveCell000023013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013121) h)
+        (by
+          have h : ((childHL thetaAboveCell000023013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023013121) h)
+        (by
+          have h : ((childHH thetaAboveCell000023013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023013121) h))
+    (by
+      have h : (thetaAboveCell000023013122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023013122 h)
+    (by
+      have h : (thetaAboveCell000023013123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023013123 h)
+
+theorem cover_subtree_9d36c543bc7e :
+    adaptiveCoverCheck 8 (childHH (childLH (childHH thetaAboveCell00002301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH thetaAboveCell00002301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013130
+        (by
+          have h : ((childLL thetaAboveCell000023013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013130) h)
+        (by
+          have h : ((childLH thetaAboveCell000023013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013130) h)
+        (by
+          have h : ((childHL thetaAboveCell000023013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023013130) h)
+        (by
+          have h : ((childHH thetaAboveCell000023013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023013130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000023013131
+        (by
+          have h : ((childLL thetaAboveCell000023013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000023013131) h)
+        (by
+          have h : ((childLH thetaAboveCell000023013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000023013131) h)
+        (by
+          have h : ((childHL thetaAboveCell000023013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000023013131) h)
+        (by
+          have h : ((childHH thetaAboveCell000023013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000023013131) h))
+    (by
+      have h : (thetaAboveCell000023013132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023013132 h)
+    (by
+      have h : (thetaAboveCell000023013133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023013133 h)
+
+theorem e24KC2ThetaAboveLeaf0000230131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00002301))
+    cover_subtree_8f3c6cde6196
+    cover_subtree_adb76ba4b661
+    cover_subtree_8b60b15919f9
+    cover_subtree_9d36c543bc7e
+theorem e24KC2ThetaAboveLeaf0000230132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00002301))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00002301))) h)
+theorem e24KC2ThetaAboveLeaf0000230133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00002301))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00002301))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00002301))) h)
+theorem e24KC2ThetaAboveLeaf0000231002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00002310))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00002310))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLL
+        thetaAboveCell00002310)))
+        (by
+          have h : (thetaAboveCell000023100220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100220 h)
+        (by
+          have h : (thetaAboveCell000023100221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100221 h)
+        (by
+          have h : (thetaAboveCell000023100222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100222 h)
+        (by
+          have h : (thetaAboveCell000023100223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childLL
+        thetaAboveCell00002310)))
+        (by
+          have h : (thetaAboveCell000023100230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100230 h)
+        (by
+          have h : (thetaAboveCell000023100231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100231 h)
+        (by
+          have h : (thetaAboveCell000023100232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100232 h)
+        (by
+          have h : (thetaAboveCell000023100233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100233 h))
+theorem e24KC2ThetaAboveLeaf0000231003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00002310))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00002310))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childLL
+        thetaAboveCell00002310)))
+        (by
+          have h : (thetaAboveCell000023100320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100320 h)
+        (by
+          have h : (thetaAboveCell000023100321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100321 h)
+        (by
+          have h : (thetaAboveCell000023100322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100322 h)
+        (by
+          have h : (thetaAboveCell000023100323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childLL
+        thetaAboveCell00002310)))
+        (by
+          have h : (thetaAboveCell000023100330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100330 h)
+        (by
+          have h : (thetaAboveCell000023100331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100331 h)
+        (by
+          have h : (thetaAboveCell000023100332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100332 h)
+        (by
+          have h : (thetaAboveCell000023100333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023100333 h))
+theorem e24KC2ThetaAboveLeaf0000231012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00002310))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00002310))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLH
+        thetaAboveCell00002310)))
+        (by
+          have h : (thetaAboveCell000023101220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101220 h)
+        (by
+          have h : (thetaAboveCell000023101221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101221 h)
+        (by
+          have h : (thetaAboveCell000023101222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101222 h)
+        (by
+          have h : (thetaAboveCell000023101223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childLH
+        thetaAboveCell00002310)))
+        (by
+          have h : (thetaAboveCell000023101230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101230 h)
+        (by
+          have h : (thetaAboveCell000023101231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101231 h)
+        (by
+          have h : (thetaAboveCell000023101232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101232 h)
+        (by
+          have h : (thetaAboveCell000023101233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101233 h))
+theorem e24KC2ThetaAboveLeaf0000231013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00002310))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00002310))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00002310))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childLH
+        thetaAboveCell00002310)))
+        (by
+          have h : (thetaAboveCell000023101320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101320 h)
+        (by
+          have h : (thetaAboveCell000023101321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101321 h)
+        (by
+          have h : (thetaAboveCell000023101322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101322 h)
+        (by
+          have h : (thetaAboveCell000023101323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childLH
+        thetaAboveCell00002310)))
+        (by
+          have h : (thetaAboveCell000023101330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101330 h)
+        (by
+          have h : (thetaAboveCell000023101331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101331 h)
+        (by
+          have h : (thetaAboveCell000023101332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101332 h)
+        (by
+          have h : (thetaAboveCell000023101333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023101333 h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle007.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle007.lean
new file mode 100644
index 0000000000..f330744d18
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle007.lean
@@ -0,0 +1,9988 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch008`.
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch026`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCells2cfecae988
+
+/-- Subcell `0100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0100 : AngleCell :=
+  childLL (childLL (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0101 : AngleCell :=
+  childLH (childLL (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `01003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0101)))
+
+end GerverSofa.PartE.CertificateCells2cfecae988
+
+namespace GerverSofa.PartE.CertificateCells6df5a7a03f
+
+/-- Subcell `0101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0101 : AngleCell :=
+  childLH (childLL (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0102 : AngleCell :=
+  childHL (childLL (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0103 : AngleCell :=
+  childHH (childLL (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0110 : AngleCell :=
+  childLL (childLH (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `01013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0110)))
+
+end GerverSofa.PartE.CertificateCells6df5a7a03f
+
+namespace GerverSofa.PartE.CertificateCells8ecd2c98f8
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133110222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110200 : AngleCell :=
+  childLL (childLL (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110201 : AngleCell :=
+  childLH (childLL (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110202 : AngleCell :=
+  childHL (childLL (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110203 : AngleCell :=
+  childHH (childLL (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110210 : AngleCell :=
+  childLL (childLH (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110211 : AngleCell :=
+  childLH (childLH (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110212 : AngleCell :=
+  childHL (childLH (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110213 : AngleCell :=
+  childHH (childLH (childHL (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110300 : AngleCell :=
+  childLL (childLL (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110301 : AngleCell :=
+  childLH (childLL (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110302 : AngleCell :=
+  childHL (childLL (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110303 : AngleCell :=
+  childHH (childLL (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110310 : AngleCell :=
+  childLL (childLH (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110311 : AngleCell :=
+  childLH (childLH (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110312 : AngleCell :=
+  childHL (childLH (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133110313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133110313 : AngleCell :=
+  childHH (childLH (childHH (childLL thetaBelowCell11113311)))
+
+/-- Subcell `111133111222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111200 : AngleCell :=
+  childLL (childLL (childHL (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111201 : AngleCell :=
+  childLH (childLL (childHL (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111202 : AngleCell :=
+  childHL (childLL (childHL (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111203 : AngleCell :=
+  childHH (childLL (childHL (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133111331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133111331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaBelowCell11113311)))
+
+/-- Subcell `111133112000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaBelowCell11113311)))
+
+/-- Subcell `111133112133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCells8ecd2c98f8
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC5TerminalBatchT204800007`.
+* `KernelOnly.PartE.E24KC5TerminalBatchT256000008`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2cfecae988
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2cfecae988
+
+open CertificateCells2cfecae988
+theorem e24KC2ThetaAboveLeaf0100330300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01003303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf0100330301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01003303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf0100330302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01003303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf0100330303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01003303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf0100330310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01003303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf0100330311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01003303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf0100330312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01003303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf0100330313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01003303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf0100330320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003303)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf0100330321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003303)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf0100330322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003303)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf0100330323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003303)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf0100330330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003303)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf0100330331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003303)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf0100330332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003303)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf0100330333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003303)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01003303)) h
+theorem e24KC2ThetaAboveLeaf010033100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003310) = true := by
+  have h : ((childLL thetaAboveCell01003310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01003310) h
+theorem e24KC2ThetaAboveLeaf010033101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003310) = true := by
+  have h : ((childLH thetaAboveCell01003310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01003310) h
+theorem e24KC2ThetaAboveLeaf0100331020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003310)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003310)) h
+theorem e24KC2ThetaAboveLeaf0100331021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003310)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003310)) h
+theorem e24KC2ThetaAboveLeaf0100331022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003310)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01003310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01003310)) h
+theorem e24KC2ThetaAboveLeaf0100331023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003310)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01003310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01003310)) h
+theorem e24KC2ThetaAboveLeaf0100331030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003310)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003310)) h
+theorem e24KC2ThetaAboveLeaf0100331031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003310)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003310)) h
+theorem e24KC2ThetaAboveLeaf0100331032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003310)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01003310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01003310)) h
+theorem e24KC2ThetaAboveLeaf0100331033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003310)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01003310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01003310)) h
+theorem e24KC2ThetaAboveLeaf010033110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003311) = true := by
+  have h : ((childLL thetaAboveCell01003311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01003311) h
+theorem e24KC2ThetaAboveLeaf010033111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003311) = true := by
+  have h : ((childLH thetaAboveCell01003311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01003311) h
+theorem e24KC2ThetaAboveLeaf0100331120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003311)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003311)) h
+theorem e24KC2ThetaAboveLeaf0100331121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003311)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003311)) h
+theorem e24KC2ThetaAboveLeaf0100331122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003311)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01003311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01003311)) h
+theorem e24KC2ThetaAboveLeaf0100331123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003311)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01003311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01003311)) h
+theorem e24KC2ThetaAboveLeaf0100331130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003311)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003311)) h
+theorem e24KC2ThetaAboveLeaf0100331131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003311)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003311)) h
+theorem e24KC2ThetaAboveLeaf0100331132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003311)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01003311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01003311)) h
+theorem e24KC2ThetaAboveLeaf0100331133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003311)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01003311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01003311)) h
+theorem e24KC2ThetaAboveLeaf0100331200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01003312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01003312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01003312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01003312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01003312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01003312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01003312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01003312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003312)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003312)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003312)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003312)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003312)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003312)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003312)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003312)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01003312)) h
+theorem e24KC2ThetaAboveLeaf0100331300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01003313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf0100331301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01003313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf0100331302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01003313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf0100331303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01003313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf0100331310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01003313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf0100331311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01003313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf0100331312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01003313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf0100331313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01003313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf0100331320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01003313)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf0100331321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01003313)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf0100331322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003313)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf0100331323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003313)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf0100331330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01003313)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf0100331331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01003313)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf0100331332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003313)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf0100331333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003313)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01003313)) h
+theorem e24KC2ThetaAboveLeaf01003320 :
+    adaptiveCoverCheck 11 thetaAboveCell01003320 = true := by
+  have h : (thetaAboveCell01003320).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003320 h
+theorem e24KC2ThetaAboveLeaf01003321 :
+    adaptiveCoverCheck 11 thetaAboveCell01003321 = true := by
+  have h : (thetaAboveCell01003321).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003321 h
+theorem e24KC2ThetaAboveLeaf01003322 :
+    adaptiveCoverCheck 11 thetaAboveCell01003322 = true := by
+  have h : (thetaAboveCell01003322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003322 h
+theorem e24KC2ThetaAboveLeaf01003323 :
+    adaptiveCoverCheck 11 thetaAboveCell01003323 = true := by
+  have h : (thetaAboveCell01003323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003323 h
+theorem e24KC2ThetaAboveLeaf010033300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003330) = true := by
+  have h : ((childLL thetaAboveCell01003330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01003330) h
+theorem e24KC2ThetaAboveLeaf010033301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003330) = true := by
+  have h : ((childLH thetaAboveCell01003330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01003330) h
+theorem e24KC2ThetaAboveLeaf010033302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003330) = true := by
+  have h : ((childHL thetaAboveCell01003330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01003330) h
+theorem e24KC2ThetaAboveLeaf010033303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003330) = true := by
+  have h : ((childHH thetaAboveCell01003330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01003330) h
+theorem e24KC2ThetaAboveLeaf010033310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003331) = true := by
+  have h : ((childLL thetaAboveCell01003331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01003331) h
+theorem e24KC2ThetaAboveLeaf010033311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003331) = true := by
+  have h : ((childLH thetaAboveCell01003331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01003331) h
+theorem e24KC2ThetaAboveLeaf010033312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003331) = true := by
+  have h : ((childHL thetaAboveCell01003331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01003331) h
+theorem e24KC2ThetaAboveLeaf010033313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003331) = true := by
+  have h : ((childHH thetaAboveCell01003331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01003331) h
+theorem e24KC2ThetaAboveLeaf01003332 :
+    adaptiveCoverCheck 11 thetaAboveCell01003332 = true := by
+  have h : (thetaAboveCell01003332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003332 h
+theorem e24KC2ThetaAboveLeaf01003333 :
+    adaptiveCoverCheck 11 thetaAboveCell01003333 = true := by
+  have h : (thetaAboveCell01003333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01003333 h
+theorem e24KC2ThetaAboveLeaf010100 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0101)) = true := by
+  have h : ((childLL (childLL thetaAboveCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell0101)) h
+theorem e24KC2ThetaAboveLeaf010101 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0101)) = true := by
+  have h : ((childLH (childLL thetaAboveCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell0101)) h
+theorem e24KC2ThetaAboveLeaf010102 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0101)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0101)) h
+theorem e24KC2ThetaAboveLeaf010103 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0101)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0101)) h
+theorem e24KC2ThetaAboveLeaf010110 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0101)) = true := by
+  have h : ((childLL (childLH thetaAboveCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell0101)) h
+theorem e24KC2ThetaAboveLeaf010111 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0101)) = true := by
+  have h : ((childLH (childLH thetaAboveCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell0101)) h
+theorem e24KC2ThetaAboveLeaf010112 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0101)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0101)) h
+theorem e24KC2ThetaAboveLeaf010113 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0101)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0101)) h
+theorem e24KC2ThetaAboveLeaf0101200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell0101))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell0101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell0101))) h
+theorem e24KC2ThetaAboveLeaf0101201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell0101))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell0101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell0101))) h
+theorem e24KC2ThetaAboveLeaf01012020 :
+    adaptiveCoverCheck 11 thetaAboveCell01012020 = true := by
+  have h : (thetaAboveCell01012020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012020 h
+theorem e24KC2ThetaAboveLeaf01012021 :
+    adaptiveCoverCheck 11 thetaAboveCell01012021 = true := by
+  have h : (thetaAboveCell01012021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012021 h
+theorem e24KC2ThetaAboveLeaf01012022 :
+    adaptiveCoverCheck 11 thetaAboveCell01012022 = true := by
+  have h : (thetaAboveCell01012022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012022 h
+theorem e24KC2ThetaAboveLeaf01012023 :
+    adaptiveCoverCheck 11 thetaAboveCell01012023 = true := by
+  have h : (thetaAboveCell01012023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012023 h
+theorem e24KC2ThetaAboveLeaf01012030 :
+    adaptiveCoverCheck 11 thetaAboveCell01012030 = true := by
+  have h : (thetaAboveCell01012030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012030 h
+theorem e24KC2ThetaAboveLeaf01012031 :
+    adaptiveCoverCheck 11 thetaAboveCell01012031 = true := by
+  have h : (thetaAboveCell01012031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012031 h
+theorem e24KC2ThetaAboveLeaf01012032 :
+    adaptiveCoverCheck 11 thetaAboveCell01012032 = true := by
+  have h : (thetaAboveCell01012032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012032 h
+theorem e24KC2ThetaAboveLeaf01012033 :
+    adaptiveCoverCheck 11 thetaAboveCell01012033 = true := by
+  have h : (thetaAboveCell01012033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012033 h
+theorem e24KC2ThetaAboveLeaf0101210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell0101))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell0101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell0101))) h
+theorem e24KC2ThetaAboveLeaf0101211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell0101))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell0101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell0101))) h
+theorem e24KC2ThetaAboveLeaf01012120 :
+    adaptiveCoverCheck 11 thetaAboveCell01012120 = true := by
+  have h : (thetaAboveCell01012120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012120 h
+theorem e24KC2ThetaAboveLeaf01012121 :
+    adaptiveCoverCheck 11 thetaAboveCell01012121 = true := by
+  have h : (thetaAboveCell01012121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012121 h
+theorem e24KC2ThetaAboveLeaf01012122 :
+    adaptiveCoverCheck 11 thetaAboveCell01012122 = true := by
+  have h : (thetaAboveCell01012122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012122 h
+theorem e24KC2ThetaAboveLeaf01012123 :
+    adaptiveCoverCheck 11 thetaAboveCell01012123 = true := by
+  have h : (thetaAboveCell01012123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012123 h
+theorem e24KC2ThetaAboveLeaf01012130 :
+    adaptiveCoverCheck 11 thetaAboveCell01012130 = true := by
+  have h : (thetaAboveCell01012130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012130 h
+theorem e24KC2ThetaAboveLeaf01012131 :
+    adaptiveCoverCheck 11 thetaAboveCell01012131 = true := by
+  have h : (thetaAboveCell01012131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012131 h
+theorem e24KC2ThetaAboveLeaf01012132 :
+    adaptiveCoverCheck 11 thetaAboveCell01012132 = true := by
+  have h : (thetaAboveCell01012132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012132 h
+theorem e24KC2ThetaAboveLeaf01012133 :
+    adaptiveCoverCheck 11 thetaAboveCell01012133 = true := by
+  have h : (thetaAboveCell01012133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012133 h
+theorem e24KC2ThetaAboveLeaf010122000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012200) = true := by
+  have h : ((childLL thetaAboveCell01012200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012200) h
+theorem e24KC2ThetaAboveLeaf010122001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012200) = true := by
+  have h : ((childLH thetaAboveCell01012200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012200) h
+theorem e24KC2ThetaAboveLeaf0101220020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012200)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012200)) h
+theorem e24KC2ThetaAboveLeaf0101220021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012200)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012200)) h
+theorem e24KC2ThetaAboveLeaf0101220022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012200)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012200)) h
+theorem e24KC2ThetaAboveLeaf0101220023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012200)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012200)) h
+theorem e24KC2ThetaAboveLeaf0101220030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012200)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012200)) h
+theorem e24KC2ThetaAboveLeaf0101220031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012200)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012200)) h
+theorem e24KC2ThetaAboveLeaf0101220032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012200)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012200)) h
+theorem e24KC2ThetaAboveLeaf0101220033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012200)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012200)) h
+theorem e24KC2ThetaAboveLeaf010122010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012201) = true := by
+  have h : ((childLL thetaAboveCell01012201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012201) h
+theorem e24KC2ThetaAboveLeaf010122011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012201) = true := by
+  have h : ((childLH thetaAboveCell01012201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012201) h
+theorem e24KC2ThetaAboveLeaf0101220120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012201)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012201)) h
+theorem e24KC2ThetaAboveLeaf0101220121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012201)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012201)) h
+theorem e24KC2ThetaAboveLeaf0101220122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012201)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012201)) h
+theorem e24KC2ThetaAboveLeaf0101220123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012201)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012201)) h
+theorem e24KC2ThetaAboveLeaf0101220130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012201)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012201)) h
+theorem e24KC2ThetaAboveLeaf0101220131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012201)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012201)) h
+theorem e24KC2ThetaAboveLeaf0101220132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012201)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012201)) h
+theorem e24KC2ThetaAboveLeaf0101220133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012201)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012201)) h
+theorem e24KC2ThetaAboveLeaf0101220200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01012202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01012202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01012202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01012202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01012202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01012202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01012202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01012202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012202)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012202)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012202)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012202)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012202)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012202)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012202)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012202)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012202)) h
+theorem e24KC2ThetaAboveLeaf0101220300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01012203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf0101220301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01012203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf0101220302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01012203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf0101220303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01012203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf0101220310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01012203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf0101220311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01012203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf0101220312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01012203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf0101220313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01012203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf0101220320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012203)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf0101220321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012203)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf0101220322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012203)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf0101220323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012203)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf0101220330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012203)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf0101220331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012203)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf0101220332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012203)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf0101220333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012203)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012203)) h
+theorem e24KC2ThetaAboveLeaf010122100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012210) = true := by
+  have h : ((childLL thetaAboveCell01012210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012210) h
+theorem e24KC2ThetaAboveLeaf010122101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012210) = true := by
+  have h : ((childLH thetaAboveCell01012210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012210) h
+theorem e24KC2ThetaAboveLeaf0101221020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012210)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012210)) h
+theorem e24KC2ThetaAboveLeaf0101221021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012210)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012210)) h
+theorem e24KC2ThetaAboveLeaf0101221022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012210)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012210)) h
+theorem e24KC2ThetaAboveLeaf0101221023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012210)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012210)) h
+theorem e24KC2ThetaAboveLeaf0101221030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012210)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012210)) h
+theorem e24KC2ThetaAboveLeaf0101221031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012210)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012210)) h
+theorem e24KC2ThetaAboveLeaf0101221032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012210)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012210)) h
+theorem e24KC2ThetaAboveLeaf0101221033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012210)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012210)) h
+theorem e24KC2ThetaAboveLeaf010122110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012211) = true := by
+  have h : ((childLL thetaAboveCell01012211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012211) h
+theorem e24KC2ThetaAboveLeaf010122111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012211) = true := by
+  have h : ((childLH thetaAboveCell01012211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012211) h
+theorem e24KC2ThetaAboveLeaf0101221120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012211)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012211)) h
+theorem e24KC2ThetaAboveLeaf0101221121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012211)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012211)) h
+theorem e24KC2ThetaAboveLeaf0101221122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012211)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012211)) h
+theorem e24KC2ThetaAboveLeaf0101221123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012211)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012211)) h
+theorem e24KC2ThetaAboveLeaf0101221130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012211)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012211)) h
+theorem e24KC2ThetaAboveLeaf0101221131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012211)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012211)) h
+theorem e24KC2ThetaAboveLeaf0101221132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012211)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012211)) h
+theorem e24KC2ThetaAboveLeaf0101221133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012211)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012211)) h
+theorem e24KC2ThetaAboveLeaf0101221200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01012212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01012212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01012212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01012212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01012212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01012212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01012212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01012212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012212)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012212)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012212)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012212)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012212)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012212)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012212)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012212)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012212)) h
+theorem e24KC2ThetaAboveLeaf0101221300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01012213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf0101221301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01012213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf0101221302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01012213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf0101221303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01012213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf0101221310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01012213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf0101221311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01012213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf0101221312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01012213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf0101221313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01012213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf0101221320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012213)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf0101221321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012213)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf0101221322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012213)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf0101221323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012213)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf0101221330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012213)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf0101221331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012213)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf0101221332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012213)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf0101221333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012213)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012213)) h
+theorem e24KC2ThetaAboveLeaf010122200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012220) = true := by
+  have h : ((childLL thetaAboveCell01012220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012220) h
+theorem e24KC2ThetaAboveLeaf010122201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012220) = true := by
+  have h : ((childLH thetaAboveCell01012220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012220) h
+theorem e24KC2ThetaAboveLeaf010122202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012220) = true := by
+  have h : ((childHL thetaAboveCell01012220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01012220) h
+theorem e24KC2ThetaAboveLeaf010122203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012220) = true := by
+  have h : ((childHH thetaAboveCell01012220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01012220) h
+theorem e24KC2ThetaAboveLeaf010122210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012221) = true := by
+  have h : ((childLL thetaAboveCell01012221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012221) h
+theorem e24KC2ThetaAboveLeaf010122211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012221) = true := by
+  have h : ((childLH thetaAboveCell01012221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012221) h
+theorem e24KC2ThetaAboveLeaf010122212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012221) = true := by
+  have h : ((childHL thetaAboveCell01012221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01012221) h
+theorem e24KC2ThetaAboveLeaf010122213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012221) = true := by
+  have h : ((childHH thetaAboveCell01012221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01012221) h
+theorem e24KC2ThetaAboveLeaf01012222 :
+    adaptiveCoverCheck 11 thetaAboveCell01012222 = true := by
+  have h : (thetaAboveCell01012222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012222 h
+theorem e24KC2ThetaAboveLeaf01012223 :
+    adaptiveCoverCheck 11 thetaAboveCell01012223 = true := by
+  have h : (thetaAboveCell01012223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012223 h
+theorem e24KC2ThetaAboveLeaf010122300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012230) = true := by
+  have h : ((childLL thetaAboveCell01012230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012230) h
+theorem e24KC2ThetaAboveLeaf010122301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012230) = true := by
+  have h : ((childLH thetaAboveCell01012230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012230) h
+theorem e24KC2ThetaAboveLeaf010122302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012230) = true := by
+  have h : ((childHL thetaAboveCell01012230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01012230) h
+theorem e24KC2ThetaAboveLeaf010122303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012230) = true := by
+  have h : ((childHH thetaAboveCell01012230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01012230) h
+theorem e24KC2ThetaAboveLeaf010122310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012231) = true := by
+  have h : ((childLL thetaAboveCell01012231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012231) h
+theorem e24KC2ThetaAboveLeaf010122311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012231) = true := by
+  have h : ((childLH thetaAboveCell01012231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012231) h
+theorem e24KC2ThetaAboveLeaf010122312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012231) = true := by
+  have h : ((childHL thetaAboveCell01012231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01012231) h
+theorem e24KC2ThetaAboveLeaf010122313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012231) = true := by
+  have h : ((childHH thetaAboveCell01012231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01012231) h
+theorem e24KC2ThetaAboveLeaf01012232 :
+    adaptiveCoverCheck 11 thetaAboveCell01012232 = true := by
+  have h : (thetaAboveCell01012232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012232 h
+theorem e24KC2ThetaAboveLeaf01012233 :
+    adaptiveCoverCheck 11 thetaAboveCell01012233 = true := by
+  have h : (thetaAboveCell01012233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012233 h
+theorem e24KC2ThetaAboveLeaf010123000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012300) = true := by
+  have h : ((childLL thetaAboveCell01012300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012300) h
+theorem e24KC2ThetaAboveLeaf010123001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012300) = true := by
+  have h : ((childLH thetaAboveCell01012300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012300) h
+theorem e24KC2ThetaAboveLeaf0101230020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012300)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012300)) h
+theorem e24KC2ThetaAboveLeaf0101230021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012300)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012300)) h
+theorem e24KC2ThetaAboveLeaf0101230022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012300)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012300)) h
+theorem e24KC2ThetaAboveLeaf0101230023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012300)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012300)) h
+theorem e24KC2ThetaAboveLeaf0101230030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012300)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012300)) h
+theorem e24KC2ThetaAboveLeaf0101230031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012300)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012300)) h
+theorem e24KC2ThetaAboveLeaf0101230032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012300)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012300)) h
+theorem e24KC2ThetaAboveLeaf0101230033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012300)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012300)) h
+theorem e24KC2ThetaAboveLeaf010123010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012301) = true := by
+  have h : ((childLL thetaAboveCell01012301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012301) h
+theorem e24KC2ThetaAboveLeaf010123011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012301) = true := by
+  have h : ((childLH thetaAboveCell01012301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012301) h
+theorem e24KC2ThetaAboveLeaf0101230120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012301)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012301)) h
+theorem e24KC2ThetaAboveLeaf0101230121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012301)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012301)) h
+theorem e24KC2ThetaAboveLeaf0101230122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012301)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012301)) h
+theorem e24KC2ThetaAboveLeaf0101230123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012301)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012301)) h
+theorem e24KC2ThetaAboveLeaf0101230130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012301)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012301)) h
+theorem e24KC2ThetaAboveLeaf0101230131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012301)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012301)) h
+theorem e24KC2ThetaAboveLeaf0101230132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012301)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012301)) h
+theorem e24KC2ThetaAboveLeaf0101230133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012301)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012301)) h
+theorem e24KC2ThetaAboveLeaf0101230200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01012302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01012302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01012302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01012302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01012302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01012302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01012302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01012302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012302)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012302)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012302)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012302)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012302)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012302)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012302)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012302)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012302)) h
+theorem e24KC2ThetaAboveLeaf0101230300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01012303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf0101230301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01012303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf0101230302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01012303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf0101230303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01012303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf0101230310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01012303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf0101230311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01012303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf0101230312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01012303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf0101230313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01012303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf0101230320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012303)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf0101230321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012303)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf0101230322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012303)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf0101230323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012303)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf0101230330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012303)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf0101230331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012303)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf0101230332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012303)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf0101230333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012303)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012303)) h
+theorem e24KC2ThetaAboveLeaf010123100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012310) = true := by
+  have h : ((childLL thetaAboveCell01012310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012310) h
+theorem e24KC2ThetaAboveLeaf010123101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012310) = true := by
+  have h : ((childLH thetaAboveCell01012310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012310) h
+theorem e24KC2ThetaAboveLeaf0101231020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012310)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012310)) h
+theorem e24KC2ThetaAboveLeaf0101231021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012310)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012310)) h
+theorem e24KC2ThetaAboveLeaf0101231022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012310)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012310)) h
+theorem e24KC2ThetaAboveLeaf0101231023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012310)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012310)) h
+theorem e24KC2ThetaAboveLeaf0101231030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012310)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012310)) h
+theorem e24KC2ThetaAboveLeaf0101231031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012310)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012310)) h
+theorem e24KC2ThetaAboveLeaf0101231032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012310)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012310)) h
+theorem e24KC2ThetaAboveLeaf0101231033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012310)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012310)) h
+theorem e24KC2ThetaAboveLeaf010123110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012311) = true := by
+  have h : ((childLL thetaAboveCell01012311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012311) h
+theorem e24KC2ThetaAboveLeaf010123111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012311) = true := by
+  have h : ((childLH thetaAboveCell01012311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012311) h
+theorem e24KC2ThetaAboveLeaf0101231120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012311)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012311)) h
+theorem e24KC2ThetaAboveLeaf0101231121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012311)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012311)) h
+theorem e24KC2ThetaAboveLeaf0101231122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012311)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012311)) h
+theorem e24KC2ThetaAboveLeaf0101231123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012311)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012311)) h
+theorem e24KC2ThetaAboveLeaf0101231130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012311)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012311)) h
+theorem e24KC2ThetaAboveLeaf0101231131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012311)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012311)) h
+theorem e24KC2ThetaAboveLeaf0101231132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012311)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012311)) h
+theorem e24KC2ThetaAboveLeaf0101231133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012311)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012311)) h
+theorem e24KC2ThetaAboveLeaf0101231200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01012312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01012312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01012312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01012312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01012312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01012312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01012312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01012312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012312)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012312)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012312)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012312)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012312)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012312)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012312)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012312)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012312)) h
+theorem e24KC2ThetaAboveLeaf0101231300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01012313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf0101231301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01012313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf0101231302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01012313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf0101231303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01012313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf0101231310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01012313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf0101231311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01012313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf0101231312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01012313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf0101231313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01012313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf0101231320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01012313)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf0101231321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01012313)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf0101231322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01012313)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf0101231323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01012313)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf0101231330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01012313)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf0101231331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01012313)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf0101231332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01012313)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf0101231333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01012313)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01012313)) h
+theorem e24KC2ThetaAboveLeaf010123200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012320) = true := by
+  have h : ((childLL thetaAboveCell01012320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012320) h
+theorem e24KC2ThetaAboveLeaf010123201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012320) = true := by
+  have h : ((childLH thetaAboveCell01012320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012320) h
+theorem e24KC2ThetaAboveLeaf010123202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012320) = true := by
+  have h : ((childHL thetaAboveCell01012320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01012320) h
+theorem e24KC2ThetaAboveLeaf010123203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012320) = true := by
+  have h : ((childHH thetaAboveCell01012320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01012320) h
+theorem e24KC2ThetaAboveLeaf010123210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012321) = true := by
+  have h : ((childLL thetaAboveCell01012321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012321) h
+theorem e24KC2ThetaAboveLeaf010123211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012321) = true := by
+  have h : ((childLH thetaAboveCell01012321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012321) h
+theorem e24KC2ThetaAboveLeaf010123212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012321) = true := by
+  have h : ((childHL thetaAboveCell01012321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01012321) h
+theorem e24KC2ThetaAboveLeaf010123213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012321) = true := by
+  have h : ((childHH thetaAboveCell01012321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01012321) h
+theorem e24KC2ThetaAboveLeaf01012322 :
+    adaptiveCoverCheck 11 thetaAboveCell01012322 = true := by
+  have h : (thetaAboveCell01012322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012322 h
+theorem e24KC2ThetaAboveLeaf01012323 :
+    adaptiveCoverCheck 11 thetaAboveCell01012323 = true := by
+  have h : (thetaAboveCell01012323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012323 h
+theorem e24KC2ThetaAboveLeaf010123300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012330) = true := by
+  have h : ((childLL thetaAboveCell01012330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012330) h
+theorem e24KC2ThetaAboveLeaf010123301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012330) = true := by
+  have h : ((childLH thetaAboveCell01012330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012330) h
+theorem e24KC2ThetaAboveLeaf010123302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012330) = true := by
+  have h : ((childHL thetaAboveCell01012330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01012330) h
+theorem e24KC2ThetaAboveLeaf010123303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012330) = true := by
+  have h : ((childHH thetaAboveCell01012330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01012330) h
+theorem e24KC2ThetaAboveLeaf010123310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012331) = true := by
+  have h : ((childLL thetaAboveCell01012331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01012331) h
+theorem e24KC2ThetaAboveLeaf010123311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012331) = true := by
+  have h : ((childLH thetaAboveCell01012331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01012331) h
+theorem e24KC2ThetaAboveLeaf010123312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012331) = true := by
+  have h : ((childHL thetaAboveCell01012331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01012331) h
+theorem e24KC2ThetaAboveLeaf010123313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012331) = true := by
+  have h : ((childHH thetaAboveCell01012331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01012331) h
+theorem e24KC2ThetaAboveLeaf01012332 :
+    adaptiveCoverCheck 11 thetaAboveCell01012332 = true := by
+  have h : (thetaAboveCell01012332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012332 h
+theorem e24KC2ThetaAboveLeaf01012333 :
+    adaptiveCoverCheck 11 thetaAboveCell01012333 = true := by
+  have h : (thetaAboveCell01012333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01012333 h
+theorem e24KC2ThetaAboveLeaf0101300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell0101))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell0101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell0101))) h
+theorem e24KC2ThetaAboveLeaf0101301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell0101))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell0101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell0101))) h
+theorem e24KC2ThetaAboveLeaf01013020 :
+    adaptiveCoverCheck 11 thetaAboveCell01013020 = true := by
+  have h : (thetaAboveCell01013020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013020 h
+theorem e24KC2ThetaAboveLeaf01013021 :
+    adaptiveCoverCheck 11 thetaAboveCell01013021 = true := by
+  have h : (thetaAboveCell01013021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013021 h
+theorem e24KC2ThetaAboveLeaf01013022 :
+    adaptiveCoverCheck 11 thetaAboveCell01013022 = true := by
+  have h : (thetaAboveCell01013022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013022 h
+theorem e24KC2ThetaAboveLeaf01013023 :
+    adaptiveCoverCheck 11 thetaAboveCell01013023 = true := by
+  have h : (thetaAboveCell01013023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013023 h
+theorem e24KC2ThetaAboveLeaf01013030 :
+    adaptiveCoverCheck 11 thetaAboveCell01013030 = true := by
+  have h : (thetaAboveCell01013030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013030 h
+theorem e24KC2ThetaAboveLeaf01013031 :
+    adaptiveCoverCheck 11 thetaAboveCell01013031 = true := by
+  have h : (thetaAboveCell01013031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013031 h
+theorem e24KC2ThetaAboveLeaf01013032 :
+    adaptiveCoverCheck 11 thetaAboveCell01013032 = true := by
+  have h : (thetaAboveCell01013032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013032 h
+theorem e24KC2ThetaAboveLeaf01013033 :
+    adaptiveCoverCheck 11 thetaAboveCell01013033 = true := by
+  have h : (thetaAboveCell01013033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013033 h
+theorem e24KC2ThetaAboveLeaf0101310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell0101))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell0101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell0101))) h
+theorem e24KC2ThetaAboveLeaf0101311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell0101))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell0101)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell0101))) h
+theorem e24KC2ThetaAboveLeaf01013120 :
+    adaptiveCoverCheck 11 thetaAboveCell01013120 = true := by
+  have h : (thetaAboveCell01013120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013120 h
+theorem e24KC2ThetaAboveLeaf01013121 :
+    adaptiveCoverCheck 11 thetaAboveCell01013121 = true := by
+  have h : (thetaAboveCell01013121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013121 h
+theorem e24KC2ThetaAboveLeaf01013122 :
+    adaptiveCoverCheck 11 thetaAboveCell01013122 = true := by
+  have h : (thetaAboveCell01013122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013122 h
+theorem e24KC2ThetaAboveLeaf01013123 :
+    adaptiveCoverCheck 11 thetaAboveCell01013123 = true := by
+  have h : (thetaAboveCell01013123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013123 h
+theorem e24KC2ThetaAboveLeaf01013130 :
+    adaptiveCoverCheck 11 thetaAboveCell01013130 = true := by
+  have h : (thetaAboveCell01013130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013130 h
+theorem e24KC2ThetaAboveLeaf01013131 :
+    adaptiveCoverCheck 11 thetaAboveCell01013131 = true := by
+  have h : (thetaAboveCell01013131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013131 h
+theorem e24KC2ThetaAboveLeaf01013132 :
+    adaptiveCoverCheck 11 thetaAboveCell01013132 = true := by
+  have h : (thetaAboveCell01013132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013132 h
+theorem e24KC2ThetaAboveLeaf01013133 :
+    adaptiveCoverCheck 11 thetaAboveCell01013133 = true := by
+  have h : (thetaAboveCell01013133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013133 h
+theorem e24KC2ThetaAboveLeaf010132000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013200) = true := by
+  have h : ((childLL thetaAboveCell01013200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013200) h
+theorem e24KC2ThetaAboveLeaf010132001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013200) = true := by
+  have h : ((childLH thetaAboveCell01013200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013200) h
+theorem e24KC2ThetaAboveLeaf0101320020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013200)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013200)) h
+theorem e24KC2ThetaAboveLeaf0101320021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013200)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013200)) h
+theorem e24KC2ThetaAboveLeaf0101320022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013200)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013200)) h
+theorem e24KC2ThetaAboveLeaf0101320023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013200)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013200)) h
+theorem e24KC2ThetaAboveLeaf0101320030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013200)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013200)) h
+theorem e24KC2ThetaAboveLeaf0101320031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013200)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013200)) h
+theorem e24KC2ThetaAboveLeaf0101320032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013200)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013200)) h
+theorem e24KC2ThetaAboveLeaf0101320033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013200)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013200)) h
+theorem e24KC2ThetaAboveLeaf010132010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013201) = true := by
+  have h : ((childLL thetaAboveCell01013201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013201) h
+theorem e24KC2ThetaAboveLeaf010132011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013201) = true := by
+  have h : ((childLH thetaAboveCell01013201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013201) h
+theorem e24KC2ThetaAboveLeaf0101320120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013201)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013201)) h
+theorem e24KC2ThetaAboveLeaf0101320121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013201)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013201)) h
+theorem e24KC2ThetaAboveLeaf0101320122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013201)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013201)) h
+theorem e24KC2ThetaAboveLeaf0101320123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013201)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013201)) h
+theorem e24KC2ThetaAboveLeaf0101320130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013201)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013201)) h
+theorem e24KC2ThetaAboveLeaf0101320131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013201)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013201)) h
+theorem e24KC2ThetaAboveLeaf0101320132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013201)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013201)) h
+theorem e24KC2ThetaAboveLeaf0101320133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013201)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013201)) h
+theorem e24KC2ThetaAboveLeaf0101320200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01013202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01013202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01013202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01013202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01013202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01013202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01013202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01013202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013202)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013202)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013202)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013202)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013202)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013202)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013202)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013202)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013202)) h
+theorem e24KC2ThetaAboveLeaf0101320300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01013203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf0101320301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01013203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf0101320302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01013203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf0101320303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01013203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf0101320310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01013203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf0101320311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01013203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf0101320312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01013203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf0101320313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01013203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf0101320320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013203)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf0101320321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013203)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf0101320322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013203)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf0101320323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013203)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf0101320330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013203)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf0101320331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013203)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf0101320332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013203)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf0101320333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013203)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013203)) h
+theorem e24KC2ThetaAboveLeaf010132100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013210) = true := by
+  have h : ((childLL thetaAboveCell01013210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013210) h
+theorem e24KC2ThetaAboveLeaf010132101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013210) = true := by
+  have h : ((childLH thetaAboveCell01013210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013210) h
+theorem e24KC2ThetaAboveLeaf0101321020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013210)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013210)) h
+theorem e24KC2ThetaAboveLeaf0101321021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013210)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013210)) h
+theorem e24KC2ThetaAboveLeaf0101321022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013210)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013210)) h
+theorem e24KC2ThetaAboveLeaf0101321023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013210)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013210)) h
+theorem e24KC2ThetaAboveLeaf0101321030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013210)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013210)) h
+theorem e24KC2ThetaAboveLeaf0101321031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013210)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013210)) h
+theorem e24KC2ThetaAboveLeaf0101321032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013210)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013210)) h
+theorem e24KC2ThetaAboveLeaf0101321033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013210)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013210)) h
+theorem e24KC2ThetaAboveLeaf010132110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013211) = true := by
+  have h : ((childLL thetaAboveCell01013211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013211) h
+theorem e24KC2ThetaAboveLeaf010132111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013211) = true := by
+  have h : ((childLH thetaAboveCell01013211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013211) h
+theorem e24KC2ThetaAboveLeaf0101321120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013211)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013211)) h
+theorem e24KC2ThetaAboveLeaf0101321121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013211)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013211)) h
+theorem e24KC2ThetaAboveLeaf0101321122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013211)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013211)) h
+theorem e24KC2ThetaAboveLeaf0101321123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013211)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013211)) h
+theorem e24KC2ThetaAboveLeaf0101321130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013211)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013211)) h
+theorem e24KC2ThetaAboveLeaf0101321131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013211)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013211)) h
+theorem e24KC2ThetaAboveLeaf0101321132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013211)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013211)) h
+theorem e24KC2ThetaAboveLeaf0101321133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013211)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013211)) h
+theorem e24KC2ThetaAboveLeaf0101321200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01013212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01013212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01013212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01013212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01013212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01013212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01013212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01013212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013212)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013212)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013212)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013212)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013212)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013212)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013212)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013212)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013212)) h
+theorem e24KC2ThetaAboveLeaf0101321300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01013213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf0101321301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01013213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf0101321302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01013213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf0101321303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01013213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf0101321310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01013213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf0101321311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01013213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf0101321312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01013213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf0101321313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01013213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf0101321320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013213)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf0101321321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013213)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf0101321322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013213)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf0101321323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013213)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf0101321330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013213)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf0101321331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013213)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf0101321332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013213)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf0101321333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013213)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013213)) h
+theorem e24KC2ThetaAboveLeaf010132200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013220) = true := by
+  have h : ((childLL thetaAboveCell01013220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013220) h
+theorem e24KC2ThetaAboveLeaf010132201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013220) = true := by
+  have h : ((childLH thetaAboveCell01013220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013220) h
+theorem e24KC2ThetaAboveLeaf010132202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013220) = true := by
+  have h : ((childHL thetaAboveCell01013220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01013220) h
+theorem e24KC2ThetaAboveLeaf010132203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013220) = true := by
+  have h : ((childHH thetaAboveCell01013220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01013220) h
+theorem e24KC2ThetaAboveLeaf010132210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013221) = true := by
+  have h : ((childLL thetaAboveCell01013221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013221) h
+theorem e24KC2ThetaAboveLeaf010132211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013221) = true := by
+  have h : ((childLH thetaAboveCell01013221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013221) h
+theorem e24KC2ThetaAboveLeaf010132212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013221) = true := by
+  have h : ((childHL thetaAboveCell01013221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01013221) h
+theorem e24KC2ThetaAboveLeaf010132213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013221) = true := by
+  have h : ((childHH thetaAboveCell01013221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01013221) h
+theorem e24KC2ThetaAboveLeaf01013222 :
+    adaptiveCoverCheck 11 thetaAboveCell01013222 = true := by
+  have h : (thetaAboveCell01013222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013222 h
+theorem e24KC2ThetaAboveLeaf01013223 :
+    adaptiveCoverCheck 11 thetaAboveCell01013223 = true := by
+  have h : (thetaAboveCell01013223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013223 h
+theorem e24KC2ThetaAboveLeaf010132300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013230) = true := by
+  have h : ((childLL thetaAboveCell01013230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013230) h
+theorem e24KC2ThetaAboveLeaf010132301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013230) = true := by
+  have h : ((childLH thetaAboveCell01013230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013230) h
+theorem e24KC2ThetaAboveLeaf010132302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013230) = true := by
+  have h : ((childHL thetaAboveCell01013230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01013230) h
+theorem e24KC2ThetaAboveLeaf010132303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013230) = true := by
+  have h : ((childHH thetaAboveCell01013230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01013230) h
+theorem e24KC2ThetaAboveLeaf010132310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013231) = true := by
+  have h : ((childLL thetaAboveCell01013231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013231) h
+theorem e24KC2ThetaAboveLeaf010132311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013231) = true := by
+  have h : ((childLH thetaAboveCell01013231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013231) h
+theorem e24KC2ThetaAboveLeaf010132312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013231) = true := by
+  have h : ((childHL thetaAboveCell01013231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01013231) h
+theorem e24KC2ThetaAboveLeaf010132313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013231) = true := by
+  have h : ((childHH thetaAboveCell01013231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01013231) h
+theorem e24KC2ThetaAboveLeaf01013232 :
+    adaptiveCoverCheck 11 thetaAboveCell01013232 = true := by
+  have h : (thetaAboveCell01013232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013232 h
+theorem e24KC2ThetaAboveLeaf01013233 :
+    adaptiveCoverCheck 11 thetaAboveCell01013233 = true := by
+  have h : (thetaAboveCell01013233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013233 h
+theorem e24KC2ThetaAboveLeaf010133000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013300) = true := by
+  have h : ((childLL thetaAboveCell01013300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013300) h
+theorem e24KC2ThetaAboveLeaf010133001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013300) = true := by
+  have h : ((childLH thetaAboveCell01013300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013300) h
+theorem e24KC2ThetaAboveLeaf0101330020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013300)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013300)) h
+theorem e24KC2ThetaAboveLeaf0101330021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013300)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013300)) h
+theorem e24KC2ThetaAboveLeaf0101330022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013300)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013300)) h
+theorem e24KC2ThetaAboveLeaf0101330023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013300)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013300)) h
+theorem e24KC2ThetaAboveLeaf0101330030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013300)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013300)) h
+theorem e24KC2ThetaAboveLeaf0101330031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013300)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013300)) h
+theorem e24KC2ThetaAboveLeaf0101330032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013300)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013300)) h
+theorem e24KC2ThetaAboveLeaf0101330033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013300)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013300)) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6df5a7a03f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6df5a7a03f
+
+open CertificateCells6df5a7a03f
+theorem e24KC2ThetaAboveLeaf010133010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013301) = true := by
+  have h : ((childLL thetaAboveCell01013301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013301) h
+theorem e24KC2ThetaAboveLeaf010133011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013301) = true := by
+  have h : ((childLH thetaAboveCell01013301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013301) h
+theorem e24KC2ThetaAboveLeaf0101330120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013301)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013301)) h
+theorem e24KC2ThetaAboveLeaf0101330121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013301)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013301)) h
+theorem e24KC2ThetaAboveLeaf0101330122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013301)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013301)) h
+theorem e24KC2ThetaAboveLeaf0101330123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013301)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013301)) h
+theorem e24KC2ThetaAboveLeaf0101330130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013301)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013301)) h
+theorem e24KC2ThetaAboveLeaf0101330131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013301)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013301)) h
+theorem e24KC2ThetaAboveLeaf0101330132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013301)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013301)) h
+theorem e24KC2ThetaAboveLeaf0101330133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013301)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013301)) h
+theorem e24KC2ThetaAboveLeaf0101330200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01013302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01013302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01013302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01013302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01013302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01013302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01013302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01013302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013302)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013302)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013302)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013302)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013302)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013302)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013302)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013302)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013302)) h
+theorem e24KC2ThetaAboveLeaf0101330300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01013303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf0101330301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01013303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf0101330302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01013303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf0101330303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01013303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf0101330310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01013303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf0101330311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01013303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf0101330312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01013303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf0101330313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01013303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf0101330320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013303)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf0101330321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013303)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf0101330322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013303)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf0101330323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013303)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf0101330330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013303)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf0101330331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013303)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf0101330332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013303)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf0101330333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013303)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013303)) h
+theorem e24KC2ThetaAboveLeaf010133100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013310) = true := by
+  have h : ((childLL thetaAboveCell01013310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013310) h
+theorem e24KC2ThetaAboveLeaf010133101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013310) = true := by
+  have h : ((childLH thetaAboveCell01013310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013310) h
+theorem e24KC2ThetaAboveLeaf0101331020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013310)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013310)) h
+theorem e24KC2ThetaAboveLeaf0101331021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013310)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013310)) h
+theorem e24KC2ThetaAboveLeaf0101331022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013310)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013310)) h
+theorem e24KC2ThetaAboveLeaf0101331023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013310)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013310)) h
+theorem e24KC2ThetaAboveLeaf0101331030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013310)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013310)) h
+theorem e24KC2ThetaAboveLeaf0101331031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013310)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013310)) h
+theorem e24KC2ThetaAboveLeaf0101331032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013310)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013310)) h
+theorem e24KC2ThetaAboveLeaf0101331033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013310)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013310)) h
+theorem e24KC2ThetaAboveLeaf010133110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013311) = true := by
+  have h : ((childLL thetaAboveCell01013311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013311) h
+theorem e24KC2ThetaAboveLeaf010133111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013311) = true := by
+  have h : ((childLH thetaAboveCell01013311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013311) h
+theorem e24KC2ThetaAboveLeaf0101331120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013311)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013311)) h
+theorem e24KC2ThetaAboveLeaf0101331121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013311)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013311)) h
+theorem e24KC2ThetaAboveLeaf0101331122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013311)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013311)) h
+theorem e24KC2ThetaAboveLeaf0101331123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013311)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013311)) h
+theorem e24KC2ThetaAboveLeaf0101331130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013311)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013311)) h
+theorem e24KC2ThetaAboveLeaf0101331131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013311)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013311)) h
+theorem e24KC2ThetaAboveLeaf0101331132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013311)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013311)) h
+theorem e24KC2ThetaAboveLeaf0101331133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013311)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013311)) h
+theorem e24KC2ThetaAboveLeaf0101331200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01013312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01013312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01013312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01013312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01013312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01013312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01013312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01013312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013312)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013312)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013312)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013312)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013312)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013312)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013312)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013312)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013312)) h
+theorem e24KC2ThetaAboveLeaf0101331300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01013313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf0101331301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01013313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf0101331302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01013313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf0101331303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01013313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf0101331310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01013313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf0101331311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01013313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf0101331312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01013313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf0101331313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01013313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf0101331320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01013313)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf0101331321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01013313)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf0101331322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01013313)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf0101331323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01013313)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf0101331330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01013313)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf0101331331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01013313)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf0101331332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01013313)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf0101331333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01013313)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01013313)) h
+theorem e24KC2ThetaAboveLeaf010133200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013320) = true := by
+  have h : ((childLL thetaAboveCell01013320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013320) h
+theorem e24KC2ThetaAboveLeaf010133201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013320) = true := by
+  have h : ((childLH thetaAboveCell01013320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013320) h
+theorem e24KC2ThetaAboveLeaf010133202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013320) = true := by
+  have h : ((childHL thetaAboveCell01013320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01013320) h
+theorem e24KC2ThetaAboveLeaf010133203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013320) = true := by
+  have h : ((childHH thetaAboveCell01013320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01013320) h
+theorem e24KC2ThetaAboveLeaf010133210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013321) = true := by
+  have h : ((childLL thetaAboveCell01013321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013321) h
+theorem e24KC2ThetaAboveLeaf010133211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013321) = true := by
+  have h : ((childLH thetaAboveCell01013321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013321) h
+theorem e24KC2ThetaAboveLeaf010133212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013321) = true := by
+  have h : ((childHL thetaAboveCell01013321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01013321) h
+theorem e24KC2ThetaAboveLeaf010133213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013321) = true := by
+  have h : ((childHH thetaAboveCell01013321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01013321) h
+theorem e24KC2ThetaAboveLeaf01013322 :
+    adaptiveCoverCheck 11 thetaAboveCell01013322 = true := by
+  have h : (thetaAboveCell01013322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013322 h
+theorem e24KC2ThetaAboveLeaf01013323 :
+    adaptiveCoverCheck 11 thetaAboveCell01013323 = true := by
+  have h : (thetaAboveCell01013323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013323 h
+theorem e24KC2ThetaAboveLeaf010133300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013330) = true := by
+  have h : ((childLL thetaAboveCell01013330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013330) h
+theorem e24KC2ThetaAboveLeaf010133301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013330) = true := by
+  have h : ((childLH thetaAboveCell01013330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013330) h
+theorem e24KC2ThetaAboveLeaf010133302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013330) = true := by
+  have h : ((childHL thetaAboveCell01013330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01013330) h
+theorem e24KC2ThetaAboveLeaf010133303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013330) = true := by
+  have h : ((childHH thetaAboveCell01013330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01013330) h
+theorem e24KC2ThetaAboveLeaf010133310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013331) = true := by
+  have h : ((childLL thetaAboveCell01013331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01013331) h
+theorem e24KC2ThetaAboveLeaf010133311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013331) = true := by
+  have h : ((childLH thetaAboveCell01013331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01013331) h
+theorem e24KC2ThetaAboveLeaf010133312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013331) = true := by
+  have h : ((childHL thetaAboveCell01013331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01013331) h
+theorem e24KC2ThetaAboveLeaf010133313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013331) = true := by
+  have h : ((childHH thetaAboveCell01013331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01013331) h
+theorem e24KC2ThetaAboveLeaf01013332 :
+    adaptiveCoverCheck 11 thetaAboveCell01013332 = true := by
+  have h : (thetaAboveCell01013332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013332 h
+theorem e24KC2ThetaAboveLeaf01013333 :
+    adaptiveCoverCheck 11 thetaAboveCell01013333 = true := by
+  have h : (thetaAboveCell01013333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01013333 h
+theorem e24KC2ThetaAboveLeaf0102000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell0102))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf0102001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell0102))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf0102002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell0102))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf0102003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell0102))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf0102010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell0102))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf0102011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell0102))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf0102012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell0102))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf0102013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell0102))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf010202 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0102)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0102)) h
+theorem e24KC2ThetaAboveLeaf010203 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0102)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0102)) h
+theorem e24KC2ThetaAboveLeaf0102100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell0102))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf0102101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell0102))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf0102102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell0102))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf0102103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell0102))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf0102110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell0102))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf0102111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell0102))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf0102112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell0102))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf0102113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell0102))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell0102)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell0102))) h
+theorem e24KC2ThetaAboveLeaf010212 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0102)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0102)) h
+theorem e24KC2ThetaAboveLeaf010213 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0102)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0102))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0102)) h
+theorem e24KC2ThetaAboveLeaf01022 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0102) = true := by
+  have h : ((childHL thetaAboveCell0102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell0102) h
+theorem e24KC2ThetaAboveLeaf01023 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0102) = true := by
+  have h : ((childHH thetaAboveCell0102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell0102) h
+theorem e24KC2ThetaAboveLeaf0103000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell0103))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf0103001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell0103))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf0103002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell0103))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf0103003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell0103))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf0103010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell0103))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf0103011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell0103))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf0103012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell0103))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf0103013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell0103))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf010302 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0103)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0103)) h
+theorem e24KC2ThetaAboveLeaf010303 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0103)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0103)) h
+theorem e24KC2ThetaAboveLeaf0103100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell0103))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf0103101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell0103))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf0103102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell0103))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf0103103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell0103))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf0103110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell0103))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf0103111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell0103))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf0103112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell0103))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf0103113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell0103))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell0103)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell0103))) h
+theorem e24KC2ThetaAboveLeaf010312 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0103)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0103)) h
+theorem e24KC2ThetaAboveLeaf010313 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0103)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0103))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0103)) h
+theorem e24KC2ThetaAboveLeaf01032 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0103) = true := by
+  have h : ((childHL thetaAboveCell0103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell0103) h
+theorem e24KC2ThetaAboveLeaf01033 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0103) = true := by
+  have h : ((childHH thetaAboveCell0103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell0103) h
+theorem e24KC2ThetaAboveLeaf011000 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0110)) = true := by
+  have h : ((childLL (childLL thetaAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell0110)) h
+theorem e24KC2ThetaAboveLeaf011001 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0110)) = true := by
+  have h : ((childLH (childLL thetaAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell0110)) h
+theorem e24KC2ThetaAboveLeaf011002 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0110)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0110)) h
+theorem e24KC2ThetaAboveLeaf011003 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0110)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0110)) h
+theorem e24KC2ThetaAboveLeaf011010 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0110)) = true := by
+  have h : ((childLL (childLH thetaAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell0110)) h
+theorem e24KC2ThetaAboveLeaf011011 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0110)) = true := by
+  have h : ((childLH (childLH thetaAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell0110)) h
+theorem e24KC2ThetaAboveLeaf011012 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0110)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0110)) h
+theorem e24KC2ThetaAboveLeaf011013 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0110)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0110)) h
+theorem e24KC2ThetaAboveLeaf0110200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell0110))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell0110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell0110))) h
+theorem e24KC2ThetaAboveLeaf0110201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell0110))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell0110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell0110))) h
+theorem e24KC2ThetaAboveLeaf01102020 :
+    adaptiveCoverCheck 11 thetaAboveCell01102020 = true := by
+  have h : (thetaAboveCell01102020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102020 h
+theorem e24KC2ThetaAboveLeaf01102021 :
+    adaptiveCoverCheck 11 thetaAboveCell01102021 = true := by
+  have h : (thetaAboveCell01102021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102021 h
+theorem e24KC2ThetaAboveLeaf01102022 :
+    adaptiveCoverCheck 11 thetaAboveCell01102022 = true := by
+  have h : (thetaAboveCell01102022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102022 h
+theorem e24KC2ThetaAboveLeaf01102023 :
+    adaptiveCoverCheck 11 thetaAboveCell01102023 = true := by
+  have h : (thetaAboveCell01102023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102023 h
+theorem e24KC2ThetaAboveLeaf01102030 :
+    adaptiveCoverCheck 11 thetaAboveCell01102030 = true := by
+  have h : (thetaAboveCell01102030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102030 h
+theorem e24KC2ThetaAboveLeaf01102031 :
+    adaptiveCoverCheck 11 thetaAboveCell01102031 = true := by
+  have h : (thetaAboveCell01102031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102031 h
+theorem e24KC2ThetaAboveLeaf01102032 :
+    adaptiveCoverCheck 11 thetaAboveCell01102032 = true := by
+  have h : (thetaAboveCell01102032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102032 h
+theorem e24KC2ThetaAboveLeaf01102033 :
+    adaptiveCoverCheck 11 thetaAboveCell01102033 = true := by
+  have h : (thetaAboveCell01102033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102033 h
+theorem e24KC2ThetaAboveLeaf0110210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell0110))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell0110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell0110))) h
+theorem e24KC2ThetaAboveLeaf0110211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell0110))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell0110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell0110))) h
+theorem e24KC2ThetaAboveLeaf01102120 :
+    adaptiveCoverCheck 11 thetaAboveCell01102120 = true := by
+  have h : (thetaAboveCell01102120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102120 h
+theorem e24KC2ThetaAboveLeaf01102121 :
+    adaptiveCoverCheck 11 thetaAboveCell01102121 = true := by
+  have h : (thetaAboveCell01102121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102121 h
+theorem e24KC2ThetaAboveLeaf01102122 :
+    adaptiveCoverCheck 11 thetaAboveCell01102122 = true := by
+  have h : (thetaAboveCell01102122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102122 h
+theorem e24KC2ThetaAboveLeaf01102123 :
+    adaptiveCoverCheck 11 thetaAboveCell01102123 = true := by
+  have h : (thetaAboveCell01102123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102123 h
+theorem e24KC2ThetaAboveLeaf01102130 :
+    adaptiveCoverCheck 11 thetaAboveCell01102130 = true := by
+  have h : (thetaAboveCell01102130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102130 h
+theorem e24KC2ThetaAboveLeaf01102131 :
+    adaptiveCoverCheck 11 thetaAboveCell01102131 = true := by
+  have h : (thetaAboveCell01102131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102131 h
+theorem e24KC2ThetaAboveLeaf01102132 :
+    adaptiveCoverCheck 11 thetaAboveCell01102132 = true := by
+  have h : (thetaAboveCell01102132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102132 h
+theorem e24KC2ThetaAboveLeaf01102133 :
+    adaptiveCoverCheck 11 thetaAboveCell01102133 = true := by
+  have h : (thetaAboveCell01102133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102133 h
+theorem e24KC2ThetaAboveLeaf011022000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102200) = true := by
+  have h : ((childLL thetaAboveCell01102200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102200) h
+theorem e24KC2ThetaAboveLeaf011022001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102200) = true := by
+  have h : ((childLH thetaAboveCell01102200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102200) h
+theorem e24KC2ThetaAboveLeaf0110220020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102200)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102200)) h
+theorem e24KC2ThetaAboveLeaf0110220021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102200)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102200)) h
+theorem e24KC2ThetaAboveLeaf0110220022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102200)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102200)) h
+theorem e24KC2ThetaAboveLeaf0110220023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102200)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102200)) h
+theorem e24KC2ThetaAboveLeaf0110220030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102200)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102200)) h
+theorem e24KC2ThetaAboveLeaf0110220031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102200)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102200)) h
+theorem e24KC2ThetaAboveLeaf0110220032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102200)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102200)) h
+theorem e24KC2ThetaAboveLeaf0110220033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102200)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102200)) h
+theorem e24KC2ThetaAboveLeaf011022010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102201) = true := by
+  have h : ((childLL thetaAboveCell01102201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102201) h
+theorem e24KC2ThetaAboveLeaf011022011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102201) = true := by
+  have h : ((childLH thetaAboveCell01102201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102201) h
+theorem e24KC2ThetaAboveLeaf0110220120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102201)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102201)) h
+theorem e24KC2ThetaAboveLeaf0110220121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102201)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102201)) h
+theorem e24KC2ThetaAboveLeaf0110220122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102201)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102201)) h
+theorem e24KC2ThetaAboveLeaf0110220123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102201)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102201)) h
+theorem e24KC2ThetaAboveLeaf0110220130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102201)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102201)) h
+theorem e24KC2ThetaAboveLeaf0110220131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102201)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102201)) h
+theorem e24KC2ThetaAboveLeaf0110220132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102201)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102201)) h
+theorem e24KC2ThetaAboveLeaf0110220133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102201)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102201)) h
+theorem e24KC2ThetaAboveLeaf0110220200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01102202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01102202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01102202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01102202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01102202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01102202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01102202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01102202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102202)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102202)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102202)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102202)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102202)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102202)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102202)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102202)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102202)) h
+theorem e24KC2ThetaAboveLeaf0110220300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01102203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf0110220301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01102203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf0110220302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01102203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf0110220303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01102203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf0110220310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01102203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf0110220311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01102203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf0110220312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01102203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf0110220313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01102203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf0110220320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102203)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf0110220321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102203)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf0110220322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102203)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf0110220323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102203)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf0110220330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102203)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf0110220331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102203)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf0110220332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102203)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf0110220333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102203)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102203)) h
+theorem e24KC2ThetaAboveLeaf011022100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102210) = true := by
+  have h : ((childLL thetaAboveCell01102210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102210) h
+theorem e24KC2ThetaAboveLeaf011022101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102210) = true := by
+  have h : ((childLH thetaAboveCell01102210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102210) h
+theorem e24KC2ThetaAboveLeaf0110221020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102210)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102210)) h
+theorem e24KC2ThetaAboveLeaf0110221021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102210)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102210)) h
+theorem e24KC2ThetaAboveLeaf0110221022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102210)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102210)) h
+theorem e24KC2ThetaAboveLeaf0110221023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102210)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102210)) h
+theorem e24KC2ThetaAboveLeaf0110221030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102210)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102210)) h
+theorem e24KC2ThetaAboveLeaf0110221031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102210)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102210)) h
+theorem e24KC2ThetaAboveLeaf0110221032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102210)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102210)) h
+theorem e24KC2ThetaAboveLeaf0110221033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102210)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102210)) h
+theorem e24KC2ThetaAboveLeaf011022110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102211) = true := by
+  have h : ((childLL thetaAboveCell01102211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102211) h
+theorem e24KC2ThetaAboveLeaf011022111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102211) = true := by
+  have h : ((childLH thetaAboveCell01102211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102211) h
+theorem e24KC2ThetaAboveLeaf0110221120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102211)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102211)) h
+theorem e24KC2ThetaAboveLeaf0110221121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102211)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102211)) h
+theorem e24KC2ThetaAboveLeaf0110221122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102211)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102211)) h
+theorem e24KC2ThetaAboveLeaf0110221123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102211)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102211)) h
+theorem e24KC2ThetaAboveLeaf0110221130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102211)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102211)) h
+theorem e24KC2ThetaAboveLeaf0110221131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102211)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102211)) h
+theorem e24KC2ThetaAboveLeaf0110221132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102211)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102211)) h
+theorem e24KC2ThetaAboveLeaf0110221133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102211)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102211)) h
+theorem e24KC2ThetaAboveLeaf0110221200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01102212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01102212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01102212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01102212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01102212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01102212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01102212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01102212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102212)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102212)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102212)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102212)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102212)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102212)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102212)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102212)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102212)) h
+theorem e24KC2ThetaAboveLeaf0110221300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01102213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf0110221301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01102213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf0110221302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01102213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf0110221303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01102213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf0110221310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01102213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf0110221311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01102213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf0110221312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01102213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf0110221313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01102213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf0110221320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102213)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf0110221321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102213)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf0110221322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102213)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf0110221323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102213)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf0110221330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102213)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf0110221331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102213)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf0110221332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102213)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf0110221333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102213)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102213)) h
+theorem e24KC2ThetaAboveLeaf011022200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102220) = true := by
+  have h : ((childLL thetaAboveCell01102220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102220) h
+theorem e24KC2ThetaAboveLeaf011022201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102220) = true := by
+  have h : ((childLH thetaAboveCell01102220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102220) h
+theorem e24KC2ThetaAboveLeaf011022202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102220) = true := by
+  have h : ((childHL thetaAboveCell01102220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01102220) h
+theorem e24KC2ThetaAboveLeaf011022203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102220) = true := by
+  have h : ((childHH thetaAboveCell01102220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01102220) h
+theorem e24KC2ThetaAboveLeaf011022210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102221) = true := by
+  have h : ((childLL thetaAboveCell01102221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102221) h
+theorem e24KC2ThetaAboveLeaf011022211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102221) = true := by
+  have h : ((childLH thetaAboveCell01102221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102221) h
+theorem e24KC2ThetaAboveLeaf011022212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102221) = true := by
+  have h : ((childHL thetaAboveCell01102221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01102221) h
+theorem e24KC2ThetaAboveLeaf011022213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102221) = true := by
+  have h : ((childHH thetaAboveCell01102221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01102221) h
+theorem e24KC2ThetaAboveLeaf01102222 :
+    adaptiveCoverCheck 11 thetaAboveCell01102222 = true := by
+  have h : (thetaAboveCell01102222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102222 h
+theorem e24KC2ThetaAboveLeaf01102223 :
+    adaptiveCoverCheck 11 thetaAboveCell01102223 = true := by
+  have h : (thetaAboveCell01102223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102223 h
+theorem e24KC2ThetaAboveLeaf011022300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102230) = true := by
+  have h : ((childLL thetaAboveCell01102230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102230) h
+theorem e24KC2ThetaAboveLeaf011022301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102230) = true := by
+  have h : ((childLH thetaAboveCell01102230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102230) h
+theorem e24KC2ThetaAboveLeaf011022302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102230) = true := by
+  have h : ((childHL thetaAboveCell01102230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01102230) h
+theorem e24KC2ThetaAboveLeaf011022303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102230) = true := by
+  have h : ((childHH thetaAboveCell01102230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01102230) h
+theorem e24KC2ThetaAboveLeaf011022310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102231) = true := by
+  have h : ((childLL thetaAboveCell01102231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102231) h
+theorem e24KC2ThetaAboveLeaf011022311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102231) = true := by
+  have h : ((childLH thetaAboveCell01102231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102231) h
+theorem e24KC2ThetaAboveLeaf011022312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102231) = true := by
+  have h : ((childHL thetaAboveCell01102231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01102231) h
+theorem e24KC2ThetaAboveLeaf011022313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102231) = true := by
+  have h : ((childHH thetaAboveCell01102231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01102231) h
+theorem e24KC2ThetaAboveLeaf01102232 :
+    adaptiveCoverCheck 11 thetaAboveCell01102232 = true := by
+  have h : (thetaAboveCell01102232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102232 h
+theorem e24KC2ThetaAboveLeaf01102233 :
+    adaptiveCoverCheck 11 thetaAboveCell01102233 = true := by
+  have h : (thetaAboveCell01102233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102233 h
+theorem e24KC2ThetaAboveLeaf011023000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102300) = true := by
+  have h : ((childLL thetaAboveCell01102300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102300) h
+theorem e24KC2ThetaAboveLeaf011023001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102300) = true := by
+  have h : ((childLH thetaAboveCell01102300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102300) h
+theorem e24KC2ThetaAboveLeaf0110230020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102300)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102300)) h
+theorem e24KC2ThetaAboveLeaf0110230021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102300)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102300)) h
+theorem e24KC2ThetaAboveLeaf0110230022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102300)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102300)) h
+theorem e24KC2ThetaAboveLeaf0110230023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102300)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102300)) h
+theorem e24KC2ThetaAboveLeaf0110230030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102300)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102300)) h
+theorem e24KC2ThetaAboveLeaf0110230031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102300)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102300)) h
+theorem e24KC2ThetaAboveLeaf0110230032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102300)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102300)) h
+theorem e24KC2ThetaAboveLeaf0110230033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102300)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102300)) h
+theorem e24KC2ThetaAboveLeaf011023010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102301) = true := by
+  have h : ((childLL thetaAboveCell01102301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102301) h
+theorem e24KC2ThetaAboveLeaf011023011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102301) = true := by
+  have h : ((childLH thetaAboveCell01102301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102301) h
+theorem e24KC2ThetaAboveLeaf0110230120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102301)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102301)) h
+theorem e24KC2ThetaAboveLeaf0110230121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102301)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102301)) h
+theorem e24KC2ThetaAboveLeaf0110230122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102301)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102301)) h
+theorem e24KC2ThetaAboveLeaf0110230123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102301)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102301)) h
+theorem e24KC2ThetaAboveLeaf0110230130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102301)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102301)) h
+theorem e24KC2ThetaAboveLeaf0110230131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102301)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102301)) h
+theorem e24KC2ThetaAboveLeaf0110230132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102301)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102301)) h
+theorem e24KC2ThetaAboveLeaf0110230133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102301)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102301)) h
+theorem e24KC2ThetaAboveLeaf0110230200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01102302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01102302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01102302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01102302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01102302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01102302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01102302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01102302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102302)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102302)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102302)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102302)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102302)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102302)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102302)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102302)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102302)) h
+theorem e24KC2ThetaAboveLeaf0110230300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01102303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf0110230301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01102303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf0110230302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01102303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf0110230303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01102303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf0110230310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01102303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf0110230311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01102303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf0110230312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01102303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf0110230313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01102303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf0110230320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102303)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf0110230321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102303)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf0110230322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102303)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf0110230323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102303)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf0110230330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102303)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf0110230331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102303)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf0110230332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102303)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf0110230333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102303)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102303)) h
+theorem e24KC2ThetaAboveLeaf011023100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102310) = true := by
+  have h : ((childLL thetaAboveCell01102310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102310) h
+theorem e24KC2ThetaAboveLeaf011023101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102310) = true := by
+  have h : ((childLH thetaAboveCell01102310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102310) h
+theorem e24KC2ThetaAboveLeaf0110231020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102310)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102310)) h
+theorem e24KC2ThetaAboveLeaf0110231021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102310)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102310)) h
+theorem e24KC2ThetaAboveLeaf0110231022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102310)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102310)) h
+theorem e24KC2ThetaAboveLeaf0110231023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102310)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102310)) h
+theorem e24KC2ThetaAboveLeaf0110231030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102310)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102310)) h
+theorem e24KC2ThetaAboveLeaf0110231031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102310)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102310)) h
+theorem e24KC2ThetaAboveLeaf0110231032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102310)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102310)) h
+theorem e24KC2ThetaAboveLeaf0110231033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102310)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102310)) h
+theorem e24KC2ThetaAboveLeaf011023110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102311) = true := by
+  have h : ((childLL thetaAboveCell01102311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102311) h
+theorem e24KC2ThetaAboveLeaf011023111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102311) = true := by
+  have h : ((childLH thetaAboveCell01102311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102311) h
+theorem e24KC2ThetaAboveLeaf0110231120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102311)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102311)) h
+theorem e24KC2ThetaAboveLeaf0110231121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102311)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102311)) h
+theorem e24KC2ThetaAboveLeaf0110231122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102311)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102311)) h
+theorem e24KC2ThetaAboveLeaf0110231123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102311)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102311)) h
+theorem e24KC2ThetaAboveLeaf0110231130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102311)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102311)) h
+theorem e24KC2ThetaAboveLeaf0110231131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102311)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102311)) h
+theorem e24KC2ThetaAboveLeaf0110231132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102311)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102311)) h
+theorem e24KC2ThetaAboveLeaf0110231133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102311)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102311)) h
+theorem e24KC2ThetaAboveLeaf0110231200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01102312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01102312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01102312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01102312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01102312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01102312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01102312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01102312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102312)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102312)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102312)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102312)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102312)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102312)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102312)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102312)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102312)) h
+theorem e24KC2ThetaAboveLeaf0110231300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01102313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf0110231301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01102313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf0110231302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01102313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf0110231303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01102313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf0110231310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01102313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf0110231311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01102313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf0110231312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01102313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf0110231313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01102313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf0110231320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01102313)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf0110231321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01102313)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf0110231322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01102313)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf0110231323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01102313)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf0110231330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01102313)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf0110231331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01102313)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf0110231332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01102313)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf0110231333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01102313)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01102313)) h
+theorem e24KC2ThetaAboveLeaf011023200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102320) = true := by
+  have h : ((childLL thetaAboveCell01102320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102320) h
+theorem e24KC2ThetaAboveLeaf011023201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102320) = true := by
+  have h : ((childLH thetaAboveCell01102320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102320) h
+theorem e24KC2ThetaAboveLeaf011023202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102320) = true := by
+  have h : ((childHL thetaAboveCell01102320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01102320) h
+theorem e24KC2ThetaAboveLeaf011023203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102320) = true := by
+  have h : ((childHH thetaAboveCell01102320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01102320) h
+theorem e24KC2ThetaAboveLeaf011023210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102321) = true := by
+  have h : ((childLL thetaAboveCell01102321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102321) h
+theorem e24KC2ThetaAboveLeaf011023211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102321) = true := by
+  have h : ((childLH thetaAboveCell01102321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102321) h
+theorem e24KC2ThetaAboveLeaf011023212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102321) = true := by
+  have h : ((childHL thetaAboveCell01102321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01102321) h
+theorem e24KC2ThetaAboveLeaf011023213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102321) = true := by
+  have h : ((childHH thetaAboveCell01102321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01102321) h
+theorem e24KC2ThetaAboveLeaf01102322 :
+    adaptiveCoverCheck 11 thetaAboveCell01102322 = true := by
+  have h : (thetaAboveCell01102322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102322 h
+theorem e24KC2ThetaAboveLeaf01102323 :
+    adaptiveCoverCheck 11 thetaAboveCell01102323 = true := by
+  have h : (thetaAboveCell01102323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102323 h
+theorem e24KC2ThetaAboveLeaf011023300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102330) = true := by
+  have h : ((childLL thetaAboveCell01102330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102330) h
+theorem e24KC2ThetaAboveLeaf011023301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102330) = true := by
+  have h : ((childLH thetaAboveCell01102330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102330) h
+theorem e24KC2ThetaAboveLeaf011023302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102330) = true := by
+  have h : ((childHL thetaAboveCell01102330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01102330) h
+theorem e24KC2ThetaAboveLeaf011023303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102330) = true := by
+  have h : ((childHH thetaAboveCell01102330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01102330) h
+theorem e24KC2ThetaAboveLeaf011023310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102331) = true := by
+  have h : ((childLL thetaAboveCell01102331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01102331) h
+theorem e24KC2ThetaAboveLeaf011023311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102331) = true := by
+  have h : ((childLH thetaAboveCell01102331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01102331) h
+theorem e24KC2ThetaAboveLeaf011023312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102331) = true := by
+  have h : ((childHL thetaAboveCell01102331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01102331) h
+theorem e24KC2ThetaAboveLeaf011023313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102331) = true := by
+  have h : ((childHH thetaAboveCell01102331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01102331) h
+theorem e24KC2ThetaAboveLeaf01102332 :
+    adaptiveCoverCheck 11 thetaAboveCell01102332 = true := by
+  have h : (thetaAboveCell01102332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102332 h
+theorem e24KC2ThetaAboveLeaf01102333 :
+    adaptiveCoverCheck 11 thetaAboveCell01102333 = true := by
+  have h : (thetaAboveCell01102333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01102333 h
+theorem e24KC2ThetaAboveLeaf0110300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell0110))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell0110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell0110))) h
+theorem e24KC2ThetaAboveLeaf0110301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell0110))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell0110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell0110))) h
+theorem e24KC2ThetaAboveLeaf01103020 :
+    adaptiveCoverCheck 11 thetaAboveCell01103020 = true := by
+  have h : (thetaAboveCell01103020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103020 h
+theorem e24KC2ThetaAboveLeaf01103021 :
+    adaptiveCoverCheck 11 thetaAboveCell01103021 = true := by
+  have h : (thetaAboveCell01103021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103021 h
+theorem e24KC2ThetaAboveLeaf01103022 :
+    adaptiveCoverCheck 11 thetaAboveCell01103022 = true := by
+  have h : (thetaAboveCell01103022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103022 h
+theorem e24KC2ThetaAboveLeaf01103023 :
+    adaptiveCoverCheck 11 thetaAboveCell01103023 = true := by
+  have h : (thetaAboveCell01103023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103023 h
+theorem e24KC2ThetaAboveLeaf01103030 :
+    adaptiveCoverCheck 11 thetaAboveCell01103030 = true := by
+  have h : (thetaAboveCell01103030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103030 h
+theorem e24KC2ThetaAboveLeaf01103031 :
+    adaptiveCoverCheck 11 thetaAboveCell01103031 = true := by
+  have h : (thetaAboveCell01103031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103031 h
+theorem e24KC2ThetaAboveLeaf01103032 :
+    adaptiveCoverCheck 11 thetaAboveCell01103032 = true := by
+  have h : (thetaAboveCell01103032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103032 h
+theorem e24KC2ThetaAboveLeaf01103033 :
+    adaptiveCoverCheck 11 thetaAboveCell01103033 = true := by
+  have h : (thetaAboveCell01103033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103033 h
+theorem e24KC2ThetaAboveLeaf0110310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell0110))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell0110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell0110))) h
+theorem e24KC2ThetaAboveLeaf0110311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell0110))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell0110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell0110))) h
+theorem e24KC2ThetaAboveLeaf01103120 :
+    adaptiveCoverCheck 11 thetaAboveCell01103120 = true := by
+  have h : (thetaAboveCell01103120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103120 h
+theorem e24KC2ThetaAboveLeaf01103121 :
+    adaptiveCoverCheck 11 thetaAboveCell01103121 = true := by
+  have h : (thetaAboveCell01103121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103121 h
+theorem e24KC2ThetaAboveLeaf01103122 :
+    adaptiveCoverCheck 11 thetaAboveCell01103122 = true := by
+  have h : (thetaAboveCell01103122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103122 h
+theorem e24KC2ThetaAboveLeaf01103123 :
+    adaptiveCoverCheck 11 thetaAboveCell01103123 = true := by
+  have h : (thetaAboveCell01103123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103123 h
+theorem e24KC2ThetaAboveLeaf01103130 :
+    adaptiveCoverCheck 11 thetaAboveCell01103130 = true := by
+  have h : (thetaAboveCell01103130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103130 h
+theorem e24KC2ThetaAboveLeaf01103131 :
+    adaptiveCoverCheck 11 thetaAboveCell01103131 = true := by
+  have h : (thetaAboveCell01103131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103131 h
+theorem e24KC2ThetaAboveLeaf01103132 :
+    adaptiveCoverCheck 11 thetaAboveCell01103132 = true := by
+  have h : (thetaAboveCell01103132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103132 h
+theorem e24KC2ThetaAboveLeaf01103133 :
+    adaptiveCoverCheck 11 thetaAboveCell01103133 = true := by
+  have h : (thetaAboveCell01103133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103133 h
+theorem e24KC2ThetaAboveLeaf011032000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103200) = true := by
+  have h : ((childLL thetaAboveCell01103200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103200) h
+theorem e24KC2ThetaAboveLeaf011032001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103200) = true := by
+  have h : ((childLH thetaAboveCell01103200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103200) h
+theorem e24KC2ThetaAboveLeaf0110320020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103200)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103200)) h
+theorem e24KC2ThetaAboveLeaf0110320021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103200)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103200)) h
+theorem e24KC2ThetaAboveLeaf0110320022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103200)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103200)) h
+theorem e24KC2ThetaAboveLeaf0110320023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103200)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103200)) h
+theorem e24KC2ThetaAboveLeaf0110320030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103200)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103200)) h
+theorem e24KC2ThetaAboveLeaf0110320031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103200)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103200)) h
+theorem e24KC2ThetaAboveLeaf0110320032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103200)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103200)) h
+theorem e24KC2ThetaAboveLeaf0110320033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103200)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103200)) h
+theorem e24KC2ThetaAboveLeaf011032010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103201) = true := by
+  have h : ((childLL thetaAboveCell01103201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103201) h
+theorem e24KC2ThetaAboveLeaf011032011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103201) = true := by
+  have h : ((childLH thetaAboveCell01103201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103201) h
+theorem e24KC2ThetaAboveLeaf0110320120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103201)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103201)) h
+theorem e24KC2ThetaAboveLeaf0110320121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103201)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103201)) h
+theorem e24KC2ThetaAboveLeaf0110320122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103201)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103201)) h
+theorem e24KC2ThetaAboveLeaf0110320123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103201)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103201)) h
+theorem e24KC2ThetaAboveLeaf0110320130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103201)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103201)) h
+theorem e24KC2ThetaAboveLeaf0110320131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103201)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103201)) h
+theorem e24KC2ThetaAboveLeaf0110320132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103201)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103201)) h
+theorem e24KC2ThetaAboveLeaf0110320133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103201)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103201)) h
+theorem e24KC2ThetaAboveLeaf0110320200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01103202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01103202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01103202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01103202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01103202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01103202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01103202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01103202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103202)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103202)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103202)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103202)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103202)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103202)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103202)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103202)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103202)) h
+theorem e24KC2ThetaAboveLeaf0110320300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01103203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf0110320301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01103203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf0110320302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01103203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf0110320303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01103203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf0110320310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01103203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf0110320311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01103203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf0110320312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01103203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf0110320313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01103203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf0110320320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103203)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf0110320321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103203)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf0110320322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103203)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf0110320323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103203)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf0110320330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103203)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf0110320331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103203)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf0110320332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103203)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf0110320333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103203)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103203)) h
+theorem e24KC2ThetaAboveLeaf011032100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103210) = true := by
+  have h : ((childLL thetaAboveCell01103210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103210) h
+theorem e24KC2ThetaAboveLeaf011032101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103210) = true := by
+  have h : ((childLH thetaAboveCell01103210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103210) h
+theorem e24KC2ThetaAboveLeaf0110321020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103210)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103210)) h
+theorem e24KC2ThetaAboveLeaf0110321021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103210)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103210)) h
+theorem e24KC2ThetaAboveLeaf0110321022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103210)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103210)) h
+theorem e24KC2ThetaAboveLeaf0110321023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103210)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103210)) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6ProofBatchAf2b60d8e2cf09a4`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8ecd2c98f8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8ecd2c98f8
+
+open CertificateCells8ecd2c98f8
+theorem cover_subtree_d3f1948c7bb2 :
+    adaptiveCoverCheck 6 thetaBelowCell111133110222 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133110222
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133110222)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133110222)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133110222)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133110222)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133110222)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133110222)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133110222)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133110222)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133110222)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133110222)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133110222)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133110222)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133110222)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133110222)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133110222)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133110222)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133110222)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133110222)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133110222)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133110222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133110222)) h))
+
+theorem cover_subtree_06756667cdcb :
+    adaptiveCoverCheck 6 thetaBelowCell111133110223 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133110223
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133110223)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133110223)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133110223)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133110223)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133110223)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133110223)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133110223)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133110223)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133110223)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133110223)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133110223)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133110223)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133110223)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133110223)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133110223)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133110223)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133110223)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133110223)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133110223)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133110223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133110223)) h))
+
+theorem cover_subtree_5fd1a813f187 :
+    adaptiveCoverCheck 7 (childHL (childHL (childLL thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHL (childLL thetaBelowCell11113311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133110220
+        (by
+          have h : ((childLL thetaBelowCell111133110220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133110220) h)
+        (by
+          have h : ((childLH thetaBelowCell111133110220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133110220) h)
+        (by
+          have h : ((childHL thetaBelowCell111133110220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133110220) h)
+        (by
+          have h : ((childHH thetaBelowCell111133110220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133110220) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133110221
+        (by
+          have h : ((childLL thetaBelowCell111133110221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133110221) h)
+        (by
+          have h : ((childLH thetaBelowCell111133110221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133110221) h)
+        (by
+          have h : ((childHL thetaBelowCell111133110221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133110221) h)
+        (by
+          have h : ((childHH thetaBelowCell111133110221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133110221) h))
+    cover_subtree_d3f1948c7bb2
+    cover_subtree_06756667cdcb
+
+theorem cover_subtree_985d36a8aada :
+    adaptiveCoverCheck 6 thetaBelowCell111133110232 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133110232
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133110232)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133110232)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133110232)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133110232)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133110232)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133110232)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133110232)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133110232)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133110232)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133110232)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133110232)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133110232)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133110232)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133110232)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133110232)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133110232)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133110232)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133110232)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133110232)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133110232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133110232)) h))
+
+theorem cover_subtree_91f684fb2a75 :
+    adaptiveCoverCheck 6 thetaBelowCell111133110233 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133110233
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133110233)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133110233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133110233)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133110233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133110233)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133110233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133110233)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133110233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133110233)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133110233)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133110233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133110233)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133110233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133110233)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133110233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133110233)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133110233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133110233)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133110233)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133110233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133110233)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133110233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133110233)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133110233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133110233)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133110233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133110233)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133110233)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133110233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133110233)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133110233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133110233)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133110233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133110233)) h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH thetaBelowCell111133110233))
+            (by
+              have h : ((childLL (childHH (childHH thetaBelowCell111133110233)))).rejected = true
+                := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHH
+                thetaBelowCell111133110233))) h)
+            (by
+              have h : ((childLH (childHH (childHH thetaBelowCell111133110233)))).rejected = true
+                := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHH
+                thetaBelowCell111133110233))) h)
+            (by
+              have h : ((childHL (childHH (childHH thetaBelowCell111133110233)))).rejected = true
+                := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHH
+                thetaBelowCell111133110233))) h)
+            (by
+              have h : ((childHH (childHH (childHH thetaBelowCell111133110233)))).rejected = true
+                := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHH
+                thetaBelowCell111133110233))) h)))
+
+theorem cover_subtree_145b055d6873 :
+    adaptiveCoverCheck 7 (childHH (childHL (childLL thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHL (childLL thetaBelowCell11113311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133110230
+        (by
+          have h : ((childLL thetaBelowCell111133110230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133110230) h)
+        (by
+          have h : ((childLH thetaBelowCell111133110230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133110230) h)
+        (by
+          have h : ((childHL thetaBelowCell111133110230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133110230) h)
+        (by
+          have h : ((childHH thetaBelowCell111133110230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133110230) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133110231
+        (by
+          have h : ((childLL thetaBelowCell111133110231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133110231) h)
+        (by
+          have h : ((childLH thetaBelowCell111133110231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133110231) h)
+        (by
+          have h : ((childHL thetaBelowCell111133110231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133110231) h)
+        (by
+          have h : ((childHH thetaBelowCell111133110231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133110231) h))
+    cover_subtree_985d36a8aada
+    cover_subtree_91f684fb2a75
+
+theorem cover_subtree_3ea9d3baa8d2 :
+    adaptiveCoverCheck 8 (childHL (childLL thetaBelowCell11113311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL thetaBelowCell11113311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHL (childLL
+        thetaBelowCell11113311)))
+        (by
+          have h : (thetaBelowCell111133110200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110200 h)
+        (by
+          have h : (thetaBelowCell111133110201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110201 h)
+        (by
+          have h : (thetaBelowCell111133110202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110202 h)
+        (by
+          have h : (thetaBelowCell111133110203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHL (childLL
+        thetaBelowCell11113311)))
+        (by
+          have h : (thetaBelowCell111133110210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110210 h)
+        (by
+          have h : (thetaBelowCell111133110211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110211 h)
+        (by
+          have h : (thetaBelowCell111133110212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110212 h)
+        (by
+          have h : (thetaBelowCell111133110213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110213 h))
+    cover_subtree_5fd1a813f187
+    cover_subtree_145b055d6873
+
+theorem cover_subtree_2d2eae393440 :
+    adaptiveCoverCheck 5 (childHL thetaBelowCell111133110322) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133110322)
+    (by
+      have h : ((childLL (childHL thetaBelowCell111133110322))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL thetaBelowCell111133110322)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL thetaBelowCell111133110322))
+        (by
+          have h : ((childLL (childLH (childHL thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childHL
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childHL
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childHL
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childHL
+            thetaBelowCell111133110322))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL thetaBelowCell111133110322))
+        (by
+          have h : ((childLL (childHL (childHL thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHL
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHL
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHL
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHL
+            thetaBelowCell111133110322))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL thetaBelowCell111133110322))
+        (by
+          have h : ((childLL (childHH (childHL thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHL
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHL
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHL
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHL
+            thetaBelowCell111133110322))) h))
+
+theorem cover_subtree_4a90a1e8c08a :
+    adaptiveCoverCheck 5 (childHH thetaBelowCell111133110322) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133110322)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH thetaBelowCell111133110322))
+        (by
+          have h : ((childLL (childLL (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childHH
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childHH
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childHH
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childHH
+            thetaBelowCell111133110322))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH thetaBelowCell111133110322))
+        (by
+          have h : ((childLL (childLH (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childHH
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childHH
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childHH
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childHH
+            thetaBelowCell111133110322))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH thetaBelowCell111133110322))
+        (by
+          have h : ((childLL (childHL (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHH
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHH
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHH
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHH
+            thetaBelowCell111133110322))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH thetaBelowCell111133110322))
+        (by
+          have h : ((childLL (childHH (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHH
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHH
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHH
+            thetaBelowCell111133110322))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaBelowCell111133110322)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHH
+            thetaBelowCell111133110322))) h))
+
+theorem cover_subtree_179b6bbb140a :
+    adaptiveCoverCheck 6 thetaBelowCell111133110322 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133110322
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133110322)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133110322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133110322)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133110322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133110322)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133110322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133110322)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133110322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133110322)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133110322)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133110322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133110322)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133110322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133110322)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133110322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133110322)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133110322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133110322)) h))
+    cover_subtree_2d2eae393440
+    cover_subtree_4a90a1e8c08a
+
+theorem cover_subtree_802a68149050 :
+    adaptiveCoverCheck 5 (childHL thetaBelowCell111133110323) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133110323)
+    (by
+      have h : ((childLL (childHL thetaBelowCell111133110323))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL thetaBelowCell111133110323)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell111133110323))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL thetaBelowCell111133110323)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL thetaBelowCell111133110323))
+        (by
+          have h : ((childLL (childHL (childHL thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHL
+            thetaBelowCell111133110323))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHL
+            thetaBelowCell111133110323))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHL
+            thetaBelowCell111133110323))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHL
+            thetaBelowCell111133110323))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL thetaBelowCell111133110323))
+        (by
+          have h : ((childLL (childHH (childHL thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHL
+            thetaBelowCell111133110323))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHL
+            thetaBelowCell111133110323))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHL
+            thetaBelowCell111133110323))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHL
+            thetaBelowCell111133110323))) h))
+
+theorem cover_subtree_15d7407001b6 :
+    adaptiveCoverCheck 5 (childHH thetaBelowCell111133110323) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133110323)
+    (by
+      have h : ((childLL (childHH thetaBelowCell111133110323))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH thetaBelowCell111133110323)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell111133110323))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH thetaBelowCell111133110323)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH thetaBelowCell111133110323))
+        (by
+          have h : ((childLL (childHL (childHH thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHH
+            thetaBelowCell111133110323))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHH
+            thetaBelowCell111133110323))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHH
+            thetaBelowCell111133110323))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHH
+            thetaBelowCell111133110323))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH thetaBelowCell111133110323))
+        (by
+          have h : ((childLL (childHH (childHH thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHH
+            thetaBelowCell111133110323))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHH
+            thetaBelowCell111133110323))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHH
+            thetaBelowCell111133110323))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaBelowCell111133110323)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHH
+            thetaBelowCell111133110323))) h))
+
+theorem cover_subtree_6849ebb64c74 :
+    adaptiveCoverCheck 6 thetaBelowCell111133110323 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133110323
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133110323)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133110323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133110323)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133110323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133110323)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133110323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133110323)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133110323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133110323)) h))
+    (by
+      have h : ((childLH thetaBelowCell111133110323)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133110323) h)
+    cover_subtree_802a68149050
+    cover_subtree_15d7407001b6
+
+theorem cover_subtree_4bd97887e2d4 :
+    adaptiveCoverCheck 7 (childHL (childHH (childLL thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHH (childLL thetaBelowCell11113311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133110320
+        (by
+          have h : ((childLL thetaBelowCell111133110320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133110320) h)
+        (by
+          have h : ((childLH thetaBelowCell111133110320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133110320) h)
+        (by
+          have h : ((childHL thetaBelowCell111133110320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133110320) h)
+        (by
+          have h : ((childHH thetaBelowCell111133110320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133110320) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133110321
+        (by
+          have h : ((childLL thetaBelowCell111133110321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133110321) h)
+        (by
+          have h : ((childLH thetaBelowCell111133110321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133110321) h)
+        (by
+          have h : ((childHL thetaBelowCell111133110321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133110321) h)
+        (by
+          have h : ((childHH thetaBelowCell111133110321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133110321) h))
+    cover_subtree_179b6bbb140a
+    cover_subtree_6849ebb64c74
+
+theorem cover_subtree_63bf927d197e :
+    adaptiveCoverCheck 5 (childHL thetaBelowCell111133110332) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133110332)
+    (by
+      have h : ((childLL (childHL thetaBelowCell111133110332))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL thetaBelowCell111133110332)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell111133110332))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL thetaBelowCell111133110332)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL thetaBelowCell111133110332))
+        (by
+          have h : ((childLL (childHL (childHL thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHL
+            thetaBelowCell111133110332))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHL
+            thetaBelowCell111133110332))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHL
+            thetaBelowCell111133110332))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHL
+            thetaBelowCell111133110332))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL thetaBelowCell111133110332))
+        (by
+          have h : ((childLL (childHH (childHL thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHL
+            thetaBelowCell111133110332))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHL
+            thetaBelowCell111133110332))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHL
+            thetaBelowCell111133110332))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHL
+            thetaBelowCell111133110332))) h))
+
+theorem cover_subtree_4b23bc69c953 :
+    adaptiveCoverCheck 5 (childHH thetaBelowCell111133110332) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133110332)
+    (by
+      have h : ((childLL (childHH thetaBelowCell111133110332))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH thetaBelowCell111133110332)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell111133110332))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH thetaBelowCell111133110332)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH thetaBelowCell111133110332))
+        (by
+          have h : ((childLL (childHL (childHH thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHH
+            thetaBelowCell111133110332))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHH
+            thetaBelowCell111133110332))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHH
+            thetaBelowCell111133110332))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHH
+            thetaBelowCell111133110332))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH thetaBelowCell111133110332))
+        (by
+          have h : ((childLL (childHH (childHH thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHH
+            thetaBelowCell111133110332))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHH
+            thetaBelowCell111133110332))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHH
+            thetaBelowCell111133110332))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaBelowCell111133110332)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHH
+            thetaBelowCell111133110332))) h))
+
+theorem cover_subtree_b2776b022cdc :
+    adaptiveCoverCheck 6 thetaBelowCell111133110332 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133110332
+    (by
+      have h : ((childLL thetaBelowCell111133110332)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133110332) h)
+    (by
+      have h : ((childLH thetaBelowCell111133110332)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133110332) h)
+    cover_subtree_63bf927d197e
+    cover_subtree_4b23bc69c953
+
+theorem cover_subtree_273cf6356f9d :
+    adaptiveCoverCheck 5 (childHL thetaBelowCell111133110333) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133110333)
+    (by
+      have h : ((childLL (childHL thetaBelowCell111133110333))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL thetaBelowCell111133110333)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell111133110333))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL thetaBelowCell111133110333)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL thetaBelowCell111133110333))
+        (by
+          have h : ((childLL (childHL (childHL thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHL
+            thetaBelowCell111133110333))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHL
+            thetaBelowCell111133110333))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHL
+            thetaBelowCell111133110333))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHL
+            thetaBelowCell111133110333))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL thetaBelowCell111133110333))
+        (by
+          have h : ((childLL (childHH (childHL thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHL
+            thetaBelowCell111133110333))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHL
+            thetaBelowCell111133110333))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHL
+            thetaBelowCell111133110333))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHL
+            thetaBelowCell111133110333))) h))
+
+theorem cover_subtree_27ddb3080e9d :
+    adaptiveCoverCheck 5 (childHH thetaBelowCell111133110333) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133110333)
+    (by
+      have h : ((childLL (childHH thetaBelowCell111133110333))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH thetaBelowCell111133110333)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell111133110333))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH thetaBelowCell111133110333)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH thetaBelowCell111133110333))
+        (by
+          have h : ((childLL (childHL (childHH thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHH
+            thetaBelowCell111133110333))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHH
+            thetaBelowCell111133110333))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHH
+            thetaBelowCell111133110333))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHH
+            thetaBelowCell111133110333))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH thetaBelowCell111133110333))
+        (by
+          have h : ((childLL (childHH (childHH thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHH
+            thetaBelowCell111133110333))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHH
+            thetaBelowCell111133110333))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHH
+            thetaBelowCell111133110333))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaBelowCell111133110333)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHH
+            thetaBelowCell111133110333))) h))
+
+theorem cover_subtree_27a44a78bf0e :
+    adaptiveCoverCheck 6 thetaBelowCell111133110333 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133110333
+    (by
+      have h : ((childLL thetaBelowCell111133110333)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133110333) h)
+    (by
+      have h : ((childLH thetaBelowCell111133110333)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133110333) h)
+    cover_subtree_273cf6356f9d
+    cover_subtree_27ddb3080e9d
+
+theorem cover_subtree_0fe512b8a712 :
+    adaptiveCoverCheck 7 (childHH (childHH (childLL thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHH (childLL thetaBelowCell11113311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133110330
+        (by
+          have h : ((childLL thetaBelowCell111133110330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133110330) h)
+        (by
+          have h : ((childLH thetaBelowCell111133110330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133110330) h)
+        (by
+          have h : ((childHL thetaBelowCell111133110330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133110330) h)
+        (by
+          have h : ((childHH thetaBelowCell111133110330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133110330) h))
+    (by
+      have h : (thetaBelowCell111133110331).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110331 h)
+    cover_subtree_b2776b022cdc
+    cover_subtree_27a44a78bf0e
+
+theorem cover_subtree_780d3c38d99a :
+    adaptiveCoverCheck 8 (childHH (childLL thetaBelowCell11113311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL thetaBelowCell11113311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHH (childLL
+        thetaBelowCell11113311)))
+        (by
+          have h : (thetaBelowCell111133110300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110300 h)
+        (by
+          have h : (thetaBelowCell111133110301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110301 h)
+        (by
+          have h : (thetaBelowCell111133110302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110302 h)
+        (by
+          have h : (thetaBelowCell111133110303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHH (childLL
+        thetaBelowCell11113311)))
+        (by
+          have h : (thetaBelowCell111133110310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110310 h)
+        (by
+          have h : (thetaBelowCell111133110311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110311 h)
+        (by
+          have h : (thetaBelowCell111133110312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110312 h)
+        (by
+          have h : (thetaBelowCell111133110313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133110313 h))
+    cover_subtree_4bd97887e2d4
+    cover_subtree_0fe512b8a712
+
+theorem e24KC2ThetaBelowLeaf111133110 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113311) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11113311)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL thetaBelowCell11113311))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLL
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLL
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLL
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLL
+            thetaBelowCell11113311))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL thetaBelowCell11113311))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLL
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLL
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLL
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLL
+            thetaBelowCell11113311))) h))
+    cover_subtree_3ea9d3baa8d2
+    cover_subtree_780d3c38d99a
+theorem cover_subtree_789cd4100198 :
+    adaptiveCoverCheck 5 (childHL thetaBelowCell111133111222) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133111222)
+    (by
+      have h : ((childLL (childHL thetaBelowCell111133111222))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL thetaBelowCell111133111222)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell111133111222))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL thetaBelowCell111133111222)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL thetaBelowCell111133111222))
+        (by
+          have h : ((childLL (childHL (childHL thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHL
+            thetaBelowCell111133111222))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHL
+            thetaBelowCell111133111222))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHL
+            thetaBelowCell111133111222))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHL
+            thetaBelowCell111133111222))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL thetaBelowCell111133111222))
+        (by
+          have h : ((childLL (childHH (childHL thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHL
+            thetaBelowCell111133111222))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHL
+            thetaBelowCell111133111222))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHL
+            thetaBelowCell111133111222))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHL
+            thetaBelowCell111133111222))) h))
+
+theorem cover_subtree_c3bc65d3adac :
+    adaptiveCoverCheck 5 (childHH thetaBelowCell111133111222) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133111222)
+    (by
+      have h : ((childLL (childHH thetaBelowCell111133111222))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH thetaBelowCell111133111222)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell111133111222))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH thetaBelowCell111133111222)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH thetaBelowCell111133111222))
+        (by
+          have h : ((childLL (childHL (childHH thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHH
+            thetaBelowCell111133111222))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHH
+            thetaBelowCell111133111222))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHH
+            thetaBelowCell111133111222))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHH
+            thetaBelowCell111133111222))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH thetaBelowCell111133111222))
+        (by
+          have h : ((childLL (childHH (childHH thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHH
+            thetaBelowCell111133111222))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHH
+            thetaBelowCell111133111222))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHH
+            thetaBelowCell111133111222))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaBelowCell111133111222)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHH
+            thetaBelowCell111133111222))) h))
+
+theorem cover_subtree_006e01dadc14 :
+    adaptiveCoverCheck 6 thetaBelowCell111133111222 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133111222
+    (by
+      have h : ((childLL thetaBelowCell111133111222)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133111222) h)
+    (by
+      have h : ((childLH thetaBelowCell111133111222)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133111222) h)
+    cover_subtree_789cd4100198
+    cover_subtree_c3bc65d3adac
+
+theorem cover_subtree_15ffb840ba9a :
+    adaptiveCoverCheck 5 (childHL thetaBelowCell111133111223) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133111223)
+    (by
+      have h : ((childLL (childHL thetaBelowCell111133111223))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL thetaBelowCell111133111223)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell111133111223))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL thetaBelowCell111133111223)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL thetaBelowCell111133111223))
+        (by
+          have h : ((childLL (childHL (childHL thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHL
+            thetaBelowCell111133111223))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHL
+            thetaBelowCell111133111223))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHL
+            thetaBelowCell111133111223))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHL
+            thetaBelowCell111133111223))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL thetaBelowCell111133111223))
+        (by
+          have h : ((childLL (childHH (childHL thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHL
+            thetaBelowCell111133111223))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHL
+            thetaBelowCell111133111223))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHL
+            thetaBelowCell111133111223))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHL
+            thetaBelowCell111133111223))) h))
+
+theorem cover_subtree_f79e20abb6dd :
+    adaptiveCoverCheck 5 (childHH thetaBelowCell111133111223) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133111223)
+    (by
+      have h : ((childLL (childHH thetaBelowCell111133111223))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH thetaBelowCell111133111223)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell111133111223))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH thetaBelowCell111133111223)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH thetaBelowCell111133111223))
+        (by
+          have h : ((childLL (childHL (childHH thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHH
+            thetaBelowCell111133111223))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHH
+            thetaBelowCell111133111223))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHH
+            thetaBelowCell111133111223))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHH
+            thetaBelowCell111133111223))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH thetaBelowCell111133111223))
+        (by
+          have h : ((childLL (childHH (childHH thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHH
+            thetaBelowCell111133111223))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHH
+            thetaBelowCell111133111223))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHH
+            thetaBelowCell111133111223))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaBelowCell111133111223)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHH
+            thetaBelowCell111133111223))) h))
+
+theorem cover_subtree_c67dbdb52005 :
+    adaptiveCoverCheck 6 thetaBelowCell111133111223 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133111223
+    (by
+      have h : ((childLL thetaBelowCell111133111223)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133111223) h)
+    (by
+      have h : ((childLH thetaBelowCell111133111223)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133111223) h)
+    cover_subtree_15ffb840ba9a
+    cover_subtree_f79e20abb6dd
+
+theorem cover_subtree_7319345fe5a3 :
+    adaptiveCoverCheck 7 (childHL (childHL (childLH thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHL (childLH thetaBelowCell11113311)))
+    (by
+      have h : (thetaBelowCell111133111220).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133111220 h)
+    (by
+      have h : (thetaBelowCell111133111221).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133111221 h)
+    cover_subtree_006e01dadc14
+    cover_subtree_c67dbdb52005
+
+theorem cover_subtree_95410d7d0fe9 :
+    adaptiveCoverCheck 5 (childHL thetaBelowCell111133111232) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133111232)
+    (by
+      have h : ((childLL (childHL thetaBelowCell111133111232))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL thetaBelowCell111133111232)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell111133111232))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL thetaBelowCell111133111232)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL thetaBelowCell111133111232))
+        (by
+          have h : ((childLL (childHL (childHL thetaBelowCell111133111232)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHL
+            thetaBelowCell111133111232))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaBelowCell111133111232)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHL
+            thetaBelowCell111133111232))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaBelowCell111133111232)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHL
+            thetaBelowCell111133111232))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaBelowCell111133111232)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHL
+            thetaBelowCell111133111232))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL thetaBelowCell111133111232))
+        (by
+          have h : ((childLL (childHH (childHL thetaBelowCell111133111232)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHL
+            thetaBelowCell111133111232))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaBelowCell111133111232)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHL
+            thetaBelowCell111133111232))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaBelowCell111133111232)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHL
+            thetaBelowCell111133111232))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaBelowCell111133111232)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHL
+            thetaBelowCell111133111232))) h))
+
+theorem cover_subtree_bb5d3c6632a7 :
+    adaptiveCoverCheck 6 thetaBelowCell111133111232 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133111232
+    (by
+      have h : ((childLL thetaBelowCell111133111232)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133111232) h)
+    (by
+      have h : ((childLH thetaBelowCell111133111232)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133111232) h)
+    cover_subtree_95410d7d0fe9
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133111232)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133111232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133111232)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133111232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133111232)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133111232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133111232)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133111232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133111232)) h))
+
+theorem cover_subtree_36f6a3ec0dcf :
+    adaptiveCoverCheck 6 thetaBelowCell111133111233 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133111233
+    (by
+      have h : ((childLL thetaBelowCell111133111233)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133111233) h)
+    (by
+      have h : ((childLH thetaBelowCell111133111233)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133111233) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133111233)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133111233)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133111233)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133111233)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133111233)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133111233)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133111233)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133111233)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133111233)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133111233)) h))
+
+theorem cover_subtree_51a12797f9e1 :
+    adaptiveCoverCheck 7 (childHH (childHL (childLH thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHL (childLH thetaBelowCell11113311)))
+    (by
+      have h : (thetaBelowCell111133111230).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133111230 h)
+    (by
+      have h : (thetaBelowCell111133111231).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133111231 h)
+    cover_subtree_bb5d3c6632a7
+    cover_subtree_36f6a3ec0dcf
+
+theorem cover_subtree_8ababed519f4 :
+    adaptiveCoverCheck 8 (childHL (childLH thetaBelowCell11113311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH thetaBelowCell11113311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHL (childLH
+        thetaBelowCell11113311)))
+        (by
+          have h : (thetaBelowCell111133111200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133111200 h)
+        (by
+          have h : (thetaBelowCell111133111201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133111201 h)
+        (by
+          have h : (thetaBelowCell111133111202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133111202 h)
+        (by
+          have h : (thetaBelowCell111133111203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133111203 h))
+    (by
+      have h : ((childLH (childHL (childLH thetaBelowCell11113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childLH
+        thetaBelowCell11113311))) h)
+    cover_subtree_7319345fe5a3
+    cover_subtree_51a12797f9e1
+
+theorem cover_subtree_b8b0e3b2827e :
+    adaptiveCoverCheck 6 thetaBelowCell111133111322 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133111322
+    (by
+      have h : ((childLL thetaBelowCell111133111322)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133111322) h)
+    (by
+      have h : ((childLH thetaBelowCell111133111322)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133111322) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133111322)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133111322)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133111322)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133111322)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133111322)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133111322)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133111322)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133111322)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133111322)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133111322)) h))
+
+theorem cover_subtree_2989b06dc516 :
+    adaptiveCoverCheck 6 thetaBelowCell111133111323 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133111323
+    (by
+      have h : ((childLL thetaBelowCell111133111323)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133111323) h)
+    (by
+      have h : ((childLH thetaBelowCell111133111323)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133111323) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133111323)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133111323)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133111323)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133111323)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133111323)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133111323)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133111323)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133111323)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133111323)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133111323)) h))
+
+theorem cover_subtree_bf01e4b37001 :
+    adaptiveCoverCheck 7 (childHL (childHH (childLH thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHH (childLH thetaBelowCell11113311)))
+    (by
+      have h : (thetaBelowCell111133111320).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133111320 h)
+    (by
+      have h : (thetaBelowCell111133111321).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133111321 h)
+    cover_subtree_b8b0e3b2827e
+    cover_subtree_2989b06dc516
+
+theorem cover_subtree_730eb8da47c4 :
+    adaptiveCoverCheck 6 thetaBelowCell111133111332 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133111332
+    (by
+      have h : ((childLL thetaBelowCell111133111332)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133111332) h)
+    (by
+      have h : ((childLH thetaBelowCell111133111332)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133111332) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133111332)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133111332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133111332)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133111332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133111332)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133111332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133111332)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133111332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133111332)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133111332)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133111332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133111332)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133111332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133111332)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133111332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133111332)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133111332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133111332)) h))
+
+theorem cover_subtree_1841de2ff0e1 :
+    adaptiveCoverCheck 6 thetaBelowCell111133111333 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133111333
+    (by
+      have h : ((childLL thetaBelowCell111133111333)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133111333) h)
+    (by
+      have h : ((childLH thetaBelowCell111133111333)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133111333) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133111333)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133111333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133111333)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133111333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133111333)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133111333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133111333)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133111333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133111333)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133111333)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133111333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133111333)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133111333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133111333)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133111333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133111333)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133111333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133111333)) h))
+
+theorem cover_subtree_245b6882ed3e :
+    adaptiveCoverCheck 7 (childHH (childHH (childLH thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHH (childLH thetaBelowCell11113311)))
+    (by
+      have h : (thetaBelowCell111133111330).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133111330 h)
+    (by
+      have h : (thetaBelowCell111133111331).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133111331 h)
+    cover_subtree_730eb8da47c4
+    cover_subtree_1841de2ff0e1
+
+theorem cover_subtree_a132a8d74d97 :
+    adaptiveCoverCheck 8 (childHH (childLH thetaBelowCell11113311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH thetaBelowCell11113311))
+    (by
+      have h : ((childLL (childHH (childLH thetaBelowCell11113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childLH
+        thetaBelowCell11113311))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaBelowCell11113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childLH
+        thetaBelowCell11113311))) h)
+    cover_subtree_bf01e4b37001
+    cover_subtree_245b6882ed3e
+
+theorem e24KC2ThetaBelowLeaf111133111 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113311) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11113311)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH thetaBelowCell11113311))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLH
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLH
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLH
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLH
+            thetaBelowCell11113311))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH thetaBelowCell11113311))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLH
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLH
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLH
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLH
+            thetaBelowCell11113311))) h))
+    cover_subtree_8ababed519f4
+    cover_subtree_a132a8d74d97
+theorem cover_subtree_d86d6d397d37 :
+    adaptiveCoverCheck 6 thetaBelowCell111133112000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133112000
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133112000)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133112000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133112000)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133112000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133112000)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133112000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133112000)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133112000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133112000)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133112000)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133112000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133112000)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133112000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133112000)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133112000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133112000)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133112000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133112000)) h))
+    (by
+      have h : ((childHL thetaBelowCell111133112000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133112000) h)
+    (by
+      have h : ((childHH thetaBelowCell111133112000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133112000) h)
+
+theorem cover_subtree_6aa6edf26d6b :
+    adaptiveCoverCheck 6 thetaBelowCell111133112001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133112001
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133112001)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133112001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133112001)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133112001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133112001)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133112001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133112001)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133112001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133112001)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133112001)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133112001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133112001)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133112001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133112001)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133112001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133112001)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133112001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133112001)) h))
+    (by
+      have h : ((childHL thetaBelowCell111133112001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133112001) h)
+    (by
+      have h : ((childHH thetaBelowCell111133112001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133112001) h)
+
+theorem cover_subtree_08f0ea41225e :
+    adaptiveCoverCheck 7 (childLL (childLL (childHL thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLL (childHL thetaBelowCell11113311)))
+    cover_subtree_d86d6d397d37
+    cover_subtree_6aa6edf26d6b
+    (by
+      have h : (thetaBelowCell111133112002).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112002 h)
+    (by
+      have h : (thetaBelowCell111133112003).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112003 h)
+
+theorem cover_subtree_9a3060df2fad :
+    adaptiveCoverCheck 6 thetaBelowCell111133112010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133112010
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133112010)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133112010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133112010)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133112010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133112010)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133112010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133112010)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133112010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133112010)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133112010)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133112010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133112010)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133112010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133112010)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133112010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133112010)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133112010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133112010)) h))
+    (by
+      have h : ((childHL thetaBelowCell111133112010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133112010) h)
+    (by
+      have h : ((childHH thetaBelowCell111133112010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133112010) h)
+
+theorem cover_subtree_2515da0e05d0 :
+    adaptiveCoverCheck 5 (childLL thetaBelowCell111133112011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133112011)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL thetaBelowCell111133112011))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLL
+            thetaBelowCell111133112011))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLL
+            thetaBelowCell111133112011))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLL
+            thetaBelowCell111133112011))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLL
+            thetaBelowCell111133112011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL thetaBelowCell111133112011))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLL
+            thetaBelowCell111133112011))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLL
+            thetaBelowCell111133112011))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLL
+            thetaBelowCell111133112011))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLL
+            thetaBelowCell111133112011))) h))
+    (by
+      have h : ((childHL (childLL thetaBelowCell111133112011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL thetaBelowCell111133112011)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell111133112011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL thetaBelowCell111133112011)) h)
+
+theorem cover_subtree_2e5826939a0b :
+    adaptiveCoverCheck 5 (childLH thetaBelowCell111133112011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133112011)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH thetaBelowCell111133112011))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLH
+            thetaBelowCell111133112011))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLH
+            thetaBelowCell111133112011))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLH
+            thetaBelowCell111133112011))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLH
+            thetaBelowCell111133112011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH thetaBelowCell111133112011))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLH
+            thetaBelowCell111133112011))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLH
+            thetaBelowCell111133112011))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLH
+            thetaBelowCell111133112011))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLH
+            thetaBelowCell111133112011))) h))
+    (by
+      have h : ((childHL (childLH thetaBelowCell111133112011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH thetaBelowCell111133112011)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH thetaBelowCell111133112011))
+        (by
+          have h : ((childLL (childHH (childLH thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childLH
+            thetaBelowCell111133112011))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childLH
+            thetaBelowCell111133112011))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childLH
+            thetaBelowCell111133112011))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaBelowCell111133112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childLH
+            thetaBelowCell111133112011))) h))
+
+theorem cover_subtree_9e7e8b4def5a :
+    adaptiveCoverCheck 6 thetaBelowCell111133112011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133112011
+    cover_subtree_2515da0e05d0
+    cover_subtree_2e5826939a0b
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133112011)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133112011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133112011)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133112011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133112011)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133112011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133112011)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133112011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133112011)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133112011)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133112011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133112011)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133112011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133112011)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133112011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133112011)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133112011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133112011)) h))
+
+theorem cover_subtree_6ec9728983f6 :
+    adaptiveCoverCheck 7 (childLH (childLL (childHL thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLH (childLL (childHL thetaBelowCell11113311)))
+    cover_subtree_9a3060df2fad
+    cover_subtree_9e7e8b4def5a
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133112012
+        (by
+          have h : ((childLL thetaBelowCell111133112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133112012) h)
+        (by
+          have h : ((childLH thetaBelowCell111133112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133112012) h)
+        (by
+          have h : ((childHL thetaBelowCell111133112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133112012) h)
+        (by
+          have h : ((childHH thetaBelowCell111133112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133112012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133112013
+        (by
+          have h : ((childLL thetaBelowCell111133112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133112013) h)
+        (by
+          have h : ((childLH thetaBelowCell111133112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133112013) h)
+        (by
+          have h : ((childHL thetaBelowCell111133112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133112013) h)
+        (by
+          have h : ((childHH thetaBelowCell111133112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133112013) h))
+
+theorem cover_subtree_35cd51a2309c :
+    adaptiveCoverCheck 8 (childLL (childHL thetaBelowCell11113311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL thetaBelowCell11113311))
+    cover_subtree_08f0ea41225e
+    cover_subtree_6ec9728983f6
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLL (childHL
+        thetaBelowCell11113311)))
+        (by
+          have h : (thetaBelowCell111133112020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112020 h)
+        (by
+          have h : (thetaBelowCell111133112021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112021 h)
+        (by
+          have h : (thetaBelowCell111133112022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112022 h)
+        (by
+          have h : (thetaBelowCell111133112023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLL (childHL
+        thetaBelowCell11113311)))
+        (by
+          have h : (thetaBelowCell111133112030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112030 h)
+        (by
+          have h : (thetaBelowCell111133112031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112031 h)
+        (by
+          have h : (thetaBelowCell111133112032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112032 h)
+        (by
+          have h : (thetaBelowCell111133112033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112033 h))
+
+theorem cover_subtree_15005e137c79 :
+    adaptiveCoverCheck 5 (childLL thetaBelowCell111133112100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133112100)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL thetaBelowCell111133112100))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLL
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLL
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLL
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLL
+            thetaBelowCell111133112100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL thetaBelowCell111133112100))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLL
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLL
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLL
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLL
+            thetaBelowCell111133112100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL thetaBelowCell111133112100))
+        (by
+          have h : ((childLL (childHL (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childLL
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childLL
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childLL
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childLL
+            thetaBelowCell111133112100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL thetaBelowCell111133112100))
+        (by
+          have h : ((childLL (childHH (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childLL
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childLL
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childLL
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childLL
+            thetaBelowCell111133112100))) h))
+
+theorem cover_subtree_28298b1a6476 :
+    adaptiveCoverCheck 5 (childLH thetaBelowCell111133112100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133112100)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH thetaBelowCell111133112100))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLH
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLH
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLH
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLH
+            thetaBelowCell111133112100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH thetaBelowCell111133112100))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLH
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLH
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLH
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLH
+            thetaBelowCell111133112100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH thetaBelowCell111133112100))
+        (by
+          have h : ((childLL (childHL (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childLH
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childLH
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childLH
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childLH
+            thetaBelowCell111133112100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH thetaBelowCell111133112100))
+        (by
+          have h : ((childLL (childHH (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childLH
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childLH
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childLH
+            thetaBelowCell111133112100))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaBelowCell111133112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childLH
+            thetaBelowCell111133112100))) h))
+
+theorem cover_subtree_dd3b58e15058 :
+    adaptiveCoverCheck 6 thetaBelowCell111133112100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133112100
+    cover_subtree_15005e137c79
+    cover_subtree_28298b1a6476
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133112100)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133112100)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133112100)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133112100)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133112100)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133112100)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133112100)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133112100)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133112100)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133112100)) h))
+
+theorem cover_subtree_e1ad3991ba4f :
+    adaptiveCoverCheck 5 (childLL thetaBelowCell111133112101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133112101)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL thetaBelowCell111133112101))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLL
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLL
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLL
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLL
+            thetaBelowCell111133112101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL thetaBelowCell111133112101))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLL
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLL
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLL
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLL
+            thetaBelowCell111133112101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL thetaBelowCell111133112101))
+        (by
+          have h : ((childLL (childHL (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childLL
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childLL
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childLL
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childLL
+            thetaBelowCell111133112101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL thetaBelowCell111133112101))
+        (by
+          have h : ((childLL (childHH (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childLL
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childLL
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childLL
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childLL
+            thetaBelowCell111133112101))) h))
+
+theorem cover_subtree_222381d65095 :
+    adaptiveCoverCheck 5 (childLH thetaBelowCell111133112101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133112101)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH thetaBelowCell111133112101))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLH
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLH
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLH
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLH
+            thetaBelowCell111133112101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH thetaBelowCell111133112101))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLH
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLH
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLH
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLH
+            thetaBelowCell111133112101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH thetaBelowCell111133112101))
+        (by
+          have h : ((childLL (childHL (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childLH
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childLH
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childLH
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childLH
+            thetaBelowCell111133112101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH thetaBelowCell111133112101))
+        (by
+          have h : ((childLL (childHH (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childLH
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childLH
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childLH
+            thetaBelowCell111133112101))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaBelowCell111133112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childLH
+            thetaBelowCell111133112101))) h))
+
+theorem cover_subtree_a500ccb0f9dd :
+    adaptiveCoverCheck 6 thetaBelowCell111133112101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133112101
+    cover_subtree_e1ad3991ba4f
+    cover_subtree_222381d65095
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133112101)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133112101)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133112101)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133112101)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133112101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133112101)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133112101)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133112101)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133112101)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133112101)) h))
+
+theorem cover_subtree_b97026ab0ee7 :
+    adaptiveCoverCheck 7 (childLL (childLH (childHL thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLH (childHL thetaBelowCell11113311)))
+    cover_subtree_dd3b58e15058
+    cover_subtree_a500ccb0f9dd
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133112102
+        (by
+          have h : ((childLL thetaBelowCell111133112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133112102) h)
+        (by
+          have h : ((childLH thetaBelowCell111133112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133112102) h)
+        (by
+          have h : ((childHL thetaBelowCell111133112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133112102) h)
+        (by
+          have h : ((childHH thetaBelowCell111133112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133112102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133112103
+        (by
+          have h : ((childLL thetaBelowCell111133112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133112103) h)
+        (by
+          have h : ((childLH thetaBelowCell111133112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133112103) h)
+        (by
+          have h : ((childHL thetaBelowCell111133112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133112103) h)
+        (by
+          have h : ((childHH thetaBelowCell111133112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133112103) h))
+
+theorem cover_subtree_0942b83515e8 :
+    adaptiveCoverCheck 5 (childLL thetaBelowCell111133112110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133112110)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL thetaBelowCell111133112110))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLL
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLL
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLL
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLL
+            thetaBelowCell111133112110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL thetaBelowCell111133112110))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLL
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLL
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLL
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLL
+            thetaBelowCell111133112110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL thetaBelowCell111133112110))
+        (by
+          have h : ((childLL (childHL (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childLL
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childLL
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childLL
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childLL
+            thetaBelowCell111133112110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL thetaBelowCell111133112110))
+        (by
+          have h : ((childLL (childHH (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childLL
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childLL
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childLL
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childLL
+            thetaBelowCell111133112110))) h))
+
+theorem cover_subtree_4ccf6d139a48 :
+    adaptiveCoverCheck 5 (childLH thetaBelowCell111133112110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133112110)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH thetaBelowCell111133112110))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLH
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLH
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLH
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLH
+            thetaBelowCell111133112110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH thetaBelowCell111133112110))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLH
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLH
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLH
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLH
+            thetaBelowCell111133112110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH thetaBelowCell111133112110))
+        (by
+          have h : ((childLL (childHL (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childLH
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childLH
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childLH
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childLH
+            thetaBelowCell111133112110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH thetaBelowCell111133112110))
+        (by
+          have h : ((childLL (childHH (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childLH
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childLH
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childLH
+            thetaBelowCell111133112110))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaBelowCell111133112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childLH
+            thetaBelowCell111133112110))) h))
+
+theorem cover_subtree_b6335e66944c :
+    adaptiveCoverCheck 6 thetaBelowCell111133112110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133112110
+    cover_subtree_0942b83515e8
+    cover_subtree_4ccf6d139a48
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133112110)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133112110)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133112110)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133112110)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133112110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133112110)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133112110)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133112110)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133112110)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133112110)) h))
+
+theorem cover_subtree_964421ffff09 :
+    adaptiveCoverCheck 5 (childLL thetaBelowCell111133112111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133112111)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL thetaBelowCell111133112111))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLL
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLL
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLL
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLL
+            thetaBelowCell111133112111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL thetaBelowCell111133112111))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLL
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLL
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLL
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLL
+            thetaBelowCell111133112111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL thetaBelowCell111133112111))
+        (by
+          have h : ((childLL (childHL (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childLL
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childLL
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childLL
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childLL
+            thetaBelowCell111133112111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL thetaBelowCell111133112111))
+        (by
+          have h : ((childLL (childHH (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childLL
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childLL
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childLL
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childLL
+            thetaBelowCell111133112111))) h))
+
+theorem cover_subtree_d60f6638c35c :
+    adaptiveCoverCheck 5 (childLH thetaBelowCell111133112111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133112111)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH thetaBelowCell111133112111))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLH
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLH
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLH
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLH
+            thetaBelowCell111133112111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH thetaBelowCell111133112111))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLH
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLH
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLH
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLH
+            thetaBelowCell111133112111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH thetaBelowCell111133112111))
+        (by
+          have h : ((childLL (childHL (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childLH
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childLH
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childLH
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childLH
+            thetaBelowCell111133112111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH thetaBelowCell111133112111))
+        (by
+          have h : ((childLL (childHH (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childLH
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childLH
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childLH
+            thetaBelowCell111133112111))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaBelowCell111133112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childLH
+            thetaBelowCell111133112111))) h))
+
+theorem cover_subtree_fd996b69dc6a :
+    adaptiveCoverCheck 6 thetaBelowCell111133112111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133112111
+    cover_subtree_964421ffff09
+    cover_subtree_d60f6638c35c
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133112111)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133112111)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133112111)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133112111)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133112111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133112111)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133112111)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133112111)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133112111)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133112111)) h))
+
+theorem cover_subtree_4fddc783c766 :
+    adaptiveCoverCheck 7 (childLH (childLH (childHL thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLH (childLH (childHL thetaBelowCell11113311)))
+    cover_subtree_b6335e66944c
+    cover_subtree_fd996b69dc6a
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133112112
+        (by
+          have h : ((childLL thetaBelowCell111133112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133112112) h)
+        (by
+          have h : ((childLH thetaBelowCell111133112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133112112) h)
+        (by
+          have h : ((childHL thetaBelowCell111133112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133112112) h)
+        (by
+          have h : ((childHH thetaBelowCell111133112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133112112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133112113
+        (by
+          have h : ((childLL thetaBelowCell111133112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133112113) h)
+        (by
+          have h : ((childLH thetaBelowCell111133112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133112113) h)
+        (by
+          have h : ((childHL thetaBelowCell111133112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133112113) h)
+        (by
+          have h : ((childHH thetaBelowCell111133112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133112113) h))
+
+theorem cover_subtree_06a38a993bbb :
+    adaptiveCoverCheck 8 (childLH (childHL thetaBelowCell11113311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL thetaBelowCell11113311))
+    cover_subtree_b97026ab0ee7
+    cover_subtree_4fddc783c766
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLH (childHL
+        thetaBelowCell11113311)))
+        (by
+          have h : (thetaBelowCell111133112120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112120 h)
+        (by
+          have h : (thetaBelowCell111133112121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112121 h)
+        (by
+          have h : (thetaBelowCell111133112122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112122 h)
+        (by
+          have h : (thetaBelowCell111133112123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLH (childHL
+        thetaBelowCell11113311)))
+        (by
+          have h : (thetaBelowCell111133112130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112130 h)
+        (by
+          have h : (thetaBelowCell111133112131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112131 h)
+        (by
+          have h : (thetaBelowCell111133112132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112132 h)
+        (by
+          have h : (thetaBelowCell111133112133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133112133 h))
+
+theorem e24KC2ThetaBelowLeaf111133112 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113311) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113311)
+    cover_subtree_35cd51a2309c
+    cover_subtree_06a38a993bbb
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL thetaBelowCell11113311))
+        (by
+          have h : ((childLL (childHL (childHL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childHL
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childHL
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childHL
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childHL
+            thetaBelowCell11113311))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL thetaBelowCell11113311))
+        (by
+          have h : ((childLL (childHH (childHL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHL
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHL
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHL
+            thetaBelowCell11113311))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaBelowCell11113311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHL
+            thetaBelowCell11113311))) h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle008.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle008.lean
new file mode 100644
index 0000000000..a3e302e6f0
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle008.lean
@@ -0,0 +1,10035 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch009`.
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch024`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCellsb518b98fbe
+
+/-- Subcell `0110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0110 : AngleCell :=
+  childLL (childLH (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0111 : AngleCell :=
+  childLH (childLH (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `01103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0111)))
+
+end GerverSofa.PartE.CertificateCellsb518b98fbe
+
+namespace GerverSofa.PartE.CertificateCells37514173c1
+
+/-- Subcell `0011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0011 : AngleCell :=
+  childLH (childLH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0100 : AngleCell :=
+  childLL (childLL (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `1110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `00113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `01002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `11103022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11112022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11113022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `111132010310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132010310 : AngleCell :=
+  childLL (childLH (childHH (childLL thetaBelowCell11113201)))
+
+/-- Subcell `111132010311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132010311 : AngleCell :=
+  childLH (childLH (childHH (childLL thetaBelowCell11113201)))
+
+/-- Subcell `111132010312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132010312 : AngleCell :=
+  childHL (childLH (childHH (childLL thetaBelowCell11113201)))
+
+/-- Subcell `111132010313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132010313 : AngleCell :=
+  childHH (childLH (childHH (childLL thetaBelowCell11113201)))
+
+/-- Subcell `111132011120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011120 : AngleCell :=
+  childLL (childHL (childLH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011121 : AngleCell :=
+  childLH (childHL (childLH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011122 : AngleCell :=
+  childHL (childHL (childLH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011123 : AngleCell :=
+  childHH (childHL (childLH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011130 : AngleCell :=
+  childLL (childHH (childLH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011131 : AngleCell :=
+  childLH (childHH (childLH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011132 : AngleCell :=
+  childHL (childHH (childLH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011133 : AngleCell :=
+  childHH (childHH (childLH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011200 : AngleCell :=
+  childLL (childLL (childHL (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011201 : AngleCell :=
+  childLH (childLL (childHL (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011202 : AngleCell :=
+  childHL (childLL (childHL (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011203 : AngleCell :=
+  childHH (childLL (childHL (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011210 : AngleCell :=
+  childLL (childLH (childHL (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011211 : AngleCell :=
+  childLH (childLH (childHL (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011212 : AngleCell :=
+  childHL (childLH (childHL (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011213 : AngleCell :=
+  childHH (childLH (childHL (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011300 : AngleCell :=
+  childLL (childLL (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011301 : AngleCell :=
+  childLH (childLL (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011302 : AngleCell :=
+  childHL (childLL (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011303 : AngleCell :=
+  childHH (childLL (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011310 : AngleCell :=
+  childLL (childLH (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011311 : AngleCell :=
+  childLH (childLH (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011312 : AngleCell :=
+  childHL (childLH (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011313 : AngleCell :=
+  childHH (childLH (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132011333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132011333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaBelowCell11113201)))
+
+/-- Subcell `111132100000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100000 : AngleCell :=
+  childLL (childLL (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100001 : AngleCell :=
+  childLH (childLL (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100002 : AngleCell :=
+  childHL (childLL (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100003 : AngleCell :=
+  childHH (childLL (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100010 : AngleCell :=
+  childLL (childLH (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100011 : AngleCell :=
+  childLH (childLH (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100012 : AngleCell :=
+  childHL (childLH (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100013 : AngleCell :=
+  childHH (childLH (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100020 : AngleCell :=
+  childLL (childHL (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100021 : AngleCell :=
+  childLH (childHL (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100022 : AngleCell :=
+  childHL (childHL (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100023 : AngleCell :=
+  childHH (childHL (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100030 : AngleCell :=
+  childLL (childHH (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100031 : AngleCell :=
+  childLH (childHH (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100032 : AngleCell :=
+  childHL (childHH (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100033 : AngleCell :=
+  childHH (childHH (childLL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100100 : AngleCell :=
+  childLL (childLL (childLH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100101 : AngleCell :=
+  childLH (childLL (childLH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100102 : AngleCell :=
+  childHL (childLL (childLH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100103 : AngleCell :=
+  childHH (childLL (childLH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100120 : AngleCell :=
+  childLL (childHL (childLH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100121 : AngleCell :=
+  childLH (childHL (childLH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100122 : AngleCell :=
+  childHL (childHL (childLH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100123 : AngleCell :=
+  childHH (childHL (childLH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100130 : AngleCell :=
+  childLL (childHH (childLH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100131 : AngleCell :=
+  childLH (childHH (childLH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100132 : AngleCell :=
+  childHL (childHH (childLH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100133 : AngleCell :=
+  childHH (childHH (childLH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100200 : AngleCell :=
+  childLL (childLL (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100201 : AngleCell :=
+  childLH (childLL (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100202 : AngleCell :=
+  childHL (childLL (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100203 : AngleCell :=
+  childHH (childLL (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100210 : AngleCell :=
+  childLL (childLH (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100211 : AngleCell :=
+  childLH (childLH (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100212 : AngleCell :=
+  childHL (childLH (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100213 : AngleCell :=
+  childHH (childLH (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100300 : AngleCell :=
+  childLL (childLL (childHH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100301 : AngleCell :=
+  childLH (childLL (childHH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100302 : AngleCell :=
+  childHL (childLL (childHH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100303 : AngleCell :=
+  childHH (childLL (childHH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100310 : AngleCell :=
+  childLL (childLH (childHH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100311 : AngleCell :=
+  childLH (childLH (childHH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100312 : AngleCell :=
+  childHL (childLH (childHH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100313 : AngleCell :=
+  childHH (childLH (childHH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaBelowCell11113210)))
+
+/-- Subcell `111132100333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111132100333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaBelowCell11113210)))
+
+end GerverSofa.PartE.CertificateCells37514173c1
+
+namespace GerverSofa.PartE.CertificateCells00c658e5bf
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaBelowCell1111)))
+
+end GerverSofa.PartE.CertificateCells00c658e5bf
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC5TerminalBatchT307200009`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb518b98fbe
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb518b98fbe
+
+open CertificateCellsb518b98fbe
+theorem e24KC2ThetaAboveLeaf0110321030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103210)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103210)) h
+theorem e24KC2ThetaAboveLeaf0110321031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103210)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103210)) h
+theorem e24KC2ThetaAboveLeaf0110321032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103210)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103210)) h
+theorem e24KC2ThetaAboveLeaf0110321033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103210)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103210)) h
+theorem e24KC2ThetaAboveLeaf011032110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103211) = true := by
+  have h : ((childLL thetaAboveCell01103211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103211) h
+theorem e24KC2ThetaAboveLeaf011032111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103211) = true := by
+  have h : ((childLH thetaAboveCell01103211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103211) h
+theorem e24KC2ThetaAboveLeaf0110321120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103211)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103211)) h
+theorem e24KC2ThetaAboveLeaf0110321121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103211)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103211)) h
+theorem e24KC2ThetaAboveLeaf0110321122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103211)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103211)) h
+theorem e24KC2ThetaAboveLeaf0110321123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103211)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103211)) h
+theorem e24KC2ThetaAboveLeaf0110321130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103211)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103211)) h
+theorem e24KC2ThetaAboveLeaf0110321131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103211)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103211)) h
+theorem e24KC2ThetaAboveLeaf0110321132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103211)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103211)) h
+theorem e24KC2ThetaAboveLeaf0110321133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103211)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103211)) h
+theorem e24KC2ThetaAboveLeaf0110321200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01103212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01103212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01103212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01103212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01103212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01103212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01103212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01103212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103212)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103212)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103212)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103212)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103212)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103212)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103212)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103212)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103212)) h
+theorem e24KC2ThetaAboveLeaf0110321300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01103213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf0110321301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01103213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf0110321302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01103213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf0110321303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01103213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf0110321310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01103213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf0110321311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01103213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf0110321312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01103213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf0110321313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01103213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf0110321320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103213)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf0110321321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103213)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf0110321322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103213)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf0110321323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103213)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf0110321330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103213)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf0110321331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103213)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf0110321332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103213)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf0110321333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103213)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103213)) h
+theorem e24KC2ThetaAboveLeaf011032200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103220) = true := by
+  have h : ((childLL thetaAboveCell01103220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103220) h
+theorem e24KC2ThetaAboveLeaf011032201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103220) = true := by
+  have h : ((childLH thetaAboveCell01103220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103220) h
+theorem e24KC2ThetaAboveLeaf011032202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103220) = true := by
+  have h : ((childHL thetaAboveCell01103220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01103220) h
+theorem e24KC2ThetaAboveLeaf011032203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103220) = true := by
+  have h : ((childHH thetaAboveCell01103220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01103220) h
+theorem e24KC2ThetaAboveLeaf011032210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103221) = true := by
+  have h : ((childLL thetaAboveCell01103221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103221) h
+theorem e24KC2ThetaAboveLeaf011032211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103221) = true := by
+  have h : ((childLH thetaAboveCell01103221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103221) h
+theorem e24KC2ThetaAboveLeaf011032212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103221) = true := by
+  have h : ((childHL thetaAboveCell01103221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01103221) h
+theorem e24KC2ThetaAboveLeaf011032213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103221) = true := by
+  have h : ((childHH thetaAboveCell01103221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01103221) h
+theorem e24KC2ThetaAboveLeaf01103222 :
+    adaptiveCoverCheck 11 thetaAboveCell01103222 = true := by
+  have h : (thetaAboveCell01103222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103222 h
+theorem e24KC2ThetaAboveLeaf01103223 :
+    adaptiveCoverCheck 11 thetaAboveCell01103223 = true := by
+  have h : (thetaAboveCell01103223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103223 h
+theorem e24KC2ThetaAboveLeaf011032300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103230) = true := by
+  have h : ((childLL thetaAboveCell01103230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103230) h
+theorem e24KC2ThetaAboveLeaf011032301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103230) = true := by
+  have h : ((childLH thetaAboveCell01103230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103230) h
+theorem e24KC2ThetaAboveLeaf011032302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103230) = true := by
+  have h : ((childHL thetaAboveCell01103230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01103230) h
+theorem e24KC2ThetaAboveLeaf011032303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103230) = true := by
+  have h : ((childHH thetaAboveCell01103230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01103230) h
+theorem e24KC2ThetaAboveLeaf011032310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103231) = true := by
+  have h : ((childLL thetaAboveCell01103231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103231) h
+theorem e24KC2ThetaAboveLeaf011032311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103231) = true := by
+  have h : ((childLH thetaAboveCell01103231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103231) h
+theorem e24KC2ThetaAboveLeaf011032312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103231) = true := by
+  have h : ((childHL thetaAboveCell01103231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01103231) h
+theorem e24KC2ThetaAboveLeaf011032313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103231) = true := by
+  have h : ((childHH thetaAboveCell01103231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01103231) h
+theorem e24KC2ThetaAboveLeaf01103232 :
+    adaptiveCoverCheck 11 thetaAboveCell01103232 = true := by
+  have h : (thetaAboveCell01103232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103232 h
+theorem e24KC2ThetaAboveLeaf01103233 :
+    adaptiveCoverCheck 11 thetaAboveCell01103233 = true := by
+  have h : (thetaAboveCell01103233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103233 h
+theorem e24KC2ThetaAboveLeaf011033000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103300) = true := by
+  have h : ((childLL thetaAboveCell01103300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103300) h
+theorem e24KC2ThetaAboveLeaf011033001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103300) = true := by
+  have h : ((childLH thetaAboveCell01103300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103300) h
+theorem e24KC2ThetaAboveLeaf0110330020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103300)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103300)) h
+theorem e24KC2ThetaAboveLeaf0110330021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103300)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103300)) h
+theorem e24KC2ThetaAboveLeaf0110330022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103300)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103300)) h
+theorem e24KC2ThetaAboveLeaf0110330023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103300)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103300)) h
+theorem e24KC2ThetaAboveLeaf0110330030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103300)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103300)) h
+theorem e24KC2ThetaAboveLeaf0110330031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103300)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103300)) h
+theorem e24KC2ThetaAboveLeaf0110330032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103300)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103300)) h
+theorem e24KC2ThetaAboveLeaf0110330033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103300)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103300)) h
+theorem e24KC2ThetaAboveLeaf011033010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103301) = true := by
+  have h : ((childLL thetaAboveCell01103301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103301) h
+theorem e24KC2ThetaAboveLeaf011033011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103301) = true := by
+  have h : ((childLH thetaAboveCell01103301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103301) h
+theorem e24KC2ThetaAboveLeaf0110330120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103301)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103301)) h
+theorem e24KC2ThetaAboveLeaf0110330121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103301)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103301)) h
+theorem e24KC2ThetaAboveLeaf0110330122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103301)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103301)) h
+theorem e24KC2ThetaAboveLeaf0110330123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103301)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103301)) h
+theorem e24KC2ThetaAboveLeaf0110330130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103301)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103301)) h
+theorem e24KC2ThetaAboveLeaf0110330131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103301)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103301)) h
+theorem e24KC2ThetaAboveLeaf0110330132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103301)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103301)) h
+theorem e24KC2ThetaAboveLeaf0110330133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103301)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103301)) h
+theorem e24KC2ThetaAboveLeaf0110330200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01103302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01103302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01103302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01103302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01103302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01103302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01103302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01103302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103302)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103302)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103302)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103302)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103302)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103302)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103302)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103302)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103302)) h
+theorem e24KC2ThetaAboveLeaf0110330300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01103303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf0110330301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01103303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf0110330302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01103303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf0110330303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01103303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf0110330310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01103303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf0110330311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01103303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf0110330312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01103303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf0110330313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01103303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf0110330320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103303)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf0110330321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103303)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf0110330322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103303)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf0110330323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103303)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf0110330330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103303)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf0110330331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103303)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf0110330332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103303)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf0110330333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103303)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103303)) h
+theorem e24KC2ThetaAboveLeaf011033100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103310) = true := by
+  have h : ((childLL thetaAboveCell01103310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103310) h
+theorem e24KC2ThetaAboveLeaf011033101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103310) = true := by
+  have h : ((childLH thetaAboveCell01103310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103310) h
+theorem e24KC2ThetaAboveLeaf0110331020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103310)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103310)) h
+theorem e24KC2ThetaAboveLeaf0110331021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103310)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103310)) h
+theorem e24KC2ThetaAboveLeaf0110331022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103310)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103310)) h
+theorem e24KC2ThetaAboveLeaf0110331023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103310)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103310)) h
+theorem e24KC2ThetaAboveLeaf0110331030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103310)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103310)) h
+theorem e24KC2ThetaAboveLeaf0110331031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103310)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103310)) h
+theorem e24KC2ThetaAboveLeaf0110331032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103310)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103310)) h
+theorem e24KC2ThetaAboveLeaf0110331033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103310)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103310)) h
+theorem e24KC2ThetaAboveLeaf011033110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103311) = true := by
+  have h : ((childLL thetaAboveCell01103311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103311) h
+theorem e24KC2ThetaAboveLeaf011033111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103311) = true := by
+  have h : ((childLH thetaAboveCell01103311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103311) h
+theorem e24KC2ThetaAboveLeaf0110331120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103311)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103311)) h
+theorem e24KC2ThetaAboveLeaf0110331121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103311)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103311)) h
+theorem e24KC2ThetaAboveLeaf0110331122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103311)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103311)) h
+theorem e24KC2ThetaAboveLeaf0110331123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103311)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103311)) h
+theorem e24KC2ThetaAboveLeaf0110331130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103311)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103311)) h
+theorem e24KC2ThetaAboveLeaf0110331131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103311)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103311)) h
+theorem e24KC2ThetaAboveLeaf0110331132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103311)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103311)) h
+theorem e24KC2ThetaAboveLeaf0110331133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103311)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103311)) h
+theorem e24KC2ThetaAboveLeaf0110331200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01103312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01103312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01103312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01103312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01103312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01103312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01103312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01103312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103312)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103312)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103312)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103312)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103312)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103312)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103312)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103312)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103312)) h
+theorem e24KC2ThetaAboveLeaf0110331300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01103313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf0110331301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01103313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf0110331302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01103313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf0110331303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01103313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf0110331310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01103313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf0110331311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01103313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf0110331312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01103313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf0110331313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01103313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf0110331320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01103313)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf0110331321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01103313)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf0110331322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01103313)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf0110331323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01103313)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf0110331330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01103313)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf0110331331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01103313)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf0110331332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01103313)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf0110331333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01103313)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01103313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01103313)) h
+theorem e24KC2ThetaAboveLeaf011033200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103320) = true := by
+  have h : ((childLL thetaAboveCell01103320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103320) h
+theorem e24KC2ThetaAboveLeaf011033201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103320) = true := by
+  have h : ((childLH thetaAboveCell01103320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103320) h
+theorem e24KC2ThetaAboveLeaf011033202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103320) = true := by
+  have h : ((childHL thetaAboveCell01103320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01103320) h
+theorem e24KC2ThetaAboveLeaf011033203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103320) = true := by
+  have h : ((childHH thetaAboveCell01103320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01103320) h
+theorem e24KC2ThetaAboveLeaf011033210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103321) = true := by
+  have h : ((childLL thetaAboveCell01103321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103321) h
+theorem e24KC2ThetaAboveLeaf011033211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103321) = true := by
+  have h : ((childLH thetaAboveCell01103321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103321) h
+theorem e24KC2ThetaAboveLeaf011033212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103321) = true := by
+  have h : ((childHL thetaAboveCell01103321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01103321) h
+theorem e24KC2ThetaAboveLeaf011033213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103321) = true := by
+  have h : ((childHH thetaAboveCell01103321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01103321) h
+theorem e24KC2ThetaAboveLeaf01103322 :
+    adaptiveCoverCheck 11 thetaAboveCell01103322 = true := by
+  have h : (thetaAboveCell01103322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103322 h
+theorem e24KC2ThetaAboveLeaf01103323 :
+    adaptiveCoverCheck 11 thetaAboveCell01103323 = true := by
+  have h : (thetaAboveCell01103323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103323 h
+theorem e24KC2ThetaAboveLeaf011033300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103330) = true := by
+  have h : ((childLL thetaAboveCell01103330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103330) h
+theorem e24KC2ThetaAboveLeaf011033301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103330) = true := by
+  have h : ((childLH thetaAboveCell01103330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103330) h
+theorem e24KC2ThetaAboveLeaf011033302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103330) = true := by
+  have h : ((childHL thetaAboveCell01103330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01103330) h
+theorem e24KC2ThetaAboveLeaf011033303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103330) = true := by
+  have h : ((childHH thetaAboveCell01103330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01103330) h
+theorem e24KC2ThetaAboveLeaf011033310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103331) = true := by
+  have h : ((childLL thetaAboveCell01103331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01103331) h
+theorem e24KC2ThetaAboveLeaf011033311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103331) = true := by
+  have h : ((childLH thetaAboveCell01103331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01103331) h
+theorem e24KC2ThetaAboveLeaf011033312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103331) = true := by
+  have h : ((childHL thetaAboveCell01103331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01103331) h
+theorem e24KC2ThetaAboveLeaf011033313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103331) = true := by
+  have h : ((childHH thetaAboveCell01103331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01103331) h
+theorem e24KC2ThetaAboveLeaf01103332 :
+    adaptiveCoverCheck 11 thetaAboveCell01103332 = true := by
+  have h : (thetaAboveCell01103332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103332 h
+theorem e24KC2ThetaAboveLeaf01103333 :
+    adaptiveCoverCheck 11 thetaAboveCell01103333 = true := by
+  have h : (thetaAboveCell01103333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01103333 h
+theorem e24KC2ThetaAboveLeaf011100 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0111)) = true := by
+  have h : ((childLL (childLL thetaAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell0111)) h
+theorem e24KC2ThetaAboveLeaf011101 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0111)) = true := by
+  have h : ((childLH (childLL thetaAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell0111)) h
+theorem e24KC2ThetaAboveLeaf011102 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0111)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0111)) h
+theorem e24KC2ThetaAboveLeaf011103 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0111)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0111)) h
+theorem e24KC2ThetaAboveLeaf011110 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0111)) = true := by
+  have h : ((childLL (childLH thetaAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell0111)) h
+theorem e24KC2ThetaAboveLeaf011111 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0111)) = true := by
+  have h : ((childLH (childLH thetaAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell0111)) h
+theorem e24KC2ThetaAboveLeaf011112 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0111)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0111)) h
+theorem e24KC2ThetaAboveLeaf011113 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0111)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0111)) h
+theorem e24KC2ThetaAboveLeaf0111200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell0111))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell0111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell0111))) h
+theorem e24KC2ThetaAboveLeaf0111201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell0111))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell0111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell0111))) h
+theorem e24KC2ThetaAboveLeaf01112020 :
+    adaptiveCoverCheck 11 thetaAboveCell01112020 = true := by
+  have h : (thetaAboveCell01112020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112020 h
+theorem e24KC2ThetaAboveLeaf01112021 :
+    adaptiveCoverCheck 11 thetaAboveCell01112021 = true := by
+  have h : (thetaAboveCell01112021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112021 h
+theorem e24KC2ThetaAboveLeaf01112022 :
+    adaptiveCoverCheck 11 thetaAboveCell01112022 = true := by
+  have h : (thetaAboveCell01112022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112022 h
+theorem e24KC2ThetaAboveLeaf01112023 :
+    adaptiveCoverCheck 11 thetaAboveCell01112023 = true := by
+  have h : (thetaAboveCell01112023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112023 h
+theorem e24KC2ThetaAboveLeaf01112030 :
+    adaptiveCoverCheck 11 thetaAboveCell01112030 = true := by
+  have h : (thetaAboveCell01112030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112030 h
+theorem e24KC2ThetaAboveLeaf01112031 :
+    adaptiveCoverCheck 11 thetaAboveCell01112031 = true := by
+  have h : (thetaAboveCell01112031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112031 h
+theorem e24KC2ThetaAboveLeaf01112032 :
+    adaptiveCoverCheck 11 thetaAboveCell01112032 = true := by
+  have h : (thetaAboveCell01112032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112032 h
+theorem e24KC2ThetaAboveLeaf01112033 :
+    adaptiveCoverCheck 11 thetaAboveCell01112033 = true := by
+  have h : (thetaAboveCell01112033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112033 h
+theorem e24KC2ThetaAboveLeaf0111210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell0111))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell0111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell0111))) h
+theorem e24KC2ThetaAboveLeaf0111211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell0111))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell0111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell0111))) h
+theorem e24KC2ThetaAboveLeaf01112120 :
+    adaptiveCoverCheck 11 thetaAboveCell01112120 = true := by
+  have h : (thetaAboveCell01112120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112120 h
+theorem e24KC2ThetaAboveLeaf01112121 :
+    adaptiveCoverCheck 11 thetaAboveCell01112121 = true := by
+  have h : (thetaAboveCell01112121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112121 h
+theorem e24KC2ThetaAboveLeaf01112122 :
+    adaptiveCoverCheck 11 thetaAboveCell01112122 = true := by
+  have h : (thetaAboveCell01112122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112122 h
+theorem e24KC2ThetaAboveLeaf01112123 :
+    adaptiveCoverCheck 11 thetaAboveCell01112123 = true := by
+  have h : (thetaAboveCell01112123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112123 h
+theorem e24KC2ThetaAboveLeaf01112130 :
+    adaptiveCoverCheck 11 thetaAboveCell01112130 = true := by
+  have h : (thetaAboveCell01112130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112130 h
+theorem e24KC2ThetaAboveLeaf01112131 :
+    adaptiveCoverCheck 11 thetaAboveCell01112131 = true := by
+  have h : (thetaAboveCell01112131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112131 h
+theorem e24KC2ThetaAboveLeaf01112132 :
+    adaptiveCoverCheck 11 thetaAboveCell01112132 = true := by
+  have h : (thetaAboveCell01112132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112132 h
+theorem e24KC2ThetaAboveLeaf01112133 :
+    adaptiveCoverCheck 11 thetaAboveCell01112133 = true := by
+  have h : (thetaAboveCell01112133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112133 h
+theorem e24KC2ThetaAboveLeaf011122000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112200) = true := by
+  have h : ((childLL thetaAboveCell01112200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112200) h
+theorem e24KC2ThetaAboveLeaf011122001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112200) = true := by
+  have h : ((childLH thetaAboveCell01112200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112200) h
+theorem e24KC2ThetaAboveLeaf0111220020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112200)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112200)) h
+theorem e24KC2ThetaAboveLeaf0111220021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112200)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112200)) h
+theorem e24KC2ThetaAboveLeaf0111220022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112200)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112200)) h
+theorem e24KC2ThetaAboveLeaf0111220023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112200)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112200)) h
+theorem e24KC2ThetaAboveLeaf0111220030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112200)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112200)) h
+theorem e24KC2ThetaAboveLeaf0111220031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112200)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112200)) h
+theorem e24KC2ThetaAboveLeaf0111220032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112200)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112200)) h
+theorem e24KC2ThetaAboveLeaf0111220033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112200)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112200)) h
+theorem e24KC2ThetaAboveLeaf011122010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112201) = true := by
+  have h : ((childLL thetaAboveCell01112201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112201) h
+theorem e24KC2ThetaAboveLeaf011122011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112201) = true := by
+  have h : ((childLH thetaAboveCell01112201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112201) h
+theorem e24KC2ThetaAboveLeaf0111220120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112201)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112201)) h
+theorem e24KC2ThetaAboveLeaf0111220121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112201)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112201)) h
+theorem e24KC2ThetaAboveLeaf0111220122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112201)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112201)) h
+theorem e24KC2ThetaAboveLeaf0111220123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112201)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112201)) h
+theorem e24KC2ThetaAboveLeaf0111220130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112201)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112201)) h
+theorem e24KC2ThetaAboveLeaf0111220131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112201)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112201)) h
+theorem e24KC2ThetaAboveLeaf0111220132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112201)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112201)) h
+theorem e24KC2ThetaAboveLeaf0111220133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112201)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112201)) h
+theorem e24KC2ThetaAboveLeaf0111220200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01112202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01112202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01112202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01112202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01112202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01112202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01112202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01112202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112202)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112202)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112202)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112202)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112202)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112202)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112202)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112202)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112202)) h
+theorem e24KC2ThetaAboveLeaf0111220300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01112203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf0111220301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01112203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf0111220302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01112203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf0111220303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01112203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf0111220310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01112203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf0111220311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01112203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf0111220312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01112203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf0111220313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01112203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf0111220320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112203)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf0111220321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112203)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf0111220322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112203)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf0111220323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112203)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf0111220330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112203)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf0111220331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112203)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf0111220332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112203)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf0111220333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112203)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112203)) h
+theorem e24KC2ThetaAboveLeaf011122100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112210) = true := by
+  have h : ((childLL thetaAboveCell01112210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112210) h
+theorem e24KC2ThetaAboveLeaf011122101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112210) = true := by
+  have h : ((childLH thetaAboveCell01112210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112210) h
+theorem e24KC2ThetaAboveLeaf0111221020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112210)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112210)) h
+theorem e24KC2ThetaAboveLeaf0111221021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112210)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112210)) h
+theorem e24KC2ThetaAboveLeaf0111221022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112210)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112210)) h
+theorem e24KC2ThetaAboveLeaf0111221023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112210)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112210)) h
+theorem e24KC2ThetaAboveLeaf0111221030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112210)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112210)) h
+theorem e24KC2ThetaAboveLeaf0111221031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112210)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112210)) h
+theorem e24KC2ThetaAboveLeaf0111221032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112210)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112210)) h
+theorem e24KC2ThetaAboveLeaf0111221033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112210)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112210)) h
+theorem e24KC2ThetaAboveLeaf011122110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112211) = true := by
+  have h : ((childLL thetaAboveCell01112211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112211) h
+theorem e24KC2ThetaAboveLeaf011122111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112211) = true := by
+  have h : ((childLH thetaAboveCell01112211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112211) h
+theorem e24KC2ThetaAboveLeaf0111221120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112211)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112211)) h
+theorem e24KC2ThetaAboveLeaf0111221121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112211)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112211)) h
+theorem e24KC2ThetaAboveLeaf0111221122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112211)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112211)) h
+theorem e24KC2ThetaAboveLeaf0111221123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112211)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112211)) h
+theorem e24KC2ThetaAboveLeaf0111221130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112211)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112211)) h
+theorem e24KC2ThetaAboveLeaf0111221131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112211)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112211)) h
+theorem e24KC2ThetaAboveLeaf0111221132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112211)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112211)) h
+theorem e24KC2ThetaAboveLeaf0111221133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112211)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112211)) h
+theorem e24KC2ThetaAboveLeaf0111221200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01112212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01112212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01112212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01112212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01112212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01112212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01112212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01112212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112212)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112212)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112212)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112212)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112212)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112212)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112212)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112212)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112212)) h
+theorem e24KC2ThetaAboveLeaf0111221300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01112213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf0111221301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01112213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf0111221302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01112213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf0111221303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01112213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf0111221310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01112213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf0111221311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01112213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf0111221312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01112213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf0111221313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01112213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf0111221320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112213)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf0111221321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112213)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf0111221322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112213)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf0111221323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112213)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf0111221330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112213)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf0111221331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112213)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf0111221332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112213)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf0111221333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112213)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112213)) h
+theorem e24KC2ThetaAboveLeaf011122200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112220) = true := by
+  have h : ((childLL thetaAboveCell01112220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112220) h
+theorem e24KC2ThetaAboveLeaf011122201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112220) = true := by
+  have h : ((childLH thetaAboveCell01112220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112220) h
+theorem e24KC2ThetaAboveLeaf011122202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112220) = true := by
+  have h : ((childHL thetaAboveCell01112220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01112220) h
+theorem e24KC2ThetaAboveLeaf011122203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112220) = true := by
+  have h : ((childHH thetaAboveCell01112220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01112220) h
+theorem e24KC2ThetaAboveLeaf011122210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112221) = true := by
+  have h : ((childLL thetaAboveCell01112221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112221) h
+theorem e24KC2ThetaAboveLeaf011122211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112221) = true := by
+  have h : ((childLH thetaAboveCell01112221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112221) h
+theorem e24KC2ThetaAboveLeaf011122212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112221) = true := by
+  have h : ((childHL thetaAboveCell01112221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01112221) h
+theorem e24KC2ThetaAboveLeaf011122213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112221) = true := by
+  have h : ((childHH thetaAboveCell01112221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01112221) h
+theorem e24KC2ThetaAboveLeaf01112222 :
+    adaptiveCoverCheck 11 thetaAboveCell01112222 = true := by
+  have h : (thetaAboveCell01112222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112222 h
+theorem e24KC2ThetaAboveLeaf01112223 :
+    adaptiveCoverCheck 11 thetaAboveCell01112223 = true := by
+  have h : (thetaAboveCell01112223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112223 h
+theorem e24KC2ThetaAboveLeaf011122300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112230) = true := by
+  have h : ((childLL thetaAboveCell01112230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112230) h
+theorem e24KC2ThetaAboveLeaf011122301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112230) = true := by
+  have h : ((childLH thetaAboveCell01112230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112230) h
+theorem e24KC2ThetaAboveLeaf011122302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112230) = true := by
+  have h : ((childHL thetaAboveCell01112230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01112230) h
+theorem e24KC2ThetaAboveLeaf011122303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112230) = true := by
+  have h : ((childHH thetaAboveCell01112230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01112230) h
+theorem e24KC2ThetaAboveLeaf011122310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112231) = true := by
+  have h : ((childLL thetaAboveCell01112231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112231) h
+theorem e24KC2ThetaAboveLeaf011122311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112231) = true := by
+  have h : ((childLH thetaAboveCell01112231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112231) h
+theorem e24KC2ThetaAboveLeaf011122312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112231) = true := by
+  have h : ((childHL thetaAboveCell01112231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01112231) h
+theorem e24KC2ThetaAboveLeaf011122313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112231) = true := by
+  have h : ((childHH thetaAboveCell01112231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01112231) h
+theorem e24KC2ThetaAboveLeaf01112232 :
+    adaptiveCoverCheck 11 thetaAboveCell01112232 = true := by
+  have h : (thetaAboveCell01112232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112232 h
+theorem e24KC2ThetaAboveLeaf01112233 :
+    adaptiveCoverCheck 11 thetaAboveCell01112233 = true := by
+  have h : (thetaAboveCell01112233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112233 h
+theorem e24KC2ThetaAboveLeaf011123000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112300) = true := by
+  have h : ((childLL thetaAboveCell01112300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112300) h
+theorem e24KC2ThetaAboveLeaf011123001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112300) = true := by
+  have h : ((childLH thetaAboveCell01112300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112300) h
+theorem e24KC2ThetaAboveLeaf0111230020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112300)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112300)) h
+theorem e24KC2ThetaAboveLeaf0111230021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112300)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112300)) h
+theorem e24KC2ThetaAboveLeaf0111230022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112300)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112300)) h
+theorem e24KC2ThetaAboveLeaf0111230023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112300)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112300)) h
+theorem e24KC2ThetaAboveLeaf0111230030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112300)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112300)) h
+theorem e24KC2ThetaAboveLeaf0111230031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112300)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112300)) h
+theorem e24KC2ThetaAboveLeaf0111230032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112300)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112300)) h
+theorem e24KC2ThetaAboveLeaf0111230033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112300)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112300)) h
+theorem e24KC2ThetaAboveLeaf011123010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112301) = true := by
+  have h : ((childLL thetaAboveCell01112301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112301) h
+theorem e24KC2ThetaAboveLeaf011123011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112301) = true := by
+  have h : ((childLH thetaAboveCell01112301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112301) h
+theorem e24KC2ThetaAboveLeaf0111230120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112301)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112301)) h
+theorem e24KC2ThetaAboveLeaf0111230121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112301)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112301)) h
+theorem e24KC2ThetaAboveLeaf0111230122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112301)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112301)) h
+theorem e24KC2ThetaAboveLeaf0111230123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112301)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112301)) h
+theorem e24KC2ThetaAboveLeaf0111230130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112301)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112301)) h
+theorem e24KC2ThetaAboveLeaf0111230131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112301)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112301)) h
+theorem e24KC2ThetaAboveLeaf0111230132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112301)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112301)) h
+theorem e24KC2ThetaAboveLeaf0111230133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112301)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112301)) h
+theorem e24KC2ThetaAboveLeaf0111230200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01112302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01112302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01112302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01112302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01112302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01112302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01112302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01112302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112302)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112302)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112302)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112302)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112302)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112302)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112302)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112302)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112302)) h
+theorem e24KC2ThetaAboveLeaf0111230300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01112303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf0111230301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01112303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf0111230302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01112303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf0111230303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01112303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf0111230310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01112303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf0111230311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01112303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf0111230312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01112303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf0111230313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01112303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf0111230320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112303)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf0111230321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112303)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf0111230322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112303)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf0111230323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112303)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf0111230330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112303)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf0111230331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112303)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf0111230332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112303)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf0111230333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112303)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112303)) h
+theorem e24KC2ThetaAboveLeaf011123100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112310) = true := by
+  have h : ((childLL thetaAboveCell01112310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112310) h
+theorem e24KC2ThetaAboveLeaf011123101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112310) = true := by
+  have h : ((childLH thetaAboveCell01112310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112310) h
+theorem e24KC2ThetaAboveLeaf0111231020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112310)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112310)) h
+theorem e24KC2ThetaAboveLeaf0111231021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112310)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112310)) h
+theorem e24KC2ThetaAboveLeaf0111231022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112310)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112310)) h
+theorem e24KC2ThetaAboveLeaf0111231023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112310)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112310)) h
+theorem e24KC2ThetaAboveLeaf0111231030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112310)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112310)) h
+theorem e24KC2ThetaAboveLeaf0111231031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112310)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112310)) h
+theorem e24KC2ThetaAboveLeaf0111231032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112310)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112310)) h
+theorem e24KC2ThetaAboveLeaf0111231033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112310)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112310)) h
+theorem e24KC2ThetaAboveLeaf011123110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112311) = true := by
+  have h : ((childLL thetaAboveCell01112311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112311) h
+theorem e24KC2ThetaAboveLeaf011123111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112311) = true := by
+  have h : ((childLH thetaAboveCell01112311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112311) h
+theorem e24KC2ThetaAboveLeaf0111231120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112311)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112311)) h
+theorem e24KC2ThetaAboveLeaf0111231121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112311)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112311)) h
+theorem e24KC2ThetaAboveLeaf0111231122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112311)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112311)) h
+theorem e24KC2ThetaAboveLeaf0111231123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112311)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112311)) h
+theorem e24KC2ThetaAboveLeaf0111231130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112311)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112311)) h
+theorem e24KC2ThetaAboveLeaf0111231131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112311)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112311)) h
+theorem e24KC2ThetaAboveLeaf0111231132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112311)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112311)) h
+theorem e24KC2ThetaAboveLeaf0111231133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112311)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112311)) h
+theorem e24KC2ThetaAboveLeaf0111231200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01112312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01112312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01112312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01112312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01112312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01112312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01112312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01112312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112312)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112312)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112312)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112312)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112312)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112312)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112312)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112312)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112312)) h
+theorem e24KC2ThetaAboveLeaf0111231300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01112313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf0111231301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01112313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf0111231302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01112313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf0111231303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01112313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf0111231310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01112313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf0111231311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01112313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf0111231312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01112313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf0111231313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01112313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf0111231320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01112313)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf0111231321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01112313)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf0111231322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01112313)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf0111231323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01112313)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf0111231330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01112313)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf0111231331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01112313)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf0111231332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01112313)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf0111231333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01112313)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01112313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01112313)) h
+theorem e24KC2ThetaAboveLeaf011123200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112320) = true := by
+  have h : ((childLL thetaAboveCell01112320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112320) h
+theorem e24KC2ThetaAboveLeaf011123201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112320) = true := by
+  have h : ((childLH thetaAboveCell01112320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112320) h
+theorem e24KC2ThetaAboveLeaf011123202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112320) = true := by
+  have h : ((childHL thetaAboveCell01112320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01112320) h
+theorem e24KC2ThetaAboveLeaf011123203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112320) = true := by
+  have h : ((childHH thetaAboveCell01112320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01112320) h
+theorem e24KC2ThetaAboveLeaf011123210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112321) = true := by
+  have h : ((childLL thetaAboveCell01112321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112321) h
+theorem e24KC2ThetaAboveLeaf011123211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112321) = true := by
+  have h : ((childLH thetaAboveCell01112321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112321) h
+theorem e24KC2ThetaAboveLeaf011123212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112321) = true := by
+  have h : ((childHL thetaAboveCell01112321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01112321) h
+theorem e24KC2ThetaAboveLeaf011123213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112321) = true := by
+  have h : ((childHH thetaAboveCell01112321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01112321) h
+theorem e24KC2ThetaAboveLeaf01112322 :
+    adaptiveCoverCheck 11 thetaAboveCell01112322 = true := by
+  have h : (thetaAboveCell01112322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112322 h
+theorem e24KC2ThetaAboveLeaf01112323 :
+    adaptiveCoverCheck 11 thetaAboveCell01112323 = true := by
+  have h : (thetaAboveCell01112323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112323 h
+theorem e24KC2ThetaAboveLeaf011123300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112330) = true := by
+  have h : ((childLL thetaAboveCell01112330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112330) h
+theorem e24KC2ThetaAboveLeaf011123301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112330) = true := by
+  have h : ((childLH thetaAboveCell01112330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112330) h
+theorem e24KC2ThetaAboveLeaf011123302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112330) = true := by
+  have h : ((childHL thetaAboveCell01112330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01112330) h
+theorem e24KC2ThetaAboveLeaf011123303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112330) = true := by
+  have h : ((childHH thetaAboveCell01112330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01112330) h
+theorem e24KC2ThetaAboveLeaf011123310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112331) = true := by
+  have h : ((childLL thetaAboveCell01112331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01112331) h
+theorem e24KC2ThetaAboveLeaf011123311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112331) = true := by
+  have h : ((childLH thetaAboveCell01112331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01112331) h
+theorem e24KC2ThetaAboveLeaf011123312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112331) = true := by
+  have h : ((childHL thetaAboveCell01112331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01112331) h
+theorem e24KC2ThetaAboveLeaf011123313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112331) = true := by
+  have h : ((childHH thetaAboveCell01112331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01112331) h
+theorem e24KC2ThetaAboveLeaf01112332 :
+    adaptiveCoverCheck 11 thetaAboveCell01112332 = true := by
+  have h : (thetaAboveCell01112332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112332 h
+theorem e24KC2ThetaAboveLeaf01112333 :
+    adaptiveCoverCheck 11 thetaAboveCell01112333 = true := by
+  have h : (thetaAboveCell01112333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01112333 h
+theorem e24KC2ThetaAboveLeaf0111300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell0111))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell0111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell0111))) h
+theorem e24KC2ThetaAboveLeaf0111301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell0111))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell0111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell0111))) h
+theorem e24KC2ThetaAboveLeaf01113020 :
+    adaptiveCoverCheck 11 thetaAboveCell01113020 = true := by
+  have h : (thetaAboveCell01113020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113020 h
+theorem e24KC2ThetaAboveLeaf01113021 :
+    adaptiveCoverCheck 11 thetaAboveCell01113021 = true := by
+  have h : (thetaAboveCell01113021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113021 h
+theorem e24KC2ThetaAboveLeaf01113022 :
+    adaptiveCoverCheck 11 thetaAboveCell01113022 = true := by
+  have h : (thetaAboveCell01113022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113022 h
+theorem e24KC2ThetaAboveLeaf01113023 :
+    adaptiveCoverCheck 11 thetaAboveCell01113023 = true := by
+  have h : (thetaAboveCell01113023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113023 h
+theorem e24KC2ThetaAboveLeaf01113030 :
+    adaptiveCoverCheck 11 thetaAboveCell01113030 = true := by
+  have h : (thetaAboveCell01113030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113030 h
+theorem e24KC2ThetaAboveLeaf01113031 :
+    adaptiveCoverCheck 11 thetaAboveCell01113031 = true := by
+  have h : (thetaAboveCell01113031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113031 h
+theorem e24KC2ThetaAboveLeaf01113032 :
+    adaptiveCoverCheck 11 thetaAboveCell01113032 = true := by
+  have h : (thetaAboveCell01113032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113032 h
+theorem e24KC2ThetaAboveLeaf01113033 :
+    adaptiveCoverCheck 11 thetaAboveCell01113033 = true := by
+  have h : (thetaAboveCell01113033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113033 h
+theorem e24KC2ThetaAboveLeaf0111310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell0111))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell0111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell0111))) h
+theorem e24KC2ThetaAboveLeaf0111311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell0111))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell0111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell0111))) h
+theorem e24KC2ThetaAboveLeaf01113120 :
+    adaptiveCoverCheck 11 thetaAboveCell01113120 = true := by
+  have h : (thetaAboveCell01113120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113120 h
+theorem e24KC2ThetaAboveLeaf01113121 :
+    adaptiveCoverCheck 11 thetaAboveCell01113121 = true := by
+  have h : (thetaAboveCell01113121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113121 h
+theorem e24KC2ThetaAboveLeaf01113122 :
+    adaptiveCoverCheck 11 thetaAboveCell01113122 = true := by
+  have h : (thetaAboveCell01113122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113122 h
+theorem e24KC2ThetaAboveLeaf01113123 :
+    adaptiveCoverCheck 11 thetaAboveCell01113123 = true := by
+  have h : (thetaAboveCell01113123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113123 h
+theorem e24KC2ThetaAboveLeaf01113130 :
+    adaptiveCoverCheck 11 thetaAboveCell01113130 = true := by
+  have h : (thetaAboveCell01113130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113130 h
+theorem e24KC2ThetaAboveLeaf01113131 :
+    adaptiveCoverCheck 11 thetaAboveCell01113131 = true := by
+  have h : (thetaAboveCell01113131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113131 h
+theorem e24KC2ThetaAboveLeaf01113132 :
+    adaptiveCoverCheck 11 thetaAboveCell01113132 = true := by
+  have h : (thetaAboveCell01113132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113132 h
+theorem e24KC2ThetaAboveLeaf01113133 :
+    adaptiveCoverCheck 11 thetaAboveCell01113133 = true := by
+  have h : (thetaAboveCell01113133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113133 h
+theorem e24KC2ThetaAboveLeaf011132000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113200) = true := by
+  have h : ((childLL thetaAboveCell01113200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113200) h
+theorem e24KC2ThetaAboveLeaf011132001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113200) = true := by
+  have h : ((childLH thetaAboveCell01113200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113200) h
+theorem e24KC2ThetaAboveLeaf0111320020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01113200)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01113200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01113200)) h
+theorem e24KC2ThetaAboveLeaf0111320021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01113200)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01113200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01113200)) h
+theorem e24KC2ThetaAboveLeaf0111320022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01113200)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01113200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01113200)) h
+theorem e24KC2ThetaAboveLeaf0111320023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01113200)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01113200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01113200)) h
+theorem e24KC2ThetaAboveLeaf0111320030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01113200)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01113200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01113200)) h
+theorem e24KC2ThetaAboveLeaf0111320031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01113200)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01113200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01113200)) h
+theorem e24KC2ThetaAboveLeaf0111320032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01113200)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01113200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01113200)) h
+theorem e24KC2ThetaAboveLeaf0111320033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01113200)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01113200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01113200)) h
+theorem e24KC2ThetaAboveLeaf011132010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113201) = true := by
+  have h : ((childLL thetaAboveCell01113201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113201) h
+theorem e24KC2ThetaAboveLeaf011132011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113201) = true := by
+  have h : ((childLH thetaAboveCell01113201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113201) h
+theorem e24KC2ThetaAboveLeaf0111320120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01113201)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01113201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01113201)) h
+theorem e24KC2ThetaAboveLeaf0111320121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01113201)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01113201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01113201)) h
+theorem e24KC2ThetaAboveLeaf0111320122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01113201)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01113201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01113201)) h
+theorem e24KC2ThetaAboveLeaf0111320123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01113201)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01113201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01113201)) h
+theorem e24KC2ThetaAboveLeaf0111320130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01113201)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01113201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01113201)) h
+theorem e24KC2ThetaAboveLeaf0111320131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01113201)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01113201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01113201)) h
+theorem e24KC2ThetaAboveLeaf0111320132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01113201)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01113201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01113201)) h
+theorem e24KC2ThetaAboveLeaf0111320133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01113201)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01113201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01113201)) h
+theorem e24KC2ThetaAboveLeaf0111320200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01113202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01113202)) h
+theorem e24KC2ThetaAboveLeaf0111320201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01113202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01113202)) h
+theorem e24KC2ThetaAboveLeaf0111320202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01113202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01113202)) h
+theorem e24KC2ThetaAboveLeaf0111320203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01113202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01113202)) h
+theorem e24KC2ThetaAboveLeaf0111320210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01113202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01113202)) h
+theorem e24KC2ThetaAboveLeaf0111320211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01113202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01113202)) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6ProofBatchA7de3be3b4ab1789`.
+* `KernelOnly.PartE.E24KC6ProofBatchAad74ad127e30429`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells37514173c1
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells37514173c1
+
+open CertificateCells37514173c1
+theorem e24KC2ThetaAboveLeaf0011331022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00113310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00113310))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00113310))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00113310))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00113310))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00113310))) h)
+theorem e24KC2ThetaAboveLeaf0011331023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00113310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00113310))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00113310))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00113310))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00113310))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00113310))) h)
+theorem e24KC2ThetaAboveLeaf0011331032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00113310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00113310))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00113310))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00113310))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00113310))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00113310))) h)
+theorem e24KC2ThetaAboveLeaf0011331033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00113310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00113310))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00113310))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00113310))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00113310))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00113310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00113310))) h)
+theorem e24KC2ThetaAboveLeaf0011331122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00113311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00113311))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00113311))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00113311))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00113311))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00113311))) h)
+theorem e24KC2ThetaAboveLeaf0011331123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00113311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00113311))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00113311))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00113311))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00113311))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00113311))) h)
+theorem e24KC2ThetaAboveLeaf0011331132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00113311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00113311))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00113311))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00113311))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00113311))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00113311))) h)
+theorem e24KC2ThetaAboveLeaf0011331133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00113311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00113311))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00113311))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00113311))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00113311))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00113311))) h)
+theorem e24KC2ThetaAboveLeaf0011331200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00113312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00113312))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00113312))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00113312))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00113312))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00113312))) h)
+theorem e24KC2ThetaAboveLeaf0011331201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00113312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00113312))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00113312))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00113312))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00113312))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00113312))) h)
+theorem e24KC2ThetaAboveLeaf0011331210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00113312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00113312))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00113312))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00113312))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00113312))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00113312))) h)
+theorem e24KC2ThetaAboveLeaf0011331211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00113312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00113312))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00113312))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00113312))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00113312))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00113312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00113312))) h)
+theorem e24KC2ThetaAboveLeaf0011331300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00113313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00113313))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00113313))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00113313))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00113313))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00113313))) h)
+theorem e24KC2ThetaAboveLeaf0011331301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00113313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00113313))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00113313))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00113313))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00113313))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00113313))) h)
+theorem e24KC2ThetaAboveLeaf0011331310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00113313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00113313))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00113313))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00113313))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00113313))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00113313))) h)
+theorem e24KC2ThetaAboveLeaf0011331311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00113313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00113313))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00113313))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00113313))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00113313))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00113313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00113313))) h)
+theorem e24KC2ThetaAboveLeaf0100220022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01002200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell01002200))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell01002200))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell01002200))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell01002200))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell01002200))) h)
+theorem e24KC2ThetaAboveLeaf0100220023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01002200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell01002200))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell01002200))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell01002200))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell01002200))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell01002200))) h)
+theorem e24KC2ThetaAboveLeaf0100220032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01002200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell01002200))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell01002200))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell01002200))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell01002200))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell01002200))) h)
+theorem e24KC2ThetaAboveLeaf0100220033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01002200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell01002200))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell01002200))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell01002200))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell01002200))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell01002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell01002200))) h)
+theorem e24KC2ThetaAboveLeaf0100220122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell01002201))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell01002201))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell01002201))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell01002201))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell01002201))) h)
+theorem e24KC2ThetaAboveLeaf0100220123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell01002201))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell01002201))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell01002201))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell01002201))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell01002201))) h)
+theorem e24KC2ThetaAboveLeaf0100220132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell01002201))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell01002201))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell01002201))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell01002201))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell01002201))) h)
+theorem e24KC2ThetaAboveLeaf0100220133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell01002201))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell01002201))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell01002201))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell01002201))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell01002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell01002201))) h)
+theorem e24KC2ThetaAboveLeaf0100220200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01002202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell01002202))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell01002202))) h)
+theorem e24KC2ThetaAboveLeaf0100220201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01002202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell01002202))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell01002202))) h)
+theorem e24KC2ThetaAboveLeaf0100220210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01002202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell01002202))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell01002202))) h)
+theorem e24KC2ThetaAboveLeaf0100220211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01002202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell01002202))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell01002202))) h)
+theorem e24KC2ThetaAboveLeaf0100220212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01002202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell01002202))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childHL (childHL (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLH
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell01002202))) h)
+theorem e24KC2ThetaAboveLeaf0100220213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01002202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell01002202))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell01002202))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell01002202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell01002202))) h)
+theorem e24KC2ThetaAboveLeaf0100220300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01002203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell01002203))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell01002203))) h)
+theorem e24KC2ThetaAboveLeaf0100220301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01002203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell01002203))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell01002203))) h)
+theorem e24KC2ThetaAboveLeaf0100220302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01002203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell01002203))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell01002203))) h)
+theorem e24KC2ThetaAboveLeaf0100220303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01002203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell01002203))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHL (childHH (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLL
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHH (childHH (childLL thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLL
+        thetaAboveCell01002203))) h)
+theorem e24KC2ThetaAboveLeaf0100220310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01002203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell01002203))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell01002203))) h)
+theorem e24KC2ThetaAboveLeaf0100220311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01002203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell01002203))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell01002203))) h)
+theorem e24KC2ThetaAboveLeaf0100220312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01002203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell01002203))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHL (childHL (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLH
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell01002203))) h)
+theorem e24KC2ThetaAboveLeaf0100220313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01002203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell01002203))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell01002203))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell01002203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell01002203))) h)
+theorem e24KC2ThetaAboveLeaf0100221022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell01002210))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell01002210))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell01002210))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell01002210))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell01002210))) h)
+theorem e24KC2ThetaAboveLeaf0100221023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell01002210))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell01002210))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell01002210))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell01002210))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell01002210))) h)
+theorem e24KC2ThetaAboveLeaf0100221032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell01002210))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell01002210))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell01002210))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell01002210))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell01002210))) h)
+theorem e24KC2ThetaAboveLeaf0100221033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell01002210))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell01002210))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell01002210))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell01002210))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell01002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell01002210))) h)
+theorem e24KC2ThetaAboveLeaf0100221122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell01002211))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell01002211))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell01002211))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell01002211))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell01002211))) h)
+theorem e24KC2ThetaAboveLeaf0100221123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell01002211))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell01002211))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell01002211))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell01002211))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell01002211))) h)
+theorem e24KC2ThetaAboveLeaf0100221132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell01002211))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell01002211))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell01002211))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell01002211))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell01002211))) h)
+theorem e24KC2ThetaAboveLeaf0100221133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell01002211))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell01002211))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell01002211))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell01002211))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell01002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell01002211))) h)
+theorem e24KC2ThetaAboveLeaf0100221200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01002212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell01002212))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell01002212))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell01002212))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell01002212))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell01002212))) h)
+theorem e24KC2ThetaAboveLeaf0100221201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01002212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell01002212))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell01002212))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell01002212))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell01002212))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell01002212))) h)
+theorem e24KC2ThetaAboveLeaf0100221202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01002212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell01002212))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell01002212))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell01002212))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell01002212))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell01002212))) h)
+theorem e24KC2ThetaAboveLeaf0100221210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01002212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell01002212))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell01002212))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell01002212))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell01002212))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell01002212))) h)
+theorem e24KC2ThetaAboveLeaf0100221211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01002212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell01002212))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell01002212))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell01002212))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell01002212))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell01002212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell01002212))) h)
+theorem e24KC2ThetaAboveLeaf0100221300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01002213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell01002213))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell01002213))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell01002213))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell01002213))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell01002213))) h)
+theorem e24KC2ThetaAboveLeaf0100221301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01002213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell01002213))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell01002213))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell01002213))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell01002213))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell01002213))) h)
+theorem e24KC2ThetaAboveLeaf0100221310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01002213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell01002213))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell01002213))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell01002213))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell01002213))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell01002213))) h)
+theorem e24KC2ThetaAboveLeaf0100221311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01002213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell01002213))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell01002213))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell01002213))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell01002213))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell01002213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell01002213))) h)
+theorem e24KC2ThetaAboveLeaf0100230022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell01002300))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell01002300))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell01002300))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell01002300))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell01002300))) h)
+theorem e24KC2ThetaAboveLeaf0100230023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell01002300))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell01002300))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell01002300))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell01002300))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell01002300))) h)
+theorem e24KC2ThetaAboveLeaf0100230032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell01002300))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell01002300))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell01002300))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell01002300))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell01002300))) h)
+theorem e24KC2ThetaAboveLeaf0100230033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell01002300))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell01002300))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell01002300))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell01002300))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell01002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell01002300))) h)
+theorem e24KC2ThetaAboveLeaf0100230122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell01002301))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell01002301))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell01002301))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell01002301))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell01002301))) h)
+theorem e24KC2ThetaAboveLeaf0100230123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell01002301))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell01002301))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell01002301))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell01002301))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell01002301))) h)
+theorem e24KC2ThetaAboveLeaf0100230132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell01002301))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell01002301))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell01002301))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell01002301))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell01002301))) h)
+theorem e24KC2ThetaAboveLeaf0100230133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01002301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell01002301))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell01002301))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell01002301))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell01002301))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell01002301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell01002301))) h)
+theorem e24KC2ThetaAboveLeaf0100230200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01002302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell01002302))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell01002302))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell01002302))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell01002302))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell01002302))) h)
+theorem e24KC2ThetaAboveLeaf0100230201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01002302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell01002302))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell01002302))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell01002302))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell01002302))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell01002302))) h)
+theorem e24KC2ThetaAboveLeaf0100230210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01002302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell01002302))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell01002302))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell01002302))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell01002302))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell01002302))) h)
+theorem e24KC2ThetaAboveLeaf0100230211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01002302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell01002302))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell01002302))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell01002302))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell01002302))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell01002302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell01002302))) h)
+theorem e24KC2ThetaAboveLeaf0100230300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01002303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell01002303))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell01002303))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell01002303))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell01002303))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell01002303))) h)
+theorem e24KC2ThetaAboveLeaf0100230301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01002303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell01002303))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell01002303))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell01002303))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell01002303))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell01002303))) h)
+theorem e24KC2ThetaAboveLeaf0100230310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01002303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell01002303))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell01002303))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell01002303))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell01002303))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell01002303))) h)
+theorem e24KC2ThetaAboveLeaf0100230311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01002303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell01002303))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell01002303))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell01002303))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell01002303))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell01002303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell01002303))) h)
+theorem e24KC2ThetaAboveLeaf0100231022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell01002310))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell01002310))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell01002310))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell01002310))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell01002310))) h)
+theorem e24KC2ThetaAboveLeaf0100231023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell01002310))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell01002310))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell01002310))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell01002310))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell01002310))) h)
+theorem e24KC2ThetaAboveLeaf0100231032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell01002310))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell01002310))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell01002310))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell01002310))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell01002310))) h)
+theorem e24KC2ThetaAboveLeaf0100231033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01002310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell01002310))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell01002310))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell01002310))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell01002310))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell01002310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell01002310))) h)
+theorem e24KC2ThetaAboveLeaf0100231122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell01002311))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell01002311))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell01002311))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell01002311))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell01002311))) h)
+theorem e24KC2ThetaAboveLeaf0100231123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell01002311))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell01002311))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell01002311))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell01002311))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell01002311))) h)
+theorem e24KC2ThetaAboveLeaf0100231132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell01002311))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell01002311))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell01002311))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell01002311))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell01002311))) h)
+theorem e24KC2ThetaAboveLeaf0100231133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01002311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell01002311))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell01002311))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell01002311))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell01002311))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell01002311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell01002311))) h)
+theorem e24KC2ThetaAboveLeaf0100320022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell01003200))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell01003200))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell01003200))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell01003200))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell01003200))) h)
+theorem e24KC2ThetaAboveLeaf0100320023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell01003200))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell01003200))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell01003200))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell01003200))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell01003200))) h)
+theorem e24KC2ThetaAboveLeaf0100320032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell01003200))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell01003200))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell01003200))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell01003200))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell01003200))) h)
+theorem e24KC2ThetaAboveLeaf0100320033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell01003200))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell01003200))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell01003200))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell01003200))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell01003200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell01003200))) h)
+theorem e24KC2ThetaAboveLeaf0100320122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell01003201))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell01003201))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell01003201))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell01003201))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell01003201))) h)
+theorem e24KC2ThetaAboveLeaf0100320123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell01003201))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell01003201))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell01003201))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell01003201))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell01003201))) h)
+theorem e24KC2ThetaAboveLeaf0100320132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell01003201))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell01003201))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell01003201))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell01003201))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell01003201))) h)
+theorem e24KC2ThetaAboveLeaf0100320133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell01003201))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell01003201))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell01003201))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell01003201))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell01003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell01003201))) h)
+theorem e24KC2ThetaBelowLeaf111030223 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103022) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11103022)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11103022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11103022)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11103022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11103022)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11103022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11103022)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11103022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11103022)) h)
+theorem e24KC2ThetaBelowLeaf111030232 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103023) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11103023)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11103023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11103023)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11103023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11103023)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11103023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11103023)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11103023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11103023)) h)
+theorem e24KC2ThetaBelowLeaf111030233 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103023) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11103023)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11103023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11103023)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11103023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11103023)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11103023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11103023)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11103023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11103023)) h)
+theorem e24KC2ThetaBelowLeaf111030312 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103031) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11103031)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11103031))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11103031)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11103031))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11103031)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11103031))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11103031)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11103031))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11103031)) h)
+theorem e24KC2ThetaBelowLeaf111030313 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103031) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11103031)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11103031))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11103031)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11103031))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11103031)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11103031))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11103031)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11103031))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11103031)) h)
+theorem e24KC2ThetaBelowLeaf111030320 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103032) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11103032)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11103032)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11103032)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11103032)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11103032)) h)
+theorem e24KC2ThetaBelowLeaf111030321 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103032) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11103032)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11103032)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11103032)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11103032)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11103032)) h)
+theorem e24KC2ThetaBelowLeaf111030322 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103032) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11103032)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11103032)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11103032)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11103032)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11103032)) h)
+theorem e24KC2ThetaBelowLeaf111030323 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103032) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11103032)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11103032)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11103032)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11103032)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11103032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11103032)) h)
+theorem e24KC2ThetaBelowLeaf111030330 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103033) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11103033)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11103033)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11103033)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11103033)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11103033)) h)
+theorem e24KC2ThetaBelowLeaf111030331 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103033) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11103033)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11103033)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11103033)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11103033)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11103033)) h)
+theorem e24KC2ThetaBelowLeaf111030332 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103033) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11103033)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11103033)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11103033)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11103033)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11103033)) h)
+theorem e24KC2ThetaBelowLeaf111030333 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103033) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11103033)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11103033)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11103033)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11103033)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11103033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11103033)) h)
+theorem e24KC2ThetaBelowLeaf111031202 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103120) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11103120)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11103120))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11103120)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11103120))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11103120)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11103120))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11103120)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11103120))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11103120)) h)
+theorem e24KC2ThetaBelowLeaf111031220 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103122) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11103122)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11103122)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11103122)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11103122)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11103122)) h)
+theorem e24KC2ThetaBelowLeaf111031221 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103122) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11103122)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11103122)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11103122)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11103122)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11103122)) h)
+theorem e24KC2ThetaBelowLeaf111031222 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103122) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11103122)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11103122)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11103122)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11103122)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11103122)) h)
+theorem e24KC2ThetaBelowLeaf111031223 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103122) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11103122)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11103122)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11103122)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11103122)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11103122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11103122)) h)
+theorem e24KC2ThetaBelowLeaf111031230 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103123) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11103123)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11103123)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11103123)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11103123)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11103123)) h)
+theorem e24KC2ThetaBelowLeaf111031231 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103123) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11103123)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11103123)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11103123)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11103123)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11103123)) h)
+theorem e24KC2ThetaBelowLeaf111031232 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103123) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11103123)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11103123)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11103123)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11103123)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11103123)) h)
+theorem e24KC2ThetaBelowLeaf111031233 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103123) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11103123)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11103123)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11103123)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11103123)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11103123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11103123)) h)
+theorem e24KC2ThetaBelowLeaf111031320 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103132) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11103132)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11103132)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11103132)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11103132)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11103132)) h)
+theorem e24KC2ThetaBelowLeaf111031321 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103132) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11103132)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11103132)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11103132)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11103132)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11103132)) h)
+theorem e24KC2ThetaBelowLeaf111031322 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103132) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11103132)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11103132)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11103132)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11103132)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11103132)) h)
+theorem e24KC2ThetaBelowLeaf111031323 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103132) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11103132)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11103132)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11103132)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11103132)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11103132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11103132)) h)
+theorem e24KC2ThetaBelowLeaf111031330 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103133) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11103133)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11103133)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11103133)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11103133)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11103133)) h)
+theorem e24KC2ThetaBelowLeaf111031331 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103133) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11103133)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11103133)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11103133)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11103133)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11103133)) h)
+theorem e24KC2ThetaBelowLeaf111031332 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103133) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11103133)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11103133)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11103133)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11103133)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11103133)) h)
+theorem e24KC2ThetaBelowLeaf111031333 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103133) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11103133)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11103133)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11103133)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11103133)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11103133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11103133)) h)
+theorem e24KC2ThetaBelowLeaf111032100 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103210) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11103210)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11103210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11103210)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11103210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11103210)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11103210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11103210)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11103210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11103210)) h)
+theorem e24KC2ThetaBelowLeaf111032101 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103210) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11103210)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11103210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11103210)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11103210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11103210)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11103210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11103210)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11103210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11103210)) h)
+theorem e24KC2ThetaBelowLeaf111032110 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103211) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11103211)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11103211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11103211)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11103211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11103211)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11103211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11103211)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11103211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11103211)) h)
+theorem e24KC2ThetaBelowLeaf111032111 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103211) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11103211)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11103211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11103211)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11103211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11103211)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11103211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11103211)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11103211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11103211)) h)
+theorem e24KC2ThetaBelowLeaf111033000 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103300) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11103300)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11103300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11103300)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11103300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11103300)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11103300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11103300)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11103300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11103300)) h)
+theorem e24KC2ThetaBelowLeaf111033001 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103300) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11103300)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11103300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11103300)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11103300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11103300)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11103300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11103300)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11103300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11103300)) h)
+theorem e24KC2ThetaBelowLeaf111033010 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103301) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11103301)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11103301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11103301)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11103301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11103301)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11103301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11103301)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11103301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11103301)) h)
+theorem e24KC2ThetaBelowLeaf111033011 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103301) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11103301)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11103301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11103301)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11103301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11103301)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11103301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11103301)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11103301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11103301)) h)
+theorem e24KC2ThetaBelowLeaf111033100 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103310) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11103310)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11103310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11103310)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11103310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11103310)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11103310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11103310)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11103310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11103310)) h)
+theorem e24KC2ThetaBelowLeaf111033101 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103310) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11103310)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11103310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11103310)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11103310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11103310)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11103310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11103310)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11103310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11103310)) h)
+theorem e24KC2ThetaBelowLeaf111033110 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103311) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11103311)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11103311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11103311)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11103311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11103311)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11103311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11103311)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11103311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11103311)) h)
+theorem e24KC2ThetaBelowLeaf111033111 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103311) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11103311)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11103311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11103311)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11103311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11103311)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11103311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11103311)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11103311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11103311)) h)
+theorem e24KC2ThetaBelowLeaf111120220 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112022) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11112022)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11112022)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11112022)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11112022)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11112022)) h)
+theorem e24KC2ThetaBelowLeaf111120221 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112022) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11112022)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11112022)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11112022)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11112022)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11112022)) h)
+theorem e24KC2ThetaBelowLeaf111120222 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112022) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11112022)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11112022)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11112022)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11112022)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11112022)) h)
+theorem e24KC2ThetaBelowLeaf111120223 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112022) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11112022)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11112022)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11112022)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11112022)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11112022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11112022)) h)
+theorem e24KC2ThetaBelowLeaf111120230 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112023) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11112023)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11112023)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11112023)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11112023)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11112023)) h)
+theorem e24KC2ThetaBelowLeaf111120231 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112023) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11112023)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11112023)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11112023)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11112023)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11112023)) h)
+theorem e24KC2ThetaBelowLeaf111120232 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112023) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11112023)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11112023)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11112023)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11112023)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11112023)) h)
+theorem e24KC2ThetaBelowLeaf111120233 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112023) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11112023)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11112023)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11112023)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11112023)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11112023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11112023)) h)
+theorem e24KC2ThetaBelowLeaf111120320 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112032) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11112032)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11112032)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11112032)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11112032)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11112032)) h)
+theorem e24KC2ThetaBelowLeaf111120321 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112032) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11112032)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11112032)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11112032)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11112032)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11112032)) h)
+theorem e24KC2ThetaBelowLeaf111120322 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112032) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11112032)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11112032)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11112032)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11112032)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11112032)) h)
+theorem e24KC2ThetaBelowLeaf111120323 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112032) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11112032)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11112032)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11112032)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11112032)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11112032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11112032)) h)
+theorem e24KC2ThetaBelowLeaf111120330 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112033) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11112033)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11112033)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11112033)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11112033)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11112033)) h)
+theorem e24KC2ThetaBelowLeaf111120331 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112033) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11112033)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11112033)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11112033)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11112033)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11112033)) h)
+theorem e24KC2ThetaBelowLeaf111120332 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112033) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11112033)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11112033)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11112033)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11112033)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11112033)) h)
+theorem e24KC2ThetaBelowLeaf111120333 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112033) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11112033)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11112033)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11112033)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11112033)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11112033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11112033)) h)
+theorem e24KC2ThetaBelowLeaf111121220 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112122) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11112122)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11112122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11112122)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11112122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11112122)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11112122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11112122)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11112122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11112122)) h)
+theorem e24KC2ThetaBelowLeaf111121221 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112122) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11112122)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11112122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11112122)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11112122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11112122)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11112122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11112122)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11112122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11112122)) h)
+theorem e24KC2ThetaBelowLeaf111121222 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112122) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11112122)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11112122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11112122)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11112122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11112122)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL thetaBelowCell11112122))
+        (by
+          have h : ((childLL (childHL (childHL thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childHL
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childHL
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childHL
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childHL
+            thetaBelowCell11112122))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL thetaBelowCell11112122))
+        (by
+          have h : ((childLL (childHH (childHL thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHL
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHL
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHL
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHL
+            thetaBelowCell11112122))) h))
+theorem e24KC2ThetaBelowLeaf111121223 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112122) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11112122)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH thetaBelowCell11112122))
+        (by
+          have h : ((childLL (childLL (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childHH
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childHH
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childHH
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childHH
+            thetaBelowCell11112122))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH thetaBelowCell11112122))
+        (by
+          have h : ((childLL (childLH (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childHH
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childHH
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childHH
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childHH
+            thetaBelowCell11112122))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH thetaBelowCell11112122))
+        (by
+          have h : ((childLL (childHL (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childHH
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childHH
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childHH
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childHH
+            thetaBelowCell11112122))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH thetaBelowCell11112122))
+        (by
+          have h : ((childLL (childHH (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHH
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHH
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHH
+            thetaBelowCell11112122))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaBelowCell11112122)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHH
+            thetaBelowCell11112122))) h))
+theorem e24KC2ThetaBelowLeaf111121230 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112123) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11112123)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11112123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11112123)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11112123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11112123)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11112123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11112123)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11112123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11112123)) h)
+theorem e24KC2ThetaBelowLeaf111121231 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112123) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11112123)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11112123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11112123)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11112123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11112123)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11112123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11112123)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11112123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11112123)) h)
+theorem e24KC2ThetaBelowLeaf111121232 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112123) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11112123)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL thetaBelowCell11112123))
+        (by
+          have h : ((childLL (childLL (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childHL
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childHL
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childHL
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childHL
+            thetaBelowCell11112123))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL thetaBelowCell11112123))
+        (by
+          have h : ((childLL (childLH (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childHL
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childHL
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childHL
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childHL
+            thetaBelowCell11112123))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL thetaBelowCell11112123))
+        (by
+          have h : ((childLL (childHL (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childHL
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childHL
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childHL
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childHL
+            thetaBelowCell11112123))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL thetaBelowCell11112123))
+        (by
+          have h : ((childLL (childHH (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHL
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHL
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHL
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHL
+            thetaBelowCell11112123))) h))
+theorem e24KC2ThetaBelowLeaf111121233 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112123) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11112123)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11112123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11112123)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11112123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11112123)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH thetaBelowCell11112123))
+        (by
+          have h : ((childLL (childHL (childHH thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childHH
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childHH
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childHH
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childHH
+            thetaBelowCell11112123))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH thetaBelowCell11112123))
+        (by
+          have h : ((childLL (childHH (childHH thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHH
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHH
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHH
+            thetaBelowCell11112123))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaBelowCell11112123)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHH
+            thetaBelowCell11112123))) h))
+theorem e24KC2ThetaBelowLeaf111121322 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112132) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11112132)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11112132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11112132)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11112132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11112132)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL thetaBelowCell11112132))
+        (by
+          have h : ((childLL (childHL (childHL thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childHL
+            thetaBelowCell11112132))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childHL
+            thetaBelowCell11112132))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childHL
+            thetaBelowCell11112132))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childHL
+            thetaBelowCell11112132))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL thetaBelowCell11112132))
+        (by
+          have h : ((childLL (childHH (childHL thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHL
+            thetaBelowCell11112132))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHL
+            thetaBelowCell11112132))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHL
+            thetaBelowCell11112132))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHL
+            thetaBelowCell11112132))) h))
+theorem e24KC2ThetaBelowLeaf111121323 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112132) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11112132)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11112132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11112132)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11112132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11112132)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH thetaBelowCell11112132))
+        (by
+          have h : ((childLL (childHL (childHH thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childHH
+            thetaBelowCell11112132))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childHH
+            thetaBelowCell11112132))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childHH
+            thetaBelowCell11112132))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childHH
+            thetaBelowCell11112132))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH thetaBelowCell11112132))
+        (by
+          have h : ((childLL (childHH (childHH thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHH
+            thetaBelowCell11112132))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHH
+            thetaBelowCell11112132))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHH
+            thetaBelowCell11112132))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaBelowCell11112132)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHH
+            thetaBelowCell11112132))) h))
+theorem e24KC2ThetaBelowLeaf111121332 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112133) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11112133)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11112133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11112133)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11112133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11112133)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL thetaBelowCell11112133))
+        (by
+          have h : ((childLL (childHL (childHL thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childHL
+            thetaBelowCell11112133))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childHL
+            thetaBelowCell11112133))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childHL
+            thetaBelowCell11112133))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childHL
+            thetaBelowCell11112133))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL thetaBelowCell11112133))
+        (by
+          have h : ((childLL (childHH (childHL thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHL
+            thetaBelowCell11112133))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHL
+            thetaBelowCell11112133))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHL
+            thetaBelowCell11112133))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHL
+            thetaBelowCell11112133))) h))
+theorem e24KC2ThetaBelowLeaf111121333 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112133) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11112133)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11112133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11112133)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11112133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11112133)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH thetaBelowCell11112133))
+        (by
+          have h : ((childLL (childHL (childHH thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childHH
+            thetaBelowCell11112133))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childHH
+            thetaBelowCell11112133))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childHH
+            thetaBelowCell11112133))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childHH
+            thetaBelowCell11112133))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH thetaBelowCell11112133))
+        (by
+          have h : ((childLL (childHH (childHH thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHH
+            thetaBelowCell11112133))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHH
+            thetaBelowCell11112133))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHH
+            thetaBelowCell11112133))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaBelowCell11112133)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHH
+            thetaBelowCell11112133))) h))
+theorem e24KC2ThetaBelowLeaf111122000 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112200) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11112200)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11112200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11112200)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11112200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11112200)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11112200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11112200)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11112200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11112200)) h)
+theorem e24KC2ThetaBelowLeaf111122001 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112200) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11112200)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11112200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11112200)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11112200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11112200)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11112200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11112200)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11112200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11112200)) h)
+theorem e24KC2ThetaBelowLeaf111122010 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112201) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11112201)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11112201))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11112201)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11112201))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11112201)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11112201))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11112201)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11112201))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11112201)) h)
+theorem e24KC2ThetaBelowLeaf111122011 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112201) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11112201)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11112201))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11112201)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11112201))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11112201)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11112201))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11112201)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11112201))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11112201)) h)
+theorem e24KC2ThetaBelowLeaf111122100 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112210) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11112210)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11112210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11112210)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11112210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11112210)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11112210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11112210)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11112210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11112210)) h)
+theorem e24KC2ThetaBelowLeaf111122101 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112210) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11112210)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11112210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11112210)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11112210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11112210)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11112210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11112210)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11112210))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11112210)) h)
+theorem e24KC2ThetaBelowLeaf111122110 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112211) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11112211)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11112211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11112211)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL thetaBelowCell11112211))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell11112211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLL
+            thetaBelowCell11112211))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell11112211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLL
+            thetaBelowCell11112211))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell11112211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLL
+            thetaBelowCell11112211))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell11112211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLL
+            thetaBelowCell11112211))) h))
+    (by
+      have h : ((childHL (childLL thetaBelowCell11112211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11112211)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11112211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11112211)) h)
+theorem e24KC2ThetaBelowLeaf111122111 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112211) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11112211)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH thetaBelowCell11112211))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell11112211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLH
+            thetaBelowCell11112211))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell11112211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLH
+            thetaBelowCell11112211))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell11112211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLH
+            thetaBelowCell11112211))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell11112211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLH
+            thetaBelowCell11112211))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH thetaBelowCell11112211))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell11112211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLH
+            thetaBelowCell11112211))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell11112211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLH
+            thetaBelowCell11112211))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell11112211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLH
+            thetaBelowCell11112211))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell11112211)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLH
+            thetaBelowCell11112211))) h))
+    (by
+      have h : ((childHL (childLH thetaBelowCell11112211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11112211)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11112211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11112211)) h)
+theorem e24KC2ThetaBelowLeaf111122113 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112211) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11112211)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11112211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11112211)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11112211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11112211)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11112211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11112211)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11112211))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11112211)) h)
+theorem e24KC2ThetaBelowLeaf111123000 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112300) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11112300)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL thetaBelowCell11112300))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLL
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLL
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLL
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLL
+            thetaBelowCell11112300))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL thetaBelowCell11112300))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLL
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLL
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLL
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLL
+            thetaBelowCell11112300))) h))
+    (by
+      have h : ((childHL (childLL thetaBelowCell11112300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11112300)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL thetaBelowCell11112300))
+        (by
+          have h : ((childLL (childHH (childLL thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childLL
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childLL
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childLL
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childLL
+            thetaBelowCell11112300))) h))
+theorem e24KC2ThetaBelowLeaf111123001 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112300) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11112300)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH thetaBelowCell11112300))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLH
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLH
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLH
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLH
+            thetaBelowCell11112300))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH thetaBelowCell11112300))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLH
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLH
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLH
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLH
+            thetaBelowCell11112300))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH thetaBelowCell11112300))
+        (by
+          have h : ((childLL (childHL (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childLH
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childLH
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childLH
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childLH
+            thetaBelowCell11112300))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH thetaBelowCell11112300))
+        (by
+          have h : ((childLL (childHH (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childLH
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childLH
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childLH
+            thetaBelowCell11112300))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaBelowCell11112300)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childLH
+            thetaBelowCell11112300))) h))
+theorem e24KC2ThetaBelowLeaf111123002 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112300) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11112300)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11112300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11112300)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11112300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11112300)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11112300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11112300)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11112300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11112300)) h)
+theorem e24KC2ThetaBelowLeaf111123003 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112300) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11112300)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11112300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11112300)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11112300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11112300)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11112300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11112300)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11112300))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11112300)) h)
+theorem e24KC2ThetaBelowLeaf111123010 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112301) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11112301)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL thetaBelowCell11112301))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLL
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLL
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLL
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLL
+            thetaBelowCell11112301))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL thetaBelowCell11112301))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLL
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLL
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLL
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLL
+            thetaBelowCell11112301))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL thetaBelowCell11112301))
+        (by
+          have h : ((childLL (childHL (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childLL
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childLL
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childLL
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childLL
+            thetaBelowCell11112301))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL thetaBelowCell11112301))
+        (by
+          have h : ((childLL (childHH (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childLL
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childLL
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childLL
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childLL
+            thetaBelowCell11112301))) h))
+theorem e24KC2ThetaBelowLeaf111123011 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112301) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11112301)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH thetaBelowCell11112301))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLH
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLH
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLH
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLH
+            thetaBelowCell11112301))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH thetaBelowCell11112301))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLH
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLH
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLH
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLH
+            thetaBelowCell11112301))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH thetaBelowCell11112301))
+        (by
+          have h : ((childLL (childHL (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childLH
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childLH
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childLH
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childLH
+            thetaBelowCell11112301))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH thetaBelowCell11112301))
+        (by
+          have h : ((childLL (childHH (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childLH
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childLH
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childLH
+            thetaBelowCell11112301))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaBelowCell11112301)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childLH
+            thetaBelowCell11112301))) h))
+theorem e24KC2ThetaBelowLeaf111123012 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112301) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11112301)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11112301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11112301)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11112301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11112301)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11112301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11112301)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11112301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11112301)) h)
+theorem e24KC2ThetaBelowLeaf111123013 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112301) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11112301)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11112301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11112301)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11112301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11112301)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11112301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11112301)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11112301))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11112301)) h)
+theorem e24KC2ThetaBelowLeaf111123100 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112310) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11112310)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL thetaBelowCell11112310))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLL
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLL
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLL
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLL
+            thetaBelowCell11112310))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL thetaBelowCell11112310))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLL
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLL
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLL
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLL
+            thetaBelowCell11112310))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL thetaBelowCell11112310))
+        (by
+          have h : ((childLL (childHL (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childLL
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childLL
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childLL
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childLL
+            thetaBelowCell11112310))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL thetaBelowCell11112310))
+        (by
+          have h : ((childLL (childHH (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childLL
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childLL
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childLL
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childLL
+            thetaBelowCell11112310))) h))
+theorem e24KC2ThetaBelowLeaf111123101 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112310) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11112310)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH thetaBelowCell11112310))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLH
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLH
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLH
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLH
+            thetaBelowCell11112310))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH thetaBelowCell11112310))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLH
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLH
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLH
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLH
+            thetaBelowCell11112310))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH thetaBelowCell11112310))
+        (by
+          have h : ((childLL (childHL (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childLH
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childLH
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childLH
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childLH
+            thetaBelowCell11112310))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH thetaBelowCell11112310))
+        (by
+          have h : ((childLL (childHH (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childLH
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childLH
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childLH
+            thetaBelowCell11112310))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaBelowCell11112310)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childLH
+            thetaBelowCell11112310))) h))
+theorem e24KC2ThetaBelowLeaf111123102 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112310) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11112310)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11112310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11112310)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11112310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11112310)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11112310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11112310)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11112310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11112310)) h)
+theorem e24KC2ThetaBelowLeaf111123103 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112310) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11112310)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11112310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11112310)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11112310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11112310)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11112310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11112310)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11112310))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11112310)) h)
+theorem e24KC2ThetaBelowLeaf111123110 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112311) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11112311)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL thetaBelowCell11112311))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLL
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLL
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLL
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLL
+            thetaBelowCell11112311))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL thetaBelowCell11112311))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLL
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLL
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLL
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLL
+            thetaBelowCell11112311))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL thetaBelowCell11112311))
+        (by
+          have h : ((childLL (childHL (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childLL
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childLL
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childLL
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childLL
+            thetaBelowCell11112311))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL thetaBelowCell11112311))
+        (by
+          have h : ((childLL (childHH (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childLL
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childLL
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childLL
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childLL
+            thetaBelowCell11112311))) h))
+theorem e24KC2ThetaBelowLeaf111123111 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112311) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11112311)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH thetaBelowCell11112311))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLH
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLH
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLH
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLH
+            thetaBelowCell11112311))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH thetaBelowCell11112311))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLH
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLH
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLH
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLH
+            thetaBelowCell11112311))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH thetaBelowCell11112311))
+        (by
+          have h : ((childLL (childHL (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childLH
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childLH
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childLH
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childLH
+            thetaBelowCell11112311))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH thetaBelowCell11112311))
+        (by
+          have h : ((childLL (childHH (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childLH
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childLH
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childLH
+            thetaBelowCell11112311))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaBelowCell11112311)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childLH
+            thetaBelowCell11112311))) h))
+theorem e24KC2ThetaBelowLeaf111123112 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112311) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11112311)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11112311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11112311)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11112311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11112311)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11112311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11112311)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11112311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11112311)) h)
+theorem e24KC2ThetaBelowLeaf111123113 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112311) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11112311)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11112311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11112311)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11112311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11112311)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11112311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11112311)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11112311))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11112311)) h)
+theorem e24KC2ThetaBelowLeaf111130222 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113022) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113022)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11113022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11113022)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11113022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11113022)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL thetaBelowCell11113022))
+        (by
+          have h : ((childLL (childHL (childHL thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childHL
+            thetaBelowCell11113022))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childHL
+            thetaBelowCell11113022))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childHL
+            thetaBelowCell11113022))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childHL
+            thetaBelowCell11113022))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL thetaBelowCell11113022))
+        (by
+          have h : ((childLL (childHH (childHL thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHL
+            thetaBelowCell11113022))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHL
+            thetaBelowCell11113022))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHL
+            thetaBelowCell11113022))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHL
+            thetaBelowCell11113022))) h))
+theorem e24KC2ThetaBelowLeaf111130223 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113022) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113022)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11113022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11113022)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11113022))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11113022)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH thetaBelowCell11113022))
+        (by
+          have h : ((childLL (childHL (childHH thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childHH
+            thetaBelowCell11113022))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childHH
+            thetaBelowCell11113022))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childHH
+            thetaBelowCell11113022))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childHH
+            thetaBelowCell11113022))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH thetaBelowCell11113022))
+        (by
+          have h : ((childLL (childHH (childHH thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHH
+            thetaBelowCell11113022))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHH
+            thetaBelowCell11113022))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHH
+            thetaBelowCell11113022))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaBelowCell11113022)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHH
+            thetaBelowCell11113022))) h))
+theorem e24KC2ThetaBelowLeaf111130232 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113023) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113023)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11113023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11113023)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11113023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11113023)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL thetaBelowCell11113023))
+        (by
+          have h : ((childLL (childHL (childHL thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childHL
+            thetaBelowCell11113023))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childHL
+            thetaBelowCell11113023))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childHL
+            thetaBelowCell11113023))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childHL
+            thetaBelowCell11113023))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL thetaBelowCell11113023))
+        (by
+          have h : ((childLL (childHH (childHL thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHL
+            thetaBelowCell11113023))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHL
+            thetaBelowCell11113023))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHL
+            thetaBelowCell11113023))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHL
+            thetaBelowCell11113023))) h))
+theorem e24KC2ThetaBelowLeaf111130233 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113023) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113023)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11113023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11113023)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11113023))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11113023)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH thetaBelowCell11113023))
+        (by
+          have h : ((childLL (childHL (childHH thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childHH
+            thetaBelowCell11113023))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childHH
+            thetaBelowCell11113023))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childHH
+            thetaBelowCell11113023))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childHH
+            thetaBelowCell11113023))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH thetaBelowCell11113023))
+        (by
+          have h : ((childLL (childHH (childHH thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHH
+            thetaBelowCell11113023))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHH
+            thetaBelowCell11113023))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHH
+            thetaBelowCell11113023))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaBelowCell11113023)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHH
+            thetaBelowCell11113023))) h))
+theorem e24KC2ThetaBelowLeaf111130322 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113032) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113032)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11113032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11113032)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11113032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11113032)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL thetaBelowCell11113032))
+        (by
+          have h : ((childLL (childHL (childHL thetaBelowCell11113032)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childHL
+            thetaBelowCell11113032))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaBelowCell11113032)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childHL
+            thetaBelowCell11113032))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaBelowCell11113032)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childHL
+            thetaBelowCell11113032))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaBelowCell11113032)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childHL
+            thetaBelowCell11113032))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL thetaBelowCell11113032))
+        (by
+          have h : ((childLL (childHH (childHL thetaBelowCell11113032)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHL
+            thetaBelowCell11113032))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaBelowCell11113032)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHL
+            thetaBelowCell11113032))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaBelowCell11113032)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHL
+            thetaBelowCell11113032))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaBelowCell11113032)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHL
+            thetaBelowCell11113032))) h))
+theorem e24KC2ThetaBelowLeaf111130323 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113032) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113032)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11113032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11113032)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11113032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11113032)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11113032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11113032)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11113032))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11113032)) h)
+theorem e24KC2ThetaBelowLeaf111130332 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113033) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113033)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11113033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11113033)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11113033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11113033)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11113033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11113033)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11113033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11113033)) h)
+theorem e24KC2ThetaBelowLeaf111130333 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113033) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113033)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11113033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11113033)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11113033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11113033)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11113033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11113033)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11113033))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11113033)) h)
+theorem e24KC2ThetaBelowLeaf111131222 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113122) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113122)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11113122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11113122)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11113122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11113122)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11113122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11113122)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11113122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11113122)) h)
+theorem e24KC2ThetaBelowLeaf111131223 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113122) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113122)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11113122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11113122)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11113122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11113122)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11113122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11113122)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11113122))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11113122)) h)
+theorem e24KC2ThetaBelowLeaf111131232 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113123) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113123)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11113123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11113123)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11113123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11113123)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11113123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11113123)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11113123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11113123)) h)
+theorem e24KC2ThetaBelowLeaf111131233 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113123) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113123)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11113123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11113123)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11113123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11113123)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11113123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11113123)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11113123))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11113123)) h)
+theorem e24KC2ThetaBelowLeaf111131322 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113132) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113132)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11113132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11113132)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11113132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11113132)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11113132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11113132)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11113132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11113132)) h)
+theorem e24KC2ThetaBelowLeaf111131323 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113132) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113132)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11113132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11113132)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11113132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11113132)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11113132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11113132)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11113132))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11113132)) h)
+theorem e24KC2ThetaBelowLeaf111131332 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113133) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113133)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11113133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11113133)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11113133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11113133)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11113133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11113133)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11113133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11113133)) h)
+theorem e24KC2ThetaBelowLeaf111131333 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113133) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113133)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11113133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11113133)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11113133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11113133)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11113133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11113133)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11113133))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11113133)) h)
+theorem e24KC2ThetaBelowLeaf111132000 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113200) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11113200)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL thetaBelowCell11113200))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLL
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLL
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLL
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLL
+            thetaBelowCell11113200))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL thetaBelowCell11113200))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLL
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLL
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLL
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLL
+            thetaBelowCell11113200))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL thetaBelowCell11113200))
+        (by
+          have h : ((childLL (childHL (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childLL
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childLL
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childLL
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childLL
+            thetaBelowCell11113200))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL thetaBelowCell11113200))
+        (by
+          have h : ((childLL (childHH (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childLL
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childLL
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childLL
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childLL
+            thetaBelowCell11113200))) h))
+theorem e24KC2ThetaBelowLeaf111132001 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113200) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11113200)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH thetaBelowCell11113200))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLH
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLH
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLH
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLH
+            thetaBelowCell11113200))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH thetaBelowCell11113200))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLH
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLH
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLH
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLH
+            thetaBelowCell11113200))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH thetaBelowCell11113200))
+        (by
+          have h : ((childLL (childHL (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childLH
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childLH
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childLH
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childLH
+            thetaBelowCell11113200))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH thetaBelowCell11113200))
+        (by
+          have h : ((childLL (childHH (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childLH
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childLH
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childLH
+            thetaBelowCell11113200))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaBelowCell11113200)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childLH
+            thetaBelowCell11113200))) h))
+theorem e24KC2ThetaBelowLeaf111132002 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113200) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113200)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11113200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11113200)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11113200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11113200)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11113200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11113200)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11113200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11113200)) h)
+theorem e24KC2ThetaBelowLeaf111132003 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113200) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113200)
+    (by
+      have h : ((childLL (childHH thetaBelowCell11113200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH thetaBelowCell11113200)) h)
+    (by
+      have h : ((childLH (childHH thetaBelowCell11113200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH thetaBelowCell11113200)) h)
+    (by
+      have h : ((childHL (childHH thetaBelowCell11113200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11113200)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11113200))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11113200)) h)
+theorem e24KC2ThetaBelowLeaf111132010 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113201) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11113201)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL thetaBelowCell11113201))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLL
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLL
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLL
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLL
+            thetaBelowCell11113201))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL thetaBelowCell11113201))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLL
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLL
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childLL
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childLL
+            thetaBelowCell11113201))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL thetaBelowCell11113201))
+        (by
+          have h : ((childLL (childHL (childLL thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childLL
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childLL
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childLL
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childLL
+            thetaBelowCell11113201))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL thetaBelowCell11113201))
+        (by
+          have h : ((childLL (childHH (childLL thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childLL
+            thetaBelowCell11113201))) h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHH (childLL
+            thetaBelowCell11113201)))
+            (by
+              have h : (thetaBelowCell111132010310).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132010310 h)
+            (by
+              have h : (thetaBelowCell111132010311).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132010311 h)
+            (by
+              have h : (thetaBelowCell111132010312).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132010312 h)
+            (by
+              have h : (thetaBelowCell111132010313).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132010313 h))
+        (by
+          have h : ((childHL (childHH (childLL thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childLL
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childLL
+            thetaBelowCell11113201))) h))
+theorem cover_subtree_b3391292f833 :
+    adaptiveCoverCheck 8 (childLH (childLH thetaBelowCell11113201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH thetaBelowCell11113201))
+    (by
+      have h : ((childLL (childLH (childLH thetaBelowCell11113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childLH
+        thetaBelowCell11113201))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaBelowCell11113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLH
+        thetaBelowCell11113201))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLH (childLH
+        thetaBelowCell11113201)))
+        (by
+          have h : (thetaBelowCell111132011120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011120 h)
+        (by
+          have h : (thetaBelowCell111132011121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011121 h)
+        (by
+          have h : (thetaBelowCell111132011122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011122 h)
+        (by
+          have h : (thetaBelowCell111132011123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLH (childLH
+        thetaBelowCell11113201)))
+        (by
+          have h : (thetaBelowCell111132011130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011130 h)
+        (by
+          have h : (thetaBelowCell111132011131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011131 h)
+        (by
+          have h : (thetaBelowCell111132011132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011132 h)
+        (by
+          have h : (thetaBelowCell111132011133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011133 h))
+
+theorem cover_subtree_3e9c61b8ab9c :
+    adaptiveCoverCheck 8 (childHL (childLH thetaBelowCell11113201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH thetaBelowCell11113201))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHL (childLH
+        thetaBelowCell11113201)))
+        (by
+          have h : (thetaBelowCell111132011200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011200 h)
+        (by
+          have h : (thetaBelowCell111132011201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011201 h)
+        (by
+          have h : (thetaBelowCell111132011202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011202 h)
+        (by
+          have h : (thetaBelowCell111132011203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHL (childLH
+        thetaBelowCell11113201)))
+        (by
+          have h : (thetaBelowCell111132011210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011210 h)
+        (by
+          have h : (thetaBelowCell111132011211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011211 h)
+        (by
+          have h : (thetaBelowCell111132011212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011212 h)
+        (by
+          have h : (thetaBelowCell111132011213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011213 h))
+    (by
+      have h : ((childHL (childHL (childLH thetaBelowCell11113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childLH
+        thetaBelowCell11113201))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaBelowCell11113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childLH
+        thetaBelowCell11113201))) h)
+
+theorem cover_subtree_466ff371217d :
+    adaptiveCoverCheck 8 (childHH (childLH thetaBelowCell11113201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH thetaBelowCell11113201))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHH (childLH
+        thetaBelowCell11113201)))
+        (by
+          have h : (thetaBelowCell111132011300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011300 h)
+        (by
+          have h : (thetaBelowCell111132011301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011301 h)
+        (by
+          have h : (thetaBelowCell111132011302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011302 h)
+        (by
+          have h : (thetaBelowCell111132011303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHH (childLH
+        thetaBelowCell11113201)))
+        (by
+          have h : (thetaBelowCell111132011310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011310 h)
+        (by
+          have h : (thetaBelowCell111132011311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011311 h)
+        (by
+          have h : (thetaBelowCell111132011312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011312 h)
+        (by
+          have h : (thetaBelowCell111132011313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHH (childLH
+        thetaBelowCell11113201)))
+        (by
+          have h : (thetaBelowCell111132011320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011320 h)
+        (by
+          have h : (thetaBelowCell111132011321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011321 h)
+        (by
+          have h : (thetaBelowCell111132011322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011322 h)
+        (by
+          have h : (thetaBelowCell111132011323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHH (childLH
+        thetaBelowCell11113201)))
+        (by
+          have h : (thetaBelowCell111132011330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011330 h)
+        (by
+          have h : (thetaBelowCell111132011331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011331 h)
+        (by
+          have h : (thetaBelowCell111132011332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011332 h)
+        (by
+          have h : (thetaBelowCell111132011333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132011333 h))
+
+theorem e24KC2ThetaBelowLeaf111132011 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113201) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11113201)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH thetaBelowCell11113201))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childLH
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childLH
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childLH
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childLH
+            thetaBelowCell11113201))) h))
+    cover_subtree_b3391292f833
+    cover_subtree_3e9c61b8ab9c
+    cover_subtree_466ff371217d
+theorem e24KC2ThetaBelowLeaf111132012 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113201) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL thetaBelowCell11113201)
+    (by
+      have h : ((childLL (childHL thetaBelowCell11113201))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL thetaBelowCell11113201)) h)
+    (by
+      have h : ((childLH (childHL thetaBelowCell11113201))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL thetaBelowCell11113201)) h)
+    (by
+      have h : ((childHL (childHL thetaBelowCell11113201))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL thetaBelowCell11113201)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell11113201))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL thetaBelowCell11113201)) h)
+theorem e24KC2ThetaBelowLeaf111132013 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113201) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113201)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH thetaBelowCell11113201))
+        (by
+          have h : ((childLL (childLL (childHH thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLL (childHH
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLL (childHH
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLL (childHH
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLL (childHH
+            thetaBelowCell11113201))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH thetaBelowCell11113201))
+        (by
+          have h : ((childLL (childLH (childHH thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childLH (childHH
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childHH
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childLH (childHH
+            thetaBelowCell11113201))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaBelowCell11113201)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childLH (childHH
+            thetaBelowCell11113201))) h))
+    (by
+      have h : ((childHL (childHH thetaBelowCell11113201))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH thetaBelowCell11113201)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell11113201))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH thetaBelowCell11113201)) h)
+theorem cover_subtree_7f4f4b92c5ab :
+    adaptiveCoverCheck 8 (childLL (childLL thetaBelowCell11113210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL thetaBelowCell11113210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLL (childLL
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132100000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100000 h)
+        (by
+          have h : (thetaBelowCell111132100001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100001 h)
+        (by
+          have h : (thetaBelowCell111132100002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100002 h)
+        (by
+          have h : (thetaBelowCell111132100003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childLL (childLL
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132100010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100010 h)
+        (by
+          have h : (thetaBelowCell111132100011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100011 h)
+        (by
+          have h : (thetaBelowCell111132100012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100012 h)
+        (by
+          have h : (thetaBelowCell111132100013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLL (childLL
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132100020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100020 h)
+        (by
+          have h : (thetaBelowCell111132100021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100021 h)
+        (by
+          have h : (thetaBelowCell111132100022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100022 h)
+        (by
+          have h : (thetaBelowCell111132100023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLL (childLL
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132100030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100030 h)
+        (by
+          have h : (thetaBelowCell111132100031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100031 h)
+        (by
+          have h : (thetaBelowCell111132100032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100032 h)
+        (by
+          have h : (thetaBelowCell111132100033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100033 h))
+
+theorem cover_subtree_c689c5c4108b :
+    adaptiveCoverCheck 8 (childLH (childLL thetaBelowCell11113210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL thetaBelowCell11113210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLH (childLL
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132100100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100100 h)
+        (by
+          have h : (thetaBelowCell111132100101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100101 h)
+        (by
+          have h : (thetaBelowCell111132100102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100102 h)
+        (by
+          have h : (thetaBelowCell111132100103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100103 h))
+    (by
+      have h : ((childLH (childLH (childLL thetaBelowCell11113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childLH (childLL
+        thetaBelowCell11113210))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLH (childLL
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132100120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100120 h)
+        (by
+          have h : (thetaBelowCell111132100121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100121 h)
+        (by
+          have h : (thetaBelowCell111132100122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100122 h)
+        (by
+          have h : (thetaBelowCell111132100123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLH (childLL
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132100130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100130 h)
+        (by
+          have h : (thetaBelowCell111132100131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100131 h)
+        (by
+          have h : (thetaBelowCell111132100132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100132 h)
+        (by
+          have h : (thetaBelowCell111132100133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100133 h))
+
+theorem cover_subtree_9df392f82d53 :
+    adaptiveCoverCheck 8 (childHL (childLL thetaBelowCell11113210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL thetaBelowCell11113210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHL (childLL
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132100200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100200 h)
+        (by
+          have h : (thetaBelowCell111132100201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100201 h)
+        (by
+          have h : (thetaBelowCell111132100202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100202 h)
+        (by
+          have h : (thetaBelowCell111132100203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHL (childLL
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132100210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100210 h)
+        (by
+          have h : (thetaBelowCell111132100211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100211 h)
+        (by
+          have h : (thetaBelowCell111132100212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100212 h)
+        (by
+          have h : (thetaBelowCell111132100213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHL (childLL
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132100220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100220 h)
+        (by
+          have h : (thetaBelowCell111132100221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100221 h)
+        (by
+          have h : (thetaBelowCell111132100222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100222 h)
+        (by
+          have h : (thetaBelowCell111132100223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHL (childLL
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132100230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100230 h)
+        (by
+          have h : (thetaBelowCell111132100231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100231 h)
+        (by
+          have h : (thetaBelowCell111132100232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100232 h)
+        (by
+          have h : (thetaBelowCell111132100233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100233 h))
+
+theorem cover_subtree_f50cdc547dbb :
+    adaptiveCoverCheck 8 (childHH (childLL thetaBelowCell11113210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL thetaBelowCell11113210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLL (childHH (childLL
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132100300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100300 h)
+        (by
+          have h : (thetaBelowCell111132100301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100301 h)
+        (by
+          have h : (thetaBelowCell111132100302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100302 h)
+        (by
+          have h : (thetaBelowCell111132100303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childLH (childHH (childLL
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132100310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100310 h)
+        (by
+          have h : (thetaBelowCell111132100311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100311 h)
+        (by
+          have h : (thetaBelowCell111132100312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100312 h)
+        (by
+          have h : (thetaBelowCell111132100313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHL (childHH (childLL
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132100320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100320 h)
+        (by
+          have h : (thetaBelowCell111132100321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100321 h)
+        (by
+          have h : (thetaBelowCell111132100322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100322 h)
+        (by
+          have h : (thetaBelowCell111132100323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 (childHH (childHH (childLL
+        thetaBelowCell11113210)))
+        (by
+          have h : (thetaBelowCell111132100330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100330 h)
+        (by
+          have h : (thetaBelowCell111132100331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100331 h)
+        (by
+          have h : (thetaBelowCell111132100332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100332 h)
+        (by
+          have h : (thetaBelowCell111132100333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111132100333 h))
+
+theorem e24KC2ThetaBelowLeaf111132100 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113210) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11113210)
+    cover_subtree_7f4f4b92c5ab
+    cover_subtree_c689c5c4108b
+    cover_subtree_9df392f82d53
+    cover_subtree_f50cdc547dbb
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells00c658e5bf
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells00c658e5bf
+
+open CertificateCells00c658e5bf
+theorem e24KC2ThetaBelowLeaf111133120 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113312) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11113312)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11113312))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11113312)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11113312))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11113312)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11113312))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11113312)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11113312))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11113312)) h)
+theorem e24KC2ThetaBelowLeaf111133121 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113312) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11113312)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11113312))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11113312)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11113312))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11113312)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11113312))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11113312)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11113312))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11113312)) h)
+theorem e24KC2ThetaBelowLeaf111133130 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113313) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL thetaBelowCell11113313)
+    (by
+      have h : ((childLL (childLL thetaBelowCell11113313))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL thetaBelowCell11113313)) h)
+    (by
+      have h : ((childLH (childLL thetaBelowCell11113313))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL thetaBelowCell11113313)) h)
+    (by
+      have h : ((childHL (childLL thetaBelowCell11113313))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL thetaBelowCell11113313)) h)
+    (by
+      have h : ((childHH (childLL thetaBelowCell11113313))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL thetaBelowCell11113313)) h)
+theorem e24KC2ThetaBelowLeaf111133131 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113313) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH thetaBelowCell11113313)
+    (by
+      have h : ((childLL (childLH thetaBelowCell11113313))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH thetaBelowCell11113313)) h)
+    (by
+      have h : ((childLH (childLH thetaBelowCell11113313))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH thetaBelowCell11113313)) h)
+    (by
+      have h : ((childHL (childLH thetaBelowCell11113313))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH thetaBelowCell11113313)) h)
+    (by
+      have h : ((childHH (childLH thetaBelowCell11113313))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH thetaBelowCell11113313)) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle009.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle009.lean
new file mode 100644
index 0000000000..58e6a751d0
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle009.lean
@@ -0,0 +1,10521 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch010`.
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch029`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCellse8870ef6ec
+
+/-- Subcell `0111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0111 : AngleCell :=
+  childLH (childLH (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0112 : AngleCell :=
+  childHL (childLH (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0113 : AngleCell :=
+  childHH (childLH (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0120 : AngleCell :=
+  childLL (childHL (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0121 : AngleCell :=
+  childLH (childHL (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0122 : AngleCell :=
+  childHL (childHL (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0123 : AngleCell :=
+  childHH (childHL (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0130 : AngleCell :=
+  childLL (childHH (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0131 : AngleCell :=
+  childLH (childHH (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0132 : AngleCell :=
+  childHL (childHH (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0133 : AngleCell :=
+  childHH (childHH (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `1000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1000 : AngleCell :=
+  childLL (childLL (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `01113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `10002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1000)))
+
+end GerverSofa.PartE.CertificateCellse8870ef6ec
+
+namespace GerverSofa.PartE.CertificateCells7b7ba6d809
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `000032012000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00003201)))
+
+/-- Subcell `000032013000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00003201)))
+
+/-- Subcell `000032102000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00003210)))
+
+/-- Subcell `000032103000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00003210)))
+
+/-- Subcell `000032112000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00003211)))
+
+/-- Subcell `000032113000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000032113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000032113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00003211)))
+
+/-- Subcell `000033002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00003300)))
+
+/-- Subcell `000033003000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00003300)))
+
+/-- Subcell `000033012000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00003301)))
+
+/-- Subcell `000033013000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00003301)))
+
+/-- Subcell `000033102000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00003310)))
+
+end GerverSofa.PartE.CertificateCells7b7ba6d809
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC5TerminalBatchT358400010`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse8870ef6ec
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse8870ef6ec
+
+open CertificateCellse8870ef6ec
+theorem e24KC2ThetaAboveLeaf0111320212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01113202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01113202)) h
+theorem e24KC2ThetaAboveLeaf0111320213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01113202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01113202)) h
+theorem e24KC2ThetaAboveLeaf0111320220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01113202)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01113202)) h
+theorem e24KC2ThetaAboveLeaf0111320221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01113202)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01113202)) h
+theorem e24KC2ThetaAboveLeaf0111320222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01113202)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01113202)) h
+theorem e24KC2ThetaAboveLeaf0111320223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01113202)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01113202)) h
+theorem e24KC2ThetaAboveLeaf0111320230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01113202)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01113202)) h
+theorem e24KC2ThetaAboveLeaf0111320231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01113202)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01113202)) h
+theorem e24KC2ThetaAboveLeaf0111320232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01113202)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01113202)) h
+theorem e24KC2ThetaAboveLeaf0111320233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01113202)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01113202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01113202)) h
+theorem e24KC2ThetaAboveLeaf0111320300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01113203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf0111320301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01113203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf0111320302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01113203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf0111320303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01113203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf0111320310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01113203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf0111320311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01113203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf0111320312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01113203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf0111320313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01113203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf0111320320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01113203)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf0111320321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01113203)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf0111320322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01113203)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf0111320323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01113203)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf0111320330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01113203)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf0111320331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01113203)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf0111320332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01113203)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf0111320333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01113203)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01113203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01113203)) h
+theorem e24KC2ThetaAboveLeaf011132100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113210) = true := by
+  have h : ((childLL thetaAboveCell01113210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113210) h
+theorem e24KC2ThetaAboveLeaf011132101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113210) = true := by
+  have h : ((childLH thetaAboveCell01113210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113210) h
+theorem e24KC2ThetaAboveLeaf011132102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113210) = true := by
+  have h : ((childHL thetaAboveCell01113210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01113210) h
+theorem e24KC2ThetaAboveLeaf011132103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113210) = true := by
+  have h : ((childHH thetaAboveCell01113210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01113210) h
+theorem e24KC2ThetaAboveLeaf011132110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113211) = true := by
+  have h : ((childLL thetaAboveCell01113211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113211) h
+theorem e24KC2ThetaAboveLeaf011132111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113211) = true := by
+  have h : ((childLH thetaAboveCell01113211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113211) h
+theorem e24KC2ThetaAboveLeaf011132112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113211) = true := by
+  have h : ((childHL thetaAboveCell01113211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01113211) h
+theorem e24KC2ThetaAboveLeaf011132113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113211) = true := by
+  have h : ((childHH thetaAboveCell01113211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01113211) h
+theorem e24KC2ThetaAboveLeaf0111321200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01113212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01113212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01113212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01113212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01113212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01113212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01113212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01113212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01113212)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01113212)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01113212)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01113212)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01113212)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01113212)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01113212)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01113212)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01113212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01113212)) h
+theorem e24KC2ThetaAboveLeaf0111321300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01113213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf0111321301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01113213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf0111321302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01113213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf0111321303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01113213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf0111321310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01113213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf0111321311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01113213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf0111321312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01113213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf0111321313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01113213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf0111321320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01113213)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf0111321321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01113213)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf0111321322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01113213)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf0111321323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01113213)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf0111321330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01113213)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf0111321331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01113213)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf0111321332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01113213)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf0111321333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01113213)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01113213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01113213)) h
+theorem e24KC2ThetaAboveLeaf011132200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113220) = true := by
+  have h : ((childLL thetaAboveCell01113220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113220) h
+theorem e24KC2ThetaAboveLeaf011132201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113220) = true := by
+  have h : ((childLH thetaAboveCell01113220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113220) h
+theorem e24KC2ThetaAboveLeaf011132202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113220) = true := by
+  have h : ((childHL thetaAboveCell01113220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01113220) h
+theorem e24KC2ThetaAboveLeaf011132203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113220) = true := by
+  have h : ((childHH thetaAboveCell01113220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01113220) h
+theorem e24KC2ThetaAboveLeaf011132210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113221) = true := by
+  have h : ((childLL thetaAboveCell01113221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113221) h
+theorem e24KC2ThetaAboveLeaf011132211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113221) = true := by
+  have h : ((childLH thetaAboveCell01113221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113221) h
+theorem e24KC2ThetaAboveLeaf011132212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113221) = true := by
+  have h : ((childHL thetaAboveCell01113221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01113221) h
+theorem e24KC2ThetaAboveLeaf011132213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113221) = true := by
+  have h : ((childHH thetaAboveCell01113221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01113221) h
+theorem e24KC2ThetaAboveLeaf01113222 :
+    adaptiveCoverCheck 11 thetaAboveCell01113222 = true := by
+  have h : (thetaAboveCell01113222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113222 h
+theorem e24KC2ThetaAboveLeaf01113223 :
+    adaptiveCoverCheck 11 thetaAboveCell01113223 = true := by
+  have h : (thetaAboveCell01113223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113223 h
+theorem e24KC2ThetaAboveLeaf011132300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113230) = true := by
+  have h : ((childLL thetaAboveCell01113230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113230) h
+theorem e24KC2ThetaAboveLeaf011132301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113230) = true := by
+  have h : ((childLH thetaAboveCell01113230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113230) h
+theorem e24KC2ThetaAboveLeaf011132302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113230) = true := by
+  have h : ((childHL thetaAboveCell01113230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01113230) h
+theorem e24KC2ThetaAboveLeaf011132303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113230) = true := by
+  have h : ((childHH thetaAboveCell01113230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01113230) h
+theorem e24KC2ThetaAboveLeaf011132310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113231) = true := by
+  have h : ((childLL thetaAboveCell01113231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113231) h
+theorem e24KC2ThetaAboveLeaf011132311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113231) = true := by
+  have h : ((childLH thetaAboveCell01113231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113231) h
+theorem e24KC2ThetaAboveLeaf011132312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113231) = true := by
+  have h : ((childHL thetaAboveCell01113231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01113231) h
+theorem e24KC2ThetaAboveLeaf011132313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113231) = true := by
+  have h : ((childHH thetaAboveCell01113231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01113231) h
+theorem e24KC2ThetaAboveLeaf01113232 :
+    adaptiveCoverCheck 11 thetaAboveCell01113232 = true := by
+  have h : (thetaAboveCell01113232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113232 h
+theorem e24KC2ThetaAboveLeaf01113233 :
+    adaptiveCoverCheck 11 thetaAboveCell01113233 = true := by
+  have h : (thetaAboveCell01113233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113233 h
+theorem e24KC2ThetaAboveLeaf011133000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113300) = true := by
+  have h : ((childLL thetaAboveCell01113300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113300) h
+theorem e24KC2ThetaAboveLeaf011133001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113300) = true := by
+  have h : ((childLH thetaAboveCell01113300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113300) h
+theorem e24KC2ThetaAboveLeaf011133002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113300) = true := by
+  have h : ((childHL thetaAboveCell01113300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01113300) h
+theorem e24KC2ThetaAboveLeaf011133003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113300) = true := by
+  have h : ((childHH thetaAboveCell01113300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01113300) h
+theorem e24KC2ThetaAboveLeaf011133010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113301) = true := by
+  have h : ((childLL thetaAboveCell01113301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113301) h
+theorem e24KC2ThetaAboveLeaf011133011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113301) = true := by
+  have h : ((childLH thetaAboveCell01113301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113301) h
+theorem e24KC2ThetaAboveLeaf011133012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113301) = true := by
+  have h : ((childHL thetaAboveCell01113301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01113301) h
+theorem e24KC2ThetaAboveLeaf011133013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113301) = true := by
+  have h : ((childHH thetaAboveCell01113301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01113301) h
+theorem e24KC2ThetaAboveLeaf0111330200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01113302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01113302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01113302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01113302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01113302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01113302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01113302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01113302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01113302)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01113302)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01113302)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01113302)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01113302)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01113302)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01113302)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01113302)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01113302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01113302)) h
+theorem e24KC2ThetaAboveLeaf0111330300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01113303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf0111330301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01113303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf0111330302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01113303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf0111330303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01113303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf0111330310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01113303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf0111330311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01113303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf0111330312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01113303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf0111330313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01113303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf0111330320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01113303)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf0111330321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01113303)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf0111330322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01113303)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf0111330323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01113303)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf0111330330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01113303)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf0111330331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01113303)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf0111330332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01113303)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf0111330333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01113303)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01113303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01113303)) h
+theorem e24KC2ThetaAboveLeaf011133100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113310) = true := by
+  have h : ((childLL thetaAboveCell01113310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113310) h
+theorem e24KC2ThetaAboveLeaf011133101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113310) = true := by
+  have h : ((childLH thetaAboveCell01113310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113310) h
+theorem e24KC2ThetaAboveLeaf011133102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113310) = true := by
+  have h : ((childHL thetaAboveCell01113310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01113310) h
+theorem e24KC2ThetaAboveLeaf011133103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113310) = true := by
+  have h : ((childHH thetaAboveCell01113310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01113310) h
+theorem e24KC2ThetaAboveLeaf011133110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113311) = true := by
+  have h : ((childLL thetaAboveCell01113311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113311) h
+theorem e24KC2ThetaAboveLeaf011133111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113311) = true := by
+  have h : ((childLH thetaAboveCell01113311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113311) h
+theorem e24KC2ThetaAboveLeaf011133112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113311) = true := by
+  have h : ((childHL thetaAboveCell01113311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01113311) h
+theorem e24KC2ThetaAboveLeaf011133113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113311) = true := by
+  have h : ((childHH thetaAboveCell01113311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01113311) h
+theorem e24KC2ThetaAboveLeaf0111331200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01113312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01113312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01113312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01113312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01113312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01113312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01113312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01113312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01113312)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01113312)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01113312)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01113312)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01113312)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01113312)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01113312)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01113312)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01113312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01113312)) h
+theorem e24KC2ThetaAboveLeaf0111331300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell01113313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf0111331301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell01113313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf0111331302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell01113313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf0111331303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell01113313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf0111331310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell01113313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf0111331311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell01113313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf0111331312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell01113313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf0111331313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell01113313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf0111331320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell01113313)) = true := by
+  have h : ((childLL (childHL thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf0111331321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell01113313)) = true := by
+  have h : ((childLH (childHL thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf0111331322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell01113313)) = true := by
+  have h : ((childHL (childHL thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf0111331323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell01113313)) = true := by
+  have h : ((childHH (childHL thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf0111331330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell01113313)) = true := by
+  have h : ((childLL (childHH thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf0111331331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell01113313)) = true := by
+  have h : ((childLH (childHH thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf0111331332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell01113313)) = true := by
+  have h : ((childHL (childHH thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf0111331333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell01113313)) = true := by
+  have h : ((childHH (childHH thetaAboveCell01113313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell01113313)) h
+theorem e24KC2ThetaAboveLeaf011133200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113320) = true := by
+  have h : ((childLL thetaAboveCell01113320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113320) h
+theorem e24KC2ThetaAboveLeaf011133201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113320) = true := by
+  have h : ((childLH thetaAboveCell01113320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113320) h
+theorem e24KC2ThetaAboveLeaf011133202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113320) = true := by
+  have h : ((childHL thetaAboveCell01113320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01113320) h
+theorem e24KC2ThetaAboveLeaf011133203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113320) = true := by
+  have h : ((childHH thetaAboveCell01113320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01113320) h
+theorem e24KC2ThetaAboveLeaf011133210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113321) = true := by
+  have h : ((childLL thetaAboveCell01113321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113321) h
+theorem e24KC2ThetaAboveLeaf011133211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113321) = true := by
+  have h : ((childLH thetaAboveCell01113321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113321) h
+theorem e24KC2ThetaAboveLeaf011133212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113321) = true := by
+  have h : ((childHL thetaAboveCell01113321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01113321) h
+theorem e24KC2ThetaAboveLeaf011133213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113321) = true := by
+  have h : ((childHH thetaAboveCell01113321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01113321) h
+theorem e24KC2ThetaAboveLeaf01113322 :
+    adaptiveCoverCheck 11 thetaAboveCell01113322 = true := by
+  have h : (thetaAboveCell01113322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113322 h
+theorem e24KC2ThetaAboveLeaf01113323 :
+    adaptiveCoverCheck 11 thetaAboveCell01113323 = true := by
+  have h : (thetaAboveCell01113323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113323 h
+theorem e24KC2ThetaAboveLeaf011133300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113330) = true := by
+  have h : ((childLL thetaAboveCell01113330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113330) h
+theorem e24KC2ThetaAboveLeaf011133301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113330) = true := by
+  have h : ((childLH thetaAboveCell01113330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113330) h
+theorem e24KC2ThetaAboveLeaf011133302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113330) = true := by
+  have h : ((childHL thetaAboveCell01113330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01113330) h
+theorem e24KC2ThetaAboveLeaf011133303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113330) = true := by
+  have h : ((childHH thetaAboveCell01113330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01113330) h
+theorem e24KC2ThetaAboveLeaf011133310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113331) = true := by
+  have h : ((childLL thetaAboveCell01113331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell01113331) h
+theorem e24KC2ThetaAboveLeaf011133311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113331) = true := by
+  have h : ((childLH thetaAboveCell01113331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell01113331) h
+theorem e24KC2ThetaAboveLeaf011133312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113331) = true := by
+  have h : ((childHL thetaAboveCell01113331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell01113331) h
+theorem e24KC2ThetaAboveLeaf011133313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113331) = true := by
+  have h : ((childHH thetaAboveCell01113331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell01113331) h
+theorem e24KC2ThetaAboveLeaf01113332 :
+    adaptiveCoverCheck 11 thetaAboveCell01113332 = true := by
+  have h : (thetaAboveCell01113332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113332 h
+theorem e24KC2ThetaAboveLeaf01113333 :
+    adaptiveCoverCheck 11 thetaAboveCell01113333 = true := by
+  have h : (thetaAboveCell01113333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell01113333 h
+theorem e24KC2ThetaAboveLeaf0112000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell0112))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf0112001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell0112))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf0112002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell0112))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf0112003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell0112))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf0112010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell0112))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf0112011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell0112))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf0112012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell0112))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf0112013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell0112))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf011202 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0112)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0112))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0112)) h
+theorem e24KC2ThetaAboveLeaf011203 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0112)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0112))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0112)) h
+theorem e24KC2ThetaAboveLeaf0112100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell0112))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf0112101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell0112))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf0112102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell0112))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf0112103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell0112))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf0112110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell0112))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf0112111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell0112))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf0112112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell0112))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf0112113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell0112))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell0112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell0112))) h
+theorem e24KC2ThetaAboveLeaf011212 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0112)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0112))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0112)) h
+theorem e24KC2ThetaAboveLeaf011213 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0112)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0112))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0112)) h
+theorem e24KC2ThetaAboveLeaf01122 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0112) = true := by
+  have h : ((childHL thetaAboveCell0112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell0112) h
+theorem e24KC2ThetaAboveLeaf01123 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0112) = true := by
+  have h : ((childHH thetaAboveCell0112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell0112) h
+theorem e24KC2ThetaAboveLeaf0113000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell0113))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf0113001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell0113))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf0113002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell0113))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf0113003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell0113))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf0113010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell0113))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf0113011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell0113))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf0113012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell0113))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf0113013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell0113))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf011302 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0113)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0113))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0113)) h
+theorem e24KC2ThetaAboveLeaf011303 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0113)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0113))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0113)) h
+theorem e24KC2ThetaAboveLeaf0113100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell0113))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf0113101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell0113))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf0113102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell0113))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf0113103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell0113))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf0113110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell0113))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf0113111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell0113))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf0113112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell0113))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf0113113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell0113))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell0113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell0113))) h
+theorem e24KC2ThetaAboveLeaf011312 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0113)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0113))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0113)) h
+theorem e24KC2ThetaAboveLeaf011313 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0113)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0113))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0113)) h
+theorem e24KC2ThetaAboveLeaf01132 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0113) = true := by
+  have h : ((childHL thetaAboveCell0113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell0113) h
+theorem e24KC2ThetaAboveLeaf01133 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0113) = true := by
+  have h : ((childHH thetaAboveCell0113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell0113) h
+theorem e24KC2ThetaAboveLeaf0120 :
+    adaptiveCoverCheck 15 thetaAboveCell0120 = true := by
+  have h : (thetaAboveCell0120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0120 h
+theorem e24KC2ThetaAboveLeaf0121 :
+    adaptiveCoverCheck 15 thetaAboveCell0121 = true := by
+  have h : (thetaAboveCell0121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0121 h
+theorem e24KC2ThetaAboveLeaf0122 :
+    adaptiveCoverCheck 15 thetaAboveCell0122 = true := by
+  have h : (thetaAboveCell0122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0122 h
+theorem e24KC2ThetaAboveLeaf0123 :
+    adaptiveCoverCheck 15 thetaAboveCell0123 = true := by
+  have h : (thetaAboveCell0123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0123 h
+theorem e24KC2ThetaAboveLeaf0130 :
+    adaptiveCoverCheck 15 thetaAboveCell0130 = true := by
+  have h : (thetaAboveCell0130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0130 h
+theorem e24KC2ThetaAboveLeaf0131 :
+    adaptiveCoverCheck 15 thetaAboveCell0131 = true := by
+  have h : (thetaAboveCell0131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0131 h
+theorem e24KC2ThetaAboveLeaf0132 :
+    adaptiveCoverCheck 15 thetaAboveCell0132 = true := by
+  have h : (thetaAboveCell0132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0132 h
+theorem e24KC2ThetaAboveLeaf0133 :
+    adaptiveCoverCheck 15 thetaAboveCell0133 = true := by
+  have h : (thetaAboveCell0133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell0133 h
+theorem e24KC2ThetaAboveLeaf020 :
+    adaptiveCoverCheck 16 (childLL (childHL (childLL e24ThetaAboveRoot))) = true := by
+  have h : ((childLL (childHL (childLL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLL (childHL (childLL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf021 :
+    adaptiveCoverCheck 16 (childLH (childHL (childLL e24ThetaAboveRoot))) = true := by
+  have h : ((childLH (childHL (childLL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLH (childHL (childLL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf022 :
+    adaptiveCoverCheck 16 (childHL (childHL (childLL e24ThetaAboveRoot))) = true := by
+  have h : ((childHL (childHL (childLL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHL (childHL (childLL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf023 :
+    adaptiveCoverCheck 16 (childHH (childHL (childLL e24ThetaAboveRoot))) = true := by
+  have h : ((childHH (childHL (childLL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHH (childHL (childLL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf030 :
+    adaptiveCoverCheck 16 (childLL (childHH (childLL e24ThetaAboveRoot))) = true := by
+  have h : ((childLL (childHH (childLL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLL (childHH (childLL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf031 :
+    adaptiveCoverCheck 16 (childLH (childHH (childLL e24ThetaAboveRoot))) = true := by
+  have h : ((childLH (childHH (childLL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLH (childHH (childLL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf032 :
+    adaptiveCoverCheck 16 (childHL (childHH (childLL e24ThetaAboveRoot))) = true := by
+  have h : ((childHL (childHH (childLL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHL (childHH (childLL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf033 :
+    adaptiveCoverCheck 16 (childHH (childHH (childLL e24ThetaAboveRoot))) = true := by
+  have h : ((childHH (childHH (childLL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHH (childHH (childLL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf100000 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1000)) = true := by
+  have h : ((childLL (childLL thetaAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell1000)) h
+theorem e24KC2ThetaAboveLeaf100001 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1000)) = true := by
+  have h : ((childLH (childLL thetaAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell1000)) h
+theorem e24KC2ThetaAboveLeaf100002 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1000)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1000)) h
+theorem e24KC2ThetaAboveLeaf100003 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1000)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1000)) h
+theorem e24KC2ThetaAboveLeaf100010 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1000)) = true := by
+  have h : ((childLL (childLH thetaAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell1000)) h
+theorem e24KC2ThetaAboveLeaf100011 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1000)) = true := by
+  have h : ((childLH (childLH thetaAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell1000)) h
+theorem e24KC2ThetaAboveLeaf100012 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1000)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1000)) h
+theorem e24KC2ThetaAboveLeaf100013 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1000)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1000)) h
+theorem e24KC2ThetaAboveLeaf1000200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell1000))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell1000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell1000))) h
+theorem e24KC2ThetaAboveLeaf1000201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell1000))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell1000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell1000))) h
+theorem e24KC2ThetaAboveLeaf10002020 :
+    adaptiveCoverCheck 11 thetaAboveCell10002020 = true := by
+  have h : (thetaAboveCell10002020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002020 h
+theorem e24KC2ThetaAboveLeaf10002021 :
+    adaptiveCoverCheck 11 thetaAboveCell10002021 = true := by
+  have h : (thetaAboveCell10002021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002021 h
+theorem e24KC2ThetaAboveLeaf10002022 :
+    adaptiveCoverCheck 11 thetaAboveCell10002022 = true := by
+  have h : (thetaAboveCell10002022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002022 h
+theorem e24KC2ThetaAboveLeaf10002023 :
+    adaptiveCoverCheck 11 thetaAboveCell10002023 = true := by
+  have h : (thetaAboveCell10002023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002023 h
+theorem e24KC2ThetaAboveLeaf10002030 :
+    adaptiveCoverCheck 11 thetaAboveCell10002030 = true := by
+  have h : (thetaAboveCell10002030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002030 h
+theorem e24KC2ThetaAboveLeaf10002031 :
+    adaptiveCoverCheck 11 thetaAboveCell10002031 = true := by
+  have h : (thetaAboveCell10002031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002031 h
+theorem e24KC2ThetaAboveLeaf10002032 :
+    adaptiveCoverCheck 11 thetaAboveCell10002032 = true := by
+  have h : (thetaAboveCell10002032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002032 h
+theorem e24KC2ThetaAboveLeaf10002033 :
+    adaptiveCoverCheck 11 thetaAboveCell10002033 = true := by
+  have h : (thetaAboveCell10002033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002033 h
+theorem e24KC2ThetaAboveLeaf1000210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell1000))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell1000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell1000))) h
+theorem e24KC2ThetaAboveLeaf1000211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell1000))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell1000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell1000))) h
+theorem e24KC2ThetaAboveLeaf10002120 :
+    adaptiveCoverCheck 11 thetaAboveCell10002120 = true := by
+  have h : (thetaAboveCell10002120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002120 h
+theorem e24KC2ThetaAboveLeaf10002121 :
+    adaptiveCoverCheck 11 thetaAboveCell10002121 = true := by
+  have h : (thetaAboveCell10002121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002121 h
+theorem e24KC2ThetaAboveLeaf10002122 :
+    adaptiveCoverCheck 11 thetaAboveCell10002122 = true := by
+  have h : (thetaAboveCell10002122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002122 h
+theorem e24KC2ThetaAboveLeaf10002123 :
+    adaptiveCoverCheck 11 thetaAboveCell10002123 = true := by
+  have h : (thetaAboveCell10002123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002123 h
+theorem e24KC2ThetaAboveLeaf10002130 :
+    adaptiveCoverCheck 11 thetaAboveCell10002130 = true := by
+  have h : (thetaAboveCell10002130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002130 h
+theorem e24KC2ThetaAboveLeaf10002131 :
+    adaptiveCoverCheck 11 thetaAboveCell10002131 = true := by
+  have h : (thetaAboveCell10002131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002131 h
+theorem e24KC2ThetaAboveLeaf10002132 :
+    adaptiveCoverCheck 11 thetaAboveCell10002132 = true := by
+  have h : (thetaAboveCell10002132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002132 h
+theorem e24KC2ThetaAboveLeaf10002133 :
+    adaptiveCoverCheck 11 thetaAboveCell10002133 = true := by
+  have h : (thetaAboveCell10002133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002133 h
+theorem e24KC2ThetaAboveLeaf100022000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002200) = true := by
+  have h : ((childLL thetaAboveCell10002200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002200) h
+theorem e24KC2ThetaAboveLeaf100022001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002200) = true := by
+  have h : ((childLH thetaAboveCell10002200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002200) h
+theorem e24KC2ThetaAboveLeaf100022002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002200) = true := by
+  have h : ((childHL thetaAboveCell10002200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002200) h
+theorem e24KC2ThetaAboveLeaf100022003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002200) = true := by
+  have h : ((childHH thetaAboveCell10002200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002200) h
+theorem e24KC2ThetaAboveLeaf100022010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002201) = true := by
+  have h : ((childLL thetaAboveCell10002201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002201) h
+theorem e24KC2ThetaAboveLeaf100022011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002201) = true := by
+  have h : ((childLH thetaAboveCell10002201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002201) h
+theorem e24KC2ThetaAboveLeaf100022012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002201) = true := by
+  have h : ((childHL thetaAboveCell10002201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002201) h
+theorem e24KC2ThetaAboveLeaf100022013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002201) = true := by
+  have h : ((childHH thetaAboveCell10002201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002201) h
+theorem e24KC2ThetaAboveLeaf1000220200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10002202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10002202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10002202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10002202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10002202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10002202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10002202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10002202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10002202)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10002202)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10002202)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10002202)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10002202)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10002202)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10002202)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10002202)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10002202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10002202)) h
+theorem e24KC2ThetaAboveLeaf1000220300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10002203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf1000220301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10002203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf1000220302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10002203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf1000220303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10002203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf1000220310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10002203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf1000220311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10002203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf1000220312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10002203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf1000220313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10002203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf1000220320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10002203)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf1000220321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10002203)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf1000220322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10002203)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf1000220323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10002203)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf1000220330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10002203)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf1000220331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10002203)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf1000220332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10002203)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf1000220333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10002203)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10002203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10002203)) h
+theorem e24KC2ThetaAboveLeaf100022100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002210) = true := by
+  have h : ((childLL thetaAboveCell10002210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002210) h
+theorem e24KC2ThetaAboveLeaf100022101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002210) = true := by
+  have h : ((childLH thetaAboveCell10002210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002210) h
+theorem e24KC2ThetaAboveLeaf100022102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002210) = true := by
+  have h : ((childHL thetaAboveCell10002210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002210) h
+theorem e24KC2ThetaAboveLeaf100022103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002210) = true := by
+  have h : ((childHH thetaAboveCell10002210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002210) h
+theorem e24KC2ThetaAboveLeaf100022110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002211) = true := by
+  have h : ((childLL thetaAboveCell10002211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002211) h
+theorem e24KC2ThetaAboveLeaf100022111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002211) = true := by
+  have h : ((childLH thetaAboveCell10002211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002211) h
+theorem e24KC2ThetaAboveLeaf100022112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002211) = true := by
+  have h : ((childHL thetaAboveCell10002211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002211) h
+theorem e24KC2ThetaAboveLeaf100022113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002211) = true := by
+  have h : ((childHH thetaAboveCell10002211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002211) h
+theorem e24KC2ThetaAboveLeaf1000221200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10002212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10002212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10002212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10002212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10002212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10002212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10002212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10002212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10002212)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10002212)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10002212)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10002212)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10002212)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10002212)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10002212)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10002212)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10002212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10002212)) h
+theorem e24KC2ThetaAboveLeaf1000221300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10002213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf1000221301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10002213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf1000221302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10002213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf1000221303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10002213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf1000221310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10002213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf1000221311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10002213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf1000221312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10002213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf1000221313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10002213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf1000221320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10002213)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf1000221321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10002213)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf1000221322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10002213)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf1000221323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10002213)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf1000221330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10002213)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf1000221331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10002213)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf1000221332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10002213)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf1000221333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10002213)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10002213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10002213)) h
+theorem e24KC2ThetaAboveLeaf100022200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002220) = true := by
+  have h : ((childLL thetaAboveCell10002220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002220) h
+theorem e24KC2ThetaAboveLeaf100022201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002220) = true := by
+  have h : ((childLH thetaAboveCell10002220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002220) h
+theorem e24KC2ThetaAboveLeaf100022202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002220) = true := by
+  have h : ((childHL thetaAboveCell10002220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002220) h
+theorem e24KC2ThetaAboveLeaf100022203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002220) = true := by
+  have h : ((childHH thetaAboveCell10002220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002220) h
+theorem e24KC2ThetaAboveLeaf100022210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002221) = true := by
+  have h : ((childLL thetaAboveCell10002221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002221) h
+theorem e24KC2ThetaAboveLeaf100022211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002221) = true := by
+  have h : ((childLH thetaAboveCell10002221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002221) h
+theorem e24KC2ThetaAboveLeaf100022212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002221) = true := by
+  have h : ((childHL thetaAboveCell10002221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002221) h
+theorem e24KC2ThetaAboveLeaf100022213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002221) = true := by
+  have h : ((childHH thetaAboveCell10002221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002221) h
+theorem e24KC2ThetaAboveLeaf10002222 :
+    adaptiveCoverCheck 11 thetaAboveCell10002222 = true := by
+  have h : (thetaAboveCell10002222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002222 h
+theorem e24KC2ThetaAboveLeaf10002223 :
+    adaptiveCoverCheck 11 thetaAboveCell10002223 = true := by
+  have h : (thetaAboveCell10002223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002223 h
+theorem e24KC2ThetaAboveLeaf100022300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002230) = true := by
+  have h : ((childLL thetaAboveCell10002230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002230) h
+theorem e24KC2ThetaAboveLeaf100022301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002230) = true := by
+  have h : ((childLH thetaAboveCell10002230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002230) h
+theorem e24KC2ThetaAboveLeaf100022302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002230) = true := by
+  have h : ((childHL thetaAboveCell10002230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002230) h
+theorem e24KC2ThetaAboveLeaf100022303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002230) = true := by
+  have h : ((childHH thetaAboveCell10002230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002230) h
+theorem e24KC2ThetaAboveLeaf100022310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002231) = true := by
+  have h : ((childLL thetaAboveCell10002231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002231) h
+theorem e24KC2ThetaAboveLeaf100022311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002231) = true := by
+  have h : ((childLH thetaAboveCell10002231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002231) h
+theorem e24KC2ThetaAboveLeaf100022312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002231) = true := by
+  have h : ((childHL thetaAboveCell10002231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002231) h
+theorem e24KC2ThetaAboveLeaf100022313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002231) = true := by
+  have h : ((childHH thetaAboveCell10002231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002231) h
+theorem e24KC2ThetaAboveLeaf10002232 :
+    adaptiveCoverCheck 11 thetaAboveCell10002232 = true := by
+  have h : (thetaAboveCell10002232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002232 h
+theorem e24KC2ThetaAboveLeaf10002233 :
+    adaptiveCoverCheck 11 thetaAboveCell10002233 = true := by
+  have h : (thetaAboveCell10002233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002233 h
+theorem e24KC2ThetaAboveLeaf100023000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002300) = true := by
+  have h : ((childLL thetaAboveCell10002300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002300) h
+theorem e24KC2ThetaAboveLeaf100023001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002300) = true := by
+  have h : ((childLH thetaAboveCell10002300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002300) h
+theorem e24KC2ThetaAboveLeaf100023002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002300) = true := by
+  have h : ((childHL thetaAboveCell10002300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002300) h
+theorem e24KC2ThetaAboveLeaf100023003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002300) = true := by
+  have h : ((childHH thetaAboveCell10002300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002300) h
+theorem e24KC2ThetaAboveLeaf100023010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002301) = true := by
+  have h : ((childLL thetaAboveCell10002301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002301) h
+theorem e24KC2ThetaAboveLeaf100023011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002301) = true := by
+  have h : ((childLH thetaAboveCell10002301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002301) h
+theorem e24KC2ThetaAboveLeaf100023012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002301) = true := by
+  have h : ((childHL thetaAboveCell10002301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002301) h
+theorem e24KC2ThetaAboveLeaf100023013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002301) = true := by
+  have h : ((childHH thetaAboveCell10002301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002301) h
+theorem e24KC2ThetaAboveLeaf1000230200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10002302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10002302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10002302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10002302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10002302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10002302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10002302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10002302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10002302)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10002302)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10002302)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10002302)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10002302)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10002302)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10002302)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10002302)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10002302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10002302)) h
+theorem e24KC2ThetaAboveLeaf1000230300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10002303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf1000230301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10002303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf1000230302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10002303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf1000230303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10002303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf1000230310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10002303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf1000230311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10002303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf1000230312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10002303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf1000230313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10002303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf1000230320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10002303)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf1000230321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10002303)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf1000230322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10002303)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf1000230323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10002303)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf1000230330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10002303)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf1000230331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10002303)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf1000230332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10002303)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf1000230333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10002303)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10002303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10002303)) h
+theorem e24KC2ThetaAboveLeaf100023100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002310) = true := by
+  have h : ((childLL thetaAboveCell10002310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002310) h
+theorem e24KC2ThetaAboveLeaf100023101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002310) = true := by
+  have h : ((childLH thetaAboveCell10002310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002310) h
+theorem e24KC2ThetaAboveLeaf100023102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002310) = true := by
+  have h : ((childHL thetaAboveCell10002310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002310) h
+theorem e24KC2ThetaAboveLeaf100023103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002310) = true := by
+  have h : ((childHH thetaAboveCell10002310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002310) h
+theorem e24KC2ThetaAboveLeaf100023110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002311) = true := by
+  have h : ((childLL thetaAboveCell10002311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002311) h
+theorem e24KC2ThetaAboveLeaf100023111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002311) = true := by
+  have h : ((childLH thetaAboveCell10002311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002311) h
+theorem e24KC2ThetaAboveLeaf100023112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002311) = true := by
+  have h : ((childHL thetaAboveCell10002311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002311) h
+theorem e24KC2ThetaAboveLeaf100023113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002311) = true := by
+  have h : ((childHH thetaAboveCell10002311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002311) h
+theorem e24KC2ThetaAboveLeaf1000231200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10002312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10002312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10002312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10002312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10002312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10002312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10002312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10002312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10002312)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10002312)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10002312)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10002312)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10002312)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10002312)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10002312)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10002312)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10002312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10002312)) h
+theorem e24KC2ThetaAboveLeaf1000231300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10002313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf1000231301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10002313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf1000231302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10002313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf1000231303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10002313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf1000231310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10002313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf1000231311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10002313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf1000231312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10002313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf1000231313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10002313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf1000231320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10002313)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf1000231321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10002313)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf1000231322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10002313)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf1000231323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10002313)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf1000231330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10002313)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf1000231331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10002313)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf1000231332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10002313)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf1000231333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10002313)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10002313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10002313)) h
+theorem e24KC2ThetaAboveLeaf100023200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002320) = true := by
+  have h : ((childLL thetaAboveCell10002320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002320) h
+theorem e24KC2ThetaAboveLeaf100023201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002320) = true := by
+  have h : ((childLH thetaAboveCell10002320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002320) h
+theorem e24KC2ThetaAboveLeaf100023202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002320) = true := by
+  have h : ((childHL thetaAboveCell10002320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002320) h
+theorem e24KC2ThetaAboveLeaf100023203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002320) = true := by
+  have h : ((childHH thetaAboveCell10002320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002320) h
+theorem e24KC2ThetaAboveLeaf100023210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002321) = true := by
+  have h : ((childLL thetaAboveCell10002321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002321) h
+theorem e24KC2ThetaAboveLeaf100023211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002321) = true := by
+  have h : ((childLH thetaAboveCell10002321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002321) h
+theorem e24KC2ThetaAboveLeaf100023212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002321) = true := by
+  have h : ((childHL thetaAboveCell10002321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002321) h
+theorem e24KC2ThetaAboveLeaf100023213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002321) = true := by
+  have h : ((childHH thetaAboveCell10002321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002321) h
+theorem e24KC2ThetaAboveLeaf10002322 :
+    adaptiveCoverCheck 11 thetaAboveCell10002322 = true := by
+  have h : (thetaAboveCell10002322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002322 h
+theorem e24KC2ThetaAboveLeaf10002323 :
+    adaptiveCoverCheck 11 thetaAboveCell10002323 = true := by
+  have h : (thetaAboveCell10002323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002323 h
+theorem e24KC2ThetaAboveLeaf100023300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002330) = true := by
+  have h : ((childLL thetaAboveCell10002330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002330) h
+theorem e24KC2ThetaAboveLeaf100023301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002330) = true := by
+  have h : ((childLH thetaAboveCell10002330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002330) h
+theorem e24KC2ThetaAboveLeaf100023302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002330) = true := by
+  have h : ((childHL thetaAboveCell10002330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002330) h
+theorem e24KC2ThetaAboveLeaf100023303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002330) = true := by
+  have h : ((childHH thetaAboveCell10002330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002330) h
+theorem e24KC2ThetaAboveLeaf100023310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002331) = true := by
+  have h : ((childLL thetaAboveCell10002331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10002331) h
+theorem e24KC2ThetaAboveLeaf100023311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002331) = true := by
+  have h : ((childLH thetaAboveCell10002331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10002331) h
+theorem e24KC2ThetaAboveLeaf100023312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002331) = true := by
+  have h : ((childHL thetaAboveCell10002331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10002331) h
+theorem e24KC2ThetaAboveLeaf100023313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002331) = true := by
+  have h : ((childHH thetaAboveCell10002331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10002331) h
+theorem e24KC2ThetaAboveLeaf10002332 :
+    adaptiveCoverCheck 11 thetaAboveCell10002332 = true := by
+  have h : (thetaAboveCell10002332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002332 h
+theorem e24KC2ThetaAboveLeaf10002333 :
+    adaptiveCoverCheck 11 thetaAboveCell10002333 = true := by
+  have h : (thetaAboveCell10002333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10002333 h
+theorem e24KC2ThetaAboveLeaf1000300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell1000))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell1000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell1000))) h
+theorem e24KC2ThetaAboveLeaf1000301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell1000))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell1000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell1000))) h
+theorem e24KC2ThetaAboveLeaf10003020 :
+    adaptiveCoverCheck 11 thetaAboveCell10003020 = true := by
+  have h : (thetaAboveCell10003020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003020 h
+theorem e24KC2ThetaAboveLeaf10003021 :
+    adaptiveCoverCheck 11 thetaAboveCell10003021 = true := by
+  have h : (thetaAboveCell10003021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003021 h
+theorem e24KC2ThetaAboveLeaf10003022 :
+    adaptiveCoverCheck 11 thetaAboveCell10003022 = true := by
+  have h : (thetaAboveCell10003022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003022 h
+theorem e24KC2ThetaAboveLeaf10003023 :
+    adaptiveCoverCheck 11 thetaAboveCell10003023 = true := by
+  have h : (thetaAboveCell10003023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003023 h
+theorem e24KC2ThetaAboveLeaf10003030 :
+    adaptiveCoverCheck 11 thetaAboveCell10003030 = true := by
+  have h : (thetaAboveCell10003030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003030 h
+theorem e24KC2ThetaAboveLeaf10003031 :
+    adaptiveCoverCheck 11 thetaAboveCell10003031 = true := by
+  have h : (thetaAboveCell10003031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003031 h
+theorem e24KC2ThetaAboveLeaf10003032 :
+    adaptiveCoverCheck 11 thetaAboveCell10003032 = true := by
+  have h : (thetaAboveCell10003032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003032 h
+theorem e24KC2ThetaAboveLeaf10003033 :
+    adaptiveCoverCheck 11 thetaAboveCell10003033 = true := by
+  have h : (thetaAboveCell10003033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003033 h
+theorem e24KC2ThetaAboveLeaf1000310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell1000))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell1000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell1000))) h
+theorem e24KC2ThetaAboveLeaf1000311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell1000))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell1000)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell1000))) h
+theorem e24KC2ThetaAboveLeaf10003120 :
+    adaptiveCoverCheck 11 thetaAboveCell10003120 = true := by
+  have h : (thetaAboveCell10003120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003120 h
+theorem e24KC2ThetaAboveLeaf10003121 :
+    adaptiveCoverCheck 11 thetaAboveCell10003121 = true := by
+  have h : (thetaAboveCell10003121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003121 h
+theorem e24KC2ThetaAboveLeaf10003122 :
+    adaptiveCoverCheck 11 thetaAboveCell10003122 = true := by
+  have h : (thetaAboveCell10003122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003122 h
+theorem e24KC2ThetaAboveLeaf10003123 :
+    adaptiveCoverCheck 11 thetaAboveCell10003123 = true := by
+  have h : (thetaAboveCell10003123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003123 h
+theorem e24KC2ThetaAboveLeaf10003130 :
+    adaptiveCoverCheck 11 thetaAboveCell10003130 = true := by
+  have h : (thetaAboveCell10003130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003130 h
+theorem e24KC2ThetaAboveLeaf10003131 :
+    adaptiveCoverCheck 11 thetaAboveCell10003131 = true := by
+  have h : (thetaAboveCell10003131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003131 h
+theorem e24KC2ThetaAboveLeaf10003132 :
+    adaptiveCoverCheck 11 thetaAboveCell10003132 = true := by
+  have h : (thetaAboveCell10003132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003132 h
+theorem e24KC2ThetaAboveLeaf10003133 :
+    adaptiveCoverCheck 11 thetaAboveCell10003133 = true := by
+  have h : (thetaAboveCell10003133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003133 h
+theorem e24KC2ThetaAboveLeaf100032000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003200) = true := by
+  have h : ((childLL thetaAboveCell10003200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003200) h
+theorem e24KC2ThetaAboveLeaf100032001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003200) = true := by
+  have h : ((childLH thetaAboveCell10003200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10003200) h
+theorem e24KC2ThetaAboveLeaf100032002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003200) = true := by
+  have h : ((childHL thetaAboveCell10003200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003200) h
+theorem e24KC2ThetaAboveLeaf100032003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003200) = true := by
+  have h : ((childHH thetaAboveCell10003200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003200) h
+theorem e24KC2ThetaAboveLeaf100032010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003201) = true := by
+  have h : ((childLL thetaAboveCell10003201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003201) h
+theorem e24KC2ThetaAboveLeaf100032011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003201) = true := by
+  have h : ((childLH thetaAboveCell10003201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10003201) h
+theorem e24KC2ThetaAboveLeaf100032012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003201) = true := by
+  have h : ((childHL thetaAboveCell10003201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003201) h
+theorem e24KC2ThetaAboveLeaf100032013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003201) = true := by
+  have h : ((childHH thetaAboveCell10003201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003201) h
+theorem e24KC2ThetaAboveLeaf1000320200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10003202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10003202)) h
+theorem e24KC2ThetaAboveLeaf1000320201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10003202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10003202)) h
+theorem e24KC2ThetaAboveLeaf1000320202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10003202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10003202)) h
+theorem e24KC2ThetaAboveLeaf1000320203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10003202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10003202)) h
+theorem e24KC2ThetaAboveLeaf1000320210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10003202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10003202)) h
+theorem e24KC2ThetaAboveLeaf1000320211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10003202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10003202)) h
+theorem e24KC2ThetaAboveLeaf1000320212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10003202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10003202)) h
+theorem e24KC2ThetaAboveLeaf1000320213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10003202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10003202)) h
+theorem e24KC2ThetaAboveLeaf1000320220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10003202)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10003202)) h
+theorem e24KC2ThetaAboveLeaf1000320221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10003202)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10003202)) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6ProofBatchDeeff549d6afce41`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells7b7ba6d809
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells7b7ba6d809
+
+open CertificateCells7b7ba6d809
+theorem cover_subtree_7c323abdd313 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00003201)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012000
+        (by
+          have h : ((childLL thetaAboveCell000032012000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012000) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012000) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012000) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012000) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012001
+        (by
+          have h : ((childLL thetaAboveCell000032012001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012001) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012001) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012001) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012001) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012002
+        (by
+          have h : ((childLL thetaAboveCell000032012002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012002) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012002) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012002) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012003
+        (by
+          have h : ((childLL thetaAboveCell000032012003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012003) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012003) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012003) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012003) h))
+
+theorem cover_subtree_2c6010c13cb7 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00003201)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012010
+        (by
+          have h : ((childLL thetaAboveCell000032012010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012010) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012010) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012010) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012010) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012011
+        (by
+          have h : ((childLL thetaAboveCell000032012011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012011) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012011) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012011) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012011) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012012
+        (by
+          have h : ((childLL thetaAboveCell000032012012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012012) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012012) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012012) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012013
+        (by
+          have h : ((childLL thetaAboveCell000032012013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012013) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012013) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012013) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012013) h))
+
+theorem cover_subtree_1c36f4e287e2 :
+    adaptiveCoverCheck 8 (childHL (childLL (childHL thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL thetaAboveCell00003201)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012020
+        (by
+          have h : ((childLL thetaAboveCell000032012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012020) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012020) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012020) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012021
+        (by
+          have h : ((childLL thetaAboveCell000032012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012021) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012021) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012021) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012021) h))
+    (by
+      have h : (thetaAboveCell000032012022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032012022 h)
+    (by
+      have h : (thetaAboveCell000032012023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032012023 h)
+
+theorem cover_subtree_2d88b84be921 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHL thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL thetaAboveCell00003201)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012030
+        (by
+          have h : ((childLL thetaAboveCell000032012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012030) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012030) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012030) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012031
+        (by
+          have h : ((childLL thetaAboveCell000032012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012031) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012031) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012031) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012031) h))
+    (by
+      have h : (thetaAboveCell000032012032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032012032 h)
+    (by
+      have h : (thetaAboveCell000032012033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032012033 h)
+
+theorem e24KC2ThetaAboveLeaf0000320120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00003201))
+    cover_subtree_7c323abdd313
+    cover_subtree_2c6010c13cb7
+    cover_subtree_1c36f4e287e2
+    cover_subtree_2d88b84be921
+theorem cover_subtree_15ce3b085062 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00003201)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012100
+        (by
+          have h : ((childLL thetaAboveCell000032012100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012100) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012100) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012100) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012100) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012101
+        (by
+          have h : ((childLL thetaAboveCell000032012101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012101) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012101) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012101) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012101) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012102
+        (by
+          have h : ((childLL thetaAboveCell000032012102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012102) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012102) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012102) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012103
+        (by
+          have h : ((childLL thetaAboveCell000032012103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012103) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012103) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012103) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012103) h))
+
+theorem cover_subtree_08caed552cac :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00003201)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012110
+        (by
+          have h : ((childLL thetaAboveCell000032012110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012110) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012110) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012110) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012110) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012111
+        (by
+          have h : ((childLL thetaAboveCell000032012111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012111) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012111) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012111) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012111) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012112
+        (by
+          have h : ((childLL thetaAboveCell000032012112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012112) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012112) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012112) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012113
+        (by
+          have h : ((childLL thetaAboveCell000032012113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012113) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012113) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012113) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012113) h))
+
+theorem cover_subtree_26481d0c715c :
+    adaptiveCoverCheck 8 (childHL (childLH (childHL thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL thetaAboveCell00003201)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012120
+        (by
+          have h : ((childLL thetaAboveCell000032012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012120) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012120) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012120) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012121
+        (by
+          have h : ((childLL thetaAboveCell000032012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012121) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012121) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012121) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012121) h))
+    (by
+      have h : (thetaAboveCell000032012122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032012122 h)
+    (by
+      have h : (thetaAboveCell000032012123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032012123 h)
+
+theorem cover_subtree_a0f2894108e0 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHL thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL thetaAboveCell00003201)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012130
+        (by
+          have h : ((childLL thetaAboveCell000032012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012130) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012130) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012130) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032012131
+        (by
+          have h : ((childLL thetaAboveCell000032012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032012131) h)
+        (by
+          have h : ((childLH thetaAboveCell000032012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032012131) h)
+        (by
+          have h : ((childHL thetaAboveCell000032012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032012131) h)
+        (by
+          have h : ((childHH thetaAboveCell000032012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032012131) h))
+    (by
+      have h : (thetaAboveCell000032012132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032012132 h)
+    (by
+      have h : (thetaAboveCell000032012133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032012133 h)
+
+theorem e24KC2ThetaAboveLeaf0000320121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00003201))
+    cover_subtree_15ce3b085062
+    cover_subtree_08caed552cac
+    cover_subtree_26481d0c715c
+    cover_subtree_a0f2894108e0
+theorem e24KC2ThetaAboveLeaf0000320122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00003201))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00003201))) h)
+theorem e24KC2ThetaAboveLeaf0000320123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00003201))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00003201))) h)
+theorem cover_subtree_69db60e02bff :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00003201)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013000
+        (by
+          have h : ((childLL thetaAboveCell000032013000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013000) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013000) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013000) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013000) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013001
+        (by
+          have h : ((childLL thetaAboveCell000032013001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013001) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013001) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013001) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013001) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013002
+        (by
+          have h : ((childLL thetaAboveCell000032013002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013002) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013002) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013002) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013003
+        (by
+          have h : ((childLL thetaAboveCell000032013003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013003) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013003) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013003) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013003) h))
+
+theorem cover_subtree_42fb80f99488 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00003201)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013010
+        (by
+          have h : ((childLL thetaAboveCell000032013010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013010) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013010) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013010) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013010) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013011
+        (by
+          have h : ((childLL thetaAboveCell000032013011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013011) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013011) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013011) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013011) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013012
+        (by
+          have h : ((childLL thetaAboveCell000032013012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013012) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013012) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013012) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013013
+        (by
+          have h : ((childLL thetaAboveCell000032013013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013013) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013013) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013013) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013013) h))
+
+theorem cover_subtree_190c09664a4f :
+    adaptiveCoverCheck 8 (childHL (childLL (childHH thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH thetaAboveCell00003201)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013020
+        (by
+          have h : ((childLL thetaAboveCell000032013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013020) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013020) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013020) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013021
+        (by
+          have h : ((childLL thetaAboveCell000032013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013021) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013021) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013021) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013021) h))
+    (by
+      have h : (thetaAboveCell000032013022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032013022 h)
+    (by
+      have h : (thetaAboveCell000032013023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032013023 h)
+
+theorem cover_subtree_f68e71dd24e5 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHH thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH thetaAboveCell00003201)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013030
+        (by
+          have h : ((childLL thetaAboveCell000032013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013030) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013030) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013030) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013031
+        (by
+          have h : ((childLL thetaAboveCell000032013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013031) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013031) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013031) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013031) h))
+    (by
+      have h : (thetaAboveCell000032013032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032013032 h)
+    (by
+      have h : (thetaAboveCell000032013033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032013033 h)
+
+theorem e24KC2ThetaAboveLeaf0000320130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00003201))
+    cover_subtree_69db60e02bff
+    cover_subtree_42fb80f99488
+    cover_subtree_190c09664a4f
+    cover_subtree_f68e71dd24e5
+theorem cover_subtree_7cb6ec704eae :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00003201)))
+    (by
+      have h : (thetaAboveCell000032013100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032013100 h)
+    (by
+      have h : (thetaAboveCell000032013101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032013101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013102
+        (by
+          have h : ((childLL thetaAboveCell000032013102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013102) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013102) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013102) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013103
+        (by
+          have h : ((childLL thetaAboveCell000032013103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013103) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013103) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013103) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013103) h))
+
+theorem cover_subtree_5ed05ff39522 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00003201)))
+    (by
+      have h : (thetaAboveCell000032013110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032013110 h)
+    (by
+      have h : (thetaAboveCell000032013111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032013111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013112
+        (by
+          have h : ((childLL thetaAboveCell000032013112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013112) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013112) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013112) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013113
+        (by
+          have h : ((childLL thetaAboveCell000032013113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013113) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013113) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013113) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013113) h))
+
+theorem cover_subtree_0c55724b42aa :
+    adaptiveCoverCheck 8 (childHL (childLH (childHH thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH thetaAboveCell00003201)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013120
+        (by
+          have h : ((childLL thetaAboveCell000032013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013120) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013120) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013120) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013121
+        (by
+          have h : ((childLL thetaAboveCell000032013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013121) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013121) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013121) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013121) h))
+    (by
+      have h : (thetaAboveCell000032013122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032013122 h)
+    (by
+      have h : (thetaAboveCell000032013123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032013123 h)
+
+theorem cover_subtree_a20be064bbc9 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHH thetaAboveCell00003201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH thetaAboveCell00003201)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013130
+        (by
+          have h : ((childLL thetaAboveCell000032013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013130) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013130) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013130) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032013131
+        (by
+          have h : ((childLL thetaAboveCell000032013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032013131) h)
+        (by
+          have h : ((childLH thetaAboveCell000032013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032013131) h)
+        (by
+          have h : ((childHL thetaAboveCell000032013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032013131) h)
+        (by
+          have h : ((childHH thetaAboveCell000032013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032013131) h))
+    (by
+      have h : (thetaAboveCell000032013132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032013132 h)
+    (by
+      have h : (thetaAboveCell000032013133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032013133 h)
+
+theorem e24KC2ThetaAboveLeaf0000320131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00003201))
+    cover_subtree_7cb6ec704eae
+    cover_subtree_5ed05ff39522
+    cover_subtree_0c55724b42aa
+    cover_subtree_a20be064bbc9
+theorem e24KC2ThetaAboveLeaf0000320132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00003201))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00003201))) h)
+theorem e24KC2ThetaAboveLeaf0000320133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00003201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00003201))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00003201))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00003201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00003201))) h)
+theorem e24KC2ThetaAboveLeaf0000321002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00003210))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell00003210))) h)
+theorem e24KC2ThetaAboveLeaf0000321003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00003210))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHL (childHH (childLL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLL
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHH (childHH (childLL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLL
+        thetaAboveCell00003210))) h)
+theorem e24KC2ThetaAboveLeaf0000321012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00003210))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHL (childHL (childLH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLH
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell00003210))) h)
+theorem e24KC2ThetaAboveLeaf0000321013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00003210))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell00003210))) h)
+theorem cover_subtree_0ba391c6ec0d :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00003210)))
+    (by
+      have h : (thetaAboveCell000032102000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102000 h)
+    (by
+      have h : (thetaAboveCell000032102001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102002
+        (by
+          have h : ((childLL thetaAboveCell000032102002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102002) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102002) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102002) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102003
+        (by
+          have h : ((childLL thetaAboveCell000032102003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102003) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102003) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102003) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102003) h))
+
+theorem cover_subtree_8fafa3f6224a :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00003210)))
+    (by
+      have h : (thetaAboveCell000032102010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102010 h)
+    (by
+      have h : (thetaAboveCell000032102011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102012
+        (by
+          have h : ((childLL thetaAboveCell000032102012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102012) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102012) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102012) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102013
+        (by
+          have h : ((childLL thetaAboveCell000032102013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102013) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102013) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102013) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102013) h))
+
+theorem cover_subtree_d5e7f8726976 :
+    adaptiveCoverCheck 8 (childHL (childLL (childHL thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL thetaAboveCell00003210)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102020
+        (by
+          have h : ((childLL thetaAboveCell000032102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102020) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102020) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102020) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102021
+        (by
+          have h : ((childLL thetaAboveCell000032102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102021) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102021) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102021) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102021) h))
+    (by
+      have h : (thetaAboveCell000032102022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102022 h)
+    (by
+      have h : (thetaAboveCell000032102023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102023 h)
+
+theorem cover_subtree_c2e91aa19513 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHL thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL thetaAboveCell00003210)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102030
+        (by
+          have h : ((childLL thetaAboveCell000032102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102030) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102030) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102030) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102031
+        (by
+          have h : ((childLL thetaAboveCell000032102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102031) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102031) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102031) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102031) h))
+    (by
+      have h : (thetaAboveCell000032102032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102032 h)
+    (by
+      have h : (thetaAboveCell000032102033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102033 h)
+
+theorem e24KC2ThetaAboveLeaf0000321020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00003210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00003210))
+    cover_subtree_0ba391c6ec0d
+    cover_subtree_8fafa3f6224a
+    cover_subtree_d5e7f8726976
+    cover_subtree_c2e91aa19513
+theorem cover_subtree_80cc442a7cb0 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00003210)))
+    (by
+      have h : (thetaAboveCell000032102100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102100 h)
+    (by
+      have h : (thetaAboveCell000032102101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102102
+        (by
+          have h : ((childLL thetaAboveCell000032102102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102102) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102102) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102102) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102103
+        (by
+          have h : ((childLL thetaAboveCell000032102103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102103) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102103) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102103) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102103) h))
+
+theorem cover_subtree_7d2d9f5abf21 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00003210)))
+    (by
+      have h : (thetaAboveCell000032102110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102110 h)
+    (by
+      have h : (thetaAboveCell000032102111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102112
+        (by
+          have h : ((childLL thetaAboveCell000032102112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102112) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102112) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102112) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102113
+        (by
+          have h : ((childLL thetaAboveCell000032102113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102113) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102113) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102113) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102113) h))
+
+theorem cover_subtree_4430ea3984ba :
+    adaptiveCoverCheck 8 (childHL (childLH (childHL thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL thetaAboveCell00003210)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102120
+        (by
+          have h : ((childLL thetaAboveCell000032102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102120) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102120) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102120) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102121
+        (by
+          have h : ((childLL thetaAboveCell000032102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102121) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102121) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102121) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102121) h))
+    (by
+      have h : (thetaAboveCell000032102122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102122 h)
+    (by
+      have h : (thetaAboveCell000032102123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102123 h)
+
+theorem cover_subtree_7e0429da25bc :
+    adaptiveCoverCheck 8 (childHH (childLH (childHL thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL thetaAboveCell00003210)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102130
+        (by
+          have h : ((childLL thetaAboveCell000032102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102130) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102130) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102130) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032102131
+        (by
+          have h : ((childLL thetaAboveCell000032102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032102131) h)
+        (by
+          have h : ((childLH thetaAboveCell000032102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032102131) h)
+        (by
+          have h : ((childHL thetaAboveCell000032102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032102131) h)
+        (by
+          have h : ((childHH thetaAboveCell000032102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032102131) h))
+    (by
+      have h : (thetaAboveCell000032102132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102132 h)
+    (by
+      have h : (thetaAboveCell000032102133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032102133 h)
+
+theorem e24KC2ThetaAboveLeaf0000321021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00003210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00003210))
+    cover_subtree_80cc442a7cb0
+    cover_subtree_7d2d9f5abf21
+    cover_subtree_4430ea3984ba
+    cover_subtree_7e0429da25bc
+theorem e24KC2ThetaAboveLeaf0000321022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00003210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00003210))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00003210))) h)
+theorem e24KC2ThetaAboveLeaf0000321023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00003210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00003210))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00003210))) h)
+theorem cover_subtree_c3fcf1053462 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00003210)))
+    (by
+      have h : (thetaAboveCell000032103000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103000 h)
+    (by
+      have h : (thetaAboveCell000032103001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103002
+        (by
+          have h : ((childLL thetaAboveCell000032103002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103002) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103002) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103002) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103003
+        (by
+          have h : ((childLL thetaAboveCell000032103003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103003) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103003) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103003) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103003) h))
+
+theorem cover_subtree_c50e8ae52b0f :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00003210)))
+    (by
+      have h : (thetaAboveCell000032103010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103010 h)
+    (by
+      have h : (thetaAboveCell000032103011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103012
+        (by
+          have h : ((childLL thetaAboveCell000032103012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103012) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103012) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103012) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103013
+        (by
+          have h : ((childLL thetaAboveCell000032103013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103013) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103013) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103013) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103013) h))
+
+theorem cover_subtree_d417fd422e6e :
+    adaptiveCoverCheck 8 (childHL (childLL (childHH thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH thetaAboveCell00003210)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103020
+        (by
+          have h : ((childLL thetaAboveCell000032103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103020) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103020) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103020) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103021
+        (by
+          have h : ((childLL thetaAboveCell000032103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103021) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103021) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103021) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103021) h))
+    (by
+      have h : (thetaAboveCell000032103022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103022 h)
+    (by
+      have h : (thetaAboveCell000032103023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103023 h)
+
+theorem cover_subtree_803a1ba0f62b :
+    adaptiveCoverCheck 8 (childHH (childLL (childHH thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH thetaAboveCell00003210)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103030
+        (by
+          have h : ((childLL thetaAboveCell000032103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103030) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103030) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103030) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103031
+        (by
+          have h : ((childLL thetaAboveCell000032103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103031) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103031) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103031) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103031) h))
+    (by
+      have h : (thetaAboveCell000032103032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103032 h)
+    (by
+      have h : (thetaAboveCell000032103033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103033 h)
+
+theorem e24KC2ThetaAboveLeaf0000321030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00003210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00003210))
+    cover_subtree_c3fcf1053462
+    cover_subtree_c50e8ae52b0f
+    cover_subtree_d417fd422e6e
+    cover_subtree_803a1ba0f62b
+theorem cover_subtree_dc9cc0fd25e0 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00003210)))
+    (by
+      have h : (thetaAboveCell000032103100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103100 h)
+    (by
+      have h : (thetaAboveCell000032103101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103102
+        (by
+          have h : ((childLL thetaAboveCell000032103102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103102) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103102) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103102) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103103
+        (by
+          have h : ((childLL thetaAboveCell000032103103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103103) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103103) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103103) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103103) h))
+
+theorem cover_subtree_4e80a706fe6e :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00003210)))
+    (by
+      have h : (thetaAboveCell000032103110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103110 h)
+    (by
+      have h : (thetaAboveCell000032103111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103112
+        (by
+          have h : ((childLL thetaAboveCell000032103112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103112) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103112) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103112) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103113
+        (by
+          have h : ((childLL thetaAboveCell000032103113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103113) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103113) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103113) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103113) h))
+
+theorem cover_subtree_06156bdf5c57 :
+    adaptiveCoverCheck 8 (childHL (childLH (childHH thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH thetaAboveCell00003210)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103120
+        (by
+          have h : ((childLL thetaAboveCell000032103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103120) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103120) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103120) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103121
+        (by
+          have h : ((childLL thetaAboveCell000032103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103121) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103121) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103121) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103121) h))
+    (by
+      have h : (thetaAboveCell000032103122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103122 h)
+    (by
+      have h : (thetaAboveCell000032103123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103123 h)
+
+theorem cover_subtree_dfe1ff6f4f14 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHH thetaAboveCell00003210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH thetaAboveCell00003210)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103130
+        (by
+          have h : ((childLL thetaAboveCell000032103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103130) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103130) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103130) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032103131
+        (by
+          have h : ((childLL thetaAboveCell000032103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032103131) h)
+        (by
+          have h : ((childLH thetaAboveCell000032103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032103131) h)
+        (by
+          have h : ((childHL thetaAboveCell000032103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032103131) h)
+        (by
+          have h : ((childHH thetaAboveCell000032103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032103131) h))
+    (by
+      have h : (thetaAboveCell000032103132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103132 h)
+    (by
+      have h : (thetaAboveCell000032103133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032103133 h)
+
+theorem e24KC2ThetaAboveLeaf0000321031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00003210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00003210))
+    cover_subtree_dc9cc0fd25e0
+    cover_subtree_4e80a706fe6e
+    cover_subtree_06156bdf5c57
+    cover_subtree_dfe1ff6f4f14
+theorem e24KC2ThetaAboveLeaf0000321032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00003210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00003210))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00003210))) h)
+theorem e24KC2ThetaAboveLeaf0000321033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00003210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00003210))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00003210))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00003210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00003210))) h)
+theorem e24KC2ThetaAboveLeaf0000321102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00003211))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell00003211))) h)
+theorem e24KC2ThetaAboveLeaf0000321103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00003211))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHL (childHH (childLL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLL
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHH (childHH (childLL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLL
+        thetaAboveCell00003211))) h)
+theorem e24KC2ThetaAboveLeaf0000321112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00003211))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHL (childHL (childLH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLH
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell00003211))) h)
+theorem e24KC2ThetaAboveLeaf0000321113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00003211))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell00003211))) h)
+theorem cover_subtree_5921e16da435 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00003211)))
+    (by
+      have h : (thetaAboveCell000032112000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112000 h)
+    (by
+      have h : (thetaAboveCell000032112001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112002
+        (by
+          have h : ((childLL thetaAboveCell000032112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112002) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112002) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112002) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112003
+        (by
+          have h : ((childLL thetaAboveCell000032112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112003) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112003) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112003) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112003) h))
+
+theorem cover_subtree_52f1b09bd9bd :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00003211)))
+    (by
+      have h : (thetaAboveCell000032112010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112010 h)
+    (by
+      have h : (thetaAboveCell000032112011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112012
+        (by
+          have h : ((childLL thetaAboveCell000032112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112012) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112012) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112012) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112013
+        (by
+          have h : ((childLL thetaAboveCell000032112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112013) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112013) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112013) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112013) h))
+
+theorem cover_subtree_0a1ef543bda5 :
+    adaptiveCoverCheck 8 (childHL (childLL (childHL thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL thetaAboveCell00003211)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112020
+        (by
+          have h : ((childLL thetaAboveCell000032112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112020) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112020) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112020) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112021
+        (by
+          have h : ((childLL thetaAboveCell000032112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112021) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112021) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112021) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112021) h))
+    (by
+      have h : (thetaAboveCell000032112022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112022 h)
+    (by
+      have h : (thetaAboveCell000032112023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112023 h)
+
+theorem cover_subtree_26949ddfa0ff :
+    adaptiveCoverCheck 8 (childHH (childLL (childHL thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL thetaAboveCell00003211)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112030
+        (by
+          have h : ((childLL thetaAboveCell000032112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112030) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112030) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112030) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112031
+        (by
+          have h : ((childLL thetaAboveCell000032112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112031) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112031) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112031) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112031) h))
+    (by
+      have h : (thetaAboveCell000032112032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112032 h)
+    (by
+      have h : (thetaAboveCell000032112033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112033 h)
+
+theorem e24KC2ThetaAboveLeaf0000321120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00003211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00003211))
+    cover_subtree_5921e16da435
+    cover_subtree_52f1b09bd9bd
+    cover_subtree_0a1ef543bda5
+    cover_subtree_26949ddfa0ff
+theorem cover_subtree_b04014698fa2 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00003211)))
+    (by
+      have h : (thetaAboveCell000032112100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112100 h)
+    (by
+      have h : (thetaAboveCell000032112101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112102
+        (by
+          have h : ((childLL thetaAboveCell000032112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112102) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112102) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112102) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112103
+        (by
+          have h : ((childLL thetaAboveCell000032112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112103) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112103) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112103) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112103) h))
+
+theorem cover_subtree_49d16aafe784 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00003211)))
+    (by
+      have h : (thetaAboveCell000032112110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112110 h)
+    (by
+      have h : (thetaAboveCell000032112111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112112
+        (by
+          have h : ((childLL thetaAboveCell000032112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112112) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112112) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112112) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112113
+        (by
+          have h : ((childLL thetaAboveCell000032112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112113) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112113) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112113) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112113) h))
+
+theorem cover_subtree_b7dcbd8da2a4 :
+    adaptiveCoverCheck 8 (childHL (childLH (childHL thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL thetaAboveCell00003211)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112120
+        (by
+          have h : ((childLL thetaAboveCell000032112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112120) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112120) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112120) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112121
+        (by
+          have h : ((childLL thetaAboveCell000032112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112121) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112121) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112121) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112121) h))
+    (by
+      have h : (thetaAboveCell000032112122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112122 h)
+    (by
+      have h : (thetaAboveCell000032112123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112123 h)
+
+theorem cover_subtree_fb1811cd44df :
+    adaptiveCoverCheck 8 (childHH (childLH (childHL thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL thetaAboveCell00003211)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112130
+        (by
+          have h : ((childLL thetaAboveCell000032112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112130) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112130) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112130) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032112131
+        (by
+          have h : ((childLL thetaAboveCell000032112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032112131) h)
+        (by
+          have h : ((childLH thetaAboveCell000032112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032112131) h)
+        (by
+          have h : ((childHL thetaAboveCell000032112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032112131) h)
+        (by
+          have h : ((childHH thetaAboveCell000032112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032112131) h))
+    (by
+      have h : (thetaAboveCell000032112132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112132 h)
+    (by
+      have h : (thetaAboveCell000032112133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032112133 h)
+
+theorem e24KC2ThetaAboveLeaf0000321121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00003211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00003211))
+    cover_subtree_b04014698fa2
+    cover_subtree_49d16aafe784
+    cover_subtree_b7dcbd8da2a4
+    cover_subtree_fb1811cd44df
+theorem e24KC2ThetaAboveLeaf0000321122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00003211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00003211))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00003211))) h)
+theorem e24KC2ThetaAboveLeaf0000321123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00003211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00003211))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00003211))) h)
+theorem cover_subtree_6f17b9dcfe31 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00003211)))
+    (by
+      have h : (thetaAboveCell000032113000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032113000 h)
+    (by
+      have h : (thetaAboveCell000032113001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032113001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113002
+        (by
+          have h : ((childLL thetaAboveCell000032113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113002) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113002) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113002) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113003
+        (by
+          have h : ((childLL thetaAboveCell000032113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113003) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113003) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113003) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113003) h))
+
+theorem cover_subtree_bf78b444527b :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00003211)))
+    (by
+      have h : (thetaAboveCell000032113010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032113010 h)
+    (by
+      have h : (thetaAboveCell000032113011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032113011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113012
+        (by
+          have h : ((childLL thetaAboveCell000032113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113012) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113012) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113012) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113013
+        (by
+          have h : ((childLL thetaAboveCell000032113013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113013) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113013) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113013) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113013) h))
+
+theorem cover_subtree_68fa0f6b74ff :
+    adaptiveCoverCheck 8 (childHL (childLL (childHH thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH thetaAboveCell00003211)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113020
+        (by
+          have h : ((childLL thetaAboveCell000032113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113020) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113020) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113020) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113021
+        (by
+          have h : ((childLL thetaAboveCell000032113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113021) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113021) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113021) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113021) h))
+    (by
+      have h : (thetaAboveCell000032113022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032113022 h)
+    (by
+      have h : (thetaAboveCell000032113023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032113023 h)
+
+theorem cover_subtree_00b90c74e0b1 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHH thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH thetaAboveCell00003211)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113030
+        (by
+          have h : ((childLL thetaAboveCell000032113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113030) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113030) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113030) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113031
+        (by
+          have h : ((childLL thetaAboveCell000032113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113031) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113031) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113031) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113031) h))
+    (by
+      have h : (thetaAboveCell000032113032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032113032 h)
+    (by
+      have h : (thetaAboveCell000032113033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032113033 h)
+
+theorem e24KC2ThetaAboveLeaf0000321130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00003211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00003211))
+    cover_subtree_6f17b9dcfe31
+    cover_subtree_bf78b444527b
+    cover_subtree_68fa0f6b74ff
+    cover_subtree_00b90c74e0b1
+theorem cover_subtree_bd2e141553af :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00003211)))
+    (by
+      have h : (thetaAboveCell000032113100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032113100 h)
+    (by
+      have h : (thetaAboveCell000032113101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032113101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113102
+        (by
+          have h : ((childLL thetaAboveCell000032113102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113102) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113102) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113102) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113103
+        (by
+          have h : ((childLL thetaAboveCell000032113103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113103) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113103) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113103) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113103) h))
+
+theorem cover_subtree_689bdaf91bf7 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00003211)))
+    (by
+      have h : (thetaAboveCell000032113110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032113110 h)
+    (by
+      have h : (thetaAboveCell000032113111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032113111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113112
+        (by
+          have h : ((childLL thetaAboveCell000032113112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113112) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113112) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113112) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113113
+        (by
+          have h : ((childLL thetaAboveCell000032113113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113113) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113113) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113113) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113113) h))
+
+theorem cover_subtree_0feb1fef76ea :
+    adaptiveCoverCheck 8 (childHL (childLH (childHH thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH thetaAboveCell00003211)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113120
+        (by
+          have h : ((childLL thetaAboveCell000032113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113120) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113120) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113120) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113121
+        (by
+          have h : ((childLL thetaAboveCell000032113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113121) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113121) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113121) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113121) h))
+    (by
+      have h : (thetaAboveCell000032113122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032113122 h)
+    (by
+      have h : (thetaAboveCell000032113123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032113123 h)
+
+theorem cover_subtree_5f404684ac08 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHH thetaAboveCell00003211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH thetaAboveCell00003211)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113130
+        (by
+          have h : ((childLL thetaAboveCell000032113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113130) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113130) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113130) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113131
+        (by
+          have h : ((childLL thetaAboveCell000032113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113131) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113131) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113131) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113131) h))
+    (by
+      have h : (thetaAboveCell000032113132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000032113132 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000032113133
+        (by
+          have h : ((childLL thetaAboveCell000032113133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000032113133) h)
+        (by
+          have h : ((childLH thetaAboveCell000032113133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000032113133) h)
+        (by
+          have h : ((childHL thetaAboveCell000032113133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000032113133) h)
+        (by
+          have h : ((childHH thetaAboveCell000032113133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000032113133) h))
+
+theorem e24KC2ThetaAboveLeaf0000321131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00003211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00003211))
+    cover_subtree_bd2e141553af
+    cover_subtree_689bdaf91bf7
+    cover_subtree_0feb1fef76ea
+    cover_subtree_5f404684ac08
+theorem e24KC2ThetaAboveLeaf0000321132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00003211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00003211))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00003211))) h)
+theorem e24KC2ThetaAboveLeaf0000321133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00003211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00003211))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00003211))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00003211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00003211))) h)
+theorem e24KC2ThetaAboveLeaf0000330002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00003300))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell00003300))) h)
+theorem e24KC2ThetaAboveLeaf0000330003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00003300))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHL (childHH (childLL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLL
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHH (childHH (childLL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLL
+        thetaAboveCell00003300))) h)
+theorem e24KC2ThetaAboveLeaf0000330012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00003300))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHL (childHL (childLH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLH
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell00003300))) h)
+theorem e24KC2ThetaAboveLeaf0000330013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00003300))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell00003300))) h)
+theorem cover_subtree_307fcc32307f :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00003300)))
+    (by
+      have h : (thetaAboveCell000033002000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033002000 h)
+    (by
+      have h : (thetaAboveCell000033002001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033002001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002002
+        (by
+          have h : ((childLL thetaAboveCell000033002002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002002) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002002) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002002) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002003
+        (by
+          have h : ((childLL thetaAboveCell000033002003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002003) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002003) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002003) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002003) h))
+
+theorem cover_subtree_d72d916d3b48 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00003300)))
+    (by
+      have h : (thetaAboveCell000033002010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033002010 h)
+    (by
+      have h : (thetaAboveCell000033002011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033002011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002012
+        (by
+          have h : ((childLL thetaAboveCell000033002012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002012) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002012) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002012) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002013
+        (by
+          have h : ((childLL thetaAboveCell000033002013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002013) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002013) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002013) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002013) h))
+
+theorem cover_subtree_68d4094c9ee3 :
+    adaptiveCoverCheck 8 (childHL (childLL (childHL thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL thetaAboveCell00003300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002020
+        (by
+          have h : ((childLL thetaAboveCell000033002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002020) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002020) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002020) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002021
+        (by
+          have h : ((childLL thetaAboveCell000033002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002021) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002021) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002021) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002022
+        (by
+          have h : ((childLL thetaAboveCell000033002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002022) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002022) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002022) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002023
+        (by
+          have h : ((childLL thetaAboveCell000033002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002023) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002023) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002023) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002023) h))
+
+theorem cover_subtree_133a2eba8f2a :
+    adaptiveCoverCheck 8 (childHH (childLL (childHL thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL thetaAboveCell00003300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002030
+        (by
+          have h : ((childLL thetaAboveCell000033002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002030) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002030) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002030) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002031
+        (by
+          have h : ((childLL thetaAboveCell000033002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002031) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002031) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002031) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002032
+        (by
+          have h : ((childLL thetaAboveCell000033002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002032) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002032) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002032) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002033
+        (by
+          have h : ((childLL thetaAboveCell000033002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002033) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002033) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002033) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002033) h))
+
+theorem e24KC2ThetaAboveLeaf0000330020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00003300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00003300))
+    cover_subtree_307fcc32307f
+    cover_subtree_d72d916d3b48
+    cover_subtree_68d4094c9ee3
+    cover_subtree_133a2eba8f2a
+theorem cover_subtree_81e0e7f5c28b :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00003300)))
+    (by
+      have h : (thetaAboveCell000033002100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033002100 h)
+    (by
+      have h : (thetaAboveCell000033002101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033002101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002102
+        (by
+          have h : ((childLL thetaAboveCell000033002102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002102) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002102) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002102) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002103
+        (by
+          have h : ((childLL thetaAboveCell000033002103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002103) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002103) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002103) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002103) h))
+
+theorem cover_subtree_db4d96344863 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00003300)))
+    (by
+      have h : (thetaAboveCell000033002110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033002110 h)
+    (by
+      have h : (thetaAboveCell000033002111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033002111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002112
+        (by
+          have h : ((childLL thetaAboveCell000033002112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002112) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002112) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002112) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002113
+        (by
+          have h : ((childLL thetaAboveCell000033002113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002113) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002113) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002113) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002113) h))
+
+theorem cover_subtree_f83decf107d3 :
+    adaptiveCoverCheck 8 (childHL (childLH (childHL thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL thetaAboveCell00003300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002120
+        (by
+          have h : ((childLL thetaAboveCell000033002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002120) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002120) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002120) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002121
+        (by
+          have h : ((childLL thetaAboveCell000033002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002121) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002121) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002121) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002122
+        (by
+          have h : ((childLL thetaAboveCell000033002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002122) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002122) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002122) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002123
+        (by
+          have h : ((childLL thetaAboveCell000033002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002123) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002123) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002123) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002123) h))
+
+theorem cover_subtree_aa2cd6339b25 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHL thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL thetaAboveCell00003300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002130
+        (by
+          have h : ((childLL thetaAboveCell000033002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002130) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002130) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002130) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002131
+        (by
+          have h : ((childLL thetaAboveCell000033002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002131) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002131) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002131) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002132
+        (by
+          have h : ((childLL thetaAboveCell000033002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002132) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002132) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002132) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033002133
+        (by
+          have h : ((childLL thetaAboveCell000033002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033002133) h)
+        (by
+          have h : ((childLH thetaAboveCell000033002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033002133) h)
+        (by
+          have h : ((childHL thetaAboveCell000033002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033002133) h)
+        (by
+          have h : ((childHH thetaAboveCell000033002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033002133) h))
+
+theorem e24KC2ThetaAboveLeaf0000330021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00003300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00003300))
+    cover_subtree_81e0e7f5c28b
+    cover_subtree_db4d96344863
+    cover_subtree_f83decf107d3
+    cover_subtree_aa2cd6339b25
+theorem e24KC2ThetaAboveLeaf0000330022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00003300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00003300))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00003300))) h)
+theorem e24KC2ThetaAboveLeaf0000330023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00003300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00003300))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00003300))) h)
+theorem cover_subtree_9d0a80f57bd8 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00003300)))
+    (by
+      have h : (thetaAboveCell000033003000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033003000 h)
+    (by
+      have h : (thetaAboveCell000033003001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033003001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003002
+        (by
+          have h : ((childLL thetaAboveCell000033003002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003002) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003002) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003002) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003003
+        (by
+          have h : ((childLL thetaAboveCell000033003003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003003) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003003) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003003) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003003) h))
+
+theorem cover_subtree_dc4f306f6518 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00003300)))
+    (by
+      have h : (thetaAboveCell000033003010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033003010 h)
+    (by
+      have h : (thetaAboveCell000033003011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033003011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003012
+        (by
+          have h : ((childLL thetaAboveCell000033003012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003012) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003012) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003012) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003013
+        (by
+          have h : ((childLL thetaAboveCell000033003013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003013) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003013) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003013) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003013) h))
+
+theorem cover_subtree_f02da567f8d1 :
+    adaptiveCoverCheck 8 (childHL (childLL (childHH thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH thetaAboveCell00003300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003020
+        (by
+          have h : ((childLL thetaAboveCell000033003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003020) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003020) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003020) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003021
+        (by
+          have h : ((childLL thetaAboveCell000033003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003021) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003021) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003021) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003022
+        (by
+          have h : ((childLL thetaAboveCell000033003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003022) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003022) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003022) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003023
+        (by
+          have h : ((childLL thetaAboveCell000033003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003023) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003023) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003023) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003023) h))
+
+theorem cover_subtree_5b244ca5864c :
+    adaptiveCoverCheck 8 (childHH (childLL (childHH thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH thetaAboveCell00003300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003030
+        (by
+          have h : ((childLL thetaAboveCell000033003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003030) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003030) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003030) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003031
+        (by
+          have h : ((childLL thetaAboveCell000033003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003031) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003031) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003031) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003032
+        (by
+          have h : ((childLL thetaAboveCell000033003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003032) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003032) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003032) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003033
+        (by
+          have h : ((childLL thetaAboveCell000033003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003033) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003033) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003033) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003033) h))
+
+theorem e24KC2ThetaAboveLeaf0000330030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00003300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00003300))
+    cover_subtree_9d0a80f57bd8
+    cover_subtree_dc4f306f6518
+    cover_subtree_f02da567f8d1
+    cover_subtree_5b244ca5864c
+theorem cover_subtree_834d2353fe6e :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00003300)))
+    (by
+      have h : (thetaAboveCell000033003100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033003100 h)
+    (by
+      have h : (thetaAboveCell000033003101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033003101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003102
+        (by
+          have h : ((childLL thetaAboveCell000033003102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003102) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003102) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003102) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003103
+        (by
+          have h : ((childLL thetaAboveCell000033003103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003103) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003103) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003103) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003103) h))
+
+theorem cover_subtree_d145e983fe8d :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00003300)))
+    (by
+      have h : (thetaAboveCell000033003110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033003110 h)
+    (by
+      have h : (thetaAboveCell000033003111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033003111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003112
+        (by
+          have h : ((childLL thetaAboveCell000033003112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003112) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003112) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003112) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003113
+        (by
+          have h : ((childLL thetaAboveCell000033003113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003113) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003113) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003113) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003113) h))
+
+theorem cover_subtree_d12e61a2b3bb :
+    adaptiveCoverCheck 8 (childHL (childLH (childHH thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH thetaAboveCell00003300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003120
+        (by
+          have h : ((childLL thetaAboveCell000033003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003120) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003120) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003120) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003121
+        (by
+          have h : ((childLL thetaAboveCell000033003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003121) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003121) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003121) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003122
+        (by
+          have h : ((childLL thetaAboveCell000033003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003122) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003122) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003122) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003123
+        (by
+          have h : ((childLL thetaAboveCell000033003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003123) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003123) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003123) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003123) h))
+
+theorem cover_subtree_19725b594124 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHH thetaAboveCell00003300))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH thetaAboveCell00003300)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003130
+        (by
+          have h : ((childLL thetaAboveCell000033003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003130) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003130) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003130) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003131
+        (by
+          have h : ((childLL thetaAboveCell000033003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003131) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003131) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003131) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003132
+        (by
+          have h : ((childLL thetaAboveCell000033003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003132) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003132) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003132) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033003133
+        (by
+          have h : ((childLL thetaAboveCell000033003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033003133) h)
+        (by
+          have h : ((childLH thetaAboveCell000033003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033003133) h)
+        (by
+          have h : ((childHL thetaAboveCell000033003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033003133) h)
+        (by
+          have h : ((childHH thetaAboveCell000033003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033003133) h))
+
+theorem e24KC2ThetaAboveLeaf0000330031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00003300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00003300))
+    cover_subtree_834d2353fe6e
+    cover_subtree_d145e983fe8d
+    cover_subtree_d12e61a2b3bb
+    cover_subtree_19725b594124
+theorem e24KC2ThetaAboveLeaf0000330032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00003300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00003300))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00003300))) h)
+theorem e24KC2ThetaAboveLeaf0000330033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00003300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00003300))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00003300))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00003300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00003300))) h)
+theorem e24KC2ThetaAboveLeaf0000330102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00003301))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell00003301))) h)
+theorem e24KC2ThetaAboveLeaf0000330103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00003301))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childHL (childHH (childLL thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLL
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childHH (childHH (childLL thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLL
+        thetaAboveCell00003301))) h)
+theorem e24KC2ThetaAboveLeaf0000330112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00003301))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childHL (childHL (childLH thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLH
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell00003301))) h)
+theorem e24KC2ThetaAboveLeaf0000330113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00003301))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell00003301))) h)
+theorem cover_subtree_8335c3bf0b1b :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00003301)))
+    (by
+      have h : (thetaAboveCell000033012000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012000 h)
+    (by
+      have h : (thetaAboveCell000033012001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012002
+        (by
+          have h : ((childLL thetaAboveCell000033012002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012002) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012002) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012002) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012003
+        (by
+          have h : ((childLL thetaAboveCell000033012003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012003) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012003) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012003) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012003) h))
+
+theorem cover_subtree_020729332332 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00003301)))
+    (by
+      have h : (thetaAboveCell000033012010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012010 h)
+    (by
+      have h : (thetaAboveCell000033012011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012012
+        (by
+          have h : ((childLL thetaAboveCell000033012012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012012) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012012) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012012) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012013
+        (by
+          have h : ((childLL thetaAboveCell000033012013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012013) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012013) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012013) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012013) h))
+
+theorem cover_subtree_f21e4f8d9787 :
+    adaptiveCoverCheck 8 (childHL (childLL (childHL thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL thetaAboveCell00003301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012020
+        (by
+          have h : ((childLL thetaAboveCell000033012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012020) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012020) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012020) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012021
+        (by
+          have h : ((childLL thetaAboveCell000033012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012021) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012021) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012021) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012022
+        (by
+          have h : ((childLL thetaAboveCell000033012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012022) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012022) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012022) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012023
+        (by
+          have h : ((childLL thetaAboveCell000033012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012023) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012023) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012023) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012023) h))
+
+theorem cover_subtree_bb534365ac4e :
+    adaptiveCoverCheck 8 (childHH (childLL (childHL thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL thetaAboveCell00003301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012030
+        (by
+          have h : ((childLL thetaAboveCell000033012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012030) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012030) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012030) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012031
+        (by
+          have h : ((childLL thetaAboveCell000033012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012031) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012031) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012031) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012032
+        (by
+          have h : ((childLL thetaAboveCell000033012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012032) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012032) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012032) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012033
+        (by
+          have h : ((childLL thetaAboveCell000033012033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012033) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012033) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012033) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012033) h))
+
+theorem e24KC2ThetaAboveLeaf0000330120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00003301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00003301))
+    cover_subtree_8335c3bf0b1b
+    cover_subtree_020729332332
+    cover_subtree_f21e4f8d9787
+    cover_subtree_bb534365ac4e
+theorem cover_subtree_2bf93d8ac027 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00003301)))
+    (by
+      have h : (thetaAboveCell000033012100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012100 h)
+    (by
+      have h : (thetaAboveCell000033012101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012102
+        (by
+          have h : ((childLL thetaAboveCell000033012102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012102) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012102) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012102) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012103
+        (by
+          have h : ((childLL thetaAboveCell000033012103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012103) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012103) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012103) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012103) h))
+
+theorem cover_subtree_1f4c8299dae5 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00003301)))
+    (by
+      have h : (thetaAboveCell000033012110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012110 h)
+    (by
+      have h : (thetaAboveCell000033012111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012112
+        (by
+          have h : ((childLL thetaAboveCell000033012112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012112) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012112) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012112) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012113
+        (by
+          have h : ((childLL thetaAboveCell000033012113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012113) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012113) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012113) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012113) h))
+
+theorem cover_subtree_6e263eab4641 :
+    adaptiveCoverCheck 8 (childHL (childLH (childHL thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL thetaAboveCell00003301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012120
+        (by
+          have h : ((childLL thetaAboveCell000033012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012120) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012120) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012120) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012121
+        (by
+          have h : ((childLL thetaAboveCell000033012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012121) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012121) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012121) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012122
+        (by
+          have h : ((childLL thetaAboveCell000033012122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012122) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012122) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012122) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012123
+        (by
+          have h : ((childLL thetaAboveCell000033012123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012123) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012123) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012123) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012123) h))
+
+theorem cover_subtree_d9fe12f58957 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHL thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL thetaAboveCell00003301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012130
+        (by
+          have h : ((childLL thetaAboveCell000033012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012130) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012130) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012130) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012131
+        (by
+          have h : ((childLL thetaAboveCell000033012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012131) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012131) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012131) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012132
+        (by
+          have h : ((childLL thetaAboveCell000033012132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012132) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012132) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012132) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033012133
+        (by
+          have h : ((childLL thetaAboveCell000033012133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033012133) h)
+        (by
+          have h : ((childLH thetaAboveCell000033012133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033012133) h)
+        (by
+          have h : ((childHL thetaAboveCell000033012133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033012133) h)
+        (by
+          have h : ((childHH thetaAboveCell000033012133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033012133) h))
+
+theorem e24KC2ThetaAboveLeaf0000330121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00003301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00003301))
+    cover_subtree_2bf93d8ac027
+    cover_subtree_1f4c8299dae5
+    cover_subtree_6e263eab4641
+    cover_subtree_d9fe12f58957
+theorem e24KC2ThetaAboveLeaf0000330122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00003301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00003301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00003301)))
+        (by
+          have h : (thetaAboveCell000033012200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012200 h)
+        (by
+          have h : (thetaAboveCell000033012201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012201 h)
+        (by
+          have h : (thetaAboveCell000033012202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012202 h)
+        (by
+          have h : (thetaAboveCell000033012203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00003301)))
+        (by
+          have h : (thetaAboveCell000033012210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012210 h)
+        (by
+          have h : (thetaAboveCell000033012211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012211 h)
+        (by
+          have h : (thetaAboveCell000033012212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012212 h)
+        (by
+          have h : (thetaAboveCell000033012213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00003301))) h)
+theorem e24KC2ThetaAboveLeaf0000330123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00003301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00003301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00003301)))
+        (by
+          have h : (thetaAboveCell000033012300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012300 h)
+        (by
+          have h : (thetaAboveCell000033012301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012301 h)
+        (by
+          have h : (thetaAboveCell000033012302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012302 h)
+        (by
+          have h : (thetaAboveCell000033012303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00003301)))
+        (by
+          have h : (thetaAboveCell000033012310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012310 h)
+        (by
+          have h : (thetaAboveCell000033012311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012311 h)
+        (by
+          have h : (thetaAboveCell000033012312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012312 h)
+        (by
+          have h : (thetaAboveCell000033012313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033012313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00003301))) h)
+theorem cover_subtree_3ee7d70a5db3 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00003301)))
+    (by
+      have h : (thetaAboveCell000033013000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013000 h)
+    (by
+      have h : (thetaAboveCell000033013001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013002
+        (by
+          have h : ((childLL thetaAboveCell000033013002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013002) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013002) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013002) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013003
+        (by
+          have h : ((childLL thetaAboveCell000033013003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013003) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013003) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013003) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013003) h))
+
+theorem cover_subtree_f6080530d2a9 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00003301)))
+    (by
+      have h : (thetaAboveCell000033013010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013010 h)
+    (by
+      have h : (thetaAboveCell000033013011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013012
+        (by
+          have h : ((childLL thetaAboveCell000033013012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013012) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013012) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013012) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013013
+        (by
+          have h : ((childLL thetaAboveCell000033013013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013013) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013013) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013013) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013013) h))
+
+theorem cover_subtree_2e5606e732f5 :
+    adaptiveCoverCheck 8 (childHL (childLL (childHH thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH thetaAboveCell00003301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013020
+        (by
+          have h : ((childLL thetaAboveCell000033013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013020) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013020) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013020) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013021
+        (by
+          have h : ((childLL thetaAboveCell000033013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013021) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013021) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013021) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013022
+        (by
+          have h : ((childLL thetaAboveCell000033013022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013022) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013022) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013022) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013023
+        (by
+          have h : ((childLL thetaAboveCell000033013023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013023) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013023) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013023) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013023) h))
+
+theorem cover_subtree_6143158a1495 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHH thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH thetaAboveCell00003301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013030
+        (by
+          have h : ((childLL thetaAboveCell000033013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013030) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013030) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013030) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013031
+        (by
+          have h : ((childLL thetaAboveCell000033013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013031) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013031) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013031) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013032
+        (by
+          have h : ((childLL thetaAboveCell000033013032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013032) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013032) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013032) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013033
+        (by
+          have h : ((childLL thetaAboveCell000033013033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013033) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013033) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013033) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013033) h))
+
+theorem e24KC2ThetaAboveLeaf0000330130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00003301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00003301))
+    cover_subtree_3ee7d70a5db3
+    cover_subtree_f6080530d2a9
+    cover_subtree_2e5606e732f5
+    cover_subtree_6143158a1495
+theorem cover_subtree_fd7379f50292 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00003301)))
+    (by
+      have h : (thetaAboveCell000033013100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013100 h)
+    (by
+      have h : (thetaAboveCell000033013101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013102
+        (by
+          have h : ((childLL thetaAboveCell000033013102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013102) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013102) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013102) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013103
+        (by
+          have h : ((childLL thetaAboveCell000033013103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013103) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013103) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013103) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013103) h))
+
+theorem cover_subtree_f974def6b3e0 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00003301)))
+    (by
+      have h : (thetaAboveCell000033013110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013110 h)
+    (by
+      have h : (thetaAboveCell000033013111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013112
+        (by
+          have h : ((childLL thetaAboveCell000033013112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013112) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013112) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013112) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013113
+        (by
+          have h : ((childLL thetaAboveCell000033013113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013113) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013113) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013113) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013113) h))
+
+theorem cover_subtree_d879dc743b47 :
+    adaptiveCoverCheck 8 (childHL (childLH (childHH thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH thetaAboveCell00003301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013120
+        (by
+          have h : ((childLL thetaAboveCell000033013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013120) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013120) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013120) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013121
+        (by
+          have h : ((childLL thetaAboveCell000033013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013121) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013121) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013121) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013122
+        (by
+          have h : ((childLL thetaAboveCell000033013122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013122) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013122) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013122) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013123
+        (by
+          have h : ((childLL thetaAboveCell000033013123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013123) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013123) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013123) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013123) h))
+
+theorem cover_subtree_ebda71a3a487 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHH thetaAboveCell00003301))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH thetaAboveCell00003301)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013130
+        (by
+          have h : ((childLL thetaAboveCell000033013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013130) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013130) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013130) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013131
+        (by
+          have h : ((childLL thetaAboveCell000033013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013131) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013131) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013131) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013132
+        (by
+          have h : ((childLL thetaAboveCell000033013132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013132) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013132) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013132) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033013133
+        (by
+          have h : ((childLL thetaAboveCell000033013133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033013133) h)
+        (by
+          have h : ((childLH thetaAboveCell000033013133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033013133) h)
+        (by
+          have h : ((childHL thetaAboveCell000033013133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033013133) h)
+        (by
+          have h : ((childHH thetaAboveCell000033013133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033013133) h))
+
+theorem e24KC2ThetaAboveLeaf0000330131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00003301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00003301))
+    cover_subtree_fd7379f50292
+    cover_subtree_f974def6b3e0
+    cover_subtree_d879dc743b47
+    cover_subtree_ebda71a3a487
+theorem e24KC2ThetaAboveLeaf0000330132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00003301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00003301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00003301)))
+        (by
+          have h : (thetaAboveCell000033013200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013200 h)
+        (by
+          have h : (thetaAboveCell000033013201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013201 h)
+        (by
+          have h : (thetaAboveCell000033013202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013202 h)
+        (by
+          have h : (thetaAboveCell000033013203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00003301)))
+        (by
+          have h : (thetaAboveCell000033013210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013210 h)
+        (by
+          have h : (thetaAboveCell000033013211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013211 h)
+        (by
+          have h : (thetaAboveCell000033013212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013212 h)
+        (by
+          have h : (thetaAboveCell000033013213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00003301))) h)
+theorem e24KC2ThetaAboveLeaf0000330133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00003301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00003301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00003301)))
+        (by
+          have h : (thetaAboveCell000033013300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013300 h)
+        (by
+          have h : (thetaAboveCell000033013301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013301 h)
+        (by
+          have h : (thetaAboveCell000033013302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013302 h)
+        (by
+          have h : (thetaAboveCell000033013303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00003301)))
+        (by
+          have h : (thetaAboveCell000033013310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013310 h)
+        (by
+          have h : (thetaAboveCell000033013311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013311 h)
+        (by
+          have h : (thetaAboveCell000033013312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013312 h)
+        (by
+          have h : (thetaAboveCell000033013313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033013313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00003301))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00003301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00003301))) h)
+theorem e24KC2ThetaAboveLeaf0000331002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00003310))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell00003310))) h)
+theorem e24KC2ThetaAboveLeaf0000331003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00003310))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childHL (childHH (childLL thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLL
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childHH (childHH (childLL thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLL
+        thetaAboveCell00003310))) h)
+theorem e24KC2ThetaAboveLeaf0000331012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00003310))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childHL (childHL (childLH thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLH
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell00003310))) h)
+theorem e24KC2ThetaAboveLeaf0000331013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00003310))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell00003310))) h)
+theorem cover_subtree_01b17a5cfced :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00003310)))
+    (by
+      have h : (thetaAboveCell000033102000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102000 h)
+    (by
+      have h : (thetaAboveCell000033102001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102002
+        (by
+          have h : ((childLL thetaAboveCell000033102002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102002) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102002) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102002) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102003
+        (by
+          have h : ((childLL thetaAboveCell000033102003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102003) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102003) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102003) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102003) h))
+
+theorem cover_subtree_26a3db88cf1e :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00003310)))
+    (by
+      have h : (thetaAboveCell000033102010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102010 h)
+    (by
+      have h : (thetaAboveCell000033102011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102012
+        (by
+          have h : ((childLL thetaAboveCell000033102012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102012) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102012) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102012) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102013
+        (by
+          have h : ((childLL thetaAboveCell000033102013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102013) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102013) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102013) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102013) h))
+
+theorem cover_subtree_afbc4e7a5775 :
+    adaptiveCoverCheck 8 (childHL (childLL (childHL thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL thetaAboveCell00003310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102020
+        (by
+          have h : ((childLL thetaAboveCell000033102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102020) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102020) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102020) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102021
+        (by
+          have h : ((childLL thetaAboveCell000033102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102021) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102021) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102021) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102022
+        (by
+          have h : ((childLL thetaAboveCell000033102022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102022) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102022) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102022) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102023
+        (by
+          have h : ((childLL thetaAboveCell000033102023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102023) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102023) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102023) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102023) h))
+
+theorem cover_subtree_0885b8dd2e23 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHL thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL thetaAboveCell00003310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102030
+        (by
+          have h : ((childLL thetaAboveCell000033102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102030) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102030) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102030) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102031
+        (by
+          have h : ((childLL thetaAboveCell000033102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102031) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102031) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102031) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102032
+        (by
+          have h : ((childLL thetaAboveCell000033102032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102032) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102032) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102032) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102033
+        (by
+          have h : ((childLL thetaAboveCell000033102033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102033) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102033) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102033) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102033) h))
+
+theorem e24KC2ThetaAboveLeaf0000331020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00003310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00003310))
+    cover_subtree_01b17a5cfced
+    cover_subtree_26a3db88cf1e
+    cover_subtree_afbc4e7a5775
+    cover_subtree_0885b8dd2e23
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle010.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle010.lean
new file mode 100644
index 0000000000..7fed58f069
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle010.lean
@@ -0,0 +1,9113 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch011`.
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch013`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCells0c0d0adfef
+
+/-- Subcell `1000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1000 : AngleCell :=
+  childLL (childLL (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1001 : AngleCell :=
+  childLH (childLL (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `10003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1001)))
+
+end GerverSofa.PartE.CertificateCells0c0d0adfef
+
+namespace GerverSofa.PartE.CertificateCells660a83a47d
+
+/-- Subcell `1001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1001 : AngleCell :=
+  childLH (childLL (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1002 : AngleCell :=
+  childHL (childLL (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1003 : AngleCell :=
+  childHH (childLL (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1010 : AngleCell :=
+  childLL (childLH (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `10013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell1010)))
+
+end GerverSofa.PartE.CertificateCells660a83a47d
+
+namespace GerverSofa.PartE.CertificateCellse876d675ca
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0001 : AngleCell :=
+  childLH (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0001)))
+
+end GerverSofa.PartE.CertificateCellse876d675ca
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC5TerminalBatchT409600011`.
+* `KernelOnly.PartE.E24KC5TerminalBatchT460800012`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0c0d0adfef
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0c0d0adfef
+
+open CertificateCells0c0d0adfef
+theorem e24KC2ThetaAboveLeaf1000320222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10003202)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10003202)) h
+theorem e24KC2ThetaAboveLeaf1000320223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10003202)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10003202)) h
+theorem e24KC2ThetaAboveLeaf1000320230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10003202)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10003202)) h
+theorem e24KC2ThetaAboveLeaf1000320231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10003202)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10003202)) h
+theorem e24KC2ThetaAboveLeaf1000320232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10003202)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10003202)) h
+theorem e24KC2ThetaAboveLeaf1000320233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10003202)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10003202)) h
+theorem e24KC2ThetaAboveLeaf1000320300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10003203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf1000320301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10003203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf1000320302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10003203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf1000320303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10003203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf1000320310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10003203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf1000320311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10003203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf1000320312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10003203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf1000320313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10003203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf1000320320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10003203)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf1000320321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10003203)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf1000320322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10003203)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf1000320323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10003203)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf1000320330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10003203)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf1000320331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10003203)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf1000320332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10003203)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf1000320333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10003203)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10003203)) h
+theorem e24KC2ThetaAboveLeaf100032100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003210) = true := by
+  have h : ((childLL thetaAboveCell10003210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003210) h
+theorem e24KC2ThetaAboveLeaf100032101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003210) = true := by
+  have h : ((childLH thetaAboveCell10003210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10003210) h
+theorem e24KC2ThetaAboveLeaf100032102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003210) = true := by
+  have h : ((childHL thetaAboveCell10003210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003210) h
+theorem e24KC2ThetaAboveLeaf100032103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003210) = true := by
+  have h : ((childHH thetaAboveCell10003210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003210) h
+theorem e24KC2ThetaAboveLeaf100032110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003211) = true := by
+  have h : ((childLL thetaAboveCell10003211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003211) h
+theorem e24KC2ThetaAboveLeaf100032111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003211) = true := by
+  have h : ((childLH thetaAboveCell10003211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10003211) h
+theorem e24KC2ThetaAboveLeaf100032112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003211) = true := by
+  have h : ((childHL thetaAboveCell10003211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003211) h
+theorem e24KC2ThetaAboveLeaf100032113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003211) = true := by
+  have h : ((childHH thetaAboveCell10003211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003211) h
+theorem e24KC2ThetaAboveLeaf1000321200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10003212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10003212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10003212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10003212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10003212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10003212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10003212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10003212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10003212)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10003212)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10003212)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10003212)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10003212)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10003212)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10003212)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10003212)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10003212)) h
+theorem e24KC2ThetaAboveLeaf1000321300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10003213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf1000321301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10003213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf1000321302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10003213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf1000321303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10003213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf1000321310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10003213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf1000321311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10003213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf1000321312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10003213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf1000321313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10003213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf1000321320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10003213)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf1000321321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10003213)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf1000321322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10003213)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf1000321323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10003213)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf1000321330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10003213)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf1000321331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10003213)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf1000321332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10003213)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf1000321333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10003213)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10003213)) h
+theorem e24KC2ThetaAboveLeaf100032200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003220) = true := by
+  have h : ((childLL thetaAboveCell10003220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003220) h
+theorem e24KC2ThetaAboveLeaf100032201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003220) = true := by
+  have h : ((childLH thetaAboveCell10003220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10003220) h
+theorem e24KC2ThetaAboveLeaf100032202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003220) = true := by
+  have h : ((childHL thetaAboveCell10003220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003220) h
+theorem e24KC2ThetaAboveLeaf100032203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003220) = true := by
+  have h : ((childHH thetaAboveCell10003220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003220) h
+theorem e24KC2ThetaAboveLeaf100032210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003221) = true := by
+  have h : ((childLL thetaAboveCell10003221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003221) h
+theorem e24KC2ThetaAboveLeaf100032211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003221) = true := by
+  have h : ((childLH thetaAboveCell10003221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10003221) h
+theorem e24KC2ThetaAboveLeaf100032212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003221) = true := by
+  have h : ((childHL thetaAboveCell10003221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003221) h
+theorem e24KC2ThetaAboveLeaf100032213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003221) = true := by
+  have h : ((childHH thetaAboveCell10003221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003221) h
+theorem e24KC2ThetaAboveLeaf10003222 :
+    adaptiveCoverCheck 11 thetaAboveCell10003222 = true := by
+  have h : (thetaAboveCell10003222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003222 h
+theorem e24KC2ThetaAboveLeaf10003223 :
+    adaptiveCoverCheck 11 thetaAboveCell10003223 = true := by
+  have h : (thetaAboveCell10003223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003223 h
+theorem e24KC2ThetaAboveLeaf100032300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003230) = true := by
+  have h : ((childLL thetaAboveCell10003230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003230) h
+theorem e24KC2ThetaAboveLeaf100032301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003230) = true := by
+  have h : ((childLH thetaAboveCell10003230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10003230) h
+theorem e24KC2ThetaAboveLeaf100032302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003230) = true := by
+  have h : ((childHL thetaAboveCell10003230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003230) h
+theorem e24KC2ThetaAboveLeaf100032303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003230) = true := by
+  have h : ((childHH thetaAboveCell10003230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003230) h
+theorem e24KC2ThetaAboveLeaf100032310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003231) = true := by
+  have h : ((childLL thetaAboveCell10003231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003231) h
+theorem e24KC2ThetaAboveLeaf100032311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003231) = true := by
+  have h : ((childLH thetaAboveCell10003231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10003231) h
+theorem e24KC2ThetaAboveLeaf100032312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003231) = true := by
+  have h : ((childHL thetaAboveCell10003231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003231) h
+theorem e24KC2ThetaAboveLeaf100032313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003231) = true := by
+  have h : ((childHH thetaAboveCell10003231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003231) h
+theorem e24KC2ThetaAboveLeaf10003232 :
+    adaptiveCoverCheck 11 thetaAboveCell10003232 = true := by
+  have h : (thetaAboveCell10003232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003232 h
+theorem e24KC2ThetaAboveLeaf10003233 :
+    adaptiveCoverCheck 11 thetaAboveCell10003233 = true := by
+  have h : (thetaAboveCell10003233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003233 h
+theorem e24KC2ThetaAboveLeaf100033000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003300) = true := by
+  have h : ((childLL thetaAboveCell10003300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003300) h
+theorem e24KC2ThetaAboveLeaf100033001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003300) = true := by
+  have h : ((childLH thetaAboveCell10003300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10003300) h
+theorem e24KC2ThetaAboveLeaf100033002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003300) = true := by
+  have h : ((childHL thetaAboveCell10003300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003300) h
+theorem e24KC2ThetaAboveLeaf100033003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003300) = true := by
+  have h : ((childHH thetaAboveCell10003300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003300) h
+theorem e24KC2ThetaAboveLeaf100033010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003301) = true := by
+  have h : ((childLL thetaAboveCell10003301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003301) h
+theorem e24KC2ThetaAboveLeaf100033011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003301) = true := by
+  have h : ((childLH thetaAboveCell10003301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10003301) h
+theorem e24KC2ThetaAboveLeaf100033012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003301) = true := by
+  have h : ((childHL thetaAboveCell10003301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003301) h
+theorem e24KC2ThetaAboveLeaf100033013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003301) = true := by
+  have h : ((childHH thetaAboveCell10003301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003301) h
+theorem e24KC2ThetaAboveLeaf1000330200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10003302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10003302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10003302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10003302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10003302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10003302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10003302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10003302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10003302)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10003302)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10003302)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10003302)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10003302)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10003302)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10003302)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10003302)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10003302)) h
+theorem e24KC2ThetaAboveLeaf1000330300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10003303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf1000330301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10003303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf1000330302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10003303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf1000330303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10003303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf1000330310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10003303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf1000330311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10003303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf1000330312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10003303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf1000330313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10003303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf1000330320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10003303)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf1000330321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10003303)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf1000330322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10003303)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf1000330323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10003303)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf1000330330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10003303)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf1000330331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10003303)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf1000330332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10003303)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf1000330333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10003303)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10003303)) h
+theorem e24KC2ThetaAboveLeaf100033100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003310) = true := by
+  have h : ((childLL thetaAboveCell10003310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003310) h
+theorem e24KC2ThetaAboveLeaf100033101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003310) = true := by
+  have h : ((childLH thetaAboveCell10003310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10003310) h
+theorem e24KC2ThetaAboveLeaf100033102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003310) = true := by
+  have h : ((childHL thetaAboveCell10003310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003310) h
+theorem e24KC2ThetaAboveLeaf100033103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003310) = true := by
+  have h : ((childHH thetaAboveCell10003310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003310) h
+theorem e24KC2ThetaAboveLeaf100033110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003311) = true := by
+  have h : ((childLL thetaAboveCell10003311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003311) h
+theorem e24KC2ThetaAboveLeaf100033111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003311) = true := by
+  have h : ((childLH thetaAboveCell10003311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10003311) h
+theorem e24KC2ThetaAboveLeaf100033112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003311) = true := by
+  have h : ((childHL thetaAboveCell10003311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003311) h
+theorem e24KC2ThetaAboveLeaf100033113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003311) = true := by
+  have h : ((childHH thetaAboveCell10003311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003311) h
+theorem e24KC2ThetaAboveLeaf1000331200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10003312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10003312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10003312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10003312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10003312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10003312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10003312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10003312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10003312)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10003312)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10003312)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10003312)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10003312)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10003312)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10003312)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10003312)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10003312)) h
+theorem e24KC2ThetaAboveLeaf1000331300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10003313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf1000331301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10003313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf1000331302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10003313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf1000331303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10003313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf1000331310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10003313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf1000331311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10003313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf1000331312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10003313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf1000331313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10003313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf1000331320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10003313)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf1000331321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10003313)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf1000331322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10003313)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf1000331323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10003313)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf1000331330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10003313)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf1000331331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10003313)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf1000331332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10003313)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf1000331333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10003313)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10003313)) h
+theorem e24KC2ThetaAboveLeaf100033200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003320) = true := by
+  have h : ((childLL thetaAboveCell10003320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003320) h
+theorem e24KC2ThetaAboveLeaf100033201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003320) = true := by
+  have h : ((childLH thetaAboveCell10003320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10003320) h
+theorem e24KC2ThetaAboveLeaf100033202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003320) = true := by
+  have h : ((childHL thetaAboveCell10003320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003320) h
+theorem e24KC2ThetaAboveLeaf100033203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003320) = true := by
+  have h : ((childHH thetaAboveCell10003320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003320) h
+theorem e24KC2ThetaAboveLeaf100033210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003321) = true := by
+  have h : ((childLL thetaAboveCell10003321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003321) h
+theorem e24KC2ThetaAboveLeaf100033211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003321) = true := by
+  have h : ((childLH thetaAboveCell10003321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10003321) h
+theorem e24KC2ThetaAboveLeaf100033212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003321) = true := by
+  have h : ((childHL thetaAboveCell10003321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003321) h
+theorem e24KC2ThetaAboveLeaf100033213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003321) = true := by
+  have h : ((childHH thetaAboveCell10003321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003321) h
+theorem e24KC2ThetaAboveLeaf10003322 :
+    adaptiveCoverCheck 11 thetaAboveCell10003322 = true := by
+  have h : (thetaAboveCell10003322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003322 h
+theorem e24KC2ThetaAboveLeaf10003323 :
+    adaptiveCoverCheck 11 thetaAboveCell10003323 = true := by
+  have h : (thetaAboveCell10003323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003323 h
+theorem e24KC2ThetaAboveLeaf100033300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003330) = true := by
+  have h : ((childLL thetaAboveCell10003330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003330) h
+theorem e24KC2ThetaAboveLeaf100033301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003330) = true := by
+  have h : ((childLH thetaAboveCell10003330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10003330) h
+theorem e24KC2ThetaAboveLeaf100033302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003330) = true := by
+  have h : ((childHL thetaAboveCell10003330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003330) h
+theorem e24KC2ThetaAboveLeaf100033303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003330) = true := by
+  have h : ((childHH thetaAboveCell10003330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003330) h
+theorem e24KC2ThetaAboveLeaf100033310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003331) = true := by
+  have h : ((childLL thetaAboveCell10003331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10003331) h
+theorem e24KC2ThetaAboveLeaf1000333110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10003331)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10003331))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10003331)) h
+theorem e24KC2ThetaAboveLeaf1000333111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10003331)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10003331))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10003331)) h
+theorem e24KC2ThetaAboveLeaf1000333112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10003331)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10003331))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10003331)) h
+theorem e24KC2ThetaAboveLeaf1000333113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10003331)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10003331))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10003331)) h
+theorem e24KC2ThetaAboveLeaf100033312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003331) = true := by
+  have h : ((childHL thetaAboveCell10003331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10003331) h
+theorem e24KC2ThetaAboveLeaf100033313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003331) = true := by
+  have h : ((childHH thetaAboveCell10003331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10003331) h
+theorem e24KC2ThetaAboveLeaf10003332 :
+    adaptiveCoverCheck 11 thetaAboveCell10003332 = true := by
+  have h : (thetaAboveCell10003332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003332 h
+theorem e24KC2ThetaAboveLeaf10003333 :
+    adaptiveCoverCheck 11 thetaAboveCell10003333 = true := by
+  have h : (thetaAboveCell10003333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10003333 h
+theorem e24KC2ThetaAboveLeaf100100 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1001)) = true := by
+  have h : ((childLL (childLL thetaAboveCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell1001)) h
+theorem e24KC2ThetaAboveLeaf100101 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1001)) = true := by
+  have h : ((childLH (childLL thetaAboveCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell1001)) h
+theorem e24KC2ThetaAboveLeaf100102 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1001)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1001)) h
+theorem e24KC2ThetaAboveLeaf100103 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1001)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1001)) h
+theorem e24KC2ThetaAboveLeaf100110 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1001)) = true := by
+  have h : ((childLL (childLH thetaAboveCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell1001)) h
+theorem e24KC2ThetaAboveLeaf100111 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1001)) = true := by
+  have h : ((childLH (childLH thetaAboveCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell1001)) h
+theorem e24KC2ThetaAboveLeaf100112 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1001)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1001)) h
+theorem e24KC2ThetaAboveLeaf100113 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1001)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1001)) h
+theorem e24KC2ThetaAboveLeaf1001200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell1001))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell1001))) h
+theorem e24KC2ThetaAboveLeaf1001201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell1001))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell1001))) h
+theorem e24KC2ThetaAboveLeaf10012020 :
+    adaptiveCoverCheck 11 thetaAboveCell10012020 = true := by
+  have h : (thetaAboveCell10012020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012020 h
+theorem e24KC2ThetaAboveLeaf10012021 :
+    adaptiveCoverCheck 11 thetaAboveCell10012021 = true := by
+  have h : (thetaAboveCell10012021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012021 h
+theorem e24KC2ThetaAboveLeaf10012022 :
+    adaptiveCoverCheck 11 thetaAboveCell10012022 = true := by
+  have h : (thetaAboveCell10012022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012022 h
+theorem e24KC2ThetaAboveLeaf10012023 :
+    adaptiveCoverCheck 11 thetaAboveCell10012023 = true := by
+  have h : (thetaAboveCell10012023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012023 h
+theorem e24KC2ThetaAboveLeaf10012030 :
+    adaptiveCoverCheck 11 thetaAboveCell10012030 = true := by
+  have h : (thetaAboveCell10012030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012030 h
+theorem e24KC2ThetaAboveLeaf10012031 :
+    adaptiveCoverCheck 11 thetaAboveCell10012031 = true := by
+  have h : (thetaAboveCell10012031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012031 h
+theorem e24KC2ThetaAboveLeaf10012032 :
+    adaptiveCoverCheck 11 thetaAboveCell10012032 = true := by
+  have h : (thetaAboveCell10012032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012032 h
+theorem e24KC2ThetaAboveLeaf10012033 :
+    adaptiveCoverCheck 11 thetaAboveCell10012033 = true := by
+  have h : (thetaAboveCell10012033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012033 h
+theorem e24KC2ThetaAboveLeaf1001210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell1001))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell1001))) h
+theorem e24KC2ThetaAboveLeaf1001211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell1001))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell1001))) h
+theorem e24KC2ThetaAboveLeaf10012120 :
+    adaptiveCoverCheck 11 thetaAboveCell10012120 = true := by
+  have h : (thetaAboveCell10012120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012120 h
+theorem e24KC2ThetaAboveLeaf10012121 :
+    adaptiveCoverCheck 11 thetaAboveCell10012121 = true := by
+  have h : (thetaAboveCell10012121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012121 h
+theorem e24KC2ThetaAboveLeaf10012122 :
+    adaptiveCoverCheck 11 thetaAboveCell10012122 = true := by
+  have h : (thetaAboveCell10012122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012122 h
+theorem e24KC2ThetaAboveLeaf10012123 :
+    adaptiveCoverCheck 11 thetaAboveCell10012123 = true := by
+  have h : (thetaAboveCell10012123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012123 h
+theorem e24KC2ThetaAboveLeaf10012130 :
+    adaptiveCoverCheck 11 thetaAboveCell10012130 = true := by
+  have h : (thetaAboveCell10012130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012130 h
+theorem e24KC2ThetaAboveLeaf10012131 :
+    adaptiveCoverCheck 11 thetaAboveCell10012131 = true := by
+  have h : (thetaAboveCell10012131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012131 h
+theorem e24KC2ThetaAboveLeaf10012132 :
+    adaptiveCoverCheck 11 thetaAboveCell10012132 = true := by
+  have h : (thetaAboveCell10012132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012132 h
+theorem e24KC2ThetaAboveLeaf10012133 :
+    adaptiveCoverCheck 11 thetaAboveCell10012133 = true := by
+  have h : (thetaAboveCell10012133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012133 h
+theorem e24KC2ThetaAboveLeaf100122000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012200) = true := by
+  have h : ((childLL thetaAboveCell10012200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10012200) h
+theorem e24KC2ThetaAboveLeaf100122001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012200) = true := by
+  have h : ((childLH thetaAboveCell10012200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10012200) h
+theorem e24KC2ThetaAboveLeaf100122002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012200) = true := by
+  have h : ((childHL thetaAboveCell10012200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012200) h
+theorem e24KC2ThetaAboveLeaf100122003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012200) = true := by
+  have h : ((childHH thetaAboveCell10012200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012200) h
+theorem e24KC2ThetaAboveLeaf100122010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012201) = true := by
+  have h : ((childLL thetaAboveCell10012201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10012201) h
+theorem e24KC2ThetaAboveLeaf100122011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012201) = true := by
+  have h : ((childLH thetaAboveCell10012201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10012201) h
+theorem e24KC2ThetaAboveLeaf100122012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012201) = true := by
+  have h : ((childHL thetaAboveCell10012201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012201) h
+theorem e24KC2ThetaAboveLeaf100122013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012201) = true := by
+  have h : ((childHH thetaAboveCell10012201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012201) h
+theorem e24KC2ThetaAboveLeaf1001220200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10012202)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10012202)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10012202)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10012202)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10012202)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10012202)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10012202)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10012202)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10012202)) h
+theorem e24KC2ThetaAboveLeaf1001220300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf1001220301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf1001220302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf1001220303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf1001220310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf1001220311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf1001220312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf1001220313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf1001220320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10012203)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf1001220321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10012203)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf1001220322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10012203)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf1001220323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10012203)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf1001220330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10012203)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf1001220331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10012203)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf1001220332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10012203)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf1001220333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10012203)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10012203)) h
+theorem e24KC2ThetaAboveLeaf100122100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012210) = true := by
+  have h : ((childLL thetaAboveCell10012210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10012210) h
+theorem e24KC2ThetaAboveLeaf100122101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012210) = true := by
+  have h : ((childLH thetaAboveCell10012210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10012210) h
+theorem e24KC2ThetaAboveLeaf100122102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012210) = true := by
+  have h : ((childHL thetaAboveCell10012210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012210) h
+theorem e24KC2ThetaAboveLeaf100122103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012210) = true := by
+  have h : ((childHH thetaAboveCell10012210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012210) h
+theorem e24KC2ThetaAboveLeaf100122110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012211) = true := by
+  have h : ((childLL thetaAboveCell10012211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10012211) h
+theorem e24KC2ThetaAboveLeaf100122111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012211) = true := by
+  have h : ((childLH thetaAboveCell10012211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10012211) h
+theorem e24KC2ThetaAboveLeaf100122112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012211) = true := by
+  have h : ((childHL thetaAboveCell10012211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012211) h
+theorem e24KC2ThetaAboveLeaf100122113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012211) = true := by
+  have h : ((childHH thetaAboveCell10012211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012211) h
+theorem e24KC2ThetaAboveLeaf1001221200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10012212)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10012212)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10012212)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10012212)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10012212)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10012212)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10012212)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10012212)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10012212)) h
+theorem e24KC2ThetaAboveLeaf1001221300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001221301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001221302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001221303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001221310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001221311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001221312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001221313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001221320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10012213)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001221321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10012213)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001221322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10012213)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001221323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10012213)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001221330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10012213)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001221331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10012213)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001221332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10012213)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001221333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10012213)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10012213)) h
+theorem e24KC2ThetaAboveLeaf1001222000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012220)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012220)) h
+theorem e24KC2ThetaAboveLeaf1001222001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012220)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012220)) h
+theorem e24KC2ThetaAboveLeaf1001222002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012220)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012220)) h
+theorem e24KC2ThetaAboveLeaf1001222003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012220)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012220)) h
+theorem e24KC2ThetaAboveLeaf1001222010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012220)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012220)) h
+theorem e24KC2ThetaAboveLeaf1001222011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012220)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012220)) h
+theorem e24KC2ThetaAboveLeaf1001222012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012220)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012220)) h
+theorem e24KC2ThetaAboveLeaf1001222013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012220)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012220)) h
+theorem e24KC2ThetaAboveLeaf100122202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012220) = true := by
+  have h : ((childHL thetaAboveCell10012220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012220) h
+theorem e24KC2ThetaAboveLeaf100122203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012220) = true := by
+  have h : ((childHH thetaAboveCell10012220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012220) h
+theorem e24KC2ThetaAboveLeaf1001222100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012221)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012221)) h
+theorem e24KC2ThetaAboveLeaf1001222101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012221)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012221)) h
+theorem e24KC2ThetaAboveLeaf1001222102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012221)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012221)) h
+theorem e24KC2ThetaAboveLeaf1001222103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012221)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012221)) h
+theorem e24KC2ThetaAboveLeaf1001222110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012221)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012221)) h
+theorem e24KC2ThetaAboveLeaf1001222111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012221)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012221)) h
+theorem e24KC2ThetaAboveLeaf1001222112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012221)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012221)) h
+theorem e24KC2ThetaAboveLeaf1001222113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012221)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012221)) h
+theorem e24KC2ThetaAboveLeaf100122212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012221) = true := by
+  have h : ((childHL thetaAboveCell10012221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012221) h
+theorem e24KC2ThetaAboveLeaf100122213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012221) = true := by
+  have h : ((childHH thetaAboveCell10012221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012221) h
+theorem e24KC2ThetaAboveLeaf10012222 :
+    adaptiveCoverCheck 11 thetaAboveCell10012222 = true := by
+  have h : (thetaAboveCell10012222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012222 h
+theorem e24KC2ThetaAboveLeaf10012223 :
+    adaptiveCoverCheck 11 thetaAboveCell10012223 = true := by
+  have h : (thetaAboveCell10012223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012223 h
+theorem e24KC2ThetaAboveLeaf1001223000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012230)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012230)) h
+theorem e24KC2ThetaAboveLeaf1001223001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012230)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012230)) h
+theorem e24KC2ThetaAboveLeaf1001223002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012230)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012230)) h
+theorem e24KC2ThetaAboveLeaf1001223003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012230)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012230)) h
+theorem e24KC2ThetaAboveLeaf1001223010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012230)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012230)) h
+theorem e24KC2ThetaAboveLeaf1001223011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012230)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012230)) h
+theorem e24KC2ThetaAboveLeaf1001223012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012230)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012230)) h
+theorem e24KC2ThetaAboveLeaf1001223013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012230)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012230)) h
+theorem e24KC2ThetaAboveLeaf100122302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012230) = true := by
+  have h : ((childHL thetaAboveCell10012230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012230) h
+theorem e24KC2ThetaAboveLeaf100122303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012230) = true := by
+  have h : ((childHH thetaAboveCell10012230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012230) h
+theorem e24KC2ThetaAboveLeaf1001223100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012231)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012231))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012231)) h
+theorem e24KC2ThetaAboveLeaf1001223101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012231)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012231))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012231)) h
+theorem e24KC2ThetaAboveLeaf1001223102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012231)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012231))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012231)) h
+theorem e24KC2ThetaAboveLeaf1001223103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012231)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012231))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012231)) h
+theorem e24KC2ThetaAboveLeaf1001223110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012231)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012231))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012231)) h
+theorem e24KC2ThetaAboveLeaf1001223111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012231)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012231))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012231)) h
+theorem e24KC2ThetaAboveLeaf1001223112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012231)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012231))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012231)) h
+theorem e24KC2ThetaAboveLeaf1001223113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012231)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012231))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012231)) h
+theorem e24KC2ThetaAboveLeaf100122312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012231) = true := by
+  have h : ((childHL thetaAboveCell10012231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012231) h
+theorem e24KC2ThetaAboveLeaf100122313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012231) = true := by
+  have h : ((childHH thetaAboveCell10012231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012231) h
+theorem e24KC2ThetaAboveLeaf10012232 :
+    adaptiveCoverCheck 11 thetaAboveCell10012232 = true := by
+  have h : (thetaAboveCell10012232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012232 h
+theorem e24KC2ThetaAboveLeaf10012233 :
+    adaptiveCoverCheck 11 thetaAboveCell10012233 = true := by
+  have h : (thetaAboveCell10012233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012233 h
+theorem e24KC2ThetaAboveLeaf100123000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012300) = true := by
+  have h : ((childLL thetaAboveCell10012300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10012300) h
+theorem e24KC2ThetaAboveLeaf100123001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012300) = true := by
+  have h : ((childLH thetaAboveCell10012300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10012300) h
+theorem e24KC2ThetaAboveLeaf100123002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012300) = true := by
+  have h : ((childHL thetaAboveCell10012300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012300) h
+theorem e24KC2ThetaAboveLeaf100123003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012300) = true := by
+  have h : ((childHH thetaAboveCell10012300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012300) h
+theorem e24KC2ThetaAboveLeaf100123010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012301) = true := by
+  have h : ((childLL thetaAboveCell10012301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10012301) h
+theorem e24KC2ThetaAboveLeaf100123011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012301) = true := by
+  have h : ((childLH thetaAboveCell10012301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10012301) h
+theorem e24KC2ThetaAboveLeaf100123012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012301) = true := by
+  have h : ((childHL thetaAboveCell10012301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012301) h
+theorem e24KC2ThetaAboveLeaf100123013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012301) = true := by
+  have h : ((childHH thetaAboveCell10012301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012301) h
+theorem e24KC2ThetaAboveLeaf1001230200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10012302)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10012302)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10012302)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10012302)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10012302)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10012302)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10012302)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10012302)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10012302)) h
+theorem e24KC2ThetaAboveLeaf1001230300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf1001230301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf1001230302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf1001230303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf1001230310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf1001230311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf1001230312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf1001230313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf1001230320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10012303)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf1001230321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10012303)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf1001230322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10012303)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf1001230323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10012303)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf1001230330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10012303)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf1001230331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10012303)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf1001230332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10012303)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf1001230333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10012303)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10012303)) h
+theorem e24KC2ThetaAboveLeaf100123100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012310) = true := by
+  have h : ((childLL thetaAboveCell10012310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10012310) h
+theorem e24KC2ThetaAboveLeaf100123101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012310) = true := by
+  have h : ((childLH thetaAboveCell10012310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10012310) h
+theorem e24KC2ThetaAboveLeaf100123102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012310) = true := by
+  have h : ((childHL thetaAboveCell10012310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012310) h
+theorem e24KC2ThetaAboveLeaf100123103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012310) = true := by
+  have h : ((childHH thetaAboveCell10012310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012310) h
+theorem e24KC2ThetaAboveLeaf100123110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012311) = true := by
+  have h : ((childLL thetaAboveCell10012311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10012311) h
+theorem e24KC2ThetaAboveLeaf100123111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012311) = true := by
+  have h : ((childLH thetaAboveCell10012311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10012311) h
+theorem e24KC2ThetaAboveLeaf100123112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012311) = true := by
+  have h : ((childHL thetaAboveCell10012311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012311) h
+theorem e24KC2ThetaAboveLeaf100123113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012311) = true := by
+  have h : ((childHH thetaAboveCell10012311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012311) h
+theorem e24KC2ThetaAboveLeaf1001231200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10012312)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10012312)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10012312)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10012312)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10012312)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10012312)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10012312)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10012312)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10012312)) h
+theorem e24KC2ThetaAboveLeaf1001231300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001231301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001231302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001231303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001231310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001231311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001231312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001231313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001231320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10012313)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001231321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10012313)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001231322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10012313)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001231323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10012313)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001231330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10012313)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001231331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10012313)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001231332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10012313)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001231333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10012313)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10012313)) h
+theorem e24KC2ThetaAboveLeaf1001232000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012320)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012320))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012320)) h
+theorem e24KC2ThetaAboveLeaf1001232001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012320)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012320))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012320)) h
+theorem e24KC2ThetaAboveLeaf1001232002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012320)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012320))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012320)) h
+theorem e24KC2ThetaAboveLeaf1001232003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012320)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012320))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012320)) h
+theorem e24KC2ThetaAboveLeaf1001232010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012320)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012320))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012320)) h
+theorem e24KC2ThetaAboveLeaf1001232011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012320)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012320))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012320)) h
+theorem e24KC2ThetaAboveLeaf1001232012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012320)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012320))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012320)) h
+theorem e24KC2ThetaAboveLeaf1001232013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012320)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012320))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012320)) h
+theorem e24KC2ThetaAboveLeaf100123202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012320) = true := by
+  have h : ((childHL thetaAboveCell10012320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012320) h
+theorem e24KC2ThetaAboveLeaf100123203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012320) = true := by
+  have h : ((childHH thetaAboveCell10012320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012320) h
+theorem e24KC2ThetaAboveLeaf1001232100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012321)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012321))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012321)) h
+theorem e24KC2ThetaAboveLeaf1001232101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012321)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012321))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012321)) h
+theorem e24KC2ThetaAboveLeaf1001232102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012321)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012321))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012321)) h
+theorem e24KC2ThetaAboveLeaf1001232103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012321)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012321))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012321)) h
+theorem e24KC2ThetaAboveLeaf1001232110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012321)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012321))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012321)) h
+theorem e24KC2ThetaAboveLeaf1001232111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012321)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012321))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012321)) h
+theorem e24KC2ThetaAboveLeaf1001232112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012321)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012321))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012321)) h
+theorem e24KC2ThetaAboveLeaf1001232113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012321)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012321))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012321)) h
+theorem e24KC2ThetaAboveLeaf100123212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012321) = true := by
+  have h : ((childHL thetaAboveCell10012321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012321) h
+theorem e24KC2ThetaAboveLeaf100123213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012321) = true := by
+  have h : ((childHH thetaAboveCell10012321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012321) h
+theorem e24KC2ThetaAboveLeaf10012322 :
+    adaptiveCoverCheck 11 thetaAboveCell10012322 = true := by
+  have h : (thetaAboveCell10012322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012322 h
+theorem e24KC2ThetaAboveLeaf10012323 :
+    adaptiveCoverCheck 11 thetaAboveCell10012323 = true := by
+  have h : (thetaAboveCell10012323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012323 h
+theorem e24KC2ThetaAboveLeaf1001233000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012330)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012330))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012330)) h
+theorem e24KC2ThetaAboveLeaf1001233001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012330)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012330))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012330)) h
+theorem e24KC2ThetaAboveLeaf1001233002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012330)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012330))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012330)) h
+theorem e24KC2ThetaAboveLeaf1001233003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012330)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012330))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012330)) h
+theorem e24KC2ThetaAboveLeaf1001233010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012330)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012330))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012330)) h
+theorem e24KC2ThetaAboveLeaf1001233011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012330)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012330))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012330)) h
+theorem e24KC2ThetaAboveLeaf1001233012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012330)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012330))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012330)) h
+theorem e24KC2ThetaAboveLeaf1001233013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012330)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012330))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012330)) h
+theorem e24KC2ThetaAboveLeaf100123302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012330) = true := by
+  have h : ((childHL thetaAboveCell10012330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012330) h
+theorem e24KC2ThetaAboveLeaf100123303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012330) = true := by
+  have h : ((childHH thetaAboveCell10012330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012330) h
+theorem e24KC2ThetaAboveLeaf1001233100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10012331)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10012331))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10012331)) h
+theorem e24KC2ThetaAboveLeaf1001233101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10012331)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10012331))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10012331)) h
+theorem e24KC2ThetaAboveLeaf1001233102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10012331)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10012331))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10012331)) h
+theorem e24KC2ThetaAboveLeaf1001233103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10012331)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10012331))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10012331)) h
+theorem e24KC2ThetaAboveLeaf1001233110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10012331)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10012331))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10012331)) h
+theorem e24KC2ThetaAboveLeaf1001233111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10012331)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10012331))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10012331)) h
+theorem e24KC2ThetaAboveLeaf1001233112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10012331)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10012331))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10012331)) h
+theorem e24KC2ThetaAboveLeaf1001233113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10012331)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10012331))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10012331)) h
+theorem e24KC2ThetaAboveLeaf100123312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012331) = true := by
+  have h : ((childHL thetaAboveCell10012331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10012331) h
+theorem e24KC2ThetaAboveLeaf100123313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012331) = true := by
+  have h : ((childHH thetaAboveCell10012331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10012331) h
+theorem e24KC2ThetaAboveLeaf10012332 :
+    adaptiveCoverCheck 11 thetaAboveCell10012332 = true := by
+  have h : (thetaAboveCell10012332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012332 h
+theorem e24KC2ThetaAboveLeaf10012333 :
+    adaptiveCoverCheck 11 thetaAboveCell10012333 = true := by
+  have h : (thetaAboveCell10012333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10012333 h
+theorem e24KC2ThetaAboveLeaf1001300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell1001))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell1001))) h
+theorem e24KC2ThetaAboveLeaf1001301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell1001))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell1001))) h
+theorem e24KC2ThetaAboveLeaf10013020 :
+    adaptiveCoverCheck 11 thetaAboveCell10013020 = true := by
+  have h : (thetaAboveCell10013020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013020 h
+theorem e24KC2ThetaAboveLeaf10013021 :
+    adaptiveCoverCheck 11 thetaAboveCell10013021 = true := by
+  have h : (thetaAboveCell10013021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013021 h
+theorem e24KC2ThetaAboveLeaf10013022 :
+    adaptiveCoverCheck 11 thetaAboveCell10013022 = true := by
+  have h : (thetaAboveCell10013022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013022 h
+theorem e24KC2ThetaAboveLeaf10013023 :
+    adaptiveCoverCheck 11 thetaAboveCell10013023 = true := by
+  have h : (thetaAboveCell10013023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013023 h
+theorem e24KC2ThetaAboveLeaf10013030 :
+    adaptiveCoverCheck 11 thetaAboveCell10013030 = true := by
+  have h : (thetaAboveCell10013030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013030 h
+theorem e24KC2ThetaAboveLeaf10013031 :
+    adaptiveCoverCheck 11 thetaAboveCell10013031 = true := by
+  have h : (thetaAboveCell10013031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013031 h
+theorem e24KC2ThetaAboveLeaf10013032 :
+    adaptiveCoverCheck 11 thetaAboveCell10013032 = true := by
+  have h : (thetaAboveCell10013032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013032 h
+theorem e24KC2ThetaAboveLeaf10013033 :
+    adaptiveCoverCheck 11 thetaAboveCell10013033 = true := by
+  have h : (thetaAboveCell10013033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013033 h
+theorem e24KC2ThetaAboveLeaf1001310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell1001))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell1001))) h
+theorem e24KC2ThetaAboveLeaf1001311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell1001))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell1001))) h
+theorem e24KC2ThetaAboveLeaf10013120 :
+    adaptiveCoverCheck 11 thetaAboveCell10013120 = true := by
+  have h : (thetaAboveCell10013120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013120 h
+theorem e24KC2ThetaAboveLeaf10013121 :
+    adaptiveCoverCheck 11 thetaAboveCell10013121 = true := by
+  have h : (thetaAboveCell10013121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013121 h
+theorem e24KC2ThetaAboveLeaf10013122 :
+    adaptiveCoverCheck 11 thetaAboveCell10013122 = true := by
+  have h : (thetaAboveCell10013122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013122 h
+theorem e24KC2ThetaAboveLeaf10013123 :
+    adaptiveCoverCheck 11 thetaAboveCell10013123 = true := by
+  have h : (thetaAboveCell10013123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013123 h
+theorem e24KC2ThetaAboveLeaf10013130 :
+    adaptiveCoverCheck 11 thetaAboveCell10013130 = true := by
+  have h : (thetaAboveCell10013130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013130 h
+theorem e24KC2ThetaAboveLeaf10013131 :
+    adaptiveCoverCheck 11 thetaAboveCell10013131 = true := by
+  have h : (thetaAboveCell10013131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013131 h
+theorem e24KC2ThetaAboveLeaf10013132 :
+    adaptiveCoverCheck 11 thetaAboveCell10013132 = true := by
+  have h : (thetaAboveCell10013132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013132 h
+theorem e24KC2ThetaAboveLeaf10013133 :
+    adaptiveCoverCheck 11 thetaAboveCell10013133 = true := by
+  have h : (thetaAboveCell10013133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013133 h
+theorem e24KC2ThetaAboveLeaf100132000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013200) = true := by
+  have h : ((childLL thetaAboveCell10013200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10013200) h
+theorem e24KC2ThetaAboveLeaf100132001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013200) = true := by
+  have h : ((childLH thetaAboveCell10013200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10013200) h
+theorem e24KC2ThetaAboveLeaf100132002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013200) = true := by
+  have h : ((childHL thetaAboveCell10013200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013200) h
+theorem e24KC2ThetaAboveLeaf100132003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013200) = true := by
+  have h : ((childHH thetaAboveCell10013200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013200) h
+theorem e24KC2ThetaAboveLeaf100132010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013201) = true := by
+  have h : ((childLL thetaAboveCell10013201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10013201) h
+theorem e24KC2ThetaAboveLeaf100132011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013201) = true := by
+  have h : ((childLH thetaAboveCell10013201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10013201) h
+theorem e24KC2ThetaAboveLeaf100132012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013201) = true := by
+  have h : ((childHL thetaAboveCell10013201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013201) h
+theorem e24KC2ThetaAboveLeaf100132013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013201) = true := by
+  have h : ((childHH thetaAboveCell10013201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013201) h
+theorem e24KC2ThetaAboveLeaf1001320200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10013202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10013202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10013202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10013202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10013202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10013202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10013202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10013202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10013202)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10013202)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10013202)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10013202)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10013202)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10013202)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10013202)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10013202)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10013202)) h
+theorem e24KC2ThetaAboveLeaf1001320300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10013203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10013203)) h
+theorem e24KC2ThetaAboveLeaf1001320301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10013203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10013203)) h
+theorem e24KC2ThetaAboveLeaf1001320302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10013203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10013203)) h
+theorem e24KC2ThetaAboveLeaf1001320303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10013203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10013203)) h
+theorem e24KC2ThetaAboveLeaf1001320310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10013203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10013203)) h
+theorem e24KC2ThetaAboveLeaf1001320311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10013203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10013203)) h
+theorem e24KC2ThetaAboveLeaf1001320312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10013203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10013203)) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells660a83a47d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells660a83a47d
+
+open CertificateCells660a83a47d
+theorem e24KC2ThetaAboveLeaf1001320313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10013203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10013203)) h
+theorem e24KC2ThetaAboveLeaf1001320320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10013203)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10013203)) h
+theorem e24KC2ThetaAboveLeaf1001320321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10013203)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10013203)) h
+theorem e24KC2ThetaAboveLeaf1001320322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10013203)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10013203)) h
+theorem e24KC2ThetaAboveLeaf1001320323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10013203)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10013203)) h
+theorem e24KC2ThetaAboveLeaf1001320330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10013203)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10013203)) h
+theorem e24KC2ThetaAboveLeaf1001320331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10013203)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10013203)) h
+theorem e24KC2ThetaAboveLeaf1001320332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10013203)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10013203)) h
+theorem e24KC2ThetaAboveLeaf1001320333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10013203)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10013203)) h
+theorem e24KC2ThetaAboveLeaf100132100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013210) = true := by
+  have h : ((childLL thetaAboveCell10013210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10013210) h
+theorem e24KC2ThetaAboveLeaf100132101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013210) = true := by
+  have h : ((childLH thetaAboveCell10013210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10013210) h
+theorem e24KC2ThetaAboveLeaf100132102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013210) = true := by
+  have h : ((childHL thetaAboveCell10013210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013210) h
+theorem e24KC2ThetaAboveLeaf100132103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013210) = true := by
+  have h : ((childHH thetaAboveCell10013210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013210) h
+theorem e24KC2ThetaAboveLeaf100132110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013211) = true := by
+  have h : ((childLL thetaAboveCell10013211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10013211) h
+theorem e24KC2ThetaAboveLeaf100132111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013211) = true := by
+  have h : ((childLH thetaAboveCell10013211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10013211) h
+theorem e24KC2ThetaAboveLeaf100132112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013211) = true := by
+  have h : ((childHL thetaAboveCell10013211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013211) h
+theorem e24KC2ThetaAboveLeaf100132113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013211) = true := by
+  have h : ((childHH thetaAboveCell10013211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013211) h
+theorem e24KC2ThetaAboveLeaf1001321200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10013212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10013212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10013212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10013212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10013212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10013212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10013212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10013212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10013212)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10013212)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10013212)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10013212)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10013212)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10013212)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10013212)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10013212)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10013212)) h
+theorem e24KC2ThetaAboveLeaf1001321300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10013213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001321301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10013213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001321302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10013213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001321303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10013213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001321310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10013213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001321311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10013213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001321312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10013213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001321313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10013213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001321320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10013213)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001321321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10013213)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001321322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10013213)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001321323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10013213)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001321330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10013213)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001321331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10013213)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001321332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10013213)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001321333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10013213)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10013213)) h
+theorem e24KC2ThetaAboveLeaf1001322000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10013220)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10013220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10013220)) h
+theorem e24KC2ThetaAboveLeaf1001322001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10013220)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10013220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10013220)) h
+theorem e24KC2ThetaAboveLeaf1001322002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10013220)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10013220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10013220)) h
+theorem e24KC2ThetaAboveLeaf1001322003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10013220)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10013220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10013220)) h
+theorem e24KC2ThetaAboveLeaf1001322010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10013220)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10013220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10013220)) h
+theorem e24KC2ThetaAboveLeaf1001322011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10013220)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10013220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10013220)) h
+theorem e24KC2ThetaAboveLeaf1001322012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10013220)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10013220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10013220)) h
+theorem e24KC2ThetaAboveLeaf1001322013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10013220)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10013220))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10013220)) h
+theorem e24KC2ThetaAboveLeaf100132202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013220) = true := by
+  have h : ((childHL thetaAboveCell10013220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013220) h
+theorem e24KC2ThetaAboveLeaf100132203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013220) = true := by
+  have h : ((childHH thetaAboveCell10013220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013220) h
+theorem e24KC2ThetaAboveLeaf1001322100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10013221)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10013221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10013221)) h
+theorem e24KC2ThetaAboveLeaf1001322101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10013221)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10013221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10013221)) h
+theorem e24KC2ThetaAboveLeaf1001322102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10013221)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10013221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10013221)) h
+theorem e24KC2ThetaAboveLeaf1001322103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10013221)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10013221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10013221)) h
+theorem e24KC2ThetaAboveLeaf1001322110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10013221)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10013221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10013221)) h
+theorem e24KC2ThetaAboveLeaf1001322111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10013221)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10013221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10013221)) h
+theorem e24KC2ThetaAboveLeaf1001322112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10013221)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10013221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10013221)) h
+theorem e24KC2ThetaAboveLeaf1001322113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10013221)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10013221))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10013221)) h
+theorem e24KC2ThetaAboveLeaf100132212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013221) = true := by
+  have h : ((childHL thetaAboveCell10013221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013221) h
+theorem e24KC2ThetaAboveLeaf100132213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013221) = true := by
+  have h : ((childHH thetaAboveCell10013221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013221) h
+theorem e24KC2ThetaAboveLeaf10013222 :
+    adaptiveCoverCheck 11 thetaAboveCell10013222 = true := by
+  have h : (thetaAboveCell10013222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013222 h
+theorem e24KC2ThetaAboveLeaf10013223 :
+    adaptiveCoverCheck 11 thetaAboveCell10013223 = true := by
+  have h : (thetaAboveCell10013223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013223 h
+theorem e24KC2ThetaAboveLeaf1001323000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10013230)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10013230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10013230)) h
+theorem e24KC2ThetaAboveLeaf1001323001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10013230)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10013230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10013230)) h
+theorem e24KC2ThetaAboveLeaf1001323002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10013230)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10013230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10013230)) h
+theorem e24KC2ThetaAboveLeaf1001323003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10013230)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10013230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10013230)) h
+theorem e24KC2ThetaAboveLeaf1001323010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10013230)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10013230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10013230)) h
+theorem e24KC2ThetaAboveLeaf1001323011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10013230)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10013230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10013230)) h
+theorem e24KC2ThetaAboveLeaf1001323012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10013230)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10013230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10013230)) h
+theorem e24KC2ThetaAboveLeaf1001323013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10013230)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10013230))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10013230)) h
+theorem e24KC2ThetaAboveLeaf100132302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013230) = true := by
+  have h : ((childHL thetaAboveCell10013230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013230) h
+theorem e24KC2ThetaAboveLeaf100132303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013230) = true := by
+  have h : ((childHH thetaAboveCell10013230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013230) h
+theorem e24KC2ThetaAboveLeaf100132310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013231) = true := by
+  have h : ((childLL thetaAboveCell10013231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10013231) h
+theorem e24KC2ThetaAboveLeaf100132311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013231) = true := by
+  have h : ((childLH thetaAboveCell10013231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10013231) h
+theorem e24KC2ThetaAboveLeaf100132312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013231) = true := by
+  have h : ((childHL thetaAboveCell10013231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013231) h
+theorem e24KC2ThetaAboveLeaf100132313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013231) = true := by
+  have h : ((childHH thetaAboveCell10013231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013231) h
+theorem e24KC2ThetaAboveLeaf10013232 :
+    adaptiveCoverCheck 11 thetaAboveCell10013232 = true := by
+  have h : (thetaAboveCell10013232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013232 h
+theorem e24KC2ThetaAboveLeaf10013233 :
+    adaptiveCoverCheck 11 thetaAboveCell10013233 = true := by
+  have h : (thetaAboveCell10013233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013233 h
+theorem e24KC2ThetaAboveLeaf100133000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013300) = true := by
+  have h : ((childLL thetaAboveCell10013300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10013300) h
+theorem e24KC2ThetaAboveLeaf100133001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013300) = true := by
+  have h : ((childLH thetaAboveCell10013300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10013300) h
+theorem e24KC2ThetaAboveLeaf100133002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013300) = true := by
+  have h : ((childHL thetaAboveCell10013300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013300) h
+theorem e24KC2ThetaAboveLeaf100133003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013300) = true := by
+  have h : ((childHH thetaAboveCell10013300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013300) h
+theorem e24KC2ThetaAboveLeaf100133010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013301) = true := by
+  have h : ((childLL thetaAboveCell10013301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10013301) h
+theorem e24KC2ThetaAboveLeaf100133011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013301) = true := by
+  have h : ((childLH thetaAboveCell10013301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10013301) h
+theorem e24KC2ThetaAboveLeaf100133012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013301) = true := by
+  have h : ((childHL thetaAboveCell10013301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013301) h
+theorem e24KC2ThetaAboveLeaf100133013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013301) = true := by
+  have h : ((childHH thetaAboveCell10013301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013301) h
+theorem e24KC2ThetaAboveLeaf1001330200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10013302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10013302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10013302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10013302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10013302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10013302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10013302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10013302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10013302)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10013302)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10013302)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10013302)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10013302)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10013302)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10013302)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10013302)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10013302)) h
+theorem e24KC2ThetaAboveLeaf1001330300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10013303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf1001330301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10013303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf1001330302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10013303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf1001330303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10013303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf1001330310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10013303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf1001330311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10013303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf1001330312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10013303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf1001330313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10013303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf1001330320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10013303)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf1001330321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10013303)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf1001330322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10013303)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf1001330323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10013303)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf1001330330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10013303)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf1001330331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10013303)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf1001330332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10013303)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf1001330333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10013303)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10013303)) h
+theorem e24KC2ThetaAboveLeaf100133100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013310) = true := by
+  have h : ((childLL thetaAboveCell10013310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10013310) h
+theorem e24KC2ThetaAboveLeaf100133101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013310) = true := by
+  have h : ((childLH thetaAboveCell10013310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10013310) h
+theorem e24KC2ThetaAboveLeaf100133102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013310) = true := by
+  have h : ((childHL thetaAboveCell10013310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013310) h
+theorem e24KC2ThetaAboveLeaf100133103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013310) = true := by
+  have h : ((childHH thetaAboveCell10013310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013310) h
+theorem e24KC2ThetaAboveLeaf100133110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013311) = true := by
+  have h : ((childLL thetaAboveCell10013311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10013311) h
+theorem e24KC2ThetaAboveLeaf100133111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013311) = true := by
+  have h : ((childLH thetaAboveCell10013311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10013311) h
+theorem e24KC2ThetaAboveLeaf100133112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013311) = true := by
+  have h : ((childHL thetaAboveCell10013311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013311) h
+theorem e24KC2ThetaAboveLeaf100133113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013311) = true := by
+  have h : ((childHH thetaAboveCell10013311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013311) h
+theorem e24KC2ThetaAboveLeaf1001331200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10013312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10013312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10013312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10013312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10013312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10013312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10013312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10013312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10013312)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10013312)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10013312)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10013312)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10013312)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10013312)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10013312)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10013312)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10013312)) h
+theorem e24KC2ThetaAboveLeaf1001331300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10013313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf1001331301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10013313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf1001331302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10013313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf1001331303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10013313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf1001331310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10013313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf1001331311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10013313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf1001331312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10013313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf1001331313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10013313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf1001331320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10013313)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf1001331321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10013313)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf1001331322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10013313)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf1001331323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10013313)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf1001331330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10013313)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf1001331331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10013313)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf1001331332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10013313)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf1001331333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10013313)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10013313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10013313)) h
+theorem e24KC2ThetaAboveLeaf100133200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013320) = true := by
+  have h : ((childLL thetaAboveCell10013320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10013320) h
+theorem e24KC2ThetaAboveLeaf100133201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013320) = true := by
+  have h : ((childLH thetaAboveCell10013320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10013320) h
+theorem e24KC2ThetaAboveLeaf100133202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013320) = true := by
+  have h : ((childHL thetaAboveCell10013320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013320) h
+theorem e24KC2ThetaAboveLeaf100133203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013320) = true := by
+  have h : ((childHH thetaAboveCell10013320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013320) h
+theorem e24KC2ThetaAboveLeaf100133210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013321) = true := by
+  have h : ((childLL thetaAboveCell10013321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10013321) h
+theorem e24KC2ThetaAboveLeaf100133211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013321) = true := by
+  have h : ((childLH thetaAboveCell10013321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10013321) h
+theorem e24KC2ThetaAboveLeaf100133212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013321) = true := by
+  have h : ((childHL thetaAboveCell10013321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013321) h
+theorem e24KC2ThetaAboveLeaf100133213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013321) = true := by
+  have h : ((childHH thetaAboveCell10013321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013321) h
+theorem e24KC2ThetaAboveLeaf10013322 :
+    adaptiveCoverCheck 11 thetaAboveCell10013322 = true := by
+  have h : (thetaAboveCell10013322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013322 h
+theorem e24KC2ThetaAboveLeaf10013323 :
+    adaptiveCoverCheck 11 thetaAboveCell10013323 = true := by
+  have h : (thetaAboveCell10013323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013323 h
+theorem e24KC2ThetaAboveLeaf100133300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013330) = true := by
+  have h : ((childLL thetaAboveCell10013330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10013330) h
+theorem e24KC2ThetaAboveLeaf100133301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013330) = true := by
+  have h : ((childLH thetaAboveCell10013330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10013330) h
+theorem e24KC2ThetaAboveLeaf100133302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013330) = true := by
+  have h : ((childHL thetaAboveCell10013330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013330) h
+theorem e24KC2ThetaAboveLeaf100133303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013330) = true := by
+  have h : ((childHH thetaAboveCell10013330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013330) h
+theorem e24KC2ThetaAboveLeaf100133310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013331) = true := by
+  have h : ((childLL thetaAboveCell10013331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10013331) h
+theorem e24KC2ThetaAboveLeaf100133311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013331) = true := by
+  have h : ((childLH thetaAboveCell10013331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10013331) h
+theorem e24KC2ThetaAboveLeaf100133312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013331) = true := by
+  have h : ((childHL thetaAboveCell10013331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10013331) h
+theorem e24KC2ThetaAboveLeaf100133313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013331) = true := by
+  have h : ((childHH thetaAboveCell10013331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10013331) h
+theorem e24KC2ThetaAboveLeaf10013332 :
+    adaptiveCoverCheck 11 thetaAboveCell10013332 = true := by
+  have h : (thetaAboveCell10013332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013332 h
+theorem e24KC2ThetaAboveLeaf10013333 :
+    adaptiveCoverCheck 11 thetaAboveCell10013333 = true := by
+  have h : (thetaAboveCell10013333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10013333 h
+theorem e24KC2ThetaAboveLeaf1002000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell1002))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf1002001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell1002))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf1002002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell1002))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf1002003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell1002))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf1002010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell1002))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf1002011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell1002))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf1002012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell1002))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf1002013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell1002))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf100202 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1002)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1002))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1002)) h
+theorem e24KC2ThetaAboveLeaf100203 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1002)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1002))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1002)) h
+theorem e24KC2ThetaAboveLeaf1002100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell1002))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf1002101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell1002))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf1002102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell1002))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf1002103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell1002))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf1002110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell1002))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf1002111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell1002))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf1002112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell1002))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf1002113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell1002))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell1002)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell1002))) h
+theorem e24KC2ThetaAboveLeaf100212 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1002)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1002))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1002)) h
+theorem e24KC2ThetaAboveLeaf100213 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1002)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1002))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1002)) h
+theorem e24KC2ThetaAboveLeaf10022 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1002) = true := by
+  have h : ((childHL thetaAboveCell1002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell1002) h
+theorem e24KC2ThetaAboveLeaf10023 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1002) = true := by
+  have h : ((childHH thetaAboveCell1002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell1002) h
+theorem e24KC2ThetaAboveLeaf1003000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell1003))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf1003001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell1003))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf1003002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell1003))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf1003003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell1003))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf1003010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell1003))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf1003011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell1003))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf1003012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell1003))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf1003013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell1003))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf100302 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1003)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1003))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1003)) h
+theorem e24KC2ThetaAboveLeaf100303 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1003)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1003))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1003)) h
+theorem e24KC2ThetaAboveLeaf1003100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell1003))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf1003101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell1003))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf1003102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell1003))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf1003103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell1003))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf1003110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell1003))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf1003111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell1003))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf1003112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell1003))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf1003113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell1003))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell1003)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell1003))) h
+theorem e24KC2ThetaAboveLeaf100312 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1003)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1003))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1003)) h
+theorem e24KC2ThetaAboveLeaf100313 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1003)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1003))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1003)) h
+theorem e24KC2ThetaAboveLeaf10032 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1003) = true := by
+  have h : ((childHL thetaAboveCell1003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell1003) h
+theorem e24KC2ThetaAboveLeaf10033 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1003) = true := by
+  have h : ((childHH thetaAboveCell1003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell1003) h
+theorem e24KC2ThetaAboveLeaf101000 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1010)) = true := by
+  have h : ((childLL (childLL thetaAboveCell1010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell1010)) h
+theorem e24KC2ThetaAboveLeaf101001 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1010)) = true := by
+  have h : ((childLH (childLL thetaAboveCell1010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell1010)) h
+theorem e24KC2ThetaAboveLeaf101002 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1010)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1010)) h
+theorem e24KC2ThetaAboveLeaf101003 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1010)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1010)) h
+theorem e24KC2ThetaAboveLeaf101010 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1010)) = true := by
+  have h : ((childLL (childLH thetaAboveCell1010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell1010)) h
+theorem e24KC2ThetaAboveLeaf101011 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1010)) = true := by
+  have h : ((childLH (childLH thetaAboveCell1010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell1010)) h
+theorem e24KC2ThetaAboveLeaf101012 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1010)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1010)) h
+theorem e24KC2ThetaAboveLeaf101013 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1010)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1010)) h
+theorem e24KC2ThetaAboveLeaf1010200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell1010))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell1010))) h
+theorem e24KC2ThetaAboveLeaf1010201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell1010))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell1010))) h
+theorem e24KC2ThetaAboveLeaf10102020 :
+    adaptiveCoverCheck 11 thetaAboveCell10102020 = true := by
+  have h : (thetaAboveCell10102020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102020 h
+theorem e24KC2ThetaAboveLeaf10102021 :
+    adaptiveCoverCheck 11 thetaAboveCell10102021 = true := by
+  have h : (thetaAboveCell10102021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102021 h
+theorem e24KC2ThetaAboveLeaf10102022 :
+    adaptiveCoverCheck 11 thetaAboveCell10102022 = true := by
+  have h : (thetaAboveCell10102022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102022 h
+theorem e24KC2ThetaAboveLeaf10102023 :
+    adaptiveCoverCheck 11 thetaAboveCell10102023 = true := by
+  have h : (thetaAboveCell10102023).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102023 h
+theorem e24KC2ThetaAboveLeaf10102030 :
+    adaptiveCoverCheck 11 thetaAboveCell10102030 = true := by
+  have h : (thetaAboveCell10102030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102030 h
+theorem e24KC2ThetaAboveLeaf10102031 :
+    adaptiveCoverCheck 11 thetaAboveCell10102031 = true := by
+  have h : (thetaAboveCell10102031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102031 h
+theorem e24KC2ThetaAboveLeaf10102032 :
+    adaptiveCoverCheck 11 thetaAboveCell10102032 = true := by
+  have h : (thetaAboveCell10102032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102032 h
+theorem e24KC2ThetaAboveLeaf10102033 :
+    adaptiveCoverCheck 11 thetaAboveCell10102033 = true := by
+  have h : (thetaAboveCell10102033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102033 h
+theorem e24KC2ThetaAboveLeaf1010210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell1010))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell1010))) h
+theorem e24KC2ThetaAboveLeaf1010211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell1010))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell1010))) h
+theorem e24KC2ThetaAboveLeaf1010212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell1010))) = true := by
+  have h : ((childHL (childLH (childHL thetaAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childHL thetaAboveCell1010))) h
+theorem e24KC2ThetaAboveLeaf1010213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell1010))) = true := by
+  have h : ((childHH (childLH (childHL thetaAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childHL thetaAboveCell1010))) h
+theorem e24KC2ThetaAboveLeaf101022000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102200) = true := by
+  have h : ((childLL thetaAboveCell10102200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102200) h
+theorem e24KC2ThetaAboveLeaf101022001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102200) = true := by
+  have h : ((childLH thetaAboveCell10102200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102200) h
+theorem e24KC2ThetaAboveLeaf101022002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102200) = true := by
+  have h : ((childHL thetaAboveCell10102200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102200) h
+theorem e24KC2ThetaAboveLeaf101022003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102200) = true := by
+  have h : ((childHH thetaAboveCell10102200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102200) h
+theorem e24KC2ThetaAboveLeaf101022010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102201) = true := by
+  have h : ((childLL thetaAboveCell10102201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102201) h
+theorem e24KC2ThetaAboveLeaf101022011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102201) = true := by
+  have h : ((childLH thetaAboveCell10102201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102201) h
+theorem e24KC2ThetaAboveLeaf101022012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102201) = true := by
+  have h : ((childHL thetaAboveCell10102201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102201) h
+theorem e24KC2ThetaAboveLeaf101022013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102201) = true := by
+  have h : ((childHH thetaAboveCell10102201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102201) h
+theorem e24KC2ThetaAboveLeaf1010220200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10102202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10102202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10102202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10102202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10102202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10102202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10102202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10102202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10102202)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10102202)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10102202)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10102202)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10102202)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10102202)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10102202)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10102202)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10102202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10102202)) h
+theorem e24KC2ThetaAboveLeaf1010220300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10102203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf1010220301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10102203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf1010220302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10102203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf1010220303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10102203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf1010220310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10102203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf1010220311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10102203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf1010220312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10102203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf1010220313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10102203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf1010220320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10102203)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf1010220321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10102203)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf1010220322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10102203)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf1010220323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10102203)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf1010220330 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10102203)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf1010220331 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10102203)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf1010220332 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10102203)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf1010220333 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10102203)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10102203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10102203)) h
+theorem e24KC2ThetaAboveLeaf101022100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102210) = true := by
+  have h : ((childLL thetaAboveCell10102210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102210) h
+theorem e24KC2ThetaAboveLeaf101022101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102210) = true := by
+  have h : ((childLH thetaAboveCell10102210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102210) h
+theorem e24KC2ThetaAboveLeaf101022102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102210) = true := by
+  have h : ((childHL thetaAboveCell10102210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102210) h
+theorem e24KC2ThetaAboveLeaf101022103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102210) = true := by
+  have h : ((childHH thetaAboveCell10102210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102210) h
+theorem e24KC2ThetaAboveLeaf101022110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102211) = true := by
+  have h : ((childLL thetaAboveCell10102211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102211) h
+theorem e24KC2ThetaAboveLeaf101022111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102211) = true := by
+  have h : ((childLH thetaAboveCell10102211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102211) h
+theorem e24KC2ThetaAboveLeaf101022112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102211) = true := by
+  have h : ((childHL thetaAboveCell10102211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102211) h
+theorem e24KC2ThetaAboveLeaf101022113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102211) = true := by
+  have h : ((childHH thetaAboveCell10102211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102211) h
+theorem e24KC2ThetaAboveLeaf1010221200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10102212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10102212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10102212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10102212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10102212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10102212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10102212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10102212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221220 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10102212)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221221 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10102212)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221222 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10102212)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221223 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10102212)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221230 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell10102212)) = true := by
+  have h : ((childLL (childHH thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHH thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221231 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell10102212)) = true := by
+  have h : ((childLH (childHH thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHH thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221232 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell10102212)) = true := by
+  have h : ((childHL (childHH thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHH thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221233 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell10102212)) = true := by
+  have h : ((childHH (childHH thetaAboveCell10102212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHH thetaAboveCell10102212)) h
+theorem e24KC2ThetaAboveLeaf1010221300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10102213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10102213)) h
+theorem e24KC2ThetaAboveLeaf1010221301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10102213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10102213)) h
+theorem e24KC2ThetaAboveLeaf1010221302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10102213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10102213)) h
+theorem e24KC2ThetaAboveLeaf1010221303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10102213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10102213)) h
+theorem e24KC2ThetaAboveLeaf1010221310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10102213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10102213)) h
+theorem e24KC2ThetaAboveLeaf1010221311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10102213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10102213)) h
+theorem e24KC2ThetaAboveLeaf1010221312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10102213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10102213)) h
+theorem e24KC2ThetaAboveLeaf1010221313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10102213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10102213)) h
+theorem e24KC2ThetaAboveLeaf1010221320 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell10102213)) = true := by
+  have h : ((childLL (childHL thetaAboveCell10102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childHL thetaAboveCell10102213)) h
+theorem e24KC2ThetaAboveLeaf1010221321 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell10102213)) = true := by
+  have h : ((childLH (childHL thetaAboveCell10102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childHL thetaAboveCell10102213)) h
+theorem e24KC2ThetaAboveLeaf1010221322 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell10102213)) = true := by
+  have h : ((childHL (childHL thetaAboveCell10102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childHL thetaAboveCell10102213)) h
+theorem e24KC2ThetaAboveLeaf1010221323 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell10102213)) = true := by
+  have h : ((childHH (childHL thetaAboveCell10102213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childHL thetaAboveCell10102213)) h
+theorem e24KC2ThetaAboveLeaf101022133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102213) = true := by
+  have h : ((childHH thetaAboveCell10102213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102213) h
+theorem e24KC2ThetaAboveLeaf101022200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102220) = true := by
+  have h : ((childLL thetaAboveCell10102220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102220) h
+theorem e24KC2ThetaAboveLeaf101022201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102220) = true := by
+  have h : ((childLH thetaAboveCell10102220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102220) h
+theorem e24KC2ThetaAboveLeaf101022202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102220) = true := by
+  have h : ((childHL thetaAboveCell10102220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102220) h
+theorem e24KC2ThetaAboveLeaf101022203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102220) = true := by
+  have h : ((childHH thetaAboveCell10102220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102220) h
+theorem e24KC2ThetaAboveLeaf101022210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102221) = true := by
+  have h : ((childLL thetaAboveCell10102221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102221) h
+theorem e24KC2ThetaAboveLeaf101022211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102221) = true := by
+  have h : ((childLH thetaAboveCell10102221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102221) h
+theorem e24KC2ThetaAboveLeaf101022212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102221) = true := by
+  have h : ((childHL thetaAboveCell10102221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102221) h
+theorem e24KC2ThetaAboveLeaf101022213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102221) = true := by
+  have h : ((childHH thetaAboveCell10102221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102221) h
+theorem e24KC2ThetaAboveLeaf10102222 :
+    adaptiveCoverCheck 11 thetaAboveCell10102222 = true := by
+  have h : (thetaAboveCell10102222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102222 h
+theorem e24KC2ThetaAboveLeaf10102223 :
+    adaptiveCoverCheck 11 thetaAboveCell10102223 = true := by
+  have h : (thetaAboveCell10102223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102223 h
+theorem e24KC2ThetaAboveLeaf101022300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102230) = true := by
+  have h : ((childLL thetaAboveCell10102230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102230) h
+theorem e24KC2ThetaAboveLeaf101022301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102230) = true := by
+  have h : ((childLH thetaAboveCell10102230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102230) h
+theorem e24KC2ThetaAboveLeaf101022302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102230) = true := by
+  have h : ((childHL thetaAboveCell10102230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102230) h
+theorem e24KC2ThetaAboveLeaf101022303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102230) = true := by
+  have h : ((childHH thetaAboveCell10102230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102230) h
+theorem e24KC2ThetaAboveLeaf101022310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102231) = true := by
+  have h : ((childLL thetaAboveCell10102231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102231) h
+theorem e24KC2ThetaAboveLeaf101022311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102231) = true := by
+  have h : ((childLH thetaAboveCell10102231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102231) h
+theorem e24KC2ThetaAboveLeaf101022312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102231) = true := by
+  have h : ((childHL thetaAboveCell10102231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102231) h
+theorem e24KC2ThetaAboveLeaf101022313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102231) = true := by
+  have h : ((childHH thetaAboveCell10102231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102231) h
+theorem e24KC2ThetaAboveLeaf10102232 :
+    adaptiveCoverCheck 11 thetaAboveCell10102232 = true := by
+  have h : (thetaAboveCell10102232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102232 h
+theorem e24KC2ThetaAboveLeaf10102233 :
+    adaptiveCoverCheck 11 thetaAboveCell10102233 = true := by
+  have h : (thetaAboveCell10102233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102233 h
+theorem e24KC2ThetaAboveLeaf101023000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102300) = true := by
+  have h : ((childLL thetaAboveCell10102300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102300) h
+theorem e24KC2ThetaAboveLeaf101023001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102300) = true := by
+  have h : ((childLH thetaAboveCell10102300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102300) h
+theorem e24KC2ThetaAboveLeaf101023002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102300) = true := by
+  have h : ((childHL thetaAboveCell10102300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102300) h
+theorem e24KC2ThetaAboveLeaf101023003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102300) = true := by
+  have h : ((childHH thetaAboveCell10102300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102300) h
+theorem e24KC2ThetaAboveLeaf101023010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102301) = true := by
+  have h : ((childLL thetaAboveCell10102301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102301) h
+theorem e24KC2ThetaAboveLeaf101023011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102301) = true := by
+  have h : ((childLH thetaAboveCell10102301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102301) h
+theorem e24KC2ThetaAboveLeaf101023012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102301) = true := by
+  have h : ((childHL thetaAboveCell10102301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102301) h
+theorem e24KC2ThetaAboveLeaf101023013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102301) = true := by
+  have h : ((childHH thetaAboveCell10102301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102301) h
+theorem e24KC2ThetaAboveLeaf1010230200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10102302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10102302)) h
+theorem e24KC2ThetaAboveLeaf1010230201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10102302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10102302)) h
+theorem e24KC2ThetaAboveLeaf1010230202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10102302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10102302)) h
+theorem e24KC2ThetaAboveLeaf1010230203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10102302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10102302)) h
+theorem e24KC2ThetaAboveLeaf1010230210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10102302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10102302)) h
+theorem e24KC2ThetaAboveLeaf1010230211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10102302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10102302)) h
+theorem e24KC2ThetaAboveLeaf1010230212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10102302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10102302)) h
+theorem e24KC2ThetaAboveLeaf1010230213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10102302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10102302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10102302)) h
+theorem e24KC2ThetaAboveLeaf101023022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102302) = true := by
+  have h : ((childHL thetaAboveCell10102302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102302) h
+theorem e24KC2ThetaAboveLeaf101023023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102302) = true := by
+  have h : ((childHH thetaAboveCell10102302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102302) h
+theorem e24KC2ThetaAboveLeaf1010230300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10102303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10102303)) h
+theorem e24KC2ThetaAboveLeaf1010230301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10102303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10102303)) h
+theorem e24KC2ThetaAboveLeaf1010230302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10102303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10102303)) h
+theorem e24KC2ThetaAboveLeaf1010230303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10102303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10102303)) h
+theorem e24KC2ThetaAboveLeaf1010230310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10102303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10102303)) h
+theorem e24KC2ThetaAboveLeaf1010230311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10102303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10102303)) h
+theorem e24KC2ThetaAboveLeaf1010230312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10102303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10102303)) h
+theorem e24KC2ThetaAboveLeaf1010230313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10102303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10102303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10102303)) h
+theorem e24KC2ThetaAboveLeaf101023032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102303) = true := by
+  have h : ((childHL thetaAboveCell10102303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102303) h
+theorem e24KC2ThetaAboveLeaf101023033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102303) = true := by
+  have h : ((childHH thetaAboveCell10102303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102303) h
+theorem e24KC2ThetaAboveLeaf101023100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102310) = true := by
+  have h : ((childLL thetaAboveCell10102310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102310) h
+theorem e24KC2ThetaAboveLeaf101023101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102310) = true := by
+  have h : ((childLH thetaAboveCell10102310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102310) h
+theorem e24KC2ThetaAboveLeaf101023102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102310) = true := by
+  have h : ((childHL thetaAboveCell10102310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102310) h
+theorem e24KC2ThetaAboveLeaf101023103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102310) = true := by
+  have h : ((childHH thetaAboveCell10102310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102310) h
+theorem e24KC2ThetaAboveLeaf101023110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102311) = true := by
+  have h : ((childLL thetaAboveCell10102311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102311) h
+theorem e24KC2ThetaAboveLeaf101023111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102311) = true := by
+  have h : ((childLH thetaAboveCell10102311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102311) h
+theorem e24KC2ThetaAboveLeaf101023112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102311) = true := by
+  have h : ((childHL thetaAboveCell10102311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102311) h
+theorem e24KC2ThetaAboveLeaf101023113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102311) = true := by
+  have h : ((childHH thetaAboveCell10102311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102311) h
+theorem e24KC2ThetaAboveLeaf1010231200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10102312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10102312)) h
+theorem e24KC2ThetaAboveLeaf1010231201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10102312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10102312)) h
+theorem e24KC2ThetaAboveLeaf1010231202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10102312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10102312)) h
+theorem e24KC2ThetaAboveLeaf1010231203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10102312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10102312)) h
+theorem e24KC2ThetaAboveLeaf1010231210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10102312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10102312)) h
+theorem e24KC2ThetaAboveLeaf1010231211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10102312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10102312)) h
+theorem e24KC2ThetaAboveLeaf1010231212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10102312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10102312)) h
+theorem e24KC2ThetaAboveLeaf1010231213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10102312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10102312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10102312)) h
+theorem e24KC2ThetaAboveLeaf101023122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102312) = true := by
+  have h : ((childHL thetaAboveCell10102312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102312) h
+theorem e24KC2ThetaAboveLeaf101023123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102312) = true := by
+  have h : ((childHH thetaAboveCell10102312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102312) h
+theorem e24KC2ThetaAboveLeaf1010231300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10102313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10102313)) h
+theorem e24KC2ThetaAboveLeaf1010231301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10102313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10102313)) h
+theorem e24KC2ThetaAboveLeaf1010231302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10102313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10102313)) h
+theorem e24KC2ThetaAboveLeaf1010231303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10102313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10102313)) h
+theorem e24KC2ThetaAboveLeaf1010231310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10102313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10102313)) h
+theorem e24KC2ThetaAboveLeaf1010231311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10102313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10102313)) h
+theorem e24KC2ThetaAboveLeaf1010231312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10102313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10102313)) h
+theorem e24KC2ThetaAboveLeaf1010231313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10102313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10102313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10102313)) h
+theorem e24KC2ThetaAboveLeaf101023132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102313) = true := by
+  have h : ((childHL thetaAboveCell10102313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102313) h
+theorem e24KC2ThetaAboveLeaf101023133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102313) = true := by
+  have h : ((childHH thetaAboveCell10102313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102313) h
+theorem e24KC2ThetaAboveLeaf101023200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102320) = true := by
+  have h : ((childLL thetaAboveCell10102320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102320) h
+theorem e24KC2ThetaAboveLeaf101023201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102320) = true := by
+  have h : ((childLH thetaAboveCell10102320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102320) h
+theorem e24KC2ThetaAboveLeaf101023202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102320) = true := by
+  have h : ((childHL thetaAboveCell10102320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102320) h
+theorem e24KC2ThetaAboveLeaf101023203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102320) = true := by
+  have h : ((childHH thetaAboveCell10102320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102320) h
+theorem e24KC2ThetaAboveLeaf101023210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102321) = true := by
+  have h : ((childLL thetaAboveCell10102321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102321) h
+theorem e24KC2ThetaAboveLeaf101023211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102321) = true := by
+  have h : ((childLH thetaAboveCell10102321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102321) h
+theorem e24KC2ThetaAboveLeaf101023212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102321) = true := by
+  have h : ((childHL thetaAboveCell10102321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102321) h
+theorem e24KC2ThetaAboveLeaf101023213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102321) = true := by
+  have h : ((childHH thetaAboveCell10102321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102321) h
+theorem e24KC2ThetaAboveLeaf10102322 :
+    adaptiveCoverCheck 11 thetaAboveCell10102322 = true := by
+  have h : (thetaAboveCell10102322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102322 h
+theorem e24KC2ThetaAboveLeaf10102323 :
+    adaptiveCoverCheck 11 thetaAboveCell10102323 = true := by
+  have h : (thetaAboveCell10102323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102323 h
+theorem e24KC2ThetaAboveLeaf101023300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102330) = true := by
+  have h : ((childLL thetaAboveCell10102330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102330) h
+theorem e24KC2ThetaAboveLeaf101023301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102330) = true := by
+  have h : ((childLH thetaAboveCell10102330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102330) h
+theorem e24KC2ThetaAboveLeaf101023302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102330) = true := by
+  have h : ((childHL thetaAboveCell10102330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102330) h
+theorem e24KC2ThetaAboveLeaf101023303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102330) = true := by
+  have h : ((childHH thetaAboveCell10102330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102330) h
+theorem e24KC2ThetaAboveLeaf101023310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102331) = true := by
+  have h : ((childLL thetaAboveCell10102331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10102331) h
+theorem e24KC2ThetaAboveLeaf101023311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102331) = true := by
+  have h : ((childLH thetaAboveCell10102331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10102331) h
+theorem e24KC2ThetaAboveLeaf101023312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102331) = true := by
+  have h : ((childHL thetaAboveCell10102331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10102331) h
+theorem e24KC2ThetaAboveLeaf101023313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102331) = true := by
+  have h : ((childHH thetaAboveCell10102331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10102331) h
+theorem e24KC2ThetaAboveLeaf10102332 :
+    adaptiveCoverCheck 11 thetaAboveCell10102332 = true := by
+  have h : (thetaAboveCell10102332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102332 h
+theorem e24KC2ThetaAboveLeaf10102333 :
+    adaptiveCoverCheck 11 thetaAboveCell10102333 = true := by
+  have h : (thetaAboveCell10102333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10102333 h
+theorem e24KC2ThetaAboveLeaf1010300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell1010))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell1010))) h
+theorem e24KC2ThetaAboveLeaf1010301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell1010))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell1010))) h
+theorem e24KC2ThetaAboveLeaf1010302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell1010))) = true := by
+  have h : ((childHL (childLL (childHH thetaAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childHH thetaAboveCell1010))) h
+theorem e24KC2ThetaAboveLeaf1010303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell1010))) = true := by
+  have h : ((childHH (childLL (childHH thetaAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childHH thetaAboveCell1010))) h
+theorem e24KC2ThetaAboveLeaf1010310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell1010))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell1010))) h
+theorem e24KC2ThetaAboveLeaf1010311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell1010))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell1010))) h
+theorem e24KC2ThetaAboveLeaf1010312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell1010))) = true := by
+  have h : ((childHL (childLH (childHH thetaAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childHH thetaAboveCell1010))) h
+theorem e24KC2ThetaAboveLeaf1010313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell1010))) = true := by
+  have h : ((childHH (childLH (childHH thetaAboveCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childHH thetaAboveCell1010))) h
+theorem e24KC2ThetaAboveLeaf101032000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103200) = true := by
+  have h : ((childLL thetaAboveCell10103200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103200) h
+theorem e24KC2ThetaAboveLeaf101032001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103200) = true := by
+  have h : ((childLH thetaAboveCell10103200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103200) h
+theorem e24KC2ThetaAboveLeaf101032002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103200) = true := by
+  have h : ((childHL thetaAboveCell10103200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103200) h
+theorem e24KC2ThetaAboveLeaf101032003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103200) = true := by
+  have h : ((childHH thetaAboveCell10103200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103200) h
+theorem e24KC2ThetaAboveLeaf101032010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103201) = true := by
+  have h : ((childLL thetaAboveCell10103201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103201) h
+theorem e24KC2ThetaAboveLeaf101032011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103201) = true := by
+  have h : ((childLH thetaAboveCell10103201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103201) h
+theorem e24KC2ThetaAboveLeaf101032012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103201) = true := by
+  have h : ((childHL thetaAboveCell10103201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103201) h
+theorem e24KC2ThetaAboveLeaf101032013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103201) = true := by
+  have h : ((childHH thetaAboveCell10103201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103201) h
+theorem e24KC2ThetaAboveLeaf1010320200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10103202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10103202)) h
+theorem e24KC2ThetaAboveLeaf1010320201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10103202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10103202)) h
+theorem e24KC2ThetaAboveLeaf1010320202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10103202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10103202)) h
+theorem e24KC2ThetaAboveLeaf1010320203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10103202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10103202)) h
+theorem e24KC2ThetaAboveLeaf1010320210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10103202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10103202)) h
+theorem e24KC2ThetaAboveLeaf1010320211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10103202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10103202)) h
+theorem e24KC2ThetaAboveLeaf1010320212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10103202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10103202)) h
+theorem e24KC2ThetaAboveLeaf1010320213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10103202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10103202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10103202)) h
+theorem e24KC2ThetaAboveLeaf101032022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103202) = true := by
+  have h : ((childHL thetaAboveCell10103202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103202) h
+theorem e24KC2ThetaAboveLeaf101032023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103202) = true := by
+  have h : ((childHH thetaAboveCell10103202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103202) h
+theorem e24KC2ThetaAboveLeaf1010320300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10103203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10103203)) h
+theorem e24KC2ThetaAboveLeaf1010320301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10103203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10103203)) h
+theorem e24KC2ThetaAboveLeaf1010320302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10103203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10103203)) h
+theorem e24KC2ThetaAboveLeaf1010320303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10103203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10103203)) h
+theorem e24KC2ThetaAboveLeaf1010320310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10103203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10103203)) h
+theorem e24KC2ThetaAboveLeaf1010320311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10103203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10103203)) h
+theorem e24KC2ThetaAboveLeaf1010320312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10103203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10103203)) h
+theorem e24KC2ThetaAboveLeaf1010320313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10103203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10103203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10103203)) h
+theorem e24KC2ThetaAboveLeaf101032032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103203) = true := by
+  have h : ((childHL thetaAboveCell10103203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103203) h
+theorem e24KC2ThetaAboveLeaf101032033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103203) = true := by
+  have h : ((childHH thetaAboveCell10103203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103203) h
+theorem e24KC2ThetaAboveLeaf101032100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103210) = true := by
+  have h : ((childLL thetaAboveCell10103210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103210) h
+theorem e24KC2ThetaAboveLeaf101032101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103210) = true := by
+  have h : ((childLH thetaAboveCell10103210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103210) h
+theorem e24KC2ThetaAboveLeaf101032102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103210) = true := by
+  have h : ((childHL thetaAboveCell10103210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103210) h
+theorem e24KC2ThetaAboveLeaf101032103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103210) = true := by
+  have h : ((childHH thetaAboveCell10103210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103210) h
+theorem e24KC2ThetaAboveLeaf101032110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103211) = true := by
+  have h : ((childLL thetaAboveCell10103211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103211) h
+theorem e24KC2ThetaAboveLeaf101032111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103211) = true := by
+  have h : ((childLH thetaAboveCell10103211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103211) h
+theorem e24KC2ThetaAboveLeaf101032112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103211) = true := by
+  have h : ((childHL thetaAboveCell10103211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103211) h
+theorem e24KC2ThetaAboveLeaf101032113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103211) = true := by
+  have h : ((childHH thetaAboveCell10103211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103211) h
+theorem e24KC2ThetaAboveLeaf1010321200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10103212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10103212)) h
+theorem e24KC2ThetaAboveLeaf1010321201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10103212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10103212)) h
+theorem e24KC2ThetaAboveLeaf1010321202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10103212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10103212)) h
+theorem e24KC2ThetaAboveLeaf1010321203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10103212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10103212)) h
+theorem e24KC2ThetaAboveLeaf1010321210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10103212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10103212)) h
+theorem e24KC2ThetaAboveLeaf1010321211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10103212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10103212)) h
+theorem e24KC2ThetaAboveLeaf1010321212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10103212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10103212)) h
+theorem e24KC2ThetaAboveLeaf1010321213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10103212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10103212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10103212)) h
+theorem e24KC2ThetaAboveLeaf101032122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103212) = true := by
+  have h : ((childHL thetaAboveCell10103212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103212) h
+theorem e24KC2ThetaAboveLeaf101032123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103212) = true := by
+  have h : ((childHH thetaAboveCell10103212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103212) h
+theorem e24KC2ThetaAboveLeaf1010321300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell10103213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell10103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell10103213)) h
+theorem e24KC2ThetaAboveLeaf1010321301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell10103213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell10103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell10103213)) h
+theorem e24KC2ThetaAboveLeaf1010321302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell10103213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell10103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell10103213)) h
+theorem e24KC2ThetaAboveLeaf1010321303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell10103213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell10103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell10103213)) h
+theorem e24KC2ThetaAboveLeaf1010321310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell10103213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell10103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell10103213)) h
+theorem e24KC2ThetaAboveLeaf1010321311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell10103213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell10103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell10103213)) h
+theorem e24KC2ThetaAboveLeaf1010321312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell10103213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell10103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell10103213)) h
+theorem e24KC2ThetaAboveLeaf1010321313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell10103213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell10103213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell10103213)) h
+theorem e24KC2ThetaAboveLeaf101032132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103213) = true := by
+  have h : ((childHL thetaAboveCell10103213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103213) h
+theorem e24KC2ThetaAboveLeaf101032133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103213) = true := by
+  have h : ((childHH thetaAboveCell10103213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103213) h
+theorem e24KC2ThetaAboveLeaf101032200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103220) = true := by
+  have h : ((childLL thetaAboveCell10103220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103220) h
+theorem e24KC2ThetaAboveLeaf101032201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103220) = true := by
+  have h : ((childLH thetaAboveCell10103220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103220) h
+theorem e24KC2ThetaAboveLeaf101032202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103220) = true := by
+  have h : ((childHL thetaAboveCell10103220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103220) h
+theorem e24KC2ThetaAboveLeaf101032203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103220) = true := by
+  have h : ((childHH thetaAboveCell10103220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103220) h
+theorem e24KC2ThetaAboveLeaf101032210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103221) = true := by
+  have h : ((childLL thetaAboveCell10103221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103221) h
+theorem e24KC2ThetaAboveLeaf101032211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103221) = true := by
+  have h : ((childLH thetaAboveCell10103221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103221) h
+theorem e24KC2ThetaAboveLeaf101032212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103221) = true := by
+  have h : ((childHL thetaAboveCell10103221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103221) h
+theorem e24KC2ThetaAboveLeaf101032213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103221) = true := by
+  have h : ((childHH thetaAboveCell10103221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103221) h
+theorem e24KC2ThetaAboveLeaf10103222 :
+    adaptiveCoverCheck 11 thetaAboveCell10103222 = true := by
+  have h : (thetaAboveCell10103222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10103222 h
+theorem e24KC2ThetaAboveLeaf10103223 :
+    adaptiveCoverCheck 11 thetaAboveCell10103223 = true := by
+  have h : (thetaAboveCell10103223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10103223 h
+theorem e24KC2ThetaAboveLeaf101032300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103230) = true := by
+  have h : ((childLL thetaAboveCell10103230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103230) h
+theorem e24KC2ThetaAboveLeaf101032301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103230) = true := by
+  have h : ((childLH thetaAboveCell10103230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103230) h
+theorem e24KC2ThetaAboveLeaf101032302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103230) = true := by
+  have h : ((childHL thetaAboveCell10103230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103230) h
+theorem e24KC2ThetaAboveLeaf101032303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103230) = true := by
+  have h : ((childHH thetaAboveCell10103230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103230) h
+theorem e24KC2ThetaAboveLeaf101032310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103231) = true := by
+  have h : ((childLL thetaAboveCell10103231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103231) h
+theorem e24KC2ThetaAboveLeaf101032311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103231) = true := by
+  have h : ((childLH thetaAboveCell10103231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103231) h
+theorem e24KC2ThetaAboveLeaf101032312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10103231) = true := by
+  have h : ((childHL thetaAboveCell10103231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell10103231) h
+theorem e24KC2ThetaAboveLeaf101032313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10103231) = true := by
+  have h : ((childHH thetaAboveCell10103231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell10103231) h
+theorem e24KC2ThetaAboveLeaf10103232 :
+    adaptiveCoverCheck 11 thetaAboveCell10103232 = true := by
+  have h : (thetaAboveCell10103232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10103232 h
+theorem e24KC2ThetaAboveLeaf10103233 :
+    adaptiveCoverCheck 11 thetaAboveCell10103233 = true := by
+  have h : (thetaAboveCell10103233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell10103233 h
+theorem e24KC2ThetaAboveLeaf101033000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103300) = true := by
+  have h : ((childLL thetaAboveCell10103300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell10103300) h
+theorem e24KC2ThetaAboveLeaf101033001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103300) = true := by
+  have h : ((childLH thetaAboveCell10103300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell10103300) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC5TerminalBatchT51200004`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse876d675ca
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse876d675ca
+
+open CertificateCellse876d675ca
+theorem e24KC2ThetaAboveLeaf00003121 :
+    adaptiveCoverCheck 11 thetaAboveCell00003121 = true := by
+  have h : (thetaAboveCell00003121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003121 h
+theorem e24KC2ThetaAboveLeaf000031220 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003122) = true := by
+  have h : ((childLL thetaAboveCell00003122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00003122) h
+theorem e24KC2ThetaAboveLeaf000031221 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003122) = true := by
+  have h : ((childLH thetaAboveCell00003122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00003122) h
+theorem e24KC2ThetaAboveLeaf000031222 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003122) = true := by
+  have h : ((childHL thetaAboveCell00003122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003122) h
+theorem e24KC2ThetaAboveLeaf000031223 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003122) = true := by
+  have h : ((childHH thetaAboveCell00003122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003122) h
+theorem e24KC2ThetaAboveLeaf000031230 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003123) = true := by
+  have h : ((childLL thetaAboveCell00003123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00003123) h
+theorem e24KC2ThetaAboveLeaf000031231 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003123) = true := by
+  have h : ((childLH thetaAboveCell00003123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00003123) h
+theorem e24KC2ThetaAboveLeaf000031232 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003123) = true := by
+  have h : ((childHL thetaAboveCell00003123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003123) h
+theorem e24KC2ThetaAboveLeaf000031233 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003123) = true := by
+  have h : ((childHH thetaAboveCell00003123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003123) h
+theorem e24KC2ThetaAboveLeaf00003130 :
+    adaptiveCoverCheck 11 thetaAboveCell00003130 = true := by
+  have h : (thetaAboveCell00003130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003130 h
+theorem e24KC2ThetaAboveLeaf00003131 :
+    adaptiveCoverCheck 11 thetaAboveCell00003131 = true := by
+  have h : (thetaAboveCell00003131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003131 h
+theorem e24KC2ThetaAboveLeaf000031320 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003132) = true := by
+  have h : ((childLL thetaAboveCell00003132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00003132) h
+theorem e24KC2ThetaAboveLeaf000031321 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003132) = true := by
+  have h : ((childLH thetaAboveCell00003132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00003132) h
+theorem e24KC2ThetaAboveLeaf000031322 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003132) = true := by
+  have h : ((childHL thetaAboveCell00003132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003132) h
+theorem e24KC2ThetaAboveLeaf000031323 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003132) = true := by
+  have h : ((childHH thetaAboveCell00003132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003132) h
+theorem e24KC2ThetaAboveLeaf000031330 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003133) = true := by
+  have h : ((childLL thetaAboveCell00003133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00003133) h
+theorem e24KC2ThetaAboveLeaf000031331 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003133) = true := by
+  have h : ((childLH thetaAboveCell00003133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00003133) h
+theorem e24KC2ThetaAboveLeaf000031332 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003133) = true := by
+  have h : ((childHL thetaAboveCell00003133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003133) h
+theorem e24KC2ThetaAboveLeaf000031333 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003133) = true := by
+  have h : ((childHH thetaAboveCell00003133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003133) h
+theorem e24KC2ThetaAboveLeaf0000320000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003200)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003200)) h
+theorem e24KC2ThetaAboveLeaf0000320001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003200)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003200)) h
+theorem e24KC2ThetaAboveLeaf0000320010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003200)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003200)) h
+theorem e24KC2ThetaAboveLeaf0000320011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003200)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003200)) h
+theorem e24KC2ThetaAboveLeaf0000320100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003201)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003201)) h
+theorem e24KC2ThetaAboveLeaf0000320101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003201)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003201)) h
+theorem e24KC2ThetaAboveLeaf0000320110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003201)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003201)) h
+theorem e24KC2ThetaAboveLeaf0000320111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003201)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003201)) h
+theorem e24KC2ThetaAboveLeaf0000320200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003202)) h
+theorem e24KC2ThetaAboveLeaf0000320201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003202)) h
+theorem e24KC2ThetaAboveLeaf0000320202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00003202)) h
+theorem e24KC2ThetaAboveLeaf0000320203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00003202)) h
+theorem e24KC2ThetaAboveLeaf0000320210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003202)) h
+theorem e24KC2ThetaAboveLeaf0000320211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003202)) h
+theorem e24KC2ThetaAboveLeaf0000320212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00003202)) h
+theorem e24KC2ThetaAboveLeaf0000320213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00003202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00003202)) h
+theorem e24KC2ThetaAboveLeaf000032022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003202) = true := by
+  have h : ((childHL thetaAboveCell00003202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003202) h
+theorem e24KC2ThetaAboveLeaf000032023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003202) = true := by
+  have h : ((childHH thetaAboveCell00003202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003202) h
+theorem e24KC2ThetaAboveLeaf0000320300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003203)) h
+theorem e24KC2ThetaAboveLeaf0000320301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003203)) h
+theorem e24KC2ThetaAboveLeaf0000320302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00003203)) h
+theorem e24KC2ThetaAboveLeaf0000320303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00003203)) h
+theorem e24KC2ThetaAboveLeaf0000320310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003203)) h
+theorem e24KC2ThetaAboveLeaf0000320311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003203)) h
+theorem e24KC2ThetaAboveLeaf0000320312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00003203)) h
+theorem e24KC2ThetaAboveLeaf0000320313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00003203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00003203)) h
+theorem e24KC2ThetaAboveLeaf000032032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003203) = true := by
+  have h : ((childHL thetaAboveCell00003203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003203) h
+theorem e24KC2ThetaAboveLeaf000032033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003203) = true := by
+  have h : ((childHH thetaAboveCell00003203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003203) h
+theorem e24KC2ThetaAboveLeaf0000321000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003210)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003210)) h
+theorem e24KC2ThetaAboveLeaf0000321001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003210)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003210)) h
+theorem e24KC2ThetaAboveLeaf0000321010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003210)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003210)) h
+theorem e24KC2ThetaAboveLeaf0000321011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003210)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003210)) h
+theorem e24KC2ThetaAboveLeaf0000321100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003211)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003211)) h
+theorem e24KC2ThetaAboveLeaf0000321101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003211)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003211)) h
+theorem e24KC2ThetaAboveLeaf0000321110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003211)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003211)) h
+theorem e24KC2ThetaAboveLeaf0000321111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003211)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003211)) h
+theorem e24KC2ThetaAboveLeaf0000321200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003212)) h
+theorem e24KC2ThetaAboveLeaf0000321201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003212)) h
+theorem e24KC2ThetaAboveLeaf0000321202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00003212)) h
+theorem e24KC2ThetaAboveLeaf0000321203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00003212)) h
+theorem e24KC2ThetaAboveLeaf0000321210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003212)) h
+theorem e24KC2ThetaAboveLeaf0000321211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003212)) h
+theorem e24KC2ThetaAboveLeaf0000321212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00003212)) h
+theorem e24KC2ThetaAboveLeaf0000321213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00003212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00003212)) h
+theorem e24KC2ThetaAboveLeaf000032122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003212) = true := by
+  have h : ((childHL thetaAboveCell00003212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003212) h
+theorem e24KC2ThetaAboveLeaf000032123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003212) = true := by
+  have h : ((childHH thetaAboveCell00003212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003212) h
+theorem e24KC2ThetaAboveLeaf0000321300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003213)) h
+theorem e24KC2ThetaAboveLeaf0000321301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003213)) h
+theorem e24KC2ThetaAboveLeaf0000321302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00003213)) h
+theorem e24KC2ThetaAboveLeaf0000321303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00003213)) h
+theorem e24KC2ThetaAboveLeaf0000321310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003213)) h
+theorem e24KC2ThetaAboveLeaf0000321311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003213)) h
+theorem e24KC2ThetaAboveLeaf0000321312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00003213)) h
+theorem e24KC2ThetaAboveLeaf0000321313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00003213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00003213)) h
+theorem e24KC2ThetaAboveLeaf000032132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003213) = true := by
+  have h : ((childHL thetaAboveCell00003213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003213) h
+theorem e24KC2ThetaAboveLeaf000032133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003213) = true := by
+  have h : ((childHH thetaAboveCell00003213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003213) h
+theorem e24KC2ThetaAboveLeaf00003220 :
+    adaptiveCoverCheck 11 thetaAboveCell00003220 = true := by
+  have h : (thetaAboveCell00003220).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003220 h
+theorem e24KC2ThetaAboveLeaf00003221 :
+    adaptiveCoverCheck 11 thetaAboveCell00003221 = true := by
+  have h : (thetaAboveCell00003221).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003221 h
+theorem e24KC2ThetaAboveLeaf00003222 :
+    adaptiveCoverCheck 11 thetaAboveCell00003222 = true := by
+  have h : (thetaAboveCell00003222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003222 h
+theorem e24KC2ThetaAboveLeaf00003223 :
+    adaptiveCoverCheck 11 thetaAboveCell00003223 = true := by
+  have h : (thetaAboveCell00003223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003223 h
+theorem e24KC2ThetaAboveLeaf00003230 :
+    adaptiveCoverCheck 11 thetaAboveCell00003230 = true := by
+  have h : (thetaAboveCell00003230).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003230 h
+theorem e24KC2ThetaAboveLeaf00003231 :
+    adaptiveCoverCheck 11 thetaAboveCell00003231 = true := by
+  have h : (thetaAboveCell00003231).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003231 h
+theorem e24KC2ThetaAboveLeaf00003232 :
+    adaptiveCoverCheck 11 thetaAboveCell00003232 = true := by
+  have h : (thetaAboveCell00003232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003232 h
+theorem e24KC2ThetaAboveLeaf00003233 :
+    adaptiveCoverCheck 11 thetaAboveCell00003233 = true := by
+  have h : (thetaAboveCell00003233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003233 h
+theorem e24KC2ThetaAboveLeaf0000330000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003300)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003300)) h
+theorem e24KC2ThetaAboveLeaf0000330001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003300)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003300)) h
+theorem e24KC2ThetaAboveLeaf0000330010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003300)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003300)) h
+theorem e24KC2ThetaAboveLeaf0000330011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003300)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003300)) h
+theorem e24KC2ThetaAboveLeaf0000330100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003301)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003301)) h
+theorem e24KC2ThetaAboveLeaf0000330101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003301)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003301)) h
+theorem e24KC2ThetaAboveLeaf0000330110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003301)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003301)) h
+theorem e24KC2ThetaAboveLeaf0000330111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003301)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003301)) h
+theorem e24KC2ThetaAboveLeaf0000330200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003302)) h
+theorem e24KC2ThetaAboveLeaf0000330201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003302)) h
+theorem e24KC2ThetaAboveLeaf0000330202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00003302)) h
+theorem e24KC2ThetaAboveLeaf0000330203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00003302)) h
+theorem e24KC2ThetaAboveLeaf0000330210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003302)) h
+theorem e24KC2ThetaAboveLeaf0000330211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003302)) h
+theorem e24KC2ThetaAboveLeaf0000330212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00003302)) h
+theorem e24KC2ThetaAboveLeaf0000330213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00003302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00003302)) h
+theorem e24KC2ThetaAboveLeaf000033022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003302) = true := by
+  have h : ((childHL thetaAboveCell00003302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003302) h
+theorem e24KC2ThetaAboveLeaf000033023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003302) = true := by
+  have h : ((childHH thetaAboveCell00003302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003302) h
+theorem e24KC2ThetaAboveLeaf0000330300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003303)) h
+theorem e24KC2ThetaAboveLeaf0000330301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003303)) h
+theorem e24KC2ThetaAboveLeaf0000330302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00003303)) h
+theorem e24KC2ThetaAboveLeaf0000330303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00003303)) h
+theorem e24KC2ThetaAboveLeaf0000330310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003303)) h
+theorem e24KC2ThetaAboveLeaf0000330311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003303)) h
+theorem e24KC2ThetaAboveLeaf0000330312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00003303)) h
+theorem e24KC2ThetaAboveLeaf0000330313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00003303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00003303)) h
+theorem e24KC2ThetaAboveLeaf000033032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003303) = true := by
+  have h : ((childHL thetaAboveCell00003303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003303) h
+theorem e24KC2ThetaAboveLeaf000033033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003303) = true := by
+  have h : ((childHH thetaAboveCell00003303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003303) h
+theorem e24KC2ThetaAboveLeaf0000331000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003310)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003310)) h
+theorem e24KC2ThetaAboveLeaf0000331001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003310)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003310)) h
+theorem e24KC2ThetaAboveLeaf0000331010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003310)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003310)) h
+theorem e24KC2ThetaAboveLeaf0000331011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003310)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003310)) h
+theorem e24KC2ThetaAboveLeaf0000331100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003311)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003311)) h
+theorem e24KC2ThetaAboveLeaf0000331101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003311)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003311)) h
+theorem e24KC2ThetaAboveLeaf0000331110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003311)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003311)) h
+theorem e24KC2ThetaAboveLeaf0000331111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003311)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003311)) h
+theorem e24KC2ThetaAboveLeaf0000331200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003312)) h
+theorem e24KC2ThetaAboveLeaf0000331201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003312)) h
+theorem e24KC2ThetaAboveLeaf0000331202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00003312)) h
+theorem e24KC2ThetaAboveLeaf0000331203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00003312)) h
+theorem e24KC2ThetaAboveLeaf0000331210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003312)) h
+theorem e24KC2ThetaAboveLeaf0000331211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003312)) h
+theorem e24KC2ThetaAboveLeaf0000331212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00003312)) h
+theorem e24KC2ThetaAboveLeaf0000331213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00003312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00003312)) h
+theorem e24KC2ThetaAboveLeaf000033122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003312) = true := by
+  have h : ((childHL thetaAboveCell00003312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003312) h
+theorem e24KC2ThetaAboveLeaf000033123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003312) = true := by
+  have h : ((childHH thetaAboveCell00003312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003312) h
+theorem e24KC2ThetaAboveLeaf0000331300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00003313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00003313)) h
+theorem e24KC2ThetaAboveLeaf0000331301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00003313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00003313)) h
+theorem e24KC2ThetaAboveLeaf0000331302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00003313)) h
+theorem e24KC2ThetaAboveLeaf0000331303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00003313)) h
+theorem e24KC2ThetaAboveLeaf0000331310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00003313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00003313)) h
+theorem e24KC2ThetaAboveLeaf0000331311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00003313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00003313)) h
+theorem e24KC2ThetaAboveLeaf0000331312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00003313)) h
+theorem e24KC2ThetaAboveLeaf0000331313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00003313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00003313)) h
+theorem e24KC2ThetaAboveLeaf000033132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003313) = true := by
+  have h : ((childHL thetaAboveCell00003313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00003313) h
+theorem e24KC2ThetaAboveLeaf000033133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003313) = true := by
+  have h : ((childHH thetaAboveCell00003313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00003313) h
+theorem e24KC2ThetaAboveLeaf00003320 :
+    adaptiveCoverCheck 11 thetaAboveCell00003320 = true := by
+  have h : (thetaAboveCell00003320).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003320 h
+theorem e24KC2ThetaAboveLeaf00003321 :
+    adaptiveCoverCheck 11 thetaAboveCell00003321 = true := by
+  have h : (thetaAboveCell00003321).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003321 h
+theorem e24KC2ThetaAboveLeaf00003322 :
+    adaptiveCoverCheck 11 thetaAboveCell00003322 = true := by
+  have h : (thetaAboveCell00003322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003322 h
+theorem e24KC2ThetaAboveLeaf00003323 :
+    adaptiveCoverCheck 11 thetaAboveCell00003323 = true := by
+  have h : (thetaAboveCell00003323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003323 h
+theorem e24KC2ThetaAboveLeaf00003330 :
+    adaptiveCoverCheck 11 thetaAboveCell00003330 = true := by
+  have h : (thetaAboveCell00003330).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003330 h
+theorem e24KC2ThetaAboveLeaf00003331 :
+    adaptiveCoverCheck 11 thetaAboveCell00003331 = true := by
+  have h : (thetaAboveCell00003331).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003331 h
+theorem e24KC2ThetaAboveLeaf00003332 :
+    adaptiveCoverCheck 11 thetaAboveCell00003332 = true := by
+  have h : (thetaAboveCell00003332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003332 h
+theorem e24KC2ThetaAboveLeaf00003333 :
+    adaptiveCoverCheck 11 thetaAboveCell00003333 = true := by
+  have h : (thetaAboveCell00003333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00003333 h
+theorem e24KC2ThetaAboveLeaf000100 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0001)) = true := by
+  have h : ((childLL (childLL thetaAboveCell0001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLL thetaAboveCell0001)) h
+theorem e24KC2ThetaAboveLeaf000101 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0001)) = true := by
+  have h : ((childLH (childLL thetaAboveCell0001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLL thetaAboveCell0001)) h
+theorem e24KC2ThetaAboveLeaf000102 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell0001)) = true := by
+  have h : ((childHL (childLL thetaAboveCell0001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell0001)) h
+theorem e24KC2ThetaAboveLeaf000103 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell0001)) = true := by
+  have h : ((childHH (childLL thetaAboveCell0001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell0001)) h
+theorem e24KC2ThetaAboveLeaf000110 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0001)) = true := by
+  have h : ((childLL (childLH thetaAboveCell0001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL (childLH thetaAboveCell0001)) h
+theorem e24KC2ThetaAboveLeaf000111 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0001)) = true := by
+  have h : ((childLH (childLH thetaAboveCell0001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH (childLH thetaAboveCell0001)) h
+theorem e24KC2ThetaAboveLeaf000112 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell0001)) = true := by
+  have h : ((childHL (childLH thetaAboveCell0001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell0001)) h
+theorem e24KC2ThetaAboveLeaf000113 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell0001)) = true := by
+  have h : ((childHH (childLH thetaAboveCell0001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell0001)) h
+theorem e24KC2ThetaAboveLeaf0001200 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHL thetaAboveCell0001))) = true := by
+  have h : ((childLL (childLL (childHL thetaAboveCell0001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHL thetaAboveCell0001))) h
+theorem e24KC2ThetaAboveLeaf0001201 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHL thetaAboveCell0001))) = true := by
+  have h : ((childLH (childLL (childHL thetaAboveCell0001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHL thetaAboveCell0001))) h
+theorem e24KC2ThetaAboveLeaf00012020 :
+    adaptiveCoverCheck 11 thetaAboveCell00012020 = true := by
+  have h : (thetaAboveCell00012020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012020 h
+theorem e24KC2ThetaAboveLeaf00012021 :
+    adaptiveCoverCheck 11 thetaAboveCell00012021 = true := by
+  have h : (thetaAboveCell00012021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012021 h
+theorem e24KC2ThetaAboveLeaf000120220 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012022) = true := by
+  have h : ((childLL thetaAboveCell00012022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00012022) h
+theorem e24KC2ThetaAboveLeaf000120221 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012022) = true := by
+  have h : ((childLH thetaAboveCell00012022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00012022) h
+theorem e24KC2ThetaAboveLeaf000120222 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012022) = true := by
+  have h : ((childHL thetaAboveCell00012022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012022) h
+theorem e24KC2ThetaAboveLeaf000120223 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012022) = true := by
+  have h : ((childHH thetaAboveCell00012022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012022) h
+theorem e24KC2ThetaAboveLeaf000120230 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012023) = true := by
+  have h : ((childLL thetaAboveCell00012023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00012023) h
+theorem e24KC2ThetaAboveLeaf000120231 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012023) = true := by
+  have h : ((childLH thetaAboveCell00012023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00012023) h
+theorem e24KC2ThetaAboveLeaf000120232 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012023) = true := by
+  have h : ((childHL thetaAboveCell00012023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012023) h
+theorem e24KC2ThetaAboveLeaf000120233 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012023) = true := by
+  have h : ((childHH thetaAboveCell00012023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012023) h
+theorem e24KC2ThetaAboveLeaf00012030 :
+    adaptiveCoverCheck 11 thetaAboveCell00012030 = true := by
+  have h : (thetaAboveCell00012030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012030 h
+theorem e24KC2ThetaAboveLeaf00012031 :
+    adaptiveCoverCheck 11 thetaAboveCell00012031 = true := by
+  have h : (thetaAboveCell00012031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012031 h
+theorem e24KC2ThetaAboveLeaf000120320 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012032) = true := by
+  have h : ((childLL thetaAboveCell00012032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00012032) h
+theorem e24KC2ThetaAboveLeaf000120321 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012032) = true := by
+  have h : ((childLH thetaAboveCell00012032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00012032) h
+theorem e24KC2ThetaAboveLeaf000120322 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012032) = true := by
+  have h : ((childHL thetaAboveCell00012032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012032) h
+theorem e24KC2ThetaAboveLeaf000120323 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012032) = true := by
+  have h : ((childHH thetaAboveCell00012032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012032) h
+theorem e24KC2ThetaAboveLeaf000120330 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012033) = true := by
+  have h : ((childLL thetaAboveCell00012033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00012033) h
+theorem e24KC2ThetaAboveLeaf000120331 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012033) = true := by
+  have h : ((childLH thetaAboveCell00012033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00012033) h
+theorem e24KC2ThetaAboveLeaf000120332 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012033) = true := by
+  have h : ((childHL thetaAboveCell00012033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012033) h
+theorem e24KC2ThetaAboveLeaf000120333 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012033) = true := by
+  have h : ((childHH thetaAboveCell00012033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012033) h
+theorem e24KC2ThetaAboveLeaf0001210 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHL thetaAboveCell0001))) = true := by
+  have h : ((childLL (childLH (childHL thetaAboveCell0001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHL thetaAboveCell0001))) h
+theorem e24KC2ThetaAboveLeaf0001211 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHL thetaAboveCell0001))) = true := by
+  have h : ((childLH (childLH (childHL thetaAboveCell0001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHL thetaAboveCell0001))) h
+theorem e24KC2ThetaAboveLeaf00012120 :
+    adaptiveCoverCheck 11 thetaAboveCell00012120 = true := by
+  have h : (thetaAboveCell00012120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012120 h
+theorem e24KC2ThetaAboveLeaf00012121 :
+    adaptiveCoverCheck 11 thetaAboveCell00012121 = true := by
+  have h : (thetaAboveCell00012121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012121 h
+theorem e24KC2ThetaAboveLeaf000121220 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012122) = true := by
+  have h : ((childLL thetaAboveCell00012122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00012122) h
+theorem e24KC2ThetaAboveLeaf000121221 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012122) = true := by
+  have h : ((childLH thetaAboveCell00012122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00012122) h
+theorem e24KC2ThetaAboveLeaf000121222 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012122) = true := by
+  have h : ((childHL thetaAboveCell00012122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012122) h
+theorem e24KC2ThetaAboveLeaf000121223 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012122) = true := by
+  have h : ((childHH thetaAboveCell00012122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012122) h
+theorem e24KC2ThetaAboveLeaf000121230 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012123) = true := by
+  have h : ((childLL thetaAboveCell00012123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00012123) h
+theorem e24KC2ThetaAboveLeaf000121231 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012123) = true := by
+  have h : ((childLH thetaAboveCell00012123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00012123) h
+theorem e24KC2ThetaAboveLeaf000121232 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012123) = true := by
+  have h : ((childHL thetaAboveCell00012123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012123) h
+theorem e24KC2ThetaAboveLeaf000121233 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012123) = true := by
+  have h : ((childHH thetaAboveCell00012123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012123) h
+theorem e24KC2ThetaAboveLeaf00012130 :
+    adaptiveCoverCheck 11 thetaAboveCell00012130 = true := by
+  have h : (thetaAboveCell00012130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012130 h
+theorem e24KC2ThetaAboveLeaf00012131 :
+    adaptiveCoverCheck 11 thetaAboveCell00012131 = true := by
+  have h : (thetaAboveCell00012131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012131 h
+theorem e24KC2ThetaAboveLeaf000121320 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012132) = true := by
+  have h : ((childLL thetaAboveCell00012132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00012132) h
+theorem e24KC2ThetaAboveLeaf000121321 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012132) = true := by
+  have h : ((childLH thetaAboveCell00012132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00012132) h
+theorem e24KC2ThetaAboveLeaf000121322 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012132) = true := by
+  have h : ((childHL thetaAboveCell00012132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012132) h
+theorem e24KC2ThetaAboveLeaf000121323 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012132) = true := by
+  have h : ((childHH thetaAboveCell00012132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012132) h
+theorem e24KC2ThetaAboveLeaf000121330 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012133) = true := by
+  have h : ((childLL thetaAboveCell00012133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00012133) h
+theorem e24KC2ThetaAboveLeaf000121331 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012133) = true := by
+  have h : ((childLH thetaAboveCell00012133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00012133) h
+theorem e24KC2ThetaAboveLeaf000121332 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012133) = true := by
+  have h : ((childHL thetaAboveCell00012133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012133) h
+theorem e24KC2ThetaAboveLeaf000121333 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012133) = true := by
+  have h : ((childHH thetaAboveCell00012133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012133) h
+theorem e24KC2ThetaAboveLeaf0001220000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012200)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012200)) h
+theorem e24KC2ThetaAboveLeaf0001220001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012200)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012200)) h
+theorem e24KC2ThetaAboveLeaf0001220010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012200)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012200)) h
+theorem e24KC2ThetaAboveLeaf0001220011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012200)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012200)) h
+theorem e24KC2ThetaAboveLeaf0001220100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012201)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012201)) h
+theorem e24KC2ThetaAboveLeaf0001220101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012201)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012201)) h
+theorem e24KC2ThetaAboveLeaf0001220110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012201)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012201)) h
+theorem e24KC2ThetaAboveLeaf0001220111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012201)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012201)) h
+theorem e24KC2ThetaAboveLeaf0001220200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012202)) h
+theorem e24KC2ThetaAboveLeaf0001220201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012202)) h
+theorem e24KC2ThetaAboveLeaf0001220202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00012202)) h
+theorem e24KC2ThetaAboveLeaf0001220203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00012202)) h
+theorem e24KC2ThetaAboveLeaf0001220210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012202)) h
+theorem e24KC2ThetaAboveLeaf0001220211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012202)) h
+theorem e24KC2ThetaAboveLeaf0001220212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00012202)) h
+theorem e24KC2ThetaAboveLeaf0001220213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00012202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00012202)) h
+theorem e24KC2ThetaAboveLeaf000122022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012202) = true := by
+  have h : ((childHL thetaAboveCell00012202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012202) h
+theorem e24KC2ThetaAboveLeaf000122023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012202) = true := by
+  have h : ((childHH thetaAboveCell00012202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012202) h
+theorem e24KC2ThetaAboveLeaf0001220300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012203)) h
+theorem e24KC2ThetaAboveLeaf0001220301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012203)) h
+theorem e24KC2ThetaAboveLeaf0001220302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00012203)) h
+theorem e24KC2ThetaAboveLeaf0001220303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00012203)) h
+theorem e24KC2ThetaAboveLeaf0001220310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012203)) h
+theorem e24KC2ThetaAboveLeaf0001220311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012203)) h
+theorem e24KC2ThetaAboveLeaf0001220312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00012203)) h
+theorem e24KC2ThetaAboveLeaf0001220313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00012203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00012203)) h
+theorem e24KC2ThetaAboveLeaf000122032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012203) = true := by
+  have h : ((childHL thetaAboveCell00012203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012203) h
+theorem e24KC2ThetaAboveLeaf000122033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012203) = true := by
+  have h : ((childHH thetaAboveCell00012203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012203) h
+theorem e24KC2ThetaAboveLeaf0001221000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012210)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012210)) h
+theorem e24KC2ThetaAboveLeaf0001221001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012210)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012210)) h
+theorem e24KC2ThetaAboveLeaf0001221010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012210)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012210)) h
+theorem e24KC2ThetaAboveLeaf0001221011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012210)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012210)) h
+theorem e24KC2ThetaAboveLeaf0001221100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012211)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012211)) h
+theorem e24KC2ThetaAboveLeaf0001221101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012211)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012211)) h
+theorem e24KC2ThetaAboveLeaf0001221110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012211)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012211)) h
+theorem e24KC2ThetaAboveLeaf0001221111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012211)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012211)) h
+theorem e24KC2ThetaAboveLeaf0001221200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012212)) h
+theorem e24KC2ThetaAboveLeaf0001221201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012212)) h
+theorem e24KC2ThetaAboveLeaf0001221202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00012212)) h
+theorem e24KC2ThetaAboveLeaf0001221203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00012212)) h
+theorem e24KC2ThetaAboveLeaf0001221210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012212)) h
+theorem e24KC2ThetaAboveLeaf0001221211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012212)) h
+theorem e24KC2ThetaAboveLeaf0001221212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00012212)) h
+theorem e24KC2ThetaAboveLeaf0001221213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00012212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00012212)) h
+theorem e24KC2ThetaAboveLeaf000122122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012212) = true := by
+  have h : ((childHL thetaAboveCell00012212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012212) h
+theorem e24KC2ThetaAboveLeaf000122123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012212) = true := by
+  have h : ((childHH thetaAboveCell00012212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012212) h
+theorem e24KC2ThetaAboveLeaf0001221300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012213)) h
+theorem e24KC2ThetaAboveLeaf0001221301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012213)) h
+theorem e24KC2ThetaAboveLeaf0001221302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00012213)) h
+theorem e24KC2ThetaAboveLeaf0001221303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00012213)) h
+theorem e24KC2ThetaAboveLeaf0001221310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012213)) h
+theorem e24KC2ThetaAboveLeaf0001221311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012213)) h
+theorem e24KC2ThetaAboveLeaf0001221312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00012213)) h
+theorem e24KC2ThetaAboveLeaf0001221313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00012213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00012213)) h
+theorem e24KC2ThetaAboveLeaf000122132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012213) = true := by
+  have h : ((childHL thetaAboveCell00012213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012213) h
+theorem e24KC2ThetaAboveLeaf000122133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012213) = true := by
+  have h : ((childHH thetaAboveCell00012213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012213) h
+theorem e24KC2ThetaAboveLeaf00012220 :
+    adaptiveCoverCheck 11 thetaAboveCell00012220 = true := by
+  have h : (thetaAboveCell00012220).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012220 h
+theorem e24KC2ThetaAboveLeaf00012221 :
+    adaptiveCoverCheck 11 thetaAboveCell00012221 = true := by
+  have h : (thetaAboveCell00012221).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012221 h
+theorem e24KC2ThetaAboveLeaf00012222 :
+    adaptiveCoverCheck 11 thetaAboveCell00012222 = true := by
+  have h : (thetaAboveCell00012222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012222 h
+theorem e24KC2ThetaAboveLeaf00012223 :
+    adaptiveCoverCheck 11 thetaAboveCell00012223 = true := by
+  have h : (thetaAboveCell00012223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012223 h
+theorem e24KC2ThetaAboveLeaf00012230 :
+    adaptiveCoverCheck 11 thetaAboveCell00012230 = true := by
+  have h : (thetaAboveCell00012230).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012230 h
+theorem e24KC2ThetaAboveLeaf00012231 :
+    adaptiveCoverCheck 11 thetaAboveCell00012231 = true := by
+  have h : (thetaAboveCell00012231).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012231 h
+theorem e24KC2ThetaAboveLeaf00012232 :
+    adaptiveCoverCheck 11 thetaAboveCell00012232 = true := by
+  have h : (thetaAboveCell00012232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012232 h
+theorem e24KC2ThetaAboveLeaf00012233 :
+    adaptiveCoverCheck 11 thetaAboveCell00012233 = true := by
+  have h : (thetaAboveCell00012233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012233 h
+theorem e24KC2ThetaAboveLeaf0001230000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012300)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012300)) h
+theorem e24KC2ThetaAboveLeaf0001230001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012300)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012300)) h
+theorem e24KC2ThetaAboveLeaf0001230010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012300)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012300)) h
+theorem e24KC2ThetaAboveLeaf0001230011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012300)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012300)) h
+theorem e24KC2ThetaAboveLeaf0001230100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012301)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012301)) h
+theorem e24KC2ThetaAboveLeaf0001230101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012301)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012301)) h
+theorem e24KC2ThetaAboveLeaf0001230110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012301)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012301)) h
+theorem e24KC2ThetaAboveLeaf0001230111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012301)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012301)) h
+theorem e24KC2ThetaAboveLeaf0001230200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012302)) h
+theorem e24KC2ThetaAboveLeaf0001230201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012302)) h
+theorem e24KC2ThetaAboveLeaf0001230202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00012302)) h
+theorem e24KC2ThetaAboveLeaf0001230203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00012302)) h
+theorem e24KC2ThetaAboveLeaf0001230210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012302)) h
+theorem e24KC2ThetaAboveLeaf0001230211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012302)) h
+theorem e24KC2ThetaAboveLeaf0001230212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00012302)) h
+theorem e24KC2ThetaAboveLeaf0001230213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00012302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00012302)) h
+theorem e24KC2ThetaAboveLeaf000123022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012302) = true := by
+  have h : ((childHL thetaAboveCell00012302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012302) h
+theorem e24KC2ThetaAboveLeaf000123023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012302) = true := by
+  have h : ((childHH thetaAboveCell00012302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012302) h
+theorem e24KC2ThetaAboveLeaf0001230300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012303)) h
+theorem e24KC2ThetaAboveLeaf0001230301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012303)) h
+theorem e24KC2ThetaAboveLeaf0001230302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00012303)) h
+theorem e24KC2ThetaAboveLeaf0001230303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00012303)) h
+theorem e24KC2ThetaAboveLeaf0001230310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012303)) h
+theorem e24KC2ThetaAboveLeaf0001230311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012303)) h
+theorem e24KC2ThetaAboveLeaf0001230312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00012303)) h
+theorem e24KC2ThetaAboveLeaf0001230313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00012303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00012303)) h
+theorem e24KC2ThetaAboveLeaf000123032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012303) = true := by
+  have h : ((childHL thetaAboveCell00012303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012303) h
+theorem e24KC2ThetaAboveLeaf000123033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012303) = true := by
+  have h : ((childHH thetaAboveCell00012303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012303) h
+theorem e24KC2ThetaAboveLeaf0001231000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012310)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012310)) h
+theorem e24KC2ThetaAboveLeaf0001231001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012310)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012310)) h
+theorem e24KC2ThetaAboveLeaf0001231003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012310)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00012310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00012310)) h
+theorem e24KC2ThetaAboveLeaf0001231010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012310)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012310)) h
+theorem e24KC2ThetaAboveLeaf0001231011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012310)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012310)) h
+theorem e24KC2ThetaAboveLeaf0001231012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012310)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00012310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00012310)) h
+theorem e24KC2ThetaAboveLeaf0001231013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012310)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00012310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00012310)) h
+theorem e24KC2ThetaAboveLeaf0001231100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012311)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012311)) h
+theorem e24KC2ThetaAboveLeaf0001231101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012311)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012311)) h
+theorem e24KC2ThetaAboveLeaf0001231102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012311)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00012311)) h
+theorem e24KC2ThetaAboveLeaf0001231103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012311)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00012311)) h
+theorem e24KC2ThetaAboveLeaf0001231110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012311)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012311)) h
+theorem e24KC2ThetaAboveLeaf0001231111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012311)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012311)) h
+theorem e24KC2ThetaAboveLeaf0001231112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012311)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00012311)) h
+theorem e24KC2ThetaAboveLeaf0001231113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012311)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00012311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00012311)) h
+theorem e24KC2ThetaAboveLeaf0001231200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012312)) h
+theorem e24KC2ThetaAboveLeaf0001231201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012312)) h
+theorem e24KC2ThetaAboveLeaf0001231202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012312)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00012312)) h
+theorem e24KC2ThetaAboveLeaf0001231203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012312)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00012312)) h
+theorem e24KC2ThetaAboveLeaf0001231210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012312)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012312)) h
+theorem e24KC2ThetaAboveLeaf0001231211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012312)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012312)) h
+theorem e24KC2ThetaAboveLeaf0001231212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012312)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00012312)) h
+theorem e24KC2ThetaAboveLeaf0001231213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012312)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00012312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00012312)) h
+theorem e24KC2ThetaAboveLeaf000123122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012312) = true := by
+  have h : ((childHL thetaAboveCell00012312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012312) h
+theorem e24KC2ThetaAboveLeaf000123123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012312) = true := by
+  have h : ((childHH thetaAboveCell00012312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012312) h
+theorem e24KC2ThetaAboveLeaf0001231300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00012313)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00012313)) h
+theorem e24KC2ThetaAboveLeaf0001231301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00012313)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00012313)) h
+theorem e24KC2ThetaAboveLeaf0001231302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012313)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00012313)) h
+theorem e24KC2ThetaAboveLeaf0001231303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012313)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00012313)) h
+theorem e24KC2ThetaAboveLeaf0001231310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00012313)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00012313)) h
+theorem e24KC2ThetaAboveLeaf0001231311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00012313)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00012313)) h
+theorem e24KC2ThetaAboveLeaf0001231312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012313)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00012313)) h
+theorem e24KC2ThetaAboveLeaf0001231313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012313)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00012313))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00012313)) h
+theorem e24KC2ThetaAboveLeaf000123132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012313) = true := by
+  have h : ((childHL thetaAboveCell00012313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00012313) h
+theorem e24KC2ThetaAboveLeaf000123133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012313) = true := by
+  have h : ((childHH thetaAboveCell00012313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00012313) h
+theorem e24KC2ThetaAboveLeaf00012320 :
+    adaptiveCoverCheck 11 thetaAboveCell00012320 = true := by
+  have h : (thetaAboveCell00012320).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012320 h
+theorem e24KC2ThetaAboveLeaf00012321 :
+    adaptiveCoverCheck 11 thetaAboveCell00012321 = true := by
+  have h : (thetaAboveCell00012321).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012321 h
+theorem e24KC2ThetaAboveLeaf00012322 :
+    adaptiveCoverCheck 11 thetaAboveCell00012322 = true := by
+  have h : (thetaAboveCell00012322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012322 h
+theorem e24KC2ThetaAboveLeaf00012323 :
+    adaptiveCoverCheck 11 thetaAboveCell00012323 = true := by
+  have h : (thetaAboveCell00012323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012323 h
+theorem e24KC2ThetaAboveLeaf00012330 :
+    adaptiveCoverCheck 11 thetaAboveCell00012330 = true := by
+  have h : (thetaAboveCell00012330).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012330 h
+theorem e24KC2ThetaAboveLeaf00012331 :
+    adaptiveCoverCheck 11 thetaAboveCell00012331 = true := by
+  have h : (thetaAboveCell00012331).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012331 h
+theorem e24KC2ThetaAboveLeaf00012332 :
+    adaptiveCoverCheck 11 thetaAboveCell00012332 = true := by
+  have h : (thetaAboveCell00012332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012332 h
+theorem e24KC2ThetaAboveLeaf00012333 :
+    adaptiveCoverCheck 11 thetaAboveCell00012333 = true := by
+  have h : (thetaAboveCell00012333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00012333 h
+theorem e24KC2ThetaAboveLeaf0001300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell0001))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell0001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell0001))) h
+theorem e24KC2ThetaAboveLeaf0001301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell0001))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell0001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell0001))) h
+theorem e24KC2ThetaAboveLeaf00013020 :
+    adaptiveCoverCheck 11 thetaAboveCell00013020 = true := by
+  have h : (thetaAboveCell00013020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013020 h
+theorem e24KC2ThetaAboveLeaf00013021 :
+    adaptiveCoverCheck 11 thetaAboveCell00013021 = true := by
+  have h : (thetaAboveCell00013021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013021 h
+theorem e24KC2ThetaAboveLeaf000130220 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013022) = true := by
+  have h : ((childLL thetaAboveCell00013022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00013022) h
+theorem e24KC2ThetaAboveLeaf000130221 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013022) = true := by
+  have h : ((childLH thetaAboveCell00013022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00013022) h
+theorem e24KC2ThetaAboveLeaf000130222 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013022) = true := by
+  have h : ((childHL thetaAboveCell00013022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013022) h
+theorem e24KC2ThetaAboveLeaf000130223 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013022) = true := by
+  have h : ((childHH thetaAboveCell00013022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013022) h
+theorem e24KC2ThetaAboveLeaf000130230 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013023) = true := by
+  have h : ((childLL thetaAboveCell00013023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00013023) h
+theorem e24KC2ThetaAboveLeaf000130231 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013023) = true := by
+  have h : ((childLH thetaAboveCell00013023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00013023) h
+theorem e24KC2ThetaAboveLeaf000130232 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013023) = true := by
+  have h : ((childHL thetaAboveCell00013023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013023) h
+theorem e24KC2ThetaAboveLeaf000130233 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013023) = true := by
+  have h : ((childHH thetaAboveCell00013023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013023) h
+theorem e24KC2ThetaAboveLeaf00013030 :
+    adaptiveCoverCheck 11 thetaAboveCell00013030 = true := by
+  have h : (thetaAboveCell00013030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013030 h
+theorem e24KC2ThetaAboveLeaf00013031 :
+    adaptiveCoverCheck 11 thetaAboveCell00013031 = true := by
+  have h : (thetaAboveCell00013031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013031 h
+theorem e24KC2ThetaAboveLeaf000130320 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013032) = true := by
+  have h : ((childLL thetaAboveCell00013032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00013032) h
+theorem e24KC2ThetaAboveLeaf000130321 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013032) = true := by
+  have h : ((childLH thetaAboveCell00013032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00013032) h
+theorem e24KC2ThetaAboveLeaf000130322 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013032) = true := by
+  have h : ((childHL thetaAboveCell00013032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013032) h
+theorem e24KC2ThetaAboveLeaf000130323 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013032) = true := by
+  have h : ((childHH thetaAboveCell00013032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013032) h
+theorem e24KC2ThetaAboveLeaf000130330 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013033) = true := by
+  have h : ((childLL thetaAboveCell00013033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00013033) h
+theorem e24KC2ThetaAboveLeaf000130331 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013033) = true := by
+  have h : ((childLH thetaAboveCell00013033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00013033) h
+theorem e24KC2ThetaAboveLeaf000130332 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013033) = true := by
+  have h : ((childHL thetaAboveCell00013033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013033) h
+theorem e24KC2ThetaAboveLeaf000130333 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013033) = true := by
+  have h : ((childHH thetaAboveCell00013033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013033) h
+theorem e24KC2ThetaAboveLeaf0001310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell0001))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell0001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell0001))) h
+theorem e24KC2ThetaAboveLeaf0001311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell0001))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell0001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell0001))) h
+theorem e24KC2ThetaAboveLeaf00013120 :
+    adaptiveCoverCheck 11 thetaAboveCell00013120 = true := by
+  have h : (thetaAboveCell00013120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013120 h
+theorem e24KC2ThetaAboveLeaf00013121 :
+    adaptiveCoverCheck 11 thetaAboveCell00013121 = true := by
+  have h : (thetaAboveCell00013121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013121 h
+theorem e24KC2ThetaAboveLeaf000131220 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013122) = true := by
+  have h : ((childLL thetaAboveCell00013122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00013122) h
+theorem e24KC2ThetaAboveLeaf000131221 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013122) = true := by
+  have h : ((childLH thetaAboveCell00013122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00013122) h
+theorem e24KC2ThetaAboveLeaf000131222 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013122) = true := by
+  have h : ((childHL thetaAboveCell00013122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013122) h
+theorem e24KC2ThetaAboveLeaf000131223 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013122) = true := by
+  have h : ((childHH thetaAboveCell00013122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013122) h
+theorem e24KC2ThetaAboveLeaf000131230 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013123) = true := by
+  have h : ((childLL thetaAboveCell00013123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00013123) h
+theorem e24KC2ThetaAboveLeaf000131231 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013123) = true := by
+  have h : ((childLH thetaAboveCell00013123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00013123) h
+theorem e24KC2ThetaAboveLeaf000131232 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013123) = true := by
+  have h : ((childHL thetaAboveCell00013123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013123) h
+theorem e24KC2ThetaAboveLeaf000131233 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013123) = true := by
+  have h : ((childHH thetaAboveCell00013123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013123) h
+theorem e24KC2ThetaAboveLeaf00013130 :
+    adaptiveCoverCheck 11 thetaAboveCell00013130 = true := by
+  have h : (thetaAboveCell00013130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013130 h
+theorem e24KC2ThetaAboveLeaf00013131 :
+    adaptiveCoverCheck 11 thetaAboveCell00013131 = true := by
+  have h : (thetaAboveCell00013131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013131 h
+theorem e24KC2ThetaAboveLeaf000131320 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013132) = true := by
+  have h : ((childLL thetaAboveCell00013132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00013132) h
+theorem e24KC2ThetaAboveLeaf000131321 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013132) = true := by
+  have h : ((childLH thetaAboveCell00013132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00013132) h
+theorem e24KC2ThetaAboveLeaf000131322 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013132) = true := by
+  have h : ((childHL thetaAboveCell00013132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013132) h
+theorem e24KC2ThetaAboveLeaf000131323 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013132) = true := by
+  have h : ((childHH thetaAboveCell00013132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013132) h
+theorem e24KC2ThetaAboveLeaf000131330 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013133) = true := by
+  have h : ((childLL thetaAboveCell00013133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell00013133) h
+theorem e24KC2ThetaAboveLeaf000131331 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013133) = true := by
+  have h : ((childLH thetaAboveCell00013133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell00013133) h
+theorem e24KC2ThetaAboveLeaf000131332 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013133) = true := by
+  have h : ((childHL thetaAboveCell00013133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013133) h
+theorem e24KC2ThetaAboveLeaf000131333 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013133) = true := by
+  have h : ((childHH thetaAboveCell00013133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013133) h
+theorem e24KC2ThetaAboveLeaf0001320000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013200)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013200)) h
+theorem e24KC2ThetaAboveLeaf0001320001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013200)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00013200)) h
+theorem e24KC2ThetaAboveLeaf0001320002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013200)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013200)) h
+theorem e24KC2ThetaAboveLeaf0001320003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013200)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013200)) h
+theorem e24KC2ThetaAboveLeaf0001320010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013200)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00013200)) h
+theorem e24KC2ThetaAboveLeaf0001320011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013200)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00013200)) h
+theorem e24KC2ThetaAboveLeaf0001320012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013200)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013200)) h
+theorem e24KC2ThetaAboveLeaf0001320013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013200)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013200))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013200)) h
+theorem e24KC2ThetaAboveLeaf0001320100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013201)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013201)) h
+theorem e24KC2ThetaAboveLeaf0001320101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013201)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00013201)) h
+theorem e24KC2ThetaAboveLeaf0001320102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013201)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013201)) h
+theorem e24KC2ThetaAboveLeaf0001320103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013201)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013201)) h
+theorem e24KC2ThetaAboveLeaf0001320110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013201)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00013201)) h
+theorem e24KC2ThetaAboveLeaf0001320111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013201)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00013201)) h
+theorem e24KC2ThetaAboveLeaf0001320112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013201)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013201)) h
+theorem e24KC2ThetaAboveLeaf0001320113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013201)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013201))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013201)) h
+theorem e24KC2ThetaAboveLeaf0001320200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013202)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013202)) h
+theorem e24KC2ThetaAboveLeaf0001320201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013202)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00013202)) h
+theorem e24KC2ThetaAboveLeaf0001320202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013202)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013202)) h
+theorem e24KC2ThetaAboveLeaf0001320203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013202)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013202)) h
+theorem e24KC2ThetaAboveLeaf0001320210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013202)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00013202)) h
+theorem e24KC2ThetaAboveLeaf0001320211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013202)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00013202)) h
+theorem e24KC2ThetaAboveLeaf0001320212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013202)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013202)) h
+theorem e24KC2ThetaAboveLeaf0001320213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013202)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013202))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013202)) h
+theorem e24KC2ThetaAboveLeaf000132022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013202) = true := by
+  have h : ((childHL thetaAboveCell00013202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013202) h
+theorem e24KC2ThetaAboveLeaf000132023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013202) = true := by
+  have h : ((childHH thetaAboveCell00013202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013202) h
+theorem e24KC2ThetaAboveLeaf0001320300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013203)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013203)) h
+theorem e24KC2ThetaAboveLeaf0001320301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013203)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00013203)) h
+theorem e24KC2ThetaAboveLeaf0001320302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013203)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013203)) h
+theorem e24KC2ThetaAboveLeaf0001320303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013203)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013203)) h
+theorem e24KC2ThetaAboveLeaf0001320310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013203)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00013203)) h
+theorem e24KC2ThetaAboveLeaf0001320311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013203)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00013203)) h
+theorem e24KC2ThetaAboveLeaf0001320312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013203)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013203)) h
+theorem e24KC2ThetaAboveLeaf0001320313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013203)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013203))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013203)) h
+theorem e24KC2ThetaAboveLeaf000132032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013203) = true := by
+  have h : ((childHL thetaAboveCell00013203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013203) h
+theorem e24KC2ThetaAboveLeaf000132033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013203) = true := by
+  have h : ((childHH thetaAboveCell00013203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013203) h
+theorem e24KC2ThetaAboveLeaf0001321000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013210)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013210)) h
+theorem e24KC2ThetaAboveLeaf0001321001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013210)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00013210)) h
+theorem e24KC2ThetaAboveLeaf0001321002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013210)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013210)) h
+theorem e24KC2ThetaAboveLeaf0001321003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013210)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013210)) h
+theorem e24KC2ThetaAboveLeaf0001321010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013210)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00013210)) h
+theorem e24KC2ThetaAboveLeaf0001321011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013210)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00013210)) h
+theorem e24KC2ThetaAboveLeaf0001321012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013210)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013210)) h
+theorem e24KC2ThetaAboveLeaf0001321013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013210)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013210))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013210)) h
+theorem e24KC2ThetaAboveLeaf0001321100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013211)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013211)) h
+theorem e24KC2ThetaAboveLeaf0001321101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013211)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00013211)) h
+theorem e24KC2ThetaAboveLeaf0001321102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013211)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013211)) h
+theorem e24KC2ThetaAboveLeaf0001321103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013211)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013211)) h
+theorem e24KC2ThetaAboveLeaf0001321110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013211)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00013211)) h
+theorem e24KC2ThetaAboveLeaf0001321111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013211)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00013211)) h
+theorem e24KC2ThetaAboveLeaf0001321112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013211)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013211)) h
+theorem e24KC2ThetaAboveLeaf0001321113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013211)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013211))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013211)) h
+theorem e24KC2ThetaAboveLeaf0001321200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013212)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013212)) h
+theorem e24KC2ThetaAboveLeaf0001321201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013212)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00013212)) h
+theorem e24KC2ThetaAboveLeaf0001321202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013212)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013212)) h
+theorem e24KC2ThetaAboveLeaf0001321203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013212)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013212)) h
+theorem e24KC2ThetaAboveLeaf0001321210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013212)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00013212)) h
+theorem e24KC2ThetaAboveLeaf0001321211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013212)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00013212)) h
+theorem e24KC2ThetaAboveLeaf0001321212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013212)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013212)) h
+theorem e24KC2ThetaAboveLeaf0001321213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013212)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013212))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013212)) h
+theorem e24KC2ThetaAboveLeaf000132122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013212) = true := by
+  have h : ((childHL thetaAboveCell00013212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013212) h
+theorem e24KC2ThetaAboveLeaf000132123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013212) = true := by
+  have h : ((childHH thetaAboveCell00013212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013212) h
+theorem e24KC2ThetaAboveLeaf0001321300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013213)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013213)) h
+theorem e24KC2ThetaAboveLeaf0001321301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013213)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00013213)) h
+theorem e24KC2ThetaAboveLeaf0001321302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013213)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013213)) h
+theorem e24KC2ThetaAboveLeaf0001321303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013213)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013213)) h
+theorem e24KC2ThetaAboveLeaf0001321310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013213)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00013213)) h
+theorem e24KC2ThetaAboveLeaf0001321311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013213)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00013213)) h
+theorem e24KC2ThetaAboveLeaf0001321312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013213)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013213)) h
+theorem e24KC2ThetaAboveLeaf0001321313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013213)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013213))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013213)) h
+theorem e24KC2ThetaAboveLeaf000132132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013213) = true := by
+  have h : ((childHL thetaAboveCell00013213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013213) h
+theorem e24KC2ThetaAboveLeaf000132133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013213) = true := by
+  have h : ((childHH thetaAboveCell00013213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013213) h
+theorem e24KC2ThetaAboveLeaf00013220 :
+    adaptiveCoverCheck 11 thetaAboveCell00013220 = true := by
+  have h : (thetaAboveCell00013220).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013220 h
+theorem e24KC2ThetaAboveLeaf00013221 :
+    adaptiveCoverCheck 11 thetaAboveCell00013221 = true := by
+  have h : (thetaAboveCell00013221).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013221 h
+theorem e24KC2ThetaAboveLeaf00013222 :
+    adaptiveCoverCheck 11 thetaAboveCell00013222 = true := by
+  have h : (thetaAboveCell00013222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013222 h
+theorem e24KC2ThetaAboveLeaf00013223 :
+    adaptiveCoverCheck 11 thetaAboveCell00013223 = true := by
+  have h : (thetaAboveCell00013223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013223 h
+theorem e24KC2ThetaAboveLeaf00013230 :
+    adaptiveCoverCheck 11 thetaAboveCell00013230 = true := by
+  have h : (thetaAboveCell00013230).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013230 h
+theorem e24KC2ThetaAboveLeaf00013231 :
+    adaptiveCoverCheck 11 thetaAboveCell00013231 = true := by
+  have h : (thetaAboveCell00013231).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013231 h
+theorem e24KC2ThetaAboveLeaf00013232 :
+    adaptiveCoverCheck 11 thetaAboveCell00013232 = true := by
+  have h : (thetaAboveCell00013232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013232 h
+theorem e24KC2ThetaAboveLeaf00013233 :
+    adaptiveCoverCheck 11 thetaAboveCell00013233 = true := by
+  have h : (thetaAboveCell00013233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell00013233 h
+theorem e24KC2ThetaAboveLeaf0001330000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013300)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013300)) h
+theorem e24KC2ThetaAboveLeaf0001330001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013300)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00013300)) h
+theorem e24KC2ThetaAboveLeaf0001330002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013300)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013300)) h
+theorem e24KC2ThetaAboveLeaf0001330003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013300)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013300)) h
+theorem e24KC2ThetaAboveLeaf0001330010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013300)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00013300)) h
+theorem e24KC2ThetaAboveLeaf0001330011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013300)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00013300)) h
+theorem e24KC2ThetaAboveLeaf0001330012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013300)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013300)) h
+theorem e24KC2ThetaAboveLeaf0001330013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013300)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013300))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013300)) h
+theorem e24KC2ThetaAboveLeaf0001330100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013301)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013301)) h
+theorem e24KC2ThetaAboveLeaf0001330101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013301)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00013301)) h
+theorem e24KC2ThetaAboveLeaf0001330102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013301)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013301)) h
+theorem e24KC2ThetaAboveLeaf0001330103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013301)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013301)) h
+theorem e24KC2ThetaAboveLeaf0001330110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013301)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00013301)) h
+theorem e24KC2ThetaAboveLeaf0001330111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013301)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00013301)) h
+theorem e24KC2ThetaAboveLeaf0001330112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013301)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013301)) h
+theorem e24KC2ThetaAboveLeaf0001330113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013301)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013301))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013301)) h
+theorem e24KC2ThetaAboveLeaf0001330200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013302)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013302)) h
+theorem e24KC2ThetaAboveLeaf0001330201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013302)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00013302)) h
+theorem e24KC2ThetaAboveLeaf0001330202 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013302)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013302)) h
+theorem e24KC2ThetaAboveLeaf0001330203 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013302)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013302)) h
+theorem e24KC2ThetaAboveLeaf0001330210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013302)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00013302)) h
+theorem e24KC2ThetaAboveLeaf0001330211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013302)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00013302)) h
+theorem e24KC2ThetaAboveLeaf0001330212 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013302)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013302)) h
+theorem e24KC2ThetaAboveLeaf0001330213 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013302)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013302))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013302)) h
+theorem e24KC2ThetaAboveLeaf000133022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013302) = true := by
+  have h : ((childHL thetaAboveCell00013302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013302) h
+theorem e24KC2ThetaAboveLeaf000133023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013302) = true := by
+  have h : ((childHH thetaAboveCell00013302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013302) h
+theorem e24KC2ThetaAboveLeaf0001330300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013303)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013303)) h
+theorem e24KC2ThetaAboveLeaf0001330301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013303)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00013303)) h
+theorem e24KC2ThetaAboveLeaf0001330302 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013303)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013303)) h
+theorem e24KC2ThetaAboveLeaf0001330303 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013303)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013303)) h
+theorem e24KC2ThetaAboveLeaf0001330310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013303)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00013303)) h
+theorem e24KC2ThetaAboveLeaf0001330311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013303)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00013303)) h
+theorem e24KC2ThetaAboveLeaf0001330312 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013303)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013303)) h
+theorem e24KC2ThetaAboveLeaf0001330313 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013303)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013303))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013303)) h
+theorem e24KC2ThetaAboveLeaf000133032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013303) = true := by
+  have h : ((childHL thetaAboveCell00013303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell00013303) h
+theorem e24KC2ThetaAboveLeaf000133033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013303) = true := by
+  have h : ((childHH thetaAboveCell00013303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell00013303) h
+theorem e24KC2ThetaAboveLeaf0001331000 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013310)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013310)) h
+theorem e24KC2ThetaAboveLeaf0001331001 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013310)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00013310)) h
+theorem e24KC2ThetaAboveLeaf0001331002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013310)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013310)) h
+theorem e24KC2ThetaAboveLeaf0001331003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013310)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013310)) h
+theorem e24KC2ThetaAboveLeaf0001331010 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013310)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00013310)) h
+theorem e24KC2ThetaAboveLeaf0001331011 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013310)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00013310)) h
+theorem e24KC2ThetaAboveLeaf0001331012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013310)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013310)) h
+theorem e24KC2ThetaAboveLeaf0001331013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013310)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013310))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013310)) h
+theorem e24KC2ThetaAboveLeaf0001331100 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013311)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013311)) h
+theorem e24KC2ThetaAboveLeaf0001331101 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013311)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00013311)) h
+theorem e24KC2ThetaAboveLeaf0001331102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00013311)) = true := by
+  have h : ((childHL (childLL thetaAboveCell00013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLL thetaAboveCell00013311)) h
+theorem e24KC2ThetaAboveLeaf0001331103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00013311)) = true := by
+  have h : ((childHH (childLL thetaAboveCell00013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLL thetaAboveCell00013311)) h
+theorem e24KC2ThetaAboveLeaf0001331110 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013311)) = true := by
+  have h : ((childLL (childLH thetaAboveCell00013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLH thetaAboveCell00013311)) h
+theorem e24KC2ThetaAboveLeaf0001331111 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013311)) = true := by
+  have h : ((childLH (childLH thetaAboveCell00013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLH thetaAboveCell00013311)) h
+theorem e24KC2ThetaAboveLeaf0001331112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00013311)) = true := by
+  have h : ((childHL (childLH thetaAboveCell00013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL (childLH thetaAboveCell00013311)) h
+theorem e24KC2ThetaAboveLeaf0001331113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00013311)) = true := by
+  have h : ((childHH (childLH thetaAboveCell00013311))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH (childLH thetaAboveCell00013311)) h
+theorem e24KC2ThetaAboveLeaf0001331200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00013312)) = true := by
+  have h : ((childLL (childLL thetaAboveCell00013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL (childLL thetaAboveCell00013312)) h
+theorem e24KC2ThetaAboveLeaf0001331201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013312)) = true := by
+  have h : ((childLH (childLL thetaAboveCell00013312))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH (childLL thetaAboveCell00013312)) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle011.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle011.lean
new file mode 100644
index 0000000000..c85ad293eb
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle011.lean
@@ -0,0 +1,9977 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch015`.
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch019`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCells2f4f8ea4fc
+
+/-- Subcell `1111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1112 : AngleCell :=
+  childHL (childLH (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1113 : AngleCell :=
+  childHH (childLH (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1120 : AngleCell :=
+  childLL (childHL (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1121 : AngleCell :=
+  childLH (childHL (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1122 : AngleCell :=
+  childHL (childHL (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1123 : AngleCell :=
+  childHH (childHL (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1130 : AngleCell :=
+  childLL (childHH (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1131 : AngleCell :=
+  childLH (childHH (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1132 : AngleCell :=
+  childHL (childHH (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1133 : AngleCell :=
+  childHH (childHH (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `0000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0001 : AngleCell :=
+  childLH (childLL (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0002 : AngleCell :=
+  childHL (childLL (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0003 : AngleCell :=
+  childHH (childLL (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0010 : AngleCell :=
+  childLL (childLH (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0011 : AngleCell :=
+  childLH (childLH (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0012 : AngleCell :=
+  childHL (childLH (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0013 : AngleCell :=
+  childHH (childLH (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0030 : AngleCell :=
+  childLL (childHH (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0031 : AngleCell :=
+  childLH (childHH (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0032 : AngleCell :=
+  childHL (childHH (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0033 : AngleCell :=
+  childHH (childHH (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0100 : AngleCell :=
+  childLL (childLL (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0101 : AngleCell :=
+  childLH (childLL (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0102 : AngleCell :=
+  childHL (childLL (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0103 : AngleCell :=
+  childHH (childLL (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0110 : AngleCell :=
+  childLL (childLH (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0111 : AngleCell :=
+  childLH (childLH (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0112 : AngleCell :=
+  childHL (childLH (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0113 : AngleCell :=
+  childHH (childLH (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0120 : AngleCell :=
+  childLL (childHL (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0121 : AngleCell :=
+  childLH (childHL (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0122 : AngleCell :=
+  childHL (childHL (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0123 : AngleCell :=
+  childHH (childHL (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0130 : AngleCell :=
+  childLL (childHH (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0131 : AngleCell :=
+  childLH (childHH (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0132 : AngleCell :=
+  childHL (childHH (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0133 : AngleCell :=
+  childHH (childHH (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `1000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1000 : AngleCell :=
+  childLL (childLL (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1001 : AngleCell :=
+  childLH (childLL (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1002 : AngleCell :=
+  childHL (childLL (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1003 : AngleCell :=
+  childHH (childLL (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1010 : AngleCell :=
+  childLL (childLH (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1011 : AngleCell :=
+  childLH (childLH (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1012 : AngleCell :=
+  childHL (childLH (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1013 : AngleCell :=
+  childHH (childLH (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1020 : AngleCell :=
+  childLL (childHL (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1021 : AngleCell :=
+  childLH (childHL (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1022 : AngleCell :=
+  childHL (childHL (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11112331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `10113030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaBelowCell1011)))
+
+/-- Subcell `10113031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaBelowCell1011)))
+
+/-- Subcell `10113032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaBelowCell1011)))
+
+/-- Subcell `10113033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaBelowCell1011)))
+
+/-- Subcell `10113110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell1011)))
+
+/-- Subcell `10113111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell1011)))
+
+/-- Subcell `10113112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaBelowCell1011)))
+
+/-- Subcell `10113113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaBelowCell1011)))
+
+/-- Subcell `10113120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaBelowCell1011)))
+
+/-- Subcell `10113121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaBelowCell1011)))
+
+/-- Subcell `10113122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaBelowCell1011)))
+
+/-- Subcell `10113123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaBelowCell1011)))
+
+/-- Subcell `10113130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaBelowCell1011)))
+
+/-- Subcell `10113131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaBelowCell1011)))
+
+/-- Subcell `10113132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaBelowCell1011)))
+
+/-- Subcell `10113133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell10113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaBelowCell1011)))
+
+end GerverSofa.PartE.CertificateCells2f4f8ea4fc
+
+namespace GerverSofa.PartE.CertificateCells2d2385ed48
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022103100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022110220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022110221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022110222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022110223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022110230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022110231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022110232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022110233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022110320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022110321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022110322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022110323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022110330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022110331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022110332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022110333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022110333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaAboveCell00002211)))
+
+/-- Subcell `000022111220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022111221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022111222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022111223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022111230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022111231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022111232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022111233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022111320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022111321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022111322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022111323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022111330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022111331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022111332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022111333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022111333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaAboveCell00002211)))
+
+/-- Subcell `000022112000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00002211)))
+
+/-- Subcell `000022113000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000022113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00002211)))
+
+/-- Subcell `000023000220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023000221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023000222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023000223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023000230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023000231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023000232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023000233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023000320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023000321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023000322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023000323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023000330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023000331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023000332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023000333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023000333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaAboveCell00002300)))
+
+/-- Subcell `000023001220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaAboveCell00002300)))
+
+/-- Subcell `000023001221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaAboveCell00002300)))
+
+/-- Subcell `000023001222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaAboveCell00002300)))
+
+/-- Subcell `000023001223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaAboveCell00002300)))
+
+/-- Subcell `000023001230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaAboveCell00002300)))
+
+/-- Subcell `000023001231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaAboveCell00002300)))
+
+/-- Subcell `000023001232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaAboveCell00002300)))
+
+/-- Subcell `000023001233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaAboveCell00002300)))
+
+/-- Subcell `000023001320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaAboveCell00002300)))
+
+/-- Subcell `000023001321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaAboveCell00002300)))
+
+/-- Subcell `000023001322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaAboveCell00002300)))
+
+/-- Subcell `000023001323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaAboveCell00002300)))
+
+/-- Subcell `000023001330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaAboveCell00002300)))
+
+/-- Subcell `000023001331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaAboveCell00002300)))
+
+/-- Subcell `000023001332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaAboveCell00002300)))
+
+/-- Subcell `000023001333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000023001333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaAboveCell00002300)))
+
+end GerverSofa.PartE.CertificateCells2d2385ed48
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC5TerminalBatchT614400015`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2f4f8ea4fc
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2f4f8ea4fc
+
+open CertificateCells2f4f8ea4fc
+theorem e24KC2ThetaAboveLeaf111123311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112331) = true := by
+  have h : ((childLH thetaAboveCell11112331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112331) h
+theorem e24KC2ThetaAboveLeaf111123312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112331) = true := by
+  have h : ((childHL thetaAboveCell11112331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112331) h
+theorem e24KC2ThetaAboveLeaf111123313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112331) = true := by
+  have h : ((childHH thetaAboveCell11112331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112331) h
+theorem e24KC2ThetaAboveLeaf111123320 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112332) = true := by
+  have h : ((childLL thetaAboveCell11112332)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112332) h
+theorem e24KC2ThetaAboveLeaf111123321 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112332) = true := by
+  have h : ((childLH thetaAboveCell11112332)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112332) h
+theorem e24KC2ThetaAboveLeaf111123322 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112332) = true := by
+  have h : ((childHL thetaAboveCell11112332)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112332) h
+theorem e24KC2ThetaAboveLeaf111123323 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112332) = true := by
+  have h : ((childHH thetaAboveCell11112332)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112332) h
+theorem e24KC2ThetaAboveLeaf111123330 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11112333) = true := by
+  have h : ((childLL thetaAboveCell11112333)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11112333) h
+theorem e24KC2ThetaAboveLeaf111123331 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11112333) = true := by
+  have h : ((childLH thetaAboveCell11112333)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11112333) h
+theorem e24KC2ThetaAboveLeaf111123332 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11112333) = true := by
+  have h : ((childHL thetaAboveCell11112333)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11112333) h
+theorem e24KC2ThetaAboveLeaf111123333 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11112333) = true := by
+  have h : ((childHH thetaAboveCell11112333)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11112333) h
+theorem e24KC2ThetaAboveLeaf1111300 :
+    adaptiveCoverCheck 12 (childLL (childLL (childHH thetaAboveCell1111))) = true := by
+  have h : ((childLL (childLL (childHH thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childHH thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf1111301 :
+    adaptiveCoverCheck 12 (childLH (childLL (childHH thetaAboveCell1111))) = true := by
+  have h : ((childLH (childLL (childHH thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childHH thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf1111302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell1111))) = true := by
+  have h : ((childHL (childLL (childHH thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childHH thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf1111303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell1111))) = true := by
+  have h : ((childHH (childLL (childHH thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childHH thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf1111310 :
+    adaptiveCoverCheck 12 (childLL (childLH (childHH thetaAboveCell1111))) = true := by
+  have h : ((childLL (childLH (childHH thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childHH thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf1111311 :
+    adaptiveCoverCheck 12 (childLH (childLH (childHH thetaAboveCell1111))) = true := by
+  have h : ((childLH (childLH (childHH thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childHH thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf1111312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell1111))) = true := by
+  have h : ((childHL (childLH (childHH thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childHH thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf1111313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell1111))) = true := by
+  have h : ((childHH (childLH (childHH thetaAboveCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childHH thetaAboveCell1111))) h
+theorem e24KC2ThetaAboveLeaf11113200 :
+    adaptiveCoverCheck 11 thetaAboveCell11113200 = true := by
+  have h : (thetaAboveCell11113200).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11113200 h
+theorem e24KC2ThetaAboveLeaf11113201 :
+    adaptiveCoverCheck 11 thetaAboveCell11113201 = true := by
+  have h : (thetaAboveCell11113201).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11113201 h
+theorem e24KC2ThetaAboveLeaf111132020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113202) = true := by
+  have h : ((childLL thetaAboveCell11113202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113202) h
+theorem e24KC2ThetaAboveLeaf111132021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113202) = true := by
+  have h : ((childLH thetaAboveCell11113202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113202) h
+theorem e24KC2ThetaAboveLeaf111132022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113202) = true := by
+  have h : ((childHL thetaAboveCell11113202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113202) h
+theorem e24KC2ThetaAboveLeaf111132023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113202) = true := by
+  have h : ((childHH thetaAboveCell11113202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113202) h
+theorem e24KC2ThetaAboveLeaf111132030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113203) = true := by
+  have h : ((childLL thetaAboveCell11113203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113203) h
+theorem e24KC2ThetaAboveLeaf111132031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113203) = true := by
+  have h : ((childLH thetaAboveCell11113203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113203) h
+theorem e24KC2ThetaAboveLeaf111132032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113203) = true := by
+  have h : ((childHL thetaAboveCell11113203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113203) h
+theorem e24KC2ThetaAboveLeaf111132033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113203) = true := by
+  have h : ((childHH thetaAboveCell11113203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113203) h
+theorem e24KC2ThetaAboveLeaf11113210 :
+    adaptiveCoverCheck 11 thetaAboveCell11113210 = true := by
+  have h : (thetaAboveCell11113210).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11113210 h
+theorem e24KC2ThetaAboveLeaf11113211 :
+    adaptiveCoverCheck 11 thetaAboveCell11113211 = true := by
+  have h : (thetaAboveCell11113211).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11113211 h
+theorem e24KC2ThetaAboveLeaf111132120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113212) = true := by
+  have h : ((childLL thetaAboveCell11113212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113212) h
+theorem e24KC2ThetaAboveLeaf111132121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113212) = true := by
+  have h : ((childLH thetaAboveCell11113212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113212) h
+theorem e24KC2ThetaAboveLeaf111132122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113212) = true := by
+  have h : ((childHL thetaAboveCell11113212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113212) h
+theorem e24KC2ThetaAboveLeaf111132123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113212) = true := by
+  have h : ((childHH thetaAboveCell11113212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113212) h
+theorem e24KC2ThetaAboveLeaf111132130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113213) = true := by
+  have h : ((childLL thetaAboveCell11113213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113213) h
+theorem e24KC2ThetaAboveLeaf111132131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113213) = true := by
+  have h : ((childLH thetaAboveCell11113213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113213) h
+theorem e24KC2ThetaAboveLeaf111132132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113213) = true := by
+  have h : ((childHL thetaAboveCell11113213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113213) h
+theorem e24KC2ThetaAboveLeaf111132133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113213) = true := by
+  have h : ((childHH thetaAboveCell11113213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113213) h
+theorem e24KC2ThetaAboveLeaf111132200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113220) = true := by
+  have h : ((childLL thetaAboveCell11113220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113220) h
+theorem e24KC2ThetaAboveLeaf111132201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113220) = true := by
+  have h : ((childLH thetaAboveCell11113220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113220) h
+theorem e24KC2ThetaAboveLeaf111132202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113220) = true := by
+  have h : ((childHL thetaAboveCell11113220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113220) h
+theorem e24KC2ThetaAboveLeaf111132203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113220) = true := by
+  have h : ((childHH thetaAboveCell11113220)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113220) h
+theorem e24KC2ThetaAboveLeaf111132210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113221) = true := by
+  have h : ((childLL thetaAboveCell11113221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113221) h
+theorem e24KC2ThetaAboveLeaf111132211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113221) = true := by
+  have h : ((childLH thetaAboveCell11113221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113221) h
+theorem e24KC2ThetaAboveLeaf111132212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113221) = true := by
+  have h : ((childHL thetaAboveCell11113221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113221) h
+theorem e24KC2ThetaAboveLeaf111132213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113221) = true := by
+  have h : ((childHH thetaAboveCell11113221)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113221) h
+theorem e24KC2ThetaAboveLeaf111132220 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113222) = true := by
+  have h : ((childLL thetaAboveCell11113222)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113222) h
+theorem e24KC2ThetaAboveLeaf111132221 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113222) = true := by
+  have h : ((childLH thetaAboveCell11113222)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113222) h
+theorem e24KC2ThetaAboveLeaf111132222 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113222) = true := by
+  have h : ((childHL thetaAboveCell11113222)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113222) h
+theorem e24KC2ThetaAboveLeaf111132223 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113222) = true := by
+  have h : ((childHH thetaAboveCell11113222)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113222) h
+theorem e24KC2ThetaAboveLeaf111132230 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113223) = true := by
+  have h : ((childLL thetaAboveCell11113223)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113223) h
+theorem e24KC2ThetaAboveLeaf111132231 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113223) = true := by
+  have h : ((childLH thetaAboveCell11113223)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113223) h
+theorem e24KC2ThetaAboveLeaf111132232 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113223) = true := by
+  have h : ((childHL thetaAboveCell11113223)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113223) h
+theorem e24KC2ThetaAboveLeaf111132233 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113223) = true := by
+  have h : ((childHH thetaAboveCell11113223)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113223) h
+theorem e24KC2ThetaAboveLeaf111132300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113230) = true := by
+  have h : ((childLL thetaAboveCell11113230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113230) h
+theorem e24KC2ThetaAboveLeaf111132301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113230) = true := by
+  have h : ((childLH thetaAboveCell11113230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113230) h
+theorem e24KC2ThetaAboveLeaf111132302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113230) = true := by
+  have h : ((childHL thetaAboveCell11113230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113230) h
+theorem e24KC2ThetaAboveLeaf111132303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113230) = true := by
+  have h : ((childHH thetaAboveCell11113230)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113230) h
+theorem e24KC2ThetaAboveLeaf111132310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113231) = true := by
+  have h : ((childLL thetaAboveCell11113231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113231) h
+theorem e24KC2ThetaAboveLeaf111132311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113231) = true := by
+  have h : ((childLH thetaAboveCell11113231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113231) h
+theorem e24KC2ThetaAboveLeaf111132312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113231) = true := by
+  have h : ((childHL thetaAboveCell11113231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113231) h
+theorem e24KC2ThetaAboveLeaf111132313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113231) = true := by
+  have h : ((childHH thetaAboveCell11113231)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113231) h
+theorem e24KC2ThetaAboveLeaf111132320 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113232) = true := by
+  have h : ((childLL thetaAboveCell11113232)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113232) h
+theorem e24KC2ThetaAboveLeaf111132321 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113232) = true := by
+  have h : ((childLH thetaAboveCell11113232)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113232) h
+theorem e24KC2ThetaAboveLeaf111132322 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113232) = true := by
+  have h : ((childHL thetaAboveCell11113232)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113232) h
+theorem e24KC2ThetaAboveLeaf111132323 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113232) = true := by
+  have h : ((childHH thetaAboveCell11113232)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113232) h
+theorem e24KC2ThetaAboveLeaf111132330 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113233) = true := by
+  have h : ((childLL thetaAboveCell11113233)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113233) h
+theorem e24KC2ThetaAboveLeaf111132331 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113233) = true := by
+  have h : ((childLH thetaAboveCell11113233)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113233) h
+theorem e24KC2ThetaAboveLeaf111132332 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113233) = true := by
+  have h : ((childHL thetaAboveCell11113233)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113233) h
+theorem e24KC2ThetaAboveLeaf111132333 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113233) = true := by
+  have h : ((childHH thetaAboveCell11113233)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113233) h
+theorem e24KC2ThetaAboveLeaf11113300 :
+    adaptiveCoverCheck 11 thetaAboveCell11113300 = true := by
+  have h : (thetaAboveCell11113300).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11113300 h
+theorem e24KC2ThetaAboveLeaf11113301 :
+    adaptiveCoverCheck 11 thetaAboveCell11113301 = true := by
+  have h : (thetaAboveCell11113301).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11113301 h
+theorem e24KC2ThetaAboveLeaf111133020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113302) = true := by
+  have h : ((childLL thetaAboveCell11113302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113302) h
+theorem e24KC2ThetaAboveLeaf111133021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113302) = true := by
+  have h : ((childLH thetaAboveCell11113302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113302) h
+theorem e24KC2ThetaAboveLeaf111133022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113302) = true := by
+  have h : ((childHL thetaAboveCell11113302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113302) h
+theorem e24KC2ThetaAboveLeaf111133023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113302) = true := by
+  have h : ((childHH thetaAboveCell11113302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113302) h
+theorem e24KC2ThetaAboveLeaf111133030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113303) = true := by
+  have h : ((childLL thetaAboveCell11113303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113303) h
+theorem e24KC2ThetaAboveLeaf111133031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113303) = true := by
+  have h : ((childLH thetaAboveCell11113303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113303) h
+theorem e24KC2ThetaAboveLeaf111133032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113303) = true := by
+  have h : ((childHL thetaAboveCell11113303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113303) h
+theorem e24KC2ThetaAboveLeaf111133033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113303) = true := by
+  have h : ((childHH thetaAboveCell11113303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113303) h
+theorem e24KC2ThetaAboveLeaf11113310 :
+    adaptiveCoverCheck 11 thetaAboveCell11113310 = true := by
+  have h : (thetaAboveCell11113310).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11113310 h
+theorem e24KC2ThetaAboveLeaf11113311 :
+    adaptiveCoverCheck 11 thetaAboveCell11113311 = true := by
+  have h : (thetaAboveCell11113311).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 thetaAboveCell11113311 h
+theorem e24KC2ThetaAboveLeaf111133120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113312) = true := by
+  have h : ((childLL thetaAboveCell11113312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113312) h
+theorem e24KC2ThetaAboveLeaf111133121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113312) = true := by
+  have h : ((childLH thetaAboveCell11113312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113312) h
+theorem e24KC2ThetaAboveLeaf111133122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113312) = true := by
+  have h : ((childHL thetaAboveCell11113312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113312) h
+theorem e24KC2ThetaAboveLeaf111133123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113312) = true := by
+  have h : ((childHH thetaAboveCell11113312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113312) h
+theorem e24KC2ThetaAboveLeaf111133130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113313) = true := by
+  have h : ((childLL thetaAboveCell11113313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113313) h
+theorem e24KC2ThetaAboveLeaf111133131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113313) = true := by
+  have h : ((childLH thetaAboveCell11113313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113313) h
+theorem e24KC2ThetaAboveLeaf111133132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113313) = true := by
+  have h : ((childHL thetaAboveCell11113313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113313) h
+theorem e24KC2ThetaAboveLeaf111133133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113313) = true := by
+  have h : ((childHH thetaAboveCell11113313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113313) h
+theorem e24KC2ThetaAboveLeaf111133200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113320) = true := by
+  have h : ((childLL thetaAboveCell11113320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113320) h
+theorem e24KC2ThetaAboveLeaf111133201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113320) = true := by
+  have h : ((childLH thetaAboveCell11113320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113320) h
+theorem e24KC2ThetaAboveLeaf111133202 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113320) = true := by
+  have h : ((childHL thetaAboveCell11113320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113320) h
+theorem e24KC2ThetaAboveLeaf111133203 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113320) = true := by
+  have h : ((childHH thetaAboveCell11113320)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113320) h
+theorem e24KC2ThetaAboveLeaf111133210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113321) = true := by
+  have h : ((childLL thetaAboveCell11113321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113321) h
+theorem e24KC2ThetaAboveLeaf111133211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113321) = true := by
+  have h : ((childLH thetaAboveCell11113321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113321) h
+theorem e24KC2ThetaAboveLeaf111133212 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113321) = true := by
+  have h : ((childHL thetaAboveCell11113321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113321) h
+theorem e24KC2ThetaAboveLeaf111133213 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113321) = true := by
+  have h : ((childHH thetaAboveCell11113321)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113321) h
+theorem e24KC2ThetaAboveLeaf111133220 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113322) = true := by
+  have h : ((childLL thetaAboveCell11113322)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113322) h
+theorem e24KC2ThetaAboveLeaf111133221 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113322) = true := by
+  have h : ((childLH thetaAboveCell11113322)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113322) h
+theorem e24KC2ThetaAboveLeaf111133222 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113322) = true := by
+  have h : ((childHL thetaAboveCell11113322)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113322) h
+theorem e24KC2ThetaAboveLeaf111133223 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113322) = true := by
+  have h : ((childHH thetaAboveCell11113322)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113322) h
+theorem e24KC2ThetaAboveLeaf111133230 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113323) = true := by
+  have h : ((childLL thetaAboveCell11113323)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113323) h
+theorem e24KC2ThetaAboveLeaf111133231 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113323) = true := by
+  have h : ((childLH thetaAboveCell11113323)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113323) h
+theorem e24KC2ThetaAboveLeaf111133232 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113323) = true := by
+  have h : ((childHL thetaAboveCell11113323)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113323) h
+theorem e24KC2ThetaAboveLeaf111133233 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113323) = true := by
+  have h : ((childHH thetaAboveCell11113323)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113323) h
+theorem e24KC2ThetaAboveLeaf111133300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113330) = true := by
+  have h : ((childLL thetaAboveCell11113330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113330) h
+theorem e24KC2ThetaAboveLeaf111133301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113330) = true := by
+  have h : ((childLH thetaAboveCell11113330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113330) h
+theorem e24KC2ThetaAboveLeaf111133302 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113330) = true := by
+  have h : ((childHL thetaAboveCell11113330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113330) h
+theorem e24KC2ThetaAboveLeaf111133303 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113330) = true := by
+  have h : ((childHH thetaAboveCell11113330)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113330) h
+theorem e24KC2ThetaAboveLeaf111133310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113331) = true := by
+  have h : ((childLL thetaAboveCell11113331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113331) h
+theorem e24KC2ThetaAboveLeaf111133311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113331) = true := by
+  have h : ((childLH thetaAboveCell11113331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113331) h
+theorem e24KC2ThetaAboveLeaf111133312 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113331) = true := by
+  have h : ((childHL thetaAboveCell11113331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113331) h
+theorem e24KC2ThetaAboveLeaf111133313 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113331) = true := by
+  have h : ((childHH thetaAboveCell11113331)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113331) h
+theorem e24KC2ThetaAboveLeaf111133320 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113332) = true := by
+  have h : ((childLL thetaAboveCell11113332)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113332) h
+theorem e24KC2ThetaAboveLeaf111133321 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113332) = true := by
+  have h : ((childLH thetaAboveCell11113332)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113332) h
+theorem e24KC2ThetaAboveLeaf111133322 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113332) = true := by
+  have h : ((childHL thetaAboveCell11113332)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113332) h
+theorem e24KC2ThetaAboveLeaf111133323 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113332) = true := by
+  have h : ((childHH thetaAboveCell11113332)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113332) h
+theorem e24KC2ThetaAboveLeaf111133330 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell11113333) = true := by
+  have h : ((childLL thetaAboveCell11113333)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLL thetaAboveCell11113333) h
+theorem e24KC2ThetaAboveLeaf111133331 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell11113333) = true := by
+  have h : ((childLH thetaAboveCell11113333)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childLH thetaAboveCell11113333) h
+theorem e24KC2ThetaAboveLeaf111133332 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell11113333) = true := by
+  have h : ((childHL thetaAboveCell11113333)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHL thetaAboveCell11113333) h
+theorem e24KC2ThetaAboveLeaf111133333 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell11113333) = true := by
+  have h : ((childHH thetaAboveCell11113333)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 (childHH thetaAboveCell11113333) h
+theorem e24KC2ThetaAboveLeaf1112000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell1112))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf1112001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell1112))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf1112002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell1112))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf1112003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell1112))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf1112010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell1112))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf1112011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell1112))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf1112012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell1112))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf1112013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell1112))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf111202 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1112)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1112))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1112)) h
+theorem e24KC2ThetaAboveLeaf111203 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1112)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1112))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1112)) h
+theorem e24KC2ThetaAboveLeaf1112100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell1112))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf1112101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell1112))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf1112102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell1112))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf1112103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell1112))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf1112110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell1112))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf1112111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell1112))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf1112112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell1112))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf1112113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell1112))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell1112))) h
+theorem e24KC2ThetaAboveLeaf111212 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1112)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1112))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1112)) h
+theorem e24KC2ThetaAboveLeaf111213 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1112)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1112))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1112)) h
+theorem e24KC2ThetaAboveLeaf11122 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1112) = true := by
+  have h : ((childHL thetaAboveCell1112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell1112) h
+theorem e24KC2ThetaAboveLeaf11123 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1112) = true := by
+  have h : ((childHH thetaAboveCell1112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell1112) h
+theorem e24KC2ThetaAboveLeaf1113000 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLL thetaAboveCell1113))) = true := by
+  have h : ((childLL (childLL (childLL thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLL thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf1113001 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLL thetaAboveCell1113))) = true := by
+  have h : ((childLH (childLL (childLL thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLL thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf1113002 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLL thetaAboveCell1113))) = true := by
+  have h : ((childHL (childLL (childLL thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLL thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf1113003 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLL thetaAboveCell1113))) = true := by
+  have h : ((childHH (childLL (childLL thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLL thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf1113010 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLL thetaAboveCell1113))) = true := by
+  have h : ((childLL (childLH (childLL thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLL thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf1113011 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLL thetaAboveCell1113))) = true := by
+  have h : ((childLH (childLH (childLL thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLL thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf1113012 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLL thetaAboveCell1113))) = true := by
+  have h : ((childHL (childLH (childLL thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLL thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf1113013 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLL thetaAboveCell1113))) = true := by
+  have h : ((childHH (childLH (childLL thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLL thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf111302 :
+    adaptiveCoverCheck 13 (childHL (childLL thetaAboveCell1113)) = true := by
+  have h : ((childHL (childLL thetaAboveCell1113))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLL thetaAboveCell1113)) h
+theorem e24KC2ThetaAboveLeaf111303 :
+    adaptiveCoverCheck 13 (childHH (childLL thetaAboveCell1113)) = true := by
+  have h : ((childHH (childLL thetaAboveCell1113))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLL thetaAboveCell1113)) h
+theorem e24KC2ThetaAboveLeaf1113100 :
+    adaptiveCoverCheck 12 (childLL (childLL (childLH thetaAboveCell1113))) = true := by
+  have h : ((childLL (childLL (childLH thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL (childLH thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf1113101 :
+    adaptiveCoverCheck 12 (childLH (childLL (childLH thetaAboveCell1113))) = true := by
+  have h : ((childLH (childLL (childLH thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL (childLH thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf1113102 :
+    adaptiveCoverCheck 12 (childHL (childLL (childLH thetaAboveCell1113))) = true := by
+  have h : ((childHL (childLL (childLH thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL (childLH thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf1113103 :
+    adaptiveCoverCheck 12 (childHH (childLL (childLH thetaAboveCell1113))) = true := by
+  have h : ((childHH (childLL (childLH thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL (childLH thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf1113110 :
+    adaptiveCoverCheck 12 (childLL (childLH (childLH thetaAboveCell1113))) = true := by
+  have h : ((childLL (childLH (childLH thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH (childLH thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf1113111 :
+    adaptiveCoverCheck 12 (childLH (childLH (childLH thetaAboveCell1113))) = true := by
+  have h : ((childLH (childLH (childLH thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH (childLH thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf1113112 :
+    adaptiveCoverCheck 12 (childHL (childLH (childLH thetaAboveCell1113))) = true := by
+  have h : ((childHL (childLH (childLH thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH (childLH thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf1113113 :
+    adaptiveCoverCheck 12 (childHH (childLH (childLH thetaAboveCell1113))) = true := by
+  have h : ((childHH (childLH (childLH thetaAboveCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH (childLH thetaAboveCell1113))) h
+theorem e24KC2ThetaAboveLeaf111312 :
+    adaptiveCoverCheck 13 (childHL (childLH thetaAboveCell1113)) = true := by
+  have h : ((childHL (childLH thetaAboveCell1113))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL (childLH thetaAboveCell1113)) h
+theorem e24KC2ThetaAboveLeaf111313 :
+    adaptiveCoverCheck 13 (childHH (childLH thetaAboveCell1113)) = true := by
+  have h : ((childHH (childLH thetaAboveCell1113))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH (childLH thetaAboveCell1113)) h
+theorem e24KC2ThetaAboveLeaf11132 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1113) = true := by
+  have h : ((childHL thetaAboveCell1113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHL thetaAboveCell1113) h
+theorem e24KC2ThetaAboveLeaf11133 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1113) = true := by
+  have h : ((childHH thetaAboveCell1113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 (childHH thetaAboveCell1113) h
+theorem e24KC2ThetaAboveLeaf1120 :
+    adaptiveCoverCheck 15 thetaAboveCell1120 = true := by
+  have h : (thetaAboveCell1120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1120 h
+theorem e24KC2ThetaAboveLeaf1121 :
+    adaptiveCoverCheck 15 thetaAboveCell1121 = true := by
+  have h : (thetaAboveCell1121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1121 h
+theorem e24KC2ThetaAboveLeaf1122 :
+    adaptiveCoverCheck 15 thetaAboveCell1122 = true := by
+  have h : (thetaAboveCell1122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1122 h
+theorem e24KC2ThetaAboveLeaf1123 :
+    adaptiveCoverCheck 15 thetaAboveCell1123 = true := by
+  have h : (thetaAboveCell1123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1123 h
+theorem e24KC2ThetaAboveLeaf1130 :
+    adaptiveCoverCheck 15 thetaAboveCell1130 = true := by
+  have h : (thetaAboveCell1130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1130 h
+theorem e24KC2ThetaAboveLeaf1131 :
+    adaptiveCoverCheck 15 thetaAboveCell1131 = true := by
+  have h : (thetaAboveCell1131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1131 h
+theorem e24KC2ThetaAboveLeaf1132 :
+    adaptiveCoverCheck 15 thetaAboveCell1132 = true := by
+  have h : (thetaAboveCell1132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1132 h
+theorem e24KC2ThetaAboveLeaf1133 :
+    adaptiveCoverCheck 15 thetaAboveCell1133 = true := by
+  have h : (thetaAboveCell1133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 thetaAboveCell1133 h
+theorem e24KC2ThetaAboveLeaf120 :
+    adaptiveCoverCheck 16 (childLL (childHL (childLH e24ThetaAboveRoot))) = true := by
+  have h : ((childLL (childHL (childLH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLL (childHL (childLH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf121 :
+    adaptiveCoverCheck 16 (childLH (childHL (childLH e24ThetaAboveRoot))) = true := by
+  have h : ((childLH (childHL (childLH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLH (childHL (childLH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf122 :
+    adaptiveCoverCheck 16 (childHL (childHL (childLH e24ThetaAboveRoot))) = true := by
+  have h : ((childHL (childHL (childLH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHL (childHL (childLH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf123 :
+    adaptiveCoverCheck 16 (childHH (childHL (childLH e24ThetaAboveRoot))) = true := by
+  have h : ((childHH (childHL (childLH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHH (childHL (childLH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf130 :
+    adaptiveCoverCheck 16 (childLL (childHH (childLH e24ThetaAboveRoot))) = true := by
+  have h : ((childLL (childHH (childLH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLL (childHH (childLH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf131 :
+    adaptiveCoverCheck 16 (childLH (childHH (childLH e24ThetaAboveRoot))) = true := by
+  have h : ((childLH (childHH (childLH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLH (childHH (childLH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf132 :
+    adaptiveCoverCheck 16 (childHL (childHH (childLH e24ThetaAboveRoot))) = true := by
+  have h : ((childHL (childHH (childLH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHL (childHH (childLH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf133 :
+    adaptiveCoverCheck 16 (childHH (childHH (childLH e24ThetaAboveRoot))) = true := by
+  have h : ((childHH (childHH (childLH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHH (childHH (childLH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf200 :
+    adaptiveCoverCheck 16 (childLL (childLL (childHL e24ThetaAboveRoot))) = true := by
+  have h : ((childLL (childLL (childHL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLL (childLL (childHL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf201 :
+    adaptiveCoverCheck 16 (childLH (childLL (childHL e24ThetaAboveRoot))) = true := by
+  have h : ((childLH (childLL (childHL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLH (childLL (childHL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf202 :
+    adaptiveCoverCheck 16 (childHL (childLL (childHL e24ThetaAboveRoot))) = true := by
+  have h : ((childHL (childLL (childHL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHL (childLL (childHL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf203 :
+    adaptiveCoverCheck 16 (childHH (childLL (childHL e24ThetaAboveRoot))) = true := by
+  have h : ((childHH (childLL (childHL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHH (childLL (childHL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf210 :
+    adaptiveCoverCheck 16 (childLL (childLH (childHL e24ThetaAboveRoot))) = true := by
+  have h : ((childLL (childLH (childHL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLL (childLH (childHL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf211 :
+    adaptiveCoverCheck 16 (childLH (childLH (childHL e24ThetaAboveRoot))) = true := by
+  have h : ((childLH (childLH (childHL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLH (childLH (childHL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf212 :
+    adaptiveCoverCheck 16 (childHL (childLH (childHL e24ThetaAboveRoot))) = true := by
+  have h : ((childHL (childLH (childHL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHL (childLH (childHL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf213 :
+    adaptiveCoverCheck 16 (childHH (childLH (childHL e24ThetaAboveRoot))) = true := by
+  have h : ((childHH (childLH (childHL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHH (childLH (childHL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf220 :
+    adaptiveCoverCheck 16 (childLL (childHL (childHL e24ThetaAboveRoot))) = true := by
+  have h : ((childLL (childHL (childHL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLL (childHL (childHL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf221 :
+    adaptiveCoverCheck 16 (childLH (childHL (childHL e24ThetaAboveRoot))) = true := by
+  have h : ((childLH (childHL (childHL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLH (childHL (childHL e24ThetaAboveRoot))) h
+
+theorem e24KC2ThetaAboveLeaf222 :
+    adaptiveCoverCheck 16 (childHL (childHL (childHL e24ThetaAboveRoot))) = true := by
+  have h : physicallyIrrelevant (childHL (childHL (childHL e24ThetaAboveRoot))) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 16 (childHL (childHL (childHL
+    e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf223 :
+    adaptiveCoverCheck 16 (childHH (childHL (childHL e24ThetaAboveRoot))) = true := by
+  have h : ((childHH (childHL (childHL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHH (childHL (childHL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf230 :
+    adaptiveCoverCheck 16 (childLL (childHH (childHL e24ThetaAboveRoot))) = true := by
+  have h : ((childLL (childHH (childHL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLL (childHH (childHL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf231 :
+    adaptiveCoverCheck 16 (childLH (childHH (childHL e24ThetaAboveRoot))) = true := by
+  have h : ((childLH (childHH (childHL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLH (childHH (childHL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf232 :
+    adaptiveCoverCheck 16 (childHL (childHH (childHL e24ThetaAboveRoot))) = true := by
+  have h : ((childHL (childHH (childHL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHL (childHH (childHL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf233 :
+    adaptiveCoverCheck 16 (childHH (childHH (childHL e24ThetaAboveRoot))) = true := by
+  have h : ((childHH (childHH (childHL e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHH (childHH (childHL e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf300 :
+    adaptiveCoverCheck 16 (childLL (childLL (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childLL (childLL (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLL (childLL (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf301 :
+    adaptiveCoverCheck 16 (childLH (childLL (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childLH (childLL (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLH (childLL (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf302 :
+    adaptiveCoverCheck 16 (childHL (childLL (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childHL (childLL (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHL (childLL (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf303 :
+    adaptiveCoverCheck 16 (childHH (childLL (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childHH (childLL (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHH (childLL (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf310 :
+    adaptiveCoverCheck 16 (childLL (childLH (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childLL (childLH (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLL (childLH (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf311 :
+    adaptiveCoverCheck 16 (childLH (childLH (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childLH (childLH (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLH (childLH (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf312 :
+    adaptiveCoverCheck 16 (childHL (childLH (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childHL (childLH (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHL (childLH (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf313 :
+    adaptiveCoverCheck 16 (childHH (childLH (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childHH (childLH (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHH (childLH (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf320 :
+    adaptiveCoverCheck 16 (childLL (childHL (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childLL (childHL (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLL (childHL (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf321 :
+    adaptiveCoverCheck 16 (childLH (childHL (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childLH (childHL (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLH (childHL (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf322 :
+    adaptiveCoverCheck 16 (childHL (childHL (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childHL (childHL (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHL (childHL (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf323 :
+    adaptiveCoverCheck 16 (childHH (childHL (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childHH (childHL (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHH (childHL (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf330 :
+    adaptiveCoverCheck 16 (childLL (childHH (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childLL (childHH (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLL (childHH (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf331 :
+    adaptiveCoverCheck 16 (childLH (childHH (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childLH (childHH (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childLH (childHH (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf332 :
+    adaptiveCoverCheck 16 (childHL (childHH (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childHL (childHH (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHL (childHH (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaAboveLeaf333 :
+    adaptiveCoverCheck 16 (childHH (childHH (childHH e24ThetaAboveRoot))) = true := by
+  have h : ((childHH (childHH (childHH e24ThetaAboveRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 16 (childHH (childHH (childHH e24ThetaAboveRoot))) h
+theorem e24KC2ThetaBelowLeaf0000 :
+    adaptiveCoverCheck 14 thetaBelowCell0000 = true := by
+  have h : (thetaBelowCell0000).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell0000 h
+theorem e24KC2ThetaBelowLeaf0001 :
+    adaptiveCoverCheck 14 thetaBelowCell0001 = true := by
+  have h : (thetaBelowCell0001).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell0001 h
+
+theorem e24KC2ThetaBelowLeaf0002 :
+    adaptiveCoverCheck 14 thetaBelowCell0002 = true := by
+  have h : physicallyIrrelevant thetaBelowCell0002 = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 14 thetaBelowCell0002 h
+theorem e24KC2ThetaBelowLeaf00030 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell0003) = true := by
+  have h : ((childLL thetaBelowCell0003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell0003) h
+theorem e24KC2ThetaBelowLeaf00031 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell0003) = true := by
+  have h : ((childLH thetaBelowCell0003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH thetaBelowCell0003) h
+
+theorem e24KC2ThetaBelowLeaf00032 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell0003) = true := by
+  have h : physicallyIrrelevant (childHL thetaBelowCell0003) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 13 (childHL thetaBelowCell0003) h
+theorem e24KC2ThetaBelowLeaf00033 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell0003) = true := by
+  have h : ((childHH thetaBelowCell0003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell0003) h
+theorem e24KC2ThetaBelowLeaf00100 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell0010) = true := by
+  have h : ((childLL thetaBelowCell0010)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell0010) h
+theorem e24KC2ThetaBelowLeaf00101 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell0010) = true := by
+  have h : ((childLH thetaBelowCell0010)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH thetaBelowCell0010) h
+theorem e24KC2ThetaBelowLeaf00102 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell0010) = true := by
+  have h : ((childHL thetaBelowCell0010)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell0010) h
+theorem e24KC2ThetaBelowLeaf00103 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell0010) = true := by
+  have h : ((childHH thetaBelowCell0010)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell0010) h
+theorem e24KC2ThetaBelowLeaf00110 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell0011) = true := by
+  have h : ((childLL thetaBelowCell0011)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell0011) h
+theorem e24KC2ThetaBelowLeaf00111 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell0011) = true := by
+  have h : ((childLH thetaBelowCell0011)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH thetaBelowCell0011) h
+theorem e24KC2ThetaBelowLeaf00112 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell0011) = true := by
+  have h : ((childHL thetaBelowCell0011)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell0011) h
+theorem e24KC2ThetaBelowLeaf00113 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell0011) = true := by
+  have h : ((childHH thetaBelowCell0011)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell0011) h
+theorem e24KC2ThetaBelowLeaf00120 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell0012) = true := by
+  have h : ((childLL thetaBelowCell0012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell0012) h
+theorem e24KC2ThetaBelowLeaf00121 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell0012) = true := by
+  have h : ((childLH thetaBelowCell0012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH thetaBelowCell0012) h
+theorem e24KC2ThetaBelowLeaf00122 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell0012) = true := by
+  have h : ((childHL thetaBelowCell0012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell0012) h
+theorem e24KC2ThetaBelowLeaf00123 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell0012) = true := by
+  have h : ((childHH thetaBelowCell0012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell0012) h
+theorem e24KC2ThetaBelowLeaf00130 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell0013) = true := by
+  have h : ((childLL thetaBelowCell0013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell0013) h
+theorem e24KC2ThetaBelowLeaf00131 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell0013) = true := by
+  have h : ((childLH thetaBelowCell0013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH thetaBelowCell0013) h
+theorem e24KC2ThetaBelowLeaf00132 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell0013) = true := by
+  have h : ((childHL thetaBelowCell0013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell0013) h
+theorem e24KC2ThetaBelowLeaf00133 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell0013) = true := by
+  have h : ((childHH thetaBelowCell0013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell0013) h
+
+theorem e24KC2ThetaBelowLeaf002 :
+    adaptiveCoverCheck 15 (childHL (childLL (childLL e24ThetaBelowRoot))) = true := by
+  have h : physicallyIrrelevant (childHL (childLL (childLL e24ThetaBelowRoot))) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 15 (childHL (childLL (childLL
+    e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf0030 :
+    adaptiveCoverCheck 14 thetaBelowCell0030 = true := by
+  have h : (thetaBelowCell0030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell0030 h
+theorem e24KC2ThetaBelowLeaf0031 :
+    adaptiveCoverCheck 14 thetaBelowCell0031 = true := by
+  have h : (thetaBelowCell0031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell0031 h
+
+theorem e24KC2ThetaBelowLeaf0032 :
+    adaptiveCoverCheck 14 thetaBelowCell0032 = true := by
+  have h : physicallyIrrelevant thetaBelowCell0032 = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 14 thetaBelowCell0032 h
+theorem e24KC2ThetaBelowLeaf0033 :
+    adaptiveCoverCheck 14 thetaBelowCell0033 = true := by
+  have h : (thetaBelowCell0033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell0033 h
+theorem e24KC2ThetaBelowLeaf01000 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell0100) = true := by
+  have h : ((childLL thetaBelowCell0100)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell0100) h
+theorem e24KC2ThetaBelowLeaf01001 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell0100) = true := by
+  have h : ((childLH thetaBelowCell0100)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH thetaBelowCell0100) h
+theorem e24KC2ThetaBelowLeaf01002 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell0100) = true := by
+  have h : ((childHL thetaBelowCell0100)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell0100) h
+theorem e24KC2ThetaBelowLeaf01003 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell0100) = true := by
+  have h : ((childHH thetaBelowCell0100)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell0100) h
+theorem e24KC2ThetaBelowLeaf01010 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell0101) = true := by
+  have h : ((childLL thetaBelowCell0101)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell0101) h
+theorem e24KC2ThetaBelowLeaf01011 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell0101) = true := by
+  have h : ((childLH thetaBelowCell0101)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH thetaBelowCell0101) h
+theorem e24KC2ThetaBelowLeaf01012 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell0101) = true := by
+  have h : ((childHL thetaBelowCell0101)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell0101) h
+theorem e24KC2ThetaBelowLeaf010130 :
+    adaptiveCoverCheck 12 (childLL (childHH thetaBelowCell0101)) = true := by
+  have h : ((childLL (childHH thetaBelowCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childHH thetaBelowCell0101)) h
+theorem e24KC2ThetaBelowLeaf010131 :
+    adaptiveCoverCheck 12 (childLH (childHH thetaBelowCell0101)) = true := by
+  have h : ((childLH (childHH thetaBelowCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childHH thetaBelowCell0101)) h
+theorem e24KC2ThetaBelowLeaf010132 :
+    adaptiveCoverCheck 12 (childHL (childHH thetaBelowCell0101)) = true := by
+  have h : ((childHL (childHH thetaBelowCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childHH thetaBelowCell0101)) h
+theorem e24KC2ThetaBelowLeaf010133 :
+    adaptiveCoverCheck 12 (childHH (childHH thetaBelowCell0101)) = true := by
+  have h : ((childHH (childHH thetaBelowCell0101))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childHH thetaBelowCell0101)) h
+theorem e24KC2ThetaBelowLeaf01020 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell0102) = true := by
+  have h : ((childLL thetaBelowCell0102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell0102) h
+theorem e24KC2ThetaBelowLeaf01021 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell0102) = true := by
+  have h : ((childLH thetaBelowCell0102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH thetaBelowCell0102) h
+theorem e24KC2ThetaBelowLeaf01022 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell0102) = true := by
+  have h : ((childHL thetaBelowCell0102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell0102) h
+theorem e24KC2ThetaBelowLeaf01023 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell0102) = true := by
+  have h : ((childHH thetaBelowCell0102)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell0102) h
+theorem e24KC2ThetaBelowLeaf01030 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell0103) = true := by
+  have h : ((childLL thetaBelowCell0103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell0103) h
+theorem e24KC2ThetaBelowLeaf01031 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell0103) = true := by
+  have h : ((childLH thetaBelowCell0103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH thetaBelowCell0103) h
+theorem e24KC2ThetaBelowLeaf01032 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell0103) = true := by
+  have h : ((childHL thetaBelowCell0103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell0103) h
+theorem e24KC2ThetaBelowLeaf01033 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell0103) = true := by
+  have h : ((childHH thetaBelowCell0103)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell0103) h
+theorem e24KC2ThetaBelowLeaf01100 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell0110) = true := by
+  have h : ((childLL thetaBelowCell0110)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell0110) h
+theorem e24KC2ThetaBelowLeaf011010 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell0110)) = true := by
+  have h : ((childLL (childLH thetaBelowCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH thetaBelowCell0110)) h
+theorem e24KC2ThetaBelowLeaf011011 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell0110)) = true := by
+  have h : ((childLH (childLH thetaBelowCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH thetaBelowCell0110)) h
+theorem e24KC2ThetaBelowLeaf011012 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell0110)) = true := by
+  have h : ((childHL (childLH thetaBelowCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH thetaBelowCell0110)) h
+theorem e24KC2ThetaBelowLeaf011013 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell0110)) = true := by
+  have h : ((childHH (childLH thetaBelowCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH thetaBelowCell0110)) h
+theorem e24KC2ThetaBelowLeaf011020 :
+    adaptiveCoverCheck 12 (childLL (childHL thetaBelowCell0110)) = true := by
+  have h : ((childLL (childHL thetaBelowCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childHL thetaBelowCell0110)) h
+theorem e24KC2ThetaBelowLeaf011021 :
+    adaptiveCoverCheck 12 (childLH (childHL thetaBelowCell0110)) = true := by
+  have h : ((childLH (childHL thetaBelowCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childHL thetaBelowCell0110)) h
+theorem e24KC2ThetaBelowLeaf011022 :
+    adaptiveCoverCheck 12 (childHL (childHL thetaBelowCell0110)) = true := by
+  have h : ((childHL (childHL thetaBelowCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childHL thetaBelowCell0110)) h
+theorem e24KC2ThetaBelowLeaf011023 :
+    adaptiveCoverCheck 12 (childHH (childHL thetaBelowCell0110)) = true := by
+  have h : ((childHH (childHL thetaBelowCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childHL thetaBelowCell0110)) h
+theorem e24KC2ThetaBelowLeaf011030 :
+    adaptiveCoverCheck 12 (childLL (childHH thetaBelowCell0110)) = true := by
+  have h : ((childLL (childHH thetaBelowCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childHH thetaBelowCell0110)) h
+theorem e24KC2ThetaBelowLeaf011031 :
+    adaptiveCoverCheck 12 (childLH (childHH thetaBelowCell0110)) = true := by
+  have h : ((childLH (childHH thetaBelowCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childHH thetaBelowCell0110)) h
+theorem e24KC2ThetaBelowLeaf011032 :
+    adaptiveCoverCheck 12 (childHL (childHH thetaBelowCell0110)) = true := by
+  have h : ((childHL (childHH thetaBelowCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childHH thetaBelowCell0110)) h
+theorem e24KC2ThetaBelowLeaf011033 :
+    adaptiveCoverCheck 12 (childHH (childHH thetaBelowCell0110)) = true := by
+  have h : ((childHH (childHH thetaBelowCell0110))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childHH thetaBelowCell0110)) h
+theorem e24KC2ThetaBelowLeaf011100 :
+    adaptiveCoverCheck 12 (childLL (childLL thetaBelowCell0111)) = true := by
+  have h : ((childLL (childLL thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf011101 :
+    adaptiveCoverCheck 12 (childLH (childLL thetaBelowCell0111)) = true := by
+  have h : ((childLH (childLL thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf011102 :
+    adaptiveCoverCheck 12 (childHL (childLL thetaBelowCell0111)) = true := by
+  have h : ((childHL (childLL thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf011103 :
+    adaptiveCoverCheck 12 (childHH (childLL thetaBelowCell0111)) = true := by
+  have h : ((childHH (childLL thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf011110 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell0111)) = true := by
+  have h : ((childLL (childLH thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf011111 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell0111)) = true := by
+  have h : ((childLH (childLH thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf011112 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell0111)) = true := by
+  have h : ((childHL (childLH thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf011113 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell0111)) = true := by
+  have h : ((childHH (childLH thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf011120 :
+    adaptiveCoverCheck 12 (childLL (childHL thetaBelowCell0111)) = true := by
+  have h : ((childLL (childHL thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childHL thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf011121 :
+    adaptiveCoverCheck 12 (childLH (childHL thetaBelowCell0111)) = true := by
+  have h : ((childLH (childHL thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childHL thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf011122 :
+    adaptiveCoverCheck 12 (childHL (childHL thetaBelowCell0111)) = true := by
+  have h : ((childHL (childHL thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childHL thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf011123 :
+    adaptiveCoverCheck 12 (childHH (childHL thetaBelowCell0111)) = true := by
+  have h : ((childHH (childHL thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childHL thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf011130 :
+    adaptiveCoverCheck 12 (childLL (childHH thetaBelowCell0111)) = true := by
+  have h : ((childLL (childHH thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childHH thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf011131 :
+    adaptiveCoverCheck 12 (childLH (childHH thetaBelowCell0111)) = true := by
+  have h : ((childLH (childHH thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childHH thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf011132 :
+    adaptiveCoverCheck 12 (childHL (childHH thetaBelowCell0111)) = true := by
+  have h : ((childHL (childHH thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childHH thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf011133 :
+    adaptiveCoverCheck 12 (childHH (childHH thetaBelowCell0111)) = true := by
+  have h : ((childHH (childHH thetaBelowCell0111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childHH thetaBelowCell0111)) h
+theorem e24KC2ThetaBelowLeaf01120 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell0112) = true := by
+  have h : ((childLL thetaBelowCell0112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell0112) h
+theorem e24KC2ThetaBelowLeaf01121 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell0112) = true := by
+  have h : ((childLH thetaBelowCell0112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH thetaBelowCell0112) h
+theorem e24KC2ThetaBelowLeaf01122 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell0112) = true := by
+  have h : ((childHL thetaBelowCell0112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell0112) h
+theorem e24KC2ThetaBelowLeaf01123 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell0112) = true := by
+  have h : ((childHH thetaBelowCell0112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell0112) h
+theorem e24KC2ThetaBelowLeaf01130 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell0113) = true := by
+  have h : ((childLL thetaBelowCell0113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell0113) h
+theorem e24KC2ThetaBelowLeaf01131 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell0113) = true := by
+  have h : ((childLH thetaBelowCell0113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH thetaBelowCell0113) h
+theorem e24KC2ThetaBelowLeaf01132 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell0113) = true := by
+  have h : ((childHL thetaBelowCell0113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell0113) h
+theorem e24KC2ThetaBelowLeaf01133 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell0113) = true := by
+  have h : ((childHH thetaBelowCell0113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell0113) h
+theorem e24KC2ThetaBelowLeaf0120 :
+    adaptiveCoverCheck 14 thetaBelowCell0120 = true := by
+  have h : (thetaBelowCell0120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell0120 h
+theorem e24KC2ThetaBelowLeaf0121 :
+    adaptiveCoverCheck 14 thetaBelowCell0121 = true := by
+  have h : (thetaBelowCell0121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell0121 h
+theorem e24KC2ThetaBelowLeaf0122 :
+    adaptiveCoverCheck 14 thetaBelowCell0122 = true := by
+  have h : (thetaBelowCell0122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell0122 h
+theorem e24KC2ThetaBelowLeaf0123 :
+    adaptiveCoverCheck 14 thetaBelowCell0123 = true := by
+  have h : (thetaBelowCell0123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell0123 h
+theorem e24KC2ThetaBelowLeaf0130 :
+    adaptiveCoverCheck 14 thetaBelowCell0130 = true := by
+  have h : (thetaBelowCell0130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell0130 h
+theorem e24KC2ThetaBelowLeaf0131 :
+    adaptiveCoverCheck 14 thetaBelowCell0131 = true := by
+  have h : (thetaBelowCell0131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell0131 h
+theorem e24KC2ThetaBelowLeaf0132 :
+    adaptiveCoverCheck 14 thetaBelowCell0132 = true := by
+  have h : (thetaBelowCell0132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell0132 h
+theorem e24KC2ThetaBelowLeaf0133 :
+    adaptiveCoverCheck 14 thetaBelowCell0133 = true := by
+  have h : (thetaBelowCell0133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell0133 h
+
+theorem e24KC2ThetaBelowLeaf02 :
+    adaptiveCoverCheck 16 (childHL (childLL e24ThetaBelowRoot)) = true := by
+  have h : physicallyIrrelevant (childHL (childLL e24ThetaBelowRoot)) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 16 (childHL (childLL e24ThetaBelowRoot)) h
+theorem e24KC2ThetaBelowLeaf030 :
+    adaptiveCoverCheck 15 (childLL (childHH (childLL e24ThetaBelowRoot))) = true := by
+  have h : ((childLL (childHH (childLL e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childLL (childHH (childLL e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf031 :
+    adaptiveCoverCheck 15 (childLH (childHH (childLL e24ThetaBelowRoot))) = true := by
+  have h : ((childLH (childHH (childLL e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childLH (childHH (childLL e24ThetaBelowRoot))) h
+
+theorem e24KC2ThetaBelowLeaf032 :
+    adaptiveCoverCheck 15 (childHL (childHH (childLL e24ThetaBelowRoot))) = true := by
+  have h : physicallyIrrelevant (childHL (childHH (childLL e24ThetaBelowRoot))) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 15 (childHL (childHH (childLL
+    e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf033 :
+    adaptiveCoverCheck 15 (childHH (childHH (childLL e24ThetaBelowRoot))) = true := by
+  have h : ((childHH (childHH (childLL e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childHH (childHH (childLL e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf100000 :
+    adaptiveCoverCheck 12 (childLL (childLL thetaBelowCell1000)) = true := by
+  have h : ((childLL (childLL thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100001 :
+    adaptiveCoverCheck 12 (childLH (childLL thetaBelowCell1000)) = true := by
+  have h : ((childLH (childLL thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100002 :
+    adaptiveCoverCheck 12 (childHL (childLL thetaBelowCell1000)) = true := by
+  have h : ((childHL (childLL thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100003 :
+    adaptiveCoverCheck 12 (childHH (childLL thetaBelowCell1000)) = true := by
+  have h : ((childHH (childLL thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100010 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell1000)) = true := by
+  have h : ((childLL (childLH thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100011 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell1000)) = true := by
+  have h : ((childLH (childLH thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100012 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell1000)) = true := by
+  have h : ((childHL (childLH thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100013 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell1000)) = true := by
+  have h : ((childHH (childLH thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100020 :
+    adaptiveCoverCheck 12 (childLL (childHL thetaBelowCell1000)) = true := by
+  have h : ((childLL (childHL thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childHL thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100021 :
+    adaptiveCoverCheck 12 (childLH (childHL thetaBelowCell1000)) = true := by
+  have h : ((childLH (childHL thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childHL thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100022 :
+    adaptiveCoverCheck 12 (childHL (childHL thetaBelowCell1000)) = true := by
+  have h : ((childHL (childHL thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childHL thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100023 :
+    adaptiveCoverCheck 12 (childHH (childHL thetaBelowCell1000)) = true := by
+  have h : ((childHH (childHL thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childHL thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100030 :
+    adaptiveCoverCheck 12 (childLL (childHH thetaBelowCell1000)) = true := by
+  have h : ((childLL (childHH thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childHH thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100031 :
+    adaptiveCoverCheck 12 (childLH (childHH thetaBelowCell1000)) = true := by
+  have h : ((childLH (childHH thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childHH thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100032 :
+    adaptiveCoverCheck 12 (childHL (childHH thetaBelowCell1000)) = true := by
+  have h : ((childHL (childHH thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childHH thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100033 :
+    adaptiveCoverCheck 12 (childHH (childHH thetaBelowCell1000)) = true := by
+  have h : ((childHH (childHH thetaBelowCell1000))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childHH thetaBelowCell1000)) h
+theorem e24KC2ThetaBelowLeaf100100 :
+    adaptiveCoverCheck 12 (childLL (childLL thetaBelowCell1001)) = true := by
+  have h : ((childLL (childLL thetaBelowCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL thetaBelowCell1001)) h
+theorem e24KC2ThetaBelowLeaf100101 :
+    adaptiveCoverCheck 12 (childLH (childLL thetaBelowCell1001)) = true := by
+  have h : ((childLH (childLL thetaBelowCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL thetaBelowCell1001)) h
+theorem e24KC2ThetaBelowLeaf100102 :
+    adaptiveCoverCheck 12 (childHL (childLL thetaBelowCell1001)) = true := by
+  have h : ((childHL (childLL thetaBelowCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL thetaBelowCell1001)) h
+theorem e24KC2ThetaBelowLeaf100103 :
+    adaptiveCoverCheck 12 (childHH (childLL thetaBelowCell1001)) = true := by
+  have h : ((childHH (childLL thetaBelowCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL thetaBelowCell1001)) h
+theorem e24KC2ThetaBelowLeaf100110 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell1001)) = true := by
+  have h : ((childLL (childLH thetaBelowCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH thetaBelowCell1001)) h
+theorem e24KC2ThetaBelowLeaf100111 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell1001)) = true := by
+  have h : ((childLH (childLH thetaBelowCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH thetaBelowCell1001)) h
+theorem e24KC2ThetaBelowLeaf100112 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell1001)) = true := by
+  have h : ((childHL (childLH thetaBelowCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH thetaBelowCell1001)) h
+theorem e24KC2ThetaBelowLeaf1001130 :
+    adaptiveCoverCheck 11 (childLL (childHH (childLH thetaBelowCell1001))) = true := by
+  have h : ((childLL (childHH (childLH thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childLH thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf1001131 :
+    adaptiveCoverCheck 11 (childLH (childHH (childLH thetaBelowCell1001))) = true := by
+  have h : ((childLH (childHH (childLH thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childLH thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf1001132 :
+    adaptiveCoverCheck 11 (childHL (childHH (childLH thetaBelowCell1001))) = true := by
+  have h : ((childHL (childHH (childLH thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childLH thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf1001133 :
+    adaptiveCoverCheck 11 (childHH (childHH (childLH thetaBelowCell1001))) = true := by
+  have h : ((childHH (childHH (childLH thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childLH thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf100120 :
+    adaptiveCoverCheck 12 (childLL (childHL thetaBelowCell1001)) = true := by
+  have h : ((childLL (childHL thetaBelowCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childHL thetaBelowCell1001)) h
+theorem e24KC2ThetaBelowLeaf1001210 :
+    adaptiveCoverCheck 11 (childLL (childLH (childHL thetaBelowCell1001))) = true := by
+  have h : ((childLL (childLH (childHL thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLH (childHL thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf1001211 :
+    adaptiveCoverCheck 11 (childLH (childLH (childHL thetaBelowCell1001))) = true := by
+  have h : ((childLH (childLH (childHL thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLH (childHL thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf1001212 :
+    adaptiveCoverCheck 11 (childHL (childLH (childHL thetaBelowCell1001))) = true := by
+  have h : ((childHL (childLH (childHL thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLH (childHL thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf1001213 :
+    adaptiveCoverCheck 11 (childHH (childLH (childHL thetaBelowCell1001))) = true := by
+  have h : ((childHH (childLH (childHL thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLH (childHL thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf100122 :
+    adaptiveCoverCheck 12 (childHL (childHL thetaBelowCell1001)) = true := by
+  have h : ((childHL (childHL thetaBelowCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childHL thetaBelowCell1001)) h
+theorem e24KC2ThetaBelowLeaf100123 :
+    adaptiveCoverCheck 12 (childHH (childHL thetaBelowCell1001)) = true := by
+  have h : ((childHH (childHL thetaBelowCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childHL thetaBelowCell1001)) h
+theorem e24KC2ThetaBelowLeaf1001300 :
+    adaptiveCoverCheck 11 (childLL (childLL (childHH thetaBelowCell1001))) = true := by
+  have h : ((childLL (childLL (childHH thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLL (childHH thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf1001301 :
+    adaptiveCoverCheck 11 (childLH (childLL (childHH thetaBelowCell1001))) = true := by
+  have h : ((childLH (childLL (childHH thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLL (childHH thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf1001302 :
+    adaptiveCoverCheck 11 (childHL (childLL (childHH thetaBelowCell1001))) = true := by
+  have h : ((childHL (childLL (childHH thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLL (childHH thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf1001303 :
+    adaptiveCoverCheck 11 (childHH (childLL (childHH thetaBelowCell1001))) = true := by
+  have h : ((childHH (childLL (childHH thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLL (childHH thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf1001310 :
+    adaptiveCoverCheck 11 (childLL (childLH (childHH thetaBelowCell1001))) = true := by
+  have h : ((childLL (childLH (childHH thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLH (childHH thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf1001311 :
+    adaptiveCoverCheck 11 (childLH (childLH (childHH thetaBelowCell1001))) = true := by
+  have h : ((childLH (childLH (childHH thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLH (childHH thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf1001312 :
+    adaptiveCoverCheck 11 (childHL (childLH (childHH thetaBelowCell1001))) = true := by
+  have h : ((childHL (childLH (childHH thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLH (childHH thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf1001313 :
+    adaptiveCoverCheck 11 (childHH (childLH (childHH thetaBelowCell1001))) = true := by
+  have h : ((childHH (childLH (childHH thetaBelowCell1001)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLH (childHH thetaBelowCell1001))) h
+theorem e24KC2ThetaBelowLeaf100132 :
+    adaptiveCoverCheck 12 (childHL (childHH thetaBelowCell1001)) = true := by
+  have h : ((childHL (childHH thetaBelowCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childHH thetaBelowCell1001)) h
+theorem e24KC2ThetaBelowLeaf100133 :
+    adaptiveCoverCheck 12 (childHH (childHH thetaBelowCell1001)) = true := by
+  have h : ((childHH (childHH thetaBelowCell1001))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childHH thetaBelowCell1001)) h
+theorem e24KC2ThetaBelowLeaf10020 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1002) = true := by
+  have h : ((childLL thetaBelowCell1002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell1002) h
+theorem e24KC2ThetaBelowLeaf10021 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1002) = true := by
+  have h : ((childLH thetaBelowCell1002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH thetaBelowCell1002) h
+theorem e24KC2ThetaBelowLeaf10022 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1002) = true := by
+  have h : ((childHL thetaBelowCell1002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell1002) h
+theorem e24KC2ThetaBelowLeaf10023 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1002) = true := by
+  have h : ((childHH thetaBelowCell1002)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell1002) h
+theorem e24KC2ThetaBelowLeaf10030 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1003) = true := by
+  have h : ((childLL thetaBelowCell1003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell1003) h
+theorem e24KC2ThetaBelowLeaf100310 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell1003)) = true := by
+  have h : ((childLL (childLH thetaBelowCell1003))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH thetaBelowCell1003)) h
+theorem e24KC2ThetaBelowLeaf100311 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell1003)) = true := by
+  have h : ((childLH (childLH thetaBelowCell1003))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH thetaBelowCell1003)) h
+theorem e24KC2ThetaBelowLeaf100312 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell1003)) = true := by
+  have h : ((childHL (childLH thetaBelowCell1003))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH thetaBelowCell1003)) h
+theorem e24KC2ThetaBelowLeaf100313 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell1003)) = true := by
+  have h : ((childHH (childLH thetaBelowCell1003))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH thetaBelowCell1003)) h
+theorem e24KC2ThetaBelowLeaf10032 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1003) = true := by
+  have h : ((childHL thetaBelowCell1003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell1003) h
+theorem e24KC2ThetaBelowLeaf10033 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1003) = true := by
+  have h : ((childHH thetaBelowCell1003)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell1003) h
+theorem e24KC2ThetaBelowLeaf101000 :
+    adaptiveCoverCheck 12 (childLL (childLL thetaBelowCell1010)) = true := by
+  have h : ((childLL (childLL thetaBelowCell1010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL thetaBelowCell1010)) h
+theorem e24KC2ThetaBelowLeaf1010010 :
+    adaptiveCoverCheck 11 (childLL (childLH (childLL thetaBelowCell1010))) = true := by
+  have h : ((childLL (childLH (childLL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLH (childLL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010011 :
+    adaptiveCoverCheck 11 (childLH (childLH (childLL thetaBelowCell1010))) = true := by
+  have h : ((childLH (childLH (childLL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLH (childLL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010012 :
+    adaptiveCoverCheck 11 (childHL (childLH (childLL thetaBelowCell1010))) = true := by
+  have h : ((childHL (childLH (childLL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLH (childLL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010013 :
+    adaptiveCoverCheck 11 (childHH (childLH (childLL thetaBelowCell1010))) = true := by
+  have h : ((childHH (childLH (childLL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLH (childLL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010020 :
+    adaptiveCoverCheck 11 (childLL (childHL (childLL thetaBelowCell1010))) = true := by
+  have h : ((childLL (childHL (childLL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childLL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010021 :
+    adaptiveCoverCheck 11 (childLH (childHL (childLL thetaBelowCell1010))) = true := by
+  have h : ((childLH (childHL (childLL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childLL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010022 :
+    adaptiveCoverCheck 11 (childHL (childHL (childLL thetaBelowCell1010))) = true := by
+  have h : ((childHL (childHL (childLL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childLL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010023 :
+    adaptiveCoverCheck 11 (childHH (childHL (childLL thetaBelowCell1010))) = true := by
+  have h : ((childHH (childHL (childLL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childLL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010030 :
+    adaptiveCoverCheck 11 (childLL (childHH (childLL thetaBelowCell1010))) = true := by
+  have h : ((childLL (childHH (childLL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childLL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010031 :
+    adaptiveCoverCheck 11 (childLH (childHH (childLL thetaBelowCell1010))) = true := by
+  have h : ((childLH (childHH (childLL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childLL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010032 :
+    adaptiveCoverCheck 11 (childHL (childHH (childLL thetaBelowCell1010))) = true := by
+  have h : ((childHL (childHH (childLL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childLL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010033 :
+    adaptiveCoverCheck 11 (childHH (childHH (childLL thetaBelowCell1010))) = true := by
+  have h : ((childHH (childHH (childLL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childLL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010100 :
+    adaptiveCoverCheck 11 (childLL (childLL (childLH thetaBelowCell1010))) = true := by
+  have h : ((childLL (childLL (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLL (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010101 :
+    adaptiveCoverCheck 11 (childLH (childLL (childLH thetaBelowCell1010))) = true := by
+  have h : ((childLH (childLL (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLL (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010102 :
+    adaptiveCoverCheck 11 (childHL (childLL (childLH thetaBelowCell1010))) = true := by
+  have h : ((childHL (childLL (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLL (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010103 :
+    adaptiveCoverCheck 11 (childHH (childLL (childLH thetaBelowCell1010))) = true := by
+  have h : ((childHH (childLL (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLL (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010110 :
+    adaptiveCoverCheck 11 (childLL (childLH (childLH thetaBelowCell1010))) = true := by
+  have h : ((childLL (childLH (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLH (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010111 :
+    adaptiveCoverCheck 11 (childLH (childLH (childLH thetaBelowCell1010))) = true := by
+  have h : ((childLH (childLH (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLH (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010112 :
+    adaptiveCoverCheck 11 (childHL (childLH (childLH thetaBelowCell1010))) = true := by
+  have h : ((childHL (childLH (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLH (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010113 :
+    adaptiveCoverCheck 11 (childHH (childLH (childLH thetaBelowCell1010))) = true := by
+  have h : ((childHH (childLH (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLH (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010120 :
+    adaptiveCoverCheck 11 (childLL (childHL (childLH thetaBelowCell1010))) = true := by
+  have h : ((childLL (childHL (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010121 :
+    adaptiveCoverCheck 11 (childLH (childHL (childLH thetaBelowCell1010))) = true := by
+  have h : ((childLH (childHL (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010122 :
+    adaptiveCoverCheck 11 (childHL (childHL (childLH thetaBelowCell1010))) = true := by
+  have h : ((childHL (childHL (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010123 :
+    adaptiveCoverCheck 11 (childHH (childHL (childLH thetaBelowCell1010))) = true := by
+  have h : ((childHH (childHL (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010130 :
+    adaptiveCoverCheck 11 (childLL (childHH (childLH thetaBelowCell1010))) = true := by
+  have h : ((childLL (childHH (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010131 :
+    adaptiveCoverCheck 11 (childLH (childHH (childLH thetaBelowCell1010))) = true := by
+  have h : ((childLH (childHH (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010132 :
+    adaptiveCoverCheck 11 (childHL (childHH (childLH thetaBelowCell1010))) = true := by
+  have h : ((childHL (childHH (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010133 :
+    adaptiveCoverCheck 11 (childHH (childHH (childLH thetaBelowCell1010))) = true := by
+  have h : ((childHH (childHH (childLH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childLH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010200 :
+    adaptiveCoverCheck 11 (childLL (childLL (childHL thetaBelowCell1010))) = true := by
+  have h : ((childLL (childLL (childHL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLL (childHL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010201 :
+    adaptiveCoverCheck 11 (childLH (childLL (childHL thetaBelowCell1010))) = true := by
+  have h : ((childLH (childLL (childHL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLL (childHL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010202 :
+    adaptiveCoverCheck 11 (childHL (childLL (childHL thetaBelowCell1010))) = true := by
+  have h : ((childHL (childLL (childHL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLL (childHL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010203 :
+    adaptiveCoverCheck 11 (childHH (childLL (childHL thetaBelowCell1010))) = true := by
+  have h : ((childHH (childLL (childHL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLL (childHL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010210 :
+    adaptiveCoverCheck 11 (childLL (childLH (childHL thetaBelowCell1010))) = true := by
+  have h : ((childLL (childLH (childHL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLH (childHL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010211 :
+    adaptiveCoverCheck 11 (childLH (childLH (childHL thetaBelowCell1010))) = true := by
+  have h : ((childLH (childLH (childHL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLH (childHL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010212 :
+    adaptiveCoverCheck 11 (childHL (childLH (childHL thetaBelowCell1010))) = true := by
+  have h : ((childHL (childLH (childHL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLH (childHL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010213 :
+    adaptiveCoverCheck 11 (childHH (childLH (childHL thetaBelowCell1010))) = true := by
+  have h : ((childHH (childLH (childHL thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLH (childHL thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf101022 :
+    adaptiveCoverCheck 12 (childHL (childHL thetaBelowCell1010)) = true := by
+  have h : ((childHL (childHL thetaBelowCell1010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childHL thetaBelowCell1010)) h
+theorem e24KC2ThetaBelowLeaf101023 :
+    adaptiveCoverCheck 12 (childHH (childHL thetaBelowCell1010)) = true := by
+  have h : ((childHH (childHL thetaBelowCell1010))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childHL thetaBelowCell1010)) h
+theorem e24KC2ThetaBelowLeaf1010300 :
+    adaptiveCoverCheck 11 (childLL (childLL (childHH thetaBelowCell1010))) = true := by
+  have h : ((childLL (childLL (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLL (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010301 :
+    adaptiveCoverCheck 11 (childLH (childLL (childHH thetaBelowCell1010))) = true := by
+  have h : ((childLH (childLL (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLL (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010302 :
+    adaptiveCoverCheck 11 (childHL (childLL (childHH thetaBelowCell1010))) = true := by
+  have h : ((childHL (childLL (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLL (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010303 :
+    adaptiveCoverCheck 11 (childHH (childLL (childHH thetaBelowCell1010))) = true := by
+  have h : ((childHH (childLL (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLL (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010310 :
+    adaptiveCoverCheck 11 (childLL (childLH (childHH thetaBelowCell1010))) = true := by
+  have h : ((childLL (childLH (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLH (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010311 :
+    adaptiveCoverCheck 11 (childLH (childLH (childHH thetaBelowCell1010))) = true := by
+  have h : ((childLH (childLH (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLH (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010312 :
+    adaptiveCoverCheck 11 (childHL (childLH (childHH thetaBelowCell1010))) = true := by
+  have h : ((childHL (childLH (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLH (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010313 :
+    adaptiveCoverCheck 11 (childHH (childLH (childHH thetaBelowCell1010))) = true := by
+  have h : ((childHH (childLH (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLH (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010320 :
+    adaptiveCoverCheck 11 (childLL (childHL (childHH thetaBelowCell1010))) = true := by
+  have h : ((childLL (childHL (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010321 :
+    adaptiveCoverCheck 11 (childLH (childHL (childHH thetaBelowCell1010))) = true := by
+  have h : ((childLH (childHL (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010322 :
+    adaptiveCoverCheck 11 (childHL (childHL (childHH thetaBelowCell1010))) = true := by
+  have h : ((childHL (childHL (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010323 :
+    adaptiveCoverCheck 11 (childHH (childHL (childHH thetaBelowCell1010))) = true := by
+  have h : ((childHH (childHL (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010330 :
+    adaptiveCoverCheck 11 (childLL (childHH (childHH thetaBelowCell1010))) = true := by
+  have h : ((childLL (childHH (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010331 :
+    adaptiveCoverCheck 11 (childLH (childHH (childHH thetaBelowCell1010))) = true := by
+  have h : ((childLH (childHH (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010332 :
+    adaptiveCoverCheck 11 (childHL (childHH (childHH thetaBelowCell1010))) = true := by
+  have h : ((childHL (childHH (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1010333 :
+    adaptiveCoverCheck 11 (childHH (childHH (childHH thetaBelowCell1010))) = true := by
+  have h : ((childHH (childHH (childHH thetaBelowCell1010)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childHH thetaBelowCell1010))) h
+theorem e24KC2ThetaBelowLeaf1011000 :
+    adaptiveCoverCheck 11 (childLL (childLL (childLL thetaBelowCell1011))) = true := by
+  have h : ((childLL (childLL (childLL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLL (childLL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011001 :
+    adaptiveCoverCheck 11 (childLH (childLL (childLL thetaBelowCell1011))) = true := by
+  have h : ((childLH (childLL (childLL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLL (childLL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011002 :
+    adaptiveCoverCheck 11 (childHL (childLL (childLL thetaBelowCell1011))) = true := by
+  have h : ((childHL (childLL (childLL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLL (childLL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011003 :
+    adaptiveCoverCheck 11 (childHH (childLL (childLL thetaBelowCell1011))) = true := by
+  have h : ((childHH (childLL (childLL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLL (childLL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf101101 :
+    adaptiveCoverCheck 12 (childLH (childLL thetaBelowCell1011)) = true := by
+  have h : ((childLH (childLL thetaBelowCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL thetaBelowCell1011)) h
+theorem e24KC2ThetaBelowLeaf1011020 :
+    adaptiveCoverCheck 11 (childLL (childHL (childLL thetaBelowCell1011))) = true := by
+  have h : ((childLL (childHL (childLL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childLL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011021 :
+    adaptiveCoverCheck 11 (childLH (childHL (childLL thetaBelowCell1011))) = true := by
+  have h : ((childLH (childHL (childLL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childLL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011022 :
+    adaptiveCoverCheck 11 (childHL (childHL (childLL thetaBelowCell1011))) = true := by
+  have h : ((childHL (childHL (childLL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childLL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011023 :
+    adaptiveCoverCheck 11 (childHH (childHL (childLL thetaBelowCell1011))) = true := by
+  have h : ((childHH (childHL (childLL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childLL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011030 :
+    adaptiveCoverCheck 11 (childLL (childHH (childLL thetaBelowCell1011))) = true := by
+  have h : ((childLL (childHH (childLL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childLL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011031 :
+    adaptiveCoverCheck 11 (childLH (childHH (childLL thetaBelowCell1011))) = true := by
+  have h : ((childLH (childHH (childLL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childLL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011032 :
+    adaptiveCoverCheck 11 (childHL (childHH (childLL thetaBelowCell1011))) = true := by
+  have h : ((childHL (childHH (childLL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childLL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011033 :
+    adaptiveCoverCheck 11 (childHH (childHH (childLL thetaBelowCell1011))) = true := by
+  have h : ((childHH (childHH (childLL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childLL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf101110 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell1011)) = true := by
+  have h : ((childLL (childLH thetaBelowCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH thetaBelowCell1011)) h
+theorem e24KC2ThetaBelowLeaf101111 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell1011)) = true := by
+  have h : ((childLH (childLH thetaBelowCell1011))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH thetaBelowCell1011)) h
+theorem e24KC2ThetaBelowLeaf1011120 :
+    adaptiveCoverCheck 11 (childLL (childHL (childLH thetaBelowCell1011))) = true := by
+  have h : ((childLL (childHL (childLH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childLH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011121 :
+    adaptiveCoverCheck 11 (childLH (childHL (childLH thetaBelowCell1011))) = true := by
+  have h : ((childLH (childHL (childLH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childLH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011122 :
+    adaptiveCoverCheck 11 (childHL (childHL (childLH thetaBelowCell1011))) = true := by
+  have h : ((childHL (childHL (childLH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childLH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011123 :
+    adaptiveCoverCheck 11 (childHH (childHL (childLH thetaBelowCell1011))) = true := by
+  have h : ((childHH (childHL (childLH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childLH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011130 :
+    adaptiveCoverCheck 11 (childLL (childHH (childLH thetaBelowCell1011))) = true := by
+  have h : ((childLL (childHH (childLH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childLH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011131 :
+    adaptiveCoverCheck 11 (childLH (childHH (childLH thetaBelowCell1011))) = true := by
+  have h : ((childLH (childHH (childLH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childLH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011132 :
+    adaptiveCoverCheck 11 (childHL (childHH (childLH thetaBelowCell1011))) = true := by
+  have h : ((childHL (childHH (childLH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childLH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011133 :
+    adaptiveCoverCheck 11 (childHH (childHH (childLH thetaBelowCell1011))) = true := by
+  have h : ((childHH (childHH (childLH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childLH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011200 :
+    adaptiveCoverCheck 11 (childLL (childLL (childHL thetaBelowCell1011))) = true := by
+  have h : ((childLL (childLL (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLL (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011201 :
+    adaptiveCoverCheck 11 (childLH (childLL (childHL thetaBelowCell1011))) = true := by
+  have h : ((childLH (childLL (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLL (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011202 :
+    adaptiveCoverCheck 11 (childHL (childLL (childHL thetaBelowCell1011))) = true := by
+  have h : ((childHL (childLL (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLL (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011203 :
+    adaptiveCoverCheck 11 (childHH (childLL (childHL thetaBelowCell1011))) = true := by
+  have h : ((childHH (childLL (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLL (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011210 :
+    adaptiveCoverCheck 11 (childLL (childLH (childHL thetaBelowCell1011))) = true := by
+  have h : ((childLL (childLH (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLH (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011211 :
+    adaptiveCoverCheck 11 (childLH (childLH (childHL thetaBelowCell1011))) = true := by
+  have h : ((childLH (childLH (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLH (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011212 :
+    adaptiveCoverCheck 11 (childHL (childLH (childHL thetaBelowCell1011))) = true := by
+  have h : ((childHL (childLH (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLH (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011213 :
+    adaptiveCoverCheck 11 (childHH (childLH (childHL thetaBelowCell1011))) = true := by
+  have h : ((childHH (childLH (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLH (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011220 :
+    adaptiveCoverCheck 11 (childLL (childHL (childHL thetaBelowCell1011))) = true := by
+  have h : ((childLL (childHL (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011221 :
+    adaptiveCoverCheck 11 (childLH (childHL (childHL thetaBelowCell1011))) = true := by
+  have h : ((childLH (childHL (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011222 :
+    adaptiveCoverCheck 11 (childHL (childHL (childHL thetaBelowCell1011))) = true := by
+  have h : ((childHL (childHL (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011223 :
+    adaptiveCoverCheck 11 (childHH (childHL (childHL thetaBelowCell1011))) = true := by
+  have h : ((childHH (childHL (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011230 :
+    adaptiveCoverCheck 11 (childLL (childHH (childHL thetaBelowCell1011))) = true := by
+  have h : ((childLL (childHH (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011231 :
+    adaptiveCoverCheck 11 (childLH (childHH (childHL thetaBelowCell1011))) = true := by
+  have h : ((childLH (childHH (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011232 :
+    adaptiveCoverCheck 11 (childHL (childHH (childHL thetaBelowCell1011))) = true := by
+  have h : ((childHL (childHH (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011233 :
+    adaptiveCoverCheck 11 (childHH (childHH (childHL thetaBelowCell1011))) = true := by
+  have h : ((childHH (childHH (childHL thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childHL thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011300 :
+    adaptiveCoverCheck 11 (childLL (childLL (childHH thetaBelowCell1011))) = true := by
+  have h : ((childLL (childLL (childHH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLL (childHH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011301 :
+    adaptiveCoverCheck 11 (childLH (childLL (childHH thetaBelowCell1011))) = true := by
+  have h : ((childLH (childLL (childHH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLL (childHH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011302 :
+    adaptiveCoverCheck 11 (childHL (childLL (childHH thetaBelowCell1011))) = true := by
+  have h : ((childHL (childLL (childHH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLL (childHH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf10113030 :
+    adaptiveCoverCheck 10 thetaBelowCell10113030 = true := by
+  have h : (thetaBelowCell10113030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113030 h
+theorem e24KC2ThetaBelowLeaf10113031 :
+    adaptiveCoverCheck 10 thetaBelowCell10113031 = true := by
+  have h : (thetaBelowCell10113031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113031 h
+theorem e24KC2ThetaBelowLeaf10113032 :
+    adaptiveCoverCheck 10 thetaBelowCell10113032 = true := by
+  have h : (thetaBelowCell10113032).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113032 h
+theorem e24KC2ThetaBelowLeaf10113033 :
+    adaptiveCoverCheck 10 thetaBelowCell10113033 = true := by
+  have h : (thetaBelowCell10113033).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113033 h
+theorem e24KC2ThetaBelowLeaf1011310 :
+    adaptiveCoverCheck 11 (childLL (childLH (childHH thetaBelowCell1011))) = true := by
+  have h : ((childLL (childLH (childHH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLH (childHH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf10113110 :
+    adaptiveCoverCheck 10 thetaBelowCell10113110 = true := by
+  have h : (thetaBelowCell10113110).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113110 h
+theorem e24KC2ThetaBelowLeaf10113111 :
+    adaptiveCoverCheck 10 thetaBelowCell10113111 = true := by
+  have h : (thetaBelowCell10113111).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113111 h
+theorem e24KC2ThetaBelowLeaf10113112 :
+    adaptiveCoverCheck 10 thetaBelowCell10113112 = true := by
+  have h : (thetaBelowCell10113112).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113112 h
+theorem e24KC2ThetaBelowLeaf10113113 :
+    adaptiveCoverCheck 10 thetaBelowCell10113113 = true := by
+  have h : (thetaBelowCell10113113).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113113 h
+theorem e24KC2ThetaBelowLeaf10113120 :
+    adaptiveCoverCheck 10 thetaBelowCell10113120 = true := by
+  have h : (thetaBelowCell10113120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113120 h
+theorem e24KC2ThetaBelowLeaf10113121 :
+    adaptiveCoverCheck 10 thetaBelowCell10113121 = true := by
+  have h : (thetaBelowCell10113121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113121 h
+theorem e24KC2ThetaBelowLeaf10113122 :
+    adaptiveCoverCheck 10 thetaBelowCell10113122 = true := by
+  have h : (thetaBelowCell10113122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113122 h
+theorem e24KC2ThetaBelowLeaf10113123 :
+    adaptiveCoverCheck 10 thetaBelowCell10113123 = true := by
+  have h : (thetaBelowCell10113123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113123 h
+theorem e24KC2ThetaBelowLeaf10113130 :
+    adaptiveCoverCheck 10 thetaBelowCell10113130 = true := by
+  have h : (thetaBelowCell10113130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113130 h
+theorem e24KC2ThetaBelowLeaf10113131 :
+    adaptiveCoverCheck 10 thetaBelowCell10113131 = true := by
+  have h : (thetaBelowCell10113131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113131 h
+theorem e24KC2ThetaBelowLeaf10113132 :
+    adaptiveCoverCheck 10 thetaBelowCell10113132 = true := by
+  have h : (thetaBelowCell10113132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113132 h
+theorem e24KC2ThetaBelowLeaf10113133 :
+    adaptiveCoverCheck 10 thetaBelowCell10113133 = true := by
+  have h : (thetaBelowCell10113133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell10113133 h
+theorem e24KC2ThetaBelowLeaf1011320 :
+    adaptiveCoverCheck 11 (childLL (childHL (childHH thetaBelowCell1011))) = true := by
+  have h : ((childLL (childHL (childHH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childHH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011321 :
+    adaptiveCoverCheck 11 (childLH (childHL (childHH thetaBelowCell1011))) = true := by
+  have h : ((childLH (childHL (childHH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childHH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011322 :
+    adaptiveCoverCheck 11 (childHL (childHL (childHH thetaBelowCell1011))) = true := by
+  have h : ((childHL (childHL (childHH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childHH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011323 :
+    adaptiveCoverCheck 11 (childHH (childHL (childHH thetaBelowCell1011))) = true := by
+  have h : ((childHH (childHL (childHH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childHH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011330 :
+    adaptiveCoverCheck 11 (childLL (childHH (childHH thetaBelowCell1011))) = true := by
+  have h : ((childLL (childHH (childHH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childHH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011331 :
+    adaptiveCoverCheck 11 (childLH (childHH (childHH thetaBelowCell1011))) = true := by
+  have h : ((childLH (childHH (childHH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childHH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011332 :
+    adaptiveCoverCheck 11 (childHL (childHH (childHH thetaBelowCell1011))) = true := by
+  have h : ((childHL (childHH (childHH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childHH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf1011333 :
+    adaptiveCoverCheck 11 (childHH (childHH (childHH thetaBelowCell1011))) = true := by
+  have h : ((childHH (childHH (childHH thetaBelowCell1011)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childHH thetaBelowCell1011))) h
+theorem e24KC2ThetaBelowLeaf101200 :
+    adaptiveCoverCheck 12 (childLL (childLL thetaBelowCell1012)) = true := by
+  have h : ((childLL (childLL thetaBelowCell1012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL thetaBelowCell1012)) h
+theorem e24KC2ThetaBelowLeaf101201 :
+    adaptiveCoverCheck 12 (childLH (childLL thetaBelowCell1012)) = true := by
+  have h : ((childLH (childLL thetaBelowCell1012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL thetaBelowCell1012)) h
+theorem e24KC2ThetaBelowLeaf101202 :
+    adaptiveCoverCheck 12 (childHL (childLL thetaBelowCell1012)) = true := by
+  have h : ((childHL (childLL thetaBelowCell1012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL thetaBelowCell1012)) h
+theorem e24KC2ThetaBelowLeaf101203 :
+    adaptiveCoverCheck 12 (childHH (childLL thetaBelowCell1012)) = true := by
+  have h : ((childHH (childLL thetaBelowCell1012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL thetaBelowCell1012)) h
+theorem e24KC2ThetaBelowLeaf101210 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell1012)) = true := by
+  have h : ((childLL (childLH thetaBelowCell1012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH thetaBelowCell1012)) h
+theorem e24KC2ThetaBelowLeaf101211 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell1012)) = true := by
+  have h : ((childLH (childLH thetaBelowCell1012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH thetaBelowCell1012)) h
+theorem e24KC2ThetaBelowLeaf101212 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell1012)) = true := by
+  have h : ((childHL (childLH thetaBelowCell1012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH thetaBelowCell1012)) h
+theorem e24KC2ThetaBelowLeaf101213 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell1012)) = true := by
+  have h : ((childHH (childLH thetaBelowCell1012))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH thetaBelowCell1012)) h
+theorem e24KC2ThetaBelowLeaf10122 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1012) = true := by
+  have h : ((childHL thetaBelowCell1012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell1012) h
+theorem e24KC2ThetaBelowLeaf10123 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1012) = true := by
+  have h : ((childHH thetaBelowCell1012)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell1012) h
+theorem e24KC2ThetaBelowLeaf101300 :
+    adaptiveCoverCheck 12 (childLL (childLL thetaBelowCell1013)) = true := by
+  have h : ((childLL (childLL thetaBelowCell1013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL thetaBelowCell1013)) h
+theorem e24KC2ThetaBelowLeaf101301 :
+    adaptiveCoverCheck 12 (childLH (childLL thetaBelowCell1013)) = true := by
+  have h : ((childLH (childLL thetaBelowCell1013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL thetaBelowCell1013)) h
+theorem e24KC2ThetaBelowLeaf101302 :
+    adaptiveCoverCheck 12 (childHL (childLL thetaBelowCell1013)) = true := by
+  have h : ((childHL (childLL thetaBelowCell1013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL thetaBelowCell1013)) h
+theorem e24KC2ThetaBelowLeaf101303 :
+    adaptiveCoverCheck 12 (childHH (childLL thetaBelowCell1013)) = true := by
+  have h : ((childHH (childLL thetaBelowCell1013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL thetaBelowCell1013)) h
+theorem e24KC2ThetaBelowLeaf101310 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell1013)) = true := by
+  have h : ((childLL (childLH thetaBelowCell1013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH thetaBelowCell1013)) h
+theorem e24KC2ThetaBelowLeaf101311 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell1013)) = true := by
+  have h : ((childLH (childLH thetaBelowCell1013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH thetaBelowCell1013)) h
+theorem e24KC2ThetaBelowLeaf101312 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell1013)) = true := by
+  have h : ((childHL (childLH thetaBelowCell1013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH thetaBelowCell1013)) h
+theorem e24KC2ThetaBelowLeaf101313 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell1013)) = true := by
+  have h : ((childHH (childLH thetaBelowCell1013))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH thetaBelowCell1013)) h
+theorem e24KC2ThetaBelowLeaf10132 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1013) = true := by
+  have h : ((childHL thetaBelowCell1013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell1013) h
+theorem e24KC2ThetaBelowLeaf10133 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1013) = true := by
+  have h : ((childHH thetaBelowCell1013)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell1013) h
+theorem e24KC2ThetaBelowLeaf1020 :
+    adaptiveCoverCheck 14 thetaBelowCell1020 = true := by
+  have h : (thetaBelowCell1020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1020 h
+theorem e24KC2ThetaBelowLeaf1021 :
+    adaptiveCoverCheck 14 thetaBelowCell1021 = true := by
+  have h : (thetaBelowCell1021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1021 h
+theorem e24KC2ThetaBelowLeaf1022 :
+    adaptiveCoverCheck 14 thetaBelowCell1022 = true := by
+  have h : (thetaBelowCell1022).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1022 h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6ProofBatch11b2bedf1dcd3516`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2d2385ed48
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2d2385ed48
+
+open CertificateCells2d2385ed48
+theorem cover_subtree_edb64f01a0c6 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022103100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022103100)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022103100))
+        (by
+          have h : ((childLL (childLL (childHL thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+            thetaAboveCell000022103100))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+            thetaAboveCell000022103100))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHL
+            thetaAboveCell000022103100))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHL
+            thetaAboveCell000022103100))) h))
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022103100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022103100)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022103100))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022103100))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022103100))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022103100))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022103100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022103100))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022103100))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022103100))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022103100))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022103100))) h))
+
+theorem cover_subtree_a40a1df9b3a1 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022103100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022103100)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022103100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022103100)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022103100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022103100)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022103100))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022103100))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022103100))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022103100))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022103100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022103100))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022103100))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022103100))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022103100))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022103100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022103100))) h))
+
+theorem cover_subtree_e870c75d402b :
+    adaptiveCoverCheck 7 thetaAboveCell000022103100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103100
+    (by
+      have h : ((childLL thetaAboveCell000022103100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022103100) h)
+    (by
+      have h : ((childLH thetaAboveCell000022103100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022103100) h)
+    cover_subtree_edb64f01a0c6
+    cover_subtree_a40a1df9b3a1
+
+theorem cover_subtree_5efea1d56268 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022103101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022103101)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022103101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022103101)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022103101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022103101)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022103101))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022103101))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022103101))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022103101))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022103101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022103101))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022103101))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022103101))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022103101))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022103101))) h))
+
+theorem cover_subtree_d2aa99fffe31 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022103101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022103101)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022103101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022103101)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022103101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022103101)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022103101))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022103101))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022103101))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022103101))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022103101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022103101))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022103101))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022103101))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022103101))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022103101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022103101))) h))
+
+theorem cover_subtree_0abbf33fdebf :
+    adaptiveCoverCheck 7 thetaAboveCell000022103101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103101
+    (by
+      have h : ((childLL thetaAboveCell000022103101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022103101) h)
+    (by
+      have h : ((childLH thetaAboveCell000022103101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022103101) h)
+    cover_subtree_5efea1d56268
+    cover_subtree_d2aa99fffe31
+
+theorem cover_subtree_1d0945e98034 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022103102) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022103102)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022103102))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022103102))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022103102))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022103102))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022103102))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022103102))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022103102))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022103102))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022103102))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022103102))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022103102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022103102)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022103102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022103102)) h)
+
+theorem cover_subtree_e19e97fa44f4 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022103102) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022103102)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022103102))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022103102))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022103102))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022103102))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022103102))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022103102))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022103102))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022103102))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022103102))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022103102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022103102))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022103102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022103102)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022103102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022103102)) h)
+
+theorem cover_subtree_92cf9ed4da50 :
+    adaptiveCoverCheck 7 thetaAboveCell000022103102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103102
+    cover_subtree_1d0945e98034
+    cover_subtree_e19e97fa44f4
+    (by
+      have h : ((childHL thetaAboveCell000022103102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022103102) h)
+    (by
+      have h : ((childHH thetaAboveCell000022103102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022103102) h)
+
+theorem cover_subtree_637f019f84f2 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022103103) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022103103)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022103103))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022103103))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022103103))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022103103))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022103103))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022103103))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022103103))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022103103))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022103103))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022103103))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022103103))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022103103)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022103103))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022103103)) h)
+
+theorem cover_subtree_2f9a3a39c9a5 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022103103) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022103103)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022103103))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022103103))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022103103))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022103103))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022103103))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022103103))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022103103))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022103103))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022103103))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022103103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022103103))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022103103))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022103103)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022103103))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022103103)) h)
+
+theorem cover_subtree_30350e948f84 :
+    adaptiveCoverCheck 7 thetaAboveCell000022103103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103103
+    cover_subtree_637f019f84f2
+    cover_subtree_2f9a3a39c9a5
+    (by
+      have h : ((childHL thetaAboveCell000022103103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022103103) h)
+    (by
+      have h : ((childHH thetaAboveCell000022103103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022103103) h)
+
+theorem cover_subtree_4a228fe99e26 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00002210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00002210)))
+    cover_subtree_e870c75d402b
+    cover_subtree_0abbf33fdebf
+    cover_subtree_92cf9ed4da50
+    cover_subtree_30350e948f84
+
+theorem cover_subtree_dec56845b136 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022103110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022103110)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022103110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022103110)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022103110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022103110)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022103110))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022103110))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022103110))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022103110))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022103110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022103110))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022103110))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022103110))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022103110))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022103110))) h))
+
+theorem cover_subtree_6de59f8bb098 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022103110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022103110)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022103110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022103110)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022103110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022103110)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022103110))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022103110))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022103110))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022103110))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022103110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022103110))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022103110))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022103110))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022103110))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022103110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022103110))) h))
+
+theorem cover_subtree_ab36c8726f5a :
+    adaptiveCoverCheck 7 thetaAboveCell000022103110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103110
+    (by
+      have h : ((childLL thetaAboveCell000022103110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022103110) h)
+    (by
+      have h : ((childLH thetaAboveCell000022103110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022103110) h)
+    cover_subtree_dec56845b136
+    cover_subtree_6de59f8bb098
+
+theorem cover_subtree_59e28493c6fc :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022103111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022103111)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022103111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022103111)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022103111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022103111)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022103111))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022103111))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022103111))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022103111))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022103111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022103111))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022103111))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022103111))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022103111))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022103111))) h))
+
+theorem cover_subtree_de44712e43a7 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022103111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022103111)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022103111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022103111)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022103111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022103111)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022103111))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022103111))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022103111))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022103111))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022103111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022103111))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022103111))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022103111))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022103111))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022103111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022103111))) h))
+
+theorem cover_subtree_526ec4bd5c8b :
+    adaptiveCoverCheck 7 thetaAboveCell000022103111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103111
+    (by
+      have h : ((childLL thetaAboveCell000022103111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022103111) h)
+    (by
+      have h : ((childLH thetaAboveCell000022103111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022103111) h)
+    cover_subtree_59e28493c6fc
+    cover_subtree_de44712e43a7
+
+theorem cover_subtree_f14f28c14e3c :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022103112) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022103112)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022103112))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022103112))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022103112))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022103112))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022103112))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022103112))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022103112))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022103112))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022103112))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022103112))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022103112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022103112)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022103112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022103112)) h)
+
+theorem cover_subtree_2e18124844df :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022103112) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022103112)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022103112))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022103112))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022103112))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022103112))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022103112))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022103112))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022103112))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022103112))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022103112))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022103112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022103112))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022103112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022103112)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022103112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022103112)) h)
+
+theorem cover_subtree_f0199f1a57d7 :
+    adaptiveCoverCheck 7 thetaAboveCell000022103112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103112
+    cover_subtree_f14f28c14e3c
+    cover_subtree_2e18124844df
+    (by
+      have h : ((childHL thetaAboveCell000022103112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022103112) h)
+    (by
+      have h : ((childHH thetaAboveCell000022103112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022103112) h)
+
+theorem cover_subtree_48a3af012806 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022103113) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022103113)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022103113))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022103113))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022103113))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022103113))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022103113))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022103113))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022103113))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022103113))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022103113))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022103113))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022103113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022103113)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022103113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022103113)) h)
+
+theorem cover_subtree_e9c96ebadfa8 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022103113) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022103113)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022103113))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022103113))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022103113))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022103113))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022103113))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022103113))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022103113))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022103113))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022103113))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022103113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022103113))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022103113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022103113)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022103113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022103113)) h)
+
+theorem cover_subtree_88cc51539fac :
+    adaptiveCoverCheck 7 thetaAboveCell000022103113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103113
+    cover_subtree_48a3af012806
+    cover_subtree_e9c96ebadfa8
+    (by
+      have h : ((childHL thetaAboveCell000022103113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022103113) h)
+    (by
+      have h : ((childHH thetaAboveCell000022103113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022103113) h)
+
+theorem cover_subtree_a1d8aa321d27 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00002210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00002210)))
+    cover_subtree_ab36c8726f5a
+    cover_subtree_526ec4bd5c8b
+    cover_subtree_f0199f1a57d7
+    cover_subtree_88cc51539fac
+
+theorem e24KC2ThetaAboveLeaf0000221031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00002210))
+    cover_subtree_4a228fe99e26
+    cover_subtree_a1d8aa321d27
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022103120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103120 h)
+        (by
+          have h : (thetaAboveCell000022103121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103121 h)
+        (by
+          have h : (thetaAboveCell000022103122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103122 h)
+        (by
+          have h : (thetaAboveCell000022103123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022103130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103130 h)
+        (by
+          have h : (thetaAboveCell000022103131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103131 h)
+        (by
+          have h : (thetaAboveCell000022103132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103132 h)
+        (by
+          have h : (thetaAboveCell000022103133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103133 h))
+theorem e24KC2ThetaAboveLeaf0000221032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00002210))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00002210))) h)
+theorem e24KC2ThetaAboveLeaf0000221033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00002210))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00002210))) h)
+theorem e24KC2ThetaAboveLeaf0000221102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00002211))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00002211))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLL
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022110220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110220 h)
+        (by
+          have h : (thetaAboveCell000022110221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110221 h)
+        (by
+          have h : (thetaAboveCell000022110222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110222 h)
+        (by
+          have h : (thetaAboveCell000022110223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childLL
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022110230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110230 h)
+        (by
+          have h : (thetaAboveCell000022110231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110231 h)
+        (by
+          have h : (thetaAboveCell000022110232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110232 h)
+        (by
+          have h : (thetaAboveCell000022110233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110233 h))
+theorem e24KC2ThetaAboveLeaf0000221103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00002211))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00002211))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childLL
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022110320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110320 h)
+        (by
+          have h : (thetaAboveCell000022110321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110321 h)
+        (by
+          have h : (thetaAboveCell000022110322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110322 h)
+        (by
+          have h : (thetaAboveCell000022110323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childLL
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022110330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110330 h)
+        (by
+          have h : (thetaAboveCell000022110331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110331 h)
+        (by
+          have h : (thetaAboveCell000022110332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110332 h)
+        (by
+          have h : (thetaAboveCell000022110333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022110333 h))
+theorem e24KC2ThetaAboveLeaf0000221112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00002211))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00002211))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLH
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022111220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111220 h)
+        (by
+          have h : (thetaAboveCell000022111221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111221 h)
+        (by
+          have h : (thetaAboveCell000022111222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111222 h)
+        (by
+          have h : (thetaAboveCell000022111223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childLH
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022111230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111230 h)
+        (by
+          have h : (thetaAboveCell000022111231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111231 h)
+        (by
+          have h : (thetaAboveCell000022111232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111232 h)
+        (by
+          have h : (thetaAboveCell000022111233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111233 h))
+theorem e24KC2ThetaAboveLeaf0000221113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00002211))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00002211))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childLH
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022111320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111320 h)
+        (by
+          have h : (thetaAboveCell000022111321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111321 h)
+        (by
+          have h : (thetaAboveCell000022111322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111322 h)
+        (by
+          have h : (thetaAboveCell000022111323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childLH
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022111330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111330 h)
+        (by
+          have h : (thetaAboveCell000022111331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111331 h)
+        (by
+          have h : (thetaAboveCell000022111332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111332 h)
+        (by
+          have h : (thetaAboveCell000022111333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022111333 h))
+theorem cover_subtree_43fd6107c581 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022112000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022112000)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022112000))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022112000)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022112000))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022112000)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022112000))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022112000))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022112000))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022112000))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022112000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022112000))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022112000))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022112000))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022112000))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022112000))) h))
+
+theorem cover_subtree_2f5d0def195d :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022112000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022112000)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022112000))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022112000)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022112000))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022112000)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022112000))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022112000))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022112000))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022112000))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022112000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022112000))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022112000))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022112000))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022112000))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022112000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022112000))) h))
+
+theorem cover_subtree_b847b0feac2d :
+    adaptiveCoverCheck 7 thetaAboveCell000022112000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112000
+    (by
+      have h : ((childLL thetaAboveCell000022112000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022112000) h)
+    (by
+      have h : ((childLH thetaAboveCell000022112000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022112000) h)
+    cover_subtree_43fd6107c581
+    cover_subtree_2f5d0def195d
+
+theorem cover_subtree_4950fac9bab9 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022112001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022112001)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022112001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022112001)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022112001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022112001)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022112001))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022112001))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022112001))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022112001))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022112001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022112001))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022112001))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022112001))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022112001))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022112001))) h))
+
+theorem cover_subtree_0e72524614ed :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022112001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022112001)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022112001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022112001)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022112001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022112001)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022112001))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022112001))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022112001))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022112001))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022112001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022112001))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022112001))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022112001))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022112001))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022112001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022112001))) h))
+
+theorem cover_subtree_b10e2b341d50 :
+    adaptiveCoverCheck 7 thetaAboveCell000022112001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112001
+    (by
+      have h : ((childLL thetaAboveCell000022112001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022112001) h)
+    (by
+      have h : ((childLH thetaAboveCell000022112001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022112001) h)
+    cover_subtree_4950fac9bab9
+    cover_subtree_0e72524614ed
+
+theorem cover_subtree_76b490d87a4e :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022112002) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022112002)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022112002))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022112002))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022112002))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022112002))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022112002))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022112002))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022112002))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022112002))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022112002))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022112002))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022112002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022112002)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022112002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022112002)) h)
+
+theorem cover_subtree_4b9c90d07695 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022112002) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022112002)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022112002))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022112002))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022112002))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022112002))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022112002))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022112002))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022112002))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022112002))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022112002))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022112002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022112002))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022112002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022112002)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022112002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022112002)) h)
+
+theorem cover_subtree_b7dd7162f0b9 :
+    adaptiveCoverCheck 7 thetaAboveCell000022112002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112002
+    cover_subtree_76b490d87a4e
+    cover_subtree_4b9c90d07695
+    (by
+      have h : ((childHL thetaAboveCell000022112002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022112002) h)
+    (by
+      have h : ((childHH thetaAboveCell000022112002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022112002) h)
+
+theorem cover_subtree_de29b5c5a11d :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022112003) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022112003)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022112003))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022112003))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022112003))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022112003))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022112003))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022112003))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022112003))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022112003))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022112003))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022112003))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022112003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022112003)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022112003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022112003)) h)
+
+theorem cover_subtree_31f4d5809705 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022112003) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022112003)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022112003))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022112003))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022112003))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022112003))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022112003))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022112003))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022112003))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022112003))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022112003))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022112003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022112003))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022112003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022112003)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022112003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022112003)) h)
+
+theorem cover_subtree_c9c566e964a9 :
+    adaptiveCoverCheck 7 thetaAboveCell000022112003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112003
+    cover_subtree_de29b5c5a11d
+    cover_subtree_31f4d5809705
+    (by
+      have h : ((childHL thetaAboveCell000022112003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022112003) h)
+    (by
+      have h : ((childHH thetaAboveCell000022112003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022112003) h)
+
+theorem cover_subtree_da67754fe633 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00002211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00002211)))
+    cover_subtree_b847b0feac2d
+    cover_subtree_b10e2b341d50
+    cover_subtree_b7dd7162f0b9
+    cover_subtree_c9c566e964a9
+
+theorem cover_subtree_95f6e6d97c24 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022112010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022112010)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022112010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022112010)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022112010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022112010)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022112010))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022112010))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022112010))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022112010))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022112010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022112010))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022112010))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022112010))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022112010))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022112010))) h))
+
+theorem cover_subtree_9c844972ea1f :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022112010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022112010)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022112010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022112010)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022112010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022112010)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022112010))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022112010))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022112010))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022112010))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022112010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022112010))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022112010))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022112010))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022112010))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022112010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022112010))) h))
+
+theorem cover_subtree_04ff1f51c7c8 :
+    adaptiveCoverCheck 7 thetaAboveCell000022112010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112010
+    (by
+      have h : ((childLL thetaAboveCell000022112010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022112010) h)
+    (by
+      have h : ((childLH thetaAboveCell000022112010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022112010) h)
+    cover_subtree_95f6e6d97c24
+    cover_subtree_9c844972ea1f
+
+theorem cover_subtree_754b681a3e26 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022112011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022112011)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022112011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022112011)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022112011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022112011)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022112011))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022112011))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022112011))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022112011))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022112011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022112011))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022112011))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022112011))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022112011))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022112011))) h))
+
+theorem cover_subtree_497a7c55810a :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022112011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022112011)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022112011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022112011)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022112011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022112011)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022112011))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022112011))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022112011))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022112011))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022112011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022112011))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022112011))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022112011))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022112011))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022112011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022112011))) h))
+
+theorem cover_subtree_92922a2edd2b :
+    adaptiveCoverCheck 7 thetaAboveCell000022112011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112011
+    (by
+      have h : ((childLL thetaAboveCell000022112011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022112011) h)
+    (by
+      have h : ((childLH thetaAboveCell000022112011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022112011) h)
+    cover_subtree_754b681a3e26
+    cover_subtree_497a7c55810a
+
+theorem cover_subtree_bfdd630a1f02 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022112012) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022112012)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022112012))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022112012))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022112012))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022112012))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022112012))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022112012))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022112012))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022112012))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022112012))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022112012))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022112012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022112012)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022112012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022112012)) h)
+
+theorem cover_subtree_d41a2b9225dd :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022112012) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022112012)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022112012))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022112012))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022112012))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022112012))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022112012))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022112012))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022112012))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022112012))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022112012))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022112012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022112012))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022112012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022112012)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022112012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022112012)) h)
+
+theorem cover_subtree_b54fc51b3ec8 :
+    adaptiveCoverCheck 7 thetaAboveCell000022112012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112012
+    cover_subtree_bfdd630a1f02
+    cover_subtree_d41a2b9225dd
+    (by
+      have h : ((childHL thetaAboveCell000022112012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022112012) h)
+    (by
+      have h : ((childHH thetaAboveCell000022112012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022112012) h)
+
+theorem cover_subtree_1ae8b8609603 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022112013) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022112013)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022112013))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022112013))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022112013))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022112013))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022112013))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022112013))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022112013))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022112013))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022112013))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022112013))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022112013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022112013)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022112013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022112013)) h)
+
+theorem cover_subtree_254e7d6a1a8d :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022112013) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022112013)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022112013))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022112013))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022112013))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022112013))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022112013))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022112013))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022112013))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022112013))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022112013))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022112013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022112013))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022112013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022112013)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022112013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022112013)) h)
+
+theorem cover_subtree_316108a366aa :
+    adaptiveCoverCheck 7 thetaAboveCell000022112013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112013
+    cover_subtree_1ae8b8609603
+    cover_subtree_254e7d6a1a8d
+    (by
+      have h : ((childHL thetaAboveCell000022112013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022112013) h)
+    (by
+      have h : ((childHH thetaAboveCell000022112013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022112013) h)
+
+theorem cover_subtree_23a35d53c9e2 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00002211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00002211)))
+    cover_subtree_04ff1f51c7c8
+    cover_subtree_92922a2edd2b
+    cover_subtree_b54fc51b3ec8
+    cover_subtree_316108a366aa
+
+theorem e24KC2ThetaAboveLeaf0000221120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00002211))
+    cover_subtree_da67754fe633
+    cover_subtree_23a35d53c9e2
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022112020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112020 h)
+        (by
+          have h : (thetaAboveCell000022112021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112021 h)
+        (by
+          have h : (thetaAboveCell000022112022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112022 h)
+        (by
+          have h : (thetaAboveCell000022112023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022112030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112030 h)
+        (by
+          have h : (thetaAboveCell000022112031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112031 h)
+        (by
+          have h : (thetaAboveCell000022112032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112032 h)
+        (by
+          have h : (thetaAboveCell000022112033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112033 h))
+theorem cover_subtree_021e44d85b50 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022112100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022112100)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022112100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022112100)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022112100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022112100)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022112100))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022112100))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022112100))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022112100))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022112100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022112100))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022112100))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022112100))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022112100))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022112100))) h))
+
+theorem cover_subtree_62fa6c2e1429 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022112100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022112100)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022112100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022112100)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022112100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022112100)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022112100))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022112100))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022112100))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022112100))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022112100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022112100))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022112100))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022112100))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022112100))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022112100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022112100))) h))
+
+theorem cover_subtree_ff28352fdcbd :
+    adaptiveCoverCheck 7 thetaAboveCell000022112100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112100
+    (by
+      have h : ((childLL thetaAboveCell000022112100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022112100) h)
+    (by
+      have h : ((childLH thetaAboveCell000022112100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022112100) h)
+    cover_subtree_021e44d85b50
+    cover_subtree_62fa6c2e1429
+
+theorem cover_subtree_4284ed2ee8d6 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022112101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022112101)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022112101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022112101)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022112101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022112101)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022112101))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022112101))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022112101))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022112101))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022112101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022112101))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022112101))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022112101))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022112101))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022112101))) h))
+
+theorem cover_subtree_21c449726701 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022112101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022112101)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022112101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022112101)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022112101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022112101)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022112101))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022112101))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022112101))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022112101))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022112101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022112101))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022112101))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022112101))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022112101))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022112101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022112101))) h))
+
+theorem cover_subtree_ecd8fe40a0e6 :
+    adaptiveCoverCheck 7 thetaAboveCell000022112101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112101
+    (by
+      have h : ((childLL thetaAboveCell000022112101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022112101) h)
+    (by
+      have h : ((childLH thetaAboveCell000022112101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022112101) h)
+    cover_subtree_4284ed2ee8d6
+    cover_subtree_21c449726701
+
+theorem cover_subtree_4e862092a132 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022112102) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022112102)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022112102))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022112102))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022112102))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022112102))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022112102))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022112102))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022112102))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022112102))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022112102))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022112102))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022112102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022112102)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022112102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022112102)) h)
+
+theorem cover_subtree_aa7a39051514 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022112102) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022112102)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022112102))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022112102))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022112102))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022112102))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022112102))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022112102))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022112102))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022112102))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022112102))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022112102))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022112102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022112102)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLH thetaAboveCell000022112102))
+        (by
+          have h : ((childLL (childHH (childLH thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLH
+            thetaAboveCell000022112102))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLH
+            thetaAboveCell000022112102))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLH
+            thetaAboveCell000022112102))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaAboveCell000022112102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLH
+            thetaAboveCell000022112102))) h))
+
+theorem cover_subtree_bcfbc1ac2009 :
+    adaptiveCoverCheck 7 thetaAboveCell000022112102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112102
+    cover_subtree_4e862092a132
+    cover_subtree_aa7a39051514
+    (by
+      have h : ((childHL thetaAboveCell000022112102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022112102) h)
+    (by
+      have h : ((childHH thetaAboveCell000022112102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022112102) h)
+
+theorem cover_subtree_340b36315559 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022112103) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022112103)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022112103))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022112103))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022112103))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022112103))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLL thetaAboveCell000022112103))
+        (by
+          have h : ((childLL (childHL (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLL
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLL
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLL
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLL
+            thetaAboveCell000022112103))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLL thetaAboveCell000022112103))
+        (by
+          have h : ((childLL (childHH (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLL
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLL
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLL
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLL
+            thetaAboveCell000022112103))) h))
+
+theorem cover_subtree_3ea1443179e3 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022112103) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022112103)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022112103))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022112103))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022112103))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022112103))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLH thetaAboveCell000022112103))
+        (by
+          have h : ((childLL (childHL (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLH
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLH
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLH
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLH
+            thetaAboveCell000022112103))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLH thetaAboveCell000022112103))
+        (by
+          have h : ((childLL (childHH (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLH
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLH
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLH
+            thetaAboveCell000022112103))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaAboveCell000022112103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLH
+            thetaAboveCell000022112103))) h))
+
+theorem cover_subtree_333ca6341b52 :
+    adaptiveCoverCheck 7 thetaAboveCell000022112103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112103
+    cover_subtree_340b36315559
+    cover_subtree_3ea1443179e3
+    (by
+      have h : ((childHL thetaAboveCell000022112103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022112103) h)
+    (by
+      have h : ((childHH thetaAboveCell000022112103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022112103) h)
+
+theorem cover_subtree_0a317c2968a3 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00002211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00002211)))
+    cover_subtree_ff28352fdcbd
+    cover_subtree_ecd8fe40a0e6
+    cover_subtree_bcfbc1ac2009
+    cover_subtree_333ca6341b52
+
+theorem cover_subtree_0f6a5316da7f :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022112110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022112110)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022112110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022112110)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022112110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022112110)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022112110))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022112110))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022112110))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022112110))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022112110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022112110))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022112110))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022112110))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022112110))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022112110))) h))
+
+theorem cover_subtree_8b21324e2245 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022112110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022112110)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022112110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022112110)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022112110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022112110)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022112110))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022112110))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022112110))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022112110))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022112110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022112110))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022112110))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022112110))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022112110))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022112110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022112110))) h))
+
+theorem cover_subtree_2b3d980fd10a :
+    adaptiveCoverCheck 7 thetaAboveCell000022112110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112110
+    (by
+      have h : ((childLL thetaAboveCell000022112110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022112110) h)
+    (by
+      have h : ((childLH thetaAboveCell000022112110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022112110) h)
+    cover_subtree_0f6a5316da7f
+    cover_subtree_8b21324e2245
+
+theorem cover_subtree_3d8086490270 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022112111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022112111)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022112111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022112111)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022112111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022112111)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022112111))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022112111))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022112111))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022112111))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022112111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022112111))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022112111))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022112111))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022112111))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022112111))) h))
+
+theorem cover_subtree_13b36b31ee50 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022112111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022112111)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022112111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022112111)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022112111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022112111)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022112111))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022112111))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022112111))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022112111))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022112111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022112111))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022112111))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022112111))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022112111))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022112111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022112111))) h))
+
+theorem cover_subtree_fe061292783d :
+    adaptiveCoverCheck 7 thetaAboveCell000022112111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112111
+    (by
+      have h : ((childLL thetaAboveCell000022112111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022112111) h)
+    (by
+      have h : ((childLH thetaAboveCell000022112111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022112111) h)
+    cover_subtree_3d8086490270
+    cover_subtree_13b36b31ee50
+
+theorem cover_subtree_acdbdc38c196 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022112112) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022112112)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022112112))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022112112))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022112112))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022112112))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLL thetaAboveCell000022112112))
+        (by
+          have h : ((childLL (childHL (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLL
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLL
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLL
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLL
+            thetaAboveCell000022112112))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLL thetaAboveCell000022112112))
+        (by
+          have h : ((childLL (childHH (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLL
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLL
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLL
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLL
+            thetaAboveCell000022112112))) h))
+
+theorem cover_subtree_f4b1386a9f86 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022112112) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022112112)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022112112))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022112112))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022112112))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022112112))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLH thetaAboveCell000022112112))
+        (by
+          have h : ((childLL (childHL (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLH
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLH
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLH
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLH
+            thetaAboveCell000022112112))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLH thetaAboveCell000022112112))
+        (by
+          have h : ((childLL (childHH (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLH
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLH
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLH
+            thetaAboveCell000022112112))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaAboveCell000022112112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLH
+            thetaAboveCell000022112112))) h))
+
+theorem cover_subtree_280eb01136d6 :
+    adaptiveCoverCheck 7 thetaAboveCell000022112112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112112
+    cover_subtree_acdbdc38c196
+    cover_subtree_f4b1386a9f86
+    (by
+      have h : ((childHL thetaAboveCell000022112112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022112112) h)
+    (by
+      have h : ((childHH thetaAboveCell000022112112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022112112) h)
+
+theorem cover_subtree_3add30d6277d :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022112113) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022112113)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022112113))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022112113))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022112113))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022112113))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLL thetaAboveCell000022112113))
+        (by
+          have h : ((childLL (childHL (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLL
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLL
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLL
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLL
+            thetaAboveCell000022112113))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLL thetaAboveCell000022112113))
+        (by
+          have h : ((childLL (childHH (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLL
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLL
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLL
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLL
+            thetaAboveCell000022112113))) h))
+
+theorem cover_subtree_4b342ee841c3 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022112113) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022112113)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022112113))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022112113))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022112113))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022112113))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLH thetaAboveCell000022112113))
+        (by
+          have h : ((childLL (childHL (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLH
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLH
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLH
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLH
+            thetaAboveCell000022112113))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLH thetaAboveCell000022112113))
+        (by
+          have h : ((childLL (childHH (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLH
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLH
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLH
+            thetaAboveCell000022112113))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaAboveCell000022112113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLH
+            thetaAboveCell000022112113))) h))
+
+theorem cover_subtree_95d32084082d :
+    adaptiveCoverCheck 7 thetaAboveCell000022112113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022112113
+    cover_subtree_3add30d6277d
+    cover_subtree_4b342ee841c3
+    (by
+      have h : ((childHL thetaAboveCell000022112113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022112113) h)
+    (by
+      have h : ((childHH thetaAboveCell000022112113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022112113) h)
+
+theorem cover_subtree_9112d0843a89 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00002211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00002211)))
+    cover_subtree_2b3d980fd10a
+    cover_subtree_fe061292783d
+    cover_subtree_280eb01136d6
+    cover_subtree_95d32084082d
+
+theorem e24KC2ThetaAboveLeaf0000221121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00002211))
+    cover_subtree_0a317c2968a3
+    cover_subtree_9112d0843a89
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022112120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112120 h)
+        (by
+          have h : (thetaAboveCell000022112121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112121 h)
+        (by
+          have h : (thetaAboveCell000022112122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112122 h)
+        (by
+          have h : (thetaAboveCell000022112123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022112130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112130 h)
+        (by
+          have h : (thetaAboveCell000022112131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112131 h)
+        (by
+          have h : (thetaAboveCell000022112132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112132 h)
+        (by
+          have h : (thetaAboveCell000022112133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022112133 h))
+theorem e24KC2ThetaAboveLeaf0000221122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00002211))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00002211))) h)
+theorem e24KC2ThetaAboveLeaf0000221123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00002211))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00002211))) h)
+theorem cover_subtree_c69725c188d3 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022113000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022113000)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022113000))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022113000)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022113000))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022113000)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022113000))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022113000))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022113000))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022113000))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022113000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022113000))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022113000))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022113000))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022113000))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022113000))) h))
+
+theorem cover_subtree_a8d579475d23 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022113000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022113000)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022113000))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022113000)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022113000))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022113000)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022113000))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022113000))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022113000))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022113000))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022113000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022113000))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022113000))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022113000))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022113000))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022113000))) h))
+
+theorem cover_subtree_560b815bd286 :
+    adaptiveCoverCheck 7 thetaAboveCell000022113000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113000
+    (by
+      have h : ((childLL thetaAboveCell000022113000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022113000) h)
+    (by
+      have h : ((childLH thetaAboveCell000022113000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022113000) h)
+    cover_subtree_c69725c188d3
+    cover_subtree_a8d579475d23
+
+theorem cover_subtree_995ab0851a0f :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022113001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022113001)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022113001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022113001)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022113001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022113001)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022113001))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022113001))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022113001))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022113001))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022113001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022113001))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022113001))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022113001))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022113001))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022113001))) h))
+
+theorem cover_subtree_f5e3b9ffae68 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022113001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022113001)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022113001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022113001)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022113001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022113001)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022113001))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022113001))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022113001))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022113001))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022113001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022113001))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022113001))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022113001))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022113001))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022113001))) h))
+
+theorem cover_subtree_e2b5567129eb :
+    adaptiveCoverCheck 7 thetaAboveCell000022113001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113001
+    (by
+      have h : ((childLL thetaAboveCell000022113001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022113001) h)
+    (by
+      have h : ((childLH thetaAboveCell000022113001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022113001) h)
+    cover_subtree_995ab0851a0f
+    cover_subtree_f5e3b9ffae68
+
+theorem cover_subtree_2a8071ef107e :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022113002) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022113002)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022113002))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022113002))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022113002))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022113002))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLL thetaAboveCell000022113002))
+        (by
+          have h : ((childLL (childHL (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLL
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLL
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLL
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLL
+            thetaAboveCell000022113002))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLL thetaAboveCell000022113002))
+        (by
+          have h : ((childLL (childHH (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLL
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLL
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLL
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLL
+            thetaAboveCell000022113002))) h))
+
+theorem cover_subtree_1f4e667d89aa :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022113002) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022113002)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022113002))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022113002))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022113002))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022113002))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLH thetaAboveCell000022113002))
+        (by
+          have h : ((childLL (childHL (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLH
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLH
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLH
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLH
+            thetaAboveCell000022113002))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLH thetaAboveCell000022113002))
+        (by
+          have h : ((childLL (childHH (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLH
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLH
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLH
+            thetaAboveCell000022113002))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaAboveCell000022113002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLH
+            thetaAboveCell000022113002))) h))
+
+theorem cover_subtree_cf43677b186a :
+    adaptiveCoverCheck 7 thetaAboveCell000022113002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113002
+    cover_subtree_2a8071ef107e
+    cover_subtree_1f4e667d89aa
+    (by
+      have h : ((childHL thetaAboveCell000022113002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022113002) h)
+    (by
+      have h : ((childHH thetaAboveCell000022113002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022113002) h)
+
+theorem cover_subtree_e06c4b38ee49 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022113003) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022113003)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022113003))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022113003))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022113003))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022113003))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLL thetaAboveCell000022113003))
+        (by
+          have h : ((childLL (childHL (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLL
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLL
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLL
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLL
+            thetaAboveCell000022113003))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLL thetaAboveCell000022113003))
+        (by
+          have h : ((childLL (childHH (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLL
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLL
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLL
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLL
+            thetaAboveCell000022113003))) h))
+
+theorem cover_subtree_5735fc3146a5 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022113003) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022113003)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022113003))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022113003))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022113003))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022113003))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022113003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022113003))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022113003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022113003)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022113003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022113003)) h)
+
+theorem cover_subtree_b9715c0ddc62 :
+    adaptiveCoverCheck 7 thetaAboveCell000022113003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113003
+    cover_subtree_e06c4b38ee49
+    cover_subtree_5735fc3146a5
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022113003)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000022113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000022113003)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000022113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000022113003)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000022113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000022113003)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000022113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000022113003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022113003)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000022113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000022113003)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000022113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000022113003)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000022113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000022113003)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000022113003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000022113003)) h))
+
+theorem cover_subtree_0338ef548092 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00002211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00002211)))
+    cover_subtree_560b815bd286
+    cover_subtree_e2b5567129eb
+    cover_subtree_cf43677b186a
+    cover_subtree_b9715c0ddc62
+
+theorem cover_subtree_8babef05283e :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022113010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022113010)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022113010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022113010)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022113010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022113010)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022113010))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022113010))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022113010))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022113010))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022113010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022113010))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022113010))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022113010))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022113010))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022113010))) h))
+
+theorem cover_subtree_453a26c2a0ce :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022113010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022113010)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022113010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022113010)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022113010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022113010)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022113010))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022113010))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022113010))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022113010))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022113010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022113010))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022113010))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022113010))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022113010))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022113010))) h))
+
+theorem cover_subtree_9a8d1e8ff5df :
+    adaptiveCoverCheck 7 thetaAboveCell000022113010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113010
+    (by
+      have h : ((childLL thetaAboveCell000022113010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022113010) h)
+    (by
+      have h : ((childLH thetaAboveCell000022113010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022113010) h)
+    cover_subtree_8babef05283e
+    cover_subtree_453a26c2a0ce
+
+theorem cover_subtree_d4f84dde625a :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022113011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022113011)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022113011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022113011)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022113011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022113011)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022113011))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022113011))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022113011))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022113011))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022113011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022113011))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022113011))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022113011))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022113011))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022113011))) h))
+
+theorem cover_subtree_a97cdaa163bd :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022113011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022113011)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022113011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022113011)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022113011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022113011)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022113011))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022113011))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022113011))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022113011))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022113011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022113011))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022113011))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022113011))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022113011))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022113011))) h))
+
+theorem cover_subtree_caad7b9ebe18 :
+    adaptiveCoverCheck 7 thetaAboveCell000022113011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113011
+    (by
+      have h : ((childLL thetaAboveCell000022113011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022113011) h)
+    (by
+      have h : ((childLH thetaAboveCell000022113011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022113011) h)
+    cover_subtree_d4f84dde625a
+    cover_subtree_a97cdaa163bd
+
+theorem cover_subtree_fbb8d3b85bf4 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022113012) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022113012)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022113012))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022113012))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022113012))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022113012))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022113012))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022113012))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022113012))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022113012))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022113012))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022113012))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022113012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022113012)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022113012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022113012)) h)
+
+theorem cover_subtree_f025663eb4cc :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022113012) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022113012)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022113012))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022113012))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022113012))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022113012))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022113012))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022113012))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022113012))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022113012))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022113012))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022113012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022113012))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022113012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022113012)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022113012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022113012)) h)
+
+theorem cover_subtree_6d9b8f942e07 :
+    adaptiveCoverCheck 7 thetaAboveCell000022113012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113012
+    cover_subtree_fbb8d3b85bf4
+    cover_subtree_f025663eb4cc
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022113012)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000022113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000022113012)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000022113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000022113012)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000022113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000022113012)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000022113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000022113012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022113012)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000022113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000022113012)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000022113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000022113012)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000022113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000022113012)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000022113012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000022113012)) h))
+
+theorem cover_subtree_c85e1b57b2c5 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022113013) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022113013)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022113013))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022113013))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022113013))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022113013))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022113013))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022113013))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022113013))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022113013))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022113013))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022113013))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022113013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022113013)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022113013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022113013)) h)
+
+theorem cover_subtree_a12be2bf4817 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022113013) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022113013)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022113013))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022113013))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022113013))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022113013))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022113013))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022113013))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022113013))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022113013))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022113013))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022113013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022113013))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022113013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022113013)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022113013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022113013)) h)
+
+theorem cover_subtree_d1382a844a10 :
+    adaptiveCoverCheck 7 thetaAboveCell000022113013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113013
+    cover_subtree_c85e1b57b2c5
+    cover_subtree_a12be2bf4817
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022113013)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000022113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000022113013)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000022113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000022113013)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000022113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000022113013)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000022113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000022113013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022113013)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000022113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000022113013)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000022113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000022113013)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000022113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000022113013)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000022113013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000022113013)) h))
+
+theorem cover_subtree_d4d29d86cd96 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00002211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00002211)))
+    cover_subtree_9a8d1e8ff5df
+    cover_subtree_caad7b9ebe18
+    cover_subtree_6d9b8f942e07
+    cover_subtree_d1382a844a10
+
+theorem e24KC2ThetaAboveLeaf0000221130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00002211))
+    cover_subtree_0338ef548092
+    cover_subtree_d4d29d86cd96
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022113020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113020 h)
+        (by
+          have h : (thetaAboveCell000022113021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113021 h)
+        (by
+          have h : (thetaAboveCell000022113022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113022 h)
+        (by
+          have h : (thetaAboveCell000022113023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022113030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113030 h)
+        (by
+          have h : (thetaAboveCell000022113031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113031 h)
+        (by
+          have h : (thetaAboveCell000022113032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113032 h)
+        (by
+          have h : (thetaAboveCell000022113033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113033 h))
+theorem cover_subtree_836d962e65ac :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022113100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022113100)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022113100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022113100)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022113100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022113100)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022113100))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022113100))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022113100))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022113100))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022113100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022113100))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022113100))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022113100))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022113100))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022113100))) h))
+
+theorem cover_subtree_ecf6c7f90571 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022113100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022113100)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022113100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022113100)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022113100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022113100)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022113100))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022113100))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022113100))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022113100))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022113100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022113100))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022113100))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022113100))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022113100))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022113100))) h))
+
+theorem cover_subtree_fa2b1e57d306 :
+    adaptiveCoverCheck 7 thetaAboveCell000022113100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113100
+    (by
+      have h : ((childLL thetaAboveCell000022113100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022113100) h)
+    (by
+      have h : ((childLH thetaAboveCell000022113100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022113100) h)
+    cover_subtree_836d962e65ac
+    cover_subtree_ecf6c7f90571
+
+theorem cover_subtree_2c032bc482bc :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022113101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022113101)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022113101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022113101)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022113101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022113101)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022113101))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022113101))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022113101))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022113101))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022113101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022113101))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022113101))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022113101))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022113101))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022113101))) h))
+
+theorem cover_subtree_3eddb0efc253 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022113101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022113101)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022113101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022113101)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022113101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022113101)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022113101))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022113101))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022113101))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022113101))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022113101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022113101))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022113101))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022113101))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022113101))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022113101))) h))
+
+theorem cover_subtree_cbdd15c92455 :
+    adaptiveCoverCheck 7 thetaAboveCell000022113101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113101
+    (by
+      have h : ((childLL thetaAboveCell000022113101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022113101) h)
+    (by
+      have h : ((childLH thetaAboveCell000022113101)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022113101) h)
+    cover_subtree_2c032bc482bc
+    cover_subtree_3eddb0efc253
+
+theorem cover_subtree_e867c54455aa :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022113102) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022113102)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022113102))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022113102))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022113102))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022113102))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022113102))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022113102))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022113102))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022113102))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022113102))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022113102))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022113102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022113102)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022113102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022113102)) h)
+
+theorem cover_subtree_d8e5977bc823 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022113102) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022113102)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022113102))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022113102))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022113102))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022113102))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022113102))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022113102))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022113102))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022113102))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022113102))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022113102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022113102))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022113102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022113102)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022113102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022113102)) h)
+
+theorem cover_subtree_81e578425a2e :
+    adaptiveCoverCheck 7 thetaAboveCell000022113102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113102
+    cover_subtree_e867c54455aa
+    cover_subtree_d8e5977bc823
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022113102)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000022113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000022113102)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000022113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000022113102)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000022113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000022113102)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000022113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000022113102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022113102)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000022113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000022113102)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000022113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000022113102)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000022113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000022113102)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000022113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000022113102)) h))
+
+theorem cover_subtree_6af7504512c8 :
+    adaptiveCoverCheck 7 thetaAboveCell000022113103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113103
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022113103)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022113103)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022113103)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022113103)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022113103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022113103)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022113103)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022113103)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022113103)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022113103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022113103)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000022113103)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000022113103)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000022113103)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000022113103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022113103)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000022113103)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000022113103)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000022113103)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000022113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000022113103)) h))
+
+theorem cover_subtree_dc05a5cb4d61 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00002211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00002211)))
+    cover_subtree_fa2b1e57d306
+    cover_subtree_cbdd15c92455
+    cover_subtree_81e578425a2e
+    cover_subtree_6af7504512c8
+
+theorem cover_subtree_fcbdf3a9e595 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022113110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022113110)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022113110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022113110)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022113110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022113110)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022113110))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022113110))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022113110))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022113110))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022113110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022113110))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022113110))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022113110))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022113110))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022113110))) h))
+
+theorem cover_subtree_2828e56e478f :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022113110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022113110)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022113110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022113110)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022113110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022113110)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022113110))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022113110))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022113110))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022113110))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022113110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022113110))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022113110))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022113110))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022113110))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022113110))) h))
+
+theorem cover_subtree_15fb86545eea :
+    adaptiveCoverCheck 7 thetaAboveCell000022113110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113110
+    (by
+      have h : ((childLL thetaAboveCell000022113110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022113110) h)
+    (by
+      have h : ((childLH thetaAboveCell000022113110)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022113110) h)
+    cover_subtree_fcbdf3a9e595
+    cover_subtree_2828e56e478f
+
+theorem cover_subtree_0f38c8258b2e :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022113111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022113111)
+    (by
+      have h : ((childLL (childHL thetaAboveCell000022113111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL thetaAboveCell000022113111)) h)
+    (by
+      have h : ((childLH (childHL thetaAboveCell000022113111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL thetaAboveCell000022113111)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022113111))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022113111))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022113111))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022113111))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022113111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022113111))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022113111))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022113111))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022113111))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022113111))) h))
+
+theorem cover_subtree_13fc46665967 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022113111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022113111)
+    (by
+      have h : ((childLL (childHH thetaAboveCell000022113111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH thetaAboveCell000022113111)) h)
+    (by
+      have h : ((childLH (childHH thetaAboveCell000022113111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH thetaAboveCell000022113111)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022113111))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022113111))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022113111))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022113111))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022113111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022113111))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022113111))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022113111))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022113111))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022113111))) h))
+
+theorem cover_subtree_83c447d13cd0 :
+    adaptiveCoverCheck 7 thetaAboveCell000022113111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113111
+    (by
+      have h : ((childLL thetaAboveCell000022113111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022113111) h)
+    (by
+      have h : ((childLH thetaAboveCell000022113111)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022113111) h)
+    cover_subtree_0f38c8258b2e
+    cover_subtree_13fc46665967
+
+theorem cover_subtree_02c32f65c578 :
+    adaptiveCoverCheck 7 thetaAboveCell000022113112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113112
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022113112)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022113112)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022113112)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022113112)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022113112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022113112)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022113112)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022113112)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022113112)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022113112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022113112)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000022113112)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000022113112)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000022113112)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000022113112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022113112)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000022113112)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000022113112)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000022113112)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000022113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000022113112)) h))
+
+theorem cover_subtree_74079318656f :
+    adaptiveCoverCheck 7 thetaAboveCell000022113113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022113113
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022113113)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022113113)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022113113)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022113113)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022113113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022113113)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022113113)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022113113)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022113113)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022113113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022113113)
+        (by
+          have h : ((childLL (childHL thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHL
+            thetaAboveCell000022113113)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHL
+            thetaAboveCell000022113113)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHL
+            thetaAboveCell000022113113)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHL
+            thetaAboveCell000022113113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022113113)
+        (by
+          have h : ((childLL (childHH thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childHH
+            thetaAboveCell000022113113)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childHH
+            thetaAboveCell000022113113)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childHH
+            thetaAboveCell000022113113)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell000022113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childHH
+            thetaAboveCell000022113113)) h))
+
+theorem cover_subtree_6a41f1f44284 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00002211))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00002211)))
+    cover_subtree_15fb86545eea
+    cover_subtree_83c447d13cd0
+    cover_subtree_02c32f65c578
+    cover_subtree_74079318656f
+
+theorem e24KC2ThetaAboveLeaf0000221131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00002211))
+    cover_subtree_dc05a5cb4d61
+    cover_subtree_6a41f1f44284
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022113120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113120 h)
+        (by
+          have h : (thetaAboveCell000022113121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113121 h)
+        (by
+          have h : (thetaAboveCell000022113122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113122 h)
+        (by
+          have h : (thetaAboveCell000022113123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00002211)))
+        (by
+          have h : (thetaAboveCell000022113130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113130 h)
+        (by
+          have h : (thetaAboveCell000022113131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113131 h)
+        (by
+          have h : (thetaAboveCell000022113132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113132 h)
+        (by
+          have h : (thetaAboveCell000022113133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022113133 h))
+theorem e24KC2ThetaAboveLeaf0000221132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00002211))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00002211))) h)
+theorem e24KC2ThetaAboveLeaf0000221133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00002211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00002211))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00002211))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00002211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00002211))) h)
+theorem e24KC2ThetaAboveLeaf0000230002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00002300))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00002300))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLL
+        thetaAboveCell00002300)))
+        (by
+          have h : (thetaAboveCell000023000220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000220 h)
+        (by
+          have h : (thetaAboveCell000023000221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000221 h)
+        (by
+          have h : (thetaAboveCell000023000222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000222 h)
+        (by
+          have h : (thetaAboveCell000023000223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childLL
+        thetaAboveCell00002300)))
+        (by
+          have h : (thetaAboveCell000023000230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000230 h)
+        (by
+          have h : (thetaAboveCell000023000231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000231 h)
+        (by
+          have h : (thetaAboveCell000023000232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000232 h)
+        (by
+          have h : (thetaAboveCell000023000233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000233 h))
+theorem e24KC2ThetaAboveLeaf0000230003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00002300))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00002300))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childLL
+        thetaAboveCell00002300)))
+        (by
+          have h : (thetaAboveCell000023000320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000320 h)
+        (by
+          have h : (thetaAboveCell000023000321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000321 h)
+        (by
+          have h : (thetaAboveCell000023000322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000322 h)
+        (by
+          have h : (thetaAboveCell000023000323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childLL
+        thetaAboveCell00002300)))
+        (by
+          have h : (thetaAboveCell000023000330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000330 h)
+        (by
+          have h : (thetaAboveCell000023000331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000331 h)
+        (by
+          have h : (thetaAboveCell000023000332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000332 h)
+        (by
+          have h : (thetaAboveCell000023000333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023000333 h))
+theorem e24KC2ThetaAboveLeaf0000230012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00002300))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00002300))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLH
+        thetaAboveCell00002300)))
+        (by
+          have h : (thetaAboveCell000023001220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001220 h)
+        (by
+          have h : (thetaAboveCell000023001221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001221 h)
+        (by
+          have h : (thetaAboveCell000023001222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001222 h)
+        (by
+          have h : (thetaAboveCell000023001223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childLH
+        thetaAboveCell00002300)))
+        (by
+          have h : (thetaAboveCell000023001230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001230 h)
+        (by
+          have h : (thetaAboveCell000023001231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001231 h)
+        (by
+          have h : (thetaAboveCell000023001232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001232 h)
+        (by
+          have h : (thetaAboveCell000023001233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001233 h))
+theorem e24KC2ThetaAboveLeaf0000230013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00002300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00002300))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00002300))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00002300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00002300))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childLH
+        thetaAboveCell00002300)))
+        (by
+          have h : (thetaAboveCell000023001320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001320 h)
+        (by
+          have h : (thetaAboveCell000023001321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001321 h)
+        (by
+          have h : (thetaAboveCell000023001322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001322 h)
+        (by
+          have h : (thetaAboveCell000023001323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childLH
+        thetaAboveCell00002300)))
+        (by
+          have h : (thetaAboveCell000023001330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001330 h)
+        (by
+          have h : (thetaAboveCell000023001331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001331 h)
+        (by
+          have h : (thetaAboveCell000023001332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001332 h)
+        (by
+          have h : (thetaAboveCell000023001333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000023001333 h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle012.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle012.lean
new file mode 100644
index 0000000000..2ca6c09200
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle012.lean
@@ -0,0 +1,11476 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch017`.
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch021`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCells2af6f575e7
+
+/-- Subcell `1110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1112 : AngleCell :=
+  childHL (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1113 : AngleCell :=
+  childHH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1120 : AngleCell :=
+  childLL (childHL (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1121 : AngleCell :=
+  childLH (childHL (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1122 : AngleCell :=
+  childHL (childHL (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1123 : AngleCell :=
+  childHH (childHL (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1130 : AngleCell :=
+  childLL (childHH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1131 : AngleCell :=
+  childLH (childHH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1132 : AngleCell :=
+  childHL (childHH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1133 : AngleCell :=
+  childHH (childHH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1300 : AngleCell :=
+  childLL (childLL (childHH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1301 : AngleCell :=
+  childLH (childLL (childHH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1302 : AngleCell :=
+  childHL (childLL (childHH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1303 : AngleCell :=
+  childHH (childLL (childHH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1310 : AngleCell :=
+  childLL (childLH (childHH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1311 : AngleCell :=
+  childLH (childLH (childHH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1312 : AngleCell :=
+  childHL (childLH (childHH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1313 : AngleCell :=
+  childHH (childLH (childHH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11103020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11112000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11113020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaBelowCell1111)))
+
+end GerverSofa.PartE.CertificateCells2af6f575e7
+
+namespace GerverSofa.PartE.CertificateCellsc9f578345f
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022003100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022010220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022010221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022010222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022010223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022010230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022010231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022010232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022010233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022010320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022010321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022010322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022010323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022010330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022010331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022010332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022010333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022010333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaAboveCell00002201)))
+
+/-- Subcell `000022011220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022011221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022011222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022011223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022011230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022011231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022011232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022011233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022011320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022011321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022011322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022011323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022011330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022011331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022011332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022011333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022011333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaAboveCell00002201)))
+
+/-- Subcell `000022012000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00002201)))
+
+/-- Subcell `0000220031002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022003100)))
+
+/-- Subcell `0000220031012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022003101)))
+
+/-- Subcell `0000220031102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022003110)))
+
+/-- Subcell `0000220031112000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220031113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220031113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022003111)))
+
+/-- Subcell `0000220120002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022012000)))
+
+/-- Subcell `0000220120012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022012001)))
+
+/-- Subcell `0000220120102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022012010)))
+
+/-- Subcell `0000220120112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell000022012011)))
+
+/-- Subcell `0000220120113333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220120113333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell000022012011)))
+
+end GerverSofa.PartE.CertificateCellsc9f578345f
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC5TerminalBatchT716800017`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC5 checkpoint-aware kernel batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2af6f575e7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2af6f575e7
+
+open CertificateCells2af6f575e7
+theorem e24KC2ThetaBelowLeaf111030203 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103020) = true := by
+  have h : ((childHH thetaBelowCell11103020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11103020) h
+theorem e24KC2ThetaBelowLeaf111030210 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103021) = true := by
+  have h : ((childLL thetaBelowCell11103021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11103021) h
+theorem e24KC2ThetaBelowLeaf111030211 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103021) = true := by
+  have h : ((childLH thetaBelowCell11103021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11103021) h
+theorem e24KC2ThetaBelowLeaf111030212 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103021) = true := by
+  have h : ((childHL thetaBelowCell11103021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11103021) h
+theorem e24KC2ThetaBelowLeaf111030213 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103021) = true := by
+  have h : ((childHH thetaBelowCell11103021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11103021) h
+theorem e24KC2ThetaBelowLeaf111030220 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103022) = true := by
+  have h : ((childLL thetaBelowCell11103022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11103022) h
+theorem e24KC2ThetaBelowLeaf111030221 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103022) = true := by
+  have h : ((childLH thetaBelowCell11103022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11103022) h
+theorem e24KC2ThetaBelowLeaf111030222 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103022) = true := by
+  have h : ((childHL thetaBelowCell11103022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11103022) h
+theorem e24KC2ThetaBelowLeaf111030230 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103023) = true := by
+  have h : ((childLL thetaBelowCell11103023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11103023) h
+theorem e24KC2ThetaBelowLeaf111030231 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103023) = true := by
+  have h : ((childLH thetaBelowCell11103023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11103023) h
+theorem e24KC2ThetaBelowLeaf111030300 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103030) = true := by
+  have h : ((childLL thetaBelowCell11103030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11103030) h
+theorem e24KC2ThetaBelowLeaf111030301 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103030) = true := by
+  have h : ((childLH thetaBelowCell11103030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11103030) h
+theorem e24KC2ThetaBelowLeaf111030302 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103030) = true := by
+  have h : ((childHL thetaBelowCell11103030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11103030) h
+theorem e24KC2ThetaBelowLeaf111030303 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103030) = true := by
+  have h : ((childHH thetaBelowCell11103030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11103030) h
+theorem e24KC2ThetaBelowLeaf111030310 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103031) = true := by
+  have h : ((childLL thetaBelowCell11103031)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11103031) h
+theorem e24KC2ThetaBelowLeaf111030311 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103031) = true := by
+  have h : ((childLH thetaBelowCell11103031)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11103031) h
+theorem e24KC2ThetaBelowLeaf11103100 :
+    adaptiveCoverCheck 10 thetaBelowCell11103100 = true := by
+  have h : (thetaBelowCell11103100).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103100 h
+theorem e24KC2ThetaBelowLeaf11103101 :
+    adaptiveCoverCheck 10 thetaBelowCell11103101 = true := by
+  have h : (thetaBelowCell11103101).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103101 h
+theorem e24KC2ThetaBelowLeaf11103102 :
+    adaptiveCoverCheck 10 thetaBelowCell11103102 = true := by
+  have h : (thetaBelowCell11103102).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103102 h
+theorem e24KC2ThetaBelowLeaf11103103 :
+    adaptiveCoverCheck 10 thetaBelowCell11103103 = true := by
+  have h : (thetaBelowCell11103103).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103103 h
+theorem e24KC2ThetaBelowLeaf11103110 :
+    adaptiveCoverCheck 10 thetaBelowCell11103110 = true := by
+  have h : (thetaBelowCell11103110).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103110 h
+theorem e24KC2ThetaBelowLeaf11103111 :
+    adaptiveCoverCheck 10 thetaBelowCell11103111 = true := by
+  have h : (thetaBelowCell11103111).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103111 h
+theorem e24KC2ThetaBelowLeaf11103112 :
+    adaptiveCoverCheck 10 thetaBelowCell11103112 = true := by
+  have h : (thetaBelowCell11103112).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103112 h
+theorem e24KC2ThetaBelowLeaf11103113 :
+    adaptiveCoverCheck 10 thetaBelowCell11103113 = true := by
+  have h : (thetaBelowCell11103113).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103113 h
+theorem e24KC2ThetaBelowLeaf111031200 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103120) = true := by
+  have h : ((childLL thetaBelowCell11103120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11103120) h
+theorem e24KC2ThetaBelowLeaf111031201 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103120) = true := by
+  have h : ((childLH thetaBelowCell11103120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11103120) h
+theorem e24KC2ThetaBelowLeaf111031203 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103120) = true := by
+  have h : ((childHH thetaBelowCell11103120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11103120) h
+theorem e24KC2ThetaBelowLeaf111031210 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103121) = true := by
+  have h : ((childLL thetaBelowCell11103121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11103121) h
+theorem e24KC2ThetaBelowLeaf111031211 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103121) = true := by
+  have h : ((childLH thetaBelowCell11103121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11103121) h
+theorem e24KC2ThetaBelowLeaf111031212 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103121) = true := by
+  have h : ((childHL thetaBelowCell11103121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11103121) h
+theorem e24KC2ThetaBelowLeaf111031213 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103121) = true := by
+  have h : ((childHH thetaBelowCell11103121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11103121) h
+theorem e24KC2ThetaBelowLeaf111031300 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103130) = true := by
+  have h : ((childLL thetaBelowCell11103130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11103130) h
+theorem e24KC2ThetaBelowLeaf111031301 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103130) = true := by
+  have h : ((childLH thetaBelowCell11103130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11103130) h
+theorem e24KC2ThetaBelowLeaf111031302 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103130) = true := by
+  have h : ((childHL thetaBelowCell11103130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11103130) h
+theorem e24KC2ThetaBelowLeaf111031303 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103130) = true := by
+  have h : ((childHH thetaBelowCell11103130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11103130) h
+theorem e24KC2ThetaBelowLeaf111031310 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103131) = true := by
+  have h : ((childLL thetaBelowCell11103131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11103131) h
+theorem e24KC2ThetaBelowLeaf111031311 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103131) = true := by
+  have h : ((childLH thetaBelowCell11103131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11103131) h
+theorem e24KC2ThetaBelowLeaf111031312 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103131) = true := by
+  have h : ((childHL thetaBelowCell11103131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11103131) h
+theorem e24KC2ThetaBelowLeaf111031313 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103131) = true := by
+  have h : ((childHH thetaBelowCell11103131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11103131) h
+theorem e24KC2ThetaBelowLeaf111032000 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103200) = true := by
+  have h : ((childLL thetaBelowCell11103200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11103200) h
+theorem e24KC2ThetaBelowLeaf111032001 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103200) = true := by
+  have h : ((childLH thetaBelowCell11103200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11103200) h
+theorem e24KC2ThetaBelowLeaf111032002 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103200) = true := by
+  have h : ((childHL thetaBelowCell11103200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11103200) h
+theorem e24KC2ThetaBelowLeaf111032003 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103200) = true := by
+  have h : ((childHH thetaBelowCell11103200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11103200) h
+theorem e24KC2ThetaBelowLeaf111032010 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11103201) = true := by
+  have h : ((childLL thetaBelowCell11103201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11103201) h
+theorem e24KC2ThetaBelowLeaf111032011 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11103201) = true := by
+  have h : ((childLH thetaBelowCell11103201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11103201) h
+theorem e24KC2ThetaBelowLeaf111032012 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103201) = true := by
+  have h : ((childHL thetaBelowCell11103201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11103201) h
+theorem e24KC2ThetaBelowLeaf111032013 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103201) = true := by
+  have h : ((childHH thetaBelowCell11103201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11103201) h
+theorem e24KC2ThetaBelowLeaf11103202 :
+    adaptiveCoverCheck 10 thetaBelowCell11103202 = true := by
+  have h : (thetaBelowCell11103202).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103202 h
+theorem e24KC2ThetaBelowLeaf11103203 :
+    adaptiveCoverCheck 10 thetaBelowCell11103203 = true := by
+  have h : (thetaBelowCell11103203).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103203 h
+theorem e24KC2ThetaBelowLeaf111032102 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103210) = true := by
+  have h : ((childHL thetaBelowCell11103210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11103210) h
+theorem e24KC2ThetaBelowLeaf111032103 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103210) = true := by
+  have h : ((childHH thetaBelowCell11103210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11103210) h
+theorem e24KC2ThetaBelowLeaf111032112 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103211) = true := by
+  have h : ((childHL thetaBelowCell11103211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11103211) h
+theorem e24KC2ThetaBelowLeaf111032113 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103211) = true := by
+  have h : ((childHH thetaBelowCell11103211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11103211) h
+theorem e24KC2ThetaBelowLeaf11103212 :
+    adaptiveCoverCheck 10 thetaBelowCell11103212 = true := by
+  have h : (thetaBelowCell11103212).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103212 h
+theorem e24KC2ThetaBelowLeaf11103213 :
+    adaptiveCoverCheck 10 thetaBelowCell11103213 = true := by
+  have h : (thetaBelowCell11103213).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103213 h
+theorem e24KC2ThetaBelowLeaf1110322 :
+    adaptiveCoverCheck 11 (childHL (childHL (childHH thetaBelowCell1110))) = true := by
+  have h : ((childHL (childHL (childHH thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childHH thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110323 :
+    adaptiveCoverCheck 11 (childHH (childHL (childHH thetaBelowCell1110))) = true := by
+  have h : ((childHH (childHL (childHH thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childHH thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf111033002 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103300) = true := by
+  have h : ((childHL thetaBelowCell11103300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11103300) h
+theorem e24KC2ThetaBelowLeaf111033003 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103300) = true := by
+  have h : ((childHH thetaBelowCell11103300)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11103300) h
+theorem e24KC2ThetaBelowLeaf111033012 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103301) = true := by
+  have h : ((childHL thetaBelowCell11103301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11103301) h
+theorem e24KC2ThetaBelowLeaf111033013 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103301) = true := by
+  have h : ((childHH thetaBelowCell11103301)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11103301) h
+theorem e24KC2ThetaBelowLeaf11103302 :
+    adaptiveCoverCheck 10 thetaBelowCell11103302 = true := by
+  have h : (thetaBelowCell11103302).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103302 h
+theorem e24KC2ThetaBelowLeaf11103303 :
+    adaptiveCoverCheck 10 thetaBelowCell11103303 = true := by
+  have h : (thetaBelowCell11103303).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103303 h
+theorem e24KC2ThetaBelowLeaf111033102 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103310) = true := by
+  have h : ((childHL thetaBelowCell11103310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11103310) h
+theorem e24KC2ThetaBelowLeaf111033103 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103310) = true := by
+  have h : ((childHH thetaBelowCell11103310)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11103310) h
+theorem e24KC2ThetaBelowLeaf111033112 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11103311) = true := by
+  have h : ((childHL thetaBelowCell11103311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11103311) h
+theorem e24KC2ThetaBelowLeaf111033113 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11103311) = true := by
+  have h : ((childHH thetaBelowCell11103311)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11103311) h
+theorem e24KC2ThetaBelowLeaf11103312 :
+    adaptiveCoverCheck 10 thetaBelowCell11103312 = true := by
+  have h : (thetaBelowCell11103312).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103312 h
+theorem e24KC2ThetaBelowLeaf11103313 :
+    adaptiveCoverCheck 10 thetaBelowCell11103313 = true := by
+  have h : (thetaBelowCell11103313).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11103313 h
+theorem e24KC2ThetaBelowLeaf1110332 :
+    adaptiveCoverCheck 11 (childHL (childHH (childHH thetaBelowCell1110))) = true := by
+  have h : ((childHL (childHH (childHH thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childHH thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf1110333 :
+    adaptiveCoverCheck 11 (childHH (childHH (childHH thetaBelowCell1110))) = true := by
+  have h : ((childHH (childHH (childHH thetaBelowCell1110)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childHH thetaBelowCell1110))) h
+theorem e24KC2ThetaBelowLeaf111100 :
+    adaptiveCoverCheck 12 (childLL (childLL thetaBelowCell1111)) = true := by
+  have h : ((childLL (childLL thetaBelowCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL thetaBelowCell1111)) h
+theorem e24KC2ThetaBelowLeaf111101 :
+    adaptiveCoverCheck 12 (childLH (childLL thetaBelowCell1111)) = true := by
+  have h : ((childLH (childLL thetaBelowCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL thetaBelowCell1111)) h
+theorem e24KC2ThetaBelowLeaf1111020 :
+    adaptiveCoverCheck 11 (childLL (childHL (childLL thetaBelowCell1111))) = true := by
+  have h : ((childLL (childHL (childLL thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childLL thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf1111021 :
+    adaptiveCoverCheck 11 (childLH (childHL (childLL thetaBelowCell1111))) = true := by
+  have h : ((childLH (childHL (childLL thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childLL thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf1111022 :
+    adaptiveCoverCheck 11 (childHL (childHL (childLL thetaBelowCell1111))) = true := by
+  have h : ((childHL (childHL (childLL thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childLL thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf1111023 :
+    adaptiveCoverCheck 11 (childHH (childHL (childLL thetaBelowCell1111))) = true := by
+  have h : ((childHH (childHL (childLL thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childLL thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf1111030 :
+    adaptiveCoverCheck 11 (childLL (childHH (childLL thetaBelowCell1111))) = true := by
+  have h : ((childLL (childHH (childLL thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHH (childLL thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf1111031 :
+    adaptiveCoverCheck 11 (childLH (childHH (childLL thetaBelowCell1111))) = true := by
+  have h : ((childLH (childHH (childLL thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHH (childLL thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf1111032 :
+    adaptiveCoverCheck 11 (childHL (childHH (childLL thetaBelowCell1111))) = true := by
+  have h : ((childHL (childHH (childLL thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHH (childLL thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf1111033 :
+    adaptiveCoverCheck 11 (childHH (childHH (childLL thetaBelowCell1111))) = true := by
+  have h : ((childHH (childHH (childLL thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHH (childLL thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf111110 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell1111)) = true := by
+  have h : ((childLL (childLH thetaBelowCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLH thetaBelowCell1111)) h
+theorem e24KC2ThetaBelowLeaf111111 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell1111)) = true := by
+  have h : ((childLH (childLH thetaBelowCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLH thetaBelowCell1111)) h
+theorem e24KC2ThetaBelowLeaf1111120 :
+    adaptiveCoverCheck 11 (childLL (childHL (childLH thetaBelowCell1111))) = true := by
+  have h : ((childLL (childHL (childLH thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childHL (childLH thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf1111121 :
+    adaptiveCoverCheck 11 (childLH (childHL (childLH thetaBelowCell1111))) = true := by
+  have h : ((childLH (childHL (childLH thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childHL (childLH thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf1111122 :
+    adaptiveCoverCheck 11 (childHL (childHL (childLH thetaBelowCell1111))) = true := by
+  have h : ((childHL (childHL (childLH thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childHL (childLH thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf1111123 :
+    adaptiveCoverCheck 11 (childHH (childHL (childLH thetaBelowCell1111))) = true := by
+  have h : ((childHH (childHL (childLH thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childHL (childLH thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf111113 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell1111)) = true := by
+  have h : ((childHH (childLH thetaBelowCell1111))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH thetaBelowCell1111)) h
+theorem e24KC2ThetaBelowLeaf11112000 :
+    adaptiveCoverCheck 10 thetaBelowCell11112000 = true := by
+  have h : (thetaBelowCell11112000).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112000 h
+theorem e24KC2ThetaBelowLeaf11112001 :
+    adaptiveCoverCheck 10 thetaBelowCell11112001 = true := by
+  have h : (thetaBelowCell11112001).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112001 h
+theorem e24KC2ThetaBelowLeaf11112002 :
+    adaptiveCoverCheck 10 thetaBelowCell11112002 = true := by
+  have h : (thetaBelowCell11112002).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112002 h
+theorem e24KC2ThetaBelowLeaf11112003 :
+    adaptiveCoverCheck 10 thetaBelowCell11112003 = true := by
+  have h : (thetaBelowCell11112003).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112003 h
+theorem e24KC2ThetaBelowLeaf11112010 :
+    adaptiveCoverCheck 10 thetaBelowCell11112010 = true := by
+  have h : (thetaBelowCell11112010).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112010 h
+theorem e24KC2ThetaBelowLeaf11112011 :
+    adaptiveCoverCheck 10 thetaBelowCell11112011 = true := by
+  have h : (thetaBelowCell11112011).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112011 h
+theorem e24KC2ThetaBelowLeaf11112012 :
+    adaptiveCoverCheck 10 thetaBelowCell11112012 = true := by
+  have h : (thetaBelowCell11112012).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112012 h
+theorem e24KC2ThetaBelowLeaf11112013 :
+    adaptiveCoverCheck 10 thetaBelowCell11112013 = true := by
+  have h : (thetaBelowCell11112013).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112013 h
+theorem e24KC2ThetaBelowLeaf111120200 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112020) = true := by
+  have h : ((childLL thetaBelowCell11112020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11112020) h
+theorem e24KC2ThetaBelowLeaf111120201 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112020) = true := by
+  have h : ((childLH thetaBelowCell11112020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11112020) h
+theorem e24KC2ThetaBelowLeaf111120202 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112020) = true := by
+  have h : ((childHL thetaBelowCell11112020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112020) h
+theorem e24KC2ThetaBelowLeaf111120203 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112020) = true := by
+  have h : ((childHH thetaBelowCell11112020)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112020) h
+theorem e24KC2ThetaBelowLeaf111120210 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112021) = true := by
+  have h : ((childLL thetaBelowCell11112021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11112021) h
+theorem e24KC2ThetaBelowLeaf111120211 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112021) = true := by
+  have h : ((childLH thetaBelowCell11112021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11112021) h
+theorem e24KC2ThetaBelowLeaf111120212 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112021) = true := by
+  have h : ((childHL thetaBelowCell11112021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112021) h
+theorem e24KC2ThetaBelowLeaf111120213 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112021) = true := by
+  have h : ((childHH thetaBelowCell11112021)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112021) h
+theorem e24KC2ThetaBelowLeaf111120300 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112030) = true := by
+  have h : ((childLL thetaBelowCell11112030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11112030) h
+theorem e24KC2ThetaBelowLeaf111120301 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112030) = true := by
+  have h : ((childLH thetaBelowCell11112030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11112030) h
+theorem e24KC2ThetaBelowLeaf111120302 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112030) = true := by
+  have h : ((childHL thetaBelowCell11112030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112030) h
+theorem e24KC2ThetaBelowLeaf111120303 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112030) = true := by
+  have h : ((childHH thetaBelowCell11112030)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112030) h
+theorem e24KC2ThetaBelowLeaf111120310 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112031) = true := by
+  have h : ((childLL thetaBelowCell11112031)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11112031) h
+theorem e24KC2ThetaBelowLeaf111120311 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112031) = true := by
+  have h : ((childLH thetaBelowCell11112031)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11112031) h
+theorem e24KC2ThetaBelowLeaf111120312 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112031) = true := by
+  have h : ((childHL thetaBelowCell11112031)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112031) h
+theorem e24KC2ThetaBelowLeaf111120313 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112031) = true := by
+  have h : ((childHH thetaBelowCell11112031)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112031) h
+theorem e24KC2ThetaBelowLeaf11112100 :
+    adaptiveCoverCheck 10 thetaBelowCell11112100 = true := by
+  have h : (thetaBelowCell11112100).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112100 h
+theorem e24KC2ThetaBelowLeaf11112101 :
+    adaptiveCoverCheck 10 thetaBelowCell11112101 = true := by
+  have h : (thetaBelowCell11112101).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112101 h
+theorem e24KC2ThetaBelowLeaf11112102 :
+    adaptiveCoverCheck 10 thetaBelowCell11112102 = true := by
+  have h : (thetaBelowCell11112102).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112102 h
+theorem e24KC2ThetaBelowLeaf11112103 :
+    adaptiveCoverCheck 10 thetaBelowCell11112103 = true := by
+  have h : (thetaBelowCell11112103).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112103 h
+theorem e24KC2ThetaBelowLeaf11112110 :
+    adaptiveCoverCheck 10 thetaBelowCell11112110 = true := by
+  have h : (thetaBelowCell11112110).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112110 h
+theorem e24KC2ThetaBelowLeaf11112111 :
+    adaptiveCoverCheck 10 thetaBelowCell11112111 = true := by
+  have h : (thetaBelowCell11112111).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112111 h
+theorem e24KC2ThetaBelowLeaf11112112 :
+    adaptiveCoverCheck 10 thetaBelowCell11112112 = true := by
+  have h : (thetaBelowCell11112112).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112112 h
+theorem e24KC2ThetaBelowLeaf11112113 :
+    adaptiveCoverCheck 10 thetaBelowCell11112113 = true := by
+  have h : (thetaBelowCell11112113).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112113 h
+theorem e24KC2ThetaBelowLeaf111121200 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112120) = true := by
+  have h : ((childLL thetaBelowCell11112120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11112120) h
+theorem e24KC2ThetaBelowLeaf111121201 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112120) = true := by
+  have h : ((childLH thetaBelowCell11112120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11112120) h
+theorem e24KC2ThetaBelowLeaf111121202 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112120) = true := by
+  have h : ((childHL thetaBelowCell11112120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112120) h
+theorem e24KC2ThetaBelowLeaf111121203 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112120) = true := by
+  have h : ((childHH thetaBelowCell11112120)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112120) h
+theorem e24KC2ThetaBelowLeaf111121210 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112121) = true := by
+  have h : ((childLL thetaBelowCell11112121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11112121) h
+theorem e24KC2ThetaBelowLeaf111121211 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112121) = true := by
+  have h : ((childLH thetaBelowCell11112121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11112121) h
+theorem e24KC2ThetaBelowLeaf111121212 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112121) = true := by
+  have h : ((childHL thetaBelowCell11112121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112121) h
+theorem e24KC2ThetaBelowLeaf111121213 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112121) = true := by
+  have h : ((childHH thetaBelowCell11112121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112121) h
+theorem e24KC2ThetaBelowLeaf111121300 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112130) = true := by
+  have h : ((childLL thetaBelowCell11112130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11112130) h
+theorem e24KC2ThetaBelowLeaf111121301 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112130) = true := by
+  have h : ((childLH thetaBelowCell11112130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11112130) h
+theorem e24KC2ThetaBelowLeaf111121302 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112130) = true := by
+  have h : ((childHL thetaBelowCell11112130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112130) h
+theorem e24KC2ThetaBelowLeaf111121303 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112130) = true := by
+  have h : ((childHH thetaBelowCell11112130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112130) h
+theorem e24KC2ThetaBelowLeaf11112131 :
+    adaptiveCoverCheck 10 thetaBelowCell11112131 = true := by
+  have h : (thetaBelowCell11112131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112131 h
+theorem e24KC2ThetaBelowLeaf111121320 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112132) = true := by
+  have h : ((childLL thetaBelowCell11112132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11112132) h
+theorem e24KC2ThetaBelowLeaf111121321 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112132) = true := by
+  have h : ((childLH thetaBelowCell11112132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11112132) h
+theorem e24KC2ThetaBelowLeaf111121330 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112133) = true := by
+  have h : ((childLL thetaBelowCell11112133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11112133) h
+theorem e24KC2ThetaBelowLeaf111121331 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112133) = true := by
+  have h : ((childLH thetaBelowCell11112133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11112133) h
+theorem e24KC2ThetaBelowLeaf111122002 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112200) = true := by
+  have h : ((childHL thetaBelowCell11112200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112200) h
+theorem e24KC2ThetaBelowLeaf111122003 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112200) = true := by
+  have h : ((childHH thetaBelowCell11112200)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112200) h
+theorem e24KC2ThetaBelowLeaf111122012 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112201) = true := by
+  have h : ((childHL thetaBelowCell11112201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112201) h
+theorem e24KC2ThetaBelowLeaf111122013 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112201) = true := by
+  have h : ((childHH thetaBelowCell11112201)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112201) h
+theorem e24KC2ThetaBelowLeaf11112202 :
+    adaptiveCoverCheck 10 thetaBelowCell11112202 = true := by
+  have h : (thetaBelowCell11112202).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112202 h
+theorem e24KC2ThetaBelowLeaf11112203 :
+    adaptiveCoverCheck 10 thetaBelowCell11112203 = true := by
+  have h : (thetaBelowCell11112203).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112203 h
+theorem e24KC2ThetaBelowLeaf111122102 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112210) = true := by
+  have h : ((childHL thetaBelowCell11112210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112210) h
+theorem e24KC2ThetaBelowLeaf111122103 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112210) = true := by
+  have h : ((childHH thetaBelowCell11112210)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112210) h
+theorem e24KC2ThetaBelowLeaf111122112 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112211) = true := by
+  have h : ((childHL thetaBelowCell11112211)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112211) h
+theorem e24KC2ThetaBelowLeaf111122120 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112212) = true := by
+  have h : ((childLL thetaBelowCell11112212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11112212) h
+theorem e24KC2ThetaBelowLeaf111122121 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112212) = true := by
+  have h : ((childLH thetaBelowCell11112212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11112212) h
+theorem e24KC2ThetaBelowLeaf111122122 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112212) = true := by
+  have h : ((childHL thetaBelowCell11112212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112212) h
+theorem e24KC2ThetaBelowLeaf111122123 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112212) = true := by
+  have h : ((childHH thetaBelowCell11112212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112212) h
+theorem e24KC2ThetaBelowLeaf111122130 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112213) = true := by
+  have h : ((childLL thetaBelowCell11112213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11112213) h
+theorem e24KC2ThetaBelowLeaf111122131 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112213) = true := by
+  have h : ((childLH thetaBelowCell11112213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11112213) h
+theorem e24KC2ThetaBelowLeaf111122132 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112213) = true := by
+  have h : ((childHL thetaBelowCell11112213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112213) h
+theorem e24KC2ThetaBelowLeaf111122133 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112213) = true := by
+  have h : ((childHH thetaBelowCell11112213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112213) h
+theorem e24KC2ThetaBelowLeaf11112220 :
+    adaptiveCoverCheck 10 thetaBelowCell11112220 = true := by
+  have h : (thetaBelowCell11112220).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112220 h
+theorem e24KC2ThetaBelowLeaf11112221 :
+    adaptiveCoverCheck 10 thetaBelowCell11112221 = true := by
+  have h : (thetaBelowCell11112221).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112221 h
+theorem e24KC2ThetaBelowLeaf11112222 :
+    adaptiveCoverCheck 10 thetaBelowCell11112222 = true := by
+  have h : (thetaBelowCell11112222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112222 h
+theorem e24KC2ThetaBelowLeaf11112223 :
+    adaptiveCoverCheck 10 thetaBelowCell11112223 = true := by
+  have h : (thetaBelowCell11112223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112223 h
+theorem e24KC2ThetaBelowLeaf11112230 :
+    adaptiveCoverCheck 10 thetaBelowCell11112230 = true := by
+  have h : (thetaBelowCell11112230).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112230 h
+theorem e24KC2ThetaBelowLeaf11112231 :
+    adaptiveCoverCheck 10 thetaBelowCell11112231 = true := by
+  have h : (thetaBelowCell11112231).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112231 h
+theorem e24KC2ThetaBelowLeaf11112232 :
+    adaptiveCoverCheck 10 thetaBelowCell11112232 = true := by
+  have h : (thetaBelowCell11112232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112232 h
+theorem e24KC2ThetaBelowLeaf11112233 :
+    adaptiveCoverCheck 10 thetaBelowCell11112233 = true := by
+  have h : (thetaBelowCell11112233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112233 h
+theorem e24KC2ThetaBelowLeaf111123020 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112302) = true := by
+  have h : ((childLL thetaBelowCell11112302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11112302) h
+theorem e24KC2ThetaBelowLeaf111123021 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112302) = true := by
+  have h : ((childLH thetaBelowCell11112302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11112302) h
+theorem e24KC2ThetaBelowLeaf111123022 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112302) = true := by
+  have h : ((childHL thetaBelowCell11112302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112302) h
+theorem e24KC2ThetaBelowLeaf111123023 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112302) = true := by
+  have h : ((childHH thetaBelowCell11112302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112302) h
+theorem e24KC2ThetaBelowLeaf111123030 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112303) = true := by
+  have h : ((childLL thetaBelowCell11112303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11112303) h
+theorem e24KC2ThetaBelowLeaf111123031 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112303) = true := by
+  have h : ((childLH thetaBelowCell11112303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11112303) h
+theorem e24KC2ThetaBelowLeaf111123032 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112303) = true := by
+  have h : ((childHL thetaBelowCell11112303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112303) h
+theorem e24KC2ThetaBelowLeaf111123033 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112303) = true := by
+  have h : ((childHH thetaBelowCell11112303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112303) h
+theorem e24KC2ThetaBelowLeaf111123120 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112312) = true := by
+  have h : ((childLL thetaBelowCell11112312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11112312) h
+theorem e24KC2ThetaBelowLeaf111123121 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112312) = true := by
+  have h : ((childLH thetaBelowCell11112312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11112312) h
+theorem e24KC2ThetaBelowLeaf111123122 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112312) = true := by
+  have h : ((childHL thetaBelowCell11112312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112312) h
+theorem e24KC2ThetaBelowLeaf111123123 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112312) = true := by
+  have h : ((childHH thetaBelowCell11112312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112312) h
+theorem e24KC2ThetaBelowLeaf111123130 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11112313) = true := by
+  have h : ((childLL thetaBelowCell11112313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11112313) h
+theorem e24KC2ThetaBelowLeaf111123131 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11112313) = true := by
+  have h : ((childLH thetaBelowCell11112313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11112313) h
+theorem e24KC2ThetaBelowLeaf111123132 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11112313) = true := by
+  have h : ((childHL thetaBelowCell11112313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11112313) h
+theorem e24KC2ThetaBelowLeaf111123133 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11112313) = true := by
+  have h : ((childHH thetaBelowCell11112313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11112313) h
+theorem e24KC2ThetaBelowLeaf11112320 :
+    adaptiveCoverCheck 10 thetaBelowCell11112320 = true := by
+  have h : (thetaBelowCell11112320).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112320 h
+theorem e24KC2ThetaBelowLeaf11112321 :
+    adaptiveCoverCheck 10 thetaBelowCell11112321 = true := by
+  have h : (thetaBelowCell11112321).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112321 h
+theorem e24KC2ThetaBelowLeaf11112322 :
+    adaptiveCoverCheck 10 thetaBelowCell11112322 = true := by
+  have h : (thetaBelowCell11112322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112322 h
+theorem e24KC2ThetaBelowLeaf11112323 :
+    adaptiveCoverCheck 10 thetaBelowCell11112323 = true := by
+  have h : (thetaBelowCell11112323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112323 h
+theorem e24KC2ThetaBelowLeaf11112330 :
+    adaptiveCoverCheck 10 thetaBelowCell11112330 = true := by
+  have h : (thetaBelowCell11112330).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112330 h
+theorem e24KC2ThetaBelowLeaf11112331 :
+    adaptiveCoverCheck 10 thetaBelowCell11112331 = true := by
+  have h : (thetaBelowCell11112331).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112331 h
+theorem e24KC2ThetaBelowLeaf11112332 :
+    adaptiveCoverCheck 10 thetaBelowCell11112332 = true := by
+  have h : (thetaBelowCell11112332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112332 h
+theorem e24KC2ThetaBelowLeaf11112333 :
+    adaptiveCoverCheck 10 thetaBelowCell11112333 = true := by
+  have h : (thetaBelowCell11112333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11112333 h
+theorem e24KC2ThetaBelowLeaf1111300 :
+    adaptiveCoverCheck 11 (childLL (childLL (childHH thetaBelowCell1111))) = true := by
+  have h : ((childLL (childLL (childHH thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLL (childHH thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf1111301 :
+    adaptiveCoverCheck 11 (childLH (childLL (childHH thetaBelowCell1111))) = true := by
+  have h : ((childLH (childLL (childHH thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLL (childHH thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf11113020 :
+    adaptiveCoverCheck 10 thetaBelowCell11113020 = true := by
+  have h : (thetaBelowCell11113020).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113020 h
+theorem e24KC2ThetaBelowLeaf11113021 :
+    adaptiveCoverCheck 10 thetaBelowCell11113021 = true := by
+  have h : (thetaBelowCell11113021).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113021 h
+theorem e24KC2ThetaBelowLeaf111130220 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113022) = true := by
+  have h : ((childLL thetaBelowCell11113022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11113022) h
+theorem e24KC2ThetaBelowLeaf111130221 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113022) = true := by
+  have h : ((childLH thetaBelowCell11113022)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11113022) h
+theorem e24KC2ThetaBelowLeaf111130230 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113023) = true := by
+  have h : ((childLL thetaBelowCell11113023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11113023) h
+theorem e24KC2ThetaBelowLeaf111130231 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113023) = true := by
+  have h : ((childLH thetaBelowCell11113023)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11113023) h
+theorem e24KC2ThetaBelowLeaf11113030 :
+    adaptiveCoverCheck 10 thetaBelowCell11113030 = true := by
+  have h : (thetaBelowCell11113030).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113030 h
+theorem e24KC2ThetaBelowLeaf11113031 :
+    adaptiveCoverCheck 10 thetaBelowCell11113031 = true := by
+  have h : (thetaBelowCell11113031).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113031 h
+theorem e24KC2ThetaBelowLeaf111130320 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113032) = true := by
+  have h : ((childLL thetaBelowCell11113032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11113032) h
+theorem e24KC2ThetaBelowLeaf111130321 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113032) = true := by
+  have h : ((childLH thetaBelowCell11113032)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11113032) h
+theorem e24KC2ThetaBelowLeaf111130330 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113033) = true := by
+  have h : ((childLL thetaBelowCell11113033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11113033) h
+theorem e24KC2ThetaBelowLeaf111130331 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113033) = true := by
+  have h : ((childLH thetaBelowCell11113033)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11113033) h
+theorem e24KC2ThetaBelowLeaf1111310 :
+    adaptiveCoverCheck 11 (childLL (childLH (childHH thetaBelowCell1111))) = true := by
+  have h : ((childLL (childLH (childHH thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLH (childHH thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf1111311 :
+    adaptiveCoverCheck 11 (childLH (childLH (childHH thetaBelowCell1111))) = true := by
+  have h : ((childLH (childLH (childHH thetaBelowCell1111)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLH (childHH thetaBelowCell1111))) h
+theorem e24KC2ThetaBelowLeaf11113120 :
+    adaptiveCoverCheck 10 thetaBelowCell11113120 = true := by
+  have h : (thetaBelowCell11113120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113120 h
+theorem e24KC2ThetaBelowLeaf11113121 :
+    adaptiveCoverCheck 10 thetaBelowCell11113121 = true := by
+  have h : (thetaBelowCell11113121).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113121 h
+theorem e24KC2ThetaBelowLeaf111131220 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113122) = true := by
+  have h : ((childLL thetaBelowCell11113122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11113122) h
+theorem e24KC2ThetaBelowLeaf111131221 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113122) = true := by
+  have h : ((childLH thetaBelowCell11113122)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11113122) h
+theorem e24KC2ThetaBelowLeaf111131230 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113123) = true := by
+  have h : ((childLL thetaBelowCell11113123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11113123) h
+theorem e24KC2ThetaBelowLeaf111131231 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113123) = true := by
+  have h : ((childLH thetaBelowCell11113123)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11113123) h
+theorem e24KC2ThetaBelowLeaf11113130 :
+    adaptiveCoverCheck 10 thetaBelowCell11113130 = true := by
+  have h : (thetaBelowCell11113130).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113130 h
+theorem e24KC2ThetaBelowLeaf11113131 :
+    adaptiveCoverCheck 10 thetaBelowCell11113131 = true := by
+  have h : (thetaBelowCell11113131).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113131 h
+theorem e24KC2ThetaBelowLeaf111131320 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113132) = true := by
+  have h : ((childLL thetaBelowCell11113132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11113132) h
+theorem e24KC2ThetaBelowLeaf111131321 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113132) = true := by
+  have h : ((childLH thetaBelowCell11113132)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11113132) h
+theorem e24KC2ThetaBelowLeaf111131330 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113133) = true := by
+  have h : ((childLL thetaBelowCell11113133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11113133) h
+theorem e24KC2ThetaBelowLeaf111131331 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113133) = true := by
+  have h : ((childLH thetaBelowCell11113133)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11113133) h
+theorem e24KC2ThetaBelowLeaf111132020 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113202) = true := by
+  have h : ((childLL thetaBelowCell11113202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11113202) h
+theorem e24KC2ThetaBelowLeaf111132021 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113202) = true := by
+  have h : ((childLH thetaBelowCell11113202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11113202) h
+theorem e24KC2ThetaBelowLeaf111132022 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113202) = true := by
+  have h : ((childHL thetaBelowCell11113202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11113202) h
+theorem e24KC2ThetaBelowLeaf111132023 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113202) = true := by
+  have h : ((childHH thetaBelowCell11113202)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11113202) h
+theorem e24KC2ThetaBelowLeaf111132030 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113203) = true := by
+  have h : ((childLL thetaBelowCell11113203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11113203) h
+theorem e24KC2ThetaBelowLeaf111132031 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113203) = true := by
+  have h : ((childLH thetaBelowCell11113203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11113203) h
+theorem e24KC2ThetaBelowLeaf111132032 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113203) = true := by
+  have h : ((childHL thetaBelowCell11113203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11113203) h
+theorem e24KC2ThetaBelowLeaf111132033 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113203) = true := by
+  have h : ((childHH thetaBelowCell11113203)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11113203) h
+theorem e24KC2ThetaBelowLeaf111132120 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113212) = true := by
+  have h : ((childLL thetaBelowCell11113212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11113212) h
+theorem e24KC2ThetaBelowLeaf111132121 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113212) = true := by
+  have h : ((childLH thetaBelowCell11113212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11113212) h
+theorem e24KC2ThetaBelowLeaf111132122 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113212) = true := by
+  have h : ((childHL thetaBelowCell11113212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11113212) h
+theorem e24KC2ThetaBelowLeaf111132123 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113212) = true := by
+  have h : ((childHH thetaBelowCell11113212)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11113212) h
+theorem e24KC2ThetaBelowLeaf111132130 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113213) = true := by
+  have h : ((childLL thetaBelowCell11113213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11113213) h
+theorem e24KC2ThetaBelowLeaf111132131 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113213) = true := by
+  have h : ((childLH thetaBelowCell11113213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11113213) h
+theorem e24KC2ThetaBelowLeaf111132132 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113213) = true := by
+  have h : ((childHL thetaBelowCell11113213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11113213) h
+theorem e24KC2ThetaBelowLeaf111132133 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113213) = true := by
+  have h : ((childHH thetaBelowCell11113213)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11113213) h
+theorem e24KC2ThetaBelowLeaf11113220 :
+    adaptiveCoverCheck 10 thetaBelowCell11113220 = true := by
+  have h : (thetaBelowCell11113220).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113220 h
+theorem e24KC2ThetaBelowLeaf11113221 :
+    adaptiveCoverCheck 10 thetaBelowCell11113221 = true := by
+  have h : (thetaBelowCell11113221).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113221 h
+theorem e24KC2ThetaBelowLeaf11113222 :
+    adaptiveCoverCheck 10 thetaBelowCell11113222 = true := by
+  have h : (thetaBelowCell11113222).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113222 h
+theorem e24KC2ThetaBelowLeaf11113223 :
+    adaptiveCoverCheck 10 thetaBelowCell11113223 = true := by
+  have h : (thetaBelowCell11113223).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113223 h
+theorem e24KC2ThetaBelowLeaf11113230 :
+    adaptiveCoverCheck 10 thetaBelowCell11113230 = true := by
+  have h : (thetaBelowCell11113230).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113230 h
+theorem e24KC2ThetaBelowLeaf11113231 :
+    adaptiveCoverCheck 10 thetaBelowCell11113231 = true := by
+  have h : (thetaBelowCell11113231).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113231 h
+theorem e24KC2ThetaBelowLeaf11113232 :
+    adaptiveCoverCheck 10 thetaBelowCell11113232 = true := by
+  have h : (thetaBelowCell11113232).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113232 h
+theorem e24KC2ThetaBelowLeaf11113233 :
+    adaptiveCoverCheck 10 thetaBelowCell11113233 = true := by
+  have h : (thetaBelowCell11113233).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113233 h
+theorem e24KC2ThetaBelowLeaf111133020 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113302) = true := by
+  have h : ((childLL thetaBelowCell11113302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11113302) h
+theorem e24KC2ThetaBelowLeaf111133021 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113302) = true := by
+  have h : ((childLH thetaBelowCell11113302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11113302) h
+theorem e24KC2ThetaBelowLeaf111133022 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113302) = true := by
+  have h : ((childHL thetaBelowCell11113302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11113302) h
+theorem e24KC2ThetaBelowLeaf111133023 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113302) = true := by
+  have h : ((childHH thetaBelowCell11113302)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11113302) h
+theorem e24KC2ThetaBelowLeaf111133030 :
+    adaptiveCoverCheck 9 (childLL thetaBelowCell11113303) = true := by
+  have h : ((childLL thetaBelowCell11113303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLL thetaBelowCell11113303) h
+theorem e24KC2ThetaBelowLeaf111133031 :
+    adaptiveCoverCheck 9 (childLH thetaBelowCell11113303) = true := by
+  have h : ((childLH thetaBelowCell11113303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childLH thetaBelowCell11113303) h
+theorem e24KC2ThetaBelowLeaf111133032 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113303) = true := by
+  have h : ((childHL thetaBelowCell11113303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11113303) h
+theorem e24KC2ThetaBelowLeaf111133033 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113303) = true := by
+  have h : ((childHH thetaBelowCell11113303)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11113303) h
+theorem e24KC2ThetaBelowLeaf111133122 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113312) = true := by
+  have h : ((childHL thetaBelowCell11113312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11113312) h
+theorem e24KC2ThetaBelowLeaf111133123 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113312) = true := by
+  have h : ((childHH thetaBelowCell11113312)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11113312) h
+theorem e24KC2ThetaBelowLeaf111133132 :
+    adaptiveCoverCheck 9 (childHL thetaBelowCell11113313) = true := by
+  have h : ((childHL thetaBelowCell11113313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHL thetaBelowCell11113313) h
+theorem e24KC2ThetaBelowLeaf111133133 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113313) = true := by
+  have h : ((childHH thetaBelowCell11113313)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 9 (childHH thetaBelowCell11113313) h
+theorem e24KC2ThetaBelowLeaf11113320 :
+    adaptiveCoverCheck 10 thetaBelowCell11113320 = true := by
+  have h : (thetaBelowCell11113320).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113320 h
+theorem e24KC2ThetaBelowLeaf11113321 :
+    adaptiveCoverCheck 10 thetaBelowCell11113321 = true := by
+  have h : (thetaBelowCell11113321).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113321 h
+theorem e24KC2ThetaBelowLeaf11113322 :
+    adaptiveCoverCheck 10 thetaBelowCell11113322 = true := by
+  have h : (thetaBelowCell11113322).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113322 h
+theorem e24KC2ThetaBelowLeaf11113323 :
+    adaptiveCoverCheck 10 thetaBelowCell11113323 = true := by
+  have h : (thetaBelowCell11113323).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113323 h
+theorem e24KC2ThetaBelowLeaf11113330 :
+    adaptiveCoverCheck 10 thetaBelowCell11113330 = true := by
+  have h : (thetaBelowCell11113330).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113330 h
+theorem e24KC2ThetaBelowLeaf11113331 :
+    adaptiveCoverCheck 10 thetaBelowCell11113331 = true := by
+  have h : (thetaBelowCell11113331).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113331 h
+theorem e24KC2ThetaBelowLeaf11113332 :
+    adaptiveCoverCheck 10 thetaBelowCell11113332 = true := by
+  have h : (thetaBelowCell11113332).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113332 h
+theorem e24KC2ThetaBelowLeaf11113333 :
+    adaptiveCoverCheck 10 thetaBelowCell11113333 = true := by
+  have h : (thetaBelowCell11113333).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 10 thetaBelowCell11113333 h
+theorem e24KC2ThetaBelowLeaf111200 :
+    adaptiveCoverCheck 12 (childLL (childLL thetaBelowCell1112)) = true := by
+  have h : ((childLL (childLL thetaBelowCell1112))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLL (childLL thetaBelowCell1112)) h
+theorem e24KC2ThetaBelowLeaf111201 :
+    adaptiveCoverCheck 12 (childLH (childLL thetaBelowCell1112)) = true := by
+  have h : ((childLH (childLL thetaBelowCell1112))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childLH (childLL thetaBelowCell1112)) h
+theorem e24KC2ThetaBelowLeaf111202 :
+    adaptiveCoverCheck 12 (childHL (childLL thetaBelowCell1112)) = true := by
+  have h : ((childHL (childLL thetaBelowCell1112))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL thetaBelowCell1112)) h
+theorem e24KC2ThetaBelowLeaf111203 :
+    adaptiveCoverCheck 12 (childHH (childLL thetaBelowCell1112)) = true := by
+  have h : ((childHH (childLL thetaBelowCell1112))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL thetaBelowCell1112)) h
+theorem e24KC2ThetaBelowLeaf1112100 :
+    adaptiveCoverCheck 11 (childLL (childLL (childLH thetaBelowCell1112))) = true := by
+  have h : ((childLL (childLL (childLH thetaBelowCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLL (childLH thetaBelowCell1112))) h
+theorem e24KC2ThetaBelowLeaf1112101 :
+    adaptiveCoverCheck 11 (childLH (childLL (childLH thetaBelowCell1112))) = true := by
+  have h : ((childLH (childLL (childLH thetaBelowCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLL (childLH thetaBelowCell1112))) h
+theorem e24KC2ThetaBelowLeaf1112102 :
+    adaptiveCoverCheck 11 (childHL (childLL (childLH thetaBelowCell1112))) = true := by
+  have h : ((childHL (childLL (childLH thetaBelowCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLL (childLH thetaBelowCell1112))) h
+theorem e24KC2ThetaBelowLeaf1112103 :
+    adaptiveCoverCheck 11 (childHH (childLL (childLH thetaBelowCell1112))) = true := by
+  have h : ((childHH (childLL (childLH thetaBelowCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLL (childLH thetaBelowCell1112))) h
+theorem e24KC2ThetaBelowLeaf1112110 :
+    adaptiveCoverCheck 11 (childLL (childLH (childLH thetaBelowCell1112))) = true := by
+  have h : ((childLL (childLH (childLH thetaBelowCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLH (childLH thetaBelowCell1112))) h
+theorem e24KC2ThetaBelowLeaf1112111 :
+    adaptiveCoverCheck 11 (childLH (childLH (childLH thetaBelowCell1112))) = true := by
+  have h : ((childLH (childLH (childLH thetaBelowCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLH (childLH thetaBelowCell1112))) h
+theorem e24KC2ThetaBelowLeaf1112112 :
+    adaptiveCoverCheck 11 (childHL (childLH (childLH thetaBelowCell1112))) = true := by
+  have h : ((childHL (childLH (childLH thetaBelowCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLH (childLH thetaBelowCell1112))) h
+theorem e24KC2ThetaBelowLeaf1112113 :
+    adaptiveCoverCheck 11 (childHH (childLH (childLH thetaBelowCell1112))) = true := by
+  have h : ((childHH (childLH (childLH thetaBelowCell1112)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLH (childLH thetaBelowCell1112))) h
+theorem e24KC2ThetaBelowLeaf111212 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell1112)) = true := by
+  have h : ((childHL (childLH thetaBelowCell1112))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH thetaBelowCell1112)) h
+theorem e24KC2ThetaBelowLeaf111213 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell1112)) = true := by
+  have h : ((childHH (childLH thetaBelowCell1112))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH thetaBelowCell1112)) h
+theorem e24KC2ThetaBelowLeaf11122 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1112) = true := by
+  have h : ((childHL thetaBelowCell1112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell1112) h
+theorem e24KC2ThetaBelowLeaf11123 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1112) = true := by
+  have h : ((childHH thetaBelowCell1112)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell1112) h
+theorem e24KC2ThetaBelowLeaf1113000 :
+    adaptiveCoverCheck 11 (childLL (childLL (childLL thetaBelowCell1113))) = true := by
+  have h : ((childLL (childLL (childLL thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLL (childLL thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf1113001 :
+    adaptiveCoverCheck 11 (childLH (childLL (childLL thetaBelowCell1113))) = true := by
+  have h : ((childLH (childLL (childLL thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLL (childLL thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf1113002 :
+    adaptiveCoverCheck 11 (childHL (childLL (childLL thetaBelowCell1113))) = true := by
+  have h : ((childHL (childLL (childLL thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLL (childLL thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf1113003 :
+    adaptiveCoverCheck 11 (childHH (childLL (childLL thetaBelowCell1113))) = true := by
+  have h : ((childHH (childLL (childLL thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLL (childLL thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf1113010 :
+    adaptiveCoverCheck 11 (childLL (childLH (childLL thetaBelowCell1113))) = true := by
+  have h : ((childLL (childLH (childLL thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLH (childLL thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf1113011 :
+    adaptiveCoverCheck 11 (childLH (childLH (childLL thetaBelowCell1113))) = true := by
+  have h : ((childLH (childLH (childLL thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLH (childLL thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf1113012 :
+    adaptiveCoverCheck 11 (childHL (childLH (childLL thetaBelowCell1113))) = true := by
+  have h : ((childHL (childLH (childLL thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLH (childLL thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf1113013 :
+    adaptiveCoverCheck 11 (childHH (childLH (childLL thetaBelowCell1113))) = true := by
+  have h : ((childHH (childLH (childLL thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLH (childLL thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf111302 :
+    adaptiveCoverCheck 12 (childHL (childLL thetaBelowCell1113)) = true := by
+  have h : ((childHL (childLL thetaBelowCell1113))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLL thetaBelowCell1113)) h
+theorem e24KC2ThetaBelowLeaf111303 :
+    adaptiveCoverCheck 12 (childHH (childLL thetaBelowCell1113)) = true := by
+  have h : ((childHH (childLL thetaBelowCell1113))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLL thetaBelowCell1113)) h
+theorem e24KC2ThetaBelowLeaf1113100 :
+    adaptiveCoverCheck 11 (childLL (childLL (childLH thetaBelowCell1113))) = true := by
+  have h : ((childLL (childLL (childLH thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLL (childLH thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf1113101 :
+    adaptiveCoverCheck 11 (childLH (childLL (childLH thetaBelowCell1113))) = true := by
+  have h : ((childLH (childLL (childLH thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLL (childLH thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf1113102 :
+    adaptiveCoverCheck 11 (childHL (childLL (childLH thetaBelowCell1113))) = true := by
+  have h : ((childHL (childLL (childLH thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLL (childLH thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf1113103 :
+    adaptiveCoverCheck 11 (childHH (childLL (childLH thetaBelowCell1113))) = true := by
+  have h : ((childHH (childLL (childLH thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLL (childLH thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf1113110 :
+    adaptiveCoverCheck 11 (childLL (childLH (childLH thetaBelowCell1113))) = true := by
+  have h : ((childLL (childLH (childLH thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLL (childLH (childLH thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf1113111 :
+    adaptiveCoverCheck 11 (childLH (childLH (childLH thetaBelowCell1113))) = true := by
+  have h : ((childLH (childLH (childLH thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childLH (childLH (childLH thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf1113112 :
+    adaptiveCoverCheck 11 (childHL (childLH (childLH thetaBelowCell1113))) = true := by
+  have h : ((childHL (childLH (childLH thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHL (childLH (childLH thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf1113113 :
+    adaptiveCoverCheck 11 (childHH (childLH (childLH thetaBelowCell1113))) = true := by
+  have h : ((childHH (childLH (childLH thetaBelowCell1113)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 11 (childHH (childLH (childLH thetaBelowCell1113))) h
+theorem e24KC2ThetaBelowLeaf111312 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell1113)) = true := by
+  have h : ((childHL (childLH thetaBelowCell1113))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHL (childLH thetaBelowCell1113)) h
+theorem e24KC2ThetaBelowLeaf111313 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell1113)) = true := by
+  have h : ((childHH (childLH thetaBelowCell1113))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 12 (childHH (childLH thetaBelowCell1113)) h
+theorem e24KC2ThetaBelowLeaf11132 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1113) = true := by
+  have h : ((childHL thetaBelowCell1113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell1113) h
+theorem e24KC2ThetaBelowLeaf11133 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1113) = true := by
+  have h : ((childHH thetaBelowCell1113)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell1113) h
+theorem e24KC2ThetaBelowLeaf1120 :
+    adaptiveCoverCheck 14 thetaBelowCell1120 = true := by
+  have h : (thetaBelowCell1120).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1120 h
+theorem e24KC2ThetaBelowLeaf11210 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1121) = true := by
+  have h : ((childLL thetaBelowCell1121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell1121) h
+theorem e24KC2ThetaBelowLeaf11211 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1121) = true := by
+  have h : ((childLH thetaBelowCell1121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH thetaBelowCell1121) h
+theorem e24KC2ThetaBelowLeaf11212 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1121) = true := by
+  have h : ((childHL thetaBelowCell1121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell1121) h
+theorem e24KC2ThetaBelowLeaf11213 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1121) = true := by
+  have h : ((childHH thetaBelowCell1121)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell1121) h
+theorem e24KC2ThetaBelowLeaf1122 :
+    adaptiveCoverCheck 14 thetaBelowCell1122 = true := by
+  have h : (thetaBelowCell1122).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1122 h
+theorem e24KC2ThetaBelowLeaf1123 :
+    adaptiveCoverCheck 14 thetaBelowCell1123 = true := by
+  have h : (thetaBelowCell1123).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1123 h
+theorem e24KC2ThetaBelowLeaf11300 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1130) = true := by
+  have h : ((childLL thetaBelowCell1130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell1130) h
+theorem e24KC2ThetaBelowLeaf11301 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1130) = true := by
+  have h : ((childLH thetaBelowCell1130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH thetaBelowCell1130) h
+theorem e24KC2ThetaBelowLeaf11302 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1130) = true := by
+  have h : ((childHL thetaBelowCell1130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell1130) h
+theorem e24KC2ThetaBelowLeaf11303 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1130) = true := by
+  have h : ((childHH thetaBelowCell1130)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell1130) h
+theorem e24KC2ThetaBelowLeaf11310 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1131) = true := by
+  have h : ((childLL thetaBelowCell1131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLL thetaBelowCell1131) h
+theorem e24KC2ThetaBelowLeaf11311 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1131) = true := by
+  have h : ((childLH thetaBelowCell1131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childLH thetaBelowCell1131) h
+theorem e24KC2ThetaBelowLeaf11312 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1131) = true := by
+  have h : ((childHL thetaBelowCell1131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHL thetaBelowCell1131) h
+theorem e24KC2ThetaBelowLeaf11313 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1131) = true := by
+  have h : ((childHH thetaBelowCell1131)).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 13 (childHH thetaBelowCell1131) h
+theorem e24KC2ThetaBelowLeaf1132 :
+    adaptiveCoverCheck 14 thetaBelowCell1132 = true := by
+  have h : (thetaBelowCell1132).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1132 h
+theorem e24KC2ThetaBelowLeaf1133 :
+    adaptiveCoverCheck 14 thetaBelowCell1133 = true := by
+  have h : (thetaBelowCell1133).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1133 h
+theorem e24KC2ThetaBelowLeaf120 :
+    adaptiveCoverCheck 15 (childLL (childHL (childLH e24ThetaBelowRoot))) = true := by
+  have h : ((childLL (childHL (childLH e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childLL (childHL (childLH e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf121 :
+    adaptiveCoverCheck 15 (childLH (childHL (childLH e24ThetaBelowRoot))) = true := by
+  have h : ((childLH (childHL (childLH e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childLH (childHL (childLH e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf122 :
+    adaptiveCoverCheck 15 (childHL (childHL (childLH e24ThetaBelowRoot))) = true := by
+  have h : ((childHL (childHL (childLH e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childHL (childHL (childLH e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf123 :
+    adaptiveCoverCheck 15 (childHH (childHL (childLH e24ThetaBelowRoot))) = true := by
+  have h : ((childHH (childHL (childLH e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childHH (childHL (childLH e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf1300 :
+    adaptiveCoverCheck 14 thetaBelowCell1300 = true := by
+  have h : (thetaBelowCell1300).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1300 h
+theorem e24KC2ThetaBelowLeaf1301 :
+    adaptiveCoverCheck 14 thetaBelowCell1301 = true := by
+  have h : (thetaBelowCell1301).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1301 h
+theorem e24KC2ThetaBelowLeaf1302 :
+    adaptiveCoverCheck 14 thetaBelowCell1302 = true := by
+  have h : (thetaBelowCell1302).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1302 h
+theorem e24KC2ThetaBelowLeaf1303 :
+    adaptiveCoverCheck 14 thetaBelowCell1303 = true := by
+  have h : (thetaBelowCell1303).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1303 h
+theorem e24KC2ThetaBelowLeaf1310 :
+    adaptiveCoverCheck 14 thetaBelowCell1310 = true := by
+  have h : (thetaBelowCell1310).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1310 h
+theorem e24KC2ThetaBelowLeaf1311 :
+    adaptiveCoverCheck 14 thetaBelowCell1311 = true := by
+  have h : (thetaBelowCell1311).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1311 h
+theorem e24KC2ThetaBelowLeaf1312 :
+    adaptiveCoverCheck 14 thetaBelowCell1312 = true := by
+  have h : (thetaBelowCell1312).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1312 h
+theorem e24KC2ThetaBelowLeaf1313 :
+    adaptiveCoverCheck 14 thetaBelowCell1313 = true := by
+  have h : (thetaBelowCell1313).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 14 thetaBelowCell1313 h
+theorem e24KC2ThetaBelowLeaf132 :
+    adaptiveCoverCheck 15 (childHL (childHH (childLH e24ThetaBelowRoot))) = true := by
+  have h : ((childHL (childHH (childLH e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childHL (childHH (childLH e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf133 :
+    adaptiveCoverCheck 15 (childHH (childHH (childLH e24ThetaBelowRoot))) = true := by
+  have h : ((childHH (childHH (childLH e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childHH (childHH (childLH e24ThetaBelowRoot))) h
+
+theorem e24KC2ThetaBelowLeaf2 :
+    adaptiveCoverCheck 17 (childHL e24ThetaBelowRoot) = true := by
+  have h : physicallyIrrelevant (childHL e24ThetaBelowRoot) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 17 (childHL e24ThetaBelowRoot) h
+theorem e24KC2ThetaBelowLeaf300 :
+    adaptiveCoverCheck 15 (childLL (childLL (childHH e24ThetaBelowRoot))) = true := by
+  have h : ((childLL (childLL (childHH e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childLL (childLL (childHH e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf301 :
+    adaptiveCoverCheck 15 (childLH (childLL (childHH e24ThetaBelowRoot))) = true := by
+  have h : ((childLH (childLL (childHH e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childLH (childLL (childHH e24ThetaBelowRoot))) h
+
+theorem e24KC2ThetaBelowLeaf302 :
+    adaptiveCoverCheck 15 (childHL (childLL (childHH e24ThetaBelowRoot))) = true := by
+  have h : physicallyIrrelevant (childHL (childLL (childHH e24ThetaBelowRoot))) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 15 (childHL (childLL (childHH
+    e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf303 :
+    adaptiveCoverCheck 15 (childHH (childLL (childHH e24ThetaBelowRoot))) = true := by
+  have h : ((childHH (childLL (childHH e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childHH (childLL (childHH e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf310 :
+    adaptiveCoverCheck 15 (childLL (childLH (childHH e24ThetaBelowRoot))) = true := by
+  have h : ((childLL (childLH (childHH e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childLL (childLH (childHH e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf311 :
+    adaptiveCoverCheck 15 (childLH (childLH (childHH e24ThetaBelowRoot))) = true := by
+  have h : ((childLH (childLH (childHH e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childLH (childLH (childHH e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf312 :
+    adaptiveCoverCheck 15 (childHL (childLH (childHH e24ThetaBelowRoot))) = true := by
+  have h : ((childHL (childLH (childHH e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childHL (childLH (childHH e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf313 :
+    adaptiveCoverCheck 15 (childHH (childLH (childHH e24ThetaBelowRoot))) = true := by
+  have h : ((childHH (childLH (childHH e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childHH (childLH (childHH e24ThetaBelowRoot))) h
+
+theorem e24KC2ThetaBelowLeaf32 :
+    adaptiveCoverCheck 16 (childHL (childHH e24ThetaBelowRoot)) = true := by
+  have h : physicallyIrrelevant (childHL (childHH e24ThetaBelowRoot)) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 16 (childHL (childHH e24ThetaBelowRoot)) h
+
+theorem e24KC2ThetaBelowLeaf330 :
+    adaptiveCoverCheck 15 (childLL (childHH (childHH e24ThetaBelowRoot))) = true := by
+  have h : physicallyIrrelevant (childLL (childHH (childHH e24ThetaBelowRoot))) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 15 (childLL (childHH (childHH
+    e24ThetaBelowRoot))) h
+theorem e24KC2ThetaBelowLeaf331 :
+    adaptiveCoverCheck 15 (childLH (childHH (childHH e24ThetaBelowRoot))) = true := by
+  have h : ((childLH (childHH (childHH e24ThetaBelowRoot)))).rejected = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_rejected 15 (childLH (childHH (childHH e24ThetaBelowRoot))) h
+
+theorem e24KC2ThetaBelowLeaf332 :
+    adaptiveCoverCheck 15 (childHL (childHH (childHH e24ThetaBelowRoot))) = true := by
+  have h : physicallyIrrelevant (childHL (childHH (childHH e24ThetaBelowRoot))) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 15 (childHL (childHH (childHH
+    e24ThetaBelowRoot))) h
+
+theorem e24KC2ThetaBelowLeaf333 :
+    adaptiveCoverCheck 15 (childHH (childHH (childHH e24ThetaBelowRoot))) = true := by
+  have h : physicallyIrrelevant (childHH (childHH (childHH e24ThetaBelowRoot))) = true := by
+    decide +kernel
+  exact adaptiveCoverCheck_true_of_physicallyIrrelevant 15 (childHH (childHH (childHH
+    e24ThetaBelowRoot))) h
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6ProofBatch38ee6acacaee22e4`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc9f578345f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc9f578345f
+
+open CertificateCellsc9f578345f
+theorem cover_subtree_27436d0eee02 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022003100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003100)
+    (by
+      have h : ((childLL (childLL thetaAboveCell000022003100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL thetaAboveCell000022003100)) h)
+    (by
+      have h : ((childLH (childLL thetaAboveCell000022003100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL thetaAboveCell000022003100)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLL thetaAboveCell000022003100))
+        (by
+          have h : ((childLL (childHL (childLL thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLL
+            thetaAboveCell000022003100))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLL
+            thetaAboveCell000022003100))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLL
+            thetaAboveCell000022003100))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLL
+            thetaAboveCell000022003100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLL thetaAboveCell000022003100))
+        (by
+          have h : ((childLL (childHH (childLL thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLL
+            thetaAboveCell000022003100))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLL
+            thetaAboveCell000022003100))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLL
+            thetaAboveCell000022003100))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLL
+            thetaAboveCell000022003100))) h))
+
+theorem cover_subtree_3bcbb01846ea :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022003100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003100)
+    (by
+      have h : ((childLL (childLH thetaAboveCell000022003100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH thetaAboveCell000022003100)) h)
+    (by
+      have h : ((childLH (childLH thetaAboveCell000022003100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH thetaAboveCell000022003100)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLH thetaAboveCell000022003100))
+        (by
+          have h : ((childLL (childHL (childLH thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLH
+            thetaAboveCell000022003100))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLH
+            thetaAboveCell000022003100))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLH
+            thetaAboveCell000022003100))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLH
+            thetaAboveCell000022003100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLH thetaAboveCell000022003100))
+        (by
+          have h : ((childLL (childHH (childLH thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLH
+            thetaAboveCell000022003100))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLH
+            thetaAboveCell000022003100))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLH
+            thetaAboveCell000022003100))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaAboveCell000022003100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLH
+            thetaAboveCell000022003100))) h))
+
+theorem cover_subtree_ffb85ef67591 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHL thetaAboveCell000022003100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+    thetaAboveCell000022003100)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002020
+        (by
+          have h : ((childLL thetaAboveCell0000220031002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002021
+        (by
+          have h : ((childLL thetaAboveCell0000220031002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002022
+        (by
+          have h : ((childLL thetaAboveCell0000220031002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002023
+        (by
+          have h : ((childLL thetaAboveCell0000220031002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002023) h))
+
+theorem cover_subtree_86bc98112499 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHL thetaAboveCell000022003100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+    thetaAboveCell000022003100)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002030
+        (by
+          have h : ((childLL thetaAboveCell0000220031002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002031
+        (by
+          have h : ((childLL thetaAboveCell0000220031002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002032
+        (by
+          have h : ((childLL thetaAboveCell0000220031002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002033
+        (by
+          have h : ((childLL thetaAboveCell0000220031002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002033) h))
+
+theorem cover_subtree_7cd81c60142e :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022003100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022003100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHL
+        thetaAboveCell000022003100)))
+        (by
+          have h : (thetaAboveCell0000220031002000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002000 h)
+        (by
+          have h : (thetaAboveCell0000220031002001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002001 h)
+        (by
+          have h : (thetaAboveCell0000220031002002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002002 h)
+        (by
+          have h : (thetaAboveCell0000220031002003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHL
+        thetaAboveCell000022003100)))
+        (by
+          have h : (thetaAboveCell0000220031002010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002010 h)
+        (by
+          have h : (thetaAboveCell0000220031002011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002011 h)
+        (by
+          have h : (thetaAboveCell0000220031002012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002012 h)
+        (by
+          have h : (thetaAboveCell0000220031002013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002013 h))
+    cover_subtree_ffb85ef67591
+    cover_subtree_86bc98112499
+
+theorem cover_subtree_5fb03cd03eba :
+    adaptiveCoverCheck 4 (childHL (childLH (childHL thetaAboveCell000022003100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+    thetaAboveCell000022003100)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002120
+        (by
+          have h : ((childLL thetaAboveCell0000220031002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002121
+        (by
+          have h : ((childLL thetaAboveCell0000220031002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002122
+        (by
+          have h : ((childLL thetaAboveCell0000220031002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002123
+        (by
+          have h : ((childLL thetaAboveCell0000220031002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002123) h))
+
+theorem cover_subtree_4ea44b529295 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHL thetaAboveCell000022003100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+    thetaAboveCell000022003100)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002130
+        (by
+          have h : ((childLL thetaAboveCell0000220031002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002131
+        (by
+          have h : ((childLL thetaAboveCell0000220031002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002132
+        (by
+          have h : ((childLL thetaAboveCell0000220031002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002133
+        (by
+          have h : ((childLL thetaAboveCell0000220031002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002133) h))
+
+theorem cover_subtree_db4866b2cd56 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022003100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022003100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHL
+        thetaAboveCell000022003100)))
+        (by
+          have h : (thetaAboveCell0000220031002100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002100 h)
+        (by
+          have h : (thetaAboveCell0000220031002101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002101 h)
+        (by
+          have h : (thetaAboveCell0000220031002102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002102 h)
+        (by
+          have h : (thetaAboveCell0000220031002103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHL
+        thetaAboveCell000022003100)))
+        (by
+          have h : (thetaAboveCell0000220031002110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002110 h)
+        (by
+          have h : (thetaAboveCell0000220031002111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002111 h)
+        (by
+          have h : (thetaAboveCell0000220031002112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002112 h)
+        (by
+          have h : (thetaAboveCell0000220031002113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002113 h))
+    cover_subtree_5fb03cd03eba
+    cover_subtree_4ea44b529295
+
+theorem cover_subtree_da1ce2a1ea10 :
+    adaptiveCoverCheck 4 (childLL (childHL (childHL thetaAboveCell000022003100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+    thetaAboveCell000022003100)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002200
+        (by
+          have h : ((childLL thetaAboveCell0000220031002200)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002200) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002200)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002200) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002200)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002200) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002200)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002200) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002201
+        (by
+          have h : ((childLL thetaAboveCell0000220031002201)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002201) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002201)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002201) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002201)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002201) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002201)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002201) h))
+    (by
+      have h : (thetaAboveCell0000220031002202).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002202 h)
+    (by
+      have h : (thetaAboveCell0000220031002203).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002203 h)
+
+theorem cover_subtree_ea6d149e7e1e :
+    adaptiveCoverCheck 4 (childLH (childHL (childHL thetaAboveCell000022003100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+    thetaAboveCell000022003100)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002210
+        (by
+          have h : ((childLL thetaAboveCell0000220031002210)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002210) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002210)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002210) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002210)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002210) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002210)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002210) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002211
+        (by
+          have h : ((childLL thetaAboveCell0000220031002211)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002211) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031002211)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002211) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031002211)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002211) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031002211)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002211) h))
+    (by
+      have h : (thetaAboveCell0000220031002212).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002212 h)
+    (by
+      have h : (thetaAboveCell0000220031002213).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002213 h)
+
+theorem cover_subtree_5106e0b16eef :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022003100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022003100))
+    cover_subtree_da1ce2a1ea10
+    cover_subtree_ea6d149e7e1e
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022003100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022003100))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022003100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022003100))) h)
+
+theorem cover_subtree_a4e3e0266d1a :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022003100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022003100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022003100)))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031002300
+            (by
+              have h : ((childLL thetaAboveCell0000220031002300)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031002300)
+                h)
+            (by
+              have h : ((childLH thetaAboveCell0000220031002300)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031002300)
+                h)
+            (by
+              have h : ((childHL thetaAboveCell0000220031002300)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031002300)
+                h)
+            (by
+              have h : ((childHH thetaAboveCell0000220031002300)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031002300)
+                h))
+        (by
+          have h : (thetaAboveCell0000220031002301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002301 h)
+        (by
+          have h : (thetaAboveCell0000220031002302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002302 h)
+        (by
+          have h : (thetaAboveCell0000220031002303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022003100)))
+        (by
+          have h : (thetaAboveCell0000220031002310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002310 h)
+        (by
+          have h : (thetaAboveCell0000220031002311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002311 h)
+        (by
+          have h : (thetaAboveCell0000220031002312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002312 h)
+        (by
+          have h : (thetaAboveCell0000220031002313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031002313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022003100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022003100))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022003100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022003100))) h)
+
+theorem cover_subtree_e3fd3e08cd04 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022003100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022003100)
+    cover_subtree_7cd81c60142e
+    cover_subtree_db4866b2cd56
+    cover_subtree_5106e0b16eef
+    cover_subtree_a4e3e0266d1a
+
+theorem cover_subtree_fed900b890f0 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHH thetaAboveCell000022003100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+    thetaAboveCell000022003100)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003020
+        (by
+          have h : ((childLL thetaAboveCell0000220031003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003021
+        (by
+          have h : ((childLL thetaAboveCell0000220031003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003022
+        (by
+          have h : ((childLL thetaAboveCell0000220031003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003023
+        (by
+          have h : ((childLL thetaAboveCell0000220031003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003023) h))
+
+theorem cover_subtree_f5a67bd75b07 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHH thetaAboveCell000022003100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+    thetaAboveCell000022003100)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003030
+        (by
+          have h : ((childLL thetaAboveCell0000220031003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003031
+        (by
+          have h : ((childLL thetaAboveCell0000220031003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003032
+        (by
+          have h : ((childLL thetaAboveCell0000220031003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003033
+        (by
+          have h : ((childLL thetaAboveCell0000220031003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003033) h))
+
+theorem cover_subtree_dc2ce2de1a67 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022003100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022003100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHH
+        thetaAboveCell000022003100)))
+        (by
+          have h : (thetaAboveCell0000220031003000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003000 h)
+        (by
+          have h : (thetaAboveCell0000220031003001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003001 h)
+        (by
+          have h : (thetaAboveCell0000220031003002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003002 h)
+        (by
+          have h : (thetaAboveCell0000220031003003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHH
+        thetaAboveCell000022003100)))
+        (by
+          have h : (thetaAboveCell0000220031003010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003010 h)
+        (by
+          have h : (thetaAboveCell0000220031003011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003011 h)
+        (by
+          have h : (thetaAboveCell0000220031003012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003012 h)
+        (by
+          have h : (thetaAboveCell0000220031003013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003013 h))
+    cover_subtree_fed900b890f0
+    cover_subtree_f5a67bd75b07
+
+theorem cover_subtree_ecb8f710d653 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHH thetaAboveCell000022003100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+    thetaAboveCell000022003100)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003120
+        (by
+          have h : ((childLL thetaAboveCell0000220031003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003121
+        (by
+          have h : ((childLL thetaAboveCell0000220031003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003122
+        (by
+          have h : ((childLL thetaAboveCell0000220031003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003123
+        (by
+          have h : ((childLL thetaAboveCell0000220031003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003123) h))
+
+theorem cover_subtree_0f60a6ff8480 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHH thetaAboveCell000022003100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+    thetaAboveCell000022003100)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003130
+        (by
+          have h : ((childLL thetaAboveCell0000220031003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003131
+        (by
+          have h : ((childLL thetaAboveCell0000220031003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003132
+        (by
+          have h : ((childLL thetaAboveCell0000220031003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031003133
+        (by
+          have h : ((childLL thetaAboveCell0000220031003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031003133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031003133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031003133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031003133) h))
+
+theorem cover_subtree_12980021aa15 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022003100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022003100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHH
+        thetaAboveCell000022003100)))
+        (by
+          have h : (thetaAboveCell0000220031003100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003100 h)
+        (by
+          have h : (thetaAboveCell0000220031003101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003101 h)
+        (by
+          have h : (thetaAboveCell0000220031003102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003102 h)
+        (by
+          have h : (thetaAboveCell0000220031003103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHH
+        thetaAboveCell000022003100)))
+        (by
+          have h : (thetaAboveCell0000220031003110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003110 h)
+        (by
+          have h : (thetaAboveCell0000220031003111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003111 h)
+        (by
+          have h : (thetaAboveCell0000220031003112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003112 h)
+        (by
+          have h : (thetaAboveCell0000220031003113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003113 h))
+    cover_subtree_ecb8f710d653
+    cover_subtree_0f60a6ff8480
+
+theorem cover_subtree_96b385ccf927 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022003100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022003100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022003100)))
+        (by
+          have h : (thetaAboveCell0000220031003200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003200 h)
+        (by
+          have h : (thetaAboveCell0000220031003201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003201 h)
+        (by
+          have h : (thetaAboveCell0000220031003202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003202 h)
+        (by
+          have h : (thetaAboveCell0000220031003203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022003100)))
+        (by
+          have h : (thetaAboveCell0000220031003210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003210 h)
+        (by
+          have h : (thetaAboveCell0000220031003211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003211 h)
+        (by
+          have h : (thetaAboveCell0000220031003212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003212 h)
+        (by
+          have h : (thetaAboveCell0000220031003213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022003100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022003100))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022003100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022003100))) h)
+
+theorem cover_subtree_22b398081cb0 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022003100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022003100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022003100)))
+        (by
+          have h : (thetaAboveCell0000220031003300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003300 h)
+        (by
+          have h : (thetaAboveCell0000220031003301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003301 h)
+        (by
+          have h : (thetaAboveCell0000220031003302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003302 h)
+        (by
+          have h : (thetaAboveCell0000220031003303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022003100)))
+        (by
+          have h : (thetaAboveCell0000220031003310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003310 h)
+        (by
+          have h : (thetaAboveCell0000220031003311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003311 h)
+        (by
+          have h : (thetaAboveCell0000220031003312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003312 h)
+        (by
+          have h : (thetaAboveCell0000220031003313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031003313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022003100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022003100))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022003100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022003100))) h)
+
+theorem cover_subtree_b9c469e743af :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022003100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022003100)
+    cover_subtree_dc2ce2de1a67
+    cover_subtree_12980021aa15
+    cover_subtree_96b385ccf927
+    cover_subtree_22b398081cb0
+
+theorem cover_subtree_2d90eed026dd :
+    adaptiveCoverCheck 7 thetaAboveCell000022003100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003100
+    cover_subtree_27436d0eee02
+    cover_subtree_3bcbb01846ea
+    cover_subtree_e3fd3e08cd04
+    cover_subtree_b9c469e743af
+
+theorem cover_subtree_e582f747675b :
+    adaptiveCoverCheck 4 (childHL (childLL (childHL thetaAboveCell000022003101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+    thetaAboveCell000022003101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031012020
+        (by
+          have h : ((childLL thetaAboveCell0000220031012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031012020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031012020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031012020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031012020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031012021
+        (by
+          have h : ((childLL thetaAboveCell0000220031012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031012021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031012021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031012021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031012021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031012022
+        (by
+          have h : ((childLL thetaAboveCell0000220031012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031012022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031012022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031012022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031012022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031012023
+        (by
+          have h : ((childLL thetaAboveCell0000220031012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031012023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031012023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031012023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031012023) h))
+
+theorem cover_subtree_9833e059fd15 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHL thetaAboveCell000022003101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+    thetaAboveCell000022003101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031012030
+        (by
+          have h : ((childLL thetaAboveCell0000220031012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031012030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031012030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031012030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031012030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031012031
+        (by
+          have h : ((childLL thetaAboveCell0000220031012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031012031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031012031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031012031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031012031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031012032
+        (by
+          have h : ((childLL thetaAboveCell0000220031012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031012032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031012032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031012032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031012032) h))
+    (by
+      have h : (thetaAboveCell0000220031012033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012033 h)
+
+theorem cover_subtree_f35cbab27ebd :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022003101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022003101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHL
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031012000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012000 h)
+        (by
+          have h : (thetaAboveCell0000220031012001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012001 h)
+        (by
+          have h : (thetaAboveCell0000220031012002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012002 h)
+        (by
+          have h : (thetaAboveCell0000220031012003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHL
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031012010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012010 h)
+        (by
+          have h : (thetaAboveCell0000220031012011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012011 h)
+        (by
+          have h : (thetaAboveCell0000220031012012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012012 h)
+        (by
+          have h : (thetaAboveCell0000220031012013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012013 h))
+    cover_subtree_e582f747675b
+    cover_subtree_9833e059fd15
+
+theorem cover_subtree_4567a99abbdc :
+    adaptiveCoverCheck 4 (childHL (childLH (childHL thetaAboveCell000022003101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+    thetaAboveCell000022003101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031012120
+        (by
+          have h : ((childLL thetaAboveCell0000220031012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031012120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031012120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031012120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031012120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031012121
+        (by
+          have h : ((childLL thetaAboveCell0000220031012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031012121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031012121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031012121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031012121) h))
+    (by
+      have h : (thetaAboveCell0000220031012122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012122 h)
+    (by
+      have h : (thetaAboveCell0000220031012123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012123 h)
+
+theorem cover_subtree_7c0ef6ceebf5 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHL thetaAboveCell000022003101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+    thetaAboveCell000022003101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031012130
+        (by
+          have h : ((childLL thetaAboveCell0000220031012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031012130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031012130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031012130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031012130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031012131
+        (by
+          have h : ((childLL thetaAboveCell0000220031012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031012131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031012131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031012131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031012131) h))
+    (by
+      have h : (thetaAboveCell0000220031012132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012132 h)
+    (by
+      have h : (thetaAboveCell0000220031012133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012133 h)
+
+theorem cover_subtree_8358923d8e75 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022003101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022003101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHL
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031012100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012100 h)
+        (by
+          have h : (thetaAboveCell0000220031012101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012101 h)
+        (by
+          have h : (thetaAboveCell0000220031012102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012102 h)
+        (by
+          have h : (thetaAboveCell0000220031012103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHL
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031012110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012110 h)
+        (by
+          have h : (thetaAboveCell0000220031012111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012111 h)
+        (by
+          have h : (thetaAboveCell0000220031012112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012112 h)
+        (by
+          have h : (thetaAboveCell0000220031012113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012113 h))
+    cover_subtree_4567a99abbdc
+    cover_subtree_7c0ef6ceebf5
+
+theorem cover_subtree_d6a0244cc9dc :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022003101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022003101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031012200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012200 h)
+        (by
+          have h : (thetaAboveCell0000220031012201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012201 h)
+        (by
+          have h : (thetaAboveCell0000220031012202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012202 h)
+        (by
+          have h : (thetaAboveCell0000220031012203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031012210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012210 h)
+        (by
+          have h : (thetaAboveCell0000220031012211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012211 h)
+        (by
+          have h : (thetaAboveCell0000220031012212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012212 h)
+        (by
+          have h : (thetaAboveCell0000220031012213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022003101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022003101))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022003101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022003101))) h)
+
+theorem cover_subtree_0a6d165e5f89 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022003101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022003101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031012300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012300 h)
+        (by
+          have h : (thetaAboveCell0000220031012301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012301 h)
+        (by
+          have h : (thetaAboveCell0000220031012302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012302 h)
+        (by
+          have h : (thetaAboveCell0000220031012303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031012310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012310 h)
+        (by
+          have h : (thetaAboveCell0000220031012311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012311 h)
+        (by
+          have h : (thetaAboveCell0000220031012312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012312 h)
+        (by
+          have h : (thetaAboveCell0000220031012313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031012313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022003101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022003101))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022003101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022003101))) h)
+
+theorem cover_subtree_0b78493a586a :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022003101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022003101)
+    cover_subtree_f35cbab27ebd
+    cover_subtree_8358923d8e75
+    cover_subtree_d6a0244cc9dc
+    cover_subtree_0a6d165e5f89
+
+theorem cover_subtree_57ad90623a2b :
+    adaptiveCoverCheck 4 (childHL (childLL (childHH thetaAboveCell000022003101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+    thetaAboveCell000022003101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031013020
+        (by
+          have h : ((childLL thetaAboveCell0000220031013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031013020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031013020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031013020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031013020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031013021
+        (by
+          have h : ((childLL thetaAboveCell0000220031013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031013021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031013021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031013021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031013021) h))
+    (by
+      have h : (thetaAboveCell0000220031013022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013022 h)
+    (by
+      have h : (thetaAboveCell0000220031013023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013023 h)
+
+theorem cover_subtree_cab424cfd961 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHH thetaAboveCell000022003101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+    thetaAboveCell000022003101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031013030
+        (by
+          have h : ((childLL thetaAboveCell0000220031013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031013030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031013030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031013030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031013030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031013031
+        (by
+          have h : ((childLL thetaAboveCell0000220031013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031013031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031013031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031013031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031013031) h))
+    (by
+      have h : (thetaAboveCell0000220031013032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013032 h)
+    (by
+      have h : (thetaAboveCell0000220031013033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013033 h)
+
+theorem cover_subtree_af6ea412af3a :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022003101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022003101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHH
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031013000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013000 h)
+        (by
+          have h : (thetaAboveCell0000220031013001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013001 h)
+        (by
+          have h : (thetaAboveCell0000220031013002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013002 h)
+        (by
+          have h : (thetaAboveCell0000220031013003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHH
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031013010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013010 h)
+        (by
+          have h : (thetaAboveCell0000220031013011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013011 h)
+        (by
+          have h : (thetaAboveCell0000220031013012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013012 h)
+        (by
+          have h : (thetaAboveCell0000220031013013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013013 h))
+    cover_subtree_57ad90623a2b
+    cover_subtree_cab424cfd961
+
+theorem cover_subtree_ab0d022d0753 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHH thetaAboveCell000022003101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+    thetaAboveCell000022003101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031013120
+        (by
+          have h : ((childLL thetaAboveCell0000220031013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031013120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031013120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031013120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031013120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031013121
+        (by
+          have h : ((childLL thetaAboveCell0000220031013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031013121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031013121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031013121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031013121) h))
+    (by
+      have h : (thetaAboveCell0000220031013122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013122 h)
+    (by
+      have h : (thetaAboveCell0000220031013123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013123 h)
+
+theorem cover_subtree_6515bfe621d1 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHH thetaAboveCell000022003101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+    thetaAboveCell000022003101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031013130
+        (by
+          have h : ((childLL thetaAboveCell0000220031013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031013130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031013130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031013130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031013130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031013131
+        (by
+          have h : ((childLL thetaAboveCell0000220031013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031013131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031013131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031013131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031013131) h))
+    (by
+      have h : (thetaAboveCell0000220031013132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013132 h)
+    (by
+      have h : (thetaAboveCell0000220031013133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013133 h)
+
+theorem cover_subtree_9d368eedb456 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022003101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022003101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHH
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031013100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013100 h)
+        (by
+          have h : (thetaAboveCell0000220031013101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013101 h)
+        (by
+          have h : (thetaAboveCell0000220031013102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013102 h)
+        (by
+          have h : (thetaAboveCell0000220031013103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHH
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031013110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013110 h)
+        (by
+          have h : (thetaAboveCell0000220031013111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013111 h)
+        (by
+          have h : (thetaAboveCell0000220031013112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013112 h)
+        (by
+          have h : (thetaAboveCell0000220031013113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013113 h))
+    cover_subtree_ab0d022d0753
+    cover_subtree_6515bfe621d1
+
+theorem cover_subtree_a19642467be9 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022003101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022003101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031013200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013200 h)
+        (by
+          have h : (thetaAboveCell0000220031013201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013201 h)
+        (by
+          have h : (thetaAboveCell0000220031013202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013202 h)
+        (by
+          have h : (thetaAboveCell0000220031013203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031013210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013210 h)
+        (by
+          have h : (thetaAboveCell0000220031013211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013211 h)
+        (by
+          have h : (thetaAboveCell0000220031013212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013212 h)
+        (by
+          have h : (thetaAboveCell0000220031013213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022003101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022003101))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022003101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022003101))) h)
+
+theorem cover_subtree_6b1d70907da0 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022003101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022003101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031013300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013300 h)
+        (by
+          have h : (thetaAboveCell0000220031013301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013301 h)
+        (by
+          have h : (thetaAboveCell0000220031013302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013302 h)
+        (by
+          have h : (thetaAboveCell0000220031013303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022003101)))
+        (by
+          have h : (thetaAboveCell0000220031013310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013310 h)
+        (by
+          have h : (thetaAboveCell0000220031013311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013311 h)
+        (by
+          have h : (thetaAboveCell0000220031013312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013312 h)
+        (by
+          have h : (thetaAboveCell0000220031013313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031013313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022003101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022003101))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022003101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022003101))) h)
+
+theorem cover_subtree_e3814f5f5798 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022003101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022003101)
+    cover_subtree_af6ea412af3a
+    cover_subtree_9d368eedb456
+    cover_subtree_a19642467be9
+    cover_subtree_6b1d70907da0
+
+theorem cover_subtree_ff20fb20b987 :
+    adaptiveCoverCheck 7 thetaAboveCell000022003101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003101
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003101)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022003101)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022003101)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022003101)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022003101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003101)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022003101)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022003101)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022003101)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022003101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022003101)) h))
+    cover_subtree_0b78493a586a
+    cover_subtree_e3814f5f5798
+
+theorem cover_subtree_2108b1583905 :
+    adaptiveCoverCheck 7 thetaAboveCell000022003102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003102
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003102)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022003102)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022003102)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022003102)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022003102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003102)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022003102)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022003102)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022003102)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022003102)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022003102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022003102) h)
+    (by
+      have h : ((childHH thetaAboveCell000022003102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022003102) h)
+
+theorem cover_subtree_c8767f5a98fb :
+    adaptiveCoverCheck 7 thetaAboveCell000022003103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003103
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003103)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022003103)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022003103)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022003103)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022003103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003103)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022003103)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022003103)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022003103)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022003103)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022003103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022003103) h)
+    (by
+      have h : ((childHH thetaAboveCell000022003103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022003103) h)
+
+theorem cover_subtree_2ae0e1bf4a6f :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00002200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00002200)))
+    cover_subtree_2d90eed026dd
+    cover_subtree_ff20fb20b987
+    cover_subtree_2108b1583905
+    cover_subtree_c8767f5a98fb
+
+theorem cover_subtree_4c6415abff0a :
+    adaptiveCoverCheck 4 (childHL (childLL (childHL thetaAboveCell000022003110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+    thetaAboveCell000022003110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031102020
+        (by
+          have h : ((childLL thetaAboveCell0000220031102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031102020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031102020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031102020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031102020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220031102021
+        (by
+          have h : ((childLL thetaAboveCell0000220031102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220031102021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220031102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220031102021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220031102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220031102021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220031102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220031102021) h))
+    (by
+      have h : (thetaAboveCell0000220031102022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102022 h)
+    (by
+      have h : (thetaAboveCell0000220031102023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102023 h)
+
+theorem cover_subtree_2eee4b5c26e8 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022003110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022003110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHL
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031102000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102000 h)
+        (by
+          have h : (thetaAboveCell0000220031102001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102001 h)
+        (by
+          have h : (thetaAboveCell0000220031102002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102002 h)
+        (by
+          have h : (thetaAboveCell0000220031102003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHL
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031102010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102010 h)
+        (by
+          have h : (thetaAboveCell0000220031102011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102011 h)
+        (by
+          have h : (thetaAboveCell0000220031102012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102012 h)
+        (by
+          have h : (thetaAboveCell0000220031102013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102013 h))
+    cover_subtree_4c6415abff0a
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031102030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102030 h)
+        (by
+          have h : (thetaAboveCell0000220031102031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102031 h)
+        (by
+          have h : (thetaAboveCell0000220031102032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102032 h)
+        (by
+          have h : (thetaAboveCell0000220031102033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102033 h))
+
+theorem cover_subtree_a1a430be962f :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022003110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022003110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHL
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031102100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102100 h)
+        (by
+          have h : (thetaAboveCell0000220031102101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102101 h)
+        (by
+          have h : (thetaAboveCell0000220031102102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102102 h)
+        (by
+          have h : (thetaAboveCell0000220031102103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHL
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031102110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102110 h)
+        (by
+          have h : (thetaAboveCell0000220031102111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102111 h)
+        (by
+          have h : (thetaAboveCell0000220031102112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102112 h)
+        (by
+          have h : (thetaAboveCell0000220031102113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031102120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102120 h)
+        (by
+          have h : (thetaAboveCell0000220031102121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102121 h)
+        (by
+          have h : (thetaAboveCell0000220031102122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102122 h)
+        (by
+          have h : (thetaAboveCell0000220031102123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031102130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102130 h)
+        (by
+          have h : (thetaAboveCell0000220031102131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102131 h)
+        (by
+          have h : (thetaAboveCell0000220031102132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102132 h)
+        (by
+          have h : (thetaAboveCell0000220031102133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102133 h))
+
+theorem cover_subtree_175aa4a9f2f2 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022003110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022003110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031102200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102200 h)
+        (by
+          have h : (thetaAboveCell0000220031102201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102201 h)
+        (by
+          have h : (thetaAboveCell0000220031102202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102202 h)
+        (by
+          have h : (thetaAboveCell0000220031102203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031102210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102210 h)
+        (by
+          have h : (thetaAboveCell0000220031102211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102211 h)
+        (by
+          have h : (thetaAboveCell0000220031102212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102212 h)
+        (by
+          have h : (thetaAboveCell0000220031102213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022003110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022003110))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022003110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022003110))) h)
+
+theorem cover_subtree_b6a561db1a8e :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022003110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022003110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031102300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102300 h)
+        (by
+          have h : (thetaAboveCell0000220031102301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102301 h)
+        (by
+          have h : (thetaAboveCell0000220031102302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102302 h)
+        (by
+          have h : (thetaAboveCell0000220031102303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031102310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102310 h)
+        (by
+          have h : (thetaAboveCell0000220031102311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102311 h)
+        (by
+          have h : (thetaAboveCell0000220031102312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102312 h)
+        (by
+          have h : (thetaAboveCell0000220031102313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031102313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022003110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022003110))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022003110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022003110))) h)
+
+theorem cover_subtree_e6198bf2e2be :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022003110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022003110)
+    cover_subtree_2eee4b5c26e8
+    cover_subtree_a1a430be962f
+    cover_subtree_175aa4a9f2f2
+    cover_subtree_b6a561db1a8e
+
+theorem cover_subtree_03c6f1190393 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022003110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022003110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHH
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031103000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103000 h)
+        (by
+          have h : (thetaAboveCell0000220031103001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103001 h)
+        (by
+          have h : (thetaAboveCell0000220031103002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103002 h)
+        (by
+          have h : (thetaAboveCell0000220031103003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHH
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031103010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103010 h)
+        (by
+          have h : (thetaAboveCell0000220031103011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103011 h)
+        (by
+          have h : (thetaAboveCell0000220031103012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103012 h)
+        (by
+          have h : (thetaAboveCell0000220031103013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031103020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103020 h)
+        (by
+          have h : (thetaAboveCell0000220031103021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103021 h)
+        (by
+          have h : (thetaAboveCell0000220031103022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103022 h)
+        (by
+          have h : (thetaAboveCell0000220031103023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031103030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103030 h)
+        (by
+          have h : (thetaAboveCell0000220031103031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103031 h)
+        (by
+          have h : (thetaAboveCell0000220031103032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103032 h)
+        (by
+          have h : (thetaAboveCell0000220031103033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103033 h))
+
+theorem cover_subtree_40c139979b71 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022003110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022003110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHH
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031103100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103100 h)
+        (by
+          have h : (thetaAboveCell0000220031103101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103101 h)
+        (by
+          have h : (thetaAboveCell0000220031103102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103102 h)
+        (by
+          have h : (thetaAboveCell0000220031103103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHH
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031103110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103110 h)
+        (by
+          have h : (thetaAboveCell0000220031103111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103111 h)
+        (by
+          have h : (thetaAboveCell0000220031103112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103112 h)
+        (by
+          have h : (thetaAboveCell0000220031103113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031103120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103120 h)
+        (by
+          have h : (thetaAboveCell0000220031103121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103121 h)
+        (by
+          have h : (thetaAboveCell0000220031103122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103122 h)
+        (by
+          have h : (thetaAboveCell0000220031103123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031103130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103130 h)
+        (by
+          have h : (thetaAboveCell0000220031103131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103131 h)
+        (by
+          have h : (thetaAboveCell0000220031103132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103132 h)
+        (by
+          have h : (thetaAboveCell0000220031103133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103133 h))
+
+theorem cover_subtree_aa0d2321f3c9 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022003110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022003110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031103200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103200 h)
+        (by
+          have h : (thetaAboveCell0000220031103201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103201 h)
+        (by
+          have h : (thetaAboveCell0000220031103202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103202 h)
+        (by
+          have h : (thetaAboveCell0000220031103203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031103210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103210 h)
+        (by
+          have h : (thetaAboveCell0000220031103211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103211 h)
+        (by
+          have h : (thetaAboveCell0000220031103212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103212 h)
+        (by
+          have h : (thetaAboveCell0000220031103213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022003110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022003110))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022003110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022003110))) h)
+
+theorem cover_subtree_fea12d0bbf6a :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022003110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022003110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031103300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103300 h)
+        (by
+          have h : (thetaAboveCell0000220031103301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103301 h)
+        (by
+          have h : (thetaAboveCell0000220031103302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103302 h)
+        (by
+          have h : (thetaAboveCell0000220031103303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022003110)))
+        (by
+          have h : (thetaAboveCell0000220031103310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103310 h)
+        (by
+          have h : (thetaAboveCell0000220031103311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103311 h)
+        (by
+          have h : (thetaAboveCell0000220031103312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103312 h)
+        (by
+          have h : (thetaAboveCell0000220031103313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031103313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022003110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022003110))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022003110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022003110))) h)
+
+theorem cover_subtree_aa69ae74ebd2 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022003110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022003110)
+    cover_subtree_03c6f1190393
+    cover_subtree_40c139979b71
+    cover_subtree_aa0d2321f3c9
+    cover_subtree_fea12d0bbf6a
+
+theorem cover_subtree_2ac73ef77180 :
+    adaptiveCoverCheck 7 thetaAboveCell000022003110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003110
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003110)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022003110)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022003110)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022003110)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022003110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003110)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022003110)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022003110)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022003110)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022003110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022003110)) h))
+    cover_subtree_e6198bf2e2be
+    cover_subtree_aa69ae74ebd2
+
+theorem cover_subtree_b28aebcb8c2e :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022003111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022003111))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHL
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031112000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112000 h)
+        (by
+          have h : (thetaAboveCell0000220031112001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112001 h)
+        (by
+          have h : (thetaAboveCell0000220031112002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112002 h)
+        (by
+          have h : (thetaAboveCell0000220031112003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHL
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031112010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112010 h)
+        (by
+          have h : (thetaAboveCell0000220031112011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112011 h)
+        (by
+          have h : (thetaAboveCell0000220031112012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112012 h)
+        (by
+          have h : (thetaAboveCell0000220031112013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031112020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112020 h)
+        (by
+          have h : (thetaAboveCell0000220031112021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112021 h)
+        (by
+          have h : (thetaAboveCell0000220031112022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112022 h)
+        (by
+          have h : (thetaAboveCell0000220031112023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031112030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112030 h)
+        (by
+          have h : (thetaAboveCell0000220031112031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112031 h)
+        (by
+          have h : (thetaAboveCell0000220031112032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112032 h)
+        (by
+          have h : (thetaAboveCell0000220031112033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112033 h))
+
+theorem cover_subtree_d16c7f3cc99a :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022003111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022003111))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022003111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022003111))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022003111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022003111))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031112120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112120 h)
+        (by
+          have h : (thetaAboveCell0000220031112121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112121 h)
+        (by
+          have h : (thetaAboveCell0000220031112122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112122 h)
+        (by
+          have h : (thetaAboveCell0000220031112123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031112130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112130 h)
+        (by
+          have h : (thetaAboveCell0000220031112131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112131 h)
+        (by
+          have h : (thetaAboveCell0000220031112132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112132 h)
+        (by
+          have h : (thetaAboveCell0000220031112133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112133 h))
+
+theorem cover_subtree_b18c7ceea6b5 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022003111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022003111))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031112200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112200 h)
+        (by
+          have h : (thetaAboveCell0000220031112201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112201 h)
+        (by
+          have h : (thetaAboveCell0000220031112202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112202 h)
+        (by
+          have h : (thetaAboveCell0000220031112203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031112210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112210 h)
+        (by
+          have h : (thetaAboveCell0000220031112211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112211 h)
+        (by
+          have h : (thetaAboveCell0000220031112212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112212 h)
+        (by
+          have h : (thetaAboveCell0000220031112213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022003111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022003111))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022003111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022003111))) h)
+
+theorem cover_subtree_acda22cf7e9d :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022003111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022003111))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031112300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112300 h)
+        (by
+          have h : (thetaAboveCell0000220031112301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112301 h)
+        (by
+          have h : (thetaAboveCell0000220031112302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112302 h)
+        (by
+          have h : (thetaAboveCell0000220031112303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031112310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112310 h)
+        (by
+          have h : (thetaAboveCell0000220031112311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112311 h)
+        (by
+          have h : (thetaAboveCell0000220031112312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112312 h)
+        (by
+          have h : (thetaAboveCell0000220031112313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031112313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022003111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022003111))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022003111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022003111))) h)
+
+theorem cover_subtree_e026b8ce03c7 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022003111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022003111)
+    cover_subtree_b28aebcb8c2e
+    cover_subtree_d16c7f3cc99a
+    cover_subtree_b18c7ceea6b5
+    cover_subtree_acda22cf7e9d
+
+theorem cover_subtree_f521b363acbf :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022003111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022003111))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022003111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022003111))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022003111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022003111))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031113020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113020 h)
+        (by
+          have h : (thetaAboveCell0000220031113021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113021 h)
+        (by
+          have h : (thetaAboveCell0000220031113022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113022 h)
+        (by
+          have h : (thetaAboveCell0000220031113023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031113030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113030 h)
+        (by
+          have h : (thetaAboveCell0000220031113031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113031 h)
+        (by
+          have h : (thetaAboveCell0000220031113032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113032 h)
+        (by
+          have h : (thetaAboveCell0000220031113033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113033 h))
+
+theorem cover_subtree_524c47da8fe4 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022003111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022003111))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell000022003111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+        thetaAboveCell000022003111))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell000022003111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+        thetaAboveCell000022003111))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031113120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113120 h)
+        (by
+          have h : (thetaAboveCell0000220031113121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113121 h)
+        (by
+          have h : (thetaAboveCell0000220031113122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113122 h)
+        (by
+          have h : (thetaAboveCell0000220031113123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031113130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113130 h)
+        (by
+          have h : (thetaAboveCell0000220031113131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113131 h)
+        (by
+          have h : (thetaAboveCell0000220031113132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113132 h)
+        (by
+          have h : (thetaAboveCell0000220031113133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113133 h))
+
+theorem cover_subtree_43c4b871b5bc :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022003111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022003111))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031113200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113200 h)
+        (by
+          have h : (thetaAboveCell0000220031113201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113201 h)
+        (by
+          have h : (thetaAboveCell0000220031113202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113202 h)
+        (by
+          have h : (thetaAboveCell0000220031113203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031113210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113210 h)
+        (by
+          have h : (thetaAboveCell0000220031113211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113211 h)
+        (by
+          have h : (thetaAboveCell0000220031113212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113212 h)
+        (by
+          have h : (thetaAboveCell0000220031113213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022003111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022003111))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022003111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022003111))) h)
+
+theorem cover_subtree_2318bf66cd31 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022003111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022003111))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031113300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113300 h)
+        (by
+          have h : (thetaAboveCell0000220031113301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113301 h)
+        (by
+          have h : (thetaAboveCell0000220031113302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113302 h)
+        (by
+          have h : (thetaAboveCell0000220031113303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022003111)))
+        (by
+          have h : (thetaAboveCell0000220031113310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113310 h)
+        (by
+          have h : (thetaAboveCell0000220031113311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113311 h)
+        (by
+          have h : (thetaAboveCell0000220031113312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113312 h)
+        (by
+          have h : (thetaAboveCell0000220031113313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220031113313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022003111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022003111))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022003111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022003111))) h)
+
+theorem cover_subtree_db44c7b5c525 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022003111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022003111)
+    cover_subtree_f521b363acbf
+    cover_subtree_524c47da8fe4
+    cover_subtree_43c4b871b5bc
+    cover_subtree_2318bf66cd31
+
+theorem cover_subtree_65fd7134399d :
+    adaptiveCoverCheck 7 thetaAboveCell000022003111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003111
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003111)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022003111)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022003111)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022003111)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022003111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003111)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022003111)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022003111)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022003111)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022003111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022003111)) h))
+    cover_subtree_e026b8ce03c7
+    cover_subtree_db44c7b5c525
+
+theorem cover_subtree_0003bc3e2951 :
+    adaptiveCoverCheck 7 thetaAboveCell000022003112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003112
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003112)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022003112)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022003112)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022003112)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022003112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003112)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022003112)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022003112)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022003112)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022003112)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022003112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022003112) h)
+    (by
+      have h : ((childHH thetaAboveCell000022003112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022003112) h)
+
+theorem cover_subtree_3bc7bdc69c3a :
+    adaptiveCoverCheck 7 thetaAboveCell000022003113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003113
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003113)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022003113)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022003113)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022003113)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022003113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003113)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022003113)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022003113)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022003113)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022003113)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022003113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022003113) h)
+    (by
+      have h : ((childHH thetaAboveCell000022003113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022003113) h)
+
+theorem cover_subtree_63491c075e54 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00002200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00002200)))
+    cover_subtree_2ac73ef77180
+    cover_subtree_65fd7134399d
+    cover_subtree_0003bc3e2951
+    cover_subtree_3bc7bdc69c3a
+
+theorem e24KC2ThetaAboveLeaf0000220031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00002200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00002200))
+    cover_subtree_2ae0e1bf4a6f
+    cover_subtree_63491c075e54
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00002200)))
+        (by
+          have h : (thetaAboveCell000022003120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003120 h)
+        (by
+          have h : (thetaAboveCell000022003121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003121 h)
+        (by
+          have h : (thetaAboveCell000022003122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003122 h)
+        (by
+          have h : (thetaAboveCell000022003123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00002200)))
+        (by
+          have h : (thetaAboveCell000022003130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003130 h)
+        (by
+          have h : (thetaAboveCell000022003131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003131 h)
+        (by
+          have h : (thetaAboveCell000022003132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003132 h)
+        (by
+          have h : (thetaAboveCell000022003133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003133 h))
+theorem e24KC2ThetaAboveLeaf0000220032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00002200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00002200))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00002200))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00002200))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00002200))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00002200))) h)
+theorem e24KC2ThetaAboveLeaf0000220033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00002200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00002200))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00002200))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00002200))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00002200))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00002200))) h)
+theorem e24KC2ThetaAboveLeaf0000220102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00002201))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00002201))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLL
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022010220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022010220 h)
+        (by
+          have h : (thetaAboveCell000022010221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022010221 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022010222
+            (by
+              have h : ((childLL thetaAboveCell000022010222)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022010222) h)
+            (by
+              have h : ((childLH thetaAboveCell000022010222)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022010222) h)
+            (by
+              have h : ((childHL thetaAboveCell000022010222)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022010222) h)
+            (by
+              have h : ((childHH thetaAboveCell000022010222)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022010222) h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022010223
+            (by
+              have h : ((childLL thetaAboveCell000022010223)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022010223) h)
+            (by
+              have h : ((childLH thetaAboveCell000022010223)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022010223) h)
+            (by
+              have h : ((childHL thetaAboveCell000022010223)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022010223) h)
+            (by
+              have h : ((childHH thetaAboveCell000022010223)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022010223) h)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childLL
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022010230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022010230 h)
+        (by
+          have h : (thetaAboveCell000022010231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022010231 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022010232
+            (by
+              have h : ((childLL thetaAboveCell000022010232)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022010232) h)
+            (by
+              have h : ((childLH thetaAboveCell000022010232)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022010232) h)
+            (by
+              have h : ((childHL thetaAboveCell000022010232)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022010232) h)
+            (by
+              have h : ((childHH thetaAboveCell000022010232)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022010232) h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022010233
+            (by
+              have h : ((childLL thetaAboveCell000022010233)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022010233) h)
+            (by
+              have h : ((childLH thetaAboveCell000022010233)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022010233) h)
+            (by
+              have h : ((childHL thetaAboveCell000022010233)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022010233) h)
+            (by
+              have h : ((childHH thetaAboveCell000022010233)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022010233) h)))
+theorem e24KC2ThetaAboveLeaf0000220103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00002201))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00002201))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childLL
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022010320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022010320 h)
+        (by
+          have h : (thetaAboveCell000022010321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022010321 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022010322
+            (by
+              have h : ((childLL thetaAboveCell000022010322)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022010322) h)
+            (by
+              have h : ((childLH thetaAboveCell000022010322)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022010322) h)
+            (by
+              have h : ((childHL thetaAboveCell000022010322)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022010322) h)
+            (by
+              have h : ((childHH thetaAboveCell000022010322)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022010322) h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022010323
+            (by
+              have h : ((childLL thetaAboveCell000022010323)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022010323) h)
+            (by
+              have h : ((childLH thetaAboveCell000022010323)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022010323) h)
+            (by
+              have h : ((childHL thetaAboveCell000022010323)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022010323) h)
+            (by
+              have h : ((childHH thetaAboveCell000022010323)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022010323) h)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childLL
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022010330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022010330 h)
+        (by
+          have h : (thetaAboveCell000022010331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022010331 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022010332
+            (by
+              have h : ((childLL thetaAboveCell000022010332)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022010332) h)
+            (by
+              have h : ((childLH thetaAboveCell000022010332)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022010332) h)
+            (by
+              have h : ((childHL thetaAboveCell000022010332)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022010332) h)
+            (by
+              have h : ((childHH thetaAboveCell000022010332)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022010332) h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022010333
+            (by
+              have h : ((childLL thetaAboveCell000022010333)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022010333) h)
+            (by
+              have h : ((childLH thetaAboveCell000022010333)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022010333) h)
+            (by
+              have h : ((childHL thetaAboveCell000022010333)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022010333) h)
+            (by
+              have h : ((childHH thetaAboveCell000022010333)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022010333) h)))
+theorem e24KC2ThetaAboveLeaf0000220112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00002201))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00002201))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLH
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022011220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022011220 h)
+        (by
+          have h : (thetaAboveCell000022011221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022011221 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022011222
+            (by
+              have h : ((childLL thetaAboveCell000022011222)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022011222) h)
+            (by
+              have h : ((childLH thetaAboveCell000022011222)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022011222) h)
+            (by
+              have h : ((childHL thetaAboveCell000022011222)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022011222) h)
+            (by
+              have h : ((childHH thetaAboveCell000022011222)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022011222) h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022011223
+            (by
+              have h : ((childLL thetaAboveCell000022011223)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022011223) h)
+            (by
+              have h : ((childLH thetaAboveCell000022011223)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022011223) h)
+            (by
+              have h : ((childHL thetaAboveCell000022011223)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022011223) h)
+            (by
+              have h : ((childHH thetaAboveCell000022011223)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022011223) h)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childLH
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022011230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022011230 h)
+        (by
+          have h : (thetaAboveCell000022011231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022011231 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022011232
+            (by
+              have h : ((childLL thetaAboveCell000022011232)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022011232) h)
+            (by
+              have h : ((childLH thetaAboveCell000022011232)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022011232) h)
+            (by
+              have h : ((childHL thetaAboveCell000022011232)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022011232) h)
+            (by
+              have h : ((childHH thetaAboveCell000022011232)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022011232) h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022011233
+            (by
+              have h : ((childLL thetaAboveCell000022011233)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022011233) h)
+            (by
+              have h : ((childLH thetaAboveCell000022011233)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022011233) h)
+            (by
+              have h : ((childHL thetaAboveCell000022011233)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022011233) h)
+            (by
+              have h : ((childHH thetaAboveCell000022011233)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022011233) h)))
+theorem e24KC2ThetaAboveLeaf0000220113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00002201))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00002201))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childLH
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022011320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022011320 h)
+        (by
+          have h : (thetaAboveCell000022011321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022011321 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022011322
+            (by
+              have h : ((childLL thetaAboveCell000022011322)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022011322) h)
+            (by
+              have h : ((childLH thetaAboveCell000022011322)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022011322) h)
+            (by
+              have h : ((childHL thetaAboveCell000022011322)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022011322) h)
+            (by
+              have h : ((childHH thetaAboveCell000022011322)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022011322) h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022011323
+            (by
+              have h : ((childLL thetaAboveCell000022011323)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022011323) h)
+            (by
+              have h : ((childLH thetaAboveCell000022011323)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022011323) h)
+            (by
+              have h : ((childHL thetaAboveCell000022011323)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022011323) h)
+            (by
+              have h : ((childHH thetaAboveCell000022011323)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022011323) h)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childLH
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022011330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022011330 h)
+        (by
+          have h : (thetaAboveCell000022011331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022011331 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022011332
+            (by
+              have h : ((childLL thetaAboveCell000022011332)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022011332) h)
+            (by
+              have h : ((childLH thetaAboveCell000022011332)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022011332) h)
+            (by
+              have h : ((childHL thetaAboveCell000022011332)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022011332) h)
+            (by
+              have h : ((childHH thetaAboveCell000022011332)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022011332) h))
+        (by
+          have h : (thetaAboveCell000022011333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022011333 h))
+theorem cover_subtree_cb61b2523d2d :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022012000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022012000))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+        thetaAboveCell000022012000))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+        thetaAboveCell000022012000))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120002020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002020 h)
+        (by
+          have h : (thetaAboveCell0000220120002021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002021 h)
+        (by
+          have h : (thetaAboveCell0000220120002022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002022 h)
+        (by
+          have h : (thetaAboveCell0000220120002023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120002030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002030 h)
+        (by
+          have h : (thetaAboveCell0000220120002031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002031 h)
+        (by
+          have h : (thetaAboveCell0000220120002032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002032 h)
+        (by
+          have h : (thetaAboveCell0000220120002033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002033 h))
+
+theorem cover_subtree_51c2832454de :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022012000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022012000))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022012000))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022012000))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120002120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002120 h)
+        (by
+          have h : (thetaAboveCell0000220120002121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002121 h)
+        (by
+          have h : (thetaAboveCell0000220120002122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002122 h)
+        (by
+          have h : (thetaAboveCell0000220120002123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120002130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002130 h)
+        (by
+          have h : (thetaAboveCell0000220120002131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002131 h)
+        (by
+          have h : (thetaAboveCell0000220120002132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002132 h)
+        (by
+          have h : (thetaAboveCell0000220120002133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002133 h))
+
+theorem cover_subtree_1ce463e93cf8 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022012000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022012000))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120002200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002200 h)
+        (by
+          have h : (thetaAboveCell0000220120002201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002201 h)
+        (by
+          have h : (thetaAboveCell0000220120002202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002202 h)
+        (by
+          have h : (thetaAboveCell0000220120002203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120002210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002210 h)
+        (by
+          have h : (thetaAboveCell0000220120002211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002211 h)
+        (by
+          have h : (thetaAboveCell0000220120002212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002212 h)
+        (by
+          have h : (thetaAboveCell0000220120002213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022012000))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022012000))) h)
+
+theorem cover_subtree_106de882d2f1 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022012000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022012000))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120002300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002300 h)
+        (by
+          have h : (thetaAboveCell0000220120002301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002301 h)
+        (by
+          have h : (thetaAboveCell0000220120002302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002302 h)
+        (by
+          have h : (thetaAboveCell0000220120002303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120002310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002310 h)
+        (by
+          have h : (thetaAboveCell0000220120002311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002311 h)
+        (by
+          have h : (thetaAboveCell0000220120002312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002312 h)
+        (by
+          have h : (thetaAboveCell0000220120002313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120002313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022012000))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022012000))) h)
+
+theorem cover_subtree_a41df9d2526f :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022012000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022012000)
+    cover_subtree_cb61b2523d2d
+    cover_subtree_51c2832454de
+    cover_subtree_1ce463e93cf8
+    cover_subtree_106de882d2f1
+
+theorem cover_subtree_850bf723be77 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022012000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022012000))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022012000))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022012000))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120003020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003020 h)
+        (by
+          have h : (thetaAboveCell0000220120003021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003021 h)
+        (by
+          have h : (thetaAboveCell0000220120003022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003022 h)
+        (by
+          have h : (thetaAboveCell0000220120003023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120003030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003030 h)
+        (by
+          have h : (thetaAboveCell0000220120003031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003031 h)
+        (by
+          have h : (thetaAboveCell0000220120003032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003032 h)
+        (by
+          have h : (thetaAboveCell0000220120003033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003033 h))
+
+theorem cover_subtree_2b3d22a9b597 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022012000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022012000))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+        thetaAboveCell000022012000))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+        thetaAboveCell000022012000))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120003120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003120 h)
+        (by
+          have h : (thetaAboveCell0000220120003121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003121 h)
+        (by
+          have h : (thetaAboveCell0000220120003122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003122 h)
+        (by
+          have h : (thetaAboveCell0000220120003123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120003130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003130 h)
+        (by
+          have h : (thetaAboveCell0000220120003131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003131 h)
+        (by
+          have h : (thetaAboveCell0000220120003132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003132 h)
+        (by
+          have h : (thetaAboveCell0000220120003133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003133 h))
+
+theorem cover_subtree_29b231fa0c5a :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022012000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022012000))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120003200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003200 h)
+        (by
+          have h : (thetaAboveCell0000220120003201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003201 h)
+        (by
+          have h : (thetaAboveCell0000220120003202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003202 h)
+        (by
+          have h : (thetaAboveCell0000220120003203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120003210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003210 h)
+        (by
+          have h : (thetaAboveCell0000220120003211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003211 h)
+        (by
+          have h : (thetaAboveCell0000220120003212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003212 h)
+        (by
+          have h : (thetaAboveCell0000220120003213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022012000))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022012000))) h)
+
+theorem cover_subtree_2f6ee77c9edb :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022012000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022012000))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120003300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003300 h)
+        (by
+          have h : (thetaAboveCell0000220120003301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003301 h)
+        (by
+          have h : (thetaAboveCell0000220120003302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003302 h)
+        (by
+          have h : (thetaAboveCell0000220120003303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022012000)))
+        (by
+          have h : (thetaAboveCell0000220120003310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003310 h)
+        (by
+          have h : (thetaAboveCell0000220120003311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003311 h)
+        (by
+          have h : (thetaAboveCell0000220120003312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003312 h)
+        (by
+          have h : (thetaAboveCell0000220120003313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120003313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022012000))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022012000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022012000))) h)
+
+theorem cover_subtree_aea54c77bf9f :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022012000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022012000)
+    cover_subtree_850bf723be77
+    cover_subtree_2b3d22a9b597
+    cover_subtree_29b231fa0c5a
+    cover_subtree_2f6ee77c9edb
+
+theorem cover_subtree_b33563914493 :
+    adaptiveCoverCheck 7 thetaAboveCell000022012000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012000
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012000)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022012000)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022012000)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022012000)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022012000)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012000)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022012000)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022012000)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022012000)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022012000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022012000)) h))
+    cover_subtree_a41df9d2526f
+    cover_subtree_aea54c77bf9f
+
+theorem cover_subtree_40103f36b622 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022012001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022012001))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+        thetaAboveCell000022012001))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+        thetaAboveCell000022012001))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120012020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012020 h)
+        (by
+          have h : (thetaAboveCell0000220120012021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012021 h)
+        (by
+          have h : (thetaAboveCell0000220120012022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012022 h)
+        (by
+          have h : (thetaAboveCell0000220120012023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120012030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012030 h)
+        (by
+          have h : (thetaAboveCell0000220120012031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012031 h)
+        (by
+          have h : (thetaAboveCell0000220120012032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012032 h)
+        (by
+          have h : (thetaAboveCell0000220120012033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012033 h))
+
+theorem cover_subtree_68b551c015fb :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022012001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022012001))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022012001))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022012001))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120012120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012120 h)
+        (by
+          have h : (thetaAboveCell0000220120012121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012121 h)
+        (by
+          have h : (thetaAboveCell0000220120012122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012122 h)
+        (by
+          have h : (thetaAboveCell0000220120012123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120012130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012130 h)
+        (by
+          have h : (thetaAboveCell0000220120012131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012131 h)
+        (by
+          have h : (thetaAboveCell0000220120012132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012132 h)
+        (by
+          have h : (thetaAboveCell0000220120012133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012133 h))
+
+theorem cover_subtree_26644cd8d850 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022012001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022012001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120012200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012200 h)
+        (by
+          have h : (thetaAboveCell0000220120012201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012201 h)
+        (by
+          have h : (thetaAboveCell0000220120012202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012202 h)
+        (by
+          have h : (thetaAboveCell0000220120012203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120012210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012210 h)
+        (by
+          have h : (thetaAboveCell0000220120012211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012211 h)
+        (by
+          have h : (thetaAboveCell0000220120012212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012212 h)
+        (by
+          have h : (thetaAboveCell0000220120012213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022012001))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022012001))) h)
+
+theorem cover_subtree_ceab235367fd :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022012001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022012001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120012300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012300 h)
+        (by
+          have h : (thetaAboveCell0000220120012301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012301 h)
+        (by
+          have h : (thetaAboveCell0000220120012302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012302 h)
+        (by
+          have h : (thetaAboveCell0000220120012303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120012310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012310 h)
+        (by
+          have h : (thetaAboveCell0000220120012311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012311 h)
+        (by
+          have h : (thetaAboveCell0000220120012312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012312 h)
+        (by
+          have h : (thetaAboveCell0000220120012313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120012313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022012001))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022012001))) h)
+
+theorem cover_subtree_4fe093f5b1c6 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022012001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022012001)
+    cover_subtree_40103f36b622
+    cover_subtree_68b551c015fb
+    cover_subtree_26644cd8d850
+    cover_subtree_ceab235367fd
+
+theorem cover_subtree_2a612fb4e5f9 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022012001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022012001))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022012001))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022012001))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120013020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013020 h)
+        (by
+          have h : (thetaAboveCell0000220120013021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013021 h)
+        (by
+          have h : (thetaAboveCell0000220120013022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013022 h)
+        (by
+          have h : (thetaAboveCell0000220120013023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120013030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013030 h)
+        (by
+          have h : (thetaAboveCell0000220120013031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013031 h)
+        (by
+          have h : (thetaAboveCell0000220120013032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013032 h)
+        (by
+          have h : (thetaAboveCell0000220120013033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013033 h))
+
+theorem cover_subtree_9768153a3fc8 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022012001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022012001))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+        thetaAboveCell000022012001))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+        thetaAboveCell000022012001))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120013120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013120 h)
+        (by
+          have h : (thetaAboveCell0000220120013121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013121 h)
+        (by
+          have h : (thetaAboveCell0000220120013122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013122 h)
+        (by
+          have h : (thetaAboveCell0000220120013123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120013130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013130 h)
+        (by
+          have h : (thetaAboveCell0000220120013131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013131 h)
+        (by
+          have h : (thetaAboveCell0000220120013132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013132 h)
+        (by
+          have h : (thetaAboveCell0000220120013133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013133 h))
+
+theorem cover_subtree_cc445576ceb2 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022012001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022012001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120013200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013200 h)
+        (by
+          have h : (thetaAboveCell0000220120013201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013201 h)
+        (by
+          have h : (thetaAboveCell0000220120013202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013202 h)
+        (by
+          have h : (thetaAboveCell0000220120013203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120013210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013210 h)
+        (by
+          have h : (thetaAboveCell0000220120013211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013211 h)
+        (by
+          have h : (thetaAboveCell0000220120013212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013212 h)
+        (by
+          have h : (thetaAboveCell0000220120013213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022012001))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022012001))) h)
+
+theorem cover_subtree_f5eaea7b07a1 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022012001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022012001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120013300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013300 h)
+        (by
+          have h : (thetaAboveCell0000220120013301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013301 h)
+        (by
+          have h : (thetaAboveCell0000220120013302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013302 h)
+        (by
+          have h : (thetaAboveCell0000220120013303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022012001)))
+        (by
+          have h : (thetaAboveCell0000220120013310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013310 h)
+        (by
+          have h : (thetaAboveCell0000220120013311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013311 h)
+        (by
+          have h : (thetaAboveCell0000220120013312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013312 h)
+        (by
+          have h : (thetaAboveCell0000220120013313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120013313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022012001))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022012001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022012001))) h)
+
+theorem cover_subtree_4bf63b1ca0fb :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022012001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022012001)
+    cover_subtree_2a612fb4e5f9
+    cover_subtree_9768153a3fc8
+    cover_subtree_cc445576ceb2
+    cover_subtree_f5eaea7b07a1
+
+theorem cover_subtree_6c90b6564adf :
+    adaptiveCoverCheck 7 thetaAboveCell000022012001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012001
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012001)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022012001)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022012001)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022012001)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022012001)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012001)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022012001)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022012001)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022012001)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022012001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022012001)) h))
+    cover_subtree_4fe093f5b1c6
+    cover_subtree_4bf63b1ca0fb
+
+theorem cover_subtree_4289605d46bf :
+    adaptiveCoverCheck 7 thetaAboveCell000022012002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012002
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012002)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022012002)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022012002)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022012002)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022012002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012002)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022012002)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022012002)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022012002)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022012002)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022012002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022012002) h)
+    (by
+      have h : ((childHH thetaAboveCell000022012002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022012002) h)
+
+theorem cover_subtree_c487c7959b39 :
+    adaptiveCoverCheck 7 thetaAboveCell000022012003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012003
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012003)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022012003)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022012003)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022012003)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022012003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012003)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022012003)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022012003)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022012003)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022012003)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022012003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022012003) h)
+    (by
+      have h : ((childHH thetaAboveCell000022012003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022012003) h)
+
+theorem cover_subtree_a0fc96b8fbeb :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00002201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00002201)))
+    cover_subtree_b33563914493
+    cover_subtree_6c90b6564adf
+    cover_subtree_4289605d46bf
+    cover_subtree_c487c7959b39
+
+theorem cover_subtree_5e121ac8879b :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022012010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022012010))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+        thetaAboveCell000022012010))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+        thetaAboveCell000022012010))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120102020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102020 h)
+        (by
+          have h : (thetaAboveCell0000220120102021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102021 h)
+        (by
+          have h : (thetaAboveCell0000220120102022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102022 h)
+        (by
+          have h : (thetaAboveCell0000220120102023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120102030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102030 h)
+        (by
+          have h : (thetaAboveCell0000220120102031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102031 h)
+        (by
+          have h : (thetaAboveCell0000220120102032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102032 h)
+        (by
+          have h : (thetaAboveCell0000220120102033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102033 h))
+
+theorem cover_subtree_85f348686687 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022012010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022012010))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022012010))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022012010))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120102120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102120 h)
+        (by
+          have h : (thetaAboveCell0000220120102121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102121 h)
+        (by
+          have h : (thetaAboveCell0000220120102122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102122 h)
+        (by
+          have h : (thetaAboveCell0000220120102123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120102130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102130 h)
+        (by
+          have h : (thetaAboveCell0000220120102131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102131 h)
+        (by
+          have h : (thetaAboveCell0000220120102132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102132 h)
+        (by
+          have h : (thetaAboveCell0000220120102133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102133 h))
+
+theorem cover_subtree_0b577b76b19a :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022012010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022012010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120102200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102200 h)
+        (by
+          have h : (thetaAboveCell0000220120102201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102201 h)
+        (by
+          have h : (thetaAboveCell0000220120102202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102202 h)
+        (by
+          have h : (thetaAboveCell0000220120102203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120102210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102210 h)
+        (by
+          have h : (thetaAboveCell0000220120102211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102211 h)
+        (by
+          have h : (thetaAboveCell0000220120102212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102212 h)
+        (by
+          have h : (thetaAboveCell0000220120102213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022012010))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022012010))) h)
+
+theorem cover_subtree_56442d1d29fc :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022012010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022012010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120102300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102300 h)
+        (by
+          have h : (thetaAboveCell0000220120102301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102301 h)
+        (by
+          have h : (thetaAboveCell0000220120102302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102302 h)
+        (by
+          have h : (thetaAboveCell0000220120102303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120102310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102310 h)
+        (by
+          have h : (thetaAboveCell0000220120102311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102311 h)
+        (by
+          have h : (thetaAboveCell0000220120102312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102312 h)
+        (by
+          have h : (thetaAboveCell0000220120102313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120102313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022012010))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022012010))) h)
+
+theorem cover_subtree_077e5d53ad41 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022012010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022012010)
+    cover_subtree_5e121ac8879b
+    cover_subtree_85f348686687
+    cover_subtree_0b577b76b19a
+    cover_subtree_56442d1d29fc
+
+theorem cover_subtree_afc343ef6d9c :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022012010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022012010))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022012010))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022012010))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120103020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103020 h)
+        (by
+          have h : (thetaAboveCell0000220120103021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103021 h)
+        (by
+          have h : (thetaAboveCell0000220120103022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103022 h)
+        (by
+          have h : (thetaAboveCell0000220120103023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120103030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103030 h)
+        (by
+          have h : (thetaAboveCell0000220120103031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103031 h)
+        (by
+          have h : (thetaAboveCell0000220120103032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103032 h)
+        (by
+          have h : (thetaAboveCell0000220120103033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103033 h))
+
+theorem cover_subtree_bd1c188bd5cc :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022012010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022012010))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+        thetaAboveCell000022012010))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+        thetaAboveCell000022012010))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120103120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103120 h)
+        (by
+          have h : (thetaAboveCell0000220120103121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103121 h)
+        (by
+          have h : (thetaAboveCell0000220120103122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103122 h)
+        (by
+          have h : (thetaAboveCell0000220120103123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120103130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103130 h)
+        (by
+          have h : (thetaAboveCell0000220120103131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103131 h)
+        (by
+          have h : (thetaAboveCell0000220120103132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103132 h)
+        (by
+          have h : (thetaAboveCell0000220120103133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103133 h))
+
+theorem cover_subtree_89214ff44665 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022012010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022012010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120103200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103200 h)
+        (by
+          have h : (thetaAboveCell0000220120103201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103201 h)
+        (by
+          have h : (thetaAboveCell0000220120103202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103202 h)
+        (by
+          have h : (thetaAboveCell0000220120103203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120103210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103210 h)
+        (by
+          have h : (thetaAboveCell0000220120103211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103211 h)
+        (by
+          have h : (thetaAboveCell0000220120103212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103212 h)
+        (by
+          have h : (thetaAboveCell0000220120103213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022012010))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022012010))) h)
+
+theorem cover_subtree_287e5af130bf :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022012010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022012010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120103300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103300 h)
+        (by
+          have h : (thetaAboveCell0000220120103301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103301 h)
+        (by
+          have h : (thetaAboveCell0000220120103302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103302 h)
+        (by
+          have h : (thetaAboveCell0000220120103303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022012010)))
+        (by
+          have h : (thetaAboveCell0000220120103310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103310 h)
+        (by
+          have h : (thetaAboveCell0000220120103311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103311 h)
+        (by
+          have h : (thetaAboveCell0000220120103312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103312 h)
+        (by
+          have h : (thetaAboveCell0000220120103313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120103313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022012010))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022012010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022012010))) h)
+
+theorem cover_subtree_bdb6f38b6019 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022012010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022012010)
+    cover_subtree_afc343ef6d9c
+    cover_subtree_bd1c188bd5cc
+    cover_subtree_89214ff44665
+    cover_subtree_287e5af130bf
+
+theorem cover_subtree_d23bef969ca7 :
+    adaptiveCoverCheck 7 thetaAboveCell000022012010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012010
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012010)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022012010)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022012010)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022012010)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022012010)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012010)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022012010)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022012010)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022012010)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022012010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022012010)) h))
+    cover_subtree_077e5d53ad41
+    cover_subtree_bdb6f38b6019
+
+theorem cover_subtree_4667900dcf78 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022012011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022012011))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell000022012011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+        thetaAboveCell000022012011))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell000022012011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+        thetaAboveCell000022012011))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120112020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112020 h)
+        (by
+          have h : (thetaAboveCell0000220120112021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112021 h)
+        (by
+          have h : (thetaAboveCell0000220120112022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112022 h)
+        (by
+          have h : (thetaAboveCell0000220120112023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120112030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112030 h)
+        (by
+          have h : (thetaAboveCell0000220120112031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112031 h)
+        (by
+          have h : (thetaAboveCell0000220120112032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112032 h)
+        (by
+          have h : (thetaAboveCell0000220120112033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112033 h))
+
+theorem cover_subtree_d9c26d139205 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022012011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022012011))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022012011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022012011))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022012011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022012011))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120112120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112120 h)
+        (by
+          have h : (thetaAboveCell0000220120112121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112121 h)
+        (by
+          have h : (thetaAboveCell0000220120112122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112122 h)
+        (by
+          have h : (thetaAboveCell0000220120112123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120112130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112130 h)
+        (by
+          have h : (thetaAboveCell0000220120112131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112131 h)
+        (by
+          have h : (thetaAboveCell0000220120112132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112132 h)
+        (by
+          have h : (thetaAboveCell0000220120112133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112133 h))
+
+theorem cover_subtree_295321b3b4d6 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022012011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022012011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120112200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112200 h)
+        (by
+          have h : (thetaAboveCell0000220120112201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112201 h)
+        (by
+          have h : (thetaAboveCell0000220120112202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112202 h)
+        (by
+          have h : (thetaAboveCell0000220120112203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120112210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112210 h)
+        (by
+          have h : (thetaAboveCell0000220120112211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112211 h)
+        (by
+          have h : (thetaAboveCell0000220120112212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112212 h)
+        (by
+          have h : (thetaAboveCell0000220120112213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022012011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022012011))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022012011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022012011))) h)
+
+theorem cover_subtree_9f2040bb35ca :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022012011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022012011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120112300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112300 h)
+        (by
+          have h : (thetaAboveCell0000220120112301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112301 h)
+        (by
+          have h : (thetaAboveCell0000220120112302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112302 h)
+        (by
+          have h : (thetaAboveCell0000220120112303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120112310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112310 h)
+        (by
+          have h : (thetaAboveCell0000220120112311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112311 h)
+        (by
+          have h : (thetaAboveCell0000220120112312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112312 h)
+        (by
+          have h : (thetaAboveCell0000220120112313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120112313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022012011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022012011))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022012011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022012011))) h)
+
+theorem cover_subtree_738fba336a14 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022012011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022012011)
+    cover_subtree_4667900dcf78
+    cover_subtree_d9c26d139205
+    cover_subtree_295321b3b4d6
+    cover_subtree_9f2040bb35ca
+
+theorem cover_subtree_4ddaba414c0c :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022012011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022012011))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022012011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022012011))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022012011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022012011))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120113020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113020 h)
+        (by
+          have h : (thetaAboveCell0000220120113021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113021 h)
+        (by
+          have h : (thetaAboveCell0000220120113022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113022 h)
+        (by
+          have h : (thetaAboveCell0000220120113023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120113030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113030 h)
+        (by
+          have h : (thetaAboveCell0000220120113031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113031 h)
+        (by
+          have h : (thetaAboveCell0000220120113032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113032 h)
+        (by
+          have h : (thetaAboveCell0000220120113033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113033 h))
+
+theorem cover_subtree_ecd6fc119970 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022012011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022012011))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell000022012011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+        thetaAboveCell000022012011))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell000022012011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+        thetaAboveCell000022012011))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120113120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113120 h)
+        (by
+          have h : (thetaAboveCell0000220120113121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113121 h)
+        (by
+          have h : (thetaAboveCell0000220120113122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113122 h)
+        (by
+          have h : (thetaAboveCell0000220120113123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120113130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113130 h)
+        (by
+          have h : (thetaAboveCell0000220120113131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113131 h)
+        (by
+          have h : (thetaAboveCell0000220120113132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113132 h)
+        (by
+          have h : (thetaAboveCell0000220120113133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113133 h))
+
+theorem cover_subtree_9104ec057e37 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022012011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022012011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120113200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113200 h)
+        (by
+          have h : (thetaAboveCell0000220120113201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113201 h)
+        (by
+          have h : (thetaAboveCell0000220120113202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113202 h)
+        (by
+          have h : (thetaAboveCell0000220120113203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120113210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113210 h)
+        (by
+          have h : (thetaAboveCell0000220120113211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113211 h)
+        (by
+          have h : (thetaAboveCell0000220120113212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113212 h)
+        (by
+          have h : (thetaAboveCell0000220120113213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHH
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120113220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113220 h)
+        (by
+          have h : (thetaAboveCell0000220120113221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113221 h)
+        (by
+          have h : (thetaAboveCell0000220120113222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113222 h)
+        (by
+          have h : (thetaAboveCell0000220120113223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHH
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120113230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113230 h)
+        (by
+          have h : (thetaAboveCell0000220120113231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113231 h)
+        (by
+          have h : (thetaAboveCell0000220120113232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113232 h)
+        (by
+          have h : (thetaAboveCell0000220120113233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113233 h))
+
+theorem cover_subtree_019584cb80e9 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022012011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022012011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120113300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113300 h)
+        (by
+          have h : (thetaAboveCell0000220120113301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113301 h)
+        (by
+          have h : (thetaAboveCell0000220120113302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113302 h)
+        (by
+          have h : (thetaAboveCell0000220120113303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120113310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113310 h)
+        (by
+          have h : (thetaAboveCell0000220120113311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113311 h)
+        (by
+          have h : (thetaAboveCell0000220120113312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113312 h)
+        (by
+          have h : (thetaAboveCell0000220120113313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHH
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120113320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113320 h)
+        (by
+          have h : (thetaAboveCell0000220120113321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113321 h)
+        (by
+          have h : (thetaAboveCell0000220120113322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113322 h)
+        (by
+          have h : (thetaAboveCell0000220120113323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHH
+        thetaAboveCell000022012011)))
+        (by
+          have h : (thetaAboveCell0000220120113330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113330 h)
+        (by
+          have h : (thetaAboveCell0000220120113331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113331 h)
+        (by
+          have h : (thetaAboveCell0000220120113332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113332 h)
+        (by
+          have h : (thetaAboveCell0000220120113333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220120113333 h))
+
+theorem cover_subtree_a099f2429294 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022012011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022012011)
+    cover_subtree_4ddaba414c0c
+    cover_subtree_ecd6fc119970
+    cover_subtree_9104ec057e37
+    cover_subtree_019584cb80e9
+
+theorem cover_subtree_3ff73ea428ff :
+    adaptiveCoverCheck 7 thetaAboveCell000022012011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012011
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012011)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022012011)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022012011)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022012011)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022012011)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012011)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022012011)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022012011)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022012011)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022012011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022012011)) h))
+    cover_subtree_738fba336a14
+    cover_subtree_a099f2429294
+
+theorem cover_subtree_9d5ad8202394 :
+    adaptiveCoverCheck 7 thetaAboveCell000022012012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012012
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012012)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022012012)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022012012)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022012012)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022012012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012012)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022012012)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022012012)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022012012)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022012012)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022012012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022012012) h)
+    (by
+      have h : ((childHH thetaAboveCell000022012012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022012012) h)
+
+theorem cover_subtree_32e4797e8235 :
+    adaptiveCoverCheck 7 thetaAboveCell000022012013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012013
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012013)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022012013)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022012013)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022012013)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022012013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012013)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022012013)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022012013)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022012013)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022012013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022012013)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022012013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022012013) h)
+    (by
+      have h : ((childHH thetaAboveCell000022012013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022012013) h)
+
+theorem cover_subtree_86e078def774 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00002201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00002201)))
+    cover_subtree_d23bef969ca7
+    cover_subtree_3ff73ea428ff
+    cover_subtree_9d5ad8202394
+    cover_subtree_32e4797e8235
+
+theorem e24KC2ThetaAboveLeaf0000220120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00002201))
+    cover_subtree_a0fc96b8fbeb
+    cover_subtree_86e078def774
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022012020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012020 h)
+        (by
+          have h : (thetaAboveCell000022012021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012021 h)
+        (by
+          have h : (thetaAboveCell000022012022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012022 h)
+        (by
+          have h : (thetaAboveCell000022012023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022012030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012030 h)
+        (by
+          have h : (thetaAboveCell000022012031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012031 h)
+        (by
+          have h : (thetaAboveCell000022012032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012032 h)
+        (by
+          have h : (thetaAboveCell000022012033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012033 h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle013.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle013.lean
new file mode 100644
index 0000000000..f8f6bcab7e
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle013.lean
@@ -0,0 +1,11302 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch018`.
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch044`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCellsfa7ef730fc
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022003000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `000022003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00002200)))
+
+/-- Subcell `0000220030002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022003000)))
+
+/-- Subcell `0000220030012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022003001)))
+
+/-- Subcell `0000220030102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022003010)))
+
+/-- Subcell `0000220030112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022003011)))
+
+/-- Subcell `0000220030113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220030113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022003011)))
+
+end GerverSofa.PartE.CertificateCellsfa7ef730fc
+
+namespace GerverSofa.PartE.CoverCertificate1bb21a49a4
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLH (childLL (childHL (childLL (childLL (childHL (childHL (childLL
+    (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate1bb21a49a4
+
+namespace GerverSofa.PartE.CoverCertificatec9db3a1aba
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLH (childLL (childHL (childLL (childLL (childHL (childHL (childLL
+    (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatec9db3a1aba
+
+namespace GerverSofa.PartE.CoverCertificatefb03004e1f
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHL (childLH (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatefb03004e1f
+
+namespace GerverSofa.PartE.CoverCertificate045cecfd86
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childHL (childLH (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate045cecfd86
+
+namespace GerverSofa.PartE.CoverCertificatef837300024
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHL (childLH (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificatef837300024
+
+namespace GerverSofa.PartE.CoverCertificate2943ba0b5c
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHL (childLH (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificate2943ba0b5c
+
+namespace GerverSofa.PartE.CoverCertificate42a60e756f
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childHL (childLH (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate42a60e756f
+
+namespace GerverSofa.PartE.CoverCertificate0cee658f4b
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHL (childLH (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate0cee658f4b
+
+namespace GerverSofa.PartE.CoverCertificateeb376f6b8e
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHL (childLH (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificateeb376f6b8e
+
+namespace GerverSofa.PartE.CoverCertificate1e208f1b6e
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHL (childLH (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificate1e208f1b6e
+
+namespace GerverSofa.PartE.CoverCertificate6fd62aabed
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childLH (childLH (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate6fd62aabed
+
+namespace GerverSofa.PartE.CoverCertificate22515ac7bd
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childLH (childLH (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate22515ac7bd
+
+namespace GerverSofa.PartE.CoverCertificatebad9a9b575
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHH (childLH (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatebad9a9b575
+
+namespace GerverSofa.PartE.CoverCertificate13a680926b
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childHH (childLH (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate13a680926b
+
+namespace GerverSofa.PartE.CoverCertificatebdefa88d14
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHH (childLH (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificatebdefa88d14
+
+namespace GerverSofa.PartE.CoverCertificateb54a5d06ec
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHH (childLH (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificateb54a5d06ec
+
+namespace GerverSofa.PartE.CertificateCells5e16eb2f1b
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells5e16eb2f1b
+
+namespace GerverSofa.PartE.CertificateCellsb7fa51f50b
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsb7fa51f50b
+
+namespace GerverSofa.PartE.CertificateCellsd618452127
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsd618452127
+
+namespace GerverSofa.PartE.CertificateCells82d3525350
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells82d3525350
+
+namespace GerverSofa.PartE.CertificateCells632f810978
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells632f810978
+
+namespace GerverSofa.PartE.CertificateCells15b8174174
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells15b8174174
+
+namespace GerverSofa.PartE.CertificateCells1f25007fe8
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells1f25007fe8
+
+namespace GerverSofa.PartE.CertificateCells1e9acaa8be
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells1e9acaa8be
+
+namespace GerverSofa.PartE.CertificateCells231d66ab0c
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells231d66ab0c
+
+namespace GerverSofa.PartE.CertificateCells2c11250698
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells2c11250698
+
+namespace GerverSofa.PartE.CertificateCells3ce31f23ce
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells3ce31f23ce
+
+namespace GerverSofa.PartE.CertificateCells3c3d48c1eb
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells3c3d48c1eb
+
+namespace GerverSofa.PartE.CertificateCells4d022b961f
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells4d022b961f
+
+namespace GerverSofa.PartE.CertificateCells9798c9c840
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells9798c9c840
+
+namespace GerverSofa.PartE.CertificateCells6832cee036
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells6832cee036
+
+namespace GerverSofa.PartE.CertificateCellsab896aa443
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsab896aa443
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6ProofBatch0bae0a82ed09fca0`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsfa7ef730fc
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsfa7ef730fc
+
+open CertificateCellsfa7ef730fc
+theorem e24KC2ThetaAboveLeaf0000220022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00002200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00002200))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00002200))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00002200))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00002200))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00002200))) h)
+theorem e24KC2ThetaAboveLeaf0000220023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00002200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00002200))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00002200))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00002200))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00002200))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00002200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00002200))) h)
+theorem cover_subtree_98370144a0ae :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022003000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003000)
+    (by
+      have h : ((childLL (childLL thetaAboveCell000022003000))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL thetaAboveCell000022003000)) h)
+    (by
+      have h : ((childLH (childLL thetaAboveCell000022003000))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL thetaAboveCell000022003000)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLL thetaAboveCell000022003000))
+        (by
+          have h : ((childLL (childHL (childLL thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLL
+            thetaAboveCell000022003000))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLL
+            thetaAboveCell000022003000))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLL
+            thetaAboveCell000022003000))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLL
+            thetaAboveCell000022003000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLL thetaAboveCell000022003000))
+        (by
+          have h : ((childLL (childHH (childLL thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLL
+            thetaAboveCell000022003000))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLL
+            thetaAboveCell000022003000))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLL
+            thetaAboveCell000022003000))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLL
+            thetaAboveCell000022003000))) h))
+
+theorem cover_subtree_2b8c79c37576 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022003000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003000)
+    (by
+      have h : ((childLL (childLH thetaAboveCell000022003000))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH thetaAboveCell000022003000)) h)
+    (by
+      have h : ((childLH (childLH thetaAboveCell000022003000))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH thetaAboveCell000022003000)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLH thetaAboveCell000022003000))
+        (by
+          have h : ((childLL (childHL (childLH thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLH
+            thetaAboveCell000022003000))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLH
+            thetaAboveCell000022003000))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLH
+            thetaAboveCell000022003000))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLH
+            thetaAboveCell000022003000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLH thetaAboveCell000022003000))
+        (by
+          have h : ((childLL (childHH (childLH thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLH
+            thetaAboveCell000022003000))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLH
+            thetaAboveCell000022003000))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLH
+            thetaAboveCell000022003000))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaAboveCell000022003000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLH
+            thetaAboveCell000022003000))) h))
+
+theorem cover_subtree_c9f10be4f818 :
+    adaptiveCoverCheck 4 (childLL (childLL (childHL thetaAboveCell000022003000))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHL
+    thetaAboveCell000022003000)))
+    (by
+      have h : (thetaAboveCell0000220030002000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002000 h)
+    (by
+      have h : (thetaAboveCell0000220030002001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002002
+        (by
+          have h : ((childLL thetaAboveCell0000220030002002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002002) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002002) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002002) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002003
+        (by
+          have h : ((childLL thetaAboveCell0000220030002003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002003) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002003) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002003) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002003) h))
+
+theorem cover_subtree_b44dc0179b82 :
+    adaptiveCoverCheck 4 (childLH (childLL (childHL thetaAboveCell000022003000))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHL
+    thetaAboveCell000022003000)))
+    (by
+      have h : (thetaAboveCell0000220030002010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002010 h)
+    (by
+      have h : (thetaAboveCell0000220030002011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002012
+        (by
+          have h : ((childLL thetaAboveCell0000220030002012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002012) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002012) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002012) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002013
+        (by
+          have h : ((childLL thetaAboveCell0000220030002013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002013) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002013) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002013) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002013) h))
+
+theorem cover_subtree_2d51606740be :
+    adaptiveCoverCheck 4 (childHL (childLL (childHL thetaAboveCell000022003000))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+    thetaAboveCell000022003000)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002020
+        (by
+          have h : ((childLL thetaAboveCell0000220030002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002021
+        (by
+          have h : ((childLL thetaAboveCell0000220030002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002022
+        (by
+          have h : ((childLL thetaAboveCell0000220030002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002023
+        (by
+          have h : ((childLL thetaAboveCell0000220030002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002023) h))
+
+theorem cover_subtree_84975093d7db :
+    adaptiveCoverCheck 4 (childHH (childLL (childHL thetaAboveCell000022003000))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+    thetaAboveCell000022003000)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002030
+        (by
+          have h : ((childLL thetaAboveCell0000220030002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002031
+        (by
+          have h : ((childLL thetaAboveCell0000220030002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002032
+        (by
+          have h : ((childLL thetaAboveCell0000220030002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002033
+        (by
+          have h : ((childLL thetaAboveCell0000220030002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002033) h))
+
+theorem cover_subtree_ba9d19bd9151 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022003000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022003000))
+    cover_subtree_c9f10be4f818
+    cover_subtree_b44dc0179b82
+    cover_subtree_2d51606740be
+    cover_subtree_84975093d7db
+
+theorem cover_subtree_625b3384c249 :
+    adaptiveCoverCheck 4 (childLL (childLH (childHL thetaAboveCell000022003000))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHL
+    thetaAboveCell000022003000)))
+    (by
+      have h : (thetaAboveCell0000220030002100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002100 h)
+    (by
+      have h : (thetaAboveCell0000220030002101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002102
+        (by
+          have h : ((childLL thetaAboveCell0000220030002102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002102) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002102) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002102) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002103
+        (by
+          have h : ((childLL thetaAboveCell0000220030002103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002103) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002103) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002103) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002103) h))
+
+theorem cover_subtree_90e4b8e577d3 :
+    adaptiveCoverCheck 4 (childLH (childLH (childHL thetaAboveCell000022003000))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHL
+    thetaAboveCell000022003000)))
+    (by
+      have h : (thetaAboveCell0000220030002110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002110 h)
+    (by
+      have h : (thetaAboveCell0000220030002111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002112
+        (by
+          have h : ((childLL thetaAboveCell0000220030002112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002112) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002112) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002112) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002113
+        (by
+          have h : ((childLL thetaAboveCell0000220030002113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002113) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002113) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002113) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002113) h))
+
+theorem cover_subtree_675556e82299 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHL thetaAboveCell000022003000))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+    thetaAboveCell000022003000)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002120
+        (by
+          have h : ((childLL thetaAboveCell0000220030002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002121
+        (by
+          have h : ((childLL thetaAboveCell0000220030002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002122
+        (by
+          have h : ((childLL thetaAboveCell0000220030002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002123
+        (by
+          have h : ((childLL thetaAboveCell0000220030002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002123) h))
+
+theorem cover_subtree_a828876ee758 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHL thetaAboveCell000022003000))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+    thetaAboveCell000022003000)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002130
+        (by
+          have h : ((childLL thetaAboveCell0000220030002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002131
+        (by
+          have h : ((childLL thetaAboveCell0000220030002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002132
+        (by
+          have h : ((childLL thetaAboveCell0000220030002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030002133
+        (by
+          have h : ((childLL thetaAboveCell0000220030002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030002133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030002133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030002133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030002133) h))
+
+theorem cover_subtree_ab8ac75e68d0 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022003000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022003000))
+    cover_subtree_625b3384c249
+    cover_subtree_90e4b8e577d3
+    cover_subtree_675556e82299
+    cover_subtree_a828876ee758
+
+theorem cover_subtree_bdc92ad9349d :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022003000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022003000))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022003000)))
+        (by
+          have h : (thetaAboveCell0000220030002200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002200 h)
+        (by
+          have h : (thetaAboveCell0000220030002201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002201 h)
+        (by
+          have h : (thetaAboveCell0000220030002202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002202 h)
+        (by
+          have h : (thetaAboveCell0000220030002203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022003000)))
+        (by
+          have h : (thetaAboveCell0000220030002210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002210 h)
+        (by
+          have h : (thetaAboveCell0000220030002211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002211 h)
+        (by
+          have h : (thetaAboveCell0000220030002212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002212 h)
+        (by
+          have h : (thetaAboveCell0000220030002213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022003000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022003000))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022003000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022003000))) h)
+
+theorem cover_subtree_d20715622b71 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022003000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022003000))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022003000)))
+        (by
+          have h : (thetaAboveCell0000220030002300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002300 h)
+        (by
+          have h : (thetaAboveCell0000220030002301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002301 h)
+        (by
+          have h : (thetaAboveCell0000220030002302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002302 h)
+        (by
+          have h : (thetaAboveCell0000220030002303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022003000)))
+        (by
+          have h : (thetaAboveCell0000220030002310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002310 h)
+        (by
+          have h : (thetaAboveCell0000220030002311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002311 h)
+        (by
+          have h : (thetaAboveCell0000220030002312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002312 h)
+        (by
+          have h : (thetaAboveCell0000220030002313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030002313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022003000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022003000))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022003000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022003000))) h)
+
+theorem cover_subtree_b30ba9777925 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022003000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022003000)
+    cover_subtree_ba9d19bd9151
+    cover_subtree_ab8ac75e68d0
+    cover_subtree_bdc92ad9349d
+    cover_subtree_d20715622b71
+
+theorem cover_subtree_ee2f6d25ebfc :
+    adaptiveCoverCheck 4 (childHL (childLL (childHH thetaAboveCell000022003000))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+    thetaAboveCell000022003000)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003020
+        (by
+          have h : ((childLL thetaAboveCell0000220030003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003021
+        (by
+          have h : ((childLL thetaAboveCell0000220030003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003022
+        (by
+          have h : ((childLL thetaAboveCell0000220030003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003023
+        (by
+          have h : ((childLL thetaAboveCell0000220030003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003023) h))
+
+theorem cover_subtree_a96993ac001b :
+    adaptiveCoverCheck 4 (childHH (childLL (childHH thetaAboveCell000022003000))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+    thetaAboveCell000022003000)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003030
+        (by
+          have h : ((childLL thetaAboveCell0000220030003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003031
+        (by
+          have h : ((childLL thetaAboveCell0000220030003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003032
+        (by
+          have h : ((childLL thetaAboveCell0000220030003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003033
+        (by
+          have h : ((childLL thetaAboveCell0000220030003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003033) h))
+
+theorem cover_subtree_7c880bebec81 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022003000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022003000))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHH
+        thetaAboveCell000022003000)))
+        (by
+          have h : (thetaAboveCell0000220030003000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003000 h)
+        (by
+          have h : (thetaAboveCell0000220030003001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003001 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003002
+            (by
+              have h : ((childLL thetaAboveCell0000220030003002)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003002)
+                h)
+            (by
+              have h : ((childLH thetaAboveCell0000220030003002)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003002)
+                h)
+            (by
+              have h : ((childHL thetaAboveCell0000220030003002)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003002)
+                h)
+            (by
+              have h : ((childHH thetaAboveCell0000220030003002)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003002)
+                h))
+        (by
+          have h : (thetaAboveCell0000220030003003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHH
+        thetaAboveCell000022003000)))
+        (by
+          have h : (thetaAboveCell0000220030003010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003010 h)
+        (by
+          have h : (thetaAboveCell0000220030003011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003011 h)
+        (by
+          have h : (thetaAboveCell0000220030003012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003012 h)
+        (by
+          have h : (thetaAboveCell0000220030003013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003013 h))
+    cover_subtree_ee2f6d25ebfc
+    cover_subtree_a96993ac001b
+
+theorem cover_subtree_cde3dc3c4679 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHH thetaAboveCell000022003000))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+    thetaAboveCell000022003000)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003120
+        (by
+          have h : ((childLL thetaAboveCell0000220030003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003121
+        (by
+          have h : ((childLL thetaAboveCell0000220030003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003122
+        (by
+          have h : ((childLL thetaAboveCell0000220030003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003123
+        (by
+          have h : ((childLL thetaAboveCell0000220030003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003123) h))
+
+theorem cover_subtree_026c8979d7e8 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHH thetaAboveCell000022003000))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+    thetaAboveCell000022003000)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003130
+        (by
+          have h : ((childLL thetaAboveCell0000220030003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003131
+        (by
+          have h : ((childLL thetaAboveCell0000220030003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003132
+        (by
+          have h : ((childLL thetaAboveCell0000220030003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030003133
+        (by
+          have h : ((childLL thetaAboveCell0000220030003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030003133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030003133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030003133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030003133) h))
+
+theorem cover_subtree_e9a5f301f08e :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022003000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022003000))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHH
+        thetaAboveCell000022003000)))
+        (by
+          have h : (thetaAboveCell0000220030003100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003100 h)
+        (by
+          have h : (thetaAboveCell0000220030003101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003101 h)
+        (by
+          have h : (thetaAboveCell0000220030003102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003102 h)
+        (by
+          have h : (thetaAboveCell0000220030003103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHH
+        thetaAboveCell000022003000)))
+        (by
+          have h : (thetaAboveCell0000220030003110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003110 h)
+        (by
+          have h : (thetaAboveCell0000220030003111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003111 h)
+        (by
+          have h : (thetaAboveCell0000220030003112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003112 h)
+        (by
+          have h : (thetaAboveCell0000220030003113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003113 h))
+    cover_subtree_cde3dc3c4679
+    cover_subtree_026c8979d7e8
+
+theorem cover_subtree_6c8da8a57403 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022003000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022003000))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022003000)))
+        (by
+          have h : (thetaAboveCell0000220030003200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003200 h)
+        (by
+          have h : (thetaAboveCell0000220030003201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003201 h)
+        (by
+          have h : (thetaAboveCell0000220030003202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003202 h)
+        (by
+          have h : (thetaAboveCell0000220030003203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022003000)))
+        (by
+          have h : (thetaAboveCell0000220030003210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003210 h)
+        (by
+          have h : (thetaAboveCell0000220030003211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003211 h)
+        (by
+          have h : (thetaAboveCell0000220030003212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003212 h)
+        (by
+          have h : (thetaAboveCell0000220030003213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022003000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022003000))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022003000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022003000))) h)
+
+theorem cover_subtree_c7667f731e87 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022003000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022003000))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022003000)))
+        (by
+          have h : (thetaAboveCell0000220030003300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003300 h)
+        (by
+          have h : (thetaAboveCell0000220030003301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003301 h)
+        (by
+          have h : (thetaAboveCell0000220030003302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003302 h)
+        (by
+          have h : (thetaAboveCell0000220030003303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022003000)))
+        (by
+          have h : (thetaAboveCell0000220030003310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003310 h)
+        (by
+          have h : (thetaAboveCell0000220030003311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003311 h)
+        (by
+          have h : (thetaAboveCell0000220030003312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003312 h)
+        (by
+          have h : (thetaAboveCell0000220030003313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030003313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022003000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022003000))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022003000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022003000))) h)
+
+theorem cover_subtree_d916df227547 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022003000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022003000)
+    cover_subtree_7c880bebec81
+    cover_subtree_e9a5f301f08e
+    cover_subtree_6c8da8a57403
+    cover_subtree_c7667f731e87
+
+theorem cover_subtree_df29b2bfe6c5 :
+    adaptiveCoverCheck 7 thetaAboveCell000022003000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003000
+    cover_subtree_98370144a0ae
+    cover_subtree_2b8c79c37576
+    cover_subtree_b30ba9777925
+    cover_subtree_d916df227547
+
+theorem cover_subtree_b0fbdd86b439 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022003001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003001)
+    (by
+      have h : ((childLL (childLL thetaAboveCell000022003001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL thetaAboveCell000022003001)) h)
+    (by
+      have h : ((childLH (childLL thetaAboveCell000022003001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL thetaAboveCell000022003001)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLL thetaAboveCell000022003001))
+        (by
+          have h : ((childLL (childHL (childLL thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLL
+            thetaAboveCell000022003001))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLL
+            thetaAboveCell000022003001))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLL
+            thetaAboveCell000022003001))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLL
+            thetaAboveCell000022003001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLL thetaAboveCell000022003001))
+        (by
+          have h : ((childLL (childHH (childLL thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLL
+            thetaAboveCell000022003001))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLL
+            thetaAboveCell000022003001))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLL
+            thetaAboveCell000022003001))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLL
+            thetaAboveCell000022003001))) h))
+
+theorem cover_subtree_f6d73cfc8f96 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022003001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003001)
+    (by
+      have h : ((childLL (childLH thetaAboveCell000022003001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH thetaAboveCell000022003001)) h)
+    (by
+      have h : ((childLH (childLH thetaAboveCell000022003001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH thetaAboveCell000022003001)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLH thetaAboveCell000022003001))
+        (by
+          have h : ((childLL (childHL (childLH thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLH
+            thetaAboveCell000022003001))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLH
+            thetaAboveCell000022003001))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLH
+            thetaAboveCell000022003001))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLH
+            thetaAboveCell000022003001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLH thetaAboveCell000022003001))
+        (by
+          have h : ((childLL (childHH (childLH thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLH
+            thetaAboveCell000022003001))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLH
+            thetaAboveCell000022003001))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLH
+            thetaAboveCell000022003001))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaAboveCell000022003001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLH
+            thetaAboveCell000022003001))) h))
+
+theorem cover_subtree_b01bf826b90b :
+    adaptiveCoverCheck 4 (childHL (childLL (childHL thetaAboveCell000022003001))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+    thetaAboveCell000022003001)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012020
+        (by
+          have h : ((childLL thetaAboveCell0000220030012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012021
+        (by
+          have h : ((childLL thetaAboveCell0000220030012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012022
+        (by
+          have h : ((childLL thetaAboveCell0000220030012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012023
+        (by
+          have h : ((childLL thetaAboveCell0000220030012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012023) h))
+
+theorem cover_subtree_97058acb92ef :
+    adaptiveCoverCheck 4 (childHH (childLL (childHL thetaAboveCell000022003001))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+    thetaAboveCell000022003001)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012030
+        (by
+          have h : ((childLL thetaAboveCell0000220030012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012031
+        (by
+          have h : ((childLL thetaAboveCell0000220030012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012032
+        (by
+          have h : ((childLL thetaAboveCell0000220030012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012033
+        (by
+          have h : ((childLL thetaAboveCell0000220030012033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012033) h))
+
+theorem cover_subtree_6f0d223fa251 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022003001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022003001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHL
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030012000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012000 h)
+        (by
+          have h : (thetaAboveCell0000220030012001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012001 h)
+        (by
+          have h : (thetaAboveCell0000220030012002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012002 h)
+        (by
+          have h : (thetaAboveCell0000220030012003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHL
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030012010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012010 h)
+        (by
+          have h : (thetaAboveCell0000220030012011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012011 h)
+        (by
+          have h : (thetaAboveCell0000220030012012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012012 h)
+        (by
+          have h : (thetaAboveCell0000220030012013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012013 h))
+    cover_subtree_b01bf826b90b
+    cover_subtree_97058acb92ef
+
+theorem cover_subtree_587871b3f6fc :
+    adaptiveCoverCheck 4 (childHL (childLH (childHL thetaAboveCell000022003001))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+    thetaAboveCell000022003001)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012120
+        (by
+          have h : ((childLL thetaAboveCell0000220030012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012121
+        (by
+          have h : ((childLL thetaAboveCell0000220030012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012122
+        (by
+          have h : ((childLL thetaAboveCell0000220030012122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012123
+        (by
+          have h : ((childLL thetaAboveCell0000220030012123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012123) h))
+
+theorem cover_subtree_665ba5a76616 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHL thetaAboveCell000022003001))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+    thetaAboveCell000022003001)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012130
+        (by
+          have h : ((childLL thetaAboveCell0000220030012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012131
+        (by
+          have h : ((childLL thetaAboveCell0000220030012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012132
+        (by
+          have h : ((childLL thetaAboveCell0000220030012132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030012133
+        (by
+          have h : ((childLL thetaAboveCell0000220030012133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030012133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030012133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030012133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030012133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030012133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030012133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030012133) h))
+
+theorem cover_subtree_17de4b0fb6aa :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022003001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022003001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHL
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030012100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012100 h)
+        (by
+          have h : (thetaAboveCell0000220030012101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012101 h)
+        (by
+          have h : (thetaAboveCell0000220030012102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012102 h)
+        (by
+          have h : (thetaAboveCell0000220030012103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHL
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030012110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012110 h)
+        (by
+          have h : (thetaAboveCell0000220030012111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012111 h)
+        (by
+          have h : (thetaAboveCell0000220030012112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012112 h)
+        (by
+          have h : (thetaAboveCell0000220030012113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012113 h))
+    cover_subtree_587871b3f6fc
+    cover_subtree_665ba5a76616
+
+theorem cover_subtree_2dc0f946809c :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022003001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022003001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030012200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012200 h)
+        (by
+          have h : (thetaAboveCell0000220030012201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012201 h)
+        (by
+          have h : (thetaAboveCell0000220030012202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012202 h)
+        (by
+          have h : (thetaAboveCell0000220030012203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030012210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012210 h)
+        (by
+          have h : (thetaAboveCell0000220030012211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012211 h)
+        (by
+          have h : (thetaAboveCell0000220030012212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012212 h)
+        (by
+          have h : (thetaAboveCell0000220030012213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022003001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022003001))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022003001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022003001))) h)
+
+theorem cover_subtree_9ae7d1f286db :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022003001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022003001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030012300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012300 h)
+        (by
+          have h : (thetaAboveCell0000220030012301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012301 h)
+        (by
+          have h : (thetaAboveCell0000220030012302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012302 h)
+        (by
+          have h : (thetaAboveCell0000220030012303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030012310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012310 h)
+        (by
+          have h : (thetaAboveCell0000220030012311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012311 h)
+        (by
+          have h : (thetaAboveCell0000220030012312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012312 h)
+        (by
+          have h : (thetaAboveCell0000220030012313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030012313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022003001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022003001))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022003001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022003001))) h)
+
+theorem cover_subtree_6705ea0fc5a0 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022003001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022003001)
+    cover_subtree_6f0d223fa251
+    cover_subtree_17de4b0fb6aa
+    cover_subtree_2dc0f946809c
+    cover_subtree_9ae7d1f286db
+
+theorem cover_subtree_542c541386a0 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHH thetaAboveCell000022003001))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+    thetaAboveCell000022003001)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013020
+        (by
+          have h : ((childLL thetaAboveCell0000220030013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013021
+        (by
+          have h : ((childLL thetaAboveCell0000220030013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013022
+        (by
+          have h : ((childLL thetaAboveCell0000220030013022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013023
+        (by
+          have h : ((childLL thetaAboveCell0000220030013023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013023) h))
+
+theorem cover_subtree_95991c39b4ae :
+    adaptiveCoverCheck 4 (childHH (childLL (childHH thetaAboveCell000022003001))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+    thetaAboveCell000022003001)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013030
+        (by
+          have h : ((childLL thetaAboveCell0000220030013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013031
+        (by
+          have h : ((childLL thetaAboveCell0000220030013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013032
+        (by
+          have h : ((childLL thetaAboveCell0000220030013032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013033
+        (by
+          have h : ((childLL thetaAboveCell0000220030013033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013033) h))
+
+theorem cover_subtree_f11aed5a1b39 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022003001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022003001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHH
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030013000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013000 h)
+        (by
+          have h : (thetaAboveCell0000220030013001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013001 h)
+        (by
+          have h : (thetaAboveCell0000220030013002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013002 h)
+        (by
+          have h : (thetaAboveCell0000220030013003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHH
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030013010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013010 h)
+        (by
+          have h : (thetaAboveCell0000220030013011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013011 h)
+        (by
+          have h : (thetaAboveCell0000220030013012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013012 h)
+        (by
+          have h : (thetaAboveCell0000220030013013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013013 h))
+    cover_subtree_542c541386a0
+    cover_subtree_95991c39b4ae
+
+theorem cover_subtree_36cc60506fd2 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHH thetaAboveCell000022003001))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+    thetaAboveCell000022003001)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013120
+        (by
+          have h : ((childLL thetaAboveCell0000220030013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013121
+        (by
+          have h : ((childLL thetaAboveCell0000220030013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013122
+        (by
+          have h : ((childLL thetaAboveCell0000220030013122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013123
+        (by
+          have h : ((childLL thetaAboveCell0000220030013123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013123) h))
+
+theorem cover_subtree_3db2622a8526 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHH thetaAboveCell000022003001))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+    thetaAboveCell000022003001)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013130
+        (by
+          have h : ((childLL thetaAboveCell0000220030013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013131
+        (by
+          have h : ((childLL thetaAboveCell0000220030013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013132
+        (by
+          have h : ((childLL thetaAboveCell0000220030013132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030013133
+        (by
+          have h : ((childLL thetaAboveCell0000220030013133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030013133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030013133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030013133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030013133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030013133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030013133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030013133) h))
+
+theorem cover_subtree_13cfd42cc1e7 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022003001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022003001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHH
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030013100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013100 h)
+        (by
+          have h : (thetaAboveCell0000220030013101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013101 h)
+        (by
+          have h : (thetaAboveCell0000220030013102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013102 h)
+        (by
+          have h : (thetaAboveCell0000220030013103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHH
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030013110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013110 h)
+        (by
+          have h : (thetaAboveCell0000220030013111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013111 h)
+        (by
+          have h : (thetaAboveCell0000220030013112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013112 h)
+        (by
+          have h : (thetaAboveCell0000220030013113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013113 h))
+    cover_subtree_36cc60506fd2
+    cover_subtree_3db2622a8526
+
+theorem cover_subtree_49af2ce5bec1 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022003001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022003001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030013200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013200 h)
+        (by
+          have h : (thetaAboveCell0000220030013201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013201 h)
+        (by
+          have h : (thetaAboveCell0000220030013202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013202 h)
+        (by
+          have h : (thetaAboveCell0000220030013203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030013210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013210 h)
+        (by
+          have h : (thetaAboveCell0000220030013211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013211 h)
+        (by
+          have h : (thetaAboveCell0000220030013212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013212 h)
+        (by
+          have h : (thetaAboveCell0000220030013213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022003001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022003001))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022003001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022003001))) h)
+
+theorem cover_subtree_8008b2d13d89 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022003001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022003001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030013300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013300 h)
+        (by
+          have h : (thetaAboveCell0000220030013301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013301 h)
+        (by
+          have h : (thetaAboveCell0000220030013302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013302 h)
+        (by
+          have h : (thetaAboveCell0000220030013303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022003001)))
+        (by
+          have h : (thetaAboveCell0000220030013310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013310 h)
+        (by
+          have h : (thetaAboveCell0000220030013311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013311 h)
+        (by
+          have h : (thetaAboveCell0000220030013312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013312 h)
+        (by
+          have h : (thetaAboveCell0000220030013313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030013313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022003001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022003001))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022003001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022003001))) h)
+
+theorem cover_subtree_bb781a080715 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022003001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022003001)
+    cover_subtree_f11aed5a1b39
+    cover_subtree_13cfd42cc1e7
+    cover_subtree_49af2ce5bec1
+    cover_subtree_8008b2d13d89
+
+theorem cover_subtree_77cd353d9ac1 :
+    adaptiveCoverCheck 7 thetaAboveCell000022003001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003001
+    cover_subtree_b0fbdd86b439
+    cover_subtree_f6d73cfc8f96
+    cover_subtree_6705ea0fc5a0
+    cover_subtree_bb781a080715
+
+theorem cover_subtree_728def59c89b :
+    adaptiveCoverCheck 7 thetaAboveCell000022003002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003002
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003002)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022003002)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022003002)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022003002)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022003002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003002)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022003002)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022003002)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022003002)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022003002)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022003002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022003002) h)
+    (by
+      have h : ((childHH thetaAboveCell000022003002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022003002) h)
+
+theorem cover_subtree_14ed04390cca :
+    adaptiveCoverCheck 7 thetaAboveCell000022003003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003003
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003003)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022003003)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022003003)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022003003)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022003003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003003)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022003003)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022003003)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022003003)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022003003)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022003003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022003003) h)
+    (by
+      have h : ((childHH thetaAboveCell000022003003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022003003) h)
+
+theorem cover_subtree_f2d4d32c5725 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00002200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00002200)))
+    cover_subtree_df29b2bfe6c5
+    cover_subtree_77cd353d9ac1
+    cover_subtree_728def59c89b
+    cover_subtree_14ed04390cca
+
+theorem cover_subtree_6de0f218ce5a :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022003010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003010)
+    (by
+      have h : ((childLL (childLL thetaAboveCell000022003010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL thetaAboveCell000022003010)) h)
+    (by
+      have h : ((childLH (childLL thetaAboveCell000022003010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL thetaAboveCell000022003010)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLL thetaAboveCell000022003010))
+        (by
+          have h : ((childLL (childHL (childLL thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLL
+            thetaAboveCell000022003010))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLL
+            thetaAboveCell000022003010))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLL
+            thetaAboveCell000022003010))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLL
+            thetaAboveCell000022003010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLL thetaAboveCell000022003010))
+        (by
+          have h : ((childLL (childHH (childLL thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLL
+            thetaAboveCell000022003010))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLL
+            thetaAboveCell000022003010))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLL
+            thetaAboveCell000022003010))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLL
+            thetaAboveCell000022003010))) h))
+
+theorem cover_subtree_09a87211ea0e :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022003010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003010)
+    (by
+      have h : ((childLL (childLH thetaAboveCell000022003010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH thetaAboveCell000022003010)) h)
+    (by
+      have h : ((childLH (childLH thetaAboveCell000022003010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH thetaAboveCell000022003010)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLH thetaAboveCell000022003010))
+        (by
+          have h : ((childLL (childHL (childLH thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLH
+            thetaAboveCell000022003010))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLH
+            thetaAboveCell000022003010))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLH
+            thetaAboveCell000022003010))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLH
+            thetaAboveCell000022003010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLH thetaAboveCell000022003010))
+        (by
+          have h : ((childLL (childHH (childLH thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLH
+            thetaAboveCell000022003010))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLH
+            thetaAboveCell000022003010))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLH
+            thetaAboveCell000022003010))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaAboveCell000022003010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLH
+            thetaAboveCell000022003010))) h))
+
+theorem cover_subtree_1222a4bbaf46 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHL thetaAboveCell000022003010))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+    thetaAboveCell000022003010)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102020
+        (by
+          have h : ((childLL thetaAboveCell0000220030102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102021
+        (by
+          have h : ((childLL thetaAboveCell0000220030102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102022
+        (by
+          have h : ((childLL thetaAboveCell0000220030102022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102023
+        (by
+          have h : ((childLL thetaAboveCell0000220030102023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102023) h))
+
+theorem cover_subtree_f813068a0ab7 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHL thetaAboveCell000022003010))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+    thetaAboveCell000022003010)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102030
+        (by
+          have h : ((childLL thetaAboveCell0000220030102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102031
+        (by
+          have h : ((childLL thetaAboveCell0000220030102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102032
+        (by
+          have h : ((childLL thetaAboveCell0000220030102032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102033
+        (by
+          have h : ((childLL thetaAboveCell0000220030102033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102033) h))
+
+theorem cover_subtree_4a16ed9c24b8 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022003010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022003010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHL
+        thetaAboveCell000022003010)))
+        (by
+          have h : (thetaAboveCell0000220030102000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102000 h)
+        (by
+          have h : (thetaAboveCell0000220030102001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102001 h)
+        (by
+          have h : (thetaAboveCell0000220030102002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102002 h)
+        (by
+          have h : (thetaAboveCell0000220030102003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHL
+        thetaAboveCell000022003010)))
+        (by
+          have h : (thetaAboveCell0000220030102010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102010 h)
+        (by
+          have h : (thetaAboveCell0000220030102011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102011 h)
+        (by
+          have h : (thetaAboveCell0000220030102012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102012 h)
+        (by
+          have h : (thetaAboveCell0000220030102013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102013 h))
+    cover_subtree_1222a4bbaf46
+    cover_subtree_f813068a0ab7
+
+theorem cover_subtree_f9b60e2c6701 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHL thetaAboveCell000022003010))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+    thetaAboveCell000022003010)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102120
+        (by
+          have h : ((childLL thetaAboveCell0000220030102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102121
+        (by
+          have h : ((childLL thetaAboveCell0000220030102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102122
+        (by
+          have h : ((childLL thetaAboveCell0000220030102122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102123
+        (by
+          have h : ((childLL thetaAboveCell0000220030102123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102123) h))
+
+theorem cover_subtree_ccf472fb2f11 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHL thetaAboveCell000022003010))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+    thetaAboveCell000022003010)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102130
+        (by
+          have h : ((childLL thetaAboveCell0000220030102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102131
+        (by
+          have h : ((childLL thetaAboveCell0000220030102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102132
+        (by
+          have h : ((childLL thetaAboveCell0000220030102132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030102133
+        (by
+          have h : ((childLL thetaAboveCell0000220030102133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030102133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030102133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030102133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030102133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030102133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030102133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030102133) h))
+
+theorem cover_subtree_7315e35a4c15 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022003010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022003010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHL
+        thetaAboveCell000022003010)))
+        (by
+          have h : (thetaAboveCell0000220030102100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102100 h)
+        (by
+          have h : (thetaAboveCell0000220030102101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102101 h)
+        (by
+          have h : (thetaAboveCell0000220030102102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102102 h)
+        (by
+          have h : (thetaAboveCell0000220030102103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHL
+        thetaAboveCell000022003010)))
+        (by
+          have h : (thetaAboveCell0000220030102110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102110 h)
+        (by
+          have h : (thetaAboveCell0000220030102111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102111 h)
+        (by
+          have h : (thetaAboveCell0000220030102112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102112 h)
+        (by
+          have h : (thetaAboveCell0000220030102113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102113 h))
+    cover_subtree_f9b60e2c6701
+    cover_subtree_ccf472fb2f11
+
+theorem cover_subtree_5d09bb0853a8 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022003010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022003010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022003010)))
+        (by
+          have h : (thetaAboveCell0000220030102200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102200 h)
+        (by
+          have h : (thetaAboveCell0000220030102201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102201 h)
+        (by
+          have h : (thetaAboveCell0000220030102202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102202 h)
+        (by
+          have h : (thetaAboveCell0000220030102203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022003010)))
+        (by
+          have h : (thetaAboveCell0000220030102210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102210 h)
+        (by
+          have h : (thetaAboveCell0000220030102211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102211 h)
+        (by
+          have h : (thetaAboveCell0000220030102212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102212 h)
+        (by
+          have h : (thetaAboveCell0000220030102213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022003010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022003010))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022003010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022003010))) h)
+
+theorem cover_subtree_b82287a58720 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022003010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022003010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022003010)))
+        (by
+          have h : (thetaAboveCell0000220030102300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102300 h)
+        (by
+          have h : (thetaAboveCell0000220030102301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102301 h)
+        (by
+          have h : (thetaAboveCell0000220030102302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102302 h)
+        (by
+          have h : (thetaAboveCell0000220030102303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022003010)))
+        (by
+          have h : (thetaAboveCell0000220030102310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102310 h)
+        (by
+          have h : (thetaAboveCell0000220030102311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102311 h)
+        (by
+          have h : (thetaAboveCell0000220030102312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102312 h)
+        (by
+          have h : (thetaAboveCell0000220030102313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030102313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022003010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022003010))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022003010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022003010))) h)
+
+theorem cover_subtree_8668bec96df6 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022003010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022003010)
+    cover_subtree_4a16ed9c24b8
+    cover_subtree_7315e35a4c15
+    cover_subtree_5d09bb0853a8
+    cover_subtree_b82287a58720
+
+theorem cover_subtree_f7763668c82e :
+    adaptiveCoverCheck 4 (childHL (childLL (childHH thetaAboveCell000022003010))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+    thetaAboveCell000022003010)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103020
+        (by
+          have h : ((childLL thetaAboveCell0000220030103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103021
+        (by
+          have h : ((childLL thetaAboveCell0000220030103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103022
+        (by
+          have h : ((childLL thetaAboveCell0000220030103022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103023
+        (by
+          have h : ((childLL thetaAboveCell0000220030103023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103023) h))
+
+theorem cover_subtree_8b7188d14517 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHH thetaAboveCell000022003010))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+    thetaAboveCell000022003010)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103030
+        (by
+          have h : ((childLL thetaAboveCell0000220030103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103031
+        (by
+          have h : ((childLL thetaAboveCell0000220030103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103032
+        (by
+          have h : ((childLL thetaAboveCell0000220030103032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103033
+        (by
+          have h : ((childLL thetaAboveCell0000220030103033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103033) h))
+
+theorem cover_subtree_7cb6055f8e6a :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022003010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022003010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHH
+        thetaAboveCell000022003010)))
+        (by
+          have h : (thetaAboveCell0000220030103000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103000 h)
+        (by
+          have h : (thetaAboveCell0000220030103001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103001 h)
+        (by
+          have h : (thetaAboveCell0000220030103002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103002 h)
+        (by
+          have h : (thetaAboveCell0000220030103003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHH
+        thetaAboveCell000022003010)))
+        (by
+          have h : (thetaAboveCell0000220030103010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103010 h)
+        (by
+          have h : (thetaAboveCell0000220030103011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103011 h)
+        (by
+          have h : (thetaAboveCell0000220030103012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103012 h)
+        (by
+          have h : (thetaAboveCell0000220030103013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103013 h))
+    cover_subtree_f7763668c82e
+    cover_subtree_8b7188d14517
+
+theorem cover_subtree_d99c2055d296 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHH thetaAboveCell000022003010))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+    thetaAboveCell000022003010)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103120
+        (by
+          have h : ((childLL thetaAboveCell0000220030103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103121
+        (by
+          have h : ((childLL thetaAboveCell0000220030103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103122
+        (by
+          have h : ((childLL thetaAboveCell0000220030103122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103123
+        (by
+          have h : ((childLL thetaAboveCell0000220030103123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103123) h))
+
+theorem cover_subtree_a6b3a4adb113 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHH thetaAboveCell000022003010))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+    thetaAboveCell000022003010)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103130
+        (by
+          have h : ((childLL thetaAboveCell0000220030103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103131
+        (by
+          have h : ((childLL thetaAboveCell0000220030103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103132
+        (by
+          have h : ((childLL thetaAboveCell0000220030103132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103133
+        (by
+          have h : ((childLL thetaAboveCell0000220030103133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103133) h))
+
+theorem cover_subtree_b64accded341 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022003010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022003010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHH
+        thetaAboveCell000022003010)))
+        (by
+          have h : (thetaAboveCell0000220030103100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103100 h)
+        (by
+          have h : (thetaAboveCell0000220030103101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103101 h)
+        (by
+          have h : (thetaAboveCell0000220030103102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103102 h)
+        (by
+          have h : (thetaAboveCell0000220030103103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHH
+        thetaAboveCell000022003010)))
+        (by
+          have h : (thetaAboveCell0000220030103110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103110 h)
+        (by
+          have h : (thetaAboveCell0000220030103111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103111 h)
+        (by
+          have h : (thetaAboveCell0000220030103112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103112 h)
+        (by
+          have h : (thetaAboveCell0000220030103113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103113 h))
+    cover_subtree_d99c2055d296
+    cover_subtree_a6b3a4adb113
+
+theorem cover_subtree_cf9e35b58ea8 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022003010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022003010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022003010)))
+        (by
+          have h : (thetaAboveCell0000220030103200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103200 h)
+        (by
+          have h : (thetaAboveCell0000220030103201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103201 h)
+        (by
+          have h : (thetaAboveCell0000220030103202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103202 h)
+        (by
+          have h : (thetaAboveCell0000220030103203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022003010)))
+        (by
+          have h : (thetaAboveCell0000220030103210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103210 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103211
+            (by
+              have h : ((childLL thetaAboveCell0000220030103211)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103211)
+                h)
+            (by
+              have h : ((childLH thetaAboveCell0000220030103211)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103211)
+                h)
+            (by
+              have h : ((childHL thetaAboveCell0000220030103211)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103211)
+                h)
+            (by
+              have h : ((childHH thetaAboveCell0000220030103211)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103211)
+                h))
+        (by
+          have h : (thetaAboveCell0000220030103212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103212 h)
+        (by
+          have h : (thetaAboveCell0000220030103213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022003010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022003010))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022003010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022003010))) h)
+
+theorem cover_subtree_f351407ea291 :
+    adaptiveCoverCheck 4 (childLL (childHH (childHH thetaAboveCell000022003010))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+    thetaAboveCell000022003010)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103300
+        (by
+          have h : ((childLL thetaAboveCell0000220030103300)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103300) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103300)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103300) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103300)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103300) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103300)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103300) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103301
+        (by
+          have h : ((childLL thetaAboveCell0000220030103301)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103301) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103301)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103301) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103301)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103301) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103301)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103301) h))
+    (by
+      have h : (thetaAboveCell0000220030103302).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103302 h)
+    (by
+      have h : (thetaAboveCell0000220030103303).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103303 h)
+
+theorem cover_subtree_0ec74c623b25 :
+    adaptiveCoverCheck 4 (childLH (childHH (childHH thetaAboveCell000022003010))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+    thetaAboveCell000022003010)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103310
+        (by
+          have h : ((childLL thetaAboveCell0000220030103310)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103310) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103310)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103310) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103310)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103310) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103310)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103310) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030103311
+        (by
+          have h : ((childLL thetaAboveCell0000220030103311)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030103311) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030103311)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030103311) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030103311)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030103311) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030103311)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030103311) h))
+    (by
+      have h : (thetaAboveCell0000220030103312).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103312 h)
+    (by
+      have h : (thetaAboveCell0000220030103313).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030103313 h)
+
+theorem cover_subtree_5b0864ff4bca :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022003010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022003010))
+    cover_subtree_f351407ea291
+    cover_subtree_0ec74c623b25
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022003010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022003010))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022003010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022003010))) h)
+
+theorem cover_subtree_deeae114ec93 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022003010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022003010)
+    cover_subtree_7cb6055f8e6a
+    cover_subtree_b64accded341
+    cover_subtree_cf9e35b58ea8
+    cover_subtree_5b0864ff4bca
+
+theorem cover_subtree_591cadb839e1 :
+    adaptiveCoverCheck 7 thetaAboveCell000022003010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003010
+    cover_subtree_6de0f218ce5a
+    cover_subtree_09a87211ea0e
+    cover_subtree_8668bec96df6
+    cover_subtree_deeae114ec93
+
+theorem cover_subtree_30fdf3821f37 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022003011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003011)
+    (by
+      have h : ((childLL (childLL thetaAboveCell000022003011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL thetaAboveCell000022003011)) h)
+    (by
+      have h : ((childLH (childLL thetaAboveCell000022003011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL thetaAboveCell000022003011)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLL thetaAboveCell000022003011))
+        (by
+          have h : ((childLL (childHL (childLL thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLL
+            thetaAboveCell000022003011))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLL
+            thetaAboveCell000022003011))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLL
+            thetaAboveCell000022003011))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLL
+            thetaAboveCell000022003011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLL thetaAboveCell000022003011))
+        (by
+          have h : ((childLL (childHH (childLL thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLL
+            thetaAboveCell000022003011))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLL
+            thetaAboveCell000022003011))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLL
+            thetaAboveCell000022003011))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLL
+            thetaAboveCell000022003011))) h))
+
+theorem cover_subtree_c04a84ca091e :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022003011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003011)
+    (by
+      have h : ((childLL (childLH thetaAboveCell000022003011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH thetaAboveCell000022003011)) h)
+    (by
+      have h : ((childLH (childLH thetaAboveCell000022003011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH thetaAboveCell000022003011)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLH thetaAboveCell000022003011))
+        (by
+          have h : ((childLL (childHL (childLH thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLH
+            thetaAboveCell000022003011))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLH
+            thetaAboveCell000022003011))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLH
+            thetaAboveCell000022003011))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLH
+            thetaAboveCell000022003011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLH thetaAboveCell000022003011))
+        (by
+          have h : ((childLL (childHH (childLH thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLH
+            thetaAboveCell000022003011))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLH
+            thetaAboveCell000022003011))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLH
+            thetaAboveCell000022003011))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaAboveCell000022003011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLH
+            thetaAboveCell000022003011))) h))
+
+theorem cover_subtree_665b5a9e2762 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHL thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112020
+        (by
+          have h : ((childLL thetaAboveCell0000220030112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112021
+        (by
+          have h : ((childLL thetaAboveCell0000220030112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112022
+        (by
+          have h : ((childLL thetaAboveCell0000220030112022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112023
+        (by
+          have h : ((childLL thetaAboveCell0000220030112023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112023) h))
+
+theorem cover_subtree_5b6db01eac76 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHL thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112030
+        (by
+          have h : ((childLL thetaAboveCell0000220030112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112031
+        (by
+          have h : ((childLL thetaAboveCell0000220030112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112032
+        (by
+          have h : ((childLL thetaAboveCell0000220030112032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112033
+        (by
+          have h : ((childLL thetaAboveCell0000220030112033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112033) h))
+
+theorem cover_subtree_f193f827132f :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022003011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022003011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHL
+        thetaAboveCell000022003011)))
+        (by
+          have h : (thetaAboveCell0000220030112000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112000 h)
+        (by
+          have h : (thetaAboveCell0000220030112001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112001 h)
+        (by
+          have h : (thetaAboveCell0000220030112002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112002 h)
+        (by
+          have h : (thetaAboveCell0000220030112003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHL
+        thetaAboveCell000022003011)))
+        (by
+          have h : (thetaAboveCell0000220030112010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112010 h)
+        (by
+          have h : (thetaAboveCell0000220030112011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112011 h)
+        (by
+          have h : (thetaAboveCell0000220030112012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112012 h)
+        (by
+          have h : (thetaAboveCell0000220030112013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112013 h))
+    cover_subtree_665b5a9e2762
+    cover_subtree_5b6db01eac76
+
+theorem cover_subtree_852b952eb7f8 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHL thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112120
+        (by
+          have h : ((childLL thetaAboveCell0000220030112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112121
+        (by
+          have h : ((childLL thetaAboveCell0000220030112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112122
+        (by
+          have h : ((childLL thetaAboveCell0000220030112122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112123
+        (by
+          have h : ((childLL thetaAboveCell0000220030112123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112123) h))
+
+theorem cover_subtree_aaaee1c4417b :
+    adaptiveCoverCheck 4 (childHH (childLH (childHL thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112130
+        (by
+          have h : ((childLL thetaAboveCell0000220030112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112131
+        (by
+          have h : ((childLL thetaAboveCell0000220030112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112132
+        (by
+          have h : ((childLL thetaAboveCell0000220030112132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112133
+        (by
+          have h : ((childLL thetaAboveCell0000220030112133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112133) h))
+
+theorem cover_subtree_351a53bd4fd0 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022003011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022003011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHL
+        thetaAboveCell000022003011)))
+        (by
+          have h : (thetaAboveCell0000220030112100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112100 h)
+        (by
+          have h : (thetaAboveCell0000220030112101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112101 h)
+        (by
+          have h : (thetaAboveCell0000220030112102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112102 h)
+        (by
+          have h : (thetaAboveCell0000220030112103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHL
+        thetaAboveCell000022003011)))
+        (by
+          have h : (thetaAboveCell0000220030112110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112110 h)
+        (by
+          have h : (thetaAboveCell0000220030112111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112111 h)
+        (by
+          have h : (thetaAboveCell0000220030112112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112112 h)
+        (by
+          have h : (thetaAboveCell0000220030112113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112113 h))
+    cover_subtree_852b952eb7f8
+    cover_subtree_aaaee1c4417b
+
+theorem cover_subtree_272316153167 :
+    adaptiveCoverCheck 4 (childLL (childHL (childHL thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112200
+        (by
+          have h : ((childLL thetaAboveCell0000220030112200)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112200) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112200)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112200) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112200)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112200) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112200)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112200) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112201
+        (by
+          have h : ((childLL thetaAboveCell0000220030112201)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112201) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112201)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112201) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112201)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112201) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112201)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112201) h))
+    (by
+      have h : (thetaAboveCell0000220030112202).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112202 h)
+    (by
+      have h : (thetaAboveCell0000220030112203).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112203 h)
+
+theorem cover_subtree_665ac8a38e4e :
+    adaptiveCoverCheck 4 (childLH (childHL (childHL thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112210
+        (by
+          have h : ((childLL thetaAboveCell0000220030112210)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112210) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112210)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112210) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112210)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112210) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112210)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112210) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112211
+        (by
+          have h : ((childLL thetaAboveCell0000220030112211)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112211) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112211)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112211) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112211)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112211) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112211)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112211) h))
+    (by
+      have h : (thetaAboveCell0000220030112212).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112212 h)
+    (by
+      have h : (thetaAboveCell0000220030112213).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112213 h)
+
+theorem cover_subtree_53d66be9fa12 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022003011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022003011))
+    cover_subtree_272316153167
+    cover_subtree_665ac8a38e4e
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022003011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022003011))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022003011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022003011))) h)
+
+theorem cover_subtree_173035df5204 :
+    adaptiveCoverCheck 4 (childLL (childHH (childHL thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112300
+        (by
+          have h : ((childLL thetaAboveCell0000220030112300)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112300) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112300)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112300) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112300)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112300) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112300)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112300) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112301
+        (by
+          have h : ((childLL thetaAboveCell0000220030112301)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112301) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112301)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112301) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112301)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112301) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112301)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112301) h))
+    (by
+      have h : (thetaAboveCell0000220030112302).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112302 h)
+    (by
+      have h : (thetaAboveCell0000220030112303).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112303 h)
+
+theorem cover_subtree_9b2979ef98c7 :
+    adaptiveCoverCheck 4 (childLH (childHH (childHL thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112310
+        (by
+          have h : ((childLL thetaAboveCell0000220030112310)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112310) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112310)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112310) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112310)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112310) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112310)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112310) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030112311
+        (by
+          have h : ((childLL thetaAboveCell0000220030112311)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030112311) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030112311)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030112311) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030112311)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030112311) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030112311)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030112311) h))
+    (by
+      have h : (thetaAboveCell0000220030112312).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112312 h)
+    (by
+      have h : (thetaAboveCell0000220030112313).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030112313 h)
+
+theorem cover_subtree_c1dab1b76c15 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022003011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022003011))
+    cover_subtree_173035df5204
+    cover_subtree_9b2979ef98c7
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022003011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022003011))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022003011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022003011))) h)
+
+theorem cover_subtree_7f164fbda87a :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022003011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022003011)
+    cover_subtree_f193f827132f
+    cover_subtree_351a53bd4fd0
+    cover_subtree_53d66be9fa12
+    cover_subtree_c1dab1b76c15
+
+theorem cover_subtree_45ff29f15e49 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHH thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113020
+        (by
+          have h : ((childLL thetaAboveCell0000220030113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113021
+        (by
+          have h : ((childLL thetaAboveCell0000220030113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113022
+        (by
+          have h : ((childLL thetaAboveCell0000220030113022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113023
+        (by
+          have h : ((childLL thetaAboveCell0000220030113023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113023) h))
+
+theorem cover_subtree_2ca50076570c :
+    adaptiveCoverCheck 4 (childHH (childLL (childHH thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113030
+        (by
+          have h : ((childLL thetaAboveCell0000220030113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113031
+        (by
+          have h : ((childLL thetaAboveCell0000220030113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113032
+        (by
+          have h : ((childLL thetaAboveCell0000220030113032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113033
+        (by
+          have h : ((childLL thetaAboveCell0000220030113033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113033) h))
+
+theorem cover_subtree_6cee8ff9ebee :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022003011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022003011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHH
+        thetaAboveCell000022003011)))
+        (by
+          have h : (thetaAboveCell0000220030113000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113000 h)
+        (by
+          have h : (thetaAboveCell0000220030113001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113001 h)
+        (by
+          have h : (thetaAboveCell0000220030113002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113002 h)
+        (by
+          have h : (thetaAboveCell0000220030113003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHH
+        thetaAboveCell000022003011)))
+        (by
+          have h : (thetaAboveCell0000220030113010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113010 h)
+        (by
+          have h : (thetaAboveCell0000220030113011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113011 h)
+        (by
+          have h : (thetaAboveCell0000220030113012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113012 h)
+        (by
+          have h : (thetaAboveCell0000220030113013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113013 h))
+    cover_subtree_45ff29f15e49
+    cover_subtree_2ca50076570c
+
+theorem cover_subtree_ce8d22ab5f51 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHH thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113120
+        (by
+          have h : ((childLL thetaAboveCell0000220030113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113121
+        (by
+          have h : ((childLL thetaAboveCell0000220030113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113122
+        (by
+          have h : ((childLL thetaAboveCell0000220030113122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113123
+        (by
+          have h : ((childLL thetaAboveCell0000220030113123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113123) h))
+
+theorem cover_subtree_23abbb95182b :
+    adaptiveCoverCheck 4 (childHH (childLH (childHH thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113130
+        (by
+          have h : ((childLL thetaAboveCell0000220030113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113131
+        (by
+          have h : ((childLL thetaAboveCell0000220030113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113132
+        (by
+          have h : ((childLL thetaAboveCell0000220030113132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113133
+        (by
+          have h : ((childLL thetaAboveCell0000220030113133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113133) h))
+
+theorem cover_subtree_7ed35f4a5f6c :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022003011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022003011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHH
+        thetaAboveCell000022003011)))
+        (by
+          have h : (thetaAboveCell0000220030113100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113100 h)
+        (by
+          have h : (thetaAboveCell0000220030113101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113101 h)
+        (by
+          have h : (thetaAboveCell0000220030113102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113102 h)
+        (by
+          have h : (thetaAboveCell0000220030113103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHH
+        thetaAboveCell000022003011)))
+        (by
+          have h : (thetaAboveCell0000220030113110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113110 h)
+        (by
+          have h : (thetaAboveCell0000220030113111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113111 h)
+        (by
+          have h : (thetaAboveCell0000220030113112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113112 h)
+        (by
+          have h : (thetaAboveCell0000220030113113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113113 h))
+    cover_subtree_ce8d22ab5f51
+    cover_subtree_23abbb95182b
+
+theorem cover_subtree_c767073053c7 :
+    adaptiveCoverCheck 4 (childLL (childHL (childHH thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113200
+        (by
+          have h : ((childLL thetaAboveCell0000220030113200)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113200) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113200)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113200) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113200)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113200) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113200)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113200) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113201
+        (by
+          have h : ((childLL thetaAboveCell0000220030113201)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113201) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113201)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113201) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113201)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113201) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113201)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113201) h))
+    (by
+      have h : (thetaAboveCell0000220030113202).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113202 h)
+    (by
+      have h : (thetaAboveCell0000220030113203).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113203 h)
+
+theorem cover_subtree_76bb187113c3 :
+    adaptiveCoverCheck 4 (childLH (childHL (childHH thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113210
+        (by
+          have h : ((childLL thetaAboveCell0000220030113210)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113210) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113210)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113210) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113210)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113210) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113210)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113210) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113211
+        (by
+          have h : ((childLL thetaAboveCell0000220030113211)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113211) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113211)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113211) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113211)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113211) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113211)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113211) h))
+    (by
+      have h : (thetaAboveCell0000220030113212).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113212 h)
+    (by
+      have h : (thetaAboveCell0000220030113213).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113213 h)
+
+theorem cover_subtree_9060e2b44288 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022003011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022003011))
+    cover_subtree_c767073053c7
+    cover_subtree_76bb187113c3
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022003011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022003011))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022003011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022003011))) h)
+
+theorem cover_subtree_15620f5ff6d8 :
+    adaptiveCoverCheck 4 (childLL (childHH (childHH thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113300
+        (by
+          have h : ((childLL thetaAboveCell0000220030113300)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113300) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113300)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113300) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113300)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113300) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113300)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113300) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113301
+        (by
+          have h : ((childLL thetaAboveCell0000220030113301)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113301) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113301)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113301) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113301)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113301) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113301)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113301) h))
+    (by
+      have h : (thetaAboveCell0000220030113302).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113302 h)
+    (by
+      have h : (thetaAboveCell0000220030113303).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113303 h)
+
+theorem cover_subtree_4b9adb2a2d0e :
+    adaptiveCoverCheck 4 (childLH (childHH (childHH thetaAboveCell000022003011))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+    thetaAboveCell000022003011)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113310
+        (by
+          have h : ((childLL thetaAboveCell0000220030113310)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113310) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113310)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113310) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113310)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113310) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113310)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113310) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220030113311
+        (by
+          have h : ((childLL thetaAboveCell0000220030113311)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220030113311) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220030113311)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220030113311) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220030113311)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220030113311) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220030113311)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220030113311) h))
+    (by
+      have h : (thetaAboveCell0000220030113312).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113312 h)
+    (by
+      have h : (thetaAboveCell0000220030113313).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220030113313 h)
+
+theorem cover_subtree_92d2cedcde3a :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022003011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022003011))
+    cover_subtree_15620f5ff6d8
+    cover_subtree_4b9adb2a2d0e
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022003011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022003011))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022003011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022003011))) h)
+
+theorem cover_subtree_f164ca59fbf6 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022003011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022003011)
+    cover_subtree_6cee8ff9ebee
+    cover_subtree_7ed35f4a5f6c
+    cover_subtree_9060e2b44288
+    cover_subtree_92d2cedcde3a
+
+theorem cover_subtree_a889ceb5a59d :
+    adaptiveCoverCheck 7 thetaAboveCell000022003011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003011
+    cover_subtree_30fdf3821f37
+    cover_subtree_c04a84ca091e
+    cover_subtree_7f164fbda87a
+    cover_subtree_f164ca59fbf6
+
+theorem cover_subtree_258ec0944afe :
+    adaptiveCoverCheck 7 thetaAboveCell000022003012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003012
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003012)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022003012)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022003012)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022003012)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022003012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003012)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022003012)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022003012)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022003012)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022003012)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022003012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022003012) h)
+    (by
+      have h : ((childHH thetaAboveCell000022003012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022003012) h)
+
+theorem cover_subtree_935d1978c95d :
+    adaptiveCoverCheck 7 thetaAboveCell000022003013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022003013
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022003013)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022003013)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022003013)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022003013)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022003013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022003013)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022003013)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022003013)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022003013)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022003013)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022003013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022003013) h)
+    (by
+      have h : ((childHH thetaAboveCell000022003013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022003013) h)
+
+theorem cover_subtree_53052734842a :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00002200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00002200)))
+    cover_subtree_591cadb839e1
+    cover_subtree_a889ceb5a59d
+    cover_subtree_258ec0944afe
+    cover_subtree_935d1978c95d
+
+theorem e24KC2ThetaAboveLeaf0000220030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00002200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00002200))
+    cover_subtree_f2d4d32c5725
+    cover_subtree_53052734842a
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00002200)))
+        (by
+          have h : (thetaAboveCell000022003020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003020 h)
+        (by
+          have h : (thetaAboveCell000022003021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003021 h)
+        (by
+          have h : (thetaAboveCell000022003022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003022 h)
+        (by
+          have h : (thetaAboveCell000022003023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00002200)))
+        (by
+          have h : (thetaAboveCell000022003030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003030 h)
+        (by
+          have h : (thetaAboveCell000022003031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003031 h)
+        (by
+          have h : (thetaAboveCell000022003032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003032 h)
+        (by
+          have h : (thetaAboveCell000022003033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022003033 h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.ThetaAbove.Leaf00642`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00643`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00646`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00647`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00648`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00649`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00652`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00653`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00654`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00655`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00657`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00658`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00662`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00663`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00664`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00665`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c0_6_00642
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5e16eb2f1b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5e16eb2f1b
+
+open CertificateCells5e16eb2f1b
+namespace CoverCertificate1bb21a49a4
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1bb21a49a4
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c0 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022002011) = true := by
+  exact CoverCertificate1bb21a49a4.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c1_6_00643
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb7fa51f50b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb7fa51f50b
+
+open CertificateCellsb7fa51f50b
+namespace CoverCertificatec9db3a1aba
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec9db3a1aba
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c1 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022002011) = true := by
+  exact CoverCertificatec9db3a1aba.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c2_c0_c0_4_00646
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd618452127
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd618452127
+
+open CertificateCellsd618452127
+namespace CoverCertificatefb03004e1f
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefb03004e1f
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c0_c0 :
+    adaptiveCoverCheck 4 (childLL (childLL (childHL thetaAboveCell000022002011))) = true := by
+  exact CoverCertificatefb03004e1f.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c2_c0_c1_4_00647
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells82d3525350
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells82d3525350
+
+open CertificateCells82d3525350
+namespace CoverCertificate045cecfd86
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate045cecfd86
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c0_c1 :
+    adaptiveCoverCheck 4 (childLH (childLL (childHL thetaAboveCell000022002011))) = true := by
+  exact CoverCertificate045cecfd86.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c2_c0_c2_4_00648
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells632f810978
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells632f810978
+
+open CertificateCells632f810978
+namespace CoverCertificatef837300024
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef837300024
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c0_c2 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHL thetaAboveCell000022002011))) = true := by
+  exact CoverCertificatef837300024.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c2_c0_c3_4_00649
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells15b8174174
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells15b8174174
+
+open CertificateCells15b8174174
+namespace CoverCertificate2943ba0b5c
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate2943ba0b5c
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c0_c3 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHL thetaAboveCell000022002011))) = true := by
+  exact CoverCertificate2943ba0b5c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c2_c1_c0_4_00652
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1f25007fe8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1f25007fe8
+
+open CertificateCells1f25007fe8
+namespace CoverCertificate42a60e756f
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate42a60e756f
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c1_c0 :
+    adaptiveCoverCheck 4 (childLL (childLH (childHL thetaAboveCell000022002011))) = true := by
+  exact CoverCertificate42a60e756f.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c2_c1_c1_4_00653
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1e9acaa8be
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1e9acaa8be
+
+open CertificateCells1e9acaa8be
+namespace CoverCertificate0cee658f4b
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0cee658f4b
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c1_c1 :
+    adaptiveCoverCheck 4 (childLH (childLH (childHL thetaAboveCell000022002011))) = true := by
+  exact CoverCertificate0cee658f4b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c2_c1_c2_4_00654
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells231d66ab0c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells231d66ab0c
+
+open CertificateCells231d66ab0c
+namespace CoverCertificateeb376f6b8e
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateeb376f6b8e
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c1_c2 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHL thetaAboveCell000022002011))) = true := by
+  exact CoverCertificateeb376f6b8e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c2_c1_c3_4_00655
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2c11250698
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2c11250698
+
+open CertificateCells2c11250698
+namespace CoverCertificate1e208f1b6e
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1e208f1b6e
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c1_c3 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHL thetaAboveCell000022002011))) = true := by
+  exact CoverCertificate1e208f1b6e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c2_c2_5_00657
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3ce31f23ce
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3ce31f23ce
+
+open CertificateCells3ce31f23ce
+namespace CoverCertificate6fd62aabed
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate6fd62aabed
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c2 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022002011)) = true := by
+  exact CoverCertificate6fd62aabed.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c2_c3_5_00658
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3c3d48c1eb
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3c3d48c1eb
+
+open CertificateCells3c3d48c1eb
+namespace CoverCertificate22515ac7bd
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate22515ac7bd
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c3 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022002011)) = true := by
+  exact CoverCertificate22515ac7bd.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c0_c0_4_00662
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4d022b961f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4d022b961f
+
+open CertificateCells4d022b961f
+namespace CoverCertificatebad9a9b575
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatebad9a9b575
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c0_c0 :
+    adaptiveCoverCheck 4 (childLL (childLL (childHH thetaAboveCell000022002011))) = true := by
+  exact CoverCertificatebad9a9b575.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c0_c1_4_00663
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9798c9c840
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9798c9c840
+
+open CertificateCells9798c9c840
+namespace CoverCertificate13a680926b
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate13a680926b
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c0_c1 :
+    adaptiveCoverCheck 4 (childLH (childLL (childHH thetaAboveCell000022002011))) = true := by
+  exact CoverCertificate13a680926b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c0_c2_4_00664
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6832cee036
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6832cee036
+
+open CertificateCells6832cee036
+namespace CoverCertificatebdefa88d14
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatebdefa88d14
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c0_c2 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHH thetaAboveCell000022002011))) = true := by
+  exact CoverCertificatebdefa88d14.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c0_c3_4_00665
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsab896aa443
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsab896aa443
+
+open CertificateCellsab896aa443
+namespace CoverCertificateb54a5d06ec
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateb54a5d06ec
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c0_c3 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHH thetaAboveCell000022002011))) = true := by
+  exact CoverCertificateb54a5d06ec.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle014.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle014.lean
new file mode 100644
index 0000000000..bb9633bce3
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle014.lean
@@ -0,0 +1,8986 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch020`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCellsea1d7ddac9
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0001 : AngleCell :=
+  childLH (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `000033102100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00003310)))
+
+/-- Subcell `000033103000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00003310)))
+
+/-- Subcell `000033112000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00003311)))
+
+/-- Subcell `000033113000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000033113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000033113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00003311)))
+
+/-- Subcell `000122002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00012200)))
+
+/-- Subcell `000122003000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00012200)))
+
+/-- Subcell `000122012000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00012201)))
+
+/-- Subcell `000122013000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00012201)))
+
+/-- Subcell `000122102000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00012210)))
+
+/-- Subcell `000122103000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00012210)))
+
+/-- Subcell `000122112000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00012211)))
+
+/-- Subcell `000122113000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000122113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000122113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00012211)))
+
+/-- Subcell `000123002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00012300)))
+
+/-- Subcell `000123003000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00012300)))
+
+/-- Subcell `000123012000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00012301)))
+
+/-- Subcell `000123013000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00012301)))
+
+end GerverSofa.PartE.CertificateCellsea1d7ddac9
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6ProofBatch32fd0d2bad1c3ab1`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsea1d7ddac9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsea1d7ddac9
+
+open CertificateCellsea1d7ddac9
+theorem cover_subtree_0099e6c88888 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00003310)))
+    (by
+      have h : (thetaAboveCell000033102100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102100 h)
+    (by
+      have h : (thetaAboveCell000033102101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102102
+        (by
+          have h : ((childLL thetaAboveCell000033102102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102102) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102102) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102102) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102103
+        (by
+          have h : ((childLL thetaAboveCell000033102103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102103) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102103) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102103) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102103) h))
+
+theorem cover_subtree_ba641df32bfd :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00003310)))
+    (by
+      have h : (thetaAboveCell000033102110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102110 h)
+    (by
+      have h : (thetaAboveCell000033102111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102112
+        (by
+          have h : ((childLL thetaAboveCell000033102112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102112) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102112) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102112) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102113
+        (by
+          have h : ((childLL thetaAboveCell000033102113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102113) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102113) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102113) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102113) h))
+
+theorem cover_subtree_7ba93df0d4c7 :
+    adaptiveCoverCheck 8 (childHL (childLH (childHL thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL thetaAboveCell00003310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102120
+        (by
+          have h : ((childLL thetaAboveCell000033102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102120) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102120) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102120) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102121
+        (by
+          have h : ((childLL thetaAboveCell000033102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102121) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102121) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102121) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102122
+        (by
+          have h : ((childLL thetaAboveCell000033102122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102122) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102122) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102122) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102123
+        (by
+          have h : ((childLL thetaAboveCell000033102123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102123) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102123) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102123) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102123) h))
+
+theorem cover_subtree_65f8fb6981b4 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHL thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL thetaAboveCell00003310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102130
+        (by
+          have h : ((childLL thetaAboveCell000033102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102130) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102130) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102130) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102131
+        (by
+          have h : ((childLL thetaAboveCell000033102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102131) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102131) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102131) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102132
+        (by
+          have h : ((childLL thetaAboveCell000033102132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102132) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102132) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102132) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033102133
+        (by
+          have h : ((childLL thetaAboveCell000033102133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033102133) h)
+        (by
+          have h : ((childLH thetaAboveCell000033102133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033102133) h)
+        (by
+          have h : ((childHL thetaAboveCell000033102133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033102133) h)
+        (by
+          have h : ((childHH thetaAboveCell000033102133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033102133) h))
+
+theorem e24KC2ThetaAboveLeaf0000331021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00003310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00003310))
+    cover_subtree_0099e6c88888
+    cover_subtree_ba641df32bfd
+    cover_subtree_7ba93df0d4c7
+    cover_subtree_65f8fb6981b4
+theorem e24KC2ThetaAboveLeaf0000331022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00003310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00003310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00003310)))
+        (by
+          have h : (thetaAboveCell000033102200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102200 h)
+        (by
+          have h : (thetaAboveCell000033102201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102201 h)
+        (by
+          have h : (thetaAboveCell000033102202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102202 h)
+        (by
+          have h : (thetaAboveCell000033102203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00003310)))
+        (by
+          have h : (thetaAboveCell000033102210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102210 h)
+        (by
+          have h : (thetaAboveCell000033102211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102211 h)
+        (by
+          have h : (thetaAboveCell000033102212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102212 h)
+        (by
+          have h : (thetaAboveCell000033102213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00003310))) h)
+theorem e24KC2ThetaAboveLeaf0000331023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00003310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00003310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00003310)))
+        (by
+          have h : (thetaAboveCell000033102300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102300 h)
+        (by
+          have h : (thetaAboveCell000033102301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102301 h)
+        (by
+          have h : (thetaAboveCell000033102302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102302 h)
+        (by
+          have h : (thetaAboveCell000033102303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00003310)))
+        (by
+          have h : (thetaAboveCell000033102310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102310 h)
+        (by
+          have h : (thetaAboveCell000033102311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102311 h)
+        (by
+          have h : (thetaAboveCell000033102312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102312 h)
+        (by
+          have h : (thetaAboveCell000033102313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033102313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00003310))) h)
+theorem cover_subtree_3709c9a249b6 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00003310)))
+    (by
+      have h : (thetaAboveCell000033103000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103000 h)
+    (by
+      have h : (thetaAboveCell000033103001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103002
+        (by
+          have h : ((childLL thetaAboveCell000033103002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103002) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103002) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103002) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103003
+        (by
+          have h : ((childLL thetaAboveCell000033103003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103003) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103003) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103003) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103003) h))
+
+theorem cover_subtree_2b5889a2610f :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00003310)))
+    (by
+      have h : (thetaAboveCell000033103010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103010 h)
+    (by
+      have h : (thetaAboveCell000033103011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103012
+        (by
+          have h : ((childLL thetaAboveCell000033103012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103012) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103012) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103012) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103013
+        (by
+          have h : ((childLL thetaAboveCell000033103013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103013) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103013) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103013) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103013) h))
+
+theorem cover_subtree_81afd77398fe :
+    adaptiveCoverCheck 8 (childHL (childLL (childHH thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH thetaAboveCell00003310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103020
+        (by
+          have h : ((childLL thetaAboveCell000033103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103020) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103020) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103020) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103021
+        (by
+          have h : ((childLL thetaAboveCell000033103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103021) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103021) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103021) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103022
+        (by
+          have h : ((childLL thetaAboveCell000033103022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103022) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103022) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103022) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103023
+        (by
+          have h : ((childLL thetaAboveCell000033103023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103023) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103023) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103023) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103023) h))
+
+theorem cover_subtree_56217b7b19f0 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHH thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH thetaAboveCell00003310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103030
+        (by
+          have h : ((childLL thetaAboveCell000033103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103030) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103030) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103030) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103031
+        (by
+          have h : ((childLL thetaAboveCell000033103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103031) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103031) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103031) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103032
+        (by
+          have h : ((childLL thetaAboveCell000033103032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103032) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103032) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103032) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103033
+        (by
+          have h : ((childLL thetaAboveCell000033103033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103033) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103033) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103033) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103033) h))
+
+theorem e24KC2ThetaAboveLeaf0000331030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00003310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00003310))
+    cover_subtree_3709c9a249b6
+    cover_subtree_2b5889a2610f
+    cover_subtree_81afd77398fe
+    cover_subtree_56217b7b19f0
+theorem cover_subtree_126d0c7909f0 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00003310)))
+    (by
+      have h : (thetaAboveCell000033103100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103100 h)
+    (by
+      have h : (thetaAboveCell000033103101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103102
+        (by
+          have h : ((childLL thetaAboveCell000033103102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103102) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103102) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103102) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103103
+        (by
+          have h : ((childLL thetaAboveCell000033103103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103103) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103103) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103103) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103103) h))
+
+theorem cover_subtree_978b69a1c1bd :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00003310)))
+    (by
+      have h : (thetaAboveCell000033103110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103110 h)
+    (by
+      have h : (thetaAboveCell000033103111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103112
+        (by
+          have h : ((childLL thetaAboveCell000033103112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103112) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103112) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103112) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103113
+        (by
+          have h : ((childLL thetaAboveCell000033103113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103113) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103113) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103113) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103113) h))
+
+theorem cover_subtree_6f4368c59bdb :
+    adaptiveCoverCheck 8 (childHL (childLH (childHH thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH thetaAboveCell00003310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103120
+        (by
+          have h : ((childLL thetaAboveCell000033103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103120) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103120) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103120) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103121
+        (by
+          have h : ((childLL thetaAboveCell000033103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103121) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103121) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103121) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103122
+        (by
+          have h : ((childLL thetaAboveCell000033103122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103122) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103122) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103122) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103123
+        (by
+          have h : ((childLL thetaAboveCell000033103123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103123) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103123) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103123) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103123) h))
+
+theorem cover_subtree_12a65271b941 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHH thetaAboveCell00003310))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH thetaAboveCell00003310)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103130
+        (by
+          have h : ((childLL thetaAboveCell000033103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103130) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103130) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103130) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103131
+        (by
+          have h : ((childLL thetaAboveCell000033103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103131) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103131) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103131) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103132
+        (by
+          have h : ((childLL thetaAboveCell000033103132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103132) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103132) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103132) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033103133
+        (by
+          have h : ((childLL thetaAboveCell000033103133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033103133) h)
+        (by
+          have h : ((childLH thetaAboveCell000033103133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033103133) h)
+        (by
+          have h : ((childHL thetaAboveCell000033103133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033103133) h)
+        (by
+          have h : ((childHH thetaAboveCell000033103133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033103133) h))
+
+theorem e24KC2ThetaAboveLeaf0000331031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00003310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00003310))
+    cover_subtree_126d0c7909f0
+    cover_subtree_978b69a1c1bd
+    cover_subtree_6f4368c59bdb
+    cover_subtree_12a65271b941
+theorem e24KC2ThetaAboveLeaf0000331032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00003310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00003310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00003310)))
+        (by
+          have h : (thetaAboveCell000033103200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103200 h)
+        (by
+          have h : (thetaAboveCell000033103201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103201 h)
+        (by
+          have h : (thetaAboveCell000033103202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103202 h)
+        (by
+          have h : (thetaAboveCell000033103203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00003310)))
+        (by
+          have h : (thetaAboveCell000033103210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103210 h)
+        (by
+          have h : (thetaAboveCell000033103211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103211 h)
+        (by
+          have h : (thetaAboveCell000033103212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103212 h)
+        (by
+          have h : (thetaAboveCell000033103213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00003310))) h)
+theorem e24KC2ThetaAboveLeaf0000331033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00003310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00003310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00003310)))
+        (by
+          have h : (thetaAboveCell000033103300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103300 h)
+        (by
+          have h : (thetaAboveCell000033103301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103301 h)
+        (by
+          have h : (thetaAboveCell000033103302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103302 h)
+        (by
+          have h : (thetaAboveCell000033103303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00003310)))
+        (by
+          have h : (thetaAboveCell000033103310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103310 h)
+        (by
+          have h : (thetaAboveCell000033103311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103311 h)
+        (by
+          have h : (thetaAboveCell000033103312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103312 h)
+        (by
+          have h : (thetaAboveCell000033103313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033103313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00003310))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00003310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00003310))) h)
+theorem e24KC2ThetaAboveLeaf0000331102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00003311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00003311))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell00003311))) h)
+theorem e24KC2ThetaAboveLeaf0000331103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00003311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00003311))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childHL (childHH (childLL thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLL
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childHH (childHH (childLL thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLL
+        thetaAboveCell00003311))) h)
+theorem e24KC2ThetaAboveLeaf0000331112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00003311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00003311))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childHL (childHL (childLH thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLH
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell00003311))) h)
+theorem e24KC2ThetaAboveLeaf0000331113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00003311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00003311))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell00003311))) h)
+theorem cover_subtree_40b30098358c :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00003311)))
+    (by
+      have h : (thetaAboveCell000033112000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112000 h)
+    (by
+      have h : (thetaAboveCell000033112001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112002
+        (by
+          have h : ((childLL thetaAboveCell000033112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112002) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112002) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112002) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112003
+        (by
+          have h : ((childLL thetaAboveCell000033112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112003) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112003) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112003) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112003) h))
+
+theorem cover_subtree_0f9e0e377596 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00003311)))
+    (by
+      have h : (thetaAboveCell000033112010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112010 h)
+    (by
+      have h : (thetaAboveCell000033112011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112012
+        (by
+          have h : ((childLL thetaAboveCell000033112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112012) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112012) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112012) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112013
+        (by
+          have h : ((childLL thetaAboveCell000033112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112013) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112013) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112013) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112013) h))
+
+theorem cover_subtree_562ee101dc47 :
+    adaptiveCoverCheck 8 (childHL (childLL (childHL thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL thetaAboveCell00003311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112020
+        (by
+          have h : ((childLL thetaAboveCell000033112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112020) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112020) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112020) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112021
+        (by
+          have h : ((childLL thetaAboveCell000033112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112021) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112021) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112021) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112022
+        (by
+          have h : ((childLL thetaAboveCell000033112022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112022) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112022) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112022) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112023
+        (by
+          have h : ((childLL thetaAboveCell000033112023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112023) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112023) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112023) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112023) h))
+
+theorem cover_subtree_bc17552a24a4 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHL thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL thetaAboveCell00003311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112030
+        (by
+          have h : ((childLL thetaAboveCell000033112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112030) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112030) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112030) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112031
+        (by
+          have h : ((childLL thetaAboveCell000033112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112031) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112031) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112031) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112032
+        (by
+          have h : ((childLL thetaAboveCell000033112032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112032) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112032) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112032) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112033
+        (by
+          have h : ((childLL thetaAboveCell000033112033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112033) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112033) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112033) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112033) h))
+
+theorem e24KC2ThetaAboveLeaf0000331120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00003311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00003311))
+    cover_subtree_40b30098358c
+    cover_subtree_0f9e0e377596
+    cover_subtree_562ee101dc47
+    cover_subtree_bc17552a24a4
+theorem cover_subtree_260aee4b9fa3 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00003311)))
+    (by
+      have h : (thetaAboveCell000033112100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112100 h)
+    (by
+      have h : (thetaAboveCell000033112101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112102
+        (by
+          have h : ((childLL thetaAboveCell000033112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112102) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112102) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112102) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112103
+        (by
+          have h : ((childLL thetaAboveCell000033112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112103) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112103) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112103) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112103) h))
+
+theorem cover_subtree_674fd6a8471c :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00003311)))
+    (by
+      have h : (thetaAboveCell000033112110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112110 h)
+    (by
+      have h : (thetaAboveCell000033112111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112112
+        (by
+          have h : ((childLL thetaAboveCell000033112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112112) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112112) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112112) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112113
+        (by
+          have h : ((childLL thetaAboveCell000033112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112113) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112113) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112113) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112113) h))
+
+theorem cover_subtree_38509f162ffa :
+    adaptiveCoverCheck 8 (childHL (childLH (childHL thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL thetaAboveCell00003311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112120
+        (by
+          have h : ((childLL thetaAboveCell000033112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112120) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112120) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112120) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112121
+        (by
+          have h : ((childLL thetaAboveCell000033112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112121) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112121) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112121) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112122
+        (by
+          have h : ((childLL thetaAboveCell000033112122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112122) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112122) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112122) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112123
+        (by
+          have h : ((childLL thetaAboveCell000033112123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112123) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112123) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112123) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112123) h))
+
+theorem cover_subtree_2b615745aa63 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHL thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL thetaAboveCell00003311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112130
+        (by
+          have h : ((childLL thetaAboveCell000033112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112130) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112130) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112130) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033112131
+        (by
+          have h : ((childLL thetaAboveCell000033112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033112131) h)
+        (by
+          have h : ((childLH thetaAboveCell000033112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033112131) h)
+        (by
+          have h : ((childHL thetaAboveCell000033112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033112131) h)
+        (by
+          have h : ((childHH thetaAboveCell000033112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033112131) h))
+    (by
+      have h : (thetaAboveCell000033112132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112132 h)
+    (by
+      have h : (thetaAboveCell000033112133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112133 h)
+
+theorem e24KC2ThetaAboveLeaf0000331121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00003311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00003311))
+    cover_subtree_260aee4b9fa3
+    cover_subtree_674fd6a8471c
+    cover_subtree_38509f162ffa
+    cover_subtree_2b615745aa63
+theorem e24KC2ThetaAboveLeaf0000331122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00003311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00003311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00003311)))
+        (by
+          have h : (thetaAboveCell000033112200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112200 h)
+        (by
+          have h : (thetaAboveCell000033112201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112201 h)
+        (by
+          have h : (thetaAboveCell000033112202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112202 h)
+        (by
+          have h : (thetaAboveCell000033112203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00003311)))
+        (by
+          have h : (thetaAboveCell000033112210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112210 h)
+        (by
+          have h : (thetaAboveCell000033112211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112211 h)
+        (by
+          have h : (thetaAboveCell000033112212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112212 h)
+        (by
+          have h : (thetaAboveCell000033112213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00003311))) h)
+theorem e24KC2ThetaAboveLeaf0000331123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00003311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00003311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00003311)))
+        (by
+          have h : (thetaAboveCell000033112300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112300 h)
+        (by
+          have h : (thetaAboveCell000033112301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112301 h)
+        (by
+          have h : (thetaAboveCell000033112302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112302 h)
+        (by
+          have h : (thetaAboveCell000033112303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00003311)))
+        (by
+          have h : (thetaAboveCell000033112310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112310 h)
+        (by
+          have h : (thetaAboveCell000033112311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112311 h)
+        (by
+          have h : (thetaAboveCell000033112312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112312 h)
+        (by
+          have h : (thetaAboveCell000033112313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033112313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00003311))) h)
+theorem cover_subtree_7b02d095f599 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00003311)))
+    (by
+      have h : (thetaAboveCell000033113000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113000 h)
+    (by
+      have h : (thetaAboveCell000033113001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113002
+        (by
+          have h : ((childLL thetaAboveCell000033113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113002) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113002) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113002) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113003
+        (by
+          have h : ((childLL thetaAboveCell000033113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113003) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113003) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113003) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113003) h))
+
+theorem cover_subtree_27bc40bbefb0 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00003311)))
+    (by
+      have h : (thetaAboveCell000033113010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113010 h)
+    (by
+      have h : (thetaAboveCell000033113011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113012
+        (by
+          have h : ((childLL thetaAboveCell000033113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113012) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113012) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113012) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113013
+        (by
+          have h : ((childLL thetaAboveCell000033113013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113013) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113013) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113013) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113013) h))
+
+theorem cover_subtree_d6ec9d4db7c3 :
+    adaptiveCoverCheck 8 (childHL (childLL (childHH thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH thetaAboveCell00003311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113020
+        (by
+          have h : ((childLL thetaAboveCell000033113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113020) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113020) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113020) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113021
+        (by
+          have h : ((childLL thetaAboveCell000033113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113021) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113021) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113021) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113021) h))
+    (by
+      have h : (thetaAboveCell000033113022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113022 h)
+    (by
+      have h : (thetaAboveCell000033113023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113023 h)
+
+theorem cover_subtree_c7a6802137f2 :
+    adaptiveCoverCheck 8 (childHH (childLL (childHH thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH thetaAboveCell00003311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113030
+        (by
+          have h : ((childLL thetaAboveCell000033113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113030) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113030) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113030) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113031
+        (by
+          have h : ((childLL thetaAboveCell000033113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113031) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113031) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113031) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113031) h))
+    (by
+      have h : (thetaAboveCell000033113032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113032 h)
+    (by
+      have h : (thetaAboveCell000033113033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113033 h)
+
+theorem e24KC2ThetaAboveLeaf0000331130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00003311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00003311))
+    cover_subtree_7b02d095f599
+    cover_subtree_27bc40bbefb0
+    cover_subtree_d6ec9d4db7c3
+    cover_subtree_c7a6802137f2
+theorem cover_subtree_525965b29b43 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00003311)))
+    (by
+      have h : (thetaAboveCell000033113100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113100 h)
+    (by
+      have h : (thetaAboveCell000033113101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113102
+        (by
+          have h : ((childLL thetaAboveCell000033113102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113102) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113102) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113102) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113103
+        (by
+          have h : ((childLL thetaAboveCell000033113103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113103) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113103) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113103) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113103) h))
+
+theorem cover_subtree_426a701a7fc0 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00003311)))
+    (by
+      have h : (thetaAboveCell000033113110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113110 h)
+    (by
+      have h : (thetaAboveCell000033113111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113112
+        (by
+          have h : ((childLL thetaAboveCell000033113112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113112) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113112) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113112) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113113
+        (by
+          have h : ((childLL thetaAboveCell000033113113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113113) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113113) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113113) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113113) h))
+
+theorem cover_subtree_c43488da1c36 :
+    adaptiveCoverCheck 8 (childHL (childLH (childHH thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH thetaAboveCell00003311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113120
+        (by
+          have h : ((childLL thetaAboveCell000033113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113120) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113120) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113120) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113121
+        (by
+          have h : ((childLL thetaAboveCell000033113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113121) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113121) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113121) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113121) h))
+    (by
+      have h : (thetaAboveCell000033113122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113122 h)
+    (by
+      have h : (thetaAboveCell000033113123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113123 h)
+
+theorem cover_subtree_f77c14a0f9b2 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHH thetaAboveCell00003311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH thetaAboveCell00003311)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113130
+        (by
+          have h : ((childLL thetaAboveCell000033113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113130) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113130) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113130) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000033113131
+        (by
+          have h : ((childLL thetaAboveCell000033113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000033113131) h)
+        (by
+          have h : ((childLH thetaAboveCell000033113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000033113131) h)
+        (by
+          have h : ((childHL thetaAboveCell000033113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000033113131) h)
+        (by
+          have h : ((childHH thetaAboveCell000033113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000033113131) h))
+    (by
+      have h : (thetaAboveCell000033113132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113132 h)
+    (by
+      have h : (thetaAboveCell000033113133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113133 h)
+
+theorem e24KC2ThetaAboveLeaf0000331131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00003311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00003311))
+    cover_subtree_525965b29b43
+    cover_subtree_426a701a7fc0
+    cover_subtree_c43488da1c36
+    cover_subtree_f77c14a0f9b2
+theorem e24KC2ThetaAboveLeaf0000331132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00003311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00003311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00003311)))
+        (by
+          have h : (thetaAboveCell000033113200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113200 h)
+        (by
+          have h : (thetaAboveCell000033113201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113201 h)
+        (by
+          have h : (thetaAboveCell000033113202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113202 h)
+        (by
+          have h : (thetaAboveCell000033113203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00003311)))
+        (by
+          have h : (thetaAboveCell000033113210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113210 h)
+        (by
+          have h : (thetaAboveCell000033113211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113211 h)
+        (by
+          have h : (thetaAboveCell000033113212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113212 h)
+        (by
+          have h : (thetaAboveCell000033113213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00003311))) h)
+theorem e24KC2ThetaAboveLeaf0000331133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00003311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00003311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00003311)))
+        (by
+          have h : (thetaAboveCell000033113300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113300 h)
+        (by
+          have h : (thetaAboveCell000033113301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113301 h)
+        (by
+          have h : (thetaAboveCell000033113302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113302 h)
+        (by
+          have h : (thetaAboveCell000033113303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00003311)))
+        (by
+          have h : (thetaAboveCell000033113310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113310 h)
+        (by
+          have h : (thetaAboveCell000033113311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113311 h)
+        (by
+          have h : (thetaAboveCell000033113312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113312 h)
+        (by
+          have h : (thetaAboveCell000033113313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000033113313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00003311))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00003311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00003311))) h)
+theorem e24KC2ThetaAboveLeaf0001220002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00012200))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell00012200))) h)
+theorem e24KC2ThetaAboveLeaf0001220003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00012200))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childHL (childHH (childLL thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLL
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childHH (childHH (childLL thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLL
+        thetaAboveCell00012200))) h)
+theorem e24KC2ThetaAboveLeaf0001220012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00012200))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childHL (childHL (childLH thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLH
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell00012200))) h)
+theorem e24KC2ThetaAboveLeaf0001220013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00012200))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell00012200))) h)
+theorem cover_subtree_bc394da66032 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00012200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00012200)))
+    (by
+      have h : (thetaAboveCell000122002000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002000 h)
+    (by
+      have h : (thetaAboveCell000122002001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122002002
+        (by
+          have h : ((childLL thetaAboveCell000122002002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122002002) h)
+        (by
+          have h : ((childLH thetaAboveCell000122002002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122002002) h)
+        (by
+          have h : ((childHL thetaAboveCell000122002002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122002002) h)
+        (by
+          have h : ((childHH thetaAboveCell000122002002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122002002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122002003
+        (by
+          have h : ((childLL thetaAboveCell000122002003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122002003) h)
+        (by
+          have h : ((childLH thetaAboveCell000122002003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122002003) h)
+        (by
+          have h : ((childHL thetaAboveCell000122002003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122002003) h)
+        (by
+          have h : ((childHH thetaAboveCell000122002003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122002003) h))
+
+theorem cover_subtree_0f99148bdc12 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00012200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00012200)))
+    (by
+      have h : (thetaAboveCell000122002010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002010 h)
+    (by
+      have h : (thetaAboveCell000122002011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122002012
+        (by
+          have h : ((childLL thetaAboveCell000122002012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122002012) h)
+        (by
+          have h : ((childLH thetaAboveCell000122002012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122002012) h)
+        (by
+          have h : ((childHL thetaAboveCell000122002012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122002012) h)
+        (by
+          have h : ((childHH thetaAboveCell000122002012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122002012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122002013
+        (by
+          have h : ((childLL thetaAboveCell000122002013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122002013) h)
+        (by
+          have h : ((childLH thetaAboveCell000122002013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122002013) h)
+        (by
+          have h : ((childHL thetaAboveCell000122002013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122002013) h)
+        (by
+          have h : ((childHH thetaAboveCell000122002013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122002013) h))
+
+theorem cover_subtree_39d9268879db :
+    adaptiveCoverCheck 8 (childHL (childLL (childHL thetaAboveCell00012200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL thetaAboveCell00012200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122002020
+        (by
+          have h : ((childLL thetaAboveCell000122002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122002020) h)
+        (by
+          have h : ((childLH thetaAboveCell000122002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122002020) h)
+        (by
+          have h : ((childHL thetaAboveCell000122002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122002020) h)
+        (by
+          have h : ((childHH thetaAboveCell000122002020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122002020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122002021
+        (by
+          have h : ((childLL thetaAboveCell000122002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122002021) h)
+        (by
+          have h : ((childLH thetaAboveCell000122002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122002021) h)
+        (by
+          have h : ((childHL thetaAboveCell000122002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122002021) h)
+        (by
+          have h : ((childHH thetaAboveCell000122002021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122002021) h))
+    (by
+      have h : (thetaAboveCell000122002022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002022 h)
+    (by
+      have h : (thetaAboveCell000122002023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002023 h)
+
+theorem cover_subtree_39d9037f050e :
+    adaptiveCoverCheck 8 (childHH (childLL (childHL thetaAboveCell00012200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL thetaAboveCell00012200)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122002030
+        (by
+          have h : ((childLL thetaAboveCell000122002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122002030) h)
+        (by
+          have h : ((childLH thetaAboveCell000122002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122002030) h)
+        (by
+          have h : ((childHL thetaAboveCell000122002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122002030) h)
+        (by
+          have h : ((childHH thetaAboveCell000122002030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122002030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122002031
+        (by
+          have h : ((childLL thetaAboveCell000122002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122002031) h)
+        (by
+          have h : ((childLH thetaAboveCell000122002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122002031) h)
+        (by
+          have h : ((childHL thetaAboveCell000122002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122002031) h)
+        (by
+          have h : ((childHH thetaAboveCell000122002031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122002031) h))
+    (by
+      have h : (thetaAboveCell000122002032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002032 h)
+    (by
+      have h : (thetaAboveCell000122002033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002033 h)
+
+theorem e24KC2ThetaAboveLeaf0001220020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00012200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00012200))
+    cover_subtree_bc394da66032
+    cover_subtree_0f99148bdc12
+    cover_subtree_39d9268879db
+    cover_subtree_39d9037f050e
+theorem e24KC2ThetaAboveLeaf0001220021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00012200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00012200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122002100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002100 h)
+        (by
+          have h : (thetaAboveCell000122002101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002101 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122002102
+            (by
+              have h : ((childLL thetaAboveCell000122002102)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122002102) h)
+            (by
+              have h : ((childLH thetaAboveCell000122002102)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122002102) h)
+            (by
+              have h : ((childHL thetaAboveCell000122002102)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122002102) h)
+            (by
+              have h : ((childHH thetaAboveCell000122002102)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122002102) h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122002103
+            (by
+              have h : ((childLL thetaAboveCell000122002103)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122002103) h)
+            (by
+              have h : ((childLH thetaAboveCell000122002103)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122002103) h)
+            (by
+              have h : ((childHL thetaAboveCell000122002103)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122002103) h)
+            (by
+              have h : ((childHH thetaAboveCell000122002103)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122002103) h)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122002110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002110 h)
+        (by
+          have h : (thetaAboveCell000122002111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002111 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122002112
+            (by
+              have h : ((childLL thetaAboveCell000122002112)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122002112) h)
+            (by
+              have h : ((childLH thetaAboveCell000122002112)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122002112) h)
+            (by
+              have h : ((childHL thetaAboveCell000122002112)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122002112) h)
+            (by
+              have h : ((childHH thetaAboveCell000122002112)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122002112) h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122002113
+            (by
+              have h : ((childLL thetaAboveCell000122002113)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122002113) h)
+            (by
+              have h : ((childLH thetaAboveCell000122002113)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122002113) h)
+            (by
+              have h : ((childHL thetaAboveCell000122002113)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122002113) h)
+            (by
+              have h : ((childHH thetaAboveCell000122002113)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122002113) h)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00012200)))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122002120
+            (by
+              have h : ((childLL thetaAboveCell000122002120)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122002120) h)
+            (by
+              have h : ((childLH thetaAboveCell000122002120)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122002120) h)
+            (by
+              have h : ((childHL thetaAboveCell000122002120)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122002120) h)
+            (by
+              have h : ((childHH thetaAboveCell000122002120)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122002120) h))
+        (by
+          have h : (thetaAboveCell000122002121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002121 h)
+        (by
+          have h : (thetaAboveCell000122002122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002122 h)
+        (by
+          have h : (thetaAboveCell000122002123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122002130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002130 h)
+        (by
+          have h : (thetaAboveCell000122002131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002131 h)
+        (by
+          have h : (thetaAboveCell000122002132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002132 h)
+        (by
+          have h : (thetaAboveCell000122002133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002133 h))
+theorem e24KC2ThetaAboveLeaf0001220022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00012200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00012200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122002200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002200 h)
+        (by
+          have h : (thetaAboveCell000122002201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002201 h)
+        (by
+          have h : (thetaAboveCell000122002202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002202 h)
+        (by
+          have h : (thetaAboveCell000122002203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122002210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002210 h)
+        (by
+          have h : (thetaAboveCell000122002211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002211 h)
+        (by
+          have h : (thetaAboveCell000122002212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002212 h)
+        (by
+          have h : (thetaAboveCell000122002213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00012200))) h)
+theorem e24KC2ThetaAboveLeaf0001220023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00012200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00012200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122002300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002300 h)
+        (by
+          have h : (thetaAboveCell000122002301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002301 h)
+        (by
+          have h : (thetaAboveCell000122002302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002302 h)
+        (by
+          have h : (thetaAboveCell000122002303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122002310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002310 h)
+        (by
+          have h : (thetaAboveCell000122002311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002311 h)
+        (by
+          have h : (thetaAboveCell000122002312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002312 h)
+        (by
+          have h : (thetaAboveCell000122002313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122002313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00012200))) h)
+theorem e24KC2ThetaAboveLeaf0001220030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00012200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00012200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122003000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003000 h)
+        (by
+          have h : (thetaAboveCell000122003001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003001 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122003002
+            (by
+              have h : ((childLL thetaAboveCell000122003002)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122003002) h)
+            (by
+              have h : ((childLH thetaAboveCell000122003002)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122003002) h)
+            (by
+              have h : ((childHL thetaAboveCell000122003002)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122003002) h)
+            (by
+              have h : ((childHH thetaAboveCell000122003002)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122003002) h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122003003
+            (by
+              have h : ((childLL thetaAboveCell000122003003)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122003003) h)
+            (by
+              have h : ((childLH thetaAboveCell000122003003)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122003003) h)
+            (by
+              have h : ((childHL thetaAboveCell000122003003)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122003003) h)
+            (by
+              have h : ((childHH thetaAboveCell000122003003)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122003003) h)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122003010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003010 h)
+        (by
+          have h : (thetaAboveCell000122003011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003011 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122003012
+            (by
+              have h : ((childLL thetaAboveCell000122003012)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122003012) h)
+            (by
+              have h : ((childLH thetaAboveCell000122003012)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122003012) h)
+            (by
+              have h : ((childHL thetaAboveCell000122003012)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122003012) h)
+            (by
+              have h : ((childHH thetaAboveCell000122003012)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122003012) h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122003013
+            (by
+              have h : ((childLL thetaAboveCell000122003013)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122003013) h)
+            (by
+              have h : ((childLH thetaAboveCell000122003013)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122003013) h)
+            (by
+              have h : ((childHL thetaAboveCell000122003013)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122003013) h)
+            (by
+              have h : ((childHH thetaAboveCell000122003013)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122003013) h)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122003020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003020 h)
+        (by
+          have h : (thetaAboveCell000122003021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003021 h)
+        (by
+          have h : (thetaAboveCell000122003022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003022 h)
+        (by
+          have h : (thetaAboveCell000122003023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122003030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003030 h)
+        (by
+          have h : (thetaAboveCell000122003031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003031 h)
+        (by
+          have h : (thetaAboveCell000122003032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003032 h)
+        (by
+          have h : (thetaAboveCell000122003033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003033 h))
+theorem e24KC2ThetaAboveLeaf0001220031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00012200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00012200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122003100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003100 h)
+        (by
+          have h : (thetaAboveCell000122003101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003101 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122003102
+            (by
+              have h : ((childLL thetaAboveCell000122003102)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122003102) h)
+            (by
+              have h : ((childLH thetaAboveCell000122003102)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122003102) h)
+            (by
+              have h : ((childHL thetaAboveCell000122003102)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122003102) h)
+            (by
+              have h : ((childHH thetaAboveCell000122003102)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122003102) h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122003103
+            (by
+              have h : ((childLL thetaAboveCell000122003103)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122003103) h)
+            (by
+              have h : ((childLH thetaAboveCell000122003103)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122003103) h)
+            (by
+              have h : ((childHL thetaAboveCell000122003103)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122003103) h)
+            (by
+              have h : ((childHH thetaAboveCell000122003103)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122003103) h)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122003110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003110 h)
+        (by
+          have h : (thetaAboveCell000122003111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003111 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122003112
+            (by
+              have h : ((childLL thetaAboveCell000122003112)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122003112) h)
+            (by
+              have h : ((childLH thetaAboveCell000122003112)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122003112) h)
+            (by
+              have h : ((childHL thetaAboveCell000122003112)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122003112) h)
+            (by
+              have h : ((childHH thetaAboveCell000122003112)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122003112) h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122003113
+            (by
+              have h : ((childLL thetaAboveCell000122003113)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122003113) h)
+            (by
+              have h : ((childLH thetaAboveCell000122003113)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122003113) h)
+            (by
+              have h : ((childHL thetaAboveCell000122003113)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122003113) h)
+            (by
+              have h : ((childHH thetaAboveCell000122003113)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122003113) h)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122003120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003120 h)
+        (by
+          have h : (thetaAboveCell000122003121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003121 h)
+        (by
+          have h : (thetaAboveCell000122003122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003122 h)
+        (by
+          have h : (thetaAboveCell000122003123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122003130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003130 h)
+        (by
+          have h : (thetaAboveCell000122003131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003131 h)
+        (by
+          have h : (thetaAboveCell000122003132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003132 h)
+        (by
+          have h : (thetaAboveCell000122003133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003133 h))
+theorem e24KC2ThetaAboveLeaf0001220032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00012200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00012200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122003200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003200 h)
+        (by
+          have h : (thetaAboveCell000122003201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003201 h)
+        (by
+          have h : (thetaAboveCell000122003202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003202 h)
+        (by
+          have h : (thetaAboveCell000122003203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122003210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003210 h)
+        (by
+          have h : (thetaAboveCell000122003211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003211 h)
+        (by
+          have h : (thetaAboveCell000122003212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003212 h)
+        (by
+          have h : (thetaAboveCell000122003213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00012200))) h)
+theorem e24KC2ThetaAboveLeaf0001220033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00012200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00012200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122003300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003300 h)
+        (by
+          have h : (thetaAboveCell000122003301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003301 h)
+        (by
+          have h : (thetaAboveCell000122003302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003302 h)
+        (by
+          have h : (thetaAboveCell000122003303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00012200)))
+        (by
+          have h : (thetaAboveCell000122003310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003310 h)
+        (by
+          have h : (thetaAboveCell000122003311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003311 h)
+        (by
+          have h : (thetaAboveCell000122003312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003312 h)
+        (by
+          have h : (thetaAboveCell000122003313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122003313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00012200))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00012200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00012200))) h)
+theorem e24KC2ThetaAboveLeaf0001220102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00012201))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell00012201))) h)
+theorem e24KC2ThetaAboveLeaf0001220103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00012201))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childHL (childHH (childLL thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLL
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childHH (childHH (childLL thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLL
+        thetaAboveCell00012201))) h)
+theorem e24KC2ThetaAboveLeaf0001220112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00012201))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childHL (childHL (childLH thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLH
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell00012201))) h)
+theorem e24KC2ThetaAboveLeaf0001220113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00012201))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell00012201))) h)
+theorem e24KC2ThetaAboveLeaf0001220120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00012201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00012201))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122012000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012000 h)
+        (by
+          have h : (thetaAboveCell000122012001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012001 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122012002
+            (by
+              have h : ((childLL thetaAboveCell000122012002)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122012002) h)
+            (by
+              have h : ((childLH thetaAboveCell000122012002)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122012002) h)
+            (by
+              have h : ((childHL thetaAboveCell000122012002)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122012002) h)
+            (by
+              have h : ((childHH thetaAboveCell000122012002)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122012002) h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122012003
+            (by
+              have h : ((childLL thetaAboveCell000122012003)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122012003) h)
+            (by
+              have h : ((childLH thetaAboveCell000122012003)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122012003) h)
+            (by
+              have h : ((childHL thetaAboveCell000122012003)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122012003) h)
+            (by
+              have h : ((childHH thetaAboveCell000122012003)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122012003) h)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122012010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012010 h)
+        (by
+          have h : (thetaAboveCell000122012011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012011 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122012012
+            (by
+              have h : ((childLL thetaAboveCell000122012012)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122012012) h)
+            (by
+              have h : ((childLH thetaAboveCell000122012012)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122012012) h)
+            (by
+              have h : ((childHL thetaAboveCell000122012012)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122012012) h)
+            (by
+              have h : ((childHH thetaAboveCell000122012012)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122012012) h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000122012013
+            (by
+              have h : ((childLL thetaAboveCell000122012013)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000122012013) h)
+            (by
+              have h : ((childLH thetaAboveCell000122012013)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000122012013) h)
+            (by
+              have h : ((childHL thetaAboveCell000122012013)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000122012013) h)
+            (by
+              have h : ((childHH thetaAboveCell000122012013)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000122012013) h)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122012020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012020 h)
+        (by
+          have h : (thetaAboveCell000122012021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012021 h)
+        (by
+          have h : (thetaAboveCell000122012022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012022 h)
+        (by
+          have h : (thetaAboveCell000122012023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122012030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012030 h)
+        (by
+          have h : (thetaAboveCell000122012031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012031 h)
+        (by
+          have h : (thetaAboveCell000122012032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012032 h)
+        (by
+          have h : (thetaAboveCell000122012033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012033 h))
+theorem e24KC2ThetaAboveLeaf0001220121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00012201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00012201))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122012100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012100 h)
+        (by
+          have h : (thetaAboveCell000122012101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012101 h)
+        (by
+          have h : (thetaAboveCell000122012102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012102 h)
+        (by
+          have h : (thetaAboveCell000122012103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122012110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012110 h)
+        (by
+          have h : (thetaAboveCell000122012111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012111 h)
+        (by
+          have h : (thetaAboveCell000122012112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012112 h)
+        (by
+          have h : (thetaAboveCell000122012113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122012120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012120 h)
+        (by
+          have h : (thetaAboveCell000122012121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012121 h)
+        (by
+          have h : (thetaAboveCell000122012122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012122 h)
+        (by
+          have h : (thetaAboveCell000122012123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122012130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012130 h)
+        (by
+          have h : (thetaAboveCell000122012131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012131 h)
+        (by
+          have h : (thetaAboveCell000122012132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012132 h)
+        (by
+          have h : (thetaAboveCell000122012133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012133 h))
+theorem e24KC2ThetaAboveLeaf0001220122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00012201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00012201))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122012200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012200 h)
+        (by
+          have h : (thetaAboveCell000122012201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012201 h)
+        (by
+          have h : (thetaAboveCell000122012202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012202 h)
+        (by
+          have h : (thetaAboveCell000122012203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122012210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012210 h)
+        (by
+          have h : (thetaAboveCell000122012211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012211 h)
+        (by
+          have h : (thetaAboveCell000122012212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012212 h)
+        (by
+          have h : (thetaAboveCell000122012213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00012201))) h)
+theorem e24KC2ThetaAboveLeaf0001220123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00012201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00012201))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122012300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012300 h)
+        (by
+          have h : (thetaAboveCell000122012301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012301 h)
+        (by
+          have h : (thetaAboveCell000122012302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012302 h)
+        (by
+          have h : (thetaAboveCell000122012303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122012310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012310 h)
+        (by
+          have h : (thetaAboveCell000122012311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012311 h)
+        (by
+          have h : (thetaAboveCell000122012312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012312 h)
+        (by
+          have h : (thetaAboveCell000122012313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122012313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00012201))) h)
+theorem e24KC2ThetaAboveLeaf0001220130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00012201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00012201))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122013000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013000 h)
+        (by
+          have h : (thetaAboveCell000122013001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013001 h)
+        (by
+          have h : (thetaAboveCell000122013002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013002 h)
+        (by
+          have h : (thetaAboveCell000122013003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122013010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013010 h)
+        (by
+          have h : (thetaAboveCell000122013011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013011 h)
+        (by
+          have h : (thetaAboveCell000122013012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013012 h)
+        (by
+          have h : (thetaAboveCell000122013013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122013020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013020 h)
+        (by
+          have h : (thetaAboveCell000122013021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013021 h)
+        (by
+          have h : (thetaAboveCell000122013022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013022 h)
+        (by
+          have h : (thetaAboveCell000122013023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122013030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013030 h)
+        (by
+          have h : (thetaAboveCell000122013031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013031 h)
+        (by
+          have h : (thetaAboveCell000122013032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013032 h)
+        (by
+          have h : (thetaAboveCell000122013033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013033 h))
+theorem e24KC2ThetaAboveLeaf0001220131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00012201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00012201))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122013100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013100 h)
+        (by
+          have h : (thetaAboveCell000122013101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013101 h)
+        (by
+          have h : (thetaAboveCell000122013102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013102 h)
+        (by
+          have h : (thetaAboveCell000122013103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122013110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013110 h)
+        (by
+          have h : (thetaAboveCell000122013111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013111 h)
+        (by
+          have h : (thetaAboveCell000122013112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013112 h)
+        (by
+          have h : (thetaAboveCell000122013113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122013120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013120 h)
+        (by
+          have h : (thetaAboveCell000122013121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013121 h)
+        (by
+          have h : (thetaAboveCell000122013122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013122 h)
+        (by
+          have h : (thetaAboveCell000122013123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122013130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013130 h)
+        (by
+          have h : (thetaAboveCell000122013131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013131 h)
+        (by
+          have h : (thetaAboveCell000122013132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013132 h)
+        (by
+          have h : (thetaAboveCell000122013133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013133 h))
+theorem e24KC2ThetaAboveLeaf0001220132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00012201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00012201))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122013200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013200 h)
+        (by
+          have h : (thetaAboveCell000122013201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013201 h)
+        (by
+          have h : (thetaAboveCell000122013202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013202 h)
+        (by
+          have h : (thetaAboveCell000122013203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122013210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013210 h)
+        (by
+          have h : (thetaAboveCell000122013211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013211 h)
+        (by
+          have h : (thetaAboveCell000122013212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013212 h)
+        (by
+          have h : (thetaAboveCell000122013213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00012201))) h)
+theorem e24KC2ThetaAboveLeaf0001220133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00012201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00012201))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122013300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013300 h)
+        (by
+          have h : (thetaAboveCell000122013301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013301 h)
+        (by
+          have h : (thetaAboveCell000122013302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013302 h)
+        (by
+          have h : (thetaAboveCell000122013303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00012201)))
+        (by
+          have h : (thetaAboveCell000122013310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013310 h)
+        (by
+          have h : (thetaAboveCell000122013311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013311 h)
+        (by
+          have h : (thetaAboveCell000122013312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013312 h)
+        (by
+          have h : (thetaAboveCell000122013313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122013313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00012201))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00012201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00012201))) h)
+theorem e24KC2ThetaAboveLeaf0001221002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00012210))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell00012210))) h)
+theorem e24KC2ThetaAboveLeaf0001221003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00012210))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childHL (childHH (childLL thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLL
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childHH (childHH (childLL thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLL
+        thetaAboveCell00012210))) h)
+theorem e24KC2ThetaAboveLeaf0001221012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00012210))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childHL (childHL (childLH thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLH
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell00012210))) h)
+theorem e24KC2ThetaAboveLeaf0001221013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00012210))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell00012210))) h)
+theorem e24KC2ThetaAboveLeaf0001221020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00012210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00012210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122102000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102000 h)
+        (by
+          have h : (thetaAboveCell000122102001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102001 h)
+        (by
+          have h : (thetaAboveCell000122102002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102002 h)
+        (by
+          have h : (thetaAboveCell000122102003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122102010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102010 h)
+        (by
+          have h : (thetaAboveCell000122102011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102011 h)
+        (by
+          have h : (thetaAboveCell000122102012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102012 h)
+        (by
+          have h : (thetaAboveCell000122102013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122102020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102020 h)
+        (by
+          have h : (thetaAboveCell000122102021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102021 h)
+        (by
+          have h : (thetaAboveCell000122102022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102022 h)
+        (by
+          have h : (thetaAboveCell000122102023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122102030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102030 h)
+        (by
+          have h : (thetaAboveCell000122102031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102031 h)
+        (by
+          have h : (thetaAboveCell000122102032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102032 h)
+        (by
+          have h : (thetaAboveCell000122102033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102033 h))
+theorem e24KC2ThetaAboveLeaf0001221021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00012210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00012210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122102100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102100 h)
+        (by
+          have h : (thetaAboveCell000122102101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102101 h)
+        (by
+          have h : (thetaAboveCell000122102102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102102 h)
+        (by
+          have h : (thetaAboveCell000122102103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122102110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102110 h)
+        (by
+          have h : (thetaAboveCell000122102111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102111 h)
+        (by
+          have h : (thetaAboveCell000122102112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102112 h)
+        (by
+          have h : (thetaAboveCell000122102113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122102120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102120 h)
+        (by
+          have h : (thetaAboveCell000122102121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102121 h)
+        (by
+          have h : (thetaAboveCell000122102122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102122 h)
+        (by
+          have h : (thetaAboveCell000122102123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122102130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102130 h)
+        (by
+          have h : (thetaAboveCell000122102131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102131 h)
+        (by
+          have h : (thetaAboveCell000122102132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102132 h)
+        (by
+          have h : (thetaAboveCell000122102133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102133 h))
+theorem e24KC2ThetaAboveLeaf0001221022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00012210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00012210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122102200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102200 h)
+        (by
+          have h : (thetaAboveCell000122102201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102201 h)
+        (by
+          have h : (thetaAboveCell000122102202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102202 h)
+        (by
+          have h : (thetaAboveCell000122102203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122102210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102210 h)
+        (by
+          have h : (thetaAboveCell000122102211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102211 h)
+        (by
+          have h : (thetaAboveCell000122102212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102212 h)
+        (by
+          have h : (thetaAboveCell000122102213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00012210))) h)
+theorem e24KC2ThetaAboveLeaf0001221023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00012210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00012210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122102300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102300 h)
+        (by
+          have h : (thetaAboveCell000122102301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102301 h)
+        (by
+          have h : (thetaAboveCell000122102302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102302 h)
+        (by
+          have h : (thetaAboveCell000122102303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122102310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102310 h)
+        (by
+          have h : (thetaAboveCell000122102311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102311 h)
+        (by
+          have h : (thetaAboveCell000122102312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102312 h)
+        (by
+          have h : (thetaAboveCell000122102313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122102313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00012210))) h)
+theorem e24KC2ThetaAboveLeaf0001221030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00012210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00012210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122103000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103000 h)
+        (by
+          have h : (thetaAboveCell000122103001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103001 h)
+        (by
+          have h : (thetaAboveCell000122103002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103002 h)
+        (by
+          have h : (thetaAboveCell000122103003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122103010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103010 h)
+        (by
+          have h : (thetaAboveCell000122103011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103011 h)
+        (by
+          have h : (thetaAboveCell000122103012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103012 h)
+        (by
+          have h : (thetaAboveCell000122103013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122103020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103020 h)
+        (by
+          have h : (thetaAboveCell000122103021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103021 h)
+        (by
+          have h : (thetaAboveCell000122103022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103022 h)
+        (by
+          have h : (thetaAboveCell000122103023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122103030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103030 h)
+        (by
+          have h : (thetaAboveCell000122103031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103031 h)
+        (by
+          have h : (thetaAboveCell000122103032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103032 h)
+        (by
+          have h : (thetaAboveCell000122103033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103033 h))
+theorem e24KC2ThetaAboveLeaf0001221031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00012210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00012210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122103100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103100 h)
+        (by
+          have h : (thetaAboveCell000122103101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103101 h)
+        (by
+          have h : (thetaAboveCell000122103102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103102 h)
+        (by
+          have h : (thetaAboveCell000122103103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122103110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103110 h)
+        (by
+          have h : (thetaAboveCell000122103111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103111 h)
+        (by
+          have h : (thetaAboveCell000122103112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103112 h)
+        (by
+          have h : (thetaAboveCell000122103113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122103120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103120 h)
+        (by
+          have h : (thetaAboveCell000122103121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103121 h)
+        (by
+          have h : (thetaAboveCell000122103122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103122 h)
+        (by
+          have h : (thetaAboveCell000122103123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122103130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103130 h)
+        (by
+          have h : (thetaAboveCell000122103131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103131 h)
+        (by
+          have h : (thetaAboveCell000122103132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103132 h)
+        (by
+          have h : (thetaAboveCell000122103133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103133 h))
+theorem e24KC2ThetaAboveLeaf0001221032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00012210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00012210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122103200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103200 h)
+        (by
+          have h : (thetaAboveCell000122103201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103201 h)
+        (by
+          have h : (thetaAboveCell000122103202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103202 h)
+        (by
+          have h : (thetaAboveCell000122103203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122103210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103210 h)
+        (by
+          have h : (thetaAboveCell000122103211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103211 h)
+        (by
+          have h : (thetaAboveCell000122103212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103212 h)
+        (by
+          have h : (thetaAboveCell000122103213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00012210))) h)
+theorem e24KC2ThetaAboveLeaf0001221033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00012210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00012210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122103300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103300 h)
+        (by
+          have h : (thetaAboveCell000122103301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103301 h)
+        (by
+          have h : (thetaAboveCell000122103302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103302 h)
+        (by
+          have h : (thetaAboveCell000122103303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00012210)))
+        (by
+          have h : (thetaAboveCell000122103310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103310 h)
+        (by
+          have h : (thetaAboveCell000122103311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103311 h)
+        (by
+          have h : (thetaAboveCell000122103312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103312 h)
+        (by
+          have h : (thetaAboveCell000122103313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122103313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00012210))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00012210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00012210))) h)
+theorem e24KC2ThetaAboveLeaf0001221102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00012211))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell00012211))) h)
+theorem e24KC2ThetaAboveLeaf0001221103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00012211))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childHL (childHH (childLL thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLL
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childHH (childHH (childLL thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLL
+        thetaAboveCell00012211))) h)
+theorem e24KC2ThetaAboveLeaf0001221112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00012211))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childHL (childHL (childLH thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLH
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell00012211))) h)
+theorem e24KC2ThetaAboveLeaf0001221113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00012211))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell00012211))) h)
+theorem e24KC2ThetaAboveLeaf0001221120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00012211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00012211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122112000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112000 h)
+        (by
+          have h : (thetaAboveCell000122112001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112001 h)
+        (by
+          have h : (thetaAboveCell000122112002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112002 h)
+        (by
+          have h : (thetaAboveCell000122112003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122112010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112010 h)
+        (by
+          have h : (thetaAboveCell000122112011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112011 h)
+        (by
+          have h : (thetaAboveCell000122112012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112012 h)
+        (by
+          have h : (thetaAboveCell000122112013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122112020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112020 h)
+        (by
+          have h : (thetaAboveCell000122112021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112021 h)
+        (by
+          have h : (thetaAboveCell000122112022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112022 h)
+        (by
+          have h : (thetaAboveCell000122112023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122112030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112030 h)
+        (by
+          have h : (thetaAboveCell000122112031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112031 h)
+        (by
+          have h : (thetaAboveCell000122112032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112032 h)
+        (by
+          have h : (thetaAboveCell000122112033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112033 h))
+theorem e24KC2ThetaAboveLeaf0001221121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00012211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00012211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122112100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112100 h)
+        (by
+          have h : (thetaAboveCell000122112101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112101 h)
+        (by
+          have h : (thetaAboveCell000122112102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112102 h)
+        (by
+          have h : (thetaAboveCell000122112103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122112110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112110 h)
+        (by
+          have h : (thetaAboveCell000122112111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112111 h)
+        (by
+          have h : (thetaAboveCell000122112112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112112 h)
+        (by
+          have h : (thetaAboveCell000122112113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122112120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112120 h)
+        (by
+          have h : (thetaAboveCell000122112121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112121 h)
+        (by
+          have h : (thetaAboveCell000122112122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112122 h)
+        (by
+          have h : (thetaAboveCell000122112123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122112130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112130 h)
+        (by
+          have h : (thetaAboveCell000122112131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112131 h)
+        (by
+          have h : (thetaAboveCell000122112132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112132 h)
+        (by
+          have h : (thetaAboveCell000122112133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112133 h))
+theorem e24KC2ThetaAboveLeaf0001221122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00012211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00012211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122112200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112200 h)
+        (by
+          have h : (thetaAboveCell000122112201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112201 h)
+        (by
+          have h : (thetaAboveCell000122112202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112202 h)
+        (by
+          have h : (thetaAboveCell000122112203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122112210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112210 h)
+        (by
+          have h : (thetaAboveCell000122112211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112211 h)
+        (by
+          have h : (thetaAboveCell000122112212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112212 h)
+        (by
+          have h : (thetaAboveCell000122112213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00012211))) h)
+theorem e24KC2ThetaAboveLeaf0001221123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00012211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00012211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122112300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112300 h)
+        (by
+          have h : (thetaAboveCell000122112301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112301 h)
+        (by
+          have h : (thetaAboveCell000122112302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112302 h)
+        (by
+          have h : (thetaAboveCell000122112303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122112310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112310 h)
+        (by
+          have h : (thetaAboveCell000122112311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112311 h)
+        (by
+          have h : (thetaAboveCell000122112312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112312 h)
+        (by
+          have h : (thetaAboveCell000122112313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122112313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00012211))) h)
+theorem e24KC2ThetaAboveLeaf0001221130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00012211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00012211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122113000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113000 h)
+        (by
+          have h : (thetaAboveCell000122113001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113001 h)
+        (by
+          have h : (thetaAboveCell000122113002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113002 h)
+        (by
+          have h : (thetaAboveCell000122113003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122113010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113010 h)
+        (by
+          have h : (thetaAboveCell000122113011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113011 h)
+        (by
+          have h : (thetaAboveCell000122113012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113012 h)
+        (by
+          have h : (thetaAboveCell000122113013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122113020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113020 h)
+        (by
+          have h : (thetaAboveCell000122113021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113021 h)
+        (by
+          have h : (thetaAboveCell000122113022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113022 h)
+        (by
+          have h : (thetaAboveCell000122113023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122113030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113030 h)
+        (by
+          have h : (thetaAboveCell000122113031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113031 h)
+        (by
+          have h : (thetaAboveCell000122113032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113032 h)
+        (by
+          have h : (thetaAboveCell000122113033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113033 h))
+theorem e24KC2ThetaAboveLeaf0001221131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00012211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00012211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122113100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113100 h)
+        (by
+          have h : (thetaAboveCell000122113101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113101 h)
+        (by
+          have h : (thetaAboveCell000122113102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113102 h)
+        (by
+          have h : (thetaAboveCell000122113103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122113110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113110 h)
+        (by
+          have h : (thetaAboveCell000122113111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113111 h)
+        (by
+          have h : (thetaAboveCell000122113112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113112 h)
+        (by
+          have h : (thetaAboveCell000122113113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122113120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113120 h)
+        (by
+          have h : (thetaAboveCell000122113121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113121 h)
+        (by
+          have h : (thetaAboveCell000122113122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113122 h)
+        (by
+          have h : (thetaAboveCell000122113123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122113130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113130 h)
+        (by
+          have h : (thetaAboveCell000122113131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113131 h)
+        (by
+          have h : (thetaAboveCell000122113132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113132 h)
+        (by
+          have h : (thetaAboveCell000122113133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113133 h))
+theorem e24KC2ThetaAboveLeaf0001221132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00012211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00012211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122113200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113200 h)
+        (by
+          have h : (thetaAboveCell000122113201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113201 h)
+        (by
+          have h : (thetaAboveCell000122113202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113202 h)
+        (by
+          have h : (thetaAboveCell000122113203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122113210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113210 h)
+        (by
+          have h : (thetaAboveCell000122113211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113211 h)
+        (by
+          have h : (thetaAboveCell000122113212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113212 h)
+        (by
+          have h : (thetaAboveCell000122113213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00012211))) h)
+theorem e24KC2ThetaAboveLeaf0001221133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00012211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00012211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122113300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113300 h)
+        (by
+          have h : (thetaAboveCell000122113301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113301 h)
+        (by
+          have h : (thetaAboveCell000122113302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113302 h)
+        (by
+          have h : (thetaAboveCell000122113303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00012211)))
+        (by
+          have h : (thetaAboveCell000122113310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113310 h)
+        (by
+          have h : (thetaAboveCell000122113311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113311 h)
+        (by
+          have h : (thetaAboveCell000122113312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113312 h)
+        (by
+          have h : (thetaAboveCell000122113313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000122113313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00012211))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00012211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00012211))) h)
+theorem e24KC2ThetaAboveLeaf0001230002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00012300))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell00012300))) h)
+theorem e24KC2ThetaAboveLeaf0001230003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00012300))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childHL (childHH (childLL thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLL
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childHH (childHH (childLL thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLL
+        thetaAboveCell00012300))) h)
+theorem e24KC2ThetaAboveLeaf0001230012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00012300))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childHL (childHL (childLH thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLH
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell00012300))) h)
+theorem e24KC2ThetaAboveLeaf0001230013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00012300))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell00012300))) h)
+theorem e24KC2ThetaAboveLeaf0001230020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00012300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00012300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123002000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002000 h)
+        (by
+          have h : (thetaAboveCell000123002001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002001 h)
+        (by
+          have h : (thetaAboveCell000123002002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002002 h)
+        (by
+          have h : (thetaAboveCell000123002003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123002010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002010 h)
+        (by
+          have h : (thetaAboveCell000123002011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002011 h)
+        (by
+          have h : (thetaAboveCell000123002012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002012 h)
+        (by
+          have h : (thetaAboveCell000123002013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123002020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002020 h)
+        (by
+          have h : (thetaAboveCell000123002021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002021 h)
+        (by
+          have h : (thetaAboveCell000123002022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002022 h)
+        (by
+          have h : (thetaAboveCell000123002023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123002030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002030 h)
+        (by
+          have h : (thetaAboveCell000123002031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002031 h)
+        (by
+          have h : (thetaAboveCell000123002032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002032 h)
+        (by
+          have h : (thetaAboveCell000123002033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002033 h))
+theorem e24KC2ThetaAboveLeaf0001230021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00012300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00012300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123002100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002100 h)
+        (by
+          have h : (thetaAboveCell000123002101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002101 h)
+        (by
+          have h : (thetaAboveCell000123002102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002102 h)
+        (by
+          have h : (thetaAboveCell000123002103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123002110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002110 h)
+        (by
+          have h : (thetaAboveCell000123002111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002111 h)
+        (by
+          have h : (thetaAboveCell000123002112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002112 h)
+        (by
+          have h : (thetaAboveCell000123002113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123002120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002120 h)
+        (by
+          have h : (thetaAboveCell000123002121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002121 h)
+        (by
+          have h : (thetaAboveCell000123002122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002122 h)
+        (by
+          have h : (thetaAboveCell000123002123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123002130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002130 h)
+        (by
+          have h : (thetaAboveCell000123002131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002131 h)
+        (by
+          have h : (thetaAboveCell000123002132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002132 h)
+        (by
+          have h : (thetaAboveCell000123002133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002133 h))
+theorem e24KC2ThetaAboveLeaf0001230022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00012300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00012300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123002200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002200 h)
+        (by
+          have h : (thetaAboveCell000123002201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002201 h)
+        (by
+          have h : (thetaAboveCell000123002202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002202 h)
+        (by
+          have h : (thetaAboveCell000123002203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123002210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002210 h)
+        (by
+          have h : (thetaAboveCell000123002211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002211 h)
+        (by
+          have h : (thetaAboveCell000123002212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002212 h)
+        (by
+          have h : (thetaAboveCell000123002213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00012300))) h)
+theorem e24KC2ThetaAboveLeaf0001230023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00012300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00012300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123002300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002300 h)
+        (by
+          have h : (thetaAboveCell000123002301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002301 h)
+        (by
+          have h : (thetaAboveCell000123002302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002302 h)
+        (by
+          have h : (thetaAboveCell000123002303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123002310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002310 h)
+        (by
+          have h : (thetaAboveCell000123002311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002311 h)
+        (by
+          have h : (thetaAboveCell000123002312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002312 h)
+        (by
+          have h : (thetaAboveCell000123002313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123002313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00012300))) h)
+theorem e24KC2ThetaAboveLeaf0001230030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00012300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00012300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123003000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003000 h)
+        (by
+          have h : (thetaAboveCell000123003001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003001 h)
+        (by
+          have h : (thetaAboveCell000123003002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003002 h)
+        (by
+          have h : (thetaAboveCell000123003003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123003010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003010 h)
+        (by
+          have h : (thetaAboveCell000123003011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003011 h)
+        (by
+          have h : (thetaAboveCell000123003012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003012 h)
+        (by
+          have h : (thetaAboveCell000123003013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123003020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003020 h)
+        (by
+          have h : (thetaAboveCell000123003021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003021 h)
+        (by
+          have h : (thetaAboveCell000123003022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003022 h)
+        (by
+          have h : (thetaAboveCell000123003023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123003030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003030 h)
+        (by
+          have h : (thetaAboveCell000123003031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003031 h)
+        (by
+          have h : (thetaAboveCell000123003032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003032 h)
+        (by
+          have h : (thetaAboveCell000123003033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003033 h))
+theorem e24KC2ThetaAboveLeaf0001230031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00012300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00012300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123003100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003100 h)
+        (by
+          have h : (thetaAboveCell000123003101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003101 h)
+        (by
+          have h : (thetaAboveCell000123003102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003102 h)
+        (by
+          have h : (thetaAboveCell000123003103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123003110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003110 h)
+        (by
+          have h : (thetaAboveCell000123003111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003111 h)
+        (by
+          have h : (thetaAboveCell000123003112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003112 h)
+        (by
+          have h : (thetaAboveCell000123003113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123003120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003120 h)
+        (by
+          have h : (thetaAboveCell000123003121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003121 h)
+        (by
+          have h : (thetaAboveCell000123003122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003122 h)
+        (by
+          have h : (thetaAboveCell000123003123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123003130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003130 h)
+        (by
+          have h : (thetaAboveCell000123003131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003131 h)
+        (by
+          have h : (thetaAboveCell000123003132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003132 h)
+        (by
+          have h : (thetaAboveCell000123003133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003133 h))
+theorem e24KC2ThetaAboveLeaf0001230032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00012300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00012300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123003200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003200 h)
+        (by
+          have h : (thetaAboveCell000123003201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003201 h)
+        (by
+          have h : (thetaAboveCell000123003202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003202 h)
+        (by
+          have h : (thetaAboveCell000123003203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123003210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003210 h)
+        (by
+          have h : (thetaAboveCell000123003211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003211 h)
+        (by
+          have h : (thetaAboveCell000123003212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003212 h)
+        (by
+          have h : (thetaAboveCell000123003213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00012300))) h)
+theorem e24KC2ThetaAboveLeaf0001230033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00012300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00012300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123003300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003300 h)
+        (by
+          have h : (thetaAboveCell000123003301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003301 h)
+        (by
+          have h : (thetaAboveCell000123003302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003302 h)
+        (by
+          have h : (thetaAboveCell000123003303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00012300)))
+        (by
+          have h : (thetaAboveCell000123003310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003310 h)
+        (by
+          have h : (thetaAboveCell000123003311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003311 h)
+        (by
+          have h : (thetaAboveCell000123003312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003312 h)
+        (by
+          have h : (thetaAboveCell000123003313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123003313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00012300))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00012300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00012300))) h)
+theorem e24KC2ThetaAboveLeaf0001230102 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00012301))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell00012301))) h)
+theorem e24KC2ThetaAboveLeaf0001230103 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00012301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00012301))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childHL (childHH (childLL thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLL
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childHH (childHH (childLL thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLL
+        thetaAboveCell00012301))) h)
+theorem e24KC2ThetaAboveLeaf0001230112 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00012301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00012301))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childHL (childHL (childLH thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLH
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childHH (childHL (childLH thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLH
+        thetaAboveCell00012301))) h)
+theorem e24KC2ThetaAboveLeaf0001230113 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00012301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00012301))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childHL (childHH (childLH thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childLH
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childHH (childHH (childLH thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childLH
+        thetaAboveCell00012301))) h)
+theorem e24KC2ThetaAboveLeaf0001230120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00012301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00012301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123012000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012000 h)
+        (by
+          have h : (thetaAboveCell000123012001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012001 h)
+        (by
+          have h : (thetaAboveCell000123012002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012002 h)
+        (by
+          have h : (thetaAboveCell000123012003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123012010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012010 h)
+        (by
+          have h : (thetaAboveCell000123012011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012011 h)
+        (by
+          have h : (thetaAboveCell000123012012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012012 h)
+        (by
+          have h : (thetaAboveCell000123012013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123012020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012020 h)
+        (by
+          have h : (thetaAboveCell000123012021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012021 h)
+        (by
+          have h : (thetaAboveCell000123012022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012022 h)
+        (by
+          have h : (thetaAboveCell000123012023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123012030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012030 h)
+        (by
+          have h : (thetaAboveCell000123012031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012031 h)
+        (by
+          have h : (thetaAboveCell000123012032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012032 h)
+        (by
+          have h : (thetaAboveCell000123012033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012033 h))
+theorem e24KC2ThetaAboveLeaf0001230121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00012301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00012301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123012100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012100 h)
+        (by
+          have h : (thetaAboveCell000123012101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012101 h)
+        (by
+          have h : (thetaAboveCell000123012102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012102 h)
+        (by
+          have h : (thetaAboveCell000123012103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123012110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012110 h)
+        (by
+          have h : (thetaAboveCell000123012111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012111 h)
+        (by
+          have h : (thetaAboveCell000123012112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012112 h)
+        (by
+          have h : (thetaAboveCell000123012113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123012120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012120 h)
+        (by
+          have h : (thetaAboveCell000123012121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012121 h)
+        (by
+          have h : (thetaAboveCell000123012122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012122 h)
+        (by
+          have h : (thetaAboveCell000123012123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123012130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012130 h)
+        (by
+          have h : (thetaAboveCell000123012131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012131 h)
+        (by
+          have h : (thetaAboveCell000123012132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012132 h)
+        (by
+          have h : (thetaAboveCell000123012133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012133 h))
+theorem e24KC2ThetaAboveLeaf0001230122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00012301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00012301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123012200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012200 h)
+        (by
+          have h : (thetaAboveCell000123012201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012201 h)
+        (by
+          have h : (thetaAboveCell000123012202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012202 h)
+        (by
+          have h : (thetaAboveCell000123012203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123012210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012210 h)
+        (by
+          have h : (thetaAboveCell000123012211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012211 h)
+        (by
+          have h : (thetaAboveCell000123012212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012212 h)
+        (by
+          have h : (thetaAboveCell000123012213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00012301))) h)
+theorem e24KC2ThetaAboveLeaf0001230123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00012301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00012301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123012300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012300 h)
+        (by
+          have h : (thetaAboveCell000123012301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012301 h)
+        (by
+          have h : (thetaAboveCell000123012302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012302 h)
+        (by
+          have h : (thetaAboveCell000123012303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123012310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012310 h)
+        (by
+          have h : (thetaAboveCell000123012311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012311 h)
+        (by
+          have h : (thetaAboveCell000123012312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012312 h)
+        (by
+          have h : (thetaAboveCell000123012313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123012313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00012301))) h)
+theorem e24KC2ThetaAboveLeaf0001230130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00012301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00012301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123013000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013000 h)
+        (by
+          have h : (thetaAboveCell000123013001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013001 h)
+        (by
+          have h : (thetaAboveCell000123013002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013002 h)
+        (by
+          have h : (thetaAboveCell000123013003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123013010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013010 h)
+        (by
+          have h : (thetaAboveCell000123013011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013011 h)
+        (by
+          have h : (thetaAboveCell000123013012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013012 h)
+        (by
+          have h : (thetaAboveCell000123013013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123013020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013020 h)
+        (by
+          have h : (thetaAboveCell000123013021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013021 h)
+        (by
+          have h : (thetaAboveCell000123013022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013022 h)
+        (by
+          have h : (thetaAboveCell000123013023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123013030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013030 h)
+        (by
+          have h : (thetaAboveCell000123013031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013031 h)
+        (by
+          have h : (thetaAboveCell000123013032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013032 h)
+        (by
+          have h : (thetaAboveCell000123013033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013033 h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle015.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle015.lean
new file mode 100644
index 0000000000..407e2b379b
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle015.lean
@@ -0,0 +1,7200 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch022`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCells9faa13f35d
+
+/-- Subcell `0010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0010 : AngleCell :=
+  childLL (childLH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0011 : AngleCell :=
+  childLH (childLH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0011)))
+
+end GerverSofa.PartE.CertificateCells9faa13f35d
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6ProofBatch3b6d17fd8ecf6129`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9faa13f35d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9faa13f35d
+
+open CertificateCells9faa13f35d
+theorem e24KC2ThetaAboveLeaf0010221121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00102211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00102211))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00102211))) h)
+theorem e24KC2ThetaAboveLeaf0010221122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00102211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00102211))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00102211))) h)
+theorem e24KC2ThetaAboveLeaf0010221123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00102211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00102211))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00102211))) h)
+theorem e24KC2ThetaAboveLeaf0010221130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00102211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00102211))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00102211))) h)
+theorem e24KC2ThetaAboveLeaf0010221131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00102211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00102211))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00102211))) h)
+theorem e24KC2ThetaAboveLeaf0010221132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00102211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00102211))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00102211))) h)
+theorem e24KC2ThetaAboveLeaf0010221133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00102211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00102211))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00102211))) h)
+theorem e24KC2ThetaAboveLeaf0010221200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00102212))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00102212))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00102212))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00102212))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00102212))) h)
+theorem e24KC2ThetaAboveLeaf0010221201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00102212))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00102212))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00102212))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00102212))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00102212))) h)
+theorem e24KC2ThetaAboveLeaf0010221210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00102212))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00102212))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00102212))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00102212))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00102212))) h)
+theorem e24KC2ThetaAboveLeaf0010221211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00102212))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00102212))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00102212))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00102212))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00102212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00102212))) h)
+theorem e24KC2ThetaAboveLeaf0010221300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00102213))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00102213))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00102213))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00102213))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00102213))) h)
+theorem e24KC2ThetaAboveLeaf0010221301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00102213))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00102213))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00102213))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00102213))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00102213))) h)
+theorem e24KC2ThetaAboveLeaf0010221310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00102213))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00102213))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00102213))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00102213))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00102213))) h)
+theorem e24KC2ThetaAboveLeaf0010221311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00102213))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00102213))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00102213))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00102213))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00102213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00102213))) h)
+theorem e24KC2ThetaAboveLeaf0010230020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00102300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00102300))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00102300))) h)
+theorem e24KC2ThetaAboveLeaf0010230021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00102300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00102300))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00102300))) h)
+theorem e24KC2ThetaAboveLeaf0010230022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00102300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00102300))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00102300))) h)
+theorem e24KC2ThetaAboveLeaf0010230023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00102300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00102300))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00102300))) h)
+theorem e24KC2ThetaAboveLeaf0010230030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00102300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00102300))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00102300))) h)
+theorem e24KC2ThetaAboveLeaf0010230031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00102300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00102300))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00102300))) h)
+theorem e24KC2ThetaAboveLeaf0010230032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00102300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00102300))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00102300))) h)
+theorem e24KC2ThetaAboveLeaf0010230033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00102300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00102300))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00102300))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00102300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00102300))) h)
+theorem e24KC2ThetaAboveLeaf0010230120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00102301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00102301))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00102301))) h)
+theorem e24KC2ThetaAboveLeaf0010230121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00102301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00102301))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00102301))) h)
+theorem e24KC2ThetaAboveLeaf0010230122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00102301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00102301))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00102301))) h)
+theorem e24KC2ThetaAboveLeaf0010230123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00102301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00102301))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00102301))) h)
+theorem e24KC2ThetaAboveLeaf0010230130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00102301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00102301))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00102301))) h)
+theorem e24KC2ThetaAboveLeaf0010230131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00102301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00102301))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00102301))) h)
+theorem e24KC2ThetaAboveLeaf0010230132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00102301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00102301))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00102301))) h)
+theorem e24KC2ThetaAboveLeaf0010230133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00102301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00102301))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00102301))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00102301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00102301))) h)
+theorem e24KC2ThetaAboveLeaf0010230200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00102302))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00102302))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00102302))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00102302))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00102302))) h)
+theorem e24KC2ThetaAboveLeaf0010230201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00102302))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00102302))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00102302))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00102302))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00102302))) h)
+theorem e24KC2ThetaAboveLeaf0010230210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00102302))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00102302))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00102302))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00102302))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00102302))) h)
+theorem e24KC2ThetaAboveLeaf0010230211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00102302))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00102302))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00102302))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00102302))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00102302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00102302))) h)
+theorem e24KC2ThetaAboveLeaf0010230300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00102303))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00102303))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00102303))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00102303))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00102303))) h)
+theorem e24KC2ThetaAboveLeaf0010230301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00102303))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00102303))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00102303))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00102303))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00102303))) h)
+theorem e24KC2ThetaAboveLeaf0010230310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00102303))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00102303))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00102303))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00102303))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00102303))) h)
+theorem e24KC2ThetaAboveLeaf0010230311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00102303))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00102303))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00102303))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00102303))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00102303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00102303))) h)
+theorem e24KC2ThetaAboveLeaf0010231020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00102310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00102310))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00102310))) h)
+theorem e24KC2ThetaAboveLeaf0010231021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00102310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00102310))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00102310))) h)
+theorem e24KC2ThetaAboveLeaf0010231022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00102310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00102310))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00102310))) h)
+theorem e24KC2ThetaAboveLeaf0010231023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00102310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00102310))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00102310))) h)
+theorem e24KC2ThetaAboveLeaf0010231030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00102310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00102310))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00102310))) h)
+theorem e24KC2ThetaAboveLeaf0010231031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00102310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00102310))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00102310))) h)
+theorem e24KC2ThetaAboveLeaf0010231032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00102310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00102310))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00102310))) h)
+theorem e24KC2ThetaAboveLeaf0010231033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00102310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00102310))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00102310))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00102310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00102310))) h)
+theorem e24KC2ThetaAboveLeaf0010231120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00102311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00102311))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00102311))) h)
+theorem e24KC2ThetaAboveLeaf0010231121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00102311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00102311))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00102311))) h)
+theorem e24KC2ThetaAboveLeaf0010231122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00102311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00102311))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00102311))) h)
+theorem e24KC2ThetaAboveLeaf0010231123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00102311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00102311))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00102311))) h)
+theorem e24KC2ThetaAboveLeaf0010231130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00102311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00102311))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00102311))) h)
+theorem e24KC2ThetaAboveLeaf0010231131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00102311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00102311))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00102311))) h)
+theorem e24KC2ThetaAboveLeaf0010231132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00102311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00102311))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00102311))) h)
+theorem e24KC2ThetaAboveLeaf0010231133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00102311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00102311))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00102311))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00102311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00102311))) h)
+theorem e24KC2ThetaAboveLeaf0010231200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00102312))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00102312))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00102312))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00102312))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00102312))) h)
+theorem e24KC2ThetaAboveLeaf0010231201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00102312))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00102312))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00102312))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00102312))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00102312))) h)
+theorem e24KC2ThetaAboveLeaf0010231210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00102312))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00102312))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00102312))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00102312))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00102312))) h)
+theorem e24KC2ThetaAboveLeaf0010231211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00102312))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00102312))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00102312))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00102312))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00102312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00102312))) h)
+theorem e24KC2ThetaAboveLeaf0010231300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00102313))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00102313))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00102313))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00102313))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00102313))) h)
+theorem e24KC2ThetaAboveLeaf0010231301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00102313))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00102313))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00102313))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00102313))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00102313))) h)
+theorem e24KC2ThetaAboveLeaf0010231310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00102313))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00102313))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00102313))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00102313))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00102313))) h)
+theorem e24KC2ThetaAboveLeaf0010231311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00102313))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00102313))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00102313))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00102313))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00102313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00102313))) h)
+theorem e24KC2ThetaAboveLeaf0010320020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00103200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00103200))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00103200))) h)
+theorem e24KC2ThetaAboveLeaf0010320021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00103200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00103200))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00103200))) h)
+theorem e24KC2ThetaAboveLeaf0010320022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00103200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00103200))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00103200))) h)
+theorem e24KC2ThetaAboveLeaf0010320023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00103200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00103200))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00103200))) h)
+theorem e24KC2ThetaAboveLeaf0010320030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00103200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00103200))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00103200))) h)
+theorem e24KC2ThetaAboveLeaf0010320031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00103200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00103200))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00103200))) h)
+theorem e24KC2ThetaAboveLeaf0010320032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00103200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00103200))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00103200))) h)
+theorem e24KC2ThetaAboveLeaf0010320033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00103200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00103200))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00103200))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00103200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00103200))) h)
+theorem e24KC2ThetaAboveLeaf0010320120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00103201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00103201))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00103201))) h)
+theorem e24KC2ThetaAboveLeaf0010320121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00103201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00103201))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00103201))) h)
+theorem e24KC2ThetaAboveLeaf0010320122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00103201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00103201))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00103201))) h)
+theorem e24KC2ThetaAboveLeaf0010320123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00103201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00103201))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00103201))) h)
+theorem e24KC2ThetaAboveLeaf0010320130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00103201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00103201))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00103201))) h)
+theorem e24KC2ThetaAboveLeaf0010320131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00103201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00103201))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00103201))) h)
+theorem e24KC2ThetaAboveLeaf0010320132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00103201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00103201))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00103201))) h)
+theorem e24KC2ThetaAboveLeaf0010320133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00103201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00103201))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00103201))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00103201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00103201))) h)
+theorem e24KC2ThetaAboveLeaf0010320200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00103202))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00103202))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00103202))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00103202))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00103202))) h)
+theorem e24KC2ThetaAboveLeaf0010320201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00103202))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00103202))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00103202))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00103202))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00103202))) h)
+theorem e24KC2ThetaAboveLeaf0010320210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00103202))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00103202))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00103202))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00103202))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00103202))) h)
+theorem e24KC2ThetaAboveLeaf0010320211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00103202))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00103202))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00103202))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00103202))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00103202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00103202))) h)
+theorem e24KC2ThetaAboveLeaf0010320300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00103203))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00103203))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00103203))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00103203))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00103203))) h)
+theorem e24KC2ThetaAboveLeaf0010320301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00103203))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00103203))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00103203))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00103203))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00103203))) h)
+theorem e24KC2ThetaAboveLeaf0010320310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00103203))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00103203))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00103203))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00103203))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00103203))) h)
+theorem e24KC2ThetaAboveLeaf0010320311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00103203))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00103203))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00103203))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00103203))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00103203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00103203))) h)
+theorem e24KC2ThetaAboveLeaf0010321020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00103210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00103210))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00103210))) h)
+theorem e24KC2ThetaAboveLeaf0010321021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00103210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00103210))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00103210))) h)
+theorem e24KC2ThetaAboveLeaf0010321022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00103210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00103210))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00103210))) h)
+theorem e24KC2ThetaAboveLeaf0010321023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00103210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00103210))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00103210))) h)
+theorem e24KC2ThetaAboveLeaf0010321030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00103210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00103210))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00103210))) h)
+theorem e24KC2ThetaAboveLeaf0010321031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00103210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00103210))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00103210))) h)
+theorem e24KC2ThetaAboveLeaf0010321032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00103210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00103210))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00103210))) h)
+theorem e24KC2ThetaAboveLeaf0010321033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00103210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00103210))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00103210))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00103210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00103210))) h)
+theorem e24KC2ThetaAboveLeaf0010321120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00103211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00103211))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00103211))) h)
+theorem e24KC2ThetaAboveLeaf0010321121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00103211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00103211))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00103211))) h)
+theorem e24KC2ThetaAboveLeaf0010321122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00103211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00103211))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00103211))) h)
+theorem e24KC2ThetaAboveLeaf0010321123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00103211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00103211))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00103211))) h)
+theorem e24KC2ThetaAboveLeaf0010321130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00103211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00103211))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00103211))) h)
+theorem e24KC2ThetaAboveLeaf0010321131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00103211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00103211))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00103211))) h)
+theorem e24KC2ThetaAboveLeaf0010321132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00103211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00103211))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00103211))) h)
+theorem e24KC2ThetaAboveLeaf0010321133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00103211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00103211))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00103211))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00103211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00103211))) h)
+theorem e24KC2ThetaAboveLeaf0010321200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00103212))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00103212))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00103212))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00103212))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00103212))) h)
+theorem e24KC2ThetaAboveLeaf0010321201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00103212))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00103212))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00103212))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00103212))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00103212))) h)
+theorem e24KC2ThetaAboveLeaf0010321210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00103212))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00103212))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00103212))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00103212))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00103212))) h)
+theorem e24KC2ThetaAboveLeaf0010321211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00103212))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00103212))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00103212))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00103212))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00103212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00103212))) h)
+theorem e24KC2ThetaAboveLeaf0010321300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00103213))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00103213))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00103213))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00103213))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00103213))) h)
+theorem e24KC2ThetaAboveLeaf0010321301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00103213))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00103213))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00103213))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00103213))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00103213))) h)
+theorem e24KC2ThetaAboveLeaf0010321310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00103213))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00103213))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00103213))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00103213))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00103213))) h)
+theorem e24KC2ThetaAboveLeaf0010321311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00103213))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00103213))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00103213))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00103213))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00103213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00103213))) h)
+theorem e24KC2ThetaAboveLeaf0010330020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00103300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00103300))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00103300))) h)
+theorem e24KC2ThetaAboveLeaf0010330021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00103300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00103300))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00103300))) h)
+theorem e24KC2ThetaAboveLeaf0010330022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00103300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00103300))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00103300))) h)
+theorem e24KC2ThetaAboveLeaf0010330023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00103300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00103300))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00103300))) h)
+theorem e24KC2ThetaAboveLeaf0010330030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00103300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00103300))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00103300))) h)
+theorem e24KC2ThetaAboveLeaf0010330031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00103300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00103300))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00103300))) h)
+theorem e24KC2ThetaAboveLeaf0010330032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00103300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00103300))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00103300))) h)
+theorem e24KC2ThetaAboveLeaf0010330033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00103300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00103300))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00103300))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00103300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00103300))) h)
+theorem e24KC2ThetaAboveLeaf0010330120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00103301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00103301))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00103301))) h)
+theorem e24KC2ThetaAboveLeaf0010330121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00103301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00103301))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00103301))) h)
+theorem e24KC2ThetaAboveLeaf0010330122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00103301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00103301))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00103301))) h)
+theorem e24KC2ThetaAboveLeaf0010330123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00103301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00103301))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00103301))) h)
+theorem e24KC2ThetaAboveLeaf0010330130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00103301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00103301))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00103301))) h)
+theorem e24KC2ThetaAboveLeaf0010330131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00103301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00103301))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00103301))) h)
+theorem e24KC2ThetaAboveLeaf0010330132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00103301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00103301))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00103301))) h)
+theorem e24KC2ThetaAboveLeaf0010330133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00103301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00103301))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00103301))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00103301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00103301))) h)
+theorem e24KC2ThetaAboveLeaf0010330200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00103302))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00103302))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00103302))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00103302))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00103302))) h)
+theorem e24KC2ThetaAboveLeaf0010330201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00103302))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00103302))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00103302))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00103302))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00103302))) h)
+theorem e24KC2ThetaAboveLeaf0010330210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00103302))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00103302))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00103302))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00103302))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00103302))) h)
+theorem e24KC2ThetaAboveLeaf0010330211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00103302))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00103302))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00103302))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00103302))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00103302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00103302))) h)
+theorem e24KC2ThetaAboveLeaf0010330300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00103303))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00103303))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00103303))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00103303))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00103303))) h)
+theorem e24KC2ThetaAboveLeaf0010330301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00103303))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00103303))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00103303))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00103303))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00103303))) h)
+theorem e24KC2ThetaAboveLeaf0010330310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00103303))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00103303))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00103303))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00103303))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00103303))) h)
+theorem e24KC2ThetaAboveLeaf0010330311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00103303))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00103303))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00103303))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00103303))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00103303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00103303))) h)
+theorem e24KC2ThetaAboveLeaf0010331020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00103310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00103310))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00103310))) h)
+theorem e24KC2ThetaAboveLeaf0010331021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00103310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00103310))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00103310))) h)
+theorem e24KC2ThetaAboveLeaf0010331022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00103310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00103310))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00103310))) h)
+theorem e24KC2ThetaAboveLeaf0010331023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00103310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00103310))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00103310))) h)
+theorem e24KC2ThetaAboveLeaf0010331030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00103310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00103310))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00103310))) h)
+theorem e24KC2ThetaAboveLeaf0010331031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00103310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00103310))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00103310))) h)
+theorem e24KC2ThetaAboveLeaf0010331032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00103310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00103310))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00103310))) h)
+theorem e24KC2ThetaAboveLeaf0010331033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00103310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00103310))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00103310))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00103310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00103310))) h)
+theorem e24KC2ThetaAboveLeaf0010331120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00103311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00103311))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00103311))) h)
+theorem e24KC2ThetaAboveLeaf0010331121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00103311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00103311))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00103311))) h)
+theorem e24KC2ThetaAboveLeaf0010331122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00103311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00103311))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00103311))) h)
+theorem e24KC2ThetaAboveLeaf0010331123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00103311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00103311))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00103311))) h)
+theorem e24KC2ThetaAboveLeaf0010331130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00103311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00103311))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00103311))) h)
+theorem e24KC2ThetaAboveLeaf0010331131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00103311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00103311))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00103311))) h)
+theorem e24KC2ThetaAboveLeaf0010331132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00103311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00103311))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00103311))) h)
+theorem e24KC2ThetaAboveLeaf0010331133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00103311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00103311))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00103311))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00103311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00103311))) h)
+theorem e24KC2ThetaAboveLeaf0010331200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00103312))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00103312))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00103312))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00103312))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00103312))) h)
+theorem e24KC2ThetaAboveLeaf0010331201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00103312))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00103312))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00103312))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00103312))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00103312))) h)
+theorem e24KC2ThetaAboveLeaf0010331210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00103312))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00103312))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00103312))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00103312))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00103312))) h)
+theorem e24KC2ThetaAboveLeaf0010331211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00103312))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00103312))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00103312))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00103312))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00103312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00103312))) h)
+theorem e24KC2ThetaAboveLeaf0010331300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00103313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00103313))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00103313))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00103313))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00103313))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00103313))) h)
+theorem e24KC2ThetaAboveLeaf0010331301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00103313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00103313))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00103313))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00103313))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00103313))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00103313))) h)
+theorem e24KC2ThetaAboveLeaf0010331310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00103313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00103313))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00103313))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00103313))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00103313))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00103313))) h)
+theorem e24KC2ThetaAboveLeaf0010331311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00103313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00103313))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00103313))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00103313))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00103313))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00103313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00103313))) h)
+theorem e24KC2ThetaAboveLeaf0011220020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00112200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00112200))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00112200))) h)
+theorem e24KC2ThetaAboveLeaf0011220021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00112200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00112200))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00112200))) h)
+theorem e24KC2ThetaAboveLeaf0011220022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00112200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00112200))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00112200))) h)
+theorem e24KC2ThetaAboveLeaf0011220023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00112200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00112200))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00112200))) h)
+theorem e24KC2ThetaAboveLeaf0011220030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00112200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00112200))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00112200))) h)
+theorem e24KC2ThetaAboveLeaf0011220031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00112200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00112200))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00112200))) h)
+theorem e24KC2ThetaAboveLeaf0011220032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00112200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00112200))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00112200))) h)
+theorem e24KC2ThetaAboveLeaf0011220033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00112200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00112200))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00112200))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00112200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00112200))) h)
+theorem e24KC2ThetaAboveLeaf0011220120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00112201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00112201))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00112201))) h)
+theorem e24KC2ThetaAboveLeaf0011220121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00112201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00112201))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00112201))) h)
+theorem e24KC2ThetaAboveLeaf0011220122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00112201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00112201))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00112201))) h)
+theorem e24KC2ThetaAboveLeaf0011220123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00112201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00112201))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00112201))) h)
+theorem e24KC2ThetaAboveLeaf0011220130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00112201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00112201))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00112201))) h)
+theorem e24KC2ThetaAboveLeaf0011220131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00112201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00112201))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00112201))) h)
+theorem e24KC2ThetaAboveLeaf0011220132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00112201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00112201))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00112201))) h)
+theorem e24KC2ThetaAboveLeaf0011220133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00112201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00112201))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00112201))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00112201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00112201))) h)
+theorem e24KC2ThetaAboveLeaf0011220200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00112202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00112202))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00112202))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00112202))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00112202))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00112202))) h)
+theorem e24KC2ThetaAboveLeaf0011220201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00112202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00112202))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00112202))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00112202))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00112202))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00112202))) h)
+theorem e24KC2ThetaAboveLeaf0011220210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00112202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00112202))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00112202))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00112202))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00112202))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00112202))) h)
+theorem e24KC2ThetaAboveLeaf0011220211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00112202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00112202))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00112202))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00112202))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00112202))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00112202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00112202))) h)
+theorem e24KC2ThetaAboveLeaf0011220300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00112203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00112203))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00112203))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00112203))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00112203))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00112203))) h)
+theorem e24KC2ThetaAboveLeaf0011220301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00112203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00112203))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00112203))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00112203))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00112203))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00112203))) h)
+theorem e24KC2ThetaAboveLeaf0011220310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00112203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00112203))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00112203))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00112203))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00112203))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00112203))) h)
+theorem e24KC2ThetaAboveLeaf0011220311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00112203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00112203))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00112203))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00112203))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00112203))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00112203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00112203))) h)
+theorem e24KC2ThetaAboveLeaf0011221020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00112210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00112210))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00112210))) h)
+theorem e24KC2ThetaAboveLeaf0011221021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00112210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00112210))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00112210))) h)
+theorem e24KC2ThetaAboveLeaf0011221022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00112210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00112210))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00112210))) h)
+theorem e24KC2ThetaAboveLeaf0011221023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00112210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00112210))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00112210))) h)
+theorem e24KC2ThetaAboveLeaf0011221030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00112210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00112210))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00112210))) h)
+theorem e24KC2ThetaAboveLeaf0011221031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00112210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00112210))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00112210))) h)
+theorem e24KC2ThetaAboveLeaf0011221032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00112210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00112210))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00112210))) h)
+theorem e24KC2ThetaAboveLeaf0011221033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00112210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00112210))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00112210))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00112210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00112210))) h)
+theorem e24KC2ThetaAboveLeaf0011221120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00112211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00112211))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00112211))) h)
+theorem e24KC2ThetaAboveLeaf0011221121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00112211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00112211))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00112211))) h)
+theorem e24KC2ThetaAboveLeaf0011221122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00112211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00112211))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00112211))) h)
+theorem e24KC2ThetaAboveLeaf0011221123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00112211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00112211))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00112211))) h)
+theorem e24KC2ThetaAboveLeaf0011221130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00112211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00112211))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00112211))) h)
+theorem e24KC2ThetaAboveLeaf0011221131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00112211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00112211))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00112211))) h)
+theorem e24KC2ThetaAboveLeaf0011221132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00112211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00112211))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00112211))) h)
+theorem e24KC2ThetaAboveLeaf0011221133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00112211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00112211))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00112211))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00112211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00112211))) h)
+theorem e24KC2ThetaAboveLeaf0011221200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00112212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00112212))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00112212))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00112212))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00112212))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00112212))) h)
+theorem e24KC2ThetaAboveLeaf0011221201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00112212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00112212))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00112212))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00112212))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00112212))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00112212))) h)
+theorem e24KC2ThetaAboveLeaf0011221210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00112212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00112212))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00112212))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00112212))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00112212))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00112212))) h)
+theorem e24KC2ThetaAboveLeaf0011221211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00112212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00112212))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00112212))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00112212))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00112212))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00112212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00112212))) h)
+theorem e24KC2ThetaAboveLeaf0011221300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00112213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00112213))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00112213))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00112213))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00112213))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00112213))) h)
+theorem e24KC2ThetaAboveLeaf0011221301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00112213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00112213))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00112213))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00112213))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00112213))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00112213))) h)
+theorem e24KC2ThetaAboveLeaf0011221310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00112213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00112213))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00112213))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00112213))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00112213))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00112213))) h)
+theorem e24KC2ThetaAboveLeaf0011221311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00112213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00112213))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00112213))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00112213))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00112213))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00112213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00112213))) h)
+theorem e24KC2ThetaAboveLeaf0011230020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00112300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00112300))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00112300))) h)
+theorem e24KC2ThetaAboveLeaf0011230021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00112300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00112300))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00112300))) h)
+theorem e24KC2ThetaAboveLeaf0011230022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00112300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00112300))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00112300))) h)
+theorem e24KC2ThetaAboveLeaf0011230023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00112300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00112300))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00112300))) h)
+theorem e24KC2ThetaAboveLeaf0011230030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00112300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00112300))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00112300))) h)
+theorem e24KC2ThetaAboveLeaf0011230031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00112300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00112300))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00112300))) h)
+theorem e24KC2ThetaAboveLeaf0011230032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00112300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00112300))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00112300))) h)
+theorem e24KC2ThetaAboveLeaf0011230033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00112300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00112300))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00112300))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00112300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00112300))) h)
+theorem e24KC2ThetaAboveLeaf0011230120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00112301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00112301))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00112301))) h)
+theorem e24KC2ThetaAboveLeaf0011230121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00112301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00112301))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00112301))) h)
+theorem e24KC2ThetaAboveLeaf0011230122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00112301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00112301))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00112301))) h)
+theorem e24KC2ThetaAboveLeaf0011230123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00112301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00112301))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00112301))) h)
+theorem e24KC2ThetaAboveLeaf0011230130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00112301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00112301))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00112301))) h)
+theorem e24KC2ThetaAboveLeaf0011230131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00112301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00112301))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00112301))) h)
+theorem e24KC2ThetaAboveLeaf0011230132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00112301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00112301))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00112301))) h)
+theorem e24KC2ThetaAboveLeaf0011230133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00112301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00112301))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00112301))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00112301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00112301))) h)
+theorem e24KC2ThetaAboveLeaf0011230200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00112302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00112302))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00112302))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00112302))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00112302))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00112302))) h)
+theorem e24KC2ThetaAboveLeaf0011230201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00112302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00112302))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00112302))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00112302))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00112302))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00112302))) h)
+theorem e24KC2ThetaAboveLeaf0011230210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00112302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00112302))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00112302))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00112302))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00112302))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00112302))) h)
+theorem e24KC2ThetaAboveLeaf0011230211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00112302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00112302))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00112302))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00112302))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00112302))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00112302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00112302))) h)
+theorem e24KC2ThetaAboveLeaf0011230300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00112303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00112303))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00112303))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00112303))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00112303))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00112303))) h)
+theorem e24KC2ThetaAboveLeaf0011230301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00112303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00112303))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00112303))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00112303))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00112303))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00112303))) h)
+theorem e24KC2ThetaAboveLeaf0011230310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00112303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00112303))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00112303))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00112303))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00112303))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00112303))) h)
+theorem e24KC2ThetaAboveLeaf0011230311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00112303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00112303))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00112303))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00112303))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00112303))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00112303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00112303))) h)
+theorem e24KC2ThetaAboveLeaf0011231020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00112310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00112310))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00112310))) h)
+theorem e24KC2ThetaAboveLeaf0011231021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00112310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00112310))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHL (childLH (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHL
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHL
+        thetaAboveCell00112310))) h)
+theorem e24KC2ThetaAboveLeaf0011231022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00112310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00112310))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00112310))) h)
+theorem e24KC2ThetaAboveLeaf0011231023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00112310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00112310))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00112310))) h)
+theorem e24KC2ThetaAboveLeaf0011231030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00112310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00112310))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHL (childLL (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHH
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHH
+        thetaAboveCell00112310))) h)
+theorem e24KC2ThetaAboveLeaf0011231031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00112310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00112310))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHL (childLH (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childHH
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childHH
+        thetaAboveCell00112310))) h)
+theorem e24KC2ThetaAboveLeaf0011231032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00112310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00112310))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00112310))) h)
+theorem e24KC2ThetaAboveLeaf0011231033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00112310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00112310))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00112310))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00112310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00112310))) h)
+theorem e24KC2ThetaAboveLeaf0011231120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00112311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00112311))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00112311))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00112311))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00112311))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00112311))) h)
+theorem e24KC2ThetaAboveLeaf0011231122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00112311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00112311))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00112311))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00112311))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00112311))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00112311))) h)
+theorem e24KC2ThetaAboveLeaf0011231123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00112311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00112311))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00112311))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00112311))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00112311))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00112311))) h)
+theorem e24KC2ThetaAboveLeaf0011231132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00112311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00112311))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00112311))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00112311))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00112311))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00112311))) h)
+theorem e24KC2ThetaAboveLeaf0011231133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00112311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00112311))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00112311))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00112311))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00112311))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00112311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00112311))) h)
+theorem e24KC2ThetaAboveLeaf0011231200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00112312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00112312))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00112312))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00112312))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00112312))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00112312))) h)
+theorem e24KC2ThetaAboveLeaf0011231201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00112312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00112312))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00112312))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00112312))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00112312))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00112312))) h)
+theorem e24KC2ThetaAboveLeaf0011231210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00112312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00112312))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00112312))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00112312))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00112312))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00112312))) h)
+theorem e24KC2ThetaAboveLeaf0011231211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00112312)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00112312))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00112312))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00112312))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00112312))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00112312)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00112312))) h)
+theorem e24KC2ThetaAboveLeaf0011231300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00112313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00112313))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00112313))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00112313))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00112313))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00112313))) h)
+theorem e24KC2ThetaAboveLeaf0011231301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00112313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00112313))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00112313))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00112313))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00112313))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00112313))) h)
+theorem e24KC2ThetaAboveLeaf0011231310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00112313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00112313))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00112313))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00112313))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00112313))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00112313))) h)
+theorem e24KC2ThetaAboveLeaf0011231311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00112313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00112313))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00112313))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00112313))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00112313))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00112313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00112313))) h)
+theorem e24KC2ThetaAboveLeaf0011320022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00113200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00113200))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00113200))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00113200))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00113200))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00113200))) h)
+theorem e24KC2ThetaAboveLeaf0011320023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00113200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00113200))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00113200))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00113200))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00113200))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00113200))) h)
+theorem e24KC2ThetaAboveLeaf0011320032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00113200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00113200))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00113200))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00113200))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00113200))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00113200))) h)
+theorem e24KC2ThetaAboveLeaf0011320033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00113200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00113200))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00113200))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00113200))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00113200))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00113200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00113200))) h)
+theorem e24KC2ThetaAboveLeaf0011320122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00113201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00113201))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00113201))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00113201))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00113201))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00113201))) h)
+theorem e24KC2ThetaAboveLeaf0011320123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00113201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00113201))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00113201))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00113201))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00113201))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00113201))) h)
+theorem e24KC2ThetaAboveLeaf0011320132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00113201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00113201))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00113201))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00113201))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00113201))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00113201))) h)
+theorem e24KC2ThetaAboveLeaf0011320133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00113201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00113201))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00113201))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00113201))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00113201))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00113201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00113201))) h)
+theorem e24KC2ThetaAboveLeaf0011320200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00113202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00113202))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00113202))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00113202))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00113202))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00113202))) h)
+theorem e24KC2ThetaAboveLeaf0011320201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00113202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00113202))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00113202))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00113202))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00113202))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00113202))) h)
+theorem e24KC2ThetaAboveLeaf0011320210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00113202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00113202))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00113202))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00113202))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00113202))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00113202))) h)
+theorem e24KC2ThetaAboveLeaf0011320211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00113202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00113202))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00113202))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00113202))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00113202))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00113202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00113202))) h)
+theorem e24KC2ThetaAboveLeaf0011320300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00113203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00113203))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00113203))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00113203))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00113203))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00113203))) h)
+theorem e24KC2ThetaAboveLeaf0011320301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00113203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00113203))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00113203))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00113203))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00113203))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00113203))) h)
+theorem e24KC2ThetaAboveLeaf0011320310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00113203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00113203))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00113203))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00113203))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00113203))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00113203))) h)
+theorem e24KC2ThetaAboveLeaf0011320311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00113203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00113203))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00113203))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00113203))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00113203))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00113203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00113203))) h)
+theorem e24KC2ThetaAboveLeaf0011321022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00113210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00113210))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00113210))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00113210))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00113210))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00113210))) h)
+theorem e24KC2ThetaAboveLeaf0011321023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00113210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00113210))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00113210))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00113210))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00113210))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00113210))) h)
+theorem e24KC2ThetaAboveLeaf0011321032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00113210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00113210))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00113210))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00113210))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00113210))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00113210))) h)
+theorem e24KC2ThetaAboveLeaf0011321033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00113210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00113210))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00113210))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00113210))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00113210))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00113210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00113210))) h)
+theorem e24KC2ThetaAboveLeaf0011321122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00113211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00113211))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00113211))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00113211))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00113211))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00113211))) h)
+theorem e24KC2ThetaAboveLeaf0011321123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00113211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00113211))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00113211))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00113211))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00113211))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00113211))) h)
+theorem e24KC2ThetaAboveLeaf0011321132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00113211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00113211))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00113211))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00113211))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00113211))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00113211))) h)
+theorem e24KC2ThetaAboveLeaf0011321133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00113211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00113211))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00113211))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00113211))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00113211))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00113211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00113211))) h)
+theorem e24KC2ThetaAboveLeaf0011321200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00113212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00113212))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00113212))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00113212))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00113212))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00113212))) h)
+theorem e24KC2ThetaAboveLeaf0011321201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00113212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00113212))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00113212))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00113212))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00113212))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00113212))) h)
+theorem e24KC2ThetaAboveLeaf0011321210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00113212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00113212))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00113212))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00113212))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00113212))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00113212))) h)
+theorem e24KC2ThetaAboveLeaf0011321211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00113212)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00113212))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00113212))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00113212))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00113212))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00113212)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00113212))) h)
+theorem e24KC2ThetaAboveLeaf0011321300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00113213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00113213))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00113213))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00113213))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00113213))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00113213))) h)
+theorem e24KC2ThetaAboveLeaf0011321301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00113213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00113213))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00113213))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00113213))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00113213))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00113213))) h)
+theorem e24KC2ThetaAboveLeaf0011321310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00113213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00113213))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00113213))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00113213))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00113213))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00113213))) h)
+theorem e24KC2ThetaAboveLeaf0011321311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00113213)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00113213))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00113213))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00113213))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00113213))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00113213)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00113213))) h)
+theorem e24KC2ThetaAboveLeaf0011330022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00113300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00113300))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00113300))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00113300))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00113300))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00113300))) h)
+theorem e24KC2ThetaAboveLeaf0011330023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00113300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00113300))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00113300))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00113300))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00113300))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00113300))) h)
+theorem e24KC2ThetaAboveLeaf0011330032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00113300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00113300))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00113300))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00113300))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00113300))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00113300))) h)
+theorem e24KC2ThetaAboveLeaf0011330033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00113300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00113300))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00113300))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00113300))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00113300))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00113300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00113300))) h)
+theorem e24KC2ThetaAboveLeaf0011330122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00113301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00113301))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00113301))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00113301))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00113301))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00113301))) h)
+theorem e24KC2ThetaAboveLeaf0011330123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00113301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00113301))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00113301))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00113301))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00113301))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00113301))) h)
+theorem e24KC2ThetaAboveLeaf0011330132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00113301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00113301))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00113301))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00113301))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00113301))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00113301))) h)
+theorem e24KC2ThetaAboveLeaf0011330133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00113301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00113301))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00113301))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00113301))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00113301))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00113301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00113301))) h)
+theorem e24KC2ThetaAboveLeaf0011330200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00113302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00113302))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00113302))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00113302))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00113302))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00113302))) h)
+theorem e24KC2ThetaAboveLeaf0011330201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00113302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00113302))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00113302))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00113302))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00113302))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00113302))) h)
+theorem e24KC2ThetaAboveLeaf0011330210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00113302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00113302))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00113302))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00113302))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00113302))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00113302))) h)
+theorem e24KC2ThetaAboveLeaf0011330211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00113302)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00113302))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00113302))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00113302))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00113302))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00113302)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00113302))) h)
+theorem e24KC2ThetaAboveLeaf0011330300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00113303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00113303))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00113303))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00113303))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00113303))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00113303))) h)
+theorem e24KC2ThetaAboveLeaf0011330301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00113303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00113303))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00113303))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00113303))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00113303))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00113303))) h)
+theorem e24KC2ThetaAboveLeaf0011330310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00113303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00113303))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00113303))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00113303))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00113303))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00113303))) h)
+theorem e24KC2ThetaAboveLeaf0011330311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00113303)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00113303))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00113303))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00113303))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00113303))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00113303)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00113303))) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle016.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle016.lean
new file mode 100644
index 0000000000..61a95aa6e6
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle016.lean
@@ -0,0 +1,10082 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch025`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCells4a1452a6c6
+
+/-- Subcell `0001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0001 : AngleCell :=
+  childLH (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0010 : AngleCell :=
+  childLL (childLH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `000123013100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00012301)))
+
+/-- Subcell `000123102000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00012310)))
+
+/-- Subcell `000123103000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00012310)))
+
+/-- Subcell `000123112000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00012311)))
+
+/-- Subcell `000123113000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000123113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000123113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00012311)))
+
+/-- Subcell `000132002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00013200)))
+
+/-- Subcell `000132003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00013200)))
+
+/-- Subcell `000132012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00013201)))
+
+/-- Subcell `000132013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00013201)))
+
+/-- Subcell `000132102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00013210)))
+
+/-- Subcell `000132103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00013210)))
+
+/-- Subcell `000132112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00013211)))
+
+/-- Subcell `000132113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000132113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000132113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00013211)))
+
+/-- Subcell `000133002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00013300)))
+
+/-- Subcell `000133003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00013300)))
+
+/-- Subcell `000133012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133012333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133012333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell00013301)))
+
+/-- Subcell `000133013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133013333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133013333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell00013301)))
+
+/-- Subcell `000133102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133102333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133102333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell00013310)))
+
+/-- Subcell `000133103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133103333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133103333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell00013310)))
+
+/-- Subcell `000133112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00013311)))
+
+/-- Subcell `000133113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `000133113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000133113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00013311)))
+
+/-- Subcell `001022002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell00102200)))
+
+/-- Subcell `001022003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell00102200)))
+
+/-- Subcell `001022012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell00102201)))
+
+/-- Subcell `001022013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00102201)))
+
+/-- Subcell `001022102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00102210)))
+
+/-- Subcell `001022103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00102210)))
+
+/-- Subcell `001022103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00102210)))
+
+/-- Subcell `001022103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00102210)))
+
+/-- Subcell `001022103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00102210)))
+
+/-- Subcell `001022103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00102210)))
+
+/-- Subcell `001022103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00102210)))
+
+/-- Subcell `001022103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00102210)))
+
+/-- Subcell `001022103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00102210)))
+
+/-- Subcell `001022103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00102210)))
+
+/-- Subcell `001022103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00102210)))
+
+/-- Subcell `001022103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00102210)))
+
+/-- Subcell `001022103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00102210)))
+
+/-- Subcell `001022103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00102210)))
+
+/-- Subcell `001022103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00102210)))
+
+/-- Subcell `001022103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00102210)))
+
+/-- Subcell `001022103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell001022103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00102210)))
+
+end GerverSofa.PartE.CertificateCells4a1452a6c6
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6ProofBatchAeb87c7b95a29e08`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4a1452a6c6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4a1452a6c6
+
+open CertificateCells4a1452a6c6
+theorem e24KC2ThetaAboveLeaf0001230131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00012301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00012301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123013100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013100 h)
+        (by
+          have h : (thetaAboveCell000123013101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013101 h)
+        (by
+          have h : (thetaAboveCell000123013102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013102 h)
+        (by
+          have h : (thetaAboveCell000123013103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123013110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013110 h)
+        (by
+          have h : (thetaAboveCell000123013111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013111 h)
+        (by
+          have h : (thetaAboveCell000123013112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013112 h)
+        (by
+          have h : (thetaAboveCell000123013113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123013120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013120 h)
+        (by
+          have h : (thetaAboveCell000123013121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013121 h)
+        (by
+          have h : (thetaAboveCell000123013122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013122 h)
+        (by
+          have h : (thetaAboveCell000123013123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123013130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013130 h)
+        (by
+          have h : (thetaAboveCell000123013131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013131 h)
+        (by
+          have h : (thetaAboveCell000123013132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013132 h)
+        (by
+          have h : (thetaAboveCell000123013133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013133 h))
+theorem e24KC2ThetaAboveLeaf0001230132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00012301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00012301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123013200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013200 h)
+        (by
+          have h : (thetaAboveCell000123013201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013201 h)
+        (by
+          have h : (thetaAboveCell000123013202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013202 h)
+        (by
+          have h : (thetaAboveCell000123013203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123013210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013210 h)
+        (by
+          have h : (thetaAboveCell000123013211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013211 h)
+        (by
+          have h : (thetaAboveCell000123013212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013212 h)
+        (by
+          have h : (thetaAboveCell000123013213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00012301))) h)
+theorem e24KC2ThetaAboveLeaf0001230133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00012301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00012301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123013300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013300 h)
+        (by
+          have h : (thetaAboveCell000123013301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013301 h)
+        (by
+          have h : (thetaAboveCell000123013302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013302 h)
+        (by
+          have h : (thetaAboveCell000123013303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00012301)))
+        (by
+          have h : (thetaAboveCell000123013310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013310 h)
+        (by
+          have h : (thetaAboveCell000123013311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013311 h)
+        (by
+          have h : (thetaAboveCell000123013312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013312 h)
+        (by
+          have h : (thetaAboveCell000123013313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123013313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00012301))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00012301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00012301))) h)
+theorem e24KC2ThetaAboveLeaf0001231002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00012310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00012310))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00012310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00012310))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00012310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00012310))) h)
+    (by
+      have h : ((childHL (childHL (childLL thetaAboveCell00012310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childLL
+        thetaAboveCell00012310))) h)
+    (by
+      have h : ((childHH (childHL (childLL thetaAboveCell00012310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childLL
+        thetaAboveCell00012310))) h)
+theorem e24KC2ThetaAboveLeaf0001231020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00012310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00012310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123102000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102000 h)
+        (by
+          have h : (thetaAboveCell000123102001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102001 h)
+        (by
+          have h : (thetaAboveCell000123102002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102002 h)
+        (by
+          have h : (thetaAboveCell000123102003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123102010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102010 h)
+        (by
+          have h : (thetaAboveCell000123102011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102011 h)
+        (by
+          have h : (thetaAboveCell000123102012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102012 h)
+        (by
+          have h : (thetaAboveCell000123102013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123102020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102020 h)
+        (by
+          have h : (thetaAboveCell000123102021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102021 h)
+        (by
+          have h : (thetaAboveCell000123102022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102022 h)
+        (by
+          have h : (thetaAboveCell000123102023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123102030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102030 h)
+        (by
+          have h : (thetaAboveCell000123102031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102031 h)
+        (by
+          have h : (thetaAboveCell000123102032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102032 h)
+        (by
+          have h : (thetaAboveCell000123102033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102033 h))
+theorem e24KC2ThetaAboveLeaf0001231021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00012310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00012310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123102100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102100 h)
+        (by
+          have h : (thetaAboveCell000123102101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102101 h)
+        (by
+          have h : (thetaAboveCell000123102102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102102 h)
+        (by
+          have h : (thetaAboveCell000123102103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123102110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102110 h)
+        (by
+          have h : (thetaAboveCell000123102111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102111 h)
+        (by
+          have h : (thetaAboveCell000123102112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102112 h)
+        (by
+          have h : (thetaAboveCell000123102113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123102120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102120 h)
+        (by
+          have h : (thetaAboveCell000123102121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102121 h)
+        (by
+          have h : (thetaAboveCell000123102122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102122 h)
+        (by
+          have h : (thetaAboveCell000123102123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123102130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102130 h)
+        (by
+          have h : (thetaAboveCell000123102131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102131 h)
+        (by
+          have h : (thetaAboveCell000123102132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102132 h)
+        (by
+          have h : (thetaAboveCell000123102133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102133 h))
+theorem e24KC2ThetaAboveLeaf0001231022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00012310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00012310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123102200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102200 h)
+        (by
+          have h : (thetaAboveCell000123102201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102201 h)
+        (by
+          have h : (thetaAboveCell000123102202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102202 h)
+        (by
+          have h : (thetaAboveCell000123102203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123102210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102210 h)
+        (by
+          have h : (thetaAboveCell000123102211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102211 h)
+        (by
+          have h : (thetaAboveCell000123102212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102212 h)
+        (by
+          have h : (thetaAboveCell000123102213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00012310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00012310))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00012310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00012310))) h)
+theorem e24KC2ThetaAboveLeaf0001231023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00012310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00012310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123102300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102300 h)
+        (by
+          have h : (thetaAboveCell000123102301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102301 h)
+        (by
+          have h : (thetaAboveCell000123102302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102302 h)
+        (by
+          have h : (thetaAboveCell000123102303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123102310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102310 h)
+        (by
+          have h : (thetaAboveCell000123102311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102311 h)
+        (by
+          have h : (thetaAboveCell000123102312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102312 h)
+        (by
+          have h : (thetaAboveCell000123102313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123102313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00012310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00012310))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00012310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00012310))) h)
+theorem e24KC2ThetaAboveLeaf0001231030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00012310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00012310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123103000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103000 h)
+        (by
+          have h : (thetaAboveCell000123103001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103001 h)
+        (by
+          have h : (thetaAboveCell000123103002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103002 h)
+        (by
+          have h : (thetaAboveCell000123103003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123103010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103010 h)
+        (by
+          have h : (thetaAboveCell000123103011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103011 h)
+        (by
+          have h : (thetaAboveCell000123103012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103012 h)
+        (by
+          have h : (thetaAboveCell000123103013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123103020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103020 h)
+        (by
+          have h : (thetaAboveCell000123103021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103021 h)
+        (by
+          have h : (thetaAboveCell000123103022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103022 h)
+        (by
+          have h : (thetaAboveCell000123103023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123103030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103030 h)
+        (by
+          have h : (thetaAboveCell000123103031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103031 h)
+        (by
+          have h : (thetaAboveCell000123103032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103032 h)
+        (by
+          have h : (thetaAboveCell000123103033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103033 h))
+theorem e24KC2ThetaAboveLeaf0001231031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00012310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00012310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123103100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103100 h)
+        (by
+          have h : (thetaAboveCell000123103101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103101 h)
+        (by
+          have h : (thetaAboveCell000123103102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103102 h)
+        (by
+          have h : (thetaAboveCell000123103103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123103110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103110 h)
+        (by
+          have h : (thetaAboveCell000123103111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103111 h)
+        (by
+          have h : (thetaAboveCell000123103112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103112 h)
+        (by
+          have h : (thetaAboveCell000123103113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123103120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103120 h)
+        (by
+          have h : (thetaAboveCell000123103121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103121 h)
+        (by
+          have h : (thetaAboveCell000123103122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103122 h)
+        (by
+          have h : (thetaAboveCell000123103123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123103130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103130 h)
+        (by
+          have h : (thetaAboveCell000123103131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103131 h)
+        (by
+          have h : (thetaAboveCell000123103132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103132 h)
+        (by
+          have h : (thetaAboveCell000123103133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103133 h))
+theorem e24KC2ThetaAboveLeaf0001231032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00012310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00012310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123103200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103200 h)
+        (by
+          have h : (thetaAboveCell000123103201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103201 h)
+        (by
+          have h : (thetaAboveCell000123103202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103202 h)
+        (by
+          have h : (thetaAboveCell000123103203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123103210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103210 h)
+        (by
+          have h : (thetaAboveCell000123103211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103211 h)
+        (by
+          have h : (thetaAboveCell000123103212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103212 h)
+        (by
+          have h : (thetaAboveCell000123103213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00012310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00012310))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00012310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00012310))) h)
+theorem e24KC2ThetaAboveLeaf0001231033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00012310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00012310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123103300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103300 h)
+        (by
+          have h : (thetaAboveCell000123103301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103301 h)
+        (by
+          have h : (thetaAboveCell000123103302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103302 h)
+        (by
+          have h : (thetaAboveCell000123103303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00012310)))
+        (by
+          have h : (thetaAboveCell000123103310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103310 h)
+        (by
+          have h : (thetaAboveCell000123103311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103311 h)
+        (by
+          have h : (thetaAboveCell000123103312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103312 h)
+        (by
+          have h : (thetaAboveCell000123103313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123103313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00012310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00012310))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00012310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00012310))) h)
+theorem e24KC2ThetaAboveLeaf0001231120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00012311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00012311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123112000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112000 h)
+        (by
+          have h : (thetaAboveCell000123112001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112001 h)
+        (by
+          have h : (thetaAboveCell000123112002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112002 h)
+        (by
+          have h : (thetaAboveCell000123112003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123112010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112010 h)
+        (by
+          have h : (thetaAboveCell000123112011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112011 h)
+        (by
+          have h : (thetaAboveCell000123112012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112012 h)
+        (by
+          have h : (thetaAboveCell000123112013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123112020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112020 h)
+        (by
+          have h : (thetaAboveCell000123112021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112021 h)
+        (by
+          have h : (thetaAboveCell000123112022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112022 h)
+        (by
+          have h : (thetaAboveCell000123112023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123112030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112030 h)
+        (by
+          have h : (thetaAboveCell000123112031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112031 h)
+        (by
+          have h : (thetaAboveCell000123112032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112032 h)
+        (by
+          have h : (thetaAboveCell000123112033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112033 h))
+theorem e24KC2ThetaAboveLeaf0001231121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00012311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00012311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123112100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112100 h)
+        (by
+          have h : (thetaAboveCell000123112101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112101 h)
+        (by
+          have h : (thetaAboveCell000123112102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112102 h)
+        (by
+          have h : (thetaAboveCell000123112103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123112110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112110 h)
+        (by
+          have h : (thetaAboveCell000123112111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112111 h)
+        (by
+          have h : (thetaAboveCell000123112112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112112 h)
+        (by
+          have h : (thetaAboveCell000123112113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123112120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112120 h)
+        (by
+          have h : (thetaAboveCell000123112121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112121 h)
+        (by
+          have h : (thetaAboveCell000123112122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112122 h)
+        (by
+          have h : (thetaAboveCell000123112123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123112130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112130 h)
+        (by
+          have h : (thetaAboveCell000123112131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112131 h)
+        (by
+          have h : (thetaAboveCell000123112132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112132 h)
+        (by
+          have h : (thetaAboveCell000123112133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112133 h))
+theorem e24KC2ThetaAboveLeaf0001231122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00012311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00012311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123112200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112200 h)
+        (by
+          have h : (thetaAboveCell000123112201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112201 h)
+        (by
+          have h : (thetaAboveCell000123112202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112202 h)
+        (by
+          have h : (thetaAboveCell000123112203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123112210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112210 h)
+        (by
+          have h : (thetaAboveCell000123112211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112211 h)
+        (by
+          have h : (thetaAboveCell000123112212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112212 h)
+        (by
+          have h : (thetaAboveCell000123112213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00012311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00012311))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00012311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00012311))) h)
+theorem e24KC2ThetaAboveLeaf0001231123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00012311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00012311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123112300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112300 h)
+        (by
+          have h : (thetaAboveCell000123112301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112301 h)
+        (by
+          have h : (thetaAboveCell000123112302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112302 h)
+        (by
+          have h : (thetaAboveCell000123112303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123112310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112310 h)
+        (by
+          have h : (thetaAboveCell000123112311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112311 h)
+        (by
+          have h : (thetaAboveCell000123112312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112312 h)
+        (by
+          have h : (thetaAboveCell000123112313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123112313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00012311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00012311))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00012311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00012311))) h)
+theorem e24KC2ThetaAboveLeaf0001231130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00012311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00012311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123113000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113000 h)
+        (by
+          have h : (thetaAboveCell000123113001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113001 h)
+        (by
+          have h : (thetaAboveCell000123113002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113002 h)
+        (by
+          have h : (thetaAboveCell000123113003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123113010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113010 h)
+        (by
+          have h : (thetaAboveCell000123113011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113011 h)
+        (by
+          have h : (thetaAboveCell000123113012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113012 h)
+        (by
+          have h : (thetaAboveCell000123113013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123113020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113020 h)
+        (by
+          have h : (thetaAboveCell000123113021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113021 h)
+        (by
+          have h : (thetaAboveCell000123113022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113022 h)
+        (by
+          have h : (thetaAboveCell000123113023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123113030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113030 h)
+        (by
+          have h : (thetaAboveCell000123113031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113031 h)
+        (by
+          have h : (thetaAboveCell000123113032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113032 h)
+        (by
+          have h : (thetaAboveCell000123113033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113033 h))
+theorem e24KC2ThetaAboveLeaf0001231131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00012311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00012311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123113100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113100 h)
+        (by
+          have h : (thetaAboveCell000123113101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113101 h)
+        (by
+          have h : (thetaAboveCell000123113102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113102 h)
+        (by
+          have h : (thetaAboveCell000123113103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123113110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113110 h)
+        (by
+          have h : (thetaAboveCell000123113111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113111 h)
+        (by
+          have h : (thetaAboveCell000123113112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113112 h)
+        (by
+          have h : (thetaAboveCell000123113113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123113120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113120 h)
+        (by
+          have h : (thetaAboveCell000123113121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113121 h)
+        (by
+          have h : (thetaAboveCell000123113122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113122 h)
+        (by
+          have h : (thetaAboveCell000123113123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123113130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113130 h)
+        (by
+          have h : (thetaAboveCell000123113131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113131 h)
+        (by
+          have h : (thetaAboveCell000123113132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113132 h)
+        (by
+          have h : (thetaAboveCell000123113133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113133 h))
+theorem e24KC2ThetaAboveLeaf0001231132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00012311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00012311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123113200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113200 h)
+        (by
+          have h : (thetaAboveCell000123113201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113201 h)
+        (by
+          have h : (thetaAboveCell000123113202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113202 h)
+        (by
+          have h : (thetaAboveCell000123113203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123113210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113210 h)
+        (by
+          have h : (thetaAboveCell000123113211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113211 h)
+        (by
+          have h : (thetaAboveCell000123113212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113212 h)
+        (by
+          have h : (thetaAboveCell000123113213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00012311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00012311))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00012311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00012311))) h)
+theorem e24KC2ThetaAboveLeaf0001231133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00012311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00012311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123113300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113300 h)
+        (by
+          have h : (thetaAboveCell000123113301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113301 h)
+        (by
+          have h : (thetaAboveCell000123113302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113302 h)
+        (by
+          have h : (thetaAboveCell000123113303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00012311)))
+        (by
+          have h : (thetaAboveCell000123113310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113310 h)
+        (by
+          have h : (thetaAboveCell000123113311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113311 h)
+        (by
+          have h : (thetaAboveCell000123113312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113312 h)
+        (by
+          have h : (thetaAboveCell000123113313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000123113313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00012311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00012311))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00012311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00012311))) h)
+theorem e24KC2ThetaAboveLeaf0001320020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00013200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00013200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132002000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002000 h)
+        (by
+          have h : (thetaAboveCell000132002001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002001 h)
+        (by
+          have h : (thetaAboveCell000132002002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002002 h)
+        (by
+          have h : (thetaAboveCell000132002003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002003 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132002010).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002010 h)
+        (by
+          have h : (thetaAboveCell000132002011).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002011 h)
+        (by
+          have h : (thetaAboveCell000132002012).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002012 h)
+        (by
+          have h : (thetaAboveCell000132002013).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002013 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132002020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002020 h)
+        (by
+          have h : (thetaAboveCell000132002021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002021 h)
+        (by
+          have h : (thetaAboveCell000132002022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002022 h)
+        (by
+          have h : (thetaAboveCell000132002023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132002030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002030 h)
+        (by
+          have h : (thetaAboveCell000132002031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002031 h)
+        (by
+          have h : (thetaAboveCell000132002032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002032 h)
+        (by
+          have h : (thetaAboveCell000132002033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002033 h))
+theorem e24KC2ThetaAboveLeaf0001320021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00013200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00013200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132002100).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002100 h)
+        (by
+          have h : (thetaAboveCell000132002101).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002101 h)
+        (by
+          have h : (thetaAboveCell000132002102).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002102 h)
+        (by
+          have h : (thetaAboveCell000132002103).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002103 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132002110).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002110 h)
+        (by
+          have h : (thetaAboveCell000132002111).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002111 h)
+        (by
+          have h : (thetaAboveCell000132002112).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002112 h)
+        (by
+          have h : (thetaAboveCell000132002113).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002113 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132002120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002120 h)
+        (by
+          have h : (thetaAboveCell000132002121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002121 h)
+        (by
+          have h : (thetaAboveCell000132002122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002122 h)
+        (by
+          have h : (thetaAboveCell000132002123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132002130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002130 h)
+        (by
+          have h : (thetaAboveCell000132002131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002131 h)
+        (by
+          have h : (thetaAboveCell000132002132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002132 h)
+        (by
+          have h : (thetaAboveCell000132002133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002133 h))
+theorem e24KC2ThetaAboveLeaf0001320022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00013200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00013200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132002200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002200 h)
+        (by
+          have h : (thetaAboveCell000132002201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002201 h)
+        (by
+          have h : (thetaAboveCell000132002202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002202 h)
+        (by
+          have h : (thetaAboveCell000132002203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132002210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002210 h)
+        (by
+          have h : (thetaAboveCell000132002211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002211 h)
+        (by
+          have h : (thetaAboveCell000132002212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002212 h)
+        (by
+          have h : (thetaAboveCell000132002213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00013200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00013200))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00013200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00013200))) h)
+theorem e24KC2ThetaAboveLeaf0001320023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00013200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00013200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132002300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002300 h)
+        (by
+          have h : (thetaAboveCell000132002301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002301 h)
+        (by
+          have h : (thetaAboveCell000132002302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002302 h)
+        (by
+          have h : (thetaAboveCell000132002303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132002310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002310 h)
+        (by
+          have h : (thetaAboveCell000132002311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002311 h)
+        (by
+          have h : (thetaAboveCell000132002312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002312 h)
+        (by
+          have h : (thetaAboveCell000132002313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132002313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00013200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00013200))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00013200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00013200))) h)
+theorem e24KC2ThetaAboveLeaf0001320030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00013200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00013200))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00013200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00013200))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00013200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00013200))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132003020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003020 h)
+        (by
+          have h : (thetaAboveCell000132003021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003021 h)
+        (by
+          have h : (thetaAboveCell000132003022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003022 h)
+        (by
+          have h : (thetaAboveCell000132003023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132003030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003030 h)
+        (by
+          have h : (thetaAboveCell000132003031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003031 h)
+        (by
+          have h : (thetaAboveCell000132003032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003032 h)
+        (by
+          have h : (thetaAboveCell000132003033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003033 h))
+theorem e24KC2ThetaAboveLeaf0001320031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00013200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00013200))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00013200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00013200))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00013200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00013200))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132003120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003120 h)
+        (by
+          have h : (thetaAboveCell000132003121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003121 h)
+        (by
+          have h : (thetaAboveCell000132003122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003122 h)
+        (by
+          have h : (thetaAboveCell000132003123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132003130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003130 h)
+        (by
+          have h : (thetaAboveCell000132003131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003131 h)
+        (by
+          have h : (thetaAboveCell000132003132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003132 h)
+        (by
+          have h : (thetaAboveCell000132003133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003133 h))
+theorem e24KC2ThetaAboveLeaf0001320032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00013200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00013200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132003200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003200 h)
+        (by
+          have h : (thetaAboveCell000132003201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003201 h)
+        (by
+          have h : (thetaAboveCell000132003202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003202 h)
+        (by
+          have h : (thetaAboveCell000132003203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132003210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003210 h)
+        (by
+          have h : (thetaAboveCell000132003211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003211 h)
+        (by
+          have h : (thetaAboveCell000132003212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003212 h)
+        (by
+          have h : (thetaAboveCell000132003213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00013200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00013200))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00013200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00013200))) h)
+theorem e24KC2ThetaAboveLeaf0001320033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00013200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00013200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132003300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003300 h)
+        (by
+          have h : (thetaAboveCell000132003301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003301 h)
+        (by
+          have h : (thetaAboveCell000132003302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003302 h)
+        (by
+          have h : (thetaAboveCell000132003303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00013200)))
+        (by
+          have h : (thetaAboveCell000132003310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003310 h)
+        (by
+          have h : (thetaAboveCell000132003311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003311 h)
+        (by
+          have h : (thetaAboveCell000132003312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003312 h)
+        (by
+          have h : (thetaAboveCell000132003313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132003313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00013200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00013200))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00013200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00013200))) h)
+theorem e24KC2ThetaAboveLeaf0001320120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00013201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00013201))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00013201))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00013201))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132012020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012020 h)
+        (by
+          have h : (thetaAboveCell000132012021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012021 h)
+        (by
+          have h : (thetaAboveCell000132012022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012022 h)
+        (by
+          have h : (thetaAboveCell000132012023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132012030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012030 h)
+        (by
+          have h : (thetaAboveCell000132012031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012031 h)
+        (by
+          have h : (thetaAboveCell000132012032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012032 h)
+        (by
+          have h : (thetaAboveCell000132012033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012033 h))
+theorem e24KC2ThetaAboveLeaf0001320121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00013201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00013201))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00013201))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00013201))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132012120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012120 h)
+        (by
+          have h : (thetaAboveCell000132012121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012121 h)
+        (by
+          have h : (thetaAboveCell000132012122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012122 h)
+        (by
+          have h : (thetaAboveCell000132012123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132012130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012130 h)
+        (by
+          have h : (thetaAboveCell000132012131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012131 h)
+        (by
+          have h : (thetaAboveCell000132012132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012132 h)
+        (by
+          have h : (thetaAboveCell000132012133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012133 h))
+theorem e24KC2ThetaAboveLeaf0001320122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00013201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00013201))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132012200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012200 h)
+        (by
+          have h : (thetaAboveCell000132012201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012201 h)
+        (by
+          have h : (thetaAboveCell000132012202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012202 h)
+        (by
+          have h : (thetaAboveCell000132012203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132012210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012210 h)
+        (by
+          have h : (thetaAboveCell000132012211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012211 h)
+        (by
+          have h : (thetaAboveCell000132012212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012212 h)
+        (by
+          have h : (thetaAboveCell000132012213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00013201))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00013201))) h)
+theorem e24KC2ThetaAboveLeaf0001320123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00013201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00013201))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132012300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012300 h)
+        (by
+          have h : (thetaAboveCell000132012301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012301 h)
+        (by
+          have h : (thetaAboveCell000132012302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012302 h)
+        (by
+          have h : (thetaAboveCell000132012303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132012310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012310 h)
+        (by
+          have h : (thetaAboveCell000132012311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012311 h)
+        (by
+          have h : (thetaAboveCell000132012312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012312 h)
+        (by
+          have h : (thetaAboveCell000132012313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132012313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00013201))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00013201))) h)
+theorem e24KC2ThetaAboveLeaf0001320130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00013201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00013201))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00013201))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00013201))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132013020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013020 h)
+        (by
+          have h : (thetaAboveCell000132013021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013021 h)
+        (by
+          have h : (thetaAboveCell000132013022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013022 h)
+        (by
+          have h : (thetaAboveCell000132013023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132013030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013030 h)
+        (by
+          have h : (thetaAboveCell000132013031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013031 h)
+        (by
+          have h : (thetaAboveCell000132013032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013032 h)
+        (by
+          have h : (thetaAboveCell000132013033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013033 h))
+theorem e24KC2ThetaAboveLeaf0001320131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00013201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00013201))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00013201))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00013201))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132013120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013120 h)
+        (by
+          have h : (thetaAboveCell000132013121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013121 h)
+        (by
+          have h : (thetaAboveCell000132013122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013122 h)
+        (by
+          have h : (thetaAboveCell000132013123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132013130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013130 h)
+        (by
+          have h : (thetaAboveCell000132013131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013131 h)
+        (by
+          have h : (thetaAboveCell000132013132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013132 h)
+        (by
+          have h : (thetaAboveCell000132013133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013133 h))
+theorem e24KC2ThetaAboveLeaf0001320132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00013201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00013201))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132013200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013200 h)
+        (by
+          have h : (thetaAboveCell000132013201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013201 h)
+        (by
+          have h : (thetaAboveCell000132013202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013202 h)
+        (by
+          have h : (thetaAboveCell000132013203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132013210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013210 h)
+        (by
+          have h : (thetaAboveCell000132013211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013211 h)
+        (by
+          have h : (thetaAboveCell000132013212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013212 h)
+        (by
+          have h : (thetaAboveCell000132013213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00013201))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00013201))) h)
+theorem e24KC2ThetaAboveLeaf0001320133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00013201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00013201))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132013300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013300 h)
+        (by
+          have h : (thetaAboveCell000132013301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013301 h)
+        (by
+          have h : (thetaAboveCell000132013302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013302 h)
+        (by
+          have h : (thetaAboveCell000132013303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00013201)))
+        (by
+          have h : (thetaAboveCell000132013310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013310 h)
+        (by
+          have h : (thetaAboveCell000132013311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013311 h)
+        (by
+          have h : (thetaAboveCell000132013312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013312 h)
+        (by
+          have h : (thetaAboveCell000132013313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132013313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00013201))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00013201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00013201))) h)
+theorem e24KC2ThetaAboveLeaf0001321020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00013210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00013210))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00013210))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00013210))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132102020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102020 h)
+        (by
+          have h : (thetaAboveCell000132102021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102021 h)
+        (by
+          have h : (thetaAboveCell000132102022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102022 h)
+        (by
+          have h : (thetaAboveCell000132102023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132102030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102030 h)
+        (by
+          have h : (thetaAboveCell000132102031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102031 h)
+        (by
+          have h : (thetaAboveCell000132102032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102032 h)
+        (by
+          have h : (thetaAboveCell000132102033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102033 h))
+theorem e24KC2ThetaAboveLeaf0001321021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00013210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00013210))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00013210))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00013210))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132102120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102120 h)
+        (by
+          have h : (thetaAboveCell000132102121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102121 h)
+        (by
+          have h : (thetaAboveCell000132102122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102122 h)
+        (by
+          have h : (thetaAboveCell000132102123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132102130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102130 h)
+        (by
+          have h : (thetaAboveCell000132102131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102131 h)
+        (by
+          have h : (thetaAboveCell000132102132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102132 h)
+        (by
+          have h : (thetaAboveCell000132102133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102133 h))
+theorem e24KC2ThetaAboveLeaf0001321022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00013210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00013210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132102200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102200 h)
+        (by
+          have h : (thetaAboveCell000132102201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102201 h)
+        (by
+          have h : (thetaAboveCell000132102202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102202 h)
+        (by
+          have h : (thetaAboveCell000132102203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132102210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102210 h)
+        (by
+          have h : (thetaAboveCell000132102211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102211 h)
+        (by
+          have h : (thetaAboveCell000132102212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102212 h)
+        (by
+          have h : (thetaAboveCell000132102213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00013210))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00013210))) h)
+theorem e24KC2ThetaAboveLeaf0001321023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00013210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00013210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132102300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102300 h)
+        (by
+          have h : (thetaAboveCell000132102301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102301 h)
+        (by
+          have h : (thetaAboveCell000132102302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102302 h)
+        (by
+          have h : (thetaAboveCell000132102303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132102310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102310 h)
+        (by
+          have h : (thetaAboveCell000132102311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102311 h)
+        (by
+          have h : (thetaAboveCell000132102312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102312 h)
+        (by
+          have h : (thetaAboveCell000132102313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132102313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00013210))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00013210))) h)
+theorem e24KC2ThetaAboveLeaf0001321030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00013210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00013210))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00013210))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00013210))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132103020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103020 h)
+        (by
+          have h : (thetaAboveCell000132103021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103021 h)
+        (by
+          have h : (thetaAboveCell000132103022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103022 h)
+        (by
+          have h : (thetaAboveCell000132103023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132103030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103030 h)
+        (by
+          have h : (thetaAboveCell000132103031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103031 h)
+        (by
+          have h : (thetaAboveCell000132103032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103032 h)
+        (by
+          have h : (thetaAboveCell000132103033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103033 h))
+theorem e24KC2ThetaAboveLeaf0001321031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00013210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00013210))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00013210))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00013210))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132103120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103120 h)
+        (by
+          have h : (thetaAboveCell000132103121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103121 h)
+        (by
+          have h : (thetaAboveCell000132103122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103122 h)
+        (by
+          have h : (thetaAboveCell000132103123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132103130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103130 h)
+        (by
+          have h : (thetaAboveCell000132103131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103131 h)
+        (by
+          have h : (thetaAboveCell000132103132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103132 h)
+        (by
+          have h : (thetaAboveCell000132103133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103133 h))
+theorem e24KC2ThetaAboveLeaf0001321032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00013210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00013210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132103200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103200 h)
+        (by
+          have h : (thetaAboveCell000132103201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103201 h)
+        (by
+          have h : (thetaAboveCell000132103202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103202 h)
+        (by
+          have h : (thetaAboveCell000132103203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132103210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103210 h)
+        (by
+          have h : (thetaAboveCell000132103211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103211 h)
+        (by
+          have h : (thetaAboveCell000132103212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103212 h)
+        (by
+          have h : (thetaAboveCell000132103213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00013210))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00013210))) h)
+theorem e24KC2ThetaAboveLeaf0001321033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00013210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00013210))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132103300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103300 h)
+        (by
+          have h : (thetaAboveCell000132103301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103301 h)
+        (by
+          have h : (thetaAboveCell000132103302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103302 h)
+        (by
+          have h : (thetaAboveCell000132103303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00013210)))
+        (by
+          have h : (thetaAboveCell000132103310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103310 h)
+        (by
+          have h : (thetaAboveCell000132103311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103311 h)
+        (by
+          have h : (thetaAboveCell000132103312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103312 h)
+        (by
+          have h : (thetaAboveCell000132103313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132103313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00013210))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00013210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00013210))) h)
+theorem e24KC2ThetaAboveLeaf0001321120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00013211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00013211))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00013211))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00013211))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132112020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112020 h)
+        (by
+          have h : (thetaAboveCell000132112021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112021 h)
+        (by
+          have h : (thetaAboveCell000132112022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112022 h)
+        (by
+          have h : (thetaAboveCell000132112023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132112030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112030 h)
+        (by
+          have h : (thetaAboveCell000132112031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112031 h)
+        (by
+          have h : (thetaAboveCell000132112032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112032 h)
+        (by
+          have h : (thetaAboveCell000132112033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112033 h))
+theorem e24KC2ThetaAboveLeaf0001321121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00013211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00013211))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00013211))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00013211))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132112120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112120 h)
+        (by
+          have h : (thetaAboveCell000132112121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112121 h)
+        (by
+          have h : (thetaAboveCell000132112122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112122 h)
+        (by
+          have h : (thetaAboveCell000132112123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132112130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112130 h)
+        (by
+          have h : (thetaAboveCell000132112131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112131 h)
+        (by
+          have h : (thetaAboveCell000132112132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112132 h)
+        (by
+          have h : (thetaAboveCell000132112133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112133 h))
+theorem e24KC2ThetaAboveLeaf0001321122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00013211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00013211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132112200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112200 h)
+        (by
+          have h : (thetaAboveCell000132112201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112201 h)
+        (by
+          have h : (thetaAboveCell000132112202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112202 h)
+        (by
+          have h : (thetaAboveCell000132112203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132112210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112210 h)
+        (by
+          have h : (thetaAboveCell000132112211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112211 h)
+        (by
+          have h : (thetaAboveCell000132112212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112212 h)
+        (by
+          have h : (thetaAboveCell000132112213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00013211))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00013211))) h)
+theorem e24KC2ThetaAboveLeaf0001321123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00013211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00013211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132112300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112300 h)
+        (by
+          have h : (thetaAboveCell000132112301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112301 h)
+        (by
+          have h : (thetaAboveCell000132112302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112302 h)
+        (by
+          have h : (thetaAboveCell000132112303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132112310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112310 h)
+        (by
+          have h : (thetaAboveCell000132112311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112311 h)
+        (by
+          have h : (thetaAboveCell000132112312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112312 h)
+        (by
+          have h : (thetaAboveCell000132112313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132112313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00013211))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00013211))) h)
+theorem e24KC2ThetaAboveLeaf0001321130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00013211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00013211))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00013211))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00013211))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132113020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113020 h)
+        (by
+          have h : (thetaAboveCell000132113021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113021 h)
+        (by
+          have h : (thetaAboveCell000132113022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113022 h)
+        (by
+          have h : (thetaAboveCell000132113023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132113030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113030 h)
+        (by
+          have h : (thetaAboveCell000132113031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113031 h)
+        (by
+          have h : (thetaAboveCell000132113032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113032 h)
+        (by
+          have h : (thetaAboveCell000132113033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113033 h))
+theorem e24KC2ThetaAboveLeaf0001321131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00013211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00013211))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00013211))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00013211))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132113120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113120 h)
+        (by
+          have h : (thetaAboveCell000132113121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113121 h)
+        (by
+          have h : (thetaAboveCell000132113122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113122 h)
+        (by
+          have h : (thetaAboveCell000132113123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132113130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113130 h)
+        (by
+          have h : (thetaAboveCell000132113131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113131 h)
+        (by
+          have h : (thetaAboveCell000132113132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113132 h)
+        (by
+          have h : (thetaAboveCell000132113133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113133 h))
+theorem e24KC2ThetaAboveLeaf0001321132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00013211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00013211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132113200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113200 h)
+        (by
+          have h : (thetaAboveCell000132113201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113201 h)
+        (by
+          have h : (thetaAboveCell000132113202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113202 h)
+        (by
+          have h : (thetaAboveCell000132113203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132113210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113210 h)
+        (by
+          have h : (thetaAboveCell000132113211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113211 h)
+        (by
+          have h : (thetaAboveCell000132113212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113212 h)
+        (by
+          have h : (thetaAboveCell000132113213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00013211))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00013211))) h)
+theorem e24KC2ThetaAboveLeaf0001321133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00013211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00013211))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132113300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113300 h)
+        (by
+          have h : (thetaAboveCell000132113301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113301 h)
+        (by
+          have h : (thetaAboveCell000132113302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113302 h)
+        (by
+          have h : (thetaAboveCell000132113303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00013211)))
+        (by
+          have h : (thetaAboveCell000132113310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113310 h)
+        (by
+          have h : (thetaAboveCell000132113311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113311 h)
+        (by
+          have h : (thetaAboveCell000132113312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113312 h)
+        (by
+          have h : (thetaAboveCell000132113313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000132113313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00013211))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00013211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00013211))) h)
+theorem e24KC2ThetaAboveLeaf0001330020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00013300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00013300))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00013300))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00013300))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133002020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002020 h)
+        (by
+          have h : (thetaAboveCell000133002021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002021 h)
+        (by
+          have h : (thetaAboveCell000133002022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002022 h)
+        (by
+          have h : (thetaAboveCell000133002023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133002030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002030 h)
+        (by
+          have h : (thetaAboveCell000133002031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002031 h)
+        (by
+          have h : (thetaAboveCell000133002032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002032 h)
+        (by
+          have h : (thetaAboveCell000133002033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002033 h))
+theorem e24KC2ThetaAboveLeaf0001330021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00013300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00013300))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00013300))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00013300))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133002120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002120 h)
+        (by
+          have h : (thetaAboveCell000133002121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002121 h)
+        (by
+          have h : (thetaAboveCell000133002122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002122 h)
+        (by
+          have h : (thetaAboveCell000133002123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133002130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002130 h)
+        (by
+          have h : (thetaAboveCell000133002131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002131 h)
+        (by
+          have h : (thetaAboveCell000133002132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002132 h)
+        (by
+          have h : (thetaAboveCell000133002133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002133 h))
+theorem e24KC2ThetaAboveLeaf0001330022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00013300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00013300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133002200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002200 h)
+        (by
+          have h : (thetaAboveCell000133002201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002201 h)
+        (by
+          have h : (thetaAboveCell000133002202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002202 h)
+        (by
+          have h : (thetaAboveCell000133002203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133002210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002210 h)
+        (by
+          have h : (thetaAboveCell000133002211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002211 h)
+        (by
+          have h : (thetaAboveCell000133002212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002212 h)
+        (by
+          have h : (thetaAboveCell000133002213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00013300))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00013300))) h)
+theorem e24KC2ThetaAboveLeaf0001330023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00013300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00013300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133002300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002300 h)
+        (by
+          have h : (thetaAboveCell000133002301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002301 h)
+        (by
+          have h : (thetaAboveCell000133002302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002302 h)
+        (by
+          have h : (thetaAboveCell000133002303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133002310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002310 h)
+        (by
+          have h : (thetaAboveCell000133002311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002311 h)
+        (by
+          have h : (thetaAboveCell000133002312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002312 h)
+        (by
+          have h : (thetaAboveCell000133002313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133002313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00013300))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00013300))) h)
+theorem e24KC2ThetaAboveLeaf0001330030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00013300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00013300))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00013300))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00013300))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133003020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003020 h)
+        (by
+          have h : (thetaAboveCell000133003021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003021 h)
+        (by
+          have h : (thetaAboveCell000133003022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003022 h)
+        (by
+          have h : (thetaAboveCell000133003023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133003030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003030 h)
+        (by
+          have h : (thetaAboveCell000133003031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003031 h)
+        (by
+          have h : (thetaAboveCell000133003032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003032 h)
+        (by
+          have h : (thetaAboveCell000133003033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003033 h))
+theorem e24KC2ThetaAboveLeaf0001330031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00013300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00013300))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00013300))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00013300))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133003120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003120 h)
+        (by
+          have h : (thetaAboveCell000133003121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003121 h)
+        (by
+          have h : (thetaAboveCell000133003122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003122 h)
+        (by
+          have h : (thetaAboveCell000133003123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133003130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003130 h)
+        (by
+          have h : (thetaAboveCell000133003131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003131 h)
+        (by
+          have h : (thetaAboveCell000133003132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003132 h)
+        (by
+          have h : (thetaAboveCell000133003133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003133 h))
+theorem e24KC2ThetaAboveLeaf0001330032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00013300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00013300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133003200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003200 h)
+        (by
+          have h : (thetaAboveCell000133003201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003201 h)
+        (by
+          have h : (thetaAboveCell000133003202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003202 h)
+        (by
+          have h : (thetaAboveCell000133003203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133003210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003210 h)
+        (by
+          have h : (thetaAboveCell000133003211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003211 h)
+        (by
+          have h : (thetaAboveCell000133003212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003212 h)
+        (by
+          have h : (thetaAboveCell000133003213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00013300))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00013300))) h)
+theorem e24KC2ThetaAboveLeaf0001330033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00013300)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00013300))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133003300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003300 h)
+        (by
+          have h : (thetaAboveCell000133003301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003301 h)
+        (by
+          have h : (thetaAboveCell000133003302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003302 h)
+        (by
+          have h : (thetaAboveCell000133003303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00013300)))
+        (by
+          have h : (thetaAboveCell000133003310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003310 h)
+        (by
+          have h : (thetaAboveCell000133003311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003311 h)
+        (by
+          have h : (thetaAboveCell000133003312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003312 h)
+        (by
+          have h : (thetaAboveCell000133003313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133003313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00013300))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00013300)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00013300))) h)
+theorem e24KC2ThetaAboveLeaf0001330120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00013301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00013301))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00013301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00013301))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00013301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00013301))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133012020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012020 h)
+        (by
+          have h : (thetaAboveCell000133012021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012021 h)
+        (by
+          have h : (thetaAboveCell000133012022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012022 h)
+        (by
+          have h : (thetaAboveCell000133012023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133012030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012030 h)
+        (by
+          have h : (thetaAboveCell000133012031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012031 h)
+        (by
+          have h : (thetaAboveCell000133012032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012032 h)
+        (by
+          have h : (thetaAboveCell000133012033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012033 h))
+theorem e24KC2ThetaAboveLeaf0001330121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00013301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00013301))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00013301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00013301))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00013301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00013301))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133012120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012120 h)
+        (by
+          have h : (thetaAboveCell000133012121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012121 h)
+        (by
+          have h : (thetaAboveCell000133012122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012122 h)
+        (by
+          have h : (thetaAboveCell000133012123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133012130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012130 h)
+        (by
+          have h : (thetaAboveCell000133012131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012131 h)
+        (by
+          have h : (thetaAboveCell000133012132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012132 h)
+        (by
+          have h : (thetaAboveCell000133012133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012133 h))
+theorem e24KC2ThetaAboveLeaf0001330122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00013301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00013301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133012200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012200 h)
+        (by
+          have h : (thetaAboveCell000133012201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012201 h)
+        (by
+          have h : (thetaAboveCell000133012202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012202 h)
+        (by
+          have h : (thetaAboveCell000133012203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133012210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012210 h)
+        (by
+          have h : (thetaAboveCell000133012211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012211 h)
+        (by
+          have h : (thetaAboveCell000133012212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012212 h)
+        (by
+          have h : (thetaAboveCell000133012213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childHL
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133012220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012220 h)
+        (by
+          have h : (thetaAboveCell000133012221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012221 h)
+        (by
+          have h : (thetaAboveCell000133012222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012222 h)
+        (by
+          have h : (thetaAboveCell000133012223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childHL
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133012230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012230 h)
+        (by
+          have h : (thetaAboveCell000133012231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012231 h)
+        (by
+          have h : (thetaAboveCell000133012232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012232 h)
+        (by
+          have h : (thetaAboveCell000133012233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012233 h))
+theorem e24KC2ThetaAboveLeaf0001330123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00013301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00013301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133012300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012300 h)
+        (by
+          have h : (thetaAboveCell000133012301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012301 h)
+        (by
+          have h : (thetaAboveCell000133012302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012302 h)
+        (by
+          have h : (thetaAboveCell000133012303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133012310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012310 h)
+        (by
+          have h : (thetaAboveCell000133012311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012311 h)
+        (by
+          have h : (thetaAboveCell000133012312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012312 h)
+        (by
+          have h : (thetaAboveCell000133012313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childHL
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133012320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012320 h)
+        (by
+          have h : (thetaAboveCell000133012321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012321 h)
+        (by
+          have h : (thetaAboveCell000133012322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012322 h)
+        (by
+          have h : (thetaAboveCell000133012323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childHL
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133012330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012330 h)
+        (by
+          have h : (thetaAboveCell000133012331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012331 h)
+        (by
+          have h : (thetaAboveCell000133012332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012332 h)
+        (by
+          have h : (thetaAboveCell000133012333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133012333 h))
+theorem e24KC2ThetaAboveLeaf0001330130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00013301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00013301))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00013301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00013301))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00013301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00013301))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133013020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013020 h)
+        (by
+          have h : (thetaAboveCell000133013021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013021 h)
+        (by
+          have h : (thetaAboveCell000133013022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013022 h)
+        (by
+          have h : (thetaAboveCell000133013023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133013030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013030 h)
+        (by
+          have h : (thetaAboveCell000133013031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013031 h)
+        (by
+          have h : (thetaAboveCell000133013032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013032 h)
+        (by
+          have h : (thetaAboveCell000133013033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013033 h))
+theorem e24KC2ThetaAboveLeaf0001330131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00013301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00013301))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00013301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00013301))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00013301)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00013301))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133013120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013120 h)
+        (by
+          have h : (thetaAboveCell000133013121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013121 h)
+        (by
+          have h : (thetaAboveCell000133013122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013122 h)
+        (by
+          have h : (thetaAboveCell000133013123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133013130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013130 h)
+        (by
+          have h : (thetaAboveCell000133013131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013131 h)
+        (by
+          have h : (thetaAboveCell000133013132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013132 h)
+        (by
+          have h : (thetaAboveCell000133013133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013133 h))
+theorem e24KC2ThetaAboveLeaf0001330132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00013301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00013301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133013200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013200 h)
+        (by
+          have h : (thetaAboveCell000133013201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013201 h)
+        (by
+          have h : (thetaAboveCell000133013202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013202 h)
+        (by
+          have h : (thetaAboveCell000133013203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133013210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013210 h)
+        (by
+          have h : (thetaAboveCell000133013211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013211 h)
+        (by
+          have h : (thetaAboveCell000133013212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013212 h)
+        (by
+          have h : (thetaAboveCell000133013213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childHH
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133013220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013220 h)
+        (by
+          have h : (thetaAboveCell000133013221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013221 h)
+        (by
+          have h : (thetaAboveCell000133013222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013222 h)
+        (by
+          have h : (thetaAboveCell000133013223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childHH
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133013230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013230 h)
+        (by
+          have h : (thetaAboveCell000133013231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013231 h)
+        (by
+          have h : (thetaAboveCell000133013232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013232 h)
+        (by
+          have h : (thetaAboveCell000133013233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013233 h))
+theorem e24KC2ThetaAboveLeaf0001330133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00013301)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00013301))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133013300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013300 h)
+        (by
+          have h : (thetaAboveCell000133013301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013301 h)
+        (by
+          have h : (thetaAboveCell000133013302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013302 h)
+        (by
+          have h : (thetaAboveCell000133013303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133013310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013310 h)
+        (by
+          have h : (thetaAboveCell000133013311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013311 h)
+        (by
+          have h : (thetaAboveCell000133013312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013312 h)
+        (by
+          have h : (thetaAboveCell000133013313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childHH
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133013320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013320 h)
+        (by
+          have h : (thetaAboveCell000133013321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013321 h)
+        (by
+          have h : (thetaAboveCell000133013322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013322 h)
+        (by
+          have h : (thetaAboveCell000133013323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childHH
+        thetaAboveCell00013301)))
+        (by
+          have h : (thetaAboveCell000133013330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013330 h)
+        (by
+          have h : (thetaAboveCell000133013331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013331 h)
+        (by
+          have h : (thetaAboveCell000133013332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013332 h)
+        (by
+          have h : (thetaAboveCell000133013333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133013333 h))
+theorem e24KC2ThetaAboveLeaf0001331020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00013310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00013310))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00013310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00013310))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00013310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00013310))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133102020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102020 h)
+        (by
+          have h : (thetaAboveCell000133102021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102021 h)
+        (by
+          have h : (thetaAboveCell000133102022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102022 h)
+        (by
+          have h : (thetaAboveCell000133102023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133102030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102030 h)
+        (by
+          have h : (thetaAboveCell000133102031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102031 h)
+        (by
+          have h : (thetaAboveCell000133102032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102032 h)
+        (by
+          have h : (thetaAboveCell000133102033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102033 h))
+theorem e24KC2ThetaAboveLeaf0001331021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00013310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00013310))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00013310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00013310))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00013310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00013310))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133102120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102120 h)
+        (by
+          have h : (thetaAboveCell000133102121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102121 h)
+        (by
+          have h : (thetaAboveCell000133102122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102122 h)
+        (by
+          have h : (thetaAboveCell000133102123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133102130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102130 h)
+        (by
+          have h : (thetaAboveCell000133102131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102131 h)
+        (by
+          have h : (thetaAboveCell000133102132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102132 h)
+        (by
+          have h : (thetaAboveCell000133102133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102133 h))
+theorem e24KC2ThetaAboveLeaf0001331022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00013310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00013310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133102200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102200 h)
+        (by
+          have h : (thetaAboveCell000133102201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102201 h)
+        (by
+          have h : (thetaAboveCell000133102202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102202 h)
+        (by
+          have h : (thetaAboveCell000133102203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133102210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102210 h)
+        (by
+          have h : (thetaAboveCell000133102211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102211 h)
+        (by
+          have h : (thetaAboveCell000133102212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102212 h)
+        (by
+          have h : (thetaAboveCell000133102213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childHL
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133102220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102220 h)
+        (by
+          have h : (thetaAboveCell000133102221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102221 h)
+        (by
+          have h : (thetaAboveCell000133102222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102222 h)
+        (by
+          have h : (thetaAboveCell000133102223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childHL
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133102230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102230 h)
+        (by
+          have h : (thetaAboveCell000133102231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102231 h)
+        (by
+          have h : (thetaAboveCell000133102232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102232 h)
+        (by
+          have h : (thetaAboveCell000133102233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102233 h))
+theorem e24KC2ThetaAboveLeaf0001331023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00013310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00013310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133102300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102300 h)
+        (by
+          have h : (thetaAboveCell000133102301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102301 h)
+        (by
+          have h : (thetaAboveCell000133102302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102302 h)
+        (by
+          have h : (thetaAboveCell000133102303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133102310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102310 h)
+        (by
+          have h : (thetaAboveCell000133102311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102311 h)
+        (by
+          have h : (thetaAboveCell000133102312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102312 h)
+        (by
+          have h : (thetaAboveCell000133102313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childHL
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133102320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102320 h)
+        (by
+          have h : (thetaAboveCell000133102321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102321 h)
+        (by
+          have h : (thetaAboveCell000133102322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102322 h)
+        (by
+          have h : (thetaAboveCell000133102323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childHL
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133102330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102330 h)
+        (by
+          have h : (thetaAboveCell000133102331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102331 h)
+        (by
+          have h : (thetaAboveCell000133102332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102332 h)
+        (by
+          have h : (thetaAboveCell000133102333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133102333 h))
+theorem e24KC2ThetaAboveLeaf0001331030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00013310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00013310))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00013310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00013310))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00013310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00013310))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133103020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103020 h)
+        (by
+          have h : (thetaAboveCell000133103021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103021 h)
+        (by
+          have h : (thetaAboveCell000133103022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103022 h)
+        (by
+          have h : (thetaAboveCell000133103023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133103030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103030 h)
+        (by
+          have h : (thetaAboveCell000133103031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103031 h)
+        (by
+          have h : (thetaAboveCell000133103032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103032 h)
+        (by
+          have h : (thetaAboveCell000133103033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103033 h))
+theorem e24KC2ThetaAboveLeaf0001331031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00013310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00013310))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00013310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00013310))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00013310)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00013310))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133103120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103120 h)
+        (by
+          have h : (thetaAboveCell000133103121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103121 h)
+        (by
+          have h : (thetaAboveCell000133103122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103122 h)
+        (by
+          have h : (thetaAboveCell000133103123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133103130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103130 h)
+        (by
+          have h : (thetaAboveCell000133103131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103131 h)
+        (by
+          have h : (thetaAboveCell000133103132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103132 h)
+        (by
+          have h : (thetaAboveCell000133103133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103133 h))
+theorem e24KC2ThetaAboveLeaf0001331032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00013310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00013310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133103200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103200 h)
+        (by
+          have h : (thetaAboveCell000133103201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103201 h)
+        (by
+          have h : (thetaAboveCell000133103202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103202 h)
+        (by
+          have h : (thetaAboveCell000133103203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133103210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103210 h)
+        (by
+          have h : (thetaAboveCell000133103211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103211 h)
+        (by
+          have h : (thetaAboveCell000133103212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103212 h)
+        (by
+          have h : (thetaAboveCell000133103213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childHH
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133103220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103220 h)
+        (by
+          have h : (thetaAboveCell000133103221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103221 h)
+        (by
+          have h : (thetaAboveCell000133103222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103222 h)
+        (by
+          have h : (thetaAboveCell000133103223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childHH
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133103230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103230 h)
+        (by
+          have h : (thetaAboveCell000133103231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103231 h)
+        (by
+          have h : (thetaAboveCell000133103232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103232 h)
+        (by
+          have h : (thetaAboveCell000133103233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103233 h))
+theorem e24KC2ThetaAboveLeaf0001331033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00013310)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00013310))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133103300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103300 h)
+        (by
+          have h : (thetaAboveCell000133103301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103301 h)
+        (by
+          have h : (thetaAboveCell000133103302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103302 h)
+        (by
+          have h : (thetaAboveCell000133103303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133103310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103310 h)
+        (by
+          have h : (thetaAboveCell000133103311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103311 h)
+        (by
+          have h : (thetaAboveCell000133103312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103312 h)
+        (by
+          have h : (thetaAboveCell000133103313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childHH
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133103320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103320 h)
+        (by
+          have h : (thetaAboveCell000133103321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103321 h)
+        (by
+          have h : (thetaAboveCell000133103322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103322 h)
+        (by
+          have h : (thetaAboveCell000133103323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childHH
+        thetaAboveCell00013310)))
+        (by
+          have h : (thetaAboveCell000133103330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103330 h)
+        (by
+          have h : (thetaAboveCell000133103331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103331 h)
+        (by
+          have h : (thetaAboveCell000133103332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103332 h)
+        (by
+          have h : (thetaAboveCell000133103333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133103333 h))
+theorem e24KC2ThetaAboveLeaf0001331120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00013311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00013311))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00013311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00013311))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00013311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00013311))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133112020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112020 h)
+        (by
+          have h : (thetaAboveCell000133112021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112021 h)
+        (by
+          have h : (thetaAboveCell000133112022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112022 h)
+        (by
+          have h : (thetaAboveCell000133112023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133112030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112030 h)
+        (by
+          have h : (thetaAboveCell000133112031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112031 h)
+        (by
+          have h : (thetaAboveCell000133112032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112032 h)
+        (by
+          have h : (thetaAboveCell000133112033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112033 h))
+theorem e24KC2ThetaAboveLeaf0001331121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00013311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00013311))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00013311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00013311))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00013311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00013311))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133112120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112120 h)
+        (by
+          have h : (thetaAboveCell000133112121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112121 h)
+        (by
+          have h : (thetaAboveCell000133112122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112122 h)
+        (by
+          have h : (thetaAboveCell000133112123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133112130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112130 h)
+        (by
+          have h : (thetaAboveCell000133112131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112131 h)
+        (by
+          have h : (thetaAboveCell000133112132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112132 h)
+        (by
+          have h : (thetaAboveCell000133112133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112133 h))
+theorem e24KC2ThetaAboveLeaf0001331122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00013311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00013311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133112200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112200 h)
+        (by
+          have h : (thetaAboveCell000133112201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112201 h)
+        (by
+          have h : (thetaAboveCell000133112202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112202 h)
+        (by
+          have h : (thetaAboveCell000133112203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133112210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112210 h)
+        (by
+          have h : (thetaAboveCell000133112211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112211 h)
+        (by
+          have h : (thetaAboveCell000133112212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112212 h)
+        (by
+          have h : (thetaAboveCell000133112213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childHL
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133112220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112220 h)
+        (by
+          have h : (thetaAboveCell000133112221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112221 h)
+        (by
+          have h : (thetaAboveCell000133112222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112222 h)
+        (by
+          have h : (thetaAboveCell000133112223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childHL
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133112230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112230 h)
+        (by
+          have h : (thetaAboveCell000133112231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112231 h)
+        (by
+          have h : (thetaAboveCell000133112232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112232 h)
+        (by
+          have h : (thetaAboveCell000133112233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112233 h))
+theorem e24KC2ThetaAboveLeaf0001331123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00013311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00013311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133112300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112300 h)
+        (by
+          have h : (thetaAboveCell000133112301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112301 h)
+        (by
+          have h : (thetaAboveCell000133112302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112302 h)
+        (by
+          have h : (thetaAboveCell000133112303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133112310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112310 h)
+        (by
+          have h : (thetaAboveCell000133112311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112311 h)
+        (by
+          have h : (thetaAboveCell000133112312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112312 h)
+        (by
+          have h : (thetaAboveCell000133112313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133112313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00013311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00013311))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00013311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00013311))) h)
+theorem e24KC2ThetaAboveLeaf0001331130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00013311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00013311))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00013311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00013311))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00013311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00013311))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133113020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113020 h)
+        (by
+          have h : (thetaAboveCell000133113021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113021 h)
+        (by
+          have h : (thetaAboveCell000133113022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113022 h)
+        (by
+          have h : (thetaAboveCell000133113023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133113030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113030 h)
+        (by
+          have h : (thetaAboveCell000133113031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113031 h)
+        (by
+          have h : (thetaAboveCell000133113032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113032 h)
+        (by
+          have h : (thetaAboveCell000133113033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113033 h))
+theorem e24KC2ThetaAboveLeaf0001331131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00013311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00013311))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00013311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00013311))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00013311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00013311))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133113120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113120 h)
+        (by
+          have h : (thetaAboveCell000133113121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113121 h)
+        (by
+          have h : (thetaAboveCell000133113122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113122 h)
+        (by
+          have h : (thetaAboveCell000133113123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133113130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113130 h)
+        (by
+          have h : (thetaAboveCell000133113131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113131 h)
+        (by
+          have h : (thetaAboveCell000133113132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113132 h)
+        (by
+          have h : (thetaAboveCell000133113133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113133 h))
+theorem e24KC2ThetaAboveLeaf0001331132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00013311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00013311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133113200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113200 h)
+        (by
+          have h : (thetaAboveCell000133113201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113201 h)
+        (by
+          have h : (thetaAboveCell000133113202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113202 h)
+        (by
+          have h : (thetaAboveCell000133113203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133113210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113210 h)
+        (by
+          have h : (thetaAboveCell000133113211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113211 h)
+        (by
+          have h : (thetaAboveCell000133113212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113212 h)
+        (by
+          have h : (thetaAboveCell000133113213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00013311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00013311))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00013311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00013311))) h)
+theorem e24KC2ThetaAboveLeaf0001331133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00013311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00013311))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133113300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113300 h)
+        (by
+          have h : (thetaAboveCell000133113301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113301 h)
+        (by
+          have h : (thetaAboveCell000133113302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113302 h)
+        (by
+          have h : (thetaAboveCell000133113303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00013311)))
+        (by
+          have h : (thetaAboveCell000133113310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113310 h)
+        (by
+          have h : (thetaAboveCell000133113311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113311 h)
+        (by
+          have h : (thetaAboveCell000133113312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113312 h)
+        (by
+          have h : (thetaAboveCell000133113313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000133113313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00013311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00013311))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00013311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00013311))) h)
+theorem e24KC2ThetaAboveLeaf0001331301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00013313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00013313))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00013313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00013313))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00013313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00013313))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00013313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00013313))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00013313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00013313))) h)
+theorem e24KC2ThetaAboveLeaf0001331310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00013313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00013313))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00013313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00013313))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00013313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00013313))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00013313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00013313))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00013313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00013313))) h)
+theorem e24KC2ThetaAboveLeaf0001331311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00013313)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00013313))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00013313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00013313))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00013313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00013313))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00013313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00013313))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00013313)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00013313))) h)
+theorem e24KC2ThetaAboveLeaf0010220020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00102200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00102200))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00102200))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00102200))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022002020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002020 h)
+        (by
+          have h : (thetaAboveCell001022002021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002021 h)
+        (by
+          have h : (thetaAboveCell001022002022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002022 h)
+        (by
+          have h : (thetaAboveCell001022002023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022002030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002030 h)
+        (by
+          have h : (thetaAboveCell001022002031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002031 h)
+        (by
+          have h : (thetaAboveCell001022002032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002032 h)
+        (by
+          have h : (thetaAboveCell001022002033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002033 h))
+theorem e24KC2ThetaAboveLeaf0010220021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00102200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00102200))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00102200))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00102200))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022002120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002120 h)
+        (by
+          have h : (thetaAboveCell001022002121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002121 h)
+        (by
+          have h : (thetaAboveCell001022002122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002122 h)
+        (by
+          have h : (thetaAboveCell001022002123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022002130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002130 h)
+        (by
+          have h : (thetaAboveCell001022002131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002131 h)
+        (by
+          have h : (thetaAboveCell001022002132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002132 h)
+        (by
+          have h : (thetaAboveCell001022002133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002133 h))
+theorem e24KC2ThetaAboveLeaf0010220022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00102200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00102200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022002200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002200 h)
+        (by
+          have h : (thetaAboveCell001022002201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002201 h)
+        (by
+          have h : (thetaAboveCell001022002202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002202 h)
+        (by
+          have h : (thetaAboveCell001022002203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHL
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022002210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002210 h)
+        (by
+          have h : (thetaAboveCell001022002211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002211 h)
+        (by
+          have h : (thetaAboveCell001022002212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002212 h)
+        (by
+          have h : (thetaAboveCell001022002213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00102200))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00102200))) h)
+theorem e24KC2ThetaAboveLeaf0010220023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00102200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00102200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHL
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022002300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002300 h)
+        (by
+          have h : (thetaAboveCell001022002301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002301 h)
+        (by
+          have h : (thetaAboveCell001022002302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002302 h)
+        (by
+          have h : (thetaAboveCell001022002303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHL
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022002310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002310 h)
+        (by
+          have h : (thetaAboveCell001022002311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002311 h)
+        (by
+          have h : (thetaAboveCell001022002312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002312 h)
+        (by
+          have h : (thetaAboveCell001022002313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022002313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00102200))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00102200))) h)
+theorem e24KC2ThetaAboveLeaf0010220030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00102200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00102200))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00102200))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00102200))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022003020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003020 h)
+        (by
+          have h : (thetaAboveCell001022003021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003021 h)
+        (by
+          have h : (thetaAboveCell001022003022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003022 h)
+        (by
+          have h : (thetaAboveCell001022003023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022003030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003030 h)
+        (by
+          have h : (thetaAboveCell001022003031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003031 h)
+        (by
+          have h : (thetaAboveCell001022003032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003032 h)
+        (by
+          have h : (thetaAboveCell001022003033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003033 h))
+theorem e24KC2ThetaAboveLeaf0010220031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00102200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00102200))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00102200))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00102200))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022003120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003120 h)
+        (by
+          have h : (thetaAboveCell001022003121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003121 h)
+        (by
+          have h : (thetaAboveCell001022003122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003122 h)
+        (by
+          have h : (thetaAboveCell001022003123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022003130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003130 h)
+        (by
+          have h : (thetaAboveCell001022003131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003131 h)
+        (by
+          have h : (thetaAboveCell001022003132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003132 h)
+        (by
+          have h : (thetaAboveCell001022003133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003133 h))
+theorem e24KC2ThetaAboveLeaf0010220032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00102200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00102200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHH
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022003200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003200 h)
+        (by
+          have h : (thetaAboveCell001022003201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003201 h)
+        (by
+          have h : (thetaAboveCell001022003202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003202 h)
+        (by
+          have h : (thetaAboveCell001022003203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHL (childHH
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022003210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003210 h)
+        (by
+          have h : (thetaAboveCell001022003211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003211 h)
+        (by
+          have h : (thetaAboveCell001022003212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003212 h)
+        (by
+          have h : (thetaAboveCell001022003213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00102200))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00102200))) h)
+theorem e24KC2ThetaAboveLeaf0010220033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00102200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00102200))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHH (childHH
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022003300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003300 h)
+        (by
+          have h : (thetaAboveCell001022003301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003301 h)
+        (by
+          have h : (thetaAboveCell001022003302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003302 h)
+        (by
+          have h : (thetaAboveCell001022003303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childHH (childHH
+        thetaAboveCell00102200)))
+        (by
+          have h : (thetaAboveCell001022003310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003310 h)
+        (by
+          have h : (thetaAboveCell001022003311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003311 h)
+        (by
+          have h : (thetaAboveCell001022003312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003312 h)
+        (by
+          have h : (thetaAboveCell001022003313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022003313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00102200))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00102200)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00102200))) h)
+theorem e24KC2ThetaAboveLeaf0010220120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00102201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00102201))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00102201))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00102201))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00102201)))
+        (by
+          have h : (thetaAboveCell001022012020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012020 h)
+        (by
+          have h : (thetaAboveCell001022012021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012021 h)
+        (by
+          have h : (thetaAboveCell001022012022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012022 h)
+        (by
+          have h : (thetaAboveCell001022012023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00102201)))
+        (by
+          have h : (thetaAboveCell001022012030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012030 h)
+        (by
+          have h : (thetaAboveCell001022012031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012031 h)
+        (by
+          have h : (thetaAboveCell001022012032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012032 h)
+        (by
+          have h : (thetaAboveCell001022012033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012033 h))
+theorem e24KC2ThetaAboveLeaf0010220121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00102201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00102201))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00102201))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00102201))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00102201)))
+        (by
+          have h : (thetaAboveCell001022012120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012120 h)
+        (by
+          have h : (thetaAboveCell001022012121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012121 h)
+        (by
+          have h : (thetaAboveCell001022012122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012122 h)
+        (by
+          have h : (thetaAboveCell001022012123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00102201)))
+        (by
+          have h : (thetaAboveCell001022012130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012130 h)
+        (by
+          have h : (thetaAboveCell001022012131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012131 h)
+        (by
+          have h : (thetaAboveCell001022012132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012132 h)
+        (by
+          have h : (thetaAboveCell001022012133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012133 h))
+theorem e24KC2ThetaAboveLeaf0010220122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00102201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00102201))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLL (childHL (childHL
+        thetaAboveCell00102201)))
+        (by
+          have h : (thetaAboveCell001022012200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012200 h)
+        (by
+          have h : (thetaAboveCell001022012201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012201 h)
+        (by
+          have h : (thetaAboveCell001022012202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012202 h)
+        (by
+          have h : (thetaAboveCell001022012203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022012203 h))
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00102201))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00102201))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00102201))) h)
+theorem e24KC2ThetaAboveLeaf0010220123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00102201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00102201))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00102201))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00102201))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00102201))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00102201))) h)
+theorem e24KC2ThetaAboveLeaf0010220130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00102201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00102201))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00102201))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00102201))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00102201)))
+        (by
+          have h : (thetaAboveCell001022013020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013020 h)
+        (by
+          have h : (thetaAboveCell001022013021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013021 h)
+        (by
+          have h : (thetaAboveCell001022013022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013022 h)
+        (by
+          have h : (thetaAboveCell001022013023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00102201)))
+        (by
+          have h : (thetaAboveCell001022013030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013030 h)
+        (by
+          have h : (thetaAboveCell001022013031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013031 h)
+        (by
+          have h : (thetaAboveCell001022013032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013032 h)
+        (by
+          have h : (thetaAboveCell001022013033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013033 h))
+theorem e24KC2ThetaAboveLeaf0010220131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00102201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00102201))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00102201))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00102201))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00102201)))
+        (by
+          have h : (thetaAboveCell001022013120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013120 h)
+        (by
+          have h : (thetaAboveCell001022013121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013121 h)
+        (by
+          have h : (thetaAboveCell001022013122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013122 h)
+        (by
+          have h : (thetaAboveCell001022013123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00102201)))
+        (by
+          have h : (thetaAboveCell001022013130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013130 h)
+        (by
+          have h : (thetaAboveCell001022013131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013131 h)
+        (by
+          have h : (thetaAboveCell001022013132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013132 h)
+        (by
+          have h : (thetaAboveCell001022013133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022013133 h))
+theorem e24KC2ThetaAboveLeaf0010220132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00102201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00102201))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00102201))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00102201))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00102201))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00102201))) h)
+theorem e24KC2ThetaAboveLeaf0010220133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00102201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00102201))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00102201))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00102201))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00102201))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00102201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00102201))) h)
+theorem e24KC2ThetaAboveLeaf0010220200 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00102202))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00102202))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00102202))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00102202))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00102202))) h)
+theorem e24KC2ThetaAboveLeaf0010220201 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00102202))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00102202))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00102202))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00102202))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00102202))) h)
+theorem e24KC2ThetaAboveLeaf0010220210 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00102202))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00102202))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00102202))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00102202))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00102202))) h)
+theorem e24KC2ThetaAboveLeaf0010220211 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102202)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00102202))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00102202))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00102202))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00102202))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00102202)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00102202))) h)
+theorem e24KC2ThetaAboveLeaf0010220300 :
+    adaptiveCoverCheck 9 (childLL (childLL thetaAboveCell00102203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLL thetaAboveCell00102203))
+    (by
+      have h : ((childLL (childLL (childLL thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLL
+        thetaAboveCell00102203))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLL
+        thetaAboveCell00102203))) h)
+    (by
+      have h : ((childHL (childLL (childLL thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLL
+        thetaAboveCell00102203))) h)
+    (by
+      have h : ((childHH (childLL (childLL thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLL
+        thetaAboveCell00102203))) h)
+theorem e24KC2ThetaAboveLeaf0010220301 :
+    adaptiveCoverCheck 9 (childLH (childLL thetaAboveCell00102203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLL thetaAboveCell00102203))
+    (by
+      have h : ((childLL (childLH (childLL thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLL
+        thetaAboveCell00102203))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLL
+        thetaAboveCell00102203))) h)
+    (by
+      have h : ((childHL (childLH (childLL thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLL
+        thetaAboveCell00102203))) h)
+    (by
+      have h : ((childHH (childLH (childLL thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLL
+        thetaAboveCell00102203))) h)
+theorem e24KC2ThetaAboveLeaf0010220310 :
+    adaptiveCoverCheck 9 (childLL (childLH thetaAboveCell00102203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childLH thetaAboveCell00102203))
+    (by
+      have h : ((childLL (childLL (childLH thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childLH
+        thetaAboveCell00102203))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childLH
+        thetaAboveCell00102203))) h)
+    (by
+      have h : ((childHL (childLL (childLH thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childLH
+        thetaAboveCell00102203))) h)
+    (by
+      have h : ((childHH (childLL (childLH thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childLH
+        thetaAboveCell00102203))) h)
+theorem e24KC2ThetaAboveLeaf0010220311 :
+    adaptiveCoverCheck 9 (childLH (childLH thetaAboveCell00102203)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childLH thetaAboveCell00102203))
+    (by
+      have h : ((childLL (childLH (childLH thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childLH
+        thetaAboveCell00102203))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childLH
+        thetaAboveCell00102203))) h)
+    (by
+      have h : ((childHL (childLH (childLH thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLH (childLH
+        thetaAboveCell00102203))) h)
+    (by
+      have h : ((childHH (childLH (childLH thetaAboveCell00102203)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLH (childLH
+        thetaAboveCell00102203))) h)
+theorem e24KC2ThetaAboveLeaf0010221020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00102210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00102210))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00102210))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00102210)))
+        (by
+          have h : (thetaAboveCell001022102020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102020 h)
+        (by
+          have h : (thetaAboveCell001022102021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102021 h)
+        (by
+          have h : (thetaAboveCell001022102022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102022 h)
+        (by
+          have h : (thetaAboveCell001022102023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00102210)))
+        (by
+          have h : (thetaAboveCell001022102030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102030 h)
+        (by
+          have h : (thetaAboveCell001022102031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102031 h)
+        (by
+          have h : (thetaAboveCell001022102032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102032 h)
+        (by
+          have h : (thetaAboveCell001022102033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102033 h))
+theorem e24KC2ThetaAboveLeaf0010221021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00102210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00102210))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHL
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHL
+        thetaAboveCell00102210))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00102210)))
+        (by
+          have h : (thetaAboveCell001022102120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102120 h)
+        (by
+          have h : (thetaAboveCell001022102121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102121 h)
+        (by
+          have h : (thetaAboveCell001022102122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102122 h)
+        (by
+          have h : (thetaAboveCell001022102123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00102210)))
+        (by
+          have h : (thetaAboveCell001022102130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102130 h)
+        (by
+          have h : (thetaAboveCell001022102131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102131 h)
+        (by
+          have h : (thetaAboveCell001022102132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102132 h)
+        (by
+          have h : (thetaAboveCell001022102133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022102133 h))
+theorem e24KC2ThetaAboveLeaf0010221022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00102210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00102210))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00102210))) h)
+theorem e24KC2ThetaAboveLeaf0010221023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00102210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00102210))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00102210))) h)
+theorem e24KC2ThetaAboveLeaf0010221030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00102210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00102210))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHH
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHH
+        thetaAboveCell00102210))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00102210)))
+        (by
+          have h : (thetaAboveCell001022103020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103020 h)
+        (by
+          have h : (thetaAboveCell001022103021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103021 h)
+        (by
+          have h : (thetaAboveCell001022103022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103022 h)
+        (by
+          have h : (thetaAboveCell001022103023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00102210)))
+        (by
+          have h : (thetaAboveCell001022103030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103030 h)
+        (by
+          have h : (thetaAboveCell001022103031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103031 h)
+        (by
+          have h : (thetaAboveCell001022103032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103032 h)
+        (by
+          have h : (thetaAboveCell001022103033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103033 h))
+theorem e24KC2ThetaAboveLeaf0010221031 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00102210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00102210))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLH (childHH
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLH (childHH
+        thetaAboveCell00102210))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00102210)))
+        (by
+          have h : (thetaAboveCell001022103120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103120 h)
+        (by
+          have h : (thetaAboveCell001022103121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103121 h)
+        (by
+          have h : (thetaAboveCell001022103122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103122 h)
+        (by
+          have h : (thetaAboveCell001022103123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00102210)))
+        (by
+          have h : (thetaAboveCell001022103130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103130 h)
+        (by
+          have h : (thetaAboveCell001022103131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103131 h)
+        (by
+          have h : (thetaAboveCell001022103132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103132 h)
+        (by
+          have h : (thetaAboveCell001022103133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell001022103133 h))
+theorem e24KC2ThetaAboveLeaf0010221032 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00102210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00102210))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00102210))) h)
+theorem e24KC2ThetaAboveLeaf0010221033 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00102210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00102210))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00102210))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00102210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00102210))) h)
+theorem e24KC2ThetaAboveLeaf0010221120 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00102211)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00102211))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childLL (childHL
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childLL (childHL
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHL (childLL (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childLL (childHL
+        thetaAboveCell00102211))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaAboveCell00102211)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childLL (childHL
+        thetaAboveCell00102211))) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle017.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle017.lean
new file mode 100644
index 0000000000..bf2391e38d
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle017.lean
@@ -0,0 +1,9055 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch030`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCellsbc9ee1179f
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022012100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00002201)))
+
+/-- Subcell `000022013000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00002201)))
+
+/-- Subcell `0000220121002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121002333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121002333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121003333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121003333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell000022012100)))
+
+/-- Subcell `0000220121012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121012333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121012333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121013333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121013333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell000022012101)))
+
+/-- Subcell `0000220121102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121102333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121102333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121103333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121103333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell000022012110)))
+
+/-- Subcell `0000220121112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121112333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121112333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220121113333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220121113333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell000022012111)))
+
+/-- Subcell `0000220130002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130002333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130002333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130003333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130003333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell000022013000)))
+
+/-- Subcell `0000220130012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130012333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130012333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130013333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130013333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell000022013001)))
+
+/-- Subcell `0000220130102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130102333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130102333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130103333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130103333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell000022013010)))
+
+/-- Subcell `0000220130112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112222 : AngleCell :=
+  childHL (childHL (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112223 : AngleCell :=
+  childHH (childHL (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112232 : AngleCell :=
+  childHL (childHH (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112233 : AngleCell :=
+  childHH (childHH (childHL (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112322 : AngleCell :=
+  childHL (childHL (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112323 : AngleCell :=
+  childHH (childHL (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130112333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130112333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell000022013011)))
+
+/-- Subcell `0000220130113323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220130113323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell000022013011)))
+
+end GerverSofa.PartE.CertificateCellsbc9ee1179f
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6ProofBatchE7cbedc8143c8e9f`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsbc9ee1179f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsbc9ee1179f
+
+open CertificateCellsbc9ee1179f
+theorem cover_subtree_c4eb81eb7e49 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022012100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022012100))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell000022012100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+        thetaAboveCell000022012100))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell000022012100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+        thetaAboveCell000022012100))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121002020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002020 h)
+        (by
+          have h : (thetaAboveCell0000220121002021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002021 h)
+        (by
+          have h : (thetaAboveCell0000220121002022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002022 h)
+        (by
+          have h : (thetaAboveCell0000220121002023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121002030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002030 h)
+        (by
+          have h : (thetaAboveCell0000220121002031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002031 h)
+        (by
+          have h : (thetaAboveCell0000220121002032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002032 h)
+        (by
+          have h : (thetaAboveCell0000220121002033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002033 h))
+
+theorem cover_subtree_20c58a0c1f65 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022012100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022012100))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022012100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022012100))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022012100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022012100))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121002120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002120 h)
+        (by
+          have h : (thetaAboveCell0000220121002121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002121 h)
+        (by
+          have h : (thetaAboveCell0000220121002122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002122 h)
+        (by
+          have h : (thetaAboveCell0000220121002123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121002130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002130 h)
+        (by
+          have h : (thetaAboveCell0000220121002131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002131 h)
+        (by
+          have h : (thetaAboveCell0000220121002132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002132 h)
+        (by
+          have h : (thetaAboveCell0000220121002133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002133 h))
+
+theorem cover_subtree_a8a40a37bd16 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022012100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022012100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121002200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002200 h)
+        (by
+          have h : (thetaAboveCell0000220121002201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002201 h)
+        (by
+          have h : (thetaAboveCell0000220121002202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002202 h)
+        (by
+          have h : (thetaAboveCell0000220121002203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121002210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002210 h)
+        (by
+          have h : (thetaAboveCell0000220121002211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002211 h)
+        (by
+          have h : (thetaAboveCell0000220121002212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002212 h)
+        (by
+          have h : (thetaAboveCell0000220121002213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHL
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121002220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002220 h)
+        (by
+          have h : (thetaAboveCell0000220121002221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002221 h)
+        (by
+          have h : (thetaAboveCell0000220121002222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002222 h)
+        (by
+          have h : (thetaAboveCell0000220121002223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHL
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121002230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002230 h)
+        (by
+          have h : (thetaAboveCell0000220121002231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002231 h)
+        (by
+          have h : (thetaAboveCell0000220121002232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002232 h)
+        (by
+          have h : (thetaAboveCell0000220121002233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002233 h))
+
+theorem cover_subtree_71af7f863f59 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022012100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022012100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121002300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002300 h)
+        (by
+          have h : (thetaAboveCell0000220121002301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002301 h)
+        (by
+          have h : (thetaAboveCell0000220121002302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002302 h)
+        (by
+          have h : (thetaAboveCell0000220121002303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121002310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002310 h)
+        (by
+          have h : (thetaAboveCell0000220121002311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002311 h)
+        (by
+          have h : (thetaAboveCell0000220121002312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002312 h)
+        (by
+          have h : (thetaAboveCell0000220121002313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHL
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121002320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002320 h)
+        (by
+          have h : (thetaAboveCell0000220121002321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002321 h)
+        (by
+          have h : (thetaAboveCell0000220121002322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002322 h)
+        (by
+          have h : (thetaAboveCell0000220121002323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHL
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121002330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002330 h)
+        (by
+          have h : (thetaAboveCell0000220121002331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002331 h)
+        (by
+          have h : (thetaAboveCell0000220121002332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002332 h)
+        (by
+          have h : (thetaAboveCell0000220121002333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121002333 h))
+
+theorem cover_subtree_e8b421b5f980 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022012100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022012100)
+    cover_subtree_c4eb81eb7e49
+    cover_subtree_20c58a0c1f65
+    cover_subtree_a8a40a37bd16
+    cover_subtree_71af7f863f59
+
+theorem cover_subtree_807256c88993 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022012100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022012100))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022012100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022012100))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022012100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022012100))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121003020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003020 h)
+        (by
+          have h : (thetaAboveCell0000220121003021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003021 h)
+        (by
+          have h : (thetaAboveCell0000220121003022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003022 h)
+        (by
+          have h : (thetaAboveCell0000220121003023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121003030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003030 h)
+        (by
+          have h : (thetaAboveCell0000220121003031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003031 h)
+        (by
+          have h : (thetaAboveCell0000220121003032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003032 h)
+        (by
+          have h : (thetaAboveCell0000220121003033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003033 h))
+
+theorem cover_subtree_8f2cea51bbf4 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022012100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022012100))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell000022012100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+        thetaAboveCell000022012100))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell000022012100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+        thetaAboveCell000022012100))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121003120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003120 h)
+        (by
+          have h : (thetaAboveCell0000220121003121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003121 h)
+        (by
+          have h : (thetaAboveCell0000220121003122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003122 h)
+        (by
+          have h : (thetaAboveCell0000220121003123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121003130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003130 h)
+        (by
+          have h : (thetaAboveCell0000220121003131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003131 h)
+        (by
+          have h : (thetaAboveCell0000220121003132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003132 h)
+        (by
+          have h : (thetaAboveCell0000220121003133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003133 h))
+
+theorem cover_subtree_c244889fccb8 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022012100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022012100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121003200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003200 h)
+        (by
+          have h : (thetaAboveCell0000220121003201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003201 h)
+        (by
+          have h : (thetaAboveCell0000220121003202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003202 h)
+        (by
+          have h : (thetaAboveCell0000220121003203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121003210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003210 h)
+        (by
+          have h : (thetaAboveCell0000220121003211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003211 h)
+        (by
+          have h : (thetaAboveCell0000220121003212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003212 h)
+        (by
+          have h : (thetaAboveCell0000220121003213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHH
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121003220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003220 h)
+        (by
+          have h : (thetaAboveCell0000220121003221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003221 h)
+        (by
+          have h : (thetaAboveCell0000220121003222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003222 h)
+        (by
+          have h : (thetaAboveCell0000220121003223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHH
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121003230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003230 h)
+        (by
+          have h : (thetaAboveCell0000220121003231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003231 h)
+        (by
+          have h : (thetaAboveCell0000220121003232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003232 h)
+        (by
+          have h : (thetaAboveCell0000220121003233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003233 h))
+
+theorem cover_subtree_2684f740078d :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022012100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022012100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121003300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003300 h)
+        (by
+          have h : (thetaAboveCell0000220121003301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003301 h)
+        (by
+          have h : (thetaAboveCell0000220121003302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003302 h)
+        (by
+          have h : (thetaAboveCell0000220121003303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121003310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003310 h)
+        (by
+          have h : (thetaAboveCell0000220121003311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003311 h)
+        (by
+          have h : (thetaAboveCell0000220121003312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003312 h)
+        (by
+          have h : (thetaAboveCell0000220121003313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHH
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121003320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003320 h)
+        (by
+          have h : (thetaAboveCell0000220121003321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003321 h)
+        (by
+          have h : (thetaAboveCell0000220121003322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003322 h)
+        (by
+          have h : (thetaAboveCell0000220121003323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHH
+        thetaAboveCell000022012100)))
+        (by
+          have h : (thetaAboveCell0000220121003330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003330 h)
+        (by
+          have h : (thetaAboveCell0000220121003331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003331 h)
+        (by
+          have h : (thetaAboveCell0000220121003332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003332 h)
+        (by
+          have h : (thetaAboveCell0000220121003333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121003333 h))
+
+theorem cover_subtree_38440f81d562 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022012100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022012100)
+    cover_subtree_807256c88993
+    cover_subtree_8f2cea51bbf4
+    cover_subtree_c244889fccb8
+    cover_subtree_2684f740078d
+
+theorem cover_subtree_c0614247fef2 :
+    adaptiveCoverCheck 7 thetaAboveCell000022012100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012100
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012100)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022012100)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022012100)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022012100)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022012100)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012100)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022012100)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022012100)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022012100)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022012100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022012100)) h))
+    cover_subtree_e8b421b5f980
+    cover_subtree_38440f81d562
+
+theorem cover_subtree_31258573aaec :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022012101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022012101))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell000022012101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+        thetaAboveCell000022012101))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell000022012101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+        thetaAboveCell000022012101))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121012020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012020 h)
+        (by
+          have h : (thetaAboveCell0000220121012021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012021 h)
+        (by
+          have h : (thetaAboveCell0000220121012022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012022 h)
+        (by
+          have h : (thetaAboveCell0000220121012023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121012030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012030 h)
+        (by
+          have h : (thetaAboveCell0000220121012031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012031 h)
+        (by
+          have h : (thetaAboveCell0000220121012032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012032 h)
+        (by
+          have h : (thetaAboveCell0000220121012033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012033 h))
+
+theorem cover_subtree_72b7b156e7ee :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022012101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022012101))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022012101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022012101))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022012101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022012101))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121012120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012120 h)
+        (by
+          have h : (thetaAboveCell0000220121012121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012121 h)
+        (by
+          have h : (thetaAboveCell0000220121012122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012122 h)
+        (by
+          have h : (thetaAboveCell0000220121012123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121012130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012130 h)
+        (by
+          have h : (thetaAboveCell0000220121012131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012131 h)
+        (by
+          have h : (thetaAboveCell0000220121012132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012132 h)
+        (by
+          have h : (thetaAboveCell0000220121012133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012133 h))
+
+theorem cover_subtree_49d59b40ac80 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022012101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022012101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121012200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012200 h)
+        (by
+          have h : (thetaAboveCell0000220121012201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012201 h)
+        (by
+          have h : (thetaAboveCell0000220121012202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012202 h)
+        (by
+          have h : (thetaAboveCell0000220121012203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121012210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012210 h)
+        (by
+          have h : (thetaAboveCell0000220121012211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012211 h)
+        (by
+          have h : (thetaAboveCell0000220121012212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012212 h)
+        (by
+          have h : (thetaAboveCell0000220121012213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHL
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121012220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012220 h)
+        (by
+          have h : (thetaAboveCell0000220121012221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012221 h)
+        (by
+          have h : (thetaAboveCell0000220121012222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012222 h)
+        (by
+          have h : (thetaAboveCell0000220121012223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHL
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121012230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012230 h)
+        (by
+          have h : (thetaAboveCell0000220121012231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012231 h)
+        (by
+          have h : (thetaAboveCell0000220121012232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012232 h)
+        (by
+          have h : (thetaAboveCell0000220121012233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012233 h))
+
+theorem cover_subtree_de88d1b7e4d0 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022012101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022012101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121012300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012300 h)
+        (by
+          have h : (thetaAboveCell0000220121012301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012301 h)
+        (by
+          have h : (thetaAboveCell0000220121012302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012302 h)
+        (by
+          have h : (thetaAboveCell0000220121012303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121012310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012310 h)
+        (by
+          have h : (thetaAboveCell0000220121012311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012311 h)
+        (by
+          have h : (thetaAboveCell0000220121012312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012312 h)
+        (by
+          have h : (thetaAboveCell0000220121012313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHL
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121012320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012320 h)
+        (by
+          have h : (thetaAboveCell0000220121012321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012321 h)
+        (by
+          have h : (thetaAboveCell0000220121012322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012322 h)
+        (by
+          have h : (thetaAboveCell0000220121012323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHL
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121012330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012330 h)
+        (by
+          have h : (thetaAboveCell0000220121012331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012331 h)
+        (by
+          have h : (thetaAboveCell0000220121012332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012332 h)
+        (by
+          have h : (thetaAboveCell0000220121012333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121012333 h))
+
+theorem cover_subtree_8fb6927013d7 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022012101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022012101)
+    cover_subtree_31258573aaec
+    cover_subtree_72b7b156e7ee
+    cover_subtree_49d59b40ac80
+    cover_subtree_de88d1b7e4d0
+
+theorem cover_subtree_02e81ee0d1ef :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022012101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022012101))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022012101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022012101))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022012101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022012101))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121013020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013020 h)
+        (by
+          have h : (thetaAboveCell0000220121013021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013021 h)
+        (by
+          have h : (thetaAboveCell0000220121013022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013022 h)
+        (by
+          have h : (thetaAboveCell0000220121013023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121013030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013030 h)
+        (by
+          have h : (thetaAboveCell0000220121013031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013031 h)
+        (by
+          have h : (thetaAboveCell0000220121013032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013032 h)
+        (by
+          have h : (thetaAboveCell0000220121013033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013033 h))
+
+theorem cover_subtree_cc00f203ac8d :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022012101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022012101))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell000022012101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+        thetaAboveCell000022012101))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell000022012101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+        thetaAboveCell000022012101))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121013120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013120 h)
+        (by
+          have h : (thetaAboveCell0000220121013121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013121 h)
+        (by
+          have h : (thetaAboveCell0000220121013122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013122 h)
+        (by
+          have h : (thetaAboveCell0000220121013123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121013130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013130 h)
+        (by
+          have h : (thetaAboveCell0000220121013131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013131 h)
+        (by
+          have h : (thetaAboveCell0000220121013132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013132 h)
+        (by
+          have h : (thetaAboveCell0000220121013133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013133 h))
+
+theorem cover_subtree_d22ca7c15af3 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022012101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022012101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121013200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013200 h)
+        (by
+          have h : (thetaAboveCell0000220121013201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013201 h)
+        (by
+          have h : (thetaAboveCell0000220121013202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013202 h)
+        (by
+          have h : (thetaAboveCell0000220121013203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121013210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013210 h)
+        (by
+          have h : (thetaAboveCell0000220121013211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013211 h)
+        (by
+          have h : (thetaAboveCell0000220121013212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013212 h)
+        (by
+          have h : (thetaAboveCell0000220121013213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHH
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121013220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013220 h)
+        (by
+          have h : (thetaAboveCell0000220121013221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013221 h)
+        (by
+          have h : (thetaAboveCell0000220121013222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013222 h)
+        (by
+          have h : (thetaAboveCell0000220121013223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHH
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121013230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013230 h)
+        (by
+          have h : (thetaAboveCell0000220121013231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013231 h)
+        (by
+          have h : (thetaAboveCell0000220121013232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013232 h)
+        (by
+          have h : (thetaAboveCell0000220121013233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013233 h))
+
+theorem cover_subtree_f04c91fe2c64 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022012101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022012101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121013300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013300 h)
+        (by
+          have h : (thetaAboveCell0000220121013301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013301 h)
+        (by
+          have h : (thetaAboveCell0000220121013302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013302 h)
+        (by
+          have h : (thetaAboveCell0000220121013303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121013310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013310 h)
+        (by
+          have h : (thetaAboveCell0000220121013311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013311 h)
+        (by
+          have h : (thetaAboveCell0000220121013312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013312 h)
+        (by
+          have h : (thetaAboveCell0000220121013313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHH
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121013320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013320 h)
+        (by
+          have h : (thetaAboveCell0000220121013321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013321 h)
+        (by
+          have h : (thetaAboveCell0000220121013322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013322 h)
+        (by
+          have h : (thetaAboveCell0000220121013323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHH
+        thetaAboveCell000022012101)))
+        (by
+          have h : (thetaAboveCell0000220121013330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013330 h)
+        (by
+          have h : (thetaAboveCell0000220121013331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013331 h)
+        (by
+          have h : (thetaAboveCell0000220121013332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013332 h)
+        (by
+          have h : (thetaAboveCell0000220121013333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121013333 h))
+
+theorem cover_subtree_893a2b14ea46 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022012101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022012101)
+    cover_subtree_02e81ee0d1ef
+    cover_subtree_cc00f203ac8d
+    cover_subtree_d22ca7c15af3
+    cover_subtree_f04c91fe2c64
+
+theorem cover_subtree_8bce29d5bf8e :
+    adaptiveCoverCheck 7 thetaAboveCell000022012101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012101
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012101)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022012101)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022012101)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022012101)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022012101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012101)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022012101)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022012101)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022012101)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022012101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022012101)) h))
+    cover_subtree_8fb6927013d7
+    cover_subtree_893a2b14ea46
+
+theorem cover_subtree_042414b8b05d :
+    adaptiveCoverCheck 7 thetaAboveCell000022012102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012102
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012102)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022012102)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022012102)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022012102)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022012102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012102)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022012102)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022012102)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022012102)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022012102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022012102)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022012102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022012102) h)
+    (by
+      have h : ((childHH thetaAboveCell000022012102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022012102) h)
+
+theorem cover_subtree_7648aff1a886 :
+    adaptiveCoverCheck 7 thetaAboveCell000022012103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012103
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012103)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022012103)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022012103)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022012103)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022012103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012103)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022012103)) h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022012103))
+            (by
+              have h : ((childLL (childLH (childLH thetaAboveCell000022012103)))).rejected = true
+                := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+                thetaAboveCell000022012103))) h)
+            (by
+              have h : ((childLH (childLH (childLH thetaAboveCell000022012103)))).rejected = true
+                := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+                thetaAboveCell000022012103))) h)
+            (by
+              have h : ((childHL (childLH (childLH thetaAboveCell000022012103)))).rejected = true
+                := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+                thetaAboveCell000022012103))) h)
+            (by
+              have h : ((childHH (childLH (childLH thetaAboveCell000022012103)))).rejected = true
+                := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+                thetaAboveCell000022012103))) h))
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022012103)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022012103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022012103)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022012103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022012103) h)
+    (by
+      have h : ((childHH thetaAboveCell000022012103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022012103) h)
+
+theorem cover_subtree_c6c6b7e63010 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00002201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00002201)))
+    cover_subtree_c0614247fef2
+    cover_subtree_8bce29d5bf8e
+    cover_subtree_042414b8b05d
+    cover_subtree_7648aff1a886
+
+theorem cover_subtree_7be42b58a89a :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022012110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022012110))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell000022012110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+        thetaAboveCell000022012110))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell000022012110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+        thetaAboveCell000022012110))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121102020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102020 h)
+        (by
+          have h : (thetaAboveCell0000220121102021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102021 h)
+        (by
+          have h : (thetaAboveCell0000220121102022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102022 h)
+        (by
+          have h : (thetaAboveCell0000220121102023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121102030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102030 h)
+        (by
+          have h : (thetaAboveCell0000220121102031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102031 h)
+        (by
+          have h : (thetaAboveCell0000220121102032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102032 h)
+        (by
+          have h : (thetaAboveCell0000220121102033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102033 h))
+
+theorem cover_subtree_b2dbe383f962 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022012110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022012110))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022012110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022012110))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022012110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022012110))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121102120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102120 h)
+        (by
+          have h : (thetaAboveCell0000220121102121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102121 h)
+        (by
+          have h : (thetaAboveCell0000220121102122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102122 h)
+        (by
+          have h : (thetaAboveCell0000220121102123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121102130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102130 h)
+        (by
+          have h : (thetaAboveCell0000220121102131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102131 h)
+        (by
+          have h : (thetaAboveCell0000220121102132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102132 h)
+        (by
+          have h : (thetaAboveCell0000220121102133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102133 h))
+
+theorem cover_subtree_62ab94416880 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022012110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022012110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121102200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102200 h)
+        (by
+          have h : (thetaAboveCell0000220121102201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102201 h)
+        (by
+          have h : (thetaAboveCell0000220121102202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102202 h)
+        (by
+          have h : (thetaAboveCell0000220121102203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121102210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102210 h)
+        (by
+          have h : (thetaAboveCell0000220121102211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102211 h)
+        (by
+          have h : (thetaAboveCell0000220121102212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102212 h)
+        (by
+          have h : (thetaAboveCell0000220121102213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHL
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121102220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102220 h)
+        (by
+          have h : (thetaAboveCell0000220121102221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102221 h)
+        (by
+          have h : (thetaAboveCell0000220121102222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102222 h)
+        (by
+          have h : (thetaAboveCell0000220121102223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHL
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121102230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102230 h)
+        (by
+          have h : (thetaAboveCell0000220121102231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102231 h)
+        (by
+          have h : (thetaAboveCell0000220121102232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102232 h)
+        (by
+          have h : (thetaAboveCell0000220121102233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102233 h))
+
+theorem cover_subtree_55ef04fe757d :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022012110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022012110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121102300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102300 h)
+        (by
+          have h : (thetaAboveCell0000220121102301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102301 h)
+        (by
+          have h : (thetaAboveCell0000220121102302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102302 h)
+        (by
+          have h : (thetaAboveCell0000220121102303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121102310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102310 h)
+        (by
+          have h : (thetaAboveCell0000220121102311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102311 h)
+        (by
+          have h : (thetaAboveCell0000220121102312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102312 h)
+        (by
+          have h : (thetaAboveCell0000220121102313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHL
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121102320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102320 h)
+        (by
+          have h : (thetaAboveCell0000220121102321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102321 h)
+        (by
+          have h : (thetaAboveCell0000220121102322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102322 h)
+        (by
+          have h : (thetaAboveCell0000220121102323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHL
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121102330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102330 h)
+        (by
+          have h : (thetaAboveCell0000220121102331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102331 h)
+        (by
+          have h : (thetaAboveCell0000220121102332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102332 h)
+        (by
+          have h : (thetaAboveCell0000220121102333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121102333 h))
+
+theorem cover_subtree_3e8ada6913b4 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022012110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022012110)
+    cover_subtree_7be42b58a89a
+    cover_subtree_b2dbe383f962
+    cover_subtree_62ab94416880
+    cover_subtree_55ef04fe757d
+
+theorem cover_subtree_9b287e7b4186 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022012110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022012110))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022012110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022012110))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022012110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022012110))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121103020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103020 h)
+        (by
+          have h : (thetaAboveCell0000220121103021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103021 h)
+        (by
+          have h : (thetaAboveCell0000220121103022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103022 h)
+        (by
+          have h : (thetaAboveCell0000220121103023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121103030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103030 h)
+        (by
+          have h : (thetaAboveCell0000220121103031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103031 h)
+        (by
+          have h : (thetaAboveCell0000220121103032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103032 h)
+        (by
+          have h : (thetaAboveCell0000220121103033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103033 h))
+
+theorem cover_subtree_6f0c3876c853 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022012110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022012110))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell000022012110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+        thetaAboveCell000022012110))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell000022012110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+        thetaAboveCell000022012110))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121103120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103120 h)
+        (by
+          have h : (thetaAboveCell0000220121103121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103121 h)
+        (by
+          have h : (thetaAboveCell0000220121103122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103122 h)
+        (by
+          have h : (thetaAboveCell0000220121103123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121103130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103130 h)
+        (by
+          have h : (thetaAboveCell0000220121103131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103131 h)
+        (by
+          have h : (thetaAboveCell0000220121103132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103132 h)
+        (by
+          have h : (thetaAboveCell0000220121103133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103133 h))
+
+theorem cover_subtree_b459e9c4ef11 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022012110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022012110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121103200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103200 h)
+        (by
+          have h : (thetaAboveCell0000220121103201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103201 h)
+        (by
+          have h : (thetaAboveCell0000220121103202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103202 h)
+        (by
+          have h : (thetaAboveCell0000220121103203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121103210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103210 h)
+        (by
+          have h : (thetaAboveCell0000220121103211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103211 h)
+        (by
+          have h : (thetaAboveCell0000220121103212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103212 h)
+        (by
+          have h : (thetaAboveCell0000220121103213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHH
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121103220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103220 h)
+        (by
+          have h : (thetaAboveCell0000220121103221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103221 h)
+        (by
+          have h : (thetaAboveCell0000220121103222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103222 h)
+        (by
+          have h : (thetaAboveCell0000220121103223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHH
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121103230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103230 h)
+        (by
+          have h : (thetaAboveCell0000220121103231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103231 h)
+        (by
+          have h : (thetaAboveCell0000220121103232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103232 h)
+        (by
+          have h : (thetaAboveCell0000220121103233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103233 h))
+
+theorem cover_subtree_59308c01291f :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022012110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022012110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121103300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103300 h)
+        (by
+          have h : (thetaAboveCell0000220121103301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103301 h)
+        (by
+          have h : (thetaAboveCell0000220121103302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103302 h)
+        (by
+          have h : (thetaAboveCell0000220121103303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121103310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103310 h)
+        (by
+          have h : (thetaAboveCell0000220121103311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103311 h)
+        (by
+          have h : (thetaAboveCell0000220121103312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103312 h)
+        (by
+          have h : (thetaAboveCell0000220121103313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHH
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121103320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103320 h)
+        (by
+          have h : (thetaAboveCell0000220121103321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103321 h)
+        (by
+          have h : (thetaAboveCell0000220121103322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103322 h)
+        (by
+          have h : (thetaAboveCell0000220121103323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHH
+        thetaAboveCell000022012110)))
+        (by
+          have h : (thetaAboveCell0000220121103330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103330 h)
+        (by
+          have h : (thetaAboveCell0000220121103331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103331 h)
+        (by
+          have h : (thetaAboveCell0000220121103332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103332 h)
+        (by
+          have h : (thetaAboveCell0000220121103333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121103333 h))
+
+theorem cover_subtree_cc7dfcac2ef6 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022012110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022012110)
+    cover_subtree_9b287e7b4186
+    cover_subtree_6f0c3876c853
+    cover_subtree_b459e9c4ef11
+    cover_subtree_59308c01291f
+
+theorem cover_subtree_4255f05ab59f :
+    adaptiveCoverCheck 7 thetaAboveCell000022012110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012110
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012110)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022012110)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022012110)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022012110)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022012110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012110)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022012110)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022012110)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022012110)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022012110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022012110)) h))
+    cover_subtree_3e8ada6913b4
+    cover_subtree_cc7dfcac2ef6
+
+theorem cover_subtree_2c166d84fe59 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022012111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022012111))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell000022012111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+        thetaAboveCell000022012111))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell000022012111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+        thetaAboveCell000022012111))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121112020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112020 h)
+        (by
+          have h : (thetaAboveCell0000220121112021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112021 h)
+        (by
+          have h : (thetaAboveCell0000220121112022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112022 h)
+        (by
+          have h : (thetaAboveCell0000220121112023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121112030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112030 h)
+        (by
+          have h : (thetaAboveCell0000220121112031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112031 h)
+        (by
+          have h : (thetaAboveCell0000220121112032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112032 h)
+        (by
+          have h : (thetaAboveCell0000220121112033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112033 h))
+
+theorem cover_subtree_d41cf2e18f87 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022012111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022012111))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022012111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022012111))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022012111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022012111))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121112120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112120 h)
+        (by
+          have h : (thetaAboveCell0000220121112121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112121 h)
+        (by
+          have h : (thetaAboveCell0000220121112122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112122 h)
+        (by
+          have h : (thetaAboveCell0000220121112123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121112130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112130 h)
+        (by
+          have h : (thetaAboveCell0000220121112131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112131 h)
+        (by
+          have h : (thetaAboveCell0000220121112132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112132 h)
+        (by
+          have h : (thetaAboveCell0000220121112133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112133 h))
+
+theorem cover_subtree_1fd2ad12dba0 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022012111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022012111))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121112200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112200 h)
+        (by
+          have h : (thetaAboveCell0000220121112201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112201 h)
+        (by
+          have h : (thetaAboveCell0000220121112202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112202 h)
+        (by
+          have h : (thetaAboveCell0000220121112203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121112210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112210 h)
+        (by
+          have h : (thetaAboveCell0000220121112211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112211 h)
+        (by
+          have h : (thetaAboveCell0000220121112212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112212 h)
+        (by
+          have h : (thetaAboveCell0000220121112213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHL
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121112220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112220 h)
+        (by
+          have h : (thetaAboveCell0000220121112221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112221 h)
+        (by
+          have h : (thetaAboveCell0000220121112222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112222 h)
+        (by
+          have h : (thetaAboveCell0000220121112223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHL
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121112230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112230 h)
+        (by
+          have h : (thetaAboveCell0000220121112231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112231 h)
+        (by
+          have h : (thetaAboveCell0000220121112232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112232 h)
+        (by
+          have h : (thetaAboveCell0000220121112233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112233 h))
+
+theorem cover_subtree_88e81007f06f :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022012111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022012111))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121112300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112300 h)
+        (by
+          have h : (thetaAboveCell0000220121112301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112301 h)
+        (by
+          have h : (thetaAboveCell0000220121112302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112302 h)
+        (by
+          have h : (thetaAboveCell0000220121112303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121112310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112310 h)
+        (by
+          have h : (thetaAboveCell0000220121112311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112311 h)
+        (by
+          have h : (thetaAboveCell0000220121112312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112312 h)
+        (by
+          have h : (thetaAboveCell0000220121112313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHL
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121112320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112320 h)
+        (by
+          have h : (thetaAboveCell0000220121112321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112321 h)
+        (by
+          have h : (thetaAboveCell0000220121112322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112322 h)
+        (by
+          have h : (thetaAboveCell0000220121112323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHL
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121112330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112330 h)
+        (by
+          have h : (thetaAboveCell0000220121112331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112331 h)
+        (by
+          have h : (thetaAboveCell0000220121112332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112332 h)
+        (by
+          have h : (thetaAboveCell0000220121112333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121112333 h))
+
+theorem cover_subtree_c6507edf1fb6 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022012111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022012111)
+    cover_subtree_2c166d84fe59
+    cover_subtree_d41cf2e18f87
+    cover_subtree_1fd2ad12dba0
+    cover_subtree_88e81007f06f
+
+theorem cover_subtree_fa22cb64c6e6 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022012111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022012111))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022012111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022012111))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022012111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022012111))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121113020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113020 h)
+        (by
+          have h : (thetaAboveCell0000220121113021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113021 h)
+        (by
+          have h : (thetaAboveCell0000220121113022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113022 h)
+        (by
+          have h : (thetaAboveCell0000220121113023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121113030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113030 h)
+        (by
+          have h : (thetaAboveCell0000220121113031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113031 h)
+        (by
+          have h : (thetaAboveCell0000220121113032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113032 h)
+        (by
+          have h : (thetaAboveCell0000220121113033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113033 h))
+
+theorem cover_subtree_8b0952b6da88 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022012111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022012111))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell000022012111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+        thetaAboveCell000022012111))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell000022012111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+        thetaAboveCell000022012111))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121113120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113120 h)
+        (by
+          have h : (thetaAboveCell0000220121113121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113121 h)
+        (by
+          have h : (thetaAboveCell0000220121113122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113122 h)
+        (by
+          have h : (thetaAboveCell0000220121113123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121113130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113130 h)
+        (by
+          have h : (thetaAboveCell0000220121113131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113131 h)
+        (by
+          have h : (thetaAboveCell0000220121113132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113132 h)
+        (by
+          have h : (thetaAboveCell0000220121113133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113133 h))
+
+theorem cover_subtree_549042ebc77b :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022012111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022012111))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121113200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113200 h)
+        (by
+          have h : (thetaAboveCell0000220121113201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113201 h)
+        (by
+          have h : (thetaAboveCell0000220121113202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113202 h)
+        (by
+          have h : (thetaAboveCell0000220121113203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121113210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113210 h)
+        (by
+          have h : (thetaAboveCell0000220121113211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113211 h)
+        (by
+          have h : (thetaAboveCell0000220121113212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113212 h)
+        (by
+          have h : (thetaAboveCell0000220121113213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHH
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121113220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113220 h)
+        (by
+          have h : (thetaAboveCell0000220121113221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113221 h)
+        (by
+          have h : (thetaAboveCell0000220121113222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113222 h)
+        (by
+          have h : (thetaAboveCell0000220121113223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHH
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121113230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113230 h)
+        (by
+          have h : (thetaAboveCell0000220121113231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113231 h)
+        (by
+          have h : (thetaAboveCell0000220121113232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113232 h)
+        (by
+          have h : (thetaAboveCell0000220121113233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113233 h))
+
+theorem cover_subtree_de6e3b296ec1 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022012111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022012111))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121113300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113300 h)
+        (by
+          have h : (thetaAboveCell0000220121113301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113301 h)
+        (by
+          have h : (thetaAboveCell0000220121113302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113302 h)
+        (by
+          have h : (thetaAboveCell0000220121113303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121113310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113310 h)
+        (by
+          have h : (thetaAboveCell0000220121113311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113311 h)
+        (by
+          have h : (thetaAboveCell0000220121113312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113312 h)
+        (by
+          have h : (thetaAboveCell0000220121113313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHH
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121113320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113320 h)
+        (by
+          have h : (thetaAboveCell0000220121113321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113321 h)
+        (by
+          have h : (thetaAboveCell0000220121113322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113322 h)
+        (by
+          have h : (thetaAboveCell0000220121113323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHH
+        thetaAboveCell000022012111)))
+        (by
+          have h : (thetaAboveCell0000220121113330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113330 h)
+        (by
+          have h : (thetaAboveCell0000220121113331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113331 h)
+        (by
+          have h : (thetaAboveCell0000220121113332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113332 h)
+        (by
+          have h : (thetaAboveCell0000220121113333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220121113333 h))
+
+theorem cover_subtree_0f2e720f6017 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022012111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022012111)
+    cover_subtree_fa22cb64c6e6
+    cover_subtree_8b0952b6da88
+    cover_subtree_549042ebc77b
+    cover_subtree_de6e3b296ec1
+
+theorem cover_subtree_35e94d0e7310 :
+    adaptiveCoverCheck 7 thetaAboveCell000022012111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012111
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012111)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022012111)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022012111)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022012111)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022012111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012111)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022012111)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022012111)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022012111)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022012111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022012111)) h))
+    cover_subtree_c6507edf1fb6
+    cover_subtree_0f2e720f6017
+
+theorem cover_subtree_65f4b19fe529 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022012112) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012112)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022012112))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022012112))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022012112))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022012112))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022012112))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022012112))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022012112))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022012112))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022012112))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022012112))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022012112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022012112)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022012112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022012112)) h)
+
+theorem cover_subtree_334b61157880 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022012112) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012112)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022012112))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022012112))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022012112))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022012112))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022012112))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022012112))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022012112))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022012112))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022012112))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022012112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022012112))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022012112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022012112)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022012112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022012112)) h)
+
+theorem cover_subtree_681fbeb68abf :
+    adaptiveCoverCheck 7 thetaAboveCell000022012112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012112
+    cover_subtree_65f4b19fe529
+    cover_subtree_334b61157880
+    (by
+      have h : ((childHL thetaAboveCell000022012112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022012112) h)
+    (by
+      have h : ((childHH thetaAboveCell000022012112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022012112) h)
+
+theorem cover_subtree_2f0ab83958c9 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022012113) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022012113)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022012113))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022012113))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022012113))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022012113))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022012113))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022012113))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022012113))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022012113))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022012113))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022012113))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022012113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022012113)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022012113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022012113)) h)
+
+theorem cover_subtree_4675766b5fc3 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022012113) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022012113)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022012113))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022012113))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022012113))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022012113))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022012113))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022012113))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022012113))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022012113))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022012113))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022012113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022012113))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022012113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022012113)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022012113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022012113)) h)
+
+theorem cover_subtree_9e9d455fd0db :
+    adaptiveCoverCheck 7 thetaAboveCell000022012113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022012113
+    cover_subtree_2f0ab83958c9
+    cover_subtree_4675766b5fc3
+    (by
+      have h : ((childHL thetaAboveCell000022012113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022012113) h)
+    (by
+      have h : ((childHH thetaAboveCell000022012113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022012113) h)
+
+theorem cover_subtree_ba2ddc618064 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00002201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00002201)))
+    cover_subtree_4255f05ab59f
+    cover_subtree_35e94d0e7310
+    cover_subtree_681fbeb68abf
+    cover_subtree_9e9d455fd0db
+
+theorem e24KC2ThetaAboveLeaf0000220121 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00002201))
+    cover_subtree_c6c6b7e63010
+    cover_subtree_ba2ddc618064
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022012120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012120 h)
+        (by
+          have h : (thetaAboveCell000022012121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012121 h)
+        (by
+          have h : (thetaAboveCell000022012122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012122 h)
+        (by
+          have h : (thetaAboveCell000022012123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022012130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012130 h)
+        (by
+          have h : (thetaAboveCell000022012131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012131 h)
+        (by
+          have h : (thetaAboveCell000022012132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012132 h)
+        (by
+          have h : (thetaAboveCell000022012133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022012133 h))
+theorem e24KC2ThetaAboveLeaf0000220122 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00002201))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00002201))) h)
+theorem e24KC2ThetaAboveLeaf0000220123 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00002201))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00002201))) h)
+theorem cover_subtree_cb54dc4a2aa0 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022013000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022013000))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell000022013000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+        thetaAboveCell000022013000))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell000022013000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+        thetaAboveCell000022013000))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130002020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002020 h)
+        (by
+          have h : (thetaAboveCell0000220130002021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002021 h)
+        (by
+          have h : (thetaAboveCell0000220130002022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002022 h)
+        (by
+          have h : (thetaAboveCell0000220130002023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130002030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002030 h)
+        (by
+          have h : (thetaAboveCell0000220130002031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002031 h)
+        (by
+          have h : (thetaAboveCell0000220130002032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002032 h)
+        (by
+          have h : (thetaAboveCell0000220130002033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002033 h))
+
+theorem cover_subtree_348d77482bf3 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022013000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022013000))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022013000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022013000))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022013000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022013000))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130002120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002120 h)
+        (by
+          have h : (thetaAboveCell0000220130002121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002121 h)
+        (by
+          have h : (thetaAboveCell0000220130002122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002122 h)
+        (by
+          have h : (thetaAboveCell0000220130002123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130002130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002130 h)
+        (by
+          have h : (thetaAboveCell0000220130002131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002131 h)
+        (by
+          have h : (thetaAboveCell0000220130002132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002132 h)
+        (by
+          have h : (thetaAboveCell0000220130002133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002133 h))
+
+theorem cover_subtree_cc712ad349df :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022013000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022013000))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130002200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002200 h)
+        (by
+          have h : (thetaAboveCell0000220130002201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002201 h)
+        (by
+          have h : (thetaAboveCell0000220130002202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002202 h)
+        (by
+          have h : (thetaAboveCell0000220130002203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130002210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002210 h)
+        (by
+          have h : (thetaAboveCell0000220130002211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002211 h)
+        (by
+          have h : (thetaAboveCell0000220130002212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002212 h)
+        (by
+          have h : (thetaAboveCell0000220130002213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHL
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130002220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002220 h)
+        (by
+          have h : (thetaAboveCell0000220130002221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002221 h)
+        (by
+          have h : (thetaAboveCell0000220130002222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002222 h)
+        (by
+          have h : (thetaAboveCell0000220130002223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHL
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130002230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002230 h)
+        (by
+          have h : (thetaAboveCell0000220130002231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002231 h)
+        (by
+          have h : (thetaAboveCell0000220130002232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002232 h)
+        (by
+          have h : (thetaAboveCell0000220130002233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002233 h))
+
+theorem cover_subtree_791e2ddd0c41 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022013000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022013000))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130002300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002300 h)
+        (by
+          have h : (thetaAboveCell0000220130002301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002301 h)
+        (by
+          have h : (thetaAboveCell0000220130002302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002302 h)
+        (by
+          have h : (thetaAboveCell0000220130002303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130002310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002310 h)
+        (by
+          have h : (thetaAboveCell0000220130002311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002311 h)
+        (by
+          have h : (thetaAboveCell0000220130002312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002312 h)
+        (by
+          have h : (thetaAboveCell0000220130002313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHL
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130002320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002320 h)
+        (by
+          have h : (thetaAboveCell0000220130002321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002321 h)
+        (by
+          have h : (thetaAboveCell0000220130002322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002322 h)
+        (by
+          have h : (thetaAboveCell0000220130002323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHL
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130002330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002330 h)
+        (by
+          have h : (thetaAboveCell0000220130002331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002331 h)
+        (by
+          have h : (thetaAboveCell0000220130002332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002332 h)
+        (by
+          have h : (thetaAboveCell0000220130002333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130002333 h))
+
+theorem cover_subtree_3a2f1fd548a5 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022013000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022013000)
+    cover_subtree_cb54dc4a2aa0
+    cover_subtree_348d77482bf3
+    cover_subtree_cc712ad349df
+    cover_subtree_791e2ddd0c41
+
+theorem cover_subtree_bd2306545933 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022013000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022013000))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022013000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022013000))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022013000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022013000))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130003020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003020 h)
+        (by
+          have h : (thetaAboveCell0000220130003021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003021 h)
+        (by
+          have h : (thetaAboveCell0000220130003022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003022 h)
+        (by
+          have h : (thetaAboveCell0000220130003023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130003030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003030 h)
+        (by
+          have h : (thetaAboveCell0000220130003031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003031 h)
+        (by
+          have h : (thetaAboveCell0000220130003032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003032 h)
+        (by
+          have h : (thetaAboveCell0000220130003033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003033 h))
+
+theorem cover_subtree_65fba8c7a135 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022013000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022013000))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell000022013000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+        thetaAboveCell000022013000))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell000022013000)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+        thetaAboveCell000022013000))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130003120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003120 h)
+        (by
+          have h : (thetaAboveCell0000220130003121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003121 h)
+        (by
+          have h : (thetaAboveCell0000220130003122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003122 h)
+        (by
+          have h : (thetaAboveCell0000220130003123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130003130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003130 h)
+        (by
+          have h : (thetaAboveCell0000220130003131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003131 h)
+        (by
+          have h : (thetaAboveCell0000220130003132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003132 h)
+        (by
+          have h : (thetaAboveCell0000220130003133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003133 h))
+
+theorem cover_subtree_f63aaf9046af :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022013000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022013000))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130003200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003200 h)
+        (by
+          have h : (thetaAboveCell0000220130003201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003201 h)
+        (by
+          have h : (thetaAboveCell0000220130003202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003202 h)
+        (by
+          have h : (thetaAboveCell0000220130003203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130003210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003210 h)
+        (by
+          have h : (thetaAboveCell0000220130003211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003211 h)
+        (by
+          have h : (thetaAboveCell0000220130003212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003212 h)
+        (by
+          have h : (thetaAboveCell0000220130003213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHH
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130003220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003220 h)
+        (by
+          have h : (thetaAboveCell0000220130003221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003221 h)
+        (by
+          have h : (thetaAboveCell0000220130003222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003222 h)
+        (by
+          have h : (thetaAboveCell0000220130003223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHH
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130003230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003230 h)
+        (by
+          have h : (thetaAboveCell0000220130003231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003231 h)
+        (by
+          have h : (thetaAboveCell0000220130003232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003232 h)
+        (by
+          have h : (thetaAboveCell0000220130003233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003233 h))
+
+theorem cover_subtree_dff62e50df3b :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022013000)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022013000))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130003300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003300 h)
+        (by
+          have h : (thetaAboveCell0000220130003301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003301 h)
+        (by
+          have h : (thetaAboveCell0000220130003302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003302 h)
+        (by
+          have h : (thetaAboveCell0000220130003303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130003310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003310 h)
+        (by
+          have h : (thetaAboveCell0000220130003311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003311 h)
+        (by
+          have h : (thetaAboveCell0000220130003312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003312 h)
+        (by
+          have h : (thetaAboveCell0000220130003313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHH
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130003320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003320 h)
+        (by
+          have h : (thetaAboveCell0000220130003321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003321 h)
+        (by
+          have h : (thetaAboveCell0000220130003322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003322 h)
+        (by
+          have h : (thetaAboveCell0000220130003323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHH
+        thetaAboveCell000022013000)))
+        (by
+          have h : (thetaAboveCell0000220130003330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003330 h)
+        (by
+          have h : (thetaAboveCell0000220130003331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003331 h)
+        (by
+          have h : (thetaAboveCell0000220130003332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003332 h)
+        (by
+          have h : (thetaAboveCell0000220130003333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130003333 h))
+
+theorem cover_subtree_c954080a5fc2 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022013000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022013000)
+    cover_subtree_bd2306545933
+    cover_subtree_65fba8c7a135
+    cover_subtree_f63aaf9046af
+    cover_subtree_dff62e50df3b
+
+theorem cover_subtree_701e722ef144 :
+    adaptiveCoverCheck 7 thetaAboveCell000022013000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013000
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013000)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022013000)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022013000)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022013000)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022013000)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013000)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022013000)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022013000)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022013000)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022013000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022013000)) h))
+    cover_subtree_3a2f1fd548a5
+    cover_subtree_c954080a5fc2
+
+theorem cover_subtree_1812a1843696 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022013001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022013001))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell000022013001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+        thetaAboveCell000022013001))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell000022013001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+        thetaAboveCell000022013001))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130012020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012020 h)
+        (by
+          have h : (thetaAboveCell0000220130012021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012021 h)
+        (by
+          have h : (thetaAboveCell0000220130012022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012022 h)
+        (by
+          have h : (thetaAboveCell0000220130012023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130012030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012030 h)
+        (by
+          have h : (thetaAboveCell0000220130012031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012031 h)
+        (by
+          have h : (thetaAboveCell0000220130012032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012032 h)
+        (by
+          have h : (thetaAboveCell0000220130012033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012033 h))
+
+theorem cover_subtree_422793f2f2c6 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022013001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022013001))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022013001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022013001))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022013001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022013001))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130012120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012120 h)
+        (by
+          have h : (thetaAboveCell0000220130012121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012121 h)
+        (by
+          have h : (thetaAboveCell0000220130012122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012122 h)
+        (by
+          have h : (thetaAboveCell0000220130012123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130012130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012130 h)
+        (by
+          have h : (thetaAboveCell0000220130012131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012131 h)
+        (by
+          have h : (thetaAboveCell0000220130012132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012132 h)
+        (by
+          have h : (thetaAboveCell0000220130012133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012133 h))
+
+theorem cover_subtree_386e2b83e45a :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022013001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022013001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130012200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012200 h)
+        (by
+          have h : (thetaAboveCell0000220130012201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012201 h)
+        (by
+          have h : (thetaAboveCell0000220130012202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012202 h)
+        (by
+          have h : (thetaAboveCell0000220130012203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130012210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012210 h)
+        (by
+          have h : (thetaAboveCell0000220130012211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012211 h)
+        (by
+          have h : (thetaAboveCell0000220130012212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012212 h)
+        (by
+          have h : (thetaAboveCell0000220130012213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHL
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130012220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012220 h)
+        (by
+          have h : (thetaAboveCell0000220130012221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012221 h)
+        (by
+          have h : (thetaAboveCell0000220130012222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012222 h)
+        (by
+          have h : (thetaAboveCell0000220130012223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHL
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130012230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012230 h)
+        (by
+          have h : (thetaAboveCell0000220130012231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012231 h)
+        (by
+          have h : (thetaAboveCell0000220130012232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012232 h)
+        (by
+          have h : (thetaAboveCell0000220130012233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012233 h))
+
+theorem cover_subtree_37513ddf0672 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022013001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022013001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130012300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012300 h)
+        (by
+          have h : (thetaAboveCell0000220130012301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012301 h)
+        (by
+          have h : (thetaAboveCell0000220130012302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012302 h)
+        (by
+          have h : (thetaAboveCell0000220130012303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130012310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012310 h)
+        (by
+          have h : (thetaAboveCell0000220130012311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012311 h)
+        (by
+          have h : (thetaAboveCell0000220130012312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012312 h)
+        (by
+          have h : (thetaAboveCell0000220130012313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHL
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130012320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012320 h)
+        (by
+          have h : (thetaAboveCell0000220130012321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012321 h)
+        (by
+          have h : (thetaAboveCell0000220130012322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012322 h)
+        (by
+          have h : (thetaAboveCell0000220130012323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHL
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130012330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012330 h)
+        (by
+          have h : (thetaAboveCell0000220130012331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012331 h)
+        (by
+          have h : (thetaAboveCell0000220130012332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012332 h)
+        (by
+          have h : (thetaAboveCell0000220130012333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130012333 h))
+
+theorem cover_subtree_6d7b5fde1ce9 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022013001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022013001)
+    cover_subtree_1812a1843696
+    cover_subtree_422793f2f2c6
+    cover_subtree_386e2b83e45a
+    cover_subtree_37513ddf0672
+
+theorem cover_subtree_dc89c8b48c1d :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022013001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022013001))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022013001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022013001))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022013001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022013001))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130013020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013020 h)
+        (by
+          have h : (thetaAboveCell0000220130013021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013021 h)
+        (by
+          have h : (thetaAboveCell0000220130013022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013022 h)
+        (by
+          have h : (thetaAboveCell0000220130013023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130013030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013030 h)
+        (by
+          have h : (thetaAboveCell0000220130013031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013031 h)
+        (by
+          have h : (thetaAboveCell0000220130013032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013032 h)
+        (by
+          have h : (thetaAboveCell0000220130013033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013033 h))
+
+theorem cover_subtree_b7e0401eaa5c :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022013001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022013001))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell000022013001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+        thetaAboveCell000022013001))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell000022013001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+        thetaAboveCell000022013001))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130013120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013120 h)
+        (by
+          have h : (thetaAboveCell0000220130013121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013121 h)
+        (by
+          have h : (thetaAboveCell0000220130013122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013122 h)
+        (by
+          have h : (thetaAboveCell0000220130013123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130013130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013130 h)
+        (by
+          have h : (thetaAboveCell0000220130013131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013131 h)
+        (by
+          have h : (thetaAboveCell0000220130013132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013132 h)
+        (by
+          have h : (thetaAboveCell0000220130013133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013133 h))
+
+theorem cover_subtree_3178ee01a645 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022013001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022013001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130013200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013200 h)
+        (by
+          have h : (thetaAboveCell0000220130013201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013201 h)
+        (by
+          have h : (thetaAboveCell0000220130013202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013202 h)
+        (by
+          have h : (thetaAboveCell0000220130013203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130013210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013210 h)
+        (by
+          have h : (thetaAboveCell0000220130013211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013211 h)
+        (by
+          have h : (thetaAboveCell0000220130013212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013212 h)
+        (by
+          have h : (thetaAboveCell0000220130013213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHH
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130013220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013220 h)
+        (by
+          have h : (thetaAboveCell0000220130013221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013221 h)
+        (by
+          have h : (thetaAboveCell0000220130013222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013222 h)
+        (by
+          have h : (thetaAboveCell0000220130013223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHH
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130013230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013230 h)
+        (by
+          have h : (thetaAboveCell0000220130013231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013231 h)
+        (by
+          have h : (thetaAboveCell0000220130013232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013232 h)
+        (by
+          have h : (thetaAboveCell0000220130013233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013233 h))
+
+theorem cover_subtree_2842986097c5 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022013001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022013001))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130013300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013300 h)
+        (by
+          have h : (thetaAboveCell0000220130013301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013301 h)
+        (by
+          have h : (thetaAboveCell0000220130013302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013302 h)
+        (by
+          have h : (thetaAboveCell0000220130013303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130013310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013310 h)
+        (by
+          have h : (thetaAboveCell0000220130013311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013311 h)
+        (by
+          have h : (thetaAboveCell0000220130013312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013312 h)
+        (by
+          have h : (thetaAboveCell0000220130013313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHH
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130013320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013320 h)
+        (by
+          have h : (thetaAboveCell0000220130013321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013321 h)
+        (by
+          have h : (thetaAboveCell0000220130013322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013322 h)
+        (by
+          have h : (thetaAboveCell0000220130013323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHH
+        thetaAboveCell000022013001)))
+        (by
+          have h : (thetaAboveCell0000220130013330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013330 h)
+        (by
+          have h : (thetaAboveCell0000220130013331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013331 h)
+        (by
+          have h : (thetaAboveCell0000220130013332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013332 h)
+        (by
+          have h : (thetaAboveCell0000220130013333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130013333 h))
+
+theorem cover_subtree_0385d11aabba :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022013001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022013001)
+    cover_subtree_dc89c8b48c1d
+    cover_subtree_b7e0401eaa5c
+    cover_subtree_3178ee01a645
+    cover_subtree_2842986097c5
+
+theorem cover_subtree_f1411005533f :
+    adaptiveCoverCheck 7 thetaAboveCell000022013001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013001
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013001)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022013001)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022013001)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022013001)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022013001)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013001)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022013001)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022013001)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022013001)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022013001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022013001)) h))
+    cover_subtree_6d7b5fde1ce9
+    cover_subtree_0385d11aabba
+
+theorem cover_subtree_1032efc16567 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022013002) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013002)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022013002))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022013002))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022013002))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022013002))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022013002))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022013002))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022013002))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022013002))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022013002))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022013002))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022013002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022013002)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022013002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022013002)) h)
+
+theorem cover_subtree_9fa6122a45d4 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022013002) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013002)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022013002))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022013002))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022013002))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022013002))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022013002))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022013002))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022013002))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022013002))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022013002))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022013002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022013002))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022013002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022013002)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022013002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022013002)) h)
+
+theorem cover_subtree_144d92380fd4 :
+    adaptiveCoverCheck 7 thetaAboveCell000022013002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013002
+    cover_subtree_1032efc16567
+    cover_subtree_9fa6122a45d4
+    (by
+      have h : ((childHL thetaAboveCell000022013002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022013002) h)
+    (by
+      have h : ((childHH thetaAboveCell000022013002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022013002) h)
+
+theorem cover_subtree_fd47657a07ff :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022013003) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013003)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022013003))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022013003))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022013003))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022013003))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022013003))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022013003))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022013003))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022013003))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022013003))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022013003))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022013003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022013003)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022013003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022013003)) h)
+
+theorem cover_subtree_e586a2e528dc :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022013003) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013003)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022013003))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022013003))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022013003))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022013003))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022013003))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022013003))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022013003))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022013003))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022013003))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022013003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022013003))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022013003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022013003)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022013003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022013003)) h)
+
+theorem cover_subtree_b0c1a95db5a1 :
+    adaptiveCoverCheck 7 thetaAboveCell000022013003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013003
+    cover_subtree_fd47657a07ff
+    cover_subtree_e586a2e528dc
+    (by
+      have h : ((childHL thetaAboveCell000022013003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022013003) h)
+    (by
+      have h : ((childHH thetaAboveCell000022013003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022013003) h)
+
+theorem cover_subtree_deb83366e758 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00002201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00002201)))
+    cover_subtree_701e722ef144
+    cover_subtree_f1411005533f
+    cover_subtree_144d92380fd4
+    cover_subtree_b0c1a95db5a1
+
+theorem cover_subtree_39d8fa6eaebd :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022013010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022013010))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell000022013010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+        thetaAboveCell000022013010))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell000022013010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+        thetaAboveCell000022013010))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130102020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102020 h)
+        (by
+          have h : (thetaAboveCell0000220130102021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102021 h)
+        (by
+          have h : (thetaAboveCell0000220130102022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102022 h)
+        (by
+          have h : (thetaAboveCell0000220130102023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130102030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102030 h)
+        (by
+          have h : (thetaAboveCell0000220130102031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102031 h)
+        (by
+          have h : (thetaAboveCell0000220130102032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102032 h)
+        (by
+          have h : (thetaAboveCell0000220130102033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102033 h))
+
+theorem cover_subtree_c9870a8382fd :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022013010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022013010))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022013010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022013010))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022013010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022013010))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130102120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102120 h)
+        (by
+          have h : (thetaAboveCell0000220130102121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102121 h)
+        (by
+          have h : (thetaAboveCell0000220130102122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102122 h)
+        (by
+          have h : (thetaAboveCell0000220130102123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130102130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102130 h)
+        (by
+          have h : (thetaAboveCell0000220130102131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102131 h)
+        (by
+          have h : (thetaAboveCell0000220130102132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102132 h)
+        (by
+          have h : (thetaAboveCell0000220130102133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102133 h))
+
+theorem cover_subtree_9ebf3cbb8048 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022013010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022013010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130102200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102200 h)
+        (by
+          have h : (thetaAboveCell0000220130102201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102201 h)
+        (by
+          have h : (thetaAboveCell0000220130102202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102202 h)
+        (by
+          have h : (thetaAboveCell0000220130102203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130102210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102210 h)
+        (by
+          have h : (thetaAboveCell0000220130102211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102211 h)
+        (by
+          have h : (thetaAboveCell0000220130102212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102212 h)
+        (by
+          have h : (thetaAboveCell0000220130102213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHL
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130102220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102220 h)
+        (by
+          have h : (thetaAboveCell0000220130102221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102221 h)
+        (by
+          have h : (thetaAboveCell0000220130102222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102222 h)
+        (by
+          have h : (thetaAboveCell0000220130102223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHL
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130102230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102230 h)
+        (by
+          have h : (thetaAboveCell0000220130102231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102231 h)
+        (by
+          have h : (thetaAboveCell0000220130102232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102232 h)
+        (by
+          have h : (thetaAboveCell0000220130102233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102233 h))
+
+theorem cover_subtree_5943bf03a25d :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022013010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022013010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130102300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102300 h)
+        (by
+          have h : (thetaAboveCell0000220130102301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102301 h)
+        (by
+          have h : (thetaAboveCell0000220130102302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102302 h)
+        (by
+          have h : (thetaAboveCell0000220130102303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130102310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102310 h)
+        (by
+          have h : (thetaAboveCell0000220130102311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102311 h)
+        (by
+          have h : (thetaAboveCell0000220130102312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102312 h)
+        (by
+          have h : (thetaAboveCell0000220130102313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHL
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130102320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102320 h)
+        (by
+          have h : (thetaAboveCell0000220130102321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102321 h)
+        (by
+          have h : (thetaAboveCell0000220130102322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102322 h)
+        (by
+          have h : (thetaAboveCell0000220130102323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHL
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130102330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102330 h)
+        (by
+          have h : (thetaAboveCell0000220130102331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102331 h)
+        (by
+          have h : (thetaAboveCell0000220130102332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102332 h)
+        (by
+          have h : (thetaAboveCell0000220130102333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130102333 h))
+
+theorem cover_subtree_6b97e89b4d72 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022013010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022013010)
+    cover_subtree_39d8fa6eaebd
+    cover_subtree_c9870a8382fd
+    cover_subtree_9ebf3cbb8048
+    cover_subtree_5943bf03a25d
+
+theorem cover_subtree_b9a006cc60d2 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022013010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022013010))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022013010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022013010))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022013010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022013010))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130103020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103020 h)
+        (by
+          have h : (thetaAboveCell0000220130103021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103021 h)
+        (by
+          have h : (thetaAboveCell0000220130103022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103022 h)
+        (by
+          have h : (thetaAboveCell0000220130103023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130103030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103030 h)
+        (by
+          have h : (thetaAboveCell0000220130103031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103031 h)
+        (by
+          have h : (thetaAboveCell0000220130103032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103032 h)
+        (by
+          have h : (thetaAboveCell0000220130103033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103033 h))
+
+theorem cover_subtree_a77e4838b8e6 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022013010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022013010))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell000022013010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+        thetaAboveCell000022013010))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell000022013010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+        thetaAboveCell000022013010))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130103120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103120 h)
+        (by
+          have h : (thetaAboveCell0000220130103121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103121 h)
+        (by
+          have h : (thetaAboveCell0000220130103122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103122 h)
+        (by
+          have h : (thetaAboveCell0000220130103123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130103130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103130 h)
+        (by
+          have h : (thetaAboveCell0000220130103131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103131 h)
+        (by
+          have h : (thetaAboveCell0000220130103132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103132 h)
+        (by
+          have h : (thetaAboveCell0000220130103133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103133 h))
+
+theorem cover_subtree_c5585ab205b2 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022013010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022013010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130103200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103200 h)
+        (by
+          have h : (thetaAboveCell0000220130103201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103201 h)
+        (by
+          have h : (thetaAboveCell0000220130103202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103202 h)
+        (by
+          have h : (thetaAboveCell0000220130103203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130103210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103210 h)
+        (by
+          have h : (thetaAboveCell0000220130103211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103211 h)
+        (by
+          have h : (thetaAboveCell0000220130103212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103212 h)
+        (by
+          have h : (thetaAboveCell0000220130103213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHH
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130103220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103220 h)
+        (by
+          have h : (thetaAboveCell0000220130103221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103221 h)
+        (by
+          have h : (thetaAboveCell0000220130103222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103222 h)
+        (by
+          have h : (thetaAboveCell0000220130103223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHH
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130103230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103230 h)
+        (by
+          have h : (thetaAboveCell0000220130103231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103231 h)
+        (by
+          have h : (thetaAboveCell0000220130103232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103232 h)
+        (by
+          have h : (thetaAboveCell0000220130103233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103233 h))
+
+theorem cover_subtree_0a8fc7348d19 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022013010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022013010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130103300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103300 h)
+        (by
+          have h : (thetaAboveCell0000220130103301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103301 h)
+        (by
+          have h : (thetaAboveCell0000220130103302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103302 h)
+        (by
+          have h : (thetaAboveCell0000220130103303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130103310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103310 h)
+        (by
+          have h : (thetaAboveCell0000220130103311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103311 h)
+        (by
+          have h : (thetaAboveCell0000220130103312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103312 h)
+        (by
+          have h : (thetaAboveCell0000220130103313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHH
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130103320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103320 h)
+        (by
+          have h : (thetaAboveCell0000220130103321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103321 h)
+        (by
+          have h : (thetaAboveCell0000220130103322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103322 h)
+        (by
+          have h : (thetaAboveCell0000220130103323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHH
+        thetaAboveCell000022013010)))
+        (by
+          have h : (thetaAboveCell0000220130103330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103330 h)
+        (by
+          have h : (thetaAboveCell0000220130103331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103331 h)
+        (by
+          have h : (thetaAboveCell0000220130103332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103332 h)
+        (by
+          have h : (thetaAboveCell0000220130103333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130103333 h))
+
+theorem cover_subtree_dffa22b1c9a1 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022013010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022013010)
+    cover_subtree_b9a006cc60d2
+    cover_subtree_a77e4838b8e6
+    cover_subtree_c5585ab205b2
+    cover_subtree_0a8fc7348d19
+
+theorem cover_subtree_3a257d8444b0 :
+    adaptiveCoverCheck 7 thetaAboveCell000022013010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013010
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013010)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022013010)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022013010)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022013010)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022013010)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013010)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022013010)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022013010)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022013010)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022013010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022013010)) h))
+    cover_subtree_6b97e89b4d72
+    cover_subtree_dffa22b1c9a1
+
+theorem cover_subtree_0ba56921169a :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022013011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022013011))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell000022013011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+        thetaAboveCell000022013011))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell000022013011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+        thetaAboveCell000022013011))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130112020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112020 h)
+        (by
+          have h : (thetaAboveCell0000220130112021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112021 h)
+        (by
+          have h : (thetaAboveCell0000220130112022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112022 h)
+        (by
+          have h : (thetaAboveCell0000220130112023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130112030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112030 h)
+        (by
+          have h : (thetaAboveCell0000220130112031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112031 h)
+        (by
+          have h : (thetaAboveCell0000220130112032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112032 h)
+        (by
+          have h : (thetaAboveCell0000220130112033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112033 h))
+
+theorem cover_subtree_7969bce7dc99 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022013011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022013011))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022013011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022013011))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022013011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022013011))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130112120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112120 h)
+        (by
+          have h : (thetaAboveCell0000220130112121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112121 h)
+        (by
+          have h : (thetaAboveCell0000220130112122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112122 h)
+        (by
+          have h : (thetaAboveCell0000220130112123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130112130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112130 h)
+        (by
+          have h : (thetaAboveCell0000220130112131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112131 h)
+        (by
+          have h : (thetaAboveCell0000220130112132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112132 h)
+        (by
+          have h : (thetaAboveCell0000220130112133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112133 h))
+
+theorem cover_subtree_6a708e0151cf :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022013011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022013011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130112200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112200 h)
+        (by
+          have h : (thetaAboveCell0000220130112201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112201 h)
+        (by
+          have h : (thetaAboveCell0000220130112202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112202 h)
+        (by
+          have h : (thetaAboveCell0000220130112203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130112210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112210 h)
+        (by
+          have h : (thetaAboveCell0000220130112211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112211 h)
+        (by
+          have h : (thetaAboveCell0000220130112212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112212 h)
+        (by
+          have h : (thetaAboveCell0000220130112213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHL
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130112220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112220 h)
+        (by
+          have h : (thetaAboveCell0000220130112221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112221 h)
+        (by
+          have h : (thetaAboveCell0000220130112222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112222 h)
+        (by
+          have h : (thetaAboveCell0000220130112223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHL
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130112230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112230 h)
+        (by
+          have h : (thetaAboveCell0000220130112231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112231 h)
+        (by
+          have h : (thetaAboveCell0000220130112232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112232 h)
+        (by
+          have h : (thetaAboveCell0000220130112233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112233 h))
+
+theorem cover_subtree_8de093d065af :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022013011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022013011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130112300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112300 h)
+        (by
+          have h : (thetaAboveCell0000220130112301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112301 h)
+        (by
+          have h : (thetaAboveCell0000220130112302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112302 h)
+        (by
+          have h : (thetaAboveCell0000220130112303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130112310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112310 h)
+        (by
+          have h : (thetaAboveCell0000220130112311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112311 h)
+        (by
+          have h : (thetaAboveCell0000220130112312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112312 h)
+        (by
+          have h : (thetaAboveCell0000220130112313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHL
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130112320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112320 h)
+        (by
+          have h : (thetaAboveCell0000220130112321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112321 h)
+        (by
+          have h : (thetaAboveCell0000220130112322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112322 h)
+        (by
+          have h : (thetaAboveCell0000220130112323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH (childHL
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130112330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112330 h)
+        (by
+          have h : (thetaAboveCell0000220130112331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112331 h)
+        (by
+          have h : (thetaAboveCell0000220130112332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112332 h)
+        (by
+          have h : (thetaAboveCell0000220130112333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130112333 h))
+
+theorem cover_subtree_03f7c5873741 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022013011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022013011)
+    cover_subtree_0ba56921169a
+    cover_subtree_7969bce7dc99
+    cover_subtree_6a708e0151cf
+    cover_subtree_8de093d065af
+
+theorem cover_subtree_b3bbbe006bfc :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022013011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022013011))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022013011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022013011))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022013011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022013011))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130113020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113020 h)
+        (by
+          have h : (thetaAboveCell0000220130113021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113021 h)
+        (by
+          have h : (thetaAboveCell0000220130113022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113022 h)
+        (by
+          have h : (thetaAboveCell0000220130113023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130113030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113030 h)
+        (by
+          have h : (thetaAboveCell0000220130113031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113031 h)
+        (by
+          have h : (thetaAboveCell0000220130113032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113032 h)
+        (by
+          have h : (thetaAboveCell0000220130113033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113033 h))
+
+theorem cover_subtree_10ce55f7ce75 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022013011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022013011))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell000022013011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+        thetaAboveCell000022013011))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell000022013011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+        thetaAboveCell000022013011))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130113120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113120 h)
+        (by
+          have h : (thetaAboveCell0000220130113121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113121 h)
+        (by
+          have h : (thetaAboveCell0000220130113122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113122 h)
+        (by
+          have h : (thetaAboveCell0000220130113123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130113130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113130 h)
+        (by
+          have h : (thetaAboveCell0000220130113131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113131 h)
+        (by
+          have h : (thetaAboveCell0000220130113132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113132 h)
+        (by
+          have h : (thetaAboveCell0000220130113133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113133 h))
+
+theorem cover_subtree_b98ded6b50f2 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022013011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022013011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130113200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113200 h)
+        (by
+          have h : (thetaAboveCell0000220130113201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113201 h)
+        (by
+          have h : (thetaAboveCell0000220130113202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113202 h)
+        (by
+          have h : (thetaAboveCell0000220130113203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130113210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113210 h)
+        (by
+          have h : (thetaAboveCell0000220130113211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113211 h)
+        (by
+          have h : (thetaAboveCell0000220130113212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113212 h)
+        (by
+          have h : (thetaAboveCell0000220130113213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL (childHH
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130113220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113220 h)
+        (by
+          have h : (thetaAboveCell0000220130113221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113221 h)
+        (by
+          have h : (thetaAboveCell0000220130113222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113222 h)
+        (by
+          have h : (thetaAboveCell0000220130113223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL (childHH
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130113230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113230 h)
+        (by
+          have h : (thetaAboveCell0000220130113231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113231 h)
+        (by
+          have h : (thetaAboveCell0000220130113232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113232 h)
+        (by
+          have h : (thetaAboveCell0000220130113233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113233 h))
+
+theorem cover_subtree_6021efadc4fa :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022013011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022013011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130113300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113300 h)
+        (by
+          have h : (thetaAboveCell0000220130113301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113301 h)
+        (by
+          have h : (thetaAboveCell0000220130113302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113302 h)
+        (by
+          have h : (thetaAboveCell0000220130113303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130113310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113310 h)
+        (by
+          have h : (thetaAboveCell0000220130113311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113311 h)
+        (by
+          have h : (thetaAboveCell0000220130113312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113312 h)
+        (by
+          have h : (thetaAboveCell0000220130113313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH (childHH
+        thetaAboveCell000022013011)))
+        (by
+          have h : (thetaAboveCell0000220130113320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113320 h)
+        (by
+          have h : (thetaAboveCell0000220130113321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113321 h)
+        (by
+          have h : (thetaAboveCell0000220130113322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113322 h)
+        (by
+          have h : (thetaAboveCell0000220130113323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220130113323 h))
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022013011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022013011))) h)
+
+theorem cover_subtree_3daa6e2b09d9 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022013011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022013011)
+    cover_subtree_b3bbbe006bfc
+    cover_subtree_10ce55f7ce75
+    cover_subtree_b98ded6b50f2
+    cover_subtree_6021efadc4fa
+
+theorem cover_subtree_adb5bf33bc5c :
+    adaptiveCoverCheck 7 thetaAboveCell000022013011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013011
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013011)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022013011)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022013011)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022013011)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022013011)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013011)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022013011)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022013011)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022013011)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022013011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022013011)) h))
+    cover_subtree_03f7c5873741
+    cover_subtree_3daa6e2b09d9
+
+theorem cover_subtree_b3708d7f6b8c :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022013012) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013012)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022013012))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022013012))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022013012))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022013012))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022013012))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022013012))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022013012))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022013012))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022013012))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022013012))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022013012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022013012)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022013012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022013012)) h)
+
+theorem cover_subtree_74cca09731bd :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022013012) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013012)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022013012))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022013012))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022013012))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022013012))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022013012))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022013012))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022013012))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022013012))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022013012))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022013012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022013012))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022013012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022013012)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022013012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022013012)) h)
+
+theorem cover_subtree_04cd22005970 :
+    adaptiveCoverCheck 7 thetaAboveCell000022013012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013012
+    cover_subtree_b3708d7f6b8c
+    cover_subtree_74cca09731bd
+    (by
+      have h : ((childHL thetaAboveCell000022013012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022013012) h)
+    (by
+      have h : ((childHH thetaAboveCell000022013012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022013012) h)
+
+theorem cover_subtree_f57488cb8c35 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022013013) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013013)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022013013))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022013013))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022013013))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022013013))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022013013))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022013013))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022013013))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022013013))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022013013))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022013013))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022013013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022013013)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022013013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022013013)) h)
+
+theorem cover_subtree_5e11cdd5ab21 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022013013) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013013)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022013013))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022013013))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022013013))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022013013))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022013013))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022013013))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022013013))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022013013))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022013013))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022013013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022013013))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022013013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022013013)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022013013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022013013)) h)
+
+theorem cover_subtree_d6f5101b1ccd :
+    adaptiveCoverCheck 7 thetaAboveCell000022013013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013013
+    cover_subtree_f57488cb8c35
+    cover_subtree_5e11cdd5ab21
+    (by
+      have h : ((childHL thetaAboveCell000022013013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022013013) h)
+    (by
+      have h : ((childHH thetaAboveCell000022013013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022013013) h)
+
+theorem cover_subtree_2784ec883b01 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00002201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00002201)))
+    cover_subtree_3a257d8444b0
+    cover_subtree_adb5bf33bc5c
+    cover_subtree_04cd22005970
+    cover_subtree_d6f5101b1ccd
+
+theorem e24KC2ThetaAboveLeaf0000220130 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00002201))
+    cover_subtree_deb83366e758
+    cover_subtree_2784ec883b01
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022013020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013020 h)
+        (by
+          have h : (thetaAboveCell000022013021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013021 h)
+        (by
+          have h : (thetaAboveCell000022013022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013022 h)
+        (by
+          have h : (thetaAboveCell000022013023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022013030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013030 h)
+        (by
+          have h : (thetaAboveCell000022013031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013031 h)
+        (by
+          have h : (thetaAboveCell000022013032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013032 h)
+        (by
+          have h : (thetaAboveCell000022013033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013033 h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle018.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle018.lean
new file mode 100644
index 0000000000..c6fffa5fd9
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle018.lean
@@ -0,0 +1,7878 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch031`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCells4f78f3c271
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022013100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell00002201)))
+
+/-- Subcell `000022100220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022100221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022100222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022100223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022100230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022100231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022100232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022100233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022100320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022100321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022100322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022100323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022100330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022100331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022100332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022100333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022100333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaAboveCell00002210)))
+
+/-- Subcell `000022101220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022101221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022101222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022101223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022101230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022101231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022101232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022101233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022101320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022101321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022101322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022101323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022101330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022101331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022101332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022101333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022101333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaAboveCell00002210)))
+
+/-- Subcell `000022102000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00002210)))
+
+/-- Subcell `000022103000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `000022103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell00002210)))
+
+/-- Subcell `0000220131002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022013100)))
+
+/-- Subcell `0000220131012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022013101)))
+
+/-- Subcell `0000220131102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022013110)))
+
+/-- Subcell `0000220131103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220131103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022013110)))
+
+end GerverSofa.PartE.CertificateCells4f78f3c271
+
+namespace GerverSofa.PartE.CertificateCellsea4abf6409
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022002100)))
+
+end GerverSofa.PartE.CertificateCellsea4abf6409
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6ProofBatchF910cf651a2033b4`.
+* `KernelOnly.PartE.E24KC6R4Subtree001eda2c378d570b`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4f78f3c271
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4f78f3c271
+
+open CertificateCells4f78f3c271
+theorem cover_subtree_be10dc5c5840 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022013100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022013100))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+        thetaAboveCell000022013100))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+        thetaAboveCell000022013100))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131002020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002020 h)
+        (by
+          have h : (thetaAboveCell0000220131002021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002021 h)
+        (by
+          have h : (thetaAboveCell0000220131002022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002022 h)
+        (by
+          have h : (thetaAboveCell0000220131002023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131002030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002030 h)
+        (by
+          have h : (thetaAboveCell0000220131002031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002031 h)
+        (by
+          have h : (thetaAboveCell0000220131002032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002032 h)
+        (by
+          have h : (thetaAboveCell0000220131002033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002033 h))
+
+theorem cover_subtree_e2664fa77be7 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022013100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022013100))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022013100))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022013100))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131002120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002120 h)
+        (by
+          have h : (thetaAboveCell0000220131002121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002121 h)
+        (by
+          have h : (thetaAboveCell0000220131002122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002122 h)
+        (by
+          have h : (thetaAboveCell0000220131002123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131002130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002130 h)
+        (by
+          have h : (thetaAboveCell0000220131002131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002131 h)
+        (by
+          have h : (thetaAboveCell0000220131002132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002132 h)
+        (by
+          have h : (thetaAboveCell0000220131002133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002133 h))
+
+theorem cover_subtree_aabbbf73f733 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022013100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022013100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131002200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002200 h)
+        (by
+          have h : (thetaAboveCell0000220131002201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002201 h)
+        (by
+          have h : (thetaAboveCell0000220131002202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002202 h)
+        (by
+          have h : (thetaAboveCell0000220131002203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131002210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002210 h)
+        (by
+          have h : (thetaAboveCell0000220131002211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002211 h)
+        (by
+          have h : (thetaAboveCell0000220131002212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002212 h)
+        (by
+          have h : (thetaAboveCell0000220131002213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022013100))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022013100))) h)
+
+theorem cover_subtree_9241fb51793c :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022013100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022013100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131002300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002300 h)
+        (by
+          have h : (thetaAboveCell0000220131002301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002301 h)
+        (by
+          have h : (thetaAboveCell0000220131002302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002302 h)
+        (by
+          have h : (thetaAboveCell0000220131002303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131002310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002310 h)
+        (by
+          have h : (thetaAboveCell0000220131002311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002311 h)
+        (by
+          have h : (thetaAboveCell0000220131002312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002312 h)
+        (by
+          have h : (thetaAboveCell0000220131002313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131002313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022013100))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022013100))) h)
+
+theorem cover_subtree_2c937348b9e5 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022013100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022013100)
+    cover_subtree_be10dc5c5840
+    cover_subtree_e2664fa77be7
+    cover_subtree_aabbbf73f733
+    cover_subtree_9241fb51793c
+
+theorem cover_subtree_ebd2440c450c :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022013100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022013100))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022013100))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022013100))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131003020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003020 h)
+        (by
+          have h : (thetaAboveCell0000220131003021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003021 h)
+        (by
+          have h : (thetaAboveCell0000220131003022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003022 h)
+        (by
+          have h : (thetaAboveCell0000220131003023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131003030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003030 h)
+        (by
+          have h : (thetaAboveCell0000220131003031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003031 h)
+        (by
+          have h : (thetaAboveCell0000220131003032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003032 h)
+        (by
+          have h : (thetaAboveCell0000220131003033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003033 h))
+
+theorem cover_subtree_7da0e2645020 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022013100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022013100))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+        thetaAboveCell000022013100))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+        thetaAboveCell000022013100))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131003120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003120 h)
+        (by
+          have h : (thetaAboveCell0000220131003121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003121 h)
+        (by
+          have h : (thetaAboveCell0000220131003122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003122 h)
+        (by
+          have h : (thetaAboveCell0000220131003123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131003130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003130 h)
+        (by
+          have h : (thetaAboveCell0000220131003131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003131 h)
+        (by
+          have h : (thetaAboveCell0000220131003132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003132 h)
+        (by
+          have h : (thetaAboveCell0000220131003133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003133 h))
+
+theorem cover_subtree_f0b1927352c1 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022013100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022013100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131003200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003200 h)
+        (by
+          have h : (thetaAboveCell0000220131003201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003201 h)
+        (by
+          have h : (thetaAboveCell0000220131003202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003202 h)
+        (by
+          have h : (thetaAboveCell0000220131003203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131003210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003210 h)
+        (by
+          have h : (thetaAboveCell0000220131003211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003211 h)
+        (by
+          have h : (thetaAboveCell0000220131003212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003212 h)
+        (by
+          have h : (thetaAboveCell0000220131003213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022013100))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022013100))) h)
+
+theorem cover_subtree_183a71e75e23 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022013100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022013100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131003300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003300 h)
+        (by
+          have h : (thetaAboveCell0000220131003301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003301 h)
+        (by
+          have h : (thetaAboveCell0000220131003302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003302 h)
+        (by
+          have h : (thetaAboveCell0000220131003303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022013100)))
+        (by
+          have h : (thetaAboveCell0000220131003310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003310 h)
+        (by
+          have h : (thetaAboveCell0000220131003311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003311 h)
+        (by
+          have h : (thetaAboveCell0000220131003312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003312 h)
+        (by
+          have h : (thetaAboveCell0000220131003313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131003313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022013100))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022013100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022013100))) h)
+
+theorem cover_subtree_fed838534b7a :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022013100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022013100)
+    cover_subtree_ebd2440c450c
+    cover_subtree_7da0e2645020
+    cover_subtree_f0b1927352c1
+    cover_subtree_183a71e75e23
+
+theorem cover_subtree_90b4414de137 :
+    adaptiveCoverCheck 7 thetaAboveCell000022013100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013100
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013100)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022013100)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022013100)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022013100)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022013100)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013100)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022013100)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022013100)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022013100)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022013100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022013100)) h))
+    cover_subtree_2c937348b9e5
+    cover_subtree_fed838534b7a
+
+theorem cover_subtree_68d45fba5878 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022013101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022013101))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell000022013101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+        thetaAboveCell000022013101))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell000022013101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+        thetaAboveCell000022013101))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022013101)))
+        (by
+          have h : (thetaAboveCell0000220131012020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012020 h)
+        (by
+          have h : (thetaAboveCell0000220131012021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012021 h)
+        (by
+          have h : (thetaAboveCell0000220131012022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012022 h)
+        (by
+          have h : (thetaAboveCell0000220131012023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022013101)))
+        (by
+          have h : (thetaAboveCell0000220131012030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012030 h)
+        (by
+          have h : (thetaAboveCell0000220131012031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012031 h)
+        (by
+          have h : (thetaAboveCell0000220131012032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012032 h)
+        (by
+          have h : (thetaAboveCell0000220131012033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012033 h))
+
+theorem cover_subtree_1378cfbe9d41 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022013101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022013101))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022013101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022013101))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022013101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022013101))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022013101)))
+        (by
+          have h : (thetaAboveCell0000220131012120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012120 h)
+        (by
+          have h : (thetaAboveCell0000220131012121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012121 h)
+        (by
+          have h : (thetaAboveCell0000220131012122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012122 h)
+        (by
+          have h : (thetaAboveCell0000220131012123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022013101)))
+        (by
+          have h : (thetaAboveCell0000220131012130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012130 h)
+        (by
+          have h : (thetaAboveCell0000220131012131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012131 h)
+        (by
+          have h : (thetaAboveCell0000220131012132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012132 h)
+        (by
+          have h : (thetaAboveCell0000220131012133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012133 h))
+
+theorem cover_subtree_d4c18a11dbdf :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022013101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022013101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022013101)))
+        (by
+          have h : (thetaAboveCell0000220131012200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012200 h)
+        (by
+          have h : (thetaAboveCell0000220131012201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012201 h)
+        (by
+          have h : (thetaAboveCell0000220131012202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012202 h)
+        (by
+          have h : (thetaAboveCell0000220131012203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022013101)))
+        (by
+          have h : (thetaAboveCell0000220131012210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012210 h)
+        (by
+          have h : (thetaAboveCell0000220131012211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012211 h)
+        (by
+          have h : (thetaAboveCell0000220131012212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012212 h)
+        (by
+          have h : (thetaAboveCell0000220131012213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022013101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022013101))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022013101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022013101))) h)
+
+theorem cover_subtree_5c02510827fa :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022013101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022013101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022013101)))
+        (by
+          have h : (thetaAboveCell0000220131012300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012300 h)
+        (by
+          have h : (thetaAboveCell0000220131012301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012301 h)
+        (by
+          have h : (thetaAboveCell0000220131012302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012302 h)
+        (by
+          have h : (thetaAboveCell0000220131012303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022013101)))
+        (by
+          have h : (thetaAboveCell0000220131012310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012310 h)
+        (by
+          have h : (thetaAboveCell0000220131012311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012311 h)
+        (by
+          have h : (thetaAboveCell0000220131012312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012312 h)
+        (by
+          have h : (thetaAboveCell0000220131012313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131012313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022013101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022013101))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022013101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022013101))) h)
+
+theorem cover_subtree_a9496679df68 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022013101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022013101)
+    cover_subtree_68d45fba5878
+    cover_subtree_1378cfbe9d41
+    cover_subtree_d4c18a11dbdf
+    cover_subtree_5c02510827fa
+
+theorem cover_subtree_489a9c102105 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022013101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022013101))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022013101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022013101))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022013101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022013101))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022013101)))
+        (by
+          have h : (thetaAboveCell0000220131013020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013020 h)
+        (by
+          have h : (thetaAboveCell0000220131013021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013021 h)
+        (by
+          have h : (thetaAboveCell0000220131013022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013022 h)
+        (by
+          have h : (thetaAboveCell0000220131013023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022013101)))
+        (by
+          have h : (thetaAboveCell0000220131013030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013030 h)
+        (by
+          have h : (thetaAboveCell0000220131013031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013031 h)
+        (by
+          have h : (thetaAboveCell0000220131013032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013032 h)
+        (by
+          have h : (thetaAboveCell0000220131013033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013033 h))
+
+theorem cover_subtree_83fbbe580c99 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022013101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022013101))
+    (by
+      have h : ((childLL (childLH (childHH thetaAboveCell000022013101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+        thetaAboveCell000022013101))) h)
+    (by
+      have h : ((childLH (childLH (childHH thetaAboveCell000022013101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+        thetaAboveCell000022013101))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+        thetaAboveCell000022013101)))
+        (by
+          have h : (thetaAboveCell0000220131013120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013120 h)
+        (by
+          have h : (thetaAboveCell0000220131013121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013121 h)
+        (by
+          have h : (thetaAboveCell0000220131013122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013122 h)
+        (by
+          have h : (thetaAboveCell0000220131013123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+        thetaAboveCell000022013101)))
+        (by
+          have h : (thetaAboveCell0000220131013130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013130 h)
+        (by
+          have h : (thetaAboveCell0000220131013131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013131 h)
+        (by
+          have h : (thetaAboveCell0000220131013132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013132 h)
+        (by
+          have h : (thetaAboveCell0000220131013133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013133 h))
+
+theorem cover_subtree_41d6d032c729 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022013101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022013101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022013101)))
+        (by
+          have h : (thetaAboveCell0000220131013200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013200 h)
+        (by
+          have h : (thetaAboveCell0000220131013201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013201 h)
+        (by
+          have h : (thetaAboveCell0000220131013202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013202 h)
+        (by
+          have h : (thetaAboveCell0000220131013203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022013101)))
+        (by
+          have h : (thetaAboveCell0000220131013210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013210 h)
+        (by
+          have h : (thetaAboveCell0000220131013211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013211 h)
+        (by
+          have h : (thetaAboveCell0000220131013212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013212 h)
+        (by
+          have h : (thetaAboveCell0000220131013213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131013213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022013101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022013101))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022013101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022013101))) h)
+
+theorem cover_subtree_51fa07c59358 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022013101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022013101)
+    cover_subtree_489a9c102105
+    cover_subtree_83fbbe580c99
+    cover_subtree_41d6d032c729
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022013101))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022013101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022013101))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022013101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022013101))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022013101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022013101))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022013101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022013101))) h))
+
+theorem cover_subtree_e719563f2a9a :
+    adaptiveCoverCheck 7 thetaAboveCell000022013101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013101
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013101)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022013101)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022013101)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022013101)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022013101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013101)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022013101)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022013101)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022013101)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022013101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022013101)) h))
+    cover_subtree_a9496679df68
+    cover_subtree_51fa07c59358
+
+theorem cover_subtree_5da232081c3f :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022013102) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013102)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022013102))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022013102))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022013102))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022013102))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022013102))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022013102))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022013102))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022013102))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022013102))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022013102))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022013102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022013102)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022013102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022013102)) h)
+
+theorem cover_subtree_5c0d2361729f :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022013102) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013102)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022013102))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022013102))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022013102))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022013102))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022013102))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022013102))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022013102))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022013102))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022013102))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022013102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022013102))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022013102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022013102)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022013102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022013102)) h)
+
+theorem cover_subtree_7dff7749156d :
+    adaptiveCoverCheck 7 thetaAboveCell000022013102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013102
+    cover_subtree_5da232081c3f
+    cover_subtree_5c0d2361729f
+    (by
+      have h : ((childHL thetaAboveCell000022013102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022013102) h)
+    (by
+      have h : ((childHH thetaAboveCell000022013102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022013102) h)
+
+theorem cover_subtree_c4dcfe2e3e24 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022013103) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013103)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022013103))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022013103))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022013103))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022013103))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022013103))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022013103))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022013103))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022013103))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022013103))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022013103))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022013103))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022013103)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022013103))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022013103)) h)
+
+theorem cover_subtree_264dbac80856 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022013103) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013103)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022013103))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022013103))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022013103))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022013103))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022013103))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022013103))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022013103))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022013103))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022013103))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022013103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022013103))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022013103))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022013103)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022013103))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022013103)) h)
+
+theorem cover_subtree_e684324b85dc :
+    adaptiveCoverCheck 7 thetaAboveCell000022013103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013103
+    cover_subtree_c4dcfe2e3e24
+    cover_subtree_264dbac80856
+    (by
+      have h : ((childHL thetaAboveCell000022013103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022013103) h)
+    (by
+      have h : ((childHH thetaAboveCell000022013103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022013103) h)
+
+theorem cover_subtree_ce7ce9f6c591 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHH thetaAboveCell00002201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHH thetaAboveCell00002201)))
+    cover_subtree_90b4414de137
+    cover_subtree_e719563f2a9a
+    cover_subtree_7dff7749156d
+    cover_subtree_e684324b85dc
+
+theorem cover_subtree_a70ca499a0b4 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022013110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022013110))
+    (by
+      have h : ((childLL (childLL (childHL thetaAboveCell000022013110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+        thetaAboveCell000022013110))) h)
+    (by
+      have h : ((childLH (childLL (childHL thetaAboveCell000022013110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+        thetaAboveCell000022013110))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+        thetaAboveCell000022013110)))
+        (by
+          have h : (thetaAboveCell0000220131102020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102020 h)
+        (by
+          have h : (thetaAboveCell0000220131102021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102021 h)
+        (by
+          have h : (thetaAboveCell0000220131102022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102022 h)
+        (by
+          have h : (thetaAboveCell0000220131102023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+        thetaAboveCell000022013110)))
+        (by
+          have h : (thetaAboveCell0000220131102030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102030 h)
+        (by
+          have h : (thetaAboveCell0000220131102031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102031 h)
+        (by
+          have h : (thetaAboveCell0000220131102032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102032 h)
+        (by
+          have h : (thetaAboveCell0000220131102033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102033 h))
+
+theorem cover_subtree_6d4fa0be4462 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022013110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022013110))
+    (by
+      have h : ((childLL (childLH (childHL thetaAboveCell000022013110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+        thetaAboveCell000022013110))) h)
+    (by
+      have h : ((childLH (childLH (childHL thetaAboveCell000022013110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+        thetaAboveCell000022013110))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+        thetaAboveCell000022013110)))
+        (by
+          have h : (thetaAboveCell0000220131102120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102120 h)
+        (by
+          have h : (thetaAboveCell0000220131102121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102121 h)
+        (by
+          have h : (thetaAboveCell0000220131102122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102122 h)
+        (by
+          have h : (thetaAboveCell0000220131102123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+        thetaAboveCell000022013110)))
+        (by
+          have h : (thetaAboveCell0000220131102130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102130 h)
+        (by
+          have h : (thetaAboveCell0000220131102131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102131 h)
+        (by
+          have h : (thetaAboveCell0000220131102132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102132 h)
+        (by
+          have h : (thetaAboveCell0000220131102133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131102133 h))
+
+theorem cover_subtree_a8973ec340e4 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022013110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022013110)
+    cover_subtree_a70ca499a0b4
+    cover_subtree_6d4fa0be4462
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022013110))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022013110))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022013110))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022013110))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022013110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022013110))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022013110))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022013110))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022013110))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022013110))) h))
+
+theorem cover_subtree_3831ad0b2d96 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022013110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022013110))
+    (by
+      have h : ((childLL (childLL (childHH thetaAboveCell000022013110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+        thetaAboveCell000022013110))) h)
+    (by
+      have h : ((childLH (childLL (childHH thetaAboveCell000022013110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+        thetaAboveCell000022013110))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+        thetaAboveCell000022013110)))
+        (by
+          have h : (thetaAboveCell0000220131103020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131103020 h)
+        (by
+          have h : (thetaAboveCell0000220131103021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131103021 h)
+        (by
+          have h : (thetaAboveCell0000220131103022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131103022 h)
+        (by
+          have h : (thetaAboveCell0000220131103023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131103023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+        thetaAboveCell000022013110)))
+        (by
+          have h : (thetaAboveCell0000220131103030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131103030 h)
+        (by
+          have h : (thetaAboveCell0000220131103031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131103031 h)
+        (by
+          have h : (thetaAboveCell0000220131103032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131103032 h)
+        (by
+          have h : (thetaAboveCell0000220131103033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220131103033 h))
+
+theorem cover_subtree_209916dfc771 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022013110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022013110)
+    cover_subtree_3831ad0b2d96
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022013110))
+        (by
+          have h : ((childLL (childLH (childHH thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+            thetaAboveCell000022013110))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+            thetaAboveCell000022013110))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHH
+            thetaAboveCell000022013110))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHH
+            thetaAboveCell000022013110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022013110))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022013110))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022013110))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022013110))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022013110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022013110))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022013110))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022013110))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022013110))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022013110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022013110))) h))
+
+theorem cover_subtree_9102dd71dbed :
+    adaptiveCoverCheck 7 thetaAboveCell000022013110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013110
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013110)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022013110)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022013110)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022013110)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022013110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013110)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022013110)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022013110)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022013110)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022013110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022013110)) h))
+    cover_subtree_a8973ec340e4
+    cover_subtree_209916dfc771
+
+theorem cover_subtree_62849ad9787b :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022013111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022013111)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022013111))
+        (by
+          have h : ((childLL (childLL (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHL
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHL
+            thetaAboveCell000022013111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022013111))
+        (by
+          have h : ((childLL (childLH (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHL
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHL
+            thetaAboveCell000022013111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022013111))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022013111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022013111))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022013111))) h))
+
+theorem cover_subtree_bfb0ae9bf369 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022013111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022013111)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022013111))
+        (by
+          have h : ((childLL (childLL (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHH
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHH
+            thetaAboveCell000022013111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022013111))
+        (by
+          have h : ((childLL (childLH (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHH
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHH
+            thetaAboveCell000022013111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022013111))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022013111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022013111))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022013111))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022013111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022013111))) h))
+
+theorem cover_subtree_eaa4ec2c0c14 :
+    adaptiveCoverCheck 7 thetaAboveCell000022013111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013111
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013111)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022013111)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022013111)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022013111)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022013111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013111)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022013111)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022013111)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022013111)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022013111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022013111)) h))
+    cover_subtree_62849ad9787b
+    cover_subtree_bfb0ae9bf369
+
+theorem cover_subtree_e17df11d5039 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022013112) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013112)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022013112))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022013112))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022013112))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022013112))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022013112))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022013112))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022013112))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022013112))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022013112))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022013112))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022013112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022013112)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022013112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022013112)) h)
+
+theorem cover_subtree_c71b0926b90d :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022013112) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013112)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022013112))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022013112))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022013112))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022013112))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022013112))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022013112))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022013112))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022013112))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022013112))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022013112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022013112))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022013112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022013112)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022013112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022013112)) h)
+
+theorem cover_subtree_9a7254ea80d1 :
+    adaptiveCoverCheck 7 thetaAboveCell000022013112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013112
+    cover_subtree_e17df11d5039
+    cover_subtree_c71b0926b90d
+    (by
+      have h : ((childHL thetaAboveCell000022013112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022013112) h)
+    (by
+      have h : ((childHH thetaAboveCell000022013112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022013112) h)
+
+theorem cover_subtree_6a20205ec3a1 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022013113) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022013113)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022013113))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022013113))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022013113))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022013113))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022013113))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022013113))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022013113))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022013113))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022013113))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022013113))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022013113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022013113)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022013113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022013113)) h)
+
+theorem cover_subtree_1b35fa72d695 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022013113) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022013113)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022013113))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022013113))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022013113))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022013113))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022013113))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022013113))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022013113))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022013113))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022013113))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022013113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022013113))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022013113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022013113)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022013113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022013113)) h)
+
+theorem cover_subtree_19dbc3a1cabf :
+    adaptiveCoverCheck 7 thetaAboveCell000022013113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022013113
+    cover_subtree_6a20205ec3a1
+    cover_subtree_1b35fa72d695
+    (by
+      have h : ((childHL thetaAboveCell000022013113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022013113) h)
+    (by
+      have h : ((childHH thetaAboveCell000022013113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022013113) h)
+
+theorem cover_subtree_306835c71900 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHH thetaAboveCell00002201))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHH thetaAboveCell00002201)))
+    cover_subtree_9102dd71dbed
+    cover_subtree_eaa4ec2c0c14
+    cover_subtree_9a7254ea80d1
+    cover_subtree_19dbc3a1cabf
+
+theorem e24KC2ThetaAboveLeaf0000220131 :
+    adaptiveCoverCheck 9 (childLH (childHH thetaAboveCell00002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHH thetaAboveCell00002201))
+    cover_subtree_ce7ce9f6c591
+    cover_subtree_306835c71900
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHH
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022013120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013120 h)
+        (by
+          have h : (thetaAboveCell000022013121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013121 h)
+        (by
+          have h : (thetaAboveCell000022013122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013122 h)
+        (by
+          have h : (thetaAboveCell000022013123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHH
+        thetaAboveCell00002201)))
+        (by
+          have h : (thetaAboveCell000022013130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013130 h)
+        (by
+          have h : (thetaAboveCell000022013131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013131 h)
+        (by
+          have h : (thetaAboveCell000022013132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013132 h)
+        (by
+          have h : (thetaAboveCell000022013133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022013133 h))
+theorem e24KC2ThetaAboveLeaf0000220132 :
+    adaptiveCoverCheck 9 (childHL (childHH thetaAboveCell00002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHH thetaAboveCell00002201))
+    (by
+      have h : ((childLL (childHL (childHH thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHH
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHH
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHH
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHH
+        thetaAboveCell00002201))) h)
+theorem e24KC2ThetaAboveLeaf0000220133 :
+    adaptiveCoverCheck 9 (childHH (childHH thetaAboveCell00002201)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHH thetaAboveCell00002201))
+    (by
+      have h : ((childLL (childHH (childHH thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHH
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHH
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHH
+        thetaAboveCell00002201))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell00002201)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHH
+        thetaAboveCell00002201))) h)
+theorem e24KC2ThetaAboveLeaf0000221002 :
+    adaptiveCoverCheck 9 (childHL (childLL thetaAboveCell00002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLL thetaAboveCell00002210))
+    (by
+      have h : ((childLL (childHL (childLL thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLL
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childLH (childHL (childLL thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLL
+        thetaAboveCell00002210))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLL
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022100220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100220 h)
+        (by
+          have h : (thetaAboveCell000022100221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100221 h)
+        (by
+          have h : (thetaAboveCell000022100222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100222 h)
+        (by
+          have h : (thetaAboveCell000022100223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childLL
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022100230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100230 h)
+        (by
+          have h : (thetaAboveCell000022100231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100231 h)
+        (by
+          have h : (thetaAboveCell000022100232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100232 h)
+        (by
+          have h : (thetaAboveCell000022100233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100233 h))
+theorem e24KC2ThetaAboveLeaf0000221003 :
+    adaptiveCoverCheck 9 (childHH (childLL thetaAboveCell00002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLL thetaAboveCell00002210))
+    (by
+      have h : ((childLL (childHH (childLL thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLL
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childLH (childHH (childLL thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLL
+        thetaAboveCell00002210))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childLL
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022100320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100320 h)
+        (by
+          have h : (thetaAboveCell000022100321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100321 h)
+        (by
+          have h : (thetaAboveCell000022100322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100322 h)
+        (by
+          have h : (thetaAboveCell000022100323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childLL
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022100330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100330 h)
+        (by
+          have h : (thetaAboveCell000022100331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100331 h)
+        (by
+          have h : (thetaAboveCell000022100332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100332 h)
+        (by
+          have h : (thetaAboveCell000022100333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022100333 h))
+theorem e24KC2ThetaAboveLeaf0000221012 :
+    adaptiveCoverCheck 9 (childHL (childLH thetaAboveCell00002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childLH thetaAboveCell00002210))
+    (by
+      have h : ((childLL (childHL (childLH thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childLH
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childLH
+        thetaAboveCell00002210))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHL (childLH
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022101220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101220 h)
+        (by
+          have h : (thetaAboveCell000022101221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101221 h)
+        (by
+          have h : (thetaAboveCell000022101222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101222 h)
+        (by
+          have h : (thetaAboveCell000022101223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHL (childLH
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022101230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101230 h)
+        (by
+          have h : (thetaAboveCell000022101231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101231 h)
+        (by
+          have h : (thetaAboveCell000022101232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101232 h)
+        (by
+          have h : (thetaAboveCell000022101233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101233 h))
+theorem e24KC2ThetaAboveLeaf0000221013 :
+    adaptiveCoverCheck 9 (childHH (childLH thetaAboveCell00002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childLH thetaAboveCell00002210))
+    (by
+      have h : ((childLL (childHH (childLH thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childLH
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childLH
+        thetaAboveCell00002210))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH (childLH
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022101320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101320 h)
+        (by
+          have h : (thetaAboveCell000022101321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101321 h)
+        (by
+          have h : (thetaAboveCell000022101322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101322 h)
+        (by
+          have h : (thetaAboveCell000022101323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH (childLH
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022101330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101330 h)
+        (by
+          have h : (thetaAboveCell000022101331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101331 h)
+        (by
+          have h : (thetaAboveCell000022101332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101332 h)
+        (by
+          have h : (thetaAboveCell000022101333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022101333 h))
+theorem cover_subtree_c6b00b0efe5e :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022102000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022102000)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022102000))
+        (by
+          have h : ((childLL (childLL (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHL
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHL
+            thetaAboveCell000022102000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022102000))
+        (by
+          have h : ((childLL (childLH (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHL
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHL
+            thetaAboveCell000022102000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022102000))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022102000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022102000))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022102000))) h))
+
+theorem cover_subtree_da57798dbd9f :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022102000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022102000)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022102000))
+        (by
+          have h : ((childLL (childLL (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHH
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHH
+            thetaAboveCell000022102000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022102000))
+        (by
+          have h : ((childLL (childLH (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHH
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHH
+            thetaAboveCell000022102000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022102000))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022102000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022102000))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022102000))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022102000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022102000))) h))
+
+theorem cover_subtree_bf93e03bf258 :
+    adaptiveCoverCheck 7 thetaAboveCell000022102000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102000
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102000)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022102000)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022102000)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022102000)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022102000)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102000)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022102000)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022102000)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022102000)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022102000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022102000)) h))
+    cover_subtree_c6b00b0efe5e
+    cover_subtree_da57798dbd9f
+
+theorem cover_subtree_77f8696b5d47 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022102001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022102001)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022102001))
+        (by
+          have h : ((childLL (childLL (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHL
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHL
+            thetaAboveCell000022102001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022102001))
+        (by
+          have h : ((childLL (childLH (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHL
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHL
+            thetaAboveCell000022102001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022102001))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022102001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022102001))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022102001))) h))
+
+theorem cover_subtree_d9ace3f5364c :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022102001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022102001)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022102001))
+        (by
+          have h : ((childLL (childLL (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHH
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHH
+            thetaAboveCell000022102001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022102001))
+        (by
+          have h : ((childLL (childLH (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHH
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHH
+            thetaAboveCell000022102001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022102001))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022102001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022102001))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022102001))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022102001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022102001))) h))
+
+theorem cover_subtree_31e6b90084eb :
+    adaptiveCoverCheck 7 thetaAboveCell000022102001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102001
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102001)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022102001)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022102001)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022102001)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022102001)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102001)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022102001)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022102001)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022102001)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022102001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022102001)) h))
+    cover_subtree_77f8696b5d47
+    cover_subtree_d9ace3f5364c
+
+theorem cover_subtree_bf46f98b974d :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022102002) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102002)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022102002))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022102002))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022102002))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022102002))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022102002))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022102002))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022102002))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022102002))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022102002))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022102002))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022102002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022102002)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022102002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022102002)) h)
+
+theorem cover_subtree_1ce6ea29c41d :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022102002) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102002)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022102002))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022102002))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022102002))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022102002))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022102002))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022102002))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022102002))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022102002))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022102002))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022102002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022102002))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022102002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022102002)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022102002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022102002)) h)
+
+theorem cover_subtree_5f2916b0629e :
+    adaptiveCoverCheck 7 thetaAboveCell000022102002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102002
+    cover_subtree_bf46f98b974d
+    cover_subtree_1ce6ea29c41d
+    (by
+      have h : ((childHL thetaAboveCell000022102002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022102002) h)
+    (by
+      have h : ((childHH thetaAboveCell000022102002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022102002) h)
+
+theorem cover_subtree_bf2a6b948a6e :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022102003) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102003)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022102003))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022102003))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022102003))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022102003))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022102003))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022102003))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022102003))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022102003))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022102003))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022102003))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022102003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022102003)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022102003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022102003)) h)
+
+theorem cover_subtree_80d3b8565aa4 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022102003) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102003)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022102003))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022102003))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022102003))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022102003))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022102003))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022102003))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022102003))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022102003))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022102003))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022102003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022102003))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022102003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022102003)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022102003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022102003)) h)
+
+theorem cover_subtree_256b0c79e2b4 :
+    adaptiveCoverCheck 7 thetaAboveCell000022102003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102003
+    cover_subtree_bf2a6b948a6e
+    cover_subtree_80d3b8565aa4
+    (by
+      have h : ((childHL thetaAboveCell000022102003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022102003) h)
+    (by
+      have h : ((childHH thetaAboveCell000022102003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022102003) h)
+
+theorem cover_subtree_2cb6fcc01da2 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00002210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00002210)))
+    cover_subtree_bf93e03bf258
+    cover_subtree_31e6b90084eb
+    cover_subtree_5f2916b0629e
+    cover_subtree_256b0c79e2b4
+
+theorem cover_subtree_9079844093a9 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022102010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022102010)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022102010))
+        (by
+          have h : ((childLL (childLL (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHL
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHL
+            thetaAboveCell000022102010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022102010))
+        (by
+          have h : ((childLL (childLH (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHL
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHL
+            thetaAboveCell000022102010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022102010))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022102010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022102010))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022102010))) h))
+
+theorem cover_subtree_dfea94ec6240 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022102010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022102010)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022102010))
+        (by
+          have h : ((childLL (childLL (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHH
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHH
+            thetaAboveCell000022102010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022102010))
+        (by
+          have h : ((childLL (childLH (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHH
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHH
+            thetaAboveCell000022102010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022102010))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022102010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022102010))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022102010))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022102010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022102010))) h))
+
+theorem cover_subtree_b660fcad9ea4 :
+    adaptiveCoverCheck 7 thetaAboveCell000022102010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102010
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102010)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022102010)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022102010)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022102010)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022102010)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102010)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022102010)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022102010)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022102010)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022102010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022102010)) h))
+    cover_subtree_9079844093a9
+    cover_subtree_dfea94ec6240
+
+theorem cover_subtree_5c3913aa5f15 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022102011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022102011)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022102011))
+        (by
+          have h : ((childLL (childLL (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHL
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHL
+            thetaAboveCell000022102011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022102011))
+        (by
+          have h : ((childLL (childLH (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHL
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHL
+            thetaAboveCell000022102011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022102011))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022102011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022102011))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022102011))) h))
+
+theorem cover_subtree_fbab0316abef :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022102011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022102011)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022102011))
+        (by
+          have h : ((childLL (childLL (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHH
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHH
+            thetaAboveCell000022102011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022102011))
+        (by
+          have h : ((childLL (childLH (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHH
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHH
+            thetaAboveCell000022102011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022102011))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022102011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022102011))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022102011))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022102011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022102011))) h))
+
+theorem cover_subtree_c4d768804304 :
+    adaptiveCoverCheck 7 thetaAboveCell000022102011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102011
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102011)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022102011)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022102011)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022102011)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022102011)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102011)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022102011)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022102011)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022102011)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022102011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022102011)) h))
+    cover_subtree_5c3913aa5f15
+    cover_subtree_fbab0316abef
+
+theorem cover_subtree_241920b95407 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022102012) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102012)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022102012))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022102012))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022102012))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022102012))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022102012))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022102012))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022102012))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022102012))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022102012))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022102012))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022102012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022102012)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022102012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022102012)) h)
+
+theorem cover_subtree_7da00eb7f7ae :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022102012) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102012)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022102012))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022102012))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022102012))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022102012))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022102012))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022102012))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022102012))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022102012))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022102012))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022102012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022102012))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022102012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022102012)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022102012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022102012)) h)
+
+theorem cover_subtree_f82bf90640b6 :
+    adaptiveCoverCheck 7 thetaAboveCell000022102012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102012
+    cover_subtree_241920b95407
+    cover_subtree_7da00eb7f7ae
+    (by
+      have h : ((childHL thetaAboveCell000022102012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022102012) h)
+    (by
+      have h : ((childHH thetaAboveCell000022102012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022102012) h)
+
+theorem cover_subtree_afdbba9126b4 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022102013) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102013)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022102013))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022102013))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022102013))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022102013))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022102013))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022102013))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022102013))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022102013))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022102013))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022102013))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022102013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022102013)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022102013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022102013)) h)
+
+theorem cover_subtree_66c22a722e78 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022102013) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102013)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022102013))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022102013))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022102013))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022102013))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022102013))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022102013))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022102013))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022102013))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022102013))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022102013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022102013))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022102013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022102013)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022102013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022102013)) h)
+
+theorem cover_subtree_21534bc80804 :
+    adaptiveCoverCheck 7 thetaAboveCell000022102013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102013
+    cover_subtree_afdbba9126b4
+    cover_subtree_66c22a722e78
+    (by
+      have h : ((childHL thetaAboveCell000022102013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022102013) h)
+    (by
+      have h : ((childHH thetaAboveCell000022102013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022102013) h)
+
+theorem cover_subtree_e39770fd6802 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHL thetaAboveCell00002210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL thetaAboveCell00002210)))
+    cover_subtree_b660fcad9ea4
+    cover_subtree_c4d768804304
+    cover_subtree_f82bf90640b6
+    cover_subtree_21534bc80804
+
+theorem e24KC2ThetaAboveLeaf0000221020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00002210))
+    cover_subtree_2cb6fcc01da2
+    cover_subtree_e39770fd6802
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022102020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102020 h)
+        (by
+          have h : (thetaAboveCell000022102021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102021 h)
+        (by
+          have h : (thetaAboveCell000022102022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102022 h)
+        (by
+          have h : (thetaAboveCell000022102023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022102030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102030 h)
+        (by
+          have h : (thetaAboveCell000022102031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102031 h)
+        (by
+          have h : (thetaAboveCell000022102032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102032 h)
+        (by
+          have h : (thetaAboveCell000022102033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102033 h))
+theorem cover_subtree_5ec7a41b8f5f :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022102100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022102100)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022102100))
+        (by
+          have h : ((childLL (childLL (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHL
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHL
+            thetaAboveCell000022102100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022102100))
+        (by
+          have h : ((childLL (childLH (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHL
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHL
+            thetaAboveCell000022102100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022102100))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022102100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022102100))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022102100))) h))
+
+theorem cover_subtree_cd20ac2a69b7 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022102100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022102100)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022102100))
+        (by
+          have h : ((childLL (childLL (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHH
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHH
+            thetaAboveCell000022102100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022102100))
+        (by
+          have h : ((childLL (childLH (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHH
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHH
+            thetaAboveCell000022102100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022102100))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022102100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022102100))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022102100))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022102100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022102100))) h))
+
+theorem cover_subtree_15babda5d60b :
+    adaptiveCoverCheck 7 thetaAboveCell000022102100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102100
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102100)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022102100)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022102100)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022102100)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022102100)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102100)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022102100)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022102100)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022102100)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022102100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022102100)) h))
+    cover_subtree_5ec7a41b8f5f
+    cover_subtree_cd20ac2a69b7
+
+theorem cover_subtree_b76fce14d656 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022102101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022102101)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022102101))
+        (by
+          have h : ((childLL (childLL (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHL
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHL
+            thetaAboveCell000022102101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022102101))
+        (by
+          have h : ((childLL (childLH (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHL
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHL
+            thetaAboveCell000022102101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022102101))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022102101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022102101))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022102101))) h))
+
+theorem cover_subtree_b7fed3746307 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022102101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022102101)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022102101))
+        (by
+          have h : ((childLL (childLL (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHH
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHH
+            thetaAboveCell000022102101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022102101))
+        (by
+          have h : ((childLL (childLH (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHH
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHH
+            thetaAboveCell000022102101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022102101))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022102101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022102101))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022102101))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022102101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022102101))) h))
+
+theorem cover_subtree_a92382a463a3 :
+    adaptiveCoverCheck 7 thetaAboveCell000022102101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102101
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102101)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022102101)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022102101)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022102101)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022102101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102101)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022102101)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022102101)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022102101)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022102101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022102101)) h))
+    cover_subtree_b76fce14d656
+    cover_subtree_b7fed3746307
+
+theorem cover_subtree_1f53d26ec077 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022102102) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102102)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022102102))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022102102))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022102102))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022102102))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022102102))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022102102))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022102102))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022102102))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022102102))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022102102))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022102102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022102102)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022102102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022102102)) h)
+
+theorem cover_subtree_d3e2670ab1ea :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022102102) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102102)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022102102))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022102102))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022102102))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022102102))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022102102))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022102102))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022102102))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022102102))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022102102))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022102102)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022102102))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022102102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022102102)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022102102))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022102102)) h)
+
+theorem cover_subtree_98de717668b3 :
+    adaptiveCoverCheck 7 thetaAboveCell000022102102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102102
+    cover_subtree_1f53d26ec077
+    cover_subtree_d3e2670ab1ea
+    (by
+      have h : ((childHL thetaAboveCell000022102102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022102102) h)
+    (by
+      have h : ((childHH thetaAboveCell000022102102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022102102) h)
+
+theorem cover_subtree_db5013993ba5 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022102103) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102103)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022102103))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022102103))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022102103))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022102103))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022102103))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022102103))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022102103))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022102103))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022102103))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022102103))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022102103))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022102103)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022102103))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022102103)) h)
+
+theorem cover_subtree_9589615e51ad :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022102103) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102103)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022102103))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022102103))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022102103))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022102103))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022102103))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022102103))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022102103))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022102103))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022102103))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022102103)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022102103))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022102103))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022102103)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022102103))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022102103)) h)
+
+theorem cover_subtree_fdef224c05a8 :
+    adaptiveCoverCheck 7 thetaAboveCell000022102103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102103
+    cover_subtree_db5013993ba5
+    cover_subtree_9589615e51ad
+    (by
+      have h : ((childHL thetaAboveCell000022102103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022102103) h)
+    (by
+      have h : ((childHH thetaAboveCell000022102103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022102103) h)
+
+theorem cover_subtree_6fb265308e16 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00002210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00002210)))
+    cover_subtree_15babda5d60b
+    cover_subtree_a92382a463a3
+    cover_subtree_98de717668b3
+    cover_subtree_fdef224c05a8
+
+theorem cover_subtree_460dcf0085c5 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022102110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022102110)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022102110))
+        (by
+          have h : ((childLL (childLL (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHL
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHL
+            thetaAboveCell000022102110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022102110))
+        (by
+          have h : ((childLL (childLH (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHL
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHL
+            thetaAboveCell000022102110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022102110))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022102110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022102110))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022102110))) h))
+
+theorem cover_subtree_51ce5942d936 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022102110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022102110)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022102110))
+        (by
+          have h : ((childLL (childLL (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHH
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHH
+            thetaAboveCell000022102110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022102110))
+        (by
+          have h : ((childLL (childLH (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHH
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHH
+            thetaAboveCell000022102110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022102110))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022102110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022102110))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022102110))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022102110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022102110))) h))
+
+theorem cover_subtree_2be474cb7a0f :
+    adaptiveCoverCheck 7 thetaAboveCell000022102110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102110
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102110)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022102110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022102110)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022102110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022102110)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022102110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022102110)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022102110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022102110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102110)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022102110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022102110)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022102110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022102110)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022102110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022102110)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022102110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022102110)) h))
+    cover_subtree_460dcf0085c5
+    cover_subtree_51ce5942d936
+
+theorem cover_subtree_4bc40a8305e2 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022102111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022102111)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022102111))
+        (by
+          have h : ((childLL (childLL (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHL
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHL
+            thetaAboveCell000022102111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022102111))
+        (by
+          have h : ((childLL (childLH (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHL
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHL
+            thetaAboveCell000022102111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022102111))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022102111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022102111))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022102111))) h))
+
+theorem cover_subtree_bc2667f250d6 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022102111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022102111)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022102111))
+        (by
+          have h : ((childLL (childLL (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHH
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHH
+            thetaAboveCell000022102111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022102111))
+        (by
+          have h : ((childLL (childLH (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHH
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHH
+            thetaAboveCell000022102111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022102111))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022102111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022102111))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022102111))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022102111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022102111))) h))
+
+theorem cover_subtree_100b7236ad12 :
+    adaptiveCoverCheck 7 thetaAboveCell000022102111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102111
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102111)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022102111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022102111)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022102111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022102111)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022102111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022102111)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022102111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022102111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102111)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022102111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022102111)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022102111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022102111)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022102111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022102111)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022102111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022102111)) h))
+    cover_subtree_4bc40a8305e2
+    cover_subtree_bc2667f250d6
+
+theorem cover_subtree_968e16bde444 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022102112) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102112)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022102112))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022102112))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022102112))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022102112))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022102112))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022102112))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022102112))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022102112))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022102112))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022102112))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022102112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022102112)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022102112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022102112)) h)
+
+theorem cover_subtree_069815638772 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022102112) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102112)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022102112))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022102112))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022102112))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022102112))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022102112))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022102112))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022102112))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022102112))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022102112))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022102112)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022102112))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022102112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022102112)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022102112))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022102112)) h)
+
+theorem cover_subtree_da19516a51b8 :
+    adaptiveCoverCheck 7 thetaAboveCell000022102112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102112
+    cover_subtree_968e16bde444
+    cover_subtree_069815638772
+    (by
+      have h : ((childHL thetaAboveCell000022102112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022102112) h)
+    (by
+      have h : ((childHH thetaAboveCell000022102112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022102112) h)
+
+theorem cover_subtree_7602df0c45bd :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022102113) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022102113)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022102113))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022102113))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022102113))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022102113))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022102113))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022102113))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022102113))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022102113))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022102113))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022102113))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022102113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022102113)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022102113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022102113)) h)
+
+theorem cover_subtree_4f04fe0498c3 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022102113) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022102113)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022102113))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022102113))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022102113))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022102113))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022102113))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022102113))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022102113))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022102113))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022102113))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022102113)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022102113))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022102113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022102113)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022102113))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022102113)) h)
+
+theorem cover_subtree_69b60a777b1c :
+    adaptiveCoverCheck 7 thetaAboveCell000022102113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022102113
+    cover_subtree_7602df0c45bd
+    cover_subtree_4f04fe0498c3
+    (by
+      have h : ((childHL thetaAboveCell000022102113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022102113) h)
+    (by
+      have h : ((childHH thetaAboveCell000022102113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022102113) h)
+
+theorem cover_subtree_13f3502e8052 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00002210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00002210)))
+    cover_subtree_2be474cb7a0f
+    cover_subtree_100b7236ad12
+    cover_subtree_da19516a51b8
+    cover_subtree_69b60a777b1c
+
+theorem e24KC2ThetaAboveLeaf0000221021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00002210))
+    cover_subtree_6fb265308e16
+    cover_subtree_13f3502e8052
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022102120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102120 h)
+        (by
+          have h : (thetaAboveCell000022102121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102121 h)
+        (by
+          have h : (thetaAboveCell000022102122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102122 h)
+        (by
+          have h : (thetaAboveCell000022102123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022102130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102130 h)
+        (by
+          have h : (thetaAboveCell000022102131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102131 h)
+        (by
+          have h : (thetaAboveCell000022102132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102132 h)
+        (by
+          have h : (thetaAboveCell000022102133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022102133 h))
+theorem e24KC2ThetaAboveLeaf0000221022 :
+    adaptiveCoverCheck 9 (childHL (childHL thetaAboveCell00002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHL (childHL thetaAboveCell00002210))
+    (by
+      have h : ((childLL (childHL (childHL thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHL (childHL
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHL (childHL
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHL (childHL
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHL (childHL
+        thetaAboveCell00002210))) h)
+theorem e24KC2ThetaAboveLeaf0000221023 :
+    adaptiveCoverCheck 9 (childHH (childHL thetaAboveCell00002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childHH (childHL thetaAboveCell00002210))
+    (by
+      have h : ((childLL (childHH (childHL thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLL (childHH (childHL
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childLH (childHH (childHL
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHL (childHH (childHL
+        thetaAboveCell00002210))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell00002210)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 8 (childHH (childHH (childHL
+        thetaAboveCell00002210))) h)
+theorem cover_subtree_4e4a856a348d :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022103000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022103000)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022103000))
+        (by
+          have h : ((childLL (childLL (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHL
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHL
+            thetaAboveCell000022103000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022103000))
+        (by
+          have h : ((childLL (childLH (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHL
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHL
+            thetaAboveCell000022103000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022103000))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022103000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022103000))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022103000))) h))
+
+theorem cover_subtree_a9ec07ce774e :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022103000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022103000)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022103000))
+        (by
+          have h : ((childLL (childLL (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHH
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHH
+            thetaAboveCell000022103000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022103000))
+        (by
+          have h : ((childLL (childLH (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHH
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHH
+            thetaAboveCell000022103000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022103000))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022103000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022103000))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022103000))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022103000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022103000))) h))
+
+theorem cover_subtree_4f8bdd1a7038 :
+    adaptiveCoverCheck 7 thetaAboveCell000022103000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103000
+    (by
+      have h : ((childLL thetaAboveCell000022103000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022103000) h)
+    (by
+      have h : ((childLH thetaAboveCell000022103000)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022103000) h)
+    cover_subtree_4e4a856a348d
+    cover_subtree_a9ec07ce774e
+
+theorem cover_subtree_45d6d513875c :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022103001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022103001)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022103001))
+        (by
+          have h : ((childLL (childLL (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHL
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHL
+            thetaAboveCell000022103001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022103001))
+        (by
+          have h : ((childLL (childLH (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHL
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHL
+            thetaAboveCell000022103001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022103001))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022103001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022103001))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022103001))) h))
+
+theorem cover_subtree_e48f2ce51ae0 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022103001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022103001)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022103001))
+        (by
+          have h : ((childLL (childLL (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHH
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHH
+            thetaAboveCell000022103001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022103001))
+        (by
+          have h : ((childLL (childLH (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHH
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHH
+            thetaAboveCell000022103001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022103001))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022103001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022103001))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022103001))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022103001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022103001))) h))
+
+theorem cover_subtree_3fbacc491dec :
+    adaptiveCoverCheck 7 thetaAboveCell000022103001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103001
+    (by
+      have h : ((childLL thetaAboveCell000022103001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022103001) h)
+    (by
+      have h : ((childLH thetaAboveCell000022103001)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022103001) h)
+    cover_subtree_45d6d513875c
+    cover_subtree_e48f2ce51ae0
+
+theorem cover_subtree_c5ade50b08f1 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022103002) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022103002)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022103002))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022103002))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022103002))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022103002))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022103002))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022103002))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022103002))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022103002))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022103002))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022103002))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022103002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022103002)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022103002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022103002)) h)
+
+theorem cover_subtree_65c0c0fb0c61 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022103002) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022103002)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022103002))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022103002))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022103002))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022103002))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022103002))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022103002))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022103002))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022103002))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022103002))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022103002)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022103002))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022103002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022103002)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022103002))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022103002)) h)
+
+theorem cover_subtree_81fc74c0ffd0 :
+    adaptiveCoverCheck 7 thetaAboveCell000022103002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103002
+    cover_subtree_c5ade50b08f1
+    cover_subtree_65c0c0fb0c61
+    (by
+      have h : ((childHL thetaAboveCell000022103002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022103002) h)
+    (by
+      have h : ((childHH thetaAboveCell000022103002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022103002) h)
+
+theorem cover_subtree_7eb489ebb172 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022103003) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022103003)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022103003))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022103003))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022103003))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022103003))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022103003))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022103003))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022103003))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022103003))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022103003))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022103003))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022103003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022103003)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022103003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022103003)) h)
+
+theorem cover_subtree_f21141f5d5d1 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022103003) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022103003)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022103003))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022103003))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022103003))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022103003))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022103003))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022103003))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022103003))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022103003))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022103003))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022103003)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022103003))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022103003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022103003)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022103003))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022103003)) h)
+
+theorem cover_subtree_ac4f9a09671e :
+    adaptiveCoverCheck 7 thetaAboveCell000022103003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103003
+    cover_subtree_7eb489ebb172
+    cover_subtree_f21141f5d5d1
+    (by
+      have h : ((childHL thetaAboveCell000022103003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022103003) h)
+    (by
+      have h : ((childHH thetaAboveCell000022103003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022103003) h)
+
+theorem cover_subtree_dc732abf341f :
+    adaptiveCoverCheck 8 (childLL (childLL (childHH thetaAboveCell00002210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHH thetaAboveCell00002210)))
+    cover_subtree_4f8bdd1a7038
+    cover_subtree_3fbacc491dec
+    cover_subtree_81fc74c0ffd0
+    cover_subtree_ac4f9a09671e
+
+theorem cover_subtree_4c23e9a4e125 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022103010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022103010)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022103010))
+        (by
+          have h : ((childLL (childLL (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHL
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHL
+            thetaAboveCell000022103010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022103010))
+        (by
+          have h : ((childLL (childLH (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHL
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHL
+            thetaAboveCell000022103010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022103010))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022103010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022103010))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022103010))) h))
+
+theorem cover_subtree_100c345ae68a :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022103010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022103010)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022103010))
+        (by
+          have h : ((childLL (childLL (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHH
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHH
+            thetaAboveCell000022103010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022103010))
+        (by
+          have h : ((childLL (childLH (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHH
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHH
+            thetaAboveCell000022103010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022103010))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022103010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022103010))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022103010))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022103010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022103010))) h))
+
+theorem cover_subtree_712a58fdc8d3 :
+    adaptiveCoverCheck 7 thetaAboveCell000022103010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103010
+    (by
+      have h : ((childLL thetaAboveCell000022103010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022103010) h)
+    (by
+      have h : ((childLH thetaAboveCell000022103010)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022103010) h)
+    cover_subtree_4c23e9a4e125
+    cover_subtree_100c345ae68a
+
+theorem cover_subtree_38c5b4ed074a :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022103011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022103011)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022103011))
+        (by
+          have h : ((childLL (childLL (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHL
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHL
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHL
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHL
+            thetaAboveCell000022103011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022103011))
+        (by
+          have h : ((childLL (childLH (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHL
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHL
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHL
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHL
+            thetaAboveCell000022103011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022103011))
+        (by
+          have h : ((childLL (childHL (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHL
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childLH (childHL (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHL
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHL (childHL (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHH (childHL (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+            thetaAboveCell000022103011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022103011))
+        (by
+          have h : ((childLL (childHH (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHL
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childLH (childHH (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHL
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHL (childHH (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHH (childHH (childHL thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+            thetaAboveCell000022103011))) h))
+
+theorem cover_subtree_0e2d12c802dd :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022103011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022103011)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022103011))
+        (by
+          have h : ((childLL (childLL (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childHH
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childHH
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childHH
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childHH
+            thetaAboveCell000022103011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022103011))
+        (by
+          have h : ((childLL (childLH (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childHH
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childHH
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childHH
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childHH
+            thetaAboveCell000022103011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022103011))
+        (by
+          have h : ((childLL (childHL (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childHH
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childHH
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+            thetaAboveCell000022103011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022103011))
+        (by
+          have h : ((childLL (childHH (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childHH
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childHH
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+            thetaAboveCell000022103011))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaAboveCell000022103011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+            thetaAboveCell000022103011))) h))
+
+theorem cover_subtree_cb9f911d2fd7 :
+    adaptiveCoverCheck 7 thetaAboveCell000022103011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103011
+    (by
+      have h : ((childLL thetaAboveCell000022103011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLL thetaAboveCell000022103011) h)
+    (by
+      have h : ((childLH thetaAboveCell000022103011)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childLH thetaAboveCell000022103011) h)
+    cover_subtree_38c5b4ed074a
+    cover_subtree_0e2d12c802dd
+
+theorem cover_subtree_f068368a2274 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022103012) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022103012)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022103012))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022103012))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022103012))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022103012))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022103012))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022103012))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022103012))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022103012))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022103012))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022103012))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022103012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022103012)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022103012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022103012)) h)
+
+theorem cover_subtree_2a95bb96602e :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022103012) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022103012)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022103012))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022103012))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022103012))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022103012))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022103012))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022103012))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022103012))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022103012))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022103012))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022103012)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022103012))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022103012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022103012)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022103012))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022103012)) h)
+
+theorem cover_subtree_75584557442b :
+    adaptiveCoverCheck 7 thetaAboveCell000022103012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103012
+    cover_subtree_f068368a2274
+    cover_subtree_2a95bb96602e
+    (by
+      have h : ((childHL thetaAboveCell000022103012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022103012) h)
+    (by
+      have h : ((childHH thetaAboveCell000022103012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022103012) h)
+
+theorem cover_subtree_5dbf852f82f6 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022103013) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022103013)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLL thetaAboveCell000022103013))
+        (by
+          have h : ((childLL (childLL (childLL thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLL
+            thetaAboveCell000022103013))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLL
+            thetaAboveCell000022103013))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLL
+            thetaAboveCell000022103013))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLL
+            thetaAboveCell000022103013))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLL thetaAboveCell000022103013))
+        (by
+          have h : ((childLL (childLH (childLL thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLL
+            thetaAboveCell000022103013))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLL
+            thetaAboveCell000022103013))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLL
+            thetaAboveCell000022103013))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLL
+            thetaAboveCell000022103013))) h))
+    (by
+      have h : ((childHL (childLL thetaAboveCell000022103013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL thetaAboveCell000022103013)) h)
+    (by
+      have h : ((childHH (childLL thetaAboveCell000022103013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL thetaAboveCell000022103013)) h)
+
+theorem cover_subtree_4ebc7849af7a :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022103013) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022103013)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL (childLH thetaAboveCell000022103013))
+        (by
+          have h : ((childLL (childLL (childLH thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL (childLH
+            thetaAboveCell000022103013))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL (childLH
+            thetaAboveCell000022103013))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL (childLH
+            thetaAboveCell000022103013))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL (childLH
+            thetaAboveCell000022103013))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH (childLH thetaAboveCell000022103013))
+        (by
+          have h : ((childLL (childLH (childLH thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH (childLH
+            thetaAboveCell000022103013))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH (childLH
+            thetaAboveCell000022103013))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH (childLH
+            thetaAboveCell000022103013))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaAboveCell000022103013)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH (childLH
+            thetaAboveCell000022103013))) h))
+    (by
+      have h : ((childHL (childLH thetaAboveCell000022103013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH thetaAboveCell000022103013)) h)
+    (by
+      have h : ((childHH (childLH thetaAboveCell000022103013))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH thetaAboveCell000022103013)) h)
+
+theorem cover_subtree_e5feda84d791 :
+    adaptiveCoverCheck 7 thetaAboveCell000022103013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022103013
+    cover_subtree_5dbf852f82f6
+    cover_subtree_4ebc7849af7a
+    (by
+      have h : ((childHL thetaAboveCell000022103013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022103013) h)
+    (by
+      have h : ((childHH thetaAboveCell000022103013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022103013) h)
+
+theorem cover_subtree_7644ff41fef4 :
+    adaptiveCoverCheck 8 (childLH (childLL (childHH thetaAboveCell00002210))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHH thetaAboveCell00002210)))
+    cover_subtree_712a58fdc8d3
+    cover_subtree_cb9f911d2fd7
+    cover_subtree_75584557442b
+    cover_subtree_e5feda84d791
+
+theorem e24KC2ThetaAboveLeaf0000221030 :
+    adaptiveCoverCheck 9 (childLL (childHH thetaAboveCell00002210)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHH thetaAboveCell00002210))
+    cover_subtree_dc732abf341f
+    cover_subtree_7644ff41fef4
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHH
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022103020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103020 h)
+        (by
+          have h : (thetaAboveCell000022103021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103021 h)
+        (by
+          have h : (thetaAboveCell000022103022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103022 h)
+        (by
+          have h : (thetaAboveCell000022103023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHH
+        thetaAboveCell00002210)))
+        (by
+          have h : (thetaAboveCell000022103030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103030 h)
+        (by
+          have h : (thetaAboveCell000022103031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103031 h)
+        (by
+          have h : (thetaAboveCell000022103032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103032 h)
+        (by
+          have h : (thetaAboveCell000022103033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022103033 h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsea4abf6409
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsea4abf6409
+
+open CertificateCellsea4abf6409
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c0_c3_c3 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022002100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022002100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022002100)))
+        (by
+          have h : (thetaAboveCell0000220021003300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003300 h)
+        (by
+          have h : (thetaAboveCell0000220021003301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003301 h)
+        (by
+          have h : (thetaAboveCell0000220021003302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003302 h)
+        (by
+          have h : (thetaAboveCell0000220021003303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022002100)))
+        (by
+          have h : (thetaAboveCell0000220021003310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003310 h)
+        (by
+          have h : (thetaAboveCell0000220021003311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003311 h)
+        (by
+          have h : (thetaAboveCell0000220021003312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003312 h)
+        (by
+          have h : (thetaAboveCell0000220021003313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022002100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022002100))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022002100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022002100))) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle019.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle019.lean
new file mode 100644
index 0000000000..849bcc3ca6
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle019.lean
@@ -0,0 +1,10233 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch032`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCells27082ce600
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331130100210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100210 : AngleCell :=
+  childLL (childLH (childHL (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100211 : AngleCell :=
+  childLH (childLH (childHL (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100212 : AngleCell :=
+  childHL (childLH (childHL (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100213 : AngleCell :=
+  childHH (childLH (childHL (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100300 : AngleCell :=
+  childLL (childLL (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100301 : AngleCell :=
+  childLH (childLL (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100302 : AngleCell :=
+  childHL (childLL (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100303 : AngleCell :=
+  childHH (childLL (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100310 : AngleCell :=
+  childLL (childLH (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100311 : AngleCell :=
+  childLH (childLH (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100312 : AngleCell :=
+  childHL (childLH (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100313 : AngleCell :=
+  childHH (childLH (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130100333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130100333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101020 : AngleCell :=
+  childLL (childHL (childLL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101021 : AngleCell :=
+  childLH (childHL (childLL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101022 : AngleCell :=
+  childHL (childHL (childLL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101023 : AngleCell :=
+  childHH (childHL (childLL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101030 : AngleCell :=
+  childLL (childHH (childLL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101031 : AngleCell :=
+  childLH (childHH (childLL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101032 : AngleCell :=
+  childHL (childHH (childLL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101033 : AngleCell :=
+  childHH (childHH (childLL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101120 : AngleCell :=
+  childLL (childHL (childLH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101121 : AngleCell :=
+  childLH (childHL (childLH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101122 : AngleCell :=
+  childHL (childHL (childLH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101123 : AngleCell :=
+  childHH (childHL (childLH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101130 : AngleCell :=
+  childLL (childHH (childLH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101131 : AngleCell :=
+  childLH (childHH (childLH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101132 : AngleCell :=
+  childHL (childHH (childLH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101133 : AngleCell :=
+  childHH (childHH (childLH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101200 : AngleCell :=
+  childLL (childLL (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101201 : AngleCell :=
+  childLH (childLL (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101202 : AngleCell :=
+  childHL (childLL (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101203 : AngleCell :=
+  childHH (childLL (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101210 : AngleCell :=
+  childLL (childLH (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101211 : AngleCell :=
+  childLH (childLH (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101212 : AngleCell :=
+  childHL (childLH (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101213 : AngleCell :=
+  childHH (childLH (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101300 : AngleCell :=
+  childLL (childLL (childHH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101301 : AngleCell :=
+  childLH (childLL (childHH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101302 : AngleCell :=
+  childHL (childLL (childHH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101303 : AngleCell :=
+  childHH (childLL (childHH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101310 : AngleCell :=
+  childLL (childLH (childHH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101311 : AngleCell :=
+  childLH (childLH (childHH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101312 : AngleCell :=
+  childHL (childLH (childHH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101313 : AngleCell :=
+  childHH (childLH (childHH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaBelowCell111133113010)))
+
+/-- Subcell `1111331130101333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130101333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaBelowCell111133113010)))
+
+end GerverSofa.PartE.CertificateCells27082ce600
+
+namespace GerverSofa.PartE.CertificateCellsc3c599388b
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `111133113121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `111133113122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `111133113123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCellsc3c599388b
+
+namespace GerverSofa.PartE.CertificateCells6f9c2931ea
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `000022002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `000022002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `000022002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells6f9c2931ea
+
+namespace GerverSofa.PartE.CertificateCells5955d51af9
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021013000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022002101)))
+
+end GerverSofa.PartE.CertificateCells5955d51af9
+
+namespace GerverSofa.PartE.CertificateCellsde04ff5fef
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131000020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000020 : AngleCell :=
+  childLL (childHL (childLL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000021 : AngleCell :=
+  childLH (childHL (childLL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000022 : AngleCell :=
+  childHL (childHL (childLL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000023 : AngleCell :=
+  childHH (childHL (childLL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000030 : AngleCell :=
+  childLL (childHH (childLL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000031 : AngleCell :=
+  childLH (childHH (childLL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000032 : AngleCell :=
+  childHL (childHH (childLL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000033 : AngleCell :=
+  childHH (childHH (childLL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000120 : AngleCell :=
+  childLL (childHL (childLH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000121 : AngleCell :=
+  childLH (childHL (childLH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000122 : AngleCell :=
+  childHL (childHL (childLH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000123 : AngleCell :=
+  childHH (childHL (childLH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000130 : AngleCell :=
+  childLL (childHH (childLH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000131 : AngleCell :=
+  childLH (childHH (childLH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000132 : AngleCell :=
+  childHL (childHH (childLH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000133 : AngleCell :=
+  childHH (childHH (childLH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000200 : AngleCell :=
+  childLL (childLL (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000201 : AngleCell :=
+  childLH (childLL (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000202 : AngleCell :=
+  childHL (childLL (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000203 : AngleCell :=
+  childHH (childLL (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000210 : AngleCell :=
+  childLL (childLH (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000211 : AngleCell :=
+  childLH (childLH (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000212 : AngleCell :=
+  childHL (childLH (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000213 : AngleCell :=
+  childHH (childLH (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000300 : AngleCell :=
+  childLL (childLL (childHH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000301 : AngleCell :=
+  childLH (childLL (childHH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000302 : AngleCell :=
+  childHL (childLL (childHH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000303 : AngleCell :=
+  childHH (childLL (childHH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000310 : AngleCell :=
+  childLL (childLH (childHH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000311 : AngleCell :=
+  childLH (childLH (childHH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000312 : AngleCell :=
+  childHL (childLH (childHH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000313 : AngleCell :=
+  childHH (childLH (childHH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131000333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131000333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaBelowCell111133113100)))
+
+end GerverSofa.PartE.CertificateCellsde04ff5fef
+
+namespace GerverSofa.PartE.CertificateCells750522796b
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCells750522796b
+
+namespace GerverSofa.PartE.CertificateCells99ba2b246f
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021012100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022002101)))
+
+end GerverSofa.PartE.CertificateCells99ba2b246f
+
+namespace GerverSofa.PartE.CertificateCells6443c5ba4b
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021112000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022002111)))
+
+end GerverSofa.PartE.CertificateCells6443c5ba4b
+
+namespace GerverSofa.PartE.CertificateCells03dbb8f506
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells03dbb8f506
+
+namespace GerverSofa.PartE.CertificateCells15b8ca5094
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022002101)))
+
+end GerverSofa.PartE.CertificateCells15b8ca5094
+
+namespace GerverSofa.PartE.CertificateCellsecf9e4e90b
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCellsecf9e4e90b
+
+namespace GerverSofa.PartE.CertificateCellsdb6ee29ee4
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021103000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022002110)))
+
+end GerverSofa.PartE.CertificateCellsdb6ee29ee4
+
+namespace GerverSofa.PartE.CertificateCells4003309a28
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `000022002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `000022002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `000022002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells4003309a28
+
+namespace GerverSofa.PartE.CertificateCellsf3f1dbc459
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022002100)))
+
+end GerverSofa.PartE.CertificateCellsf3f1dbc459
+
+namespace GerverSofa.PartE.CertificateCells441cc2b0d8
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131113100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaBelowCell111133113111)))
+
+end GerverSofa.PartE.CertificateCells441cc2b0d8
+
+namespace GerverSofa.PartE.CertificateCells5980303e93
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells5980303e93
+
+namespace GerverSofa.PartE.CertificateCellsa9276513a2
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCellsa9276513a2
+
+namespace GerverSofa.PartE.CertificateCells9972cada5a
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+end GerverSofa.PartE.CertificateCells9972cada5a
+
+namespace GerverSofa.PartE.CertificateCellsa6306b8bb7
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131112000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaBelowCell111133113111)))
+
+end GerverSofa.PartE.CertificateCellsa6306b8bb7
+
+namespace GerverSofa.PartE.CertificateCellsc1e1941644
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131012000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131012001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131012002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131012003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131012010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131012011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131012012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131012013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131012100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131012101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131012102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131012103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131012110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131012111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131012112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131012113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131012113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaBelowCell111133113101)))
+
+end GerverSofa.PartE.CertificateCellsc1e1941644
+
+namespace GerverSofa.PartE.CertificateCells5a169f617b
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells5a169f617b
+
+namespace GerverSofa.PartE.CertificateCellsc15ab9b9a5
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCellsc15ab9b9a5
+
+namespace GerverSofa.PartE.CertificateCells861ec8c785
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells861ec8c785
+
+namespace GerverSofa.PartE.CertificateCells854f59bf16
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaBelowCell11113311)))
+
+/-- Subcell `111133113001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaBelowCell11113311)))
+
+/-- Subcell `111133113002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaBelowCell11113311)))
+
+/-- Subcell `111133113003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331130011320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130011320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaBelowCell111133113001)))
+
+/-- Subcell `1111331130011321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130011321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaBelowCell111133113001)))
+
+/-- Subcell `1111331130011322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130011322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaBelowCell111133113001)))
+
+/-- Subcell `1111331130011323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130011323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaBelowCell111133113001)))
+
+/-- Subcell `1111331130011330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130011330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaBelowCell111133113001)))
+
+/-- Subcell `1111331130011331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130011331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaBelowCell111133113001)))
+
+/-- Subcell `1111331130011332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130011332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaBelowCell111133113001)))
+
+/-- Subcell `1111331130011333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130011333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaBelowCell111133113001)))
+
+end GerverSofa.PartE.CertificateCells854f59bf16
+
+namespace GerverSofa.PartE.CertificateCells5153122198
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131112100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131112133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaBelowCell111133113111)))
+
+end GerverSofa.PartE.CertificateCells5153122198
+
+namespace GerverSofa.PartE.CertificateCells0a969d2c42
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131102000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131102001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131102002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131102003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131102010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131102011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131102012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131102013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131102100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131102101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131102102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131102103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131102110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131102111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131102112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131102113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131102113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaBelowCell111133113110)))
+
+end GerverSofa.PartE.CertificateCells0a969d2c42
+
+namespace GerverSofa.PartE.CertificateCells50ae5a428f
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131101220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101200 : AngleCell :=
+  childLL (childLL (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101201 : AngleCell :=
+  childLH (childLL (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101202 : AngleCell :=
+  childHL (childLL (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101203 : AngleCell :=
+  childHH (childLL (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101210 : AngleCell :=
+  childLL (childLH (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101211 : AngleCell :=
+  childLH (childLH (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101212 : AngleCell :=
+  childHL (childLH (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101213 : AngleCell :=
+  childHH (childLH (childHL (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101300 : AngleCell :=
+  childLL (childLL (childHH (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101301 : AngleCell :=
+  childLH (childLL (childHH (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101302 : AngleCell :=
+  childHL (childLL (childHH (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101303 : AngleCell :=
+  childHH (childLL (childHH (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101310 : AngleCell :=
+  childLL (childLH (childHH (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101311 : AngleCell :=
+  childLH (childLH (childHH (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101312 : AngleCell :=
+  childHL (childLH (childHH (childLH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131101313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131101313 : AngleCell :=
+  childHH (childLH (childHH (childLH thetaBelowCell111133113110)))
+
+end GerverSofa.PartE.CertificateCells50ae5a428f
+
+namespace GerverSofa.PartE.CertificateCells754567fdeb
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022002101)))
+
+end GerverSofa.PartE.CertificateCells754567fdeb
+
+namespace GerverSofa.PartE.CertificateCells146a7460c7
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022002100)))
+
+end GerverSofa.PartE.CertificateCells146a7460c7
+
+namespace GerverSofa.PartE.CertificateCells5446bfae87
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131100220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100200 : AngleCell :=
+  childLL (childLL (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100201 : AngleCell :=
+  childLH (childLL (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100202 : AngleCell :=
+  childHL (childLL (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100203 : AngleCell :=
+  childHH (childLL (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100210 : AngleCell :=
+  childLL (childLH (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100211 : AngleCell :=
+  childLH (childLH (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100212 : AngleCell :=
+  childHL (childLH (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100213 : AngleCell :=
+  childHH (childLH (childHL (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100300 : AngleCell :=
+  childLL (childLL (childHH (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100301 : AngleCell :=
+  childLH (childLL (childHH (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100302 : AngleCell :=
+  childHL (childLL (childHH (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100303 : AngleCell :=
+  childHH (childLL (childHH (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100310 : AngleCell :=
+  childLL (childLH (childHH (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100311 : AngleCell :=
+  childLH (childLH (childHH (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100312 : AngleCell :=
+  childHL (childLH (childHH (childLL thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131100313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131100313 : AngleCell :=
+  childHH (childLH (childHH (childLL thetaBelowCell111133113110)))
+
+end GerverSofa.PartE.CertificateCells5446bfae87
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6R4Subtree01d19858ee032aa8`.
+* `KernelOnly.PartE.E24KC6R4Subtree04c8a4da4d3d19a1`.
+* `KernelOnly.PartE.E24KC6R4Subtree1033056e03edbb99`.
+* `KernelOnly.PartE.E24KC6R4Subtree1952c351f5a890d0`.
+* `KernelOnly.PartE.E24KC6R4Subtree1974d41b8023f47d`.
+* `KernelOnly.PartE.E24KC6R4Subtree19ea3647d9c4385d`.
+* `KernelOnly.PartE.E24KC6R4Subtree1df7325990c509b9`.
+* `KernelOnly.PartE.E24KC6R4Subtree1ec47ff132608c89`.
+* `KernelOnly.PartE.E24KC6R4Subtree23aa11edc1c0beab`.
+* `KernelOnly.PartE.E24KC6R4Subtree291bec528ce09a45`.
+* `KernelOnly.PartE.E24KC6R4Subtree31a1bf2506de3a6b`.
+* `KernelOnly.PartE.E24KC6R4Subtree35d8a6b024243cfb`.
+* `KernelOnly.PartE.E24KC6R4Subtree369cc1bf8a73ac93`.
+* `KernelOnly.PartE.E24KC6R4Subtree3bdce31885f964fb`.
+* `KernelOnly.PartE.E24KC6R4Subtree3e7c6f0a436cdb12`.
+* `KernelOnly.PartE.E24KC6R4Subtree403b497f85c69738`.
+* `KernelOnly.PartE.E24KC6R4Subtree428fdb9c294c5957`.
+* `KernelOnly.PartE.E24KC6R4Subtree4417485270da2542`.
+* `KernelOnly.PartE.E24KC6R4Subtree55d98d161eaa82bf`.
+* `KernelOnly.PartE.E24KC6R4Subtree60715e5435db178e`.
+* `KernelOnly.PartE.E24KC6R4Subtree60cca7d0a876716e`.
+* `KernelOnly.PartE.E24KC6R4Subtree639075a28aed7212`.
+* `KernelOnly.PartE.E24KC6R4Subtree6ec7ee50125cb2b8`.
+* `KernelOnly.PartE.E24KC6R4Subtree70c72a172c9cfd93`.
+* `KernelOnly.PartE.E24KC6R4Subtree70d61ab175d41a4a`.
+* `KernelOnly.PartE.E24KC6R4Subtree7734a7147db71699`.
+* `KernelOnly.PartE.E24KC6R4Subtree77fe1988ff8cd192`.
+* `KernelOnly.PartE.E24KC6R4Subtree7a80e3dfcc4f34f3`.
+* `KernelOnly.PartE.E24KC6R4Subtree7a9ec2fb66ec8862`.
+* `KernelOnly.PartE.E24KC6R4Subtree7ae79e99a9b79eaf`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells27082ce600
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells27082ce600
+
+open CertificateCells27082ce600
+theorem cover_subtree_49690a55e534 :
+    adaptiveCoverCheck 4 (childHL (childLL thetaBelowCell111133113010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL thetaBelowCell111133113010))
+    (by
+      have h : ((childLL (childHL (childLL thetaBelowCell111133113010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childLL
+        thetaBelowCell111133113010))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHL (childLL
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130100210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100210 h)
+        (by
+          have h : (thetaBelowCell1111331130100211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100211 h)
+        (by
+          have h : (thetaBelowCell1111331130100212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100212 h)
+        (by
+          have h : (thetaBelowCell1111331130100213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHL (childLL
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130100220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100220 h)
+        (by
+          have h : (thetaBelowCell1111331130100221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100221 h)
+        (by
+          have h : (thetaBelowCell1111331130100222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100222 h)
+        (by
+          have h : (thetaBelowCell1111331130100223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHL (childLL
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130100230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100230 h)
+        (by
+          have h : (thetaBelowCell1111331130100231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100231 h)
+        (by
+          have h : (thetaBelowCell1111331130100232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100232 h)
+        (by
+          have h : (thetaBelowCell1111331130100233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100233 h))
+
+theorem cover_subtree_ca9ffb091c53 :
+    adaptiveCoverCheck 4 (childHH (childLL thetaBelowCell111133113010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL thetaBelowCell111133113010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHH (childLL
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130100300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100300 h)
+        (by
+          have h : (thetaBelowCell1111331130100301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100301 h)
+        (by
+          have h : (thetaBelowCell1111331130100302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100302 h)
+        (by
+          have h : (thetaBelowCell1111331130100303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHH (childLL
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130100310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100310 h)
+        (by
+          have h : (thetaBelowCell1111331130100311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100311 h)
+        (by
+          have h : (thetaBelowCell1111331130100312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100312 h)
+        (by
+          have h : (thetaBelowCell1111331130100313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHH (childLL
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130100320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100320 h)
+        (by
+          have h : (thetaBelowCell1111331130100321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100321 h)
+        (by
+          have h : (thetaBelowCell1111331130100322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100322 h)
+        (by
+          have h : (thetaBelowCell1111331130100323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHH (childLL
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130100330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100330 h)
+        (by
+          have h : (thetaBelowCell1111331130100331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100331 h)
+        (by
+          have h : (thetaBelowCell1111331130100332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100332 h)
+        (by
+          have h : (thetaBelowCell1111331130100333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130100333 h))
+
+theorem cover_subtree_afcf5ef458d7 :
+    adaptiveCoverCheck 5 (childLL thetaBelowCell111133113010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133113010)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL thetaBelowCell111133113010))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLL
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLL
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLL
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLL
+            thetaBelowCell111133113010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL thetaBelowCell111133113010))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLL
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLL
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLL
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLL
+            thetaBelowCell111133113010))) h))
+    cover_subtree_49690a55e534
+    cover_subtree_ca9ffb091c53
+
+theorem cover_subtree_51fb51ed5b56 :
+    adaptiveCoverCheck 4 (childLL (childLH thetaBelowCell111133113010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH thetaBelowCell111133113010))
+    (by
+      have h : ((childLL (childLL (childLH thetaBelowCell111133113010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLH
+        thetaBelowCell111133113010))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaBelowCell111133113010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLH
+        thetaBelowCell111133113010))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childLL (childLH
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130101020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101020 h)
+        (by
+          have h : (thetaBelowCell1111331130101021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101021 h)
+        (by
+          have h : (thetaBelowCell1111331130101022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101022 h)
+        (by
+          have h : (thetaBelowCell1111331130101023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childLL (childLH
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130101030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101030 h)
+        (by
+          have h : (thetaBelowCell1111331130101031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101031 h)
+        (by
+          have h : (thetaBelowCell1111331130101032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101032 h)
+        (by
+          have h : (thetaBelowCell1111331130101033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101033 h))
+
+theorem cover_subtree_f305258c7368 :
+    adaptiveCoverCheck 4 (childLH (childLH thetaBelowCell111133113010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH thetaBelowCell111133113010))
+    (by
+      have h : ((childLL (childLH (childLH thetaBelowCell111133113010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLH
+        thetaBelowCell111133113010))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaBelowCell111133113010)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLH
+        thetaBelowCell111133113010))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childLH (childLH
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130101120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101120 h)
+        (by
+          have h : (thetaBelowCell1111331130101121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101121 h)
+        (by
+          have h : (thetaBelowCell1111331130101122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101122 h)
+        (by
+          have h : (thetaBelowCell1111331130101123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childLH (childLH
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130101130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101130 h)
+        (by
+          have h : (thetaBelowCell1111331130101131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101131 h)
+        (by
+          have h : (thetaBelowCell1111331130101132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101132 h)
+        (by
+          have h : (thetaBelowCell1111331130101133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101133 h))
+
+theorem cover_subtree_5599471dbfbf :
+    adaptiveCoverCheck 4 (childHL (childLH thetaBelowCell111133113010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH thetaBelowCell111133113010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHL (childLH
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130101200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101200 h)
+        (by
+          have h : (thetaBelowCell1111331130101201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101201 h)
+        (by
+          have h : (thetaBelowCell1111331130101202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101202 h)
+        (by
+          have h : (thetaBelowCell1111331130101203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHL (childLH
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130101210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101210 h)
+        (by
+          have h : (thetaBelowCell1111331130101211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101211 h)
+        (by
+          have h : (thetaBelowCell1111331130101212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101212 h)
+        (by
+          have h : (thetaBelowCell1111331130101213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHL (childLH
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130101220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101220 h)
+        (by
+          have h : (thetaBelowCell1111331130101221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101221 h)
+        (by
+          have h : (thetaBelowCell1111331130101222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101222 h)
+        (by
+          have h : (thetaBelowCell1111331130101223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHL (childLH
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130101230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101230 h)
+        (by
+          have h : (thetaBelowCell1111331130101231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101231 h)
+        (by
+          have h : (thetaBelowCell1111331130101232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101232 h)
+        (by
+          have h : (thetaBelowCell1111331130101233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101233 h))
+
+theorem cover_subtree_5b939cb0e15c :
+    adaptiveCoverCheck 4 (childHH (childLH thetaBelowCell111133113010)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH thetaBelowCell111133113010))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHH (childLH
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130101300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101300 h)
+        (by
+          have h : (thetaBelowCell1111331130101301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101301 h)
+        (by
+          have h : (thetaBelowCell1111331130101302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101302 h)
+        (by
+          have h : (thetaBelowCell1111331130101303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHH (childLH
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130101310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101310 h)
+        (by
+          have h : (thetaBelowCell1111331130101311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101311 h)
+        (by
+          have h : (thetaBelowCell1111331130101312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101312 h)
+        (by
+          have h : (thetaBelowCell1111331130101313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHH (childLH
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130101320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101320 h)
+        (by
+          have h : (thetaBelowCell1111331130101321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101321 h)
+        (by
+          have h : (thetaBelowCell1111331130101322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101322 h)
+        (by
+          have h : (thetaBelowCell1111331130101323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHH (childLH
+        thetaBelowCell111133113010)))
+        (by
+          have h : (thetaBelowCell1111331130101330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101330 h)
+        (by
+          have h : (thetaBelowCell1111331130101331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101331 h)
+        (by
+          have h : (thetaBelowCell1111331130101332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101332 h)
+        (by
+          have h : (thetaBelowCell1111331130101333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130101333 h))
+
+theorem cover_subtree_2f4f93c67946 :
+    adaptiveCoverCheck 5 (childLH thetaBelowCell111133113010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133113010)
+    cover_subtree_51fb51ed5b56
+    cover_subtree_f305258c7368
+    cover_subtree_5599471dbfbf
+    cover_subtree_5b939cb0e15c
+
+theorem cover_subtree_1374b272a307 :
+    adaptiveCoverCheck 5 (childHL thetaBelowCell111133113010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133113010)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL thetaBelowCell111133113010))
+        (by
+          have h : ((childLL (childLL (childHL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childHL
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childHL
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childHL
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childHL
+            thetaBelowCell111133113010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL thetaBelowCell111133113010))
+        (by
+          have h : ((childLL (childLH (childHL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childHL
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childHL
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childHL
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childHL
+            thetaBelowCell111133113010))) h))
+    (by
+      have h : ((childHL (childHL thetaBelowCell111133113010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL thetaBelowCell111133113010)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell111133113010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL thetaBelowCell111133113010)) h)
+
+theorem cover_subtree_91c9f04664e0 :
+    adaptiveCoverCheck 5 (childHH thetaBelowCell111133113010) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133113010)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH thetaBelowCell111133113010))
+        (by
+          have h : ((childLL (childLL (childHH thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childHH
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childHH
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childHH
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childHH
+            thetaBelowCell111133113010))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH thetaBelowCell111133113010))
+        (by
+          have h : ((childLL (childLH (childHH thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childHH
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childHH
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childHH
+            thetaBelowCell111133113010))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaBelowCell111133113010)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childHH
+            thetaBelowCell111133113010))) h))
+    (by
+      have h : ((childHL (childHH thetaBelowCell111133113010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH thetaBelowCell111133113010)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell111133113010))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH thetaBelowCell111133113010)) h)
+
+theorem e24KC2ThetaBelowLeaf111133113_c0_c1_c0 :
+    adaptiveCoverCheck 6 thetaBelowCell111133113010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113010
+    cover_subtree_afcf5ef458d7
+    cover_subtree_2f4f93c67946
+    cover_subtree_1374b272a307
+    cover_subtree_91c9f04664e0
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc3c599388b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc3c599388b
+
+open CertificateCellsc3c599388b
+theorem e24KC2ThetaBelowLeaf111133113_c1_c2 :
+    adaptiveCoverCheck 7 (childHL (childLH (childHH thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLH (childHH thetaBelowCell11113311)))
+    (by
+      have h : (thetaBelowCell111133113120).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113120 h)
+    (by
+      have h : (thetaBelowCell111133113121).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113121 h)
+    (by
+      have h : (thetaBelowCell111133113122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113122 h)
+    (by
+      have h : (thetaBelowCell111133113123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113123 h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6f9c2931ea
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6f9c2931ea
+
+open CertificateCells6f9c2931ea
+theorem e24KC2ThetaAboveLeaf0000220021_c2 :
+    adaptiveCoverCheck 8 (childHL (childLH (childHL thetaAboveCell00002200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLH (childHL thetaAboveCell00002200)))
+    (by
+      have h : (thetaAboveCell000022002120).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002120 h)
+    (by
+      have h : (thetaAboveCell000022002121).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002121 h)
+    (by
+      have h : (thetaAboveCell000022002122).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002122 h)
+    (by
+      have h : (thetaAboveCell000022002123).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002123 h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5955d51af9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5955d51af9
+
+open CertificateCells5955d51af9
+theorem cover_subtree_f2e068b42d23 :
+    adaptiveCoverCheck 4 (childLL (childLL (childHH thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHH
+    thetaAboveCell000022002101)))
+    (by
+      have h : (thetaAboveCell0000220021013000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013000 h)
+    (by
+      have h : (thetaAboveCell0000220021013001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013002
+        (by
+          have h : ((childLL thetaAboveCell0000220021013002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013002) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013002) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013002) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013003
+        (by
+          have h : ((childLL thetaAboveCell0000220021013003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013003) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013003) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013003) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013003) h))
+
+theorem cover_subtree_6c2a359fbe9a :
+    adaptiveCoverCheck 4 (childLH (childLL (childHH thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHH
+    thetaAboveCell000022002101)))
+    (by
+      have h : (thetaAboveCell0000220021013010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013010 h)
+    (by
+      have h : (thetaAboveCell0000220021013011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013012
+        (by
+          have h : ((childLL thetaAboveCell0000220021013012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013012) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013012) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013012) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013013
+        (by
+          have h : ((childLL thetaAboveCell0000220021013013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013013) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013013) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013013) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013013) h))
+
+theorem cover_subtree_fe0ef8f0f647 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHH thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+    thetaAboveCell000022002101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013020
+        (by
+          have h : ((childLL thetaAboveCell0000220021013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013021
+        (by
+          have h : ((childLL thetaAboveCell0000220021013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013022
+        (by
+          have h : ((childLL thetaAboveCell0000220021013022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013023
+        (by
+          have h : ((childLL thetaAboveCell0000220021013023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013023) h))
+
+theorem cover_subtree_fdabca9a4955 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHH thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+    thetaAboveCell000022002101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013030
+        (by
+          have h : ((childLL thetaAboveCell0000220021013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013031
+        (by
+          have h : ((childLL thetaAboveCell0000220021013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013032
+        (by
+          have h : ((childLL thetaAboveCell0000220021013032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013033
+        (by
+          have h : ((childLL thetaAboveCell0000220021013033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013033) h))
+
+theorem cover_subtree_c4b02e6e9e8a :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022002101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022002101))
+    cover_subtree_f2e068b42d23
+    cover_subtree_6c2a359fbe9a
+    cover_subtree_fe0ef8f0f647
+    cover_subtree_fdabca9a4955
+
+theorem cover_subtree_6859a502ecc6 :
+    adaptiveCoverCheck 4 (childLL (childLH (childHH thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHH
+    thetaAboveCell000022002101)))
+    (by
+      have h : (thetaAboveCell0000220021013100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013100 h)
+    (by
+      have h : (thetaAboveCell0000220021013101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013102
+        (by
+          have h : ((childLL thetaAboveCell0000220021013102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013102) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013102) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013102) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013103
+        (by
+          have h : ((childLL thetaAboveCell0000220021013103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013103) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013103) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013103) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013103) h))
+
+theorem cover_subtree_f4d742b10a6f :
+    adaptiveCoverCheck 4 (childLH (childLH (childHH thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHH
+    thetaAboveCell000022002101)))
+    (by
+      have h : (thetaAboveCell0000220021013110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013110 h)
+    (by
+      have h : (thetaAboveCell0000220021013111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013112
+        (by
+          have h : ((childLL thetaAboveCell0000220021013112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013112) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013112) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013112) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013113
+        (by
+          have h : ((childLL thetaAboveCell0000220021013113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013113) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013113) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013113) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013113) h))
+
+theorem cover_subtree_659abecde61a :
+    adaptiveCoverCheck 4 (childHL (childLH (childHH thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+    thetaAboveCell000022002101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013120
+        (by
+          have h : ((childLL thetaAboveCell0000220021013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013121
+        (by
+          have h : ((childLL thetaAboveCell0000220021013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013122
+        (by
+          have h : ((childLL thetaAboveCell0000220021013122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013123
+        (by
+          have h : ((childLL thetaAboveCell0000220021013123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013123) h))
+
+theorem cover_subtree_ad8d4e6209b9 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHH thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+    thetaAboveCell000022002101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013130
+        (by
+          have h : ((childLL thetaAboveCell0000220021013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013131
+        (by
+          have h : ((childLL thetaAboveCell0000220021013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013132
+        (by
+          have h : ((childLL thetaAboveCell0000220021013132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021013133
+        (by
+          have h : ((childLL thetaAboveCell0000220021013133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021013133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021013133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021013133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021013133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021013133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021013133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021013133) h))
+
+theorem cover_subtree_465622ba8fb4 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022002101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022002101))
+    cover_subtree_6859a502ecc6
+    cover_subtree_f4d742b10a6f
+    cover_subtree_659abecde61a
+    cover_subtree_ad8d4e6209b9
+
+theorem cover_subtree_f5a0481cebfc :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022002101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022002101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022002101)))
+        (by
+          have h : (thetaAboveCell0000220021013200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013200 h)
+        (by
+          have h : (thetaAboveCell0000220021013201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013201 h)
+        (by
+          have h : (thetaAboveCell0000220021013202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013202 h)
+        (by
+          have h : (thetaAboveCell0000220021013203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022002101)))
+        (by
+          have h : (thetaAboveCell0000220021013210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013210 h)
+        (by
+          have h : (thetaAboveCell0000220021013211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013211 h)
+        (by
+          have h : (thetaAboveCell0000220021013212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013212 h)
+        (by
+          have h : (thetaAboveCell0000220021013213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022002101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022002101))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022002101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022002101))) h)
+
+theorem cover_subtree_c6ead7987dc9 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022002101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022002101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022002101)))
+        (by
+          have h : (thetaAboveCell0000220021013300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013300 h)
+        (by
+          have h : (thetaAboveCell0000220021013301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013301 h)
+        (by
+          have h : (thetaAboveCell0000220021013302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013302 h)
+        (by
+          have h : (thetaAboveCell0000220021013303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022002101)))
+        (by
+          have h : (thetaAboveCell0000220021013310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013310 h)
+        (by
+          have h : (thetaAboveCell0000220021013311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013311 h)
+        (by
+          have h : (thetaAboveCell0000220021013312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013312 h)
+        (by
+          have h : (thetaAboveCell0000220021013313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021013313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022002101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022002101))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022002101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022002101))) h)
+
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c1_c3 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022002101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022002101)
+    cover_subtree_c4b02e6e9e8a
+    cover_subtree_465622ba8fb4
+    cover_subtree_f5a0481cebfc
+    cover_subtree_c6ead7987dc9
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsde04ff5fef
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsde04ff5fef
+
+open CertificateCellsde04ff5fef
+theorem cover_subtree_0d52430fd500 :
+    adaptiveCoverCheck 4 (childLL (childLL thetaBelowCell111133113100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL thetaBelowCell111133113100))
+    (by
+      have h : ((childLL (childLL (childLL thetaBelowCell111133113100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLL
+        thetaBelowCell111133113100))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaBelowCell111133113100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLL
+        thetaBelowCell111133113100))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childLL (childLL
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131000020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000020 h)
+        (by
+          have h : (thetaBelowCell1111331131000021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000021 h)
+        (by
+          have h : (thetaBelowCell1111331131000022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000022 h)
+        (by
+          have h : (thetaBelowCell1111331131000023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childLL (childLL
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131000030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000030 h)
+        (by
+          have h : (thetaBelowCell1111331131000031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000031 h)
+        (by
+          have h : (thetaBelowCell1111331131000032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000032 h)
+        (by
+          have h : (thetaBelowCell1111331131000033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000033 h))
+
+theorem cover_subtree_f0f2a49a1c42 :
+    adaptiveCoverCheck 4 (childLH (childLL thetaBelowCell111133113100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL thetaBelowCell111133113100))
+    (by
+      have h : ((childLL (childLH (childLL thetaBelowCell111133113100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLL
+        thetaBelowCell111133113100))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaBelowCell111133113100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLL
+        thetaBelowCell111133113100))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childLH (childLL
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131000120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000120 h)
+        (by
+          have h : (thetaBelowCell1111331131000121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000121 h)
+        (by
+          have h : (thetaBelowCell1111331131000122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000122 h)
+        (by
+          have h : (thetaBelowCell1111331131000123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childLH (childLL
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131000130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000130 h)
+        (by
+          have h : (thetaBelowCell1111331131000131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000131 h)
+        (by
+          have h : (thetaBelowCell1111331131000132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000132 h)
+        (by
+          have h : (thetaBelowCell1111331131000133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000133 h))
+
+theorem cover_subtree_ac65d29ed71a :
+    adaptiveCoverCheck 4 (childHL (childLL thetaBelowCell111133113100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL thetaBelowCell111133113100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHL (childLL
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131000200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000200 h)
+        (by
+          have h : (thetaBelowCell1111331131000201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000201 h)
+        (by
+          have h : (thetaBelowCell1111331131000202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000202 h)
+        (by
+          have h : (thetaBelowCell1111331131000203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHL (childLL
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131000210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000210 h)
+        (by
+          have h : (thetaBelowCell1111331131000211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000211 h)
+        (by
+          have h : (thetaBelowCell1111331131000212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000212 h)
+        (by
+          have h : (thetaBelowCell1111331131000213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHL (childLL
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131000220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000220 h)
+        (by
+          have h : (thetaBelowCell1111331131000221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000221 h)
+        (by
+          have h : (thetaBelowCell1111331131000222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000222 h)
+        (by
+          have h : (thetaBelowCell1111331131000223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHL (childLL
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131000230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000230 h)
+        (by
+          have h : (thetaBelowCell1111331131000231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000231 h)
+        (by
+          have h : (thetaBelowCell1111331131000232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000232 h)
+        (by
+          have h : (thetaBelowCell1111331131000233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000233 h))
+
+theorem cover_subtree_b80d2217aa18 :
+    adaptiveCoverCheck 4 (childHH (childLL thetaBelowCell111133113100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL thetaBelowCell111133113100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHH (childLL
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131000300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000300 h)
+        (by
+          have h : (thetaBelowCell1111331131000301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000301 h)
+        (by
+          have h : (thetaBelowCell1111331131000302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000302 h)
+        (by
+          have h : (thetaBelowCell1111331131000303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHH (childLL
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131000310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000310 h)
+        (by
+          have h : (thetaBelowCell1111331131000311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000311 h)
+        (by
+          have h : (thetaBelowCell1111331131000312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000312 h)
+        (by
+          have h : (thetaBelowCell1111331131000313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHH (childLL
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131000320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000320 h)
+        (by
+          have h : (thetaBelowCell1111331131000321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000321 h)
+        (by
+          have h : (thetaBelowCell1111331131000322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000322 h)
+        (by
+          have h : (thetaBelowCell1111331131000323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHH (childLL
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131000330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000330 h)
+        (by
+          have h : (thetaBelowCell1111331131000331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000331 h)
+        (by
+          have h : (thetaBelowCell1111331131000332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000332 h)
+        (by
+          have h : (thetaBelowCell1111331131000333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131000333 h))
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c0_c0_c0 :
+    adaptiveCoverCheck 5 (childLL thetaBelowCell111133113100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133113100)
+    cover_subtree_0d52430fd500
+    cover_subtree_f0f2a49a1c42
+    cover_subtree_ac65d29ed71a
+    cover_subtree_b80d2217aa18
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells750522796b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells750522796b
+
+open CertificateCells750522796b
+theorem e24KC2ThetaBelowLeaf111133113_c1_c0_c2 :
+    adaptiveCoverCheck 6 thetaBelowCell111133113102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113102
+    (by
+      have h : ((childLL thetaBelowCell111133113102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133113102) h)
+    (by
+      have h : ((childLH thetaBelowCell111133113102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133113102) h)
+    (by
+      have h : ((childHL thetaBelowCell111133113102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133113102) h)
+    (by
+      have h : ((childHH thetaBelowCell111133113102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133113102) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells99ba2b246f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells99ba2b246f
+
+open CertificateCells99ba2b246f
+theorem cover_subtree_bda1c51066a4 :
+    adaptiveCoverCheck 4 (childLL (childLH (childHL thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHL
+    thetaAboveCell000022002101)))
+    (by
+      have h : (thetaAboveCell0000220021012100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012100 h)
+    (by
+      have h : (thetaAboveCell0000220021012101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012102
+        (by
+          have h : ((childLL thetaAboveCell0000220021012102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012102) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021012102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012102) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021012102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012102) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021012102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012103
+        (by
+          have h : ((childLL thetaAboveCell0000220021012103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012103) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021012103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012103) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021012103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012103) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021012103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012103) h))
+
+theorem cover_subtree_c820aa5287c7 :
+    adaptiveCoverCheck 4 (childLH (childLH (childHL thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHL
+    thetaAboveCell000022002101)))
+    (by
+      have h : (thetaAboveCell0000220021012110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012110 h)
+    (by
+      have h : (thetaAboveCell0000220021012111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012112
+        (by
+          have h : ((childLL thetaAboveCell0000220021012112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012112) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021012112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012112) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021012112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012112) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021012112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012113
+        (by
+          have h : ((childLL thetaAboveCell0000220021012113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012113) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021012113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012113) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021012113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012113) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021012113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012113) h))
+
+theorem cover_subtree_538d3ba3b92e :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021012120 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012120
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021012120)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021012120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021012120)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021012120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021012120)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021012120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021012120)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021012120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021012120)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021012120)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021012120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021012120)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021012120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021012120)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021012120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021012120)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021012120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021012120)) h))
+    (by
+      have h : ((childHL thetaAboveCell0000220021012120)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012120) h)
+    (by
+      have h : ((childHH thetaAboveCell0000220021012120)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012120) h)
+
+theorem cover_subtree_ba0b8d4abb41 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021012121 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012121
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021012121)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021012121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021012121)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021012121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021012121)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021012121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021012121)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021012121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021012121)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021012121)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021012121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021012121)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021012121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021012121)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021012121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021012121)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021012121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021012121)) h))
+    (by
+      have h : ((childHL thetaAboveCell0000220021012121)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012121) h)
+    (by
+      have h : ((childHH thetaAboveCell0000220021012121)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012121) h)
+
+theorem cover_subtree_d11d13dd2be9 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHL thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+    thetaAboveCell000022002101)))
+    cover_subtree_538d3ba3b92e
+    cover_subtree_ba0b8d4abb41
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012122
+        (by
+          have h : ((childLL thetaAboveCell0000220021012122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021012122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021012122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021012122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012123
+        (by
+          have h : ((childLL thetaAboveCell0000220021012123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021012123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021012123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021012123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012123) h))
+
+theorem cover_subtree_60696ffddf60 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021012130 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012130
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021012130)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021012130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021012130)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021012130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021012130)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021012130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021012130)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021012130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021012130)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021012130)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021012130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021012130)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021012130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021012130)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021012130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021012130)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021012130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021012130)) h))
+    (by
+      have h : ((childHL thetaAboveCell0000220021012130)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012130) h)
+    (by
+      have h : ((childHH thetaAboveCell0000220021012130)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012130) h)
+
+theorem cover_subtree_90c18ca6ca12 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHL thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+    thetaAboveCell000022002101)))
+    cover_subtree_60696ffddf60
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012131
+        (by
+          have h : ((childLL thetaAboveCell0000220021012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021012131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012132
+        (by
+          have h : ((childLL thetaAboveCell0000220021012132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021012132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021012132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021012132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012133
+        (by
+          have h : ((childLL thetaAboveCell0000220021012133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021012133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021012133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021012133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012133) h))
+
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c1_c2_c1 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022002101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022002101))
+    cover_subtree_bda1c51066a4
+    cover_subtree_c820aa5287c7
+    cover_subtree_d11d13dd2be9
+    cover_subtree_90c18ca6ca12
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6443c5ba4b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6443c5ba4b
+
+open CertificateCells6443c5ba4b
+theorem cover_subtree_100284623361 :
+    adaptiveCoverCheck 4 (childLL (childLL (childHL thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHL
+    thetaAboveCell000022002111)))
+    (by
+      have h : (thetaAboveCell0000220021112000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112000 h)
+    (by
+      have h : (thetaAboveCell0000220021112001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112002
+        (by
+          have h : ((childLL thetaAboveCell0000220021112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112002) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112002) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112002) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112003
+        (by
+          have h : ((childLL thetaAboveCell0000220021112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112003) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112003) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112003) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112003) h))
+
+theorem cover_subtree_06468370fe05 :
+    adaptiveCoverCheck 4 (childLH (childLL (childHL thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHL
+    thetaAboveCell000022002111)))
+    (by
+      have h : (thetaAboveCell0000220021112010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112010 h)
+    (by
+      have h : (thetaAboveCell0000220021112011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112012
+        (by
+          have h : ((childLL thetaAboveCell0000220021112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112012) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112012) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112012) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112013
+        (by
+          have h : ((childLL thetaAboveCell0000220021112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112013) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112013) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112013) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112013) h))
+
+theorem cover_subtree_a6aeae3aff5e :
+    adaptiveCoverCheck 4 (childHL (childLL (childHL thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+    thetaAboveCell000022002111)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112020
+        (by
+          have h : ((childLL thetaAboveCell0000220021112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112021
+        (by
+          have h : ((childLL thetaAboveCell0000220021112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112022
+        (by
+          have h : ((childLL thetaAboveCell0000220021112022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112023
+        (by
+          have h : ((childLL thetaAboveCell0000220021112023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112023) h))
+
+theorem cover_subtree_e6a5a6a1c8c7 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHL thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+    thetaAboveCell000022002111)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112030
+        (by
+          have h : ((childLL thetaAboveCell0000220021112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112031
+        (by
+          have h : ((childLL thetaAboveCell0000220021112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112032
+        (by
+          have h : ((childLL thetaAboveCell0000220021112032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112033
+        (by
+          have h : ((childLL thetaAboveCell0000220021112033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112033) h))
+
+theorem cover_subtree_b7bd7dd08527 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022002111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022002111))
+    cover_subtree_100284623361
+    cover_subtree_06468370fe05
+    cover_subtree_a6aeae3aff5e
+    cover_subtree_e6a5a6a1c8c7
+
+theorem cover_subtree_19177aff5ede :
+    adaptiveCoverCheck 4 (childLL (childLH (childHL thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHL
+    thetaAboveCell000022002111)))
+    (by
+      have h : (thetaAboveCell0000220021112100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112100 h)
+    (by
+      have h : (thetaAboveCell0000220021112101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112102
+        (by
+          have h : ((childLL thetaAboveCell0000220021112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112102) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112102) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112102) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112103
+        (by
+          have h : ((childLL thetaAboveCell0000220021112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112103) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112103) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112103) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112103) h))
+
+theorem cover_subtree_a6e5897577a8 :
+    adaptiveCoverCheck 4 (childLH (childLH (childHL thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHL
+    thetaAboveCell000022002111)))
+    (by
+      have h : (thetaAboveCell0000220021112110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112110 h)
+    (by
+      have h : (thetaAboveCell0000220021112111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112112
+        (by
+          have h : ((childLL thetaAboveCell0000220021112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112112) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112112) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112112) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112113
+        (by
+          have h : ((childLL thetaAboveCell0000220021112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112113) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112113) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112113) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112113) h))
+
+theorem cover_subtree_c0e11192bd72 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHL thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+    thetaAboveCell000022002111)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112120
+        (by
+          have h : ((childLL thetaAboveCell0000220021112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112121
+        (by
+          have h : ((childLL thetaAboveCell0000220021112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112122
+        (by
+          have h : ((childLL thetaAboveCell0000220021112122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112123
+        (by
+          have h : ((childLL thetaAboveCell0000220021112123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112123) h))
+
+theorem cover_subtree_3544788b12c5 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHL thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+    thetaAboveCell000022002111)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112130
+        (by
+          have h : ((childLL thetaAboveCell0000220021112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112131
+        (by
+          have h : ((childLL thetaAboveCell0000220021112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112132
+        (by
+          have h : ((childLL thetaAboveCell0000220021112132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021112133
+        (by
+          have h : ((childLL thetaAboveCell0000220021112133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021112133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021112133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021112133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021112133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021112133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021112133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021112133) h))
+
+theorem cover_subtree_4f053fea53cc :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022002111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022002111))
+    cover_subtree_19177aff5ede
+    cover_subtree_a6e5897577a8
+    cover_subtree_c0e11192bd72
+    cover_subtree_3544788b12c5
+
+theorem cover_subtree_ad3586fc5d64 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022002111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022002111))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022002111)))
+        (by
+          have h : (thetaAboveCell0000220021112200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112200 h)
+        (by
+          have h : (thetaAboveCell0000220021112201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112201 h)
+        (by
+          have h : (thetaAboveCell0000220021112202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112202 h)
+        (by
+          have h : (thetaAboveCell0000220021112203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022002111)))
+        (by
+          have h : (thetaAboveCell0000220021112210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112210 h)
+        (by
+          have h : (thetaAboveCell0000220021112211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112211 h)
+        (by
+          have h : (thetaAboveCell0000220021112212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112212 h)
+        (by
+          have h : (thetaAboveCell0000220021112213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022002111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022002111))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022002111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022002111))) h)
+
+theorem cover_subtree_a8dfb5281914 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022002111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022002111))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022002111)))
+        (by
+          have h : (thetaAboveCell0000220021112300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112300 h)
+        (by
+          have h : (thetaAboveCell0000220021112301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112301 h)
+        (by
+          have h : (thetaAboveCell0000220021112302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112302 h)
+        (by
+          have h : (thetaAboveCell0000220021112303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022002111)))
+        (by
+          have h : (thetaAboveCell0000220021112310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112310 h)
+        (by
+          have h : (thetaAboveCell0000220021112311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112311 h)
+        (by
+          have h : (thetaAboveCell0000220021112312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112312 h)
+        (by
+          have h : (thetaAboveCell0000220021112313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021112313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022002111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022002111))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022002111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022002111))) h)
+
+theorem e24KC2ThetaAboveLeaf0000220021_c1_c1_c2 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022002111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022002111)
+    cover_subtree_b7bd7dd08527
+    cover_subtree_4f053fea53cc
+    cover_subtree_ad3586fc5d64
+    cover_subtree_a8dfb5281914
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells03dbb8f506
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells03dbb8f506
+
+open CertificateCells03dbb8f506
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c1_c1 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022002101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022002101)
+    (by
+      have h : ((childLL (childLH thetaAboveCell000022002101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH thetaAboveCell000022002101)) h)
+    (by
+      have h : ((childLH (childLH thetaAboveCell000022002101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH thetaAboveCell000022002101)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLH thetaAboveCell000022002101))
+        (by
+          have h : ((childLL (childHL (childLH thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLH
+            thetaAboveCell000022002101))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLH
+            thetaAboveCell000022002101))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLH
+            thetaAboveCell000022002101))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLH
+            thetaAboveCell000022002101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLH thetaAboveCell000022002101))
+        (by
+          have h : ((childLL (childHH (childLH thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLH
+            thetaAboveCell000022002101))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLH
+            thetaAboveCell000022002101))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLH
+            thetaAboveCell000022002101))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLH
+            thetaAboveCell000022002101))) h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells15b8ca5094
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells15b8ca5094
+
+open CertificateCells15b8ca5094
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c1_c2_c3 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022002101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022002101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022002101)))
+        (by
+          have h : (thetaAboveCell0000220021012300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012300 h)
+        (by
+          have h : (thetaAboveCell0000220021012301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012301 h)
+        (by
+          have h : (thetaAboveCell0000220021012302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012302 h)
+        (by
+          have h : (thetaAboveCell0000220021012303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022002101)))
+        (by
+          have h : (thetaAboveCell0000220021012310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012310 h)
+        (by
+          have h : (thetaAboveCell0000220021012311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012311 h)
+        (by
+          have h : (thetaAboveCell0000220021012312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012312 h)
+        (by
+          have h : (thetaAboveCell0000220021012313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022002101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022002101))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022002101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022002101))) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsecf9e4e90b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsecf9e4e90b
+
+open CertificateCellsecf9e4e90b
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c2_c2 :
+    adaptiveCoverCheck 4 (childHL (childHL thetaBelowCell111133113111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHL thetaBelowCell111133113111))
+    (by
+      have h : ((childLL (childHL (childHL thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHL
+        thetaBelowCell111133113111))) h)
+    (by
+      have h : ((childLH (childHL (childHL thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHL
+        thetaBelowCell111133113111))) h)
+    (by
+      have h : ((childHL (childHL (childHL thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHL
+        thetaBelowCell111133113111))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHL
+        thetaBelowCell111133113111))) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdb6ee29ee4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdb6ee29ee4
+
+open CertificateCellsdb6ee29ee4
+theorem cover_subtree_4cc417ccb089 :
+    adaptiveCoverCheck 4 (childLL (childLL (childHH thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHH
+    thetaAboveCell000022002110)))
+    (by
+      have h : (thetaAboveCell0000220021103000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103000 h)
+    (by
+      have h : (thetaAboveCell0000220021103001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103002
+        (by
+          have h : ((childLL thetaAboveCell0000220021103002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103002) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103002) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103002) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103003
+        (by
+          have h : ((childLL thetaAboveCell0000220021103003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103003) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103003) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103003) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103003) h))
+
+theorem cover_subtree_3b70ffafef44 :
+    adaptiveCoverCheck 4 (childLH (childLL (childHH thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHH
+    thetaAboveCell000022002110)))
+    (by
+      have h : (thetaAboveCell0000220021103010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103010 h)
+    (by
+      have h : (thetaAboveCell0000220021103011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103012
+        (by
+          have h : ((childLL thetaAboveCell0000220021103012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103012) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103012) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103012) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103013
+        (by
+          have h : ((childLL thetaAboveCell0000220021103013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103013) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103013) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103013) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103013) h))
+
+theorem cover_subtree_dcf1d830bfb7 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHH thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+    thetaAboveCell000022002110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103020
+        (by
+          have h : ((childLL thetaAboveCell0000220021103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103021
+        (by
+          have h : ((childLL thetaAboveCell0000220021103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103022
+        (by
+          have h : ((childLL thetaAboveCell0000220021103022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103023
+        (by
+          have h : ((childLL thetaAboveCell0000220021103023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103023) h))
+
+theorem cover_subtree_8808b93a2a5b :
+    adaptiveCoverCheck 4 (childHH (childLL (childHH thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+    thetaAboveCell000022002110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103030
+        (by
+          have h : ((childLL thetaAboveCell0000220021103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103031
+        (by
+          have h : ((childLL thetaAboveCell0000220021103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103032
+        (by
+          have h : ((childLL thetaAboveCell0000220021103032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103033
+        (by
+          have h : ((childLL thetaAboveCell0000220021103033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103033) h))
+
+theorem cover_subtree_adc568740138 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022002110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022002110))
+    cover_subtree_4cc417ccb089
+    cover_subtree_3b70ffafef44
+    cover_subtree_dcf1d830bfb7
+    cover_subtree_8808b93a2a5b
+
+theorem cover_subtree_9d7ffe00b1e7 :
+    adaptiveCoverCheck 4 (childLL (childLH (childHH thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHH
+    thetaAboveCell000022002110)))
+    (by
+      have h : (thetaAboveCell0000220021103100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103100 h)
+    (by
+      have h : (thetaAboveCell0000220021103101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103102
+        (by
+          have h : ((childLL thetaAboveCell0000220021103102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103102) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103102) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103102) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103103
+        (by
+          have h : ((childLL thetaAboveCell0000220021103103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103103) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103103) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103103) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103103) h))
+
+theorem cover_subtree_0d9aaf9c725c :
+    adaptiveCoverCheck 4 (childLH (childLH (childHH thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHH
+    thetaAboveCell000022002110)))
+    (by
+      have h : (thetaAboveCell0000220021103110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103110 h)
+    (by
+      have h : (thetaAboveCell0000220021103111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103112
+        (by
+          have h : ((childLL thetaAboveCell0000220021103112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103112) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103112) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103112) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103113
+        (by
+          have h : ((childLL thetaAboveCell0000220021103113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103113) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103113) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103113) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103113) h))
+
+theorem cover_subtree_d26de62ce5c2 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHH thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+    thetaAboveCell000022002110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103120
+        (by
+          have h : ((childLL thetaAboveCell0000220021103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103121
+        (by
+          have h : ((childLL thetaAboveCell0000220021103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103122
+        (by
+          have h : ((childLL thetaAboveCell0000220021103122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103123
+        (by
+          have h : ((childLL thetaAboveCell0000220021103123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103123) h))
+
+theorem cover_subtree_6e084944e1b7 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHH thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+    thetaAboveCell000022002110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103130
+        (by
+          have h : ((childLL thetaAboveCell0000220021103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103131
+        (by
+          have h : ((childLL thetaAboveCell0000220021103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103132
+        (by
+          have h : ((childLL thetaAboveCell0000220021103132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021103133
+        (by
+          have h : ((childLL thetaAboveCell0000220021103133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021103133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021103133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021103133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021103133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021103133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021103133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021103133) h))
+
+theorem cover_subtree_e73271ac6053 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022002110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022002110))
+    cover_subtree_9d7ffe00b1e7
+    cover_subtree_0d9aaf9c725c
+    cover_subtree_d26de62ce5c2
+    cover_subtree_6e084944e1b7
+
+theorem cover_subtree_0ec46983a3e4 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022002110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022002110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022002110)))
+        (by
+          have h : (thetaAboveCell0000220021103200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103200 h)
+        (by
+          have h : (thetaAboveCell0000220021103201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103201 h)
+        (by
+          have h : (thetaAboveCell0000220021103202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103202 h)
+        (by
+          have h : (thetaAboveCell0000220021103203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022002110)))
+        (by
+          have h : (thetaAboveCell0000220021103210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103210 h)
+        (by
+          have h : (thetaAboveCell0000220021103211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103211 h)
+        (by
+          have h : (thetaAboveCell0000220021103212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103212 h)
+        (by
+          have h : (thetaAboveCell0000220021103213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022002110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022002110))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022002110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022002110))) h)
+
+theorem cover_subtree_5cbb0735d9b5 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022002110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022002110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022002110)))
+        (by
+          have h : (thetaAboveCell0000220021103300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103300 h)
+        (by
+          have h : (thetaAboveCell0000220021103301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103301 h)
+        (by
+          have h : (thetaAboveCell0000220021103302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103302 h)
+        (by
+          have h : (thetaAboveCell0000220021103303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022002110)))
+        (by
+          have h : (thetaAboveCell0000220021103310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103310 h)
+        (by
+          have h : (thetaAboveCell0000220021103311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103311 h)
+        (by
+          have h : (thetaAboveCell0000220021103312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103312 h)
+        (by
+          have h : (thetaAboveCell0000220021103313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021103313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022002110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022002110))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022002110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022002110))) h)
+
+theorem e24KC2ThetaAboveLeaf0000220021_c1_c0_c3 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022002110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022002110)
+    cover_subtree_adc568740138
+    cover_subtree_e73271ac6053
+    cover_subtree_0ec46983a3e4
+    cover_subtree_5cbb0735d9b5
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4003309a28
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4003309a28
+
+open CertificateCells4003309a28
+theorem e24KC2ThetaAboveLeaf0000220021_c3 :
+    adaptiveCoverCheck 8 (childHH (childLH (childHL thetaAboveCell00002200))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLH (childHL thetaAboveCell00002200)))
+    (by
+      have h : (thetaAboveCell000022002130).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002130 h)
+    (by
+      have h : (thetaAboveCell000022002131).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002131 h)
+    (by
+      have h : (thetaAboveCell000022002132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002132 h)
+    (by
+      have h : (thetaAboveCell000022002133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002133 h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf3f1dbc459
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf3f1dbc459
+
+open CertificateCellsf3f1dbc459
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c0_c2_c3 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022002100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022002100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022002100)))
+        (by
+          have h : (thetaAboveCell0000220021002300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002300 h)
+        (by
+          have h : (thetaAboveCell0000220021002301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002301 h)
+        (by
+          have h : (thetaAboveCell0000220021002302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002302 h)
+        (by
+          have h : (thetaAboveCell0000220021002303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022002100)))
+        (by
+          have h : (thetaAboveCell0000220021002310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002310 h)
+        (by
+          have h : (thetaAboveCell0000220021002311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002311 h)
+        (by
+          have h : (thetaAboveCell0000220021002312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002312 h)
+        (by
+          have h : (thetaAboveCell0000220021002313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022002100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022002100))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022002100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022002100))) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells441cc2b0d8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells441cc2b0d8
+
+open CertificateCells441cc2b0d8
+theorem cover_subtree_9a19b7b479a6 :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131113100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113100
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLL thetaBelowCell1111331131113100)
+        (by
+          have h : ((childLL (childLL thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLL
+            thetaBelowCell1111331131113100)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLL
+            thetaBelowCell1111331131113100)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLL
+            thetaBelowCell1111331131113100)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLL
+            thetaBelowCell1111331131113100)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131113100)
+        (by
+          have h : ((childLL (childLH thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+            thetaBelowCell1111331131113100)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+            thetaBelowCell1111331131113100)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+            thetaBelowCell1111331131113100)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+            thetaBelowCell1111331131113100)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131113100)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131113100)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131113100)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131113100)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131113100)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131113100)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131113100)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131113100)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131113100)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131113100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131113100)) h))
+
+theorem cover_subtree_d45287132b6e :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131113101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113101
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLL thetaBelowCell1111331131113101)
+        (by
+          have h : ((childLL (childLL thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLL
+            thetaBelowCell1111331131113101)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLL
+            thetaBelowCell1111331131113101)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLL
+            thetaBelowCell1111331131113101)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLL
+            thetaBelowCell1111331131113101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131113101)
+        (by
+          have h : ((childLL (childLH thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+            thetaBelowCell1111331131113101)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+            thetaBelowCell1111331131113101)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+            thetaBelowCell1111331131113101)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+            thetaBelowCell1111331131113101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131113101)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131113101)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131113101)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131113101)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131113101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131113101)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131113101)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131113101)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131113101)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131113101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131113101)) h))
+
+theorem cover_subtree_e155163f58e6 :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131113102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113102
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLL thetaBelowCell1111331131113102)
+        (by
+          have h : ((childLL (childLL thetaBelowCell1111331131113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLL
+            thetaBelowCell1111331131113102)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell1111331131113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLL
+            thetaBelowCell1111331131113102)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell1111331131113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLL
+            thetaBelowCell1111331131113102)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell1111331131113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLL
+            thetaBelowCell1111331131113102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131113102)
+        (by
+          have h : ((childLL (childLH thetaBelowCell1111331131113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+            thetaBelowCell1111331131113102)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell1111331131113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+            thetaBelowCell1111331131113102)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell1111331131113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+            thetaBelowCell1111331131113102)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell1111331131113102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+            thetaBelowCell1111331131113102)) h))
+    (by
+      have h : ((childHL thetaBelowCell1111331131113102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131113102) h)
+    (by
+      have h : ((childHH thetaBelowCell1111331131113102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131113102) h)
+
+theorem cover_subtree_8c5c60491e76 :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131113103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113103
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLL thetaBelowCell1111331131113103)
+        (by
+          have h : ((childLL (childLL thetaBelowCell1111331131113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLL
+            thetaBelowCell1111331131113103)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell1111331131113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLL
+            thetaBelowCell1111331131113103)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell1111331131113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLL
+            thetaBelowCell1111331131113103)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell1111331131113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLL
+            thetaBelowCell1111331131113103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131113103)
+        (by
+          have h : ((childLL (childLH thetaBelowCell1111331131113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+            thetaBelowCell1111331131113103)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell1111331131113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+            thetaBelowCell1111331131113103)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell1111331131113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+            thetaBelowCell1111331131113103)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell1111331131113103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+            thetaBelowCell1111331131113103)) h))
+    (by
+      have h : ((childHL thetaBelowCell1111331131113103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131113103) h)
+    (by
+      have h : ((childHH thetaBelowCell1111331131113103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131113103) h)
+
+theorem cover_subtree_deda1d8f689c :
+    adaptiveCoverCheck 3 (childLL (childLH (childHH thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLL (childLH (childHH
+    thetaBelowCell111133113111)))
+    cover_subtree_9a19b7b479a6
+    cover_subtree_d45287132b6e
+    cover_subtree_e155163f58e6
+    cover_subtree_8c5c60491e76
+
+theorem cover_subtree_afe933400d68 :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131113110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113110
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLL thetaBelowCell1111331131113110)
+        (by
+          have h : ((childLL (childLL thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLL
+            thetaBelowCell1111331131113110)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLL
+            thetaBelowCell1111331131113110)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLL
+            thetaBelowCell1111331131113110)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLL
+            thetaBelowCell1111331131113110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131113110)
+        (by
+          have h : ((childLL (childLH thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+            thetaBelowCell1111331131113110)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+            thetaBelowCell1111331131113110)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+            thetaBelowCell1111331131113110)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+            thetaBelowCell1111331131113110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131113110)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131113110)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131113110)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131113110)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131113110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131113110)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131113110)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131113110)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131113110)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131113110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131113110)) h))
+
+theorem cover_subtree_3d26797fa2c0 :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131113111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113111
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLL thetaBelowCell1111331131113111)
+        (by
+          have h : ((childLL (childLL thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLL
+            thetaBelowCell1111331131113111)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLL
+            thetaBelowCell1111331131113111)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLL
+            thetaBelowCell1111331131113111)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLL
+            thetaBelowCell1111331131113111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131113111)
+        (by
+          have h : ((childLL (childLH thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+            thetaBelowCell1111331131113111)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+            thetaBelowCell1111331131113111)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+            thetaBelowCell1111331131113111)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+            thetaBelowCell1111331131113111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131113111)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131113111)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131113111)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131113111)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131113111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131113111)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131113111)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131113111)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131113111)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131113111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131113111)) h))
+
+theorem cover_subtree_2306090dc6e1 :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131113112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113112
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLL thetaBelowCell1111331131113112)
+        (by
+          have h : ((childLL (childLL thetaBelowCell1111331131113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLL
+            thetaBelowCell1111331131113112)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell1111331131113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLL
+            thetaBelowCell1111331131113112)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell1111331131113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLL
+            thetaBelowCell1111331131113112)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell1111331131113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLL
+            thetaBelowCell1111331131113112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131113112)
+        (by
+          have h : ((childLL (childLH thetaBelowCell1111331131113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+            thetaBelowCell1111331131113112)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell1111331131113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+            thetaBelowCell1111331131113112)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell1111331131113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+            thetaBelowCell1111331131113112)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell1111331131113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+            thetaBelowCell1111331131113112)) h))
+    (by
+      have h : ((childHL thetaBelowCell1111331131113112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131113112) h)
+    (by
+      have h : ((childHH thetaBelowCell1111331131113112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131113112) h)
+
+theorem cover_subtree_f50a24eff089 :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131113113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113113
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLL thetaBelowCell1111331131113113)
+        (by
+          have h : ((childLL (childLL thetaBelowCell1111331131113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLL
+            thetaBelowCell1111331131113113)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell1111331131113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLL
+            thetaBelowCell1111331131113113)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell1111331131113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLL
+            thetaBelowCell1111331131113113)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell1111331131113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLL
+            thetaBelowCell1111331131113113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131113113)
+        (by
+          have h : ((childLL (childLH thetaBelowCell1111331131113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+            thetaBelowCell1111331131113113)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell1111331131113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+            thetaBelowCell1111331131113113)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell1111331131113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+            thetaBelowCell1111331131113113)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell1111331131113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+            thetaBelowCell1111331131113113)) h))
+    (by
+      have h : ((childHL thetaBelowCell1111331131113113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131113113) h)
+    (by
+      have h : ((childHH thetaBelowCell1111331131113113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131113113) h)
+
+theorem cover_subtree_5a8fcf48e536 :
+    adaptiveCoverCheck 3 (childLH (childLH (childHH thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLH (childHH
+    thetaBelowCell111133113111)))
+    cover_subtree_afe933400d68
+    cover_subtree_3d26797fa2c0
+    cover_subtree_2306090dc6e1
+    cover_subtree_f50a24eff089
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c3_c1 :
+    adaptiveCoverCheck 4 (childLH (childHH thetaBelowCell111133113111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH thetaBelowCell111133113111))
+    cover_subtree_deda1d8f689c
+    cover_subtree_5a8fcf48e536
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childLH (childHH
+        thetaBelowCell111133113111)))
+        (by
+          have h : (thetaBelowCell1111331131113120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113120 h)
+        (by
+          have h : (thetaBelowCell1111331131113121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113121 h)
+        (by
+          have h : (thetaBelowCell1111331131113122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113122 h)
+        (by
+          have h : (thetaBelowCell1111331131113123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childLH (childHH
+        thetaBelowCell111133113111)))
+        (by
+          have h : (thetaBelowCell1111331131113130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113130 h)
+        (by
+          have h : (thetaBelowCell1111331131113131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113131 h)
+        (by
+          have h : (thetaBelowCell1111331131113132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113132 h)
+        (by
+          have h : (thetaBelowCell1111331131113133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113133 h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5980303e93
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5980303e93
+
+open CertificateCells5980303e93
+theorem e24KC2ThetaAboveLeaf0000220021_c1_c0_c1 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022002110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022002110)
+    (by
+      have h : ((childLL (childLH thetaAboveCell000022002110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH thetaAboveCell000022002110)) h)
+    (by
+      have h : ((childLH (childLH thetaAboveCell000022002110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH thetaAboveCell000022002110)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLH thetaAboveCell000022002110))
+        (by
+          have h : ((childLL (childHL (childLH thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLH
+            thetaAboveCell000022002110))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLH
+            thetaAboveCell000022002110))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLH
+            thetaAboveCell000022002110))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLH
+            thetaAboveCell000022002110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLH thetaAboveCell000022002110))
+        (by
+          have h : ((childLL (childHH (childLH thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLH
+            thetaAboveCell000022002110))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLH
+            thetaAboveCell000022002110))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLH
+            thetaAboveCell000022002110))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLH
+            thetaAboveCell000022002110))) h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa9276513a2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa9276513a2
+
+open CertificateCellsa9276513a2
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c2 :
+    adaptiveCoverCheck 6 thetaBelowCell111133113112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113112
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133113112)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133113112)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133113112)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133113112)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133113112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133113112)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133113112)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133113112)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133113112)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133113112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133113112)) h))
+    (by
+      have h : ((childHL thetaBelowCell111133113112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133113112) h)
+    (by
+      have h : ((childHH thetaBelowCell111133113112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133113112) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9972cada5a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9972cada5a
+
+open CertificateCells9972cada5a
+theorem e24KC2ThetaBelowLeaf111133113_c2 :
+    adaptiveCoverCheck 8 (childHL (childHH thetaBelowCell11113311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHL (childHH thetaBelowCell11113311))
+    (by
+      have h : ((childLL (childHL (childHH thetaBelowCell11113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHL (childHH
+        thetaBelowCell11113311))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaBelowCell11113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHL (childHH
+        thetaBelowCell11113311))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaBelowCell11113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHL (childHH
+        thetaBelowCell11113311))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaBelowCell11113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHL (childHH
+        thetaBelowCell11113311))) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa6306b8bb7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa6306b8bb7
+
+open CertificateCellsa6306b8bb7
+theorem cover_subtree_64ca6a23a782 :
+    adaptiveCoverCheck 3 (childLL (childLL (childHL thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLL (childLL (childHL
+    thetaBelowCell111133113111)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112000
+        (by
+          have h : ((childLL thetaBelowCell1111331131112000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131112000) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131112000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131112000) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131112000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131112000) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131112000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131112000) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112001
+        (by
+          have h : ((childLL thetaBelowCell1111331131112001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131112001) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131112001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131112001) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131112001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131112001) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131112001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131112001) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112002
+        (by
+          have h : ((childLL thetaBelowCell1111331131112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131112002) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131112002) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131112002) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131112002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131112002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112003
+        (by
+          have h : ((childLL thetaBelowCell1111331131112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131112003) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131112003) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131112003) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131112003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131112003) h))
+
+theorem cover_subtree_5f702c8c435f :
+    adaptiveCoverCheck 3 (childLH (childLL (childHL thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLL (childHL
+    thetaBelowCell111133113111)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112010
+        (by
+          have h : ((childLL thetaBelowCell1111331131112010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131112010) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131112010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131112010) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131112010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131112010) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131112010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131112010) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112011
+        (by
+          have h : ((childLL thetaBelowCell1111331131112011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131112011) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131112011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131112011) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131112011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131112011) h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131112011)
+            (by
+              have h : ((childLL (childHH thetaBelowCell1111331131112011))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+                thetaBelowCell1111331131112011)) h)
+            (by
+              have h : ((childLH (childHH thetaBelowCell1111331131112011))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+                thetaBelowCell1111331131112011)) h)
+            (by
+              have h : ((childHL (childHH thetaBelowCell1111331131112011))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+                thetaBelowCell1111331131112011)) h)
+            (by
+              have h : ((childHH (childHH thetaBelowCell1111331131112011))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+                thetaBelowCell1111331131112011)) h)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112012
+        (by
+          have h : ((childLL thetaBelowCell1111331131112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131112012) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131112012) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131112012) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131112012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131112012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112013
+        (by
+          have h : ((childLL thetaBelowCell1111331131112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131112013) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131112013) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131112013) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131112013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131112013) h))
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c2_c0 :
+    adaptiveCoverCheck 4 (childLL (childHL thetaBelowCell111133113111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL thetaBelowCell111133113111))
+    cover_subtree_64ca6a23a782
+    cover_subtree_5f702c8c435f
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childLL (childHL
+        thetaBelowCell111133113111)))
+        (by
+          have h : (thetaBelowCell1111331131112020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112020 h)
+        (by
+          have h : (thetaBelowCell1111331131112021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112021 h)
+        (by
+          have h : (thetaBelowCell1111331131112022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112022 h)
+        (by
+          have h : (thetaBelowCell1111331131112023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childLL (childHL
+        thetaBelowCell111133113111)))
+        (by
+          have h : (thetaBelowCell1111331131112030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112030 h)
+        (by
+          have h : (thetaBelowCell1111331131112031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112031 h)
+        (by
+          have h : (thetaBelowCell1111331131112032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112032 h)
+        (by
+          have h : (thetaBelowCell1111331131112033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112033 h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc1e1941644
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc1e1941644
+
+open CertificateCellsc1e1941644
+theorem e24KC2ThetaBelowLeaf111133113_c1_c0_c1_c2 :
+    adaptiveCoverCheck 5 (childHL thetaBelowCell111133113101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133113101)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL thetaBelowCell111133113101))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 2 (childLL (childLL (childHL
+            thetaBelowCell111133113101)))
+            (by
+              have h : (thetaBelowCell1111331131012000).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012000 h)
+            (by
+              have h : (thetaBelowCell1111331131012001).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012001 h)
+            (by
+              have h : (thetaBelowCell1111331131012002).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012002 h)
+            (by
+              have h : (thetaBelowCell1111331131012003).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012003 h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLL (childHL
+            thetaBelowCell111133113101)))
+            (by
+              have h : (thetaBelowCell1111331131012010).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012010 h)
+            (by
+              have h : (thetaBelowCell1111331131012011).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012011 h)
+            (by
+              have h : (thetaBelowCell1111331131012012).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012012 h)
+            (by
+              have h : (thetaBelowCell1111331131012013).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012013 h))
+        (by
+          have h : ((childHL (childLL (childHL thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childHL
+            thetaBelowCell111133113101))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childHL
+            thetaBelowCell111133113101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL thetaBelowCell111133113101))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 2 (childLL (childLH (childHL
+            thetaBelowCell111133113101)))
+            (by
+              have h : (thetaBelowCell1111331131012100).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012100 h)
+            (by
+              have h : (thetaBelowCell1111331131012101).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012101 h)
+            (by
+              have h : (thetaBelowCell1111331131012102).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012102 h)
+            (by
+              have h : (thetaBelowCell1111331131012103).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012103 h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLH (childHL
+            thetaBelowCell111133113101)))
+            (by
+              have h : (thetaBelowCell1111331131012110).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012110 h)
+            (by
+              have h : (thetaBelowCell1111331131012111).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012111 h)
+            (by
+              have h : (thetaBelowCell1111331131012112).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012112 h)
+            (by
+              have h : (thetaBelowCell1111331131012113).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131012113 h))
+        (by
+          have h : ((childHL (childLH (childHL thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childHL
+            thetaBelowCell111133113101))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childHL
+            thetaBelowCell111133113101))) h))
+    (by
+      have h : ((childHL (childHL thetaBelowCell111133113101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL thetaBelowCell111133113101)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell111133113101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL thetaBelowCell111133113101)) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5a169f617b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5a169f617b
+
+open CertificateCells5a169f617b
+theorem e24KC2ThetaAboveLeaf0000220021_c1_c1_c1 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022002111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022002111)
+    (by
+      have h : ((childLL (childLH thetaAboveCell000022002111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH thetaAboveCell000022002111)) h)
+    (by
+      have h : ((childLH (childLH thetaAboveCell000022002111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH thetaAboveCell000022002111)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLH thetaAboveCell000022002111))
+        (by
+          have h : ((childLL (childHL (childLH thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLH
+            thetaAboveCell000022002111))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLH
+            thetaAboveCell000022002111))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLH
+            thetaAboveCell000022002111))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLH
+            thetaAboveCell000022002111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLH thetaAboveCell000022002111))
+        (by
+          have h : ((childLL (childHH (childLH thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLH
+            thetaAboveCell000022002111))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLH
+            thetaAboveCell000022002111))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLH
+            thetaAboveCell000022002111))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLH
+            thetaAboveCell000022002111))) h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc15ab9b9a5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc15ab9b9a5
+
+open CertificateCellsc15ab9b9a5
+theorem e24KC2ThetaBelowLeaf111133113_c0_c1_c3 :
+    adaptiveCoverCheck 6 thetaBelowCell111133113013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113013
+    (by
+      have h : ((childLL thetaBelowCell111133113013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133113013) h)
+    (by
+      have h : ((childLH thetaBelowCell111133113013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133113013) h)
+    (by
+      have h : ((childHL thetaBelowCell111133113013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133113013) h)
+    (by
+      have h : ((childHH thetaBelowCell111133113013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133113013) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells861ec8c785
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells861ec8c785
+
+open CertificateCells861ec8c785
+theorem e24KC2ThetaAboveLeaf0000220021_c1_c0_c0 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022002110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022002110)
+    (by
+      have h : ((childLL (childLL thetaAboveCell000022002110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL thetaAboveCell000022002110)) h)
+    (by
+      have h : ((childLH (childLL thetaAboveCell000022002110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL thetaAboveCell000022002110)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLL thetaAboveCell000022002110))
+        (by
+          have h : ((childLL (childHL (childLL thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLL
+            thetaAboveCell000022002110))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLL
+            thetaAboveCell000022002110))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLL
+            thetaAboveCell000022002110))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLL
+            thetaAboveCell000022002110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLL thetaAboveCell000022002110))
+        (by
+          have h : ((childLL (childHH (childLL thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLL
+            thetaAboveCell000022002110))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLL
+            thetaAboveCell000022002110))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLL
+            thetaAboveCell000022002110))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaAboveCell000022002110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLL
+            thetaAboveCell000022002110))) h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells854f59bf16
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells854f59bf16
+
+open CertificateCells854f59bf16
+theorem cover_subtree_868906559995 :
+    adaptiveCoverCheck 5 (childLL thetaBelowCell111133113000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133113000)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL thetaBelowCell111133113000))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLL
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLL
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLL
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLL
+            thetaBelowCell111133113000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL thetaBelowCell111133113000))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLL
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLL
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLL
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLL
+            thetaBelowCell111133113000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL thetaBelowCell111133113000))
+        (by
+          have h : ((childLL (childHL (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childLL
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childLL
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childLL
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childLL
+            thetaBelowCell111133113000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL thetaBelowCell111133113000))
+        (by
+          have h : ((childLL (childHH (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childLL
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childLL
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childLL
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childLL
+            thetaBelowCell111133113000))) h))
+
+theorem cover_subtree_cf661e32fd3c :
+    adaptiveCoverCheck 5 (childLH thetaBelowCell111133113000) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133113000)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH thetaBelowCell111133113000))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLH
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLH
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLH
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLH
+            thetaBelowCell111133113000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH thetaBelowCell111133113000))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLH
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLH
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLH
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLH
+            thetaBelowCell111133113000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH thetaBelowCell111133113000))
+        (by
+          have h : ((childLL (childHL (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childLH
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childLH
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childLH
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childLH
+            thetaBelowCell111133113000))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH thetaBelowCell111133113000))
+        (by
+          have h : ((childLL (childHH (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childLH
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childLH
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childLH
+            thetaBelowCell111133113000))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaBelowCell111133113000)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childLH
+            thetaBelowCell111133113000))) h))
+
+theorem cover_subtree_852412a1da1f :
+    adaptiveCoverCheck 6 thetaBelowCell111133113000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113000
+    cover_subtree_868906559995
+    cover_subtree_cf661e32fd3c
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133113000)
+        (by
+          have h : ((childLL (childHL thetaBelowCell111133113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL
+            thetaBelowCell111133113000)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell111133113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL
+            thetaBelowCell111133113000)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell111133113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL
+            thetaBelowCell111133113000)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell111133113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL
+            thetaBelowCell111133113000)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133113000)
+        (by
+          have h : ((childLL (childHH thetaBelowCell111133113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH
+            thetaBelowCell111133113000)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell111133113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH
+            thetaBelowCell111133113000)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell111133113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH
+            thetaBelowCell111133113000)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell111133113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH
+            thetaBelowCell111133113000)) h))
+
+theorem cover_subtree_5c713f8b8f83 :
+    adaptiveCoverCheck 5 (childLL thetaBelowCell111133113001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133113001)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL thetaBelowCell111133113001))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLL
+            thetaBelowCell111133113001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL thetaBelowCell111133113001))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLL
+            thetaBelowCell111133113001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL thetaBelowCell111133113001))
+        (by
+          have h : ((childLL (childHL (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childLL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childLL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childLL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childLL
+            thetaBelowCell111133113001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL thetaBelowCell111133113001))
+        (by
+          have h : ((childLL (childHH (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childLL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childLL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childLL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childLL
+            thetaBelowCell111133113001))) h))
+
+theorem cover_subtree_9ce321b0f1ea :
+    adaptiveCoverCheck 4 (childHH (childLH thetaBelowCell111133113001)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH thetaBelowCell111133113001))
+    (by
+      have h : ((childLL (childHH (childLH thetaBelowCell111133113001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childLH
+        thetaBelowCell111133113001))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaBelowCell111133113001)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childLH
+        thetaBelowCell111133113001))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHH (childLH
+        thetaBelowCell111133113001)))
+        (by
+          have h : (thetaBelowCell1111331130011320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130011320 h)
+        (by
+          have h : (thetaBelowCell1111331130011321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130011321 h)
+        (by
+          have h : (thetaBelowCell1111331130011322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130011322 h)
+        (by
+          have h : (thetaBelowCell1111331130011323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130011323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHH (childLH
+        thetaBelowCell111133113001)))
+        (by
+          have h : (thetaBelowCell1111331130011330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130011330 h)
+        (by
+          have h : (thetaBelowCell1111331130011331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130011331 h)
+        (by
+          have h : (thetaBelowCell1111331130011332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130011332 h)
+        (by
+          have h : (thetaBelowCell1111331130011333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130011333 h))
+
+theorem cover_subtree_5f41a0b225f7 :
+    adaptiveCoverCheck 5 (childLH thetaBelowCell111133113001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133113001)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH thetaBelowCell111133113001))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLH
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLH
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLH
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLH
+            thetaBelowCell111133113001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH thetaBelowCell111133113001))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLH
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLH
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLH
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLH
+            thetaBelowCell111133113001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH thetaBelowCell111133113001))
+        (by
+          have h : ((childLL (childHL (childLH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childLH
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childLH
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childLH
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childLH
+            thetaBelowCell111133113001))) h))
+    cover_subtree_9ce321b0f1ea
+
+theorem cover_subtree_dce958c46e4b :
+    adaptiveCoverCheck 5 (childHL thetaBelowCell111133113001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133113001)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL thetaBelowCell111133113001))
+        (by
+          have h : ((childLL (childLL (childHL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childHL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childHL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childHL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childHL
+            thetaBelowCell111133113001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL thetaBelowCell111133113001))
+        (by
+          have h : ((childLL (childLH (childHL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childHL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childHL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childHL
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childHL
+            thetaBelowCell111133113001))) h))
+    (by
+      have h : ((childHL (childHL thetaBelowCell111133113001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL thetaBelowCell111133113001)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell111133113001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL thetaBelowCell111133113001)) h)
+
+theorem cover_subtree_5f44219dd9e0 :
+    adaptiveCoverCheck 5 (childHH thetaBelowCell111133113001) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133113001)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH thetaBelowCell111133113001))
+        (by
+          have h : ((childLL (childLL (childHH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childHH
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childHH
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childHH
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childHH
+            thetaBelowCell111133113001))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH thetaBelowCell111133113001))
+        (by
+          have h : ((childLL (childLH (childHH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childHH
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childHH
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childHH
+            thetaBelowCell111133113001))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaBelowCell111133113001)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childHH
+            thetaBelowCell111133113001))) h))
+    (by
+      have h : ((childHL (childHH thetaBelowCell111133113001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH thetaBelowCell111133113001)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell111133113001))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH thetaBelowCell111133113001)) h)
+
+theorem cover_subtree_f79674f0cd5e :
+    adaptiveCoverCheck 6 thetaBelowCell111133113001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113001
+    cover_subtree_5c713f8b8f83
+    cover_subtree_5f41a0b225f7
+    cover_subtree_dce958c46e4b
+    cover_subtree_5f44219dd9e0
+
+theorem e24KC2ThetaBelowLeaf111133113_c0_c0 :
+    adaptiveCoverCheck 7 (childLL (childLL (childHH thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childLL (childLL (childHH thetaBelowCell11113311)))
+    cover_subtree_852412a1da1f
+    cover_subtree_f79674f0cd5e
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113002
+        (by
+          have h : ((childLL thetaBelowCell111133113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133113002) h)
+        (by
+          have h : ((childLH thetaBelowCell111133113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133113002) h)
+        (by
+          have h : ((childHL thetaBelowCell111133113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133113002) h)
+        (by
+          have h : ((childHH thetaBelowCell111133113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133113002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113003
+        (by
+          have h : ((childLL thetaBelowCell111133113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133113003) h)
+        (by
+          have h : ((childLH thetaBelowCell111133113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133113003) h)
+        (by
+          have h : ((childHL thetaBelowCell111133113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133113003) h)
+        (by
+          have h : ((childHH thetaBelowCell111133113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133113003) h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5153122198
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5153122198
+
+open CertificateCells5153122198
+theorem cover_subtree_4d0609015f95 :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131112100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112100
+    (by
+      have h : ((childLL thetaBelowCell1111331131112100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131112100) h)
+    (by
+      have h : ((childLH thetaBelowCell1111331131112100)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131112100) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131112100)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131112100)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131112100)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131112100)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131112100)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131112100)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131112100)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131112100)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131112100)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131112100))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131112100)) h))
+
+theorem cover_subtree_dc10eb0281eb :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131112101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112101
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLL thetaBelowCell1111331131112101)
+        (by
+          have h : ((childLL (childLL thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLL
+            thetaBelowCell1111331131112101)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLL
+            thetaBelowCell1111331131112101)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLL
+            thetaBelowCell1111331131112101)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLL
+            thetaBelowCell1111331131112101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131112101)
+        (by
+          have h : ((childLL (childLH thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+            thetaBelowCell1111331131112101)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+            thetaBelowCell1111331131112101)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+            thetaBelowCell1111331131112101)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+            thetaBelowCell1111331131112101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131112101)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131112101)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131112101)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131112101)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131112101)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131112101)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131112101)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131112101)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131112101)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131112101))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131112101)) h))
+
+theorem cover_subtree_115058b1a8cb :
+    adaptiveCoverCheck 3 (childLL (childLH (childHL thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLL (childLH (childHL
+    thetaBelowCell111133113111)))
+    cover_subtree_4d0609015f95
+    cover_subtree_dc10eb0281eb
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112102
+        (by
+          have h : ((childLL thetaBelowCell1111331131112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131112102) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131112102) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131112102) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131112102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131112102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112103
+        (by
+          have h : ((childLL thetaBelowCell1111331131112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131112103) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131112103) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131112103) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131112103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131112103) h))
+
+theorem cover_subtree_73eb4fcea78d :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131112110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112110
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLL thetaBelowCell1111331131112110)
+        (by
+          have h : ((childLL (childLL thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLL
+            thetaBelowCell1111331131112110)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLL
+            thetaBelowCell1111331131112110)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLL
+            thetaBelowCell1111331131112110)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLL
+            thetaBelowCell1111331131112110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131112110)
+        (by
+          have h : ((childLL (childLH thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+            thetaBelowCell1111331131112110)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+            thetaBelowCell1111331131112110)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+            thetaBelowCell1111331131112110)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+            thetaBelowCell1111331131112110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131112110)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131112110)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131112110)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131112110)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131112110)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131112110)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131112110)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131112110)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131112110)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131112110))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131112110)) h))
+
+theorem cover_subtree_44584246e90e :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131112111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112111
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLL thetaBelowCell1111331131112111)
+        (by
+          have h : ((childLL (childLL thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLL
+            thetaBelowCell1111331131112111)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLL
+            thetaBelowCell1111331131112111)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLL
+            thetaBelowCell1111331131112111)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLL
+            thetaBelowCell1111331131112111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131112111)
+        (by
+          have h : ((childLL (childLH thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+            thetaBelowCell1111331131112111)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+            thetaBelowCell1111331131112111)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+            thetaBelowCell1111331131112111)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+            thetaBelowCell1111331131112111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131112111)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131112111)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131112111)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131112111)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131112111)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131112111)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131112111)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131112111)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131112111)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131112111))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131112111)) h))
+
+theorem cover_subtree_0e57e75e5954 :
+    adaptiveCoverCheck 3 (childLH (childLH (childHL thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLH (childHL
+    thetaBelowCell111133113111)))
+    cover_subtree_73eb4fcea78d
+    cover_subtree_44584246e90e
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112112
+        (by
+          have h : ((childLL thetaBelowCell1111331131112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131112112) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131112112) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131112112) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131112112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131112112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131112113
+        (by
+          have h : ((childLL thetaBelowCell1111331131112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131112113) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131112113) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131112113) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131112113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131112113) h))
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c2_c1 :
+    adaptiveCoverCheck 4 (childLH (childHL thetaBelowCell111133113111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL thetaBelowCell111133113111))
+    cover_subtree_115058b1a8cb
+    cover_subtree_0e57e75e5954
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childLH (childHL
+        thetaBelowCell111133113111)))
+        (by
+          have h : (thetaBelowCell1111331131112120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112120 h)
+        (by
+          have h : (thetaBelowCell1111331131112121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112121 h)
+        (by
+          have h : (thetaBelowCell1111331131112122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112122 h)
+        (by
+          have h : (thetaBelowCell1111331131112123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childLH (childHL
+        thetaBelowCell111133113111)))
+        (by
+          have h : (thetaBelowCell1111331131112130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112130 h)
+        (by
+          have h : (thetaBelowCell1111331131112131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112131 h)
+        (by
+          have h : (thetaBelowCell1111331131112132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112132 h)
+        (by
+          have h : (thetaBelowCell1111331131112133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131112133 h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0a969d2c42
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0a969d2c42
+
+open CertificateCells0a969d2c42
+theorem cover_subtree_6164650f4936 :
+    adaptiveCoverCheck 3 (childLL (childLL (childHL thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLL (childLL (childHL
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131102000
+        (by
+          have h : ((childLL thetaBelowCell1111331131102000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131102000) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131102000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131102000) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131102000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131102000) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131102000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131102000) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131102001
+        (by
+          have h : ((childLL thetaBelowCell1111331131102001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131102001) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131102001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131102001) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131102001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131102001) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131102001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131102001) h))
+    (by
+      have h : (thetaBelowCell1111331131102002).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131102002 h)
+    (by
+      have h : (thetaBelowCell1111331131102003).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131102003 h)
+
+theorem cover_subtree_3db66bc1eb1b :
+    adaptiveCoverCheck 3 (childLH (childLL (childHL thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLL (childHL
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131102010
+        (by
+          have h : ((childLL thetaBelowCell1111331131102010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131102010) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131102010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131102010) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131102010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131102010) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131102010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131102010) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131102011
+        (by
+          have h : ((childLL thetaBelowCell1111331131102011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131102011) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131102011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131102011) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131102011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131102011) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131102011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131102011) h))
+    (by
+      have h : (thetaBelowCell1111331131102012).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131102012 h)
+    (by
+      have h : (thetaBelowCell1111331131102013).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131102013 h)
+
+theorem cover_subtree_3f70d11b4e7e :
+    adaptiveCoverCheck 4 (childLL (childHL thetaBelowCell111133113110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL thetaBelowCell111133113110))
+    cover_subtree_6164650f4936
+    cover_subtree_3db66bc1eb1b
+    (by
+      have h : ((childHL (childLL (childHL thetaBelowCell111133113110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childHL
+        thetaBelowCell111133113110))) h)
+    (by
+      have h : ((childHH (childLL (childHL thetaBelowCell111133113110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childHL
+        thetaBelowCell111133113110))) h)
+
+theorem cover_subtree_261557c5db75 :
+    adaptiveCoverCheck 3 (childLL (childLH (childHL thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLL (childLH (childHL
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131102100
+        (by
+          have h : ((childLL thetaBelowCell1111331131102100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131102100) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131102100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131102100) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131102100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131102100) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131102100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131102100) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131102101
+        (by
+          have h : ((childLL thetaBelowCell1111331131102101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131102101) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131102101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131102101) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131102101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131102101) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131102101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131102101) h))
+    (by
+      have h : (thetaBelowCell1111331131102102).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131102102 h)
+    (by
+      have h : (thetaBelowCell1111331131102103).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131102103 h)
+
+theorem cover_subtree_12428cd495c4 :
+    adaptiveCoverCheck 3 (childLH (childLH (childHL thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLH (childHL
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131102110
+        (by
+          have h : ((childLL thetaBelowCell1111331131102110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131102110) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131102110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131102110) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131102110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131102110) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131102110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131102110) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131102111
+        (by
+          have h : ((childLL thetaBelowCell1111331131102111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131102111) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131102111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131102111) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131102111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131102111) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131102111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131102111) h))
+    (by
+      have h : (thetaBelowCell1111331131102112).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131102112 h)
+    (by
+      have h : (thetaBelowCell1111331131102113).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131102113 h)
+
+theorem cover_subtree_adb043cdb732 :
+    adaptiveCoverCheck 4 (childLH (childHL thetaBelowCell111133113110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL thetaBelowCell111133113110))
+    cover_subtree_261557c5db75
+    cover_subtree_12428cd495c4
+    (by
+      have h : ((childHL (childLH (childHL thetaBelowCell111133113110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childHL
+        thetaBelowCell111133113110))) h)
+    (by
+      have h : ((childHH (childLH (childHL thetaBelowCell111133113110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childHL
+        thetaBelowCell111133113110))) h)
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c0_c2 :
+    adaptiveCoverCheck 5 (childHL thetaBelowCell111133113110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133113110)
+    cover_subtree_3f70d11b4e7e
+    cover_subtree_adb043cdb732
+    (by
+      have h : ((childHL (childHL thetaBelowCell111133113110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL thetaBelowCell111133113110)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell111133113110))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL thetaBelowCell111133113110)) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells50ae5a428f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells50ae5a428f
+
+open CertificateCells50ae5a428f
+theorem cover_subtree_2e97d4844408 :
+    adaptiveCoverCheck 3 (childHL (childHL (childLH thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHL (childLH
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101220
+        (by
+          have h : ((childLL thetaBelowCell1111331131101220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101220) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101220) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101220) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101220) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101221
+        (by
+          have h : ((childLL thetaBelowCell1111331131101221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101221) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101221) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101221) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101221) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101222
+        (by
+          have h : ((childLL thetaBelowCell1111331131101222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101222) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101222) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101222) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101222) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101223
+        (by
+          have h : ((childLL thetaBelowCell1111331131101223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101223) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101223) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101223) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101223) h))
+
+theorem cover_subtree_0dc0f2664d18 :
+    adaptiveCoverCheck 3 (childHH (childHL (childLH thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHL (childLH
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101230
+        (by
+          have h : ((childLL thetaBelowCell1111331131101230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101230) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101230) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101230) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101230) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101231
+        (by
+          have h : ((childLL thetaBelowCell1111331131101231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101231) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101231) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101231) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101231) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101232
+        (by
+          have h : ((childLL thetaBelowCell1111331131101232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101232) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101232) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101232) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101232) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101233
+        (by
+          have h : ((childLL thetaBelowCell1111331131101233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101233) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101233) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101233) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101233) h))
+
+theorem cover_subtree_60a2aff6cef2 :
+    adaptiveCoverCheck 4 (childHL (childLH thetaBelowCell111133113110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH thetaBelowCell111133113110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHL (childLH
+        thetaBelowCell111133113110)))
+        (by
+          have h : (thetaBelowCell1111331131101200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101200 h)
+        (by
+          have h : (thetaBelowCell1111331131101201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101201 h)
+        (by
+          have h : (thetaBelowCell1111331131101202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101202 h)
+        (by
+          have h : (thetaBelowCell1111331131101203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHL (childLH
+        thetaBelowCell111133113110)))
+        (by
+          have h : (thetaBelowCell1111331131101210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101210 h)
+        (by
+          have h : (thetaBelowCell1111331131101211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101211 h)
+        (by
+          have h : (thetaBelowCell1111331131101212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101212 h)
+        (by
+          have h : (thetaBelowCell1111331131101213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101213 h))
+    cover_subtree_2e97d4844408
+    cover_subtree_0dc0f2664d18
+
+theorem cover_subtree_ab4066e0c111 :
+    adaptiveCoverCheck 3 (childHL (childHH (childLH thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHH (childLH
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101320
+        (by
+          have h : ((childLL thetaBelowCell1111331131101320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101320) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101320) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101320) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101320) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101321
+        (by
+          have h : ((childLL thetaBelowCell1111331131101321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101321) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101321) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101321) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101321) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101322
+        (by
+          have h : ((childLL thetaBelowCell1111331131101322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101322) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101322) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101322) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101322) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101323
+        (by
+          have h : ((childLL thetaBelowCell1111331131101323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101323) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101323) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101323) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101323) h))
+
+theorem cover_subtree_8607f2894d1c :
+    adaptiveCoverCheck 3 (childHH (childHH (childLH thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHH (childLH
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101330
+        (by
+          have h : ((childLL thetaBelowCell1111331131101330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101330) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101330) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101330) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101330) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101331
+        (by
+          have h : ((childLL thetaBelowCell1111331131101331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101331) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101331) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101331) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101331) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101332
+        (by
+          have h : ((childLL thetaBelowCell1111331131101332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101332) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101332) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101332) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101332) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131101333
+        (by
+          have h : ((childLL thetaBelowCell1111331131101333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131101333) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131101333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131101333) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131101333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131101333) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131101333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131101333) h))
+
+theorem cover_subtree_dca1503025ec :
+    adaptiveCoverCheck 4 (childHH (childLH thetaBelowCell111133113110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH thetaBelowCell111133113110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHH (childLH
+        thetaBelowCell111133113110)))
+        (by
+          have h : (thetaBelowCell1111331131101300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101300 h)
+        (by
+          have h : (thetaBelowCell1111331131101301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101301 h)
+        (by
+          have h : (thetaBelowCell1111331131101302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101302 h)
+        (by
+          have h : (thetaBelowCell1111331131101303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHH (childLH
+        thetaBelowCell111133113110)))
+        (by
+          have h : (thetaBelowCell1111331131101310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101310 h)
+        (by
+          have h : (thetaBelowCell1111331131101311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101311 h)
+        (by
+          have h : (thetaBelowCell1111331131101312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101312 h)
+        (by
+          have h : (thetaBelowCell1111331131101313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131101313 h))
+    cover_subtree_ab4066e0c111
+    cover_subtree_8607f2894d1c
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c0_c1 :
+    adaptiveCoverCheck 5 (childLH thetaBelowCell111133113110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133113110)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH thetaBelowCell111133113110))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLH
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLH
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLH
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLH
+            thetaBelowCell111133113110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH thetaBelowCell111133113110))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLH
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLH
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLH
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLH
+            thetaBelowCell111133113110))) h))
+    cover_subtree_60a2aff6cef2
+    cover_subtree_dca1503025ec
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells754567fdeb
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells754567fdeb
+
+open CertificateCells754567fdeb
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c1_c2_c2 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022002101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022002101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022002101)))
+        (by
+          have h : (thetaAboveCell0000220021012200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012200 h)
+        (by
+          have h : (thetaAboveCell0000220021012201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012201 h)
+        (by
+          have h : (thetaAboveCell0000220021012202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012202 h)
+        (by
+          have h : (thetaAboveCell0000220021012203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022002101)))
+        (by
+          have h : (thetaAboveCell0000220021012210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012210 h)
+        (by
+          have h : (thetaAboveCell0000220021012211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012211 h)
+        (by
+          have h : (thetaAboveCell0000220021012212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012212 h)
+        (by
+          have h : (thetaAboveCell0000220021012213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022002101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022002101))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022002101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022002101))) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells146a7460c7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells146a7460c7
+
+open CertificateCells146a7460c7
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c0_c2_c2 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022002100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022002100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022002100)))
+        (by
+          have h : (thetaAboveCell0000220021002200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002200 h)
+        (by
+          have h : (thetaAboveCell0000220021002201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002201 h)
+        (by
+          have h : (thetaAboveCell0000220021002202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002202 h)
+        (by
+          have h : (thetaAboveCell0000220021002203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022002100)))
+        (by
+          have h : (thetaAboveCell0000220021002210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002210 h)
+        (by
+          have h : (thetaAboveCell0000220021002211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002211 h)
+        (by
+          have h : (thetaAboveCell0000220021002212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002212 h)
+        (by
+          have h : (thetaAboveCell0000220021002213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022002100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022002100))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022002100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022002100))) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5446bfae87
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5446bfae87
+
+open CertificateCells5446bfae87
+theorem cover_subtree_847043a66fa6 :
+    adaptiveCoverCheck 3 (childHL (childHL (childLL thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHL (childLL
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100220
+        (by
+          have h : ((childLL thetaBelowCell1111331131100220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100220) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100220) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100220) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100220) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100221
+        (by
+          have h : ((childLL thetaBelowCell1111331131100221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100221) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100221) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100221) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100221) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100222
+        (by
+          have h : ((childLL thetaBelowCell1111331131100222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100222) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100222) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100222) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100222) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100223
+        (by
+          have h : ((childLL thetaBelowCell1111331131100223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100223) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100223) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100223) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100223) h))
+
+theorem cover_subtree_9cd5da4b1592 :
+    adaptiveCoverCheck 3 (childHH (childHL (childLL thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHL (childLL
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100230
+        (by
+          have h : ((childLL thetaBelowCell1111331131100230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100230) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100230) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100230) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100230) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100231
+        (by
+          have h : ((childLL thetaBelowCell1111331131100231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100231) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100231) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100231) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100231) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100232
+        (by
+          have h : ((childLL thetaBelowCell1111331131100232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100232) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100232) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100232) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100232) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100233
+        (by
+          have h : ((childLL thetaBelowCell1111331131100233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100233) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100233) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100233) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100233) h))
+
+theorem cover_subtree_49612604cfd8 :
+    adaptiveCoverCheck 4 (childHL (childLL thetaBelowCell111133113110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL thetaBelowCell111133113110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHL (childLL
+        thetaBelowCell111133113110)))
+        (by
+          have h : (thetaBelowCell1111331131100200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100200 h)
+        (by
+          have h : (thetaBelowCell1111331131100201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100201 h)
+        (by
+          have h : (thetaBelowCell1111331131100202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100202 h)
+        (by
+          have h : (thetaBelowCell1111331131100203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHL (childLL
+        thetaBelowCell111133113110)))
+        (by
+          have h : (thetaBelowCell1111331131100210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100210 h)
+        (by
+          have h : (thetaBelowCell1111331131100211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100211 h)
+        (by
+          have h : (thetaBelowCell1111331131100212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100212 h)
+        (by
+          have h : (thetaBelowCell1111331131100213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100213 h))
+    cover_subtree_847043a66fa6
+    cover_subtree_9cd5da4b1592
+
+theorem cover_subtree_e94de792d061 :
+    adaptiveCoverCheck 3 (childHL (childHH (childLL thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHH (childLL
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100320
+        (by
+          have h : ((childLL thetaBelowCell1111331131100320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100320) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100320) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100320) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100320) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100321
+        (by
+          have h : ((childLL thetaBelowCell1111331131100321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100321) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100321) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100321) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100321) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100322
+        (by
+          have h : ((childLL thetaBelowCell1111331131100322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100322) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100322) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100322) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100322) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100323
+        (by
+          have h : ((childLL thetaBelowCell1111331131100323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100323) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100323) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100323) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100323) h))
+
+theorem cover_subtree_0fbb14a9cae8 :
+    adaptiveCoverCheck 3 (childHH (childHH (childLL thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHH (childLL
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100330
+        (by
+          have h : ((childLL thetaBelowCell1111331131100330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100330) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100330) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100330) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100330) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100331
+        (by
+          have h : ((childLL thetaBelowCell1111331131100331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100331) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100331) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100331) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100331) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100332
+        (by
+          have h : ((childLL thetaBelowCell1111331131100332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100332) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100332) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100332) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100332) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131100333
+        (by
+          have h : ((childLL thetaBelowCell1111331131100333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131100333) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131100333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131100333) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131100333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131100333) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131100333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131100333) h))
+
+theorem cover_subtree_ac6486a45a05 :
+    adaptiveCoverCheck 4 (childHH (childLL thetaBelowCell111133113110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL thetaBelowCell111133113110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHH (childLL
+        thetaBelowCell111133113110)))
+        (by
+          have h : (thetaBelowCell1111331131100300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100300 h)
+        (by
+          have h : (thetaBelowCell1111331131100301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100301 h)
+        (by
+          have h : (thetaBelowCell1111331131100302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100302 h)
+        (by
+          have h : (thetaBelowCell1111331131100303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHH (childLL
+        thetaBelowCell111133113110)))
+        (by
+          have h : (thetaBelowCell1111331131100310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100310 h)
+        (by
+          have h : (thetaBelowCell1111331131100311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100311 h)
+        (by
+          have h : (thetaBelowCell1111331131100312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100312 h)
+        (by
+          have h : (thetaBelowCell1111331131100313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131100313 h))
+    cover_subtree_e94de792d061
+    cover_subtree_0fbb14a9cae8
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c0_c0 :
+    adaptiveCoverCheck 5 (childLL thetaBelowCell111133113110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133113110)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL thetaBelowCell111133113110))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLL
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLL
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLL
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLL
+            thetaBelowCell111133113110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL thetaBelowCell111133113110))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLL
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLL
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLL
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLL
+            thetaBelowCell111133113110))) h))
+    cover_subtree_49612604cfd8
+    cover_subtree_ac6486a45a05
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle020.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle020.lean
new file mode 100644
index 0000000000..14911cdd02
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle020.lean
@@ -0,0 +1,13044 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle019
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch033`.
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch043`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCellsefec3b1324
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131111222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131111223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131111220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131111221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131111232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131111233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131111230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131111231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131111322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131111323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131111320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131111321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131111330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131111331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131111332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131111333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131111333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaBelowCell111133113111)))
+
+end GerverSofa.PartE.CertificateCellsefec3b1324
+
+namespace GerverSofa.PartE.CertificateCells11cd3db1c7
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131103000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaBelowCell111133113110)))
+
+/-- Subcell `1111331131103133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaBelowCell111133113110)))
+
+end GerverSofa.PartE.CertificateCells11cd3db1c7
+
+namespace GerverSofa.PartE.CertificateCellse82c2dced4
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCellse82c2dced4
+
+namespace GerverSofa.PartE.CertificateCells7a1b24c2da
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131011220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011200 : AngleCell :=
+  childLL (childLL (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011201 : AngleCell :=
+  childLH (childLL (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011202 : AngleCell :=
+  childHL (childLL (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011203 : AngleCell :=
+  childHH (childLL (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011210 : AngleCell :=
+  childLL (childLH (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011211 : AngleCell :=
+  childLH (childLH (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011212 : AngleCell :=
+  childHL (childLH (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011213 : AngleCell :=
+  childHH (childLH (childHL (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011300 : AngleCell :=
+  childLL (childLL (childHH (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011301 : AngleCell :=
+  childLH (childLL (childHH (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011302 : AngleCell :=
+  childHL (childLL (childHH (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011303 : AngleCell :=
+  childHH (childLL (childHH (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011310 : AngleCell :=
+  childLL (childLH (childHH (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011311 : AngleCell :=
+  childLH (childLH (childHH (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011312 : AngleCell :=
+  childHL (childLH (childHH (childLH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131011313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131011313 : AngleCell :=
+  childHH (childLH (childHH (childLH thetaBelowCell111133113101)))
+
+end GerverSofa.PartE.CertificateCells7a1b24c2da
+
+namespace GerverSofa.PartE.CertificateCells3846b81fdf
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021003002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022002100)))
+
+end GerverSofa.PartE.CertificateCells3846b81fdf
+
+namespace GerverSofa.PartE.CertificateCells8d8f6e86b1
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells8d8f6e86b1
+
+namespace GerverSofa.PartE.CertificateCells67d4ab08b2
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022002100)))
+
+end GerverSofa.PartE.CertificateCells67d4ab08b2
+
+namespace GerverSofa.PartE.CertificateCells37e4000123
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCells37e4000123
+
+namespace GerverSofa.PartE.CertificateCells61d6915992
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCells61d6915992
+
+namespace GerverSofa.PartE.CertificateCells49a6e2db55
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCells49a6e2db55
+
+namespace GerverSofa.PartE.CertificateCells23cc985c42
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCells23cc985c42
+
+namespace GerverSofa.PartE.CoverCertificate19031d0370
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHL (childLH (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell0000 : AngleCell :=
+  childLL cell000
+
+private abbrev cell0001 : AngleCell :=
+  childLH cell000
+
+private abbrev cell0002 : AngleCell :=
+  childHL cell000
+
+private abbrev cell0003 : AngleCell :=
+  childHH cell000
+
+private abbrev cell0010 : AngleCell :=
+  childLL cell001
+
+private abbrev cell0011 : AngleCell :=
+  childLH cell001
+
+private abbrev cell0012 : AngleCell :=
+  childHL cell001
+
+private abbrev cell0013 : AngleCell :=
+  childHH cell001
+
+private abbrev cell0100 : AngleCell :=
+  childLL cell010
+
+private abbrev cell0101 : AngleCell :=
+  childLH cell010
+
+private abbrev cell0102 : AngleCell :=
+  childHL cell010
+
+private abbrev cell0103 : AngleCell :=
+  childHH cell010
+
+private abbrev cell0110 : AngleCell :=
+  childLL cell011
+
+private abbrev cell0111 : AngleCell :=
+  childLH cell011
+
+private abbrev cell0112 : AngleCell :=
+  childHL cell011
+
+private abbrev cell0113 : AngleCell :=
+  childHH cell011
+
+private abbrev cell1000 : AngleCell :=
+  childLL cell100
+
+private abbrev cell1001 : AngleCell :=
+  childLH cell100
+
+private abbrev cell1002 : AngleCell :=
+  childHL cell100
+
+private abbrev cell1003 : AngleCell :=
+  childHH cell100
+
+private abbrev cell1010 : AngleCell :=
+  childLL cell101
+
+private abbrev cell1011 : AngleCell :=
+  childLH cell101
+
+private abbrev cell1012 : AngleCell :=
+  childHL cell101
+
+private abbrev cell1013 : AngleCell :=
+  childHH cell101
+
+end GerverSofa.PartE.CoverCertificate19031d0370
+
+namespace GerverSofa.PartE.CoverCertificate30936f26ba
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHL (childLH (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificate30936f26ba
+
+namespace GerverSofa.PartE.CoverCertificate953a140331
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childHL (childLH (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate953a140331
+
+namespace GerverSofa.PartE.CoverCertificatec0f3040bb9
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childHL (childLH (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificatec0f3040bb9
+
+namespace GerverSofa.PartE.CoverCertificatec5d26eacb3
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHL (childLH (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+private abbrev cell2200 : AngleCell :=
+  childLL cell220
+
+private abbrev cell2201 : AngleCell :=
+  childLH cell220
+
+private abbrev cell2202 : AngleCell :=
+  childHL cell220
+
+private abbrev cell2203 : AngleCell :=
+  childHH cell220
+
+private abbrev cell2210 : AngleCell :=
+  childLL cell221
+
+private abbrev cell2211 : AngleCell :=
+  childLH cell221
+
+private abbrev cell2212 : AngleCell :=
+  childHL cell221
+
+private abbrev cell2213 : AngleCell :=
+  childHH cell221
+
+private abbrev cell2300 : AngleCell :=
+  childLL cell230
+
+private abbrev cell2301 : AngleCell :=
+  childLH cell230
+
+private abbrev cell2302 : AngleCell :=
+  childHL cell230
+
+private abbrev cell2303 : AngleCell :=
+  childHH cell230
+
+private abbrev cell2310 : AngleCell :=
+  childLL cell231
+
+private abbrev cell2311 : AngleCell :=
+  childLH cell231
+
+private abbrev cell2312 : AngleCell :=
+  childHL cell231
+
+private abbrev cell2313 : AngleCell :=
+  childHH cell231
+
+private abbrev cell3200 : AngleCell :=
+  childLL cell320
+
+private abbrev cell3201 : AngleCell :=
+  childLH cell320
+
+private abbrev cell3202 : AngleCell :=
+  childHL cell320
+
+private abbrev cell3203 : AngleCell :=
+  childHH cell320
+
+private abbrev cell3210 : AngleCell :=
+  childLL cell321
+
+private abbrev cell3211 : AngleCell :=
+  childLH cell321
+
+private abbrev cell3212 : AngleCell :=
+  childHL cell321
+
+private abbrev cell3213 : AngleCell :=
+  childHH cell321
+
+private abbrev cell3300 : AngleCell :=
+  childLL cell330
+
+private abbrev cell3301 : AngleCell :=
+  childLH cell330
+
+private abbrev cell3302 : AngleCell :=
+  childHL cell330
+
+private abbrev cell3303 : AngleCell :=
+  childHH cell330
+
+private abbrev cell3310 : AngleCell :=
+  childLL cell331
+
+private abbrev cell3311 : AngleCell :=
+  childLH cell331
+
+private abbrev cell3312 : AngleCell :=
+  childHL cell331
+
+private abbrev cell3313 : AngleCell :=
+  childHH cell331
+
+end GerverSofa.PartE.CoverCertificatec5d26eacb3
+
+namespace GerverSofa.PartE.CoverCertificate3612aa63be
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHL (childLH (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificate3612aa63be
+
+namespace GerverSofa.PartE.CoverCertificateba32fc9e0a
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHL (childLH (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificateba32fc9e0a
+
+namespace GerverSofa.PartE.CoverCertificate2170ad4ca0
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childLH (childLL (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate2170ad4ca0
+
+namespace GerverSofa.PartE.CoverCertificate513511aa15
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childLH (childLL (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate513511aa15
+
+namespace GerverSofa.PartE.CoverCertificate3743e2f873
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHH (childLH (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+private abbrev cell2200 : AngleCell :=
+  childLL cell220
+
+private abbrev cell2201 : AngleCell :=
+  childLH cell220
+
+private abbrev cell2202 : AngleCell :=
+  childHL cell220
+
+private abbrev cell2203 : AngleCell :=
+  childHH cell220
+
+private abbrev cell2210 : AngleCell :=
+  childLL cell221
+
+private abbrev cell2211 : AngleCell :=
+  childLH cell221
+
+private abbrev cell2212 : AngleCell :=
+  childHL cell221
+
+private abbrev cell2213 : AngleCell :=
+  childHH cell221
+
+private abbrev cell2300 : AngleCell :=
+  childLL cell230
+
+private abbrev cell2301 : AngleCell :=
+  childLH cell230
+
+private abbrev cell2302 : AngleCell :=
+  childHL cell230
+
+private abbrev cell2303 : AngleCell :=
+  childHH cell230
+
+end GerverSofa.PartE.CoverCertificate3743e2f873
+
+namespace GerverSofa.PartE.CoverCertificate3e2742ca77
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childHH (childLH (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate3e2742ca77
+
+namespace GerverSofa.PartE.CoverCertificate05d280259b
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHH (childLH (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificate05d280259b
+
+namespace GerverSofa.PartE.CoverCertificate73ec5d89dc
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHH (childLH (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificate73ec5d89dc
+
+namespace GerverSofa.PartE.CoverCertificated1fc30835b
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childHH (childLH (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificated1fc30835b
+
+namespace GerverSofa.PartE.CoverCertificatec03bec1842
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHH (childLH (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatec03bec1842
+
+namespace GerverSofa.PartE.CoverCertificate854ce4fcce
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHH (childLH (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificate854ce4fcce
+
+namespace GerverSofa.PartE.CoverCertificate756063c6fb
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHH (childLH (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificate756063c6fb
+
+namespace GerverSofa.PartE.CoverCertificatebb637b4eef
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childLH (childLL (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatebb637b4eef
+
+namespace GerverSofa.PartE.CoverCertificate7fa5919613
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childLH (childLL (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate7fa5919613
+
+namespace GerverSofa.PartE.CoverCertificateb766a036ce
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childHL (childLL (childLL (childHL (childHL (childLL (childLL
+    (childLL (childLL (e24ThetaAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificateb766a036ce
+
+namespace GerverSofa.PartE.CoverCertificate61295956d2
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childHL (childLL (childLL (childHL (childHL (childLL (childLL
+    (childLL (childLL (e24ThetaAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate61295956d2
+
+namespace GerverSofa.PartE.CoverCertificate828511f0b8
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLH (childLL (childHL (childLL (childLL (childHL (childHL (childLL
+    (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate828511f0b8
+
+namespace GerverSofa.PartE.CoverCertificated54c339469
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLH (childLL (childHL (childLL (childLL (childHL (childHL (childLL
+    (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificated54c339469
+
+namespace GerverSofa.PartE.CoverCertificate4dd8dcfac0
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHL (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate4dd8dcfac0
+
+namespace GerverSofa.PartE.CoverCertificate4ee27c37fd
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childHL (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate4ee27c37fd
+
+namespace GerverSofa.PartE.CoverCertificated7b655454c
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHL (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificated7b655454c
+
+namespace GerverSofa.PartE.CoverCertificate34bc1a6525
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHL (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificate34bc1a6525
+
+namespace GerverSofa.PartE.CoverCertificate6e40ca0512
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childHL (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate6e40ca0512
+
+namespace GerverSofa.PartE.CoverCertificateefd035cf65
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHL (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificateefd035cf65
+
+namespace GerverSofa.PartE.CoverCertificatea4b667aaa0
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHL (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificatea4b667aaa0
+
+namespace GerverSofa.PartE.CoverCertificateccad0dc048
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHL (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificateccad0dc048
+
+namespace GerverSofa.PartE.CoverCertificate89deca0334
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childLL (childLH (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate89deca0334
+
+namespace GerverSofa.PartE.CoverCertificate8778f820bd
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childLL (childLH (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate8778f820bd
+
+namespace GerverSofa.PartE.CoverCertificatef14d24461a
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHH (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatef14d24461a
+
+namespace GerverSofa.PartE.CoverCertificate79980b2e6c
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childHH (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate79980b2e6c
+
+namespace GerverSofa.PartE.CoverCertificateed2ff1f24e
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHH (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificateed2ff1f24e
+
+namespace GerverSofa.PartE.CoverCertificate4bcf50db3e
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHH (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificate4bcf50db3e
+
+namespace GerverSofa.PartE.CoverCertificate1e4788169b
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childHH (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate1e4788169b
+
+namespace GerverSofa.PartE.CoverCertificate4f5a63a5df
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHH (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate4f5a63a5df
+
+namespace GerverSofa.PartE.CoverCertificate89bb9fd21e
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHH (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificate89bb9fd21e
+
+namespace GerverSofa.PartE.CoverCertificate5c80136736
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHH (childLL (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificate5c80136736
+
+namespace GerverSofa.PartE.CoverCertificate645f79320a
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childLL (childLH (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate645f79320a
+
+namespace GerverSofa.PartE.CoverCertificate3e0a4a2c6f
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childLL (childLH (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate3e0a4a2c6f
+
+namespace GerverSofa.PartE.CertificateCells6deb17ad88
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells6deb17ad88
+
+namespace GerverSofa.PartE.CertificateCells21b7c7dcb8
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells21b7c7dcb8
+
+namespace GerverSofa.PartE.CertificateCells51629e8703
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020012100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020012100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell000022002001)))
+
+end GerverSofa.PartE.CertificateCells51629e8703
+
+namespace GerverSofa.PartE.CertificateCellse4c4271ea3
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020012101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020012101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell000022002001)))
+
+end GerverSofa.PartE.CertificateCellse4c4271ea3
+
+namespace GerverSofa.PartE.CertificateCells07d8bdd7d9
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020012102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020012102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell000022002001)))
+
+end GerverSofa.PartE.CertificateCells07d8bdd7d9
+
+namespace GerverSofa.PartE.CertificateCells2bf9eb5708
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020012103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020012103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell000022002001)))
+
+end GerverSofa.PartE.CertificateCells2bf9eb5708
+
+namespace GerverSofa.PartE.CertificateCells7eefa0b856
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells7eefa0b856
+
+namespace GerverSofa.PartE.CertificateCellsdb6ef42c13
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsdb6ef42c13
+
+namespace GerverSofa.PartE.CertificateCells91885e0374
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells91885e0374
+
+namespace GerverSofa.PartE.CertificateCellsba71b1d520
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsba71b1d520
+
+namespace GerverSofa.PartE.CertificateCells54caf0dec0
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells54caf0dec0
+
+namespace GerverSofa.PartE.CertificateCells22ecf07105
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells22ecf07105
+
+namespace GerverSofa.PartE.CertificateCells2d236bc977
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells2d236bc977
+
+namespace GerverSofa.PartE.CertificateCellsfbf08c6815
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsfbf08c6815
+
+namespace GerverSofa.PartE.CertificateCellsb2588049db
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsb2588049db
+
+namespace GerverSofa.PartE.CertificateCells80702a4342
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells80702a4342
+
+namespace GerverSofa.PartE.CertificateCellsfcacd99e70
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsfcacd99e70
+
+namespace GerverSofa.PartE.CertificateCellsc63c49e043
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsc63c49e043
+
+namespace GerverSofa.PartE.CertificateCellse2c2881509
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellse2c2881509
+
+namespace GerverSofa.PartE.CertificateCells1273bfc545
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells1273bfc545
+
+namespace GerverSofa.PartE.CertificateCells758d72a450
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells758d72a450
+
+namespace GerverSofa.PartE.CertificateCells898eeb07c6
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells898eeb07c6
+
+namespace GerverSofa.PartE.CertificateCells3b9ec898a9
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells3b9ec898a9
+
+namespace GerverSofa.PartE.CertificateCellscbef0a66a7
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellscbef0a66a7
+
+namespace GerverSofa.PartE.CertificateCells0e30799078
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells0e30799078
+
+namespace GerverSofa.PartE.CertificateCellsaa7ab31314
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsaa7ab31314
+
+namespace GerverSofa.PartE.CertificateCellsa4a7cbe7b2
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsa4a7cbe7b2
+
+namespace GerverSofa.PartE.CertificateCells9f358cc74d
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells9f358cc74d
+
+namespace GerverSofa.PartE.CertificateCellsf2bd9cd8f9
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsf2bd9cd8f9
+
+namespace GerverSofa.PartE.CertificateCellse30fe464b6
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellse30fe464b6
+
+namespace GerverSofa.PartE.CertificateCells50e57ae8a7
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells50e57ae8a7
+
+namespace GerverSofa.PartE.CertificateCells6831b5295c
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells6831b5295c
+
+namespace GerverSofa.PartE.CertificateCells6ec7ee0b32
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells6ec7ee0b32
+
+namespace GerverSofa.PartE.CertificateCellscdb025ab65
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellscdb025ab65
+
+namespace GerverSofa.PartE.CertificateCells251eb00cd5
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells251eb00cd5
+
+namespace GerverSofa.PartE.CertificateCellsb46fd93863
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsb46fd93863
+
+namespace GerverSofa.PartE.CertificateCells02c02b7fd3
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells02c02b7fd3
+
+namespace GerverSofa.PartE.CertificateCells969e2199d5
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells969e2199d5
+
+namespace GerverSofa.PartE.CertificateCells99f6185557
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells99f6185557
+
+namespace GerverSofa.PartE.CertificateCellsea3eb05dc6
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsea3eb05dc6
+
+namespace GerverSofa.PartE.CertificateCells1daa3566d0
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells1daa3566d0
+
+namespace GerverSofa.PartE.CertificateCells04b8a575e1
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells04b8a575e1
+
+namespace GerverSofa.PartE.CertificateCellsb52fa9450d
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsb52fa9450d
+
+namespace GerverSofa.PartE.CertificateCells75262755f1
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells75262755f1
+
+namespace GerverSofa.PartE.CertificateCells88775d9be2
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells88775d9be2
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6R4Subtree7c075da8e9f0577b`.
+* `KernelOnly.PartE.E24KC6R4Subtree7e83efce8bd8caa9`.
+* `KernelOnly.PartE.E24KC6R4Join57ced70670a400e2`.
+* `KernelOnly.PartE.E24KC6R4Subtree81ebe3f900bc76a5`.
+* `KernelOnly.PartE.E24KC6R4Subtree84bebec26b36007b`.
+* `KernelOnly.PartE.E24KC6R4Subtree879f3ccc81ca9b24`.
+* `KernelOnly.PartE.E24KC6R4Subtree88f80fd4db374a2d`.
+* `KernelOnly.PartE.E24KC6R4Subtree8921da6a3153d018`.
+* `KernelOnly.PartE.E24KC6R4Subtree89badd31312e459d`.
+* `KernelOnly.PartE.E24KC6R4Subtree8b6172e52c41dded`.
+* `KernelOnly.PartE.E24KC6R4Join0f0434a94ae699a2`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsefec3b1324
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsefec3b1324
+
+open CertificateCellsefec3b1324
+theorem cover_subtree_7aa3d52e724a :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131111222 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131111222
+    (by
+      have h : ((childLL thetaBelowCell1111331131111222)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131111222) h)
+    (by
+      have h : ((childLH thetaBelowCell1111331131111222)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131111222) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131111222)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131111222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131111222)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131111222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131111222)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131111222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131111222)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131111222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131111222)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131111222)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131111222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131111222)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131111222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131111222)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131111222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131111222)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131111222))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131111222)) h))
+
+theorem cover_subtree_818c6b64504a :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131111223 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131111223
+    (by
+      have h : ((childLL thetaBelowCell1111331131111223)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131111223) h)
+    (by
+      have h : ((childLH thetaBelowCell1111331131111223)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131111223) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131111223)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131111223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131111223)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131111223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131111223)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131111223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131111223)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131111223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131111223)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131111223)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131111223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131111223)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131111223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131111223)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131111223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131111223)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131111223))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131111223)) h))
+
+theorem cover_subtree_75546d36bae0 :
+    adaptiveCoverCheck 3 (childHL (childHL (childLH thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHL (childLH
+    thetaBelowCell111133113111)))
+    (by
+      have h : (thetaBelowCell1111331131111220).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131111220 h)
+    (by
+      have h : (thetaBelowCell1111331131111221).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131111221 h)
+    cover_subtree_7aa3d52e724a
+    cover_subtree_818c6b64504a
+
+theorem cover_subtree_38aeeec8d754 :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131111232 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131111232
+    (by
+      have h : ((childLL thetaBelowCell1111331131111232)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131111232) h)
+    (by
+      have h : ((childLH thetaBelowCell1111331131111232)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131111232) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131111232)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131111232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131111232)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131111232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131111232)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131111232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131111232)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131111232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131111232)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131111232)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131111232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131111232)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131111232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131111232)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131111232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131111232)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131111232))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131111232)) h))
+
+theorem cover_subtree_7653e3da7d41 :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131111233 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131111233
+    (by
+      have h : ((childLL thetaBelowCell1111331131111233)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131111233) h)
+    (by
+      have h : ((childLH thetaBelowCell1111331131111233)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131111233) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131111233)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131111233)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131111233)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131111233)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131111233)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131111233)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131111233)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131111233)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131111233)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131111233))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131111233)) h))
+
+theorem cover_subtree_c48c3d0e1082 :
+    adaptiveCoverCheck 3 (childHH (childHL (childLH thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHL (childLH
+    thetaBelowCell111133113111)))
+    (by
+      have h : (thetaBelowCell1111331131111230).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131111230 h)
+    (by
+      have h : (thetaBelowCell1111331131111231).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131111231 h)
+    cover_subtree_38aeeec8d754
+    cover_subtree_7653e3da7d41
+
+theorem cover_subtree_850575b22b27 :
+    adaptiveCoverCheck 4 (childHL (childLH thetaBelowCell111133113111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH thetaBelowCell111133113111))
+    (by
+      have h : ((childLL (childHL (childLH thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childLH
+        thetaBelowCell111133113111))) h)
+    (by
+      have h : ((childLH (childHL (childLH thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childLH
+        thetaBelowCell111133113111))) h)
+    cover_subtree_75546d36bae0
+    cover_subtree_c48c3d0e1082
+
+theorem cover_subtree_8b97f2fa7857 :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131111322 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131111322
+    (by
+      have h : ((childLL thetaBelowCell1111331131111322)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131111322) h)
+    (by
+      have h : ((childLH thetaBelowCell1111331131111322)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131111322) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131111322)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131111322)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131111322)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131111322)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131111322)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131111322)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131111322)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131111322)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131111322)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131111322))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131111322)) h))
+
+theorem cover_subtree_4e37d42ba06c :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131111323 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131111323
+    (by
+      have h : ((childLL thetaBelowCell1111331131111323)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131111323) h)
+    (by
+      have h : ((childLH thetaBelowCell1111331131111323)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131111323) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131111323)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131111323)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131111323)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131111323)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131111323)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131111323)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131111323)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131111323)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131111323)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131111323))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131111323)) h))
+
+theorem cover_subtree_27a9e6ae3bf9 :
+    adaptiveCoverCheck 3 (childHL (childHH (childLH thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHH (childLH
+    thetaBelowCell111133113111)))
+    (by
+      have h : (thetaBelowCell1111331131111320).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131111320 h)
+    (by
+      have h : (thetaBelowCell1111331131111321).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131111321 h)
+    cover_subtree_8b97f2fa7857
+    cover_subtree_4e37d42ba06c
+
+theorem cover_subtree_73b432883d7f :
+    adaptiveCoverCheck 3 (childHH (childHH (childLH thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHH (childLH
+    thetaBelowCell111133113111)))
+    (by
+      have h : (thetaBelowCell1111331131111330).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131111330 h)
+    (by
+      have h : (thetaBelowCell1111331131111331).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131111331 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131111332
+        (by
+          have h : ((childLL thetaBelowCell1111331131111332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131111332) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131111332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131111332) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131111332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131111332) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131111332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131111332) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131111333
+        (by
+          have h : ((childLL thetaBelowCell1111331131111333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131111333) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131111333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131111333) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131111333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131111333) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131111333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131111333) h))
+
+theorem cover_subtree_cff1fffcbf04 :
+    adaptiveCoverCheck 4 (childHH (childLH thetaBelowCell111133113111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH thetaBelowCell111133113111))
+    (by
+      have h : ((childLL (childHH (childLH thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childLH
+        thetaBelowCell111133113111))) h)
+    (by
+      have h : ((childLH (childHH (childLH thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childLH
+        thetaBelowCell111133113111))) h)
+    cover_subtree_27a9e6ae3bf9
+    cover_subtree_73b432883d7f
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c1 :
+    adaptiveCoverCheck 5 (childLH thetaBelowCell111133113111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133113111)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH thetaBelowCell111133113111))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell111133113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLH
+            thetaBelowCell111133113111))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell111133113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLH
+            thetaBelowCell111133113111))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell111133113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLH
+            thetaBelowCell111133113111))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell111133113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLH
+            thetaBelowCell111133113111))) h))
+    (by
+      have h : ((childLH (childLH thetaBelowCell111133113111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH thetaBelowCell111133113111)) h)
+    cover_subtree_850575b22b27
+    cover_subtree_cff1fffcbf04
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells11cd3db1c7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells11cd3db1c7
+
+open CertificateCells11cd3db1c7
+theorem cover_subtree_36a7bfb19453 :
+    adaptiveCoverCheck 3 (childLL (childLL (childHH thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLL (childLL (childHH
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131103000
+        (by
+          have h : ((childLL thetaBelowCell1111331131103000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131103000) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131103000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131103000) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131103000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131103000) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131103000)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131103000) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131103001
+        (by
+          have h : ((childLL thetaBelowCell1111331131103001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131103001) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131103001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131103001) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131103001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131103001) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131103001)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131103001) h))
+    (by
+      have h : (thetaBelowCell1111331131103002).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131103002 h)
+    (by
+      have h : (thetaBelowCell1111331131103003).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131103003 h)
+
+theorem cover_subtree_05a08e0bea5f :
+    adaptiveCoverCheck 3 (childLH (childLL (childHH thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLL (childHH
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131103010
+        (by
+          have h : ((childLL thetaBelowCell1111331131103010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131103010) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131103010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131103010) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131103010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131103010) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131103010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131103010) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131103011
+        (by
+          have h : ((childLL thetaBelowCell1111331131103011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131103011) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131103011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131103011) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131103011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131103011) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131103011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131103011) h))
+    (by
+      have h : (thetaBelowCell1111331131103012).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131103012 h)
+    (by
+      have h : (thetaBelowCell1111331131103013).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131103013 h)
+
+theorem cover_subtree_e2abbc5cba3c :
+    adaptiveCoverCheck 4 (childLL (childHH thetaBelowCell111133113110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH thetaBelowCell111133113110))
+    cover_subtree_36a7bfb19453
+    cover_subtree_05a08e0bea5f
+    (by
+      have h : ((childHL (childLL (childHH thetaBelowCell111133113110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childHH
+        thetaBelowCell111133113110))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaBelowCell111133113110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childHH
+        thetaBelowCell111133113110))) h)
+
+theorem cover_subtree_09ec3c616243 :
+    adaptiveCoverCheck 3 (childLL (childLH (childHH thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLL (childLH (childHH
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131103100
+        (by
+          have h : ((childLL thetaBelowCell1111331131103100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131103100) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131103100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131103100) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131103100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131103100) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131103100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131103100) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131103101
+        (by
+          have h : ((childLL thetaBelowCell1111331131103101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131103101) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131103101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131103101) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131103101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131103101) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131103101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131103101) h))
+    (by
+      have h : (thetaBelowCell1111331131103102).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131103102 h)
+    (by
+      have h : (thetaBelowCell1111331131103103).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131103103 h)
+
+theorem cover_subtree_6c69a3f38b7c :
+    adaptiveCoverCheck 3 (childLH (childLH (childHH thetaBelowCell111133113110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLH (childHH
+    thetaBelowCell111133113110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131103110
+        (by
+          have h : ((childLL thetaBelowCell1111331131103110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131103110) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131103110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131103110) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131103110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131103110) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131103110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131103110) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131103111
+        (by
+          have h : ((childLL thetaBelowCell1111331131103111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131103111) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131103111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131103111) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131103111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131103111) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131103111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131103111) h))
+    (by
+      have h : (thetaBelowCell1111331131103112).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131103112 h)
+    (by
+      have h : (thetaBelowCell1111331131103113).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131103113 h)
+
+theorem cover_subtree_5b1de072a154 :
+    adaptiveCoverCheck 4 (childLH (childHH thetaBelowCell111133113110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH thetaBelowCell111133113110))
+    cover_subtree_09ec3c616243
+    cover_subtree_6c69a3f38b7c
+    (by
+      have h : ((childHL (childLH (childHH thetaBelowCell111133113110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childHH
+        thetaBelowCell111133113110))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childLH (childHH
+        thetaBelowCell111133113110)))
+        (by
+          have h : (thetaBelowCell1111331131103130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131103130 h)
+        (by
+          have h : (thetaBelowCell1111331131103131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131103131 h)
+        (by
+          have h : (thetaBelowCell1111331131103132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131103132 h)
+        (by
+          have h : (thetaBelowCell1111331131103133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131103133 h))
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c0_c3 :
+    adaptiveCoverCheck 5 (childHH thetaBelowCell111133113110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133113110)
+    cover_subtree_e2abbc5cba3c
+    cover_subtree_5b1de072a154
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH thetaBelowCell111133113110))
+        (by
+          have h : ((childLL (childHL (childHH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHH
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childLH (childHL (childHH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHH
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childHL (childHL (childHH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHH
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childHH (childHL (childHH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHH
+            thetaBelowCell111133113110))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH thetaBelowCell111133113110))
+        (by
+          have h : ((childLL (childHH (childHH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHH
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childLH (childHH (childHH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHH
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childHL (childHH (childHH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHH
+            thetaBelowCell111133113110))) h)
+        (by
+          have h : ((childHH (childHH (childHH thetaBelowCell111133113110)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHH
+            thetaBelowCell111133113110))) h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse82c2dced4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse82c2dced4
+
+open CertificateCellse82c2dced4
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c0 :
+    adaptiveCoverCheck 6 thetaBelowCell111133113110 = true :=
+  adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113110
+    e24KC2ThetaBelowLeaf111133113_c1_c1_c0_c0 e24KC2ThetaBelowLeaf111133113_c1_c1_c0_c1
+      e24KC2ThetaBelowLeaf111133113_c1_c1_c0_c2 e24KC2ThetaBelowLeaf111133113_c1_c1_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells7a1b24c2da
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells7a1b24c2da
+
+open CertificateCells7a1b24c2da
+theorem cover_subtree_9615f1192754 :
+    adaptiveCoverCheck 3 (childHL (childHL (childLH thetaBelowCell111133113101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHL (childLH
+    thetaBelowCell111133113101)))
+    (by
+      have h : (thetaBelowCell1111331131011220).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011220 h)
+    (by
+      have h : (thetaBelowCell1111331131011221).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011221 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131011222
+        (by
+          have h : ((childLL thetaBelowCell1111331131011222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131011222) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131011222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131011222) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131011222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131011222) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131011222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131011222) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131011223
+        (by
+          have h : ((childLL thetaBelowCell1111331131011223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131011223) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131011223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131011223) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131011223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131011223) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131011223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131011223) h))
+
+theorem cover_subtree_1f726b3116b0 :
+    adaptiveCoverCheck 3 (childHH (childHL (childLH thetaBelowCell111133113101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHL (childLH
+    thetaBelowCell111133113101)))
+    (by
+      have h : (thetaBelowCell1111331131011230).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011230 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131011231
+        (by
+          have h : ((childLL thetaBelowCell1111331131011231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131011231) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131011231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131011231) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131011231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131011231) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131011231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131011231) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131011232
+        (by
+          have h : ((childLL thetaBelowCell1111331131011232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131011232) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131011232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131011232) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131011232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131011232) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131011232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131011232) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131011233
+        (by
+          have h : ((childLL thetaBelowCell1111331131011233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131011233) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131011233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131011233) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131011233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131011233) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131011233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131011233) h))
+
+theorem cover_subtree_cac2e3b1401d :
+    adaptiveCoverCheck 4 (childHL (childLH thetaBelowCell111133113101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH thetaBelowCell111133113101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHL (childLH
+        thetaBelowCell111133113101)))
+        (by
+          have h : (thetaBelowCell1111331131011200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011200 h)
+        (by
+          have h : (thetaBelowCell1111331131011201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011201 h)
+        (by
+          have h : (thetaBelowCell1111331131011202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011202 h)
+        (by
+          have h : (thetaBelowCell1111331131011203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHL (childLH
+        thetaBelowCell111133113101)))
+        (by
+          have h : (thetaBelowCell1111331131011210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011210 h)
+        (by
+          have h : (thetaBelowCell1111331131011211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011211 h)
+        (by
+          have h : (thetaBelowCell1111331131011212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011212 h)
+        (by
+          have h : (thetaBelowCell1111331131011213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011213 h))
+    cover_subtree_9615f1192754
+    cover_subtree_1f726b3116b0
+
+theorem cover_subtree_4f3cdd0c257d :
+    adaptiveCoverCheck 3 (childHL (childHH (childLH thetaBelowCell111133113101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHH (childLH
+    thetaBelowCell111133113101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131011320
+        (by
+          have h : ((childLL thetaBelowCell1111331131011320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131011320) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131011320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131011320) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131011320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131011320) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131011320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131011320) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131011321
+        (by
+          have h : ((childLL thetaBelowCell1111331131011321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131011321) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131011321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131011321) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131011321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131011321) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131011321)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131011321) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131011322
+        (by
+          have h : ((childLL thetaBelowCell1111331131011322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131011322) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131011322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131011322) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131011322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131011322) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131011322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131011322) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131011323
+        (by
+          have h : ((childLL thetaBelowCell1111331131011323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131011323) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131011323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131011323) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131011323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131011323) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131011323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131011323) h))
+
+theorem cover_subtree_8b305093aef1 :
+    adaptiveCoverCheck 3 (childHH (childHH (childLH thetaBelowCell111133113101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHH (childLH
+    thetaBelowCell111133113101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131011330
+        (by
+          have h : ((childLL thetaBelowCell1111331131011330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131011330) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131011330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131011330) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131011330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131011330) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131011330)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131011330) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131011331
+        (by
+          have h : ((childLL thetaBelowCell1111331131011331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131011331) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131011331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131011331) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131011331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131011331) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131011331)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131011331) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131011332
+        (by
+          have h : ((childLL thetaBelowCell1111331131011332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131011332) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131011332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131011332) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131011332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131011332) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131011332)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131011332) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131011333
+        (by
+          have h : ((childLL thetaBelowCell1111331131011333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131011333) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131011333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131011333) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131011333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131011333) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131011333)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131011333) h))
+
+theorem cover_subtree_38619e7e6aac :
+    adaptiveCoverCheck 4 (childHH (childLH thetaBelowCell111133113101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH thetaBelowCell111133113101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHH (childLH
+        thetaBelowCell111133113101)))
+        (by
+          have h : (thetaBelowCell1111331131011300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011300 h)
+        (by
+          have h : (thetaBelowCell1111331131011301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011301 h)
+        (by
+          have h : (thetaBelowCell1111331131011302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011302 h)
+        (by
+          have h : (thetaBelowCell1111331131011303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHH (childLH
+        thetaBelowCell111133113101)))
+        (by
+          have h : (thetaBelowCell1111331131011310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011310 h)
+        (by
+          have h : (thetaBelowCell1111331131011311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011311 h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131011312
+            (by
+              have h : ((childLL thetaBelowCell1111331131011312)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131011312)
+                h)
+            (by
+              have h : ((childLH thetaBelowCell1111331131011312)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131011312)
+                h)
+            (by
+              have h : ((childHL thetaBelowCell1111331131011312)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131011312)
+                h)
+            (by
+              have h : ((childHH thetaBelowCell1111331131011312)).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131011312)
+                h))
+        (by
+          have h : (thetaBelowCell1111331131011313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131011313 h))
+    cover_subtree_4f3cdd0c257d
+    cover_subtree_8b305093aef1
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c0_c1_c1 :
+    adaptiveCoverCheck 5 (childLH thetaBelowCell111133113101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133113101)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH thetaBelowCell111133113101))
+        (by
+          have h : ((childLL (childLL (childLH thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLH
+            thetaBelowCell111133113101))) h)
+        (by
+          have h : ((childLH (childLL (childLH thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLH
+            thetaBelowCell111133113101))) h)
+        (by
+          have h : ((childHL (childLL (childLH thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLH
+            thetaBelowCell111133113101))) h)
+        (by
+          have h : ((childHH (childLL (childLH thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLH
+            thetaBelowCell111133113101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH thetaBelowCell111133113101))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLH
+            thetaBelowCell111133113101))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLH
+            thetaBelowCell111133113101))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLH
+            thetaBelowCell111133113101))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLH
+            thetaBelowCell111133113101))) h))
+    cover_subtree_cac2e3b1401d
+    cover_subtree_38619e7e6aac
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3846b81fdf
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3846b81fdf
+
+open CertificateCells3846b81fdf
+theorem cover_subtree_c9cdbb673d76 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003002
+    (by
+      have h : ((childLL thetaAboveCell0000220021003002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003002) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021003002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003002) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003002)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003002)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003002)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003002)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021003002)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021003002)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021003002)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021003002)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021003002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021003002)) h))
+
+theorem cover_subtree_0d25773c4d7a :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003003
+    (by
+      have h : ((childLL thetaAboveCell0000220021003003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003003) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021003003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003003) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003003)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003003)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003003)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003003)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021003003)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021003003)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021003003)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021003003)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021003003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021003003)) h))
+
+theorem cover_subtree_d91230c1595a :
+    adaptiveCoverCheck 4 (childLL (childLL (childHH thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHH
+    thetaAboveCell000022002100)))
+    (by
+      have h : (thetaAboveCell0000220021003000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003000 h)
+    (by
+      have h : (thetaAboveCell0000220021003001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003001 h)
+    cover_subtree_c9cdbb673d76
+    cover_subtree_0d25773c4d7a
+
+theorem cover_subtree_3b00f6acc230 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003012
+    (by
+      have h : ((childLL thetaAboveCell0000220021003012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003012) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021003012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003012) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003012)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003012)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003012)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003012)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021003012)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021003012)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021003012)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021003012)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021003012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021003012)) h))
+
+theorem cover_subtree_7ad163274380 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003013
+    (by
+      have h : ((childLL thetaAboveCell0000220021003013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003013) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021003013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003013) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003013)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003013)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003013)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003013)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021003013)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021003013)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021003013)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021003013)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021003013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021003013)) h))
+
+theorem cover_subtree_5685dda3095f :
+    adaptiveCoverCheck 4 (childLH (childLL (childHH thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHH
+    thetaAboveCell000022002100)))
+    (by
+      have h : (thetaAboveCell0000220021003010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003010 h)
+    (by
+      have h : (thetaAboveCell0000220021003011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003011 h)
+    cover_subtree_3b00f6acc230
+    cover_subtree_7ad163274380
+
+theorem cover_subtree_f7f8133fac9a :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003020 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003020
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021003020)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021003020)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021003020)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021003020)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021003020)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021003020)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021003020)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021003020)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021003020)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021003020)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003020)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003020)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003020)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003020)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003020)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021003020)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021003020)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021003020)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021003020)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021003020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021003020)) h))
+
+theorem cover_subtree_34eb89665f6e :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003021 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003021
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021003021)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021003021)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021003021)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021003021)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021003021)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021003021)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021003021)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021003021)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021003021)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021003021)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003021)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003021)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003021)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003021)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003021)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021003021)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021003021)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021003021)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021003021)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021003021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021003021)) h))
+
+theorem cover_subtree_854759339591 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHH thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+    thetaAboveCell000022002100)))
+    cover_subtree_f7f8133fac9a
+    cover_subtree_34eb89665f6e
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003022
+        (by
+          have h : ((childLL thetaAboveCell0000220021003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021003022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021003022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021003022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003023
+        (by
+          have h : ((childLL thetaAboveCell0000220021003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021003023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021003023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021003023) h))
+
+theorem cover_subtree_a49deaaa8f12 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003030 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003030
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021003030)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021003030)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021003030)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021003030)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021003030)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021003030)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021003030)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021003030)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021003030)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021003030)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003030)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003030)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003030)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003030)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003030)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021003030)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021003030)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021003030)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021003030)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021003030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021003030)) h))
+
+theorem cover_subtree_2236023309b5 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003031 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003031
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021003031)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021003031)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021003031)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021003031)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021003031)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021003031)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021003031)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021003031)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021003031)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021003031)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003031)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003031)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003031)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003031)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003031)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021003031)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021003031)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021003031)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021003031)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021003031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021003031)) h))
+
+theorem cover_subtree_2c168101607b :
+    adaptiveCoverCheck 4 (childHH (childLL (childHH thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+    thetaAboveCell000022002100)))
+    cover_subtree_a49deaaa8f12
+    cover_subtree_2236023309b5
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003032
+        (by
+          have h : ((childLL thetaAboveCell0000220021003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021003032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021003032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021003032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003033
+        (by
+          have h : ((childLL thetaAboveCell0000220021003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021003033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021003033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021003033) h))
+
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c0_c3_c0 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022002100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022002100))
+    cover_subtree_d91230c1595a
+    cover_subtree_5685dda3095f
+    cover_subtree_854759339591
+    cover_subtree_2c168101607b
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8d8f6e86b1
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8d8f6e86b1
+
+open CertificateCells8d8f6e86b1
+theorem e24KC2ThetaAboveLeaf0000220021_c1_c2 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022002112
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022002112)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022002112)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022002112)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022002112)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022002112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022002112)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022002112)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022002112)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022002112)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022002112)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022002112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022002112) h)
+    (by
+      have h : ((childHH thetaAboveCell000022002112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022002112) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells67d4ab08b2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells67d4ab08b2
+
+open CertificateCells67d4ab08b2
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c0_c3_c2 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022002100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022002100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022002100)))
+        (by
+          have h : (thetaAboveCell0000220021003200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003200 h)
+        (by
+          have h : (thetaAboveCell0000220021003201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003201 h)
+        (by
+          have h : (thetaAboveCell0000220021003202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003202 h)
+        (by
+          have h : (thetaAboveCell0000220021003203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022002100)))
+        (by
+          have h : (thetaAboveCell0000220021003210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003210 h)
+        (by
+          have h : (thetaAboveCell0000220021003211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003211 h)
+        (by
+          have h : (thetaAboveCell0000220021003212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003212 h)
+        (by
+          have h : (thetaAboveCell0000220021003213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022002100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022002100))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022002100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022002100))) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells37e4000123
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells37e4000123
+
+open CertificateCells37e4000123
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c3 :
+    adaptiveCoverCheck 6 thetaBelowCell111133113113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113113
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133113113)
+        (by
+          have h : ((childLL (childLL thetaBelowCell111133113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLL
+            thetaBelowCell111133113113)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell111133113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLL
+            thetaBelowCell111133113113)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell111133113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLL
+            thetaBelowCell111133113113)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell111133113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLL
+            thetaBelowCell111133113113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133113113)
+        (by
+          have h : ((childLL (childLH thetaBelowCell111133113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childLH
+            thetaBelowCell111133113113)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell111133113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childLH
+            thetaBelowCell111133113113)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell111133113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childLH
+            thetaBelowCell111133113113)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell111133113113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childLH
+            thetaBelowCell111133113113)) h))
+    (by
+      have h : ((childHL thetaBelowCell111133113113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133113113) h)
+    (by
+      have h : ((childHH thetaBelowCell111133113113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133113113) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells61d6915992
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells61d6915992
+
+open CertificateCells61d6915992
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c3_c2 :
+    adaptiveCoverCheck 4 (childHL (childHH thetaBelowCell111133113111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childHH thetaBelowCell111133113111))
+    (by
+      have h : ((childLL (childHL (childHH thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHL (childHH
+        thetaBelowCell111133113111))) h)
+    (by
+      have h : ((childLH (childHL (childHH thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHL (childHH
+        thetaBelowCell111133113111))) h)
+    (by
+      have h : ((childHL (childHL (childHH thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHL (childHH
+        thetaBelowCell111133113111))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHL (childHH
+        thetaBelowCell111133113111))) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells49a6e2db55
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells49a6e2db55
+
+open CertificateCells49a6e2db55
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c2_c3 :
+    adaptiveCoverCheck 4 (childHH (childHL thetaBelowCell111133113111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHL thetaBelowCell111133113111))
+    (by
+      have h : ((childLL (childHH (childHL thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHL
+        thetaBelowCell111133113111))) h)
+    (by
+      have h : ((childLH (childHH (childHL thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHL
+        thetaBelowCell111133113111))) h)
+    (by
+      have h : ((childHL (childHH (childHL thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHL
+        thetaBelowCell111133113111))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHL
+        thetaBelowCell111133113111))) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells23cc985c42
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells23cc985c42
+
+open CertificateCells23cc985c42
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c2 :
+    adaptiveCoverCheck 5 (childHL thetaBelowCell111133113111) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133113111)
+    e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c2_c0 e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c2_c1
+      e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c2_c2 e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.ThetaAbove.Leaf00567`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00568`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00572`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00573`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00574`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00575`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00577`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00578`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00579`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00581`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00582`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00586`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00587`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00588`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00589`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00592`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00593`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00594`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00595`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00597`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00598`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00601`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00602`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00606`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00607`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00610`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00611`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00612`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00613`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00616`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00617`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00618`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00619`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00621`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00622`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00626`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00627`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00628`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00629`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00632`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00633`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00634`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00635`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00637`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00638`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c0_c2_4_00567
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6deb17ad88
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6deb17ad88
+
+open CertificateCells6deb17ad88
+namespace CoverCertificate19031d0370
+
+private theorem checked0000 : adaptiveCoverCheck 0 cell0000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0000 (by decide +kernel)
+
+private theorem checked0001 : adaptiveCoverCheck 0 cell0001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0001 (by decide +kernel)
+
+private theorem checked0002 : adaptiveCoverCheck 0 cell0002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0002 (by decide +kernel)
+
+private theorem checked0003 : adaptiveCoverCheck 0 cell0003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0003 (by decide +kernel)
+
+private theorem checked0010 : adaptiveCoverCheck 0 cell0010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0010 (by decide +kernel)
+
+private theorem checked0011 : adaptiveCoverCheck 0 cell0011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0011 (by decide +kernel)
+
+private theorem checked0012 : adaptiveCoverCheck 0 cell0012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0012 (by decide +kernel)
+
+private theorem checked0013 : adaptiveCoverCheck 0 cell0013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0013 (by decide +kernel)
+
+private theorem checked0100 : adaptiveCoverCheck 0 cell0100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0100 (by decide +kernel)
+
+private theorem checked0101 : adaptiveCoverCheck 0 cell0101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0101 (by decide +kernel)
+
+private theorem checked0102 : adaptiveCoverCheck 0 cell0102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0102 (by decide +kernel)
+
+private theorem checked0103 : adaptiveCoverCheck 0 cell0103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0103 (by decide +kernel)
+
+private theorem checked0110 : adaptiveCoverCheck 0 cell0110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0110 (by decide +kernel)
+
+private theorem checked0111 : adaptiveCoverCheck 0 cell0111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0111 (by decide +kernel)
+
+private theorem checked0112 : adaptiveCoverCheck 0 cell0112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0112 (by decide +kernel)
+
+private theorem checked0113 : adaptiveCoverCheck 0 cell0113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0113 (by decide +kernel)
+
+private theorem checked1000 : adaptiveCoverCheck 0 cell1000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1000 (by decide +kernel)
+
+private theorem checked1001 : adaptiveCoverCheck 0 cell1001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1001 (by decide +kernel)
+
+private theorem checked1002 : adaptiveCoverCheck 0 cell1002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1002 (by decide +kernel)
+
+private theorem checked1003 : adaptiveCoverCheck 0 cell1003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1003 (by decide +kernel)
+
+private theorem checked1010 : adaptiveCoverCheck 0 cell1010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1010 (by decide +kernel)
+
+private theorem checked1011 : adaptiveCoverCheck 0 cell1011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1011 (by decide +kernel)
+
+private theorem checked1012 : adaptiveCoverCheck 0 cell1012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1012 (by decide +kernel)
+
+private theorem checked1013 : adaptiveCoverCheck 0 cell1013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1013 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell000
+    checked0000 checked0001 checked0002 checked0003
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell001
+    checked0010 checked0011 checked0012 checked0013
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell010
+    checked0100 checked0101 checked0102 checked0103
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell011
+    checked0110 checked0111 checked0112 checked0113
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell100
+    checked1000 checked1001 checked1002 checked1003
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell101
+    checked1010 checked1011 checked1012 checked1013
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate19031d0370
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c2 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHL thetaAboveCell000022002001))) = true := by
+  exact CoverCertificate19031d0370.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c0_c3_4_00568
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells21b7c7dcb8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells21b7c7dcb8
+
+open CertificateCells21b7c7dcb8
+namespace CoverCertificate30936f26ba
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate30936f26ba
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c3 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHL thetaAboveCell000022002001))) = true := by
+  exact CoverCertificate30936f26ba.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c1_c0_c0_3_00572
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells51629e8703
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells51629e8703
+
+open CertificateCells51629e8703
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c0_c0 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020012100 = true := by
+  decide +kernel
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c1_c0_c1_3_00573
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse4c4271ea3
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse4c4271ea3
+
+open CertificateCellse4c4271ea3
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c0_c1 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020012101 = true := by
+  decide +kernel
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c1_c0_c2_3_00574
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells07d8bdd7d9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells07d8bdd7d9
+
+open CertificateCells07d8bdd7d9
+namespace CoverCertificate953a140331
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate953a140331
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c0_c2 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020012102 = true := by
+  exact CoverCertificate953a140331.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c1_c0_c3_3_00575
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2bf9eb5708
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2bf9eb5708
+
+open CertificateCells2bf9eb5708
+namespace CoverCertificatec0f3040bb9
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec0f3040bb9
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c0_c3 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020012103 = true := by
+  exact CoverCertificatec0f3040bb9.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c1_c1_4_00577
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells7eefa0b856
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells7eefa0b856
+
+open CertificateCells7eefa0b856
+namespace CoverCertificatec5d26eacb3
+
+private theorem checked2200 : adaptiveCoverCheck 0 cell2200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2200 (by decide +kernel)
+
+private theorem checked2201 : adaptiveCoverCheck 0 cell2201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2201 (by decide +kernel)
+
+private theorem checked2202 : adaptiveCoverCheck 0 cell2202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2202 (by decide +kernel)
+
+private theorem checked2203 : adaptiveCoverCheck 0 cell2203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2203 (by decide +kernel)
+
+private theorem checked2210 : adaptiveCoverCheck 0 cell2210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2210 (by decide +kernel)
+
+private theorem checked2211 : adaptiveCoverCheck 0 cell2211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2211 (by decide +kernel)
+
+private theorem checked2212 : adaptiveCoverCheck 0 cell2212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2212 (by decide +kernel)
+
+private theorem checked2213 : adaptiveCoverCheck 0 cell2213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2213 (by decide +kernel)
+
+private theorem checked2300 : adaptiveCoverCheck 0 cell2300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2300 (by decide +kernel)
+
+private theorem checked2301 : adaptiveCoverCheck 0 cell2301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2301 (by decide +kernel)
+
+private theorem checked2302 : adaptiveCoverCheck 0 cell2302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2302 (by decide +kernel)
+
+private theorem checked2303 : adaptiveCoverCheck 0 cell2303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2303 (by decide +kernel)
+
+private theorem checked2310 : adaptiveCoverCheck 0 cell2310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2310 (by decide +kernel)
+
+private theorem checked2311 : adaptiveCoverCheck 0 cell2311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2311 (by decide +kernel)
+
+private theorem checked2312 : adaptiveCoverCheck 0 cell2312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2312 (by decide +kernel)
+
+private theorem checked2313 : adaptiveCoverCheck 0 cell2313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2313 (by decide +kernel)
+
+private theorem checked3200 : adaptiveCoverCheck 0 cell3200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3200 (by decide +kernel)
+
+private theorem checked3201 : adaptiveCoverCheck 0 cell3201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3201 (by decide +kernel)
+
+private theorem checked3202 : adaptiveCoverCheck 0 cell3202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3202 (by decide +kernel)
+
+private theorem checked3203 : adaptiveCoverCheck 0 cell3203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3203 (by decide +kernel)
+
+private theorem checked3210 : adaptiveCoverCheck 0 cell3210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3210 (by decide +kernel)
+
+private theorem checked3211 : adaptiveCoverCheck 0 cell3211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3211 (by decide +kernel)
+
+private theorem checked3212 : adaptiveCoverCheck 0 cell3212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3212 (by decide +kernel)
+
+private theorem checked3213 : adaptiveCoverCheck 0 cell3213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3213 (by decide +kernel)
+
+private theorem checked3300 : adaptiveCoverCheck 0 cell3300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3300 (by decide +kernel)
+
+private theorem checked3301 : adaptiveCoverCheck 0 cell3301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3301 (by decide +kernel)
+
+private theorem checked3302 : adaptiveCoverCheck 0 cell3302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3302 (by decide +kernel)
+
+private theorem checked3303 : adaptiveCoverCheck 0 cell3303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3303 (by decide +kernel)
+
+private theorem checked3310 : adaptiveCoverCheck 0 cell3310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3310 (by decide +kernel)
+
+private theorem checked3311 : adaptiveCoverCheck 0 cell3311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3311 (by decide +kernel)
+
+private theorem checked3312 : adaptiveCoverCheck 0 cell3312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3312 (by decide +kernel)
+
+private theorem checked3313 : adaptiveCoverCheck 0 cell3313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell3313 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell220
+    checked2200 checked2201 checked2202 checked2203
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell221
+    checked2210 checked2211 checked2212 checked2213
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell230
+    checked2300 checked2301 checked2302 checked2303
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell231
+    checked2310 checked2311 checked2312 checked2313
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell320
+    checked3200 checked3201 checked3202 checked3203
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell321
+    checked3210 checked3211 checked3212 checked3213
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell330
+    checked3300 checked3301 checked3302 checked3303
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell331
+    checked3310 checked3311 checked3312 checked3313
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec5d26eacb3
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c1 :
+    adaptiveCoverCheck 4 (childLH (childLH (childHL thetaAboveCell000022002001))) = true := by
+  exact CoverCertificatec5d26eacb3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c1_c2_4_00578
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdb6ef42c13
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdb6ef42c13
+
+open CertificateCellsdb6ef42c13
+namespace CoverCertificate3612aa63be
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3612aa63be
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c2 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHL thetaAboveCell000022002001))) = true := by
+  exact CoverCertificate3612aa63be.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c1_c3_4_00579
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells91885e0374
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells91885e0374
+
+open CertificateCells91885e0374
+namespace CoverCertificateba32fc9e0a
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateba32fc9e0a
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c3 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHL thetaAboveCell000022002001))) = true := by
+  exact CoverCertificateba32fc9e0a.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c2_5_00581
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsba71b1d520
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsba71b1d520
+
+open CertificateCellsba71b1d520
+namespace CoverCertificate2170ad4ca0
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate2170ad4ca0
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c2 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022002001)) = true := by
+  exact CoverCertificate2170ad4ca0.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c3_5_00582
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells54caf0dec0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells54caf0dec0
+
+open CertificateCells54caf0dec0
+namespace CoverCertificate513511aa15
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate513511aa15
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c3 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022002001)) = true := by
+  exact CoverCertificate513511aa15.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c3_c0_c0_4_00586
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells22ecf07105
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells22ecf07105
+
+open CertificateCells22ecf07105
+namespace CoverCertificate3743e2f873
+
+private theorem checked2200 : adaptiveCoverCheck 0 cell2200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2200 (by decide +kernel)
+
+private theorem checked2201 : adaptiveCoverCheck 0 cell2201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2201 (by decide +kernel)
+
+private theorem checked2202 : adaptiveCoverCheck 0 cell2202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2202 (by decide +kernel)
+
+private theorem checked2203 : adaptiveCoverCheck 0 cell2203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2203 (by decide +kernel)
+
+private theorem checked2210 : adaptiveCoverCheck 0 cell2210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2210 (by decide +kernel)
+
+private theorem checked2211 : adaptiveCoverCheck 0 cell2211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2211 (by decide +kernel)
+
+private theorem checked2212 : adaptiveCoverCheck 0 cell2212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2212 (by decide +kernel)
+
+private theorem checked2213 : adaptiveCoverCheck 0 cell2213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2213 (by decide +kernel)
+
+private theorem checked2300 : adaptiveCoverCheck 0 cell2300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2300 (by decide +kernel)
+
+private theorem checked2301 : adaptiveCoverCheck 0 cell2301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2301 (by decide +kernel)
+
+private theorem checked2302 : adaptiveCoverCheck 0 cell2302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2302 (by decide +kernel)
+
+private theorem checked2303 : adaptiveCoverCheck 0 cell2303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2303 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell220
+    checked2200 checked2201 checked2202 checked2203
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell221
+    checked2210 checked2211 checked2212 checked2213
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell230
+    checked2300 checked2301 checked2302 checked2303
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3743e2f873
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c0_c0 :
+    adaptiveCoverCheck 4 (childLL (childLL (childHH thetaAboveCell000022002001))) = true := by
+  exact CoverCertificate3743e2f873.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c3_c0_c1_4_00587
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2d236bc977
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2d236bc977
+
+open CertificateCells2d236bc977
+namespace CoverCertificate3e2742ca77
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3e2742ca77
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c0_c1 :
+    adaptiveCoverCheck 4 (childLH (childLL (childHH thetaAboveCell000022002001))) = true := by
+  exact CoverCertificate3e2742ca77.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c3_c0_c2_4_00588
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsfbf08c6815
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsfbf08c6815
+
+open CertificateCellsfbf08c6815
+namespace CoverCertificate05d280259b
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate05d280259b
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c0_c2 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHH thetaAboveCell000022002001))) = true := by
+  exact CoverCertificate05d280259b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c3_c0_c3_4_00589
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb2588049db
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb2588049db
+
+open CertificateCellsb2588049db
+namespace CoverCertificate73ec5d89dc
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate73ec5d89dc
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c0_c3 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHH thetaAboveCell000022002001))) = true := by
+  exact CoverCertificate73ec5d89dc.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c3_c1_c0_4_00592
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells80702a4342
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells80702a4342
+
+open CertificateCells80702a4342
+namespace CoverCertificated1fc30835b
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated1fc30835b
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c1_c0 :
+    adaptiveCoverCheck 4 (childLL (childLH (childHH thetaAboveCell000022002001))) = true := by
+  exact CoverCertificated1fc30835b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c3_c1_c1_4_00593
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsfcacd99e70
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsfcacd99e70
+
+open CertificateCellsfcacd99e70
+namespace CoverCertificatec03bec1842
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec03bec1842
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c1_c1 :
+    adaptiveCoverCheck 4 (childLH (childLH (childHH thetaAboveCell000022002001))) = true := by
+  exact CoverCertificatec03bec1842.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c3_c1_c2_4_00594
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc63c49e043
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc63c49e043
+
+open CertificateCellsc63c49e043
+namespace CoverCertificate854ce4fcce
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate854ce4fcce
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c1_c2 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHH thetaAboveCell000022002001))) = true := by
+  exact CoverCertificate854ce4fcce.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c3_c1_c3_4_00595
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse2c2881509
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse2c2881509
+
+open CertificateCellse2c2881509
+namespace CoverCertificate756063c6fb
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate756063c6fb
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c1_c3 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHH thetaAboveCell000022002001))) = true := by
+  exact CoverCertificate756063c6fb.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c3_c2_5_00597
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1273bfc545
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1273bfc545
+
+open CertificateCells1273bfc545
+namespace CoverCertificatebb637b4eef
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatebb637b4eef
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c2 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022002001)) = true := by
+  exact CoverCertificatebb637b4eef.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c3_c3_5_00598
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells758d72a450
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells758d72a450
+
+open CertificateCells758d72a450
+namespace CoverCertificate7fa5919613
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate7fa5919613
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c3 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022002001)) = true := by
+  exact CoverCertificate7fa5919613.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above Leaf0000220020_c0_c2_7_00601
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells898eeb07c6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells898eeb07c6
+
+open CertificateCells898eeb07c6
+namespace CoverCertificateb766a036ce
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateb766a036ce
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c2 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002002 = true := by
+  exact CoverCertificateb766a036ce.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above Leaf0000220020_c0_c3_7_00602
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3b9ec898a9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3b9ec898a9
+
+open CertificateCells3b9ec898a9
+namespace CoverCertificate61295956d2
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate61295956d2
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c3 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002003 = true := by
+  exact CoverCertificate61295956d2.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c0_6_00606
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellscbef0a66a7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellscbef0a66a7
+
+open CertificateCellscbef0a66a7
+namespace CoverCertificate828511f0b8
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate828511f0b8
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c0 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022002010) = true := by
+  exact CoverCertificate828511f0b8.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c1_6_00607
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0e30799078
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0e30799078
+
+open CertificateCells0e30799078
+namespace CoverCertificated54c339469
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated54c339469
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c1 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022002010) = true := by
+  exact CoverCertificated54c339469.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c2_c0_c0_4_00610
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsaa7ab31314
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsaa7ab31314
+
+open CertificateCellsaa7ab31314
+namespace CoverCertificate4dd8dcfac0
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4dd8dcfac0
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c0_c0 :
+    adaptiveCoverCheck 4 (childLL (childLL (childHL thetaAboveCell000022002010))) = true := by
+  exact CoverCertificate4dd8dcfac0.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c2_c0_c1_4_00611
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa4a7cbe7b2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa4a7cbe7b2
+
+open CertificateCellsa4a7cbe7b2
+namespace CoverCertificate4ee27c37fd
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4ee27c37fd
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c0_c1 :
+    adaptiveCoverCheck 4 (childLH (childLL (childHL thetaAboveCell000022002010))) = true := by
+  exact CoverCertificate4ee27c37fd.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c2_c0_c2_4_00612
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9f358cc74d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9f358cc74d
+
+open CertificateCells9f358cc74d
+namespace CoverCertificated7b655454c
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated7b655454c
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c0_c2 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHL thetaAboveCell000022002010))) = true := by
+  exact CoverCertificated7b655454c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c2_c0_c3_4_00613
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf2bd9cd8f9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf2bd9cd8f9
+
+open CertificateCellsf2bd9cd8f9
+namespace CoverCertificate34bc1a6525
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate34bc1a6525
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c0_c3 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHL thetaAboveCell000022002010))) = true := by
+  exact CoverCertificate34bc1a6525.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c2_c1_c0_4_00616
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse30fe464b6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse30fe464b6
+
+open CertificateCellse30fe464b6
+namespace CoverCertificate6e40ca0512
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate6e40ca0512
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c1_c0 :
+    adaptiveCoverCheck 4 (childLL (childLH (childHL thetaAboveCell000022002010))) = true := by
+  exact CoverCertificate6e40ca0512.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c2_c1_c1_4_00617
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells50e57ae8a7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells50e57ae8a7
+
+open CertificateCells50e57ae8a7
+namespace CoverCertificateefd035cf65
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateefd035cf65
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c1_c1 :
+    adaptiveCoverCheck 4 (childLH (childLH (childHL thetaAboveCell000022002010))) = true := by
+  exact CoverCertificateefd035cf65.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c2_c1_c2_4_00618
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6831b5295c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6831b5295c
+
+open CertificateCells6831b5295c
+namespace CoverCertificatea4b667aaa0
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea4b667aaa0
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c1_c2 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHL thetaAboveCell000022002010))) = true := by
+  exact CoverCertificatea4b667aaa0.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c2_c1_c3_4_00619
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6ec7ee0b32
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6ec7ee0b32
+
+open CertificateCells6ec7ee0b32
+namespace CoverCertificateccad0dc048
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateccad0dc048
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c1_c3 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHL thetaAboveCell000022002010))) = true := by
+  exact CoverCertificateccad0dc048.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c2_c2_5_00621
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellscdb025ab65
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellscdb025ab65
+
+open CertificateCellscdb025ab65
+namespace CoverCertificate89deca0334
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate89deca0334
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c2 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022002010)) = true := by
+  exact CoverCertificate89deca0334.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c2_c3_5_00622
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells251eb00cd5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells251eb00cd5
+
+open CertificateCells251eb00cd5
+namespace CoverCertificate8778f820bd
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate8778f820bd
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c3 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022002010)) = true := by
+  exact CoverCertificate8778f820bd.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c3_c0_c0_4_00626
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb46fd93863
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb46fd93863
+
+open CertificateCellsb46fd93863
+namespace CoverCertificatef14d24461a
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef14d24461a
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c0_c0 :
+    adaptiveCoverCheck 4 (childLL (childLL (childHH thetaAboveCell000022002010))) = true := by
+  exact CoverCertificatef14d24461a.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c3_c0_c1_4_00627
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells02c02b7fd3
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells02c02b7fd3
+
+open CertificateCells02c02b7fd3
+namespace CoverCertificate79980b2e6c
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate79980b2e6c
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c0_c1 :
+    adaptiveCoverCheck 4 (childLH (childLL (childHH thetaAboveCell000022002010))) = true := by
+  exact CoverCertificate79980b2e6c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c3_c0_c2_4_00628
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells969e2199d5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells969e2199d5
+
+open CertificateCells969e2199d5
+namespace CoverCertificateed2ff1f24e
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateed2ff1f24e
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c0_c2 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHH thetaAboveCell000022002010))) = true := by
+  exact CoverCertificateed2ff1f24e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c3_c0_c3_4_00629
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells99f6185557
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells99f6185557
+
+open CertificateCells99f6185557
+namespace CoverCertificate4bcf50db3e
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4bcf50db3e
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c0_c3 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHH thetaAboveCell000022002010))) = true := by
+  exact CoverCertificate4bcf50db3e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c3_c1_c0_4_00632
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsea3eb05dc6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsea3eb05dc6
+
+open CertificateCellsea3eb05dc6
+namespace CoverCertificate1e4788169b
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1e4788169b
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c1_c0 :
+    adaptiveCoverCheck 4 (childLL (childLH (childHH thetaAboveCell000022002010))) = true := by
+  exact CoverCertificate1e4788169b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c3_c1_c1_4_00633
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1daa3566d0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1daa3566d0
+
+open CertificateCells1daa3566d0
+namespace CoverCertificate4f5a63a5df
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4f5a63a5df
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c1_c1 :
+    adaptiveCoverCheck 4 (childLH (childLH (childHH thetaAboveCell000022002010))) = true := by
+  exact CoverCertificate4f5a63a5df.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c3_c1_c2_4_00634
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells04b8a575e1
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells04b8a575e1
+
+open CertificateCells04b8a575e1
+namespace CoverCertificate89bb9fd21e
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate89bb9fd21e
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c1_c2 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHH thetaAboveCell000022002010))) = true := by
+  exact CoverCertificate89bb9fd21e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c3_c1_c3_4_00635
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb52fa9450d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb52fa9450d
+
+open CertificateCellsb52fa9450d
+namespace CoverCertificate5c80136736
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5c80136736
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c1_c3 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHH thetaAboveCell000022002010))) = true := by
+  exact CoverCertificate5c80136736.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c3_c2_5_00637
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells75262755f1
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells75262755f1
+
+open CertificateCells75262755f1
+namespace CoverCertificate645f79320a
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate645f79320a
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c2 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022002010)) = true := by
+  exact CoverCertificate645f79320a.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c0_c3_c3_5_00638
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells88775d9be2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells88775d9be2
+
+open CertificateCells88775d9be2
+namespace CoverCertificate3e0a4a2c6f
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3e0a4a2c6f
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c3 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022002010)) = true := by
+  exact CoverCertificate3e0a4a2c6f.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle021.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle021.lean
new file mode 100644
index 0000000000..8078acacaf
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle021.lean
@@ -0,0 +1,10989 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle018
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle019
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle020
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch034`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCells84b6dd5403
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331130110020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110020 : AngleCell :=
+  childLL (childHL (childLL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110021 : AngleCell :=
+  childLH (childHL (childLL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110022 : AngleCell :=
+  childHL (childHL (childLL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110023 : AngleCell :=
+  childHH (childHL (childLL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110030 : AngleCell :=
+  childLL (childHH (childLL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110031 : AngleCell :=
+  childLH (childHH (childLL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110032 : AngleCell :=
+  childHL (childHH (childLL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110033 : AngleCell :=
+  childHH (childHH (childLL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110120 : AngleCell :=
+  childLL (childHL (childLH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110121 : AngleCell :=
+  childLH (childHL (childLH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110122 : AngleCell :=
+  childHL (childHL (childLH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110123 : AngleCell :=
+  childHH (childHL (childLH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110130 : AngleCell :=
+  childLL (childHH (childLH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110131 : AngleCell :=
+  childLH (childHH (childLH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110132 : AngleCell :=
+  childHL (childHH (childLH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110133 : AngleCell :=
+  childHH (childHH (childLH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110200 : AngleCell :=
+  childLL (childLL (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110201 : AngleCell :=
+  childLH (childLL (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110202 : AngleCell :=
+  childHL (childLL (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110203 : AngleCell :=
+  childHH (childLL (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110210 : AngleCell :=
+  childLL (childLH (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110211 : AngleCell :=
+  childLH (childLH (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110212 : AngleCell :=
+  childHL (childLH (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110213 : AngleCell :=
+  childHH (childLH (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110300 : AngleCell :=
+  childLL (childLL (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110301 : AngleCell :=
+  childLH (childLL (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110302 : AngleCell :=
+  childHL (childLL (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110303 : AngleCell :=
+  childHH (childLL (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110310 : AngleCell :=
+  childLL (childLH (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110311 : AngleCell :=
+  childLH (childLH (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110312 : AngleCell :=
+  childHL (childLH (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110313 : AngleCell :=
+  childHH (childLH (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130110333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130110333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111020 : AngleCell :=
+  childLL (childHL (childLL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111021 : AngleCell :=
+  childLH (childHL (childLL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111022 : AngleCell :=
+  childHL (childHL (childLL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111023 : AngleCell :=
+  childHH (childHL (childLL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111030 : AngleCell :=
+  childLL (childHH (childLL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111031 : AngleCell :=
+  childLH (childHH (childLL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111032 : AngleCell :=
+  childHL (childHH (childLL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111033 : AngleCell :=
+  childHH (childHH (childLL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111120 : AngleCell :=
+  childLL (childHL (childLH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111121 : AngleCell :=
+  childLH (childHL (childLH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111122 : AngleCell :=
+  childHL (childHL (childLH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111123 : AngleCell :=
+  childHH (childHL (childLH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111130 : AngleCell :=
+  childLL (childHH (childLH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111131 : AngleCell :=
+  childLH (childHH (childLH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111132 : AngleCell :=
+  childHL (childHH (childLH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111133 : AngleCell :=
+  childHH (childHH (childLH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111200 : AngleCell :=
+  childLL (childLL (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111201 : AngleCell :=
+  childLH (childLL (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111202 : AngleCell :=
+  childHL (childLL (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111203 : AngleCell :=
+  childHH (childLL (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111210 : AngleCell :=
+  childLL (childLH (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111211 : AngleCell :=
+  childLH (childLH (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111212 : AngleCell :=
+  childHL (childLH (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111213 : AngleCell :=
+  childHH (childLH (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111300 : AngleCell :=
+  childLL (childLL (childHH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111301 : AngleCell :=
+  childLH (childLL (childHH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111302 : AngleCell :=
+  childHL (childLL (childHH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111303 : AngleCell :=
+  childHH (childLL (childHH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111310 : AngleCell :=
+  childLL (childLH (childHH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111311 : AngleCell :=
+  childLH (childLH (childHH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111312 : AngleCell :=
+  childHL (childLH (childHH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111313 : AngleCell :=
+  childHH (childLH (childHH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaBelowCell111133113011)))
+
+/-- Subcell `1111331130111333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331130111333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaBelowCell111133113011)))
+
+end GerverSofa.PartE.CertificateCells84b6dd5403
+
+namespace GerverSofa.PartE.CertificateCellsa837c94d6b
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsa837c94d6b
+
+namespace GerverSofa.PartE.CertificateCellsad9f83220e
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+end GerverSofa.PartE.CertificateCellsad9f83220e
+
+namespace GerverSofa.PartE.CertificateCells9fb4000a9c
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells9fb4000a9c
+
+namespace GerverSofa.PartE.CertificateCells39e4b63459
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells39e4b63459
+
+namespace GerverSofa.PartE.CertificateCells1f04a5458e
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021003102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022002100)))
+
+end GerverSofa.PartE.CertificateCells1f04a5458e
+
+namespace GerverSofa.PartE.CertificateCells6fd31b8688
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells6fd31b8688
+
+namespace GerverSofa.PartE.CertificateCells228f0f8fd4
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021113000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell000022002111)))
+
+/-- Subcell `0000220021113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell000022002111)))
+
+end GerverSofa.PartE.CertificateCells228f0f8fd4
+
+namespace GerverSofa.PartE.CertificateCells0a19d30544
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells0a19d30544
+
+namespace GerverSofa.PartE.CertificateCells2ee5d31193
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCells2ee5d31193
+
+namespace GerverSofa.PartE.CertificateCells41213b9d19
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells41213b9d19
+
+namespace GerverSofa.PartE.CertificateCellsb64c3c15bb
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsb64c3c15bb
+
+namespace GerverSofa.PartE.CertificateCells32e697bee5
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131001020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001020 : AngleCell :=
+  childLL (childHL (childLL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001021 : AngleCell :=
+  childLH (childHL (childLL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001022 : AngleCell :=
+  childHL (childHL (childLL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001023 : AngleCell :=
+  childHH (childHL (childLL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001030 : AngleCell :=
+  childLL (childHH (childLL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001031 : AngleCell :=
+  childLH (childHH (childLL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001032 : AngleCell :=
+  childHL (childHH (childLL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001033 : AngleCell :=
+  childHH (childHH (childLL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001200 : AngleCell :=
+  childLL (childLL (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001201 : AngleCell :=
+  childLH (childLL (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001202 : AngleCell :=
+  childHL (childLL (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001203 : AngleCell :=
+  childHH (childLL (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001210 : AngleCell :=
+  childLL (childLH (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001211 : AngleCell :=
+  childLH (childLH (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001212 : AngleCell :=
+  childHL (childLH (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001213 : AngleCell :=
+  childHH (childLH (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001220 : AngleCell :=
+  childLL (childHL (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001221 : AngleCell :=
+  childLH (childHL (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001222 : AngleCell :=
+  childHL (childHL (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001223 : AngleCell :=
+  childHH (childHL (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001230 : AngleCell :=
+  childLL (childHH (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001231 : AngleCell :=
+  childLH (childHH (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001232 : AngleCell :=
+  childHL (childHH (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001233 : AngleCell :=
+  childHH (childHH (childHL (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001300 : AngleCell :=
+  childLL (childLL (childHH (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001301 : AngleCell :=
+  childLH (childLL (childHH (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001302 : AngleCell :=
+  childHL (childLL (childHH (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001303 : AngleCell :=
+  childHH (childLL (childHH (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001310 : AngleCell :=
+  childLL (childLH (childHH (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001311 : AngleCell :=
+  childLH (childLH (childHH (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001312 : AngleCell :=
+  childHL (childLH (childHH (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001313 : AngleCell :=
+  childHH (childLH (childHH (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001320 : AngleCell :=
+  childLL (childHL (childHH (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001321 : AngleCell :=
+  childLH (childHL (childHH (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001322 : AngleCell :=
+  childHL (childHL (childHH (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001323 : AngleCell :=
+  childHH (childHL (childHH (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001330 : AngleCell :=
+  childLL (childHH (childHH (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001331 : AngleCell :=
+  childLH (childHH (childHH (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001332 : AngleCell :=
+  childHL (childHH (childHH (childLH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131001333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131001333 : AngleCell :=
+  childHH (childHH (childHH (childLH thetaBelowCell111133113100)))
+
+end GerverSofa.PartE.CertificateCells32e697bee5
+
+namespace GerverSofa.PartE.CertificateCells64ee0fdc2a
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131002010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131002011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131002012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131002012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131002013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131002013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131002100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131002101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131002102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131002102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131002103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131002103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131002110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131002111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131002112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131002112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131002113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131002113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaBelowCell111133113100)))
+
+end GerverSofa.PartE.CertificateCells64ee0fdc2a
+
+namespace GerverSofa.PartE.CertificateCells30e97e201e
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021002002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022002100)))
+
+end GerverSofa.PartE.CertificateCells30e97e201e
+
+namespace GerverSofa.PartE.CertificateCellsd48981846f
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCellsd48981846f
+
+namespace GerverSofa.PartE.CertificateCells2454b18c1c
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCells2454b18c1c
+
+namespace GerverSofa.PartE.CertificateCellsd87c25aa46
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+end GerverSofa.PartE.CertificateCellsd87c25aa46
+
+namespace GerverSofa.PartE.CertificateCells90183e831a
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131003000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131003001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131003002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131003003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131003010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131003011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131003012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131003013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131003100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131003101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131003102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131003103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131003110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131003111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131003112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaBelowCell111133113100)))
+
+/-- Subcell `1111331131003113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131003113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaBelowCell111133113100)))
+
+end GerverSofa.PartE.CertificateCells90183e831a
+
+namespace GerverSofa.PartE.CertificateCells47753ffbe9
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCells47753ffbe9
+
+namespace GerverSofa.PartE.CertificateCellse21bcf4574
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellse21bcf4574
+
+namespace GerverSofa.PartE.CertificateCellsda9a873863
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsda9a873863
+
+namespace GerverSofa.PartE.CertificateCells7d865d7559
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131113000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131113033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaBelowCell111133113111)))
+
+end GerverSofa.PartE.CertificateCells7d865d7559
+
+namespace GerverSofa.PartE.CertificateCellsca42d19ac6
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCellsca42d19ac6
+
+namespace GerverSofa.PartE.CertificateCells257a1a1faa
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131010200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010200 : AngleCell :=
+  childLL (childLL (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010201 : AngleCell :=
+  childLH (childLL (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010202 : AngleCell :=
+  childHL (childLL (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010203 : AngleCell :=
+  childHH (childLL (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010210 : AngleCell :=
+  childLL (childLH (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010211 : AngleCell :=
+  childLH (childLH (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010212 : AngleCell :=
+  childHL (childLH (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010213 : AngleCell :=
+  childHH (childLH (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010300 : AngleCell :=
+  childLL (childLL (childHH (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010301 : AngleCell :=
+  childLH (childLL (childHH (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010302 : AngleCell :=
+  childHL (childLL (childHH (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010303 : AngleCell :=
+  childHH (childLL (childHH (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010310 : AngleCell :=
+  childLL (childLH (childHH (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010311 : AngleCell :=
+  childLH (childLH (childHH (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010312 : AngleCell :=
+  childHL (childLH (childHH (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010313 : AngleCell :=
+  childHH (childLH (childHH (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131010333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131010333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaBelowCell111133113101)))
+
+end GerverSofa.PartE.CertificateCells257a1a1faa
+
+namespace GerverSofa.PartE.CertificateCellsb187906d24
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021012002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022002101)))
+
+/-- Subcell `0000220021012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022002101)))
+
+end GerverSofa.PartE.CertificateCellsb187906d24
+
+namespace GerverSofa.PartE.CertificateCellseb98bd469c
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellseb98bd469c
+
+namespace GerverSofa.PartE.CertificateCells0fb8590a1f
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells0fb8590a1f
+
+namespace GerverSofa.PartE.CertificateCells21614274e2
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `111133113131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `111133113132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `111133113133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCells21614274e2
+
+namespace GerverSofa.PartE.CertificateCells075667848d
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021002102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022002100)))
+
+/-- Subcell `0000220021002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022002100)))
+
+end GerverSofa.PartE.CertificateCells075667848d
+
+namespace GerverSofa.PartE.CertificateCells135b119fc2
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells135b119fc2
+
+namespace GerverSofa.PartE.CertificateCellse53a016898
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellse53a016898
+
+namespace GerverSofa.PartE.CertificateCells8b72842cc0
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+end GerverSofa.PartE.CertificateCells8b72842cc0
+
+namespace GerverSofa.PartE.CertificateCells3d04a04d91
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaBelowCell11113311)))
+
+/-- Subcell `111133113031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaBelowCell11113311)))
+
+/-- Subcell `111133113032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaBelowCell11113311)))
+
+/-- Subcell `111133113033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCells3d04a04d91
+
+namespace GerverSofa.PartE.CertificateCells665e67ba22
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131013010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131013011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131013012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131013013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131013000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131013001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131013002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131013003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131013100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131013101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131013102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131013103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131013110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131013111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131013112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaBelowCell111133113101)))
+
+/-- Subcell `1111331131013113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131013113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaBelowCell111133113101)))
+
+end GerverSofa.PartE.CertificateCells665e67ba22
+
+namespace GerverSofa.PartE.CertificateCellsa275032fe9
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCellsa275032fe9
+
+namespace GerverSofa.PartE.CertificateCells09124dcbf9
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+end GerverSofa.PartE.CertificateCells09124dcbf9
+
+namespace GerverSofa.PartE.CertificateCellse133bf0420
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220021102000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell000022002110)))
+
+/-- Subcell `0000220021102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220021102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell000022002110)))
+
+end GerverSofa.PartE.CertificateCellse133bf0420
+
+namespace GerverSofa.PartE.CertificateCellsf1833757d0
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsf1833757d0
+
+namespace GerverSofa.PartE.CertificateCells21f94a2daf
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+end GerverSofa.PartE.CertificateCells21f94a2daf
+
+namespace GerverSofa.PartE.CertificateCellsbb4f00460c
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+end GerverSofa.PartE.CertificateCellsbb4f00460c
+
+namespace GerverSofa.PartE.CertificateCells232e7e9a76
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaBelowCell11113311)))
+
+/-- Subcell `111133113021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaBelowCell11113311)))
+
+/-- Subcell `111133113022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaBelowCell11113311)))
+
+/-- Subcell `111133113023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCells232e7e9a76
+
+namespace GerverSofa.PartE.CertificateCellse657ecadb5
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+end GerverSofa.PartE.CertificateCellse657ecadb5
+
+namespace GerverSofa.PartE.CertificateCells78b824f474
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113311)))
+
+/-- Subcell `1111331131110220` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110220 : AngleCell :=
+  childLL (childHL (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110221` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110221 : AngleCell :=
+  childLH (childHL (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110222` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110222 : AngleCell :=
+  childHL (childHL (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110223` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110223 : AngleCell :=
+  childHH (childHL (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110230` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110230 : AngleCell :=
+  childLL (childHH (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110231` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110231 : AngleCell :=
+  childLH (childHH (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110232` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110232 : AngleCell :=
+  childHL (childHH (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110233` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110233 : AngleCell :=
+  childHH (childHH (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110200 : AngleCell :=
+  childLL (childLL (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110201 : AngleCell :=
+  childLH (childLL (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110202 : AngleCell :=
+  childHL (childLL (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110203 : AngleCell :=
+  childHH (childLL (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110210 : AngleCell :=
+  childLL (childLH (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110211 : AngleCell :=
+  childLH (childLH (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110212 : AngleCell :=
+  childHL (childLH (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110213 : AngleCell :=
+  childHH (childLH (childHL (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110320` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110320 : AngleCell :=
+  childLL (childHL (childHH (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110321` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110321 : AngleCell :=
+  childLH (childHL (childHH (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110322` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110322 : AngleCell :=
+  childHL (childHL (childHH (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110323` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110323 : AngleCell :=
+  childHH (childHL (childHH (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110332` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110332 : AngleCell :=
+  childHL (childHH (childHH (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110333` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110333 : AngleCell :=
+  childHH (childHH (childHH (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110330` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110330 : AngleCell :=
+  childLL (childHH (childHH (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110331` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110331 : AngleCell :=
+  childLH (childHH (childHH (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110300 : AngleCell :=
+  childLL (childLL (childHH (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110301 : AngleCell :=
+  childLH (childLL (childHH (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110302 : AngleCell :=
+  childHL (childLL (childHH (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110303 : AngleCell :=
+  childHH (childLL (childHH (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110310 : AngleCell :=
+  childLL (childLH (childHH (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110311 : AngleCell :=
+  childLH (childLH (childHH (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110312 : AngleCell :=
+  childHL (childLH (childHH (childLL thetaBelowCell111133113111)))
+
+/-- Subcell `1111331131110313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111331131110313 : AngleCell :=
+  childHH (childLH (childHH (childLL thetaBelowCell111133113111)))
+
+end GerverSofa.PartE.CertificateCells78b824f474
+
+namespace GerverSofa.PartE.CertificateCellsccc2f90403
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `111133113111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell111133113111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaBelowCell11113311)))
+
+end GerverSofa.PartE.CertificateCellsccc2f90403
+
+namespace GerverSofa.PartE.CertificateCellse5aa25a242
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+end GerverSofa.PartE.CertificateCellse5aa25a242
+
+namespace GerverSofa.PartE.CertificateCells0d1c7d48ff
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+end GerverSofa.PartE.CertificateCells0d1c7d48ff
+
+namespace GerverSofa.PartE.CertificateCells1d418dae20
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+end GerverSofa.PartE.CertificateCells1d418dae20
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.E24KC6R4Subtree8dd4d0d86bb3ff9f`.
+* `KernelOnly.PartE.E24KC6R4Subtree8e967390514267d6`.
+* `KernelOnly.PartE.E24KC6R4Subtree91992483bb6a8f04`.
+* `KernelOnly.PartE.E24KC6R4Subtree924080aa784e817a`.
+* `KernelOnly.PartE.E24KC6R4Subtree9eb1f025174c3177`.
+* `KernelOnly.PartE.E24KC6R4SubtreeA095c7ff4c6ad6b6`.
+* `KernelOnly.PartE.E24KC6R4Join32513267a6450294`.
+* `KernelOnly.PartE.E24KC6R4SubtreeA80f1d406f4d338d`.
+* `KernelOnly.PartE.E24KC6R4SubtreeAea2c9f5ded50576`.
+* `KernelOnly.PartE.E24KC6R4SubtreeB93b1c054dba72bb`.
+* `KernelOnly.PartE.E24KC6R4SubtreeC309c137435a4a8a`.
+* `KernelOnly.PartE.E24KC6R4Join1c93c00761c21de7`.
+* `KernelOnly.PartE.E24KC6R4SubtreeC8b63043186f1b44`.
+* `KernelOnly.PartE.E24KC6R4SubtreeCbd4f9934ee17393`.
+* `KernelOnly.PartE.E24KC6R4SubtreeD658abc468986ad2`.
+* `KernelOnly.PartE.E24KC6R4SubtreeDc47723dc66dfda8`.
+* `KernelOnly.PartE.E24KC6R4SubtreeE07cf48cb1b589f9`.
+* `KernelOnly.PartE.E24KC6R4JoinA1daf15b91aa7155`.
+* `KernelOnly.PartE.E24KC6R4SubtreeE1aad79037162af3`.
+* `KernelOnly.PartE.E24KC6R4JoinB6430364d883da5c`.
+* `KernelOnly.PartE.E24KC6R4SubtreeE3bcfcc291fdb5b4`.
+* `KernelOnly.PartE.E24KC6R4SubtreeE5b6fc380e82e47e`.
+* `KernelOnly.PartE.E24KC6R4SubtreeE8870b13405ffc13`.
+* `KernelOnly.PartE.E24KC6R4Join3b19a6737a79b00f`.
+* `KernelOnly.PartE.E24KC6R4SubtreeE91aff9c114b660b`.
+* `KernelOnly.PartE.E24KC6R4SubtreeEc06b97bc924fb9a`.
+* `KernelOnly.PartE.E24KC6R4Join62d0e072891eacfc`.
+* `KernelOnly.PartE.E24KC6R4Join45f1be0097db6fa8`.
+* `KernelOnly.PartE.E24KC6R4SubtreeEd73d41bc0ae6f27`.
+* `KernelOnly.PartE.E24KC6R4SubtreeEdd4c8d57c5a9ded`.
+* `KernelOnly.PartE.E24KC6R4Join96d3935117f8c848`.
+* `KernelOnly.PartE.E24KC6R4Join7c9f3e2241f40b54`.
+* `KernelOnly.PartE.E24KC6R4Join489f93824ca49e04`.
+* `KernelOnly.PartE.E24KC6R4SubtreeF553a12d6978535b`.
+* `KernelOnly.PartE.E24KC6R4SubtreeF60ae8a1c3b6076a`.
+* `KernelOnly.PartE.E24KC6R4Join978735cf4a10b0c4`.
+* `KernelOnly.PartE.E24KC6R4JoinAf86a73e06a6d340`.
+* `KernelOnly.PartE.E24KC6R4SubtreeF8d6ff3220b818df`.
+* `KernelOnly.PartE.E24KC6R4JoinB118d49025de8b76`.
+* `KernelOnly.PartE.E24KC6R4JoinE6a27f6b8bf66074`.
+* `KernelOnly.PartE.E24KC6R4Join1480e3e518aaa8b9`.
+* `KernelOnly.PartE.E24KC6R4SubtreeF9a00b60dcd28a42`.
+* `KernelOnly.PartE.E24KC6R4Join68100c5f054c8ea2`.
+* `KernelOnly.PartE.E24KC6R4SubtreeFfa75b98dddd4b5e`.
+* `KernelOnly.PartE.E24KC6R4Join25ba04fb03bf9d4a`.
+* `KernelOnly.PartE.E24KC6R4JoinB0c26eb36c2c399d`.
+* `KernelOnly.PartE.E24KC6R4JoinC502233200399653`.
+* `KernelOnly.PartE.E24KC6R4Join459e5b5f3815ebd2`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells84b6dd5403
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells84b6dd5403
+
+open CertificateCells84b6dd5403
+theorem cover_subtree_ae17a6be47c3 :
+    adaptiveCoverCheck 4 (childLL (childLL thetaBelowCell111133113011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL thetaBelowCell111133113011))
+    (by
+      have h : ((childLL (childLL (childLL thetaBelowCell111133113011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLL
+        thetaBelowCell111133113011))) h)
+    (by
+      have h : ((childLH (childLL (childLL thetaBelowCell111133113011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLL
+        thetaBelowCell111133113011))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childLL (childLL
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130110020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110020 h)
+        (by
+          have h : (thetaBelowCell1111331130110021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110021 h)
+        (by
+          have h : (thetaBelowCell1111331130110022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110022 h)
+        (by
+          have h : (thetaBelowCell1111331130110023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childLL (childLL
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130110030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110030 h)
+        (by
+          have h : (thetaBelowCell1111331130110031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110031 h)
+        (by
+          have h : (thetaBelowCell1111331130110032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110032 h)
+        (by
+          have h : (thetaBelowCell1111331130110033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110033 h))
+
+theorem cover_subtree_370884aab783 :
+    adaptiveCoverCheck 4 (childLH (childLL thetaBelowCell111133113011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL thetaBelowCell111133113011))
+    (by
+      have h : ((childLL (childLH (childLL thetaBelowCell111133113011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLL
+        thetaBelowCell111133113011))) h)
+    (by
+      have h : ((childLH (childLH (childLL thetaBelowCell111133113011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLL
+        thetaBelowCell111133113011))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childLH (childLL
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130110120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110120 h)
+        (by
+          have h : (thetaBelowCell1111331130110121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110121 h)
+        (by
+          have h : (thetaBelowCell1111331130110122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110122 h)
+        (by
+          have h : (thetaBelowCell1111331130110123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childLH (childLL
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130110130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110130 h)
+        (by
+          have h : (thetaBelowCell1111331130110131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110131 h)
+        (by
+          have h : (thetaBelowCell1111331130110132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110132 h)
+        (by
+          have h : (thetaBelowCell1111331130110133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110133 h))
+
+theorem cover_subtree_73e4cb204c73 :
+    adaptiveCoverCheck 4 (childHL (childLL thetaBelowCell111133113011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL thetaBelowCell111133113011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHL (childLL
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130110200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110200 h)
+        (by
+          have h : (thetaBelowCell1111331130110201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110201 h)
+        (by
+          have h : (thetaBelowCell1111331130110202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110202 h)
+        (by
+          have h : (thetaBelowCell1111331130110203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHL (childLL
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130110210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110210 h)
+        (by
+          have h : (thetaBelowCell1111331130110211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110211 h)
+        (by
+          have h : (thetaBelowCell1111331130110212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110212 h)
+        (by
+          have h : (thetaBelowCell1111331130110213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHL (childLL
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130110220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110220 h)
+        (by
+          have h : (thetaBelowCell1111331130110221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110221 h)
+        (by
+          have h : (thetaBelowCell1111331130110222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110222 h)
+        (by
+          have h : (thetaBelowCell1111331130110223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHL (childLL
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130110230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110230 h)
+        (by
+          have h : (thetaBelowCell1111331130110231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110231 h)
+        (by
+          have h : (thetaBelowCell1111331130110232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110232 h)
+        (by
+          have h : (thetaBelowCell1111331130110233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110233 h))
+
+theorem cover_subtree_c1c14ad83b45 :
+    adaptiveCoverCheck 4 (childHH (childLL thetaBelowCell111133113011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL thetaBelowCell111133113011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHH (childLL
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130110300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110300 h)
+        (by
+          have h : (thetaBelowCell1111331130110301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110301 h)
+        (by
+          have h : (thetaBelowCell1111331130110302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110302 h)
+        (by
+          have h : (thetaBelowCell1111331130110303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHH (childLL
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130110310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110310 h)
+        (by
+          have h : (thetaBelowCell1111331130110311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110311 h)
+        (by
+          have h : (thetaBelowCell1111331130110312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110312 h)
+        (by
+          have h : (thetaBelowCell1111331130110313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHH (childLL
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130110320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110320 h)
+        (by
+          have h : (thetaBelowCell1111331130110321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110321 h)
+        (by
+          have h : (thetaBelowCell1111331130110322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110322 h)
+        (by
+          have h : (thetaBelowCell1111331130110323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHH (childLL
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130110330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110330 h)
+        (by
+          have h : (thetaBelowCell1111331130110331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110331 h)
+        (by
+          have h : (thetaBelowCell1111331130110332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110332 h)
+        (by
+          have h : (thetaBelowCell1111331130110333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130110333 h))
+
+theorem cover_subtree_f30e12efeb1a :
+    adaptiveCoverCheck 5 (childLL thetaBelowCell111133113011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133113011)
+    cover_subtree_ae17a6be47c3
+    cover_subtree_370884aab783
+    cover_subtree_73e4cb204c73
+    cover_subtree_c1c14ad83b45
+
+theorem cover_subtree_98d9e5389017 :
+    adaptiveCoverCheck 4 (childLL (childLH thetaBelowCell111133113011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH thetaBelowCell111133113011))
+    (by
+      have h : ((childLL (childLL (childLH thetaBelowCell111133113011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLH
+        thetaBelowCell111133113011))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaBelowCell111133113011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLH
+        thetaBelowCell111133113011))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childLL (childLH
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130111020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111020 h)
+        (by
+          have h : (thetaBelowCell1111331130111021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111021 h)
+        (by
+          have h : (thetaBelowCell1111331130111022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111022 h)
+        (by
+          have h : (thetaBelowCell1111331130111023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childLL (childLH
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130111030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111030 h)
+        (by
+          have h : (thetaBelowCell1111331130111031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111031 h)
+        (by
+          have h : (thetaBelowCell1111331130111032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111032 h)
+        (by
+          have h : (thetaBelowCell1111331130111033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111033 h))
+
+theorem cover_subtree_0bc86d85f103 :
+    adaptiveCoverCheck 4 (childLH (childLH thetaBelowCell111133113011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH thetaBelowCell111133113011))
+    (by
+      have h : ((childLL (childLH (childLH thetaBelowCell111133113011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLH
+        thetaBelowCell111133113011))) h)
+    (by
+      have h : ((childLH (childLH (childLH thetaBelowCell111133113011)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLH
+        thetaBelowCell111133113011))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childLH (childLH
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130111120).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111120 h)
+        (by
+          have h : (thetaBelowCell1111331130111121).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111121 h)
+        (by
+          have h : (thetaBelowCell1111331130111122).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111122 h)
+        (by
+          have h : (thetaBelowCell1111331130111123).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111123 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childLH (childLH
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130111130).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111130 h)
+        (by
+          have h : (thetaBelowCell1111331130111131).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111131 h)
+        (by
+          have h : (thetaBelowCell1111331130111132).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111132 h)
+        (by
+          have h : (thetaBelowCell1111331130111133).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111133 h))
+
+theorem cover_subtree_29f3a4c72110 :
+    adaptiveCoverCheck 4 (childHL (childLH thetaBelowCell111133113011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH thetaBelowCell111133113011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHL (childLH
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130111200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111200 h)
+        (by
+          have h : (thetaBelowCell1111331130111201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111201 h)
+        (by
+          have h : (thetaBelowCell1111331130111202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111202 h)
+        (by
+          have h : (thetaBelowCell1111331130111203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHL (childLH
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130111210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111210 h)
+        (by
+          have h : (thetaBelowCell1111331130111211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111211 h)
+        (by
+          have h : (thetaBelowCell1111331130111212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111212 h)
+        (by
+          have h : (thetaBelowCell1111331130111213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHL (childLH
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130111220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111220 h)
+        (by
+          have h : (thetaBelowCell1111331130111221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111221 h)
+        (by
+          have h : (thetaBelowCell1111331130111222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111222 h)
+        (by
+          have h : (thetaBelowCell1111331130111223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHL (childLH
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130111230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111230 h)
+        (by
+          have h : (thetaBelowCell1111331130111231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111231 h)
+        (by
+          have h : (thetaBelowCell1111331130111232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111232 h)
+        (by
+          have h : (thetaBelowCell1111331130111233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111233 h))
+
+theorem cover_subtree_71904a1bf379 :
+    adaptiveCoverCheck 4 (childHH (childLH thetaBelowCell111133113011)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH thetaBelowCell111133113011))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHH (childLH
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130111300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111300 h)
+        (by
+          have h : (thetaBelowCell1111331130111301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111301 h)
+        (by
+          have h : (thetaBelowCell1111331130111302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111302 h)
+        (by
+          have h : (thetaBelowCell1111331130111303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHH (childLH
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130111310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111310 h)
+        (by
+          have h : (thetaBelowCell1111331130111311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111311 h)
+        (by
+          have h : (thetaBelowCell1111331130111312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111312 h)
+        (by
+          have h : (thetaBelowCell1111331130111313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHH (childLH
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130111320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111320 h)
+        (by
+          have h : (thetaBelowCell1111331130111321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111321 h)
+        (by
+          have h : (thetaBelowCell1111331130111322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111322 h)
+        (by
+          have h : (thetaBelowCell1111331130111323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHH (childLH
+        thetaBelowCell111133113011)))
+        (by
+          have h : (thetaBelowCell1111331130111330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111330 h)
+        (by
+          have h : (thetaBelowCell1111331130111331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111331 h)
+        (by
+          have h : (thetaBelowCell1111331130111332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111332 h)
+        (by
+          have h : (thetaBelowCell1111331130111333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331130111333 h))
+
+theorem cover_subtree_c4f09e9f0953 :
+    adaptiveCoverCheck 5 (childLH thetaBelowCell111133113011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133113011)
+    cover_subtree_98d9e5389017
+    cover_subtree_0bc86d85f103
+    cover_subtree_29f3a4c72110
+    cover_subtree_71904a1bf379
+
+theorem cover_subtree_f6e8aaad1d7c :
+    adaptiveCoverCheck 5 (childHL thetaBelowCell111133113011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133113011)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL thetaBelowCell111133113011))
+        (by
+          have h : ((childLL (childLL (childHL thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childHL
+            thetaBelowCell111133113011))) h)
+        (by
+          have h : ((childLH (childLL (childHL thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childHL
+            thetaBelowCell111133113011))) h)
+        (by
+          have h : ((childHL (childLL (childHL thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childHL
+            thetaBelowCell111133113011))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childHL
+            thetaBelowCell111133113011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL thetaBelowCell111133113011))
+        (by
+          have h : ((childLL (childLH (childHL thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childHL
+            thetaBelowCell111133113011))) h)
+        (by
+          have h : ((childLH (childLH (childHL thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childHL
+            thetaBelowCell111133113011))) h)
+        (by
+          have h : ((childHL (childLH (childHL thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childHL
+            thetaBelowCell111133113011))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childHL
+            thetaBelowCell111133113011))) h))
+    (by
+      have h : ((childHL (childHL thetaBelowCell111133113011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL thetaBelowCell111133113011)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell111133113011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL thetaBelowCell111133113011)) h)
+
+theorem cover_subtree_29b6f04811a1 :
+    adaptiveCoverCheck 5 (childHH thetaBelowCell111133113011) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133113011)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH thetaBelowCell111133113011))
+        (by
+          have h : ((childLL (childLL (childHH thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childHH
+            thetaBelowCell111133113011))) h)
+        (by
+          have h : ((childLH (childLL (childHH thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childHH
+            thetaBelowCell111133113011))) h)
+        (by
+          have h : ((childHL (childLL (childHH thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childHH
+            thetaBelowCell111133113011))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childHH
+            thetaBelowCell111133113011))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH thetaBelowCell111133113011))
+        (by
+          have h : ((childLL (childLH (childHH thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childHH
+            thetaBelowCell111133113011))) h)
+        (by
+          have h : ((childLH (childLH (childHH thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childHH
+            thetaBelowCell111133113011))) h)
+        (by
+          have h : ((childHL (childLH (childHH thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childHH
+            thetaBelowCell111133113011))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaBelowCell111133113011)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childHH
+            thetaBelowCell111133113011))) h))
+    (by
+      have h : ((childHL (childHH thetaBelowCell111133113011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH thetaBelowCell111133113011)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell111133113011))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH thetaBelowCell111133113011)) h)
+
+theorem e24KC2ThetaBelowLeaf111133113_c0_c1_c1 :
+    adaptiveCoverCheck 6 thetaBelowCell111133113011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113011
+    cover_subtree_f30e12efeb1a
+    cover_subtree_c4f09e9f0953
+    cover_subtree_f6e8aaad1d7c
+    cover_subtree_29b6f04811a1
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa837c94d6b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa837c94d6b
+
+open CertificateCellsa837c94d6b
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c0_c1 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022002100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022002100)
+    (by
+      have h : ((childLL (childLH thetaAboveCell000022002100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH thetaAboveCell000022002100)) h)
+    (by
+      have h : ((childLH (childLH thetaAboveCell000022002100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH thetaAboveCell000022002100)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLH thetaAboveCell000022002100))
+        (by
+          have h : ((childLL (childHL (childLH thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLH
+            thetaAboveCell000022002100))) h)
+        (by
+          have h : ((childLH (childHL (childLH thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLH
+            thetaAboveCell000022002100))) h)
+        (by
+          have h : ((childHL (childHL (childLH thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLH
+            thetaAboveCell000022002100))) h)
+        (by
+          have h : ((childHH (childHL (childLH thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLH
+            thetaAboveCell000022002100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLH thetaAboveCell000022002100))
+        (by
+          have h : ((childLL (childHH (childLH thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLH
+            thetaAboveCell000022002100))) h)
+        (by
+          have h : ((childLH (childHH (childLH thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLH
+            thetaAboveCell000022002100))) h)
+        (by
+          have h : ((childHL (childHH (childLH thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLH
+            thetaAboveCell000022002100))) h)
+        (by
+          have h : ((childHH (childHH (childLH thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLH
+            thetaAboveCell000022002100))) h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsad9f83220e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsad9f83220e
+
+open CertificateCellsad9f83220e
+theorem e24KC2ThetaBelowLeaf111133113_c3 :
+    adaptiveCoverCheck 8 (childHH (childHH thetaBelowCell11113311)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 (childHH (childHH thetaBelowCell11113311))
+    (by
+      have h : ((childLL (childHH (childHH thetaBelowCell11113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLL (childHH (childHH
+        thetaBelowCell11113311))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaBelowCell11113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childLH (childHH (childHH
+        thetaBelowCell11113311))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaBelowCell11113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childHL (childHH (childHH
+        thetaBelowCell11113311))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaBelowCell11113311)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 7 (childHH (childHH (childHH
+        thetaBelowCell11113311))) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9fb4000a9c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9fb4000a9c
+
+open CertificateCells9fb4000a9c
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c0_c0 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022002100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022002100)
+    (by
+      have h : ((childLL (childLL thetaAboveCell000022002100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL thetaAboveCell000022002100)) h)
+    (by
+      have h : ((childLH (childLL thetaAboveCell000022002100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL thetaAboveCell000022002100)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLL thetaAboveCell000022002100))
+        (by
+          have h : ((childLL (childHL (childLL thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLL
+            thetaAboveCell000022002100))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLL
+            thetaAboveCell000022002100))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLL
+            thetaAboveCell000022002100))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLL
+            thetaAboveCell000022002100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLL thetaAboveCell000022002100))
+        (by
+          have h : ((childLL (childHH (childLL thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLL
+            thetaAboveCell000022002100))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLL
+            thetaAboveCell000022002100))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLL
+            thetaAboveCell000022002100))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaAboveCell000022002100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLL
+            thetaAboveCell000022002100))) h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells39e4b63459
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells39e4b63459
+
+open CertificateCells39e4b63459
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c3 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022002103
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022002103)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022002103)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022002103)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022002103)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022002103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022002103)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022002103)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022002103)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022002103)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022002103)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022002103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022002103) h)
+    (by
+      have h : ((childHH thetaAboveCell000022002103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022002103) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1f04a5458e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1f04a5458e
+
+open CertificateCells1f04a5458e
+theorem cover_subtree_ac4ab9e2280b :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003102
+    (by
+      have h : ((childLL thetaAboveCell0000220021003102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003102) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021003102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003102) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003102)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003102)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003102)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003102)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021003102)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021003102)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021003102)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021003102)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021003102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021003102)) h))
+
+theorem cover_subtree_f32049ef12fd :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003103
+    (by
+      have h : ((childLL thetaAboveCell0000220021003103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003103) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021003103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003103) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003103)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003103)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003103)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003103)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021003103)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021003103)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021003103)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021003103)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021003103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021003103)) h))
+
+theorem cover_subtree_f8ee2b69a6a4 :
+    adaptiveCoverCheck 4 (childLL (childLH (childHH thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHH
+    thetaAboveCell000022002100)))
+    (by
+      have h : (thetaAboveCell0000220021003100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003100 h)
+    (by
+      have h : (thetaAboveCell0000220021003101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003101 h)
+    cover_subtree_ac4ab9e2280b
+    cover_subtree_f32049ef12fd
+
+theorem cover_subtree_81e656cd512a :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003112
+    (by
+      have h : ((childLL thetaAboveCell0000220021003112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003112) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021003112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003112) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003112)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003112)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003112)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003112)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021003112)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021003112)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021003112)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021003112)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021003112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021003112)) h))
+
+theorem cover_subtree_4d1806a48386 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003113
+    (by
+      have h : ((childLL thetaAboveCell0000220021003113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003113) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021003113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003113) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003113)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003113)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003113)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003113)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021003113)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021003113)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021003113)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021003113)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021003113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021003113)) h))
+
+theorem cover_subtree_08c961b8fea6 :
+    adaptiveCoverCheck 4 (childLH (childLH (childHH thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHH
+    thetaAboveCell000022002100)))
+    (by
+      have h : (thetaAboveCell0000220021003110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003110 h)
+    (by
+      have h : (thetaAboveCell0000220021003111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021003111 h)
+    cover_subtree_81e656cd512a
+    cover_subtree_4d1806a48386
+
+theorem cover_subtree_ac21c862ec3a :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003120 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003120
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021003120)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021003120)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021003120)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021003120)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021003120)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021003120)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021003120)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021003120)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021003120)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021003120)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003120)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003120)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003120)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003120)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003120)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021003120)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021003120)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021003120)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021003120)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021003120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021003120)) h))
+
+theorem cover_subtree_535d1cff404c :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003121 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003121
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021003121)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021003121)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021003121)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021003121)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021003121)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021003121)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021003121)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021003121)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021003121)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021003121)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003121)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003121)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003121)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003121)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003121)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021003121)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021003121)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021003121)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021003121)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021003121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021003121)) h))
+
+theorem cover_subtree_c98fab407e07 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHH thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+    thetaAboveCell000022002100)))
+    cover_subtree_ac21c862ec3a
+    cover_subtree_535d1cff404c
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003122
+        (by
+          have h : ((childLL thetaAboveCell0000220021003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021003122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021003122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021003122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003123
+        (by
+          have h : ((childLL thetaAboveCell0000220021003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021003123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021003123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021003123) h))
+
+theorem cover_subtree_6d1eab6c80cc :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003130 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003130
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021003130)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021003130)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021003130)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021003130)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021003130)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021003130)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021003130)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021003130)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021003130)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021003130)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003130)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003130)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003130)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003130)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003130)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021003130)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021003130)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021003130)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021003130)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021003130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021003130)) h))
+
+theorem cover_subtree_d7741bcc2937 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021003131 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003131
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021003131)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021003131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021003131)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021003131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021003131)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021003131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021003131)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021003131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021003131)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021003131)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021003131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021003131)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021003131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021003131)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021003131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021003131)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021003131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021003131)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021003131)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021003131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021003131)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021003131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021003131)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021003131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021003131)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021003131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021003131)) h))
+    (by
+      have h : ((childHH thetaAboveCell0000220021003131)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021003131) h)
+
+theorem cover_subtree_db05fb36c692 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHH thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+    thetaAboveCell000022002100)))
+    cover_subtree_6d1eab6c80cc
+    cover_subtree_d7741bcc2937
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003132
+        (by
+          have h : ((childLL thetaAboveCell0000220021003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021003132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021003132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021003132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021003133
+        (by
+          have h : ((childLL thetaAboveCell0000220021003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021003133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021003133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021003133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021003133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021003133) h))
+
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c0_c3_c1 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022002100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022002100))
+    cover_subtree_f8ee2b69a6a4
+    cover_subtree_08c961b8fea6
+    cover_subtree_c98fab407e07
+    cover_subtree_db05fb36c692
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6fd31b8688
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6fd31b8688
+
+open CertificateCells6fd31b8688
+
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c0_c3 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022002100) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022002100)
+    e24KC2ThetaAboveLeaf0000220021_c0_c0_c3_c0 e24KC2ThetaAboveLeaf0000220021_c0_c0_c3_c1
+      e24KC2ThetaAboveLeaf0000220021_c0_c0_c3_c2 e24KC2ThetaAboveLeaf0000220021_c0_c0_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells228f0f8fd4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells228f0f8fd4
+
+open CertificateCells228f0f8fd4
+theorem cover_subtree_d5a5e06650ce :
+    adaptiveCoverCheck 4 (childLL (childLL (childHH thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHH
+    thetaAboveCell000022002111)))
+    (by
+      have h : (thetaAboveCell0000220021113000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113000 h)
+    (by
+      have h : (thetaAboveCell0000220021113001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113002
+        (by
+          have h : ((childLL thetaAboveCell0000220021113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113002) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113002) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113002) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113003
+        (by
+          have h : ((childLL thetaAboveCell0000220021113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113003) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113003) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113003) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113003) h))
+
+theorem cover_subtree_cc0b6dcb9698 :
+    adaptiveCoverCheck 4 (childLH (childLL (childHH thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHH
+    thetaAboveCell000022002111)))
+    (by
+      have h : (thetaAboveCell0000220021113010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113010 h)
+    (by
+      have h : (thetaAboveCell0000220021113011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113012
+        (by
+          have h : ((childLL thetaAboveCell0000220021113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113012) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113012) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113012) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113013
+        (by
+          have h : ((childLL thetaAboveCell0000220021113013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113013) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113013) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113013) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113013) h))
+
+theorem cover_subtree_4b6a6f343911 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHH thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHH
+    thetaAboveCell000022002111)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113020
+        (by
+          have h : ((childLL thetaAboveCell0000220021113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113021
+        (by
+          have h : ((childLL thetaAboveCell0000220021113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113022
+        (by
+          have h : ((childLL thetaAboveCell0000220021113022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113023
+        (by
+          have h : ((childLL thetaAboveCell0000220021113023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113023) h))
+
+theorem cover_subtree_3b5c8a093b1d :
+    adaptiveCoverCheck 4 (childHH (childLL (childHH thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHH
+    thetaAboveCell000022002111)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113030
+        (by
+          have h : ((childLL thetaAboveCell0000220021113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113031
+        (by
+          have h : ((childLL thetaAboveCell0000220021113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113032
+        (by
+          have h : ((childLL thetaAboveCell0000220021113032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113033
+        (by
+          have h : ((childLL thetaAboveCell0000220021113033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113033) h))
+
+theorem cover_subtree_536b999011e3 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022002111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022002111))
+    cover_subtree_d5a5e06650ce
+    cover_subtree_cc0b6dcb9698
+    cover_subtree_4b6a6f343911
+    cover_subtree_3b5c8a093b1d
+
+theorem cover_subtree_34d62deb6720 :
+    adaptiveCoverCheck 4 (childLL (childLH (childHH thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHH
+    thetaAboveCell000022002111)))
+    (by
+      have h : (thetaAboveCell0000220021113100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113100 h)
+    (by
+      have h : (thetaAboveCell0000220021113101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113102
+        (by
+          have h : ((childLL thetaAboveCell0000220021113102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113102) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113102) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113102) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113103
+        (by
+          have h : ((childLL thetaAboveCell0000220021113103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113103) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113103) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113103) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113103) h))
+
+theorem cover_subtree_e8b5cb8c6a09 :
+    adaptiveCoverCheck 4 (childLH (childLH (childHH thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHH
+    thetaAboveCell000022002111)))
+    (by
+      have h : (thetaAboveCell0000220021113110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113110 h)
+    (by
+      have h : (thetaAboveCell0000220021113111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113112
+        (by
+          have h : ((childLL thetaAboveCell0000220021113112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113112) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113112) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113112) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113113
+        (by
+          have h : ((childLL thetaAboveCell0000220021113113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113113) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113113) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113113) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113113) h))
+
+theorem cover_subtree_ef6f45c91ec2 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHH thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH
+    thetaAboveCell000022002111)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113120
+        (by
+          have h : ((childLL thetaAboveCell0000220021113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113121
+        (by
+          have h : ((childLL thetaAboveCell0000220021113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113122
+        (by
+          have h : ((childLL thetaAboveCell0000220021113122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113123
+        (by
+          have h : ((childLL thetaAboveCell0000220021113123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113123) h))
+
+theorem cover_subtree_a2fe78f78c5a :
+    adaptiveCoverCheck 4 (childHH (childLH (childHH thetaAboveCell000022002111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH
+    thetaAboveCell000022002111)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113130
+        (by
+          have h : ((childLL thetaAboveCell0000220021113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113131
+        (by
+          have h : ((childLL thetaAboveCell0000220021113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113132
+        (by
+          have h : ((childLL thetaAboveCell0000220021113132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021113133
+        (by
+          have h : ((childLL thetaAboveCell0000220021113133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021113133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021113133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021113133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021113133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021113133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021113133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021113133) h))
+
+theorem cover_subtree_42bf432f3a23 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022002111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022002111))
+    cover_subtree_34d62deb6720
+    cover_subtree_e8b5cb8c6a09
+    cover_subtree_ef6f45c91ec2
+    cover_subtree_a2fe78f78c5a
+
+theorem cover_subtree_a6728f6d4a12 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022002111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHH thetaAboveCell000022002111))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHH
+        thetaAboveCell000022002111)))
+        (by
+          have h : (thetaAboveCell0000220021113200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113200 h)
+        (by
+          have h : (thetaAboveCell0000220021113201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113201 h)
+        (by
+          have h : (thetaAboveCell0000220021113202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113202 h)
+        (by
+          have h : (thetaAboveCell0000220021113203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHH
+        thetaAboveCell000022002111)))
+        (by
+          have h : (thetaAboveCell0000220021113210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113210 h)
+        (by
+          have h : (thetaAboveCell0000220021113211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113211 h)
+        (by
+          have h : (thetaAboveCell0000220021113212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113212 h)
+        (by
+          have h : (thetaAboveCell0000220021113213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113213 h))
+    (by
+      have h : ((childHL (childHL (childHH thetaAboveCell000022002111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHH
+        thetaAboveCell000022002111))) h)
+    (by
+      have h : ((childHH (childHL (childHH thetaAboveCell000022002111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHH
+        thetaAboveCell000022002111))) h)
+
+theorem cover_subtree_fafcbbc2f9f1 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022002111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHH thetaAboveCell000022002111))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHH
+        thetaAboveCell000022002111)))
+        (by
+          have h : (thetaAboveCell0000220021113300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113300 h)
+        (by
+          have h : (thetaAboveCell0000220021113301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113301 h)
+        (by
+          have h : (thetaAboveCell0000220021113302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113302 h)
+        (by
+          have h : (thetaAboveCell0000220021113303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHH
+        thetaAboveCell000022002111)))
+        (by
+          have h : (thetaAboveCell0000220021113310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113310 h)
+        (by
+          have h : (thetaAboveCell0000220021113311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113311 h)
+        (by
+          have h : (thetaAboveCell0000220021113312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113312 h)
+        (by
+          have h : (thetaAboveCell0000220021113313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021113313 h))
+    (by
+      have h : ((childHL (childHH (childHH thetaAboveCell000022002111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHH
+        thetaAboveCell000022002111))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaAboveCell000022002111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHH
+        thetaAboveCell000022002111))) h)
+
+theorem e24KC2ThetaAboveLeaf0000220021_c1_c1_c3 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022002111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022002111)
+    cover_subtree_536b999011e3
+    cover_subtree_42bf432f3a23
+    cover_subtree_a6728f6d4a12
+    cover_subtree_fafcbbc2f9f1
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0a19d30544
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0a19d30544
+
+open CertificateCells0a19d30544
+theorem e24KC2ThetaAboveLeaf0000220021_c1_c3 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022002113
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022002113)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022002113)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022002113)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022002113)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022002113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022002113)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022002113)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022002113)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022002113)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022002113)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022002113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022002113) h)
+    (by
+      have h : ((childHH thetaAboveCell000022002113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022002113) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2ee5d31193
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2ee5d31193
+
+open CertificateCells2ee5d31193
+theorem e24KC2ThetaBelowLeaf111133113_c1_c0_c3 :
+    adaptiveCoverCheck 6 thetaBelowCell111133113103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113103
+    (by
+      have h : ((childLL thetaBelowCell111133113103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133113103) h)
+    (by
+      have h : ((childLH thetaBelowCell111133113103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133113103) h)
+    (by
+      have h : ((childHL thetaBelowCell111133113103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133113103) h)
+    (by
+      have h : ((childHH thetaBelowCell111133113103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133113103) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells41213b9d19
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells41213b9d19
+
+open CertificateCells41213b9d19
+theorem e24KC2ThetaAboveLeaf0000220021_c1_c1_c0 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022002111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022002111)
+    (by
+      have h : ((childLL (childLL thetaAboveCell000022002111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL thetaAboveCell000022002111)) h)
+    (by
+      have h : ((childLH (childLL thetaAboveCell000022002111))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL thetaAboveCell000022002111)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLL thetaAboveCell000022002111))
+        (by
+          have h : ((childLL (childHL (childLL thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLL
+            thetaAboveCell000022002111))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLL
+            thetaAboveCell000022002111))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLL
+            thetaAboveCell000022002111))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLL
+            thetaAboveCell000022002111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLL thetaAboveCell000022002111))
+        (by
+          have h : ((childLL (childHH (childLL thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLL
+            thetaAboveCell000022002111))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLL
+            thetaAboveCell000022002111))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLL
+            thetaAboveCell000022002111))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaAboveCell000022002111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLL
+            thetaAboveCell000022002111))) h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb64c3c15bb
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb64c3c15bb
+
+open CertificateCellsb64c3c15bb
+
+theorem e24KC2ThetaAboveLeaf0000220021_c1_c1 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002111 = true :=
+  adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022002111
+    e24KC2ThetaAboveLeaf0000220021_c1_c1_c0 e24KC2ThetaAboveLeaf0000220021_c1_c1_c1
+      e24KC2ThetaAboveLeaf0000220021_c1_c1_c2 e24KC2ThetaAboveLeaf0000220021_c1_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells32e697bee5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells32e697bee5
+
+open CertificateCells32e697bee5
+theorem cover_subtree_401efdae503c :
+    adaptiveCoverCheck 4 (childLL (childLH thetaBelowCell111133113100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH thetaBelowCell111133113100))
+    (by
+      have h : ((childLL (childLL (childLH thetaBelowCell111133113100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLH
+        thetaBelowCell111133113100))) h)
+    (by
+      have h : ((childLH (childLL (childLH thetaBelowCell111133113100)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLH
+        thetaBelowCell111133113100))) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childLL (childLH
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131001020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001020 h)
+        (by
+          have h : (thetaBelowCell1111331131001021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001021 h)
+        (by
+          have h : (thetaBelowCell1111331131001022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001022 h)
+        (by
+          have h : (thetaBelowCell1111331131001023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childLL (childLH
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131001030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001030 h)
+        (by
+          have h : (thetaBelowCell1111331131001031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001031 h)
+        (by
+          have h : (thetaBelowCell1111331131001032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001032 h)
+        (by
+          have h : (thetaBelowCell1111331131001033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001033 h))
+
+theorem cover_subtree_d2432b059763 :
+    adaptiveCoverCheck 4 (childHL (childLH thetaBelowCell111133113100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH thetaBelowCell111133113100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHL (childLH
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131001200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001200 h)
+        (by
+          have h : (thetaBelowCell1111331131001201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001201 h)
+        (by
+          have h : (thetaBelowCell1111331131001202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001202 h)
+        (by
+          have h : (thetaBelowCell1111331131001203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHL (childLH
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131001210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001210 h)
+        (by
+          have h : (thetaBelowCell1111331131001211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001211 h)
+        (by
+          have h : (thetaBelowCell1111331131001212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001212 h)
+        (by
+          have h : (thetaBelowCell1111331131001213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHL (childLH
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131001220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001220 h)
+        (by
+          have h : (thetaBelowCell1111331131001221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001221 h)
+        (by
+          have h : (thetaBelowCell1111331131001222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001222 h)
+        (by
+          have h : (thetaBelowCell1111331131001223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHL (childLH
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131001230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001230 h)
+        (by
+          have h : (thetaBelowCell1111331131001231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001231 h)
+        (by
+          have h : (thetaBelowCell1111331131001232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001232 h)
+        (by
+          have h : (thetaBelowCell1111331131001233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001233 h))
+
+theorem cover_subtree_a9ad7d61343c :
+    adaptiveCoverCheck 4 (childHH (childLH thetaBelowCell111133113100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH thetaBelowCell111133113100))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHH (childLH
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131001300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001300 h)
+        (by
+          have h : (thetaBelowCell1111331131001301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001301 h)
+        (by
+          have h : (thetaBelowCell1111331131001302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001302 h)
+        (by
+          have h : (thetaBelowCell1111331131001303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHH (childLH
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131001310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001310 h)
+        (by
+          have h : (thetaBelowCell1111331131001311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001311 h)
+        (by
+          have h : (thetaBelowCell1111331131001312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001312 h)
+        (by
+          have h : (thetaBelowCell1111331131001313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHH (childLH
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131001320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001320 h)
+        (by
+          have h : (thetaBelowCell1111331131001321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001321 h)
+        (by
+          have h : (thetaBelowCell1111331131001322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001322 h)
+        (by
+          have h : (thetaBelowCell1111331131001323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHH (childLH
+        thetaBelowCell111133113100)))
+        (by
+          have h : (thetaBelowCell1111331131001330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001330 h)
+        (by
+          have h : (thetaBelowCell1111331131001331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001331 h)
+        (by
+          have h : (thetaBelowCell1111331131001332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001332 h)
+        (by
+          have h : (thetaBelowCell1111331131001333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131001333 h))
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c0_c0_c1 :
+    adaptiveCoverCheck 5 (childLH thetaBelowCell111133113100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH thetaBelowCell111133113100)
+    cover_subtree_401efdae503c
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH thetaBelowCell111133113100))
+        (by
+          have h : ((childLL (childLH (childLH thetaBelowCell111133113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLH
+            thetaBelowCell111133113100))) h)
+        (by
+          have h : ((childLH (childLH (childLH thetaBelowCell111133113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLH
+            thetaBelowCell111133113100))) h)
+        (by
+          have h : ((childHL (childLH (childLH thetaBelowCell111133113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLH
+            thetaBelowCell111133113100))) h)
+        (by
+          have h : ((childHH (childLH (childLH thetaBelowCell111133113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLH
+            thetaBelowCell111133113100))) h))
+    cover_subtree_d2432b059763
+    cover_subtree_a9ad7d61343c
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells64ee0fdc2a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells64ee0fdc2a
+
+open CertificateCells64ee0fdc2a
+theorem e24KC2ThetaBelowLeaf111133113_c1_c0_c0_c2 :
+    adaptiveCoverCheck 5 (childHL thetaBelowCell111133113100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL thetaBelowCell111133113100)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL thetaBelowCell111133113100))
+        (by
+          have h : ((childLL (childLL (childHL thetaBelowCell111133113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childHL
+            thetaBelowCell111133113100))) h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLL (childHL
+            thetaBelowCell111133113100)))
+            (by
+              have h : (thetaBelowCell1111331131002010).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131002010 h)
+            (by
+              have h : (thetaBelowCell1111331131002011).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131002011 h)
+            (by
+              have h : (thetaBelowCell1111331131002012).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131002012 h)
+            (by
+              have h : (thetaBelowCell1111331131002013).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131002013 h))
+        (by
+          have h : ((childHL (childLL (childHL thetaBelowCell111133113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childHL
+            thetaBelowCell111133113100))) h)
+        (by
+          have h : ((childHH (childLL (childHL thetaBelowCell111133113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childHL
+            thetaBelowCell111133113100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL thetaBelowCell111133113100))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 2 (childLL (childLH (childHL
+            thetaBelowCell111133113100)))
+            (by
+              have h : (thetaBelowCell1111331131002100).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131002100 h)
+            (by
+              have h : (thetaBelowCell1111331131002101).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131002101 h)
+            (by
+              have h : (thetaBelowCell1111331131002102).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131002102 h)
+            (by
+              have h : (thetaBelowCell1111331131002103).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131002103 h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLH (childHL
+            thetaBelowCell111133113100)))
+            (by
+              have h : (thetaBelowCell1111331131002110).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131002110 h)
+            (by
+              have h : (thetaBelowCell1111331131002111).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131002111 h)
+            (by
+              have h : (thetaBelowCell1111331131002112).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131002112 h)
+            (by
+              have h : (thetaBelowCell1111331131002113).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131002113 h))
+        (by
+          have h : ((childHL (childLH (childHL thetaBelowCell111133113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childHL
+            thetaBelowCell111133113100))) h)
+        (by
+          have h : ((childHH (childLH (childHL thetaBelowCell111133113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childHL
+            thetaBelowCell111133113100))) h))
+    (by
+      have h : ((childHL (childHL thetaBelowCell111133113100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL thetaBelowCell111133113100)) h)
+    (by
+      have h : ((childHH (childHL thetaBelowCell111133113100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL thetaBelowCell111133113100)) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells30e97e201e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells30e97e201e
+
+open CertificateCells30e97e201e
+theorem cover_subtree_5271df7a6cfb :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002002
+    (by
+      have h : ((childLL thetaAboveCell0000220021002002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002002) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021002002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002002) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002002)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002002)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002002)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002002)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002002)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002002)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002002)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002002)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002002)) h))
+
+theorem cover_subtree_ecb69e009e0c :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002003
+    (by
+      have h : ((childLL thetaAboveCell0000220021002003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002003) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021002003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002003) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002003)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002003)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002003)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002003)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002003)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002003)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002003)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002003)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002003)) h))
+
+theorem cover_subtree_3d1aed828393 :
+    adaptiveCoverCheck 4 (childLL (childLL (childHL thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHL
+    thetaAboveCell000022002100)))
+    (by
+      have h : (thetaAboveCell0000220021002000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002000 h)
+    (by
+      have h : (thetaAboveCell0000220021002001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002001 h)
+    cover_subtree_5271df7a6cfb
+    cover_subtree_ecb69e009e0c
+
+theorem cover_subtree_b9bf7e1adc6d :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002012
+    (by
+      have h : ((childLL thetaAboveCell0000220021002012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002012) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021002012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002012) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002012)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002012)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002012)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002012)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002012)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002012)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002012)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002012)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002012)) h))
+
+theorem cover_subtree_e49d7d8f5134 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002013 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002013
+    (by
+      have h : ((childLL thetaAboveCell0000220021002013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002013) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021002013)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002013) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002013)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002013)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002013)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002013)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002013)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002013)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002013)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002013)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002013)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002013))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002013)) h))
+
+theorem cover_subtree_e1d0aa057fae :
+    adaptiveCoverCheck 4 (childLH (childLL (childHL thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHL
+    thetaAboveCell000022002100)))
+    (by
+      have h : (thetaAboveCell0000220021002010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002010 h)
+    (by
+      have h : (thetaAboveCell0000220021002011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002011 h)
+    cover_subtree_b9bf7e1adc6d
+    cover_subtree_e49d7d8f5134
+
+theorem cover_subtree_da176857c63b :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002020 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002020
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021002020)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021002020)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021002020)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021002020)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021002020)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021002020)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021002020)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021002020)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021002020)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021002020)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002020)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002020)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002020)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002020)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002020)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002020)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002020)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002020)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002020)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002020)) h))
+
+theorem cover_subtree_6fcd4b6c5e8d :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002021 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002021
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021002021)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021002021)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021002021)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021002021)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021002021)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021002021)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021002021)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021002021)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021002021)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021002021)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002021)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002021)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002021)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002021)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002021)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002021)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002021)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002021)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002021)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002021)) h))
+
+theorem cover_subtree_6d0fa2f909d3 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHL thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+    thetaAboveCell000022002100)))
+    cover_subtree_da176857c63b
+    cover_subtree_6fcd4b6c5e8d
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002022
+        (by
+          have h : ((childLL thetaAboveCell0000220021002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021002022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021002022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021002022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002023
+        (by
+          have h : ((childLL thetaAboveCell0000220021002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021002023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021002023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021002023) h))
+
+theorem cover_subtree_db002a1c8e4f :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002030 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002030
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021002030)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021002030)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021002030)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021002030)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021002030)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021002030)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021002030)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021002030)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021002030)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021002030)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002030)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002030)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002030)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002030)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002030)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002030)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002030)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002030)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002030)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002030)) h))
+
+theorem cover_subtree_b3dca0ca0dcf :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002031 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002031
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021002031)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021002031)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021002031)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021002031)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021002031)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021002031)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021002031)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021002031)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021002031)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021002031)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002031)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002031)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002031)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002031)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002031)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002031)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002031)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002031)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002031)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002031)) h))
+
+theorem cover_subtree_78fc8a461c2f :
+    adaptiveCoverCheck 4 (childHH (childLL (childHL thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+    thetaAboveCell000022002100)))
+    cover_subtree_db002a1c8e4f
+    cover_subtree_b3dca0ca0dcf
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002032
+        (by
+          have h : ((childLL thetaAboveCell0000220021002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021002032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021002032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021002032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002033
+        (by
+          have h : ((childLL thetaAboveCell0000220021002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021002033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021002033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021002033) h))
+
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c0_c2_c0 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022002100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022002100))
+    cover_subtree_3d1aed828393
+    cover_subtree_e1d0aa057fae
+    cover_subtree_6d0fa2f909d3
+    cover_subtree_78fc8a461c2f
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd48981846f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd48981846f
+
+open CertificateCellsd48981846f
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c3_c3 :
+    adaptiveCoverCheck 4 (childHH (childHH thetaBelowCell111133113111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childHH thetaBelowCell111133113111))
+    (by
+      have h : ((childLL (childHH (childHH thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childHH (childHH
+        thetaBelowCell111133113111))) h)
+    (by
+      have h : ((childLH (childHH (childHH thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childHH (childHH
+        thetaBelowCell111133113111))) h)
+    (by
+      have h : ((childHL (childHH (childHH thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childHH (childHH
+        thetaBelowCell111133113111))) h)
+    (by
+      have h : ((childHH (childHH (childHH thetaBelowCell111133113111)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childHH (childHH
+        thetaBelowCell111133113111))) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2454b18c1c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2454b18c1c
+
+open CertificateCells2454b18c1c
+theorem e24KC2ThetaBelowLeaf111133113_c0_c1_c2 :
+    adaptiveCoverCheck 6 thetaBelowCell111133113012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113012
+    (by
+      have h : ((childLL thetaBelowCell111133113012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL thetaBelowCell111133113012) h)
+    (by
+      have h : ((childLH thetaBelowCell111133113012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH thetaBelowCell111133113012) h)
+    (by
+      have h : ((childHL thetaBelowCell111133113012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHL thetaBelowCell111133113012) h)
+    (by
+      have h : ((childHH thetaBelowCell111133113012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childHH thetaBelowCell111133113012) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd87c25aa46
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd87c25aa46
+
+open CertificateCellsd87c25aa46
+
+theorem e24KC2ThetaBelowLeaf111133113_c0_c1 :
+    adaptiveCoverCheck 7 (childLH (childLL (childHH thetaBelowCell11113311))) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childLH (childLL (childHH thetaBelowCell11113311)))
+    e24KC2ThetaBelowLeaf111133113_c0_c1_c0 e24KC2ThetaBelowLeaf111133113_c0_c1_c1
+      e24KC2ThetaBelowLeaf111133113_c0_c1_c2 e24KC2ThetaBelowLeaf111133113_c0_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells90183e831a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells90183e831a
+
+open CertificateCells90183e831a
+theorem e24KC2ThetaBelowLeaf111133113_c1_c0_c0_c3 :
+    adaptiveCoverCheck 5 (childHH thetaBelowCell111133113100) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133113100)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH thetaBelowCell111133113100))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 2 (childLL (childLL (childHH
+            thetaBelowCell111133113100)))
+            (by
+              have h : (thetaBelowCell1111331131003000).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003000 h)
+            (by
+              have h : (thetaBelowCell1111331131003001).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003001 h)
+            (by
+              have h : (thetaBelowCell1111331131003002).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003002 h)
+            (by
+              have h : (thetaBelowCell1111331131003003).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003003 h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLL (childHH
+            thetaBelowCell111133113100)))
+            (by
+              have h : (thetaBelowCell1111331131003010).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003010 h)
+            (by
+              have h : (thetaBelowCell1111331131003011).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003011 h)
+            (by
+              have h : (thetaBelowCell1111331131003012).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003012 h)
+            (by
+              have h : (thetaBelowCell1111331131003013).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003013 h))
+        (by
+          have h : ((childHL (childLL (childHH thetaBelowCell111133113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childHH
+            thetaBelowCell111133113100))) h)
+        (by
+          have h : ((childHH (childLL (childHH thetaBelowCell111133113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childHH
+            thetaBelowCell111133113100))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH thetaBelowCell111133113100))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 2 (childLL (childLH (childHH
+            thetaBelowCell111133113100)))
+            (by
+              have h : (thetaBelowCell1111331131003100).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003100 h)
+            (by
+              have h : (thetaBelowCell1111331131003101).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003101 h)
+            (by
+              have h : (thetaBelowCell1111331131003102).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003102 h)
+            (by
+              have h : (thetaBelowCell1111331131003103).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003103 h))
+        (by
+          exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLH (childHH
+            thetaBelowCell111133113100)))
+            (by
+              have h : (thetaBelowCell1111331131003110).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003110 h)
+            (by
+              have h : (thetaBelowCell1111331131003111).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003111 h)
+            (by
+              have h : (thetaBelowCell1111331131003112).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003112 h)
+            (by
+              have h : (thetaBelowCell1111331131003113).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131003113 h))
+        (by
+          have h : ((childHL (childLH (childHH thetaBelowCell111133113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childHH
+            thetaBelowCell111133113100))) h)
+        (by
+          have h : ((childHH (childLH (childHH thetaBelowCell111133113100)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childHH
+            thetaBelowCell111133113100))) h))
+    (by
+      have h : ((childHL (childHH thetaBelowCell111133113100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH thetaBelowCell111133113100)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell111133113100))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH thetaBelowCell111133113100)) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells47753ffbe9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells47753ffbe9
+
+open CertificateCells47753ffbe9
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c0_c0 :
+    adaptiveCoverCheck 6 thetaBelowCell111133113100 = true :=
+  adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113100
+    e24KC2ThetaBelowLeaf111133113_c1_c0_c0_c0 e24KC2ThetaBelowLeaf111133113_c1_c0_c0_c1
+      e24KC2ThetaBelowLeaf111133113_c1_c0_c0_c2 e24KC2ThetaBelowLeaf111133113_c1_c0_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse21bcf4574
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse21bcf4574
+
+open CertificateCellse21bcf4574
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c1_c0 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022002101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022002101)
+    (by
+      have h : ((childLL (childLL thetaAboveCell000022002101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL thetaAboveCell000022002101)) h)
+    (by
+      have h : ((childLH (childLL thetaAboveCell000022002101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL thetaAboveCell000022002101)) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHL (childLL thetaAboveCell000022002101))
+        (by
+          have h : ((childLL (childHL (childLL thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHL (childLL
+            thetaAboveCell000022002101))) h)
+        (by
+          have h : ((childLH (childHL (childLL thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHL (childLL
+            thetaAboveCell000022002101))) h)
+        (by
+          have h : ((childHL (childHL (childLL thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childLL
+            thetaAboveCell000022002101))) h)
+        (by
+          have h : ((childHH (childHL (childLL thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childLL
+            thetaAboveCell000022002101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 4 (childHH (childLL thetaAboveCell000022002101))
+        (by
+          have h : ((childLL (childHH (childLL thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLL (childHH (childLL
+            thetaAboveCell000022002101))) h)
+        (by
+          have h : ((childLH (childHH (childLL thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childLH (childHH (childLL
+            thetaAboveCell000022002101))) h)
+        (by
+          have h : ((childHL (childHH (childLL thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childLL
+            thetaAboveCell000022002101))) h)
+        (by
+          have h : ((childHH (childHH (childLL thetaAboveCell000022002101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childLL
+            thetaAboveCell000022002101))) h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsda9a873863
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsda9a873863
+
+open CertificateCellsda9a873863
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c2 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022002102
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLL thetaAboveCell000022002102)
+        (by
+          have h : ((childLL (childLL thetaAboveCell000022002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLL
+            thetaAboveCell000022002102)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell000022002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLL
+            thetaAboveCell000022002102)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell000022002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLL
+            thetaAboveCell000022002102)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell000022002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLL
+            thetaAboveCell000022002102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 5 (childLH thetaAboveCell000022002102)
+        (by
+          have h : ((childLL (childLH thetaAboveCell000022002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLL (childLH
+            thetaAboveCell000022002102)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell000022002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childLH (childLH
+            thetaAboveCell000022002102)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell000022002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHL (childLH
+            thetaAboveCell000022002102)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell000022002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 5 (childHH (childLH
+            thetaAboveCell000022002102)) h))
+    (by
+      have h : ((childHL thetaAboveCell000022002102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHL thetaAboveCell000022002102) h)
+    (by
+      have h : ((childHH thetaAboveCell000022002102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 (childHH thetaAboveCell000022002102) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells7d865d7559
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells7d865d7559
+
+open CertificateCells7d865d7559
+theorem cover_subtree_6a2ae51f01eb :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131113000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113000
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLL thetaBelowCell1111331131113000)
+        (by
+          have h : ((childLL (childLL thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLL
+            thetaBelowCell1111331131113000)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLL
+            thetaBelowCell1111331131113000)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLL
+            thetaBelowCell1111331131113000)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLL
+            thetaBelowCell1111331131113000)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131113000)
+        (by
+          have h : ((childLL (childLH thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+            thetaBelowCell1111331131113000)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+            thetaBelowCell1111331131113000)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+            thetaBelowCell1111331131113000)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+            thetaBelowCell1111331131113000)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131113000)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131113000)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131113000)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131113000)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131113000)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131113000)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131113000)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131113000)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131113000)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131113000))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131113000)) h))
+
+theorem cover_subtree_a45c44028853 :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131113001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113001
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLL thetaBelowCell1111331131113001)
+        (by
+          have h : ((childLL (childLL thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLL
+            thetaBelowCell1111331131113001)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLL
+            thetaBelowCell1111331131113001)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLL
+            thetaBelowCell1111331131113001)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLL
+            thetaBelowCell1111331131113001)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131113001)
+        (by
+          have h : ((childLL (childLH thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+            thetaBelowCell1111331131113001)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+            thetaBelowCell1111331131113001)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+            thetaBelowCell1111331131113001)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+            thetaBelowCell1111331131113001)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131113001)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131113001)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131113001)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131113001)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131113001)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131113001)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131113001)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131113001)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131113001)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131113001))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131113001)) h))
+
+theorem cover_subtree_d2590f3569cd :
+    adaptiveCoverCheck 3 (childLL (childLL (childHH thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLL (childLL (childHH
+    thetaBelowCell111133113111)))
+    cover_subtree_6a2ae51f01eb
+    cover_subtree_a45c44028853
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113002
+        (by
+          have h : ((childLL thetaBelowCell1111331131113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131113002) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131113002) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131113002) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131113002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131113002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113003
+        (by
+          have h : ((childLL thetaBelowCell1111331131113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131113003) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131113003) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131113003) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131113003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131113003) h))
+
+theorem cover_subtree_903db5f2747c :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131113010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113010
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLL thetaBelowCell1111331131113010)
+        (by
+          have h : ((childLL (childLL thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLL
+            thetaBelowCell1111331131113010)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLL
+            thetaBelowCell1111331131113010)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLL
+            thetaBelowCell1111331131113010)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLL
+            thetaBelowCell1111331131113010)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131113010)
+        (by
+          have h : ((childLL (childLH thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+            thetaBelowCell1111331131113010)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+            thetaBelowCell1111331131113010)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+            thetaBelowCell1111331131113010)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+            thetaBelowCell1111331131113010)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131113010)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131113010)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131113010)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131113010)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131113010)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131113010)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131113010)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131113010)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131113010)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131113010))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131113010)) h))
+
+theorem cover_subtree_4ba95efb8e56 :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131113011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113011
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLL thetaBelowCell1111331131113011)
+        (by
+          have h : ((childLL (childLL thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLL
+            thetaBelowCell1111331131113011)) h)
+        (by
+          have h : ((childLH (childLL thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLL
+            thetaBelowCell1111331131113011)) h)
+        (by
+          have h : ((childHL (childLL thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLL
+            thetaBelowCell1111331131113011)) h)
+        (by
+          have h : ((childHH (childLL thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLL
+            thetaBelowCell1111331131113011)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131113011)
+        (by
+          have h : ((childLL (childLH thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+            thetaBelowCell1111331131113011)) h)
+        (by
+          have h : ((childLH (childLH thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+            thetaBelowCell1111331131113011)) h)
+        (by
+          have h : ((childHL (childLH thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+            thetaBelowCell1111331131113011)) h)
+        (by
+          have h : ((childHH (childLH thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+            thetaBelowCell1111331131113011)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131113011)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131113011)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131113011)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131113011)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131113011)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131113011)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131113011)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131113011)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131113011)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131113011))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131113011)) h))
+
+theorem cover_subtree_189c3ed220c3 :
+    adaptiveCoverCheck 3 (childLH (childLL (childHH thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLL (childHH
+    thetaBelowCell111133113111)))
+    cover_subtree_903db5f2747c
+    cover_subtree_4ba95efb8e56
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113012
+        (by
+          have h : ((childLL thetaBelowCell1111331131113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131113012) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131113012) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131113012) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131113012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131113012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131113013
+        (by
+          have h : ((childLL thetaBelowCell1111331131113013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131113013) h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 0 (childLH thetaBelowCell1111331131113013)
+            (by
+              have h : ((childLL (childLH thetaBelowCell1111331131113013))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childLH
+                thetaBelowCell1111331131113013)) h)
+            (by
+              have h : ((childLH (childLH thetaBelowCell1111331131113013))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childLH
+                thetaBelowCell1111331131113013)) h)
+            (by
+              have h : ((childHL (childLH thetaBelowCell1111331131113013))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childLH
+                thetaBelowCell1111331131113013)) h)
+            (by
+              have h : ((childHH (childLH thetaBelowCell1111331131113013))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childLH
+                thetaBelowCell1111331131113013)) h))
+        (by
+          have h : ((childHL thetaBelowCell1111331131113013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131113013) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131113013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131113013) h))
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c3_c0 :
+    adaptiveCoverCheck 4 (childLL (childHH thetaBelowCell111133113111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH thetaBelowCell111133113111))
+    cover_subtree_d2590f3569cd
+    cover_subtree_189c3ed220c3
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childLL (childHH
+        thetaBelowCell111133113111)))
+        (by
+          have h : (thetaBelowCell1111331131113020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113020 h)
+        (by
+          have h : (thetaBelowCell1111331131113021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113021 h)
+        (by
+          have h : (thetaBelowCell1111331131113022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113022 h)
+        (by
+          have h : (thetaBelowCell1111331131113023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childLL (childHH
+        thetaBelowCell111133113111)))
+        (by
+          have h : (thetaBelowCell1111331131113030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113030 h)
+        (by
+          have h : (thetaBelowCell1111331131113031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113031 h)
+        (by
+          have h : (thetaBelowCell1111331131113032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113032 h)
+        (by
+          have h : (thetaBelowCell1111331131113033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131113033 h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsca42d19ac6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsca42d19ac6
+
+open CertificateCellsca42d19ac6
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c3 :
+    adaptiveCoverCheck 5 (childHH thetaBelowCell111133113111) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133113111)
+    e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c3_c0 e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c3_c1
+      e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c3_c2 e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells257a1a1faa
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells257a1a1faa
+
+open CertificateCells257a1a1faa
+theorem cover_subtree_596b54980499 :
+    adaptiveCoverCheck 4 (childHL (childLL thetaBelowCell111133113101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL thetaBelowCell111133113101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHL (childLL
+        thetaBelowCell111133113101)))
+        (by
+          have h : (thetaBelowCell1111331131010200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010200 h)
+        (by
+          have h : (thetaBelowCell1111331131010201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010201 h)
+        (by
+          have h : (thetaBelowCell1111331131010202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010202 h)
+        (by
+          have h : (thetaBelowCell1111331131010203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHL (childLL
+        thetaBelowCell111133113101)))
+        (by
+          have h : (thetaBelowCell1111331131010210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010210 h)
+        (by
+          have h : (thetaBelowCell1111331131010211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010211 h)
+        (by
+          have h : (thetaBelowCell1111331131010212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010212 h)
+        (by
+          have h : (thetaBelowCell1111331131010213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010213 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHL (childLL
+        thetaBelowCell111133113101)))
+        (by
+          have h : (thetaBelowCell1111331131010220).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010220 h)
+        (by
+          have h : (thetaBelowCell1111331131010221).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010221 h)
+        (by
+          have h : (thetaBelowCell1111331131010222).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010222 h)
+        (by
+          have h : (thetaBelowCell1111331131010223).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010223 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHL (childLL
+        thetaBelowCell111133113101)))
+        (by
+          have h : (thetaBelowCell1111331131010230).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010230 h)
+        (by
+          have h : (thetaBelowCell1111331131010231).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010231 h)
+        (by
+          have h : (thetaBelowCell1111331131010232).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010232 h)
+        (by
+          have h : (thetaBelowCell1111331131010233).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010233 h))
+
+theorem cover_subtree_e230aaf0d2f3 :
+    adaptiveCoverCheck 4 (childHH (childLL thetaBelowCell111133113101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL thetaBelowCell111133113101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHH (childLL
+        thetaBelowCell111133113101)))
+        (by
+          have h : (thetaBelowCell1111331131010300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010300 h)
+        (by
+          have h : (thetaBelowCell1111331131010301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010301 h)
+        (by
+          have h : (thetaBelowCell1111331131010302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010302 h)
+        (by
+          have h : (thetaBelowCell1111331131010303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHH (childLL
+        thetaBelowCell111133113101)))
+        (by
+          have h : (thetaBelowCell1111331131010310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010310 h)
+        (by
+          have h : (thetaBelowCell1111331131010311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010311 h)
+        (by
+          have h : (thetaBelowCell1111331131010312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010312 h)
+        (by
+          have h : (thetaBelowCell1111331131010313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010313 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHH (childLL
+        thetaBelowCell111133113101)))
+        (by
+          have h : (thetaBelowCell1111331131010320).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010320 h)
+        (by
+          have h : (thetaBelowCell1111331131010321).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010321 h)
+        (by
+          have h : (thetaBelowCell1111331131010322).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010322 h)
+        (by
+          have h : (thetaBelowCell1111331131010323).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010323 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHH (childLL
+        thetaBelowCell111133113101)))
+        (by
+          have h : (thetaBelowCell1111331131010330).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010330 h)
+        (by
+          have h : (thetaBelowCell1111331131010331).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010331 h)
+        (by
+          have h : (thetaBelowCell1111331131010332).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010332 h)
+        (by
+          have h : (thetaBelowCell1111331131010333).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131010333 h))
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c0_c1_c0 :
+    adaptiveCoverCheck 5 (childLL thetaBelowCell111133113101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133113101)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL thetaBelowCell111133113101))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLL
+            thetaBelowCell111133113101))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLL
+            thetaBelowCell111133113101))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLL
+            thetaBelowCell111133113101))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLL
+            thetaBelowCell111133113101))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL thetaBelowCell111133113101))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLL
+            thetaBelowCell111133113101))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLL
+            thetaBelowCell111133113101))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLL
+            thetaBelowCell111133113101))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell111133113101)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLL
+            thetaBelowCell111133113101))) h))
+    cover_subtree_596b54980499
+    cover_subtree_e230aaf0d2f3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb187906d24
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb187906d24
+
+open CertificateCellsb187906d24
+theorem cover_subtree_504c637fe057 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021012002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012002
+    (by
+      have h : ((childLL thetaAboveCell0000220021012002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012002) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021012002)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012002) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021012002)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021012002)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021012002)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021012002)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021012002)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021012002)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021012002)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021012002)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021012002)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021012002))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021012002)) h))
+
+theorem cover_subtree_2a357587fe2c :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021012003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012003
+    (by
+      have h : ((childLL thetaAboveCell0000220021012003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012003) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021012003)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012003) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021012003)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021012003)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021012003)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021012003)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021012003)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021012003)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021012003)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021012003)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021012003)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021012003))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021012003)) h))
+
+theorem cover_subtree_e90aeadfa33e :
+    adaptiveCoverCheck 4 (childLL (childLL (childHL thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHL
+    thetaAboveCell000022002101)))
+    (by
+      have h : (thetaAboveCell0000220021012000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012000 h)
+    (by
+      have h : (thetaAboveCell0000220021012001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012001 h)
+    cover_subtree_504c637fe057
+    cover_subtree_2a357587fe2c
+
+theorem cover_subtree_fb92afbe7b07 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021012012 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012012
+    (by
+      have h : ((childLL thetaAboveCell0000220021012012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012012) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021012012)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012012) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021012012)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021012012)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021012012)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021012012)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021012012)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021012012)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021012012)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021012012)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021012012)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021012012))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021012012)) h))
+
+theorem cover_subtree_0e32fa63735b :
+    adaptiveCoverCheck 4 (childLH (childLL (childHL thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHL
+    thetaAboveCell000022002101)))
+    (by
+      have h : (thetaAboveCell0000220021012010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012010 h)
+    (by
+      have h : (thetaAboveCell0000220021012011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021012011 h)
+    cover_subtree_fb92afbe7b07
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012013
+        (by
+          have h : ((childLL thetaAboveCell0000220021012013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012013) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021012013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012013) h)
+        (by
+          exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021012013)
+            (by
+              have h : ((childLL (childHL thetaAboveCell0000220021012013))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+                thetaAboveCell0000220021012013)) h)
+            (by
+              have h : ((childLH (childHL thetaAboveCell0000220021012013))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+                thetaAboveCell0000220021012013)) h)
+            (by
+              have h : ((childHL (childHL thetaAboveCell0000220021012013))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+                thetaAboveCell0000220021012013)) h)
+            (by
+              have h : ((childHH (childHL thetaAboveCell0000220021012013))).rejected = true := by
+                decide +kernel
+              exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+                thetaAboveCell0000220021012013)) h))
+        (by
+          have h : ((childHH thetaAboveCell0000220021012013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012013) h))
+
+theorem cover_subtree_2d2c3f59bbe5 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021012020 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012020
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021012020)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021012020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021012020)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021012020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021012020)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021012020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021012020)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021012020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021012020)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021012020)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021012020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021012020)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021012020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021012020)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021012020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021012020)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021012020))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021012020)) h))
+    (by
+      have h : ((childHL thetaAboveCell0000220021012020)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012020) h)
+    (by
+      have h : ((childHH thetaAboveCell0000220021012020)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012020) h)
+
+theorem cover_subtree_118ca13c27e0 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021012021 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012021
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021012021)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021012021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021012021)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021012021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021012021)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021012021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021012021)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021012021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021012021)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021012021)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021012021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021012021)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021012021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021012021)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021012021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021012021)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021012021))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021012021)) h))
+    (by
+      have h : ((childHL thetaAboveCell0000220021012021)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012021) h)
+    (by
+      have h : ((childHH thetaAboveCell0000220021012021)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012021) h)
+
+theorem cover_subtree_1582f7d2c473 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHL thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+    thetaAboveCell000022002101)))
+    cover_subtree_2d2c3f59bbe5
+    cover_subtree_118ca13c27e0
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012022
+        (by
+          have h : ((childLL thetaAboveCell0000220021012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021012022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012023
+        (by
+          have h : ((childLL thetaAboveCell0000220021012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021012023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012023) h))
+
+theorem cover_subtree_b689532944b1 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021012030 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012030
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021012030)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021012030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021012030)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021012030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021012030)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021012030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021012030)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021012030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021012030)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021012030)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021012030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021012030)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021012030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021012030)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021012030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021012030)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021012030))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021012030)) h))
+    (by
+      have h : ((childHL thetaAboveCell0000220021012030)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012030) h)
+    (by
+      have h : ((childHH thetaAboveCell0000220021012030)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012030) h)
+
+theorem cover_subtree_229b7653f1bc :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021012031 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012031
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021012031)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021012031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021012031)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021012031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021012031)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021012031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021012031)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021012031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021012031)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021012031)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021012031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021012031)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021012031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021012031)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021012031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021012031)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021012031))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021012031)) h))
+    (by
+      have h : ((childHL thetaAboveCell0000220021012031)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012031) h)
+    (by
+      have h : ((childHH thetaAboveCell0000220021012031)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012031) h)
+
+theorem cover_subtree_b1c2b8600845 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHL thetaAboveCell000022002101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+    thetaAboveCell000022002101)))
+    cover_subtree_b689532944b1
+    cover_subtree_229b7653f1bc
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012032
+        (by
+          have h : ((childLL thetaAboveCell0000220021012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021012032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021012033
+        (by
+          have h : ((childLL thetaAboveCell0000220021012033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021012033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021012033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021012033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021012033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021012033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021012033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021012033) h))
+
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c1_c2_c0 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022002101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022002101))
+    cover_subtree_e90aeadfa33e
+    cover_subtree_0e32fa63735b
+    cover_subtree_1582f7d2c473
+    cover_subtree_b1c2b8600845
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellseb98bd469c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellseb98bd469c
+
+open CertificateCellseb98bd469c
+
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c1_c2 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022002101) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022002101)
+    e24KC2ThetaAboveLeaf0000220021_c0_c1_c2_c0 e24KC2ThetaAboveLeaf0000220021_c0_c1_c2_c1
+      e24KC2ThetaAboveLeaf0000220021_c0_c1_c2_c2 e24KC2ThetaAboveLeaf0000220021_c0_c1_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0fb8590a1f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0fb8590a1f
+
+open CertificateCells0fb8590a1f
+
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c1 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002101 = true :=
+  adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022002101
+    e24KC2ThetaAboveLeaf0000220021_c0_c1_c0 e24KC2ThetaAboveLeaf0000220021_c0_c1_c1
+      e24KC2ThetaAboveLeaf0000220021_c0_c1_c2 e24KC2ThetaAboveLeaf0000220021_c0_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells21614274e2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells21614274e2
+
+open CertificateCells21614274e2
+theorem e24KC2ThetaBelowLeaf111133113_c1_c3 :
+    adaptiveCoverCheck 7 (childHH (childLH (childHH thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLH (childHH thetaBelowCell11113311)))
+    (by
+      have h : (thetaBelowCell111133113130).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113130 h)
+    (by
+      have h : (thetaBelowCell111133113131).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113131 h)
+    (by
+      have h : (thetaBelowCell111133113132).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113132 h)
+    (by
+      have h : (thetaBelowCell111133113133).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113133 h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells075667848d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells075667848d
+
+open CertificateCells075667848d
+theorem cover_subtree_162fb6940e90 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002102 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002102
+    (by
+      have h : ((childLL thetaAboveCell0000220021002102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002102) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021002102)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002102) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002102)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002102)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002102)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002102)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002102)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002102)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002102)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002102)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002102)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002102))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002102)) h))
+
+theorem cover_subtree_2b38e4b2557d :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002103 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002103
+    (by
+      have h : ((childLL thetaAboveCell0000220021002103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002103) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021002103)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002103) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002103)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002103)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002103)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002103)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002103)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002103)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002103)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002103)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002103)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002103))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002103)) h))
+
+theorem cover_subtree_ee2829c05e95 :
+    adaptiveCoverCheck 4 (childLL (childLH (childHL thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHL
+    thetaAboveCell000022002100)))
+    (by
+      have h : (thetaAboveCell0000220021002100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002100 h)
+    (by
+      have h : (thetaAboveCell0000220021002101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002101 h)
+    cover_subtree_162fb6940e90
+    cover_subtree_2b38e4b2557d
+
+theorem cover_subtree_d302433685b0 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002112
+    (by
+      have h : ((childLL thetaAboveCell0000220021002112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002112) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021002112)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002112) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002112)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002112)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002112)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002112)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002112)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002112)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002112)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002112)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002112)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002112))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002112)) h))
+
+theorem cover_subtree_549e46a5d3e3 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002113
+    (by
+      have h : ((childLL thetaAboveCell0000220021002113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002113) h)
+    (by
+      have h : ((childLH thetaAboveCell0000220021002113)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002113) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002113)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002113)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002113)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002113)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002113)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002113)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002113)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002113)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002113)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002113))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002113)) h))
+
+theorem cover_subtree_4c9680a92e0c :
+    adaptiveCoverCheck 4 (childLH (childLH (childHL thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHL
+    thetaAboveCell000022002100)))
+    (by
+      have h : (thetaAboveCell0000220021002110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002110 h)
+    (by
+      have h : (thetaAboveCell0000220021002111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021002111 h)
+    cover_subtree_d302433685b0
+    cover_subtree_549e46a5d3e3
+
+theorem cover_subtree_78071866416e :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002120 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002120
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021002120)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021002120)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021002120)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021002120)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021002120)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021002120)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021002120)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021002120)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021002120)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021002120)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002120)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002120)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002120)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002120)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002120)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002120)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002120)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002120)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002120)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002120))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002120)) h))
+
+theorem cover_subtree_088dc6ec9713 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002121 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002121
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021002121)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021002121)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021002121)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021002121)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021002121)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021002121)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021002121)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021002121)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021002121)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021002121)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002121)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002121)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002121)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002121)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002121)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002121)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002121)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002121)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002121)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002121))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002121)) h))
+
+theorem cover_subtree_e0b48be7f1b2 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHL thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+    thetaAboveCell000022002100)))
+    cover_subtree_78071866416e
+    cover_subtree_088dc6ec9713
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002122
+        (by
+          have h : ((childLL thetaAboveCell0000220021002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021002122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021002122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021002122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002123
+        (by
+          have h : ((childLL thetaAboveCell0000220021002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021002123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021002123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021002123) h))
+
+theorem cover_subtree_cb8fede25e6d :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002130 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002130
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021002130)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021002130)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021002130)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021002130)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021002130)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021002130)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021002130)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021002130)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021002130)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021002130)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002130)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002130)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002130)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002130)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002130)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002130)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002130)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002130)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002130)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002130))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002130)) h))
+
+theorem cover_subtree_9df38978a25b :
+    adaptiveCoverCheck 3 thetaAboveCell0000220021002131 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002131
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLL thetaAboveCell0000220021002131)
+        (by
+          have h : ((childLL (childLL thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLL
+            thetaAboveCell0000220021002131)) h)
+        (by
+          have h : ((childLH (childLL thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLL
+            thetaAboveCell0000220021002131)) h)
+        (by
+          have h : ((childHL (childLL thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLL
+            thetaAboveCell0000220021002131)) h)
+        (by
+          have h : ((childHH (childLL thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLL
+            thetaAboveCell0000220021002131)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childLH thetaAboveCell0000220021002131)
+        (by
+          have h : ((childLL (childLH thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childLH
+            thetaAboveCell0000220021002131)) h)
+        (by
+          have h : ((childLH (childLH thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childLH
+            thetaAboveCell0000220021002131)) h)
+        (by
+          have h : ((childHL (childLH thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childLH
+            thetaAboveCell0000220021002131)) h)
+        (by
+          have h : ((childHH (childLH thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childLH
+            thetaAboveCell0000220021002131)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHL thetaAboveCell0000220021002131)
+        (by
+          have h : ((childLL (childHL thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHL
+            thetaAboveCell0000220021002131)) h)
+        (by
+          have h : ((childLH (childHL thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHL
+            thetaAboveCell0000220021002131)) h)
+        (by
+          have h : ((childHL (childHL thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHL
+            thetaAboveCell0000220021002131)) h)
+        (by
+          have h : ((childHH (childHL thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHL
+            thetaAboveCell0000220021002131)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 (childHH thetaAboveCell0000220021002131)
+        (by
+          have h : ((childLL (childHH thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL (childHH
+            thetaAboveCell0000220021002131)) h)
+        (by
+          have h : ((childLH (childHH thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH (childHH
+            thetaAboveCell0000220021002131)) h)
+        (by
+          have h : ((childHL (childHH thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL (childHH
+            thetaAboveCell0000220021002131)) h)
+        (by
+          have h : ((childHH (childHH thetaAboveCell0000220021002131))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH (childHH
+            thetaAboveCell0000220021002131)) h))
+
+theorem cover_subtree_413edc037905 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHL thetaAboveCell000022002100))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+    thetaAboveCell000022002100)))
+    cover_subtree_cb8fede25e6d
+    cover_subtree_9df38978a25b
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002132
+        (by
+          have h : ((childLL thetaAboveCell0000220021002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021002132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021002132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021002132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021002133
+        (by
+          have h : ((childLL thetaAboveCell0000220021002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021002133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021002133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021002133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021002133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021002133) h))
+
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c0_c2_c1 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022002100)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022002100))
+    cover_subtree_ee2829c05e95
+    cover_subtree_4c9680a92e0c
+    cover_subtree_e0b48be7f1b2
+    cover_subtree_413edc037905
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells135b119fc2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells135b119fc2
+
+open CertificateCells135b119fc2
+
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c0_c2 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022002100) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022002100)
+    e24KC2ThetaAboveLeaf0000220021_c0_c0_c2_c0 e24KC2ThetaAboveLeaf0000220021_c0_c0_c2_c1
+      e24KC2ThetaAboveLeaf0000220021_c0_c0_c2_c2 e24KC2ThetaAboveLeaf0000220021_c0_c0_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse53a016898
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse53a016898
+
+open CertificateCellse53a016898
+
+theorem e24KC2ThetaAboveLeaf0000220021_c0_c0 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002100 = true :=
+  adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022002100
+    e24KC2ThetaAboveLeaf0000220021_c0_c0_c0 e24KC2ThetaAboveLeaf0000220021_c0_c0_c1
+      e24KC2ThetaAboveLeaf0000220021_c0_c0_c2 e24KC2ThetaAboveLeaf0000220021_c0_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8b72842cc0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8b72842cc0
+
+open CertificateCells8b72842cc0
+
+theorem e24KC2ThetaAboveLeaf0000220021_c0 :
+    adaptiveCoverCheck 8 (childLL (childLH (childHL thetaAboveCell00002200))) = true :=
+  adaptiveCoverCheck_succ_of_children 7 (childLL (childLH (childHL thetaAboveCell00002200)))
+    e24KC2ThetaAboveLeaf0000220021_c0_c0 e24KC2ThetaAboveLeaf0000220021_c0_c1
+      e24KC2ThetaAboveLeaf0000220021_c0_c2 e24KC2ThetaAboveLeaf0000220021_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3d04a04d91
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3d04a04d91
+
+open CertificateCells3d04a04d91
+theorem e24KC2ThetaBelowLeaf111133113_c0_c3 :
+    adaptiveCoverCheck 7 (childHH (childLL (childHH thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHH (childLL (childHH thetaBelowCell11113311)))
+    (by
+      have h : (thetaBelowCell111133113030).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113030 h)
+    (by
+      have h : (thetaBelowCell111133113031).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113031 h)
+    (by
+      have h : (thetaBelowCell111133113032).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113032 h)
+    (by
+      have h : (thetaBelowCell111133113033).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113033 h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells665e67ba22
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells665e67ba22
+
+open CertificateCells665e67ba22
+theorem cover_subtree_1c545d2d332e :
+    adaptiveCoverCheck 3 (childLH (childLL (childHH thetaBelowCell111133113101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLL (childHH
+    thetaBelowCell111133113101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131013010
+        (by
+          have h : ((childLL thetaBelowCell1111331131013010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131013010) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131013010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131013010) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131013010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131013010) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131013010)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131013010) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131013011
+        (by
+          have h : ((childLL thetaBelowCell1111331131013011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131013011) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131013011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131013011) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131013011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131013011) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131013011)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131013011) h))
+    (by
+      have h : (thetaBelowCell1111331131013012).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131013012 h)
+    (by
+      have h : (thetaBelowCell1111331131013013).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131013013 h)
+
+theorem cover_subtree_a6892a976bdc :
+    adaptiveCoverCheck 4 (childLL (childHH thetaBelowCell111133113101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH thetaBelowCell111133113101))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childLL (childHH
+        thetaBelowCell111133113101)))
+        (by
+          have h : (thetaBelowCell1111331131013000).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131013000 h)
+        (by
+          have h : (thetaBelowCell1111331131013001).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131013001 h)
+        (by
+          have h : (thetaBelowCell1111331131013002).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131013002 h)
+        (by
+          have h : (thetaBelowCell1111331131013003).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131013003 h))
+    cover_subtree_1c545d2d332e
+    (by
+      have h : ((childHL (childLL (childHH thetaBelowCell111133113101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childHH
+        thetaBelowCell111133113101))) h)
+    (by
+      have h : ((childHH (childLL (childHH thetaBelowCell111133113101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childHH
+        thetaBelowCell111133113101))) h)
+
+theorem cover_subtree_8f27cd7d62f9 :
+    adaptiveCoverCheck 3 (childLL (childLH (childHH thetaBelowCell111133113101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLL (childLH (childHH
+    thetaBelowCell111133113101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131013100
+        (by
+          have h : ((childLL thetaBelowCell1111331131013100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131013100) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131013100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131013100) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131013100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131013100) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131013100)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131013100) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131013101
+        (by
+          have h : ((childLL thetaBelowCell1111331131013101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131013101) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131013101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131013101) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131013101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131013101) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131013101)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131013101) h))
+    (by
+      have h : (thetaBelowCell1111331131013102).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131013102 h)
+    (by
+      have h : (thetaBelowCell1111331131013103).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131013103 h)
+
+theorem cover_subtree_3a8993d4b677 :
+    adaptiveCoverCheck 3 (childLH (childLH (childHH thetaBelowCell111133113101))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childLH (childLH (childHH
+    thetaBelowCell111133113101)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131013110
+        (by
+          have h : ((childLL thetaBelowCell1111331131013110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131013110) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131013110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131013110) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131013110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131013110) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131013110)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131013110) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131013111
+        (by
+          have h : ((childLL thetaBelowCell1111331131013111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131013111) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131013111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131013111) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131013111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131013111) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131013111)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131013111) h))
+    (by
+      have h : (thetaBelowCell1111331131013112).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131013112 h)
+    (by
+      have h : (thetaBelowCell1111331131013113).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131013113 h)
+
+theorem cover_subtree_f225800535b1 :
+    adaptiveCoverCheck 4 (childLH (childHH thetaBelowCell111133113101)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH thetaBelowCell111133113101))
+    cover_subtree_8f27cd7d62f9
+    cover_subtree_3a8993d4b677
+    (by
+      have h : ((childHL (childLH (childHH thetaBelowCell111133113101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childHH
+        thetaBelowCell111133113101))) h)
+    (by
+      have h : ((childHH (childLH (childHH thetaBelowCell111133113101)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childHH
+        thetaBelowCell111133113101))) h)
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c0_c1_c3 :
+    adaptiveCoverCheck 5 (childHH thetaBelowCell111133113101) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH thetaBelowCell111133113101)
+    cover_subtree_a6892a976bdc
+    cover_subtree_f225800535b1
+    (by
+      have h : ((childHL (childHH thetaBelowCell111133113101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH thetaBelowCell111133113101)) h)
+    (by
+      have h : ((childHH (childHH thetaBelowCell111133113101))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH thetaBelowCell111133113101)) h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa275032fe9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa275032fe9
+
+open CertificateCellsa275032fe9
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c0_c1 :
+    adaptiveCoverCheck 6 thetaBelowCell111133113101 = true :=
+  adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113101
+    e24KC2ThetaBelowLeaf111133113_c1_c0_c1_c0 e24KC2ThetaBelowLeaf111133113_c1_c0_c1_c1
+      e24KC2ThetaBelowLeaf111133113_c1_c0_c1_c2 e24KC2ThetaBelowLeaf111133113_c1_c0_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells09124dcbf9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells09124dcbf9
+
+open CertificateCells09124dcbf9
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c0 :
+    adaptiveCoverCheck 7 (childLL (childLH (childHH thetaBelowCell11113311))) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childLL (childLH (childHH thetaBelowCell11113311)))
+    e24KC2ThetaBelowLeaf111133113_c1_c0_c0 e24KC2ThetaBelowLeaf111133113_c1_c0_c1
+      e24KC2ThetaBelowLeaf111133113_c1_c0_c2 e24KC2ThetaBelowLeaf111133113_c1_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse133bf0420
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse133bf0420
+
+open CertificateCellse133bf0420
+theorem cover_subtree_8c4e3e00c778 :
+    adaptiveCoverCheck 4 (childLL (childLL (childHL thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHL
+    thetaAboveCell000022002110)))
+    (by
+      have h : (thetaAboveCell0000220021102000).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102000 h)
+    (by
+      have h : (thetaAboveCell0000220021102001).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102001 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102002
+        (by
+          have h : ((childLL thetaAboveCell0000220021102002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102002) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102002) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102002) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102002)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102002) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102003
+        (by
+          have h : ((childLL thetaAboveCell0000220021102003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102003) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102003) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102003) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102003)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102003) h))
+
+theorem cover_subtree_49f6eb3ee7e0 :
+    adaptiveCoverCheck 4 (childLH (childLL (childHL thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHL
+    thetaAboveCell000022002110)))
+    (by
+      have h : (thetaAboveCell0000220021102010).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102010 h)
+    (by
+      have h : (thetaAboveCell0000220021102011).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102011 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102012
+        (by
+          have h : ((childLL thetaAboveCell0000220021102012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102012) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102012) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102012) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102012)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102012) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102013
+        (by
+          have h : ((childLL thetaAboveCell0000220021102013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102013) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102013) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102013) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102013)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102013) h))
+
+theorem cover_subtree_adebd26c440a :
+    adaptiveCoverCheck 4 (childHL (childLL (childHL thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL (childHL
+    thetaAboveCell000022002110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102020
+        (by
+          have h : ((childLL thetaAboveCell0000220021102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102020) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102020) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102020) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102020)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102020) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102021
+        (by
+          have h : ((childLL thetaAboveCell0000220021102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102021) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102021) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102021) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102021)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102021) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102022
+        (by
+          have h : ((childLL thetaAboveCell0000220021102022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102022) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102022) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102022) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102022)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102022) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102023
+        (by
+          have h : ((childLL thetaAboveCell0000220021102023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102023) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102023) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102023) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102023)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102023) h))
+
+theorem cover_subtree_cfe74f698d24 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHL thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL (childHL
+    thetaAboveCell000022002110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102030
+        (by
+          have h : ((childLL thetaAboveCell0000220021102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102030) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102030) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102030) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102030)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102030) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102031
+        (by
+          have h : ((childLL thetaAboveCell0000220021102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102031) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102031) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102031) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102031)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102031) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102032
+        (by
+          have h : ((childLL thetaAboveCell0000220021102032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102032) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102032) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102032) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102032)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102032) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102033
+        (by
+          have h : ((childLL thetaAboveCell0000220021102033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102033) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102033) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102033) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102033)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102033) h))
+
+theorem cover_subtree_64b5e3c15e8d :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022002110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022002110))
+    cover_subtree_8c4e3e00c778
+    cover_subtree_49f6eb3ee7e0
+    cover_subtree_adebd26c440a
+    cover_subtree_cfe74f698d24
+
+theorem cover_subtree_aeac3ba6168f :
+    adaptiveCoverCheck 4 (childLL (childLH (childHL thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHL
+    thetaAboveCell000022002110)))
+    (by
+      have h : (thetaAboveCell0000220021102100).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102100 h)
+    (by
+      have h : (thetaAboveCell0000220021102101).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102101 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102102
+        (by
+          have h : ((childLL thetaAboveCell0000220021102102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102102) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102102) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102102) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102102)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102102) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102103
+        (by
+          have h : ((childLL thetaAboveCell0000220021102103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102103) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102103) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102103) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102103)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102103) h))
+
+theorem cover_subtree_325d50012465 :
+    adaptiveCoverCheck 4 (childLH (childLH (childHL thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHL
+    thetaAboveCell000022002110)))
+    (by
+      have h : (thetaAboveCell0000220021102110).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102110 h)
+    (by
+      have h : (thetaAboveCell0000220021102111).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102111 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102112
+        (by
+          have h : ((childLL thetaAboveCell0000220021102112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102112) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102112) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102112) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102112)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102112) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102113
+        (by
+          have h : ((childLL thetaAboveCell0000220021102113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102113) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102113) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102113) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102113)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102113) h))
+
+theorem cover_subtree_b265451cd007 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHL thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHL
+    thetaAboveCell000022002110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102120
+        (by
+          have h : ((childLL thetaAboveCell0000220021102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102120) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102120) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102120) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102120)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102120) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102121
+        (by
+          have h : ((childLL thetaAboveCell0000220021102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102121) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102121) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102121) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102121)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102121) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102122
+        (by
+          have h : ((childLL thetaAboveCell0000220021102122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102122) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102122) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102122) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102122)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102122) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102123
+        (by
+          have h : ((childLL thetaAboveCell0000220021102123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102123) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102123) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102123) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102123)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102123) h))
+
+theorem cover_subtree_613b8841c9c0 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHL thetaAboveCell000022002110))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHL
+    thetaAboveCell000022002110)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102130
+        (by
+          have h : ((childLL thetaAboveCell0000220021102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102130) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102130) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102130) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102130)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102130) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102131
+        (by
+          have h : ((childLL thetaAboveCell0000220021102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102131) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102131) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102131) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102131)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102131) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102132
+        (by
+          have h : ((childLL thetaAboveCell0000220021102132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102132) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102132) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102132) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102132)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102132) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 thetaAboveCell0000220021102133
+        (by
+          have h : ((childLL thetaAboveCell0000220021102133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLL thetaAboveCell0000220021102133) h)
+        (by
+          have h : ((childLH thetaAboveCell0000220021102133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childLH thetaAboveCell0000220021102133) h)
+        (by
+          have h : ((childHL thetaAboveCell0000220021102133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHL thetaAboveCell0000220021102133) h)
+        (by
+          have h : ((childHH thetaAboveCell0000220021102133)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 (childHH thetaAboveCell0000220021102133) h))
+
+theorem cover_subtree_52c3ab758725 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022002110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022002110))
+    cover_subtree_aeac3ba6168f
+    cover_subtree_325d50012465
+    cover_subtree_b265451cd007
+    cover_subtree_613b8841c9c0
+
+theorem cover_subtree_1c021cc633a3 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022002110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHL (childHL thetaAboveCell000022002110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHL (childHL
+        thetaAboveCell000022002110)))
+        (by
+          have h : (thetaAboveCell0000220021102200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102200 h)
+        (by
+          have h : (thetaAboveCell0000220021102201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102201 h)
+        (by
+          have h : (thetaAboveCell0000220021102202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102202 h)
+        (by
+          have h : (thetaAboveCell0000220021102203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHL (childHL
+        thetaAboveCell000022002110)))
+        (by
+          have h : (thetaAboveCell0000220021102210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102210 h)
+        (by
+          have h : (thetaAboveCell0000220021102211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102211 h)
+        (by
+          have h : (thetaAboveCell0000220021102212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102212 h)
+        (by
+          have h : (thetaAboveCell0000220021102213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102213 h))
+    (by
+      have h : ((childHL (childHL (childHL thetaAboveCell000022002110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHL (childHL
+        thetaAboveCell000022002110))) h)
+    (by
+      have h : ((childHH (childHL (childHL thetaAboveCell000022002110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHL (childHL
+        thetaAboveCell000022002110))) h)
+
+theorem cover_subtree_191cbd679a04 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022002110)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childHH (childHL thetaAboveCell000022002110))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childHH (childHL
+        thetaAboveCell000022002110)))
+        (by
+          have h : (thetaAboveCell0000220021102300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102300 h)
+        (by
+          have h : (thetaAboveCell0000220021102301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102301 h)
+        (by
+          have h : (thetaAboveCell0000220021102302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102302 h)
+        (by
+          have h : (thetaAboveCell0000220021102303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childHH (childHL
+        thetaAboveCell000022002110)))
+        (by
+          have h : (thetaAboveCell0000220021102310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102310 h)
+        (by
+          have h : (thetaAboveCell0000220021102311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102311 h)
+        (by
+          have h : (thetaAboveCell0000220021102312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102312 h)
+        (by
+          have h : (thetaAboveCell0000220021102313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 thetaAboveCell0000220021102313 h))
+    (by
+      have h : ((childHL (childHH (childHL thetaAboveCell000022002110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHL (childHH (childHL
+        thetaAboveCell000022002110))) h)
+    (by
+      have h : ((childHH (childHH (childHL thetaAboveCell000022002110)))).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 4 (childHH (childHH (childHL
+        thetaAboveCell000022002110))) h)
+
+theorem e24KC2ThetaAboveLeaf0000220021_c1_c0_c2 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022002110) = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022002110)
+    cover_subtree_64b5e3c15e8d
+    cover_subtree_52c3ab758725
+    cover_subtree_1c021cc633a3
+    cover_subtree_191cbd679a04
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf1833757d0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf1833757d0
+
+open CertificateCellsf1833757d0
+
+theorem e24KC2ThetaAboveLeaf0000220021_c1_c0 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002110 = true :=
+  adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022002110
+    e24KC2ThetaAboveLeaf0000220021_c1_c0_c0 e24KC2ThetaAboveLeaf0000220021_c1_c0_c1
+      e24KC2ThetaAboveLeaf0000220021_c1_c0_c2 e24KC2ThetaAboveLeaf0000220021_c1_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells21f94a2daf
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells21f94a2daf
+
+open CertificateCells21f94a2daf
+
+theorem e24KC2ThetaAboveLeaf0000220021_c1 :
+    adaptiveCoverCheck 8 (childLH (childLH (childHL thetaAboveCell00002200))) = true :=
+  adaptiveCoverCheck_succ_of_children 7 (childLH (childLH (childHL thetaAboveCell00002200)))
+    e24KC2ThetaAboveLeaf0000220021_c1_c0 e24KC2ThetaAboveLeaf0000220021_c1_c1
+      e24KC2ThetaAboveLeaf0000220021_c1_c2 e24KC2ThetaAboveLeaf0000220021_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsbb4f00460c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsbb4f00460c
+
+open CertificateCellsbb4f00460c
+
+theorem e24KC2ThetaAboveLeaf0000220021 :
+    adaptiveCoverCheck 9 (childLH (childHL thetaAboveCell00002200)) = true :=
+  adaptiveCoverCheck_succ_of_children 8 (childLH (childHL thetaAboveCell00002200))
+    e24KC2ThetaAboveLeaf0000220021_c0 e24KC2ThetaAboveLeaf0000220021_c1
+      e24KC2ThetaAboveLeaf0000220021_c2 e24KC2ThetaAboveLeaf0000220021_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells232e7e9a76
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells232e7e9a76
+
+open CertificateCells232e7e9a76
+theorem e24KC2ThetaBelowLeaf111133113_c0_c2 :
+    adaptiveCoverCheck 7 (childHL (childLL (childHH thetaBelowCell11113311))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 (childHL (childLL (childHH thetaBelowCell11113311)))
+    (by
+      have h : (thetaBelowCell111133113020).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113020 h)
+    (by
+      have h : (thetaBelowCell111133113021).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113021 h)
+    (by
+      have h : (thetaBelowCell111133113022).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113022 h)
+    (by
+      have h : (thetaBelowCell111133113023).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 6 thetaBelowCell111133113023 h)
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse657ecadb5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse657ecadb5
+
+open CertificateCellse657ecadb5
+
+theorem e24KC2ThetaBelowLeaf111133113_c0 :
+    adaptiveCoverCheck 8 (childLL (childHH thetaBelowCell11113311)) = true :=
+  adaptiveCoverCheck_succ_of_children 7 (childLL (childHH thetaBelowCell11113311))
+    e24KC2ThetaBelowLeaf111133113_c0_c0 e24KC2ThetaBelowLeaf111133113_c0_c1
+      e24KC2ThetaBelowLeaf111133113_c0_c2 e24KC2ThetaBelowLeaf111133113_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 explicit terminal-certificate subtree. No adaptive search. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells78b824f474
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells78b824f474
+
+open CertificateCells78b824f474
+theorem cover_subtree_7d1a4629139c :
+    adaptiveCoverCheck 3 (childHL (childHL (childLL thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHL (childLL
+    thetaBelowCell111133113111)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131110220
+        (by
+          have h : ((childLL thetaBelowCell1111331131110220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131110220) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131110220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131110220) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131110220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131110220) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131110220)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131110220) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131110221
+        (by
+          have h : ((childLL thetaBelowCell1111331131110221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131110221) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131110221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131110221) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131110221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131110221) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131110221)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131110221) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131110222
+        (by
+          have h : ((childLL thetaBelowCell1111331131110222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131110222) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131110222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131110222) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131110222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131110222) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131110222)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131110222) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131110223
+        (by
+          have h : ((childLL thetaBelowCell1111331131110223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131110223) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131110223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131110223) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131110223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131110223) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131110223)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131110223) h))
+
+theorem cover_subtree_271c266eebff :
+    adaptiveCoverCheck 3 (childHH (childHL (childLL thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHL (childLL
+    thetaBelowCell111133113111)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131110230
+        (by
+          have h : ((childLL thetaBelowCell1111331131110230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131110230) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131110230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131110230) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131110230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131110230) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131110230)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131110230) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131110231
+        (by
+          have h : ((childLL thetaBelowCell1111331131110231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131110231) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131110231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131110231) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131110231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131110231) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131110231)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131110231) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131110232
+        (by
+          have h : ((childLL thetaBelowCell1111331131110232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131110232) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131110232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131110232) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131110232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131110232) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131110232)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131110232) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131110233
+        (by
+          have h : ((childLL thetaBelowCell1111331131110233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131110233) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131110233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131110233) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131110233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131110233) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131110233)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131110233) h))
+
+theorem cover_subtree_f2e98bb9d295 :
+    adaptiveCoverCheck 4 (childHL (childLL thetaBelowCell111133113111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHL (childLL thetaBelowCell111133113111))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHL (childLL
+        thetaBelowCell111133113111)))
+        (by
+          have h : (thetaBelowCell1111331131110200).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110200 h)
+        (by
+          have h : (thetaBelowCell1111331131110201).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110201 h)
+        (by
+          have h : (thetaBelowCell1111331131110202).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110202 h)
+        (by
+          have h : (thetaBelowCell1111331131110203).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110203 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHL (childLL
+        thetaBelowCell111133113111)))
+        (by
+          have h : (thetaBelowCell1111331131110210).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110210 h)
+        (by
+          have h : (thetaBelowCell1111331131110211).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110211 h)
+        (by
+          have h : (thetaBelowCell1111331131110212).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110212 h)
+        (by
+          have h : (thetaBelowCell1111331131110213).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110213 h))
+    cover_subtree_7d1a4629139c
+    cover_subtree_271c266eebff
+
+theorem cover_subtree_9fadcdfbc6fb :
+    adaptiveCoverCheck 3 (childHL (childHH (childLL thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHL (childHH (childLL
+    thetaBelowCell111133113111)))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131110320
+        (by
+          have h : ((childLL thetaBelowCell1111331131110320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131110320) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131110320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131110320) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131110320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131110320) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131110320)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131110320) h))
+    (by
+      have h : (thetaBelowCell1111331131110321).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110321 h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131110322
+        (by
+          have h : ((childLL thetaBelowCell1111331131110322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131110322) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131110322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131110322) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131110322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131110322) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131110322)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131110322) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131110323
+        (by
+          have h : ((childLL thetaBelowCell1111331131110323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131110323) h)
+        (by
+          have h : ((childLH thetaBelowCell1111331131110323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131110323) h)
+        (by
+          have h : ((childHL thetaBelowCell1111331131110323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHL thetaBelowCell1111331131110323) h)
+        (by
+          have h : ((childHH thetaBelowCell1111331131110323)).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 1 (childHH thetaBelowCell1111331131110323) h))
+
+theorem cover_subtree_3e826769a774 :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131110332 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131110332
+    (by
+      have h : ((childLL thetaBelowCell1111331131110332)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131110332) h)
+    (by
+      have h : ((childLH thetaBelowCell1111331131110332)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131110332) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131110332)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131110332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131110332)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131110332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131110332)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131110332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131110332)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131110332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131110332)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131110332)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131110332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131110332)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131110332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131110332)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131110332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131110332)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131110332))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131110332)) h))
+
+theorem cover_subtree_647c14158ebb :
+    adaptiveCoverCheck 2 thetaBelowCell1111331131110333 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 thetaBelowCell1111331131110333
+    (by
+      have h : ((childLL thetaBelowCell1111331131110333)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLL thetaBelowCell1111331131110333) h)
+    (by
+      have h : ((childLH thetaBelowCell1111331131110333)).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 1 (childLH thetaBelowCell1111331131110333) h)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHL thetaBelowCell1111331131110333)
+        (by
+          have h : ((childLL (childHL thetaBelowCell1111331131110333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHL
+            thetaBelowCell1111331131110333)) h)
+        (by
+          have h : ((childLH (childHL thetaBelowCell1111331131110333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHL
+            thetaBelowCell1111331131110333)) h)
+        (by
+          have h : ((childHL (childHL thetaBelowCell1111331131110333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHL
+            thetaBelowCell1111331131110333)) h)
+        (by
+          have h : ((childHH (childHL thetaBelowCell1111331131110333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHL
+            thetaBelowCell1111331131110333)) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 0 (childHH thetaBelowCell1111331131110333)
+        (by
+          have h : ((childLL (childHH thetaBelowCell1111331131110333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLL (childHH
+            thetaBelowCell1111331131110333)) h)
+        (by
+          have h : ((childLH (childHH thetaBelowCell1111331131110333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childLH (childHH
+            thetaBelowCell1111331131110333)) h)
+        (by
+          have h : ((childHL (childHH thetaBelowCell1111331131110333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHL (childHH
+            thetaBelowCell1111331131110333)) h)
+        (by
+          have h : ((childHH (childHH thetaBelowCell1111331131110333))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 0 (childHH (childHH
+            thetaBelowCell1111331131110333)) h))
+
+theorem cover_subtree_7bb13e3867b3 :
+    adaptiveCoverCheck 3 (childHH (childHH (childLL thetaBelowCell111133113111))) = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 (childHH (childHH (childLL
+    thetaBelowCell111133113111)))
+    (by
+      have h : (thetaBelowCell1111331131110330).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110330 h)
+    (by
+      have h : (thetaBelowCell1111331131110331).rejected = true := by
+        decide +kernel
+      exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110331 h)
+    cover_subtree_3e826769a774
+    cover_subtree_647c14158ebb
+
+theorem cover_subtree_108f8990274f :
+    adaptiveCoverCheck 4 (childHH (childLL thetaBelowCell111133113111)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 (childHH (childLL thetaBelowCell111133113111))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLL (childHH (childLL
+        thetaBelowCell111133113111)))
+        (by
+          have h : (thetaBelowCell1111331131110300).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110300 h)
+        (by
+          have h : (thetaBelowCell1111331131110301).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110301 h)
+        (by
+          have h : (thetaBelowCell1111331131110302).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110302 h)
+        (by
+          have h : (thetaBelowCell1111331131110303).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110303 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 2 (childLH (childHH (childLL
+        thetaBelowCell111133113111)))
+        (by
+          have h : (thetaBelowCell1111331131110310).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110310 h)
+        (by
+          have h : (thetaBelowCell1111331131110311).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110311 h)
+        (by
+          have h : (thetaBelowCell1111331131110312).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110312 h)
+        (by
+          have h : (thetaBelowCell1111331131110313).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 2 thetaBelowCell1111331131110313 h))
+    cover_subtree_9fadcdfbc6fb
+    cover_subtree_7bb13e3867b3
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c0 :
+    adaptiveCoverCheck 5 (childLL thetaBelowCell111133113111) = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 (childLL thetaBelowCell111133113111)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLL (childLL thetaBelowCell111133113111))
+        (by
+          have h : ((childLL (childLL (childLL thetaBelowCell111133113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLL (childLL
+            thetaBelowCell111133113111))) h)
+        (by
+          have h : ((childLH (childLL (childLL thetaBelowCell111133113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLL (childLL
+            thetaBelowCell111133113111))) h)
+        (by
+          have h : ((childHL (childLL (childLL thetaBelowCell111133113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLL (childLL
+            thetaBelowCell111133113111))) h)
+        (by
+          have h : ((childHH (childLL (childLL thetaBelowCell111133113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLL (childLL
+            thetaBelowCell111133113111))) h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 3 (childLH (childLL thetaBelowCell111133113111))
+        (by
+          have h : ((childLL (childLH (childLL thetaBelowCell111133113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLL (childLH (childLL
+            thetaBelowCell111133113111))) h)
+        (by
+          have h : ((childLH (childLH (childLL thetaBelowCell111133113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childLH (childLH (childLL
+            thetaBelowCell111133113111))) h)
+        (by
+          have h : ((childHL (childLH (childLL thetaBelowCell111133113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHL (childLH (childLL
+            thetaBelowCell111133113111))) h)
+        (by
+          have h : ((childHH (childLH (childLL thetaBelowCell111133113111)))).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 3 (childHH (childLH (childLL
+            thetaBelowCell111133113111))) h))
+    cover_subtree_f2e98bb9d295
+    cover_subtree_108f8990274f
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsccc2f90403
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsccc2f90403
+
+open CertificateCellsccc2f90403
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1_c1 :
+    adaptiveCoverCheck 6 thetaBelowCell111133113111 = true :=
+  adaptiveCoverCheck_succ_of_children 5 thetaBelowCell111133113111
+    e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c0 e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c1
+      e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c2 e24KC2ThetaBelowLeaf111133113_c1_c1_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse5aa25a242
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse5aa25a242
+
+open CertificateCellse5aa25a242
+
+theorem e24KC2ThetaBelowLeaf111133113_c1_c1 :
+    adaptiveCoverCheck 7 (childLH (childLH (childHH thetaBelowCell11113311))) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childLH (childLH (childHH thetaBelowCell11113311)))
+    e24KC2ThetaBelowLeaf111133113_c1_c1_c0 e24KC2ThetaBelowLeaf111133113_c1_c1_c1
+      e24KC2ThetaBelowLeaf111133113_c1_c1_c2 e24KC2ThetaBelowLeaf111133113_c1_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0d1c7d48ff
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0d1c7d48ff
+
+open CertificateCells0d1c7d48ff
+
+theorem e24KC2ThetaBelowLeaf111133113_c1 :
+    adaptiveCoverCheck 8 (childLH (childHH thetaBelowCell11113311)) = true :=
+  adaptiveCoverCheck_succ_of_children 7 (childLH (childHH thetaBelowCell11113311))
+    e24KC2ThetaBelowLeaf111133113_c1_c0 e24KC2ThetaBelowLeaf111133113_c1_c1
+      e24KC2ThetaBelowLeaf111133113_c1_c2 e24KC2ThetaBelowLeaf111133113_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! KC6R4 pure logical subtree join. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1d418dae20
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1d418dae20
+
+open CertificateCells1d418dae20
+
+theorem e24KC2ThetaBelowLeaf111133113 :
+    adaptiveCoverCheck 9 (childHH thetaBelowCell11113311) = true :=
+  adaptiveCoverCheck_succ_of_children 8 (childHH thetaBelowCell11113311)
+    e24KC2ThetaBelowLeaf111133113_c0 e24KC2ThetaBelowLeaf111133113_c1
+      e24KC2ThetaBelowLeaf111133113_c2 e24KC2ThetaBelowLeaf111133113_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle022.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle022.lean
new file mode 100644
index 0000000000..62de062094
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle022.lean
@@ -0,0 +1,7154 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch035`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CoverCertificatefdb2e5aa72
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLL (childLL (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificatefdb2e5aa72
+
+namespace GerverSofa.PartE.CoverCertificatefeab482a88
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLL (childLL (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell0000 : AngleCell :=
+  childLL cell000
+
+private abbrev cell0001 : AngleCell :=
+  childLH cell000
+
+private abbrev cell0002 : AngleCell :=
+  childHL cell000
+
+private abbrev cell0003 : AngleCell :=
+  childHH cell000
+
+private abbrev cell0010 : AngleCell :=
+  childLL cell001
+
+private abbrev cell0011 : AngleCell :=
+  childLH cell001
+
+private abbrev cell0012 : AngleCell :=
+  childHL cell001
+
+private abbrev cell0013 : AngleCell :=
+  childHH cell001
+
+private abbrev cell0100 : AngleCell :=
+  childLL cell010
+
+private abbrev cell0101 : AngleCell :=
+  childLH cell010
+
+private abbrev cell0102 : AngleCell :=
+  childHL cell010
+
+private abbrev cell0103 : AngleCell :=
+  childHH cell010
+
+private abbrev cell0110 : AngleCell :=
+  childLL cell011
+
+private abbrev cell0111 : AngleCell :=
+  childLH cell011
+
+private abbrev cell0112 : AngleCell :=
+  childHL cell011
+
+private abbrev cell0113 : AngleCell :=
+  childHH cell011
+
+private abbrev cell1000 : AngleCell :=
+  childLL cell100
+
+private abbrev cell1001 : AngleCell :=
+  childLH cell100
+
+private abbrev cell1002 : AngleCell :=
+  childHL cell100
+
+private abbrev cell1003 : AngleCell :=
+  childHH cell100
+
+private abbrev cell1010 : AngleCell :=
+  childLL cell101
+
+private abbrev cell1011 : AngleCell :=
+  childLH cell101
+
+private abbrev cell1012 : AngleCell :=
+  childHL cell101
+
+private abbrev cell1013 : AngleCell :=
+  childHH cell101
+
+private abbrev cell1100 : AngleCell :=
+  childLL cell110
+
+private abbrev cell1101 : AngleCell :=
+  childLH cell110
+
+private abbrev cell1102 : AngleCell :=
+  childHL cell110
+
+private abbrev cell1103 : AngleCell :=
+  childHH cell110
+
+private abbrev cell1110 : AngleCell :=
+  childLL cell111
+
+private abbrev cell1111 : AngleCell :=
+  childLH cell111
+
+private abbrev cell1112 : AngleCell :=
+  childHL cell111
+
+private abbrev cell1113 : AngleCell :=
+  childHH cell111
+
+end GerverSofa.PartE.CoverCertificatefeab482a88
+
+namespace GerverSofa.PartE.CoverCertificatebf967458da
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childLL (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatebf967458da
+
+namespace GerverSofa.PartE.CoverCertificate0ae56857be
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childLL (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate0ae56857be
+
+namespace GerverSofa.PartE.CoverCertificate2ea7c8ddef
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLH (childLL (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell1000 : AngleCell :=
+  childLL cell100
+
+private abbrev cell1001 : AngleCell :=
+  childLH cell100
+
+private abbrev cell1002 : AngleCell :=
+  childHL cell100
+
+private abbrev cell1003 : AngleCell :=
+  childHH cell100
+
+private abbrev cell1010 : AngleCell :=
+  childLL cell101
+
+private abbrev cell1011 : AngleCell :=
+  childLH cell101
+
+private abbrev cell1012 : AngleCell :=
+  childHL cell101
+
+private abbrev cell1013 : AngleCell :=
+  childHH cell101
+
+private abbrev cell1100 : AngleCell :=
+  childLL cell110
+
+private abbrev cell1101 : AngleCell :=
+  childLH cell110
+
+private abbrev cell1102 : AngleCell :=
+  childHL cell110
+
+private abbrev cell1103 : AngleCell :=
+  childHH cell110
+
+private abbrev cell1110 : AngleCell :=
+  childLL cell111
+
+private abbrev cell1111 : AngleCell :=
+  childLH cell111
+
+private abbrev cell1112 : AngleCell :=
+  childHL cell111
+
+private abbrev cell1113 : AngleCell :=
+  childHH cell111
+
+end GerverSofa.PartE.CoverCertificate2ea7c8ddef
+
+namespace GerverSofa.PartE.CoverCertificate75cf82dc9c
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLH (childLL (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell0000 : AngleCell :=
+  childLL cell000
+
+private abbrev cell0001 : AngleCell :=
+  childLH cell000
+
+private abbrev cell0002 : AngleCell :=
+  childHL cell000
+
+private abbrev cell0003 : AngleCell :=
+  childHH cell000
+
+private abbrev cell0010 : AngleCell :=
+  childLL cell001
+
+private abbrev cell0011 : AngleCell :=
+  childLH cell001
+
+private abbrev cell0012 : AngleCell :=
+  childHL cell001
+
+private abbrev cell0013 : AngleCell :=
+  childHH cell001
+
+private abbrev cell0100 : AngleCell :=
+  childLL cell010
+
+private abbrev cell0101 : AngleCell :=
+  childLH cell010
+
+private abbrev cell0102 : AngleCell :=
+  childHL cell010
+
+private abbrev cell0103 : AngleCell :=
+  childHH cell010
+
+private abbrev cell0110 : AngleCell :=
+  childLL cell011
+
+private abbrev cell0111 : AngleCell :=
+  childLH cell011
+
+private abbrev cell0112 : AngleCell :=
+  childHL cell011
+
+private abbrev cell0113 : AngleCell :=
+  childHH cell011
+
+private abbrev cell1000 : AngleCell :=
+  childLL cell100
+
+private abbrev cell1001 : AngleCell :=
+  childLH cell100
+
+private abbrev cell1002 : AngleCell :=
+  childHL cell100
+
+private abbrev cell1003 : AngleCell :=
+  childHH cell100
+
+private abbrev cell1010 : AngleCell :=
+  childLL cell101
+
+private abbrev cell1011 : AngleCell :=
+  childLH cell101
+
+private abbrev cell1012 : AngleCell :=
+  childHL cell101
+
+private abbrev cell1013 : AngleCell :=
+  childHH cell101
+
+private abbrev cell1100 : AngleCell :=
+  childLL cell110
+
+private abbrev cell1101 : AngleCell :=
+  childLH cell110
+
+private abbrev cell1102 : AngleCell :=
+  childHL cell110
+
+private abbrev cell1103 : AngleCell :=
+  childHH cell110
+
+end GerverSofa.PartE.CoverCertificate75cf82dc9c
+
+namespace GerverSofa.PartE.CoverCertificate94c7306624
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childLL (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate94c7306624
+
+namespace GerverSofa.PartE.CoverCertificatefc8969e3d2
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childLL (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatefc8969e3d2
+
+namespace GerverSofa.PartE.CoverCertificate0773e6576d
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLH (childLL (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificate0773e6576d
+
+namespace GerverSofa.PartE.CoverCertificate6cd163be74
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLH (childLL (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificate6cd163be74
+
+namespace GerverSofa.PartE.CoverCertificate105a276d33
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLH (childLL (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate105a276d33
+
+namespace GerverSofa.PartE.CoverCertificate7b7ada8a92
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLH (childLL (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate7b7ada8a92
+
+namespace GerverSofa.PartE.CoverCertificate216e1a86b9
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childLL (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate216e1a86b9
+
+namespace GerverSofa.PartE.CoverCertificatecbce86617c
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childLL (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatecbce86617c
+
+namespace GerverSofa.PartE.CoverCertificate5b9a8c2094
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childLH (childLL (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate5b9a8c2094
+
+namespace GerverSofa.PartE.CoverCertificatee500b98dc1
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childLH (childLL (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatee500b98dc1
+
+namespace GerverSofa.PartE.CoverCertificatec4a9c16244
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLL (childLH (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificatec4a9c16244
+
+namespace GerverSofa.PartE.CoverCertificated58785d59b
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLL (childLH (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+end GerverSofa.PartE.CoverCertificated58785d59b
+
+namespace GerverSofa.PartE.CoverCertificate770e3357d2
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childLH (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate770e3357d2
+
+namespace GerverSofa.PartE.CoverCertificate4042eb5094
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childLH (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate4042eb5094
+
+namespace GerverSofa.PartE.CoverCertificate9283dd1f18
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLL (childLH (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificate9283dd1f18
+
+namespace GerverSofa.PartE.CoverCertificate5607aeacca
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLL (childLH (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+end GerverSofa.PartE.CoverCertificate5607aeacca
+
+namespace GerverSofa.PartE.CoverCertificate093986cbf5
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childLH (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate093986cbf5
+
+namespace GerverSofa.PartE.CoverCertificate4478c1ce2d
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childLH (childLL (childLH (childLL (childLH (childLH (childLL
+    (childLH (childLH (e24PhiAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate4478c1ce2d
+
+namespace GerverSofa.PartE.CoverCertificate52d1f78193
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childLL (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate52d1f78193
+
+namespace GerverSofa.PartE.CoverCertificate263f6f95b0
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childLL (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate263f6f95b0
+
+namespace GerverSofa.PartE.CoverCertificatefad4f5904e
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLH (childLL (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell0000 : AngleCell :=
+  childLL cell000
+
+private abbrev cell0001 : AngleCell :=
+  childLH cell000
+
+private abbrev cell0002 : AngleCell :=
+  childHL cell000
+
+private abbrev cell0003 : AngleCell :=
+  childHH cell000
+
+end GerverSofa.PartE.CoverCertificatefad4f5904e
+
+namespace GerverSofa.PartE.CoverCertificatec88446cc5e
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLH (childLL (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificatec88446cc5e
+
+namespace GerverSofa.PartE.CoverCertificate8053b84bb6
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLH (childLL (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate8053b84bb6
+
+namespace GerverSofa.PartE.CoverCertificatea83bf6868e
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLH (childLL (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatea83bf6868e
+
+namespace GerverSofa.PartE.CoverCertificatec458632307
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childLH (childLL (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatec458632307
+
+namespace GerverSofa.PartE.CoverCertificate5ac90060dc
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childLH (childLL (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate5ac90060dc
+
+namespace GerverSofa.PartE.CoverCertificatee2e2523660
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childLL (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatee2e2523660
+
+namespace GerverSofa.PartE.CoverCertificate2c610dcf32
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childLL (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate2c610dcf32
+
+namespace GerverSofa.PartE.CoverCertificate7a43bce112
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLL (childLH (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificate7a43bce112
+
+namespace GerverSofa.PartE.CertificateCells9e0a1773d8
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells9e0a1773d8
+
+namespace GerverSofa.PartE.CertificateCells5867c25b03
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells5867c25b03
+
+namespace GerverSofa.PartE.CertificateCellsca01649193
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsca01649193
+
+namespace GerverSofa.PartE.CertificateCells469c6912b9
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells469c6912b9
+
+namespace GerverSofa.PartE.CertificateCellscf0a854a36
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110100100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110100100 : AngleCell :=
+  childLL (childLL (childLH (childLL phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCellscf0a854a36
+
+namespace GerverSofa.PartE.CertificateCellseb0fd3147a
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110100101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110100101 : AngleCell :=
+  childLH (childLL (childLH (childLL phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCellseb0fd3147a
+
+namespace GerverSofa.PartE.CertificateCellsa5ae349f6a
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110100102` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110100102 : AngleCell :=
+  childHL (childLL (childLH (childLL phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCellsa5ae349f6a
+
+namespace GerverSofa.PartE.CertificateCellsed6c91cc2c
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110100103` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110100103 : AngleCell :=
+  childHH (childLL (childLH (childLL phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCellsed6c91cc2c
+
+namespace GerverSofa.PartE.CertificateCellsd58d6f2902
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110100110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110100110 : AngleCell :=
+  childLL (childLH (childLH (childLL phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCellsd58d6f2902
+
+namespace GerverSofa.PartE.CertificateCellsaa730af275
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110100111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110100111 : AngleCell :=
+  childLH (childLH (childLH (childLL phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCellsaa730af275
+
+namespace GerverSofa.PartE.CertificateCellse47b06e09b
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110100112` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110100112 : AngleCell :=
+  childHL (childLH (childLH (childLL phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCellse47b06e09b
+
+namespace GerverSofa.PartE.CertificateCells6d37377041
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110100113` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110100113 : AngleCell :=
+  childHH (childLH (childLH (childLL phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCells6d37377041
+
+namespace GerverSofa.PartE.CertificateCellse508f838c4
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellse508f838c4
+
+namespace GerverSofa.PartE.CertificateCells1cda533a6e
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells1cda533a6e
+
+namespace GerverSofa.PartE.CertificateCells3babcd7448
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells3babcd7448
+
+namespace GerverSofa.PartE.CertificateCellsbbdc1f9510
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsbbdc1f9510
+
+namespace GerverSofa.PartE.CertificateCells44a650755e
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110101000` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110101000 : AngleCell :=
+  childLL (childLL (childLL (childLH phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCells44a650755e
+
+namespace GerverSofa.PartE.CertificateCells9dff8fd68c
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110101001` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110101001 : AngleCell :=
+  childLH (childLL (childLL (childLH phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCells9dff8fd68c
+
+namespace GerverSofa.PartE.CertificateCells0eb9c7dd4e
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110101002` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110101002 : AngleCell :=
+  childHL (childLL (childLL (childLH phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCells0eb9c7dd4e
+
+namespace GerverSofa.PartE.CertificateCellsa1f8281244
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110101003` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110101003 : AngleCell :=
+  childHH (childLL (childLL (childLH phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCellsa1f8281244
+
+namespace GerverSofa.PartE.CertificateCellsa31bfb0e48
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110101010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110101010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCellsa31bfb0e48
+
+namespace GerverSofa.PartE.CertificateCellsba54477f25
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110101011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110101011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCellsba54477f25
+
+namespace GerverSofa.PartE.CertificateCells40378f08af
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110101012` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110101012 : AngleCell :=
+  childHL (childLH (childLL (childLH phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCells40378f08af
+
+namespace GerverSofa.PartE.CertificateCellsbef31c9887
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+/-- Subcell `110110101013` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell110110101013 : AngleCell :=
+  childHH (childLH (childLL (childLH phiAboveCell11011010)))
+
+end GerverSofa.PartE.CertificateCellsbef31c9887
+
+namespace GerverSofa.PartE.CertificateCells6837eba069
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells6837eba069
+
+namespace GerverSofa.PartE.CertificateCells21829695a9
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells21829695a9
+
+namespace GerverSofa.PartE.CertificateCellsd170a2bed6
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsd170a2bed6
+
+namespace GerverSofa.PartE.CertificateCells3fed941c0c
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells3fed941c0c
+
+namespace GerverSofa.PartE.CertificateCellsb780467ddc
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsb780467ddc
+
+namespace GerverSofa.PartE.CertificateCellse5752d8306
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellse5752d8306
+
+namespace GerverSofa.PartE.CertificateCells18ee7077e5
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells18ee7077e5
+
+namespace GerverSofa.PartE.CertificateCells481c5d2cb5
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells481c5d2cb5
+
+namespace GerverSofa.PartE.CertificateCellsc1762ca3ac
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsc1762ca3ac
+
+namespace GerverSofa.PartE.CertificateCells105558b317
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells105558b317
+
+namespace GerverSofa.PartE.CertificateCells411895cea3
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells411895cea3
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.PhiAbove.Leaf00025`.
+* `KernelOnly.PartE.PhiAbove.Leaf00026`.
+* `KernelOnly.PartE.PhiAbove.Leaf00027`.
+* `KernelOnly.PartE.PhiAbove.Leaf00028`.
+* `KernelOnly.PartE.PhiAbove.Leaf00032`.
+* `KernelOnly.PartE.PhiAbove.Leaf00033`.
+* `KernelOnly.PartE.PhiAbove.Leaf00034`.
+* `KernelOnly.PartE.PhiAbove.Leaf00035`.
+* `KernelOnly.PartE.PhiAbove.Leaf00038`.
+* `KernelOnly.PartE.PhiAbove.Leaf00039`.
+* `KernelOnly.PartE.PhiAbove.Leaf00040`.
+* `KernelOnly.PartE.PhiAbove.Leaf00041`.
+* `KernelOnly.PartE.PhiAbove.Leaf00043`.
+* `KernelOnly.PartE.PhiAbove.Leaf00044`.
+* `KernelOnly.PartE.PhiAbove.Leaf00046`.
+* `KernelOnly.PartE.PhiAbove.Leaf00047`.
+* `KernelOnly.PartE.PhiAbove.Leaf00052`.
+* `KernelOnly.PartE.PhiAbove.Leaf00053`.
+* `KernelOnly.PartE.PhiAbove.Leaf00054`.
+* `KernelOnly.PartE.PhiAbove.Leaf00055`.
+* `KernelOnly.PartE.PhiAbove.Leaf00058`.
+* `KernelOnly.PartE.PhiAbove.Leaf00059`.
+* `KernelOnly.PartE.PhiAbove.Leaf00060`.
+* `KernelOnly.PartE.PhiAbove.Leaf00061`.
+* `KernelOnly.PartE.PhiAbove.Leaf00063`.
+* `KernelOnly.PartE.PhiAbove.Leaf00064`.
+* `KernelOnly.PartE.PhiAbove.Leaf00067`.
+* `KernelOnly.PartE.PhiAbove.Leaf00068`.
+* `KernelOnly.PartE.PhiAbove.Leaf00069`.
+* `KernelOnly.PartE.PhiAbove.Leaf00070`.
+* `KernelOnly.PartE.PhiAbove.Leaf00072`.
+* `KernelOnly.PartE.PhiAbove.Leaf00073`.
+* `KernelOnly.PartE.PhiAbove.Leaf00075`.
+* `KernelOnly.PartE.PhiAbove.Leaf00076`.
+* `KernelOnly.PartE.PhiAbove.Leaf00081`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c0_c0_c0_5_00025
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9e0a1773d8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9e0a1773d8
+
+open CertificateCells9e0a1773d8
+namespace CoverCertificatefdb2e5aa72
+
+private theorem checked000 : adaptiveCoverCheck 2 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 2 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 2 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 2 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 2 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 2 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 2 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 2 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 2 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 2 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 2 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 2 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 2 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 2 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 2 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 2 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefdb2e5aa72
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c0_c0 :
+    adaptiveCoverCheck 5 (childLL (childLL (childLL phiAboveCell11011010))) = true := by
+  exact CoverCertificatefdb2e5aa72.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c0_c0_c1_5_00026
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5867c25b03
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5867c25b03
+
+open CertificateCells5867c25b03
+namespace CoverCertificatefeab482a88
+
+private theorem checked0000 : adaptiveCoverCheck 1 cell0000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0000 (by decide +kernel)
+
+private theorem checked0001 : adaptiveCoverCheck 1 cell0001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0001 (by decide +kernel)
+
+private theorem checked0002 : adaptiveCoverCheck 1 cell0002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0002 (by decide +kernel)
+
+private theorem checked0003 : adaptiveCoverCheck 1 cell0003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0003 (by decide +kernel)
+
+private theorem checked0010 : adaptiveCoverCheck 1 cell0010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0010 (by decide +kernel)
+
+private theorem checked0011 : adaptiveCoverCheck 1 cell0011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0011 (by decide +kernel)
+
+private theorem checked0012 : adaptiveCoverCheck 1 cell0012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0012 (by decide +kernel)
+
+private theorem checked0013 : adaptiveCoverCheck 1 cell0013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0013 (by decide +kernel)
+
+private theorem checked0100 : adaptiveCoverCheck 1 cell0100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0100 (by decide +kernel)
+
+private theorem checked0101 : adaptiveCoverCheck 1 cell0101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0101 (by decide +kernel)
+
+private theorem checked0102 : adaptiveCoverCheck 1 cell0102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0102 (by decide +kernel)
+
+private theorem checked0103 : adaptiveCoverCheck 1 cell0103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0103 (by decide +kernel)
+
+private theorem checked0110 : adaptiveCoverCheck 1 cell0110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0110 (by decide +kernel)
+
+private theorem checked0111 : adaptiveCoverCheck 1 cell0111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0111 (by decide +kernel)
+
+private theorem checked0112 : adaptiveCoverCheck 1 cell0112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0112 (by decide +kernel)
+
+private theorem checked0113 : adaptiveCoverCheck 1 cell0113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0113 (by decide +kernel)
+
+private theorem checked1000 : adaptiveCoverCheck 1 cell1000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1000 (by decide +kernel)
+
+private theorem checked1001 : adaptiveCoverCheck 1 cell1001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1001 (by decide +kernel)
+
+private theorem checked1002 : adaptiveCoverCheck 1 cell1002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1002 (by decide +kernel)
+
+private theorem checked1003 : adaptiveCoverCheck 1 cell1003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1003 (by decide +kernel)
+
+private theorem checked1010 : adaptiveCoverCheck 1 cell1010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1010 (by decide +kernel)
+
+private theorem checked1011 : adaptiveCoverCheck 1 cell1011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1011 (by decide +kernel)
+
+private theorem checked1012 : adaptiveCoverCheck 1 cell1012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1012 (by decide +kernel)
+
+private theorem checked1013 : adaptiveCoverCheck 1 cell1013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1013 (by decide +kernel)
+
+private theorem checked1100 : adaptiveCoverCheck 1 cell1100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1100 (by decide +kernel)
+
+private theorem checked1101 : adaptiveCoverCheck 1 cell1101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1101 (by decide +kernel)
+
+private theorem checked1102 : adaptiveCoverCheck 1 cell1102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1102 (by decide +kernel)
+
+private theorem checked1103 : adaptiveCoverCheck 1 cell1103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1103 (by decide +kernel)
+
+private theorem checked1110 : adaptiveCoverCheck 1 cell1110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1110 (by decide +kernel)
+
+private theorem checked1111 : adaptiveCoverCheck 1 cell1111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1111 (by decide +kernel)
+
+private theorem checked1112 : adaptiveCoverCheck 1 cell1112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1112 (by decide +kernel)
+
+private theorem checked1113 : adaptiveCoverCheck 1 cell1113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1113 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 2 cell000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell000
+    checked0000 checked0001 checked0002 checked0003
+
+private theorem checked001 : adaptiveCoverCheck 2 cell001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell001
+    checked0010 checked0011 checked0012 checked0013
+
+private theorem checked002 : adaptiveCoverCheck 2 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 2 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 2 cell010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell010
+    checked0100 checked0101 checked0102 checked0103
+
+private theorem checked011 : adaptiveCoverCheck 2 cell011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell011
+    checked0110 checked0111 checked0112 checked0113
+
+private theorem checked012 : adaptiveCoverCheck 2 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 2 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 2 cell100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell100
+    checked1000 checked1001 checked1002 checked1003
+
+private theorem checked101 : adaptiveCoverCheck 2 cell101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell101
+    checked1010 checked1011 checked1012 checked1013
+
+private theorem checked102 : adaptiveCoverCheck 2 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 2 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 2 cell110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell110
+    checked1100 checked1101 checked1102 checked1103
+
+private theorem checked111 : adaptiveCoverCheck 2 cell111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell111
+    checked1110 checked1111 checked1112 checked1113
+
+private theorem checked112 : adaptiveCoverCheck 2 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 2 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefeab482a88
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c0_c1 :
+    adaptiveCoverCheck 5 (childLH (childLL (childLL phiAboveCell11011010))) = true := by
+  exact CoverCertificatefeab482a88.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c0_c0_c2_5_00027
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsca01649193
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsca01649193
+
+open CertificateCellsca01649193
+namespace CoverCertificatebf967458da
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatebf967458da
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c0_c2 :
+    adaptiveCoverCheck 5 (childHL (childLL (childLL phiAboveCell11011010))) = true := by
+  exact CoverCertificatebf967458da.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c0_c0_c3_5_00028
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells469c6912b9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells469c6912b9
+
+open CertificateCells469c6912b9
+namespace CoverCertificate0ae56857be
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0ae56857be
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c0_c3 :
+    adaptiveCoverCheck 5 (childHH (childLL (childLL phiAboveCell11011010))) = true := by
+  exact CoverCertificate0ae56857be.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c0_c1_c0_c0_4_00032
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellscf0a854a36
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellscf0a854a36
+
+open CertificateCellscf0a854a36
+namespace CoverCertificate2ea7c8ddef
+
+private theorem checked1000 : adaptiveCoverCheck 0 cell1000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1000 (by decide +kernel)
+
+private theorem checked1001 : adaptiveCoverCheck 0 cell1001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1001 (by decide +kernel)
+
+private theorem checked1002 : adaptiveCoverCheck 0 cell1002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1002 (by decide +kernel)
+
+private theorem checked1003 : adaptiveCoverCheck 0 cell1003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1003 (by decide +kernel)
+
+private theorem checked1010 : adaptiveCoverCheck 0 cell1010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1010 (by decide +kernel)
+
+private theorem checked1011 : adaptiveCoverCheck 0 cell1011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1011 (by decide +kernel)
+
+private theorem checked1012 : adaptiveCoverCheck 0 cell1012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1012 (by decide +kernel)
+
+private theorem checked1013 : adaptiveCoverCheck 0 cell1013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1013 (by decide +kernel)
+
+private theorem checked1100 : adaptiveCoverCheck 0 cell1100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1100 (by decide +kernel)
+
+private theorem checked1101 : adaptiveCoverCheck 0 cell1101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1101 (by decide +kernel)
+
+private theorem checked1102 : adaptiveCoverCheck 0 cell1102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1102 (by decide +kernel)
+
+private theorem checked1103 : adaptiveCoverCheck 0 cell1103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1103 (by decide +kernel)
+
+private theorem checked1110 : adaptiveCoverCheck 0 cell1110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1110 (by decide +kernel)
+
+private theorem checked1111 : adaptiveCoverCheck 0 cell1111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1111 (by decide +kernel)
+
+private theorem checked1112 : adaptiveCoverCheck 0 cell1112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1112 (by decide +kernel)
+
+private theorem checked1113 : adaptiveCoverCheck 0 cell1113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1113 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell100
+    checked1000 checked1001 checked1002 checked1003
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell101
+    checked1010 checked1011 checked1012 checked1013
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell110
+    checked1100 checked1101 checked1102 checked1103
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell111
+    checked1110 checked1111 checked1112 checked1113
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate2ea7c8ddef
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c1_c0_c0 :
+    adaptiveCoverCheck 4 phiAboveCell110110100100 = true := by
+  exact CoverCertificate2ea7c8ddef.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c0_c1_c0_c1_4_00033
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellseb0fd3147a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellseb0fd3147a
+
+open CertificateCellseb0fd3147a
+namespace CoverCertificate75cf82dc9c
+
+private theorem checked0000 : adaptiveCoverCheck 0 cell0000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0000 (by decide +kernel)
+
+private theorem checked0001 : adaptiveCoverCheck 0 cell0001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0001 (by decide +kernel)
+
+private theorem checked0002 : adaptiveCoverCheck 0 cell0002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0002 (by decide +kernel)
+
+private theorem checked0003 : adaptiveCoverCheck 0 cell0003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0003 (by decide +kernel)
+
+private theorem checked0010 : adaptiveCoverCheck 0 cell0010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0010 (by decide +kernel)
+
+private theorem checked0011 : adaptiveCoverCheck 0 cell0011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0011 (by decide +kernel)
+
+private theorem checked0012 : adaptiveCoverCheck 0 cell0012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0012 (by decide +kernel)
+
+private theorem checked0013 : adaptiveCoverCheck 0 cell0013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0013 (by decide +kernel)
+
+private theorem checked0100 : adaptiveCoverCheck 0 cell0100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0100 (by decide +kernel)
+
+private theorem checked0101 : adaptiveCoverCheck 0 cell0101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0101 (by decide +kernel)
+
+private theorem checked0102 : adaptiveCoverCheck 0 cell0102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0102 (by decide +kernel)
+
+private theorem checked0103 : adaptiveCoverCheck 0 cell0103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0103 (by decide +kernel)
+
+private theorem checked0110 : adaptiveCoverCheck 0 cell0110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0110 (by decide +kernel)
+
+private theorem checked0111 : adaptiveCoverCheck 0 cell0111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0111 (by decide +kernel)
+
+private theorem checked0112 : adaptiveCoverCheck 0 cell0112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0112 (by decide +kernel)
+
+private theorem checked0113 : adaptiveCoverCheck 0 cell0113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0113 (by decide +kernel)
+
+private theorem checked1000 : adaptiveCoverCheck 0 cell1000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1000 (by decide +kernel)
+
+private theorem checked1001 : adaptiveCoverCheck 0 cell1001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1001 (by decide +kernel)
+
+private theorem checked1002 : adaptiveCoverCheck 0 cell1002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1002 (by decide +kernel)
+
+private theorem checked1003 : adaptiveCoverCheck 0 cell1003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1003 (by decide +kernel)
+
+private theorem checked1010 : adaptiveCoverCheck 0 cell1010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1010 (by decide +kernel)
+
+private theorem checked1011 : adaptiveCoverCheck 0 cell1011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1011 (by decide +kernel)
+
+private theorem checked1012 : adaptiveCoverCheck 0 cell1012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1012 (by decide +kernel)
+
+private theorem checked1013 : adaptiveCoverCheck 0 cell1013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1013 (by decide +kernel)
+
+private theorem checked1100 : adaptiveCoverCheck 0 cell1100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1100 (by decide +kernel)
+
+private theorem checked1101 : adaptiveCoverCheck 0 cell1101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1101 (by decide +kernel)
+
+private theorem checked1102 : adaptiveCoverCheck 0 cell1102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1102 (by decide +kernel)
+
+private theorem checked1103 : adaptiveCoverCheck 0 cell1103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1103 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell000
+    checked0000 checked0001 checked0002 checked0003
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell001
+    checked0010 checked0011 checked0012 checked0013
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell010
+    checked0100 checked0101 checked0102 checked0103
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell011
+    checked0110 checked0111 checked0112 checked0113
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell100
+    checked1000 checked1001 checked1002 checked1003
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell101
+    checked1010 checked1011 checked1012 checked1013
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell110
+    checked1100 checked1101 checked1102 checked1103
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate75cf82dc9c
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c1_c0_c1 :
+    adaptiveCoverCheck 4 phiAboveCell110110100101 = true := by
+  exact CoverCertificate75cf82dc9c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c0_c1_c0_c2_4_00034
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa5ae349f6a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa5ae349f6a
+
+open CertificateCellsa5ae349f6a
+namespace CoverCertificate94c7306624
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate94c7306624
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c1_c0_c2 :
+    adaptiveCoverCheck 4 phiAboveCell110110100102 = true := by
+  exact CoverCertificate94c7306624.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c0_c1_c0_c3_4_00035
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsed6c91cc2c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsed6c91cc2c
+
+open CertificateCellsed6c91cc2c
+namespace CoverCertificatefc8969e3d2
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefc8969e3d2
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c1_c0_c3 :
+    adaptiveCoverCheck 4 phiAboveCell110110100103 = true := by
+  exact CoverCertificatefc8969e3d2.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c0_c1_c1_c0_4_00038
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd58d6f2902
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd58d6f2902
+
+open CertificateCellsd58d6f2902
+namespace CoverCertificate0773e6576d
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0773e6576d
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c1_c1_c0 :
+    adaptiveCoverCheck 4 phiAboveCell110110100110 = true := by
+  exact CoverCertificate0773e6576d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c0_c1_c1_c1_4_00039
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsaa730af275
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsaa730af275
+
+open CertificateCellsaa730af275
+namespace CoverCertificate6cd163be74
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate6cd163be74
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c1_c1_c1 :
+    adaptiveCoverCheck 4 phiAboveCell110110100111 = true := by
+  exact CoverCertificate6cd163be74.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c0_c1_c1_c2_4_00040
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse47b06e09b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse47b06e09b
+
+open CertificateCellse47b06e09b
+namespace CoverCertificate105a276d33
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate105a276d33
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c1_c1_c2 :
+    adaptiveCoverCheck 4 phiAboveCell110110100112 = true := by
+  exact CoverCertificate105a276d33.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c0_c1_c1_c3_4_00041
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6d37377041
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6d37377041
+
+open CertificateCells6d37377041
+namespace CoverCertificate7b7ada8a92
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate7b7ada8a92
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c1_c1_c3 :
+    adaptiveCoverCheck 4 phiAboveCell110110100113 = true := by
+  exact CoverCertificate7b7ada8a92.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c0_c1_c2_5_00043
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse508f838c4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse508f838c4
+
+open CertificateCellse508f838c4
+namespace CoverCertificate216e1a86b9
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate216e1a86b9
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c1_c2 :
+    adaptiveCoverCheck 5 (childHL (childLH (childLL phiAboveCell11011010))) = true := by
+  exact CoverCertificate216e1a86b9.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c0_c1_c3_5_00044
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1cda533a6e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1cda533a6e
+
+open CertificateCells1cda533a6e
+namespace CoverCertificatecbce86617c
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatecbce86617c
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c1_c3 :
+    adaptiveCoverCheck 5 (childHH (childLH (childLL phiAboveCell11011010))) = true := by
+  exact CoverCertificatecbce86617c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c0_c2_6_00046
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3babcd7448
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3babcd7448
+
+open CertificateCells3babcd7448
+namespace CoverCertificate5b9a8c2094
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5b9a8c2094
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c2 :
+    adaptiveCoverCheck 6 (childHL (childLL phiAboveCell11011010)) = true := by
+  exact CoverCertificate5b9a8c2094.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c0_c3_6_00047
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsbbdc1f9510
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsbbdc1f9510
+
+open CertificateCellsbbdc1f9510
+namespace CoverCertificatee500b98dc1
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee500b98dc1
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c3 :
+    adaptiveCoverCheck 6 (childHH (childLL phiAboveCell11011010)) = true := by
+  exact CoverCertificatee500b98dc1.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c1_c0_c0_c0_4_00052
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells44a650755e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells44a650755e
+
+open CertificateCells44a650755e
+namespace CoverCertificatec4a9c16244
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec4a9c16244
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c0_c0_c0 :
+    adaptiveCoverCheck 4 phiAboveCell110110101000 = true := by
+  exact CoverCertificatec4a9c16244.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c1_c0_c0_c1_4_00053
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9dff8fd68c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9dff8fd68c
+
+open CertificateCells9dff8fd68c
+namespace CoverCertificated58785d59b
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated58785d59b
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c0_c0_c1 :
+    adaptiveCoverCheck 4 phiAboveCell110110101001 = true := by
+  exact CoverCertificated58785d59b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c1_c0_c0_c2_4_00054
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0eb9c7dd4e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0eb9c7dd4e
+
+open CertificateCells0eb9c7dd4e
+namespace CoverCertificate770e3357d2
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate770e3357d2
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c0_c0_c2 :
+    adaptiveCoverCheck 4 phiAboveCell110110101002 = true := by
+  exact CoverCertificate770e3357d2.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c1_c0_c0_c3_4_00055
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa1f8281244
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa1f8281244
+
+open CertificateCellsa1f8281244
+namespace CoverCertificate4042eb5094
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4042eb5094
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c0_c0_c3 :
+    adaptiveCoverCheck 4 phiAboveCell110110101003 = true := by
+  exact CoverCertificate4042eb5094.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c1_c0_c1_c0_4_00058
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa31bfb0e48
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa31bfb0e48
+
+open CertificateCellsa31bfb0e48
+namespace CoverCertificate9283dd1f18
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 1 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 1 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 1 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 1 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate9283dd1f18
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c0_c1_c0 :
+    adaptiveCoverCheck 4 phiAboveCell110110101010 = true := by
+  exact CoverCertificate9283dd1f18.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c1_c0_c1_c1_4_00059
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsba54477f25
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsba54477f25
+
+open CertificateCellsba54477f25
+namespace CoverCertificate5607aeacca
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 1 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 1 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 1 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 1 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell203 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5607aeacca
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c0_c1_c1 :
+    adaptiveCoverCheck 4 phiAboveCell110110101011 = true := by
+  exact CoverCertificate5607aeacca.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c1_c0_c1_c2_4_00060
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells40378f08af
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells40378f08af
+
+open CertificateCells40378f08af
+namespace CoverCertificate093986cbf5
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate093986cbf5
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c0_c1_c2 :
+    adaptiveCoverCheck 4 phiAboveCell110110101012 = true := by
+  exact CoverCertificate093986cbf5.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above
+Leaf1101101_c0_c1_c0_c1_c3_4_00061
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsbef31c9887
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsbef31c9887
+
+open CertificateCellsbef31c9887
+namespace CoverCertificate4478c1ce2d
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4478c1ce2d
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c0_c1_c3 :
+    adaptiveCoverCheck 4 phiAboveCell110110101013 = true := by
+  exact CoverCertificate4478c1ce2d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c1_c0_c2_5_00063
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6837eba069
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6837eba069
+
+open CertificateCells6837eba069
+namespace CoverCertificate52d1f78193
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate52d1f78193
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c0_c2 :
+    adaptiveCoverCheck 5 (childHL (childLL (childLH phiAboveCell11011010))) = true := by
+  exact CoverCertificate52d1f78193.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c1_c0_c3_5_00064
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells21829695a9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells21829695a9
+
+open CertificateCells21829695a9
+namespace CoverCertificate263f6f95b0
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate263f6f95b0
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c0_c3 :
+    adaptiveCoverCheck 5 (childHH (childLL (childLH phiAboveCell11011010))) = true := by
+  exact CoverCertificate263f6f95b0.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c1_c1_c0_5_00067
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd170a2bed6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd170a2bed6
+
+open CertificateCellsd170a2bed6
+namespace CoverCertificatefad4f5904e
+
+private theorem checked0000 : adaptiveCoverCheck 1 cell0000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0000 (by decide +kernel)
+
+private theorem checked0001 : adaptiveCoverCheck 1 cell0001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0001 (by decide +kernel)
+
+private theorem checked0002 : adaptiveCoverCheck 1 cell0002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0002 (by decide +kernel)
+
+private theorem checked0003 : adaptiveCoverCheck 1 cell0003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0003 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 2 cell000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell000
+    checked0000 checked0001 checked0002 checked0003
+
+private theorem checked001 : adaptiveCoverCheck 2 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 2 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 2 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 2 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 2 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 2 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 2 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 2 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 2 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 2 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 2 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 2 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 2 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 2 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 2 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 2 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 2 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 2 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 2 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 2 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 2 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 2 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 2 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 2 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 2 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 2 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 2 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 2 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 2 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 2 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 2 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell133 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefad4f5904e
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c1_c0 :
+    adaptiveCoverCheck 5 (childLL (childLH (childLH phiAboveCell11011010))) = true := by
+  exact CoverCertificatefad4f5904e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c1_c1_c1_5_00068
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3fed941c0c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3fed941c0c
+
+open CertificateCells3fed941c0c
+namespace CoverCertificatec88446cc5e
+
+private theorem checked000 : adaptiveCoverCheck 2 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 2 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 2 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 2 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 2 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 2 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 2 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 2 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 2 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 2 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 2 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 2 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 2 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 2 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 2 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 2 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 2 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 2 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 2 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 2 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 2 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 2 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 2 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 2 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 2 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 2 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 2 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 2 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 2 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 2 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 2 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 2 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 2 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 2 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 2 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 2 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 2 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 2 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 2 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 2 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 2 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 2 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 2 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 2 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 2 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 2 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 2 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 2 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec88446cc5e
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c1_c1 :
+    adaptiveCoverCheck 5 (childLH (childLH (childLH phiAboveCell11011010))) = true := by
+  exact CoverCertificatec88446cc5e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c1_c1_c2_5_00069
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb780467ddc
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb780467ddc
+
+open CertificateCellsb780467ddc
+namespace CoverCertificate8053b84bb6
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate8053b84bb6
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c1_c2 :
+    adaptiveCoverCheck 5 (childHL (childLH (childLH phiAboveCell11011010))) = true := by
+  exact CoverCertificate8053b84bb6.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c1_c1_c3_5_00070
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse5752d8306
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse5752d8306
+
+open CertificateCellse5752d8306
+namespace CoverCertificatea83bf6868e
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea83bf6868e
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c1_c3 :
+    adaptiveCoverCheck 5 (childHH (childLH (childLH phiAboveCell11011010))) = true := by
+  exact CoverCertificatea83bf6868e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c1_c2_6_00072
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells18ee7077e5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells18ee7077e5
+
+open CertificateCells18ee7077e5
+namespace CoverCertificatec458632307
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec458632307
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c2 :
+    adaptiveCoverCheck 6 (childHL (childLH phiAboveCell11011010)) = true := by
+  exact CoverCertificatec458632307.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c1_c3_6_00073
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells481c5d2cb5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells481c5d2cb5
+
+open CertificateCells481c5d2cb5
+namespace CoverCertificate5ac90060dc
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5ac90060dc
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c3 :
+    adaptiveCoverCheck 6 (childHH (childLH phiAboveCell11011010)) = true := by
+  exact CoverCertificate5ac90060dc.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c2_7_00075
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc1762ca3ac
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc1762ca3ac
+
+open CertificateCellsc1762ca3ac
+namespace CoverCertificatee2e2523660
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee2e2523660
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c2 :
+    adaptiveCoverCheck 7 (childHL phiAboveCell11011010) = true := by
+  exact CoverCertificatee2e2523660.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c0_c3_7_00076
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells105558b317
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells105558b317
+
+open CertificateCells105558b317
+namespace CoverCertificate2c610dcf32
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate2c610dcf32
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c3 :
+    adaptiveCoverCheck 7 (childHH phiAboveCell11011010) = true := by
+  exact CoverCertificate2c610dcf32.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c0_c0_c0_5_00081
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells411895cea3
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells411895cea3
+
+open CertificateCells411895cea3
+namespace CoverCertificate7a43bce112
+
+private theorem checked000 : adaptiveCoverCheck 2 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 2 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 2 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 2 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 2 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 2 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 2 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 2 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 2 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 2 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 2 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 2 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 2 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 2 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 2 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 2 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 2 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 2 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 2 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 2 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 2 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 2 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 2 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 2 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 2 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 2 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 2 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 2 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 2 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 2 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 2 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 2 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 2 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 2 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 2 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 2 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 2 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 2 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 2 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 2 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 2 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 2 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 2 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 2 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 2 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 2 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 2 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 2 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate7a43bce112
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c0_c0_c0 :
+    adaptiveCoverCheck 5 (childLL (childLL (childLL phiAboveCell11011011))) = true := by
+  exact CoverCertificate7a43bce112.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle023.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle023.lean
new file mode 100644
index 0000000000..d2f714e2ab
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle023.lean
@@ -0,0 +1,9566 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch036`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CoverCertificate33f50162e9
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLL (childLH (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificate33f50162e9
+
+namespace GerverSofa.PartE.CoverCertificatea52e53d66e
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childLH (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatea52e53d66e
+
+namespace GerverSofa.PartE.CoverCertificate68f30819ed
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childLH (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate68f30819ed
+
+namespace GerverSofa.PartE.CoverCertificatef50e05a2d9
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLL (childLH (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificatef50e05a2d9
+
+namespace GerverSofa.PartE.CoverCertificate886cd459b6
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLL (childLH (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+end GerverSofa.PartE.CoverCertificate886cd459b6
+
+namespace GerverSofa.PartE.CoverCertificate6334bf560d
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childLH (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate6334bf560d
+
+namespace GerverSofa.PartE.CoverCertificate40b0f107ea
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childLH (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate40b0f107ea
+
+namespace GerverSofa.PartE.CoverCertificatee77541b394
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childLH (childLL (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatee77541b394
+
+namespace GerverSofa.PartE.CoverCertificate5ffc90202d
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childLH (childLL (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate5ffc90202d
+
+namespace GerverSofa.PartE.CoverCertificatee265233880
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLH (childLH (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+end GerverSofa.PartE.CoverCertificatee265233880
+
+namespace GerverSofa.PartE.CoverCertificatef18591c0b6
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLH (childLH (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatef18591c0b6
+
+namespace GerverSofa.PartE.CoverCertificatef7582d4176
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childLH (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificatef7582d4176
+
+namespace GerverSofa.PartE.CoverCertificate042241a8e3
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childLH (childLH (childLL (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate042241a8e3
+
+namespace GerverSofa.PartE.CoverCertificateb30a576a6c
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLH (childLH (childLL (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificateb30a576a6c
+
+namespace GerverSofa.PartE.CoverCertificatee29f1ec14c
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLH (childLH (childLL (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatee29f1ec14c
+
+namespace GerverSofa.PartE.CoverCertificate7c835f5bcd
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLH (childLH (childLL (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate7c835f5bcd
+
+namespace GerverSofa.PartE.CoverCertificate1847743b04
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLH (childLL (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate1847743b04
+
+namespace GerverSofa.PartE.CoverCertificated22de6b6b3
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLH (childLL (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificated22de6b6b3
+
+namespace GerverSofa.PartE.CoverCertificatec9d844cfa1
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childLH (childLH (childLL (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatec9d844cfa1
+
+namespace GerverSofa.PartE.CoverCertificate0b1bb49c5e
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childLH (childLH (childLL (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate0b1bb49c5e
+
+namespace GerverSofa.PartE.CoverCertificatedcf0833de3
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLL (childLL (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificatedcf0833de3
+
+namespace GerverSofa.PartE.CoverCertificate292a59b81b
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLL (childLL (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate292a59b81b
+
+namespace GerverSofa.PartE.CoverCertificate38c27d4b72
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childLL (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate38c27d4b72
+
+namespace GerverSofa.PartE.CoverCertificate80ac67346e
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childLL (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate80ac67346e
+
+namespace GerverSofa.PartE.CoverCertificate6a3da1aa29
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLH (childLL (childLL (childLH (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate6a3da1aa29
+
+namespace GerverSofa.PartE.CoverCertificatec0de2b5230
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLH (childLL (childLL (childLH (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatec0de2b5230
+
+namespace GerverSofa.PartE.CoverCertificateeb67261010
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childLL (childLL (childLH (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificateeb67261010
+
+namespace GerverSofa.PartE.CoverCertificate4ceb0bf432
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childLL (childLL (childLH (childLH (childLH (childLL (childLH
+    (childLH (e24PhiAboveRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate4ceb0bf432
+
+namespace GerverSofa.PartE.CoverCertificate1407c62e79
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLL (childLL (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificate1407c62e79
+
+namespace GerverSofa.PartE.CoverCertificate3835f13788
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childLL (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate3835f13788
+
+namespace GerverSofa.PartE.CoverCertificate2b398d0240
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childLL (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate2b398d0240
+
+namespace GerverSofa.PartE.CoverCertificate46690a85d6
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate46690a85d6
+
+namespace GerverSofa.PartE.CoverCertificate79d55c50a4
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate79d55c50a4
+
+namespace GerverSofa.PartE.CoverCertificate3882c70981
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLL (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell0020 : AngleCell :=
+  childLL cell002
+
+private abbrev cell0021 : AngleCell :=
+  childLH cell002
+
+private abbrev cell0022 : AngleCell :=
+  childHL cell002
+
+private abbrev cell0023 : AngleCell :=
+  childHH cell002
+
+private abbrev cell0030 : AngleCell :=
+  childLL cell003
+
+private abbrev cell0031 : AngleCell :=
+  childLH cell003
+
+private abbrev cell0032 : AngleCell :=
+  childHL cell003
+
+private abbrev cell0033 : AngleCell :=
+  childHH cell003
+
+end GerverSofa.PartE.CoverCertificate3882c70981
+
+namespace GerverSofa.PartE.CoverCertificate749186c43c
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLL (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificate749186c43c
+
+namespace GerverSofa.PartE.CoverCertificate69709d6517
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate69709d6517
+
+namespace GerverSofa.PartE.CoverCertificate9b05e6137c
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate9b05e6137c
+
+namespace GerverSofa.PartE.CoverCertificatea08e6229dd
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childLH (childLH (childLL (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatea08e6229dd
+
+namespace GerverSofa.PartE.CoverCertificate80957e0c30
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childLH (childLH (childLL (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate80957e0c30
+
+namespace GerverSofa.PartE.CoverCertificate6b5d6c54cf
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLH (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificate6b5d6c54cf
+
+namespace GerverSofa.PartE.CoverCertificate5e2d2d50e9
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLL (childLH (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate5e2d2d50e9
+
+namespace GerverSofa.PartE.CoverCertificate4eafbde6a7
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLL (childLH (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate4eafbde6a7
+
+namespace GerverSofa.PartE.CoverCertificateb57436bcb4
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childLH (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificateb57436bcb4
+
+namespace GerverSofa.PartE.CoverCertificate8e406c3a19
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childLH (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate8e406c3a19
+
+namespace GerverSofa.PartE.CoverCertificate11dc9c1c2a
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate11dc9c1c2a
+
+namespace GerverSofa.PartE.CoverCertificate52298353c4
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate52298353c4
+
+namespace GerverSofa.PartE.CertificateCells02f8aca2f2
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells02f8aca2f2
+
+namespace GerverSofa.PartE.CertificateCellsca6ca7b54c
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsca6ca7b54c
+
+namespace GerverSofa.PartE.CertificateCells8161bed319
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells8161bed319
+
+namespace GerverSofa.PartE.CertificateCells5f2ef3defa
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells5f2ef3defa
+
+namespace GerverSofa.PartE.CertificateCellsd4f6865820
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsd4f6865820
+
+namespace GerverSofa.PartE.CertificateCellsaca7f76cfa
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsaca7f76cfa
+
+namespace GerverSofa.PartE.CertificateCellsa7ad4ac263
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsa7ad4ac263
+
+namespace GerverSofa.PartE.CertificateCells53f8862830
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells53f8862830
+
+namespace GerverSofa.PartE.CertificateCellsb6a37b0947
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsb6a37b0947
+
+namespace GerverSofa.PartE.CertificateCellsbc3b77fd62
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsbc3b77fd62
+
+namespace GerverSofa.PartE.CertificateCellsec2a80bcb9
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsec2a80bcb9
+
+namespace GerverSofa.PartE.CertificateCells69b4ed96bb
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells69b4ed96bb
+
+namespace GerverSofa.PartE.CertificateCells81024818f0
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells81024818f0
+
+namespace GerverSofa.PartE.CertificateCellsbaba88fb03
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsbaba88fb03
+
+namespace GerverSofa.PartE.CertificateCells4431add919
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells4431add919
+
+namespace GerverSofa.PartE.CertificateCells0c4c476da0
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells0c4c476da0
+
+namespace GerverSofa.PartE.CertificateCells4cc4ec35a2
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells4cc4ec35a2
+
+namespace GerverSofa.PartE.CertificateCells35cb6fddbf
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells35cb6fddbf
+
+namespace GerverSofa.PartE.CertificateCells23ac2ec29f
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011012` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011012 : AngleCell :=
+  childHL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells23ac2ec29f
+
+namespace GerverSofa.PartE.CertificateCellsfeaf82eaaf
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011013` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011013 : AngleCell :=
+  childHH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsfeaf82eaaf
+
+namespace GerverSofa.PartE.CertificateCellsb3a5df470b
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsb3a5df470b
+
+namespace GerverSofa.PartE.CertificateCells347f84d8ab
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells347f84d8ab
+
+namespace GerverSofa.PartE.CertificateCells7c0e2d8793
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells7c0e2d8793
+
+namespace GerverSofa.PartE.CertificateCells7ce200bf6e
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells7ce200bf6e
+
+namespace GerverSofa.PartE.CertificateCells947c600e46
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells947c600e46
+
+namespace GerverSofa.PartE.CertificateCells8f0ac2a0b1
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells8f0ac2a0b1
+
+namespace GerverSofa.PartE.CertificateCells3019c743c4
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells3019c743c4
+
+namespace GerverSofa.PartE.CertificateCellsa28fd06134
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsa28fd06134
+
+namespace GerverSofa.PartE.CertificateCells77eee4a3e4
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells77eee4a3e4
+
+namespace GerverSofa.PartE.CertificateCells49eb854006
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells49eb854006
+
+namespace GerverSofa.PartE.CertificateCellsfd1ec1f302
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsfd1ec1f302
+
+namespace GerverSofa.PartE.CertificateCellsc041f3b547
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsc041f3b547
+
+namespace GerverSofa.PartE.CertificateCells4859223aba
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells4859223aba
+
+namespace GerverSofa.PartE.CertificateCellsfd6149c12c
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011101 : AngleCell :=
+  childLH (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsfd6149c12c
+
+namespace GerverSofa.PartE.CertificateCells858f68cb01
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011101 : AngleCell :=
+  childLH (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells858f68cb01
+
+namespace GerverSofa.PartE.CertificateCellsfd737d0524
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011101 : AngleCell :=
+  childLH (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsfd737d0524
+
+namespace GerverSofa.PartE.CertificateCells8872ea9267
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011101 : AngleCell :=
+  childLH (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells8872ea9267
+
+namespace GerverSofa.PartE.CertificateCells911bdd7a98
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011102` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011102 : AngleCell :=
+  childHL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells911bdd7a98
+
+namespace GerverSofa.PartE.CertificateCells6b2d569c3f
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011103` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011103 : AngleCell :=
+  childHH (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells6b2d569c3f
+
+namespace GerverSofa.PartE.CertificateCells692b3d6bf8
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011110 : AngleCell :=
+  childLL (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells692b3d6bf8
+
+namespace GerverSofa.PartE.CertificateCells6503060a21
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011110 : AngleCell :=
+  childLL (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells6503060a21
+
+namespace GerverSofa.PartE.CertificateCells262d88f1dc
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011110 : AngleCell :=
+  childLL (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells262d88f1dc
+
+namespace GerverSofa.PartE.CertificateCellse46caf457c
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011110 : AngleCell :=
+  childLL (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellse46caf457c
+
+namespace GerverSofa.PartE.CertificateCells16ed6aeb9e
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011110 : AngleCell :=
+  childLL (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells16ed6aeb9e
+
+namespace GerverSofa.PartE.CertificateCellsde6b2a5e55
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011110 : AngleCell :=
+  childLL (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsde6b2a5e55
+
+namespace GerverSofa.PartE.CertificateCells522fe632f2
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011110 : AngleCell :=
+  childLL (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells522fe632f2
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.PhiAbove.Leaf00082`.
+* `KernelOnly.PartE.PhiAbove.Leaf00083`.
+* `KernelOnly.PartE.PhiAbove.Leaf00084`.
+* `KernelOnly.PartE.PhiAbove.Leaf00087`.
+* `KernelOnly.PartE.PhiAbove.Leaf00088`.
+* `KernelOnly.PartE.PhiAbove.Leaf00089`.
+* `KernelOnly.PartE.PhiAbove.Leaf00090`.
+* `KernelOnly.PartE.PhiAbove.Leaf00092`.
+* `KernelOnly.PartE.PhiAbove.Leaf00093`.
+* `KernelOnly.PartE.PhiAbove.Leaf00097`.
+* `KernelOnly.PartE.PhiAbove.Leaf00098`.
+* `KernelOnly.PartE.PhiAbove.Leaf00099`.
+* `KernelOnly.PartE.PhiAbove.Leaf00100`.
+* `KernelOnly.PartE.PhiAbove.Leaf00102`.
+* `KernelOnly.PartE.PhiAbove.Leaf00103`.
+* `KernelOnly.PartE.PhiAbove.Leaf00104`.
+* `KernelOnly.PartE.PhiAbove.Leaf00106`.
+* `KernelOnly.PartE.PhiAbove.Leaf00107`.
+* `KernelOnly.PartE.PhiAbove.Leaf00109`.
+* `KernelOnly.PartE.PhiAbove.Leaf00110`.
+* `KernelOnly.PartE.PhiAbove.Leaf00118`.
+* `KernelOnly.PartE.PhiAbove.Leaf00119`.
+* `KernelOnly.PartE.PhiAbove.Leaf00120`.
+* `KernelOnly.PartE.PhiAbove.Leaf00121`.
+* `KernelOnly.PartE.PhiAbove.Leaf00125`.
+* `KernelOnly.PartE.PhiAbove.Leaf00126`.
+* `KernelOnly.PartE.PhiAbove.Leaf00127`.
+* `KernelOnly.PartE.PhiAbove.Leaf00128`.
+* `KernelOnly.PartE.PhiAbove.Leaf00130`.
+* `KernelOnly.PartE.PhiAbove.Leaf00131`.
+* `KernelOnly.PartE.PhiAbove.Leaf00132`.
+* `KernelOnly.PartE.PhiAbove.Leaf00134`.
+* `KernelOnly.PartE.PhiAbove.Leaf00135`.
+* `KernelOnly.PartE.PhiAbove.Leaf00138`.
+* `KernelOnly.PartE.PhiAbove.Leaf00139`.
+* `KernelOnly.PartE.PhiAbove.Leaf00140`.
+* `KernelOnly.PartE.PhiAbove.Leaf00141`.
+* `KernelOnly.PartE.PhiAbove.Leaf00143`.
+* `KernelOnly.PartE.PhiAbove.Leaf00144`.
+* `KernelOnly.PartE.PhiAbove.Leaf00150`.
+* `KernelOnly.PartE.PhiAbove.Leaf00152`.
+* `KernelOnly.PartE.PhiAbove.Leaf00153`.
+* `KernelOnly.PartE.PhiAbove.Leaf00154`.
+* `KernelOnly.PartE.PhiAbove.Leaf00155`.
+* `KernelOnly.PartE.PhiAbove.Leaf00157`.
+* `KernelOnly.PartE.PhiAbove.Leaf00158`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c0_c0_c1_5_00082
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells02f8aca2f2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells02f8aca2f2
+
+open CertificateCells02f8aca2f2
+namespace CoverCertificate33f50162e9
+
+private theorem checked000 : adaptiveCoverCheck 2 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 2 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 2 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 2 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 2 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 2 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 2 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 2 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 2 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 2 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 2 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 2 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 2 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 2 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 2 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 2 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 2 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 2 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 2 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 2 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 2 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 2 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 2 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 2 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 2 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 2 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 2 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 2 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 2 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 2 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 2 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 2 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 2 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 2 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 2 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 2 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 2 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 2 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 2 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 2 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 2 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 2 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 2 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 2 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 2 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 2 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 2 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 2 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate33f50162e9
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c0_c0_c1 :
+    adaptiveCoverCheck 5 (childLH (childLL (childLL phiAboveCell11011011))) = true := by
+  exact CoverCertificate33f50162e9.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c0_c0_c2_5_00083
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsca6ca7b54c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsca6ca7b54c
+
+open CertificateCellsca6ca7b54c
+namespace CoverCertificatea52e53d66e
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea52e53d66e
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c0_c0_c2 :
+    adaptiveCoverCheck 5 (childHL (childLL (childLL phiAboveCell11011011))) = true := by
+  exact CoverCertificatea52e53d66e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c0_c0_c3_5_00084
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8161bed319
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8161bed319
+
+open CertificateCells8161bed319
+namespace CoverCertificate68f30819ed
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate68f30819ed
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c0_c0_c3 :
+    adaptiveCoverCheck 5 (childHH (childLL (childLL phiAboveCell11011011))) = true := by
+  exact CoverCertificate68f30819ed.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c0_c1_c0_5_00087
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5f2ef3defa
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5f2ef3defa
+
+open CertificateCells5f2ef3defa
+namespace CoverCertificatef50e05a2d9
+
+private theorem checked000 : adaptiveCoverCheck 2 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 2 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 2 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 2 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 2 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 2 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 2 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 2 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 2 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 2 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 2 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 2 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 2 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 2 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 2 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 2 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 2 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 2 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 2 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 2 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 2 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 2 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 2 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 2 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 2 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 2 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 2 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 2 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 2 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 2 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 2 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 2 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 2 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 2 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 2 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 2 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 2 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 2 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 2 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 2 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 2 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 2 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 2 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 2 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 2 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 2 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 2 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 2 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef50e05a2d9
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c0_c1_c0 :
+    adaptiveCoverCheck 5 (childLL (childLH (childLL phiAboveCell11011011))) = true := by
+  exact CoverCertificatef50e05a2d9.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c0_c1_c1_5_00088
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd4f6865820
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd4f6865820
+
+open CertificateCellsd4f6865820
+namespace CoverCertificate886cd459b6
+
+private theorem checked000 : adaptiveCoverCheck 2 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 2 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 2 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 2 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 2 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 2 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 2 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 2 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 2 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 2 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 2 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 2 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 2 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 2 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 2 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 2 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 2 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 2 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 2 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 2 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 2 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 2 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 2 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 2 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 2 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 2 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 2 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 2 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 2 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 2 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 2 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 2 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 2 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 2 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 2 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 2 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 2 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 2 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 2 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 2 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 2 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 2 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 2 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 2 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell303 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate886cd459b6
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c0_c1_c1 :
+    adaptiveCoverCheck 5 (childLH (childLH (childLL phiAboveCell11011011))) = true := by
+  exact CoverCertificate886cd459b6.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c0_c1_c2_5_00089
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsaca7f76cfa
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsaca7f76cfa
+
+open CertificateCellsaca7f76cfa
+namespace CoverCertificate6334bf560d
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate6334bf560d
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c0_c1_c2 :
+    adaptiveCoverCheck 5 (childHL (childLH (childLL phiAboveCell11011011))) = true := by
+  exact CoverCertificate6334bf560d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c0_c1_c3_5_00090
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa7ad4ac263
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa7ad4ac263
+
+open CertificateCellsa7ad4ac263
+namespace CoverCertificate40b0f107ea
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate40b0f107ea
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c0_c1_c3 :
+    adaptiveCoverCheck 5 (childHH (childLH (childLL phiAboveCell11011011))) = true := by
+  exact CoverCertificate40b0f107ea.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c0_c2_6_00092
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells53f8862830
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells53f8862830
+
+open CertificateCells53f8862830
+namespace CoverCertificatee77541b394
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee77541b394
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c0_c2 :
+    adaptiveCoverCheck 6 (childHL (childLL phiAboveCell11011011)) = true := by
+  exact CoverCertificatee77541b394.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c0_c3_6_00093
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb6a37b0947
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb6a37b0947
+
+open CertificateCellsb6a37b0947
+namespace CoverCertificate5ffc90202d
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5ffc90202d
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c0_c3 :
+    adaptiveCoverCheck 6 (childHH (childLL phiAboveCell11011011)) = true := by
+  exact CoverCertificate5ffc90202d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c1_c0_c0_5_00097
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsbc3b77fd62
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsbc3b77fd62
+
+open CertificateCellsbc3b77fd62
+namespace CoverCertificatee265233880
+
+private theorem checked000 : adaptiveCoverCheck 2 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 2 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 2 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 2 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 2 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 2 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 2 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 2 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 2 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 2 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 2 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 2 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell023 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 2 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 2 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 2 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 2 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell103 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee265233880
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c1_c0_c0 :
+    adaptiveCoverCheck 5 (childLL (childLL (childLH phiAboveCell11011011))) = true := by
+  exact CoverCertificatee265233880.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c1_c0_c1_5_00098
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsec2a80bcb9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsec2a80bcb9
+
+open CertificateCellsec2a80bcb9
+namespace CoverCertificatef18591c0b6
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef18591c0b6
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c1_c0_c1 :
+    adaptiveCoverCheck 5 (childLH (childLL (childLH phiAboveCell11011011))) = true := by
+  exact CoverCertificatef18591c0b6.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c1_c0_c2_5_00099
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells69b4ed96bb
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells69b4ed96bb
+
+open CertificateCells69b4ed96bb
+namespace CoverCertificatef7582d4176
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef7582d4176
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c1_c0_c2 :
+    adaptiveCoverCheck 5 (childHL (childLL (childLH phiAboveCell11011011))) = true := by
+  exact CoverCertificatef7582d4176.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c1_c0_c3_5_00100
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells81024818f0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells81024818f0
+
+open CertificateCells81024818f0
+namespace CoverCertificate042241a8e3
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate042241a8e3
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c1_c0_c3 :
+    adaptiveCoverCheck 5 (childHH (childLL (childLH phiAboveCell11011011))) = true := by
+  exact CoverCertificate042241a8e3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c1_c1_6_00102
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsbaba88fb03
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsbaba88fb03
+
+open CertificateCellsbaba88fb03
+namespace CoverCertificateb30a576a6c
+
+private theorem checked000 : adaptiveCoverCheck 3 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 3 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 3 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 3 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 3 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 3 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 3 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 3 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 3 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 3 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 3 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 3 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 3 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 3 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 3 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 3 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 3 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 3 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 3 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 3 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 3 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 3 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 3 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 3 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 3 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 3 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 3 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 3 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 3 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 3 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 3 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 3 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 3 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 3 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 3 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 3 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 3 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 3 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 3 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 3 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 3 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 3 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 3 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 3 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 3 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 3 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 3 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 3 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateb30a576a6c
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c1_c1 :
+    adaptiveCoverCheck 6 (childLH (childLH phiAboveCell11011011)) = true := by
+  exact CoverCertificateb30a576a6c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c1_c2_6_00103
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4431add919
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4431add919
+
+open CertificateCells4431add919
+namespace CoverCertificatee29f1ec14c
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee29f1ec14c
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c1_c2 :
+    adaptiveCoverCheck 6 (childHL (childLH phiAboveCell11011011)) = true := by
+  exact CoverCertificatee29f1ec14c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c1_c3_6_00104
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0c4c476da0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0c4c476da0
+
+open CertificateCells0c4c476da0
+namespace CoverCertificate7c835f5bcd
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate7c835f5bcd
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c1_c3 :
+    adaptiveCoverCheck 6 (childHH (childLH phiAboveCell11011011)) = true := by
+  exact CoverCertificate7c835f5bcd.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c2_7_00106
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4cc4ec35a2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4cc4ec35a2
+
+open CertificateCells4cc4ec35a2
+namespace CoverCertificate1847743b04
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1847743b04
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c2 :
+    adaptiveCoverCheck 7 (childHL phiAboveCell11011011) = true := by
+  exact CoverCertificate1847743b04.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c1_c3_7_00107
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells35cb6fddbf
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells35cb6fddbf
+
+open CertificateCells35cb6fddbf
+namespace CoverCertificated22de6b6b3
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated22de6b6b3
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c3 :
+    adaptiveCoverCheck 7 (childHH phiAboveCell11011011) = true := by
+  exact CoverCertificated22de6b6b3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c2_8_00109
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells23ac2ec29f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells23ac2ec29f
+
+open CertificateCells23ac2ec29f
+namespace CoverCertificatec9d844cfa1
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec9d844cfa1
+
+theorem e24KC2PhiAboveLeaf1101101_c2 :
+    adaptiveCoverCheck 8 phiAboveCell11011012 = true := by
+  exact CoverCertificatec9d844cfa1.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101101_c3_8_00110
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsfeaf82eaaf
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsfeaf82eaaf
+
+open CertificateCellsfeaf82eaaf
+namespace CoverCertificate0b1bb49c5e
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0b1bb49c5e
+
+theorem e24KC2PhiAboveLeaf1101101_c3 :
+    adaptiveCoverCheck 8 phiAboveCell11011013 = true := by
+  exact CoverCertificate0b1bb49c5e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c0_c0_c0_6_00118
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb3a5df470b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb3a5df470b
+
+open CertificateCellsb3a5df470b
+namespace CoverCertificatedcf0833de3
+
+private theorem checked000 : adaptiveCoverCheck 3 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 3 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 3 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 3 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 3 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 3 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 3 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 3 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 3 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 3 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 3 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 3 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 3 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 3 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 3 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 3 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 3 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 3 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 3 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 3 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 3 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 3 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 3 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 3 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 3 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 3 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 3 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 3 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 3 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 3 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 3 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 3 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 3 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 3 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 3 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 3 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 3 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 3 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 3 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 3 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 3 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 3 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 3 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 3 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 3 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 3 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 3 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 3 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatedcf0833de3
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c0_c0 :
+    adaptiveCoverCheck 6 (childLL (childLL phiAboveCell11011100)) = true := by
+  exact CoverCertificatedcf0833de3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c0_c0_c1_6_00119
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells347f84d8ab
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells347f84d8ab
+
+open CertificateCells347f84d8ab
+namespace CoverCertificate292a59b81b
+
+private theorem checked000 : adaptiveCoverCheck 3 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 3 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 3 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 3 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 3 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 3 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 3 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 3 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 3 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 3 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 3 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 3 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 3 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 3 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 3 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 3 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 3 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 3 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 3 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 3 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 3 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 3 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 3 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 3 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 3 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 3 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 3 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 3 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 3 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 3 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 3 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 3 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 3 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 3 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 3 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 3 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 3 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 3 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 3 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 3 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 3 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 3 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 3 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 3 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 3 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 3 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 3 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 3 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell313 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 3 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 3 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 3 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 3 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate292a59b81b
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c0_c1 :
+    adaptiveCoverCheck 6 (childLH (childLL phiAboveCell11011100)) = true := by
+  exact CoverCertificate292a59b81b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c0_c0_c2_6_00120
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells7c0e2d8793
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells7c0e2d8793
+
+open CertificateCells7c0e2d8793
+namespace CoverCertificate38c27d4b72
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate38c27d4b72
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c0_c2 :
+    adaptiveCoverCheck 6 (childHL (childLL phiAboveCell11011100)) = true := by
+  exact CoverCertificate38c27d4b72.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c0_c0_c3_6_00121
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells7ce200bf6e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells7ce200bf6e
+
+open CertificateCells7ce200bf6e
+namespace CoverCertificate80ac67346e
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate80ac67346e
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c0_c3 :
+    adaptiveCoverCheck 6 (childHH (childLL phiAboveCell11011100)) = true := by
+  exact CoverCertificate80ac67346e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c0_c1_c0_c0_5_00125
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells947c600e46
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells947c600e46
+
+open CertificateCells947c600e46
+namespace CoverCertificate6a3da1aa29
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate6a3da1aa29
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c1_c0_c0 :
+    adaptiveCoverCheck 5 (childLL (childLL (childLH phiAboveCell11011100))) = true := by
+  exact CoverCertificate6a3da1aa29.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c0_c1_c0_c1_5_00126
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8f0ac2a0b1
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8f0ac2a0b1
+
+open CertificateCells8f0ac2a0b1
+namespace CoverCertificatec0de2b5230
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec0de2b5230
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c1_c0_c1 :
+    adaptiveCoverCheck 5 (childLH (childLL (childLH phiAboveCell11011100))) = true := by
+  exact CoverCertificatec0de2b5230.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c0_c1_c0_c2_5_00127
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3019c743c4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3019c743c4
+
+open CertificateCells3019c743c4
+namespace CoverCertificateeb67261010
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateeb67261010
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c1_c0_c2 :
+    adaptiveCoverCheck 5 (childHL (childLL (childLH phiAboveCell11011100))) = true := by
+  exact CoverCertificateeb67261010.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c0_c1_c0_c3_5_00128
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa28fd06134
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa28fd06134
+
+open CertificateCellsa28fd06134
+namespace CoverCertificate4ceb0bf432
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4ceb0bf432
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c1_c0_c3 :
+    adaptiveCoverCheck 5 (childHH (childLL (childLH phiAboveCell11011100))) = true := by
+  exact CoverCertificate4ceb0bf432.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c0_c1_c1_6_00130
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells77eee4a3e4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells77eee4a3e4
+
+open CertificateCells77eee4a3e4
+namespace CoverCertificate1407c62e79
+
+private theorem checked000 : adaptiveCoverCheck 3 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 3 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 3 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 3 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 3 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 3 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 3 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 3 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 3 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 3 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 3 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 3 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 3 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 3 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 3 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 3 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 3 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 3 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 3 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 3 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 3 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 3 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 3 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 3 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 3 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 3 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 3 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 3 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 3 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 3 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 3 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 3 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 3 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 3 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 3 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 3 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 3 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 3 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 3 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 3 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 3 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 3 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 3 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 3 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 3 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 3 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 3 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 3 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1407c62e79
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c1_c1 :
+    adaptiveCoverCheck 6 (childLH (childLH phiAboveCell11011100)) = true := by
+  exact CoverCertificate1407c62e79.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c0_c1_c2_6_00131
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells49eb854006
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells49eb854006
+
+open CertificateCells49eb854006
+namespace CoverCertificate3835f13788
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3835f13788
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c1_c2 :
+    adaptiveCoverCheck 6 (childHL (childLH phiAboveCell11011100)) = true := by
+  exact CoverCertificate3835f13788.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c0_c1_c3_6_00132
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsfd1ec1f302
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsfd1ec1f302
+
+open CertificateCellsfd1ec1f302
+namespace CoverCertificate2b398d0240
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate2b398d0240
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c1_c3 :
+    adaptiveCoverCheck 6 (childHH (childLH phiAboveCell11011100)) = true := by
+  exact CoverCertificate2b398d0240.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c0_c2_7_00134
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc041f3b547
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc041f3b547
+
+open CertificateCellsc041f3b547
+namespace CoverCertificate46690a85d6
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate46690a85d6
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c2 :
+    adaptiveCoverCheck 7 (childHL phiAboveCell11011100) = true := by
+  exact CoverCertificate46690a85d6.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c0_c3_7_00135
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4859223aba
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4859223aba
+
+open CertificateCells4859223aba
+namespace CoverCertificate79d55c50a4
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate79d55c50a4
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c3 :
+    adaptiveCoverCheck 7 (childHH phiAboveCell11011100) = true := by
+  exact CoverCertificate79d55c50a4.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c1_c0_7_00138
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsfd6149c12c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsfd6149c12c
+
+open CertificateCellsfd6149c12c
+namespace CoverCertificate3882c70981
+
+private theorem checked0020 : adaptiveCoverCheck 3 cell0020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0020 (by decide +kernel)
+
+private theorem checked0021 : adaptiveCoverCheck 3 cell0021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0021 (by decide +kernel)
+
+private theorem checked0022 : adaptiveCoverCheck 3 cell0022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0022 (by decide +kernel)
+
+private theorem checked0023 : adaptiveCoverCheck 3 cell0023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0023 (by decide +kernel)
+
+private theorem checked0030 : adaptiveCoverCheck 3 cell0030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0030 (by decide +kernel)
+
+private theorem checked0031 : adaptiveCoverCheck 3 cell0031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0031 (by decide +kernel)
+
+private theorem checked0032 : adaptiveCoverCheck 3 cell0032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0032 (by decide +kernel)
+
+private theorem checked0033 : adaptiveCoverCheck 3 cell0033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0033 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 4 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 4 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 4 cell002 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell002
+    checked0020 checked0021 checked0022 checked0023
+
+private theorem checked003 : adaptiveCoverCheck 4 cell003 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell003
+    checked0030 checked0031 checked0032 checked0033
+
+private theorem checked010 : adaptiveCoverCheck 4 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 4 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 4 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 4 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 4 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 4 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 4 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 4 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 4 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 4 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 4 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 4 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 4 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 4 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 4 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 4 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 4 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 4 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 4 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 4 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 4 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 4 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 4 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 4 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 4 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 4 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 4 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 4 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 4 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 4 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 4 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 4 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 4 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 4 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 4 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 4 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 4 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 4 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 4 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 4 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 4 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 4 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 4 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 4 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3882c70981
+
+theorem e24KC2PhiAboveLeaf1101110_c1_c0 :
+    adaptiveCoverCheck 7 (childLL phiAboveCell11011101) = true := by
+  exact CoverCertificate3882c70981.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c1_c1_7_00139
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells858f68cb01
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells858f68cb01
+
+open CertificateCells858f68cb01
+namespace CoverCertificate749186c43c
+
+private theorem checked000 : adaptiveCoverCheck 4 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 4 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 4 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 4 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 4 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 4 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 4 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 4 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 4 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 4 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 4 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 4 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 4 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 4 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 4 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 4 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 4 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 4 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 4 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 4 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 4 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 4 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 4 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 4 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 4 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 4 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 4 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 4 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 4 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 4 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 4 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 4 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 4 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 4 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 4 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 4 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 4 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 4 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 4 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 4 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 4 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 4 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 4 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 4 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 4 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 4 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 4 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 4 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate749186c43c
+
+theorem e24KC2PhiAboveLeaf1101110_c1_c1 :
+    adaptiveCoverCheck 7 (childLH phiAboveCell11011101) = true := by
+  exact CoverCertificate749186c43c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c1_c2_7_00140
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsfd737d0524
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsfd737d0524
+
+open CertificateCellsfd737d0524
+namespace CoverCertificate69709d6517
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate69709d6517
+
+theorem e24KC2PhiAboveLeaf1101110_c1_c2 :
+    adaptiveCoverCheck 7 (childHL phiAboveCell11011101) = true := by
+  exact CoverCertificate69709d6517.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c1_c3_7_00141
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8872ea9267
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8872ea9267
+
+open CertificateCells8872ea9267
+namespace CoverCertificate9b05e6137c
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate9b05e6137c
+
+theorem e24KC2PhiAboveLeaf1101110_c1_c3 :
+    adaptiveCoverCheck 7 (childHH phiAboveCell11011101) = true := by
+  exact CoverCertificate9b05e6137c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c2_8_00143
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells911bdd7a98
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells911bdd7a98
+
+open CertificateCells911bdd7a98
+namespace CoverCertificatea08e6229dd
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea08e6229dd
+
+theorem e24KC2PhiAboveLeaf1101110_c2 :
+    adaptiveCoverCheck 8 phiAboveCell11011102 = true := by
+  exact CoverCertificatea08e6229dd.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101110_c3_8_00144
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6b2d569c3f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6b2d569c3f
+
+open CertificateCells6b2d569c3f
+namespace CoverCertificate80957e0c30
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate80957e0c30
+
+theorem e24KC2PhiAboveLeaf1101110_c3 :
+    adaptiveCoverCheck 8 phiAboveCell11011103 = true := by
+  exact CoverCertificate80957e0c30.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101111_c0_c0_7_00150
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells692b3d6bf8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells692b3d6bf8
+
+open CertificateCells692b3d6bf8
+namespace CoverCertificate6b5d6c54cf
+
+private theorem checked000 : adaptiveCoverCheck 4 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 4 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 4 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 4 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 4 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 4 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 4 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 4 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 4 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 4 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 4 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 4 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 4 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 4 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 4 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 4 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 4 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 4 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 4 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 4 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 4 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 4 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 4 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 4 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 4 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 4 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 4 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 4 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 4 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 4 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 4 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 4 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 4 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 4 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 4 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 4 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 4 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 4 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 4 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 4 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 4 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 4 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 4 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 4 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 4 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 4 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 4 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 4 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate6b5d6c54cf
+
+theorem e24KC2PhiAboveLeaf1101111_c0_c0 :
+    adaptiveCoverCheck 7 (childLL phiAboveCell11011110) = true := by
+  exact CoverCertificate6b5d6c54cf.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101111_c0_c1_c0_6_00152
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6503060a21
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6503060a21
+
+open CertificateCells6503060a21
+namespace CoverCertificate5e2d2d50e9
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5e2d2d50e9
+
+theorem e24KC2PhiAboveLeaf1101111_c0_c1_c0 :
+    adaptiveCoverCheck 6 (childLL (childLH phiAboveCell11011110)) = true := by
+  exact CoverCertificate5e2d2d50e9.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101111_c0_c1_c1_6_00153
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells262d88f1dc
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells262d88f1dc
+
+open CertificateCells262d88f1dc
+namespace CoverCertificate4eafbde6a7
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4eafbde6a7
+
+theorem e24KC2PhiAboveLeaf1101111_c0_c1_c1 :
+    adaptiveCoverCheck 6 (childLH (childLH phiAboveCell11011110)) = true := by
+  exact CoverCertificate4eafbde6a7.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101111_c0_c1_c2_6_00154
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse46caf457c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse46caf457c
+
+open CertificateCellse46caf457c
+namespace CoverCertificateb57436bcb4
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateb57436bcb4
+
+theorem e24KC2PhiAboveLeaf1101111_c0_c1_c2 :
+    adaptiveCoverCheck 6 (childHL (childLH phiAboveCell11011110)) = true := by
+  exact CoverCertificateb57436bcb4.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101111_c0_c1_c3_6_00155
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells16ed6aeb9e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells16ed6aeb9e
+
+open CertificateCells16ed6aeb9e
+namespace CoverCertificate8e406c3a19
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate8e406c3a19
+
+theorem e24KC2PhiAboveLeaf1101111_c0_c1_c3 :
+    adaptiveCoverCheck 6 (childHH (childLH phiAboveCell11011110)) = true := by
+  exact CoverCertificate8e406c3a19.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101111_c0_c2_7_00157
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsde6b2a5e55
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsde6b2a5e55
+
+open CertificateCellsde6b2a5e55
+namespace CoverCertificate11dc9c1c2a
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate11dc9c1c2a
+
+theorem e24KC2PhiAboveLeaf1101111_c0_c2 :
+    adaptiveCoverCheck 7 (childHL phiAboveCell11011110) = true := by
+  exact CoverCertificate11dc9c1c2a.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101111_c0_c3_7_00158
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells522fe632f2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells522fe632f2
+
+open CertificateCells522fe632f2
+namespace CoverCertificate52298353c4
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate52298353c4
+
+theorem e24KC2PhiAboveLeaf1101111_c0_c3 :
+    adaptiveCoverCheck 7 (childHH phiAboveCell11011110) = true := by
+  exact CoverCertificate52298353c4.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle024.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle024.lean
new file mode 100644
index 0000000000..32c18f00e4
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle024.lean
@@ -0,0 +1,8995 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch037`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CoverCertificate276a139af5
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLH (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate276a139af5
+
+namespace GerverSofa.PartE.CoverCertificatee02eecb06a
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLH (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatee02eecb06a
+
+namespace GerverSofa.PartE.CoverCertificate783b4cee82
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLH (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate783b4cee82
+
+namespace GerverSofa.PartE.CoverCertificate90fa7d5883
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLH (childLH (childLH (childLH (childLL (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate90fa7d5883
+
+namespace GerverSofa.PartE.CoverCertificate0133fea31e
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLH (childLH (childLH (childLL (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate0133fea31e
+
+namespace GerverSofa.PartE.CoverCertificate3a1ac6bacd
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLH (childLH (childLH (childLL (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate3a1ac6bacd
+
+namespace GerverSofa.PartE.CoverCertificateff206b6a85
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLL (childLL (childLL (childLL (childLH (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+end GerverSofa.PartE.CoverCertificateff206b6a85
+
+namespace GerverSofa.PartE.CoverCertificate71299415d1
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLL (childLL (childLL (childLL (childLH (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate71299415d1
+
+namespace GerverSofa.PartE.CoverCertificate176775891f
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childLL (childLL (childLL (childLH (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate176775891f
+
+namespace GerverSofa.PartE.CoverCertificate09737eefc3
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childLL (childLL (childLL (childLH (childLH (childLH
+    (e24PhiAboveRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate09737eefc3
+
+namespace GerverSofa.PartE.CoverCertificatea812690e7e
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLL (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificatea812690e7e
+
+namespace GerverSofa.PartE.CoverCertificate35de9fca65
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate35de9fca65
+
+namespace GerverSofa.PartE.CoverCertificatee6cd0593fa
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatee6cd0593fa
+
+namespace GerverSofa.PartE.CoverCertificate30c4e37720
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLL (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate30c4e37720
+
+namespace GerverSofa.PartE.CoverCertificatedfaf53edbb
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLL (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatedfaf53edbb
+
+namespace GerverSofa.PartE.CoverCertificatea7d865eaeb
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificatea7d865eaeb
+
+namespace GerverSofa.PartE.CoverCertificate59a502ab21
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate59a502ab21
+
+namespace GerverSofa.PartE.CoverCertificatedb9da39caf
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLH (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatedb9da39caf
+
+namespace GerverSofa.PartE.CoverCertificateefd199c221
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLH (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificateefd199c221
+
+namespace GerverSofa.PartE.CoverCertificatedb27ad4afe
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificatedb27ad4afe
+
+namespace GerverSofa.PartE.CoverCertificate05565992a9
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate05565992a9
+
+namespace GerverSofa.PartE.CoverCertificatec7fa9ff5d0
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLH (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatec7fa9ff5d0
+
+namespace GerverSofa.PartE.CoverCertificatef60369d605
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLH (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+end GerverSofa.PartE.CoverCertificatef60369d605
+
+namespace GerverSofa.PartE.CoverCertificate5759c94c52
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLH (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate5759c94c52
+
+namespace GerverSofa.PartE.CoverCertificate03a553a361
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLH (childLL (childLL (childLH (childLH (childLH (e24PhiAboveRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate03a553a361
+
+namespace GerverSofa.PartE.CoverCertificated3212f4a8b
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childLL (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificated3212f4a8b
+
+namespace GerverSofa.PartE.CoverCertificate92ae3ca2d2
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childLL (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate92ae3ca2d2
+
+namespace GerverSofa.PartE.CoverCertificate126af0396e
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childLL (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate126af0396e
+
+namespace GerverSofa.PartE.CoverCertificate7a0799a8ec
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childLL (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate7a0799a8ec
+
+namespace GerverSofa.PartE.CoverCertificate0c897fb4ca
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childHL (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate0c897fb4ca
+
+namespace GerverSofa.PartE.CoverCertificateac87260947
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHL (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificateac87260947
+
+namespace GerverSofa.PartE.CoverCertificatee4e72882c5
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHL (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatee4e72882c5
+
+namespace GerverSofa.PartE.CoverCertificateeab8fed9d7
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHL (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificateeab8fed9d7
+
+namespace GerverSofa.PartE.CertificateCells018fd5e668
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011111 : AngleCell :=
+  childLH (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells018fd5e668
+
+namespace GerverSofa.PartE.CertificateCells63f0679649
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011111 : AngleCell :=
+  childLH (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells63f0679649
+
+namespace GerverSofa.PartE.CertificateCellsa5edf5ee06
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011111 : AngleCell :=
+  childLH (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsa5edf5ee06
+
+namespace GerverSofa.PartE.CertificateCellsd0bb651e18
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011111 : AngleCell :=
+  childLH (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsd0bb651e18
+
+namespace GerverSofa.PartE.CertificateCells347e84a45d
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011112` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011112 : AngleCell :=
+  childHL (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells347e84a45d
+
+namespace GerverSofa.PartE.CertificateCells06372696b5
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011113` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011113 : AngleCell :=
+  childHH (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells06372696b5
+
+namespace GerverSofa.PartE.CertificateCellsdc3bb009fb
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100000` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100000 : AngleCell :=
+  childLL (childLL (childLL (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCellsdc3bb009fb
+
+namespace GerverSofa.PartE.CertificateCellsa17c45a931
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100000` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100000 : AngleCell :=
+  childLL (childLL (childLL (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCellsa17c45a931
+
+namespace GerverSofa.PartE.CertificateCells11ac46d6b2
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100000` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100000 : AngleCell :=
+  childLL (childLL (childLL (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCells11ac46d6b2
+
+namespace GerverSofa.PartE.CertificateCells4bb3cae3bb
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100000` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100000 : AngleCell :=
+  childLL (childLL (childLL (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCells4bb3cae3bb
+
+namespace GerverSofa.PartE.CertificateCells517a8caf86
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100001` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100001 : AngleCell :=
+  childLH (childLL (childLL (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCells517a8caf86
+
+namespace GerverSofa.PartE.CertificateCells5795a3d5ea
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100002` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100002 : AngleCell :=
+  childHL (childLL (childLL (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCells5795a3d5ea
+
+namespace GerverSofa.PartE.CertificateCellsa32d189193
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100003` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100003 : AngleCell :=
+  childHH (childLL (childLL (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCellsa32d189193
+
+namespace GerverSofa.PartE.CertificateCellsf628113fc1
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100010 : AngleCell :=
+  childLL (childLH (childLL (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCellsf628113fc1
+
+namespace GerverSofa.PartE.CertificateCells740eeaba32
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100011 : AngleCell :=
+  childLH (childLH (childLL (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCells740eeaba32
+
+namespace GerverSofa.PartE.CertificateCells626f3fc028
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100012` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100012 : AngleCell :=
+  childHL (childLH (childLL (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCells626f3fc028
+
+namespace GerverSofa.PartE.CertificateCells15f9582e4b
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100013` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100013 : AngleCell :=
+  childHH (childLH (childLL (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCells15f9582e4b
+
+namespace GerverSofa.PartE.CertificateCellsc156062d1b
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100100 : AngleCell :=
+  childLL (childLL (childLH (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCellsc156062d1b
+
+namespace GerverSofa.PartE.CertificateCellsdb4a23ce5a
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100101 : AngleCell :=
+  childLH (childLL (childLH (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCellsdb4a23ce5a
+
+namespace GerverSofa.PartE.CertificateCells8ced9a3143
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100102` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100102 : AngleCell :=
+  childHL (childLL (childLH (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCells8ced9a3143
+
+namespace GerverSofa.PartE.CertificateCells1c46ec9a00
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100103` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100103 : AngleCell :=
+  childHH (childLL (childLH (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCells1c46ec9a00
+
+namespace GerverSofa.PartE.CertificateCellsc9ea34c02b
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100110 : AngleCell :=
+  childLL (childLH (childLH (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCellsc9ea34c02b
+
+namespace GerverSofa.PartE.CertificateCells7575776df3
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100111 : AngleCell :=
+  childLH (childLH (childLH (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCells7575776df3
+
+namespace GerverSofa.PartE.CertificateCells8b638b7054
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100112` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100112 : AngleCell :=
+  childHL (childLH (childLH (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCells8b638b7054
+
+namespace GerverSofa.PartE.CertificateCells3bd9cbb8df
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100113` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100113 : AngleCell :=
+  childHH (childLH (childLH (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCells3bd9cbb8df
+
+namespace GerverSofa.PartE.CertificateCells41c704a1f1
+
+/-- Subcell `3320` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3320 : AngleCell :=
+  childLL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells41c704a1f1
+
+namespace GerverSofa.PartE.CertificateCells67d940c92a
+
+/-- Subcell `3320` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3320 : AngleCell :=
+  childLL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells67d940c92a
+
+namespace GerverSofa.PartE.CertificateCells8d68909634
+
+/-- Subcell `3320` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3320 : AngleCell :=
+  childLL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells8d68909634
+
+namespace GerverSofa.PartE.CertificateCellsc208943a95
+
+/-- Subcell `3320` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3320 : AngleCell :=
+  childLL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsc208943a95
+
+namespace GerverSofa.PartE.CertificateCells42a66228ec
+
+/-- Subcell `3322` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3322 : AngleCell :=
+  childHL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells42a66228ec
+
+namespace GerverSofa.PartE.CertificateCells67af310b24
+
+/-- Subcell `3322` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3322 : AngleCell :=
+  childHL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells67af310b24
+
+namespace GerverSofa.PartE.CertificateCells9b91503b21
+
+/-- Subcell `3322` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3322 : AngleCell :=
+  childHL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells9b91503b21
+
+namespace GerverSofa.PartE.CertificateCellsb36cfca8f0
+
+/-- Subcell `3322` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3322 : AngleCell :=
+  childHL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsb36cfca8f0
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.PhiAbove.Leaf00161`.
+* `KernelOnly.PartE.PhiAbove.Leaf00162`.
+* `KernelOnly.PartE.PhiAbove.Leaf00163`.
+* `KernelOnly.PartE.PhiAbove.Leaf00164`.
+* `KernelOnly.PartE.PhiAbove.Leaf00166`.
+* `KernelOnly.PartE.PhiAbove.Leaf00167`.
+* `KernelOnly.PartE.PhiAbove.Leaf00174`.
+* `KernelOnly.PartE.PhiAbove.Leaf00175`.
+* `KernelOnly.PartE.PhiAbove.Leaf00176`.
+* `KernelOnly.PartE.PhiAbove.Leaf00177`.
+* `KernelOnly.PartE.PhiAbove.Leaf00179`.
+* `KernelOnly.PartE.PhiAbove.Leaf00180`.
+* `KernelOnly.PartE.PhiAbove.Leaf00181`.
+* `KernelOnly.PartE.PhiAbove.Leaf00187`.
+* `KernelOnly.PartE.PhiAbove.Leaf00188`.
+* `KernelOnly.PartE.PhiAbove.Leaf00189`.
+* `KernelOnly.PartE.PhiAbove.Leaf00190`.
+* `KernelOnly.PartE.PhiAbove.Leaf00196`.
+* `KernelOnly.PartE.PhiAbove.Leaf00197`.
+* `KernelOnly.PartE.PhiAbove.Leaf00198`.
+* `KernelOnly.PartE.PhiAbove.Leaf00199`.
+* `KernelOnly.PartE.PhiAbove.Leaf00204`.
+* `KernelOnly.PartE.PhiAbove.Leaf00205`.
+* `KernelOnly.PartE.PhiAbove.Leaf00206`.
+* `KernelOnly.PartE.PhiAbove.Leaf00207`.
+* `KernelOnly.PartE.PhiBelow.Leaf00242`.
+* `KernelOnly.PartE.PhiBelow.Leaf00243`.
+* `KernelOnly.PartE.PhiBelow.Leaf00244`.
+* `KernelOnly.PartE.PhiBelow.Leaf00245`.
+* `KernelOnly.PartE.PhiBelow.Leaf00256`.
+* `KernelOnly.PartE.PhiBelow.Leaf00257`.
+* `KernelOnly.PartE.PhiBelow.Leaf00258`.
+* `KernelOnly.PartE.PhiBelow.Leaf00259`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101111_c1_c0_7_00161
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells018fd5e668
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells018fd5e668
+
+open CertificateCells018fd5e668
+namespace CoverCertificate276a139af5
+
+private theorem checked000 : adaptiveCoverCheck 4 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 4 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 4 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 4 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 4 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 4 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 4 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 4 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 4 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 4 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 4 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 4 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 4 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 4 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 4 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 4 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 4 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 4 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 4 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 4 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell103 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 4 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 4 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 4 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 4 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 4 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 4 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 4 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 4 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 4 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 4 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 4 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 4 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 4 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 4 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 4 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 4 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 4 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 4 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 4 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 4 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 4 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 4 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 4 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 4 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 4 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 4 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 4 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 4 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 4 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 4 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 4 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 4 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 4 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 4 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 4 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 4 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 4 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 4 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 4 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 4 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate276a139af5
+
+theorem e24KC2PhiAboveLeaf1101111_c1_c0 :
+    adaptiveCoverCheck 7 (childLL phiAboveCell11011111) = true := by
+  exact CoverCertificate276a139af5.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101111_c1_c1_7_00162
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells63f0679649
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells63f0679649
+
+open CertificateCells63f0679649
+namespace CoverCertificatee02eecb06a
+
+private theorem checked020 : adaptiveCoverCheck 4 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 4 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 4 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 4 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 4 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 4 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 4 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 4 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 4 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 4 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 4 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 4 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 4 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 4 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 4 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 4 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 4 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 4 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 4 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 4 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 4 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 4 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 4 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 4 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 4 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 4 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 4 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 4 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 4 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 4 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 4 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 4 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 4 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 4 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 4 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 4 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 4 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 4 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 4 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 4 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 4 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 4 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 4 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 4 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 4 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 4 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 4 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 4 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee02eecb06a
+
+theorem e24KC2PhiAboveLeaf1101111_c1_c1 :
+    adaptiveCoverCheck 7 (childLH phiAboveCell11011111) = true := by
+  exact CoverCertificatee02eecb06a.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101111_c1_c2_7_00163
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa5edf5ee06
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa5edf5ee06
+
+open CertificateCellsa5edf5ee06
+namespace CoverCertificate783b4cee82
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate783b4cee82
+
+theorem e24KC2PhiAboveLeaf1101111_c1_c2 :
+    adaptiveCoverCheck 7 (childHL phiAboveCell11011111) = true := by
+  exact CoverCertificate783b4cee82.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101111_c1_c3_7_00164
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd0bb651e18
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd0bb651e18
+
+open CertificateCellsd0bb651e18
+namespace CoverCertificate90fa7d5883
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate90fa7d5883
+
+theorem e24KC2PhiAboveLeaf1101111_c1_c3 :
+    adaptiveCoverCheck 7 (childHH phiAboveCell11011111) = true := by
+  exact CoverCertificate90fa7d5883.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101111_c2_8_00166
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells347e84a45d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells347e84a45d
+
+open CertificateCells347e84a45d
+namespace CoverCertificate0133fea31e
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0133fea31e
+
+theorem e24KC2PhiAboveLeaf1101111_c2 :
+    adaptiveCoverCheck 8 phiAboveCell11011112 = true := by
+  exact CoverCertificate0133fea31e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1101111_c3_8_00167
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells06372696b5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells06372696b5
+
+open CertificateCells06372696b5
+namespace CoverCertificate3a1ac6bacd
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3a1ac6bacd
+
+theorem e24KC2PhiAboveLeaf1101111_c3 :
+    adaptiveCoverCheck 8 phiAboveCell11011113 = true := by
+  exact CoverCertificate3a1ac6bacd.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110000_c0_c0_7_00174
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdc3bb009fb
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdc3bb009fb
+
+open CertificateCellsdc3bb009fb
+namespace CoverCertificateff206b6a85
+
+private theorem checked020 : adaptiveCoverCheck 4 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 4 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 4 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 4 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 4 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 4 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 4 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 4 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 4 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 4 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 4 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 4 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 4 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 4 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 4 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 4 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 4 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 4 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 4 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 4 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 4 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 4 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 4 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 4 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell213 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateff206b6a85
+
+theorem e24KC2PhiAboveLeaf1110000_c0_c0 :
+    adaptiveCoverCheck 7 (childLL phiAboveCell11100000) = true := by
+  exact CoverCertificateff206b6a85.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110000_c0_c1_7_00175
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa17c45a931
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa17c45a931
+
+open CertificateCellsa17c45a931
+namespace CoverCertificate71299415d1
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate71299415d1
+
+theorem e24KC2PhiAboveLeaf1110000_c0_c1 :
+    adaptiveCoverCheck 7 (childLH phiAboveCell11100000) = true := by
+  exact CoverCertificate71299415d1.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110000_c0_c2_7_00176
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells11ac46d6b2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells11ac46d6b2
+
+open CertificateCells11ac46d6b2
+namespace CoverCertificate176775891f
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate176775891f
+
+theorem e24KC2PhiAboveLeaf1110000_c0_c2 :
+    adaptiveCoverCheck 7 (childHL phiAboveCell11100000) = true := by
+  exact CoverCertificate176775891f.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110000_c0_c3_7_00177
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4bb3cae3bb
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4bb3cae3bb
+
+open CertificateCells4bb3cae3bb
+namespace CoverCertificate09737eefc3
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate09737eefc3
+
+theorem e24KC2PhiAboveLeaf1110000_c0_c3 :
+    adaptiveCoverCheck 7 (childHH phiAboveCell11100000) = true := by
+  exact CoverCertificate09737eefc3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110000_c1_8_00179
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells517a8caf86
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells517a8caf86
+
+open CertificateCells517a8caf86
+namespace CoverCertificatea812690e7e
+
+private theorem checked000 : adaptiveCoverCheck 5 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 5 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 5 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 5 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 5 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 5 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 5 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 5 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 5 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 5 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 5 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 5 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 5 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 5 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 5 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 5 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 5 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 5 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 5 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 5 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 5 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 5 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 5 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 5 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 5 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 5 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 5 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 5 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 5 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 5 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 5 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 5 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 5 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 5 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 5 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 5 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 5 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 5 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 5 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 5 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 5 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 5 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 5 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 5 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 5 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 5 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 5 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 5 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea812690e7e
+
+theorem e24KC2PhiAboveLeaf1110000_c1 :
+    adaptiveCoverCheck 8 phiAboveCell11100001 = true := by
+  exact CoverCertificatea812690e7e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110000_c2_8_00180
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5795a3d5ea
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5795a3d5ea
+
+open CertificateCells5795a3d5ea
+namespace CoverCertificate35de9fca65
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate35de9fca65
+
+theorem e24KC2PhiAboveLeaf1110000_c2 :
+    adaptiveCoverCheck 8 phiAboveCell11100002 = true := by
+  exact CoverCertificate35de9fca65.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110000_c3_8_00181
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa32d189193
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa32d189193
+
+open CertificateCellsa32d189193
+namespace CoverCertificatee6cd0593fa
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee6cd0593fa
+
+theorem e24KC2PhiAboveLeaf1110000_c3 :
+    adaptiveCoverCheck 8 phiAboveCell11100003 = true := by
+  exact CoverCertificatee6cd0593fa.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110001_c0_8_00187
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf628113fc1
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf628113fc1
+
+open CertificateCellsf628113fc1
+namespace CoverCertificate30c4e37720
+
+private theorem checked000 : adaptiveCoverCheck 5 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 5 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 5 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 5 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 5 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 5 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 5 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 5 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 5 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 5 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 5 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 5 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 5 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 5 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 5 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 5 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 5 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 5 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 5 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 5 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 5 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 5 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 5 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 5 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 5 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 5 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 5 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 5 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 5 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 5 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 5 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 5 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 5 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 5 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 5 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 5 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 5 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 5 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 5 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 5 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 5 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 5 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 5 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 5 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 5 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 5 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 5 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 5 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell313 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 5 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 5 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 5 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 5 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate30c4e37720
+
+theorem e24KC2PhiAboveLeaf1110001_c0 :
+    adaptiveCoverCheck 8 phiAboveCell11100010 = true := by
+  exact CoverCertificate30c4e37720.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110001_c1_8_00188
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells740eeaba32
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells740eeaba32
+
+open CertificateCells740eeaba32
+namespace CoverCertificatedfaf53edbb
+
+private theorem checked000 : adaptiveCoverCheck 5 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 5 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 5 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 5 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 5 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 5 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 5 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 5 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 5 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 5 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 5 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 5 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 5 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 5 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 5 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 5 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 5 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 5 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 5 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 5 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 5 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 5 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 5 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 5 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 5 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 5 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 5 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 5 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 5 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 5 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 5 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 5 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 5 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 5 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 5 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 5 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 5 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 5 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 5 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 5 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 5 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 5 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 5 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 5 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 5 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 5 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 5 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 5 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 5 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 5 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 5 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 5 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 5 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 5 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 5 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 5 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 5 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 5 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 5 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 5 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 5 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 5 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 5 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 5 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatedfaf53edbb
+
+theorem e24KC2PhiAboveLeaf1110001_c1 :
+    adaptiveCoverCheck 8 phiAboveCell11100011 = true := by
+  exact CoverCertificatedfaf53edbb.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110001_c2_8_00189
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells626f3fc028
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells626f3fc028
+
+open CertificateCells626f3fc028
+namespace CoverCertificatea7d865eaeb
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea7d865eaeb
+
+theorem e24KC2PhiAboveLeaf1110001_c2 :
+    adaptiveCoverCheck 8 phiAboveCell11100012 = true := by
+  exact CoverCertificatea7d865eaeb.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110001_c3_8_00190
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells15f9582e4b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells15f9582e4b
+
+open CertificateCells15f9582e4b
+namespace CoverCertificate59a502ab21
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate59a502ab21
+
+theorem e24KC2PhiAboveLeaf1110001_c3 :
+    adaptiveCoverCheck 8 phiAboveCell11100013 = true := by
+  exact CoverCertificate59a502ab21.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110010_c0_8_00196
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc156062d1b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc156062d1b
+
+open CertificateCellsc156062d1b
+namespace CoverCertificatedb9da39caf
+
+private theorem checked000 : adaptiveCoverCheck 5 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 5 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 5 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 5 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 5 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 5 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 5 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 5 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 5 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 5 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 5 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 5 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 5 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 5 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 5 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 5 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 5 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 5 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 5 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 5 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 5 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 5 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 5 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 5 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 5 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 5 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 5 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 5 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 5 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 5 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 5 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 5 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 5 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 5 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 5 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 5 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 5 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 5 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 5 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 5 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 5 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 5 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 5 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 5 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 5 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 5 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 5 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 5 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 5 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 5 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 5 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 5 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 5 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 5 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 5 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 5 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatedb9da39caf
+
+theorem e24KC2PhiAboveLeaf1110010_c0 :
+    adaptiveCoverCheck 8 phiAboveCell11100100 = true := by
+  exact CoverCertificatedb9da39caf.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110010_c1_8_00197
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdb4a23ce5a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdb4a23ce5a
+
+open CertificateCellsdb4a23ce5a
+namespace CoverCertificateefd199c221
+
+private theorem checked020 : adaptiveCoverCheck 5 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 5 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 5 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 5 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 5 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 5 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 5 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 5 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 5 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 5 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 5 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 5 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 5 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 5 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 5 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 5 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 5 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 5 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 5 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 5 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 5 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 5 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 5 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 5 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 5 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 5 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 5 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 5 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 5 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 5 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 5 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 5 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 5 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 5 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 5 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 5 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 5 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 5 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 5 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 5 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 5 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 5 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 5 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 5 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 5 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 5 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 5 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 5 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateefd199c221
+
+theorem e24KC2PhiAboveLeaf1110010_c1 :
+    adaptiveCoverCheck 8 phiAboveCell11100101 = true := by
+  exact CoverCertificateefd199c221.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110010_c2_8_00198
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8ced9a3143
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8ced9a3143
+
+open CertificateCells8ced9a3143
+namespace CoverCertificatedb27ad4afe
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatedb27ad4afe
+
+theorem e24KC2PhiAboveLeaf1110010_c2 :
+    adaptiveCoverCheck 8 phiAboveCell11100102 = true := by
+  exact CoverCertificatedb27ad4afe.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110010_c3_8_00199
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1c46ec9a00
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1c46ec9a00
+
+open CertificateCells1c46ec9a00
+namespace CoverCertificate05565992a9
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate05565992a9
+
+theorem e24KC2PhiAboveLeaf1110010_c3 :
+    adaptiveCoverCheck 8 phiAboveCell11100103 = true := by
+  exact CoverCertificate05565992a9.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110011_c0_8_00204
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc9ea34c02b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc9ea34c02b
+
+open CertificateCellsc9ea34c02b
+namespace CoverCertificatec7fa9ff5d0
+
+private theorem checked020 : adaptiveCoverCheck 5 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 5 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 5 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 5 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 5 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 5 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 5 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 5 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 5 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 5 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 5 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 5 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 5 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 5 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 5 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 5 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 5 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 5 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 5 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 5 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 5 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 5 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 5 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 5 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 5 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 5 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 5 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 5 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 5 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 5 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 5 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 5 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 5 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 5 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 5 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 5 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 5 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 5 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 5 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 5 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 5 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 5 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 5 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 5 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 5 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 5 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 5 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 5 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec7fa9ff5d0
+
+theorem e24KC2PhiAboveLeaf1110011_c0 :
+    adaptiveCoverCheck 8 phiAboveCell11100110 = true := by
+  exact CoverCertificatec7fa9ff5d0.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110011_c1_8_00205
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells7575776df3
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells7575776df3
+
+open CertificateCells7575776df3
+namespace CoverCertificatef60369d605
+
+private theorem checked020 : adaptiveCoverCheck 5 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 5 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 5 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 5 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 5 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 5 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 5 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 5 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 5 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 5 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 5 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 5 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 5 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 5 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 5 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 5 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 5 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 5 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 5 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 5 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 5 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 5 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 5 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 5 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell213 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef60369d605
+
+theorem e24KC2PhiAboveLeaf1110011_c1 :
+    adaptiveCoverCheck 8 phiAboveCell11100111 = true := by
+  exact CoverCertificatef60369d605.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110011_c2_8_00206
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8b638b7054
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8b638b7054
+
+open CertificateCells8b638b7054
+namespace CoverCertificate5759c94c52
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5759c94c52
+
+theorem e24KC2PhiAboveLeaf1110011_c2 :
+    adaptiveCoverCheck 8 phiAboveCell11100112 = true := by
+  exact CoverCertificate5759c94c52.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Above Leaf1110011_c3_8_00207
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3bd9cbb8df
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3bd9cbb8df
+
+open CertificateCells3bd9cbb8df
+namespace CoverCertificate03a553a361
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate03a553a361
+
+theorem e24KC2PhiAboveLeaf1110011_c3 :
+    adaptiveCoverCheck 8 phiAboveCell11100113 = true := by
+  exact CoverCertificate03a553a361.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33203_c0_8_00242
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells41c704a1f1
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells41c704a1f1
+
+open CertificateCells41c704a1f1
+namespace CoverCertificated3212f4a8b
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated3212f4a8b
+
+theorem e24KC2PhiBelowLeaf33203_c0 :
+    adaptiveCoverCheck 8 (childLL (childHH phiBelowCell3320)) = true := by
+  exact CoverCertificated3212f4a8b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33203_c1_8_00243
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells67d940c92a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells67d940c92a
+
+open CertificateCells67d940c92a
+namespace CoverCertificate92ae3ca2d2
+
+private theorem checked300 : adaptiveCoverCheck 5 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 5 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 5 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 5 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell303 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 5 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 5 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 5 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 5 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 5 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 5 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 5 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 5 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate92ae3ca2d2
+
+theorem e24KC2PhiBelowLeaf33203_c1 :
+    adaptiveCoverCheck 8 (childLH (childHH phiBelowCell3320)) = true := by
+  exact CoverCertificate92ae3ca2d2.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33203_c2_8_00244
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8d68909634
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8d68909634
+
+open CertificateCells8d68909634
+namespace CoverCertificate126af0396e
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate126af0396e
+
+theorem e24KC2PhiBelowLeaf33203_c2 :
+    adaptiveCoverCheck 8 (childHL (childHH phiBelowCell3320)) = true := by
+  exact CoverCertificate126af0396e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33203_c3_8_00245
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc208943a95
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc208943a95
+
+open CertificateCellsc208943a95
+namespace CoverCertificate7a0799a8ec
+
+private theorem checked100 : adaptiveCoverCheck 5 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 5 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 5 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 5 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 5 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 5 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 5 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 5 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 5 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 5 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 5 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 5 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 5 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 5 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 5 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 5 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell133 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 5 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 5 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 5 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 5 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 5 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 5 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 5 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 5 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 5 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 5 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 5 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 5 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 5 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 5 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 5 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 5 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate7a0799a8ec
+
+theorem e24KC2PhiBelowLeaf33203_c3 :
+    adaptiveCoverCheck 8 (childHH (childHH phiBelowCell3320)) = true := by
+  exact CoverCertificate7a0799a8ec.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33221_c0_8_00256
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells42a66228ec
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells42a66228ec
+
+open CertificateCells42a66228ec
+namespace CoverCertificate0c897fb4ca
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0c897fb4ca
+
+theorem e24KC2PhiBelowLeaf33221_c0 :
+    adaptiveCoverCheck 8 (childLL (childLH phiBelowCell3322)) = true := by
+  exact CoverCertificate0c897fb4ca.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33221_c1_8_00257
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells67af310b24
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells67af310b24
+
+open CertificateCells67af310b24
+namespace CoverCertificateac87260947
+
+private theorem checked100 : adaptiveCoverCheck 5 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 5 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 5 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 5 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 5 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 5 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 5 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 5 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 5 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 5 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 5 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 5 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 5 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 5 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 5 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 5 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell133 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 5 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 5 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 5 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 5 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell213 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 5 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 5 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 5 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 5 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 5 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 5 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 5 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 5 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 5 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 5 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 5 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 5 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 5 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 5 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 5 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 5 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 5 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 5 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 5 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 5 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateac87260947
+
+theorem e24KC2PhiBelowLeaf33221_c1 :
+    adaptiveCoverCheck 8 (childLH (childLH phiBelowCell3322)) = true := by
+  exact CoverCertificateac87260947.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33221_c2_8_00258
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9b91503b21
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9b91503b21
+
+open CertificateCells9b91503b21
+namespace CoverCertificatee4e72882c5
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee4e72882c5
+
+theorem e24KC2PhiBelowLeaf33221_c2 :
+    adaptiveCoverCheck 8 (childHL (childLH phiBelowCell3322)) = true := by
+  exact CoverCertificatee4e72882c5.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33221_c3_8_00259
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb36cfca8f0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb36cfca8f0
+
+open CertificateCellsb36cfca8f0
+namespace CoverCertificateeab8fed9d7
+
+private theorem checked010 : adaptiveCoverCheck 5 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 5 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 5 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 5 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell013 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 5 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 5 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 5 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 5 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 5 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 5 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 5 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 5 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 5 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 5 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 5 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 5 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 5 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 5 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 5 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 5 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 5 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 5 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 5 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 5 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell133 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 5 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 5 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 5 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 5 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell213 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 5 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 5 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 5 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 5 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 5 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 5 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 5 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 5 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 5 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 5 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 5 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 5 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 5 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 5 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 5 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 5 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 5 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 5 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 5 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 5 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateeab8fed9d7
+
+theorem e24KC2PhiBelowLeaf33221_c3 :
+    adaptiveCoverCheck 8 (childHH (childLH phiBelowCell3322)) = true := by
+  exact CoverCertificateeab8fed9d7.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle025.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle025.lean
new file mode 100644
index 0000000000..793c240b01
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle025.lean
@@ -0,0 +1,9692 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch038`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CoverCertificate23bfa411d5
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childHL (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate23bfa411d5
+
+namespace GerverSofa.PartE.CoverCertificate6b64856605
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childHL (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate6b64856605
+
+namespace GerverSofa.PartE.CoverCertificate1cb4de1b4d
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childHL (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate1cb4de1b4d
+
+namespace GerverSofa.PartE.CoverCertificate3f82ba6920
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childHL (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificate3f82ba6920
+
+namespace GerverSofa.PartE.CoverCertificate01ce83adcb
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate01ce83adcb
+
+namespace GerverSofa.PartE.CoverCertificatea44c879c6c
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+end GerverSofa.PartE.CoverCertificatea44c879c6c
+
+namespace GerverSofa.PartE.CoverCertificateed570ec179
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childLL (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificateed570ec179
+
+namespace GerverSofa.PartE.CoverCertificate40a2855a31
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childLL (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate40a2855a31
+
+namespace GerverSofa.PartE.CoverCertificate1f5c554cb5
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childLL (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate1f5c554cb5
+
+namespace GerverSofa.PartE.CoverCertificate4c26cf7a20
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childLL (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate4c26cf7a20
+
+namespace GerverSofa.PartE.CoverCertificate3fd78c5005
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childLL (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+end GerverSofa.PartE.CoverCertificate3fd78c5005
+
+namespace GerverSofa.PartE.CoverCertificateb53ccd0a15
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childLL (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificateb53ccd0a15
+
+namespace GerverSofa.PartE.CoverCertificate1f18ab38ed
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childLL (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate1f18ab38ed
+
+namespace GerverSofa.PartE.CoverCertificatef023150b0c
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childLL (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatef023150b0c
+
+namespace GerverSofa.PartE.CoverCertificateaf9fa2570b
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificateaf9fa2570b
+
+namespace GerverSofa.PartE.CoverCertificate837ca9ea44
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate837ca9ea44
+
+namespace GerverSofa.PartE.CoverCertificatebe04593b3c
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatebe04593b3c
+
+namespace GerverSofa.PartE.CoverCertificate0452312845
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate0452312845
+
+namespace GerverSofa.PartE.CoverCertificate3947a065a0
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate3947a065a0
+
+namespace GerverSofa.PartE.CoverCertificate4722a46e95
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate4722a46e95
+
+namespace GerverSofa.PartE.CoverCertificate31c2c5878b
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate31c2c5878b
+
+namespace GerverSofa.PartE.CoverCertificate5543096fd0
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate5543096fd0
+
+namespace GerverSofa.PartE.CoverCertificate0e40621a3c
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate0e40621a3c
+
+namespace GerverSofa.PartE.CoverCertificate63d38ae319
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childLH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate63d38ae319
+
+namespace GerverSofa.PartE.CoverCertificated27c8c9957
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childLH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificated27c8c9957
+
+namespace GerverSofa.PartE.CoverCertificatedd39c23c21
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childLH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatedd39c23c21
+
+namespace GerverSofa.PartE.CoverCertificate93b82d2646
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childLH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate93b82d2646
+
+namespace GerverSofa.PartE.CoverCertificateb0fb1ed898
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childLH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificateb0fb1ed898
+
+namespace GerverSofa.PartE.CoverCertificatee4cd586f88
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childLH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatee4cd586f88
+
+namespace GerverSofa.PartE.CoverCertificate6f132ca828
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childLH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate6f132ca828
+
+namespace GerverSofa.PartE.CoverCertificate3f0c4cdced
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childLH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+end GerverSofa.PartE.CoverCertificate3f0c4cdced
+
+namespace GerverSofa.PartE.CoverCertificatee4839d9dd1
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+private abbrev cell1100 : AngleCell :=
+  childLL cell110
+
+private abbrev cell1101 : AngleCell :=
+  childLH cell110
+
+private abbrev cell1102 : AngleCell :=
+  childHL cell110
+
+private abbrev cell1103 : AngleCell :=
+  childHH cell110
+
+private abbrev cell1110 : AngleCell :=
+  childLL cell111
+
+private abbrev cell1111 : AngleCell :=
+  childLH cell111
+
+private abbrev cell1112 : AngleCell :=
+  childHL cell111
+
+private abbrev cell1113 : AngleCell :=
+  childHH cell111
+
+private abbrev cell1120 : AngleCell :=
+  childLL cell112
+
+private abbrev cell1121 : AngleCell :=
+  childLH cell112
+
+private abbrev cell1122 : AngleCell :=
+  childHL cell112
+
+private abbrev cell1123 : AngleCell :=
+  childHH cell112
+
+private abbrev cell1130 : AngleCell :=
+  childLL cell113
+
+private abbrev cell1131 : AngleCell :=
+  childLH cell113
+
+private abbrev cell1132 : AngleCell :=
+  childHL cell113
+
+private abbrev cell1133 : AngleCell :=
+  childHH cell113
+
+private abbrev cell1310 : AngleCell :=
+  childLL cell131
+
+private abbrev cell1311 : AngleCell :=
+  childLH cell131
+
+private abbrev cell1312 : AngleCell :=
+  childHL cell131
+
+private abbrev cell1313 : AngleCell :=
+  childHH cell131
+
+end GerverSofa.PartE.CoverCertificatee4839d9dd1
+
+namespace GerverSofa.PartE.CoverCertificateb46d993ac8
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificateb46d993ac8
+
+namespace GerverSofa.PartE.CoverCertificate98e57d34e4
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childHH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate98e57d34e4
+
+namespace GerverSofa.PartE.CoverCertificate27d7b561c3
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLH (childHH (childHL (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate27d7b561c3
+
+namespace GerverSofa.PartE.CoverCertificate25f92715ca
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLH (childHH (childHL (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate25f92715ca
+
+namespace GerverSofa.PartE.CertificateCells8d1955d9db
+
+/-- Subcell `3322` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3322 : AngleCell :=
+  childHL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells8d1955d9db
+
+namespace GerverSofa.PartE.CertificateCells5a22a388aa
+
+/-- Subcell `3322` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3322 : AngleCell :=
+  childHL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells5a22a388aa
+
+namespace GerverSofa.PartE.CertificateCells24ba63b604
+
+/-- Subcell `3322` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3322 : AngleCell :=
+  childHL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells24ba63b604
+
+namespace GerverSofa.PartE.CertificateCells78208ba1a7
+
+/-- Subcell `3322` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3322 : AngleCell :=
+  childHL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells78208ba1a7
+
+namespace GerverSofa.PartE.CertificateCellsc3455a44b7
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsc3455a44b7
+
+namespace GerverSofa.PartE.CertificateCells9c4f656d63
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells9c4f656d63
+
+namespace GerverSofa.PartE.CertificateCells37244c6e24
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells37244c6e24
+
+namespace GerverSofa.PartE.CertificateCells3e4038f4ea
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells3e4038f4ea
+
+namespace GerverSofa.PartE.CertificateCells25be007753
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells25be007753
+
+namespace GerverSofa.PartE.CertificateCells9371db9199
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells9371db9199
+
+namespace GerverSofa.PartE.CertificateCells2426e39946
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells2426e39946
+
+namespace GerverSofa.PartE.CertificateCellsa09b62ac68
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa09b62ac68
+
+namespace GerverSofa.PartE.CertificateCells296b03de2f
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells296b03de2f
+
+namespace GerverSofa.PartE.CertificateCells230653113b
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells230653113b
+
+namespace GerverSofa.PartE.CertificateCells02421cb4a8
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells02421cb4a8
+
+namespace GerverSofa.PartE.CertificateCells8af559ef49
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells8af559ef49
+
+namespace GerverSofa.PartE.CertificateCells874482b603
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells874482b603
+
+namespace GerverSofa.PartE.CertificateCellse5b788289b
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellse5b788289b
+
+namespace GerverSofa.PartE.CertificateCellsc2714ba804
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232100` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232100 : AngleCell :=
+  childLL (childLL (childLH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsc2714ba804
+
+namespace GerverSofa.PartE.CertificateCellscab6c01cd3
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232101` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232101 : AngleCell :=
+  childLH (childLL (childLH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellscab6c01cd3
+
+namespace GerverSofa.PartE.CertificateCells50cb464285
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232102` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232102 : AngleCell :=
+  childHL (childLL (childLH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells50cb464285
+
+namespace GerverSofa.PartE.CertificateCellsb0d9885c34
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232103` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232103 : AngleCell :=
+  childHH (childLL (childLH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsb0d9885c34
+
+namespace GerverSofa.PartE.CertificateCellsa6cd731e2a
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa6cd731e2a
+
+namespace GerverSofa.PartE.CertificateCells786c9d28c6
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232120` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232120 : AngleCell :=
+  childLL (childHL (childLH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells786c9d28c6
+
+namespace GerverSofa.PartE.CertificateCellsf9e248a6ed
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232121` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232121 : AngleCell :=
+  childLH (childHL (childLH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsf9e248a6ed
+
+namespace GerverSofa.PartE.CertificateCells576f3a7876
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232122` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232122 : AngleCell :=
+  childHL (childHL (childLH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells576f3a7876
+
+namespace GerverSofa.PartE.CertificateCellsf783690d6f
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232123` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232123 : AngleCell :=
+  childHH (childHL (childLH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsf783690d6f
+
+namespace GerverSofa.PartE.CertificateCellsde47ee26bd
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232130` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232130 : AngleCell :=
+  childLL (childHH (childLH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsde47ee26bd
+
+namespace GerverSofa.PartE.CertificateCells05c25def9b
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232131` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232131 : AngleCell :=
+  childLH (childHH (childLH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells05c25def9b
+
+namespace GerverSofa.PartE.CertificateCellsc4f0c2a140
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232132` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232132 : AngleCell :=
+  childHL (childHH (childLH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsc4f0c2a140
+
+namespace GerverSofa.PartE.CertificateCellsc119914926
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232133` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232133 : AngleCell :=
+  childHH (childHH (childLH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsc119914926
+
+namespace GerverSofa.PartE.CertificateCells70881db6a6
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells70881db6a6
+
+namespace GerverSofa.PartE.CertificateCellsfed0e78e01
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsfed0e78e01
+
+namespace GerverSofa.PartE.CertificateCells4781918595
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232310` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232310 : AngleCell :=
+  childLL (childLH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells4781918595
+
+namespace GerverSofa.PartE.CertificateCellsa584418aa1
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232311` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232311 : AngleCell :=
+  childLH (childLH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsa584418aa1
+
+namespace GerverSofa.PartE.CertificateCells09c85e6b71
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232311` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232311 : AngleCell :=
+  childLH (childLH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells09c85e6b71
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.PhiBelow.Leaf00265`.
+* `KernelOnly.PartE.PhiBelow.Leaf00266`.
+* `KernelOnly.PartE.PhiBelow.Leaf00267`.
+* `KernelOnly.PartE.PhiBelow.Leaf00268`.
+* `KernelOnly.PartE.PhiBelow.Leaf00271`.
+* `KernelOnly.PartE.PhiBelow.Leaf00272`.
+* `KernelOnly.PartE.PhiBelow.Leaf00274`.
+* `KernelOnly.PartE.PhiBelow.Leaf00275`.
+* `KernelOnly.PartE.PhiBelow.Leaf00276`.
+* `KernelOnly.PartE.PhiBelow.Leaf00277`.
+* `KernelOnly.PartE.PhiBelow.Leaf00280`.
+* `KernelOnly.PartE.PhiBelow.Leaf00281`.
+* `KernelOnly.PartE.PhiBelow.Leaf00282`.
+* `KernelOnly.PartE.PhiBelow.Leaf00283`.
+* `KernelOnly.PartE.PhiBelow.Leaf00290`.
+* `KernelOnly.PartE.PhiBelow.Leaf00291`.
+* `KernelOnly.PartE.PhiBelow.Leaf00292`.
+* `KernelOnly.PartE.PhiBelow.Leaf00293`.
+* `KernelOnly.PartE.PhiBelow.Leaf00297`.
+* `KernelOnly.PartE.PhiBelow.Leaf00298`.
+* `KernelOnly.PartE.PhiBelow.Leaf00299`.
+* `KernelOnly.PartE.PhiBelow.Leaf00300`.
+* `KernelOnly.PartE.PhiBelow.Leaf00302`.
+* `KernelOnly.PartE.PhiBelow.Leaf00304`.
+* `KernelOnly.PartE.PhiBelow.Leaf00305`.
+* `KernelOnly.PartE.PhiBelow.Leaf00306`.
+* `KernelOnly.PartE.PhiBelow.Leaf00307`.
+* `KernelOnly.PartE.PhiBelow.Leaf00310`.
+* `KernelOnly.PartE.PhiBelow.Leaf00311`.
+* `KernelOnly.PartE.PhiBelow.Leaf00312`.
+* `KernelOnly.PartE.PhiBelow.Leaf00313`.
+* `KernelOnly.PartE.PhiBelow.Leaf00316`.
+* `KernelOnly.PartE.PhiBelow.Leaf00318`.
+* `KernelOnly.PartE.PhiBelow.Leaf00320`.
+* `KernelOnly.PartE.PhiBelow.Leaf00322`.
+* `KernelOnly.PartE.PhiBelow.Leaf00323`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33223_c0_8_00265
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8d1955d9db
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8d1955d9db
+
+open CertificateCells8d1955d9db
+namespace CoverCertificate23bfa411d5
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate23bfa411d5
+
+theorem e24KC2PhiBelowLeaf33223_c0 :
+    adaptiveCoverCheck 8 (childLL (childHH phiBelowCell3322)) = true := by
+  exact CoverCertificate23bfa411d5.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33223_c1_8_00266
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5a22a388aa
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5a22a388aa
+
+open CertificateCells5a22a388aa
+namespace CoverCertificate6b64856605
+
+private theorem checked010 : adaptiveCoverCheck 5 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 5 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 5 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 5 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell013 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 5 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 5 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 5 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 5 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 5 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 5 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 5 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 5 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 5 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 5 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 5 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 5 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 5 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 5 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 5 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 5 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 5 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 5 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 5 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 5 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell133 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 5 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 5 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 5 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 5 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 5 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 5 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 5 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 5 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 5 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 5 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 5 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 5 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 5 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 5 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 5 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 5 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 5 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 5 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 5 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 5 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate6b64856605
+
+theorem e24KC2PhiBelowLeaf33223_c1 :
+    adaptiveCoverCheck 8 (childLH (childHH phiBelowCell3322)) = true := by
+  exact CoverCertificate6b64856605.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33223_c2_8_00267
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells24ba63b604
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells24ba63b604
+
+open CertificateCells24ba63b604
+namespace CoverCertificate1cb4de1b4d
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 7 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1cb4de1b4d
+
+theorem e24KC2PhiBelowLeaf33223_c2 :
+    adaptiveCoverCheck 8 (childHL (childHH phiBelowCell3322)) = true := by
+  exact CoverCertificate1cb4de1b4d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33223_c3_8_00268
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells78208ba1a7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells78208ba1a7
+
+open CertificateCells78208ba1a7
+namespace CoverCertificate3f82ba6920
+
+private theorem checked110 : adaptiveCoverCheck 5 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 5 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 5 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 5 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3f82ba6920
+
+theorem e24KC2PhiBelowLeaf33223_c3 :
+    adaptiveCoverCheck 8 (childHH (childHH phiBelowCell3322)) = true := by
+  exact CoverCertificate3f82ba6920.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33230_c0_8_00271
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc3455a44b7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc3455a44b7
+
+open CertificateCellsc3455a44b7
+namespace CoverCertificate01ce83adcb
+
+private theorem checked000 : adaptiveCoverCheck 5 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 5 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 5 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 5 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 5 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 5 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 5 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 5 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 5 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 5 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 5 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 5 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 5 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 5 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 5 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 5 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 5 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 5 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 5 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 5 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 5 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 5 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 5 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 5 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 5 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 5 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 5 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 5 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 5 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 5 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 5 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 5 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 5 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 5 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 5 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 5 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 5 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 5 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 5 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 5 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 5 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 5 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 5 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 5 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 5 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 5 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 5 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 5 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 5 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 5 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 5 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 5 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 5 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 5 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 5 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 5 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 5 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 5 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 5 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 5 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 5 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 5 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 5 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 5 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate01ce83adcb
+
+theorem e24KC2PhiBelowLeaf33230_c0 :
+    adaptiveCoverCheck 8 (childLL (childLL phiBelowCell3323)) = true := by
+  exact CoverCertificate01ce83adcb.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33230_c1_8_00272
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9c4f656d63
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9c4f656d63
+
+open CertificateCells9c4f656d63
+namespace CoverCertificatea44c879c6c
+
+private theorem checked020 : adaptiveCoverCheck 5 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 5 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 5 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 5 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell023 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 5 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 5 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 5 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 5 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell203 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 5 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 5 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 5 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 5 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 5 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 5 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 5 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 5 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell233 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea44c879c6c
+
+theorem e24KC2PhiBelowLeaf33230_c1 :
+    adaptiveCoverCheck 8 (childLH (childLL phiBelowCell3323)) = true := by
+  exact CoverCertificatea44c879c6c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33230_c2_c0_7_00274
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells37244c6e24
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells37244c6e24
+
+open CertificateCells37244c6e24
+namespace CoverCertificateed570ec179
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateed570ec179
+
+theorem e24KC2PhiBelowLeaf33230_c2_c0 :
+    adaptiveCoverCheck 7 (childLL (childHL (childLL phiBelowCell3323))) = true := by
+  exact CoverCertificateed570ec179.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33230_c2_c1_7_00275
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3e4038f4ea
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3e4038f4ea
+
+open CertificateCells3e4038f4ea
+namespace CoverCertificate40a2855a31
+
+private theorem checked330 : adaptiveCoverCheck 4 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 4 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 4 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 4 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate40a2855a31
+
+theorem e24KC2PhiBelowLeaf33230_c2_c1 :
+    adaptiveCoverCheck 7 (childLH (childHL (childLL phiBelowCell3323))) = true := by
+  exact CoverCertificate40a2855a31.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33230_c2_c2_7_00276
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells25be007753
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells25be007753
+
+open CertificateCells25be007753
+namespace CoverCertificate1f5c554cb5
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1f5c554cb5
+
+theorem e24KC2PhiBelowLeaf33230_c2_c2 :
+    adaptiveCoverCheck 7 (childHL (childHL (childLL phiBelowCell3323))) = true := by
+  exact CoverCertificate1f5c554cb5.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33230_c2_c3_7_00277
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9371db9199
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9371db9199
+
+open CertificateCells9371db9199
+namespace CoverCertificate4c26cf7a20
+
+private theorem checked110 : adaptiveCoverCheck 4 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 4 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 4 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 4 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell113 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 4 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 4 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 4 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 4 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell133 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 4 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 4 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 4 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 4 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell313 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 4 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 4 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 4 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 4 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4c26cf7a20
+
+theorem e24KC2PhiBelowLeaf33230_c2_c3 :
+    adaptiveCoverCheck 7 (childHH (childHL (childLL phiBelowCell3323))) = true := by
+  exact CoverCertificate4c26cf7a20.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33230_c3_c0_7_00280
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2426e39946
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2426e39946
+
+open CertificateCells2426e39946
+namespace CoverCertificate3fd78c5005
+
+private theorem checked220 : adaptiveCoverCheck 4 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 4 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 4 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 4 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell223 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3fd78c5005
+
+theorem e24KC2PhiBelowLeaf33230_c3_c0 :
+    adaptiveCoverCheck 7 (childLL (childHH (childLL phiBelowCell3323))) = true := by
+  exact CoverCertificate3fd78c5005.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33230_c3_c1_7_00281
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa09b62ac68
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa09b62ac68
+
+open CertificateCellsa09b62ac68
+namespace CoverCertificateb53ccd0a15
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateb53ccd0a15
+
+theorem e24KC2PhiBelowLeaf33230_c3_c1 :
+    adaptiveCoverCheck 7 (childLH (childHH (childLL phiBelowCell3323))) = true := by
+  exact CoverCertificateb53ccd0a15.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33230_c3_c2_7_00282
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells296b03de2f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells296b03de2f
+
+open CertificateCells296b03de2f
+namespace CoverCertificate1f18ab38ed
+
+private theorem checked000 : adaptiveCoverCheck 4 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 4 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 4 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 4 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 4 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 4 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 4 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 4 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 4 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 4 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 4 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 4 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 4 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 4 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 4 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 4 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 4 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 4 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 4 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 4 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell123 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 4 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 4 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 4 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 4 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 4 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 4 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 4 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 4 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 4 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 4 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 4 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 4 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 4 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 4 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 4 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 4 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 4 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 4 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 4 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 4 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell303 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 4 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 4 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 4 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 4 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 4 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 4 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 4 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 4 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1f18ab38ed
+
+theorem e24KC2PhiBelowLeaf33230_c3_c2 :
+    adaptiveCoverCheck 7 (childHL (childHH (childLL phiBelowCell3323))) = true := by
+  exact CoverCertificate1f18ab38ed.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33230_c3_c3_7_00283
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells230653113b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells230653113b
+
+open CertificateCells230653113b
+namespace CoverCertificatef023150b0c
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef023150b0c
+
+theorem e24KC2PhiBelowLeaf33230_c3_c3 :
+    adaptiveCoverCheck 7 (childHH (childHH (childLL phiBelowCell3323))) = true := by
+  exact CoverCertificatef023150b0c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c0_c0_7_00290
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells02421cb4a8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells02421cb4a8
+
+open CertificateCells02421cb4a8
+namespace CoverCertificateaf9fa2570b
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateaf9fa2570b
+
+theorem e24KC2PhiBelowLeaf33232_c0_c0 :
+    adaptiveCoverCheck 7 (childLL (childLL (childHL phiBelowCell3323))) = true := by
+  exact CoverCertificateaf9fa2570b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c0_c1_7_00291
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8af559ef49
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8af559ef49
+
+open CertificateCells8af559ef49
+namespace CoverCertificate837ca9ea44
+
+private theorem checked110 : adaptiveCoverCheck 4 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 4 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 4 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 4 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell113 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 4 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 4 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 4 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 4 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell133 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 4 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 4 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 4 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 4 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell313 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 4 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 4 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 4 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 4 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate837ca9ea44
+
+theorem e24KC2PhiBelowLeaf33232_c0_c1 :
+    adaptiveCoverCheck 7 (childLH (childLL (childHL phiBelowCell3323))) = true := by
+  exact CoverCertificate837ca9ea44.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c0_c2_7_00292
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells874482b603
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells874482b603
+
+open CertificateCells874482b603
+namespace CoverCertificatebe04593b3c
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatebe04593b3c
+
+theorem e24KC2PhiBelowLeaf33232_c0_c2 :
+    adaptiveCoverCheck 7 (childHL (childLL (childHL phiBelowCell3323))) = true := by
+  exact CoverCertificatebe04593b3c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c0_c3_7_00293
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse5b788289b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse5b788289b
+
+open CertificateCellse5b788289b
+namespace CoverCertificate0452312845
+
+private theorem checked110 : adaptiveCoverCheck 4 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 4 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 4 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 4 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 4 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 4 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 4 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 4 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 4 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 4 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 4 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 4 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell133 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 4 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 4 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 4 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 4 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 4 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 4 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 4 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 4 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 4 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 4 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 4 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 4 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 4 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 4 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 4 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 4 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0452312845
+
+theorem e24KC2PhiBelowLeaf33232_c0_c3 :
+    adaptiveCoverCheck 7 (childHH (childLL (childHL phiBelowCell3323))) = true := by
+  exact CoverCertificate0452312845.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c1_c0_c0_6_00297
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc2714ba804
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc2714ba804
+
+open CertificateCellsc2714ba804
+namespace CoverCertificate3947a065a0
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3947a065a0
+
+theorem e24KC2PhiBelowLeaf33232_c1_c0_c0 :
+    adaptiveCoverCheck 6 phiBelowCell33232100 = true := by
+  exact CoverCertificate3947a065a0.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c1_c0_c1_6_00298
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellscab6c01cd3
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellscab6c01cd3
+
+open CertificateCellscab6c01cd3
+namespace CoverCertificate4722a46e95
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4722a46e95
+
+theorem e24KC2PhiBelowLeaf33232_c1_c0_c1 :
+    adaptiveCoverCheck 6 phiBelowCell33232101 = true := by
+  exact CoverCertificate4722a46e95.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c1_c0_c2_6_00299
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells50cb464285
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells50cb464285
+
+open CertificateCells50cb464285
+namespace CoverCertificate31c2c5878b
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate31c2c5878b
+
+theorem e24KC2PhiBelowLeaf33232_c1_c0_c2 :
+    adaptiveCoverCheck 6 phiBelowCell33232102 = true := by
+  exact CoverCertificate31c2c5878b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c1_c0_c3_6_00300
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb0d9885c34
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb0d9885c34
+
+open CertificateCellsb0d9885c34
+namespace CoverCertificate5543096fd0
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5543096fd0
+
+theorem e24KC2PhiBelowLeaf33232_c1_c0_c3 :
+    adaptiveCoverCheck 6 phiBelowCell33232103 = true := by
+  exact CoverCertificate5543096fd0.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c1_c1_7_00302
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa6cd731e2a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa6cd731e2a
+
+open CertificateCellsa6cd731e2a
+namespace CoverCertificate0e40621a3c
+
+private theorem checked000 : adaptiveCoverCheck 4 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 4 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 4 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 4 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell003 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 4 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 4 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 4 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 4 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 4 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 4 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 4 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 4 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell033 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 4 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 4 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 4 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 4 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 4 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 4 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 4 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 4 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 4 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 4 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 4 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 4 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 4 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 4 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 4 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 4 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 4 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 4 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 4 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 4 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell303 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 4 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 4 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 4 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 4 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 4 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 4 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 4 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 4 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0e40621a3c
+
+theorem e24KC2PhiBelowLeaf33232_c1_c1 :
+    adaptiveCoverCheck 7 (childLH (childLH (childHL phiBelowCell3323))) = true := by
+  exact CoverCertificate0e40621a3c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c1_c2_c0_6_00304
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells786c9d28c6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells786c9d28c6
+
+open CertificateCells786c9d28c6
+namespace CoverCertificate63d38ae319
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate63d38ae319
+
+theorem e24KC2PhiBelowLeaf33232_c1_c2_c0 :
+    adaptiveCoverCheck 6 phiBelowCell33232120 = true := by
+  exact CoverCertificate63d38ae319.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c1_c2_c1_6_00305
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf9e248a6ed
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf9e248a6ed
+
+open CertificateCellsf9e248a6ed
+namespace CoverCertificated27c8c9957
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated27c8c9957
+
+theorem e24KC2PhiBelowLeaf33232_c1_c2_c1 :
+    adaptiveCoverCheck 6 phiBelowCell33232121 = true := by
+  exact CoverCertificated27c8c9957.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c1_c2_c2_6_00306
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells576f3a7876
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells576f3a7876
+
+open CertificateCells576f3a7876
+namespace CoverCertificatedd39c23c21
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatedd39c23c21
+
+theorem e24KC2PhiBelowLeaf33232_c1_c2_c2 :
+    adaptiveCoverCheck 6 phiBelowCell33232122 = true := by
+  exact CoverCertificatedd39c23c21.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c1_c2_c3_6_00307
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf783690d6f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf783690d6f
+
+open CertificateCellsf783690d6f
+namespace CoverCertificate93b82d2646
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate93b82d2646
+
+theorem e24KC2PhiBelowLeaf33232_c1_c2_c3 :
+    adaptiveCoverCheck 6 phiBelowCell33232123 = true := by
+  exact CoverCertificate93b82d2646.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c1_c3_c0_6_00310
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsde47ee26bd
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsde47ee26bd
+
+open CertificateCellsde47ee26bd
+namespace CoverCertificateb0fb1ed898
+
+private theorem checked330 : adaptiveCoverCheck 3 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 3 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 3 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 3 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateb0fb1ed898
+
+theorem e24KC2PhiBelowLeaf33232_c1_c3_c0 :
+    adaptiveCoverCheck 6 phiBelowCell33232130 = true := by
+  exact CoverCertificateb0fb1ed898.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c1_c3_c1_6_00311
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells05c25def9b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells05c25def9b
+
+open CertificateCells05c25def9b
+namespace CoverCertificatee4cd586f88
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee4cd586f88
+
+theorem e24KC2PhiBelowLeaf33232_c1_c3_c1 :
+    adaptiveCoverCheck 6 phiBelowCell33232131 = true := by
+  exact CoverCertificatee4cd586f88.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c1_c3_c2_6_00312
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc4f0c2a140
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc4f0c2a140
+
+open CertificateCellsc4f0c2a140
+namespace CoverCertificate6f132ca828
+
+private theorem checked110 : adaptiveCoverCheck 3 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 3 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 3 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 3 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell113 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 3 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 3 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 3 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 3 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell133 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 3 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 3 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 3 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 3 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell313 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 3 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 3 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 3 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 3 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate6f132ca828
+
+theorem e24KC2PhiBelowLeaf33232_c1_c3_c2 :
+    adaptiveCoverCheck 6 phiBelowCell33232132 = true := by
+  exact CoverCertificate6f132ca828.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c1_c3_c3_6_00313
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc119914926
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc119914926
+
+open CertificateCellsc119914926
+namespace CoverCertificate3f0c4cdced
+
+private theorem checked000 : adaptiveCoverCheck 3 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 3 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 3 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 3 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell003 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 3 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 3 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 3 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 3 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 3 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 3 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 3 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 3 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell033 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 3 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 3 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 3 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 3 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 3 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 3 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 3 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 3 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 3 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 3 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 3 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 3 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 3 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 3 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 3 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 3 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 3 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 3 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 3 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 3 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell303 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 3 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 3 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 3 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 3 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell323 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3f0c4cdced
+
+theorem e24KC2PhiBelowLeaf33232_c1_c3_c3 :
+    adaptiveCoverCheck 6 phiBelowCell33232133 = true := by
+  exact CoverCertificate3f0c4cdced.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c2_8_00316
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells70881db6a6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells70881db6a6
+
+open CertificateCells70881db6a6
+namespace CoverCertificatee4839d9dd1
+
+private theorem checked1100 : adaptiveCoverCheck 4 cell1100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1100 (by decide +kernel)
+
+private theorem checked1101 : adaptiveCoverCheck 4 cell1101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1101 (by decide +kernel)
+
+private theorem checked1102 : adaptiveCoverCheck 4 cell1102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1102 (by decide +kernel)
+
+private theorem checked1103 : adaptiveCoverCheck 4 cell1103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1103 (by decide +kernel)
+
+private theorem checked1110 : adaptiveCoverCheck 4 cell1110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1110 (by decide +kernel)
+
+private theorem checked1111 : adaptiveCoverCheck 4 cell1111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1111 (by decide +kernel)
+
+private theorem checked1112 : adaptiveCoverCheck 4 cell1112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1112 (by decide +kernel)
+
+private theorem checked1113 : adaptiveCoverCheck 4 cell1113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1113 (by decide +kernel)
+
+private theorem checked1120 : adaptiveCoverCheck 4 cell1120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1120 (by decide +kernel)
+
+private theorem checked1121 : adaptiveCoverCheck 4 cell1121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1121 (by decide +kernel)
+
+private theorem checked1122 : adaptiveCoverCheck 4 cell1122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1122 (by decide +kernel)
+
+private theorem checked1123 : adaptiveCoverCheck 4 cell1123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1123 (by decide +kernel)
+
+private theorem checked1130 : adaptiveCoverCheck 4 cell1130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1130 (by decide +kernel)
+
+private theorem checked1131 : adaptiveCoverCheck 4 cell1131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1131 (by decide +kernel)
+
+private theorem checked1132 : adaptiveCoverCheck 4 cell1132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1132 (by decide +kernel)
+
+private theorem checked1133 : adaptiveCoverCheck 4 cell1133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1133 (by decide +kernel)
+
+private theorem checked1310 : adaptiveCoverCheck 4 cell1310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1310 (by decide +kernel)
+
+private theorem checked1311 : adaptiveCoverCheck 4 cell1311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1311 (by decide +kernel)
+
+private theorem checked1312 : adaptiveCoverCheck 4 cell1312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1312 (by decide +kernel)
+
+private theorem checked1313 : adaptiveCoverCheck 4 cell1313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1313 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 5 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 5 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 5 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 5 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 5 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 5 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 5 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 5 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 5 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 5 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 5 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 5 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 5 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 5 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 5 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 5 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 5 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 5 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 5 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 5 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 5 cell110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell110
+    checked1100 checked1101 checked1102 checked1103
+
+private theorem checked111 : adaptiveCoverCheck 5 cell111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell111
+    checked1110 checked1111 checked1112 checked1113
+
+private theorem checked112 : adaptiveCoverCheck 5 cell112 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell112
+    checked1120 checked1121 checked1122 checked1123
+
+private theorem checked113 : adaptiveCoverCheck 5 cell113 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell113
+    checked1130 checked1131 checked1132 checked1133
+
+private theorem checked120 : adaptiveCoverCheck 5 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 5 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 5 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 5 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 5 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 5 cell131 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell131
+    checked1310 checked1311 checked1312 checked1313
+
+private theorem checked132 : adaptiveCoverCheck 5 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 5 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell133 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 5 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 5 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 5 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 5 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 5 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 5 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 5 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 5 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 5 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 5 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 5 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 5 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 5 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 5 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 5 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 5 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 5 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 5 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 5 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 5 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee4839d9dd1
+
+theorem e24KC2PhiBelowLeaf33232_c2 :
+    adaptiveCoverCheck 8 (childHL (childHL phiBelowCell3323)) = true := by
+  exact CoverCertificatee4839d9dd1.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c0_7_00318
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsfed0e78e01
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsfed0e78e01
+
+open CertificateCellsfed0e78e01
+namespace CoverCertificateb46d993ac8
+
+private theorem checked000 : adaptiveCoverCheck 4 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 4 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 4 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 4 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 4 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 4 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 4 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 4 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 4 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 4 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 4 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 4 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 4 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 4 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 4 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 4 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 4 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 4 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 4 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 4 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 4 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 4 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 4 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 4 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 4 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 4 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 4 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 4 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 4 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 4 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 4 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 4 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 4 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 4 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 4 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 4 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 4 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 4 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 4 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 4 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 4 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 4 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 4 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 4 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 4 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 4 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 4 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 4 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 4 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 4 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 4 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 4 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 4 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 4 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 4 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 4 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 4 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 4 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 4 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 4 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 4 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 4 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 4 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 4 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateb46d993ac8
+
+theorem e24KC2PhiBelowLeaf33232_c3_c0 :
+    adaptiveCoverCheck 7 (childLL (childHH (childHL phiBelowCell3323))) = true := by
+  exact CoverCertificateb46d993ac8.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c1_c0_6_00320
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4781918595
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4781918595
+
+open CertificateCells4781918595
+namespace CoverCertificate98e57d34e4
+
+private theorem checked110 : adaptiveCoverCheck 3 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 3 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 3 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 3 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell113 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 3 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 3 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 3 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 3 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell133 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 3 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 3 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 3 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 3 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 3 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 3 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 3 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 3 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 3 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 3 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 3 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 3 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 3 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 3 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 3 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 3 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate98e57d34e4
+
+theorem e24KC2PhiBelowLeaf33232_c3_c1_c0 :
+    adaptiveCoverCheck 6 phiBelowCell33232310 = true := by
+  exact CoverCertificate98e57d34e4.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c1_c1_c0_5_00322
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa584418aa1
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa584418aa1
+
+open CertificateCellsa584418aa1
+namespace CoverCertificate27d7b561c3
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate27d7b561c3
+
+theorem e24KC2PhiBelowLeaf33232_c3_c1_c1_c0 :
+    adaptiveCoverCheck 5 (childLL phiBelowCell33232311) = true := by
+  exact CoverCertificate27d7b561c3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c1_c1_c1_5_00323
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells09c85e6b71
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells09c85e6b71
+
+open CertificateCells09c85e6b71
+namespace CoverCertificate25f92715ca
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate25f92715ca
+
+theorem e24KC2PhiBelowLeaf33232_c3_c1_c1_c1 :
+    adaptiveCoverCheck 5 (childLH phiBelowCell33232311) = true := by
+  exact CoverCertificate25f92715ca.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle026.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle026.lean
new file mode 100644
index 0000000000..51c152e716
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle026.lean
@@ -0,0 +1,9304 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch039`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CoverCertificatecb0042e211
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLH (childHH (childHL (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatecb0042e211
+
+namespace GerverSofa.PartE.CoverCertificated3159d9210
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLH (childHH (childHL (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificated3159d9210
+
+namespace GerverSofa.PartE.CoverCertificate738ac21cf7
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate738ac21cf7
+
+namespace GerverSofa.PartE.CoverCertificate0da9d99aab
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childLH (childHH (childHL (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate0da9d99aab
+
+namespace GerverSofa.PartE.CoverCertificate04249ac52d
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childLH (childHH (childHL (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate04249ac52d
+
+namespace GerverSofa.PartE.CoverCertificate09e47636f9
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childLH (childHH (childHL (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate09e47636f9
+
+namespace GerverSofa.PartE.CoverCertificatedaf0eef81a
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childLH (childHH (childHL (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatedaf0eef81a
+
+namespace GerverSofa.PartE.CoverCertificatec3b1c42cf8
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatec3b1c42cf8
+
+namespace GerverSofa.PartE.CoverCertificate5d5fa4761b
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childHH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate5d5fa4761b
+
+namespace GerverSofa.PartE.CoverCertificate05931abc25
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childHH (childHH (childHL (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate05931abc25
+
+namespace GerverSofa.PartE.CoverCertificate954b4350f7
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHH (childHH (childHL (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate954b4350f7
+
+namespace GerverSofa.PartE.CoverCertificate1011a55380
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHH (childHH (childHL (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate1011a55380
+
+namespace GerverSofa.PartE.CoverCertificate8bfb4ead96
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHH (childHH (childHL (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate8bfb4ead96
+
+namespace GerverSofa.PartE.CoverCertificated8d96e8252
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childHH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificated8d96e8252
+
+namespace GerverSofa.PartE.CoverCertificate0df7da9c96
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childHH (childHL (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate0df7da9c96
+
+namespace GerverSofa.PartE.CoverCertificate264c040c94
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHH (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate264c040c94
+
+namespace GerverSofa.PartE.CoverCertificate603f728f09
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childHH (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate603f728f09
+
+namespace GerverSofa.PartE.CoverCertificate2808f47ef6
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHH (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+end GerverSofa.PartE.CoverCertificate2808f47ef6
+
+namespace GerverSofa.PartE.CoverCertificate233b768290
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHH (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate233b768290
+
+namespace GerverSofa.PartE.CoverCertificate857c2cf2d4
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+end GerverSofa.PartE.CoverCertificate857c2cf2d4
+
+namespace GerverSofa.PartE.CoverCertificated5abd49324
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHL (childHH (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+end GerverSofa.PartE.CoverCertificated5abd49324
+
+namespace GerverSofa.PartE.CoverCertificatea1a8b4adcc
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childHL (childHH (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatea1a8b4adcc
+
+namespace GerverSofa.PartE.CoverCertificate77d08095a5
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childLL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate77d08095a5
+
+namespace GerverSofa.PartE.CoverCertificate74ace5aa8c
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childLL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate74ace5aa8c
+
+namespace GerverSofa.PartE.CoverCertificatea0eb5f4855
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childLL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatea0eb5f4855
+
+namespace GerverSofa.PartE.CoverCertificate9f8da47b45
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childLL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+end GerverSofa.PartE.CoverCertificate9f8da47b45
+
+namespace GerverSofa.PartE.CoverCertificate9e0a37c11f
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHL (childHH (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+end GerverSofa.PartE.CoverCertificate9e0a37c11f
+
+namespace GerverSofa.PartE.CoverCertificate0e160b9225
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childHH (childHH (childHL (childHH (childHH (e24PhiBelowRoot))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+end GerverSofa.PartE.CoverCertificate0e160b9225
+
+namespace GerverSofa.PartE.CoverCertificate684c483ac7
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate684c483ac7
+
+namespace GerverSofa.PartE.CoverCertificatef33b36c2e8
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatef33b36c2e8
+
+namespace GerverSofa.PartE.CoverCertificate31a518d7cc
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate31a518d7cc
+
+namespace GerverSofa.PartE.CoverCertificateaad2dcb95d
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childLL (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificateaad2dcb95d
+
+namespace GerverSofa.PartE.CoverCertificate282e3cce1c
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childLL (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate282e3cce1c
+
+namespace GerverSofa.PartE.CoverCertificate81553dd700
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childLL (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate81553dd700
+
+namespace GerverSofa.PartE.CoverCertificatef041c45414
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childLL (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatef041c45414
+
+namespace GerverSofa.PartE.CoverCertificatea6cedf179d
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+end GerverSofa.PartE.CoverCertificatea6cedf179d
+
+namespace GerverSofa.PartE.CoverCertificated71b8c6f81
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificated71b8c6f81
+
+namespace GerverSofa.PartE.CertificateCells88f33db9e4
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232311` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232311 : AngleCell :=
+  childLH (childLH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells88f33db9e4
+
+namespace GerverSofa.PartE.CertificateCellsbe3ab504ac
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232311` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232311 : AngleCell :=
+  childLH (childLH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsbe3ab504ac
+
+namespace GerverSofa.PartE.CertificateCellsc46bd88a54
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232312` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232312 : AngleCell :=
+  childHL (childLH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsc46bd88a54
+
+namespace GerverSofa.PartE.CertificateCellse089a8f9ae
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232313` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232313 : AngleCell :=
+  childHH (childLH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellse089a8f9ae
+
+namespace GerverSofa.PartE.CertificateCellsf5ba6a429c
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232313` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232313 : AngleCell :=
+  childHH (childLH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsf5ba6a429c
+
+namespace GerverSofa.PartE.CertificateCellsff1f5625e7
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232313` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232313 : AngleCell :=
+  childHH (childLH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsff1f5625e7
+
+namespace GerverSofa.PartE.CertificateCells2a20ecd725
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232313` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232313 : AngleCell :=
+  childHH (childLH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells2a20ecd725
+
+namespace GerverSofa.PartE.CertificateCellsa4ad8d7233
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa4ad8d7233
+
+namespace GerverSofa.PartE.CertificateCells6d4400643e
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232330` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232330 : AngleCell :=
+  childLL (childHH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells6d4400643e
+
+namespace GerverSofa.PartE.CertificateCellsa7c3c8a0e8
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232331` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232331 : AngleCell :=
+  childLH (childHH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsa7c3c8a0e8
+
+namespace GerverSofa.PartE.CertificateCellsee53a497bf
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232331` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232331 : AngleCell :=
+  childLH (childHH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsee53a497bf
+
+namespace GerverSofa.PartE.CertificateCells672c985771
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232331` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232331 : AngleCell :=
+  childLH (childHH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells672c985771
+
+namespace GerverSofa.PartE.CertificateCellsbdc9b1cb1f
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232331` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232331 : AngleCell :=
+  childLH (childHH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsbdc9b1cb1f
+
+namespace GerverSofa.PartE.CertificateCells6f8943a51a
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232332` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232332 : AngleCell :=
+  childHL (childHH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells6f8943a51a
+
+namespace GerverSofa.PartE.CertificateCells3c06cca83d
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232333` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232333 : AngleCell :=
+  childHH (childHH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells3c06cca83d
+
+namespace GerverSofa.PartE.CertificateCells9a18487959
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells9a18487959
+
+namespace GerverSofa.PartE.CertificateCells5679b6d0b8
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells5679b6d0b8
+
+namespace GerverSofa.PartE.CertificateCells94565979ef
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells94565979ef
+
+namespace GerverSofa.PartE.CertificateCells1a32aff68c
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells1a32aff68c
+
+namespace GerverSofa.PartE.CertificateCells170ce03ef2
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells170ce03ef2
+
+namespace GerverSofa.PartE.CertificateCells8706070b49
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233200` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233200 : AngleCell :=
+  childLL (childLL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells8706070b49
+
+namespace GerverSofa.PartE.CertificateCells29218b6dfb
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233201` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233201 : AngleCell :=
+  childLH (childLL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells29218b6dfb
+
+namespace GerverSofa.PartE.CertificateCellscd85cede4b
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233202` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233202 : AngleCell :=
+  childHL (childLL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellscd85cede4b
+
+namespace GerverSofa.PartE.CertificateCells14bc720168
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233202` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233202 : AngleCell :=
+  childHL (childLL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells14bc720168
+
+namespace GerverSofa.PartE.CertificateCellsc20163bbb0
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233202` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233202 : AngleCell :=
+  childHL (childLL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsc20163bbb0
+
+namespace GerverSofa.PartE.CertificateCellsfb4e8d760d
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233202` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233202 : AngleCell :=
+  childHL (childLL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsfb4e8d760d
+
+namespace GerverSofa.PartE.CertificateCells1167e65ac9
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233203` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233203 : AngleCell :=
+  childHH (childLL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells1167e65ac9
+
+namespace GerverSofa.PartE.CertificateCellsa97cd5b1f9
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsa97cd5b1f9
+
+namespace GerverSofa.PartE.CertificateCells33181810b4
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233220` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233220 : AngleCell :=
+  childLL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells33181810b4
+
+namespace GerverSofa.PartE.CertificateCellseb15d79203
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233220` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233220 : AngleCell :=
+  childLL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellseb15d79203
+
+namespace GerverSofa.PartE.CertificateCellse0133145f0
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233220` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233220 : AngleCell :=
+  childLL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellse0133145f0
+
+namespace GerverSofa.PartE.CertificateCellsdabae3eeeb
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233220` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233220 : AngleCell :=
+  childLL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsdabae3eeeb
+
+namespace GerverSofa.PartE.CertificateCells18d43d056c
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233220` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233220 : AngleCell :=
+  childLL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells18d43d056c
+
+namespace GerverSofa.PartE.CertificateCellse49667421f
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233220` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233220 : AngleCell :=
+  childLL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellse49667421f
+
+namespace GerverSofa.PartE.CertificateCellsd4600f4c86
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233220` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233220 : AngleCell :=
+  childLL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsd4600f4c86
+
+namespace GerverSofa.PartE.CertificateCells368be09a97
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233221` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233221 : AngleCell :=
+  childLH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells368be09a97
+
+namespace GerverSofa.PartE.CertificateCells1f54f397e7
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233221` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233221 : AngleCell :=
+  childLH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells1f54f397e7
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.PhiBelow.Leaf00324`.
+* `KernelOnly.PartE.PhiBelow.Leaf00325`.
+* `KernelOnly.PartE.PhiBelow.Leaf00327`.
+* `KernelOnly.PartE.PhiBelow.Leaf00329`.
+* `KernelOnly.PartE.PhiBelow.Leaf00330`.
+* `KernelOnly.PartE.PhiBelow.Leaf00331`.
+* `KernelOnly.PartE.PhiBelow.Leaf00332`.
+* `KernelOnly.PartE.PhiBelow.Leaf00335`.
+* `KernelOnly.PartE.PhiBelow.Leaf00337`.
+* `KernelOnly.PartE.PhiBelow.Leaf00339`.
+* `KernelOnly.PartE.PhiBelow.Leaf00340`.
+* `KernelOnly.PartE.PhiBelow.Leaf00341`.
+* `KernelOnly.PartE.PhiBelow.Leaf00342`.
+* `KernelOnly.PartE.PhiBelow.Leaf00344`.
+* `KernelOnly.PartE.PhiBelow.Leaf00345`.
+* `KernelOnly.PartE.PhiBelow.Leaf00354`.
+* `KernelOnly.PartE.PhiBelow.Leaf00355`.
+* `KernelOnly.PartE.PhiBelow.Leaf00356`.
+* `KernelOnly.PartE.PhiBelow.Leaf00357`.
+* `KernelOnly.PartE.PhiBelow.Leaf00359`.
+* `KernelOnly.PartE.PhiBelow.Leaf00362`.
+* `KernelOnly.PartE.PhiBelow.Leaf00363`.
+* `KernelOnly.PartE.PhiBelow.Leaf00365`.
+* `KernelOnly.PartE.PhiBelow.Leaf00366`.
+* `KernelOnly.PartE.PhiBelow.Leaf00367`.
+* `KernelOnly.PartE.PhiBelow.Leaf00368`.
+* `KernelOnly.PartE.PhiBelow.Leaf00370`.
+* `KernelOnly.PartE.PhiBelow.Leaf00372`.
+* `KernelOnly.PartE.PhiBelow.Leaf00375`.
+* `KernelOnly.PartE.PhiBelow.Leaf00376`.
+* `KernelOnly.PartE.PhiBelow.Leaf00377`.
+* `KernelOnly.PartE.PhiBelow.Leaf00379`.
+* `KernelOnly.PartE.PhiBelow.Leaf00380`.
+* `KernelOnly.PartE.PhiBelow.Leaf00381`.
+* `KernelOnly.PartE.PhiBelow.Leaf00382`.
+* `KernelOnly.PartE.PhiBelow.Leaf00386`.
+* `KernelOnly.PartE.PhiBelow.Leaf00387`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c1_c1_c2_5_00324
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells88f33db9e4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells88f33db9e4
+
+open CertificateCells88f33db9e4
+namespace CoverCertificatecb0042e211
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatecb0042e211
+
+theorem e24KC2PhiBelowLeaf33232_c3_c1_c1_c2 :
+    adaptiveCoverCheck 5 (childHL phiBelowCell33232311) = true := by
+  exact CoverCertificatecb0042e211.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c1_c1_c3_5_00325
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsbe3ab504ac
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsbe3ab504ac
+
+open CertificateCellsbe3ab504ac
+namespace CoverCertificated3159d9210
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated3159d9210
+
+theorem e24KC2PhiBelowLeaf33232_c3_c1_c1_c3 :
+    adaptiveCoverCheck 5 (childHH phiBelowCell33232311) = true := by
+  exact CoverCertificated3159d9210.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c1_c2_6_00327
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc46bd88a54
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc46bd88a54
+
+open CertificateCellsc46bd88a54
+namespace CoverCertificate738ac21cf7
+
+private theorem checked100 : adaptiveCoverCheck 3 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 3 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 3 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 3 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 3 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 3 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 3 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 3 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 3 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 3 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 3 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 3 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 3 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 3 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 3 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 3 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell133 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 3 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 3 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 3 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 3 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 3 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 3 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 3 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 3 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 3 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 3 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 3 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 3 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 3 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 3 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 3 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 3 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate738ac21cf7
+
+theorem e24KC2PhiBelowLeaf33232_c3_c1_c2 :
+    adaptiveCoverCheck 6 phiBelowCell33232312 = true := by
+  exact CoverCertificate738ac21cf7.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c1_c3_c0_5_00329
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse089a8f9ae
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse089a8f9ae
+
+open CertificateCellse089a8f9ae
+namespace CoverCertificate0da9d99aab
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0da9d99aab
+
+theorem e24KC2PhiBelowLeaf33232_c3_c1_c3_c0 :
+    adaptiveCoverCheck 5 (childLL phiBelowCell33232313) = true := by
+  exact CoverCertificate0da9d99aab.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c1_c3_c1_5_00330
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf5ba6a429c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf5ba6a429c
+
+open CertificateCellsf5ba6a429c
+namespace CoverCertificate04249ac52d
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate04249ac52d
+
+theorem e24KC2PhiBelowLeaf33232_c3_c1_c3_c1 :
+    adaptiveCoverCheck 5 (childLH phiBelowCell33232313) = true := by
+  exact CoverCertificate04249ac52d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c1_c3_c2_5_00331
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsff1f5625e7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsff1f5625e7
+
+open CertificateCellsff1f5625e7
+namespace CoverCertificate09e47636f9
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate09e47636f9
+
+theorem e24KC2PhiBelowLeaf33232_c3_c1_c3_c2 :
+    adaptiveCoverCheck 5 (childHL phiBelowCell33232313) = true := by
+  exact CoverCertificate09e47636f9.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c1_c3_c3_5_00332
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2a20ecd725
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2a20ecd725
+
+open CertificateCells2a20ecd725
+namespace CoverCertificatedaf0eef81a
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatedaf0eef81a
+
+theorem e24KC2PhiBelowLeaf33232_c3_c1_c3_c3 :
+    adaptiveCoverCheck 5 (childHH phiBelowCell33232313) = true := by
+  exact CoverCertificatedaf0eef81a.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c2_7_00335
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa4ad8d7233
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa4ad8d7233
+
+open CertificateCellsa4ad8d7233
+namespace CoverCertificatec3b1c42cf8
+
+private theorem checked000 : adaptiveCoverCheck 4 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 4 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 4 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 4 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 4 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 4 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 4 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 4 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell013 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 4 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 4 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 4 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 4 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 4 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 4 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 4 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 4 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 4 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 4 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 4 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 4 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 4 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 4 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 4 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 4 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 4 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 4 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 4 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 4 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell133 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 4 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 4 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 4 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 4 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 4 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 4 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 4 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 4 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell313 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 4 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 4 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 4 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 4 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec3b1c42cf8
+
+theorem e24KC2PhiBelowLeaf33232_c3_c2 :
+    adaptiveCoverCheck 7 (childHL (childHH (childHL phiBelowCell3323))) = true := by
+  exact CoverCertificatec3b1c42cf8.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c3_c0_6_00337
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6d4400643e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6d4400643e
+
+open CertificateCells6d4400643e
+namespace CoverCertificate5d5fa4761b
+
+private theorem checked100 : adaptiveCoverCheck 3 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 3 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 3 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 3 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 3 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 3 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 3 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 3 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 3 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 3 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 3 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 3 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 3 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 3 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 3 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 3 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell133 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 3 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 3 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 3 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 3 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 3 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 3 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 3 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 3 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell313 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 3 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 3 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 3 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 3 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5d5fa4761b
+
+theorem e24KC2PhiBelowLeaf33232_c3_c3_c0 :
+    adaptiveCoverCheck 6 phiBelowCell33232330 = true := by
+  exact CoverCertificate5d5fa4761b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c3_c1_c0_5_00339
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa7c3c8a0e8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa7c3c8a0e8
+
+open CertificateCellsa7c3c8a0e8
+namespace CoverCertificate05931abc25
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate05931abc25
+
+theorem e24KC2PhiBelowLeaf33232_c3_c3_c1_c0 :
+    adaptiveCoverCheck 5 (childLL phiBelowCell33232331) = true := by
+  exact CoverCertificate05931abc25.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c3_c1_c1_5_00340
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsee53a497bf
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsee53a497bf
+
+open CertificateCellsee53a497bf
+namespace CoverCertificate954b4350f7
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate954b4350f7
+
+theorem e24KC2PhiBelowLeaf33232_c3_c3_c1_c1 :
+    adaptiveCoverCheck 5 (childLH phiBelowCell33232331) = true := by
+  exact CoverCertificate954b4350f7.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c3_c1_c2_5_00341
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells672c985771
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells672c985771
+
+open CertificateCells672c985771
+namespace CoverCertificate1011a55380
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1011a55380
+
+theorem e24KC2PhiBelowLeaf33232_c3_c3_c1_c2 :
+    adaptiveCoverCheck 5 (childHL phiBelowCell33232331) = true := by
+  exact CoverCertificate1011a55380.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c3_c1_c3_5_00342
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsbdc9b1cb1f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsbdc9b1cb1f
+
+open CertificateCellsbdc9b1cb1f
+namespace CoverCertificate8bfb4ead96
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate8bfb4ead96
+
+theorem e24KC2PhiBelowLeaf33232_c3_c3_c1_c3 :
+    adaptiveCoverCheck 5 (childHH phiBelowCell33232331) = true := by
+  exact CoverCertificate8bfb4ead96.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c3_c2_6_00344
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6f8943a51a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6f8943a51a
+
+open CertificateCells6f8943a51a
+namespace CoverCertificated8d96e8252
+
+private theorem checked110 : adaptiveCoverCheck 3 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 3 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 3 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 3 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated8d96e8252
+
+theorem e24KC2PhiBelowLeaf33232_c3_c3_c2 :
+    adaptiveCoverCheck 6 phiBelowCell33232332 = true := by
+  exact CoverCertificated8d96e8252.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33232_c3_c3_c3_6_00345
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3c06cca83d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3c06cca83d
+
+open CertificateCells3c06cca83d
+namespace CoverCertificate0df7da9c96
+
+private theorem checked000 : adaptiveCoverCheck 3 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 3 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 3 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 3 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 3 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 3 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 3 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 3 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 3 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 3 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 3 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 3 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 3 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 3 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 3 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 3 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 3 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 3 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 3 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 3 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 3 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 3 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 3 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 3 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 3 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 3 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 3 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 3 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 3 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 3 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 3 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 3 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell133 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 3 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 3 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 3 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 3 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell213 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 3 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 3 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 3 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 3 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 3 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 3 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 3 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 3 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 3 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 3 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 3 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 3 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 3 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 3 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 3 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 3 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0df7da9c96
+
+theorem e24KC2PhiBelowLeaf33232_c3_c3_c3 :
+    adaptiveCoverCheck 6 phiBelowCell33232333 = true := by
+  exact CoverCertificate0df7da9c96.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c0_c0_7_00354
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9a18487959
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9a18487959
+
+open CertificateCells9a18487959
+namespace CoverCertificate264c040c94
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate264c040c94
+
+theorem e24KC2PhiBelowLeaf33233_c0_c0 :
+    adaptiveCoverCheck 7 (childLL (childLL (childHH phiBelowCell3323))) = true := by
+  exact CoverCertificate264c040c94.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c0_c1_7_00355
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5679b6d0b8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5679b6d0b8
+
+open CertificateCells5679b6d0b8
+namespace CoverCertificate603f728f09
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate603f728f09
+
+theorem e24KC2PhiBelowLeaf33233_c0_c1 :
+    adaptiveCoverCheck 7 (childLH (childLL (childHH phiBelowCell3323))) = true := by
+  exact CoverCertificate603f728f09.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c0_c2_7_00356
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells94565979ef
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells94565979ef
+
+open CertificateCells94565979ef
+namespace CoverCertificate2808f47ef6
+
+private theorem checked000 : adaptiveCoverCheck 4 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 4 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 4 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 4 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell003 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 4 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 4 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 4 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 4 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 4 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 4 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 4 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 4 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell033 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 4 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 4 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 4 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 4 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 4 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 4 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 4 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 4 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 4 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 4 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 4 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 4 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 4 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 4 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 4 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 4 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 4 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 4 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 4 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 4 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell323 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate2808f47ef6
+
+theorem e24KC2PhiBelowLeaf33233_c0_c2 :
+    adaptiveCoverCheck 7 (childHL (childLL (childHH phiBelowCell3323))) = true := by
+  exact CoverCertificate2808f47ef6.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c0_c3_7_00357
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1a32aff68c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1a32aff68c
+
+open CertificateCells1a32aff68c
+namespace CoverCertificate233b768290
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate233b768290
+
+theorem e24KC2PhiBelowLeaf33233_c0_c3 :
+    adaptiveCoverCheck 7 (childHH (childLL (childHH phiBelowCell3323))) = true := by
+  exact CoverCertificate233b768290.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c1_8_00359
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells170ce03ef2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells170ce03ef2
+
+open CertificateCells170ce03ef2
+namespace CoverCertificate857c2cf2d4
+
+private theorem checked220 : adaptiveCoverCheck 5 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 5 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 5 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 5 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell223 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate857c2cf2d4
+
+theorem e24KC2PhiBelowLeaf33233_c1 :
+    adaptiveCoverCheck 8 (childLH (childHH phiBelowCell3323)) = true := by
+  exact CoverCertificate857c2cf2d4.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c0_c0_6_00362
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8706070b49
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8706070b49
+
+open CertificateCells8706070b49
+namespace CoverCertificated5abd49324
+
+private theorem checked020 : adaptiveCoverCheck 3 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 3 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 3 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 3 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell023 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 3 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 3 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 3 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 3 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 3 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 3 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 3 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 3 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 3 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 3 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 3 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 3 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 3 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 3 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 3 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 3 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 3 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 3 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 3 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 3 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell323 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated5abd49324
+
+theorem e24KC2PhiBelowLeaf33233_c2_c0_c0 :
+    adaptiveCoverCheck 6 phiBelowCell33233200 = true := by
+  exact CoverCertificated5abd49324.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c0_c1_6_00363
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells29218b6dfb
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells29218b6dfb
+
+open CertificateCells29218b6dfb
+namespace CoverCertificatea1a8b4adcc
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea1a8b4adcc
+
+theorem e24KC2PhiBelowLeaf33233_c2_c0_c1 :
+    adaptiveCoverCheck 6 phiBelowCell33233201 = true := by
+  exact CoverCertificatea1a8b4adcc.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c0_c2_c0_5_00365
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellscd85cede4b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellscd85cede4b
+
+open CertificateCellscd85cede4b
+namespace CoverCertificate77d08095a5
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate77d08095a5
+
+theorem e24KC2PhiBelowLeaf33233_c2_c0_c2_c0 :
+    adaptiveCoverCheck 5 (childLL phiBelowCell33233202) = true := by
+  exact CoverCertificate77d08095a5.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c0_c2_c1_5_00366
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells14bc720168
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells14bc720168
+
+open CertificateCells14bc720168
+namespace CoverCertificate74ace5aa8c
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate74ace5aa8c
+
+theorem e24KC2PhiBelowLeaf33233_c2_c0_c2_c1 :
+    adaptiveCoverCheck 5 (childLH phiBelowCell33233202) = true := by
+  exact CoverCertificate74ace5aa8c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c0_c2_c2_5_00367
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc20163bbb0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc20163bbb0
+
+open CertificateCellsc20163bbb0
+namespace CoverCertificatea0eb5f4855
+
+private theorem checked130 : adaptiveCoverCheck 2 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 2 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 2 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 2 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell133 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 2 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 2 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 2 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 2 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell313 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 2 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 2 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 2 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 2 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea0eb5f4855
+
+theorem e24KC2PhiBelowLeaf33233_c2_c0_c2_c2 :
+    adaptiveCoverCheck 5 (childHL phiBelowCell33233202) = true := by
+  exact CoverCertificatea0eb5f4855.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c0_c2_c3_5_00368
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsfb4e8d760d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsfb4e8d760d
+
+open CertificateCellsfb4e8d760d
+namespace CoverCertificate9f8da47b45
+
+private theorem checked200 : adaptiveCoverCheck 2 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 2 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 2 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 2 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell203 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 2 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 2 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 2 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 2 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 2 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 2 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 2 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 2 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell233 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate9f8da47b45
+
+theorem e24KC2PhiBelowLeaf33233_c2_c0_c2_c3 :
+    adaptiveCoverCheck 5 (childHH phiBelowCell33233202) = true := by
+  exact CoverCertificate9f8da47b45.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c0_c3_6_00370
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1167e65ac9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1167e65ac9
+
+open CertificateCells1167e65ac9
+namespace CoverCertificate9e0a37c11f
+
+private theorem checked020 : adaptiveCoverCheck 3 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 3 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 3 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 3 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell023 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 3 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 3 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 3 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 3 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 3 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 3 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 3 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 3 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 3 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 3 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 3 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 3 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 3 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 3 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 3 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 3 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell233 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate9e0a37c11f
+
+theorem e24KC2PhiBelowLeaf33233_c2_c0_c3 :
+    adaptiveCoverCheck 6 phiBelowCell33233203 = true := by
+  exact CoverCertificate9e0a37c11f.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c1_7_00372
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa97cd5b1f9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa97cd5b1f9
+
+open CertificateCellsa97cd5b1f9
+namespace CoverCertificate0e160b9225
+
+private theorem checked020 : adaptiveCoverCheck 4 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 4 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 4 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 4 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell023 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 4 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 4 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 4 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 4 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 4 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 4 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 4 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 4 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 4 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 4 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 4 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 4 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 4 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 4 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 4 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 4 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 4 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 4 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 4 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 4 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell323 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 5 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 5 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 5 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 5 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 5 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 5 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 5 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 5 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0e160b9225
+
+theorem e24KC2PhiBelowLeaf33233_c2_c1 :
+    adaptiveCoverCheck 7 (childLH (childHL (childHH phiBelowCell3323))) = true := by
+  exact CoverCertificate0e160b9225.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c0_c0_5_00375
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells33181810b4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells33181810b4
+
+open CertificateCells33181810b4
+namespace CoverCertificate684c483ac7
+
+private theorem checked110 : adaptiveCoverCheck 2 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 2 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 2 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 2 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell113 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 2 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 2 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 2 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 2 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell133 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 2 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 2 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 2 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 2 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 2 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 2 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 2 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 2 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 2 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 2 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 2 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 2 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 2 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 2 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 2 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 2 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate684c483ac7
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c0_c0 :
+    adaptiveCoverCheck 5 (childLL phiBelowCell33233220) = true := by
+  exact CoverCertificate684c483ac7.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c0_c1_5_00376
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellseb15d79203
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellseb15d79203
+
+open CertificateCellseb15d79203
+namespace CoverCertificatef33b36c2e8
+
+private theorem checked000 : adaptiveCoverCheck 2 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 2 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 2 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 2 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 2 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 2 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 2 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 2 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 2 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 2 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 2 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 2 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 2 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 2 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 2 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 2 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 2 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 2 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 2 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 2 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell123 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 2 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 2 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 2 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 2 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 2 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 2 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 2 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 2 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 2 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 2 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 2 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 2 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 2 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 2 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 2 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 2 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 2 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 2 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 2 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 2 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 2 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 2 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 2 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 2 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 2 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 2 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 2 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 2 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 2 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 2 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 2 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 2 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef33b36c2e8
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c0_c1 :
+    adaptiveCoverCheck 5 (childLH phiBelowCell33233220) = true := by
+  exact CoverCertificatef33b36c2e8.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c0_c2_5_00377
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse0133145f0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse0133145f0
+
+open CertificateCellse0133145f0
+namespace CoverCertificate31a518d7cc
+
+private theorem checked100 : adaptiveCoverCheck 2 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 2 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 2 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 2 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 2 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 2 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 2 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 2 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 2 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 2 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 2 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 2 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 2 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 2 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 2 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 2 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell133 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 2 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 2 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 2 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 2 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 2 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 2 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 2 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 2 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 2 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 2 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 2 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 2 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 2 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 2 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 2 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 2 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate31a518d7cc
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c0_c2 :
+    adaptiveCoverCheck 5 (childHL phiBelowCell33233220) = true := by
+  exact CoverCertificate31a518d7cc.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c0_c3_c0_4_00379
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdabae3eeeb
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdabae3eeeb
+
+open CertificateCellsdabae3eeeb
+namespace CoverCertificateaad2dcb95d
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateaad2dcb95d
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c0_c3_c0 :
+    adaptiveCoverCheck 4 (childLL (childHH phiBelowCell33233220)) = true := by
+  exact CoverCertificateaad2dcb95d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c0_c3_c1_4_00380
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells18d43d056c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells18d43d056c
+
+open CertificateCells18d43d056c
+namespace CoverCertificate282e3cce1c
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate282e3cce1c
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c0_c3_c1 :
+    adaptiveCoverCheck 4 (childLH (childHH phiBelowCell33233220)) = true := by
+  exact CoverCertificate282e3cce1c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c0_c3_c2_4_00381
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse49667421f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse49667421f
+
+open CertificateCellse49667421f
+namespace CoverCertificate81553dd700
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate81553dd700
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c0_c3_c2 :
+    adaptiveCoverCheck 4 (childHL (childHH phiBelowCell33233220)) = true := by
+  exact CoverCertificate81553dd700.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c0_c3_c3_4_00382
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd4600f4c86
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd4600f4c86
+
+open CertificateCellsd4600f4c86
+namespace CoverCertificatef041c45414
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef041c45414
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c0_c3_c3 :
+    adaptiveCoverCheck 4 (childHH (childHH phiBelowCell33233220)) = true := by
+  exact CoverCertificatef041c45414.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c1_c0_5_00386
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells368be09a97
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells368be09a97
+
+open CertificateCells368be09a97
+namespace CoverCertificatea6cedf179d
+
+private theorem checked220 : adaptiveCoverCheck 2 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 2 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 2 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 2 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell223 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea6cedf179d
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c1_c0 :
+    adaptiveCoverCheck 5 (childLL phiBelowCell33233221) = true := by
+  exact CoverCertificatea6cedf179d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c1_c1_5_00387
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1f54f397e7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1f54f397e7
+
+open CertificateCells1f54f397e7
+namespace CoverCertificated71b8c6f81
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated71b8c6f81
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c1_c1 :
+    adaptiveCoverCheck 5 (childLH phiBelowCell33233221) = true := by
+  exact CoverCertificated71b8c6f81.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle027.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle027.lean
new file mode 100644
index 0000000000..6d4b1387d1
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle027.lean
@@ -0,0 +1,9396 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch040`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CoverCertificate1d94560d47
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate1d94560d47
+
+namespace GerverSofa.PartE.CoverCertificatefa0ec53117
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+end GerverSofa.PartE.CoverCertificatefa0ec53117
+
+namespace GerverSofa.PartE.CoverCertificate38482fd67b
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate38482fd67b
+
+namespace GerverSofa.PartE.CoverCertificate774f03a3cd
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childHL (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate774f03a3cd
+
+namespace GerverSofa.PartE.CoverCertificate141819f8d7
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHL (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate141819f8d7
+
+namespace GerverSofa.PartE.CoverCertificated9624571ac
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHL (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificated9624571ac
+
+namespace GerverSofa.PartE.CoverCertificate6840dd8bca
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHL (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate6840dd8bca
+
+namespace GerverSofa.PartE.CoverCertificate781ead79bd
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+end GerverSofa.PartE.CoverCertificate781ead79bd
+
+namespace GerverSofa.PartE.CoverCertificatefcce63a7f6
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatefcce63a7f6
+
+namespace GerverSofa.PartE.CoverCertificate84a4f6988f
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childHH (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate84a4f6988f
+
+namespace GerverSofa.PartE.CoverCertificate03b35dd0cf
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childHH (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+end GerverSofa.PartE.CoverCertificate03b35dd0cf
+
+namespace GerverSofa.PartE.CoverCertificate2781da35b5
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHH (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate2781da35b5
+
+namespace GerverSofa.PartE.CoverCertificate7c9b16d986
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHH (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate7c9b16d986
+
+namespace GerverSofa.PartE.CoverCertificate7a97434462
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+private abbrev cell2220 : AngleCell :=
+  childLL cell222
+
+private abbrev cell2221 : AngleCell :=
+  childLH cell222
+
+private abbrev cell2222 : AngleCell :=
+  childHL cell222
+
+private abbrev cell2223 : AngleCell :=
+  childHH cell222
+
+end GerverSofa.PartE.CoverCertificate7a97434462
+
+namespace GerverSofa.PartE.CoverCertificate993722881e
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childHH (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate993722881e
+
+namespace GerverSofa.PartE.CoverCertificate8f03b44673
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childHL (childHH (childHL (childHL (childHH (childHH (childHL (childHH
+    (childHH (e24PhiBelowRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate8f03b44673
+
+namespace GerverSofa.PartE.CoverCertificatefffe180fcd
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHL (childHH (childHL (childHL (childHH (childHH (childHL (childHH
+    (childHH (e24PhiBelowRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatefffe180fcd
+
+namespace GerverSofa.PartE.CoverCertificated2a9415b82
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childLH (childHL (childHH (childHL (childHL (childHH (childHH (childHL
+    (childHH (childHH (e24PhiBelowRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificated2a9415b82
+
+namespace GerverSofa.PartE.CoverCertificatef93658a45e
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childLH (childHL (childHH (childHL (childHL (childHH (childHH (childHL
+    (childHH (childHH (e24PhiBelowRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatef93658a45e
+
+namespace GerverSofa.PartE.CoverCertificate3d3c2d6223
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childLH (childHL (childHH (childHL (childHL (childHH (childHH (childHL
+    (childHH (childHH (e24PhiBelowRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate3d3c2d6223
+
+namespace GerverSofa.PartE.CoverCertificatea6d03c9687
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHL (childHH (childHL (childHL (childHH (childHH (childHL (childHH
+    (childHH (e24PhiBelowRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatea6d03c9687
+
+namespace GerverSofa.PartE.CoverCertificatefce6b2559b
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childHH (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatefce6b2559b
+
+namespace GerverSofa.PartE.CoverCertificate0ac0b8c68f
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childHL (childHH (childHL (childHL (childHH (childHH (childHL (childHH
+    (childHH (e24PhiBelowRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate0ac0b8c68f
+
+namespace GerverSofa.PartE.CoverCertificate9ea4339e1a
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childHL (childHH (childHL (childHL (childHH (childHH (childHL (childHH
+    (childHH (e24PhiBelowRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate9ea4339e1a
+
+namespace GerverSofa.PartE.CoverCertificatec65ef0b520
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childHL (childHH (childHL (childHL (childHH (childHH (childHL (childHH
+    (childHH (e24PhiBelowRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatec65ef0b520
+
+namespace GerverSofa.PartE.CoverCertificate64e89cb34c
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childHL (childHH (childHL (childHL (childHH (childHH (childHL (childHH
+    (childHH (e24PhiBelowRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate64e89cb34c
+
+namespace GerverSofa.PartE.CoverCertificatef71635c89a
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childHH (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatef71635c89a
+
+namespace GerverSofa.PartE.CoverCertificate4d208d4148
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childHH (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+end GerverSofa.PartE.CoverCertificate4d208d4148
+
+namespace GerverSofa.PartE.CoverCertificatebbc2bd14f5
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childHH (childHH (childHL (childHL (childHH (childHH (childHL (childHH
+    (childHH (e24PhiBelowRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatebbc2bd14f5
+
+namespace GerverSofa.PartE.CoverCertificate0f826c338a
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childHH (childHH (childHL (childHL (childHH (childHH (childHL (childHH
+    (childHH (e24PhiBelowRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+end GerverSofa.PartE.CoverCertificate0f826c338a
+
+namespace GerverSofa.PartE.CoverCertificate8518e81351
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childHH (childHH (childHL (childHL (childHH (childHH (childHL (childHH
+    (childHH (e24PhiBelowRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate8518e81351
+
+namespace GerverSofa.PartE.CoverCertificate4ecdb55cf7
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childHH (childHH (childHL (childHL (childHH (childHH (childHL (childHH
+    (childHH (e24PhiBelowRoot))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell130 : AngleCell :=
+  childLL cell13
+
+private abbrev cell131 : AngleCell :=
+  childLH cell13
+
+private abbrev cell132 : AngleCell :=
+  childHL cell13
+
+private abbrev cell133 : AngleCell :=
+  childHH cell13
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate4ecdb55cf7
+
+namespace GerverSofa.PartE.CertificateCells6309ffd285
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233221` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233221 : AngleCell :=
+  childLH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells6309ffd285
+
+namespace GerverSofa.PartE.CertificateCells48d3b8afd4
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233221` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233221 : AngleCell :=
+  childLH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells48d3b8afd4
+
+namespace GerverSofa.PartE.CertificateCellsf95c4e9f9f
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233222` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233222 : AngleCell :=
+  childHL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsf95c4e9f9f
+
+namespace GerverSofa.PartE.CertificateCells38449ac605
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233222` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233222 : AngleCell :=
+  childHL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells38449ac605
+
+namespace GerverSofa.PartE.CertificateCells43ee04fc11
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233222` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233222 : AngleCell :=
+  childHL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells43ee04fc11
+
+namespace GerverSofa.PartE.CertificateCells303915a97c
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233222` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233222 : AngleCell :=
+  childHL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells303915a97c
+
+namespace GerverSofa.PartE.CertificateCells88a95e6b6b
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233222` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233222 : AngleCell :=
+  childHL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells88a95e6b6b
+
+namespace GerverSofa.PartE.CertificateCells4611c2250a
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233222` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233222 : AngleCell :=
+  childHL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells4611c2250a
+
+namespace GerverSofa.PartE.CertificateCellscbfda2707a
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233222` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233222 : AngleCell :=
+  childHL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellscbfda2707a
+
+namespace GerverSofa.PartE.CertificateCells003531f536
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells003531f536
+
+namespace GerverSofa.PartE.CertificateCells1f3e328824
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells1f3e328824
+
+namespace GerverSofa.PartE.CertificateCells47ff5dcac0
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells47ff5dcac0
+
+namespace GerverSofa.PartE.CertificateCells65a3665007
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells65a3665007
+
+namespace GerverSofa.PartE.CertificateCells45a72ee869
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells45a72ee869
+
+namespace GerverSofa.PartE.CertificateCells3d2f3b2d12
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells3d2f3b2d12
+
+namespace GerverSofa.PartE.CertificateCellsce18f6e886
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsce18f6e886
+
+namespace GerverSofa.PartE.CertificateCellsb5cebfd5b6
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsb5cebfd5b6
+
+namespace GerverSofa.PartE.CertificateCells2aad13d936
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+/-- Subcell `332332232120` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell332332232120 : AngleCell :=
+  childLL (childHL (childLH (childHL phiBelowCell33233223)))
+
+end GerverSofa.PartE.CertificateCells2aad13d936
+
+namespace GerverSofa.PartE.CertificateCells6a90bc785d
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+/-- Subcell `332332232120` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell332332232120 : AngleCell :=
+  childLL (childHL (childLH (childHL phiBelowCell33233223)))
+
+end GerverSofa.PartE.CertificateCells6a90bc785d
+
+namespace GerverSofa.PartE.CertificateCellsdb781a1ec8
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+/-- Subcell `332332232120` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell332332232120 : AngleCell :=
+  childLL (childHL (childLH (childHL phiBelowCell33233223)))
+
+end GerverSofa.PartE.CertificateCellsdb781a1ec8
+
+namespace GerverSofa.PartE.CertificateCells1cfc9442b0
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+/-- Subcell `332332232120` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell332332232120 : AngleCell :=
+  childLL (childHL (childLH (childHL phiBelowCell33233223)))
+
+end GerverSofa.PartE.CertificateCells1cfc9442b0
+
+namespace GerverSofa.PartE.CertificateCells2dfa44feb0
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+/-- Subcell `332332232121` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell332332232121 : AngleCell :=
+  childLH (childHL (childLH (childHL phiBelowCell33233223)))
+
+end GerverSofa.PartE.CertificateCells2dfa44feb0
+
+namespace GerverSofa.PartE.CertificateCells0c5519e954
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+/-- Subcell `332332232122` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell332332232122 : AngleCell :=
+  childHL (childHL (childLH (childHL phiBelowCell33233223)))
+
+end GerverSofa.PartE.CertificateCells0c5519e954
+
+namespace GerverSofa.PartE.CertificateCells06dff71a35
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+/-- Subcell `332332232123` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell332332232123 : AngleCell :=
+  childHH (childHL (childLH (childHL phiBelowCell33233223)))
+
+end GerverSofa.PartE.CertificateCells06dff71a35
+
+namespace GerverSofa.PartE.CertificateCells529c7df50a
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells529c7df50a
+
+namespace GerverSofa.PartE.CertificateCells25a838feeb
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells25a838feeb
+
+namespace GerverSofa.PartE.CertificateCellsbc23e0e6af
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsbc23e0e6af
+
+namespace GerverSofa.PartE.CertificateCells13f9420722
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells13f9420722
+
+namespace GerverSofa.PartE.CertificateCells8f726fb04b
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells8f726fb04b
+
+namespace GerverSofa.PartE.CertificateCells7e3f428846
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells7e3f428846
+
+namespace GerverSofa.PartE.CertificateCells4caf1a7a91
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells4caf1a7a91
+
+namespace GerverSofa.PartE.CertificateCells2907beb942
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells2907beb942
+
+namespace GerverSofa.PartE.CertificateCellsb81ac6c2c5
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsb81ac6c2c5
+
+namespace GerverSofa.PartE.CertificateCells6afa02519f
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells6afa02519f
+
+namespace GerverSofa.PartE.CertificateCells27566e6112
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells27566e6112
+
+namespace GerverSofa.PartE.CertificateCellsc20116b798
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsc20116b798
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.PhiBelow.Leaf00388`.
+* `KernelOnly.PartE.PhiBelow.Leaf00389`.
+* `KernelOnly.PartE.PhiBelow.Leaf00392`.
+* `KernelOnly.PartE.PhiBelow.Leaf00394`.
+* `KernelOnly.PartE.PhiBelow.Leaf00395`.
+* `KernelOnly.PartE.PhiBelow.Leaf00396`.
+* `KernelOnly.PartE.PhiBelow.Leaf00397`.
+* `KernelOnly.PartE.PhiBelow.Leaf00399`.
+* `KernelOnly.PartE.PhiBelow.Leaf00400`.
+* `KernelOnly.PartE.PhiBelow.Leaf00404`.
+* `KernelOnly.PartE.PhiBelow.Leaf00405`.
+* `KernelOnly.PartE.PhiBelow.Leaf00406`.
+* `KernelOnly.PartE.PhiBelow.Leaf00407`.
+* `KernelOnly.PartE.PhiBelow.Leaf00409`.
+* `KernelOnly.PartE.PhiBelow.Leaf00411`.
+* `KernelOnly.PartE.PhiBelow.Leaf00413`.
+* `KernelOnly.PartE.PhiBelow.Leaf00414`.
+* `KernelOnly.PartE.PhiBelow.Leaf00417`.
+* `KernelOnly.PartE.PhiBelow.Leaf00418`.
+* `KernelOnly.PartE.PhiBelow.Leaf00419`.
+* `KernelOnly.PartE.PhiBelow.Leaf00420`.
+* `KernelOnly.PartE.PhiBelow.Leaf00422`.
+* `KernelOnly.PartE.PhiBelow.Leaf00423`.
+* `KernelOnly.PartE.PhiBelow.Leaf00424`.
+* `KernelOnly.PartE.PhiBelow.Leaf00426`.
+* `KernelOnly.PartE.PhiBelow.Leaf00428`.
+* `KernelOnly.PartE.PhiBelow.Leaf00430`.
+* `KernelOnly.PartE.PhiBelow.Leaf00431`.
+* `KernelOnly.PartE.PhiBelow.Leaf00432`.
+* `KernelOnly.PartE.PhiBelow.Leaf00433`.
+* `KernelOnly.PartE.PhiBelow.Leaf00437`.
+* `KernelOnly.PartE.PhiBelow.Leaf00438`.
+* `KernelOnly.PartE.PhiBelow.Leaf00440`.
+* `KernelOnly.PartE.PhiBelow.Leaf00441`.
+* `KernelOnly.PartE.PhiBelow.Leaf00442`.
+* `KernelOnly.PartE.PhiBelow.Leaf00443`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c1_c2_5_00388
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6309ffd285
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6309ffd285
+
+open CertificateCells6309ffd285
+namespace CoverCertificate1d94560d47
+
+private theorem checked000 : adaptiveCoverCheck 2 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 2 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 2 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 2 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 2 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 2 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 2 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 2 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 2 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 2 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 2 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 2 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 2 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 2 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 2 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 2 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 2 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 2 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 2 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 2 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell123 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 2 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 2 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 2 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 2 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 2 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 2 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 2 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 2 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 2 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 2 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 2 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 2 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 2 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 2 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 2 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 2 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 2 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 2 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 2 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 2 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 2 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 2 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 2 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 2 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 2 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 2 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 2 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 2 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 2 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 2 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 2 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 2 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1d94560d47
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c1_c2 :
+    adaptiveCoverCheck 5 (childHL phiBelowCell33233221) = true := by
+  exact CoverCertificate1d94560d47.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c1_c3_5_00389
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells48d3b8afd4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells48d3b8afd4
+
+open CertificateCells48d3b8afd4
+namespace CoverCertificatefa0ec53117
+
+private theorem checked220 : adaptiveCoverCheck 2 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 2 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 2 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 2 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell223 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefa0ec53117
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c1_c3 :
+    adaptiveCoverCheck 5 (childHH phiBelowCell33233221) = true := by
+  exact CoverCertificatefa0ec53117.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c2_c0_5_00392
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf95c4e9f9f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf95c4e9f9f
+
+open CertificateCellsf95c4e9f9f
+namespace CoverCertificate38482fd67b
+
+private theorem checked100 : adaptiveCoverCheck 2 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 2 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 2 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 2 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 2 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 2 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 2 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 2 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 2 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 2 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 2 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 2 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 2 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 2 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 2 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 2 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell133 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 2 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 2 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 2 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 2 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 2 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 2 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 2 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 2 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 2 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 2 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 2 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 2 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 2 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 2 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 2 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 2 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 2 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 2 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 2 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 2 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate38482fd67b
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c2_c0 :
+    adaptiveCoverCheck 5 (childLL phiBelowCell33233222) = true := by
+  exact CoverCertificate38482fd67b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c2_c1_c0_4_00394
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells38449ac605
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells38449ac605
+
+open CertificateCells38449ac605
+namespace CoverCertificate774f03a3cd
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate774f03a3cd
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c2_c1_c0 :
+    adaptiveCoverCheck 4 (childLL (childLH phiBelowCell33233222)) = true := by
+  exact CoverCertificate774f03a3cd.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c2_c1_c1_4_00395
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells43ee04fc11
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells43ee04fc11
+
+open CertificateCells43ee04fc11
+namespace CoverCertificate141819f8d7
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate141819f8d7
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c2_c1_c1 :
+    adaptiveCoverCheck 4 (childLH (childLH phiBelowCell33233222)) = true := by
+  exact CoverCertificate141819f8d7.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c2_c1_c2_4_00396
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells303915a97c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells303915a97c
+
+open CertificateCells303915a97c
+namespace CoverCertificated9624571ac
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated9624571ac
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c2_c1_c2 :
+    adaptiveCoverCheck 4 (childHL (childLH phiBelowCell33233222)) = true := by
+  exact CoverCertificated9624571ac.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c2_c1_c3_4_00397
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells88a95e6b6b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells88a95e6b6b
+
+open CertificateCells88a95e6b6b
+namespace CoverCertificate6840dd8bca
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate6840dd8bca
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c2_c1_c3 :
+    adaptiveCoverCheck 4 (childHH (childLH phiBelowCell33233222)) = true := by
+  exact CoverCertificate6840dd8bca.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c2_c2_5_00399
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4611c2250a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4611c2250a
+
+open CertificateCells4611c2250a
+namespace CoverCertificate781ead79bd
+
+private theorem checked100 : adaptiveCoverCheck 2 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 2 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 2 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 2 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 2 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 2 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 2 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 2 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 2 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 2 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 2 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 2 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 2 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 2 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 2 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 2 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell133 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 2 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 2 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 2 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 2 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell313 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate781ead79bd
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c2_c2 :
+    adaptiveCoverCheck 5 (childHL phiBelowCell33233222) = true := by
+  exact CoverCertificate781ead79bd.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c2_c3_5_00400
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellscbfda2707a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellscbfda2707a
+
+open CertificateCellscbfda2707a
+namespace CoverCertificatefcce63a7f6
+
+private theorem checked000 : adaptiveCoverCheck 2 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 2 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 2 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 2 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 2 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 2 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 2 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 2 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 2 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 2 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 2 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 2 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 2 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 2 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 2 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 2 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 2 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 2 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 2 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 2 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 2 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 2 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 2 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 2 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 2 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 2 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 2 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 2 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 2 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 2 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 2 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 2 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 2 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 2 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 2 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 2 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 2 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 2 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 2 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 2 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 2 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 2 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 2 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 2 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 2 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 2 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 2 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 2 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 2 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 2 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 2 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 2 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 2 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 2 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 2 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 2 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 2 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 2 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 2 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 2 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 2 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 2 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 2 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 2 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefcce63a7f6
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c2_c3 :
+    adaptiveCoverCheck 5 (childHH phiBelowCell33233222) = true := by
+  exact CoverCertificatefcce63a7f6.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c3_c0_c0_4_00404
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells003531f536
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells003531f536
+
+open CertificateCells003531f536
+namespace CoverCertificate84a4f6988f
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 1 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 1 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 1 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 1 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell313 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate84a4f6988f
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c0_c0 :
+    adaptiveCoverCheck 4 (childLL (childLL phiBelowCell33233223)) = true := by
+  exact CoverCertificate84a4f6988f.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c3_c0_c1_4_00405
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1f3e328824
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1f3e328824
+
+open CertificateCells1f3e328824
+namespace CoverCertificate03b35dd0cf
+
+private theorem checked200 : adaptiveCoverCheck 1 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 1 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 1 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 1 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell203 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate03b35dd0cf
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c0_c1 :
+    adaptiveCoverCheck 4 (childLH (childLL phiBelowCell33233223)) = true := by
+  exact CoverCertificate03b35dd0cf.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c3_c0_c2_4_00406
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells47ff5dcac0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells47ff5dcac0
+
+open CertificateCells47ff5dcac0
+namespace CoverCertificate2781da35b5
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 1 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 1 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 1 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 1 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 1 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 1 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 1 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 1 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate2781da35b5
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c0_c2 :
+    adaptiveCoverCheck 4 (childHL (childLL phiBelowCell33233223)) = true := by
+  exact CoverCertificate2781da35b5.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c3_c0_c3_4_00407
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells65a3665007
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells65a3665007
+
+open CertificateCells65a3665007
+namespace CoverCertificate7c9b16d986
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 1 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 1 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 1 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 1 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 1 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 1 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 1 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 1 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 1 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 1 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 1 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 1 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 1 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 1 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 1 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 1 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate7c9b16d986
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c0_c3 :
+    adaptiveCoverCheck 4 (childHH (childLL phiBelowCell33233223)) = true := by
+  exact CoverCertificate7c9b16d986.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c3_c1_5_00409
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells45a72ee869
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells45a72ee869
+
+open CertificateCells45a72ee869
+namespace CoverCertificate7a97434462
+
+private theorem checked2220 : adaptiveCoverCheck 1 cell2220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell2220 (by decide +kernel)
+
+private theorem checked2221 : adaptiveCoverCheck 1 cell2221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell2221 (by decide +kernel)
+
+private theorem checked2222 : adaptiveCoverCheck 1 cell2222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell2222 (by decide +kernel)
+
+private theorem checked2223 : adaptiveCoverCheck 1 cell2223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell2223 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 2 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 2 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 2 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 2 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 2 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 2 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 2 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 2 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 2 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 2 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 2 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 2 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 2 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 2 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 2 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 2 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 2 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 2 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 2 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 2 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell123 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 2 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 2 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 2 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 2 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 2 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 2 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 2 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 2 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 2 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 2 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 2 cell222 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell222
+    checked2220 checked2221 checked2222 checked2223
+
+private theorem checked223 : adaptiveCoverCheck 2 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 2 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 2 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 2 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 2 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 2 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 2 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 2 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 2 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell303 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 2 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 2 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 2 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 2 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 2 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 2 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 2 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 2 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate7a97434462
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c1 :
+    adaptiveCoverCheck 5 (childLH phiBelowCell33233223) = true := by
+  exact CoverCertificate7a97434462.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c3_c2_c0_4_00411
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3d2f3b2d12
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3d2f3b2d12
+
+open CertificateCells3d2f3b2d12
+namespace CoverCertificate993722881e
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 1 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 1 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 1 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 1 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 1 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 1 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 1 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 1 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate993722881e
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c0 :
+    adaptiveCoverCheck 4 (childLL (childHL phiBelowCell33233223)) = true := by
+  exact CoverCertificate993722881e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c1_c0_3_00413
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsce18f6e886
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsce18f6e886
+
+open CertificateCellsce18f6e886
+namespace CoverCertificate8f03b44673
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate8f03b44673
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c0 :
+    adaptiveCoverCheck 3 (childLL (childLH (childHL phiBelowCell33233223))) = true := by
+  exact CoverCertificate8f03b44673.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c1_c1_3_00414
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb5cebfd5b6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb5cebfd5b6
+
+open CertificateCellsb5cebfd5b6
+namespace CoverCertificatefffe180fcd
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefffe180fcd
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c1 :
+    adaptiveCoverCheck 3 (childLH (childLH (childHL phiBelowCell33233223))) = true := by
+  exact CoverCertificatefffe180fcd.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c1_c2_c0_c0_1_00417
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2aad13d936
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2aad13d936
+
+open CertificateCells2aad13d936
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c0_c0 :
+    adaptiveCoverCheck 1 (childLL phiBelowCell332332232120) = true := by
+  decide +kernel
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c1_c2_c0_c1_1_00418
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6a90bc785d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6a90bc785d
+
+open CertificateCells6a90bc785d
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c0_c1 :
+    adaptiveCoverCheck 1 (childLH phiBelowCell332332232120) = true := by
+  decide +kernel
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c1_c2_c0_c2_1_00419
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdb781a1ec8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdb781a1ec8
+
+open CertificateCellsdb781a1ec8
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c0_c2 :
+    adaptiveCoverCheck 1 (childHL phiBelowCell332332232120) = true := by
+  decide +kernel
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c1_c2_c0_c3_1_00420
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1cfc9442b0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1cfc9442b0
+
+open CertificateCells1cfc9442b0
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c0_c3 :
+    adaptiveCoverCheck 1 (childHH phiBelowCell332332232120) = true := by
+  decide +kernel
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c1_c2_c1_2_00422
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2dfa44feb0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2dfa44feb0
+
+open CertificateCells2dfa44feb0
+namespace CoverCertificated2a9415b82
+
+private theorem checked0 : adaptiveCoverCheck 1 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 1 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 1 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 1 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 2 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated2a9415b82
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c1 :
+    adaptiveCoverCheck 2 phiBelowCell332332232121 = true := by
+  exact CoverCertificated2a9415b82.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c1_c2_c2_2_00423
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0c5519e954
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0c5519e954
+
+open CertificateCells0c5519e954
+namespace CoverCertificatef93658a45e
+
+private theorem checked0 : adaptiveCoverCheck 1 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 1 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 1 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 1 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 2 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef93658a45e
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c2 :
+    adaptiveCoverCheck 2 phiBelowCell332332232122 = true := by
+  exact CoverCertificatef93658a45e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c1_c2_c3_2_00424
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells06dff71a35
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells06dff71a35
+
+open CertificateCells06dff71a35
+namespace CoverCertificate3d3c2d6223
+
+private theorem checked0 : adaptiveCoverCheck 1 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 1 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 1 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 1 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 2 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3d3c2d6223
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c3 :
+    adaptiveCoverCheck 2 phiBelowCell332332232123 = true := by
+  exact CoverCertificate3d3c2d6223.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c1_c3_3_00426
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells529c7df50a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells529c7df50a
+
+open CertificateCells529c7df50a
+namespace CoverCertificatea6d03c9687
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea6d03c9687
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c3 :
+    adaptiveCoverCheck 3 (childHH (childLH (childHL phiBelowCell33233223))) = true := by
+  exact CoverCertificatea6d03c9687.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c3_c2_c2_4_00428
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells25a838feeb
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells25a838feeb
+
+open CertificateCells25a838feeb
+namespace CoverCertificatefce6b2559b
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 1 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 1 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 1 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 1 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell133 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 1 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 1 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 1 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 1 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 1 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 1 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 1 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 1 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefce6b2559b
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c2 :
+    adaptiveCoverCheck 4 (childHL (childHL phiBelowCell33233223)) = true := by
+  exact CoverCertificatefce6b2559b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c3_c0_3_00430
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsbc23e0e6af
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsbc23e0e6af
+
+open CertificateCellsbc23e0e6af
+namespace CoverCertificate0ac0b8c68f
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0ac0b8c68f
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c3_c0 :
+    adaptiveCoverCheck 3 (childLL (childHH (childHL phiBelowCell33233223))) = true := by
+  exact CoverCertificate0ac0b8c68f.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c3_c1_3_00431
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells13f9420722
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells13f9420722
+
+open CertificateCells13f9420722
+namespace CoverCertificate9ea4339e1a
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate9ea4339e1a
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c3_c1 :
+    adaptiveCoverCheck 3 (childLH (childHH (childHL phiBelowCell33233223))) = true := by
+  exact CoverCertificate9ea4339e1a.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c3_c2_3_00432
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8f726fb04b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8f726fb04b
+
+open CertificateCells8f726fb04b
+namespace CoverCertificatec65ef0b520
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec65ef0b520
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c3_c2 :
+    adaptiveCoverCheck 3 (childHL (childHH (childHL phiBelowCell33233223))) = true := by
+  exact CoverCertificatec65ef0b520.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c3_c3_3_00433
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells7e3f428846
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells7e3f428846
+
+open CertificateCells7e3f428846
+namespace CoverCertificate64e89cb34c
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate64e89cb34c
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c3_c3 :
+    adaptiveCoverCheck 3 (childHH (childHH (childHL phiBelowCell33233223))) = true := by
+  exact CoverCertificate64e89cb34c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c3_c3_c0_4_00437
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4caf1a7a91
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4caf1a7a91
+
+open CertificateCells4caf1a7a91
+namespace CoverCertificatef71635c89a
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 1 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 1 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 1 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 1 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 1 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 1 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 1 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 1 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 1 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 1 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 1 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 1 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 1 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 1 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 1 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 1 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef71635c89a
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c0 :
+    adaptiveCoverCheck 4 (childLL (childHH phiBelowCell33233223)) = true := by
+  exact CoverCertificatef71635c89a.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c3_c3_c1_4_00438
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2907beb942
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2907beb942
+
+open CertificateCells2907beb942
+namespace CoverCertificate4d208d4148
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4d208d4148
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c1 :
+    adaptiveCoverCheck 4 (childLH (childHH phiBelowCell33233223)) = true := by
+  exact CoverCertificate4d208d4148.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c3_c2_c0_3_00440
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb81ac6c2c5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb81ac6c2c5
+
+open CertificateCellsb81ac6c2c5
+namespace CoverCertificatebbc2bd14f5
+
+private theorem checked130 : adaptiveCoverCheck 0 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 0 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 0 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 0 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell133 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatebbc2bd14f5
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c2_c0 :
+    adaptiveCoverCheck 3 (childLL (childHL (childHH phiBelowCell33233223))) = true := by
+  exact CoverCertificatebbc2bd14f5.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c3_c2_c1_3_00441
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6afa02519f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6afa02519f
+
+open CertificateCells6afa02519f
+namespace CoverCertificate0f826c338a
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0f826c338a
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c2_c1 :
+    adaptiveCoverCheck 3 (childLH (childHL (childHH phiBelowCell33233223))) = true := by
+  exact CoverCertificate0f826c338a.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c3_c2_c2_3_00442
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells27566e6112
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells27566e6112
+
+open CertificateCells27566e6112
+namespace CoverCertificate8518e81351
+
+private theorem checked110 : adaptiveCoverCheck 0 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 0 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 0 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 0 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell113 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 0 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 0 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 0 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 0 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell133 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate8518e81351
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c2_c2 :
+    adaptiveCoverCheck 3 (childHL (childHL (childHH phiBelowCell33233223))) = true := by
+  exact CoverCertificate8518e81351.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c3_c2_c3_3_00443
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc20116b798
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc20116b798
+
+open CertificateCellsc20116b798
+namespace CoverCertificate4ecdb55cf7
+
+private theorem checked000 : adaptiveCoverCheck 0 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 0 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 0 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 0 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 0 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 0 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 0 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 0 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 0 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 0 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 0 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 0 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 0 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 0 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 0 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 0 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell033 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 0 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 0 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 0 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 0 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell103 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 0 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 0 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 0 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 0 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell123 (by decide +kernel)
+
+private theorem checked130 : adaptiveCoverCheck 0 cell130 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell130 (by decide +kernel)
+
+private theorem checked131 : adaptiveCoverCheck 0 cell131 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell131 (by decide +kernel)
+
+private theorem checked132 : adaptiveCoverCheck 0 cell132 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell132 (by decide +kernel)
+
+private theorem checked133 : adaptiveCoverCheck 0 cell133 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell133 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell13
+    checked130 checked131 checked132 checked133
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4ecdb55cf7
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c2_c3 :
+    adaptiveCoverCheck 3 (childHH (childHL (childHH phiBelowCell33233223))) = true := by
+  exact CoverCertificate4ecdb55cf7.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle028.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle028.lean
new file mode 100644
index 0000000000..df46e5e903
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle028.lean
@@ -0,0 +1,8979 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle007
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch041`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CoverCertificate68b045bc3f
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childHH (childHL (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot)))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+private abbrev cell2220 : AngleCell :=
+  childLL cell222
+
+private abbrev cell2221 : AngleCell :=
+  childLH cell222
+
+private abbrev cell2222 : AngleCell :=
+  childHL cell222
+
+private abbrev cell2223 : AngleCell :=
+  childHH cell222
+
+end GerverSofa.PartE.CoverCertificate68b045bc3f
+
+namespace GerverSofa.PartE.CoverCertificate56f1cababc
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childHL (childHH (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+end GerverSofa.PartE.CoverCertificate56f1cababc
+
+namespace GerverSofa.PartE.CoverCertificateb43d9cc590
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childHL (childHH (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificateb43d9cc590
+
+namespace GerverSofa.PartE.CoverCertificate3c8c170ddf
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childHH (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate3c8c170ddf
+
+namespace GerverSofa.PartE.CoverCertificate31d8f6a3a1
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childHH (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate31d8f6a3a1
+
+namespace GerverSofa.PartE.CoverCertificate3406557cb6
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childHH (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate3406557cb6
+
+namespace GerverSofa.PartE.CoverCertificated5f4efe538
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childHH (childHL (childHH (childHH (childHL (childHH (childHH
+    (e24PhiBelowRoot))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificated5f4efe538
+
+namespace GerverSofa.PartE.CoverCertificatecb887b449d
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childHL (childHH (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+end GerverSofa.PartE.CoverCertificatecb887b449d
+
+namespace GerverSofa.PartE.CoverCertificateb1645a3287
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childHH (childHL (childHH (childHH (e24PhiBelowRoot)))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell020 : AngleCell :=
+  childLL cell02
+
+private abbrev cell021 : AngleCell :=
+  childLH cell02
+
+private abbrev cell022 : AngleCell :=
+  childHL cell02
+
+private abbrev cell023 : AngleCell :=
+  childHH cell02
+
+private abbrev cell030 : AngleCell :=
+  childLL cell03
+
+private abbrev cell031 : AngleCell :=
+  childLH cell03
+
+private abbrev cell032 : AngleCell :=
+  childHL cell03
+
+private abbrev cell033 : AngleCell :=
+  childHH cell03
+
+private abbrev cell120 : AngleCell :=
+  childLL cell12
+
+private abbrev cell121 : AngleCell :=
+  childLH cell12
+
+private abbrev cell122 : AngleCell :=
+  childHL cell12
+
+private abbrev cell123 : AngleCell :=
+  childHH cell12
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+private abbrev cell2020 : AngleCell :=
+  childLL cell202
+
+private abbrev cell2021 : AngleCell :=
+  childLH cell202
+
+private abbrev cell2022 : AngleCell :=
+  childHL cell202
+
+private abbrev cell2023 : AngleCell :=
+  childHH cell202
+
+private abbrev cell2200 : AngleCell :=
+  childLL cell220
+
+private abbrev cell2201 : AngleCell :=
+  childLH cell220
+
+private abbrev cell2202 : AngleCell :=
+  childHL cell220
+
+private abbrev cell2203 : AngleCell :=
+  childHH cell220
+
+private abbrev cell2210 : AngleCell :=
+  childLL cell221
+
+private abbrev cell2211 : AngleCell :=
+  childLH cell221
+
+private abbrev cell2212 : AngleCell :=
+  childHL cell221
+
+private abbrev cell2213 : AngleCell :=
+  childHH cell221
+
+private abbrev cell2220 : AngleCell :=
+  childLL cell222
+
+private abbrev cell2221 : AngleCell :=
+  childLH cell222
+
+private abbrev cell2222 : AngleCell :=
+  childHL cell222
+
+private abbrev cell2223 : AngleCell :=
+  childHH cell222
+
+private abbrev cell2230 : AngleCell :=
+  childLL cell223
+
+private abbrev cell2231 : AngleCell :=
+  childLH cell223
+
+private abbrev cell2232 : AngleCell :=
+  childHL cell223
+
+private abbrev cell2233 : AngleCell :=
+  childHH cell223
+
+private abbrev cell2320 : AngleCell :=
+  childLL cell232
+
+private abbrev cell2321 : AngleCell :=
+  childLH cell232
+
+private abbrev cell2322 : AngleCell :=
+  childHL cell232
+
+private abbrev cell2323 : AngleCell :=
+  childHH cell232
+
+end GerverSofa.PartE.CoverCertificateb1645a3287
+
+namespace GerverSofa.PartE.CoverCertificatef6f28b19e8
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childHH (childLH (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatef6f28b19e8
+
+namespace GerverSofa.PartE.CoverCertificate15902c79ec
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childHH (childLH (childLH (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate15902c79ec
+
+namespace GerverSofa.PartE.CoverCertificate36f48ed6fa
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHL (childLH (childHH (childLH (childLH (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate36f48ed6fa
+
+namespace GerverSofa.PartE.CoverCertificate80e1d488d9
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHL (childLH (childHH (childLH (childLH (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate80e1d488d9
+
+namespace GerverSofa.PartE.CoverCertificate24ddeb8618
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childLH (childHH (childLH (childLH (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate24ddeb8618
+
+namespace GerverSofa.PartE.CoverCertificatea600465144
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childLH (childHH (childLH (childLH (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatea600465144
+
+namespace GerverSofa.PartE.CoverCertificatefdcad14d68
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childHH (childLH (childHH (childLH (childLH (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatefdcad14d68
+
+namespace GerverSofa.PartE.CoverCertificate3ad1ac00a7
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childHH (childLH (childHH (childLH (childLH (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate3ad1ac00a7
+
+namespace GerverSofa.PartE.CoverCertificatea3bd708439
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childLH (childHH (childLH (childLH (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatea3bd708439
+
+namespace GerverSofa.PartE.CoverCertificateaf7d47a996
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childLH (childHH (childLH (childLH (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificateaf7d47a996
+
+namespace GerverSofa.PartE.CoverCertificatec662ef9997
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childLH (childLH (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatec662ef9997
+
+namespace GerverSofa.PartE.CoverCertificate32e68ad46a
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childLH (childLH (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate32e68ad46a
+
+namespace GerverSofa.PartE.CoverCertificate713bbdaba3
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childHL (childLL (childLL (childHL (childHL (childLL (childLL
+    (childLL (childLL (e24ThetaAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate713bbdaba3
+
+namespace GerverSofa.PartE.CoverCertificate5832573e8a
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childHL (childLL (childLL (childHL (childHL (childLL (childLL
+    (childLL (childLL (e24ThetaAboveRoot)))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+end GerverSofa.PartE.CoverCertificate5832573e8a
+
+namespace GerverSofa.PartE.CoverCertificatec47ffcf2f2
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLL (childLL (childHL (childLL (childLL (childHL (childHL (childLL
+    (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatec47ffcf2f2
+
+namespace GerverSofa.PartE.CoverCertificate535ad25afd
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLL (childLL (childHL (childLL (childLL (childHL (childHL (childLL
+    (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate535ad25afd
+
+namespace GerverSofa.PartE.CoverCertificate746010fa88
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLL (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate746010fa88
+
+namespace GerverSofa.PartE.CoverCertificate1f629121b3
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLL (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate1f629121b3
+
+namespace GerverSofa.PartE.CoverCertificatec4d5c9280e
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatec4d5c9280e
+
+namespace GerverSofa.PartE.CoverCertificate68a7675d0f
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate68a7675d0f
+
+namespace GerverSofa.PartE.CoverCertificateaaac1ef041
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLL (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificateaaac1ef041
+
+namespace GerverSofa.PartE.CoverCertificateeac308265d
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLL (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificateeac308265d
+
+namespace GerverSofa.PartE.CoverCertificateee7e82afcc
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificateee7e82afcc
+
+namespace GerverSofa.PartE.CoverCertificatec74fdabb9c
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatec74fdabb9c
+
+namespace GerverSofa.PartE.CoverCertificateb0390aea72
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHL (childLL (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+end GerverSofa.PartE.CoverCertificateb0390aea72
+
+namespace GerverSofa.PartE.CoverCertificate1f25ee1c58
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHL (childLL (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell1000 : AngleCell :=
+  childLL cell100
+
+private abbrev cell1001 : AngleCell :=
+  childLH cell100
+
+private abbrev cell1002 : AngleCell :=
+  childHL cell100
+
+private abbrev cell1003 : AngleCell :=
+  childHH cell100
+
+private abbrev cell1010 : AngleCell :=
+  childLL cell101
+
+private abbrev cell1011 : AngleCell :=
+  childLH cell101
+
+private abbrev cell1012 : AngleCell :=
+  childHL cell101
+
+private abbrev cell1013 : AngleCell :=
+  childHH cell101
+
+private abbrev cell1100 : AngleCell :=
+  childLL cell110
+
+private abbrev cell1101 : AngleCell :=
+  childLH cell110
+
+private abbrev cell1102 : AngleCell :=
+  childHL cell110
+
+private abbrev cell1103 : AngleCell :=
+  childHH cell110
+
+private abbrev cell1110 : AngleCell :=
+  childLL cell111
+
+private abbrev cell1111 : AngleCell :=
+  childLH cell111
+
+private abbrev cell1112 : AngleCell :=
+  childHL cell111
+
+private abbrev cell1113 : AngleCell :=
+  childHH cell111
+
+end GerverSofa.PartE.CoverCertificate1f25ee1c58
+
+namespace GerverSofa.PartE.CoverCertificate64365f01f0
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLH (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate64365f01f0
+
+namespace GerverSofa.PartE.CoverCertificatea308b70bb8
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLH (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatea308b70bb8
+
+namespace GerverSofa.PartE.CoverCertificate181aa5f0fe
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate181aa5f0fe
+
+namespace GerverSofa.PartE.CoverCertificatee52fd52368
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatee52fd52368
+
+namespace GerverSofa.PartE.CoverCertificate5b77908b1d
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLH (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate5b77908b1d
+
+namespace GerverSofa.PartE.CoverCertificatea6ac82148c
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLH (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatea6ac82148c
+
+namespace GerverSofa.PartE.CertificateCellsd112ae8b34
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsd112ae8b34
+
+namespace GerverSofa.PartE.CertificateCells005204f093
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233230` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233230 : AngleCell :=
+  childLL (childHH (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells005204f093
+
+namespace GerverSofa.PartE.CertificateCells1db372ce58
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233231` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233231 : AngleCell :=
+  childLH (childHH (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells1db372ce58
+
+namespace GerverSofa.PartE.CertificateCells747d0e3511
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233232` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233232 : AngleCell :=
+  childHL (childHH (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells747d0e3511
+
+namespace GerverSofa.PartE.CertificateCells2c20a17ff6
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233232` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233232 : AngleCell :=
+  childHL (childHH (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells2c20a17ff6
+
+namespace GerverSofa.PartE.CertificateCells6d9a408f54
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233232` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233232 : AngleCell :=
+  childHL (childHH (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells6d9a408f54
+
+namespace GerverSofa.PartE.CertificateCells26c5fe31ab
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233232` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233232 : AngleCell :=
+  childHL (childHH (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells26c5fe31ab
+
+namespace GerverSofa.PartE.CertificateCells8faba9c99a
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233233` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233233 : AngleCell :=
+  childHH (childHH (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells8faba9c99a
+
+namespace GerverSofa.PartE.CertificateCellsd0acf0005e
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsd0acf0005e
+
+namespace GerverSofa.PartE.CertificateCellsd9f8181f0f
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsd9f8181f0f
+
+namespace GerverSofa.PartE.CertificateCellsc3ac49754f
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsc3ac49754f
+
+namespace GerverSofa.PartE.CertificateCellsa708f81b81
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020113120` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020113120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaAboveCell000022002011)))
+
+end GerverSofa.PartE.CertificateCellsa708f81b81
+
+namespace GerverSofa.PartE.CertificateCellsd0cc56b48c
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020113121` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020113121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaAboveCell000022002011)))
+
+end GerverSofa.PartE.CertificateCellsd0cc56b48c
+
+namespace GerverSofa.PartE.CertificateCellsecda937af3
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020113122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell000022002011)))
+
+end GerverSofa.PartE.CertificateCellsecda937af3
+
+namespace GerverSofa.PartE.CertificateCellsefa0cf83b9
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020113123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell000022002011)))
+
+end GerverSofa.PartE.CertificateCellsefa0cf83b9
+
+namespace GerverSofa.PartE.CertificateCells38fe0cae80
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020113130` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020113130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaAboveCell000022002011)))
+
+end GerverSofa.PartE.CertificateCells38fe0cae80
+
+namespace GerverSofa.PartE.CertificateCells32c7b8b1fa
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020113131` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020113131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaAboveCell000022002011)))
+
+end GerverSofa.PartE.CertificateCells32c7b8b1fa
+
+namespace GerverSofa.PartE.CertificateCellsb136ba3941
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020113132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell000022002011)))
+
+end GerverSofa.PartE.CertificateCellsb136ba3941
+
+namespace GerverSofa.PartE.CertificateCells8ac48dda13
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020113133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell000022002011)))
+
+end GerverSofa.PartE.CertificateCells8ac48dda13
+
+namespace GerverSofa.PartE.CertificateCellsde5c2f11fb
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsde5c2f11fb
+
+namespace GerverSofa.PartE.CertificateCellsc44d408296
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsc44d408296
+
+namespace GerverSofa.PartE.CertificateCells057d5ed82d
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells057d5ed82d
+
+namespace GerverSofa.PartE.CertificateCellsc56edf0af2
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsc56edf0af2
+
+namespace GerverSofa.PartE.CertificateCells38f587bebf
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells38f587bebf
+
+namespace GerverSofa.PartE.CertificateCellsc330c6667b
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsc330c6667b
+
+namespace GerverSofa.PartE.CertificateCells8addf3ec0f
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells8addf3ec0f
+
+namespace GerverSofa.PartE.CertificateCells06b0cdd59e
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells06b0cdd59e
+
+namespace GerverSofa.PartE.CertificateCellsf1a599f1e1
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCellsf1a599f1e1
+
+namespace GerverSofa.PartE.CertificateCells260222ce28
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells260222ce28
+
+namespace GerverSofa.PartE.CertificateCells5c03b20b81
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells5c03b20b81
+
+namespace GerverSofa.PartE.CertificateCells36e0f7722b
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells36e0f7722b
+
+namespace GerverSofa.PartE.CertificateCellsb7dd59a80a
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCellsb7dd59a80a
+
+namespace GerverSofa.PartE.CertificateCellsef0079dc28
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCellsef0079dc28
+
+namespace GerverSofa.PartE.CertificateCells58fa53de1e
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells58fa53de1e
+
+namespace GerverSofa.PartE.CertificateCells5c48ba0580
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells5c48ba0580
+
+namespace GerverSofa.PartE.CertificateCells35af4df13c
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002100 : AngleCell :=
+  childLL (childLL (childLH (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells35af4df13c
+
+namespace GerverSofa.PartE.CertificateCellsbb596d2a3e
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002101 : AngleCell :=
+  childLH (childLL (childLH (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCellsbb596d2a3e
+
+namespace GerverSofa.PartE.CertificateCells8645eca0c5
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002102 : AngleCell :=
+  childHL (childLL (childLH (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells8645eca0c5
+
+namespace GerverSofa.PartE.CertificateCellsbe94e44cff
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002103 : AngleCell :=
+  childHH (childLL (childLH (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCellsbe94e44cff
+
+namespace GerverSofa.PartE.CertificateCellsd01cbbb894
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002110 : AngleCell :=
+  childLL (childLH (childLH (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCellsd01cbbb894
+
+namespace GerverSofa.PartE.CertificateCells768f9f5a16
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002111 : AngleCell :=
+  childLH (childLH (childLH (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells768f9f5a16
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.PhiBelow.Leaf00445`.
+* `KernelOnly.PartE.PhiBelow.Leaf00450`.
+* `KernelOnly.PartE.PhiBelow.Leaf00451`.
+* `KernelOnly.PartE.PhiBelow.Leaf00453`.
+* `KernelOnly.PartE.PhiBelow.Leaf00454`.
+* `KernelOnly.PartE.PhiBelow.Leaf00455`.
+* `KernelOnly.PartE.PhiBelow.Leaf00456`.
+* `KernelOnly.PartE.PhiBelow.Leaf00458`.
+* `KernelOnly.PartE.PhiBelow.Leaf00461`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00009`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00010`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00012`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00013`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00014`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00015`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00018`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00019`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00020`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00021`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00024`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00025`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00028`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00029`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00475`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00476`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00480`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00481`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00482`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00483`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00486`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00487`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00488`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00489`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00491`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00492`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00496`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00497`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00498`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00499`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00502`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00503`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c2_c3_c3_c3_4_00445
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd112ae8b34
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd112ae8b34
+
+open CertificateCellsd112ae8b34
+namespace CoverCertificate68b045bc3f
+
+private theorem checked2220 : adaptiveCoverCheck 0 cell2220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2220 (by decide +kernel)
+
+private theorem checked2221 : adaptiveCoverCheck 0 cell2221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2221 (by decide +kernel)
+
+private theorem checked2222 : adaptiveCoverCheck 0 cell2222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2222 (by decide +kernel)
+
+private theorem checked2223 : adaptiveCoverCheck 0 cell2223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell2223 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 1 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 1 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 1 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 1 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 1 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 1 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 1 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 1 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 1 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 1 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 1 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 1 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell123 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 1 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 1 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 1 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 1 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 1 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 1 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 1 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 1 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell222
+    checked2220 checked2221 checked2222 checked2223
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 1 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 1 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 1 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 1 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell303 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate68b045bc3f
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c3 :
+    adaptiveCoverCheck 4 (childHH (childHH phiBelowCell33233223)) = true := by
+  exact CoverCertificate68b045bc3f.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c3_c0_6_00450
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells005204f093
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells005204f093
+
+open CertificateCells005204f093
+namespace CoverCertificate56f1cababc
+
+private theorem checked200 : adaptiveCoverCheck 3 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 3 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 3 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 3 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell203 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 3 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 3 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 3 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 3 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 3 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 3 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 3 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 3 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell233 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate56f1cababc
+
+theorem e24KC2PhiBelowLeaf33233_c2_c3_c0 :
+    adaptiveCoverCheck 6 phiBelowCell33233230 = true := by
+  exact CoverCertificate56f1cababc.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c3_c1_6_00451
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1db372ce58
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1db372ce58
+
+open CertificateCells1db372ce58
+namespace CoverCertificateb43d9cc590
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateb43d9cc590
+
+theorem e24KC2PhiBelowLeaf33233_c2_c3_c1 :
+    adaptiveCoverCheck 6 phiBelowCell33233231 = true := by
+  exact CoverCertificateb43d9cc590.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c3_c2_c0_5_00453
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells747d0e3511
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells747d0e3511
+
+open CertificateCells747d0e3511
+namespace CoverCertificate3c8c170ddf
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3c8c170ddf
+
+theorem e24KC2PhiBelowLeaf33233_c2_c3_c2_c0 :
+    adaptiveCoverCheck 5 (childLL phiBelowCell33233232) = true := by
+  exact CoverCertificate3c8c170ddf.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c3_c2_c1_5_00454
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2c20a17ff6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2c20a17ff6
+
+open CertificateCells2c20a17ff6
+namespace CoverCertificate31d8f6a3a1
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate31d8f6a3a1
+
+theorem e24KC2PhiBelowLeaf33233_c2_c3_c2_c1 :
+    adaptiveCoverCheck 5 (childLH phiBelowCell33233232) = true := by
+  exact CoverCertificate31d8f6a3a1.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c3_c2_c2_5_00455
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6d9a408f54
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6d9a408f54
+
+open CertificateCells6d9a408f54
+namespace CoverCertificate3406557cb6
+
+private theorem checked000 : adaptiveCoverCheck 2 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 2 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 2 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 2 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell003 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 2 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 2 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 2 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 2 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 2 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 2 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 2 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 2 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell033 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 2 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 2 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 2 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 2 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 2 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 2 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 2 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 2 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 2 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 2 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 2 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 2 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 2 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 2 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 2 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 2 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 2 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 2 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 2 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 2 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell303 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 2 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 2 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 2 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 2 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 2 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 2 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 2 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 2 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3406557cb6
+
+theorem e24KC2PhiBelowLeaf33233_c2_c3_c2_c2 :
+    adaptiveCoverCheck 5 (childHL phiBelowCell33233232) = true := by
+  exact CoverCertificate3406557cb6.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c3_c2_c3_5_00456
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells26c5fe31ab
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells26c5fe31ab
+
+open CertificateCells26c5fe31ab
+namespace CoverCertificated5f4efe538
+
+private theorem checked00 : adaptiveCoverCheck 3 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 3 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 3 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 3 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 3 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 3 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 3 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 3 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 3 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 3 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 3 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 3 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 3 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 3 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 3 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 3 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated5f4efe538
+
+theorem e24KC2PhiBelowLeaf33233_c2_c3_c2_c3 :
+    adaptiveCoverCheck 5 (childHH phiBelowCell33233232) = true := by
+  exact CoverCertificated5f4efe538.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c2_c3_c3_6_00458
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8faba9c99a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8faba9c99a
+
+open CertificateCells8faba9c99a
+namespace CoverCertificatecb887b449d
+
+private theorem checked200 : adaptiveCoverCheck 3 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 3 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 3 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 3 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell203 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 3 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 3 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 3 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 3 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 3 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 3 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 3 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 3 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell233 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 4 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 4 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 4 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 4 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 4 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 4 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 4 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 4 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatecb887b449d
+
+theorem e24KC2PhiBelowLeaf33233_c2_c3_c3 :
+    adaptiveCoverCheck 6 phiBelowCell33233233 = true := by
+  exact CoverCertificatecb887b449d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Phi Below Leaf33233_c3_8_00461
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd0acf0005e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd0acf0005e
+
+open CertificateCellsd0acf0005e
+namespace CoverCertificateb1645a3287
+
+private theorem checked2020 : adaptiveCoverCheck 4 cell2020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2020 (by decide +kernel)
+
+private theorem checked2021 : adaptiveCoverCheck 4 cell2021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2021 (by decide +kernel)
+
+private theorem checked2022 : adaptiveCoverCheck 4 cell2022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2022 (by decide +kernel)
+
+private theorem checked2023 : adaptiveCoverCheck 4 cell2023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2023 (by decide +kernel)
+
+private theorem checked2200 : adaptiveCoverCheck 4 cell2200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2200 (by decide +kernel)
+
+private theorem checked2201 : adaptiveCoverCheck 4 cell2201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2201 (by decide +kernel)
+
+private theorem checked2202 : adaptiveCoverCheck 4 cell2202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2202 (by decide +kernel)
+
+private theorem checked2203 : adaptiveCoverCheck 4 cell2203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2203 (by decide +kernel)
+
+private theorem checked2210 : adaptiveCoverCheck 4 cell2210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2210 (by decide +kernel)
+
+private theorem checked2211 : adaptiveCoverCheck 4 cell2211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2211 (by decide +kernel)
+
+private theorem checked2212 : adaptiveCoverCheck 4 cell2212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2212 (by decide +kernel)
+
+private theorem checked2213 : adaptiveCoverCheck 4 cell2213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2213 (by decide +kernel)
+
+private theorem checked2220 : adaptiveCoverCheck 4 cell2220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2220 (by decide +kernel)
+
+private theorem checked2221 : adaptiveCoverCheck 4 cell2221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2221 (by decide +kernel)
+
+private theorem checked2222 : adaptiveCoverCheck 4 cell2222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2222 (by decide +kernel)
+
+private theorem checked2223 : adaptiveCoverCheck 4 cell2223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2223 (by decide +kernel)
+
+private theorem checked2230 : adaptiveCoverCheck 4 cell2230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2230 (by decide +kernel)
+
+private theorem checked2231 : adaptiveCoverCheck 4 cell2231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2231 (by decide +kernel)
+
+private theorem checked2232 : adaptiveCoverCheck 4 cell2232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2232 (by decide +kernel)
+
+private theorem checked2233 : adaptiveCoverCheck 4 cell2233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2233 (by decide +kernel)
+
+private theorem checked2320 : adaptiveCoverCheck 4 cell2320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2320 (by decide +kernel)
+
+private theorem checked2321 : adaptiveCoverCheck 4 cell2321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2321 (by decide +kernel)
+
+private theorem checked2322 : adaptiveCoverCheck 4 cell2322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2322 (by decide +kernel)
+
+private theorem checked2323 : adaptiveCoverCheck 4 cell2323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2323 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 5 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 5 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 5 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 5 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 5 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 5 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 5 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 5 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell013 (by decide +kernel)
+
+private theorem checked020 : adaptiveCoverCheck 5 cell020 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell020 (by decide +kernel)
+
+private theorem checked021 : adaptiveCoverCheck 5 cell021 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell021 (by decide +kernel)
+
+private theorem checked022 : adaptiveCoverCheck 5 cell022 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell022 (by decide +kernel)
+
+private theorem checked023 : adaptiveCoverCheck 5 cell023 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell023 (by decide +kernel)
+
+private theorem checked030 : adaptiveCoverCheck 5 cell030 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell030 (by decide +kernel)
+
+private theorem checked031 : adaptiveCoverCheck 5 cell031 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell031 (by decide +kernel)
+
+private theorem checked032 : adaptiveCoverCheck 5 cell032 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell032 (by decide +kernel)
+
+private theorem checked033 : adaptiveCoverCheck 5 cell033 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell033 (by decide +kernel)
+
+private theorem checked120 : adaptiveCoverCheck 5 cell120 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell120 (by decide +kernel)
+
+private theorem checked121 : adaptiveCoverCheck 5 cell121 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell121 (by decide +kernel)
+
+private theorem checked122 : adaptiveCoverCheck 5 cell122 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell122 (by decide +kernel)
+
+private theorem checked123 : adaptiveCoverCheck 5 cell123 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell123 (by decide +kernel)
+
+private theorem checked200 : adaptiveCoverCheck 5 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 5 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 5 cell202 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell202
+    checked2020 checked2021 checked2022 checked2023
+
+private theorem checked203 : adaptiveCoverCheck 5 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 5 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 5 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 5 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 5 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 5 cell220 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell220
+    checked2200 checked2201 checked2202 checked2203
+
+private theorem checked221 : adaptiveCoverCheck 5 cell221 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell221
+    checked2210 checked2211 checked2212 checked2213
+
+private theorem checked222 : adaptiveCoverCheck 5 cell222 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell222
+    checked2220 checked2221 checked2222 checked2223
+
+private theorem checked223 : adaptiveCoverCheck 5 cell223 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell223
+    checked2230 checked2231 checked2232 checked2233
+
+private theorem checked230 : adaptiveCoverCheck 5 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 5 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 5 cell232 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell232
+    checked2320 checked2321 checked2322 checked2323
+
+private theorem checked233 : adaptiveCoverCheck 5 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 5 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 5 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 5 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 5 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 5 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 5 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 5 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 5 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 5 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 5 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 5 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 5 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 5 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 5 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 5 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 5 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 6 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 6 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 6 cell02 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell02
+    checked020 checked021 checked022 checked023
+
+private theorem checked03 : adaptiveCoverCheck 6 cell03 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell03
+    checked030 checked031 checked032 checked033
+
+private theorem checked10 : adaptiveCoverCheck 6 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 6 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 6 cell12 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell12
+    checked120 checked121 checked122 checked123
+
+private theorem checked13 : adaptiveCoverCheck 6 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 6 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 6 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 6 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 6 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 6 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 6 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 6 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 6 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 7 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 7 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 7 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 7 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 8 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 7 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateb1645a3287
+
+theorem e24KC2PhiBelowLeaf33233_c3 :
+    adaptiveCoverCheck 8 (childHH (childHH phiBelowCell3323)) = true := by
+  exact CoverCertificateb1645a3287.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c1_c0_4_00009
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd9f8181f0f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd9f8181f0f
+
+open CertificateCellsd9f8181f0f
+namespace CoverCertificatef6f28b19e8
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatef6f28b19e8
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c0 :
+    adaptiveCoverCheck 4 (childLL (childLH (childHH thetaAboveCell000022002011))) = true := by
+  exact CoverCertificatef6f28b19e8.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c1_c1_4_00010
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc3ac49754f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc3ac49754f
+
+open CertificateCellsc3ac49754f
+namespace CoverCertificate15902c79ec
+
+private theorem checked220 : adaptiveCoverCheck 1 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 1 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 1 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 1 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 1 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 1 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 1 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 1 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell233 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 1 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 1 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 1 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 1 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 1 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 1 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 1 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 1 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell333 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 2 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 2 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 2 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 2 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 2 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 2 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 2 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 2 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate15902c79ec
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c1 :
+    adaptiveCoverCheck 4 (childLH (childLH (childHH thetaAboveCell000022002011))) = true := by
+  exact CoverCertificate15902c79ec.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c1_c2_c0_3_00012
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa708f81b81
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa708f81b81
+
+open CertificateCellsa708f81b81
+namespace CoverCertificate36f48ed6fa
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate36f48ed6fa
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c2_c0 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020113120 = true := by
+  exact CoverCertificate36f48ed6fa.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c1_c2_c1_3_00013
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd0cc56b48c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd0cc56b48c
+
+open CertificateCellsd0cc56b48c
+namespace CoverCertificate80e1d488d9
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate80e1d488d9
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c2_c1 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020113121 = true := by
+  exact CoverCertificate80e1d488d9.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c1_c2_c2_3_00014
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsecda937af3
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsecda937af3
+
+open CertificateCellsecda937af3
+namespace CoverCertificate24ddeb8618
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate24ddeb8618
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c2_c2 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020113122 = true := by
+  exact CoverCertificate24ddeb8618.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c1_c2_c3_3_00015
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsefa0cf83b9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsefa0cf83b9
+
+open CertificateCellsefa0cf83b9
+namespace CoverCertificatea600465144
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea600465144
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c2_c3 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020113123 = true := by
+  exact CoverCertificatea600465144.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c1_c3_c0_3_00018
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells38fe0cae80
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells38fe0cae80
+
+open CertificateCells38fe0cae80
+namespace CoverCertificatefdcad14d68
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatefdcad14d68
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c3_c0 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020113130 = true := by
+  exact CoverCertificatefdcad14d68.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c1_c3_c1_3_00019
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells32c7b8b1fa
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells32c7b8b1fa
+
+open CertificateCells32c7b8b1fa
+namespace CoverCertificate3ad1ac00a7
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate3ad1ac00a7
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c3_c1 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020113131 = true := by
+  exact CoverCertificate3ad1ac00a7.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c1_c3_c2_3_00020
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb136ba3941
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb136ba3941
+
+open CertificateCellsb136ba3941
+namespace CoverCertificatea3bd708439
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea3bd708439
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c3_c2 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020113132 = true := by
+  exact CoverCertificatea3bd708439.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c1_c3_c3_3_00021
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8ac48dda13
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8ac48dda13
+
+open CertificateCells8ac48dda13
+namespace CoverCertificateaf7d47a996
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateaf7d47a996
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c3_c3 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020113133 = true := by
+  exact CoverCertificateaf7d47a996.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c2_5_00024
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsde5c2f11fb
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsde5c2f11fb
+
+open CertificateCellsde5c2f11fb
+namespace CoverCertificatec662ef9997
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec662ef9997
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c2 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022002011)) = true := by
+  exact CoverCertificatec662ef9997.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c3_5_00025
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc44d408296
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc44d408296
+
+open CertificateCellsc44d408296
+namespace CoverCertificate32e68ad46a
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate32e68ad46a
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c3 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022002011)) = true := by
+  exact CoverCertificate32e68ad46a.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above Leaf0000220020_c1_c2_7_00028
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells057d5ed82d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells057d5ed82d
+
+open CertificateCells057d5ed82d
+namespace CoverCertificate713bbdaba3
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate713bbdaba3
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c2 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002012 = true := by
+  exact CoverCertificate713bbdaba3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above Leaf0000220020_c1_c3_7_00029
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc56edf0af2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc56edf0af2
+
+open CertificateCellsc56edf0af2
+namespace CoverCertificate5832573e8a
+
+private theorem checked00 : adaptiveCoverCheck 5 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 5 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 5 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 5 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 5 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 5 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 5 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 5 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 6 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 6 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 6 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 6 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 6 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 7 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 6 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5832573e8a
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c3 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002013 = true := by
+  exact CoverCertificate5832573e8a.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c0_6_00475
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells38f587bebf
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells38f587bebf
+
+open CertificateCells38f587bebf
+namespace CoverCertificatec47ffcf2f2
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec47ffcf2f2
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c0 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022002000) = true := by
+  exact CoverCertificatec47ffcf2f2.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c1_6_00476
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc330c6667b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc330c6667b
+
+open CertificateCellsc330c6667b
+namespace CoverCertificate535ad25afd
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate535ad25afd
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c1 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022002000) = true := by
+  exact CoverCertificate535ad25afd.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c0_c0_c0_3_00480
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8addf3ec0f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8addf3ec0f
+
+open CertificateCells8addf3ec0f
+namespace CoverCertificate746010fa88
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate746010fa88
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c0_c0 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002000 = true := by
+  exact CoverCertificate746010fa88.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c0_c0_c1_3_00481
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells06b0cdd59e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells06b0cdd59e
+
+open CertificateCells06b0cdd59e
+namespace CoverCertificate1f629121b3
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1f629121b3
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c0_c1 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002001 = true := by
+  exact CoverCertificate1f629121b3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c0_c0_c2_3_00482
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf1a599f1e1
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf1a599f1e1
+
+open CertificateCellsf1a599f1e1
+namespace CoverCertificatec4d5c9280e
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec4d5c9280e
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c0_c2 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002002 = true := by
+  exact CoverCertificatec4d5c9280e.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c0_c0_c3_3_00483
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells260222ce28
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells260222ce28
+
+open CertificateCells260222ce28
+namespace CoverCertificate68a7675d0f
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate68a7675d0f
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c0_c3 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002003 = true := by
+  exact CoverCertificate68a7675d0f.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c0_c1_c0_3_00486
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5c03b20b81
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5c03b20b81
+
+open CertificateCells5c03b20b81
+namespace CoverCertificateaaac1ef041
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateaaac1ef041
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c1_c0 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002010 = true := by
+  exact CoverCertificateaaac1ef041.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c0_c1_c1_3_00487
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells36e0f7722b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells36e0f7722b
+
+open CertificateCells36e0f7722b
+namespace CoverCertificateeac308265d
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateeac308265d
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c1_c1 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002011 = true := by
+  exact CoverCertificateeac308265d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c0_c1_c2_3_00488
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb7dd59a80a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb7dd59a80a
+
+open CertificateCellsb7dd59a80a
+namespace CoverCertificateee7e82afcc
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateee7e82afcc
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c1_c2 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002012 = true := by
+  exact CoverCertificateee7e82afcc.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c0_c1_c3_3_00489
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsef0079dc28
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsef0079dc28
+
+open CertificateCellsef0079dc28
+namespace CoverCertificatec74fdabb9c
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec74fdabb9c
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c1_c3 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002013 = true := by
+  exact CoverCertificatec74fdabb9c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c0_c2_4_00491
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells58fa53de1e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells58fa53de1e
+
+open CertificateCells58fa53de1e
+namespace CoverCertificateb0390aea72
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell100 (by decide +kernel)
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell101 (by decide +kernel)
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell110 (by decide +kernel)
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell111 (by decide +kernel)
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateb0390aea72
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c2 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHL thetaAboveCell000022002000))) = true := by
+  exact CoverCertificateb0390aea72.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c0_c3_4_00492
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5c48ba0580
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5c48ba0580
+
+open CertificateCells5c48ba0580
+namespace CoverCertificate1f25ee1c58
+
+private theorem checked1000 : adaptiveCoverCheck 0 cell1000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1000 (by decide +kernel)
+
+private theorem checked1001 : adaptiveCoverCheck 0 cell1001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1001 (by decide +kernel)
+
+private theorem checked1002 : adaptiveCoverCheck 0 cell1002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1002 (by decide +kernel)
+
+private theorem checked1003 : adaptiveCoverCheck 0 cell1003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1003 (by decide +kernel)
+
+private theorem checked1010 : adaptiveCoverCheck 0 cell1010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1010 (by decide +kernel)
+
+private theorem checked1011 : adaptiveCoverCheck 0 cell1011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1011 (by decide +kernel)
+
+private theorem checked1012 : adaptiveCoverCheck 0 cell1012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1012 (by decide +kernel)
+
+private theorem checked1013 : adaptiveCoverCheck 0 cell1013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1013 (by decide +kernel)
+
+private theorem checked1100 : adaptiveCoverCheck 0 cell1100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1100 (by decide +kernel)
+
+private theorem checked1101 : adaptiveCoverCheck 0 cell1101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1101 (by decide +kernel)
+
+private theorem checked1102 : adaptiveCoverCheck 0 cell1102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1102 (by decide +kernel)
+
+private theorem checked1103 : adaptiveCoverCheck 0 cell1103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1103 (by decide +kernel)
+
+private theorem checked1110 : adaptiveCoverCheck 0 cell1110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1110 (by decide +kernel)
+
+private theorem checked1111 : adaptiveCoverCheck 0 cell1111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1111 (by decide +kernel)
+
+private theorem checked1112 : adaptiveCoverCheck 0 cell1112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1112 (by decide +kernel)
+
+private theorem checked1113 : adaptiveCoverCheck 0 cell1113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1113 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell000 (by decide +kernel)
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell001 (by decide +kernel)
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell010 (by decide +kernel)
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell011 (by decide +kernel)
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell100
+    checked1000 checked1001 checked1002 checked1003
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell101
+    checked1010 checked1011 checked1012 checked1013
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell110
+    checked1100 checked1101 checked1102 checked1103
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell111
+    checked1110 checked1111 checked1112 checked1113
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1f25ee1c58
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c3 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHL thetaAboveCell000022002000))) = true := by
+  exact CoverCertificate1f25ee1c58.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c1_c0_c0_3_00496
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells35af4df13c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells35af4df13c
+
+open CertificateCells35af4df13c
+namespace CoverCertificate64365f01f0
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate64365f01f0
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c0_c0 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002100 = true := by
+  exact CoverCertificate64365f01f0.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c1_c0_c1_3_00497
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsbb596d2a3e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsbb596d2a3e
+
+open CertificateCellsbb596d2a3e
+namespace CoverCertificatea308b70bb8
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea308b70bb8
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c0_c1 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002101 = true := by
+  exact CoverCertificatea308b70bb8.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c1_c0_c2_3_00498
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8645eca0c5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8645eca0c5
+
+open CertificateCells8645eca0c5
+namespace CoverCertificate181aa5f0fe
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate181aa5f0fe
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c0_c2 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002102 = true := by
+  exact CoverCertificate181aa5f0fe.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c1_c0_c3_3_00499
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsbe94e44cff
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsbe94e44cff
+
+open CertificateCellsbe94e44cff
+namespace CoverCertificatee52fd52368
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee52fd52368
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c0_c3 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002103 = true := by
+  exact CoverCertificatee52fd52368.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c1_c1_c0_3_00502
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd01cbbb894
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd01cbbb894
+
+open CertificateCellsd01cbbb894
+namespace CoverCertificate5b77908b1d
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate5b77908b1d
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c1_c0 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002110 = true := by
+  exact CoverCertificate5b77908b1d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c1_c1_c1_3_00503
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells768f9f5a16
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells768f9f5a16
+
+open CertificateCells768f9f5a16
+namespace CoverCertificatea6ac82148c
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea6ac82148c
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c1_c1 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002111 = true := by
+  exact CoverCertificatea6ac82148c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle029.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle029.lean
new file mode 100644
index 0000000000..334f934a27
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle029.lean
@@ -0,0 +1,9721 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch042`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CoverCertificate09deb4be62
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLH (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate09deb4be62
+
+namespace GerverSofa.PartE.CoverCertificate80b49a8860
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLH (childHL (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate80b49a8860
+
+namespace GerverSofa.PartE.CoverCertificate8fa79bb108
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHL (childLL (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell0000 : AngleCell :=
+  childLL cell000
+
+private abbrev cell0001 : AngleCell :=
+  childLH cell000
+
+private abbrev cell0002 : AngleCell :=
+  childHL cell000
+
+private abbrev cell0003 : AngleCell :=
+  childHH cell000
+
+private abbrev cell0010 : AngleCell :=
+  childLL cell001
+
+private abbrev cell0011 : AngleCell :=
+  childLH cell001
+
+private abbrev cell0012 : AngleCell :=
+  childHL cell001
+
+private abbrev cell0013 : AngleCell :=
+  childHH cell001
+
+private abbrev cell0100 : AngleCell :=
+  childLL cell010
+
+private abbrev cell0101 : AngleCell :=
+  childLH cell010
+
+private abbrev cell0102 : AngleCell :=
+  childHL cell010
+
+private abbrev cell0103 : AngleCell :=
+  childHH cell010
+
+private abbrev cell0110 : AngleCell :=
+  childLL cell011
+
+private abbrev cell0111 : AngleCell :=
+  childLH cell011
+
+private abbrev cell0112 : AngleCell :=
+  childHL cell011
+
+private abbrev cell0113 : AngleCell :=
+  childHH cell011
+
+private abbrev cell1000 : AngleCell :=
+  childLL cell100
+
+private abbrev cell1001 : AngleCell :=
+  childLH cell100
+
+private abbrev cell1002 : AngleCell :=
+  childHL cell100
+
+private abbrev cell1003 : AngleCell :=
+  childHH cell100
+
+private abbrev cell1010 : AngleCell :=
+  childLL cell101
+
+private abbrev cell1011 : AngleCell :=
+  childLH cell101
+
+private abbrev cell1012 : AngleCell :=
+  childHL cell101
+
+private abbrev cell1013 : AngleCell :=
+  childHH cell101
+
+private abbrev cell1100 : AngleCell :=
+  childLL cell110
+
+private abbrev cell1101 : AngleCell :=
+  childLH cell110
+
+private abbrev cell1102 : AngleCell :=
+  childHL cell110
+
+private abbrev cell1103 : AngleCell :=
+  childHH cell110
+
+private abbrev cell1110 : AngleCell :=
+  childLL cell111
+
+private abbrev cell1111 : AngleCell :=
+  childLH cell111
+
+private abbrev cell1112 : AngleCell :=
+  childHL cell111
+
+private abbrev cell1113 : AngleCell :=
+  childHH cell111
+
+end GerverSofa.PartE.CoverCertificate8fa79bb108
+
+namespace GerverSofa.PartE.CoverCertificate9198440329
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHL (childLL (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell0000 : AngleCell :=
+  childLL cell000
+
+private abbrev cell0001 : AngleCell :=
+  childLH cell000
+
+private abbrev cell0002 : AngleCell :=
+  childHL cell000
+
+private abbrev cell0003 : AngleCell :=
+  childHH cell000
+
+private abbrev cell0010 : AngleCell :=
+  childLL cell001
+
+private abbrev cell0011 : AngleCell :=
+  childLH cell001
+
+private abbrev cell0012 : AngleCell :=
+  childHL cell001
+
+private abbrev cell0013 : AngleCell :=
+  childHH cell001
+
+private abbrev cell0100 : AngleCell :=
+  childLL cell010
+
+private abbrev cell0101 : AngleCell :=
+  childLH cell010
+
+private abbrev cell0102 : AngleCell :=
+  childHL cell010
+
+private abbrev cell0103 : AngleCell :=
+  childHH cell010
+
+private abbrev cell0110 : AngleCell :=
+  childLL cell011
+
+private abbrev cell0111 : AngleCell :=
+  childLH cell011
+
+private abbrev cell0112 : AngleCell :=
+  childHL cell011
+
+private abbrev cell0113 : AngleCell :=
+  childHH cell011
+
+private abbrev cell1000 : AngleCell :=
+  childLL cell100
+
+private abbrev cell1001 : AngleCell :=
+  childLH cell100
+
+private abbrev cell1002 : AngleCell :=
+  childHL cell100
+
+private abbrev cell1003 : AngleCell :=
+  childHH cell100
+
+private abbrev cell1010 : AngleCell :=
+  childLL cell101
+
+private abbrev cell1011 : AngleCell :=
+  childLH cell101
+
+private abbrev cell1012 : AngleCell :=
+  childHL cell101
+
+private abbrev cell1013 : AngleCell :=
+  childHH cell101
+
+private abbrev cell1100 : AngleCell :=
+  childLL cell110
+
+private abbrev cell1101 : AngleCell :=
+  childLH cell110
+
+private abbrev cell1102 : AngleCell :=
+  childHL cell110
+
+private abbrev cell1103 : AngleCell :=
+  childHH cell110
+
+private abbrev cell1110 : AngleCell :=
+  childLL cell111
+
+private abbrev cell1111 : AngleCell :=
+  childLH cell111
+
+private abbrev cell1112 : AngleCell :=
+  childHL cell111
+
+private abbrev cell1113 : AngleCell :=
+  childHH cell111
+
+end GerverSofa.PartE.CoverCertificate9198440329
+
+namespace GerverSofa.PartE.CoverCertificate54d2ea68c4
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHL (childLL (childLL (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate54d2ea68c4
+
+namespace GerverSofa.PartE.CoverCertificate9a90f66090
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHL (childLL (childLL (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate9a90f66090
+
+namespace GerverSofa.PartE.CoverCertificate9a54be935a
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLL (childHH (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate9a54be935a
+
+namespace GerverSofa.PartE.CoverCertificatea35db1f021
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLL (childLL (childHH (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificatea35db1f021
+
+namespace GerverSofa.PartE.CoverCertificate8333a307a4
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childHH (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate8333a307a4
+
+namespace GerverSofa.PartE.CoverCertificated46e97b4d3
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childHH (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificated46e97b4d3
+
+namespace GerverSofa.PartE.CoverCertificate39bdb9b52b
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLL (childHH (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate39bdb9b52b
+
+namespace GerverSofa.PartE.CoverCertificate21662757e3
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLL (childHH (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate21662757e3
+
+namespace GerverSofa.PartE.CoverCertificate7ec8971130
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childHH (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate7ec8971130
+
+namespace GerverSofa.PartE.CoverCertificatec012db59ec
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childHH (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatec012db59ec
+
+namespace GerverSofa.PartE.CoverCertificate418c92a9a1
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childHH (childLL (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell0000 : AngleCell :=
+  childLL cell000
+
+private abbrev cell0001 : AngleCell :=
+  childLH cell000
+
+private abbrev cell0002 : AngleCell :=
+  childHL cell000
+
+private abbrev cell0003 : AngleCell :=
+  childHH cell000
+
+private abbrev cell0010 : AngleCell :=
+  childLL cell001
+
+private abbrev cell0011 : AngleCell :=
+  childLH cell001
+
+private abbrev cell0012 : AngleCell :=
+  childHL cell001
+
+private abbrev cell0013 : AngleCell :=
+  childHH cell001
+
+private abbrev cell0100 : AngleCell :=
+  childLL cell010
+
+private abbrev cell0101 : AngleCell :=
+  childLH cell010
+
+private abbrev cell0102 : AngleCell :=
+  childHL cell010
+
+private abbrev cell0103 : AngleCell :=
+  childHH cell010
+
+private abbrev cell0110 : AngleCell :=
+  childLL cell011
+
+private abbrev cell0111 : AngleCell :=
+  childLH cell011
+
+private abbrev cell0112 : AngleCell :=
+  childHL cell011
+
+private abbrev cell0113 : AngleCell :=
+  childHH cell011
+
+private abbrev cell1000 : AngleCell :=
+  childLL cell100
+
+private abbrev cell1001 : AngleCell :=
+  childLH cell100
+
+private abbrev cell1002 : AngleCell :=
+  childHL cell100
+
+private abbrev cell1003 : AngleCell :=
+  childHH cell100
+
+private abbrev cell1010 : AngleCell :=
+  childLL cell101
+
+private abbrev cell1011 : AngleCell :=
+  childLH cell101
+
+private abbrev cell1012 : AngleCell :=
+  childHL cell101
+
+private abbrev cell1013 : AngleCell :=
+  childHH cell101
+
+private abbrev cell1100 : AngleCell :=
+  childLL cell110
+
+private abbrev cell1101 : AngleCell :=
+  childLH cell110
+
+private abbrev cell1102 : AngleCell :=
+  childHL cell110
+
+private abbrev cell1103 : AngleCell :=
+  childHH cell110
+
+private abbrev cell1110 : AngleCell :=
+  childLL cell111
+
+private abbrev cell1111 : AngleCell :=
+  childLH cell111
+
+private abbrev cell1112 : AngleCell :=
+  childHL cell111
+
+private abbrev cell1113 : AngleCell :=
+  childHH cell111
+
+end GerverSofa.PartE.CoverCertificate418c92a9a1
+
+namespace GerverSofa.PartE.CoverCertificate69c64655e6
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childHH (childLL (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell0000 : AngleCell :=
+  childLL cell000
+
+private abbrev cell0001 : AngleCell :=
+  childLH cell000
+
+private abbrev cell0002 : AngleCell :=
+  childHL cell000
+
+private abbrev cell0003 : AngleCell :=
+  childHH cell000
+
+private abbrev cell0010 : AngleCell :=
+  childLL cell001
+
+private abbrev cell0011 : AngleCell :=
+  childLH cell001
+
+private abbrev cell0012 : AngleCell :=
+  childHL cell001
+
+private abbrev cell0013 : AngleCell :=
+  childHH cell001
+
+private abbrev cell0100 : AngleCell :=
+  childLL cell010
+
+private abbrev cell0101 : AngleCell :=
+  childLH cell010
+
+private abbrev cell0102 : AngleCell :=
+  childHL cell010
+
+private abbrev cell0103 : AngleCell :=
+  childHH cell010
+
+private abbrev cell0110 : AngleCell :=
+  childLL cell011
+
+private abbrev cell0111 : AngleCell :=
+  childLH cell011
+
+private abbrev cell0112 : AngleCell :=
+  childHL cell011
+
+private abbrev cell0113 : AngleCell :=
+  childHH cell011
+
+private abbrev cell1000 : AngleCell :=
+  childLL cell100
+
+private abbrev cell1001 : AngleCell :=
+  childLH cell100
+
+private abbrev cell1002 : AngleCell :=
+  childHL cell100
+
+private abbrev cell1003 : AngleCell :=
+  childHH cell100
+
+private abbrev cell1010 : AngleCell :=
+  childLL cell101
+
+private abbrev cell1011 : AngleCell :=
+  childLH cell101
+
+private abbrev cell1012 : AngleCell :=
+  childHL cell101
+
+private abbrev cell1013 : AngleCell :=
+  childHH cell101
+
+private abbrev cell1100 : AngleCell :=
+  childLL cell110
+
+private abbrev cell1101 : AngleCell :=
+  childLH cell110
+
+private abbrev cell1102 : AngleCell :=
+  childHL cell110
+
+private abbrev cell1103 : AngleCell :=
+  childHH cell110
+
+private abbrev cell1110 : AngleCell :=
+  childLL cell111
+
+private abbrev cell1111 : AngleCell :=
+  childLH cell111
+
+private abbrev cell1112 : AngleCell :=
+  childHL cell111
+
+private abbrev cell1113 : AngleCell :=
+  childHH cell111
+
+end GerverSofa.PartE.CoverCertificate69c64655e6
+
+namespace GerverSofa.PartE.CoverCertificateca69cb479d
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLL (childLH (childHH (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificateca69cb479d
+
+namespace GerverSofa.PartE.CoverCertificate25b8e68ebd
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLH (childHH (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate25b8e68ebd
+
+namespace GerverSofa.PartE.CoverCertificate0b5492bc32
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLH (childHH (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate0b5492bc32
+
+namespace GerverSofa.PartE.CoverCertificatec2a3e401f5
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLH (childHH (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatec2a3e401f5
+
+namespace GerverSofa.PartE.CoverCertificatee2dbedfd13
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLH (childHH (childLL (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatee2dbedfd13
+
+namespace GerverSofa.PartE.CoverCertificate558b871c6c
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childHH (childLL (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell0000 : AngleCell :=
+  childLL cell000
+
+private abbrev cell0001 : AngleCell :=
+  childLH cell000
+
+private abbrev cell0002 : AngleCell :=
+  childHL cell000
+
+private abbrev cell0003 : AngleCell :=
+  childHH cell000
+
+private abbrev cell0010 : AngleCell :=
+  childLL cell001
+
+private abbrev cell0011 : AngleCell :=
+  childLH cell001
+
+private abbrev cell0012 : AngleCell :=
+  childHL cell001
+
+private abbrev cell0013 : AngleCell :=
+  childHH cell001
+
+private abbrev cell0100 : AngleCell :=
+  childLL cell010
+
+private abbrev cell0101 : AngleCell :=
+  childLH cell010
+
+private abbrev cell0102 : AngleCell :=
+  childHL cell010
+
+private abbrev cell0103 : AngleCell :=
+  childHH cell010
+
+private abbrev cell0110 : AngleCell :=
+  childLL cell011
+
+private abbrev cell0111 : AngleCell :=
+  childLH cell011
+
+private abbrev cell0112 : AngleCell :=
+  childHL cell011
+
+private abbrev cell0113 : AngleCell :=
+  childHH cell011
+
+private abbrev cell1000 : AngleCell :=
+  childLL cell100
+
+private abbrev cell1001 : AngleCell :=
+  childLH cell100
+
+private abbrev cell1002 : AngleCell :=
+  childHL cell100
+
+private abbrev cell1003 : AngleCell :=
+  childHH cell100
+
+private abbrev cell1010 : AngleCell :=
+  childLL cell101
+
+private abbrev cell1011 : AngleCell :=
+  childLH cell101
+
+private abbrev cell1012 : AngleCell :=
+  childHL cell101
+
+private abbrev cell1013 : AngleCell :=
+  childHH cell101
+
+private abbrev cell1100 : AngleCell :=
+  childLL cell110
+
+private abbrev cell1101 : AngleCell :=
+  childLH cell110
+
+private abbrev cell1102 : AngleCell :=
+  childHL cell110
+
+private abbrev cell1103 : AngleCell :=
+  childHH cell110
+
+private abbrev cell1110 : AngleCell :=
+  childLL cell111
+
+private abbrev cell1111 : AngleCell :=
+  childLH cell111
+
+private abbrev cell1112 : AngleCell :=
+  childHL cell111
+
+private abbrev cell1113 : AngleCell :=
+  childHH cell111
+
+end GerverSofa.PartE.CoverCertificate558b871c6c
+
+namespace GerverSofa.PartE.CoverCertificatec90af2cbcd
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childHH (childLL (childLL (childLL (childHL (childLL (childLL (childHL
+    (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell000 : AngleCell :=
+  childLL cell00
+
+private abbrev cell001 : AngleCell :=
+  childLH cell00
+
+private abbrev cell002 : AngleCell :=
+  childHL cell00
+
+private abbrev cell003 : AngleCell :=
+  childHH cell00
+
+private abbrev cell010 : AngleCell :=
+  childLL cell01
+
+private abbrev cell011 : AngleCell :=
+  childLH cell01
+
+private abbrev cell012 : AngleCell :=
+  childHL cell01
+
+private abbrev cell013 : AngleCell :=
+  childHH cell01
+
+private abbrev cell100 : AngleCell :=
+  childLL cell10
+
+private abbrev cell101 : AngleCell :=
+  childLH cell10
+
+private abbrev cell102 : AngleCell :=
+  childHL cell10
+
+private abbrev cell103 : AngleCell :=
+  childHH cell10
+
+private abbrev cell110 : AngleCell :=
+  childLL cell11
+
+private abbrev cell111 : AngleCell :=
+  childLH cell11
+
+private abbrev cell112 : AngleCell :=
+  childHL cell11
+
+private abbrev cell113 : AngleCell :=
+  childHH cell11
+
+private abbrev cell0000 : AngleCell :=
+  childLL cell000
+
+private abbrev cell0001 : AngleCell :=
+  childLH cell000
+
+private abbrev cell0002 : AngleCell :=
+  childHL cell000
+
+private abbrev cell0003 : AngleCell :=
+  childHH cell000
+
+private abbrev cell0010 : AngleCell :=
+  childLL cell001
+
+private abbrev cell0011 : AngleCell :=
+  childLH cell001
+
+private abbrev cell0012 : AngleCell :=
+  childHL cell001
+
+private abbrev cell0013 : AngleCell :=
+  childHH cell001
+
+private abbrev cell0100 : AngleCell :=
+  childLL cell010
+
+private abbrev cell0101 : AngleCell :=
+  childLH cell010
+
+private abbrev cell0102 : AngleCell :=
+  childHL cell010
+
+private abbrev cell0103 : AngleCell :=
+  childHH cell010
+
+private abbrev cell0110 : AngleCell :=
+  childLL cell011
+
+private abbrev cell0111 : AngleCell :=
+  childLH cell011
+
+private abbrev cell0112 : AngleCell :=
+  childHL cell011
+
+private abbrev cell0113 : AngleCell :=
+  childHH cell011
+
+private abbrev cell1000 : AngleCell :=
+  childLL cell100
+
+private abbrev cell1001 : AngleCell :=
+  childLH cell100
+
+private abbrev cell1002 : AngleCell :=
+  childHL cell100
+
+private abbrev cell1003 : AngleCell :=
+  childHH cell100
+
+private abbrev cell1010 : AngleCell :=
+  childLL cell101
+
+private abbrev cell1011 : AngleCell :=
+  childLH cell101
+
+private abbrev cell1012 : AngleCell :=
+  childHL cell101
+
+private abbrev cell1013 : AngleCell :=
+  childHH cell101
+
+private abbrev cell1100 : AngleCell :=
+  childLL cell110
+
+private abbrev cell1101 : AngleCell :=
+  childLH cell110
+
+private abbrev cell1102 : AngleCell :=
+  childHL cell110
+
+private abbrev cell1103 : AngleCell :=
+  childHH cell110
+
+private abbrev cell1110 : AngleCell :=
+  childLL cell111
+
+private abbrev cell1111 : AngleCell :=
+  childLH cell111
+
+private abbrev cell1112 : AngleCell :=
+  childHL cell111
+
+private abbrev cell1113 : AngleCell :=
+  childHH cell111
+
+end GerverSofa.PartE.CoverCertificatec90af2cbcd
+
+namespace GerverSofa.PartE.CoverCertificate850c59f3da
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childHH (childLL (childLL (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate850c59f3da
+
+namespace GerverSofa.PartE.CoverCertificate1c095a90b4
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childHH (childLL (childLL (childLL (childHL (childLL (childLL (childHL (childHL
+    (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+end GerverSofa.PartE.CoverCertificate1c095a90b4
+
+namespace GerverSofa.PartE.CoverCertificate0341d165df
+
+private abbrev cellRoot : AngleCell :=
+  (childLL (childLH (childLL (childLL (childHL (childLL (childLL (childHL (childHL (childLL
+    (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificate0341d165df
+
+namespace GerverSofa.PartE.CoverCertificatedfaaa383f3
+
+private abbrev cellRoot : AngleCell :=
+  (childLH (childLH (childLL (childLL (childHL (childLL (childLL (childHL (childHL (childLL
+    (childLL (childLL (childLL (e24ThetaAboveRoot))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+end GerverSofa.PartE.CoverCertificatedfaaa383f3
+
+namespace GerverSofa.PartE.CoverCertificatea8d5b0d3f5
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLL (childLL (childHL (childLH (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificatea8d5b0d3f5
+
+namespace GerverSofa.PartE.CoverCertificate0573161232
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLL (childLL (childHL (childLH (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate0573161232
+
+namespace GerverSofa.PartE.CoverCertificate4fc67a95c3
+
+private abbrev cellRoot : AngleCell :=
+  (childHL (childLH (childLL (childHL (childLH (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificate4fc67a95c3
+
+namespace GerverSofa.PartE.CoverCertificateba1f6e5865
+
+private abbrev cellRoot : AngleCell :=
+  (childHH (childLH (childLL (childHL (childLH (childLL (childLL (childHL (childLL (childLL
+    (childHL (childHL (childLL (childLL (childLL (childLL (e24ThetaAboveRoot)))))))))))))))))
+
+private abbrev cell0 : AngleCell :=
+  childLL cellRoot
+
+private abbrev cell1 : AngleCell :=
+  childLH cellRoot
+
+private abbrev cell2 : AngleCell :=
+  childHL cellRoot
+
+private abbrev cell3 : AngleCell :=
+  childHH cellRoot
+
+private abbrev cell00 : AngleCell :=
+  childLL cell0
+
+private abbrev cell01 : AngleCell :=
+  childLH cell0
+
+private abbrev cell02 : AngleCell :=
+  childHL cell0
+
+private abbrev cell03 : AngleCell :=
+  childHH cell0
+
+private abbrev cell10 : AngleCell :=
+  childLL cell1
+
+private abbrev cell11 : AngleCell :=
+  childLH cell1
+
+private abbrev cell12 : AngleCell :=
+  childHL cell1
+
+private abbrev cell13 : AngleCell :=
+  childHH cell1
+
+private abbrev cell20 : AngleCell :=
+  childLL cell2
+
+private abbrev cell21 : AngleCell :=
+  childLH cell2
+
+private abbrev cell22 : AngleCell :=
+  childHL cell2
+
+private abbrev cell23 : AngleCell :=
+  childHH cell2
+
+private abbrev cell30 : AngleCell :=
+  childLL cell3
+
+private abbrev cell31 : AngleCell :=
+  childLH cell3
+
+private abbrev cell32 : AngleCell :=
+  childHL cell3
+
+private abbrev cell33 : AngleCell :=
+  childHH cell3
+
+private abbrev cell200 : AngleCell :=
+  childLL cell20
+
+private abbrev cell201 : AngleCell :=
+  childLH cell20
+
+private abbrev cell202 : AngleCell :=
+  childHL cell20
+
+private abbrev cell203 : AngleCell :=
+  childHH cell20
+
+private abbrev cell210 : AngleCell :=
+  childLL cell21
+
+private abbrev cell211 : AngleCell :=
+  childLH cell21
+
+private abbrev cell212 : AngleCell :=
+  childHL cell21
+
+private abbrev cell213 : AngleCell :=
+  childHH cell21
+
+private abbrev cell220 : AngleCell :=
+  childLL cell22
+
+private abbrev cell221 : AngleCell :=
+  childLH cell22
+
+private abbrev cell222 : AngleCell :=
+  childHL cell22
+
+private abbrev cell223 : AngleCell :=
+  childHH cell22
+
+private abbrev cell230 : AngleCell :=
+  childLL cell23
+
+private abbrev cell231 : AngleCell :=
+  childLH cell23
+
+private abbrev cell232 : AngleCell :=
+  childHL cell23
+
+private abbrev cell233 : AngleCell :=
+  childHH cell23
+
+private abbrev cell300 : AngleCell :=
+  childLL cell30
+
+private abbrev cell301 : AngleCell :=
+  childLH cell30
+
+private abbrev cell302 : AngleCell :=
+  childHL cell30
+
+private abbrev cell303 : AngleCell :=
+  childHH cell30
+
+private abbrev cell310 : AngleCell :=
+  childLL cell31
+
+private abbrev cell311 : AngleCell :=
+  childLH cell31
+
+private abbrev cell312 : AngleCell :=
+  childHL cell31
+
+private abbrev cell313 : AngleCell :=
+  childHH cell31
+
+private abbrev cell320 : AngleCell :=
+  childLL cell32
+
+private abbrev cell321 : AngleCell :=
+  childLH cell32
+
+private abbrev cell322 : AngleCell :=
+  childHL cell32
+
+private abbrev cell323 : AngleCell :=
+  childHH cell32
+
+private abbrev cell330 : AngleCell :=
+  childLL cell33
+
+private abbrev cell331 : AngleCell :=
+  childLH cell33
+
+private abbrev cell332 : AngleCell :=
+  childHL cell33
+
+private abbrev cell333 : AngleCell :=
+  childHH cell33
+
+end GerverSofa.PartE.CoverCertificateba1f6e5865
+
+namespace GerverSofa.PartE.CertificateCells508bd70e72
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002112 : AngleCell :=
+  childHL (childLH (childLH (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells508bd70e72
+
+namespace GerverSofa.PartE.CertificateCells2ebb3ae727
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020002113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020002113 : AngleCell :=
+  childHH (childLH (childLH (childHL thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells2ebb3ae727
+
+namespace GerverSofa.PartE.CertificateCells808549b800
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells808549b800
+
+namespace GerverSofa.PartE.CertificateCellsd717193e27
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsd717193e27
+
+namespace GerverSofa.PartE.CertificateCellsc9eea5ae86
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsc9eea5ae86
+
+namespace GerverSofa.PartE.CertificateCells883d9c2359
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells883d9c2359
+
+namespace GerverSofa.PartE.CertificateCells81adac8f59
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003000 : AngleCell :=
+  childLL (childLL (childLL (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells81adac8f59
+
+namespace GerverSofa.PartE.CertificateCellsd02c0d99aa
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003001 : AngleCell :=
+  childLH (childLL (childLL (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCellsd02c0d99aa
+
+namespace GerverSofa.PartE.CertificateCellsb3e2fb602b
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003002 : AngleCell :=
+  childHL (childLL (childLL (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCellsb3e2fb602b
+
+namespace GerverSofa.PartE.CertificateCells4e515808c8
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003003 : AngleCell :=
+  childHH (childLL (childLL (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells4e515808c8
+
+namespace GerverSofa.PartE.CertificateCellsc37a4f0e7c
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003010 : AngleCell :=
+  childLL (childLH (childLL (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCellsc37a4f0e7c
+
+namespace GerverSofa.PartE.CertificateCells6662eefeaf
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003011 : AngleCell :=
+  childLH (childLH (childLL (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells6662eefeaf
+
+namespace GerverSofa.PartE.CertificateCells9cbe967e81
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003012 : AngleCell :=
+  childHL (childLH (childLL (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells9cbe967e81
+
+namespace GerverSofa.PartE.CertificateCells868d9f8f69
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells868d9f8f69
+
+namespace GerverSofa.PartE.CertificateCells3c1ed7e077
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells3c1ed7e077
+
+namespace GerverSofa.PartE.CertificateCellsda1171cd1c
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsda1171cd1c
+
+namespace GerverSofa.PartE.CertificateCells00487431af
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003100 : AngleCell :=
+  childLL (childLL (childLH (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells00487431af
+
+namespace GerverSofa.PartE.CertificateCellsebfda2d52e
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003101 : AngleCell :=
+  childLH (childLL (childLH (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCellsebfda2d52e
+
+namespace GerverSofa.PartE.CertificateCells0d7543751e
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells0d7543751e
+
+namespace GerverSofa.PartE.CertificateCells3d57dcf8b1
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells3d57dcf8b1
+
+namespace GerverSofa.PartE.CertificateCells280bbc9838
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003110 : AngleCell :=
+  childLL (childLH (childLH (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells280bbc9838
+
+namespace GerverSofa.PartE.CertificateCells86fd793561
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003111 : AngleCell :=
+  childLH (childLH (childLH (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells86fd793561
+
+namespace GerverSofa.PartE.CertificateCellsc0e73f28b6
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCellsc0e73f28b6
+
+namespace GerverSofa.PartE.CertificateCells97ab098fbd
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020003113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020003113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaAboveCell000022002000)))
+
+end GerverSofa.PartE.CertificateCells97ab098fbd
+
+namespace GerverSofa.PartE.CertificateCells90296eb122
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells90296eb122
+
+namespace GerverSofa.PartE.CertificateCells48c4f5a54f
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells48c4f5a54f
+
+namespace GerverSofa.PartE.CertificateCellsd47d097993
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsd47d097993
+
+namespace GerverSofa.PartE.CertificateCells861c7fa81d
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells861c7fa81d
+
+namespace GerverSofa.PartE.CertificateCells96a538a32c
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells96a538a32c
+
+namespace GerverSofa.PartE.CertificateCells867b92c925
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells867b92c925
+
+namespace GerverSofa.PartE.CertificateCellse8beb6dc5c
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020012000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020012000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell000022002001)))
+
+end GerverSofa.PartE.CertificateCellse8beb6dc5c
+
+namespace GerverSofa.PartE.CertificateCellsa4fa334623
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020012001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020012001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell000022002001)))
+
+end GerverSofa.PartE.CertificateCellsa4fa334623
+
+namespace GerverSofa.PartE.CertificateCellsd70a3df0a0
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020012002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020012002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaAboveCell000022002001)))
+
+end GerverSofa.PartE.CertificateCellsd70a3df0a0
+
+namespace GerverSofa.PartE.CertificateCells02a2748806
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020012003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020012003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaAboveCell000022002001)))
+
+end GerverSofa.PartE.CertificateCells02a2748806
+
+namespace GerverSofa.PartE.CertificateCells3863a30667
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020012010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020012010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell000022002001)))
+
+end GerverSofa.PartE.CertificateCells3863a30667
+
+namespace GerverSofa.PartE.CertificateCellsd6d3f4989a
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020012011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020012011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell000022002001)))
+
+end GerverSofa.PartE.CertificateCellsd6d3f4989a
+
+namespace GerverSofa.PartE.CertificateCellsb420376865
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020012012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020012012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaAboveCell000022002001)))
+
+end GerverSofa.PartE.CertificateCellsb420376865
+
+namespace GerverSofa.PartE.CertificateCellsa9306cab0b
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `0000220020012013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000220020012013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaAboveCell000022002001)))
+
+end GerverSofa.PartE.CertificateCellsa9306cab0b
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.ThetaAbove.Leaf00504`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00505`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00507`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00508`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00510`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00511`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00516`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00517`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00518`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00519`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00522`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00523`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00524`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00525`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00527`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00528`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00532`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00533`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00534`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00535`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00538`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00539`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00540`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00541`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00543`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00544`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00546`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00547`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00551`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00552`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00556`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00557`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00558`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00559`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00562`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00563`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00564`.
+* `KernelOnly.PartE.ThetaAbove.Leaf00565`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c1_c1_c2_3_00504
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells508bd70e72
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells508bd70e72
+
+open CertificateCells508bd70e72
+namespace CoverCertificate09deb4be62
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate09deb4be62
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c1_c2 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002112 = true := by
+  exact CoverCertificate09deb4be62.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c1_c1_c3_3_00505
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2ebb3ae727
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2ebb3ae727
+
+open CertificateCells2ebb3ae727
+namespace CoverCertificate80b49a8860
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate80b49a8860
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c1_c3 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020002113 = true := by
+  exact CoverCertificate80b49a8860.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c1_c2_4_00507
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells808549b800
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells808549b800
+
+open CertificateCells808549b800
+namespace CoverCertificate8fa79bb108
+
+private theorem checked0000 : adaptiveCoverCheck 0 cell0000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0000 (by decide +kernel)
+
+private theorem checked0001 : adaptiveCoverCheck 0 cell0001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0001 (by decide +kernel)
+
+private theorem checked0002 : adaptiveCoverCheck 0 cell0002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0002 (by decide +kernel)
+
+private theorem checked0003 : adaptiveCoverCheck 0 cell0003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0003 (by decide +kernel)
+
+private theorem checked0010 : adaptiveCoverCheck 0 cell0010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0010 (by decide +kernel)
+
+private theorem checked0011 : adaptiveCoverCheck 0 cell0011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0011 (by decide +kernel)
+
+private theorem checked0012 : adaptiveCoverCheck 0 cell0012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0012 (by decide +kernel)
+
+private theorem checked0013 : adaptiveCoverCheck 0 cell0013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0013 (by decide +kernel)
+
+private theorem checked0100 : adaptiveCoverCheck 0 cell0100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0100 (by decide +kernel)
+
+private theorem checked0101 : adaptiveCoverCheck 0 cell0101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0101 (by decide +kernel)
+
+private theorem checked0102 : adaptiveCoverCheck 0 cell0102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0102 (by decide +kernel)
+
+private theorem checked0103 : adaptiveCoverCheck 0 cell0103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0103 (by decide +kernel)
+
+private theorem checked0110 : adaptiveCoverCheck 0 cell0110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0110 (by decide +kernel)
+
+private theorem checked0111 : adaptiveCoverCheck 0 cell0111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0111 (by decide +kernel)
+
+private theorem checked0112 : adaptiveCoverCheck 0 cell0112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0112 (by decide +kernel)
+
+private theorem checked0113 : adaptiveCoverCheck 0 cell0113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0113 (by decide +kernel)
+
+private theorem checked1000 : adaptiveCoverCheck 0 cell1000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1000 (by decide +kernel)
+
+private theorem checked1001 : adaptiveCoverCheck 0 cell1001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1001 (by decide +kernel)
+
+private theorem checked1002 : adaptiveCoverCheck 0 cell1002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1002 (by decide +kernel)
+
+private theorem checked1003 : adaptiveCoverCheck 0 cell1003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1003 (by decide +kernel)
+
+private theorem checked1010 : adaptiveCoverCheck 0 cell1010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1010 (by decide +kernel)
+
+private theorem checked1011 : adaptiveCoverCheck 0 cell1011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1011 (by decide +kernel)
+
+private theorem checked1012 : adaptiveCoverCheck 0 cell1012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1012 (by decide +kernel)
+
+private theorem checked1013 : adaptiveCoverCheck 0 cell1013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1013 (by decide +kernel)
+
+private theorem checked1100 : adaptiveCoverCheck 0 cell1100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1100 (by decide +kernel)
+
+private theorem checked1101 : adaptiveCoverCheck 0 cell1101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1101 (by decide +kernel)
+
+private theorem checked1102 : adaptiveCoverCheck 0 cell1102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1102 (by decide +kernel)
+
+private theorem checked1103 : adaptiveCoverCheck 0 cell1103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1103 (by decide +kernel)
+
+private theorem checked1110 : adaptiveCoverCheck 0 cell1110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1110 (by decide +kernel)
+
+private theorem checked1111 : adaptiveCoverCheck 0 cell1111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1111 (by decide +kernel)
+
+private theorem checked1112 : adaptiveCoverCheck 0 cell1112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1112 (by decide +kernel)
+
+private theorem checked1113 : adaptiveCoverCheck 0 cell1113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1113 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell000
+    checked0000 checked0001 checked0002 checked0003
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell001
+    checked0010 checked0011 checked0012 checked0013
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell010
+    checked0100 checked0101 checked0102 checked0103
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell011
+    checked0110 checked0111 checked0112 checked0113
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell100
+    checked1000 checked1001 checked1002 checked1003
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell101
+    checked1010 checked1011 checked1012 checked1013
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell110
+    checked1100 checked1101 checked1102 checked1103
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell111
+    checked1110 checked1111 checked1112 checked1113
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate8fa79bb108
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c2 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHL thetaAboveCell000022002000))) = true := by
+  exact CoverCertificate8fa79bb108.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c1_c3_4_00508
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd717193e27
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd717193e27
+
+open CertificateCellsd717193e27
+namespace CoverCertificate9198440329
+
+private theorem checked0000 : adaptiveCoverCheck 0 cell0000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0000 (by decide +kernel)
+
+private theorem checked0001 : adaptiveCoverCheck 0 cell0001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0001 (by decide +kernel)
+
+private theorem checked0002 : adaptiveCoverCheck 0 cell0002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0002 (by decide +kernel)
+
+private theorem checked0003 : adaptiveCoverCheck 0 cell0003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0003 (by decide +kernel)
+
+private theorem checked0010 : adaptiveCoverCheck 0 cell0010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0010 (by decide +kernel)
+
+private theorem checked0011 : adaptiveCoverCheck 0 cell0011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0011 (by decide +kernel)
+
+private theorem checked0012 : adaptiveCoverCheck 0 cell0012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0012 (by decide +kernel)
+
+private theorem checked0013 : adaptiveCoverCheck 0 cell0013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0013 (by decide +kernel)
+
+private theorem checked0100 : adaptiveCoverCheck 0 cell0100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0100 (by decide +kernel)
+
+private theorem checked0101 : adaptiveCoverCheck 0 cell0101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0101 (by decide +kernel)
+
+private theorem checked0102 : adaptiveCoverCheck 0 cell0102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0102 (by decide +kernel)
+
+private theorem checked0103 : adaptiveCoverCheck 0 cell0103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0103 (by decide +kernel)
+
+private theorem checked0110 : adaptiveCoverCheck 0 cell0110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0110 (by decide +kernel)
+
+private theorem checked0111 : adaptiveCoverCheck 0 cell0111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0111 (by decide +kernel)
+
+private theorem checked0112 : adaptiveCoverCheck 0 cell0112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0112 (by decide +kernel)
+
+private theorem checked0113 : adaptiveCoverCheck 0 cell0113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0113 (by decide +kernel)
+
+private theorem checked1000 : adaptiveCoverCheck 0 cell1000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1000 (by decide +kernel)
+
+private theorem checked1001 : adaptiveCoverCheck 0 cell1001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1001 (by decide +kernel)
+
+private theorem checked1002 : adaptiveCoverCheck 0 cell1002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1002 (by decide +kernel)
+
+private theorem checked1003 : adaptiveCoverCheck 0 cell1003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1003 (by decide +kernel)
+
+private theorem checked1010 : adaptiveCoverCheck 0 cell1010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1010 (by decide +kernel)
+
+private theorem checked1011 : adaptiveCoverCheck 0 cell1011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1011 (by decide +kernel)
+
+private theorem checked1012 : adaptiveCoverCheck 0 cell1012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1012 (by decide +kernel)
+
+private theorem checked1013 : adaptiveCoverCheck 0 cell1013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1013 (by decide +kernel)
+
+private theorem checked1100 : adaptiveCoverCheck 0 cell1100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1100 (by decide +kernel)
+
+private theorem checked1101 : adaptiveCoverCheck 0 cell1101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1101 (by decide +kernel)
+
+private theorem checked1102 : adaptiveCoverCheck 0 cell1102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1102 (by decide +kernel)
+
+private theorem checked1103 : adaptiveCoverCheck 0 cell1103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1103 (by decide +kernel)
+
+private theorem checked1110 : adaptiveCoverCheck 0 cell1110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1110 (by decide +kernel)
+
+private theorem checked1111 : adaptiveCoverCheck 0 cell1111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1111 (by decide +kernel)
+
+private theorem checked1112 : adaptiveCoverCheck 0 cell1112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1112 (by decide +kernel)
+
+private theorem checked1113 : adaptiveCoverCheck 0 cell1113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1113 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell000
+    checked0000 checked0001 checked0002 checked0003
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell001
+    checked0010 checked0011 checked0012 checked0013
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell010
+    checked0100 checked0101 checked0102 checked0103
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell011
+    checked0110 checked0111 checked0112 checked0113
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell100
+    checked1000 checked1001 checked1002 checked1003
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell101
+    checked1010 checked1011 checked1012 checked1013
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell110
+    checked1100 checked1101 checked1102 checked1103
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell111
+    checked1110 checked1111 checked1112 checked1113
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate9198440329
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c3 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHL thetaAboveCell000022002000))) = true := by
+  exact CoverCertificate9198440329.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c2_5_00510
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc9eea5ae86
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc9eea5ae86
+
+open CertificateCellsc9eea5ae86
+namespace CoverCertificate54d2ea68c4
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate54d2ea68c4
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c2 :
+    adaptiveCoverCheck 5 (childHL (childHL thetaAboveCell000022002000)) = true := by
+  exact CoverCertificate54d2ea68c4.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c3_5_00511
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells883d9c2359
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells883d9c2359
+
+open CertificateCells883d9c2359
+namespace CoverCertificate9a90f66090
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate9a90f66090
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c3 :
+    adaptiveCoverCheck 5 (childHH (childHL thetaAboveCell000022002000)) = true := by
+  exact CoverCertificate9a90f66090.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c0_c0_c0_3_00516
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells81adac8f59
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells81adac8f59
+
+open CertificateCells81adac8f59
+namespace CoverCertificate9a54be935a
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate9a54be935a
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c0_c0 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003000 = true := by
+  exact CoverCertificate9a54be935a.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c0_c0_c1_3_00517
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd02c0d99aa
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd02c0d99aa
+
+open CertificateCellsd02c0d99aa
+namespace CoverCertificatea35db1f021
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea35db1f021
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c0_c1 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003001 = true := by
+  exact CoverCertificatea35db1f021.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c0_c0_c2_3_00518
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb3e2fb602b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb3e2fb602b
+
+open CertificateCellsb3e2fb602b
+namespace CoverCertificate8333a307a4
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate8333a307a4
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c0_c2 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003002 = true := by
+  exact CoverCertificate8333a307a4.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c0_c0_c3_3_00519
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4e515808c8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4e515808c8
+
+open CertificateCells4e515808c8
+namespace CoverCertificated46e97b4d3
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificated46e97b4d3
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c0_c3 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003003 = true := by
+  exact CoverCertificated46e97b4d3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c0_c1_c0_3_00522
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc37a4f0e7c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc37a4f0e7c
+
+open CertificateCellsc37a4f0e7c
+namespace CoverCertificate39bdb9b52b
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate39bdb9b52b
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c1_c0 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003010 = true := by
+  exact CoverCertificate39bdb9b52b.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c0_c1_c1_3_00523
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6662eefeaf
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6662eefeaf
+
+open CertificateCells6662eefeaf
+namespace CoverCertificate21662757e3
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate21662757e3
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c1_c1 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003011 = true := by
+  exact CoverCertificate21662757e3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c0_c1_c2_3_00524
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9cbe967e81
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9cbe967e81
+
+open CertificateCells9cbe967e81
+namespace CoverCertificate7ec8971130
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate7ec8971130
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c1_c2 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003012 = true := by
+  exact CoverCertificate7ec8971130.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c0_c1_c3_3_00525
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells868d9f8f69
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells868d9f8f69
+
+open CertificateCells868d9f8f69
+namespace CoverCertificatec012db59ec
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec012db59ec
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c1_c3 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003013 = true := by
+  exact CoverCertificatec012db59ec.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c0_c2_4_00527
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3c1ed7e077
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3c1ed7e077
+
+open CertificateCells3c1ed7e077
+namespace CoverCertificate418c92a9a1
+
+private theorem checked0000 : adaptiveCoverCheck 0 cell0000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0000 (by decide +kernel)
+
+private theorem checked0001 : adaptiveCoverCheck 0 cell0001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0001 (by decide +kernel)
+
+private theorem checked0002 : adaptiveCoverCheck 0 cell0002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0002 (by decide +kernel)
+
+private theorem checked0003 : adaptiveCoverCheck 0 cell0003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0003 (by decide +kernel)
+
+private theorem checked0010 : adaptiveCoverCheck 0 cell0010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0010 (by decide +kernel)
+
+private theorem checked0011 : adaptiveCoverCheck 0 cell0011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0011 (by decide +kernel)
+
+private theorem checked0012 : adaptiveCoverCheck 0 cell0012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0012 (by decide +kernel)
+
+private theorem checked0013 : adaptiveCoverCheck 0 cell0013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0013 (by decide +kernel)
+
+private theorem checked0100 : adaptiveCoverCheck 0 cell0100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0100 (by decide +kernel)
+
+private theorem checked0101 : adaptiveCoverCheck 0 cell0101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0101 (by decide +kernel)
+
+private theorem checked0102 : adaptiveCoverCheck 0 cell0102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0102 (by decide +kernel)
+
+private theorem checked0103 : adaptiveCoverCheck 0 cell0103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0103 (by decide +kernel)
+
+private theorem checked0110 : adaptiveCoverCheck 0 cell0110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0110 (by decide +kernel)
+
+private theorem checked0111 : adaptiveCoverCheck 0 cell0111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0111 (by decide +kernel)
+
+private theorem checked0112 : adaptiveCoverCheck 0 cell0112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0112 (by decide +kernel)
+
+private theorem checked0113 : adaptiveCoverCheck 0 cell0113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0113 (by decide +kernel)
+
+private theorem checked1000 : adaptiveCoverCheck 0 cell1000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1000 (by decide +kernel)
+
+private theorem checked1001 : adaptiveCoverCheck 0 cell1001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1001 (by decide +kernel)
+
+private theorem checked1002 : adaptiveCoverCheck 0 cell1002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1002 (by decide +kernel)
+
+private theorem checked1003 : adaptiveCoverCheck 0 cell1003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1003 (by decide +kernel)
+
+private theorem checked1010 : adaptiveCoverCheck 0 cell1010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1010 (by decide +kernel)
+
+private theorem checked1011 : adaptiveCoverCheck 0 cell1011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1011 (by decide +kernel)
+
+private theorem checked1012 : adaptiveCoverCheck 0 cell1012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1012 (by decide +kernel)
+
+private theorem checked1013 : adaptiveCoverCheck 0 cell1013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1013 (by decide +kernel)
+
+private theorem checked1100 : adaptiveCoverCheck 0 cell1100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1100 (by decide +kernel)
+
+private theorem checked1101 : adaptiveCoverCheck 0 cell1101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1101 (by decide +kernel)
+
+private theorem checked1102 : adaptiveCoverCheck 0 cell1102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1102 (by decide +kernel)
+
+private theorem checked1103 : adaptiveCoverCheck 0 cell1103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1103 (by decide +kernel)
+
+private theorem checked1110 : adaptiveCoverCheck 0 cell1110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1110 (by decide +kernel)
+
+private theorem checked1111 : adaptiveCoverCheck 0 cell1111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1111 (by decide +kernel)
+
+private theorem checked1112 : adaptiveCoverCheck 0 cell1112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1112 (by decide +kernel)
+
+private theorem checked1113 : adaptiveCoverCheck 0 cell1113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1113 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell000
+    checked0000 checked0001 checked0002 checked0003
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell001
+    checked0010 checked0011 checked0012 checked0013
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell010
+    checked0100 checked0101 checked0102 checked0103
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell011
+    checked0110 checked0111 checked0112 checked0113
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell100
+    checked1000 checked1001 checked1002 checked1003
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell101
+    checked1010 checked1011 checked1012 checked1013
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell110
+    checked1100 checked1101 checked1102 checked1103
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell111
+    checked1110 checked1111 checked1112 checked1113
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate418c92a9a1
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c2 :
+    adaptiveCoverCheck 4 (childHL (childLL (childHH thetaAboveCell000022002000))) = true := by
+  exact CoverCertificate418c92a9a1.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c0_c3_4_00528
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsda1171cd1c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsda1171cd1c
+
+open CertificateCellsda1171cd1c
+namespace CoverCertificate69c64655e6
+
+private theorem checked0000 : adaptiveCoverCheck 0 cell0000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0000 (by decide +kernel)
+
+private theorem checked0001 : adaptiveCoverCheck 0 cell0001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0001 (by decide +kernel)
+
+private theorem checked0002 : adaptiveCoverCheck 0 cell0002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0002 (by decide +kernel)
+
+private theorem checked0003 : adaptiveCoverCheck 0 cell0003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0003 (by decide +kernel)
+
+private theorem checked0010 : adaptiveCoverCheck 0 cell0010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0010 (by decide +kernel)
+
+private theorem checked0011 : adaptiveCoverCheck 0 cell0011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0011 (by decide +kernel)
+
+private theorem checked0012 : adaptiveCoverCheck 0 cell0012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0012 (by decide +kernel)
+
+private theorem checked0013 : adaptiveCoverCheck 0 cell0013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0013 (by decide +kernel)
+
+private theorem checked0100 : adaptiveCoverCheck 0 cell0100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0100 (by decide +kernel)
+
+private theorem checked0101 : adaptiveCoverCheck 0 cell0101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0101 (by decide +kernel)
+
+private theorem checked0102 : adaptiveCoverCheck 0 cell0102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0102 (by decide +kernel)
+
+private theorem checked0103 : adaptiveCoverCheck 0 cell0103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0103 (by decide +kernel)
+
+private theorem checked0110 : adaptiveCoverCheck 0 cell0110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0110 (by decide +kernel)
+
+private theorem checked0111 : adaptiveCoverCheck 0 cell0111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0111 (by decide +kernel)
+
+private theorem checked0112 : adaptiveCoverCheck 0 cell0112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0112 (by decide +kernel)
+
+private theorem checked0113 : adaptiveCoverCheck 0 cell0113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0113 (by decide +kernel)
+
+private theorem checked1000 : adaptiveCoverCheck 0 cell1000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1000 (by decide +kernel)
+
+private theorem checked1001 : adaptiveCoverCheck 0 cell1001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1001 (by decide +kernel)
+
+private theorem checked1002 : adaptiveCoverCheck 0 cell1002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1002 (by decide +kernel)
+
+private theorem checked1003 : adaptiveCoverCheck 0 cell1003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1003 (by decide +kernel)
+
+private theorem checked1010 : adaptiveCoverCheck 0 cell1010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1010 (by decide +kernel)
+
+private theorem checked1011 : adaptiveCoverCheck 0 cell1011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1011 (by decide +kernel)
+
+private theorem checked1012 : adaptiveCoverCheck 0 cell1012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1012 (by decide +kernel)
+
+private theorem checked1013 : adaptiveCoverCheck 0 cell1013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1013 (by decide +kernel)
+
+private theorem checked1100 : adaptiveCoverCheck 0 cell1100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1100 (by decide +kernel)
+
+private theorem checked1101 : adaptiveCoverCheck 0 cell1101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1101 (by decide +kernel)
+
+private theorem checked1102 : adaptiveCoverCheck 0 cell1102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1102 (by decide +kernel)
+
+private theorem checked1103 : adaptiveCoverCheck 0 cell1103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1103 (by decide +kernel)
+
+private theorem checked1110 : adaptiveCoverCheck 0 cell1110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1110 (by decide +kernel)
+
+private theorem checked1111 : adaptiveCoverCheck 0 cell1111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1111 (by decide +kernel)
+
+private theorem checked1112 : adaptiveCoverCheck 0 cell1112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1112 (by decide +kernel)
+
+private theorem checked1113 : adaptiveCoverCheck 0 cell1113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1113 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell000
+    checked0000 checked0001 checked0002 checked0003
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell001
+    checked0010 checked0011 checked0012 checked0013
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell010
+    checked0100 checked0101 checked0102 checked0103
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell011
+    checked0110 checked0111 checked0112 checked0113
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell100
+    checked1000 checked1001 checked1002 checked1003
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell101
+    checked1010 checked1011 checked1012 checked1013
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell110
+    checked1100 checked1101 checked1102 checked1103
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell111
+    checked1110 checked1111 checked1112 checked1113
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate69c64655e6
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c3 :
+    adaptiveCoverCheck 4 (childHH (childLL (childHH thetaAboveCell000022002000))) = true := by
+  exact CoverCertificate69c64655e6.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c1_c0_c0_3_00532
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells00487431af
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells00487431af
+
+open CertificateCells00487431af
+namespace CoverCertificateca69cb479d
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateca69cb479d
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c0_c0 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003100 = true := by
+  exact CoverCertificateca69cb479d.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c1_c0_c1_3_00533
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsebfda2d52e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsebfda2d52e
+
+open CertificateCellsebfda2d52e
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c0_c1 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003101 = true := by
+  decide +kernel
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c1_c0_c2_3_00534
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0d7543751e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0d7543751e
+
+open CertificateCells0d7543751e
+namespace CoverCertificate25b8e68ebd
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate25b8e68ebd
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c0_c2 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003102 = true := by
+  exact CoverCertificate25b8e68ebd.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c1_c0_c3_3_00535
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3d57dcf8b1
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3d57dcf8b1
+
+open CertificateCells3d57dcf8b1
+namespace CoverCertificate0b5492bc32
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0b5492bc32
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c0_c3 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003103 = true := by
+  exact CoverCertificate0b5492bc32.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c1_c1_c0_3_00538
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells280bbc9838
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells280bbc9838
+
+open CertificateCells280bbc9838
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c1_c0 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003110 = true := by
+  decide +kernel
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c1_c1_c1_3_00539
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells86fd793561
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells86fd793561
+
+open CertificateCells86fd793561
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c1_c1 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003111 = true := by
+  decide +kernel
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c1_c1_c2_3_00540
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc0e73f28b6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc0e73f28b6
+
+open CertificateCellsc0e73f28b6
+namespace CoverCertificatec2a3e401f5
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec2a3e401f5
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c1_c2 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003112 = true := by
+  exact CoverCertificatec2a3e401f5.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c1_c1_c3_3_00541
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells97ab098fbd
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells97ab098fbd
+
+open CertificateCells97ab098fbd
+namespace CoverCertificatee2dbedfd13
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatee2dbedfd13
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c1_c3 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020003113 = true := by
+  exact CoverCertificatee2dbedfd13.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c1_c2_4_00543
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells90296eb122
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells90296eb122
+
+open CertificateCells90296eb122
+namespace CoverCertificate558b871c6c
+
+private theorem checked0000 : adaptiveCoverCheck 0 cell0000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0000 (by decide +kernel)
+
+private theorem checked0001 : adaptiveCoverCheck 0 cell0001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0001 (by decide +kernel)
+
+private theorem checked0002 : adaptiveCoverCheck 0 cell0002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0002 (by decide +kernel)
+
+private theorem checked0003 : adaptiveCoverCheck 0 cell0003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0003 (by decide +kernel)
+
+private theorem checked0010 : adaptiveCoverCheck 0 cell0010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0010 (by decide +kernel)
+
+private theorem checked0011 : adaptiveCoverCheck 0 cell0011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0011 (by decide +kernel)
+
+private theorem checked0012 : adaptiveCoverCheck 0 cell0012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0012 (by decide +kernel)
+
+private theorem checked0013 : adaptiveCoverCheck 0 cell0013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0013 (by decide +kernel)
+
+private theorem checked0100 : adaptiveCoverCheck 0 cell0100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0100 (by decide +kernel)
+
+private theorem checked0101 : adaptiveCoverCheck 0 cell0101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0101 (by decide +kernel)
+
+private theorem checked0102 : adaptiveCoverCheck 0 cell0102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0102 (by decide +kernel)
+
+private theorem checked0103 : adaptiveCoverCheck 0 cell0103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0103 (by decide +kernel)
+
+private theorem checked0110 : adaptiveCoverCheck 0 cell0110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0110 (by decide +kernel)
+
+private theorem checked0111 : adaptiveCoverCheck 0 cell0111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0111 (by decide +kernel)
+
+private theorem checked0112 : adaptiveCoverCheck 0 cell0112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0112 (by decide +kernel)
+
+private theorem checked0113 : adaptiveCoverCheck 0 cell0113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0113 (by decide +kernel)
+
+private theorem checked1000 : adaptiveCoverCheck 0 cell1000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1000 (by decide +kernel)
+
+private theorem checked1001 : adaptiveCoverCheck 0 cell1001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1001 (by decide +kernel)
+
+private theorem checked1002 : adaptiveCoverCheck 0 cell1002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1002 (by decide +kernel)
+
+private theorem checked1003 : adaptiveCoverCheck 0 cell1003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1003 (by decide +kernel)
+
+private theorem checked1010 : adaptiveCoverCheck 0 cell1010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1010 (by decide +kernel)
+
+private theorem checked1011 : adaptiveCoverCheck 0 cell1011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1011 (by decide +kernel)
+
+private theorem checked1012 : adaptiveCoverCheck 0 cell1012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1012 (by decide +kernel)
+
+private theorem checked1013 : adaptiveCoverCheck 0 cell1013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1013 (by decide +kernel)
+
+private theorem checked1100 : adaptiveCoverCheck 0 cell1100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1100 (by decide +kernel)
+
+private theorem checked1101 : adaptiveCoverCheck 0 cell1101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1101 (by decide +kernel)
+
+private theorem checked1102 : adaptiveCoverCheck 0 cell1102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1102 (by decide +kernel)
+
+private theorem checked1103 : adaptiveCoverCheck 0 cell1103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1103 (by decide +kernel)
+
+private theorem checked1110 : adaptiveCoverCheck 0 cell1110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1110 (by decide +kernel)
+
+private theorem checked1111 : adaptiveCoverCheck 0 cell1111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1111 (by decide +kernel)
+
+private theorem checked1112 : adaptiveCoverCheck 0 cell1112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1112 (by decide +kernel)
+
+private theorem checked1113 : adaptiveCoverCheck 0 cell1113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1113 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell000
+    checked0000 checked0001 checked0002 checked0003
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell001
+    checked0010 checked0011 checked0012 checked0013
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell010
+    checked0100 checked0101 checked0102 checked0103
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell011
+    checked0110 checked0111 checked0112 checked0113
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell100
+    checked1000 checked1001 checked1002 checked1003
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell101
+    checked1010 checked1011 checked1012 checked1013
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell110
+    checked1100 checked1101 checked1102 checked1103
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell111
+    checked1110 checked1111 checked1112 checked1113
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate558b871c6c
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c2 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHH thetaAboveCell000022002000))) = true := by
+  exact CoverCertificate558b871c6c.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c1_c3_4_00544
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells48c4f5a54f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells48c4f5a54f
+
+open CertificateCells48c4f5a54f
+namespace CoverCertificatec90af2cbcd
+
+private theorem checked0000 : adaptiveCoverCheck 0 cell0000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0000 (by decide +kernel)
+
+private theorem checked0001 : adaptiveCoverCheck 0 cell0001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0001 (by decide +kernel)
+
+private theorem checked0002 : adaptiveCoverCheck 0 cell0002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0002 (by decide +kernel)
+
+private theorem checked0003 : adaptiveCoverCheck 0 cell0003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0003 (by decide +kernel)
+
+private theorem checked0010 : adaptiveCoverCheck 0 cell0010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0010 (by decide +kernel)
+
+private theorem checked0011 : adaptiveCoverCheck 0 cell0011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0011 (by decide +kernel)
+
+private theorem checked0012 : adaptiveCoverCheck 0 cell0012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0012 (by decide +kernel)
+
+private theorem checked0013 : adaptiveCoverCheck 0 cell0013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0013 (by decide +kernel)
+
+private theorem checked0100 : adaptiveCoverCheck 0 cell0100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0100 (by decide +kernel)
+
+private theorem checked0101 : adaptiveCoverCheck 0 cell0101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0101 (by decide +kernel)
+
+private theorem checked0102 : adaptiveCoverCheck 0 cell0102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0102 (by decide +kernel)
+
+private theorem checked0103 : adaptiveCoverCheck 0 cell0103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0103 (by decide +kernel)
+
+private theorem checked0110 : adaptiveCoverCheck 0 cell0110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0110 (by decide +kernel)
+
+private theorem checked0111 : adaptiveCoverCheck 0 cell0111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0111 (by decide +kernel)
+
+private theorem checked0112 : adaptiveCoverCheck 0 cell0112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0112 (by decide +kernel)
+
+private theorem checked0113 : adaptiveCoverCheck 0 cell0113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell0113 (by decide +kernel)
+
+private theorem checked1000 : adaptiveCoverCheck 0 cell1000 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1000 (by decide +kernel)
+
+private theorem checked1001 : adaptiveCoverCheck 0 cell1001 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1001 (by decide +kernel)
+
+private theorem checked1002 : adaptiveCoverCheck 0 cell1002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1002 (by decide +kernel)
+
+private theorem checked1003 : adaptiveCoverCheck 0 cell1003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1003 (by decide +kernel)
+
+private theorem checked1010 : adaptiveCoverCheck 0 cell1010 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1010 (by decide +kernel)
+
+private theorem checked1011 : adaptiveCoverCheck 0 cell1011 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1011 (by decide +kernel)
+
+private theorem checked1012 : adaptiveCoverCheck 0 cell1012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1012 (by decide +kernel)
+
+private theorem checked1013 : adaptiveCoverCheck 0 cell1013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1013 (by decide +kernel)
+
+private theorem checked1100 : adaptiveCoverCheck 0 cell1100 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1100 (by decide +kernel)
+
+private theorem checked1101 : adaptiveCoverCheck 0 cell1101 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1101 (by decide +kernel)
+
+private theorem checked1102 : adaptiveCoverCheck 0 cell1102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1102 (by decide +kernel)
+
+private theorem checked1103 : adaptiveCoverCheck 0 cell1103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1103 (by decide +kernel)
+
+private theorem checked1110 : adaptiveCoverCheck 0 cell1110 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1110 (by decide +kernel)
+
+private theorem checked1111 : adaptiveCoverCheck 0 cell1111 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1111 (by decide +kernel)
+
+private theorem checked1112 : adaptiveCoverCheck 0 cell1112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1112 (by decide +kernel)
+
+private theorem checked1113 : adaptiveCoverCheck 0 cell1113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell1113 (by decide +kernel)
+
+private theorem checked000 : adaptiveCoverCheck 1 cell000 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell000
+    checked0000 checked0001 checked0002 checked0003
+
+private theorem checked001 : adaptiveCoverCheck 1 cell001 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell001
+    checked0010 checked0011 checked0012 checked0013
+
+private theorem checked002 : adaptiveCoverCheck 1 cell002 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell002 (by decide +kernel)
+
+private theorem checked003 : adaptiveCoverCheck 1 cell003 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell003 (by decide +kernel)
+
+private theorem checked010 : adaptiveCoverCheck 1 cell010 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell010
+    checked0100 checked0101 checked0102 checked0103
+
+private theorem checked011 : adaptiveCoverCheck 1 cell011 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell011
+    checked0110 checked0111 checked0112 checked0113
+
+private theorem checked012 : adaptiveCoverCheck 1 cell012 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell012 (by decide +kernel)
+
+private theorem checked013 : adaptiveCoverCheck 1 cell013 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell013 (by decide +kernel)
+
+private theorem checked100 : adaptiveCoverCheck 1 cell100 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell100
+    checked1000 checked1001 checked1002 checked1003
+
+private theorem checked101 : adaptiveCoverCheck 1 cell101 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell101
+    checked1010 checked1011 checked1012 checked1013
+
+private theorem checked102 : adaptiveCoverCheck 1 cell102 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell102 (by decide +kernel)
+
+private theorem checked103 : adaptiveCoverCheck 1 cell103 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell103 (by decide +kernel)
+
+private theorem checked110 : adaptiveCoverCheck 1 cell110 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell110
+    checked1100 checked1101 checked1102 checked1103
+
+private theorem checked111 : adaptiveCoverCheck 1 cell111 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell111
+    checked1110 checked1111 checked1112 checked1113
+
+private theorem checked112 : adaptiveCoverCheck 1 cell112 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell112 (by decide +kernel)
+
+private theorem checked113 : adaptiveCoverCheck 1 cell113 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell113 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 2 cell00 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell00
+    checked000 checked001 checked002 checked003
+
+private theorem checked01 : adaptiveCoverCheck 2 cell01 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell01
+    checked010 checked011 checked012 checked013
+
+private theorem checked02 : adaptiveCoverCheck 2 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 2 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 2 cell10 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell10
+    checked100 checked101 checked102 checked103
+
+private theorem checked11 : adaptiveCoverCheck 2 cell11 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell11
+    checked110 checked111 checked112 checked113
+
+private theorem checked12 : adaptiveCoverCheck 2 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 2 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 2 cell13 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 3 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 3 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 3 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 3 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 3 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 4 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 3 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatec90af2cbcd
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c3 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHH thetaAboveCell000022002000))) = true := by
+  exact CoverCertificatec90af2cbcd.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c2_5_00546
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd47d097993
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd47d097993
+
+open CertificateCellsd47d097993
+namespace CoverCertificate850c59f3da
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate850c59f3da
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c2 :
+    adaptiveCoverCheck 5 (childHL (childHH thetaAboveCell000022002000)) = true := by
+  exact CoverCertificate850c59f3da.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c3_5_00547
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells861c7fa81d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells861c7fa81d
+
+open CertificateCells861c7fa81d
+namespace CoverCertificate1c095a90b4
+
+private theorem checked0 : adaptiveCoverCheck 4 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 4 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 4 cell2 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell2 (by decide +kernel)
+
+private theorem checked3 : adaptiveCoverCheck 4 cell3 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell3 (by decide +kernel)
+
+private theorem checkedRoot : adaptiveCoverCheck 5 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate1c095a90b4
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c3 :
+    adaptiveCoverCheck 5 (childHH (childHH thetaAboveCell000022002000)) = true := by
+  exact CoverCertificate1c095a90b4.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c0_6_00551
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells96a538a32c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells96a538a32c
+
+open CertificateCells96a538a32c
+namespace CoverCertificate0341d165df
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0341d165df
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c0 :
+    adaptiveCoverCheck 6 (childLL thetaAboveCell000022002001) = true := by
+  exact CoverCertificate0341d165df.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c1_6_00552
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells867b92c925
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells867b92c925
+
+open CertificateCells867b92c925
+namespace CoverCertificatedfaaa383f3
+
+private theorem checked20 : adaptiveCoverCheck 4 cell20 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell20 (by decide +kernel)
+
+private theorem checked21 : adaptiveCoverCheck 4 cell21 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell21 (by decide +kernel)
+
+private theorem checked22 : adaptiveCoverCheck 4 cell22 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell22 (by decide +kernel)
+
+private theorem checked23 : adaptiveCoverCheck 4 cell23 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell23 (by decide +kernel)
+
+private theorem checked30 : adaptiveCoverCheck 4 cell30 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell30 (by decide +kernel)
+
+private theorem checked31 : adaptiveCoverCheck 4 cell31 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell31 (by decide +kernel)
+
+private theorem checked32 : adaptiveCoverCheck 4 cell32 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell32 (by decide +kernel)
+
+private theorem checked33 : adaptiveCoverCheck 4 cell33 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 4 cell33 (by decide +kernel)
+
+private theorem checked0 : adaptiveCoverCheck 5 cell0 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell0 (by decide +kernel)
+
+private theorem checked1 : adaptiveCoverCheck 5 cell1 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 5 cell1 (by decide +kernel)
+
+private theorem checked2 : adaptiveCoverCheck 5 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 5 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 4 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 6 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 5 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatedfaaa383f3
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c1 :
+    adaptiveCoverCheck 6 (childLH thetaAboveCell000022002001) = true := by
+  exact CoverCertificatedfaaa383f3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c0_c0_c0_3_00556
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse8beb6dc5c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse8beb6dc5c
+
+open CertificateCellse8beb6dc5c
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c0_c0 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020012000 = true := by
+  decide +kernel
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c0_c0_c1_3_00557
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa4fa334623
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa4fa334623
+
+open CertificateCellsa4fa334623
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c0_c1 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020012001 = true := by
+  decide +kernel
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c0_c0_c2_3_00558
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd70a3df0a0
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd70a3df0a0
+
+open CertificateCellsd70a3df0a0
+namespace CoverCertificatea8d5b0d3f5
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificatea8d5b0d3f5
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c0_c2 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020012002 = true := by
+  exact CoverCertificatea8d5b0d3f5.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c0_c0_c3_3_00559
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells02a2748806
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells02a2748806
+
+open CertificateCells02a2748806
+namespace CoverCertificate0573161232
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate0573161232
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c0_c3 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020012003 = true := by
+  exact CoverCertificate0573161232.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c0_c1_c0_3_00562
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3863a30667
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3863a30667
+
+open CertificateCells3863a30667
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c1_c0 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020012010 = true := by
+  decide +kernel
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c0_c1_c1_3_00563
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd6d3f4989a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd6d3f4989a
+
+open CertificateCellsd6d3f4989a
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c1_c1 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020012011 = true := by
+  decide +kernel
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c0_c1_c2_3_00564
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb420376865
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb420376865
+
+open CertificateCellsb420376865
+namespace CoverCertificate4fc67a95c3
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificate4fc67a95c3
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c1_c2 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020012012 = true := by
+  exact CoverCertificate4fc67a95c3.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Direct_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c0_c1_c3_3_00565
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa9306cab0b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa9306cab0b
+
+open CertificateCellsa9306cab0b
+namespace CoverCertificateba1f6e5865
+
+private theorem checked200 : adaptiveCoverCheck 0 cell200 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell200 (by decide +kernel)
+
+private theorem checked201 : adaptiveCoverCheck 0 cell201 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell201 (by decide +kernel)
+
+private theorem checked202 : adaptiveCoverCheck 0 cell202 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell202 (by decide +kernel)
+
+private theorem checked203 : adaptiveCoverCheck 0 cell203 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell203 (by decide +kernel)
+
+private theorem checked210 : adaptiveCoverCheck 0 cell210 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell210 (by decide +kernel)
+
+private theorem checked211 : adaptiveCoverCheck 0 cell211 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell211 (by decide +kernel)
+
+private theorem checked212 : adaptiveCoverCheck 0 cell212 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell212 (by decide +kernel)
+
+private theorem checked213 : adaptiveCoverCheck 0 cell213 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell213 (by decide +kernel)
+
+private theorem checked220 : adaptiveCoverCheck 0 cell220 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell220 (by decide +kernel)
+
+private theorem checked221 : adaptiveCoverCheck 0 cell221 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell221 (by decide +kernel)
+
+private theorem checked222 : adaptiveCoverCheck 0 cell222 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell222 (by decide +kernel)
+
+private theorem checked223 : adaptiveCoverCheck 0 cell223 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell223 (by decide +kernel)
+
+private theorem checked230 : adaptiveCoverCheck 0 cell230 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell230 (by decide +kernel)
+
+private theorem checked231 : adaptiveCoverCheck 0 cell231 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell231 (by decide +kernel)
+
+private theorem checked232 : adaptiveCoverCheck 0 cell232 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell232 (by decide +kernel)
+
+private theorem checked233 : adaptiveCoverCheck 0 cell233 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell233 (by decide +kernel)
+
+private theorem checked300 : adaptiveCoverCheck 0 cell300 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell300 (by decide +kernel)
+
+private theorem checked301 : adaptiveCoverCheck 0 cell301 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell301 (by decide +kernel)
+
+private theorem checked302 : adaptiveCoverCheck 0 cell302 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell302 (by decide +kernel)
+
+private theorem checked303 : adaptiveCoverCheck 0 cell303 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell303 (by decide +kernel)
+
+private theorem checked310 : adaptiveCoverCheck 0 cell310 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell310 (by decide +kernel)
+
+private theorem checked311 : adaptiveCoverCheck 0 cell311 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell311 (by decide +kernel)
+
+private theorem checked312 : adaptiveCoverCheck 0 cell312 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell312 (by decide +kernel)
+
+private theorem checked313 : adaptiveCoverCheck 0 cell313 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell313 (by decide +kernel)
+
+private theorem checked320 : adaptiveCoverCheck 0 cell320 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell320 (by decide +kernel)
+
+private theorem checked321 : adaptiveCoverCheck 0 cell321 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell321 (by decide +kernel)
+
+private theorem checked322 : adaptiveCoverCheck 0 cell322 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell322 (by decide +kernel)
+
+private theorem checked323 : adaptiveCoverCheck 0 cell323 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell323 (by decide +kernel)
+
+private theorem checked330 : adaptiveCoverCheck 0 cell330 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell330 (by decide +kernel)
+
+private theorem checked331 : adaptiveCoverCheck 0 cell331 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell331 (by decide +kernel)
+
+private theorem checked332 : adaptiveCoverCheck 0 cell332 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell332 (by decide +kernel)
+
+private theorem checked333 : adaptiveCoverCheck 0 cell333 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 0 cell333 (by decide +kernel)
+
+private theorem checked00 : adaptiveCoverCheck 1 cell00 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell00 (by decide +kernel)
+
+private theorem checked01 : adaptiveCoverCheck 1 cell01 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell01 (by decide +kernel)
+
+private theorem checked02 : adaptiveCoverCheck 1 cell02 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell02 (by decide +kernel)
+
+private theorem checked03 : adaptiveCoverCheck 1 cell03 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell03 (by decide +kernel)
+
+private theorem checked10 : adaptiveCoverCheck 1 cell10 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell10 (by decide +kernel)
+
+private theorem checked11 : adaptiveCoverCheck 1 cell11 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell11 (by decide +kernel)
+
+private theorem checked12 : adaptiveCoverCheck 1 cell12 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell12 (by decide +kernel)
+
+private theorem checked13 : adaptiveCoverCheck 1 cell13 = true := by
+  exact adaptiveCoverCheck_true_of_rejected 1 cell13 (by decide +kernel)
+
+private theorem checked20 : adaptiveCoverCheck 1 cell20 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell20
+    checked200 checked201 checked202 checked203
+
+private theorem checked21 : adaptiveCoverCheck 1 cell21 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell21
+    checked210 checked211 checked212 checked213
+
+private theorem checked22 : adaptiveCoverCheck 1 cell22 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell22
+    checked220 checked221 checked222 checked223
+
+private theorem checked23 : adaptiveCoverCheck 1 cell23 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell23
+    checked230 checked231 checked232 checked233
+
+private theorem checked30 : adaptiveCoverCheck 1 cell30 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell30
+    checked300 checked301 checked302 checked303
+
+private theorem checked31 : adaptiveCoverCheck 1 cell31 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell31
+    checked310 checked311 checked312 checked313
+
+private theorem checked32 : adaptiveCoverCheck 1 cell32 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell32
+    checked320 checked321 checked322 checked323
+
+private theorem checked33 : adaptiveCoverCheck 1 cell33 = true := by
+  exact adaptiveCoverCheck_succ_of_children 0 cell33
+    checked330 checked331 checked332 checked333
+
+private theorem checked0 : adaptiveCoverCheck 2 cell0 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell0
+    checked00 checked01 checked02 checked03
+
+private theorem checked1 : adaptiveCoverCheck 2 cell1 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell1
+    checked10 checked11 checked12 checked13
+
+private theorem checked2 : adaptiveCoverCheck 2 cell2 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell2
+    checked20 checked21 checked22 checked23
+
+private theorem checked3 : adaptiveCoverCheck 2 cell3 = true := by
+  exact adaptiveCoverCheck_succ_of_children 1 cell3
+    checked30 checked31 checked32 checked33
+
+private theorem checkedRoot : adaptiveCoverCheck 3 cellRoot = true := by
+  exact adaptiveCoverCheck_succ_of_children 2 cellRoot
+    checked0 checked1 checked2 checked3
+
+end CoverCertificateba1f6e5865
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c1_c3 :
+    adaptiveCoverCheck 3 thetaAboveCell0000220020012013 = true := by
+  exact CoverCertificateba1f6e5865.checkedRoot
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle030.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle030.lean
new file mode 100644
index 0000000000..39a05442cf
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle030.lean
@@ -0,0 +1,10735 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle001
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle002
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle003
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle004
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle005
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle006
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle007
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle008
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle009
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle010
+
+
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle011
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle012
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle013
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle014
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle015
+
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle016
+
+
+
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle017
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle018
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle019
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle020
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle021
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle022
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle023
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle024
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle025
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle026
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle027
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle028
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle029
+
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle005
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.Core.Bundle007
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Reconstruction`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCells6754c8bb3e
+
+/-- Subcell `1101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1011 : AngleCell :=
+  childLH (childLH (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1100 : AngleCell :=
+  childLL (childLL (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1001` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1001 : AngleCell :=
+  childLH (childLL (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1010 : AngleCell :=
+  childLL (childLH (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1112 : AngleCell :=
+  childHL (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1113 : AngleCell :=
+  childHH (childLH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `0101` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0101 : AngleCell :=
+  childLH (childLL (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0110` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0110 : AngleCell :=
+  childLL (childLH (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0111` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0111 : AngleCell :=
+  childLH (childLH (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `1000` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1000 : AngleCell :=
+  childLL (childLL (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1003 : AngleCell :=
+  childHH (childLL (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1012 : AngleCell :=
+  childHL (childLH (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1013 : AngleCell :=
+  childHH (childLH (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1102 : AngleCell :=
+  childHL (childLL (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1103 : AngleCell :=
+  childHH (childLL (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `0003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0003 : AngleCell :=
+  childHH (childLL (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0010` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0010 : AngleCell :=
+  childLL (childLH (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0011` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0011 : AngleCell :=
+  childLH (childLH (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0012 : AngleCell :=
+  childHL (childLH (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0013 : AngleCell :=
+  childHH (childLH (childLL (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0100` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0100 : AngleCell :=
+  childLL (childLL (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0102 : AngleCell :=
+  childHL (childLL (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0103 : AngleCell :=
+  childHH (childLL (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0112 : AngleCell :=
+  childHL (childLH (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `0113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell0113 : AngleCell :=
+  childHH (childLH (childLH (childLL e24ThetaBelowRoot)))
+
+/-- Subcell `1002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1002 : AngleCell :=
+  childHL (childLL (childLL (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1121 : AngleCell :=
+  childLH (childHL (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1130 : AngleCell :=
+  childLL (childHH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `1131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell1131 : AngleCell :=
+  childLH (childHH (childLH (childLH e24ThetaBelowRoot)))
+
+/-- Subcell `11012132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11012133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaBelowCell1101)))
+
+/-- Subcell `11013013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013013 : AngleCell :=
+  childHH (childLH (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013102` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013102 : AngleCell :=
+  childHL (childLL (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013103` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013103 : AngleCell :=
+  childHH (childLL (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013112` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013112 : AngleCell :=
+  childHL (childLH (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013113` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013113 : AngleCell :=
+  childHH (childLH (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11013133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaBelowCell1101)))
+
+/-- Subcell `11102002` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102002 : AngleCell :=
+  childHL (childLL (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102003` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102003 : AngleCell :=
+  childHH (childLL (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102012` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102012 : AngleCell :=
+  childHL (childLH (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102013` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102013 : AngleCell :=
+  childHH (childLH (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102131 : AngleCell :=
+  childLH (childHH (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11102311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaBelowCell1110)))
+
+/-- Subcell `11103020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103020 : AngleCell :=
+  childLL (childHL (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103021 : AngleCell :=
+  childLH (childHL (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103030 : AngleCell :=
+  childLL (childHH (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103031 : AngleCell :=
+  childLH (childHH (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103120 : AngleCell :=
+  childLL (childHL (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103121 : AngleCell :=
+  childLH (childHL (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103130 : AngleCell :=
+  childLL (childHH (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103131` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103131 : AngleCell :=
+  childLH (childHH (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11103311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1110)))
+
+/-- Subcell `11112020` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112021` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112030` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112031` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112120` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112120 : AngleCell :=
+  childLL (childHL (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112121` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112121 : AngleCell :=
+  childLH (childHL (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112130` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112130 : AngleCell :=
+  childLL (childHH (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11112313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaBelowCell1111)))
+
+/-- Subcell `11113022` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113023` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113032` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113033` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113122` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113123` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113132` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113133` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113200` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113201` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113202` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113203` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113210` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113211` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113212` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113213` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113300` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113301` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113302` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113303` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113310` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113311` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113312` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaBelowCell1111)))
+
+/-- Subcell `11113313` of the theta-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaBelowCell11113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaBelowCell1111)))
+
+end GerverSofa.PartE.CertificateCells6754c8bb3e
+
+namespace GerverSofa.PartE.CertificateCellse9ea903ef1
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellse9ea903ef1
+
+namespace GerverSofa.PartE.CertificateCells89d797401e
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells89d797401e
+
+namespace GerverSofa.PartE.CertificateCellsafdf6e4305
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsafdf6e4305
+
+namespace GerverSofa.PartE.CertificateCells0d2c374949
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells0d2c374949
+
+namespace GerverSofa.PartE.CertificateCells3df86e4955
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells3df86e4955
+
+namespace GerverSofa.PartE.CertificateCells3298a326ef
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells3298a326ef
+
+namespace GerverSofa.PartE.CertificateCells47980ca575
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells47980ca575
+
+namespace GerverSofa.PartE.CertificateCellsfed39385df
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsfed39385df
+
+namespace GerverSofa.PartE.CertificateCellsb1b33fecca
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsb1b33fecca
+
+namespace GerverSofa.PartE.CertificateCellsc427a8578f
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsc427a8578f
+
+namespace GerverSofa.PartE.CertificateCells25978ec18d
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011010 : AngleCell :=
+  childLL (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells25978ec18d
+
+namespace GerverSofa.PartE.CertificateCellse6161a42be
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellse6161a42be
+
+namespace GerverSofa.PartE.CertificateCells1f6ffcc997
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells1f6ffcc997
+
+namespace GerverSofa.PartE.CertificateCells1275d3adc4
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells1275d3adc4
+
+namespace GerverSofa.PartE.CertificateCells021b870816
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells021b870816
+
+namespace GerverSofa.PartE.CertificateCells9ade3f586f
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells9ade3f586f
+
+namespace GerverSofa.PartE.CertificateCellsfd8221930a
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011011 : AngleCell :=
+  childLH (childLH (childLL (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsfd8221930a
+
+namespace GerverSofa.PartE.CertificateCells5aaf24f73d
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells5aaf24f73d
+
+namespace GerverSofa.PartE.CertificateCells31cca9f2c5
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells31cca9f2c5
+
+namespace GerverSofa.PartE.CertificateCells0a8495a296
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells0a8495a296
+
+namespace GerverSofa.PartE.CertificateCells4cabb07dbd
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells4cabb07dbd
+
+namespace GerverSofa.PartE.CertificateCellsc6c93e8f5c
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011100 : AngleCell :=
+  childLL (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsc6c93e8f5c
+
+namespace GerverSofa.PartE.CertificateCellsaa40da30f6
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011101 : AngleCell :=
+  childLH (childLL (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsaa40da30f6
+
+namespace GerverSofa.PartE.CertificateCells919c5d7ca9
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells919c5d7ca9
+
+namespace GerverSofa.PartE.CertificateCells87d4f5e678
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011110 : AngleCell :=
+  childLL (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCells87d4f5e678
+
+namespace GerverSofa.PartE.CertificateCellsefc5f924ee
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011110 : AngleCell :=
+  childLL (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsefc5f924ee
+
+namespace GerverSofa.PartE.CertificateCellsac583a159d
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11011111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11011111 : AngleCell :=
+  childLH (childLH (childLH (childLH phiAboveCell1101)))
+
+end GerverSofa.PartE.CertificateCellsac583a159d
+
+namespace GerverSofa.PartE.CertificateCellsf317a94cb3
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCellsf317a94cb3
+
+namespace GerverSofa.PartE.CertificateCellsf7dce840c8
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `11100000` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell11100000 : AngleCell :=
+  childLL (childLL (childLL (childLL phiAboveCell1110)))
+
+end GerverSofa.PartE.CertificateCellsf7dce840c8
+
+namespace GerverSofa.PartE.CertificateCells5fe660c4fc
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells5fe660c4fc
+
+namespace GerverSofa.PartE.CertificateCells7e3a6c7b6b
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells7e3a6c7b6b
+
+namespace GerverSofa.PartE.CertificateCellsa5efdb03f3
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCellsa5efdb03f3
+
+namespace GerverSofa.PartE.CertificateCells4f05b2093e
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCells4f05b2093e
+
+namespace GerverSofa.PartE.CertificateCellsa15368d7db
+
+/-- Subcell `1001` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1001 : AngleCell :=
+  childLH (childLL (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1010 : AngleCell :=
+  childLL (childLH (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1011 : AngleCell :=
+  childLH (childLH (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1100 : AngleCell :=
+  childLL (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `0101` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0101 : AngleCell :=
+  childLH (childLL (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0110` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0110 : AngleCell :=
+  childLL (childLH (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0111` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0111 : AngleCell :=
+  childLH (childLH (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `1000` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1000 : AngleCell :=
+  childLL (childLL (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `0000` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0001` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0001 : AngleCell :=
+  childLH (childLL (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0010` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0010 : AngleCell :=
+  childLL (childLH (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0011` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0011 : AngleCell :=
+  childLH (childLH (childLL (childLL e24PhiAboveRoot)))
+
+/-- Subcell `0100` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell0100 : AngleCell :=
+  childLL (childLL (childLH (childLL e24PhiAboveRoot)))
+
+/-- Subcell `1002` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1002 : AngleCell :=
+  childHL (childLL (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1003` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1003 : AngleCell :=
+  childHH (childLL (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1012` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1012 : AngleCell :=
+  childHL (childLH (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1013` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1013 : AngleCell :=
+  childHH (childLH (childLL (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1102` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1102 : AngleCell :=
+  childHL (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1103` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1103 : AngleCell :=
+  childHH (childLL (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1112` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1112 : AngleCell :=
+  childHL (childLH (childLH (childLH e24PhiAboveRoot)))
+
+/-- Subcell `1113` of the phi-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiAboveCell1113 : AngleCell :=
+  childHH (childLH (childLH (childLH e24PhiAboveRoot)))
+
+end GerverSofa.PartE.CertificateCellsa15368d7db
+
+namespace GerverSofa.PartE.CertificateCells12ecbb1984
+
+/-- Subcell `3320` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3320 : AngleCell :=
+  childLL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells12ecbb1984
+
+namespace GerverSofa.PartE.CertificateCellsf9b1d22c2c
+
+/-- Subcell `3322` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3322 : AngleCell :=
+  childHL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsf9b1d22c2c
+
+namespace GerverSofa.PartE.CertificateCells5481d87062
+
+/-- Subcell `3322` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3322 : AngleCell :=
+  childHL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells5481d87062
+
+namespace GerverSofa.PartE.CertificateCellsc99c114252
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsc99c114252
+
+namespace GerverSofa.PartE.CertificateCells85102f4bff
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells85102f4bff
+
+namespace GerverSofa.PartE.CertificateCellsd7423e4c3b
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsd7423e4c3b
+
+namespace GerverSofa.PartE.CertificateCells8ae9088aab
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells8ae9088aab
+
+namespace GerverSofa.PartE.CertificateCellsb8a79b56b3
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsb8a79b56b3
+
+namespace GerverSofa.PartE.CertificateCells13ae325d6d
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells13ae325d6d
+
+namespace GerverSofa.PartE.CertificateCells6add7bae04
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells6add7bae04
+
+namespace GerverSofa.PartE.CertificateCellscd1d17c9b4
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellscd1d17c9b4
+
+namespace GerverSofa.PartE.CertificateCells709ea69226
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232311` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232311 : AngleCell :=
+  childLH (childLH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells709ea69226
+
+namespace GerverSofa.PartE.CertificateCellsade8569b0a
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232313` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232313 : AngleCell :=
+  childHH (childLH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsade8569b0a
+
+namespace GerverSofa.PartE.CertificateCellsfece773e85
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsfece773e85
+
+namespace GerverSofa.PartE.CertificateCellse593ec4a84
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33232331` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33232331 : AngleCell :=
+  childLH (childHH (childHH (childHL phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellse593ec4a84
+
+namespace GerverSofa.PartE.CertificateCells0ccc164bd1
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells0ccc164bd1
+
+namespace GerverSofa.PartE.CertificateCells072df6b25f
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells072df6b25f
+
+namespace GerverSofa.PartE.CertificateCells215f6c9ebd
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells215f6c9ebd
+
+namespace GerverSofa.PartE.CertificateCellsef857ed1c7
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCellsef857ed1c7
+
+namespace GerverSofa.PartE.CertificateCells2ebe0c2db9
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233202` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233202 : AngleCell :=
+  childHL (childLL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells2ebe0c2db9
+
+namespace GerverSofa.PartE.CertificateCells27c0e8c594
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells27c0e8c594
+
+namespace GerverSofa.PartE.CertificateCells4d117fb26d
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233220` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233220 : AngleCell :=
+  childLL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells4d117fb26d
+
+namespace GerverSofa.PartE.CertificateCells848f180c5a
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233220` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233220 : AngleCell :=
+  childLL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells848f180c5a
+
+namespace GerverSofa.PartE.CertificateCells090e12bc76
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233221` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233221 : AngleCell :=
+  childLH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells090e12bc76
+
+namespace GerverSofa.PartE.CertificateCells857be56025
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233222` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233222 : AngleCell :=
+  childHL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells857be56025
+
+namespace GerverSofa.PartE.CertificateCellsea76f7562d
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233222` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233222 : AngleCell :=
+  childHL (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsea76f7562d
+
+namespace GerverSofa.PartE.CertificateCellsdd77c2320b
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsdd77c2320b
+
+namespace GerverSofa.PartE.CertificateCells08c6706994
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+/-- Subcell `332332232120` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell332332232120 : AngleCell :=
+  childLL (childHL (childLH (childHL phiBelowCell33233223)))
+
+end GerverSofa.PartE.CertificateCells08c6706994
+
+namespace GerverSofa.PartE.CertificateCells96d537e773
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells96d537e773
+
+namespace GerverSofa.PartE.CertificateCells0abe1d75a7
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells0abe1d75a7
+
+namespace GerverSofa.PartE.CertificateCells4849d1c13b
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells4849d1c13b
+
+namespace GerverSofa.PartE.CertificateCellsf2740dc787
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsf2740dc787
+
+namespace GerverSofa.PartE.CertificateCells3f18540de9
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells3f18540de9
+
+namespace GerverSofa.PartE.CertificateCellsa21caf16f2
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCellsa21caf16f2
+
+namespace GerverSofa.PartE.CertificateCells21943a9b83
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233223 : AngleCell :=
+  childHH (childHL (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells21943a9b83
+
+namespace GerverSofa.PartE.CertificateCells62ff50a4e8
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells62ff50a4e8
+
+namespace GerverSofa.PartE.CertificateCells6677fc4b7c
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `33233232` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell33233232 : AngleCell :=
+  childHL (childHH (childHL (childHH phiBelowCell3323)))
+
+end GerverSofa.PartE.CertificateCells6677fc4b7c
+
+namespace GerverSofa.PartE.CertificateCells07875eeb47
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells07875eeb47
+
+namespace GerverSofa.PartE.CertificateCells5ed2775b48
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells5ed2775b48
+
+namespace GerverSofa.PartE.CertificateCells83d9ff7319
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells83d9ff7319
+
+namespace GerverSofa.PartE.CertificateCells240a9bf558
+
+/-- Subcell `3000` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3000 : AngleCell :=
+  childLL (childLL (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3001` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3001 : AngleCell :=
+  childLH (childLL (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3002` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3002 : AngleCell :=
+  childHL (childLL (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3003` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3003 : AngleCell :=
+  childHH (childLL (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3010` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3010 : AngleCell :=
+  childLL (childLH (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3011` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3011 : AngleCell :=
+  childLH (childLH (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3012` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3012 : AngleCell :=
+  childHL (childLH (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3013` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3013 : AngleCell :=
+  childHH (childLH (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3020` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3020 : AngleCell :=
+  childLL (childHL (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3021` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3021 : AngleCell :=
+  childLH (childHL (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3022` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3022 : AngleCell :=
+  childHL (childHL (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3023` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3023 : AngleCell :=
+  childHH (childHL (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3030` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3030 : AngleCell :=
+  childLL (childHH (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3031` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3031 : AngleCell :=
+  childLH (childHH (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3032` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3032 : AngleCell :=
+  childHL (childHH (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3033` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3033 : AngleCell :=
+  childHH (childHH (childLL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3100` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3100 : AngleCell :=
+  childLL (childLL (childLH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3101` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3101 : AngleCell :=
+  childLH (childLL (childLH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3102` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3102 : AngleCell :=
+  childHL (childLL (childLH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3103` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3103 : AngleCell :=
+  childHH (childLL (childLH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3112` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3112 : AngleCell :=
+  childHL (childLH (childLH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3120` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3120 : AngleCell :=
+  childLL (childHL (childLH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3121` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3121 : AngleCell :=
+  childLH (childHL (childLH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3122` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3122 : AngleCell :=
+  childHL (childHL (childLH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3123` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3123 : AngleCell :=
+  childHH (childHL (childLH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3130` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3130 : AngleCell :=
+  childLL (childHH (childLH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3131` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3131 : AngleCell :=
+  childLH (childHH (childLH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3132` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3132 : AngleCell :=
+  childHL (childHH (childLH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3133` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3133 : AngleCell :=
+  childHH (childHH (childLH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3200` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3200 : AngleCell :=
+  childLL (childLL (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3201` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3201 : AngleCell :=
+  childLH (childLL (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3202` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3202 : AngleCell :=
+  childHL (childLL (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3203` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3203 : AngleCell :=
+  childHH (childLL (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3210` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3210 : AngleCell :=
+  childLL (childLH (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3211` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3211 : AngleCell :=
+  childLH (childLH (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3212` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3212 : AngleCell :=
+  childHL (childLH (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3213` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3213 : AngleCell :=
+  childHH (childLH (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3220` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3220 : AngleCell :=
+  childLL (childHL (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3221` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3221 : AngleCell :=
+  childLH (childHL (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3222` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3222 : AngleCell :=
+  childHL (childHL (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3223` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3223 : AngleCell :=
+  childHH (childHL (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3230` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3230 : AngleCell :=
+  childLL (childHH (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3231` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3231 : AngleCell :=
+  childLH (childHH (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3232` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3232 : AngleCell :=
+  childHL (childHH (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3233` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3233 : AngleCell :=
+  childHH (childHH (childHL (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3300` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3300 : AngleCell :=
+  childLL (childLL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3301` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3301 : AngleCell :=
+  childLH (childLL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3302` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3302 : AngleCell :=
+  childHL (childLL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3303` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3303 : AngleCell :=
+  childHH (childLL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3310` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3310 : AngleCell :=
+  childLL (childLH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3311` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3311 : AngleCell :=
+  childLH (childLH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3312` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3312 : AngleCell :=
+  childHL (childLH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3313` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3313 : AngleCell :=
+  childHH (childLH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3320` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3320 : AngleCell :=
+  childLL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3321` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3321 : AngleCell :=
+  childLH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3322` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3322 : AngleCell :=
+  childHL (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3323` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3323 : AngleCell :=
+  childHH (childHL (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3330` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3330 : AngleCell :=
+  childLL (childHH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3331` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3331 : AngleCell :=
+  childLH (childHH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3332` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3332 : AngleCell :=
+  childHL (childHH (childHH (childHH e24PhiBelowRoot)))
+
+/-- Subcell `3333` of the phi-below root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev phiBelowCell3333 : AngleCell :=
+  childHH (childHH (childHH (childHH e24PhiBelowRoot)))
+
+end GerverSofa.PartE.CertificateCells240a9bf558
+
+namespace GerverSofa.PartE.CertificateCells22695df69a
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells22695df69a
+
+namespace GerverSofa.PartE.CertificateCells782ac45202
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells782ac45202
+
+namespace GerverSofa.PartE.CertificateCellsb8f48433cf
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsb8f48433cf
+
+namespace GerverSofa.PartE.CertificateCells5bc1aa1a31
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells5bc1aa1a31
+
+namespace GerverSofa.PartE.CertificateCells29b43c6eb7
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells29b43c6eb7
+
+namespace GerverSofa.PartE.CertificateCells64bf70ad53
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells64bf70ad53
+
+namespace GerverSofa.PartE.CertificateCellsc935d550f9
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsc935d550f9
+
+namespace GerverSofa.PartE.CertificateCells0a80d69c2b
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells0a80d69c2b
+
+namespace GerverSofa.PartE.CertificateCellsfd35b16333
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsfd35b16333
+
+namespace GerverSofa.PartE.CertificateCellsd8bbac8cc5
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsd8bbac8cc5
+
+namespace GerverSofa.PartE.CertificateCells2eb3bc7e90
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells2eb3bc7e90
+
+namespace GerverSofa.PartE.CertificateCellsbc99de789e
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsbc99de789e
+
+namespace GerverSofa.PartE.CertificateCellsdc52d78b64
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsdc52d78b64
+
+namespace GerverSofa.PartE.CertificateCells69f98d6dd4
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells69f98d6dd4
+
+namespace GerverSofa.PartE.CertificateCells675835e7b9
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells675835e7b9
+
+namespace GerverSofa.PartE.CertificateCellsda5cde5aa9
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsda5cde5aa9
+
+namespace GerverSofa.PartE.CertificateCellsed33eab041
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsed33eab041
+
+namespace GerverSofa.PartE.CertificateCellsf997ca54bf
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002000 : AngleCell :=
+  childLL (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsf997ca54bf
+
+namespace GerverSofa.PartE.CertificateCellsba35077fe5
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsba35077fe5
+
+namespace GerverSofa.PartE.CertificateCells5d0bfdcc89
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells5d0bfdcc89
+
+namespace GerverSofa.PartE.CertificateCellsb9eba9ae01
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsb9eba9ae01
+
+namespace GerverSofa.PartE.CertificateCells5ae8bbaf2e
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells5ae8bbaf2e
+
+namespace GerverSofa.PartE.CertificateCellsbec28e0304
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsbec28e0304
+
+namespace GerverSofa.PartE.CertificateCells1b578c3d10
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells1b578c3d10
+
+namespace GerverSofa.PartE.CertificateCells2bc65b7190
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells2bc65b7190
+
+namespace GerverSofa.PartE.CertificateCellsc46dbfc77b
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsc46dbfc77b
+
+namespace GerverSofa.PartE.CertificateCells752b604750
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells752b604750
+
+namespace GerverSofa.PartE.CertificateCellse66f3ce3f5
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002001 : AngleCell :=
+  childLH (childLL (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellse66f3ce3f5
+
+namespace GerverSofa.PartE.CertificateCells45a373af38
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+end GerverSofa.PartE.CertificateCells45a373af38
+
+namespace GerverSofa.PartE.CertificateCellsdabf824b24
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsdabf824b24
+
+namespace GerverSofa.PartE.CertificateCells11c2c46ea8
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells11c2c46ea8
+
+namespace GerverSofa.PartE.CertificateCellsfc2738963a
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsfc2738963a
+
+namespace GerverSofa.PartE.CertificateCells2c3b2a68a2
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells2c3b2a68a2
+
+namespace GerverSofa.PartE.CertificateCells024fe9d391
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells024fe9d391
+
+namespace GerverSofa.PartE.CertificateCells95bf7ab700
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells95bf7ab700
+
+namespace GerverSofa.PartE.CertificateCellsb2f77f238f
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002010 : AngleCell :=
+  childLL (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsb2f77f238f
+
+namespace GerverSofa.PartE.CertificateCells8179cb7dd8
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells8179cb7dd8
+
+namespace GerverSofa.PartE.CertificateCellsdb257f0c7c
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsdb257f0c7c
+
+namespace GerverSofa.PartE.CertificateCells3a304b7358
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells3a304b7358
+
+namespace GerverSofa.PartE.CertificateCells6cdc8e3bc2
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells6cdc8e3bc2
+
+namespace GerverSofa.PartE.CertificateCells0157659d62
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells0157659d62
+
+namespace GerverSofa.PartE.CertificateCells834e9932ec
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002011 : AngleCell :=
+  childLH (childLH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCells834e9932ec
+
+namespace GerverSofa.PartE.CertificateCellsb02c7d0e39
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `000022002020` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002020 : AngleCell :=
+  childLL (childHL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `000022002021` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002021 : AngleCell :=
+  childLH (childHL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `000022002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `000022002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `000022002030` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002030 : AngleCell :=
+  childLL (childHH (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `000022002031` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002031 : AngleCell :=
+  childLH (childHH (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `000022002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell00002200)))
+
+/-- Subcell `000022002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell000022002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell00002200)))
+
+end GerverSofa.PartE.CertificateCellsb02c7d0e39
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Combining adaptive-cover certificates
+
+The numerical leaf checks are imported separately so that they can compile in parallel.
+This module combines them using the adaptive-cover soundness lemmas.
+
+* `KernelOnly.PartE.E24KC6ThetaBelowReconstruct`.
+* `KernelOnly.PartE.PhiAbove.Join00029`.
+* `KernelOnly.PartE.PhiAbove.Join00036`.
+* `KernelOnly.PartE.PhiAbove.Join00042`.
+* `KernelOnly.PartE.PhiAbove.Join00045`.
+* `KernelOnly.PartE.PhiAbove.Join00048`.
+* `KernelOnly.PartE.PhiAbove.Join00056`.
+* `KernelOnly.PartE.PhiAbove.Join00062`.
+* `KernelOnly.PartE.PhiAbove.Join00065`.
+* `KernelOnly.PartE.PhiAbove.Join00071`.
+* `KernelOnly.PartE.PhiAbove.Join00074`.
+* `KernelOnly.PartE.PhiAbove.Join00077`.
+* `KernelOnly.PartE.PhiAbove.Join00085`.
+* `KernelOnly.PartE.PhiAbove.Join00091`.
+* `KernelOnly.PartE.PhiAbove.Join00094`.
+* `KernelOnly.PartE.PhiAbove.Join00101`.
+* `KernelOnly.PartE.PhiAbove.Join00105`.
+* `KernelOnly.PartE.PhiAbove.Join00108`.
+* `KernelOnly.PartE.PhiAbove.Join00111`.
+* `KernelOnly.PartE.PhiAbove.Join00122`.
+* `KernelOnly.PartE.PhiAbove.Join00129`.
+* `KernelOnly.PartE.PhiAbove.Join00133`.
+* `KernelOnly.PartE.PhiAbove.Join00136`.
+* `KernelOnly.PartE.PhiAbove.Join00142`.
+* `KernelOnly.PartE.PhiAbove.Join00145`.
+* `KernelOnly.PartE.PhiAbove.Join00156`.
+* `KernelOnly.PartE.PhiAbove.Join00159`.
+* `KernelOnly.PartE.PhiAbove.Join00165`.
+* `KernelOnly.PartE.PhiAbove.Join00168`.
+* `KernelOnly.PartE.PhiAbove.Join00178`.
+* `KernelOnly.PartE.PhiAbove.Join00182`.
+* `KernelOnly.PartE.PhiAbove.Join00191`.
+* `KernelOnly.PartE.PhiAbove.Join00200`.
+* `KernelOnly.PartE.PhiAbove.Join00208`.
+* `KernelOnly.PartE.E24KC6PhiAboveReconstruct`.
+* `KernelOnly.PartE.PhiBelow.Join00246`.
+* `KernelOnly.PartE.PhiBelow.Join00260`.
+* `KernelOnly.PartE.PhiBelow.Join00269`.
+* `KernelOnly.PartE.PhiBelow.Join00278`.
+* `KernelOnly.PartE.PhiBelow.Join00284`.
+* `KernelOnly.PartE.PhiBelow.Join00285`.
+* `KernelOnly.PartE.PhiBelow.Join00294`.
+* `KernelOnly.PartE.PhiBelow.Join00301`.
+* `KernelOnly.PartE.PhiBelow.Join00308`.
+* `KernelOnly.PartE.PhiBelow.Join00314`.
+* `KernelOnly.PartE.PhiBelow.Join00315`.
+* `KernelOnly.PartE.PhiBelow.Join00326`.
+* `KernelOnly.PartE.PhiBelow.Join00333`.
+* `KernelOnly.PartE.PhiBelow.Join00334`.
+* `KernelOnly.PartE.PhiBelow.Join00343`.
+* `KernelOnly.PartE.PhiBelow.Join00346`.
+* `KernelOnly.PartE.PhiBelow.Join00347`.
+* `KernelOnly.PartE.PhiBelow.Join00348`.
+* `KernelOnly.PartE.PhiBelow.Join00358`.
+* `KernelOnly.PartE.PhiBelow.Join00369`.
+* `KernelOnly.PartE.PhiBelow.Join00371`.
+* `KernelOnly.PartE.PhiBelow.Join00383`.
+* `KernelOnly.PartE.PhiBelow.Join00384`.
+* `KernelOnly.PartE.PhiBelow.Join00390`.
+* `KernelOnly.PartE.PhiBelow.Join00398`.
+* `KernelOnly.PartE.PhiBelow.Join00401`.
+* `KernelOnly.PartE.PhiBelow.Join00408`.
+* `KernelOnly.PartE.PhiBelow.Join00421`.
+* `KernelOnly.PartE.PhiBelow.Join00425`.
+* `KernelOnly.PartE.PhiBelow.Join00427`.
+* `KernelOnly.PartE.PhiBelow.Join00434`.
+* `KernelOnly.PartE.PhiBelow.Join00435`.
+* `KernelOnly.PartE.PhiBelow.Join00444`.
+* `KernelOnly.PartE.PhiBelow.Join00446`.
+* `KernelOnly.PartE.PhiBelow.Join00447`.
+* `KernelOnly.PartE.PhiBelow.Join00448`.
+* `KernelOnly.PartE.PhiBelow.Join00457`.
+* `KernelOnly.PartE.PhiBelow.Join00459`.
+* `KernelOnly.PartE.PhiBelow.Join00460`.
+* `KernelOnly.PartE.PhiBelow.Join00462`.
+* `KernelOnly.PartE.E24KC6PhiBelowReconstruct`.
+* `KernelOnly.PartE.ThetaAbove.Join00016`.
+* `KernelOnly.PartE.ThetaAbove.Join00022`.
+* `KernelOnly.PartE.ThetaAbove.Join00023`.
+* `KernelOnly.PartE.ThetaAbove.Join00484`.
+* `KernelOnly.PartE.ThetaAbove.Join00490`.
+* `KernelOnly.PartE.ThetaAbove.Join00493`.
+* `KernelOnly.PartE.ThetaAbove.Join00500`.
+* `KernelOnly.PartE.ThetaAbove.Join00506`.
+* `KernelOnly.PartE.ThetaAbove.Join00509`.
+* `KernelOnly.PartE.ThetaAbove.Join00512`.
+* `KernelOnly.PartE.ThetaAbove.Join00520`.
+* `KernelOnly.PartE.ThetaAbove.Join00526`.
+* `KernelOnly.PartE.ThetaAbove.Join00529`.
+* `KernelOnly.PartE.ThetaAbove.Join00536`.
+* `KernelOnly.PartE.ThetaAbove.Join00542`.
+* `KernelOnly.PartE.ThetaAbove.Join00545`.
+* `KernelOnly.PartE.ThetaAbove.Join00548`.
+* `KernelOnly.PartE.ThetaAbove.Join00549`.
+* `KernelOnly.PartE.ThetaAbove.Join00560`.
+* `KernelOnly.PartE.ThetaAbove.Join00566`.
+* `KernelOnly.PartE.ThetaAbove.Join00569`.
+* `KernelOnly.PartE.ThetaAbove.Join00576`.
+* `KernelOnly.PartE.ThetaAbove.Join00580`.
+* `KernelOnly.PartE.ThetaAbove.Join00583`.
+* `KernelOnly.PartE.ThetaAbove.Join00590`.
+* `KernelOnly.PartE.ThetaAbove.Join00596`.
+* `KernelOnly.PartE.ThetaAbove.Join00599`.
+* `KernelOnly.PartE.ThetaAbove.Join00600`.
+* `KernelOnly.PartE.ThetaAbove.Join00603`.
+* `KernelOnly.PartE.ThetaAbove.Join00614`.
+* `KernelOnly.PartE.ThetaAbove.Join00620`.
+* `KernelOnly.PartE.ThetaAbove.Join00623`.
+* `KernelOnly.PartE.ThetaAbove.Join00630`.
+* `KernelOnly.PartE.ThetaAbove.Join00636`.
+* `KernelOnly.PartE.ThetaAbove.Join00639`.
+* `KernelOnly.PartE.ThetaAbove.Join00640`.
+* `KernelOnly.PartE.ThetaAbove.Join00650`.
+* `KernelOnly.PartE.ThetaAbove.Join00656`.
+* `KernelOnly.PartE.ThetaAbove.Join00659`.
+* `KernelOnly.PartE.ThetaAbove.Join00666`.
+* `KernelOnly.PartE.ThetaAbove.Join00026`.
+* `KernelOnly.PartE.ThetaAbove.Join00027`.
+* `KernelOnly.PartE.E24KC6ProofBatch9b503cfa8ccc3ab1`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC6Theta Below Reconstruct
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6754c8bb3e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6754c8bb3e
+
+open CertificateCells6754c8bb3e
+
+theorem e24KC2ThetaBelowNode11012132 :
+    adaptiveCoverCheck 10 thetaBelowCell11012132 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11012132
+    e24KC2ThetaBelowLeaf110121320 e24KC2ThetaBelowLeaf110121321 e24KC2ThetaBelowLeaf110121322
+      e24KC2ThetaBelowLeaf110121323
+
+theorem e24KC2ThetaBelowNode11012133 :
+    adaptiveCoverCheck 10 thetaBelowCell11012133 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11012133
+    e24KC2ThetaBelowLeaf110121330 e24KC2ThetaBelowLeaf110121331 e24KC2ThetaBelowLeaf110121332
+      e24KC2ThetaBelowLeaf110121333
+
+theorem e24KC2ThetaBelowNode11013013 :
+    adaptiveCoverCheck 10 thetaBelowCell11013013 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013013
+    e24KC2ThetaBelowLeaf110130130 e24KC2ThetaBelowLeaf110130131 e24KC2ThetaBelowLeaf110130132
+      e24KC2ThetaBelowLeaf110130133
+
+theorem e24KC2ThetaBelowNode11013021 :
+    adaptiveCoverCheck 10 thetaBelowCell11013021 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013021
+    e24KC2ThetaBelowLeaf110130210 e24KC2ThetaBelowLeaf110130211 e24KC2ThetaBelowLeaf110130212
+      e24KC2ThetaBelowLeaf110130213
+
+theorem e24KC2ThetaBelowNode11013022 :
+    adaptiveCoverCheck 10 thetaBelowCell11013022 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013022
+    e24KC2ThetaBelowLeaf110130220 e24KC2ThetaBelowLeaf110130221 e24KC2ThetaBelowLeaf110130222
+      e24KC2ThetaBelowLeaf110130223
+
+theorem e24KC2ThetaBelowNode11013023 :
+    adaptiveCoverCheck 10 thetaBelowCell11013023 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013023
+    e24KC2ThetaBelowLeaf110130230 e24KC2ThetaBelowLeaf110130231 e24KC2ThetaBelowLeaf110130232
+      e24KC2ThetaBelowLeaf110130233
+
+theorem e24KC2ThetaBelowNode11013030 :
+    adaptiveCoverCheck 10 thetaBelowCell11013030 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013030
+    e24KC2ThetaBelowLeaf110130300 e24KC2ThetaBelowLeaf110130301 e24KC2ThetaBelowLeaf110130302
+      e24KC2ThetaBelowLeaf110130303
+
+theorem e24KC2ThetaBelowNode11013031 :
+    adaptiveCoverCheck 10 thetaBelowCell11013031 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013031
+    e24KC2ThetaBelowLeaf110130310 e24KC2ThetaBelowLeaf110130311 e24KC2ThetaBelowLeaf110130312
+      e24KC2ThetaBelowLeaf110130313
+
+theorem e24KC2ThetaBelowNode11013032 :
+    adaptiveCoverCheck 10 thetaBelowCell11013032 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013032
+    e24KC2ThetaBelowLeaf110130320 e24KC2ThetaBelowLeaf110130321 e24KC2ThetaBelowLeaf110130322
+      e24KC2ThetaBelowLeaf110130323
+
+theorem e24KC2ThetaBelowNode11013033 :
+    adaptiveCoverCheck 10 thetaBelowCell11013033 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013033
+    e24KC2ThetaBelowLeaf110130330 e24KC2ThetaBelowLeaf110130331 e24KC2ThetaBelowLeaf110130332
+      e24KC2ThetaBelowLeaf110130333
+
+theorem e24KC2ThetaBelowNode11013102 :
+    adaptiveCoverCheck 10 thetaBelowCell11013102 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013102
+    e24KC2ThetaBelowLeaf110131020 e24KC2ThetaBelowLeaf110131021 e24KC2ThetaBelowLeaf110131022
+      e24KC2ThetaBelowLeaf110131023
+
+theorem e24KC2ThetaBelowNode11013103 :
+    adaptiveCoverCheck 10 thetaBelowCell11013103 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013103
+    e24KC2ThetaBelowLeaf110131030 e24KC2ThetaBelowLeaf110131031 e24KC2ThetaBelowLeaf110131032
+      e24KC2ThetaBelowLeaf110131033
+
+theorem e24KC2ThetaBelowNode11013112 :
+    adaptiveCoverCheck 10 thetaBelowCell11013112 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013112
+    e24KC2ThetaBelowLeaf110131120 e24KC2ThetaBelowLeaf110131121 e24KC2ThetaBelowLeaf110131122
+      e24KC2ThetaBelowLeaf110131123
+
+theorem e24KC2ThetaBelowNode11013113 :
+    adaptiveCoverCheck 10 thetaBelowCell11013113 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013113
+    e24KC2ThetaBelowLeaf110131130 e24KC2ThetaBelowLeaf110131131 e24KC2ThetaBelowLeaf110131132
+      e24KC2ThetaBelowLeaf110131133
+
+theorem e24KC2ThetaBelowNode11013120 :
+    adaptiveCoverCheck 10 thetaBelowCell11013120 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013120
+    e24KC2ThetaBelowLeaf110131200 e24KC2ThetaBelowLeaf110131201 e24KC2ThetaBelowLeaf110131202
+      e24KC2ThetaBelowLeaf110131203
+
+theorem e24KC2ThetaBelowNode11013121 :
+    adaptiveCoverCheck 10 thetaBelowCell11013121 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013121
+    e24KC2ThetaBelowLeaf110131210 e24KC2ThetaBelowLeaf110131211 e24KC2ThetaBelowLeaf110131212
+      e24KC2ThetaBelowLeaf110131213
+
+theorem e24KC2ThetaBelowNode11013122 :
+    adaptiveCoverCheck 10 thetaBelowCell11013122 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013122
+    e24KC2ThetaBelowLeaf110131220 e24KC2ThetaBelowLeaf110131221 e24KC2ThetaBelowLeaf110131222
+      e24KC2ThetaBelowLeaf110131223
+
+theorem e24KC2ThetaBelowNode11013123 :
+    adaptiveCoverCheck 10 thetaBelowCell11013123 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013123
+    e24KC2ThetaBelowLeaf110131230 e24KC2ThetaBelowLeaf110131231 e24KC2ThetaBelowLeaf110131232
+      e24KC2ThetaBelowLeaf110131233
+
+theorem e24KC2ThetaBelowNode11013130 :
+    adaptiveCoverCheck 10 thetaBelowCell11013130 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013130
+    e24KC2ThetaBelowLeaf110131300 e24KC2ThetaBelowLeaf110131301 e24KC2ThetaBelowLeaf110131302
+      e24KC2ThetaBelowLeaf110131303
+
+theorem e24KC2ThetaBelowNode11013131 :
+    adaptiveCoverCheck 10 thetaBelowCell11013131 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013131
+    e24KC2ThetaBelowLeaf110131310 e24KC2ThetaBelowLeaf110131311 e24KC2ThetaBelowLeaf110131312
+      e24KC2ThetaBelowLeaf110131313
+
+theorem e24KC2ThetaBelowNode11013132 :
+    adaptiveCoverCheck 10 thetaBelowCell11013132 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013132
+    e24KC2ThetaBelowLeaf110131320 e24KC2ThetaBelowLeaf110131321 e24KC2ThetaBelowLeaf110131322
+      e24KC2ThetaBelowLeaf110131323
+
+theorem e24KC2ThetaBelowNode11013133 :
+    adaptiveCoverCheck 10 thetaBelowCell11013133 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11013133
+    e24KC2ThetaBelowLeaf110131330 e24KC2ThetaBelowLeaf110131331 e24KC2ThetaBelowLeaf110131332
+      e24KC2ThetaBelowLeaf110131333
+
+theorem e24KC2ThetaBelowNode11102002 :
+    adaptiveCoverCheck 10 thetaBelowCell11102002 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102002
+    e24KC2ThetaBelowLeaf111020020 e24KC2ThetaBelowLeaf111020021 e24KC2ThetaBelowLeaf111020022
+      e24KC2ThetaBelowLeaf111020023
+
+theorem e24KC2ThetaBelowNode11102003 :
+    adaptiveCoverCheck 10 thetaBelowCell11102003 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102003
+    e24KC2ThetaBelowLeaf111020030 e24KC2ThetaBelowLeaf111020031 e24KC2ThetaBelowLeaf111020032
+      e24KC2ThetaBelowLeaf111020033
+
+theorem e24KC2ThetaBelowNode11102012 :
+    adaptiveCoverCheck 10 thetaBelowCell11102012 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102012
+    e24KC2ThetaBelowLeaf111020120 e24KC2ThetaBelowLeaf111020121 e24KC2ThetaBelowLeaf111020122
+      e24KC2ThetaBelowLeaf111020123
+
+theorem e24KC2ThetaBelowNode11102013 :
+    adaptiveCoverCheck 10 thetaBelowCell11102013 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102013
+    e24KC2ThetaBelowLeaf111020130 e24KC2ThetaBelowLeaf111020131 e24KC2ThetaBelowLeaf111020132
+      e24KC2ThetaBelowLeaf111020133
+
+theorem e24KC2ThetaBelowNode11102020 :
+    adaptiveCoverCheck 10 thetaBelowCell11102020 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102020
+    e24KC2ThetaBelowLeaf111020200 e24KC2ThetaBelowLeaf111020201 e24KC2ThetaBelowLeaf111020202
+      e24KC2ThetaBelowLeaf111020203
+
+theorem e24KC2ThetaBelowNode11102021 :
+    adaptiveCoverCheck 10 thetaBelowCell11102021 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102021
+    e24KC2ThetaBelowLeaf111020210 e24KC2ThetaBelowLeaf111020211 e24KC2ThetaBelowLeaf111020212
+      e24KC2ThetaBelowLeaf111020213
+
+theorem e24KC2ThetaBelowNode11102022 :
+    adaptiveCoverCheck 10 thetaBelowCell11102022 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102022
+    e24KC2ThetaBelowLeaf111020220 e24KC2ThetaBelowLeaf111020221 e24KC2ThetaBelowLeaf111020222
+      e24KC2ThetaBelowLeaf111020223
+
+theorem e24KC2ThetaBelowNode11102023 :
+    adaptiveCoverCheck 10 thetaBelowCell11102023 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102023
+    e24KC2ThetaBelowLeaf111020230 e24KC2ThetaBelowLeaf111020231 e24KC2ThetaBelowLeaf111020232
+      e24KC2ThetaBelowLeaf111020233
+
+theorem e24KC2ThetaBelowNode11102030 :
+    adaptiveCoverCheck 10 thetaBelowCell11102030 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102030
+    e24KC2ThetaBelowLeaf111020300 e24KC2ThetaBelowLeaf111020301 e24KC2ThetaBelowLeaf111020302
+      e24KC2ThetaBelowLeaf111020303
+
+theorem e24KC2ThetaBelowNode11102031 :
+    adaptiveCoverCheck 10 thetaBelowCell11102031 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102031
+    e24KC2ThetaBelowLeaf111020310 e24KC2ThetaBelowLeaf111020311 e24KC2ThetaBelowLeaf111020312
+      e24KC2ThetaBelowLeaf111020313
+
+theorem e24KC2ThetaBelowNode11102032 :
+    adaptiveCoverCheck 10 thetaBelowCell11102032 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102032
+    e24KC2ThetaBelowLeaf111020320 e24KC2ThetaBelowLeaf111020321 e24KC2ThetaBelowLeaf111020322
+      e24KC2ThetaBelowLeaf111020323
+
+theorem e24KC2ThetaBelowNode11102033 :
+    adaptiveCoverCheck 10 thetaBelowCell11102033 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102033
+    e24KC2ThetaBelowLeaf111020330 e24KC2ThetaBelowLeaf111020331 e24KC2ThetaBelowLeaf111020332
+      e24KC2ThetaBelowLeaf111020333
+
+theorem e24KC2ThetaBelowNode11102120 :
+    adaptiveCoverCheck 10 thetaBelowCell11102120 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102120
+    e24KC2ThetaBelowLeaf111021200 e24KC2ThetaBelowLeaf111021201 e24KC2ThetaBelowLeaf111021202
+      e24KC2ThetaBelowLeaf111021203
+
+theorem e24KC2ThetaBelowNode11102121 :
+    adaptiveCoverCheck 10 thetaBelowCell11102121 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102121
+    e24KC2ThetaBelowLeaf111021210 e24KC2ThetaBelowLeaf111021211 e24KC2ThetaBelowLeaf111021212
+      e24KC2ThetaBelowLeaf111021213
+
+theorem e24KC2ThetaBelowNode11102122 :
+    adaptiveCoverCheck 10 thetaBelowCell11102122 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102122
+    e24KC2ThetaBelowLeaf111021220 e24KC2ThetaBelowLeaf111021221 e24KC2ThetaBelowLeaf111021222
+      e24KC2ThetaBelowLeaf111021223
+
+theorem e24KC2ThetaBelowNode11102123 :
+    adaptiveCoverCheck 10 thetaBelowCell11102123 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102123
+    e24KC2ThetaBelowLeaf111021230 e24KC2ThetaBelowLeaf111021231 e24KC2ThetaBelowLeaf111021232
+      e24KC2ThetaBelowLeaf111021233
+
+theorem e24KC2ThetaBelowNode11102130 :
+    adaptiveCoverCheck 10 thetaBelowCell11102130 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102130
+    e24KC2ThetaBelowLeaf111021300 e24KC2ThetaBelowLeaf111021301 e24KC2ThetaBelowLeaf111021302
+      e24KC2ThetaBelowLeaf111021303
+
+theorem e24KC2ThetaBelowNode11102131 :
+    adaptiveCoverCheck 10 thetaBelowCell11102131 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102131
+    e24KC2ThetaBelowLeaf111021310 e24KC2ThetaBelowLeaf111021311 e24KC2ThetaBelowLeaf111021312
+      e24KC2ThetaBelowLeaf111021313
+
+theorem e24KC2ThetaBelowNode11102132 :
+    adaptiveCoverCheck 10 thetaBelowCell11102132 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102132
+    e24KC2ThetaBelowLeaf111021320 e24KC2ThetaBelowLeaf111021321 e24KC2ThetaBelowLeaf111021322
+      e24KC2ThetaBelowLeaf111021323
+
+theorem e24KC2ThetaBelowNode11102133 :
+    adaptiveCoverCheck 10 thetaBelowCell11102133 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102133
+    e24KC2ThetaBelowLeaf111021330 e24KC2ThetaBelowLeaf111021331 e24KC2ThetaBelowLeaf111021332
+      e24KC2ThetaBelowLeaf111021333
+
+theorem e24KC2ThetaBelowNode11102200 :
+    adaptiveCoverCheck 10 thetaBelowCell11102200 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102200
+    e24KC2ThetaBelowLeaf111022000 e24KC2ThetaBelowLeaf111022001 e24KC2ThetaBelowLeaf111022002
+      e24KC2ThetaBelowLeaf111022003
+
+theorem e24KC2ThetaBelowNode11102201 :
+    adaptiveCoverCheck 10 thetaBelowCell11102201 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102201
+    e24KC2ThetaBelowLeaf111022010 e24KC2ThetaBelowLeaf111022011 e24KC2ThetaBelowLeaf111022012
+      e24KC2ThetaBelowLeaf111022013
+
+theorem e24KC2ThetaBelowNode11102210 :
+    adaptiveCoverCheck 10 thetaBelowCell11102210 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102210
+    e24KC2ThetaBelowLeaf111022100 e24KC2ThetaBelowLeaf111022101 e24KC2ThetaBelowLeaf111022102
+      e24KC2ThetaBelowLeaf111022103
+
+theorem e24KC2ThetaBelowNode11102211 :
+    adaptiveCoverCheck 10 thetaBelowCell11102211 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102211
+    e24KC2ThetaBelowLeaf111022110 e24KC2ThetaBelowLeaf111022111 e24KC2ThetaBelowLeaf111022112
+      e24KC2ThetaBelowLeaf111022113
+
+theorem e24KC2ThetaBelowNode11102300 :
+    adaptiveCoverCheck 10 thetaBelowCell11102300 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102300
+    e24KC2ThetaBelowLeaf111023000 e24KC2ThetaBelowLeaf111023001 e24KC2ThetaBelowLeaf111023002
+      e24KC2ThetaBelowLeaf111023003
+
+theorem e24KC2ThetaBelowNode11102301 :
+    adaptiveCoverCheck 10 thetaBelowCell11102301 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102301
+    e24KC2ThetaBelowLeaf111023010 e24KC2ThetaBelowLeaf111023011 e24KC2ThetaBelowLeaf111023012
+      e24KC2ThetaBelowLeaf111023013
+
+theorem e24KC2ThetaBelowNode11102310 :
+    adaptiveCoverCheck 10 thetaBelowCell11102310 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102310
+    e24KC2ThetaBelowLeaf111023100 e24KC2ThetaBelowLeaf111023101 e24KC2ThetaBelowLeaf111023102
+      e24KC2ThetaBelowLeaf111023103
+
+theorem e24KC2ThetaBelowNode11102311 :
+    adaptiveCoverCheck 10 thetaBelowCell11102311 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11102311
+    e24KC2ThetaBelowLeaf111023110 e24KC2ThetaBelowLeaf111023111 e24KC2ThetaBelowLeaf111023112
+      e24KC2ThetaBelowLeaf111023113
+
+theorem e24KC2ThetaBelowNode11103020 :
+    adaptiveCoverCheck 10 thetaBelowCell11103020 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103020
+    e24KC2ThetaBelowLeaf111030200 e24KC2ThetaBelowLeaf111030201 e24KC2ThetaBelowLeaf111030202
+      e24KC2ThetaBelowLeaf111030203
+
+theorem e24KC2ThetaBelowNode11103021 :
+    adaptiveCoverCheck 10 thetaBelowCell11103021 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103021
+    e24KC2ThetaBelowLeaf111030210 e24KC2ThetaBelowLeaf111030211 e24KC2ThetaBelowLeaf111030212
+      e24KC2ThetaBelowLeaf111030213
+
+theorem e24KC2ThetaBelowNode11103022 :
+    adaptiveCoverCheck 10 thetaBelowCell11103022 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103022
+    e24KC2ThetaBelowLeaf111030220 e24KC2ThetaBelowLeaf111030221 e24KC2ThetaBelowLeaf111030222
+      e24KC2ThetaBelowLeaf111030223
+
+theorem e24KC2ThetaBelowNode11103023 :
+    adaptiveCoverCheck 10 thetaBelowCell11103023 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103023
+    e24KC2ThetaBelowLeaf111030230 e24KC2ThetaBelowLeaf111030231 e24KC2ThetaBelowLeaf111030232
+      e24KC2ThetaBelowLeaf111030233
+
+theorem e24KC2ThetaBelowNode11103030 :
+    adaptiveCoverCheck 10 thetaBelowCell11103030 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103030
+    e24KC2ThetaBelowLeaf111030300 e24KC2ThetaBelowLeaf111030301 e24KC2ThetaBelowLeaf111030302
+      e24KC2ThetaBelowLeaf111030303
+
+theorem e24KC2ThetaBelowNode11103031 :
+    adaptiveCoverCheck 10 thetaBelowCell11103031 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103031
+    e24KC2ThetaBelowLeaf111030310 e24KC2ThetaBelowLeaf111030311 e24KC2ThetaBelowLeaf111030312
+      e24KC2ThetaBelowLeaf111030313
+
+theorem e24KC2ThetaBelowNode11103032 :
+    adaptiveCoverCheck 10 thetaBelowCell11103032 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103032
+    e24KC2ThetaBelowLeaf111030320 e24KC2ThetaBelowLeaf111030321 e24KC2ThetaBelowLeaf111030322
+      e24KC2ThetaBelowLeaf111030323
+
+theorem e24KC2ThetaBelowNode11103033 :
+    adaptiveCoverCheck 10 thetaBelowCell11103033 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103033
+    e24KC2ThetaBelowLeaf111030330 e24KC2ThetaBelowLeaf111030331 e24KC2ThetaBelowLeaf111030332
+      e24KC2ThetaBelowLeaf111030333
+
+theorem e24KC2ThetaBelowNode11103120 :
+    adaptiveCoverCheck 10 thetaBelowCell11103120 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103120
+    e24KC2ThetaBelowLeaf111031200 e24KC2ThetaBelowLeaf111031201 e24KC2ThetaBelowLeaf111031202
+      e24KC2ThetaBelowLeaf111031203
+
+theorem e24KC2ThetaBelowNode11103121 :
+    adaptiveCoverCheck 10 thetaBelowCell11103121 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103121
+    e24KC2ThetaBelowLeaf111031210 e24KC2ThetaBelowLeaf111031211 e24KC2ThetaBelowLeaf111031212
+      e24KC2ThetaBelowLeaf111031213
+
+theorem e24KC2ThetaBelowNode11103122 :
+    adaptiveCoverCheck 10 thetaBelowCell11103122 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103122
+    e24KC2ThetaBelowLeaf111031220 e24KC2ThetaBelowLeaf111031221 e24KC2ThetaBelowLeaf111031222
+      e24KC2ThetaBelowLeaf111031223
+
+theorem e24KC2ThetaBelowNode11103123 :
+    adaptiveCoverCheck 10 thetaBelowCell11103123 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103123
+    e24KC2ThetaBelowLeaf111031230 e24KC2ThetaBelowLeaf111031231 e24KC2ThetaBelowLeaf111031232
+      e24KC2ThetaBelowLeaf111031233
+
+theorem e24KC2ThetaBelowNode11103130 :
+    adaptiveCoverCheck 10 thetaBelowCell11103130 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103130
+    e24KC2ThetaBelowLeaf111031300 e24KC2ThetaBelowLeaf111031301 e24KC2ThetaBelowLeaf111031302
+      e24KC2ThetaBelowLeaf111031303
+
+theorem e24KC2ThetaBelowNode11103131 :
+    adaptiveCoverCheck 10 thetaBelowCell11103131 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103131
+    e24KC2ThetaBelowLeaf111031310 e24KC2ThetaBelowLeaf111031311 e24KC2ThetaBelowLeaf111031312
+      e24KC2ThetaBelowLeaf111031313
+
+theorem e24KC2ThetaBelowNode11103132 :
+    adaptiveCoverCheck 10 thetaBelowCell11103132 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103132
+    e24KC2ThetaBelowLeaf111031320 e24KC2ThetaBelowLeaf111031321 e24KC2ThetaBelowLeaf111031322
+      e24KC2ThetaBelowLeaf111031323
+
+theorem e24KC2ThetaBelowNode11103133 :
+    adaptiveCoverCheck 10 thetaBelowCell11103133 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103133
+    e24KC2ThetaBelowLeaf111031330 e24KC2ThetaBelowLeaf111031331 e24KC2ThetaBelowLeaf111031332
+      e24KC2ThetaBelowLeaf111031333
+
+theorem e24KC2ThetaBelowNode11103200 :
+    adaptiveCoverCheck 10 thetaBelowCell11103200 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103200
+    e24KC2ThetaBelowLeaf111032000 e24KC2ThetaBelowLeaf111032001 e24KC2ThetaBelowLeaf111032002
+      e24KC2ThetaBelowLeaf111032003
+
+theorem e24KC2ThetaBelowNode11103201 :
+    adaptiveCoverCheck 10 thetaBelowCell11103201 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103201
+    e24KC2ThetaBelowLeaf111032010 e24KC2ThetaBelowLeaf111032011 e24KC2ThetaBelowLeaf111032012
+      e24KC2ThetaBelowLeaf111032013
+
+theorem e24KC2ThetaBelowNode11103210 :
+    adaptiveCoverCheck 10 thetaBelowCell11103210 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103210
+    e24KC2ThetaBelowLeaf111032100 e24KC2ThetaBelowLeaf111032101 e24KC2ThetaBelowLeaf111032102
+      e24KC2ThetaBelowLeaf111032103
+
+theorem e24KC2ThetaBelowNode11103211 :
+    adaptiveCoverCheck 10 thetaBelowCell11103211 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103211
+    e24KC2ThetaBelowLeaf111032110 e24KC2ThetaBelowLeaf111032111 e24KC2ThetaBelowLeaf111032112
+      e24KC2ThetaBelowLeaf111032113
+
+theorem e24KC2ThetaBelowNode11103300 :
+    adaptiveCoverCheck 10 thetaBelowCell11103300 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103300
+    e24KC2ThetaBelowLeaf111033000 e24KC2ThetaBelowLeaf111033001 e24KC2ThetaBelowLeaf111033002
+      e24KC2ThetaBelowLeaf111033003
+
+theorem e24KC2ThetaBelowNode11103301 :
+    adaptiveCoverCheck 10 thetaBelowCell11103301 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103301
+    e24KC2ThetaBelowLeaf111033010 e24KC2ThetaBelowLeaf111033011 e24KC2ThetaBelowLeaf111033012
+      e24KC2ThetaBelowLeaf111033013
+
+theorem e24KC2ThetaBelowNode11103310 :
+    adaptiveCoverCheck 10 thetaBelowCell11103310 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103310
+    e24KC2ThetaBelowLeaf111033100 e24KC2ThetaBelowLeaf111033101 e24KC2ThetaBelowLeaf111033102
+      e24KC2ThetaBelowLeaf111033103
+
+theorem e24KC2ThetaBelowNode11103311 :
+    adaptiveCoverCheck 10 thetaBelowCell11103311 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11103311
+    e24KC2ThetaBelowLeaf111033110 e24KC2ThetaBelowLeaf111033111 e24KC2ThetaBelowLeaf111033112
+      e24KC2ThetaBelowLeaf111033113
+
+theorem e24KC2ThetaBelowNode11112020 :
+    adaptiveCoverCheck 10 thetaBelowCell11112020 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112020
+    e24KC2ThetaBelowLeaf111120200 e24KC2ThetaBelowLeaf111120201 e24KC2ThetaBelowLeaf111120202
+      e24KC2ThetaBelowLeaf111120203
+
+theorem e24KC2ThetaBelowNode11112021 :
+    adaptiveCoverCheck 10 thetaBelowCell11112021 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112021
+    e24KC2ThetaBelowLeaf111120210 e24KC2ThetaBelowLeaf111120211 e24KC2ThetaBelowLeaf111120212
+      e24KC2ThetaBelowLeaf111120213
+
+theorem e24KC2ThetaBelowNode11112022 :
+    adaptiveCoverCheck 10 thetaBelowCell11112022 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112022
+    e24KC2ThetaBelowLeaf111120220 e24KC2ThetaBelowLeaf111120221 e24KC2ThetaBelowLeaf111120222
+      e24KC2ThetaBelowLeaf111120223
+
+theorem e24KC2ThetaBelowNode11112023 :
+    adaptiveCoverCheck 10 thetaBelowCell11112023 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112023
+    e24KC2ThetaBelowLeaf111120230 e24KC2ThetaBelowLeaf111120231 e24KC2ThetaBelowLeaf111120232
+      e24KC2ThetaBelowLeaf111120233
+
+theorem e24KC2ThetaBelowNode11112030 :
+    adaptiveCoverCheck 10 thetaBelowCell11112030 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112030
+    e24KC2ThetaBelowLeaf111120300 e24KC2ThetaBelowLeaf111120301 e24KC2ThetaBelowLeaf111120302
+      e24KC2ThetaBelowLeaf111120303
+
+theorem e24KC2ThetaBelowNode11112031 :
+    adaptiveCoverCheck 10 thetaBelowCell11112031 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112031
+    e24KC2ThetaBelowLeaf111120310 e24KC2ThetaBelowLeaf111120311 e24KC2ThetaBelowLeaf111120312
+      e24KC2ThetaBelowLeaf111120313
+
+theorem e24KC2ThetaBelowNode11112032 :
+    adaptiveCoverCheck 10 thetaBelowCell11112032 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112032
+    e24KC2ThetaBelowLeaf111120320 e24KC2ThetaBelowLeaf111120321 e24KC2ThetaBelowLeaf111120322
+      e24KC2ThetaBelowLeaf111120323
+
+theorem e24KC2ThetaBelowNode11112033 :
+    adaptiveCoverCheck 10 thetaBelowCell11112033 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112033
+    e24KC2ThetaBelowLeaf111120330 e24KC2ThetaBelowLeaf111120331 e24KC2ThetaBelowLeaf111120332
+      e24KC2ThetaBelowLeaf111120333
+
+theorem e24KC2ThetaBelowNode11112120 :
+    adaptiveCoverCheck 10 thetaBelowCell11112120 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112120
+    e24KC2ThetaBelowLeaf111121200 e24KC2ThetaBelowLeaf111121201 e24KC2ThetaBelowLeaf111121202
+      e24KC2ThetaBelowLeaf111121203
+
+theorem e24KC2ThetaBelowNode11112121 :
+    adaptiveCoverCheck 10 thetaBelowCell11112121 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112121
+    e24KC2ThetaBelowLeaf111121210 e24KC2ThetaBelowLeaf111121211 e24KC2ThetaBelowLeaf111121212
+      e24KC2ThetaBelowLeaf111121213
+
+theorem e24KC2ThetaBelowNode11112122 :
+    adaptiveCoverCheck 10 thetaBelowCell11112122 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112122
+    e24KC2ThetaBelowLeaf111121220 e24KC2ThetaBelowLeaf111121221 e24KC2ThetaBelowLeaf111121222
+      e24KC2ThetaBelowLeaf111121223
+
+theorem e24KC2ThetaBelowNode11112123 :
+    adaptiveCoverCheck 10 thetaBelowCell11112123 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112123
+    e24KC2ThetaBelowLeaf111121230 e24KC2ThetaBelowLeaf111121231 e24KC2ThetaBelowLeaf111121232
+      e24KC2ThetaBelowLeaf111121233
+
+theorem e24KC2ThetaBelowNode11112130 :
+    adaptiveCoverCheck 10 thetaBelowCell11112130 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112130
+    e24KC2ThetaBelowLeaf111121300 e24KC2ThetaBelowLeaf111121301 e24KC2ThetaBelowLeaf111121302
+      e24KC2ThetaBelowLeaf111121303
+
+theorem e24KC2ThetaBelowNode11112132 :
+    adaptiveCoverCheck 10 thetaBelowCell11112132 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112132
+    e24KC2ThetaBelowLeaf111121320 e24KC2ThetaBelowLeaf111121321 e24KC2ThetaBelowLeaf111121322
+      e24KC2ThetaBelowLeaf111121323
+
+theorem e24KC2ThetaBelowNode11112133 :
+    adaptiveCoverCheck 10 thetaBelowCell11112133 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112133
+    e24KC2ThetaBelowLeaf111121330 e24KC2ThetaBelowLeaf111121331 e24KC2ThetaBelowLeaf111121332
+      e24KC2ThetaBelowLeaf111121333
+
+theorem e24KC2ThetaBelowNode11112200 :
+    adaptiveCoverCheck 10 thetaBelowCell11112200 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112200
+    e24KC2ThetaBelowLeaf111122000 e24KC2ThetaBelowLeaf111122001 e24KC2ThetaBelowLeaf111122002
+      e24KC2ThetaBelowLeaf111122003
+
+theorem e24KC2ThetaBelowNode11112201 :
+    adaptiveCoverCheck 10 thetaBelowCell11112201 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112201
+    e24KC2ThetaBelowLeaf111122010 e24KC2ThetaBelowLeaf111122011 e24KC2ThetaBelowLeaf111122012
+      e24KC2ThetaBelowLeaf111122013
+
+theorem e24KC2ThetaBelowNode11112210 :
+    adaptiveCoverCheck 10 thetaBelowCell11112210 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112210
+    e24KC2ThetaBelowLeaf111122100 e24KC2ThetaBelowLeaf111122101 e24KC2ThetaBelowLeaf111122102
+      e24KC2ThetaBelowLeaf111122103
+
+theorem e24KC2ThetaBelowNode11112211 :
+    adaptiveCoverCheck 10 thetaBelowCell11112211 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112211
+    e24KC2ThetaBelowLeaf111122110 e24KC2ThetaBelowLeaf111122111 e24KC2ThetaBelowLeaf111122112
+      e24KC2ThetaBelowLeaf111122113
+
+theorem e24KC2ThetaBelowNode11112212 :
+    adaptiveCoverCheck 10 thetaBelowCell11112212 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112212
+    e24KC2ThetaBelowLeaf111122120 e24KC2ThetaBelowLeaf111122121 e24KC2ThetaBelowLeaf111122122
+      e24KC2ThetaBelowLeaf111122123
+
+theorem e24KC2ThetaBelowNode11112213 :
+    adaptiveCoverCheck 10 thetaBelowCell11112213 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112213
+    e24KC2ThetaBelowLeaf111122130 e24KC2ThetaBelowLeaf111122131 e24KC2ThetaBelowLeaf111122132
+      e24KC2ThetaBelowLeaf111122133
+
+theorem e24KC2ThetaBelowNode11112300 :
+    adaptiveCoverCheck 10 thetaBelowCell11112300 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112300
+    e24KC2ThetaBelowLeaf111123000 e24KC2ThetaBelowLeaf111123001 e24KC2ThetaBelowLeaf111123002
+      e24KC2ThetaBelowLeaf111123003
+
+theorem e24KC2ThetaBelowNode11112301 :
+    adaptiveCoverCheck 10 thetaBelowCell11112301 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112301
+    e24KC2ThetaBelowLeaf111123010 e24KC2ThetaBelowLeaf111123011 e24KC2ThetaBelowLeaf111123012
+      e24KC2ThetaBelowLeaf111123013
+
+theorem e24KC2ThetaBelowNode11112302 :
+    adaptiveCoverCheck 10 thetaBelowCell11112302 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112302
+    e24KC2ThetaBelowLeaf111123020 e24KC2ThetaBelowLeaf111123021 e24KC2ThetaBelowLeaf111123022
+      e24KC2ThetaBelowLeaf111123023
+
+theorem e24KC2ThetaBelowNode11112303 :
+    adaptiveCoverCheck 10 thetaBelowCell11112303 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112303
+    e24KC2ThetaBelowLeaf111123030 e24KC2ThetaBelowLeaf111123031 e24KC2ThetaBelowLeaf111123032
+      e24KC2ThetaBelowLeaf111123033
+
+theorem e24KC2ThetaBelowNode11112310 :
+    adaptiveCoverCheck 10 thetaBelowCell11112310 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112310
+    e24KC2ThetaBelowLeaf111123100 e24KC2ThetaBelowLeaf111123101 e24KC2ThetaBelowLeaf111123102
+      e24KC2ThetaBelowLeaf111123103
+
+theorem e24KC2ThetaBelowNode11112311 :
+    adaptiveCoverCheck 10 thetaBelowCell11112311 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112311
+    e24KC2ThetaBelowLeaf111123110 e24KC2ThetaBelowLeaf111123111 e24KC2ThetaBelowLeaf111123112
+      e24KC2ThetaBelowLeaf111123113
+
+theorem e24KC2ThetaBelowNode11112312 :
+    adaptiveCoverCheck 10 thetaBelowCell11112312 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112312
+    e24KC2ThetaBelowLeaf111123120 e24KC2ThetaBelowLeaf111123121 e24KC2ThetaBelowLeaf111123122
+      e24KC2ThetaBelowLeaf111123123
+
+theorem e24KC2ThetaBelowNode11112313 :
+    adaptiveCoverCheck 10 thetaBelowCell11112313 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11112313
+    e24KC2ThetaBelowLeaf111123130 e24KC2ThetaBelowLeaf111123131 e24KC2ThetaBelowLeaf111123132
+      e24KC2ThetaBelowLeaf111123133
+
+theorem e24KC2ThetaBelowNode11113022 :
+    adaptiveCoverCheck 10 thetaBelowCell11113022 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113022
+    e24KC2ThetaBelowLeaf111130220 e24KC2ThetaBelowLeaf111130221 e24KC2ThetaBelowLeaf111130222
+      e24KC2ThetaBelowLeaf111130223
+
+theorem e24KC2ThetaBelowNode11113023 :
+    adaptiveCoverCheck 10 thetaBelowCell11113023 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113023
+    e24KC2ThetaBelowLeaf111130230 e24KC2ThetaBelowLeaf111130231 e24KC2ThetaBelowLeaf111130232
+      e24KC2ThetaBelowLeaf111130233
+
+theorem e24KC2ThetaBelowNode11113032 :
+    adaptiveCoverCheck 10 thetaBelowCell11113032 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113032
+    e24KC2ThetaBelowLeaf111130320 e24KC2ThetaBelowLeaf111130321 e24KC2ThetaBelowLeaf111130322
+      e24KC2ThetaBelowLeaf111130323
+
+theorem e24KC2ThetaBelowNode11113033 :
+    adaptiveCoverCheck 10 thetaBelowCell11113033 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113033
+    e24KC2ThetaBelowLeaf111130330 e24KC2ThetaBelowLeaf111130331 e24KC2ThetaBelowLeaf111130332
+      e24KC2ThetaBelowLeaf111130333
+
+theorem e24KC2ThetaBelowNode11113122 :
+    adaptiveCoverCheck 10 thetaBelowCell11113122 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113122
+    e24KC2ThetaBelowLeaf111131220 e24KC2ThetaBelowLeaf111131221 e24KC2ThetaBelowLeaf111131222
+      e24KC2ThetaBelowLeaf111131223
+
+theorem e24KC2ThetaBelowNode11113123 :
+    adaptiveCoverCheck 10 thetaBelowCell11113123 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113123
+    e24KC2ThetaBelowLeaf111131230 e24KC2ThetaBelowLeaf111131231 e24KC2ThetaBelowLeaf111131232
+      e24KC2ThetaBelowLeaf111131233
+
+theorem e24KC2ThetaBelowNode11113132 :
+    adaptiveCoverCheck 10 thetaBelowCell11113132 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113132
+    e24KC2ThetaBelowLeaf111131320 e24KC2ThetaBelowLeaf111131321 e24KC2ThetaBelowLeaf111131322
+      e24KC2ThetaBelowLeaf111131323
+
+theorem e24KC2ThetaBelowNode11113133 :
+    adaptiveCoverCheck 10 thetaBelowCell11113133 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113133
+    e24KC2ThetaBelowLeaf111131330 e24KC2ThetaBelowLeaf111131331 e24KC2ThetaBelowLeaf111131332
+      e24KC2ThetaBelowLeaf111131333
+
+theorem e24KC2ThetaBelowNode11113200 :
+    adaptiveCoverCheck 10 thetaBelowCell11113200 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113200
+    e24KC2ThetaBelowLeaf111132000 e24KC2ThetaBelowLeaf111132001 e24KC2ThetaBelowLeaf111132002
+      e24KC2ThetaBelowLeaf111132003
+
+theorem e24KC2ThetaBelowNode11113201 :
+    adaptiveCoverCheck 10 thetaBelowCell11113201 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113201
+    e24KC2ThetaBelowLeaf111132010 e24KC2ThetaBelowLeaf111132011 e24KC2ThetaBelowLeaf111132012
+      e24KC2ThetaBelowLeaf111132013
+
+theorem e24KC2ThetaBelowNode11113202 :
+    adaptiveCoverCheck 10 thetaBelowCell11113202 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113202
+    e24KC2ThetaBelowLeaf111132020 e24KC2ThetaBelowLeaf111132021 e24KC2ThetaBelowLeaf111132022
+      e24KC2ThetaBelowLeaf111132023
+
+theorem e24KC2ThetaBelowNode11113203 :
+    adaptiveCoverCheck 10 thetaBelowCell11113203 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113203
+    e24KC2ThetaBelowLeaf111132030 e24KC2ThetaBelowLeaf111132031 e24KC2ThetaBelowLeaf111132032
+      e24KC2ThetaBelowLeaf111132033
+
+theorem e24KC2ThetaBelowNode11113210 :
+    adaptiveCoverCheck 10 thetaBelowCell11113210 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113210
+    e24KC2ThetaBelowLeaf111132100 e24KC2ThetaBelowLeaf111132101 e24KC2ThetaBelowLeaf111132102
+      e24KC2ThetaBelowLeaf111132103
+
+theorem e24KC2ThetaBelowNode11113211 :
+    adaptiveCoverCheck 10 thetaBelowCell11113211 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113211
+    e24KC2ThetaBelowLeaf111132110 e24KC2ThetaBelowLeaf111132111 e24KC2ThetaBelowLeaf111132112
+      e24KC2ThetaBelowLeaf111132113
+
+theorem e24KC2ThetaBelowNode11113212 :
+    adaptiveCoverCheck 10 thetaBelowCell11113212 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113212
+    e24KC2ThetaBelowLeaf111132120 e24KC2ThetaBelowLeaf111132121 e24KC2ThetaBelowLeaf111132122
+      e24KC2ThetaBelowLeaf111132123
+
+theorem e24KC2ThetaBelowNode11113213 :
+    adaptiveCoverCheck 10 thetaBelowCell11113213 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113213
+    e24KC2ThetaBelowLeaf111132130 e24KC2ThetaBelowLeaf111132131 e24KC2ThetaBelowLeaf111132132
+      e24KC2ThetaBelowLeaf111132133
+
+theorem e24KC2ThetaBelowNode11113300 :
+    adaptiveCoverCheck 10 thetaBelowCell11113300 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113300
+    e24KC2ThetaBelowLeaf111133000 e24KC2ThetaBelowLeaf111133001 e24KC2ThetaBelowLeaf111133002
+      e24KC2ThetaBelowLeaf111133003
+
+theorem e24KC2ThetaBelowNode11113301 :
+    adaptiveCoverCheck 10 thetaBelowCell11113301 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113301
+    e24KC2ThetaBelowLeaf111133010 e24KC2ThetaBelowLeaf111133011 e24KC2ThetaBelowLeaf111133012
+      e24KC2ThetaBelowLeaf111133013
+
+theorem e24KC2ThetaBelowNode11113302 :
+    adaptiveCoverCheck 10 thetaBelowCell11113302 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113302
+    e24KC2ThetaBelowLeaf111133020 e24KC2ThetaBelowLeaf111133021 e24KC2ThetaBelowLeaf111133022
+      e24KC2ThetaBelowLeaf111133023
+
+theorem e24KC2ThetaBelowNode11113303 :
+    adaptiveCoverCheck 10 thetaBelowCell11113303 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113303
+    e24KC2ThetaBelowLeaf111133030 e24KC2ThetaBelowLeaf111133031 e24KC2ThetaBelowLeaf111133032
+      e24KC2ThetaBelowLeaf111133033
+
+theorem e24KC2ThetaBelowNode11113310 :
+    adaptiveCoverCheck 10 thetaBelowCell11113310 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113310
+    e24KC2ThetaBelowLeaf111133100 e24KC2ThetaBelowLeaf111133101 e24KC2ThetaBelowLeaf111133102
+      e24KC2ThetaBelowLeaf111133103
+
+theorem e24KC2ThetaBelowNode11113311 :
+    adaptiveCoverCheck 10 thetaBelowCell11113311 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113311
+    e24KC2ThetaBelowLeaf111133110 e24KC2ThetaBelowLeaf111133111 e24KC2ThetaBelowLeaf111133112
+      e24KC2ThetaBelowLeaf111133113
+
+theorem e24KC2ThetaBelowNode11113312 :
+    adaptiveCoverCheck 10 thetaBelowCell11113312 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113312
+    e24KC2ThetaBelowLeaf111133120 e24KC2ThetaBelowLeaf111133121 e24KC2ThetaBelowLeaf111133122
+      e24KC2ThetaBelowLeaf111133123
+
+theorem e24KC2ThetaBelowNode11113313 :
+    adaptiveCoverCheck 10 thetaBelowCell11113313 = true :=
+  adaptiveCoverCheck_succ_of_children 9 thetaBelowCell11113313
+    e24KC2ThetaBelowLeaf111133130 e24KC2ThetaBelowLeaf111133131 e24KC2ThetaBelowLeaf111133132
+      e24KC2ThetaBelowLeaf111133133
+
+theorem e24KC2ThetaBelowNode1011303 :
+    adaptiveCoverCheck 11 (childHH (childLL (childHH thetaBelowCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLL (childHH thetaBelowCell1011)))
+    e24KC2ThetaBelowLeaf10113030 e24KC2ThetaBelowLeaf10113031 e24KC2ThetaBelowLeaf10113032
+      e24KC2ThetaBelowLeaf10113033
+
+theorem e24KC2ThetaBelowNode1011311 :
+    adaptiveCoverCheck 11 (childLH (childLH (childHH thetaBelowCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLH (childHH thetaBelowCell1011)))
+    e24KC2ThetaBelowLeaf10113110 e24KC2ThetaBelowLeaf10113111 e24KC2ThetaBelowLeaf10113112
+      e24KC2ThetaBelowLeaf10113113
+
+theorem e24KC2ThetaBelowNode1011312 :
+    adaptiveCoverCheck 11 (childHL (childLH (childHH thetaBelowCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLH (childHH thetaBelowCell1011)))
+    e24KC2ThetaBelowLeaf10113120 e24KC2ThetaBelowLeaf10113121 e24KC2ThetaBelowLeaf10113122
+      e24KC2ThetaBelowLeaf10113123
+
+theorem e24KC2ThetaBelowNode1011313 :
+    adaptiveCoverCheck 11 (childHH (childLH (childHH thetaBelowCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLH (childHH thetaBelowCell1011)))
+    e24KC2ThetaBelowLeaf10113130 e24KC2ThetaBelowLeaf10113131 e24KC2ThetaBelowLeaf10113132
+      e24KC2ThetaBelowLeaf10113133
+
+theorem e24KC2ThetaBelowNode1100032 :
+    adaptiveCoverCheck 11 (childHL (childHH (childLL thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childHH (childLL thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11000320 e24KC2ThetaBelowLeaf11000321 e24KC2ThetaBelowLeaf11000322
+      e24KC2ThetaBelowLeaf11000323
+
+theorem e24KC2ThetaBelowNode1100033 :
+    adaptiveCoverCheck 11 (childHH (childHH (childLL thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childHH (childLL thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11000330 e24KC2ThetaBelowLeaf11000331 e24KC2ThetaBelowLeaf11000332
+      e24KC2ThetaBelowLeaf11000333
+
+theorem e24KC2ThetaBelowNode1100122 :
+    adaptiveCoverCheck 11 (childHL (childHL (childLH thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childHL (childLH thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11001220 e24KC2ThetaBelowLeaf11001221 e24KC2ThetaBelowLeaf11001222
+      e24KC2ThetaBelowLeaf11001223
+
+theorem e24KC2ThetaBelowNode1100123 :
+    adaptiveCoverCheck 11 (childHH (childHL (childLH thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childHL (childLH thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11001230 e24KC2ThetaBelowLeaf11001231 e24KC2ThetaBelowLeaf11001232
+      e24KC2ThetaBelowLeaf11001233
+
+theorem e24KC2ThetaBelowNode1100132 :
+    adaptiveCoverCheck 11 (childHL (childHH (childLH thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childHH (childLH thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11001320 e24KC2ThetaBelowLeaf11001321 e24KC2ThetaBelowLeaf11001322
+      e24KC2ThetaBelowLeaf11001323
+
+theorem e24KC2ThetaBelowNode1100133 :
+    adaptiveCoverCheck 11 (childHH (childHH (childLH thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childHH (childLH thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11001330 e24KC2ThetaBelowLeaf11001331 e24KC2ThetaBelowLeaf11001332
+      e24KC2ThetaBelowLeaf11001333
+
+theorem e24KC2ThetaBelowNode1100200 :
+    adaptiveCoverCheck 11 (childLL (childLL (childHL thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLL (childHL thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11002000 e24KC2ThetaBelowLeaf11002001 e24KC2ThetaBelowLeaf11002002
+      e24KC2ThetaBelowLeaf11002003
+
+theorem e24KC2ThetaBelowNode1100201 :
+    adaptiveCoverCheck 11 (childLH (childLL (childHL thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLL (childHL thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11002010 e24KC2ThetaBelowLeaf11002011 e24KC2ThetaBelowLeaf11002012
+      e24KC2ThetaBelowLeaf11002013
+
+theorem e24KC2ThetaBelowNode1100202 :
+    adaptiveCoverCheck 11 (childHL (childLL (childHL thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLL (childHL thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11002020 e24KC2ThetaBelowLeaf11002021 e24KC2ThetaBelowLeaf11002022
+      e24KC2ThetaBelowLeaf11002023
+
+theorem e24KC2ThetaBelowNode1100203 :
+    adaptiveCoverCheck 11 (childHH (childLL (childHL thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLL (childHL thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11002030 e24KC2ThetaBelowLeaf11002031 e24KC2ThetaBelowLeaf11002032
+      e24KC2ThetaBelowLeaf11002033
+
+theorem e24KC2ThetaBelowNode1100210 :
+    adaptiveCoverCheck 11 (childLL (childLH (childHL thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLH (childHL thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11002100 e24KC2ThetaBelowLeaf11002101 e24KC2ThetaBelowLeaf11002102
+      e24KC2ThetaBelowLeaf11002103
+
+theorem e24KC2ThetaBelowNode1100211 :
+    adaptiveCoverCheck 11 (childLH (childLH (childHL thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLH (childHL thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11002110 e24KC2ThetaBelowLeaf11002111 e24KC2ThetaBelowLeaf11002112
+      e24KC2ThetaBelowLeaf11002113
+
+theorem e24KC2ThetaBelowNode1100212 :
+    adaptiveCoverCheck 11 (childHL (childLH (childHL thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLH (childHL thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11002120 e24KC2ThetaBelowLeaf11002121 e24KC2ThetaBelowLeaf11002122
+      e24KC2ThetaBelowLeaf11002123
+
+theorem e24KC2ThetaBelowNode1100213 :
+    adaptiveCoverCheck 11 (childHH (childLH (childHL thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLH (childHL thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11002130 e24KC2ThetaBelowLeaf11002131 e24KC2ThetaBelowLeaf11002132
+      e24KC2ThetaBelowLeaf11002133
+
+theorem e24KC2ThetaBelowNode1100300 :
+    adaptiveCoverCheck 11 (childLL (childLL (childHH thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLL (childHH thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11003000 e24KC2ThetaBelowLeaf11003001 e24KC2ThetaBelowLeaf11003002
+      e24KC2ThetaBelowLeaf11003003
+
+theorem e24KC2ThetaBelowNode1100301 :
+    adaptiveCoverCheck 11 (childLH (childLL (childHH thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLL (childHH thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11003010 e24KC2ThetaBelowLeaf11003011 e24KC2ThetaBelowLeaf11003012
+      e24KC2ThetaBelowLeaf11003013
+
+theorem e24KC2ThetaBelowNode1100302 :
+    adaptiveCoverCheck 11 (childHL (childLL (childHH thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLL (childHH thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11003020 e24KC2ThetaBelowLeaf11003021 e24KC2ThetaBelowLeaf11003022
+      e24KC2ThetaBelowLeaf11003023
+
+theorem e24KC2ThetaBelowNode1100303 :
+    adaptiveCoverCheck 11 (childHH (childLL (childHH thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLL (childHH thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11003030 e24KC2ThetaBelowLeaf11003031 e24KC2ThetaBelowLeaf11003032
+      e24KC2ThetaBelowLeaf11003033
+
+theorem e24KC2ThetaBelowNode1100310 :
+    adaptiveCoverCheck 11 (childLL (childLH (childHH thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLH (childHH thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11003100 e24KC2ThetaBelowLeaf11003101 e24KC2ThetaBelowLeaf11003102
+      e24KC2ThetaBelowLeaf11003103
+
+theorem e24KC2ThetaBelowNode1100311 :
+    adaptiveCoverCheck 11 (childLH (childLH (childHH thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLH (childHH thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11003110 e24KC2ThetaBelowLeaf11003111 e24KC2ThetaBelowLeaf11003112
+      e24KC2ThetaBelowLeaf11003113
+
+theorem e24KC2ThetaBelowNode1100312 :
+    adaptiveCoverCheck 11 (childHL (childLH (childHH thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLH (childHH thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11003120 e24KC2ThetaBelowLeaf11003121 e24KC2ThetaBelowLeaf11003122
+      e24KC2ThetaBelowLeaf11003123
+
+theorem e24KC2ThetaBelowNode1100313 :
+    adaptiveCoverCheck 11 (childHH (childLH (childHH thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLH (childHH thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11003130 e24KC2ThetaBelowLeaf11003131 e24KC2ThetaBelowLeaf11003132
+      e24KC2ThetaBelowLeaf11003133
+
+theorem e24KC2ThetaBelowNode1100330 :
+    adaptiveCoverCheck 11 (childLL (childHH (childHH thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childHH (childHH thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11003300 e24KC2ThetaBelowLeaf11003301 e24KC2ThetaBelowLeaf11003302
+      e24KC2ThetaBelowLeaf11003303
+
+theorem e24KC2ThetaBelowNode1100331 :
+    adaptiveCoverCheck 11 (childLH (childHH (childHH thetaBelowCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childHH (childHH thetaBelowCell1100)))
+    e24KC2ThetaBelowLeaf11003310 e24KC2ThetaBelowLeaf11003311 e24KC2ThetaBelowLeaf11003312
+      e24KC2ThetaBelowLeaf11003313
+
+theorem e24KC2ThetaBelowNode1101022 :
+    adaptiveCoverCheck 11 (childHL (childHL (childLL thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childHL (childLL thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11010220 e24KC2ThetaBelowLeaf11010221 e24KC2ThetaBelowLeaf11010222
+      e24KC2ThetaBelowLeaf11010223
+
+theorem e24KC2ThetaBelowNode1101023 :
+    adaptiveCoverCheck 11 (childHH (childHL (childLL thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childHL (childLL thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11010230 e24KC2ThetaBelowLeaf11010231 e24KC2ThetaBelowLeaf11010232
+      e24KC2ThetaBelowLeaf11010233
+
+theorem e24KC2ThetaBelowNode1101200 :
+    adaptiveCoverCheck 11 (childLL (childLL (childHL thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLL (childHL thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11012000 e24KC2ThetaBelowLeaf11012001 e24KC2ThetaBelowLeaf11012002
+      e24KC2ThetaBelowLeaf11012003
+
+theorem e24KC2ThetaBelowNode1101201 :
+    adaptiveCoverCheck 11 (childLH (childLL (childHL thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLL (childHL thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11012010 e24KC2ThetaBelowLeaf11012011 e24KC2ThetaBelowLeaf11012012
+      e24KC2ThetaBelowLeaf11012013
+
+theorem e24KC2ThetaBelowNode1101202 :
+    adaptiveCoverCheck 11 (childHL (childLL (childHL thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLL (childHL thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11012020 e24KC2ThetaBelowLeaf11012021 e24KC2ThetaBelowLeaf11012022
+      e24KC2ThetaBelowLeaf11012023
+
+theorem e24KC2ThetaBelowNode1101203 :
+    adaptiveCoverCheck 11 (childHH (childLL (childHL thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLL (childHL thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11012030 e24KC2ThetaBelowLeaf11012031 e24KC2ThetaBelowLeaf11012032
+      e24KC2ThetaBelowLeaf11012033
+
+theorem e24KC2ThetaBelowNode1101210 :
+    adaptiveCoverCheck 11 (childLL (childLH (childHL thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLH (childHL thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11012100 e24KC2ThetaBelowLeaf11012101 e24KC2ThetaBelowLeaf11012102
+      e24KC2ThetaBelowLeaf11012103
+
+theorem e24KC2ThetaBelowNode1101211 :
+    adaptiveCoverCheck 11 (childLH (childLH (childHL thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLH (childHL thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11012110 e24KC2ThetaBelowLeaf11012111 e24KC2ThetaBelowLeaf11012112
+      e24KC2ThetaBelowLeaf11012113
+
+theorem e24KC2ThetaBelowNode1101212 :
+    adaptiveCoverCheck 11 (childHL (childLH (childHL thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLH (childHL thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11012120 e24KC2ThetaBelowLeaf11012121 e24KC2ThetaBelowLeaf11012122
+      e24KC2ThetaBelowLeaf11012123
+
+theorem e24KC2ThetaBelowNode1101213 :
+    adaptiveCoverCheck 11 (childHH (childLH (childHL thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLH (childHL thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11012130 e24KC2ThetaBelowLeaf11012131 e24KC2ThetaBelowNode11012132
+      e24KC2ThetaBelowNode11012133
+
+theorem e24KC2ThetaBelowNode1101220 :
+    adaptiveCoverCheck 11 (childLL (childHL (childHL thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childHL (childHL thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11012200 e24KC2ThetaBelowLeaf11012201 e24KC2ThetaBelowLeaf11012202
+      e24KC2ThetaBelowLeaf11012203
+
+theorem e24KC2ThetaBelowNode1101221 :
+    adaptiveCoverCheck 11 (childLH (childHL (childHL thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childHL (childHL thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11012210 e24KC2ThetaBelowLeaf11012211 e24KC2ThetaBelowLeaf11012212
+      e24KC2ThetaBelowLeaf11012213
+
+theorem e24KC2ThetaBelowNode1101230 :
+    adaptiveCoverCheck 11 (childLL (childHH (childHL thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childHH (childHL thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11012300 e24KC2ThetaBelowLeaf11012301 e24KC2ThetaBelowLeaf11012302
+      e24KC2ThetaBelowLeaf11012303
+
+theorem e24KC2ThetaBelowNode1101231 :
+    adaptiveCoverCheck 11 (childLH (childHH (childHL thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childHH (childHL thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11012310 e24KC2ThetaBelowLeaf11012311 e24KC2ThetaBelowLeaf11012312
+      e24KC2ThetaBelowLeaf11012313
+
+theorem e24KC2ThetaBelowNode1101300 :
+    adaptiveCoverCheck 11 (childLL (childLL (childHH thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLL (childHH thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11013000 e24KC2ThetaBelowLeaf11013001 e24KC2ThetaBelowLeaf11013002
+      e24KC2ThetaBelowLeaf11013003
+
+theorem e24KC2ThetaBelowNode1101301 :
+    adaptiveCoverCheck 11 (childLH (childLL (childHH thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLL (childHH thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11013010 e24KC2ThetaBelowLeaf11013011 e24KC2ThetaBelowLeaf11013012
+      e24KC2ThetaBelowNode11013013
+
+theorem e24KC2ThetaBelowNode1101302 :
+    adaptiveCoverCheck 11 (childHL (childLL (childHH thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLL (childHH thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11013020 e24KC2ThetaBelowNode11013021 e24KC2ThetaBelowNode11013022
+      e24KC2ThetaBelowNode11013023
+
+theorem e24KC2ThetaBelowNode1101303 :
+    adaptiveCoverCheck 11 (childHH (childLL (childHH thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLL (childHH thetaBelowCell1101)))
+    e24KC2ThetaBelowNode11013030 e24KC2ThetaBelowNode11013031 e24KC2ThetaBelowNode11013032
+      e24KC2ThetaBelowNode11013033
+
+theorem e24KC2ThetaBelowNode1101310 :
+    adaptiveCoverCheck 11 (childLL (childLH (childHH thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLH (childHH thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11013100 e24KC2ThetaBelowLeaf11013101 e24KC2ThetaBelowNode11013102
+      e24KC2ThetaBelowNode11013103
+
+theorem e24KC2ThetaBelowNode1101311 :
+    adaptiveCoverCheck 11 (childLH (childLH (childHH thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLH (childHH thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11013110 e24KC2ThetaBelowLeaf11013111 e24KC2ThetaBelowNode11013112
+      e24KC2ThetaBelowNode11013113
+
+theorem e24KC2ThetaBelowNode1101312 :
+    adaptiveCoverCheck 11 (childHL (childLH (childHH thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLH (childHH thetaBelowCell1101)))
+    e24KC2ThetaBelowNode11013120 e24KC2ThetaBelowNode11013121 e24KC2ThetaBelowNode11013122
+      e24KC2ThetaBelowNode11013123
+
+theorem e24KC2ThetaBelowNode1101313 :
+    adaptiveCoverCheck 11 (childHH (childLH (childHH thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLH (childHH thetaBelowCell1101)))
+    e24KC2ThetaBelowNode11013130 e24KC2ThetaBelowNode11013131 e24KC2ThetaBelowNode11013132
+      e24KC2ThetaBelowNode11013133
+
+theorem e24KC2ThetaBelowNode1101320 :
+    adaptiveCoverCheck 11 (childLL (childHL (childHH thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childHL (childHH thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11013200 e24KC2ThetaBelowLeaf11013201 e24KC2ThetaBelowLeaf11013202
+      e24KC2ThetaBelowLeaf11013203
+
+theorem e24KC2ThetaBelowNode1101321 :
+    adaptiveCoverCheck 11 (childLH (childHL (childHH thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childHL (childHH thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11013210 e24KC2ThetaBelowLeaf11013211 e24KC2ThetaBelowLeaf11013212
+      e24KC2ThetaBelowLeaf11013213
+
+theorem e24KC2ThetaBelowNode1101330 :
+    adaptiveCoverCheck 11 (childLL (childHH (childHH thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childHH (childHH thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11013300 e24KC2ThetaBelowLeaf11013301 e24KC2ThetaBelowLeaf11013302
+      e24KC2ThetaBelowLeaf11013303
+
+theorem e24KC2ThetaBelowNode1101331 :
+    adaptiveCoverCheck 11 (childLH (childHH (childHH thetaBelowCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childHH (childHH thetaBelowCell1101)))
+    e24KC2ThetaBelowLeaf11013310 e24KC2ThetaBelowLeaf11013311 e24KC2ThetaBelowLeaf11013312
+      e24KC2ThetaBelowLeaf11013313
+
+theorem e24KC2ThetaBelowNode1110200 :
+    adaptiveCoverCheck 11 (childLL (childLL (childHL thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLL (childHL thetaBelowCell1110)))
+    e24KC2ThetaBelowLeaf11102000 e24KC2ThetaBelowLeaf11102001 e24KC2ThetaBelowNode11102002
+      e24KC2ThetaBelowNode11102003
+
+theorem e24KC2ThetaBelowNode1110201 :
+    adaptiveCoverCheck 11 (childLH (childLL (childHL thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLL (childHL thetaBelowCell1110)))
+    e24KC2ThetaBelowLeaf11102010 e24KC2ThetaBelowLeaf11102011 e24KC2ThetaBelowNode11102012
+      e24KC2ThetaBelowNode11102013
+
+theorem e24KC2ThetaBelowNode1110202 :
+    adaptiveCoverCheck 11 (childHL (childLL (childHL thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLL (childHL thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11102020 e24KC2ThetaBelowNode11102021 e24KC2ThetaBelowNode11102022
+      e24KC2ThetaBelowNode11102023
+
+theorem e24KC2ThetaBelowNode1110203 :
+    adaptiveCoverCheck 11 (childHH (childLL (childHL thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLL (childHL thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11102030 e24KC2ThetaBelowNode11102031 e24KC2ThetaBelowNode11102032
+      e24KC2ThetaBelowNode11102033
+
+theorem e24KC2ThetaBelowNode1110210 :
+    adaptiveCoverCheck 11 (childLL (childLH (childHL thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLH (childHL thetaBelowCell1110)))
+    e24KC2ThetaBelowLeaf11102100 e24KC2ThetaBelowLeaf11102101 e24KC2ThetaBelowLeaf11102102
+      e24KC2ThetaBelowLeaf11102103
+
+theorem e24KC2ThetaBelowNode1110211 :
+    adaptiveCoverCheck 11 (childLH (childLH (childHL thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLH (childHL thetaBelowCell1110)))
+    e24KC2ThetaBelowLeaf11102110 e24KC2ThetaBelowLeaf11102111 e24KC2ThetaBelowLeaf11102112
+      e24KC2ThetaBelowLeaf11102113
+
+theorem e24KC2ThetaBelowNode1110212 :
+    adaptiveCoverCheck 11 (childHL (childLH (childHL thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLH (childHL thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11102120 e24KC2ThetaBelowNode11102121 e24KC2ThetaBelowNode11102122
+      e24KC2ThetaBelowNode11102123
+
+theorem e24KC2ThetaBelowNode1110213 :
+    adaptiveCoverCheck 11 (childHH (childLH (childHL thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLH (childHL thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11102130 e24KC2ThetaBelowNode11102131 e24KC2ThetaBelowNode11102132
+      e24KC2ThetaBelowNode11102133
+
+theorem e24KC2ThetaBelowNode1110220 :
+    adaptiveCoverCheck 11 (childLL (childHL (childHL thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childHL (childHL thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11102200 e24KC2ThetaBelowNode11102201 e24KC2ThetaBelowLeaf11102202
+      e24KC2ThetaBelowLeaf11102203
+
+theorem e24KC2ThetaBelowNode1110221 :
+    adaptiveCoverCheck 11 (childLH (childHL (childHL thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childHL (childHL thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11102210 e24KC2ThetaBelowNode11102211 e24KC2ThetaBelowLeaf11102212
+      e24KC2ThetaBelowLeaf11102213
+
+theorem e24KC2ThetaBelowNode1110230 :
+    adaptiveCoverCheck 11 (childLL (childHH (childHL thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childHH (childHL thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11102300 e24KC2ThetaBelowNode11102301 e24KC2ThetaBelowLeaf11102302
+      e24KC2ThetaBelowLeaf11102303
+
+theorem e24KC2ThetaBelowNode1110231 :
+    adaptiveCoverCheck 11 (childLH (childHH (childHL thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childHH (childHL thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11102310 e24KC2ThetaBelowNode11102311 e24KC2ThetaBelowLeaf11102312
+      e24KC2ThetaBelowLeaf11102313
+
+theorem e24KC2ThetaBelowNode1110300 :
+    adaptiveCoverCheck 11 (childLL (childLL (childHH thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLL (childHH thetaBelowCell1110)))
+    e24KC2ThetaBelowLeaf11103000 e24KC2ThetaBelowLeaf11103001 e24KC2ThetaBelowLeaf11103002
+      e24KC2ThetaBelowLeaf11103003
+
+theorem e24KC2ThetaBelowNode1110301 :
+    adaptiveCoverCheck 11 (childLH (childLL (childHH thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLL (childHH thetaBelowCell1110)))
+    e24KC2ThetaBelowLeaf11103010 e24KC2ThetaBelowLeaf11103011 e24KC2ThetaBelowLeaf11103012
+      e24KC2ThetaBelowLeaf11103013
+
+theorem e24KC2ThetaBelowNode1110302 :
+    adaptiveCoverCheck 11 (childHL (childLL (childHH thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLL (childHH thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11103020 e24KC2ThetaBelowNode11103021 e24KC2ThetaBelowNode11103022
+      e24KC2ThetaBelowNode11103023
+
+theorem e24KC2ThetaBelowNode1110303 :
+    adaptiveCoverCheck 11 (childHH (childLL (childHH thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLL (childHH thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11103030 e24KC2ThetaBelowNode11103031 e24KC2ThetaBelowNode11103032
+      e24KC2ThetaBelowNode11103033
+
+theorem e24KC2ThetaBelowNode1110310 :
+    adaptiveCoverCheck 11 (childLL (childLH (childHH thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLH (childHH thetaBelowCell1110)))
+    e24KC2ThetaBelowLeaf11103100 e24KC2ThetaBelowLeaf11103101 e24KC2ThetaBelowLeaf11103102
+      e24KC2ThetaBelowLeaf11103103
+
+theorem e24KC2ThetaBelowNode1110311 :
+    adaptiveCoverCheck 11 (childLH (childLH (childHH thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLH (childHH thetaBelowCell1110)))
+    e24KC2ThetaBelowLeaf11103110 e24KC2ThetaBelowLeaf11103111 e24KC2ThetaBelowLeaf11103112
+      e24KC2ThetaBelowLeaf11103113
+
+theorem e24KC2ThetaBelowNode1110312 :
+    adaptiveCoverCheck 11 (childHL (childLH (childHH thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLH (childHH thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11103120 e24KC2ThetaBelowNode11103121 e24KC2ThetaBelowNode11103122
+      e24KC2ThetaBelowNode11103123
+
+theorem e24KC2ThetaBelowNode1110313 :
+    adaptiveCoverCheck 11 (childHH (childLH (childHH thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLH (childHH thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11103130 e24KC2ThetaBelowNode11103131 e24KC2ThetaBelowNode11103132
+      e24KC2ThetaBelowNode11103133
+
+theorem e24KC2ThetaBelowNode1110320 :
+    adaptiveCoverCheck 11 (childLL (childHL (childHH thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childHL (childHH thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11103200 e24KC2ThetaBelowNode11103201 e24KC2ThetaBelowLeaf11103202
+      e24KC2ThetaBelowLeaf11103203
+
+theorem e24KC2ThetaBelowNode1110321 :
+    adaptiveCoverCheck 11 (childLH (childHL (childHH thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childHL (childHH thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11103210 e24KC2ThetaBelowNode11103211 e24KC2ThetaBelowLeaf11103212
+      e24KC2ThetaBelowLeaf11103213
+
+theorem e24KC2ThetaBelowNode1110330 :
+    adaptiveCoverCheck 11 (childLL (childHH (childHH thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childHH (childHH thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11103300 e24KC2ThetaBelowNode11103301 e24KC2ThetaBelowLeaf11103302
+      e24KC2ThetaBelowLeaf11103303
+
+theorem e24KC2ThetaBelowNode1110331 :
+    adaptiveCoverCheck 11 (childLH (childHH (childHH thetaBelowCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childHH (childHH thetaBelowCell1110)))
+    e24KC2ThetaBelowNode11103310 e24KC2ThetaBelowNode11103311 e24KC2ThetaBelowLeaf11103312
+      e24KC2ThetaBelowLeaf11103313
+
+theorem e24KC2ThetaBelowNode1111200 :
+    adaptiveCoverCheck 11 (childLL (childLL (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLL (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11112000 e24KC2ThetaBelowLeaf11112001 e24KC2ThetaBelowLeaf11112002
+      e24KC2ThetaBelowLeaf11112003
+
+theorem e24KC2ThetaBelowNode1111201 :
+    adaptiveCoverCheck 11 (childLH (childLL (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLL (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11112010 e24KC2ThetaBelowLeaf11112011 e24KC2ThetaBelowLeaf11112012
+      e24KC2ThetaBelowLeaf11112013
+
+theorem e24KC2ThetaBelowNode1111202 :
+    adaptiveCoverCheck 11 (childHL (childLL (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLL (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowNode11112020 e24KC2ThetaBelowNode11112021 e24KC2ThetaBelowNode11112022
+      e24KC2ThetaBelowNode11112023
+
+theorem e24KC2ThetaBelowNode1111203 :
+    adaptiveCoverCheck 11 (childHH (childLL (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLL (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowNode11112030 e24KC2ThetaBelowNode11112031 e24KC2ThetaBelowNode11112032
+      e24KC2ThetaBelowNode11112033
+
+theorem e24KC2ThetaBelowNode1111210 :
+    adaptiveCoverCheck 11 (childLL (childLH (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLH (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11112100 e24KC2ThetaBelowLeaf11112101 e24KC2ThetaBelowLeaf11112102
+      e24KC2ThetaBelowLeaf11112103
+
+theorem e24KC2ThetaBelowNode1111211 :
+    adaptiveCoverCheck 11 (childLH (childLH (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLH (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11112110 e24KC2ThetaBelowLeaf11112111 e24KC2ThetaBelowLeaf11112112
+      e24KC2ThetaBelowLeaf11112113
+
+theorem e24KC2ThetaBelowNode1111212 :
+    adaptiveCoverCheck 11 (childHL (childLH (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLH (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowNode11112120 e24KC2ThetaBelowNode11112121 e24KC2ThetaBelowNode11112122
+      e24KC2ThetaBelowNode11112123
+
+theorem e24KC2ThetaBelowNode1111213 :
+    adaptiveCoverCheck 11 (childHH (childLH (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLH (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowNode11112130 e24KC2ThetaBelowLeaf11112131 e24KC2ThetaBelowNode11112132
+      e24KC2ThetaBelowNode11112133
+
+theorem e24KC2ThetaBelowNode1111220 :
+    adaptiveCoverCheck 11 (childLL (childHL (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childHL (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowNode11112200 e24KC2ThetaBelowNode11112201 e24KC2ThetaBelowLeaf11112202
+      e24KC2ThetaBelowLeaf11112203
+
+theorem e24KC2ThetaBelowNode1111221 :
+    adaptiveCoverCheck 11 (childLH (childHL (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childHL (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowNode11112210 e24KC2ThetaBelowNode11112211 e24KC2ThetaBelowNode11112212
+      e24KC2ThetaBelowNode11112213
+
+theorem e24KC2ThetaBelowNode1111222 :
+    adaptiveCoverCheck 11 (childHL (childHL (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childHL (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11112220 e24KC2ThetaBelowLeaf11112221 e24KC2ThetaBelowLeaf11112222
+      e24KC2ThetaBelowLeaf11112223
+
+theorem e24KC2ThetaBelowNode1111223 :
+    adaptiveCoverCheck 11 (childHH (childHL (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childHL (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11112230 e24KC2ThetaBelowLeaf11112231 e24KC2ThetaBelowLeaf11112232
+      e24KC2ThetaBelowLeaf11112233
+
+theorem e24KC2ThetaBelowNode1111230 :
+    adaptiveCoverCheck 11 (childLL (childHH (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childHH (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowNode11112300 e24KC2ThetaBelowNode11112301 e24KC2ThetaBelowNode11112302
+      e24KC2ThetaBelowNode11112303
+
+theorem e24KC2ThetaBelowNode1111231 :
+    adaptiveCoverCheck 11 (childLH (childHH (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childHH (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowNode11112310 e24KC2ThetaBelowNode11112311 e24KC2ThetaBelowNode11112312
+      e24KC2ThetaBelowNode11112313
+
+theorem e24KC2ThetaBelowNode1111232 :
+    adaptiveCoverCheck 11 (childHL (childHH (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childHH (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11112320 e24KC2ThetaBelowLeaf11112321 e24KC2ThetaBelowLeaf11112322
+      e24KC2ThetaBelowLeaf11112323
+
+theorem e24KC2ThetaBelowNode1111233 :
+    adaptiveCoverCheck 11 (childHH (childHH (childHL thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childHH (childHL thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11112330 e24KC2ThetaBelowLeaf11112331 e24KC2ThetaBelowLeaf11112332
+      e24KC2ThetaBelowLeaf11112333
+
+theorem e24KC2ThetaBelowNode1111302 :
+    adaptiveCoverCheck 11 (childHL (childLL (childHH thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLL (childHH thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11113020 e24KC2ThetaBelowLeaf11113021 e24KC2ThetaBelowNode11113022
+      e24KC2ThetaBelowNode11113023
+
+theorem e24KC2ThetaBelowNode1111303 :
+    adaptiveCoverCheck 11 (childHH (childLL (childHH thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLL (childHH thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11113030 e24KC2ThetaBelowLeaf11113031 e24KC2ThetaBelowNode11113032
+      e24KC2ThetaBelowNode11113033
+
+theorem e24KC2ThetaBelowNode1111312 :
+    adaptiveCoverCheck 11 (childHL (childLH (childHH thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLH (childHH thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11113120 e24KC2ThetaBelowLeaf11113121 e24KC2ThetaBelowNode11113122
+      e24KC2ThetaBelowNode11113123
+
+theorem e24KC2ThetaBelowNode1111313 :
+    adaptiveCoverCheck 11 (childHH (childLH (childHH thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLH (childHH thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11113130 e24KC2ThetaBelowLeaf11113131 e24KC2ThetaBelowNode11113132
+      e24KC2ThetaBelowNode11113133
+
+theorem e24KC2ThetaBelowNode1111320 :
+    adaptiveCoverCheck 11 (childLL (childHL (childHH thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childHL (childHH thetaBelowCell1111)))
+    e24KC2ThetaBelowNode11113200 e24KC2ThetaBelowNode11113201 e24KC2ThetaBelowNode11113202
+      e24KC2ThetaBelowNode11113203
+
+theorem e24KC2ThetaBelowNode1111321 :
+    adaptiveCoverCheck 11 (childLH (childHL (childHH thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childHL (childHH thetaBelowCell1111)))
+    e24KC2ThetaBelowNode11113210 e24KC2ThetaBelowNode11113211 e24KC2ThetaBelowNode11113212
+      e24KC2ThetaBelowNode11113213
+
+theorem e24KC2ThetaBelowNode1111322 :
+    adaptiveCoverCheck 11 (childHL (childHL (childHH thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childHL (childHH thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11113220 e24KC2ThetaBelowLeaf11113221 e24KC2ThetaBelowLeaf11113222
+      e24KC2ThetaBelowLeaf11113223
+
+theorem e24KC2ThetaBelowNode1111323 :
+    adaptiveCoverCheck 11 (childHH (childHL (childHH thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childHL (childHH thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11113230 e24KC2ThetaBelowLeaf11113231 e24KC2ThetaBelowLeaf11113232
+      e24KC2ThetaBelowLeaf11113233
+
+theorem e24KC2ThetaBelowNode1111330 :
+    adaptiveCoverCheck 11 (childLL (childHH (childHH thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childHH (childHH thetaBelowCell1111)))
+    e24KC2ThetaBelowNode11113300 e24KC2ThetaBelowNode11113301 e24KC2ThetaBelowNode11113302
+      e24KC2ThetaBelowNode11113303
+
+theorem e24KC2ThetaBelowNode1111331 :
+    adaptiveCoverCheck 11 (childLH (childHH (childHH thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childHH (childHH thetaBelowCell1111)))
+    e24KC2ThetaBelowNode11113310 e24KC2ThetaBelowNode11113311 e24KC2ThetaBelowNode11113312
+      e24KC2ThetaBelowNode11113313
+
+theorem e24KC2ThetaBelowNode1111332 :
+    adaptiveCoverCheck 11 (childHL (childHH (childHH thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childHH (childHH thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11113320 e24KC2ThetaBelowLeaf11113321 e24KC2ThetaBelowLeaf11113322
+      e24KC2ThetaBelowLeaf11113323
+
+theorem e24KC2ThetaBelowNode1111333 :
+    adaptiveCoverCheck 11 (childHH (childHH (childHH thetaBelowCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childHH (childHH thetaBelowCell1111)))
+    e24KC2ThetaBelowLeaf11113330 e24KC2ThetaBelowLeaf11113331 e24KC2ThetaBelowLeaf11113332
+      e24KC2ThetaBelowLeaf11113333
+
+theorem e24KC2ThetaBelowNode100113 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell1001)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH thetaBelowCell1001))
+    e24KC2ThetaBelowLeaf1001130 e24KC2ThetaBelowLeaf1001131 e24KC2ThetaBelowLeaf1001132
+      e24KC2ThetaBelowLeaf1001133
+
+theorem e24KC2ThetaBelowNode100121 :
+    adaptiveCoverCheck 12 (childLH (childHL thetaBelowCell1001)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL thetaBelowCell1001))
+    e24KC2ThetaBelowLeaf1001210 e24KC2ThetaBelowLeaf1001211 e24KC2ThetaBelowLeaf1001212
+      e24KC2ThetaBelowLeaf1001213
+
+theorem e24KC2ThetaBelowNode100130 :
+    adaptiveCoverCheck 12 (childLL (childHH thetaBelowCell1001)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH thetaBelowCell1001))
+    e24KC2ThetaBelowLeaf1001300 e24KC2ThetaBelowLeaf1001301 e24KC2ThetaBelowLeaf1001302
+      e24KC2ThetaBelowLeaf1001303
+
+theorem e24KC2ThetaBelowNode100131 :
+    adaptiveCoverCheck 12 (childLH (childHH thetaBelowCell1001)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH thetaBelowCell1001))
+    e24KC2ThetaBelowLeaf1001310 e24KC2ThetaBelowLeaf1001311 e24KC2ThetaBelowLeaf1001312
+      e24KC2ThetaBelowLeaf1001313
+
+theorem e24KC2ThetaBelowNode101001 :
+    adaptiveCoverCheck 12 (childLH (childLL thetaBelowCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childLL thetaBelowCell1010))
+    e24KC2ThetaBelowLeaf1010010 e24KC2ThetaBelowLeaf1010011 e24KC2ThetaBelowLeaf1010012
+      e24KC2ThetaBelowLeaf1010013
+
+theorem e24KC2ThetaBelowNode101002 :
+    adaptiveCoverCheck 12 (childHL (childLL thetaBelowCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL thetaBelowCell1010))
+    e24KC2ThetaBelowLeaf1010020 e24KC2ThetaBelowLeaf1010021 e24KC2ThetaBelowLeaf1010022
+      e24KC2ThetaBelowLeaf1010023
+
+theorem e24KC2ThetaBelowNode101003 :
+    adaptiveCoverCheck 12 (childHH (childLL thetaBelowCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL thetaBelowCell1010))
+    e24KC2ThetaBelowLeaf1010030 e24KC2ThetaBelowLeaf1010031 e24KC2ThetaBelowLeaf1010032
+      e24KC2ThetaBelowLeaf1010033
+
+theorem e24KC2ThetaBelowNode101010 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childLH thetaBelowCell1010))
+    e24KC2ThetaBelowLeaf1010100 e24KC2ThetaBelowLeaf1010101 e24KC2ThetaBelowLeaf1010102
+      e24KC2ThetaBelowLeaf1010103
+
+theorem e24KC2ThetaBelowNode101011 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childLH thetaBelowCell1010))
+    e24KC2ThetaBelowLeaf1010110 e24KC2ThetaBelowLeaf1010111 e24KC2ThetaBelowLeaf1010112
+      e24KC2ThetaBelowLeaf1010113
+
+theorem e24KC2ThetaBelowNode101012 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH thetaBelowCell1010))
+    e24KC2ThetaBelowLeaf1010120 e24KC2ThetaBelowLeaf1010121 e24KC2ThetaBelowLeaf1010122
+      e24KC2ThetaBelowLeaf1010123
+
+theorem e24KC2ThetaBelowNode101013 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH thetaBelowCell1010))
+    e24KC2ThetaBelowLeaf1010130 e24KC2ThetaBelowLeaf1010131 e24KC2ThetaBelowLeaf1010132
+      e24KC2ThetaBelowLeaf1010133
+
+theorem e24KC2ThetaBelowNode101020 :
+    adaptiveCoverCheck 12 (childLL (childHL thetaBelowCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL thetaBelowCell1010))
+    e24KC2ThetaBelowLeaf1010200 e24KC2ThetaBelowLeaf1010201 e24KC2ThetaBelowLeaf1010202
+      e24KC2ThetaBelowLeaf1010203
+
+theorem e24KC2ThetaBelowNode101021 :
+    adaptiveCoverCheck 12 (childLH (childHL thetaBelowCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL thetaBelowCell1010))
+    e24KC2ThetaBelowLeaf1010210 e24KC2ThetaBelowLeaf1010211 e24KC2ThetaBelowLeaf1010212
+      e24KC2ThetaBelowLeaf1010213
+
+theorem e24KC2ThetaBelowNode101030 :
+    adaptiveCoverCheck 12 (childLL (childHH thetaBelowCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH thetaBelowCell1010))
+    e24KC2ThetaBelowLeaf1010300 e24KC2ThetaBelowLeaf1010301 e24KC2ThetaBelowLeaf1010302
+      e24KC2ThetaBelowLeaf1010303
+
+theorem e24KC2ThetaBelowNode101031 :
+    adaptiveCoverCheck 12 (childLH (childHH thetaBelowCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH thetaBelowCell1010))
+    e24KC2ThetaBelowLeaf1010310 e24KC2ThetaBelowLeaf1010311 e24KC2ThetaBelowLeaf1010312
+      e24KC2ThetaBelowLeaf1010313
+
+theorem e24KC2ThetaBelowNode101032 :
+    adaptiveCoverCheck 12 (childHL (childHH thetaBelowCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH thetaBelowCell1010))
+    e24KC2ThetaBelowLeaf1010320 e24KC2ThetaBelowLeaf1010321 e24KC2ThetaBelowLeaf1010322
+      e24KC2ThetaBelowLeaf1010323
+
+theorem e24KC2ThetaBelowNode101033 :
+    adaptiveCoverCheck 12 (childHH (childHH thetaBelowCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH thetaBelowCell1010))
+    e24KC2ThetaBelowLeaf1010330 e24KC2ThetaBelowLeaf1010331 e24KC2ThetaBelowLeaf1010332
+      e24KC2ThetaBelowLeaf1010333
+
+theorem e24KC2ThetaBelowNode101100 :
+    adaptiveCoverCheck 12 (childLL (childLL thetaBelowCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childLL thetaBelowCell1011))
+    e24KC2ThetaBelowLeaf1011000 e24KC2ThetaBelowLeaf1011001 e24KC2ThetaBelowLeaf1011002
+      e24KC2ThetaBelowLeaf1011003
+
+theorem e24KC2ThetaBelowNode101102 :
+    adaptiveCoverCheck 12 (childHL (childLL thetaBelowCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL thetaBelowCell1011))
+    e24KC2ThetaBelowLeaf1011020 e24KC2ThetaBelowLeaf1011021 e24KC2ThetaBelowLeaf1011022
+      e24KC2ThetaBelowLeaf1011023
+
+theorem e24KC2ThetaBelowNode101103 :
+    adaptiveCoverCheck 12 (childHH (childLL thetaBelowCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL thetaBelowCell1011))
+    e24KC2ThetaBelowLeaf1011030 e24KC2ThetaBelowLeaf1011031 e24KC2ThetaBelowLeaf1011032
+      e24KC2ThetaBelowLeaf1011033
+
+theorem e24KC2ThetaBelowNode101112 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH thetaBelowCell1011))
+    e24KC2ThetaBelowLeaf1011120 e24KC2ThetaBelowLeaf1011121 e24KC2ThetaBelowLeaf1011122
+      e24KC2ThetaBelowLeaf1011123
+
+theorem e24KC2ThetaBelowNode101113 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH thetaBelowCell1011))
+    e24KC2ThetaBelowLeaf1011130 e24KC2ThetaBelowLeaf1011131 e24KC2ThetaBelowLeaf1011132
+      e24KC2ThetaBelowLeaf1011133
+
+theorem e24KC2ThetaBelowNode101120 :
+    adaptiveCoverCheck 12 (childLL (childHL thetaBelowCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL thetaBelowCell1011))
+    e24KC2ThetaBelowLeaf1011200 e24KC2ThetaBelowLeaf1011201 e24KC2ThetaBelowLeaf1011202
+      e24KC2ThetaBelowLeaf1011203
+
+theorem e24KC2ThetaBelowNode101121 :
+    adaptiveCoverCheck 12 (childLH (childHL thetaBelowCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL thetaBelowCell1011))
+    e24KC2ThetaBelowLeaf1011210 e24KC2ThetaBelowLeaf1011211 e24KC2ThetaBelowLeaf1011212
+      e24KC2ThetaBelowLeaf1011213
+
+theorem e24KC2ThetaBelowNode101122 :
+    adaptiveCoverCheck 12 (childHL (childHL thetaBelowCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL thetaBelowCell1011))
+    e24KC2ThetaBelowLeaf1011220 e24KC2ThetaBelowLeaf1011221 e24KC2ThetaBelowLeaf1011222
+      e24KC2ThetaBelowLeaf1011223
+
+theorem e24KC2ThetaBelowNode101123 :
+    adaptiveCoverCheck 12 (childHH (childHL thetaBelowCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL thetaBelowCell1011))
+    e24KC2ThetaBelowLeaf1011230 e24KC2ThetaBelowLeaf1011231 e24KC2ThetaBelowLeaf1011232
+      e24KC2ThetaBelowLeaf1011233
+
+theorem e24KC2ThetaBelowNode101130 :
+    adaptiveCoverCheck 12 (childLL (childHH thetaBelowCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH thetaBelowCell1011))
+    e24KC2ThetaBelowLeaf1011300 e24KC2ThetaBelowLeaf1011301 e24KC2ThetaBelowLeaf1011302
+      e24KC2ThetaBelowNode1011303
+
+theorem e24KC2ThetaBelowNode101131 :
+    adaptiveCoverCheck 12 (childLH (childHH thetaBelowCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH thetaBelowCell1011))
+    e24KC2ThetaBelowLeaf1011310 e24KC2ThetaBelowNode1011311 e24KC2ThetaBelowNode1011312
+      e24KC2ThetaBelowNode1011313
+
+theorem e24KC2ThetaBelowNode101132 :
+    adaptiveCoverCheck 12 (childHL (childHH thetaBelowCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH thetaBelowCell1011))
+    e24KC2ThetaBelowLeaf1011320 e24KC2ThetaBelowLeaf1011321 e24KC2ThetaBelowLeaf1011322
+      e24KC2ThetaBelowLeaf1011323
+
+theorem e24KC2ThetaBelowNode101133 :
+    adaptiveCoverCheck 12 (childHH (childHH thetaBelowCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH thetaBelowCell1011))
+    e24KC2ThetaBelowLeaf1011330 e24KC2ThetaBelowLeaf1011331 e24KC2ThetaBelowLeaf1011332
+      e24KC2ThetaBelowLeaf1011333
+
+theorem e24KC2ThetaBelowNode110002 :
+    adaptiveCoverCheck 12 (childHL (childLL thetaBelowCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL thetaBelowCell1100))
+    e24KC2ThetaBelowLeaf1100020 e24KC2ThetaBelowLeaf1100021 e24KC2ThetaBelowLeaf1100022
+      e24KC2ThetaBelowLeaf1100023
+
+theorem e24KC2ThetaBelowNode110003 :
+    adaptiveCoverCheck 12 (childHH (childLL thetaBelowCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL thetaBelowCell1100))
+    e24KC2ThetaBelowLeaf1100030 e24KC2ThetaBelowLeaf1100031 e24KC2ThetaBelowNode1100032
+      e24KC2ThetaBelowNode1100033
+
+theorem e24KC2ThetaBelowNode110012 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH thetaBelowCell1100))
+    e24KC2ThetaBelowLeaf1100120 e24KC2ThetaBelowLeaf1100121 e24KC2ThetaBelowNode1100122
+      e24KC2ThetaBelowNode1100123
+
+theorem e24KC2ThetaBelowNode110013 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH thetaBelowCell1100))
+    e24KC2ThetaBelowLeaf1100130 e24KC2ThetaBelowLeaf1100131 e24KC2ThetaBelowNode1100132
+      e24KC2ThetaBelowNode1100133
+
+theorem e24KC2ThetaBelowNode110020 :
+    adaptiveCoverCheck 12 (childLL (childHL thetaBelowCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL thetaBelowCell1100))
+    e24KC2ThetaBelowNode1100200 e24KC2ThetaBelowNode1100201 e24KC2ThetaBelowNode1100202
+      e24KC2ThetaBelowNode1100203
+
+theorem e24KC2ThetaBelowNode110021 :
+    adaptiveCoverCheck 12 (childLH (childHL thetaBelowCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL thetaBelowCell1100))
+    e24KC2ThetaBelowNode1100210 e24KC2ThetaBelowNode1100211 e24KC2ThetaBelowNode1100212
+      e24KC2ThetaBelowNode1100213
+
+theorem e24KC2ThetaBelowNode110022 :
+    adaptiveCoverCheck 12 (childHL (childHL thetaBelowCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL thetaBelowCell1100))
+    e24KC2ThetaBelowLeaf1100220 e24KC2ThetaBelowLeaf1100221 e24KC2ThetaBelowLeaf1100222
+      e24KC2ThetaBelowLeaf1100223
+
+theorem e24KC2ThetaBelowNode110023 :
+    adaptiveCoverCheck 12 (childHH (childHL thetaBelowCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL thetaBelowCell1100))
+    e24KC2ThetaBelowLeaf1100230 e24KC2ThetaBelowLeaf1100231 e24KC2ThetaBelowLeaf1100232
+      e24KC2ThetaBelowLeaf1100233
+
+theorem e24KC2ThetaBelowNode110030 :
+    adaptiveCoverCheck 12 (childLL (childHH thetaBelowCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH thetaBelowCell1100))
+    e24KC2ThetaBelowNode1100300 e24KC2ThetaBelowNode1100301 e24KC2ThetaBelowNode1100302
+      e24KC2ThetaBelowNode1100303
+
+theorem e24KC2ThetaBelowNode110031 :
+    adaptiveCoverCheck 12 (childLH (childHH thetaBelowCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH thetaBelowCell1100))
+    e24KC2ThetaBelowNode1100310 e24KC2ThetaBelowNode1100311 e24KC2ThetaBelowNode1100312
+      e24KC2ThetaBelowNode1100313
+
+theorem e24KC2ThetaBelowNode110032 :
+    adaptiveCoverCheck 12 (childHL (childHH thetaBelowCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH thetaBelowCell1100))
+    e24KC2ThetaBelowLeaf1100320 e24KC2ThetaBelowLeaf1100321 e24KC2ThetaBelowLeaf1100322
+      e24KC2ThetaBelowLeaf1100323
+
+theorem e24KC2ThetaBelowNode110033 :
+    adaptiveCoverCheck 12 (childHH (childHH thetaBelowCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH thetaBelowCell1100))
+    e24KC2ThetaBelowNode1100330 e24KC2ThetaBelowNode1100331 e24KC2ThetaBelowLeaf1100332
+      e24KC2ThetaBelowLeaf1100333
+
+theorem e24KC2ThetaBelowNode110102 :
+    adaptiveCoverCheck 12 (childHL (childLL thetaBelowCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL thetaBelowCell1101))
+    e24KC2ThetaBelowLeaf1101020 e24KC2ThetaBelowLeaf1101021 e24KC2ThetaBelowNode1101022
+      e24KC2ThetaBelowNode1101023
+
+theorem e24KC2ThetaBelowNode110103 :
+    adaptiveCoverCheck 12 (childHH (childLL thetaBelowCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL thetaBelowCell1101))
+    e24KC2ThetaBelowLeaf1101030 e24KC2ThetaBelowLeaf1101031 e24KC2ThetaBelowLeaf1101032
+      e24KC2ThetaBelowLeaf1101033
+
+theorem e24KC2ThetaBelowNode110112 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH thetaBelowCell1101))
+    e24KC2ThetaBelowLeaf1101120 e24KC2ThetaBelowLeaf1101121 e24KC2ThetaBelowLeaf1101122
+      e24KC2ThetaBelowLeaf1101123
+
+theorem e24KC2ThetaBelowNode110113 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH thetaBelowCell1101))
+    e24KC2ThetaBelowLeaf1101130 e24KC2ThetaBelowLeaf1101131 e24KC2ThetaBelowLeaf1101132
+      e24KC2ThetaBelowLeaf1101133
+
+theorem e24KC2ThetaBelowNode110120 :
+    adaptiveCoverCheck 12 (childLL (childHL thetaBelowCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL thetaBelowCell1101))
+    e24KC2ThetaBelowNode1101200 e24KC2ThetaBelowNode1101201 e24KC2ThetaBelowNode1101202
+      e24KC2ThetaBelowNode1101203
+
+theorem e24KC2ThetaBelowNode110121 :
+    adaptiveCoverCheck 12 (childLH (childHL thetaBelowCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL thetaBelowCell1101))
+    e24KC2ThetaBelowNode1101210 e24KC2ThetaBelowNode1101211 e24KC2ThetaBelowNode1101212
+      e24KC2ThetaBelowNode1101213
+
+theorem e24KC2ThetaBelowNode110122 :
+    adaptiveCoverCheck 12 (childHL (childHL thetaBelowCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL thetaBelowCell1101))
+    e24KC2ThetaBelowNode1101220 e24KC2ThetaBelowNode1101221 e24KC2ThetaBelowLeaf1101222
+      e24KC2ThetaBelowLeaf1101223
+
+theorem e24KC2ThetaBelowNode110123 :
+    adaptiveCoverCheck 12 (childHH (childHL thetaBelowCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL thetaBelowCell1101))
+    e24KC2ThetaBelowNode1101230 e24KC2ThetaBelowNode1101231 e24KC2ThetaBelowLeaf1101232
+      e24KC2ThetaBelowLeaf1101233
+
+theorem e24KC2ThetaBelowNode110130 :
+    adaptiveCoverCheck 12 (childLL (childHH thetaBelowCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH thetaBelowCell1101))
+    e24KC2ThetaBelowNode1101300 e24KC2ThetaBelowNode1101301 e24KC2ThetaBelowNode1101302
+      e24KC2ThetaBelowNode1101303
+
+theorem e24KC2ThetaBelowNode110131 :
+    adaptiveCoverCheck 12 (childLH (childHH thetaBelowCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH thetaBelowCell1101))
+    e24KC2ThetaBelowNode1101310 e24KC2ThetaBelowNode1101311 e24KC2ThetaBelowNode1101312
+      e24KC2ThetaBelowNode1101313
+
+theorem e24KC2ThetaBelowNode110132 :
+    adaptiveCoverCheck 12 (childHL (childHH thetaBelowCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH thetaBelowCell1101))
+    e24KC2ThetaBelowNode1101320 e24KC2ThetaBelowNode1101321 e24KC2ThetaBelowLeaf1101322
+      e24KC2ThetaBelowLeaf1101323
+
+theorem e24KC2ThetaBelowNode110133 :
+    adaptiveCoverCheck 12 (childHH (childHH thetaBelowCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH thetaBelowCell1101))
+    e24KC2ThetaBelowNode1101330 e24KC2ThetaBelowNode1101331 e24KC2ThetaBelowLeaf1101332
+      e24KC2ThetaBelowLeaf1101333
+
+theorem e24KC2ThetaBelowNode111002 :
+    adaptiveCoverCheck 12 (childHL (childLL thetaBelowCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL thetaBelowCell1110))
+    e24KC2ThetaBelowLeaf1110020 e24KC2ThetaBelowLeaf1110021 e24KC2ThetaBelowLeaf1110022
+      e24KC2ThetaBelowLeaf1110023
+
+theorem e24KC2ThetaBelowNode111003 :
+    adaptiveCoverCheck 12 (childHH (childLL thetaBelowCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL thetaBelowCell1110))
+    e24KC2ThetaBelowLeaf1110030 e24KC2ThetaBelowLeaf1110031 e24KC2ThetaBelowLeaf1110032
+      e24KC2ThetaBelowLeaf1110033
+
+theorem e24KC2ThetaBelowNode111012 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH thetaBelowCell1110))
+    e24KC2ThetaBelowLeaf1110120 e24KC2ThetaBelowLeaf1110121 e24KC2ThetaBelowLeaf1110122
+      e24KC2ThetaBelowLeaf1110123
+
+theorem e24KC2ThetaBelowNode111013 :
+    adaptiveCoverCheck 12 (childHH (childLH thetaBelowCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH thetaBelowCell1110))
+    e24KC2ThetaBelowLeaf1110130 e24KC2ThetaBelowLeaf1110131 e24KC2ThetaBelowLeaf1110132
+      e24KC2ThetaBelowLeaf1110133
+
+theorem e24KC2ThetaBelowNode111020 :
+    adaptiveCoverCheck 12 (childLL (childHL thetaBelowCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL thetaBelowCell1110))
+    e24KC2ThetaBelowNode1110200 e24KC2ThetaBelowNode1110201 e24KC2ThetaBelowNode1110202
+      e24KC2ThetaBelowNode1110203
+
+theorem e24KC2ThetaBelowNode111021 :
+    adaptiveCoverCheck 12 (childLH (childHL thetaBelowCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL thetaBelowCell1110))
+    e24KC2ThetaBelowNode1110210 e24KC2ThetaBelowNode1110211 e24KC2ThetaBelowNode1110212
+      e24KC2ThetaBelowNode1110213
+
+theorem e24KC2ThetaBelowNode111022 :
+    adaptiveCoverCheck 12 (childHL (childHL thetaBelowCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL thetaBelowCell1110))
+    e24KC2ThetaBelowNode1110220 e24KC2ThetaBelowNode1110221 e24KC2ThetaBelowLeaf1110222
+      e24KC2ThetaBelowLeaf1110223
+
+theorem e24KC2ThetaBelowNode111023 :
+    adaptiveCoverCheck 12 (childHH (childHL thetaBelowCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL thetaBelowCell1110))
+    e24KC2ThetaBelowNode1110230 e24KC2ThetaBelowNode1110231 e24KC2ThetaBelowLeaf1110232
+      e24KC2ThetaBelowLeaf1110233
+
+theorem e24KC2ThetaBelowNode111030 :
+    adaptiveCoverCheck 12 (childLL (childHH thetaBelowCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH thetaBelowCell1110))
+    e24KC2ThetaBelowNode1110300 e24KC2ThetaBelowNode1110301 e24KC2ThetaBelowNode1110302
+      e24KC2ThetaBelowNode1110303
+
+theorem e24KC2ThetaBelowNode111031 :
+    adaptiveCoverCheck 12 (childLH (childHH thetaBelowCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH thetaBelowCell1110))
+    e24KC2ThetaBelowNode1110310 e24KC2ThetaBelowNode1110311 e24KC2ThetaBelowNode1110312
+      e24KC2ThetaBelowNode1110313
+
+theorem e24KC2ThetaBelowNode111032 :
+    adaptiveCoverCheck 12 (childHL (childHH thetaBelowCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH thetaBelowCell1110))
+    e24KC2ThetaBelowNode1110320 e24KC2ThetaBelowNode1110321 e24KC2ThetaBelowLeaf1110322
+      e24KC2ThetaBelowLeaf1110323
+
+theorem e24KC2ThetaBelowNode111033 :
+    adaptiveCoverCheck 12 (childHH (childHH thetaBelowCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH thetaBelowCell1110))
+    e24KC2ThetaBelowNode1110330 e24KC2ThetaBelowNode1110331 e24KC2ThetaBelowLeaf1110332
+      e24KC2ThetaBelowLeaf1110333
+
+theorem e24KC2ThetaBelowNode111102 :
+    adaptiveCoverCheck 12 (childHL (childLL thetaBelowCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL thetaBelowCell1111))
+    e24KC2ThetaBelowLeaf1111020 e24KC2ThetaBelowLeaf1111021 e24KC2ThetaBelowLeaf1111022
+      e24KC2ThetaBelowLeaf1111023
+
+theorem e24KC2ThetaBelowNode111103 :
+    adaptiveCoverCheck 12 (childHH (childLL thetaBelowCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL thetaBelowCell1111))
+    e24KC2ThetaBelowLeaf1111030 e24KC2ThetaBelowLeaf1111031 e24KC2ThetaBelowLeaf1111032
+      e24KC2ThetaBelowLeaf1111033
+
+theorem e24KC2ThetaBelowNode111112 :
+    adaptiveCoverCheck 12 (childHL (childLH thetaBelowCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH thetaBelowCell1111))
+    e24KC2ThetaBelowLeaf1111120 e24KC2ThetaBelowLeaf1111121 e24KC2ThetaBelowLeaf1111122
+      e24KC2ThetaBelowLeaf1111123
+
+theorem e24KC2ThetaBelowNode111120 :
+    adaptiveCoverCheck 12 (childLL (childHL thetaBelowCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL thetaBelowCell1111))
+    e24KC2ThetaBelowNode1111200 e24KC2ThetaBelowNode1111201 e24KC2ThetaBelowNode1111202
+      e24KC2ThetaBelowNode1111203
+
+theorem e24KC2ThetaBelowNode111121 :
+    adaptiveCoverCheck 12 (childLH (childHL thetaBelowCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL thetaBelowCell1111))
+    e24KC2ThetaBelowNode1111210 e24KC2ThetaBelowNode1111211 e24KC2ThetaBelowNode1111212
+      e24KC2ThetaBelowNode1111213
+
+theorem e24KC2ThetaBelowNode111122 :
+    adaptiveCoverCheck 12 (childHL (childHL thetaBelowCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL thetaBelowCell1111))
+    e24KC2ThetaBelowNode1111220 e24KC2ThetaBelowNode1111221 e24KC2ThetaBelowNode1111222
+      e24KC2ThetaBelowNode1111223
+
+theorem e24KC2ThetaBelowNode111123 :
+    adaptiveCoverCheck 12 (childHH (childHL thetaBelowCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL thetaBelowCell1111))
+    e24KC2ThetaBelowNode1111230 e24KC2ThetaBelowNode1111231 e24KC2ThetaBelowNode1111232
+      e24KC2ThetaBelowNode1111233
+
+theorem e24KC2ThetaBelowNode111130 :
+    adaptiveCoverCheck 12 (childLL (childHH thetaBelowCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH thetaBelowCell1111))
+    e24KC2ThetaBelowLeaf1111300 e24KC2ThetaBelowLeaf1111301 e24KC2ThetaBelowNode1111302
+      e24KC2ThetaBelowNode1111303
+
+theorem e24KC2ThetaBelowNode111131 :
+    adaptiveCoverCheck 12 (childLH (childHH thetaBelowCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH thetaBelowCell1111))
+    e24KC2ThetaBelowLeaf1111310 e24KC2ThetaBelowLeaf1111311 e24KC2ThetaBelowNode1111312
+      e24KC2ThetaBelowNode1111313
+
+theorem e24KC2ThetaBelowNode111132 :
+    adaptiveCoverCheck 12 (childHL (childHH thetaBelowCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH thetaBelowCell1111))
+    e24KC2ThetaBelowNode1111320 e24KC2ThetaBelowNode1111321 e24KC2ThetaBelowNode1111322
+      e24KC2ThetaBelowNode1111323
+
+theorem e24KC2ThetaBelowNode111133 :
+    adaptiveCoverCheck 12 (childHH (childHH thetaBelowCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH thetaBelowCell1111))
+    e24KC2ThetaBelowNode1111330 e24KC2ThetaBelowNode1111331 e24KC2ThetaBelowNode1111332
+      e24KC2ThetaBelowNode1111333
+
+theorem e24KC2ThetaBelowNode111210 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell1112)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childLH thetaBelowCell1112))
+    e24KC2ThetaBelowLeaf1112100 e24KC2ThetaBelowLeaf1112101 e24KC2ThetaBelowLeaf1112102
+      e24KC2ThetaBelowLeaf1112103
+
+theorem e24KC2ThetaBelowNode111211 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell1112)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childLH thetaBelowCell1112))
+    e24KC2ThetaBelowLeaf1112110 e24KC2ThetaBelowLeaf1112111 e24KC2ThetaBelowLeaf1112112
+      e24KC2ThetaBelowLeaf1112113
+
+theorem e24KC2ThetaBelowNode111300 :
+    adaptiveCoverCheck 12 (childLL (childLL thetaBelowCell1113)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childLL thetaBelowCell1113))
+    e24KC2ThetaBelowLeaf1113000 e24KC2ThetaBelowLeaf1113001 e24KC2ThetaBelowLeaf1113002
+      e24KC2ThetaBelowLeaf1113003
+
+theorem e24KC2ThetaBelowNode111301 :
+    adaptiveCoverCheck 12 (childLH (childLL thetaBelowCell1113)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childLL thetaBelowCell1113))
+    e24KC2ThetaBelowLeaf1113010 e24KC2ThetaBelowLeaf1113011 e24KC2ThetaBelowLeaf1113012
+      e24KC2ThetaBelowLeaf1113013
+
+theorem e24KC2ThetaBelowNode111310 :
+    adaptiveCoverCheck 12 (childLL (childLH thetaBelowCell1113)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childLH thetaBelowCell1113))
+    e24KC2ThetaBelowLeaf1113100 e24KC2ThetaBelowLeaf1113101 e24KC2ThetaBelowLeaf1113102
+      e24KC2ThetaBelowLeaf1113103
+
+theorem e24KC2ThetaBelowNode111311 :
+    adaptiveCoverCheck 12 (childLH (childLH thetaBelowCell1113)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childLH thetaBelowCell1113))
+    e24KC2ThetaBelowLeaf1113110 e24KC2ThetaBelowLeaf1113111 e24KC2ThetaBelowLeaf1113112
+      e24KC2ThetaBelowLeaf1113113
+
+theorem e24KC2ThetaBelowNode01013 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell0101) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH thetaBelowCell0101)
+    e24KC2ThetaBelowLeaf010130 e24KC2ThetaBelowLeaf010131 e24KC2ThetaBelowLeaf010132
+      e24KC2ThetaBelowLeaf010133
+
+theorem e24KC2ThetaBelowNode01101 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell0110) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell0110)
+    e24KC2ThetaBelowLeaf011010 e24KC2ThetaBelowLeaf011011 e24KC2ThetaBelowLeaf011012
+      e24KC2ThetaBelowLeaf011013
+
+theorem e24KC2ThetaBelowNode01102 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell0110) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL thetaBelowCell0110)
+    e24KC2ThetaBelowLeaf011020 e24KC2ThetaBelowLeaf011021 e24KC2ThetaBelowLeaf011022
+      e24KC2ThetaBelowLeaf011023
+
+theorem e24KC2ThetaBelowNode01103 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell0110) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH thetaBelowCell0110)
+    e24KC2ThetaBelowLeaf011030 e24KC2ThetaBelowLeaf011031 e24KC2ThetaBelowLeaf011032
+      e24KC2ThetaBelowLeaf011033
+
+theorem e24KC2ThetaBelowNode01110 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell0111) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL thetaBelowCell0111)
+    e24KC2ThetaBelowLeaf011100 e24KC2ThetaBelowLeaf011101 e24KC2ThetaBelowLeaf011102
+      e24KC2ThetaBelowLeaf011103
+
+theorem e24KC2ThetaBelowNode01111 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell0111) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell0111)
+    e24KC2ThetaBelowLeaf011110 e24KC2ThetaBelowLeaf011111 e24KC2ThetaBelowLeaf011112
+      e24KC2ThetaBelowLeaf011113
+
+theorem e24KC2ThetaBelowNode01112 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell0111) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL thetaBelowCell0111)
+    e24KC2ThetaBelowLeaf011120 e24KC2ThetaBelowLeaf011121 e24KC2ThetaBelowLeaf011122
+      e24KC2ThetaBelowLeaf011123
+
+theorem e24KC2ThetaBelowNode01113 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell0111) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH thetaBelowCell0111)
+    e24KC2ThetaBelowLeaf011130 e24KC2ThetaBelowLeaf011131 e24KC2ThetaBelowLeaf011132
+      e24KC2ThetaBelowLeaf011133
+
+theorem e24KC2ThetaBelowNode10000 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1000) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL thetaBelowCell1000)
+    e24KC2ThetaBelowLeaf100000 e24KC2ThetaBelowLeaf100001 e24KC2ThetaBelowLeaf100002
+      e24KC2ThetaBelowLeaf100003
+
+theorem e24KC2ThetaBelowNode10001 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1000) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell1000)
+    e24KC2ThetaBelowLeaf100010 e24KC2ThetaBelowLeaf100011 e24KC2ThetaBelowLeaf100012
+      e24KC2ThetaBelowLeaf100013
+
+theorem e24KC2ThetaBelowNode10002 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1000) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL thetaBelowCell1000)
+    e24KC2ThetaBelowLeaf100020 e24KC2ThetaBelowLeaf100021 e24KC2ThetaBelowLeaf100022
+      e24KC2ThetaBelowLeaf100023
+
+theorem e24KC2ThetaBelowNode10003 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1000) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH thetaBelowCell1000)
+    e24KC2ThetaBelowLeaf100030 e24KC2ThetaBelowLeaf100031 e24KC2ThetaBelowLeaf100032
+      e24KC2ThetaBelowLeaf100033
+
+theorem e24KC2ThetaBelowNode10010 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1001) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL thetaBelowCell1001)
+    e24KC2ThetaBelowLeaf100100 e24KC2ThetaBelowLeaf100101 e24KC2ThetaBelowLeaf100102
+      e24KC2ThetaBelowLeaf100103
+
+theorem e24KC2ThetaBelowNode10011 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1001) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell1001)
+    e24KC2ThetaBelowLeaf100110 e24KC2ThetaBelowLeaf100111 e24KC2ThetaBelowLeaf100112
+      e24KC2ThetaBelowNode100113
+
+theorem e24KC2ThetaBelowNode10012 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1001) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL thetaBelowCell1001)
+    e24KC2ThetaBelowLeaf100120 e24KC2ThetaBelowNode100121 e24KC2ThetaBelowLeaf100122
+      e24KC2ThetaBelowLeaf100123
+
+theorem e24KC2ThetaBelowNode10013 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1001) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH thetaBelowCell1001)
+    e24KC2ThetaBelowNode100130 e24KC2ThetaBelowNode100131 e24KC2ThetaBelowLeaf100132
+      e24KC2ThetaBelowLeaf100133
+
+theorem e24KC2ThetaBelowNode10031 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1003) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell1003)
+    e24KC2ThetaBelowLeaf100310 e24KC2ThetaBelowLeaf100311 e24KC2ThetaBelowLeaf100312
+      e24KC2ThetaBelowLeaf100313
+
+theorem e24KC2ThetaBelowNode10100 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1010) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL thetaBelowCell1010)
+    e24KC2ThetaBelowLeaf101000 e24KC2ThetaBelowNode101001 e24KC2ThetaBelowNode101002
+      e24KC2ThetaBelowNode101003
+
+theorem e24KC2ThetaBelowNode10101 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1010) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell1010)
+    e24KC2ThetaBelowNode101010 e24KC2ThetaBelowNode101011 e24KC2ThetaBelowNode101012
+      e24KC2ThetaBelowNode101013
+
+theorem e24KC2ThetaBelowNode10102 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1010) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL thetaBelowCell1010)
+    e24KC2ThetaBelowNode101020 e24KC2ThetaBelowNode101021 e24KC2ThetaBelowLeaf101022
+      e24KC2ThetaBelowLeaf101023
+
+theorem e24KC2ThetaBelowNode10103 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1010) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH thetaBelowCell1010)
+    e24KC2ThetaBelowNode101030 e24KC2ThetaBelowNode101031 e24KC2ThetaBelowNode101032
+      e24KC2ThetaBelowNode101033
+
+theorem e24KC2ThetaBelowNode10110 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1011) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL thetaBelowCell1011)
+    e24KC2ThetaBelowNode101100 e24KC2ThetaBelowLeaf101101 e24KC2ThetaBelowNode101102
+      e24KC2ThetaBelowNode101103
+
+theorem e24KC2ThetaBelowNode10111 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1011) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell1011)
+    e24KC2ThetaBelowLeaf101110 e24KC2ThetaBelowLeaf101111 e24KC2ThetaBelowNode101112
+      e24KC2ThetaBelowNode101113
+
+theorem e24KC2ThetaBelowNode10112 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1011) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL thetaBelowCell1011)
+    e24KC2ThetaBelowNode101120 e24KC2ThetaBelowNode101121 e24KC2ThetaBelowNode101122
+      e24KC2ThetaBelowNode101123
+
+theorem e24KC2ThetaBelowNode10113 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1011) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH thetaBelowCell1011)
+    e24KC2ThetaBelowNode101130 e24KC2ThetaBelowNode101131 e24KC2ThetaBelowNode101132
+      e24KC2ThetaBelowNode101133
+
+theorem e24KC2ThetaBelowNode10120 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1012) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL thetaBelowCell1012)
+    e24KC2ThetaBelowLeaf101200 e24KC2ThetaBelowLeaf101201 e24KC2ThetaBelowLeaf101202
+      e24KC2ThetaBelowLeaf101203
+
+theorem e24KC2ThetaBelowNode10121 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1012) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell1012)
+    e24KC2ThetaBelowLeaf101210 e24KC2ThetaBelowLeaf101211 e24KC2ThetaBelowLeaf101212
+      e24KC2ThetaBelowLeaf101213
+
+theorem e24KC2ThetaBelowNode10130 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1013) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL thetaBelowCell1013)
+    e24KC2ThetaBelowLeaf101300 e24KC2ThetaBelowLeaf101301 e24KC2ThetaBelowLeaf101302
+      e24KC2ThetaBelowLeaf101303
+
+theorem e24KC2ThetaBelowNode10131 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1013) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell1013)
+    e24KC2ThetaBelowLeaf101310 e24KC2ThetaBelowLeaf101311 e24KC2ThetaBelowLeaf101312
+      e24KC2ThetaBelowLeaf101313
+
+theorem e24KC2ThetaBelowNode11000 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1100) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL thetaBelowCell1100)
+    e24KC2ThetaBelowLeaf110000 e24KC2ThetaBelowLeaf110001 e24KC2ThetaBelowNode110002
+      e24KC2ThetaBelowNode110003
+
+theorem e24KC2ThetaBelowNode11001 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1100) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell1100)
+    e24KC2ThetaBelowLeaf110010 e24KC2ThetaBelowLeaf110011 e24KC2ThetaBelowNode110012
+      e24KC2ThetaBelowNode110013
+
+theorem e24KC2ThetaBelowNode11002 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1100) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL thetaBelowCell1100)
+    e24KC2ThetaBelowNode110020 e24KC2ThetaBelowNode110021 e24KC2ThetaBelowNode110022
+      e24KC2ThetaBelowNode110023
+
+theorem e24KC2ThetaBelowNode11003 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1100) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH thetaBelowCell1100)
+    e24KC2ThetaBelowNode110030 e24KC2ThetaBelowNode110031 e24KC2ThetaBelowNode110032
+      e24KC2ThetaBelowNode110033
+
+theorem e24KC2ThetaBelowNode11010 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1101) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL thetaBelowCell1101)
+    e24KC2ThetaBelowLeaf110100 e24KC2ThetaBelowLeaf110101 e24KC2ThetaBelowNode110102
+      e24KC2ThetaBelowNode110103
+
+theorem e24KC2ThetaBelowNode11011 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1101) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell1101)
+    e24KC2ThetaBelowLeaf110110 e24KC2ThetaBelowLeaf110111 e24KC2ThetaBelowNode110112
+      e24KC2ThetaBelowNode110113
+
+theorem e24KC2ThetaBelowNode11012 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1101) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL thetaBelowCell1101)
+    e24KC2ThetaBelowNode110120 e24KC2ThetaBelowNode110121 e24KC2ThetaBelowNode110122
+      e24KC2ThetaBelowNode110123
+
+theorem e24KC2ThetaBelowNode11013 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1101) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH thetaBelowCell1101)
+    e24KC2ThetaBelowNode110130 e24KC2ThetaBelowNode110131 e24KC2ThetaBelowNode110132
+      e24KC2ThetaBelowNode110133
+
+theorem e24KC2ThetaBelowNode11020 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1102) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL thetaBelowCell1102)
+    e24KC2ThetaBelowLeaf110200 e24KC2ThetaBelowLeaf110201 e24KC2ThetaBelowLeaf110202
+      e24KC2ThetaBelowLeaf110203
+
+theorem e24KC2ThetaBelowNode11021 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1102) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell1102)
+    e24KC2ThetaBelowLeaf110210 e24KC2ThetaBelowLeaf110211 e24KC2ThetaBelowLeaf110212
+      e24KC2ThetaBelowLeaf110213
+
+theorem e24KC2ThetaBelowNode11030 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1103) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL thetaBelowCell1103)
+    e24KC2ThetaBelowLeaf110300 e24KC2ThetaBelowLeaf110301 e24KC2ThetaBelowLeaf110302
+      e24KC2ThetaBelowLeaf110303
+
+theorem e24KC2ThetaBelowNode11031 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1103) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell1103)
+    e24KC2ThetaBelowLeaf110310 e24KC2ThetaBelowLeaf110311 e24KC2ThetaBelowLeaf110312
+      e24KC2ThetaBelowLeaf110313
+
+theorem e24KC2ThetaBelowNode11100 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1110) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL thetaBelowCell1110)
+    e24KC2ThetaBelowLeaf111000 e24KC2ThetaBelowLeaf111001 e24KC2ThetaBelowNode111002
+      e24KC2ThetaBelowNode111003
+
+theorem e24KC2ThetaBelowNode11101 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1110) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell1110)
+    e24KC2ThetaBelowLeaf111010 e24KC2ThetaBelowLeaf111011 e24KC2ThetaBelowNode111012
+      e24KC2ThetaBelowNode111013
+
+theorem e24KC2ThetaBelowNode11102 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1110) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL thetaBelowCell1110)
+    e24KC2ThetaBelowNode111020 e24KC2ThetaBelowNode111021 e24KC2ThetaBelowNode111022
+      e24KC2ThetaBelowNode111023
+
+theorem e24KC2ThetaBelowNode11103 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1110) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH thetaBelowCell1110)
+    e24KC2ThetaBelowNode111030 e24KC2ThetaBelowNode111031 e24KC2ThetaBelowNode111032
+      e24KC2ThetaBelowNode111033
+
+theorem e24KC2ThetaBelowNode11110 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1111) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL thetaBelowCell1111)
+    e24KC2ThetaBelowLeaf111100 e24KC2ThetaBelowLeaf111101 e24KC2ThetaBelowNode111102
+      e24KC2ThetaBelowNode111103
+
+theorem e24KC2ThetaBelowNode11111 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1111) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell1111)
+    e24KC2ThetaBelowLeaf111110 e24KC2ThetaBelowLeaf111111 e24KC2ThetaBelowNode111112
+      e24KC2ThetaBelowLeaf111113
+
+theorem e24KC2ThetaBelowNode11112 :
+    adaptiveCoverCheck 13 (childHL thetaBelowCell1111) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL thetaBelowCell1111)
+    e24KC2ThetaBelowNode111120 e24KC2ThetaBelowNode111121 e24KC2ThetaBelowNode111122
+      e24KC2ThetaBelowNode111123
+
+theorem e24KC2ThetaBelowNode11113 :
+    adaptiveCoverCheck 13 (childHH thetaBelowCell1111) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH thetaBelowCell1111)
+    e24KC2ThetaBelowNode111130 e24KC2ThetaBelowNode111131 e24KC2ThetaBelowNode111132
+      e24KC2ThetaBelowNode111133
+
+theorem e24KC2ThetaBelowNode11120 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1112) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL thetaBelowCell1112)
+    e24KC2ThetaBelowLeaf111200 e24KC2ThetaBelowLeaf111201 e24KC2ThetaBelowLeaf111202
+      e24KC2ThetaBelowLeaf111203
+
+theorem e24KC2ThetaBelowNode11121 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1112) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell1112)
+    e24KC2ThetaBelowNode111210 e24KC2ThetaBelowNode111211 e24KC2ThetaBelowLeaf111212
+      e24KC2ThetaBelowLeaf111213
+
+theorem e24KC2ThetaBelowNode11130 :
+    adaptiveCoverCheck 13 (childLL thetaBelowCell1113) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL thetaBelowCell1113)
+    e24KC2ThetaBelowNode111300 e24KC2ThetaBelowNode111301 e24KC2ThetaBelowLeaf111302
+      e24KC2ThetaBelowLeaf111303
+
+theorem e24KC2ThetaBelowNode11131 :
+    adaptiveCoverCheck 13 (childLH thetaBelowCell1113) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH thetaBelowCell1113)
+    e24KC2ThetaBelowNode111310 e24KC2ThetaBelowNode111311 e24KC2ThetaBelowLeaf111312
+      e24KC2ThetaBelowLeaf111313
+
+theorem e24KC2ThetaBelowNode0003 :
+    adaptiveCoverCheck 14 thetaBelowCell0003 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell0003
+    e24KC2ThetaBelowLeaf00030 e24KC2ThetaBelowLeaf00031 e24KC2ThetaBelowLeaf00032
+      e24KC2ThetaBelowLeaf00033
+
+theorem e24KC2ThetaBelowNode0010 :
+    adaptiveCoverCheck 14 thetaBelowCell0010 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell0010
+    e24KC2ThetaBelowLeaf00100 e24KC2ThetaBelowLeaf00101 e24KC2ThetaBelowLeaf00102
+      e24KC2ThetaBelowLeaf00103
+
+theorem e24KC2ThetaBelowNode0011 :
+    adaptiveCoverCheck 14 thetaBelowCell0011 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell0011
+    e24KC2ThetaBelowLeaf00110 e24KC2ThetaBelowLeaf00111 e24KC2ThetaBelowLeaf00112
+      e24KC2ThetaBelowLeaf00113
+
+theorem e24KC2ThetaBelowNode0012 :
+    adaptiveCoverCheck 14 thetaBelowCell0012 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell0012
+    e24KC2ThetaBelowLeaf00120 e24KC2ThetaBelowLeaf00121 e24KC2ThetaBelowLeaf00122
+      e24KC2ThetaBelowLeaf00123
+
+theorem e24KC2ThetaBelowNode0013 :
+    adaptiveCoverCheck 14 thetaBelowCell0013 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell0013
+    e24KC2ThetaBelowLeaf00130 e24KC2ThetaBelowLeaf00131 e24KC2ThetaBelowLeaf00132
+      e24KC2ThetaBelowLeaf00133
+
+theorem e24KC2ThetaBelowNode0100 :
+    adaptiveCoverCheck 14 thetaBelowCell0100 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell0100
+    e24KC2ThetaBelowLeaf01000 e24KC2ThetaBelowLeaf01001 e24KC2ThetaBelowLeaf01002
+      e24KC2ThetaBelowLeaf01003
+
+theorem e24KC2ThetaBelowNode0101 :
+    adaptiveCoverCheck 14 thetaBelowCell0101 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell0101
+    e24KC2ThetaBelowLeaf01010 e24KC2ThetaBelowLeaf01011 e24KC2ThetaBelowLeaf01012
+      e24KC2ThetaBelowNode01013
+
+theorem e24KC2ThetaBelowNode0102 :
+    adaptiveCoverCheck 14 thetaBelowCell0102 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell0102
+    e24KC2ThetaBelowLeaf01020 e24KC2ThetaBelowLeaf01021 e24KC2ThetaBelowLeaf01022
+      e24KC2ThetaBelowLeaf01023
+
+theorem e24KC2ThetaBelowNode0103 :
+    adaptiveCoverCheck 14 thetaBelowCell0103 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell0103
+    e24KC2ThetaBelowLeaf01030 e24KC2ThetaBelowLeaf01031 e24KC2ThetaBelowLeaf01032
+      e24KC2ThetaBelowLeaf01033
+
+theorem e24KC2ThetaBelowNode0110 :
+    adaptiveCoverCheck 14 thetaBelowCell0110 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell0110
+    e24KC2ThetaBelowLeaf01100 e24KC2ThetaBelowNode01101 e24KC2ThetaBelowNode01102
+      e24KC2ThetaBelowNode01103
+
+theorem e24KC2ThetaBelowNode0111 :
+    adaptiveCoverCheck 14 thetaBelowCell0111 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell0111
+    e24KC2ThetaBelowNode01110 e24KC2ThetaBelowNode01111 e24KC2ThetaBelowNode01112
+      e24KC2ThetaBelowNode01113
+
+theorem e24KC2ThetaBelowNode0112 :
+    adaptiveCoverCheck 14 thetaBelowCell0112 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell0112
+    e24KC2ThetaBelowLeaf01120 e24KC2ThetaBelowLeaf01121 e24KC2ThetaBelowLeaf01122
+      e24KC2ThetaBelowLeaf01123
+
+theorem e24KC2ThetaBelowNode0113 :
+    adaptiveCoverCheck 14 thetaBelowCell0113 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell0113
+    e24KC2ThetaBelowLeaf01130 e24KC2ThetaBelowLeaf01131 e24KC2ThetaBelowLeaf01132
+      e24KC2ThetaBelowLeaf01133
+
+theorem e24KC2ThetaBelowNode1000 :
+    adaptiveCoverCheck 14 thetaBelowCell1000 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1000
+    e24KC2ThetaBelowNode10000 e24KC2ThetaBelowNode10001 e24KC2ThetaBelowNode10002
+      e24KC2ThetaBelowNode10003
+
+theorem e24KC2ThetaBelowNode1001 :
+    adaptiveCoverCheck 14 thetaBelowCell1001 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1001
+    e24KC2ThetaBelowNode10010 e24KC2ThetaBelowNode10011 e24KC2ThetaBelowNode10012
+      e24KC2ThetaBelowNode10013
+
+theorem e24KC2ThetaBelowNode1002 :
+    adaptiveCoverCheck 14 thetaBelowCell1002 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1002
+    e24KC2ThetaBelowLeaf10020 e24KC2ThetaBelowLeaf10021 e24KC2ThetaBelowLeaf10022
+      e24KC2ThetaBelowLeaf10023
+
+theorem e24KC2ThetaBelowNode1003 :
+    adaptiveCoverCheck 14 thetaBelowCell1003 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1003
+    e24KC2ThetaBelowLeaf10030 e24KC2ThetaBelowNode10031 e24KC2ThetaBelowLeaf10032
+      e24KC2ThetaBelowLeaf10033
+
+theorem e24KC2ThetaBelowNode1010 :
+    adaptiveCoverCheck 14 thetaBelowCell1010 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1010
+    e24KC2ThetaBelowNode10100 e24KC2ThetaBelowNode10101 e24KC2ThetaBelowNode10102
+      e24KC2ThetaBelowNode10103
+
+theorem e24KC2ThetaBelowNode1011 :
+    adaptiveCoverCheck 14 thetaBelowCell1011 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1011
+    e24KC2ThetaBelowNode10110 e24KC2ThetaBelowNode10111 e24KC2ThetaBelowNode10112
+      e24KC2ThetaBelowNode10113
+
+theorem e24KC2ThetaBelowNode1012 :
+    adaptiveCoverCheck 14 thetaBelowCell1012 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1012
+    e24KC2ThetaBelowNode10120 e24KC2ThetaBelowNode10121 e24KC2ThetaBelowLeaf10122
+      e24KC2ThetaBelowLeaf10123
+
+theorem e24KC2ThetaBelowNode1013 :
+    adaptiveCoverCheck 14 thetaBelowCell1013 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1013
+    e24KC2ThetaBelowNode10130 e24KC2ThetaBelowNode10131 e24KC2ThetaBelowLeaf10132
+      e24KC2ThetaBelowLeaf10133
+
+theorem e24KC2ThetaBelowNode1100 :
+    adaptiveCoverCheck 14 thetaBelowCell1100 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1100
+    e24KC2ThetaBelowNode11000 e24KC2ThetaBelowNode11001 e24KC2ThetaBelowNode11002
+      e24KC2ThetaBelowNode11003
+
+theorem e24KC2ThetaBelowNode1101 :
+    adaptiveCoverCheck 14 thetaBelowCell1101 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1101
+    e24KC2ThetaBelowNode11010 e24KC2ThetaBelowNode11011 e24KC2ThetaBelowNode11012
+      e24KC2ThetaBelowNode11013
+
+theorem e24KC2ThetaBelowNode1102 :
+    adaptiveCoverCheck 14 thetaBelowCell1102 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1102
+    e24KC2ThetaBelowNode11020 e24KC2ThetaBelowNode11021 e24KC2ThetaBelowLeaf11022
+      e24KC2ThetaBelowLeaf11023
+
+theorem e24KC2ThetaBelowNode1103 :
+    adaptiveCoverCheck 14 thetaBelowCell1103 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1103
+    e24KC2ThetaBelowNode11030 e24KC2ThetaBelowNode11031 e24KC2ThetaBelowLeaf11032
+      e24KC2ThetaBelowLeaf11033
+
+theorem e24KC2ThetaBelowNode1110 :
+    adaptiveCoverCheck 14 thetaBelowCell1110 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1110
+    e24KC2ThetaBelowNode11100 e24KC2ThetaBelowNode11101 e24KC2ThetaBelowNode11102
+      e24KC2ThetaBelowNode11103
+
+theorem e24KC2ThetaBelowNode1111 :
+    adaptiveCoverCheck 14 thetaBelowCell1111 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1111
+    e24KC2ThetaBelowNode11110 e24KC2ThetaBelowNode11111 e24KC2ThetaBelowNode11112
+      e24KC2ThetaBelowNode11113
+
+theorem e24KC2ThetaBelowNode1112 :
+    adaptiveCoverCheck 14 thetaBelowCell1112 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1112
+    e24KC2ThetaBelowNode11120 e24KC2ThetaBelowNode11121 e24KC2ThetaBelowLeaf11122
+      e24KC2ThetaBelowLeaf11123
+
+theorem e24KC2ThetaBelowNode1113 :
+    adaptiveCoverCheck 14 thetaBelowCell1113 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1113
+    e24KC2ThetaBelowNode11130 e24KC2ThetaBelowNode11131 e24KC2ThetaBelowLeaf11132
+      e24KC2ThetaBelowLeaf11133
+
+theorem e24KC2ThetaBelowNode1121 :
+    adaptiveCoverCheck 14 thetaBelowCell1121 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1121
+    e24KC2ThetaBelowLeaf11210 e24KC2ThetaBelowLeaf11211 e24KC2ThetaBelowLeaf11212
+      e24KC2ThetaBelowLeaf11213
+
+theorem e24KC2ThetaBelowNode1130 :
+    adaptiveCoverCheck 14 thetaBelowCell1130 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1130
+    e24KC2ThetaBelowLeaf11300 e24KC2ThetaBelowLeaf11301 e24KC2ThetaBelowLeaf11302
+      e24KC2ThetaBelowLeaf11303
+
+theorem e24KC2ThetaBelowNode1131 :
+    adaptiveCoverCheck 14 thetaBelowCell1131 = true :=
+  adaptiveCoverCheck_succ_of_children 13 thetaBelowCell1131
+    e24KC2ThetaBelowLeaf11310 e24KC2ThetaBelowLeaf11311 e24KC2ThetaBelowLeaf11312
+      e24KC2ThetaBelowLeaf11313
+
+theorem e24KC2ThetaBelowNode000 :
+    adaptiveCoverCheck 15 (childLL (childLL (childLL e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childLL (childLL (childLL e24ThetaBelowRoot)))
+    e24KC2ThetaBelowLeaf0000 e24KC2ThetaBelowLeaf0001 e24KC2ThetaBelowLeaf0002
+      e24KC2ThetaBelowNode0003
+
+theorem e24KC2ThetaBelowNode001 :
+    adaptiveCoverCheck 15 (childLH (childLL (childLL e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childLH (childLL (childLL e24ThetaBelowRoot)))
+    e24KC2ThetaBelowNode0010 e24KC2ThetaBelowNode0011 e24KC2ThetaBelowNode0012
+      e24KC2ThetaBelowNode0013
+
+theorem e24KC2ThetaBelowNode003 :
+    adaptiveCoverCheck 15 (childHH (childLL (childLL e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childHH (childLL (childLL e24ThetaBelowRoot)))
+    e24KC2ThetaBelowLeaf0030 e24KC2ThetaBelowLeaf0031 e24KC2ThetaBelowLeaf0032
+      e24KC2ThetaBelowLeaf0033
+
+theorem e24KC2ThetaBelowNode010 :
+    adaptiveCoverCheck 15 (childLL (childLH (childLL e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childLL (childLH (childLL e24ThetaBelowRoot)))
+    e24KC2ThetaBelowNode0100 e24KC2ThetaBelowNode0101 e24KC2ThetaBelowNode0102
+      e24KC2ThetaBelowNode0103
+
+theorem e24KC2ThetaBelowNode011 :
+    adaptiveCoverCheck 15 (childLH (childLH (childLL e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childLH (childLH (childLL e24ThetaBelowRoot)))
+    e24KC2ThetaBelowNode0110 e24KC2ThetaBelowNode0111 e24KC2ThetaBelowNode0112
+      e24KC2ThetaBelowNode0113
+
+theorem e24KC2ThetaBelowNode012 :
+    adaptiveCoverCheck 15 (childHL (childLH (childLL e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childHL (childLH (childLL e24ThetaBelowRoot)))
+    e24KC2ThetaBelowLeaf0120 e24KC2ThetaBelowLeaf0121 e24KC2ThetaBelowLeaf0122
+      e24KC2ThetaBelowLeaf0123
+
+theorem e24KC2ThetaBelowNode013 :
+    adaptiveCoverCheck 15 (childHH (childLH (childLL e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childHH (childLH (childLL e24ThetaBelowRoot)))
+    e24KC2ThetaBelowLeaf0130 e24KC2ThetaBelowLeaf0131 e24KC2ThetaBelowLeaf0132
+      e24KC2ThetaBelowLeaf0133
+
+theorem e24KC2ThetaBelowNode100 :
+    adaptiveCoverCheck 15 (childLL (childLL (childLH e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childLL (childLL (childLH e24ThetaBelowRoot)))
+    e24KC2ThetaBelowNode1000 e24KC2ThetaBelowNode1001 e24KC2ThetaBelowNode1002
+      e24KC2ThetaBelowNode1003
+
+theorem e24KC2ThetaBelowNode101 :
+    adaptiveCoverCheck 15 (childLH (childLL (childLH e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childLH (childLL (childLH e24ThetaBelowRoot)))
+    e24KC2ThetaBelowNode1010 e24KC2ThetaBelowNode1011 e24KC2ThetaBelowNode1012
+      e24KC2ThetaBelowNode1013
+
+theorem e24KC2ThetaBelowNode102 :
+    adaptiveCoverCheck 15 (childHL (childLL (childLH e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childHL (childLL (childLH e24ThetaBelowRoot)))
+    e24KC2ThetaBelowLeaf1020 e24KC2ThetaBelowLeaf1021 e24KC2ThetaBelowLeaf1022
+      e24KC2ThetaBelowLeaf1023
+
+theorem e24KC2ThetaBelowNode103 :
+    adaptiveCoverCheck 15 (childHH (childLL (childLH e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childHH (childLL (childLH e24ThetaBelowRoot)))
+    e24KC2ThetaBelowLeaf1030 e24KC2ThetaBelowLeaf1031 e24KC2ThetaBelowLeaf1032
+      e24KC2ThetaBelowLeaf1033
+
+theorem e24KC2ThetaBelowNode110 :
+    adaptiveCoverCheck 15 (childLL (childLH (childLH e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childLL (childLH (childLH e24ThetaBelowRoot)))
+    e24KC2ThetaBelowNode1100 e24KC2ThetaBelowNode1101 e24KC2ThetaBelowNode1102
+      e24KC2ThetaBelowNode1103
+
+theorem e24KC2ThetaBelowNode111 :
+    adaptiveCoverCheck 15 (childLH (childLH (childLH e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childLH (childLH (childLH e24ThetaBelowRoot)))
+    e24KC2ThetaBelowNode1110 e24KC2ThetaBelowNode1111 e24KC2ThetaBelowNode1112
+      e24KC2ThetaBelowNode1113
+
+theorem e24KC2ThetaBelowNode112 :
+    adaptiveCoverCheck 15 (childHL (childLH (childLH e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childHL (childLH (childLH e24ThetaBelowRoot)))
+    e24KC2ThetaBelowLeaf1120 e24KC2ThetaBelowNode1121 e24KC2ThetaBelowLeaf1122
+      e24KC2ThetaBelowLeaf1123
+
+theorem e24KC2ThetaBelowNode113 :
+    adaptiveCoverCheck 15 (childHH (childLH (childLH e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childHH (childLH (childLH e24ThetaBelowRoot)))
+    e24KC2ThetaBelowNode1130 e24KC2ThetaBelowNode1131 e24KC2ThetaBelowLeaf1132
+      e24KC2ThetaBelowLeaf1133
+
+theorem e24KC2ThetaBelowNode130 :
+    adaptiveCoverCheck 15 (childLL (childHH (childLH e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childLL (childHH (childLH e24ThetaBelowRoot)))
+    e24KC2ThetaBelowLeaf1300 e24KC2ThetaBelowLeaf1301 e24KC2ThetaBelowLeaf1302
+      e24KC2ThetaBelowLeaf1303
+
+theorem e24KC2ThetaBelowNode131 :
+    adaptiveCoverCheck 15 (childLH (childHH (childLH e24ThetaBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childLH (childHH (childLH e24ThetaBelowRoot)))
+    e24KC2ThetaBelowLeaf1310 e24KC2ThetaBelowLeaf1311 e24KC2ThetaBelowLeaf1312
+      e24KC2ThetaBelowLeaf1313
+
+theorem e24KC2ThetaBelowNode00 :
+    adaptiveCoverCheck 16 (childLL (childLL e24ThetaBelowRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childLL (childLL e24ThetaBelowRoot))
+    e24KC2ThetaBelowNode000 e24KC2ThetaBelowNode001 e24KC2ThetaBelowLeaf002 e24KC2ThetaBelowNode003
+
+theorem e24KC2ThetaBelowNode01 :
+    adaptiveCoverCheck 16 (childLH (childLL e24ThetaBelowRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childLH (childLL e24ThetaBelowRoot))
+    e24KC2ThetaBelowNode010 e24KC2ThetaBelowNode011 e24KC2ThetaBelowNode012 e24KC2ThetaBelowNode013
+
+theorem e24KC2ThetaBelowNode03 :
+    adaptiveCoverCheck 16 (childHH (childLL e24ThetaBelowRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childHH (childLL e24ThetaBelowRoot))
+    e24KC2ThetaBelowLeaf030 e24KC2ThetaBelowLeaf031 e24KC2ThetaBelowLeaf032 e24KC2ThetaBelowLeaf033
+
+theorem e24KC2ThetaBelowNode10 :
+    adaptiveCoverCheck 16 (childLL (childLH e24ThetaBelowRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childLL (childLH e24ThetaBelowRoot))
+    e24KC2ThetaBelowNode100 e24KC2ThetaBelowNode101 e24KC2ThetaBelowNode102 e24KC2ThetaBelowNode103
+
+theorem e24KC2ThetaBelowNode11 :
+    adaptiveCoverCheck 16 (childLH (childLH e24ThetaBelowRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childLH (childLH e24ThetaBelowRoot))
+    e24KC2ThetaBelowNode110 e24KC2ThetaBelowNode111 e24KC2ThetaBelowNode112 e24KC2ThetaBelowNode113
+
+theorem e24KC2ThetaBelowNode12 :
+    adaptiveCoverCheck 16 (childHL (childLH e24ThetaBelowRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childHL (childLH e24ThetaBelowRoot))
+    e24KC2ThetaBelowLeaf120 e24KC2ThetaBelowLeaf121 e24KC2ThetaBelowLeaf122 e24KC2ThetaBelowLeaf123
+
+theorem e24KC2ThetaBelowNode13 :
+    adaptiveCoverCheck 16 (childHH (childLH e24ThetaBelowRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childHH (childLH e24ThetaBelowRoot))
+    e24KC2ThetaBelowNode130 e24KC2ThetaBelowNode131 e24KC2ThetaBelowLeaf132 e24KC2ThetaBelowLeaf133
+
+theorem e24KC2ThetaBelowNode30 :
+    adaptiveCoverCheck 16 (childLL (childHH e24ThetaBelowRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childLL (childHH e24ThetaBelowRoot))
+    e24KC2ThetaBelowLeaf300 e24KC2ThetaBelowLeaf301 e24KC2ThetaBelowLeaf302 e24KC2ThetaBelowLeaf303
+
+theorem e24KC2ThetaBelowNode31 :
+    adaptiveCoverCheck 16 (childLH (childHH e24ThetaBelowRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childLH (childHH e24ThetaBelowRoot))
+    e24KC2ThetaBelowLeaf310 e24KC2ThetaBelowLeaf311 e24KC2ThetaBelowLeaf312 e24KC2ThetaBelowLeaf313
+
+theorem e24KC2ThetaBelowNode33 :
+    adaptiveCoverCheck 16 (childHH (childHH e24ThetaBelowRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childHH (childHH e24ThetaBelowRoot))
+    e24KC2ThetaBelowLeaf330 e24KC2ThetaBelowLeaf331 e24KC2ThetaBelowLeaf332 e24KC2ThetaBelowLeaf333
+
+theorem e24KC2ThetaBelowNode0 :
+    adaptiveCoverCheck 17 (childLL e24ThetaBelowRoot) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childLL e24ThetaBelowRoot)
+    e24KC2ThetaBelowNode00 e24KC2ThetaBelowNode01 e24KC2ThetaBelowLeaf02 e24KC2ThetaBelowNode03
+
+theorem e24KC2ThetaBelowNode1 :
+    adaptiveCoverCheck 17 (childLH e24ThetaBelowRoot) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childLH e24ThetaBelowRoot)
+    e24KC2ThetaBelowNode10 e24KC2ThetaBelowNode11 e24KC2ThetaBelowNode12 e24KC2ThetaBelowNode13
+
+theorem e24KC2ThetaBelowNode3 :
+    adaptiveCoverCheck 17 (childHH e24ThetaBelowRoot) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childHH e24ThetaBelowRoot)
+    e24KC2ThetaBelowNode30 e24KC2ThetaBelowNode31 e24KC2ThetaBelowLeaf32 e24KC2ThetaBelowNode33
+
+theorem e24KC2ThetaBelowNodeROOT :
+    adaptiveCoverCheck 18 e24ThetaBelowRoot = true :=
+  adaptiveCoverCheck_succ_of_children 17 e24ThetaBelowRoot
+    e24KC2ThetaBelowNode0 e24KC2ThetaBelowNode1 e24KC2ThetaBelowLeaf2 e24KC2ThetaBelowNode3
+
+theorem e24ThetaBelowKernelCheck :
+    adaptiveCoverCheck 18 e24ThetaBelowRoot = true :=
+  e24KC2ThetaBelowNodeROOT
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c0_c0_c0_6_00029
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse9ea903ef1
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse9ea903ef1
+
+open CertificateCellse9ea903ef1
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c0 :
+    adaptiveCoverCheck 6 (childLL (childLL phiAboveCell11011010)) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childLL (childLL phiAboveCell11011010))
+    e24KC2PhiAboveLeaf1101101_c0_c0_c0_c0 e24KC2PhiAboveLeaf1101101_c0_c0_c0_c1
+      e24KC2PhiAboveLeaf1101101_c0_c0_c0_c2 e24KC2PhiAboveLeaf1101101_c0_c0_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c0_c0_c1_c0_5_00036
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells89d797401e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells89d797401e
+
+open CertificateCells89d797401e
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c1_c0 :
+    adaptiveCoverCheck 5 (childLL (childLH (childLL phiAboveCell11011010))) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLL (childLH (childLL phiAboveCell11011010)))
+    e24KC2PhiAboveLeaf1101101_c0_c0_c1_c0_c0 e24KC2PhiAboveLeaf1101101_c0_c0_c1_c0_c1
+      e24KC2PhiAboveLeaf1101101_c0_c0_c1_c0_c2 e24KC2PhiAboveLeaf1101101_c0_c0_c1_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c0_c0_c1_c1_5_00042
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsafdf6e4305
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsafdf6e4305
+
+open CertificateCellsafdf6e4305
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c1_c1 :
+    adaptiveCoverCheck 5 (childLH (childLH (childLL phiAboveCell11011010))) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLH (childLH (childLL phiAboveCell11011010)))
+    e24KC2PhiAboveLeaf1101101_c0_c0_c1_c1_c0 e24KC2PhiAboveLeaf1101101_c0_c0_c1_c1_c1
+      e24KC2PhiAboveLeaf1101101_c0_c0_c1_c1_c2 e24KC2PhiAboveLeaf1101101_c0_c0_c1_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c0_c0_c1_6_00045
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0d2c374949
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0d2c374949
+
+open CertificateCells0d2c374949
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0_c1 :
+    adaptiveCoverCheck 6 (childLH (childLL phiAboveCell11011010)) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childLH (childLL phiAboveCell11011010))
+    e24KC2PhiAboveLeaf1101101_c0_c0_c1_c0 e24KC2PhiAboveLeaf1101101_c0_c0_c1_c1
+      e24KC2PhiAboveLeaf1101101_c0_c0_c1_c2 e24KC2PhiAboveLeaf1101101_c0_c0_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c0_c0_7_00048
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3df86e4955
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3df86e4955
+
+open CertificateCells3df86e4955
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c0 :
+    adaptiveCoverCheck 7 (childLL phiAboveCell11011010) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childLL phiAboveCell11011010)
+    e24KC2PhiAboveLeaf1101101_c0_c0_c0 e24KC2PhiAboveLeaf1101101_c0_c0_c1
+      e24KC2PhiAboveLeaf1101101_c0_c0_c2 e24KC2PhiAboveLeaf1101101_c0_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c0_c1_c0_c0_5_00056
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3298a326ef
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3298a326ef
+
+open CertificateCells3298a326ef
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c0_c0 :
+    adaptiveCoverCheck 5 (childLL (childLL (childLH phiAboveCell11011010))) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLL (childLL (childLH phiAboveCell11011010)))
+    e24KC2PhiAboveLeaf1101101_c0_c1_c0_c0_c0 e24KC2PhiAboveLeaf1101101_c0_c1_c0_c0_c1
+      e24KC2PhiAboveLeaf1101101_c0_c1_c0_c0_c2 e24KC2PhiAboveLeaf1101101_c0_c1_c0_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c0_c1_c0_c1_5_00062
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells47980ca575
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells47980ca575
+
+open CertificateCells47980ca575
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c0_c1 :
+    adaptiveCoverCheck 5 (childLH (childLL (childLH phiAboveCell11011010))) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLH (childLL (childLH phiAboveCell11011010)))
+    e24KC2PhiAboveLeaf1101101_c0_c1_c0_c1_c0 e24KC2PhiAboveLeaf1101101_c0_c1_c0_c1_c1
+      e24KC2PhiAboveLeaf1101101_c0_c1_c0_c1_c2 e24KC2PhiAboveLeaf1101101_c0_c1_c0_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c0_c1_c0_6_00065
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsfed39385df
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsfed39385df
+
+open CertificateCellsfed39385df
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c0 :
+    adaptiveCoverCheck 6 (childLL (childLH phiAboveCell11011010)) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childLL (childLH phiAboveCell11011010))
+    e24KC2PhiAboveLeaf1101101_c0_c1_c0_c0 e24KC2PhiAboveLeaf1101101_c0_c1_c0_c1
+      e24KC2PhiAboveLeaf1101101_c0_c1_c0_c2 e24KC2PhiAboveLeaf1101101_c0_c1_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c0_c1_c1_6_00071
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb1b33fecca
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb1b33fecca
+
+open CertificateCellsb1b33fecca
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1_c1 :
+    adaptiveCoverCheck 6 (childLH (childLH phiAboveCell11011010)) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childLH (childLH phiAboveCell11011010))
+    e24KC2PhiAboveLeaf1101101_c0_c1_c1_c0 e24KC2PhiAboveLeaf1101101_c0_c1_c1_c1
+      e24KC2PhiAboveLeaf1101101_c0_c1_c1_c2 e24KC2PhiAboveLeaf1101101_c0_c1_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c0_c1_7_00074
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc427a8578f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc427a8578f
+
+open CertificateCellsc427a8578f
+
+theorem e24KC2PhiAboveLeaf1101101_c0_c1 :
+    adaptiveCoverCheck 7 (childLH phiAboveCell11011010) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childLH phiAboveCell11011010)
+    e24KC2PhiAboveLeaf1101101_c0_c1_c0 e24KC2PhiAboveLeaf1101101_c0_c1_c1
+      e24KC2PhiAboveLeaf1101101_c0_c1_c2 e24KC2PhiAboveLeaf1101101_c0_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c0_8_00077
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells25978ec18d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells25978ec18d
+
+open CertificateCells25978ec18d
+
+theorem e24KC2PhiAboveLeaf1101101_c0 :
+    adaptiveCoverCheck 8 phiAboveCell11011010 = true :=
+  adaptiveCoverCheck_succ_of_children 7 phiAboveCell11011010
+    e24KC2PhiAboveLeaf1101101_c0_c0 e24KC2PhiAboveLeaf1101101_c0_c1
+      e24KC2PhiAboveLeaf1101101_c0_c2 e24KC2PhiAboveLeaf1101101_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c1_c0_c0_6_00085
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse6161a42be
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse6161a42be
+
+open CertificateCellse6161a42be
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c0_c0 :
+    adaptiveCoverCheck 6 (childLL (childLL phiAboveCell11011011)) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childLL (childLL phiAboveCell11011011))
+    e24KC2PhiAboveLeaf1101101_c1_c0_c0_c0 e24KC2PhiAboveLeaf1101101_c1_c0_c0_c1
+      e24KC2PhiAboveLeaf1101101_c1_c0_c0_c2 e24KC2PhiAboveLeaf1101101_c1_c0_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c1_c0_c1_6_00091
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1f6ffcc997
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1f6ffcc997
+
+open CertificateCells1f6ffcc997
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c0_c1 :
+    adaptiveCoverCheck 6 (childLH (childLL phiAboveCell11011011)) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childLH (childLL phiAboveCell11011011))
+    e24KC2PhiAboveLeaf1101101_c1_c0_c1_c0 e24KC2PhiAboveLeaf1101101_c1_c0_c1_c1
+      e24KC2PhiAboveLeaf1101101_c1_c0_c1_c2 e24KC2PhiAboveLeaf1101101_c1_c0_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c1_c0_7_00094
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1275d3adc4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1275d3adc4
+
+open CertificateCells1275d3adc4
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c0 :
+    adaptiveCoverCheck 7 (childLL phiAboveCell11011011) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childLL phiAboveCell11011011)
+    e24KC2PhiAboveLeaf1101101_c1_c0_c0 e24KC2PhiAboveLeaf1101101_c1_c0_c1
+      e24KC2PhiAboveLeaf1101101_c1_c0_c2 e24KC2PhiAboveLeaf1101101_c1_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c1_c1_c0_6_00101
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells021b870816
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells021b870816
+
+open CertificateCells021b870816
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c1_c0 :
+    adaptiveCoverCheck 6 (childLL (childLH phiAboveCell11011011)) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childLL (childLH phiAboveCell11011011))
+    e24KC2PhiAboveLeaf1101101_c1_c1_c0_c0 e24KC2PhiAboveLeaf1101101_c1_c1_c0_c1
+      e24KC2PhiAboveLeaf1101101_c1_c1_c0_c2 e24KC2PhiAboveLeaf1101101_c1_c1_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c1_c1_7_00105
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells9ade3f586f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells9ade3f586f
+
+open CertificateCells9ade3f586f
+
+theorem e24KC2PhiAboveLeaf1101101_c1_c1 :
+    adaptiveCoverCheck 7 (childLH phiAboveCell11011011) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childLH phiAboveCell11011011)
+    e24KC2PhiAboveLeaf1101101_c1_c1_c0 e24KC2PhiAboveLeaf1101101_c1_c1_c1
+      e24KC2PhiAboveLeaf1101101_c1_c1_c2 e24KC2PhiAboveLeaf1101101_c1_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_c1_8_00108
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsfd8221930a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsfd8221930a
+
+open CertificateCellsfd8221930a
+
+theorem e24KC2PhiAboveLeaf1101101_c1 :
+    adaptiveCoverCheck 8 phiAboveCell11011011 = true :=
+  adaptiveCoverCheck_succ_of_children 7 phiAboveCell11011011
+    e24KC2PhiAboveLeaf1101101_c1_c0 e24KC2PhiAboveLeaf1101101_c1_c1
+      e24KC2PhiAboveLeaf1101101_c1_c2 e24KC2PhiAboveLeaf1101101_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101101_9_00111
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5aaf24f73d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5aaf24f73d
+
+open CertificateCells5aaf24f73d
+
+theorem e24KC2PhiAboveLeaf1101101 :
+    adaptiveCoverCheck 9 (childLH (childLL (childLH phiAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 8 (childLH (childLL (childLH phiAboveCell1101)))
+    e24KC2PhiAboveLeaf1101101_c0 e24KC2PhiAboveLeaf1101101_c1 e24KC2PhiAboveLeaf1101101_c2
+      e24KC2PhiAboveLeaf1101101_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101110_c0_c0_7_00122
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells31cca9f2c5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells31cca9f2c5
+
+open CertificateCells31cca9f2c5
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c0 :
+    adaptiveCoverCheck 7 (childLL phiAboveCell11011100) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childLL phiAboveCell11011100)
+    e24KC2PhiAboveLeaf1101110_c0_c0_c0 e24KC2PhiAboveLeaf1101110_c0_c0_c1
+      e24KC2PhiAboveLeaf1101110_c0_c0_c2 e24KC2PhiAboveLeaf1101110_c0_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101110_c0_c1_c0_6_00129
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0a8495a296
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0a8495a296
+
+open CertificateCells0a8495a296
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c1_c0 :
+    adaptiveCoverCheck 6 (childLL (childLH phiAboveCell11011100)) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childLL (childLH phiAboveCell11011100))
+    e24KC2PhiAboveLeaf1101110_c0_c1_c0_c0 e24KC2PhiAboveLeaf1101110_c0_c1_c0_c1
+      e24KC2PhiAboveLeaf1101110_c0_c1_c0_c2 e24KC2PhiAboveLeaf1101110_c0_c1_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101110_c0_c1_7_00133
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4cabb07dbd
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4cabb07dbd
+
+open CertificateCells4cabb07dbd
+
+theorem e24KC2PhiAboveLeaf1101110_c0_c1 :
+    adaptiveCoverCheck 7 (childLH phiAboveCell11011100) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childLH phiAboveCell11011100)
+    e24KC2PhiAboveLeaf1101110_c0_c1_c0 e24KC2PhiAboveLeaf1101110_c0_c1_c1
+      e24KC2PhiAboveLeaf1101110_c0_c1_c2 e24KC2PhiAboveLeaf1101110_c0_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101110_c0_8_00136
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc6c93e8f5c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc6c93e8f5c
+
+open CertificateCellsc6c93e8f5c
+
+theorem e24KC2PhiAboveLeaf1101110_c0 :
+    adaptiveCoverCheck 8 phiAboveCell11011100 = true :=
+  adaptiveCoverCheck_succ_of_children 7 phiAboveCell11011100
+    e24KC2PhiAboveLeaf1101110_c0_c0 e24KC2PhiAboveLeaf1101110_c0_c1
+      e24KC2PhiAboveLeaf1101110_c0_c2 e24KC2PhiAboveLeaf1101110_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101110_c1_8_00142
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsaa40da30f6
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsaa40da30f6
+
+open CertificateCellsaa40da30f6
+
+theorem e24KC2PhiAboveLeaf1101110_c1 :
+    adaptiveCoverCheck 8 phiAboveCell11011101 = true :=
+  adaptiveCoverCheck_succ_of_children 7 phiAboveCell11011101
+    e24KC2PhiAboveLeaf1101110_c1_c0 e24KC2PhiAboveLeaf1101110_c1_c1
+      e24KC2PhiAboveLeaf1101110_c1_c2 e24KC2PhiAboveLeaf1101110_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101110_9_00145
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells919c5d7ca9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells919c5d7ca9
+
+open CertificateCells919c5d7ca9
+
+theorem e24KC2PhiAboveLeaf1101110 :
+    adaptiveCoverCheck 9 (childLL (childLH (childLH phiAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 8 (childLL (childLH (childLH phiAboveCell1101)))
+    e24KC2PhiAboveLeaf1101110_c0 e24KC2PhiAboveLeaf1101110_c1 e24KC2PhiAboveLeaf1101110_c2
+      e24KC2PhiAboveLeaf1101110_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101111_c0_c1_7_00156
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells87d4f5e678
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells87d4f5e678
+
+open CertificateCells87d4f5e678
+
+theorem e24KC2PhiAboveLeaf1101111_c0_c1 :
+    adaptiveCoverCheck 7 (childLH phiAboveCell11011110) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childLH phiAboveCell11011110)
+    e24KC2PhiAboveLeaf1101111_c0_c1_c0 e24KC2PhiAboveLeaf1101111_c0_c1_c1
+      e24KC2PhiAboveLeaf1101111_c0_c1_c2 e24KC2PhiAboveLeaf1101111_c0_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101111_c0_8_00159
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsefc5f924ee
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsefc5f924ee
+
+open CertificateCellsefc5f924ee
+
+theorem e24KC2PhiAboveLeaf1101111_c0 :
+    adaptiveCoverCheck 8 phiAboveCell11011110 = true :=
+  adaptiveCoverCheck_succ_of_children 7 phiAboveCell11011110
+    e24KC2PhiAboveLeaf1101111_c0_c0 e24KC2PhiAboveLeaf1101111_c0_c1
+      e24KC2PhiAboveLeaf1101111_c0_c2 e24KC2PhiAboveLeaf1101111_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101111_c1_8_00165
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsac583a159d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsac583a159d
+
+open CertificateCellsac583a159d
+
+theorem e24KC2PhiAboveLeaf1101111_c1 :
+    adaptiveCoverCheck 8 phiAboveCell11011111 = true :=
+  adaptiveCoverCheck_succ_of_children 7 phiAboveCell11011111
+    e24KC2PhiAboveLeaf1101111_c1_c0 e24KC2PhiAboveLeaf1101111_c1_c1
+      e24KC2PhiAboveLeaf1101111_c1_c2 e24KC2PhiAboveLeaf1101111_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1101111_9_00168
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf317a94cb3
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf317a94cb3
+
+open CertificateCellsf317a94cb3
+
+theorem e24KC2PhiAboveLeaf1101111 :
+    adaptiveCoverCheck 9 (childLH (childLH (childLH phiAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 8 (childLH (childLH (childLH phiAboveCell1101)))
+    e24KC2PhiAboveLeaf1101111_c0 e24KC2PhiAboveLeaf1101111_c1 e24KC2PhiAboveLeaf1101111_c2
+      e24KC2PhiAboveLeaf1101111_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1110000_c0_8_00178
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf7dce840c8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf7dce840c8
+
+open CertificateCellsf7dce840c8
+
+theorem e24KC2PhiAboveLeaf1110000_c0 :
+    adaptiveCoverCheck 8 phiAboveCell11100000 = true :=
+  adaptiveCoverCheck_succ_of_children 7 phiAboveCell11100000
+    e24KC2PhiAboveLeaf1110000_c0_c0 e24KC2PhiAboveLeaf1110000_c0_c1
+      e24KC2PhiAboveLeaf1110000_c0_c2 e24KC2PhiAboveLeaf1110000_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1110000_9_00182
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5fe660c4fc
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5fe660c4fc
+
+open CertificateCells5fe660c4fc
+
+theorem e24KC2PhiAboveLeaf1110000 :
+    adaptiveCoverCheck 9 (childLL (childLL (childLL phiAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 8 (childLL (childLL (childLL phiAboveCell1110)))
+    e24KC2PhiAboveLeaf1110000_c0 e24KC2PhiAboveLeaf1110000_c1 e24KC2PhiAboveLeaf1110000_c2
+      e24KC2PhiAboveLeaf1110000_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1110001_9_00191
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells7e3a6c7b6b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells7e3a6c7b6b
+
+open CertificateCells7e3a6c7b6b
+
+theorem e24KC2PhiAboveLeaf1110001 :
+    adaptiveCoverCheck 9 (childLH (childLL (childLL phiAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 8 (childLH (childLL (childLL phiAboveCell1110)))
+    e24KC2PhiAboveLeaf1110001_c0 e24KC2PhiAboveLeaf1110001_c1 e24KC2PhiAboveLeaf1110001_c2
+      e24KC2PhiAboveLeaf1110001_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1110010_9_00200
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa5efdb03f3
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa5efdb03f3
+
+open CertificateCellsa5efdb03f3
+
+theorem e24KC2PhiAboveLeaf1110010 :
+    adaptiveCoverCheck 9 (childLL (childLH (childLL phiAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 8 (childLL (childLH (childLL phiAboveCell1110)))
+    e24KC2PhiAboveLeaf1110010_c0 e24KC2PhiAboveLeaf1110010_c1 e24KC2PhiAboveLeaf1110010_c2
+      e24KC2PhiAboveLeaf1110010_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Above Leaf1110011_9_00208
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4f05b2093e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4f05b2093e
+
+open CertificateCells4f05b2093e
+
+theorem e24KC2PhiAboveLeaf1110011 :
+    adaptiveCoverCheck 9 (childLH (childLH (childLL phiAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 8 (childLH (childLH (childLL phiAboveCell1110)))
+    e24KC2PhiAboveLeaf1110011_c0 e24KC2PhiAboveLeaf1110011_c1 e24KC2PhiAboveLeaf1110011_c2
+      e24KC2PhiAboveLeaf1110011_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC6Phi Above Reconstruct
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa15368d7db
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa15368d7db
+
+open CertificateCellsa15368d7db
+
+theorem e24KC2PhiAboveNode100101 :
+    adaptiveCoverCheck 10 (childLH (childLL phiAboveCell1001)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH (childLL phiAboveCell1001))
+    e24KC2PhiAboveLeaf1001010 e24KC2PhiAboveLeaf1001011 e24KC2PhiAboveLeaf1001012
+      e24KC2PhiAboveLeaf1001013
+
+theorem e24KC2PhiAboveNode100110 :
+    adaptiveCoverCheck 10 (childLL (childLH phiAboveCell1001)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL (childLH phiAboveCell1001))
+    e24KC2PhiAboveLeaf1001100 e24KC2PhiAboveLeaf1001101 e24KC2PhiAboveLeaf1001102
+      e24KC2PhiAboveLeaf1001103
+
+theorem e24KC2PhiAboveNode100111 :
+    adaptiveCoverCheck 10 (childLH (childLH phiAboveCell1001)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH (childLH phiAboveCell1001))
+    e24KC2PhiAboveLeaf1001110 e24KC2PhiAboveLeaf1001111 e24KC2PhiAboveLeaf1001112
+      e24KC2PhiAboveLeaf1001113
+
+theorem e24KC2PhiAboveNode101000 :
+    adaptiveCoverCheck 10 (childLL (childLL phiAboveCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL (childLL phiAboveCell1010))
+    e24KC2PhiAboveLeaf1010000 e24KC2PhiAboveLeaf1010001 e24KC2PhiAboveLeaf1010002
+      e24KC2PhiAboveLeaf1010003
+
+theorem e24KC2PhiAboveNode101001 :
+    adaptiveCoverCheck 10 (childLH (childLL phiAboveCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH (childLL phiAboveCell1010))
+    e24KC2PhiAboveLeaf1010010 e24KC2PhiAboveLeaf1010011 e24KC2PhiAboveLeaf1010012
+      e24KC2PhiAboveLeaf1010013
+
+theorem e24KC2PhiAboveNode101010 :
+    adaptiveCoverCheck 10 (childLL (childLH phiAboveCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL (childLH phiAboveCell1010))
+    e24KC2PhiAboveLeaf1010100 e24KC2PhiAboveLeaf1010101 e24KC2PhiAboveLeaf1010102
+      e24KC2PhiAboveLeaf1010103
+
+theorem e24KC2PhiAboveNode101011 :
+    adaptiveCoverCheck 10 (childLH (childLH phiAboveCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH (childLH phiAboveCell1010))
+    e24KC2PhiAboveLeaf1010110 e24KC2PhiAboveLeaf1010111 e24KC2PhiAboveLeaf1010112
+      e24KC2PhiAboveLeaf1010113
+
+theorem e24KC2PhiAboveNode101100 :
+    adaptiveCoverCheck 10 (childLL (childLL phiAboveCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL (childLL phiAboveCell1011))
+    e24KC2PhiAboveLeaf1011000 e24KC2PhiAboveLeaf1011001 e24KC2PhiAboveLeaf1011002
+      e24KC2PhiAboveLeaf1011003
+
+theorem e24KC2PhiAboveNode101101 :
+    adaptiveCoverCheck 10 (childLH (childLL phiAboveCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH (childLL phiAboveCell1011))
+    e24KC2PhiAboveLeaf1011010 e24KC2PhiAboveLeaf1011011 e24KC2PhiAboveLeaf1011012
+      e24KC2PhiAboveLeaf1011013
+
+theorem e24KC2PhiAboveNode101110 :
+    adaptiveCoverCheck 10 (childLL (childLH phiAboveCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL (childLH phiAboveCell1011))
+    e24KC2PhiAboveLeaf1011100 e24KC2PhiAboveLeaf1011101 e24KC2PhiAboveLeaf1011102
+      e24KC2PhiAboveLeaf1011103
+
+theorem e24KC2PhiAboveNode101111 :
+    adaptiveCoverCheck 10 (childLH (childLH phiAboveCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH (childLH phiAboveCell1011))
+    e24KC2PhiAboveLeaf1011110 e24KC2PhiAboveLeaf1011111 e24KC2PhiAboveLeaf1011112
+      e24KC2PhiAboveLeaf1011113
+
+theorem e24KC2PhiAboveNode110000 :
+    adaptiveCoverCheck 10 (childLL (childLL phiAboveCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL (childLL phiAboveCell1100))
+    e24KC2PhiAboveLeaf1100000 e24KC2PhiAboveLeaf1100001 e24KC2PhiAboveLeaf1100002
+      e24KC2PhiAboveLeaf1100003
+
+theorem e24KC2PhiAboveNode110001 :
+    adaptiveCoverCheck 10 (childLH (childLL phiAboveCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH (childLL phiAboveCell1100))
+    e24KC2PhiAboveLeaf1100010 e24KC2PhiAboveLeaf1100011 e24KC2PhiAboveLeaf1100012
+      e24KC2PhiAboveLeaf1100013
+
+theorem e24KC2PhiAboveNode110010 :
+    adaptiveCoverCheck 10 (childLL (childLH phiAboveCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL (childLH phiAboveCell1100))
+    e24KC2PhiAboveLeaf1100100 e24KC2PhiAboveLeaf1100101 e24KC2PhiAboveLeaf1100102
+      e24KC2PhiAboveLeaf1100103
+
+theorem e24KC2PhiAboveNode110011 :
+    adaptiveCoverCheck 10 (childLH (childLH phiAboveCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH (childLH phiAboveCell1100))
+    e24KC2PhiAboveLeaf1100110 e24KC2PhiAboveLeaf1100111 e24KC2PhiAboveLeaf1100112
+      e24KC2PhiAboveLeaf1100113
+
+theorem e24KC2PhiAboveNode110013 :
+    adaptiveCoverCheck 10 (childHH (childLH phiAboveCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH (childLH phiAboveCell1100))
+    e24KC2PhiAboveLeaf1100130 e24KC2PhiAboveLeaf1100131 e24KC2PhiAboveLeaf1100132
+      e24KC2PhiAboveLeaf1100133
+
+theorem e24KC2PhiAboveNode110100 :
+    adaptiveCoverCheck 10 (childLL (childLL phiAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL (childLL phiAboveCell1101))
+    e24KC2PhiAboveLeaf1101000 e24KC2PhiAboveLeaf1101001 e24KC2PhiAboveLeaf1101002
+      e24KC2PhiAboveLeaf1101003
+
+theorem e24KC2PhiAboveNode110101 :
+    adaptiveCoverCheck 10 (childLH (childLL phiAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH (childLL phiAboveCell1101))
+    e24KC2PhiAboveLeaf1101010 e24KC2PhiAboveLeaf1101011 e24KC2PhiAboveLeaf1101012
+      e24KC2PhiAboveLeaf1101013
+
+theorem e24KC2PhiAboveNode110102 :
+    adaptiveCoverCheck 10 (childHL (childLL phiAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL (childLL phiAboveCell1101))
+    e24KC2PhiAboveLeaf1101020 e24KC2PhiAboveLeaf1101021 e24KC2PhiAboveLeaf1101022
+      e24KC2PhiAboveLeaf1101023
+
+theorem e24KC2PhiAboveNode110103 :
+    adaptiveCoverCheck 10 (childHH (childLL phiAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH (childLL phiAboveCell1101))
+    e24KC2PhiAboveLeaf1101030 e24KC2PhiAboveLeaf1101031 e24KC2PhiAboveLeaf1101032
+      e24KC2PhiAboveLeaf1101033
+
+theorem e24KC2PhiAboveNode110110 :
+    adaptiveCoverCheck 10 (childLL (childLH phiAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL (childLH phiAboveCell1101))
+    e24KC2PhiAboveLeaf1101100 e24KC2PhiAboveLeaf1101101 e24KC2PhiAboveLeaf1101102
+      e24KC2PhiAboveLeaf1101103
+
+theorem e24KC2PhiAboveNode110111 :
+    adaptiveCoverCheck 10 (childLH (childLH phiAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH (childLH phiAboveCell1101))
+    e24KC2PhiAboveLeaf1101110 e24KC2PhiAboveLeaf1101111 e24KC2PhiAboveLeaf1101112
+      e24KC2PhiAboveLeaf1101113
+
+theorem e24KC2PhiAboveNode110112 :
+    adaptiveCoverCheck 10 (childHL (childLH phiAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL (childLH phiAboveCell1101))
+    e24KC2PhiAboveLeaf1101120 e24KC2PhiAboveLeaf1101121 e24KC2PhiAboveLeaf1101122
+      e24KC2PhiAboveLeaf1101123
+
+theorem e24KC2PhiAboveNode110113 :
+    adaptiveCoverCheck 10 (childHH (childLH phiAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH (childLH phiAboveCell1101))
+    e24KC2PhiAboveLeaf1101130 e24KC2PhiAboveLeaf1101131 e24KC2PhiAboveLeaf1101132
+      e24KC2PhiAboveLeaf1101133
+
+theorem e24KC2PhiAboveNode111000 :
+    adaptiveCoverCheck 10 (childLL (childLL phiAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL (childLL phiAboveCell1110))
+    e24KC2PhiAboveLeaf1110000 e24KC2PhiAboveLeaf1110001 e24KC2PhiAboveLeaf1110002
+      e24KC2PhiAboveLeaf1110003
+
+theorem e24KC2PhiAboveNode111001 :
+    adaptiveCoverCheck 10 (childLH (childLL phiAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH (childLL phiAboveCell1110))
+    e24KC2PhiAboveLeaf1110010 e24KC2PhiAboveLeaf1110011 e24KC2PhiAboveLeaf1110012
+      e24KC2PhiAboveLeaf1110013
+
+theorem e24KC2PhiAboveNode111002 :
+    adaptiveCoverCheck 10 (childHL (childLL phiAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL (childLL phiAboveCell1110))
+    e24KC2PhiAboveLeaf1110020 e24KC2PhiAboveLeaf1110021 e24KC2PhiAboveLeaf1110022
+      e24KC2PhiAboveLeaf1110023
+
+theorem e24KC2PhiAboveNode111003 :
+    adaptiveCoverCheck 10 (childHH (childLL phiAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH (childLL phiAboveCell1110))
+    e24KC2PhiAboveLeaf1110030 e24KC2PhiAboveLeaf1110031 e24KC2PhiAboveLeaf1110032
+      e24KC2PhiAboveLeaf1110033
+
+theorem e24KC2PhiAboveNode111010 :
+    adaptiveCoverCheck 10 (childLL (childLH phiAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL (childLH phiAboveCell1110))
+    e24KC2PhiAboveLeaf1110100 e24KC2PhiAboveLeaf1110101 e24KC2PhiAboveLeaf1110102
+      e24KC2PhiAboveLeaf1110103
+
+theorem e24KC2PhiAboveNode111011 :
+    adaptiveCoverCheck 10 (childLH (childLH phiAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH (childLH phiAboveCell1110))
+    e24KC2PhiAboveLeaf1110110 e24KC2PhiAboveLeaf1110111 e24KC2PhiAboveLeaf1110112
+      e24KC2PhiAboveLeaf1110113
+
+theorem e24KC2PhiAboveNode111012 :
+    adaptiveCoverCheck 10 (childHL (childLH phiAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL (childLH phiAboveCell1110))
+    e24KC2PhiAboveLeaf1110120 e24KC2PhiAboveLeaf1110121 e24KC2PhiAboveLeaf1110122
+      e24KC2PhiAboveLeaf1110123
+
+theorem e24KC2PhiAboveNode111013 :
+    adaptiveCoverCheck 10 (childHH (childLH phiAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH (childLH phiAboveCell1110))
+    e24KC2PhiAboveLeaf1110130 e24KC2PhiAboveLeaf1110131 e24KC2PhiAboveLeaf1110132
+      e24KC2PhiAboveLeaf1110133
+
+theorem e24KC2PhiAboveNode111100 :
+    adaptiveCoverCheck 10 (childLL (childLL phiAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL (childLL phiAboveCell1111))
+    e24KC2PhiAboveLeaf1111000 e24KC2PhiAboveLeaf1111001 e24KC2PhiAboveLeaf1111002
+      e24KC2PhiAboveLeaf1111003
+
+theorem e24KC2PhiAboveNode111101 :
+    adaptiveCoverCheck 10 (childLH (childLL phiAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH (childLL phiAboveCell1111))
+    e24KC2PhiAboveLeaf1111010 e24KC2PhiAboveLeaf1111011 e24KC2PhiAboveLeaf1111012
+      e24KC2PhiAboveLeaf1111013
+
+theorem e24KC2PhiAboveNode111102 :
+    adaptiveCoverCheck 10 (childHL (childLL phiAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL (childLL phiAboveCell1111))
+    e24KC2PhiAboveLeaf1111020 e24KC2PhiAboveLeaf1111021 e24KC2PhiAboveLeaf1111022
+      e24KC2PhiAboveLeaf1111023
+
+theorem e24KC2PhiAboveNode111103 :
+    adaptiveCoverCheck 10 (childHH (childLL phiAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH (childLL phiAboveCell1111))
+    e24KC2PhiAboveLeaf1111030 e24KC2PhiAboveLeaf1111031 e24KC2PhiAboveLeaf1111032
+      e24KC2PhiAboveLeaf1111033
+
+theorem e24KC2PhiAboveNode111110 :
+    adaptiveCoverCheck 10 (childLL (childLH phiAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL (childLH phiAboveCell1111))
+    e24KC2PhiAboveLeaf1111100 e24KC2PhiAboveLeaf1111101 e24KC2PhiAboveLeaf1111102
+      e24KC2PhiAboveLeaf1111103
+
+theorem e24KC2PhiAboveNode111111 :
+    adaptiveCoverCheck 10 (childLH (childLH phiAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH (childLH phiAboveCell1111))
+    e24KC2PhiAboveLeaf1111110 e24KC2PhiAboveLeaf1111111 e24KC2PhiAboveLeaf1111112
+      e24KC2PhiAboveLeaf1111113
+
+theorem e24KC2PhiAboveNode111112 :
+    adaptiveCoverCheck 10 (childHL (childLH phiAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL (childLH phiAboveCell1111))
+    e24KC2PhiAboveLeaf1111120 e24KC2PhiAboveLeaf1111121 e24KC2PhiAboveLeaf1111122
+      e24KC2PhiAboveLeaf1111123
+
+theorem e24KC2PhiAboveNode111113 :
+    adaptiveCoverCheck 10 (childHH (childLH phiAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH (childLH phiAboveCell1111))
+    e24KC2PhiAboveLeaf1111130 e24KC2PhiAboveLeaf1111131 e24KC2PhiAboveLeaf1111132
+      e24KC2PhiAboveLeaf1111133
+
+theorem e24KC2PhiAboveNode01011 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell0101) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH phiAboveCell0101)
+    e24KC2PhiAboveLeaf010110 e24KC2PhiAboveLeaf010111 e24KC2PhiAboveLeaf010112
+      e24KC2PhiAboveLeaf010113
+
+theorem e24KC2PhiAboveNode01100 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell0110) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL phiAboveCell0110)
+    e24KC2PhiAboveLeaf011000 e24KC2PhiAboveLeaf011001 e24KC2PhiAboveLeaf011002
+      e24KC2PhiAboveLeaf011003
+
+theorem e24KC2PhiAboveNode01101 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell0110) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH phiAboveCell0110)
+    e24KC2PhiAboveLeaf011010 e24KC2PhiAboveLeaf011011 e24KC2PhiAboveLeaf011012
+      e24KC2PhiAboveLeaf011013
+
+theorem e24KC2PhiAboveNode01110 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell0111) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL phiAboveCell0111)
+    e24KC2PhiAboveLeaf011100 e24KC2PhiAboveLeaf011101 e24KC2PhiAboveLeaf011102
+      e24KC2PhiAboveLeaf011103
+
+theorem e24KC2PhiAboveNode01111 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell0111) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH phiAboveCell0111)
+    e24KC2PhiAboveLeaf011110 e24KC2PhiAboveLeaf011111 e24KC2PhiAboveLeaf011112
+      e24KC2PhiAboveLeaf011113
+
+theorem e24KC2PhiAboveNode10000 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1000) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL phiAboveCell1000)
+    e24KC2PhiAboveLeaf100000 e24KC2PhiAboveLeaf100001 e24KC2PhiAboveLeaf100002
+      e24KC2PhiAboveLeaf100003
+
+theorem e24KC2PhiAboveNode10001 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1000) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH phiAboveCell1000)
+    e24KC2PhiAboveLeaf100010 e24KC2PhiAboveLeaf100011 e24KC2PhiAboveLeaf100012
+      e24KC2PhiAboveLeaf100013
+
+theorem e24KC2PhiAboveNode10010 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1001) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL phiAboveCell1001)
+    e24KC2PhiAboveLeaf100100 e24KC2PhiAboveNode100101 e24KC2PhiAboveLeaf100102
+      e24KC2PhiAboveLeaf100103
+
+theorem e24KC2PhiAboveNode10011 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1001) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH phiAboveCell1001)
+    e24KC2PhiAboveNode100110 e24KC2PhiAboveNode100111 e24KC2PhiAboveLeaf100112
+      e24KC2PhiAboveLeaf100113
+
+theorem e24KC2PhiAboveNode10100 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1010) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL phiAboveCell1010)
+    e24KC2PhiAboveNode101000 e24KC2PhiAboveNode101001 e24KC2PhiAboveLeaf101002
+      e24KC2PhiAboveLeaf101003
+
+theorem e24KC2PhiAboveNode10101 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1010) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH phiAboveCell1010)
+    e24KC2PhiAboveNode101010 e24KC2PhiAboveNode101011 e24KC2PhiAboveLeaf101012
+      e24KC2PhiAboveLeaf101013
+
+theorem e24KC2PhiAboveNode10110 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1011) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL phiAboveCell1011)
+    e24KC2PhiAboveNode101100 e24KC2PhiAboveNode101101 e24KC2PhiAboveLeaf101102
+      e24KC2PhiAboveLeaf101103
+
+theorem e24KC2PhiAboveNode10111 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1011) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH phiAboveCell1011)
+    e24KC2PhiAboveNode101110 e24KC2PhiAboveNode101111 e24KC2PhiAboveLeaf101112
+      e24KC2PhiAboveLeaf101113
+
+theorem e24KC2PhiAboveNode10113 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1011) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH phiAboveCell1011)
+    e24KC2PhiAboveLeaf101130 e24KC2PhiAboveLeaf101131 e24KC2PhiAboveLeaf101132
+      e24KC2PhiAboveLeaf101133
+
+theorem e24KC2PhiAboveNode11000 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1100) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL phiAboveCell1100)
+    e24KC2PhiAboveNode110000 e24KC2PhiAboveNode110001 e24KC2PhiAboveLeaf110002
+      e24KC2PhiAboveLeaf110003
+
+theorem e24KC2PhiAboveNode11001 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1100) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH phiAboveCell1100)
+    e24KC2PhiAboveNode110010 e24KC2PhiAboveNode110011 e24KC2PhiAboveLeaf110012
+      e24KC2PhiAboveNode110013
+
+theorem e24KC2PhiAboveNode11002 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1100) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL phiAboveCell1100)
+    e24KC2PhiAboveLeaf110020 e24KC2PhiAboveLeaf110021 e24KC2PhiAboveLeaf110022
+      e24KC2PhiAboveLeaf110023
+
+theorem e24KC2PhiAboveNode11003 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1100) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH phiAboveCell1100)
+    e24KC2PhiAboveLeaf110030 e24KC2PhiAboveLeaf110031 e24KC2PhiAboveLeaf110032
+      e24KC2PhiAboveLeaf110033
+
+theorem e24KC2PhiAboveNode11010 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1101) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL phiAboveCell1101)
+    e24KC2PhiAboveNode110100 e24KC2PhiAboveNode110101 e24KC2PhiAboveNode110102
+      e24KC2PhiAboveNode110103
+
+theorem e24KC2PhiAboveNode11011 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1101) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH phiAboveCell1101)
+    e24KC2PhiAboveNode110110 e24KC2PhiAboveNode110111 e24KC2PhiAboveNode110112
+      e24KC2PhiAboveNode110113
+
+theorem e24KC2PhiAboveNode11012 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1101) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL phiAboveCell1101)
+    e24KC2PhiAboveLeaf110120 e24KC2PhiAboveLeaf110121 e24KC2PhiAboveLeaf110122
+      e24KC2PhiAboveLeaf110123
+
+theorem e24KC2PhiAboveNode11013 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1101) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH phiAboveCell1101)
+    e24KC2PhiAboveLeaf110130 e24KC2PhiAboveLeaf110131 e24KC2PhiAboveLeaf110132
+      e24KC2PhiAboveLeaf110133
+
+theorem e24KC2PhiAboveNode11100 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1110) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL phiAboveCell1110)
+    e24KC2PhiAboveNode111000 e24KC2PhiAboveNode111001 e24KC2PhiAboveNode111002
+      e24KC2PhiAboveNode111003
+
+theorem e24KC2PhiAboveNode11101 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1110) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH phiAboveCell1110)
+    e24KC2PhiAboveNode111010 e24KC2PhiAboveNode111011 e24KC2PhiAboveNode111012
+      e24KC2PhiAboveNode111013
+
+theorem e24KC2PhiAboveNode11102 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1110) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL phiAboveCell1110)
+    e24KC2PhiAboveLeaf111020 e24KC2PhiAboveLeaf111021 e24KC2PhiAboveLeaf111022
+      e24KC2PhiAboveLeaf111023
+
+theorem e24KC2PhiAboveNode11103 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1110) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH phiAboveCell1110)
+    e24KC2PhiAboveLeaf111030 e24KC2PhiAboveLeaf111031 e24KC2PhiAboveLeaf111032
+      e24KC2PhiAboveLeaf111033
+
+theorem e24KC2PhiAboveNode11110 :
+    adaptiveCoverCheck 11 (childLL phiAboveCell1111) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL phiAboveCell1111)
+    e24KC2PhiAboveNode111100 e24KC2PhiAboveNode111101 e24KC2PhiAboveNode111102
+      e24KC2PhiAboveNode111103
+
+theorem e24KC2PhiAboveNode11111 :
+    adaptiveCoverCheck 11 (childLH phiAboveCell1111) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH phiAboveCell1111)
+    e24KC2PhiAboveNode111110 e24KC2PhiAboveNode111111 e24KC2PhiAboveNode111112
+      e24KC2PhiAboveNode111113
+
+theorem e24KC2PhiAboveNode11112 :
+    adaptiveCoverCheck 11 (childHL phiAboveCell1111) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL phiAboveCell1111)
+    e24KC2PhiAboveLeaf111120 e24KC2PhiAboveLeaf111121 e24KC2PhiAboveLeaf111122
+      e24KC2PhiAboveLeaf111123
+
+theorem e24KC2PhiAboveNode11113 :
+    adaptiveCoverCheck 11 (childHH phiAboveCell1111) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH phiAboveCell1111)
+    e24KC2PhiAboveLeaf111130 e24KC2PhiAboveLeaf111131 e24KC2PhiAboveLeaf111132
+      e24KC2PhiAboveLeaf111133
+
+theorem e24KC2PhiAboveNode0000 :
+    adaptiveCoverCheck 12 phiAboveCell0000 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell0000
+    e24KC2PhiAboveLeaf00000 e24KC2PhiAboveLeaf00001 e24KC2PhiAboveLeaf00002 e24KC2PhiAboveLeaf00003
+
+theorem e24KC2PhiAboveNode0001 :
+    adaptiveCoverCheck 12 phiAboveCell0001 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell0001
+    e24KC2PhiAboveLeaf00010 e24KC2PhiAboveLeaf00011 e24KC2PhiAboveLeaf00012 e24KC2PhiAboveLeaf00013
+
+theorem e24KC2PhiAboveNode0010 :
+    adaptiveCoverCheck 12 phiAboveCell0010 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell0010
+    e24KC2PhiAboveLeaf00100 e24KC2PhiAboveLeaf00101 e24KC2PhiAboveLeaf00102 e24KC2PhiAboveLeaf00103
+
+theorem e24KC2PhiAboveNode0011 :
+    adaptiveCoverCheck 12 phiAboveCell0011 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell0011
+    e24KC2PhiAboveLeaf00110 e24KC2PhiAboveLeaf00111 e24KC2PhiAboveLeaf00112 e24KC2PhiAboveLeaf00113
+
+theorem e24KC2PhiAboveNode0100 :
+    adaptiveCoverCheck 12 phiAboveCell0100 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell0100
+    e24KC2PhiAboveLeaf01000 e24KC2PhiAboveLeaf01001 e24KC2PhiAboveLeaf01002 e24KC2PhiAboveLeaf01003
+
+theorem e24KC2PhiAboveNode0101 :
+    adaptiveCoverCheck 12 phiAboveCell0101 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell0101
+    e24KC2PhiAboveLeaf01010 e24KC2PhiAboveNode01011 e24KC2PhiAboveLeaf01012 e24KC2PhiAboveLeaf01013
+
+theorem e24KC2PhiAboveNode0110 :
+    adaptiveCoverCheck 12 phiAboveCell0110 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell0110
+    e24KC2PhiAboveNode01100 e24KC2PhiAboveNode01101 e24KC2PhiAboveLeaf01102 e24KC2PhiAboveLeaf01103
+
+theorem e24KC2PhiAboveNode0111 :
+    adaptiveCoverCheck 12 phiAboveCell0111 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell0111
+    e24KC2PhiAboveNode01110 e24KC2PhiAboveNode01111 e24KC2PhiAboveLeaf01112 e24KC2PhiAboveLeaf01113
+
+theorem e24KC2PhiAboveNode1000 :
+    adaptiveCoverCheck 12 phiAboveCell1000 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1000
+    e24KC2PhiAboveNode10000 e24KC2PhiAboveNode10001 e24KC2PhiAboveLeaf10002 e24KC2PhiAboveLeaf10003
+
+theorem e24KC2PhiAboveNode1001 :
+    adaptiveCoverCheck 12 phiAboveCell1001 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1001
+    e24KC2PhiAboveNode10010 e24KC2PhiAboveNode10011 e24KC2PhiAboveLeaf10012 e24KC2PhiAboveLeaf10013
+
+theorem e24KC2PhiAboveNode1002 :
+    adaptiveCoverCheck 12 phiAboveCell1002 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1002
+    e24KC2PhiAboveLeaf10020 e24KC2PhiAboveLeaf10021 e24KC2PhiAboveLeaf10022 e24KC2PhiAboveLeaf10023
+
+theorem e24KC2PhiAboveNode1003 :
+    adaptiveCoverCheck 12 phiAboveCell1003 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1003
+    e24KC2PhiAboveLeaf10030 e24KC2PhiAboveLeaf10031 e24KC2PhiAboveLeaf10032 e24KC2PhiAboveLeaf10033
+
+theorem e24KC2PhiAboveNode1010 :
+    adaptiveCoverCheck 12 phiAboveCell1010 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1010
+    e24KC2PhiAboveNode10100 e24KC2PhiAboveNode10101 e24KC2PhiAboveLeaf10102 e24KC2PhiAboveLeaf10103
+
+theorem e24KC2PhiAboveNode1011 :
+    adaptiveCoverCheck 12 phiAboveCell1011 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1011
+    e24KC2PhiAboveNode10110 e24KC2PhiAboveNode10111 e24KC2PhiAboveLeaf10112 e24KC2PhiAboveNode10113
+
+theorem e24KC2PhiAboveNode1012 :
+    adaptiveCoverCheck 12 phiAboveCell1012 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1012
+    e24KC2PhiAboveLeaf10120 e24KC2PhiAboveLeaf10121 e24KC2PhiAboveLeaf10122 e24KC2PhiAboveLeaf10123
+
+theorem e24KC2PhiAboveNode1013 :
+    adaptiveCoverCheck 12 phiAboveCell1013 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1013
+    e24KC2PhiAboveLeaf10130 e24KC2PhiAboveLeaf10131 e24KC2PhiAboveLeaf10132 e24KC2PhiAboveLeaf10133
+
+theorem e24KC2PhiAboveNode1100 :
+    adaptiveCoverCheck 12 phiAboveCell1100 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1100
+    e24KC2PhiAboveNode11000 e24KC2PhiAboveNode11001 e24KC2PhiAboveNode11002 e24KC2PhiAboveNode11003
+
+theorem e24KC2PhiAboveNode1101 :
+    adaptiveCoverCheck 12 phiAboveCell1101 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1101
+    e24KC2PhiAboveNode11010 e24KC2PhiAboveNode11011 e24KC2PhiAboveNode11012 e24KC2PhiAboveNode11013
+
+theorem e24KC2PhiAboveNode1102 :
+    adaptiveCoverCheck 12 phiAboveCell1102 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1102
+    e24KC2PhiAboveLeaf11020 e24KC2PhiAboveLeaf11021 e24KC2PhiAboveLeaf11022 e24KC2PhiAboveLeaf11023
+
+theorem e24KC2PhiAboveNode1103 :
+    adaptiveCoverCheck 12 phiAboveCell1103 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1103
+    e24KC2PhiAboveLeaf11030 e24KC2PhiAboveLeaf11031 e24KC2PhiAboveLeaf11032 e24KC2PhiAboveLeaf11033
+
+theorem e24KC2PhiAboveNode1110 :
+    adaptiveCoverCheck 12 phiAboveCell1110 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1110
+    e24KC2PhiAboveNode11100 e24KC2PhiAboveNode11101 e24KC2PhiAboveNode11102 e24KC2PhiAboveNode11103
+
+theorem e24KC2PhiAboveNode1111 :
+    adaptiveCoverCheck 12 phiAboveCell1111 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1111
+    e24KC2PhiAboveNode11110 e24KC2PhiAboveNode11111 e24KC2PhiAboveNode11112 e24KC2PhiAboveNode11113
+
+theorem e24KC2PhiAboveNode1112 :
+    adaptiveCoverCheck 12 phiAboveCell1112 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1112
+    e24KC2PhiAboveLeaf11120 e24KC2PhiAboveLeaf11121 e24KC2PhiAboveLeaf11122 e24KC2PhiAboveLeaf11123
+
+theorem e24KC2PhiAboveNode1113 :
+    adaptiveCoverCheck 12 phiAboveCell1113 = true :=
+  adaptiveCoverCheck_succ_of_children 11 phiAboveCell1113
+    e24KC2PhiAboveLeaf11130 e24KC2PhiAboveLeaf11131 e24KC2PhiAboveLeaf11132 e24KC2PhiAboveLeaf11133
+
+theorem e24KC2PhiAboveNode000 :
+    adaptiveCoverCheck 13 (childLL (childLL (childLL e24PhiAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL (childLL e24PhiAboveRoot)))
+    e24KC2PhiAboveNode0000 e24KC2PhiAboveNode0001 e24KC2PhiAboveLeaf0002 e24KC2PhiAboveLeaf0003
+
+theorem e24KC2PhiAboveNode001 :
+    adaptiveCoverCheck 13 (childLH (childLL (childLL e24PhiAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL (childLL e24PhiAboveRoot)))
+    e24KC2PhiAboveNode0010 e24KC2PhiAboveNode0011 e24KC2PhiAboveLeaf0012 e24KC2PhiAboveLeaf0013
+
+theorem e24KC2PhiAboveNode003 :
+    adaptiveCoverCheck 13 (childHH (childLL (childLL e24PhiAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childLL (childLL e24PhiAboveRoot)))
+    e24KC2PhiAboveLeaf0030 e24KC2PhiAboveLeaf0031 e24KC2PhiAboveLeaf0032 e24KC2PhiAboveLeaf0033
+
+theorem e24KC2PhiAboveNode010 :
+    adaptiveCoverCheck 13 (childLL (childLH (childLL e24PhiAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH (childLL e24PhiAboveRoot)))
+    e24KC2PhiAboveNode0100 e24KC2PhiAboveNode0101 e24KC2PhiAboveLeaf0102 e24KC2PhiAboveLeaf0103
+
+theorem e24KC2PhiAboveNode011 :
+    adaptiveCoverCheck 13 (childLH (childLH (childLL e24PhiAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH (childLL e24PhiAboveRoot)))
+    e24KC2PhiAboveNode0110 e24KC2PhiAboveNode0111 e24KC2PhiAboveLeaf0112 e24KC2PhiAboveLeaf0113
+
+theorem e24KC2PhiAboveNode012 :
+    adaptiveCoverCheck 13 (childHL (childLH (childLL e24PhiAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childLH (childLL e24PhiAboveRoot)))
+    e24KC2PhiAboveLeaf0120 e24KC2PhiAboveLeaf0121 e24KC2PhiAboveLeaf0122 e24KC2PhiAboveLeaf0123
+
+theorem e24KC2PhiAboveNode013 :
+    adaptiveCoverCheck 13 (childHH (childLH (childLL e24PhiAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childLH (childLL e24PhiAboveRoot)))
+    e24KC2PhiAboveLeaf0130 e24KC2PhiAboveLeaf0131 e24KC2PhiAboveLeaf0132 e24KC2PhiAboveLeaf0133
+
+theorem e24KC2PhiAboveNode100 :
+    adaptiveCoverCheck 13 (childLL (childLL (childLH e24PhiAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL (childLH e24PhiAboveRoot)))
+    e24KC2PhiAboveNode1000 e24KC2PhiAboveNode1001 e24KC2PhiAboveNode1002 e24KC2PhiAboveNode1003
+
+theorem e24KC2PhiAboveNode101 :
+    adaptiveCoverCheck 13 (childLH (childLL (childLH e24PhiAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL (childLH e24PhiAboveRoot)))
+    e24KC2PhiAboveNode1010 e24KC2PhiAboveNode1011 e24KC2PhiAboveNode1012 e24KC2PhiAboveNode1013
+
+theorem e24KC2PhiAboveNode102 :
+    adaptiveCoverCheck 13 (childHL (childLL (childLH e24PhiAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childLL (childLH e24PhiAboveRoot)))
+    e24KC2PhiAboveLeaf1020 e24KC2PhiAboveLeaf1021 e24KC2PhiAboveLeaf1022 e24KC2PhiAboveLeaf1023
+
+theorem e24KC2PhiAboveNode103 :
+    adaptiveCoverCheck 13 (childHH (childLL (childLH e24PhiAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childLL (childLH e24PhiAboveRoot)))
+    e24KC2PhiAboveLeaf1030 e24KC2PhiAboveLeaf1031 e24KC2PhiAboveLeaf1032 e24KC2PhiAboveLeaf1033
+
+theorem e24KC2PhiAboveNode110 :
+    adaptiveCoverCheck 13 (childLL (childLH (childLH e24PhiAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH (childLH e24PhiAboveRoot)))
+    e24KC2PhiAboveNode1100 e24KC2PhiAboveNode1101 e24KC2PhiAboveNode1102 e24KC2PhiAboveNode1103
+
+theorem e24KC2PhiAboveNode111 :
+    adaptiveCoverCheck 13 (childLH (childLH (childLH e24PhiAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH (childLH e24PhiAboveRoot)))
+    e24KC2PhiAboveNode1110 e24KC2PhiAboveNode1111 e24KC2PhiAboveNode1112 e24KC2PhiAboveNode1113
+
+theorem e24KC2PhiAboveNode112 :
+    adaptiveCoverCheck 13 (childHL (childLH (childLH e24PhiAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childLH (childLH e24PhiAboveRoot)))
+    e24KC2PhiAboveLeaf1120 e24KC2PhiAboveLeaf1121 e24KC2PhiAboveLeaf1122 e24KC2PhiAboveLeaf1123
+
+theorem e24KC2PhiAboveNode113 :
+    adaptiveCoverCheck 13 (childHH (childLH (childLH e24PhiAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childLH (childLH e24PhiAboveRoot)))
+    e24KC2PhiAboveLeaf1130 e24KC2PhiAboveLeaf1131 e24KC2PhiAboveLeaf1132 e24KC2PhiAboveLeaf1133
+
+theorem e24KC2PhiAboveNode00 :
+    adaptiveCoverCheck 14 (childLL (childLL e24PhiAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL (childLL e24PhiAboveRoot))
+    e24KC2PhiAboveNode000 e24KC2PhiAboveNode001 e24KC2PhiAboveLeaf002 e24KC2PhiAboveNode003
+
+theorem e24KC2PhiAboveNode01 :
+    adaptiveCoverCheck 14 (childLH (childLL e24PhiAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH (childLL e24PhiAboveRoot))
+    e24KC2PhiAboveNode010 e24KC2PhiAboveNode011 e24KC2PhiAboveNode012 e24KC2PhiAboveNode013
+
+theorem e24KC2PhiAboveNode03 :
+    adaptiveCoverCheck 14 (childHH (childLL e24PhiAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH (childLL e24PhiAboveRoot))
+    e24KC2PhiAboveLeaf030 e24KC2PhiAboveLeaf031 e24KC2PhiAboveLeaf032 e24KC2PhiAboveLeaf033
+
+theorem e24KC2PhiAboveNode10 :
+    adaptiveCoverCheck 14 (childLL (childLH e24PhiAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL (childLH e24PhiAboveRoot))
+    e24KC2PhiAboveNode100 e24KC2PhiAboveNode101 e24KC2PhiAboveNode102 e24KC2PhiAboveNode103
+
+theorem e24KC2PhiAboveNode11 :
+    adaptiveCoverCheck 14 (childLH (childLH e24PhiAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH (childLH e24PhiAboveRoot))
+    e24KC2PhiAboveNode110 e24KC2PhiAboveNode111 e24KC2PhiAboveNode112 e24KC2PhiAboveNode113
+
+theorem e24KC2PhiAboveNode12 :
+    adaptiveCoverCheck 14 (childHL (childLH e24PhiAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL (childLH e24PhiAboveRoot))
+    e24KC2PhiAboveLeaf120 e24KC2PhiAboveLeaf121 e24KC2PhiAboveLeaf122 e24KC2PhiAboveLeaf123
+
+theorem e24KC2PhiAboveNode13 :
+    adaptiveCoverCheck 14 (childHH (childLH e24PhiAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH (childLH e24PhiAboveRoot))
+    e24KC2PhiAboveLeaf130 e24KC2PhiAboveLeaf131 e24KC2PhiAboveLeaf132 e24KC2PhiAboveLeaf133
+
+theorem e24KC2PhiAboveNode30 :
+    adaptiveCoverCheck 14 (childLL (childHH e24PhiAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL (childHH e24PhiAboveRoot))
+    e24KC2PhiAboveLeaf300 e24KC2PhiAboveLeaf301 e24KC2PhiAboveLeaf302 e24KC2PhiAboveLeaf303
+
+theorem e24KC2PhiAboveNode31 :
+    adaptiveCoverCheck 14 (childLH (childHH e24PhiAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH (childHH e24PhiAboveRoot))
+    e24KC2PhiAboveLeaf310 e24KC2PhiAboveLeaf311 e24KC2PhiAboveLeaf312 e24KC2PhiAboveLeaf313
+
+theorem e24KC2PhiAboveNode33 :
+    adaptiveCoverCheck 14 (childHH (childHH e24PhiAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH (childHH e24PhiAboveRoot))
+    e24KC2PhiAboveLeaf330 e24KC2PhiAboveLeaf331 e24KC2PhiAboveLeaf332 e24KC2PhiAboveLeaf333
+
+theorem e24KC2PhiAboveNode0 :
+    adaptiveCoverCheck 15 (childLL e24PhiAboveRoot) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childLL e24PhiAboveRoot)
+    e24KC2PhiAboveNode00 e24KC2PhiAboveNode01 e24KC2PhiAboveLeaf02 e24KC2PhiAboveNode03
+
+theorem e24KC2PhiAboveNode1 :
+    adaptiveCoverCheck 15 (childLH e24PhiAboveRoot) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childLH e24PhiAboveRoot)
+    e24KC2PhiAboveNode10 e24KC2PhiAboveNode11 e24KC2PhiAboveNode12 e24KC2PhiAboveNode13
+
+theorem e24KC2PhiAboveNode3 :
+    adaptiveCoverCheck 15 (childHH e24PhiAboveRoot) = true :=
+  adaptiveCoverCheck_succ_of_children 14 (childHH e24PhiAboveRoot)
+    e24KC2PhiAboveNode30 e24KC2PhiAboveNode31 e24KC2PhiAboveLeaf32 e24KC2PhiAboveNode33
+
+theorem e24KC2PhiAboveNodeROOT :
+    adaptiveCoverCheck 16 e24PhiAboveRoot = true :=
+  adaptiveCoverCheck_succ_of_children 15 e24PhiAboveRoot
+    e24KC2PhiAboveNode0 e24KC2PhiAboveNode1 e24KC2PhiAboveLeaf2 e24KC2PhiAboveNode3
+
+theorem e24PhiAboveKernelCheck :
+    adaptiveCoverCheck 16 e24PhiAboveRoot = true :=
+  e24KC2PhiAboveNodeROOT
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33203_9_00246
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells12ecbb1984
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells12ecbb1984
+
+open CertificateCells12ecbb1984
+
+theorem e24KC2PhiBelowLeaf33203 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3320) = true :=
+  adaptiveCoverCheck_succ_of_children 8 (childHH phiBelowCell3320)
+    e24KC2PhiBelowLeaf33203_c0 e24KC2PhiBelowLeaf33203_c1 e24KC2PhiBelowLeaf33203_c2
+      e24KC2PhiBelowLeaf33203_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33221_9_00260
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf9b1d22c2c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf9b1d22c2c
+
+open CertificateCellsf9b1d22c2c
+
+theorem e24KC2PhiBelowLeaf33221 :
+    adaptiveCoverCheck 9 (childLH phiBelowCell3322) = true :=
+  adaptiveCoverCheck_succ_of_children 8 (childLH phiBelowCell3322)
+    e24KC2PhiBelowLeaf33221_c0 e24KC2PhiBelowLeaf33221_c1 e24KC2PhiBelowLeaf33221_c2
+      e24KC2PhiBelowLeaf33221_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33223_9_00269
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5481d87062
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5481d87062
+
+open CertificateCells5481d87062
+
+theorem e24KC2PhiBelowLeaf33223 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3322) = true :=
+  adaptiveCoverCheck_succ_of_children 8 (childHH phiBelowCell3322)
+    e24KC2PhiBelowLeaf33223_c0 e24KC2PhiBelowLeaf33223_c1 e24KC2PhiBelowLeaf33223_c2
+      e24KC2PhiBelowLeaf33223_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33230_c2_8_00278
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc99c114252
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc99c114252
+
+open CertificateCellsc99c114252
+
+theorem e24KC2PhiBelowLeaf33230_c2 :
+    adaptiveCoverCheck 8 (childHL (childLL phiBelowCell3323)) = true :=
+  adaptiveCoverCheck_succ_of_children 7 (childHL (childLL phiBelowCell3323))
+    e24KC2PhiBelowLeaf33230_c2_c0 e24KC2PhiBelowLeaf33230_c2_c1 e24KC2PhiBelowLeaf33230_c2_c2
+      e24KC2PhiBelowLeaf33230_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33230_c3_8_00284
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells85102f4bff
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells85102f4bff
+
+open CertificateCells85102f4bff
+
+theorem e24KC2PhiBelowLeaf33230_c3 :
+    adaptiveCoverCheck 8 (childHH (childLL phiBelowCell3323)) = true :=
+  adaptiveCoverCheck_succ_of_children 7 (childHH (childLL phiBelowCell3323))
+    e24KC2PhiBelowLeaf33230_c3_c0 e24KC2PhiBelowLeaf33230_c3_c1 e24KC2PhiBelowLeaf33230_c3_c2
+      e24KC2PhiBelowLeaf33230_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33230_9_00285
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd7423e4c3b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd7423e4c3b
+
+open CertificateCellsd7423e4c3b
+
+theorem e24KC2PhiBelowLeaf33230 :
+    adaptiveCoverCheck 9 (childLL phiBelowCell3323) = true :=
+  adaptiveCoverCheck_succ_of_children 8 (childLL phiBelowCell3323)
+    e24KC2PhiBelowLeaf33230_c0 e24KC2PhiBelowLeaf33230_c1 e24KC2PhiBelowLeaf33230_c2
+      e24KC2PhiBelowLeaf33230_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33232_c0_8_00294
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8ae9088aab
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8ae9088aab
+
+open CertificateCells8ae9088aab
+
+theorem e24KC2PhiBelowLeaf33232_c0 :
+    adaptiveCoverCheck 8 (childLL (childHL phiBelowCell3323)) = true :=
+  adaptiveCoverCheck_succ_of_children 7 (childLL (childHL phiBelowCell3323))
+    e24KC2PhiBelowLeaf33232_c0_c0 e24KC2PhiBelowLeaf33232_c0_c1 e24KC2PhiBelowLeaf33232_c0_c2
+      e24KC2PhiBelowLeaf33232_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33232_c1_c0_7_00301
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb8a79b56b3
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb8a79b56b3
+
+open CertificateCellsb8a79b56b3
+
+theorem e24KC2PhiBelowLeaf33232_c1_c0 :
+    adaptiveCoverCheck 7 (childLL (childLH (childHL phiBelowCell3323))) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childLL (childLH (childHL phiBelowCell3323)))
+    e24KC2PhiBelowLeaf33232_c1_c0_c0 e24KC2PhiBelowLeaf33232_c1_c0_c1
+      e24KC2PhiBelowLeaf33232_c1_c0_c2 e24KC2PhiBelowLeaf33232_c1_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33232_c1_c2_7_00308
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells13ae325d6d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells13ae325d6d
+
+open CertificateCells13ae325d6d
+
+theorem e24KC2PhiBelowLeaf33232_c1_c2 :
+    adaptiveCoverCheck 7 (childHL (childLH (childHL phiBelowCell3323))) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childHL (childLH (childHL phiBelowCell3323)))
+    e24KC2PhiBelowLeaf33232_c1_c2_c0 e24KC2PhiBelowLeaf33232_c1_c2_c1
+      e24KC2PhiBelowLeaf33232_c1_c2_c2 e24KC2PhiBelowLeaf33232_c1_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33232_c1_c3_7_00314
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6add7bae04
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6add7bae04
+
+open CertificateCells6add7bae04
+
+theorem e24KC2PhiBelowLeaf33232_c1_c3 :
+    adaptiveCoverCheck 7 (childHH (childLH (childHL phiBelowCell3323))) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childHH (childLH (childHL phiBelowCell3323)))
+    e24KC2PhiBelowLeaf33232_c1_c3_c0 e24KC2PhiBelowLeaf33232_c1_c3_c1
+      e24KC2PhiBelowLeaf33232_c1_c3_c2 e24KC2PhiBelowLeaf33232_c1_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33232_c1_8_00315
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellscd1d17c9b4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellscd1d17c9b4
+
+open CertificateCellscd1d17c9b4
+
+theorem e24KC2PhiBelowLeaf33232_c1 :
+    adaptiveCoverCheck 8 (childLH (childHL phiBelowCell3323)) = true :=
+  adaptiveCoverCheck_succ_of_children 7 (childLH (childHL phiBelowCell3323))
+    e24KC2PhiBelowLeaf33232_c1_c0 e24KC2PhiBelowLeaf33232_c1_c1 e24KC2PhiBelowLeaf33232_c1_c2
+      e24KC2PhiBelowLeaf33232_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33232_c3_c1_c1_6_00326
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells709ea69226
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells709ea69226
+
+open CertificateCells709ea69226
+
+theorem e24KC2PhiBelowLeaf33232_c3_c1_c1 :
+    adaptiveCoverCheck 6 phiBelowCell33232311 = true :=
+  adaptiveCoverCheck_succ_of_children 5 phiBelowCell33232311
+    e24KC2PhiBelowLeaf33232_c3_c1_c1_c0 e24KC2PhiBelowLeaf33232_c3_c1_c1_c1
+      e24KC2PhiBelowLeaf33232_c3_c1_c1_c2 e24KC2PhiBelowLeaf33232_c3_c1_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33232_c3_c1_c3_6_00333
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsade8569b0a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsade8569b0a
+
+open CertificateCellsade8569b0a
+
+theorem e24KC2PhiBelowLeaf33232_c3_c1_c3 :
+    adaptiveCoverCheck 6 phiBelowCell33232313 = true :=
+  adaptiveCoverCheck_succ_of_children 5 phiBelowCell33232313
+    e24KC2PhiBelowLeaf33232_c3_c1_c3_c0 e24KC2PhiBelowLeaf33232_c3_c1_c3_c1
+      e24KC2PhiBelowLeaf33232_c3_c1_c3_c2 e24KC2PhiBelowLeaf33232_c3_c1_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33232_c3_c1_7_00334
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsfece773e85
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsfece773e85
+
+open CertificateCellsfece773e85
+
+theorem e24KC2PhiBelowLeaf33232_c3_c1 :
+    adaptiveCoverCheck 7 (childLH (childHH (childHL phiBelowCell3323))) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childLH (childHH (childHL phiBelowCell3323)))
+    e24KC2PhiBelowLeaf33232_c3_c1_c0 e24KC2PhiBelowLeaf33232_c3_c1_c1
+      e24KC2PhiBelowLeaf33232_c3_c1_c2 e24KC2PhiBelowLeaf33232_c3_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33232_c3_c3_c1_6_00343
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse593ec4a84
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse593ec4a84
+
+open CertificateCellse593ec4a84
+
+theorem e24KC2PhiBelowLeaf33232_c3_c3_c1 :
+    adaptiveCoverCheck 6 phiBelowCell33232331 = true :=
+  adaptiveCoverCheck_succ_of_children 5 phiBelowCell33232331
+    e24KC2PhiBelowLeaf33232_c3_c3_c1_c0 e24KC2PhiBelowLeaf33232_c3_c3_c1_c1
+      e24KC2PhiBelowLeaf33232_c3_c3_c1_c2 e24KC2PhiBelowLeaf33232_c3_c3_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33232_c3_c3_7_00346
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0ccc164bd1
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0ccc164bd1
+
+open CertificateCells0ccc164bd1
+
+theorem e24KC2PhiBelowLeaf33232_c3_c3 :
+    adaptiveCoverCheck 7 (childHH (childHH (childHL phiBelowCell3323))) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childHH (childHH (childHL phiBelowCell3323)))
+    e24KC2PhiBelowLeaf33232_c3_c3_c0 e24KC2PhiBelowLeaf33232_c3_c3_c1
+      e24KC2PhiBelowLeaf33232_c3_c3_c2 e24KC2PhiBelowLeaf33232_c3_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33232_c3_8_00347
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells072df6b25f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells072df6b25f
+
+open CertificateCells072df6b25f
+
+theorem e24KC2PhiBelowLeaf33232_c3 :
+    adaptiveCoverCheck 8 (childHH (childHL phiBelowCell3323)) = true :=
+  adaptiveCoverCheck_succ_of_children 7 (childHH (childHL phiBelowCell3323))
+    e24KC2PhiBelowLeaf33232_c3_c0 e24KC2PhiBelowLeaf33232_c3_c1 e24KC2PhiBelowLeaf33232_c3_c2
+      e24KC2PhiBelowLeaf33232_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33232_9_00348
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells215f6c9ebd
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells215f6c9ebd
+
+open CertificateCells215f6c9ebd
+
+theorem e24KC2PhiBelowLeaf33232 :
+    adaptiveCoverCheck 9 (childHL phiBelowCell3323) = true :=
+  adaptiveCoverCheck_succ_of_children 8 (childHL phiBelowCell3323)
+    e24KC2PhiBelowLeaf33232_c0 e24KC2PhiBelowLeaf33232_c1 e24KC2PhiBelowLeaf33232_c2
+      e24KC2PhiBelowLeaf33232_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c0_8_00358
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsef857ed1c7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsef857ed1c7
+
+open CertificateCellsef857ed1c7
+
+theorem e24KC2PhiBelowLeaf33233_c0 :
+    adaptiveCoverCheck 8 (childLL (childHH phiBelowCell3323)) = true :=
+  adaptiveCoverCheck_succ_of_children 7 (childLL (childHH phiBelowCell3323))
+    e24KC2PhiBelowLeaf33233_c0_c0 e24KC2PhiBelowLeaf33233_c0_c1 e24KC2PhiBelowLeaf33233_c0_c2
+      e24KC2PhiBelowLeaf33233_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c0_c2_6_00369
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2ebe0c2db9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2ebe0c2db9
+
+open CertificateCells2ebe0c2db9
+
+theorem e24KC2PhiBelowLeaf33233_c2_c0_c2 :
+    adaptiveCoverCheck 6 phiBelowCell33233202 = true :=
+  adaptiveCoverCheck_succ_of_children 5 phiBelowCell33233202
+    e24KC2PhiBelowLeaf33233_c2_c0_c2_c0 e24KC2PhiBelowLeaf33233_c2_c0_c2_c1
+      e24KC2PhiBelowLeaf33233_c2_c0_c2_c2 e24KC2PhiBelowLeaf33233_c2_c0_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c0_7_00371
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells27c0e8c594
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells27c0e8c594
+
+open CertificateCells27c0e8c594
+
+theorem e24KC2PhiBelowLeaf33233_c2_c0 :
+    adaptiveCoverCheck 7 (childLL (childHL (childHH phiBelowCell3323))) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childLL (childHL (childHH phiBelowCell3323)))
+    e24KC2PhiBelowLeaf33233_c2_c0_c0 e24KC2PhiBelowLeaf33233_c2_c0_c1
+      e24KC2PhiBelowLeaf33233_c2_c0_c2 e24KC2PhiBelowLeaf33233_c2_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c2_c0_c3_5_00383
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4d117fb26d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4d117fb26d
+
+open CertificateCells4d117fb26d
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c0_c3 :
+    adaptiveCoverCheck 5 (childHH phiBelowCell33233220) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childHH phiBelowCell33233220)
+    e24KC2PhiBelowLeaf33233_c2_c2_c0_c3_c0 e24KC2PhiBelowLeaf33233_c2_c2_c0_c3_c1
+      e24KC2PhiBelowLeaf33233_c2_c2_c0_c3_c2 e24KC2PhiBelowLeaf33233_c2_c2_c0_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c2_c0_6_00384
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells848f180c5a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells848f180c5a
+
+open CertificateCells848f180c5a
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c0 :
+    adaptiveCoverCheck 6 phiBelowCell33233220 = true :=
+  adaptiveCoverCheck_succ_of_children 5 phiBelowCell33233220
+    e24KC2PhiBelowLeaf33233_c2_c2_c0_c0 e24KC2PhiBelowLeaf33233_c2_c2_c0_c1
+      e24KC2PhiBelowLeaf33233_c2_c2_c0_c2 e24KC2PhiBelowLeaf33233_c2_c2_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c2_c1_6_00390
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells090e12bc76
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells090e12bc76
+
+open CertificateCells090e12bc76
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c1 :
+    adaptiveCoverCheck 6 phiBelowCell33233221 = true :=
+  adaptiveCoverCheck_succ_of_children 5 phiBelowCell33233221
+    e24KC2PhiBelowLeaf33233_c2_c2_c1_c0 e24KC2PhiBelowLeaf33233_c2_c2_c1_c1
+      e24KC2PhiBelowLeaf33233_c2_c2_c1_c2 e24KC2PhiBelowLeaf33233_c2_c2_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c2_c2_c1_5_00398
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells857be56025
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells857be56025
+
+open CertificateCells857be56025
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c2_c1 :
+    adaptiveCoverCheck 5 (childLH phiBelowCell33233222) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLH phiBelowCell33233222)
+    e24KC2PhiBelowLeaf33233_c2_c2_c2_c1_c0 e24KC2PhiBelowLeaf33233_c2_c2_c2_c1_c1
+      e24KC2PhiBelowLeaf33233_c2_c2_c2_c1_c2 e24KC2PhiBelowLeaf33233_c2_c2_c2_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c2_c2_6_00401
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsea76f7562d
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsea76f7562d
+
+open CertificateCellsea76f7562d
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c2 :
+    adaptiveCoverCheck 6 phiBelowCell33233222 = true :=
+  adaptiveCoverCheck_succ_of_children 5 phiBelowCell33233222
+    e24KC2PhiBelowLeaf33233_c2_c2_c2_c0 e24KC2PhiBelowLeaf33233_c2_c2_c2_c1
+      e24KC2PhiBelowLeaf33233_c2_c2_c2_c2 e24KC2PhiBelowLeaf33233_c2_c2_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c2_c3_c0_5_00408
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdd77c2320b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdd77c2320b
+
+open CertificateCellsdd77c2320b
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c0 :
+    adaptiveCoverCheck 5 (childLL phiBelowCell33233223) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLL phiBelowCell33233223)
+    e24KC2PhiBelowLeaf33233_c2_c2_c3_c0_c0 e24KC2PhiBelowLeaf33233_c2_c2_c3_c0_c1
+      e24KC2PhiBelowLeaf33233_c2_c2_c3_c0_c2 e24KC2PhiBelowLeaf33233_c2_c2_c3_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c1_c2_c0_2_00421
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells08c6706994
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells08c6706994
+
+open CertificateCells08c6706994
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c0 :
+    adaptiveCoverCheck 2 phiBelowCell332332232120 = true :=
+  adaptiveCoverCheck_succ_of_children 1 phiBelowCell332332232120
+    e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c0_c0
+      e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c0_c1
+      e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c0_c2
+      e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below
+Leaf33233_c2_c2_c3_c2_c1_c2_3_00425
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells96d537e773
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells96d537e773
+
+open CertificateCells96d537e773
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2 :
+    adaptiveCoverCheck 3 (childHL (childLH (childHL phiBelowCell33233223))) = true :=
+  adaptiveCoverCheck_succ_of_children 2 (childHL (childLH (childHL phiBelowCell33233223)))
+    e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c0 e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c1
+      e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c2 e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c2_c3_c2_c1_4_00427
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0abe1d75a7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0abe1d75a7
+
+open CertificateCells0abe1d75a7
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1 :
+    adaptiveCoverCheck 4 (childLH (childHL phiBelowCell33233223)) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childLH (childHL phiBelowCell33233223))
+    e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c0 e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c1
+      e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c2 e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c2_c3_c2_c3_4_00434
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells4849d1c13b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells4849d1c13b
+
+open CertificateCells4849d1c13b
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c3 :
+    adaptiveCoverCheck 4 (childHH (childHL phiBelowCell33233223)) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childHH (childHL phiBelowCell33233223))
+    e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c3_c0 e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c3_c1
+      e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c3_c2 e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c2_c3_c2_5_00435
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf2740dc787
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf2740dc787
+
+open CertificateCellsf2740dc787
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c2 :
+    adaptiveCoverCheck 5 (childHL phiBelowCell33233223) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childHL phiBelowCell33233223)
+    e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c0 e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c1
+      e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c2 e24KC2PhiBelowLeaf33233_c2_c2_c3_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c2_c3_c3_c2_4_00444
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3f18540de9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3f18540de9
+
+open CertificateCells3f18540de9
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c2 :
+    adaptiveCoverCheck 4 (childHL (childHH phiBelowCell33233223)) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childHL (childHH phiBelowCell33233223))
+    e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c2_c0 e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c2_c1
+      e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c2_c2 e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c2_c3_c3_5_00446
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsa21caf16f2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsa21caf16f2
+
+open CertificateCellsa21caf16f2
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3_c3 :
+    adaptiveCoverCheck 5 (childHH phiBelowCell33233223) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childHH phiBelowCell33233223)
+    e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c0 e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c1
+      e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c2 e24KC2PhiBelowLeaf33233_c2_c2_c3_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c2_c3_6_00447
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells21943a9b83
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells21943a9b83
+
+open CertificateCells21943a9b83
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2_c3 :
+    adaptiveCoverCheck 6 phiBelowCell33233223 = true :=
+  adaptiveCoverCheck_succ_of_children 5 phiBelowCell33233223
+    e24KC2PhiBelowLeaf33233_c2_c2_c3_c0 e24KC2PhiBelowLeaf33233_c2_c2_c3_c1
+      e24KC2PhiBelowLeaf33233_c2_c2_c3_c2 e24KC2PhiBelowLeaf33233_c2_c2_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c2_7_00448
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells62ff50a4e8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells62ff50a4e8
+
+open CertificateCells62ff50a4e8
+
+theorem e24KC2PhiBelowLeaf33233_c2_c2 :
+    adaptiveCoverCheck 7 (childHL (childHL (childHH phiBelowCell3323))) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childHL (childHL (childHH phiBelowCell3323)))
+    e24KC2PhiBelowLeaf33233_c2_c2_c0 e24KC2PhiBelowLeaf33233_c2_c2_c1
+      e24KC2PhiBelowLeaf33233_c2_c2_c2 e24KC2PhiBelowLeaf33233_c2_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c3_c2_6_00457
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6677fc4b7c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6677fc4b7c
+
+open CertificateCells6677fc4b7c
+
+theorem e24KC2PhiBelowLeaf33233_c2_c3_c2 :
+    adaptiveCoverCheck 6 phiBelowCell33233232 = true :=
+  adaptiveCoverCheck_succ_of_children 5 phiBelowCell33233232
+    e24KC2PhiBelowLeaf33233_c2_c3_c2_c0 e24KC2PhiBelowLeaf33233_c2_c3_c2_c1
+      e24KC2PhiBelowLeaf33233_c2_c3_c2_c2 e24KC2PhiBelowLeaf33233_c2_c3_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_c3_7_00459
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells07875eeb47
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells07875eeb47
+
+open CertificateCells07875eeb47
+
+theorem e24KC2PhiBelowLeaf33233_c2_c3 :
+    adaptiveCoverCheck 7 (childHH (childHL (childHH phiBelowCell3323))) = true :=
+  adaptiveCoverCheck_succ_of_children 6 (childHH (childHL (childHH phiBelowCell3323)))
+    e24KC2PhiBelowLeaf33233_c2_c3_c0 e24KC2PhiBelowLeaf33233_c2_c3_c1
+      e24KC2PhiBelowLeaf33233_c2_c3_c2 e24KC2PhiBelowLeaf33233_c2_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_c2_8_00460
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5ed2775b48
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5ed2775b48
+
+open CertificateCells5ed2775b48
+
+theorem e24KC2PhiBelowLeaf33233_c2 :
+    adaptiveCoverCheck 8 (childHL (childHH phiBelowCell3323)) = true :=
+  adaptiveCoverCheck_succ_of_children 7 (childHL (childHH phiBelowCell3323))
+    e24KC2PhiBelowLeaf33233_c2_c0 e24KC2PhiBelowLeaf33233_c2_c1 e24KC2PhiBelowLeaf33233_c2_c2
+      e24KC2PhiBelowLeaf33233_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Phi Below Leaf33233_9_00462
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells83d9ff7319
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells83d9ff7319
+
+open CertificateCells83d9ff7319
+
+theorem e24KC2PhiBelowLeaf33233 :
+    adaptiveCoverCheck 9 (childHH phiBelowCell3323) = true :=
+  adaptiveCoverCheck_succ_of_children 8 (childHH phiBelowCell3323)
+    e24KC2PhiBelowLeaf33233_c0 e24KC2PhiBelowLeaf33233_c1 e24KC2PhiBelowLeaf33233_c2
+      e24KC2PhiBelowLeaf33233_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC6Phi Below Reconstruct
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells240a9bf558
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells240a9bf558
+
+open CertificateCells240a9bf558
+
+theorem e24KC2PhiBelowNode3000 :
+    adaptiveCoverCheck 10 phiBelowCell3000 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3000
+    e24KC2PhiBelowLeaf30000 e24KC2PhiBelowLeaf30001 e24KC2PhiBelowLeaf30002 e24KC2PhiBelowLeaf30003
+
+theorem e24KC2PhiBelowNode3001 :
+    adaptiveCoverCheck 10 phiBelowCell3001 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3001
+    e24KC2PhiBelowLeaf30010 e24KC2PhiBelowLeaf30011 e24KC2PhiBelowLeaf30012 e24KC2PhiBelowLeaf30013
+
+theorem e24KC2PhiBelowNode3002 :
+    adaptiveCoverCheck 10 phiBelowCell3002 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3002
+    e24KC2PhiBelowLeaf30020 e24KC2PhiBelowLeaf30021 e24KC2PhiBelowLeaf30022 e24KC2PhiBelowLeaf30023
+
+theorem e24KC2PhiBelowNode3003 :
+    adaptiveCoverCheck 10 phiBelowCell3003 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3003
+    e24KC2PhiBelowLeaf30030 e24KC2PhiBelowLeaf30031 e24KC2PhiBelowLeaf30032 e24KC2PhiBelowLeaf30033
+
+theorem e24KC2PhiBelowNode3010 :
+    adaptiveCoverCheck 10 phiBelowCell3010 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3010
+    e24KC2PhiBelowLeaf30100 e24KC2PhiBelowLeaf30101 e24KC2PhiBelowLeaf30102 e24KC2PhiBelowLeaf30103
+
+theorem e24KC2PhiBelowNode3011 :
+    adaptiveCoverCheck 10 phiBelowCell3011 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3011
+    e24KC2PhiBelowLeaf30110 e24KC2PhiBelowLeaf30111 e24KC2PhiBelowLeaf30112 e24KC2PhiBelowLeaf30113
+
+theorem e24KC2PhiBelowNode3012 :
+    adaptiveCoverCheck 10 phiBelowCell3012 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3012
+    e24KC2PhiBelowLeaf30120 e24KC2PhiBelowLeaf30121 e24KC2PhiBelowLeaf30122 e24KC2PhiBelowLeaf30123
+
+theorem e24KC2PhiBelowNode3013 :
+    adaptiveCoverCheck 10 phiBelowCell3013 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3013
+    e24KC2PhiBelowLeaf30130 e24KC2PhiBelowLeaf30131 e24KC2PhiBelowLeaf30132 e24KC2PhiBelowLeaf30133
+
+theorem e24KC2PhiBelowNode3020 :
+    adaptiveCoverCheck 10 phiBelowCell3020 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3020
+    e24KC2PhiBelowLeaf30200 e24KC2PhiBelowLeaf30201 e24KC2PhiBelowLeaf30202 e24KC2PhiBelowLeaf30203
+
+theorem e24KC2PhiBelowNode3021 :
+    adaptiveCoverCheck 10 phiBelowCell3021 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3021
+    e24KC2PhiBelowLeaf30210 e24KC2PhiBelowLeaf30211 e24KC2PhiBelowLeaf30212 e24KC2PhiBelowLeaf30213
+
+theorem e24KC2PhiBelowNode3022 :
+    adaptiveCoverCheck 10 phiBelowCell3022 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3022
+    e24KC2PhiBelowLeaf30220 e24KC2PhiBelowLeaf30221 e24KC2PhiBelowLeaf30222 e24KC2PhiBelowLeaf30223
+
+theorem e24KC2PhiBelowNode3023 :
+    adaptiveCoverCheck 10 phiBelowCell3023 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3023
+    e24KC2PhiBelowLeaf30230 e24KC2PhiBelowLeaf30231 e24KC2PhiBelowLeaf30232 e24KC2PhiBelowLeaf30233
+
+theorem e24KC2PhiBelowNode3030 :
+    adaptiveCoverCheck 10 phiBelowCell3030 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3030
+    e24KC2PhiBelowLeaf30300 e24KC2PhiBelowLeaf30301 e24KC2PhiBelowLeaf30302 e24KC2PhiBelowLeaf30303
+
+theorem e24KC2PhiBelowNode3031 :
+    adaptiveCoverCheck 10 phiBelowCell3031 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3031
+    e24KC2PhiBelowLeaf30310 e24KC2PhiBelowLeaf30311 e24KC2PhiBelowLeaf30312 e24KC2PhiBelowLeaf30313
+
+theorem e24KC2PhiBelowNode3032 :
+    adaptiveCoverCheck 10 phiBelowCell3032 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3032
+    e24KC2PhiBelowLeaf30320 e24KC2PhiBelowLeaf30321 e24KC2PhiBelowLeaf30322 e24KC2PhiBelowLeaf30323
+
+theorem e24KC2PhiBelowNode3033 :
+    adaptiveCoverCheck 10 phiBelowCell3033 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3033
+    e24KC2PhiBelowLeaf30330 e24KC2PhiBelowLeaf30331 e24KC2PhiBelowLeaf30332 e24KC2PhiBelowLeaf30333
+
+theorem e24KC2PhiBelowNode3100 :
+    adaptiveCoverCheck 10 phiBelowCell3100 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3100
+    e24KC2PhiBelowLeaf31000 e24KC2PhiBelowLeaf31001 e24KC2PhiBelowLeaf31002 e24KC2PhiBelowLeaf31003
+
+theorem e24KC2PhiBelowNode3101 :
+    adaptiveCoverCheck 10 phiBelowCell3101 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3101
+    e24KC2PhiBelowLeaf31010 e24KC2PhiBelowLeaf31011 e24KC2PhiBelowLeaf31012 e24KC2PhiBelowLeaf31013
+
+theorem e24KC2PhiBelowNode3102 :
+    adaptiveCoverCheck 10 phiBelowCell3102 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3102
+    e24KC2PhiBelowLeaf31020 e24KC2PhiBelowLeaf31021 e24KC2PhiBelowLeaf31022 e24KC2PhiBelowLeaf31023
+
+theorem e24KC2PhiBelowNode3103 :
+    adaptiveCoverCheck 10 phiBelowCell3103 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3103
+    e24KC2PhiBelowLeaf31030 e24KC2PhiBelowLeaf31031 e24KC2PhiBelowLeaf31032 e24KC2PhiBelowLeaf31033
+
+theorem e24KC2PhiBelowNode3112 :
+    adaptiveCoverCheck 10 phiBelowCell3112 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3112
+    e24KC2PhiBelowLeaf31120 e24KC2PhiBelowLeaf31121 e24KC2PhiBelowLeaf31122 e24KC2PhiBelowLeaf31123
+
+theorem e24KC2PhiBelowNode3120 :
+    adaptiveCoverCheck 10 phiBelowCell3120 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3120
+    e24KC2PhiBelowLeaf31200 e24KC2PhiBelowLeaf31201 e24KC2PhiBelowLeaf31202 e24KC2PhiBelowLeaf31203
+
+theorem e24KC2PhiBelowNode3121 :
+    adaptiveCoverCheck 10 phiBelowCell3121 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3121
+    e24KC2PhiBelowLeaf31210 e24KC2PhiBelowLeaf31211 e24KC2PhiBelowLeaf31212 e24KC2PhiBelowLeaf31213
+
+theorem e24KC2PhiBelowNode3122 :
+    adaptiveCoverCheck 10 phiBelowCell3122 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3122
+    e24KC2PhiBelowLeaf31220 e24KC2PhiBelowLeaf31221 e24KC2PhiBelowLeaf31222 e24KC2PhiBelowLeaf31223
+
+theorem e24KC2PhiBelowNode3123 :
+    adaptiveCoverCheck 10 phiBelowCell3123 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3123
+    e24KC2PhiBelowLeaf31230 e24KC2PhiBelowLeaf31231 e24KC2PhiBelowLeaf31232 e24KC2PhiBelowLeaf31233
+
+theorem e24KC2PhiBelowNode3130 :
+    adaptiveCoverCheck 10 phiBelowCell3130 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3130
+    e24KC2PhiBelowLeaf31300 e24KC2PhiBelowLeaf31301 e24KC2PhiBelowLeaf31302 e24KC2PhiBelowLeaf31303
+
+theorem e24KC2PhiBelowNode3131 :
+    adaptiveCoverCheck 10 phiBelowCell3131 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3131
+    e24KC2PhiBelowLeaf31310 e24KC2PhiBelowLeaf31311 e24KC2PhiBelowLeaf31312 e24KC2PhiBelowLeaf31313
+
+theorem e24KC2PhiBelowNode3132 :
+    adaptiveCoverCheck 10 phiBelowCell3132 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3132
+    e24KC2PhiBelowLeaf31320 e24KC2PhiBelowLeaf31321 e24KC2PhiBelowLeaf31322 e24KC2PhiBelowLeaf31323
+
+theorem e24KC2PhiBelowNode3133 :
+    adaptiveCoverCheck 10 phiBelowCell3133 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3133
+    e24KC2PhiBelowLeaf31330 e24KC2PhiBelowLeaf31331 e24KC2PhiBelowLeaf31332 e24KC2PhiBelowLeaf31333
+
+theorem e24KC2PhiBelowNode3200 :
+    adaptiveCoverCheck 10 phiBelowCell3200 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3200
+    e24KC2PhiBelowLeaf32000 e24KC2PhiBelowLeaf32001 e24KC2PhiBelowLeaf32002 e24KC2PhiBelowLeaf32003
+
+theorem e24KC2PhiBelowNode3201 :
+    adaptiveCoverCheck 10 phiBelowCell3201 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3201
+    e24KC2PhiBelowLeaf32010 e24KC2PhiBelowLeaf32011 e24KC2PhiBelowLeaf32012 e24KC2PhiBelowLeaf32013
+
+theorem e24KC2PhiBelowNode3202 :
+    adaptiveCoverCheck 10 phiBelowCell3202 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3202
+    e24KC2PhiBelowLeaf32020 e24KC2PhiBelowLeaf32021 e24KC2PhiBelowLeaf32022 e24KC2PhiBelowLeaf32023
+
+theorem e24KC2PhiBelowNode3203 :
+    adaptiveCoverCheck 10 phiBelowCell3203 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3203
+    e24KC2PhiBelowLeaf32030 e24KC2PhiBelowLeaf32031 e24KC2PhiBelowLeaf32032 e24KC2PhiBelowLeaf32033
+
+theorem e24KC2PhiBelowNode3210 :
+    adaptiveCoverCheck 10 phiBelowCell3210 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3210
+    e24KC2PhiBelowLeaf32100 e24KC2PhiBelowLeaf32101 e24KC2PhiBelowLeaf32102 e24KC2PhiBelowLeaf32103
+
+theorem e24KC2PhiBelowNode3211 :
+    adaptiveCoverCheck 10 phiBelowCell3211 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3211
+    e24KC2PhiBelowLeaf32110 e24KC2PhiBelowLeaf32111 e24KC2PhiBelowLeaf32112 e24KC2PhiBelowLeaf32113
+
+theorem e24KC2PhiBelowNode3212 :
+    adaptiveCoverCheck 10 phiBelowCell3212 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3212
+    e24KC2PhiBelowLeaf32120 e24KC2PhiBelowLeaf32121 e24KC2PhiBelowLeaf32122 e24KC2PhiBelowLeaf32123
+
+theorem e24KC2PhiBelowNode3213 :
+    adaptiveCoverCheck 10 phiBelowCell3213 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3213
+    e24KC2PhiBelowLeaf32130 e24KC2PhiBelowLeaf32131 e24KC2PhiBelowLeaf32132 e24KC2PhiBelowLeaf32133
+
+theorem e24KC2PhiBelowNode3220 :
+    adaptiveCoverCheck 10 phiBelowCell3220 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3220
+    e24KC2PhiBelowLeaf32200 e24KC2PhiBelowLeaf32201 e24KC2PhiBelowLeaf32202 e24KC2PhiBelowLeaf32203
+
+theorem e24KC2PhiBelowNode3221 :
+    adaptiveCoverCheck 10 phiBelowCell3221 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3221
+    e24KC2PhiBelowLeaf32210 e24KC2PhiBelowLeaf32211 e24KC2PhiBelowLeaf32212 e24KC2PhiBelowLeaf32213
+
+theorem e24KC2PhiBelowNode3222 :
+    adaptiveCoverCheck 10 phiBelowCell3222 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3222
+    e24KC2PhiBelowLeaf32220 e24KC2PhiBelowLeaf32221 e24KC2PhiBelowLeaf32222 e24KC2PhiBelowLeaf32223
+
+theorem e24KC2PhiBelowNode3223 :
+    adaptiveCoverCheck 10 phiBelowCell3223 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3223
+    e24KC2PhiBelowLeaf32230 e24KC2PhiBelowLeaf32231 e24KC2PhiBelowLeaf32232 e24KC2PhiBelowLeaf32233
+
+theorem e24KC2PhiBelowNode3230 :
+    adaptiveCoverCheck 10 phiBelowCell3230 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3230
+    e24KC2PhiBelowLeaf32300 e24KC2PhiBelowLeaf32301 e24KC2PhiBelowLeaf32302 e24KC2PhiBelowLeaf32303
+
+theorem e24KC2PhiBelowNode3231 :
+    adaptiveCoverCheck 10 phiBelowCell3231 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3231
+    e24KC2PhiBelowLeaf32310 e24KC2PhiBelowLeaf32311 e24KC2PhiBelowLeaf32312 e24KC2PhiBelowLeaf32313
+
+theorem e24KC2PhiBelowNode3232 :
+    adaptiveCoverCheck 10 phiBelowCell3232 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3232
+    e24KC2PhiBelowLeaf32320 e24KC2PhiBelowLeaf32321 e24KC2PhiBelowLeaf32322 e24KC2PhiBelowLeaf32323
+
+theorem e24KC2PhiBelowNode3233 :
+    adaptiveCoverCheck 10 phiBelowCell3233 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3233
+    e24KC2PhiBelowLeaf32330 e24KC2PhiBelowLeaf32331 e24KC2PhiBelowLeaf32332 e24KC2PhiBelowLeaf32333
+
+theorem e24KC2PhiBelowNode3300 :
+    adaptiveCoverCheck 10 phiBelowCell3300 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3300
+    e24KC2PhiBelowLeaf33000 e24KC2PhiBelowLeaf33001 e24KC2PhiBelowLeaf33002 e24KC2PhiBelowLeaf33003
+
+theorem e24KC2PhiBelowNode3301 :
+    adaptiveCoverCheck 10 phiBelowCell3301 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3301
+    e24KC2PhiBelowLeaf33010 e24KC2PhiBelowLeaf33011 e24KC2PhiBelowLeaf33012 e24KC2PhiBelowLeaf33013
+
+theorem e24KC2PhiBelowNode3302 :
+    adaptiveCoverCheck 10 phiBelowCell3302 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3302
+    e24KC2PhiBelowLeaf33020 e24KC2PhiBelowLeaf33021 e24KC2PhiBelowLeaf33022 e24KC2PhiBelowLeaf33023
+
+theorem e24KC2PhiBelowNode3303 :
+    adaptiveCoverCheck 10 phiBelowCell3303 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3303
+    e24KC2PhiBelowLeaf33030 e24KC2PhiBelowLeaf33031 e24KC2PhiBelowLeaf33032 e24KC2PhiBelowLeaf33033
+
+theorem e24KC2PhiBelowNode3310 :
+    adaptiveCoverCheck 10 phiBelowCell3310 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3310
+    e24KC2PhiBelowLeaf33100 e24KC2PhiBelowLeaf33101 e24KC2PhiBelowLeaf33102 e24KC2PhiBelowLeaf33103
+
+theorem e24KC2PhiBelowNode3311 :
+    adaptiveCoverCheck 10 phiBelowCell3311 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3311
+    e24KC2PhiBelowLeaf33110 e24KC2PhiBelowLeaf33111 e24KC2PhiBelowLeaf33112 e24KC2PhiBelowLeaf33113
+
+theorem e24KC2PhiBelowNode3312 :
+    adaptiveCoverCheck 10 phiBelowCell3312 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3312
+    e24KC2PhiBelowLeaf33120 e24KC2PhiBelowLeaf33121 e24KC2PhiBelowLeaf33122 e24KC2PhiBelowLeaf33123
+
+theorem e24KC2PhiBelowNode3313 :
+    adaptiveCoverCheck 10 phiBelowCell3313 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3313
+    e24KC2PhiBelowLeaf33130 e24KC2PhiBelowLeaf33131 e24KC2PhiBelowLeaf33132 e24KC2PhiBelowLeaf33133
+
+theorem e24KC2PhiBelowNode3320 :
+    adaptiveCoverCheck 10 phiBelowCell3320 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3320
+    e24KC2PhiBelowLeaf33200 e24KC2PhiBelowLeaf33201 e24KC2PhiBelowLeaf33202 e24KC2PhiBelowLeaf33203
+
+theorem e24KC2PhiBelowNode3321 :
+    adaptiveCoverCheck 10 phiBelowCell3321 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3321
+    e24KC2PhiBelowLeaf33210 e24KC2PhiBelowLeaf33211 e24KC2PhiBelowLeaf33212 e24KC2PhiBelowLeaf33213
+
+theorem e24KC2PhiBelowNode3322 :
+    adaptiveCoverCheck 10 phiBelowCell3322 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3322
+    e24KC2PhiBelowLeaf33220 e24KC2PhiBelowLeaf33221 e24KC2PhiBelowLeaf33222 e24KC2PhiBelowLeaf33223
+
+theorem e24KC2PhiBelowNode3323 :
+    adaptiveCoverCheck 10 phiBelowCell3323 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3323
+    e24KC2PhiBelowLeaf33230 e24KC2PhiBelowLeaf33231 e24KC2PhiBelowLeaf33232 e24KC2PhiBelowLeaf33233
+
+theorem e24KC2PhiBelowNode3330 :
+    adaptiveCoverCheck 10 phiBelowCell3330 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3330
+    e24KC2PhiBelowLeaf33300 e24KC2PhiBelowLeaf33301 e24KC2PhiBelowLeaf33302 e24KC2PhiBelowLeaf33303
+
+theorem e24KC2PhiBelowNode3331 :
+    adaptiveCoverCheck 10 phiBelowCell3331 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3331
+    e24KC2PhiBelowLeaf33310 e24KC2PhiBelowLeaf33311 e24KC2PhiBelowLeaf33312 e24KC2PhiBelowLeaf33313
+
+theorem e24KC2PhiBelowNode3332 :
+    adaptiveCoverCheck 10 phiBelowCell3332 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3332
+    e24KC2PhiBelowLeaf33320 e24KC2PhiBelowLeaf33321 e24KC2PhiBelowLeaf33322 e24KC2PhiBelowLeaf33323
+
+theorem e24KC2PhiBelowNode3333 :
+    adaptiveCoverCheck 10 phiBelowCell3333 = true :=
+  adaptiveCoverCheck_succ_of_children 9 phiBelowCell3333
+    e24KC2PhiBelowLeaf33330 e24KC2PhiBelowLeaf33331 e24KC2PhiBelowLeaf33332 e24KC2PhiBelowLeaf33333
+
+theorem e24KC2PhiBelowNode300 :
+    adaptiveCoverCheck 11 (childLL (childLL (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLL (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowNode3000 e24KC2PhiBelowNode3001 e24KC2PhiBelowNode3002 e24KC2PhiBelowNode3003
+
+theorem e24KC2PhiBelowNode301 :
+    adaptiveCoverCheck 11 (childLH (childLL (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLL (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowNode3010 e24KC2PhiBelowNode3011 e24KC2PhiBelowNode3012 e24KC2PhiBelowNode3013
+
+theorem e24KC2PhiBelowNode302 :
+    adaptiveCoverCheck 11 (childHL (childLL (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLL (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowNode3020 e24KC2PhiBelowNode3021 e24KC2PhiBelowNode3022 e24KC2PhiBelowNode3023
+
+theorem e24KC2PhiBelowNode303 :
+    adaptiveCoverCheck 11 (childHH (childLL (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLL (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowNode3030 e24KC2PhiBelowNode3031 e24KC2PhiBelowNode3032 e24KC2PhiBelowNode3033
+
+theorem e24KC2PhiBelowNode310 :
+    adaptiveCoverCheck 11 (childLL (childLH (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childLH (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowNode3100 e24KC2PhiBelowNode3101 e24KC2PhiBelowNode3102 e24KC2PhiBelowNode3103
+
+theorem e24KC2PhiBelowNode311 :
+    adaptiveCoverCheck 11 (childLH (childLH (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childLH (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowLeaf3110 e24KC2PhiBelowLeaf3111 e24KC2PhiBelowNode3112 e24KC2PhiBelowLeaf3113
+
+theorem e24KC2PhiBelowNode312 :
+    adaptiveCoverCheck 11 (childHL (childLH (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childLH (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowNode3120 e24KC2PhiBelowNode3121 e24KC2PhiBelowNode3122 e24KC2PhiBelowNode3123
+
+theorem e24KC2PhiBelowNode313 :
+    adaptiveCoverCheck 11 (childHH (childLH (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childLH (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowNode3130 e24KC2PhiBelowNode3131 e24KC2PhiBelowNode3132 e24KC2PhiBelowNode3133
+
+theorem e24KC2PhiBelowNode320 :
+    adaptiveCoverCheck 11 (childLL (childHL (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childHL (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowNode3200 e24KC2PhiBelowNode3201 e24KC2PhiBelowNode3202 e24KC2PhiBelowNode3203
+
+theorem e24KC2PhiBelowNode321 :
+    adaptiveCoverCheck 11 (childLH (childHL (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childHL (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowNode3210 e24KC2PhiBelowNode3211 e24KC2PhiBelowNode3212 e24KC2PhiBelowNode3213
+
+theorem e24KC2PhiBelowNode322 :
+    adaptiveCoverCheck 11 (childHL (childHL (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childHL (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowNode3220 e24KC2PhiBelowNode3221 e24KC2PhiBelowNode3222 e24KC2PhiBelowNode3223
+
+theorem e24KC2PhiBelowNode323 :
+    adaptiveCoverCheck 11 (childHH (childHL (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childHL (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowNode3230 e24KC2PhiBelowNode3231 e24KC2PhiBelowNode3232 e24KC2PhiBelowNode3233
+
+theorem e24KC2PhiBelowNode330 :
+    adaptiveCoverCheck 11 (childLL (childHH (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLL (childHH (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowNode3300 e24KC2PhiBelowNode3301 e24KC2PhiBelowNode3302 e24KC2PhiBelowNode3303
+
+theorem e24KC2PhiBelowNode331 :
+    adaptiveCoverCheck 11 (childLH (childHH (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childLH (childHH (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowNode3310 e24KC2PhiBelowNode3311 e24KC2PhiBelowNode3312 e24KC2PhiBelowNode3313
+
+theorem e24KC2PhiBelowNode332 :
+    adaptiveCoverCheck 11 (childHL (childHH (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHL (childHH (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowNode3320 e24KC2PhiBelowNode3321 e24KC2PhiBelowNode3322 e24KC2PhiBelowNode3323
+
+theorem e24KC2PhiBelowNode333 :
+    adaptiveCoverCheck 11 (childHH (childHH (childHH e24PhiBelowRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 10 (childHH (childHH (childHH e24PhiBelowRoot)))
+    e24KC2PhiBelowNode3330 e24KC2PhiBelowNode3331 e24KC2PhiBelowNode3332 e24KC2PhiBelowNode3333
+
+theorem e24KC2PhiBelowNode30 :
+    adaptiveCoverCheck 12 (childLL (childHH e24PhiBelowRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH e24PhiBelowRoot))
+    e24KC2PhiBelowNode300 e24KC2PhiBelowNode301 e24KC2PhiBelowNode302 e24KC2PhiBelowNode303
+
+theorem e24KC2PhiBelowNode31 :
+    adaptiveCoverCheck 12 (childLH (childHH e24PhiBelowRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH e24PhiBelowRoot))
+    e24KC2PhiBelowNode310 e24KC2PhiBelowNode311 e24KC2PhiBelowNode312 e24KC2PhiBelowNode313
+
+theorem e24KC2PhiBelowNode32 :
+    adaptiveCoverCheck 12 (childHL (childHH e24PhiBelowRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH e24PhiBelowRoot))
+    e24KC2PhiBelowNode320 e24KC2PhiBelowNode321 e24KC2PhiBelowNode322 e24KC2PhiBelowNode323
+
+theorem e24KC2PhiBelowNode33 :
+    adaptiveCoverCheck 12 (childHH (childHH e24PhiBelowRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH e24PhiBelowRoot))
+    e24KC2PhiBelowNode330 e24KC2PhiBelowNode331 e24KC2PhiBelowNode332 e24KC2PhiBelowNode333
+
+theorem e24KC2PhiBelowNode3 :
+    adaptiveCoverCheck 13 (childHH e24PhiBelowRoot) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH e24PhiBelowRoot)
+    e24KC2PhiBelowNode30 e24KC2PhiBelowNode31 e24KC2PhiBelowNode32 e24KC2PhiBelowNode33
+
+theorem e24KC2PhiBelowNodeROOT :
+    adaptiveCoverCheck 14 e24PhiBelowRoot = true :=
+  adaptiveCoverCheck_succ_of_children 13 e24PhiBelowRoot
+    e24PhiBelowKernelLL e24PhiBelowKernelLH e24PhiBelowKernelHL e24KC2PhiBelowNode3
+
+theorem e24PhiBelowKernelCheck :
+    adaptiveCoverCheck 14 e24PhiBelowRoot = true :=
+  e24KC2PhiBelowNodeROOT
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c1_c2_4_00016
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells22695df69a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells22695df69a
+
+open CertificateCells22695df69a
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c2 :
+    adaptiveCoverCheck 4 (childHL (childLH (childHH thetaAboveCell000022002011))) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childHL (childLH (childHH thetaAboveCell000022002011)))
+    e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c2_c0
+      e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c2_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c2_c2
+      e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c1_c3_4_00022
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells782ac45202
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells782ac45202
+
+open CertificateCells782ac45202
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c3 :
+    adaptiveCoverCheck 4 (childHH (childLH (childHH thetaAboveCell000022002011))) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childHH (childLH (childHH thetaAboveCell000022002011)))
+    e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c3_c0
+      e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c3_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c3_c2
+      e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c1_5_00023
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb8f48433cf
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb8f48433cf
+
+open CertificateCellsb8f48433cf
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022002011)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022002011))
+    e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c0 e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c2 e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c0_c0_4_00484
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5bc1aa1a31
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5bc1aa1a31
+
+open CertificateCells5bc1aa1a31
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c0 :
+    adaptiveCoverCheck 4 (childLL (childLL (childHL thetaAboveCell000022002000))) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHL thetaAboveCell000022002000)))
+    e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c0_c0
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c0_c2
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c0_c1_4_00490
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells29b43c6eb7
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells29b43c6eb7
+
+open CertificateCells29b43c6eb7
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c1 :
+    adaptiveCoverCheck 4 (childLH (childLL (childHL thetaAboveCell000022002000))) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHL thetaAboveCell000022002000)))
+    e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c1_c0
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c1_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c1_c2
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c0_5_00493
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells64bf70ad53
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells64bf70ad53
+
+open CertificateCells64bf70ad53
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022002000)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022002000))
+    e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c0 e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c2 e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c1_c0_4_00500
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc935d550f9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc935d550f9
+
+open CertificateCellsc935d550f9
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c0 :
+    adaptiveCoverCheck 4 (childLL (childLH (childHL thetaAboveCell000022002000))) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHL thetaAboveCell000022002000)))
+    e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c0_c0
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c0_c2
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c1_c1_4_00506
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0a80d69c2b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0a80d69c2b
+
+open CertificateCells0a80d69c2b
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c1 :
+    adaptiveCoverCheck 4 (childLH (childLH (childHL thetaAboveCell000022002000))) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHL thetaAboveCell000022002000)))
+    e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c1_c0
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c1_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c1_c2
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c0_c2_c1_5_00509
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsfd35b16333
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsfd35b16333
+
+open CertificateCellsfd35b16333
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022002000)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022002000))
+    e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c0 e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c2 e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above Leaf0000220020_c0_c0_c2_6_00512
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsd8bbac8cc5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsd8bbac8cc5
+
+open CertificateCellsd8bbac8cc5
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c2 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022002000) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022002000)
+    e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c0 e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c2 e24KC2ThetaAboveLeaf0000220020_c0_c0_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c0_c0_4_00520
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2eb3bc7e90
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2eb3bc7e90
+
+open CertificateCells2eb3bc7e90
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c0 :
+    adaptiveCoverCheck 4 (childLL (childLL (childHH thetaAboveCell000022002000))) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHH thetaAboveCell000022002000)))
+    e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c0_c0
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c0_c2
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c0_c1_4_00526
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsbc99de789e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsbc99de789e
+
+open CertificateCellsbc99de789e
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c1 :
+    adaptiveCoverCheck 4 (childLH (childLL (childHH thetaAboveCell000022002000))) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHH thetaAboveCell000022002000)))
+    e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c1_c0
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c1_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c1_c2
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c0_5_00529
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdc52d78b64
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdc52d78b64
+
+open CertificateCellsdc52d78b64
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022002000)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022002000))
+    e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c0 e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c2 e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c1_c0_4_00536
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells69f98d6dd4
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells69f98d6dd4
+
+open CertificateCells69f98d6dd4
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c0 :
+    adaptiveCoverCheck 4 (childLL (childLH (childHH thetaAboveCell000022002000))) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHH thetaAboveCell000022002000)))
+    e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c0_c0
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c0_c2
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c1_c1_4_00542
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells675835e7b9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells675835e7b9
+
+open CertificateCells675835e7b9
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c1 :
+    adaptiveCoverCheck 4 (childLH (childLH (childHH thetaAboveCell000022002000))) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childLH (childLH (childHH thetaAboveCell000022002000)))
+    e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c1_c0
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c1_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c1_c2
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c0_c3_c1_5_00545
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsda5cde5aa9
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsda5cde5aa9
+
+open CertificateCellsda5cde5aa9
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022002000)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022002000))
+    e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c0 e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c2 e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above Leaf0000220020_c0_c0_c3_6_00548
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsed33eab041
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsed33eab041
+
+open CertificateCellsed33eab041
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0_c3 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022002000) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022002000)
+    e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c0 e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c2 e24KC2ThetaAboveLeaf0000220020_c0_c0_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above Leaf0000220020_c0_c0_7_00549
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsf997ca54bf
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsf997ca54bf
+
+open CertificateCellsf997ca54bf
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c0 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002000 = true :=
+  adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022002000
+    e24KC2ThetaAboveLeaf0000220020_c0_c0_c0 e24KC2ThetaAboveLeaf0000220020_c0_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c0_c2 e24KC2ThetaAboveLeaf0000220020_c0_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c0_c0_4_00560
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsba35077fe5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsba35077fe5
+
+open CertificateCellsba35077fe5
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c0 :
+    adaptiveCoverCheck 4 (childLL (childLL (childHL thetaAboveCell000022002001))) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childLL (childLL (childHL thetaAboveCell000022002001)))
+    e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c0_c0
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c0_c2
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c0_c1_4_00566
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5d0bfdcc89
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5d0bfdcc89
+
+open CertificateCells5d0bfdcc89
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c1 :
+    adaptiveCoverCheck 4 (childLH (childLL (childHL thetaAboveCell000022002001))) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childLH (childLL (childHL thetaAboveCell000022002001)))
+    e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c1_c0
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c1_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c1_c2
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c0_5_00569
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb9eba9ae01
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb9eba9ae01
+
+open CertificateCellsb9eba9ae01
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022002001)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022002001))
+    e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c0 e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c2 e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c1_c0_4_00576
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells5ae8bbaf2e
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells5ae8bbaf2e
+
+open CertificateCells5ae8bbaf2e
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c0 :
+    adaptiveCoverCheck 4 (childLL (childLH (childHL thetaAboveCell000022002001))) = true :=
+  adaptiveCoverCheck_succ_of_children 3 (childLL (childLH (childHL thetaAboveCell000022002001)))
+    e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c0_c0
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c0_c2
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c1_c2_c1_5_00580
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsbec28e0304
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsbec28e0304
+
+open CertificateCellsbec28e0304
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022002001)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022002001))
+    e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c0 e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c2 e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above Leaf0000220020_c0_c1_c2_6_00583
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1b578c3d10
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1b578c3d10
+
+open CertificateCells1b578c3d10
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c2 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022002001) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022002001)
+    e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c0 e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c2 e24KC2ThetaAboveLeaf0000220020_c0_c1_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c1_c3_c0_5_00590
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2bc65b7190
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2bc65b7190
+
+open CertificateCells2bc65b7190
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c0 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022002001)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022002001))
+    e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c0_c0 e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c0_c2 e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c0_c1_c3_c1_5_00596
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsc46dbfc77b
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsc46dbfc77b
+
+open CertificateCellsc46dbfc77b
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c1 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022002001)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022002001))
+    e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c1_c0 e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c1_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c1_c2 e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above Leaf0000220020_c0_c1_c3_6_00599
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells752b604750
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells752b604750
+
+open CertificateCells752b604750
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1_c3 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022002001) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022002001)
+    e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c0 e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c2 e24KC2ThetaAboveLeaf0000220020_c0_c1_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above Leaf0000220020_c0_c1_7_00600
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellse66f3ce3f5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellse66f3ce3f5
+
+open CertificateCellse66f3ce3f5
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0_c1 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002001 = true :=
+  adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022002001
+    e24KC2ThetaAboveLeaf0000220020_c0_c1_c0 e24KC2ThetaAboveLeaf0000220020_c0_c1_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c1_c2 e24KC2ThetaAboveLeaf0000220020_c0_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above Leaf0000220020_c0_8_00603
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells45a373af38
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells45a373af38
+
+open CertificateCells45a373af38
+
+theorem e24KC2ThetaAboveLeaf0000220020_c0 :
+    adaptiveCoverCheck 8 (childLL (childLL (childHL thetaAboveCell00002200))) = true :=
+  adaptiveCoverCheck_succ_of_children 7 (childLL (childLL (childHL thetaAboveCell00002200)))
+    e24KC2ThetaAboveLeaf0000220020_c0_c0 e24KC2ThetaAboveLeaf0000220020_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c0_c2 e24KC2ThetaAboveLeaf0000220020_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c1_c0_c2_c0_5_00614
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdabf824b24
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdabf824b24
+
+open CertificateCellsdabf824b24
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c0 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022002010)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022002010))
+    e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c0_c0 e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c0_c2 e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c1_c0_c2_c1_5_00620
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells11c2c46ea8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells11c2c46ea8
+
+open CertificateCells11c2c46ea8
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c1 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022002010)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022002010))
+    e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c1_c0 e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c1_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c1_c2 e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above Leaf0000220020_c1_c0_c2_6_00623
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsfc2738963a
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsfc2738963a
+
+open CertificateCellsfc2738963a
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c2 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022002010) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022002010)
+    e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c0 e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c2 e24KC2ThetaAboveLeaf0000220020_c1_c0_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c1_c0_c3_c0_5_00630
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells2c3b2a68a2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells2c3b2a68a2
+
+open CertificateCells2c3b2a68a2
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c0 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022002010)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022002010))
+    e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c0_c0 e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c0_c2 e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c1_c0_c3_c1_5_00636
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells024fe9d391
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells024fe9d391
+
+open CertificateCells024fe9d391
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c1 :
+    adaptiveCoverCheck 5 (childLH (childHH thetaAboveCell000022002010)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLH (childHH thetaAboveCell000022002010))
+    e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c1_c0 e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c1_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c1_c2 e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above Leaf0000220020_c1_c0_c3_6_00639
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells95bf7ab700
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells95bf7ab700
+
+open CertificateCells95bf7ab700
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0_c3 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022002010) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022002010)
+    e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c0 e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c2 e24KC2ThetaAboveLeaf0000220020_c1_c0_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above Leaf0000220020_c1_c0_7_00640
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb2f77f238f
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb2f77f238f
+
+open CertificateCellsb2f77f238f
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c0 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002010 = true :=
+  adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022002010
+    e24KC2ThetaAboveLeaf0000220020_c1_c0_c0 e24KC2ThetaAboveLeaf0000220020_c1_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c0_c2 e24KC2ThetaAboveLeaf0000220020_c1_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c1_c1_c2_c0_5_00650
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells8179cb7dd8
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells8179cb7dd8
+
+open CertificateCells8179cb7dd8
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c0 :
+    adaptiveCoverCheck 5 (childLL (childHL thetaAboveCell000022002011)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLL (childHL thetaAboveCell000022002011))
+    e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c0_c0 e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c0_c2 e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c1_c1_c2_c1_5_00656
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsdb257f0c7c
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsdb257f0c7c
+
+open CertificateCellsdb257f0c7c
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c1 :
+    adaptiveCoverCheck 5 (childLH (childHL thetaAboveCell000022002011)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLH (childHL thetaAboveCell000022002011))
+    e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c1_c0 e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c1_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c1_c2 e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above Leaf0000220020_c1_c1_c2_6_00659
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells3a304b7358
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells3a304b7358
+
+open CertificateCells3a304b7358
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c2 :
+    adaptiveCoverCheck 6 (childHL thetaAboveCell000022002011) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childHL thetaAboveCell000022002011)
+    e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c0 e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c2 e24KC2ThetaAboveLeaf0000220020_c1_c1_c2_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above
+Leaf0000220020_c1_c1_c3_c0_5_00666
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells6cdc8e3bc2
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells6cdc8e3bc2
+
+open CertificateCells6cdc8e3bc2
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c0 :
+    adaptiveCoverCheck 5 (childLL (childHH thetaAboveCell000022002011)) = true :=
+  adaptiveCoverCheck_succ_of_children 4 (childLL (childHH thetaAboveCell000022002011))
+    e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c0_c0 e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c0_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c0_c2 e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c0_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above Leaf0000220020_c1_c1_c3_6_00026
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells0157659d62
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells0157659d62
+
+open CertificateCells0157659d62
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1_c3 :
+    adaptiveCoverCheck 6 (childHH thetaAboveCell000022002011) = true :=
+  adaptiveCoverCheck_succ_of_children 5 (childHH thetaAboveCell000022002011)
+    e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c0 e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c2 e24KC2ThetaAboveLeaf0000220020_c1_c1_c3_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC5Join_e24KC2Theta Above Leaf0000220020_c1_c1_7_00027
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells834e9932ec
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells834e9932ec
+
+open CertificateCells834e9932ec
+
+theorem e24KC2ThetaAboveLeaf0000220020_c1_c1 :
+    adaptiveCoverCheck 7 thetaAboveCell000022002011 = true :=
+  adaptiveCoverCheck_succ_of_children 6 thetaAboveCell000022002011
+    e24KC2ThetaAboveLeaf0000220020_c1_c1_c0 e24KC2ThetaAboveLeaf0000220020_c1_c1_c1
+      e24KC2ThetaAboveLeaf0000220020_c1_c1_c2 e24KC2ThetaAboveLeaf0000220020_c1_c1_c3
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-! E24KC6 explicit proof-producing certificate batch. -/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCellsb02c7d0e39
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCellsb02c7d0e39
+
+open CertificateCellsb02c7d0e39
+theorem e24KC2ThetaAboveLeaf0000220020 :
+    adaptiveCoverCheck 9 (childLL (childHL thetaAboveCell00002200)) = true := by
+  exact adaptiveCoverCheck_succ_of_children 8 (childLL (childHL thetaAboveCell00002200))
+    (by
+      exact e24KC2ThetaAboveLeaf0000220020_c0)
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childLH (childLL (childHL
+        thetaAboveCell00002200)))
+        (by
+          exact e24KC2ThetaAboveLeaf0000220020_c1_c0)
+        (by
+          exact e24KC2ThetaAboveLeaf0000220020_c1_c1)
+        (by
+          exact e24KC2ThetaAboveLeaf0000220020_c1_c2)
+        (by
+          exact e24KC2ThetaAboveLeaf0000220020_c1_c3))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHL (childLL (childHL
+        thetaAboveCell00002200)))
+        (by
+          have h : (thetaAboveCell000022002020).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002020 h)
+        (by
+          have h : (thetaAboveCell000022002021).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002021 h)
+        (by
+          have h : (thetaAboveCell000022002022).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002022 h)
+        (by
+          have h : (thetaAboveCell000022002023).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002023 h))
+    (by
+      exact adaptiveCoverCheck_succ_of_children 7 (childHH (childLL (childHL
+        thetaAboveCell00002200)))
+        (by
+          have h : (thetaAboveCell000022002030).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002030 h)
+        (by
+          have h : (thetaAboveCell000022002031).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002031 h)
+        (by
+          have h : (thetaAboveCell000022002032).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002032 h)
+        (by
+          have h : (thetaAboveCell000022002033).rejected = true := by
+            decide +kernel
+          exact adaptiveCoverCheck_true_of_rejected 7 thetaAboveCell000022002033 h))
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle031.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle031.lean
new file mode 100644
index 0000000000..352247b91f
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle031.lean
@@ -0,0 +1,6932 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle030
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch045`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa.PartE.CertificateCells1a75b359b5
+
+/-- Subcell `0000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0000 : AngleCell :=
+  childLL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0001 : AngleCell :=
+  childLH (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0010 : AngleCell :=
+  childLL (childLH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0011 : AngleCell :=
+  childLH (childLH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0100 : AngleCell :=
+  childLL (childLL (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0101 : AngleCell :=
+  childLH (childLL (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0110 : AngleCell :=
+  childLL (childLH (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0111 : AngleCell :=
+  childLH (childLH (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `1000` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1000 : AngleCell :=
+  childLL (childLL (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1001` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1001 : AngleCell :=
+  childLH (childLL (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1010` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1010 : AngleCell :=
+  childLL (childLH (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1011` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1011 : AngleCell :=
+  childLH (childLH (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1100` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1100 : AngleCell :=
+  childLL (childLL (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1101` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1101 : AngleCell :=
+  childLH (childLL (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1110` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1110 : AngleCell :=
+  childLL (childLH (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1111` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1111 : AngleCell :=
+  childLH (childLH (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `0002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0002 : AngleCell :=
+  childHL (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0003 : AngleCell :=
+  childHH (childLL (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0012 : AngleCell :=
+  childHL (childLH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0013 : AngleCell :=
+  childHH (childLH (childLL (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0102 : AngleCell :=
+  childHL (childLL (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0103 : AngleCell :=
+  childHH (childLL (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0112 : AngleCell :=
+  childHL (childLH (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `0113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell0113 : AngleCell :=
+  childHH (childLH (childLH (childLL e24ThetaAboveRoot)))
+
+/-- Subcell `1002` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1002 : AngleCell :=
+  childHL (childLL (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1003` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1003 : AngleCell :=
+  childHH (childLL (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1012` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1012 : AngleCell :=
+  childHL (childLH (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1013` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1013 : AngleCell :=
+  childHH (childLH (childLL (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1102` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1102 : AngleCell :=
+  childHL (childLL (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1103` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1103 : AngleCell :=
+  childHH (childLL (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1112` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1112 : AngleCell :=
+  childHL (childLH (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `1113` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell1113 : AngleCell :=
+  childHH (childLH (childLH (childLH e24ThetaAboveRoot)))
+
+/-- Subcell `00002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0010)))
+
+/-- Subcell `00112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0011)))
+
+/-- Subcell `00113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `00113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0011)))
+
+/-- Subcell `01002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0100)))
+
+/-- Subcell `01003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `10002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `00002022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell0000)))
+
+/-- Subcell `00003022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00003133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00003133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell0000)))
+
+/-- Subcell `00012022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00012133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00012133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell0001)))
+
+/-- Subcell `00013022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013022 : AngleCell :=
+  childHL (childHL (childLL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013023 : AngleCell :=
+  childHH (childHL (childLL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013032 : AngleCell :=
+  childHL (childHH (childLL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013033 : AngleCell :=
+  childHH (childHH (childLL (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013122 : AngleCell :=
+  childHL (childHL (childLH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013123 : AngleCell :=
+  childHH (childHL (childLH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013132 : AngleCell :=
+  childHL (childHH (childLH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00013133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00013133 : AngleCell :=
+  childHH (childHH (childLH (childHH thetaAboveCell0001)))
+
+/-- Subcell `00102022` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102022 : AngleCell :=
+  childHL (childHL (childLL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102023` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102023 : AngleCell :=
+  childHH (childHL (childLL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102032` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102032 : AngleCell :=
+  childHL (childHH (childLL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102033` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102033 : AngleCell :=
+  childHH (childHH (childLL (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102122` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102122 : AngleCell :=
+  childHL (childHL (childLH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102123` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102123 : AngleCell :=
+  childHH (childHL (childLH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102132` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102132 : AngleCell :=
+  childHL (childHH (childLH (childHL thetaAboveCell0010)))
+
+/-- Subcell `00102133` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell00102133 : AngleCell :=
+  childHH (childHH (childLH (childHL thetaAboveCell0010)))
+
+/-- Subcell `01003330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01003331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01003331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell0100)))
+
+/-- Subcell `01012220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01012331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01012331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell0101)))
+
+/-- Subcell `01013220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01013331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01013331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell0101)))
+
+/-- Subcell `01102220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01102331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01102331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell0110)))
+
+/-- Subcell `01103220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01103331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01103331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell0110)))
+
+/-- Subcell `01112220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01112331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01112331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell0111)))
+
+/-- Subcell `01113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `01113331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell01113331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell0111)))
+
+/-- Subcell `10002200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10002331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10002331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1000)))
+
+/-- Subcell `10003200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10003330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10003330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1000)))
+
+/-- Subcell `10012200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10012311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10012311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell1001)))
+
+/-- Subcell `10013200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10013331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10013331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1001)))
+
+/-- Subcell `10102200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10102331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10102331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1010)))
+
+/-- Subcell `10103200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10103331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10103331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1010)))
+
+/-- Subcell `10112200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112200 : AngleCell :=
+  childLL (childLL (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112201 : AngleCell :=
+  childLH (childLL (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112210 : AngleCell :=
+  childLL (childLH (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112211 : AngleCell :=
+  childLH (childLH (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112300 : AngleCell :=
+  childLL (childLL (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112301 : AngleCell :=
+  childLH (childLL (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112310 : AngleCell :=
+  childLL (childLH (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112311 : AngleCell :=
+  childLH (childLH (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10112331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10112331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1011)))
+
+/-- Subcell `10113200` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113200 : AngleCell :=
+  childLL (childLL (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113201` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113201 : AngleCell :=
+  childLH (childLL (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113210` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113210 : AngleCell :=
+  childLL (childLH (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113211` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113211 : AngleCell :=
+  childLH (childLH (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113300` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113300 : AngleCell :=
+  childLL (childLL (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113301` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113301 : AngleCell :=
+  childLH (childLL (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113310` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113310 : AngleCell :=
+  childLL (childLH (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113311` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113311 : AngleCell :=
+  childLH (childLH (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `10113331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell10113331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1011)))
+
+/-- Subcell `11002202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11002331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11002331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1100)))
+
+/-- Subcell `11003202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11003331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11003331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1100)))
+
+/-- Subcell `11012202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11012331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11012331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1101)))
+
+/-- Subcell `11013202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11013331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11013331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1101)))
+
+/-- Subcell `11102202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11102331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11102331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1110)))
+
+/-- Subcell `11103202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11103331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11103331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1110)))
+
+/-- Subcell `11112202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112202 : AngleCell :=
+  childHL (childLL (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112203 : AngleCell :=
+  childHH (childLL (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112212 : AngleCell :=
+  childHL (childLH (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112213 : AngleCell :=
+  childHH (childLH (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112220 : AngleCell :=
+  childLL (childHL (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112221 : AngleCell :=
+  childLH (childHL (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112230 : AngleCell :=
+  childLL (childHH (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112231 : AngleCell :=
+  childLH (childHH (childHL (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112302 : AngleCell :=
+  childHL (childLL (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112303 : AngleCell :=
+  childHH (childLL (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112312 : AngleCell :=
+  childHL (childLH (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112313 : AngleCell :=
+  childHH (childLH (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112320 : AngleCell :=
+  childLL (childHL (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112321 : AngleCell :=
+  childLH (childHL (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112330 : AngleCell :=
+  childLL (childHH (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112331 : AngleCell :=
+  childLH (childHH (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112332 : AngleCell :=
+  childHL (childHH (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11112333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11112333 : AngleCell :=
+  childHH (childHH (childHH (childHL thetaAboveCell1111)))
+
+/-- Subcell `11113202` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113202 : AngleCell :=
+  childHL (childLL (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113203` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113203 : AngleCell :=
+  childHH (childLL (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113212` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113212 : AngleCell :=
+  childHL (childLH (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113213` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113213 : AngleCell :=
+  childHH (childLH (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113220` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113220 : AngleCell :=
+  childLL (childHL (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113221` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113221 : AngleCell :=
+  childLH (childHL (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113222` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113222 : AngleCell :=
+  childHL (childHL (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113223` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113223 : AngleCell :=
+  childHH (childHL (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113230` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113230 : AngleCell :=
+  childLL (childHH (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113231` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113231 : AngleCell :=
+  childLH (childHH (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113232` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113232 : AngleCell :=
+  childHL (childHH (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113233` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113233 : AngleCell :=
+  childHH (childHH (childHL (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113302` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113302 : AngleCell :=
+  childHL (childLL (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113303` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113303 : AngleCell :=
+  childHH (childLL (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113312` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113312 : AngleCell :=
+  childHL (childLH (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113313` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113313 : AngleCell :=
+  childHH (childLH (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113320` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113320 : AngleCell :=
+  childLL (childHL (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113321` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113321 : AngleCell :=
+  childLH (childHL (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113322` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113322 : AngleCell :=
+  childHL (childHL (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113323` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113323 : AngleCell :=
+  childHH (childHL (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113330` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113330 : AngleCell :=
+  childLL (childHH (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113331` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113331 : AngleCell :=
+  childLH (childHH (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113332` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113332 : AngleCell :=
+  childHL (childHH (childHH (childHH thetaAboveCell1111)))
+
+/-- Subcell `11113333` of the theta-above root; digits 0–3 mean LL, LH, HL, HH. -/
+abbrev thetaAboveCell11113333 : AngleCell :=
+  childHH (childHH (childHH (childHH thetaAboveCell1111)))
+
+end GerverSofa.PartE.CertificateCells1a75b359b5
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.ThetaAbove.ReconstructionBase`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC6Theta Above Reconstruct
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+namespace CertificateCells1a75b359b5
+
+-- Base-four digits encode LL, LH, HL, HH subdivisions of the named root.
+
+end CertificateCells1a75b359b5
+
+open CertificateCells1a75b359b5
+
+theorem e24KC2ThetaAboveNode000022000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00002200)
+    e24KC2ThetaAboveLeaf0000220000 e24KC2ThetaAboveLeaf0000220001 e24KC2ThetaAboveLeaf0000220002
+      e24KC2ThetaAboveLeaf0000220003
+
+theorem e24KC2ThetaAboveNode000022001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00002200)
+    e24KC2ThetaAboveLeaf0000220010 e24KC2ThetaAboveLeaf0000220011 e24KC2ThetaAboveLeaf0000220012
+      e24KC2ThetaAboveLeaf0000220013
+
+theorem e24KC2ThetaAboveNode000022002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00002200)
+    e24KC2ThetaAboveLeaf0000220020 e24KC2ThetaAboveLeaf0000220021 e24KC2ThetaAboveLeaf0000220022
+      e24KC2ThetaAboveLeaf0000220023
+
+theorem e24KC2ThetaAboveNode000022003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00002200)
+    e24KC2ThetaAboveLeaf0000220030 e24KC2ThetaAboveLeaf0000220031 e24KC2ThetaAboveLeaf0000220032
+      e24KC2ThetaAboveLeaf0000220033
+
+theorem e24KC2ThetaAboveNode000022010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00002201)
+    e24KC2ThetaAboveLeaf0000220100 e24KC2ThetaAboveLeaf0000220101 e24KC2ThetaAboveLeaf0000220102
+      e24KC2ThetaAboveLeaf0000220103
+
+theorem e24KC2ThetaAboveNode000022011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00002201)
+    e24KC2ThetaAboveLeaf0000220110 e24KC2ThetaAboveLeaf0000220111 e24KC2ThetaAboveLeaf0000220112
+      e24KC2ThetaAboveLeaf0000220113
+
+theorem e24KC2ThetaAboveNode000022012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00002201)
+    e24KC2ThetaAboveLeaf0000220120 e24KC2ThetaAboveLeaf0000220121 e24KC2ThetaAboveLeaf0000220122
+      e24KC2ThetaAboveLeaf0000220123
+
+theorem e24KC2ThetaAboveNode000022013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00002201)
+    e24KC2ThetaAboveLeaf0000220130 e24KC2ThetaAboveLeaf0000220131 e24KC2ThetaAboveLeaf0000220132
+      e24KC2ThetaAboveLeaf0000220133
+
+theorem e24KC2ThetaAboveNode000022100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00002210)
+    e24KC2ThetaAboveLeaf0000221000 e24KC2ThetaAboveLeaf0000221001 e24KC2ThetaAboveLeaf0000221002
+      e24KC2ThetaAboveLeaf0000221003
+
+theorem e24KC2ThetaAboveNode000022101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00002210)
+    e24KC2ThetaAboveLeaf0000221010 e24KC2ThetaAboveLeaf0000221011 e24KC2ThetaAboveLeaf0000221012
+      e24KC2ThetaAboveLeaf0000221013
+
+theorem e24KC2ThetaAboveNode000022102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00002210)
+    e24KC2ThetaAboveLeaf0000221020 e24KC2ThetaAboveLeaf0000221021 e24KC2ThetaAboveLeaf0000221022
+      e24KC2ThetaAboveLeaf0000221023
+
+theorem e24KC2ThetaAboveNode000022103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00002210)
+    e24KC2ThetaAboveLeaf0000221030 e24KC2ThetaAboveLeaf0000221031 e24KC2ThetaAboveLeaf0000221032
+      e24KC2ThetaAboveLeaf0000221033
+
+theorem e24KC2ThetaAboveNode000022110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00002211)
+    e24KC2ThetaAboveLeaf0000221100 e24KC2ThetaAboveLeaf0000221101 e24KC2ThetaAboveLeaf0000221102
+      e24KC2ThetaAboveLeaf0000221103
+
+theorem e24KC2ThetaAboveNode000022111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00002211)
+    e24KC2ThetaAboveLeaf0000221110 e24KC2ThetaAboveLeaf0000221111 e24KC2ThetaAboveLeaf0000221112
+      e24KC2ThetaAboveLeaf0000221113
+
+theorem e24KC2ThetaAboveNode000022112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00002211)
+    e24KC2ThetaAboveLeaf0000221120 e24KC2ThetaAboveLeaf0000221121 e24KC2ThetaAboveLeaf0000221122
+      e24KC2ThetaAboveLeaf0000221123
+
+theorem e24KC2ThetaAboveNode000022113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00002211)
+    e24KC2ThetaAboveLeaf0000221130 e24KC2ThetaAboveLeaf0000221131 e24KC2ThetaAboveLeaf0000221132
+      e24KC2ThetaAboveLeaf0000221133
+
+theorem e24KC2ThetaAboveNode000022131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00002213)
+    e24KC2ThetaAboveLeaf0000221310 e24KC2ThetaAboveLeaf0000221311 e24KC2ThetaAboveLeaf0000221312
+      e24KC2ThetaAboveLeaf0000221313
+
+theorem e24KC2ThetaAboveNode000023000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00002300)
+    e24KC2ThetaAboveLeaf0000230000 e24KC2ThetaAboveLeaf0000230001 e24KC2ThetaAboveLeaf0000230002
+      e24KC2ThetaAboveLeaf0000230003
+
+theorem e24KC2ThetaAboveNode000023001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00002300)
+    e24KC2ThetaAboveLeaf0000230010 e24KC2ThetaAboveLeaf0000230011 e24KC2ThetaAboveLeaf0000230012
+      e24KC2ThetaAboveLeaf0000230013
+
+theorem e24KC2ThetaAboveNode000023002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00002300)
+    e24KC2ThetaAboveLeaf0000230020 e24KC2ThetaAboveLeaf0000230021 e24KC2ThetaAboveLeaf0000230022
+      e24KC2ThetaAboveLeaf0000230023
+
+theorem e24KC2ThetaAboveNode000023003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00002300)
+    e24KC2ThetaAboveLeaf0000230030 e24KC2ThetaAboveLeaf0000230031 e24KC2ThetaAboveLeaf0000230032
+      e24KC2ThetaAboveLeaf0000230033
+
+theorem e24KC2ThetaAboveNode000023010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00002301)
+    e24KC2ThetaAboveLeaf0000230100 e24KC2ThetaAboveLeaf0000230101 e24KC2ThetaAboveLeaf0000230102
+      e24KC2ThetaAboveLeaf0000230103
+
+theorem e24KC2ThetaAboveNode000023011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00002301)
+    e24KC2ThetaAboveLeaf0000230110 e24KC2ThetaAboveLeaf0000230111 e24KC2ThetaAboveLeaf0000230112
+      e24KC2ThetaAboveLeaf0000230113
+
+theorem e24KC2ThetaAboveNode000023012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00002301)
+    e24KC2ThetaAboveLeaf0000230120 e24KC2ThetaAboveLeaf0000230121 e24KC2ThetaAboveLeaf0000230122
+      e24KC2ThetaAboveLeaf0000230123
+
+theorem e24KC2ThetaAboveNode000023013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00002301)
+    e24KC2ThetaAboveLeaf0000230130 e24KC2ThetaAboveLeaf0000230131 e24KC2ThetaAboveLeaf0000230132
+      e24KC2ThetaAboveLeaf0000230133
+
+theorem e24KC2ThetaAboveNode000023020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00002302)
+    e24KC2ThetaAboveLeaf0000230200 e24KC2ThetaAboveLeaf0000230201 e24KC2ThetaAboveLeaf0000230202
+      e24KC2ThetaAboveLeaf0000230203
+
+theorem e24KC2ThetaAboveNode000023021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00002302)
+    e24KC2ThetaAboveLeaf0000230210 e24KC2ThetaAboveLeaf0000230211 e24KC2ThetaAboveLeaf0000230212
+      e24KC2ThetaAboveLeaf0000230213
+
+theorem e24KC2ThetaAboveNode000023030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00002303)
+    e24KC2ThetaAboveLeaf0000230300 e24KC2ThetaAboveLeaf0000230301 e24KC2ThetaAboveLeaf0000230302
+      e24KC2ThetaAboveLeaf0000230303
+
+theorem e24KC2ThetaAboveNode000023031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00002303)
+    e24KC2ThetaAboveLeaf0000230310 e24KC2ThetaAboveLeaf0000230311 e24KC2ThetaAboveLeaf0000230312
+      e24KC2ThetaAboveLeaf0000230313
+
+theorem e24KC2ThetaAboveNode000023100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00002310)
+    e24KC2ThetaAboveLeaf0000231000 e24KC2ThetaAboveLeaf0000231001 e24KC2ThetaAboveLeaf0000231002
+      e24KC2ThetaAboveLeaf0000231003
+
+theorem e24KC2ThetaAboveNode000023101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00002310)
+    e24KC2ThetaAboveLeaf0000231010 e24KC2ThetaAboveLeaf0000231011 e24KC2ThetaAboveLeaf0000231012
+      e24KC2ThetaAboveLeaf0000231013
+
+theorem e24KC2ThetaAboveNode000023102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00002310)
+    e24KC2ThetaAboveLeaf0000231020 e24KC2ThetaAboveLeaf0000231021 e24KC2ThetaAboveLeaf0000231022
+      e24KC2ThetaAboveLeaf0000231023
+
+theorem e24KC2ThetaAboveNode000023103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00002310)
+    e24KC2ThetaAboveLeaf0000231030 e24KC2ThetaAboveLeaf0000231031 e24KC2ThetaAboveLeaf0000231032
+      e24KC2ThetaAboveLeaf0000231033
+
+theorem e24KC2ThetaAboveNode000023110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00002311)
+    e24KC2ThetaAboveLeaf0000231100 e24KC2ThetaAboveLeaf0000231101 e24KC2ThetaAboveLeaf0000231102
+      e24KC2ThetaAboveLeaf0000231103
+
+theorem e24KC2ThetaAboveNode000023111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00002311)
+    e24KC2ThetaAboveLeaf0000231110 e24KC2ThetaAboveLeaf0000231111 e24KC2ThetaAboveLeaf0000231112
+      e24KC2ThetaAboveLeaf0000231113
+
+theorem e24KC2ThetaAboveNode000023112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00002311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00002311)
+    e24KC2ThetaAboveLeaf0000231120 e24KC2ThetaAboveLeaf0000231121 e24KC2ThetaAboveLeaf0000231122
+      e24KC2ThetaAboveLeaf0000231123
+
+theorem e24KC2ThetaAboveNode000023113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00002311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00002311)
+    e24KC2ThetaAboveLeaf0000231130 e24KC2ThetaAboveLeaf0000231131 e24KC2ThetaAboveLeaf0000231132
+      e24KC2ThetaAboveLeaf0000231133
+
+theorem e24KC2ThetaAboveNode000023120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00002312)
+    e24KC2ThetaAboveLeaf0000231200 e24KC2ThetaAboveLeaf0000231201 e24KC2ThetaAboveLeaf0000231202
+      e24KC2ThetaAboveLeaf0000231203
+
+theorem e24KC2ThetaAboveNode000023121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00002312)
+    e24KC2ThetaAboveLeaf0000231210 e24KC2ThetaAboveLeaf0000231211 e24KC2ThetaAboveLeaf0000231212
+      e24KC2ThetaAboveLeaf0000231213
+
+theorem e24KC2ThetaAboveNode000023130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00002313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00002313)
+    e24KC2ThetaAboveLeaf0000231300 e24KC2ThetaAboveLeaf0000231301 e24KC2ThetaAboveLeaf0000231302
+      e24KC2ThetaAboveLeaf0000231303
+
+theorem e24KC2ThetaAboveNode000023131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00002313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00002313)
+    e24KC2ThetaAboveLeaf0000231310 e24KC2ThetaAboveLeaf0000231311 e24KC2ThetaAboveLeaf0000231312
+      e24KC2ThetaAboveLeaf0000231313
+
+theorem e24KC2ThetaAboveNode000032000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003200)
+    e24KC2ThetaAboveLeaf0000320000 e24KC2ThetaAboveLeaf0000320001 e24KC2ThetaAboveLeaf0000320002
+      e24KC2ThetaAboveLeaf0000320003
+
+theorem e24KC2ThetaAboveNode000032001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003200)
+    e24KC2ThetaAboveLeaf0000320010 e24KC2ThetaAboveLeaf0000320011 e24KC2ThetaAboveLeaf0000320012
+      e24KC2ThetaAboveLeaf0000320013
+
+theorem e24KC2ThetaAboveNode000032002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00003200)
+    e24KC2ThetaAboveLeaf0000320020 e24KC2ThetaAboveLeaf0000320021 e24KC2ThetaAboveLeaf0000320022
+      e24KC2ThetaAboveLeaf0000320023
+
+theorem e24KC2ThetaAboveNode000032003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00003200)
+    e24KC2ThetaAboveLeaf0000320030 e24KC2ThetaAboveLeaf0000320031 e24KC2ThetaAboveLeaf0000320032
+      e24KC2ThetaAboveLeaf0000320033
+
+theorem e24KC2ThetaAboveNode000032010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003201)
+    e24KC2ThetaAboveLeaf0000320100 e24KC2ThetaAboveLeaf0000320101 e24KC2ThetaAboveLeaf0000320102
+      e24KC2ThetaAboveLeaf0000320103
+
+theorem e24KC2ThetaAboveNode000032011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003201)
+    e24KC2ThetaAboveLeaf0000320110 e24KC2ThetaAboveLeaf0000320111 e24KC2ThetaAboveLeaf0000320112
+      e24KC2ThetaAboveLeaf0000320113
+
+theorem e24KC2ThetaAboveNode000032012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00003201)
+    e24KC2ThetaAboveLeaf0000320120 e24KC2ThetaAboveLeaf0000320121 e24KC2ThetaAboveLeaf0000320122
+      e24KC2ThetaAboveLeaf0000320123
+
+theorem e24KC2ThetaAboveNode000032013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00003201)
+    e24KC2ThetaAboveLeaf0000320130 e24KC2ThetaAboveLeaf0000320131 e24KC2ThetaAboveLeaf0000320132
+      e24KC2ThetaAboveLeaf0000320133
+
+theorem e24KC2ThetaAboveNode000032020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003202)
+    e24KC2ThetaAboveLeaf0000320200 e24KC2ThetaAboveLeaf0000320201 e24KC2ThetaAboveLeaf0000320202
+      e24KC2ThetaAboveLeaf0000320203
+
+theorem e24KC2ThetaAboveNode000032021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003202)
+    e24KC2ThetaAboveLeaf0000320210 e24KC2ThetaAboveLeaf0000320211 e24KC2ThetaAboveLeaf0000320212
+      e24KC2ThetaAboveLeaf0000320213
+
+theorem e24KC2ThetaAboveNode000032030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003203)
+    e24KC2ThetaAboveLeaf0000320300 e24KC2ThetaAboveLeaf0000320301 e24KC2ThetaAboveLeaf0000320302
+      e24KC2ThetaAboveLeaf0000320303
+
+theorem e24KC2ThetaAboveNode000032031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003203)
+    e24KC2ThetaAboveLeaf0000320310 e24KC2ThetaAboveLeaf0000320311 e24KC2ThetaAboveLeaf0000320312
+      e24KC2ThetaAboveLeaf0000320313
+
+theorem e24KC2ThetaAboveNode000032100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003210)
+    e24KC2ThetaAboveLeaf0000321000 e24KC2ThetaAboveLeaf0000321001 e24KC2ThetaAboveLeaf0000321002
+      e24KC2ThetaAboveLeaf0000321003
+
+theorem e24KC2ThetaAboveNode000032101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003210)
+    e24KC2ThetaAboveLeaf0000321010 e24KC2ThetaAboveLeaf0000321011 e24KC2ThetaAboveLeaf0000321012
+      e24KC2ThetaAboveLeaf0000321013
+
+theorem e24KC2ThetaAboveNode000032102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00003210)
+    e24KC2ThetaAboveLeaf0000321020 e24KC2ThetaAboveLeaf0000321021 e24KC2ThetaAboveLeaf0000321022
+      e24KC2ThetaAboveLeaf0000321023
+
+theorem e24KC2ThetaAboveNode000032103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00003210)
+    e24KC2ThetaAboveLeaf0000321030 e24KC2ThetaAboveLeaf0000321031 e24KC2ThetaAboveLeaf0000321032
+      e24KC2ThetaAboveLeaf0000321033
+
+theorem e24KC2ThetaAboveNode000032110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003211)
+    e24KC2ThetaAboveLeaf0000321100 e24KC2ThetaAboveLeaf0000321101 e24KC2ThetaAboveLeaf0000321102
+      e24KC2ThetaAboveLeaf0000321103
+
+theorem e24KC2ThetaAboveNode000032111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003211)
+    e24KC2ThetaAboveLeaf0000321110 e24KC2ThetaAboveLeaf0000321111 e24KC2ThetaAboveLeaf0000321112
+      e24KC2ThetaAboveLeaf0000321113
+
+theorem e24KC2ThetaAboveNode000032112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00003211)
+    e24KC2ThetaAboveLeaf0000321120 e24KC2ThetaAboveLeaf0000321121 e24KC2ThetaAboveLeaf0000321122
+      e24KC2ThetaAboveLeaf0000321123
+
+theorem e24KC2ThetaAboveNode000032113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00003211)
+    e24KC2ThetaAboveLeaf0000321130 e24KC2ThetaAboveLeaf0000321131 e24KC2ThetaAboveLeaf0000321132
+      e24KC2ThetaAboveLeaf0000321133
+
+theorem e24KC2ThetaAboveNode000032120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003212)
+    e24KC2ThetaAboveLeaf0000321200 e24KC2ThetaAboveLeaf0000321201 e24KC2ThetaAboveLeaf0000321202
+      e24KC2ThetaAboveLeaf0000321203
+
+theorem e24KC2ThetaAboveNode000032121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003212)
+    e24KC2ThetaAboveLeaf0000321210 e24KC2ThetaAboveLeaf0000321211 e24KC2ThetaAboveLeaf0000321212
+      e24KC2ThetaAboveLeaf0000321213
+
+theorem e24KC2ThetaAboveNode000032130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003213)
+    e24KC2ThetaAboveLeaf0000321300 e24KC2ThetaAboveLeaf0000321301 e24KC2ThetaAboveLeaf0000321302
+      e24KC2ThetaAboveLeaf0000321303
+
+theorem e24KC2ThetaAboveNode000032131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003213)
+    e24KC2ThetaAboveLeaf0000321310 e24KC2ThetaAboveLeaf0000321311 e24KC2ThetaAboveLeaf0000321312
+      e24KC2ThetaAboveLeaf0000321313
+
+theorem e24KC2ThetaAboveNode000033000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003300)
+    e24KC2ThetaAboveLeaf0000330000 e24KC2ThetaAboveLeaf0000330001 e24KC2ThetaAboveLeaf0000330002
+      e24KC2ThetaAboveLeaf0000330003
+
+theorem e24KC2ThetaAboveNode000033001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003300)
+    e24KC2ThetaAboveLeaf0000330010 e24KC2ThetaAboveLeaf0000330011 e24KC2ThetaAboveLeaf0000330012
+      e24KC2ThetaAboveLeaf0000330013
+
+theorem e24KC2ThetaAboveNode000033002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00003300)
+    e24KC2ThetaAboveLeaf0000330020 e24KC2ThetaAboveLeaf0000330021 e24KC2ThetaAboveLeaf0000330022
+      e24KC2ThetaAboveLeaf0000330023
+
+theorem e24KC2ThetaAboveNode000033003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00003300)
+    e24KC2ThetaAboveLeaf0000330030 e24KC2ThetaAboveLeaf0000330031 e24KC2ThetaAboveLeaf0000330032
+      e24KC2ThetaAboveLeaf0000330033
+
+theorem e24KC2ThetaAboveNode000033010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003301)
+    e24KC2ThetaAboveLeaf0000330100 e24KC2ThetaAboveLeaf0000330101 e24KC2ThetaAboveLeaf0000330102
+      e24KC2ThetaAboveLeaf0000330103
+
+theorem e24KC2ThetaAboveNode000033011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003301)
+    e24KC2ThetaAboveLeaf0000330110 e24KC2ThetaAboveLeaf0000330111 e24KC2ThetaAboveLeaf0000330112
+      e24KC2ThetaAboveLeaf0000330113
+
+theorem e24KC2ThetaAboveNode000033012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00003301)
+    e24KC2ThetaAboveLeaf0000330120 e24KC2ThetaAboveLeaf0000330121 e24KC2ThetaAboveLeaf0000330122
+      e24KC2ThetaAboveLeaf0000330123
+
+theorem e24KC2ThetaAboveNode000033013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00003301)
+    e24KC2ThetaAboveLeaf0000330130 e24KC2ThetaAboveLeaf0000330131 e24KC2ThetaAboveLeaf0000330132
+      e24KC2ThetaAboveLeaf0000330133
+
+theorem e24KC2ThetaAboveNode000033020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003302)
+    e24KC2ThetaAboveLeaf0000330200 e24KC2ThetaAboveLeaf0000330201 e24KC2ThetaAboveLeaf0000330202
+      e24KC2ThetaAboveLeaf0000330203
+
+theorem e24KC2ThetaAboveNode000033021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003302)
+    e24KC2ThetaAboveLeaf0000330210 e24KC2ThetaAboveLeaf0000330211 e24KC2ThetaAboveLeaf0000330212
+      e24KC2ThetaAboveLeaf0000330213
+
+theorem e24KC2ThetaAboveNode000033030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003303)
+    e24KC2ThetaAboveLeaf0000330300 e24KC2ThetaAboveLeaf0000330301 e24KC2ThetaAboveLeaf0000330302
+      e24KC2ThetaAboveLeaf0000330303
+
+theorem e24KC2ThetaAboveNode000033031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003303)
+    e24KC2ThetaAboveLeaf0000330310 e24KC2ThetaAboveLeaf0000330311 e24KC2ThetaAboveLeaf0000330312
+      e24KC2ThetaAboveLeaf0000330313
+
+theorem e24KC2ThetaAboveNode000033100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003310)
+    e24KC2ThetaAboveLeaf0000331000 e24KC2ThetaAboveLeaf0000331001 e24KC2ThetaAboveLeaf0000331002
+      e24KC2ThetaAboveLeaf0000331003
+
+theorem e24KC2ThetaAboveNode000033101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003310)
+    e24KC2ThetaAboveLeaf0000331010 e24KC2ThetaAboveLeaf0000331011 e24KC2ThetaAboveLeaf0000331012
+      e24KC2ThetaAboveLeaf0000331013
+
+theorem e24KC2ThetaAboveNode000033102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00003310)
+    e24KC2ThetaAboveLeaf0000331020 e24KC2ThetaAboveLeaf0000331021 e24KC2ThetaAboveLeaf0000331022
+      e24KC2ThetaAboveLeaf0000331023
+
+theorem e24KC2ThetaAboveNode000033103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00003310)
+    e24KC2ThetaAboveLeaf0000331030 e24KC2ThetaAboveLeaf0000331031 e24KC2ThetaAboveLeaf0000331032
+      e24KC2ThetaAboveLeaf0000331033
+
+theorem e24KC2ThetaAboveNode000033110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003311)
+    e24KC2ThetaAboveLeaf0000331100 e24KC2ThetaAboveLeaf0000331101 e24KC2ThetaAboveLeaf0000331102
+      e24KC2ThetaAboveLeaf0000331103
+
+theorem e24KC2ThetaAboveNode000033111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003311)
+    e24KC2ThetaAboveLeaf0000331110 e24KC2ThetaAboveLeaf0000331111 e24KC2ThetaAboveLeaf0000331112
+      e24KC2ThetaAboveLeaf0000331113
+
+theorem e24KC2ThetaAboveNode000033112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00003311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00003311)
+    e24KC2ThetaAboveLeaf0000331120 e24KC2ThetaAboveLeaf0000331121 e24KC2ThetaAboveLeaf0000331122
+      e24KC2ThetaAboveLeaf0000331123
+
+theorem e24KC2ThetaAboveNode000033113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00003311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00003311)
+    e24KC2ThetaAboveLeaf0000331130 e24KC2ThetaAboveLeaf0000331131 e24KC2ThetaAboveLeaf0000331132
+      e24KC2ThetaAboveLeaf0000331133
+
+theorem e24KC2ThetaAboveNode000033120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003312)
+    e24KC2ThetaAboveLeaf0000331200 e24KC2ThetaAboveLeaf0000331201 e24KC2ThetaAboveLeaf0000331202
+      e24KC2ThetaAboveLeaf0000331203
+
+theorem e24KC2ThetaAboveNode000033121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003312)
+    e24KC2ThetaAboveLeaf0000331210 e24KC2ThetaAboveLeaf0000331211 e24KC2ThetaAboveLeaf0000331212
+      e24KC2ThetaAboveLeaf0000331213
+
+theorem e24KC2ThetaAboveNode000033130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00003313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00003313)
+    e24KC2ThetaAboveLeaf0000331300 e24KC2ThetaAboveLeaf0000331301 e24KC2ThetaAboveLeaf0000331302
+      e24KC2ThetaAboveLeaf0000331303
+
+theorem e24KC2ThetaAboveNode000033131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00003313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00003313)
+    e24KC2ThetaAboveLeaf0000331310 e24KC2ThetaAboveLeaf0000331311 e24KC2ThetaAboveLeaf0000331312
+      e24KC2ThetaAboveLeaf0000331313
+
+theorem e24KC2ThetaAboveNode000122000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012200)
+    e24KC2ThetaAboveLeaf0001220000 e24KC2ThetaAboveLeaf0001220001 e24KC2ThetaAboveLeaf0001220002
+      e24KC2ThetaAboveLeaf0001220003
+
+theorem e24KC2ThetaAboveNode000122001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012200)
+    e24KC2ThetaAboveLeaf0001220010 e24KC2ThetaAboveLeaf0001220011 e24KC2ThetaAboveLeaf0001220012
+      e24KC2ThetaAboveLeaf0001220013
+
+theorem e24KC2ThetaAboveNode000122002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00012200)
+    e24KC2ThetaAboveLeaf0001220020 e24KC2ThetaAboveLeaf0001220021 e24KC2ThetaAboveLeaf0001220022
+      e24KC2ThetaAboveLeaf0001220023
+
+theorem e24KC2ThetaAboveNode000122003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00012200)
+    e24KC2ThetaAboveLeaf0001220030 e24KC2ThetaAboveLeaf0001220031 e24KC2ThetaAboveLeaf0001220032
+      e24KC2ThetaAboveLeaf0001220033
+
+theorem e24KC2ThetaAboveNode000122010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012201)
+    e24KC2ThetaAboveLeaf0001220100 e24KC2ThetaAboveLeaf0001220101 e24KC2ThetaAboveLeaf0001220102
+      e24KC2ThetaAboveLeaf0001220103
+
+theorem e24KC2ThetaAboveNode000122011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012201)
+    e24KC2ThetaAboveLeaf0001220110 e24KC2ThetaAboveLeaf0001220111 e24KC2ThetaAboveLeaf0001220112
+      e24KC2ThetaAboveLeaf0001220113
+
+theorem e24KC2ThetaAboveNode000122012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00012201)
+    e24KC2ThetaAboveLeaf0001220120 e24KC2ThetaAboveLeaf0001220121 e24KC2ThetaAboveLeaf0001220122
+      e24KC2ThetaAboveLeaf0001220123
+
+theorem e24KC2ThetaAboveNode000122013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00012201)
+    e24KC2ThetaAboveLeaf0001220130 e24KC2ThetaAboveLeaf0001220131 e24KC2ThetaAboveLeaf0001220132
+      e24KC2ThetaAboveLeaf0001220133
+
+theorem e24KC2ThetaAboveNode000122020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012202)
+    e24KC2ThetaAboveLeaf0001220200 e24KC2ThetaAboveLeaf0001220201 e24KC2ThetaAboveLeaf0001220202
+      e24KC2ThetaAboveLeaf0001220203
+
+theorem e24KC2ThetaAboveNode000122021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012202)
+    e24KC2ThetaAboveLeaf0001220210 e24KC2ThetaAboveLeaf0001220211 e24KC2ThetaAboveLeaf0001220212
+      e24KC2ThetaAboveLeaf0001220213
+
+theorem e24KC2ThetaAboveNode000122030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012203)
+    e24KC2ThetaAboveLeaf0001220300 e24KC2ThetaAboveLeaf0001220301 e24KC2ThetaAboveLeaf0001220302
+      e24KC2ThetaAboveLeaf0001220303
+
+theorem e24KC2ThetaAboveNode000122031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012203)
+    e24KC2ThetaAboveLeaf0001220310 e24KC2ThetaAboveLeaf0001220311 e24KC2ThetaAboveLeaf0001220312
+      e24KC2ThetaAboveLeaf0001220313
+
+theorem e24KC2ThetaAboveNode000122100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012210)
+    e24KC2ThetaAboveLeaf0001221000 e24KC2ThetaAboveLeaf0001221001 e24KC2ThetaAboveLeaf0001221002
+      e24KC2ThetaAboveLeaf0001221003
+
+theorem e24KC2ThetaAboveNode000122101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012210)
+    e24KC2ThetaAboveLeaf0001221010 e24KC2ThetaAboveLeaf0001221011 e24KC2ThetaAboveLeaf0001221012
+      e24KC2ThetaAboveLeaf0001221013
+
+theorem e24KC2ThetaAboveNode000122102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00012210)
+    e24KC2ThetaAboveLeaf0001221020 e24KC2ThetaAboveLeaf0001221021 e24KC2ThetaAboveLeaf0001221022
+      e24KC2ThetaAboveLeaf0001221023
+
+theorem e24KC2ThetaAboveNode000122103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00012210)
+    e24KC2ThetaAboveLeaf0001221030 e24KC2ThetaAboveLeaf0001221031 e24KC2ThetaAboveLeaf0001221032
+      e24KC2ThetaAboveLeaf0001221033
+
+theorem e24KC2ThetaAboveNode000122110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012211)
+    e24KC2ThetaAboveLeaf0001221100 e24KC2ThetaAboveLeaf0001221101 e24KC2ThetaAboveLeaf0001221102
+      e24KC2ThetaAboveLeaf0001221103
+
+theorem e24KC2ThetaAboveNode000122111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012211)
+    e24KC2ThetaAboveLeaf0001221110 e24KC2ThetaAboveLeaf0001221111 e24KC2ThetaAboveLeaf0001221112
+      e24KC2ThetaAboveLeaf0001221113
+
+theorem e24KC2ThetaAboveNode000122112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00012211)
+    e24KC2ThetaAboveLeaf0001221120 e24KC2ThetaAboveLeaf0001221121 e24KC2ThetaAboveLeaf0001221122
+      e24KC2ThetaAboveLeaf0001221123
+
+theorem e24KC2ThetaAboveNode000122113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00012211)
+    e24KC2ThetaAboveLeaf0001221130 e24KC2ThetaAboveLeaf0001221131 e24KC2ThetaAboveLeaf0001221132
+      e24KC2ThetaAboveLeaf0001221133
+
+theorem e24KC2ThetaAboveNode000122120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012212)
+    e24KC2ThetaAboveLeaf0001221200 e24KC2ThetaAboveLeaf0001221201 e24KC2ThetaAboveLeaf0001221202
+      e24KC2ThetaAboveLeaf0001221203
+
+theorem e24KC2ThetaAboveNode000122121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012212)
+    e24KC2ThetaAboveLeaf0001221210 e24KC2ThetaAboveLeaf0001221211 e24KC2ThetaAboveLeaf0001221212
+      e24KC2ThetaAboveLeaf0001221213
+
+theorem e24KC2ThetaAboveNode000122130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012213)
+    e24KC2ThetaAboveLeaf0001221300 e24KC2ThetaAboveLeaf0001221301 e24KC2ThetaAboveLeaf0001221302
+      e24KC2ThetaAboveLeaf0001221303
+
+theorem e24KC2ThetaAboveNode000122131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012213)
+    e24KC2ThetaAboveLeaf0001221310 e24KC2ThetaAboveLeaf0001221311 e24KC2ThetaAboveLeaf0001221312
+      e24KC2ThetaAboveLeaf0001221313
+
+theorem e24KC2ThetaAboveNode000123000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012300)
+    e24KC2ThetaAboveLeaf0001230000 e24KC2ThetaAboveLeaf0001230001 e24KC2ThetaAboveLeaf0001230002
+      e24KC2ThetaAboveLeaf0001230003
+
+theorem e24KC2ThetaAboveNode000123001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012300)
+    e24KC2ThetaAboveLeaf0001230010 e24KC2ThetaAboveLeaf0001230011 e24KC2ThetaAboveLeaf0001230012
+      e24KC2ThetaAboveLeaf0001230013
+
+theorem e24KC2ThetaAboveNode000123002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00012300)
+    e24KC2ThetaAboveLeaf0001230020 e24KC2ThetaAboveLeaf0001230021 e24KC2ThetaAboveLeaf0001230022
+      e24KC2ThetaAboveLeaf0001230023
+
+theorem e24KC2ThetaAboveNode000123003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00012300)
+    e24KC2ThetaAboveLeaf0001230030 e24KC2ThetaAboveLeaf0001230031 e24KC2ThetaAboveLeaf0001230032
+      e24KC2ThetaAboveLeaf0001230033
+
+theorem e24KC2ThetaAboveNode000123010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012301)
+    e24KC2ThetaAboveLeaf0001230100 e24KC2ThetaAboveLeaf0001230101 e24KC2ThetaAboveLeaf0001230102
+      e24KC2ThetaAboveLeaf0001230103
+
+theorem e24KC2ThetaAboveNode000123011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012301)
+    e24KC2ThetaAboveLeaf0001230110 e24KC2ThetaAboveLeaf0001230111 e24KC2ThetaAboveLeaf0001230112
+      e24KC2ThetaAboveLeaf0001230113
+
+theorem e24KC2ThetaAboveNode000123012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00012301)
+    e24KC2ThetaAboveLeaf0001230120 e24KC2ThetaAboveLeaf0001230121 e24KC2ThetaAboveLeaf0001230122
+      e24KC2ThetaAboveLeaf0001230123
+
+theorem e24KC2ThetaAboveNode000123013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00012301)
+    e24KC2ThetaAboveLeaf0001230130 e24KC2ThetaAboveLeaf0001230131 e24KC2ThetaAboveLeaf0001230132
+      e24KC2ThetaAboveLeaf0001230133
+
+theorem e24KC2ThetaAboveNode000123020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012302)
+    e24KC2ThetaAboveLeaf0001230200 e24KC2ThetaAboveLeaf0001230201 e24KC2ThetaAboveLeaf0001230202
+      e24KC2ThetaAboveLeaf0001230203
+
+theorem e24KC2ThetaAboveNode000123021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012302)
+    e24KC2ThetaAboveLeaf0001230210 e24KC2ThetaAboveLeaf0001230211 e24KC2ThetaAboveLeaf0001230212
+      e24KC2ThetaAboveLeaf0001230213
+
+theorem e24KC2ThetaAboveNode000123030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012303)
+    e24KC2ThetaAboveLeaf0001230300 e24KC2ThetaAboveLeaf0001230301 e24KC2ThetaAboveLeaf0001230302
+      e24KC2ThetaAboveLeaf0001230303
+
+theorem e24KC2ThetaAboveNode000123031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012303)
+    e24KC2ThetaAboveLeaf0001230310 e24KC2ThetaAboveLeaf0001230311 e24KC2ThetaAboveLeaf0001230312
+      e24KC2ThetaAboveLeaf0001230313
+
+theorem e24KC2ThetaAboveNode000123100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012310)
+    e24KC2ThetaAboveLeaf0001231000 e24KC2ThetaAboveLeaf0001231001 e24KC2ThetaAboveLeaf0001231002
+      e24KC2ThetaAboveLeaf0001231003
+
+theorem e24KC2ThetaAboveNode000123101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012310)
+    e24KC2ThetaAboveLeaf0001231010 e24KC2ThetaAboveLeaf0001231011 e24KC2ThetaAboveLeaf0001231012
+      e24KC2ThetaAboveLeaf0001231013
+
+theorem e24KC2ThetaAboveNode000123102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00012310)
+    e24KC2ThetaAboveLeaf0001231020 e24KC2ThetaAboveLeaf0001231021 e24KC2ThetaAboveLeaf0001231022
+      e24KC2ThetaAboveLeaf0001231023
+
+theorem e24KC2ThetaAboveNode000123103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00012310)
+    e24KC2ThetaAboveLeaf0001231030 e24KC2ThetaAboveLeaf0001231031 e24KC2ThetaAboveLeaf0001231032
+      e24KC2ThetaAboveLeaf0001231033
+
+theorem e24KC2ThetaAboveNode000123110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012311)
+    e24KC2ThetaAboveLeaf0001231100 e24KC2ThetaAboveLeaf0001231101 e24KC2ThetaAboveLeaf0001231102
+      e24KC2ThetaAboveLeaf0001231103
+
+theorem e24KC2ThetaAboveNode000123111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012311)
+    e24KC2ThetaAboveLeaf0001231110 e24KC2ThetaAboveLeaf0001231111 e24KC2ThetaAboveLeaf0001231112
+      e24KC2ThetaAboveLeaf0001231113
+
+theorem e24KC2ThetaAboveNode000123112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00012311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00012311)
+    e24KC2ThetaAboveLeaf0001231120 e24KC2ThetaAboveLeaf0001231121 e24KC2ThetaAboveLeaf0001231122
+      e24KC2ThetaAboveLeaf0001231123
+
+theorem e24KC2ThetaAboveNode000123113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00012311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00012311)
+    e24KC2ThetaAboveLeaf0001231130 e24KC2ThetaAboveLeaf0001231131 e24KC2ThetaAboveLeaf0001231132
+      e24KC2ThetaAboveLeaf0001231133
+
+theorem e24KC2ThetaAboveNode000123120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012312)
+    e24KC2ThetaAboveLeaf0001231200 e24KC2ThetaAboveLeaf0001231201 e24KC2ThetaAboveLeaf0001231202
+      e24KC2ThetaAboveLeaf0001231203
+
+theorem e24KC2ThetaAboveNode000123121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012312)
+    e24KC2ThetaAboveLeaf0001231210 e24KC2ThetaAboveLeaf0001231211 e24KC2ThetaAboveLeaf0001231212
+      e24KC2ThetaAboveLeaf0001231213
+
+theorem e24KC2ThetaAboveNode000123130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00012313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00012313)
+    e24KC2ThetaAboveLeaf0001231300 e24KC2ThetaAboveLeaf0001231301 e24KC2ThetaAboveLeaf0001231302
+      e24KC2ThetaAboveLeaf0001231303
+
+theorem e24KC2ThetaAboveNode000123131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00012313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00012313)
+    e24KC2ThetaAboveLeaf0001231310 e24KC2ThetaAboveLeaf0001231311 e24KC2ThetaAboveLeaf0001231312
+      e24KC2ThetaAboveLeaf0001231313
+
+theorem e24KC2ThetaAboveNode000132000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013200)
+    e24KC2ThetaAboveLeaf0001320000 e24KC2ThetaAboveLeaf0001320001 e24KC2ThetaAboveLeaf0001320002
+      e24KC2ThetaAboveLeaf0001320003
+
+theorem e24KC2ThetaAboveNode000132001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013200)
+    e24KC2ThetaAboveLeaf0001320010 e24KC2ThetaAboveLeaf0001320011 e24KC2ThetaAboveLeaf0001320012
+      e24KC2ThetaAboveLeaf0001320013
+
+theorem e24KC2ThetaAboveNode000132002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00013200)
+    e24KC2ThetaAboveLeaf0001320020 e24KC2ThetaAboveLeaf0001320021 e24KC2ThetaAboveLeaf0001320022
+      e24KC2ThetaAboveLeaf0001320023
+
+theorem e24KC2ThetaAboveNode000132003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00013200)
+    e24KC2ThetaAboveLeaf0001320030 e24KC2ThetaAboveLeaf0001320031 e24KC2ThetaAboveLeaf0001320032
+      e24KC2ThetaAboveLeaf0001320033
+
+theorem e24KC2ThetaAboveNode000132010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013201)
+    e24KC2ThetaAboveLeaf0001320100 e24KC2ThetaAboveLeaf0001320101 e24KC2ThetaAboveLeaf0001320102
+      e24KC2ThetaAboveLeaf0001320103
+
+theorem e24KC2ThetaAboveNode000132011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013201)
+    e24KC2ThetaAboveLeaf0001320110 e24KC2ThetaAboveLeaf0001320111 e24KC2ThetaAboveLeaf0001320112
+      e24KC2ThetaAboveLeaf0001320113
+
+theorem e24KC2ThetaAboveNode000132012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00013201)
+    e24KC2ThetaAboveLeaf0001320120 e24KC2ThetaAboveLeaf0001320121 e24KC2ThetaAboveLeaf0001320122
+      e24KC2ThetaAboveLeaf0001320123
+
+theorem e24KC2ThetaAboveNode000132013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00013201)
+    e24KC2ThetaAboveLeaf0001320130 e24KC2ThetaAboveLeaf0001320131 e24KC2ThetaAboveLeaf0001320132
+      e24KC2ThetaAboveLeaf0001320133
+
+theorem e24KC2ThetaAboveNode000132020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013202)
+    e24KC2ThetaAboveLeaf0001320200 e24KC2ThetaAboveLeaf0001320201 e24KC2ThetaAboveLeaf0001320202
+      e24KC2ThetaAboveLeaf0001320203
+
+theorem e24KC2ThetaAboveNode000132021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013202)
+    e24KC2ThetaAboveLeaf0001320210 e24KC2ThetaAboveLeaf0001320211 e24KC2ThetaAboveLeaf0001320212
+      e24KC2ThetaAboveLeaf0001320213
+
+theorem e24KC2ThetaAboveNode000132030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013203)
+    e24KC2ThetaAboveLeaf0001320300 e24KC2ThetaAboveLeaf0001320301 e24KC2ThetaAboveLeaf0001320302
+      e24KC2ThetaAboveLeaf0001320303
+
+theorem e24KC2ThetaAboveNode000132031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013203)
+    e24KC2ThetaAboveLeaf0001320310 e24KC2ThetaAboveLeaf0001320311 e24KC2ThetaAboveLeaf0001320312
+      e24KC2ThetaAboveLeaf0001320313
+
+theorem e24KC2ThetaAboveNode000132100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013210)
+    e24KC2ThetaAboveLeaf0001321000 e24KC2ThetaAboveLeaf0001321001 e24KC2ThetaAboveLeaf0001321002
+      e24KC2ThetaAboveLeaf0001321003
+
+theorem e24KC2ThetaAboveNode000132101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013210)
+    e24KC2ThetaAboveLeaf0001321010 e24KC2ThetaAboveLeaf0001321011 e24KC2ThetaAboveLeaf0001321012
+      e24KC2ThetaAboveLeaf0001321013
+
+theorem e24KC2ThetaAboveNode000132102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00013210)
+    e24KC2ThetaAboveLeaf0001321020 e24KC2ThetaAboveLeaf0001321021 e24KC2ThetaAboveLeaf0001321022
+      e24KC2ThetaAboveLeaf0001321023
+
+theorem e24KC2ThetaAboveNode000132103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00013210)
+    e24KC2ThetaAboveLeaf0001321030 e24KC2ThetaAboveLeaf0001321031 e24KC2ThetaAboveLeaf0001321032
+      e24KC2ThetaAboveLeaf0001321033
+
+theorem e24KC2ThetaAboveNode000132110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013211)
+    e24KC2ThetaAboveLeaf0001321100 e24KC2ThetaAboveLeaf0001321101 e24KC2ThetaAboveLeaf0001321102
+      e24KC2ThetaAboveLeaf0001321103
+
+theorem e24KC2ThetaAboveNode000132111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013211)
+    e24KC2ThetaAboveLeaf0001321110 e24KC2ThetaAboveLeaf0001321111 e24KC2ThetaAboveLeaf0001321112
+      e24KC2ThetaAboveLeaf0001321113
+
+theorem e24KC2ThetaAboveNode000132112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00013211)
+    e24KC2ThetaAboveLeaf0001321120 e24KC2ThetaAboveLeaf0001321121 e24KC2ThetaAboveLeaf0001321122
+      e24KC2ThetaAboveLeaf0001321123
+
+theorem e24KC2ThetaAboveNode000132113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00013211)
+    e24KC2ThetaAboveLeaf0001321130 e24KC2ThetaAboveLeaf0001321131 e24KC2ThetaAboveLeaf0001321132
+      e24KC2ThetaAboveLeaf0001321133
+
+theorem e24KC2ThetaAboveNode000132120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013212)
+    e24KC2ThetaAboveLeaf0001321200 e24KC2ThetaAboveLeaf0001321201 e24KC2ThetaAboveLeaf0001321202
+      e24KC2ThetaAboveLeaf0001321203
+
+theorem e24KC2ThetaAboveNode000132121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013212)
+    e24KC2ThetaAboveLeaf0001321210 e24KC2ThetaAboveLeaf0001321211 e24KC2ThetaAboveLeaf0001321212
+      e24KC2ThetaAboveLeaf0001321213
+
+theorem e24KC2ThetaAboveNode000132130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013213)
+    e24KC2ThetaAboveLeaf0001321300 e24KC2ThetaAboveLeaf0001321301 e24KC2ThetaAboveLeaf0001321302
+      e24KC2ThetaAboveLeaf0001321303
+
+theorem e24KC2ThetaAboveNode000132131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013213)
+    e24KC2ThetaAboveLeaf0001321310 e24KC2ThetaAboveLeaf0001321311 e24KC2ThetaAboveLeaf0001321312
+      e24KC2ThetaAboveLeaf0001321313
+
+theorem e24KC2ThetaAboveNode000133000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013300)
+    e24KC2ThetaAboveLeaf0001330000 e24KC2ThetaAboveLeaf0001330001 e24KC2ThetaAboveLeaf0001330002
+      e24KC2ThetaAboveLeaf0001330003
+
+theorem e24KC2ThetaAboveNode000133001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013300)
+    e24KC2ThetaAboveLeaf0001330010 e24KC2ThetaAboveLeaf0001330011 e24KC2ThetaAboveLeaf0001330012
+      e24KC2ThetaAboveLeaf0001330013
+
+theorem e24KC2ThetaAboveNode000133002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00013300)
+    e24KC2ThetaAboveLeaf0001330020 e24KC2ThetaAboveLeaf0001330021 e24KC2ThetaAboveLeaf0001330022
+      e24KC2ThetaAboveLeaf0001330023
+
+theorem e24KC2ThetaAboveNode000133003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00013300)
+    e24KC2ThetaAboveLeaf0001330030 e24KC2ThetaAboveLeaf0001330031 e24KC2ThetaAboveLeaf0001330032
+      e24KC2ThetaAboveLeaf0001330033
+
+theorem e24KC2ThetaAboveNode000133010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013301)
+    e24KC2ThetaAboveLeaf0001330100 e24KC2ThetaAboveLeaf0001330101 e24KC2ThetaAboveLeaf0001330102
+      e24KC2ThetaAboveLeaf0001330103
+
+theorem e24KC2ThetaAboveNode000133011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013301)
+    e24KC2ThetaAboveLeaf0001330110 e24KC2ThetaAboveLeaf0001330111 e24KC2ThetaAboveLeaf0001330112
+      e24KC2ThetaAboveLeaf0001330113
+
+theorem e24KC2ThetaAboveNode000133012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00013301)
+    e24KC2ThetaAboveLeaf0001330120 e24KC2ThetaAboveLeaf0001330121 e24KC2ThetaAboveLeaf0001330122
+      e24KC2ThetaAboveLeaf0001330123
+
+theorem e24KC2ThetaAboveNode000133013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00013301)
+    e24KC2ThetaAboveLeaf0001330130 e24KC2ThetaAboveLeaf0001330131 e24KC2ThetaAboveLeaf0001330132
+      e24KC2ThetaAboveLeaf0001330133
+
+theorem e24KC2ThetaAboveNode000133020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013302)
+    e24KC2ThetaAboveLeaf0001330200 e24KC2ThetaAboveLeaf0001330201 e24KC2ThetaAboveLeaf0001330202
+      e24KC2ThetaAboveLeaf0001330203
+
+theorem e24KC2ThetaAboveNode000133021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013302)
+    e24KC2ThetaAboveLeaf0001330210 e24KC2ThetaAboveLeaf0001330211 e24KC2ThetaAboveLeaf0001330212
+      e24KC2ThetaAboveLeaf0001330213
+
+theorem e24KC2ThetaAboveNode000133030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013303)
+    e24KC2ThetaAboveLeaf0001330300 e24KC2ThetaAboveLeaf0001330301 e24KC2ThetaAboveLeaf0001330302
+      e24KC2ThetaAboveLeaf0001330303
+
+theorem e24KC2ThetaAboveNode000133031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013303)
+    e24KC2ThetaAboveLeaf0001330310 e24KC2ThetaAboveLeaf0001330311 e24KC2ThetaAboveLeaf0001330312
+      e24KC2ThetaAboveLeaf0001330313
+
+theorem e24KC2ThetaAboveNode000133100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013310)
+    e24KC2ThetaAboveLeaf0001331000 e24KC2ThetaAboveLeaf0001331001 e24KC2ThetaAboveLeaf0001331002
+      e24KC2ThetaAboveLeaf0001331003
+
+theorem e24KC2ThetaAboveNode000133101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013310)
+    e24KC2ThetaAboveLeaf0001331010 e24KC2ThetaAboveLeaf0001331011 e24KC2ThetaAboveLeaf0001331012
+      e24KC2ThetaAboveLeaf0001331013
+
+theorem e24KC2ThetaAboveNode000133102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00013310)
+    e24KC2ThetaAboveLeaf0001331020 e24KC2ThetaAboveLeaf0001331021 e24KC2ThetaAboveLeaf0001331022
+      e24KC2ThetaAboveLeaf0001331023
+
+theorem e24KC2ThetaAboveNode000133103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00013310)
+    e24KC2ThetaAboveLeaf0001331030 e24KC2ThetaAboveLeaf0001331031 e24KC2ThetaAboveLeaf0001331032
+      e24KC2ThetaAboveLeaf0001331033
+
+theorem e24KC2ThetaAboveNode000133110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013311)
+    e24KC2ThetaAboveLeaf0001331100 e24KC2ThetaAboveLeaf0001331101 e24KC2ThetaAboveLeaf0001331102
+      e24KC2ThetaAboveLeaf0001331103
+
+theorem e24KC2ThetaAboveNode000133111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013311)
+    e24KC2ThetaAboveLeaf0001331110 e24KC2ThetaAboveLeaf0001331111 e24KC2ThetaAboveLeaf0001331112
+      e24KC2ThetaAboveLeaf0001331113
+
+theorem e24KC2ThetaAboveNode000133112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00013311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00013311)
+    e24KC2ThetaAboveLeaf0001331120 e24KC2ThetaAboveLeaf0001331121 e24KC2ThetaAboveLeaf0001331122
+      e24KC2ThetaAboveLeaf0001331123
+
+theorem e24KC2ThetaAboveNode000133113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00013311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00013311)
+    e24KC2ThetaAboveLeaf0001331130 e24KC2ThetaAboveLeaf0001331131 e24KC2ThetaAboveLeaf0001331132
+      e24KC2ThetaAboveLeaf0001331133
+
+theorem e24KC2ThetaAboveNode000133120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013312)
+    e24KC2ThetaAboveLeaf0001331200 e24KC2ThetaAboveLeaf0001331201 e24KC2ThetaAboveLeaf0001331202
+      e24KC2ThetaAboveLeaf0001331203
+
+theorem e24KC2ThetaAboveNode000133121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013312)
+    e24KC2ThetaAboveLeaf0001331210 e24KC2ThetaAboveLeaf0001331211 e24KC2ThetaAboveLeaf0001331212
+      e24KC2ThetaAboveLeaf0001331213
+
+theorem e24KC2ThetaAboveNode000133130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00013313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00013313)
+    e24KC2ThetaAboveLeaf0001331300 e24KC2ThetaAboveLeaf0001331301 e24KC2ThetaAboveLeaf0001331302
+      e24KC2ThetaAboveLeaf0001331303
+
+theorem e24KC2ThetaAboveNode000133131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00013313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00013313)
+    e24KC2ThetaAboveLeaf0001331310 e24KC2ThetaAboveLeaf0001331311 e24KC2ThetaAboveLeaf0001331312
+      e24KC2ThetaAboveLeaf0001331313
+
+theorem e24KC2ThetaAboveNode001022000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102200)
+    e24KC2ThetaAboveLeaf0010220000 e24KC2ThetaAboveLeaf0010220001 e24KC2ThetaAboveLeaf0010220002
+      e24KC2ThetaAboveLeaf0010220003
+
+theorem e24KC2ThetaAboveNode001022001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102200)
+    e24KC2ThetaAboveLeaf0010220010 e24KC2ThetaAboveLeaf0010220011 e24KC2ThetaAboveLeaf0010220012
+      e24KC2ThetaAboveLeaf0010220013
+
+theorem e24KC2ThetaAboveNode001022002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00102200)
+    e24KC2ThetaAboveLeaf0010220020 e24KC2ThetaAboveLeaf0010220021 e24KC2ThetaAboveLeaf0010220022
+      e24KC2ThetaAboveLeaf0010220023
+
+theorem e24KC2ThetaAboveNode001022003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00102200)
+    e24KC2ThetaAboveLeaf0010220030 e24KC2ThetaAboveLeaf0010220031 e24KC2ThetaAboveLeaf0010220032
+      e24KC2ThetaAboveLeaf0010220033
+
+theorem e24KC2ThetaAboveNode001022010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102201)
+    e24KC2ThetaAboveLeaf0010220100 e24KC2ThetaAboveLeaf0010220101 e24KC2ThetaAboveLeaf0010220102
+      e24KC2ThetaAboveLeaf0010220103
+
+theorem e24KC2ThetaAboveNode001022011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102201)
+    e24KC2ThetaAboveLeaf0010220110 e24KC2ThetaAboveLeaf0010220111 e24KC2ThetaAboveLeaf0010220112
+      e24KC2ThetaAboveLeaf0010220113
+
+theorem e24KC2ThetaAboveNode001022012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00102201)
+    e24KC2ThetaAboveLeaf0010220120 e24KC2ThetaAboveLeaf0010220121 e24KC2ThetaAboveLeaf0010220122
+      e24KC2ThetaAboveLeaf0010220123
+
+theorem e24KC2ThetaAboveNode001022013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00102201)
+    e24KC2ThetaAboveLeaf0010220130 e24KC2ThetaAboveLeaf0010220131 e24KC2ThetaAboveLeaf0010220132
+      e24KC2ThetaAboveLeaf0010220133
+
+theorem e24KC2ThetaAboveNode001022020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102202)
+    e24KC2ThetaAboveLeaf0010220200 e24KC2ThetaAboveLeaf0010220201 e24KC2ThetaAboveLeaf0010220202
+      e24KC2ThetaAboveLeaf0010220203
+
+theorem e24KC2ThetaAboveNode001022021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102202)
+    e24KC2ThetaAboveLeaf0010220210 e24KC2ThetaAboveLeaf0010220211 e24KC2ThetaAboveLeaf0010220212
+      e24KC2ThetaAboveLeaf0010220213
+
+theorem e24KC2ThetaAboveNode001022030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102203)
+    e24KC2ThetaAboveLeaf0010220300 e24KC2ThetaAboveLeaf0010220301 e24KC2ThetaAboveLeaf0010220302
+      e24KC2ThetaAboveLeaf0010220303
+
+theorem e24KC2ThetaAboveNode001022031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102203)
+    e24KC2ThetaAboveLeaf0010220310 e24KC2ThetaAboveLeaf0010220311 e24KC2ThetaAboveLeaf0010220312
+      e24KC2ThetaAboveLeaf0010220313
+
+theorem e24KC2ThetaAboveNode001022100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102210)
+    e24KC2ThetaAboveLeaf0010221000 e24KC2ThetaAboveLeaf0010221001 e24KC2ThetaAboveLeaf0010221002
+      e24KC2ThetaAboveLeaf0010221003
+
+theorem e24KC2ThetaAboveNode001022101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102210)
+    e24KC2ThetaAboveLeaf0010221010 e24KC2ThetaAboveLeaf0010221011 e24KC2ThetaAboveLeaf0010221012
+      e24KC2ThetaAboveLeaf0010221013
+
+theorem e24KC2ThetaAboveNode001022102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00102210)
+    e24KC2ThetaAboveLeaf0010221020 e24KC2ThetaAboveLeaf0010221021 e24KC2ThetaAboveLeaf0010221022
+      e24KC2ThetaAboveLeaf0010221023
+
+theorem e24KC2ThetaAboveNode001022103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00102210)
+    e24KC2ThetaAboveLeaf0010221030 e24KC2ThetaAboveLeaf0010221031 e24KC2ThetaAboveLeaf0010221032
+      e24KC2ThetaAboveLeaf0010221033
+
+theorem e24KC2ThetaAboveNode001022110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102211)
+    e24KC2ThetaAboveLeaf0010221100 e24KC2ThetaAboveLeaf0010221101 e24KC2ThetaAboveLeaf0010221102
+      e24KC2ThetaAboveLeaf0010221103
+
+theorem e24KC2ThetaAboveNode001022111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102211)
+    e24KC2ThetaAboveLeaf0010221110 e24KC2ThetaAboveLeaf0010221111 e24KC2ThetaAboveLeaf0010221112
+      e24KC2ThetaAboveLeaf0010221113
+
+theorem e24KC2ThetaAboveNode001022112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00102211)
+    e24KC2ThetaAboveLeaf0010221120 e24KC2ThetaAboveLeaf0010221121 e24KC2ThetaAboveLeaf0010221122
+      e24KC2ThetaAboveLeaf0010221123
+
+theorem e24KC2ThetaAboveNode001022113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00102211)
+    e24KC2ThetaAboveLeaf0010221130 e24KC2ThetaAboveLeaf0010221131 e24KC2ThetaAboveLeaf0010221132
+      e24KC2ThetaAboveLeaf0010221133
+
+theorem e24KC2ThetaAboveNode001022120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102212)
+    e24KC2ThetaAboveLeaf0010221200 e24KC2ThetaAboveLeaf0010221201 e24KC2ThetaAboveLeaf0010221202
+      e24KC2ThetaAboveLeaf0010221203
+
+theorem e24KC2ThetaAboveNode001022121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102212)
+    e24KC2ThetaAboveLeaf0010221210 e24KC2ThetaAboveLeaf0010221211 e24KC2ThetaAboveLeaf0010221212
+      e24KC2ThetaAboveLeaf0010221213
+
+theorem e24KC2ThetaAboveNode001022130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102213)
+    e24KC2ThetaAboveLeaf0010221300 e24KC2ThetaAboveLeaf0010221301 e24KC2ThetaAboveLeaf0010221302
+      e24KC2ThetaAboveLeaf0010221303
+
+theorem e24KC2ThetaAboveNode001022131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102213)
+    e24KC2ThetaAboveLeaf0010221310 e24KC2ThetaAboveLeaf0010221311 e24KC2ThetaAboveLeaf0010221312
+      e24KC2ThetaAboveLeaf0010221313
+
+theorem e24KC2ThetaAboveNode001023000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102300)
+    e24KC2ThetaAboveLeaf0010230000 e24KC2ThetaAboveLeaf0010230001 e24KC2ThetaAboveLeaf0010230002
+      e24KC2ThetaAboveLeaf0010230003
+
+theorem e24KC2ThetaAboveNode001023001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102300)
+    e24KC2ThetaAboveLeaf0010230010 e24KC2ThetaAboveLeaf0010230011 e24KC2ThetaAboveLeaf0010230012
+      e24KC2ThetaAboveLeaf0010230013
+
+theorem e24KC2ThetaAboveNode001023002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00102300)
+    e24KC2ThetaAboveLeaf0010230020 e24KC2ThetaAboveLeaf0010230021 e24KC2ThetaAboveLeaf0010230022
+      e24KC2ThetaAboveLeaf0010230023
+
+theorem e24KC2ThetaAboveNode001023003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00102300)
+    e24KC2ThetaAboveLeaf0010230030 e24KC2ThetaAboveLeaf0010230031 e24KC2ThetaAboveLeaf0010230032
+      e24KC2ThetaAboveLeaf0010230033
+
+theorem e24KC2ThetaAboveNode001023010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102301)
+    e24KC2ThetaAboveLeaf0010230100 e24KC2ThetaAboveLeaf0010230101 e24KC2ThetaAboveLeaf0010230102
+      e24KC2ThetaAboveLeaf0010230103
+
+theorem e24KC2ThetaAboveNode001023011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102301)
+    e24KC2ThetaAboveLeaf0010230110 e24KC2ThetaAboveLeaf0010230111 e24KC2ThetaAboveLeaf0010230112
+      e24KC2ThetaAboveLeaf0010230113
+
+theorem e24KC2ThetaAboveNode001023012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00102301)
+    e24KC2ThetaAboveLeaf0010230120 e24KC2ThetaAboveLeaf0010230121 e24KC2ThetaAboveLeaf0010230122
+      e24KC2ThetaAboveLeaf0010230123
+
+theorem e24KC2ThetaAboveNode001023013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00102301)
+    e24KC2ThetaAboveLeaf0010230130 e24KC2ThetaAboveLeaf0010230131 e24KC2ThetaAboveLeaf0010230132
+      e24KC2ThetaAboveLeaf0010230133
+
+theorem e24KC2ThetaAboveNode001023020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102302)
+    e24KC2ThetaAboveLeaf0010230200 e24KC2ThetaAboveLeaf0010230201 e24KC2ThetaAboveLeaf0010230202
+      e24KC2ThetaAboveLeaf0010230203
+
+theorem e24KC2ThetaAboveNode001023021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102302)
+    e24KC2ThetaAboveLeaf0010230210 e24KC2ThetaAboveLeaf0010230211 e24KC2ThetaAboveLeaf0010230212
+      e24KC2ThetaAboveLeaf0010230213
+
+theorem e24KC2ThetaAboveNode001023030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102303)
+    e24KC2ThetaAboveLeaf0010230300 e24KC2ThetaAboveLeaf0010230301 e24KC2ThetaAboveLeaf0010230302
+      e24KC2ThetaAboveLeaf0010230303
+
+theorem e24KC2ThetaAboveNode001023031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102303)
+    e24KC2ThetaAboveLeaf0010230310 e24KC2ThetaAboveLeaf0010230311 e24KC2ThetaAboveLeaf0010230312
+      e24KC2ThetaAboveLeaf0010230313
+
+theorem e24KC2ThetaAboveNode001023100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102310)
+    e24KC2ThetaAboveLeaf0010231000 e24KC2ThetaAboveLeaf0010231001 e24KC2ThetaAboveLeaf0010231002
+      e24KC2ThetaAboveLeaf0010231003
+
+theorem e24KC2ThetaAboveNode001023101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102310)
+    e24KC2ThetaAboveLeaf0010231010 e24KC2ThetaAboveLeaf0010231011 e24KC2ThetaAboveLeaf0010231012
+      e24KC2ThetaAboveLeaf0010231013
+
+theorem e24KC2ThetaAboveNode001023102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00102310)
+    e24KC2ThetaAboveLeaf0010231020 e24KC2ThetaAboveLeaf0010231021 e24KC2ThetaAboveLeaf0010231022
+      e24KC2ThetaAboveLeaf0010231023
+
+theorem e24KC2ThetaAboveNode001023103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00102310)
+    e24KC2ThetaAboveLeaf0010231030 e24KC2ThetaAboveLeaf0010231031 e24KC2ThetaAboveLeaf0010231032
+      e24KC2ThetaAboveLeaf0010231033
+
+theorem e24KC2ThetaAboveNode001023110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102311)
+    e24KC2ThetaAboveLeaf0010231100 e24KC2ThetaAboveLeaf0010231101 e24KC2ThetaAboveLeaf0010231102
+      e24KC2ThetaAboveLeaf0010231103
+
+theorem e24KC2ThetaAboveNode001023111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102311)
+    e24KC2ThetaAboveLeaf0010231110 e24KC2ThetaAboveLeaf0010231111 e24KC2ThetaAboveLeaf0010231112
+      e24KC2ThetaAboveLeaf0010231113
+
+theorem e24KC2ThetaAboveNode001023112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00102311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00102311)
+    e24KC2ThetaAboveLeaf0010231120 e24KC2ThetaAboveLeaf0010231121 e24KC2ThetaAboveLeaf0010231122
+      e24KC2ThetaAboveLeaf0010231123
+
+theorem e24KC2ThetaAboveNode001023113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00102311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00102311)
+    e24KC2ThetaAboveLeaf0010231130 e24KC2ThetaAboveLeaf0010231131 e24KC2ThetaAboveLeaf0010231132
+      e24KC2ThetaAboveLeaf0010231133
+
+theorem e24KC2ThetaAboveNode001023120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102312)
+    e24KC2ThetaAboveLeaf0010231200 e24KC2ThetaAboveLeaf0010231201 e24KC2ThetaAboveLeaf0010231202
+      e24KC2ThetaAboveLeaf0010231203
+
+theorem e24KC2ThetaAboveNode001023121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102312)
+    e24KC2ThetaAboveLeaf0010231210 e24KC2ThetaAboveLeaf0010231211 e24KC2ThetaAboveLeaf0010231212
+      e24KC2ThetaAboveLeaf0010231213
+
+theorem e24KC2ThetaAboveNode001023130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00102313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00102313)
+    e24KC2ThetaAboveLeaf0010231300 e24KC2ThetaAboveLeaf0010231301 e24KC2ThetaAboveLeaf0010231302
+      e24KC2ThetaAboveLeaf0010231303
+
+theorem e24KC2ThetaAboveNode001023131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00102313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00102313)
+    e24KC2ThetaAboveLeaf0010231310 e24KC2ThetaAboveLeaf0010231311 e24KC2ThetaAboveLeaf0010231312
+      e24KC2ThetaAboveLeaf0010231313
+
+theorem e24KC2ThetaAboveNode001032000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103200)
+    e24KC2ThetaAboveLeaf0010320000 e24KC2ThetaAboveLeaf0010320001 e24KC2ThetaAboveLeaf0010320002
+      e24KC2ThetaAboveLeaf0010320003
+
+theorem e24KC2ThetaAboveNode001032001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103200)
+    e24KC2ThetaAboveLeaf0010320010 e24KC2ThetaAboveLeaf0010320011 e24KC2ThetaAboveLeaf0010320012
+      e24KC2ThetaAboveLeaf0010320013
+
+theorem e24KC2ThetaAboveNode001032002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00103200)
+    e24KC2ThetaAboveLeaf0010320020 e24KC2ThetaAboveLeaf0010320021 e24KC2ThetaAboveLeaf0010320022
+      e24KC2ThetaAboveLeaf0010320023
+
+theorem e24KC2ThetaAboveNode001032003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00103200)
+    e24KC2ThetaAboveLeaf0010320030 e24KC2ThetaAboveLeaf0010320031 e24KC2ThetaAboveLeaf0010320032
+      e24KC2ThetaAboveLeaf0010320033
+
+theorem e24KC2ThetaAboveNode001032010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103201)
+    e24KC2ThetaAboveLeaf0010320100 e24KC2ThetaAboveLeaf0010320101 e24KC2ThetaAboveLeaf0010320102
+      e24KC2ThetaAboveLeaf0010320103
+
+theorem e24KC2ThetaAboveNode001032011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103201)
+    e24KC2ThetaAboveLeaf0010320110 e24KC2ThetaAboveLeaf0010320111 e24KC2ThetaAboveLeaf0010320112
+      e24KC2ThetaAboveLeaf0010320113
+
+theorem e24KC2ThetaAboveNode001032012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00103201)
+    e24KC2ThetaAboveLeaf0010320120 e24KC2ThetaAboveLeaf0010320121 e24KC2ThetaAboveLeaf0010320122
+      e24KC2ThetaAboveLeaf0010320123
+
+theorem e24KC2ThetaAboveNode001032013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00103201)
+    e24KC2ThetaAboveLeaf0010320130 e24KC2ThetaAboveLeaf0010320131 e24KC2ThetaAboveLeaf0010320132
+      e24KC2ThetaAboveLeaf0010320133
+
+theorem e24KC2ThetaAboveNode001032020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103202)
+    e24KC2ThetaAboveLeaf0010320200 e24KC2ThetaAboveLeaf0010320201 e24KC2ThetaAboveLeaf0010320202
+      e24KC2ThetaAboveLeaf0010320203
+
+theorem e24KC2ThetaAboveNode001032021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103202)
+    e24KC2ThetaAboveLeaf0010320210 e24KC2ThetaAboveLeaf0010320211 e24KC2ThetaAboveLeaf0010320212
+      e24KC2ThetaAboveLeaf0010320213
+
+theorem e24KC2ThetaAboveNode001032030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103203)
+    e24KC2ThetaAboveLeaf0010320300 e24KC2ThetaAboveLeaf0010320301 e24KC2ThetaAboveLeaf0010320302
+      e24KC2ThetaAboveLeaf0010320303
+
+theorem e24KC2ThetaAboveNode001032031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103203)
+    e24KC2ThetaAboveLeaf0010320310 e24KC2ThetaAboveLeaf0010320311 e24KC2ThetaAboveLeaf0010320312
+      e24KC2ThetaAboveLeaf0010320313
+
+theorem e24KC2ThetaAboveNode001032100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103210)
+    e24KC2ThetaAboveLeaf0010321000 e24KC2ThetaAboveLeaf0010321001 e24KC2ThetaAboveLeaf0010321002
+      e24KC2ThetaAboveLeaf0010321003
+
+theorem e24KC2ThetaAboveNode001032101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103210)
+    e24KC2ThetaAboveLeaf0010321010 e24KC2ThetaAboveLeaf0010321011 e24KC2ThetaAboveLeaf0010321012
+      e24KC2ThetaAboveLeaf0010321013
+
+theorem e24KC2ThetaAboveNode001032102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00103210)
+    e24KC2ThetaAboveLeaf0010321020 e24KC2ThetaAboveLeaf0010321021 e24KC2ThetaAboveLeaf0010321022
+      e24KC2ThetaAboveLeaf0010321023
+
+theorem e24KC2ThetaAboveNode001032103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00103210)
+    e24KC2ThetaAboveLeaf0010321030 e24KC2ThetaAboveLeaf0010321031 e24KC2ThetaAboveLeaf0010321032
+      e24KC2ThetaAboveLeaf0010321033
+
+theorem e24KC2ThetaAboveNode001032110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103211)
+    e24KC2ThetaAboveLeaf0010321100 e24KC2ThetaAboveLeaf0010321101 e24KC2ThetaAboveLeaf0010321102
+      e24KC2ThetaAboveLeaf0010321103
+
+theorem e24KC2ThetaAboveNode001032111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103211)
+    e24KC2ThetaAboveLeaf0010321110 e24KC2ThetaAboveLeaf0010321111 e24KC2ThetaAboveLeaf0010321112
+      e24KC2ThetaAboveLeaf0010321113
+
+theorem e24KC2ThetaAboveNode001032112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00103211)
+    e24KC2ThetaAboveLeaf0010321120 e24KC2ThetaAboveLeaf0010321121 e24KC2ThetaAboveLeaf0010321122
+      e24KC2ThetaAboveLeaf0010321123
+
+theorem e24KC2ThetaAboveNode001032113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00103211)
+    e24KC2ThetaAboveLeaf0010321130 e24KC2ThetaAboveLeaf0010321131 e24KC2ThetaAboveLeaf0010321132
+      e24KC2ThetaAboveLeaf0010321133
+
+theorem e24KC2ThetaAboveNode001032120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103212)
+    e24KC2ThetaAboveLeaf0010321200 e24KC2ThetaAboveLeaf0010321201 e24KC2ThetaAboveLeaf0010321202
+      e24KC2ThetaAboveLeaf0010321203
+
+theorem e24KC2ThetaAboveNode001032121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103212)
+    e24KC2ThetaAboveLeaf0010321210 e24KC2ThetaAboveLeaf0010321211 e24KC2ThetaAboveLeaf0010321212
+      e24KC2ThetaAboveLeaf0010321213
+
+theorem e24KC2ThetaAboveNode001032130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103213)
+    e24KC2ThetaAboveLeaf0010321300 e24KC2ThetaAboveLeaf0010321301 e24KC2ThetaAboveLeaf0010321302
+      e24KC2ThetaAboveLeaf0010321303
+
+theorem e24KC2ThetaAboveNode001032131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103213)
+    e24KC2ThetaAboveLeaf0010321310 e24KC2ThetaAboveLeaf0010321311 e24KC2ThetaAboveLeaf0010321312
+      e24KC2ThetaAboveLeaf0010321313
+
+theorem e24KC2ThetaAboveNode001033000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103300)
+    e24KC2ThetaAboveLeaf0010330000 e24KC2ThetaAboveLeaf0010330001 e24KC2ThetaAboveLeaf0010330002
+      e24KC2ThetaAboveLeaf0010330003
+
+theorem e24KC2ThetaAboveNode001033001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103300)
+    e24KC2ThetaAboveLeaf0010330010 e24KC2ThetaAboveLeaf0010330011 e24KC2ThetaAboveLeaf0010330012
+      e24KC2ThetaAboveLeaf0010330013
+
+theorem e24KC2ThetaAboveNode001033002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00103300)
+    e24KC2ThetaAboveLeaf0010330020 e24KC2ThetaAboveLeaf0010330021 e24KC2ThetaAboveLeaf0010330022
+      e24KC2ThetaAboveLeaf0010330023
+
+theorem e24KC2ThetaAboveNode001033003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00103300)
+    e24KC2ThetaAboveLeaf0010330030 e24KC2ThetaAboveLeaf0010330031 e24KC2ThetaAboveLeaf0010330032
+      e24KC2ThetaAboveLeaf0010330033
+
+theorem e24KC2ThetaAboveNode001033010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103301)
+    e24KC2ThetaAboveLeaf0010330100 e24KC2ThetaAboveLeaf0010330101 e24KC2ThetaAboveLeaf0010330102
+      e24KC2ThetaAboveLeaf0010330103
+
+theorem e24KC2ThetaAboveNode001033011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103301)
+    e24KC2ThetaAboveLeaf0010330110 e24KC2ThetaAboveLeaf0010330111 e24KC2ThetaAboveLeaf0010330112
+      e24KC2ThetaAboveLeaf0010330113
+
+theorem e24KC2ThetaAboveNode001033012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00103301)
+    e24KC2ThetaAboveLeaf0010330120 e24KC2ThetaAboveLeaf0010330121 e24KC2ThetaAboveLeaf0010330122
+      e24KC2ThetaAboveLeaf0010330123
+
+theorem e24KC2ThetaAboveNode001033013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00103301)
+    e24KC2ThetaAboveLeaf0010330130 e24KC2ThetaAboveLeaf0010330131 e24KC2ThetaAboveLeaf0010330132
+      e24KC2ThetaAboveLeaf0010330133
+
+theorem e24KC2ThetaAboveNode001033020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103302)
+    e24KC2ThetaAboveLeaf0010330200 e24KC2ThetaAboveLeaf0010330201 e24KC2ThetaAboveLeaf0010330202
+      e24KC2ThetaAboveLeaf0010330203
+
+theorem e24KC2ThetaAboveNode001033021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103302)
+    e24KC2ThetaAboveLeaf0010330210 e24KC2ThetaAboveLeaf0010330211 e24KC2ThetaAboveLeaf0010330212
+      e24KC2ThetaAboveLeaf0010330213
+
+theorem e24KC2ThetaAboveNode001033030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103303)
+    e24KC2ThetaAboveLeaf0010330300 e24KC2ThetaAboveLeaf0010330301 e24KC2ThetaAboveLeaf0010330302
+      e24KC2ThetaAboveLeaf0010330303
+
+theorem e24KC2ThetaAboveNode001033031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103303)
+    e24KC2ThetaAboveLeaf0010330310 e24KC2ThetaAboveLeaf0010330311 e24KC2ThetaAboveLeaf0010330312
+      e24KC2ThetaAboveLeaf0010330313
+
+theorem e24KC2ThetaAboveNode001033100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103310)
+    e24KC2ThetaAboveLeaf0010331000 e24KC2ThetaAboveLeaf0010331001 e24KC2ThetaAboveLeaf0010331002
+      e24KC2ThetaAboveLeaf0010331003
+
+theorem e24KC2ThetaAboveNode001033101 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103310)
+    e24KC2ThetaAboveLeaf0010331010 e24KC2ThetaAboveLeaf0010331011 e24KC2ThetaAboveLeaf0010331012
+      e24KC2ThetaAboveLeaf0010331013
+
+theorem e24KC2ThetaAboveNode001033102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00103310)
+    e24KC2ThetaAboveLeaf0010331020 e24KC2ThetaAboveLeaf0010331021 e24KC2ThetaAboveLeaf0010331022
+      e24KC2ThetaAboveLeaf0010331023
+
+theorem e24KC2ThetaAboveNode001033103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00103310)
+    e24KC2ThetaAboveLeaf0010331030 e24KC2ThetaAboveLeaf0010331031 e24KC2ThetaAboveLeaf0010331032
+      e24KC2ThetaAboveLeaf0010331033
+
+theorem e24KC2ThetaAboveNode001033110 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103311)
+    e24KC2ThetaAboveLeaf0010331100 e24KC2ThetaAboveLeaf0010331101 e24KC2ThetaAboveLeaf0010331102
+      e24KC2ThetaAboveLeaf0010331103
+
+theorem e24KC2ThetaAboveNode001033111 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103311)
+    e24KC2ThetaAboveLeaf0010331110 e24KC2ThetaAboveLeaf0010331111 e24KC2ThetaAboveLeaf0010331112
+      e24KC2ThetaAboveLeaf0010331113
+
+theorem e24KC2ThetaAboveNode001033112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00103311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00103311)
+    e24KC2ThetaAboveLeaf0010331120 e24KC2ThetaAboveLeaf0010331121 e24KC2ThetaAboveLeaf0010331122
+      e24KC2ThetaAboveLeaf0010331123
+
+theorem e24KC2ThetaAboveNode001033113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00103311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00103311)
+    e24KC2ThetaAboveLeaf0010331130 e24KC2ThetaAboveLeaf0010331131 e24KC2ThetaAboveLeaf0010331132
+      e24KC2ThetaAboveLeaf0010331133
+
+theorem e24KC2ThetaAboveNode001033120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103312)
+    e24KC2ThetaAboveLeaf0010331200 e24KC2ThetaAboveLeaf0010331201 e24KC2ThetaAboveLeaf0010331202
+      e24KC2ThetaAboveLeaf0010331203
+
+theorem e24KC2ThetaAboveNode001033121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103312)
+    e24KC2ThetaAboveLeaf0010331210 e24KC2ThetaAboveLeaf0010331211 e24KC2ThetaAboveLeaf0010331212
+      e24KC2ThetaAboveLeaf0010331213
+
+theorem e24KC2ThetaAboveNode001033130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00103313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00103313)
+    e24KC2ThetaAboveLeaf0010331300 e24KC2ThetaAboveLeaf0010331301 e24KC2ThetaAboveLeaf0010331302
+      e24KC2ThetaAboveLeaf0010331303
+
+theorem e24KC2ThetaAboveNode001033131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00103313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00103313)
+    e24KC2ThetaAboveLeaf0010331310 e24KC2ThetaAboveLeaf0010331311 e24KC2ThetaAboveLeaf0010331312
+      e24KC2ThetaAboveLeaf0010331313
+
+theorem e24KC2ThetaAboveNode001122000 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00112200)
+    e24KC2ThetaAboveLeaf0011220000 e24KC2ThetaAboveLeaf0011220001 e24KC2ThetaAboveLeaf0011220002
+      e24KC2ThetaAboveLeaf0011220003
+
+theorem e24KC2ThetaAboveNode001122001 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00112200)
+    e24KC2ThetaAboveLeaf0011220010 e24KC2ThetaAboveLeaf0011220011 e24KC2ThetaAboveLeaf0011220012
+      e24KC2ThetaAboveLeaf0011220013
+
+theorem e24KC2ThetaAboveNode001122002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00112200)
+    e24KC2ThetaAboveLeaf0011220020 e24KC2ThetaAboveLeaf0011220021 e24KC2ThetaAboveLeaf0011220022
+      e24KC2ThetaAboveLeaf0011220023
+
+theorem e24KC2ThetaAboveNode001122003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00112200)
+    e24KC2ThetaAboveLeaf0011220030 e24KC2ThetaAboveLeaf0011220031 e24KC2ThetaAboveLeaf0011220032
+      e24KC2ThetaAboveLeaf0011220033
+
+theorem e24KC2ThetaAboveNode001122010 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00112201)
+    e24KC2ThetaAboveLeaf0011220100 e24KC2ThetaAboveLeaf0011220101 e24KC2ThetaAboveLeaf0011220102
+      e24KC2ThetaAboveLeaf0011220103
+
+theorem e24KC2ThetaAboveNode001122011 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00112201)
+    e24KC2ThetaAboveLeaf0011220110 e24KC2ThetaAboveLeaf0011220111 e24KC2ThetaAboveLeaf0011220112
+      e24KC2ThetaAboveLeaf0011220113
+
+theorem e24KC2ThetaAboveNode001122012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00112201)
+    e24KC2ThetaAboveLeaf0011220120 e24KC2ThetaAboveLeaf0011220121 e24KC2ThetaAboveLeaf0011220122
+      e24KC2ThetaAboveLeaf0011220123
+
+theorem e24KC2ThetaAboveNode001122013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00112201)
+    e24KC2ThetaAboveLeaf0011220130 e24KC2ThetaAboveLeaf0011220131 e24KC2ThetaAboveLeaf0011220132
+      e24KC2ThetaAboveLeaf0011220133
+
+theorem e24KC2ThetaAboveNode001122020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00112202)
+    e24KC2ThetaAboveLeaf0011220200 e24KC2ThetaAboveLeaf0011220201 e24KC2ThetaAboveLeaf0011220202
+      e24KC2ThetaAboveLeaf0011220203
+
+theorem e24KC2ThetaAboveNode001122021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00112202)
+    e24KC2ThetaAboveLeaf0011220210 e24KC2ThetaAboveLeaf0011220211 e24KC2ThetaAboveLeaf0011220212
+      e24KC2ThetaAboveLeaf0011220213
+
+theorem e24KC2ThetaAboveNode001122030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00112203)
+    e24KC2ThetaAboveLeaf0011220300 e24KC2ThetaAboveLeaf0011220301 e24KC2ThetaAboveLeaf0011220302
+      e24KC2ThetaAboveLeaf0011220303
+
+theorem e24KC2ThetaAboveNode001122031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00112203)
+    e24KC2ThetaAboveLeaf0011220310 e24KC2ThetaAboveLeaf0011220311 e24KC2ThetaAboveLeaf0011220312
+      e24KC2ThetaAboveLeaf0011220313
+
+theorem e24KC2ThetaAboveNode001122100 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00112210)
+    e24KC2ThetaAboveLeaf0011221000 e24KC2ThetaAboveLeaf0011221001 e24KC2ThetaAboveLeaf0011221002
+      e24KC2ThetaAboveLeaf0011221003
+
+theorem e24KC2ThetaAboveNode001122102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00112210)
+    e24KC2ThetaAboveLeaf0011221020 e24KC2ThetaAboveLeaf0011221021 e24KC2ThetaAboveLeaf0011221022
+      e24KC2ThetaAboveLeaf0011221023
+
+theorem e24KC2ThetaAboveNode001122103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00112210)
+    e24KC2ThetaAboveLeaf0011221030 e24KC2ThetaAboveLeaf0011221031 e24KC2ThetaAboveLeaf0011221032
+      e24KC2ThetaAboveLeaf0011221033
+
+theorem e24KC2ThetaAboveNode001122112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00112211)
+    e24KC2ThetaAboveLeaf0011221120 e24KC2ThetaAboveLeaf0011221121 e24KC2ThetaAboveLeaf0011221122
+      e24KC2ThetaAboveLeaf0011221123
+
+theorem e24KC2ThetaAboveNode001122113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00112211)
+    e24KC2ThetaAboveLeaf0011221130 e24KC2ThetaAboveLeaf0011221131 e24KC2ThetaAboveLeaf0011221132
+      e24KC2ThetaAboveLeaf0011221133
+
+theorem e24KC2ThetaAboveNode001122120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00112212)
+    e24KC2ThetaAboveLeaf0011221200 e24KC2ThetaAboveLeaf0011221201 e24KC2ThetaAboveLeaf0011221202
+      e24KC2ThetaAboveLeaf0011221203
+
+theorem e24KC2ThetaAboveNode001122121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00112212)
+    e24KC2ThetaAboveLeaf0011221210 e24KC2ThetaAboveLeaf0011221211 e24KC2ThetaAboveLeaf0011221212
+      e24KC2ThetaAboveLeaf0011221213
+
+theorem e24KC2ThetaAboveNode001122130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00112213)
+    e24KC2ThetaAboveLeaf0011221300 e24KC2ThetaAboveLeaf0011221301 e24KC2ThetaAboveLeaf0011221302
+      e24KC2ThetaAboveLeaf0011221303
+
+theorem e24KC2ThetaAboveNode001122131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00112213)
+    e24KC2ThetaAboveLeaf0011221310 e24KC2ThetaAboveLeaf0011221311 e24KC2ThetaAboveLeaf0011221312
+      e24KC2ThetaAboveLeaf0011221313
+
+theorem e24KC2ThetaAboveNode001123002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00112300)
+    e24KC2ThetaAboveLeaf0011230020 e24KC2ThetaAboveLeaf0011230021 e24KC2ThetaAboveLeaf0011230022
+      e24KC2ThetaAboveLeaf0011230023
+
+theorem e24KC2ThetaAboveNode001123003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00112300)
+    e24KC2ThetaAboveLeaf0011230030 e24KC2ThetaAboveLeaf0011230031 e24KC2ThetaAboveLeaf0011230032
+      e24KC2ThetaAboveLeaf0011230033
+
+theorem e24KC2ThetaAboveNode001123012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00112301)
+    e24KC2ThetaAboveLeaf0011230120 e24KC2ThetaAboveLeaf0011230121 e24KC2ThetaAboveLeaf0011230122
+      e24KC2ThetaAboveLeaf0011230123
+
+theorem e24KC2ThetaAboveNode001123013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00112301)
+    e24KC2ThetaAboveLeaf0011230130 e24KC2ThetaAboveLeaf0011230131 e24KC2ThetaAboveLeaf0011230132
+      e24KC2ThetaAboveLeaf0011230133
+
+theorem e24KC2ThetaAboveNode001123020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00112302)
+    e24KC2ThetaAboveLeaf0011230200 e24KC2ThetaAboveLeaf0011230201 e24KC2ThetaAboveLeaf0011230202
+      e24KC2ThetaAboveLeaf0011230203
+
+theorem e24KC2ThetaAboveNode001123021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00112302)
+    e24KC2ThetaAboveLeaf0011230210 e24KC2ThetaAboveLeaf0011230211 e24KC2ThetaAboveLeaf0011230212
+      e24KC2ThetaAboveLeaf0011230213
+
+theorem e24KC2ThetaAboveNode001123030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00112303)
+    e24KC2ThetaAboveLeaf0011230300 e24KC2ThetaAboveLeaf0011230301 e24KC2ThetaAboveLeaf0011230302
+      e24KC2ThetaAboveLeaf0011230303
+
+theorem e24KC2ThetaAboveNode001123031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00112303)
+    e24KC2ThetaAboveLeaf0011230310 e24KC2ThetaAboveLeaf0011230311 e24KC2ThetaAboveLeaf0011230312
+      e24KC2ThetaAboveLeaf0011230313
+
+theorem e24KC2ThetaAboveNode001123102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00112310)
+    e24KC2ThetaAboveLeaf0011231020 e24KC2ThetaAboveLeaf0011231021 e24KC2ThetaAboveLeaf0011231022
+      e24KC2ThetaAboveLeaf0011231023
+
+theorem e24KC2ThetaAboveNode001123103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00112310)
+    e24KC2ThetaAboveLeaf0011231030 e24KC2ThetaAboveLeaf0011231031 e24KC2ThetaAboveLeaf0011231032
+      e24KC2ThetaAboveLeaf0011231033
+
+theorem e24KC2ThetaAboveNode001123112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00112311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00112311)
+    e24KC2ThetaAboveLeaf0011231120 e24KC2ThetaAboveLeaf0011231121 e24KC2ThetaAboveLeaf0011231122
+      e24KC2ThetaAboveLeaf0011231123
+
+theorem e24KC2ThetaAboveNode001123113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00112311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00112311)
+    e24KC2ThetaAboveLeaf0011231130 e24KC2ThetaAboveLeaf0011231131 e24KC2ThetaAboveLeaf0011231132
+      e24KC2ThetaAboveLeaf0011231133
+
+theorem e24KC2ThetaAboveNode001123120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00112312)
+    e24KC2ThetaAboveLeaf0011231200 e24KC2ThetaAboveLeaf0011231201 e24KC2ThetaAboveLeaf0011231202
+      e24KC2ThetaAboveLeaf0011231203
+
+theorem e24KC2ThetaAboveNode001123121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00112312)
+    e24KC2ThetaAboveLeaf0011231210 e24KC2ThetaAboveLeaf0011231211 e24KC2ThetaAboveLeaf0011231212
+      e24KC2ThetaAboveLeaf0011231213
+
+theorem e24KC2ThetaAboveNode001123130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00112313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00112313)
+    e24KC2ThetaAboveLeaf0011231300 e24KC2ThetaAboveLeaf0011231301 e24KC2ThetaAboveLeaf0011231302
+      e24KC2ThetaAboveLeaf0011231303
+
+theorem e24KC2ThetaAboveNode001123131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00112313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00112313)
+    e24KC2ThetaAboveLeaf0011231310 e24KC2ThetaAboveLeaf0011231311 e24KC2ThetaAboveLeaf0011231312
+      e24KC2ThetaAboveLeaf0011231313
+
+theorem e24KC2ThetaAboveNode001132002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00113200)
+    e24KC2ThetaAboveLeaf0011320020 e24KC2ThetaAboveLeaf0011320021 e24KC2ThetaAboveLeaf0011320022
+      e24KC2ThetaAboveLeaf0011320023
+
+theorem e24KC2ThetaAboveNode001132003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00113200)
+    e24KC2ThetaAboveLeaf0011320030 e24KC2ThetaAboveLeaf0011320031 e24KC2ThetaAboveLeaf0011320032
+      e24KC2ThetaAboveLeaf0011320033
+
+theorem e24KC2ThetaAboveNode001132012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00113201)
+    e24KC2ThetaAboveLeaf0011320120 e24KC2ThetaAboveLeaf0011320121 e24KC2ThetaAboveLeaf0011320122
+      e24KC2ThetaAboveLeaf0011320123
+
+theorem e24KC2ThetaAboveNode001132013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00113201)
+    e24KC2ThetaAboveLeaf0011320130 e24KC2ThetaAboveLeaf0011320131 e24KC2ThetaAboveLeaf0011320132
+      e24KC2ThetaAboveLeaf0011320133
+
+theorem e24KC2ThetaAboveNode001132020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00113202)
+    e24KC2ThetaAboveLeaf0011320200 e24KC2ThetaAboveLeaf0011320201 e24KC2ThetaAboveLeaf0011320202
+      e24KC2ThetaAboveLeaf0011320203
+
+theorem e24KC2ThetaAboveNode001132021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00113202)
+    e24KC2ThetaAboveLeaf0011320210 e24KC2ThetaAboveLeaf0011320211 e24KC2ThetaAboveLeaf0011320212
+      e24KC2ThetaAboveLeaf0011320213
+
+theorem e24KC2ThetaAboveNode001132030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00113203)
+    e24KC2ThetaAboveLeaf0011320300 e24KC2ThetaAboveLeaf0011320301 e24KC2ThetaAboveLeaf0011320302
+      e24KC2ThetaAboveLeaf0011320303
+
+theorem e24KC2ThetaAboveNode001132031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00113203)
+    e24KC2ThetaAboveLeaf0011320310 e24KC2ThetaAboveLeaf0011320311 e24KC2ThetaAboveLeaf0011320312
+      e24KC2ThetaAboveLeaf0011320313
+
+theorem e24KC2ThetaAboveNode001132102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00113210)
+    e24KC2ThetaAboveLeaf0011321020 e24KC2ThetaAboveLeaf0011321021 e24KC2ThetaAboveLeaf0011321022
+      e24KC2ThetaAboveLeaf0011321023
+
+theorem e24KC2ThetaAboveNode001132103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00113210)
+    e24KC2ThetaAboveLeaf0011321030 e24KC2ThetaAboveLeaf0011321031 e24KC2ThetaAboveLeaf0011321032
+      e24KC2ThetaAboveLeaf0011321033
+
+theorem e24KC2ThetaAboveNode001132112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00113211)
+    e24KC2ThetaAboveLeaf0011321120 e24KC2ThetaAboveLeaf0011321121 e24KC2ThetaAboveLeaf0011321122
+      e24KC2ThetaAboveLeaf0011321123
+
+theorem e24KC2ThetaAboveNode001132113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00113211)
+    e24KC2ThetaAboveLeaf0011321130 e24KC2ThetaAboveLeaf0011321131 e24KC2ThetaAboveLeaf0011321132
+      e24KC2ThetaAboveLeaf0011321133
+
+theorem e24KC2ThetaAboveNode001132120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00113212)
+    e24KC2ThetaAboveLeaf0011321200 e24KC2ThetaAboveLeaf0011321201 e24KC2ThetaAboveLeaf0011321202
+      e24KC2ThetaAboveLeaf0011321203
+
+theorem e24KC2ThetaAboveNode001132121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00113212)
+    e24KC2ThetaAboveLeaf0011321210 e24KC2ThetaAboveLeaf0011321211 e24KC2ThetaAboveLeaf0011321212
+      e24KC2ThetaAboveLeaf0011321213
+
+theorem e24KC2ThetaAboveNode001132130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00113213)
+    e24KC2ThetaAboveLeaf0011321300 e24KC2ThetaAboveLeaf0011321301 e24KC2ThetaAboveLeaf0011321302
+      e24KC2ThetaAboveLeaf0011321303
+
+theorem e24KC2ThetaAboveNode001132131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00113213)
+    e24KC2ThetaAboveLeaf0011321310 e24KC2ThetaAboveLeaf0011321311 e24KC2ThetaAboveLeaf0011321312
+      e24KC2ThetaAboveLeaf0011321313
+
+theorem e24KC2ThetaAboveNode001133002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00113300)
+    e24KC2ThetaAboveLeaf0011330020 e24KC2ThetaAboveLeaf0011330021 e24KC2ThetaAboveLeaf0011330022
+      e24KC2ThetaAboveLeaf0011330023
+
+theorem e24KC2ThetaAboveNode001133003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00113300)
+    e24KC2ThetaAboveLeaf0011330030 e24KC2ThetaAboveLeaf0011330031 e24KC2ThetaAboveLeaf0011330032
+      e24KC2ThetaAboveLeaf0011330033
+
+theorem e24KC2ThetaAboveNode001133012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00113301)
+    e24KC2ThetaAboveLeaf0011330120 e24KC2ThetaAboveLeaf0011330121 e24KC2ThetaAboveLeaf0011330122
+      e24KC2ThetaAboveLeaf0011330123
+
+theorem e24KC2ThetaAboveNode001133013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00113301)
+    e24KC2ThetaAboveLeaf0011330130 e24KC2ThetaAboveLeaf0011330131 e24KC2ThetaAboveLeaf0011330132
+      e24KC2ThetaAboveLeaf0011330133
+
+theorem e24KC2ThetaAboveNode001133020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00113302)
+    e24KC2ThetaAboveLeaf0011330200 e24KC2ThetaAboveLeaf0011330201 e24KC2ThetaAboveLeaf0011330202
+      e24KC2ThetaAboveLeaf0011330203
+
+theorem e24KC2ThetaAboveNode001133021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00113302)
+    e24KC2ThetaAboveLeaf0011330210 e24KC2ThetaAboveLeaf0011330211 e24KC2ThetaAboveLeaf0011330212
+      e24KC2ThetaAboveLeaf0011330213
+
+theorem e24KC2ThetaAboveNode001133030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00113303)
+    e24KC2ThetaAboveLeaf0011330300 e24KC2ThetaAboveLeaf0011330301 e24KC2ThetaAboveLeaf0011330302
+      e24KC2ThetaAboveLeaf0011330303
+
+theorem e24KC2ThetaAboveNode001133031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00113303)
+    e24KC2ThetaAboveLeaf0011330310 e24KC2ThetaAboveLeaf0011330311 e24KC2ThetaAboveLeaf0011330312
+      e24KC2ThetaAboveLeaf0011330313
+
+theorem e24KC2ThetaAboveNode001133102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00113310)
+    e24KC2ThetaAboveLeaf0011331020 e24KC2ThetaAboveLeaf0011331021 e24KC2ThetaAboveLeaf0011331022
+      e24KC2ThetaAboveLeaf0011331023
+
+theorem e24KC2ThetaAboveNode001133103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00113310)
+    e24KC2ThetaAboveLeaf0011331030 e24KC2ThetaAboveLeaf0011331031 e24KC2ThetaAboveLeaf0011331032
+      e24KC2ThetaAboveLeaf0011331033
+
+theorem e24KC2ThetaAboveNode001133112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell00113311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell00113311)
+    e24KC2ThetaAboveLeaf0011331120 e24KC2ThetaAboveLeaf0011331121 e24KC2ThetaAboveLeaf0011331122
+      e24KC2ThetaAboveLeaf0011331123
+
+theorem e24KC2ThetaAboveNode001133113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell00113311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell00113311)
+    e24KC2ThetaAboveLeaf0011331130 e24KC2ThetaAboveLeaf0011331131 e24KC2ThetaAboveLeaf0011331132
+      e24KC2ThetaAboveLeaf0011331133
+
+theorem e24KC2ThetaAboveNode001133120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00113312)
+    e24KC2ThetaAboveLeaf0011331200 e24KC2ThetaAboveLeaf0011331201 e24KC2ThetaAboveLeaf0011331202
+      e24KC2ThetaAboveLeaf0011331203
+
+theorem e24KC2ThetaAboveNode001133121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00113312)
+    e24KC2ThetaAboveLeaf0011331210 e24KC2ThetaAboveLeaf0011331211 e24KC2ThetaAboveLeaf0011331212
+      e24KC2ThetaAboveLeaf0011331213
+
+theorem e24KC2ThetaAboveNode001133130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell00113313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell00113313)
+    e24KC2ThetaAboveLeaf0011331300 e24KC2ThetaAboveLeaf0011331301 e24KC2ThetaAboveLeaf0011331302
+      e24KC2ThetaAboveLeaf0011331303
+
+theorem e24KC2ThetaAboveNode001133131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell00113313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell00113313)
+    e24KC2ThetaAboveLeaf0011331310 e24KC2ThetaAboveLeaf0011331311 e24KC2ThetaAboveLeaf0011331312
+      e24KC2ThetaAboveLeaf0011331313
+
+theorem e24KC2ThetaAboveNode010022002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01002200)
+    e24KC2ThetaAboveLeaf0100220020 e24KC2ThetaAboveLeaf0100220021 e24KC2ThetaAboveLeaf0100220022
+      e24KC2ThetaAboveLeaf0100220023
+
+theorem e24KC2ThetaAboveNode010022003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01002200)
+    e24KC2ThetaAboveLeaf0100220030 e24KC2ThetaAboveLeaf0100220031 e24KC2ThetaAboveLeaf0100220032
+      e24KC2ThetaAboveLeaf0100220033
+
+theorem e24KC2ThetaAboveNode010022012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01002201)
+    e24KC2ThetaAboveLeaf0100220120 e24KC2ThetaAboveLeaf0100220121 e24KC2ThetaAboveLeaf0100220122
+      e24KC2ThetaAboveLeaf0100220123
+
+theorem e24KC2ThetaAboveNode010022013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01002201)
+    e24KC2ThetaAboveLeaf0100220130 e24KC2ThetaAboveLeaf0100220131 e24KC2ThetaAboveLeaf0100220132
+      e24KC2ThetaAboveLeaf0100220133
+
+theorem e24KC2ThetaAboveNode010022020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01002202)
+    e24KC2ThetaAboveLeaf0100220200 e24KC2ThetaAboveLeaf0100220201 e24KC2ThetaAboveLeaf0100220202
+      e24KC2ThetaAboveLeaf0100220203
+
+theorem e24KC2ThetaAboveNode010022021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01002202)
+    e24KC2ThetaAboveLeaf0100220210 e24KC2ThetaAboveLeaf0100220211 e24KC2ThetaAboveLeaf0100220212
+      e24KC2ThetaAboveLeaf0100220213
+
+theorem e24KC2ThetaAboveNode010022030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01002203)
+    e24KC2ThetaAboveLeaf0100220300 e24KC2ThetaAboveLeaf0100220301 e24KC2ThetaAboveLeaf0100220302
+      e24KC2ThetaAboveLeaf0100220303
+
+theorem e24KC2ThetaAboveNode010022031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01002203)
+    e24KC2ThetaAboveLeaf0100220310 e24KC2ThetaAboveLeaf0100220311 e24KC2ThetaAboveLeaf0100220312
+      e24KC2ThetaAboveLeaf0100220313
+
+theorem e24KC2ThetaAboveNode010022102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01002210)
+    e24KC2ThetaAboveLeaf0100221020 e24KC2ThetaAboveLeaf0100221021 e24KC2ThetaAboveLeaf0100221022
+      e24KC2ThetaAboveLeaf0100221023
+
+theorem e24KC2ThetaAboveNode010022103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01002210)
+    e24KC2ThetaAboveLeaf0100221030 e24KC2ThetaAboveLeaf0100221031 e24KC2ThetaAboveLeaf0100221032
+      e24KC2ThetaAboveLeaf0100221033
+
+theorem e24KC2ThetaAboveNode010022112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01002211)
+    e24KC2ThetaAboveLeaf0100221120 e24KC2ThetaAboveLeaf0100221121 e24KC2ThetaAboveLeaf0100221122
+      e24KC2ThetaAboveLeaf0100221123
+
+theorem e24KC2ThetaAboveNode010022113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01002211)
+    e24KC2ThetaAboveLeaf0100221130 e24KC2ThetaAboveLeaf0100221131 e24KC2ThetaAboveLeaf0100221132
+      e24KC2ThetaAboveLeaf0100221133
+
+theorem e24KC2ThetaAboveNode010022120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01002212)
+    e24KC2ThetaAboveLeaf0100221200 e24KC2ThetaAboveLeaf0100221201 e24KC2ThetaAboveLeaf0100221202
+      e24KC2ThetaAboveLeaf0100221203
+
+theorem e24KC2ThetaAboveNode010022121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01002212)
+    e24KC2ThetaAboveLeaf0100221210 e24KC2ThetaAboveLeaf0100221211 e24KC2ThetaAboveLeaf0100221212
+      e24KC2ThetaAboveLeaf0100221213
+
+theorem e24KC2ThetaAboveNode010022130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01002213)
+    e24KC2ThetaAboveLeaf0100221300 e24KC2ThetaAboveLeaf0100221301 e24KC2ThetaAboveLeaf0100221302
+      e24KC2ThetaAboveLeaf0100221303
+
+theorem e24KC2ThetaAboveNode010022131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01002213)
+    e24KC2ThetaAboveLeaf0100221310 e24KC2ThetaAboveLeaf0100221311 e24KC2ThetaAboveLeaf0100221312
+      e24KC2ThetaAboveLeaf0100221313
+
+theorem e24KC2ThetaAboveNode010023002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01002300)
+    e24KC2ThetaAboveLeaf0100230020 e24KC2ThetaAboveLeaf0100230021 e24KC2ThetaAboveLeaf0100230022
+      e24KC2ThetaAboveLeaf0100230023
+
+theorem e24KC2ThetaAboveNode010023003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01002300)
+    e24KC2ThetaAboveLeaf0100230030 e24KC2ThetaAboveLeaf0100230031 e24KC2ThetaAboveLeaf0100230032
+      e24KC2ThetaAboveLeaf0100230033
+
+theorem e24KC2ThetaAboveNode010023012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01002301)
+    e24KC2ThetaAboveLeaf0100230120 e24KC2ThetaAboveLeaf0100230121 e24KC2ThetaAboveLeaf0100230122
+      e24KC2ThetaAboveLeaf0100230123
+
+theorem e24KC2ThetaAboveNode010023013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01002301)
+    e24KC2ThetaAboveLeaf0100230130 e24KC2ThetaAboveLeaf0100230131 e24KC2ThetaAboveLeaf0100230132
+      e24KC2ThetaAboveLeaf0100230133
+
+theorem e24KC2ThetaAboveNode010023020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01002302)
+    e24KC2ThetaAboveLeaf0100230200 e24KC2ThetaAboveLeaf0100230201 e24KC2ThetaAboveLeaf0100230202
+      e24KC2ThetaAboveLeaf0100230203
+
+theorem e24KC2ThetaAboveNode010023021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01002302)
+    e24KC2ThetaAboveLeaf0100230210 e24KC2ThetaAboveLeaf0100230211 e24KC2ThetaAboveLeaf0100230212
+      e24KC2ThetaAboveLeaf0100230213
+
+theorem e24KC2ThetaAboveNode010023030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01002303)
+    e24KC2ThetaAboveLeaf0100230300 e24KC2ThetaAboveLeaf0100230301 e24KC2ThetaAboveLeaf0100230302
+      e24KC2ThetaAboveLeaf0100230303
+
+theorem e24KC2ThetaAboveNode010023031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01002303)
+    e24KC2ThetaAboveLeaf0100230310 e24KC2ThetaAboveLeaf0100230311 e24KC2ThetaAboveLeaf0100230312
+      e24KC2ThetaAboveLeaf0100230313
+
+theorem e24KC2ThetaAboveNode010023032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01002303)
+    e24KC2ThetaAboveLeaf0100230320 e24KC2ThetaAboveLeaf0100230321 e24KC2ThetaAboveLeaf0100230322
+      e24KC2ThetaAboveLeaf0100230323
+
+theorem e24KC2ThetaAboveNode010023033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01002303)
+    e24KC2ThetaAboveLeaf0100230330 e24KC2ThetaAboveLeaf0100230331 e24KC2ThetaAboveLeaf0100230332
+      e24KC2ThetaAboveLeaf0100230333
+
+theorem e24KC2ThetaAboveNode010023102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01002310)
+    e24KC2ThetaAboveLeaf0100231020 e24KC2ThetaAboveLeaf0100231021 e24KC2ThetaAboveLeaf0100231022
+      e24KC2ThetaAboveLeaf0100231023
+
+theorem e24KC2ThetaAboveNode010023103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01002310)
+    e24KC2ThetaAboveLeaf0100231030 e24KC2ThetaAboveLeaf0100231031 e24KC2ThetaAboveLeaf0100231032
+      e24KC2ThetaAboveLeaf0100231033
+
+theorem e24KC2ThetaAboveNode010023112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01002311)
+    e24KC2ThetaAboveLeaf0100231120 e24KC2ThetaAboveLeaf0100231121 e24KC2ThetaAboveLeaf0100231122
+      e24KC2ThetaAboveLeaf0100231123
+
+theorem e24KC2ThetaAboveNode010023113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01002311)
+    e24KC2ThetaAboveLeaf0100231130 e24KC2ThetaAboveLeaf0100231131 e24KC2ThetaAboveLeaf0100231132
+      e24KC2ThetaAboveLeaf0100231133
+
+theorem e24KC2ThetaAboveNode010023120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01002312)
+    e24KC2ThetaAboveLeaf0100231200 e24KC2ThetaAboveLeaf0100231201 e24KC2ThetaAboveLeaf0100231202
+      e24KC2ThetaAboveLeaf0100231203
+
+theorem e24KC2ThetaAboveNode010023121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01002312)
+    e24KC2ThetaAboveLeaf0100231210 e24KC2ThetaAboveLeaf0100231211 e24KC2ThetaAboveLeaf0100231212
+      e24KC2ThetaAboveLeaf0100231213
+
+theorem e24KC2ThetaAboveNode010023122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01002312)
+    e24KC2ThetaAboveLeaf0100231220 e24KC2ThetaAboveLeaf0100231221 e24KC2ThetaAboveLeaf0100231222
+      e24KC2ThetaAboveLeaf0100231223
+
+theorem e24KC2ThetaAboveNode010023123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01002312)
+    e24KC2ThetaAboveLeaf0100231230 e24KC2ThetaAboveLeaf0100231231 e24KC2ThetaAboveLeaf0100231232
+      e24KC2ThetaAboveLeaf0100231233
+
+theorem e24KC2ThetaAboveNode010023130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01002313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01002313)
+    e24KC2ThetaAboveLeaf0100231300 e24KC2ThetaAboveLeaf0100231301 e24KC2ThetaAboveLeaf0100231302
+      e24KC2ThetaAboveLeaf0100231303
+
+theorem e24KC2ThetaAboveNode010023131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01002313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01002313)
+    e24KC2ThetaAboveLeaf0100231310 e24KC2ThetaAboveLeaf0100231311 e24KC2ThetaAboveLeaf0100231312
+      e24KC2ThetaAboveLeaf0100231313
+
+theorem e24KC2ThetaAboveNode010023132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01002313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01002313)
+    e24KC2ThetaAboveLeaf0100231320 e24KC2ThetaAboveLeaf0100231321 e24KC2ThetaAboveLeaf0100231322
+      e24KC2ThetaAboveLeaf0100231323
+
+theorem e24KC2ThetaAboveNode010023133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01002313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01002313)
+    e24KC2ThetaAboveLeaf0100231330 e24KC2ThetaAboveLeaf0100231331 e24KC2ThetaAboveLeaf0100231332
+      e24KC2ThetaAboveLeaf0100231333
+
+theorem e24KC2ThetaAboveNode010032002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003200)
+    e24KC2ThetaAboveLeaf0100320020 e24KC2ThetaAboveLeaf0100320021 e24KC2ThetaAboveLeaf0100320022
+      e24KC2ThetaAboveLeaf0100320023
+
+theorem e24KC2ThetaAboveNode010032003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003200)
+    e24KC2ThetaAboveLeaf0100320030 e24KC2ThetaAboveLeaf0100320031 e24KC2ThetaAboveLeaf0100320032
+      e24KC2ThetaAboveLeaf0100320033
+
+theorem e24KC2ThetaAboveNode010032012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003201)
+    e24KC2ThetaAboveLeaf0100320120 e24KC2ThetaAboveLeaf0100320121 e24KC2ThetaAboveLeaf0100320122
+      e24KC2ThetaAboveLeaf0100320123
+
+theorem e24KC2ThetaAboveNode010032013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003201)
+    e24KC2ThetaAboveLeaf0100320130 e24KC2ThetaAboveLeaf0100320131 e24KC2ThetaAboveLeaf0100320132
+      e24KC2ThetaAboveLeaf0100320133
+
+theorem e24KC2ThetaAboveNode010032020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01003202)
+    e24KC2ThetaAboveLeaf0100320200 e24KC2ThetaAboveLeaf0100320201 e24KC2ThetaAboveLeaf0100320202
+      e24KC2ThetaAboveLeaf0100320203
+
+theorem e24KC2ThetaAboveNode010032021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01003202)
+    e24KC2ThetaAboveLeaf0100320210 e24KC2ThetaAboveLeaf0100320211 e24KC2ThetaAboveLeaf0100320212
+      e24KC2ThetaAboveLeaf0100320213
+
+theorem e24KC2ThetaAboveNode010032022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003202)
+    e24KC2ThetaAboveLeaf0100320220 e24KC2ThetaAboveLeaf0100320221 e24KC2ThetaAboveLeaf0100320222
+      e24KC2ThetaAboveLeaf0100320223
+
+theorem e24KC2ThetaAboveNode010032023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003202)
+    e24KC2ThetaAboveLeaf0100320230 e24KC2ThetaAboveLeaf0100320231 e24KC2ThetaAboveLeaf0100320232
+      e24KC2ThetaAboveLeaf0100320233
+
+theorem e24KC2ThetaAboveNode010032030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01003203)
+    e24KC2ThetaAboveLeaf0100320300 e24KC2ThetaAboveLeaf0100320301 e24KC2ThetaAboveLeaf0100320302
+      e24KC2ThetaAboveLeaf0100320303
+
+theorem e24KC2ThetaAboveNode010032031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01003203)
+    e24KC2ThetaAboveLeaf0100320310 e24KC2ThetaAboveLeaf0100320311 e24KC2ThetaAboveLeaf0100320312
+      e24KC2ThetaAboveLeaf0100320313
+
+theorem e24KC2ThetaAboveNode010032032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003203)
+    e24KC2ThetaAboveLeaf0100320320 e24KC2ThetaAboveLeaf0100320321 e24KC2ThetaAboveLeaf0100320322
+      e24KC2ThetaAboveLeaf0100320323
+
+theorem e24KC2ThetaAboveNode010032033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003203)
+    e24KC2ThetaAboveLeaf0100320330 e24KC2ThetaAboveLeaf0100320331 e24KC2ThetaAboveLeaf0100320332
+      e24KC2ThetaAboveLeaf0100320333
+
+theorem e24KC2ThetaAboveNode010032102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003210)
+    e24KC2ThetaAboveLeaf0100321020 e24KC2ThetaAboveLeaf0100321021 e24KC2ThetaAboveLeaf0100321022
+      e24KC2ThetaAboveLeaf0100321023
+
+theorem e24KC2ThetaAboveNode010032103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003210)
+    e24KC2ThetaAboveLeaf0100321030 e24KC2ThetaAboveLeaf0100321031 e24KC2ThetaAboveLeaf0100321032
+      e24KC2ThetaAboveLeaf0100321033
+
+theorem e24KC2ThetaAboveNode010032112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003211)
+    e24KC2ThetaAboveLeaf0100321120 e24KC2ThetaAboveLeaf0100321121 e24KC2ThetaAboveLeaf0100321122
+      e24KC2ThetaAboveLeaf0100321123
+
+theorem e24KC2ThetaAboveNode010032113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003211)
+    e24KC2ThetaAboveLeaf0100321130 e24KC2ThetaAboveLeaf0100321131 e24KC2ThetaAboveLeaf0100321132
+      e24KC2ThetaAboveLeaf0100321133
+
+theorem e24KC2ThetaAboveNode010032120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01003212)
+    e24KC2ThetaAboveLeaf0100321200 e24KC2ThetaAboveLeaf0100321201 e24KC2ThetaAboveLeaf0100321202
+      e24KC2ThetaAboveLeaf0100321203
+
+theorem e24KC2ThetaAboveNode010032121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01003212)
+    e24KC2ThetaAboveLeaf0100321210 e24KC2ThetaAboveLeaf0100321211 e24KC2ThetaAboveLeaf0100321212
+      e24KC2ThetaAboveLeaf0100321213
+
+theorem e24KC2ThetaAboveNode010032122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003212)
+    e24KC2ThetaAboveLeaf0100321220 e24KC2ThetaAboveLeaf0100321221 e24KC2ThetaAboveLeaf0100321222
+      e24KC2ThetaAboveLeaf0100321223
+
+theorem e24KC2ThetaAboveNode010032123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003212)
+    e24KC2ThetaAboveLeaf0100321230 e24KC2ThetaAboveLeaf0100321231 e24KC2ThetaAboveLeaf0100321232
+      e24KC2ThetaAboveLeaf0100321233
+
+theorem e24KC2ThetaAboveNode010032130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01003213)
+    e24KC2ThetaAboveLeaf0100321300 e24KC2ThetaAboveLeaf0100321301 e24KC2ThetaAboveLeaf0100321302
+      e24KC2ThetaAboveLeaf0100321303
+
+theorem e24KC2ThetaAboveNode010032131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01003213)
+    e24KC2ThetaAboveLeaf0100321310 e24KC2ThetaAboveLeaf0100321311 e24KC2ThetaAboveLeaf0100321312
+      e24KC2ThetaAboveLeaf0100321313
+
+theorem e24KC2ThetaAboveNode010032132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003213)
+    e24KC2ThetaAboveLeaf0100321320 e24KC2ThetaAboveLeaf0100321321 e24KC2ThetaAboveLeaf0100321322
+      e24KC2ThetaAboveLeaf0100321323
+
+theorem e24KC2ThetaAboveNode010032133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003213)
+    e24KC2ThetaAboveLeaf0100321330 e24KC2ThetaAboveLeaf0100321331 e24KC2ThetaAboveLeaf0100321332
+      e24KC2ThetaAboveLeaf0100321333
+
+theorem e24KC2ThetaAboveNode010033002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003300)
+    e24KC2ThetaAboveLeaf0100330020 e24KC2ThetaAboveLeaf0100330021 e24KC2ThetaAboveLeaf0100330022
+      e24KC2ThetaAboveLeaf0100330023
+
+theorem e24KC2ThetaAboveNode010033003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003300)
+    e24KC2ThetaAboveLeaf0100330030 e24KC2ThetaAboveLeaf0100330031 e24KC2ThetaAboveLeaf0100330032
+      e24KC2ThetaAboveLeaf0100330033
+
+theorem e24KC2ThetaAboveNode010033012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003301)
+    e24KC2ThetaAboveLeaf0100330120 e24KC2ThetaAboveLeaf0100330121 e24KC2ThetaAboveLeaf0100330122
+      e24KC2ThetaAboveLeaf0100330123
+
+theorem e24KC2ThetaAboveNode010033013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003301)
+    e24KC2ThetaAboveLeaf0100330130 e24KC2ThetaAboveLeaf0100330131 e24KC2ThetaAboveLeaf0100330132
+      e24KC2ThetaAboveLeaf0100330133
+
+theorem e24KC2ThetaAboveNode010033020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01003302)
+    e24KC2ThetaAboveLeaf0100330200 e24KC2ThetaAboveLeaf0100330201 e24KC2ThetaAboveLeaf0100330202
+      e24KC2ThetaAboveLeaf0100330203
+
+theorem e24KC2ThetaAboveNode010033021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01003302)
+    e24KC2ThetaAboveLeaf0100330210 e24KC2ThetaAboveLeaf0100330211 e24KC2ThetaAboveLeaf0100330212
+      e24KC2ThetaAboveLeaf0100330213
+
+theorem e24KC2ThetaAboveNode010033022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003302)
+    e24KC2ThetaAboveLeaf0100330220 e24KC2ThetaAboveLeaf0100330221 e24KC2ThetaAboveLeaf0100330222
+      e24KC2ThetaAboveLeaf0100330223
+
+theorem e24KC2ThetaAboveNode010033023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003302)
+    e24KC2ThetaAboveLeaf0100330230 e24KC2ThetaAboveLeaf0100330231 e24KC2ThetaAboveLeaf0100330232
+      e24KC2ThetaAboveLeaf0100330233
+
+theorem e24KC2ThetaAboveNode010033030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01003303)
+    e24KC2ThetaAboveLeaf0100330300 e24KC2ThetaAboveLeaf0100330301 e24KC2ThetaAboveLeaf0100330302
+      e24KC2ThetaAboveLeaf0100330303
+
+theorem e24KC2ThetaAboveNode010033031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01003303)
+    e24KC2ThetaAboveLeaf0100330310 e24KC2ThetaAboveLeaf0100330311 e24KC2ThetaAboveLeaf0100330312
+      e24KC2ThetaAboveLeaf0100330313
+
+theorem e24KC2ThetaAboveNode010033032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003303)
+    e24KC2ThetaAboveLeaf0100330320 e24KC2ThetaAboveLeaf0100330321 e24KC2ThetaAboveLeaf0100330322
+      e24KC2ThetaAboveLeaf0100330323
+
+theorem e24KC2ThetaAboveNode010033033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003303)
+    e24KC2ThetaAboveLeaf0100330330 e24KC2ThetaAboveLeaf0100330331 e24KC2ThetaAboveLeaf0100330332
+      e24KC2ThetaAboveLeaf0100330333
+
+theorem e24KC2ThetaAboveNode010033102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003310)
+    e24KC2ThetaAboveLeaf0100331020 e24KC2ThetaAboveLeaf0100331021 e24KC2ThetaAboveLeaf0100331022
+      e24KC2ThetaAboveLeaf0100331023
+
+theorem e24KC2ThetaAboveNode010033103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003310)
+    e24KC2ThetaAboveLeaf0100331030 e24KC2ThetaAboveLeaf0100331031 e24KC2ThetaAboveLeaf0100331032
+      e24KC2ThetaAboveLeaf0100331033
+
+theorem e24KC2ThetaAboveNode010033112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003311)
+    e24KC2ThetaAboveLeaf0100331120 e24KC2ThetaAboveLeaf0100331121 e24KC2ThetaAboveLeaf0100331122
+      e24KC2ThetaAboveLeaf0100331123
+
+theorem e24KC2ThetaAboveNode010033113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003311)
+    e24KC2ThetaAboveLeaf0100331130 e24KC2ThetaAboveLeaf0100331131 e24KC2ThetaAboveLeaf0100331132
+      e24KC2ThetaAboveLeaf0100331133
+
+theorem e24KC2ThetaAboveNode010033120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01003312)
+    e24KC2ThetaAboveLeaf0100331200 e24KC2ThetaAboveLeaf0100331201 e24KC2ThetaAboveLeaf0100331202
+      e24KC2ThetaAboveLeaf0100331203
+
+theorem e24KC2ThetaAboveNode010033121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01003312)
+    e24KC2ThetaAboveLeaf0100331210 e24KC2ThetaAboveLeaf0100331211 e24KC2ThetaAboveLeaf0100331212
+      e24KC2ThetaAboveLeaf0100331213
+
+theorem e24KC2ThetaAboveNode010033122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003312)
+    e24KC2ThetaAboveLeaf0100331220 e24KC2ThetaAboveLeaf0100331221 e24KC2ThetaAboveLeaf0100331222
+      e24KC2ThetaAboveLeaf0100331223
+
+theorem e24KC2ThetaAboveNode010033123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003312)
+    e24KC2ThetaAboveLeaf0100331230 e24KC2ThetaAboveLeaf0100331231 e24KC2ThetaAboveLeaf0100331232
+      e24KC2ThetaAboveLeaf0100331233
+
+theorem e24KC2ThetaAboveNode010033130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01003313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01003313)
+    e24KC2ThetaAboveLeaf0100331300 e24KC2ThetaAboveLeaf0100331301 e24KC2ThetaAboveLeaf0100331302
+      e24KC2ThetaAboveLeaf0100331303
+
+theorem e24KC2ThetaAboveNode010033131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01003313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01003313)
+    e24KC2ThetaAboveLeaf0100331310 e24KC2ThetaAboveLeaf0100331311 e24KC2ThetaAboveLeaf0100331312
+      e24KC2ThetaAboveLeaf0100331313
+
+theorem e24KC2ThetaAboveNode010033132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01003313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01003313)
+    e24KC2ThetaAboveLeaf0100331320 e24KC2ThetaAboveLeaf0100331321 e24KC2ThetaAboveLeaf0100331322
+      e24KC2ThetaAboveLeaf0100331323
+
+theorem e24KC2ThetaAboveNode010033133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01003313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01003313)
+    e24KC2ThetaAboveLeaf0100331330 e24KC2ThetaAboveLeaf0100331331 e24KC2ThetaAboveLeaf0100331332
+      e24KC2ThetaAboveLeaf0100331333
+
+theorem e24KC2ThetaAboveNode010122002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012200)
+    e24KC2ThetaAboveLeaf0101220020 e24KC2ThetaAboveLeaf0101220021 e24KC2ThetaAboveLeaf0101220022
+      e24KC2ThetaAboveLeaf0101220023
+
+theorem e24KC2ThetaAboveNode010122003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012200)
+    e24KC2ThetaAboveLeaf0101220030 e24KC2ThetaAboveLeaf0101220031 e24KC2ThetaAboveLeaf0101220032
+      e24KC2ThetaAboveLeaf0101220033
+
+theorem e24KC2ThetaAboveNode010122012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012201)
+    e24KC2ThetaAboveLeaf0101220120 e24KC2ThetaAboveLeaf0101220121 e24KC2ThetaAboveLeaf0101220122
+      e24KC2ThetaAboveLeaf0101220123
+
+theorem e24KC2ThetaAboveNode010122013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012201)
+    e24KC2ThetaAboveLeaf0101220130 e24KC2ThetaAboveLeaf0101220131 e24KC2ThetaAboveLeaf0101220132
+      e24KC2ThetaAboveLeaf0101220133
+
+theorem e24KC2ThetaAboveNode010122020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01012202)
+    e24KC2ThetaAboveLeaf0101220200 e24KC2ThetaAboveLeaf0101220201 e24KC2ThetaAboveLeaf0101220202
+      e24KC2ThetaAboveLeaf0101220203
+
+theorem e24KC2ThetaAboveNode010122021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01012202)
+    e24KC2ThetaAboveLeaf0101220210 e24KC2ThetaAboveLeaf0101220211 e24KC2ThetaAboveLeaf0101220212
+      e24KC2ThetaAboveLeaf0101220213
+
+theorem e24KC2ThetaAboveNode010122022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012202)
+    e24KC2ThetaAboveLeaf0101220220 e24KC2ThetaAboveLeaf0101220221 e24KC2ThetaAboveLeaf0101220222
+      e24KC2ThetaAboveLeaf0101220223
+
+theorem e24KC2ThetaAboveNode010122023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012202)
+    e24KC2ThetaAboveLeaf0101220230 e24KC2ThetaAboveLeaf0101220231 e24KC2ThetaAboveLeaf0101220232
+      e24KC2ThetaAboveLeaf0101220233
+
+theorem e24KC2ThetaAboveNode010122030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01012203)
+    e24KC2ThetaAboveLeaf0101220300 e24KC2ThetaAboveLeaf0101220301 e24KC2ThetaAboveLeaf0101220302
+      e24KC2ThetaAboveLeaf0101220303
+
+theorem e24KC2ThetaAboveNode010122031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01012203)
+    e24KC2ThetaAboveLeaf0101220310 e24KC2ThetaAboveLeaf0101220311 e24KC2ThetaAboveLeaf0101220312
+      e24KC2ThetaAboveLeaf0101220313
+
+theorem e24KC2ThetaAboveNode010122032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012203)
+    e24KC2ThetaAboveLeaf0101220320 e24KC2ThetaAboveLeaf0101220321 e24KC2ThetaAboveLeaf0101220322
+      e24KC2ThetaAboveLeaf0101220323
+
+theorem e24KC2ThetaAboveNode010122033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012203)
+    e24KC2ThetaAboveLeaf0101220330 e24KC2ThetaAboveLeaf0101220331 e24KC2ThetaAboveLeaf0101220332
+      e24KC2ThetaAboveLeaf0101220333
+
+theorem e24KC2ThetaAboveNode010122102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012210)
+    e24KC2ThetaAboveLeaf0101221020 e24KC2ThetaAboveLeaf0101221021 e24KC2ThetaAboveLeaf0101221022
+      e24KC2ThetaAboveLeaf0101221023
+
+theorem e24KC2ThetaAboveNode010122103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012210)
+    e24KC2ThetaAboveLeaf0101221030 e24KC2ThetaAboveLeaf0101221031 e24KC2ThetaAboveLeaf0101221032
+      e24KC2ThetaAboveLeaf0101221033
+
+theorem e24KC2ThetaAboveNode010122112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012211)
+    e24KC2ThetaAboveLeaf0101221120 e24KC2ThetaAboveLeaf0101221121 e24KC2ThetaAboveLeaf0101221122
+      e24KC2ThetaAboveLeaf0101221123
+
+theorem e24KC2ThetaAboveNode010122113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012211)
+    e24KC2ThetaAboveLeaf0101221130 e24KC2ThetaAboveLeaf0101221131 e24KC2ThetaAboveLeaf0101221132
+      e24KC2ThetaAboveLeaf0101221133
+
+theorem e24KC2ThetaAboveNode010122120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01012212)
+    e24KC2ThetaAboveLeaf0101221200 e24KC2ThetaAboveLeaf0101221201 e24KC2ThetaAboveLeaf0101221202
+      e24KC2ThetaAboveLeaf0101221203
+
+theorem e24KC2ThetaAboveNode010122121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01012212)
+    e24KC2ThetaAboveLeaf0101221210 e24KC2ThetaAboveLeaf0101221211 e24KC2ThetaAboveLeaf0101221212
+      e24KC2ThetaAboveLeaf0101221213
+
+theorem e24KC2ThetaAboveNode010122122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012212)
+    e24KC2ThetaAboveLeaf0101221220 e24KC2ThetaAboveLeaf0101221221 e24KC2ThetaAboveLeaf0101221222
+      e24KC2ThetaAboveLeaf0101221223
+
+theorem e24KC2ThetaAboveNode010122123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012212)
+    e24KC2ThetaAboveLeaf0101221230 e24KC2ThetaAboveLeaf0101221231 e24KC2ThetaAboveLeaf0101221232
+      e24KC2ThetaAboveLeaf0101221233
+
+theorem e24KC2ThetaAboveNode010122130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01012213)
+    e24KC2ThetaAboveLeaf0101221300 e24KC2ThetaAboveLeaf0101221301 e24KC2ThetaAboveLeaf0101221302
+      e24KC2ThetaAboveLeaf0101221303
+
+theorem e24KC2ThetaAboveNode010122131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01012213)
+    e24KC2ThetaAboveLeaf0101221310 e24KC2ThetaAboveLeaf0101221311 e24KC2ThetaAboveLeaf0101221312
+      e24KC2ThetaAboveLeaf0101221313
+
+theorem e24KC2ThetaAboveNode010122132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012213)
+    e24KC2ThetaAboveLeaf0101221320 e24KC2ThetaAboveLeaf0101221321 e24KC2ThetaAboveLeaf0101221322
+      e24KC2ThetaAboveLeaf0101221323
+
+theorem e24KC2ThetaAboveNode010122133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012213)
+    e24KC2ThetaAboveLeaf0101221330 e24KC2ThetaAboveLeaf0101221331 e24KC2ThetaAboveLeaf0101221332
+      e24KC2ThetaAboveLeaf0101221333
+
+theorem e24KC2ThetaAboveNode010123002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012300)
+    e24KC2ThetaAboveLeaf0101230020 e24KC2ThetaAboveLeaf0101230021 e24KC2ThetaAboveLeaf0101230022
+      e24KC2ThetaAboveLeaf0101230023
+
+theorem e24KC2ThetaAboveNode010123003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012300)
+    e24KC2ThetaAboveLeaf0101230030 e24KC2ThetaAboveLeaf0101230031 e24KC2ThetaAboveLeaf0101230032
+      e24KC2ThetaAboveLeaf0101230033
+
+theorem e24KC2ThetaAboveNode010123012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012301)
+    e24KC2ThetaAboveLeaf0101230120 e24KC2ThetaAboveLeaf0101230121 e24KC2ThetaAboveLeaf0101230122
+      e24KC2ThetaAboveLeaf0101230123
+
+theorem e24KC2ThetaAboveNode010123013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012301)
+    e24KC2ThetaAboveLeaf0101230130 e24KC2ThetaAboveLeaf0101230131 e24KC2ThetaAboveLeaf0101230132
+      e24KC2ThetaAboveLeaf0101230133
+
+theorem e24KC2ThetaAboveNode010123020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01012302)
+    e24KC2ThetaAboveLeaf0101230200 e24KC2ThetaAboveLeaf0101230201 e24KC2ThetaAboveLeaf0101230202
+      e24KC2ThetaAboveLeaf0101230203
+
+theorem e24KC2ThetaAboveNode010123021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01012302)
+    e24KC2ThetaAboveLeaf0101230210 e24KC2ThetaAboveLeaf0101230211 e24KC2ThetaAboveLeaf0101230212
+      e24KC2ThetaAboveLeaf0101230213
+
+theorem e24KC2ThetaAboveNode010123022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012302)
+    e24KC2ThetaAboveLeaf0101230220 e24KC2ThetaAboveLeaf0101230221 e24KC2ThetaAboveLeaf0101230222
+      e24KC2ThetaAboveLeaf0101230223
+
+theorem e24KC2ThetaAboveNode010123023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012302)
+    e24KC2ThetaAboveLeaf0101230230 e24KC2ThetaAboveLeaf0101230231 e24KC2ThetaAboveLeaf0101230232
+      e24KC2ThetaAboveLeaf0101230233
+
+theorem e24KC2ThetaAboveNode010123030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01012303)
+    e24KC2ThetaAboveLeaf0101230300 e24KC2ThetaAboveLeaf0101230301 e24KC2ThetaAboveLeaf0101230302
+      e24KC2ThetaAboveLeaf0101230303
+
+theorem e24KC2ThetaAboveNode010123031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01012303)
+    e24KC2ThetaAboveLeaf0101230310 e24KC2ThetaAboveLeaf0101230311 e24KC2ThetaAboveLeaf0101230312
+      e24KC2ThetaAboveLeaf0101230313
+
+theorem e24KC2ThetaAboveNode010123032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012303)
+    e24KC2ThetaAboveLeaf0101230320 e24KC2ThetaAboveLeaf0101230321 e24KC2ThetaAboveLeaf0101230322
+      e24KC2ThetaAboveLeaf0101230323
+
+theorem e24KC2ThetaAboveNode010123033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012303)
+    e24KC2ThetaAboveLeaf0101230330 e24KC2ThetaAboveLeaf0101230331 e24KC2ThetaAboveLeaf0101230332
+      e24KC2ThetaAboveLeaf0101230333
+
+theorem e24KC2ThetaAboveNode010123102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012310)
+    e24KC2ThetaAboveLeaf0101231020 e24KC2ThetaAboveLeaf0101231021 e24KC2ThetaAboveLeaf0101231022
+      e24KC2ThetaAboveLeaf0101231023
+
+theorem e24KC2ThetaAboveNode010123103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012310)
+    e24KC2ThetaAboveLeaf0101231030 e24KC2ThetaAboveLeaf0101231031 e24KC2ThetaAboveLeaf0101231032
+      e24KC2ThetaAboveLeaf0101231033
+
+theorem e24KC2ThetaAboveNode010123112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012311)
+    e24KC2ThetaAboveLeaf0101231120 e24KC2ThetaAboveLeaf0101231121 e24KC2ThetaAboveLeaf0101231122
+      e24KC2ThetaAboveLeaf0101231123
+
+theorem e24KC2ThetaAboveNode010123113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012311)
+    e24KC2ThetaAboveLeaf0101231130 e24KC2ThetaAboveLeaf0101231131 e24KC2ThetaAboveLeaf0101231132
+      e24KC2ThetaAboveLeaf0101231133
+
+theorem e24KC2ThetaAboveNode010123120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01012312)
+    e24KC2ThetaAboveLeaf0101231200 e24KC2ThetaAboveLeaf0101231201 e24KC2ThetaAboveLeaf0101231202
+      e24KC2ThetaAboveLeaf0101231203
+
+theorem e24KC2ThetaAboveNode010123121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01012312)
+    e24KC2ThetaAboveLeaf0101231210 e24KC2ThetaAboveLeaf0101231211 e24KC2ThetaAboveLeaf0101231212
+      e24KC2ThetaAboveLeaf0101231213
+
+theorem e24KC2ThetaAboveNode010123122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012312)
+    e24KC2ThetaAboveLeaf0101231220 e24KC2ThetaAboveLeaf0101231221 e24KC2ThetaAboveLeaf0101231222
+      e24KC2ThetaAboveLeaf0101231223
+
+theorem e24KC2ThetaAboveNode010123123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012312)
+    e24KC2ThetaAboveLeaf0101231230 e24KC2ThetaAboveLeaf0101231231 e24KC2ThetaAboveLeaf0101231232
+      e24KC2ThetaAboveLeaf0101231233
+
+theorem e24KC2ThetaAboveNode010123130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01012313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01012313)
+    e24KC2ThetaAboveLeaf0101231300 e24KC2ThetaAboveLeaf0101231301 e24KC2ThetaAboveLeaf0101231302
+      e24KC2ThetaAboveLeaf0101231303
+
+theorem e24KC2ThetaAboveNode010123131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01012313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01012313)
+    e24KC2ThetaAboveLeaf0101231310 e24KC2ThetaAboveLeaf0101231311 e24KC2ThetaAboveLeaf0101231312
+      e24KC2ThetaAboveLeaf0101231313
+
+theorem e24KC2ThetaAboveNode010123132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01012313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01012313)
+    e24KC2ThetaAboveLeaf0101231320 e24KC2ThetaAboveLeaf0101231321 e24KC2ThetaAboveLeaf0101231322
+      e24KC2ThetaAboveLeaf0101231323
+
+theorem e24KC2ThetaAboveNode010123133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01012313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01012313)
+    e24KC2ThetaAboveLeaf0101231330 e24KC2ThetaAboveLeaf0101231331 e24KC2ThetaAboveLeaf0101231332
+      e24KC2ThetaAboveLeaf0101231333
+
+theorem e24KC2ThetaAboveNode010132002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013200)
+    e24KC2ThetaAboveLeaf0101320020 e24KC2ThetaAboveLeaf0101320021 e24KC2ThetaAboveLeaf0101320022
+      e24KC2ThetaAboveLeaf0101320023
+
+theorem e24KC2ThetaAboveNode010132003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013200)
+    e24KC2ThetaAboveLeaf0101320030 e24KC2ThetaAboveLeaf0101320031 e24KC2ThetaAboveLeaf0101320032
+      e24KC2ThetaAboveLeaf0101320033
+
+theorem e24KC2ThetaAboveNode010132012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013201)
+    e24KC2ThetaAboveLeaf0101320120 e24KC2ThetaAboveLeaf0101320121 e24KC2ThetaAboveLeaf0101320122
+      e24KC2ThetaAboveLeaf0101320123
+
+theorem e24KC2ThetaAboveNode010132013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013201)
+    e24KC2ThetaAboveLeaf0101320130 e24KC2ThetaAboveLeaf0101320131 e24KC2ThetaAboveLeaf0101320132
+      e24KC2ThetaAboveLeaf0101320133
+
+theorem e24KC2ThetaAboveNode010132020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01013202)
+    e24KC2ThetaAboveLeaf0101320200 e24KC2ThetaAboveLeaf0101320201 e24KC2ThetaAboveLeaf0101320202
+      e24KC2ThetaAboveLeaf0101320203
+
+theorem e24KC2ThetaAboveNode010132021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01013202)
+    e24KC2ThetaAboveLeaf0101320210 e24KC2ThetaAboveLeaf0101320211 e24KC2ThetaAboveLeaf0101320212
+      e24KC2ThetaAboveLeaf0101320213
+
+theorem e24KC2ThetaAboveNode010132022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013202)
+    e24KC2ThetaAboveLeaf0101320220 e24KC2ThetaAboveLeaf0101320221 e24KC2ThetaAboveLeaf0101320222
+      e24KC2ThetaAboveLeaf0101320223
+
+theorem e24KC2ThetaAboveNode010132023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013202)
+    e24KC2ThetaAboveLeaf0101320230 e24KC2ThetaAboveLeaf0101320231 e24KC2ThetaAboveLeaf0101320232
+      e24KC2ThetaAboveLeaf0101320233
+
+theorem e24KC2ThetaAboveNode010132030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01013203)
+    e24KC2ThetaAboveLeaf0101320300 e24KC2ThetaAboveLeaf0101320301 e24KC2ThetaAboveLeaf0101320302
+      e24KC2ThetaAboveLeaf0101320303
+
+theorem e24KC2ThetaAboveNode010132031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01013203)
+    e24KC2ThetaAboveLeaf0101320310 e24KC2ThetaAboveLeaf0101320311 e24KC2ThetaAboveLeaf0101320312
+      e24KC2ThetaAboveLeaf0101320313
+
+theorem e24KC2ThetaAboveNode010132032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013203)
+    e24KC2ThetaAboveLeaf0101320320 e24KC2ThetaAboveLeaf0101320321 e24KC2ThetaAboveLeaf0101320322
+      e24KC2ThetaAboveLeaf0101320323
+
+theorem e24KC2ThetaAboveNode010132033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013203)
+    e24KC2ThetaAboveLeaf0101320330 e24KC2ThetaAboveLeaf0101320331 e24KC2ThetaAboveLeaf0101320332
+      e24KC2ThetaAboveLeaf0101320333
+
+theorem e24KC2ThetaAboveNode010132102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013210)
+    e24KC2ThetaAboveLeaf0101321020 e24KC2ThetaAboveLeaf0101321021 e24KC2ThetaAboveLeaf0101321022
+      e24KC2ThetaAboveLeaf0101321023
+
+theorem e24KC2ThetaAboveNode010132103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013210)
+    e24KC2ThetaAboveLeaf0101321030 e24KC2ThetaAboveLeaf0101321031 e24KC2ThetaAboveLeaf0101321032
+      e24KC2ThetaAboveLeaf0101321033
+
+theorem e24KC2ThetaAboveNode010132112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013211)
+    e24KC2ThetaAboveLeaf0101321120 e24KC2ThetaAboveLeaf0101321121 e24KC2ThetaAboveLeaf0101321122
+      e24KC2ThetaAboveLeaf0101321123
+
+theorem e24KC2ThetaAboveNode010132113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013211)
+    e24KC2ThetaAboveLeaf0101321130 e24KC2ThetaAboveLeaf0101321131 e24KC2ThetaAboveLeaf0101321132
+      e24KC2ThetaAboveLeaf0101321133
+
+theorem e24KC2ThetaAboveNode010132120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01013212)
+    e24KC2ThetaAboveLeaf0101321200 e24KC2ThetaAboveLeaf0101321201 e24KC2ThetaAboveLeaf0101321202
+      e24KC2ThetaAboveLeaf0101321203
+
+theorem e24KC2ThetaAboveNode010132121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01013212)
+    e24KC2ThetaAboveLeaf0101321210 e24KC2ThetaAboveLeaf0101321211 e24KC2ThetaAboveLeaf0101321212
+      e24KC2ThetaAboveLeaf0101321213
+
+theorem e24KC2ThetaAboveNode010132122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013212)
+    e24KC2ThetaAboveLeaf0101321220 e24KC2ThetaAboveLeaf0101321221 e24KC2ThetaAboveLeaf0101321222
+      e24KC2ThetaAboveLeaf0101321223
+
+theorem e24KC2ThetaAboveNode010132123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013212)
+    e24KC2ThetaAboveLeaf0101321230 e24KC2ThetaAboveLeaf0101321231 e24KC2ThetaAboveLeaf0101321232
+      e24KC2ThetaAboveLeaf0101321233
+
+theorem e24KC2ThetaAboveNode010132130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01013213)
+    e24KC2ThetaAboveLeaf0101321300 e24KC2ThetaAboveLeaf0101321301 e24KC2ThetaAboveLeaf0101321302
+      e24KC2ThetaAboveLeaf0101321303
+
+theorem e24KC2ThetaAboveNode010132131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01013213)
+    e24KC2ThetaAboveLeaf0101321310 e24KC2ThetaAboveLeaf0101321311 e24KC2ThetaAboveLeaf0101321312
+      e24KC2ThetaAboveLeaf0101321313
+
+theorem e24KC2ThetaAboveNode010132132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013213)
+    e24KC2ThetaAboveLeaf0101321320 e24KC2ThetaAboveLeaf0101321321 e24KC2ThetaAboveLeaf0101321322
+      e24KC2ThetaAboveLeaf0101321323
+
+theorem e24KC2ThetaAboveNode010132133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013213)
+    e24KC2ThetaAboveLeaf0101321330 e24KC2ThetaAboveLeaf0101321331 e24KC2ThetaAboveLeaf0101321332
+      e24KC2ThetaAboveLeaf0101321333
+
+theorem e24KC2ThetaAboveNode010133002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013300)
+    e24KC2ThetaAboveLeaf0101330020 e24KC2ThetaAboveLeaf0101330021 e24KC2ThetaAboveLeaf0101330022
+      e24KC2ThetaAboveLeaf0101330023
+
+theorem e24KC2ThetaAboveNode010133003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013300)
+    e24KC2ThetaAboveLeaf0101330030 e24KC2ThetaAboveLeaf0101330031 e24KC2ThetaAboveLeaf0101330032
+      e24KC2ThetaAboveLeaf0101330033
+
+theorem e24KC2ThetaAboveNode010133012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013301)
+    e24KC2ThetaAboveLeaf0101330120 e24KC2ThetaAboveLeaf0101330121 e24KC2ThetaAboveLeaf0101330122
+      e24KC2ThetaAboveLeaf0101330123
+
+theorem e24KC2ThetaAboveNode010133013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013301)
+    e24KC2ThetaAboveLeaf0101330130 e24KC2ThetaAboveLeaf0101330131 e24KC2ThetaAboveLeaf0101330132
+      e24KC2ThetaAboveLeaf0101330133
+
+theorem e24KC2ThetaAboveNode010133020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01013302)
+    e24KC2ThetaAboveLeaf0101330200 e24KC2ThetaAboveLeaf0101330201 e24KC2ThetaAboveLeaf0101330202
+      e24KC2ThetaAboveLeaf0101330203
+
+theorem e24KC2ThetaAboveNode010133021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01013302)
+    e24KC2ThetaAboveLeaf0101330210 e24KC2ThetaAboveLeaf0101330211 e24KC2ThetaAboveLeaf0101330212
+      e24KC2ThetaAboveLeaf0101330213
+
+theorem e24KC2ThetaAboveNode010133022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013302)
+    e24KC2ThetaAboveLeaf0101330220 e24KC2ThetaAboveLeaf0101330221 e24KC2ThetaAboveLeaf0101330222
+      e24KC2ThetaAboveLeaf0101330223
+
+theorem e24KC2ThetaAboveNode010133023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013302)
+    e24KC2ThetaAboveLeaf0101330230 e24KC2ThetaAboveLeaf0101330231 e24KC2ThetaAboveLeaf0101330232
+      e24KC2ThetaAboveLeaf0101330233
+
+theorem e24KC2ThetaAboveNode010133030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01013303)
+    e24KC2ThetaAboveLeaf0101330300 e24KC2ThetaAboveLeaf0101330301 e24KC2ThetaAboveLeaf0101330302
+      e24KC2ThetaAboveLeaf0101330303
+
+theorem e24KC2ThetaAboveNode010133031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01013303)
+    e24KC2ThetaAboveLeaf0101330310 e24KC2ThetaAboveLeaf0101330311 e24KC2ThetaAboveLeaf0101330312
+      e24KC2ThetaAboveLeaf0101330313
+
+theorem e24KC2ThetaAboveNode010133032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013303)
+    e24KC2ThetaAboveLeaf0101330320 e24KC2ThetaAboveLeaf0101330321 e24KC2ThetaAboveLeaf0101330322
+      e24KC2ThetaAboveLeaf0101330323
+
+theorem e24KC2ThetaAboveNode010133033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013303)
+    e24KC2ThetaAboveLeaf0101330330 e24KC2ThetaAboveLeaf0101330331 e24KC2ThetaAboveLeaf0101330332
+      e24KC2ThetaAboveLeaf0101330333
+
+theorem e24KC2ThetaAboveNode010133102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013310)
+    e24KC2ThetaAboveLeaf0101331020 e24KC2ThetaAboveLeaf0101331021 e24KC2ThetaAboveLeaf0101331022
+      e24KC2ThetaAboveLeaf0101331023
+
+theorem e24KC2ThetaAboveNode010133103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013310)
+    e24KC2ThetaAboveLeaf0101331030 e24KC2ThetaAboveLeaf0101331031 e24KC2ThetaAboveLeaf0101331032
+      e24KC2ThetaAboveLeaf0101331033
+
+theorem e24KC2ThetaAboveNode010133112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013311)
+    e24KC2ThetaAboveLeaf0101331120 e24KC2ThetaAboveLeaf0101331121 e24KC2ThetaAboveLeaf0101331122
+      e24KC2ThetaAboveLeaf0101331123
+
+theorem e24KC2ThetaAboveNode010133113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013311)
+    e24KC2ThetaAboveLeaf0101331130 e24KC2ThetaAboveLeaf0101331131 e24KC2ThetaAboveLeaf0101331132
+      e24KC2ThetaAboveLeaf0101331133
+
+theorem e24KC2ThetaAboveNode010133120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01013312)
+    e24KC2ThetaAboveLeaf0101331200 e24KC2ThetaAboveLeaf0101331201 e24KC2ThetaAboveLeaf0101331202
+      e24KC2ThetaAboveLeaf0101331203
+
+theorem e24KC2ThetaAboveNode010133121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01013312)
+    e24KC2ThetaAboveLeaf0101331210 e24KC2ThetaAboveLeaf0101331211 e24KC2ThetaAboveLeaf0101331212
+      e24KC2ThetaAboveLeaf0101331213
+
+theorem e24KC2ThetaAboveNode010133122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013312)
+    e24KC2ThetaAboveLeaf0101331220 e24KC2ThetaAboveLeaf0101331221 e24KC2ThetaAboveLeaf0101331222
+      e24KC2ThetaAboveLeaf0101331223
+
+theorem e24KC2ThetaAboveNode010133123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013312)
+    e24KC2ThetaAboveLeaf0101331230 e24KC2ThetaAboveLeaf0101331231 e24KC2ThetaAboveLeaf0101331232
+      e24KC2ThetaAboveLeaf0101331233
+
+theorem e24KC2ThetaAboveNode010133130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01013313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01013313)
+    e24KC2ThetaAboveLeaf0101331300 e24KC2ThetaAboveLeaf0101331301 e24KC2ThetaAboveLeaf0101331302
+      e24KC2ThetaAboveLeaf0101331303
+
+theorem e24KC2ThetaAboveNode010133131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01013313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01013313)
+    e24KC2ThetaAboveLeaf0101331310 e24KC2ThetaAboveLeaf0101331311 e24KC2ThetaAboveLeaf0101331312
+      e24KC2ThetaAboveLeaf0101331313
+
+theorem e24KC2ThetaAboveNode010133132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01013313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01013313)
+    e24KC2ThetaAboveLeaf0101331320 e24KC2ThetaAboveLeaf0101331321 e24KC2ThetaAboveLeaf0101331322
+      e24KC2ThetaAboveLeaf0101331323
+
+theorem e24KC2ThetaAboveNode010133133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01013313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01013313)
+    e24KC2ThetaAboveLeaf0101331330 e24KC2ThetaAboveLeaf0101331331 e24KC2ThetaAboveLeaf0101331332
+      e24KC2ThetaAboveLeaf0101331333
+
+theorem e24KC2ThetaAboveNode011022002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102200)
+    e24KC2ThetaAboveLeaf0110220020 e24KC2ThetaAboveLeaf0110220021 e24KC2ThetaAboveLeaf0110220022
+      e24KC2ThetaAboveLeaf0110220023
+
+theorem e24KC2ThetaAboveNode011022003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102200)
+    e24KC2ThetaAboveLeaf0110220030 e24KC2ThetaAboveLeaf0110220031 e24KC2ThetaAboveLeaf0110220032
+      e24KC2ThetaAboveLeaf0110220033
+
+theorem e24KC2ThetaAboveNode011022012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102201)
+    e24KC2ThetaAboveLeaf0110220120 e24KC2ThetaAboveLeaf0110220121 e24KC2ThetaAboveLeaf0110220122
+      e24KC2ThetaAboveLeaf0110220123
+
+theorem e24KC2ThetaAboveNode011022013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102201)
+    e24KC2ThetaAboveLeaf0110220130 e24KC2ThetaAboveLeaf0110220131 e24KC2ThetaAboveLeaf0110220132
+      e24KC2ThetaAboveLeaf0110220133
+
+theorem e24KC2ThetaAboveNode011022020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01102202)
+    e24KC2ThetaAboveLeaf0110220200 e24KC2ThetaAboveLeaf0110220201 e24KC2ThetaAboveLeaf0110220202
+      e24KC2ThetaAboveLeaf0110220203
+
+theorem e24KC2ThetaAboveNode011022021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01102202)
+    e24KC2ThetaAboveLeaf0110220210 e24KC2ThetaAboveLeaf0110220211 e24KC2ThetaAboveLeaf0110220212
+      e24KC2ThetaAboveLeaf0110220213
+
+theorem e24KC2ThetaAboveNode011022022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102202)
+    e24KC2ThetaAboveLeaf0110220220 e24KC2ThetaAboveLeaf0110220221 e24KC2ThetaAboveLeaf0110220222
+      e24KC2ThetaAboveLeaf0110220223
+
+theorem e24KC2ThetaAboveNode011022023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102202)
+    e24KC2ThetaAboveLeaf0110220230 e24KC2ThetaAboveLeaf0110220231 e24KC2ThetaAboveLeaf0110220232
+      e24KC2ThetaAboveLeaf0110220233
+
+theorem e24KC2ThetaAboveNode011022030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01102203)
+    e24KC2ThetaAboveLeaf0110220300 e24KC2ThetaAboveLeaf0110220301 e24KC2ThetaAboveLeaf0110220302
+      e24KC2ThetaAboveLeaf0110220303
+
+theorem e24KC2ThetaAboveNode011022031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01102203)
+    e24KC2ThetaAboveLeaf0110220310 e24KC2ThetaAboveLeaf0110220311 e24KC2ThetaAboveLeaf0110220312
+      e24KC2ThetaAboveLeaf0110220313
+
+theorem e24KC2ThetaAboveNode011022032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102203)
+    e24KC2ThetaAboveLeaf0110220320 e24KC2ThetaAboveLeaf0110220321 e24KC2ThetaAboveLeaf0110220322
+      e24KC2ThetaAboveLeaf0110220323
+
+theorem e24KC2ThetaAboveNode011022033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102203)
+    e24KC2ThetaAboveLeaf0110220330 e24KC2ThetaAboveLeaf0110220331 e24KC2ThetaAboveLeaf0110220332
+      e24KC2ThetaAboveLeaf0110220333
+
+theorem e24KC2ThetaAboveNode011022102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102210)
+    e24KC2ThetaAboveLeaf0110221020 e24KC2ThetaAboveLeaf0110221021 e24KC2ThetaAboveLeaf0110221022
+      e24KC2ThetaAboveLeaf0110221023
+
+theorem e24KC2ThetaAboveNode011022103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102210)
+    e24KC2ThetaAboveLeaf0110221030 e24KC2ThetaAboveLeaf0110221031 e24KC2ThetaAboveLeaf0110221032
+      e24KC2ThetaAboveLeaf0110221033
+
+theorem e24KC2ThetaAboveNode011022112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102211)
+    e24KC2ThetaAboveLeaf0110221120 e24KC2ThetaAboveLeaf0110221121 e24KC2ThetaAboveLeaf0110221122
+      e24KC2ThetaAboveLeaf0110221123
+
+theorem e24KC2ThetaAboveNode011022113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102211)
+    e24KC2ThetaAboveLeaf0110221130 e24KC2ThetaAboveLeaf0110221131 e24KC2ThetaAboveLeaf0110221132
+      e24KC2ThetaAboveLeaf0110221133
+
+theorem e24KC2ThetaAboveNode011022120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01102212)
+    e24KC2ThetaAboveLeaf0110221200 e24KC2ThetaAboveLeaf0110221201 e24KC2ThetaAboveLeaf0110221202
+      e24KC2ThetaAboveLeaf0110221203
+
+theorem e24KC2ThetaAboveNode011022121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01102212)
+    e24KC2ThetaAboveLeaf0110221210 e24KC2ThetaAboveLeaf0110221211 e24KC2ThetaAboveLeaf0110221212
+      e24KC2ThetaAboveLeaf0110221213
+
+theorem e24KC2ThetaAboveNode011022122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102212)
+    e24KC2ThetaAboveLeaf0110221220 e24KC2ThetaAboveLeaf0110221221 e24KC2ThetaAboveLeaf0110221222
+      e24KC2ThetaAboveLeaf0110221223
+
+theorem e24KC2ThetaAboveNode011022123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102212)
+    e24KC2ThetaAboveLeaf0110221230 e24KC2ThetaAboveLeaf0110221231 e24KC2ThetaAboveLeaf0110221232
+      e24KC2ThetaAboveLeaf0110221233
+
+theorem e24KC2ThetaAboveNode011022130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01102213)
+    e24KC2ThetaAboveLeaf0110221300 e24KC2ThetaAboveLeaf0110221301 e24KC2ThetaAboveLeaf0110221302
+      e24KC2ThetaAboveLeaf0110221303
+
+theorem e24KC2ThetaAboveNode011022131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01102213)
+    e24KC2ThetaAboveLeaf0110221310 e24KC2ThetaAboveLeaf0110221311 e24KC2ThetaAboveLeaf0110221312
+      e24KC2ThetaAboveLeaf0110221313
+
+theorem e24KC2ThetaAboveNode011022132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102213)
+    e24KC2ThetaAboveLeaf0110221320 e24KC2ThetaAboveLeaf0110221321 e24KC2ThetaAboveLeaf0110221322
+      e24KC2ThetaAboveLeaf0110221323
+
+theorem e24KC2ThetaAboveNode011022133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102213)
+    e24KC2ThetaAboveLeaf0110221330 e24KC2ThetaAboveLeaf0110221331 e24KC2ThetaAboveLeaf0110221332
+      e24KC2ThetaAboveLeaf0110221333
+
+theorem e24KC2ThetaAboveNode011023002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102300)
+    e24KC2ThetaAboveLeaf0110230020 e24KC2ThetaAboveLeaf0110230021 e24KC2ThetaAboveLeaf0110230022
+      e24KC2ThetaAboveLeaf0110230023
+
+theorem e24KC2ThetaAboveNode011023003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102300)
+    e24KC2ThetaAboveLeaf0110230030 e24KC2ThetaAboveLeaf0110230031 e24KC2ThetaAboveLeaf0110230032
+      e24KC2ThetaAboveLeaf0110230033
+
+theorem e24KC2ThetaAboveNode011023012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102301)
+    e24KC2ThetaAboveLeaf0110230120 e24KC2ThetaAboveLeaf0110230121 e24KC2ThetaAboveLeaf0110230122
+      e24KC2ThetaAboveLeaf0110230123
+
+theorem e24KC2ThetaAboveNode011023013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102301)
+    e24KC2ThetaAboveLeaf0110230130 e24KC2ThetaAboveLeaf0110230131 e24KC2ThetaAboveLeaf0110230132
+      e24KC2ThetaAboveLeaf0110230133
+
+theorem e24KC2ThetaAboveNode011023020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01102302)
+    e24KC2ThetaAboveLeaf0110230200 e24KC2ThetaAboveLeaf0110230201 e24KC2ThetaAboveLeaf0110230202
+      e24KC2ThetaAboveLeaf0110230203
+
+theorem e24KC2ThetaAboveNode011023021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01102302)
+    e24KC2ThetaAboveLeaf0110230210 e24KC2ThetaAboveLeaf0110230211 e24KC2ThetaAboveLeaf0110230212
+      e24KC2ThetaAboveLeaf0110230213
+
+theorem e24KC2ThetaAboveNode011023022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102302)
+    e24KC2ThetaAboveLeaf0110230220 e24KC2ThetaAboveLeaf0110230221 e24KC2ThetaAboveLeaf0110230222
+      e24KC2ThetaAboveLeaf0110230223
+
+theorem e24KC2ThetaAboveNode011023023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102302)
+    e24KC2ThetaAboveLeaf0110230230 e24KC2ThetaAboveLeaf0110230231 e24KC2ThetaAboveLeaf0110230232
+      e24KC2ThetaAboveLeaf0110230233
+
+theorem e24KC2ThetaAboveNode011023030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01102303)
+    e24KC2ThetaAboveLeaf0110230300 e24KC2ThetaAboveLeaf0110230301 e24KC2ThetaAboveLeaf0110230302
+      e24KC2ThetaAboveLeaf0110230303
+
+theorem e24KC2ThetaAboveNode011023031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01102303)
+    e24KC2ThetaAboveLeaf0110230310 e24KC2ThetaAboveLeaf0110230311 e24KC2ThetaAboveLeaf0110230312
+      e24KC2ThetaAboveLeaf0110230313
+
+theorem e24KC2ThetaAboveNode011023032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102303)
+    e24KC2ThetaAboveLeaf0110230320 e24KC2ThetaAboveLeaf0110230321 e24KC2ThetaAboveLeaf0110230322
+      e24KC2ThetaAboveLeaf0110230323
+
+theorem e24KC2ThetaAboveNode011023033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102303)
+    e24KC2ThetaAboveLeaf0110230330 e24KC2ThetaAboveLeaf0110230331 e24KC2ThetaAboveLeaf0110230332
+      e24KC2ThetaAboveLeaf0110230333
+
+theorem e24KC2ThetaAboveNode011023102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102310)
+    e24KC2ThetaAboveLeaf0110231020 e24KC2ThetaAboveLeaf0110231021 e24KC2ThetaAboveLeaf0110231022
+      e24KC2ThetaAboveLeaf0110231023
+
+theorem e24KC2ThetaAboveNode011023103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102310)
+    e24KC2ThetaAboveLeaf0110231030 e24KC2ThetaAboveLeaf0110231031 e24KC2ThetaAboveLeaf0110231032
+      e24KC2ThetaAboveLeaf0110231033
+
+theorem e24KC2ThetaAboveNode011023112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102311)
+    e24KC2ThetaAboveLeaf0110231120 e24KC2ThetaAboveLeaf0110231121 e24KC2ThetaAboveLeaf0110231122
+      e24KC2ThetaAboveLeaf0110231123
+
+theorem e24KC2ThetaAboveNode011023113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102311)
+    e24KC2ThetaAboveLeaf0110231130 e24KC2ThetaAboveLeaf0110231131 e24KC2ThetaAboveLeaf0110231132
+      e24KC2ThetaAboveLeaf0110231133
+
+theorem e24KC2ThetaAboveNode011023120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01102312)
+    e24KC2ThetaAboveLeaf0110231200 e24KC2ThetaAboveLeaf0110231201 e24KC2ThetaAboveLeaf0110231202
+      e24KC2ThetaAboveLeaf0110231203
+
+theorem e24KC2ThetaAboveNode011023121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01102312)
+    e24KC2ThetaAboveLeaf0110231210 e24KC2ThetaAboveLeaf0110231211 e24KC2ThetaAboveLeaf0110231212
+      e24KC2ThetaAboveLeaf0110231213
+
+theorem e24KC2ThetaAboveNode011023122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102312)
+    e24KC2ThetaAboveLeaf0110231220 e24KC2ThetaAboveLeaf0110231221 e24KC2ThetaAboveLeaf0110231222
+      e24KC2ThetaAboveLeaf0110231223
+
+theorem e24KC2ThetaAboveNode011023123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102312)
+    e24KC2ThetaAboveLeaf0110231230 e24KC2ThetaAboveLeaf0110231231 e24KC2ThetaAboveLeaf0110231232
+      e24KC2ThetaAboveLeaf0110231233
+
+theorem e24KC2ThetaAboveNode011023130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01102313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01102313)
+    e24KC2ThetaAboveLeaf0110231300 e24KC2ThetaAboveLeaf0110231301 e24KC2ThetaAboveLeaf0110231302
+      e24KC2ThetaAboveLeaf0110231303
+
+theorem e24KC2ThetaAboveNode011023131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01102313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01102313)
+    e24KC2ThetaAboveLeaf0110231310 e24KC2ThetaAboveLeaf0110231311 e24KC2ThetaAboveLeaf0110231312
+      e24KC2ThetaAboveLeaf0110231313
+
+theorem e24KC2ThetaAboveNode011023132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01102313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01102313)
+    e24KC2ThetaAboveLeaf0110231320 e24KC2ThetaAboveLeaf0110231321 e24KC2ThetaAboveLeaf0110231322
+      e24KC2ThetaAboveLeaf0110231323
+
+theorem e24KC2ThetaAboveNode011023133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01102313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01102313)
+    e24KC2ThetaAboveLeaf0110231330 e24KC2ThetaAboveLeaf0110231331 e24KC2ThetaAboveLeaf0110231332
+      e24KC2ThetaAboveLeaf0110231333
+
+theorem e24KC2ThetaAboveNode011032002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103200)
+    e24KC2ThetaAboveLeaf0110320020 e24KC2ThetaAboveLeaf0110320021 e24KC2ThetaAboveLeaf0110320022
+      e24KC2ThetaAboveLeaf0110320023
+
+theorem e24KC2ThetaAboveNode011032003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103200)
+    e24KC2ThetaAboveLeaf0110320030 e24KC2ThetaAboveLeaf0110320031 e24KC2ThetaAboveLeaf0110320032
+      e24KC2ThetaAboveLeaf0110320033
+
+theorem e24KC2ThetaAboveNode011032012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103201)
+    e24KC2ThetaAboveLeaf0110320120 e24KC2ThetaAboveLeaf0110320121 e24KC2ThetaAboveLeaf0110320122
+      e24KC2ThetaAboveLeaf0110320123
+
+theorem e24KC2ThetaAboveNode011032013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103201)
+    e24KC2ThetaAboveLeaf0110320130 e24KC2ThetaAboveLeaf0110320131 e24KC2ThetaAboveLeaf0110320132
+      e24KC2ThetaAboveLeaf0110320133
+
+theorem e24KC2ThetaAboveNode011032020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01103202)
+    e24KC2ThetaAboveLeaf0110320200 e24KC2ThetaAboveLeaf0110320201 e24KC2ThetaAboveLeaf0110320202
+      e24KC2ThetaAboveLeaf0110320203
+
+theorem e24KC2ThetaAboveNode011032021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01103202)
+    e24KC2ThetaAboveLeaf0110320210 e24KC2ThetaAboveLeaf0110320211 e24KC2ThetaAboveLeaf0110320212
+      e24KC2ThetaAboveLeaf0110320213
+
+theorem e24KC2ThetaAboveNode011032022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103202)
+    e24KC2ThetaAboveLeaf0110320220 e24KC2ThetaAboveLeaf0110320221 e24KC2ThetaAboveLeaf0110320222
+      e24KC2ThetaAboveLeaf0110320223
+
+theorem e24KC2ThetaAboveNode011032023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103202)
+    e24KC2ThetaAboveLeaf0110320230 e24KC2ThetaAboveLeaf0110320231 e24KC2ThetaAboveLeaf0110320232
+      e24KC2ThetaAboveLeaf0110320233
+
+theorem e24KC2ThetaAboveNode011032030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01103203)
+    e24KC2ThetaAboveLeaf0110320300 e24KC2ThetaAboveLeaf0110320301 e24KC2ThetaAboveLeaf0110320302
+      e24KC2ThetaAboveLeaf0110320303
+
+theorem e24KC2ThetaAboveNode011032031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01103203)
+    e24KC2ThetaAboveLeaf0110320310 e24KC2ThetaAboveLeaf0110320311 e24KC2ThetaAboveLeaf0110320312
+      e24KC2ThetaAboveLeaf0110320313
+
+theorem e24KC2ThetaAboveNode011032032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103203)
+    e24KC2ThetaAboveLeaf0110320320 e24KC2ThetaAboveLeaf0110320321 e24KC2ThetaAboveLeaf0110320322
+      e24KC2ThetaAboveLeaf0110320323
+
+theorem e24KC2ThetaAboveNode011032033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103203)
+    e24KC2ThetaAboveLeaf0110320330 e24KC2ThetaAboveLeaf0110320331 e24KC2ThetaAboveLeaf0110320332
+      e24KC2ThetaAboveLeaf0110320333
+
+theorem e24KC2ThetaAboveNode011032102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103210)
+    e24KC2ThetaAboveLeaf0110321020 e24KC2ThetaAboveLeaf0110321021 e24KC2ThetaAboveLeaf0110321022
+      e24KC2ThetaAboveLeaf0110321023
+
+theorem e24KC2ThetaAboveNode011032103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103210)
+    e24KC2ThetaAboveLeaf0110321030 e24KC2ThetaAboveLeaf0110321031 e24KC2ThetaAboveLeaf0110321032
+      e24KC2ThetaAboveLeaf0110321033
+
+theorem e24KC2ThetaAboveNode011032112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103211)
+    e24KC2ThetaAboveLeaf0110321120 e24KC2ThetaAboveLeaf0110321121 e24KC2ThetaAboveLeaf0110321122
+      e24KC2ThetaAboveLeaf0110321123
+
+theorem e24KC2ThetaAboveNode011032113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103211)
+    e24KC2ThetaAboveLeaf0110321130 e24KC2ThetaAboveLeaf0110321131 e24KC2ThetaAboveLeaf0110321132
+      e24KC2ThetaAboveLeaf0110321133
+
+theorem e24KC2ThetaAboveNode011032120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01103212)
+    e24KC2ThetaAboveLeaf0110321200 e24KC2ThetaAboveLeaf0110321201 e24KC2ThetaAboveLeaf0110321202
+      e24KC2ThetaAboveLeaf0110321203
+
+theorem e24KC2ThetaAboveNode011032121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01103212)
+    e24KC2ThetaAboveLeaf0110321210 e24KC2ThetaAboveLeaf0110321211 e24KC2ThetaAboveLeaf0110321212
+      e24KC2ThetaAboveLeaf0110321213
+
+theorem e24KC2ThetaAboveNode011032122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103212)
+    e24KC2ThetaAboveLeaf0110321220 e24KC2ThetaAboveLeaf0110321221 e24KC2ThetaAboveLeaf0110321222
+      e24KC2ThetaAboveLeaf0110321223
+
+theorem e24KC2ThetaAboveNode011032123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103212)
+    e24KC2ThetaAboveLeaf0110321230 e24KC2ThetaAboveLeaf0110321231 e24KC2ThetaAboveLeaf0110321232
+      e24KC2ThetaAboveLeaf0110321233
+
+theorem e24KC2ThetaAboveNode011032130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01103213)
+    e24KC2ThetaAboveLeaf0110321300 e24KC2ThetaAboveLeaf0110321301 e24KC2ThetaAboveLeaf0110321302
+      e24KC2ThetaAboveLeaf0110321303
+
+theorem e24KC2ThetaAboveNode011032131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01103213)
+    e24KC2ThetaAboveLeaf0110321310 e24KC2ThetaAboveLeaf0110321311 e24KC2ThetaAboveLeaf0110321312
+      e24KC2ThetaAboveLeaf0110321313
+
+theorem e24KC2ThetaAboveNode011032132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103213)
+    e24KC2ThetaAboveLeaf0110321320 e24KC2ThetaAboveLeaf0110321321 e24KC2ThetaAboveLeaf0110321322
+      e24KC2ThetaAboveLeaf0110321323
+
+theorem e24KC2ThetaAboveNode011032133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103213)
+    e24KC2ThetaAboveLeaf0110321330 e24KC2ThetaAboveLeaf0110321331 e24KC2ThetaAboveLeaf0110321332
+      e24KC2ThetaAboveLeaf0110321333
+
+theorem e24KC2ThetaAboveNode011033002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103300)
+    e24KC2ThetaAboveLeaf0110330020 e24KC2ThetaAboveLeaf0110330021 e24KC2ThetaAboveLeaf0110330022
+      e24KC2ThetaAboveLeaf0110330023
+
+theorem e24KC2ThetaAboveNode011033003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103300)
+    e24KC2ThetaAboveLeaf0110330030 e24KC2ThetaAboveLeaf0110330031 e24KC2ThetaAboveLeaf0110330032
+      e24KC2ThetaAboveLeaf0110330033
+
+theorem e24KC2ThetaAboveNode011033012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103301)
+    e24KC2ThetaAboveLeaf0110330120 e24KC2ThetaAboveLeaf0110330121 e24KC2ThetaAboveLeaf0110330122
+      e24KC2ThetaAboveLeaf0110330123
+
+theorem e24KC2ThetaAboveNode011033013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103301)
+    e24KC2ThetaAboveLeaf0110330130 e24KC2ThetaAboveLeaf0110330131 e24KC2ThetaAboveLeaf0110330132
+      e24KC2ThetaAboveLeaf0110330133
+
+theorem e24KC2ThetaAboveNode011033020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01103302)
+    e24KC2ThetaAboveLeaf0110330200 e24KC2ThetaAboveLeaf0110330201 e24KC2ThetaAboveLeaf0110330202
+      e24KC2ThetaAboveLeaf0110330203
+
+theorem e24KC2ThetaAboveNode011033021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01103302)
+    e24KC2ThetaAboveLeaf0110330210 e24KC2ThetaAboveLeaf0110330211 e24KC2ThetaAboveLeaf0110330212
+      e24KC2ThetaAboveLeaf0110330213
+
+theorem e24KC2ThetaAboveNode011033022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103302)
+    e24KC2ThetaAboveLeaf0110330220 e24KC2ThetaAboveLeaf0110330221 e24KC2ThetaAboveLeaf0110330222
+      e24KC2ThetaAboveLeaf0110330223
+
+theorem e24KC2ThetaAboveNode011033023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103302)
+    e24KC2ThetaAboveLeaf0110330230 e24KC2ThetaAboveLeaf0110330231 e24KC2ThetaAboveLeaf0110330232
+      e24KC2ThetaAboveLeaf0110330233
+
+theorem e24KC2ThetaAboveNode011033030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01103303)
+    e24KC2ThetaAboveLeaf0110330300 e24KC2ThetaAboveLeaf0110330301 e24KC2ThetaAboveLeaf0110330302
+      e24KC2ThetaAboveLeaf0110330303
+
+theorem e24KC2ThetaAboveNode011033031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01103303)
+    e24KC2ThetaAboveLeaf0110330310 e24KC2ThetaAboveLeaf0110330311 e24KC2ThetaAboveLeaf0110330312
+      e24KC2ThetaAboveLeaf0110330313
+
+theorem e24KC2ThetaAboveNode011033032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103303)
+    e24KC2ThetaAboveLeaf0110330320 e24KC2ThetaAboveLeaf0110330321 e24KC2ThetaAboveLeaf0110330322
+      e24KC2ThetaAboveLeaf0110330323
+
+theorem e24KC2ThetaAboveNode011033033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103303)
+    e24KC2ThetaAboveLeaf0110330330 e24KC2ThetaAboveLeaf0110330331 e24KC2ThetaAboveLeaf0110330332
+      e24KC2ThetaAboveLeaf0110330333
+
+theorem e24KC2ThetaAboveNode011033102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103310)
+    e24KC2ThetaAboveLeaf0110331020 e24KC2ThetaAboveLeaf0110331021 e24KC2ThetaAboveLeaf0110331022
+      e24KC2ThetaAboveLeaf0110331023
+
+theorem e24KC2ThetaAboveNode011033103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103310)
+    e24KC2ThetaAboveLeaf0110331030 e24KC2ThetaAboveLeaf0110331031 e24KC2ThetaAboveLeaf0110331032
+      e24KC2ThetaAboveLeaf0110331033
+
+theorem e24KC2ThetaAboveNode011033112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103311)
+    e24KC2ThetaAboveLeaf0110331120 e24KC2ThetaAboveLeaf0110331121 e24KC2ThetaAboveLeaf0110331122
+      e24KC2ThetaAboveLeaf0110331123
+
+theorem e24KC2ThetaAboveNode011033113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103311)
+    e24KC2ThetaAboveLeaf0110331130 e24KC2ThetaAboveLeaf0110331131 e24KC2ThetaAboveLeaf0110331132
+      e24KC2ThetaAboveLeaf0110331133
+
+theorem e24KC2ThetaAboveNode011033120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01103312)
+    e24KC2ThetaAboveLeaf0110331200 e24KC2ThetaAboveLeaf0110331201 e24KC2ThetaAboveLeaf0110331202
+      e24KC2ThetaAboveLeaf0110331203
+
+theorem e24KC2ThetaAboveNode011033121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01103312)
+    e24KC2ThetaAboveLeaf0110331210 e24KC2ThetaAboveLeaf0110331211 e24KC2ThetaAboveLeaf0110331212
+      e24KC2ThetaAboveLeaf0110331213
+
+theorem e24KC2ThetaAboveNode011033122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103312)
+    e24KC2ThetaAboveLeaf0110331220 e24KC2ThetaAboveLeaf0110331221 e24KC2ThetaAboveLeaf0110331222
+      e24KC2ThetaAboveLeaf0110331223
+
+theorem e24KC2ThetaAboveNode011033123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103312)
+    e24KC2ThetaAboveLeaf0110331230 e24KC2ThetaAboveLeaf0110331231 e24KC2ThetaAboveLeaf0110331232
+      e24KC2ThetaAboveLeaf0110331233
+
+theorem e24KC2ThetaAboveNode011033130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01103313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01103313)
+    e24KC2ThetaAboveLeaf0110331300 e24KC2ThetaAboveLeaf0110331301 e24KC2ThetaAboveLeaf0110331302
+      e24KC2ThetaAboveLeaf0110331303
+
+theorem e24KC2ThetaAboveNode011033131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01103313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01103313)
+    e24KC2ThetaAboveLeaf0110331310 e24KC2ThetaAboveLeaf0110331311 e24KC2ThetaAboveLeaf0110331312
+      e24KC2ThetaAboveLeaf0110331313
+
+theorem e24KC2ThetaAboveNode011033132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01103313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01103313)
+    e24KC2ThetaAboveLeaf0110331320 e24KC2ThetaAboveLeaf0110331321 e24KC2ThetaAboveLeaf0110331322
+      e24KC2ThetaAboveLeaf0110331323
+
+theorem e24KC2ThetaAboveNode011033133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01103313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01103313)
+    e24KC2ThetaAboveLeaf0110331330 e24KC2ThetaAboveLeaf0110331331 e24KC2ThetaAboveLeaf0110331332
+      e24KC2ThetaAboveLeaf0110331333
+
+theorem e24KC2ThetaAboveNode011122002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112200)
+    e24KC2ThetaAboveLeaf0111220020 e24KC2ThetaAboveLeaf0111220021 e24KC2ThetaAboveLeaf0111220022
+      e24KC2ThetaAboveLeaf0111220023
+
+theorem e24KC2ThetaAboveNode011122003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112200)
+    e24KC2ThetaAboveLeaf0111220030 e24KC2ThetaAboveLeaf0111220031 e24KC2ThetaAboveLeaf0111220032
+      e24KC2ThetaAboveLeaf0111220033
+
+theorem e24KC2ThetaAboveNode011122012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112201)
+    e24KC2ThetaAboveLeaf0111220120 e24KC2ThetaAboveLeaf0111220121 e24KC2ThetaAboveLeaf0111220122
+      e24KC2ThetaAboveLeaf0111220123
+
+theorem e24KC2ThetaAboveNode011122013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112201)
+    e24KC2ThetaAboveLeaf0111220130 e24KC2ThetaAboveLeaf0111220131 e24KC2ThetaAboveLeaf0111220132
+      e24KC2ThetaAboveLeaf0111220133
+
+theorem e24KC2ThetaAboveNode011122020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01112202)
+    e24KC2ThetaAboveLeaf0111220200 e24KC2ThetaAboveLeaf0111220201 e24KC2ThetaAboveLeaf0111220202
+      e24KC2ThetaAboveLeaf0111220203
+
+theorem e24KC2ThetaAboveNode011122021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01112202)
+    e24KC2ThetaAboveLeaf0111220210 e24KC2ThetaAboveLeaf0111220211 e24KC2ThetaAboveLeaf0111220212
+      e24KC2ThetaAboveLeaf0111220213
+
+theorem e24KC2ThetaAboveNode011122022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112202)
+    e24KC2ThetaAboveLeaf0111220220 e24KC2ThetaAboveLeaf0111220221 e24KC2ThetaAboveLeaf0111220222
+      e24KC2ThetaAboveLeaf0111220223
+
+theorem e24KC2ThetaAboveNode011122023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112202)
+    e24KC2ThetaAboveLeaf0111220230 e24KC2ThetaAboveLeaf0111220231 e24KC2ThetaAboveLeaf0111220232
+      e24KC2ThetaAboveLeaf0111220233
+
+theorem e24KC2ThetaAboveNode011122030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01112203)
+    e24KC2ThetaAboveLeaf0111220300 e24KC2ThetaAboveLeaf0111220301 e24KC2ThetaAboveLeaf0111220302
+      e24KC2ThetaAboveLeaf0111220303
+
+theorem e24KC2ThetaAboveNode011122031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01112203)
+    e24KC2ThetaAboveLeaf0111220310 e24KC2ThetaAboveLeaf0111220311 e24KC2ThetaAboveLeaf0111220312
+      e24KC2ThetaAboveLeaf0111220313
+
+theorem e24KC2ThetaAboveNode011122032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112203)
+    e24KC2ThetaAboveLeaf0111220320 e24KC2ThetaAboveLeaf0111220321 e24KC2ThetaAboveLeaf0111220322
+      e24KC2ThetaAboveLeaf0111220323
+
+theorem e24KC2ThetaAboveNode011122033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112203)
+    e24KC2ThetaAboveLeaf0111220330 e24KC2ThetaAboveLeaf0111220331 e24KC2ThetaAboveLeaf0111220332
+      e24KC2ThetaAboveLeaf0111220333
+
+theorem e24KC2ThetaAboveNode011122102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112210)
+    e24KC2ThetaAboveLeaf0111221020 e24KC2ThetaAboveLeaf0111221021 e24KC2ThetaAboveLeaf0111221022
+      e24KC2ThetaAboveLeaf0111221023
+
+theorem e24KC2ThetaAboveNode011122103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112210) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112210)
+    e24KC2ThetaAboveLeaf0111221030 e24KC2ThetaAboveLeaf0111221031 e24KC2ThetaAboveLeaf0111221032
+      e24KC2ThetaAboveLeaf0111221033
+
+theorem e24KC2ThetaAboveNode011122112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112211)
+    e24KC2ThetaAboveLeaf0111221120 e24KC2ThetaAboveLeaf0111221121 e24KC2ThetaAboveLeaf0111221122
+      e24KC2ThetaAboveLeaf0111221123
+
+theorem e24KC2ThetaAboveNode011122113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112211) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112211)
+    e24KC2ThetaAboveLeaf0111221130 e24KC2ThetaAboveLeaf0111221131 e24KC2ThetaAboveLeaf0111221132
+      e24KC2ThetaAboveLeaf0111221133
+
+theorem e24KC2ThetaAboveNode011122120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01112212)
+    e24KC2ThetaAboveLeaf0111221200 e24KC2ThetaAboveLeaf0111221201 e24KC2ThetaAboveLeaf0111221202
+      e24KC2ThetaAboveLeaf0111221203
+
+theorem e24KC2ThetaAboveNode011122121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01112212)
+    e24KC2ThetaAboveLeaf0111221210 e24KC2ThetaAboveLeaf0111221211 e24KC2ThetaAboveLeaf0111221212
+      e24KC2ThetaAboveLeaf0111221213
+
+theorem e24KC2ThetaAboveNode011122122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112212)
+    e24KC2ThetaAboveLeaf0111221220 e24KC2ThetaAboveLeaf0111221221 e24KC2ThetaAboveLeaf0111221222
+      e24KC2ThetaAboveLeaf0111221223
+
+theorem e24KC2ThetaAboveNode011122123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112212)
+    e24KC2ThetaAboveLeaf0111221230 e24KC2ThetaAboveLeaf0111221231 e24KC2ThetaAboveLeaf0111221232
+      e24KC2ThetaAboveLeaf0111221233
+
+theorem e24KC2ThetaAboveNode011122130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01112213)
+    e24KC2ThetaAboveLeaf0111221300 e24KC2ThetaAboveLeaf0111221301 e24KC2ThetaAboveLeaf0111221302
+      e24KC2ThetaAboveLeaf0111221303
+
+theorem e24KC2ThetaAboveNode011122131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01112213)
+    e24KC2ThetaAboveLeaf0111221310 e24KC2ThetaAboveLeaf0111221311 e24KC2ThetaAboveLeaf0111221312
+      e24KC2ThetaAboveLeaf0111221313
+
+theorem e24KC2ThetaAboveNode011122132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112213)
+    e24KC2ThetaAboveLeaf0111221320 e24KC2ThetaAboveLeaf0111221321 e24KC2ThetaAboveLeaf0111221322
+      e24KC2ThetaAboveLeaf0111221323
+
+theorem e24KC2ThetaAboveNode011122133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112213)
+    e24KC2ThetaAboveLeaf0111221330 e24KC2ThetaAboveLeaf0111221331 e24KC2ThetaAboveLeaf0111221332
+      e24KC2ThetaAboveLeaf0111221333
+
+theorem e24KC2ThetaAboveNode011123002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112300)
+    e24KC2ThetaAboveLeaf0111230020 e24KC2ThetaAboveLeaf0111230021 e24KC2ThetaAboveLeaf0111230022
+      e24KC2ThetaAboveLeaf0111230023
+
+theorem e24KC2ThetaAboveNode011123003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112300) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112300)
+    e24KC2ThetaAboveLeaf0111230030 e24KC2ThetaAboveLeaf0111230031 e24KC2ThetaAboveLeaf0111230032
+      e24KC2ThetaAboveLeaf0111230033
+
+theorem e24KC2ThetaAboveNode011123012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112301)
+    e24KC2ThetaAboveLeaf0111230120 e24KC2ThetaAboveLeaf0111230121 e24KC2ThetaAboveLeaf0111230122
+      e24KC2ThetaAboveLeaf0111230123
+
+theorem e24KC2ThetaAboveNode011123013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112301) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112301)
+    e24KC2ThetaAboveLeaf0111230130 e24KC2ThetaAboveLeaf0111230131 e24KC2ThetaAboveLeaf0111230132
+      e24KC2ThetaAboveLeaf0111230133
+
+theorem e24KC2ThetaAboveNode011123020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01112302)
+    e24KC2ThetaAboveLeaf0111230200 e24KC2ThetaAboveLeaf0111230201 e24KC2ThetaAboveLeaf0111230202
+      e24KC2ThetaAboveLeaf0111230203
+
+theorem e24KC2ThetaAboveNode011123021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01112302)
+    e24KC2ThetaAboveLeaf0111230210 e24KC2ThetaAboveLeaf0111230211 e24KC2ThetaAboveLeaf0111230212
+      e24KC2ThetaAboveLeaf0111230213
+
+theorem e24KC2ThetaAboveNode011123022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112302)
+    e24KC2ThetaAboveLeaf0111230220 e24KC2ThetaAboveLeaf0111230221 e24KC2ThetaAboveLeaf0111230222
+      e24KC2ThetaAboveLeaf0111230223
+
+theorem e24KC2ThetaAboveNode011123023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112302)
+    e24KC2ThetaAboveLeaf0111230230 e24KC2ThetaAboveLeaf0111230231 e24KC2ThetaAboveLeaf0111230232
+      e24KC2ThetaAboveLeaf0111230233
+
+theorem e24KC2ThetaAboveNode011123030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01112303)
+    e24KC2ThetaAboveLeaf0111230300 e24KC2ThetaAboveLeaf0111230301 e24KC2ThetaAboveLeaf0111230302
+      e24KC2ThetaAboveLeaf0111230303
+
+theorem e24KC2ThetaAboveNode011123031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01112303)
+    e24KC2ThetaAboveLeaf0111230310 e24KC2ThetaAboveLeaf0111230311 e24KC2ThetaAboveLeaf0111230312
+      e24KC2ThetaAboveLeaf0111230313
+
+theorem e24KC2ThetaAboveNode011123032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112303)
+    e24KC2ThetaAboveLeaf0111230320 e24KC2ThetaAboveLeaf0111230321 e24KC2ThetaAboveLeaf0111230322
+      e24KC2ThetaAboveLeaf0111230323
+
+theorem e24KC2ThetaAboveNode011123033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112303)
+    e24KC2ThetaAboveLeaf0111230330 e24KC2ThetaAboveLeaf0111230331 e24KC2ThetaAboveLeaf0111230332
+      e24KC2ThetaAboveLeaf0111230333
+
+theorem e24KC2ThetaAboveNode011123102 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112310)
+    e24KC2ThetaAboveLeaf0111231020 e24KC2ThetaAboveLeaf0111231021 e24KC2ThetaAboveLeaf0111231022
+      e24KC2ThetaAboveLeaf0111231023
+
+theorem e24KC2ThetaAboveNode011123103 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112310) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112310)
+    e24KC2ThetaAboveLeaf0111231030 e24KC2ThetaAboveLeaf0111231031 e24KC2ThetaAboveLeaf0111231032
+      e24KC2ThetaAboveLeaf0111231033
+
+theorem e24KC2ThetaAboveNode011123112 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112311)
+    e24KC2ThetaAboveLeaf0111231120 e24KC2ThetaAboveLeaf0111231121 e24KC2ThetaAboveLeaf0111231122
+      e24KC2ThetaAboveLeaf0111231123
+
+theorem e24KC2ThetaAboveNode011123113 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112311) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112311)
+    e24KC2ThetaAboveLeaf0111231130 e24KC2ThetaAboveLeaf0111231131 e24KC2ThetaAboveLeaf0111231132
+      e24KC2ThetaAboveLeaf0111231133
+
+theorem e24KC2ThetaAboveNode011123120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01112312)
+    e24KC2ThetaAboveLeaf0111231200 e24KC2ThetaAboveLeaf0111231201 e24KC2ThetaAboveLeaf0111231202
+      e24KC2ThetaAboveLeaf0111231203
+
+theorem e24KC2ThetaAboveNode011123121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01112312)
+    e24KC2ThetaAboveLeaf0111231210 e24KC2ThetaAboveLeaf0111231211 e24KC2ThetaAboveLeaf0111231212
+      e24KC2ThetaAboveLeaf0111231213
+
+theorem e24KC2ThetaAboveNode011123122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112312)
+    e24KC2ThetaAboveLeaf0111231220 e24KC2ThetaAboveLeaf0111231221 e24KC2ThetaAboveLeaf0111231222
+      e24KC2ThetaAboveLeaf0111231223
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle032.lean b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle032.lean
new file mode 100644
index 0000000000..d895be5ff0
--- /dev/null
+++ b/LeanPool/MovingSofa/GerverSofa/KernelOnly/PartE/Certificates/Bundle032.lean
@@ -0,0 +1,9678 @@
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+module
+
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle022
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle024
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle028
+public import LeanPool.MovingSofa.GerverSofa.KernelOnly.PartE.Certificates.Bundle031
+/-!
+# Gerver sofa: related certificate and semantic modules
+
+* `GerverSofa.KernelOnly.PartE.Certificates.Batch046`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-
+Copyright (c) 2026 Dawid Trela. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dawid Trela
+-/
+/-!
+# Gerver sofa dependency batch
+
+* `KernelOnly.PartE.ThetaAbove.ReconstructionMiddle`.
+* `KernelOnly.PartE.E24KC6ThetaAboveReconstruct`.
+* `KernelOnly.PartE.E24KC6KernelFinalClosure`.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+section
+
+/-!
+# Intermediate reconstruction of the upper theta cover
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+open CertificateCells1a75b359b5
+
+theorem e24KC2ThetaAboveNode011123123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112312)
+    e24KC2ThetaAboveLeaf0111231230 e24KC2ThetaAboveLeaf0111231231 e24KC2ThetaAboveLeaf0111231232
+      e24KC2ThetaAboveLeaf0111231233
+
+theorem e24KC2ThetaAboveNode011123130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01112313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01112313)
+    e24KC2ThetaAboveLeaf0111231300 e24KC2ThetaAboveLeaf0111231301 e24KC2ThetaAboveLeaf0111231302
+      e24KC2ThetaAboveLeaf0111231303
+
+theorem e24KC2ThetaAboveNode011123131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01112313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01112313)
+    e24KC2ThetaAboveLeaf0111231310 e24KC2ThetaAboveLeaf0111231311 e24KC2ThetaAboveLeaf0111231312
+      e24KC2ThetaAboveLeaf0111231313
+
+theorem e24KC2ThetaAboveNode011123132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01112313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01112313)
+    e24KC2ThetaAboveLeaf0111231320 e24KC2ThetaAboveLeaf0111231321 e24KC2ThetaAboveLeaf0111231322
+      e24KC2ThetaAboveLeaf0111231323
+
+theorem e24KC2ThetaAboveNode011123133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01112313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01112313)
+    e24KC2ThetaAboveLeaf0111231330 e24KC2ThetaAboveLeaf0111231331 e24KC2ThetaAboveLeaf0111231332
+      e24KC2ThetaAboveLeaf0111231333
+
+theorem e24KC2ThetaAboveNode011132002 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01113200)
+    e24KC2ThetaAboveLeaf0111320020 e24KC2ThetaAboveLeaf0111320021 e24KC2ThetaAboveLeaf0111320022
+      e24KC2ThetaAboveLeaf0111320023
+
+theorem e24KC2ThetaAboveNode011132003 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113200) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01113200)
+    e24KC2ThetaAboveLeaf0111320030 e24KC2ThetaAboveLeaf0111320031 e24KC2ThetaAboveLeaf0111320032
+      e24KC2ThetaAboveLeaf0111320033
+
+theorem e24KC2ThetaAboveNode011132012 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01113201)
+    e24KC2ThetaAboveLeaf0111320120 e24KC2ThetaAboveLeaf0111320121 e24KC2ThetaAboveLeaf0111320122
+      e24KC2ThetaAboveLeaf0111320123
+
+theorem e24KC2ThetaAboveNode011132013 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113201) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01113201)
+    e24KC2ThetaAboveLeaf0111320130 e24KC2ThetaAboveLeaf0111320131 e24KC2ThetaAboveLeaf0111320132
+      e24KC2ThetaAboveLeaf0111320133
+
+theorem e24KC2ThetaAboveNode011132020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01113202)
+    e24KC2ThetaAboveLeaf0111320200 e24KC2ThetaAboveLeaf0111320201 e24KC2ThetaAboveLeaf0111320202
+      e24KC2ThetaAboveLeaf0111320203
+
+theorem e24KC2ThetaAboveNode011132021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01113202)
+    e24KC2ThetaAboveLeaf0111320210 e24KC2ThetaAboveLeaf0111320211 e24KC2ThetaAboveLeaf0111320212
+      e24KC2ThetaAboveLeaf0111320213
+
+theorem e24KC2ThetaAboveNode011132022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01113202)
+    e24KC2ThetaAboveLeaf0111320220 e24KC2ThetaAboveLeaf0111320221 e24KC2ThetaAboveLeaf0111320222
+      e24KC2ThetaAboveLeaf0111320223
+
+theorem e24KC2ThetaAboveNode011132023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01113202)
+    e24KC2ThetaAboveLeaf0111320230 e24KC2ThetaAboveLeaf0111320231 e24KC2ThetaAboveLeaf0111320232
+      e24KC2ThetaAboveLeaf0111320233
+
+theorem e24KC2ThetaAboveNode011132030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01113203)
+    e24KC2ThetaAboveLeaf0111320300 e24KC2ThetaAboveLeaf0111320301 e24KC2ThetaAboveLeaf0111320302
+      e24KC2ThetaAboveLeaf0111320303
+
+theorem e24KC2ThetaAboveNode011132031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01113203)
+    e24KC2ThetaAboveLeaf0111320310 e24KC2ThetaAboveLeaf0111320311 e24KC2ThetaAboveLeaf0111320312
+      e24KC2ThetaAboveLeaf0111320313
+
+theorem e24KC2ThetaAboveNode011132032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01113203)
+    e24KC2ThetaAboveLeaf0111320320 e24KC2ThetaAboveLeaf0111320321 e24KC2ThetaAboveLeaf0111320322
+      e24KC2ThetaAboveLeaf0111320323
+
+theorem e24KC2ThetaAboveNode011132033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01113203)
+    e24KC2ThetaAboveLeaf0111320330 e24KC2ThetaAboveLeaf0111320331 e24KC2ThetaAboveLeaf0111320332
+      e24KC2ThetaAboveLeaf0111320333
+
+theorem e24KC2ThetaAboveNode011132120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01113212)
+    e24KC2ThetaAboveLeaf0111321200 e24KC2ThetaAboveLeaf0111321201 e24KC2ThetaAboveLeaf0111321202
+      e24KC2ThetaAboveLeaf0111321203
+
+theorem e24KC2ThetaAboveNode011132121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01113212)
+    e24KC2ThetaAboveLeaf0111321210 e24KC2ThetaAboveLeaf0111321211 e24KC2ThetaAboveLeaf0111321212
+      e24KC2ThetaAboveLeaf0111321213
+
+theorem e24KC2ThetaAboveNode011132122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01113212)
+    e24KC2ThetaAboveLeaf0111321220 e24KC2ThetaAboveLeaf0111321221 e24KC2ThetaAboveLeaf0111321222
+      e24KC2ThetaAboveLeaf0111321223
+
+theorem e24KC2ThetaAboveNode011132123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01113212)
+    e24KC2ThetaAboveLeaf0111321230 e24KC2ThetaAboveLeaf0111321231 e24KC2ThetaAboveLeaf0111321232
+      e24KC2ThetaAboveLeaf0111321233
+
+theorem e24KC2ThetaAboveNode011132130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01113213)
+    e24KC2ThetaAboveLeaf0111321300 e24KC2ThetaAboveLeaf0111321301 e24KC2ThetaAboveLeaf0111321302
+      e24KC2ThetaAboveLeaf0111321303
+
+theorem e24KC2ThetaAboveNode011132131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01113213)
+    e24KC2ThetaAboveLeaf0111321310 e24KC2ThetaAboveLeaf0111321311 e24KC2ThetaAboveLeaf0111321312
+      e24KC2ThetaAboveLeaf0111321313
+
+theorem e24KC2ThetaAboveNode011132132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01113213)
+    e24KC2ThetaAboveLeaf0111321320 e24KC2ThetaAboveLeaf0111321321 e24KC2ThetaAboveLeaf0111321322
+      e24KC2ThetaAboveLeaf0111321323
+
+theorem e24KC2ThetaAboveNode011132133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01113213)
+    e24KC2ThetaAboveLeaf0111321330 e24KC2ThetaAboveLeaf0111321331 e24KC2ThetaAboveLeaf0111321332
+      e24KC2ThetaAboveLeaf0111321333
+
+theorem e24KC2ThetaAboveNode011133020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01113302)
+    e24KC2ThetaAboveLeaf0111330200 e24KC2ThetaAboveLeaf0111330201 e24KC2ThetaAboveLeaf0111330202
+      e24KC2ThetaAboveLeaf0111330203
+
+theorem e24KC2ThetaAboveNode011133021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01113302)
+    e24KC2ThetaAboveLeaf0111330210 e24KC2ThetaAboveLeaf0111330211 e24KC2ThetaAboveLeaf0111330212
+      e24KC2ThetaAboveLeaf0111330213
+
+theorem e24KC2ThetaAboveNode011133022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01113302)
+    e24KC2ThetaAboveLeaf0111330220 e24KC2ThetaAboveLeaf0111330221 e24KC2ThetaAboveLeaf0111330222
+      e24KC2ThetaAboveLeaf0111330223
+
+theorem e24KC2ThetaAboveNode011133023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01113302)
+    e24KC2ThetaAboveLeaf0111330230 e24KC2ThetaAboveLeaf0111330231 e24KC2ThetaAboveLeaf0111330232
+      e24KC2ThetaAboveLeaf0111330233
+
+theorem e24KC2ThetaAboveNode011133030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01113303)
+    e24KC2ThetaAboveLeaf0111330300 e24KC2ThetaAboveLeaf0111330301 e24KC2ThetaAboveLeaf0111330302
+      e24KC2ThetaAboveLeaf0111330303
+
+theorem e24KC2ThetaAboveNode011133031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01113303)
+    e24KC2ThetaAboveLeaf0111330310 e24KC2ThetaAboveLeaf0111330311 e24KC2ThetaAboveLeaf0111330312
+      e24KC2ThetaAboveLeaf0111330313
+
+theorem e24KC2ThetaAboveNode011133032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01113303)
+    e24KC2ThetaAboveLeaf0111330320 e24KC2ThetaAboveLeaf0111330321 e24KC2ThetaAboveLeaf0111330322
+      e24KC2ThetaAboveLeaf0111330323
+
+theorem e24KC2ThetaAboveNode011133033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01113303)
+    e24KC2ThetaAboveLeaf0111330330 e24KC2ThetaAboveLeaf0111330331 e24KC2ThetaAboveLeaf0111330332
+      e24KC2ThetaAboveLeaf0111330333
+
+theorem e24KC2ThetaAboveNode011133120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01113312)
+    e24KC2ThetaAboveLeaf0111331200 e24KC2ThetaAboveLeaf0111331201 e24KC2ThetaAboveLeaf0111331202
+      e24KC2ThetaAboveLeaf0111331203
+
+theorem e24KC2ThetaAboveNode011133121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01113312)
+    e24KC2ThetaAboveLeaf0111331210 e24KC2ThetaAboveLeaf0111331211 e24KC2ThetaAboveLeaf0111331212
+      e24KC2ThetaAboveLeaf0111331213
+
+theorem e24KC2ThetaAboveNode011133122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01113312)
+    e24KC2ThetaAboveLeaf0111331220 e24KC2ThetaAboveLeaf0111331221 e24KC2ThetaAboveLeaf0111331222
+      e24KC2ThetaAboveLeaf0111331223
+
+theorem e24KC2ThetaAboveNode011133123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01113312)
+    e24KC2ThetaAboveLeaf0111331230 e24KC2ThetaAboveLeaf0111331231 e24KC2ThetaAboveLeaf0111331232
+      e24KC2ThetaAboveLeaf0111331233
+
+theorem e24KC2ThetaAboveNode011133130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell01113313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell01113313)
+    e24KC2ThetaAboveLeaf0111331300 e24KC2ThetaAboveLeaf0111331301 e24KC2ThetaAboveLeaf0111331302
+      e24KC2ThetaAboveLeaf0111331303
+
+theorem e24KC2ThetaAboveNode011133131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell01113313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell01113313)
+    e24KC2ThetaAboveLeaf0111331310 e24KC2ThetaAboveLeaf0111331311 e24KC2ThetaAboveLeaf0111331312
+      e24KC2ThetaAboveLeaf0111331313
+
+theorem e24KC2ThetaAboveNode011133132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell01113313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell01113313)
+    e24KC2ThetaAboveLeaf0111331320 e24KC2ThetaAboveLeaf0111331321 e24KC2ThetaAboveLeaf0111331322
+      e24KC2ThetaAboveLeaf0111331323
+
+theorem e24KC2ThetaAboveNode011133133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell01113313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell01113313)
+    e24KC2ThetaAboveLeaf0111331330 e24KC2ThetaAboveLeaf0111331331 e24KC2ThetaAboveLeaf0111331332
+      e24KC2ThetaAboveLeaf0111331333
+
+theorem e24KC2ThetaAboveNode100022020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10002202)
+    e24KC2ThetaAboveLeaf1000220200 e24KC2ThetaAboveLeaf1000220201 e24KC2ThetaAboveLeaf1000220202
+      e24KC2ThetaAboveLeaf1000220203
+
+theorem e24KC2ThetaAboveNode100022021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10002202)
+    e24KC2ThetaAboveLeaf1000220210 e24KC2ThetaAboveLeaf1000220211 e24KC2ThetaAboveLeaf1000220212
+      e24KC2ThetaAboveLeaf1000220213
+
+theorem e24KC2ThetaAboveNode100022022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10002202)
+    e24KC2ThetaAboveLeaf1000220220 e24KC2ThetaAboveLeaf1000220221 e24KC2ThetaAboveLeaf1000220222
+      e24KC2ThetaAboveLeaf1000220223
+
+theorem e24KC2ThetaAboveNode100022023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10002202)
+    e24KC2ThetaAboveLeaf1000220230 e24KC2ThetaAboveLeaf1000220231 e24KC2ThetaAboveLeaf1000220232
+      e24KC2ThetaAboveLeaf1000220233
+
+theorem e24KC2ThetaAboveNode100022030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10002203)
+    e24KC2ThetaAboveLeaf1000220300 e24KC2ThetaAboveLeaf1000220301 e24KC2ThetaAboveLeaf1000220302
+      e24KC2ThetaAboveLeaf1000220303
+
+theorem e24KC2ThetaAboveNode100022031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10002203)
+    e24KC2ThetaAboveLeaf1000220310 e24KC2ThetaAboveLeaf1000220311 e24KC2ThetaAboveLeaf1000220312
+      e24KC2ThetaAboveLeaf1000220313
+
+theorem e24KC2ThetaAboveNode100022032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10002203)
+    e24KC2ThetaAboveLeaf1000220320 e24KC2ThetaAboveLeaf1000220321 e24KC2ThetaAboveLeaf1000220322
+      e24KC2ThetaAboveLeaf1000220323
+
+theorem e24KC2ThetaAboveNode100022033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10002203)
+    e24KC2ThetaAboveLeaf1000220330 e24KC2ThetaAboveLeaf1000220331 e24KC2ThetaAboveLeaf1000220332
+      e24KC2ThetaAboveLeaf1000220333
+
+theorem e24KC2ThetaAboveNode100022120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10002212)
+    e24KC2ThetaAboveLeaf1000221200 e24KC2ThetaAboveLeaf1000221201 e24KC2ThetaAboveLeaf1000221202
+      e24KC2ThetaAboveLeaf1000221203
+
+theorem e24KC2ThetaAboveNode100022121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10002212)
+    e24KC2ThetaAboveLeaf1000221210 e24KC2ThetaAboveLeaf1000221211 e24KC2ThetaAboveLeaf1000221212
+      e24KC2ThetaAboveLeaf1000221213
+
+theorem e24KC2ThetaAboveNode100022122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10002212)
+    e24KC2ThetaAboveLeaf1000221220 e24KC2ThetaAboveLeaf1000221221 e24KC2ThetaAboveLeaf1000221222
+      e24KC2ThetaAboveLeaf1000221223
+
+theorem e24KC2ThetaAboveNode100022123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10002212)
+    e24KC2ThetaAboveLeaf1000221230 e24KC2ThetaAboveLeaf1000221231 e24KC2ThetaAboveLeaf1000221232
+      e24KC2ThetaAboveLeaf1000221233
+
+theorem e24KC2ThetaAboveNode100022130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10002213)
+    e24KC2ThetaAboveLeaf1000221300 e24KC2ThetaAboveLeaf1000221301 e24KC2ThetaAboveLeaf1000221302
+      e24KC2ThetaAboveLeaf1000221303
+
+theorem e24KC2ThetaAboveNode100022131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10002213)
+    e24KC2ThetaAboveLeaf1000221310 e24KC2ThetaAboveLeaf1000221311 e24KC2ThetaAboveLeaf1000221312
+      e24KC2ThetaAboveLeaf1000221313
+
+theorem e24KC2ThetaAboveNode100022132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10002213)
+    e24KC2ThetaAboveLeaf1000221320 e24KC2ThetaAboveLeaf1000221321 e24KC2ThetaAboveLeaf1000221322
+      e24KC2ThetaAboveLeaf1000221323
+
+theorem e24KC2ThetaAboveNode100022133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10002213)
+    e24KC2ThetaAboveLeaf1000221330 e24KC2ThetaAboveLeaf1000221331 e24KC2ThetaAboveLeaf1000221332
+      e24KC2ThetaAboveLeaf1000221333
+
+theorem e24KC2ThetaAboveNode100023020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10002302)
+    e24KC2ThetaAboveLeaf1000230200 e24KC2ThetaAboveLeaf1000230201 e24KC2ThetaAboveLeaf1000230202
+      e24KC2ThetaAboveLeaf1000230203
+
+theorem e24KC2ThetaAboveNode100023021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10002302)
+    e24KC2ThetaAboveLeaf1000230210 e24KC2ThetaAboveLeaf1000230211 e24KC2ThetaAboveLeaf1000230212
+      e24KC2ThetaAboveLeaf1000230213
+
+theorem e24KC2ThetaAboveNode100023022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10002302)
+    e24KC2ThetaAboveLeaf1000230220 e24KC2ThetaAboveLeaf1000230221 e24KC2ThetaAboveLeaf1000230222
+      e24KC2ThetaAboveLeaf1000230223
+
+theorem e24KC2ThetaAboveNode100023023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10002302)
+    e24KC2ThetaAboveLeaf1000230230 e24KC2ThetaAboveLeaf1000230231 e24KC2ThetaAboveLeaf1000230232
+      e24KC2ThetaAboveLeaf1000230233
+
+theorem e24KC2ThetaAboveNode100023030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10002303)
+    e24KC2ThetaAboveLeaf1000230300 e24KC2ThetaAboveLeaf1000230301 e24KC2ThetaAboveLeaf1000230302
+      e24KC2ThetaAboveLeaf1000230303
+
+theorem e24KC2ThetaAboveNode100023031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10002303)
+    e24KC2ThetaAboveLeaf1000230310 e24KC2ThetaAboveLeaf1000230311 e24KC2ThetaAboveLeaf1000230312
+      e24KC2ThetaAboveLeaf1000230313
+
+theorem e24KC2ThetaAboveNode100023032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10002303)
+    e24KC2ThetaAboveLeaf1000230320 e24KC2ThetaAboveLeaf1000230321 e24KC2ThetaAboveLeaf1000230322
+      e24KC2ThetaAboveLeaf1000230323
+
+theorem e24KC2ThetaAboveNode100023033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10002303)
+    e24KC2ThetaAboveLeaf1000230330 e24KC2ThetaAboveLeaf1000230331 e24KC2ThetaAboveLeaf1000230332
+      e24KC2ThetaAboveLeaf1000230333
+
+theorem e24KC2ThetaAboveNode100023120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10002312)
+    e24KC2ThetaAboveLeaf1000231200 e24KC2ThetaAboveLeaf1000231201 e24KC2ThetaAboveLeaf1000231202
+      e24KC2ThetaAboveLeaf1000231203
+
+theorem e24KC2ThetaAboveNode100023121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10002312)
+    e24KC2ThetaAboveLeaf1000231210 e24KC2ThetaAboveLeaf1000231211 e24KC2ThetaAboveLeaf1000231212
+      e24KC2ThetaAboveLeaf1000231213
+
+theorem e24KC2ThetaAboveNode100023122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10002312)
+    e24KC2ThetaAboveLeaf1000231220 e24KC2ThetaAboveLeaf1000231221 e24KC2ThetaAboveLeaf1000231222
+      e24KC2ThetaAboveLeaf1000231223
+
+theorem e24KC2ThetaAboveNode100023123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10002312)
+    e24KC2ThetaAboveLeaf1000231230 e24KC2ThetaAboveLeaf1000231231 e24KC2ThetaAboveLeaf1000231232
+      e24KC2ThetaAboveLeaf1000231233
+
+theorem e24KC2ThetaAboveNode100023130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10002313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10002313)
+    e24KC2ThetaAboveLeaf1000231300 e24KC2ThetaAboveLeaf1000231301 e24KC2ThetaAboveLeaf1000231302
+      e24KC2ThetaAboveLeaf1000231303
+
+theorem e24KC2ThetaAboveNode100023131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10002313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10002313)
+    e24KC2ThetaAboveLeaf1000231310 e24KC2ThetaAboveLeaf1000231311 e24KC2ThetaAboveLeaf1000231312
+      e24KC2ThetaAboveLeaf1000231313
+
+theorem e24KC2ThetaAboveNode100023132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10002313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10002313)
+    e24KC2ThetaAboveLeaf1000231320 e24KC2ThetaAboveLeaf1000231321 e24KC2ThetaAboveLeaf1000231322
+      e24KC2ThetaAboveLeaf1000231323
+
+theorem e24KC2ThetaAboveNode100023133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10002313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10002313)
+    e24KC2ThetaAboveLeaf1000231330 e24KC2ThetaAboveLeaf1000231331 e24KC2ThetaAboveLeaf1000231332
+      e24KC2ThetaAboveLeaf1000231333
+
+theorem e24KC2ThetaAboveNode100032020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10003202)
+    e24KC2ThetaAboveLeaf1000320200 e24KC2ThetaAboveLeaf1000320201 e24KC2ThetaAboveLeaf1000320202
+      e24KC2ThetaAboveLeaf1000320203
+
+theorem e24KC2ThetaAboveNode100032021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10003202)
+    e24KC2ThetaAboveLeaf1000320210 e24KC2ThetaAboveLeaf1000320211 e24KC2ThetaAboveLeaf1000320212
+      e24KC2ThetaAboveLeaf1000320213
+
+theorem e24KC2ThetaAboveNode100032022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10003202)
+    e24KC2ThetaAboveLeaf1000320220 e24KC2ThetaAboveLeaf1000320221 e24KC2ThetaAboveLeaf1000320222
+      e24KC2ThetaAboveLeaf1000320223
+
+theorem e24KC2ThetaAboveNode100032023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10003202)
+    e24KC2ThetaAboveLeaf1000320230 e24KC2ThetaAboveLeaf1000320231 e24KC2ThetaAboveLeaf1000320232
+      e24KC2ThetaAboveLeaf1000320233
+
+theorem e24KC2ThetaAboveNode100032030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10003203)
+    e24KC2ThetaAboveLeaf1000320300 e24KC2ThetaAboveLeaf1000320301 e24KC2ThetaAboveLeaf1000320302
+      e24KC2ThetaAboveLeaf1000320303
+
+theorem e24KC2ThetaAboveNode100032031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10003203)
+    e24KC2ThetaAboveLeaf1000320310 e24KC2ThetaAboveLeaf1000320311 e24KC2ThetaAboveLeaf1000320312
+      e24KC2ThetaAboveLeaf1000320313
+
+theorem e24KC2ThetaAboveNode100032032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10003203)
+    e24KC2ThetaAboveLeaf1000320320 e24KC2ThetaAboveLeaf1000320321 e24KC2ThetaAboveLeaf1000320322
+      e24KC2ThetaAboveLeaf1000320323
+
+theorem e24KC2ThetaAboveNode100032033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10003203)
+    e24KC2ThetaAboveLeaf1000320330 e24KC2ThetaAboveLeaf1000320331 e24KC2ThetaAboveLeaf1000320332
+      e24KC2ThetaAboveLeaf1000320333
+
+theorem e24KC2ThetaAboveNode100032120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10003212)
+    e24KC2ThetaAboveLeaf1000321200 e24KC2ThetaAboveLeaf1000321201 e24KC2ThetaAboveLeaf1000321202
+      e24KC2ThetaAboveLeaf1000321203
+
+theorem e24KC2ThetaAboveNode100032121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10003212)
+    e24KC2ThetaAboveLeaf1000321210 e24KC2ThetaAboveLeaf1000321211 e24KC2ThetaAboveLeaf1000321212
+      e24KC2ThetaAboveLeaf1000321213
+
+theorem e24KC2ThetaAboveNode100032122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10003212)
+    e24KC2ThetaAboveLeaf1000321220 e24KC2ThetaAboveLeaf1000321221 e24KC2ThetaAboveLeaf1000321222
+      e24KC2ThetaAboveLeaf1000321223
+
+theorem e24KC2ThetaAboveNode100032123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10003212)
+    e24KC2ThetaAboveLeaf1000321230 e24KC2ThetaAboveLeaf1000321231 e24KC2ThetaAboveLeaf1000321232
+      e24KC2ThetaAboveLeaf1000321233
+
+theorem e24KC2ThetaAboveNode100032130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10003213)
+    e24KC2ThetaAboveLeaf1000321300 e24KC2ThetaAboveLeaf1000321301 e24KC2ThetaAboveLeaf1000321302
+      e24KC2ThetaAboveLeaf1000321303
+
+theorem e24KC2ThetaAboveNode100032131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10003213)
+    e24KC2ThetaAboveLeaf1000321310 e24KC2ThetaAboveLeaf1000321311 e24KC2ThetaAboveLeaf1000321312
+      e24KC2ThetaAboveLeaf1000321313
+
+theorem e24KC2ThetaAboveNode100032132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10003213)
+    e24KC2ThetaAboveLeaf1000321320 e24KC2ThetaAboveLeaf1000321321 e24KC2ThetaAboveLeaf1000321322
+      e24KC2ThetaAboveLeaf1000321323
+
+theorem e24KC2ThetaAboveNode100032133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10003213)
+    e24KC2ThetaAboveLeaf1000321330 e24KC2ThetaAboveLeaf1000321331 e24KC2ThetaAboveLeaf1000321332
+      e24KC2ThetaAboveLeaf1000321333
+
+theorem e24KC2ThetaAboveNode100033020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10003302)
+    e24KC2ThetaAboveLeaf1000330200 e24KC2ThetaAboveLeaf1000330201 e24KC2ThetaAboveLeaf1000330202
+      e24KC2ThetaAboveLeaf1000330203
+
+theorem e24KC2ThetaAboveNode100033021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10003302)
+    e24KC2ThetaAboveLeaf1000330210 e24KC2ThetaAboveLeaf1000330211 e24KC2ThetaAboveLeaf1000330212
+      e24KC2ThetaAboveLeaf1000330213
+
+theorem e24KC2ThetaAboveNode100033022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10003302)
+    e24KC2ThetaAboveLeaf1000330220 e24KC2ThetaAboveLeaf1000330221 e24KC2ThetaAboveLeaf1000330222
+      e24KC2ThetaAboveLeaf1000330223
+
+theorem e24KC2ThetaAboveNode100033023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10003302)
+    e24KC2ThetaAboveLeaf1000330230 e24KC2ThetaAboveLeaf1000330231 e24KC2ThetaAboveLeaf1000330232
+      e24KC2ThetaAboveLeaf1000330233
+
+theorem e24KC2ThetaAboveNode100033030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10003303)
+    e24KC2ThetaAboveLeaf1000330300 e24KC2ThetaAboveLeaf1000330301 e24KC2ThetaAboveLeaf1000330302
+      e24KC2ThetaAboveLeaf1000330303
+
+theorem e24KC2ThetaAboveNode100033031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10003303)
+    e24KC2ThetaAboveLeaf1000330310 e24KC2ThetaAboveLeaf1000330311 e24KC2ThetaAboveLeaf1000330312
+      e24KC2ThetaAboveLeaf1000330313
+
+theorem e24KC2ThetaAboveNode100033032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10003303)
+    e24KC2ThetaAboveLeaf1000330320 e24KC2ThetaAboveLeaf1000330321 e24KC2ThetaAboveLeaf1000330322
+      e24KC2ThetaAboveLeaf1000330323
+
+theorem e24KC2ThetaAboveNode100033033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10003303)
+    e24KC2ThetaAboveLeaf1000330330 e24KC2ThetaAboveLeaf1000330331 e24KC2ThetaAboveLeaf1000330332
+      e24KC2ThetaAboveLeaf1000330333
+
+theorem e24KC2ThetaAboveNode100033120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10003312)
+    e24KC2ThetaAboveLeaf1000331200 e24KC2ThetaAboveLeaf1000331201 e24KC2ThetaAboveLeaf1000331202
+      e24KC2ThetaAboveLeaf1000331203
+
+theorem e24KC2ThetaAboveNode100033121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10003312)
+    e24KC2ThetaAboveLeaf1000331210 e24KC2ThetaAboveLeaf1000331211 e24KC2ThetaAboveLeaf1000331212
+      e24KC2ThetaAboveLeaf1000331213
+
+theorem e24KC2ThetaAboveNode100033122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10003312)
+    e24KC2ThetaAboveLeaf1000331220 e24KC2ThetaAboveLeaf1000331221 e24KC2ThetaAboveLeaf1000331222
+      e24KC2ThetaAboveLeaf1000331223
+
+theorem e24KC2ThetaAboveNode100033123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10003312)
+    e24KC2ThetaAboveLeaf1000331230 e24KC2ThetaAboveLeaf1000331231 e24KC2ThetaAboveLeaf1000331232
+      e24KC2ThetaAboveLeaf1000331233
+
+theorem e24KC2ThetaAboveNode100033130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10003313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10003313)
+    e24KC2ThetaAboveLeaf1000331300 e24KC2ThetaAboveLeaf1000331301 e24KC2ThetaAboveLeaf1000331302
+      e24KC2ThetaAboveLeaf1000331303
+
+theorem e24KC2ThetaAboveNode100033131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10003313)
+    e24KC2ThetaAboveLeaf1000331310 e24KC2ThetaAboveLeaf1000331311 e24KC2ThetaAboveLeaf1000331312
+      e24KC2ThetaAboveLeaf1000331313
+
+theorem e24KC2ThetaAboveNode100033132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10003313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10003313)
+    e24KC2ThetaAboveLeaf1000331320 e24KC2ThetaAboveLeaf1000331321 e24KC2ThetaAboveLeaf1000331322
+      e24KC2ThetaAboveLeaf1000331323
+
+theorem e24KC2ThetaAboveNode100033133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10003313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10003313)
+    e24KC2ThetaAboveLeaf1000331330 e24KC2ThetaAboveLeaf1000331331 e24KC2ThetaAboveLeaf1000331332
+      e24KC2ThetaAboveLeaf1000331333
+
+theorem e24KC2ThetaAboveNode100033311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10003331) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10003331)
+    e24KC2ThetaAboveLeaf1000333110 e24KC2ThetaAboveLeaf1000333111 e24KC2ThetaAboveLeaf1000333112
+      e24KC2ThetaAboveLeaf1000333113
+
+theorem e24KC2ThetaAboveNode100122020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012202)
+    e24KC2ThetaAboveLeaf1001220200 e24KC2ThetaAboveLeaf1001220201 e24KC2ThetaAboveLeaf1001220202
+      e24KC2ThetaAboveLeaf1001220203
+
+theorem e24KC2ThetaAboveNode100122021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012202)
+    e24KC2ThetaAboveLeaf1001220210 e24KC2ThetaAboveLeaf1001220211 e24KC2ThetaAboveLeaf1001220212
+      e24KC2ThetaAboveLeaf1001220213
+
+theorem e24KC2ThetaAboveNode100122022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10012202)
+    e24KC2ThetaAboveLeaf1001220220 e24KC2ThetaAboveLeaf1001220221 e24KC2ThetaAboveLeaf1001220222
+      e24KC2ThetaAboveLeaf1001220223
+
+theorem e24KC2ThetaAboveNode100122023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10012202)
+    e24KC2ThetaAboveLeaf1001220230 e24KC2ThetaAboveLeaf1001220231 e24KC2ThetaAboveLeaf1001220232
+      e24KC2ThetaAboveLeaf1001220233
+
+theorem e24KC2ThetaAboveNode100122030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012203)
+    e24KC2ThetaAboveLeaf1001220300 e24KC2ThetaAboveLeaf1001220301 e24KC2ThetaAboveLeaf1001220302
+      e24KC2ThetaAboveLeaf1001220303
+
+theorem e24KC2ThetaAboveNode100122031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012203)
+    e24KC2ThetaAboveLeaf1001220310 e24KC2ThetaAboveLeaf1001220311 e24KC2ThetaAboveLeaf1001220312
+      e24KC2ThetaAboveLeaf1001220313
+
+theorem e24KC2ThetaAboveNode100122032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10012203)
+    e24KC2ThetaAboveLeaf1001220320 e24KC2ThetaAboveLeaf1001220321 e24KC2ThetaAboveLeaf1001220322
+      e24KC2ThetaAboveLeaf1001220323
+
+theorem e24KC2ThetaAboveNode100122033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10012203)
+    e24KC2ThetaAboveLeaf1001220330 e24KC2ThetaAboveLeaf1001220331 e24KC2ThetaAboveLeaf1001220332
+      e24KC2ThetaAboveLeaf1001220333
+
+theorem e24KC2ThetaAboveNode100122120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012212)
+    e24KC2ThetaAboveLeaf1001221200 e24KC2ThetaAboveLeaf1001221201 e24KC2ThetaAboveLeaf1001221202
+      e24KC2ThetaAboveLeaf1001221203
+
+theorem e24KC2ThetaAboveNode100122121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012212)
+    e24KC2ThetaAboveLeaf1001221210 e24KC2ThetaAboveLeaf1001221211 e24KC2ThetaAboveLeaf1001221212
+      e24KC2ThetaAboveLeaf1001221213
+
+theorem e24KC2ThetaAboveNode100122122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10012212)
+    e24KC2ThetaAboveLeaf1001221220 e24KC2ThetaAboveLeaf1001221221 e24KC2ThetaAboveLeaf1001221222
+      e24KC2ThetaAboveLeaf1001221223
+
+theorem e24KC2ThetaAboveNode100122123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10012212)
+    e24KC2ThetaAboveLeaf1001221230 e24KC2ThetaAboveLeaf1001221231 e24KC2ThetaAboveLeaf1001221232
+      e24KC2ThetaAboveLeaf1001221233
+
+theorem e24KC2ThetaAboveNode100122130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012213)
+    e24KC2ThetaAboveLeaf1001221300 e24KC2ThetaAboveLeaf1001221301 e24KC2ThetaAboveLeaf1001221302
+      e24KC2ThetaAboveLeaf1001221303
+
+theorem e24KC2ThetaAboveNode100122131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012213)
+    e24KC2ThetaAboveLeaf1001221310 e24KC2ThetaAboveLeaf1001221311 e24KC2ThetaAboveLeaf1001221312
+      e24KC2ThetaAboveLeaf1001221313
+
+theorem e24KC2ThetaAboveNode100122132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10012213)
+    e24KC2ThetaAboveLeaf1001221320 e24KC2ThetaAboveLeaf1001221321 e24KC2ThetaAboveLeaf1001221322
+      e24KC2ThetaAboveLeaf1001221323
+
+theorem e24KC2ThetaAboveNode100122133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10012213)
+    e24KC2ThetaAboveLeaf1001221330 e24KC2ThetaAboveLeaf1001221331 e24KC2ThetaAboveLeaf1001221332
+      e24KC2ThetaAboveLeaf1001221333
+
+theorem e24KC2ThetaAboveNode100122200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012220) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012220)
+    e24KC2ThetaAboveLeaf1001222000 e24KC2ThetaAboveLeaf1001222001 e24KC2ThetaAboveLeaf1001222002
+      e24KC2ThetaAboveLeaf1001222003
+
+theorem e24KC2ThetaAboveNode100122201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012220) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012220)
+    e24KC2ThetaAboveLeaf1001222010 e24KC2ThetaAboveLeaf1001222011 e24KC2ThetaAboveLeaf1001222012
+      e24KC2ThetaAboveLeaf1001222013
+
+theorem e24KC2ThetaAboveNode100122210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012221) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012221)
+    e24KC2ThetaAboveLeaf1001222100 e24KC2ThetaAboveLeaf1001222101 e24KC2ThetaAboveLeaf1001222102
+      e24KC2ThetaAboveLeaf1001222103
+
+theorem e24KC2ThetaAboveNode100122211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012221) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012221)
+    e24KC2ThetaAboveLeaf1001222110 e24KC2ThetaAboveLeaf1001222111 e24KC2ThetaAboveLeaf1001222112
+      e24KC2ThetaAboveLeaf1001222113
+
+theorem e24KC2ThetaAboveNode100122300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012230) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012230)
+    e24KC2ThetaAboveLeaf1001223000 e24KC2ThetaAboveLeaf1001223001 e24KC2ThetaAboveLeaf1001223002
+      e24KC2ThetaAboveLeaf1001223003
+
+theorem e24KC2ThetaAboveNode100122301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012230) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012230)
+    e24KC2ThetaAboveLeaf1001223010 e24KC2ThetaAboveLeaf1001223011 e24KC2ThetaAboveLeaf1001223012
+      e24KC2ThetaAboveLeaf1001223013
+
+theorem e24KC2ThetaAboveNode100122310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012231) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012231)
+    e24KC2ThetaAboveLeaf1001223100 e24KC2ThetaAboveLeaf1001223101 e24KC2ThetaAboveLeaf1001223102
+      e24KC2ThetaAboveLeaf1001223103
+
+theorem e24KC2ThetaAboveNode100122311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012231) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012231)
+    e24KC2ThetaAboveLeaf1001223110 e24KC2ThetaAboveLeaf1001223111 e24KC2ThetaAboveLeaf1001223112
+      e24KC2ThetaAboveLeaf1001223113
+
+theorem e24KC2ThetaAboveNode100123020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012302)
+    e24KC2ThetaAboveLeaf1001230200 e24KC2ThetaAboveLeaf1001230201 e24KC2ThetaAboveLeaf1001230202
+      e24KC2ThetaAboveLeaf1001230203
+
+theorem e24KC2ThetaAboveNode100123021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012302)
+    e24KC2ThetaAboveLeaf1001230210 e24KC2ThetaAboveLeaf1001230211 e24KC2ThetaAboveLeaf1001230212
+      e24KC2ThetaAboveLeaf1001230213
+
+theorem e24KC2ThetaAboveNode100123022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10012302)
+    e24KC2ThetaAboveLeaf1001230220 e24KC2ThetaAboveLeaf1001230221 e24KC2ThetaAboveLeaf1001230222
+      e24KC2ThetaAboveLeaf1001230223
+
+theorem e24KC2ThetaAboveNode100123023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10012302)
+    e24KC2ThetaAboveLeaf1001230230 e24KC2ThetaAboveLeaf1001230231 e24KC2ThetaAboveLeaf1001230232
+      e24KC2ThetaAboveLeaf1001230233
+
+theorem e24KC2ThetaAboveNode100123030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012303)
+    e24KC2ThetaAboveLeaf1001230300 e24KC2ThetaAboveLeaf1001230301 e24KC2ThetaAboveLeaf1001230302
+      e24KC2ThetaAboveLeaf1001230303
+
+theorem e24KC2ThetaAboveNode100123031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012303)
+    e24KC2ThetaAboveLeaf1001230310 e24KC2ThetaAboveLeaf1001230311 e24KC2ThetaAboveLeaf1001230312
+      e24KC2ThetaAboveLeaf1001230313
+
+theorem e24KC2ThetaAboveNode100123032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10012303)
+    e24KC2ThetaAboveLeaf1001230320 e24KC2ThetaAboveLeaf1001230321 e24KC2ThetaAboveLeaf1001230322
+      e24KC2ThetaAboveLeaf1001230323
+
+theorem e24KC2ThetaAboveNode100123033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10012303)
+    e24KC2ThetaAboveLeaf1001230330 e24KC2ThetaAboveLeaf1001230331 e24KC2ThetaAboveLeaf1001230332
+      e24KC2ThetaAboveLeaf1001230333
+
+theorem e24KC2ThetaAboveNode100123120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012312)
+    e24KC2ThetaAboveLeaf1001231200 e24KC2ThetaAboveLeaf1001231201 e24KC2ThetaAboveLeaf1001231202
+      e24KC2ThetaAboveLeaf1001231203
+
+theorem e24KC2ThetaAboveNode100123121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012312)
+    e24KC2ThetaAboveLeaf1001231210 e24KC2ThetaAboveLeaf1001231211 e24KC2ThetaAboveLeaf1001231212
+      e24KC2ThetaAboveLeaf1001231213
+
+theorem e24KC2ThetaAboveNode100123122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10012312)
+    e24KC2ThetaAboveLeaf1001231220 e24KC2ThetaAboveLeaf1001231221 e24KC2ThetaAboveLeaf1001231222
+      e24KC2ThetaAboveLeaf1001231223
+
+theorem e24KC2ThetaAboveNode100123123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10012312)
+    e24KC2ThetaAboveLeaf1001231230 e24KC2ThetaAboveLeaf1001231231 e24KC2ThetaAboveLeaf1001231232
+      e24KC2ThetaAboveLeaf1001231233
+
+theorem e24KC2ThetaAboveNode100123130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012313)
+    e24KC2ThetaAboveLeaf1001231300 e24KC2ThetaAboveLeaf1001231301 e24KC2ThetaAboveLeaf1001231302
+      e24KC2ThetaAboveLeaf1001231303
+
+theorem e24KC2ThetaAboveNode100123131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012313)
+    e24KC2ThetaAboveLeaf1001231310 e24KC2ThetaAboveLeaf1001231311 e24KC2ThetaAboveLeaf1001231312
+      e24KC2ThetaAboveLeaf1001231313
+
+theorem e24KC2ThetaAboveNode100123132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10012313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10012313)
+    e24KC2ThetaAboveLeaf1001231320 e24KC2ThetaAboveLeaf1001231321 e24KC2ThetaAboveLeaf1001231322
+      e24KC2ThetaAboveLeaf1001231323
+
+theorem e24KC2ThetaAboveNode100123133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10012313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10012313)
+    e24KC2ThetaAboveLeaf1001231330 e24KC2ThetaAboveLeaf1001231331 e24KC2ThetaAboveLeaf1001231332
+      e24KC2ThetaAboveLeaf1001231333
+
+theorem e24KC2ThetaAboveNode100123200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012320) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012320)
+    e24KC2ThetaAboveLeaf1001232000 e24KC2ThetaAboveLeaf1001232001 e24KC2ThetaAboveLeaf1001232002
+      e24KC2ThetaAboveLeaf1001232003
+
+theorem e24KC2ThetaAboveNode100123201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012320) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012320)
+    e24KC2ThetaAboveLeaf1001232010 e24KC2ThetaAboveLeaf1001232011 e24KC2ThetaAboveLeaf1001232012
+      e24KC2ThetaAboveLeaf1001232013
+
+theorem e24KC2ThetaAboveNode100123210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012321) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012321)
+    e24KC2ThetaAboveLeaf1001232100 e24KC2ThetaAboveLeaf1001232101 e24KC2ThetaAboveLeaf1001232102
+      e24KC2ThetaAboveLeaf1001232103
+
+theorem e24KC2ThetaAboveNode100123211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012321) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012321)
+    e24KC2ThetaAboveLeaf1001232110 e24KC2ThetaAboveLeaf1001232111 e24KC2ThetaAboveLeaf1001232112
+      e24KC2ThetaAboveLeaf1001232113
+
+theorem e24KC2ThetaAboveNode100123300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012330) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012330)
+    e24KC2ThetaAboveLeaf1001233000 e24KC2ThetaAboveLeaf1001233001 e24KC2ThetaAboveLeaf1001233002
+      e24KC2ThetaAboveLeaf1001233003
+
+theorem e24KC2ThetaAboveNode100123301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012330) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012330)
+    e24KC2ThetaAboveLeaf1001233010 e24KC2ThetaAboveLeaf1001233011 e24KC2ThetaAboveLeaf1001233012
+      e24KC2ThetaAboveLeaf1001233013
+
+theorem e24KC2ThetaAboveNode100123310 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10012331) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10012331)
+    e24KC2ThetaAboveLeaf1001233100 e24KC2ThetaAboveLeaf1001233101 e24KC2ThetaAboveLeaf1001233102
+      e24KC2ThetaAboveLeaf1001233103
+
+theorem e24KC2ThetaAboveNode100123311 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10012331) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10012331)
+    e24KC2ThetaAboveLeaf1001233110 e24KC2ThetaAboveLeaf1001233111 e24KC2ThetaAboveLeaf1001233112
+      e24KC2ThetaAboveLeaf1001233113
+
+theorem e24KC2ThetaAboveNode100132020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10013202)
+    e24KC2ThetaAboveLeaf1001320200 e24KC2ThetaAboveLeaf1001320201 e24KC2ThetaAboveLeaf1001320202
+      e24KC2ThetaAboveLeaf1001320203
+
+theorem e24KC2ThetaAboveNode100132021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10013202)
+    e24KC2ThetaAboveLeaf1001320210 e24KC2ThetaAboveLeaf1001320211 e24KC2ThetaAboveLeaf1001320212
+      e24KC2ThetaAboveLeaf1001320213
+
+theorem e24KC2ThetaAboveNode100132022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10013202)
+    e24KC2ThetaAboveLeaf1001320220 e24KC2ThetaAboveLeaf1001320221 e24KC2ThetaAboveLeaf1001320222
+      e24KC2ThetaAboveLeaf1001320223
+
+theorem e24KC2ThetaAboveNode100132023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10013202)
+    e24KC2ThetaAboveLeaf1001320230 e24KC2ThetaAboveLeaf1001320231 e24KC2ThetaAboveLeaf1001320232
+      e24KC2ThetaAboveLeaf1001320233
+
+theorem e24KC2ThetaAboveNode100132030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10013203)
+    e24KC2ThetaAboveLeaf1001320300 e24KC2ThetaAboveLeaf1001320301 e24KC2ThetaAboveLeaf1001320302
+      e24KC2ThetaAboveLeaf1001320303
+
+theorem e24KC2ThetaAboveNode100132031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10013203)
+    e24KC2ThetaAboveLeaf1001320310 e24KC2ThetaAboveLeaf1001320311 e24KC2ThetaAboveLeaf1001320312
+      e24KC2ThetaAboveLeaf1001320313
+
+theorem e24KC2ThetaAboveNode100132032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10013203)
+    e24KC2ThetaAboveLeaf1001320320 e24KC2ThetaAboveLeaf1001320321 e24KC2ThetaAboveLeaf1001320322
+      e24KC2ThetaAboveLeaf1001320323
+
+theorem e24KC2ThetaAboveNode100132033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10013203)
+    e24KC2ThetaAboveLeaf1001320330 e24KC2ThetaAboveLeaf1001320331 e24KC2ThetaAboveLeaf1001320332
+      e24KC2ThetaAboveLeaf1001320333
+
+theorem e24KC2ThetaAboveNode100132120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10013212)
+    e24KC2ThetaAboveLeaf1001321200 e24KC2ThetaAboveLeaf1001321201 e24KC2ThetaAboveLeaf1001321202
+      e24KC2ThetaAboveLeaf1001321203
+
+theorem e24KC2ThetaAboveNode100132121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10013212)
+    e24KC2ThetaAboveLeaf1001321210 e24KC2ThetaAboveLeaf1001321211 e24KC2ThetaAboveLeaf1001321212
+      e24KC2ThetaAboveLeaf1001321213
+
+theorem e24KC2ThetaAboveNode100132122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10013212)
+    e24KC2ThetaAboveLeaf1001321220 e24KC2ThetaAboveLeaf1001321221 e24KC2ThetaAboveLeaf1001321222
+      e24KC2ThetaAboveLeaf1001321223
+
+theorem e24KC2ThetaAboveNode100132123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10013212)
+    e24KC2ThetaAboveLeaf1001321230 e24KC2ThetaAboveLeaf1001321231 e24KC2ThetaAboveLeaf1001321232
+      e24KC2ThetaAboveLeaf1001321233
+
+theorem e24KC2ThetaAboveNode100132130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10013213)
+    e24KC2ThetaAboveLeaf1001321300 e24KC2ThetaAboveLeaf1001321301 e24KC2ThetaAboveLeaf1001321302
+      e24KC2ThetaAboveLeaf1001321303
+
+theorem e24KC2ThetaAboveNode100132131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10013213)
+    e24KC2ThetaAboveLeaf1001321310 e24KC2ThetaAboveLeaf1001321311 e24KC2ThetaAboveLeaf1001321312
+      e24KC2ThetaAboveLeaf1001321313
+
+theorem e24KC2ThetaAboveNode100132132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10013213)
+    e24KC2ThetaAboveLeaf1001321320 e24KC2ThetaAboveLeaf1001321321 e24KC2ThetaAboveLeaf1001321322
+      e24KC2ThetaAboveLeaf1001321323
+
+theorem e24KC2ThetaAboveNode100132133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10013213)
+    e24KC2ThetaAboveLeaf1001321330 e24KC2ThetaAboveLeaf1001321331 e24KC2ThetaAboveLeaf1001321332
+      e24KC2ThetaAboveLeaf1001321333
+
+theorem e24KC2ThetaAboveNode100132200 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013220) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10013220)
+    e24KC2ThetaAboveLeaf1001322000 e24KC2ThetaAboveLeaf1001322001 e24KC2ThetaAboveLeaf1001322002
+      e24KC2ThetaAboveLeaf1001322003
+
+theorem e24KC2ThetaAboveNode100132201 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013220) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10013220)
+    e24KC2ThetaAboveLeaf1001322010 e24KC2ThetaAboveLeaf1001322011 e24KC2ThetaAboveLeaf1001322012
+      e24KC2ThetaAboveLeaf1001322013
+
+theorem e24KC2ThetaAboveNode100132210 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013221) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10013221)
+    e24KC2ThetaAboveLeaf1001322100 e24KC2ThetaAboveLeaf1001322101 e24KC2ThetaAboveLeaf1001322102
+      e24KC2ThetaAboveLeaf1001322103
+
+theorem e24KC2ThetaAboveNode100132211 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013221) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10013221)
+    e24KC2ThetaAboveLeaf1001322110 e24KC2ThetaAboveLeaf1001322111 e24KC2ThetaAboveLeaf1001322112
+      e24KC2ThetaAboveLeaf1001322113
+
+theorem e24KC2ThetaAboveNode100132300 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013230) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10013230)
+    e24KC2ThetaAboveLeaf1001323000 e24KC2ThetaAboveLeaf1001323001 e24KC2ThetaAboveLeaf1001323002
+      e24KC2ThetaAboveLeaf1001323003
+
+theorem e24KC2ThetaAboveNode100132301 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013230) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10013230)
+    e24KC2ThetaAboveLeaf1001323010 e24KC2ThetaAboveLeaf1001323011 e24KC2ThetaAboveLeaf1001323012
+      e24KC2ThetaAboveLeaf1001323013
+
+theorem e24KC2ThetaAboveNode100133020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10013302)
+    e24KC2ThetaAboveLeaf1001330200 e24KC2ThetaAboveLeaf1001330201 e24KC2ThetaAboveLeaf1001330202
+      e24KC2ThetaAboveLeaf1001330203
+
+theorem e24KC2ThetaAboveNode100133021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10013302)
+    e24KC2ThetaAboveLeaf1001330210 e24KC2ThetaAboveLeaf1001330211 e24KC2ThetaAboveLeaf1001330212
+      e24KC2ThetaAboveLeaf1001330213
+
+theorem e24KC2ThetaAboveNode100133022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10013302)
+    e24KC2ThetaAboveLeaf1001330220 e24KC2ThetaAboveLeaf1001330221 e24KC2ThetaAboveLeaf1001330222
+      e24KC2ThetaAboveLeaf1001330223
+
+theorem e24KC2ThetaAboveNode100133023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10013302)
+    e24KC2ThetaAboveLeaf1001330230 e24KC2ThetaAboveLeaf1001330231 e24KC2ThetaAboveLeaf1001330232
+      e24KC2ThetaAboveLeaf1001330233
+
+theorem e24KC2ThetaAboveNode100133030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10013303)
+    e24KC2ThetaAboveLeaf1001330300 e24KC2ThetaAboveLeaf1001330301 e24KC2ThetaAboveLeaf1001330302
+      e24KC2ThetaAboveLeaf1001330303
+
+theorem e24KC2ThetaAboveNode100133031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10013303)
+    e24KC2ThetaAboveLeaf1001330310 e24KC2ThetaAboveLeaf1001330311 e24KC2ThetaAboveLeaf1001330312
+      e24KC2ThetaAboveLeaf1001330313
+
+theorem e24KC2ThetaAboveNode100133032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10013303)
+    e24KC2ThetaAboveLeaf1001330320 e24KC2ThetaAboveLeaf1001330321 e24KC2ThetaAboveLeaf1001330322
+      e24KC2ThetaAboveLeaf1001330323
+
+theorem e24KC2ThetaAboveNode100133033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10013303)
+    e24KC2ThetaAboveLeaf1001330330 e24KC2ThetaAboveLeaf1001330331 e24KC2ThetaAboveLeaf1001330332
+      e24KC2ThetaAboveLeaf1001330333
+
+theorem e24KC2ThetaAboveNode100133120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10013312)
+    e24KC2ThetaAboveLeaf1001331200 e24KC2ThetaAboveLeaf1001331201 e24KC2ThetaAboveLeaf1001331202
+      e24KC2ThetaAboveLeaf1001331203
+
+theorem e24KC2ThetaAboveNode100133121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10013312)
+    e24KC2ThetaAboveLeaf1001331210 e24KC2ThetaAboveLeaf1001331211 e24KC2ThetaAboveLeaf1001331212
+      e24KC2ThetaAboveLeaf1001331213
+
+theorem e24KC2ThetaAboveNode100133122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10013312)
+    e24KC2ThetaAboveLeaf1001331220 e24KC2ThetaAboveLeaf1001331221 e24KC2ThetaAboveLeaf1001331222
+      e24KC2ThetaAboveLeaf1001331223
+
+theorem e24KC2ThetaAboveNode100133123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10013312)
+    e24KC2ThetaAboveLeaf1001331230 e24KC2ThetaAboveLeaf1001331231 e24KC2ThetaAboveLeaf1001331232
+      e24KC2ThetaAboveLeaf1001331233
+
+theorem e24KC2ThetaAboveNode100133130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10013313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10013313)
+    e24KC2ThetaAboveLeaf1001331300 e24KC2ThetaAboveLeaf1001331301 e24KC2ThetaAboveLeaf1001331302
+      e24KC2ThetaAboveLeaf1001331303
+
+theorem e24KC2ThetaAboveNode100133131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10013313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10013313)
+    e24KC2ThetaAboveLeaf1001331310 e24KC2ThetaAboveLeaf1001331311 e24KC2ThetaAboveLeaf1001331312
+      e24KC2ThetaAboveLeaf1001331313
+
+theorem e24KC2ThetaAboveNode100133132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10013313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10013313)
+    e24KC2ThetaAboveLeaf1001331320 e24KC2ThetaAboveLeaf1001331321 e24KC2ThetaAboveLeaf1001331322
+      e24KC2ThetaAboveLeaf1001331323
+
+theorem e24KC2ThetaAboveNode100133133 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10013313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10013313)
+    e24KC2ThetaAboveLeaf1001331330 e24KC2ThetaAboveLeaf1001331331 e24KC2ThetaAboveLeaf1001331332
+      e24KC2ThetaAboveLeaf1001331333
+
+theorem e24KC2ThetaAboveNode101022020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10102202)
+    e24KC2ThetaAboveLeaf1010220200 e24KC2ThetaAboveLeaf1010220201 e24KC2ThetaAboveLeaf1010220202
+      e24KC2ThetaAboveLeaf1010220203
+
+theorem e24KC2ThetaAboveNode101022021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10102202)
+    e24KC2ThetaAboveLeaf1010220210 e24KC2ThetaAboveLeaf1010220211 e24KC2ThetaAboveLeaf1010220212
+      e24KC2ThetaAboveLeaf1010220213
+
+theorem e24KC2ThetaAboveNode101022022 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10102202)
+    e24KC2ThetaAboveLeaf1010220220 e24KC2ThetaAboveLeaf1010220221 e24KC2ThetaAboveLeaf1010220222
+      e24KC2ThetaAboveLeaf1010220223
+
+theorem e24KC2ThetaAboveNode101022023 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10102202)
+    e24KC2ThetaAboveLeaf1010220230 e24KC2ThetaAboveLeaf1010220231 e24KC2ThetaAboveLeaf1010220232
+      e24KC2ThetaAboveLeaf1010220233
+
+theorem e24KC2ThetaAboveNode101022030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10102203)
+    e24KC2ThetaAboveLeaf1010220300 e24KC2ThetaAboveLeaf1010220301 e24KC2ThetaAboveLeaf1010220302
+      e24KC2ThetaAboveLeaf1010220303
+
+theorem e24KC2ThetaAboveNode101022031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10102203)
+    e24KC2ThetaAboveLeaf1010220310 e24KC2ThetaAboveLeaf1010220311 e24KC2ThetaAboveLeaf1010220312
+      e24KC2ThetaAboveLeaf1010220313
+
+theorem e24KC2ThetaAboveNode101022032 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10102203)
+    e24KC2ThetaAboveLeaf1010220320 e24KC2ThetaAboveLeaf1010220321 e24KC2ThetaAboveLeaf1010220322
+      e24KC2ThetaAboveLeaf1010220323
+
+theorem e24KC2ThetaAboveNode101022033 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10102203)
+    e24KC2ThetaAboveLeaf1010220330 e24KC2ThetaAboveLeaf1010220331 e24KC2ThetaAboveLeaf1010220332
+      e24KC2ThetaAboveLeaf1010220333
+
+theorem e24KC2ThetaAboveNode101022120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10102212)
+    e24KC2ThetaAboveLeaf1010221200 e24KC2ThetaAboveLeaf1010221201 e24KC2ThetaAboveLeaf1010221202
+      e24KC2ThetaAboveLeaf1010221203
+
+theorem e24KC2ThetaAboveNode101022121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10102212)
+    e24KC2ThetaAboveLeaf1010221210 e24KC2ThetaAboveLeaf1010221211 e24KC2ThetaAboveLeaf1010221212
+      e24KC2ThetaAboveLeaf1010221213
+
+theorem e24KC2ThetaAboveNode101022122 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10102212)
+    e24KC2ThetaAboveLeaf1010221220 e24KC2ThetaAboveLeaf1010221221 e24KC2ThetaAboveLeaf1010221222
+      e24KC2ThetaAboveLeaf1010221223
+
+theorem e24KC2ThetaAboveNode101022123 :
+    adaptiveCoverCheck 10 (childHH thetaAboveCell10102212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHH thetaAboveCell10102212)
+    e24KC2ThetaAboveLeaf1010221230 e24KC2ThetaAboveLeaf1010221231 e24KC2ThetaAboveLeaf1010221232
+      e24KC2ThetaAboveLeaf1010221233
+
+theorem e24KC2ThetaAboveNode101022130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10102213)
+    e24KC2ThetaAboveLeaf1010221300 e24KC2ThetaAboveLeaf1010221301 e24KC2ThetaAboveLeaf1010221302
+      e24KC2ThetaAboveLeaf1010221303
+
+theorem e24KC2ThetaAboveNode101022131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10102213)
+    e24KC2ThetaAboveLeaf1010221310 e24KC2ThetaAboveLeaf1010221311 e24KC2ThetaAboveLeaf1010221312
+      e24KC2ThetaAboveLeaf1010221313
+
+theorem e24KC2ThetaAboveNode101022132 :
+    adaptiveCoverCheck 10 (childHL thetaAboveCell10102213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childHL thetaAboveCell10102213)
+    e24KC2ThetaAboveLeaf1010221320 e24KC2ThetaAboveLeaf1010221321 e24KC2ThetaAboveLeaf1010221322
+      e24KC2ThetaAboveLeaf1010221323
+
+theorem e24KC2ThetaAboveNode101023020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10102302)
+    e24KC2ThetaAboveLeaf1010230200 e24KC2ThetaAboveLeaf1010230201 e24KC2ThetaAboveLeaf1010230202
+      e24KC2ThetaAboveLeaf1010230203
+
+theorem e24KC2ThetaAboveNode101023021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10102302)
+    e24KC2ThetaAboveLeaf1010230210 e24KC2ThetaAboveLeaf1010230211 e24KC2ThetaAboveLeaf1010230212
+      e24KC2ThetaAboveLeaf1010230213
+
+theorem e24KC2ThetaAboveNode101023030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10102303)
+    e24KC2ThetaAboveLeaf1010230300 e24KC2ThetaAboveLeaf1010230301 e24KC2ThetaAboveLeaf1010230302
+      e24KC2ThetaAboveLeaf1010230303
+
+theorem e24KC2ThetaAboveNode101023031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102303) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10102303)
+    e24KC2ThetaAboveLeaf1010230310 e24KC2ThetaAboveLeaf1010230311 e24KC2ThetaAboveLeaf1010230312
+      e24KC2ThetaAboveLeaf1010230313
+
+theorem e24KC2ThetaAboveNode101023120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10102312)
+    e24KC2ThetaAboveLeaf1010231200 e24KC2ThetaAboveLeaf1010231201 e24KC2ThetaAboveLeaf1010231202
+      e24KC2ThetaAboveLeaf1010231203
+
+theorem e24KC2ThetaAboveNode101023121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102312) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10102312)
+    e24KC2ThetaAboveLeaf1010231210 e24KC2ThetaAboveLeaf1010231211 e24KC2ThetaAboveLeaf1010231212
+      e24KC2ThetaAboveLeaf1010231213
+
+theorem e24KC2ThetaAboveNode101023130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10102313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10102313)
+    e24KC2ThetaAboveLeaf1010231300 e24KC2ThetaAboveLeaf1010231301 e24KC2ThetaAboveLeaf1010231302
+      e24KC2ThetaAboveLeaf1010231303
+
+theorem e24KC2ThetaAboveNode101023131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10102313) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10102313)
+    e24KC2ThetaAboveLeaf1010231310 e24KC2ThetaAboveLeaf1010231311 e24KC2ThetaAboveLeaf1010231312
+      e24KC2ThetaAboveLeaf1010231313
+
+theorem e24KC2ThetaAboveNode101032020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10103202)
+    e24KC2ThetaAboveLeaf1010320200 e24KC2ThetaAboveLeaf1010320201 e24KC2ThetaAboveLeaf1010320202
+      e24KC2ThetaAboveLeaf1010320203
+
+theorem e24KC2ThetaAboveNode101032021 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103202) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10103202)
+    e24KC2ThetaAboveLeaf1010320210 e24KC2ThetaAboveLeaf1010320211 e24KC2ThetaAboveLeaf1010320212
+      e24KC2ThetaAboveLeaf1010320213
+
+theorem e24KC2ThetaAboveNode101032030 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10103203)
+    e24KC2ThetaAboveLeaf1010320300 e24KC2ThetaAboveLeaf1010320301 e24KC2ThetaAboveLeaf1010320302
+      e24KC2ThetaAboveLeaf1010320303
+
+theorem e24KC2ThetaAboveNode101032031 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103203) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10103203)
+    e24KC2ThetaAboveLeaf1010320310 e24KC2ThetaAboveLeaf1010320311 e24KC2ThetaAboveLeaf1010320312
+      e24KC2ThetaAboveLeaf1010320313
+
+theorem e24KC2ThetaAboveNode101032120 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10103212)
+    e24KC2ThetaAboveLeaf1010321200 e24KC2ThetaAboveLeaf1010321201 e24KC2ThetaAboveLeaf1010321202
+      e24KC2ThetaAboveLeaf1010321203
+
+theorem e24KC2ThetaAboveNode101032121 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103212) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10103212)
+    e24KC2ThetaAboveLeaf1010321210 e24KC2ThetaAboveLeaf1010321211 e24KC2ThetaAboveLeaf1010321212
+      e24KC2ThetaAboveLeaf1010321213
+
+theorem e24KC2ThetaAboveNode101032130 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10103213)
+    e24KC2ThetaAboveLeaf1010321300 e24KC2ThetaAboveLeaf1010321301 e24KC2ThetaAboveLeaf1010321302
+      e24KC2ThetaAboveLeaf1010321303
+
+theorem e24KC2ThetaAboveNode101032131 :
+    adaptiveCoverCheck 10 (childLH thetaAboveCell10103213) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLH thetaAboveCell10103213)
+    e24KC2ThetaAboveLeaf1010321310 e24KC2ThetaAboveLeaf1010321311 e24KC2ThetaAboveLeaf1010321312
+      e24KC2ThetaAboveLeaf1010321313
+
+theorem e24KC2ThetaAboveNode101033020 :
+    adaptiveCoverCheck 10 (childLL thetaAboveCell10103302) = true :=
+  adaptiveCoverCheck_succ_of_children 9 (childLL thetaAboveCell10103302)
+    e24KC2ThetaAboveLeaf1010330200 e24KC2ThetaAboveLeaf1010330201 e24KC2ThetaAboveLeaf1010330202
+      e24KC2ThetaAboveLeaf1010330203
+
+theorem e24KC2ThetaAboveNode00002022 :
+    adaptiveCoverCheck 11 thetaAboveCell00002022 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002022
+    e24KC2ThetaAboveLeaf000020220 e24KC2ThetaAboveLeaf000020221 e24KC2ThetaAboveLeaf000020222
+      e24KC2ThetaAboveLeaf000020223
+
+theorem e24KC2ThetaAboveNode00002023 :
+    adaptiveCoverCheck 11 thetaAboveCell00002023 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002023
+    e24KC2ThetaAboveLeaf000020230 e24KC2ThetaAboveLeaf000020231 e24KC2ThetaAboveLeaf000020232
+      e24KC2ThetaAboveLeaf000020233
+
+theorem e24KC2ThetaAboveNode00002032 :
+    adaptiveCoverCheck 11 thetaAboveCell00002032 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002032
+    e24KC2ThetaAboveLeaf000020320 e24KC2ThetaAboveLeaf000020321 e24KC2ThetaAboveLeaf000020322
+      e24KC2ThetaAboveLeaf000020323
+
+theorem e24KC2ThetaAboveNode00002033 :
+    adaptiveCoverCheck 11 thetaAboveCell00002033 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002033
+    e24KC2ThetaAboveLeaf000020330 e24KC2ThetaAboveLeaf000020331 e24KC2ThetaAboveLeaf000020332
+      e24KC2ThetaAboveLeaf000020333
+
+theorem e24KC2ThetaAboveNode00002122 :
+    adaptiveCoverCheck 11 thetaAboveCell00002122 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002122
+    e24KC2ThetaAboveLeaf000021220 e24KC2ThetaAboveLeaf000021221 e24KC2ThetaAboveLeaf000021222
+      e24KC2ThetaAboveLeaf000021223
+
+theorem e24KC2ThetaAboveNode00002123 :
+    adaptiveCoverCheck 11 thetaAboveCell00002123 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002123
+    e24KC2ThetaAboveLeaf000021230 e24KC2ThetaAboveLeaf000021231 e24KC2ThetaAboveLeaf000021232
+      e24KC2ThetaAboveLeaf000021233
+
+theorem e24KC2ThetaAboveNode00002132 :
+    adaptiveCoverCheck 11 thetaAboveCell00002132 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002132
+    e24KC2ThetaAboveLeaf000021320 e24KC2ThetaAboveLeaf000021321 e24KC2ThetaAboveLeaf000021322
+      e24KC2ThetaAboveLeaf000021323
+
+theorem e24KC2ThetaAboveNode00002133 :
+    adaptiveCoverCheck 11 thetaAboveCell00002133 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002133
+    e24KC2ThetaAboveLeaf000021330 e24KC2ThetaAboveLeaf000021331 e24KC2ThetaAboveLeaf000021332
+      e24KC2ThetaAboveLeaf000021333
+
+theorem e24KC2ThetaAboveNode00002200 :
+    adaptiveCoverCheck 11 thetaAboveCell00002200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002200
+    e24KC2ThetaAboveNode000022000 e24KC2ThetaAboveNode000022001 e24KC2ThetaAboveNode000022002
+      e24KC2ThetaAboveNode000022003
+
+theorem e24KC2ThetaAboveNode00002201 :
+    adaptiveCoverCheck 11 thetaAboveCell00002201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002201
+    e24KC2ThetaAboveNode000022010 e24KC2ThetaAboveNode000022011 e24KC2ThetaAboveNode000022012
+      e24KC2ThetaAboveNode000022013
+
+theorem e24KC2ThetaAboveNode00002202 :
+    adaptiveCoverCheck 11 thetaAboveCell00002202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002202
+    e24KC2ThetaAboveLeaf000022020 e24KC2ThetaAboveLeaf000022021 e24KC2ThetaAboveLeaf000022022
+      e24KC2ThetaAboveLeaf000022023
+
+theorem e24KC2ThetaAboveNode00002203 :
+    adaptiveCoverCheck 11 thetaAboveCell00002203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002203
+    e24KC2ThetaAboveLeaf000022030 e24KC2ThetaAboveLeaf000022031 e24KC2ThetaAboveLeaf000022032
+      e24KC2ThetaAboveLeaf000022033
+
+theorem e24KC2ThetaAboveNode00002210 :
+    adaptiveCoverCheck 11 thetaAboveCell00002210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002210
+    e24KC2ThetaAboveNode000022100 e24KC2ThetaAboveNode000022101 e24KC2ThetaAboveNode000022102
+      e24KC2ThetaAboveNode000022103
+
+theorem e24KC2ThetaAboveNode00002211 :
+    adaptiveCoverCheck 11 thetaAboveCell00002211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002211
+    e24KC2ThetaAboveNode000022110 e24KC2ThetaAboveNode000022111 e24KC2ThetaAboveNode000022112
+      e24KC2ThetaAboveNode000022113
+
+theorem e24KC2ThetaAboveNode00002212 :
+    adaptiveCoverCheck 11 thetaAboveCell00002212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002212
+    e24KC2ThetaAboveLeaf000022120 e24KC2ThetaAboveLeaf000022121 e24KC2ThetaAboveLeaf000022122
+      e24KC2ThetaAboveLeaf000022123
+
+theorem e24KC2ThetaAboveNode00002213 :
+    adaptiveCoverCheck 11 thetaAboveCell00002213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002213
+    e24KC2ThetaAboveLeaf000022130 e24KC2ThetaAboveNode000022131 e24KC2ThetaAboveLeaf000022132
+      e24KC2ThetaAboveLeaf000022133
+
+theorem e24KC2ThetaAboveNode00002300 :
+    adaptiveCoverCheck 11 thetaAboveCell00002300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002300
+    e24KC2ThetaAboveNode000023000 e24KC2ThetaAboveNode000023001 e24KC2ThetaAboveNode000023002
+      e24KC2ThetaAboveNode000023003
+
+theorem e24KC2ThetaAboveNode00002301 :
+    adaptiveCoverCheck 11 thetaAboveCell00002301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002301
+    e24KC2ThetaAboveNode000023010 e24KC2ThetaAboveNode000023011 e24KC2ThetaAboveNode000023012
+      e24KC2ThetaAboveNode000023013
+
+theorem e24KC2ThetaAboveNode00002302 :
+    adaptiveCoverCheck 11 thetaAboveCell00002302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002302
+    e24KC2ThetaAboveNode000023020 e24KC2ThetaAboveNode000023021 e24KC2ThetaAboveLeaf000023022
+      e24KC2ThetaAboveLeaf000023023
+
+theorem e24KC2ThetaAboveNode00002303 :
+    adaptiveCoverCheck 11 thetaAboveCell00002303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002303
+    e24KC2ThetaAboveNode000023030 e24KC2ThetaAboveNode000023031 e24KC2ThetaAboveLeaf000023032
+      e24KC2ThetaAboveLeaf000023033
+
+theorem e24KC2ThetaAboveNode00002310 :
+    adaptiveCoverCheck 11 thetaAboveCell00002310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002310
+    e24KC2ThetaAboveNode000023100 e24KC2ThetaAboveNode000023101 e24KC2ThetaAboveNode000023102
+      e24KC2ThetaAboveNode000023103
+
+theorem e24KC2ThetaAboveNode00002311 :
+    adaptiveCoverCheck 11 thetaAboveCell00002311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002311
+    e24KC2ThetaAboveNode000023110 e24KC2ThetaAboveNode000023111 e24KC2ThetaAboveNode000023112
+      e24KC2ThetaAboveNode000023113
+
+theorem e24KC2ThetaAboveNode00002312 :
+    adaptiveCoverCheck 11 thetaAboveCell00002312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002312
+    e24KC2ThetaAboveNode000023120 e24KC2ThetaAboveNode000023121 e24KC2ThetaAboveLeaf000023122
+      e24KC2ThetaAboveLeaf000023123
+
+theorem e24KC2ThetaAboveNode00002313 :
+    adaptiveCoverCheck 11 thetaAboveCell00002313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00002313
+    e24KC2ThetaAboveNode000023130 e24KC2ThetaAboveNode000023131 e24KC2ThetaAboveLeaf000023132
+      e24KC2ThetaAboveLeaf000023133
+
+theorem e24KC2ThetaAboveNode00003022 :
+    adaptiveCoverCheck 11 thetaAboveCell00003022 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003022
+    e24KC2ThetaAboveLeaf000030220 e24KC2ThetaAboveLeaf000030221 e24KC2ThetaAboveLeaf000030222
+      e24KC2ThetaAboveLeaf000030223
+
+theorem e24KC2ThetaAboveNode00003023 :
+    adaptiveCoverCheck 11 thetaAboveCell00003023 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003023
+    e24KC2ThetaAboveLeaf000030230 e24KC2ThetaAboveLeaf000030231 e24KC2ThetaAboveLeaf000030232
+      e24KC2ThetaAboveLeaf000030233
+
+theorem e24KC2ThetaAboveNode00003032 :
+    adaptiveCoverCheck 11 thetaAboveCell00003032 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003032
+    e24KC2ThetaAboveLeaf000030320 e24KC2ThetaAboveLeaf000030321 e24KC2ThetaAboveLeaf000030322
+      e24KC2ThetaAboveLeaf000030323
+
+theorem e24KC2ThetaAboveNode00003033 :
+    adaptiveCoverCheck 11 thetaAboveCell00003033 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003033
+    e24KC2ThetaAboveLeaf000030330 e24KC2ThetaAboveLeaf000030331 e24KC2ThetaAboveLeaf000030332
+      e24KC2ThetaAboveLeaf000030333
+
+theorem e24KC2ThetaAboveNode00003122 :
+    adaptiveCoverCheck 11 thetaAboveCell00003122 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003122
+    e24KC2ThetaAboveLeaf000031220 e24KC2ThetaAboveLeaf000031221 e24KC2ThetaAboveLeaf000031222
+      e24KC2ThetaAboveLeaf000031223
+
+theorem e24KC2ThetaAboveNode00003123 :
+    adaptiveCoverCheck 11 thetaAboveCell00003123 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003123
+    e24KC2ThetaAboveLeaf000031230 e24KC2ThetaAboveLeaf000031231 e24KC2ThetaAboveLeaf000031232
+      e24KC2ThetaAboveLeaf000031233
+
+theorem e24KC2ThetaAboveNode00003132 :
+    adaptiveCoverCheck 11 thetaAboveCell00003132 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003132
+    e24KC2ThetaAboveLeaf000031320 e24KC2ThetaAboveLeaf000031321 e24KC2ThetaAboveLeaf000031322
+      e24KC2ThetaAboveLeaf000031323
+
+theorem e24KC2ThetaAboveNode00003133 :
+    adaptiveCoverCheck 11 thetaAboveCell00003133 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003133
+    e24KC2ThetaAboveLeaf000031330 e24KC2ThetaAboveLeaf000031331 e24KC2ThetaAboveLeaf000031332
+      e24KC2ThetaAboveLeaf000031333
+
+theorem e24KC2ThetaAboveNode00003200 :
+    adaptiveCoverCheck 11 thetaAboveCell00003200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003200
+    e24KC2ThetaAboveNode000032000 e24KC2ThetaAboveNode000032001 e24KC2ThetaAboveNode000032002
+      e24KC2ThetaAboveNode000032003
+
+theorem e24KC2ThetaAboveNode00003201 :
+    adaptiveCoverCheck 11 thetaAboveCell00003201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003201
+    e24KC2ThetaAboveNode000032010 e24KC2ThetaAboveNode000032011 e24KC2ThetaAboveNode000032012
+      e24KC2ThetaAboveNode000032013
+
+theorem e24KC2ThetaAboveNode00003202 :
+    adaptiveCoverCheck 11 thetaAboveCell00003202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003202
+    e24KC2ThetaAboveNode000032020 e24KC2ThetaAboveNode000032021 e24KC2ThetaAboveLeaf000032022
+      e24KC2ThetaAboveLeaf000032023
+
+theorem e24KC2ThetaAboveNode00003203 :
+    adaptiveCoverCheck 11 thetaAboveCell00003203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003203
+    e24KC2ThetaAboveNode000032030 e24KC2ThetaAboveNode000032031 e24KC2ThetaAboveLeaf000032032
+      e24KC2ThetaAboveLeaf000032033
+
+theorem e24KC2ThetaAboveNode00003210 :
+    adaptiveCoverCheck 11 thetaAboveCell00003210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003210
+    e24KC2ThetaAboveNode000032100 e24KC2ThetaAboveNode000032101 e24KC2ThetaAboveNode000032102
+      e24KC2ThetaAboveNode000032103
+
+theorem e24KC2ThetaAboveNode00003211 :
+    adaptiveCoverCheck 11 thetaAboveCell00003211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003211
+    e24KC2ThetaAboveNode000032110 e24KC2ThetaAboveNode000032111 e24KC2ThetaAboveNode000032112
+      e24KC2ThetaAboveNode000032113
+
+theorem e24KC2ThetaAboveNode00003212 :
+    adaptiveCoverCheck 11 thetaAboveCell00003212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003212
+    e24KC2ThetaAboveNode000032120 e24KC2ThetaAboveNode000032121 e24KC2ThetaAboveLeaf000032122
+      e24KC2ThetaAboveLeaf000032123
+
+theorem e24KC2ThetaAboveNode00003213 :
+    adaptiveCoverCheck 11 thetaAboveCell00003213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003213
+    e24KC2ThetaAboveNode000032130 e24KC2ThetaAboveNode000032131 e24KC2ThetaAboveLeaf000032132
+      e24KC2ThetaAboveLeaf000032133
+
+theorem e24KC2ThetaAboveNode00003300 :
+    adaptiveCoverCheck 11 thetaAboveCell00003300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003300
+    e24KC2ThetaAboveNode000033000 e24KC2ThetaAboveNode000033001 e24KC2ThetaAboveNode000033002
+      e24KC2ThetaAboveNode000033003
+
+theorem e24KC2ThetaAboveNode00003301 :
+    adaptiveCoverCheck 11 thetaAboveCell00003301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003301
+    e24KC2ThetaAboveNode000033010 e24KC2ThetaAboveNode000033011 e24KC2ThetaAboveNode000033012
+      e24KC2ThetaAboveNode000033013
+
+theorem e24KC2ThetaAboveNode00003302 :
+    adaptiveCoverCheck 11 thetaAboveCell00003302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003302
+    e24KC2ThetaAboveNode000033020 e24KC2ThetaAboveNode000033021 e24KC2ThetaAboveLeaf000033022
+      e24KC2ThetaAboveLeaf000033023
+
+theorem e24KC2ThetaAboveNode00003303 :
+    adaptiveCoverCheck 11 thetaAboveCell00003303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003303
+    e24KC2ThetaAboveNode000033030 e24KC2ThetaAboveNode000033031 e24KC2ThetaAboveLeaf000033032
+      e24KC2ThetaAboveLeaf000033033
+
+theorem e24KC2ThetaAboveNode00003310 :
+    adaptiveCoverCheck 11 thetaAboveCell00003310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003310
+    e24KC2ThetaAboveNode000033100 e24KC2ThetaAboveNode000033101 e24KC2ThetaAboveNode000033102
+      e24KC2ThetaAboveNode000033103
+
+theorem e24KC2ThetaAboveNode00003311 :
+    adaptiveCoverCheck 11 thetaAboveCell00003311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003311
+    e24KC2ThetaAboveNode000033110 e24KC2ThetaAboveNode000033111 e24KC2ThetaAboveNode000033112
+      e24KC2ThetaAboveNode000033113
+
+theorem e24KC2ThetaAboveNode00003312 :
+    adaptiveCoverCheck 11 thetaAboveCell00003312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003312
+    e24KC2ThetaAboveNode000033120 e24KC2ThetaAboveNode000033121 e24KC2ThetaAboveLeaf000033122
+      e24KC2ThetaAboveLeaf000033123
+
+theorem e24KC2ThetaAboveNode00003313 :
+    adaptiveCoverCheck 11 thetaAboveCell00003313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00003313
+    e24KC2ThetaAboveNode000033130 e24KC2ThetaAboveNode000033131 e24KC2ThetaAboveLeaf000033132
+      e24KC2ThetaAboveLeaf000033133
+
+theorem e24KC2ThetaAboveNode00012022 :
+    adaptiveCoverCheck 11 thetaAboveCell00012022 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012022
+    e24KC2ThetaAboveLeaf000120220 e24KC2ThetaAboveLeaf000120221 e24KC2ThetaAboveLeaf000120222
+      e24KC2ThetaAboveLeaf000120223
+
+theorem e24KC2ThetaAboveNode00012023 :
+    adaptiveCoverCheck 11 thetaAboveCell00012023 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012023
+    e24KC2ThetaAboveLeaf000120230 e24KC2ThetaAboveLeaf000120231 e24KC2ThetaAboveLeaf000120232
+      e24KC2ThetaAboveLeaf000120233
+
+theorem e24KC2ThetaAboveNode00012032 :
+    adaptiveCoverCheck 11 thetaAboveCell00012032 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012032
+    e24KC2ThetaAboveLeaf000120320 e24KC2ThetaAboveLeaf000120321 e24KC2ThetaAboveLeaf000120322
+      e24KC2ThetaAboveLeaf000120323
+
+theorem e24KC2ThetaAboveNode00012033 :
+    adaptiveCoverCheck 11 thetaAboveCell00012033 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012033
+    e24KC2ThetaAboveLeaf000120330 e24KC2ThetaAboveLeaf000120331 e24KC2ThetaAboveLeaf000120332
+      e24KC2ThetaAboveLeaf000120333
+
+theorem e24KC2ThetaAboveNode00012122 :
+    adaptiveCoverCheck 11 thetaAboveCell00012122 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012122
+    e24KC2ThetaAboveLeaf000121220 e24KC2ThetaAboveLeaf000121221 e24KC2ThetaAboveLeaf000121222
+      e24KC2ThetaAboveLeaf000121223
+
+theorem e24KC2ThetaAboveNode00012123 :
+    adaptiveCoverCheck 11 thetaAboveCell00012123 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012123
+    e24KC2ThetaAboveLeaf000121230 e24KC2ThetaAboveLeaf000121231 e24KC2ThetaAboveLeaf000121232
+      e24KC2ThetaAboveLeaf000121233
+
+theorem e24KC2ThetaAboveNode00012132 :
+    adaptiveCoverCheck 11 thetaAboveCell00012132 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012132
+    e24KC2ThetaAboveLeaf000121320 e24KC2ThetaAboveLeaf000121321 e24KC2ThetaAboveLeaf000121322
+      e24KC2ThetaAboveLeaf000121323
+
+theorem e24KC2ThetaAboveNode00012133 :
+    adaptiveCoverCheck 11 thetaAboveCell00012133 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012133
+    e24KC2ThetaAboveLeaf000121330 e24KC2ThetaAboveLeaf000121331 e24KC2ThetaAboveLeaf000121332
+      e24KC2ThetaAboveLeaf000121333
+
+theorem e24KC2ThetaAboveNode00012200 :
+    adaptiveCoverCheck 11 thetaAboveCell00012200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012200
+    e24KC2ThetaAboveNode000122000 e24KC2ThetaAboveNode000122001 e24KC2ThetaAboveNode000122002
+      e24KC2ThetaAboveNode000122003
+
+theorem e24KC2ThetaAboveNode00012201 :
+    adaptiveCoverCheck 11 thetaAboveCell00012201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012201
+    e24KC2ThetaAboveNode000122010 e24KC2ThetaAboveNode000122011 e24KC2ThetaAboveNode000122012
+      e24KC2ThetaAboveNode000122013
+
+theorem e24KC2ThetaAboveNode00012202 :
+    adaptiveCoverCheck 11 thetaAboveCell00012202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012202
+    e24KC2ThetaAboveNode000122020 e24KC2ThetaAboveNode000122021 e24KC2ThetaAboveLeaf000122022
+      e24KC2ThetaAboveLeaf000122023
+
+theorem e24KC2ThetaAboveNode00012203 :
+    adaptiveCoverCheck 11 thetaAboveCell00012203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012203
+    e24KC2ThetaAboveNode000122030 e24KC2ThetaAboveNode000122031 e24KC2ThetaAboveLeaf000122032
+      e24KC2ThetaAboveLeaf000122033
+
+theorem e24KC2ThetaAboveNode00012210 :
+    adaptiveCoverCheck 11 thetaAboveCell00012210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012210
+    e24KC2ThetaAboveNode000122100 e24KC2ThetaAboveNode000122101 e24KC2ThetaAboveNode000122102
+      e24KC2ThetaAboveNode000122103
+
+theorem e24KC2ThetaAboveNode00012211 :
+    adaptiveCoverCheck 11 thetaAboveCell00012211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012211
+    e24KC2ThetaAboveNode000122110 e24KC2ThetaAboveNode000122111 e24KC2ThetaAboveNode000122112
+      e24KC2ThetaAboveNode000122113
+
+theorem e24KC2ThetaAboveNode00012212 :
+    adaptiveCoverCheck 11 thetaAboveCell00012212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012212
+    e24KC2ThetaAboveNode000122120 e24KC2ThetaAboveNode000122121 e24KC2ThetaAboveLeaf000122122
+      e24KC2ThetaAboveLeaf000122123
+
+theorem e24KC2ThetaAboveNode00012213 :
+    adaptiveCoverCheck 11 thetaAboveCell00012213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012213
+    e24KC2ThetaAboveNode000122130 e24KC2ThetaAboveNode000122131 e24KC2ThetaAboveLeaf000122132
+      e24KC2ThetaAboveLeaf000122133
+
+theorem e24KC2ThetaAboveNode00012300 :
+    adaptiveCoverCheck 11 thetaAboveCell00012300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012300
+    e24KC2ThetaAboveNode000123000 e24KC2ThetaAboveNode000123001 e24KC2ThetaAboveNode000123002
+      e24KC2ThetaAboveNode000123003
+
+theorem e24KC2ThetaAboveNode00012301 :
+    adaptiveCoverCheck 11 thetaAboveCell00012301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012301
+    e24KC2ThetaAboveNode000123010 e24KC2ThetaAboveNode000123011 e24KC2ThetaAboveNode000123012
+      e24KC2ThetaAboveNode000123013
+
+theorem e24KC2ThetaAboveNode00012302 :
+    adaptiveCoverCheck 11 thetaAboveCell00012302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012302
+    e24KC2ThetaAboveNode000123020 e24KC2ThetaAboveNode000123021 e24KC2ThetaAboveLeaf000123022
+      e24KC2ThetaAboveLeaf000123023
+
+theorem e24KC2ThetaAboveNode00012303 :
+    adaptiveCoverCheck 11 thetaAboveCell00012303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012303
+    e24KC2ThetaAboveNode000123030 e24KC2ThetaAboveNode000123031 e24KC2ThetaAboveLeaf000123032
+      e24KC2ThetaAboveLeaf000123033
+
+theorem e24KC2ThetaAboveNode00012310 :
+    adaptiveCoverCheck 11 thetaAboveCell00012310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012310
+    e24KC2ThetaAboveNode000123100 e24KC2ThetaAboveNode000123101 e24KC2ThetaAboveNode000123102
+      e24KC2ThetaAboveNode000123103
+
+theorem e24KC2ThetaAboveNode00012311 :
+    adaptiveCoverCheck 11 thetaAboveCell00012311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012311
+    e24KC2ThetaAboveNode000123110 e24KC2ThetaAboveNode000123111 e24KC2ThetaAboveNode000123112
+      e24KC2ThetaAboveNode000123113
+
+theorem e24KC2ThetaAboveNode00012312 :
+    adaptiveCoverCheck 11 thetaAboveCell00012312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012312
+    e24KC2ThetaAboveNode000123120 e24KC2ThetaAboveNode000123121 e24KC2ThetaAboveLeaf000123122
+      e24KC2ThetaAboveLeaf000123123
+
+theorem e24KC2ThetaAboveNode00012313 :
+    adaptiveCoverCheck 11 thetaAboveCell00012313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00012313
+    e24KC2ThetaAboveNode000123130 e24KC2ThetaAboveNode000123131 e24KC2ThetaAboveLeaf000123132
+      e24KC2ThetaAboveLeaf000123133
+
+theorem e24KC2ThetaAboveNode00013022 :
+    adaptiveCoverCheck 11 thetaAboveCell00013022 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013022
+    e24KC2ThetaAboveLeaf000130220 e24KC2ThetaAboveLeaf000130221 e24KC2ThetaAboveLeaf000130222
+      e24KC2ThetaAboveLeaf000130223
+
+theorem e24KC2ThetaAboveNode00013023 :
+    adaptiveCoverCheck 11 thetaAboveCell00013023 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013023
+    e24KC2ThetaAboveLeaf000130230 e24KC2ThetaAboveLeaf000130231 e24KC2ThetaAboveLeaf000130232
+      e24KC2ThetaAboveLeaf000130233
+
+theorem e24KC2ThetaAboveNode00013032 :
+    adaptiveCoverCheck 11 thetaAboveCell00013032 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013032
+    e24KC2ThetaAboveLeaf000130320 e24KC2ThetaAboveLeaf000130321 e24KC2ThetaAboveLeaf000130322
+      e24KC2ThetaAboveLeaf000130323
+
+theorem e24KC2ThetaAboveNode00013033 :
+    adaptiveCoverCheck 11 thetaAboveCell00013033 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013033
+    e24KC2ThetaAboveLeaf000130330 e24KC2ThetaAboveLeaf000130331 e24KC2ThetaAboveLeaf000130332
+      e24KC2ThetaAboveLeaf000130333
+
+theorem e24KC2ThetaAboveNode00013122 :
+    adaptiveCoverCheck 11 thetaAboveCell00013122 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013122
+    e24KC2ThetaAboveLeaf000131220 e24KC2ThetaAboveLeaf000131221 e24KC2ThetaAboveLeaf000131222
+      e24KC2ThetaAboveLeaf000131223
+
+theorem e24KC2ThetaAboveNode00013123 :
+    adaptiveCoverCheck 11 thetaAboveCell00013123 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013123
+    e24KC2ThetaAboveLeaf000131230 e24KC2ThetaAboveLeaf000131231 e24KC2ThetaAboveLeaf000131232
+      e24KC2ThetaAboveLeaf000131233
+
+theorem e24KC2ThetaAboveNode00013132 :
+    adaptiveCoverCheck 11 thetaAboveCell00013132 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013132
+    e24KC2ThetaAboveLeaf000131320 e24KC2ThetaAboveLeaf000131321 e24KC2ThetaAboveLeaf000131322
+      e24KC2ThetaAboveLeaf000131323
+
+theorem e24KC2ThetaAboveNode00013133 :
+    adaptiveCoverCheck 11 thetaAboveCell00013133 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013133
+    e24KC2ThetaAboveLeaf000131330 e24KC2ThetaAboveLeaf000131331 e24KC2ThetaAboveLeaf000131332
+      e24KC2ThetaAboveLeaf000131333
+
+theorem e24KC2ThetaAboveNode00013200 :
+    adaptiveCoverCheck 11 thetaAboveCell00013200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013200
+    e24KC2ThetaAboveNode000132000 e24KC2ThetaAboveNode000132001 e24KC2ThetaAboveNode000132002
+      e24KC2ThetaAboveNode000132003
+
+theorem e24KC2ThetaAboveNode00013201 :
+    adaptiveCoverCheck 11 thetaAboveCell00013201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013201
+    e24KC2ThetaAboveNode000132010 e24KC2ThetaAboveNode000132011 e24KC2ThetaAboveNode000132012
+      e24KC2ThetaAboveNode000132013
+
+theorem e24KC2ThetaAboveNode00013202 :
+    adaptiveCoverCheck 11 thetaAboveCell00013202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013202
+    e24KC2ThetaAboveNode000132020 e24KC2ThetaAboveNode000132021 e24KC2ThetaAboveLeaf000132022
+      e24KC2ThetaAboveLeaf000132023
+
+theorem e24KC2ThetaAboveNode00013203 :
+    adaptiveCoverCheck 11 thetaAboveCell00013203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013203
+    e24KC2ThetaAboveNode000132030 e24KC2ThetaAboveNode000132031 e24KC2ThetaAboveLeaf000132032
+      e24KC2ThetaAboveLeaf000132033
+
+theorem e24KC2ThetaAboveNode00013210 :
+    adaptiveCoverCheck 11 thetaAboveCell00013210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013210
+    e24KC2ThetaAboveNode000132100 e24KC2ThetaAboveNode000132101 e24KC2ThetaAboveNode000132102
+      e24KC2ThetaAboveNode000132103
+
+theorem e24KC2ThetaAboveNode00013211 :
+    adaptiveCoverCheck 11 thetaAboveCell00013211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013211
+    e24KC2ThetaAboveNode000132110 e24KC2ThetaAboveNode000132111 e24KC2ThetaAboveNode000132112
+      e24KC2ThetaAboveNode000132113
+
+theorem e24KC2ThetaAboveNode00013212 :
+    adaptiveCoverCheck 11 thetaAboveCell00013212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013212
+    e24KC2ThetaAboveNode000132120 e24KC2ThetaAboveNode000132121 e24KC2ThetaAboveLeaf000132122
+      e24KC2ThetaAboveLeaf000132123
+
+theorem e24KC2ThetaAboveNode00013213 :
+    adaptiveCoverCheck 11 thetaAboveCell00013213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013213
+    e24KC2ThetaAboveNode000132130 e24KC2ThetaAboveNode000132131 e24KC2ThetaAboveLeaf000132132
+      e24KC2ThetaAboveLeaf000132133
+
+theorem e24KC2ThetaAboveNode00013300 :
+    adaptiveCoverCheck 11 thetaAboveCell00013300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013300
+    e24KC2ThetaAboveNode000133000 e24KC2ThetaAboveNode000133001 e24KC2ThetaAboveNode000133002
+      e24KC2ThetaAboveNode000133003
+
+theorem e24KC2ThetaAboveNode00013301 :
+    adaptiveCoverCheck 11 thetaAboveCell00013301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013301
+    e24KC2ThetaAboveNode000133010 e24KC2ThetaAboveNode000133011 e24KC2ThetaAboveNode000133012
+      e24KC2ThetaAboveNode000133013
+
+theorem e24KC2ThetaAboveNode00013302 :
+    adaptiveCoverCheck 11 thetaAboveCell00013302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013302
+    e24KC2ThetaAboveNode000133020 e24KC2ThetaAboveNode000133021 e24KC2ThetaAboveLeaf000133022
+      e24KC2ThetaAboveLeaf000133023
+
+theorem e24KC2ThetaAboveNode00013303 :
+    adaptiveCoverCheck 11 thetaAboveCell00013303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013303
+    e24KC2ThetaAboveNode000133030 e24KC2ThetaAboveNode000133031 e24KC2ThetaAboveLeaf000133032
+      e24KC2ThetaAboveLeaf000133033
+
+theorem e24KC2ThetaAboveNode00013310 :
+    adaptiveCoverCheck 11 thetaAboveCell00013310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013310
+    e24KC2ThetaAboveNode000133100 e24KC2ThetaAboveNode000133101 e24KC2ThetaAboveNode000133102
+      e24KC2ThetaAboveNode000133103
+
+theorem e24KC2ThetaAboveNode00013311 :
+    adaptiveCoverCheck 11 thetaAboveCell00013311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013311
+    e24KC2ThetaAboveNode000133110 e24KC2ThetaAboveNode000133111 e24KC2ThetaAboveNode000133112
+      e24KC2ThetaAboveNode000133113
+
+theorem e24KC2ThetaAboveNode00013312 :
+    adaptiveCoverCheck 11 thetaAboveCell00013312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013312
+    e24KC2ThetaAboveNode000133120 e24KC2ThetaAboveNode000133121 e24KC2ThetaAboveLeaf000133122
+      e24KC2ThetaAboveLeaf000133123
+
+theorem e24KC2ThetaAboveNode00013313 :
+    adaptiveCoverCheck 11 thetaAboveCell00013313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00013313
+    e24KC2ThetaAboveNode000133130 e24KC2ThetaAboveNode000133131 e24KC2ThetaAboveLeaf000133132
+      e24KC2ThetaAboveLeaf000133133
+
+theorem e24KC2ThetaAboveNode00102022 :
+    adaptiveCoverCheck 11 thetaAboveCell00102022 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102022
+    e24KC2ThetaAboveLeaf001020220 e24KC2ThetaAboveLeaf001020221 e24KC2ThetaAboveLeaf001020222
+      e24KC2ThetaAboveLeaf001020223
+
+theorem e24KC2ThetaAboveNode00102023 :
+    adaptiveCoverCheck 11 thetaAboveCell00102023 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102023
+    e24KC2ThetaAboveLeaf001020230 e24KC2ThetaAboveLeaf001020231 e24KC2ThetaAboveLeaf001020232
+      e24KC2ThetaAboveLeaf001020233
+
+theorem e24KC2ThetaAboveNode00102032 :
+    adaptiveCoverCheck 11 thetaAboveCell00102032 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102032
+    e24KC2ThetaAboveLeaf001020320 e24KC2ThetaAboveLeaf001020321 e24KC2ThetaAboveLeaf001020322
+      e24KC2ThetaAboveLeaf001020323
+
+theorem e24KC2ThetaAboveNode00102033 :
+    adaptiveCoverCheck 11 thetaAboveCell00102033 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102033
+    e24KC2ThetaAboveLeaf001020330 e24KC2ThetaAboveLeaf001020331 e24KC2ThetaAboveLeaf001020332
+      e24KC2ThetaAboveLeaf001020333
+
+theorem e24KC2ThetaAboveNode00102122 :
+    adaptiveCoverCheck 11 thetaAboveCell00102122 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102122
+    e24KC2ThetaAboveLeaf001021220 e24KC2ThetaAboveLeaf001021221 e24KC2ThetaAboveLeaf001021222
+      e24KC2ThetaAboveLeaf001021223
+
+theorem e24KC2ThetaAboveNode00102123 :
+    adaptiveCoverCheck 11 thetaAboveCell00102123 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102123
+    e24KC2ThetaAboveLeaf001021230 e24KC2ThetaAboveLeaf001021231 e24KC2ThetaAboveLeaf001021232
+      e24KC2ThetaAboveLeaf001021233
+
+theorem e24KC2ThetaAboveNode00102132 :
+    adaptiveCoverCheck 11 thetaAboveCell00102132 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102132
+    e24KC2ThetaAboveLeaf001021320 e24KC2ThetaAboveLeaf001021321 e24KC2ThetaAboveLeaf001021322
+      e24KC2ThetaAboveLeaf001021323
+
+theorem e24KC2ThetaAboveNode00102133 :
+    adaptiveCoverCheck 11 thetaAboveCell00102133 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102133
+    e24KC2ThetaAboveLeaf001021330 e24KC2ThetaAboveLeaf001021331 e24KC2ThetaAboveLeaf001021332
+      e24KC2ThetaAboveLeaf001021333
+
+theorem e24KC2ThetaAboveNode00102200 :
+    adaptiveCoverCheck 11 thetaAboveCell00102200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102200
+    e24KC2ThetaAboveNode001022000 e24KC2ThetaAboveNode001022001 e24KC2ThetaAboveNode001022002
+      e24KC2ThetaAboveNode001022003
+
+theorem e24KC2ThetaAboveNode00102201 :
+    adaptiveCoverCheck 11 thetaAboveCell00102201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102201
+    e24KC2ThetaAboveNode001022010 e24KC2ThetaAboveNode001022011 e24KC2ThetaAboveNode001022012
+      e24KC2ThetaAboveNode001022013
+
+theorem e24KC2ThetaAboveNode00102202 :
+    adaptiveCoverCheck 11 thetaAboveCell00102202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102202
+    e24KC2ThetaAboveNode001022020 e24KC2ThetaAboveNode001022021 e24KC2ThetaAboveLeaf001022022
+      e24KC2ThetaAboveLeaf001022023
+
+theorem e24KC2ThetaAboveNode00102203 :
+    adaptiveCoverCheck 11 thetaAboveCell00102203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102203
+    e24KC2ThetaAboveNode001022030 e24KC2ThetaAboveNode001022031 e24KC2ThetaAboveLeaf001022032
+      e24KC2ThetaAboveLeaf001022033
+
+theorem e24KC2ThetaAboveNode00102210 :
+    adaptiveCoverCheck 11 thetaAboveCell00102210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102210
+    e24KC2ThetaAboveNode001022100 e24KC2ThetaAboveNode001022101 e24KC2ThetaAboveNode001022102
+      e24KC2ThetaAboveNode001022103
+
+theorem e24KC2ThetaAboveNode00102211 :
+    adaptiveCoverCheck 11 thetaAboveCell00102211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102211
+    e24KC2ThetaAboveNode001022110 e24KC2ThetaAboveNode001022111 e24KC2ThetaAboveNode001022112
+      e24KC2ThetaAboveNode001022113
+
+theorem e24KC2ThetaAboveNode00102212 :
+    adaptiveCoverCheck 11 thetaAboveCell00102212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102212
+    e24KC2ThetaAboveNode001022120 e24KC2ThetaAboveNode001022121 e24KC2ThetaAboveLeaf001022122
+      e24KC2ThetaAboveLeaf001022123
+
+theorem e24KC2ThetaAboveNode00102213 :
+    adaptiveCoverCheck 11 thetaAboveCell00102213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102213
+    e24KC2ThetaAboveNode001022130 e24KC2ThetaAboveNode001022131 e24KC2ThetaAboveLeaf001022132
+      e24KC2ThetaAboveLeaf001022133
+
+theorem e24KC2ThetaAboveNode00102300 :
+    adaptiveCoverCheck 11 thetaAboveCell00102300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102300
+    e24KC2ThetaAboveNode001023000 e24KC2ThetaAboveNode001023001 e24KC2ThetaAboveNode001023002
+      e24KC2ThetaAboveNode001023003
+
+theorem e24KC2ThetaAboveNode00102301 :
+    adaptiveCoverCheck 11 thetaAboveCell00102301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102301
+    e24KC2ThetaAboveNode001023010 e24KC2ThetaAboveNode001023011 e24KC2ThetaAboveNode001023012
+      e24KC2ThetaAboveNode001023013
+
+theorem e24KC2ThetaAboveNode00102302 :
+    adaptiveCoverCheck 11 thetaAboveCell00102302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102302
+    e24KC2ThetaAboveNode001023020 e24KC2ThetaAboveNode001023021 e24KC2ThetaAboveLeaf001023022
+      e24KC2ThetaAboveLeaf001023023
+
+theorem e24KC2ThetaAboveNode00102303 :
+    adaptiveCoverCheck 11 thetaAboveCell00102303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102303
+    e24KC2ThetaAboveNode001023030 e24KC2ThetaAboveNode001023031 e24KC2ThetaAboveLeaf001023032
+      e24KC2ThetaAboveLeaf001023033
+
+theorem e24KC2ThetaAboveNode00102310 :
+    adaptiveCoverCheck 11 thetaAboveCell00102310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102310
+    e24KC2ThetaAboveNode001023100 e24KC2ThetaAboveNode001023101 e24KC2ThetaAboveNode001023102
+      e24KC2ThetaAboveNode001023103
+
+theorem e24KC2ThetaAboveNode00102311 :
+    adaptiveCoverCheck 11 thetaAboveCell00102311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102311
+    e24KC2ThetaAboveNode001023110 e24KC2ThetaAboveNode001023111 e24KC2ThetaAboveNode001023112
+      e24KC2ThetaAboveNode001023113
+
+theorem e24KC2ThetaAboveNode00102312 :
+    adaptiveCoverCheck 11 thetaAboveCell00102312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102312
+    e24KC2ThetaAboveNode001023120 e24KC2ThetaAboveNode001023121 e24KC2ThetaAboveLeaf001023122
+      e24KC2ThetaAboveLeaf001023123
+
+theorem e24KC2ThetaAboveNode00102313 :
+    adaptiveCoverCheck 11 thetaAboveCell00102313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00102313
+    e24KC2ThetaAboveNode001023130 e24KC2ThetaAboveNode001023131 e24KC2ThetaAboveLeaf001023132
+      e24KC2ThetaAboveLeaf001023133
+
+theorem e24KC2ThetaAboveNode00103200 :
+    adaptiveCoverCheck 11 thetaAboveCell00103200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103200
+    e24KC2ThetaAboveNode001032000 e24KC2ThetaAboveNode001032001 e24KC2ThetaAboveNode001032002
+      e24KC2ThetaAboveNode001032003
+
+theorem e24KC2ThetaAboveNode00103201 :
+    adaptiveCoverCheck 11 thetaAboveCell00103201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103201
+    e24KC2ThetaAboveNode001032010 e24KC2ThetaAboveNode001032011 e24KC2ThetaAboveNode001032012
+      e24KC2ThetaAboveNode001032013
+
+theorem e24KC2ThetaAboveNode00103202 :
+    adaptiveCoverCheck 11 thetaAboveCell00103202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103202
+    e24KC2ThetaAboveNode001032020 e24KC2ThetaAboveNode001032021 e24KC2ThetaAboveLeaf001032022
+      e24KC2ThetaAboveLeaf001032023
+
+theorem e24KC2ThetaAboveNode00103203 :
+    adaptiveCoverCheck 11 thetaAboveCell00103203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103203
+    e24KC2ThetaAboveNode001032030 e24KC2ThetaAboveNode001032031 e24KC2ThetaAboveLeaf001032032
+      e24KC2ThetaAboveLeaf001032033
+
+theorem e24KC2ThetaAboveNode00103210 :
+    adaptiveCoverCheck 11 thetaAboveCell00103210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103210
+    e24KC2ThetaAboveNode001032100 e24KC2ThetaAboveNode001032101 e24KC2ThetaAboveNode001032102
+      e24KC2ThetaAboveNode001032103
+
+theorem e24KC2ThetaAboveNode00103211 :
+    adaptiveCoverCheck 11 thetaAboveCell00103211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103211
+    e24KC2ThetaAboveNode001032110 e24KC2ThetaAboveNode001032111 e24KC2ThetaAboveNode001032112
+      e24KC2ThetaAboveNode001032113
+
+theorem e24KC2ThetaAboveNode00103212 :
+    adaptiveCoverCheck 11 thetaAboveCell00103212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103212
+    e24KC2ThetaAboveNode001032120 e24KC2ThetaAboveNode001032121 e24KC2ThetaAboveLeaf001032122
+      e24KC2ThetaAboveLeaf001032123
+
+theorem e24KC2ThetaAboveNode00103213 :
+    adaptiveCoverCheck 11 thetaAboveCell00103213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103213
+    e24KC2ThetaAboveNode001032130 e24KC2ThetaAboveNode001032131 e24KC2ThetaAboveLeaf001032132
+      e24KC2ThetaAboveLeaf001032133
+
+theorem e24KC2ThetaAboveNode00103300 :
+    adaptiveCoverCheck 11 thetaAboveCell00103300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103300
+    e24KC2ThetaAboveNode001033000 e24KC2ThetaAboveNode001033001 e24KC2ThetaAboveNode001033002
+      e24KC2ThetaAboveNode001033003
+
+theorem e24KC2ThetaAboveNode00103301 :
+    adaptiveCoverCheck 11 thetaAboveCell00103301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103301
+    e24KC2ThetaAboveNode001033010 e24KC2ThetaAboveNode001033011 e24KC2ThetaAboveNode001033012
+      e24KC2ThetaAboveNode001033013
+
+theorem e24KC2ThetaAboveNode00103302 :
+    adaptiveCoverCheck 11 thetaAboveCell00103302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103302
+    e24KC2ThetaAboveNode001033020 e24KC2ThetaAboveNode001033021 e24KC2ThetaAboveLeaf001033022
+      e24KC2ThetaAboveLeaf001033023
+
+theorem e24KC2ThetaAboveNode00103303 :
+    adaptiveCoverCheck 11 thetaAboveCell00103303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103303
+    e24KC2ThetaAboveNode001033030 e24KC2ThetaAboveNode001033031 e24KC2ThetaAboveLeaf001033032
+      e24KC2ThetaAboveLeaf001033033
+
+theorem e24KC2ThetaAboveNode00103310 :
+    adaptiveCoverCheck 11 thetaAboveCell00103310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103310
+    e24KC2ThetaAboveNode001033100 e24KC2ThetaAboveNode001033101 e24KC2ThetaAboveNode001033102
+      e24KC2ThetaAboveNode001033103
+
+theorem e24KC2ThetaAboveNode00103311 :
+    adaptiveCoverCheck 11 thetaAboveCell00103311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103311
+    e24KC2ThetaAboveNode001033110 e24KC2ThetaAboveNode001033111 e24KC2ThetaAboveNode001033112
+      e24KC2ThetaAboveNode001033113
+
+theorem e24KC2ThetaAboveNode00103312 :
+    adaptiveCoverCheck 11 thetaAboveCell00103312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103312
+    e24KC2ThetaAboveNode001033120 e24KC2ThetaAboveNode001033121 e24KC2ThetaAboveLeaf001033122
+      e24KC2ThetaAboveLeaf001033123
+
+theorem e24KC2ThetaAboveNode00103313 :
+    adaptiveCoverCheck 11 thetaAboveCell00103313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00103313
+    e24KC2ThetaAboveNode001033130 e24KC2ThetaAboveNode001033131 e24KC2ThetaAboveLeaf001033132
+      e24KC2ThetaAboveLeaf001033133
+
+theorem e24KC2ThetaAboveNode00112200 :
+    adaptiveCoverCheck 11 thetaAboveCell00112200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112200
+    e24KC2ThetaAboveNode001122000 e24KC2ThetaAboveNode001122001 e24KC2ThetaAboveNode001122002
+      e24KC2ThetaAboveNode001122003
+
+theorem e24KC2ThetaAboveNode00112201 :
+    adaptiveCoverCheck 11 thetaAboveCell00112201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112201
+    e24KC2ThetaAboveNode001122010 e24KC2ThetaAboveNode001122011 e24KC2ThetaAboveNode001122012
+      e24KC2ThetaAboveNode001122013
+
+theorem e24KC2ThetaAboveNode00112202 :
+    adaptiveCoverCheck 11 thetaAboveCell00112202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112202
+    e24KC2ThetaAboveNode001122020 e24KC2ThetaAboveNode001122021 e24KC2ThetaAboveLeaf001122022
+      e24KC2ThetaAboveLeaf001122023
+
+theorem e24KC2ThetaAboveNode00112203 :
+    adaptiveCoverCheck 11 thetaAboveCell00112203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112203
+    e24KC2ThetaAboveNode001122030 e24KC2ThetaAboveNode001122031 e24KC2ThetaAboveLeaf001122032
+      e24KC2ThetaAboveLeaf001122033
+
+theorem e24KC2ThetaAboveNode00112210 :
+    adaptiveCoverCheck 11 thetaAboveCell00112210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112210
+    e24KC2ThetaAboveNode001122100 e24KC2ThetaAboveLeaf001122101 e24KC2ThetaAboveNode001122102
+      e24KC2ThetaAboveNode001122103
+
+theorem e24KC2ThetaAboveNode00112211 :
+    adaptiveCoverCheck 11 thetaAboveCell00112211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112211
+    e24KC2ThetaAboveLeaf001122110 e24KC2ThetaAboveLeaf001122111 e24KC2ThetaAboveNode001122112
+      e24KC2ThetaAboveNode001122113
+
+theorem e24KC2ThetaAboveNode00112212 :
+    adaptiveCoverCheck 11 thetaAboveCell00112212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112212
+    e24KC2ThetaAboveNode001122120 e24KC2ThetaAboveNode001122121 e24KC2ThetaAboveLeaf001122122
+      e24KC2ThetaAboveLeaf001122123
+
+theorem e24KC2ThetaAboveNode00112213 :
+    adaptiveCoverCheck 11 thetaAboveCell00112213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112213
+    e24KC2ThetaAboveNode001122130 e24KC2ThetaAboveNode001122131 e24KC2ThetaAboveLeaf001122132
+      e24KC2ThetaAboveLeaf001122133
+
+theorem e24KC2ThetaAboveNode00112300 :
+    adaptiveCoverCheck 11 thetaAboveCell00112300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112300
+    e24KC2ThetaAboveLeaf001123000 e24KC2ThetaAboveLeaf001123001 e24KC2ThetaAboveNode001123002
+      e24KC2ThetaAboveNode001123003
+
+theorem e24KC2ThetaAboveNode00112301 :
+    adaptiveCoverCheck 11 thetaAboveCell00112301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112301
+    e24KC2ThetaAboveLeaf001123010 e24KC2ThetaAboveLeaf001123011 e24KC2ThetaAboveNode001123012
+      e24KC2ThetaAboveNode001123013
+
+theorem e24KC2ThetaAboveNode00112302 :
+    adaptiveCoverCheck 11 thetaAboveCell00112302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112302
+    e24KC2ThetaAboveNode001123020 e24KC2ThetaAboveNode001123021 e24KC2ThetaAboveLeaf001123022
+      e24KC2ThetaAboveLeaf001123023
+
+theorem e24KC2ThetaAboveNode00112303 :
+    adaptiveCoverCheck 11 thetaAboveCell00112303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112303
+    e24KC2ThetaAboveNode001123030 e24KC2ThetaAboveNode001123031 e24KC2ThetaAboveLeaf001123032
+      e24KC2ThetaAboveLeaf001123033
+
+theorem e24KC2ThetaAboveNode00112310 :
+    adaptiveCoverCheck 11 thetaAboveCell00112310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112310
+    e24KC2ThetaAboveLeaf001123100 e24KC2ThetaAboveLeaf001123101 e24KC2ThetaAboveNode001123102
+      e24KC2ThetaAboveNode001123103
+
+theorem e24KC2ThetaAboveNode00112311 :
+    adaptiveCoverCheck 11 thetaAboveCell00112311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112311
+    e24KC2ThetaAboveLeaf001123110 e24KC2ThetaAboveLeaf001123111 e24KC2ThetaAboveNode001123112
+      e24KC2ThetaAboveNode001123113
+
+theorem e24KC2ThetaAboveNode00112312 :
+    adaptiveCoverCheck 11 thetaAboveCell00112312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112312
+    e24KC2ThetaAboveNode001123120 e24KC2ThetaAboveNode001123121 e24KC2ThetaAboveLeaf001123122
+      e24KC2ThetaAboveLeaf001123123
+
+theorem e24KC2ThetaAboveNode00112313 :
+    adaptiveCoverCheck 11 thetaAboveCell00112313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00112313
+    e24KC2ThetaAboveNode001123130 e24KC2ThetaAboveNode001123131 e24KC2ThetaAboveLeaf001123132
+      e24KC2ThetaAboveLeaf001123133
+
+theorem e24KC2ThetaAboveNode00113200 :
+    adaptiveCoverCheck 11 thetaAboveCell00113200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113200
+    e24KC2ThetaAboveLeaf001132000 e24KC2ThetaAboveLeaf001132001 e24KC2ThetaAboveNode001132002
+      e24KC2ThetaAboveNode001132003
+
+theorem e24KC2ThetaAboveNode00113201 :
+    adaptiveCoverCheck 11 thetaAboveCell00113201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113201
+    e24KC2ThetaAboveLeaf001132010 e24KC2ThetaAboveLeaf001132011 e24KC2ThetaAboveNode001132012
+      e24KC2ThetaAboveNode001132013
+
+theorem e24KC2ThetaAboveNode00113202 :
+    adaptiveCoverCheck 11 thetaAboveCell00113202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113202
+    e24KC2ThetaAboveNode001132020 e24KC2ThetaAboveNode001132021 e24KC2ThetaAboveLeaf001132022
+      e24KC2ThetaAboveLeaf001132023
+
+theorem e24KC2ThetaAboveNode00113203 :
+    adaptiveCoverCheck 11 thetaAboveCell00113203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113203
+    e24KC2ThetaAboveNode001132030 e24KC2ThetaAboveNode001132031 e24KC2ThetaAboveLeaf001132032
+      e24KC2ThetaAboveLeaf001132033
+
+theorem e24KC2ThetaAboveNode00113210 :
+    adaptiveCoverCheck 11 thetaAboveCell00113210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113210
+    e24KC2ThetaAboveLeaf001132100 e24KC2ThetaAboveLeaf001132101 e24KC2ThetaAboveNode001132102
+      e24KC2ThetaAboveNode001132103
+
+theorem e24KC2ThetaAboveNode00113211 :
+    adaptiveCoverCheck 11 thetaAboveCell00113211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113211
+    e24KC2ThetaAboveLeaf001132110 e24KC2ThetaAboveLeaf001132111 e24KC2ThetaAboveNode001132112
+      e24KC2ThetaAboveNode001132113
+
+theorem e24KC2ThetaAboveNode00113212 :
+    adaptiveCoverCheck 11 thetaAboveCell00113212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113212
+    e24KC2ThetaAboveNode001132120 e24KC2ThetaAboveNode001132121 e24KC2ThetaAboveLeaf001132122
+      e24KC2ThetaAboveLeaf001132123
+
+theorem e24KC2ThetaAboveNode00113213 :
+    adaptiveCoverCheck 11 thetaAboveCell00113213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113213
+    e24KC2ThetaAboveNode001132130 e24KC2ThetaAboveNode001132131 e24KC2ThetaAboveLeaf001132132
+      e24KC2ThetaAboveLeaf001132133
+
+theorem e24KC2ThetaAboveNode00113300 :
+    adaptiveCoverCheck 11 thetaAboveCell00113300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113300
+    e24KC2ThetaAboveLeaf001133000 e24KC2ThetaAboveLeaf001133001 e24KC2ThetaAboveNode001133002
+      e24KC2ThetaAboveNode001133003
+
+theorem e24KC2ThetaAboveNode00113301 :
+    adaptiveCoverCheck 11 thetaAboveCell00113301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113301
+    e24KC2ThetaAboveLeaf001133010 e24KC2ThetaAboveLeaf001133011 e24KC2ThetaAboveNode001133012
+      e24KC2ThetaAboveNode001133013
+
+theorem e24KC2ThetaAboveNode00113302 :
+    adaptiveCoverCheck 11 thetaAboveCell00113302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113302
+    e24KC2ThetaAboveNode001133020 e24KC2ThetaAboveNode001133021 e24KC2ThetaAboveLeaf001133022
+      e24KC2ThetaAboveLeaf001133023
+
+theorem e24KC2ThetaAboveNode00113303 :
+    adaptiveCoverCheck 11 thetaAboveCell00113303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113303
+    e24KC2ThetaAboveNode001133030 e24KC2ThetaAboveNode001133031 e24KC2ThetaAboveLeaf001133032
+      e24KC2ThetaAboveLeaf001133033
+
+theorem e24KC2ThetaAboveNode00113310 :
+    adaptiveCoverCheck 11 thetaAboveCell00113310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113310
+    e24KC2ThetaAboveLeaf001133100 e24KC2ThetaAboveLeaf001133101 e24KC2ThetaAboveNode001133102
+      e24KC2ThetaAboveNode001133103
+
+theorem e24KC2ThetaAboveNode00113311 :
+    adaptiveCoverCheck 11 thetaAboveCell00113311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113311
+    e24KC2ThetaAboveLeaf001133110 e24KC2ThetaAboveLeaf001133111 e24KC2ThetaAboveNode001133112
+      e24KC2ThetaAboveNode001133113
+
+theorem e24KC2ThetaAboveNode00113312 :
+    adaptiveCoverCheck 11 thetaAboveCell00113312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113312
+    e24KC2ThetaAboveNode001133120 e24KC2ThetaAboveNode001133121 e24KC2ThetaAboveLeaf001133122
+      e24KC2ThetaAboveLeaf001133123
+
+theorem e24KC2ThetaAboveNode00113313 :
+    adaptiveCoverCheck 11 thetaAboveCell00113313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell00113313
+    e24KC2ThetaAboveNode001133130 e24KC2ThetaAboveNode001133131 e24KC2ThetaAboveLeaf001133132
+      e24KC2ThetaAboveLeaf001133133
+
+theorem e24KC2ThetaAboveNode01002200 :
+    adaptiveCoverCheck 11 thetaAboveCell01002200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002200
+    e24KC2ThetaAboveLeaf010022000 e24KC2ThetaAboveLeaf010022001 e24KC2ThetaAboveNode010022002
+      e24KC2ThetaAboveNode010022003
+
+theorem e24KC2ThetaAboveNode01002201 :
+    adaptiveCoverCheck 11 thetaAboveCell01002201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002201
+    e24KC2ThetaAboveLeaf010022010 e24KC2ThetaAboveLeaf010022011 e24KC2ThetaAboveNode010022012
+      e24KC2ThetaAboveNode010022013
+
+theorem e24KC2ThetaAboveNode01002202 :
+    adaptiveCoverCheck 11 thetaAboveCell01002202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002202
+    e24KC2ThetaAboveNode010022020 e24KC2ThetaAboveNode010022021 e24KC2ThetaAboveLeaf010022022
+      e24KC2ThetaAboveLeaf010022023
+
+theorem e24KC2ThetaAboveNode01002203 :
+    adaptiveCoverCheck 11 thetaAboveCell01002203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002203
+    e24KC2ThetaAboveNode010022030 e24KC2ThetaAboveNode010022031 e24KC2ThetaAboveLeaf010022032
+      e24KC2ThetaAboveLeaf010022033
+
+theorem e24KC2ThetaAboveNode01002210 :
+    adaptiveCoverCheck 11 thetaAboveCell01002210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002210
+    e24KC2ThetaAboveLeaf010022100 e24KC2ThetaAboveLeaf010022101 e24KC2ThetaAboveNode010022102
+      e24KC2ThetaAboveNode010022103
+
+theorem e24KC2ThetaAboveNode01002211 :
+    adaptiveCoverCheck 11 thetaAboveCell01002211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002211
+    e24KC2ThetaAboveLeaf010022110 e24KC2ThetaAboveLeaf010022111 e24KC2ThetaAboveNode010022112
+      e24KC2ThetaAboveNode010022113
+
+theorem e24KC2ThetaAboveNode01002212 :
+    adaptiveCoverCheck 11 thetaAboveCell01002212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002212
+    e24KC2ThetaAboveNode010022120 e24KC2ThetaAboveNode010022121 e24KC2ThetaAboveLeaf010022122
+      e24KC2ThetaAboveLeaf010022123
+
+theorem e24KC2ThetaAboveNode01002213 :
+    adaptiveCoverCheck 11 thetaAboveCell01002213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002213
+    e24KC2ThetaAboveNode010022130 e24KC2ThetaAboveNode010022131 e24KC2ThetaAboveLeaf010022132
+      e24KC2ThetaAboveLeaf010022133
+
+theorem e24KC2ThetaAboveNode01002300 :
+    adaptiveCoverCheck 11 thetaAboveCell01002300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002300
+    e24KC2ThetaAboveLeaf010023000 e24KC2ThetaAboveLeaf010023001 e24KC2ThetaAboveNode010023002
+      e24KC2ThetaAboveNode010023003
+
+theorem e24KC2ThetaAboveNode01002301 :
+    adaptiveCoverCheck 11 thetaAboveCell01002301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002301
+    e24KC2ThetaAboveLeaf010023010 e24KC2ThetaAboveLeaf010023011 e24KC2ThetaAboveNode010023012
+      e24KC2ThetaAboveNode010023013
+
+theorem e24KC2ThetaAboveNode01002302 :
+    adaptiveCoverCheck 11 thetaAboveCell01002302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002302
+    e24KC2ThetaAboveNode010023020 e24KC2ThetaAboveNode010023021 e24KC2ThetaAboveLeaf010023022
+      e24KC2ThetaAboveLeaf010023023
+
+theorem e24KC2ThetaAboveNode01002303 :
+    adaptiveCoverCheck 11 thetaAboveCell01002303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002303
+    e24KC2ThetaAboveNode010023030 e24KC2ThetaAboveNode010023031 e24KC2ThetaAboveNode010023032
+      e24KC2ThetaAboveNode010023033
+
+theorem e24KC2ThetaAboveNode01002310 :
+    adaptiveCoverCheck 11 thetaAboveCell01002310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002310
+    e24KC2ThetaAboveLeaf010023100 e24KC2ThetaAboveLeaf010023101 e24KC2ThetaAboveNode010023102
+      e24KC2ThetaAboveNode010023103
+
+theorem e24KC2ThetaAboveNode01002311 :
+    adaptiveCoverCheck 11 thetaAboveCell01002311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002311
+    e24KC2ThetaAboveLeaf010023110 e24KC2ThetaAboveLeaf010023111 e24KC2ThetaAboveNode010023112
+      e24KC2ThetaAboveNode010023113
+
+theorem e24KC2ThetaAboveNode01002312 :
+    adaptiveCoverCheck 11 thetaAboveCell01002312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002312
+    e24KC2ThetaAboveNode010023120 e24KC2ThetaAboveNode010023121 e24KC2ThetaAboveNode010023122
+      e24KC2ThetaAboveNode010023123
+
+theorem e24KC2ThetaAboveNode01002313 :
+    adaptiveCoverCheck 11 thetaAboveCell01002313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01002313
+    e24KC2ThetaAboveNode010023130 e24KC2ThetaAboveNode010023131 e24KC2ThetaAboveNode010023132
+      e24KC2ThetaAboveNode010023133
+
+theorem e24KC2ThetaAboveNode01003200 :
+    adaptiveCoverCheck 11 thetaAboveCell01003200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003200
+    e24KC2ThetaAboveLeaf010032000 e24KC2ThetaAboveLeaf010032001 e24KC2ThetaAboveNode010032002
+      e24KC2ThetaAboveNode010032003
+
+theorem e24KC2ThetaAboveNode01003201 :
+    adaptiveCoverCheck 11 thetaAboveCell01003201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003201
+    e24KC2ThetaAboveLeaf010032010 e24KC2ThetaAboveLeaf010032011 e24KC2ThetaAboveNode010032012
+      e24KC2ThetaAboveNode010032013
+
+theorem e24KC2ThetaAboveNode01003202 :
+    adaptiveCoverCheck 11 thetaAboveCell01003202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003202
+    e24KC2ThetaAboveNode010032020 e24KC2ThetaAboveNode010032021 e24KC2ThetaAboveNode010032022
+      e24KC2ThetaAboveNode010032023
+
+theorem e24KC2ThetaAboveNode01003203 :
+    adaptiveCoverCheck 11 thetaAboveCell01003203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003203
+    e24KC2ThetaAboveNode010032030 e24KC2ThetaAboveNode010032031 e24KC2ThetaAboveNode010032032
+      e24KC2ThetaAboveNode010032033
+
+theorem e24KC2ThetaAboveNode01003210 :
+    adaptiveCoverCheck 11 thetaAboveCell01003210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003210
+    e24KC2ThetaAboveLeaf010032100 e24KC2ThetaAboveLeaf010032101 e24KC2ThetaAboveNode010032102
+      e24KC2ThetaAboveNode010032103
+
+theorem e24KC2ThetaAboveNode01003211 :
+    adaptiveCoverCheck 11 thetaAboveCell01003211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003211
+    e24KC2ThetaAboveLeaf010032110 e24KC2ThetaAboveLeaf010032111 e24KC2ThetaAboveNode010032112
+      e24KC2ThetaAboveNode010032113
+
+theorem e24KC2ThetaAboveNode01003212 :
+    adaptiveCoverCheck 11 thetaAboveCell01003212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003212
+    e24KC2ThetaAboveNode010032120 e24KC2ThetaAboveNode010032121 e24KC2ThetaAboveNode010032122
+      e24KC2ThetaAboveNode010032123
+
+theorem e24KC2ThetaAboveNode01003213 :
+    adaptiveCoverCheck 11 thetaAboveCell01003213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003213
+    e24KC2ThetaAboveNode010032130 e24KC2ThetaAboveNode010032131 e24KC2ThetaAboveNode010032132
+      e24KC2ThetaAboveNode010032133
+
+theorem e24KC2ThetaAboveNode01003300 :
+    adaptiveCoverCheck 11 thetaAboveCell01003300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003300
+    e24KC2ThetaAboveLeaf010033000 e24KC2ThetaAboveLeaf010033001 e24KC2ThetaAboveNode010033002
+      e24KC2ThetaAboveNode010033003
+
+theorem e24KC2ThetaAboveNode01003301 :
+    adaptiveCoverCheck 11 thetaAboveCell01003301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003301
+    e24KC2ThetaAboveLeaf010033010 e24KC2ThetaAboveLeaf010033011 e24KC2ThetaAboveNode010033012
+      e24KC2ThetaAboveNode010033013
+
+theorem e24KC2ThetaAboveNode01003302 :
+    adaptiveCoverCheck 11 thetaAboveCell01003302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003302
+    e24KC2ThetaAboveNode010033020 e24KC2ThetaAboveNode010033021 e24KC2ThetaAboveNode010033022
+      e24KC2ThetaAboveNode010033023
+
+theorem e24KC2ThetaAboveNode01003303 :
+    adaptiveCoverCheck 11 thetaAboveCell01003303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003303
+    e24KC2ThetaAboveNode010033030 e24KC2ThetaAboveNode010033031 e24KC2ThetaAboveNode010033032
+      e24KC2ThetaAboveNode010033033
+
+theorem e24KC2ThetaAboveNode01003310 :
+    adaptiveCoverCheck 11 thetaAboveCell01003310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003310
+    e24KC2ThetaAboveLeaf010033100 e24KC2ThetaAboveLeaf010033101 e24KC2ThetaAboveNode010033102
+      e24KC2ThetaAboveNode010033103
+
+theorem e24KC2ThetaAboveNode01003311 :
+    adaptiveCoverCheck 11 thetaAboveCell01003311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003311
+    e24KC2ThetaAboveLeaf010033110 e24KC2ThetaAboveLeaf010033111 e24KC2ThetaAboveNode010033112
+      e24KC2ThetaAboveNode010033113
+
+theorem e24KC2ThetaAboveNode01003312 :
+    adaptiveCoverCheck 11 thetaAboveCell01003312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003312
+    e24KC2ThetaAboveNode010033120 e24KC2ThetaAboveNode010033121 e24KC2ThetaAboveNode010033122
+      e24KC2ThetaAboveNode010033123
+
+theorem e24KC2ThetaAboveNode01003313 :
+    adaptiveCoverCheck 11 thetaAboveCell01003313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003313
+    e24KC2ThetaAboveNode010033130 e24KC2ThetaAboveNode010033131 e24KC2ThetaAboveNode010033132
+      e24KC2ThetaAboveNode010033133
+
+theorem e24KC2ThetaAboveNode01003330 :
+    adaptiveCoverCheck 11 thetaAboveCell01003330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003330
+    e24KC2ThetaAboveLeaf010033300 e24KC2ThetaAboveLeaf010033301 e24KC2ThetaAboveLeaf010033302
+      e24KC2ThetaAboveLeaf010033303
+
+theorem e24KC2ThetaAboveNode01003331 :
+    adaptiveCoverCheck 11 thetaAboveCell01003331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01003331
+    e24KC2ThetaAboveLeaf010033310 e24KC2ThetaAboveLeaf010033311 e24KC2ThetaAboveLeaf010033312
+      e24KC2ThetaAboveLeaf010033313
+
+theorem e24KC2ThetaAboveNode01012200 :
+    adaptiveCoverCheck 11 thetaAboveCell01012200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012200
+    e24KC2ThetaAboveLeaf010122000 e24KC2ThetaAboveLeaf010122001 e24KC2ThetaAboveNode010122002
+      e24KC2ThetaAboveNode010122003
+
+theorem e24KC2ThetaAboveNode01012201 :
+    adaptiveCoverCheck 11 thetaAboveCell01012201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012201
+    e24KC2ThetaAboveLeaf010122010 e24KC2ThetaAboveLeaf010122011 e24KC2ThetaAboveNode010122012
+      e24KC2ThetaAboveNode010122013
+
+theorem e24KC2ThetaAboveNode01012202 :
+    adaptiveCoverCheck 11 thetaAboveCell01012202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012202
+    e24KC2ThetaAboveNode010122020 e24KC2ThetaAboveNode010122021 e24KC2ThetaAboveNode010122022
+      e24KC2ThetaAboveNode010122023
+
+theorem e24KC2ThetaAboveNode01012203 :
+    adaptiveCoverCheck 11 thetaAboveCell01012203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012203
+    e24KC2ThetaAboveNode010122030 e24KC2ThetaAboveNode010122031 e24KC2ThetaAboveNode010122032
+      e24KC2ThetaAboveNode010122033
+
+theorem e24KC2ThetaAboveNode01012210 :
+    adaptiveCoverCheck 11 thetaAboveCell01012210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012210
+    e24KC2ThetaAboveLeaf010122100 e24KC2ThetaAboveLeaf010122101 e24KC2ThetaAboveNode010122102
+      e24KC2ThetaAboveNode010122103
+
+theorem e24KC2ThetaAboveNode01012211 :
+    adaptiveCoverCheck 11 thetaAboveCell01012211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012211
+    e24KC2ThetaAboveLeaf010122110 e24KC2ThetaAboveLeaf010122111 e24KC2ThetaAboveNode010122112
+      e24KC2ThetaAboveNode010122113
+
+theorem e24KC2ThetaAboveNode01012212 :
+    adaptiveCoverCheck 11 thetaAboveCell01012212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012212
+    e24KC2ThetaAboveNode010122120 e24KC2ThetaAboveNode010122121 e24KC2ThetaAboveNode010122122
+      e24KC2ThetaAboveNode010122123
+
+theorem e24KC2ThetaAboveNode01012213 :
+    adaptiveCoverCheck 11 thetaAboveCell01012213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012213
+    e24KC2ThetaAboveNode010122130 e24KC2ThetaAboveNode010122131 e24KC2ThetaAboveNode010122132
+      e24KC2ThetaAboveNode010122133
+
+theorem e24KC2ThetaAboveNode01012220 :
+    adaptiveCoverCheck 11 thetaAboveCell01012220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012220
+    e24KC2ThetaAboveLeaf010122200 e24KC2ThetaAboveLeaf010122201 e24KC2ThetaAboveLeaf010122202
+      e24KC2ThetaAboveLeaf010122203
+
+theorem e24KC2ThetaAboveNode01012221 :
+    adaptiveCoverCheck 11 thetaAboveCell01012221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012221
+    e24KC2ThetaAboveLeaf010122210 e24KC2ThetaAboveLeaf010122211 e24KC2ThetaAboveLeaf010122212
+      e24KC2ThetaAboveLeaf010122213
+
+theorem e24KC2ThetaAboveNode01012230 :
+    adaptiveCoverCheck 11 thetaAboveCell01012230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012230
+    e24KC2ThetaAboveLeaf010122300 e24KC2ThetaAboveLeaf010122301 e24KC2ThetaAboveLeaf010122302
+      e24KC2ThetaAboveLeaf010122303
+
+theorem e24KC2ThetaAboveNode01012231 :
+    adaptiveCoverCheck 11 thetaAboveCell01012231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012231
+    e24KC2ThetaAboveLeaf010122310 e24KC2ThetaAboveLeaf010122311 e24KC2ThetaAboveLeaf010122312
+      e24KC2ThetaAboveLeaf010122313
+
+theorem e24KC2ThetaAboveNode01012300 :
+    adaptiveCoverCheck 11 thetaAboveCell01012300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012300
+    e24KC2ThetaAboveLeaf010123000 e24KC2ThetaAboveLeaf010123001 e24KC2ThetaAboveNode010123002
+      e24KC2ThetaAboveNode010123003
+
+theorem e24KC2ThetaAboveNode01012301 :
+    adaptiveCoverCheck 11 thetaAboveCell01012301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012301
+    e24KC2ThetaAboveLeaf010123010 e24KC2ThetaAboveLeaf010123011 e24KC2ThetaAboveNode010123012
+      e24KC2ThetaAboveNode010123013
+
+theorem e24KC2ThetaAboveNode01012302 :
+    adaptiveCoverCheck 11 thetaAboveCell01012302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012302
+    e24KC2ThetaAboveNode010123020 e24KC2ThetaAboveNode010123021 e24KC2ThetaAboveNode010123022
+      e24KC2ThetaAboveNode010123023
+
+theorem e24KC2ThetaAboveNode01012303 :
+    adaptiveCoverCheck 11 thetaAboveCell01012303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012303
+    e24KC2ThetaAboveNode010123030 e24KC2ThetaAboveNode010123031 e24KC2ThetaAboveNode010123032
+      e24KC2ThetaAboveNode010123033
+
+theorem e24KC2ThetaAboveNode01012310 :
+    adaptiveCoverCheck 11 thetaAboveCell01012310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012310
+    e24KC2ThetaAboveLeaf010123100 e24KC2ThetaAboveLeaf010123101 e24KC2ThetaAboveNode010123102
+      e24KC2ThetaAboveNode010123103
+
+theorem e24KC2ThetaAboveNode01012311 :
+    adaptiveCoverCheck 11 thetaAboveCell01012311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012311
+    e24KC2ThetaAboveLeaf010123110 e24KC2ThetaAboveLeaf010123111 e24KC2ThetaAboveNode010123112
+      e24KC2ThetaAboveNode010123113
+
+theorem e24KC2ThetaAboveNode01012312 :
+    adaptiveCoverCheck 11 thetaAboveCell01012312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012312
+    e24KC2ThetaAboveNode010123120 e24KC2ThetaAboveNode010123121 e24KC2ThetaAboveNode010123122
+      e24KC2ThetaAboveNode010123123
+
+theorem e24KC2ThetaAboveNode01012313 :
+    adaptiveCoverCheck 11 thetaAboveCell01012313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012313
+    e24KC2ThetaAboveNode010123130 e24KC2ThetaAboveNode010123131 e24KC2ThetaAboveNode010123132
+      e24KC2ThetaAboveNode010123133
+
+theorem e24KC2ThetaAboveNode01012320 :
+    adaptiveCoverCheck 11 thetaAboveCell01012320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012320
+    e24KC2ThetaAboveLeaf010123200 e24KC2ThetaAboveLeaf010123201 e24KC2ThetaAboveLeaf010123202
+      e24KC2ThetaAboveLeaf010123203
+
+theorem e24KC2ThetaAboveNode01012321 :
+    adaptiveCoverCheck 11 thetaAboveCell01012321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012321
+    e24KC2ThetaAboveLeaf010123210 e24KC2ThetaAboveLeaf010123211 e24KC2ThetaAboveLeaf010123212
+      e24KC2ThetaAboveLeaf010123213
+
+theorem e24KC2ThetaAboveNode01012330 :
+    adaptiveCoverCheck 11 thetaAboveCell01012330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012330
+    e24KC2ThetaAboveLeaf010123300 e24KC2ThetaAboveLeaf010123301 e24KC2ThetaAboveLeaf010123302
+      e24KC2ThetaAboveLeaf010123303
+
+theorem e24KC2ThetaAboveNode01012331 :
+    adaptiveCoverCheck 11 thetaAboveCell01012331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01012331
+    e24KC2ThetaAboveLeaf010123310 e24KC2ThetaAboveLeaf010123311 e24KC2ThetaAboveLeaf010123312
+      e24KC2ThetaAboveLeaf010123313
+
+theorem e24KC2ThetaAboveNode01013200 :
+    adaptiveCoverCheck 11 thetaAboveCell01013200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013200
+    e24KC2ThetaAboveLeaf010132000 e24KC2ThetaAboveLeaf010132001 e24KC2ThetaAboveNode010132002
+      e24KC2ThetaAboveNode010132003
+
+theorem e24KC2ThetaAboveNode01013201 :
+    adaptiveCoverCheck 11 thetaAboveCell01013201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013201
+    e24KC2ThetaAboveLeaf010132010 e24KC2ThetaAboveLeaf010132011 e24KC2ThetaAboveNode010132012
+      e24KC2ThetaAboveNode010132013
+
+theorem e24KC2ThetaAboveNode01013202 :
+    adaptiveCoverCheck 11 thetaAboveCell01013202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013202
+    e24KC2ThetaAboveNode010132020 e24KC2ThetaAboveNode010132021 e24KC2ThetaAboveNode010132022
+      e24KC2ThetaAboveNode010132023
+
+theorem e24KC2ThetaAboveNode01013203 :
+    adaptiveCoverCheck 11 thetaAboveCell01013203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013203
+    e24KC2ThetaAboveNode010132030 e24KC2ThetaAboveNode010132031 e24KC2ThetaAboveNode010132032
+      e24KC2ThetaAboveNode010132033
+
+theorem e24KC2ThetaAboveNode01013210 :
+    adaptiveCoverCheck 11 thetaAboveCell01013210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013210
+    e24KC2ThetaAboveLeaf010132100 e24KC2ThetaAboveLeaf010132101 e24KC2ThetaAboveNode010132102
+      e24KC2ThetaAboveNode010132103
+
+theorem e24KC2ThetaAboveNode01013211 :
+    adaptiveCoverCheck 11 thetaAboveCell01013211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013211
+    e24KC2ThetaAboveLeaf010132110 e24KC2ThetaAboveLeaf010132111 e24KC2ThetaAboveNode010132112
+      e24KC2ThetaAboveNode010132113
+
+theorem e24KC2ThetaAboveNode01013212 :
+    adaptiveCoverCheck 11 thetaAboveCell01013212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013212
+    e24KC2ThetaAboveNode010132120 e24KC2ThetaAboveNode010132121 e24KC2ThetaAboveNode010132122
+      e24KC2ThetaAboveNode010132123
+
+theorem e24KC2ThetaAboveNode01013213 :
+    adaptiveCoverCheck 11 thetaAboveCell01013213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013213
+    e24KC2ThetaAboveNode010132130 e24KC2ThetaAboveNode010132131 e24KC2ThetaAboveNode010132132
+      e24KC2ThetaAboveNode010132133
+
+theorem e24KC2ThetaAboveNode01013220 :
+    adaptiveCoverCheck 11 thetaAboveCell01013220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013220
+    e24KC2ThetaAboveLeaf010132200 e24KC2ThetaAboveLeaf010132201 e24KC2ThetaAboveLeaf010132202
+      e24KC2ThetaAboveLeaf010132203
+
+theorem e24KC2ThetaAboveNode01013221 :
+    adaptiveCoverCheck 11 thetaAboveCell01013221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013221
+    e24KC2ThetaAboveLeaf010132210 e24KC2ThetaAboveLeaf010132211 e24KC2ThetaAboveLeaf010132212
+      e24KC2ThetaAboveLeaf010132213
+
+theorem e24KC2ThetaAboveNode01013230 :
+    adaptiveCoverCheck 11 thetaAboveCell01013230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013230
+    e24KC2ThetaAboveLeaf010132300 e24KC2ThetaAboveLeaf010132301 e24KC2ThetaAboveLeaf010132302
+      e24KC2ThetaAboveLeaf010132303
+
+theorem e24KC2ThetaAboveNode01013231 :
+    adaptiveCoverCheck 11 thetaAboveCell01013231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013231
+    e24KC2ThetaAboveLeaf010132310 e24KC2ThetaAboveLeaf010132311 e24KC2ThetaAboveLeaf010132312
+      e24KC2ThetaAboveLeaf010132313
+
+theorem e24KC2ThetaAboveNode01013300 :
+    adaptiveCoverCheck 11 thetaAboveCell01013300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013300
+    e24KC2ThetaAboveLeaf010133000 e24KC2ThetaAboveLeaf010133001 e24KC2ThetaAboveNode010133002
+      e24KC2ThetaAboveNode010133003
+
+theorem e24KC2ThetaAboveNode01013301 :
+    adaptiveCoverCheck 11 thetaAboveCell01013301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013301
+    e24KC2ThetaAboveLeaf010133010 e24KC2ThetaAboveLeaf010133011 e24KC2ThetaAboveNode010133012
+      e24KC2ThetaAboveNode010133013
+
+theorem e24KC2ThetaAboveNode01013302 :
+    adaptiveCoverCheck 11 thetaAboveCell01013302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013302
+    e24KC2ThetaAboveNode010133020 e24KC2ThetaAboveNode010133021 e24KC2ThetaAboveNode010133022
+      e24KC2ThetaAboveNode010133023
+
+theorem e24KC2ThetaAboveNode01013303 :
+    adaptiveCoverCheck 11 thetaAboveCell01013303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013303
+    e24KC2ThetaAboveNode010133030 e24KC2ThetaAboveNode010133031 e24KC2ThetaAboveNode010133032
+      e24KC2ThetaAboveNode010133033
+
+theorem e24KC2ThetaAboveNode01013310 :
+    adaptiveCoverCheck 11 thetaAboveCell01013310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013310
+    e24KC2ThetaAboveLeaf010133100 e24KC2ThetaAboveLeaf010133101 e24KC2ThetaAboveNode010133102
+      e24KC2ThetaAboveNode010133103
+
+theorem e24KC2ThetaAboveNode01013311 :
+    adaptiveCoverCheck 11 thetaAboveCell01013311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013311
+    e24KC2ThetaAboveLeaf010133110 e24KC2ThetaAboveLeaf010133111 e24KC2ThetaAboveNode010133112
+      e24KC2ThetaAboveNode010133113
+
+theorem e24KC2ThetaAboveNode01013312 :
+    adaptiveCoverCheck 11 thetaAboveCell01013312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013312
+    e24KC2ThetaAboveNode010133120 e24KC2ThetaAboveNode010133121 e24KC2ThetaAboveNode010133122
+      e24KC2ThetaAboveNode010133123
+
+theorem e24KC2ThetaAboveNode01013313 :
+    adaptiveCoverCheck 11 thetaAboveCell01013313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013313
+    e24KC2ThetaAboveNode010133130 e24KC2ThetaAboveNode010133131 e24KC2ThetaAboveNode010133132
+      e24KC2ThetaAboveNode010133133
+
+theorem e24KC2ThetaAboveNode01013320 :
+    adaptiveCoverCheck 11 thetaAboveCell01013320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013320
+    e24KC2ThetaAboveLeaf010133200 e24KC2ThetaAboveLeaf010133201 e24KC2ThetaAboveLeaf010133202
+      e24KC2ThetaAboveLeaf010133203
+
+theorem e24KC2ThetaAboveNode01013321 :
+    adaptiveCoverCheck 11 thetaAboveCell01013321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013321
+    e24KC2ThetaAboveLeaf010133210 e24KC2ThetaAboveLeaf010133211 e24KC2ThetaAboveLeaf010133212
+      e24KC2ThetaAboveLeaf010133213
+
+theorem e24KC2ThetaAboveNode01013330 :
+    adaptiveCoverCheck 11 thetaAboveCell01013330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013330
+    e24KC2ThetaAboveLeaf010133300 e24KC2ThetaAboveLeaf010133301 e24KC2ThetaAboveLeaf010133302
+      e24KC2ThetaAboveLeaf010133303
+
+theorem e24KC2ThetaAboveNode01013331 :
+    adaptiveCoverCheck 11 thetaAboveCell01013331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01013331
+    e24KC2ThetaAboveLeaf010133310 e24KC2ThetaAboveLeaf010133311 e24KC2ThetaAboveLeaf010133312
+      e24KC2ThetaAboveLeaf010133313
+
+theorem e24KC2ThetaAboveNode01102200 :
+    adaptiveCoverCheck 11 thetaAboveCell01102200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102200
+    e24KC2ThetaAboveLeaf011022000 e24KC2ThetaAboveLeaf011022001 e24KC2ThetaAboveNode011022002
+      e24KC2ThetaAboveNode011022003
+
+theorem e24KC2ThetaAboveNode01102201 :
+    adaptiveCoverCheck 11 thetaAboveCell01102201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102201
+    e24KC2ThetaAboveLeaf011022010 e24KC2ThetaAboveLeaf011022011 e24KC2ThetaAboveNode011022012
+      e24KC2ThetaAboveNode011022013
+
+theorem e24KC2ThetaAboveNode01102202 :
+    adaptiveCoverCheck 11 thetaAboveCell01102202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102202
+    e24KC2ThetaAboveNode011022020 e24KC2ThetaAboveNode011022021 e24KC2ThetaAboveNode011022022
+      e24KC2ThetaAboveNode011022023
+
+theorem e24KC2ThetaAboveNode01102203 :
+    adaptiveCoverCheck 11 thetaAboveCell01102203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102203
+    e24KC2ThetaAboveNode011022030 e24KC2ThetaAboveNode011022031 e24KC2ThetaAboveNode011022032
+      e24KC2ThetaAboveNode011022033
+
+theorem e24KC2ThetaAboveNode01102210 :
+    adaptiveCoverCheck 11 thetaAboveCell01102210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102210
+    e24KC2ThetaAboveLeaf011022100 e24KC2ThetaAboveLeaf011022101 e24KC2ThetaAboveNode011022102
+      e24KC2ThetaAboveNode011022103
+
+theorem e24KC2ThetaAboveNode01102211 :
+    adaptiveCoverCheck 11 thetaAboveCell01102211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102211
+    e24KC2ThetaAboveLeaf011022110 e24KC2ThetaAboveLeaf011022111 e24KC2ThetaAboveNode011022112
+      e24KC2ThetaAboveNode011022113
+
+theorem e24KC2ThetaAboveNode01102212 :
+    adaptiveCoverCheck 11 thetaAboveCell01102212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102212
+    e24KC2ThetaAboveNode011022120 e24KC2ThetaAboveNode011022121 e24KC2ThetaAboveNode011022122
+      e24KC2ThetaAboveNode011022123
+
+theorem e24KC2ThetaAboveNode01102213 :
+    adaptiveCoverCheck 11 thetaAboveCell01102213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102213
+    e24KC2ThetaAboveNode011022130 e24KC2ThetaAboveNode011022131 e24KC2ThetaAboveNode011022132
+      e24KC2ThetaAboveNode011022133
+
+theorem e24KC2ThetaAboveNode01102220 :
+    adaptiveCoverCheck 11 thetaAboveCell01102220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102220
+    e24KC2ThetaAboveLeaf011022200 e24KC2ThetaAboveLeaf011022201 e24KC2ThetaAboveLeaf011022202
+      e24KC2ThetaAboveLeaf011022203
+
+theorem e24KC2ThetaAboveNode01102221 :
+    adaptiveCoverCheck 11 thetaAboveCell01102221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102221
+    e24KC2ThetaAboveLeaf011022210 e24KC2ThetaAboveLeaf011022211 e24KC2ThetaAboveLeaf011022212
+      e24KC2ThetaAboveLeaf011022213
+
+theorem e24KC2ThetaAboveNode01102230 :
+    adaptiveCoverCheck 11 thetaAboveCell01102230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102230
+    e24KC2ThetaAboveLeaf011022300 e24KC2ThetaAboveLeaf011022301 e24KC2ThetaAboveLeaf011022302
+      e24KC2ThetaAboveLeaf011022303
+
+theorem e24KC2ThetaAboveNode01102231 :
+    adaptiveCoverCheck 11 thetaAboveCell01102231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102231
+    e24KC2ThetaAboveLeaf011022310 e24KC2ThetaAboveLeaf011022311 e24KC2ThetaAboveLeaf011022312
+      e24KC2ThetaAboveLeaf011022313
+
+theorem e24KC2ThetaAboveNode01102300 :
+    adaptiveCoverCheck 11 thetaAboveCell01102300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102300
+    e24KC2ThetaAboveLeaf011023000 e24KC2ThetaAboveLeaf011023001 e24KC2ThetaAboveNode011023002
+      e24KC2ThetaAboveNode011023003
+
+theorem e24KC2ThetaAboveNode01102301 :
+    adaptiveCoverCheck 11 thetaAboveCell01102301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102301
+    e24KC2ThetaAboveLeaf011023010 e24KC2ThetaAboveLeaf011023011 e24KC2ThetaAboveNode011023012
+      e24KC2ThetaAboveNode011023013
+
+theorem e24KC2ThetaAboveNode01102302 :
+    adaptiveCoverCheck 11 thetaAboveCell01102302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102302
+    e24KC2ThetaAboveNode011023020 e24KC2ThetaAboveNode011023021 e24KC2ThetaAboveNode011023022
+      e24KC2ThetaAboveNode011023023
+
+theorem e24KC2ThetaAboveNode01102303 :
+    adaptiveCoverCheck 11 thetaAboveCell01102303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102303
+    e24KC2ThetaAboveNode011023030 e24KC2ThetaAboveNode011023031 e24KC2ThetaAboveNode011023032
+      e24KC2ThetaAboveNode011023033
+
+theorem e24KC2ThetaAboveNode01102310 :
+    adaptiveCoverCheck 11 thetaAboveCell01102310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102310
+    e24KC2ThetaAboveLeaf011023100 e24KC2ThetaAboveLeaf011023101 e24KC2ThetaAboveNode011023102
+      e24KC2ThetaAboveNode011023103
+
+theorem e24KC2ThetaAboveNode01102311 :
+    adaptiveCoverCheck 11 thetaAboveCell01102311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102311
+    e24KC2ThetaAboveLeaf011023110 e24KC2ThetaAboveLeaf011023111 e24KC2ThetaAboveNode011023112
+      e24KC2ThetaAboveNode011023113
+
+theorem e24KC2ThetaAboveNode01102312 :
+    adaptiveCoverCheck 11 thetaAboveCell01102312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102312
+    e24KC2ThetaAboveNode011023120 e24KC2ThetaAboveNode011023121 e24KC2ThetaAboveNode011023122
+      e24KC2ThetaAboveNode011023123
+
+theorem e24KC2ThetaAboveNode01102313 :
+    adaptiveCoverCheck 11 thetaAboveCell01102313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102313
+    e24KC2ThetaAboveNode011023130 e24KC2ThetaAboveNode011023131 e24KC2ThetaAboveNode011023132
+      e24KC2ThetaAboveNode011023133
+
+theorem e24KC2ThetaAboveNode01102320 :
+    adaptiveCoverCheck 11 thetaAboveCell01102320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102320
+    e24KC2ThetaAboveLeaf011023200 e24KC2ThetaAboveLeaf011023201 e24KC2ThetaAboveLeaf011023202
+      e24KC2ThetaAboveLeaf011023203
+
+theorem e24KC2ThetaAboveNode01102321 :
+    adaptiveCoverCheck 11 thetaAboveCell01102321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102321
+    e24KC2ThetaAboveLeaf011023210 e24KC2ThetaAboveLeaf011023211 e24KC2ThetaAboveLeaf011023212
+      e24KC2ThetaAboveLeaf011023213
+
+theorem e24KC2ThetaAboveNode01102330 :
+    adaptiveCoverCheck 11 thetaAboveCell01102330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102330
+    e24KC2ThetaAboveLeaf011023300 e24KC2ThetaAboveLeaf011023301 e24KC2ThetaAboveLeaf011023302
+      e24KC2ThetaAboveLeaf011023303
+
+theorem e24KC2ThetaAboveNode01102331 :
+    adaptiveCoverCheck 11 thetaAboveCell01102331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01102331
+    e24KC2ThetaAboveLeaf011023310 e24KC2ThetaAboveLeaf011023311 e24KC2ThetaAboveLeaf011023312
+      e24KC2ThetaAboveLeaf011023313
+
+theorem e24KC2ThetaAboveNode01103200 :
+    adaptiveCoverCheck 11 thetaAboveCell01103200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103200
+    e24KC2ThetaAboveLeaf011032000 e24KC2ThetaAboveLeaf011032001 e24KC2ThetaAboveNode011032002
+      e24KC2ThetaAboveNode011032003
+
+theorem e24KC2ThetaAboveNode01103201 :
+    adaptiveCoverCheck 11 thetaAboveCell01103201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103201
+    e24KC2ThetaAboveLeaf011032010 e24KC2ThetaAboveLeaf011032011 e24KC2ThetaAboveNode011032012
+      e24KC2ThetaAboveNode011032013
+
+theorem e24KC2ThetaAboveNode01103202 :
+    adaptiveCoverCheck 11 thetaAboveCell01103202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103202
+    e24KC2ThetaAboveNode011032020 e24KC2ThetaAboveNode011032021 e24KC2ThetaAboveNode011032022
+      e24KC2ThetaAboveNode011032023
+
+theorem e24KC2ThetaAboveNode01103203 :
+    adaptiveCoverCheck 11 thetaAboveCell01103203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103203
+    e24KC2ThetaAboveNode011032030 e24KC2ThetaAboveNode011032031 e24KC2ThetaAboveNode011032032
+      e24KC2ThetaAboveNode011032033
+
+theorem e24KC2ThetaAboveNode01103210 :
+    adaptiveCoverCheck 11 thetaAboveCell01103210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103210
+    e24KC2ThetaAboveLeaf011032100 e24KC2ThetaAboveLeaf011032101 e24KC2ThetaAboveNode011032102
+      e24KC2ThetaAboveNode011032103
+
+theorem e24KC2ThetaAboveNode01103211 :
+    adaptiveCoverCheck 11 thetaAboveCell01103211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103211
+    e24KC2ThetaAboveLeaf011032110 e24KC2ThetaAboveLeaf011032111 e24KC2ThetaAboveNode011032112
+      e24KC2ThetaAboveNode011032113
+
+theorem e24KC2ThetaAboveNode01103212 :
+    adaptiveCoverCheck 11 thetaAboveCell01103212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103212
+    e24KC2ThetaAboveNode011032120 e24KC2ThetaAboveNode011032121 e24KC2ThetaAboveNode011032122
+      e24KC2ThetaAboveNode011032123
+
+theorem e24KC2ThetaAboveNode01103213 :
+    adaptiveCoverCheck 11 thetaAboveCell01103213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103213
+    e24KC2ThetaAboveNode011032130 e24KC2ThetaAboveNode011032131 e24KC2ThetaAboveNode011032132
+      e24KC2ThetaAboveNode011032133
+
+theorem e24KC2ThetaAboveNode01103220 :
+    adaptiveCoverCheck 11 thetaAboveCell01103220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103220
+    e24KC2ThetaAboveLeaf011032200 e24KC2ThetaAboveLeaf011032201 e24KC2ThetaAboveLeaf011032202
+      e24KC2ThetaAboveLeaf011032203
+
+theorem e24KC2ThetaAboveNode01103221 :
+    adaptiveCoverCheck 11 thetaAboveCell01103221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103221
+    e24KC2ThetaAboveLeaf011032210 e24KC2ThetaAboveLeaf011032211 e24KC2ThetaAboveLeaf011032212
+      e24KC2ThetaAboveLeaf011032213
+
+theorem e24KC2ThetaAboveNode01103230 :
+    adaptiveCoverCheck 11 thetaAboveCell01103230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103230
+    e24KC2ThetaAboveLeaf011032300 e24KC2ThetaAboveLeaf011032301 e24KC2ThetaAboveLeaf011032302
+      e24KC2ThetaAboveLeaf011032303
+
+theorem e24KC2ThetaAboveNode01103231 :
+    adaptiveCoverCheck 11 thetaAboveCell01103231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103231
+    e24KC2ThetaAboveLeaf011032310 e24KC2ThetaAboveLeaf011032311 e24KC2ThetaAboveLeaf011032312
+      e24KC2ThetaAboveLeaf011032313
+
+theorem e24KC2ThetaAboveNode01103300 :
+    adaptiveCoverCheck 11 thetaAboveCell01103300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103300
+    e24KC2ThetaAboveLeaf011033000 e24KC2ThetaAboveLeaf011033001 e24KC2ThetaAboveNode011033002
+      e24KC2ThetaAboveNode011033003
+
+theorem e24KC2ThetaAboveNode01103301 :
+    adaptiveCoverCheck 11 thetaAboveCell01103301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103301
+    e24KC2ThetaAboveLeaf011033010 e24KC2ThetaAboveLeaf011033011 e24KC2ThetaAboveNode011033012
+      e24KC2ThetaAboveNode011033013
+
+theorem e24KC2ThetaAboveNode01103302 :
+    adaptiveCoverCheck 11 thetaAboveCell01103302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103302
+    e24KC2ThetaAboveNode011033020 e24KC2ThetaAboveNode011033021 e24KC2ThetaAboveNode011033022
+      e24KC2ThetaAboveNode011033023
+
+theorem e24KC2ThetaAboveNode01103303 :
+    adaptiveCoverCheck 11 thetaAboveCell01103303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103303
+    e24KC2ThetaAboveNode011033030 e24KC2ThetaAboveNode011033031 e24KC2ThetaAboveNode011033032
+      e24KC2ThetaAboveNode011033033
+
+theorem e24KC2ThetaAboveNode01103310 :
+    adaptiveCoverCheck 11 thetaAboveCell01103310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103310
+    e24KC2ThetaAboveLeaf011033100 e24KC2ThetaAboveLeaf011033101 e24KC2ThetaAboveNode011033102
+      e24KC2ThetaAboveNode011033103
+
+theorem e24KC2ThetaAboveNode01103311 :
+    adaptiveCoverCheck 11 thetaAboveCell01103311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103311
+    e24KC2ThetaAboveLeaf011033110 e24KC2ThetaAboveLeaf011033111 e24KC2ThetaAboveNode011033112
+      e24KC2ThetaAboveNode011033113
+
+theorem e24KC2ThetaAboveNode01103312 :
+    adaptiveCoverCheck 11 thetaAboveCell01103312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103312
+    e24KC2ThetaAboveNode011033120 e24KC2ThetaAboveNode011033121 e24KC2ThetaAboveNode011033122
+      e24KC2ThetaAboveNode011033123
+
+theorem e24KC2ThetaAboveNode01103313 :
+    adaptiveCoverCheck 11 thetaAboveCell01103313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103313
+    e24KC2ThetaAboveNode011033130 e24KC2ThetaAboveNode011033131 e24KC2ThetaAboveNode011033132
+      e24KC2ThetaAboveNode011033133
+
+theorem e24KC2ThetaAboveNode01103320 :
+    adaptiveCoverCheck 11 thetaAboveCell01103320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103320
+    e24KC2ThetaAboveLeaf011033200 e24KC2ThetaAboveLeaf011033201 e24KC2ThetaAboveLeaf011033202
+      e24KC2ThetaAboveLeaf011033203
+
+theorem e24KC2ThetaAboveNode01103321 :
+    adaptiveCoverCheck 11 thetaAboveCell01103321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103321
+    e24KC2ThetaAboveLeaf011033210 e24KC2ThetaAboveLeaf011033211 e24KC2ThetaAboveLeaf011033212
+      e24KC2ThetaAboveLeaf011033213
+
+theorem e24KC2ThetaAboveNode01103330 :
+    adaptiveCoverCheck 11 thetaAboveCell01103330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103330
+    e24KC2ThetaAboveLeaf011033300 e24KC2ThetaAboveLeaf011033301 e24KC2ThetaAboveLeaf011033302
+      e24KC2ThetaAboveLeaf011033303
+
+theorem e24KC2ThetaAboveNode01103331 :
+    adaptiveCoverCheck 11 thetaAboveCell01103331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01103331
+    e24KC2ThetaAboveLeaf011033310 e24KC2ThetaAboveLeaf011033311 e24KC2ThetaAboveLeaf011033312
+      e24KC2ThetaAboveLeaf011033313
+
+theorem e24KC2ThetaAboveNode01112200 :
+    adaptiveCoverCheck 11 thetaAboveCell01112200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112200
+    e24KC2ThetaAboveLeaf011122000 e24KC2ThetaAboveLeaf011122001 e24KC2ThetaAboveNode011122002
+      e24KC2ThetaAboveNode011122003
+
+theorem e24KC2ThetaAboveNode01112201 :
+    adaptiveCoverCheck 11 thetaAboveCell01112201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112201
+    e24KC2ThetaAboveLeaf011122010 e24KC2ThetaAboveLeaf011122011 e24KC2ThetaAboveNode011122012
+      e24KC2ThetaAboveNode011122013
+
+theorem e24KC2ThetaAboveNode01112202 :
+    adaptiveCoverCheck 11 thetaAboveCell01112202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112202
+    e24KC2ThetaAboveNode011122020 e24KC2ThetaAboveNode011122021 e24KC2ThetaAboveNode011122022
+      e24KC2ThetaAboveNode011122023
+
+theorem e24KC2ThetaAboveNode01112203 :
+    adaptiveCoverCheck 11 thetaAboveCell01112203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112203
+    e24KC2ThetaAboveNode011122030 e24KC2ThetaAboveNode011122031 e24KC2ThetaAboveNode011122032
+      e24KC2ThetaAboveNode011122033
+
+theorem e24KC2ThetaAboveNode01112210 :
+    adaptiveCoverCheck 11 thetaAboveCell01112210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112210
+    e24KC2ThetaAboveLeaf011122100 e24KC2ThetaAboveLeaf011122101 e24KC2ThetaAboveNode011122102
+      e24KC2ThetaAboveNode011122103
+
+theorem e24KC2ThetaAboveNode01112211 :
+    adaptiveCoverCheck 11 thetaAboveCell01112211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112211
+    e24KC2ThetaAboveLeaf011122110 e24KC2ThetaAboveLeaf011122111 e24KC2ThetaAboveNode011122112
+      e24KC2ThetaAboveNode011122113
+
+theorem e24KC2ThetaAboveNode01112212 :
+    adaptiveCoverCheck 11 thetaAboveCell01112212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112212
+    e24KC2ThetaAboveNode011122120 e24KC2ThetaAboveNode011122121 e24KC2ThetaAboveNode011122122
+      e24KC2ThetaAboveNode011122123
+
+theorem e24KC2ThetaAboveNode01112213 :
+    adaptiveCoverCheck 11 thetaAboveCell01112213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112213
+    e24KC2ThetaAboveNode011122130 e24KC2ThetaAboveNode011122131 e24KC2ThetaAboveNode011122132
+      e24KC2ThetaAboveNode011122133
+
+theorem e24KC2ThetaAboveNode01112220 :
+    adaptiveCoverCheck 11 thetaAboveCell01112220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112220
+    e24KC2ThetaAboveLeaf011122200 e24KC2ThetaAboveLeaf011122201 e24KC2ThetaAboveLeaf011122202
+      e24KC2ThetaAboveLeaf011122203
+
+theorem e24KC2ThetaAboveNode01112221 :
+    adaptiveCoverCheck 11 thetaAboveCell01112221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112221
+    e24KC2ThetaAboveLeaf011122210 e24KC2ThetaAboveLeaf011122211 e24KC2ThetaAboveLeaf011122212
+      e24KC2ThetaAboveLeaf011122213
+
+theorem e24KC2ThetaAboveNode01112230 :
+    adaptiveCoverCheck 11 thetaAboveCell01112230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112230
+    e24KC2ThetaAboveLeaf011122300 e24KC2ThetaAboveLeaf011122301 e24KC2ThetaAboveLeaf011122302
+      e24KC2ThetaAboveLeaf011122303
+
+theorem e24KC2ThetaAboveNode01112231 :
+    adaptiveCoverCheck 11 thetaAboveCell01112231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112231
+    e24KC2ThetaAboveLeaf011122310 e24KC2ThetaAboveLeaf011122311 e24KC2ThetaAboveLeaf011122312
+      e24KC2ThetaAboveLeaf011122313
+
+theorem e24KC2ThetaAboveNode01112300 :
+    adaptiveCoverCheck 11 thetaAboveCell01112300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112300
+    e24KC2ThetaAboveLeaf011123000 e24KC2ThetaAboveLeaf011123001 e24KC2ThetaAboveNode011123002
+      e24KC2ThetaAboveNode011123003
+
+theorem e24KC2ThetaAboveNode01112301 :
+    adaptiveCoverCheck 11 thetaAboveCell01112301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112301
+    e24KC2ThetaAboveLeaf011123010 e24KC2ThetaAboveLeaf011123011 e24KC2ThetaAboveNode011123012
+      e24KC2ThetaAboveNode011123013
+
+theorem e24KC2ThetaAboveNode01112302 :
+    adaptiveCoverCheck 11 thetaAboveCell01112302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112302
+    e24KC2ThetaAboveNode011123020 e24KC2ThetaAboveNode011123021 e24KC2ThetaAboveNode011123022
+      e24KC2ThetaAboveNode011123023
+
+theorem e24KC2ThetaAboveNode01112303 :
+    adaptiveCoverCheck 11 thetaAboveCell01112303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112303
+    e24KC2ThetaAboveNode011123030 e24KC2ThetaAboveNode011123031 e24KC2ThetaAboveNode011123032
+      e24KC2ThetaAboveNode011123033
+
+theorem e24KC2ThetaAboveNode01112310 :
+    adaptiveCoverCheck 11 thetaAboveCell01112310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112310
+    e24KC2ThetaAboveLeaf011123100 e24KC2ThetaAboveLeaf011123101 e24KC2ThetaAboveNode011123102
+      e24KC2ThetaAboveNode011123103
+
+theorem e24KC2ThetaAboveNode01112311 :
+    adaptiveCoverCheck 11 thetaAboveCell01112311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112311
+    e24KC2ThetaAboveLeaf011123110 e24KC2ThetaAboveLeaf011123111 e24KC2ThetaAboveNode011123112
+      e24KC2ThetaAboveNode011123113
+
+theorem e24KC2ThetaAboveNode01112312 :
+    adaptiveCoverCheck 11 thetaAboveCell01112312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112312
+    e24KC2ThetaAboveNode011123120 e24KC2ThetaAboveNode011123121 e24KC2ThetaAboveNode011123122
+      e24KC2ThetaAboveNode011123123
+
+theorem e24KC2ThetaAboveNode01112313 :
+    adaptiveCoverCheck 11 thetaAboveCell01112313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112313
+    e24KC2ThetaAboveNode011123130 e24KC2ThetaAboveNode011123131 e24KC2ThetaAboveNode011123132
+      e24KC2ThetaAboveNode011123133
+
+theorem e24KC2ThetaAboveNode01112320 :
+    adaptiveCoverCheck 11 thetaAboveCell01112320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112320
+    e24KC2ThetaAboveLeaf011123200 e24KC2ThetaAboveLeaf011123201 e24KC2ThetaAboveLeaf011123202
+      e24KC2ThetaAboveLeaf011123203
+
+theorem e24KC2ThetaAboveNode01112321 :
+    adaptiveCoverCheck 11 thetaAboveCell01112321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112321
+    e24KC2ThetaAboveLeaf011123210 e24KC2ThetaAboveLeaf011123211 e24KC2ThetaAboveLeaf011123212
+      e24KC2ThetaAboveLeaf011123213
+
+theorem e24KC2ThetaAboveNode01112330 :
+    adaptiveCoverCheck 11 thetaAboveCell01112330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112330
+    e24KC2ThetaAboveLeaf011123300 e24KC2ThetaAboveLeaf011123301 e24KC2ThetaAboveLeaf011123302
+      e24KC2ThetaAboveLeaf011123303
+
+theorem e24KC2ThetaAboveNode01112331 :
+    adaptiveCoverCheck 11 thetaAboveCell01112331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01112331
+    e24KC2ThetaAboveLeaf011123310 e24KC2ThetaAboveLeaf011123311 e24KC2ThetaAboveLeaf011123312
+      e24KC2ThetaAboveLeaf011123313
+
+theorem e24KC2ThetaAboveNode01113200 :
+    adaptiveCoverCheck 11 thetaAboveCell01113200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113200
+    e24KC2ThetaAboveLeaf011132000 e24KC2ThetaAboveLeaf011132001 e24KC2ThetaAboveNode011132002
+      e24KC2ThetaAboveNode011132003
+
+theorem e24KC2ThetaAboveNode01113201 :
+    adaptiveCoverCheck 11 thetaAboveCell01113201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113201
+    e24KC2ThetaAboveLeaf011132010 e24KC2ThetaAboveLeaf011132011 e24KC2ThetaAboveNode011132012
+      e24KC2ThetaAboveNode011132013
+
+theorem e24KC2ThetaAboveNode01113202 :
+    adaptiveCoverCheck 11 thetaAboveCell01113202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113202
+    e24KC2ThetaAboveNode011132020 e24KC2ThetaAboveNode011132021 e24KC2ThetaAboveNode011132022
+      e24KC2ThetaAboveNode011132023
+
+theorem e24KC2ThetaAboveNode01113203 :
+    adaptiveCoverCheck 11 thetaAboveCell01113203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113203
+    e24KC2ThetaAboveNode011132030 e24KC2ThetaAboveNode011132031 e24KC2ThetaAboveNode011132032
+      e24KC2ThetaAboveNode011132033
+
+theorem e24KC2ThetaAboveNode01113210 :
+    adaptiveCoverCheck 11 thetaAboveCell01113210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113210
+    e24KC2ThetaAboveLeaf011132100 e24KC2ThetaAboveLeaf011132101 e24KC2ThetaAboveLeaf011132102
+      e24KC2ThetaAboveLeaf011132103
+
+theorem e24KC2ThetaAboveNode01113211 :
+    adaptiveCoverCheck 11 thetaAboveCell01113211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113211
+    e24KC2ThetaAboveLeaf011132110 e24KC2ThetaAboveLeaf011132111 e24KC2ThetaAboveLeaf011132112
+      e24KC2ThetaAboveLeaf011132113
+
+theorem e24KC2ThetaAboveNode01113212 :
+    adaptiveCoverCheck 11 thetaAboveCell01113212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113212
+    e24KC2ThetaAboveNode011132120 e24KC2ThetaAboveNode011132121 e24KC2ThetaAboveNode011132122
+      e24KC2ThetaAboveNode011132123
+
+theorem e24KC2ThetaAboveNode01113213 :
+    adaptiveCoverCheck 11 thetaAboveCell01113213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113213
+    e24KC2ThetaAboveNode011132130 e24KC2ThetaAboveNode011132131 e24KC2ThetaAboveNode011132132
+      e24KC2ThetaAboveNode011132133
+
+theorem e24KC2ThetaAboveNode01113220 :
+    adaptiveCoverCheck 11 thetaAboveCell01113220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113220
+    e24KC2ThetaAboveLeaf011132200 e24KC2ThetaAboveLeaf011132201 e24KC2ThetaAboveLeaf011132202
+      e24KC2ThetaAboveLeaf011132203
+
+theorem e24KC2ThetaAboveNode01113221 :
+    adaptiveCoverCheck 11 thetaAboveCell01113221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113221
+    e24KC2ThetaAboveLeaf011132210 e24KC2ThetaAboveLeaf011132211 e24KC2ThetaAboveLeaf011132212
+      e24KC2ThetaAboveLeaf011132213
+
+theorem e24KC2ThetaAboveNode01113230 :
+    adaptiveCoverCheck 11 thetaAboveCell01113230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113230
+    e24KC2ThetaAboveLeaf011132300 e24KC2ThetaAboveLeaf011132301 e24KC2ThetaAboveLeaf011132302
+      e24KC2ThetaAboveLeaf011132303
+
+theorem e24KC2ThetaAboveNode01113231 :
+    adaptiveCoverCheck 11 thetaAboveCell01113231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113231
+    e24KC2ThetaAboveLeaf011132310 e24KC2ThetaAboveLeaf011132311 e24KC2ThetaAboveLeaf011132312
+      e24KC2ThetaAboveLeaf011132313
+
+theorem e24KC2ThetaAboveNode01113300 :
+    adaptiveCoverCheck 11 thetaAboveCell01113300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113300
+    e24KC2ThetaAboveLeaf011133000 e24KC2ThetaAboveLeaf011133001 e24KC2ThetaAboveLeaf011133002
+      e24KC2ThetaAboveLeaf011133003
+
+theorem e24KC2ThetaAboveNode01113301 :
+    adaptiveCoverCheck 11 thetaAboveCell01113301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113301
+    e24KC2ThetaAboveLeaf011133010 e24KC2ThetaAboveLeaf011133011 e24KC2ThetaAboveLeaf011133012
+      e24KC2ThetaAboveLeaf011133013
+
+theorem e24KC2ThetaAboveNode01113302 :
+    adaptiveCoverCheck 11 thetaAboveCell01113302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113302
+    e24KC2ThetaAboveNode011133020 e24KC2ThetaAboveNode011133021 e24KC2ThetaAboveNode011133022
+      e24KC2ThetaAboveNode011133023
+
+theorem e24KC2ThetaAboveNode01113303 :
+    adaptiveCoverCheck 11 thetaAboveCell01113303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113303
+    e24KC2ThetaAboveNode011133030 e24KC2ThetaAboveNode011133031 e24KC2ThetaAboveNode011133032
+      e24KC2ThetaAboveNode011133033
+
+theorem e24KC2ThetaAboveNode01113310 :
+    adaptiveCoverCheck 11 thetaAboveCell01113310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113310
+    e24KC2ThetaAboveLeaf011133100 e24KC2ThetaAboveLeaf011133101 e24KC2ThetaAboveLeaf011133102
+      e24KC2ThetaAboveLeaf011133103
+
+theorem e24KC2ThetaAboveNode01113311 :
+    adaptiveCoverCheck 11 thetaAboveCell01113311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113311
+    e24KC2ThetaAboveLeaf011133110 e24KC2ThetaAboveLeaf011133111 e24KC2ThetaAboveLeaf011133112
+      e24KC2ThetaAboveLeaf011133113
+
+theorem e24KC2ThetaAboveNode01113312 :
+    adaptiveCoverCheck 11 thetaAboveCell01113312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113312
+    e24KC2ThetaAboveNode011133120 e24KC2ThetaAboveNode011133121 e24KC2ThetaAboveNode011133122
+      e24KC2ThetaAboveNode011133123
+
+theorem e24KC2ThetaAboveNode01113313 :
+    adaptiveCoverCheck 11 thetaAboveCell01113313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113313
+    e24KC2ThetaAboveNode011133130 e24KC2ThetaAboveNode011133131 e24KC2ThetaAboveNode011133132
+      e24KC2ThetaAboveNode011133133
+
+theorem e24KC2ThetaAboveNode01113320 :
+    adaptiveCoverCheck 11 thetaAboveCell01113320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113320
+    e24KC2ThetaAboveLeaf011133200 e24KC2ThetaAboveLeaf011133201 e24KC2ThetaAboveLeaf011133202
+      e24KC2ThetaAboveLeaf011133203
+
+theorem e24KC2ThetaAboveNode01113321 :
+    adaptiveCoverCheck 11 thetaAboveCell01113321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113321
+    e24KC2ThetaAboveLeaf011133210 e24KC2ThetaAboveLeaf011133211 e24KC2ThetaAboveLeaf011133212
+      e24KC2ThetaAboveLeaf011133213
+
+theorem e24KC2ThetaAboveNode01113330 :
+    adaptiveCoverCheck 11 thetaAboveCell01113330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113330
+    e24KC2ThetaAboveLeaf011133300 e24KC2ThetaAboveLeaf011133301 e24KC2ThetaAboveLeaf011133302
+      e24KC2ThetaAboveLeaf011133303
+
+theorem e24KC2ThetaAboveNode01113331 :
+    adaptiveCoverCheck 11 thetaAboveCell01113331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell01113331
+    e24KC2ThetaAboveLeaf011133310 e24KC2ThetaAboveLeaf011133311 e24KC2ThetaAboveLeaf011133312
+      e24KC2ThetaAboveLeaf011133313
+
+theorem e24KC2ThetaAboveNode10002200 :
+    adaptiveCoverCheck 11 thetaAboveCell10002200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002200
+    e24KC2ThetaAboveLeaf100022000 e24KC2ThetaAboveLeaf100022001 e24KC2ThetaAboveLeaf100022002
+      e24KC2ThetaAboveLeaf100022003
+
+theorem e24KC2ThetaAboveNode10002201 :
+    adaptiveCoverCheck 11 thetaAboveCell10002201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002201
+    e24KC2ThetaAboveLeaf100022010 e24KC2ThetaAboveLeaf100022011 e24KC2ThetaAboveLeaf100022012
+      e24KC2ThetaAboveLeaf100022013
+
+theorem e24KC2ThetaAboveNode10002202 :
+    adaptiveCoverCheck 11 thetaAboveCell10002202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002202
+    e24KC2ThetaAboveNode100022020 e24KC2ThetaAboveNode100022021 e24KC2ThetaAboveNode100022022
+      e24KC2ThetaAboveNode100022023
+
+theorem e24KC2ThetaAboveNode10002203 :
+    adaptiveCoverCheck 11 thetaAboveCell10002203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002203
+    e24KC2ThetaAboveNode100022030 e24KC2ThetaAboveNode100022031 e24KC2ThetaAboveNode100022032
+      e24KC2ThetaAboveNode100022033
+
+theorem e24KC2ThetaAboveNode10002210 :
+    adaptiveCoverCheck 11 thetaAboveCell10002210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002210
+    e24KC2ThetaAboveLeaf100022100 e24KC2ThetaAboveLeaf100022101 e24KC2ThetaAboveLeaf100022102
+      e24KC2ThetaAboveLeaf100022103
+
+theorem e24KC2ThetaAboveNode10002211 :
+    adaptiveCoverCheck 11 thetaAboveCell10002211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002211
+    e24KC2ThetaAboveLeaf100022110 e24KC2ThetaAboveLeaf100022111 e24KC2ThetaAboveLeaf100022112
+      e24KC2ThetaAboveLeaf100022113
+
+theorem e24KC2ThetaAboveNode10002212 :
+    adaptiveCoverCheck 11 thetaAboveCell10002212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002212
+    e24KC2ThetaAboveNode100022120 e24KC2ThetaAboveNode100022121 e24KC2ThetaAboveNode100022122
+      e24KC2ThetaAboveNode100022123
+
+theorem e24KC2ThetaAboveNode10002213 :
+    adaptiveCoverCheck 11 thetaAboveCell10002213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002213
+    e24KC2ThetaAboveNode100022130 e24KC2ThetaAboveNode100022131 e24KC2ThetaAboveNode100022132
+      e24KC2ThetaAboveNode100022133
+
+theorem e24KC2ThetaAboveNode10002220 :
+    adaptiveCoverCheck 11 thetaAboveCell10002220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002220
+    e24KC2ThetaAboveLeaf100022200 e24KC2ThetaAboveLeaf100022201 e24KC2ThetaAboveLeaf100022202
+      e24KC2ThetaAboveLeaf100022203
+
+theorem e24KC2ThetaAboveNode10002221 :
+    adaptiveCoverCheck 11 thetaAboveCell10002221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002221
+    e24KC2ThetaAboveLeaf100022210 e24KC2ThetaAboveLeaf100022211 e24KC2ThetaAboveLeaf100022212
+      e24KC2ThetaAboveLeaf100022213
+
+theorem e24KC2ThetaAboveNode10002230 :
+    adaptiveCoverCheck 11 thetaAboveCell10002230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002230
+    e24KC2ThetaAboveLeaf100022300 e24KC2ThetaAboveLeaf100022301 e24KC2ThetaAboveLeaf100022302
+      e24KC2ThetaAboveLeaf100022303
+
+theorem e24KC2ThetaAboveNode10002231 :
+    adaptiveCoverCheck 11 thetaAboveCell10002231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002231
+    e24KC2ThetaAboveLeaf100022310 e24KC2ThetaAboveLeaf100022311 e24KC2ThetaAboveLeaf100022312
+      e24KC2ThetaAboveLeaf100022313
+
+theorem e24KC2ThetaAboveNode10002300 :
+    adaptiveCoverCheck 11 thetaAboveCell10002300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002300
+    e24KC2ThetaAboveLeaf100023000 e24KC2ThetaAboveLeaf100023001 e24KC2ThetaAboveLeaf100023002
+      e24KC2ThetaAboveLeaf100023003
+
+theorem e24KC2ThetaAboveNode10002301 :
+    adaptiveCoverCheck 11 thetaAboveCell10002301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002301
+    e24KC2ThetaAboveLeaf100023010 e24KC2ThetaAboveLeaf100023011 e24KC2ThetaAboveLeaf100023012
+      e24KC2ThetaAboveLeaf100023013
+
+theorem e24KC2ThetaAboveNode10002302 :
+    adaptiveCoverCheck 11 thetaAboveCell10002302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002302
+    e24KC2ThetaAboveNode100023020 e24KC2ThetaAboveNode100023021 e24KC2ThetaAboveNode100023022
+      e24KC2ThetaAboveNode100023023
+
+theorem e24KC2ThetaAboveNode10002303 :
+    adaptiveCoverCheck 11 thetaAboveCell10002303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002303
+    e24KC2ThetaAboveNode100023030 e24KC2ThetaAboveNode100023031 e24KC2ThetaAboveNode100023032
+      e24KC2ThetaAboveNode100023033
+
+theorem e24KC2ThetaAboveNode10002310 :
+    adaptiveCoverCheck 11 thetaAboveCell10002310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002310
+    e24KC2ThetaAboveLeaf100023100 e24KC2ThetaAboveLeaf100023101 e24KC2ThetaAboveLeaf100023102
+      e24KC2ThetaAboveLeaf100023103
+
+theorem e24KC2ThetaAboveNode10002311 :
+    adaptiveCoverCheck 11 thetaAboveCell10002311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002311
+    e24KC2ThetaAboveLeaf100023110 e24KC2ThetaAboveLeaf100023111 e24KC2ThetaAboveLeaf100023112
+      e24KC2ThetaAboveLeaf100023113
+
+theorem e24KC2ThetaAboveNode10002312 :
+    adaptiveCoverCheck 11 thetaAboveCell10002312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002312
+    e24KC2ThetaAboveNode100023120 e24KC2ThetaAboveNode100023121 e24KC2ThetaAboveNode100023122
+      e24KC2ThetaAboveNode100023123
+
+theorem e24KC2ThetaAboveNode10002313 :
+    adaptiveCoverCheck 11 thetaAboveCell10002313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002313
+    e24KC2ThetaAboveNode100023130 e24KC2ThetaAboveNode100023131 e24KC2ThetaAboveNode100023132
+      e24KC2ThetaAboveNode100023133
+
+theorem e24KC2ThetaAboveNode10002320 :
+    adaptiveCoverCheck 11 thetaAboveCell10002320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002320
+    e24KC2ThetaAboveLeaf100023200 e24KC2ThetaAboveLeaf100023201 e24KC2ThetaAboveLeaf100023202
+      e24KC2ThetaAboveLeaf100023203
+
+theorem e24KC2ThetaAboveNode10002321 :
+    adaptiveCoverCheck 11 thetaAboveCell10002321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002321
+    e24KC2ThetaAboveLeaf100023210 e24KC2ThetaAboveLeaf100023211 e24KC2ThetaAboveLeaf100023212
+      e24KC2ThetaAboveLeaf100023213
+
+theorem e24KC2ThetaAboveNode10002330 :
+    adaptiveCoverCheck 11 thetaAboveCell10002330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002330
+    e24KC2ThetaAboveLeaf100023300 e24KC2ThetaAboveLeaf100023301 e24KC2ThetaAboveLeaf100023302
+      e24KC2ThetaAboveLeaf100023303
+
+theorem e24KC2ThetaAboveNode10002331 :
+    adaptiveCoverCheck 11 thetaAboveCell10002331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10002331
+    e24KC2ThetaAboveLeaf100023310 e24KC2ThetaAboveLeaf100023311 e24KC2ThetaAboveLeaf100023312
+      e24KC2ThetaAboveLeaf100023313
+
+theorem e24KC2ThetaAboveNode10003200 :
+    adaptiveCoverCheck 11 thetaAboveCell10003200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003200
+    e24KC2ThetaAboveLeaf100032000 e24KC2ThetaAboveLeaf100032001 e24KC2ThetaAboveLeaf100032002
+      e24KC2ThetaAboveLeaf100032003
+
+theorem e24KC2ThetaAboveNode10003201 :
+    adaptiveCoverCheck 11 thetaAboveCell10003201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003201
+    e24KC2ThetaAboveLeaf100032010 e24KC2ThetaAboveLeaf100032011 e24KC2ThetaAboveLeaf100032012
+      e24KC2ThetaAboveLeaf100032013
+
+theorem e24KC2ThetaAboveNode10003202 :
+    adaptiveCoverCheck 11 thetaAboveCell10003202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003202
+    e24KC2ThetaAboveNode100032020 e24KC2ThetaAboveNode100032021 e24KC2ThetaAboveNode100032022
+      e24KC2ThetaAboveNode100032023
+
+theorem e24KC2ThetaAboveNode10003203 :
+    adaptiveCoverCheck 11 thetaAboveCell10003203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003203
+    e24KC2ThetaAboveNode100032030 e24KC2ThetaAboveNode100032031 e24KC2ThetaAboveNode100032032
+      e24KC2ThetaAboveNode100032033
+
+theorem e24KC2ThetaAboveNode10003210 :
+    adaptiveCoverCheck 11 thetaAboveCell10003210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003210
+    e24KC2ThetaAboveLeaf100032100 e24KC2ThetaAboveLeaf100032101 e24KC2ThetaAboveLeaf100032102
+      e24KC2ThetaAboveLeaf100032103
+
+theorem e24KC2ThetaAboveNode10003211 :
+    adaptiveCoverCheck 11 thetaAboveCell10003211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003211
+    e24KC2ThetaAboveLeaf100032110 e24KC2ThetaAboveLeaf100032111 e24KC2ThetaAboveLeaf100032112
+      e24KC2ThetaAboveLeaf100032113
+
+theorem e24KC2ThetaAboveNode10003212 :
+    adaptiveCoverCheck 11 thetaAboveCell10003212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003212
+    e24KC2ThetaAboveNode100032120 e24KC2ThetaAboveNode100032121 e24KC2ThetaAboveNode100032122
+      e24KC2ThetaAboveNode100032123
+
+theorem e24KC2ThetaAboveNode10003213 :
+    adaptiveCoverCheck 11 thetaAboveCell10003213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003213
+    e24KC2ThetaAboveNode100032130 e24KC2ThetaAboveNode100032131 e24KC2ThetaAboveNode100032132
+      e24KC2ThetaAboveNode100032133
+
+theorem e24KC2ThetaAboveNode10003220 :
+    adaptiveCoverCheck 11 thetaAboveCell10003220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003220
+    e24KC2ThetaAboveLeaf100032200 e24KC2ThetaAboveLeaf100032201 e24KC2ThetaAboveLeaf100032202
+      e24KC2ThetaAboveLeaf100032203
+
+theorem e24KC2ThetaAboveNode10003221 :
+    adaptiveCoverCheck 11 thetaAboveCell10003221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003221
+    e24KC2ThetaAboveLeaf100032210 e24KC2ThetaAboveLeaf100032211 e24KC2ThetaAboveLeaf100032212
+      e24KC2ThetaAboveLeaf100032213
+
+theorem e24KC2ThetaAboveNode10003230 :
+    adaptiveCoverCheck 11 thetaAboveCell10003230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003230
+    e24KC2ThetaAboveLeaf100032300 e24KC2ThetaAboveLeaf100032301 e24KC2ThetaAboveLeaf100032302
+      e24KC2ThetaAboveLeaf100032303
+
+theorem e24KC2ThetaAboveNode10003231 :
+    adaptiveCoverCheck 11 thetaAboveCell10003231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003231
+    e24KC2ThetaAboveLeaf100032310 e24KC2ThetaAboveLeaf100032311 e24KC2ThetaAboveLeaf100032312
+      e24KC2ThetaAboveLeaf100032313
+
+theorem e24KC2ThetaAboveNode10003300 :
+    adaptiveCoverCheck 11 thetaAboveCell10003300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003300
+    e24KC2ThetaAboveLeaf100033000 e24KC2ThetaAboveLeaf100033001 e24KC2ThetaAboveLeaf100033002
+      e24KC2ThetaAboveLeaf100033003
+
+theorem e24KC2ThetaAboveNode10003301 :
+    adaptiveCoverCheck 11 thetaAboveCell10003301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003301
+    e24KC2ThetaAboveLeaf100033010 e24KC2ThetaAboveLeaf100033011 e24KC2ThetaAboveLeaf100033012
+      e24KC2ThetaAboveLeaf100033013
+
+theorem e24KC2ThetaAboveNode10003302 :
+    adaptiveCoverCheck 11 thetaAboveCell10003302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003302
+    e24KC2ThetaAboveNode100033020 e24KC2ThetaAboveNode100033021 e24KC2ThetaAboveNode100033022
+      e24KC2ThetaAboveNode100033023
+
+theorem e24KC2ThetaAboveNode10003303 :
+    adaptiveCoverCheck 11 thetaAboveCell10003303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003303
+    e24KC2ThetaAboveNode100033030 e24KC2ThetaAboveNode100033031 e24KC2ThetaAboveNode100033032
+      e24KC2ThetaAboveNode100033033
+
+theorem e24KC2ThetaAboveNode10003310 :
+    adaptiveCoverCheck 11 thetaAboveCell10003310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003310
+    e24KC2ThetaAboveLeaf100033100 e24KC2ThetaAboveLeaf100033101 e24KC2ThetaAboveLeaf100033102
+      e24KC2ThetaAboveLeaf100033103
+
+theorem e24KC2ThetaAboveNode10003311 :
+    adaptiveCoverCheck 11 thetaAboveCell10003311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003311
+    e24KC2ThetaAboveLeaf100033110 e24KC2ThetaAboveLeaf100033111 e24KC2ThetaAboveLeaf100033112
+      e24KC2ThetaAboveLeaf100033113
+
+theorem e24KC2ThetaAboveNode10003312 :
+    adaptiveCoverCheck 11 thetaAboveCell10003312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003312
+    e24KC2ThetaAboveNode100033120 e24KC2ThetaAboveNode100033121 e24KC2ThetaAboveNode100033122
+      e24KC2ThetaAboveNode100033123
+
+theorem e24KC2ThetaAboveNode10003313 :
+    adaptiveCoverCheck 11 thetaAboveCell10003313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003313
+    e24KC2ThetaAboveNode100033130 e24KC2ThetaAboveNode100033131 e24KC2ThetaAboveNode100033132
+      e24KC2ThetaAboveNode100033133
+
+theorem e24KC2ThetaAboveNode10003320 :
+    adaptiveCoverCheck 11 thetaAboveCell10003320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003320
+    e24KC2ThetaAboveLeaf100033200 e24KC2ThetaAboveLeaf100033201 e24KC2ThetaAboveLeaf100033202
+      e24KC2ThetaAboveLeaf100033203
+
+theorem e24KC2ThetaAboveNode10003321 :
+    adaptiveCoverCheck 11 thetaAboveCell10003321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003321
+    e24KC2ThetaAboveLeaf100033210 e24KC2ThetaAboveLeaf100033211 e24KC2ThetaAboveLeaf100033212
+      e24KC2ThetaAboveLeaf100033213
+
+theorem e24KC2ThetaAboveNode10003330 :
+    adaptiveCoverCheck 11 thetaAboveCell10003330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003330
+    e24KC2ThetaAboveLeaf100033300 e24KC2ThetaAboveLeaf100033301 e24KC2ThetaAboveLeaf100033302
+      e24KC2ThetaAboveLeaf100033303
+
+theorem e24KC2ThetaAboveNode10003331 :
+    adaptiveCoverCheck 11 thetaAboveCell10003331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10003331
+    e24KC2ThetaAboveLeaf100033310 e24KC2ThetaAboveNode100033311 e24KC2ThetaAboveLeaf100033312
+      e24KC2ThetaAboveLeaf100033313
+
+theorem e24KC2ThetaAboveNode10012200 :
+    adaptiveCoverCheck 11 thetaAboveCell10012200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012200
+    e24KC2ThetaAboveLeaf100122000 e24KC2ThetaAboveLeaf100122001 e24KC2ThetaAboveLeaf100122002
+      e24KC2ThetaAboveLeaf100122003
+
+theorem e24KC2ThetaAboveNode10012201 :
+    adaptiveCoverCheck 11 thetaAboveCell10012201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012201
+    e24KC2ThetaAboveLeaf100122010 e24KC2ThetaAboveLeaf100122011 e24KC2ThetaAboveLeaf100122012
+      e24KC2ThetaAboveLeaf100122013
+
+theorem e24KC2ThetaAboveNode10012202 :
+    adaptiveCoverCheck 11 thetaAboveCell10012202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012202
+    e24KC2ThetaAboveNode100122020 e24KC2ThetaAboveNode100122021 e24KC2ThetaAboveNode100122022
+      e24KC2ThetaAboveNode100122023
+
+theorem e24KC2ThetaAboveNode10012203 :
+    adaptiveCoverCheck 11 thetaAboveCell10012203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012203
+    e24KC2ThetaAboveNode100122030 e24KC2ThetaAboveNode100122031 e24KC2ThetaAboveNode100122032
+      e24KC2ThetaAboveNode100122033
+
+theorem e24KC2ThetaAboveNode10012210 :
+    adaptiveCoverCheck 11 thetaAboveCell10012210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012210
+    e24KC2ThetaAboveLeaf100122100 e24KC2ThetaAboveLeaf100122101 e24KC2ThetaAboveLeaf100122102
+      e24KC2ThetaAboveLeaf100122103
+
+theorem e24KC2ThetaAboveNode10012211 :
+    adaptiveCoverCheck 11 thetaAboveCell10012211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012211
+    e24KC2ThetaAboveLeaf100122110 e24KC2ThetaAboveLeaf100122111 e24KC2ThetaAboveLeaf100122112
+      e24KC2ThetaAboveLeaf100122113
+
+theorem e24KC2ThetaAboveNode10012212 :
+    adaptiveCoverCheck 11 thetaAboveCell10012212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012212
+    e24KC2ThetaAboveNode100122120 e24KC2ThetaAboveNode100122121 e24KC2ThetaAboveNode100122122
+      e24KC2ThetaAboveNode100122123
+
+theorem e24KC2ThetaAboveNode10012213 :
+    adaptiveCoverCheck 11 thetaAboveCell10012213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012213
+    e24KC2ThetaAboveNode100122130 e24KC2ThetaAboveNode100122131 e24KC2ThetaAboveNode100122132
+      e24KC2ThetaAboveNode100122133
+
+theorem e24KC2ThetaAboveNode10012220 :
+    adaptiveCoverCheck 11 thetaAboveCell10012220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012220
+    e24KC2ThetaAboveNode100122200 e24KC2ThetaAboveNode100122201 e24KC2ThetaAboveLeaf100122202
+      e24KC2ThetaAboveLeaf100122203
+
+theorem e24KC2ThetaAboveNode10012221 :
+    adaptiveCoverCheck 11 thetaAboveCell10012221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012221
+    e24KC2ThetaAboveNode100122210 e24KC2ThetaAboveNode100122211 e24KC2ThetaAboveLeaf100122212
+      e24KC2ThetaAboveLeaf100122213
+
+theorem e24KC2ThetaAboveNode10012230 :
+    adaptiveCoverCheck 11 thetaAboveCell10012230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012230
+    e24KC2ThetaAboveNode100122300 e24KC2ThetaAboveNode100122301 e24KC2ThetaAboveLeaf100122302
+      e24KC2ThetaAboveLeaf100122303
+
+theorem e24KC2ThetaAboveNode10012231 :
+    adaptiveCoverCheck 11 thetaAboveCell10012231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012231
+    e24KC2ThetaAboveNode100122310 e24KC2ThetaAboveNode100122311 e24KC2ThetaAboveLeaf100122312
+      e24KC2ThetaAboveLeaf100122313
+
+theorem e24KC2ThetaAboveNode10012300 :
+    adaptiveCoverCheck 11 thetaAboveCell10012300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012300
+    e24KC2ThetaAboveLeaf100123000 e24KC2ThetaAboveLeaf100123001 e24KC2ThetaAboveLeaf100123002
+      e24KC2ThetaAboveLeaf100123003
+
+theorem e24KC2ThetaAboveNode10012301 :
+    adaptiveCoverCheck 11 thetaAboveCell10012301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012301
+    e24KC2ThetaAboveLeaf100123010 e24KC2ThetaAboveLeaf100123011 e24KC2ThetaAboveLeaf100123012
+      e24KC2ThetaAboveLeaf100123013
+
+theorem e24KC2ThetaAboveNode10012302 :
+    adaptiveCoverCheck 11 thetaAboveCell10012302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012302
+    e24KC2ThetaAboveNode100123020 e24KC2ThetaAboveNode100123021 e24KC2ThetaAboveNode100123022
+      e24KC2ThetaAboveNode100123023
+
+theorem e24KC2ThetaAboveNode10012303 :
+    adaptiveCoverCheck 11 thetaAboveCell10012303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012303
+    e24KC2ThetaAboveNode100123030 e24KC2ThetaAboveNode100123031 e24KC2ThetaAboveNode100123032
+      e24KC2ThetaAboveNode100123033
+
+theorem e24KC2ThetaAboveNode10012310 :
+    adaptiveCoverCheck 11 thetaAboveCell10012310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012310
+    e24KC2ThetaAboveLeaf100123100 e24KC2ThetaAboveLeaf100123101 e24KC2ThetaAboveLeaf100123102
+      e24KC2ThetaAboveLeaf100123103
+
+theorem e24KC2ThetaAboveNode10012311 :
+    adaptiveCoverCheck 11 thetaAboveCell10012311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012311
+    e24KC2ThetaAboveLeaf100123110 e24KC2ThetaAboveLeaf100123111 e24KC2ThetaAboveLeaf100123112
+      e24KC2ThetaAboveLeaf100123113
+
+theorem e24KC2ThetaAboveNode10012312 :
+    adaptiveCoverCheck 11 thetaAboveCell10012312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012312
+    e24KC2ThetaAboveNode100123120 e24KC2ThetaAboveNode100123121 e24KC2ThetaAboveNode100123122
+      e24KC2ThetaAboveNode100123123
+
+theorem e24KC2ThetaAboveNode10012313 :
+    adaptiveCoverCheck 11 thetaAboveCell10012313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012313
+    e24KC2ThetaAboveNode100123130 e24KC2ThetaAboveNode100123131 e24KC2ThetaAboveNode100123132
+      e24KC2ThetaAboveNode100123133
+
+theorem e24KC2ThetaAboveNode10012320 :
+    adaptiveCoverCheck 11 thetaAboveCell10012320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012320
+    e24KC2ThetaAboveNode100123200 e24KC2ThetaAboveNode100123201 e24KC2ThetaAboveLeaf100123202
+      e24KC2ThetaAboveLeaf100123203
+
+theorem e24KC2ThetaAboveNode10012321 :
+    adaptiveCoverCheck 11 thetaAboveCell10012321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012321
+    e24KC2ThetaAboveNode100123210 e24KC2ThetaAboveNode100123211 e24KC2ThetaAboveLeaf100123212
+      e24KC2ThetaAboveLeaf100123213
+
+theorem e24KC2ThetaAboveNode10012330 :
+    adaptiveCoverCheck 11 thetaAboveCell10012330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012330
+    e24KC2ThetaAboveNode100123300 e24KC2ThetaAboveNode100123301 e24KC2ThetaAboveLeaf100123302
+      e24KC2ThetaAboveLeaf100123303
+
+theorem e24KC2ThetaAboveNode10012331 :
+    adaptiveCoverCheck 11 thetaAboveCell10012331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10012331
+    e24KC2ThetaAboveNode100123310 e24KC2ThetaAboveNode100123311 e24KC2ThetaAboveLeaf100123312
+      e24KC2ThetaAboveLeaf100123313
+
+theorem e24KC2ThetaAboveNode10013200 :
+    adaptiveCoverCheck 11 thetaAboveCell10013200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013200
+    e24KC2ThetaAboveLeaf100132000 e24KC2ThetaAboveLeaf100132001 e24KC2ThetaAboveLeaf100132002
+      e24KC2ThetaAboveLeaf100132003
+
+theorem e24KC2ThetaAboveNode10013201 :
+    adaptiveCoverCheck 11 thetaAboveCell10013201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013201
+    e24KC2ThetaAboveLeaf100132010 e24KC2ThetaAboveLeaf100132011 e24KC2ThetaAboveLeaf100132012
+      e24KC2ThetaAboveLeaf100132013
+
+theorem e24KC2ThetaAboveNode10013202 :
+    adaptiveCoverCheck 11 thetaAboveCell10013202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013202
+    e24KC2ThetaAboveNode100132020 e24KC2ThetaAboveNode100132021 e24KC2ThetaAboveNode100132022
+      e24KC2ThetaAboveNode100132023
+
+theorem e24KC2ThetaAboveNode10013203 :
+    adaptiveCoverCheck 11 thetaAboveCell10013203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013203
+    e24KC2ThetaAboveNode100132030 e24KC2ThetaAboveNode100132031 e24KC2ThetaAboveNode100132032
+      e24KC2ThetaAboveNode100132033
+
+theorem e24KC2ThetaAboveNode10013210 :
+    adaptiveCoverCheck 11 thetaAboveCell10013210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013210
+    e24KC2ThetaAboveLeaf100132100 e24KC2ThetaAboveLeaf100132101 e24KC2ThetaAboveLeaf100132102
+      e24KC2ThetaAboveLeaf100132103
+
+theorem e24KC2ThetaAboveNode10013211 :
+    adaptiveCoverCheck 11 thetaAboveCell10013211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013211
+    e24KC2ThetaAboveLeaf100132110 e24KC2ThetaAboveLeaf100132111 e24KC2ThetaAboveLeaf100132112
+      e24KC2ThetaAboveLeaf100132113
+
+theorem e24KC2ThetaAboveNode10013212 :
+    adaptiveCoverCheck 11 thetaAboveCell10013212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013212
+    e24KC2ThetaAboveNode100132120 e24KC2ThetaAboveNode100132121 e24KC2ThetaAboveNode100132122
+      e24KC2ThetaAboveNode100132123
+
+theorem e24KC2ThetaAboveNode10013213 :
+    adaptiveCoverCheck 11 thetaAboveCell10013213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013213
+    e24KC2ThetaAboveNode100132130 e24KC2ThetaAboveNode100132131 e24KC2ThetaAboveNode100132132
+      e24KC2ThetaAboveNode100132133
+
+theorem e24KC2ThetaAboveNode10013220 :
+    adaptiveCoverCheck 11 thetaAboveCell10013220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013220
+    e24KC2ThetaAboveNode100132200 e24KC2ThetaAboveNode100132201 e24KC2ThetaAboveLeaf100132202
+      e24KC2ThetaAboveLeaf100132203
+
+theorem e24KC2ThetaAboveNode10013221 :
+    adaptiveCoverCheck 11 thetaAboveCell10013221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013221
+    e24KC2ThetaAboveNode100132210 e24KC2ThetaAboveNode100132211 e24KC2ThetaAboveLeaf100132212
+      e24KC2ThetaAboveLeaf100132213
+
+theorem e24KC2ThetaAboveNode10013230 :
+    adaptiveCoverCheck 11 thetaAboveCell10013230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013230
+    e24KC2ThetaAboveNode100132300 e24KC2ThetaAboveNode100132301 e24KC2ThetaAboveLeaf100132302
+      e24KC2ThetaAboveLeaf100132303
+
+theorem e24KC2ThetaAboveNode10013231 :
+    adaptiveCoverCheck 11 thetaAboveCell10013231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013231
+    e24KC2ThetaAboveLeaf100132310 e24KC2ThetaAboveLeaf100132311 e24KC2ThetaAboveLeaf100132312
+      e24KC2ThetaAboveLeaf100132313
+
+theorem e24KC2ThetaAboveNode10013300 :
+    adaptiveCoverCheck 11 thetaAboveCell10013300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013300
+    e24KC2ThetaAboveLeaf100133000 e24KC2ThetaAboveLeaf100133001 e24KC2ThetaAboveLeaf100133002
+      e24KC2ThetaAboveLeaf100133003
+
+theorem e24KC2ThetaAboveNode10013301 :
+    adaptiveCoverCheck 11 thetaAboveCell10013301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013301
+    e24KC2ThetaAboveLeaf100133010 e24KC2ThetaAboveLeaf100133011 e24KC2ThetaAboveLeaf100133012
+      e24KC2ThetaAboveLeaf100133013
+
+theorem e24KC2ThetaAboveNode10013302 :
+    adaptiveCoverCheck 11 thetaAboveCell10013302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013302
+    e24KC2ThetaAboveNode100133020 e24KC2ThetaAboveNode100133021 e24KC2ThetaAboveNode100133022
+      e24KC2ThetaAboveNode100133023
+
+theorem e24KC2ThetaAboveNode10013303 :
+    adaptiveCoverCheck 11 thetaAboveCell10013303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013303
+    e24KC2ThetaAboveNode100133030 e24KC2ThetaAboveNode100133031 e24KC2ThetaAboveNode100133032
+      e24KC2ThetaAboveNode100133033
+
+theorem e24KC2ThetaAboveNode10013310 :
+    adaptiveCoverCheck 11 thetaAboveCell10013310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013310
+    e24KC2ThetaAboveLeaf100133100 e24KC2ThetaAboveLeaf100133101 e24KC2ThetaAboveLeaf100133102
+      e24KC2ThetaAboveLeaf100133103
+
+theorem e24KC2ThetaAboveNode10013311 :
+    adaptiveCoverCheck 11 thetaAboveCell10013311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013311
+    e24KC2ThetaAboveLeaf100133110 e24KC2ThetaAboveLeaf100133111 e24KC2ThetaAboveLeaf100133112
+      e24KC2ThetaAboveLeaf100133113
+
+theorem e24KC2ThetaAboveNode10013312 :
+    adaptiveCoverCheck 11 thetaAboveCell10013312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013312
+    e24KC2ThetaAboveNode100133120 e24KC2ThetaAboveNode100133121 e24KC2ThetaAboveNode100133122
+      e24KC2ThetaAboveNode100133123
+
+theorem e24KC2ThetaAboveNode10013313 :
+    adaptiveCoverCheck 11 thetaAboveCell10013313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013313
+    e24KC2ThetaAboveNode100133130 e24KC2ThetaAboveNode100133131 e24KC2ThetaAboveNode100133132
+      e24KC2ThetaAboveNode100133133
+
+theorem e24KC2ThetaAboveNode10013320 :
+    adaptiveCoverCheck 11 thetaAboveCell10013320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013320
+    e24KC2ThetaAboveLeaf100133200 e24KC2ThetaAboveLeaf100133201 e24KC2ThetaAboveLeaf100133202
+      e24KC2ThetaAboveLeaf100133203
+
+theorem e24KC2ThetaAboveNode10013321 :
+    adaptiveCoverCheck 11 thetaAboveCell10013321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013321
+    e24KC2ThetaAboveLeaf100133210 e24KC2ThetaAboveLeaf100133211 e24KC2ThetaAboveLeaf100133212
+      e24KC2ThetaAboveLeaf100133213
+
+theorem e24KC2ThetaAboveNode10013330 :
+    adaptiveCoverCheck 11 thetaAboveCell10013330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013330
+    e24KC2ThetaAboveLeaf100133300 e24KC2ThetaAboveLeaf100133301 e24KC2ThetaAboveLeaf100133302
+      e24KC2ThetaAboveLeaf100133303
+
+theorem e24KC2ThetaAboveNode10013331 :
+    adaptiveCoverCheck 11 thetaAboveCell10013331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10013331
+    e24KC2ThetaAboveLeaf100133310 e24KC2ThetaAboveLeaf100133311 e24KC2ThetaAboveLeaf100133312
+      e24KC2ThetaAboveLeaf100133313
+
+theorem e24KC2ThetaAboveNode10102200 :
+    adaptiveCoverCheck 11 thetaAboveCell10102200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102200
+    e24KC2ThetaAboveLeaf101022000 e24KC2ThetaAboveLeaf101022001 e24KC2ThetaAboveLeaf101022002
+      e24KC2ThetaAboveLeaf101022003
+
+theorem e24KC2ThetaAboveNode10102201 :
+    adaptiveCoverCheck 11 thetaAboveCell10102201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102201
+    e24KC2ThetaAboveLeaf101022010 e24KC2ThetaAboveLeaf101022011 e24KC2ThetaAboveLeaf101022012
+      e24KC2ThetaAboveLeaf101022013
+
+theorem e24KC2ThetaAboveNode10102202 :
+    adaptiveCoverCheck 11 thetaAboveCell10102202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102202
+    e24KC2ThetaAboveNode101022020 e24KC2ThetaAboveNode101022021 e24KC2ThetaAboveNode101022022
+      e24KC2ThetaAboveNode101022023
+
+theorem e24KC2ThetaAboveNode10102203 :
+    adaptiveCoverCheck 11 thetaAboveCell10102203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102203
+    e24KC2ThetaAboveNode101022030 e24KC2ThetaAboveNode101022031 e24KC2ThetaAboveNode101022032
+      e24KC2ThetaAboveNode101022033
+
+theorem e24KC2ThetaAboveNode10102210 :
+    adaptiveCoverCheck 11 thetaAboveCell10102210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102210
+    e24KC2ThetaAboveLeaf101022100 e24KC2ThetaAboveLeaf101022101 e24KC2ThetaAboveLeaf101022102
+      e24KC2ThetaAboveLeaf101022103
+
+theorem e24KC2ThetaAboveNode10102211 :
+    adaptiveCoverCheck 11 thetaAboveCell10102211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102211
+    e24KC2ThetaAboveLeaf101022110 e24KC2ThetaAboveLeaf101022111 e24KC2ThetaAboveLeaf101022112
+      e24KC2ThetaAboveLeaf101022113
+
+theorem e24KC2ThetaAboveNode10102212 :
+    adaptiveCoverCheck 11 thetaAboveCell10102212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102212
+    e24KC2ThetaAboveNode101022120 e24KC2ThetaAboveNode101022121 e24KC2ThetaAboveNode101022122
+      e24KC2ThetaAboveNode101022123
+
+theorem e24KC2ThetaAboveNode10102213 :
+    adaptiveCoverCheck 11 thetaAboveCell10102213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102213
+    e24KC2ThetaAboveNode101022130 e24KC2ThetaAboveNode101022131 e24KC2ThetaAboveNode101022132
+      e24KC2ThetaAboveLeaf101022133
+
+theorem e24KC2ThetaAboveNode10102220 :
+    adaptiveCoverCheck 11 thetaAboveCell10102220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102220
+    e24KC2ThetaAboveLeaf101022200 e24KC2ThetaAboveLeaf101022201 e24KC2ThetaAboveLeaf101022202
+      e24KC2ThetaAboveLeaf101022203
+
+theorem e24KC2ThetaAboveNode10102221 :
+    adaptiveCoverCheck 11 thetaAboveCell10102221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102221
+    e24KC2ThetaAboveLeaf101022210 e24KC2ThetaAboveLeaf101022211 e24KC2ThetaAboveLeaf101022212
+      e24KC2ThetaAboveLeaf101022213
+
+theorem e24KC2ThetaAboveNode10102230 :
+    adaptiveCoverCheck 11 thetaAboveCell10102230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102230
+    e24KC2ThetaAboveLeaf101022300 e24KC2ThetaAboveLeaf101022301 e24KC2ThetaAboveLeaf101022302
+      e24KC2ThetaAboveLeaf101022303
+
+theorem e24KC2ThetaAboveNode10102231 :
+    adaptiveCoverCheck 11 thetaAboveCell10102231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102231
+    e24KC2ThetaAboveLeaf101022310 e24KC2ThetaAboveLeaf101022311 e24KC2ThetaAboveLeaf101022312
+      e24KC2ThetaAboveLeaf101022313
+
+theorem e24KC2ThetaAboveNode10102300 :
+    adaptiveCoverCheck 11 thetaAboveCell10102300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102300
+    e24KC2ThetaAboveLeaf101023000 e24KC2ThetaAboveLeaf101023001 e24KC2ThetaAboveLeaf101023002
+      e24KC2ThetaAboveLeaf101023003
+
+theorem e24KC2ThetaAboveNode10102301 :
+    adaptiveCoverCheck 11 thetaAboveCell10102301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102301
+    e24KC2ThetaAboveLeaf101023010 e24KC2ThetaAboveLeaf101023011 e24KC2ThetaAboveLeaf101023012
+      e24KC2ThetaAboveLeaf101023013
+
+theorem e24KC2ThetaAboveNode10102302 :
+    adaptiveCoverCheck 11 thetaAboveCell10102302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102302
+    e24KC2ThetaAboveNode101023020 e24KC2ThetaAboveNode101023021 e24KC2ThetaAboveLeaf101023022
+      e24KC2ThetaAboveLeaf101023023
+
+theorem e24KC2ThetaAboveNode10102303 :
+    adaptiveCoverCheck 11 thetaAboveCell10102303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102303
+    e24KC2ThetaAboveNode101023030 e24KC2ThetaAboveNode101023031 e24KC2ThetaAboveLeaf101023032
+      e24KC2ThetaAboveLeaf101023033
+
+theorem e24KC2ThetaAboveNode10102310 :
+    adaptiveCoverCheck 11 thetaAboveCell10102310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102310
+    e24KC2ThetaAboveLeaf101023100 e24KC2ThetaAboveLeaf101023101 e24KC2ThetaAboveLeaf101023102
+      e24KC2ThetaAboveLeaf101023103
+
+theorem e24KC2ThetaAboveNode10102311 :
+    adaptiveCoverCheck 11 thetaAboveCell10102311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102311
+    e24KC2ThetaAboveLeaf101023110 e24KC2ThetaAboveLeaf101023111 e24KC2ThetaAboveLeaf101023112
+      e24KC2ThetaAboveLeaf101023113
+
+theorem e24KC2ThetaAboveNode10102312 :
+    adaptiveCoverCheck 11 thetaAboveCell10102312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102312
+    e24KC2ThetaAboveNode101023120 e24KC2ThetaAboveNode101023121 e24KC2ThetaAboveLeaf101023122
+      e24KC2ThetaAboveLeaf101023123
+
+theorem e24KC2ThetaAboveNode10102313 :
+    adaptiveCoverCheck 11 thetaAboveCell10102313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102313
+    e24KC2ThetaAboveNode101023130 e24KC2ThetaAboveNode101023131 e24KC2ThetaAboveLeaf101023132
+      e24KC2ThetaAboveLeaf101023133
+
+theorem e24KC2ThetaAboveNode10102320 :
+    adaptiveCoverCheck 11 thetaAboveCell10102320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102320
+    e24KC2ThetaAboveLeaf101023200 e24KC2ThetaAboveLeaf101023201 e24KC2ThetaAboveLeaf101023202
+      e24KC2ThetaAboveLeaf101023203
+
+theorem e24KC2ThetaAboveNode10102321 :
+    adaptiveCoverCheck 11 thetaAboveCell10102321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102321
+    e24KC2ThetaAboveLeaf101023210 e24KC2ThetaAboveLeaf101023211 e24KC2ThetaAboveLeaf101023212
+      e24KC2ThetaAboveLeaf101023213
+
+theorem e24KC2ThetaAboveNode10102330 :
+    adaptiveCoverCheck 11 thetaAboveCell10102330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102330
+    e24KC2ThetaAboveLeaf101023300 e24KC2ThetaAboveLeaf101023301 e24KC2ThetaAboveLeaf101023302
+      e24KC2ThetaAboveLeaf101023303
+
+theorem e24KC2ThetaAboveNode10102331 :
+    adaptiveCoverCheck 11 thetaAboveCell10102331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10102331
+    e24KC2ThetaAboveLeaf101023310 e24KC2ThetaAboveLeaf101023311 e24KC2ThetaAboveLeaf101023312
+      e24KC2ThetaAboveLeaf101023313
+
+theorem e24KC2ThetaAboveNode10103200 :
+    adaptiveCoverCheck 11 thetaAboveCell10103200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103200
+    e24KC2ThetaAboveLeaf101032000 e24KC2ThetaAboveLeaf101032001 e24KC2ThetaAboveLeaf101032002
+      e24KC2ThetaAboveLeaf101032003
+
+theorem e24KC2ThetaAboveNode10103201 :
+    adaptiveCoverCheck 11 thetaAboveCell10103201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103201
+    e24KC2ThetaAboveLeaf101032010 e24KC2ThetaAboveLeaf101032011 e24KC2ThetaAboveLeaf101032012
+      e24KC2ThetaAboveLeaf101032013
+
+theorem e24KC2ThetaAboveNode10103202 :
+    adaptiveCoverCheck 11 thetaAboveCell10103202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103202
+    e24KC2ThetaAboveNode101032020 e24KC2ThetaAboveNode101032021 e24KC2ThetaAboveLeaf101032022
+      e24KC2ThetaAboveLeaf101032023
+
+theorem e24KC2ThetaAboveNode10103203 :
+    adaptiveCoverCheck 11 thetaAboveCell10103203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103203
+    e24KC2ThetaAboveNode101032030 e24KC2ThetaAboveNode101032031 e24KC2ThetaAboveLeaf101032032
+      e24KC2ThetaAboveLeaf101032033
+
+theorem e24KC2ThetaAboveNode10103210 :
+    adaptiveCoverCheck 11 thetaAboveCell10103210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103210
+    e24KC2ThetaAboveLeaf101032100 e24KC2ThetaAboveLeaf101032101 e24KC2ThetaAboveLeaf101032102
+      e24KC2ThetaAboveLeaf101032103
+
+theorem e24KC2ThetaAboveNode10103211 :
+    adaptiveCoverCheck 11 thetaAboveCell10103211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103211
+    e24KC2ThetaAboveLeaf101032110 e24KC2ThetaAboveLeaf101032111 e24KC2ThetaAboveLeaf101032112
+      e24KC2ThetaAboveLeaf101032113
+
+theorem e24KC2ThetaAboveNode10103212 :
+    adaptiveCoverCheck 11 thetaAboveCell10103212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103212
+    e24KC2ThetaAboveNode101032120 e24KC2ThetaAboveNode101032121 e24KC2ThetaAboveLeaf101032122
+      e24KC2ThetaAboveLeaf101032123
+
+theorem e24KC2ThetaAboveNode10103213 :
+    adaptiveCoverCheck 11 thetaAboveCell10103213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103213
+    e24KC2ThetaAboveNode101032130 e24KC2ThetaAboveNode101032131 e24KC2ThetaAboveLeaf101032132
+      e24KC2ThetaAboveLeaf101032133
+
+theorem e24KC2ThetaAboveNode10103220 :
+    adaptiveCoverCheck 11 thetaAboveCell10103220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103220
+    e24KC2ThetaAboveLeaf101032200 e24KC2ThetaAboveLeaf101032201 e24KC2ThetaAboveLeaf101032202
+      e24KC2ThetaAboveLeaf101032203
+
+theorem e24KC2ThetaAboveNode10103221 :
+    adaptiveCoverCheck 11 thetaAboveCell10103221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103221
+    e24KC2ThetaAboveLeaf101032210 e24KC2ThetaAboveLeaf101032211 e24KC2ThetaAboveLeaf101032212
+      e24KC2ThetaAboveLeaf101032213
+
+theorem e24KC2ThetaAboveNode10103230 :
+    adaptiveCoverCheck 11 thetaAboveCell10103230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103230
+    e24KC2ThetaAboveLeaf101032300 e24KC2ThetaAboveLeaf101032301 e24KC2ThetaAboveLeaf101032302
+      e24KC2ThetaAboveLeaf101032303
+
+theorem e24KC2ThetaAboveNode10103231 :
+    adaptiveCoverCheck 11 thetaAboveCell10103231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103231
+    e24KC2ThetaAboveLeaf101032310 e24KC2ThetaAboveLeaf101032311 e24KC2ThetaAboveLeaf101032312
+      e24KC2ThetaAboveLeaf101032313
+
+theorem e24KC2ThetaAboveNode10103300 :
+    adaptiveCoverCheck 11 thetaAboveCell10103300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103300
+    e24KC2ThetaAboveLeaf101033000 e24KC2ThetaAboveLeaf101033001 e24KC2ThetaAboveLeaf101033002
+      e24KC2ThetaAboveLeaf101033003
+
+theorem e24KC2ThetaAboveNode10103301 :
+    adaptiveCoverCheck 11 thetaAboveCell10103301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103301
+    e24KC2ThetaAboveLeaf101033010 e24KC2ThetaAboveLeaf101033011 e24KC2ThetaAboveLeaf101033012
+      e24KC2ThetaAboveLeaf101033013
+
+theorem e24KC2ThetaAboveNode10103302 :
+    adaptiveCoverCheck 11 thetaAboveCell10103302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103302
+    e24KC2ThetaAboveNode101033020 e24KC2ThetaAboveLeaf101033021 e24KC2ThetaAboveLeaf101033022
+      e24KC2ThetaAboveLeaf101033023
+
+theorem e24KC2ThetaAboveNode10103303 :
+    adaptiveCoverCheck 11 thetaAboveCell10103303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103303
+    e24KC2ThetaAboveLeaf101033030 e24KC2ThetaAboveLeaf101033031 e24KC2ThetaAboveLeaf101033032
+      e24KC2ThetaAboveLeaf101033033
+
+theorem e24KC2ThetaAboveNode10103310 :
+    adaptiveCoverCheck 11 thetaAboveCell10103310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103310
+    e24KC2ThetaAboveLeaf101033100 e24KC2ThetaAboveLeaf101033101 e24KC2ThetaAboveLeaf101033102
+      e24KC2ThetaAboveLeaf101033103
+
+theorem e24KC2ThetaAboveNode10103311 :
+    adaptiveCoverCheck 11 thetaAboveCell10103311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103311
+    e24KC2ThetaAboveLeaf101033110 e24KC2ThetaAboveLeaf101033111 e24KC2ThetaAboveLeaf101033112
+      e24KC2ThetaAboveLeaf101033113
+
+theorem e24KC2ThetaAboveNode10103312 :
+    adaptiveCoverCheck 11 thetaAboveCell10103312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103312
+    e24KC2ThetaAboveLeaf101033120 e24KC2ThetaAboveLeaf101033121 e24KC2ThetaAboveLeaf101033122
+      e24KC2ThetaAboveLeaf101033123
+
+theorem e24KC2ThetaAboveNode10103313 :
+    adaptiveCoverCheck 11 thetaAboveCell10103313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103313
+    e24KC2ThetaAboveLeaf101033130 e24KC2ThetaAboveLeaf101033131 e24KC2ThetaAboveLeaf101033132
+      e24KC2ThetaAboveLeaf101033133
+
+theorem e24KC2ThetaAboveNode10103320 :
+    adaptiveCoverCheck 11 thetaAboveCell10103320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103320
+    e24KC2ThetaAboveLeaf101033200 e24KC2ThetaAboveLeaf101033201 e24KC2ThetaAboveLeaf101033202
+      e24KC2ThetaAboveLeaf101033203
+
+theorem e24KC2ThetaAboveNode10103321 :
+    adaptiveCoverCheck 11 thetaAboveCell10103321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103321
+    e24KC2ThetaAboveLeaf101033210 e24KC2ThetaAboveLeaf101033211 e24KC2ThetaAboveLeaf101033212
+      e24KC2ThetaAboveLeaf101033213
+
+theorem e24KC2ThetaAboveNode10103330 :
+    adaptiveCoverCheck 11 thetaAboveCell10103330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103330
+    e24KC2ThetaAboveLeaf101033300 e24KC2ThetaAboveLeaf101033301 e24KC2ThetaAboveLeaf101033302
+      e24KC2ThetaAboveLeaf101033303
+
+theorem e24KC2ThetaAboveNode10103331 :
+    adaptiveCoverCheck 11 thetaAboveCell10103331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10103331
+    e24KC2ThetaAboveLeaf101033310 e24KC2ThetaAboveLeaf101033311 e24KC2ThetaAboveLeaf101033312
+      e24KC2ThetaAboveLeaf101033313
+
+theorem e24KC2ThetaAboveNode10112200 :
+    adaptiveCoverCheck 11 thetaAboveCell10112200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112200
+    e24KC2ThetaAboveLeaf101122000 e24KC2ThetaAboveLeaf101122001 e24KC2ThetaAboveLeaf101122002
+      e24KC2ThetaAboveLeaf101122003
+
+theorem e24KC2ThetaAboveNode10112201 :
+    adaptiveCoverCheck 11 thetaAboveCell10112201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112201
+    e24KC2ThetaAboveLeaf101122010 e24KC2ThetaAboveLeaf101122011 e24KC2ThetaAboveLeaf101122012
+      e24KC2ThetaAboveLeaf101122013
+
+theorem e24KC2ThetaAboveNode10112202 :
+    adaptiveCoverCheck 11 thetaAboveCell10112202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112202
+    e24KC2ThetaAboveLeaf101122020 e24KC2ThetaAboveLeaf101122021 e24KC2ThetaAboveLeaf101122022
+      e24KC2ThetaAboveLeaf101122023
+
+theorem e24KC2ThetaAboveNode10112203 :
+    adaptiveCoverCheck 11 thetaAboveCell10112203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112203
+    e24KC2ThetaAboveLeaf101122030 e24KC2ThetaAboveLeaf101122031 e24KC2ThetaAboveLeaf101122032
+      e24KC2ThetaAboveLeaf101122033
+
+theorem e24KC2ThetaAboveNode10112210 :
+    adaptiveCoverCheck 11 thetaAboveCell10112210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112210
+    e24KC2ThetaAboveLeaf101122100 e24KC2ThetaAboveLeaf101122101 e24KC2ThetaAboveLeaf101122102
+      e24KC2ThetaAboveLeaf101122103
+
+theorem e24KC2ThetaAboveNode10112211 :
+    adaptiveCoverCheck 11 thetaAboveCell10112211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112211
+    e24KC2ThetaAboveLeaf101122110 e24KC2ThetaAboveLeaf101122111 e24KC2ThetaAboveLeaf101122112
+      e24KC2ThetaAboveLeaf101122113
+
+theorem e24KC2ThetaAboveNode10112212 :
+    adaptiveCoverCheck 11 thetaAboveCell10112212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112212
+    e24KC2ThetaAboveLeaf101122120 e24KC2ThetaAboveLeaf101122121 e24KC2ThetaAboveLeaf101122122
+      e24KC2ThetaAboveLeaf101122123
+
+theorem e24KC2ThetaAboveNode10112213 :
+    adaptiveCoverCheck 11 thetaAboveCell10112213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112213
+    e24KC2ThetaAboveLeaf101122130 e24KC2ThetaAboveLeaf101122131 e24KC2ThetaAboveLeaf101122132
+      e24KC2ThetaAboveLeaf101122133
+
+theorem e24KC2ThetaAboveNode10112220 :
+    adaptiveCoverCheck 11 thetaAboveCell10112220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112220
+    e24KC2ThetaAboveLeaf101122200 e24KC2ThetaAboveLeaf101122201 e24KC2ThetaAboveLeaf101122202
+      e24KC2ThetaAboveLeaf101122203
+
+theorem e24KC2ThetaAboveNode10112221 :
+    adaptiveCoverCheck 11 thetaAboveCell10112221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112221
+    e24KC2ThetaAboveLeaf101122210 e24KC2ThetaAboveLeaf101122211 e24KC2ThetaAboveLeaf101122212
+      e24KC2ThetaAboveLeaf101122213
+
+theorem e24KC2ThetaAboveNode10112230 :
+    adaptiveCoverCheck 11 thetaAboveCell10112230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112230
+    e24KC2ThetaAboveLeaf101122300 e24KC2ThetaAboveLeaf101122301 e24KC2ThetaAboveLeaf101122302
+      e24KC2ThetaAboveLeaf101122303
+
+theorem e24KC2ThetaAboveNode10112231 :
+    adaptiveCoverCheck 11 thetaAboveCell10112231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112231
+    e24KC2ThetaAboveLeaf101122310 e24KC2ThetaAboveLeaf101122311 e24KC2ThetaAboveLeaf101122312
+      e24KC2ThetaAboveLeaf101122313
+
+theorem e24KC2ThetaAboveNode10112300 :
+    adaptiveCoverCheck 11 thetaAboveCell10112300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112300
+    e24KC2ThetaAboveLeaf101123000 e24KC2ThetaAboveLeaf101123001 e24KC2ThetaAboveLeaf101123002
+      e24KC2ThetaAboveLeaf101123003
+
+theorem e24KC2ThetaAboveNode10112301 :
+    adaptiveCoverCheck 11 thetaAboveCell10112301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112301
+    e24KC2ThetaAboveLeaf101123010 e24KC2ThetaAboveLeaf101123011 e24KC2ThetaAboveLeaf101123012
+      e24KC2ThetaAboveLeaf101123013
+
+theorem e24KC2ThetaAboveNode10112302 :
+    adaptiveCoverCheck 11 thetaAboveCell10112302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112302
+    e24KC2ThetaAboveLeaf101123020 e24KC2ThetaAboveLeaf101123021 e24KC2ThetaAboveLeaf101123022
+      e24KC2ThetaAboveLeaf101123023
+
+theorem e24KC2ThetaAboveNode10112303 :
+    adaptiveCoverCheck 11 thetaAboveCell10112303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112303
+    e24KC2ThetaAboveLeaf101123030 e24KC2ThetaAboveLeaf101123031 e24KC2ThetaAboveLeaf101123032
+      e24KC2ThetaAboveLeaf101123033
+
+theorem e24KC2ThetaAboveNode10112310 :
+    adaptiveCoverCheck 11 thetaAboveCell10112310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112310
+    e24KC2ThetaAboveLeaf101123100 e24KC2ThetaAboveLeaf101123101 e24KC2ThetaAboveLeaf101123102
+      e24KC2ThetaAboveLeaf101123103
+
+theorem e24KC2ThetaAboveNode10112311 :
+    adaptiveCoverCheck 11 thetaAboveCell10112311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112311
+    e24KC2ThetaAboveLeaf101123110 e24KC2ThetaAboveLeaf101123111 e24KC2ThetaAboveLeaf101123112
+      e24KC2ThetaAboveLeaf101123113
+
+theorem e24KC2ThetaAboveNode10112312 :
+    adaptiveCoverCheck 11 thetaAboveCell10112312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112312
+    e24KC2ThetaAboveLeaf101123120 e24KC2ThetaAboveLeaf101123121 e24KC2ThetaAboveLeaf101123122
+      e24KC2ThetaAboveLeaf101123123
+
+theorem e24KC2ThetaAboveNode10112313 :
+    adaptiveCoverCheck 11 thetaAboveCell10112313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112313
+    e24KC2ThetaAboveLeaf101123130 e24KC2ThetaAboveLeaf101123131 e24KC2ThetaAboveLeaf101123132
+      e24KC2ThetaAboveLeaf101123133
+
+theorem e24KC2ThetaAboveNode10112320 :
+    adaptiveCoverCheck 11 thetaAboveCell10112320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112320
+    e24KC2ThetaAboveLeaf101123200 e24KC2ThetaAboveLeaf101123201 e24KC2ThetaAboveLeaf101123202
+      e24KC2ThetaAboveLeaf101123203
+
+theorem e24KC2ThetaAboveNode10112321 :
+    adaptiveCoverCheck 11 thetaAboveCell10112321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112321
+    e24KC2ThetaAboveLeaf101123210 e24KC2ThetaAboveLeaf101123211 e24KC2ThetaAboveLeaf101123212
+      e24KC2ThetaAboveLeaf101123213
+
+theorem e24KC2ThetaAboveNode10112330 :
+    adaptiveCoverCheck 11 thetaAboveCell10112330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112330
+    e24KC2ThetaAboveLeaf101123300 e24KC2ThetaAboveLeaf101123301 e24KC2ThetaAboveLeaf101123302
+      e24KC2ThetaAboveLeaf101123303
+
+theorem e24KC2ThetaAboveNode10112331 :
+    adaptiveCoverCheck 11 thetaAboveCell10112331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10112331
+    e24KC2ThetaAboveLeaf101123310 e24KC2ThetaAboveLeaf101123311 e24KC2ThetaAboveLeaf101123312
+      e24KC2ThetaAboveLeaf101123313
+
+theorem e24KC2ThetaAboveNode10113200 :
+    adaptiveCoverCheck 11 thetaAboveCell10113200 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113200
+    e24KC2ThetaAboveLeaf101132000 e24KC2ThetaAboveLeaf101132001 e24KC2ThetaAboveLeaf101132002
+      e24KC2ThetaAboveLeaf101132003
+
+theorem e24KC2ThetaAboveNode10113201 :
+    adaptiveCoverCheck 11 thetaAboveCell10113201 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113201
+    e24KC2ThetaAboveLeaf101132010 e24KC2ThetaAboveLeaf101132011 e24KC2ThetaAboveLeaf101132012
+      e24KC2ThetaAboveLeaf101132013
+
+theorem e24KC2ThetaAboveNode10113202 :
+    adaptiveCoverCheck 11 thetaAboveCell10113202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113202
+    e24KC2ThetaAboveLeaf101132020 e24KC2ThetaAboveLeaf101132021 e24KC2ThetaAboveLeaf101132022
+      e24KC2ThetaAboveLeaf101132023
+
+theorem e24KC2ThetaAboveNode10113203 :
+    adaptiveCoverCheck 11 thetaAboveCell10113203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113203
+    e24KC2ThetaAboveLeaf101132030 e24KC2ThetaAboveLeaf101132031 e24KC2ThetaAboveLeaf101132032
+      e24KC2ThetaAboveLeaf101132033
+
+theorem e24KC2ThetaAboveNode10113210 :
+    adaptiveCoverCheck 11 thetaAboveCell10113210 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113210
+    e24KC2ThetaAboveLeaf101132100 e24KC2ThetaAboveLeaf101132101 e24KC2ThetaAboveLeaf101132102
+      e24KC2ThetaAboveLeaf101132103
+
+theorem e24KC2ThetaAboveNode10113211 :
+    adaptiveCoverCheck 11 thetaAboveCell10113211 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113211
+    e24KC2ThetaAboveLeaf101132110 e24KC2ThetaAboveLeaf101132111 e24KC2ThetaAboveLeaf101132112
+      e24KC2ThetaAboveLeaf101132113
+
+theorem e24KC2ThetaAboveNode10113212 :
+    adaptiveCoverCheck 11 thetaAboveCell10113212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113212
+    e24KC2ThetaAboveLeaf101132120 e24KC2ThetaAboveLeaf101132121 e24KC2ThetaAboveLeaf101132122
+      e24KC2ThetaAboveLeaf101132123
+
+theorem e24KC2ThetaAboveNode10113213 :
+    adaptiveCoverCheck 11 thetaAboveCell10113213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113213
+    e24KC2ThetaAboveLeaf101132130 e24KC2ThetaAboveLeaf101132131 e24KC2ThetaAboveLeaf101132132
+      e24KC2ThetaAboveLeaf101132133
+
+theorem e24KC2ThetaAboveNode10113220 :
+    adaptiveCoverCheck 11 thetaAboveCell10113220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113220
+    e24KC2ThetaAboveLeaf101132200 e24KC2ThetaAboveLeaf101132201 e24KC2ThetaAboveLeaf101132202
+      e24KC2ThetaAboveLeaf101132203
+
+theorem e24KC2ThetaAboveNode10113221 :
+    adaptiveCoverCheck 11 thetaAboveCell10113221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113221
+    e24KC2ThetaAboveLeaf101132210 e24KC2ThetaAboveLeaf101132211 e24KC2ThetaAboveLeaf101132212
+      e24KC2ThetaAboveLeaf101132213
+
+theorem e24KC2ThetaAboveNode10113230 :
+    adaptiveCoverCheck 11 thetaAboveCell10113230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113230
+    e24KC2ThetaAboveLeaf101132300 e24KC2ThetaAboveLeaf101132301 e24KC2ThetaAboveLeaf101132302
+      e24KC2ThetaAboveLeaf101132303
+
+theorem e24KC2ThetaAboveNode10113231 :
+    adaptiveCoverCheck 11 thetaAboveCell10113231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113231
+    e24KC2ThetaAboveLeaf101132310 e24KC2ThetaAboveLeaf101132311 e24KC2ThetaAboveLeaf101132312
+      e24KC2ThetaAboveLeaf101132313
+
+theorem e24KC2ThetaAboveNode10113300 :
+    adaptiveCoverCheck 11 thetaAboveCell10113300 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113300
+    e24KC2ThetaAboveLeaf101133000 e24KC2ThetaAboveLeaf101133001 e24KC2ThetaAboveLeaf101133002
+      e24KC2ThetaAboveLeaf101133003
+
+theorem e24KC2ThetaAboveNode10113301 :
+    adaptiveCoverCheck 11 thetaAboveCell10113301 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113301
+    e24KC2ThetaAboveLeaf101133010 e24KC2ThetaAboveLeaf101133011 e24KC2ThetaAboveLeaf101133012
+      e24KC2ThetaAboveLeaf101133013
+
+theorem e24KC2ThetaAboveNode10113302 :
+    adaptiveCoverCheck 11 thetaAboveCell10113302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113302
+    e24KC2ThetaAboveLeaf101133020 e24KC2ThetaAboveLeaf101133021 e24KC2ThetaAboveLeaf101133022
+      e24KC2ThetaAboveLeaf101133023
+
+theorem e24KC2ThetaAboveNode10113303 :
+    adaptiveCoverCheck 11 thetaAboveCell10113303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113303
+    e24KC2ThetaAboveLeaf101133030 e24KC2ThetaAboveLeaf101133031 e24KC2ThetaAboveLeaf101133032
+      e24KC2ThetaAboveLeaf101133033
+
+theorem e24KC2ThetaAboveNode10113310 :
+    adaptiveCoverCheck 11 thetaAboveCell10113310 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113310
+    e24KC2ThetaAboveLeaf101133100 e24KC2ThetaAboveLeaf101133101 e24KC2ThetaAboveLeaf101133102
+      e24KC2ThetaAboveLeaf101133103
+
+theorem e24KC2ThetaAboveNode10113311 :
+    adaptiveCoverCheck 11 thetaAboveCell10113311 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113311
+    e24KC2ThetaAboveLeaf101133110 e24KC2ThetaAboveLeaf101133111 e24KC2ThetaAboveLeaf101133112
+      e24KC2ThetaAboveLeaf101133113
+
+theorem e24KC2ThetaAboveNode10113312 :
+    adaptiveCoverCheck 11 thetaAboveCell10113312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113312
+    e24KC2ThetaAboveLeaf101133120 e24KC2ThetaAboveLeaf101133121 e24KC2ThetaAboveLeaf101133122
+      e24KC2ThetaAboveLeaf101133123
+
+theorem e24KC2ThetaAboveNode10113313 :
+    adaptiveCoverCheck 11 thetaAboveCell10113313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113313
+    e24KC2ThetaAboveLeaf101133130 e24KC2ThetaAboveLeaf101133131 e24KC2ThetaAboveLeaf101133132
+      e24KC2ThetaAboveLeaf101133133
+
+theorem e24KC2ThetaAboveNode10113320 :
+    adaptiveCoverCheck 11 thetaAboveCell10113320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113320
+    e24KC2ThetaAboveLeaf101133200 e24KC2ThetaAboveLeaf101133201 e24KC2ThetaAboveLeaf101133202
+      e24KC2ThetaAboveLeaf101133203
+
+theorem e24KC2ThetaAboveNode10113321 :
+    adaptiveCoverCheck 11 thetaAboveCell10113321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113321
+    e24KC2ThetaAboveLeaf101133210 e24KC2ThetaAboveLeaf101133211 e24KC2ThetaAboveLeaf101133212
+      e24KC2ThetaAboveLeaf101133213
+
+theorem e24KC2ThetaAboveNode10113330 :
+    adaptiveCoverCheck 11 thetaAboveCell10113330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113330
+    e24KC2ThetaAboveLeaf101133300 e24KC2ThetaAboveLeaf101133301 e24KC2ThetaAboveLeaf101133302
+      e24KC2ThetaAboveLeaf101133303
+
+theorem e24KC2ThetaAboveNode10113331 :
+    adaptiveCoverCheck 11 thetaAboveCell10113331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell10113331
+    e24KC2ThetaAboveLeaf101133310 e24KC2ThetaAboveLeaf101133311 e24KC2ThetaAboveLeaf101133312
+      e24KC2ThetaAboveLeaf101133313
+
+theorem e24KC2ThetaAboveNode11002202 :
+    adaptiveCoverCheck 11 thetaAboveCell11002202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002202
+    e24KC2ThetaAboveLeaf110022020 e24KC2ThetaAboveLeaf110022021 e24KC2ThetaAboveLeaf110022022
+      e24KC2ThetaAboveLeaf110022023
+
+theorem e24KC2ThetaAboveNode11002203 :
+    adaptiveCoverCheck 11 thetaAboveCell11002203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002203
+    e24KC2ThetaAboveLeaf110022030 e24KC2ThetaAboveLeaf110022031 e24KC2ThetaAboveLeaf110022032
+      e24KC2ThetaAboveLeaf110022033
+
+theorem e24KC2ThetaAboveNode11002212 :
+    adaptiveCoverCheck 11 thetaAboveCell11002212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002212
+    e24KC2ThetaAboveLeaf110022120 e24KC2ThetaAboveLeaf110022121 e24KC2ThetaAboveLeaf110022122
+      e24KC2ThetaAboveLeaf110022123
+
+theorem e24KC2ThetaAboveNode11002213 :
+    adaptiveCoverCheck 11 thetaAboveCell11002213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002213
+    e24KC2ThetaAboveLeaf110022130 e24KC2ThetaAboveLeaf110022131 e24KC2ThetaAboveLeaf110022132
+      e24KC2ThetaAboveLeaf110022133
+
+theorem e24KC2ThetaAboveNode11002220 :
+    adaptiveCoverCheck 11 thetaAboveCell11002220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002220
+    e24KC2ThetaAboveLeaf110022200 e24KC2ThetaAboveLeaf110022201 e24KC2ThetaAboveLeaf110022202
+      e24KC2ThetaAboveLeaf110022203
+
+theorem e24KC2ThetaAboveNode11002221 :
+    adaptiveCoverCheck 11 thetaAboveCell11002221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002221
+    e24KC2ThetaAboveLeaf110022210 e24KC2ThetaAboveLeaf110022211 e24KC2ThetaAboveLeaf110022212
+      e24KC2ThetaAboveLeaf110022213
+
+theorem e24KC2ThetaAboveNode11002230 :
+    adaptiveCoverCheck 11 thetaAboveCell11002230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002230
+    e24KC2ThetaAboveLeaf110022300 e24KC2ThetaAboveLeaf110022301 e24KC2ThetaAboveLeaf110022302
+      e24KC2ThetaAboveLeaf110022303
+
+theorem e24KC2ThetaAboveNode11002231 :
+    adaptiveCoverCheck 11 thetaAboveCell11002231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002231
+    e24KC2ThetaAboveLeaf110022310 e24KC2ThetaAboveLeaf110022311 e24KC2ThetaAboveLeaf110022312
+      e24KC2ThetaAboveLeaf110022313
+
+theorem e24KC2ThetaAboveNode11002302 :
+    adaptiveCoverCheck 11 thetaAboveCell11002302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002302
+    e24KC2ThetaAboveLeaf110023020 e24KC2ThetaAboveLeaf110023021 e24KC2ThetaAboveLeaf110023022
+      e24KC2ThetaAboveLeaf110023023
+
+theorem e24KC2ThetaAboveNode11002303 :
+    adaptiveCoverCheck 11 thetaAboveCell11002303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002303
+    e24KC2ThetaAboveLeaf110023030 e24KC2ThetaAboveLeaf110023031 e24KC2ThetaAboveLeaf110023032
+      e24KC2ThetaAboveLeaf110023033
+
+theorem e24KC2ThetaAboveNode11002312 :
+    adaptiveCoverCheck 11 thetaAboveCell11002312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002312
+    e24KC2ThetaAboveLeaf110023120 e24KC2ThetaAboveLeaf110023121 e24KC2ThetaAboveLeaf110023122
+      e24KC2ThetaAboveLeaf110023123
+
+theorem e24KC2ThetaAboveNode11002313 :
+    adaptiveCoverCheck 11 thetaAboveCell11002313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002313
+    e24KC2ThetaAboveLeaf110023130 e24KC2ThetaAboveLeaf110023131 e24KC2ThetaAboveLeaf110023132
+      e24KC2ThetaAboveLeaf110023133
+
+theorem e24KC2ThetaAboveNode11002320 :
+    adaptiveCoverCheck 11 thetaAboveCell11002320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002320
+    e24KC2ThetaAboveLeaf110023200 e24KC2ThetaAboveLeaf110023201 e24KC2ThetaAboveLeaf110023202
+      e24KC2ThetaAboveLeaf110023203
+
+theorem e24KC2ThetaAboveNode11002321 :
+    adaptiveCoverCheck 11 thetaAboveCell11002321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002321
+    e24KC2ThetaAboveLeaf110023210 e24KC2ThetaAboveLeaf110023211 e24KC2ThetaAboveLeaf110023212
+      e24KC2ThetaAboveLeaf110023213
+
+theorem e24KC2ThetaAboveNode11002330 :
+    adaptiveCoverCheck 11 thetaAboveCell11002330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002330
+    e24KC2ThetaAboveLeaf110023300 e24KC2ThetaAboveLeaf110023301 e24KC2ThetaAboveLeaf110023302
+      e24KC2ThetaAboveLeaf110023303
+
+theorem e24KC2ThetaAboveNode11002331 :
+    adaptiveCoverCheck 11 thetaAboveCell11002331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11002331
+    e24KC2ThetaAboveLeaf110023310 e24KC2ThetaAboveLeaf110023311 e24KC2ThetaAboveLeaf110023312
+      e24KC2ThetaAboveLeaf110023313
+
+theorem e24KC2ThetaAboveNode11003202 :
+    adaptiveCoverCheck 11 thetaAboveCell11003202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003202
+    e24KC2ThetaAboveLeaf110032020 e24KC2ThetaAboveLeaf110032021 e24KC2ThetaAboveLeaf110032022
+      e24KC2ThetaAboveLeaf110032023
+
+theorem e24KC2ThetaAboveNode11003203 :
+    adaptiveCoverCheck 11 thetaAboveCell11003203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003203
+    e24KC2ThetaAboveLeaf110032030 e24KC2ThetaAboveLeaf110032031 e24KC2ThetaAboveLeaf110032032
+      e24KC2ThetaAboveLeaf110032033
+
+theorem e24KC2ThetaAboveNode11003212 :
+    adaptiveCoverCheck 11 thetaAboveCell11003212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003212
+    e24KC2ThetaAboveLeaf110032120 e24KC2ThetaAboveLeaf110032121 e24KC2ThetaAboveLeaf110032122
+      e24KC2ThetaAboveLeaf110032123
+
+theorem e24KC2ThetaAboveNode11003213 :
+    adaptiveCoverCheck 11 thetaAboveCell11003213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003213
+    e24KC2ThetaAboveLeaf110032130 e24KC2ThetaAboveLeaf110032131 e24KC2ThetaAboveLeaf110032132
+      e24KC2ThetaAboveLeaf110032133
+
+theorem e24KC2ThetaAboveNode11003220 :
+    adaptiveCoverCheck 11 thetaAboveCell11003220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003220
+    e24KC2ThetaAboveLeaf110032200 e24KC2ThetaAboveLeaf110032201 e24KC2ThetaAboveLeaf110032202
+      e24KC2ThetaAboveLeaf110032203
+
+theorem e24KC2ThetaAboveNode11003221 :
+    adaptiveCoverCheck 11 thetaAboveCell11003221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003221
+    e24KC2ThetaAboveLeaf110032210 e24KC2ThetaAboveLeaf110032211 e24KC2ThetaAboveLeaf110032212
+      e24KC2ThetaAboveLeaf110032213
+
+theorem e24KC2ThetaAboveNode11003230 :
+    adaptiveCoverCheck 11 thetaAboveCell11003230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003230
+    e24KC2ThetaAboveLeaf110032300 e24KC2ThetaAboveLeaf110032301 e24KC2ThetaAboveLeaf110032302
+      e24KC2ThetaAboveLeaf110032303
+
+theorem e24KC2ThetaAboveNode11003231 :
+    adaptiveCoverCheck 11 thetaAboveCell11003231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003231
+    e24KC2ThetaAboveLeaf110032310 e24KC2ThetaAboveLeaf110032311 e24KC2ThetaAboveLeaf110032312
+      e24KC2ThetaAboveLeaf110032313
+
+theorem e24KC2ThetaAboveNode11003302 :
+    adaptiveCoverCheck 11 thetaAboveCell11003302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003302
+    e24KC2ThetaAboveLeaf110033020 e24KC2ThetaAboveLeaf110033021 e24KC2ThetaAboveLeaf110033022
+      e24KC2ThetaAboveLeaf110033023
+
+theorem e24KC2ThetaAboveNode11003303 :
+    adaptiveCoverCheck 11 thetaAboveCell11003303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003303
+    e24KC2ThetaAboveLeaf110033030 e24KC2ThetaAboveLeaf110033031 e24KC2ThetaAboveLeaf110033032
+      e24KC2ThetaAboveLeaf110033033
+
+theorem e24KC2ThetaAboveNode11003312 :
+    adaptiveCoverCheck 11 thetaAboveCell11003312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003312
+    e24KC2ThetaAboveLeaf110033120 e24KC2ThetaAboveLeaf110033121 e24KC2ThetaAboveLeaf110033122
+      e24KC2ThetaAboveLeaf110033123
+
+theorem e24KC2ThetaAboveNode11003313 :
+    adaptiveCoverCheck 11 thetaAboveCell11003313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003313
+    e24KC2ThetaAboveLeaf110033130 e24KC2ThetaAboveLeaf110033131 e24KC2ThetaAboveLeaf110033132
+      e24KC2ThetaAboveLeaf110033133
+
+theorem e24KC2ThetaAboveNode11003320 :
+    adaptiveCoverCheck 11 thetaAboveCell11003320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003320
+    e24KC2ThetaAboveLeaf110033200 e24KC2ThetaAboveLeaf110033201 e24KC2ThetaAboveLeaf110033202
+      e24KC2ThetaAboveLeaf110033203
+
+theorem e24KC2ThetaAboveNode11003321 :
+    adaptiveCoverCheck 11 thetaAboveCell11003321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003321
+    e24KC2ThetaAboveLeaf110033210 e24KC2ThetaAboveLeaf110033211 e24KC2ThetaAboveLeaf110033212
+      e24KC2ThetaAboveLeaf110033213
+
+theorem e24KC2ThetaAboveNode11003330 :
+    adaptiveCoverCheck 11 thetaAboveCell11003330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003330
+    e24KC2ThetaAboveLeaf110033300 e24KC2ThetaAboveLeaf110033301 e24KC2ThetaAboveLeaf110033302
+      e24KC2ThetaAboveLeaf110033303
+
+theorem e24KC2ThetaAboveNode11003331 :
+    adaptiveCoverCheck 11 thetaAboveCell11003331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11003331
+    e24KC2ThetaAboveLeaf110033310 e24KC2ThetaAboveLeaf110033311 e24KC2ThetaAboveLeaf110033312
+      e24KC2ThetaAboveLeaf110033313
+
+theorem e24KC2ThetaAboveNode11012202 :
+    adaptiveCoverCheck 11 thetaAboveCell11012202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012202
+    e24KC2ThetaAboveLeaf110122020 e24KC2ThetaAboveLeaf110122021 e24KC2ThetaAboveLeaf110122022
+      e24KC2ThetaAboveLeaf110122023
+
+theorem e24KC2ThetaAboveNode11012203 :
+    adaptiveCoverCheck 11 thetaAboveCell11012203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012203
+    e24KC2ThetaAboveLeaf110122030 e24KC2ThetaAboveLeaf110122031 e24KC2ThetaAboveLeaf110122032
+      e24KC2ThetaAboveLeaf110122033
+
+theorem e24KC2ThetaAboveNode11012212 :
+    adaptiveCoverCheck 11 thetaAboveCell11012212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012212
+    e24KC2ThetaAboveLeaf110122120 e24KC2ThetaAboveLeaf110122121 e24KC2ThetaAboveLeaf110122122
+      e24KC2ThetaAboveLeaf110122123
+
+theorem e24KC2ThetaAboveNode11012213 :
+    adaptiveCoverCheck 11 thetaAboveCell11012213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012213
+    e24KC2ThetaAboveLeaf110122130 e24KC2ThetaAboveLeaf110122131 e24KC2ThetaAboveLeaf110122132
+      e24KC2ThetaAboveLeaf110122133
+
+theorem e24KC2ThetaAboveNode11012220 :
+    adaptiveCoverCheck 11 thetaAboveCell11012220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012220
+    e24KC2ThetaAboveLeaf110122200 e24KC2ThetaAboveLeaf110122201 e24KC2ThetaAboveLeaf110122202
+      e24KC2ThetaAboveLeaf110122203
+
+theorem e24KC2ThetaAboveNode11012221 :
+    adaptiveCoverCheck 11 thetaAboveCell11012221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012221
+    e24KC2ThetaAboveLeaf110122210 e24KC2ThetaAboveLeaf110122211 e24KC2ThetaAboveLeaf110122212
+      e24KC2ThetaAboveLeaf110122213
+
+theorem e24KC2ThetaAboveNode11012230 :
+    adaptiveCoverCheck 11 thetaAboveCell11012230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012230
+    e24KC2ThetaAboveLeaf110122300 e24KC2ThetaAboveLeaf110122301 e24KC2ThetaAboveLeaf110122302
+      e24KC2ThetaAboveLeaf110122303
+
+theorem e24KC2ThetaAboveNode11012231 :
+    adaptiveCoverCheck 11 thetaAboveCell11012231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012231
+    e24KC2ThetaAboveLeaf110122310 e24KC2ThetaAboveLeaf110122311 e24KC2ThetaAboveLeaf110122312
+      e24KC2ThetaAboveLeaf110122313
+
+theorem e24KC2ThetaAboveNode11012302 :
+    adaptiveCoverCheck 11 thetaAboveCell11012302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012302
+    e24KC2ThetaAboveLeaf110123020 e24KC2ThetaAboveLeaf110123021 e24KC2ThetaAboveLeaf110123022
+      e24KC2ThetaAboveLeaf110123023
+
+theorem e24KC2ThetaAboveNode11012303 :
+    adaptiveCoverCheck 11 thetaAboveCell11012303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012303
+    e24KC2ThetaAboveLeaf110123030 e24KC2ThetaAboveLeaf110123031 e24KC2ThetaAboveLeaf110123032
+      e24KC2ThetaAboveLeaf110123033
+
+theorem e24KC2ThetaAboveNode11012312 :
+    adaptiveCoverCheck 11 thetaAboveCell11012312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012312
+    e24KC2ThetaAboveLeaf110123120 e24KC2ThetaAboveLeaf110123121 e24KC2ThetaAboveLeaf110123122
+      e24KC2ThetaAboveLeaf110123123
+
+theorem e24KC2ThetaAboveNode11012313 :
+    adaptiveCoverCheck 11 thetaAboveCell11012313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012313
+    e24KC2ThetaAboveLeaf110123130 e24KC2ThetaAboveLeaf110123131 e24KC2ThetaAboveLeaf110123132
+      e24KC2ThetaAboveLeaf110123133
+
+theorem e24KC2ThetaAboveNode11012320 :
+    adaptiveCoverCheck 11 thetaAboveCell11012320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012320
+    e24KC2ThetaAboveLeaf110123200 e24KC2ThetaAboveLeaf110123201 e24KC2ThetaAboveLeaf110123202
+      e24KC2ThetaAboveLeaf110123203
+
+theorem e24KC2ThetaAboveNode11012321 :
+    adaptiveCoverCheck 11 thetaAboveCell11012321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012321
+    e24KC2ThetaAboveLeaf110123210 e24KC2ThetaAboveLeaf110123211 e24KC2ThetaAboveLeaf110123212
+      e24KC2ThetaAboveLeaf110123213
+
+theorem e24KC2ThetaAboveNode11012330 :
+    adaptiveCoverCheck 11 thetaAboveCell11012330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012330
+    e24KC2ThetaAboveLeaf110123300 e24KC2ThetaAboveLeaf110123301 e24KC2ThetaAboveLeaf110123302
+      e24KC2ThetaAboveLeaf110123303
+
+theorem e24KC2ThetaAboveNode11012331 :
+    adaptiveCoverCheck 11 thetaAboveCell11012331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11012331
+    e24KC2ThetaAboveLeaf110123310 e24KC2ThetaAboveLeaf110123311 e24KC2ThetaAboveLeaf110123312
+      e24KC2ThetaAboveLeaf110123313
+
+theorem e24KC2ThetaAboveNode11013202 :
+    adaptiveCoverCheck 11 thetaAboveCell11013202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013202
+    e24KC2ThetaAboveLeaf110132020 e24KC2ThetaAboveLeaf110132021 e24KC2ThetaAboveLeaf110132022
+      e24KC2ThetaAboveLeaf110132023
+
+theorem e24KC2ThetaAboveNode11013203 :
+    adaptiveCoverCheck 11 thetaAboveCell11013203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013203
+    e24KC2ThetaAboveLeaf110132030 e24KC2ThetaAboveLeaf110132031 e24KC2ThetaAboveLeaf110132032
+      e24KC2ThetaAboveLeaf110132033
+
+theorem e24KC2ThetaAboveNode11013212 :
+    adaptiveCoverCheck 11 thetaAboveCell11013212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013212
+    e24KC2ThetaAboveLeaf110132120 e24KC2ThetaAboveLeaf110132121 e24KC2ThetaAboveLeaf110132122
+      e24KC2ThetaAboveLeaf110132123
+
+theorem e24KC2ThetaAboveNode11013213 :
+    adaptiveCoverCheck 11 thetaAboveCell11013213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013213
+    e24KC2ThetaAboveLeaf110132130 e24KC2ThetaAboveLeaf110132131 e24KC2ThetaAboveLeaf110132132
+      e24KC2ThetaAboveLeaf110132133
+
+theorem e24KC2ThetaAboveNode11013220 :
+    adaptiveCoverCheck 11 thetaAboveCell11013220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013220
+    e24KC2ThetaAboveLeaf110132200 e24KC2ThetaAboveLeaf110132201 e24KC2ThetaAboveLeaf110132202
+      e24KC2ThetaAboveLeaf110132203
+
+theorem e24KC2ThetaAboveNode11013221 :
+    adaptiveCoverCheck 11 thetaAboveCell11013221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013221
+    e24KC2ThetaAboveLeaf110132210 e24KC2ThetaAboveLeaf110132211 e24KC2ThetaAboveLeaf110132212
+      e24KC2ThetaAboveLeaf110132213
+
+theorem e24KC2ThetaAboveNode11013230 :
+    adaptiveCoverCheck 11 thetaAboveCell11013230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013230
+    e24KC2ThetaAboveLeaf110132300 e24KC2ThetaAboveLeaf110132301 e24KC2ThetaAboveLeaf110132302
+      e24KC2ThetaAboveLeaf110132303
+
+theorem e24KC2ThetaAboveNode11013231 :
+    adaptiveCoverCheck 11 thetaAboveCell11013231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013231
+    e24KC2ThetaAboveLeaf110132310 e24KC2ThetaAboveLeaf110132311 e24KC2ThetaAboveLeaf110132312
+      e24KC2ThetaAboveLeaf110132313
+
+theorem e24KC2ThetaAboveNode11013302 :
+    adaptiveCoverCheck 11 thetaAboveCell11013302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013302
+    e24KC2ThetaAboveLeaf110133020 e24KC2ThetaAboveLeaf110133021 e24KC2ThetaAboveLeaf110133022
+      e24KC2ThetaAboveLeaf110133023
+
+theorem e24KC2ThetaAboveNode11013303 :
+    adaptiveCoverCheck 11 thetaAboveCell11013303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013303
+    e24KC2ThetaAboveLeaf110133030 e24KC2ThetaAboveLeaf110133031 e24KC2ThetaAboveLeaf110133032
+      e24KC2ThetaAboveLeaf110133033
+
+theorem e24KC2ThetaAboveNode11013312 :
+    adaptiveCoverCheck 11 thetaAboveCell11013312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013312
+    e24KC2ThetaAboveLeaf110133120 e24KC2ThetaAboveLeaf110133121 e24KC2ThetaAboveLeaf110133122
+      e24KC2ThetaAboveLeaf110133123
+
+theorem e24KC2ThetaAboveNode11013313 :
+    adaptiveCoverCheck 11 thetaAboveCell11013313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013313
+    e24KC2ThetaAboveLeaf110133130 e24KC2ThetaAboveLeaf110133131 e24KC2ThetaAboveLeaf110133132
+      e24KC2ThetaAboveLeaf110133133
+
+theorem e24KC2ThetaAboveNode11013320 :
+    adaptiveCoverCheck 11 thetaAboveCell11013320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013320
+    e24KC2ThetaAboveLeaf110133200 e24KC2ThetaAboveLeaf110133201 e24KC2ThetaAboveLeaf110133202
+      e24KC2ThetaAboveLeaf110133203
+
+theorem e24KC2ThetaAboveNode11013321 :
+    adaptiveCoverCheck 11 thetaAboveCell11013321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013321
+    e24KC2ThetaAboveLeaf110133210 e24KC2ThetaAboveLeaf110133211 e24KC2ThetaAboveLeaf110133212
+      e24KC2ThetaAboveLeaf110133213
+
+theorem e24KC2ThetaAboveNode11013330 :
+    adaptiveCoverCheck 11 thetaAboveCell11013330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013330
+    e24KC2ThetaAboveLeaf110133300 e24KC2ThetaAboveLeaf110133301 e24KC2ThetaAboveLeaf110133302
+      e24KC2ThetaAboveLeaf110133303
+
+theorem e24KC2ThetaAboveNode11013331 :
+    adaptiveCoverCheck 11 thetaAboveCell11013331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11013331
+    e24KC2ThetaAboveLeaf110133310 e24KC2ThetaAboveLeaf110133311 e24KC2ThetaAboveLeaf110133312
+      e24KC2ThetaAboveLeaf110133313
+
+theorem e24KC2ThetaAboveNode11102202 :
+    adaptiveCoverCheck 11 thetaAboveCell11102202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102202
+    e24KC2ThetaAboveLeaf111022020 e24KC2ThetaAboveLeaf111022021 e24KC2ThetaAboveLeaf111022022
+      e24KC2ThetaAboveLeaf111022023
+
+theorem e24KC2ThetaAboveNode11102203 :
+    adaptiveCoverCheck 11 thetaAboveCell11102203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102203
+    e24KC2ThetaAboveLeaf111022030 e24KC2ThetaAboveLeaf111022031 e24KC2ThetaAboveLeaf111022032
+      e24KC2ThetaAboveLeaf111022033
+
+theorem e24KC2ThetaAboveNode11102212 :
+    adaptiveCoverCheck 11 thetaAboveCell11102212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102212
+    e24KC2ThetaAboveLeaf111022120 e24KC2ThetaAboveLeaf111022121 e24KC2ThetaAboveLeaf111022122
+      e24KC2ThetaAboveLeaf111022123
+
+theorem e24KC2ThetaAboveNode11102213 :
+    adaptiveCoverCheck 11 thetaAboveCell11102213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102213
+    e24KC2ThetaAboveLeaf111022130 e24KC2ThetaAboveLeaf111022131 e24KC2ThetaAboveLeaf111022132
+      e24KC2ThetaAboveLeaf111022133
+
+theorem e24KC2ThetaAboveNode11102220 :
+    adaptiveCoverCheck 11 thetaAboveCell11102220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102220
+    e24KC2ThetaAboveLeaf111022200 e24KC2ThetaAboveLeaf111022201 e24KC2ThetaAboveLeaf111022202
+      e24KC2ThetaAboveLeaf111022203
+
+theorem e24KC2ThetaAboveNode11102221 :
+    adaptiveCoverCheck 11 thetaAboveCell11102221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102221
+    e24KC2ThetaAboveLeaf111022210 e24KC2ThetaAboveLeaf111022211 e24KC2ThetaAboveLeaf111022212
+      e24KC2ThetaAboveLeaf111022213
+
+theorem e24KC2ThetaAboveNode11102230 :
+    adaptiveCoverCheck 11 thetaAboveCell11102230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102230
+    e24KC2ThetaAboveLeaf111022300 e24KC2ThetaAboveLeaf111022301 e24KC2ThetaAboveLeaf111022302
+      e24KC2ThetaAboveLeaf111022303
+
+theorem e24KC2ThetaAboveNode11102231 :
+    adaptiveCoverCheck 11 thetaAboveCell11102231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102231
+    e24KC2ThetaAboveLeaf111022310 e24KC2ThetaAboveLeaf111022311 e24KC2ThetaAboveLeaf111022312
+      e24KC2ThetaAboveLeaf111022313
+
+theorem e24KC2ThetaAboveNode11102302 :
+    adaptiveCoverCheck 11 thetaAboveCell11102302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102302
+    e24KC2ThetaAboveLeaf111023020 e24KC2ThetaAboveLeaf111023021 e24KC2ThetaAboveLeaf111023022
+      e24KC2ThetaAboveLeaf111023023
+
+theorem e24KC2ThetaAboveNode11102303 :
+    adaptiveCoverCheck 11 thetaAboveCell11102303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102303
+    e24KC2ThetaAboveLeaf111023030 e24KC2ThetaAboveLeaf111023031 e24KC2ThetaAboveLeaf111023032
+      e24KC2ThetaAboveLeaf111023033
+
+theorem e24KC2ThetaAboveNode11102312 :
+    adaptiveCoverCheck 11 thetaAboveCell11102312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102312
+    e24KC2ThetaAboveLeaf111023120 e24KC2ThetaAboveLeaf111023121 e24KC2ThetaAboveLeaf111023122
+      e24KC2ThetaAboveLeaf111023123
+
+theorem e24KC2ThetaAboveNode11102313 :
+    adaptiveCoverCheck 11 thetaAboveCell11102313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102313
+    e24KC2ThetaAboveLeaf111023130 e24KC2ThetaAboveLeaf111023131 e24KC2ThetaAboveLeaf111023132
+      e24KC2ThetaAboveLeaf111023133
+
+theorem e24KC2ThetaAboveNode11102320 :
+    adaptiveCoverCheck 11 thetaAboveCell11102320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102320
+    e24KC2ThetaAboveLeaf111023200 e24KC2ThetaAboveLeaf111023201 e24KC2ThetaAboveLeaf111023202
+      e24KC2ThetaAboveLeaf111023203
+
+theorem e24KC2ThetaAboveNode11102321 :
+    adaptiveCoverCheck 11 thetaAboveCell11102321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102321
+    e24KC2ThetaAboveLeaf111023210 e24KC2ThetaAboveLeaf111023211 e24KC2ThetaAboveLeaf111023212
+      e24KC2ThetaAboveLeaf111023213
+
+theorem e24KC2ThetaAboveNode11102330 :
+    adaptiveCoverCheck 11 thetaAboveCell11102330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102330
+    e24KC2ThetaAboveLeaf111023300 e24KC2ThetaAboveLeaf111023301 e24KC2ThetaAboveLeaf111023302
+      e24KC2ThetaAboveLeaf111023303
+
+theorem e24KC2ThetaAboveNode11102331 :
+    adaptiveCoverCheck 11 thetaAboveCell11102331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11102331
+    e24KC2ThetaAboveLeaf111023310 e24KC2ThetaAboveLeaf111023311 e24KC2ThetaAboveLeaf111023312
+      e24KC2ThetaAboveLeaf111023313
+
+theorem e24KC2ThetaAboveNode11103202 :
+    adaptiveCoverCheck 11 thetaAboveCell11103202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103202
+    e24KC2ThetaAboveLeaf111032020 e24KC2ThetaAboveLeaf111032021 e24KC2ThetaAboveLeaf111032022
+      e24KC2ThetaAboveLeaf111032023
+
+theorem e24KC2ThetaAboveNode11103203 :
+    adaptiveCoverCheck 11 thetaAboveCell11103203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103203
+    e24KC2ThetaAboveLeaf111032030 e24KC2ThetaAboveLeaf111032031 e24KC2ThetaAboveLeaf111032032
+      e24KC2ThetaAboveLeaf111032033
+
+theorem e24KC2ThetaAboveNode11103212 :
+    adaptiveCoverCheck 11 thetaAboveCell11103212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103212
+    e24KC2ThetaAboveLeaf111032120 e24KC2ThetaAboveLeaf111032121 e24KC2ThetaAboveLeaf111032122
+      e24KC2ThetaAboveLeaf111032123
+
+theorem e24KC2ThetaAboveNode11103213 :
+    adaptiveCoverCheck 11 thetaAboveCell11103213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103213
+    e24KC2ThetaAboveLeaf111032130 e24KC2ThetaAboveLeaf111032131 e24KC2ThetaAboveLeaf111032132
+      e24KC2ThetaAboveLeaf111032133
+
+theorem e24KC2ThetaAboveNode11103220 :
+    adaptiveCoverCheck 11 thetaAboveCell11103220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103220
+    e24KC2ThetaAboveLeaf111032200 e24KC2ThetaAboveLeaf111032201 e24KC2ThetaAboveLeaf111032202
+      e24KC2ThetaAboveLeaf111032203
+
+theorem e24KC2ThetaAboveNode11103221 :
+    adaptiveCoverCheck 11 thetaAboveCell11103221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103221
+    e24KC2ThetaAboveLeaf111032210 e24KC2ThetaAboveLeaf111032211 e24KC2ThetaAboveLeaf111032212
+      e24KC2ThetaAboveLeaf111032213
+
+theorem e24KC2ThetaAboveNode11103230 :
+    adaptiveCoverCheck 11 thetaAboveCell11103230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103230
+    e24KC2ThetaAboveLeaf111032300 e24KC2ThetaAboveLeaf111032301 e24KC2ThetaAboveLeaf111032302
+      e24KC2ThetaAboveLeaf111032303
+
+theorem e24KC2ThetaAboveNode11103231 :
+    adaptiveCoverCheck 11 thetaAboveCell11103231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103231
+    e24KC2ThetaAboveLeaf111032310 e24KC2ThetaAboveLeaf111032311 e24KC2ThetaAboveLeaf111032312
+      e24KC2ThetaAboveLeaf111032313
+
+theorem e24KC2ThetaAboveNode11103302 :
+    adaptiveCoverCheck 11 thetaAboveCell11103302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103302
+    e24KC2ThetaAboveLeaf111033020 e24KC2ThetaAboveLeaf111033021 e24KC2ThetaAboveLeaf111033022
+      e24KC2ThetaAboveLeaf111033023
+
+theorem e24KC2ThetaAboveNode11103303 :
+    adaptiveCoverCheck 11 thetaAboveCell11103303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103303
+    e24KC2ThetaAboveLeaf111033030 e24KC2ThetaAboveLeaf111033031 e24KC2ThetaAboveLeaf111033032
+      e24KC2ThetaAboveLeaf111033033
+
+theorem e24KC2ThetaAboveNode11103312 :
+    adaptiveCoverCheck 11 thetaAboveCell11103312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103312
+    e24KC2ThetaAboveLeaf111033120 e24KC2ThetaAboveLeaf111033121 e24KC2ThetaAboveLeaf111033122
+      e24KC2ThetaAboveLeaf111033123
+
+theorem e24KC2ThetaAboveNode11103313 :
+    adaptiveCoverCheck 11 thetaAboveCell11103313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103313
+    e24KC2ThetaAboveLeaf111033130 e24KC2ThetaAboveLeaf111033131 e24KC2ThetaAboveLeaf111033132
+      e24KC2ThetaAboveLeaf111033133
+
+theorem e24KC2ThetaAboveNode11103320 :
+    adaptiveCoverCheck 11 thetaAboveCell11103320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103320
+    e24KC2ThetaAboveLeaf111033200 e24KC2ThetaAboveLeaf111033201 e24KC2ThetaAboveLeaf111033202
+      e24KC2ThetaAboveLeaf111033203
+
+theorem e24KC2ThetaAboveNode11103321 :
+    adaptiveCoverCheck 11 thetaAboveCell11103321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103321
+    e24KC2ThetaAboveLeaf111033210 e24KC2ThetaAboveLeaf111033211 e24KC2ThetaAboveLeaf111033212
+      e24KC2ThetaAboveLeaf111033213
+
+theorem e24KC2ThetaAboveNode11103330 :
+    adaptiveCoverCheck 11 thetaAboveCell11103330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103330
+    e24KC2ThetaAboveLeaf111033300 e24KC2ThetaAboveLeaf111033301 e24KC2ThetaAboveLeaf111033302
+      e24KC2ThetaAboveLeaf111033303
+
+theorem e24KC2ThetaAboveNode11103331 :
+    adaptiveCoverCheck 11 thetaAboveCell11103331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11103331
+    e24KC2ThetaAboveLeaf111033310 e24KC2ThetaAboveLeaf111033311 e24KC2ThetaAboveLeaf111033312
+      e24KC2ThetaAboveLeaf111033313
+
+theorem e24KC2ThetaAboveNode11112202 :
+    adaptiveCoverCheck 11 thetaAboveCell11112202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112202
+    e24KC2ThetaAboveLeaf111122020 e24KC2ThetaAboveLeaf111122021 e24KC2ThetaAboveLeaf111122022
+      e24KC2ThetaAboveLeaf111122023
+
+theorem e24KC2ThetaAboveNode11112203 :
+    adaptiveCoverCheck 11 thetaAboveCell11112203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112203
+    e24KC2ThetaAboveLeaf111122030 e24KC2ThetaAboveLeaf111122031 e24KC2ThetaAboveLeaf111122032
+      e24KC2ThetaAboveLeaf111122033
+
+theorem e24KC2ThetaAboveNode11112212 :
+    adaptiveCoverCheck 11 thetaAboveCell11112212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112212
+    e24KC2ThetaAboveLeaf111122120 e24KC2ThetaAboveLeaf111122121 e24KC2ThetaAboveLeaf111122122
+      e24KC2ThetaAboveLeaf111122123
+
+theorem e24KC2ThetaAboveNode11112213 :
+    adaptiveCoverCheck 11 thetaAboveCell11112213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112213
+    e24KC2ThetaAboveLeaf111122130 e24KC2ThetaAboveLeaf111122131 e24KC2ThetaAboveLeaf111122132
+      e24KC2ThetaAboveLeaf111122133
+
+theorem e24KC2ThetaAboveNode11112220 :
+    adaptiveCoverCheck 11 thetaAboveCell11112220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112220
+    e24KC2ThetaAboveLeaf111122200 e24KC2ThetaAboveLeaf111122201 e24KC2ThetaAboveLeaf111122202
+      e24KC2ThetaAboveLeaf111122203
+
+theorem e24KC2ThetaAboveNode11112221 :
+    adaptiveCoverCheck 11 thetaAboveCell11112221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112221
+    e24KC2ThetaAboveLeaf111122210 e24KC2ThetaAboveLeaf111122211 e24KC2ThetaAboveLeaf111122212
+      e24KC2ThetaAboveLeaf111122213
+
+theorem e24KC2ThetaAboveNode11112230 :
+    adaptiveCoverCheck 11 thetaAboveCell11112230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112230
+    e24KC2ThetaAboveLeaf111122300 e24KC2ThetaAboveLeaf111122301 e24KC2ThetaAboveLeaf111122302
+      e24KC2ThetaAboveLeaf111122303
+
+theorem e24KC2ThetaAboveNode11112231 :
+    adaptiveCoverCheck 11 thetaAboveCell11112231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112231
+    e24KC2ThetaAboveLeaf111122310 e24KC2ThetaAboveLeaf111122311 e24KC2ThetaAboveLeaf111122312
+      e24KC2ThetaAboveLeaf111122313
+
+theorem e24KC2ThetaAboveNode11112302 :
+    adaptiveCoverCheck 11 thetaAboveCell11112302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112302
+    e24KC2ThetaAboveLeaf111123020 e24KC2ThetaAboveLeaf111123021 e24KC2ThetaAboveLeaf111123022
+      e24KC2ThetaAboveLeaf111123023
+
+theorem e24KC2ThetaAboveNode11112303 :
+    adaptiveCoverCheck 11 thetaAboveCell11112303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112303
+    e24KC2ThetaAboveLeaf111123030 e24KC2ThetaAboveLeaf111123031 e24KC2ThetaAboveLeaf111123032
+      e24KC2ThetaAboveLeaf111123033
+
+theorem e24KC2ThetaAboveNode11112312 :
+    adaptiveCoverCheck 11 thetaAboveCell11112312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112312
+    e24KC2ThetaAboveLeaf111123120 e24KC2ThetaAboveLeaf111123121 e24KC2ThetaAboveLeaf111123122
+      e24KC2ThetaAboveLeaf111123123
+
+theorem e24KC2ThetaAboveNode11112313 :
+    adaptiveCoverCheck 11 thetaAboveCell11112313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112313
+    e24KC2ThetaAboveLeaf111123130 e24KC2ThetaAboveLeaf111123131 e24KC2ThetaAboveLeaf111123132
+      e24KC2ThetaAboveLeaf111123133
+
+theorem e24KC2ThetaAboveNode11112320 :
+    adaptiveCoverCheck 11 thetaAboveCell11112320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112320
+    e24KC2ThetaAboveLeaf111123200 e24KC2ThetaAboveLeaf111123201 e24KC2ThetaAboveLeaf111123202
+      e24KC2ThetaAboveLeaf111123203
+
+theorem e24KC2ThetaAboveNode11112321 :
+    adaptiveCoverCheck 11 thetaAboveCell11112321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112321
+    e24KC2ThetaAboveLeaf111123210 e24KC2ThetaAboveLeaf111123211 e24KC2ThetaAboveLeaf111123212
+      e24KC2ThetaAboveLeaf111123213
+
+theorem e24KC2ThetaAboveNode11112330 :
+    adaptiveCoverCheck 11 thetaAboveCell11112330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112330
+    e24KC2ThetaAboveLeaf111123300 e24KC2ThetaAboveLeaf111123301 e24KC2ThetaAboveLeaf111123302
+      e24KC2ThetaAboveLeaf111123303
+
+theorem e24KC2ThetaAboveNode11112331 :
+    adaptiveCoverCheck 11 thetaAboveCell11112331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112331
+    e24KC2ThetaAboveLeaf111123310 e24KC2ThetaAboveLeaf111123311 e24KC2ThetaAboveLeaf111123312
+      e24KC2ThetaAboveLeaf111123313
+
+theorem e24KC2ThetaAboveNode11112332 :
+    adaptiveCoverCheck 11 thetaAboveCell11112332 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112332
+    e24KC2ThetaAboveLeaf111123320 e24KC2ThetaAboveLeaf111123321 e24KC2ThetaAboveLeaf111123322
+      e24KC2ThetaAboveLeaf111123323
+
+theorem e24KC2ThetaAboveNode11112333 :
+    adaptiveCoverCheck 11 thetaAboveCell11112333 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11112333
+    e24KC2ThetaAboveLeaf111123330 e24KC2ThetaAboveLeaf111123331 e24KC2ThetaAboveLeaf111123332
+      e24KC2ThetaAboveLeaf111123333
+
+theorem e24KC2ThetaAboveNode11113202 :
+    adaptiveCoverCheck 11 thetaAboveCell11113202 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113202
+    e24KC2ThetaAboveLeaf111132020 e24KC2ThetaAboveLeaf111132021 e24KC2ThetaAboveLeaf111132022
+      e24KC2ThetaAboveLeaf111132023
+
+theorem e24KC2ThetaAboveNode11113203 :
+    adaptiveCoverCheck 11 thetaAboveCell11113203 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113203
+    e24KC2ThetaAboveLeaf111132030 e24KC2ThetaAboveLeaf111132031 e24KC2ThetaAboveLeaf111132032
+      e24KC2ThetaAboveLeaf111132033
+
+theorem e24KC2ThetaAboveNode11113212 :
+    adaptiveCoverCheck 11 thetaAboveCell11113212 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113212
+    e24KC2ThetaAboveLeaf111132120 e24KC2ThetaAboveLeaf111132121 e24KC2ThetaAboveLeaf111132122
+      e24KC2ThetaAboveLeaf111132123
+
+theorem e24KC2ThetaAboveNode11113213 :
+    adaptiveCoverCheck 11 thetaAboveCell11113213 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113213
+    e24KC2ThetaAboveLeaf111132130 e24KC2ThetaAboveLeaf111132131 e24KC2ThetaAboveLeaf111132132
+      e24KC2ThetaAboveLeaf111132133
+
+theorem e24KC2ThetaAboveNode11113220 :
+    adaptiveCoverCheck 11 thetaAboveCell11113220 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113220
+    e24KC2ThetaAboveLeaf111132200 e24KC2ThetaAboveLeaf111132201 e24KC2ThetaAboveLeaf111132202
+      e24KC2ThetaAboveLeaf111132203
+
+theorem e24KC2ThetaAboveNode11113221 :
+    adaptiveCoverCheck 11 thetaAboveCell11113221 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113221
+    e24KC2ThetaAboveLeaf111132210 e24KC2ThetaAboveLeaf111132211 e24KC2ThetaAboveLeaf111132212
+      e24KC2ThetaAboveLeaf111132213
+
+theorem e24KC2ThetaAboveNode11113222 :
+    adaptiveCoverCheck 11 thetaAboveCell11113222 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113222
+    e24KC2ThetaAboveLeaf111132220 e24KC2ThetaAboveLeaf111132221 e24KC2ThetaAboveLeaf111132222
+      e24KC2ThetaAboveLeaf111132223
+
+theorem e24KC2ThetaAboveNode11113223 :
+    adaptiveCoverCheck 11 thetaAboveCell11113223 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113223
+    e24KC2ThetaAboveLeaf111132230 e24KC2ThetaAboveLeaf111132231 e24KC2ThetaAboveLeaf111132232
+      e24KC2ThetaAboveLeaf111132233
+
+theorem e24KC2ThetaAboveNode11113230 :
+    adaptiveCoverCheck 11 thetaAboveCell11113230 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113230
+    e24KC2ThetaAboveLeaf111132300 e24KC2ThetaAboveLeaf111132301 e24KC2ThetaAboveLeaf111132302
+      e24KC2ThetaAboveLeaf111132303
+
+theorem e24KC2ThetaAboveNode11113231 :
+    adaptiveCoverCheck 11 thetaAboveCell11113231 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113231
+    e24KC2ThetaAboveLeaf111132310 e24KC2ThetaAboveLeaf111132311 e24KC2ThetaAboveLeaf111132312
+      e24KC2ThetaAboveLeaf111132313
+
+theorem e24KC2ThetaAboveNode11113232 :
+    adaptiveCoverCheck 11 thetaAboveCell11113232 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113232
+    e24KC2ThetaAboveLeaf111132320 e24KC2ThetaAboveLeaf111132321 e24KC2ThetaAboveLeaf111132322
+      e24KC2ThetaAboveLeaf111132323
+
+theorem e24KC2ThetaAboveNode11113233 :
+    adaptiveCoverCheck 11 thetaAboveCell11113233 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113233
+    e24KC2ThetaAboveLeaf111132330 e24KC2ThetaAboveLeaf111132331 e24KC2ThetaAboveLeaf111132332
+      e24KC2ThetaAboveLeaf111132333
+
+theorem e24KC2ThetaAboveNode11113302 :
+    adaptiveCoverCheck 11 thetaAboveCell11113302 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113302
+    e24KC2ThetaAboveLeaf111133020 e24KC2ThetaAboveLeaf111133021 e24KC2ThetaAboveLeaf111133022
+      e24KC2ThetaAboveLeaf111133023
+
+theorem e24KC2ThetaAboveNode11113303 :
+    adaptiveCoverCheck 11 thetaAboveCell11113303 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113303
+    e24KC2ThetaAboveLeaf111133030 e24KC2ThetaAboveLeaf111133031 e24KC2ThetaAboveLeaf111133032
+      e24KC2ThetaAboveLeaf111133033
+
+theorem e24KC2ThetaAboveNode11113312 :
+    adaptiveCoverCheck 11 thetaAboveCell11113312 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113312
+    e24KC2ThetaAboveLeaf111133120 e24KC2ThetaAboveLeaf111133121 e24KC2ThetaAboveLeaf111133122
+      e24KC2ThetaAboveLeaf111133123
+
+theorem e24KC2ThetaAboveNode11113313 :
+    adaptiveCoverCheck 11 thetaAboveCell11113313 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113313
+    e24KC2ThetaAboveLeaf111133130 e24KC2ThetaAboveLeaf111133131 e24KC2ThetaAboveLeaf111133132
+      e24KC2ThetaAboveLeaf111133133
+
+theorem e24KC2ThetaAboveNode11113320 :
+    adaptiveCoverCheck 11 thetaAboveCell11113320 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113320
+    e24KC2ThetaAboveLeaf111133200 e24KC2ThetaAboveLeaf111133201 e24KC2ThetaAboveLeaf111133202
+      e24KC2ThetaAboveLeaf111133203
+
+theorem e24KC2ThetaAboveNode11113321 :
+    adaptiveCoverCheck 11 thetaAboveCell11113321 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113321
+    e24KC2ThetaAboveLeaf111133210 e24KC2ThetaAboveLeaf111133211 e24KC2ThetaAboveLeaf111133212
+      e24KC2ThetaAboveLeaf111133213
+
+theorem e24KC2ThetaAboveNode11113322 :
+    adaptiveCoverCheck 11 thetaAboveCell11113322 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113322
+    e24KC2ThetaAboveLeaf111133220 e24KC2ThetaAboveLeaf111133221 e24KC2ThetaAboveLeaf111133222
+      e24KC2ThetaAboveLeaf111133223
+
+theorem e24KC2ThetaAboveNode11113323 :
+    adaptiveCoverCheck 11 thetaAboveCell11113323 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113323
+    e24KC2ThetaAboveLeaf111133230 e24KC2ThetaAboveLeaf111133231 e24KC2ThetaAboveLeaf111133232
+      e24KC2ThetaAboveLeaf111133233
+
+theorem e24KC2ThetaAboveNode11113330 :
+    adaptiveCoverCheck 11 thetaAboveCell11113330 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113330
+    e24KC2ThetaAboveLeaf111133300 e24KC2ThetaAboveLeaf111133301 e24KC2ThetaAboveLeaf111133302
+      e24KC2ThetaAboveLeaf111133303
+
+theorem e24KC2ThetaAboveNode11113331 :
+    adaptiveCoverCheck 11 thetaAboveCell11113331 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113331
+    e24KC2ThetaAboveLeaf111133310 e24KC2ThetaAboveLeaf111133311 e24KC2ThetaAboveLeaf111133312
+      e24KC2ThetaAboveLeaf111133313
+
+theorem e24KC2ThetaAboveNode11113332 :
+    adaptiveCoverCheck 11 thetaAboveCell11113332 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113332
+    e24KC2ThetaAboveLeaf111133320 e24KC2ThetaAboveLeaf111133321 e24KC2ThetaAboveLeaf111133322
+      e24KC2ThetaAboveLeaf111133323
+
+theorem e24KC2ThetaAboveNode11113333 :
+    adaptiveCoverCheck 11 thetaAboveCell11113333 = true :=
+  adaptiveCoverCheck_succ_of_children 10 thetaAboveCell11113333
+    e24KC2ThetaAboveLeaf111133330 e24KC2ThetaAboveLeaf111133331 e24KC2ThetaAboveLeaf111133332
+      e24KC2ThetaAboveLeaf111133333
+
+theorem e24KC2ThetaAboveNode0000202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHL thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00002020 e24KC2ThetaAboveLeaf00002021 e24KC2ThetaAboveNode00002022
+      e24KC2ThetaAboveNode00002023
+
+theorem e24KC2ThetaAboveNode0000203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHL thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00002030 e24KC2ThetaAboveLeaf00002031 e24KC2ThetaAboveNode00002032
+      e24KC2ThetaAboveNode00002033
+
+theorem e24KC2ThetaAboveNode0000212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHL thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00002120 e24KC2ThetaAboveLeaf00002121 e24KC2ThetaAboveNode00002122
+      e24KC2ThetaAboveNode00002123
+
+theorem e24KC2ThetaAboveNode0000213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHL thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00002130 e24KC2ThetaAboveLeaf00002131 e24KC2ThetaAboveNode00002132
+      e24KC2ThetaAboveNode00002133
+
+theorem e24KC2ThetaAboveNode0000220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell0000)))
+    e24KC2ThetaAboveNode00002200 e24KC2ThetaAboveNode00002201 e24KC2ThetaAboveNode00002202
+      e24KC2ThetaAboveNode00002203
+
+theorem e24KC2ThetaAboveNode0000221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell0000)))
+    e24KC2ThetaAboveNode00002210 e24KC2ThetaAboveNode00002211 e24KC2ThetaAboveNode00002212
+      e24KC2ThetaAboveNode00002213
+
+theorem e24KC2ThetaAboveNode0000222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00002220 e24KC2ThetaAboveLeaf00002221 e24KC2ThetaAboveLeaf00002222
+      e24KC2ThetaAboveLeaf00002223
+
+theorem e24KC2ThetaAboveNode0000223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00002230 e24KC2ThetaAboveLeaf00002231 e24KC2ThetaAboveLeaf00002232
+      e24KC2ThetaAboveLeaf00002233
+
+theorem e24KC2ThetaAboveNode0000230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell0000)))
+    e24KC2ThetaAboveNode00002300 e24KC2ThetaAboveNode00002301 e24KC2ThetaAboveNode00002302
+      e24KC2ThetaAboveNode00002303
+
+theorem e24KC2ThetaAboveNode0000231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell0000)))
+    e24KC2ThetaAboveNode00002310 e24KC2ThetaAboveNode00002311 e24KC2ThetaAboveNode00002312
+      e24KC2ThetaAboveNode00002313
+
+theorem e24KC2ThetaAboveNode0000232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00002320 e24KC2ThetaAboveLeaf00002321 e24KC2ThetaAboveLeaf00002322
+      e24KC2ThetaAboveLeaf00002323
+
+theorem e24KC2ThetaAboveNode0000233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00002330 e24KC2ThetaAboveLeaf00002331 e24KC2ThetaAboveLeaf00002332
+      e24KC2ThetaAboveLeaf00002333
+
+theorem e24KC2ThetaAboveNode0000302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHH thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00003020 e24KC2ThetaAboveLeaf00003021 e24KC2ThetaAboveNode00003022
+      e24KC2ThetaAboveNode00003023
+
+theorem e24KC2ThetaAboveNode0000303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHH thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00003030 e24KC2ThetaAboveLeaf00003031 e24KC2ThetaAboveNode00003032
+      e24KC2ThetaAboveNode00003033
+
+theorem e24KC2ThetaAboveNode0000312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHH thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00003120 e24KC2ThetaAboveLeaf00003121 e24KC2ThetaAboveNode00003122
+      e24KC2ThetaAboveNode00003123
+
+theorem e24KC2ThetaAboveNode0000313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHH thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00003130 e24KC2ThetaAboveLeaf00003131 e24KC2ThetaAboveNode00003132
+      e24KC2ThetaAboveNode00003133
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Reconstruction of the upper theta cover
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+open CertificateCells1a75b359b5
+
+theorem e24KC2ThetaAboveNode0000320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell0000)))
+    e24KC2ThetaAboveNode00003200 e24KC2ThetaAboveNode00003201 e24KC2ThetaAboveNode00003202
+      e24KC2ThetaAboveNode00003203
+
+theorem e24KC2ThetaAboveNode0000321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell0000)))
+    e24KC2ThetaAboveNode00003210 e24KC2ThetaAboveNode00003211 e24KC2ThetaAboveNode00003212
+      e24KC2ThetaAboveNode00003213
+
+theorem e24KC2ThetaAboveNode0000322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00003220 e24KC2ThetaAboveLeaf00003221 e24KC2ThetaAboveLeaf00003222
+      e24KC2ThetaAboveLeaf00003223
+
+theorem e24KC2ThetaAboveNode0000323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00003230 e24KC2ThetaAboveLeaf00003231 e24KC2ThetaAboveLeaf00003232
+      e24KC2ThetaAboveLeaf00003233
+
+theorem e24KC2ThetaAboveNode0000330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell0000)))
+    e24KC2ThetaAboveNode00003300 e24KC2ThetaAboveNode00003301 e24KC2ThetaAboveNode00003302
+      e24KC2ThetaAboveNode00003303
+
+theorem e24KC2ThetaAboveNode0000331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell0000)))
+    e24KC2ThetaAboveNode00003310 e24KC2ThetaAboveNode00003311 e24KC2ThetaAboveNode00003312
+      e24KC2ThetaAboveNode00003313
+
+theorem e24KC2ThetaAboveNode0000332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00003320 e24KC2ThetaAboveLeaf00003321 e24KC2ThetaAboveLeaf00003322
+      e24KC2ThetaAboveLeaf00003323
+
+theorem e24KC2ThetaAboveNode0000333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell0000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell0000)))
+    e24KC2ThetaAboveLeaf00003330 e24KC2ThetaAboveLeaf00003331 e24KC2ThetaAboveLeaf00003332
+      e24KC2ThetaAboveLeaf00003333
+
+theorem e24KC2ThetaAboveNode0001202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHL thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00012020 e24KC2ThetaAboveLeaf00012021 e24KC2ThetaAboveNode00012022
+      e24KC2ThetaAboveNode00012023
+
+theorem e24KC2ThetaAboveNode0001203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHL thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00012030 e24KC2ThetaAboveLeaf00012031 e24KC2ThetaAboveNode00012032
+      e24KC2ThetaAboveNode00012033
+
+theorem e24KC2ThetaAboveNode0001212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHL thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00012120 e24KC2ThetaAboveLeaf00012121 e24KC2ThetaAboveNode00012122
+      e24KC2ThetaAboveNode00012123
+
+theorem e24KC2ThetaAboveNode0001213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHL thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00012130 e24KC2ThetaAboveLeaf00012131 e24KC2ThetaAboveNode00012132
+      e24KC2ThetaAboveNode00012133
+
+theorem e24KC2ThetaAboveNode0001220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell0001)))
+    e24KC2ThetaAboveNode00012200 e24KC2ThetaAboveNode00012201 e24KC2ThetaAboveNode00012202
+      e24KC2ThetaAboveNode00012203
+
+theorem e24KC2ThetaAboveNode0001221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell0001)))
+    e24KC2ThetaAboveNode00012210 e24KC2ThetaAboveNode00012211 e24KC2ThetaAboveNode00012212
+      e24KC2ThetaAboveNode00012213
+
+theorem e24KC2ThetaAboveNode0001222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00012220 e24KC2ThetaAboveLeaf00012221 e24KC2ThetaAboveLeaf00012222
+      e24KC2ThetaAboveLeaf00012223
+
+theorem e24KC2ThetaAboveNode0001223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00012230 e24KC2ThetaAboveLeaf00012231 e24KC2ThetaAboveLeaf00012232
+      e24KC2ThetaAboveLeaf00012233
+
+theorem e24KC2ThetaAboveNode0001230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell0001)))
+    e24KC2ThetaAboveNode00012300 e24KC2ThetaAboveNode00012301 e24KC2ThetaAboveNode00012302
+      e24KC2ThetaAboveNode00012303
+
+theorem e24KC2ThetaAboveNode0001231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell0001)))
+    e24KC2ThetaAboveNode00012310 e24KC2ThetaAboveNode00012311 e24KC2ThetaAboveNode00012312
+      e24KC2ThetaAboveNode00012313
+
+theorem e24KC2ThetaAboveNode0001232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00012320 e24KC2ThetaAboveLeaf00012321 e24KC2ThetaAboveLeaf00012322
+      e24KC2ThetaAboveLeaf00012323
+
+theorem e24KC2ThetaAboveNode0001233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00012330 e24KC2ThetaAboveLeaf00012331 e24KC2ThetaAboveLeaf00012332
+      e24KC2ThetaAboveLeaf00012333
+
+theorem e24KC2ThetaAboveNode0001302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHH thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00013020 e24KC2ThetaAboveLeaf00013021 e24KC2ThetaAboveNode00013022
+      e24KC2ThetaAboveNode00013023
+
+theorem e24KC2ThetaAboveNode0001303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHH thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00013030 e24KC2ThetaAboveLeaf00013031 e24KC2ThetaAboveNode00013032
+      e24KC2ThetaAboveNode00013033
+
+theorem e24KC2ThetaAboveNode0001312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHH thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00013120 e24KC2ThetaAboveLeaf00013121 e24KC2ThetaAboveNode00013122
+      e24KC2ThetaAboveNode00013123
+
+theorem e24KC2ThetaAboveNode0001313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHH thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00013130 e24KC2ThetaAboveLeaf00013131 e24KC2ThetaAboveNode00013132
+      e24KC2ThetaAboveNode00013133
+
+theorem e24KC2ThetaAboveNode0001320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell0001)))
+    e24KC2ThetaAboveNode00013200 e24KC2ThetaAboveNode00013201 e24KC2ThetaAboveNode00013202
+      e24KC2ThetaAboveNode00013203
+
+theorem e24KC2ThetaAboveNode0001321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell0001)))
+    e24KC2ThetaAboveNode00013210 e24KC2ThetaAboveNode00013211 e24KC2ThetaAboveNode00013212
+      e24KC2ThetaAboveNode00013213
+
+theorem e24KC2ThetaAboveNode0001322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00013220 e24KC2ThetaAboveLeaf00013221 e24KC2ThetaAboveLeaf00013222
+      e24KC2ThetaAboveLeaf00013223
+
+theorem e24KC2ThetaAboveNode0001323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00013230 e24KC2ThetaAboveLeaf00013231 e24KC2ThetaAboveLeaf00013232
+      e24KC2ThetaAboveLeaf00013233
+
+theorem e24KC2ThetaAboveNode0001330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell0001)))
+    e24KC2ThetaAboveNode00013300 e24KC2ThetaAboveNode00013301 e24KC2ThetaAboveNode00013302
+      e24KC2ThetaAboveNode00013303
+
+theorem e24KC2ThetaAboveNode0001331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell0001)))
+    e24KC2ThetaAboveNode00013310 e24KC2ThetaAboveNode00013311 e24KC2ThetaAboveNode00013312
+      e24KC2ThetaAboveNode00013313
+
+theorem e24KC2ThetaAboveNode0001332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00013320 e24KC2ThetaAboveLeaf00013321 e24KC2ThetaAboveLeaf00013322
+      e24KC2ThetaAboveLeaf00013323
+
+theorem e24KC2ThetaAboveNode0001333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell0001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell0001)))
+    e24KC2ThetaAboveLeaf00013330 e24KC2ThetaAboveLeaf00013331 e24KC2ThetaAboveLeaf00013332
+      e24KC2ThetaAboveLeaf00013333
+
+theorem e24KC2ThetaAboveNode0010202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHL thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00102020 e24KC2ThetaAboveLeaf00102021 e24KC2ThetaAboveNode00102022
+      e24KC2ThetaAboveNode00102023
+
+theorem e24KC2ThetaAboveNode0010203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHL thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00102030 e24KC2ThetaAboveLeaf00102031 e24KC2ThetaAboveNode00102032
+      e24KC2ThetaAboveNode00102033
+
+theorem e24KC2ThetaAboveNode0010212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHL thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00102120 e24KC2ThetaAboveLeaf00102121 e24KC2ThetaAboveNode00102122
+      e24KC2ThetaAboveNode00102123
+
+theorem e24KC2ThetaAboveNode0010213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHL thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00102130 e24KC2ThetaAboveLeaf00102131 e24KC2ThetaAboveNode00102132
+      e24KC2ThetaAboveNode00102133
+
+theorem e24KC2ThetaAboveNode0010220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell0010)))
+    e24KC2ThetaAboveNode00102200 e24KC2ThetaAboveNode00102201 e24KC2ThetaAboveNode00102202
+      e24KC2ThetaAboveNode00102203
+
+theorem e24KC2ThetaAboveNode0010221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell0010)))
+    e24KC2ThetaAboveNode00102210 e24KC2ThetaAboveNode00102211 e24KC2ThetaAboveNode00102212
+      e24KC2ThetaAboveNode00102213
+
+theorem e24KC2ThetaAboveNode0010222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00102220 e24KC2ThetaAboveLeaf00102221 e24KC2ThetaAboveLeaf00102222
+      e24KC2ThetaAboveLeaf00102223
+
+theorem e24KC2ThetaAboveNode0010223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00102230 e24KC2ThetaAboveLeaf00102231 e24KC2ThetaAboveLeaf00102232
+      e24KC2ThetaAboveLeaf00102233
+
+theorem e24KC2ThetaAboveNode0010230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell0010)))
+    e24KC2ThetaAboveNode00102300 e24KC2ThetaAboveNode00102301 e24KC2ThetaAboveNode00102302
+      e24KC2ThetaAboveNode00102303
+
+theorem e24KC2ThetaAboveNode0010231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell0010)))
+    e24KC2ThetaAboveNode00102310 e24KC2ThetaAboveNode00102311 e24KC2ThetaAboveNode00102312
+      e24KC2ThetaAboveNode00102313
+
+theorem e24KC2ThetaAboveNode0010232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00102320 e24KC2ThetaAboveLeaf00102321 e24KC2ThetaAboveLeaf00102322
+      e24KC2ThetaAboveLeaf00102323
+
+theorem e24KC2ThetaAboveNode0010233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00102330 e24KC2ThetaAboveLeaf00102331 e24KC2ThetaAboveLeaf00102332
+      e24KC2ThetaAboveLeaf00102333
+
+theorem e24KC2ThetaAboveNode0010302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHH thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00103020 e24KC2ThetaAboveLeaf00103021 e24KC2ThetaAboveLeaf00103022
+      e24KC2ThetaAboveLeaf00103023
+
+theorem e24KC2ThetaAboveNode0010303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHH thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00103030 e24KC2ThetaAboveLeaf00103031 e24KC2ThetaAboveLeaf00103032
+      e24KC2ThetaAboveLeaf00103033
+
+theorem e24KC2ThetaAboveNode0010312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHH thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00103120 e24KC2ThetaAboveLeaf00103121 e24KC2ThetaAboveLeaf00103122
+      e24KC2ThetaAboveLeaf00103123
+
+theorem e24KC2ThetaAboveNode0010313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHH thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00103130 e24KC2ThetaAboveLeaf00103131 e24KC2ThetaAboveLeaf00103132
+      e24KC2ThetaAboveLeaf00103133
+
+theorem e24KC2ThetaAboveNode0010320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell0010)))
+    e24KC2ThetaAboveNode00103200 e24KC2ThetaAboveNode00103201 e24KC2ThetaAboveNode00103202
+      e24KC2ThetaAboveNode00103203
+
+theorem e24KC2ThetaAboveNode0010321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell0010)))
+    e24KC2ThetaAboveNode00103210 e24KC2ThetaAboveNode00103211 e24KC2ThetaAboveNode00103212
+      e24KC2ThetaAboveNode00103213
+
+theorem e24KC2ThetaAboveNode0010322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00103220 e24KC2ThetaAboveLeaf00103221 e24KC2ThetaAboveLeaf00103222
+      e24KC2ThetaAboveLeaf00103223
+
+theorem e24KC2ThetaAboveNode0010323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00103230 e24KC2ThetaAboveLeaf00103231 e24KC2ThetaAboveLeaf00103232
+      e24KC2ThetaAboveLeaf00103233
+
+theorem e24KC2ThetaAboveNode0010330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell0010)))
+    e24KC2ThetaAboveNode00103300 e24KC2ThetaAboveNode00103301 e24KC2ThetaAboveNode00103302
+      e24KC2ThetaAboveNode00103303
+
+theorem e24KC2ThetaAboveNode0010331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell0010)))
+    e24KC2ThetaAboveNode00103310 e24KC2ThetaAboveNode00103311 e24KC2ThetaAboveNode00103312
+      e24KC2ThetaAboveNode00103313
+
+theorem e24KC2ThetaAboveNode0010332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00103320 e24KC2ThetaAboveLeaf00103321 e24KC2ThetaAboveLeaf00103322
+      e24KC2ThetaAboveLeaf00103323
+
+theorem e24KC2ThetaAboveNode0010333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell0010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell0010)))
+    e24KC2ThetaAboveLeaf00103330 e24KC2ThetaAboveLeaf00103331 e24KC2ThetaAboveLeaf00103332
+      e24KC2ThetaAboveLeaf00103333
+
+theorem e24KC2ThetaAboveNode0011202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHL thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00112020 e24KC2ThetaAboveLeaf00112021 e24KC2ThetaAboveLeaf00112022
+      e24KC2ThetaAboveLeaf00112023
+
+theorem e24KC2ThetaAboveNode0011203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHL thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00112030 e24KC2ThetaAboveLeaf00112031 e24KC2ThetaAboveLeaf00112032
+      e24KC2ThetaAboveLeaf00112033
+
+theorem e24KC2ThetaAboveNode0011212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHL thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00112120 e24KC2ThetaAboveLeaf00112121 e24KC2ThetaAboveLeaf00112122
+      e24KC2ThetaAboveLeaf00112123
+
+theorem e24KC2ThetaAboveNode0011213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHL thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00112130 e24KC2ThetaAboveLeaf00112131 e24KC2ThetaAboveLeaf00112132
+      e24KC2ThetaAboveLeaf00112133
+
+theorem e24KC2ThetaAboveNode0011220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell0011)))
+    e24KC2ThetaAboveNode00112200 e24KC2ThetaAboveNode00112201 e24KC2ThetaAboveNode00112202
+      e24KC2ThetaAboveNode00112203
+
+theorem e24KC2ThetaAboveNode0011221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell0011)))
+    e24KC2ThetaAboveNode00112210 e24KC2ThetaAboveNode00112211 e24KC2ThetaAboveNode00112212
+      e24KC2ThetaAboveNode00112213
+
+theorem e24KC2ThetaAboveNode0011222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00112220 e24KC2ThetaAboveLeaf00112221 e24KC2ThetaAboveLeaf00112222
+      e24KC2ThetaAboveLeaf00112223
+
+theorem e24KC2ThetaAboveNode0011223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00112230 e24KC2ThetaAboveLeaf00112231 e24KC2ThetaAboveLeaf00112232
+      e24KC2ThetaAboveLeaf00112233
+
+theorem e24KC2ThetaAboveNode0011230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell0011)))
+    e24KC2ThetaAboveNode00112300 e24KC2ThetaAboveNode00112301 e24KC2ThetaAboveNode00112302
+      e24KC2ThetaAboveNode00112303
+
+theorem e24KC2ThetaAboveNode0011231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell0011)))
+    e24KC2ThetaAboveNode00112310 e24KC2ThetaAboveNode00112311 e24KC2ThetaAboveNode00112312
+      e24KC2ThetaAboveNode00112313
+
+theorem e24KC2ThetaAboveNode0011232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00112320 e24KC2ThetaAboveLeaf00112321 e24KC2ThetaAboveLeaf00112322
+      e24KC2ThetaAboveLeaf00112323
+
+theorem e24KC2ThetaAboveNode0011233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00112330 e24KC2ThetaAboveLeaf00112331 e24KC2ThetaAboveLeaf00112332
+      e24KC2ThetaAboveLeaf00112333
+
+theorem e24KC2ThetaAboveNode0011302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHH thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00113020 e24KC2ThetaAboveLeaf00113021 e24KC2ThetaAboveLeaf00113022
+      e24KC2ThetaAboveLeaf00113023
+
+theorem e24KC2ThetaAboveNode0011303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHH thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00113030 e24KC2ThetaAboveLeaf00113031 e24KC2ThetaAboveLeaf00113032
+      e24KC2ThetaAboveLeaf00113033
+
+theorem e24KC2ThetaAboveNode0011312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHH thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00113120 e24KC2ThetaAboveLeaf00113121 e24KC2ThetaAboveLeaf00113122
+      e24KC2ThetaAboveLeaf00113123
+
+theorem e24KC2ThetaAboveNode0011313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHH thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00113130 e24KC2ThetaAboveLeaf00113131 e24KC2ThetaAboveLeaf00113132
+      e24KC2ThetaAboveLeaf00113133
+
+theorem e24KC2ThetaAboveNode0011320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell0011)))
+    e24KC2ThetaAboveNode00113200 e24KC2ThetaAboveNode00113201 e24KC2ThetaAboveNode00113202
+      e24KC2ThetaAboveNode00113203
+
+theorem e24KC2ThetaAboveNode0011321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell0011)))
+    e24KC2ThetaAboveNode00113210 e24KC2ThetaAboveNode00113211 e24KC2ThetaAboveNode00113212
+      e24KC2ThetaAboveNode00113213
+
+theorem e24KC2ThetaAboveNode0011322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00113220 e24KC2ThetaAboveLeaf00113221 e24KC2ThetaAboveLeaf00113222
+      e24KC2ThetaAboveLeaf00113223
+
+theorem e24KC2ThetaAboveNode0011323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00113230 e24KC2ThetaAboveLeaf00113231 e24KC2ThetaAboveLeaf00113232
+      e24KC2ThetaAboveLeaf00113233
+
+theorem e24KC2ThetaAboveNode0011330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell0011)))
+    e24KC2ThetaAboveNode00113300 e24KC2ThetaAboveNode00113301 e24KC2ThetaAboveNode00113302
+      e24KC2ThetaAboveNode00113303
+
+theorem e24KC2ThetaAboveNode0011331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell0011)))
+    e24KC2ThetaAboveNode00113310 e24KC2ThetaAboveNode00113311 e24KC2ThetaAboveNode00113312
+      e24KC2ThetaAboveNode00113313
+
+theorem e24KC2ThetaAboveNode0011332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00113320 e24KC2ThetaAboveLeaf00113321 e24KC2ThetaAboveLeaf00113322
+      e24KC2ThetaAboveLeaf00113323
+
+theorem e24KC2ThetaAboveNode0011333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell0011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell0011)))
+    e24KC2ThetaAboveLeaf00113330 e24KC2ThetaAboveLeaf00113331 e24KC2ThetaAboveLeaf00113332
+      e24KC2ThetaAboveLeaf00113333
+
+theorem e24KC2ThetaAboveNode0100202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHL thetaAboveCell0100)))
+    e24KC2ThetaAboveLeaf01002020 e24KC2ThetaAboveLeaf01002021 e24KC2ThetaAboveLeaf01002022
+      e24KC2ThetaAboveLeaf01002023
+
+theorem e24KC2ThetaAboveNode0100203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHL thetaAboveCell0100)))
+    e24KC2ThetaAboveLeaf01002030 e24KC2ThetaAboveLeaf01002031 e24KC2ThetaAboveLeaf01002032
+      e24KC2ThetaAboveLeaf01002033
+
+theorem e24KC2ThetaAboveNode0100212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHL thetaAboveCell0100)))
+    e24KC2ThetaAboveLeaf01002120 e24KC2ThetaAboveLeaf01002121 e24KC2ThetaAboveLeaf01002122
+      e24KC2ThetaAboveLeaf01002123
+
+theorem e24KC2ThetaAboveNode0100213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHL thetaAboveCell0100)))
+    e24KC2ThetaAboveLeaf01002130 e24KC2ThetaAboveLeaf01002131 e24KC2ThetaAboveLeaf01002132
+      e24KC2ThetaAboveLeaf01002133
+
+theorem e24KC2ThetaAboveNode0100220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell0100)))
+    e24KC2ThetaAboveNode01002200 e24KC2ThetaAboveNode01002201 e24KC2ThetaAboveNode01002202
+      e24KC2ThetaAboveNode01002203
+
+theorem e24KC2ThetaAboveNode0100221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell0100)))
+    e24KC2ThetaAboveNode01002210 e24KC2ThetaAboveNode01002211 e24KC2ThetaAboveNode01002212
+      e24KC2ThetaAboveNode01002213
+
+theorem e24KC2ThetaAboveNode0100222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell0100)))
+    e24KC2ThetaAboveLeaf01002220 e24KC2ThetaAboveLeaf01002221 e24KC2ThetaAboveLeaf01002222
+      e24KC2ThetaAboveLeaf01002223
+
+theorem e24KC2ThetaAboveNode0100223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell0100)))
+    e24KC2ThetaAboveLeaf01002230 e24KC2ThetaAboveLeaf01002231 e24KC2ThetaAboveLeaf01002232
+      e24KC2ThetaAboveLeaf01002233
+
+theorem e24KC2ThetaAboveNode0100230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell0100)))
+    e24KC2ThetaAboveNode01002300 e24KC2ThetaAboveNode01002301 e24KC2ThetaAboveNode01002302
+      e24KC2ThetaAboveNode01002303
+
+theorem e24KC2ThetaAboveNode0100231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell0100)))
+    e24KC2ThetaAboveNode01002310 e24KC2ThetaAboveNode01002311 e24KC2ThetaAboveNode01002312
+      e24KC2ThetaAboveNode01002313
+
+theorem e24KC2ThetaAboveNode0100232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell0100)))
+    e24KC2ThetaAboveLeaf01002320 e24KC2ThetaAboveLeaf01002321 e24KC2ThetaAboveLeaf01002322
+      e24KC2ThetaAboveLeaf01002323
+
+theorem e24KC2ThetaAboveNode0100233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell0100)))
+    e24KC2ThetaAboveLeaf01002330 e24KC2ThetaAboveLeaf01002331 e24KC2ThetaAboveLeaf01002332
+      e24KC2ThetaAboveLeaf01002333
+
+theorem e24KC2ThetaAboveNode0100302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHH thetaAboveCell0100)))
+    e24KC2ThetaAboveLeaf01003020 e24KC2ThetaAboveLeaf01003021 e24KC2ThetaAboveLeaf01003022
+      e24KC2ThetaAboveLeaf01003023
+
+theorem e24KC2ThetaAboveNode0100303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHH thetaAboveCell0100)))
+    e24KC2ThetaAboveLeaf01003030 e24KC2ThetaAboveLeaf01003031 e24KC2ThetaAboveLeaf01003032
+      e24KC2ThetaAboveLeaf01003033
+
+theorem e24KC2ThetaAboveNode0100312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHH thetaAboveCell0100)))
+    e24KC2ThetaAboveLeaf01003120 e24KC2ThetaAboveLeaf01003121 e24KC2ThetaAboveLeaf01003122
+      e24KC2ThetaAboveLeaf01003123
+
+theorem e24KC2ThetaAboveNode0100313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHH thetaAboveCell0100)))
+    e24KC2ThetaAboveLeaf01003130 e24KC2ThetaAboveLeaf01003131 e24KC2ThetaAboveLeaf01003132
+      e24KC2ThetaAboveLeaf01003133
+
+theorem e24KC2ThetaAboveNode0100320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell0100)))
+    e24KC2ThetaAboveNode01003200 e24KC2ThetaAboveNode01003201 e24KC2ThetaAboveNode01003202
+      e24KC2ThetaAboveNode01003203
+
+theorem e24KC2ThetaAboveNode0100321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell0100)))
+    e24KC2ThetaAboveNode01003210 e24KC2ThetaAboveNode01003211 e24KC2ThetaAboveNode01003212
+      e24KC2ThetaAboveNode01003213
+
+theorem e24KC2ThetaAboveNode0100322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell0100)))
+    e24KC2ThetaAboveLeaf01003220 e24KC2ThetaAboveLeaf01003221 e24KC2ThetaAboveLeaf01003222
+      e24KC2ThetaAboveLeaf01003223
+
+theorem e24KC2ThetaAboveNode0100323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell0100)))
+    e24KC2ThetaAboveLeaf01003230 e24KC2ThetaAboveLeaf01003231 e24KC2ThetaAboveLeaf01003232
+      e24KC2ThetaAboveLeaf01003233
+
+theorem e24KC2ThetaAboveNode0100330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell0100)))
+    e24KC2ThetaAboveNode01003300 e24KC2ThetaAboveNode01003301 e24KC2ThetaAboveNode01003302
+      e24KC2ThetaAboveNode01003303
+
+theorem e24KC2ThetaAboveNode0100331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell0100)))
+    e24KC2ThetaAboveNode01003310 e24KC2ThetaAboveNode01003311 e24KC2ThetaAboveNode01003312
+      e24KC2ThetaAboveNode01003313
+
+theorem e24KC2ThetaAboveNode0100332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell0100)))
+    e24KC2ThetaAboveLeaf01003320 e24KC2ThetaAboveLeaf01003321 e24KC2ThetaAboveLeaf01003322
+      e24KC2ThetaAboveLeaf01003323
+
+theorem e24KC2ThetaAboveNode0100333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell0100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell0100)))
+    e24KC2ThetaAboveNode01003330 e24KC2ThetaAboveNode01003331 e24KC2ThetaAboveLeaf01003332
+      e24KC2ThetaAboveLeaf01003333
+
+theorem e24KC2ThetaAboveNode0101202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHL thetaAboveCell0101)))
+    e24KC2ThetaAboveLeaf01012020 e24KC2ThetaAboveLeaf01012021 e24KC2ThetaAboveLeaf01012022
+      e24KC2ThetaAboveLeaf01012023
+
+theorem e24KC2ThetaAboveNode0101203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHL thetaAboveCell0101)))
+    e24KC2ThetaAboveLeaf01012030 e24KC2ThetaAboveLeaf01012031 e24KC2ThetaAboveLeaf01012032
+      e24KC2ThetaAboveLeaf01012033
+
+theorem e24KC2ThetaAboveNode0101212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHL thetaAboveCell0101)))
+    e24KC2ThetaAboveLeaf01012120 e24KC2ThetaAboveLeaf01012121 e24KC2ThetaAboveLeaf01012122
+      e24KC2ThetaAboveLeaf01012123
+
+theorem e24KC2ThetaAboveNode0101213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHL thetaAboveCell0101)))
+    e24KC2ThetaAboveLeaf01012130 e24KC2ThetaAboveLeaf01012131 e24KC2ThetaAboveLeaf01012132
+      e24KC2ThetaAboveLeaf01012133
+
+theorem e24KC2ThetaAboveNode0101220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01012200 e24KC2ThetaAboveNode01012201 e24KC2ThetaAboveNode01012202
+      e24KC2ThetaAboveNode01012203
+
+theorem e24KC2ThetaAboveNode0101221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01012210 e24KC2ThetaAboveNode01012211 e24KC2ThetaAboveNode01012212
+      e24KC2ThetaAboveNode01012213
+
+theorem e24KC2ThetaAboveNode0101222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01012220 e24KC2ThetaAboveNode01012221 e24KC2ThetaAboveLeaf01012222
+      e24KC2ThetaAboveLeaf01012223
+
+theorem e24KC2ThetaAboveNode0101223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01012230 e24KC2ThetaAboveNode01012231 e24KC2ThetaAboveLeaf01012232
+      e24KC2ThetaAboveLeaf01012233
+
+theorem e24KC2ThetaAboveNode0101230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01012300 e24KC2ThetaAboveNode01012301 e24KC2ThetaAboveNode01012302
+      e24KC2ThetaAboveNode01012303
+
+theorem e24KC2ThetaAboveNode0101231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01012310 e24KC2ThetaAboveNode01012311 e24KC2ThetaAboveNode01012312
+      e24KC2ThetaAboveNode01012313
+
+theorem e24KC2ThetaAboveNode0101232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01012320 e24KC2ThetaAboveNode01012321 e24KC2ThetaAboveLeaf01012322
+      e24KC2ThetaAboveLeaf01012323
+
+theorem e24KC2ThetaAboveNode0101233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01012330 e24KC2ThetaAboveNode01012331 e24KC2ThetaAboveLeaf01012332
+      e24KC2ThetaAboveLeaf01012333
+
+theorem e24KC2ThetaAboveNode0101302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHH thetaAboveCell0101)))
+    e24KC2ThetaAboveLeaf01013020 e24KC2ThetaAboveLeaf01013021 e24KC2ThetaAboveLeaf01013022
+      e24KC2ThetaAboveLeaf01013023
+
+theorem e24KC2ThetaAboveNode0101303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHH thetaAboveCell0101)))
+    e24KC2ThetaAboveLeaf01013030 e24KC2ThetaAboveLeaf01013031 e24KC2ThetaAboveLeaf01013032
+      e24KC2ThetaAboveLeaf01013033
+
+theorem e24KC2ThetaAboveNode0101312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHH thetaAboveCell0101)))
+    e24KC2ThetaAboveLeaf01013120 e24KC2ThetaAboveLeaf01013121 e24KC2ThetaAboveLeaf01013122
+      e24KC2ThetaAboveLeaf01013123
+
+theorem e24KC2ThetaAboveNode0101313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHH thetaAboveCell0101)))
+    e24KC2ThetaAboveLeaf01013130 e24KC2ThetaAboveLeaf01013131 e24KC2ThetaAboveLeaf01013132
+      e24KC2ThetaAboveLeaf01013133
+
+theorem e24KC2ThetaAboveNode0101320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01013200 e24KC2ThetaAboveNode01013201 e24KC2ThetaAboveNode01013202
+      e24KC2ThetaAboveNode01013203
+
+theorem e24KC2ThetaAboveNode0101321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01013210 e24KC2ThetaAboveNode01013211 e24KC2ThetaAboveNode01013212
+      e24KC2ThetaAboveNode01013213
+
+theorem e24KC2ThetaAboveNode0101322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01013220 e24KC2ThetaAboveNode01013221 e24KC2ThetaAboveLeaf01013222
+      e24KC2ThetaAboveLeaf01013223
+
+theorem e24KC2ThetaAboveNode0101323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01013230 e24KC2ThetaAboveNode01013231 e24KC2ThetaAboveLeaf01013232
+      e24KC2ThetaAboveLeaf01013233
+
+theorem e24KC2ThetaAboveNode0101330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01013300 e24KC2ThetaAboveNode01013301 e24KC2ThetaAboveNode01013302
+      e24KC2ThetaAboveNode01013303
+
+theorem e24KC2ThetaAboveNode0101331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01013310 e24KC2ThetaAboveNode01013311 e24KC2ThetaAboveNode01013312
+      e24KC2ThetaAboveNode01013313
+
+theorem e24KC2ThetaAboveNode0101332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01013320 e24KC2ThetaAboveNode01013321 e24KC2ThetaAboveLeaf01013322
+      e24KC2ThetaAboveLeaf01013323
+
+theorem e24KC2ThetaAboveNode0101333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell0101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell0101)))
+    e24KC2ThetaAboveNode01013330 e24KC2ThetaAboveNode01013331 e24KC2ThetaAboveLeaf01013332
+      e24KC2ThetaAboveLeaf01013333
+
+theorem e24KC2ThetaAboveNode0110202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHL thetaAboveCell0110)))
+    e24KC2ThetaAboveLeaf01102020 e24KC2ThetaAboveLeaf01102021 e24KC2ThetaAboveLeaf01102022
+      e24KC2ThetaAboveLeaf01102023
+
+theorem e24KC2ThetaAboveNode0110203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHL thetaAboveCell0110)))
+    e24KC2ThetaAboveLeaf01102030 e24KC2ThetaAboveLeaf01102031 e24KC2ThetaAboveLeaf01102032
+      e24KC2ThetaAboveLeaf01102033
+
+theorem e24KC2ThetaAboveNode0110212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHL thetaAboveCell0110)))
+    e24KC2ThetaAboveLeaf01102120 e24KC2ThetaAboveLeaf01102121 e24KC2ThetaAboveLeaf01102122
+      e24KC2ThetaAboveLeaf01102123
+
+theorem e24KC2ThetaAboveNode0110213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHL thetaAboveCell0110)))
+    e24KC2ThetaAboveLeaf01102130 e24KC2ThetaAboveLeaf01102131 e24KC2ThetaAboveLeaf01102132
+      e24KC2ThetaAboveLeaf01102133
+
+theorem e24KC2ThetaAboveNode0110220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01102200 e24KC2ThetaAboveNode01102201 e24KC2ThetaAboveNode01102202
+      e24KC2ThetaAboveNode01102203
+
+theorem e24KC2ThetaAboveNode0110221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01102210 e24KC2ThetaAboveNode01102211 e24KC2ThetaAboveNode01102212
+      e24KC2ThetaAboveNode01102213
+
+theorem e24KC2ThetaAboveNode0110222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01102220 e24KC2ThetaAboveNode01102221 e24KC2ThetaAboveLeaf01102222
+      e24KC2ThetaAboveLeaf01102223
+
+theorem e24KC2ThetaAboveNode0110223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01102230 e24KC2ThetaAboveNode01102231 e24KC2ThetaAboveLeaf01102232
+      e24KC2ThetaAboveLeaf01102233
+
+theorem e24KC2ThetaAboveNode0110230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01102300 e24KC2ThetaAboveNode01102301 e24KC2ThetaAboveNode01102302
+      e24KC2ThetaAboveNode01102303
+
+theorem e24KC2ThetaAboveNode0110231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01102310 e24KC2ThetaAboveNode01102311 e24KC2ThetaAboveNode01102312
+      e24KC2ThetaAboveNode01102313
+
+theorem e24KC2ThetaAboveNode0110232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01102320 e24KC2ThetaAboveNode01102321 e24KC2ThetaAboveLeaf01102322
+      e24KC2ThetaAboveLeaf01102323
+
+theorem e24KC2ThetaAboveNode0110233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01102330 e24KC2ThetaAboveNode01102331 e24KC2ThetaAboveLeaf01102332
+      e24KC2ThetaAboveLeaf01102333
+
+theorem e24KC2ThetaAboveNode0110302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHH thetaAboveCell0110)))
+    e24KC2ThetaAboveLeaf01103020 e24KC2ThetaAboveLeaf01103021 e24KC2ThetaAboveLeaf01103022
+      e24KC2ThetaAboveLeaf01103023
+
+theorem e24KC2ThetaAboveNode0110303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHH thetaAboveCell0110)))
+    e24KC2ThetaAboveLeaf01103030 e24KC2ThetaAboveLeaf01103031 e24KC2ThetaAboveLeaf01103032
+      e24KC2ThetaAboveLeaf01103033
+
+theorem e24KC2ThetaAboveNode0110312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHH thetaAboveCell0110)))
+    e24KC2ThetaAboveLeaf01103120 e24KC2ThetaAboveLeaf01103121 e24KC2ThetaAboveLeaf01103122
+      e24KC2ThetaAboveLeaf01103123
+
+theorem e24KC2ThetaAboveNode0110313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHH thetaAboveCell0110)))
+    e24KC2ThetaAboveLeaf01103130 e24KC2ThetaAboveLeaf01103131 e24KC2ThetaAboveLeaf01103132
+      e24KC2ThetaAboveLeaf01103133
+
+theorem e24KC2ThetaAboveNode0110320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01103200 e24KC2ThetaAboveNode01103201 e24KC2ThetaAboveNode01103202
+      e24KC2ThetaAboveNode01103203
+
+theorem e24KC2ThetaAboveNode0110321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01103210 e24KC2ThetaAboveNode01103211 e24KC2ThetaAboveNode01103212
+      e24KC2ThetaAboveNode01103213
+
+theorem e24KC2ThetaAboveNode0110322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01103220 e24KC2ThetaAboveNode01103221 e24KC2ThetaAboveLeaf01103222
+      e24KC2ThetaAboveLeaf01103223
+
+theorem e24KC2ThetaAboveNode0110323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01103230 e24KC2ThetaAboveNode01103231 e24KC2ThetaAboveLeaf01103232
+      e24KC2ThetaAboveLeaf01103233
+
+theorem e24KC2ThetaAboveNode0110330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01103300 e24KC2ThetaAboveNode01103301 e24KC2ThetaAboveNode01103302
+      e24KC2ThetaAboveNode01103303
+
+theorem e24KC2ThetaAboveNode0110331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01103310 e24KC2ThetaAboveNode01103311 e24KC2ThetaAboveNode01103312
+      e24KC2ThetaAboveNode01103313
+
+theorem e24KC2ThetaAboveNode0110332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01103320 e24KC2ThetaAboveNode01103321 e24KC2ThetaAboveLeaf01103322
+      e24KC2ThetaAboveLeaf01103323
+
+theorem e24KC2ThetaAboveNode0110333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell0110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell0110)))
+    e24KC2ThetaAboveNode01103330 e24KC2ThetaAboveNode01103331 e24KC2ThetaAboveLeaf01103332
+      e24KC2ThetaAboveLeaf01103333
+
+theorem e24KC2ThetaAboveNode0111202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHL thetaAboveCell0111)))
+    e24KC2ThetaAboveLeaf01112020 e24KC2ThetaAboveLeaf01112021 e24KC2ThetaAboveLeaf01112022
+      e24KC2ThetaAboveLeaf01112023
+
+theorem e24KC2ThetaAboveNode0111203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHL thetaAboveCell0111)))
+    e24KC2ThetaAboveLeaf01112030 e24KC2ThetaAboveLeaf01112031 e24KC2ThetaAboveLeaf01112032
+      e24KC2ThetaAboveLeaf01112033
+
+theorem e24KC2ThetaAboveNode0111212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHL thetaAboveCell0111)))
+    e24KC2ThetaAboveLeaf01112120 e24KC2ThetaAboveLeaf01112121 e24KC2ThetaAboveLeaf01112122
+      e24KC2ThetaAboveLeaf01112123
+
+theorem e24KC2ThetaAboveNode0111213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHL thetaAboveCell0111)))
+    e24KC2ThetaAboveLeaf01112130 e24KC2ThetaAboveLeaf01112131 e24KC2ThetaAboveLeaf01112132
+      e24KC2ThetaAboveLeaf01112133
+
+theorem e24KC2ThetaAboveNode0111220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01112200 e24KC2ThetaAboveNode01112201 e24KC2ThetaAboveNode01112202
+      e24KC2ThetaAboveNode01112203
+
+theorem e24KC2ThetaAboveNode0111221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01112210 e24KC2ThetaAboveNode01112211 e24KC2ThetaAboveNode01112212
+      e24KC2ThetaAboveNode01112213
+
+theorem e24KC2ThetaAboveNode0111222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01112220 e24KC2ThetaAboveNode01112221 e24KC2ThetaAboveLeaf01112222
+      e24KC2ThetaAboveLeaf01112223
+
+theorem e24KC2ThetaAboveNode0111223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01112230 e24KC2ThetaAboveNode01112231 e24KC2ThetaAboveLeaf01112232
+      e24KC2ThetaAboveLeaf01112233
+
+theorem e24KC2ThetaAboveNode0111230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01112300 e24KC2ThetaAboveNode01112301 e24KC2ThetaAboveNode01112302
+      e24KC2ThetaAboveNode01112303
+
+theorem e24KC2ThetaAboveNode0111231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01112310 e24KC2ThetaAboveNode01112311 e24KC2ThetaAboveNode01112312
+      e24KC2ThetaAboveNode01112313
+
+theorem e24KC2ThetaAboveNode0111232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01112320 e24KC2ThetaAboveNode01112321 e24KC2ThetaAboveLeaf01112322
+      e24KC2ThetaAboveLeaf01112323
+
+theorem e24KC2ThetaAboveNode0111233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01112330 e24KC2ThetaAboveNode01112331 e24KC2ThetaAboveLeaf01112332
+      e24KC2ThetaAboveLeaf01112333
+
+theorem e24KC2ThetaAboveNode0111302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHH thetaAboveCell0111)))
+    e24KC2ThetaAboveLeaf01113020 e24KC2ThetaAboveLeaf01113021 e24KC2ThetaAboveLeaf01113022
+      e24KC2ThetaAboveLeaf01113023
+
+theorem e24KC2ThetaAboveNode0111303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHH thetaAboveCell0111)))
+    e24KC2ThetaAboveLeaf01113030 e24KC2ThetaAboveLeaf01113031 e24KC2ThetaAboveLeaf01113032
+      e24KC2ThetaAboveLeaf01113033
+
+theorem e24KC2ThetaAboveNode0111312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHH thetaAboveCell0111)))
+    e24KC2ThetaAboveLeaf01113120 e24KC2ThetaAboveLeaf01113121 e24KC2ThetaAboveLeaf01113122
+      e24KC2ThetaAboveLeaf01113123
+
+theorem e24KC2ThetaAboveNode0111313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHH thetaAboveCell0111)))
+    e24KC2ThetaAboveLeaf01113130 e24KC2ThetaAboveLeaf01113131 e24KC2ThetaAboveLeaf01113132
+      e24KC2ThetaAboveLeaf01113133
+
+theorem e24KC2ThetaAboveNode0111320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01113200 e24KC2ThetaAboveNode01113201 e24KC2ThetaAboveNode01113202
+      e24KC2ThetaAboveNode01113203
+
+theorem e24KC2ThetaAboveNode0111321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01113210 e24KC2ThetaAboveNode01113211 e24KC2ThetaAboveNode01113212
+      e24KC2ThetaAboveNode01113213
+
+theorem e24KC2ThetaAboveNode0111322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01113220 e24KC2ThetaAboveNode01113221 e24KC2ThetaAboveLeaf01113222
+      e24KC2ThetaAboveLeaf01113223
+
+theorem e24KC2ThetaAboveNode0111323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01113230 e24KC2ThetaAboveNode01113231 e24KC2ThetaAboveLeaf01113232
+      e24KC2ThetaAboveLeaf01113233
+
+theorem e24KC2ThetaAboveNode0111330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01113300 e24KC2ThetaAboveNode01113301 e24KC2ThetaAboveNode01113302
+      e24KC2ThetaAboveNode01113303
+
+theorem e24KC2ThetaAboveNode0111331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01113310 e24KC2ThetaAboveNode01113311 e24KC2ThetaAboveNode01113312
+      e24KC2ThetaAboveNode01113313
+
+theorem e24KC2ThetaAboveNode0111332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01113320 e24KC2ThetaAboveNode01113321 e24KC2ThetaAboveLeaf01113322
+      e24KC2ThetaAboveLeaf01113323
+
+theorem e24KC2ThetaAboveNode0111333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell0111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell0111)))
+    e24KC2ThetaAboveNode01113330 e24KC2ThetaAboveNode01113331 e24KC2ThetaAboveLeaf01113332
+      e24KC2ThetaAboveLeaf01113333
+
+theorem e24KC2ThetaAboveNode1000202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHL thetaAboveCell1000)))
+    e24KC2ThetaAboveLeaf10002020 e24KC2ThetaAboveLeaf10002021 e24KC2ThetaAboveLeaf10002022
+      e24KC2ThetaAboveLeaf10002023
+
+theorem e24KC2ThetaAboveNode1000203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHL thetaAboveCell1000)))
+    e24KC2ThetaAboveLeaf10002030 e24KC2ThetaAboveLeaf10002031 e24KC2ThetaAboveLeaf10002032
+      e24KC2ThetaAboveLeaf10002033
+
+theorem e24KC2ThetaAboveNode1000212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHL thetaAboveCell1000)))
+    e24KC2ThetaAboveLeaf10002120 e24KC2ThetaAboveLeaf10002121 e24KC2ThetaAboveLeaf10002122
+      e24KC2ThetaAboveLeaf10002123
+
+theorem e24KC2ThetaAboveNode1000213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHL thetaAboveCell1000)))
+    e24KC2ThetaAboveLeaf10002130 e24KC2ThetaAboveLeaf10002131 e24KC2ThetaAboveLeaf10002132
+      e24KC2ThetaAboveLeaf10002133
+
+theorem e24KC2ThetaAboveNode1000220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10002200 e24KC2ThetaAboveNode10002201 e24KC2ThetaAboveNode10002202
+      e24KC2ThetaAboveNode10002203
+
+theorem e24KC2ThetaAboveNode1000221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10002210 e24KC2ThetaAboveNode10002211 e24KC2ThetaAboveNode10002212
+      e24KC2ThetaAboveNode10002213
+
+theorem e24KC2ThetaAboveNode1000222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10002220 e24KC2ThetaAboveNode10002221 e24KC2ThetaAboveLeaf10002222
+      e24KC2ThetaAboveLeaf10002223
+
+theorem e24KC2ThetaAboveNode1000223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10002230 e24KC2ThetaAboveNode10002231 e24KC2ThetaAboveLeaf10002232
+      e24KC2ThetaAboveLeaf10002233
+
+theorem e24KC2ThetaAboveNode1000230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10002300 e24KC2ThetaAboveNode10002301 e24KC2ThetaAboveNode10002302
+      e24KC2ThetaAboveNode10002303
+
+theorem e24KC2ThetaAboveNode1000231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10002310 e24KC2ThetaAboveNode10002311 e24KC2ThetaAboveNode10002312
+      e24KC2ThetaAboveNode10002313
+
+theorem e24KC2ThetaAboveNode1000232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10002320 e24KC2ThetaAboveNode10002321 e24KC2ThetaAboveLeaf10002322
+      e24KC2ThetaAboveLeaf10002323
+
+theorem e24KC2ThetaAboveNode1000233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10002330 e24KC2ThetaAboveNode10002331 e24KC2ThetaAboveLeaf10002332
+      e24KC2ThetaAboveLeaf10002333
+
+theorem e24KC2ThetaAboveNode1000302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHH thetaAboveCell1000)))
+    e24KC2ThetaAboveLeaf10003020 e24KC2ThetaAboveLeaf10003021 e24KC2ThetaAboveLeaf10003022
+      e24KC2ThetaAboveLeaf10003023
+
+theorem e24KC2ThetaAboveNode1000303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHH thetaAboveCell1000)))
+    e24KC2ThetaAboveLeaf10003030 e24KC2ThetaAboveLeaf10003031 e24KC2ThetaAboveLeaf10003032
+      e24KC2ThetaAboveLeaf10003033
+
+theorem e24KC2ThetaAboveNode1000312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHH thetaAboveCell1000)))
+    e24KC2ThetaAboveLeaf10003120 e24KC2ThetaAboveLeaf10003121 e24KC2ThetaAboveLeaf10003122
+      e24KC2ThetaAboveLeaf10003123
+
+theorem e24KC2ThetaAboveNode1000313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHH thetaAboveCell1000)))
+    e24KC2ThetaAboveLeaf10003130 e24KC2ThetaAboveLeaf10003131 e24KC2ThetaAboveLeaf10003132
+      e24KC2ThetaAboveLeaf10003133
+
+theorem e24KC2ThetaAboveNode1000320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10003200 e24KC2ThetaAboveNode10003201 e24KC2ThetaAboveNode10003202
+      e24KC2ThetaAboveNode10003203
+
+theorem e24KC2ThetaAboveNode1000321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10003210 e24KC2ThetaAboveNode10003211 e24KC2ThetaAboveNode10003212
+      e24KC2ThetaAboveNode10003213
+
+theorem e24KC2ThetaAboveNode1000322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10003220 e24KC2ThetaAboveNode10003221 e24KC2ThetaAboveLeaf10003222
+      e24KC2ThetaAboveLeaf10003223
+
+theorem e24KC2ThetaAboveNode1000323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10003230 e24KC2ThetaAboveNode10003231 e24KC2ThetaAboveLeaf10003232
+      e24KC2ThetaAboveLeaf10003233
+
+theorem e24KC2ThetaAboveNode1000330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10003300 e24KC2ThetaAboveNode10003301 e24KC2ThetaAboveNode10003302
+      e24KC2ThetaAboveNode10003303
+
+theorem e24KC2ThetaAboveNode1000331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10003310 e24KC2ThetaAboveNode10003311 e24KC2ThetaAboveNode10003312
+      e24KC2ThetaAboveNode10003313
+
+theorem e24KC2ThetaAboveNode1000332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10003320 e24KC2ThetaAboveNode10003321 e24KC2ThetaAboveLeaf10003322
+      e24KC2ThetaAboveLeaf10003323
+
+theorem e24KC2ThetaAboveNode1000333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell1000))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell1000)))
+    e24KC2ThetaAboveNode10003330 e24KC2ThetaAboveNode10003331 e24KC2ThetaAboveLeaf10003332
+      e24KC2ThetaAboveLeaf10003333
+
+theorem e24KC2ThetaAboveNode1001202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHL thetaAboveCell1001)))
+    e24KC2ThetaAboveLeaf10012020 e24KC2ThetaAboveLeaf10012021 e24KC2ThetaAboveLeaf10012022
+      e24KC2ThetaAboveLeaf10012023
+
+theorem e24KC2ThetaAboveNode1001203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHL thetaAboveCell1001)))
+    e24KC2ThetaAboveLeaf10012030 e24KC2ThetaAboveLeaf10012031 e24KC2ThetaAboveLeaf10012032
+      e24KC2ThetaAboveLeaf10012033
+
+theorem e24KC2ThetaAboveNode1001212 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHL thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHL thetaAboveCell1001)))
+    e24KC2ThetaAboveLeaf10012120 e24KC2ThetaAboveLeaf10012121 e24KC2ThetaAboveLeaf10012122
+      e24KC2ThetaAboveLeaf10012123
+
+theorem e24KC2ThetaAboveNode1001213 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHL thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHL thetaAboveCell1001)))
+    e24KC2ThetaAboveLeaf10012130 e24KC2ThetaAboveLeaf10012131 e24KC2ThetaAboveLeaf10012132
+      e24KC2ThetaAboveLeaf10012133
+
+theorem e24KC2ThetaAboveNode1001220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10012200 e24KC2ThetaAboveNode10012201 e24KC2ThetaAboveNode10012202
+      e24KC2ThetaAboveNode10012203
+
+theorem e24KC2ThetaAboveNode1001221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10012210 e24KC2ThetaAboveNode10012211 e24KC2ThetaAboveNode10012212
+      e24KC2ThetaAboveNode10012213
+
+theorem e24KC2ThetaAboveNode1001222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10012220 e24KC2ThetaAboveNode10012221 e24KC2ThetaAboveLeaf10012222
+      e24KC2ThetaAboveLeaf10012223
+
+theorem e24KC2ThetaAboveNode1001223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10012230 e24KC2ThetaAboveNode10012231 e24KC2ThetaAboveLeaf10012232
+      e24KC2ThetaAboveLeaf10012233
+
+theorem e24KC2ThetaAboveNode1001230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10012300 e24KC2ThetaAboveNode10012301 e24KC2ThetaAboveNode10012302
+      e24KC2ThetaAboveNode10012303
+
+theorem e24KC2ThetaAboveNode1001231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10012310 e24KC2ThetaAboveNode10012311 e24KC2ThetaAboveNode10012312
+      e24KC2ThetaAboveNode10012313
+
+theorem e24KC2ThetaAboveNode1001232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10012320 e24KC2ThetaAboveNode10012321 e24KC2ThetaAboveLeaf10012322
+      e24KC2ThetaAboveLeaf10012323
+
+theorem e24KC2ThetaAboveNode1001233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10012330 e24KC2ThetaAboveNode10012331 e24KC2ThetaAboveLeaf10012332
+      e24KC2ThetaAboveLeaf10012333
+
+theorem e24KC2ThetaAboveNode1001302 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHH thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHH thetaAboveCell1001)))
+    e24KC2ThetaAboveLeaf10013020 e24KC2ThetaAboveLeaf10013021 e24KC2ThetaAboveLeaf10013022
+      e24KC2ThetaAboveLeaf10013023
+
+theorem e24KC2ThetaAboveNode1001303 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHH thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHH thetaAboveCell1001)))
+    e24KC2ThetaAboveLeaf10013030 e24KC2ThetaAboveLeaf10013031 e24KC2ThetaAboveLeaf10013032
+      e24KC2ThetaAboveLeaf10013033
+
+theorem e24KC2ThetaAboveNode1001312 :
+    adaptiveCoverCheck 12 (childHL (childLH (childHH thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLH (childHH thetaAboveCell1001)))
+    e24KC2ThetaAboveLeaf10013120 e24KC2ThetaAboveLeaf10013121 e24KC2ThetaAboveLeaf10013122
+      e24KC2ThetaAboveLeaf10013123
+
+theorem e24KC2ThetaAboveNode1001313 :
+    adaptiveCoverCheck 12 (childHH (childLH (childHH thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLH (childHH thetaAboveCell1001)))
+    e24KC2ThetaAboveLeaf10013130 e24KC2ThetaAboveLeaf10013131 e24KC2ThetaAboveLeaf10013132
+      e24KC2ThetaAboveLeaf10013133
+
+theorem e24KC2ThetaAboveNode1001320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10013200 e24KC2ThetaAboveNode10013201 e24KC2ThetaAboveNode10013202
+      e24KC2ThetaAboveNode10013203
+
+theorem e24KC2ThetaAboveNode1001321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10013210 e24KC2ThetaAboveNode10013211 e24KC2ThetaAboveNode10013212
+      e24KC2ThetaAboveNode10013213
+
+theorem e24KC2ThetaAboveNode1001322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10013220 e24KC2ThetaAboveNode10013221 e24KC2ThetaAboveLeaf10013222
+      e24KC2ThetaAboveLeaf10013223
+
+theorem e24KC2ThetaAboveNode1001323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10013230 e24KC2ThetaAboveNode10013231 e24KC2ThetaAboveLeaf10013232
+      e24KC2ThetaAboveLeaf10013233
+
+theorem e24KC2ThetaAboveNode1001330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10013300 e24KC2ThetaAboveNode10013301 e24KC2ThetaAboveNode10013302
+      e24KC2ThetaAboveNode10013303
+
+theorem e24KC2ThetaAboveNode1001331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10013310 e24KC2ThetaAboveNode10013311 e24KC2ThetaAboveNode10013312
+      e24KC2ThetaAboveNode10013313
+
+theorem e24KC2ThetaAboveNode1001332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10013320 e24KC2ThetaAboveNode10013321 e24KC2ThetaAboveLeaf10013322
+      e24KC2ThetaAboveLeaf10013323
+
+theorem e24KC2ThetaAboveNode1001333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell1001))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell1001)))
+    e24KC2ThetaAboveNode10013330 e24KC2ThetaAboveNode10013331 e24KC2ThetaAboveLeaf10013332
+      e24KC2ThetaAboveLeaf10013333
+
+theorem e24KC2ThetaAboveNode1010202 :
+    adaptiveCoverCheck 12 (childHL (childLL (childHL thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childLL (childHL thetaAboveCell1010)))
+    e24KC2ThetaAboveLeaf10102020 e24KC2ThetaAboveLeaf10102021 e24KC2ThetaAboveLeaf10102022
+      e24KC2ThetaAboveLeaf10102023
+
+theorem e24KC2ThetaAboveNode1010203 :
+    adaptiveCoverCheck 12 (childHH (childLL (childHL thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childLL (childHL thetaAboveCell1010)))
+    e24KC2ThetaAboveLeaf10102030 e24KC2ThetaAboveLeaf10102031 e24KC2ThetaAboveLeaf10102032
+      e24KC2ThetaAboveLeaf10102033
+
+theorem e24KC2ThetaAboveNode1010220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10102200 e24KC2ThetaAboveNode10102201 e24KC2ThetaAboveNode10102202
+      e24KC2ThetaAboveNode10102203
+
+theorem e24KC2ThetaAboveNode1010221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10102210 e24KC2ThetaAboveNode10102211 e24KC2ThetaAboveNode10102212
+      e24KC2ThetaAboveNode10102213
+
+theorem e24KC2ThetaAboveNode1010222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10102220 e24KC2ThetaAboveNode10102221 e24KC2ThetaAboveLeaf10102222
+      e24KC2ThetaAboveLeaf10102223
+
+theorem e24KC2ThetaAboveNode1010223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10102230 e24KC2ThetaAboveNode10102231 e24KC2ThetaAboveLeaf10102232
+      e24KC2ThetaAboveLeaf10102233
+
+theorem e24KC2ThetaAboveNode1010230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10102300 e24KC2ThetaAboveNode10102301 e24KC2ThetaAboveNode10102302
+      e24KC2ThetaAboveNode10102303
+
+theorem e24KC2ThetaAboveNode1010231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10102310 e24KC2ThetaAboveNode10102311 e24KC2ThetaAboveNode10102312
+      e24KC2ThetaAboveNode10102313
+
+theorem e24KC2ThetaAboveNode1010232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10102320 e24KC2ThetaAboveNode10102321 e24KC2ThetaAboveLeaf10102322
+      e24KC2ThetaAboveLeaf10102323
+
+theorem e24KC2ThetaAboveNode1010233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10102330 e24KC2ThetaAboveNode10102331 e24KC2ThetaAboveLeaf10102332
+      e24KC2ThetaAboveLeaf10102333
+
+theorem e24KC2ThetaAboveNode1010320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10103200 e24KC2ThetaAboveNode10103201 e24KC2ThetaAboveNode10103202
+      e24KC2ThetaAboveNode10103203
+
+theorem e24KC2ThetaAboveNode1010321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10103210 e24KC2ThetaAboveNode10103211 e24KC2ThetaAboveNode10103212
+      e24KC2ThetaAboveNode10103213
+
+theorem e24KC2ThetaAboveNode1010322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10103220 e24KC2ThetaAboveNode10103221 e24KC2ThetaAboveLeaf10103222
+      e24KC2ThetaAboveLeaf10103223
+
+theorem e24KC2ThetaAboveNode1010323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10103230 e24KC2ThetaAboveNode10103231 e24KC2ThetaAboveLeaf10103232
+      e24KC2ThetaAboveLeaf10103233
+
+theorem e24KC2ThetaAboveNode1010330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10103300 e24KC2ThetaAboveNode10103301 e24KC2ThetaAboveNode10103302
+      e24KC2ThetaAboveNode10103303
+
+theorem e24KC2ThetaAboveNode1010331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10103310 e24KC2ThetaAboveNode10103311 e24KC2ThetaAboveNode10103312
+      e24KC2ThetaAboveNode10103313
+
+theorem e24KC2ThetaAboveNode1010332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10103320 e24KC2ThetaAboveNode10103321 e24KC2ThetaAboveLeaf10103322
+      e24KC2ThetaAboveLeaf10103323
+
+theorem e24KC2ThetaAboveNode1010333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell1010))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell1010)))
+    e24KC2ThetaAboveNode10103330 e24KC2ThetaAboveNode10103331 e24KC2ThetaAboveLeaf10103332
+      e24KC2ThetaAboveLeaf10103333
+
+theorem e24KC2ThetaAboveNode1011220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10112200 e24KC2ThetaAboveNode10112201 e24KC2ThetaAboveNode10112202
+      e24KC2ThetaAboveNode10112203
+
+theorem e24KC2ThetaAboveNode1011221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10112210 e24KC2ThetaAboveNode10112211 e24KC2ThetaAboveNode10112212
+      e24KC2ThetaAboveNode10112213
+
+theorem e24KC2ThetaAboveNode1011222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10112220 e24KC2ThetaAboveNode10112221 e24KC2ThetaAboveLeaf10112222
+      e24KC2ThetaAboveLeaf10112223
+
+theorem e24KC2ThetaAboveNode1011223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10112230 e24KC2ThetaAboveNode10112231 e24KC2ThetaAboveLeaf10112232
+      e24KC2ThetaAboveLeaf10112233
+
+theorem e24KC2ThetaAboveNode1011230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10112300 e24KC2ThetaAboveNode10112301 e24KC2ThetaAboveNode10112302
+      e24KC2ThetaAboveNode10112303
+
+theorem e24KC2ThetaAboveNode1011231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10112310 e24KC2ThetaAboveNode10112311 e24KC2ThetaAboveNode10112312
+      e24KC2ThetaAboveNode10112313
+
+theorem e24KC2ThetaAboveNode1011232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10112320 e24KC2ThetaAboveNode10112321 e24KC2ThetaAboveLeaf10112322
+      e24KC2ThetaAboveLeaf10112323
+
+theorem e24KC2ThetaAboveNode1011233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10112330 e24KC2ThetaAboveNode10112331 e24KC2ThetaAboveLeaf10112332
+      e24KC2ThetaAboveLeaf10112333
+
+theorem e24KC2ThetaAboveNode1011320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10113200 e24KC2ThetaAboveNode10113201 e24KC2ThetaAboveNode10113202
+      e24KC2ThetaAboveNode10113203
+
+theorem e24KC2ThetaAboveNode1011321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10113210 e24KC2ThetaAboveNode10113211 e24KC2ThetaAboveNode10113212
+      e24KC2ThetaAboveNode10113213
+
+theorem e24KC2ThetaAboveNode1011322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10113220 e24KC2ThetaAboveNode10113221 e24KC2ThetaAboveLeaf10113222
+      e24KC2ThetaAboveLeaf10113223
+
+theorem e24KC2ThetaAboveNode1011323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10113230 e24KC2ThetaAboveNode10113231 e24KC2ThetaAboveLeaf10113232
+      e24KC2ThetaAboveLeaf10113233
+
+theorem e24KC2ThetaAboveNode1011330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10113300 e24KC2ThetaAboveNode10113301 e24KC2ThetaAboveNode10113302
+      e24KC2ThetaAboveNode10113303
+
+theorem e24KC2ThetaAboveNode1011331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10113310 e24KC2ThetaAboveNode10113311 e24KC2ThetaAboveNode10113312
+      e24KC2ThetaAboveNode10113313
+
+theorem e24KC2ThetaAboveNode1011332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10113320 e24KC2ThetaAboveNode10113321 e24KC2ThetaAboveLeaf10113322
+      e24KC2ThetaAboveLeaf10113323
+
+theorem e24KC2ThetaAboveNode1011333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell1011))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell1011)))
+    e24KC2ThetaAboveNode10113330 e24KC2ThetaAboveNode10113331 e24KC2ThetaAboveLeaf10113332
+      e24KC2ThetaAboveLeaf10113333
+
+theorem e24KC2ThetaAboveNode1100220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell1100)))
+    e24KC2ThetaAboveLeaf11002200 e24KC2ThetaAboveLeaf11002201 e24KC2ThetaAboveNode11002202
+      e24KC2ThetaAboveNode11002203
+
+theorem e24KC2ThetaAboveNode1100221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell1100)))
+    e24KC2ThetaAboveLeaf11002210 e24KC2ThetaAboveLeaf11002211 e24KC2ThetaAboveNode11002212
+      e24KC2ThetaAboveNode11002213
+
+theorem e24KC2ThetaAboveNode1100222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell1100)))
+    e24KC2ThetaAboveNode11002220 e24KC2ThetaAboveNode11002221 e24KC2ThetaAboveLeaf11002222
+      e24KC2ThetaAboveLeaf11002223
+
+theorem e24KC2ThetaAboveNode1100223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell1100)))
+    e24KC2ThetaAboveNode11002230 e24KC2ThetaAboveNode11002231 e24KC2ThetaAboveLeaf11002232
+      e24KC2ThetaAboveLeaf11002233
+
+theorem e24KC2ThetaAboveNode1100230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell1100)))
+    e24KC2ThetaAboveLeaf11002300 e24KC2ThetaAboveLeaf11002301 e24KC2ThetaAboveNode11002302
+      e24KC2ThetaAboveNode11002303
+
+theorem e24KC2ThetaAboveNode1100231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell1100)))
+    e24KC2ThetaAboveLeaf11002310 e24KC2ThetaAboveLeaf11002311 e24KC2ThetaAboveNode11002312
+      e24KC2ThetaAboveNode11002313
+
+theorem e24KC2ThetaAboveNode1100232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell1100)))
+    e24KC2ThetaAboveNode11002320 e24KC2ThetaAboveNode11002321 e24KC2ThetaAboveLeaf11002322
+      e24KC2ThetaAboveLeaf11002323
+
+theorem e24KC2ThetaAboveNode1100233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell1100)))
+    e24KC2ThetaAboveNode11002330 e24KC2ThetaAboveNode11002331 e24KC2ThetaAboveLeaf11002332
+      e24KC2ThetaAboveLeaf11002333
+
+theorem e24KC2ThetaAboveNode1100320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell1100)))
+    e24KC2ThetaAboveLeaf11003200 e24KC2ThetaAboveLeaf11003201 e24KC2ThetaAboveNode11003202
+      e24KC2ThetaAboveNode11003203
+
+theorem e24KC2ThetaAboveNode1100321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell1100)))
+    e24KC2ThetaAboveLeaf11003210 e24KC2ThetaAboveLeaf11003211 e24KC2ThetaAboveNode11003212
+      e24KC2ThetaAboveNode11003213
+
+theorem e24KC2ThetaAboveNode1100322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell1100)))
+    e24KC2ThetaAboveNode11003220 e24KC2ThetaAboveNode11003221 e24KC2ThetaAboveLeaf11003222
+      e24KC2ThetaAboveLeaf11003223
+
+theorem e24KC2ThetaAboveNode1100323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell1100)))
+    e24KC2ThetaAboveNode11003230 e24KC2ThetaAboveNode11003231 e24KC2ThetaAboveLeaf11003232
+      e24KC2ThetaAboveLeaf11003233
+
+theorem e24KC2ThetaAboveNode1100330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell1100)))
+    e24KC2ThetaAboveLeaf11003300 e24KC2ThetaAboveLeaf11003301 e24KC2ThetaAboveNode11003302
+      e24KC2ThetaAboveNode11003303
+
+theorem e24KC2ThetaAboveNode1100331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell1100)))
+    e24KC2ThetaAboveLeaf11003310 e24KC2ThetaAboveLeaf11003311 e24KC2ThetaAboveNode11003312
+      e24KC2ThetaAboveNode11003313
+
+theorem e24KC2ThetaAboveNode1100332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell1100)))
+    e24KC2ThetaAboveNode11003320 e24KC2ThetaAboveNode11003321 e24KC2ThetaAboveLeaf11003322
+      e24KC2ThetaAboveLeaf11003323
+
+theorem e24KC2ThetaAboveNode1100333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell1100))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell1100)))
+    e24KC2ThetaAboveNode11003330 e24KC2ThetaAboveNode11003331 e24KC2ThetaAboveLeaf11003332
+      e24KC2ThetaAboveLeaf11003333
+
+theorem e24KC2ThetaAboveNode1101220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell1101)))
+    e24KC2ThetaAboveLeaf11012200 e24KC2ThetaAboveLeaf11012201 e24KC2ThetaAboveNode11012202
+      e24KC2ThetaAboveNode11012203
+
+theorem e24KC2ThetaAboveNode1101221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell1101)))
+    e24KC2ThetaAboveLeaf11012210 e24KC2ThetaAboveLeaf11012211 e24KC2ThetaAboveNode11012212
+      e24KC2ThetaAboveNode11012213
+
+theorem e24KC2ThetaAboveNode1101222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell1101)))
+    e24KC2ThetaAboveNode11012220 e24KC2ThetaAboveNode11012221 e24KC2ThetaAboveLeaf11012222
+      e24KC2ThetaAboveLeaf11012223
+
+theorem e24KC2ThetaAboveNode1101223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell1101)))
+    e24KC2ThetaAboveNode11012230 e24KC2ThetaAboveNode11012231 e24KC2ThetaAboveLeaf11012232
+      e24KC2ThetaAboveLeaf11012233
+
+theorem e24KC2ThetaAboveNode1101230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell1101)))
+    e24KC2ThetaAboveLeaf11012300 e24KC2ThetaAboveLeaf11012301 e24KC2ThetaAboveNode11012302
+      e24KC2ThetaAboveNode11012303
+
+theorem e24KC2ThetaAboveNode1101231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell1101)))
+    e24KC2ThetaAboveLeaf11012310 e24KC2ThetaAboveLeaf11012311 e24KC2ThetaAboveNode11012312
+      e24KC2ThetaAboveNode11012313
+
+theorem e24KC2ThetaAboveNode1101232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell1101)))
+    e24KC2ThetaAboveNode11012320 e24KC2ThetaAboveNode11012321 e24KC2ThetaAboveLeaf11012322
+      e24KC2ThetaAboveLeaf11012323
+
+theorem e24KC2ThetaAboveNode1101233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell1101)))
+    e24KC2ThetaAboveNode11012330 e24KC2ThetaAboveNode11012331 e24KC2ThetaAboveLeaf11012332
+      e24KC2ThetaAboveLeaf11012333
+
+theorem e24KC2ThetaAboveNode1101320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell1101)))
+    e24KC2ThetaAboveLeaf11013200 e24KC2ThetaAboveLeaf11013201 e24KC2ThetaAboveNode11013202
+      e24KC2ThetaAboveNode11013203
+
+theorem e24KC2ThetaAboveNode1101321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell1101)))
+    e24KC2ThetaAboveLeaf11013210 e24KC2ThetaAboveLeaf11013211 e24KC2ThetaAboveNode11013212
+      e24KC2ThetaAboveNode11013213
+
+theorem e24KC2ThetaAboveNode1101322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell1101)))
+    e24KC2ThetaAboveNode11013220 e24KC2ThetaAboveNode11013221 e24KC2ThetaAboveLeaf11013222
+      e24KC2ThetaAboveLeaf11013223
+
+theorem e24KC2ThetaAboveNode1101323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell1101)))
+    e24KC2ThetaAboveNode11013230 e24KC2ThetaAboveNode11013231 e24KC2ThetaAboveLeaf11013232
+      e24KC2ThetaAboveLeaf11013233
+
+theorem e24KC2ThetaAboveNode1101330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell1101)))
+    e24KC2ThetaAboveLeaf11013300 e24KC2ThetaAboveLeaf11013301 e24KC2ThetaAboveNode11013302
+      e24KC2ThetaAboveNode11013303
+
+theorem e24KC2ThetaAboveNode1101331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell1101)))
+    e24KC2ThetaAboveLeaf11013310 e24KC2ThetaAboveLeaf11013311 e24KC2ThetaAboveNode11013312
+      e24KC2ThetaAboveNode11013313
+
+theorem e24KC2ThetaAboveNode1101332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell1101)))
+    e24KC2ThetaAboveNode11013320 e24KC2ThetaAboveNode11013321 e24KC2ThetaAboveLeaf11013322
+      e24KC2ThetaAboveLeaf11013323
+
+theorem e24KC2ThetaAboveNode1101333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell1101))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell1101)))
+    e24KC2ThetaAboveNode11013330 e24KC2ThetaAboveNode11013331 e24KC2ThetaAboveLeaf11013332
+      e24KC2ThetaAboveLeaf11013333
+
+theorem e24KC2ThetaAboveNode1110220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell1110)))
+    e24KC2ThetaAboveLeaf11102200 e24KC2ThetaAboveLeaf11102201 e24KC2ThetaAboveNode11102202
+      e24KC2ThetaAboveNode11102203
+
+theorem e24KC2ThetaAboveNode1110221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell1110)))
+    e24KC2ThetaAboveLeaf11102210 e24KC2ThetaAboveLeaf11102211 e24KC2ThetaAboveNode11102212
+      e24KC2ThetaAboveNode11102213
+
+theorem e24KC2ThetaAboveNode1110222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell1110)))
+    e24KC2ThetaAboveNode11102220 e24KC2ThetaAboveNode11102221 e24KC2ThetaAboveLeaf11102222
+      e24KC2ThetaAboveLeaf11102223
+
+theorem e24KC2ThetaAboveNode1110223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell1110)))
+    e24KC2ThetaAboveNode11102230 e24KC2ThetaAboveNode11102231 e24KC2ThetaAboveLeaf11102232
+      e24KC2ThetaAboveLeaf11102233
+
+theorem e24KC2ThetaAboveNode1110230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell1110)))
+    e24KC2ThetaAboveLeaf11102300 e24KC2ThetaAboveLeaf11102301 e24KC2ThetaAboveNode11102302
+      e24KC2ThetaAboveNode11102303
+
+theorem e24KC2ThetaAboveNode1110231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell1110)))
+    e24KC2ThetaAboveLeaf11102310 e24KC2ThetaAboveLeaf11102311 e24KC2ThetaAboveNode11102312
+      e24KC2ThetaAboveNode11102313
+
+theorem e24KC2ThetaAboveNode1110232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell1110)))
+    e24KC2ThetaAboveNode11102320 e24KC2ThetaAboveNode11102321 e24KC2ThetaAboveLeaf11102322
+      e24KC2ThetaAboveLeaf11102323
+
+theorem e24KC2ThetaAboveNode1110233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell1110)))
+    e24KC2ThetaAboveNode11102330 e24KC2ThetaAboveNode11102331 e24KC2ThetaAboveLeaf11102332
+      e24KC2ThetaAboveLeaf11102333
+
+theorem e24KC2ThetaAboveNode1110320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell1110)))
+    e24KC2ThetaAboveLeaf11103200 e24KC2ThetaAboveLeaf11103201 e24KC2ThetaAboveNode11103202
+      e24KC2ThetaAboveNode11103203
+
+theorem e24KC2ThetaAboveNode1110321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell1110)))
+    e24KC2ThetaAboveLeaf11103210 e24KC2ThetaAboveLeaf11103211 e24KC2ThetaAboveNode11103212
+      e24KC2ThetaAboveNode11103213
+
+theorem e24KC2ThetaAboveNode1110322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell1110)))
+    e24KC2ThetaAboveNode11103220 e24KC2ThetaAboveNode11103221 e24KC2ThetaAboveLeaf11103222
+      e24KC2ThetaAboveLeaf11103223
+
+theorem e24KC2ThetaAboveNode1110323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell1110)))
+    e24KC2ThetaAboveNode11103230 e24KC2ThetaAboveNode11103231 e24KC2ThetaAboveLeaf11103232
+      e24KC2ThetaAboveLeaf11103233
+
+theorem e24KC2ThetaAboveNode1110330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell1110)))
+    e24KC2ThetaAboveLeaf11103300 e24KC2ThetaAboveLeaf11103301 e24KC2ThetaAboveNode11103302
+      e24KC2ThetaAboveNode11103303
+
+theorem e24KC2ThetaAboveNode1110331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell1110)))
+    e24KC2ThetaAboveLeaf11103310 e24KC2ThetaAboveLeaf11103311 e24KC2ThetaAboveNode11103312
+      e24KC2ThetaAboveNode11103313
+
+theorem e24KC2ThetaAboveNode1110332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell1110)))
+    e24KC2ThetaAboveNode11103320 e24KC2ThetaAboveNode11103321 e24KC2ThetaAboveLeaf11103322
+      e24KC2ThetaAboveLeaf11103323
+
+theorem e24KC2ThetaAboveNode1110333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell1110))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell1110)))
+    e24KC2ThetaAboveNode11103330 e24KC2ThetaAboveNode11103331 e24KC2ThetaAboveLeaf11103332
+      e24KC2ThetaAboveLeaf11103333
+
+theorem e24KC2ThetaAboveNode1111220 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHL thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHL thetaAboveCell1111)))
+    e24KC2ThetaAboveLeaf11112200 e24KC2ThetaAboveLeaf11112201 e24KC2ThetaAboveNode11112202
+      e24KC2ThetaAboveNode11112203
+
+theorem e24KC2ThetaAboveNode1111221 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHL thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHL thetaAboveCell1111)))
+    e24KC2ThetaAboveLeaf11112210 e24KC2ThetaAboveLeaf11112211 e24KC2ThetaAboveNode11112212
+      e24KC2ThetaAboveNode11112213
+
+theorem e24KC2ThetaAboveNode1111222 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHL thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHL thetaAboveCell1111)))
+    e24KC2ThetaAboveNode11112220 e24KC2ThetaAboveNode11112221 e24KC2ThetaAboveLeaf11112222
+      e24KC2ThetaAboveLeaf11112223
+
+theorem e24KC2ThetaAboveNode1111223 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHL thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHL thetaAboveCell1111)))
+    e24KC2ThetaAboveNode11112230 e24KC2ThetaAboveNode11112231 e24KC2ThetaAboveLeaf11112232
+      e24KC2ThetaAboveLeaf11112233
+
+theorem e24KC2ThetaAboveNode1111230 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHL thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHL thetaAboveCell1111)))
+    e24KC2ThetaAboveLeaf11112300 e24KC2ThetaAboveLeaf11112301 e24KC2ThetaAboveNode11112302
+      e24KC2ThetaAboveNode11112303
+
+theorem e24KC2ThetaAboveNode1111231 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHL thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHL thetaAboveCell1111)))
+    e24KC2ThetaAboveLeaf11112310 e24KC2ThetaAboveLeaf11112311 e24KC2ThetaAboveNode11112312
+      e24KC2ThetaAboveNode11112313
+
+theorem e24KC2ThetaAboveNode1111232 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHL thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHL thetaAboveCell1111)))
+    e24KC2ThetaAboveNode11112320 e24KC2ThetaAboveNode11112321 e24KC2ThetaAboveLeaf11112322
+      e24KC2ThetaAboveLeaf11112323
+
+theorem e24KC2ThetaAboveNode1111233 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHL thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHL thetaAboveCell1111)))
+    e24KC2ThetaAboveNode11112330 e24KC2ThetaAboveNode11112331 e24KC2ThetaAboveNode11112332
+      e24KC2ThetaAboveNode11112333
+
+theorem e24KC2ThetaAboveNode1111320 :
+    adaptiveCoverCheck 12 (childLL (childHL (childHH thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHL (childHH thetaAboveCell1111)))
+    e24KC2ThetaAboveLeaf11113200 e24KC2ThetaAboveLeaf11113201 e24KC2ThetaAboveNode11113202
+      e24KC2ThetaAboveNode11113203
+
+theorem e24KC2ThetaAboveNode1111321 :
+    adaptiveCoverCheck 12 (childLH (childHL (childHH thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHL (childHH thetaAboveCell1111)))
+    e24KC2ThetaAboveLeaf11113210 e24KC2ThetaAboveLeaf11113211 e24KC2ThetaAboveNode11113212
+      e24KC2ThetaAboveNode11113213
+
+theorem e24KC2ThetaAboveNode1111322 :
+    adaptiveCoverCheck 12 (childHL (childHL (childHH thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHL (childHH thetaAboveCell1111)))
+    e24KC2ThetaAboveNode11113220 e24KC2ThetaAboveNode11113221 e24KC2ThetaAboveNode11113222
+      e24KC2ThetaAboveNode11113223
+
+theorem e24KC2ThetaAboveNode1111323 :
+    adaptiveCoverCheck 12 (childHH (childHL (childHH thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHL (childHH thetaAboveCell1111)))
+    e24KC2ThetaAboveNode11113230 e24KC2ThetaAboveNode11113231 e24KC2ThetaAboveNode11113232
+      e24KC2ThetaAboveNode11113233
+
+theorem e24KC2ThetaAboveNode1111330 :
+    adaptiveCoverCheck 12 (childLL (childHH (childHH thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLL (childHH (childHH thetaAboveCell1111)))
+    e24KC2ThetaAboveLeaf11113300 e24KC2ThetaAboveLeaf11113301 e24KC2ThetaAboveNode11113302
+      e24KC2ThetaAboveNode11113303
+
+theorem e24KC2ThetaAboveNode1111331 :
+    adaptiveCoverCheck 12 (childLH (childHH (childHH thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childLH (childHH (childHH thetaAboveCell1111)))
+    e24KC2ThetaAboveLeaf11113310 e24KC2ThetaAboveLeaf11113311 e24KC2ThetaAboveNode11113312
+      e24KC2ThetaAboveNode11113313
+
+theorem e24KC2ThetaAboveNode1111332 :
+    adaptiveCoverCheck 12 (childHL (childHH (childHH thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHL (childHH (childHH thetaAboveCell1111)))
+    e24KC2ThetaAboveNode11113320 e24KC2ThetaAboveNode11113321 e24KC2ThetaAboveNode11113322
+      e24KC2ThetaAboveNode11113323
+
+theorem e24KC2ThetaAboveNode1111333 :
+    adaptiveCoverCheck 12 (childHH (childHH (childHH thetaAboveCell1111))) = true :=
+  adaptiveCoverCheck_succ_of_children 11 (childHH (childHH (childHH thetaAboveCell1111)))
+    e24KC2ThetaAboveNode11113330 e24KC2ThetaAboveNode11113331 e24KC2ThetaAboveNode11113332
+      e24KC2ThetaAboveNode11113333
+
+theorem e24KC2ThetaAboveNode000020 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell0000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell0000))
+    e24KC2ThetaAboveLeaf0000200 e24KC2ThetaAboveLeaf0000201 e24KC2ThetaAboveNode0000202
+      e24KC2ThetaAboveNode0000203
+
+theorem e24KC2ThetaAboveNode000021 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell0000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell0000))
+    e24KC2ThetaAboveLeaf0000210 e24KC2ThetaAboveLeaf0000211 e24KC2ThetaAboveNode0000212
+      e24KC2ThetaAboveNode0000213
+
+theorem e24KC2ThetaAboveNode000022 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell0000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell0000))
+    e24KC2ThetaAboveNode0000220 e24KC2ThetaAboveNode0000221 e24KC2ThetaAboveNode0000222
+      e24KC2ThetaAboveNode0000223
+
+theorem e24KC2ThetaAboveNode000023 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell0000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell0000))
+    e24KC2ThetaAboveNode0000230 e24KC2ThetaAboveNode0000231 e24KC2ThetaAboveNode0000232
+      e24KC2ThetaAboveNode0000233
+
+theorem e24KC2ThetaAboveNode000030 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell0000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell0000))
+    e24KC2ThetaAboveLeaf0000300 e24KC2ThetaAboveLeaf0000301 e24KC2ThetaAboveNode0000302
+      e24KC2ThetaAboveNode0000303
+
+theorem e24KC2ThetaAboveNode000031 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell0000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell0000))
+    e24KC2ThetaAboveLeaf0000310 e24KC2ThetaAboveLeaf0000311 e24KC2ThetaAboveNode0000312
+      e24KC2ThetaAboveNode0000313
+
+theorem e24KC2ThetaAboveNode000032 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell0000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell0000))
+    e24KC2ThetaAboveNode0000320 e24KC2ThetaAboveNode0000321 e24KC2ThetaAboveNode0000322
+      e24KC2ThetaAboveNode0000323
+
+theorem e24KC2ThetaAboveNode000033 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell0000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell0000))
+    e24KC2ThetaAboveNode0000330 e24KC2ThetaAboveNode0000331 e24KC2ThetaAboveNode0000332
+      e24KC2ThetaAboveNode0000333
+
+theorem e24KC2ThetaAboveNode000120 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell0001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell0001))
+    e24KC2ThetaAboveLeaf0001200 e24KC2ThetaAboveLeaf0001201 e24KC2ThetaAboveNode0001202
+      e24KC2ThetaAboveNode0001203
+
+theorem e24KC2ThetaAboveNode000121 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell0001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell0001))
+    e24KC2ThetaAboveLeaf0001210 e24KC2ThetaAboveLeaf0001211 e24KC2ThetaAboveNode0001212
+      e24KC2ThetaAboveNode0001213
+
+theorem e24KC2ThetaAboveNode000122 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell0001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell0001))
+    e24KC2ThetaAboveNode0001220 e24KC2ThetaAboveNode0001221 e24KC2ThetaAboveNode0001222
+      e24KC2ThetaAboveNode0001223
+
+theorem e24KC2ThetaAboveNode000123 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell0001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell0001))
+    e24KC2ThetaAboveNode0001230 e24KC2ThetaAboveNode0001231 e24KC2ThetaAboveNode0001232
+      e24KC2ThetaAboveNode0001233
+
+theorem e24KC2ThetaAboveNode000130 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell0001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell0001))
+    e24KC2ThetaAboveLeaf0001300 e24KC2ThetaAboveLeaf0001301 e24KC2ThetaAboveNode0001302
+      e24KC2ThetaAboveNode0001303
+
+theorem e24KC2ThetaAboveNode000131 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell0001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell0001))
+    e24KC2ThetaAboveLeaf0001310 e24KC2ThetaAboveLeaf0001311 e24KC2ThetaAboveNode0001312
+      e24KC2ThetaAboveNode0001313
+
+theorem e24KC2ThetaAboveNode000132 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell0001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell0001))
+    e24KC2ThetaAboveNode0001320 e24KC2ThetaAboveNode0001321 e24KC2ThetaAboveNode0001322
+      e24KC2ThetaAboveNode0001323
+
+theorem e24KC2ThetaAboveNode000133 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell0001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell0001))
+    e24KC2ThetaAboveNode0001330 e24KC2ThetaAboveNode0001331 e24KC2ThetaAboveNode0001332
+      e24KC2ThetaAboveNode0001333
+
+theorem e24KC2ThetaAboveNode000200 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0002)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell0002))
+    e24KC2ThetaAboveLeaf0002000 e24KC2ThetaAboveLeaf0002001 e24KC2ThetaAboveLeaf0002002
+      e24KC2ThetaAboveLeaf0002003
+
+theorem e24KC2ThetaAboveNode000201 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0002)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell0002))
+    e24KC2ThetaAboveLeaf0002010 e24KC2ThetaAboveLeaf0002011 e24KC2ThetaAboveLeaf0002012
+      e24KC2ThetaAboveLeaf0002013
+
+theorem e24KC2ThetaAboveNode000210 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0002)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell0002))
+    e24KC2ThetaAboveLeaf0002100 e24KC2ThetaAboveLeaf0002101 e24KC2ThetaAboveLeaf0002102
+      e24KC2ThetaAboveLeaf0002103
+
+theorem e24KC2ThetaAboveNode000211 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0002)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell0002))
+    e24KC2ThetaAboveLeaf0002110 e24KC2ThetaAboveLeaf0002111 e24KC2ThetaAboveLeaf0002112
+      e24KC2ThetaAboveLeaf0002113
+
+theorem e24KC2ThetaAboveNode000300 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0003)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell0003))
+    e24KC2ThetaAboveLeaf0003000 e24KC2ThetaAboveLeaf0003001 e24KC2ThetaAboveLeaf0003002
+      e24KC2ThetaAboveLeaf0003003
+
+theorem e24KC2ThetaAboveNode000301 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0003)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell0003))
+    e24KC2ThetaAboveLeaf0003010 e24KC2ThetaAboveLeaf0003011 e24KC2ThetaAboveLeaf0003012
+      e24KC2ThetaAboveLeaf0003013
+
+theorem e24KC2ThetaAboveNode000310 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0003)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell0003))
+    e24KC2ThetaAboveLeaf0003100 e24KC2ThetaAboveLeaf0003101 e24KC2ThetaAboveLeaf0003102
+      e24KC2ThetaAboveLeaf0003103
+
+theorem e24KC2ThetaAboveNode000311 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0003)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell0003))
+    e24KC2ThetaAboveLeaf0003110 e24KC2ThetaAboveLeaf0003111 e24KC2ThetaAboveLeaf0003112
+      e24KC2ThetaAboveLeaf0003113
+
+theorem e24KC2ThetaAboveNode001020 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell0010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell0010))
+    e24KC2ThetaAboveLeaf0010200 e24KC2ThetaAboveLeaf0010201 e24KC2ThetaAboveNode0010202
+      e24KC2ThetaAboveNode0010203
+
+theorem e24KC2ThetaAboveNode001021 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell0010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell0010))
+    e24KC2ThetaAboveLeaf0010210 e24KC2ThetaAboveLeaf0010211 e24KC2ThetaAboveNode0010212
+      e24KC2ThetaAboveNode0010213
+
+theorem e24KC2ThetaAboveNode001022 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell0010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell0010))
+    e24KC2ThetaAboveNode0010220 e24KC2ThetaAboveNode0010221 e24KC2ThetaAboveNode0010222
+      e24KC2ThetaAboveNode0010223
+
+theorem e24KC2ThetaAboveNode001023 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell0010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell0010))
+    e24KC2ThetaAboveNode0010230 e24KC2ThetaAboveNode0010231 e24KC2ThetaAboveNode0010232
+      e24KC2ThetaAboveNode0010233
+
+theorem e24KC2ThetaAboveNode001030 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell0010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell0010))
+    e24KC2ThetaAboveLeaf0010300 e24KC2ThetaAboveLeaf0010301 e24KC2ThetaAboveNode0010302
+      e24KC2ThetaAboveNode0010303
+
+theorem e24KC2ThetaAboveNode001031 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell0010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell0010))
+    e24KC2ThetaAboveLeaf0010310 e24KC2ThetaAboveLeaf0010311 e24KC2ThetaAboveNode0010312
+      e24KC2ThetaAboveNode0010313
+
+theorem e24KC2ThetaAboveNode001032 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell0010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell0010))
+    e24KC2ThetaAboveNode0010320 e24KC2ThetaAboveNode0010321 e24KC2ThetaAboveNode0010322
+      e24KC2ThetaAboveNode0010323
+
+theorem e24KC2ThetaAboveNode001033 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell0010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell0010))
+    e24KC2ThetaAboveNode0010330 e24KC2ThetaAboveNode0010331 e24KC2ThetaAboveNode0010332
+      e24KC2ThetaAboveNode0010333
+
+theorem e24KC2ThetaAboveNode001120 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell0011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell0011))
+    e24KC2ThetaAboveLeaf0011200 e24KC2ThetaAboveLeaf0011201 e24KC2ThetaAboveNode0011202
+      e24KC2ThetaAboveNode0011203
+
+theorem e24KC2ThetaAboveNode001121 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell0011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell0011))
+    e24KC2ThetaAboveLeaf0011210 e24KC2ThetaAboveLeaf0011211 e24KC2ThetaAboveNode0011212
+      e24KC2ThetaAboveNode0011213
+
+theorem e24KC2ThetaAboveNode001122 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell0011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell0011))
+    e24KC2ThetaAboveNode0011220 e24KC2ThetaAboveNode0011221 e24KC2ThetaAboveNode0011222
+      e24KC2ThetaAboveNode0011223
+
+theorem e24KC2ThetaAboveNode001123 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell0011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell0011))
+    e24KC2ThetaAboveNode0011230 e24KC2ThetaAboveNode0011231 e24KC2ThetaAboveNode0011232
+      e24KC2ThetaAboveNode0011233
+
+theorem e24KC2ThetaAboveNode001130 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell0011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell0011))
+    e24KC2ThetaAboveLeaf0011300 e24KC2ThetaAboveLeaf0011301 e24KC2ThetaAboveNode0011302
+      e24KC2ThetaAboveNode0011303
+
+theorem e24KC2ThetaAboveNode001131 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell0011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell0011))
+    e24KC2ThetaAboveLeaf0011310 e24KC2ThetaAboveLeaf0011311 e24KC2ThetaAboveNode0011312
+      e24KC2ThetaAboveNode0011313
+
+theorem e24KC2ThetaAboveNode001132 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell0011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell0011))
+    e24KC2ThetaAboveNode0011320 e24KC2ThetaAboveNode0011321 e24KC2ThetaAboveNode0011322
+      e24KC2ThetaAboveNode0011323
+
+theorem e24KC2ThetaAboveNode001133 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell0011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell0011))
+    e24KC2ThetaAboveNode0011330 e24KC2ThetaAboveNode0011331 e24KC2ThetaAboveNode0011332
+      e24KC2ThetaAboveNode0011333
+
+theorem e24KC2ThetaAboveNode001200 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0012)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell0012))
+    e24KC2ThetaAboveLeaf0012000 e24KC2ThetaAboveLeaf0012001 e24KC2ThetaAboveLeaf0012002
+      e24KC2ThetaAboveLeaf0012003
+
+theorem e24KC2ThetaAboveNode001201 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0012)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell0012))
+    e24KC2ThetaAboveLeaf0012010 e24KC2ThetaAboveLeaf0012011 e24KC2ThetaAboveLeaf0012012
+      e24KC2ThetaAboveLeaf0012013
+
+theorem e24KC2ThetaAboveNode001210 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0012)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell0012))
+    e24KC2ThetaAboveLeaf0012100 e24KC2ThetaAboveLeaf0012101 e24KC2ThetaAboveLeaf0012102
+      e24KC2ThetaAboveLeaf0012103
+
+theorem e24KC2ThetaAboveNode001211 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0012)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell0012))
+    e24KC2ThetaAboveLeaf0012110 e24KC2ThetaAboveLeaf0012111 e24KC2ThetaAboveLeaf0012112
+      e24KC2ThetaAboveLeaf0012113
+
+theorem e24KC2ThetaAboveNode001300 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0013)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell0013))
+    e24KC2ThetaAboveLeaf0013000 e24KC2ThetaAboveLeaf0013001 e24KC2ThetaAboveLeaf0013002
+      e24KC2ThetaAboveLeaf0013003
+
+theorem e24KC2ThetaAboveNode001301 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0013)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell0013))
+    e24KC2ThetaAboveLeaf0013010 e24KC2ThetaAboveLeaf0013011 e24KC2ThetaAboveLeaf0013012
+      e24KC2ThetaAboveLeaf0013013
+
+theorem e24KC2ThetaAboveNode001310 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0013)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell0013))
+    e24KC2ThetaAboveLeaf0013100 e24KC2ThetaAboveLeaf0013101 e24KC2ThetaAboveLeaf0013102
+      e24KC2ThetaAboveLeaf0013103
+
+theorem e24KC2ThetaAboveNode001311 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0013)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell0013))
+    e24KC2ThetaAboveLeaf0013110 e24KC2ThetaAboveLeaf0013111 e24KC2ThetaAboveLeaf0013112
+      e24KC2ThetaAboveLeaf0013113
+
+theorem e24KC2ThetaAboveNode010020 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell0100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell0100))
+    e24KC2ThetaAboveLeaf0100200 e24KC2ThetaAboveLeaf0100201 e24KC2ThetaAboveNode0100202
+      e24KC2ThetaAboveNode0100203
+
+theorem e24KC2ThetaAboveNode010021 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell0100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell0100))
+    e24KC2ThetaAboveLeaf0100210 e24KC2ThetaAboveLeaf0100211 e24KC2ThetaAboveNode0100212
+      e24KC2ThetaAboveNode0100213
+
+theorem e24KC2ThetaAboveNode010022 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell0100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell0100))
+    e24KC2ThetaAboveNode0100220 e24KC2ThetaAboveNode0100221 e24KC2ThetaAboveNode0100222
+      e24KC2ThetaAboveNode0100223
+
+theorem e24KC2ThetaAboveNode010023 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell0100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell0100))
+    e24KC2ThetaAboveNode0100230 e24KC2ThetaAboveNode0100231 e24KC2ThetaAboveNode0100232
+      e24KC2ThetaAboveNode0100233
+
+theorem e24KC2ThetaAboveNode010030 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell0100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell0100))
+    e24KC2ThetaAboveLeaf0100300 e24KC2ThetaAboveLeaf0100301 e24KC2ThetaAboveNode0100302
+      e24KC2ThetaAboveNode0100303
+
+theorem e24KC2ThetaAboveNode010031 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell0100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell0100))
+    e24KC2ThetaAboveLeaf0100310 e24KC2ThetaAboveLeaf0100311 e24KC2ThetaAboveNode0100312
+      e24KC2ThetaAboveNode0100313
+
+theorem e24KC2ThetaAboveNode010032 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell0100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell0100))
+    e24KC2ThetaAboveNode0100320 e24KC2ThetaAboveNode0100321 e24KC2ThetaAboveNode0100322
+      e24KC2ThetaAboveNode0100323
+
+theorem e24KC2ThetaAboveNode010033 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell0100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell0100))
+    e24KC2ThetaAboveNode0100330 e24KC2ThetaAboveNode0100331 e24KC2ThetaAboveNode0100332
+      e24KC2ThetaAboveNode0100333
+
+theorem e24KC2ThetaAboveNode010120 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell0101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell0101))
+    e24KC2ThetaAboveLeaf0101200 e24KC2ThetaAboveLeaf0101201 e24KC2ThetaAboveNode0101202
+      e24KC2ThetaAboveNode0101203
+
+theorem e24KC2ThetaAboveNode010121 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell0101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell0101))
+    e24KC2ThetaAboveLeaf0101210 e24KC2ThetaAboveLeaf0101211 e24KC2ThetaAboveNode0101212
+      e24KC2ThetaAboveNode0101213
+
+theorem e24KC2ThetaAboveNode010122 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell0101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell0101))
+    e24KC2ThetaAboveNode0101220 e24KC2ThetaAboveNode0101221 e24KC2ThetaAboveNode0101222
+      e24KC2ThetaAboveNode0101223
+
+theorem e24KC2ThetaAboveNode010123 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell0101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell0101))
+    e24KC2ThetaAboveNode0101230 e24KC2ThetaAboveNode0101231 e24KC2ThetaAboveNode0101232
+      e24KC2ThetaAboveNode0101233
+
+theorem e24KC2ThetaAboveNode010130 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell0101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell0101))
+    e24KC2ThetaAboveLeaf0101300 e24KC2ThetaAboveLeaf0101301 e24KC2ThetaAboveNode0101302
+      e24KC2ThetaAboveNode0101303
+
+theorem e24KC2ThetaAboveNode010131 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell0101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell0101))
+    e24KC2ThetaAboveLeaf0101310 e24KC2ThetaAboveLeaf0101311 e24KC2ThetaAboveNode0101312
+      e24KC2ThetaAboveNode0101313
+
+theorem e24KC2ThetaAboveNode010132 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell0101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell0101))
+    e24KC2ThetaAboveNode0101320 e24KC2ThetaAboveNode0101321 e24KC2ThetaAboveNode0101322
+      e24KC2ThetaAboveNode0101323
+
+theorem e24KC2ThetaAboveNode010133 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell0101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell0101))
+    e24KC2ThetaAboveNode0101330 e24KC2ThetaAboveNode0101331 e24KC2ThetaAboveNode0101332
+      e24KC2ThetaAboveNode0101333
+
+theorem e24KC2ThetaAboveNode010200 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0102)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell0102))
+    e24KC2ThetaAboveLeaf0102000 e24KC2ThetaAboveLeaf0102001 e24KC2ThetaAboveLeaf0102002
+      e24KC2ThetaAboveLeaf0102003
+
+theorem e24KC2ThetaAboveNode010201 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0102)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell0102))
+    e24KC2ThetaAboveLeaf0102010 e24KC2ThetaAboveLeaf0102011 e24KC2ThetaAboveLeaf0102012
+      e24KC2ThetaAboveLeaf0102013
+
+theorem e24KC2ThetaAboveNode010210 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0102)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell0102))
+    e24KC2ThetaAboveLeaf0102100 e24KC2ThetaAboveLeaf0102101 e24KC2ThetaAboveLeaf0102102
+      e24KC2ThetaAboveLeaf0102103
+
+theorem e24KC2ThetaAboveNode010211 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0102)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell0102))
+    e24KC2ThetaAboveLeaf0102110 e24KC2ThetaAboveLeaf0102111 e24KC2ThetaAboveLeaf0102112
+      e24KC2ThetaAboveLeaf0102113
+
+theorem e24KC2ThetaAboveNode010300 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0103)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell0103))
+    e24KC2ThetaAboveLeaf0103000 e24KC2ThetaAboveLeaf0103001 e24KC2ThetaAboveLeaf0103002
+      e24KC2ThetaAboveLeaf0103003
+
+theorem e24KC2ThetaAboveNode010301 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0103)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell0103))
+    e24KC2ThetaAboveLeaf0103010 e24KC2ThetaAboveLeaf0103011 e24KC2ThetaAboveLeaf0103012
+      e24KC2ThetaAboveLeaf0103013
+
+theorem e24KC2ThetaAboveNode010310 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0103)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell0103))
+    e24KC2ThetaAboveLeaf0103100 e24KC2ThetaAboveLeaf0103101 e24KC2ThetaAboveLeaf0103102
+      e24KC2ThetaAboveLeaf0103103
+
+theorem e24KC2ThetaAboveNode010311 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0103)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell0103))
+    e24KC2ThetaAboveLeaf0103110 e24KC2ThetaAboveLeaf0103111 e24KC2ThetaAboveLeaf0103112
+      e24KC2ThetaAboveLeaf0103113
+
+theorem e24KC2ThetaAboveNode011020 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell0110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell0110))
+    e24KC2ThetaAboveLeaf0110200 e24KC2ThetaAboveLeaf0110201 e24KC2ThetaAboveNode0110202
+      e24KC2ThetaAboveNode0110203
+
+theorem e24KC2ThetaAboveNode011021 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell0110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell0110))
+    e24KC2ThetaAboveLeaf0110210 e24KC2ThetaAboveLeaf0110211 e24KC2ThetaAboveNode0110212
+      e24KC2ThetaAboveNode0110213
+
+theorem e24KC2ThetaAboveNode011022 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell0110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell0110))
+    e24KC2ThetaAboveNode0110220 e24KC2ThetaAboveNode0110221 e24KC2ThetaAboveNode0110222
+      e24KC2ThetaAboveNode0110223
+
+theorem e24KC2ThetaAboveNode011023 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell0110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell0110))
+    e24KC2ThetaAboveNode0110230 e24KC2ThetaAboveNode0110231 e24KC2ThetaAboveNode0110232
+      e24KC2ThetaAboveNode0110233
+
+theorem e24KC2ThetaAboveNode011030 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell0110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell0110))
+    e24KC2ThetaAboveLeaf0110300 e24KC2ThetaAboveLeaf0110301 e24KC2ThetaAboveNode0110302
+      e24KC2ThetaAboveNode0110303
+
+theorem e24KC2ThetaAboveNode011031 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell0110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell0110))
+    e24KC2ThetaAboveLeaf0110310 e24KC2ThetaAboveLeaf0110311 e24KC2ThetaAboveNode0110312
+      e24KC2ThetaAboveNode0110313
+
+theorem e24KC2ThetaAboveNode011032 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell0110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell0110))
+    e24KC2ThetaAboveNode0110320 e24KC2ThetaAboveNode0110321 e24KC2ThetaAboveNode0110322
+      e24KC2ThetaAboveNode0110323
+
+theorem e24KC2ThetaAboveNode011033 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell0110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell0110))
+    e24KC2ThetaAboveNode0110330 e24KC2ThetaAboveNode0110331 e24KC2ThetaAboveNode0110332
+      e24KC2ThetaAboveNode0110333
+
+theorem e24KC2ThetaAboveNode011120 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell0111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell0111))
+    e24KC2ThetaAboveLeaf0111200 e24KC2ThetaAboveLeaf0111201 e24KC2ThetaAboveNode0111202
+      e24KC2ThetaAboveNode0111203
+
+theorem e24KC2ThetaAboveNode011121 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell0111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell0111))
+    e24KC2ThetaAboveLeaf0111210 e24KC2ThetaAboveLeaf0111211 e24KC2ThetaAboveNode0111212
+      e24KC2ThetaAboveNode0111213
+
+theorem e24KC2ThetaAboveNode011122 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell0111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell0111))
+    e24KC2ThetaAboveNode0111220 e24KC2ThetaAboveNode0111221 e24KC2ThetaAboveNode0111222
+      e24KC2ThetaAboveNode0111223
+
+theorem e24KC2ThetaAboveNode011123 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell0111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell0111))
+    e24KC2ThetaAboveNode0111230 e24KC2ThetaAboveNode0111231 e24KC2ThetaAboveNode0111232
+      e24KC2ThetaAboveNode0111233
+
+theorem e24KC2ThetaAboveNode011130 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell0111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell0111))
+    e24KC2ThetaAboveLeaf0111300 e24KC2ThetaAboveLeaf0111301 e24KC2ThetaAboveNode0111302
+      e24KC2ThetaAboveNode0111303
+
+theorem e24KC2ThetaAboveNode011131 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell0111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell0111))
+    e24KC2ThetaAboveLeaf0111310 e24KC2ThetaAboveLeaf0111311 e24KC2ThetaAboveNode0111312
+      e24KC2ThetaAboveNode0111313
+
+theorem e24KC2ThetaAboveNode011132 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell0111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell0111))
+    e24KC2ThetaAboveNode0111320 e24KC2ThetaAboveNode0111321 e24KC2ThetaAboveNode0111322
+      e24KC2ThetaAboveNode0111323
+
+theorem e24KC2ThetaAboveNode011133 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell0111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell0111))
+    e24KC2ThetaAboveNode0111330 e24KC2ThetaAboveNode0111331 e24KC2ThetaAboveNode0111332
+      e24KC2ThetaAboveNode0111333
+
+theorem e24KC2ThetaAboveNode011200 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0112)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell0112))
+    e24KC2ThetaAboveLeaf0112000 e24KC2ThetaAboveLeaf0112001 e24KC2ThetaAboveLeaf0112002
+      e24KC2ThetaAboveLeaf0112003
+
+theorem e24KC2ThetaAboveNode011201 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0112)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell0112))
+    e24KC2ThetaAboveLeaf0112010 e24KC2ThetaAboveLeaf0112011 e24KC2ThetaAboveLeaf0112012
+      e24KC2ThetaAboveLeaf0112013
+
+theorem e24KC2ThetaAboveNode011210 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0112)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell0112))
+    e24KC2ThetaAboveLeaf0112100 e24KC2ThetaAboveLeaf0112101 e24KC2ThetaAboveLeaf0112102
+      e24KC2ThetaAboveLeaf0112103
+
+theorem e24KC2ThetaAboveNode011211 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0112)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell0112))
+    e24KC2ThetaAboveLeaf0112110 e24KC2ThetaAboveLeaf0112111 e24KC2ThetaAboveLeaf0112112
+      e24KC2ThetaAboveLeaf0112113
+
+theorem e24KC2ThetaAboveNode011300 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell0113)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell0113))
+    e24KC2ThetaAboveLeaf0113000 e24KC2ThetaAboveLeaf0113001 e24KC2ThetaAboveLeaf0113002
+      e24KC2ThetaAboveLeaf0113003
+
+theorem e24KC2ThetaAboveNode011301 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell0113)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell0113))
+    e24KC2ThetaAboveLeaf0113010 e24KC2ThetaAboveLeaf0113011 e24KC2ThetaAboveLeaf0113012
+      e24KC2ThetaAboveLeaf0113013
+
+theorem e24KC2ThetaAboveNode011310 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell0113)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell0113))
+    e24KC2ThetaAboveLeaf0113100 e24KC2ThetaAboveLeaf0113101 e24KC2ThetaAboveLeaf0113102
+      e24KC2ThetaAboveLeaf0113103
+
+theorem e24KC2ThetaAboveNode011311 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell0113)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell0113))
+    e24KC2ThetaAboveLeaf0113110 e24KC2ThetaAboveLeaf0113111 e24KC2ThetaAboveLeaf0113112
+      e24KC2ThetaAboveLeaf0113113
+
+theorem e24KC2ThetaAboveNode100020 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell1000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell1000))
+    e24KC2ThetaAboveLeaf1000200 e24KC2ThetaAboveLeaf1000201 e24KC2ThetaAboveNode1000202
+      e24KC2ThetaAboveNode1000203
+
+theorem e24KC2ThetaAboveNode100021 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell1000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell1000))
+    e24KC2ThetaAboveLeaf1000210 e24KC2ThetaAboveLeaf1000211 e24KC2ThetaAboveNode1000212
+      e24KC2ThetaAboveNode1000213
+
+theorem e24KC2ThetaAboveNode100022 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell1000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell1000))
+    e24KC2ThetaAboveNode1000220 e24KC2ThetaAboveNode1000221 e24KC2ThetaAboveNode1000222
+      e24KC2ThetaAboveNode1000223
+
+theorem e24KC2ThetaAboveNode100023 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell1000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell1000))
+    e24KC2ThetaAboveNode1000230 e24KC2ThetaAboveNode1000231 e24KC2ThetaAboveNode1000232
+      e24KC2ThetaAboveNode1000233
+
+theorem e24KC2ThetaAboveNode100030 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell1000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell1000))
+    e24KC2ThetaAboveLeaf1000300 e24KC2ThetaAboveLeaf1000301 e24KC2ThetaAboveNode1000302
+      e24KC2ThetaAboveNode1000303
+
+theorem e24KC2ThetaAboveNode100031 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell1000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell1000))
+    e24KC2ThetaAboveLeaf1000310 e24KC2ThetaAboveLeaf1000311 e24KC2ThetaAboveNode1000312
+      e24KC2ThetaAboveNode1000313
+
+theorem e24KC2ThetaAboveNode100032 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell1000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell1000))
+    e24KC2ThetaAboveNode1000320 e24KC2ThetaAboveNode1000321 e24KC2ThetaAboveNode1000322
+      e24KC2ThetaAboveNode1000323
+
+theorem e24KC2ThetaAboveNode100033 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell1000)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell1000))
+    e24KC2ThetaAboveNode1000330 e24KC2ThetaAboveNode1000331 e24KC2ThetaAboveNode1000332
+      e24KC2ThetaAboveNode1000333
+
+theorem e24KC2ThetaAboveNode100120 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell1001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell1001))
+    e24KC2ThetaAboveLeaf1001200 e24KC2ThetaAboveLeaf1001201 e24KC2ThetaAboveNode1001202
+      e24KC2ThetaAboveNode1001203
+
+theorem e24KC2ThetaAboveNode100121 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell1001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell1001))
+    e24KC2ThetaAboveLeaf1001210 e24KC2ThetaAboveLeaf1001211 e24KC2ThetaAboveNode1001212
+      e24KC2ThetaAboveNode1001213
+
+theorem e24KC2ThetaAboveNode100122 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell1001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell1001))
+    e24KC2ThetaAboveNode1001220 e24KC2ThetaAboveNode1001221 e24KC2ThetaAboveNode1001222
+      e24KC2ThetaAboveNode1001223
+
+theorem e24KC2ThetaAboveNode100123 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell1001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell1001))
+    e24KC2ThetaAboveNode1001230 e24KC2ThetaAboveNode1001231 e24KC2ThetaAboveNode1001232
+      e24KC2ThetaAboveNode1001233
+
+theorem e24KC2ThetaAboveNode100130 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell1001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell1001))
+    e24KC2ThetaAboveLeaf1001300 e24KC2ThetaAboveLeaf1001301 e24KC2ThetaAboveNode1001302
+      e24KC2ThetaAboveNode1001303
+
+theorem e24KC2ThetaAboveNode100131 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell1001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell1001))
+    e24KC2ThetaAboveLeaf1001310 e24KC2ThetaAboveLeaf1001311 e24KC2ThetaAboveNode1001312
+      e24KC2ThetaAboveNode1001313
+
+theorem e24KC2ThetaAboveNode100132 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell1001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell1001))
+    e24KC2ThetaAboveNode1001320 e24KC2ThetaAboveNode1001321 e24KC2ThetaAboveNode1001322
+      e24KC2ThetaAboveNode1001323
+
+theorem e24KC2ThetaAboveNode100133 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell1001)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell1001))
+    e24KC2ThetaAboveNode1001330 e24KC2ThetaAboveNode1001331 e24KC2ThetaAboveNode1001332
+      e24KC2ThetaAboveNode1001333
+
+theorem e24KC2ThetaAboveNode100200 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1002)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell1002))
+    e24KC2ThetaAboveLeaf1002000 e24KC2ThetaAboveLeaf1002001 e24KC2ThetaAboveLeaf1002002
+      e24KC2ThetaAboveLeaf1002003
+
+theorem e24KC2ThetaAboveNode100201 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1002)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell1002))
+    e24KC2ThetaAboveLeaf1002010 e24KC2ThetaAboveLeaf1002011 e24KC2ThetaAboveLeaf1002012
+      e24KC2ThetaAboveLeaf1002013
+
+theorem e24KC2ThetaAboveNode100210 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1002)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell1002))
+    e24KC2ThetaAboveLeaf1002100 e24KC2ThetaAboveLeaf1002101 e24KC2ThetaAboveLeaf1002102
+      e24KC2ThetaAboveLeaf1002103
+
+theorem e24KC2ThetaAboveNode100211 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1002)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell1002))
+    e24KC2ThetaAboveLeaf1002110 e24KC2ThetaAboveLeaf1002111 e24KC2ThetaAboveLeaf1002112
+      e24KC2ThetaAboveLeaf1002113
+
+theorem e24KC2ThetaAboveNode100300 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1003)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell1003))
+    e24KC2ThetaAboveLeaf1003000 e24KC2ThetaAboveLeaf1003001 e24KC2ThetaAboveLeaf1003002
+      e24KC2ThetaAboveLeaf1003003
+
+theorem e24KC2ThetaAboveNode100301 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1003)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell1003))
+    e24KC2ThetaAboveLeaf1003010 e24KC2ThetaAboveLeaf1003011 e24KC2ThetaAboveLeaf1003012
+      e24KC2ThetaAboveLeaf1003013
+
+theorem e24KC2ThetaAboveNode100310 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1003)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell1003))
+    e24KC2ThetaAboveLeaf1003100 e24KC2ThetaAboveLeaf1003101 e24KC2ThetaAboveLeaf1003102
+      e24KC2ThetaAboveLeaf1003103
+
+theorem e24KC2ThetaAboveNode100311 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1003)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell1003))
+    e24KC2ThetaAboveLeaf1003110 e24KC2ThetaAboveLeaf1003111 e24KC2ThetaAboveLeaf1003112
+      e24KC2ThetaAboveLeaf1003113
+
+theorem e24KC2ThetaAboveNode101020 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell1010))
+    e24KC2ThetaAboveLeaf1010200 e24KC2ThetaAboveLeaf1010201 e24KC2ThetaAboveNode1010202
+      e24KC2ThetaAboveNode1010203
+
+theorem e24KC2ThetaAboveNode101021 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell1010))
+    e24KC2ThetaAboveLeaf1010210 e24KC2ThetaAboveLeaf1010211 e24KC2ThetaAboveLeaf1010212
+      e24KC2ThetaAboveLeaf1010213
+
+theorem e24KC2ThetaAboveNode101022 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell1010))
+    e24KC2ThetaAboveNode1010220 e24KC2ThetaAboveNode1010221 e24KC2ThetaAboveNode1010222
+      e24KC2ThetaAboveNode1010223
+
+theorem e24KC2ThetaAboveNode101023 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell1010))
+    e24KC2ThetaAboveNode1010230 e24KC2ThetaAboveNode1010231 e24KC2ThetaAboveNode1010232
+      e24KC2ThetaAboveNode1010233
+
+theorem e24KC2ThetaAboveNode101030 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell1010))
+    e24KC2ThetaAboveLeaf1010300 e24KC2ThetaAboveLeaf1010301 e24KC2ThetaAboveLeaf1010302
+      e24KC2ThetaAboveLeaf1010303
+
+theorem e24KC2ThetaAboveNode101031 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell1010))
+    e24KC2ThetaAboveLeaf1010310 e24KC2ThetaAboveLeaf1010311 e24KC2ThetaAboveLeaf1010312
+      e24KC2ThetaAboveLeaf1010313
+
+theorem e24KC2ThetaAboveNode101032 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell1010))
+    e24KC2ThetaAboveNode1010320 e24KC2ThetaAboveNode1010321 e24KC2ThetaAboveNode1010322
+      e24KC2ThetaAboveNode1010323
+
+theorem e24KC2ThetaAboveNode101033 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell1010)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell1010))
+    e24KC2ThetaAboveNode1010330 e24KC2ThetaAboveNode1010331 e24KC2ThetaAboveNode1010332
+      e24KC2ThetaAboveNode1010333
+
+theorem e24KC2ThetaAboveNode101120 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell1011))
+    e24KC2ThetaAboveLeaf1011200 e24KC2ThetaAboveLeaf1011201 e24KC2ThetaAboveLeaf1011202
+      e24KC2ThetaAboveLeaf1011203
+
+theorem e24KC2ThetaAboveNode101121 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell1011))
+    e24KC2ThetaAboveLeaf1011210 e24KC2ThetaAboveLeaf1011211 e24KC2ThetaAboveLeaf1011212
+      e24KC2ThetaAboveLeaf1011213
+
+theorem e24KC2ThetaAboveNode101122 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell1011))
+    e24KC2ThetaAboveNode1011220 e24KC2ThetaAboveNode1011221 e24KC2ThetaAboveNode1011222
+      e24KC2ThetaAboveNode1011223
+
+theorem e24KC2ThetaAboveNode101123 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell1011))
+    e24KC2ThetaAboveNode1011230 e24KC2ThetaAboveNode1011231 e24KC2ThetaAboveNode1011232
+      e24KC2ThetaAboveNode1011233
+
+theorem e24KC2ThetaAboveNode101130 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell1011))
+    e24KC2ThetaAboveLeaf1011300 e24KC2ThetaAboveLeaf1011301 e24KC2ThetaAboveLeaf1011302
+      e24KC2ThetaAboveLeaf1011303
+
+theorem e24KC2ThetaAboveNode101131 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell1011))
+    e24KC2ThetaAboveLeaf1011310 e24KC2ThetaAboveLeaf1011311 e24KC2ThetaAboveLeaf1011312
+      e24KC2ThetaAboveLeaf1011313
+
+theorem e24KC2ThetaAboveNode101132 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell1011))
+    e24KC2ThetaAboveNode1011320 e24KC2ThetaAboveNode1011321 e24KC2ThetaAboveNode1011322
+      e24KC2ThetaAboveNode1011323
+
+theorem e24KC2ThetaAboveNode101133 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell1011)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell1011))
+    e24KC2ThetaAboveNode1011330 e24KC2ThetaAboveNode1011331 e24KC2ThetaAboveNode1011332
+      e24KC2ThetaAboveNode1011333
+
+theorem e24KC2ThetaAboveNode101200 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1012)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell1012))
+    e24KC2ThetaAboveLeaf1012000 e24KC2ThetaAboveLeaf1012001 e24KC2ThetaAboveLeaf1012002
+      e24KC2ThetaAboveLeaf1012003
+
+theorem e24KC2ThetaAboveNode101201 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1012)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell1012))
+    e24KC2ThetaAboveLeaf1012010 e24KC2ThetaAboveLeaf1012011 e24KC2ThetaAboveLeaf1012012
+      e24KC2ThetaAboveLeaf1012013
+
+theorem e24KC2ThetaAboveNode101210 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1012)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell1012))
+    e24KC2ThetaAboveLeaf1012100 e24KC2ThetaAboveLeaf1012101 e24KC2ThetaAboveLeaf1012102
+      e24KC2ThetaAboveLeaf1012103
+
+theorem e24KC2ThetaAboveNode101211 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1012)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell1012))
+    e24KC2ThetaAboveLeaf1012110 e24KC2ThetaAboveLeaf1012111 e24KC2ThetaAboveLeaf1012112
+      e24KC2ThetaAboveLeaf1012113
+
+theorem e24KC2ThetaAboveNode101300 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1013)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell1013))
+    e24KC2ThetaAboveLeaf1013000 e24KC2ThetaAboveLeaf1013001 e24KC2ThetaAboveLeaf1013002
+      e24KC2ThetaAboveLeaf1013003
+
+theorem e24KC2ThetaAboveNode101301 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1013)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell1013))
+    e24KC2ThetaAboveLeaf1013010 e24KC2ThetaAboveLeaf1013011 e24KC2ThetaAboveLeaf1013012
+      e24KC2ThetaAboveLeaf1013013
+
+theorem e24KC2ThetaAboveNode101310 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1013)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell1013))
+    e24KC2ThetaAboveLeaf1013100 e24KC2ThetaAboveLeaf1013101 e24KC2ThetaAboveLeaf1013102
+      e24KC2ThetaAboveLeaf1013103
+
+theorem e24KC2ThetaAboveNode101311 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1013)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell1013))
+    e24KC2ThetaAboveLeaf1013110 e24KC2ThetaAboveLeaf1013111 e24KC2ThetaAboveLeaf1013112
+      e24KC2ThetaAboveLeaf1013113
+
+theorem e24KC2ThetaAboveNode110020 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell1100))
+    e24KC2ThetaAboveLeaf1100200 e24KC2ThetaAboveLeaf1100201 e24KC2ThetaAboveLeaf1100202
+      e24KC2ThetaAboveLeaf1100203
+
+theorem e24KC2ThetaAboveNode110021 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell1100))
+    e24KC2ThetaAboveLeaf1100210 e24KC2ThetaAboveLeaf1100211 e24KC2ThetaAboveLeaf1100212
+      e24KC2ThetaAboveLeaf1100213
+
+theorem e24KC2ThetaAboveNode110022 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell1100))
+    e24KC2ThetaAboveNode1100220 e24KC2ThetaAboveNode1100221 e24KC2ThetaAboveNode1100222
+      e24KC2ThetaAboveNode1100223
+
+theorem e24KC2ThetaAboveNode110023 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell1100))
+    e24KC2ThetaAboveNode1100230 e24KC2ThetaAboveNode1100231 e24KC2ThetaAboveNode1100232
+      e24KC2ThetaAboveNode1100233
+
+theorem e24KC2ThetaAboveNode110030 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell1100))
+    e24KC2ThetaAboveLeaf1100300 e24KC2ThetaAboveLeaf1100301 e24KC2ThetaAboveLeaf1100302
+      e24KC2ThetaAboveLeaf1100303
+
+theorem e24KC2ThetaAboveNode110031 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell1100))
+    e24KC2ThetaAboveLeaf1100310 e24KC2ThetaAboveLeaf1100311 e24KC2ThetaAboveLeaf1100312
+      e24KC2ThetaAboveLeaf1100313
+
+theorem e24KC2ThetaAboveNode110032 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell1100))
+    e24KC2ThetaAboveNode1100320 e24KC2ThetaAboveNode1100321 e24KC2ThetaAboveNode1100322
+      e24KC2ThetaAboveNode1100323
+
+theorem e24KC2ThetaAboveNode110033 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell1100)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell1100))
+    e24KC2ThetaAboveNode1100330 e24KC2ThetaAboveNode1100331 e24KC2ThetaAboveNode1100332
+      e24KC2ThetaAboveNode1100333
+
+theorem e24KC2ThetaAboveNode110120 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell1101))
+    e24KC2ThetaAboveLeaf1101200 e24KC2ThetaAboveLeaf1101201 e24KC2ThetaAboveLeaf1101202
+      e24KC2ThetaAboveLeaf1101203
+
+theorem e24KC2ThetaAboveNode110121 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell1101))
+    e24KC2ThetaAboveLeaf1101210 e24KC2ThetaAboveLeaf1101211 e24KC2ThetaAboveLeaf1101212
+      e24KC2ThetaAboveLeaf1101213
+
+theorem e24KC2ThetaAboveNode110122 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell1101))
+    e24KC2ThetaAboveNode1101220 e24KC2ThetaAboveNode1101221 e24KC2ThetaAboveNode1101222
+      e24KC2ThetaAboveNode1101223
+
+theorem e24KC2ThetaAboveNode110123 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell1101))
+    e24KC2ThetaAboveNode1101230 e24KC2ThetaAboveNode1101231 e24KC2ThetaAboveNode1101232
+      e24KC2ThetaAboveNode1101233
+
+theorem e24KC2ThetaAboveNode110130 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell1101))
+    e24KC2ThetaAboveLeaf1101300 e24KC2ThetaAboveLeaf1101301 e24KC2ThetaAboveLeaf1101302
+      e24KC2ThetaAboveLeaf1101303
+
+theorem e24KC2ThetaAboveNode110131 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell1101))
+    e24KC2ThetaAboveLeaf1101310 e24KC2ThetaAboveLeaf1101311 e24KC2ThetaAboveLeaf1101312
+      e24KC2ThetaAboveLeaf1101313
+
+theorem e24KC2ThetaAboveNode110132 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell1101))
+    e24KC2ThetaAboveNode1101320 e24KC2ThetaAboveNode1101321 e24KC2ThetaAboveNode1101322
+      e24KC2ThetaAboveNode1101323
+
+theorem e24KC2ThetaAboveNode110133 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell1101)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell1101))
+    e24KC2ThetaAboveNode1101330 e24KC2ThetaAboveNode1101331 e24KC2ThetaAboveNode1101332
+      e24KC2ThetaAboveNode1101333
+
+theorem e24KC2ThetaAboveNode110200 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1102)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell1102))
+    e24KC2ThetaAboveLeaf1102000 e24KC2ThetaAboveLeaf1102001 e24KC2ThetaAboveLeaf1102002
+      e24KC2ThetaAboveLeaf1102003
+
+theorem e24KC2ThetaAboveNode110201 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1102)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell1102))
+    e24KC2ThetaAboveLeaf1102010 e24KC2ThetaAboveLeaf1102011 e24KC2ThetaAboveLeaf1102012
+      e24KC2ThetaAboveLeaf1102013
+
+theorem e24KC2ThetaAboveNode110210 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1102)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell1102))
+    e24KC2ThetaAboveLeaf1102100 e24KC2ThetaAboveLeaf1102101 e24KC2ThetaAboveLeaf1102102
+      e24KC2ThetaAboveLeaf1102103
+
+theorem e24KC2ThetaAboveNode110211 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1102)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell1102))
+    e24KC2ThetaAboveLeaf1102110 e24KC2ThetaAboveLeaf1102111 e24KC2ThetaAboveLeaf1102112
+      e24KC2ThetaAboveLeaf1102113
+
+theorem e24KC2ThetaAboveNode110300 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1103)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell1103))
+    e24KC2ThetaAboveLeaf1103000 e24KC2ThetaAboveLeaf1103001 e24KC2ThetaAboveLeaf1103002
+      e24KC2ThetaAboveLeaf1103003
+
+theorem e24KC2ThetaAboveNode110301 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1103)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell1103))
+    e24KC2ThetaAboveLeaf1103010 e24KC2ThetaAboveLeaf1103011 e24KC2ThetaAboveLeaf1103012
+      e24KC2ThetaAboveLeaf1103013
+
+theorem e24KC2ThetaAboveNode110310 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1103)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell1103))
+    e24KC2ThetaAboveLeaf1103100 e24KC2ThetaAboveLeaf1103101 e24KC2ThetaAboveLeaf1103102
+      e24KC2ThetaAboveLeaf1103103
+
+theorem e24KC2ThetaAboveNode110311 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1103)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell1103))
+    e24KC2ThetaAboveLeaf1103110 e24KC2ThetaAboveLeaf1103111 e24KC2ThetaAboveLeaf1103112
+      e24KC2ThetaAboveLeaf1103113
+
+theorem e24KC2ThetaAboveNode111020 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell1110))
+    e24KC2ThetaAboveLeaf1110200 e24KC2ThetaAboveLeaf1110201 e24KC2ThetaAboveLeaf1110202
+      e24KC2ThetaAboveLeaf1110203
+
+theorem e24KC2ThetaAboveNode111021 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell1110))
+    e24KC2ThetaAboveLeaf1110210 e24KC2ThetaAboveLeaf1110211 e24KC2ThetaAboveLeaf1110212
+      e24KC2ThetaAboveLeaf1110213
+
+theorem e24KC2ThetaAboveNode111022 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell1110))
+    e24KC2ThetaAboveNode1110220 e24KC2ThetaAboveNode1110221 e24KC2ThetaAboveNode1110222
+      e24KC2ThetaAboveNode1110223
+
+theorem e24KC2ThetaAboveNode111023 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell1110))
+    e24KC2ThetaAboveNode1110230 e24KC2ThetaAboveNode1110231 e24KC2ThetaAboveNode1110232
+      e24KC2ThetaAboveNode1110233
+
+theorem e24KC2ThetaAboveNode111030 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell1110))
+    e24KC2ThetaAboveLeaf1110300 e24KC2ThetaAboveLeaf1110301 e24KC2ThetaAboveLeaf1110302
+      e24KC2ThetaAboveLeaf1110303
+
+theorem e24KC2ThetaAboveNode111031 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell1110))
+    e24KC2ThetaAboveLeaf1110310 e24KC2ThetaAboveLeaf1110311 e24KC2ThetaAboveLeaf1110312
+      e24KC2ThetaAboveLeaf1110313
+
+theorem e24KC2ThetaAboveNode111032 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell1110))
+    e24KC2ThetaAboveNode1110320 e24KC2ThetaAboveNode1110321 e24KC2ThetaAboveNode1110322
+      e24KC2ThetaAboveNode1110323
+
+theorem e24KC2ThetaAboveNode111033 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell1110)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell1110))
+    e24KC2ThetaAboveNode1110330 e24KC2ThetaAboveNode1110331 e24KC2ThetaAboveNode1110332
+      e24KC2ThetaAboveNode1110333
+
+theorem e24KC2ThetaAboveNode111120 :
+    adaptiveCoverCheck 13 (childLL (childHL thetaAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHL thetaAboveCell1111))
+    e24KC2ThetaAboveLeaf1111200 e24KC2ThetaAboveLeaf1111201 e24KC2ThetaAboveLeaf1111202
+      e24KC2ThetaAboveLeaf1111203
+
+theorem e24KC2ThetaAboveNode111121 :
+    adaptiveCoverCheck 13 (childLH (childHL thetaAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHL thetaAboveCell1111))
+    e24KC2ThetaAboveLeaf1111210 e24KC2ThetaAboveLeaf1111211 e24KC2ThetaAboveLeaf1111212
+      e24KC2ThetaAboveLeaf1111213
+
+theorem e24KC2ThetaAboveNode111122 :
+    adaptiveCoverCheck 13 (childHL (childHL thetaAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHL thetaAboveCell1111))
+    e24KC2ThetaAboveNode1111220 e24KC2ThetaAboveNode1111221 e24KC2ThetaAboveNode1111222
+      e24KC2ThetaAboveNode1111223
+
+theorem e24KC2ThetaAboveNode111123 :
+    adaptiveCoverCheck 13 (childHH (childHL thetaAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHL thetaAboveCell1111))
+    e24KC2ThetaAboveNode1111230 e24KC2ThetaAboveNode1111231 e24KC2ThetaAboveNode1111232
+      e24KC2ThetaAboveNode1111233
+
+theorem e24KC2ThetaAboveNode111130 :
+    adaptiveCoverCheck 13 (childLL (childHH thetaAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childHH thetaAboveCell1111))
+    e24KC2ThetaAboveLeaf1111300 e24KC2ThetaAboveLeaf1111301 e24KC2ThetaAboveLeaf1111302
+      e24KC2ThetaAboveLeaf1111303
+
+theorem e24KC2ThetaAboveNode111131 :
+    adaptiveCoverCheck 13 (childLH (childHH thetaAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childHH thetaAboveCell1111))
+    e24KC2ThetaAboveLeaf1111310 e24KC2ThetaAboveLeaf1111311 e24KC2ThetaAboveLeaf1111312
+      e24KC2ThetaAboveLeaf1111313
+
+theorem e24KC2ThetaAboveNode111132 :
+    adaptiveCoverCheck 13 (childHL (childHH thetaAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHL (childHH thetaAboveCell1111))
+    e24KC2ThetaAboveNode1111320 e24KC2ThetaAboveNode1111321 e24KC2ThetaAboveNode1111322
+      e24KC2ThetaAboveNode1111323
+
+theorem e24KC2ThetaAboveNode111133 :
+    adaptiveCoverCheck 13 (childHH (childHH thetaAboveCell1111)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childHH (childHH thetaAboveCell1111))
+    e24KC2ThetaAboveNode1111330 e24KC2ThetaAboveNode1111331 e24KC2ThetaAboveNode1111332
+      e24KC2ThetaAboveNode1111333
+
+theorem e24KC2ThetaAboveNode111200 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1112)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell1112))
+    e24KC2ThetaAboveLeaf1112000 e24KC2ThetaAboveLeaf1112001 e24KC2ThetaAboveLeaf1112002
+      e24KC2ThetaAboveLeaf1112003
+
+theorem e24KC2ThetaAboveNode111201 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1112)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell1112))
+    e24KC2ThetaAboveLeaf1112010 e24KC2ThetaAboveLeaf1112011 e24KC2ThetaAboveLeaf1112012
+      e24KC2ThetaAboveLeaf1112013
+
+theorem e24KC2ThetaAboveNode111210 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1112)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell1112))
+    e24KC2ThetaAboveLeaf1112100 e24KC2ThetaAboveLeaf1112101 e24KC2ThetaAboveLeaf1112102
+      e24KC2ThetaAboveLeaf1112103
+
+theorem e24KC2ThetaAboveNode111211 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1112)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell1112))
+    e24KC2ThetaAboveLeaf1112110 e24KC2ThetaAboveLeaf1112111 e24KC2ThetaAboveLeaf1112112
+      e24KC2ThetaAboveLeaf1112113
+
+theorem e24KC2ThetaAboveNode111300 :
+    adaptiveCoverCheck 13 (childLL (childLL thetaAboveCell1113)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLL thetaAboveCell1113))
+    e24KC2ThetaAboveLeaf1113000 e24KC2ThetaAboveLeaf1113001 e24KC2ThetaAboveLeaf1113002
+      e24KC2ThetaAboveLeaf1113003
+
+theorem e24KC2ThetaAboveNode111301 :
+    adaptiveCoverCheck 13 (childLH (childLL thetaAboveCell1113)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLL thetaAboveCell1113))
+    e24KC2ThetaAboveLeaf1113010 e24KC2ThetaAboveLeaf1113011 e24KC2ThetaAboveLeaf1113012
+      e24KC2ThetaAboveLeaf1113013
+
+theorem e24KC2ThetaAboveNode111310 :
+    adaptiveCoverCheck 13 (childLL (childLH thetaAboveCell1113)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLL (childLH thetaAboveCell1113))
+    e24KC2ThetaAboveLeaf1113100 e24KC2ThetaAboveLeaf1113101 e24KC2ThetaAboveLeaf1113102
+      e24KC2ThetaAboveLeaf1113103
+
+theorem e24KC2ThetaAboveNode111311 :
+    adaptiveCoverCheck 13 (childLH (childLH thetaAboveCell1113)) = true :=
+  adaptiveCoverCheck_succ_of_children 12 (childLH (childLH thetaAboveCell1113))
+    e24KC2ThetaAboveLeaf1113110 e24KC2ThetaAboveLeaf1113111 e24KC2ThetaAboveLeaf1113112
+      e24KC2ThetaAboveLeaf1113113
+
+theorem e24KC2ThetaAboveNode00000 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0000) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0000)
+    e24KC2ThetaAboveLeaf000000 e24KC2ThetaAboveLeaf000001 e24KC2ThetaAboveLeaf000002
+      e24KC2ThetaAboveLeaf000003
+
+theorem e24KC2ThetaAboveNode00001 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0000) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0000)
+    e24KC2ThetaAboveLeaf000010 e24KC2ThetaAboveLeaf000011 e24KC2ThetaAboveLeaf000012
+      e24KC2ThetaAboveLeaf000013
+
+theorem e24KC2ThetaAboveNode00002 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0000) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell0000)
+    e24KC2ThetaAboveNode000020 e24KC2ThetaAboveNode000021 e24KC2ThetaAboveNode000022
+      e24KC2ThetaAboveNode000023
+
+theorem e24KC2ThetaAboveNode00003 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0000) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell0000)
+    e24KC2ThetaAboveNode000030 e24KC2ThetaAboveNode000031 e24KC2ThetaAboveNode000032
+      e24KC2ThetaAboveNode000033
+
+theorem e24KC2ThetaAboveNode00010 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0001) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0001)
+    e24KC2ThetaAboveLeaf000100 e24KC2ThetaAboveLeaf000101 e24KC2ThetaAboveLeaf000102
+      e24KC2ThetaAboveLeaf000103
+
+theorem e24KC2ThetaAboveNode00011 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0001) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0001)
+    e24KC2ThetaAboveLeaf000110 e24KC2ThetaAboveLeaf000111 e24KC2ThetaAboveLeaf000112
+      e24KC2ThetaAboveLeaf000113
+
+theorem e24KC2ThetaAboveNode00012 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0001) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell0001)
+    e24KC2ThetaAboveNode000120 e24KC2ThetaAboveNode000121 e24KC2ThetaAboveNode000122
+      e24KC2ThetaAboveNode000123
+
+theorem e24KC2ThetaAboveNode00013 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0001) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell0001)
+    e24KC2ThetaAboveNode000130 e24KC2ThetaAboveNode000131 e24KC2ThetaAboveNode000132
+      e24KC2ThetaAboveNode000133
+
+theorem e24KC2ThetaAboveNode00020 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0002) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0002)
+    e24KC2ThetaAboveNode000200 e24KC2ThetaAboveNode000201 e24KC2ThetaAboveLeaf000202
+      e24KC2ThetaAboveLeaf000203
+
+theorem e24KC2ThetaAboveNode00021 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0002) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0002)
+    e24KC2ThetaAboveNode000210 e24KC2ThetaAboveNode000211 e24KC2ThetaAboveLeaf000212
+      e24KC2ThetaAboveLeaf000213
+
+theorem e24KC2ThetaAboveNode00030 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0003) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0003)
+    e24KC2ThetaAboveNode000300 e24KC2ThetaAboveNode000301 e24KC2ThetaAboveLeaf000302
+      e24KC2ThetaAboveLeaf000303
+
+theorem e24KC2ThetaAboveNode00031 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0003) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0003)
+    e24KC2ThetaAboveNode000310 e24KC2ThetaAboveNode000311 e24KC2ThetaAboveLeaf000312
+      e24KC2ThetaAboveLeaf000313
+
+theorem e24KC2ThetaAboveNode00100 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0010) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0010)
+    e24KC2ThetaAboveLeaf001000 e24KC2ThetaAboveLeaf001001 e24KC2ThetaAboveLeaf001002
+      e24KC2ThetaAboveLeaf001003
+
+theorem e24KC2ThetaAboveNode00101 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0010) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0010)
+    e24KC2ThetaAboveLeaf001010 e24KC2ThetaAboveLeaf001011 e24KC2ThetaAboveLeaf001012
+      e24KC2ThetaAboveLeaf001013
+
+theorem e24KC2ThetaAboveNode00102 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0010) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell0010)
+    e24KC2ThetaAboveNode001020 e24KC2ThetaAboveNode001021 e24KC2ThetaAboveNode001022
+      e24KC2ThetaAboveNode001023
+
+theorem e24KC2ThetaAboveNode00103 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0010) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell0010)
+    e24KC2ThetaAboveNode001030 e24KC2ThetaAboveNode001031 e24KC2ThetaAboveNode001032
+      e24KC2ThetaAboveNode001033
+
+theorem e24KC2ThetaAboveNode00110 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0011) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0011)
+    e24KC2ThetaAboveLeaf001100 e24KC2ThetaAboveLeaf001101 e24KC2ThetaAboveLeaf001102
+      e24KC2ThetaAboveLeaf001103
+
+theorem e24KC2ThetaAboveNode00111 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0011) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0011)
+    e24KC2ThetaAboveLeaf001110 e24KC2ThetaAboveLeaf001111 e24KC2ThetaAboveLeaf001112
+      e24KC2ThetaAboveLeaf001113
+
+theorem e24KC2ThetaAboveNode00112 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0011) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell0011)
+    e24KC2ThetaAboveNode001120 e24KC2ThetaAboveNode001121 e24KC2ThetaAboveNode001122
+      e24KC2ThetaAboveNode001123
+
+theorem e24KC2ThetaAboveNode00113 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0011) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell0011)
+    e24KC2ThetaAboveNode001130 e24KC2ThetaAboveNode001131 e24KC2ThetaAboveNode001132
+      e24KC2ThetaAboveNode001133
+
+theorem e24KC2ThetaAboveNode00120 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0012) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0012)
+    e24KC2ThetaAboveNode001200 e24KC2ThetaAboveNode001201 e24KC2ThetaAboveLeaf001202
+      e24KC2ThetaAboveLeaf001203
+
+theorem e24KC2ThetaAboveNode00121 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0012) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0012)
+    e24KC2ThetaAboveNode001210 e24KC2ThetaAboveNode001211 e24KC2ThetaAboveLeaf001212
+      e24KC2ThetaAboveLeaf001213
+
+theorem e24KC2ThetaAboveNode00130 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0013) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0013)
+    e24KC2ThetaAboveNode001300 e24KC2ThetaAboveNode001301 e24KC2ThetaAboveLeaf001302
+      e24KC2ThetaAboveLeaf001303
+
+theorem e24KC2ThetaAboveNode00131 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0013) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0013)
+    e24KC2ThetaAboveNode001310 e24KC2ThetaAboveNode001311 e24KC2ThetaAboveLeaf001312
+      e24KC2ThetaAboveLeaf001313
+
+theorem e24KC2ThetaAboveNode01000 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0100) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0100)
+    e24KC2ThetaAboveLeaf010000 e24KC2ThetaAboveLeaf010001 e24KC2ThetaAboveLeaf010002
+      e24KC2ThetaAboveLeaf010003
+
+theorem e24KC2ThetaAboveNode01001 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0100) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0100)
+    e24KC2ThetaAboveLeaf010010 e24KC2ThetaAboveLeaf010011 e24KC2ThetaAboveLeaf010012
+      e24KC2ThetaAboveLeaf010013
+
+theorem e24KC2ThetaAboveNode01002 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0100) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell0100)
+    e24KC2ThetaAboveNode010020 e24KC2ThetaAboveNode010021 e24KC2ThetaAboveNode010022
+      e24KC2ThetaAboveNode010023
+
+theorem e24KC2ThetaAboveNode01003 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0100) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell0100)
+    e24KC2ThetaAboveNode010030 e24KC2ThetaAboveNode010031 e24KC2ThetaAboveNode010032
+      e24KC2ThetaAboveNode010033
+
+theorem e24KC2ThetaAboveNode01010 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0101) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0101)
+    e24KC2ThetaAboveLeaf010100 e24KC2ThetaAboveLeaf010101 e24KC2ThetaAboveLeaf010102
+      e24KC2ThetaAboveLeaf010103
+
+theorem e24KC2ThetaAboveNode01011 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0101) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0101)
+    e24KC2ThetaAboveLeaf010110 e24KC2ThetaAboveLeaf010111 e24KC2ThetaAboveLeaf010112
+      e24KC2ThetaAboveLeaf010113
+
+theorem e24KC2ThetaAboveNode01012 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0101) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell0101)
+    e24KC2ThetaAboveNode010120 e24KC2ThetaAboveNode010121 e24KC2ThetaAboveNode010122
+      e24KC2ThetaAboveNode010123
+
+theorem e24KC2ThetaAboveNode01013 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0101) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell0101)
+    e24KC2ThetaAboveNode010130 e24KC2ThetaAboveNode010131 e24KC2ThetaAboveNode010132
+      e24KC2ThetaAboveNode010133
+
+theorem e24KC2ThetaAboveNode01020 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0102) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0102)
+    e24KC2ThetaAboveNode010200 e24KC2ThetaAboveNode010201 e24KC2ThetaAboveLeaf010202
+      e24KC2ThetaAboveLeaf010203
+
+theorem e24KC2ThetaAboveNode01021 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0102) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0102)
+    e24KC2ThetaAboveNode010210 e24KC2ThetaAboveNode010211 e24KC2ThetaAboveLeaf010212
+      e24KC2ThetaAboveLeaf010213
+
+theorem e24KC2ThetaAboveNode01030 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0103) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0103)
+    e24KC2ThetaAboveNode010300 e24KC2ThetaAboveNode010301 e24KC2ThetaAboveLeaf010302
+      e24KC2ThetaAboveLeaf010303
+
+theorem e24KC2ThetaAboveNode01031 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0103) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0103)
+    e24KC2ThetaAboveNode010310 e24KC2ThetaAboveNode010311 e24KC2ThetaAboveLeaf010312
+      e24KC2ThetaAboveLeaf010313
+
+theorem e24KC2ThetaAboveNode01100 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0110) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0110)
+    e24KC2ThetaAboveLeaf011000 e24KC2ThetaAboveLeaf011001 e24KC2ThetaAboveLeaf011002
+      e24KC2ThetaAboveLeaf011003
+
+theorem e24KC2ThetaAboveNode01101 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0110) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0110)
+    e24KC2ThetaAboveLeaf011010 e24KC2ThetaAboveLeaf011011 e24KC2ThetaAboveLeaf011012
+      e24KC2ThetaAboveLeaf011013
+
+theorem e24KC2ThetaAboveNode01102 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0110) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell0110)
+    e24KC2ThetaAboveNode011020 e24KC2ThetaAboveNode011021 e24KC2ThetaAboveNode011022
+      e24KC2ThetaAboveNode011023
+
+theorem e24KC2ThetaAboveNode01103 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0110) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell0110)
+    e24KC2ThetaAboveNode011030 e24KC2ThetaAboveNode011031 e24KC2ThetaAboveNode011032
+      e24KC2ThetaAboveNode011033
+
+theorem e24KC2ThetaAboveNode01110 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0111) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0111)
+    e24KC2ThetaAboveLeaf011100 e24KC2ThetaAboveLeaf011101 e24KC2ThetaAboveLeaf011102
+      e24KC2ThetaAboveLeaf011103
+
+theorem e24KC2ThetaAboveNode01111 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0111) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0111)
+    e24KC2ThetaAboveLeaf011110 e24KC2ThetaAboveLeaf011111 e24KC2ThetaAboveLeaf011112
+      e24KC2ThetaAboveLeaf011113
+
+theorem e24KC2ThetaAboveNode01112 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell0111) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell0111)
+    e24KC2ThetaAboveNode011120 e24KC2ThetaAboveNode011121 e24KC2ThetaAboveNode011122
+      e24KC2ThetaAboveNode011123
+
+theorem e24KC2ThetaAboveNode01113 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell0111) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell0111)
+    e24KC2ThetaAboveNode011130 e24KC2ThetaAboveNode011131 e24KC2ThetaAboveNode011132
+      e24KC2ThetaAboveNode011133
+
+theorem e24KC2ThetaAboveNode01120 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0112) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0112)
+    e24KC2ThetaAboveNode011200 e24KC2ThetaAboveNode011201 e24KC2ThetaAboveLeaf011202
+      e24KC2ThetaAboveLeaf011203
+
+theorem e24KC2ThetaAboveNode01121 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0112) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0112)
+    e24KC2ThetaAboveNode011210 e24KC2ThetaAboveNode011211 e24KC2ThetaAboveLeaf011212
+      e24KC2ThetaAboveLeaf011213
+
+theorem e24KC2ThetaAboveNode01130 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell0113) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell0113)
+    e24KC2ThetaAboveNode011300 e24KC2ThetaAboveNode011301 e24KC2ThetaAboveLeaf011302
+      e24KC2ThetaAboveLeaf011303
+
+theorem e24KC2ThetaAboveNode01131 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell0113) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell0113)
+    e24KC2ThetaAboveNode011310 e24KC2ThetaAboveNode011311 e24KC2ThetaAboveLeaf011312
+      e24KC2ThetaAboveLeaf011313
+
+theorem e24KC2ThetaAboveNode10000 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1000) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1000)
+    e24KC2ThetaAboveLeaf100000 e24KC2ThetaAboveLeaf100001 e24KC2ThetaAboveLeaf100002
+      e24KC2ThetaAboveLeaf100003
+
+theorem e24KC2ThetaAboveNode10001 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1000) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1000)
+    e24KC2ThetaAboveLeaf100010 e24KC2ThetaAboveLeaf100011 e24KC2ThetaAboveLeaf100012
+      e24KC2ThetaAboveLeaf100013
+
+theorem e24KC2ThetaAboveNode10002 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1000) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell1000)
+    e24KC2ThetaAboveNode100020 e24KC2ThetaAboveNode100021 e24KC2ThetaAboveNode100022
+      e24KC2ThetaAboveNode100023
+
+theorem e24KC2ThetaAboveNode10003 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1000) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell1000)
+    e24KC2ThetaAboveNode100030 e24KC2ThetaAboveNode100031 e24KC2ThetaAboveNode100032
+      e24KC2ThetaAboveNode100033
+
+theorem e24KC2ThetaAboveNode10010 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1001) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1001)
+    e24KC2ThetaAboveLeaf100100 e24KC2ThetaAboveLeaf100101 e24KC2ThetaAboveLeaf100102
+      e24KC2ThetaAboveLeaf100103
+
+theorem e24KC2ThetaAboveNode10011 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1001) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1001)
+    e24KC2ThetaAboveLeaf100110 e24KC2ThetaAboveLeaf100111 e24KC2ThetaAboveLeaf100112
+      e24KC2ThetaAboveLeaf100113
+
+theorem e24KC2ThetaAboveNode10012 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1001) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell1001)
+    e24KC2ThetaAboveNode100120 e24KC2ThetaAboveNode100121 e24KC2ThetaAboveNode100122
+      e24KC2ThetaAboveNode100123
+
+theorem e24KC2ThetaAboveNode10013 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1001) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell1001)
+    e24KC2ThetaAboveNode100130 e24KC2ThetaAboveNode100131 e24KC2ThetaAboveNode100132
+      e24KC2ThetaAboveNode100133
+
+theorem e24KC2ThetaAboveNode10020 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1002) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1002)
+    e24KC2ThetaAboveNode100200 e24KC2ThetaAboveNode100201 e24KC2ThetaAboveLeaf100202
+      e24KC2ThetaAboveLeaf100203
+
+theorem e24KC2ThetaAboveNode10021 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1002) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1002)
+    e24KC2ThetaAboveNode100210 e24KC2ThetaAboveNode100211 e24KC2ThetaAboveLeaf100212
+      e24KC2ThetaAboveLeaf100213
+
+theorem e24KC2ThetaAboveNode10030 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1003) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1003)
+    e24KC2ThetaAboveNode100300 e24KC2ThetaAboveNode100301 e24KC2ThetaAboveLeaf100302
+      e24KC2ThetaAboveLeaf100303
+
+theorem e24KC2ThetaAboveNode10031 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1003) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1003)
+    e24KC2ThetaAboveNode100310 e24KC2ThetaAboveNode100311 e24KC2ThetaAboveLeaf100312
+      e24KC2ThetaAboveLeaf100313
+
+theorem e24KC2ThetaAboveNode10100 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1010) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1010)
+    e24KC2ThetaAboveLeaf101000 e24KC2ThetaAboveLeaf101001 e24KC2ThetaAboveLeaf101002
+      e24KC2ThetaAboveLeaf101003
+
+theorem e24KC2ThetaAboveNode10101 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1010) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1010)
+    e24KC2ThetaAboveLeaf101010 e24KC2ThetaAboveLeaf101011 e24KC2ThetaAboveLeaf101012
+      e24KC2ThetaAboveLeaf101013
+
+theorem e24KC2ThetaAboveNode10102 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1010) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell1010)
+    e24KC2ThetaAboveNode101020 e24KC2ThetaAboveNode101021 e24KC2ThetaAboveNode101022
+      e24KC2ThetaAboveNode101023
+
+theorem e24KC2ThetaAboveNode10103 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1010) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell1010)
+    e24KC2ThetaAboveNode101030 e24KC2ThetaAboveNode101031 e24KC2ThetaAboveNode101032
+      e24KC2ThetaAboveNode101033
+
+theorem e24KC2ThetaAboveNode10110 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1011) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1011)
+    e24KC2ThetaAboveLeaf101100 e24KC2ThetaAboveLeaf101101 e24KC2ThetaAboveLeaf101102
+      e24KC2ThetaAboveLeaf101103
+
+theorem e24KC2ThetaAboveNode10111 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1011) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1011)
+    e24KC2ThetaAboveLeaf101110 e24KC2ThetaAboveLeaf101111 e24KC2ThetaAboveLeaf101112
+      e24KC2ThetaAboveLeaf101113
+
+theorem e24KC2ThetaAboveNode10112 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1011) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell1011)
+    e24KC2ThetaAboveNode101120 e24KC2ThetaAboveNode101121 e24KC2ThetaAboveNode101122
+      e24KC2ThetaAboveNode101123
+
+theorem e24KC2ThetaAboveNode10113 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1011) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell1011)
+    e24KC2ThetaAboveNode101130 e24KC2ThetaAboveNode101131 e24KC2ThetaAboveNode101132
+      e24KC2ThetaAboveNode101133
+
+theorem e24KC2ThetaAboveNode10120 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1012) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1012)
+    e24KC2ThetaAboveNode101200 e24KC2ThetaAboveNode101201 e24KC2ThetaAboveLeaf101202
+      e24KC2ThetaAboveLeaf101203
+
+theorem e24KC2ThetaAboveNode10121 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1012) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1012)
+    e24KC2ThetaAboveNode101210 e24KC2ThetaAboveNode101211 e24KC2ThetaAboveLeaf101212
+      e24KC2ThetaAboveLeaf101213
+
+theorem e24KC2ThetaAboveNode10130 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1013) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1013)
+    e24KC2ThetaAboveNode101300 e24KC2ThetaAboveNode101301 e24KC2ThetaAboveLeaf101302
+      e24KC2ThetaAboveLeaf101303
+
+theorem e24KC2ThetaAboveNode10131 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1013) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1013)
+    e24KC2ThetaAboveNode101310 e24KC2ThetaAboveNode101311 e24KC2ThetaAboveLeaf101312
+      e24KC2ThetaAboveLeaf101313
+
+theorem e24KC2ThetaAboveNode11000 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1100) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1100)
+    e24KC2ThetaAboveLeaf110000 e24KC2ThetaAboveLeaf110001 e24KC2ThetaAboveLeaf110002
+      e24KC2ThetaAboveLeaf110003
+
+theorem e24KC2ThetaAboveNode11001 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1100) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1100)
+    e24KC2ThetaAboveLeaf110010 e24KC2ThetaAboveLeaf110011 e24KC2ThetaAboveLeaf110012
+      e24KC2ThetaAboveLeaf110013
+
+theorem e24KC2ThetaAboveNode11002 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1100) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell1100)
+    e24KC2ThetaAboveNode110020 e24KC2ThetaAboveNode110021 e24KC2ThetaAboveNode110022
+      e24KC2ThetaAboveNode110023
+
+theorem e24KC2ThetaAboveNode11003 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1100) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell1100)
+    e24KC2ThetaAboveNode110030 e24KC2ThetaAboveNode110031 e24KC2ThetaAboveNode110032
+      e24KC2ThetaAboveNode110033
+
+theorem e24KC2ThetaAboveNode11010 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1101) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1101)
+    e24KC2ThetaAboveLeaf110100 e24KC2ThetaAboveLeaf110101 e24KC2ThetaAboveLeaf110102
+      e24KC2ThetaAboveLeaf110103
+
+theorem e24KC2ThetaAboveNode11011 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1101) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1101)
+    e24KC2ThetaAboveLeaf110110 e24KC2ThetaAboveLeaf110111 e24KC2ThetaAboveLeaf110112
+      e24KC2ThetaAboveLeaf110113
+
+theorem e24KC2ThetaAboveNode11012 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1101) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell1101)
+    e24KC2ThetaAboveNode110120 e24KC2ThetaAboveNode110121 e24KC2ThetaAboveNode110122
+      e24KC2ThetaAboveNode110123
+
+theorem e24KC2ThetaAboveNode11013 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1101) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell1101)
+    e24KC2ThetaAboveNode110130 e24KC2ThetaAboveNode110131 e24KC2ThetaAboveNode110132
+      e24KC2ThetaAboveNode110133
+
+theorem e24KC2ThetaAboveNode11020 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1102) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1102)
+    e24KC2ThetaAboveNode110200 e24KC2ThetaAboveNode110201 e24KC2ThetaAboveLeaf110202
+      e24KC2ThetaAboveLeaf110203
+
+theorem e24KC2ThetaAboveNode11021 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1102) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1102)
+    e24KC2ThetaAboveNode110210 e24KC2ThetaAboveNode110211 e24KC2ThetaAboveLeaf110212
+      e24KC2ThetaAboveLeaf110213
+
+theorem e24KC2ThetaAboveNode11030 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1103) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1103)
+    e24KC2ThetaAboveNode110300 e24KC2ThetaAboveNode110301 e24KC2ThetaAboveLeaf110302
+      e24KC2ThetaAboveLeaf110303
+
+theorem e24KC2ThetaAboveNode11031 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1103) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1103)
+    e24KC2ThetaAboveNode110310 e24KC2ThetaAboveNode110311 e24KC2ThetaAboveLeaf110312
+      e24KC2ThetaAboveLeaf110313
+
+theorem e24KC2ThetaAboveNode11100 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1110) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1110)
+    e24KC2ThetaAboveLeaf111000 e24KC2ThetaAboveLeaf111001 e24KC2ThetaAboveLeaf111002
+      e24KC2ThetaAboveLeaf111003
+
+theorem e24KC2ThetaAboveNode11101 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1110) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1110)
+    e24KC2ThetaAboveLeaf111010 e24KC2ThetaAboveLeaf111011 e24KC2ThetaAboveLeaf111012
+      e24KC2ThetaAboveLeaf111013
+
+theorem e24KC2ThetaAboveNode11102 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1110) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell1110)
+    e24KC2ThetaAboveNode111020 e24KC2ThetaAboveNode111021 e24KC2ThetaAboveNode111022
+      e24KC2ThetaAboveNode111023
+
+theorem e24KC2ThetaAboveNode11103 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1110) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell1110)
+    e24KC2ThetaAboveNode111030 e24KC2ThetaAboveNode111031 e24KC2ThetaAboveNode111032
+      e24KC2ThetaAboveNode111033
+
+theorem e24KC2ThetaAboveNode11110 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1111) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1111)
+    e24KC2ThetaAboveLeaf111100 e24KC2ThetaAboveLeaf111101 e24KC2ThetaAboveLeaf111102
+      e24KC2ThetaAboveLeaf111103
+
+theorem e24KC2ThetaAboveNode11111 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1111) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1111)
+    e24KC2ThetaAboveLeaf111110 e24KC2ThetaAboveLeaf111111 e24KC2ThetaAboveLeaf111112
+      e24KC2ThetaAboveLeaf111113
+
+theorem e24KC2ThetaAboveNode11112 :
+    adaptiveCoverCheck 14 (childHL thetaAboveCell1111) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHL thetaAboveCell1111)
+    e24KC2ThetaAboveNode111120 e24KC2ThetaAboveNode111121 e24KC2ThetaAboveNode111122
+      e24KC2ThetaAboveNode111123
+
+theorem e24KC2ThetaAboveNode11113 :
+    adaptiveCoverCheck 14 (childHH thetaAboveCell1111) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childHH thetaAboveCell1111)
+    e24KC2ThetaAboveNode111130 e24KC2ThetaAboveNode111131 e24KC2ThetaAboveNode111132
+      e24KC2ThetaAboveNode111133
+
+theorem e24KC2ThetaAboveNode11120 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1112) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1112)
+    e24KC2ThetaAboveNode111200 e24KC2ThetaAboveNode111201 e24KC2ThetaAboveLeaf111202
+      e24KC2ThetaAboveLeaf111203
+
+theorem e24KC2ThetaAboveNode11121 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1112) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1112)
+    e24KC2ThetaAboveNode111210 e24KC2ThetaAboveNode111211 e24KC2ThetaAboveLeaf111212
+      e24KC2ThetaAboveLeaf111213
+
+theorem e24KC2ThetaAboveNode11130 :
+    adaptiveCoverCheck 14 (childLL thetaAboveCell1113) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLL thetaAboveCell1113)
+    e24KC2ThetaAboveNode111300 e24KC2ThetaAboveNode111301 e24KC2ThetaAboveLeaf111302
+      e24KC2ThetaAboveLeaf111303
+
+theorem e24KC2ThetaAboveNode11131 :
+    adaptiveCoverCheck 14 (childLH thetaAboveCell1113) = true :=
+  adaptiveCoverCheck_succ_of_children 13 (childLH thetaAboveCell1113)
+    e24KC2ThetaAboveNode111310 e24KC2ThetaAboveNode111311 e24KC2ThetaAboveLeaf111312
+      e24KC2ThetaAboveLeaf111313
+
+theorem e24KC2ThetaAboveNode0000 :
+    adaptiveCoverCheck 15 thetaAboveCell0000 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0000
+    e24KC2ThetaAboveNode00000 e24KC2ThetaAboveNode00001 e24KC2ThetaAboveNode00002
+      e24KC2ThetaAboveNode00003
+
+theorem e24KC2ThetaAboveNode0001 :
+    adaptiveCoverCheck 15 thetaAboveCell0001 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0001
+    e24KC2ThetaAboveNode00010 e24KC2ThetaAboveNode00011 e24KC2ThetaAboveNode00012
+      e24KC2ThetaAboveNode00013
+
+theorem e24KC2ThetaAboveNode0002 :
+    adaptiveCoverCheck 15 thetaAboveCell0002 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0002
+    e24KC2ThetaAboveNode00020 e24KC2ThetaAboveNode00021 e24KC2ThetaAboveLeaf00022
+      e24KC2ThetaAboveLeaf00023
+
+theorem e24KC2ThetaAboveNode0003 :
+    adaptiveCoverCheck 15 thetaAboveCell0003 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0003
+    e24KC2ThetaAboveNode00030 e24KC2ThetaAboveNode00031 e24KC2ThetaAboveLeaf00032
+      e24KC2ThetaAboveLeaf00033
+
+theorem e24KC2ThetaAboveNode0010 :
+    adaptiveCoverCheck 15 thetaAboveCell0010 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0010
+    e24KC2ThetaAboveNode00100 e24KC2ThetaAboveNode00101 e24KC2ThetaAboveNode00102
+      e24KC2ThetaAboveNode00103
+
+theorem e24KC2ThetaAboveNode0011 :
+    adaptiveCoverCheck 15 thetaAboveCell0011 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0011
+    e24KC2ThetaAboveNode00110 e24KC2ThetaAboveNode00111 e24KC2ThetaAboveNode00112
+      e24KC2ThetaAboveNode00113
+
+theorem e24KC2ThetaAboveNode0012 :
+    adaptiveCoverCheck 15 thetaAboveCell0012 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0012
+    e24KC2ThetaAboveNode00120 e24KC2ThetaAboveNode00121 e24KC2ThetaAboveLeaf00122
+      e24KC2ThetaAboveLeaf00123
+
+theorem e24KC2ThetaAboveNode0013 :
+    adaptiveCoverCheck 15 thetaAboveCell0013 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0013
+    e24KC2ThetaAboveNode00130 e24KC2ThetaAboveNode00131 e24KC2ThetaAboveLeaf00132
+      e24KC2ThetaAboveLeaf00133
+
+theorem e24KC2ThetaAboveNode0100 :
+    adaptiveCoverCheck 15 thetaAboveCell0100 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0100
+    e24KC2ThetaAboveNode01000 e24KC2ThetaAboveNode01001 e24KC2ThetaAboveNode01002
+      e24KC2ThetaAboveNode01003
+
+theorem e24KC2ThetaAboveNode0101 :
+    adaptiveCoverCheck 15 thetaAboveCell0101 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0101
+    e24KC2ThetaAboveNode01010 e24KC2ThetaAboveNode01011 e24KC2ThetaAboveNode01012
+      e24KC2ThetaAboveNode01013
+
+theorem e24KC2ThetaAboveNode0102 :
+    adaptiveCoverCheck 15 thetaAboveCell0102 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0102
+    e24KC2ThetaAboveNode01020 e24KC2ThetaAboveNode01021 e24KC2ThetaAboveLeaf01022
+      e24KC2ThetaAboveLeaf01023
+
+theorem e24KC2ThetaAboveNode0103 :
+    adaptiveCoverCheck 15 thetaAboveCell0103 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0103
+    e24KC2ThetaAboveNode01030 e24KC2ThetaAboveNode01031 e24KC2ThetaAboveLeaf01032
+      e24KC2ThetaAboveLeaf01033
+
+theorem e24KC2ThetaAboveNode0110 :
+    adaptiveCoverCheck 15 thetaAboveCell0110 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0110
+    e24KC2ThetaAboveNode01100 e24KC2ThetaAboveNode01101 e24KC2ThetaAboveNode01102
+      e24KC2ThetaAboveNode01103
+
+theorem e24KC2ThetaAboveNode0111 :
+    adaptiveCoverCheck 15 thetaAboveCell0111 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0111
+    e24KC2ThetaAboveNode01110 e24KC2ThetaAboveNode01111 e24KC2ThetaAboveNode01112
+      e24KC2ThetaAboveNode01113
+
+theorem e24KC2ThetaAboveNode0112 :
+    adaptiveCoverCheck 15 thetaAboveCell0112 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0112
+    e24KC2ThetaAboveNode01120 e24KC2ThetaAboveNode01121 e24KC2ThetaAboveLeaf01122
+      e24KC2ThetaAboveLeaf01123
+
+theorem e24KC2ThetaAboveNode0113 :
+    adaptiveCoverCheck 15 thetaAboveCell0113 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell0113
+    e24KC2ThetaAboveNode01130 e24KC2ThetaAboveNode01131 e24KC2ThetaAboveLeaf01132
+      e24KC2ThetaAboveLeaf01133
+
+theorem e24KC2ThetaAboveNode1000 :
+    adaptiveCoverCheck 15 thetaAboveCell1000 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1000
+    e24KC2ThetaAboveNode10000 e24KC2ThetaAboveNode10001 e24KC2ThetaAboveNode10002
+      e24KC2ThetaAboveNode10003
+
+theorem e24KC2ThetaAboveNode1001 :
+    adaptiveCoverCheck 15 thetaAboveCell1001 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1001
+    e24KC2ThetaAboveNode10010 e24KC2ThetaAboveNode10011 e24KC2ThetaAboveNode10012
+      e24KC2ThetaAboveNode10013
+
+theorem e24KC2ThetaAboveNode1002 :
+    adaptiveCoverCheck 15 thetaAboveCell1002 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1002
+    e24KC2ThetaAboveNode10020 e24KC2ThetaAboveNode10021 e24KC2ThetaAboveLeaf10022
+      e24KC2ThetaAboveLeaf10023
+
+theorem e24KC2ThetaAboveNode1003 :
+    adaptiveCoverCheck 15 thetaAboveCell1003 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1003
+    e24KC2ThetaAboveNode10030 e24KC2ThetaAboveNode10031 e24KC2ThetaAboveLeaf10032
+      e24KC2ThetaAboveLeaf10033
+
+theorem e24KC2ThetaAboveNode1010 :
+    adaptiveCoverCheck 15 thetaAboveCell1010 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1010
+    e24KC2ThetaAboveNode10100 e24KC2ThetaAboveNode10101 e24KC2ThetaAboveNode10102
+      e24KC2ThetaAboveNode10103
+
+theorem e24KC2ThetaAboveNode1011 :
+    adaptiveCoverCheck 15 thetaAboveCell1011 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1011
+    e24KC2ThetaAboveNode10110 e24KC2ThetaAboveNode10111 e24KC2ThetaAboveNode10112
+      e24KC2ThetaAboveNode10113
+
+theorem e24KC2ThetaAboveNode1012 :
+    adaptiveCoverCheck 15 thetaAboveCell1012 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1012
+    e24KC2ThetaAboveNode10120 e24KC2ThetaAboveNode10121 e24KC2ThetaAboveLeaf10122
+      e24KC2ThetaAboveLeaf10123
+
+theorem e24KC2ThetaAboveNode1013 :
+    adaptiveCoverCheck 15 thetaAboveCell1013 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1013
+    e24KC2ThetaAboveNode10130 e24KC2ThetaAboveNode10131 e24KC2ThetaAboveLeaf10132
+      e24KC2ThetaAboveLeaf10133
+
+theorem e24KC2ThetaAboveNode1100 :
+    adaptiveCoverCheck 15 thetaAboveCell1100 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1100
+    e24KC2ThetaAboveNode11000 e24KC2ThetaAboveNode11001 e24KC2ThetaAboveNode11002
+      e24KC2ThetaAboveNode11003
+
+theorem e24KC2ThetaAboveNode1101 :
+    adaptiveCoverCheck 15 thetaAboveCell1101 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1101
+    e24KC2ThetaAboveNode11010 e24KC2ThetaAboveNode11011 e24KC2ThetaAboveNode11012
+      e24KC2ThetaAboveNode11013
+
+theorem e24KC2ThetaAboveNode1102 :
+    adaptiveCoverCheck 15 thetaAboveCell1102 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1102
+    e24KC2ThetaAboveNode11020 e24KC2ThetaAboveNode11021 e24KC2ThetaAboveLeaf11022
+      e24KC2ThetaAboveLeaf11023
+
+theorem e24KC2ThetaAboveNode1103 :
+    adaptiveCoverCheck 15 thetaAboveCell1103 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1103
+    e24KC2ThetaAboveNode11030 e24KC2ThetaAboveNode11031 e24KC2ThetaAboveLeaf11032
+      e24KC2ThetaAboveLeaf11033
+
+theorem e24KC2ThetaAboveNode1110 :
+    adaptiveCoverCheck 15 thetaAboveCell1110 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1110
+    e24KC2ThetaAboveNode11100 e24KC2ThetaAboveNode11101 e24KC2ThetaAboveNode11102
+      e24KC2ThetaAboveNode11103
+
+theorem e24KC2ThetaAboveNode1111 :
+    adaptiveCoverCheck 15 thetaAboveCell1111 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1111
+    e24KC2ThetaAboveNode11110 e24KC2ThetaAboveNode11111 e24KC2ThetaAboveNode11112
+      e24KC2ThetaAboveNode11113
+
+theorem e24KC2ThetaAboveNode1112 :
+    adaptiveCoverCheck 15 thetaAboveCell1112 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1112
+    e24KC2ThetaAboveNode11120 e24KC2ThetaAboveNode11121 e24KC2ThetaAboveLeaf11122
+      e24KC2ThetaAboveLeaf11123
+
+theorem e24KC2ThetaAboveNode1113 :
+    adaptiveCoverCheck 15 thetaAboveCell1113 = true :=
+  adaptiveCoverCheck_succ_of_children 14 thetaAboveCell1113
+    e24KC2ThetaAboveNode11130 e24KC2ThetaAboveNode11131 e24KC2ThetaAboveLeaf11132
+      e24KC2ThetaAboveLeaf11133
+
+theorem e24KC2ThetaAboveNode000 :
+    adaptiveCoverCheck 16 (childLL (childLL (childLL e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childLL (childLL (childLL e24ThetaAboveRoot)))
+    e24KC2ThetaAboveNode0000 e24KC2ThetaAboveNode0001 e24KC2ThetaAboveNode0002
+      e24KC2ThetaAboveNode0003
+
+theorem e24KC2ThetaAboveNode001 :
+    adaptiveCoverCheck 16 (childLH (childLL (childLL e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childLH (childLL (childLL e24ThetaAboveRoot)))
+    e24KC2ThetaAboveNode0010 e24KC2ThetaAboveNode0011 e24KC2ThetaAboveNode0012
+      e24KC2ThetaAboveNode0013
+
+theorem e24KC2ThetaAboveNode002 :
+    adaptiveCoverCheck 16 (childHL (childLL (childLL e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childHL (childLL (childLL e24ThetaAboveRoot)))
+    e24KC2ThetaAboveLeaf0020 e24KC2ThetaAboveLeaf0021 e24KC2ThetaAboveLeaf0022
+      e24KC2ThetaAboveLeaf0023
+
+theorem e24KC2ThetaAboveNode003 :
+    adaptiveCoverCheck 16 (childHH (childLL (childLL e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childHH (childLL (childLL e24ThetaAboveRoot)))
+    e24KC2ThetaAboveLeaf0030 e24KC2ThetaAboveLeaf0031 e24KC2ThetaAboveLeaf0032
+      e24KC2ThetaAboveLeaf0033
+
+theorem e24KC2ThetaAboveNode010 :
+    adaptiveCoverCheck 16 (childLL (childLH (childLL e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childLL (childLH (childLL e24ThetaAboveRoot)))
+    e24KC2ThetaAboveNode0100 e24KC2ThetaAboveNode0101 e24KC2ThetaAboveNode0102
+      e24KC2ThetaAboveNode0103
+
+theorem e24KC2ThetaAboveNode011 :
+    adaptiveCoverCheck 16 (childLH (childLH (childLL e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childLH (childLH (childLL e24ThetaAboveRoot)))
+    e24KC2ThetaAboveNode0110 e24KC2ThetaAboveNode0111 e24KC2ThetaAboveNode0112
+      e24KC2ThetaAboveNode0113
+
+theorem e24KC2ThetaAboveNode012 :
+    adaptiveCoverCheck 16 (childHL (childLH (childLL e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childHL (childLH (childLL e24ThetaAboveRoot)))
+    e24KC2ThetaAboveLeaf0120 e24KC2ThetaAboveLeaf0121 e24KC2ThetaAboveLeaf0122
+      e24KC2ThetaAboveLeaf0123
+
+theorem e24KC2ThetaAboveNode013 :
+    adaptiveCoverCheck 16 (childHH (childLH (childLL e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childHH (childLH (childLL e24ThetaAboveRoot)))
+    e24KC2ThetaAboveLeaf0130 e24KC2ThetaAboveLeaf0131 e24KC2ThetaAboveLeaf0132
+      e24KC2ThetaAboveLeaf0133
+
+theorem e24KC2ThetaAboveNode100 :
+    adaptiveCoverCheck 16 (childLL (childLL (childLH e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childLL (childLL (childLH e24ThetaAboveRoot)))
+    e24KC2ThetaAboveNode1000 e24KC2ThetaAboveNode1001 e24KC2ThetaAboveNode1002
+      e24KC2ThetaAboveNode1003
+
+theorem e24KC2ThetaAboveNode101 :
+    adaptiveCoverCheck 16 (childLH (childLL (childLH e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childLH (childLL (childLH e24ThetaAboveRoot)))
+    e24KC2ThetaAboveNode1010 e24KC2ThetaAboveNode1011 e24KC2ThetaAboveNode1012
+      e24KC2ThetaAboveNode1013
+
+theorem e24KC2ThetaAboveNode102 :
+    adaptiveCoverCheck 16 (childHL (childLL (childLH e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childHL (childLL (childLH e24ThetaAboveRoot)))
+    e24KC2ThetaAboveLeaf1020 e24KC2ThetaAboveLeaf1021 e24KC2ThetaAboveLeaf1022
+      e24KC2ThetaAboveLeaf1023
+
+theorem e24KC2ThetaAboveNode103 :
+    adaptiveCoverCheck 16 (childHH (childLL (childLH e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childHH (childLL (childLH e24ThetaAboveRoot)))
+    e24KC2ThetaAboveLeaf1030 e24KC2ThetaAboveLeaf1031 e24KC2ThetaAboveLeaf1032
+      e24KC2ThetaAboveLeaf1033
+
+theorem e24KC2ThetaAboveNode110 :
+    adaptiveCoverCheck 16 (childLL (childLH (childLH e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childLL (childLH (childLH e24ThetaAboveRoot)))
+    e24KC2ThetaAboveNode1100 e24KC2ThetaAboveNode1101 e24KC2ThetaAboveNode1102
+      e24KC2ThetaAboveNode1103
+
+theorem e24KC2ThetaAboveNode111 :
+    adaptiveCoverCheck 16 (childLH (childLH (childLH e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childLH (childLH (childLH e24ThetaAboveRoot)))
+    e24KC2ThetaAboveNode1110 e24KC2ThetaAboveNode1111 e24KC2ThetaAboveNode1112
+      e24KC2ThetaAboveNode1113
+
+theorem e24KC2ThetaAboveNode112 :
+    adaptiveCoverCheck 16 (childHL (childLH (childLH e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childHL (childLH (childLH e24ThetaAboveRoot)))
+    e24KC2ThetaAboveLeaf1120 e24KC2ThetaAboveLeaf1121 e24KC2ThetaAboveLeaf1122
+      e24KC2ThetaAboveLeaf1123
+
+theorem e24KC2ThetaAboveNode113 :
+    adaptiveCoverCheck 16 (childHH (childLH (childLH e24ThetaAboveRoot))) = true :=
+  adaptiveCoverCheck_succ_of_children 15 (childHH (childLH (childLH e24ThetaAboveRoot)))
+    e24KC2ThetaAboveLeaf1130 e24KC2ThetaAboveLeaf1131 e24KC2ThetaAboveLeaf1132
+      e24KC2ThetaAboveLeaf1133
+
+theorem e24KC2ThetaAboveNode00 :
+    adaptiveCoverCheck 17 (childLL (childLL e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childLL (childLL e24ThetaAboveRoot))
+    e24KC2ThetaAboveNode000 e24KC2ThetaAboveNode001 e24KC2ThetaAboveNode002 e24KC2ThetaAboveNode003
+
+theorem e24KC2ThetaAboveNode01 :
+    adaptiveCoverCheck 17 (childLH (childLL e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childLH (childLL e24ThetaAboveRoot))
+    e24KC2ThetaAboveNode010 e24KC2ThetaAboveNode011 e24KC2ThetaAboveNode012 e24KC2ThetaAboveNode013
+
+theorem e24KC2ThetaAboveNode02 :
+    adaptiveCoverCheck 17 (childHL (childLL e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childHL (childLL e24ThetaAboveRoot))
+    e24KC2ThetaAboveLeaf020 e24KC2ThetaAboveLeaf021 e24KC2ThetaAboveLeaf022 e24KC2ThetaAboveLeaf023
+
+theorem e24KC2ThetaAboveNode03 :
+    adaptiveCoverCheck 17 (childHH (childLL e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childHH (childLL e24ThetaAboveRoot))
+    e24KC2ThetaAboveLeaf030 e24KC2ThetaAboveLeaf031 e24KC2ThetaAboveLeaf032 e24KC2ThetaAboveLeaf033
+
+theorem e24KC2ThetaAboveNode10 :
+    adaptiveCoverCheck 17 (childLL (childLH e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childLL (childLH e24ThetaAboveRoot))
+    e24KC2ThetaAboveNode100 e24KC2ThetaAboveNode101 e24KC2ThetaAboveNode102 e24KC2ThetaAboveNode103
+
+theorem e24KC2ThetaAboveNode11 :
+    adaptiveCoverCheck 17 (childLH (childLH e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childLH (childLH e24ThetaAboveRoot))
+    e24KC2ThetaAboveNode110 e24KC2ThetaAboveNode111 e24KC2ThetaAboveNode112 e24KC2ThetaAboveNode113
+
+theorem e24KC2ThetaAboveNode12 :
+    adaptiveCoverCheck 17 (childHL (childLH e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childHL (childLH e24ThetaAboveRoot))
+    e24KC2ThetaAboveLeaf120 e24KC2ThetaAboveLeaf121 e24KC2ThetaAboveLeaf122 e24KC2ThetaAboveLeaf123
+
+theorem e24KC2ThetaAboveNode13 :
+    adaptiveCoverCheck 17 (childHH (childLH e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childHH (childLH e24ThetaAboveRoot))
+    e24KC2ThetaAboveLeaf130 e24KC2ThetaAboveLeaf131 e24KC2ThetaAboveLeaf132 e24KC2ThetaAboveLeaf133
+
+theorem e24KC2ThetaAboveNode20 :
+    adaptiveCoverCheck 17 (childLL (childHL e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childLL (childHL e24ThetaAboveRoot))
+    e24KC2ThetaAboveLeaf200 e24KC2ThetaAboveLeaf201 e24KC2ThetaAboveLeaf202 e24KC2ThetaAboveLeaf203
+
+theorem e24KC2ThetaAboveNode21 :
+    adaptiveCoverCheck 17 (childLH (childHL e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childLH (childHL e24ThetaAboveRoot))
+    e24KC2ThetaAboveLeaf210 e24KC2ThetaAboveLeaf211 e24KC2ThetaAboveLeaf212 e24KC2ThetaAboveLeaf213
+
+theorem e24KC2ThetaAboveNode22 :
+    adaptiveCoverCheck 17 (childHL (childHL e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childHL (childHL e24ThetaAboveRoot))
+    e24KC2ThetaAboveLeaf220 e24KC2ThetaAboveLeaf221 e24KC2ThetaAboveLeaf222 e24KC2ThetaAboveLeaf223
+
+theorem e24KC2ThetaAboveNode23 :
+    adaptiveCoverCheck 17 (childHH (childHL e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childHH (childHL e24ThetaAboveRoot))
+    e24KC2ThetaAboveLeaf230 e24KC2ThetaAboveLeaf231 e24KC2ThetaAboveLeaf232 e24KC2ThetaAboveLeaf233
+
+theorem e24KC2ThetaAboveNode30 :
+    adaptiveCoverCheck 17 (childLL (childHH e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childLL (childHH e24ThetaAboveRoot))
+    e24KC2ThetaAboveLeaf300 e24KC2ThetaAboveLeaf301 e24KC2ThetaAboveLeaf302 e24KC2ThetaAboveLeaf303
+
+theorem e24KC2ThetaAboveNode31 :
+    adaptiveCoverCheck 17 (childLH (childHH e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childLH (childHH e24ThetaAboveRoot))
+    e24KC2ThetaAboveLeaf310 e24KC2ThetaAboveLeaf311 e24KC2ThetaAboveLeaf312 e24KC2ThetaAboveLeaf313
+
+theorem e24KC2ThetaAboveNode32 :
+    adaptiveCoverCheck 17 (childHL (childHH e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childHL (childHH e24ThetaAboveRoot))
+    e24KC2ThetaAboveLeaf320 e24KC2ThetaAboveLeaf321 e24KC2ThetaAboveLeaf322 e24KC2ThetaAboveLeaf323
+
+theorem e24KC2ThetaAboveNode33 :
+    adaptiveCoverCheck 17 (childHH (childHH e24ThetaAboveRoot)) = true :=
+  adaptiveCoverCheck_succ_of_children 16 (childHH (childHH e24ThetaAboveRoot))
+    e24KC2ThetaAboveLeaf330 e24KC2ThetaAboveLeaf331 e24KC2ThetaAboveLeaf332 e24KC2ThetaAboveLeaf333
+
+theorem e24KC2ThetaAboveNode0 :
+    adaptiveCoverCheck 18 (childLL e24ThetaAboveRoot) = true :=
+  adaptiveCoverCheck_succ_of_children 17 (childLL e24ThetaAboveRoot)
+    e24KC2ThetaAboveNode00 e24KC2ThetaAboveNode01 e24KC2ThetaAboveNode02 e24KC2ThetaAboveNode03
+
+theorem e24KC2ThetaAboveNode1 :
+    adaptiveCoverCheck 18 (childLH e24ThetaAboveRoot) = true :=
+  adaptiveCoverCheck_succ_of_children 17 (childLH e24ThetaAboveRoot)
+    e24KC2ThetaAboveNode10 e24KC2ThetaAboveNode11 e24KC2ThetaAboveNode12 e24KC2ThetaAboveNode13
+
+theorem e24KC2ThetaAboveNode2 :
+    adaptiveCoverCheck 18 (childHL e24ThetaAboveRoot) = true :=
+  adaptiveCoverCheck_succ_of_children 17 (childHL e24ThetaAboveRoot)
+    e24KC2ThetaAboveNode20 e24KC2ThetaAboveNode21 e24KC2ThetaAboveNode22 e24KC2ThetaAboveNode23
+
+theorem e24KC2ThetaAboveNode3 :
+    adaptiveCoverCheck 18 (childHH e24ThetaAboveRoot) = true :=
+  adaptiveCoverCheck_succ_of_children 17 (childHH e24ThetaAboveRoot)
+    e24KC2ThetaAboveNode30 e24KC2ThetaAboveNode31 e24KC2ThetaAboveNode32 e24KC2ThetaAboveNode33
+
+theorem e24KC2ThetaAboveNodeROOT :
+    adaptiveCoverCheck 19 e24ThetaAboveRoot = true :=
+  adaptiveCoverCheck_succ_of_children 18 e24ThetaAboveRoot
+    e24KC2ThetaAboveNode0 e24KC2ThetaAboveNode1 e24KC2ThetaAboveNode2 e24KC2ThetaAboveNode3
+
+theorem e24ThetaAboveKernelCheck :
+    adaptiveCoverCheck 19 e24ThetaAboveRoot = true :=
+  e24KC2ThetaAboveNodeROOT
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+section
+
+/-!
+# Gerver Sofa / Kernel Only / Part E / E24KC6Kernel Final Closure
+-/
+
+@[expose] public section
+
+noncomputable section
+
+namespace GerverSofa
+namespace PartE
+
+theorem deepMindABPhiTheta_existsUnique :
+    ∃! ABphiTheta : ℝ × ℝ × ℝ × ℝ,
+      DeepMindABPhiThetaSpec
+        ABphiTheta.1 ABphiTheta.2.1 ABphiTheta.2.2.1 ABphiTheta.2.2.2 :=
+  deepMindABPhiTheta_existsUnique_of_alignedChecks
+    e24PhiBelowKernelCheck
+    e24PhiAboveKernelCheck
+    e24ThetaBelowKernelCheck
+    e24ThetaAboveKernelCheck
+
+end PartE
+end GerverSofa
+
+end
+
+end
+
+end
+
+end
+
+end
+
+end
diff --git a/LeanPool/projects.yml b/LeanPool/projects.yml
index 0214805fce..637c3322bb 100644
--- a/LeanPool/projects.yml
+++ b/LeanPool/projects.yml
@@ -1,4 +1,65 @@
 projects:
+  - slug: moving-sofa
+    title: Optimality of Gerver's sofa
+    summary: >-
+      Formalizes Baek's solution of the moving sofa problem: Gerver's sofa attains
+      the supremum of the areas of closed connected shapes that can move continuously
+      around a right-angled corner in a unit-width hallway. Includes certified Gerver
+      parameters, admissibility, convex cap geometry, Stieltjes curve area, Jordan
+      separation, polygon approximation, and the concave quadratic upper bound.
+    branch: convex geometry
+    entry_module: LeanPool.MovingSofa
+    authors:
+      - Dean Cureton
+      - Dawid Trela
+      - Rado Kirov
+      - Kirill Alkhimov
+    source:
+      title: Optimality of Gerver's Sofa
+      authors:
+        - Jineon Baek
+      arxiv: '2411.19826'
+      url: https://github.com/deancureton/MovingSofa/tree/4d5569131940815f47a9ccf3e90a4c5043c56127
+      github_repo: deancureton/MovingSofa
+      commit: 4d5569131940815f47a9ccf3e90a4c5043c56127
+    license: Apache-2.0
+    status: verified
+    provenance: AI
+    main_declarations:
+      - MovingSofa.sofaConstant_eq_volume_gerversSofa
+    main_results:
+      - declaration: MovingSofa.sofaConstant_eq_volume_gerversSofa
+        informal: >-
+          The supremum, in extended nonnegative real numbers, of the areas of all
+          admissible moving sofas equals the area of the explicitly defined Gerver sofa.
+        source_ref: >-
+          Baek, Optimality of Gerver's Sofa, Theorem 1.1. Upstream Lean completion
+          a0753860f1d2b4424e059c35e64ad611be71a415 (2026-09-20 10:55:24 UTC), announced
+          2026-09-20 18:42:40 UTC at https://x.com/deancureton/status/2101743865392926967.
+          The imported commit has identical Lean sources; its later change is documentation only.
+      - declaration: MovingSofa.GerversSofa.ABφθSpec.existsUnique
+        informal: >-
+          Gerver's four defining equations have exactly one quadruple of parameters
+          with A and B nonnegative and 0 ≤ φ ≤ θ ≤ π/4.
+        source_ref: >-
+          Romik, Differential equations and exact solutions in the moving sofa problem,
+          equations (1)-(4), arXiv:1606.08111. The GerverSofaLean v1.1.0 release archive
+          (MIT, Dawid Trela) has SHA-256
+          7c496150f709ac709ae01f4836e9a702a18f752450efa69f15cf44264c0c2f77.
+      - declaration: MovingSofa.isMovingSofa_gerversSofa
+        informal: >-
+          Gerver's sofa is closed and connected and admits a continuous rigid motion
+          from the horizontal arm to the vertical arm, starting at the identity and
+          remaining within the unit-width hallway throughout.
+    tags:
+      - convex-geometry
+      - moving-sofa-problem
+      - geometric-optimization
+      - interval-arithmetic
+    msc:
+      - 52A40
+      - 52A10
+      - 49Q10
   - slug: krohn-rhodes
     title: Aperiodic and simple-group wreath-division towers
     summary: >-